id	sid	tid	token	lemma	pos
ejpam-4793	1	1	european	european	PROPN
ejpam-4793	1	2	journal	journal	PROPN
ejpam-4793	1	3	of	of	ADP
ejpam-4793	1	4	pure	pure	ADJ
ejpam-4793	1	5	and	and	CCONJ
ejpam-4793	1	6	applied	apply	VERB
ejpam-4793	1	7	mathematics	mathematic	NOUN
ejpam-4793	1	8	vol	vol	NOUN
ejpam-4793	1	9	.	.	PUNCT
ejpam-4793	2	1	16	16	NUM
ejpam-4793	2	2	,	,	PUNCT
ejpam-4793	2	3	no	no	INTJ
ejpam-4793	2	4	.	.	NOUN
ejpam-4793	2	5	3	3	NUM
ejpam-4793	2	6	,	,	PUNCT
ejpam-4793	2	7	2023	2023	NUM
ejpam-4793	2	8	,	,	PUNCT
ejpam-4793	2	9	1772	1772	NUM
ejpam-4793	2	10	-	-	SYM
ejpam-4793	2	11	1793	1793	NUM
ejpam-4793	2	12	issn	issn	PROPN
ejpam-4793	2	13	1307	1307	NUM
ejpam-4793	2	14	-	-	SYM
ejpam-4793	2	15	5543	5543	NUM
ejpam-4793	2	16	–	–	PUNCT
ejpam-4793	2	17	ejpam.com	ejpam.com	X
ejpam-4793	2	18	published	publish	VERB
ejpam-4793	2	19	by	by	ADP
ejpam-4793	2	20	new	new	PROPN
ejpam-4793	2	21	york	york	PROPN
ejpam-4793	2	22	business	business	PROPN
ejpam-4793	2	23	global	global	PROPN
ejpam-4793	2	24	on	on	ADP
ejpam-4793	2	25	γ	γ	NOUN
ejpam-4793	2	26	-	-	PUNCT
ejpam-4793	2	27	ideals	ideal	NOUN
ejpam-4793	2	28	,	,	PUNCT
ejpam-4793	2	29	γ	γ	NOUN
ejpam-4793	2	30	-	-	ADJ
ejpam-4793	2	31	submonoids	submonoid	NOUN
ejpam-4793	2	32	and	and	CCONJ
ejpam-4793	2	33	isomorphism	isomorphism	NOUN
ejpam-4793	2	34	theorems	theorem	NOUN
ejpam-4793	2	35	of	of	ADP
ejpam-4793	2	36	γ	γ	NOUN
ejpam-4793	2	37	-	-	PUNCT
ejpam-4793	2	38	monoids	monoid	NOUN
ejpam-4793	2	39	via	via	ADP
ejpam-4793	2	40	γ	γ	PROPN
ejpam-4793	2	41	-	-	PUNCT
ejpam-4793	2	42	submonoids	submonoids	ADJ
ejpam-4793	2	43	hulsen	hulsen	NOUN
ejpam-4793	2	44	t.	t.	PROPN
ejpam-4793	2	45	sarapuddin1,∗	sarapuddin1,∗	PROPN
ejpam-4793	2	46	,	,	PUNCT
ejpam-4793	2	47	jocelyn	jocelyn	PROPN
ejpam-4793	2	48	p.	p.	PROPN
ejpam-4793	2	49	vilela1	vilela1	NOUN
ejpam-4793	3	1	1	1	NUM
ejpam-4793	3	2	department	department	NOUN
ejpam-4793	3	3	of	of	ADP
ejpam-4793	3	4	mathematics	mathematic	NOUN
ejpam-4793	3	5	and	and	CCONJ
ejpam-4793	3	6	statistics	statistic	NOUN
ejpam-4793	3	7	,	,	PUNCT
ejpam-4793	3	8	college	college	NOUN
ejpam-4793	3	9	of	of	ADP
ejpam-4793	3	10	science	science	NOUN
ejpam-4793	3	11	and	and	CCONJ
ejpam-4793	3	12	mathematics	mathematic	NOUN
ejpam-4793	3	13	,	,	PUNCT
ejpam-4793	3	14	center	center	NOUN
ejpam-4793	3	15	of	of	ADP
ejpam-4793	3	16	mathematical	mathematical	ADJ
ejpam-4793	3	17	and	and	CCONJ
ejpam-4793	3	18	theoretical	theoretical	ADJ
ejpam-4793	3	19	physical	physical	ADJ
ejpam-4793	3	20	sciences	science	NOUN
ejpam-4793	3	21	-	-	PUNCT
ejpam-4793	3	22	prism	prism	NOUN
ejpam-4793	3	23	,	,	PUNCT
ejpam-4793	3	24	msu	msu	PROPN
ejpam-4793	3	25	-	-	PUNCT
ejpam-4793	3	26	iligan	iligan	PROPN
ejpam-4793	3	27	institute	institute	PROPN
ejpam-4793	3	28	of	of	ADP
ejpam-4793	3	29	technology	technology	PROPN
ejpam-4793	3	30	,	,	PUNCT
ejpam-4793	3	31	9200	9200	NUM
ejpam-4793	3	32	iligan	iligan	ADJ
ejpam-4793	3	33	city	city	NOUN
ejpam-4793	3	34	,	,	PUNCT
ejpam-4793	3	35	philippines	philippine	NOUN
ejpam-4793	3	36	abstract	abstract	ADJ
ejpam-4793	3	37	.	.	PUNCT
ejpam-4793	4	1	this	this	DET
ejpam-4793	4	2	study	study	NOUN
ejpam-4793	4	3	introduces	introduce	VERB
ejpam-4793	4	4	the	the	DET
ejpam-4793	4	5	concept	concept	NOUN
ejpam-4793	4	6	of	of	ADP
ejpam-4793	4	7	γ	γ	NOUN
ejpam-4793	4	8	-	-	NOUN
ejpam-4793	4	9	ideals	ideal	NOUN
ejpam-4793	4	10	and	and	CCONJ
ejpam-4793	4	11	γ	γ	NOUN
ejpam-4793	4	12	-	-	NOUN
ejpam-4793	4	13	submonoids	submonoid	NOUN
ejpam-4793	4	14	of	of	ADP
ejpam-4793	4	15	γ	γ	NOUN
ejpam-4793	4	16	-	-	PUNCT
ejpam-4793	4	17	monoids	monoid	NOUN
ejpam-4793	4	18	and	and	CCONJ
ejpam-4793	4	19	investigates	investigate	VERB
ejpam-4793	4	20	their	their	PRON
ejpam-4793	4	21	relationships	relationship	NOUN
ejpam-4793	4	22	with	with	ADP
ejpam-4793	4	23	the	the	DET
ejpam-4793	4	24	existing	exist	VERB
ejpam-4793	4	25	γ	γ	NOUN
ejpam-4793	4	26	-	-	PUNCT
ejpam-4793	4	27	order	order	NOUN
ejpam-4793	4	28	-	-	PUNCT
ejpam-4793	4	29	ideals	ideal	NOUN
ejpam-4793	4	30	.	.	PUNCT
ejpam-4793	5	1	moreover	moreover	ADV
ejpam-4793	5	2	,	,	PUNCT
ejpam-4793	5	3	quotient	quotient	NOUN
ejpam-4793	5	4	of	of	ADP
ejpam-4793	5	5	γ	γ	NOUN
ejpam-4793	5	6	-	-	PUNCT
ejpam-4793	5	7	monoids	monoids	PROPN
ejpam-4793	5	8	and	and	CCONJ
ejpam-4793	5	9	isomorphism	isomorphism	NOUN
ejpam-4793	5	10	theorems	theorem	NOUN
ejpam-4793	5	11	via	via	ADP
ejpam-4793	5	12	γ	γ	NOUN
ejpam-4793	5	13	-	-	PUNCT
ejpam-4793	5	14	submonoids	submonoid	NOUN
ejpam-4793	5	15	are	be	AUX
ejpam-4793	5	16	proved	prove	VERB
ejpam-4793	5	17	.	.	PUNCT
ejpam-4793	6	1	2020	2020	NUM
ejpam-4793	6	2	mathematics	mathematic	NOUN
ejpam-4793	6	3	subject	subject	NOUN
ejpam-4793	6	4	classifications	classification	NOUN
ejpam-4793	6	5	:	:	PUNCT
ejpam-4793	6	6	20m32	20m32	NUM
ejpam-4793	6	7	key	key	ADJ
ejpam-4793	6	8	words	word	NOUN
ejpam-4793	6	9	and	and	CCONJ
ejpam-4793	6	10	phrases	phrase	NOUN
ejpam-4793	6	11	:	:	PUNCT
ejpam-4793	6	12	γ	γ	NOUN
ejpam-4793	6	13	-	-	PUNCT
ejpam-4793	6	14	monoids	monoids	PROPN
ejpam-4793	6	15	,	,	PUNCT
ejpam-4793	6	16	γ	γ	PROPN
ejpam-4793	6	17	-	-	PUNCT
ejpam-4793	6	18	monoid	monoid	NOUN
ejpam-4793	6	19	homomorphism	homomorphism	NOUN
ejpam-4793	6	20	,	,	PUNCT
ejpam-4793	6	21	γ	γ	NOUN
ejpam-4793	6	22	-	-	PUNCT
ejpam-4793	6	23	order	order	NOUN
ejpam-4793	6	24	-	-	PUNCT
ejpam-4793	6	25	ideals	ideal	NOUN
ejpam-4793	6	26	,	,	PUNCT
ejpam-4793	6	27	γ	γ	NOUN
ejpam-4793	6	28	-	-	PUNCT
ejpam-4793	6	29	ideals	ideal	NOUN
ejpam-4793	6	30	,	,	PUNCT
ejpam-4793	6	31	γsubmonoids	γsubmonoid	NOUN
ejpam-4793	6	32	,	,	PUNCT
ejpam-4793	6	33	isomorphism	isomorphism	NOUN
ejpam-4793	6	34	theorem	theorem	VERB
ejpam-4793	6	35	1	1	NUM
ejpam-4793	6	36	.	.	PUNCT
ejpam-4793	6	37	introduction	introduction	NOUN
ejpam-4793	6	38	the	the	DET
ejpam-4793	6	39	talented	talented	ADJ
ejpam-4793	6	40	monoid	monoid	NOUN
ejpam-4793	6	41	of	of	ADP
ejpam-4793	6	42	a	a	DET
ejpam-4793	6	43	row	row	NOUN
ejpam-4793	6	44	-	-	PUNCT
ejpam-4793	6	45	finite	finite	ADJ
ejpam-4793	6	46	directed	direct	VERB
ejpam-4793	6	47	graph	graph	NOUN
ejpam-4793	6	48	e	e	NOUN
ejpam-4793	6	49	=	=	PUNCT
ejpam-4793	6	50	(	(	PUNCT
ejpam-4793	6	51	e0	e0	PROPN
ejpam-4793	6	52	,	,	PUNCT
ejpam-4793	6	53	e1	e1	PROPN
ejpam-4793	6	54	,	,	PUNCT
ejpam-4793	6	55	r	r	NOUN
ejpam-4793	6	56	,	,	PUNCT
ejpam-4793	6	57	s	s	PART
ejpam-4793	6	58	)	)	PUNCT
ejpam-4793	6	59	,	,	PUNCT
ejpam-4793	6	60	denoted	denote	VERB
ejpam-4793	6	61	by	by	ADP
ejpam-4793	6	62	te	te	PROPN
ejpam-4793	6	63	,	,	PUNCT
ejpam-4793	6	64	is	be	AUX
ejpam-4793	6	65	the	the	DET
ejpam-4793	6	66	commutative	commutative	ADJ
ejpam-4793	6	67	monoid	monoid	NOUN
ejpam-4793	6	68	generated	generate	VERB
ejpam-4793	6	69	by	by	ADP
ejpam-4793	6	70	{	{	PUNCT
ejpam-4793	6	71	v(i	v(i	NUM
ejpam-4793	6	72	)	)	PUNCT
ejpam-4793	6	73	:	:	PUNCT
ejpam-4793	6	74	v	v	X
ejpam-4793	6	75	∈	∈	PROPN
ejpam-4793	6	76	e0	e0	NOUN
ejpam-4793	6	77	,	,	PUNCT
ejpam-4793	7	1	i	i	PROPN
ejpam-4793	7	2	∈	∈	PROPN
ejpam-4793	8	1	z	z	AUX
ejpam-4793	8	2	}	}	PUNCT
ejpam-4793	8	3	such	such	ADJ
ejpam-4793	8	4	that	that	DET
ejpam-4793	8	5	v(i	v(i	NOUN
ejpam-4793	8	6	)	)	PUNCT
ejpam-4793	9	1	=	=	NOUN
ejpam-4793	9	2	∑	∑	PUNCT
ejpam-4793	9	3	e∈s−1(v	e∈s−1(v	PROPN
ejpam-4793	9	4	)	)	PUNCT
ejpam-4793	9	5	r(e)(i	r(e)(i	NOUN
ejpam-4793	10	1	+	+	CCONJ
ejpam-4793	10	2	1	1	X
ejpam-4793	10	3	)	)	PUNCT
ejpam-4793	10	4	for	for	ADP
ejpam-4793	10	5	every	every	DET
ejpam-4793	10	6	i	i	PROPN
ejpam-4793	10	7	∈	∈	PROPN
ejpam-4793	10	8	z	z	NOUN
ejpam-4793	10	9	and	and	CCONJ
ejpam-4793	10	10	every	every	DET
ejpam-4793	10	11	v	v	NOUN
ejpam-4793	10	12	∈	∈	PROPN
ejpam-4793	10	13	e0	e0	NOUN
ejpam-4793	10	14	that	that	PRON
ejpam-4793	10	15	is	be	AUX
ejpam-4793	10	16	not	not	PART
ejpam-4793	10	17	a	a	DET
ejpam-4793	10	18	sink	sink	NOUN
ejpam-4793	10	19	.	.	PUNCT
ejpam-4793	11	1	the	the	DET
ejpam-4793	11	2	additive	additive	ADJ
ejpam-4793	11	3	group	group	NOUN
ejpam-4793	11	4	z	z	PROPN
ejpam-4793	11	5	of	of	ADP
ejpam-4793	11	6	integers	integer	NOUN
ejpam-4793	11	7	acts	act	VERB
ejpam-4793	11	8	on	on	ADP
ejpam-4793	11	9	te	te	ADP
ejpam-4793	11	10	by	by	ADP
ejpam-4793	11	11	monoid	monoid	NOUN
ejpam-4793	11	12	automorphisms	automorphisms	PROPN
ejpam-4793	11	13	by	by	ADP
ejpam-4793	11	14	shifting	shift	VERB
ejpam-4793	11	15	indices	index	NOUN
ejpam-4793	11	16	:	:	PUNCT
ejpam-4793	11	17	for	for	ADP
ejpam-4793	11	18	each	each	DET
ejpam-4793	11	19	n	n	NOUN
ejpam-4793	11	20	,	,	PUNCT
ejpam-4793	11	21	i	i	PRON
ejpam-4793	11	22	∈	∈	PROPN
ejpam-4793	11	23	z	z	NOUN
ejpam-4793	11	24	and	and	CCONJ
ejpam-4793	11	25	v	v	ADP
ejpam-4793	11	26	∈	∈	PROPN
ejpam-4793	11	27	e0	e0	NOUN
ejpam-4793	11	28	,	,	PUNCT
ejpam-4793	11	29	define	define	VERB
ejpam-4793	11	30	nv(i	nv(i	NOUN
ejpam-4793	11	31	)	)	PUNCT
ejpam-4793	11	32	=	=	PUNCT
ejpam-4793	12	1	v(i	v(i	ADJ
ejpam-4793	12	2	+	+	CCONJ
ejpam-4793	12	3	n	n	CCONJ
ejpam-4793	12	4	)	)	PUNCT
ejpam-4793	12	5	,	,	PUNCT
ejpam-4793	12	6	which	which	PRON
ejpam-4793	12	7	extends	extend	VERB
ejpam-4793	12	8	to	to	ADP
ejpam-4793	12	9	an	an	DET
ejpam-4793	12	10	action	action	NOUN
ejpam-4793	12	11	of	of	ADP
ejpam-4793	12	12	z	z	NOUN
ejpam-4793	12	13	on	on	ADP
ejpam-4793	12	14	te	te	PROPN
ejpam-4793	12	15	[	[	X
ejpam-4793	12	16	3	3	NUM
ejpam-4793	12	17	]	]	PUNCT
ejpam-4793	12	18	.	.	PUNCT
ejpam-4793	13	1	monoids	monoid	NOUN
ejpam-4793	13	2	with	with	ADP
ejpam-4793	13	3	a	a	DET
ejpam-4793	13	4	group	group	NOUN
ejpam-4793	13	5	γ	γ	NOUN
ejpam-4793	13	6	acting	act	VERB
ejpam-4793	13	7	(	(	PUNCT
ejpam-4793	13	8	by	by	ADP
ejpam-4793	13	9	monoid	monoid	NOUN
ejpam-4793	13	10	automorphisms	automorphisms	PROPN
ejpam-4793	13	11	)	)	PUNCT
ejpam-4793	13	12	on	on	ADP
ejpam-4793	13	13	it	it	PRON
ejpam-4793	13	14	,	,	PUNCT
ejpam-4793	13	15	called	call	VERB
ejpam-4793	13	16	γ	γ	NOUN
ejpam-4793	13	17	-	-	PUNCT
ejpam-4793	13	18	monoids	monoid	NOUN
ejpam-4793	13	19	,	,	PUNCT
ejpam-4793	13	20	was	be	AUX
ejpam-4793	13	21	first	first	ADV
ejpam-4793	13	22	introduced	introduce	VERB
ejpam-4793	13	23	in	in	ADP
ejpam-4793	13	24	the	the	DET
ejpam-4793	13	25	paper	paper	NOUN
ejpam-4793	13	26	of	of	ADP
ejpam-4793	13	27	hazrat	hazrat	NOUN
ejpam-4793	13	28	and	and	CCONJ
ejpam-4793	13	29	li	li	NOUN
ejpam-4793	14	1	[	[	X
ejpam-4793	14	2	1	1	NUM
ejpam-4793	14	3	]	]	PUNCT
ejpam-4793	14	4	as	as	ADP
ejpam-4793	14	5	a	a	DET
ejpam-4793	14	6	tool	tool	NOUN
ejpam-4793	14	7	in	in	ADP
ejpam-4793	14	8	the	the	DET
ejpam-4793	14	9	study	study	NOUN
ejpam-4793	14	10	of	of	ADP
ejpam-4793	14	11	talented	talented	ADJ
ejpam-4793	14	12	monoids	monoid	NOUN
ejpam-4793	14	13	.	.	PUNCT
ejpam-4793	15	1	in	in	ADP
ejpam-4793	15	2	the	the	DET
ejpam-4793	15	3	same	same	ADJ
ejpam-4793	15	4	paper	paper	NOUN
ejpam-4793	15	5	,	,	PUNCT
ejpam-4793	15	6	γ	γ	NOUN
ejpam-4793	15	7	-	-	PUNCT
ejpam-4793	15	8	order	order	NOUN
ejpam-4793	15	9	-	-	PUNCT
ejpam-4793	15	10	ideals	ideal	NOUN
ejpam-4793	15	11	of	of	ADP
ejpam-4793	15	12	γ	γ	NOUN
ejpam-4793	15	13	-	-	PUNCT
ejpam-4793	15	14	monoids	monoid	NOUN
ejpam-4793	15	15	are	be	AUX
ejpam-4793	15	16	also	also	ADV
ejpam-4793	15	17	introduced	introduce	VERB
ejpam-4793	15	18	.	.	PUNCT
ejpam-4793	16	1	sebandal	sebandal	NOUN
ejpam-4793	16	2	and	and	CCONJ
ejpam-4793	16	3	vilela	vilela	PROPN
ejpam-4793	17	1	[	[	X
ejpam-4793	17	2	5	5	X
ejpam-4793	17	3	]	]	PUNCT
ejpam-4793	17	4	prove	prove	VERB
ejpam-4793	17	5	some	some	DET
ejpam-4793	17	6	properties	property	NOUN
ejpam-4793	17	7	,	,	PUNCT
ejpam-4793	17	8	including	include	VERB
ejpam-4793	17	9	the	the	DET
ejpam-4793	17	10	isomorphism	isomorphism	NOUN
ejpam-4793	17	11	theorems	theorem	NOUN
ejpam-4793	17	12	for	for	ADP
ejpam-4793	17	13	γ	γ	NOUN
ejpam-4793	17	14	-	-	PUNCT
ejpam-4793	17	15	monoids	monoid	NOUN
ejpam-4793	17	16	and	and	CCONJ
ejpam-4793	17	17	γ	γ	NOUN
ejpam-4793	17	18	-	-	PUNCT
ejpam-4793	17	19	order	order	NOUN
ejpam-4793	17	20	-	-	PUNCT
ejpam-4793	17	21	ideals	ideal	NOUN
ejpam-4793	17	22	are	be	AUX
ejpam-4793	17	23	established	establish	VERB
ejpam-4793	17	24	.	.	PUNCT
ejpam-4793	18	1	this	this	DET
ejpam-4793	18	2	paper	paper	NOUN
ejpam-4793	18	3	extends	extend	VERB
ejpam-4793	18	4	the	the	DET
ejpam-4793	18	5	study	study	NOUN
ejpam-4793	18	6	of	of	ADP
ejpam-4793	18	7	γ	γ	NOUN
ejpam-4793	18	8	-	-	PUNCT
ejpam-4793	18	9	monoids	monoid	NOUN
ejpam-4793	18	10	by	by	ADP
ejpam-4793	18	11	defining	define	VERB
ejpam-4793	18	12	the	the	DET
ejpam-4793	18	13	concept	concept	NOUN
ejpam-4793	18	14	of	of	ADP
ejpam-4793	18	15	γ	γ	NOUN
ejpam-4793	18	16	-	-	NOUN
ejpam-4793	18	17	ideals	ideal	NOUN
ejpam-4793	18	18	and	and	CCONJ
ejpam-4793	18	19	γ	γ	NOUN
ejpam-4793	18	20	-	-	NOUN
ejpam-4793	18	21	submonoids	submonoid	NOUN
ejpam-4793	18	22	and	and	CCONJ
ejpam-4793	18	23	establishing	establish	VERB
ejpam-4793	18	24	some	some	PRON
ejpam-4793	18	25	of	of	ADP
ejpam-4793	18	26	their	their	PRON
ejpam-4793	18	27	properties	property	NOUN
ejpam-4793	18	28	.	.	PUNCT
ejpam-4793	19	1	moreover	moreover	ADV
ejpam-4793	19	2	,	,	PUNCT
ejpam-4793	19	3	this	this	DET
ejpam-4793	19	4	paper	paper	NOUN
ejpam-4793	19	5	studies	study	NOUN
ejpam-4793	19	6	quotient	quotient	VERB
ejpam-4793	19	7	of	of	ADP
ejpam-4793	19	8	γ	γ	NOUN
ejpam-4793	19	9	-	-	PUNCT
ejpam-4793	19	10	monoids	monoid	NOUN
ejpam-4793	19	11	via	via	ADP
ejpam-4793	19	12	equivalence	equivalence	NOUN
ejpam-4793	19	13	classes	class	NOUN
ejpam-4793	19	14	of	of	ADP
ejpam-4793	19	15	γ	γ	NOUN
ejpam-4793	19	16	-	-	NOUN
ejpam-4793	19	17	submonoids	submonoid	NOUN
ejpam-4793	19	18	and	and	CCONJ
ejpam-4793	19	19	proves	prove	VERB
ejpam-4793	19	20	isomorphism	isomorphism	NOUN
ejpam-4793	19	21	theorems	theorem	NOUN
ejpam-4793	19	22	.	.	PUNCT
ejpam-4793	20	1	∗corresponding	∗corresponde	VERB
ejpam-4793	20	2	author	author	NOUN
ejpam-4793	20	3	.	.	PUNCT
ejpam-4793	21	1	doi	doi	NOUN
ejpam-4793	21	2	:	:	PUNCT
ejpam-4793	21	3	https://doi.org/10.29020/nybg.ejpam.v16i3.4793	https://doi.org/10.29020/nybg.ejpam.v16i3.4793	ADJ
ejpam-4793	21	4	email	email	NOUN
ejpam-4793	21	5	addresses	address	NOUN
ejpam-4793	21	6	:	:	PUNCT
ejpam-4793	21	7	hulsen.sarapuddin@g.msuiit.edu.ph	hulsen.sarapuddin@g.msuiit.edu.ph	PROPN
ejpam-4793	21	8	(	(	PUNCT
ejpam-4793	21	9	h.	h.	PROPN
ejpam-4793	21	10	t.	t.	PROPN
ejpam-4793	21	11	sarapuddin	sarapuddin	PROPN
ejpam-4793	21	12	)	)	PUNCT
ejpam-4793	21	13	,	,	PUNCT
ejpam-4793	21	14	jocelyn.vilela@g.msuiit.edu.ph	jocelyn.vilela@g.msuiit.edu.ph	PROPN
ejpam-4793	21	15	(	(	PUNCT
ejpam-4793	21	16	j.	j.	PROPN
ejpam-4793	21	17	p.	p.	PROPN
ejpam-4793	21	18	vilela	vilela	PROPN
ejpam-4793	21	19	)	)	PUNCT
ejpam-4793	21	20	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4793	21	21	1772	1772	NUM
ejpam-4793	22	1	©	©	PROPN
ejpam-4793	22	2	2023	2023	NUM
ejpam-4793	22	3	ejpam	ejpam	NOUN
ejpam-4793	22	4	all	all	DET
ejpam-4793	22	5	rights	right	NOUN
ejpam-4793	22	6	reserved	reserve	VERB
ejpam-4793	22	7	.	.	PUNCT
ejpam-4793	23	1	h.	h.	PROPN
ejpam-4793	23	2	sarapuddin	sarapuddin	PROPN
ejpam-4793	23	3	,	,	PUNCT
ejpam-4793	23	4	j.	j.	PROPN
ejpam-4793	23	5	vilela	vilela	PROPN
ejpam-4793	23	6	/	/	SYM
ejpam-4793	23	7	eur	eur	PROPN
ejpam-4793	23	8	.	.	PUNCT
ejpam-4793	24	1	j.	j.	PROPN
ejpam-4793	24	2	pure	pure	PROPN
ejpam-4793	24	3	appl	appl	PROPN
ejpam-4793	24	4	.	.	PROPN
ejpam-4793	24	5	math	math	PROPN
ejpam-4793	24	6	,	,	PUNCT
ejpam-4793	24	7	16	16	NUM
ejpam-4793	24	8	(	(	PUNCT
ejpam-4793	24	9	3	3	NUM
ejpam-4793	24	10	)	)	PUNCT
ejpam-4793	24	11	(	(	PUNCT
ejpam-4793	24	12	2023	2023	NUM
ejpam-4793	24	13	)	)	PUNCT
ejpam-4793	24	14	,	,	PUNCT
ejpam-4793	24	15	1772	1772	NUM
ejpam-4793	24	16	-	-	SYM
ejpam-4793	24	17	1793	1793	NUM
ejpam-4793	24	18	1773	1773	NUM
ejpam-4793	24	19	2	2	NUM
ejpam-4793	24	20	.	.	PUNCT
ejpam-4793	24	21	preliminaries	preliminary	NOUN
ejpam-4793	24	22	in	in	ADP
ejpam-4793	24	23	this	this	DET
ejpam-4793	24	24	section	section	NOUN
ejpam-4793	25	1	,	,	PUNCT
ejpam-4793	25	2	we	we	PRON
ejpam-4793	25	3	present	present	VERB
ejpam-4793	25	4	some	some	DET
ejpam-4793	25	5	basic	basic	ADJ
ejpam-4793	25	6	concepts	concept	NOUN
ejpam-4793	25	7	and	and	CCONJ
ejpam-4793	25	8	known	know	VERB
ejpam-4793	25	9	results	result	NOUN
ejpam-4793	25	10	that	that	PRON
ejpam-4793	25	11	are	be	AUX
ejpam-4793	25	12	useful	useful	ADJ
ejpam-4793	25	13	in	in	ADP
ejpam-4793	25	14	this	this	DET
ejpam-4793	25	15	study	study	NOUN
ejpam-4793	25	16	.	.	PUNCT
ejpam-4793	26	1	definition	definition	NOUN
ejpam-4793	26	2	1	1	NUM
ejpam-4793	26	3	.	.	PUNCT
ejpam-4793	27	1	[	[	X
ejpam-4793	27	2	2	2	X
ejpam-4793	27	3	]	]	PUNCT
ejpam-4793	27	4	a	a	DET
ejpam-4793	27	5	semigroup	semigroup	NOUN
ejpam-4793	27	6	is	be	AUX
ejpam-4793	27	7	a	a	DET
ejpam-4793	27	8	nonempty	nonempty	ADJ
ejpam-4793	27	9	set	set	VERB
ejpam-4793	27	10	m	m	PRON
ejpam-4793	27	11	together	together	ADV
ejpam-4793	27	12	with	with	ADP
ejpam-4793	27	13	a	a	DET
ejpam-4793	27	14	binary	binary	ADJ
ejpam-4793	27	15	operation	operation	NOUN
ejpam-4793	27	16	∗	∗	NOUN
ejpam-4793	27	17	on	on	ADP
ejpam-4793	27	18	m	m	PROPN
ejpam-4793	27	19	which	which	PRON
ejpam-4793	27	20	is	be	AUX
ejpam-4793	27	21	associative	associative	ADJ
ejpam-4793	27	22	,	,	PUNCT
ejpam-4793	27	23	that	that	ADV
ejpam-4793	27	24	is	is	ADV
ejpam-4793	27	25	,	,	PUNCT
ejpam-4793	27	26	for	for	ADP
ejpam-4793	27	27	all	all	DET
ejpam-4793	27	28	a	a	DET
ejpam-4793	27	29	,	,	PUNCT
ejpam-4793	27	30	b	b	NOUN
ejpam-4793	27	31	,	,	PUNCT
ejpam-4793	27	32	c	c	PROPN
ejpam-4793	27	33	∈	∈	PROPN
ejpam-4793	27	34	m	m	PROPN
ejpam-4793	27	35	,	,	PUNCT
ejpam-4793	27	36	a	a	DET
ejpam-4793	27	37	∗	∗	NOUN
ejpam-4793	27	38	(	(	PUNCT
ejpam-4793	27	39	b	b	NOUN
ejpam-4793	27	40	∗	∗	NOUN
ejpam-4793	27	41	c	c	NOUN
ejpam-4793	27	42	)	)	PUNCT
ejpam-4793	27	43	=	=	NOUN
ejpam-4793	27	44	(	(	PUNCT
ejpam-4793	27	45	a	a	DET
ejpam-4793	27	46	∗	∗	NOUN
ejpam-4793	27	47	b	b	NOUN
ejpam-4793	27	48	)	)	PUNCT
ejpam-4793	27	49	∗	∗	NOUN
ejpam-4793	27	50	c.	c.	NOUN
ejpam-4793	27	51	definition	definition	NOUN
ejpam-4793	27	52	2	2	NUM
ejpam-4793	27	53	.	.	PUNCT
ejpam-4793	28	1	[	[	X
ejpam-4793	28	2	2	2	X
ejpam-4793	28	3	]	]	PUNCT
ejpam-4793	28	4	a	a	DET
ejpam-4793	28	5	monoid	monoid	NOUN
ejpam-4793	28	6	is	be	AUX
ejpam-4793	28	7	a	a	DET
ejpam-4793	28	8	semigroupm	semigroupm	NOUN
ejpam-4793	28	9	which	which	PRON
ejpam-4793	28	10	contains	contain	VERB
ejpam-4793	28	11	an	an	DET
ejpam-4793	28	12	identity	identity	NOUN
ejpam-4793	28	13	element	element	NOUN
ejpam-4793	28	14	1	1	NUM
ejpam-4793	28	15	m	m	NOUN
ejpam-4793	28	16	∈	∈	NOUN
ejpam-4793	28	17	m	m	VERB
ejpam-4793	28	18	such	such	ADJ
ejpam-4793	28	19	that	that	SCONJ
ejpam-4793	28	20	1	1	NUM
ejpam-4793	28	21	m	m	NOUN
ejpam-4793	28	22	∗m	∗m	NOUN
ejpam-4793	28	23	=	=	PUNCT
ejpam-4793	28	24	m	m	NOUN
ejpam-4793	28	25	∗	∗	NOUN
ejpam-4793	28	26	1	1	NUM
ejpam-4793	28	27	m	m	NOUN
ejpam-4793	28	28	=	=	NOUN
ejpam-4793	28	29	m	m	VERB
ejpam-4793	28	30	for	for	ADP
ejpam-4793	28	31	all	all	DET
ejpam-4793	28	32	m	m	NOUN
ejpam-4793	28	33	∈	∈	NOUN
ejpam-4793	28	34	m	m	NOUN
ejpam-4793	28	35	.	.	PUNCT
ejpam-4793	29	1	for	for	ADP
ejpam-4793	29	2	a	a	DET
ejpam-4793	29	3	monoid	monoid	NOUN
ejpam-4793	29	4	m	m	NOUN
ejpam-4793	29	5	with	with	ADP
ejpam-4793	29	6	the	the	DET
ejpam-4793	29	7	binary	binary	PROPN
ejpam-4793	29	8	operation	operation	NOUN
ejpam-4793	29	9	∗	∗	NOUN
ejpam-4793	29	10	,	,	PUNCT
ejpam-4793	29	11	we	we	PRON
ejpam-4793	29	12	may	may	AUX
ejpam-4793	29	13	also	also	ADV
ejpam-4793	29	14	say	say	VERB
ejpam-4793	29	15	that	that	SCONJ
ejpam-4793	29	16	m	m	PROPN
ejpam-4793	29	17	is	be	AUX
ejpam-4793	29	18	a	a	DET
ejpam-4793	29	19	monoid	monoid	NOUN
ejpam-4793	29	20	under	under	ADP
ejpam-4793	29	21	∗.	∗.	PROPN
ejpam-4793	29	22	a	a	DET
ejpam-4793	29	23	monoid	monoid	NOUN
ejpam-4793	29	24	m	m	NOUN
ejpam-4793	29	25	is	be	AUX
ejpam-4793	29	26	said	say	VERB
ejpam-4793	29	27	to	to	PART
ejpam-4793	29	28	be	be	AUX
ejpam-4793	29	29	commutative	commutative	ADJ
ejpam-4793	29	30	if	if	SCONJ
ejpam-4793	29	31	for	for	ADP
ejpam-4793	29	32	all	all	DET
ejpam-4793	29	33	x	x	NOUN
ejpam-4793	29	34	,	,	PUNCT
ejpam-4793	29	35	y	y	PROPN
ejpam-4793	29	36	∈	∈	PROPN
ejpam-4793	29	37	m	m	VERB
ejpam-4793	29	38	,	,	PUNCT
ejpam-4793	29	39	x	x	PROPN
ejpam-4793	29	40	∗	∗	NOUN
ejpam-4793	29	41	y	y	NOUN
ejpam-4793	29	42	=	=	SYM
ejpam-4793	29	43	y	y	PROPN
ejpam-4793	29	44	∗	∗	NOUN
ejpam-4793	29	45	x.	x.	NOUN
ejpam-4793	30	1	if	if	SCONJ
ejpam-4793	30	2	no	no	DET
ejpam-4793	30	3	confusion	confusion	NOUN
ejpam-4793	30	4	arises	arise	VERB
ejpam-4793	30	5	,	,	PUNCT
ejpam-4793	30	6	by	by	ADP
ejpam-4793	30	7	a	a	DET
ejpam-4793	30	8	monoid	monoid	NOUN
ejpam-4793	30	9	m	m	VERB
ejpam-4793	30	10	,	,	PUNCT
ejpam-4793	30	11	we	we	PRON
ejpam-4793	30	12	shall	shall	AUX
ejpam-4793	30	13	mean	mean	VERB
ejpam-4793	30	14	a	a	DET
ejpam-4793	30	15	triple	triple	ADJ
ejpam-4793	30	16	(	(	PUNCT
ejpam-4793	30	17	m	m	PROPN
ejpam-4793	30	18	,	,	PUNCT
ejpam-4793	30	19	1	1	NUM
ejpam-4793	30	20	m	m	NOUN
ejpam-4793	30	21	,	,	PUNCT
ejpam-4793	30	22	∗	∗	NOUN
ejpam-4793	30	23	)	)	PUNCT
ejpam-4793	30	24	unless	unless	SCONJ
ejpam-4793	30	25	otherwise	otherwise	ADV
ejpam-4793	30	26	specified	specify	VERB
ejpam-4793	30	27	.	.	PUNCT
ejpam-4793	31	1	definition	definition	NOUN
ejpam-4793	31	2	3	3	NUM
ejpam-4793	31	3	.	.	PUNCT
ejpam-4793	32	1	[	[	X
ejpam-4793	32	2	6	6	NUM
ejpam-4793	32	3	]	]	X
ejpam-4793	32	4	let	let	VERB
ejpam-4793	32	5	(	(	PUNCT
ejpam-4793	32	6	m	m	NOUN
ejpam-4793	32	7	,	,	PUNCT
ejpam-4793	32	8	∗	∗	NOUN
ejpam-4793	32	9	)	)	PUNCT
ejpam-4793	32	10	be	be	VERB
ejpam-4793	32	11	a	a	DET
ejpam-4793	32	12	monoid	monoid	NOUN
ejpam-4793	32	13	.	.	PUNCT
ejpam-4793	33	1	a	a	DET
ejpam-4793	33	2	submonoid	submonoid	NOUN
ejpam-4793	33	3	is	be	AUX
ejpam-4793	33	4	a	a	DET
ejpam-4793	33	5	subset	subset	NOUN
ejpam-4793	33	6	s	s	NOUN
ejpam-4793	33	7	of	of	ADP
ejpam-4793	33	8	m	m	PRON
ejpam-4793	33	9	which	which	PRON
ejpam-4793	33	10	is	be	AUX
ejpam-4793	33	11	closed	close	VERB
ejpam-4793	33	12	under	under	ADP
ejpam-4793	33	13	the	the	DET
ejpam-4793	33	14	binary	binary	ADJ
ejpam-4793	33	15	operation	operation	NOUN
ejpam-4793	33	16	on	on	ADP
ejpam-4793	33	17	m	m	NOUN
ejpam-4793	33	18	and	and	CCONJ
ejpam-4793	33	19	contains	contain	VERB
ejpam-4793	33	20	the	the	DET
ejpam-4793	33	21	identity	identity	NOUN
ejpam-4793	33	22	1	1	NUM
ejpam-4793	33	23	m	m	NOUN
ejpam-4793	33	24	of	of	ADP
ejpam-4793	33	25	m	m	PROPN
ejpam-4793	33	26	.	.	PUNCT
ejpam-4793	34	1	definition	definition	NOUN
ejpam-4793	34	2	4	4	NUM
ejpam-4793	34	3	.	.	PUNCT
ejpam-4793	35	1	[	[	X
ejpam-4793	35	2	6	6	NUM
ejpam-4793	35	3	]	]	X
ejpam-4793	35	4	let	let	VERB
ejpam-4793	35	5	(	(	PUNCT
ejpam-4793	35	6	m	m	NOUN
ejpam-4793	35	7	,	,	PUNCT
ejpam-4793	35	8	∗	∗	NOUN
ejpam-4793	35	9	)	)	PUNCT
ejpam-4793	35	10	and	and	CCONJ
ejpam-4793	35	11	(	(	PUNCT
ejpam-4793	35	12	n	n	CCONJ
ejpam-4793	35	13	,	,	PUNCT
ejpam-4793	35	14	·	·	PUNCT
ejpam-4793	35	15	)	)	PUNCT
ejpam-4793	35	16	be	be	AUX
ejpam-4793	35	17	monoids	monoid	NOUN
ejpam-4793	35	18	.	.	PUNCT
ejpam-4793	36	1	a	a	DET
ejpam-4793	36	2	monoid	monoid	NOUN
ejpam-4793	36	3	homomorphism	homomorphism	NOUN
ejpam-4793	36	4	is	be	AUX
ejpam-4793	36	5	a	a	DET
ejpam-4793	36	6	mapping	mapping	NOUN
ejpam-4793	36	7	φ	φ	NOUN
ejpam-4793	36	8	:	:	PUNCT
ejpam-4793	36	9	m	m	VERB
ejpam-4793	36	10	→	→	SYM
ejpam-4793	36	11	n	n	CCONJ
ejpam-4793	36	12	such	such	ADJ
ejpam-4793	36	13	that	that	SCONJ
ejpam-4793	36	14	φ(a	φ(a	ADJ
ejpam-4793	36	15	∗	∗	X
ejpam-4793	36	16	b	b	NOUN
ejpam-4793	36	17	)	)	PUNCT
ejpam-4793	36	18	=	=	SYM
ejpam-4793	36	19	φ(a	φ(a	ADJ
ejpam-4793	36	20	)	)	PUNCT
ejpam-4793	36	21	·	·	PUNCT
ejpam-4793	37	1	φ(b	φ(b	ADV
ejpam-4793	37	2	)	)	PUNCT
ejpam-4793	37	3	and	and	CCONJ
ejpam-4793	37	4	φ(1	φ(1	PROPN
ejpam-4793	37	5	m	m	NOUN
ejpam-4793	37	6	)	)	PUNCT
ejpam-4793	38	1	=	=	SYM
ejpam-4793	38	2	1n	1n	NUM
ejpam-4793	38	3	for	for	ADP
ejpam-4793	38	4	all	all	DET
ejpam-4793	38	5	a	a	PRON
ejpam-4793	38	6	,	,	PUNCT
ejpam-4793	38	7	b	b	X
ejpam-4793	38	8	∈	∈	PROPN
ejpam-4793	38	9	m	m	VERB
ejpam-4793	38	10	where	where	SCONJ
ejpam-4793	38	11	1	1	NUM
ejpam-4793	38	12	m	m	NOUN
ejpam-4793	38	13	and	and	CCONJ
ejpam-4793	38	14	1n	1n	NUM
ejpam-4793	38	15	are	be	AUX
ejpam-4793	38	16	the	the	DET
ejpam-4793	38	17	identities	identity	NOUN
ejpam-4793	38	18	in	in	ADP
ejpam-4793	38	19	m	m	PROPN
ejpam-4793	38	20	and	and	CCONJ
ejpam-4793	38	21	n	n	CCONJ
ejpam-4793	38	22	,	,	PUNCT
ejpam-4793	38	23	respectively	respectively	ADV
ejpam-4793	38	24	.	.	PUNCT
ejpam-4793	38	25	example	example	NOUN
ejpam-4793	39	1	1	1	NUM
ejpam-4793	39	2	.	.	X
ejpam-4793	39	3	consider	consider	VERB
ejpam-4793	39	4	the	the	DET
ejpam-4793	39	5	monoids	monoid	NOUN
ejpam-4793	39	6	m	m	VERB
ejpam-4793	39	7	=	=	SYM
ejpam-4793	39	8	(	(	PUNCT
ejpam-4793	39	9	n,+	n,+	NUM
ejpam-4793	39	10	)	)	PUNCT
ejpam-4793	39	11	and	and	CCONJ
ejpam-4793	39	12	n	n	CCONJ
ejpam-4793	39	13	=	=	SYM
ejpam-4793	39	14	(	(	PUNCT
ejpam-4793	39	15	n	n	CCONJ
ejpam-4793	39	16	,	,	PUNCT
ejpam-4793	39	17	·	·	PUNCT
ejpam-4793	39	18	)	)	PUNCT
ejpam-4793	39	19	and	and	CCONJ
ejpam-4793	39	20	the	the	DET
ejpam-4793	39	21	mapping	mapping	NOUN
ejpam-4793	39	22	φ	φ	NOUN
ejpam-4793	39	23	:	:	PUNCT
ejpam-4793	39	24	m	m	PROPN
ejpam-4793	39	25	→	→	SYM
ejpam-4793	39	26	n	n	PRON
ejpam-4793	39	27	defined	define	VERB
ejpam-4793	39	28	by	by	ADP
ejpam-4793	39	29	φ(x	φ(x	NOUN
ejpam-4793	39	30	)	)	PUNCT
ejpam-4793	39	31	=	=	SYM
ejpam-4793	39	32	bx	bx	PROPN
ejpam-4793	39	33	,	,	PUNCT
ejpam-4793	39	34	where	where	SCONJ
ejpam-4793	39	35	b	b	X
ejpam-4793	39	36	∈	∈	PROPN
ejpam-4793	39	37	n	n	PRON
ejpam-4793	39	38	\	\	NOUN
ejpam-4793	39	39	{	{	PUNCT
ejpam-4793	39	40	0	0	NUM
ejpam-4793	39	41	}	}	PUNCT
ejpam-4793	39	42	.	.	PUNCT
ejpam-4793	40	1	for	for	ADP
ejpam-4793	40	2	any	any	DET
ejpam-4793	40	3	x	x	NOUN
ejpam-4793	40	4	,	,	PUNCT
ejpam-4793	40	5	y	y	PROPN
ejpam-4793	40	6	∈	∈	PROPN
ejpam-4793	40	7	m	m	VERB
ejpam-4793	40	8	,	,	PUNCT
ejpam-4793	40	9	we	we	PRON
ejpam-4793	40	10	have	have	VERB
ejpam-4793	40	11	φ(x+	φ(x+	PROPN
ejpam-4793	40	12	y	y	X
ejpam-4793	40	13	)	)	PUNCT
ejpam-4793	41	1	=	=	SYM
ejpam-4793	41	2	bx+y	bx+y	NUM
ejpam-4793	41	3	=	=	PUNCT
ejpam-4793	41	4	bx	bx	X
ejpam-4793	41	5	·	·	PUNCT
ejpam-4793	41	6	by	by	ADP
ejpam-4793	41	7	=	=	ADJ
ejpam-4793	41	8	φ(x	φ(x	X
ejpam-4793	41	9	)	)	PUNCT
ejpam-4793	41	10	·	·	PUNCT
ejpam-4793	41	11	φ(y	φ(y	NOUN
ejpam-4793	41	12	)	)	PUNCT
ejpam-4793	41	13	and	and	CCONJ
ejpam-4793	41	14	φ(0	φ(0	ADJ
ejpam-4793	41	15	)	)	PUNCT
ejpam-4793	41	16	=	=	SYM
ejpam-4793	41	17	b0	b0	NOUN
ejpam-4793	41	18	=	=	SYM
ejpam-4793	41	19	1	1	NUM
ejpam-4793	41	20	.	.	PUNCT
ejpam-4793	41	21	therefore	therefore	ADV
ejpam-4793	41	22	,	,	PUNCT
ejpam-4793	41	23	φ	φ	PROPN
ejpam-4793	41	24	is	be	AUX
ejpam-4793	41	25	a	a	DET
ejpam-4793	41	26	monoid	monoid	NOUN
ejpam-4793	41	27	homomorphism	homomorphism	NOUN
ejpam-4793	41	28	.	.	PUNCT
ejpam-4793	42	1	definition	definition	NOUN
ejpam-4793	42	2	5	5	NUM
ejpam-4793	42	3	.	.	PUNCT
ejpam-4793	43	1	[	[	X
ejpam-4793	43	2	6	6	NUM
ejpam-4793	43	3	]	]	PUNCT
ejpam-4793	43	4	a	a	DET
ejpam-4793	43	5	congruence	congruence	NOUN
ejpam-4793	43	6	on	on	ADP
ejpam-4793	43	7	a	a	DET
ejpam-4793	43	8	monoid	monoid	NOUN
ejpam-4793	43	9	m	m	NOUN
ejpam-4793	43	10	is	be	AUX
ejpam-4793	43	11	an	an	DET
ejpam-4793	43	12	equivalence	equivalence	NOUN
ejpam-4793	43	13	relation	relation	NOUN
ejpam-4793	43	14	ρ	ρ	PROPN
ejpam-4793	43	15	on	on	ADP
ejpam-4793	43	16	m	m	PROPN
ejpam-4793	43	17	which	which	PRON
ejpam-4793	43	18	satisfies	satisfy	VERB
ejpam-4793	43	19	the	the	DET
ejpam-4793	43	20	condition	condition	NOUN
ejpam-4793	43	21	:	:	PUNCT
ejpam-4793	43	22	for	for	ADP
ejpam-4793	43	23	all	all	DET
ejpam-4793	43	24	u	u	NOUN
ejpam-4793	43	25	,	,	PUNCT
ejpam-4793	43	26	v	v	NOUN
ejpam-4793	43	27	,	,	PUNCT
ejpam-4793	43	28	x	x	PRON
ejpam-4793	43	29	,	,	PUNCT
ejpam-4793	43	30	y	y	PROPN
ejpam-4793	43	31	∈	∈	PROPN
ejpam-4793	43	32	m	m	INTJ
ejpam-4793	43	33	,	,	PUNCT
ejpam-4793	43	34	if	if	SCONJ
ejpam-4793	43	35	xρy	xρy	PROPN
ejpam-4793	43	36	,	,	PUNCT
ejpam-4793	43	37	then	then	ADV
ejpam-4793	43	38	(	(	PUNCT
ejpam-4793	43	39	u	u	NOUN
ejpam-4793	43	40	∗	∗	X
ejpam-4793	43	41	x	x	SYM
ejpam-4793	43	42	∗	∗	X
ejpam-4793	43	43	v)ρ(u	v)ρ(u	NOUN
ejpam-4793	43	44	∗	∗	NOUN
ejpam-4793	43	45	y	y	PROPN
ejpam-4793	43	46	∗	∗	NOUN
ejpam-4793	43	47	v	v	NOUN
ejpam-4793	43	48	)	)	PUNCT
ejpam-4793	43	49	.	.	PUNCT
ejpam-4793	44	1	proposition	proposition	NOUN
ejpam-4793	44	2	1	1	NUM
ejpam-4793	44	3	.	.	PUNCT
ejpam-4793	45	1	[	[	X
ejpam-4793	45	2	6	6	NUM
ejpam-4793	45	3	]	]	PUNCT
ejpam-4793	45	4	let	let	VERB
ejpam-4793	45	5	ρ	ρ	NOUN
ejpam-4793	45	6	be	be	AUX
ejpam-4793	45	7	a	a	DET
ejpam-4793	45	8	congruence	congruence	NOUN
ejpam-4793	45	9	on	on	ADP
ejpam-4793	45	10	a	a	DET
ejpam-4793	45	11	monoid	monoid	NOUN
ejpam-4793	45	12	m	m	NOUN
ejpam-4793	45	13	.	.	PUNCT
ejpam-4793	46	1	then	then	ADV
ejpam-4793	46	2	m	m	PROPN
ejpam-4793	46	3	/	/	SYM
ejpam-4793	46	4	ρ	ρ	PROPN
ejpam-4793	46	5	is	be	AUX
ejpam-4793	46	6	a	a	DET
ejpam-4793	46	7	monoid	monoid	NOUN
ejpam-4793	46	8	with	with	ADP
ejpam-4793	46	9	binary	binary	ADJ
ejpam-4793	46	10	operation	operation	NOUN
ejpam-4793	46	11	◦	◦	NOUN
ejpam-4793	46	12	given	give	VERB
ejpam-4793	46	13	by	by	ADP
ejpam-4793	46	14	ρ(x	ρ(x	NUM
ejpam-4793	46	15	)	)	PUNCT
ejpam-4793	46	16	◦	◦	NOUN
ejpam-4793	46	17	ρ(y	ρ(y	NOUN
ejpam-4793	46	18	)	)	PUNCT
ejpam-4793	47	1	=	=	PUNCT
ejpam-4793	47	2	ρ(x	ρ(x	PROPN
ejpam-4793	47	3	∗	∗	NOUN
ejpam-4793	47	4	y	y	NOUN
ejpam-4793	47	5	)	)	PUNCT
ejpam-4793	47	6	for	for	ADP
ejpam-4793	47	7	all	all	DET
ejpam-4793	47	8	x	x	NOUN
ejpam-4793	47	9	,	,	PUNCT
ejpam-4793	47	10	y	y	PROPN
ejpam-4793	47	11	∈	∈	PROPN
ejpam-4793	47	12	m	m	VERB
ejpam-4793	47	13	.	.	PUNCT
ejpam-4793	48	1	definition	definition	NOUN
ejpam-4793	48	2	6	6	NUM
ejpam-4793	48	3	.	.	PUNCT
ejpam-4793	49	1	[	[	X
ejpam-4793	49	2	4	4	X
ejpam-4793	49	3	]	]	PUNCT
ejpam-4793	49	4	let	let	AUX
ejpam-4793	49	5	m	m	PRON
ejpam-4793	49	6	be	be	AUX
ejpam-4793	49	7	a	a	DET
ejpam-4793	49	8	commutative	commutative	ADJ
ejpam-4793	49	9	monoid	monoid	NOUN
ejpam-4793	49	10	.	.	PUNCT
ejpam-4793	50	1	for	for	ADP
ejpam-4793	50	2	any	any	DET
ejpam-4793	50	3	submonoid	submonoid	ADJ
ejpam-4793	50	4	h	h	NOUN
ejpam-4793	50	5	of	of	ADP
ejpam-4793	50	6	m	m	PROPN
ejpam-4793	50	7	,	,	PUNCT
ejpam-4793	50	8	we	we	PRON
ejpam-4793	50	9	define	define	VERB
ejpam-4793	50	10	a	a	DET
ejpam-4793	50	11	binary	binary	ADJ
ejpam-4793	50	12	relation	relation	NOUN
ejpam-4793	50	13	ρh	ρh	VERB
ejpam-4793	50	14	in	in	ADP
ejpam-4793	50	15	m	m	PROPN
ejpam-4793	50	16	by	by	ADP
ejpam-4793	50	17	xρhy	xρhy	PROPN
ejpam-4793	50	18	if	if	SCONJ
ejpam-4793	50	19	and	and	CCONJ
ejpam-4793	50	20	only	only	ADV
ejpam-4793	50	21	if	if	SCONJ
ejpam-4793	50	22	(	(	PUNCT
ejpam-4793	50	23	x	x	NOUN
ejpam-4793	50	24	∗h	∗h	NOUN
ejpam-4793	50	25	)	)	PUNCT
ejpam-4793	50	26	∩	∩	NOUN
ejpam-4793	50	27	(	(	PUNCT
ejpam-4793	50	28	y	y	PROPN
ejpam-4793	50	29	∗h	∗h	PROPN
ejpam-4793	50	30	)	)	PUNCT
ejpam-4793	50	31	̸=	̸=	PROPN
ejpam-4793	50	32	∅.	∅.	ADV
ejpam-4793	50	33	remark	remark	NOUN
ejpam-4793	50	34	1	1	NUM
ejpam-4793	50	35	.	.	PUNCT
ejpam-4793	51	1	[	[	X
ejpam-4793	51	2	4	4	X
ejpam-4793	51	3	]	]	PUNCT
ejpam-4793	51	4	for	for	ADP
ejpam-4793	51	5	any	any	DET
ejpam-4793	51	6	submonoid	submonoid	ADJ
ejpam-4793	51	7	h	h	NOUN
ejpam-4793	51	8	of	of	ADP
ejpam-4793	51	9	a	a	DET
ejpam-4793	51	10	commutative	commutative	ADJ
ejpam-4793	51	11	monoid	monoid	NOUN
ejpam-4793	51	12	m	m	PROPN
ejpam-4793	51	13	,	,	PUNCT
ejpam-4793	51	14	ρh	ρh	PROPN
ejpam-4793	51	15	is	be	AUX
ejpam-4793	51	16	an	an	DET
ejpam-4793	51	17	equivalence	equivalence	NOUN
ejpam-4793	51	18	relation	relation	NOUN
ejpam-4793	51	19	on	on	ADP
ejpam-4793	51	20	m	m	PROPN
ejpam-4793	51	21	.	.	PUNCT
ejpam-4793	52	1	definition	definition	NOUN
ejpam-4793	52	2	7	7	NUM
ejpam-4793	52	3	.	.	PUNCT
ejpam-4793	53	1	[	[	X
ejpam-4793	53	2	2	2	X
ejpam-4793	53	3	]	]	PUNCT
ejpam-4793	53	4	an	an	DET
ejpam-4793	53	5	action	action	NOUN
ejpam-4793	53	6	of	of	ADP
ejpam-4793	53	7	a	a	DET
ejpam-4793	53	8	group	group	NOUN
ejpam-4793	53	9	(	(	PUNCT
ejpam-4793	53	10	g	g	NOUN
ejpam-4793	53	11	,	,	PUNCT
ejpam-4793	53	12	◦	◦	NOUN
ejpam-4793	53	13	)	)	PUNCT
ejpam-4793	53	14	in	in	ADP
ejpam-4793	53	15	a	a	DET
ejpam-4793	53	16	set	set	NOUN
ejpam-4793	53	17	s	s	PART
ejpam-4793	53	18	is	be	AUX
ejpam-4793	53	19	a	a	DET
ejpam-4793	53	20	function	function	NOUN
ejpam-4793	53	21	ϕ	ϕ	NOUN
ejpam-4793	53	22	:	:	PUNCT
ejpam-4793	53	23	g×s	g×s	PROPN
ejpam-4793	53	24	−→	−→	NOUN
ejpam-4793	53	25	s	s	VERB
ejpam-4793	53	26	such	such	ADJ
ejpam-4793	53	27	that	that	PRON
ejpam-4793	53	28	for	for	ADP
ejpam-4793	53	29	all	all	DET
ejpam-4793	53	30	x	x	SYM
ejpam-4793	53	31	∈	∈	PROPN
ejpam-4793	53	32	s	s	NOUN
ejpam-4793	53	33	,	,	PUNCT
ejpam-4793	53	34	and	and	CCONJ
ejpam-4793	53	35	g1	g1	NOUN
ejpam-4793	53	36	,	,	PUNCT
ejpam-4793	53	37	g2	g2	PROPN
ejpam-4793	53	38	∈	∈	PROPN
ejpam-4793	54	1	g	g	NOUN
ejpam-4793	54	2	:	:	PUNCT
ejpam-4793	54	3	ϕ((1	ϕ((1	PROPN
ejpam-4793	54	4	g	g	PROPN
ejpam-4793	54	5	,	,	PUNCT
ejpam-4793	54	6	x	x	NOUN
ejpam-4793	54	7	)	)	PUNCT
ejpam-4793	54	8	)	)	PUNCT
ejpam-4793	55	1	=	=	PUNCT
ejpam-4793	55	2	x	x	PUNCT
ejpam-4793	55	3	and	and	CCONJ
ejpam-4793	55	4	ϕ((g1	ϕ((g1	PROPN
ejpam-4793	55	5	◦	◦	NOUN
ejpam-4793	55	6	g2	g2	PROPN
ejpam-4793	55	7	,	,	PUNCT
ejpam-4793	55	8	x	x	NOUN
ejpam-4793	55	9	)	)	PUNCT
ejpam-4793	55	10	)	)	PUNCT
ejpam-4793	56	1	=	=	PUNCT
ejpam-4793	56	2	ϕ((g1	ϕ((g1	ADJ
ejpam-4793	56	3	,	,	PUNCT
ejpam-4793	56	4	ϕ((g2	ϕ((g2	PROPN
ejpam-4793	56	5	,	,	PUNCT
ejpam-4793	56	6	x	x	NOUN
ejpam-4793	56	7	)	)	PUNCT
ejpam-4793	56	8	)	)	PUNCT
ejpam-4793	56	9	)	)	PUNCT
ejpam-4793	56	10	)	)	PUNCT
ejpam-4793	56	11	.	.	PUNCT
ejpam-4793	57	1	when	when	SCONJ
ejpam-4793	57	2	such	such	DET
ejpam-4793	57	3	an	an	DET
ejpam-4793	57	4	action	action	NOUN
ejpam-4793	57	5	is	be	AUX
ejpam-4793	57	6	given	give	VERB
ejpam-4793	57	7	,	,	PUNCT
ejpam-4793	57	8	g	g	PROPN
ejpam-4793	57	9	is	be	AUX
ejpam-4793	57	10	said	say	VERB
ejpam-4793	57	11	to	to	PART
ejpam-4793	57	12	act	act	VERB
ejpam-4793	57	13	on	on	ADP
ejpam-4793	57	14	the	the	DET
ejpam-4793	57	15	set	set	NOUN
ejpam-4793	57	16	s.	s.	PROPN
ejpam-4793	57	17	example	example	NOUN
ejpam-4793	57	18	2	2	X
ejpam-4793	57	19	.	.	X
ejpam-4793	57	20	consider	consider	VERB
ejpam-4793	57	21	the	the	DET
ejpam-4793	57	22	group	group	NOUN
ejpam-4793	57	23	g	g	NOUN
ejpam-4793	57	24	=	=	PROPN
ejpam-4793	57	25	z	z	PROPN
ejpam-4793	57	26	under	under	ADP
ejpam-4793	57	27	the	the	DET
ejpam-4793	57	28	usual	usual	ADJ
ejpam-4793	57	29	addition	addition	NOUN
ejpam-4793	57	30	and	and	CCONJ
ejpam-4793	57	31	the	the	DET
ejpam-4793	57	32	set	set	NOUN
ejpam-4793	57	33	s	s	PART
ejpam-4793	57	34	=	=	NOUN
ejpam-4793	57	35	r	r	NOUN
ejpam-4793	57	36	of	of	ADP
ejpam-4793	57	37	real	real	ADJ
ejpam-4793	57	38	numbers	number	NOUN
ejpam-4793	57	39	and	and	CCONJ
ejpam-4793	57	40	the	the	DET
ejpam-4793	57	41	mapping	mapping	NOUN
ejpam-4793	57	42	ϕ	ϕ	NOUN
ejpam-4793	57	43	:	:	PUNCT
ejpam-4793	57	44	g	g	PROPN
ejpam-4793	57	45	×	×	PROPN
ejpam-4793	57	46	s	s	X
ejpam-4793	57	47	→	→	SYM
ejpam-4793	57	48	s	s	AUX
ejpam-4793	57	49	given	give	VERB
ejpam-4793	57	50	by	by	ADP
ejpam-4793	57	51	ϕ((g	ϕ((g	ADJ
ejpam-4793	57	52	,	,	PUNCT
ejpam-4793	57	53	x	x	NOUN
ejpam-4793	57	54	)	)	PUNCT
ejpam-4793	57	55	)	)	PUNCT
ejpam-4793	58	1	=	=	SYM
ejpam-4793	58	2	2gx	2gx	NOUN
ejpam-4793	58	3	.	.	PUNCT
ejpam-4793	59	1	let	let	VERB
ejpam-4793	59	2	(	(	PUNCT
ejpam-4793	59	3	g	g	NOUN
ejpam-4793	59	4	,	,	PUNCT
ejpam-4793	59	5	x	x	NOUN
ejpam-4793	59	6	)	)	PUNCT
ejpam-4793	59	7	,	,	PUNCT
ejpam-4793	59	8	(	(	PUNCT
ejpam-4793	59	9	h	h	NOUN
ejpam-4793	59	10	,	,	PUNCT
ejpam-4793	59	11	y	y	NOUN
ejpam-4793	59	12	)	)	PUNCT
ejpam-4793	59	13	∈	∈	PROPN
ejpam-4793	60	1	g	g	ADP
ejpam-4793	60	2	×	×	PROPN
ejpam-4793	60	3	s	s	VERB
ejpam-4793	60	4	such	such	ADJ
ejpam-4793	60	5	that	that	SCONJ
ejpam-4793	60	6	(	(	PUNCT
ejpam-4793	60	7	g	g	NOUN
ejpam-4793	60	8	,	,	PUNCT
ejpam-4793	60	9	x	x	NOUN
ejpam-4793	60	10	)	)	PUNCT
ejpam-4793	60	11	=	=	SYM
ejpam-4793	60	12	(	(	PUNCT
ejpam-4793	60	13	h	h	NOUN
ejpam-4793	60	14	,	,	PUNCT
ejpam-4793	60	15	y	y	PROPN
ejpam-4793	60	16	)	)	PUNCT
ejpam-4793	60	17	.	.	PUNCT
ejpam-4793	61	1	then	then	ADV
ejpam-4793	61	2	g	g	PROPN
ejpam-4793	61	3	=	=	PROPN
ejpam-4793	61	4	h	h	PROPN
ejpam-4793	61	5	and	and	CCONJ
ejpam-4793	61	6	x	x	X
ejpam-4793	62	1	=	=	SYM
ejpam-4793	62	2	y.	y.	PROPN
ejpam-4793	62	3	thus	thus	ADV
ejpam-4793	62	4	,	,	PUNCT
ejpam-4793	62	5	we	we	PRON
ejpam-4793	62	6	have	have	VERB
ejpam-4793	62	7	ϕ((g	ϕ((g	ADJ
ejpam-4793	62	8	,	,	PUNCT
ejpam-4793	62	9	x	x	NOUN
ejpam-4793	62	10	)	)	PUNCT
ejpam-4793	62	11	)	)	PUNCT
ejpam-4793	63	1	=	=	SYM
ejpam-4793	63	2	2gx	2gx	NOUN
ejpam-4793	63	3	=	=	PUNCT
ejpam-4793	63	4	2hy	2hy	NOUN
ejpam-4793	63	5	=	=	SYM
ejpam-4793	63	6	ϕ((h	ϕ((h	PROPN
ejpam-4793	63	7	,	,	PUNCT
ejpam-4793	63	8	y	y	PROPN
ejpam-4793	63	9	)	)	PUNCT
ejpam-4793	63	10	)	)	PUNCT
ejpam-4793	63	11	and	and	CCONJ
ejpam-4793	63	12	ϕ	ϕ	NOUN
ejpam-4793	63	13	is	be	AUX
ejpam-4793	63	14	well	well	ADV
ejpam-4793	63	15	-	-	PUNCT
ejpam-4793	63	16	defined	define	VERB
ejpam-4793	63	17	.	.	PUNCT
ejpam-4793	64	1	now	now	ADV
ejpam-4793	64	2	,	,	PUNCT
ejpam-4793	64	3	for	for	ADP
ejpam-4793	64	4	any	any	DET
ejpam-4793	64	5	g1	g1	NOUN
ejpam-4793	64	6	,	,	PUNCT
ejpam-4793	64	7	g2	g2	PROPN
ejpam-4793	64	8	∈	∈	PROPN
ejpam-4793	64	9	g	g	PROPN
ejpam-4793	64	10	and	and	CCONJ
ejpam-4793	64	11	x	x	SYM
ejpam-4793	64	12	∈	∈	PROPN
ejpam-4793	64	13	s	s	X
ejpam-4793	64	14	,	,	PUNCT
ejpam-4793	64	15	we	we	PRON
ejpam-4793	64	16	have	have	VERB
ejpam-4793	64	17	ϕ((0	ϕ((0	PROPN
ejpam-4793	64	18	,	,	PUNCT
ejpam-4793	64	19	x	x	NOUN
ejpam-4793	64	20	)	)	PUNCT
ejpam-4793	64	21	)	)	PUNCT
ejpam-4793	65	1	=	=	SYM
ejpam-4793	65	2	20x	20x	NOUN
ejpam-4793	65	3	=	=	SYM
ejpam-4793	65	4	x	x	X
ejpam-4793	65	5	and	and	CCONJ
ejpam-4793	65	6	ϕ((g1	ϕ((g1	X
ejpam-4793	65	7	+	+	CCONJ
ejpam-4793	65	8	g2	g2	PROPN
ejpam-4793	65	9	,	,	PUNCT
ejpam-4793	65	10	x	x	NOUN
ejpam-4793	65	11	)	)	PUNCT
ejpam-4793	65	12	)	)	PUNCT
ejpam-4793	66	1	=	=	PUNCT
ejpam-4793	66	2	2g1+g2x	2g1+g2x	NUM
ejpam-4793	66	3	=	=	SYM
ejpam-4793	66	4	2g12g2x	2g12g2x	NOUN
ejpam-4793	66	5	=	=	SYM
ejpam-4793	66	6	ϕ((g1	ϕ((g1	ADJ
ejpam-4793	66	7	,	,	PUNCT
ejpam-4793	66	8	ϕ((g2	ϕ((g2	PROPN
ejpam-4793	66	9	,	,	PUNCT
ejpam-4793	66	10	x	x	NOUN
ejpam-4793	66	11	)	)	PUNCT
ejpam-4793	66	12	)	)	PUNCT
ejpam-4793	66	13	)	)	PUNCT
ejpam-4793	66	14	)	)	PUNCT
ejpam-4793	66	15	.	.	PUNCT
ejpam-4793	67	1	therefore	therefore	ADV
ejpam-4793	67	2	,	,	PUNCT
ejpam-4793	67	3	ϕ	ϕ	PROPN
ejpam-4793	67	4	is	be	AUX
ejpam-4793	67	5	an	an	DET
ejpam-4793	67	6	action	action	NOUN
ejpam-4793	67	7	.	.	PUNCT
ejpam-4793	68	1	h.	h.	PROPN
ejpam-4793	68	2	sarapuddin	sarapuddin	PROPN
ejpam-4793	68	3	,	,	PUNCT
ejpam-4793	68	4	j.	j.	PROPN
ejpam-4793	68	5	vilela	vilela	PROPN
ejpam-4793	68	6	/	/	SYM
ejpam-4793	68	7	eur	eur	PROPN
ejpam-4793	68	8	.	.	PUNCT
ejpam-4793	69	1	j.	j.	PROPN
ejpam-4793	69	2	pure	pure	PROPN
ejpam-4793	69	3	appl	appl	PROPN
ejpam-4793	69	4	.	.	PROPN
ejpam-4793	69	5	math	math	PROPN
ejpam-4793	69	6	,	,	PUNCT
ejpam-4793	69	7	16	16	NUM
ejpam-4793	69	8	(	(	PUNCT
ejpam-4793	69	9	3	3	NUM
ejpam-4793	69	10	)	)	PUNCT
ejpam-4793	69	11	(	(	PUNCT
ejpam-4793	69	12	2023	2023	NUM
ejpam-4793	69	13	)	)	PUNCT
ejpam-4793	69	14	,	,	PUNCT
ejpam-4793	69	15	1772	1772	NUM
ejpam-4793	69	16	-	-	SYM
ejpam-4793	69	17	1793	1793	NUM
ejpam-4793	69	18	1774	1774	NUM
ejpam-4793	69	19	definition	definition	NOUN
ejpam-4793	69	20	8	8	NUM
ejpam-4793	69	21	.	.	PUNCT
ejpam-4793	70	1	[	[	X
ejpam-4793	70	2	3	3	X
ejpam-4793	70	3	]	]	X
ejpam-4793	70	4	let	let	VERB
ejpam-4793	70	5	m	m	PRON
ejpam-4793	70	6	be	be	AUX
ejpam-4793	70	7	a	a	DET
ejpam-4793	70	8	monoid	monoid	NOUN
ejpam-4793	70	9	and	and	CCONJ
ejpam-4793	70	10	γ	γ	X
ejpam-4793	70	11	a	a	DET
ejpam-4793	70	12	group	group	NOUN
ejpam-4793	70	13	.	.	PUNCT
ejpam-4793	71	1	m	m	PROPN
ejpam-4793	71	2	is	be	AUX
ejpam-4793	71	3	said	say	VERB
ejpam-4793	71	4	to	to	PART
ejpam-4793	71	5	be	be	AUX
ejpam-4793	71	6	a	a	DET
ejpam-4793	71	7	γ	γ	NOUN
ejpam-4793	71	8	-	-	PUNCT
ejpam-4793	71	9	monoid	monoid	NOUN
ejpam-4793	71	10	if	if	SCONJ
ejpam-4793	71	11	there	there	PRON
ejpam-4793	71	12	is	be	VERB
ejpam-4793	71	13	an	an	DET
ejpam-4793	71	14	action	action	NOUN
ejpam-4793	71	15	ϕ	ϕ	NOUN
ejpam-4793	71	16	:	:	PUNCT
ejpam-4793	71	17	γ×m	γ×m	PROPN
ejpam-4793	71	18	→	→	SYM
ejpam-4793	71	19	m	m	NOUN
ejpam-4793	71	20	of	of	ADP
ejpam-4793	71	21	γ	γ	NOUN
ejpam-4793	71	22	on	on	ADP
ejpam-4793	71	23	m	m	PROPN
ejpam-4793	71	24	via	via	ADP
ejpam-4793	71	25	monoid	monoid	PROPN
ejpam-4793	71	26	automorphism	automorphism	NOUN
ejpam-4793	71	27	,	,	PUNCT
ejpam-4793	71	28	that	that	ADV
ejpam-4793	71	29	is	is	ADV
ejpam-4793	71	30	,	,	PUNCT
ejpam-4793	71	31	ϕ	ϕ	NOUN
ejpam-4793	71	32	is	be	AUX
ejpam-4793	71	33	an	an	DET
ejpam-4793	71	34	action	action	NOUN
ejpam-4793	71	35	which	which	PRON
ejpam-4793	71	36	satisfies	satisfy	VERB
ejpam-4793	71	37	:	:	PUNCT
ejpam-4793	71	38	for	for	ADP
ejpam-4793	71	39	all	all	PRON
ejpam-4793	71	40	α	α	DET
ejpam-4793	71	41	∈	∈	NOUN
ejpam-4793	71	42	γ	γ	NOUN
ejpam-4793	71	43	and	and	CCONJ
ejpam-4793	71	44	x	x	NOUN
ejpam-4793	71	45	,	,	PUNCT
ejpam-4793	71	46	y	y	PROPN
ejpam-4793	71	47	∈	∈	PROPN
ejpam-4793	71	48	m	m	PROPN
ejpam-4793	71	49	,	,	PUNCT
ejpam-4793	71	50	ϕ((α	ϕ((α	PROPN
ejpam-4793	71	51	,	,	PUNCT
ejpam-4793	71	52	x	x	PROPN
ejpam-4793	71	53	∗	∗	PROPN
ejpam-4793	71	54	y	y	NOUN
ejpam-4793	71	55	)	)	PUNCT
ejpam-4793	71	56	)	)	PUNCT
ejpam-4793	72	1	=	=	SYM
ejpam-4793	72	2	ϕ((α	ϕ((α	PROPN
ejpam-4793	72	3	,	,	PUNCT
ejpam-4793	72	4	x	x	NOUN
ejpam-4793	72	5	)	)	PUNCT
ejpam-4793	72	6	)	)	PUNCT
ejpam-4793	72	7	∗	∗	NOUN
ejpam-4793	72	8	ϕ((α	ϕ((α	PROPN
ejpam-4793	72	9	,	,	PUNCT
ejpam-4793	72	10	y	y	NOUN
ejpam-4793	72	11	)	)	PUNCT
ejpam-4793	72	12	)	)	PUNCT
ejpam-4793	72	13	.	.	PUNCT
ejpam-4793	73	1	for	for	ADP
ejpam-4793	73	2	α	α	DET
ejpam-4793	73	3	∈	∈	PROPN
ejpam-4793	73	4	γ	γ	NOUN
ejpam-4793	73	5	and	and	CCONJ
ejpam-4793	73	6	a	a	DET
ejpam-4793	73	7	∈	∈	NOUN
ejpam-4793	73	8	m	m	NOUN
ejpam-4793	73	9	,	,	PUNCT
ejpam-4793	73	10	the	the	DET
ejpam-4793	73	11	action	action	NOUN
ejpam-4793	73	12	of	of	ADP
ejpam-4793	73	13	α	α	NOUN
ejpam-4793	73	14	on	on	ADP
ejpam-4793	73	15	a	a	PRON
ejpam-4793	73	16	shall	shall	AUX
ejpam-4793	73	17	be	be	AUX
ejpam-4793	73	18	denoted	denote	VERB
ejpam-4793	73	19	by	by	ADP
ejpam-4793	73	20	αa	αa	PROPN
ejpam-4793	73	21	.	.	PROPN
ejpam-4793	73	22	example	example	NOUN
ejpam-4793	74	1	3	3	X
ejpam-4793	74	2	.	.	PUNCT
ejpam-4793	74	3	consider	consider	VERB
ejpam-4793	74	4	γ	γ	X
ejpam-4793	74	5	=	=	PROPN
ejpam-4793	74	6	z	z	PROPN
ejpam-4793	74	7	a	a	DET
ejpam-4793	74	8	group	group	NOUN
ejpam-4793	74	9	of	of	ADP
ejpam-4793	74	10	integers	integer	NOUN
ejpam-4793	74	11	under	under	ADP
ejpam-4793	74	12	the	the	DET
ejpam-4793	74	13	usual	usual	ADJ
ejpam-4793	74	14	addition	addition	NOUN
ejpam-4793	74	15	and	and	CCONJ
ejpam-4793	74	16	the	the	DET
ejpam-4793	74	17	set	set	NOUN
ejpam-4793	74	18	m	m	NOUN
ejpam-4793	74	19	=	=	NOUN
ejpam-4793	74	20	r	r	NOUN
ejpam-4793	74	21	with	with	ADP
ejpam-4793	74	22	the	the	DET
ejpam-4793	74	23	usual	usual	ADJ
ejpam-4793	74	24	addition	addition	NOUN
ejpam-4793	74	25	as	as	ADP
ejpam-4793	74	26	its	its	PRON
ejpam-4793	74	27	binary	binary	ADJ
ejpam-4793	74	28	operation	operation	NOUN
ejpam-4793	74	29	.	.	PUNCT
ejpam-4793	75	1	then	then	ADV
ejpam-4793	75	2	,	,	PUNCT
ejpam-4793	75	3	(	(	PUNCT
ejpam-4793	75	4	m,+	m,+	INTJ
ejpam-4793	75	5	)	)	PUNCT
ejpam-4793	75	6	is	be	AUX
ejpam-4793	75	7	a	a	DET
ejpam-4793	75	8	monoid	monoid	NOUN
ejpam-4793	75	9	with	with	ADP
ejpam-4793	75	10	identity	identity	NOUN
ejpam-4793	75	11	0	0	NUM
ejpam-4793	75	12	.	.	PUNCT
ejpam-4793	76	1	consider	consider	VERB
ejpam-4793	76	2	the	the	DET
ejpam-4793	76	3	action	action	NOUN
ejpam-4793	76	4	ϕ	ϕ	NOUN
ejpam-4793	76	5	:	:	PUNCT
ejpam-4793	76	6	γ	γ	PROPN
ejpam-4793	76	7	×m	×m	PROPN
ejpam-4793	76	8	→	→	SYM
ejpam-4793	76	9	m	m	AUX
ejpam-4793	76	10	given	give	VERB
ejpam-4793	76	11	by	by	ADP
ejpam-4793	76	12	ϕ((α	ϕ((α	PROPN
ejpam-4793	76	13	,	,	PUNCT
ejpam-4793	76	14	x	x	NOUN
ejpam-4793	76	15	)	)	PUNCT
ejpam-4793	76	16	)	)	PUNCT
ejpam-4793	77	1	=	=	SYM
ejpam-4793	77	2	2αx	2αx	NOUN
ejpam-4793	77	3	in	in	ADP
ejpam-4793	77	4	example	example	NOUN
ejpam-4793	77	5	2	2	X
ejpam-4793	77	6	.	.	PUNCT
ejpam-4793	77	7	now	now	ADV
ejpam-4793	77	8	,	,	PUNCT
ejpam-4793	77	9	let	let	VERB
ejpam-4793	77	10	α	α	PRON
ejpam-4793	77	11	∈	∈	PROPN
ejpam-4793	77	12	γ	γ	X
ejpam-4793	77	13	and	and	CCONJ
ejpam-4793	77	14	x	x	NOUN
ejpam-4793	77	15	,	,	PUNCT
ejpam-4793	77	16	y	y	PROPN
ejpam-4793	77	17	∈	∈	PROPN
ejpam-4793	77	18	m	m	VERB
ejpam-4793	77	19	.	.	PUNCT
ejpam-4793	78	1	then	then	ADV
ejpam-4793	78	2	we	we	PRON
ejpam-4793	78	3	have	have	VERB
ejpam-4793	78	4	ϕ((α	ϕ((α	NOUN
ejpam-4793	78	5	,	,	PUNCT
ejpam-4793	78	6	x	x	PUNCT
ejpam-4793	79	1	+	+	NUM
ejpam-4793	79	2	y	y	NOUN
ejpam-4793	79	3	)	)	PUNCT
ejpam-4793	79	4	)	)	PUNCT
ejpam-4793	80	1	=	=	PUNCT
ejpam-4793	81	1	2α(x	2α(x	NUM
ejpam-4793	81	2	+	+	CCONJ
ejpam-4793	81	3	y	y	NOUN
ejpam-4793	81	4	)	)	PUNCT
ejpam-4793	82	1	=	=	SYM
ejpam-4793	82	2	2αx	2αx	NOUN
ejpam-4793	83	1	+	+	CCONJ
ejpam-4793	83	2	2αy	2αy	ADJ
ejpam-4793	83	3	=	=	PUNCT
ejpam-4793	83	4	ϕ((α	ϕ((α	NOUN
ejpam-4793	83	5	,	,	PUNCT
ejpam-4793	83	6	x	x	NOUN
ejpam-4793	83	7	)	)	PUNCT
ejpam-4793	83	8	)	)	PUNCT
ejpam-4793	84	1	+	+	CCONJ
ejpam-4793	84	2	ϕ((α	ϕ((α	PROPN
ejpam-4793	84	3	,	,	PUNCT
ejpam-4793	84	4	y	y	NOUN
ejpam-4793	84	5	)	)	PUNCT
ejpam-4793	84	6	)	)	PUNCT
ejpam-4793	84	7	.	.	PUNCT
ejpam-4793	85	1	therefore	therefore	ADV
ejpam-4793	85	2	,	,	PUNCT
ejpam-4793	85	3	m	m	VERB
ejpam-4793	85	4	is	be	AUX
ejpam-4793	85	5	a	a	DET
ejpam-4793	85	6	γ	γ	X
ejpam-4793	85	7	-	-	PUNCT
ejpam-4793	85	8	monoid	monoid	NOUN
ejpam-4793	85	9	.	.	PUNCT
ejpam-4793	85	10	example	example	NOUN
ejpam-4793	86	1	4	4	X
ejpam-4793	86	2	.	.	PUNCT
ejpam-4793	86	3	let	let	VERB
ejpam-4793	86	4	γ	γ	NOUN
ejpam-4793	86	5	be	be	AUX
ejpam-4793	86	6	a	a	DET
ejpam-4793	86	7	group	group	NOUN
ejpam-4793	86	8	of	of	ADP
ejpam-4793	86	9	integers	integer	NOUN
ejpam-4793	86	10	under	under	ADP
ejpam-4793	86	11	addition	addition	NOUN
ejpam-4793	86	12	and	and	CCONJ
ejpam-4793	86	13	let	let	VERB
ejpam-4793	86	14	t	t	NOUN
ejpam-4793	86	15	=	=	SYM
ejpam-4793	86	16	m2(r	m2(r	PROPN
ejpam-4793	86	17	)	)	PUNCT
ejpam-4793	86	18	under	under	ADP
ejpam-4793	86	19	matrix	matrix	NOUN
ejpam-4793	86	20	addition	addition	NOUN
ejpam-4793	86	21	.	.	PUNCT
ejpam-4793	87	1	consider	consider	VERB
ejpam-4793	87	2	the	the	DET
ejpam-4793	87	3	mapping	mapping	NOUN
ejpam-4793	87	4	ϕ	ϕ	NOUN
ejpam-4793	87	5	:	:	PUNCT
ejpam-4793	87	6	γ	γ	X
ejpam-4793	87	7	×	×	PROPN
ejpam-4793	87	8	t	t	PROPN
ejpam-4793	87	9	→	→	SYM
ejpam-4793	87	10	t	t	PROPN
ejpam-4793	87	11	given	give	VERB
ejpam-4793	87	12	by	by	ADP
ejpam-4793	87	13	(	(	PUNCT
ejpam-4793	87	14	α	α	X
ejpam-4793	87	15	,	,	PUNCT
ejpam-4793	87	16	(	(	PUNCT
ejpam-4793	87	17	a	a	DET
ejpam-4793	87	18	b	b	NOUN
ejpam-4793	87	19	c	c	NOUN
ejpam-4793	87	20	d	d	NOUN
ejpam-4793	87	21	)	)	PUNCT
ejpam-4793	87	22	)	)	PUNCT
ejpam-4793	88	1	7→	7→	NUM
ejpam-4793	88	2	α	α	NOUN
ejpam-4793	88	3	(	(	PUNCT
ejpam-4793	88	4	a	a	DET
ejpam-4793	88	5	b	b	NOUN
ejpam-4793	88	6	c	c	PROPN
ejpam-4793	88	7	d	d	PROPN
ejpam-4793	88	8	)	)	PUNCT
ejpam-4793	88	9	=(	=(	NOUN
ejpam-4793	88	10	2αa	2αa	ADJ
ejpam-4793	88	11	2αb	2αb	ADJ
ejpam-4793	88	12	2αc	2αc	ADJ
ejpam-4793	88	13	2αd	2αd	NOUN
ejpam-4793	88	14	)	)	PUNCT
ejpam-4793	88	15	.	.	PUNCT
ejpam-4793	89	1	let	let	VERB
ejpam-4793	89	2	(	(	PUNCT
ejpam-4793	89	3	α	α	X
ejpam-4793	89	4	,	,	PUNCT
ejpam-4793	89	5	(	(	PUNCT
ejpam-4793	89	6	a	a	DET
ejpam-4793	89	7	b	b	NOUN
ejpam-4793	89	8	c	c	NOUN
ejpam-4793	89	9	d	d	NOUN
ejpam-4793	89	10	)	)	PUNCT
ejpam-4793	89	11	)	)	PUNCT
ejpam-4793	89	12	,	,	PUNCT
ejpam-4793	89	13	(	(	PUNCT
ejpam-4793	89	14	β	β	X
ejpam-4793	89	15	,	,	PUNCT
ejpam-4793	89	16	(	(	PUNCT
ejpam-4793	89	17	e	e	NOUN
ejpam-4793	89	18	f	f	PROPN
ejpam-4793	89	19	g	g	PROPN
ejpam-4793	89	20	h	h	PROPN
ejpam-4793	89	21	)	)	PUNCT
ejpam-4793	89	22	)	)	PUNCT
ejpam-4793	90	1	∈	∈	PROPN
ejpam-4793	91	1	γ×	γ×	PROPN
ejpam-4793	91	2	t	t	X
ejpam-4793	91	3	such	such	ADJ
ejpam-4793	91	4	that	that	SCONJ
ejpam-4793	91	5	(	(	PUNCT
ejpam-4793	91	6	α	α	NOUN
ejpam-4793	91	7	,	,	PUNCT
ejpam-4793	91	8	(	(	PUNCT
ejpam-4793	91	9	a	a	DET
ejpam-4793	91	10	b	b	NOUN
ejpam-4793	91	11	c	c	NOUN
ejpam-4793	91	12	d	d	NOUN
ejpam-4793	91	13	)	)	PUNCT
ejpam-4793	91	14	)	)	PUNCT
ejpam-4793	92	1	=	=	PRON
ejpam-4793	92	2	(	(	PUNCT
ejpam-4793	92	3	β	β	X
ejpam-4793	92	4	,	,	PUNCT
ejpam-4793	92	5	(	(	PUNCT
ejpam-4793	92	6	e	e	NOUN
ejpam-4793	92	7	f	f	PROPN
ejpam-4793	92	8	g	g	PROPN
ejpam-4793	92	9	h	h	PROPN
ejpam-4793	92	10	)	)	PUNCT
ejpam-4793	92	11	)	)	PUNCT
ejpam-4793	92	12	.	.	PUNCT
ejpam-4793	93	1	then	then	ADV
ejpam-4793	93	2	α	α	X
ejpam-4793	93	3	=	=	PUNCT
ejpam-4793	93	4	β	β	X
ejpam-4793	93	5	and	and	CCONJ
ejpam-4793	93	6	(	(	PUNCT
ejpam-4793	93	7	a	a	DET
ejpam-4793	93	8	b	b	NOUN
ejpam-4793	93	9	c	c	NOUN
ejpam-4793	93	10	d	d	NOUN
ejpam-4793	93	11	)	)	PUNCT
ejpam-4793	93	12	=	=	PUNCT
ejpam-4793	94	1	(	(	PUNCT
ejpam-4793	94	2	e	e	X
ejpam-4793	94	3	f	f	PROPN
ejpam-4793	94	4	g	g	PROPN
ejpam-4793	94	5	h	h	PROPN
ejpam-4793	94	6	)	)	PUNCT
ejpam-4793	94	7	.	.	PUNCT
ejpam-4793	95	1	thus	thus	ADV
ejpam-4793	95	2	,	,	PUNCT
ejpam-4793	95	3	(	(	PUNCT
ejpam-4793	95	4	2αa	2αa	ADJ
ejpam-4793	95	5	2αb	2αb	ADJ
ejpam-4793	95	6	2αc	2αc	ADJ
ejpam-4793	95	7	2αd	2αd	NOUN
ejpam-4793	95	8	)	)	PUNCT
ejpam-4793	95	9	=(	=(	PROPN
ejpam-4793	95	10	2βe	2βe	NOUN
ejpam-4793	95	11	2βf	2βf	ADJ
ejpam-4793	95	12	2βg	2βg	ADJ
ejpam-4793	95	13	2βh	2βh	NOUN
ejpam-4793	95	14	)	)	PUNCT
ejpam-4793	95	15	and	and	CCONJ
ejpam-4793	95	16	ϕ	ϕ	NOUN
ejpam-4793	95	17	is	be	AUX
ejpam-4793	95	18	well	well	ADV
ejpam-4793	95	19	-	-	PUNCT
ejpam-4793	95	20	defined	define	VERB
ejpam-4793	95	21	.	.	PUNCT
ejpam-4793	96	1	now	now	ADV
ejpam-4793	96	2	,	,	PUNCT
ejpam-4793	96	3	for	for	ADP
ejpam-4793	96	4	any	any	DET
ejpam-4793	96	5	α	α	NOUN
ejpam-4793	96	6	,	,	PUNCT
ejpam-4793	96	7	β	β	X
ejpam-4793	96	8	∈	∈	PROPN
ejpam-4793	96	9	γ	γ	NOUN
ejpam-4793	96	10	and	and	CCONJ
ejpam-4793	96	11	a	a	DET
ejpam-4793	96	12	,	,	PUNCT
ejpam-4793	96	13	b	b	NOUN
ejpam-4793	96	14	,	,	PUNCT
ejpam-4793	96	15	c	c	NOUN
ejpam-4793	96	16	,	,	PUNCT
ejpam-4793	96	17	d	d	PROPN
ejpam-4793	96	18	∈	∈	PROPN
ejpam-4793	96	19	r	r	NOUN
ejpam-4793	96	20	,	,	PUNCT
ejpam-4793	96	21	we	we	PRON
ejpam-4793	96	22	have	have	VERB
ejpam-4793	96	23	ϕ	ϕ	NOUN
ejpam-4793	96	24	(	(	PUNCT
ejpam-4793	96	25	(	(	PUNCT
ejpam-4793	96	26	0	0	NUM
ejpam-4793	96	27	,	,	PUNCT
ejpam-4793	96	28	(	(	PUNCT
ejpam-4793	96	29	a	a	DET
ejpam-4793	96	30	b	b	NOUN
ejpam-4793	96	31	c	c	NOUN
ejpam-4793	96	32	d	d	NOUN
ejpam-4793	96	33	)	)	PUNCT
ejpam-4793	96	34	)	)	PUNCT
ejpam-4793	96	35	)	)	PUNCT
ejpam-4793	97	1	=	=	SYM
ejpam-4793	97	2	0	0	PUNCT
ejpam-4793	97	3	(	(	PUNCT
ejpam-4793	97	4	a	a	DET
ejpam-4793	97	5	b	b	NOUN
ejpam-4793	97	6	c	c	NOUN
ejpam-4793	97	7	d	d	NOUN
ejpam-4793	97	8	)	)	PUNCT
ejpam-4793	97	9	=	=	SYM
ejpam-4793	97	10	(	(	PUNCT
ejpam-4793	97	11	20a	20a	NOUN
ejpam-4793	97	12	20b	20b	NOUN
ejpam-4793	97	13	20c	20c	NOUN
ejpam-4793	97	14	20d	20d	NOUN
ejpam-4793	97	15	)	)	PUNCT
ejpam-4793	97	16	=	=	SYM
ejpam-4793	97	17	(	(	PUNCT
ejpam-4793	97	18	a	a	DET
ejpam-4793	97	19	b	b	NOUN
ejpam-4793	97	20	c	c	NOUN
ejpam-4793	97	21	d	d	NOUN
ejpam-4793	97	22	)	)	PUNCT
ejpam-4793	97	23	and	and	CCONJ
ejpam-4793	97	24	ϕ	ϕ	X
ejpam-4793	97	25	(	(	PUNCT
ejpam-4793	97	26	(	(	PUNCT
ejpam-4793	97	27	α+	α+	X
ejpam-4793	97	28	β	β	X
ejpam-4793	97	29	,	,	PUNCT
ejpam-4793	97	30	(	(	PUNCT
ejpam-4793	97	31	a	a	DET
ejpam-4793	97	32	b	b	NOUN
ejpam-4793	97	33	c	c	NOUN
ejpam-4793	97	34	d	d	NOUN
ejpam-4793	97	35	)	)	PUNCT
ejpam-4793	97	36	)	)	PUNCT
ejpam-4793	97	37	)	)	PUNCT
ejpam-4793	98	1	=	=	SYM
ejpam-4793	98	2	α+β	α+β	PROPN
ejpam-4793	98	3	(	(	PUNCT
ejpam-4793	98	4	a	a	DET
ejpam-4793	98	5	b	b	NOUN
ejpam-4793	98	6	c	c	NOUN
ejpam-4793	98	7	d	d	NOUN
ejpam-4793	98	8	)	)	PUNCT
ejpam-4793	98	9	=	=	SYM
ejpam-4793	98	10	(	(	PUNCT
ejpam-4793	98	11	2α+βa	2α+βa	NUM
ejpam-4793	98	12	2α+βb	2α+βb	NUM
ejpam-4793	98	13	2α+βc	2α+βc	NUM
ejpam-4793	98	14	2α+βd	2α+βd	NUM
ejpam-4793	98	15	)	)	PUNCT
ejpam-4793	98	16	=	=	SYM
ejpam-4793	98	17	(	(	PUNCT
ejpam-4793	98	18	2α2βa	2α2βa	NUM
ejpam-4793	98	19	2α3βb	2α3βb	NUM
ejpam-4793	98	20	2α4βc	2α4βc	NUM
ejpam-4793	98	21	2α5βd	2α5βd	NUM
ejpam-4793	98	22	)	)	PUNCT
ejpam-4793	99	1	=	=	SYM
ejpam-4793	99	2	ϕ	ϕ	X
ejpam-4793	99	3	(	(	PUNCT
ejpam-4793	99	4	(	(	PUNCT
ejpam-4793	99	5	α	α	X
ejpam-4793	99	6	,	,	PUNCT
ejpam-4793	99	7	(	(	PUNCT
ejpam-4793	99	8	2βa	2βa	ADJ
ejpam-4793	99	9	2βb	2βb	ADJ
ejpam-4793	99	10	2βc	2βc	NOUN
ejpam-4793	99	11	2βd	2βd	NOUN
ejpam-4793	99	12	)	)	PUNCT
ejpam-4793	99	13	)	)	PUNCT
ejpam-4793	99	14	)	)	PUNCT
ejpam-4793	100	1	=	=	SYM
ejpam-4793	101	1	ϕ	ϕ	X
ejpam-4793	101	2	(	(	PUNCT
ejpam-4793	101	3	(	(	PUNCT
ejpam-4793	101	4	α	α	X
ejpam-4793	101	5	,	,	PUNCT
ejpam-4793	101	6	φ	φ	PROPN
ejpam-4793	101	7	(	(	PUNCT
ejpam-4793	101	8	(	(	PUNCT
ejpam-4793	101	9	β	β	X
ejpam-4793	101	10	,	,	PUNCT
ejpam-4793	101	11	(	(	PUNCT
ejpam-4793	101	12	a	a	DET
ejpam-4793	101	13	b	b	NOUN
ejpam-4793	101	14	c	c	NOUN
ejpam-4793	101	15	d	d	NOUN
ejpam-4793	101	16	)	)	PUNCT
ejpam-4793	101	17	)	)	PUNCT
ejpam-4793	101	18	)	)	PUNCT
ejpam-4793	101	19	)	)	PUNCT
ejpam-4793	101	20	)	)	PUNCT
ejpam-4793	101	21	.	.	PUNCT
ejpam-4793	102	1	thus	thus	ADV
ejpam-4793	102	2	,	,	PUNCT
ejpam-4793	102	3	ϕ	ϕ	PROPN
ejpam-4793	102	4	is	be	AUX
ejpam-4793	102	5	an	an	DET
ejpam-4793	102	6	action	action	NOUN
ejpam-4793	102	7	.	.	PUNCT
ejpam-4793	103	1	now	now	ADV
ejpam-4793	103	2	,	,	PUNCT
ejpam-4793	103	3	let	let	VERB
ejpam-4793	103	4	α	α	PRON
ejpam-4793	103	5	∈	∈	PROPN
ejpam-4793	103	6	γ	γ	X
ejpam-4793	103	7	and	and	CCONJ
ejpam-4793	103	8	(	(	PUNCT
ejpam-4793	103	9	a	a	DET
ejpam-4793	103	10	b	b	NOUN
ejpam-4793	103	11	c	c	NOUN
ejpam-4793	103	12	d	d	NOUN
ejpam-4793	103	13	)	)	PUNCT
ejpam-4793	103	14	,	,	PUNCT
ejpam-4793	103	15	(	(	PUNCT
ejpam-4793	103	16	e	e	NOUN
ejpam-4793	103	17	f	f	PROPN
ejpam-4793	103	18	g	g	PROPN
ejpam-4793	103	19	h	h	PROPN
ejpam-4793	104	1	)	)	PUNCT
ejpam-4793	104	2	∈	∈	PROPN
ejpam-4793	105	1	t.	t.	NOUN
ejpam-4793	105	2	then	then	ADV
ejpam-4793	105	3	we	we	PRON
ejpam-4793	105	4	have	have	VERB
ejpam-4793	105	5	ϕ	ϕ	NOUN
ejpam-4793	105	6	(	(	PUNCT
ejpam-4793	105	7	(	(	PUNCT
ejpam-4793	105	8	α	α	X
ejpam-4793	105	9	,	,	PUNCT
ejpam-4793	105	10	(	(	PUNCT
ejpam-4793	105	11	a	a	DET
ejpam-4793	105	12	b	b	NOUN
ejpam-4793	105	13	c	c	NOUN
ejpam-4793	105	14	d	d	NOUN
ejpam-4793	105	15	)	)	PUNCT
ejpam-4793	106	1	+	+	CCONJ
ejpam-4793	106	2	(	(	PUNCT
ejpam-4793	106	3	e	e	X
ejpam-4793	106	4	f	f	PROPN
ejpam-4793	106	5	g	g	PROPN
ejpam-4793	106	6	h	h	PROPN
ejpam-4793	106	7	)	)	PUNCT
ejpam-4793	106	8	)	)	PUNCT
ejpam-4793	106	9	)	)	PUNCT
ejpam-4793	107	1	=	=	SYM
ejpam-4793	107	2	ϕ	ϕ	X
ejpam-4793	107	3	(	(	PUNCT
ejpam-4793	107	4	(	(	PUNCT
ejpam-4793	107	5	α	α	X
ejpam-4793	107	6	,	,	PUNCT
ejpam-4793	107	7	(	(	PUNCT
ejpam-4793	107	8	a+	a+	X
ejpam-4793	107	9	e	e	X
ejpam-4793	107	10	b+	b+	X
ejpam-4793	107	11	f	f	X
ejpam-4793	107	12	c+	c+	VERB
ejpam-4793	107	13	g	g	PROPN
ejpam-4793	107	14	d+	d+	PROPN
ejpam-4793	107	15	h	h	NOUN
ejpam-4793	107	16	)	)	PUNCT
ejpam-4793	107	17	)	)	PUNCT
ejpam-4793	107	18	)	)	PUNCT
ejpam-4793	108	1	=	=	PRON
ejpam-4793	108	2	(	(	PUNCT
ejpam-4793	108	3	2α(a+	2α(a+	PROPN
ejpam-4793	108	4	e	e	NOUN
ejpam-4793	108	5	)	)	PUNCT
ejpam-4793	108	6	2α(b+	2α(b+	PROPN
ejpam-4793	108	7	f	f	X
ejpam-4793	108	8	)	)	PUNCT
ejpam-4793	108	9	2α(c+	2α(c+	NUM
ejpam-4793	108	10	g	g	NOUN
ejpam-4793	108	11	)	)	PUNCT
ejpam-4793	108	12	2α(d+	2α(d+	NUM
ejpam-4793	108	13	h	h	NOUN
ejpam-4793	108	14	)	)	PUNCT
ejpam-4793	108	15	)	)	PUNCT
ejpam-4793	109	1	h.	h.	PROPN
ejpam-4793	109	2	sarapuddin	sarapuddin	PROPN
ejpam-4793	109	3	,	,	PUNCT
ejpam-4793	109	4	j.	j.	PROPN
ejpam-4793	109	5	vilela	vilela	PROPN
ejpam-4793	109	6	/	/	SYM
ejpam-4793	109	7	eur	eur	PROPN
ejpam-4793	109	8	.	.	PUNCT
ejpam-4793	110	1	j.	j.	PROPN
ejpam-4793	110	2	pure	pure	PROPN
ejpam-4793	110	3	appl	appl	PROPN
ejpam-4793	110	4	.	.	PROPN
ejpam-4793	110	5	math	math	PROPN
ejpam-4793	110	6	,	,	PUNCT
ejpam-4793	110	7	16	16	NUM
ejpam-4793	110	8	(	(	PUNCT
ejpam-4793	110	9	3	3	NUM
ejpam-4793	110	10	)	)	PUNCT
ejpam-4793	110	11	(	(	PUNCT
ejpam-4793	110	12	2023	2023	NUM
ejpam-4793	110	13	)	)	PUNCT
ejpam-4793	110	14	,	,	PUNCT
ejpam-4793	110	15	1772	1772	NUM
ejpam-4793	110	16	-	-	SYM
ejpam-4793	110	17	1793	1793	NUM
ejpam-4793	110	18	1775	1775	NUM
ejpam-4793	110	19	=	=	SYM
ejpam-4793	110	20	(	(	PUNCT
ejpam-4793	110	21	2αa+	2αa+	NUM
ejpam-4793	110	22	2αe	2αe	ADJ
ejpam-4793	110	23	2αb+	2αb+	NUM
ejpam-4793	110	24	2αf	2αf	NOUN
ejpam-4793	110	25	2αc+	2αc+	NUM
ejpam-4793	110	26	2αg	2αg	ADJ
ejpam-4793	110	27	2αd+	2αd+	NUM
ejpam-4793	110	28	2αh	2αh	NOUN
ejpam-4793	110	29	)	)	PUNCT
ejpam-4793	111	1	=	=	PUNCT
ejpam-4793	111	2	(	(	PUNCT
ejpam-4793	111	3	2αa	2αa	ADJ
ejpam-4793	111	4	2αb	2αb	ADJ
ejpam-4793	111	5	2αc	2αc	ADJ
ejpam-4793	111	6	2αd	2αd	NOUN
ejpam-4793	111	7	)	)	PUNCT
ejpam-4793	112	1	+	+	CCONJ
ejpam-4793	112	2	(	(	PUNCT
ejpam-4793	112	3	2αe	2αe	ADJ
ejpam-4793	112	4	2αf	2αf	ADJ
ejpam-4793	112	5	2αg	2αg	ADJ
ejpam-4793	112	6	2αh	2αh	NOUN
ejpam-4793	112	7	)	)	PUNCT
ejpam-4793	113	1	=	=	SYM
ejpam-4793	113	2	ϕ	ϕ	X
ejpam-4793	113	3	(	(	PUNCT
ejpam-4793	113	4	(	(	PUNCT
ejpam-4793	113	5	α	α	X
ejpam-4793	113	6	,	,	PUNCT
ejpam-4793	113	7	(	(	PUNCT
ejpam-4793	113	8	a	a	DET
ejpam-4793	113	9	b	b	NOUN
ejpam-4793	113	10	c	c	NOUN
ejpam-4793	113	11	d	d	NOUN
ejpam-4793	113	12	)	)	PUNCT
ejpam-4793	113	13	)	)	PUNCT
ejpam-4793	113	14	)	)	PUNCT
ejpam-4793	114	1	+	+	CCONJ
ejpam-4793	114	2	ϕ	ϕ	X
ejpam-4793	114	3	(	(	PUNCT
ejpam-4793	114	4	(	(	PUNCT
ejpam-4793	114	5	α	α	X
ejpam-4793	114	6	,	,	PUNCT
ejpam-4793	114	7	(	(	PUNCT
ejpam-4793	114	8	e	e	NOUN
ejpam-4793	114	9	f	f	PROPN
ejpam-4793	114	10	g	g	PROPN
ejpam-4793	114	11	h	h	PROPN
ejpam-4793	114	12	)	)	PUNCT
ejpam-4793	114	13	)	)	PUNCT
ejpam-4793	114	14	)	)	PUNCT
ejpam-4793	114	15	.	.	PUNCT
ejpam-4793	115	1	therefore	therefore	ADV
ejpam-4793	115	2	,	,	PUNCT
ejpam-4793	115	3	t	t	PROPN
ejpam-4793	115	4	is	be	AUX
ejpam-4793	115	5	a	a	DET
ejpam-4793	115	6	γ	γ	X
ejpam-4793	115	7	-	-	PUNCT
ejpam-4793	115	8	monoid	monoid	NOUN
ejpam-4793	115	9	.	.	PUNCT
ejpam-4793	115	10	example	example	NOUN
ejpam-4793	116	1	5	5	NUM
ejpam-4793	116	2	.	.	PUNCT
ejpam-4793	116	3	consider	consider	VERB
ejpam-4793	116	4	the	the	DET
ejpam-4793	116	5	set	set	NOUN
ejpam-4793	116	6	m	m	NOUN
ejpam-4793	116	7	=	=	PUNCT
ejpam-4793	116	8	{	{	PUNCT
ejpam-4793	116	9	1	1	NUM
ejpam-4793	116	10	,	,	PUNCT
ejpam-4793	116	11	a	a	DET
ejpam-4793	116	12	,	,	PUNCT
ejpam-4793	116	13	b	b	NOUN
ejpam-4793	116	14	,	,	PUNCT
ejpam-4793	116	15	c	c	NOUN
ejpam-4793	116	16	,	,	PUNCT
ejpam-4793	116	17	d	d	NOUN
ejpam-4793	116	18	,	,	PUNCT
ejpam-4793	116	19	e	e	NOUN
ejpam-4793	116	20	}	}	PUNCT
ejpam-4793	116	21	and	and	CCONJ
ejpam-4793	116	22	an	an	DET
ejpam-4793	116	23	operation	operation	NOUN
ejpam-4793	116	24	∗	∗	NOUN
ejpam-4793	116	25	given	give	VERB
ejpam-4793	116	26	by	by	ADP
ejpam-4793	116	27	∗	∗	NOUN
ejpam-4793	116	28	1	1	NUM
ejpam-4793	116	29	a	a	DET
ejpam-4793	116	30	b	b	NOUN
ejpam-4793	116	31	c	c	NOUN
ejpam-4793	116	32	d	d	X
ejpam-4793	116	33	e	e	PROPN
ejpam-4793	116	34	1	1	NUM
ejpam-4793	116	35	1	1	NUM
ejpam-4793	116	36	a	a	DET
ejpam-4793	116	37	b	b	NOUN
ejpam-4793	116	38	c	c	NOUN
ejpam-4793	116	39	d	d	PROPN
ejpam-4793	116	40	e	e	X
ejpam-4793	116	41	a	a	PRON
ejpam-4793	116	42	a	a	DET
ejpam-4793	116	43	a	a	DET
ejpam-4793	116	44	a	a	DET
ejpam-4793	116	45	a	a	PRON
ejpam-4793	116	46	a	a	PRON
ejpam-4793	116	47	a	a	DET
ejpam-4793	116	48	b	b	PROPN
ejpam-4793	116	49	b	b	PROPN
ejpam-4793	116	50	b	b	PROPN
ejpam-4793	116	51	b	b	PROPN
ejpam-4793	116	52	b	b	PROPN
ejpam-4793	116	53	b	b	PROPN
ejpam-4793	116	54	b	b	PROPN
ejpam-4793	116	55	c	c	NOUN
ejpam-4793	116	56	c	c	NOUN
ejpam-4793	117	1	c	c	NOUN
ejpam-4793	117	2	c	c	NOUN
ejpam-4793	117	3	c	c	NOUN
ejpam-4793	117	4	c	c	NOUN
ejpam-4793	117	5	c	c	NOUN
ejpam-4793	118	1	d	d	PUNCT
ejpam-4793	118	2	d	d	PROPN
ejpam-4793	118	3	d	d	PROPN
ejpam-4793	118	4	d	d	PROPN
ejpam-4793	118	5	d	d	PROPN
ejpam-4793	118	6	d	d	X
ejpam-4793	118	7	d	d	X
ejpam-4793	118	8	e	e	X
ejpam-4793	118	9	e	e	X
ejpam-4793	118	10	e	e	X
ejpam-4793	118	11	e	e	X
ejpam-4793	118	12	e	e	X
ejpam-4793	118	13	e	e	X
ejpam-4793	118	14	e	e	PROPN
ejpam-4793	118	15	the	the	DET
ejpam-4793	118	16	operation	operation	NOUN
ejpam-4793	118	17	∗	∗	NOUN
ejpam-4793	118	18	is	be	AUX
ejpam-4793	118	19	closed	close	VERB
ejpam-4793	118	20	and	and	CCONJ
ejpam-4793	118	21	associative	associative	ADJ
ejpam-4793	118	22	since	since	SCONJ
ejpam-4793	118	23	for	for	ADP
ejpam-4793	118	24	all	all	DET
ejpam-4793	118	25	x	x	NOUN
ejpam-4793	118	26	,	,	PUNCT
ejpam-4793	118	27	y	y	PROPN
ejpam-4793	118	28	∈	∈	PROPN
ejpam-4793	118	29	m	m	VERB
ejpam-4793	118	30	,	,	PUNCT
ejpam-4793	119	1	x	x	PROPN
ejpam-4793	119	2	∗	∗	NOUN
ejpam-4793	119	3	y	y	NOUN
ejpam-4793	119	4	=	=	PUNCT
ejpam-4793	119	5	x	x	PUNCT
ejpam-4793	119	6	holds	hold	VERB
ejpam-4793	119	7	for	for	ADP
ejpam-4793	119	8	all	all	DET
ejpam-4793	119	9	x	x	PUNCT
ejpam-4793	119	10	̸=	̸=	PROPN
ejpam-4793	119	11	1	1	NUM
ejpam-4793	119	12	.	.	PUNCT
ejpam-4793	120	1	clearly	clearly	ADV
ejpam-4793	120	2	,	,	PUNCT
ejpam-4793	120	3	1	1	NUM
ejpam-4793	120	4	is	be	AUX
ejpam-4793	120	5	an	an	DET
ejpam-4793	120	6	identity	identity	NOUN
ejpam-4793	120	7	in	in	ADP
ejpam-4793	120	8	m	m	PROPN
ejpam-4793	120	9	.	.	PUNCT
ejpam-4793	121	1	thus	thus	ADV
ejpam-4793	121	2	,	,	PUNCT
ejpam-4793	121	3	m	m	VERB
ejpam-4793	121	4	is	be	AUX
ejpam-4793	121	5	a	a	DET
ejpam-4793	121	6	monoid	monoid	NOUN
ejpam-4793	121	7	.	.	PUNCT
ejpam-4793	122	1	with	with	ADP
ejpam-4793	122	2	a	a	DET
ejpam-4793	122	3	group	group	NOUN
ejpam-4793	122	4	γ	γ	NOUN
ejpam-4793	122	5	acting	act	VERB
ejpam-4793	122	6	trivially	trivially	ADV
ejpam-4793	122	7	on	on	ADP
ejpam-4793	122	8	m	m	PROPN
ejpam-4793	122	9	,	,	PUNCT
ejpam-4793	122	10	we	we	PRON
ejpam-4793	122	11	obtain	obtain	VERB
ejpam-4793	122	12	that	that	SCONJ
ejpam-4793	122	13	m	m	PROPN
ejpam-4793	122	14	is	be	AUX
ejpam-4793	122	15	a	a	DET
ejpam-4793	122	16	γ	γ	X
ejpam-4793	122	17	-	-	PUNCT
ejpam-4793	122	18	monoid	monoid	NOUN
ejpam-4793	122	19	.	.	PUNCT
ejpam-4793	122	20	definition	definition	NOUN
ejpam-4793	122	21	9	9	NUM
ejpam-4793	122	22	.	.	PUNCT
ejpam-4793	123	1	[	[	X
ejpam-4793	123	2	1	1	X
ejpam-4793	123	3	]	]	X
ejpam-4793	123	4	let	let	VERB
ejpam-4793	123	5	m	m	PRON
ejpam-4793	123	6	,	,	PUNCT
ejpam-4793	123	7	m1	m1	PROPN
ejpam-4793	123	8	and	and	CCONJ
ejpam-4793	123	9	m2	m2	PROPN
ejpam-4793	123	10	be	be	AUX
ejpam-4793	123	11	monoids	monoid	NOUN
ejpam-4793	123	12	and	and	CCONJ
ejpam-4793	123	13	let	let	VERB
ejpam-4793	123	14	γ	γ	NOUN
ejpam-4793	123	15	be	be	AUX
ejpam-4793	123	16	a	a	DET
ejpam-4793	123	17	group	group	NOUN
ejpam-4793	123	18	acting	act	VERB
ejpam-4793	123	19	on	on	ADP
ejpam-4793	123	20	m	m	PROPN
ejpam-4793	123	21	,	,	PUNCT
ejpam-4793	123	22	m1	m1	PROPN
ejpam-4793	123	23	and	and	CCONJ
ejpam-4793	123	24	m2	m2	PROPN
ejpam-4793	123	25	.	.	PUNCT
ejpam-4793	124	1	(	(	PUNCT
ejpam-4793	124	2	i	i	NOUN
ejpam-4793	124	3	)	)	PUNCT
ejpam-4793	124	4	a	a	DET
ejpam-4793	124	5	γ	γ	PROPN
ejpam-4793	124	6	-	-	PUNCT
ejpam-4793	124	7	monoid	monoid	NOUN
ejpam-4793	124	8	homomorphism	homomorphism	NOUN
ejpam-4793	124	9	is	be	AUX
ejpam-4793	124	10	a	a	DET
ejpam-4793	124	11	monoid	monoid	NOUN
ejpam-4793	124	12	homomorphism	homomorphism	NOUN
ejpam-4793	124	13	ϕ	ϕ	NOUN
ejpam-4793	124	14	:	:	PUNCT
ejpam-4793	124	15	m1	m1	PROPN
ejpam-4793	124	16	−→	−→	NOUN
ejpam-4793	124	17	m2	m2	PROPN
ejpam-4793	124	18	that	that	PRON
ejpam-4793	124	19	respects	respect	VERB
ejpam-4793	124	20	the	the	DET
ejpam-4793	124	21	action	action	NOUN
ejpam-4793	124	22	of	of	ADP
ejpam-4793	124	23	γ	γ	PROPN
ejpam-4793	124	24	,	,	PUNCT
ejpam-4793	124	25	this	this	PRON
ejpam-4793	124	26	means	mean	VERB
ejpam-4793	124	27	ϕ(αa	ϕ(αa	NUM
ejpam-4793	124	28	)	)	PUNCT
ejpam-4793	124	29	=	=	PUNCT
ejpam-4793	124	30	αϕ(a	αϕ(a	NOUN
ejpam-4793	124	31	)	)	PUNCT
ejpam-4793	124	32	.	.	PUNCT
ejpam-4793	125	1	(	(	PUNCT
ejpam-4793	125	2	ii	ii	NOUN
ejpam-4793	125	3	)	)	PUNCT
ejpam-4793	125	4	a	a	DET
ejpam-4793	125	5	γ	γ	NOUN
ejpam-4793	125	6	-	-	PUNCT
ejpam-4793	125	7	order	order	NOUN
ejpam-4793	125	8	-	-	PUNCT
ejpam-4793	125	9	ideal	ideal	NOUN
ejpam-4793	125	10	of	of	ADP
ejpam-4793	125	11	a	a	DET
ejpam-4793	125	12	monoid	monoid	NOUN
ejpam-4793	125	13	m	m	NOUN
ejpam-4793	125	14	is	be	AUX
ejpam-4793	125	15	a	a	DET
ejpam-4793	125	16	subset	subset	NOUN
ejpam-4793	125	17	i	i	PRON
ejpam-4793	125	18	of	of	ADP
ejpam-4793	125	19	m	m	PRON
ejpam-4793	125	20	such	such	ADJ
ejpam-4793	125	21	that	that	PRON
ejpam-4793	125	22	for	for	ADP
ejpam-4793	125	23	any	any	DET
ejpam-4793	125	24	α	α	NOUN
ejpam-4793	125	25	,	,	PUNCT
ejpam-4793	125	26	β	β	PROPN
ejpam-4793	125	27	∈	∈	PROPN
ejpam-4793	125	28	γ	γ	X
ejpam-4793	125	29	,	,	PUNCT
ejpam-4793	125	30	αa	αa	ADV
ejpam-4793	125	31	∗	∗	VERB
ejpam-4793	125	32	βb	βb	DET
ejpam-4793	125	33	∈	∈	PROPN
ejpam-4793	126	1	i	i	PRON
ejpam-4793	126	2	if	if	SCONJ
ejpam-4793	126	3	and	and	CCONJ
ejpam-4793	126	4	only	only	ADV
ejpam-4793	126	5	if	if	SCONJ
ejpam-4793	126	6	a	a	DET
ejpam-4793	126	7	,	,	PUNCT
ejpam-4793	126	8	b	b	PROPN
ejpam-4793	126	9	∈	∈	PROPN
ejpam-4793	126	10	i.	i.	NOUN
ejpam-4793	126	11	remark	remark	NOUN
ejpam-4793	126	12	2	2	NUM
ejpam-4793	126	13	.	.	PUNCT
ejpam-4793	127	1	[	[	X
ejpam-4793	127	2	1	1	X
ejpam-4793	127	3	]	]	PUNCT
ejpam-4793	127	4	a	a	DET
ejpam-4793	127	5	γ	γ	NOUN
ejpam-4793	127	6	-	-	PUNCT
ejpam-4793	127	7	order	order	NOUN
ejpam-4793	127	8	-	-	PUNCT
ejpam-4793	127	9	ideal	ideal	NOUN
ejpam-4793	127	10	is	be	AUX
ejpam-4793	127	11	a	a	DET
ejpam-4793	127	12	submonoid	submonoid	ADJ
ejpam-4793	127	13	i	i	PRON
ejpam-4793	127	14	of	of	ADP
ejpam-4793	127	15	m	m	PRON
ejpam-4793	127	16	which	which	PRON
ejpam-4793	127	17	is	be	AUX
ejpam-4793	127	18	closed	close	VERB
ejpam-4793	127	19	under	under	ADP
ejpam-4793	127	20	the	the	DET
ejpam-4793	127	21	action	action	NOUN
ejpam-4793	127	22	of	of	ADP
ejpam-4793	127	23	γ	γ	PROPN
ejpam-4793	127	24	.	.	PROPN
ejpam-4793	127	25	example	example	NOUN
ejpam-4793	127	26	6	6	NUM
ejpam-4793	127	27	.	.	PUNCT
ejpam-4793	128	1	let	let	VERB
ejpam-4793	128	2	a	a	DET
ejpam-4793	128	3	group	group	NOUN
ejpam-4793	128	4	γ	γ	PROPN
ejpam-4793	128	5	acts	act	VERB
ejpam-4793	128	6	trivially	trivially	ADV
ejpam-4793	128	7	on	on	ADP
ejpam-4793	128	8	both	both	DET
ejpam-4793	128	9	monoids	monoid	NOUN
ejpam-4793	128	10	m	m	VERB
ejpam-4793	128	11	=	=	SYM
ejpam-4793	128	12	(	(	PUNCT
ejpam-4793	128	13	n,+	n,+	NUM
ejpam-4793	128	14	)	)	PUNCT
ejpam-4793	128	15	and	and	CCONJ
ejpam-4793	128	16	n	n	CCONJ
ejpam-4793	128	17	=	=	SYM
ejpam-4793	128	18	(	(	PUNCT
ejpam-4793	128	19	n	n	CCONJ
ejpam-4793	128	20	,	,	PUNCT
ejpam-4793	128	21	·	·	PUNCT
ejpam-4793	128	22	)	)	PUNCT
ejpam-4793	128	23	,	,	PUNCT
ejpam-4793	128	24	that	that	ADV
ejpam-4793	128	25	is	is	ADV
ejpam-4793	128	26	,	,	PUNCT
ejpam-4793	128	27	for	for	ADP
ejpam-4793	128	28	all	all	PRON
ejpam-4793	128	29	α	α	PRON
ejpam-4793	128	30	∈	∈	PROPN
ejpam-4793	128	31	γ	γ	X
ejpam-4793	128	32	,	,	PUNCT
ejpam-4793	128	33	we	we	PRON
ejpam-4793	128	34	have	have	VERB
ejpam-4793	128	35	ϕ((α	ϕ((α	NOUN
ejpam-4793	128	36	,	,	PUNCT
ejpam-4793	128	37	m	m	NOUN
ejpam-4793	128	38	)	)	PUNCT
ejpam-4793	128	39	)	)	PUNCT
ejpam-4793	129	1	=	=	PUNCT
ejpam-4793	129	2	αm	αm	NOUN
ejpam-4793	129	3	=	=	PUNCT
ejpam-4793	129	4	m	m	PROPN
ejpam-4793	129	5	and	and	CCONJ
ejpam-4793	129	6	ϕ((α	ϕ((α	PROPN
ejpam-4793	129	7	,	,	PUNCT
ejpam-4793	129	8	n	n	CCONJ
ejpam-4793	129	9	)	)	PUNCT
ejpam-4793	129	10	)	)	PUNCT
ejpam-4793	130	1	=	=	PUNCT
ejpam-4793	130	2	αn	αn	NOUN
ejpam-4793	131	1	=	=	SYM
ejpam-4793	132	1	n	n	PROPN
ejpam-4793	132	2	for	for	ADP
ejpam-4793	132	3	all	all	DET
ejpam-4793	132	4	m	m	NOUN
ejpam-4793	132	5	∈	∈	NOUN
ejpam-4793	132	6	m	m	NOUN
ejpam-4793	132	7	and	and	CCONJ
ejpam-4793	132	8	n	n	PRON
ejpam-4793	132	9	∈	∈	PROPN
ejpam-4793	132	10	n	n	NOUN
ejpam-4793	132	11	.	.	PUNCT
ejpam-4793	133	1	now	now	ADV
ejpam-4793	133	2	,	,	PUNCT
ejpam-4793	133	3	let	let	VERB
ejpam-4793	133	4	α	α	PRON
ejpam-4793	133	5	∈	∈	PROPN
ejpam-4793	133	6	γ	γ	X
ejpam-4793	133	7	and	and	CCONJ
ejpam-4793	133	8	x	x	NOUN
ejpam-4793	133	9	,	,	PUNCT
ejpam-4793	133	10	y	y	PROPN
ejpam-4793	133	11	∈	∈	PROPN
ejpam-4793	133	12	m	m	VERB
ejpam-4793	133	13	.	.	PUNCT
ejpam-4793	134	1	then	then	ADV
ejpam-4793	134	2	,	,	PUNCT
ejpam-4793	134	3	ϕ((α	ϕ((α	PROPN
ejpam-4793	134	4	,	,	PUNCT
ejpam-4793	134	5	x	x	PUNCT
ejpam-4793	135	1	+	+	NUM
ejpam-4793	135	2	y	y	NOUN
ejpam-4793	135	3	)	)	PUNCT
ejpam-4793	135	4	)	)	PUNCT
ejpam-4793	136	1	=	=	PUNCT
ejpam-4793	137	1	α(x+	α(x+	NUM
ejpam-4793	137	2	y	y	X
ejpam-4793	137	3	)	)	PUNCT
ejpam-4793	137	4	=	=	PUNCT
ejpam-4793	138	1	x	x	PUNCT
ejpam-4793	139	1	+	+	NUM
ejpam-4793	139	2	y	y	NOUN
ejpam-4793	139	3	=	=	SYM
ejpam-4793	139	4	αx	αx	PROPN
ejpam-4793	140	1	+	+	CCONJ
ejpam-4793	140	2	αy	αy	ADJ
ejpam-4793	140	3	=	=	PUNCT
ejpam-4793	140	4	ϕ((α	ϕ((α	PROPN
ejpam-4793	140	5	,	,	PUNCT
ejpam-4793	140	6	x	x	NOUN
ejpam-4793	140	7	)	)	PUNCT
ejpam-4793	140	8	)	)	PUNCT
ejpam-4793	141	1	+	+	CCONJ
ejpam-4793	141	2	ϕ((α	ϕ((α	PROPN
ejpam-4793	141	3	,	,	PUNCT
ejpam-4793	141	4	y	y	NOUN
ejpam-4793	141	5	)	)	PUNCT
ejpam-4793	141	6	)	)	PUNCT
ejpam-4793	141	7	.	.	PUNCT
ejpam-4793	142	1	thus	thus	ADV
ejpam-4793	142	2	,	,	PUNCT
ejpam-4793	142	3	m	m	VERB
ejpam-4793	142	4	and	and	CCONJ
ejpam-4793	142	5	n	n	PROPN
ejpam-4793	142	6	are	be	AUX
ejpam-4793	142	7	γ	γ	NOUN
ejpam-4793	142	8	-	-	PUNCT
ejpam-4793	142	9	monoids	monoid	NOUN
ejpam-4793	142	10	.	.	PUNCT
ejpam-4793	143	1	consider	consider	VERB
ejpam-4793	143	2	the	the	DET
ejpam-4793	143	3	monoid	monoid	NOUN
ejpam-4793	143	4	homomorphism	homomorphism	PROPN
ejpam-4793	143	5	φ	φ	X
ejpam-4793	143	6	:	:	PUNCT
ejpam-4793	143	7	m	m	VERB
ejpam-4793	143	8	→	→	SYM
ejpam-4793	143	9	n	n	PRON
ejpam-4793	143	10	defined	define	VERB
ejpam-4793	143	11	by	by	ADP
ejpam-4793	143	12	φ(x	φ(x	NOUN
ejpam-4793	143	13	)	)	PUNCT
ejpam-4793	144	1	=	=	SYM
ejpam-4793	144	2	bx	bx	PROPN
ejpam-4793	144	3	,	,	PUNCT
ejpam-4793	144	4	where	where	SCONJ
ejpam-4793	144	5	b	b	X
ejpam-4793	144	6	∈	∈	PROPN
ejpam-4793	144	7	n	n	PRON
ejpam-4793	144	8	\	\	NOUN
ejpam-4793	144	9	{	{	PUNCT
ejpam-4793	144	10	0	0	NUM
ejpam-4793	144	11	}	}	PUNCT
ejpam-4793	144	12	in	in	ADP
ejpam-4793	144	13	example	example	NOUN
ejpam-4793	144	14	1	1	NUM
ejpam-4793	144	15	.	.	PUNCT
ejpam-4793	144	16	for	for	ADP
ejpam-4793	144	17	all	all	PRON
ejpam-4793	144	18	α	α	DET
ejpam-4793	144	19	∈	∈	NOUN
ejpam-4793	144	20	γ	γ	NOUN
ejpam-4793	144	21	and	and	CCONJ
ejpam-4793	144	22	a	a	DET
ejpam-4793	144	23	∈	∈	NOUN
ejpam-4793	144	24	m	m	VERB
ejpam-4793	144	25	,	,	PUNCT
ejpam-4793	144	26	we	we	PRON
ejpam-4793	144	27	have	have	VERB
ejpam-4793	144	28	φ(αa	φ(αa	NOUN
ejpam-4793	144	29	)	)	PUNCT
ejpam-4793	144	30	=	=	SYM
ejpam-4793	144	31	φ(a	φ(a	ADJ
ejpam-4793	144	32	)	)	PUNCT
ejpam-4793	144	33	=	=	PUNCT
ejpam-4793	144	34	αφ(a	αφ(a	NUM
ejpam-4793	144	35	)	)	PUNCT
ejpam-4793	144	36	.	.	PUNCT
ejpam-4793	145	1	thus	thus	ADV
ejpam-4793	145	2	,	,	PUNCT
ejpam-4793	145	3	by	by	ADP
ejpam-4793	145	4	definition	definition	NOUN
ejpam-4793	145	5	9(ii	9(ii	PROPN
ejpam-4793	145	6	)	)	PUNCT
ejpam-4793	145	7	,	,	PUNCT
ejpam-4793	145	8	φ	φ	PROPN
ejpam-4793	145	9	is	be	AUX
ejpam-4793	145	10	a	a	DET
ejpam-4793	145	11	γ	γ	NOUN
ejpam-4793	145	12	-	-	PUNCT
ejpam-4793	145	13	monoid	monoid	NOUN
ejpam-4793	145	14	homomorphism	homomorphism	NOUN
ejpam-4793	145	15	.	.	PUNCT
ejpam-4793	146	1	example	example	NOUN
ejpam-4793	146	2	7	7	NUM
ejpam-4793	146	3	.	.	PUNCT
ejpam-4793	146	4	consider	consider	VERB
ejpam-4793	146	5	the	the	DET
ejpam-4793	146	6	γ	γ	NOUN
ejpam-4793	146	7	-	-	PUNCT
ejpam-4793	146	8	monoid	monoid	NOUN
ejpam-4793	146	9	m	m	NOUN
ejpam-4793	146	10	=	=	NOUN
ejpam-4793	146	11	r	r	NOUN
ejpam-4793	146	12	under	under	ADP
ejpam-4793	146	13	the	the	DET
ejpam-4793	146	14	usual	usual	ADJ
ejpam-4793	146	15	addition	addition	NOUN
ejpam-4793	146	16	in	in	ADP
ejpam-4793	146	17	example	example	NOUN
ejpam-4793	146	18	3	3	NUM
ejpam-4793	146	19	and	and	CCONJ
ejpam-4793	146	20	the	the	DET
ejpam-4793	146	21	γ	γ	PROPN
ejpam-4793	146	22	-	-	PUNCT
ejpam-4793	146	23	monoid	monoid	PROPN
ejpam-4793	146	24	t	t	PROPN
ejpam-4793	146	25	=	=	SYM
ejpam-4793	146	26	m2(r	m2(r	PROPN
ejpam-4793	146	27	)	)	PUNCT
ejpam-4793	146	28	under	under	ADP
ejpam-4793	146	29	matrix	matrix	NOUN
ejpam-4793	146	30	addition	addition	NOUN
ejpam-4793	146	31	in	in	ADP
ejpam-4793	146	32	example	example	NOUN
ejpam-4793	146	33	4	4	NUM
ejpam-4793	146	34	.	.	PUNCT
ejpam-4793	146	35	define	define	VERB
ejpam-4793	146	36	a	a	DET
ejpam-4793	146	37	mapping	mapping	NOUN
ejpam-4793	146	38	h.	h.	NOUN
ejpam-4793	146	39	sarapuddin	sarapuddin	PROPN
ejpam-4793	146	40	,	,	PUNCT
ejpam-4793	146	41	j.	j.	PROPN
ejpam-4793	146	42	vilela	vilela	PROPN
ejpam-4793	146	43	/	/	SYM
ejpam-4793	146	44	eur	eur	PROPN
ejpam-4793	146	45	.	.	PUNCT
ejpam-4793	147	1	j.	j.	PROPN
ejpam-4793	147	2	pure	pure	PROPN
ejpam-4793	147	3	appl	appl	PROPN
ejpam-4793	147	4	.	.	PROPN
ejpam-4793	147	5	math	math	PROPN
ejpam-4793	147	6	,	,	PUNCT
ejpam-4793	147	7	16	16	NUM
ejpam-4793	147	8	(	(	PUNCT
ejpam-4793	147	9	3	3	NUM
ejpam-4793	147	10	)	)	PUNCT
ejpam-4793	147	11	(	(	PUNCT
ejpam-4793	147	12	2023	2023	NUM
ejpam-4793	147	13	)	)	PUNCT
ejpam-4793	147	14	,	,	PUNCT
ejpam-4793	147	15	1772	1772	NUM
ejpam-4793	147	16	-	-	SYM
ejpam-4793	147	17	1793	1793	NUM
ejpam-4793	147	18	1776	1776	NUM
ejpam-4793	147	19	ϕ	ϕ	NOUN
ejpam-4793	147	20	:	:	PUNCT
ejpam-4793	147	21	t	t	PROPN
ejpam-4793	147	22	→	→	SYM
ejpam-4793	147	23	m	m	VERB
ejpam-4793	147	24	by	by	ADP
ejpam-4793	147	25	ϕ	ϕ	X
ejpam-4793	147	26	(	(	PUNCT
ejpam-4793	147	27	(	(	PUNCT
ejpam-4793	147	28	a	a	DET
ejpam-4793	147	29	b	b	NOUN
ejpam-4793	147	30	c	c	NOUN
ejpam-4793	147	31	d	d	NOUN
ejpam-4793	147	32	)	)	PUNCT
ejpam-4793	147	33	)	)	PUNCT
ejpam-4793	148	1	=	=	PUNCT
ejpam-4793	149	1	2(a	2(a	NUM
ejpam-4793	149	2	+	+	SYM
ejpam-4793	149	3	b	b	NOUN
ejpam-4793	150	1	+	+	CCONJ
ejpam-4793	150	2	c	c	NOUN
ejpam-4793	150	3	+	+	CCONJ
ejpam-4793	150	4	d	d	NOUN
ejpam-4793	150	5	)	)	PUNCT
ejpam-4793	150	6	.	.	PUNCT
ejpam-4793	151	1	let	let	VERB
ejpam-4793	151	2	(	(	PUNCT
ejpam-4793	151	3	a	a	DET
ejpam-4793	151	4	b	b	NOUN
ejpam-4793	151	5	c	c	NOUN
ejpam-4793	151	6	d	d	NOUN
ejpam-4793	151	7	)	)	PUNCT
ejpam-4793	151	8	,	,	PUNCT
ejpam-4793	151	9	(	(	PUNCT
ejpam-4793	151	10	e	e	NOUN
ejpam-4793	151	11	f	f	PROPN
ejpam-4793	151	12	g	g	PROPN
ejpam-4793	151	13	h	h	PROPN
ejpam-4793	151	14	)	)	PUNCT
ejpam-4793	151	15	∈	∈	PROPN
ejpam-4793	151	16	t	t	NOUN
ejpam-4793	151	17	such	such	ADJ
ejpam-4793	151	18	that	that	SCONJ
ejpam-4793	151	19	(	(	PUNCT
ejpam-4793	151	20	a	a	DET
ejpam-4793	151	21	b	b	NOUN
ejpam-4793	151	22	c	c	NOUN
ejpam-4793	151	23	d	d	NOUN
ejpam-4793	151	24	)	)	PUNCT
ejpam-4793	151	25	=	=	PUNCT
ejpam-4793	152	1	(	(	PUNCT
ejpam-4793	152	2	e	e	X
ejpam-4793	152	3	f	f	PROPN
ejpam-4793	152	4	g	g	PROPN
ejpam-4793	152	5	h	h	PROPN
ejpam-4793	152	6	)	)	PUNCT
ejpam-4793	152	7	.	.	PUNCT
ejpam-4793	153	1	then	then	ADV
ejpam-4793	153	2	a	a	DET
ejpam-4793	153	3	=	=	SYM
ejpam-4793	153	4	e	e	NOUN
ejpam-4793	153	5	,	,	PUNCT
ejpam-4793	153	6	b	b	PROPN
ejpam-4793	153	7	=	=	SYM
ejpam-4793	153	8	f	f	PROPN
ejpam-4793	153	9	,	,	PUNCT
ejpam-4793	153	10	c	c	PROPN
ejpam-4793	153	11	=	=	SYM
ejpam-4793	153	12	g	g	PROPN
ejpam-4793	153	13	and	and	CCONJ
ejpam-4793	153	14	d	d	PROPN
ejpam-4793	153	15	=	=	PROPN
ejpam-4793	153	16	h.	h.	PROPN
ejpam-4793	153	17	thus	thus	ADV
ejpam-4793	153	18	,	,	PUNCT
ejpam-4793	153	19	2(a	2(a	NUM
ejpam-4793	153	20	+	+	SYM
ejpam-4793	153	21	b	b	NOUN
ejpam-4793	154	1	+	+	CCONJ
ejpam-4793	154	2	c	c	NOUN
ejpam-4793	154	3	+	+	CCONJ
ejpam-4793	154	4	d	d	NOUN
ejpam-4793	154	5	)	)	PUNCT
ejpam-4793	155	1	=	=	PUNCT
ejpam-4793	155	2	2(e	2(e	NUM
ejpam-4793	156	1	+	+	CCONJ
ejpam-4793	156	2	f	f	X
ejpam-4793	156	3	+	+	CCONJ
ejpam-4793	156	4	g	g	PROPN
ejpam-4793	156	5	+	+	NOUN
ejpam-4793	156	6	h	h	NOUN
ejpam-4793	156	7	)	)	PUNCT
ejpam-4793	156	8	and	and	CCONJ
ejpam-4793	156	9	ϕ	ϕ	NOUN
ejpam-4793	156	10	is	be	AUX
ejpam-4793	156	11	well	well	ADV
ejpam-4793	156	12	-	-	PUNCT
ejpam-4793	156	13	defined	define	VERB
ejpam-4793	156	14	.	.	PUNCT
ejpam-4793	157	1	now	now	ADV
ejpam-4793	157	2	,	,	PUNCT
ejpam-4793	157	3	for	for	ADP
ejpam-4793	157	4	any	any	DET
ejpam-4793	157	5	(	(	PUNCT
ejpam-4793	157	6	a	a	DET
ejpam-4793	157	7	b	b	NOUN
ejpam-4793	157	8	c	c	NOUN
ejpam-4793	157	9	d	d	NOUN
ejpam-4793	157	10	)	)	PUNCT
ejpam-4793	157	11	,	,	PUNCT
ejpam-4793	157	12	(	(	PUNCT
ejpam-4793	157	13	e	e	NOUN
ejpam-4793	157	14	f	f	PROPN
ejpam-4793	157	15	g	g	PROPN
ejpam-4793	157	16	h	h	PROPN
ejpam-4793	157	17	)	)	PUNCT
ejpam-4793	157	18	∈	∈	PROPN
ejpam-4793	157	19	t	t	PROPN
ejpam-4793	157	20	,	,	PUNCT
ejpam-4793	157	21	we	we	PRON
ejpam-4793	157	22	have	have	VERB
ejpam-4793	157	23	ϕ	ϕ	NOUN
ejpam-4793	157	24	(	(	PUNCT
ejpam-4793	157	25	(	(	PUNCT
ejpam-4793	157	26	0	0	NUM
ejpam-4793	157	27	0	0	NUM
ejpam-4793	157	28	0	0	NUM
ejpam-4793	157	29	0	0	NUM
ejpam-4793	157	30	)	)	PUNCT
ejpam-4793	157	31	)	)	PUNCT
ejpam-4793	158	1	=	=	PUNCT
ejpam-4793	158	2	2(0	2(0	NUM
ejpam-4793	158	3	+	+	CCONJ
ejpam-4793	158	4	0	0	NUM
ejpam-4793	159	1	+	+	CCONJ
ejpam-4793	159	2	0	0	NUM
ejpam-4793	160	1	+	+	CCONJ
ejpam-4793	160	2	0	0	NUM
ejpam-4793	160	3	)	)	PUNCT
ejpam-4793	160	4	=	=	SYM
ejpam-4793	160	5	2(0	2(0	NUM
ejpam-4793	160	6	)	)	PUNCT
ejpam-4793	160	7	=	=	SYM
ejpam-4793	160	8	0	0	NUM
ejpam-4793	160	9	and	and	CCONJ
ejpam-4793	160	10	ϕ	ϕ	X
ejpam-4793	160	11	(	(	PUNCT
ejpam-4793	160	12	(	(	PUNCT
ejpam-4793	160	13	a	a	DET
ejpam-4793	160	14	b	b	NOUN
ejpam-4793	160	15	c	c	NOUN
ejpam-4793	160	16	d	d	NOUN
ejpam-4793	160	17	)	)	PUNCT
ejpam-4793	161	1	+	+	CCONJ
ejpam-4793	161	2	(	(	PUNCT
ejpam-4793	161	3	e	e	X
ejpam-4793	161	4	f	f	PROPN
ejpam-4793	161	5	g	g	PROPN
ejpam-4793	161	6	h	h	PROPN
ejpam-4793	161	7	)	)	PUNCT
ejpam-4793	161	8	)	)	PUNCT
ejpam-4793	162	1	=	=	SYM
ejpam-4793	162	2	ϕ	ϕ	X
ejpam-4793	162	3	(	(	PUNCT
ejpam-4793	162	4	(	(	PUNCT
ejpam-4793	162	5	a+	a+	X
ejpam-4793	162	6	e	e	X
ejpam-4793	162	7	b+	b+	X
ejpam-4793	162	8	f	f	X
ejpam-4793	162	9	c+	c+	VERB
ejpam-4793	162	10	g	g	PROPN
ejpam-4793	162	11	d+	d+	PROPN
ejpam-4793	162	12	h	h	NOUN
ejpam-4793	162	13	)	)	PUNCT
ejpam-4793	162	14	)	)	PUNCT
ejpam-4793	163	1	=	=	PUNCT
ejpam-4793	164	1	2((a+	2((a+	NUM
ejpam-4793	164	2	e	e	NOUN
ejpam-4793	164	3	)	)	PUNCT
ejpam-4793	165	1	+	+	CCONJ
ejpam-4793	165	2	(	(	PUNCT
ejpam-4793	165	3	b+	b+	X
ejpam-4793	165	4	f	f	X
ejpam-4793	165	5	)	)	PUNCT
ejpam-4793	166	1	+	+	CCONJ
ejpam-4793	166	2	(	(	PUNCT
ejpam-4793	166	3	c+	c+	VERB
ejpam-4793	166	4	g	g	NOUN
ejpam-4793	166	5	)	)	PUNCT
ejpam-4793	167	1	+	+	CCONJ
ejpam-4793	167	2	(	(	PUNCT
ejpam-4793	167	3	d+	d+	NOUN
ejpam-4793	167	4	h	h	NOUN
ejpam-4793	167	5	)	)	PUNCT
ejpam-4793	167	6	)	)	PUNCT
ejpam-4793	168	1	=	=	SYM
ejpam-4793	169	1	2((a+	2((a+	NUM
ejpam-4793	169	2	b+	b+	AUX
ejpam-4793	169	3	c+	c+	VERB
ejpam-4793	169	4	d	d	NOUN
ejpam-4793	169	5	)	)	PUNCT
ejpam-4793	170	1	+	+	CCONJ
ejpam-4793	170	2	(	(	PUNCT
ejpam-4793	170	3	e+	e+	PUNCT
ejpam-4793	170	4	f	f	PROPN
ejpam-4793	170	5	+	+	CCONJ
ejpam-4793	170	6	g	g	PROPN
ejpam-4793	170	7	+	+	NUM
ejpam-4793	170	8	h	h	NOUN
ejpam-4793	170	9	)	)	PUNCT
ejpam-4793	170	10	)	)	PUNCT
ejpam-4793	171	1	=	=	SYM
ejpam-4793	171	2	2(a+	2(a+	NUM
ejpam-4793	171	3	b+	b+	X
ejpam-4793	171	4	c+	c+	VERB
ejpam-4793	171	5	d	d	NOUN
ejpam-4793	171	6	)	)	PUNCT
ejpam-4793	172	1	+	+	CCONJ
ejpam-4793	172	2	2(e+	2(e+	NUM
ejpam-4793	172	3	f	f	NOUN
ejpam-4793	172	4	+	+	CCONJ
ejpam-4793	172	5	g	g	PROPN
ejpam-4793	172	6	+	+	NOUN
ejpam-4793	172	7	h	h	NOUN
ejpam-4793	172	8	)	)	PUNCT
ejpam-4793	172	9	=	=	SYM
ejpam-4793	172	10	ϕ	ϕ	X
ejpam-4793	172	11	(	(	PUNCT
ejpam-4793	172	12	(	(	PUNCT
ejpam-4793	172	13	a	a	DET
ejpam-4793	172	14	b	b	NOUN
ejpam-4793	172	15	c	c	NOUN
ejpam-4793	172	16	d	d	NOUN
ejpam-4793	172	17	)	)	PUNCT
ejpam-4793	172	18	)	)	PUNCT
ejpam-4793	173	1	+	+	CCONJ
ejpam-4793	173	2	ϕ	ϕ	X
ejpam-4793	173	3	(	(	PUNCT
ejpam-4793	173	4	(	(	PUNCT
ejpam-4793	173	5	e	e	X
ejpam-4793	173	6	f	f	PROPN
ejpam-4793	173	7	g	g	PROPN
ejpam-4793	173	8	h	h	PROPN
ejpam-4793	173	9	)	)	PUNCT
ejpam-4793	173	10	)	)	PUNCT
ejpam-4793	173	11	.	.	PUNCT
ejpam-4793	174	1	thus	thus	ADV
ejpam-4793	174	2	,	,	PUNCT
ejpam-4793	174	3	ϕ	ϕ	PROPN
ejpam-4793	174	4	is	be	AUX
ejpam-4793	174	5	a	a	DET
ejpam-4793	174	6	monoid	monoid	NOUN
ejpam-4793	174	7	homomorphism	homomorphism	NOUN
ejpam-4793	174	8	.	.	PUNCT
ejpam-4793	175	1	also	also	ADV
ejpam-4793	175	2	,	,	PUNCT
ejpam-4793	175	3	for	for	ADP
ejpam-4793	175	4	all	all	PRON
ejpam-4793	175	5	α	α	DET
ejpam-4793	175	6	∈	∈	NOUN
ejpam-4793	175	7	γ	γ	X
ejpam-4793	175	8	and	and	CCONJ
ejpam-4793	175	9	(	(	PUNCT
ejpam-4793	175	10	a	a	DET
ejpam-4793	175	11	b	b	NOUN
ejpam-4793	175	12	c	c	PROPN
ejpam-4793	175	13	d	d	NOUN
ejpam-4793	175	14	)	)	PUNCT
ejpam-4793	175	15	∈	∈	PROPN
ejpam-4793	175	16	t	t	PROPN
ejpam-4793	175	17	,	,	PUNCT
ejpam-4793	175	18	we	we	PRON
ejpam-4793	175	19	have	have	VERB
ejpam-4793	175	20	ϕ	ϕ	NOUN
ejpam-4793	175	21	(	(	PUNCT
ejpam-4793	175	22	α	α	X
ejpam-4793	175	23	(	(	PUNCT
ejpam-4793	175	24	a	a	DET
ejpam-4793	175	25	b	b	NOUN
ejpam-4793	175	26	c	c	NOUN
ejpam-4793	175	27	d	d	NOUN
ejpam-4793	175	28	)	)	PUNCT
ejpam-4793	175	29	)	)	PUNCT
ejpam-4793	176	1	=	=	SYM
ejpam-4793	176	2	ϕ	ϕ	X
ejpam-4793	176	3	(	(	PUNCT
ejpam-4793	176	4	(	(	PUNCT
ejpam-4793	176	5	2αa	2αa	ADJ
ejpam-4793	176	6	2αb	2αb	ADJ
ejpam-4793	176	7	2αc	2αc	ADJ
ejpam-4793	176	8	2αd	2αd	NOUN
ejpam-4793	176	9	)	)	PUNCT
ejpam-4793	176	10	)	)	PUNCT
ejpam-4793	177	1	=	=	SYM
ejpam-4793	177	2	2(2αa+	2(2αa+	NUM
ejpam-4793	177	3	2αb+	2αb+	NUM
ejpam-4793	177	4	2αc+	2αc+	NUM
ejpam-4793	177	5	2αd	2αd	ADV
ejpam-4793	177	6	)	)	PUNCT
ejpam-4793	178	1	=	=	SYM
ejpam-4793	178	2	2α2(a+	2α2(a+	NOUN
ejpam-4793	178	3	b+	b+	AUX
ejpam-4793	178	4	c+	c+	VERB
ejpam-4793	178	5	d	d	NOUN
ejpam-4793	178	6	)	)	PUNCT
ejpam-4793	178	7	=	=	SYM
ejpam-4793	178	8	αϕ	αϕ	INTJ
ejpam-4793	178	9	(	(	PUNCT
ejpam-4793	178	10	(	(	PUNCT
ejpam-4793	178	11	a	a	DET
ejpam-4793	178	12	b	b	NOUN
ejpam-4793	178	13	c	c	NOUN
ejpam-4793	178	14	d	d	NOUN
ejpam-4793	178	15	)	)	PUNCT
ejpam-4793	178	16	)	)	PUNCT
ejpam-4793	178	17	.	.	PUNCT
ejpam-4793	179	1	hence	hence	ADV
ejpam-4793	179	2	,	,	PUNCT
ejpam-4793	179	3	ϕ	ϕ	PROPN
ejpam-4793	179	4	is	be	AUX
ejpam-4793	179	5	a	a	DET
ejpam-4793	179	6	γ	γ	NOUN
ejpam-4793	179	7	-	-	PUNCT
ejpam-4793	179	8	monoid	monoid	NOUN
ejpam-4793	179	9	homomorphism	homomorphism	NOUN
ejpam-4793	179	10	.	.	PUNCT
ejpam-4793	180	1	theorem	theorem	NOUN
ejpam-4793	180	2	1	1	NUM
ejpam-4793	180	3	.	.	PUNCT
ejpam-4793	181	1	[	[	X
ejpam-4793	181	2	4	4	X
ejpam-4793	181	3	]	]	PUNCT
ejpam-4793	181	4	let	let	VERB
ejpam-4793	181	5	m1	m1	PROPN
ejpam-4793	181	6	and	and	CCONJ
ejpam-4793	181	7	m2	m2	PROPN
ejpam-4793	181	8	be	be	AUX
ejpam-4793	181	9	commutative	commutative	ADJ
ejpam-4793	181	10	monoids	monoid	NOUN
ejpam-4793	181	11	and	and	CCONJ
ejpam-4793	181	12	let	let	VERB
ejpam-4793	181	13	f	f	NOUN
ejpam-4793	181	14	:	:	PUNCT
ejpam-4793	181	15	m1	m1	PROPN
ejpam-4793	181	16	−→	−→	NOUN
ejpam-4793	181	17	m2	m2	PROPN
ejpam-4793	181	18	be	be	VERB
ejpam-4793	181	19	a	a	DET
ejpam-4793	181	20	homomorphism	homomorphism	NOUN
ejpam-4793	181	21	.	.	PUNCT
ejpam-4793	182	1	there	there	PRON
ejpam-4793	182	2	exists	exist	VERB
ejpam-4793	182	3	a	a	DET
ejpam-4793	182	4	unique	unique	ADJ
ejpam-4793	182	5	homomorphism	homomorphism	NOUN
ejpam-4793	182	6	φ	φ	NOUN
ejpam-4793	182	7	:	:	PUNCT
ejpam-4793	182	8	m1/	m1/	PROPN
ejpam-4793	182	9	ker	ker	PROPN
ejpam-4793	183	1	f	f	PROPN
ejpam-4793	184	1	−→	−→	NOUN
ejpam-4793	184	2	m2	m2	PROPN
ejpam-4793	184	3	such	such	ADJ
ejpam-4793	184	4	that	that	SCONJ
ejpam-4793	184	5	the	the	DET
ejpam-4793	184	6	following	follow	VERB
ejpam-4793	184	7	diagram	diagram	NOUN
ejpam-4793	184	8	is	be	AUX
ejpam-4793	184	9	commutative	commutative	ADJ
ejpam-4793	184	10	m1	m1	PROPN
ejpam-4793	184	11	m2	m2	PROPN
ejpam-4793	184	12	m1/	m1/	PROPN
ejpam-4793	184	13	ker	ker	PROPN
ejpam-4793	184	14	f	f	PROPN
ejpam-4793	184	15	rker	rker	PROPN
ejpam-4793	184	16	f	f	PROPN
ejpam-4793	184	17	f	f	PROPN
ejpam-4793	184	18	φ	φ	PROPN
ejpam-4793	184	19	that	that	PRON
ejpam-4793	184	20	is	is	ADV
ejpam-4793	184	21	,	,	PUNCT
ejpam-4793	184	22	φ	φ	PROPN
ejpam-4793	184	23	◦	◦	PROPN
ejpam-4793	184	24	rker	rker	NOUN
ejpam-4793	184	25	f	f	PROPN
ejpam-4793	184	26	=	=	SYM
ejpam-4793	184	27	f	f	PROPN
ejpam-4793	184	28	,	,	PUNCT
ejpam-4793	184	29	where	where	SCONJ
ejpam-4793	184	30	rker	rker	NOUN
ejpam-4793	184	31	f	f	PROPN
ejpam-4793	184	32	(	(	PUNCT
ejpam-4793	184	33	x	x	NOUN
ejpam-4793	184	34	)	)	PUNCT
ejpam-4793	184	35	:	:	PUNCT
ejpam-4793	184	36	=	=	SYM
ejpam-4793	184	37	ρker	ρker	NOUN
ejpam-4793	184	38	f	f	PROPN
ejpam-4793	184	39	(	(	PUNCT
ejpam-4793	184	40	x	x	NOUN
ejpam-4793	184	41	)	)	PUNCT
ejpam-4793	184	42	.	.	PUNCT
ejpam-4793	185	1	moreover	moreover	ADV
ejpam-4793	185	2	,	,	PUNCT
ejpam-4793	185	3	φ	φ	PROPN
ejpam-4793	185	4	is	be	AUX
ejpam-4793	185	5	onto	onto	ADP
ejpam-4793	185	6	and	and	CCONJ
ejpam-4793	185	7	it	it	PRON
ejpam-4793	185	8	has	have	VERB
ejpam-4793	185	9	a	a	DET
ejpam-4793	185	10	trivial	trivial	ADJ
ejpam-4793	185	11	kernel	kernel	NOUN
ejpam-4793	185	12	,	,	PUNCT
ejpam-4793	185	13	namely	namely	ADV
ejpam-4793	185	14	,	,	PUNCT
ejpam-4793	185	15	kerφ	kerφ	PROPN
ejpam-4793	185	16	=	=	PUNCT
ejpam-4793	185	17	{	{	PUNCT
ejpam-4793	185	18	ker	ker	NOUN
ejpam-4793	185	19	f	f	X
ejpam-4793	185	20	}	}	PUNCT
ejpam-4793	185	21	.	.	PUNCT
ejpam-4793	186	1	however	however	ADV
ejpam-4793	186	2	,	,	PUNCT
ejpam-4793	186	3	φ	φ	PROPN
ejpam-4793	186	4	is	be	AUX
ejpam-4793	186	5	an	an	DET
ejpam-4793	186	6	isomorphism	isomorphism	NOUN
ejpam-4793	186	7	if	if	SCONJ
ejpam-4793	186	8	and	and	CCONJ
ejpam-4793	186	9	only	only	ADV
ejpam-4793	186	10	if	if	SCONJ
ejpam-4793	186	11	ρf	ρf	PROPN
ejpam-4793	186	12	=	=	SYM
ejpam-4793	186	13	ρker	ρker	NOUN
ejpam-4793	186	14	f	f	PROPN
ejpam-4793	186	15	.	.	PUNCT
ejpam-4793	187	1	h.	h.	PROPN
ejpam-4793	187	2	sarapuddin	sarapuddin	PROPN
ejpam-4793	187	3	,	,	PUNCT
ejpam-4793	187	4	j.	j.	PROPN
ejpam-4793	187	5	vilela	vilela	PROPN
ejpam-4793	187	6	/	/	SYM
ejpam-4793	187	7	eur	eur	PROPN
ejpam-4793	187	8	.	.	PUNCT
ejpam-4793	188	1	j.	j.	PROPN
ejpam-4793	188	2	pure	pure	PROPN
ejpam-4793	188	3	appl	appl	PROPN
ejpam-4793	188	4	.	.	PROPN
ejpam-4793	188	5	math	math	PROPN
ejpam-4793	188	6	,	,	PUNCT
ejpam-4793	188	7	16	16	NUM
ejpam-4793	188	8	(	(	PUNCT
ejpam-4793	188	9	3	3	NUM
ejpam-4793	188	10	)	)	PUNCT
ejpam-4793	188	11	(	(	PUNCT
ejpam-4793	188	12	2023	2023	NUM
ejpam-4793	188	13	)	)	PUNCT
ejpam-4793	188	14	,	,	PUNCT
ejpam-4793	188	15	1772	1772	NUM
ejpam-4793	188	16	-	-	SYM
ejpam-4793	188	17	1793	1793	NUM
ejpam-4793	188	18	1777	1777	NUM
ejpam-4793	188	19	3	3	NUM
ejpam-4793	188	20	.	.	PUNCT
ejpam-4793	189	1	γ	γ	NOUN
ejpam-4793	189	2	-	-	NOUN
ejpam-4793	189	3	ideals	ideal	NOUN
ejpam-4793	189	4	in	in	ADP
ejpam-4793	189	5	this	this	DET
ejpam-4793	189	6	section	section	NOUN
ejpam-4793	189	7	,	,	PUNCT
ejpam-4793	189	8	we	we	PRON
ejpam-4793	189	9	discuss	discuss	VERB
ejpam-4793	189	10	the	the	DET
ejpam-4793	189	11	properties	property	NOUN
ejpam-4793	189	12	of	of	ADP
ejpam-4793	189	13	γ	γ	NOUN
ejpam-4793	189	14	-	-	NOUN
ejpam-4793	189	15	ideals	ideal	NOUN
ejpam-4793	189	16	of	of	ADP
ejpam-4793	189	17	γ	γ	NOUN
ejpam-4793	189	18	-	-	PUNCT
ejpam-4793	189	19	monoids	monoid	NOUN
ejpam-4793	189	20	.	.	PUNCT
ejpam-4793	190	1	let	let	VERB
ejpam-4793	190	2	m	m	PRON
ejpam-4793	190	3	be	be	AUX
ejpam-4793	190	4	a	a	DET
ejpam-4793	190	5	γ	γ	X
ejpam-4793	190	6	-	-	PUNCT
ejpam-4793	190	7	monoid	monoid	NOUN
ejpam-4793	190	8	and	and	CCONJ
ejpam-4793	190	9	x	x	SYM
ejpam-4793	190	10	∈	∈	PROPN
ejpam-4793	190	11	m	m	VERB
ejpam-4793	190	12	.	.	PUNCT
ejpam-4793	191	1	by	by	ADP
ejpam-4793	191	2	definition	definition	NOUN
ejpam-4793	191	3	8	8	NUM
ejpam-4793	191	4	,	,	PUNCT
ejpam-4793	191	5	for	for	ADP
ejpam-4793	191	6	all	all	PRON
ejpam-4793	191	7	α	α	PRON
ejpam-4793	191	8	∈	∈	PROPN
ejpam-4793	191	9	γ	γ	X
ejpam-4793	191	10	,	,	PUNCT
ejpam-4793	191	11	αx	αx	PROPN
ejpam-4793	191	12	∗	∗	NOUN
ejpam-4793	191	13	α1	α1	PROPN
ejpam-4793	191	14	m	m	PROPN
ejpam-4793	191	15	=	=	ADJ
ejpam-4793	191	16	α(x	α(x	PROPN
ejpam-4793	191	17	∗	∗	NOUN
ejpam-4793	191	18	1	1	NUM
ejpam-4793	191	19	m	m	NOUN
ejpam-4793	191	20	)	)	PUNCT
ejpam-4793	191	21	=	=	PUNCT
ejpam-4793	191	22	αx	αx	NOUN
ejpam-4793	191	23	and	and	CCONJ
ejpam-4793	191	24	α1	α1	PROPN
ejpam-4793	191	25	m	m	PROPN
ejpam-4793	191	26	∗	∗	NOUN
ejpam-4793	191	27	αx	αx	NOUN
ejpam-4793	191	28	=	=	SYM
ejpam-4793	191	29	α(1	α(1	PROPN
ejpam-4793	191	30	m	m	NOUN
ejpam-4793	191	31	∗	∗	NOUN
ejpam-4793	191	32	x	x	NOUN
ejpam-4793	191	33	)	)	PUNCT
ejpam-4793	191	34	=	=	PUNCT
ejpam-4793	191	35	αx	αx	NOUN
ejpam-4793	191	36	.	.	PUNCT
ejpam-4793	191	37	by	by	ADP
ejpam-4793	191	38	uniqueness	uniqueness	NOUN
ejpam-4793	191	39	of	of	ADP
ejpam-4793	191	40	the	the	DET
ejpam-4793	191	41	identity	identity	NOUN
ejpam-4793	191	42	element	element	NOUN
ejpam-4793	191	43	in	in	ADP
ejpam-4793	191	44	m	m	PROPN
ejpam-4793	191	45	,	,	PUNCT
ejpam-4793	191	46	α1	α1	PROPN
ejpam-4793	191	47	m	m	NOUN
ejpam-4793	191	48	=	=	SYM
ejpam-4793	191	49	1	1	NUM
ejpam-4793	191	50	m	m	NOUN
ejpam-4793	191	51	.	.	PUNCT
ejpam-4793	191	52	remark	remark	PROPN
ejpam-4793	191	53	3	3	NUM
ejpam-4793	191	54	.	.	PUNCT
ejpam-4793	192	1	for	for	ADP
ejpam-4793	192	2	a	a	DET
ejpam-4793	192	3	γ	γ	X
ejpam-4793	192	4	-	-	PUNCT
ejpam-4793	192	5	monoid	monoid	NOUN
ejpam-4793	192	6	m	m	PROPN
ejpam-4793	192	7	and	and	CCONJ
ejpam-4793	192	8	α	α	PROPN
ejpam-4793	192	9	∈	∈	PROPN
ejpam-4793	192	10	γ	γ	X
ejpam-4793	192	11	,	,	PUNCT
ejpam-4793	192	12	α1	α1	PROPN
ejpam-4793	192	13	m	m	NOUN
ejpam-4793	192	14	=	=	NOUN
ejpam-4793	192	15	1	1	NUM
ejpam-4793	192	16	m	m	NOUN
ejpam-4793	192	17	.	.	PUNCT
ejpam-4793	193	1	definition	definition	NOUN
ejpam-4793	193	2	10	10	NUM
ejpam-4793	193	3	.	.	PUNCT
ejpam-4793	194	1	let	let	VERB
ejpam-4793	194	2	m	m	PRON
ejpam-4793	194	3	be	be	AUX
ejpam-4793	194	4	a	a	DET
ejpam-4793	194	5	γ	γ	X
ejpam-4793	194	6	-	-	PUNCT
ejpam-4793	194	7	monoid	monoid	NOUN
ejpam-4793	194	8	.	.	PUNCT
ejpam-4793	195	1	a	a	DET
ejpam-4793	195	2	left	left	ADJ
ejpam-4793	195	3	γ	γ	NOUN
ejpam-4793	195	4	-	-	NOUN
ejpam-4793	195	5	ideal	ideal	NOUN
ejpam-4793	195	6	(	(	PUNCT
ejpam-4793	195	7	respectively	respectively	ADV
ejpam-4793	195	8	,	,	PUNCT
ejpam-4793	195	9	right	right	ADJ
ejpam-4793	195	10	γ	γ	X
ejpam-4793	195	11	-	-	PUNCT
ejpam-4793	195	12	ideal	ideal	NOUN
ejpam-4793	195	13	)	)	PUNCT
ejpam-4793	195	14	of	of	ADP
ejpam-4793	195	15	m	m	PROPN
ejpam-4793	195	16	is	be	AUX
ejpam-4793	195	17	a	a	DET
ejpam-4793	195	18	subset	subset	NOUN
ejpam-4793	195	19	i	i	PRON
ejpam-4793	195	20	of	of	ADP
ejpam-4793	195	21	m	m	PRON
ejpam-4793	195	22	such	such	ADJ
ejpam-4793	195	23	that	that	PRON
ejpam-4793	195	24	for	for	ADP
ejpam-4793	195	25	any	any	DET
ejpam-4793	195	26	α	α	NOUN
ejpam-4793	195	27	,	,	PUNCT
ejpam-4793	195	28	β	β	PROPN
ejpam-4793	195	29	∈	∈	PROPN
ejpam-4793	195	30	γ	γ	X
ejpam-4793	195	31	,	,	PUNCT
ejpam-4793	195	32	for	for	ADP
ejpam-4793	195	33	all	all	DET
ejpam-4793	195	34	a	a	DET
ejpam-4793	195	35	∈	∈	NOUN
ejpam-4793	196	1	i	i	PRON
ejpam-4793	196	2	and	and	CCONJ
ejpam-4793	196	3	m	m	PROPN
ejpam-4793	196	4	∈	∈	PROPN
ejpam-4793	196	5	m	m	NOUN
ejpam-4793	196	6	,	,	PUNCT
ejpam-4793	196	7	αm	αm	NOUN
ejpam-4793	196	8	∗	∗	NOUN
ejpam-4793	196	9	βa	βa	INTJ
ejpam-4793	197	1	∈	∈	PROPN
ejpam-4793	197	2	i	i	PRON
ejpam-4793	197	3	(	(	PUNCT
ejpam-4793	197	4	respectively	respectively	ADV
ejpam-4793	197	5	,	,	PUNCT
ejpam-4793	197	6	αa	αa	PROPN
ejpam-4793	197	7	∗	∗	NOUN
ejpam-4793	197	8	βm	βm	VERB
ejpam-4793	197	9	∈	∈	PROPN
ejpam-4793	197	10	i	i	PROPN
ejpam-4793	197	11	)	)	PUNCT
ejpam-4793	197	12	.	.	PUNCT
ejpam-4793	198	1	a	a	DET
ejpam-4793	198	2	γ	γ	NOUN
ejpam-4793	198	3	-	-	NOUN
ejpam-4793	198	4	ideal	ideal	NOUN
ejpam-4793	198	5	of	of	ADP
ejpam-4793	198	6	m	m	PROPN
ejpam-4793	198	7	is	be	AUX
ejpam-4793	198	8	a	a	DET
ejpam-4793	198	9	subset	subset	NOUN
ejpam-4793	198	10	i	i	PRON
ejpam-4793	198	11	of	of	ADP
ejpam-4793	198	12	m	m	PRON
ejpam-4793	198	13	such	such	ADJ
ejpam-4793	198	14	that	that	SCONJ
ejpam-4793	198	15	i	i	PRON
ejpam-4793	198	16	is	be	AUX
ejpam-4793	198	17	both	both	CCONJ
ejpam-4793	198	18	a	a	DET
ejpam-4793	198	19	left	left	ADJ
ejpam-4793	198	20	and	and	CCONJ
ejpam-4793	198	21	right	right	ADJ
ejpam-4793	198	22	γ	γ	NOUN
ejpam-4793	198	23	-	-	NOUN
ejpam-4793	198	24	ideal	ideal	NOUN
ejpam-4793	198	25	of	of	ADP
ejpam-4793	198	26	m	m	PROPN
ejpam-4793	198	27	.	.	PUNCT
ejpam-4793	199	1	let	let	AUX
ejpam-4793	199	2	(	(	PUNCT
ejpam-4793	199	3	m	m	NOUN
ejpam-4793	199	4	,	,	PUNCT
ejpam-4793	199	5	∗	∗	NOUN
ejpam-4793	199	6	)	)	PUNCT
ejpam-4793	199	7	be	be	VERB
ejpam-4793	199	8	a	a	DET
ejpam-4793	199	9	γ	γ	NOUN
ejpam-4793	199	10	-	-	PUNCT
ejpam-4793	199	11	monoid	monoid	NOUN
ejpam-4793	199	12	and	and	CCONJ
ejpam-4793	199	13	a	a	DET
ejpam-4793	199	14	a	a	DET
ejpam-4793	199	15	γ	γ	NOUN
ejpam-4793	199	16	-	-	NOUN
ejpam-4793	199	17	ideal	ideal	NOUN
ejpam-4793	199	18	of	of	ADP
ejpam-4793	199	19	m	m	PROPN
ejpam-4793	199	20	with	with	ADP
ejpam-4793	199	21	a	a	DET
ejpam-4793	199	22	∈	∈	PROPN
ejpam-4793	199	23	a.	a.	NOUN
ejpam-4793	199	24	then	then	ADV
ejpam-4793	199	25	for	for	ADP
ejpam-4793	199	26	all	all	DET
ejpam-4793	199	27	α	α	NOUN
ejpam-4793	199	28	,	,	PUNCT
ejpam-4793	199	29	β	β	PROPN
ejpam-4793	199	30	∈	∈	PROPN
ejpam-4793	199	31	γ	γ	X
ejpam-4793	199	32	,	,	PUNCT
ejpam-4793	199	33	we	we	PRON
ejpam-4793	199	34	have	have	VERB
ejpam-4793	199	35	αa	αa	NOUN
ejpam-4793	199	36	=	=	SYM
ejpam-4793	199	37	αa	αa	PROPN
ejpam-4793	199	38	∗	∗	PROPN
ejpam-4793	199	39	α1	α1	PROPN
ejpam-4793	199	40	m	m	PROPN
ejpam-4793	199	41	∈	∈	NOUN
ejpam-4793	199	42	a.	a.	NOUN
ejpam-4793	199	43	thus	thus	ADV
ejpam-4793	199	44	,	,	PUNCT
ejpam-4793	199	45	we	we	PRON
ejpam-4793	199	46	have	have	VERB
ejpam-4793	199	47	the	the	DET
ejpam-4793	199	48	following	follow	VERB
ejpam-4793	199	49	remark	remark	NOUN
ejpam-4793	199	50	.	.	PUNCT
ejpam-4793	200	1	remark	remark	PROPN
ejpam-4793	200	2	4	4	NUM
ejpam-4793	200	3	.	.	PUNCT
ejpam-4793	201	1	let	let	AUX
ejpam-4793	201	2	(	(	PUNCT
ejpam-4793	201	3	m	m	NOUN
ejpam-4793	201	4	,	,	PUNCT
ejpam-4793	201	5	∗	∗	NOUN
ejpam-4793	201	6	)	)	PUNCT
ejpam-4793	201	7	be	be	VERB
ejpam-4793	201	8	a	a	DET
ejpam-4793	201	9	γ	γ	NOUN
ejpam-4793	201	10	-	-	PUNCT
ejpam-4793	201	11	monoid	monoid	NOUN
ejpam-4793	201	12	and	and	CCONJ
ejpam-4793	201	13	a	a	DET
ejpam-4793	201	14	be	be	AUX
ejpam-4793	201	15	a	a	DET
ejpam-4793	201	16	γ	γ	NOUN
ejpam-4793	201	17	-	-	NOUN
ejpam-4793	201	18	ideal	ideal	NOUN
ejpam-4793	201	19	of	of	ADP
ejpam-4793	201	20	m	m	PROPN
ejpam-4793	201	21	.	.	PUNCT
ejpam-4793	202	1	(	(	PUNCT
ejpam-4793	202	2	i	i	NOUN
ejpam-4793	202	3	)	)	PUNCT
ejpam-4793	202	4	m	m	VERB
ejpam-4793	202	5	is	be	AUX
ejpam-4793	202	6	a	a	DET
ejpam-4793	202	7	γ	γ	NOUN
ejpam-4793	202	8	-	-	PUNCT
ejpam-4793	202	9	ideal	ideal	NOUN
ejpam-4793	202	10	.	.	PUNCT
ejpam-4793	203	1	(	(	PUNCT
ejpam-4793	203	2	ii	ii	NOUN
ejpam-4793	203	3	)	)	PUNCT
ejpam-4793	203	4	for	for	ADP
ejpam-4793	203	5	all	all	PRON
ejpam-4793	203	6	α	α	DET
ejpam-4793	203	7	∈	∈	NOUN
ejpam-4793	203	8	γ	γ	NOUN
ejpam-4793	203	9	and	and	CCONJ
ejpam-4793	203	10	for	for	ADP
ejpam-4793	203	11	all	all	DET
ejpam-4793	203	12	a	a	DET
ejpam-4793	203	13	∈	∈	PROPN
ejpam-4793	203	14	a	a	DET
ejpam-4793	203	15	,	,	PUNCT
ejpam-4793	203	16	αa	αa	PROPN
ejpam-4793	203	17	∈	∈	PROPN
ejpam-4793	203	18	a.	a.	NOUN
ejpam-4793	203	19	lemma	lemma	PROPN
ejpam-4793	204	1	1	1	X
ejpam-4793	204	2	.	.	PUNCT
ejpam-4793	205	1	let	let	VERB
ejpam-4793	205	2	a	a	PRON
ejpam-4793	205	3	and	and	CCONJ
ejpam-4793	205	4	b	b	NOUN
ejpam-4793	205	5	be	be	AUX
ejpam-4793	205	6	γ	γ	NOUN
ejpam-4793	205	7	-	-	NOUN
ejpam-4793	205	8	ideals	ideal	NOUN
ejpam-4793	205	9	of	of	ADP
ejpam-4793	205	10	a	a	DET
ejpam-4793	205	11	γ	γ	X
ejpam-4793	205	12	-	-	PUNCT
ejpam-4793	205	13	monoid	monoid	NOUN
ejpam-4793	205	14	m	m	PROPN
ejpam-4793	205	15	.	.	PUNCT
ejpam-4793	206	1	then	then	ADV
ejpam-4793	206	2	a	a	DET
ejpam-4793	206	3	∗b	∗b	PROPN
ejpam-4793	206	4	is	be	AUX
ejpam-4793	206	5	a	a	DET
ejpam-4793	206	6	γ	γ	NOUN
ejpam-4793	206	7	-	-	NOUN
ejpam-4793	206	8	ideal	ideal	NOUN
ejpam-4793	206	9	of	of	ADP
ejpam-4793	206	10	m	m	PROPN
ejpam-4793	206	11	.	.	PUNCT
ejpam-4793	207	1	proof	proof	NOUN
ejpam-4793	207	2	.	.	PUNCT
ejpam-4793	208	1	let	let	VERB
ejpam-4793	208	2	a	a	PRON
ejpam-4793	208	3	and	and	CCONJ
ejpam-4793	208	4	b	b	NOUN
ejpam-4793	208	5	be	be	AUX
ejpam-4793	208	6	γ	γ	NOUN
ejpam-4793	208	7	-	-	NOUN
ejpam-4793	208	8	ideals	ideal	NOUN
ejpam-4793	208	9	of	of	ADP
ejpam-4793	208	10	a	a	DET
ejpam-4793	208	11	γ	γ	X
ejpam-4793	208	12	-	-	PUNCT
ejpam-4793	208	13	monoid	monoid	NOUN
ejpam-4793	208	14	m	m	PROPN
ejpam-4793	208	15	.	.	PUNCT
ejpam-4793	209	1	clearly	clearly	ADV
ejpam-4793	209	2	,	,	PUNCT
ejpam-4793	209	3	a	a	DET
ejpam-4793	209	4	∗b	∗b	PROPN
ejpam-4793	209	5	⊆	⊆	NUM
ejpam-4793	209	6	m	m	NOUN
ejpam-4793	209	7	.	.	PUNCT
ejpam-4793	210	1	let	let	VERB
ejpam-4793	210	2	x	x	SYM
ejpam-4793	210	3	∈	∈	PROPN
ejpam-4793	210	4	a	a	DET
ejpam-4793	210	5	∗b	∗b	PROPN
ejpam-4793	210	6	and	and	CCONJ
ejpam-4793	210	7	m	m	PROPN
ejpam-4793	210	8	∈	∈	ADJ
ejpam-4793	210	9	m	m	NOUN
ejpam-4793	210	10	.	.	PUNCT
ejpam-4793	211	1	then	then	ADV
ejpam-4793	211	2	x	x	X
ejpam-4793	211	3	=	=	PUNCT
ejpam-4793	211	4	a	a	DET
ejpam-4793	211	5	∗	∗	NOUN
ejpam-4793	211	6	b	b	NOUN
ejpam-4793	211	7	for	for	ADP
ejpam-4793	211	8	some	some	DET
ejpam-4793	211	9	a	a	DET
ejpam-4793	211	10	∈	∈	PROPN
ejpam-4793	211	11	a	a	DET
ejpam-4793	211	12	and	and	CCONJ
ejpam-4793	211	13	b	b	PROPN
ejpam-4793	211	14	∈	∈	PROPN
ejpam-4793	211	15	b.	b.	PROPN
ejpam-4793	212	1	now	now	ADV
ejpam-4793	212	2	,	,	PUNCT
ejpam-4793	212	3	for	for	ADP
ejpam-4793	212	4	all	all	DET
ejpam-4793	212	5	α	α	NOUN
ejpam-4793	212	6	,	,	PUNCT
ejpam-4793	212	7	β	β	PROPN
ejpam-4793	212	8	∈	∈	PROPN
ejpam-4793	212	9	γ	γ	X
ejpam-4793	212	10	,	,	PUNCT
ejpam-4793	212	11	αx∗βm	αx∗βm	PROPN
ejpam-4793	212	12	=	=	SYM
ejpam-4793	212	13	α(a	α(a	NOUN
ejpam-4793	212	14	∗	∗	NOUN
ejpam-4793	212	15	b)∗βm	b)∗βm	NOUN
ejpam-4793	212	16	=	=	SYM
ejpam-4793	212	17	αa∗αb∗βm	αa∗αb∗βm	NOUN
ejpam-4793	212	18	=	=	SYM
ejpam-4793	212	19	αa∗(αb∗βm	αa∗(αb∗βm	NOUN
ejpam-4793	212	20	)	)	PUNCT
ejpam-4793	212	21	∈	∈	NOUN
ejpam-4793	212	22	a∗b	a∗b	NUM
ejpam-4793	212	23	by	by	ADP
ejpam-4793	212	24	remark	remark	NOUN
ejpam-4793	212	25	4(ii	4(ii	NUM
ejpam-4793	212	26	)	)	PUNCT
ejpam-4793	212	27	and	and	CCONJ
ejpam-4793	212	28	definition	definition	NOUN
ejpam-4793	212	29	10	10	NUM
ejpam-4793	212	30	.	.	PUNCT
ejpam-4793	213	1	similarly	similarly	ADV
ejpam-4793	213	2	,	,	PUNCT
ejpam-4793	213	3	for	for	ADP
ejpam-4793	213	4	all	all	DET
ejpam-4793	213	5	α	α	NOUN
ejpam-4793	213	6	,	,	PUNCT
ejpam-4793	213	7	β	β	PROPN
ejpam-4793	213	8	∈	∈	PROPN
ejpam-4793	213	9	γ	γ	X
ejpam-4793	213	10	,	,	PUNCT
ejpam-4793	213	11	αm∗βx	αm∗βx	PROPN
ejpam-4793	213	12	∈	∈	PROPN
ejpam-4793	213	13	a∗b	a∗b	PROPN
ejpam-4793	213	14	.	.	PUNCT
ejpam-4793	214	1	therefore	therefore	ADV
ejpam-4793	214	2	,	,	PUNCT
ejpam-4793	214	3	a∗b	a∗b	PROPN
ejpam-4793	214	4	is	be	AUX
ejpam-4793	214	5	a	a	DET
ejpam-4793	214	6	γ	γ	NOUN
ejpam-4793	214	7	-	-	NOUN
ejpam-4793	214	8	ideal	ideal	NOUN
ejpam-4793	214	9	of	of	ADP
ejpam-4793	214	10	m	m	PROPN
ejpam-4793	214	11	.	.	PUNCT
ejpam-4793	215	1	the	the	DET
ejpam-4793	215	2	following	follow	VERB
ejpam-4793	215	3	example	example	NOUN
ejpam-4793	215	4	shows	show	VERB
ejpam-4793	215	5	that	that	SCONJ
ejpam-4793	215	6	a	a	DET
ejpam-4793	215	7	γ	γ	X
ejpam-4793	215	8	-	-	PUNCT
ejpam-4793	215	9	ideal	ideal	NOUN
ejpam-4793	215	10	is	be	AUX
ejpam-4793	215	11	not	not	PART
ejpam-4793	215	12	necessarily	necessarily	ADV
ejpam-4793	215	13	a	a	DET
ejpam-4793	215	14	γ	γ	NOUN
ejpam-4793	215	15	-	-	PUNCT
ejpam-4793	215	16	order	order	NOUN
ejpam-4793	215	17	-	-	PUNCT
ejpam-4793	215	18	ideal	ideal	NOUN
ejpam-4793	215	19	.	.	PUNCT
ejpam-4793	216	1	example	example	NOUN
ejpam-4793	216	2	8	8	NUM
ejpam-4793	216	3	.	.	PUNCT
ejpam-4793	217	1	consider	consider	VERB
ejpam-4793	217	2	the	the	DET
ejpam-4793	217	3	set	set	NOUN
ejpam-4793	217	4	m	m	NOUN
ejpam-4793	217	5	=	=	PUNCT
ejpam-4793	217	6	{	{	PUNCT
ejpam-4793	217	7	1	1	NUM
ejpam-4793	217	8	,	,	PUNCT
ejpam-4793	217	9	n	n	CCONJ
ejpam-4793	217	10	,	,	PUNCT
ejpam-4793	217	11	h	h	NOUN
ejpam-4793	217	12	,	,	PUNCT
ejpam-4793	217	13	s	s	PART
ejpam-4793	217	14	}	}	PUNCT
ejpam-4793	217	15	and	and	CCONJ
ejpam-4793	217	16	operation	operation	NOUN
ejpam-4793	217	17	∗	∗	NOUN
ejpam-4793	217	18	given	give	VERB
ejpam-4793	217	19	by	by	ADP
ejpam-4793	217	20	∗	∗	NOUN
ejpam-4793	217	21	1	1	NUM
ejpam-4793	218	1	n	n	NUM
ejpam-4793	218	2	h	h	NOUN
ejpam-4793	218	3	s	s	NOUN
ejpam-4793	218	4	1	1	NUM
ejpam-4793	218	5	1	1	NUM
ejpam-4793	218	6	n	n	NUM
ejpam-4793	218	7	h	h	NOUN
ejpam-4793	218	8	s	s	NOUN
ejpam-4793	218	9	n	n	CCONJ
ejpam-4793	218	10	n	n	CCONJ
ejpam-4793	218	11	n	n	NOUN
ejpam-4793	218	12	h	h	NOUN
ejpam-4793	218	13	s	s	NOUN
ejpam-4793	218	14	h	h	NOUN
ejpam-4793	218	15	h	h	NOUN
ejpam-4793	218	16	h	h	NOUN
ejpam-4793	219	1	h	h	NOUN
ejpam-4793	219	2	s	s	PART
ejpam-4793	219	3	s	s	X
ejpam-4793	219	4	s	s	X
ejpam-4793	219	5	s	s	X
ejpam-4793	219	6	s	s	X
ejpam-4793	219	7	s	s	X
ejpam-4793	219	8	clearly	clearly	ADV
ejpam-4793	219	9	,	,	PUNCT
ejpam-4793	219	10	the	the	DET
ejpam-4793	219	11	operation	operation	NOUN
ejpam-4793	219	12	is	be	AUX
ejpam-4793	219	13	commutative	commutative	ADJ
ejpam-4793	219	14	.	.	PUNCT
ejpam-4793	220	1	it	it	PRON
ejpam-4793	220	2	can	can	AUX
ejpam-4793	220	3	be	be	AUX
ejpam-4793	220	4	verified	verify	VERB
ejpam-4793	220	5	that	that	SCONJ
ejpam-4793	220	6	∗	∗	NOUN
ejpam-4793	220	7	is	be	AUX
ejpam-4793	220	8	associative	associative	ADJ
ejpam-4793	220	9	.	.	PUNCT
ejpam-4793	221	1	since	since	SCONJ
ejpam-4793	221	2	1	1	NUM
ejpam-4793	221	3	∗	∗	NOUN
ejpam-4793	221	4	1	1	NUM
ejpam-4793	221	5	=	=	SYM
ejpam-4793	221	6	1	1	NUM
ejpam-4793	221	7	,	,	PUNCT
ejpam-4793	221	8	1	1	NUM
ejpam-4793	221	9	∗	∗	NOUN
ejpam-4793	221	10	n	n	NOUN
ejpam-4793	221	11	=	=	SYM
ejpam-4793	221	12	n	n	CCONJ
ejpam-4793	221	13	,	,	PUNCT
ejpam-4793	221	14	1	1	NUM
ejpam-4793	221	15	∗	∗	NOUN
ejpam-4793	221	16	h	h	NOUN
ejpam-4793	221	17	=	=	NOUN
ejpam-4793	221	18	h	h	NOUN
ejpam-4793	221	19	and	and	CCONJ
ejpam-4793	221	20	1	1	NUM
ejpam-4793	221	21	∗	∗	NOUN
ejpam-4793	221	22	s	s	PART
ejpam-4793	221	23	=	=	SYM
ejpam-4793	221	24	s	s	X
ejpam-4793	221	25	,	,	PUNCT
ejpam-4793	221	26	it	it	PRON
ejpam-4793	221	27	follows	follow	VERB
ejpam-4793	221	28	that	that	SCONJ
ejpam-4793	221	29	1	1	NUM
ejpam-4793	221	30	is	be	AUX
ejpam-4793	221	31	the	the	DET
ejpam-4793	221	32	identity	identity	NOUN
ejpam-4793	221	33	in	in	ADP
ejpam-4793	221	34	m	m	PROPN
ejpam-4793	221	35	.	.	PUNCT
ejpam-4793	222	1	thus	thus	ADV
ejpam-4793	222	2	,	,	PUNCT
ejpam-4793	222	3	m	m	VERB
ejpam-4793	222	4	is	be	AUX
ejpam-4793	222	5	a	a	DET
ejpam-4793	222	6	commutative	commutative	ADJ
ejpam-4793	222	7	monoid	monoid	NOUN
ejpam-4793	222	8	.	.	PUNCT
ejpam-4793	223	1	let	let	VERB
ejpam-4793	223	2	γ	γ	NOUN
ejpam-4793	223	3	be	be	AUX
ejpam-4793	223	4	a	a	DET
ejpam-4793	223	5	group	group	NOUN
ejpam-4793	223	6	and	and	CCONJ
ejpam-4793	223	7	the	the	DET
ejpam-4793	223	8	mapping	mapping	NOUN
ejpam-4793	223	9	ϕ	ϕ	NOUN
ejpam-4793	223	10	:	:	PUNCT
ejpam-4793	223	11	γ×m	γ×m	PROPN
ejpam-4793	223	12	−→	−→	NOUN
ejpam-4793	223	13	m	m	AUX
ejpam-4793	223	14	given	give	VERB
ejpam-4793	223	15	by	by	ADP
ejpam-4793	223	16	(	(	PUNCT
ejpam-4793	223	17	α	α	NOUN
ejpam-4793	223	18	,	,	PUNCT
ejpam-4793	223	19	a	a	PRON
ejpam-4793	223	20	)	)	PUNCT
ejpam-4793	223	21	7→	7→	NUM
ejpam-4793	223	22	αa	αa	NOUN
ejpam-4793	223	23	=	=	NOUN
ejpam-4793	223	24	a.	a.	NOUN
ejpam-4793	223	25	for	for	ADP
ejpam-4793	223	26	any	any	DET
ejpam-4793	223	27	α	α	NOUN
ejpam-4793	223	28	,	,	PUNCT
ejpam-4793	223	29	β	β	X
ejpam-4793	223	30	∈	∈	PROPN
ejpam-4793	223	31	γ	γ	NOUN
ejpam-4793	223	32	and	and	CCONJ
ejpam-4793	223	33	a	a	DET
ejpam-4793	223	34	∈	∈	NOUN
ejpam-4793	223	35	m	m	VERB
ejpam-4793	223	36	,	,	PUNCT
ejpam-4793	223	37	we	we	PRON
ejpam-4793	223	38	have	have	VERB
ejpam-4793	223	39	ϕ((0	ϕ((0	PROPN
ejpam-4793	223	40	,	,	PUNCT
ejpam-4793	223	41	a	a	PRON
ejpam-4793	223	42	)	)	PUNCT
ejpam-4793	223	43	)	)	PUNCT
ejpam-4793	224	1	=	=	SYM
ejpam-4793	224	2	0a	0a	PROPN
ejpam-4793	224	3	=	=	PUNCT
ejpam-4793	224	4	a	a	PROPN
ejpam-4793	224	5	and	and	CCONJ
ejpam-4793	224	6	ϕ((α+	ϕ((α+	PRON
ejpam-4793	224	7	β	β	NOUN
ejpam-4793	224	8	,	,	PUNCT
ejpam-4793	224	9	a	a	PRON
ejpam-4793	224	10	)	)	PUNCT
ejpam-4793	224	11	)	)	PUNCT
ejpam-4793	225	1	=	=	PUNCT
ejpam-4793	225	2	α+βa	α+βa	PROPN
ejpam-4793	225	3	=	=	PUNCT
ejpam-4793	225	4	a	a	PROPN
ejpam-4793	225	5	=	=	SYM
ejpam-4793	225	6	ϕ(β	ϕ(β	PROPN
ejpam-4793	225	7	,	,	PUNCT
ejpam-4793	225	8	a	a	PRON
ejpam-4793	225	9	)	)	PUNCT
ejpam-4793	225	10	=	=	SYM
ejpam-4793	225	11	βa	βa	NOUN
ejpam-4793	225	12	=	=	PUNCT
ejpam-4793	225	13	ϕ((α	ϕ((α	PROPN
ejpam-4793	225	14	,	,	PUNCT
ejpam-4793	225	15	βa	βa	NUM
ejpam-4793	225	16	)	)	PUNCT
ejpam-4793	225	17	)	)	PUNCT
ejpam-4793	226	1	=	=	SYM
ejpam-4793	226	2	ϕ((α	ϕ((α	PROPN
ejpam-4793	226	3	,	,	PUNCT
ejpam-4793	226	4	ϕ((β	ϕ((β	PROPN
ejpam-4793	226	5	,	,	PUNCT
ejpam-4793	226	6	a	a	PRON
ejpam-4793	226	7	)	)	PUNCT
ejpam-4793	226	8	)	)	PUNCT
ejpam-4793	226	9	)	)	PUNCT
ejpam-4793	226	10	)	)	PUNCT
ejpam-4793	226	11	.	.	PUNCT
ejpam-4793	227	1	h.	h.	PROPN
ejpam-4793	227	2	sarapuddin	sarapuddin	PROPN
ejpam-4793	227	3	,	,	PUNCT
ejpam-4793	227	4	j.	j.	PROPN
ejpam-4793	227	5	vilela	vilela	PROPN
ejpam-4793	227	6	/	/	SYM
ejpam-4793	227	7	eur	eur	PROPN
ejpam-4793	227	8	.	.	PUNCT
ejpam-4793	228	1	j.	j.	PROPN
ejpam-4793	228	2	pure	pure	PROPN
ejpam-4793	228	3	appl	appl	PROPN
ejpam-4793	228	4	.	.	PROPN
ejpam-4793	228	5	math	math	PROPN
ejpam-4793	228	6	,	,	PUNCT
ejpam-4793	228	7	16	16	NUM
ejpam-4793	228	8	(	(	PUNCT
ejpam-4793	228	9	3	3	NUM
ejpam-4793	228	10	)	)	PUNCT
ejpam-4793	228	11	(	(	PUNCT
ejpam-4793	228	12	2023	2023	NUM
ejpam-4793	228	13	)	)	PUNCT
ejpam-4793	228	14	,	,	PUNCT
ejpam-4793	228	15	1772	1772	NUM
ejpam-4793	228	16	-	-	SYM
ejpam-4793	228	17	1793	1793	NUM
ejpam-4793	228	18	1778	1778	NUM
ejpam-4793	228	19	thus	thus	ADV
ejpam-4793	228	20	,	,	PUNCT
ejpam-4793	228	21	ϕ	ϕ	PROPN
ejpam-4793	228	22	is	be	AUX
ejpam-4793	228	23	an	an	DET
ejpam-4793	228	24	action	action	NOUN
ejpam-4793	228	25	.	.	PUNCT
ejpam-4793	229	1	now	now	ADV
ejpam-4793	229	2	,	,	PUNCT
ejpam-4793	229	3	let	let	VERB
ejpam-4793	229	4	α	α	PRON
ejpam-4793	229	5	∈	∈	PROPN
ejpam-4793	229	6	γ	γ	NOUN
ejpam-4793	229	7	and	and	CCONJ
ejpam-4793	229	8	a	a	PRON
ejpam-4793	229	9	,	,	PUNCT
ejpam-4793	229	10	b	b	X
ejpam-4793	229	11	∈	∈	ADV
ejpam-4793	229	12	m	m	VERB
ejpam-4793	229	13	.	.	PUNCT
ejpam-4793	230	1	then	then	ADV
ejpam-4793	230	2	ϕ((α	ϕ((α	PROPN
ejpam-4793	230	3	,	,	PUNCT
ejpam-4793	230	4	a	a	DET
ejpam-4793	230	5	∗	∗	NOUN
ejpam-4793	230	6	b	b	NOUN
ejpam-4793	230	7	)	)	PUNCT
ejpam-4793	230	8	)	)	PUNCT
ejpam-4793	231	1	=	=	PUNCT
ejpam-4793	231	2	α(a	α(a	NOUN
ejpam-4793	231	3	∗	∗	NOUN
ejpam-4793	231	4	b	b	NOUN
ejpam-4793	231	5	)	)	PUNCT
ejpam-4793	231	6	=	=	PUNCT
ejpam-4793	232	1	a	a	DET
ejpam-4793	232	2	∗	∗	NOUN
ejpam-4793	232	3	b	b	NOUN
ejpam-4793	232	4	=	=	SYM
ejpam-4793	232	5	αa	αa	PROPN
ejpam-4793	232	6	∗	∗	NOUN
ejpam-4793	232	7	αb	αb	ADP
ejpam-4793	232	8	=	=	SYM
ejpam-4793	232	9	ϕ((α	ϕ((α	PROPN
ejpam-4793	232	10	,	,	PUNCT
ejpam-4793	232	11	a	a	PRON
ejpam-4793	232	12	)	)	PUNCT
ejpam-4793	232	13	)	)	PUNCT
ejpam-4793	232	14	∗	∗	NOUN
ejpam-4793	232	15	ϕ((α	ϕ((α	NOUN
ejpam-4793	232	16	,	,	PUNCT
ejpam-4793	232	17	b	b	NOUN
ejpam-4793	232	18	)	)	PUNCT
ejpam-4793	232	19	)	)	PUNCT
ejpam-4793	232	20	.	.	PUNCT
ejpam-4793	233	1	hence	hence	ADV
ejpam-4793	233	2	,	,	PUNCT
ejpam-4793	233	3	m	m	VERB
ejpam-4793	233	4	is	be	AUX
ejpam-4793	233	5	a	a	DET
ejpam-4793	233	6	γ	γ	X
ejpam-4793	233	7	-	-	PUNCT
ejpam-4793	233	8	monoid	monoid	NOUN
ejpam-4793	233	9	.	.	PUNCT
ejpam-4793	234	1	let	let	VERB
ejpam-4793	234	2	c	c	NOUN
ejpam-4793	234	3	=	=	SYM
ejpam-4793	234	4	{	{	PUNCT
ejpam-4793	234	5	n	n	CCONJ
ejpam-4793	234	6	,	,	PUNCT
ejpam-4793	234	7	h	h	NOUN
ejpam-4793	234	8	,	,	PUNCT
ejpam-4793	234	9	s	s	PART
ejpam-4793	234	10	}	}	PUNCT
ejpam-4793	234	11	.	.	PUNCT
ejpam-4793	235	1	then	then	ADV
ejpam-4793	235	2	for	for	ADP
ejpam-4793	235	3	any	any	DET
ejpam-4793	235	4	α	α	NOUN
ejpam-4793	235	5	,	,	PUNCT
ejpam-4793	235	6	β	β	PROPN
ejpam-4793	235	7	∈	∈	PROPN
ejpam-4793	235	8	γ	γ	X
ejpam-4793	235	9	,	,	PUNCT
ejpam-4793	235	10	we	we	PRON
ejpam-4793	235	11	have	have	VERB
ejpam-4793	235	12	for	for	ADP
ejpam-4793	235	13	all	all	DET
ejpam-4793	235	14	a	a	DET
ejpam-4793	235	15	∈	∈	PROPN
ejpam-4793	235	16	c	c	NOUN
ejpam-4793	235	17	and	and	CCONJ
ejpam-4793	235	18	m	m	PROPN
ejpam-4793	235	19	∈	∈	PROPN
ejpam-4793	235	20	m	m	NOUN
ejpam-4793	235	21	,	,	PUNCT
ejpam-4793	235	22	αa	αa	ADV
ejpam-4793	235	23	∗	∗	NOUN
ejpam-4793	235	24	βm	βm	VERB
ejpam-4793	236	1	=	=	SYM
ejpam-4793	236	2	αn	αn	NOUN
ejpam-4793	236	3	∗	∗	NOUN
ejpam-4793	236	4	β1	β1	NOUN
ejpam-4793	236	5	=	=	PUNCT
ejpam-4793	236	6	n	n	PROPN
ejpam-4793	236	7	∗	∗	NOUN
ejpam-4793	236	8	1	1	NUM
ejpam-4793	236	9	=	=	SYM
ejpam-4793	236	10	n	n	SYM
ejpam-4793	236	11	∈	∈	NOUN
ejpam-4793	236	12	c	c	X
ejpam-4793	236	13	,	,	PUNCT
ejpam-4793	236	14	αa	αa	PROPN
ejpam-4793	236	15	∗	∗	NOUN
ejpam-4793	236	16	βm	βm	VERB
ejpam-4793	237	1	=	=	SYM
ejpam-4793	237	2	αn	αn	NOUN
ejpam-4793	237	3	∗	∗	NOUN
ejpam-4793	237	4	βh	βh	ADP
ejpam-4793	238	1	=	=	SYM
ejpam-4793	238	2	n	n	NOUN
ejpam-4793	238	3	∗	∗	NOUN
ejpam-4793	238	4	h	h	NOUN
ejpam-4793	239	1	=	=	NOUN
ejpam-4793	239	2	h	h	NOUN
ejpam-4793	240	1	∈	∈	PROPN
ejpam-4793	240	2	c	c	X
ejpam-4793	240	3	,	,	PUNCT
ejpam-4793	240	4	αa	αa	PROPN
ejpam-4793	240	5	∗	∗	NOUN
ejpam-4793	240	6	βm	βm	VERB
ejpam-4793	241	1	=	=	VERB
ejpam-4793	241	2	αh	αh	NOUN
ejpam-4793	241	3	∗	∗	NOUN
ejpam-4793	241	4	β1	β1	NOUN
ejpam-4793	241	5	=	=	PUNCT
ejpam-4793	241	6	h	h	PROPN
ejpam-4793	241	7	∗	∗	NOUN
ejpam-4793	241	8	1	1	NUM
ejpam-4793	242	1	=	=	SYM
ejpam-4793	242	2	h	h	NOUN
ejpam-4793	242	3	∈	∈	PROPN
ejpam-4793	242	4	c	c	X
ejpam-4793	242	5	,	,	PUNCT
ejpam-4793	242	6	αa	αa	PROPN
ejpam-4793	242	7	∗	∗	NOUN
ejpam-4793	242	8	βm	βm	VERB
ejpam-4793	242	9	=	=	VERB
ejpam-4793	242	10	αh	αh	PART
ejpam-4793	242	11	∗	∗	NOUN
ejpam-4793	242	12	βh	βh	PUNCT
ejpam-4793	243	1	=	=	SYM
ejpam-4793	243	2	h	h	NOUN
ejpam-4793	243	3	∗	∗	NOUN
ejpam-4793	243	4	h	h	NOUN
ejpam-4793	244	1	=	=	NOUN
ejpam-4793	244	2	h	h	NOUN
ejpam-4793	245	1	∈	∈	PROPN
ejpam-4793	245	2	c	c	X
ejpam-4793	245	3	,	,	PUNCT
ejpam-4793	245	4	αa	αa	PROPN
ejpam-4793	245	5	∗	∗	NOUN
ejpam-4793	245	6	βm	βm	NOUN
ejpam-4793	246	1	=	=	SYM
ejpam-4793	246	2	αs	αs	INTJ
ejpam-4793	246	3	∗	∗	NOUN
ejpam-4793	246	4	β1	β1	PROPN
ejpam-4793	246	5	=	=	PUNCT
ejpam-4793	246	6	s	s	PROPN
ejpam-4793	246	7	∗	∗	NOUN
ejpam-4793	246	8	1	1	NUM
ejpam-4793	246	9	=	=	SYM
ejpam-4793	246	10	s	s	PART
ejpam-4793	246	11	∈	∈	PROPN
ejpam-4793	246	12	c	c	NOUN
ejpam-4793	246	13	,	,	PUNCT
ejpam-4793	246	14	αa	αa	PROPN
ejpam-4793	246	15	∗	∗	NOUN
ejpam-4793	246	16	βm	βm	NOUN
ejpam-4793	247	1	=	=	SYM
ejpam-4793	247	2	αs	αs	INTJ
ejpam-4793	247	3	∗	∗	NOUN
ejpam-4793	247	4	βh	βh	ADP
ejpam-4793	248	1	=	=	SYM
ejpam-4793	248	2	s	s	NOUN
ejpam-4793	248	3	∗	∗	NOUN
ejpam-4793	248	4	h	h	NOUN
ejpam-4793	248	5	=	=	SYM
ejpam-4793	248	6	s	s	PART
ejpam-4793	248	7	∈	∈	PROPN
ejpam-4793	248	8	c	c	NOUN
ejpam-4793	248	9	,	,	PUNCT
ejpam-4793	248	10	αa	αa	PROPN
ejpam-4793	248	11	∗	∗	NOUN
ejpam-4793	248	12	βm	βm	VERB
ejpam-4793	249	1	=	=	SYM
ejpam-4793	249	2	αn	αn	NOUN
ejpam-4793	249	3	∗	∗	NOUN
ejpam-4793	249	4	βn	βn	NOUN
ejpam-4793	250	1	=	=	SYM
ejpam-4793	250	2	n	n	NOUN
ejpam-4793	250	3	∗	∗	NOUN
ejpam-4793	250	4	n	n	NOUN
ejpam-4793	250	5	=	=	SYM
ejpam-4793	250	6	n	n	CCONJ
ejpam-4793	250	7	∈	∈	PROPN
ejpam-4793	250	8	c	c	NOUN
ejpam-4793	250	9	;	;	PUNCT
ejpam-4793	250	10	αa	αa	PROPN
ejpam-4793	250	11	∗	∗	NOUN
ejpam-4793	250	12	βm	βm	VERB
ejpam-4793	251	1	=	=	SYM
ejpam-4793	251	2	αn	αn	NOUN
ejpam-4793	251	3	∗	∗	NOUN
ejpam-4793	251	4	βs	βs	PUNCT
ejpam-4793	252	1	=	=	SYM
ejpam-4793	252	2	n	n	PART
ejpam-4793	252	3	∗	∗	NOUN
ejpam-4793	252	4	s	s	PART
ejpam-4793	252	5	=	=	SYM
ejpam-4793	252	6	s	s	PART
ejpam-4793	252	7	∈	∈	PROPN
ejpam-4793	252	8	c	c	NOUN
ejpam-4793	252	9	;	;	PUNCT
ejpam-4793	252	10	αa	αa	PROPN
ejpam-4793	252	11	∗	∗	NOUN
ejpam-4793	252	12	βm	βm	VERB
ejpam-4793	252	13	=	=	VERB
ejpam-4793	252	14	αh	αh	NOUN
ejpam-4793	252	15	∗	∗	NOUN
ejpam-4793	252	16	βn	βn	NOUN
ejpam-4793	253	1	=	=	PUNCT
ejpam-4793	253	2	h	h	NOUN
ejpam-4793	253	3	∗	∗	NOUN
ejpam-4793	253	4	n	n	NOUN
ejpam-4793	253	5	=	=	SYM
ejpam-4793	253	6	h	h	NOUN
ejpam-4793	253	7	∈	∈	PROPN
ejpam-4793	253	8	c	c	X
ejpam-4793	253	9	;	;	PUNCT
ejpam-4793	253	10	αa	αa	PROPN
ejpam-4793	253	11	∗	∗	NOUN
ejpam-4793	253	12	βm	βm	VERB
ejpam-4793	253	13	=	=	VERB
ejpam-4793	253	14	αh	αh	PROPN
ejpam-4793	253	15	∗	∗	NOUN
ejpam-4793	253	16	βs	βs	PUNCT
ejpam-4793	254	1	=	=	SYM
ejpam-4793	254	2	h	h	NOUN
ejpam-4793	254	3	∗	∗	NOUN
ejpam-4793	254	4	s	s	PART
ejpam-4793	254	5	=	=	X
ejpam-4793	254	6	s	s	PART
ejpam-4793	254	7	∈	∈	PROPN
ejpam-4793	254	8	c	c	NOUN
ejpam-4793	254	9	;	;	PUNCT
ejpam-4793	254	10	αa	αa	PROPN
ejpam-4793	254	11	∗	∗	NOUN
ejpam-4793	254	12	βm	βm	VERB
ejpam-4793	255	1	=	=	SYM
ejpam-4793	255	2	αs	αs	INTJ
ejpam-4793	255	3	∗	∗	NOUN
ejpam-4793	255	4	βn	βn	PUNCT
ejpam-4793	256	1	=	=	SYM
ejpam-4793	256	2	s	s	NOUN
ejpam-4793	256	3	∗	∗	NOUN
ejpam-4793	256	4	n	n	NOUN
ejpam-4793	256	5	=	=	SYM
ejpam-4793	256	6	s	s	PART
ejpam-4793	256	7	∈	∈	PROPN
ejpam-4793	256	8	c	c	NOUN
ejpam-4793	256	9	;	;	PUNCT
ejpam-4793	256	10	αa	αa	PROPN
ejpam-4793	256	11	∗	∗	NOUN
ejpam-4793	256	12	βm	βm	VERB
ejpam-4793	257	1	=	=	SYM
ejpam-4793	257	2	αs	αs	INTJ
ejpam-4793	257	3	∗	∗	NOUN
ejpam-4793	257	4	βs	βs	PUNCT
ejpam-4793	258	1	=	=	SYM
ejpam-4793	258	2	s	s	NOUN
ejpam-4793	258	3	∗	∗	NOUN
ejpam-4793	258	4	s	s	PART
ejpam-4793	258	5	=	=	SYM
ejpam-4793	258	6	s	s	PROPN
ejpam-4793	258	7	∈	∈	PROPN
ejpam-4793	258	8	c.	c.	NOUN
ejpam-4793	258	9	since	since	SCONJ
ejpam-4793	258	10	m	m	PROPN
ejpam-4793	258	11	is	be	AUX
ejpam-4793	258	12	commutative	commutative	ADJ
ejpam-4793	258	13	,	,	PUNCT
ejpam-4793	258	14	βm	βm	VERB
ejpam-4793	258	15	∗	∗	NOUN
ejpam-4793	258	16	αa	αa	NOUN
ejpam-4793	259	1	=	=	SYM
ejpam-4793	259	2	αa	αa	PROPN
ejpam-4793	259	3	∗	∗	NOUN
ejpam-4793	259	4	βm	βm	PROPN
ejpam-4793	259	5	∈	∈	PROPN
ejpam-4793	259	6	c.	c.	PROPN
ejpam-4793	259	7	thus	thus	ADV
ejpam-4793	259	8	,	,	PUNCT
ejpam-4793	259	9	by	by	ADP
ejpam-4793	259	10	definition	definition	NOUN
ejpam-4793	259	11	10	10	NUM
ejpam-4793	259	12	,	,	PUNCT
ejpam-4793	259	13	c	c	PROPN
ejpam-4793	259	14	is	be	AUX
ejpam-4793	259	15	a	a	DET
ejpam-4793	259	16	γ	γ	NOUN
ejpam-4793	259	17	-	-	PUNCT
ejpam-4793	259	18	ideal	ideal	NOUN
ejpam-4793	259	19	.	.	PUNCT
ejpam-4793	260	1	however	however	ADV
ejpam-4793	260	2	,	,	PUNCT
ejpam-4793	260	3	the	the	DET
ejpam-4793	260	4	identity	identity	NOUN
ejpam-4793	260	5	1	1	NUM
ejpam-4793	260	6	/∈	/∈	PUNCT
ejpam-4793	260	7	c.	c.	PROPN
ejpam-4793	260	8	thus	thus	ADV
ejpam-4793	260	9	,	,	PUNCT
ejpam-4793	260	10	c	c	PROPN
ejpam-4793	260	11	is	be	AUX
ejpam-4793	260	12	not	not	PART
ejpam-4793	260	13	a	a	DET
ejpam-4793	260	14	γ	γ	NOUN
ejpam-4793	260	15	-	-	PUNCT
ejpam-4793	260	16	order	order	NOUN
ejpam-4793	260	17	-	-	PUNCT
ejpam-4793	260	18	ideal	ideal	NOUN
ejpam-4793	260	19	of	of	ADP
ejpam-4793	260	20	m	m	PROPN
ejpam-4793	260	21	.	.	PUNCT
ejpam-4793	261	1	the	the	DET
ejpam-4793	261	2	following	follow	VERB
ejpam-4793	261	3	example	example	NOUN
ejpam-4793	261	4	shows	show	VERB
ejpam-4793	261	5	that	that	SCONJ
ejpam-4793	261	6	γ	γ	ADJ
ejpam-4793	261	7	-	-	PUNCT
ejpam-4793	261	8	order	order	NOUN
ejpam-4793	261	9	-	-	PUNCT
ejpam-4793	261	10	ideal	ideal	NOUN
ejpam-4793	261	11	is	be	AUX
ejpam-4793	261	12	not	not	PART
ejpam-4793	261	13	necessarily	necessarily	ADV
ejpam-4793	261	14	a	a	DET
ejpam-4793	261	15	γ	γ	NOUN
ejpam-4793	261	16	-	-	PUNCT
ejpam-4793	261	17	ideal	ideal	NOUN
ejpam-4793	261	18	.	.	PUNCT
ejpam-4793	262	1	example	example	NOUN
ejpam-4793	262	2	9	9	NUM
ejpam-4793	262	3	.	.	X
ejpam-4793	262	4	consider	consider	VERB
ejpam-4793	262	5	the	the	DET
ejpam-4793	262	6	γ	γ	NOUN
ejpam-4793	262	7	-	-	PUNCT
ejpam-4793	262	8	monoid	monoid	NOUN
ejpam-4793	262	9	m	m	NOUN
ejpam-4793	262	10	=	=	PUNCT
ejpam-4793	262	11	{	{	PUNCT
ejpam-4793	262	12	1	1	NUM
ejpam-4793	262	13	,	,	PUNCT
ejpam-4793	262	14	n	n	CCONJ
ejpam-4793	262	15	,	,	PUNCT
ejpam-4793	262	16	h	h	NOUN
ejpam-4793	262	17	,	,	PUNCT
ejpam-4793	262	18	s	s	AUX
ejpam-4793	262	19	}	}	PUNCT
ejpam-4793	262	20	in	in	ADP
ejpam-4793	262	21	example	example	NOUN
ejpam-4793	262	22	8	8	NUM
ejpam-4793	262	23	.	.	PUNCT
ejpam-4793	263	1	let	let	VERB
ejpam-4793	263	2	a	a	DET
ejpam-4793	263	3	=	=	PUNCT
ejpam-4793	263	4	{	{	PUNCT
ejpam-4793	263	5	1	1	NUM
ejpam-4793	263	6	,	,	PUNCT
ejpam-4793	263	7	n	n	CCONJ
ejpam-4793	263	8	,	,	PUNCT
ejpam-4793	263	9	h	h	NOUN
ejpam-4793	263	10	}	}	PUNCT
ejpam-4793	263	11	.	.	PUNCT
ejpam-4793	264	1	now	now	ADV
ejpam-4793	264	2	,	,	PUNCT
ejpam-4793	264	3	suppose	suppose	VERB
ejpam-4793	264	4	that	that	SCONJ
ejpam-4793	264	5	for	for	ADP
ejpam-4793	264	6	all	all	DET
ejpam-4793	264	7	a	a	PRON
ejpam-4793	264	8	,	,	PUNCT
ejpam-4793	264	9	b	b	X
ejpam-4793	264	10	∈	∈	NOUN
ejpam-4793	264	11	m	m	X
ejpam-4793	264	12	and	and	CCONJ
ejpam-4793	264	13	for	for	ADP
ejpam-4793	264	14	all	all	DET
ejpam-4793	264	15	α	α	NOUN
ejpam-4793	264	16	,	,	PUNCT
ejpam-4793	264	17	β	β	PROPN
ejpam-4793	264	18	∈	∈	PROPN
ejpam-4793	264	19	γ	γ	X
ejpam-4793	264	20	,	,	PUNCT
ejpam-4793	264	21	αa	αa	ADV
ejpam-4793	264	22	∗	∗	VERB
ejpam-4793	264	23	βb	βb	PROPN
ejpam-4793	264	24	∈	∈	PROPN
ejpam-4793	264	25	a.	a.	NOUN
ejpam-4793	264	26	then	then	ADV
ejpam-4793	264	27	a	a	DET
ejpam-4793	264	28	∗	∗	NOUN
ejpam-4793	264	29	b	b	NOUN
ejpam-4793	264	30	∈	∈	NOUN
ejpam-4793	264	31	a.	a.	NOUN
ejpam-4793	264	32	we	we	PRON
ejpam-4793	264	33	consider	consider	VERB
ejpam-4793	264	34	the	the	DET
ejpam-4793	264	35	following	follow	VERB
ejpam-4793	264	36	three	three	NUM
ejpam-4793	264	37	cases	case	NOUN
ejpam-4793	264	38	.	.	PUNCT
ejpam-4793	265	1	case	case	NOUN
ejpam-4793	265	2	1	1	NUM
ejpam-4793	265	3	.	.	PUNCT
ejpam-4793	266	1	a	a	DET
ejpam-4793	266	2	∗	∗	NOUN
ejpam-4793	266	3	b	b	NOUN
ejpam-4793	266	4	=	=	SYM
ejpam-4793	266	5	1	1	NUM
ejpam-4793	266	6	.	.	PUNCT
ejpam-4793	267	1	then	then	ADV
ejpam-4793	267	2	a	a	PRON
ejpam-4793	267	3	=	=	SYM
ejpam-4793	267	4	1	1	NUM
ejpam-4793	267	5	and	and	CCONJ
ejpam-4793	267	6	b	b	NOUN
ejpam-4793	267	7	=	=	SYM
ejpam-4793	267	8	1	1	NUM
ejpam-4793	267	9	.	.	PUNCT
ejpam-4793	267	10	thus	thus	ADV
ejpam-4793	267	11	a	a	DET
ejpam-4793	267	12	,	,	PUNCT
ejpam-4793	267	13	b	b	X
ejpam-4793	267	14	∈	∈	PROPN
ejpam-4793	267	15	a.	a.	NOUN
ejpam-4793	267	16	case	case	NOUN
ejpam-4793	267	17	2	2	NUM
ejpam-4793	267	18	.	.	PUNCT
ejpam-4793	268	1	a	a	DET
ejpam-4793	268	2	∗	∗	NOUN
ejpam-4793	268	3	b	b	X
ejpam-4793	268	4	=	=	X
ejpam-4793	268	5	n.	n.	NOUN
ejpam-4793	268	6	then	then	ADV
ejpam-4793	269	1	a	a	DET
ejpam-4793	269	2	∗	∗	NOUN
ejpam-4793	269	3	b	b	NOUN
ejpam-4793	269	4	=	=	SYM
ejpam-4793	269	5	1	1	NUM
ejpam-4793	269	6	∗	∗	NOUN
ejpam-4793	269	7	n	n	NOUN
ejpam-4793	269	8	=	=	SYM
ejpam-4793	269	9	n	n	NOUN
ejpam-4793	269	10	∗	∗	NOUN
ejpam-4793	269	11	1	1	NUM
ejpam-4793	269	12	=	=	SYM
ejpam-4793	269	13	n	n	PROPN
ejpam-4793	269	14	∗	∗	NOUN
ejpam-4793	269	15	n.	n.	NOUN
ejpam-4793	269	16	clearly	clearly	ADV
ejpam-4793	269	17	,	,	PUNCT
ejpam-4793	269	18	a	a	DET
ejpam-4793	269	19	,	,	PUNCT
ejpam-4793	269	20	b	b	X
ejpam-4793	269	21	∈	∈	PROPN
ejpam-4793	269	22	a.	a.	NOUN
ejpam-4793	269	23	case	case	NOUN
ejpam-4793	269	24	3	3	X
ejpam-4793	269	25	.	.	PUNCT
ejpam-4793	269	26	a	a	DET
ejpam-4793	269	27	∗	∗	NOUN
ejpam-4793	269	28	b	b	NOUN
ejpam-4793	269	29	=	=	PUNCT
ejpam-4793	269	30	h.	h.	PROPN
ejpam-4793	269	31	then	then	ADV
ejpam-4793	269	32	a	a	DET
ejpam-4793	269	33	∗	∗	NOUN
ejpam-4793	269	34	b	b	NOUN
ejpam-4793	269	35	=	=	SYM
ejpam-4793	269	36	1	1	NUM
ejpam-4793	269	37	∗	∗	NOUN
ejpam-4793	269	38	h	h	NOUN
ejpam-4793	269	39	=	=	SYM
ejpam-4793	269	40	n	n	NOUN
ejpam-4793	269	41	∗	∗	NOUN
ejpam-4793	269	42	h	h	NOUN
ejpam-4793	270	1	=	=	NOUN
ejpam-4793	270	2	h	h	PROPN
ejpam-4793	270	3	∗	∗	NOUN
ejpam-4793	270	4	1	1	NUM
ejpam-4793	270	5	=	=	SYM
ejpam-4793	270	6	h	h	NOUN
ejpam-4793	270	7	∗	∗	NOUN
ejpam-4793	270	8	n.	n.	PROPN
ejpam-4793	270	9	clearly	clearly	ADV
ejpam-4793	270	10	,	,	PUNCT
ejpam-4793	270	11	a	a	DET
ejpam-4793	270	12	,	,	PUNCT
ejpam-4793	270	13	b	b	X
ejpam-4793	270	14	∈	∈	PROPN
ejpam-4793	270	15	a.	a.	NOUN
ejpam-4793	270	16	thus	thus	ADV
ejpam-4793	270	17	,	,	PUNCT
ejpam-4793	270	18	a	a	DET
ejpam-4793	270	19	,	,	PUNCT
ejpam-4793	270	20	b	b	X
ejpam-4793	270	21	∈	∈	PROPN
ejpam-4793	270	22	a.	a.	NOUN
ejpam-4793	270	23	now	now	ADV
ejpam-4793	270	24	,	,	PUNCT
ejpam-4793	270	25	suppose	suppose	VERB
ejpam-4793	270	26	that	that	SCONJ
ejpam-4793	270	27	a	a	DET
ejpam-4793	270	28	,	,	PUNCT
ejpam-4793	270	29	b	b	X
ejpam-4793	270	30	∈	∈	PROPN
ejpam-4793	270	31	a.	a.	NOUN
ejpam-4793	270	32	then	then	ADV
ejpam-4793	270	33	,	,	PUNCT
ejpam-4793	270	34	we	we	PRON
ejpam-4793	270	35	have	have	VERB
ejpam-4793	270	36	αa	αa	PROPN
ejpam-4793	270	37	∗	∗	NOUN
ejpam-4793	270	38	βb	βb	DET
ejpam-4793	270	39	=	=	ADJ
ejpam-4793	270	40	α1	α1	PROPN
ejpam-4793	270	41	∗	∗	NOUN
ejpam-4793	270	42	β1	β1	NOUN
ejpam-4793	270	43	=	=	PUNCT
ejpam-4793	270	44	1	1	NUM
ejpam-4793	270	45	∗	∗	NOUN
ejpam-4793	270	46	1	1	NUM
ejpam-4793	270	47	=	=	SYM
ejpam-4793	270	48	1	1	NUM
ejpam-4793	270	49	∈	∈	NOUN
ejpam-4793	270	50	a	a	PRON
ejpam-4793	270	51	;	;	PUNCT
ejpam-4793	270	52	αa	αa	NUM
ejpam-4793	270	53	∗	∗	NOUN
ejpam-4793	270	54	βb	βb	DET
ejpam-4793	270	55	=	=	ADJ
ejpam-4793	270	56	α1	α1	PROPN
ejpam-4793	270	57	∗	∗	NOUN
ejpam-4793	270	58	βn	βn	NOUN
ejpam-4793	271	1	=	=	SYM
ejpam-4793	271	2	1	1	NUM
ejpam-4793	271	3	∗	∗	NOUN
ejpam-4793	271	4	n	n	NOUN
ejpam-4793	271	5	=	=	SYM
ejpam-4793	271	6	n	n	SYM
ejpam-4793	271	7	∈	∈	PROPN
ejpam-4793	271	8	a	a	PRON
ejpam-4793	271	9	;	;	PUNCT
ejpam-4793	271	10	αa	αa	NUM
ejpam-4793	271	11	∗	∗	NOUN
ejpam-4793	271	12	βb	βb	DET
ejpam-4793	271	13	=	=	ADJ
ejpam-4793	271	14	α1	α1	PROPN
ejpam-4793	271	15	∗	∗	NOUN
ejpam-4793	271	16	βh	βh	ADP
ejpam-4793	271	17	=	=	SYM
ejpam-4793	271	18	1	1	NUM
ejpam-4793	271	19	∗	∗	NOUN
ejpam-4793	271	20	h	h	NOUN
ejpam-4793	272	1	=	=	NOUN
ejpam-4793	272	2	h	h	NOUN
ejpam-4793	272	3	∈	∈	PROPN
ejpam-4793	273	1	a	a	PRON
ejpam-4793	273	2	;	;	PUNCT
ejpam-4793	273	3	αa	αa	NUM
ejpam-4793	273	4	∗	∗	NOUN
ejpam-4793	273	5	βb	βb	DET
ejpam-4793	273	6	=	=	NOUN
ejpam-4793	273	7	αn	αn	NOUN
ejpam-4793	273	8	∗	∗	NOUN
ejpam-4793	273	9	βn	βn	NOUN
ejpam-4793	274	1	=	=	SYM
ejpam-4793	274	2	n	n	NOUN
ejpam-4793	274	3	∗	∗	NOUN
ejpam-4793	274	4	n	n	NOUN
ejpam-4793	274	5	=	=	SYM
ejpam-4793	274	6	n	n	SYM
ejpam-4793	274	7	∈	∈	PROPN
ejpam-4793	274	8	a	a	PRON
ejpam-4793	274	9	;	;	PUNCT
ejpam-4793	274	10	αa	αa	NUM
ejpam-4793	274	11	∗	∗	NOUN
ejpam-4793	274	12	βb	βb	DET
ejpam-4793	274	13	=	=	NOUN
ejpam-4793	274	14	αn	αn	NOUN
ejpam-4793	274	15	∗	∗	NOUN
ejpam-4793	274	16	βh	βh	ADP
ejpam-4793	274	17	=	=	SYM
ejpam-4793	274	18	n	n	NOUN
ejpam-4793	274	19	∗	∗	NOUN
ejpam-4793	274	20	h	h	NOUN
ejpam-4793	275	1	=	=	NOUN
ejpam-4793	275	2	h	h	NOUN
ejpam-4793	275	3	∈	∈	PROPN
ejpam-4793	276	1	a	a	PRON
ejpam-4793	276	2	;	;	PUNCT
ejpam-4793	276	3	αa	αa	NUM
ejpam-4793	276	4	∗	∗	NOUN
ejpam-4793	276	5	βb	βb	PRON
ejpam-4793	276	6	=	=	NOUN
ejpam-4793	276	7	αh	αh	NOUN
ejpam-4793	276	8	∗	∗	NOUN
ejpam-4793	276	9	βh	βh	PUNCT
ejpam-4793	277	1	=	=	SYM
ejpam-4793	277	2	h	h	NOUN
ejpam-4793	277	3	∗	∗	NOUN
ejpam-4793	277	4	h	h	NOUN
ejpam-4793	278	1	=	=	NOUN
ejpam-4793	278	2	h	h	NOUN
ejpam-4793	278	3	∈	∈	PROPN
ejpam-4793	278	4	a.	a.	NOUN
ejpam-4793	278	5	thus	thus	ADV
ejpam-4793	278	6	,	,	PUNCT
ejpam-4793	278	7	αa	αa	ADV
ejpam-4793	278	8	∗	∗	VERB
ejpam-4793	278	9	βb	βb	PRON
ejpam-4793	278	10	∈	∈	PROPN
ejpam-4793	278	11	a.	a.	NOUN
ejpam-4793	278	12	hence	hence	ADV
ejpam-4793	278	13	,	,	PUNCT
ejpam-4793	278	14	a	a	PRON
ejpam-4793	278	15	is	be	AUX
ejpam-4793	278	16	a	a	DET
ejpam-4793	278	17	γ	γ	NOUN
ejpam-4793	278	18	-	-	PUNCT
ejpam-4793	278	19	order	order	NOUN
ejpam-4793	278	20	-	-	PUNCT
ejpam-4793	278	21	ideal	ideal	NOUN
ejpam-4793	278	22	of	of	ADP
ejpam-4793	278	23	m	m	PROPN
ejpam-4793	278	24	.	.	PUNCT
ejpam-4793	279	1	observe	observe	VERB
ejpam-4793	279	2	that	that	SCONJ
ejpam-4793	279	3	there	there	PRON
ejpam-4793	279	4	exist	exist	VERB
ejpam-4793	279	5	n	n	PRON
ejpam-4793	279	6	∈	∈	PROPN
ejpam-4793	279	7	a	a	PRON
ejpam-4793	279	8	and	and	CCONJ
ejpam-4793	279	9	s	s	PROPN
ejpam-4793	279	10	∈	∈	NOUN
ejpam-4793	279	11	m	m	VERB
ejpam-4793	279	12	such	such	ADJ
ejpam-4793	279	13	that	that	PRON
ejpam-4793	279	14	for	for	ADP
ejpam-4793	279	15	any	any	DET
ejpam-4793	279	16	α	α	NOUN
ejpam-4793	279	17	,	,	PUNCT
ejpam-4793	279	18	β	β	PROPN
ejpam-4793	279	19	∈	∈	PROPN
ejpam-4793	279	20	γ	γ	X
ejpam-4793	279	21	,	,	PUNCT
ejpam-4793	279	22	αn∗βs	αn∗βs	ADJ
ejpam-4793	279	23	=	=	PUNCT
ejpam-4793	279	24	n∗s	n∗s	NUM
ejpam-4793	279	25	=	=	SYM
ejpam-4793	279	26	s	s	NOUN
ejpam-4793	279	27	/∈	/∈	NOUN
ejpam-4793	279	28	a.	a.	NOUN
ejpam-4793	279	29	thus	thus	ADV
ejpam-4793	279	30	,	,	PUNCT
ejpam-4793	279	31	by	by	ADP
ejpam-4793	279	32	definition	definition	NOUN
ejpam-4793	279	33	10	10	NUM
ejpam-4793	279	34	,	,	PUNCT
ejpam-4793	279	35	a	a	PRON
ejpam-4793	279	36	is	be	AUX
ejpam-4793	279	37	not	not	PART
ejpam-4793	279	38	a	a	DET
ejpam-4793	279	39	γ	γ	NOUN
ejpam-4793	279	40	-	-	PUNCT
ejpam-4793	279	41	ideal	ideal	ADJ
ejpam-4793	279	42	.	.	PUNCT
ejpam-4793	280	1	remark	remark	NOUN
ejpam-4793	280	2	5	5	NUM
ejpam-4793	280	3	.	.	PUNCT
ejpam-4793	281	1	if	if	SCONJ
ejpam-4793	281	2	i	i	PRON
ejpam-4793	281	3	is	be	AUX
ejpam-4793	281	4	a	a	DET
ejpam-4793	281	5	γ	γ	NOUN
ejpam-4793	281	6	-	-	PUNCT
ejpam-4793	281	7	ideal	ideal	ADJ
ejpam-4793	281	8	,	,	PUNCT
ejpam-4793	281	9	in	in	ADP
ejpam-4793	281	10	general	general	ADJ
ejpam-4793	281	11	i	i	PRON
ejpam-4793	281	12	is	be	AUX
ejpam-4793	281	13	not	not	PART
ejpam-4793	281	14	necessarily	necessarily	ADV
ejpam-4793	281	15	a	a	DET
ejpam-4793	281	16	γ	γ	NOUN
ejpam-4793	281	17	-	-	PUNCT
ejpam-4793	281	18	order	order	NOUN
ejpam-4793	281	19	-	-	PUNCT
ejpam-4793	281	20	ideal	ideal	NOUN
ejpam-4793	281	21	.	.	PUNCT
ejpam-4793	282	1	similarly	similarly	ADV
ejpam-4793	282	2	,	,	PUNCT
ejpam-4793	282	3	if	if	SCONJ
ejpam-4793	282	4	i	i	PRON
ejpam-4793	282	5	is	be	AUX
ejpam-4793	282	6	a	a	DET
ejpam-4793	282	7	γ	γ	NOUN
ejpam-4793	282	8	-	-	PUNCT
ejpam-4793	282	9	order	order	NOUN
ejpam-4793	282	10	-	-	PUNCT
ejpam-4793	282	11	ideal	ideal	ADJ
ejpam-4793	282	12	,	,	PUNCT
ejpam-4793	282	13	in	in	ADP
ejpam-4793	282	14	general	general	ADJ
ejpam-4793	282	15	i	i	PRON
ejpam-4793	282	16	is	be	AUX
ejpam-4793	282	17	not	not	PART
ejpam-4793	282	18	necessarily	necessarily	ADV
ejpam-4793	282	19	a	a	DET
ejpam-4793	282	20	γ	γ	NOUN
ejpam-4793	282	21	-	-	PUNCT
ejpam-4793	282	22	ideal	ideal	NOUN
ejpam-4793	282	23	.	.	PUNCT
ejpam-4793	283	1	lemma	lemma	PROPN
ejpam-4793	283	2	2	2	X
ejpam-4793	283	3	.	.	PUNCT
ejpam-4793	284	1	let	let	VERB
ejpam-4793	284	2	i	i	PRON
ejpam-4793	284	3	be	be	AUX
ejpam-4793	284	4	a	a	DET
ejpam-4793	284	5	γ	γ	NOUN
ejpam-4793	284	6	-	-	NOUN
ejpam-4793	284	7	ideal	ideal	NOUN
ejpam-4793	284	8	of	of	ADP
ejpam-4793	284	9	a	a	DET
ejpam-4793	284	10	γ	γ	X
ejpam-4793	284	11	-	-	PUNCT
ejpam-4793	284	12	monoid	monoid	NOUN
ejpam-4793	284	13	m	m	PROPN
ejpam-4793	284	14	.	.	PUNCT
ejpam-4793	285	1	then	then	ADV
ejpam-4793	285	2	the	the	DET
ejpam-4793	285	3	identity	identity	NOUN
ejpam-4793	285	4	1	1	NUM
ejpam-4793	285	5	m	m	NOUN
ejpam-4793	285	6	∈	∈	NOUN
ejpam-4793	285	7	i	i	PRON
ejpam-4793	285	8	if	if	SCONJ
ejpam-4793	285	9	and	and	CCONJ
ejpam-4793	285	10	only	only	ADV
ejpam-4793	285	11	if	if	SCONJ
ejpam-4793	285	12	i	i	PRON
ejpam-4793	285	13	=	=	NOUN
ejpam-4793	285	14	m	m	VERB
ejpam-4793	285	15	.	.	PUNCT
ejpam-4793	286	1	proof	proof	NOUN
ejpam-4793	286	2	.	.	PUNCT
ejpam-4793	287	1	let	let	VERB
ejpam-4793	287	2	i	i	PRON
ejpam-4793	287	3	is	be	AUX
ejpam-4793	287	4	a	a	DET
ejpam-4793	287	5	γ	γ	NOUN
ejpam-4793	287	6	-	-	NOUN
ejpam-4793	287	7	ideal	ideal	NOUN
ejpam-4793	287	8	of	of	ADP
ejpam-4793	287	9	m	m	PROPN
ejpam-4793	287	10	.	.	PUNCT
ejpam-4793	288	1	suppose	suppose	VERB
ejpam-4793	288	2	that	that	SCONJ
ejpam-4793	288	3	the	the	DET
ejpam-4793	288	4	identity	identity	NOUN
ejpam-4793	288	5	1	1	NUM
ejpam-4793	288	6	m	m	NOUN
ejpam-4793	288	7	∈	∈	NOUN
ejpam-4793	288	8	i	i	PRON
ejpam-4793	288	9	and	and	CCONJ
ejpam-4793	288	10	m	m	PROPN
ejpam-4793	288	11	∈	∈	PROPN
ejpam-4793	288	12	m	m	NOUN
ejpam-4793	288	13	.	.	PUNCT
ejpam-4793	289	1	then	then	ADV
ejpam-4793	289	2	for	for	ADP
ejpam-4793	289	3	any	any	DET
ejpam-4793	289	4	α	α	NOUN
ejpam-4793	289	5	,	,	PUNCT
ejpam-4793	289	6	β	β	PROPN
ejpam-4793	289	7	∈	∈	PROPN
ejpam-4793	289	8	γ	γ	X
ejpam-4793	289	9	,	,	PUNCT
ejpam-4793	289	10	we	we	PRON
ejpam-4793	289	11	have	have	VERB
ejpam-4793	289	12	α1	α1	PROPN
ejpam-4793	289	13	m	m	NOUN
ejpam-4793	289	14	∗βm	∗βm	ADP
ejpam-4793	289	15	∈	∈	PROPN
ejpam-4793	289	16	i.	i.	NOUN
ejpam-4793	289	17	for	for	ADP
ejpam-4793	289	18	α	α	NOUN
ejpam-4793	289	19	=	=	SYM
ejpam-4793	289	20	β	β	X
ejpam-4793	289	21	=	=	SYM
ejpam-4793	289	22	0	0	NUM
ejpam-4793	289	23	,	,	PUNCT
ejpam-4793	289	24	we	we	PRON
ejpam-4793	289	25	have	have	VERB
ejpam-4793	289	26	01	01	NUM
ejpam-4793	289	27	m	m	PROPN
ejpam-4793	289	28	∗	∗	NOUN
ejpam-4793	289	29	0	0	NUM
ejpam-4793	289	30	m	m	NOUN
ejpam-4793	289	31	=	=	ADJ
ejpam-4793	289	32	1	1	NUM
ejpam-4793	289	33	m	m	NOUN
ejpam-4793	289	34	∗m	∗m	NOUN
ejpam-4793	289	35	=	=	PUNCT
ejpam-4793	289	36	m	m	PROPN
ejpam-4793	289	37	∈	∈	PROPN
ejpam-4793	289	38	i.	i.	NOUN
ejpam-4793	289	39	thus	thus	ADV
ejpam-4793	289	40	,	,	PUNCT
ejpam-4793	289	41	m	m	PROPN
ejpam-4793	289	42	⊆	⊆	NUM
ejpam-4793	289	43	i.	i.	NOUN
ejpam-4793	289	44	consequently	consequently	ADV
ejpam-4793	289	45	,	,	PUNCT
ejpam-4793	289	46	i	i	PRON
ejpam-4793	289	47	=	=	NOUN
ejpam-4793	289	48	m	m	VERB
ejpam-4793	289	49	.	.	PUNCT
ejpam-4793	290	1	conversely	conversely	ADV
ejpam-4793	290	2	,	,	PUNCT
ejpam-4793	290	3	suppose	suppose	VERB
ejpam-4793	290	4	that	that	SCONJ
ejpam-4793	290	5	i	i	PRON
ejpam-4793	290	6	=	=	NOUN
ejpam-4793	290	7	m	m	VERB
ejpam-4793	290	8	.	.	PUNCT
ejpam-4793	291	1	thus	thus	ADV
ejpam-4793	291	2	,	,	PUNCT
ejpam-4793	291	3	the	the	DET
ejpam-4793	291	4	identity	identity	NOUN
ejpam-4793	291	5	1	1	NUM
ejpam-4793	291	6	m	m	NOUN
ejpam-4793	291	7	∈	∈	PROPN
ejpam-4793	291	8	i.	i.	PROPN
ejpam-4793	291	9	h.	h.	PROPN
ejpam-4793	291	10	sarapuddin	sarapuddin	PROPN
ejpam-4793	291	11	,	,	PUNCT
ejpam-4793	291	12	j.	j.	PROPN
ejpam-4793	291	13	vilela	vilela	PROPN
ejpam-4793	291	14	/	/	SYM
ejpam-4793	291	15	eur	eur	PROPN
ejpam-4793	291	16	.	.	PUNCT
ejpam-4793	292	1	j.	j.	PROPN
ejpam-4793	292	2	pure	pure	PROPN
ejpam-4793	292	3	appl	appl	PROPN
ejpam-4793	292	4	.	.	PROPN
ejpam-4793	292	5	math	math	PROPN
ejpam-4793	292	6	,	,	PUNCT
ejpam-4793	292	7	16	16	NUM
ejpam-4793	292	8	(	(	PUNCT
ejpam-4793	292	9	3	3	NUM
ejpam-4793	292	10	)	)	PUNCT
ejpam-4793	292	11	(	(	PUNCT
ejpam-4793	292	12	2023	2023	NUM
ejpam-4793	292	13	)	)	PUNCT
ejpam-4793	292	14	,	,	PUNCT
ejpam-4793	292	15	1772	1772	NUM
ejpam-4793	292	16	-	-	SYM
ejpam-4793	292	17	1793	1793	NUM
ejpam-4793	292	18	1779	1779	NUM
ejpam-4793	292	19	theorems	theorem	NOUN
ejpam-4793	292	20	2	2	NUM
ejpam-4793	292	21	and	and	CCONJ
ejpam-4793	292	22	3	3	NUM
ejpam-4793	292	23	imply	imply	VERB
ejpam-4793	292	24	that	that	SCONJ
ejpam-4793	292	25	there	there	PRON
ejpam-4793	292	26	exists	exist	VERB
ejpam-4793	292	27	no	no	DET
ejpam-4793	292	28	proper	proper	ADJ
ejpam-4793	292	29	γ	γ	ADJ
ejpam-4793	292	30	-	-	PUNCT
ejpam-4793	292	31	order	order	NOUN
ejpam-4793	292	32	-	-	PUNCT
ejpam-4793	292	33	ideal	ideal	NOUN
ejpam-4793	292	34	which	which	PRON
ejpam-4793	292	35	is	be	AUX
ejpam-4793	292	36	also	also	ADV
ejpam-4793	292	37	a	a	DET
ejpam-4793	292	38	γ	γ	X
ejpam-4793	292	39	-	-	PUNCT
ejpam-4793	292	40	ideal	ideal	NOUN
ejpam-4793	292	41	and	and	CCONJ
ejpam-4793	292	42	vice	vice	ADV
ejpam-4793	292	43	versa	versa	ADV
ejpam-4793	292	44	.	.	PUNCT
ejpam-4793	293	1	theorem	theorem	NOUN
ejpam-4793	293	2	2	2	NUM
ejpam-4793	293	3	.	.	PUNCT
ejpam-4793	294	1	let	let	VERB
ejpam-4793	294	2	i	i	PRON
ejpam-4793	294	3	be	be	AUX
ejpam-4793	294	4	a	a	DET
ejpam-4793	294	5	γ	γ	NOUN
ejpam-4793	294	6	-	-	NOUN
ejpam-4793	294	7	ideal	ideal	NOUN
ejpam-4793	294	8	of	of	ADP
ejpam-4793	294	9	a	a	DET
ejpam-4793	294	10	γ	γ	X
ejpam-4793	294	11	-	-	PUNCT
ejpam-4793	294	12	monoid	monoid	NOUN
ejpam-4793	294	13	m	m	PROPN
ejpam-4793	294	14	.	.	PUNCT
ejpam-4793	295	1	then	then	ADV
ejpam-4793	295	2	i	i	PRON
ejpam-4793	295	3	is	be	AUX
ejpam-4793	295	4	a	a	DET
ejpam-4793	295	5	γ	γ	NOUN
ejpam-4793	295	6	-	-	PUNCT
ejpam-4793	295	7	order	order	NOUN
ejpam-4793	295	8	-	-	PUNCT
ejpam-4793	295	9	ideal	ideal	NOUN
ejpam-4793	295	10	of	of	ADP
ejpam-4793	295	11	m	m	PRON
ejpam-4793	295	12	if	if	SCONJ
ejpam-4793	296	1	and	and	CCONJ
ejpam-4793	296	2	only	only	ADV
ejpam-4793	296	3	if	if	SCONJ
ejpam-4793	296	4	i	i	PRON
ejpam-4793	296	5	=	=	NOUN
ejpam-4793	296	6	m	m	VERB
ejpam-4793	296	7	.	.	PUNCT
ejpam-4793	297	1	proof	proof	NOUN
ejpam-4793	297	2	.	.	PUNCT
ejpam-4793	298	1	let	let	VERB
ejpam-4793	298	2	i	i	PRON
ejpam-4793	298	3	be	be	AUX
ejpam-4793	298	4	a	a	DET
ejpam-4793	298	5	γ	γ	NOUN
ejpam-4793	298	6	-	-	NOUN
ejpam-4793	298	7	ideal	ideal	NOUN
ejpam-4793	298	8	of	of	ADP
ejpam-4793	298	9	m	m	PROPN
ejpam-4793	298	10	.	.	PUNCT
ejpam-4793	299	1	suppose	suppose	VERB
ejpam-4793	299	2	that	that	SCONJ
ejpam-4793	299	3	i	i	PRON
ejpam-4793	299	4	is	be	AUX
ejpam-4793	299	5	a	a	DET
ejpam-4793	299	6	γ	γ	NOUN
ejpam-4793	299	7	-	-	PUNCT
ejpam-4793	299	8	order	order	NOUN
ejpam-4793	299	9	-	-	PUNCT
ejpam-4793	299	10	ideal	ideal	NOUN
ejpam-4793	299	11	of	of	ADP
ejpam-4793	299	12	m	m	PROPN
ejpam-4793	299	13	.	.	PUNCT
ejpam-4793	300	1	then	then	ADV
ejpam-4793	300	2	the	the	DET
ejpam-4793	300	3	identity	identity	NOUN
ejpam-4793	300	4	1	1	NUM
ejpam-4793	300	5	m	m	NOUN
ejpam-4793	300	6	∈	∈	PROPN
ejpam-4793	300	7	i.	i.	NOUN
ejpam-4793	300	8	by	by	ADP
ejpam-4793	300	9	lemma	lemma	PROPN
ejpam-4793	300	10	2	2	NUM
ejpam-4793	300	11	,	,	PUNCT
ejpam-4793	300	12	i	i	PRON
ejpam-4793	300	13	=	=	NOUN
ejpam-4793	300	14	m	m	VERB
ejpam-4793	300	15	.	.	PUNCT
ejpam-4793	301	1	conversely	conversely	ADV
ejpam-4793	301	2	,	,	PUNCT
ejpam-4793	301	3	suppose	suppose	VERB
ejpam-4793	301	4	that	that	SCONJ
ejpam-4793	301	5	i	i	PRON
ejpam-4793	301	6	=	=	NOUN
ejpam-4793	301	7	m	m	VERB
ejpam-4793	301	8	.	.	PUNCT
ejpam-4793	302	1	thus	thus	ADV
ejpam-4793	302	2	,	,	PUNCT
ejpam-4793	302	3	i	i	PRON
ejpam-4793	302	4	is	be	AUX
ejpam-4793	302	5	a	a	DET
ejpam-4793	302	6	γ	γ	NOUN
ejpam-4793	302	7	-	-	PUNCT
ejpam-4793	302	8	order	order	NOUN
ejpam-4793	302	9	-	-	PUNCT
ejpam-4793	302	10	ideal	ideal	NOUN
ejpam-4793	302	11	.	.	PUNCT
ejpam-4793	303	1	theorem	theorem	NOUN
ejpam-4793	303	2	3	3	X
ejpam-4793	303	3	.	.	PUNCT
ejpam-4793	304	1	let	let	VERB
ejpam-4793	304	2	i	i	PRON
ejpam-4793	304	3	be	be	AUX
ejpam-4793	304	4	a	a	DET
ejpam-4793	304	5	γ	γ	NOUN
ejpam-4793	304	6	-	-	PUNCT
ejpam-4793	304	7	order	order	NOUN
ejpam-4793	304	8	-	-	PUNCT
ejpam-4793	304	9	ideal	ideal	NOUN
ejpam-4793	304	10	of	of	ADP
ejpam-4793	304	11	a	a	DET
ejpam-4793	304	12	γ	γ	X
ejpam-4793	304	13	-	-	PUNCT
ejpam-4793	304	14	monoid	monoid	NOUN
ejpam-4793	304	15	m	m	PROPN
ejpam-4793	304	16	.	.	PUNCT
ejpam-4793	305	1	then	then	ADV
ejpam-4793	305	2	i	i	PRON
ejpam-4793	305	3	is	be	AUX
ejpam-4793	305	4	a	a	DET
ejpam-4793	305	5	γ	γ	NOUN
ejpam-4793	305	6	-	-	NOUN
ejpam-4793	305	7	ideal	ideal	NOUN
ejpam-4793	305	8	of	of	ADP
ejpam-4793	305	9	m	m	PRON
ejpam-4793	305	10	if	if	SCONJ
ejpam-4793	306	1	and	and	CCONJ
ejpam-4793	306	2	only	only	ADV
ejpam-4793	306	3	if	if	SCONJ
ejpam-4793	306	4	i	i	PRON
ejpam-4793	306	5	=	=	NOUN
ejpam-4793	306	6	m	m	VERB
ejpam-4793	306	7	.	.	PUNCT
ejpam-4793	307	1	proof	proof	NOUN
ejpam-4793	307	2	.	.	PUNCT
ejpam-4793	308	1	let	let	VERB
ejpam-4793	308	2	i	i	PRON
ejpam-4793	308	3	be	be	AUX
ejpam-4793	308	4	a	a	DET
ejpam-4793	308	5	γ	γ	NOUN
ejpam-4793	308	6	-	-	PUNCT
ejpam-4793	308	7	order	order	NOUN
ejpam-4793	308	8	-	-	PUNCT
ejpam-4793	308	9	ideal	ideal	NOUN
ejpam-4793	308	10	of	of	ADP
ejpam-4793	308	11	a	a	DET
ejpam-4793	308	12	γ	γ	X
ejpam-4793	308	13	-	-	PUNCT
ejpam-4793	308	14	monoid	monoid	NOUN
ejpam-4793	308	15	m	m	PROPN
ejpam-4793	308	16	.	.	PUNCT
ejpam-4793	309	1	then	then	ADV
ejpam-4793	309	2	1	1	NUM
ejpam-4793	309	3	m	m	NOUN
ejpam-4793	309	4	∈	∈	NOUN
ejpam-4793	309	5	i	i	PRON
ejpam-4793	309	6	since	since	SCONJ
ejpam-4793	309	7	i	i	PRON
ejpam-4793	309	8	is	be	AUX
ejpam-4793	309	9	also	also	ADV
ejpam-4793	309	10	a	a	DET
ejpam-4793	309	11	submonoid	submonoid	NOUN
ejpam-4793	309	12	.	.	PUNCT
ejpam-4793	309	13	suppose	suppose	VERB
ejpam-4793	309	14	that	that	SCONJ
ejpam-4793	309	15	i	i	PRON
ejpam-4793	309	16	is	be	AUX
ejpam-4793	309	17	a	a	DET
ejpam-4793	309	18	γ	γ	NOUN
ejpam-4793	309	19	-	-	NOUN
ejpam-4793	309	20	ideal	ideal	NOUN
ejpam-4793	309	21	of	of	ADP
ejpam-4793	309	22	m	m	PRON
ejpam-4793	309	23	.	.	PUNCT
ejpam-4793	310	1	by	by	ADP
ejpam-4793	310	2	lemma	lemma	PROPN
ejpam-4793	310	3	2	2	NUM
ejpam-4793	310	4	,	,	PUNCT
ejpam-4793	310	5	i	i	PRON
ejpam-4793	310	6	=	=	NOUN
ejpam-4793	310	7	m	m	VERB
ejpam-4793	310	8	.	.	PUNCT
ejpam-4793	311	1	conversely	conversely	ADV
ejpam-4793	311	2	,	,	PUNCT
ejpam-4793	311	3	suppose	suppose	VERB
ejpam-4793	311	4	that	that	SCONJ
ejpam-4793	311	5	i	i	PRON
ejpam-4793	311	6	=	=	NOUN
ejpam-4793	311	7	m	m	VERB
ejpam-4793	311	8	.	.	PUNCT
ejpam-4793	312	1	thus	thus	ADV
ejpam-4793	312	2	,	,	PUNCT
ejpam-4793	312	3	by	by	ADP
ejpam-4793	312	4	remark	remark	NOUN
ejpam-4793	312	5	4(i	4(i	NUM
ejpam-4793	312	6	)	)	PUNCT
ejpam-4793	312	7	,	,	PUNCT
ejpam-4793	312	8	i	i	PRON
ejpam-4793	312	9	is	be	AUX
ejpam-4793	312	10	a	a	DET
ejpam-4793	312	11	γ	γ	NOUN
ejpam-4793	312	12	-	-	PUNCT
ejpam-4793	312	13	ideal	ideal	NOUN
ejpam-4793	312	14	.	.	PUNCT
ejpam-4793	313	1	lemma	lemma	PROPN
ejpam-4793	313	2	3	3	X
ejpam-4793	313	3	.	.	PUNCT
ejpam-4793	314	1	let	let	VERB
ejpam-4793	314	2	a	a	PRON
ejpam-4793	314	3	and	and	CCONJ
ejpam-4793	314	4	b	b	NOUN
ejpam-4793	314	5	be	be	AUX
ejpam-4793	314	6	γ	γ	NOUN
ejpam-4793	314	7	-	-	NOUN
ejpam-4793	314	8	ideals	ideal	NOUN
ejpam-4793	314	9	of	of	ADP
ejpam-4793	314	10	a	a	DET
ejpam-4793	314	11	γ	γ	X
ejpam-4793	314	12	-	-	PUNCT
ejpam-4793	314	13	monoid	monoid	NOUN
ejpam-4793	314	14	m	m	PROPN
ejpam-4793	314	15	.	.	PUNCT
ejpam-4793	315	1	then	then	ADV
ejpam-4793	315	2	a∩b	a∩b	PROPN
ejpam-4793	315	3	and	and	CCONJ
ejpam-4793	315	4	a∪b	a∪b	NOUN
ejpam-4793	315	5	are	be	AUX
ejpam-4793	315	6	γ	γ	NOUN
ejpam-4793	315	7	-	-	NOUN
ejpam-4793	315	8	ideals	ideal	NOUN
ejpam-4793	315	9	of	of	ADP
ejpam-4793	315	10	m	m	PROPN
ejpam-4793	315	11	.	.	PUNCT
ejpam-4793	316	1	proof	proof	NOUN
ejpam-4793	316	2	.	.	PUNCT
ejpam-4793	317	1	let	let	VERB
ejpam-4793	317	2	a	a	PRON
ejpam-4793	317	3	and	and	CCONJ
ejpam-4793	317	4	b	b	NOUN
ejpam-4793	317	5	be	be	AUX
ejpam-4793	317	6	γ	γ	NOUN
ejpam-4793	317	7	-	-	NOUN
ejpam-4793	317	8	ideals	ideal	NOUN
ejpam-4793	317	9	of	of	ADP
ejpam-4793	317	10	m	m	PROPN
ejpam-4793	317	11	.	.	PUNCT
ejpam-4793	318	1	let	let	VERB
ejpam-4793	318	2	x	x	SYM
ejpam-4793	318	3	∈	∈	PROPN
ejpam-4793	318	4	a	a	DET
ejpam-4793	318	5	∩	∩	ADJ
ejpam-4793	318	6	b	b	NOUN
ejpam-4793	318	7	and	and	CCONJ
ejpam-4793	318	8	m	m	PROPN
ejpam-4793	318	9	∈	∈	NOUN
ejpam-4793	318	10	m	m	NOUN
ejpam-4793	318	11	.	.	PUNCT
ejpam-4793	319	1	then	then	ADV
ejpam-4793	319	2	x	x	SYM
ejpam-4793	319	3	∈	∈	PROPN
ejpam-4793	319	4	a	a	PRON
ejpam-4793	319	5	and	and	CCONJ
ejpam-4793	319	6	x	x	PROPN
ejpam-4793	319	7	∈	∈	PROPN
ejpam-4793	319	8	b.	b.	PROPN
ejpam-4793	319	9	since	since	SCONJ
ejpam-4793	319	10	a	a	PRON
ejpam-4793	319	11	and	and	CCONJ
ejpam-4793	319	12	b	b	NOUN
ejpam-4793	319	13	are	be	AUX
ejpam-4793	319	14	γ	γ	NOUN
ejpam-4793	319	15	-	-	NOUN
ejpam-4793	319	16	ideals	ideal	NOUN
ejpam-4793	319	17	of	of	ADP
ejpam-4793	319	18	m	m	PRON
ejpam-4793	319	19	,	,	PUNCT
ejpam-4793	319	20	for	for	ADP
ejpam-4793	319	21	all	all	DET
ejpam-4793	319	22	α	α	NOUN
ejpam-4793	319	23	,	,	PUNCT
ejpam-4793	319	24	β	β	PROPN
ejpam-4793	319	25	∈	∈	PROPN
ejpam-4793	319	26	γ	γ	X
ejpam-4793	319	27	,	,	PUNCT
ejpam-4793	319	28	we	we	PRON
ejpam-4793	319	29	have	have	VERB
ejpam-4793	319	30	αx∗βm	αx∗βm	NOUN
ejpam-4793	319	31	,	,	PUNCT
ejpam-4793	319	32	αm∗βx	αm∗βx	PROPN
ejpam-4793	319	33	∈	∈	PROPN
ejpam-4793	319	34	a	a	PRON
ejpam-4793	319	35	and	and	CCONJ
ejpam-4793	319	36	αx	αx	PROPN
ejpam-4793	319	37	∗	∗	NOUN
ejpam-4793	319	38	βm	βm	VERB
ejpam-4793	319	39	,	,	PUNCT
ejpam-4793	319	40	αm	αm	NOUN
ejpam-4793	319	41	∗	∗	NOUN
ejpam-4793	319	42	βx	βx	ADP
ejpam-4793	319	43	∈	∈	PROPN
ejpam-4793	319	44	b.	b.	PROPN
ejpam-4793	319	45	hence	hence	ADV
ejpam-4793	319	46	,	,	PUNCT
ejpam-4793	319	47	for	for	ADP
ejpam-4793	319	48	all	all	DET
ejpam-4793	319	49	α	α	NOUN
ejpam-4793	319	50	,	,	PUNCT
ejpam-4793	319	51	β	β	PROPN
ejpam-4793	319	52	∈	∈	PROPN
ejpam-4793	319	53	γ	γ	X
ejpam-4793	319	54	,	,	PUNCT
ejpam-4793	319	55	αx	αx	ADV
ejpam-4793	319	56	∗	∗	NOUN
ejpam-4793	319	57	βm	βm	NOUN
ejpam-4793	319	58	,	,	PUNCT
ejpam-4793	319	59	αm	αm	NOUN
ejpam-4793	319	60	∗	∗	NOUN
ejpam-4793	319	61	βx	βx	ADP
ejpam-4793	319	62	∈	∈	PROPN
ejpam-4793	319	63	a∩b	a∩b	PROPN
ejpam-4793	319	64	.	.	PUNCT
ejpam-4793	320	1	therefore	therefore	ADV
ejpam-4793	320	2	,	,	PUNCT
ejpam-4793	320	3	a∩b	a∩b	PROPN
ejpam-4793	320	4	is	be	AUX
ejpam-4793	320	5	a	a	DET
ejpam-4793	320	6	γ	γ	NOUN
ejpam-4793	320	7	-	-	NOUN
ejpam-4793	320	8	ideal	ideal	NOUN
ejpam-4793	320	9	of	of	ADP
ejpam-4793	320	10	m	m	PROPN
ejpam-4793	320	11	.	.	PUNCT
ejpam-4793	321	1	now	now	ADV
ejpam-4793	321	2	,	,	PUNCT
ejpam-4793	321	3	let	let	VERB
ejpam-4793	321	4	x	x	X
ejpam-4793	321	5	∈	∈	PROPN
ejpam-4793	321	6	a∪b	a∪b	NOUN
ejpam-4793	321	7	and	and	CCONJ
ejpam-4793	321	8	m	m	PROPN
ejpam-4793	321	9	∈	∈	PROPN
ejpam-4793	321	10	m	m	NOUN
ejpam-4793	321	11	.	.	PUNCT
ejpam-4793	322	1	then	then	ADV
ejpam-4793	322	2	x	x	SYM
ejpam-4793	322	3	∈	∈	PROPN
ejpam-4793	322	4	a	a	PRON
ejpam-4793	322	5	or	or	CCONJ
ejpam-4793	322	6	x	x	PROPN
ejpam-4793	322	7	∈	∈	PROPN
ejpam-4793	322	8	b.	b.	PROPN
ejpam-4793	322	9	since	since	SCONJ
ejpam-4793	322	10	a	a	PRON
ejpam-4793	322	11	and	and	CCONJ
ejpam-4793	322	12	b	b	NOUN
ejpam-4793	322	13	are	be	AUX
ejpam-4793	322	14	γ	γ	NOUN
ejpam-4793	322	15	-	-	NOUN
ejpam-4793	322	16	ideals	ideal	NOUN
ejpam-4793	322	17	of	of	ADP
ejpam-4793	322	18	m	m	PRON
ejpam-4793	322	19	,	,	PUNCT
ejpam-4793	322	20	for	for	ADP
ejpam-4793	322	21	all	all	DET
ejpam-4793	322	22	α	α	NOUN
ejpam-4793	322	23	,	,	PUNCT
ejpam-4793	322	24	β	β	PROPN
ejpam-4793	322	25	∈	∈	PROPN
ejpam-4793	322	26	γ	γ	X
ejpam-4793	322	27	,	,	PUNCT
ejpam-4793	322	28	we	we	PRON
ejpam-4793	322	29	have	have	AUX
ejpam-4793	322	30	αx	αx	PROPN
ejpam-4793	322	31	∗	∗	NOUN
ejpam-4793	322	32	βm	βm	VERB
ejpam-4793	322	33	,	,	PUNCT
ejpam-4793	323	1	αm	αm	CCONJ
ejpam-4793	323	2	∗	∗	NOUN
ejpam-4793	323	3	βx	βx	ADP
ejpam-4793	323	4	∈	∈	PROPN
ejpam-4793	323	5	a	a	DET
ejpam-4793	323	6	or	or	CCONJ
ejpam-4793	323	7	αx	αx	ADV
ejpam-4793	323	8	∗	∗	NOUN
ejpam-4793	323	9	βm	βm	VERB
ejpam-4793	323	10	,	,	PUNCT
ejpam-4793	323	11	αm	αm	NOUN
ejpam-4793	323	12	∗	∗	NOUN
ejpam-4793	323	13	βx	βx	ADP
ejpam-4793	323	14	∈	∈	PROPN
ejpam-4793	323	15	b.	b.	PROPN
ejpam-4793	323	16	hence	hence	ADV
ejpam-4793	323	17	,	,	PUNCT
ejpam-4793	323	18	for	for	ADP
ejpam-4793	323	19	all	all	DET
ejpam-4793	323	20	α	α	NOUN
ejpam-4793	323	21	,	,	PUNCT
ejpam-4793	323	22	β	β	X
ejpam-4793	323	23	∈	∈	PROPN
ejpam-4793	323	24	γ	γ	PROPN
ejpam-4793	323	25	αx	αx	PROPN
ejpam-4793	323	26	∗	∗	NOUN
ejpam-4793	323	27	βm	βm	PROPN
ejpam-4793	323	28	,	,	PUNCT
ejpam-4793	323	29	αm	αm	NOUN
ejpam-4793	323	30	∗	∗	NOUN
ejpam-4793	323	31	βx	βx	ADP
ejpam-4793	323	32	∈	∈	PROPN
ejpam-4793	323	33	a∪b	a∪b	NOUN
ejpam-4793	323	34	.	.	PUNCT
ejpam-4793	324	1	therefore	therefore	ADV
ejpam-4793	324	2	,	,	PUNCT
ejpam-4793	324	3	a∪b	a∪b	PRON
ejpam-4793	324	4	is	be	AUX
ejpam-4793	324	5	a	a	DET
ejpam-4793	324	6	γ	γ	NOUN
ejpam-4793	324	7	-	-	NOUN
ejpam-4793	324	8	ideal	ideal	NOUN
ejpam-4793	324	9	of	of	ADP
ejpam-4793	324	10	m	m	PROPN
ejpam-4793	324	11	.	.	PUNCT
ejpam-4793	325	1	theorem	theorem	ADJ
ejpam-4793	325	2	4	4	NUM
ejpam-4793	325	3	.	.	PUNCT
ejpam-4793	326	1	let	let	VERB
ejpam-4793	326	2	i	i	PRON
ejpam-4793	326	3	be	be	AUX
ejpam-4793	326	4	a	a	DET
ejpam-4793	326	5	γ	γ	NOUN
ejpam-4793	326	6	-	-	PUNCT
ejpam-4793	326	7	order	order	NOUN
ejpam-4793	326	8	-	-	PUNCT
ejpam-4793	326	9	ideal	ideal	NOUN
ejpam-4793	326	10	of	of	ADP
ejpam-4793	326	11	a	a	DET
ejpam-4793	326	12	γ	γ	X
ejpam-4793	326	13	-	-	PUNCT
ejpam-4793	326	14	monoid	monoid	NOUN
ejpam-4793	326	15	m	m	PROPN
ejpam-4793	326	16	and	and	CCONJ
ejpam-4793	326	17	j	j	PROPN
ejpam-4793	326	18	a	a	DET
ejpam-4793	326	19	γ	γ	NOUN
ejpam-4793	326	20	-	-	NOUN
ejpam-4793	326	21	ideal	ideal	NOUN
ejpam-4793	326	22	of	of	ADP
ejpam-4793	326	23	m	m	PROPN
ejpam-4793	326	24	.	.	PUNCT
ejpam-4793	327	1	(	(	PUNCT
ejpam-4793	327	2	i	i	NOUN
ejpam-4793	327	3	)	)	PUNCT
ejpam-4793	327	4	if	if	SCONJ
ejpam-4793	327	5	j	j	PROPN
ejpam-4793	327	6	∩	∩	NOUN
ejpam-4793	327	7	i	i	PROPN
ejpam-4793	327	8	̸=	̸=	PROPN
ejpam-4793	327	9	∅	∅	NOUN
ejpam-4793	327	10	,	,	PUNCT
ejpam-4793	327	11	then	then	ADV
ejpam-4793	327	12	j	j	PROPN
ejpam-4793	327	13	∩	∩	NOUN
ejpam-4793	327	14	i	i	PRON
ejpam-4793	327	15	is	be	AUX
ejpam-4793	327	16	a	a	DET
ejpam-4793	327	17	γ	γ	NOUN
ejpam-4793	327	18	-	-	PUNCT
ejpam-4793	327	19	ideal	ideal	NOUN
ejpam-4793	327	20	of	of	ADP
ejpam-4793	327	21	i.	i.	PROPN
ejpam-4793	327	22	(	(	PUNCT
ejpam-4793	327	23	ii	ii	PROPN
ejpam-4793	327	24	)	)	PUNCT
ejpam-4793	327	25	if	if	SCONJ
ejpam-4793	327	26	m	m	NOUN
ejpam-4793	327	27	is	be	AUX
ejpam-4793	327	28	commutative	commutative	ADJ
ejpam-4793	327	29	,	,	PUNCT
ejpam-4793	327	30	then	then	ADV
ejpam-4793	327	31	j	j	PROPN
ejpam-4793	327	32	∪	∪	PROPN
ejpam-4793	327	33	i	i	PRON
ejpam-4793	327	34	is	be	AUX
ejpam-4793	327	35	a	a	DET
ejpam-4793	327	36	γ	γ	NOUN
ejpam-4793	327	37	-	-	PUNCT
ejpam-4793	327	38	order	order	NOUN
ejpam-4793	327	39	-	-	PUNCT
ejpam-4793	327	40	ideal	ideal	NOUN
ejpam-4793	327	41	of	of	ADP
ejpam-4793	327	42	m	m	PROPN
ejpam-4793	327	43	.	.	PUNCT
ejpam-4793	328	1	proof	proof	NOUN
ejpam-4793	328	2	.	.	PUNCT
ejpam-4793	329	1	let	let	VERB
ejpam-4793	329	2	i	i	PRON
ejpam-4793	329	3	be	be	AUX
ejpam-4793	329	4	a	a	DET
ejpam-4793	329	5	γ	γ	NOUN
ejpam-4793	329	6	-	-	PUNCT
ejpam-4793	329	7	order	order	NOUN
ejpam-4793	329	8	-	-	PUNCT
ejpam-4793	329	9	ideal	ideal	NOUN
ejpam-4793	329	10	of	of	ADP
ejpam-4793	329	11	m	m	PROPN
ejpam-4793	329	12	and	and	CCONJ
ejpam-4793	329	13	j	j	PROPN
ejpam-4793	329	14	a	a	DET
ejpam-4793	329	15	γ	γ	NOUN
ejpam-4793	329	16	-	-	NOUN
ejpam-4793	329	17	ideal	ideal	NOUN
ejpam-4793	329	18	of	of	ADP
ejpam-4793	329	19	m	m	PROPN
ejpam-4793	329	20	.	.	PUNCT
ejpam-4793	330	1	(	(	PUNCT
ejpam-4793	330	2	i	i	NOUN
ejpam-4793	330	3	)	)	PUNCT
ejpam-4793	330	4	let	let	VERB
ejpam-4793	330	5	x	x	PUNCT
ejpam-4793	330	6	∈	∈	PROPN
ejpam-4793	330	7	j	j	PROPN
ejpam-4793	330	8	∩	∩	PROPN
ejpam-4793	330	9	i	i	PRON
ejpam-4793	330	10	and	and	CCONJ
ejpam-4793	330	11	a	a	DET
ejpam-4793	330	12	∈	∈	PROPN
ejpam-4793	330	13	i.	i.	NOUN
ejpam-4793	330	14	then	then	ADV
ejpam-4793	330	15	x	x	SYM
ejpam-4793	330	16	∈	∈	PROPN
ejpam-4793	330	17	j	j	PROPN
ejpam-4793	330	18	and	and	CCONJ
ejpam-4793	330	19	x	x	PROPN
ejpam-4793	330	20	∈	∈	PROPN
ejpam-4793	330	21	i.	i.	NOUN
ejpam-4793	330	22	since	since	SCONJ
ejpam-4793	330	23	j	j	PROPN
ejpam-4793	330	24	is	be	AUX
ejpam-4793	330	25	a	a	DET
ejpam-4793	330	26	γ	γ	NOUN
ejpam-4793	330	27	-	-	NOUN
ejpam-4793	330	28	ideal	ideal	NOUN
ejpam-4793	330	29	of	of	ADP
ejpam-4793	330	30	m	m	PRON
ejpam-4793	330	31	,	,	PUNCT
ejpam-4793	330	32	for	for	ADP
ejpam-4793	330	33	all	all	DET
ejpam-4793	330	34	α	α	NOUN
ejpam-4793	330	35	,	,	PUNCT
ejpam-4793	330	36	β	β	PROPN
ejpam-4793	330	37	∈	∈	PROPN
ejpam-4793	330	38	γ	γ	X
ejpam-4793	330	39	,	,	PUNCT
ejpam-4793	330	40	αx	αx	ADV
ejpam-4793	330	41	∗	∗	NOUN
ejpam-4793	330	42	βa	βa	PROPN
ejpam-4793	330	43	,	,	PUNCT
ejpam-4793	330	44	αa	αa	ADV
ejpam-4793	330	45	∗	∗	NOUN
ejpam-4793	330	46	βx	βx	ADP
ejpam-4793	330	47	∈	∈	PROPN
ejpam-4793	330	48	j	j	PROPN
ejpam-4793	330	49	.	.	PUNCT
ejpam-4793	331	1	also	also	ADV
ejpam-4793	331	2	,	,	PUNCT
ejpam-4793	331	3	since	since	SCONJ
ejpam-4793	331	4	i	i	PRON
ejpam-4793	331	5	is	be	AUX
ejpam-4793	331	6	a	a	DET
ejpam-4793	331	7	γ	γ	NOUN
ejpam-4793	331	8	-	-	PUNCT
ejpam-4793	331	9	order	order	NOUN
ejpam-4793	331	10	-	-	PUNCT
ejpam-4793	331	11	ideal	ideal	NOUN
ejpam-4793	331	12	of	of	ADP
ejpam-4793	331	13	m	m	PRON
ejpam-4793	331	14	,	,	PUNCT
ejpam-4793	331	15	for	for	ADP
ejpam-4793	331	16	all	all	DET
ejpam-4793	331	17	α	α	NOUN
ejpam-4793	331	18	,	,	PUNCT
ejpam-4793	331	19	β	β	PROPN
ejpam-4793	331	20	∈	∈	PROPN
ejpam-4793	331	21	γ	γ	X
ejpam-4793	331	22	,	,	PUNCT
ejpam-4793	331	23	αx	αx	ADV
ejpam-4793	331	24	∗	∗	NOUN
ejpam-4793	331	25	βa	βa	PROPN
ejpam-4793	331	26	,	,	PUNCT
ejpam-4793	331	27	αa	αa	PROPN
ejpam-4793	331	28	∗	∗	NOUN
ejpam-4793	331	29	βx	βx	ADP
ejpam-4793	331	30	∈	∈	PROPN
ejpam-4793	331	31	i.	i.	NOUN
ejpam-4793	331	32	thus	thus	ADV
ejpam-4793	331	33	,	,	PUNCT
ejpam-4793	331	34	for	for	ADP
ejpam-4793	331	35	all	all	DET
ejpam-4793	331	36	α	α	NOUN
ejpam-4793	331	37	,	,	PUNCT
ejpam-4793	331	38	β	β	PROPN
ejpam-4793	331	39	∈	∈	PROPN
ejpam-4793	331	40	γ	γ	X
ejpam-4793	331	41	,	,	PUNCT
ejpam-4793	331	42	αx	αx	ADV
ejpam-4793	331	43	∗	∗	NOUN
ejpam-4793	331	44	βa	βa	PROPN
ejpam-4793	331	45	,	,	PUNCT
ejpam-4793	331	46	αa	αa	ADV
ejpam-4793	331	47	∗	∗	NOUN
ejpam-4793	331	48	βx	βx	ADP
ejpam-4793	331	49	∈	∈	PROPN
ejpam-4793	331	50	j	j	PROPN
ejpam-4793	331	51	∩	∩	PROPN
ejpam-4793	331	52	i.	i.	PROPN
ejpam-4793	331	53	therefore	therefore	ADV
ejpam-4793	331	54	,	,	PUNCT
ejpam-4793	331	55	j	j	PROPN
ejpam-4793	331	56	∩	∩	NOUN
ejpam-4793	331	57	i	i	PRON
ejpam-4793	331	58	is	be	AUX
ejpam-4793	331	59	a	a	DET
ejpam-4793	331	60	γ	γ	NOUN
ejpam-4793	331	61	-	-	PUNCT
ejpam-4793	331	62	ideal	ideal	NOUN
ejpam-4793	331	63	of	of	ADP
ejpam-4793	331	64	i.	i.	PROPN
ejpam-4793	331	65	(	(	PUNCT
ejpam-4793	331	66	ii	ii	PROPN
ejpam-4793	331	67	)	)	PUNCT
ejpam-4793	331	68	suppose	suppose	VERB
ejpam-4793	331	69	that	that	SCONJ
ejpam-4793	331	70	αx	αx	PRON
ejpam-4793	331	71	∗	∗	VERB
ejpam-4793	331	72	βa	βa	INTJ
ejpam-4793	331	73	∈	∈	PROPN
ejpam-4793	331	74	j	j	PROPN
ejpam-4793	331	75	∪	∪	VERB
ejpam-4793	331	76	i	i	PRON
ejpam-4793	331	77	for	for	ADP
ejpam-4793	331	78	all	all	DET
ejpam-4793	331	79	α	α	NOUN
ejpam-4793	331	80	,	,	PUNCT
ejpam-4793	331	81	β	β	PROPN
ejpam-4793	331	82	∈	∈	PROPN
ejpam-4793	331	83	γ	γ	X
ejpam-4793	331	84	.	.	PROPN
ejpam-4793	331	85	then	then	ADV
ejpam-4793	331	86	,	,	PUNCT
ejpam-4793	331	87	αx	αx	ADV
ejpam-4793	331	88	∗	∗	VERB
ejpam-4793	331	89	βa	βa	INTJ
ejpam-4793	331	90	∈	∈	PROPN
ejpam-4793	331	91	j	j	PROPN
ejpam-4793	331	92	or	or	CCONJ
ejpam-4793	331	93	αx	αx	PROPN
ejpam-4793	331	94	∗	∗	NOUN
ejpam-4793	331	95	βa	βa	INTJ
ejpam-4793	332	1	∈	∈	PROPN
ejpam-4793	332	2	i.	i.	NOUN
ejpam-4793	332	3	since	since	SCONJ
ejpam-4793	332	4	i	i	PRON
ejpam-4793	332	5	is	be	AUX
ejpam-4793	332	6	a	a	DET
ejpam-4793	332	7	γ	γ	NOUN
ejpam-4793	332	8	-	-	PUNCT
ejpam-4793	332	9	order	order	NOUN
ejpam-4793	332	10	-	-	PUNCT
ejpam-4793	332	11	ideal	ideal	NOUN
ejpam-4793	332	12	of	of	ADP
ejpam-4793	332	13	m	m	PRON
ejpam-4793	332	14	,	,	PUNCT
ejpam-4793	332	15	it	it	PRON
ejpam-4793	332	16	follows	follow	VERB
ejpam-4793	332	17	that	that	SCONJ
ejpam-4793	332	18	x	x	X
ejpam-4793	332	19	,	,	PUNCT
ejpam-4793	332	20	a	a	DET
ejpam-4793	332	21	∈	∈	NOUN
ejpam-4793	332	22	i	i	NOUN
ejpam-4793	332	23	⊆	⊆	NUM
ejpam-4793	332	24	j	j	PROPN
ejpam-4793	332	25	∪	∪	PROPN
ejpam-4793	332	26	i.	i.	PROPN
ejpam-4793	332	27	now	now	ADV
ejpam-4793	332	28	,	,	PUNCT
ejpam-4793	332	29	suppose	suppose	VERB
ejpam-4793	332	30	that	that	SCONJ
ejpam-4793	332	31	x	x	X
ejpam-4793	332	32	,	,	PUNCT
ejpam-4793	332	33	a	a	DET
ejpam-4793	332	34	∈	∈	PROPN
ejpam-4793	332	35	j	j	PROPN
ejpam-4793	332	36	∪	∪	PROPN
ejpam-4793	332	37	i.	i.	PROPN
ejpam-4793	332	38	consider	consider	VERB
ejpam-4793	332	39	the	the	DET
ejpam-4793	332	40	following	follow	VERB
ejpam-4793	332	41	cases	case	NOUN
ejpam-4793	332	42	.	.	PUNCT
ejpam-4793	333	1	case	case	NOUN
ejpam-4793	333	2	1	1	NUM
ejpam-4793	333	3	.	.	NUM
ejpam-4793	334	1	x	x	X
ejpam-4793	334	2	,	,	PUNCT
ejpam-4793	334	3	a	a	DET
ejpam-4793	334	4	∈	∈	PROPN
ejpam-4793	334	5	i.	i.	NOUN
ejpam-4793	334	6	then	then	ADV
ejpam-4793	334	7	,	,	PUNCT
ejpam-4793	334	8	since	since	SCONJ
ejpam-4793	334	9	i	i	PRON
ejpam-4793	334	10	is	be	AUX
ejpam-4793	334	11	a	a	DET
ejpam-4793	334	12	γ	γ	NOUN
ejpam-4793	334	13	-	-	PUNCT
ejpam-4793	334	14	order	order	NOUN
ejpam-4793	334	15	-	-	PUNCT
ejpam-4793	334	16	ideal	ideal	NOUN
ejpam-4793	334	17	of	of	ADP
ejpam-4793	334	18	m	m	PRON
ejpam-4793	334	19	,	,	PUNCT
ejpam-4793	334	20	for	for	ADP
ejpam-4793	334	21	all	all	DET
ejpam-4793	334	22	α	α	NOUN
ejpam-4793	334	23	,	,	PUNCT
ejpam-4793	334	24	β	β	PROPN
ejpam-4793	334	25	∈	∈	PROPN
ejpam-4793	334	26	γ	γ	X
ejpam-4793	334	27	,	,	PUNCT
ejpam-4793	334	28	αx	αx	ADV
ejpam-4793	334	29	∗	∗	NOUN
ejpam-4793	334	30	βa	βa	INTJ
ejpam-4793	334	31	∈	∈	PROPN
ejpam-4793	335	1	i	i	NOUN
ejpam-4793	335	2	⊆	⊆	NUM
ejpam-4793	335	3	j	j	PROPN
ejpam-4793	335	4	∪	∪	PROPN
ejpam-4793	335	5	i.	i.	PROPN
ejpam-4793	335	6	case	case	NOUN
ejpam-4793	335	7	2	2	NUM
ejpam-4793	335	8	.	.	PUNCT
ejpam-4793	335	9	x	x	SYM
ejpam-4793	336	1	∈	∈	PROPN
ejpam-4793	336	2	i	i	PRON
ejpam-4793	336	3	,	,	PUNCT
ejpam-4793	336	4	a	a	DET
ejpam-4793	336	5	∈	∈	PROPN
ejpam-4793	336	6	j	j	PROPN
ejpam-4793	336	7	.	.	PUNCT
ejpam-4793	337	1	then	then	ADV
ejpam-4793	337	2	,	,	PUNCT
ejpam-4793	337	3	since	since	SCONJ
ejpam-4793	337	4	j	j	PROPN
ejpam-4793	337	5	is	be	AUX
ejpam-4793	337	6	a	a	DET
ejpam-4793	337	7	γ	γ	NOUN
ejpam-4793	337	8	-	-	NOUN
ejpam-4793	337	9	ideal	ideal	NOUN
ejpam-4793	337	10	of	of	ADP
ejpam-4793	337	11	m	m	PROPN
ejpam-4793	337	12	and	and	CCONJ
ejpam-4793	337	13	m	m	PROPN
ejpam-4793	337	14	is	be	AUX
ejpam-4793	337	15	commutative	commutative	ADJ
ejpam-4793	337	16	,	,	PUNCT
ejpam-4793	337	17	for	for	ADP
ejpam-4793	337	18	all	all	DET
ejpam-4793	337	19	α	α	NOUN
ejpam-4793	337	20	,	,	PUNCT
ejpam-4793	337	21	β	β	PROPN
ejpam-4793	337	22	∈	∈	PROPN
ejpam-4793	337	23	γ	γ	X
ejpam-4793	337	24	,	,	PUNCT
ejpam-4793	337	25	we	we	PRON
ejpam-4793	337	26	have	have	VERB
ejpam-4793	337	27	αx	αx	PROPN
ejpam-4793	337	28	∗	∗	NOUN
ejpam-4793	337	29	βa	βa	INTJ
ejpam-4793	338	1	=	=	NOUN
ejpam-4793	338	2	βa	βa	NOUN
ejpam-4793	338	3	∗	∗	NOUN
ejpam-4793	338	4	αx	αx	NOUN
ejpam-4793	339	1	∈	∈	PROPN
ejpam-4793	339	2	j	j	PROPN
ejpam-4793	339	3	⊆	⊆	NUM
ejpam-4793	339	4	j	j	PROPN
ejpam-4793	339	5	∪	∪	PROPN
ejpam-4793	339	6	i.	i.	PROPN
ejpam-4793	339	7	h.	h.	PROPN
ejpam-4793	339	8	sarapuddin	sarapuddin	PROPN
ejpam-4793	339	9	,	,	PUNCT
ejpam-4793	339	10	j.	j.	PROPN
ejpam-4793	339	11	vilela	vilela	PROPN
ejpam-4793	339	12	/	/	SYM
ejpam-4793	339	13	eur	eur	PROPN
ejpam-4793	339	14	.	.	PUNCT
ejpam-4793	340	1	j.	j.	PROPN
ejpam-4793	340	2	pure	pure	PROPN
ejpam-4793	340	3	appl	appl	PROPN
ejpam-4793	340	4	.	.	PROPN
ejpam-4793	340	5	math	math	PROPN
ejpam-4793	340	6	,	,	PUNCT
ejpam-4793	340	7	16	16	NUM
ejpam-4793	340	8	(	(	PUNCT
ejpam-4793	340	9	3	3	NUM
ejpam-4793	340	10	)	)	PUNCT
ejpam-4793	340	11	(	(	PUNCT
ejpam-4793	340	12	2023	2023	NUM
ejpam-4793	340	13	)	)	PUNCT
ejpam-4793	340	14	,	,	PUNCT
ejpam-4793	340	15	1772	1772	NUM
ejpam-4793	340	16	-	-	SYM
ejpam-4793	340	17	1793	1793	NUM
ejpam-4793	340	18	1780	1780	NUM
ejpam-4793	340	19	case	case	NOUN
ejpam-4793	340	20	3	3	NUM
ejpam-4793	340	21	.	.	PUNCT
ejpam-4793	340	22	x	x	SYM
ejpam-4793	340	23	∈	∈	PROPN
ejpam-4793	340	24	j	j	PROPN
ejpam-4793	340	25	,	,	PUNCT
ejpam-4793	340	26	a	a	DET
ejpam-4793	340	27	∈	∈	PROPN
ejpam-4793	340	28	i.	i.	NOUN
ejpam-4793	340	29	then	then	ADV
ejpam-4793	340	30	,	,	PUNCT
ejpam-4793	340	31	since	since	SCONJ
ejpam-4793	340	32	j	j	PROPN
ejpam-4793	340	33	is	be	AUX
ejpam-4793	340	34	a	a	DET
ejpam-4793	340	35	γ	γ	NOUN
ejpam-4793	340	36	-	-	NOUN
ejpam-4793	340	37	ideal	ideal	NOUN
ejpam-4793	340	38	of	of	ADP
ejpam-4793	340	39	m	m	PRON
ejpam-4793	340	40	,	,	PUNCT
ejpam-4793	340	41	for	for	ADP
ejpam-4793	340	42	all	all	DET
ejpam-4793	340	43	α	α	NOUN
ejpam-4793	340	44	,	,	PUNCT
ejpam-4793	340	45	β	β	PROPN
ejpam-4793	340	46	∈	∈	PROPN
ejpam-4793	340	47	γ	γ	X
ejpam-4793	340	48	,	,	PUNCT
ejpam-4793	340	49	we	we	PRON
ejpam-4793	340	50	have	have	VERB
ejpam-4793	340	51	αx	αx	PROPN
ejpam-4793	340	52	∗	∗	NOUN
ejpam-4793	340	53	βa	βa	INTJ
ejpam-4793	341	1	∈	∈	PROPN
ejpam-4793	341	2	j	j	PROPN
ejpam-4793	342	1	⊆	⊆	NUM
ejpam-4793	342	2	j	j	PROPN
ejpam-4793	342	3	∪	∪	PROPN
ejpam-4793	342	4	i.	i.	PROPN
ejpam-4793	342	5	case	case	NOUN
ejpam-4793	342	6	4	4	NUM
ejpam-4793	342	7	.	.	X
ejpam-4793	342	8	x	x	X
ejpam-4793	342	9	,	,	PUNCT
ejpam-4793	342	10	a	a	DET
ejpam-4793	342	11	∈	∈	PROPN
ejpam-4793	342	12	j	j	PROPN
ejpam-4793	342	13	.	.	PUNCT
ejpam-4793	343	1	then	then	ADV
ejpam-4793	343	2	,	,	PUNCT
ejpam-4793	343	3	since	since	SCONJ
ejpam-4793	343	4	j	j	PROPN
ejpam-4793	343	5	is	be	AUX
ejpam-4793	343	6	a	a	DET
ejpam-4793	343	7	γ	γ	NOUN
ejpam-4793	343	8	-	-	NOUN
ejpam-4793	343	9	ideal	ideal	NOUN
ejpam-4793	343	10	of	of	ADP
ejpam-4793	343	11	m	m	PRON
ejpam-4793	343	12	,	,	PUNCT
ejpam-4793	343	13	for	for	ADP
ejpam-4793	343	14	all	all	DET
ejpam-4793	343	15	α	α	NOUN
ejpam-4793	343	16	,	,	PUNCT
ejpam-4793	343	17	β	β	PROPN
ejpam-4793	343	18	∈	∈	PROPN
ejpam-4793	343	19	γ	γ	X
ejpam-4793	343	20	,	,	PUNCT
ejpam-4793	343	21	we	we	PRON
ejpam-4793	343	22	have	have	VERB
ejpam-4793	343	23	αx	αx	PROPN
ejpam-4793	343	24	∗	∗	NOUN
ejpam-4793	343	25	βa	βa	INTJ
ejpam-4793	344	1	∈	∈	PROPN
ejpam-4793	344	2	j	j	PROPN
ejpam-4793	344	3	⊆	⊆	NUM
ejpam-4793	344	4	j	j	PROPN
ejpam-4793	344	5	∪	∪	PROPN
ejpam-4793	344	6	i.	i.	PROPN
ejpam-4793	344	7	thus	thus	ADV
ejpam-4793	344	8	,	,	PUNCT
ejpam-4793	344	9	j	j	PROPN
ejpam-4793	344	10	∪	∪	ADP
ejpam-4793	344	11	i	i	PRON
ejpam-4793	344	12	is	be	AUX
ejpam-4793	344	13	a	a	DET
ejpam-4793	344	14	γ	γ	NOUN
ejpam-4793	344	15	-	-	PUNCT
ejpam-4793	344	16	order	order	NOUN
ejpam-4793	344	17	-	-	PUNCT
ejpam-4793	344	18	ideal	ideal	NOUN
ejpam-4793	344	19	of	of	ADP
ejpam-4793	344	20	m	m	PROPN
ejpam-4793	344	21	.	.	PUNCT
ejpam-4793	345	1	definition	definition	NOUN
ejpam-4793	345	2	11	11	NUM
ejpam-4793	345	3	.	.	PUNCT
ejpam-4793	346	1	let	let	VERB
ejpam-4793	346	2	(	(	PUNCT
ejpam-4793	346	3	m	m	NOUN
ejpam-4793	346	4	,	,	PUNCT
ejpam-4793	346	5	∗	∗	NOUN
ejpam-4793	346	6	)	)	PUNCT
ejpam-4793	346	7	and	and	CCONJ
ejpam-4793	346	8	(	(	PUNCT
ejpam-4793	346	9	n	n	CCONJ
ejpam-4793	346	10	,	,	PUNCT
ejpam-4793	346	11	·	·	PUNCT
ejpam-4793	346	12	)	)	PUNCT
ejpam-4793	346	13	be	be	AUX
ejpam-4793	346	14	γ	γ	NOUN
ejpam-4793	346	15	-	-	PUNCT
ejpam-4793	346	16	monoids	monoid	NOUN
ejpam-4793	346	17	and	and	CCONJ
ejpam-4793	346	18	φ	φ	NOUN
ejpam-4793	346	19	:	:	PUNCT
ejpam-4793	346	20	m	m	VERB
ejpam-4793	346	21	→	→	SYM
ejpam-4793	346	22	n	n	CCONJ
ejpam-4793	346	23	a	a	DET
ejpam-4793	346	24	γ	γ	PROPN
ejpam-4793	346	25	-	-	PUNCT
ejpam-4793	346	26	monoid	monoid	NOUN
ejpam-4793	346	27	homomorphism	homomorphism	NOUN
ejpam-4793	346	28	.	.	PUNCT
ejpam-4793	347	1	the	the	DET
ejpam-4793	347	2	kernel	kernel	NOUN
ejpam-4793	347	3	of	of	ADP
ejpam-4793	347	4	φ	φ	PROPN
ejpam-4793	347	5	is	be	AUX
ejpam-4793	347	6	denoted	denote	VERB
ejpam-4793	347	7	and	and	CCONJ
ejpam-4793	347	8	defined	define	VERB
ejpam-4793	347	9	by	by	ADP
ejpam-4793	347	10	kerφ	kerφ	PROPN
ejpam-4793	347	11	=	=	PUNCT
ejpam-4793	347	12	{	{	PUNCT
ejpam-4793	347	13	m	m	VERB
ejpam-4793	347	14	∈	∈	ADJ
ejpam-4793	347	15	m	m	VERB
ejpam-4793	347	16	:	:	PUNCT
ejpam-4793	347	17	φ(m	φ(m	ADJ
ejpam-4793	347	18	)	)	PUNCT
ejpam-4793	347	19	=	=	SYM
ejpam-4793	347	20	1n	1n	NUM
ejpam-4793	347	21	}	}	PUNCT
ejpam-4793	347	22	.	.	PUNCT
ejpam-4793	348	1	proposition	proposition	NOUN
ejpam-4793	348	2	2	2	NUM
ejpam-4793	348	3	.	.	PUNCT
ejpam-4793	349	1	let	let	VERB
ejpam-4793	349	2	(	(	PUNCT
ejpam-4793	349	3	m	m	NOUN
ejpam-4793	349	4	,	,	PUNCT
ejpam-4793	349	5	∗	∗	NOUN
ejpam-4793	349	6	)	)	PUNCT
ejpam-4793	349	7	and	and	CCONJ
ejpam-4793	349	8	(	(	PUNCT
ejpam-4793	349	9	n	n	CCONJ
ejpam-4793	349	10	,	,	PUNCT
ejpam-4793	349	11	·	·	PUNCT
ejpam-4793	349	12	)	)	PUNCT
ejpam-4793	349	13	be	be	AUX
ejpam-4793	349	14	γ	γ	NOUN
ejpam-4793	349	15	-	-	PUNCT
ejpam-4793	349	16	monoids	monoid	NOUN
ejpam-4793	349	17	and	and	CCONJ
ejpam-4793	349	18	φ	φ	NOUN
ejpam-4793	349	19	:	:	PUNCT
ejpam-4793	349	20	m	m	VERB
ejpam-4793	349	21	→	→	SYM
ejpam-4793	349	22	n	n	CCONJ
ejpam-4793	349	23	a	a	DET
ejpam-4793	349	24	γ	γ	PROPN
ejpam-4793	349	25	-	-	PUNCT
ejpam-4793	349	26	monoid	monoid	NOUN
ejpam-4793	349	27	homomorphism	homomorphism	NOUN
ejpam-4793	349	28	.	.	PUNCT
ejpam-4793	350	1	(	(	PUNCT
ejpam-4793	350	2	i	i	NOUN
ejpam-4793	350	3	)	)	PUNCT
ejpam-4793	350	4	if	if	SCONJ
ejpam-4793	350	5	φ	φ	PROPN
ejpam-4793	350	6	is	be	AUX
ejpam-4793	350	7	surjective	surjective	ADJ
ejpam-4793	351	1	and	and	CCONJ
ejpam-4793	351	2	i	i	PRON
ejpam-4793	351	3	is	be	AUX
ejpam-4793	351	4	a	a	DET
ejpam-4793	351	5	γ	γ	NOUN
ejpam-4793	351	6	-	-	NOUN
ejpam-4793	351	7	ideal	ideal	NOUN
ejpam-4793	351	8	of	of	ADP
ejpam-4793	351	9	m	m	PROPN
ejpam-4793	351	10	,	,	PUNCT
ejpam-4793	351	11	then	then	ADV
ejpam-4793	351	12	φ(i	φ(i	NUM
ejpam-4793	351	13	)	)	PUNCT
ejpam-4793	352	1	is	be	AUX
ejpam-4793	352	2	a	a	DET
ejpam-4793	352	3	γ	γ	NOUN
ejpam-4793	352	4	-	-	NOUN
ejpam-4793	352	5	ideal	ideal	NOUN
ejpam-4793	352	6	of	of	ADP
ejpam-4793	352	7	n	n	PROPN
ejpam-4793	352	8	.	.	PUNCT
ejpam-4793	353	1	(	(	PUNCT
ejpam-4793	353	2	ii	ii	NOUN
ejpam-4793	353	3	)	)	PUNCT
ejpam-4793	353	4	if	if	SCONJ
ejpam-4793	353	5	j	j	PROPN
ejpam-4793	353	6	is	be	AUX
ejpam-4793	353	7	a	a	DET
ejpam-4793	353	8	γ	γ	NOUN
ejpam-4793	353	9	-	-	NOUN
ejpam-4793	353	10	ideal	ideal	NOUN
ejpam-4793	353	11	of	of	ADP
ejpam-4793	353	12	n	n	PROPN
ejpam-4793	353	13	,	,	PUNCT
ejpam-4793	353	14	then	then	ADV
ejpam-4793	353	15	φ−1(j	φ−1(j	PROPN
ejpam-4793	353	16	)	)	PUNCT
ejpam-4793	353	17	is	be	AUX
ejpam-4793	353	18	a	a	DET
ejpam-4793	353	19	γ	γ	NOUN
ejpam-4793	353	20	-	-	NOUN
ejpam-4793	353	21	ideal	ideal	NOUN
ejpam-4793	353	22	of	of	ADP
ejpam-4793	353	23	m	m	PROPN
ejpam-4793	353	24	.	.	PUNCT
ejpam-4793	354	1	proof	proof	NOUN
ejpam-4793	354	2	.	.	PUNCT
ejpam-4793	355	1	let	let	VERB
ejpam-4793	355	2	φ	φ	NOUN
ejpam-4793	355	3	:	:	PUNCT
ejpam-4793	355	4	m	m	PROPN
ejpam-4793	355	5	→	→	SYM
ejpam-4793	355	6	n	n	CCONJ
ejpam-4793	355	7	be	be	AUX
ejpam-4793	355	8	a	a	DET
ejpam-4793	355	9	γ	γ	NOUN
ejpam-4793	355	10	-	-	PUNCT
ejpam-4793	355	11	monoid	monoid	NOUN
ejpam-4793	355	12	homomorphism	homomorphism	NOUN
ejpam-4793	355	13	.	.	PUNCT
ejpam-4793	356	1	(	(	PUNCT
ejpam-4793	356	2	i	i	NOUN
ejpam-4793	356	3	)	)	PUNCT
ejpam-4793	356	4	let	let	VERB
ejpam-4793	356	5	x	x	X
ejpam-4793	356	6	∈	∈	PROPN
ejpam-4793	356	7	φ(i	φ(i	PROPN
ejpam-4793	356	8	)	)	PUNCT
ejpam-4793	357	1	and	and	CCONJ
ejpam-4793	357	2	z	z	NOUN
ejpam-4793	357	3	∈	∈	PROPN
ejpam-4793	357	4	n	n	ADV
ejpam-4793	357	5	.	.	PUNCT
ejpam-4793	358	1	since	since	SCONJ
ejpam-4793	358	2	φ	φ	PROPN
ejpam-4793	358	3	is	be	AUX
ejpam-4793	358	4	surjective	surjective	ADJ
ejpam-4793	358	5	,	,	PUNCT
ejpam-4793	358	6	z	z	NOUN
ejpam-4793	358	7	=	=	PUNCT
ejpam-4793	358	8	φ(n	φ(n	NOUN
ejpam-4793	358	9	)	)	PUNCT
ejpam-4793	358	10	for	for	ADP
ejpam-4793	358	11	some	some	DET
ejpam-4793	358	12	n	n	PRON
ejpam-4793	358	13	∈	∈	NOUN
ejpam-4793	358	14	m	m	NOUN
ejpam-4793	358	15	and	and	CCONJ
ejpam-4793	358	16	x	x	X
ejpam-4793	358	17	=	=	SYM
ejpam-4793	358	18	φ(y	φ(y	PROPN
ejpam-4793	358	19	)	)	PUNCT
ejpam-4793	358	20	for	for	ADP
ejpam-4793	358	21	some	some	DET
ejpam-4793	358	22	y	y	PROPN
ejpam-4793	358	23	∈	∈	PROPN
ejpam-4793	358	24	i.	i.	NOUN
ejpam-4793	358	25	then	then	ADV
ejpam-4793	358	26	for	for	ADP
ejpam-4793	358	27	all	all	DET
ejpam-4793	358	28	α	α	NOUN
ejpam-4793	358	29	,	,	PUNCT
ejpam-4793	358	30	β	β	PROPN
ejpam-4793	358	31	∈	∈	PROPN
ejpam-4793	358	32	γ	γ	X
ejpam-4793	358	33	,	,	PUNCT
ejpam-4793	358	34	αx	αx	ADV
ejpam-4793	358	35	∗	∗	NOUN
ejpam-4793	358	36	βz	βz	ADP
ejpam-4793	358	37	=	=	SYM
ejpam-4793	358	38	αφ(y	αφ(y	NUM
ejpam-4793	358	39	)	)	PUNCT
ejpam-4793	358	40	·	·	PUNCT
ejpam-4793	359	1	βφ(n	βφ(n	PUNCT
ejpam-4793	359	2	)	)	PUNCT
ejpam-4793	359	3	=	=	PUNCT
ejpam-4793	359	4	φ(αy	φ(αy	X
ejpam-4793	359	5	)	)	PUNCT
ejpam-4793	359	6	·	·	PUNCT
ejpam-4793	359	7	φ(βn	φ(βn	NUM
ejpam-4793	359	8	)	)	PUNCT
ejpam-4793	359	9	=	=	PUNCT
ejpam-4793	359	10	φ(αy	φ(αy	X
ejpam-4793	359	11	∗	∗	NOUN
ejpam-4793	359	12	βn	βn	NOUN
ejpam-4793	359	13	)	)	PUNCT
ejpam-4793	359	14	.	.	PUNCT
ejpam-4793	360	1	since	since	SCONJ
ejpam-4793	360	2	i	i	PRON
ejpam-4793	360	3	is	be	AUX
ejpam-4793	360	4	a	a	DET
ejpam-4793	360	5	γ	γ	NOUN
ejpam-4793	360	6	-	-	NOUN
ejpam-4793	360	7	ideal	ideal	NOUN
ejpam-4793	360	8	of	of	ADP
ejpam-4793	360	9	m	m	PROPN
ejpam-4793	360	10	,	,	PUNCT
ejpam-4793	360	11	αy	αy	ADP
ejpam-4793	360	12	∗	∗	NOUN
ejpam-4793	360	13	βn	βn	PUNCT
ejpam-4793	361	1	∈	∈	PROPN
ejpam-4793	362	1	i	i	PRON
ejpam-4793	362	2	,	,	PUNCT
ejpam-4793	362	3	so	so	ADV
ejpam-4793	362	4	,	,	PUNCT
ejpam-4793	362	5	αx	αx	ADV
ejpam-4793	362	6	∗	∗	VERB
ejpam-4793	362	7	βz	βz	ADP
ejpam-4793	362	8	∈	∈	PROPN
ejpam-4793	362	9	φ(i	φ(i	PROPN
ejpam-4793	362	10	)	)	PUNCT
ejpam-4793	362	11	.	.	PUNCT
ejpam-4793	363	1	similarly	similarly	ADV
ejpam-4793	363	2	,	,	PUNCT
ejpam-4793	363	3	for	for	ADP
ejpam-4793	363	4	all	all	DET
ejpam-4793	363	5	α	α	NOUN
ejpam-4793	363	6	,	,	PUNCT
ejpam-4793	363	7	β	β	PROPN
ejpam-4793	363	8	∈	∈	PROPN
ejpam-4793	363	9	γ	γ	X
ejpam-4793	363	10	,	,	PUNCT
ejpam-4793	363	11	αz	αz	ADP
ejpam-4793	363	12	∗	∗	NOUN
ejpam-4793	363	13	βx	βx	ADP
ejpam-4793	363	14	∈	∈	PROPN
ejpam-4793	363	15	φ(i	φ(i	PROPN
ejpam-4793	363	16	)	)	PUNCT
ejpam-4793	363	17	.	.	PUNCT
ejpam-4793	364	1	therefore	therefore	ADV
ejpam-4793	364	2	,	,	PUNCT
ejpam-4793	364	3	φ(i	φ(i	PROPN
ejpam-4793	364	4	)	)	PUNCT
ejpam-4793	364	5	is	be	AUX
ejpam-4793	364	6	a	a	DET
ejpam-4793	364	7	γ	γ	NOUN
ejpam-4793	364	8	-	-	NOUN
ejpam-4793	364	9	ideal	ideal	NOUN
ejpam-4793	364	10	of	of	ADP
ejpam-4793	364	11	n	n	PROPN
ejpam-4793	364	12	.	.	PUNCT
ejpam-4793	365	1	(	(	PUNCT
ejpam-4793	365	2	ii	ii	NOUN
ejpam-4793	365	3	)	)	PUNCT
ejpam-4793	365	4	let	let	VERB
ejpam-4793	365	5	y	y	PROPN
ejpam-4793	365	6	∈	∈	PROPN
ejpam-4793	365	7	φ−1(j	φ−1(j	PROPN
ejpam-4793	365	8	)	)	PUNCT
ejpam-4793	365	9	and	and	CCONJ
ejpam-4793	365	10	m	m	PROPN
ejpam-4793	365	11	∈	∈	ADJ
ejpam-4793	365	12	m	m	VERB
ejpam-4793	365	13	.	.	PUNCT
ejpam-4793	366	1	then	then	ADV
ejpam-4793	366	2	φ(y	φ(y	VERB
ejpam-4793	366	3	)	)	PUNCT
ejpam-4793	366	4	∈	∈	PROPN
ejpam-4793	366	5	j	j	PROPN
ejpam-4793	366	6	and	and	CCONJ
ejpam-4793	366	7	φ(m	φ(m	PROPN
ejpam-4793	366	8	)	)	PUNCT
ejpam-4793	366	9	∈	∈	PROPN
ejpam-4793	366	10	n	n	NOUN
ejpam-4793	366	11	.	.	PUNCT
ejpam-4793	367	1	thus	thus	ADV
ejpam-4793	367	2	,	,	PUNCT
ejpam-4793	367	3	for	for	ADP
ejpam-4793	367	4	all	all	DET
ejpam-4793	367	5	α	α	NOUN
ejpam-4793	367	6	,	,	PUNCT
ejpam-4793	367	7	β	β	PROPN
ejpam-4793	367	8	∈	∈	PROPN
ejpam-4793	367	9	γ	γ	X
ejpam-4793	367	10	,	,	PUNCT
ejpam-4793	367	11	φ(αy	φ(αy	ADV
ejpam-4793	367	12	∗	∗	NOUN
ejpam-4793	367	13	βm	βm	NOUN
ejpam-4793	367	14	)	)	PUNCT
ejpam-4793	367	15	=	=	PUNCT
ejpam-4793	367	16	φ(αy	φ(αy	X
ejpam-4793	367	17	)	)	PUNCT
ejpam-4793	367	18	·	·	PUNCT
ejpam-4793	367	19	φ(βm	φ(βm	X
ejpam-4793	367	20	)	)	PUNCT
ejpam-4793	367	21	=	=	SYM
ejpam-4793	367	22	αφ(y	αφ(y	NUM
ejpam-4793	367	23	)	)	PUNCT
ejpam-4793	367	24	·	·	PUNCT
ejpam-4793	367	25	βφ(m	βφ(m	NUM
ejpam-4793	367	26	)	)	PUNCT
ejpam-4793	368	1	∈	∈	PROPN
ejpam-4793	368	2	j	j	PROPN
ejpam-4793	368	3	,	,	PUNCT
ejpam-4793	368	4	since	since	SCONJ
ejpam-4793	368	5	j	j	PROPN
ejpam-4793	368	6	is	be	AUX
ejpam-4793	368	7	a	a	DET
ejpam-4793	368	8	γ	γ	NOUN
ejpam-4793	368	9	-	-	NOUN
ejpam-4793	368	10	ideal	ideal	NOUN
ejpam-4793	368	11	of	of	ADP
ejpam-4793	368	12	n	n	PROPN
ejpam-4793	368	13	.	.	PUNCT
ejpam-4793	369	1	hence	hence	ADV
ejpam-4793	369	2	,	,	PUNCT
ejpam-4793	369	3	αy	αy	ADP
ejpam-4793	369	4	∗	∗	NOUN
ejpam-4793	369	5	βm	βm	PROPN
ejpam-4793	369	6	∈	∈	PROPN
ejpam-4793	369	7	φ−1(j	φ−1(j	PROPN
ejpam-4793	369	8	)	)	PUNCT
ejpam-4793	369	9	for	for	ADP
ejpam-4793	369	10	all	all	DET
ejpam-4793	369	11	α	α	NOUN
ejpam-4793	369	12	,	,	PUNCT
ejpam-4793	369	13	β	β	PROPN
ejpam-4793	369	14	∈	∈	PROPN
ejpam-4793	369	15	γ	γ	X
ejpam-4793	369	16	.	.	PROPN
ejpam-4793	369	17	similarly	similarly	ADV
ejpam-4793	369	18	,	,	PUNCT
ejpam-4793	369	19	for	for	ADP
ejpam-4793	369	20	all	all	DET
ejpam-4793	369	21	α	α	NOUN
ejpam-4793	369	22	,	,	PUNCT
ejpam-4793	369	23	β	β	PROPN
ejpam-4793	369	24	∈	∈	PROPN
ejpam-4793	369	25	γ	γ	X
ejpam-4793	369	26	,	,	PUNCT
ejpam-4793	369	27	αm	αm	NOUN
ejpam-4793	369	28	∗	∗	NOUN
ejpam-4793	369	29	βy	βy	PRON
ejpam-4793	369	30	∈	∈	PROPN
ejpam-4793	369	31	φ−1(j	φ−1(j	PROPN
ejpam-4793	369	32	)	)	PUNCT
ejpam-4793	369	33	.	.	PUNCT
ejpam-4793	370	1	therefore	therefore	ADV
ejpam-4793	370	2	,	,	PUNCT
ejpam-4793	370	3	φ−1(j	φ−1(j	PROPN
ejpam-4793	370	4	)	)	PUNCT
ejpam-4793	370	5	is	be	AUX
ejpam-4793	370	6	a	a	DET
ejpam-4793	370	7	γ	γ	NOUN
ejpam-4793	370	8	-	-	NOUN
ejpam-4793	370	9	ideal	ideal	NOUN
ejpam-4793	370	10	of	of	ADP
ejpam-4793	370	11	m	m	PROPN
ejpam-4793	370	12	.	.	PUNCT
ejpam-4793	370	13	example	example	NOUN
ejpam-4793	371	1	10	10	NUM
ejpam-4793	371	2	.	.	PUNCT
ejpam-4793	372	1	consider	consider	VERB
ejpam-4793	372	2	the	the	DET
ejpam-4793	372	3	γ	γ	NOUN
ejpam-4793	372	4	-	-	PUNCT
ejpam-4793	372	5	monoid	monoid	NOUN
ejpam-4793	372	6	homomorphism	homomorphism	PROPN
ejpam-4793	372	7	φ	φ	X
ejpam-4793	372	8	:	:	PUNCT
ejpam-4793	372	9	m	m	VERB
ejpam-4793	372	10	→	→	SYM
ejpam-4793	372	11	n	n	PRON
ejpam-4793	372	12	defined	define	VERB
ejpam-4793	372	13	by	by	ADP
ejpam-4793	372	14	φ(x	φ(x	NOUN
ejpam-4793	372	15	)	)	PUNCT
ejpam-4793	373	1	=	=	SYM
ejpam-4793	373	2	bx	bx	PROPN
ejpam-4793	373	3	,	,	PUNCT
ejpam-4793	373	4	where	where	SCONJ
ejpam-4793	373	5	b	b	X
ejpam-4793	373	6	̸=	̸=	PROPN
ejpam-4793	373	7	0	0	NUM
ejpam-4793	373	8	in	in	ADP
ejpam-4793	373	9	example	example	NOUN
ejpam-4793	373	10	6	6	NUM
ejpam-4793	373	11	.	.	X
ejpam-4793	373	12	note	note	VERB
ejpam-4793	373	13	that	that	SCONJ
ejpam-4793	373	14	kerφ	kerφ	PROPN
ejpam-4793	373	15	=	=	PUNCT
ejpam-4793	373	16	{	{	PUNCT
ejpam-4793	373	17	x	x	SYM
ejpam-4793	373	18	∈	∈	NOUN
ejpam-4793	373	19	m	m	VERB
ejpam-4793	373	20	:	:	PUNCT
ejpam-4793	373	21	φ(x	φ(x	VERB
ejpam-4793	373	22	)	)	PUNCT
ejpam-4793	373	23	=	=	SYM
ejpam-4793	374	1	1	1	X
ejpam-4793	374	2	}	}	PUNCT
ejpam-4793	374	3	=	=	PRON
ejpam-4793	374	4	{	{	PUNCT
ejpam-4793	374	5	x	x	SYM
ejpam-4793	374	6	∈	∈	NOUN
ejpam-4793	374	7	m	m	VERB
ejpam-4793	374	8	:	:	PUNCT
ejpam-4793	374	9	bx	bx	X
ejpam-4793	374	10	=	=	NOUN
ejpam-4793	374	11	1	1	X
ejpam-4793	374	12	}	}	PUNCT
ejpam-4793	374	13	=	=	PRON
ejpam-4793	374	14	{	{	PUNCT
ejpam-4793	374	15	x	x	SYM
ejpam-4793	374	16	∈	∈	NOUN
ejpam-4793	374	17	m	m	VERB
ejpam-4793	374	18	:	:	PUNCT
ejpam-4793	374	19	b	b	X
ejpam-4793	374	20	=	=	SYM
ejpam-4793	374	21	1	1	NUM
ejpam-4793	374	22	or	or	CCONJ
ejpam-4793	374	23	x	x	SYM
ejpam-4793	374	24	=	=	NOUN
ejpam-4793	374	25	0	0	NUM
ejpam-4793	374	26	}	}	PUNCT
ejpam-4793	374	27	.	.	PUNCT
ejpam-4793	375	1	take	take	VERB
ejpam-4793	375	2	x	x	PUNCT
ejpam-4793	375	3	=	=	SYM
ejpam-4793	375	4	0	0	NUM
ejpam-4793	375	5	∈	∈	PROPN
ejpam-4793	375	6	kerφ	kerφ	PROPN
ejpam-4793	375	7	,	,	PUNCT
ejpam-4793	375	8	m	m	VERB
ejpam-4793	375	9	=	=	SYM
ejpam-4793	375	10	2	2	NUM
ejpam-4793	375	11	∈	∈	NOUN
ejpam-4793	375	12	m	m	NOUN
ejpam-4793	375	13	,	,	PUNCT
ejpam-4793	375	14	and	and	CCONJ
ejpam-4793	375	15	b	b	X
ejpam-4793	375	16	=	=	SYM
ejpam-4793	375	17	2	2	NUM
ejpam-4793	375	18	.	.	PUNCT
ejpam-4793	375	19	then	then	ADV
ejpam-4793	375	20	for	for	ADP
ejpam-4793	375	21	all	all	DET
ejpam-4793	375	22	α	α	NOUN
ejpam-4793	375	23	,	,	PUNCT
ejpam-4793	375	24	β	β	PROPN
ejpam-4793	375	25	∈	∈	PROPN
ejpam-4793	375	26	γ	γ	X
ejpam-4793	375	27	,	,	PUNCT
ejpam-4793	375	28	φ(αx	φ(αx	X
ejpam-4793	375	29	+	+	CCONJ
ejpam-4793	375	30	βm	βm	ADJ
ejpam-4793	375	31	)	)	PUNCT
ejpam-4793	375	32	=	=	SYM
ejpam-4793	375	33	φ(α0	φ(α0	X
ejpam-4793	376	1	+	+	SYM
ejpam-4793	376	2	β2	β2	VERB
ejpam-4793	376	3	)	)	PUNCT
ejpam-4793	376	4	=	=	PUNCT
ejpam-4793	377	1	φ(0	φ(0	ADJ
ejpam-4793	378	1	+	+	CCONJ
ejpam-4793	378	2	2	2	X
ejpam-4793	378	3	)	)	PUNCT
ejpam-4793	378	4	=	=	SYM
ejpam-4793	378	5	φ(2	φ(2	PROPN
ejpam-4793	378	6	)	)	PUNCT
ejpam-4793	379	1	=	=	SYM
ejpam-4793	380	1	22	22	NUM
ejpam-4793	380	2	̸=	̸=	PROPN
ejpam-4793	380	3	1	1	NUM
ejpam-4793	380	4	.	.	PUNCT
ejpam-4793	381	1	this	this	PRON
ejpam-4793	381	2	implies	imply	VERB
ejpam-4793	381	3	that	that	SCONJ
ejpam-4793	381	4	αx	αx	PRON
ejpam-4793	381	5	+	+	CCONJ
ejpam-4793	381	6	βm	βm	VERB
ejpam-4793	381	7	/∈	/∈	PROPN
ejpam-4793	381	8	kerφ	kerφ	PROPN
ejpam-4793	381	9	.	.	PUNCT
ejpam-4793	382	1	by	by	ADP
ejpam-4793	382	2	definition	definition	NOUN
ejpam-4793	382	3	10	10	NUM
ejpam-4793	382	4	,	,	PUNCT
ejpam-4793	382	5	kerφ	kerφ	PROPN
ejpam-4793	382	6	is	be	AUX
ejpam-4793	382	7	not	not	PART
ejpam-4793	382	8	a	a	DET
ejpam-4793	382	9	γ	γ	NOUN
ejpam-4793	382	10	-	-	NOUN
ejpam-4793	382	11	ideal	ideal	NOUN
ejpam-4793	382	12	of	of	ADP
ejpam-4793	382	13	m	m	PROPN
ejpam-4793	382	14	.	.	PUNCT
ejpam-4793	383	1	remark	remark	PROPN
ejpam-4793	383	2	6	6	NUM
ejpam-4793	383	3	.	.	PUNCT
ejpam-4793	384	1	for	for	ADP
ejpam-4793	384	2	any	any	DET
ejpam-4793	384	3	γ	γ	NOUN
ejpam-4793	384	4	-	-	PUNCT
ejpam-4793	384	5	monoids	monoid	NOUN
ejpam-4793	384	6	m	m	VERB
ejpam-4793	384	7	and	and	CCONJ
ejpam-4793	384	8	n	n	CCONJ
ejpam-4793	384	9	,	,	PUNCT
ejpam-4793	384	10	the	the	DET
ejpam-4793	384	11	kernel	kernel	NOUN
ejpam-4793	384	12	of	of	ADP
ejpam-4793	384	13	a	a	DET
ejpam-4793	384	14	γ	γ	X
ejpam-4793	384	15	-	-	PUNCT
ejpam-4793	384	16	monoid	monoid	NOUN
ejpam-4793	384	17	homomorphism	homomorphism	PROPN
ejpam-4793	384	18	φ	φ	X
ejpam-4793	384	19	:	:	PUNCT
ejpam-4793	384	20	m	m	PROPN
ejpam-4793	384	21	→	→	SYM
ejpam-4793	384	22	n	n	X
ejpam-4793	384	23	is	be	AUX
ejpam-4793	384	24	not	not	PART
ejpam-4793	384	25	necessarily	necessarily	ADV
ejpam-4793	384	26	a	a	DET
ejpam-4793	384	27	γ	γ	NOUN
ejpam-4793	384	28	-	-	NOUN
ejpam-4793	384	29	ideal	ideal	NOUN
ejpam-4793	384	30	of	of	ADP
ejpam-4793	384	31	m	m	PROPN
ejpam-4793	384	32	.	.	PUNCT
ejpam-4793	385	1	proposition	proposition	NOUN
ejpam-4793	385	2	3	3	X
ejpam-4793	385	3	.	.	PUNCT
ejpam-4793	386	1	let	let	VERB
ejpam-4793	386	2	(	(	PUNCT
ejpam-4793	386	3	m	m	NOUN
ejpam-4793	386	4	,	,	PUNCT
ejpam-4793	386	5	∗	∗	NOUN
ejpam-4793	386	6	)	)	PUNCT
ejpam-4793	386	7	and	and	CCONJ
ejpam-4793	386	8	(	(	PUNCT
ejpam-4793	386	9	n	n	CCONJ
ejpam-4793	386	10	,	,	PUNCT
ejpam-4793	386	11	·	·	PUNCT
ejpam-4793	386	12	)	)	PUNCT
ejpam-4793	386	13	be	be	AUX
ejpam-4793	386	14	γ	γ	NOUN
ejpam-4793	386	15	-	-	PUNCT
ejpam-4793	386	16	monoids	monoid	NOUN
ejpam-4793	386	17	and	and	CCONJ
ejpam-4793	386	18	φ	φ	NOUN
ejpam-4793	386	19	:	:	PUNCT
ejpam-4793	386	20	m	m	VERB
ejpam-4793	386	21	→	→	SYM
ejpam-4793	386	22	n	n	CCONJ
ejpam-4793	386	23	a	a	DET
ejpam-4793	386	24	γ	γ	PROPN
ejpam-4793	386	25	-	-	PUNCT
ejpam-4793	386	26	monoid	monoid	NOUN
ejpam-4793	386	27	homomorphism	homomorphism	NOUN
ejpam-4793	386	28	.	.	PUNCT
ejpam-4793	387	1	then	then	ADV
ejpam-4793	387	2	kerφ	kerφ	PROPN
ejpam-4793	387	3	is	be	AUX
ejpam-4793	387	4	a	a	DET
ejpam-4793	387	5	γ	γ	NOUN
ejpam-4793	387	6	-	-	NOUN
ejpam-4793	387	7	ideal	ideal	NOUN
ejpam-4793	387	8	of	of	ADP
ejpam-4793	387	9	m	m	PRON
ejpam-4793	387	10	if	if	SCONJ
ejpam-4793	388	1	and	and	CCONJ
ejpam-4793	388	2	only	only	ADV
ejpam-4793	388	3	if	if	SCONJ
ejpam-4793	388	4	kerφ	kerφ	PROPN
ejpam-4793	388	5	=	=	PUNCT
ejpam-4793	388	6	m.	m.	NOUN
ejpam-4793	388	7	h.	h.	PROPN
ejpam-4793	388	8	sarapuddin	sarapuddin	PROPN
ejpam-4793	388	9	,	,	PUNCT
ejpam-4793	388	10	j.	j.	PROPN
ejpam-4793	388	11	vilela	vilela	PROPN
ejpam-4793	388	12	/	/	SYM
ejpam-4793	388	13	eur	eur	PROPN
ejpam-4793	388	14	.	.	PUNCT
ejpam-4793	389	1	j.	j.	PROPN
ejpam-4793	389	2	pure	pure	PROPN
ejpam-4793	389	3	appl	appl	PROPN
ejpam-4793	389	4	.	.	PROPN
ejpam-4793	389	5	math	math	PROPN
ejpam-4793	389	6	,	,	PUNCT
ejpam-4793	389	7	16	16	NUM
ejpam-4793	389	8	(	(	PUNCT
ejpam-4793	389	9	3	3	NUM
ejpam-4793	389	10	)	)	PUNCT
ejpam-4793	389	11	(	(	PUNCT
ejpam-4793	389	12	2023	2023	NUM
ejpam-4793	389	13	)	)	PUNCT
ejpam-4793	389	14	,	,	PUNCT
ejpam-4793	389	15	1772	1772	NUM
ejpam-4793	389	16	-	-	SYM
ejpam-4793	389	17	1793	1793	NUM
ejpam-4793	389	18	1781	1781	NUM
ejpam-4793	389	19	proof	proof	NOUN
ejpam-4793	389	20	.	.	PUNCT
ejpam-4793	390	1	let	let	VERB
ejpam-4793	390	2	φ	φ	NOUN
ejpam-4793	390	3	:	:	PUNCT
ejpam-4793	390	4	m	m	PROPN
ejpam-4793	390	5	→	→	SYM
ejpam-4793	390	6	n	n	CCONJ
ejpam-4793	390	7	be	be	AUX
ejpam-4793	390	8	a	a	DET
ejpam-4793	390	9	γ	γ	NOUN
ejpam-4793	390	10	-	-	PUNCT
ejpam-4793	390	11	monoid	monoid	NOUN
ejpam-4793	390	12	homomorphism	homomorphism	NOUN
ejpam-4793	390	13	.	.	PUNCT
ejpam-4793	391	1	then	then	ADV
ejpam-4793	391	2	1	1	NUM
ejpam-4793	391	3	m	m	NOUN
ejpam-4793	391	4	∈	∈	PROPN
ejpam-4793	391	5	kerφ	kerφ	PROPN
ejpam-4793	391	6	.	.	PUNCT
ejpam-4793	392	1	suppose	suppose	VERB
ejpam-4793	392	2	that	that	SCONJ
ejpam-4793	392	3	kerφ	kerφ	PROPN
ejpam-4793	392	4	is	be	AUX
ejpam-4793	392	5	a	a	DET
ejpam-4793	392	6	γ	γ	NOUN
ejpam-4793	392	7	-	-	PUNCT
ejpam-4793	392	8	ideal	ideal	ADJ
ejpam-4793	392	9	ofm	ofm	PROPN
ejpam-4793	392	10	.	.	PUNCT
ejpam-4793	393	1	then	then	ADV
ejpam-4793	393	2	by	by	ADP
ejpam-4793	393	3	lemma	lemma	PROPN
ejpam-4793	393	4	2	2	NUM
ejpam-4793	393	5	,	,	PUNCT
ejpam-4793	393	6	kerφ	kerφ	PROPN
ejpam-4793	393	7	=	=	PUNCT
ejpam-4793	394	1	m	m	PROPN
ejpam-4793	394	2	.	.	PUNCT
ejpam-4793	395	1	now	now	ADV
ejpam-4793	395	2	,	,	PUNCT
ejpam-4793	395	3	suppose	suppose	VERB
ejpam-4793	395	4	that	that	SCONJ
ejpam-4793	395	5	kerφ	kerφ	PROPN
ejpam-4793	395	6	=	=	PROPN
ejpam-4793	395	7	m	m	VERB
ejpam-4793	395	8	.	.	PUNCT
ejpam-4793	396	1	then	then	ADV
ejpam-4793	396	2	by	by	ADP
ejpam-4793	396	3	remark	remark	NOUN
ejpam-4793	396	4	4(i	4(i	NUM
ejpam-4793	396	5	)	)	PUNCT
ejpam-4793	396	6	,	,	PUNCT
ejpam-4793	396	7	kerφ	kerφ	PROPN
ejpam-4793	396	8	is	be	AUX
ejpam-4793	396	9	a	a	DET
ejpam-4793	396	10	γ	γ	NOUN
ejpam-4793	396	11	-	-	NOUN
ejpam-4793	396	12	ideal	ideal	NOUN
ejpam-4793	396	13	of	of	ADP
ejpam-4793	396	14	m	m	PRON
ejpam-4793	396	15	.	.	PUNCT
ejpam-4793	397	1	by	by	ADP
ejpam-4793	397	2	proposition	proposition	NOUN
ejpam-4793	397	3	3	3	NUM
ejpam-4793	397	4	,	,	PUNCT
ejpam-4793	397	5	kerφ	kerφ	PROPN
ejpam-4793	397	6	is	be	AUX
ejpam-4793	397	7	a	a	DET
ejpam-4793	397	8	γ	γ	NOUN
ejpam-4793	397	9	-	-	PUNCT
ejpam-4793	397	10	ideal	ideal	NOUN
ejpam-4793	397	11	if	if	SCONJ
ejpam-4793	398	1	and	and	CCONJ
ejpam-4793	398	2	only	only	ADV
ejpam-4793	398	3	if	if	SCONJ
ejpam-4793	398	4	φ	φ	PROPN
ejpam-4793	398	5	is	be	AUX
ejpam-4793	398	6	a	a	DET
ejpam-4793	398	7	zero	zero	NUM
ejpam-4793	398	8	map	map	NOUN
ejpam-4793	398	9	.	.	PUNCT
ejpam-4793	399	1	thus	thus	ADV
ejpam-4793	399	2	,	,	PUNCT
ejpam-4793	399	3	isomorphism	isomorphism	NOUN
ejpam-4793	399	4	theorems	theorem	NOUN
ejpam-4793	399	5	via	via	ADP
ejpam-4793	399	6	γ	γ	NOUN
ejpam-4793	399	7	-	-	PUNCT
ejpam-4793	399	8	ideals	ideal	NOUN
ejpam-4793	399	9	are	be	AUX
ejpam-4793	399	10	irrelevant	irrelevant	ADJ
ejpam-4793	399	11	.	.	PUNCT
ejpam-4793	400	1	4	4	X
ejpam-4793	400	2	.	.	X
ejpam-4793	400	3	γ	γ	NOUN
ejpam-4793	400	4	-	-	AUX
ejpam-4793	400	5	submonoids	submonoid	NOUN
ejpam-4793	400	6	this	this	DET
ejpam-4793	400	7	section	section	NOUN
ejpam-4793	400	8	presents	present	VERB
ejpam-4793	400	9	the	the	DET
ejpam-4793	400	10	discussions	discussion	NOUN
ejpam-4793	400	11	on	on	ADP
ejpam-4793	400	12	γ	γ	NOUN
ejpam-4793	400	13	-	-	NOUN
ejpam-4793	400	14	submonoids	submonoid	NOUN
ejpam-4793	400	15	of	of	ADP
ejpam-4793	400	16	γ	γ	NOUN
ejpam-4793	400	17	-	-	PUNCT
ejpam-4793	400	18	monoids	monoid	NOUN
ejpam-4793	400	19	.	.	PUNCT
ejpam-4793	401	1	definition	definition	NOUN
ejpam-4793	401	2	12	12	NUM
ejpam-4793	401	3	.	.	PUNCT
ejpam-4793	402	1	let	let	AUX
ejpam-4793	402	2	(	(	PUNCT
ejpam-4793	402	3	m	m	NOUN
ejpam-4793	402	4	,	,	PUNCT
ejpam-4793	402	5	∗	∗	NOUN
ejpam-4793	402	6	)	)	PUNCT
ejpam-4793	402	7	be	be	VERB
ejpam-4793	402	8	a	a	DET
ejpam-4793	402	9	γ	γ	X
ejpam-4793	402	10	-	-	PUNCT
ejpam-4793	402	11	monoid	monoid	NOUN
ejpam-4793	402	12	.	.	PUNCT
ejpam-4793	403	1	a	a	DET
ejpam-4793	403	2	γ	γ	X
ejpam-4793	403	3	-	-	ADJ
ejpam-4793	403	4	submonoid	submonoid	ADJ
ejpam-4793	403	5	is	be	AUX
ejpam-4793	403	6	a	a	DET
ejpam-4793	403	7	subset	subset	NOUN
ejpam-4793	403	8	s	s	NOUN
ejpam-4793	403	9	of	of	ADP
ejpam-4793	403	10	m	m	PRON
ejpam-4793	403	11	such	such	ADJ
ejpam-4793	403	12	that	that	SCONJ
ejpam-4793	403	13	the	the	DET
ejpam-4793	403	14	identity	identity	NOUN
ejpam-4793	403	15	1	1	NUM
ejpam-4793	403	16	m	m	NOUN
ejpam-4793	403	17	∈	∈	NOUN
ejpam-4793	403	18	s	s	NOUN
ejpam-4793	403	19	and	and	CCONJ
ejpam-4793	403	20	,	,	PUNCT
ejpam-4793	403	21	for	for	ADP
ejpam-4793	403	22	all	all	DET
ejpam-4793	403	23	α	α	NOUN
ejpam-4793	403	24	,	,	PUNCT
ejpam-4793	403	25	β	β	X
ejpam-4793	403	26	∈	∈	NOUN
ejpam-4793	403	27	γ	γ	NOUN
ejpam-4793	403	28	and	and	CCONJ
ejpam-4793	403	29	for	for	ADP
ejpam-4793	403	30	all	all	DET
ejpam-4793	403	31	s	s	PROPN
ejpam-4793	403	32	,	,	PUNCT
ejpam-4793	403	33	t	t	PROPN
ejpam-4793	403	34	∈	∈	PROPN
ejpam-4793	403	35	s	s	PROPN
ejpam-4793	403	36	,	,	PUNCT
ejpam-4793	403	37	αs	αs	ADP
ejpam-4793	403	38	∗	∗	NOUN
ejpam-4793	403	39	βt	βt	PUNCT
ejpam-4793	404	1	∈	∈	PROPN
ejpam-4793	404	2	s.	s.	PROPN
ejpam-4793	404	3	let	let	VERB
ejpam-4793	404	4	s	s	PRON
ejpam-4793	404	5	be	be	AUX
ejpam-4793	404	6	a	a	DET
ejpam-4793	404	7	γ	γ	NOUN
ejpam-4793	404	8	-	-	ADJ
ejpam-4793	404	9	submonoid	submonoid	NOUN
ejpam-4793	404	10	of	of	ADP
ejpam-4793	404	11	m	m	PROPN
ejpam-4793	404	12	.	.	PUNCT
ejpam-4793	405	1	then	then	ADV
ejpam-4793	405	2	1	1	NUM
ejpam-4793	405	3	m	m	NOUN
ejpam-4793	405	4	∈	∈	NOUN
ejpam-4793	405	5	s	s	NOUN
ejpam-4793	405	6	and	and	CCONJ
ejpam-4793	405	7	for	for	ADP
ejpam-4793	405	8	all	all	DET
ejpam-4793	405	9	α	α	NOUN
ejpam-4793	405	10	,	,	PUNCT
ejpam-4793	405	11	β	β	X
ejpam-4793	405	12	∈	∈	NOUN
ejpam-4793	405	13	γ	γ	NOUN
ejpam-4793	405	14	and	and	CCONJ
ejpam-4793	405	15	for	for	ADP
ejpam-4793	405	16	all	all	DET
ejpam-4793	405	17	s	s	PROPN
ejpam-4793	405	18	,	,	PUNCT
ejpam-4793	405	19	t	t	PROPN
ejpam-4793	405	20	∈	∈	PROPN
ejpam-4793	405	21	s	s	X
ejpam-4793	405	22	,	,	PUNCT
ejpam-4793	405	23	we	we	PRON
ejpam-4793	405	24	have	have	VERB
ejpam-4793	405	25	αs	αs	INTJ
ejpam-4793	405	26	∗	∗	NOUN
ejpam-4793	405	27	βt	βt	ADP
ejpam-4793	406	1	∈	∈	PROPN
ejpam-4793	406	2	s.	s.	PROPN
ejpam-4793	406	3	take	take	VERB
ejpam-4793	406	4	α	α	NOUN
ejpam-4793	406	5	=	=	SYM
ejpam-4793	406	6	β	β	X
ejpam-4793	406	7	=	=	SYM
ejpam-4793	406	8	0	0	X
ejpam-4793	406	9	.	.	PUNCT
ejpam-4793	407	1	thus	thus	ADV
ejpam-4793	407	2	,	,	PUNCT
ejpam-4793	407	3	we	we	PRON
ejpam-4793	407	4	have	have	VERB
ejpam-4793	407	5	s	s	NOUN
ejpam-4793	407	6	∗	∗	NOUN
ejpam-4793	407	7	t	t	NOUN
ejpam-4793	408	1	=	=	SYM
ejpam-4793	409	1	0s	0s	NOUN
ejpam-4793	409	2	∗	∗	NOUN
ejpam-4793	409	3	0	0	NUM
ejpam-4793	409	4	t	t	PROPN
ejpam-4793	409	5	∈	∈	PROPN
ejpam-4793	409	6	s.	s.	PROPN
ejpam-4793	409	7	hence	hence	ADV
ejpam-4793	409	8	,	,	PUNCT
ejpam-4793	409	9	s	s	VERB
ejpam-4793	409	10	is	be	AUX
ejpam-4793	409	11	a	a	DET
ejpam-4793	409	12	submonoid	submonoid	NOUN
ejpam-4793	409	13	of	of	ADP
ejpam-4793	409	14	m	m	PROPN
ejpam-4793	409	15	.	.	PUNCT
ejpam-4793	410	1	remark	remark	PROPN
ejpam-4793	410	2	7	7	NUM
ejpam-4793	410	3	.	.	PUNCT
ejpam-4793	411	1	let	let	VERB
ejpam-4793	411	2	s	s	PRON
ejpam-4793	411	3	be	be	AUX
ejpam-4793	411	4	a	a	DET
ejpam-4793	411	5	γ	γ	NOUN
ejpam-4793	411	6	-	-	ADJ
ejpam-4793	411	7	submonoid	submonoid	NOUN
ejpam-4793	411	8	of	of	ADP
ejpam-4793	411	9	a	a	DET
ejpam-4793	411	10	γ	γ	X
ejpam-4793	411	11	-	-	PUNCT
ejpam-4793	411	12	monoid	monoid	NOUN
ejpam-4793	411	13	m	m	NOUN
ejpam-4793	411	14	.	.	PUNCT
ejpam-4793	412	1	(	(	PUNCT
ejpam-4793	412	2	i	i	NOUN
ejpam-4793	412	3	)	)	PUNCT
ejpam-4793	412	4	s	s	AUX
ejpam-4793	412	5	is	be	AUX
ejpam-4793	412	6	a	a	DET
ejpam-4793	412	7	submonoid	submonoid	NOUN
ejpam-4793	412	8	of	of	ADP
ejpam-4793	412	9	m	m	PRON
ejpam-4793	412	10	,	,	PUNCT
ejpam-4793	412	11	hence	hence	ADV
ejpam-4793	412	12	a	a	DET
ejpam-4793	412	13	monoid	monoid	NOUN
ejpam-4793	412	14	itself	itself	PRON
ejpam-4793	412	15	.	.	PUNCT
ejpam-4793	413	1	(	(	PUNCT
ejpam-4793	413	2	ii	ii	NOUN
ejpam-4793	413	3	)	)	PUNCT
ejpam-4793	413	4	for	for	ADP
ejpam-4793	413	5	all	all	DET
ejpam-4793	413	6	s	s	PART
ejpam-4793	413	7	∈	∈	NOUN
ejpam-4793	413	8	s	s	NOUN
ejpam-4793	413	9	and	and	CCONJ
ejpam-4793	413	10	for	for	ADP
ejpam-4793	413	11	all	all	PRON
ejpam-4793	413	12	α	α	PRON
ejpam-4793	413	13	∈	∈	PROPN
ejpam-4793	413	14	γ	γ	X
ejpam-4793	413	15	,	,	PUNCT
ejpam-4793	413	16	αs	αs	PROPN
ejpam-4793	413	17	∈	∈	PROPN
ejpam-4793	413	18	s.	s.	PROPN
ejpam-4793	413	19	(	(	PUNCT
ejpam-4793	413	20	iii	iii	X
ejpam-4793	413	21	)	)	PUNCT
ejpam-4793	413	22	m	m	VERB
ejpam-4793	413	23	is	be	AUX
ejpam-4793	413	24	a	a	DET
ejpam-4793	413	25	γ	γ	NOUN
ejpam-4793	413	26	-	-	ADJ
ejpam-4793	413	27	submonoid	submonoid	NOUN
ejpam-4793	413	28	of	of	ADP
ejpam-4793	413	29	m	m	PROPN
ejpam-4793	413	30	.	.	PUNCT
ejpam-4793	414	1	let	let	VERB
ejpam-4793	414	2	s	s	PRON
ejpam-4793	414	3	be	be	AUX
ejpam-4793	414	4	a	a	DET
ejpam-4793	414	5	γ	γ	NOUN
ejpam-4793	414	6	-	-	ADJ
ejpam-4793	414	7	submonoid	submonoid	NOUN
ejpam-4793	414	8	of	of	ADP
ejpam-4793	414	9	a	a	DET
ejpam-4793	414	10	γ	γ	X
ejpam-4793	414	11	-	-	PUNCT
ejpam-4793	414	12	monoid	monoid	NOUN
ejpam-4793	414	13	m	m	NOUN
ejpam-4793	414	14	and	and	CCONJ
ejpam-4793	414	15	let	let	VERB
ejpam-4793	414	16	ϕ	ϕ	NOUN
ejpam-4793	414	17	:	:	PUNCT
ejpam-4793	414	18	γ	γ	X
ejpam-4793	414	19	×	×	PROPN
ejpam-4793	414	20	m	m	INTJ
ejpam-4793	414	21	→	→	VERB
ejpam-4793	414	22	m	m	VERB
ejpam-4793	414	23	be	be	VERB
ejpam-4793	414	24	the	the	DET
ejpam-4793	414	25	action	action	NOUN
ejpam-4793	414	26	(	(	PUNCT
ejpam-4793	414	27	by	by	ADP
ejpam-4793	414	28	monoid	monoid	NOUN
ejpam-4793	414	29	automorphism	automorphism	NOUN
ejpam-4793	414	30	)	)	PUNCT
ejpam-4793	414	31	of	of	ADP
ejpam-4793	414	32	a	a	DET
ejpam-4793	414	33	group	group	NOUN
ejpam-4793	414	34	γ	γ	NOUN
ejpam-4793	414	35	on	on	ADP
ejpam-4793	414	36	m	m	PRON
ejpam-4793	414	37	.	.	PUNCT
ejpam-4793	415	1	by	by	ADP
ejpam-4793	415	2	remark	remark	NOUN
ejpam-4793	415	3	7	7	NUM
ejpam-4793	415	4	,	,	PUNCT
ejpam-4793	415	5	s	s	VERB
ejpam-4793	415	6	is	be	AUX
ejpam-4793	415	7	a	a	DET
ejpam-4793	415	8	monoid	monoid	NOUN
ejpam-4793	415	9	.	.	PUNCT
ejpam-4793	416	1	moreover	moreover	ADV
ejpam-4793	416	2	,	,	PUNCT
ejpam-4793	416	3	by	by	ADP
ejpam-4793	416	4	restricting	restrict	VERB
ejpam-4793	416	5	the	the	DET
ejpam-4793	416	6	action	action	NOUN
ejpam-4793	416	7	ϕ	ϕ	NOUN
ejpam-4793	416	8	to	to	ADP
ejpam-4793	416	9	s	s	PROPN
ejpam-4793	416	10	,	,	PUNCT
ejpam-4793	416	11	ϕ	ϕ	NOUN
ejpam-4793	416	12	acts	act	NOUN
ejpam-4793	416	13	on	on	ADP
ejpam-4793	416	14	s	s	PRON
ejpam-4793	416	15	by	by	ADP
ejpam-4793	416	16	monoid	monoid	NOUN
ejpam-4793	416	17	automorphism	automorphism	NOUN
ejpam-4793	416	18	and	and	CCONJ
ejpam-4793	416	19	hence	hence	ADV
ejpam-4793	416	20	,	,	PUNCT
ejpam-4793	416	21	s	s	VERB
ejpam-4793	416	22	is	be	AUX
ejpam-4793	416	23	a	a	DET
ejpam-4793	416	24	γ	γ	PROPN
ejpam-4793	416	25	-	-	PUNCT
ejpam-4793	416	26	monoid	monoid	NOUN
ejpam-4793	416	27	.	.	PUNCT
ejpam-4793	416	28	remark	remark	PROPN
ejpam-4793	416	29	8	8	NUM
ejpam-4793	416	30	.	.	PUNCT
ejpam-4793	417	1	a	a	DET
ejpam-4793	417	2	γ	γ	NOUN
ejpam-4793	417	3	-	-	ADJ
ejpam-4793	417	4	submonoid	submonoid	NOUN
ejpam-4793	417	5	of	of	ADP
ejpam-4793	417	6	a	a	DET
ejpam-4793	417	7	γ	γ	X
ejpam-4793	417	8	-	-	PUNCT
ejpam-4793	417	9	monoid	monoid	NOUN
ejpam-4793	417	10	is	be	AUX
ejpam-4793	417	11	itself	itself	PRON
ejpam-4793	417	12	a	a	DET
ejpam-4793	417	13	γ	γ	NOUN
ejpam-4793	417	14	-	-	PUNCT
ejpam-4793	417	15	monoid	monoid	NOUN
ejpam-4793	417	16	.	.	PUNCT
ejpam-4793	417	17	example	example	NOUN
ejpam-4793	418	1	11	11	NUM
ejpam-4793	418	2	.	.	PUNCT
ejpam-4793	419	1	consider	consider	VERB
ejpam-4793	419	2	the	the	DET
ejpam-4793	419	3	set	set	NOUN
ejpam-4793	419	4	m	m	NOUN
ejpam-4793	419	5	=	=	PUNCT
ejpam-4793	419	6	{	{	PUNCT
ejpam-4793	419	7	0	0	NUM
ejpam-4793	419	8	,	,	PUNCT
ejpam-4793	419	9	1	1	NUM
ejpam-4793	419	10	,	,	PUNCT
ejpam-4793	419	11	x	x	NOUN
ejpam-4793	419	12	,	,	PUNCT
ejpam-4793	419	13	y	y	PROPN
ejpam-4793	419	14	,	,	PUNCT
ejpam-4793	419	15	z	z	PROPN
ejpam-4793	419	16	,	,	PUNCT
ejpam-4793	419	17	s	s	PROPN
ejpam-4793	419	18	,	,	PUNCT
ejpam-4793	419	19	b	b	NOUN
ejpam-4793	419	20	}	}	PUNCT
ejpam-4793	419	21	and	and	CCONJ
ejpam-4793	419	22	an	an	DET
ejpam-4793	419	23	operation	operation	NOUN
ejpam-4793	419	24	+	+	CCONJ
ejpam-4793	419	25	given	give	VERB
ejpam-4793	419	26	by	by	ADP
ejpam-4793	419	27	+	+	PROPN
ejpam-4793	419	28	0	0	NUM
ejpam-4793	419	29	1	1	NUM
ejpam-4793	419	30	x	x	SYM
ejpam-4793	419	31	y	y	PROPN
ejpam-4793	419	32	z	z	PROPN
ejpam-4793	419	33	s	s	PROPN
ejpam-4793	419	34	b	b	PROPN
ejpam-4793	419	35	0	0	NUM
ejpam-4793	419	36	0	0	NUM
ejpam-4793	419	37	1	1	NUM
ejpam-4793	419	38	x	x	SYM
ejpam-4793	419	39	y	y	PROPN
ejpam-4793	419	40	z	z	PROPN
ejpam-4793	419	41	s	s	PROPN
ejpam-4793	419	42	b	b	PROPN
ejpam-4793	419	43	1	1	NUM
ejpam-4793	419	44	1	1	NUM
ejpam-4793	419	45	1	1	NUM
ejpam-4793	419	46	1	1	NUM
ejpam-4793	419	47	s	s	NOUN
ejpam-4793	419	48	s	s	NOUN
ejpam-4793	419	49	s	s	NOUN
ejpam-4793	419	50	b	b	NOUN
ejpam-4793	419	51	x	x	SYM
ejpam-4793	419	52	x	x	SYM
ejpam-4793	419	53	1	1	NUM
ejpam-4793	419	54	1	1	NUM
ejpam-4793	419	55	s	s	PART
ejpam-4793	419	56	s	s	NOUN
ejpam-4793	419	57	s	s	X
ejpam-4793	419	58	b	b	X
ejpam-4793	419	59	y	y	PROPN
ejpam-4793	419	60	y	y	PROPN
ejpam-4793	419	61	s	s	PROPN
ejpam-4793	419	62	s	s	PROPN
ejpam-4793	419	63	y	y	PROPN
ejpam-4793	420	1	y	y	PROPN
ejpam-4793	420	2	s	s	PROPN
ejpam-4793	420	3	b	b	PROPN
ejpam-4793	420	4	z	z	PROPN
ejpam-4793	420	5	z	z	PROPN
ejpam-4793	420	6	s	s	NOUN
ejpam-4793	420	7	s	s	X
ejpam-4793	420	8	y	y	PROPN
ejpam-4793	420	9	y	y	PROPN
ejpam-4793	420	10	s	s	PROPN
ejpam-4793	420	11	b	b	PROPN
ejpam-4793	420	12	s	s	X
ejpam-4793	420	13	s	s	X
ejpam-4793	420	14	s	s	X
ejpam-4793	420	15	s	s	X
ejpam-4793	420	16	s	s	X
ejpam-4793	420	17	s	s	X
ejpam-4793	420	18	s	s	PROPN
ejpam-4793	420	19	b	b	PROPN
ejpam-4793	420	20	b	b	PROPN
ejpam-4793	420	21	b	b	PROPN
ejpam-4793	420	22	b	b	PROPN
ejpam-4793	420	23	b	b	PROPN
ejpam-4793	420	24	b	b	PROPN
ejpam-4793	420	25	b	b	PROPN
ejpam-4793	420	26	b	b	PROPN
ejpam-4793	420	27	s	s	PROPN
ejpam-4793	420	28	h.	h.	PROPN
ejpam-4793	420	29	sarapuddin	sarapuddin	PROPN
ejpam-4793	420	30	,	,	PUNCT
ejpam-4793	420	31	j.	j.	PROPN
ejpam-4793	420	32	vilela	vilela	PROPN
ejpam-4793	420	33	/	/	SYM
ejpam-4793	420	34	eur	eur	PROPN
ejpam-4793	420	35	.	.	PUNCT
ejpam-4793	421	1	j.	j.	PROPN
ejpam-4793	421	2	pure	pure	PROPN
ejpam-4793	421	3	appl	appl	PROPN
ejpam-4793	421	4	.	.	PROPN
ejpam-4793	421	5	math	math	PROPN
ejpam-4793	421	6	,	,	PUNCT
ejpam-4793	421	7	16	16	NUM
ejpam-4793	421	8	(	(	PUNCT
ejpam-4793	421	9	3	3	NUM
ejpam-4793	421	10	)	)	PUNCT
ejpam-4793	421	11	(	(	PUNCT
ejpam-4793	421	12	2023	2023	NUM
ejpam-4793	421	13	)	)	PUNCT
ejpam-4793	421	14	,	,	PUNCT
ejpam-4793	421	15	1772	1772	NUM
ejpam-4793	421	16	-	-	SYM
ejpam-4793	421	17	1793	1793	NUM
ejpam-4793	421	18	1782	1782	NUM
ejpam-4793	421	19	it	it	PRON
ejpam-4793	421	20	was	be	AUX
ejpam-4793	421	21	shown	show	VERB
ejpam-4793	421	22	in	in	ADP
ejpam-4793	421	23	[	[	X
ejpam-4793	421	24	5	5	NUM
ejpam-4793	421	25	]	]	X
ejpam-4793	421	26	thatm	thatm	NOUN
ejpam-4793	421	27	is	be	AUX
ejpam-4793	421	28	a	a	DET
ejpam-4793	421	29	commutative	commutative	ADJ
ejpam-4793	421	30	γ	γ	X
ejpam-4793	421	31	-	-	NOUN
ejpam-4793	421	32	monoid	monoid	NOUN
ejpam-4793	421	33	with	with	ADP
ejpam-4793	421	34	identity	identity	NOUN
ejpam-4793	421	35	0	0	NUM
ejpam-4793	421	36	,	,	PUNCT
ejpam-4793	421	37	where	where	SCONJ
ejpam-4793	421	38	the	the	DET
ejpam-4793	421	39	trivial	trivial	ADJ
ejpam-4793	421	40	group	group	NOUN
ejpam-4793	421	41	γ	γ	X
ejpam-4793	421	42	=	=	SYM
ejpam-4793	421	43	{	{	PUNCT
ejpam-4793	421	44	0	0	NUM
ejpam-4793	421	45	}	}	PUNCT
ejpam-4793	421	46	acts	act	VERB
ejpam-4793	421	47	trivially	trivially	ADV
ejpam-4793	421	48	on	on	ADP
ejpam-4793	421	49	m	m	PRON
ejpam-4793	421	50	.	.	PUNCT
ejpam-4793	422	1	let	let	VERB
ejpam-4793	422	2	s	s	VERB
ejpam-4793	422	3	=	=	X
ejpam-4793	422	4	{	{	PUNCT
ejpam-4793	422	5	0	0	NUM
ejpam-4793	422	6	,	,	PUNCT
ejpam-4793	422	7	y	y	PROPN
ejpam-4793	422	8	,	,	PUNCT
ejpam-4793	422	9	s	s	PROPN
ejpam-4793	422	10	,	,	PUNCT
ejpam-4793	422	11	b	b	NOUN
ejpam-4793	422	12	}	}	PUNCT
ejpam-4793	422	13	,	,	PUNCT
ejpam-4793	422	14	u	u	NOUN
ejpam-4793	422	15	=	=	PUNCT
ejpam-4793	422	16	{	{	PUNCT
ejpam-4793	422	17	0	0	NUM
ejpam-4793	422	18	,	,	PUNCT
ejpam-4793	422	19	1	1	NUM
ejpam-4793	422	20	,	,	PUNCT
ejpam-4793	422	21	y	y	PROPN
ejpam-4793	422	22	,	,	PUNCT
ejpam-4793	422	23	s	s	PROPN
ejpam-4793	422	24	,	,	PUNCT
ejpam-4793	422	25	b	b	NOUN
ejpam-4793	422	26	}	}	PUNCT
ejpam-4793	422	27	,	,	PUNCT
ejpam-4793	422	28	v	v	NOUN
ejpam-4793	422	29	=	=	SYM
ejpam-4793	422	30	{	{	PUNCT
ejpam-4793	422	31	0	0	NUM
ejpam-4793	422	32	,	,	PUNCT
ejpam-4793	422	33	1	1	NUM
ejpam-4793	422	34	,	,	PUNCT
ejpam-4793	422	35	x	x	NOUN
ejpam-4793	422	36	}	}	PUNCT
ejpam-4793	422	37	and	and	CCONJ
ejpam-4793	422	38	w	w	NOUN
ejpam-4793	422	39	=	=	PUNCT
ejpam-4793	422	40	{	{	PUNCT
ejpam-4793	422	41	0	0	NUM
ejpam-4793	422	42	,	,	PUNCT
ejpam-4793	422	43	y	y	NOUN
ejpam-4793	422	44	}	}	PUNCT
ejpam-4793	422	45	.	.	PUNCT
ejpam-4793	423	1	note	note	VERB
ejpam-4793	423	2	that	that	SCONJ
ejpam-4793	423	3	the	the	DET
ejpam-4793	423	4	identity	identity	NOUN
ejpam-4793	423	5	0	0	NUM
ejpam-4793	423	6	is	be	AUX
ejpam-4793	423	7	in	in	ADP
ejpam-4793	423	8	s	s	PROPN
ejpam-4793	423	9	,	,	PUNCT
ejpam-4793	423	10	u	u	NOUN
ejpam-4793	423	11	,	,	PUNCT
ejpam-4793	423	12	v	v	NOUN
ejpam-4793	423	13	and	and	CCONJ
ejpam-4793	423	14	w	w	NOUN
ejpam-4793	423	15	.	.	PUNCT
ejpam-4793	424	1	now	now	ADV
ejpam-4793	424	2	,	,	PUNCT
ejpam-4793	424	3	we	we	PRON
ejpam-4793	424	4	have	have	VERB
ejpam-4793	424	5	00	00	NUM
ejpam-4793	425	1	+	+	NUM
ejpam-4793	425	2	00	00	NUM
ejpam-4793	426	1	=	=	SYM
ejpam-4793	426	2	0	0	PUNCT
ejpam-4793	427	1	+	+	CCONJ
ejpam-4793	427	2	0	0	NUM
ejpam-4793	427	3	=	=	SYM
ejpam-4793	427	4	0	0	NUM
ejpam-4793	427	5	∈	∈	PROPN
ejpam-4793	427	6	s	s	NOUN
ejpam-4793	427	7	,	,	PUNCT
ejpam-4793	427	8	00	00	PUNCT
ejpam-4793	428	1	+	+	NUM
ejpam-4793	428	2	0s	0s	NUM
ejpam-4793	428	3	=	=	SYM
ejpam-4793	428	4	0	0	PUNCT
ejpam-4793	429	1	+	+	NUM
ejpam-4793	429	2	s	s	X
ejpam-4793	429	3	=	=	SYM
ejpam-4793	429	4	s	s	PART
ejpam-4793	429	5	∈	∈	PROPN
ejpam-4793	429	6	s	s	NOUN
ejpam-4793	429	7	,	,	PUNCT
ejpam-4793	429	8	0y	0y	X
ejpam-4793	429	9	+	+	CCONJ
ejpam-4793	429	10	0y	0y	NOUN
ejpam-4793	429	11	=	=	SYM
ejpam-4793	429	12	y	y	PROPN
ejpam-4793	430	1	+	+	NOUN
ejpam-4793	430	2	y	y	PROPN
ejpam-4793	430	3	=	=	PUNCT
ejpam-4793	430	4	y	y	PROPN
ejpam-4793	430	5	∈	∈	PROPN
ejpam-4793	430	6	s	s	NOUN
ejpam-4793	430	7	,	,	PUNCT
ejpam-4793	430	8	0y	0y	NOUN
ejpam-4793	430	9	+	+	CCONJ
ejpam-4793	430	10	0b	0b	NOUN
ejpam-4793	430	11	=	=	SYM
ejpam-4793	430	12	y	y	PROPN
ejpam-4793	430	13	+	+	NUM
ejpam-4793	430	14	b	b	PROPN
ejpam-4793	430	15	=	=	SYM
ejpam-4793	430	16	b	b	PROPN
ejpam-4793	430	17	∈	∈	PROPN
ejpam-4793	430	18	s	s	NOUN
ejpam-4793	430	19	,	,	PUNCT
ejpam-4793	430	20	0s+	0s+	X
ejpam-4793	430	21	0b	0b	NOUN
ejpam-4793	430	22	=	=	PUNCT
ejpam-4793	430	23	s+	s+	PUNCT
ejpam-4793	430	24	b	b	X
ejpam-4793	430	25	=	=	SYM
ejpam-4793	430	26	b	b	PROPN
ejpam-4793	430	27	∈	∈	PROPN
ejpam-4793	430	28	s	s	NOUN
ejpam-4793	430	29	,	,	PUNCT
ejpam-4793	430	30	00	00	PUNCT
ejpam-4793	431	1	+	+	NUM
ejpam-4793	431	2	0y	0y	NUM
ejpam-4793	431	3	=	=	SYM
ejpam-4793	431	4	0	0	PUNCT
ejpam-4793	431	5	+	+	CCONJ
ejpam-4793	431	6	y	y	PROPN
ejpam-4793	431	7	=	=	SYM
ejpam-4793	431	8	y	y	PROPN
ejpam-4793	431	9	∈	∈	PROPN
ejpam-4793	431	10	s	s	PART
ejpam-4793	431	11	;	;	PUNCT
ejpam-4793	431	12	00	00	PUNCT
ejpam-4793	431	13	+	+	NUM
ejpam-4793	431	14	0b	0b	NOUN
ejpam-4793	431	15	=	=	SYM
ejpam-4793	431	16	0	0	PUNCT
ejpam-4793	432	1	+	+	NUM
ejpam-4793	432	2	b	b	X
ejpam-4793	432	3	=	=	SYM
ejpam-4793	432	4	b	b	PROPN
ejpam-4793	432	5	∈	∈	PROPN
ejpam-4793	432	6	s	s	NOUN
ejpam-4793	432	7	;	;	PUNCT
ejpam-4793	432	8	0y	0y	NUM
ejpam-4793	432	9	+	+	CCONJ
ejpam-4793	432	10	0s	0s	NUM
ejpam-4793	432	11	=	=	SYM
ejpam-4793	432	12	y	y	PROPN
ejpam-4793	432	13	+	+	SYM
ejpam-4793	432	14	s	s	PART
ejpam-4793	432	15	=	=	SYM
ejpam-4793	432	16	s	s	PART
ejpam-4793	432	17	∈	∈	PROPN
ejpam-4793	432	18	s	s	X
ejpam-4793	432	19	;	;	PUNCT
ejpam-4793	432	20	0s+	0s+	NUM
ejpam-4793	432	21	0s	0s	NUM
ejpam-4793	433	1	=	=	PUNCT
ejpam-4793	433	2	s+	s+	PUNCT
ejpam-4793	433	3	s	s	PART
ejpam-4793	433	4	=	=	SYM
ejpam-4793	433	5	s	s	PART
ejpam-4793	433	6	∈	∈	PROPN
ejpam-4793	433	7	s	s	NOUN
ejpam-4793	433	8	;	;	PUNCT
ejpam-4793	433	9	0b+	0b+	NUM
ejpam-4793	433	10	0b	0b	NOUN
ejpam-4793	433	11	=	=	PUNCT
ejpam-4793	433	12	b+	b+	PUNCT
ejpam-4793	433	13	b	b	X
ejpam-4793	433	14	=	=	SYM
ejpam-4793	433	15	b	b	PROPN
ejpam-4793	433	16	∈	∈	PROPN
ejpam-4793	433	17	s.	s.	PROPN
ejpam-4793	433	18	thus	thus	ADV
ejpam-4793	433	19	,	,	PUNCT
ejpam-4793	433	20	by	by	ADP
ejpam-4793	433	21	definition	definition	NOUN
ejpam-4793	433	22	12	12	NUM
ejpam-4793	433	23	,	,	PUNCT
ejpam-4793	433	24	s	s	X
ejpam-4793	433	25	is	be	AUX
ejpam-4793	433	26	γ	γ	X
ejpam-4793	433	27	-	-	ADJ
ejpam-4793	433	28	submonoid	submonoid	NOUN
ejpam-4793	433	29	of	of	ADP
ejpam-4793	433	30	m	m	PROPN
ejpam-4793	433	31	.	.	PUNCT
ejpam-4793	434	1	similarly	similarly	ADV
ejpam-4793	434	2	,	,	PUNCT
ejpam-4793	434	3	u	u	NOUN
ejpam-4793	434	4	,	,	PUNCT
ejpam-4793	434	5	v	v	NOUN
ejpam-4793	434	6	and	and	CCONJ
ejpam-4793	434	7	w	w	PROPN
ejpam-4793	434	8	are	be	AUX
ejpam-4793	434	9	γ	γ	NOUN
ejpam-4793	434	10	-	-	NOUN
ejpam-4793	434	11	submonoids	submonoid	NOUN
ejpam-4793	434	12	of	of	ADP
ejpam-4793	434	13	m	m	PROPN
ejpam-4793	434	14	.	.	PUNCT
ejpam-4793	435	1	consider	consider	VERB
ejpam-4793	435	2	the	the	DET
ejpam-4793	435	3	γ	γ	NOUN
ejpam-4793	435	4	-	-	ADJ
ejpam-4793	435	5	submonoid	submonoid	ADJ
ejpam-4793	435	6	s	s	PART
ejpam-4793	435	7	=	=	PUNCT
ejpam-4793	435	8	{	{	PUNCT
ejpam-4793	435	9	0	0	NUM
ejpam-4793	435	10	,	,	PUNCT
ejpam-4793	435	11	y	y	PROPN
ejpam-4793	435	12	,	,	PUNCT
ejpam-4793	435	13	s	s	PROPN
ejpam-4793	435	14	,	,	PUNCT
ejpam-4793	435	15	b	b	NOUN
ejpam-4793	435	16	}	}	PUNCT
ejpam-4793	435	17	.	.	PUNCT
ejpam-4793	436	1	now	now	ADV
ejpam-4793	436	2	,	,	PUNCT
ejpam-4793	436	3	take	take	VERB
ejpam-4793	436	4	0	0	NUM
ejpam-4793	436	5	∈	∈	NOUN
ejpam-4793	436	6	s	s	PART
ejpam-4793	436	7	and	and	CCONJ
ejpam-4793	436	8	z	z	NOUN
ejpam-4793	436	9	∈	∈	PROPN
ejpam-4793	436	10	m	m	VERB
ejpam-4793	436	11	.	.	PUNCT
ejpam-4793	437	1	then	then	ADV
ejpam-4793	437	2	0	0	NUM
ejpam-4793	437	3	∗	∗	NOUN
ejpam-4793	437	4	z	z	NOUN
ejpam-4793	438	1	=	=	SYM
ejpam-4793	438	2	z	z	NOUN
ejpam-4793	438	3	/∈	/∈	PUNCT
ejpam-4793	439	1	s.	s.	PROPN
ejpam-4793	439	2	thus	thus	ADV
ejpam-4793	439	3	,	,	PUNCT
ejpam-4793	439	4	s	s	VERB
ejpam-4793	439	5	is	be	AUX
ejpam-4793	439	6	not	not	PART
ejpam-4793	439	7	a	a	DET
ejpam-4793	439	8	γ	γ	NOUN
ejpam-4793	439	9	-	-	NOUN
ejpam-4793	439	10	ideal	ideal	NOUN
ejpam-4793	439	11	of	of	ADP
ejpam-4793	439	12	m	m	PROPN
ejpam-4793	439	13	.	.	PUNCT
ejpam-4793	440	1	remark	remark	PROPN
ejpam-4793	440	2	9	9	NUM
ejpam-4793	440	3	.	.	PUNCT
ejpam-4793	441	1	let	let	VERB
ejpam-4793	441	2	m	m	PRON
ejpam-4793	441	3	be	be	AUX
ejpam-4793	441	4	a	a	DET
ejpam-4793	441	5	γ	γ	X
ejpam-4793	441	6	-	-	PUNCT
ejpam-4793	441	7	monoid	monoid	NOUN
ejpam-4793	441	8	.	.	PUNCT
ejpam-4793	442	1	a	a	DET
ejpam-4793	442	2	γ	γ	NOUN
ejpam-4793	442	3	-	-	PUNCT
ejpam-4793	442	4	submonoid	submonoid	NOUN
ejpam-4793	442	5	of	of	ADP
ejpam-4793	442	6	m	m	PROPN
ejpam-4793	442	7	is	be	AUX
ejpam-4793	442	8	not	not	PART
ejpam-4793	442	9	necessarily	necessarily	ADV
ejpam-4793	442	10	a	a	DET
ejpam-4793	442	11	γ	γ	NOUN
ejpam-4793	442	12	-	-	NOUN
ejpam-4793	442	13	ideal	ideal	NOUN
ejpam-4793	442	14	of	of	ADP
ejpam-4793	442	15	m	m	PROPN
ejpam-4793	442	16	.	.	PUNCT
ejpam-4793	443	1	theorems	theorem	NOUN
ejpam-4793	443	2	5	5	NUM
ejpam-4793	443	3	and	and	CCONJ
ejpam-4793	443	4	6	6	NUM
ejpam-4793	443	5	imply	imply	NOUN
ejpam-4793	443	6	that	that	SCONJ
ejpam-4793	443	7	there	there	PRON
ejpam-4793	443	8	is	be	VERB
ejpam-4793	443	9	no	no	DET
ejpam-4793	443	10	proper	proper	ADJ
ejpam-4793	443	11	γ	γ	X
ejpam-4793	443	12	-	-	ADJ
ejpam-4793	443	13	submonoid	submonoid	ADJ
ejpam-4793	443	14	which	which	PRON
ejpam-4793	443	15	is	be	AUX
ejpam-4793	443	16	also	also	ADV
ejpam-4793	443	17	a	a	DET
ejpam-4793	443	18	γ	γ	X
ejpam-4793	443	19	-	-	PUNCT
ejpam-4793	443	20	ideal	ideal	NOUN
ejpam-4793	443	21	and	and	CCONJ
ejpam-4793	443	22	vice	vice	ADV
ejpam-4793	443	23	versa	versa	ADV
ejpam-4793	443	24	.	.	PUNCT
ejpam-4793	444	1	theorem	theorem	NOUN
ejpam-4793	444	2	5	5	NUM
ejpam-4793	444	3	.	.	PUNCT
ejpam-4793	445	1	let	let	VERB
ejpam-4793	445	2	s	s	PRON
ejpam-4793	445	3	be	be	AUX
ejpam-4793	445	4	a	a	DET
ejpam-4793	445	5	γ	γ	NOUN
ejpam-4793	445	6	-	-	ADJ
ejpam-4793	445	7	submonoid	submonoid	NOUN
ejpam-4793	445	8	of	of	ADP
ejpam-4793	445	9	a	a	DET
ejpam-4793	445	10	γ	γ	X
ejpam-4793	445	11	-	-	PUNCT
ejpam-4793	445	12	monoid	monoid	NOUN
ejpam-4793	445	13	m	m	PROPN
ejpam-4793	445	14	.	.	PUNCT
ejpam-4793	446	1	then	then	ADV
ejpam-4793	446	2	s	s	VERB
ejpam-4793	446	3	is	be	AUX
ejpam-4793	446	4	a	a	DET
ejpam-4793	446	5	γ	γ	NOUN
ejpam-4793	446	6	-	-	NOUN
ejpam-4793	446	7	ideal	ideal	NOUN
ejpam-4793	446	8	of	of	ADP
ejpam-4793	446	9	m	m	PRON
ejpam-4793	446	10	if	if	SCONJ
ejpam-4793	447	1	and	and	CCONJ
ejpam-4793	447	2	only	only	ADV
ejpam-4793	447	3	if	if	SCONJ
ejpam-4793	447	4	s	s	VERB
ejpam-4793	447	5	=	=	NOUN
ejpam-4793	447	6	m	m	VERB
ejpam-4793	447	7	.	.	PUNCT
ejpam-4793	448	1	proof	proof	NOUN
ejpam-4793	448	2	.	.	PUNCT
ejpam-4793	449	1	let	let	VERB
ejpam-4793	449	2	s	s	PRON
ejpam-4793	449	3	be	be	AUX
ejpam-4793	449	4	a	a	DET
ejpam-4793	449	5	γ	γ	NOUN
ejpam-4793	449	6	-	-	ADJ
ejpam-4793	449	7	submonoid	submonoid	NOUN
ejpam-4793	449	8	of	of	ADP
ejpam-4793	449	9	m	m	PROPN
ejpam-4793	449	10	.	.	PUNCT
ejpam-4793	450	1	suppose	suppose	VERB
ejpam-4793	450	2	that	that	SCONJ
ejpam-4793	450	3	s	s	VERB
ejpam-4793	450	4	is	be	AUX
ejpam-4793	450	5	a	a	DET
ejpam-4793	450	6	γ	γ	NOUN
ejpam-4793	450	7	-	-	NOUN
ejpam-4793	450	8	ideal	ideal	NOUN
ejpam-4793	450	9	of	of	ADP
ejpam-4793	450	10	m	m	PROPN
ejpam-4793	450	11	.	.	PUNCT
ejpam-4793	451	1	since	since	SCONJ
ejpam-4793	451	2	s	s	PROPN
ejpam-4793	451	3	is	be	AUX
ejpam-4793	451	4	a	a	DET
ejpam-4793	451	5	γ	γ	NOUN
ejpam-4793	451	6	-	-	ADJ
ejpam-4793	451	7	submonoid	submonoid	ADJ
ejpam-4793	451	8	,	,	PUNCT
ejpam-4793	451	9	1	1	NUM
ejpam-4793	451	10	m	m	NOUN
ejpam-4793	451	11	∈	∈	NOUN
ejpam-4793	451	12	s	s	NOUN
ejpam-4793	451	13	and	and	CCONJ
ejpam-4793	451	14	thus	thus	ADV
ejpam-4793	451	15	,	,	PUNCT
ejpam-4793	451	16	by	by	ADP
ejpam-4793	451	17	lemma	lemma	PROPN
ejpam-4793	451	18	2	2	NUM
ejpam-4793	451	19	,	,	PUNCT
ejpam-4793	451	20	s	s	PART
ejpam-4793	451	21	=	=	NOUN
ejpam-4793	451	22	m	m	VERB
ejpam-4793	451	23	.	.	PUNCT
ejpam-4793	452	1	conversely	conversely	ADV
ejpam-4793	452	2	,	,	PUNCT
ejpam-4793	452	3	suppose	suppose	VERB
ejpam-4793	452	4	that	that	SCONJ
ejpam-4793	452	5	s	s	VERB
ejpam-4793	452	6	=	=	NOUN
ejpam-4793	452	7	m	m	VERB
ejpam-4793	452	8	.	.	PUNCT
ejpam-4793	453	1	then	then	ADV
ejpam-4793	453	2	,	,	PUNCT
ejpam-4793	453	3	by	by	ADP
ejpam-4793	453	4	remark	remark	NOUN
ejpam-4793	453	5	4(i	4(i	NUM
ejpam-4793	453	6	)	)	PUNCT
ejpam-4793	453	7	,	,	PUNCT
ejpam-4793	453	8	s	s	VERB
ejpam-4793	453	9	is	be	AUX
ejpam-4793	453	10	a	a	DET
ejpam-4793	453	11	γ	γ	NOUN
ejpam-4793	453	12	-	-	NOUN
ejpam-4793	453	13	ideal	ideal	NOUN
ejpam-4793	453	14	of	of	ADP
ejpam-4793	453	15	m	m	PROPN
ejpam-4793	453	16	.	.	PUNCT
ejpam-4793	454	1	theorem	theorem	ADJ
ejpam-4793	454	2	6	6	NUM
ejpam-4793	454	3	.	.	PUNCT
ejpam-4793	455	1	let	let	VERB
ejpam-4793	455	2	i	i	PRON
ejpam-4793	455	3	be	be	AUX
ejpam-4793	455	4	a	a	DET
ejpam-4793	455	5	γ	γ	NOUN
ejpam-4793	455	6	-	-	NOUN
ejpam-4793	455	7	ideal	ideal	NOUN
ejpam-4793	455	8	of	of	ADP
ejpam-4793	455	9	a	a	DET
ejpam-4793	455	10	γ	γ	X
ejpam-4793	455	11	-	-	PUNCT
ejpam-4793	455	12	monoid	monoid	NOUN
ejpam-4793	455	13	m	m	PROPN
ejpam-4793	455	14	.	.	PUNCT
ejpam-4793	456	1	then	then	ADV
ejpam-4793	456	2	i	i	PRON
ejpam-4793	456	3	is	be	AUX
ejpam-4793	456	4	a	a	DET
ejpam-4793	456	5	γ	γ	NOUN
ejpam-4793	456	6	-	-	ADJ
ejpam-4793	456	7	submonoid	submonoid	NOUN
ejpam-4793	456	8	of	of	ADP
ejpam-4793	456	9	m	m	PROPN
ejpam-4793	456	10	if	if	SCONJ
ejpam-4793	457	1	and	and	CCONJ
ejpam-4793	457	2	only	only	ADV
ejpam-4793	457	3	if	if	SCONJ
ejpam-4793	457	4	i	i	PRON
ejpam-4793	457	5	=	=	NOUN
ejpam-4793	457	6	m	m	VERB
ejpam-4793	457	7	.	.	PUNCT
ejpam-4793	458	1	proof	proof	NOUN
ejpam-4793	458	2	.	.	PUNCT
ejpam-4793	459	1	let	let	VERB
ejpam-4793	459	2	i	i	PRON
ejpam-4793	459	3	be	be	AUX
ejpam-4793	459	4	a	a	DET
ejpam-4793	459	5	γ	γ	NOUN
ejpam-4793	459	6	-	-	NOUN
ejpam-4793	459	7	ideal	ideal	NOUN
ejpam-4793	459	8	of	of	ADP
ejpam-4793	459	9	a	a	DET
ejpam-4793	459	10	γ	γ	X
ejpam-4793	459	11	-	-	PUNCT
ejpam-4793	459	12	monoid	monoid	NOUN
ejpam-4793	459	13	m	m	PROPN
ejpam-4793	459	14	.	.	PUNCT
ejpam-4793	460	1	suppose	suppose	VERB
ejpam-4793	460	2	that	that	SCONJ
ejpam-4793	460	3	i	i	PRON
ejpam-4793	460	4	is	be	AUX
ejpam-4793	460	5	a	a	DET
ejpam-4793	460	6	γ	γ	NOUN
ejpam-4793	460	7	-	-	ADJ
ejpam-4793	460	8	submonoid	submonoid	NOUN
ejpam-4793	460	9	of	of	ADP
ejpam-4793	460	10	m	m	PROPN
ejpam-4793	460	11	.	.	PUNCT
ejpam-4793	461	1	then	then	ADV
ejpam-4793	461	2	1	1	NUM
ejpam-4793	461	3	m	m	NOUN
ejpam-4793	461	4	∈	∈	NOUN
ejpam-4793	462	1	i	i	PRON
ejpam-4793	462	2	and	and	CCONJ
ejpam-4793	462	3	i	i	PRON
ejpam-4793	462	4	=	=	NOUN
ejpam-4793	462	5	m	m	VERB
ejpam-4793	462	6	.	.	PUNCT
ejpam-4793	463	1	conversely	conversely	ADV
ejpam-4793	463	2	,	,	PUNCT
ejpam-4793	463	3	suppose	suppose	VERB
ejpam-4793	463	4	that	that	SCONJ
ejpam-4793	463	5	i	i	PRON
ejpam-4793	463	6	=	=	NOUN
ejpam-4793	463	7	m	m	VERB
ejpam-4793	463	8	.	.	PUNCT
ejpam-4793	464	1	by	by	ADP
ejpam-4793	464	2	remark	remark	NOUN
ejpam-4793	464	3	7(iii	7(iii	NUM
ejpam-4793	464	4	)	)	PUNCT
ejpam-4793	464	5	,	,	PUNCT
ejpam-4793	464	6	i	i	PRON
ejpam-4793	464	7	is	be	AUX
ejpam-4793	464	8	a	a	DET
ejpam-4793	464	9	γ	γ	NOUN
ejpam-4793	464	10	-	-	ADJ
ejpam-4793	464	11	submonoid	submonoid	NOUN
ejpam-4793	464	12	of	of	ADP
ejpam-4793	464	13	m	m	PROPN
ejpam-4793	464	14	.	.	PUNCT
ejpam-4793	465	1	example	example	NOUN
ejpam-4793	465	2	12	12	NUM
ejpam-4793	465	3	.	.	PUNCT
ejpam-4793	466	1	consider	consider	VERB
ejpam-4793	466	2	the	the	DET
ejpam-4793	466	3	γ	γ	NOUN
ejpam-4793	466	4	-	-	ADJ
ejpam-4793	466	5	submonoid	submonoid	ADJ
ejpam-4793	466	6	s	s	PART
ejpam-4793	466	7	=	=	PUNCT
ejpam-4793	466	8	{	{	PUNCT
ejpam-4793	466	9	0	0	NUM
ejpam-4793	466	10	,	,	PUNCT
ejpam-4793	466	11	y	y	PROPN
ejpam-4793	466	12	,	,	PUNCT
ejpam-4793	466	13	s	s	PROPN
ejpam-4793	466	14	,	,	PUNCT
ejpam-4793	466	15	b	b	NOUN
ejpam-4793	466	16	}	}	PUNCT
ejpam-4793	466	17	in	in	ADP
ejpam-4793	466	18	example	example	NOUN
ejpam-4793	466	19	11	11	NUM
ejpam-4793	466	20	.	.	PUNCT
ejpam-4793	467	1	note	note	VERB
ejpam-4793	467	2	that	that	SCONJ
ejpam-4793	467	3	x	x	NOUN
ejpam-4793	467	4	∗	∗	NOUN
ejpam-4793	467	5	z	z	NOUN
ejpam-4793	467	6	=	=	SYM
ejpam-4793	467	7	s	s	PART
ejpam-4793	467	8	∈	∈	PROPN
ejpam-4793	467	9	s.	s.	PROPN
ejpam-4793	467	10	however	however	ADV
ejpam-4793	467	11	,	,	PUNCT
ejpam-4793	467	12	x	x	X
ejpam-4793	467	13	,	,	PUNCT
ejpam-4793	467	14	z	z	PROPN
ejpam-4793	467	15	/∈	/∈	PUNCT
ejpam-4793	468	1	s.	s.	PROPN
ejpam-4793	468	2	thus	thus	ADV
ejpam-4793	468	3	,	,	PUNCT
ejpam-4793	468	4	s	s	VERB
ejpam-4793	468	5	is	be	AUX
ejpam-4793	468	6	not	not	PART
ejpam-4793	468	7	a	a	DET
ejpam-4793	468	8	γ	γ	NOUN
ejpam-4793	468	9	-	-	PUNCT
ejpam-4793	468	10	order	order	NOUN
ejpam-4793	468	11	-	-	PUNCT
ejpam-4793	468	12	ideal	ideal	NOUN
ejpam-4793	468	13	of	of	ADP
ejpam-4793	468	14	m	m	PROPN
ejpam-4793	468	15	.	.	PUNCT
ejpam-4793	469	1	note	note	VERB
ejpam-4793	469	2	that	that	SCONJ
ejpam-4793	469	3	if	if	SCONJ
ejpam-4793	469	4	s	s	NOUN
ejpam-4793	469	5	is	be	AUX
ejpam-4793	469	6	a	a	DET
ejpam-4793	469	7	γ	γ	NOUN
ejpam-4793	469	8	-	-	PUNCT
ejpam-4793	469	9	order	order	NOUN
ejpam-4793	469	10	-	-	PUNCT
ejpam-4793	469	11	ideal	ideal	NOUN
ejpam-4793	469	12	of	of	ADP
ejpam-4793	469	13	a	a	DET
ejpam-4793	469	14	γ	γ	X
ejpam-4793	469	15	-	-	PUNCT
ejpam-4793	469	16	monoid	monoid	NOUN
ejpam-4793	469	17	m	m	NOUN
ejpam-4793	469	18	,	,	PUNCT
ejpam-4793	469	19	then	then	ADV
ejpam-4793	469	20	by	by	ADP
ejpam-4793	469	21	remark	remark	NOUN
ejpam-4793	469	22	2	2	NUM
ejpam-4793	469	23	,	,	PUNCT
ejpam-4793	469	24	s	s	VERB
ejpam-4793	469	25	is	be	AUX
ejpam-4793	469	26	a	a	DET
ejpam-4793	469	27	submonoid	submonoid	ADJ
ejpam-4793	469	28	and	and	CCONJ
ejpam-4793	469	29	1	1	NUM
ejpam-4793	469	30	m	m	NOUN
ejpam-4793	469	31	∈	∈	PROPN
ejpam-4793	469	32	s.	s.	PROPN
ejpam-4793	469	33	also	also	ADV
ejpam-4793	469	34	,	,	PUNCT
ejpam-4793	469	35	since	since	SCONJ
ejpam-4793	469	36	s	s	NOUN
ejpam-4793	469	37	is	be	AUX
ejpam-4793	469	38	a	a	DET
ejpam-4793	469	39	γ	γ	NOUN
ejpam-4793	469	40	-	-	PUNCT
ejpam-4793	469	41	order	order	NOUN
ejpam-4793	469	42	-	-	PUNCT
ejpam-4793	469	43	ideal	ideal	ADJ
ejpam-4793	469	44	,	,	PUNCT
ejpam-4793	469	45	for	for	ADP
ejpam-4793	469	46	all	all	DET
ejpam-4793	469	47	α	α	NOUN
ejpam-4793	469	48	,	,	PUNCT
ejpam-4793	469	49	β	β	X
ejpam-4793	469	50	∈	∈	NOUN
ejpam-4793	469	51	γ	γ	NOUN
ejpam-4793	469	52	and	and	CCONJ
ejpam-4793	469	53	for	for	ADP
ejpam-4793	469	54	all	all	DET
ejpam-4793	469	55	s	s	PROPN
ejpam-4793	469	56	,	,	PUNCT
ejpam-4793	469	57	t	t	PROPN
ejpam-4793	469	58	∈	∈	PROPN
ejpam-4793	469	59	s	s	X
ejpam-4793	469	60	,	,	PUNCT
ejpam-4793	469	61	we	we	PRON
ejpam-4793	469	62	have	have	VERB
ejpam-4793	469	63	αs	αs	INTJ
ejpam-4793	469	64	∗	∗	NOUN
ejpam-4793	469	65	βt	βt	ADP
ejpam-4793	470	1	∈	∈	PROPN
ejpam-4793	470	2	s.	s.	PROPN
ejpam-4793	470	3	thus	thus	ADV
ejpam-4793	470	4	,	,	PUNCT
ejpam-4793	470	5	s	s	VERB
ejpam-4793	470	6	is	be	AUX
ejpam-4793	470	7	a	a	DET
ejpam-4793	470	8	γ	γ	NOUN
ejpam-4793	470	9	-	-	ADJ
ejpam-4793	470	10	submonoid	submonoid	NOUN
ejpam-4793	470	11	of	of	ADP
ejpam-4793	470	12	m	m	PROPN
ejpam-4793	470	13	and	and	CCONJ
ejpam-4793	470	14	the	the	DET
ejpam-4793	470	15	following	follow	VERB
ejpam-4793	470	16	remark	remark	NOUN
ejpam-4793	470	17	holds	hold	VERB
ejpam-4793	470	18	.	.	PUNCT
ejpam-4793	471	1	remark	remark	PROPN
ejpam-4793	471	2	10	10	NUM
ejpam-4793	471	3	.	.	PUNCT
ejpam-4793	472	1	every	every	DET
ejpam-4793	472	2	γ	γ	PROPN
ejpam-4793	472	3	-	-	PUNCT
ejpam-4793	472	4	order	order	NOUN
ejpam-4793	472	5	-	-	PUNCT
ejpam-4793	472	6	ideal	ideal	NOUN
ejpam-4793	472	7	of	of	ADP
ejpam-4793	472	8	a	a	DET
ejpam-4793	472	9	γ	γ	PROPN
ejpam-4793	472	10	-	-	PUNCT
ejpam-4793	472	11	monoid	monoid	NOUN
ejpam-4793	472	12	m	m	NOUN
ejpam-4793	472	13	is	be	AUX
ejpam-4793	472	14	a	a	DET
ejpam-4793	472	15	γ	γ	NOUN
ejpam-4793	472	16	-	-	ADJ
ejpam-4793	472	17	submonoid	submonoid	NOUN
ejpam-4793	472	18	of	of	ADP
ejpam-4793	472	19	m	m	PROPN
ejpam-4793	472	20	.	.	PUNCT
ejpam-4793	473	1	however	however	ADV
ejpam-4793	473	2	,	,	PUNCT
ejpam-4793	473	3	a	a	DET
ejpam-4793	473	4	γ	γ	NOUN
ejpam-4793	473	5	-	-	ADJ
ejpam-4793	473	6	submonoid	submonoid	NOUN
ejpam-4793	473	7	of	of	ADP
ejpam-4793	473	8	m	m	PROPN
ejpam-4793	473	9	is	be	AUX
ejpam-4793	473	10	not	not	PART
ejpam-4793	473	11	necessarily	necessarily	ADV
ejpam-4793	473	12	a	a	DET
ejpam-4793	473	13	γ	γ	NOUN
ejpam-4793	473	14	-	-	PUNCT
ejpam-4793	473	15	order	order	NOUN
ejpam-4793	473	16	-	-	PUNCT
ejpam-4793	473	17	ideal	ideal	NOUN
ejpam-4793	473	18	of	of	ADP
ejpam-4793	473	19	m	m	PROPN
ejpam-4793	473	20	.	.	PUNCT
ejpam-4793	474	1	the	the	DET
ejpam-4793	474	2	following	follow	VERB
ejpam-4793	474	3	example	example	NOUN
ejpam-4793	474	4	shows	show	VERB
ejpam-4793	474	5	that	that	SCONJ
ejpam-4793	474	6	a	a	DET
ejpam-4793	474	7	γ	γ	X
ejpam-4793	474	8	-	-	ADJ
ejpam-4793	474	9	submonoid	submonoid	ADJ
ejpam-4793	474	10	is	be	AUX
ejpam-4793	474	11	not	not	PART
ejpam-4793	474	12	necessarily	necessarily	ADV
ejpam-4793	474	13	a	a	DET
ejpam-4793	474	14	normal	normal	ADJ
ejpam-4793	474	15	submonoid	submonoid	NOUN
ejpam-4793	474	16	.	.	PUNCT
ejpam-4793	475	1	h.	h.	PROPN
ejpam-4793	475	2	sarapuddin	sarapuddin	PROPN
ejpam-4793	475	3	,	,	PUNCT
ejpam-4793	475	4	j.	j.	PROPN
ejpam-4793	475	5	vilela	vilela	PROPN
ejpam-4793	475	6	/	/	SYM
ejpam-4793	475	7	eur	eur	PROPN
ejpam-4793	475	8	.	.	PUNCT
ejpam-4793	476	1	j.	j.	PROPN
ejpam-4793	476	2	pure	pure	PROPN
ejpam-4793	476	3	appl	appl	PROPN
ejpam-4793	476	4	.	.	PROPN
ejpam-4793	476	5	math	math	PROPN
ejpam-4793	476	6	,	,	PUNCT
ejpam-4793	476	7	16	16	NUM
ejpam-4793	476	8	(	(	PUNCT
ejpam-4793	476	9	3	3	NUM
ejpam-4793	476	10	)	)	PUNCT
ejpam-4793	476	11	(	(	PUNCT
ejpam-4793	476	12	2023	2023	NUM
ejpam-4793	476	13	)	)	PUNCT
ejpam-4793	476	14	,	,	PUNCT
ejpam-4793	476	15	1772	1772	NUM
ejpam-4793	476	16	-	-	SYM
ejpam-4793	476	17	1793	1793	NUM
ejpam-4793	476	18	1783	1783	NUM
ejpam-4793	476	19	example	example	NOUN
ejpam-4793	476	20	13	13	NUM
ejpam-4793	476	21	.	.	PUNCT
ejpam-4793	476	22	consider	consider	VERB
ejpam-4793	476	23	the	the	DET
ejpam-4793	476	24	γ	γ	X
ejpam-4793	476	25	-	-	ADJ
ejpam-4793	476	26	submonoid	submonoid	ADJ
ejpam-4793	476	27	u	u	NOUN
ejpam-4793	476	28	=	=	PUNCT
ejpam-4793	476	29	{	{	PUNCT
ejpam-4793	476	30	0	0	NUM
ejpam-4793	476	31	,	,	PUNCT
ejpam-4793	476	32	1	1	NUM
ejpam-4793	476	33	,	,	PUNCT
ejpam-4793	476	34	y	y	PROPN
ejpam-4793	476	35	,	,	PUNCT
ejpam-4793	476	36	s	s	PROPN
ejpam-4793	476	37	,	,	PUNCT
ejpam-4793	476	38	b	b	NOUN
ejpam-4793	476	39	}	}	PUNCT
ejpam-4793	476	40	in	in	ADP
ejpam-4793	476	41	example	example	NOUN
ejpam-4793	476	42	11	11	NUM
ejpam-4793	476	43	which	which	PRON
ejpam-4793	476	44	is	be	AUX
ejpam-4793	476	45	also	also	ADV
ejpam-4793	476	46	commutative	commutative	ADJ
ejpam-4793	476	47	.	.	PUNCT
ejpam-4793	477	1	observe	observe	VERB
ejpam-4793	477	2	that	that	SCONJ
ejpam-4793	477	3	y	y	PROPN
ejpam-4793	477	4	,	,	PUNCT
ejpam-4793	477	5	z	z	PROPN
ejpam-4793	477	6	∈	∈	PROPN
ejpam-4793	477	7	m	m	VERB
ejpam-4793	477	8	such	such	ADJ
ejpam-4793	477	9	that	that	SCONJ
ejpam-4793	477	10	y	y	PROPN
ejpam-4793	477	11	,	,	PUNCT
ejpam-4793	477	12	y	y	PROPN
ejpam-4793	477	13	∗	∗	NOUN
ejpam-4793	477	14	z	z	NOUN
ejpam-4793	477	15	=	=	PUNCT
ejpam-4793	477	16	y	y	PROPN
ejpam-4793	477	17	∈	∈	PROPN
ejpam-4793	477	18	u	u	PROPN
ejpam-4793	477	19	.	.	PUNCT
ejpam-4793	478	1	however	however	ADV
ejpam-4793	478	2	,	,	PUNCT
ejpam-4793	478	3	z	z	NOUN
ejpam-4793	478	4	/∈	/∈	PUNCT
ejpam-4793	478	5	u	u	INTJ
ejpam-4793	478	6	.	.	PUNCT
ejpam-4793	479	1	thus	thus	ADV
ejpam-4793	479	2	,	,	PUNCT
ejpam-4793	479	3	u	u	NOUN
ejpam-4793	479	4	is	be	AUX
ejpam-4793	479	5	not	not	PART
ejpam-4793	479	6	a	a	DET
ejpam-4793	479	7	normal	normal	ADJ
ejpam-4793	479	8	submonoid	submonoid	NOUN
ejpam-4793	479	9	of	of	ADP
ejpam-4793	479	10	m	m	PROPN
ejpam-4793	479	11	.	.	PUNCT
ejpam-4793	480	1	remark	remark	PROPN
ejpam-4793	480	2	11	11	NUM
ejpam-4793	480	3	.	.	PUNCT
ejpam-4793	481	1	in	in	ADP
ejpam-4793	481	2	general	general	ADJ
ejpam-4793	481	3	,	,	PUNCT
ejpam-4793	481	4	a	a	DET
ejpam-4793	481	5	γ	γ	NOUN
ejpam-4793	481	6	-	-	ADJ
ejpam-4793	481	7	submonoid	submonoid	NOUN
ejpam-4793	481	8	of	of	ADP
ejpam-4793	481	9	a	a	DET
ejpam-4793	481	10	γ	γ	PROPN
ejpam-4793	481	11	-	-	PUNCT
ejpam-4793	481	12	monoid	monoid	NOUN
ejpam-4793	481	13	m	m	NOUN
ejpam-4793	481	14	is	be	AUX
ejpam-4793	481	15	not	not	PART
ejpam-4793	481	16	necessarily	necessarily	ADV
ejpam-4793	481	17	a	a	DET
ejpam-4793	481	18	normal	normal	ADJ
ejpam-4793	481	19	submonoid	submonoid	NOUN
ejpam-4793	481	20	of	of	ADP
ejpam-4793	481	21	m	m	PROPN
ejpam-4793	481	22	.	.	PUNCT
ejpam-4793	482	1	theorem	theorem	ADJ
ejpam-4793	482	2	7	7	NUM
ejpam-4793	482	3	.	.	PUNCT
ejpam-4793	483	1	let	let	VERB
ejpam-4793	483	2	s	s	PRON
ejpam-4793	483	3	be	be	AUX
ejpam-4793	483	4	a	a	DET
ejpam-4793	483	5	subset	subset	NOUN
ejpam-4793	483	6	of	of	ADP
ejpam-4793	483	7	a	a	DET
ejpam-4793	483	8	γ	γ	X
ejpam-4793	483	9	-	-	PUNCT
ejpam-4793	483	10	monoid	monoid	NOUN
ejpam-4793	483	11	m	m	PROPN
ejpam-4793	483	12	.	.	PUNCT
ejpam-4793	484	1	then	then	ADV
ejpam-4793	484	2	s	s	VERB
ejpam-4793	484	3	is	be	AUX
ejpam-4793	484	4	a	a	DET
ejpam-4793	484	5	γ	γ	NOUN
ejpam-4793	484	6	-	-	PUNCT
ejpam-4793	484	7	order	order	NOUN
ejpam-4793	484	8	-	-	PUNCT
ejpam-4793	484	9	ideal	ideal	NOUN
ejpam-4793	484	10	if	if	SCONJ
ejpam-4793	485	1	and	and	CCONJ
ejpam-4793	485	2	only	only	ADV
ejpam-4793	485	3	if	if	SCONJ
ejpam-4793	485	4	s	s	NOUN
ejpam-4793	485	5	is	be	AUX
ejpam-4793	485	6	a	a	DET
ejpam-4793	485	7	γ	γ	NOUN
ejpam-4793	485	8	-	-	ADJ
ejpam-4793	485	9	submonoid	submonoid	ADJ
ejpam-4793	485	10	such	such	ADJ
ejpam-4793	485	11	that	that	SCONJ
ejpam-4793	485	12	x	x	PROPN
ejpam-4793	485	13	∗	∗	NOUN
ejpam-4793	485	14	y	y	PROPN
ejpam-4793	485	15	∈	∈	PROPN
ejpam-4793	485	16	s	s	PART
ejpam-4793	485	17	implies	imply	VERB
ejpam-4793	485	18	x	x	X
ejpam-4793	485	19	,	,	PUNCT
ejpam-4793	485	20	y	y	PROPN
ejpam-4793	485	21	∈	∈	PROPN
ejpam-4793	485	22	s.	s.	PROPN
ejpam-4793	485	23	proof	proof	PROPN
ejpam-4793	485	24	.	.	PUNCT
ejpam-4793	486	1	let	let	VERB
ejpam-4793	486	2	s	s	PRON
ejpam-4793	486	3	be	be	AUX
ejpam-4793	486	4	a	a	DET
ejpam-4793	486	5	subset	subset	NOUN
ejpam-4793	486	6	of	of	ADP
ejpam-4793	486	7	a	a	DET
ejpam-4793	486	8	γ	γ	X
ejpam-4793	486	9	-	-	PUNCT
ejpam-4793	486	10	monoid	monoid	NOUN
ejpam-4793	486	11	m	m	PROPN
ejpam-4793	486	12	.	.	PUNCT
ejpam-4793	487	1	suppose	suppose	VERB
ejpam-4793	487	2	s	s	PRON
ejpam-4793	487	3	is	be	AUX
ejpam-4793	487	4	a	a	DET
ejpam-4793	487	5	γ	γ	NOUN
ejpam-4793	487	6	-	-	PUNCT
ejpam-4793	487	7	order	order	NOUN
ejpam-4793	487	8	-	-	PUNCT
ejpam-4793	487	9	ideal	ideal	NOUN
ejpam-4793	487	10	of	of	ADP
ejpam-4793	487	11	m	m	PROPN
ejpam-4793	487	12	.	.	PUNCT
ejpam-4793	488	1	then	then	ADV
ejpam-4793	488	2	by	by	ADP
ejpam-4793	488	3	remark	remark	NOUN
ejpam-4793	488	4	10	10	NUM
ejpam-4793	488	5	,	,	PUNCT
ejpam-4793	488	6	s	s	VERB
ejpam-4793	488	7	is	be	AUX
ejpam-4793	488	8	a	a	DET
ejpam-4793	488	9	γ	γ	NOUN
ejpam-4793	488	10	-	-	ADJ
ejpam-4793	488	11	submonoid	submonoid	ADJ
ejpam-4793	488	12	and	and	CCONJ
ejpam-4793	488	13	for	for	ADP
ejpam-4793	488	14	α	α	NOUN
ejpam-4793	488	15	=	=	SYM
ejpam-4793	488	16	β	β	X
ejpam-4793	488	17	=	=	SYM
ejpam-4793	488	18	0	0	NUM
ejpam-4793	488	19	,	,	PUNCT
ejpam-4793	488	20	we	we	PRON
ejpam-4793	488	21	have	have	VERB
ejpam-4793	488	22	x	x	NOUN
ejpam-4793	488	23	∗	∗	NOUN
ejpam-4793	488	24	y	y	NOUN
ejpam-4793	488	25	=	=	SYM
ejpam-4793	488	26	0x	0x	NOUN
ejpam-4793	488	27	∗	∗	NOUN
ejpam-4793	488	28	0y	0y	X
ejpam-4793	488	29	∈	∈	PROPN
ejpam-4793	488	30	s	s	PART
ejpam-4793	488	31	implies	imply	VERB
ejpam-4793	488	32	x	x	X
ejpam-4793	488	33	,	,	PUNCT
ejpam-4793	488	34	y	y	PROPN
ejpam-4793	488	35	∈	∈	PROPN
ejpam-4793	488	36	s	s	AUX
ejpam-4793	488	37	since	since	SCONJ
ejpam-4793	488	38	s	s	PROPN
ejpam-4793	488	39	is	be	AUX
ejpam-4793	488	40	a	a	DET
ejpam-4793	488	41	γ	γ	NOUN
ejpam-4793	488	42	-	-	PUNCT
ejpam-4793	488	43	order	order	NOUN
ejpam-4793	488	44	-	-	PUNCT
ejpam-4793	488	45	ideal	ideal	NOUN
ejpam-4793	488	46	.	.	PUNCT
ejpam-4793	489	1	now	now	ADV
ejpam-4793	489	2	,	,	PUNCT
ejpam-4793	489	3	suppose	suppose	VERB
ejpam-4793	489	4	s	s	NOUN
ejpam-4793	489	5	is	be	AUX
ejpam-4793	489	6	a	a	DET
ejpam-4793	489	7	γ	γ	NOUN
ejpam-4793	489	8	-	-	ADJ
ejpam-4793	489	9	submonoid	submonoid	ADJ
ejpam-4793	489	10	such	such	ADJ
ejpam-4793	489	11	that	that	SCONJ
ejpam-4793	489	12	x	x	PROPN
ejpam-4793	489	13	∗	∗	NOUN
ejpam-4793	489	14	y	y	PROPN
ejpam-4793	489	15	∈	∈	PROPN
ejpam-4793	489	16	s	s	PART
ejpam-4793	489	17	implies	imply	VERB
ejpam-4793	489	18	x	x	X
ejpam-4793	489	19	,	,	PUNCT
ejpam-4793	489	20	y	y	PROPN
ejpam-4793	489	21	∈	∈	PROPN
ejpam-4793	489	22	s.	s.	PROPN
ejpam-4793	489	23	then	then	ADV
ejpam-4793	489	24	for	for	ADP
ejpam-4793	489	25	all	all	DET
ejpam-4793	489	26	α	α	NOUN
ejpam-4793	489	27	,	,	PUNCT
ejpam-4793	489	28	β	β	X
ejpam-4793	489	29	∈	∈	NOUN
ejpam-4793	489	30	γ	γ	NOUN
ejpam-4793	489	31	and	and	CCONJ
ejpam-4793	489	32	for	for	ADP
ejpam-4793	489	33	all	all	DET
ejpam-4793	489	34	x	x	NOUN
ejpam-4793	489	35	,	,	PUNCT
ejpam-4793	489	36	y	y	PROPN
ejpam-4793	489	37	∈	∈	PROPN
ejpam-4793	489	38	s	s	PART
ejpam-4793	489	39	,	,	PUNCT
ejpam-4793	489	40	αx	αx	ADV
ejpam-4793	489	41	∗	∗	VERB
ejpam-4793	489	42	βy	βy	PRON
ejpam-4793	490	1	∈	∈	PROPN
ejpam-4793	490	2	s.	s.	PROPN
ejpam-4793	490	3	suppose	suppose	VERB
ejpam-4793	490	4	for	for	ADP
ejpam-4793	490	5	all	all	DET
ejpam-4793	490	6	α	α	NOUN
ejpam-4793	490	7	,	,	PUNCT
ejpam-4793	490	8	β	β	PROPN
ejpam-4793	490	9	∈	∈	PROPN
ejpam-4793	490	10	γ	γ	X
ejpam-4793	490	11	,	,	PUNCT
ejpam-4793	490	12	αx	αx	ADV
ejpam-4793	490	13	∗	∗	VERB
ejpam-4793	490	14	βy	βy	PRON
ejpam-4793	490	15	∈	∈	PROPN
ejpam-4793	490	16	s.	s.	PROPN
ejpam-4793	490	17	take	take	VERB
ejpam-4793	490	18	α	α	NOUN
ejpam-4793	490	19	=	=	SYM
ejpam-4793	490	20	β	β	X
ejpam-4793	490	21	=	=	SYM
ejpam-4793	491	1	0	0	X
ejpam-4793	491	2	.	.	PUNCT
ejpam-4793	492	1	then	then	ADV
ejpam-4793	492	2	x	x	X
ejpam-4793	492	3	∗	∗	NOUN
ejpam-4793	492	4	y	y	NOUN
ejpam-4793	492	5	=	=	SYM
ejpam-4793	492	6	0x	0x	NOUN
ejpam-4793	492	7	∗	∗	NOUN
ejpam-4793	492	8	0y	0y	X
ejpam-4793	492	9	∈	∈	PROPN
ejpam-4793	492	10	s	s	X
ejpam-4793	492	11	which	which	PRON
ejpam-4793	492	12	implies	imply	VERB
ejpam-4793	492	13	that	that	SCONJ
ejpam-4793	492	14	x	x	X
ejpam-4793	492	15	,	,	PUNCT
ejpam-4793	492	16	y	y	PROPN
ejpam-4793	492	17	∈	∈	PROPN
ejpam-4793	492	18	s.	s.	PROPN
ejpam-4793	492	19	therefore	therefore	ADV
ejpam-4793	492	20	,	,	PUNCT
ejpam-4793	492	21	s	s	VERB
ejpam-4793	492	22	is	be	AUX
ejpam-4793	492	23	a	a	DET
ejpam-4793	492	24	γ	γ	NOUN
ejpam-4793	492	25	-	-	PUNCT
ejpam-4793	492	26	order	order	NOUN
ejpam-4793	492	27	-	-	PUNCT
ejpam-4793	492	28	ideal	ideal	NOUN
ejpam-4793	492	29	.	.	PUNCT
ejpam-4793	493	1	lemma	lemma	PROPN
ejpam-4793	493	2	4	4	X
ejpam-4793	493	3	.	.	PUNCT
ejpam-4793	494	1	let	let	VERB
ejpam-4793	494	2	a	a	PRON
ejpam-4793	494	3	and	and	CCONJ
ejpam-4793	494	4	b	b	NOUN
ejpam-4793	494	5	be	be	AUX
ejpam-4793	494	6	γ	γ	NOUN
ejpam-4793	494	7	-	-	NOUN
ejpam-4793	494	8	submonoids	submonoid	NOUN
ejpam-4793	494	9	of	of	ADP
ejpam-4793	494	10	a	a	DET
ejpam-4793	494	11	γ	γ	X
ejpam-4793	494	12	-	-	PUNCT
ejpam-4793	494	13	monoid	monoid	NOUN
ejpam-4793	494	14	m	m	PROPN
ejpam-4793	494	15	.	.	PUNCT
ejpam-4793	495	1	then	then	ADV
ejpam-4793	495	2	(	(	PUNCT
ejpam-4793	495	3	i	i	NOUN
ejpam-4793	495	4	)	)	PUNCT
ejpam-4793	495	5	a	a	DET
ejpam-4793	495	6	∩b	∩b	NOUN
ejpam-4793	495	7	is	be	AUX
ejpam-4793	495	8	a	a	DET
ejpam-4793	495	9	γ	γ	NOUN
ejpam-4793	495	10	-	-	ADJ
ejpam-4793	495	11	submonoid	submonoid	NOUN
ejpam-4793	495	12	of	of	ADP
ejpam-4793	495	13	m	m	PROPN
ejpam-4793	495	14	.	.	PUNCT
ejpam-4793	496	1	(	(	PUNCT
ejpam-4793	496	2	ii	ii	NOUN
ejpam-4793	496	3	)	)	PUNCT
ejpam-4793	496	4	if	if	SCONJ
ejpam-4793	496	5	m	m	NOUN
ejpam-4793	496	6	is	be	AUX
ejpam-4793	496	7	commutative	commutative	ADJ
ejpam-4793	496	8	and	and	CCONJ
ejpam-4793	496	9	a	a	DET
ejpam-4793	496	10	,	,	PUNCT
ejpam-4793	496	11	b	b	NOUN
ejpam-4793	496	12	are	be	AUX
ejpam-4793	496	13	normal	normal	ADJ
ejpam-4793	496	14	,	,	PUNCT
ejpam-4793	496	15	then	then	ADV
ejpam-4793	496	16	a∩b	a∩b	PROPN
ejpam-4793	496	17	is	be	AUX
ejpam-4793	496	18	a	a	DET
ejpam-4793	496	19	normal	normal	ADJ
ejpam-4793	496	20	γ	γ	NOUN
ejpam-4793	496	21	-	-	NOUN
ejpam-4793	496	22	submonoid	submonoid	NOUN
ejpam-4793	496	23	of	of	ADP
ejpam-4793	496	24	m	m	PROPN
ejpam-4793	496	25	.	.	PUNCT
ejpam-4793	497	1	proof	proof	NOUN
ejpam-4793	497	2	.	.	PUNCT
ejpam-4793	498	1	let	let	VERB
ejpam-4793	498	2	a	a	PRON
ejpam-4793	498	3	and	and	CCONJ
ejpam-4793	498	4	b	b	NOUN
ejpam-4793	498	5	be	be	AUX
ejpam-4793	498	6	γ	γ	NOUN
ejpam-4793	498	7	-	-	NOUN
ejpam-4793	498	8	submonoids	submonoid	NOUN
ejpam-4793	498	9	of	of	ADP
ejpam-4793	498	10	a	a	DET
ejpam-4793	498	11	γ	γ	X
ejpam-4793	498	12	-	-	PUNCT
ejpam-4793	498	13	monoid	monoid	NOUN
ejpam-4793	498	14	m	m	NOUN
ejpam-4793	498	15	.	.	PUNCT
ejpam-4793	499	1	(	(	PUNCT
ejpam-4793	499	2	i	i	NOUN
ejpam-4793	499	3	)	)	PUNCT
ejpam-4793	499	4	since	since	SCONJ
ejpam-4793	499	5	a	a	PRON
ejpam-4793	499	6	and	and	CCONJ
ejpam-4793	499	7	b	b	NOUN
ejpam-4793	499	8	are	be	AUX
ejpam-4793	499	9	γ	γ	NOUN
ejpam-4793	499	10	-	-	NOUN
ejpam-4793	499	11	submonoids	submonoid	NOUN
ejpam-4793	499	12	of	of	ADP
ejpam-4793	499	13	m	m	PRON
ejpam-4793	499	14	,	,	PUNCT
ejpam-4793	499	15	the	the	DET
ejpam-4793	499	16	identity	identity	NOUN
ejpam-4793	499	17	1	1	NUM
ejpam-4793	499	18	m	m	NOUN
ejpam-4793	499	19	∈	∈	NOUN
ejpam-4793	499	20	a	a	DET
ejpam-4793	499	21	and	and	CCONJ
ejpam-4793	499	22	1	1	NUM
ejpam-4793	499	23	m	m	PROPN
ejpam-4793	499	24	∈	∈	PROPN
ejpam-4793	499	25	b.	b.	PROPN
ejpam-4793	500	1	thus	thus	ADV
ejpam-4793	500	2	,	,	PUNCT
ejpam-4793	500	3	1	1	NUM
ejpam-4793	500	4	m	m	NOUN
ejpam-4793	500	5	∈	∈	PROPN
ejpam-4793	500	6	a∩b	a∩b	PROPN
ejpam-4793	500	7	.	.	PUNCT
ejpam-4793	501	1	now	now	ADV
ejpam-4793	501	2	,	,	PUNCT
ejpam-4793	501	3	let	let	VERB
ejpam-4793	501	4	a	a	DET
ejpam-4793	501	5	,	,	PUNCT
ejpam-4793	501	6	b	b	PROPN
ejpam-4793	501	7	∈	∈	PROPN
ejpam-4793	501	8	a∩b	a∩b	PROPN
ejpam-4793	501	9	.	.	PUNCT
ejpam-4793	502	1	then	then	ADV
ejpam-4793	502	2	,	,	PUNCT
ejpam-4793	502	3	a	a	DET
ejpam-4793	502	4	,	,	PUNCT
ejpam-4793	502	5	b	b	X
ejpam-4793	502	6	∈	∈	PROPN
ejpam-4793	502	7	a	a	PRON
ejpam-4793	502	8	and	and	CCONJ
ejpam-4793	502	9	a	a	PRON
ejpam-4793	502	10	,	,	PUNCT
ejpam-4793	502	11	b	b	PROPN
ejpam-4793	502	12	∈	∈	PROPN
ejpam-4793	502	13	b.	b.	PROPN
ejpam-4793	502	14	since	since	SCONJ
ejpam-4793	502	15	a	a	PRON
ejpam-4793	502	16	and	and	CCONJ
ejpam-4793	502	17	b	b	NOUN
ejpam-4793	502	18	are	be	AUX
ejpam-4793	502	19	γ	γ	NOUN
ejpam-4793	502	20	-	-	NOUN
ejpam-4793	502	21	submonoids	submonoid	NOUN
ejpam-4793	502	22	,	,	PUNCT
ejpam-4793	502	23	for	for	ADP
ejpam-4793	502	24	all	all	DET
ejpam-4793	502	25	α	α	NOUN
ejpam-4793	502	26	,	,	PUNCT
ejpam-4793	502	27	β	β	PROPN
ejpam-4793	502	28	∈	∈	PROPN
ejpam-4793	502	29	γ	γ	X
ejpam-4793	502	30	,	,	PUNCT
ejpam-4793	502	31	αa	αa	ADV
ejpam-4793	502	32	∗	∗	VERB
ejpam-4793	502	33	βb	βb	DET
ejpam-4793	502	34	∈	∈	PROPN
ejpam-4793	502	35	a	a	DET
ejpam-4793	502	36	and	and	CCONJ
ejpam-4793	502	37	αa	αa	NOUN
ejpam-4793	502	38	∗	∗	NOUN
ejpam-4793	502	39	βb	βb	PROPN
ejpam-4793	502	40	∈	∈	PROPN
ejpam-4793	502	41	b.	b.	PROPN
ejpam-4793	503	1	hence	hence	ADV
ejpam-4793	503	2	,	,	PUNCT
ejpam-4793	503	3	αa	αa	ADV
ejpam-4793	503	4	∗	∗	VERB
ejpam-4793	503	5	βb	βb	DET
ejpam-4793	503	6	∈	∈	PROPN
ejpam-4793	503	7	a∩b	a∩b	PROPN
ejpam-4793	503	8	.	.	PUNCT
ejpam-4793	504	1	therefore	therefore	ADV
ejpam-4793	504	2	,	,	PUNCT
ejpam-4793	504	3	a	a	DET
ejpam-4793	504	4	∩b	∩b	NOUN
ejpam-4793	504	5	is	be	AUX
ejpam-4793	504	6	a	a	DET
ejpam-4793	504	7	γ	γ	NOUN
ejpam-4793	504	8	-	-	ADJ
ejpam-4793	504	9	submonoid	submonoid	NOUN
ejpam-4793	504	10	of	of	ADP
ejpam-4793	504	11	m	m	PROPN
ejpam-4793	504	12	.	.	PUNCT
ejpam-4793	505	1	(	(	PUNCT
ejpam-4793	505	2	ii	ii	NOUN
ejpam-4793	505	3	)	)	PUNCT
ejpam-4793	505	4	by	by	ADP
ejpam-4793	505	5	(	(	PUNCT
ejpam-4793	505	6	i	i	NOUN
ejpam-4793	505	7	)	)	PUNCT
ejpam-4793	505	8	,	,	PUNCT
ejpam-4793	505	9	a∩b	a∩b	PROPN
ejpam-4793	505	10	is	be	AUX
ejpam-4793	505	11	a	a	DET
ejpam-4793	505	12	γ	γ	NOUN
ejpam-4793	505	13	-	-	ADJ
ejpam-4793	505	14	submonoid	submonoid	NOUN
ejpam-4793	505	15	of	of	ADP
ejpam-4793	505	16	m	m	PROPN
ejpam-4793	505	17	.	.	PUNCT
ejpam-4793	506	1	it	it	PRON
ejpam-4793	506	2	remains	remain	VERB
ejpam-4793	506	3	to	to	PART
ejpam-4793	506	4	show	show	VERB
ejpam-4793	506	5	that	that	SCONJ
ejpam-4793	506	6	a∩b	a∩b	PROPN
ejpam-4793	506	7	is	be	AUX
ejpam-4793	506	8	normal	normal	ADJ
ejpam-4793	506	9	.	.	PUNCT
ejpam-4793	507	1	let	let	VERB
ejpam-4793	507	2	x	x	PRON
ejpam-4793	507	3	,	,	PUNCT
ejpam-4793	507	4	x	x	SYM
ejpam-4793	507	5	∗	∗	NOUN
ejpam-4793	507	6	y	y	PROPN
ejpam-4793	507	7	∈	∈	PROPN
ejpam-4793	507	8	a	a	DET
ejpam-4793	507	9	∩	∩	ADJ
ejpam-4793	507	10	b.	b.	NOUN
ejpam-4793	507	11	then	then	ADV
ejpam-4793	507	12	x	x	X
ejpam-4793	507	13	,	,	PUNCT
ejpam-4793	507	14	x	x	SYM
ejpam-4793	507	15	∗	∗	NOUN
ejpam-4793	507	16	y	y	PROPN
ejpam-4793	507	17	∈	∈	PROPN
ejpam-4793	507	18	a	a	PRON
ejpam-4793	507	19	and	and	CCONJ
ejpam-4793	507	20	x	x	NOUN
ejpam-4793	507	21	,	,	PUNCT
ejpam-4793	507	22	x	x	PUNCT
ejpam-4793	507	23	∗	∗	NOUN
ejpam-4793	507	24	y	y	PROPN
ejpam-4793	507	25	∈	∈	PROPN
ejpam-4793	507	26	b.	b.	PROPN
ejpam-4793	507	27	since	since	SCONJ
ejpam-4793	507	28	a	a	PRON
ejpam-4793	507	29	and	and	CCONJ
ejpam-4793	507	30	b	b	NOUN
ejpam-4793	507	31	are	be	AUX
ejpam-4793	507	32	normal	normal	ADJ
ejpam-4793	507	33	,	,	PUNCT
ejpam-4793	507	34	y	y	PROPN
ejpam-4793	507	35	∈	∈	PROPN
ejpam-4793	507	36	a	a	PRON
ejpam-4793	507	37	and	and	CCONJ
ejpam-4793	507	38	y	y	PROPN
ejpam-4793	507	39	∈	∈	PROPN
ejpam-4793	507	40	b.	b.	PROPN
ejpam-4793	507	41	therefore	therefore	ADV
ejpam-4793	507	42	,	,	PUNCT
ejpam-4793	507	43	y	y	PROPN
ejpam-4793	507	44	∈	∈	PROPN
ejpam-4793	507	45	a	a	DET
ejpam-4793	507	46	∩	∩	ADJ
ejpam-4793	507	47	b	b	NOUN
ejpam-4793	507	48	and	and	CCONJ
ejpam-4793	507	49	a	a	DET
ejpam-4793	507	50	∩	∩	ADJ
ejpam-4793	507	51	b	b	NOUN
ejpam-4793	507	52	is	be	AUX
ejpam-4793	507	53	a	a	DET
ejpam-4793	507	54	normal	normal	ADJ
ejpam-4793	507	55	γ	γ	NOUN
ejpam-4793	507	56	-	-	NOUN
ejpam-4793	507	57	submonoid	submonoid	NOUN
ejpam-4793	507	58	of	of	ADP
ejpam-4793	507	59	m	m	PROPN
ejpam-4793	507	60	.	.	PUNCT
ejpam-4793	508	1	example	example	NOUN
ejpam-4793	508	2	14	14	NUM
ejpam-4793	508	3	.	.	PUNCT
ejpam-4793	509	1	consider	consider	VERB
ejpam-4793	509	2	the	the	DET
ejpam-4793	509	3	γ	γ	NOUN
ejpam-4793	509	4	-	-	ADJ
ejpam-4793	509	5	submonoids	submonoid	NOUN
ejpam-4793	509	6	v	v	NOUN
ejpam-4793	509	7	=	=	SYM
ejpam-4793	509	8	{	{	PUNCT
ejpam-4793	509	9	0	0	NUM
ejpam-4793	509	10	,	,	PUNCT
ejpam-4793	509	11	1	1	NUM
ejpam-4793	509	12	,	,	PUNCT
ejpam-4793	509	13	x	x	NOUN
ejpam-4793	509	14	}	}	PUNCT
ejpam-4793	509	15	and	and	CCONJ
ejpam-4793	509	16	w	w	NOUN
ejpam-4793	509	17	=	=	PUNCT
ejpam-4793	509	18	{	{	PUNCT
ejpam-4793	509	19	0	0	NUM
ejpam-4793	509	20	,	,	PUNCT
ejpam-4793	509	21	y	y	NOUN
ejpam-4793	509	22	}	}	PUNCT
ejpam-4793	509	23	in	in	ADP
ejpam-4793	509	24	example	example	NOUN
ejpam-4793	509	25	11	11	NUM
ejpam-4793	509	26	.	.	PUNCT
ejpam-4793	510	1	then	then	ADV
ejpam-4793	510	2	,	,	PUNCT
ejpam-4793	510	3	v	v	ADP
ejpam-4793	510	4	∪w	∪w	PROPN
ejpam-4793	510	5	=	=	PUNCT
ejpam-4793	510	6	{	{	PUNCT
ejpam-4793	510	7	0	0	NUM
ejpam-4793	510	8	,	,	PUNCT
ejpam-4793	510	9	1	1	NUM
ejpam-4793	510	10	,	,	PUNCT
ejpam-4793	510	11	x	x	NOUN
ejpam-4793	510	12	,	,	PUNCT
ejpam-4793	510	13	y	y	PROPN
ejpam-4793	510	14	}	}	PUNCT
ejpam-4793	510	15	.	.	PUNCT
ejpam-4793	511	1	now	now	ADV
ejpam-4793	511	2	,	,	PUNCT
ejpam-4793	511	3	for	for	ADP
ejpam-4793	511	4	x	x	X
ejpam-4793	511	5	,	,	PUNCT
ejpam-4793	511	6	y	y	PROPN
ejpam-4793	511	7	∈	∈	PROPN
ejpam-4793	511	8	v	v	AUX
ejpam-4793	511	9	∪w	∪w	PROPN
ejpam-4793	511	10	,	,	PUNCT
ejpam-4793	511	11	we	we	PRON
ejpam-4793	511	12	have	have	VERB
ejpam-4793	511	13	x	x	NOUN
ejpam-4793	511	14	∗	∗	NOUN
ejpam-4793	511	15	y	y	NOUN
ejpam-4793	511	16	=	=	SYM
ejpam-4793	511	17	s	s	PROPN
ejpam-4793	511	18	/∈	/∈	X
ejpam-4793	511	19	v	v	ADP
ejpam-4793	511	20	∪w	∪w	PROPN
ejpam-4793	511	21	.	.	PUNCT
ejpam-4793	512	1	thus	thus	ADV
ejpam-4793	512	2	,	,	PUNCT
ejpam-4793	512	3	v	v	X
ejpam-4793	512	4	∪w	∪w	PROPN
ejpam-4793	512	5	is	be	AUX
ejpam-4793	512	6	not	not	PART
ejpam-4793	512	7	a	a	DET
ejpam-4793	512	8	γ	γ	NOUN
ejpam-4793	512	9	-	-	ADJ
ejpam-4793	512	10	submonoid	submonoid	NOUN
ejpam-4793	512	11	of	of	ADP
ejpam-4793	512	12	m	m	PROPN
ejpam-4793	512	13	.	.	PUNCT
ejpam-4793	513	1	remark	remark	PROPN
ejpam-4793	513	2	12	12	NUM
ejpam-4793	513	3	.	.	PUNCT
ejpam-4793	514	1	the	the	DET
ejpam-4793	514	2	union	union	NOUN
ejpam-4793	514	3	of	of	ADP
ejpam-4793	514	4	two	two	NUM
ejpam-4793	514	5	γ	γ	NOUN
ejpam-4793	514	6	-	-	PUNCT
ejpam-4793	514	7	submonoids	submonoid	NOUN
ejpam-4793	514	8	of	of	ADP
ejpam-4793	514	9	a	a	DET
ejpam-4793	514	10	γ	γ	PROPN
ejpam-4793	514	11	-	-	PUNCT
ejpam-4793	514	12	monoid	monoid	NOUN
ejpam-4793	514	13	m	m	NOUN
ejpam-4793	514	14	is	be	AUX
ejpam-4793	514	15	not	not	PART
ejpam-4793	514	16	necessarily	necessarily	ADV
ejpam-4793	514	17	a	a	DET
ejpam-4793	514	18	γsubmonoid	γsubmonoid	NOUN
ejpam-4793	514	19	of	of	ADP
ejpam-4793	514	20	m	m	PROPN
ejpam-4793	514	21	.	.	PUNCT
ejpam-4793	515	1	theorem	theorem	ADJ
ejpam-4793	515	2	8	8	NUM
ejpam-4793	515	3	.	.	PUNCT
ejpam-4793	516	1	let	let	VERB
ejpam-4793	516	2	(	(	PUNCT
ejpam-4793	516	3	m	m	NOUN
ejpam-4793	516	4	,	,	PUNCT
ejpam-4793	516	5	∗	∗	NOUN
ejpam-4793	516	6	)	)	PUNCT
ejpam-4793	516	7	and	and	CCONJ
ejpam-4793	516	8	(	(	PUNCT
ejpam-4793	516	9	n	n	CCONJ
ejpam-4793	516	10	,	,	PUNCT
ejpam-4793	516	11	·	·	PUNCT
ejpam-4793	516	12	)	)	PUNCT
ejpam-4793	516	13	be	be	AUX
ejpam-4793	516	14	γ	γ	NOUN
ejpam-4793	516	15	-	-	PUNCT
ejpam-4793	516	16	monoids	monoid	NOUN
ejpam-4793	516	17	and	and	CCONJ
ejpam-4793	516	18	φ	φ	NOUN
ejpam-4793	516	19	:	:	PUNCT
ejpam-4793	516	20	m	m	VERB
ejpam-4793	516	21	→	→	SYM
ejpam-4793	516	22	n	n	CCONJ
ejpam-4793	516	23	a	a	DET
ejpam-4793	516	24	γ	γ	PROPN
ejpam-4793	516	25	-	-	PUNCT
ejpam-4793	516	26	monoid	monoid	NOUN
ejpam-4793	516	27	homomorphism	homomorphism	NOUN
ejpam-4793	516	28	.	.	PUNCT
ejpam-4793	517	1	(	(	PUNCT
ejpam-4793	517	2	i	i	NOUN
ejpam-4793	517	3	)	)	PUNCT
ejpam-4793	517	4	if	if	SCONJ
ejpam-4793	517	5	s	s	PROPN
ejpam-4793	517	6	is	be	AUX
ejpam-4793	517	7	a	a	DET
ejpam-4793	517	8	γ	γ	NOUN
ejpam-4793	517	9	-	-	ADJ
ejpam-4793	517	10	submonoid	submonoid	NOUN
ejpam-4793	517	11	of	of	ADP
ejpam-4793	517	12	m	m	PRON
ejpam-4793	517	13	,	,	PUNCT
ejpam-4793	517	14	then	then	ADV
ejpam-4793	517	15	φ(s	φ(s	NOUN
ejpam-4793	517	16	)	)	PUNCT
ejpam-4793	517	17	is	be	AUX
ejpam-4793	517	18	a	a	DET
ejpam-4793	517	19	γ	γ	NOUN
ejpam-4793	517	20	-	-	ADJ
ejpam-4793	517	21	submonoid	submonoid	NOUN
ejpam-4793	517	22	of	of	ADP
ejpam-4793	517	23	n	n	PROPN
ejpam-4793	517	24	.	.	PUNCT
ejpam-4793	518	1	in	in	ADP
ejpam-4793	518	2	particular	particular	ADJ
ejpam-4793	518	3	,	,	PUNCT
ejpam-4793	518	4	φ(m	φ(m	NOUN
ejpam-4793	518	5	)	)	PUNCT
ejpam-4793	518	6	is	be	AUX
ejpam-4793	518	7	a	a	DET
ejpam-4793	518	8	γ	γ	NOUN
ejpam-4793	518	9	-	-	ADJ
ejpam-4793	518	10	submonoid	submonoid	NOUN
ejpam-4793	518	11	of	of	ADP
ejpam-4793	518	12	n	n	PROPN
ejpam-4793	518	13	.	.	PUNCT
ejpam-4793	519	1	h.	h.	PROPN
ejpam-4793	519	2	sarapuddin	sarapuddin	PROPN
ejpam-4793	519	3	,	,	PUNCT
ejpam-4793	519	4	j.	j.	PROPN
ejpam-4793	519	5	vilela	vilela	PROPN
ejpam-4793	519	6	/	/	SYM
ejpam-4793	519	7	eur	eur	PROPN
ejpam-4793	519	8	.	.	PUNCT
ejpam-4793	520	1	j.	j.	PROPN
ejpam-4793	520	2	pure	pure	PROPN
ejpam-4793	520	3	appl	appl	PROPN
ejpam-4793	520	4	.	.	PROPN
ejpam-4793	520	5	math	math	PROPN
ejpam-4793	520	6	,	,	PUNCT
ejpam-4793	520	7	16	16	NUM
ejpam-4793	520	8	(	(	PUNCT
ejpam-4793	520	9	3	3	NUM
ejpam-4793	520	10	)	)	PUNCT
ejpam-4793	520	11	(	(	PUNCT
ejpam-4793	520	12	2023	2023	NUM
ejpam-4793	520	13	)	)	PUNCT
ejpam-4793	520	14	,	,	PUNCT
ejpam-4793	520	15	1772	1772	NUM
ejpam-4793	520	16	-	-	SYM
ejpam-4793	520	17	1793	1793	NUM
ejpam-4793	520	18	1784	1784	NUM
ejpam-4793	520	19	(	(	PUNCT
ejpam-4793	520	20	ii	ii	NOUN
ejpam-4793	520	21	)	)	PUNCT
ejpam-4793	520	22	if	if	SCONJ
ejpam-4793	520	23	t	t	PROPN
ejpam-4793	520	24	is	be	AUX
ejpam-4793	520	25	a	a	DET
ejpam-4793	520	26	γ	γ	NOUN
ejpam-4793	520	27	-	-	ADJ
ejpam-4793	520	28	submonoid	submonoid	NOUN
ejpam-4793	520	29	of	of	ADP
ejpam-4793	520	30	n	n	PROPN
ejpam-4793	520	31	,	,	PUNCT
ejpam-4793	520	32	then	then	ADV
ejpam-4793	520	33	φ−1(t	φ−1(t	NOUN
ejpam-4793	520	34	)	)	PUNCT
ejpam-4793	520	35	is	be	AUX
ejpam-4793	520	36	a	a	DET
ejpam-4793	520	37	γ	γ	NOUN
ejpam-4793	520	38	-	-	ADJ
ejpam-4793	520	39	submonoid	submonoid	NOUN
ejpam-4793	520	40	of	of	ADP
ejpam-4793	520	41	m	m	PROPN
ejpam-4793	520	42	.	.	PUNCT
ejpam-4793	521	1	(	(	PUNCT
ejpam-4793	521	2	iii	iii	X
ejpam-4793	521	3	)	)	PUNCT
ejpam-4793	521	4	kerφ	kerφ	PROPN
ejpam-4793	521	5	is	be	AUX
ejpam-4793	521	6	a	a	DET
ejpam-4793	521	7	γ	γ	NOUN
ejpam-4793	521	8	-	-	ADJ
ejpam-4793	521	9	submonoid	submonoid	NOUN
ejpam-4793	521	10	of	of	ADP
ejpam-4793	521	11	m	m	PROPN
ejpam-4793	521	12	.	.	PUNCT
ejpam-4793	522	1	(	(	PUNCT
ejpam-4793	522	2	iv	iv	X
ejpam-4793	522	3	)	)	PUNCT
ejpam-4793	522	4	if	if	SCONJ
ejpam-4793	522	5	m	m	NOUN
ejpam-4793	522	6	is	be	AUX
ejpam-4793	522	7	commutative	commutative	ADJ
ejpam-4793	522	8	,	,	PUNCT
ejpam-4793	522	9	then	then	ADV
ejpam-4793	522	10	kerφ	kerφ	PROPN
ejpam-4793	522	11	is	be	AUX
ejpam-4793	522	12	normal	normal	ADJ
ejpam-4793	522	13	.	.	PUNCT
ejpam-4793	523	1	proof	proof	NOUN
ejpam-4793	523	2	.	.	PUNCT
ejpam-4793	524	1	let	let	VERB
ejpam-4793	524	2	φ	φ	NOUN
ejpam-4793	524	3	:	:	PUNCT
ejpam-4793	524	4	m	m	PROPN
ejpam-4793	524	5	→	→	SYM
ejpam-4793	524	6	n	n	CCONJ
ejpam-4793	524	7	be	be	AUX
ejpam-4793	524	8	a	a	DET
ejpam-4793	524	9	γ	γ	NOUN
ejpam-4793	524	10	-	-	PUNCT
ejpam-4793	524	11	monoid	monoid	NOUN
ejpam-4793	524	12	homomorphism	homomorphism	NOUN
ejpam-4793	524	13	.	.	PUNCT
ejpam-4793	525	1	(	(	PUNCT
ejpam-4793	525	2	i	i	NOUN
ejpam-4793	525	3	)	)	PUNCT
ejpam-4793	525	4	let	let	VERB
ejpam-4793	525	5	s	s	PRON
ejpam-4793	525	6	be	be	AUX
ejpam-4793	525	7	a	a	DET
ejpam-4793	525	8	γ	γ	NOUN
ejpam-4793	525	9	-	-	ADJ
ejpam-4793	525	10	submonoid	submonoid	NOUN
ejpam-4793	525	11	of	of	ADP
ejpam-4793	525	12	m	m	PROPN
ejpam-4793	525	13	.	.	PUNCT
ejpam-4793	526	1	then	then	ADV
ejpam-4793	526	2	1	1	NUM
ejpam-4793	526	3	m	m	NOUN
ejpam-4793	526	4	∈	∈	NOUN
ejpam-4793	526	5	s	s	NOUN
ejpam-4793	526	6	and	and	CCONJ
ejpam-4793	526	7	1n	1n	NUM
ejpam-4793	526	8	=	=	SYM
ejpam-4793	526	9	φ(1	φ(1	PROPN
ejpam-4793	526	10	m	m	NOUN
ejpam-4793	526	11	)	)	PUNCT
ejpam-4793	526	12	∈	∈	PROPN
ejpam-4793	526	13	φ(s	φ(s	NOUN
ejpam-4793	526	14	)	)	PUNCT
ejpam-4793	526	15	.	.	PUNCT
ejpam-4793	527	1	let	let	VERB
ejpam-4793	527	2	x	x	PRON
ejpam-4793	527	3	,	,	PUNCT
ejpam-4793	527	4	y	y	PROPN
ejpam-4793	527	5	∈	∈	PROPN
ejpam-4793	527	6	φ(s	φ(s	NOUN
ejpam-4793	527	7	)	)	PUNCT
ejpam-4793	527	8	.	.	PUNCT
ejpam-4793	528	1	then	then	ADV
ejpam-4793	528	2	x	x	X
ejpam-4793	528	3	=	=	PUNCT
ejpam-4793	528	4	φ(a	φ(a	ADJ
ejpam-4793	528	5	)	)	PUNCT
ejpam-4793	528	6	and	and	CCONJ
ejpam-4793	528	7	y	y	PROPN
ejpam-4793	528	8	=	=	PUNCT
ejpam-4793	528	9	φ(b	φ(b	PROPN
ejpam-4793	528	10	)	)	PUNCT
ejpam-4793	528	11	for	for	ADP
ejpam-4793	528	12	some	some	PRON
ejpam-4793	528	13	a	a	PRON
ejpam-4793	528	14	,	,	PUNCT
ejpam-4793	528	15	b	b	PROPN
ejpam-4793	528	16	∈	∈	PROPN
ejpam-4793	528	17	s.	s.	PROPN
ejpam-4793	528	18	since	since	SCONJ
ejpam-4793	528	19	s	s	PROPN
ejpam-4793	528	20	is	be	AUX
ejpam-4793	528	21	a	a	DET
ejpam-4793	528	22	γsubmonoid	γsubmonoid	NOUN
ejpam-4793	528	23	,	,	PUNCT
ejpam-4793	528	24	for	for	ADP
ejpam-4793	528	25	all	all	DET
ejpam-4793	528	26	α	α	NOUN
ejpam-4793	528	27	,	,	PUNCT
ejpam-4793	528	28	β	β	PROPN
ejpam-4793	528	29	∈	∈	PROPN
ejpam-4793	528	30	γ	γ	X
ejpam-4793	528	31	,	,	PUNCT
ejpam-4793	528	32	αa	αa	ADV
ejpam-4793	528	33	∗	∗	VERB
ejpam-4793	528	34	βb	βb	DET
ejpam-4793	528	35	∈	∈	PROPN
ejpam-4793	528	36	s.	s.	PROPN
ejpam-4793	528	37	now	now	ADV
ejpam-4793	528	38	,	,	PUNCT
ejpam-4793	528	39	for	for	ADP
ejpam-4793	528	40	all	all	DET
ejpam-4793	528	41	α	α	NOUN
ejpam-4793	528	42	,	,	PUNCT
ejpam-4793	528	43	β	β	PROPN
ejpam-4793	528	44	∈	∈	PROPN
ejpam-4793	528	45	γ	γ	X
ejpam-4793	528	46	,	,	PUNCT
ejpam-4793	528	47	we	we	PRON
ejpam-4793	528	48	have	have	VERB
ejpam-4793	528	49	αx	αx	ADV
ejpam-4793	528	50	·	·	PUNCT
ejpam-4793	528	51	βy	βy	PRON
ejpam-4793	528	52	=	=	PUNCT
ejpam-4793	528	53	αφ(a	αφ(a	NUM
ejpam-4793	528	54	)	)	PUNCT
ejpam-4793	528	55	·	·	PUNCT
ejpam-4793	528	56	βφ(b	βφ(b	NUM
ejpam-4793	528	57	)	)	PUNCT
ejpam-4793	528	58	=	=	SYM
ejpam-4793	528	59	φ(αa	φ(αa	X
ejpam-4793	528	60	)	)	PUNCT
ejpam-4793	528	61	·	·	PUNCT
ejpam-4793	528	62	φ(βb	φ(βb	ADJ
ejpam-4793	528	63	)	)	PUNCT
ejpam-4793	528	64	=	=	SYM
ejpam-4793	528	65	φ(αa	φ(αa	X
ejpam-4793	528	66	∗	∗	NOUN
ejpam-4793	528	67	βb	βb	ADJ
ejpam-4793	528	68	)	)	PUNCT
ejpam-4793	528	69	.	.	PUNCT
ejpam-4793	529	1	since	since	SCONJ
ejpam-4793	529	2	αa	αa	NOUN
ejpam-4793	529	3	∗	∗	VERB
ejpam-4793	529	4	βb	βb	DET
ejpam-4793	529	5	∈	∈	PROPN
ejpam-4793	529	6	s	s	PROPN
ejpam-4793	529	7	,	,	PUNCT
ejpam-4793	529	8	it	it	PRON
ejpam-4793	529	9	follows	follow	VERB
ejpam-4793	529	10	that	that	SCONJ
ejpam-4793	529	11	αx	αx	ADV
ejpam-4793	529	12	·	·	PUNCT
ejpam-4793	529	13	βy	βy	NOUN
ejpam-4793	529	14	=	=	PUNCT
ejpam-4793	529	15	φ(αa	φ(αa	X
ejpam-4793	529	16	∗	∗	NOUN
ejpam-4793	529	17	βb	βb	PRON
ejpam-4793	529	18	)	)	PUNCT
ejpam-4793	529	19	∈	∈	PROPN
ejpam-4793	529	20	φ(s	φ(s	NOUN
ejpam-4793	529	21	)	)	PUNCT
ejpam-4793	529	22	.	.	PUNCT
ejpam-4793	530	1	thus	thus	ADV
ejpam-4793	530	2	,	,	PUNCT
ejpam-4793	530	3	φ(s	φ(s	NOUN
ejpam-4793	530	4	)	)	PUNCT
ejpam-4793	530	5	is	be	AUX
ejpam-4793	530	6	a	a	DET
ejpam-4793	530	7	γ	γ	NOUN
ejpam-4793	530	8	-	-	ADJ
ejpam-4793	530	9	submonoid	submonoid	NOUN
ejpam-4793	530	10	of	of	ADP
ejpam-4793	530	11	n	n	PROPN
ejpam-4793	530	12	.	.	PUNCT
ejpam-4793	531	1	(	(	PUNCT
ejpam-4793	531	2	ii	ii	NOUN
ejpam-4793	531	3	)	)	PUNCT
ejpam-4793	531	4	let	let	VERB
ejpam-4793	531	5	t	t	NOUN
ejpam-4793	531	6	be	be	AUX
ejpam-4793	531	7	a	a	DET
ejpam-4793	531	8	γ	γ	NOUN
ejpam-4793	531	9	-	-	ADJ
ejpam-4793	531	10	submonoid	submonoid	NOUN
ejpam-4793	531	11	of	of	ADP
ejpam-4793	531	12	n	n	PROPN
ejpam-4793	531	13	.	.	PUNCT
ejpam-4793	532	1	then	then	ADV
ejpam-4793	532	2	,	,	PUNCT
ejpam-4793	532	3	φ(1	φ(1	PROPN
ejpam-4793	532	4	m	m	NOUN
ejpam-4793	532	5	)	)	PUNCT
ejpam-4793	533	1	=	=	SYM
ejpam-4793	533	2	1n	1n	NUM
ejpam-4793	533	3	∈	∈	PROPN
ejpam-4793	533	4	t	t	NOUN
ejpam-4793	533	5	and	and	CCONJ
ejpam-4793	533	6	1	1	NUM
ejpam-4793	533	7	m	m	NOUN
ejpam-4793	533	8	∈	∈	NOUN
ejpam-4793	533	9	φ−1(t	φ−1(t	NOUN
ejpam-4793	533	10	)	)	PUNCT
ejpam-4793	533	11	.	.	PUNCT
ejpam-4793	534	1	let	let	VERB
ejpam-4793	534	2	x	x	PRON
ejpam-4793	534	3	,	,	PUNCT
ejpam-4793	534	4	y	y	PROPN
ejpam-4793	534	5	∈	∈	PROPN
ejpam-4793	534	6	φ−1(t	φ−1(t	NOUN
ejpam-4793	534	7	)	)	PUNCT
ejpam-4793	534	8	.	.	PUNCT
ejpam-4793	535	1	then	then	ADV
ejpam-4793	535	2	φ(x	φ(x	PROPN
ejpam-4793	535	3	)	)	PUNCT
ejpam-4793	535	4	,	,	PUNCT
ejpam-4793	535	5	φ(y	φ(y	NOUN
ejpam-4793	535	6	)	)	PUNCT
ejpam-4793	535	7	∈	∈	PROPN
ejpam-4793	535	8	t	t	PROPN
ejpam-4793	535	9	.	.	PUNCT
ejpam-4793	536	1	now	now	ADV
ejpam-4793	536	2	,	,	PUNCT
ejpam-4793	536	3	for	for	ADP
ejpam-4793	536	4	all	all	DET
ejpam-4793	536	5	α	α	NOUN
ejpam-4793	536	6	,	,	PUNCT
ejpam-4793	536	7	β	β	PROPN
ejpam-4793	536	8	∈	∈	PROPN
ejpam-4793	536	9	γ	γ	X
ejpam-4793	536	10	,	,	PUNCT
ejpam-4793	536	11	we	we	PRON
ejpam-4793	536	12	have	have	VERB
ejpam-4793	536	13	φ(αx	φ(αx	NOUN
ejpam-4793	536	14	∗	∗	NOUN
ejpam-4793	536	15	βy	βy	NOUN
ejpam-4793	536	16	)	)	PUNCT
ejpam-4793	536	17	=	=	SYM
ejpam-4793	536	18	φ(αx	φ(αx	NOUN
ejpam-4793	536	19	)	)	PUNCT
ejpam-4793	536	20	·	·	PUNCT
ejpam-4793	536	21	φ(βy	φ(βy	X
ejpam-4793	536	22	)	)	PUNCT
ejpam-4793	536	23	=	=	SYM
ejpam-4793	536	24	αφ(x	αφ(x	X
ejpam-4793	536	25	)	)	PUNCT
ejpam-4793	536	26	·	·	PUNCT
ejpam-4793	536	27	βφ(y	βφ(y	NUM
ejpam-4793	536	28	)	)	PUNCT
ejpam-4793	536	29	∈	∈	PROPN
ejpam-4793	536	30	t	t	PROPN
ejpam-4793	536	31	since	since	SCONJ
ejpam-4793	536	32	t	t	PROPN
ejpam-4793	536	33	is	be	AUX
ejpam-4793	536	34	a	a	DET
ejpam-4793	536	35	γ	γ	NOUN
ejpam-4793	536	36	-	-	ADJ
ejpam-4793	536	37	submonoid	submonoid	NOUN
ejpam-4793	536	38	of	of	ADP
ejpam-4793	536	39	n	n	PROPN
ejpam-4793	536	40	.	.	PUNCT
ejpam-4793	537	1	this	this	PRON
ejpam-4793	537	2	implies	imply	VERB
ejpam-4793	537	3	that	that	SCONJ
ejpam-4793	537	4	for	for	ADP
ejpam-4793	537	5	all	all	DET
ejpam-4793	537	6	α	α	NOUN
ejpam-4793	537	7	,	,	PUNCT
ejpam-4793	537	8	β	β	PROPN
ejpam-4793	537	9	∈	∈	PROPN
ejpam-4793	537	10	γ	γ	X
ejpam-4793	537	11	,	,	PUNCT
ejpam-4793	537	12	we	we	PRON
ejpam-4793	537	13	have	have	AUX
ejpam-4793	537	14	αx	αx	PROPN
ejpam-4793	537	15	∗	∗	VERB
ejpam-4793	537	16	βy	βy	PRON
ejpam-4793	537	17	∈	∈	PROPN
ejpam-4793	537	18	φ−1(t	φ−1(t	NOUN
ejpam-4793	537	19	)	)	PUNCT
ejpam-4793	537	20	.	.	PUNCT
ejpam-4793	538	1	therefore	therefore	ADV
ejpam-4793	538	2	,	,	PUNCT
ejpam-4793	538	3	φ−1(t	φ−1(t	X
ejpam-4793	538	4	)	)	PUNCT
ejpam-4793	538	5	is	be	AUX
ejpam-4793	538	6	a	a	DET
ejpam-4793	538	7	γ	γ	NOUN
ejpam-4793	538	8	-	-	ADJ
ejpam-4793	538	9	submonoid	submonoid	NOUN
ejpam-4793	538	10	of	of	ADP
ejpam-4793	538	11	m	m	PROPN
ejpam-4793	538	12	.	.	PUNCT
ejpam-4793	539	1	(	(	PUNCT
ejpam-4793	539	2	iii	iii	X
ejpam-4793	539	3	)	)	PUNCT
ejpam-4793	539	4	since	since	SCONJ
ejpam-4793	539	5	φ	φ	PROPN
ejpam-4793	539	6	is	be	AUX
ejpam-4793	539	7	a	a	DET
ejpam-4793	539	8	γ	γ	PROPN
ejpam-4793	539	9	-	-	PUNCT
ejpam-4793	539	10	monoid	monoid	NOUN
ejpam-4793	539	11	homomorphism	homomorphism	NOUN
ejpam-4793	539	12	,	,	PUNCT
ejpam-4793	539	13	φ(1	φ(1	PROPN
ejpam-4793	539	14	m	m	NOUN
ejpam-4793	539	15	)	)	PUNCT
ejpam-4793	540	1	=	=	SYM
ejpam-4793	540	2	1n	1n	NUM
ejpam-4793	540	3	.	.	PUNCT
ejpam-4793	541	1	thus	thus	ADV
ejpam-4793	541	2	,	,	PUNCT
ejpam-4793	541	3	1	1	NUM
ejpam-4793	541	4	m	m	NOUN
ejpam-4793	541	5	∈	∈	PROPN
ejpam-4793	541	6	kerφ	kerφ	PROPN
ejpam-4793	541	7	.	.	PUNCT
ejpam-4793	542	1	now	now	ADV
ejpam-4793	542	2	,	,	PUNCT
ejpam-4793	542	3	let	let	VERB
ejpam-4793	542	4	x	x	PRON
ejpam-4793	542	5	,	,	PUNCT
ejpam-4793	542	6	y	y	PROPN
ejpam-4793	542	7	∈	∈	PROPN
ejpam-4793	542	8	kerφ	kerφ	PROPN
ejpam-4793	542	9	.	.	PUNCT
ejpam-4793	543	1	then	then	ADV
ejpam-4793	543	2	φ(x	φ(x	NOUN
ejpam-4793	543	3	)	)	PUNCT
ejpam-4793	543	4	=	=	SYM
ejpam-4793	543	5	1n	1n	NUM
ejpam-4793	543	6	and	and	CCONJ
ejpam-4793	543	7	φ(y	φ(y	ADJ
ejpam-4793	543	8	)	)	PUNCT
ejpam-4793	543	9	=	=	SYM
ejpam-4793	543	10	1n	1n	NUM
ejpam-4793	543	11	.	.	PUNCT
ejpam-4793	544	1	thus	thus	ADV
ejpam-4793	544	2	,	,	PUNCT
ejpam-4793	544	3	by	by	ADP
ejpam-4793	544	4	remark	remark	NOUN
ejpam-4793	544	5	3	3	NUM
ejpam-4793	544	6	,	,	PUNCT
ejpam-4793	544	7	for	for	ADP
ejpam-4793	544	8	all	all	DET
ejpam-4793	544	9	α	α	NOUN
ejpam-4793	544	10	,	,	PUNCT
ejpam-4793	544	11	β	β	PROPN
ejpam-4793	544	12	∈	∈	PROPN
ejpam-4793	544	13	γ	γ	X
ejpam-4793	544	14	,	,	PUNCT
ejpam-4793	544	15	φ(αx	φ(αx	X
ejpam-4793	544	16	∗	∗	NOUN
ejpam-4793	544	17	βy	βy	NOUN
ejpam-4793	544	18	)	)	PUNCT
ejpam-4793	544	19	=	=	SYM
ejpam-4793	544	20	φ(αx	φ(αx	NOUN
ejpam-4793	544	21	)	)	PUNCT
ejpam-4793	544	22	·	·	PUNCT
ejpam-4793	544	23	φ(βy	φ(βy	NOUN
ejpam-4793	544	24	)	)	PUNCT
ejpam-4793	544	25	=	=	SYM
ejpam-4793	544	26	αφ(x	αφ(x	X
ejpam-4793	544	27	)	)	PUNCT
ejpam-4793	544	28	·	·	PUNCT
ejpam-4793	544	29	βφ(y	βφ(y	NUM
ejpam-4793	544	30	)	)	PUNCT
ejpam-4793	545	1	=	=	SYM
ejpam-4793	545	2	α1n	α1n	NOUN
ejpam-4793	545	3	·	·	PUNCT
ejpam-4793	545	4	β1n	β1n	PUNCT
ejpam-4793	546	1	=	=	SYM
ejpam-4793	546	2	1n	1n	NUM
ejpam-4793	546	3	·	·	PUNCT
ejpam-4793	546	4	1n	1n	NUM
ejpam-4793	546	5	=	=	SYM
ejpam-4793	546	6	1n	1n	NUM
ejpam-4793	546	7	.	.	PUNCT
ejpam-4793	547	1	hence	hence	ADV
ejpam-4793	547	2	,	,	PUNCT
ejpam-4793	547	3	for	for	ADP
ejpam-4793	547	4	all	all	DET
ejpam-4793	547	5	α	α	NOUN
ejpam-4793	547	6	,	,	PUNCT
ejpam-4793	547	7	β	β	PROPN
ejpam-4793	547	8	∈	∈	PROPN
ejpam-4793	547	9	γ	γ	X
ejpam-4793	547	10	,	,	PUNCT
ejpam-4793	547	11	αx	αx	ADV
ejpam-4793	547	12	∗	∗	VERB
ejpam-4793	547	13	βy	βy	PRON
ejpam-4793	548	1	∈	∈	PROPN
ejpam-4793	548	2	kerφ	kerφ	PROPN
ejpam-4793	548	3	.	.	PUNCT
ejpam-4793	549	1	therefore	therefore	ADV
ejpam-4793	549	2	,	,	PUNCT
ejpam-4793	549	3	kerφ	kerφ	PROPN
ejpam-4793	549	4	is	be	AUX
ejpam-4793	549	5	a	a	DET
ejpam-4793	549	6	γ	γ	NOUN
ejpam-4793	549	7	-	-	ADJ
ejpam-4793	549	8	submonoid	submonoid	NOUN
ejpam-4793	549	9	of	of	ADP
ejpam-4793	549	10	m	m	PROPN
ejpam-4793	549	11	.	.	PUNCT
ejpam-4793	550	1	(	(	PUNCT
ejpam-4793	550	2	iv	iv	X
ejpam-4793	550	3	)	)	PUNCT
ejpam-4793	550	4	let	let	VERB
ejpam-4793	550	5	x	x	X
ejpam-4793	550	6	,	,	PUNCT
ejpam-4793	550	7	x	x	SYM
ejpam-4793	550	8	∗	∗	NOUN
ejpam-4793	550	9	y	y	PROPN
ejpam-4793	550	10	∈	∈	PROPN
ejpam-4793	550	11	kerφ	kerφ	PROPN
ejpam-4793	550	12	.	.	PUNCT
ejpam-4793	551	1	then	then	ADV
ejpam-4793	551	2	φ(x	φ(x	NOUN
ejpam-4793	551	3	)	)	PUNCT
ejpam-4793	551	4	=	=	SYM
ejpam-4793	551	5	1n	1n	NUM
ejpam-4793	551	6	and	and	CCONJ
ejpam-4793	551	7	φ(x	φ(x	PROPN
ejpam-4793	551	8	∗	∗	PROPN
ejpam-4793	551	9	y	y	NOUN
ejpam-4793	551	10	)	)	PUNCT
ejpam-4793	551	11	=	=	SYM
ejpam-4793	552	1	1n	1n	NUM
ejpam-4793	552	2	.	.	PUNCT
ejpam-4793	553	1	thus	thus	ADV
ejpam-4793	553	2	,	,	PUNCT
ejpam-4793	553	3	φ(y	φ(y	NOUN
ejpam-4793	553	4	)	)	PUNCT
ejpam-4793	553	5	=	=	SYM
ejpam-4793	554	1	1n	1n	NUM
ejpam-4793	554	2	·	·	PUNCT
ejpam-4793	554	3	φ(y	φ(y	NOUN
ejpam-4793	554	4	)	)	PUNCT
ejpam-4793	554	5	=	=	SYM
ejpam-4793	554	6	φ(x	φ(x	X
ejpam-4793	554	7	)	)	PUNCT
ejpam-4793	554	8	·	·	PUNCT
ejpam-4793	554	9	φ(y	φ(y	X
ejpam-4793	554	10	)	)	PUNCT
ejpam-4793	554	11	=	=	SYM
ejpam-4793	555	1	φ(x	φ(x	PROPN
ejpam-4793	555	2	∗	∗	NOUN
ejpam-4793	555	3	y	y	NOUN
ejpam-4793	555	4	)	)	PUNCT
ejpam-4793	556	1	=	=	SYM
ejpam-4793	556	2	1n	1n	NUM
ejpam-4793	556	3	.	.	PUNCT
ejpam-4793	557	1	this	this	PRON
ejpam-4793	557	2	implies	imply	VERB
ejpam-4793	557	3	that	that	SCONJ
ejpam-4793	557	4	y	y	PROPN
ejpam-4793	557	5	∈	∈	PROPN
ejpam-4793	557	6	kerφ	kerφ	PROPN
ejpam-4793	557	7	and	and	CCONJ
ejpam-4793	557	8	thus	thus	ADV
ejpam-4793	557	9	,	,	PUNCT
ejpam-4793	557	10	kerφ	kerφ	PROPN
ejpam-4793	557	11	is	be	AUX
ejpam-4793	557	12	normal	normal	ADJ
ejpam-4793	557	13	.	.	PUNCT
ejpam-4793	558	1	theorem	theorem	NOUN
ejpam-4793	558	2	9	9	NUM
ejpam-4793	558	3	.	.	PUNCT
ejpam-4793	559	1	let	let	VERB
ejpam-4793	559	2	j	j	PROPN
ejpam-4793	559	3	be	be	AUX
ejpam-4793	559	4	a	a	DET
ejpam-4793	559	5	γ	γ	NOUN
ejpam-4793	559	6	-	-	ADJ
ejpam-4793	559	7	ideal	ideal	NOUN
ejpam-4793	559	8	and	and	CCONJ
ejpam-4793	559	9	s	s	VERB
ejpam-4793	559	10	a	a	DET
ejpam-4793	559	11	γ	γ	NOUN
ejpam-4793	559	12	-	-	ADJ
ejpam-4793	559	13	submonoid	submonoid	NOUN
ejpam-4793	559	14	of	of	ADP
ejpam-4793	559	15	a	a	DET
ejpam-4793	559	16	γ	γ	X
ejpam-4793	559	17	-	-	PUNCT
ejpam-4793	559	18	monoid	monoid	NOUN
ejpam-4793	559	19	m	m	NOUN
ejpam-4793	559	20	such	such	ADJ
ejpam-4793	559	21	that	that	SCONJ
ejpam-4793	559	22	j	j	PROPN
ejpam-4793	559	23	∩	∩	PROPN
ejpam-4793	559	24	s	s	PART
ejpam-4793	559	25	̸=	̸=	PROPN
ejpam-4793	559	26	∅.	∅.	NOUN
ejpam-4793	559	27	then	then	ADV
ejpam-4793	559	28	(	(	PUNCT
ejpam-4793	559	29	i	i	NOUN
ejpam-4793	559	30	)	)	PUNCT
ejpam-4793	559	31	j	j	PROPN
ejpam-4793	559	32	∩	∩	NOUN
ejpam-4793	559	33	s	s	PART
ejpam-4793	559	34	is	be	AUX
ejpam-4793	559	35	a	a	DET
ejpam-4793	559	36	γ	γ	NOUN
ejpam-4793	559	37	-	-	NOUN
ejpam-4793	559	38	ideal	ideal	NOUN
ejpam-4793	559	39	of	of	ADP
ejpam-4793	559	40	s	s	PROPN
ejpam-4793	559	41	;	;	PUNCT
ejpam-4793	559	42	(	(	PUNCT
ejpam-4793	559	43	ii	ii	NOUN
ejpam-4793	559	44	)	)	PUNCT
ejpam-4793	559	45	j	j	PROPN
ejpam-4793	559	46	∪	∪	ADP
ejpam-4793	559	47	s	s	PART
ejpam-4793	559	48	is	be	AUX
ejpam-4793	559	49	a	a	DET
ejpam-4793	559	50	γ	γ	NOUN
ejpam-4793	559	51	-	-	ADJ
ejpam-4793	559	52	submonoid	submonoid	NOUN
ejpam-4793	559	53	of	of	ADP
ejpam-4793	559	54	m	m	PROPN
ejpam-4793	559	55	.	.	PUNCT
ejpam-4793	560	1	proof	proof	NOUN
ejpam-4793	560	2	.	.	PUNCT
ejpam-4793	561	1	let	let	VERB
ejpam-4793	561	2	j	j	PROPN
ejpam-4793	561	3	be	be	AUX
ejpam-4793	561	4	a	a	DET
ejpam-4793	561	5	γ	γ	NOUN
ejpam-4793	561	6	-	-	ADJ
ejpam-4793	561	7	ideal	ideal	NOUN
ejpam-4793	561	8	and	and	CCONJ
ejpam-4793	561	9	s	s	VERB
ejpam-4793	561	10	a	a	DET
ejpam-4793	561	11	γ	γ	NOUN
ejpam-4793	561	12	-	-	ADJ
ejpam-4793	561	13	submonoid	submonoid	NOUN
ejpam-4793	561	14	of	of	ADP
ejpam-4793	561	15	m	m	PRON
ejpam-4793	561	16	such	such	ADJ
ejpam-4793	561	17	that	that	SCONJ
ejpam-4793	561	18	j	j	PROPN
ejpam-4793	561	19	∩	∩	PROPN
ejpam-4793	561	20	s	s	PART
ejpam-4793	561	21	̸=	̸=	PROPN
ejpam-4793	561	22	∅.	∅.	X
ejpam-4793	561	23	(	(	PUNCT
ejpam-4793	561	24	i	i	NOUN
ejpam-4793	561	25	)	)	PUNCT
ejpam-4793	561	26	let	let	VERB
ejpam-4793	561	27	x	x	SYM
ejpam-4793	561	28	∈	∈	PROPN
ejpam-4793	561	29	j	j	PROPN
ejpam-4793	561	30	∩s	∩s	PROPN
ejpam-4793	561	31	and	and	CCONJ
ejpam-4793	561	32	s	s	PROPN
ejpam-4793	561	33	∈	∈	PROPN
ejpam-4793	561	34	s.	s.	PROPN
ejpam-4793	561	35	then	then	ADV
ejpam-4793	561	36	x	x	PROPN
ejpam-4793	561	37	∈	∈	PROPN
ejpam-4793	561	38	j	j	PROPN
ejpam-4793	561	39	and	and	CCONJ
ejpam-4793	561	40	x	x	NOUN
ejpam-4793	561	41	,	,	PUNCT
ejpam-4793	561	42	s	s	VERB
ejpam-4793	561	43	∈	∈	PROPN
ejpam-4793	561	44	s.	s.	PROPN
ejpam-4793	561	45	since	since	SCONJ
ejpam-4793	561	46	j	j	PROPN
ejpam-4793	561	47	is	be	AUX
ejpam-4793	561	48	a	a	DET
ejpam-4793	561	49	γ	γ	NOUN
ejpam-4793	561	50	-	-	NOUN
ejpam-4793	561	51	ideal	ideal	NOUN
ejpam-4793	561	52	of	of	ADP
ejpam-4793	561	53	m	m	PRON
ejpam-4793	561	54	,	,	PUNCT
ejpam-4793	561	55	for	for	ADP
ejpam-4793	561	56	all	all	DET
ejpam-4793	561	57	α	α	NOUN
ejpam-4793	561	58	,	,	PUNCT
ejpam-4793	561	59	β	β	PROPN
ejpam-4793	561	60	∈	∈	PROPN
ejpam-4793	561	61	γ	γ	X
ejpam-4793	561	62	,	,	PUNCT
ejpam-4793	561	63	αx	αx	ADV
ejpam-4793	561	64	∗	∗	NOUN
ejpam-4793	561	65	βs	βs	ADP
ejpam-4793	561	66	,	,	PUNCT
ejpam-4793	561	67	αs	αs	ADP
ejpam-4793	561	68	∗	∗	NOUN
ejpam-4793	561	69	βx	βx	ADP
ejpam-4793	562	1	∈	∈	PROPN
ejpam-4793	563	1	j	j	PROPN
ejpam-4793	563	2	.	.	PUNCT
ejpam-4793	564	1	also	also	ADV
ejpam-4793	564	2	,	,	PUNCT
ejpam-4793	564	3	since	since	SCONJ
ejpam-4793	564	4	s	s	NOUN
ejpam-4793	564	5	is	be	AUX
ejpam-4793	564	6	a	a	DET
ejpam-4793	564	7	γ	γ	NOUN
ejpam-4793	564	8	-	-	ADJ
ejpam-4793	564	9	submonoid	submonoid	NOUN
ejpam-4793	564	10	of	of	ADP
ejpam-4793	564	11	m	m	PRON
ejpam-4793	564	12	,	,	PUNCT
ejpam-4793	564	13	for	for	ADP
ejpam-4793	564	14	all	all	DET
ejpam-4793	564	15	α	α	NOUN
ejpam-4793	564	16	,	,	PUNCT
ejpam-4793	564	17	β	β	PROPN
ejpam-4793	564	18	∈	∈	PROPN
ejpam-4793	564	19	γ	γ	X
ejpam-4793	564	20	,	,	PUNCT
ejpam-4793	564	21	αx	αx	ADV
ejpam-4793	564	22	∗	∗	NOUN
ejpam-4793	564	23	βs	βs	ADP
ejpam-4793	564	24	,	,	PUNCT
ejpam-4793	564	25	αs	αs	ADP
ejpam-4793	564	26	∗	∗	NOUN
ejpam-4793	564	27	βx	βx	ADP
ejpam-4793	564	28	∈	∈	PROPN
ejpam-4793	564	29	s.	s.	PROPN
ejpam-4793	564	30	thus	thus	ADV
ejpam-4793	564	31	,	,	PUNCT
ejpam-4793	564	32	for	for	ADP
ejpam-4793	564	33	all	all	DET
ejpam-4793	564	34	α	α	NOUN
ejpam-4793	564	35	,	,	PUNCT
ejpam-4793	564	36	β	β	PROPN
ejpam-4793	564	37	∈	∈	PROPN
ejpam-4793	564	38	γ	γ	X
ejpam-4793	564	39	,	,	PUNCT
ejpam-4793	564	40	αx	αx	ADV
ejpam-4793	564	41	∗	∗	NOUN
ejpam-4793	564	42	βs	βs	ADP
ejpam-4793	564	43	,	,	PUNCT
ejpam-4793	564	44	αs	αs	ADP
ejpam-4793	564	45	∗	∗	NOUN
ejpam-4793	564	46	βx	βx	ADP
ejpam-4793	564	47	∈	∈	PROPN
ejpam-4793	564	48	j	j	PROPN
ejpam-4793	564	49	∩	∩	PROPN
ejpam-4793	564	50	s	s	PART
ejpam-4793	564	51	and	and	CCONJ
ejpam-4793	564	52	so	so	ADV
ejpam-4793	564	53	,	,	PUNCT
ejpam-4793	564	54	j	j	PROPN
ejpam-4793	564	55	∩	∩	PROPN
ejpam-4793	564	56	s	s	PART
ejpam-4793	564	57	is	be	AUX
ejpam-4793	564	58	a	a	DET
ejpam-4793	564	59	γ	γ	NOUN
ejpam-4793	564	60	-	-	NOUN
ejpam-4793	564	61	ideal	ideal	NOUN
ejpam-4793	564	62	of	of	ADP
ejpam-4793	564	63	s.	s.	PROPN
ejpam-4793	564	64	(	(	PUNCT
ejpam-4793	564	65	ii	ii	PROPN
ejpam-4793	564	66	)	)	PUNCT
ejpam-4793	564	67	let	let	VERB
ejpam-4793	564	68	x	x	PRON
ejpam-4793	564	69	,	,	PUNCT
ejpam-4793	564	70	y	y	PROPN
ejpam-4793	564	71	∈	∈	PROPN
ejpam-4793	564	72	j	j	PROPN
ejpam-4793	564	73	∪	∪	ADP
ejpam-4793	564	74	s.	s.	PROPN
ejpam-4793	564	75	we	we	PRON
ejpam-4793	564	76	consider	consider	VERB
ejpam-4793	564	77	the	the	DET
ejpam-4793	564	78	following	follow	VERB
ejpam-4793	564	79	cases	case	NOUN
ejpam-4793	564	80	.	.	PUNCT
ejpam-4793	565	1	case	case	NOUN
ejpam-4793	565	2	1	1	NUM
ejpam-4793	565	3	.	.	NUM
ejpam-4793	566	1	x	x	X
ejpam-4793	566	2	,	,	PUNCT
ejpam-4793	566	3	y	y	PROPN
ejpam-4793	566	4	∈	∈	PROPN
ejpam-4793	566	5	j	j	PROPN
ejpam-4793	566	6	.	.	PUNCT
ejpam-4793	567	1	since	since	SCONJ
ejpam-4793	567	2	j	j	PROPN
ejpam-4793	567	3	is	be	AUX
ejpam-4793	567	4	a	a	DET
ejpam-4793	567	5	γ	γ	NOUN
ejpam-4793	567	6	-	-	NOUN
ejpam-4793	567	7	ideal	ideal	NOUN
ejpam-4793	567	8	of	of	ADP
ejpam-4793	567	9	m	m	PRON
ejpam-4793	567	10	,	,	PUNCT
ejpam-4793	567	11	for	for	ADP
ejpam-4793	567	12	all	all	DET
ejpam-4793	567	13	α	α	NOUN
ejpam-4793	567	14	,	,	PUNCT
ejpam-4793	567	15	β	β	PROPN
ejpam-4793	567	16	∈	∈	PROPN
ejpam-4793	567	17	γ	γ	X
ejpam-4793	567	18	,	,	PUNCT
ejpam-4793	567	19	αx	αx	ADV
ejpam-4793	567	20	∗	∗	VERB
ejpam-4793	567	21	βy	βy	ADP
ejpam-4793	568	1	∈	∈	PROPN
ejpam-4793	568	2	j	j	PROPN
ejpam-4793	568	3	⊆	⊆	NUM
ejpam-4793	568	4	j	j	PROPN
ejpam-4793	568	5	∪	∪	ADP
ejpam-4793	568	6	s.	s.	PROPN
ejpam-4793	568	7	case	case	NOUN
ejpam-4793	568	8	2	2	NUM
ejpam-4793	568	9	.	.	PUNCT
ejpam-4793	568	10	x	x	SYM
ejpam-4793	568	11	∈	∈	PROPN
ejpam-4793	568	12	j	j	PROPN
ejpam-4793	568	13	,	,	PUNCT
ejpam-4793	568	14	y	y	PROPN
ejpam-4793	568	15	∈	∈	PROPN
ejpam-4793	568	16	s.	s.	PROPN
ejpam-4793	568	17	since	since	SCONJ
ejpam-4793	568	18	j	j	PROPN
ejpam-4793	568	19	is	be	AUX
ejpam-4793	568	20	a	a	DET
ejpam-4793	568	21	γ	γ	NOUN
ejpam-4793	568	22	-	-	NOUN
ejpam-4793	568	23	ideal	ideal	NOUN
ejpam-4793	568	24	of	of	ADP
ejpam-4793	568	25	m	m	PRON
ejpam-4793	568	26	,	,	PUNCT
ejpam-4793	568	27	for	for	ADP
ejpam-4793	568	28	all	all	DET
ejpam-4793	568	29	α	α	NOUN
ejpam-4793	568	30	,	,	PUNCT
ejpam-4793	568	31	β	β	PROPN
ejpam-4793	568	32	∈	∈	PROPN
ejpam-4793	568	33	γ	γ	X
ejpam-4793	568	34	,	,	PUNCT
ejpam-4793	568	35	αx	αx	ADV
ejpam-4793	568	36	∗	∗	VERB
ejpam-4793	568	37	βy	βy	ADP
ejpam-4793	569	1	∈	∈	PROPN
ejpam-4793	569	2	j	j	PROPN
ejpam-4793	569	3	⊆	⊆	NUM
ejpam-4793	569	4	j	j	PROPN
ejpam-4793	569	5	∪	∪	ADP
ejpam-4793	569	6	s.	s.	PROPN
ejpam-4793	569	7	case	case	NOUN
ejpam-4793	569	8	3	3	NUM
ejpam-4793	569	9	.	.	NUM
ejpam-4793	569	10	x	x	X
ejpam-4793	569	11	,	,	PUNCT
ejpam-4793	569	12	y	y	PROPN
ejpam-4793	569	13	∈	∈	PROPN
ejpam-4793	569	14	s.	s.	PROPN
ejpam-4793	569	15	since	since	SCONJ
ejpam-4793	569	16	s	s	PROPN
ejpam-4793	569	17	is	be	AUX
ejpam-4793	569	18	a	a	DET
ejpam-4793	569	19	γ	γ	NOUN
ejpam-4793	569	20	-	-	ADJ
ejpam-4793	569	21	submonoid	submonoid	NOUN
ejpam-4793	569	22	of	of	ADP
ejpam-4793	569	23	m	m	PRON
ejpam-4793	569	24	,	,	PUNCT
ejpam-4793	569	25	for	for	ADP
ejpam-4793	569	26	all	all	DET
ejpam-4793	569	27	α	α	NOUN
ejpam-4793	569	28	,	,	PUNCT
ejpam-4793	569	29	β	β	PROPN
ejpam-4793	569	30	∈	∈	PROPN
ejpam-4793	569	31	γ	γ	X
ejpam-4793	569	32	,	,	PUNCT
ejpam-4793	569	33	αx	αx	ADV
ejpam-4793	569	34	∗	∗	VERB
ejpam-4793	569	35	βy	βy	PRON
ejpam-4793	570	1	∈	∈	PROPN
ejpam-4793	570	2	s	s	PART
ejpam-4793	570	3	⊆	⊆	NUM
ejpam-4793	570	4	j	j	PROPN
ejpam-4793	570	5	∪	∪	ADP
ejpam-4793	570	6	s.	s.	PROPN
ejpam-4793	570	7	case	case	PROPN
ejpam-4793	570	8	4	4	NUM
ejpam-4793	570	9	.	.	PUNCT
ejpam-4793	570	10	y	y	PROPN
ejpam-4793	570	11	∈	∈	PROPN
ejpam-4793	570	12	j	j	PROPN
ejpam-4793	570	13	,	,	PUNCT
ejpam-4793	570	14	x	x	PROPN
ejpam-4793	570	15	∈	∈	PROPN
ejpam-4793	570	16	s.	s.	PROPN
ejpam-4793	570	17	since	since	SCONJ
ejpam-4793	570	18	j	j	PROPN
ejpam-4793	570	19	is	be	AUX
ejpam-4793	570	20	a	a	DET
ejpam-4793	570	21	γ	γ	NOUN
ejpam-4793	570	22	-	-	NOUN
ejpam-4793	570	23	ideal	ideal	NOUN
ejpam-4793	570	24	of	of	ADP
ejpam-4793	570	25	m	m	PRON
ejpam-4793	570	26	,	,	PUNCT
ejpam-4793	570	27	for	for	ADP
ejpam-4793	570	28	all	all	DET
ejpam-4793	570	29	α	α	NOUN
ejpam-4793	570	30	,	,	PUNCT
ejpam-4793	570	31	β	β	PROPN
ejpam-4793	570	32	∈	∈	PROPN
ejpam-4793	570	33	γ	γ	X
ejpam-4793	570	34	,	,	PUNCT
ejpam-4793	570	35	αx	αx	ADV
ejpam-4793	570	36	∗	∗	VERB
ejpam-4793	570	37	βy	βy	ADP
ejpam-4793	571	1	∈	∈	PROPN
ejpam-4793	571	2	j	j	PROPN
ejpam-4793	571	3	⊆	⊆	NUM
ejpam-4793	571	4	j	j	PROPN
ejpam-4793	571	5	∪	∪	PROPN
ejpam-4793	571	6	s.	s.	PROPN
ejpam-4793	571	7	h.	h.	PROPN
ejpam-4793	571	8	sarapuddin	sarapuddin	PROPN
ejpam-4793	571	9	,	,	PUNCT
ejpam-4793	571	10	j.	j.	PROPN
ejpam-4793	571	11	vilela	vilela	PROPN
ejpam-4793	571	12	/	/	SYM
ejpam-4793	571	13	eur	eur	PROPN
ejpam-4793	571	14	.	.	PUNCT
ejpam-4793	572	1	j.	j.	PROPN
ejpam-4793	572	2	pure	pure	PROPN
ejpam-4793	572	3	appl	appl	PROPN
ejpam-4793	572	4	.	.	PROPN
ejpam-4793	572	5	math	math	PROPN
ejpam-4793	572	6	,	,	PUNCT
ejpam-4793	572	7	16	16	NUM
ejpam-4793	572	8	(	(	PUNCT
ejpam-4793	572	9	3	3	NUM
ejpam-4793	572	10	)	)	PUNCT
ejpam-4793	572	11	(	(	PUNCT
ejpam-4793	572	12	2023	2023	NUM
ejpam-4793	572	13	)	)	PUNCT
ejpam-4793	572	14	,	,	PUNCT
ejpam-4793	572	15	1772	1772	NUM
ejpam-4793	572	16	-	-	SYM
ejpam-4793	572	17	1793	1793	NUM
ejpam-4793	572	18	1785	1785	NUM
ejpam-4793	572	19	also	also	ADV
ejpam-4793	572	20	,	,	PUNCT
ejpam-4793	572	21	since	since	SCONJ
ejpam-4793	572	22	s	s	NOUN
ejpam-4793	572	23	is	be	AUX
ejpam-4793	572	24	a	a	DET
ejpam-4793	572	25	γ	γ	NOUN
ejpam-4793	572	26	-	-	ADJ
ejpam-4793	572	27	submonoid	submonoid	NOUN
ejpam-4793	572	28	of	of	ADP
ejpam-4793	572	29	m	m	PRON
ejpam-4793	572	30	,	,	PUNCT
ejpam-4793	572	31	1	1	NUM
ejpam-4793	572	32	m	m	NOUN
ejpam-4793	572	33	∈	∈	NOUN
ejpam-4793	572	34	s	s	VERB
ejpam-4793	572	35	⊆	⊆	NUM
ejpam-4793	572	36	j	j	PROPN
ejpam-4793	572	37	∪s	∪s	NOUN
ejpam-4793	572	38	.	.	PUNCT
ejpam-4793	573	1	therefore	therefore	ADV
ejpam-4793	573	2	,	,	PUNCT
ejpam-4793	573	3	j	j	PROPN
ejpam-4793	573	4	∪s	∪s	PROPN
ejpam-4793	573	5	is	be	AUX
ejpam-4793	573	6	a	a	DET
ejpam-4793	573	7	γ	γ	NOUN
ejpam-4793	573	8	-	-	ADJ
ejpam-4793	573	9	submonoid	submonoid	NOUN
ejpam-4793	573	10	of	of	ADP
ejpam-4793	573	11	m	m	PROPN
ejpam-4793	573	12	.	.	PUNCT
ejpam-4793	574	1	remark	remark	PROPN
ejpam-4793	574	2	13	13	NUM
ejpam-4793	574	3	.	.	PUNCT
ejpam-4793	575	1	theorem	theorem	PROPN
ejpam-4793	575	2	4(i	4(i	NUM
ejpam-4793	575	3	)	)	PUNCT
ejpam-4793	576	1	is	be	AUX
ejpam-4793	576	2	also	also	ADV
ejpam-4793	576	3	a	a	DET
ejpam-4793	576	4	consequence	consequence	NOUN
ejpam-4793	576	5	of	of	ADP
ejpam-4793	576	6	theorem9(i	theorem9(i	NOUN
ejpam-4793	576	7	)	)	PUNCT
ejpam-4793	576	8	.	.	PUNCT
ejpam-4793	577	1	lemma	lemma	PROPN
ejpam-4793	577	2	5	5	X
ejpam-4793	577	3	.	.	PUNCT
ejpam-4793	578	1	let	let	VERB
ejpam-4793	578	2	a	a	PRON
ejpam-4793	578	3	and	and	CCONJ
ejpam-4793	578	4	b	b	NOUN
ejpam-4793	578	5	be	be	AUX
ejpam-4793	578	6	γ	γ	NOUN
ejpam-4793	578	7	-	-	NOUN
ejpam-4793	578	8	submonoids	submonoid	NOUN
ejpam-4793	578	9	of	of	ADP
ejpam-4793	578	10	a	a	DET
ejpam-4793	578	11	commutative	commutative	ADJ
ejpam-4793	578	12	γ	γ	X
ejpam-4793	578	13	-	-	PUNCT
ejpam-4793	578	14	monoid	monoid	NOUN
ejpam-4793	578	15	m	m	PROPN
ejpam-4793	578	16	.	.	PUNCT
ejpam-4793	579	1	then	then	ADV
ejpam-4793	579	2	a	a	DET
ejpam-4793	579	3	∗	∗	NOUN
ejpam-4793	579	4	b	b	NOUN
ejpam-4793	579	5	is	be	AUX
ejpam-4793	579	6	a	a	DET
ejpam-4793	579	7	γ	γ	NOUN
ejpam-4793	579	8	-	-	ADJ
ejpam-4793	579	9	submonoid	submonoid	NOUN
ejpam-4793	579	10	of	of	ADP
ejpam-4793	579	11	m	m	PROPN
ejpam-4793	579	12	.	.	PUNCT
ejpam-4793	580	1	proof	proof	NOUN
ejpam-4793	580	2	.	.	PUNCT
ejpam-4793	581	1	let	let	VERB
ejpam-4793	581	2	x	x	PRON
ejpam-4793	581	3	,	,	PUNCT
ejpam-4793	581	4	y	y	PROPN
ejpam-4793	581	5	∈	∈	PROPN
ejpam-4793	581	6	a∗b	a∗b	NUM
ejpam-4793	581	7	and	and	CCONJ
ejpam-4793	581	8	α	α	NOUN
ejpam-4793	581	9	,	,	PUNCT
ejpam-4793	581	10	β	β	PROPN
ejpam-4793	581	11	∈	∈	PROPN
ejpam-4793	581	12	γ	γ	X
ejpam-4793	581	13	.	.	PROPN
ejpam-4793	582	1	then	then	ADV
ejpam-4793	582	2	x	x	X
ejpam-4793	582	3	=	=	PUNCT
ejpam-4793	582	4	a1∗b1	a1∗b1	PROPN
ejpam-4793	582	5	and	and	CCONJ
ejpam-4793	582	6	y	y	PROPN
ejpam-4793	582	7	=	=	PUNCT
ejpam-4793	582	8	a2∗b2	a2∗b2	PROPN
ejpam-4793	582	9	for	for	ADP
ejpam-4793	582	10	some	some	DET
ejpam-4793	582	11	a1	a1	NOUN
ejpam-4793	582	12	,	,	PUNCT
ejpam-4793	582	13	a2	a2	PROPN
ejpam-4793	582	14	∈	∈	PROPN
ejpam-4793	582	15	a	a	PRON
ejpam-4793	582	16	and	and	CCONJ
ejpam-4793	582	17	b1	b1	NOUN
ejpam-4793	582	18	,	,	PUNCT
ejpam-4793	582	19	b2	b2	NOUN
ejpam-4793	582	20	∈	∈	PROPN
ejpam-4793	582	21	b.	b.	PROPN
ejpam-4793	582	22	since	since	SCONJ
ejpam-4793	582	23	a	a	PRON
ejpam-4793	582	24	and	and	CCONJ
ejpam-4793	582	25	b	b	NOUN
ejpam-4793	582	26	are	be	AUX
ejpam-4793	582	27	γ	γ	NOUN
ejpam-4793	582	28	-	-	PUNCT
ejpam-4793	582	29	submonoids	submonoid	NOUN
ejpam-4793	582	30	,	,	PUNCT
ejpam-4793	582	31	αa1	αa1	PROPN
ejpam-4793	582	32	∗	∗	NOUN
ejpam-4793	582	33	βa2	βa2	X
ejpam-4793	582	34	∈	∈	PROPN
ejpam-4793	582	35	a	a	PRON
ejpam-4793	582	36	and	and	CCONJ
ejpam-4793	582	37	αb1	αb1	NOUN
ejpam-4793	582	38	∗	∗	NOUN
ejpam-4793	582	39	βb2	βb2	X
ejpam-4793	582	40	∈	∈	PROPN
ejpam-4793	582	41	b.	b.	PROPN
ejpam-4793	582	42	note	note	VERB
ejpam-4793	582	43	that	that	SCONJ
ejpam-4793	582	44	1	1	NUM
ejpam-4793	582	45	m	m	NOUN
ejpam-4793	582	46	=	=	NUM
ejpam-4793	582	47	1	1	NUM
ejpam-4793	582	48	m	m	NOUN
ejpam-4793	582	49	∗	∗	NOUN
ejpam-4793	582	50	1	1	NUM
ejpam-4793	582	51	m	m	NOUN
ejpam-4793	582	52	∈	∈	NOUN
ejpam-4793	582	53	a	a	DET
ejpam-4793	582	54	∗b	∗b	NOUN
ejpam-4793	582	55	.	.	PUNCT
ejpam-4793	583	1	since	since	SCONJ
ejpam-4793	583	2	m	m	PROPN
ejpam-4793	583	3	is	be	AUX
ejpam-4793	583	4	commutative	commutative	ADJ
ejpam-4793	583	5	,	,	PUNCT
ejpam-4793	583	6	αx	αx	ADV
ejpam-4793	583	7	∗	∗	VERB
ejpam-4793	583	8	βy	βy	VERB
ejpam-4793	584	1	=	=	PUNCT
ejpam-4793	584	2	α(a1	α(a1	PROPN
ejpam-4793	584	3	∗	∗	NOUN
ejpam-4793	584	4	b1	b1	PROPN
ejpam-4793	584	5	)	)	PUNCT
ejpam-4793	584	6	∗	∗	NOUN
ejpam-4793	584	7	β(a2	β(a2	NUM
ejpam-4793	584	8	∗	∗	NOUN
ejpam-4793	584	9	b2	b2	NOUN
ejpam-4793	584	10	)	)	PUNCT
ejpam-4793	584	11	=	=	SYM
ejpam-4793	584	12	(	(	PUNCT
ejpam-4793	584	13	αa1	αa1	PROPN
ejpam-4793	584	14	∗	∗	NOUN
ejpam-4793	584	15	αb1	αb1	NOUN
ejpam-4793	584	16	)	)	PUNCT
ejpam-4793	585	1	∗	∗	NOUN
ejpam-4793	585	2	(	(	PUNCT
ejpam-4793	585	3	βa2	βa2	PROPN
ejpam-4793	585	4	∗	∗	NOUN
ejpam-4793	585	5	βb2	βb2	NOUN
ejpam-4793	585	6	)	)	PUNCT
ejpam-4793	585	7	=	=	PUNCT
ejpam-4793	585	8	(	(	PUNCT
ejpam-4793	585	9	αa1	αa1	PROPN
ejpam-4793	585	10	∗	∗	NOUN
ejpam-4793	585	11	βa2	βa2	NOUN
ejpam-4793	585	12	)	)	PUNCT
ejpam-4793	585	13	∗	∗	NOUN
ejpam-4793	585	14	(	(	PUNCT
ejpam-4793	585	15	αb1	αb1	NOUN
ejpam-4793	585	16	∗	∗	NOUN
ejpam-4793	585	17	βb2	βb2	NOUN
ejpam-4793	585	18	)	)	PUNCT
ejpam-4793	585	19	.	.	PUNCT
ejpam-4793	586	1	this	this	PRON
ejpam-4793	586	2	implies	imply	VERB
ejpam-4793	586	3	that	that	SCONJ
ejpam-4793	586	4	αx	αx	ADV
ejpam-4793	586	5	∗	∗	VERB
ejpam-4793	586	6	βy	βy	PRON
ejpam-4793	586	7	∈	∈	PROPN
ejpam-4793	586	8	a	a	DET
ejpam-4793	586	9	∗b	∗b	PROPN
ejpam-4793	586	10	.	.	PUNCT
ejpam-4793	587	1	therefore	therefore	ADV
ejpam-4793	587	2	,	,	PUNCT
ejpam-4793	587	3	a	a	DET
ejpam-4793	587	4	∗b	∗b	PROPN
ejpam-4793	587	5	is	be	AUX
ejpam-4793	587	6	a	a	DET
ejpam-4793	587	7	γ	γ	NOUN
ejpam-4793	587	8	-	-	ADJ
ejpam-4793	587	9	submonoid	submonoid	NOUN
ejpam-4793	587	10	of	of	ADP
ejpam-4793	587	11	m	m	PROPN
ejpam-4793	587	12	.	.	PUNCT
ejpam-4793	588	1	lemma	lemma	PROPN
ejpam-4793	588	2	6	6	NUM
ejpam-4793	588	3	.	.	PUNCT
ejpam-4793	589	1	let	let	VERB
ejpam-4793	589	2	a	a	PRON
ejpam-4793	589	3	and	and	CCONJ
ejpam-4793	589	4	b	b	NOUN
ejpam-4793	589	5	be	be	AUX
ejpam-4793	589	6	γ	γ	NOUN
ejpam-4793	589	7	-	-	NOUN
ejpam-4793	589	8	submonoids	submonoid	NOUN
ejpam-4793	589	9	of	of	ADP
ejpam-4793	589	10	a	a	DET
ejpam-4793	589	11	commutative	commutative	ADJ
ejpam-4793	589	12	γ	γ	X
ejpam-4793	589	13	-	-	PUNCT
ejpam-4793	589	14	monoid	monoid	NOUN
ejpam-4793	589	15	m	m	PROPN
ejpam-4793	589	16	.	.	PUNCT
ejpam-4793	590	1	then	then	ADV
ejpam-4793	590	2	the	the	DET
ejpam-4793	590	3	map	map	NOUN
ejpam-4793	590	4	f	f	X
ejpam-4793	590	5	:	:	PUNCT
ejpam-4793	590	6	a	a	DET
ejpam-4793	590	7	→	→	PUNCT
ejpam-4793	590	8	a	a	DET
ejpam-4793	590	9	∗b	∗b	PROPN
ejpam-4793	590	10	defined	define	VERB
ejpam-4793	590	11	by	by	ADP
ejpam-4793	590	12	f(a	f(a	PROPN
ejpam-4793	590	13	)	)	PUNCT
ejpam-4793	590	14	=	=	PUNCT
ejpam-4793	590	15	a	a	DET
ejpam-4793	590	16	∗	∗	NOUN
ejpam-4793	590	17	1	1	NUM
ejpam-4793	590	18	m	m	NOUN
ejpam-4793	590	19	is	be	AUX
ejpam-4793	590	20	a	a	DET
ejpam-4793	590	21	γ	γ	NOUN
ejpam-4793	590	22	-	-	PUNCT
ejpam-4793	590	23	monoid	monoid	NOUN
ejpam-4793	590	24	homomorphism	homomorphism	NOUN
ejpam-4793	590	25	.	.	PUNCT
ejpam-4793	591	1	proof	proof	NOUN
ejpam-4793	591	2	.	.	PUNCT
ejpam-4793	592	1	let	let	VERB
ejpam-4793	592	2	x	x	PRON
ejpam-4793	592	3	,	,	PUNCT
ejpam-4793	592	4	y	y	PROPN
ejpam-4793	592	5	∈	∈	PROPN
ejpam-4793	592	6	a	a	DET
ejpam-4793	592	7	such	such	ADJ
ejpam-4793	592	8	that	that	PRON
ejpam-4793	592	9	x	x	X
ejpam-4793	592	10	=	=	PUNCT
ejpam-4793	592	11	y.	y.	NOUN
ejpam-4793	592	12	then	then	ADV
ejpam-4793	592	13	f(x	f(x	PROPN
ejpam-4793	592	14	)	)	PUNCT
ejpam-4793	593	1	=	=	PUNCT
ejpam-4793	594	1	x	x	SYM
ejpam-4793	594	2	∗	∗	NOUN
ejpam-4793	594	3	1	1	NUM
ejpam-4793	594	4	m	m	NOUN
ejpam-4793	594	5	=	=	NOUN
ejpam-4793	594	6	x	x	PUNCT
ejpam-4793	594	7	=	=	PUNCT
ejpam-4793	594	8	y	y	PROPN
ejpam-4793	594	9	=	=	SYM
ejpam-4793	594	10	y	y	PROPN
ejpam-4793	594	11	∗	∗	PROPN
ejpam-4793	594	12	1	1	NUM
ejpam-4793	594	13	m	m	PROPN
ejpam-4793	594	14	=	=	SYM
ejpam-4793	594	15	f(y	f(y	NOUN
ejpam-4793	594	16	)	)	PUNCT
ejpam-4793	594	17	and	and	CCONJ
ejpam-4793	594	18	f	f	PROPN
ejpam-4793	594	19	is	be	AUX
ejpam-4793	594	20	well	well	ADV
ejpam-4793	594	21	-	-	PUNCT
ejpam-4793	594	22	defined	define	VERB
ejpam-4793	594	23	.	.	PUNCT
ejpam-4793	595	1	let	let	VERB
ejpam-4793	595	2	x	x	PRON
ejpam-4793	595	3	,	,	PUNCT
ejpam-4793	595	4	y	y	PROPN
ejpam-4793	595	5	∈	∈	PROPN
ejpam-4793	595	6	a.	a.	NOUN
ejpam-4793	595	7	then	then	ADV
ejpam-4793	596	1	(	(	PUNCT
ejpam-4793	596	2	i	i	NOUN
ejpam-4793	596	3	)	)	PUNCT
ejpam-4793	596	4	f(x	f(x	PROPN
ejpam-4793	596	5	∗	∗	VERB
ejpam-4793	596	6	y	y	NOUN
ejpam-4793	596	7	)	)	PUNCT
ejpam-4793	596	8	=	=	PUNCT
ejpam-4793	597	1	x	x	SYM
ejpam-4793	597	2	∗	∗	NOUN
ejpam-4793	597	3	y	y	PROPN
ejpam-4793	597	4	∗	∗	PROPN
ejpam-4793	597	5	1	1	NUM
ejpam-4793	597	6	m	m	NOUN
ejpam-4793	597	7	=	=	NOUN
ejpam-4793	597	8	x	x	SYM
ejpam-4793	597	9	∗	∗	NOUN
ejpam-4793	597	10	y	y	NOUN
ejpam-4793	597	11	=	=	SYM
ejpam-4793	597	12	(	(	PUNCT
ejpam-4793	597	13	x	x	SYM
ejpam-4793	597	14	∗	∗	NOUN
ejpam-4793	597	15	1	1	NUM
ejpam-4793	597	16	m	m	NOUN
ejpam-4793	597	17	)	)	PUNCT
ejpam-4793	597	18	∗	∗	NOUN
ejpam-4793	597	19	(	(	PUNCT
ejpam-4793	597	20	y	y	PROPN
ejpam-4793	597	21	∗	∗	PROPN
ejpam-4793	597	22	1	1	NUM
ejpam-4793	597	23	m	m	NOUN
ejpam-4793	597	24	)	)	PUNCT
ejpam-4793	598	1	=	=	SYM
ejpam-4793	598	2	f(x	f(x	PROPN
ejpam-4793	598	3	)	)	PUNCT
ejpam-4793	598	4	∗	∗	NOUN
ejpam-4793	598	5	f(y	f(y	NOUN
ejpam-4793	598	6	)	)	PUNCT
ejpam-4793	598	7	,	,	PUNCT
ejpam-4793	598	8	(	(	PUNCT
ejpam-4793	598	9	ii	ii	NOUN
ejpam-4793	598	10	)	)	PUNCT
ejpam-4793	598	11	f(1	f(1	PROPN
ejpam-4793	598	12	m	m	NOUN
ejpam-4793	598	13	)	)	PUNCT
ejpam-4793	599	1	=	=	PUNCT
ejpam-4793	599	2	1	1	NUM
ejpam-4793	599	3	m	m	NOUN
ejpam-4793	599	4	∗	∗	NOUN
ejpam-4793	599	5	1	1	NUM
ejpam-4793	599	6	m	m	NOUN
ejpam-4793	599	7	,	,	PUNCT
ejpam-4793	599	8	the	the	DET
ejpam-4793	599	9	identity	identity	NOUN
ejpam-4793	599	10	in	in	ADP
ejpam-4793	599	11	a	a	DET
ejpam-4793	599	12	∗b	∗b	NOUN
ejpam-4793	599	13	.	.	PUNCT
ejpam-4793	600	1	thus	thus	ADV
ejpam-4793	600	2	,	,	PUNCT
ejpam-4793	600	3	f	f	PROPN
ejpam-4793	600	4	is	be	AUX
ejpam-4793	600	5	a	a	DET
ejpam-4793	600	6	monoid	monoid	NOUN
ejpam-4793	600	7	homomorphism	homomorphism	NOUN
ejpam-4793	600	8	.	.	PUNCT
ejpam-4793	601	1	now	now	ADV
ejpam-4793	601	2	,	,	PUNCT
ejpam-4793	601	3	for	for	ADP
ejpam-4793	601	4	all	all	PRON
ejpam-4793	601	5	α	α	DET
ejpam-4793	601	6	∈	∈	NOUN
ejpam-4793	601	7	γ	γ	NOUN
ejpam-4793	601	8	and	and	CCONJ
ejpam-4793	601	9	x	x	PROPN
ejpam-4793	601	10	∈	∈	PROPN
ejpam-4793	601	11	a	a	PRON
ejpam-4793	601	12	,	,	PUNCT
ejpam-4793	601	13	f(αx	f(αx	PROPN
ejpam-4793	601	14	)	)	PUNCT
ejpam-4793	601	15	=	=	PUNCT
ejpam-4793	601	16	αx	αx	NOUN
ejpam-4793	601	17	∗	∗	NOUN
ejpam-4793	601	18	1	1	NUM
ejpam-4793	601	19	m	m	NOUN
ejpam-4793	601	20	=	=	VERB
ejpam-4793	601	21	αx	αx	PROPN
ejpam-4793	601	22	∗	∗	PROPN
ejpam-4793	601	23	α1	α1	PROPN
ejpam-4793	601	24	m	m	PROPN
ejpam-4793	601	25	=	=	ADJ
ejpam-4793	601	26	α(x	α(x	PROPN
ejpam-4793	601	27	∗	∗	NOUN
ejpam-4793	601	28	1	1	NUM
ejpam-4793	601	29	m	m	NOUN
ejpam-4793	601	30	)	)	PUNCT
ejpam-4793	601	31	=	=	SYM
ejpam-4793	601	32	αf(x	αf(x	NUM
ejpam-4793	601	33	)	)	PUNCT
ejpam-4793	601	34	.	.	PUNCT
ejpam-4793	602	1	thus	thus	ADV
ejpam-4793	602	2	,	,	PUNCT
ejpam-4793	602	3	f	f	PROPN
ejpam-4793	602	4	is	be	AUX
ejpam-4793	602	5	a	a	DET
ejpam-4793	602	6	γ	γ	NOUN
ejpam-4793	602	7	-	-	PUNCT
ejpam-4793	602	8	monoid	monoid	NOUN
ejpam-4793	602	9	homomorphism	homomorphism	NOUN
ejpam-4793	602	10	.	.	PUNCT
ejpam-4793	603	1	5	5	X
ejpam-4793	603	2	.	.	X
ejpam-4793	603	3	quotient	quotient	VERB
ejpam-4793	603	4	γ	γ	NOUN
ejpam-4793	603	5	-	-	PUNCT
ejpam-4793	603	6	monoids	monoid	NOUN
ejpam-4793	603	7	in	in	ADP
ejpam-4793	603	8	[	[	X
ejpam-4793	603	9	5	5	NUM
ejpam-4793	603	10	]	]	PUNCT
ejpam-4793	603	11	,	,	PUNCT
ejpam-4793	603	12	the	the	DET
ejpam-4793	603	13	quotient	quotient	NOUN
ejpam-4793	603	14	γ	γ	PROPN
ejpam-4793	603	15	-	-	PUNCT
ejpam-4793	603	16	monoid	monoid	NOUN
ejpam-4793	603	17	m	m	PROPN
ejpam-4793	603	18	/	/	SYM
ejpam-4793	603	19	s	s	VERB
ejpam-4793	603	20	was	be	AUX
ejpam-4793	603	21	established	establish	VERB
ejpam-4793	603	22	using	use	VERB
ejpam-4793	603	23	the	the	DET
ejpam-4793	603	24	equivalence	equivalence	NOUN
ejpam-4793	603	25	relation	relation	NOUN
ejpam-4793	603	26	in	in	ADP
ejpam-4793	603	27	definition	definition	NOUN
ejpam-4793	603	28	6	6	NUM
ejpam-4793	603	29	such	such	ADJ
ejpam-4793	603	30	that	that	SCONJ
ejpam-4793	603	31	the	the	DET
ejpam-4793	603	32	commutative	commutative	ADJ
ejpam-4793	603	33	γ	γ	NOUN
ejpam-4793	603	34	-	-	PUNCT
ejpam-4793	603	35	monoidm	monoidm	ADJ
ejpam-4793	603	36	and	and	CCONJ
ejpam-4793	603	37	γ	γ	NOUN
ejpam-4793	603	38	-	-	PUNCT
ejpam-4793	603	39	order	order	NOUN
ejpam-4793	603	40	-	-	PUNCT
ejpam-4793	603	41	ideal	ideal	NOUN
ejpam-4793	603	42	s	s	PART
ejpam-4793	603	43	ofm	ofm	PROPN
ejpam-4793	603	44	were	be	AUX
ejpam-4793	603	45	treated	treat	VERB
ejpam-4793	603	46	as	as	ADP
ejpam-4793	603	47	commutative	commutative	ADJ
ejpam-4793	603	48	monoid	monoid	NOUN
ejpam-4793	603	49	and	and	CCONJ
ejpam-4793	603	50	submonoid	submonoid	ADJ
ejpam-4793	603	51	,	,	PUNCT
ejpam-4793	603	52	respectively	respectively	ADV
ejpam-4793	603	53	.	.	PUNCT
ejpam-4793	604	1	further	far	ADV
ejpam-4793	604	2	,	,	PUNCT
ejpam-4793	604	3	the	the	DET
ejpam-4793	604	4	third	third	ADJ
ejpam-4793	604	5	isomorphism	isomorphism	NOUN
ejpam-4793	604	6	theorem	theorem	NOUN
ejpam-4793	604	7	for	for	ADP
ejpam-4793	604	8	γ	γ	NOUN
ejpam-4793	604	9	-	-	PUNCT
ejpam-4793	604	10	monoids	monoid	NOUN
ejpam-4793	604	11	via	via	ADP
ejpam-4793	604	12	γ	γ	NOUN
ejpam-4793	604	13	-	-	PUNCT
ejpam-4793	604	14	order	order	NOUN
ejpam-4793	604	15	-	-	PUNCT
ejpam-4793	604	16	ideals	ideal	NOUN
ejpam-4793	604	17	was	be	AUX
ejpam-4793	604	18	proved	prove	VERB
ejpam-4793	604	19	.	.	PUNCT
ejpam-4793	605	1	here	here	ADV
ejpam-4793	605	2	,	,	PUNCT
ejpam-4793	605	3	we	we	PRON
ejpam-4793	605	4	define	define	VERB
ejpam-4793	605	5	an	an	DET
ejpam-4793	605	6	equivalence	equivalence	NOUN
ejpam-4793	605	7	relation	relation	NOUN
ejpam-4793	605	8	and	and	CCONJ
ejpam-4793	605	9	construct	construct	VERB
ejpam-4793	605	10	quotient	quotient	NOUN
ejpam-4793	605	11	γ	γ	NOUN
ejpam-4793	605	12	-	-	PUNCT
ejpam-4793	605	13	monoids	monoid	NOUN
ejpam-4793	605	14	via	via	ADP
ejpam-4793	605	15	γsubmonoids	γsubmonoid	NOUN
ejpam-4793	605	16	.	.	PUNCT
ejpam-4793	606	1	moreover	moreover	ADV
ejpam-4793	606	2	,	,	PUNCT
ejpam-4793	606	3	we	we	PRON
ejpam-4793	606	4	prove	prove	VERB
ejpam-4793	606	5	the	the	DET
ejpam-4793	606	6	isomorphism	isomorphism	NOUN
ejpam-4793	606	7	theorems	theorem	NOUN
ejpam-4793	606	8	.	.	PUNCT
ejpam-4793	607	1	definition	definition	NOUN
ejpam-4793	607	2	13	13	NUM
ejpam-4793	607	3	.	.	PUNCT
ejpam-4793	608	1	let	let	VERB
ejpam-4793	608	2	m	m	PRON
ejpam-4793	608	3	be	be	AUX
ejpam-4793	608	4	a	a	DET
ejpam-4793	608	5	γ	γ	X
ejpam-4793	608	6	-	-	PUNCT
ejpam-4793	608	7	monoid	monoid	NOUN
ejpam-4793	608	8	.	.	PUNCT
ejpam-4793	609	1	for	for	ADP
ejpam-4793	609	2	any	any	DET
ejpam-4793	609	3	γ	γ	NOUN
ejpam-4793	609	4	-	-	ADJ
ejpam-4793	609	5	submonoid	submonoid	ADJ
ejpam-4793	609	6	s	s	PROPN
ejpam-4793	609	7	of	of	ADP
ejpam-4793	609	8	m	m	PROPN
ejpam-4793	609	9	and	and	CCONJ
ejpam-4793	609	10	for	for	ADP
ejpam-4793	609	11	all	all	DET
ejpam-4793	609	12	x	x	NOUN
ejpam-4793	609	13	,	,	PUNCT
ejpam-4793	609	14	y	y	PROPN
ejpam-4793	609	15	∈	∈	PROPN
ejpam-4793	609	16	m	m	VERB
ejpam-4793	609	17	,	,	PUNCT
ejpam-4793	609	18	we	we	PRON
ejpam-4793	609	19	define	define	VERB
ejpam-4793	609	20	a	a	DET
ejpam-4793	609	21	binary	binary	ADJ
ejpam-4793	609	22	relation	relation	NOUN
ejpam-4793	609	23	ρs	ρs	PROPN
ejpam-4793	609	24	in	in	ADP
ejpam-4793	609	25	m	m	PROPN
ejpam-4793	609	26	by	by	ADP
ejpam-4793	609	27	xρsy	xρsy	PROPN
ejpam-4793	609	28	if	if	SCONJ
ejpam-4793	609	29	and	and	CCONJ
ejpam-4793	609	30	only	only	ADV
ejpam-4793	609	31	if	if	SCONJ
ejpam-4793	609	32	for	for	ADP
ejpam-4793	609	33	all	all	PRON
ejpam-4793	609	34	α	α	PRON
ejpam-4793	609	35	∈	∈	PROPN
ejpam-4793	609	36	γ	γ	X
ejpam-4793	609	37	,	,	PUNCT
ejpam-4793	609	38	(	(	PUNCT
ejpam-4793	609	39	αx∗s)∩(αy∗s	αx∗s)∩(αy∗s	PROPN
ejpam-4793	609	40	)	)	PUNCT
ejpam-4793	610	1	̸=	̸=	PROPN
ejpam-4793	610	2	∅.	∅.	ADP
ejpam-4793	610	3	the	the	DET
ejpam-4793	610	4	next	next	ADJ
ejpam-4793	610	5	example	example	NOUN
ejpam-4793	610	6	shows	show	VERB
ejpam-4793	610	7	that	that	SCONJ
ejpam-4793	610	8	if	if	SCONJ
ejpam-4793	610	9	a	a	DET
ejpam-4793	610	10	γ	γ	NOUN
ejpam-4793	610	11	-	-	ADJ
ejpam-4793	610	12	submonoid	submonoid	ADJ
ejpam-4793	610	13	s	s	PROPN
ejpam-4793	610	14	of	of	ADP
ejpam-4793	610	15	a	a	DET
ejpam-4793	610	16	γ	γ	PROPN
ejpam-4793	610	17	-	-	PUNCT
ejpam-4793	610	18	monoid	monoid	NOUN
ejpam-4793	610	19	m	m	NOUN
ejpam-4793	610	20	is	be	AUX
ejpam-4793	610	21	not	not	PART
ejpam-4793	610	22	commutative	commutative	ADJ
ejpam-4793	610	23	,	,	PUNCT
ejpam-4793	610	24	then	then	ADV
ejpam-4793	610	25	ρs	ρs	ADV
ejpam-4793	610	26	is	be	AUX
ejpam-4793	610	27	not	not	PART
ejpam-4793	610	28	an	an	DET
ejpam-4793	610	29	equivalence	equivalence	NOUN
ejpam-4793	610	30	relation	relation	NOUN
ejpam-4793	610	31	.	.	PUNCT
ejpam-4793	611	1	h.	h.	PROPN
ejpam-4793	611	2	sarapuddin	sarapuddin	PROPN
ejpam-4793	611	3	,	,	PUNCT
ejpam-4793	611	4	j.	j.	PROPN
ejpam-4793	611	5	vilela	vilela	PROPN
ejpam-4793	611	6	/	/	SYM
ejpam-4793	611	7	eur	eur	PROPN
ejpam-4793	611	8	.	.	PUNCT
ejpam-4793	612	1	j.	j.	PROPN
ejpam-4793	612	2	pure	pure	PROPN
ejpam-4793	612	3	appl	appl	PROPN
ejpam-4793	612	4	.	.	PROPN
ejpam-4793	612	5	math	math	PROPN
ejpam-4793	612	6	,	,	PUNCT
ejpam-4793	612	7	16	16	NUM
ejpam-4793	612	8	(	(	PUNCT
ejpam-4793	612	9	3	3	NUM
ejpam-4793	612	10	)	)	PUNCT
ejpam-4793	612	11	(	(	PUNCT
ejpam-4793	612	12	2023	2023	NUM
ejpam-4793	612	13	)	)	PUNCT
ejpam-4793	612	14	,	,	PUNCT
ejpam-4793	612	15	1772	1772	NUM
ejpam-4793	612	16	-	-	SYM
ejpam-4793	612	17	1793	1793	NUM
ejpam-4793	612	18	1786	1786	NUM
ejpam-4793	612	19	example	example	NOUN
ejpam-4793	612	20	15	15	NUM
ejpam-4793	612	21	.	.	PUNCT
ejpam-4793	612	22	consider	consider	VERB
ejpam-4793	612	23	the	the	DET
ejpam-4793	612	24	γ	γ	NOUN
ejpam-4793	612	25	-	-	PUNCT
ejpam-4793	612	26	monoid	monoid	NOUN
ejpam-4793	612	27	m	m	NOUN
ejpam-4793	612	28	=	=	PUNCT
ejpam-4793	612	29	{	{	PUNCT
ejpam-4793	612	30	1	1	NUM
ejpam-4793	612	31	,	,	PUNCT
ejpam-4793	612	32	a	a	DET
ejpam-4793	612	33	,	,	PUNCT
ejpam-4793	612	34	b	b	NOUN
ejpam-4793	612	35	,	,	PUNCT
ejpam-4793	612	36	c	c	NOUN
ejpam-4793	612	37	,	,	PUNCT
ejpam-4793	612	38	d	d	NOUN
ejpam-4793	612	39	,	,	PUNCT
ejpam-4793	612	40	e	e	NOUN
ejpam-4793	612	41	}	}	PUNCT
ejpam-4793	612	42	in	in	ADP
ejpam-4793	612	43	example	example	NOUN
ejpam-4793	612	44	5	5	NUM
ejpam-4793	612	45	with	with	ADP
ejpam-4793	612	46	operation	operation	NOUN
ejpam-4793	612	47	∗	∗	NOUN
ejpam-4793	612	48	given	give	VERB
ejpam-4793	612	49	by	by	ADP
ejpam-4793	612	50	∗	∗	NOUN
ejpam-4793	612	51	1	1	NUM
ejpam-4793	612	52	a	a	DET
ejpam-4793	612	53	b	b	NOUN
ejpam-4793	612	54	c	c	NOUN
ejpam-4793	612	55	d	d	X
ejpam-4793	612	56	e	e	PROPN
ejpam-4793	612	57	1	1	NUM
ejpam-4793	612	58	1	1	NUM
ejpam-4793	612	59	a	a	DET
ejpam-4793	612	60	b	b	NOUN
ejpam-4793	612	61	c	c	NOUN
ejpam-4793	612	62	d	d	PROPN
ejpam-4793	612	63	e	e	X
ejpam-4793	612	64	a	a	PRON
ejpam-4793	612	65	a	a	PRON
ejpam-4793	612	66	a	a	DET
ejpam-4793	612	67	a	a	PRON
ejpam-4793	612	68	a	a	PRON
ejpam-4793	612	69	a	a	PRON
ejpam-4793	612	70	a	a	DET
ejpam-4793	612	71	b	b	PROPN
ejpam-4793	612	72	b	b	PROPN
ejpam-4793	612	73	b	b	PROPN
ejpam-4793	612	74	b	b	PROPN
ejpam-4793	612	75	b	b	PROPN
ejpam-4793	612	76	b	b	PROPN
ejpam-4793	612	77	b	b	PROPN
ejpam-4793	612	78	c	c	NOUN
ejpam-4793	612	79	c	c	NOUN
ejpam-4793	612	80	c	c	NOUN
ejpam-4793	612	81	c	c	NOUN
ejpam-4793	612	82	c	c	NOUN
ejpam-4793	612	83	c	c	NOUN
ejpam-4793	612	84	c	c	NOUN
ejpam-4793	613	1	d	d	PUNCT
ejpam-4793	613	2	d	d	PROPN
ejpam-4793	613	3	d	d	PROPN
ejpam-4793	613	4	d	d	PROPN
ejpam-4793	613	5	d	d	PROPN
ejpam-4793	613	6	d	d	X
ejpam-4793	613	7	d	d	X
ejpam-4793	613	8	e	e	X
ejpam-4793	613	9	e	e	X
ejpam-4793	613	10	e	e	X
ejpam-4793	613	11	e	e	X
ejpam-4793	613	12	e	e	X
ejpam-4793	613	13	e	e	X
ejpam-4793	613	14	e	e	NOUN
ejpam-4793	613	15	let	let	VERB
ejpam-4793	613	16	s	s	VERB
ejpam-4793	613	17	=	=	VERB
ejpam-4793	613	18	{	{	PUNCT
ejpam-4793	613	19	1	1	NUM
ejpam-4793	613	20	,	,	PUNCT
ejpam-4793	613	21	a	a	DET
ejpam-4793	613	22	,	,	PUNCT
ejpam-4793	613	23	b	b	NOUN
ejpam-4793	613	24	}	}	PUNCT
ejpam-4793	613	25	.	.	PUNCT
ejpam-4793	614	1	then	then	ADV
ejpam-4793	614	2	,	,	PUNCT
ejpam-4793	614	3	by	by	ADP
ejpam-4793	614	4	routine	routine	ADJ
ejpam-4793	614	5	calculation	calculation	NOUN
ejpam-4793	614	6	,	,	PUNCT
ejpam-4793	614	7	s	s	VERB
ejpam-4793	614	8	is	be	AUX
ejpam-4793	614	9	a	a	DET
ejpam-4793	614	10	γ	γ	NOUN
ejpam-4793	614	11	-	-	ADJ
ejpam-4793	614	12	submonoid	submonoid	NOUN
ejpam-4793	614	13	of	of	ADP
ejpam-4793	614	14	m	m	PROPN
ejpam-4793	614	15	.	.	PUNCT
ejpam-4793	615	1	also	also	ADV
ejpam-4793	615	2	,	,	PUNCT
ejpam-4793	615	3	s	s	VERB
ejpam-4793	615	4	is	be	AUX
ejpam-4793	615	5	not	not	PART
ejpam-4793	615	6	commutative	commutative	ADJ
ejpam-4793	615	7	since	since	SCONJ
ejpam-4793	615	8	a∗b	a∗b	PROPN
ejpam-4793	615	9	=	=	PUNCT
ejpam-4793	615	10	a	a	DET
ejpam-4793	615	11	̸=	̸=	PROPN
ejpam-4793	615	12	b	b	NOUN
ejpam-4793	615	13	=	=	SYM
ejpam-4793	615	14	b∗a	b∗a	PROPN
ejpam-4793	615	15	.	.	PUNCT
ejpam-4793	616	1	now	now	ADV
ejpam-4793	616	2	,	,	PUNCT
ejpam-4793	616	3	for	for	ADP
ejpam-4793	616	4	all	all	PRON
ejpam-4793	616	5	α	α	PRON
ejpam-4793	616	6	∈	∈	PROPN
ejpam-4793	616	7	γ	γ	X
ejpam-4793	616	8	,	,	PUNCT
ejpam-4793	616	9	we	we	PRON
ejpam-4793	616	10	have	have	VERB
ejpam-4793	616	11	α1∗s	α1∗s	X
ejpam-4793	616	12	=	=	SYM
ejpam-4793	616	13	1∗s	1∗s	NUM
ejpam-4793	616	14	=	=	SYM
ejpam-4793	616	15	{	{	PUNCT
ejpam-4793	616	16	1	1	NUM
ejpam-4793	616	17	,	,	PUNCT
ejpam-4793	616	18	a	a	DET
ejpam-4793	616	19	,	,	PUNCT
ejpam-4793	616	20	b	b	NOUN
ejpam-4793	616	21	}	}	PUNCT
ejpam-4793	616	22	,	,	PUNCT
ejpam-4793	616	23	αa	αa	ADV
ejpam-4793	616	24	∗	∗	NOUN
ejpam-4793	616	25	s	s	PART
ejpam-4793	616	26	=	=	NOUN
ejpam-4793	616	27	a	a	DET
ejpam-4793	616	28	∗	∗	NOUN
ejpam-4793	616	29	s	s	PART
ejpam-4793	616	30	=	=	X
ejpam-4793	616	31	{	{	PUNCT
ejpam-4793	616	32	a	a	NOUN
ejpam-4793	616	33	}	}	PUNCT
ejpam-4793	616	34	and	and	CCONJ
ejpam-4793	616	35	αb	αb	ADP
ejpam-4793	616	36	∗	∗	NOUN
ejpam-4793	616	37	s	s	PART
ejpam-4793	616	38	=	=	SYM
ejpam-4793	616	39	b	b	PROPN
ejpam-4793	616	40	∗	∗	NOUN
ejpam-4793	616	41	s	s	PART
ejpam-4793	616	42	=	=	X
ejpam-4793	616	43	{	{	PUNCT
ejpam-4793	616	44	b	b	NOUN
ejpam-4793	616	45	}	}	PUNCT
ejpam-4793	616	46	.	.	PUNCT
ejpam-4793	617	1	thus	thus	ADV
ejpam-4793	617	2	,	,	PUNCT
ejpam-4793	617	3	(	(	PUNCT
ejpam-4793	617	4	αa	αa	NOUN
ejpam-4793	617	5	∗	∗	NUM
ejpam-4793	617	6	s)∩	s)∩	NOUN
ejpam-4793	617	7	(	(	PUNCT
ejpam-4793	617	8	α1	α1	PROPN
ejpam-4793	617	9	∗	∗	NOUN
ejpam-4793	617	10	s	s	PART
ejpam-4793	617	11	)	)	PUNCT
ejpam-4793	617	12	=	=	SYM
ejpam-4793	617	13	{	{	PUNCT
ejpam-4793	617	14	a	a	NOUN
ejpam-4793	617	15	}	}	PUNCT
ejpam-4793	617	16	=	=	NOUN
ejpam-4793	617	17	̸	̸	ADJ
ejpam-4793	617	18	∅	∅	NOUN
ejpam-4793	617	19	which	which	PRON
ejpam-4793	617	20	implies	imply	VERB
ejpam-4793	617	21	that	that	DET
ejpam-4793	617	22	aρs1	aρs1	PROPN
ejpam-4793	617	23	.	.	PUNCT
ejpam-4793	618	1	also	also	ADV
ejpam-4793	618	2	,	,	PUNCT
ejpam-4793	618	3	(	(	PUNCT
ejpam-4793	618	4	α1	α1	PROPN
ejpam-4793	618	5	∗	∗	NOUN
ejpam-4793	618	6	s	s	NOUN
ejpam-4793	618	7	)	)	PUNCT
ejpam-4793	618	8	∩	∩	NOUN
ejpam-4793	618	9	(	(	PUNCT
ejpam-4793	618	10	αb	αb	ADP
ejpam-4793	618	11	∗	∗	NOUN
ejpam-4793	618	12	s	s	PART
ejpam-4793	618	13	)	)	PUNCT
ejpam-4793	618	14	=	=	SYM
ejpam-4793	618	15	{	{	PUNCT
ejpam-4793	618	16	b	b	NOUN
ejpam-4793	618	17	}	}	PUNCT
ejpam-4793	618	18	=	=	NOUN
ejpam-4793	618	19	̸	̸	ADJ
ejpam-4793	618	20	∅	∅	NOUN
ejpam-4793	618	21	which	which	PRON
ejpam-4793	618	22	implies	imply	VERB
ejpam-4793	618	23	that	that	SCONJ
ejpam-4793	618	24	1ρsb	1ρsb	NUM
ejpam-4793	618	25	.	.	PUNCT
ejpam-4793	619	1	however	however	ADV
ejpam-4793	619	2	,	,	PUNCT
ejpam-4793	619	3	(	(	PUNCT
ejpam-4793	619	4	αa	αa	PROPN
ejpam-4793	619	5	∗s)∩	∗s)∩	PROPN
ejpam-4793	619	6	(	(	PUNCT
ejpam-4793	619	7	βb	βb	PROPN
ejpam-4793	619	8	∗s	∗s	NOUN
ejpam-4793	619	9	)	)	PUNCT
ejpam-4793	619	10	=	=	NOUN
ejpam-4793	619	11	∅	∅	NOUN
ejpam-4793	619	12	which	which	PRON
ejpam-4793	619	13	implies	imply	VERB
ejpam-4793	619	14	that	that	SCONJ
ejpam-4793	619	15	a	a	PRON
ejpam-4793	619	16	is	be	AUX
ejpam-4793	619	17	not	not	PART
ejpam-4793	619	18	related	relate	VERB
ejpam-4793	619	19	to	to	ADP
ejpam-4793	619	20	b	b	NOUN
ejpam-4793	619	21	under	under	ADP
ejpam-4793	619	22	ρs	ρs	ADV
ejpam-4793	619	23	,	,	PUNCT
ejpam-4793	619	24	that	that	ADV
ejpam-4793	619	25	is	is	ADV
ejpam-4793	619	26	,	,	PUNCT
ejpam-4793	619	27	ρs	ρs	ADV
ejpam-4793	619	28	is	be	AUX
ejpam-4793	619	29	not	not	PART
ejpam-4793	619	30	transitive	transitive	ADJ
ejpam-4793	619	31	,	,	PUNCT
ejpam-4793	619	32	hence	hence	ADV
ejpam-4793	619	33	not	not	PART
ejpam-4793	619	34	an	an	DET
ejpam-4793	619	35	equivalence	equivalence	NOUN
ejpam-4793	619	36	relation	relation	NOUN
ejpam-4793	619	37	.	.	PUNCT
ejpam-4793	620	1	the	the	DET
ejpam-4793	620	2	following	following	ADJ
ejpam-4793	620	3	result	result	NOUN
ejpam-4793	620	4	tells	tell	VERB
ejpam-4793	620	5	us	we	PRON
ejpam-4793	620	6	that	that	PRON
ejpam-4793	620	7	ρs	ρs	NOUN
ejpam-4793	620	8	is	be	AUX
ejpam-4793	620	9	an	an	DET
ejpam-4793	620	10	equivalence	equivalence	NOUN
ejpam-4793	620	11	relation	relation	NOUN
ejpam-4793	620	12	for	for	ADP
ejpam-4793	620	13	any	any	DET
ejpam-4793	620	14	commutative	commutative	ADJ
ejpam-4793	620	15	γ	γ	NOUN
ejpam-4793	620	16	-	-	ADJ
ejpam-4793	620	17	submonoid	submonoid	ADJ
ejpam-4793	620	18	s	s	PROPN
ejpam-4793	620	19	of	of	ADP
ejpam-4793	620	20	a	a	DET
ejpam-4793	620	21	γ	γ	X
ejpam-4793	620	22	-	-	PUNCT
ejpam-4793	620	23	monoid	monoid	NOUN
ejpam-4793	620	24	m	m	NOUN
ejpam-4793	620	25	.	.	PUNCT
ejpam-4793	621	1	further	far	ADV
ejpam-4793	621	2	,	,	PUNCT
ejpam-4793	621	3	if	if	SCONJ
ejpam-4793	621	4	m	m	NOUN
ejpam-4793	621	5	is	be	AUX
ejpam-4793	621	6	commutative	commutative	ADJ
ejpam-4793	621	7	,	,	PUNCT
ejpam-4793	621	8	then	then	ADV
ejpam-4793	621	9	ρs	ρs	ADV
ejpam-4793	621	10	is	be	AUX
ejpam-4793	621	11	a	a	DET
ejpam-4793	621	12	congruence	congruence	NOUN
ejpam-4793	621	13	relation	relation	NOUN
ejpam-4793	621	14	on	on	ADP
ejpam-4793	621	15	m	m	PROPN
ejpam-4793	621	16	.	.	PUNCT
ejpam-4793	622	1	theorem	theorem	ADJ
ejpam-4793	622	2	10	10	NUM
ejpam-4793	622	3	.	.	PUNCT
ejpam-4793	623	1	let	let	VERB
ejpam-4793	623	2	s	s	PRON
ejpam-4793	623	3	be	be	AUX
ejpam-4793	623	4	a	a	DET
ejpam-4793	623	5	commutative	commutative	ADJ
ejpam-4793	623	6	γ	γ	NOUN
ejpam-4793	623	7	-	-	ADJ
ejpam-4793	623	8	submonoid	submonoid	NOUN
ejpam-4793	623	9	of	of	ADP
ejpam-4793	623	10	a	a	DET
ejpam-4793	623	11	γ	γ	X
ejpam-4793	623	12	-	-	PUNCT
ejpam-4793	623	13	monoid	monoid	NOUN
ejpam-4793	623	14	m	m	PROPN
ejpam-4793	623	15	.	.	PUNCT
ejpam-4793	624	1	then	then	ADV
ejpam-4793	624	2	(	(	PUNCT
ejpam-4793	624	3	i	i	NOUN
ejpam-4793	624	4	)	)	PUNCT
ejpam-4793	624	5	ρs	ρs	ADV
ejpam-4793	624	6	is	be	AUX
ejpam-4793	624	7	an	an	DET
ejpam-4793	624	8	equivalence	equivalence	NOUN
ejpam-4793	624	9	relation	relation	NOUN
ejpam-4793	624	10	on	on	ADP
ejpam-4793	624	11	m	m	PROPN
ejpam-4793	624	12	.	.	PUNCT
ejpam-4793	625	1	(	(	PUNCT
ejpam-4793	625	2	ii	ii	NOUN
ejpam-4793	625	3	)	)	PUNCT
ejpam-4793	625	4	if	if	SCONJ
ejpam-4793	625	5	m	m	NOUN
ejpam-4793	625	6	is	be	AUX
ejpam-4793	625	7	commutative	commutative	ADJ
ejpam-4793	625	8	,	,	PUNCT
ejpam-4793	625	9	then	then	ADV
ejpam-4793	625	10	ρs	ρs	ADV
ejpam-4793	625	11	is	be	AUX
ejpam-4793	625	12	a	a	DET
ejpam-4793	625	13	congruence	congruence	NOUN
ejpam-4793	625	14	relation	relation	NOUN
ejpam-4793	625	15	on	on	ADP
ejpam-4793	625	16	m	m	PROPN
ejpam-4793	625	17	.	.	PUNCT
ejpam-4793	626	1	proof	proof	NOUN
ejpam-4793	626	2	.	.	PUNCT
ejpam-4793	627	1	let	let	VERB
ejpam-4793	627	2	s	s	PRON
ejpam-4793	627	3	be	be	AUX
ejpam-4793	627	4	a	a	DET
ejpam-4793	627	5	commutative	commutative	ADJ
ejpam-4793	627	6	γ	γ	NOUN
ejpam-4793	627	7	-	-	ADJ
ejpam-4793	627	8	submonoid	submonoid	NOUN
ejpam-4793	627	9	of	of	ADP
ejpam-4793	627	10	a	a	DET
ejpam-4793	627	11	γ	γ	X
ejpam-4793	627	12	-	-	PUNCT
ejpam-4793	627	13	monoid	monoid	NOUN
ejpam-4793	627	14	m	m	NOUN
ejpam-4793	627	15	.	.	PUNCT
ejpam-4793	628	1	(	(	PUNCT
ejpam-4793	628	2	i	i	NOUN
ejpam-4793	628	3	)	)	PUNCT
ejpam-4793	628	4	let	let	VERB
ejpam-4793	628	5	x	x	PUNCT
ejpam-4793	628	6	∈	∈	VERB
ejpam-4793	628	7	m	m	PROPN
ejpam-4793	628	8	and	and	CCONJ
ejpam-4793	628	9	s	s	VERB
ejpam-4793	628	10	a	a	DET
ejpam-4793	628	11	γ	γ	NOUN
ejpam-4793	628	12	-	-	ADJ
ejpam-4793	628	13	submonoid	submonoid	NOUN
ejpam-4793	628	14	of	of	ADP
ejpam-4793	628	15	m	m	PROPN
ejpam-4793	628	16	.	.	PUNCT
ejpam-4793	629	1	then	then	ADV
ejpam-4793	629	2	,	,	PUNCT
ejpam-4793	629	3	for	for	ADP
ejpam-4793	629	4	α	α	DET
ejpam-4793	629	5	∈	∈	PROPN
ejpam-4793	629	6	γ	γ	X
ejpam-4793	629	7	,	,	PUNCT
ejpam-4793	629	8	we	we	PRON
ejpam-4793	629	9	have	have	VERB
ejpam-4793	629	10	(	(	PUNCT
ejpam-4793	629	11	αx∗s)∩(αx∗s	αx∗s)∩(αx∗s	ADJ
ejpam-4793	629	12	)	)	PUNCT
ejpam-4793	629	13	=	=	SYM
ejpam-4793	630	1	αx	αx	NOUN
ejpam-4793	630	2	∗	∗	NOUN
ejpam-4793	630	3	s	s	PART
ejpam-4793	630	4	̸=	̸=	PROPN
ejpam-4793	630	5	∅	∅	NOUN
ejpam-4793	630	6	since	since	SCONJ
ejpam-4793	630	7	αx	αx	PROPN
ejpam-4793	630	8	=	=	PUNCT
ejpam-4793	630	9	αx	αx	PROPN
ejpam-4793	630	10	∗	∗	NOUN
ejpam-4793	630	11	1	1	NUM
ejpam-4793	630	12	m	m	NOUN
ejpam-4793	630	13	∈	∈	NOUN
ejpam-4793	630	14	αx	αx	NOUN
ejpam-4793	630	15	∗	∗	NOUN
ejpam-4793	630	16	s.	s.	PROPN
ejpam-4793	630	17	thus	thus	ADV
ejpam-4793	630	18	,	,	PUNCT
ejpam-4793	630	19	xρsx	xρsx	PROPN
ejpam-4793	630	20	and	and	CCONJ
ejpam-4793	630	21	ρs	ρs	ADV
ejpam-4793	630	22	is	be	AUX
ejpam-4793	630	23	reflexive	reflexive	ADJ
ejpam-4793	630	24	.	.	PUNCT
ejpam-4793	631	1	let	let	VERB
ejpam-4793	631	2	xρsy	xρsy	PROPN
ejpam-4793	631	3	.	.	PUNCT
ejpam-4793	632	1	then	then	ADV
ejpam-4793	632	2	,	,	PUNCT
ejpam-4793	632	3	for	for	ADP
ejpam-4793	632	4	all	all	PRON
ejpam-4793	632	5	α	α	PRON
ejpam-4793	632	6	∈	∈	PROPN
ejpam-4793	632	7	γ	γ	X
ejpam-4793	632	8	,	,	PUNCT
ejpam-4793	632	9	(	(	PUNCT
ejpam-4793	632	10	αx	αx	ADV
ejpam-4793	632	11	∗	∗	PROPN
ejpam-4793	632	12	s	s	PART
ejpam-4793	632	13	)	)	PUNCT
ejpam-4793	632	14	∩	∩	NOUN
ejpam-4793	632	15	(	(	PUNCT
ejpam-4793	632	16	αy	αy	ADP
ejpam-4793	632	17	∗	∗	NOUN
ejpam-4793	632	18	s	s	PART
ejpam-4793	632	19	)	)	PUNCT
ejpam-4793	632	20	̸=	̸=	PROPN
ejpam-4793	632	21	∅.	∅.	ADV
ejpam-4793	632	22	thus	thus	ADV
ejpam-4793	632	23	,	,	PUNCT
ejpam-4793	632	24	(	(	PUNCT
ejpam-4793	632	25	αy	αy	ADP
ejpam-4793	632	26	∗	∗	NUM
ejpam-4793	632	27	s	s	NOUN
ejpam-4793	632	28	)	)	PUNCT
ejpam-4793	632	29	∩	∩	NOUN
ejpam-4793	632	30	(	(	PUNCT
ejpam-4793	632	31	αx	αx	ADV
ejpam-4793	632	32	∗	∗	NOUN
ejpam-4793	632	33	s	s	PART
ejpam-4793	632	34	)	)	PUNCT
ejpam-4793	632	35	=	=	SYM
ejpam-4793	632	36	(	(	PUNCT
ejpam-4793	632	37	αx	αx	ADV
ejpam-4793	632	38	∗	∗	PROPN
ejpam-4793	632	39	s	s	PART
ejpam-4793	632	40	)	)	PUNCT
ejpam-4793	632	41	∩	∩	NOUN
ejpam-4793	632	42	(	(	PUNCT
ejpam-4793	632	43	αy	αy	ADP
ejpam-4793	632	44	∗	∗	NOUN
ejpam-4793	632	45	s	s	PART
ejpam-4793	632	46	)	)	PUNCT
ejpam-4793	632	47	̸=	̸=	PROPN
ejpam-4793	632	48	∅.	∅.	PRON
ejpam-4793	632	49	hence	hence	ADV
ejpam-4793	632	50	,	,	PUNCT
ejpam-4793	632	51	yρsx	yρsx	ADJ
ejpam-4793	632	52	and	and	CCONJ
ejpam-4793	632	53	ρs	ρs	ADV
ejpam-4793	632	54	is	be	AUX
ejpam-4793	632	55	symmetric	symmetric	ADJ
ejpam-4793	632	56	.	.	PUNCT
ejpam-4793	633	1	now	now	ADV
ejpam-4793	633	2	,	,	PUNCT
ejpam-4793	633	3	let	let	VERB
ejpam-4793	633	4	xρsy	xρsy	PROPN
ejpam-4793	633	5	and	and	CCONJ
ejpam-4793	633	6	yρsz	yρsz	NOUN
ejpam-4793	633	7	.	.	PUNCT
ejpam-4793	634	1	then	then	ADV
ejpam-4793	634	2	,	,	PUNCT
ejpam-4793	634	3	for	for	ADP
ejpam-4793	634	4	all	all	DET
ejpam-4793	634	5	α	α	NOUN
ejpam-4793	634	6	,	,	PUNCT
ejpam-4793	634	7	β	β	PROPN
ejpam-4793	634	8	∈	∈	PROPN
ejpam-4793	634	9	γ	γ	X
ejpam-4793	634	10	,	,	PUNCT
ejpam-4793	634	11	(	(	PUNCT
ejpam-4793	634	12	αx	αx	ADV
ejpam-4793	634	13	∗	∗	PROPN
ejpam-4793	634	14	s	s	PART
ejpam-4793	634	15	)	)	PUNCT
ejpam-4793	634	16	∩	∩	NOUN
ejpam-4793	634	17	(	(	PUNCT
ejpam-4793	634	18	αy	αy	ADP
ejpam-4793	634	19	∗	∗	NOUN
ejpam-4793	634	20	s	s	PART
ejpam-4793	634	21	)	)	PUNCT
ejpam-4793	634	22	̸=	̸=	PROPN
ejpam-4793	634	23	∅	∅	NOUN
ejpam-4793	634	24	and	and	CCONJ
ejpam-4793	634	25	(	(	PUNCT
ejpam-4793	634	26	βy	βy	INTJ
ejpam-4793	634	27	∗	∗	PROPN
ejpam-4793	634	28	s	s	NOUN
ejpam-4793	634	29	)	)	PUNCT
ejpam-4793	634	30	∩	∩	NOUN
ejpam-4793	634	31	(	(	PUNCT
ejpam-4793	634	32	βz	βz	ADP
ejpam-4793	634	33	∗	∗	PRON
ejpam-4793	634	34	s	s	PART
ejpam-4793	634	35	)	)	PUNCT
ejpam-4793	634	36	̸=	̸=	PROPN
ejpam-4793	634	37	∅.	∅.	ADV
ejpam-4793	634	38	thus	thus	ADV
ejpam-4793	634	39	,	,	PUNCT
ejpam-4793	634	40	we	we	PRON
ejpam-4793	634	41	have	have	VERB
ejpam-4793	634	42	αx	αx	PROPN
ejpam-4793	634	43	∗	∗	NOUN
ejpam-4793	634	44	s1	s1	NOUN
ejpam-4793	634	45	=	=	PUNCT
ejpam-4793	634	46	αy	αy	PROPN
ejpam-4793	634	47	∗	∗	NOUN
ejpam-4793	634	48	s2	s2	PROPN
ejpam-4793	634	49	and	and	CCONJ
ejpam-4793	634	50	βy	βy	DET
ejpam-4793	634	51	∗	∗	NOUN
ejpam-4793	634	52	s3	s3	NOUN
ejpam-4793	634	53	=	=	PROPN
ejpam-4793	634	54	βz	βz	NOUN
ejpam-4793	634	55	∗	∗	NOUN
ejpam-4793	634	56	s4	s4	NOUN
ejpam-4793	634	57	for	for	ADP
ejpam-4793	634	58	some	some	DET
ejpam-4793	634	59	s1	s1	NOUN
ejpam-4793	634	60	,	,	PUNCT
ejpam-4793	634	61	s2	s2	PROPN
ejpam-4793	634	62	,	,	PUNCT
ejpam-4793	634	63	s3	s3	PROPN
ejpam-4793	634	64	,	,	PUNCT
ejpam-4793	634	65	s4	s4	PROPN
ejpam-4793	634	66	∈	∈	PROPN
ejpam-4793	634	67	s.	s.	PROPN
ejpam-4793	634	68	hence	hence	ADV
ejpam-4793	634	69	,	,	PUNCT
ejpam-4793	634	70	for	for	ADP
ejpam-4793	634	71	all	all	PRON
ejpam-4793	634	72	α	α	PRON
ejpam-4793	634	73	∈	∈	PROPN
ejpam-4793	634	74	γ	γ	X
ejpam-4793	634	75	,	,	PUNCT
ejpam-4793	634	76	αx	αx	ADV
ejpam-4793	634	77	∗	∗	NOUN
ejpam-4793	634	78	s1	s1	PROPN
ejpam-4793	634	79	∗	∗	NOUN
ejpam-4793	634	80	s3	s3	NOUN
ejpam-4793	634	81	=	=	SYM
ejpam-4793	634	82	αy	αy	PROPN
ejpam-4793	634	83	∗	∗	NOUN
ejpam-4793	634	84	s2	s2	PROPN
ejpam-4793	634	85	∗	∗	NOUN
ejpam-4793	634	86	s3	s3	NOUN
ejpam-4793	634	87	=	=	SYM
ejpam-4793	634	88	αz	αz	PROPN
ejpam-4793	634	89	∗	∗	NOUN
ejpam-4793	634	90	s2	s2	PROPN
ejpam-4793	634	91	∗	∗	NOUN
ejpam-4793	634	92	s4	s4	NOUN
ejpam-4793	634	93	and	and	CCONJ
ejpam-4793	634	94	s1	s1	PROPN
ejpam-4793	634	95	∗	∗	NOUN
ejpam-4793	634	96	s3	s3	PROPN
ejpam-4793	634	97	,	,	PUNCT
ejpam-4793	634	98	s2	s2	PROPN
ejpam-4793	634	99	∗	∗	NOUN
ejpam-4793	634	100	s4	s4	PROPN
ejpam-4793	634	101	∈	∈	PROPN
ejpam-4793	634	102	s	s	PART
ejpam-4793	634	103	since	since	SCONJ
ejpam-4793	634	104	s	s	PROPN
ejpam-4793	634	105	is	be	AUX
ejpam-4793	634	106	a	a	DET
ejpam-4793	634	107	γ	γ	NOUN
ejpam-4793	634	108	-	-	ADJ
ejpam-4793	634	109	submonoid	submonoid	ADJ
ejpam-4793	634	110	.	.	PUNCT
ejpam-4793	635	1	hence	hence	ADV
ejpam-4793	635	2	,	,	PUNCT
ejpam-4793	635	3	(	(	PUNCT
ejpam-4793	635	4	αx	αx	ADV
ejpam-4793	635	5	∗s)∩	∗s)∩	PROPN
ejpam-4793	635	6	(	(	PUNCT
ejpam-4793	635	7	αz	αz	ADP
ejpam-4793	635	8	∗s	∗s	ADV
ejpam-4793	635	9	)	)	PUNCT
ejpam-4793	635	10	̸=	̸=	PROPN
ejpam-4793	635	11	∅	∅	NOUN
ejpam-4793	635	12	and	and	CCONJ
ejpam-4793	635	13	xρsz	xρsz	PROPN
ejpam-4793	635	14	.	.	PUNCT
ejpam-4793	636	1	therefore	therefore	ADV
ejpam-4793	636	2	,	,	PUNCT
ejpam-4793	636	3	ρs	ρs	ADV
ejpam-4793	636	4	is	be	AUX
ejpam-4793	636	5	transitive	transitive	ADJ
ejpam-4793	636	6	.	.	PUNCT
ejpam-4793	637	1	consequently	consequently	ADV
ejpam-4793	637	2	,	,	PUNCT
ejpam-4793	637	3	ρs	ρs	ADV
ejpam-4793	637	4	is	be	AUX
ejpam-4793	637	5	an	an	DET
ejpam-4793	637	6	equivalence	equivalence	NOUN
ejpam-4793	637	7	relation	relation	NOUN
ejpam-4793	637	8	on	on	ADP
ejpam-4793	637	9	m	m	PROPN
ejpam-4793	637	10	.	.	PUNCT
ejpam-4793	638	1	(	(	PUNCT
ejpam-4793	638	2	ii	ii	NOUN
ejpam-4793	638	3	)	)	PUNCT
ejpam-4793	638	4	let	let	VERB
ejpam-4793	638	5	m	m	PRON
ejpam-4793	638	6	be	be	AUX
ejpam-4793	638	7	a	a	DET
ejpam-4793	638	8	commutative	commutative	ADJ
ejpam-4793	638	9	γ	γ	X
ejpam-4793	638	10	-	-	PUNCT
ejpam-4793	638	11	monoid	monoid	NOUN
ejpam-4793	638	12	.	.	PUNCT
ejpam-4793	638	13	suppose	suppose	VERB
ejpam-4793	638	14	that	that	SCONJ
ejpam-4793	638	15	xρsy	xρsy	PROPN
ejpam-4793	638	16	and	and	CCONJ
ejpam-4793	638	17	u	u	PROPN
ejpam-4793	638	18	,	,	PUNCT
ejpam-4793	638	19	v	v	ADP
ejpam-4793	638	20	∈	∈	NOUN
ejpam-4793	638	21	m	m	NOUN
ejpam-4793	638	22	.	.	PUNCT
ejpam-4793	639	1	then	then	ADV
ejpam-4793	639	2	,	,	PUNCT
ejpam-4793	639	3	we	we	PRON
ejpam-4793	639	4	have	have	VERB
ejpam-4793	639	5	for	for	ADP
ejpam-4793	639	6	all	all	DET
ejpam-4793	639	7	α	α	NOUN
ejpam-4793	639	8	,	,	PUNCT
ejpam-4793	639	9	β	β	PROPN
ejpam-4793	639	10	∈	∈	PROPN
ejpam-4793	639	11	γ	γ	X
ejpam-4793	639	12	,	,	PUNCT
ejpam-4793	639	13	(	(	PUNCT
ejpam-4793	639	14	αx	αx	ADV
ejpam-4793	639	15	∗	∗	PROPN
ejpam-4793	639	16	s	s	PART
ejpam-4793	639	17	)	)	PUNCT
ejpam-4793	639	18	∩	∩	NOUN
ejpam-4793	639	19	(	(	PUNCT
ejpam-4793	639	20	αy	αy	ADP
ejpam-4793	639	21	∗	∗	NOUN
ejpam-4793	639	22	s	s	PART
ejpam-4793	639	23	)	)	PUNCT
ejpam-4793	639	24	̸=	̸=	PROPN
ejpam-4793	639	25	∅	∅	NOUN
ejpam-4793	639	26	and	and	CCONJ
ejpam-4793	639	27	thus	thus	ADV
ejpam-4793	639	28	,	,	PUNCT
ejpam-4793	639	29	αx	αx	ADV
ejpam-4793	639	30	∗	∗	VERB
ejpam-4793	639	31	s1	s1	NOUN
ejpam-4793	639	32	=	=	PUNCT
ejpam-4793	639	33	αy	αy	PROPN
ejpam-4793	639	34	∗	∗	NOUN
ejpam-4793	639	35	s2	s2	NOUN
ejpam-4793	639	36	for	for	ADP
ejpam-4793	639	37	some	some	DET
ejpam-4793	639	38	s1	s1	NOUN
ejpam-4793	639	39	,	,	PUNCT
ejpam-4793	639	40	s2	s2	PROPN
ejpam-4793	639	41	∈	∈	PROPN
ejpam-4793	639	42	s.	s.	PROPN
ejpam-4793	639	43	hence	hence	ADV
ejpam-4793	639	44	,	,	PUNCT
ejpam-4793	639	45	(	(	PUNCT
ejpam-4793	639	46	αx	αx	ADV
ejpam-4793	639	47	∗	∗	NOUN
ejpam-4793	639	48	s1	s1	NOUN
ejpam-4793	639	49	)	)	PUNCT
ejpam-4793	639	50	∗	∗	VERB
ejpam-4793	639	51	α(u	α(u	PROPN
ejpam-4793	639	52	∗	∗	NOUN
ejpam-4793	639	53	v	v	NOUN
ejpam-4793	639	54	)	)	PUNCT
ejpam-4793	639	55	=	=	PUNCT
ejpam-4793	639	56	(	(	PUNCT
ejpam-4793	639	57	αy	αy	NOUN
ejpam-4793	639	58	∗	∗	NUM
ejpam-4793	639	59	s2	s2	PROPN
ejpam-4793	639	60	)	)	PUNCT
ejpam-4793	639	61	∗	∗	VERB
ejpam-4793	639	62	α(u	α(u	PROPN
ejpam-4793	639	63	∗	∗	NOUN
ejpam-4793	639	64	v	v	NOUN
ejpam-4793	639	65	)	)	PUNCT
ejpam-4793	639	66	.	.	PUNCT
ejpam-4793	640	1	since	since	SCONJ
ejpam-4793	640	2	m	m	PROPN
ejpam-4793	640	3	is	be	AUX
ejpam-4793	640	4	commutative	commutative	ADJ
ejpam-4793	640	5	,	,	PUNCT
ejpam-4793	640	6	for	for	ADP
ejpam-4793	640	7	all	all	PRON
ejpam-4793	640	8	α	α	PRON
ejpam-4793	640	9	∈	∈	PROPN
ejpam-4793	640	10	γ	γ	X
ejpam-4793	640	11	,	,	PUNCT
ejpam-4793	640	12	α(u	α(u	PROPN
ejpam-4793	640	13	∗	∗	NOUN
ejpam-4793	640	14	x	x	PROPN
ejpam-4793	640	15	∗	∗	NOUN
ejpam-4793	640	16	v	v	NOUN
ejpam-4793	640	17	)	)	PUNCT
ejpam-4793	640	18	∗	∗	NOUN
ejpam-4793	640	19	s1	s1	NOUN
ejpam-4793	640	20	=	=	PUNCT
ejpam-4793	640	21	α(u	α(u	PROPN
ejpam-4793	640	22	∗	∗	VERB
ejpam-4793	640	23	y	y	PROPN
ejpam-4793	640	24	∗	∗	PROPN
ejpam-4793	640	25	v	v	NOUN
ejpam-4793	640	26	)	)	PUNCT
ejpam-4793	640	27	∗	∗	NOUN
ejpam-4793	640	28	s2	s2	NOUN
ejpam-4793	640	29	and	and	CCONJ
ejpam-4793	640	30	(	(	PUNCT
ejpam-4793	640	31	u	u	NOUN
ejpam-4793	640	32	∗	∗	NOUN
ejpam-4793	640	33	x	x	PUNCT
ejpam-4793	640	34	∗	∗	NOUN
ejpam-4793	640	35	v)ρs(u	v)ρs(u	X
ejpam-4793	640	36	∗	∗	PROPN
ejpam-4793	640	37	y	y	PROPN
ejpam-4793	640	38	∗	∗	NOUN
ejpam-4793	640	39	v	v	NOUN
ejpam-4793	640	40	)	)	PUNCT
ejpam-4793	640	41	.	.	PUNCT
ejpam-4793	641	1	thus	thus	ADV
ejpam-4793	641	2	,	,	PUNCT
ejpam-4793	641	3	ρs	ρs	ADV
ejpam-4793	641	4	is	be	AUX
ejpam-4793	641	5	a	a	DET
ejpam-4793	641	6	congruence	congruence	NOUN
ejpam-4793	641	7	relation	relation	NOUN
ejpam-4793	641	8	on	on	ADP
ejpam-4793	641	9	m	m	PROPN
ejpam-4793	641	10	.	.	PUNCT
ejpam-4793	642	1	h.	h.	PROPN
ejpam-4793	642	2	sarapuddin	sarapuddin	PROPN
ejpam-4793	642	3	,	,	PUNCT
ejpam-4793	642	4	j.	j.	PROPN
ejpam-4793	642	5	vilela	vilela	PROPN
ejpam-4793	642	6	/	/	SYM
ejpam-4793	642	7	eur	eur	PROPN
ejpam-4793	642	8	.	.	PUNCT
ejpam-4793	643	1	j.	j.	PROPN
ejpam-4793	643	2	pure	pure	PROPN
ejpam-4793	643	3	appl	appl	PROPN
ejpam-4793	643	4	.	.	PROPN
ejpam-4793	643	5	math	math	PROPN
ejpam-4793	643	6	,	,	PUNCT
ejpam-4793	643	7	16	16	NUM
ejpam-4793	643	8	(	(	PUNCT
ejpam-4793	643	9	3	3	NUM
ejpam-4793	643	10	)	)	PUNCT
ejpam-4793	643	11	(	(	PUNCT
ejpam-4793	643	12	2023	2023	NUM
ejpam-4793	643	13	)	)	PUNCT
ejpam-4793	643	14	,	,	PUNCT
ejpam-4793	643	15	1772	1772	NUM
ejpam-4793	643	16	-	-	SYM
ejpam-4793	643	17	1793	1793	NUM
ejpam-4793	643	18	1787	1787	NUM
ejpam-4793	643	19	definition	definition	NOUN
ejpam-4793	643	20	14	14	NUM
ejpam-4793	643	21	.	.	PUNCT
ejpam-4793	644	1	let	let	VERB
ejpam-4793	644	2	s	s	PRON
ejpam-4793	644	3	be	be	AUX
ejpam-4793	644	4	a	a	DET
ejpam-4793	644	5	commutative	commutative	ADJ
ejpam-4793	644	6	γ	γ	NOUN
ejpam-4793	644	7	-	-	ADJ
ejpam-4793	644	8	submonoid	submonoid	NOUN
ejpam-4793	644	9	of	of	ADP
ejpam-4793	644	10	a	a	DET
ejpam-4793	644	11	γ	γ	X
ejpam-4793	644	12	-	-	PUNCT
ejpam-4793	644	13	monoid	monoid	NOUN
ejpam-4793	644	14	m	m	PROPN
ejpam-4793	644	15	.	.	PUNCT
ejpam-4793	645	1	then	then	ADV
ejpam-4793	645	2	for	for	ADP
ejpam-4793	645	3	all	all	PRON
ejpam-4793	645	4	x	x	SYM
ejpam-4793	645	5	∈	∈	PROPN
ejpam-4793	645	6	m	m	NOUN
ejpam-4793	645	7	,	,	PUNCT
ejpam-4793	645	8	the	the	DET
ejpam-4793	645	9	equivalence	equivalence	NOUN
ejpam-4793	645	10	class	class	NOUN
ejpam-4793	645	11	of	of	ADP
ejpam-4793	645	12	x	x	PROPN
ejpam-4793	645	13	is	be	AUX
ejpam-4793	645	14	denoted	denote	VERB
ejpam-4793	645	15	and	and	CCONJ
ejpam-4793	645	16	defined	define	VERB
ejpam-4793	645	17	by	by	ADP
ejpam-4793	645	18	ρs(x	ρs(x	NOUN
ejpam-4793	645	19	)	)	PUNCT
ejpam-4793	646	1	=	=	PRON
ejpam-4793	646	2	{	{	PUNCT
ejpam-4793	646	3	y	y	PROPN
ejpam-4793	646	4	∈	∈	PROPN
ejpam-4793	646	5	m	m	VERB
ejpam-4793	646	6	:	:	PUNCT
ejpam-4793	646	7	xρsy	xρsy	PROPN
ejpam-4793	646	8	}	}	PUNCT
ejpam-4793	646	9	.	.	PUNCT
ejpam-4793	647	1	let	let	VERB
ejpam-4793	647	2	s	s	PRON
ejpam-4793	647	3	be	be	AUX
ejpam-4793	647	4	a	a	DET
ejpam-4793	647	5	commutative	commutative	ADJ
ejpam-4793	647	6	γ	γ	NOUN
ejpam-4793	647	7	-	-	ADJ
ejpam-4793	647	8	submonoid	submonoid	NOUN
ejpam-4793	647	9	of	of	ADP
ejpam-4793	647	10	a	a	DET
ejpam-4793	647	11	γ	γ	X
ejpam-4793	647	12	-	-	PUNCT
ejpam-4793	647	13	monoid	monoid	NOUN
ejpam-4793	647	14	m	m	NOUN
ejpam-4793	647	15	and	and	CCONJ
ejpam-4793	647	16	let	let	VERB
ejpam-4793	647	17	m	m	PRON
ejpam-4793	647	18	∈	∈	VERB
ejpam-4793	647	19	m	m	NOUN
ejpam-4793	647	20	.	.	PUNCT
ejpam-4793	648	1	then	then	ADV
ejpam-4793	648	2	for	for	ADP
ejpam-4793	648	3	all	all	PRON
ejpam-4793	648	4	α	α	PRON
ejpam-4793	648	5	∈	∈	PROPN
ejpam-4793	648	6	γ	γ	X
ejpam-4793	648	7	,	,	PUNCT
ejpam-4793	648	8	(	(	PUNCT
ejpam-4793	648	9	αm	αm	NOUN
ejpam-4793	648	10	∗	∗	NOUN
ejpam-4793	648	11	s	s	NOUN
ejpam-4793	648	12	)	)	PUNCT
ejpam-4793	648	13	∩	∩	NOUN
ejpam-4793	648	14	(	(	PUNCT
ejpam-4793	648	15	αm	αm	NOUN
ejpam-4793	648	16	∗	∗	NOUN
ejpam-4793	648	17	s	s	PART
ejpam-4793	648	18	)	)	PUNCT
ejpam-4793	648	19	=	=	VERB
ejpam-4793	648	20	αm	αm	NOUN
ejpam-4793	648	21	∗	∗	NOUN
ejpam-4793	648	22	s	s	PART
ejpam-4793	648	23	̸=	̸=	PROPN
ejpam-4793	648	24	∅	∅	NOUN
ejpam-4793	648	25	since	since	SCONJ
ejpam-4793	648	26	for	for	ADP
ejpam-4793	648	27	α	α	NOUN
ejpam-4793	648	28	=	=	SYM
ejpam-4793	648	29	0	0	NUM
ejpam-4793	648	30	,	,	PUNCT
ejpam-4793	648	31	m	m	VERB
ejpam-4793	648	32	=	=	NOUN
ejpam-4793	648	33	m	m	NOUN
ejpam-4793	648	34	∗	∗	NOUN
ejpam-4793	649	1	1	1	NUM
ejpam-4793	649	2	m	m	NOUN
ejpam-4793	649	3	∈	∈	NOUN
ejpam-4793	649	4	m	m	NOUN
ejpam-4793	649	5	∗	∗	NOUN
ejpam-4793	649	6	s.	s.	PROPN
ejpam-4793	649	7	thus	thus	ADV
ejpam-4793	649	8	,	,	PUNCT
ejpam-4793	649	9	m	m	PROPN
ejpam-4793	649	10	∈	∈	NOUN
ejpam-4793	649	11	ρs(m	ρs(m	NUM
ejpam-4793	649	12	)	)	PUNCT
ejpam-4793	649	13	.	.	PUNCT
ejpam-4793	650	1	hence	hence	ADV
ejpam-4793	650	2	,	,	PUNCT
ejpam-4793	650	3	the	the	DET
ejpam-4793	650	4	following	follow	VERB
ejpam-4793	650	5	remark	remark	NOUN
ejpam-4793	650	6	holds	hold	VERB
ejpam-4793	650	7	.	.	PUNCT
ejpam-4793	651	1	remark	remark	PROPN
ejpam-4793	651	2	14	14	NUM
ejpam-4793	651	3	.	.	PUNCT
ejpam-4793	652	1	let	let	VERB
ejpam-4793	652	2	s	s	PRON
ejpam-4793	652	3	be	be	AUX
ejpam-4793	652	4	a	a	DET
ejpam-4793	652	5	commutative	commutative	ADJ
ejpam-4793	652	6	γ	γ	NOUN
ejpam-4793	652	7	-	-	ADJ
ejpam-4793	652	8	submonoid	submonoid	NOUN
ejpam-4793	652	9	of	of	ADP
ejpam-4793	652	10	a	a	DET
ejpam-4793	652	11	γ	γ	NOUN
ejpam-4793	652	12	-	-	PUNCT
ejpam-4793	652	13	monoidm	monoidm	ADJ
ejpam-4793	652	14	and	and	CCONJ
ejpam-4793	652	15	letm1,m2	letm1,m2	NOUN
ejpam-4793	652	16	∈	∈	PROPN
ejpam-4793	652	17	m	m	NOUN
ejpam-4793	652	18	.	.	PUNCT
ejpam-4793	653	1	(	(	PUNCT
ejpam-4793	653	2	i	i	NOUN
ejpam-4793	653	3	)	)	PUNCT
ejpam-4793	653	4	for	for	ADP
ejpam-4793	653	5	all	all	DET
ejpam-4793	653	6	m	m	NOUN
ejpam-4793	653	7	∈	∈	ADJ
ejpam-4793	653	8	m	m	NOUN
ejpam-4793	653	9	,	,	PUNCT
ejpam-4793	653	10	m	m	VERB
ejpam-4793	653	11	∈	∈	NOUN
ejpam-4793	653	12	ρs(m	ρs(m	NUM
ejpam-4793	653	13	)	)	PUNCT
ejpam-4793	653	14	.	.	PUNCT
ejpam-4793	654	1	(	(	PUNCT
ejpam-4793	654	2	ii	ii	NOUN
ejpam-4793	654	3	)	)	PUNCT
ejpam-4793	654	4	ρs(m1	ρs(m1	NOUN
ejpam-4793	654	5	)	)	PUNCT
ejpam-4793	654	6	=	=	SYM
ejpam-4793	655	1	ρs(m2	ρs(m2	X
ejpam-4793	655	2	)	)	PUNCT
ejpam-4793	656	1	if	if	SCONJ
ejpam-4793	656	2	and	and	CCONJ
ejpam-4793	656	3	only	only	ADV
ejpam-4793	656	4	if	if	SCONJ
ejpam-4793	656	5	(	(	PUNCT
ejpam-4793	656	6	αm1	αm1	NOUN
ejpam-4793	656	7	∗	∗	NOUN
ejpam-4793	656	8	s	s	NOUN
ejpam-4793	656	9	)	)	PUNCT
ejpam-4793	656	10	∩	∩	NOUN
ejpam-4793	656	11	(	(	PUNCT
ejpam-4793	656	12	αm2	αm2	X
ejpam-4793	656	13	∗	∗	NOUN
ejpam-4793	656	14	s	s	PART
ejpam-4793	656	15	)	)	PUNCT
ejpam-4793	656	16	̸=	̸=	PROPN
ejpam-4793	656	17	∅	∅	NOUN
ejpam-4793	656	18	for	for	ADP
ejpam-4793	656	19	all	all	PRON
ejpam-4793	656	20	α	α	PRON
ejpam-4793	656	21	∈	∈	PROPN
ejpam-4793	656	22	γ	γ	X
ejpam-4793	656	23	.	.	PUNCT
ejpam-4793	657	1	the	the	DET
ejpam-4793	657	2	quotient	quotient	NOUN
ejpam-4793	657	3	m	m	PROPN
ejpam-4793	657	4	/	/	SYM
ejpam-4793	657	5	s	s	AUX
ejpam-4793	657	6	using	use	VERB
ejpam-4793	657	7	equivalence	equivalence	NOUN
ejpam-4793	657	8	relation	relation	NOUN
ejpam-4793	657	9	in	in	ADP
ejpam-4793	657	10	definition	definition	NOUN
ejpam-4793	657	11	6	6	NUM
ejpam-4793	657	12	,	,	PUNCT
ejpam-4793	657	13	where	where	SCONJ
ejpam-4793	657	14	m	m	NOUN
ejpam-4793	657	15	is	be	AUX
ejpam-4793	657	16	a	a	DET
ejpam-4793	657	17	monoid	monoid	NOUN
ejpam-4793	657	18	and	and	CCONJ
ejpam-4793	657	19	s	s	NOUN
ejpam-4793	657	20	is	be	AUX
ejpam-4793	657	21	a	a	DET
ejpam-4793	657	22	submonoid	submonoid	ADJ
ejpam-4793	657	23	ofm	ofm	PROPN
ejpam-4793	657	24	is	be	AUX
ejpam-4793	657	25	different	different	ADJ
ejpam-4793	657	26	fromm	fromm	PROPN
ejpam-4793	657	27	/	/	SYM
ejpam-4793	657	28	s	s	AUX
ejpam-4793	657	29	using	use	VERB
ejpam-4793	657	30	the	the	DET
ejpam-4793	657	31	equivalence	equivalence	NOUN
ejpam-4793	657	32	relation	relation	NOUN
ejpam-4793	657	33	in	in	ADP
ejpam-4793	657	34	definition	definition	NOUN
ejpam-4793	657	35	13	13	NUM
ejpam-4793	657	36	,	,	PUNCT
ejpam-4793	657	37	where	where	SCONJ
ejpam-4793	657	38	m	m	NOUN
ejpam-4793	657	39	is	be	AUX
ejpam-4793	657	40	a	a	DET
ejpam-4793	657	41	γ	γ	X
ejpam-4793	657	42	-	-	PUNCT
ejpam-4793	657	43	monoid	monoid	NOUN
ejpam-4793	657	44	and	and	CCONJ
ejpam-4793	657	45	s	s	NOUN
ejpam-4793	657	46	is	be	AUX
ejpam-4793	657	47	a	a	DET
ejpam-4793	657	48	γ	γ	NOUN
ejpam-4793	657	49	-	-	ADJ
ejpam-4793	657	50	submonoid	submonoid	ADJ
ejpam-4793	657	51	as	as	SCONJ
ejpam-4793	657	52	shown	show	VERB
ejpam-4793	657	53	in	in	ADP
ejpam-4793	657	54	the	the	DET
ejpam-4793	657	55	following	follow	VERB
ejpam-4793	657	56	example	example	NOUN
ejpam-4793	657	57	.	.	PUNCT
ejpam-4793	658	1	example	example	NOUN
ejpam-4793	659	1	16	16	NUM
ejpam-4793	659	2	.	.	PUNCT
ejpam-4793	660	1	let	let	VERB
ejpam-4793	660	2	γ	γ	X
ejpam-4793	660	3	=	=	PROPN
ejpam-4793	660	4	z	z	X
ejpam-4793	660	5	the	the	DET
ejpam-4793	660	6	additive	additive	ADJ
ejpam-4793	660	7	group	group	NOUN
ejpam-4793	660	8	of	of	ADP
ejpam-4793	660	9	integers	integer	NOUN
ejpam-4793	660	10	and	and	CCONJ
ejpam-4793	660	11	m	m	PROPN
ejpam-4793	660	12	=	=	ADJ
ejpam-4793	660	13	z8	z8	NOUN
ejpam-4793	660	14	=	=	SYM
ejpam-4793	660	15	{	{	PUNCT
ejpam-4793	660	16	0	0	NUM
ejpam-4793	660	17	,	,	PUNCT
ejpam-4793	660	18	1	1	NUM
ejpam-4793	660	19	,	,	PUNCT
ejpam-4793	660	20	2	2	NUM
ejpam-4793	660	21	,	,	PUNCT
ejpam-4793	660	22	3	3	NUM
ejpam-4793	660	23	,	,	PUNCT
ejpam-4793	660	24	4	4	NUM
ejpam-4793	660	25	,	,	PUNCT
ejpam-4793	660	26	5	5	NUM
ejpam-4793	660	27	,	,	PUNCT
ejpam-4793	660	28	6	6	NUM
ejpam-4793	660	29	,	,	PUNCT
ejpam-4793	660	30	7	7	NUM
ejpam-4793	660	31	}	}	PUNCT
ejpam-4793	660	32	under	under	ADP
ejpam-4793	660	33	addition	addition	NOUN
ejpam-4793	660	34	modulo	modulo	NOUN
ejpam-4793	660	35	8	8	NUM
ejpam-4793	660	36	.	.	PUNCT
ejpam-4793	661	1	then	then	ADV
ejpam-4793	661	2	m	m	PROPN
ejpam-4793	661	3	is	be	AUX
ejpam-4793	661	4	a	a	DET
ejpam-4793	661	5	monoid	monoid	NOUN
ejpam-4793	661	6	with	with	ADP
ejpam-4793	661	7	identity	identity	NOUN
ejpam-4793	661	8	0	0	NUM
ejpam-4793	661	9	.	.	PUNCT
ejpam-4793	662	1	consider	consider	VERB
ejpam-4793	662	2	a	a	DET
ejpam-4793	662	3	mapping	mapping	NOUN
ejpam-4793	662	4	ϕ	ϕ	NOUN
ejpam-4793	662	5	:	:	PUNCT
ejpam-4793	662	6	γ	γ	X
ejpam-4793	662	7	×	×	PROPN
ejpam-4793	662	8	m	m	INTJ
ejpam-4793	662	9	→	→	AUX
ejpam-4793	662	10	m	m	AUX
ejpam-4793	662	11	given	give	VERB
ejpam-4793	662	12	by	by	ADP
ejpam-4793	662	13	ϕ((α	ϕ((α	PROPN
ejpam-4793	662	14	,	,	PUNCT
ejpam-4793	662	15	m	m	NOUN
ejpam-4793	662	16	)	)	PUNCT
ejpam-4793	662	17	)	)	PUNCT
ejpam-4793	663	1	=	=	SYM
ejpam-4793	663	2	7αm	7αm	PROPN
ejpam-4793	663	3	.	.	PUNCT
ejpam-4793	664	1	let	let	VERB
ejpam-4793	664	2	(	(	PUNCT
ejpam-4793	664	3	α	α	NOUN
ejpam-4793	664	4	,	,	PUNCT
ejpam-4793	664	5	x	x	NOUN
ejpam-4793	664	6	)	)	PUNCT
ejpam-4793	664	7	,	,	PUNCT
ejpam-4793	664	8	(	(	PUNCT
ejpam-4793	664	9	β	β	X
ejpam-4793	664	10	,	,	PUNCT
ejpam-4793	664	11	y	y	NOUN
ejpam-4793	664	12	)	)	PUNCT
ejpam-4793	664	13	∈	∈	PROPN
ejpam-4793	664	14	γ	γ	X
ejpam-4793	664	15	×	×	PROPN
ejpam-4793	664	16	m	m	VERB
ejpam-4793	664	17	such	such	ADJ
ejpam-4793	664	18	that	that	SCONJ
ejpam-4793	664	19	(	(	PUNCT
ejpam-4793	664	20	α	α	NOUN
ejpam-4793	664	21	,	,	PUNCT
ejpam-4793	664	22	x	x	NOUN
ejpam-4793	664	23	)	)	PUNCT
ejpam-4793	664	24	=	=	SYM
ejpam-4793	664	25	(	(	PUNCT
ejpam-4793	664	26	β	β	X
ejpam-4793	664	27	,	,	PUNCT
ejpam-4793	664	28	y	y	PROPN
ejpam-4793	664	29	)	)	PUNCT
ejpam-4793	664	30	.	.	PUNCT
ejpam-4793	665	1	then	then	ADV
ejpam-4793	665	2	α	α	X
ejpam-4793	665	3	=	=	PUNCT
ejpam-4793	665	4	β	β	X
ejpam-4793	665	5	and	and	CCONJ
ejpam-4793	665	6	x	x	X
ejpam-4793	665	7	=	=	PUNCT
ejpam-4793	665	8	y.	y.	PROPN
ejpam-4793	665	9	thus	thus	ADV
ejpam-4793	665	10	,	,	PUNCT
ejpam-4793	665	11	7αx	7αx	NOUN
ejpam-4793	665	12	=	=	SYM
ejpam-4793	665	13	7βy	7βy	NOUN
ejpam-4793	665	14	and	and	CCONJ
ejpam-4793	665	15	ϕ	ϕ	NOUN
ejpam-4793	665	16	is	be	AUX
ejpam-4793	665	17	well	well	ADV
ejpam-4793	665	18	-	-	PUNCT
ejpam-4793	665	19	defined	define	VERB
ejpam-4793	665	20	.	.	PUNCT
ejpam-4793	666	1	now	now	ADV
ejpam-4793	666	2	,	,	PUNCT
ejpam-4793	666	3	let	let	VERB
ejpam-4793	666	4	α	α	PRON
ejpam-4793	666	5	,	,	PUNCT
ejpam-4793	666	6	β	β	X
ejpam-4793	666	7	∈	∈	PROPN
ejpam-4793	666	8	γ	γ	NOUN
ejpam-4793	666	9	and	and	CCONJ
ejpam-4793	666	10	m	m	PROPN
ejpam-4793	666	11	∈	∈	PROPN
ejpam-4793	666	12	m	m	VERB
ejpam-4793	666	13	.	.	PUNCT
ejpam-4793	667	1	observe	observe	VERB
ejpam-4793	667	2	that	that	SCONJ
ejpam-4793	667	3	(	(	PUNCT
ejpam-4793	667	4	i	i	NOUN
ejpam-4793	667	5	)	)	PUNCT
ejpam-4793	667	6	ϕ((0,m	ϕ((0,m	PROPN
ejpam-4793	667	7	)	)	PUNCT
ejpam-4793	667	8	)	)	PUNCT
ejpam-4793	668	1	=	=	PUNCT
ejpam-4793	669	1	70	70	NUM
ejpam-4793	669	2	m	m	NOUN
ejpam-4793	669	3	=	=	NOUN
ejpam-4793	669	4	m	m	PROPN
ejpam-4793	669	5	;	;	PUNCT
ejpam-4793	669	6	(	(	PUNCT
ejpam-4793	669	7	ii	ii	NOUN
ejpam-4793	669	8	)	)	PUNCT
ejpam-4793	669	9	ϕ((α+	ϕ((α+	PRON
ejpam-4793	669	10	β	β	X
ejpam-4793	669	11	,	,	PUNCT
ejpam-4793	669	12	m	m	NOUN
ejpam-4793	669	13	)	)	PUNCT
ejpam-4793	669	14	)	)	PUNCT
ejpam-4793	670	1	=	=	SYM
ejpam-4793	670	2	7α+βm	7α+βm	NOUN
ejpam-4793	670	3	=	=	SYM
ejpam-4793	670	4	7α7βm	7α7βm	PROPN
ejpam-4793	670	5	=	=	SYM
ejpam-4793	670	6	ϕ((α	ϕ((α	PROPN
ejpam-4793	670	7	,	,	PUNCT
ejpam-4793	670	8	ϕ((β	ϕ((β	PROPN
ejpam-4793	670	9	,	,	PUNCT
ejpam-4793	670	10	m	m	PROPN
ejpam-4793	670	11	)	)	PUNCT
ejpam-4793	670	12	)	)	PUNCT
ejpam-4793	670	13	)	)	PUNCT
ejpam-4793	670	14	)	)	PUNCT
ejpam-4793	670	15	.	.	PUNCT
ejpam-4793	671	1	this	this	PRON
ejpam-4793	671	2	implies	imply	VERB
ejpam-4793	671	3	that	that	SCONJ
ejpam-4793	671	4	ϕ	ϕ	NOUN
ejpam-4793	671	5	is	be	AUX
ejpam-4793	671	6	an	an	DET
ejpam-4793	671	7	action	action	NOUN
ejpam-4793	671	8	.	.	PUNCT
ejpam-4793	672	1	now	now	ADV
ejpam-4793	672	2	,	,	PUNCT
ejpam-4793	672	3	let	let	VERB
ejpam-4793	672	4	α	α	PRON
ejpam-4793	672	5	∈	∈	PROPN
ejpam-4793	672	6	γ	γ	X
ejpam-4793	672	7	and	and	CCONJ
ejpam-4793	672	8	x	x	NOUN
ejpam-4793	672	9	,	,	PUNCT
ejpam-4793	672	10	y	y	PROPN
ejpam-4793	672	11	∈	∈	PROPN
ejpam-4793	672	12	m	m	VERB
ejpam-4793	672	13	.	.	PUNCT
ejpam-4793	673	1	then	then	ADV
ejpam-4793	673	2	ϕ((α	ϕ((α	PROPN
ejpam-4793	673	3	,	,	PUNCT
ejpam-4793	673	4	x+8	x+8	PROPN
ejpam-4793	673	5	y	y	NOUN
ejpam-4793	673	6	)	)	PUNCT
ejpam-4793	673	7	)	)	PUNCT
ejpam-4793	674	1	=	=	SYM
ejpam-4793	674	2	ϕ((α	ϕ((α	PROPN
ejpam-4793	674	3	,	,	PUNCT
ejpam-4793	674	4	x+8	x+8	PROPN
ejpam-4793	674	5	y	y	NOUN
ejpam-4793	674	6	)	)	PUNCT
ejpam-4793	674	7	)	)	PUNCT
ejpam-4793	675	1	=	=	PUNCT
ejpam-4793	676	1	7α(x+8	7α(x+8	NUM
ejpam-4793	676	2	y	y	X
ejpam-4793	676	3	)	)	PUNCT
ejpam-4793	676	4	=	=	PUNCT
ejpam-4793	677	1	7αx+8	7αx+8	NUM
ejpam-4793	677	2	7αy	7αy	NOUN
ejpam-4793	677	3	=	=	SYM
ejpam-4793	677	4	ϕ((α	ϕ((α	PROPN
ejpam-4793	677	5	,	,	PUNCT
ejpam-4793	677	6	x	x	NOUN
ejpam-4793	677	7	)	)	PUNCT
ejpam-4793	677	8	)	)	PUNCT
ejpam-4793	678	1	+8	+8	PROPN
ejpam-4793	678	2	ϕ((α	ϕ((α	PROPN
ejpam-4793	678	3	,	,	PUNCT
ejpam-4793	678	4	y	y	NOUN
ejpam-4793	678	5	)	)	PUNCT
ejpam-4793	678	6	)	)	PUNCT
ejpam-4793	678	7	.	.	PUNCT
ejpam-4793	679	1	therefore	therefore	ADV
ejpam-4793	679	2	,	,	PUNCT
ejpam-4793	679	3	m	m	VERB
ejpam-4793	679	4	is	be	AUX
ejpam-4793	679	5	a	a	DET
ejpam-4793	679	6	γ	γ	X
ejpam-4793	679	7	-	-	PUNCT
ejpam-4793	679	8	monoid	monoid	NOUN
ejpam-4793	679	9	.	.	PUNCT
ejpam-4793	680	1	let	let	VERB
ejpam-4793	680	2	s	s	PRON
ejpam-4793	680	3	=	=	X
ejpam-4793	680	4	{	{	PUNCT
ejpam-4793	680	5	0	0	NUM
ejpam-4793	680	6	,	,	PUNCT
ejpam-4793	680	7	4	4	NUM
ejpam-4793	680	8	}	}	PUNCT
ejpam-4793	680	9	.	.	PUNCT
ejpam-4793	681	1	observe	observe	VERB
ejpam-4793	681	2	that	that	SCONJ
ejpam-4793	681	3	the	the	DET
ejpam-4793	681	4	identity	identity	NOUN
ejpam-4793	681	5	0	0	NUM
ejpam-4793	681	6	∈	∈	PROPN
ejpam-4793	681	7	s	s	PART
ejpam-4793	681	8	and	and	CCONJ
ejpam-4793	682	1	0	0	NUM
ejpam-4793	682	2	+8	+8	NOUN
ejpam-4793	682	3	0	0	NUM
ejpam-4793	683	1	=	=	SYM
ejpam-4793	683	2	0	0	NUM
ejpam-4793	683	3	,	,	PUNCT
ejpam-4793	683	4	0	0	NUM
ejpam-4793	684	1	+8	+8	NOUN
ejpam-4793	684	2	4	4	NUM
ejpam-4793	684	3	=	=	SYM
ejpam-4793	684	4	4	4	NUM
ejpam-4793	684	5	+8	+8	NOUN
ejpam-4793	684	6	0	0	NUM
ejpam-4793	685	1	=	=	SYM
ejpam-4793	685	2	4	4	NUM
ejpam-4793	685	3	,	,	PUNCT
ejpam-4793	685	4	4	4	NUM
ejpam-4793	685	5	+8	+8	NOUN
ejpam-4793	685	6	4	4	NUM
ejpam-4793	685	7	=	=	SYM
ejpam-4793	685	8	0	0	NUM
ejpam-4793	685	9	∈	∈	PROPN
ejpam-4793	685	10	s.	s.	PROPN
ejpam-4793	685	11	this	this	PRON
ejpam-4793	685	12	implies	imply	VERB
ejpam-4793	685	13	that	that	SCONJ
ejpam-4793	685	14	s	s	VERB
ejpam-4793	685	15	is	be	AUX
ejpam-4793	685	16	a	a	DET
ejpam-4793	685	17	submonoid	submonoid	NOUN
ejpam-4793	685	18	of	of	ADP
ejpam-4793	685	19	m	m	PROPN
ejpam-4793	685	20	.	.	PUNCT
ejpam-4793	686	1	now	now	ADV
ejpam-4793	686	2	,	,	PUNCT
ejpam-4793	686	3	note	note	VERB
ejpam-4793	686	4	that	that	SCONJ
ejpam-4793	686	5	0	0	PUNCT
ejpam-4793	687	1	+8	+8	PRON
ejpam-4793	687	2	s	s	PART
ejpam-4793	687	3	=	=	SYM
ejpam-4793	687	4	0	0	NUM
ejpam-4793	687	5	+8	+8	NOUN
ejpam-4793	687	6	{	{	PUNCT
ejpam-4793	687	7	0	0	NUM
ejpam-4793	687	8	,	,	PUNCT
ejpam-4793	687	9	4	4	NUM
ejpam-4793	687	10	}	}	PUNCT
ejpam-4793	687	11	=	=	SYM
ejpam-4793	687	12	{	{	PUNCT
ejpam-4793	687	13	0	0	NUM
ejpam-4793	687	14	,	,	PUNCT
ejpam-4793	687	15	4	4	NUM
ejpam-4793	687	16	}	}	PUNCT
ejpam-4793	687	17	,	,	PUNCT
ejpam-4793	687	18	4	4	NUM
ejpam-4793	687	19	+8	+8	NOUN
ejpam-4793	687	20	s	s	PART
ejpam-4793	687	21	=	=	SYM
ejpam-4793	687	22	4	4	NUM
ejpam-4793	687	23	+8	+8	NOUN
ejpam-4793	687	24	{	{	PUNCT
ejpam-4793	687	25	0	0	NUM
ejpam-4793	687	26	,	,	PUNCT
ejpam-4793	687	27	4	4	NUM
ejpam-4793	687	28	}	}	PUNCT
ejpam-4793	687	29	=	=	SYM
ejpam-4793	687	30	{	{	PUNCT
ejpam-4793	687	31	0	0	NUM
ejpam-4793	687	32	,	,	PUNCT
ejpam-4793	687	33	4	4	NUM
ejpam-4793	687	34	}	}	PUNCT
ejpam-4793	687	35	;	;	PUNCT
ejpam-4793	687	36	1	1	NUM
ejpam-4793	687	37	+8	+8	NOUN
ejpam-4793	687	38	s	s	PART
ejpam-4793	687	39	=	=	SYM
ejpam-4793	687	40	1	1	NUM
ejpam-4793	687	41	+8	+8	NOUN
ejpam-4793	687	42	{	{	PUNCT
ejpam-4793	687	43	0	0	NUM
ejpam-4793	687	44	,	,	PUNCT
ejpam-4793	687	45	4	4	NUM
ejpam-4793	687	46	}	}	PUNCT
ejpam-4793	687	47	=	=	SYM
ejpam-4793	687	48	{	{	PUNCT
ejpam-4793	687	49	1	1	NUM
ejpam-4793	687	50	,	,	PUNCT
ejpam-4793	687	51	5	5	NUM
ejpam-4793	687	52	}	}	PUNCT
ejpam-4793	687	53	,	,	PUNCT
ejpam-4793	687	54	5	5	NUM
ejpam-4793	687	55	+8	+8	NOUN
ejpam-4793	687	56	s	s	PART
ejpam-4793	687	57	=	=	SYM
ejpam-4793	687	58	5	5	NUM
ejpam-4793	687	59	+8	+8	NOUN
ejpam-4793	687	60	{	{	PUNCT
ejpam-4793	687	61	0	0	NUM
ejpam-4793	687	62	,	,	PUNCT
ejpam-4793	687	63	4	4	NUM
ejpam-4793	687	64	}	}	PUNCT
ejpam-4793	687	65	=	=	SYM
ejpam-4793	687	66	{	{	PUNCT
ejpam-4793	687	67	1	1	NUM
ejpam-4793	687	68	,	,	PUNCT
ejpam-4793	687	69	5	5	NUM
ejpam-4793	687	70	}	}	PUNCT
ejpam-4793	687	71	;	;	PUNCT
ejpam-4793	687	72	2	2	NUM
ejpam-4793	687	73	+8	+8	NOUN
ejpam-4793	687	74	s	s	PART
ejpam-4793	687	75	=	=	SYM
ejpam-4793	687	76	2	2	NUM
ejpam-4793	687	77	+8	+8	NOUN
ejpam-4793	687	78	{	{	PUNCT
ejpam-4793	687	79	0	0	NUM
ejpam-4793	687	80	,	,	PUNCT
ejpam-4793	687	81	4	4	NUM
ejpam-4793	687	82	}	}	PUNCT
ejpam-4793	687	83	=	=	SYM
ejpam-4793	687	84	{	{	PUNCT
ejpam-4793	687	85	2	2	NUM
ejpam-4793	687	86	,	,	PUNCT
ejpam-4793	687	87	6	6	NUM
ejpam-4793	687	88	}	}	PUNCT
ejpam-4793	687	89	,	,	PUNCT
ejpam-4793	687	90	6	6	NUM
ejpam-4793	687	91	+8	+8	NOUN
ejpam-4793	687	92	s	s	PART
ejpam-4793	687	93	=	=	SYM
ejpam-4793	687	94	6	6	NUM
ejpam-4793	687	95	+8	+8	NOUN
ejpam-4793	687	96	{	{	PUNCT
ejpam-4793	687	97	0	0	NUM
ejpam-4793	687	98	,	,	PUNCT
ejpam-4793	687	99	4	4	NUM
ejpam-4793	687	100	}	}	PUNCT
ejpam-4793	687	101	=	=	SYM
ejpam-4793	687	102	{	{	PUNCT
ejpam-4793	687	103	2	2	NUM
ejpam-4793	687	104	,	,	PUNCT
ejpam-4793	687	105	6	6	NUM
ejpam-4793	687	106	}	}	PUNCT
ejpam-4793	687	107	;	;	PUNCT
ejpam-4793	687	108	4	4	NUM
ejpam-4793	687	109	+8	+8	NOUN
ejpam-4793	687	110	s	s	PART
ejpam-4793	687	111	=	=	SYM
ejpam-4793	687	112	4	4	NUM
ejpam-4793	687	113	+8	+8	NOUN
ejpam-4793	687	114	{	{	PUNCT
ejpam-4793	687	115	0	0	NUM
ejpam-4793	687	116	,	,	PUNCT
ejpam-4793	687	117	4	4	NUM
ejpam-4793	687	118	}	}	PUNCT
ejpam-4793	687	119	=	=	SYM
ejpam-4793	687	120	{	{	PUNCT
ejpam-4793	687	121	0	0	NUM
ejpam-4793	687	122	,	,	PUNCT
ejpam-4793	687	123	4	4	NUM
ejpam-4793	687	124	}	}	PUNCT
ejpam-4793	687	125	,	,	PUNCT
ejpam-4793	687	126	7	7	NUM
ejpam-4793	687	127	+8	+8	NOUN
ejpam-4793	687	128	s	s	PART
ejpam-4793	687	129	=	=	SYM
ejpam-4793	687	130	7	7	NUM
ejpam-4793	687	131	+8	+8	NOUN
ejpam-4793	687	132	{	{	PUNCT
ejpam-4793	687	133	0	0	NUM
ejpam-4793	687	134	,	,	PUNCT
ejpam-4793	687	135	4	4	NUM
ejpam-4793	687	136	}	}	PUNCT
ejpam-4793	687	137	=	=	SYM
ejpam-4793	687	138	{	{	PUNCT
ejpam-4793	687	139	3	3	NUM
ejpam-4793	687	140	,	,	PUNCT
ejpam-4793	687	141	7	7	NUM
ejpam-4793	687	142	}	}	PUNCT
ejpam-4793	687	143	.	.	PUNCT
ejpam-4793	688	1	moreover	moreover	ADV
ejpam-4793	688	2	,	,	PUNCT
ejpam-4793	688	3	ρs(0	ρs(0	NOUN
ejpam-4793	688	4	)	)	PUNCT
ejpam-4793	688	5	=	=	SYM
ejpam-4793	688	6	{	{	PUNCT
ejpam-4793	688	7	0	0	NUM
ejpam-4793	688	8	,	,	PUNCT
ejpam-4793	688	9	4	4	NUM
ejpam-4793	688	10	}	}	PUNCT
ejpam-4793	688	11	,	,	PUNCT
ejpam-4793	688	12	ρs(1	ρs(1	X
ejpam-4793	688	13	)	)	PUNCT
ejpam-4793	688	14	=	=	PUNCT
ejpam-4793	688	15	{	{	PUNCT
ejpam-4793	688	16	1	1	NUM
ejpam-4793	688	17	,	,	PUNCT
ejpam-4793	688	18	5	5	NUM
ejpam-4793	688	19	}	}	PUNCT
ejpam-4793	688	20	,	,	PUNCT
ejpam-4793	688	21	ρs(2	ρs(2	NOUN
ejpam-4793	688	22	)	)	PUNCT
ejpam-4793	688	23	=	=	NOUN
ejpam-4793	688	24	{	{	PUNCT
ejpam-4793	688	25	2	2	NUM
ejpam-4793	688	26	,	,	PUNCT
ejpam-4793	688	27	6	6	NUM
ejpam-4793	688	28	}	}	PUNCT
ejpam-4793	688	29	,	,	PUNCT
ejpam-4793	688	30	ρs(3	ρs(3	NOUN
ejpam-4793	688	31	)	)	PUNCT
ejpam-4793	688	32	=	=	SYM
ejpam-4793	688	33	{	{	PUNCT
ejpam-4793	688	34	3	3	NUM
ejpam-4793	688	35	,	,	PUNCT
ejpam-4793	688	36	7	7	NUM
ejpam-4793	688	37	}	}	PUNCT
ejpam-4793	688	38	,	,	PUNCT
ejpam-4793	688	39	ρs(4	ρs(4	NOUN
ejpam-4793	688	40	)	)	PUNCT
ejpam-4793	688	41	=	=	PUNCT
ejpam-4793	688	42	{	{	PUNCT
ejpam-4793	688	43	0	0	NUM
ejpam-4793	688	44	,	,	PUNCT
ejpam-4793	688	45	4	4	NUM
ejpam-4793	688	46	}	}	PUNCT
ejpam-4793	688	47	,	,	PUNCT
ejpam-4793	688	48	ρs(5	ρs(5	NOUN
ejpam-4793	688	49	)	)	PUNCT
ejpam-4793	688	50	=	=	SYM
ejpam-4793	688	51	{	{	PUNCT
ejpam-4793	688	52	1	1	NUM
ejpam-4793	688	53	,	,	PUNCT
ejpam-4793	688	54	5	5	NUM
ejpam-4793	688	55	}	}	PUNCT
ejpam-4793	688	56	,	,	PUNCT
ejpam-4793	688	57	ρs(6	ρs(6	NOUN
ejpam-4793	688	58	)	)	PUNCT
ejpam-4793	688	59	=	=	SYM
ejpam-4793	688	60	{	{	PUNCT
ejpam-4793	688	61	2	2	NUM
ejpam-4793	688	62	,	,	PUNCT
ejpam-4793	688	63	6	6	NUM
ejpam-4793	688	64	}	}	PUNCT
ejpam-4793	688	65	,	,	PUNCT
ejpam-4793	688	66	and	and	CCONJ
ejpam-4793	688	67	ρs(7	ρs(7	NOUN
ejpam-4793	688	68	)	)	PUNCT
ejpam-4793	688	69	=	=	NOUN
ejpam-4793	688	70	{	{	PUNCT
ejpam-4793	688	71	3	3	NUM
ejpam-4793	688	72	,	,	PUNCT
ejpam-4793	688	73	7	7	NUM
ejpam-4793	688	74	}	}	PUNCT
ejpam-4793	688	75	.	.	PUNCT
ejpam-4793	689	1	thus	thus	ADV
ejpam-4793	689	2	,	,	PUNCT
ejpam-4793	689	3	the	the	DET
ejpam-4793	689	4	quotient	quotient	NOUN
ejpam-4793	689	5	m	m	PROPN
ejpam-4793	689	6	/	/	SYM
ejpam-4793	689	7	s	s	PART
ejpam-4793	689	8	=	=	X
ejpam-4793	689	9	{	{	PUNCT
ejpam-4793	689	10	ρs(0	ρs(0	NOUN
ejpam-4793	689	11	)	)	PUNCT
ejpam-4793	689	12	,	,	PUNCT
ejpam-4793	689	13	ρs(1	ρs(1	PROPN
ejpam-4793	689	14	)	)	PUNCT
ejpam-4793	689	15	,	,	PUNCT
ejpam-4793	689	16	ρs(2	ρs(2	NOUN
ejpam-4793	689	17	)	)	PUNCT
ejpam-4793	689	18	,	,	PUNCT
ejpam-4793	689	19	ρs(3	ρs(3	NOUN
ejpam-4793	689	20	)	)	PUNCT
ejpam-4793	689	21	}	}	PUNCT
ejpam-4793	689	22	using	use	VERB
ejpam-4793	689	23	the	the	DET
ejpam-4793	689	24	equivalence	equivalence	NOUN
ejpam-4793	689	25	relation	relation	NOUN
ejpam-4793	689	26	in	in	ADP
ejpam-4793	689	27	definition	definition	NOUN
ejpam-4793	689	28	6	6	NUM
ejpam-4793	689	29	.	.	PUNCT
ejpam-4793	690	1	now	now	ADV
ejpam-4793	690	2	,	,	PUNCT
ejpam-4793	690	3	observe	observe	VERB
ejpam-4793	690	4	that	that	SCONJ
ejpam-4793	690	5	for	for	ADP
ejpam-4793	690	6	all	all	PRON
ejpam-4793	690	7	α	α	PRON
ejpam-4793	690	8	∈	∈	PROPN
ejpam-4793	690	9	γ	γ	X
ejpam-4793	690	10	,	,	PUNCT
ejpam-4793	690	11	7α	7α	NOUN
ejpam-4793	690	12	=	=	SYM
ejpam-4793	690	13	1	1	NUM
ejpam-4793	690	14	or	or	CCONJ
ejpam-4793	690	15	7α	7α	NOUN
ejpam-4793	690	16	=	=	SYM
ejpam-4793	690	17	7	7	X
ejpam-4793	690	18	.	.	X
ejpam-4793	690	19	note	note	VERB
ejpam-4793	690	20	that	that	SCONJ
ejpam-4793	690	21	the	the	DET
ejpam-4793	690	22	identity	identity	NOUN
ejpam-4793	690	23	0	0	NUM
ejpam-4793	690	24	∈	∈	PROPN
ejpam-4793	690	25	s	s	NOUN
ejpam-4793	690	26	and	and	CCONJ
ejpam-4793	690	27	for	for	ADP
ejpam-4793	690	28	all	all	DET
ejpam-4793	690	29	α	α	NOUN
ejpam-4793	690	30	,	,	PUNCT
ejpam-4793	690	31	β	β	PROPN
ejpam-4793	690	32	∈	∈	PROPN
ejpam-4793	690	33	γ	γ	X
ejpam-4793	690	34	,	,	PUNCT
ejpam-4793	690	35	α0	α0	ADJ
ejpam-4793	691	1	+8	+8	ADJ
ejpam-4793	691	2	β0	β0	NOUN
ejpam-4793	691	3	=	=	NOUN
ejpam-4793	691	4	7α0	7α0	NUM
ejpam-4793	692	1	+8	+8	NOUN
ejpam-4793	692	2	7β0	7β0	NUM
ejpam-4793	692	3	=	=	SYM
ejpam-4793	692	4	0	0	PUNCT
ejpam-4793	693	1	+8	+8	NOUN
ejpam-4793	693	2	0	0	NUM
ejpam-4793	694	1	∈	∈	PROPN
ejpam-4793	694	2	s	s	PROPN
ejpam-4793	694	3	;	;	PUNCT
ejpam-4793	694	4	h.	h.	PROPN
ejpam-4793	694	5	sarapuddin	sarapuddin	PROPN
ejpam-4793	694	6	,	,	PUNCT
ejpam-4793	694	7	j.	j.	PROPN
ejpam-4793	694	8	vilela	vilela	PROPN
ejpam-4793	694	9	/	/	SYM
ejpam-4793	694	10	eur	eur	PROPN
ejpam-4793	694	11	.	.	PUNCT
ejpam-4793	695	1	j.	j.	PROPN
ejpam-4793	695	2	pure	pure	PROPN
ejpam-4793	695	3	appl	appl	PROPN
ejpam-4793	695	4	.	.	PROPN
ejpam-4793	695	5	math	math	PROPN
ejpam-4793	695	6	,	,	PUNCT
ejpam-4793	695	7	16	16	NUM
ejpam-4793	695	8	(	(	PUNCT
ejpam-4793	695	9	3	3	NUM
ejpam-4793	695	10	)	)	PUNCT
ejpam-4793	695	11	(	(	PUNCT
ejpam-4793	695	12	2023	2023	NUM
ejpam-4793	695	13	)	)	PUNCT
ejpam-4793	695	14	,	,	PUNCT
ejpam-4793	695	15	1772	1772	NUM
ejpam-4793	695	16	-	-	SYM
ejpam-4793	695	17	1793	1793	NUM
ejpam-4793	695	18	1788	1788	NUM
ejpam-4793	695	19	α0	α0	ADJ
ejpam-4793	696	1	+8	+8	PROPN
ejpam-4793	696	2	β4	β4	PROPN
ejpam-4793	696	3	=	=	PUNCT
ejpam-4793	696	4	7α0	7α0	NUM
ejpam-4793	697	1	+8	+8	NOUN
ejpam-4793	697	2	7β4	7β4	NUM
ejpam-4793	697	3	=	=	SYM
ejpam-4793	698	1	7β4	7β4	NUM
ejpam-4793	698	2	=	=	SYM
ejpam-4793	698	3	4	4	NUM
ejpam-4793	698	4	∈	∈	NOUN
ejpam-4793	698	5	s	s	NOUN
ejpam-4793	698	6	;	;	PUNCT
ejpam-4793	698	7	α4	α4	NOUN
ejpam-4793	698	8	+8	+8	NOUN
ejpam-4793	698	9	β4	β4	PROPN
ejpam-4793	698	10	=	=	PROPN
ejpam-4793	699	1	7α4	7α4	NUM
ejpam-4793	699	2	+8	+8	NOUN
ejpam-4793	699	3	7β4	7β4	NUM
ejpam-4793	700	1	=	=	NOUN
ejpam-4793	700	2	0	0	NUM
ejpam-4793	700	3	or	or	CCONJ
ejpam-4793	700	4	4	4	NUM
ejpam-4793	700	5	∈	∈	PROPN
ejpam-4793	700	6	s.	s.	PROPN
ejpam-4793	700	7	this	this	PRON
ejpam-4793	700	8	implies	imply	VERB
ejpam-4793	700	9	that	that	SCONJ
ejpam-4793	700	10	s	s	VERB
ejpam-4793	700	11	is	be	AUX
ejpam-4793	700	12	a	a	DET
ejpam-4793	700	13	γ	γ	NOUN
ejpam-4793	700	14	-	-	ADJ
ejpam-4793	700	15	submonoid	submonoid	NOUN
ejpam-4793	700	16	of	of	ADP
ejpam-4793	700	17	m	m	PROPN
ejpam-4793	700	18	.	.	PUNCT
ejpam-4793	701	1	now	now	ADV
ejpam-4793	701	2	,	,	PUNCT
ejpam-4793	701	3	note	note	VERB
ejpam-4793	701	4	that	that	SCONJ
ejpam-4793	701	5	for	for	SCONJ
ejpam-4793	701	6	all	all	PRON
ejpam-4793	701	7	α	α	PRON
ejpam-4793	701	8	∈	∈	PROPN
ejpam-4793	701	9	γ	γ	X
ejpam-4793	701	10	,	,	PUNCT
ejpam-4793	701	11	α0	α0	PROPN
ejpam-4793	701	12	+8	+8	PROPN
ejpam-4793	701	13	s	s	PART
ejpam-4793	701	14	=	=	SYM
ejpam-4793	702	1	α0	α0	ADJ
ejpam-4793	702	2	+8	+8	PROPN
ejpam-4793	702	3	{	{	PUNCT
ejpam-4793	702	4	0	0	NUM
ejpam-4793	702	5	,	,	PUNCT
ejpam-4793	702	6	4	4	NUM
ejpam-4793	702	7	}	}	PUNCT
ejpam-4793	702	8	=	=	SYM
ejpam-4793	702	9	7α0	7α0	NUM
ejpam-4793	702	10	+8	+8	NOUN
ejpam-4793	702	11	{	{	PUNCT
ejpam-4793	702	12	0	0	NUM
ejpam-4793	702	13	,	,	PUNCT
ejpam-4793	702	14	4	4	NUM
ejpam-4793	702	15	}	}	PUNCT
ejpam-4793	702	16	=	=	SYM
ejpam-4793	702	17	{	{	PUNCT
ejpam-4793	702	18	0	0	NUM
ejpam-4793	702	19	,	,	PUNCT
ejpam-4793	702	20	4	4	NUM
ejpam-4793	702	21	}	}	PUNCT
ejpam-4793	702	22	;	;	PUNCT
ejpam-4793	702	23	α1	α1	PROPN
ejpam-4793	702	24	+8	+8	PROPN
ejpam-4793	702	25	s	s	PART
ejpam-4793	702	26	=	=	PROPN
ejpam-4793	702	27	α1	α1	PROPN
ejpam-4793	702	28	+8	+8	PROPN
ejpam-4793	702	29	{	{	PUNCT
ejpam-4793	702	30	0	0	NUM
ejpam-4793	702	31	,	,	PUNCT
ejpam-4793	702	32	4	4	NUM
ejpam-4793	702	33	}	}	PUNCT
ejpam-4793	702	34	=	=	SYM
ejpam-4793	702	35	7α1	7α1	NUM
ejpam-4793	702	36	+8	+8	NOUN
ejpam-4793	702	37	{	{	PUNCT
ejpam-4793	702	38	0	0	NUM
ejpam-4793	702	39	,	,	PUNCT
ejpam-4793	702	40	4	4	NUM
ejpam-4793	702	41	}	}	PUNCT
ejpam-4793	702	42	=	=	SYM
ejpam-4793	702	43	{	{	PUNCT
ejpam-4793	702	44	1	1	NUM
ejpam-4793	702	45	,	,	PUNCT
ejpam-4793	702	46	5	5	NUM
ejpam-4793	702	47	}	}	PUNCT
ejpam-4793	702	48	or	or	CCONJ
ejpam-4793	702	49	{	{	PUNCT
ejpam-4793	702	50	3	3	NUM
ejpam-4793	702	51	,	,	PUNCT
ejpam-4793	702	52	7	7	NUM
ejpam-4793	702	53	}	}	PUNCT
ejpam-4793	702	54	;	;	PUNCT
ejpam-4793	702	55	α2	α2	ADJ
ejpam-4793	702	56	+8	+8	PROPN
ejpam-4793	702	57	s	s	PART
ejpam-4793	702	58	=	=	X
ejpam-4793	702	59	α2	α2	ADJ
ejpam-4793	702	60	+8	+8	PROPN
ejpam-4793	702	61	{	{	PUNCT
ejpam-4793	702	62	0	0	NUM
ejpam-4793	702	63	,	,	PUNCT
ejpam-4793	702	64	4	4	NUM
ejpam-4793	702	65	}	}	PUNCT
ejpam-4793	702	66	=	=	SYM
ejpam-4793	702	67	7α2	7α2	NUM
ejpam-4793	703	1	+8	+8	NOUN
ejpam-4793	703	2	{	{	PUNCT
ejpam-4793	703	3	0	0	NUM
ejpam-4793	703	4	,	,	PUNCT
ejpam-4793	703	5	4	4	NUM
ejpam-4793	703	6	}	}	PUNCT
ejpam-4793	703	7	=	=	SYM
ejpam-4793	703	8	{	{	PUNCT
ejpam-4793	703	9	2	2	NUM
ejpam-4793	703	10	,	,	PUNCT
ejpam-4793	703	11	6	6	NUM
ejpam-4793	703	12	}	}	PUNCT
ejpam-4793	703	13	;	;	PUNCT
ejpam-4793	703	14	α3	α3	ADV
ejpam-4793	703	15	+8	+8	PROPN
ejpam-4793	703	16	s	s	PART
ejpam-4793	703	17	=	=	X
ejpam-4793	703	18	α3	α3	NOUN
ejpam-4793	703	19	+8	+8	PROPN
ejpam-4793	703	20	{	{	PUNCT
ejpam-4793	703	21	0	0	NUM
ejpam-4793	703	22	,	,	PUNCT
ejpam-4793	703	23	4	4	NUM
ejpam-4793	703	24	}	}	PUNCT
ejpam-4793	703	25	=	=	SYM
ejpam-4793	703	26	7α3	7α3	NUM
ejpam-4793	703	27	+8	+8	NOUN
ejpam-4793	703	28	{	{	PUNCT
ejpam-4793	703	29	0	0	NUM
ejpam-4793	703	30	,	,	PUNCT
ejpam-4793	703	31	4	4	NUM
ejpam-4793	703	32	}	}	PUNCT
ejpam-4793	703	33	=	=	SYM
ejpam-4793	703	34	{	{	PUNCT
ejpam-4793	703	35	1	1	NUM
ejpam-4793	703	36	,	,	PUNCT
ejpam-4793	703	37	5	5	NUM
ejpam-4793	703	38	}	}	PUNCT
ejpam-4793	703	39	or	or	CCONJ
ejpam-4793	703	40	{	{	PUNCT
ejpam-4793	703	41	3	3	NUM
ejpam-4793	703	42	,	,	PUNCT
ejpam-4793	703	43	7	7	NUM
ejpam-4793	703	44	}	}	PUNCT
ejpam-4793	703	45	;	;	PUNCT
ejpam-4793	703	46	α4	α4	VERB
ejpam-4793	703	47	+8	+8	NOUN
ejpam-4793	703	48	s	s	PART
ejpam-4793	703	49	=	=	NOUN
ejpam-4793	703	50	α4	α4	NOUN
ejpam-4793	703	51	+8	+8	NOUN
ejpam-4793	703	52	{	{	PUNCT
ejpam-4793	703	53	0	0	NUM
ejpam-4793	703	54	,	,	PUNCT
ejpam-4793	703	55	4	4	NUM
ejpam-4793	703	56	}	}	PUNCT
ejpam-4793	703	57	=	=	PUNCT
ejpam-4793	703	58	7α4	7α4	NUM
ejpam-4793	703	59	+8	+8	NOUN
ejpam-4793	703	60	{	{	PUNCT
ejpam-4793	703	61	0	0	NUM
ejpam-4793	703	62	,	,	PUNCT
ejpam-4793	703	63	4	4	NUM
ejpam-4793	703	64	}	}	PUNCT
ejpam-4793	703	65	=	=	SYM
ejpam-4793	703	66	{	{	PUNCT
ejpam-4793	703	67	0	0	NUM
ejpam-4793	703	68	,	,	PUNCT
ejpam-4793	703	69	4	4	NUM
ejpam-4793	703	70	}	}	PUNCT
ejpam-4793	703	71	;	;	PUNCT
ejpam-4793	703	72	α5	α5	NOUN
ejpam-4793	703	73	+8	+8	PROPN
ejpam-4793	703	74	s	s	PART
ejpam-4793	703	75	=	=	SYM
ejpam-4793	703	76	α5	α5	PROPN
ejpam-4793	703	77	+8	+8	PROPN
ejpam-4793	703	78	{	{	PUNCT
ejpam-4793	703	79	0	0	NUM
ejpam-4793	703	80	,	,	PUNCT
ejpam-4793	703	81	4	4	NUM
ejpam-4793	703	82	}	}	PUNCT
ejpam-4793	703	83	=	=	SYM
ejpam-4793	703	84	7α5	7α5	NUM
ejpam-4793	703	85	+8	+8	NOUN
ejpam-4793	703	86	{	{	PUNCT
ejpam-4793	703	87	0	0	NUM
ejpam-4793	703	88	,	,	PUNCT
ejpam-4793	703	89	4	4	NUM
ejpam-4793	703	90	}	}	PUNCT
ejpam-4793	703	91	=	=	SYM
ejpam-4793	703	92	{	{	PUNCT
ejpam-4793	703	93	1	1	NUM
ejpam-4793	703	94	,	,	PUNCT
ejpam-4793	703	95	5	5	NUM
ejpam-4793	703	96	}	}	PUNCT
ejpam-4793	703	97	or	or	CCONJ
ejpam-4793	703	98	{	{	PUNCT
ejpam-4793	703	99	3	3	NUM
ejpam-4793	703	100	,	,	PUNCT
ejpam-4793	703	101	7	7	NUM
ejpam-4793	703	102	}	}	PUNCT
ejpam-4793	703	103	;	;	PUNCT
ejpam-4793	703	104	α6	α6	NOUN
ejpam-4793	703	105	+8	+8	PROPN
ejpam-4793	703	106	s	s	PART
ejpam-4793	703	107	=	=	SYM
ejpam-4793	703	108	α6	α6	PROPN
ejpam-4793	703	109	+8	+8	PROPN
ejpam-4793	703	110	{	{	PUNCT
ejpam-4793	703	111	0	0	NUM
ejpam-4793	703	112	,	,	PUNCT
ejpam-4793	703	113	4	4	NUM
ejpam-4793	703	114	}	}	PUNCT
ejpam-4793	703	115	=	=	SYM
ejpam-4793	703	116	7α6	7α6	NUM
ejpam-4793	703	117	+8	+8	NOUN
ejpam-4793	703	118	{	{	PUNCT
ejpam-4793	703	119	0	0	NUM
ejpam-4793	703	120	,	,	PUNCT
ejpam-4793	703	121	4	4	NUM
ejpam-4793	703	122	}	}	PUNCT
ejpam-4793	703	123	=	=	SYM
ejpam-4793	703	124	{	{	PUNCT
ejpam-4793	703	125	2	2	NUM
ejpam-4793	703	126	,	,	PUNCT
ejpam-4793	703	127	6	6	NUM
ejpam-4793	703	128	}	}	PUNCT
ejpam-4793	703	129	;	;	PUNCT
ejpam-4793	703	130	α7	α7	NOUN
ejpam-4793	703	131	+8	+8	PROPN
ejpam-4793	703	132	s	s	PART
ejpam-4793	703	133	=	=	X
ejpam-4793	703	134	α7	α7	NOUN
ejpam-4793	703	135	+8	+8	NOUN
ejpam-4793	703	136	{	{	PUNCT
ejpam-4793	703	137	0	0	NUM
ejpam-4793	703	138	,	,	PUNCT
ejpam-4793	703	139	4	4	NUM
ejpam-4793	703	140	}	}	PUNCT
ejpam-4793	703	141	=	=	SYM
ejpam-4793	704	1	7α7	7α7	NUM
ejpam-4793	704	2	+8	+8	NOUN
ejpam-4793	704	3	{	{	PUNCT
ejpam-4793	704	4	0	0	NUM
ejpam-4793	704	5	,	,	PUNCT
ejpam-4793	704	6	4	4	NUM
ejpam-4793	704	7	}	}	PUNCT
ejpam-4793	704	8	=	=	SYM
ejpam-4793	704	9	{	{	PUNCT
ejpam-4793	704	10	1	1	NUM
ejpam-4793	704	11	,	,	PUNCT
ejpam-4793	704	12	5	5	NUM
ejpam-4793	704	13	}	}	PUNCT
ejpam-4793	704	14	or	or	CCONJ
ejpam-4793	704	15	{	{	PUNCT
ejpam-4793	704	16	3	3	NUM
ejpam-4793	704	17	,	,	PUNCT
ejpam-4793	704	18	7	7	NUM
ejpam-4793	704	19	}	}	PUNCT
ejpam-4793	704	20	.	.	PUNCT
ejpam-4793	705	1	moreover	moreover	ADV
ejpam-4793	705	2	,	,	PUNCT
ejpam-4793	705	3	we	we	PRON
ejpam-4793	705	4	have	have	VERB
ejpam-4793	705	5	ρs(0	ρs(0	NOUN
ejpam-4793	705	6	)	)	PUNCT
ejpam-4793	705	7	=	=	SYM
ejpam-4793	705	8	ρs(4	ρs(4	NOUN
ejpam-4793	705	9	)	)	PUNCT
ejpam-4793	705	10	=	=	PUNCT
ejpam-4793	705	11	{	{	PUNCT
ejpam-4793	705	12	0	0	NUM
ejpam-4793	705	13	,	,	PUNCT
ejpam-4793	705	14	4	4	NUM
ejpam-4793	705	15	}	}	PUNCT
ejpam-4793	705	16	,	,	PUNCT
ejpam-4793	705	17	ρs(2	ρs(2	NOUN
ejpam-4793	705	18	)	)	PUNCT
ejpam-4793	705	19	=	=	SYM
ejpam-4793	705	20	ρs(6	ρs(6	PROPN
ejpam-4793	705	21	)	)	PUNCT
ejpam-4793	705	22	=	=	PUNCT
ejpam-4793	705	23	{	{	PUNCT
ejpam-4793	705	24	2	2	NUM
ejpam-4793	705	25	,	,	PUNCT
ejpam-4793	705	26	6	6	NUM
ejpam-4793	705	27	}	}	PUNCT
ejpam-4793	705	28	,	,	PUNCT
ejpam-4793	705	29	and	and	CCONJ
ejpam-4793	705	30	ρs(1	ρs(1	X
ejpam-4793	705	31	)	)	PUNCT
ejpam-4793	705	32	=	=	SYM
ejpam-4793	705	33	ρs(3	ρs(3	NOUN
ejpam-4793	705	34	)	)	PUNCT
ejpam-4793	706	1	=	=	SYM
ejpam-4793	706	2	ρs(5	ρs(5	NOUN
ejpam-4793	706	3	)	)	PUNCT
ejpam-4793	706	4	=	=	SYM
ejpam-4793	706	5	ρs(7	ρs(7	X
ejpam-4793	706	6	)	)	PUNCT
ejpam-4793	706	7	=	=	NOUN
ejpam-4793	706	8	{	{	PUNCT
ejpam-4793	706	9	1	1	NUM
ejpam-4793	706	10	,	,	PUNCT
ejpam-4793	706	11	3	3	NUM
ejpam-4793	706	12	,	,	PUNCT
ejpam-4793	706	13	5	5	NUM
ejpam-4793	706	14	,	,	PUNCT
ejpam-4793	706	15	7	7	NUM
ejpam-4793	706	16	}	}	PUNCT
ejpam-4793	706	17	.	.	PUNCT
ejpam-4793	707	1	thus	thus	ADV
ejpam-4793	707	2	,	,	PUNCT
ejpam-4793	707	3	the	the	DET
ejpam-4793	707	4	quotient	quotient	NOUN
ejpam-4793	707	5	m	m	PROPN
ejpam-4793	707	6	/	/	SYM
ejpam-4793	707	7	s	s	PART
ejpam-4793	707	8	=	=	X
ejpam-4793	707	9	{	{	PUNCT
ejpam-4793	707	10	ρs(0	ρs(0	NOUN
ejpam-4793	707	11	)	)	PUNCT
ejpam-4793	707	12	,	,	PUNCT
ejpam-4793	707	13	ρs(1	ρs(1	PROPN
ejpam-4793	707	14	)	)	PUNCT
ejpam-4793	707	15	,	,	PUNCT
ejpam-4793	707	16	ρs(2	ρs(2	NOUN
ejpam-4793	707	17	)	)	PUNCT
ejpam-4793	707	18	}	}	PUNCT
ejpam-4793	707	19	using	use	VERB
ejpam-4793	707	20	the	the	DET
ejpam-4793	707	21	definition	definition	NOUN
ejpam-4793	707	22	13	13	NUM
ejpam-4793	707	23	.	.	PUNCT
ejpam-4793	707	24	observe	observe	VERB
ejpam-4793	707	25	that	that	SCONJ
ejpam-4793	707	26	m	m	PROPN
ejpam-4793	707	27	/	/	SYM
ejpam-4793	707	28	s	s	PART
ejpam-4793	707	29	yield	yield	NOUN
ejpam-4793	707	30	is	be	AUX
ejpam-4793	707	31	not	not	PART
ejpam-4793	707	32	equal	equal	ADJ
ejpam-4793	707	33	to	to	ADP
ejpam-4793	707	34	m	m	PRON
ejpam-4793	707	35	/	/	SYM
ejpam-4793	707	36	s	s	VERB
ejpam-4793	707	37	above	above	ADV
ejpam-4793	707	38	.	.	PUNCT
ejpam-4793	708	1	moreover	moreover	ADV
ejpam-4793	708	2	,	,	PUNCT
ejpam-4793	708	3	ρs(0	ρs(0	NOUN
ejpam-4793	708	4	)	)	PUNCT
ejpam-4793	708	5	is	be	AUX
ejpam-4793	708	6	the	the	DET
ejpam-4793	708	7	same	same	ADJ
ejpam-4793	708	8	with	with	ADP
ejpam-4793	708	9	ρs(0	ρs(0	NOUN
ejpam-4793	708	10	)	)	PUNCT
ejpam-4793	708	11	above	above	ADV
ejpam-4793	708	12	,	,	PUNCT
ejpam-4793	708	13	however	however	ADV
ejpam-4793	708	14	,	,	PUNCT
ejpam-4793	708	15	ρs(1)s	ρs(1)s	PROPN
ejpam-4793	708	16	are	be	AUX
ejpam-4793	708	17	different	different	ADJ
ejpam-4793	708	18	.	.	PUNCT
ejpam-4793	709	1	this	this	PRON
ejpam-4793	709	2	implies	imply	VERB
ejpam-4793	709	3	that	that	SCONJ
ejpam-4793	709	4	their	their	PRON
ejpam-4793	709	5	equivalence	equivalence	NOUN
ejpam-4793	709	6	classes	class	NOUN
ejpam-4793	709	7	are	be	AUX
ejpam-4793	709	8	not	not	PART
ejpam-4793	709	9	equal	equal	ADJ
ejpam-4793	709	10	.	.	PUNCT
ejpam-4793	710	1	hence	hence	ADV
ejpam-4793	710	2	,	,	PUNCT
ejpam-4793	710	3	m	m	PROPN
ejpam-4793	710	4	/	/	SYM
ejpam-4793	710	5	s	s	PROPN
ejpam-4793	710	6	via	via	ADP
ejpam-4793	710	7	γ	γ	NOUN
ejpam-4793	710	8	-	-	ADJ
ejpam-4793	710	9	submonoid	submonoid	ADJ
ejpam-4793	710	10	is	be	AUX
ejpam-4793	710	11	different	different	ADJ
ejpam-4793	710	12	from	from	ADP
ejpam-4793	710	13	m	m	PROPN
ejpam-4793	710	14	/	/	SYM
ejpam-4793	710	15	s	s	PROPN
ejpam-4793	710	16	via	via	ADP
ejpam-4793	710	17	submonoid	submonoid	ADJ
ejpam-4793	710	18	,	,	PUNCT
ejpam-4793	710	19	where	where	SCONJ
ejpam-4793	710	20	m	m	NOUN
ejpam-4793	710	21	is	be	AUX
ejpam-4793	710	22	a	a	DET
ejpam-4793	710	23	monoid	monoid	NOUN
ejpam-4793	710	24	.	.	PUNCT
ejpam-4793	711	1	theorem	theorem	NOUN
ejpam-4793	711	2	11	11	NUM
ejpam-4793	711	3	.	.	PUNCT
ejpam-4793	712	1	if	if	SCONJ
ejpam-4793	712	2	m	m	NOUN
ejpam-4793	712	3	is	be	AUX
ejpam-4793	712	4	a	a	DET
ejpam-4793	712	5	commutative	commutative	ADJ
ejpam-4793	712	6	γ	γ	X
ejpam-4793	712	7	-	-	PUNCT
ejpam-4793	712	8	monoid	monoid	NOUN
ejpam-4793	712	9	and	and	CCONJ
ejpam-4793	712	10	s	s	VERB
ejpam-4793	712	11	a	a	DET
ejpam-4793	712	12	γ	γ	NOUN
ejpam-4793	712	13	-	-	ADJ
ejpam-4793	712	14	submonoid	submonoid	NOUN
ejpam-4793	712	15	of	of	ADP
ejpam-4793	712	16	m	m	PROPN
ejpam-4793	712	17	,	,	PUNCT
ejpam-4793	712	18	then	then	ADV
ejpam-4793	712	19	m	m	PROPN
ejpam-4793	712	20	/	/	SYM
ejpam-4793	712	21	s	s	PART
ejpam-4793	712	22	is	be	AUX
ejpam-4793	712	23	a	a	DET
ejpam-4793	712	24	γ	γ	X
ejpam-4793	712	25	-	-	PUNCT
ejpam-4793	712	26	monoid	monoid	NOUN
ejpam-4793	712	27	.	.	PUNCT
ejpam-4793	713	1	proof	proof	NOUN
ejpam-4793	713	2	.	.	PUNCT
ejpam-4793	714	1	let	let	VERB
ejpam-4793	714	2	m	m	PRON
ejpam-4793	714	3	be	be	AUX
ejpam-4793	714	4	a	a	DET
ejpam-4793	714	5	commutative	commutative	ADJ
ejpam-4793	714	6	γ	γ	X
ejpam-4793	714	7	-	-	PUNCT
ejpam-4793	714	8	monoid	monoid	NOUN
ejpam-4793	714	9	and	and	CCONJ
ejpam-4793	714	10	s	s	VERB
ejpam-4793	714	11	a	a	DET
ejpam-4793	714	12	γ	γ	NOUN
ejpam-4793	714	13	-	-	ADJ
ejpam-4793	714	14	submonoid	submonoid	NOUN
ejpam-4793	714	15	of	of	ADP
ejpam-4793	714	16	m	m	PROPN
ejpam-4793	714	17	.	.	PUNCT
ejpam-4793	715	1	by	by	ADP
ejpam-4793	715	2	proposition	proposition	NOUN
ejpam-4793	715	3	1	1	NUM
ejpam-4793	715	4	,	,	PUNCT
ejpam-4793	715	5	since	since	SCONJ
ejpam-4793	715	6	ρs	ρs	ADV
ejpam-4793	715	7	is	be	VERB
ejpam-4793	715	8	a	a	DET
ejpam-4793	715	9	congruence	congruence	NOUN
ejpam-4793	715	10	on	on	ADP
ejpam-4793	715	11	m	m	PROPN
ejpam-4793	715	12	,	,	PUNCT
ejpam-4793	715	13	we	we	PRON
ejpam-4793	715	14	have	have	VERB
ejpam-4793	715	15	m	m	PRON
ejpam-4793	715	16	/	/	SYM
ejpam-4793	715	17	ρs	ρs	NOUN
ejpam-4793	715	18	=	=	NOUN
ejpam-4793	715	19	m	m	NOUN
ejpam-4793	715	20	/	/	SYM
ejpam-4793	715	21	s	s	VERB
ejpam-4793	715	22	is	be	AUX
ejpam-4793	715	23	a	a	DET
ejpam-4793	715	24	monoid	monoid	NOUN
ejpam-4793	715	25	with	with	ADP
ejpam-4793	715	26	binary	binary	ADJ
ejpam-4793	715	27	operation	operation	NOUN
ejpam-4793	715	28	◦	◦	NOUN
ejpam-4793	715	29	given	give	VERB
ejpam-4793	715	30	by	by	ADP
ejpam-4793	715	31	ρs(x	ρs(x	NOUN
ejpam-4793	715	32	)	)	PUNCT
ejpam-4793	715	33	◦	◦	NOUN
ejpam-4793	715	34	ρs(y	ρs(y	NUM
ejpam-4793	715	35	)	)	PUNCT
ejpam-4793	715	36	=	=	NOUN
ejpam-4793	715	37	ρs(x	ρs(x	X
ejpam-4793	715	38	∗	∗	X
ejpam-4793	715	39	y	y	NOUN
ejpam-4793	715	40	)	)	PUNCT
ejpam-4793	715	41	with	with	ADP
ejpam-4793	715	42	identity	identity	NOUN
ejpam-4793	715	43	ρs(1	ρs(1	NOUN
ejpam-4793	715	44	m	m	PROPN
ejpam-4793	715	45	)	)	PUNCT
ejpam-4793	715	46	.	.	PUNCT
ejpam-4793	716	1	consider	consider	VERB
ejpam-4793	716	2	a	a	DET
ejpam-4793	716	3	mapping	mapping	NOUN
ejpam-4793	716	4	ϕ	ϕ	NOUN
ejpam-4793	716	5	:	:	PUNCT
ejpam-4793	716	6	γ	γ	X
ejpam-4793	716	7	×	×	PROPN
ejpam-4793	716	8	m	m	PROPN
ejpam-4793	716	9	/	/	SYM
ejpam-4793	716	10	s	s	PART
ejpam-4793	716	11	−→	−→	NOUN
ejpam-4793	716	12	m	m	NOUN
ejpam-4793	716	13	/	/	SYM
ejpam-4793	716	14	s	s	AUX
ejpam-4793	716	15	given	give	VERB
ejpam-4793	716	16	by	by	ADP
ejpam-4793	716	17	(	(	PUNCT
ejpam-4793	716	18	α	α	NOUN
ejpam-4793	716	19	,	,	PUNCT
ejpam-4793	716	20	ρs(x	ρs(x	ADJ
ejpam-4793	716	21	)	)	PUNCT
ejpam-4793	716	22	)	)	PUNCT
ejpam-4793	717	1	7→	7→	NUM
ejpam-4793	718	1	αρs(x	αρs(x	X
ejpam-4793	718	2	)	)	PUNCT
ejpam-4793	718	3	=	=	SYM
ejpam-4793	718	4	ρs	ρs	PROPN
ejpam-4793	718	5	(	(	PUNCT
ejpam-4793	718	6	αx	αx	NOUN
ejpam-4793	718	7	)	)	PUNCT
ejpam-4793	718	8	for	for	ADP
ejpam-4793	718	9	all	all	PRON
ejpam-4793	718	10	α	α	DET
ejpam-4793	718	11	∈	∈	NOUN
ejpam-4793	718	12	γ	γ	NOUN
ejpam-4793	718	13	and	and	CCONJ
ejpam-4793	718	14	x	x	SYM
ejpam-4793	718	15	∈	∈	PROPN
ejpam-4793	718	16	m	m	VERB
ejpam-4793	718	17	.	.	PUNCT
ejpam-4793	719	1	let	let	VERB
ejpam-4793	719	2	(	(	PUNCT
ejpam-4793	719	3	α	α	NOUN
ejpam-4793	719	4	,	,	PUNCT
ejpam-4793	719	5	ρs(x	ρs(x	ADJ
ejpam-4793	719	6	)	)	PUNCT
ejpam-4793	719	7	)	)	PUNCT
ejpam-4793	719	8	,	,	PUNCT
ejpam-4793	719	9	(	(	PUNCT
ejpam-4793	719	10	β	β	X
ejpam-4793	719	11	,	,	PUNCT
ejpam-4793	719	12	ρs(y	ρs(y	NUM
ejpam-4793	719	13	)	)	PUNCT
ejpam-4793	719	14	)	)	PUNCT
ejpam-4793	720	1	∈	∈	PROPN
ejpam-4793	720	2	γ	γ	X
ejpam-4793	720	3	×	×	PROPN
ejpam-4793	720	4	m	m	PROPN
ejpam-4793	720	5	/	/	SYM
ejpam-4793	720	6	s	s	VERB
ejpam-4793	720	7	such	such	ADJ
ejpam-4793	720	8	that	that	SCONJ
ejpam-4793	720	9	(	(	PUNCT
ejpam-4793	720	10	α	α	NOUN
ejpam-4793	720	11	,	,	PUNCT
ejpam-4793	720	12	ρs(x	ρs(x	ADJ
ejpam-4793	720	13	)	)	PUNCT
ejpam-4793	720	14	)	)	PUNCT
ejpam-4793	721	1	=	=	SYM
ejpam-4793	721	2	(	(	PUNCT
ejpam-4793	721	3	β	β	X
ejpam-4793	721	4	,	,	PUNCT
ejpam-4793	721	5	ρs(y	ρs(y	NUM
ejpam-4793	721	6	)	)	PUNCT
ejpam-4793	721	7	)	)	PUNCT
ejpam-4793	721	8	.	.	PUNCT
ejpam-4793	722	1	then	then	ADV
ejpam-4793	722	2	α	α	X
ejpam-4793	722	3	=	=	SYM
ejpam-4793	722	4	β	β	X
ejpam-4793	722	5	and	and	CCONJ
ejpam-4793	722	6	ρs(x	ρs(x	X
ejpam-4793	722	7	)	)	PUNCT
ejpam-4793	722	8	=	=	SYM
ejpam-4793	722	9	ρs(y	ρs(y	NUM
ejpam-4793	722	10	)	)	PUNCT
ejpam-4793	722	11	.	.	PUNCT
ejpam-4793	723	1	thus	thus	ADV
ejpam-4793	723	2	,	,	PUNCT
ejpam-4793	723	3	by	by	ADP
ejpam-4793	723	4	remark	remark	NOUN
ejpam-4793	723	5	14(ii	14(ii	NUM
ejpam-4793	723	6	)	)	PUNCT
ejpam-4793	723	7	,	,	PUNCT
ejpam-4793	723	8	(	(	PUNCT
ejpam-4793	723	9	α	α	NOUN
ejpam-4793	723	10	′	′	NUM
ejpam-4793	723	11	x	x	PUNCT
ejpam-4793	723	12	∗	∗	NOUN
ejpam-4793	723	13	s	s	NOUN
ejpam-4793	723	14	)	)	PUNCT
ejpam-4793	723	15	∩	∩	NOUN
ejpam-4793	723	16	(	(	PUNCT
ejpam-4793	723	17	α	α	NOUN
ejpam-4793	723	18	′	′	NUM
ejpam-4793	723	19	y	y	PROPN
ejpam-4793	723	20	∗	∗	NOUN
ejpam-4793	723	21	s	s	PART
ejpam-4793	723	22	)	)	PUNCT
ejpam-4793	723	23	̸=	̸=	PROPN
ejpam-4793	723	24	∅	∅	NOUN
ejpam-4793	723	25	for	for	ADP
ejpam-4793	723	26	all	all	DET
ejpam-4793	723	27	α′	α′	NUM
ejpam-4793	723	28	∈	∈	PROPN
ejpam-4793	723	29	γ	γ	NOUN
ejpam-4793	723	30	,	,	PUNCT
ejpam-4793	723	31	which	which	PRON
ejpam-4793	723	32	implies	imply	VERB
ejpam-4793	723	33	that	that	SCONJ
ejpam-4793	723	34	α′	α′	NUM
ejpam-4793	723	35	x	x	SYM
ejpam-4793	723	36	∗	∗	NOUN
ejpam-4793	723	37	s1	s1	NOUN
ejpam-4793	723	38	=	=	PUNCT
ejpam-4793	723	39	α′	α′	NUM
ejpam-4793	723	40	y	y	PROPN
ejpam-4793	723	41	∗	∗	NOUN
ejpam-4793	723	42	s2	s2	NOUN
ejpam-4793	723	43	for	for	ADP
ejpam-4793	723	44	some	some	DET
ejpam-4793	723	45	s1	s1	NOUN
ejpam-4793	723	46	,	,	PUNCT
ejpam-4793	723	47	s2	s2	PROPN
ejpam-4793	723	48	∈	∈	PROPN
ejpam-4793	723	49	s.	s.	PROPN
ejpam-4793	723	50	accordingly	accordingly	ADV
ejpam-4793	723	51	,	,	PUNCT
ejpam-4793	723	52	α(α	α(α	PROPN
ejpam-4793	723	53	′	′	NUM
ejpam-4793	723	54	x	x	SYM
ejpam-4793	723	55	∗	∗	NOUN
ejpam-4793	723	56	s1	s1	NOUN
ejpam-4793	723	57	)	)	PUNCT
ejpam-4793	723	58	=	=	PUNCT
ejpam-4793	724	1	α(α	α(α	NOUN
ejpam-4793	724	2	′	′	NUM
ejpam-4793	724	3	x	x	X
ejpam-4793	724	4	)	)	PUNCT
ejpam-4793	724	5	∗	∗	NOUN
ejpam-4793	724	6	αs1	αs1	NOUN
ejpam-4793	725	1	=	=	SYM
ejpam-4793	725	2	α(α	α(α	NOUN
ejpam-4793	725	3	′	′	NUM
ejpam-4793	726	1	y	y	PROPN
ejpam-4793	726	2	∗	∗	X
ejpam-4793	726	3	s2	s2	PROPN
ejpam-4793	726	4	)	)	PUNCT
ejpam-4793	726	5	=	=	PUNCT
ejpam-4793	727	1	α(α	α(α	PROPN
ejpam-4793	727	2	′	′	NUM
ejpam-4793	728	1	y	y	NOUN
ejpam-4793	728	2	)	)	PUNCT
ejpam-4793	728	3	∗	∗	NOUN
ejpam-4793	728	4	αs2	αs2	VERB
ejpam-4793	728	5	.	.	PUNCT
ejpam-4793	729	1	since	since	SCONJ
ejpam-4793	729	2	s	s	PROPN
ejpam-4793	729	3	is	be	AUX
ejpam-4793	729	4	a	a	DET
ejpam-4793	729	5	γ	γ	NOUN
ejpam-4793	729	6	-	-	PUNCT
ejpam-4793	729	7	submonoid	submonoid	ADJ
ejpam-4793	729	8	,	,	PUNCT
ejpam-4793	729	9	αs1	αs1	NOUN
ejpam-4793	729	10	,	,	PUNCT
ejpam-4793	729	11	αs2	αs2	VERB
ejpam-4793	729	12	∈	∈	PROPN
ejpam-4793	729	13	s	s	PART
ejpam-4793	729	14	and	and	CCONJ
ejpam-4793	729	15	(	(	PUNCT
ejpam-4793	729	16	α+α′	α+α′	X
ejpam-4793	729	17	x	x	X
ejpam-4793	729	18	∗	∗	X
ejpam-4793	729	19	s	s	NOUN
ejpam-4793	729	20	)	)	PUNCT
ejpam-4793	729	21	∩	∩	NOUN
ejpam-4793	729	22	(	(	PUNCT
ejpam-4793	729	23	α+α′	α+α′	X
ejpam-4793	729	24	y	y	PROPN
ejpam-4793	729	25	∗	∗	PROPN
ejpam-4793	729	26	s	s	PART
ejpam-4793	729	27	)	)	PUNCT
ejpam-4793	729	28	̸=	̸=	PROPN
ejpam-4793	729	29	∅.	∅.	ADP
ejpam-4793	729	30	this	this	PRON
ejpam-4793	729	31	means	mean	VERB
ejpam-4793	729	32	that	that	SCONJ
ejpam-4793	729	33	ϕ(α	ϕ(α	PROPN
ejpam-4793	729	34	,	,	PUNCT
ejpam-4793	729	35	ρs(x	ρs(x	ADJ
ejpam-4793	729	36	)	)	PUNCT
ejpam-4793	729	37	)	)	PUNCT
ejpam-4793	730	1	=	=	PUNCT
ejpam-4793	730	2	αρs(x	αρs(x	X
ejpam-4793	730	3	)	)	PUNCT
ejpam-4793	730	4	=	=	SYM
ejpam-4793	730	5	ρs	ρs	PROPN
ejpam-4793	730	6	(	(	PUNCT
ejpam-4793	730	7	αx	αx	ADV
ejpam-4793	730	8	)	)	PUNCT
ejpam-4793	730	9	=	=	SYM
ejpam-4793	730	10	ρs	ρs	PROPN
ejpam-4793	730	11	(	(	PUNCT
ejpam-4793	730	12	βy	βy	ADJ
ejpam-4793	730	13	)	)	PUNCT
ejpam-4793	730	14	=	=	SYM
ejpam-4793	730	15	βρs(y	βρs(y	PROPN
ejpam-4793	730	16	)	)	PUNCT
ejpam-4793	730	17	=	=	SYM
ejpam-4793	730	18	ϕ(β	ϕ(β	PROPN
ejpam-4793	730	19	,	,	PUNCT
ejpam-4793	730	20	ρs(y	ρs(y	NUM
ejpam-4793	730	21	)	)	PUNCT
ejpam-4793	730	22	)	)	PUNCT
ejpam-4793	730	23	.	.	PUNCT
ejpam-4793	731	1	hence	hence	ADV
ejpam-4793	731	2	,	,	PUNCT
ejpam-4793	731	3	ϕ	ϕ	PROPN
ejpam-4793	731	4	is	be	AUX
ejpam-4793	731	5	well	well	ADV
ejpam-4793	731	6	-	-	PUNCT
ejpam-4793	731	7	defined	define	VERB
ejpam-4793	731	8	.	.	PUNCT
ejpam-4793	732	1	now	now	ADV
ejpam-4793	732	2	,	,	PUNCT
ejpam-4793	732	3	for	for	ADP
ejpam-4793	732	4	any	any	DET
ejpam-4793	732	5	α	α	NOUN
ejpam-4793	732	6	,	,	PUNCT
ejpam-4793	732	7	β	β	X
ejpam-4793	732	8	∈	∈	PROPN
ejpam-4793	732	9	γ	γ	NOUN
ejpam-4793	732	10	and	and	CCONJ
ejpam-4793	732	11	x	x	SYM
ejpam-4793	732	12	∈	∈	PROPN
ejpam-4793	732	13	m	m	NOUN
ejpam-4793	732	14	,	,	PUNCT
ejpam-4793	732	15	ϕ((0	ϕ((0	PROPN
ejpam-4793	732	16	,	,	PUNCT
ejpam-4793	732	17	ρs(x	ρs(x	ADJ
ejpam-4793	732	18	)	)	PUNCT
ejpam-4793	732	19	)	)	PUNCT
ejpam-4793	732	20	)	)	PUNCT
ejpam-4793	733	1	=	=	SYM
ejpam-4793	733	2	0ρs(x	0ρs(x	NUM
ejpam-4793	733	3	)	)	PUNCT
ejpam-4793	733	4	=	=	SYM
ejpam-4793	733	5	ρs	ρs	PROPN
ejpam-4793	733	6	(	(	PUNCT
ejpam-4793	733	7	0x	0x	NOUN
ejpam-4793	733	8	)	)	PUNCT
ejpam-4793	733	9	=	=	SYM
ejpam-4793	733	10	ρs(x	ρs(x	X
ejpam-4793	733	11	)	)	PUNCT
ejpam-4793	733	12	and	and	CCONJ
ejpam-4793	733	13	ϕ((α+	ϕ((α+	DET
ejpam-4793	733	14	β	β	NOUN
ejpam-4793	733	15	,	,	PUNCT
ejpam-4793	733	16	ρs(x	ρs(x	PROPN
ejpam-4793	733	17	)	)	PUNCT
ejpam-4793	733	18	)	)	PUNCT
ejpam-4793	733	19	)	)	PUNCT
ejpam-4793	734	1	=	=	SYM
ejpam-4793	734	2	α+βρs(x	α+βρs(x	X
ejpam-4793	734	3	)	)	PUNCT
ejpam-4793	734	4	=	=	SYM
ejpam-4793	734	5	α(βρs(x	α(βρs(x	NOUN
ejpam-4793	734	6	)	)	PUNCT
ejpam-4793	734	7	)	)	PUNCT
ejpam-4793	735	1	=	=	SYM
ejpam-4793	735	2	ϕ((α	ϕ((α	PROPN
ejpam-4793	735	3	,	,	PUNCT
ejpam-4793	735	4	ϕ((β	ϕ((β	PROPN
ejpam-4793	735	5	,	,	PUNCT
ejpam-4793	735	6	ρs(x	ρs(x	PROPN
ejpam-4793	735	7	)	)	PUNCT
ejpam-4793	735	8	)	)	PUNCT
ejpam-4793	735	9	)	)	PUNCT
ejpam-4793	735	10	)	)	PUNCT
ejpam-4793	735	11	)	)	PUNCT
ejpam-4793	735	12	.	.	PUNCT
ejpam-4793	736	1	thus	thus	ADV
ejpam-4793	736	2	,	,	PUNCT
ejpam-4793	736	3	ϕ	ϕ	PROPN
ejpam-4793	736	4	is	be	AUX
ejpam-4793	736	5	an	an	DET
ejpam-4793	736	6	action	action	NOUN
ejpam-4793	736	7	.	.	PUNCT
ejpam-4793	737	1	now	now	ADV
ejpam-4793	737	2	,	,	PUNCT
ejpam-4793	737	3	let	let	VERB
ejpam-4793	737	4	α	α	PRON
ejpam-4793	737	5	∈	∈	PROPN
ejpam-4793	737	6	γ	γ	X
ejpam-4793	737	7	and	and	CCONJ
ejpam-4793	737	8	x	x	NOUN
ejpam-4793	737	9	,	,	PUNCT
ejpam-4793	737	10	y	y	PROPN
ejpam-4793	737	11	∈	∈	PROPN
ejpam-4793	737	12	m	m	VERB
ejpam-4793	737	13	.	.	PUNCT
ejpam-4793	738	1	then	then	ADV
ejpam-4793	738	2	ϕ((α	ϕ((α	PROPN
ejpam-4793	738	3	,	,	PUNCT
ejpam-4793	738	4	ρs(x	ρs(x	X
ejpam-4793	738	5	)	)	PUNCT
ejpam-4793	738	6	◦	◦	NOUN
ejpam-4793	738	7	ρs(y	ρs(y	NUM
ejpam-4793	738	8	)	)	PUNCT
ejpam-4793	738	9	)	)	PUNCT
ejpam-4793	738	10	)	)	PUNCT
ejpam-4793	739	1	=	=	PUNCT
ejpam-4793	739	2	α(ρs(x	α(ρs(x	X
ejpam-4793	739	3	)	)	PUNCT
ejpam-4793	739	4	◦	◦	NOUN
ejpam-4793	739	5	ρs(y	ρs(y	NUM
ejpam-4793	739	6	)	)	PUNCT
ejpam-4793	739	7	)	)	PUNCT
ejpam-4793	740	1	=	=	PUNCT
ejpam-4793	740	2	α(ρs(x	α(ρs(x	PROPN
ejpam-4793	740	3	∗	∗	NOUN
ejpam-4793	740	4	y	y	NOUN
ejpam-4793	740	5	)	)	PUNCT
ejpam-4793	740	6	)	)	PUNCT
ejpam-4793	741	1	h.	h.	PROPN
ejpam-4793	741	2	sarapuddin	sarapuddin	PROPN
ejpam-4793	741	3	,	,	PUNCT
ejpam-4793	741	4	j.	j.	PROPN
ejpam-4793	741	5	vilela	vilela	PROPN
ejpam-4793	741	6	/	/	SYM
ejpam-4793	741	7	eur	eur	PROPN
ejpam-4793	741	8	.	.	PUNCT
ejpam-4793	742	1	j.	j.	PROPN
ejpam-4793	742	2	pure	pure	PROPN
ejpam-4793	742	3	appl	appl	PROPN
ejpam-4793	742	4	.	.	PROPN
ejpam-4793	742	5	math	math	PROPN
ejpam-4793	742	6	,	,	PUNCT
ejpam-4793	742	7	16	16	NUM
ejpam-4793	742	8	(	(	PUNCT
ejpam-4793	742	9	3	3	NUM
ejpam-4793	742	10	)	)	PUNCT
ejpam-4793	742	11	(	(	PUNCT
ejpam-4793	742	12	2023	2023	NUM
ejpam-4793	742	13	)	)	PUNCT
ejpam-4793	742	14	,	,	PUNCT
ejpam-4793	742	15	1772	1772	NUM
ejpam-4793	742	16	-	-	SYM
ejpam-4793	742	17	1793	1793	NUM
ejpam-4793	742	18	1789	1789	NUM
ejpam-4793	742	19	=	=	SYM
ejpam-4793	742	20	ρs	ρs	PROPN
ejpam-4793	742	21	(	(	PUNCT
ejpam-4793	742	22	α(x	α(x	PROPN
ejpam-4793	742	23	∗	∗	NOUN
ejpam-4793	742	24	y	y	NOUN
ejpam-4793	742	25	)	)	PUNCT
ejpam-4793	742	26	)	)	PUNCT
ejpam-4793	743	1	=	=	SYM
ejpam-4793	743	2	ρs	ρs	PROPN
ejpam-4793	743	3	(	(	PUNCT
ejpam-4793	743	4	αx	αx	ADV
ejpam-4793	743	5	∗	∗	NOUN
ejpam-4793	743	6	αy	αy	NOUN
ejpam-4793	743	7	)	)	PUNCT
ejpam-4793	743	8	=	=	SYM
ejpam-4793	743	9	ρs	ρs	PROPN
ejpam-4793	743	10	(	(	PUNCT
ejpam-4793	743	11	αx	αx	NOUN
ejpam-4793	743	12	)	)	PUNCT
ejpam-4793	743	13	◦	◦	NOUN
ejpam-4793	743	14	ρs(αy	ρs(αy	NUM
ejpam-4793	743	15	)	)	PUNCT
ejpam-4793	743	16	=	=	PUNCT
ejpam-4793	744	1	αρs(x	αρs(x	X
ejpam-4793	744	2	)	)	PUNCT
ejpam-4793	744	3	◦	◦	NOUN
ejpam-4793	744	4	αρs(y	αρs(y	NUM
ejpam-4793	744	5	)	)	PUNCT
ejpam-4793	744	6	=	=	SYM
ejpam-4793	744	7	ϕ((α	ϕ((α	NOUN
ejpam-4793	744	8	,	,	PUNCT
ejpam-4793	744	9	ρs(x	ρs(x	PROPN
ejpam-4793	744	10	)	)	PUNCT
ejpam-4793	744	11	)	)	PUNCT
ejpam-4793	744	12	)	)	PUNCT
ejpam-4793	745	1	◦	◦	NOUN
ejpam-4793	745	2	ϕ((α	ϕ((α	NOUN
ejpam-4793	745	3	,	,	PUNCT
ejpam-4793	745	4	ρs(y	ρs(y	NUM
ejpam-4793	745	5	)	)	PUNCT
ejpam-4793	745	6	)	)	PUNCT
ejpam-4793	745	7	)	)	PUNCT
ejpam-4793	745	8	.	.	PUNCT
ejpam-4793	746	1	therefore	therefore	ADV
ejpam-4793	746	2	,	,	PUNCT
ejpam-4793	746	3	m	m	PROPN
ejpam-4793	746	4	/	/	SYM
ejpam-4793	746	5	s	s	PART
ejpam-4793	746	6	is	be	AUX
ejpam-4793	746	7	a	a	DET
ejpam-4793	746	8	γ	γ	X
ejpam-4793	746	9	-	-	PUNCT
ejpam-4793	746	10	monoid	monoid	NOUN
ejpam-4793	746	11	.	.	PUNCT
ejpam-4793	747	1	proposition	proposition	NOUN
ejpam-4793	747	2	4	4	NUM
ejpam-4793	747	3	.	.	PUNCT
ejpam-4793	748	1	let	let	VERB
ejpam-4793	748	2	s	s	PRON
ejpam-4793	748	3	be	be	AUX
ejpam-4793	748	4	a	a	DET
ejpam-4793	748	5	normal	normal	ADJ
ejpam-4793	748	6	γ	γ	NOUN
ejpam-4793	748	7	-	-	NOUN
ejpam-4793	748	8	submonoid	submonoid	NOUN
ejpam-4793	748	9	of	of	ADP
ejpam-4793	748	10	a	a	DET
ejpam-4793	748	11	commutative	commutative	ADJ
ejpam-4793	748	12	γ	γ	X
ejpam-4793	748	13	-	-	PUNCT
ejpam-4793	748	14	monoid	monoid	NOUN
ejpam-4793	748	15	m	m	PROPN
ejpam-4793	748	16	.	.	PUNCT
ejpam-4793	749	1	then	then	ADV
ejpam-4793	749	2	ρs(h	ρs(h	NUM
ejpam-4793	749	3	)	)	PUNCT
ejpam-4793	749	4	=	=	PUNCT
ejpam-4793	750	1	ρs(1	ρs(1	PROPN
ejpam-4793	750	2	m	m	PROPN
ejpam-4793	750	3	)	)	PUNCT
ejpam-4793	751	1	if	if	SCONJ
ejpam-4793	751	2	and	and	CCONJ
ejpam-4793	751	3	only	only	ADV
ejpam-4793	751	4	if	if	SCONJ
ejpam-4793	751	5	h	h	PROPN
ejpam-4793	751	6	∈	∈	PROPN
ejpam-4793	751	7	s.	s.	PROPN
ejpam-4793	751	8	proof	proof	PROPN
ejpam-4793	751	9	.	.	PUNCT
ejpam-4793	752	1	suppose	suppose	VERB
ejpam-4793	752	2	h	h	PROPN
ejpam-4793	752	3	∈	∈	PROPN
ejpam-4793	752	4	s.	s.	PROPN
ejpam-4793	752	5	let	let	VERB
ejpam-4793	752	6	x	x	SYM
ejpam-4793	752	7	∈	∈	PROPN
ejpam-4793	752	8	ρs(h	ρs(h	NUM
ejpam-4793	752	9	)	)	PUNCT
ejpam-4793	752	10	.	.	PUNCT
ejpam-4793	753	1	then	then	ADV
ejpam-4793	753	2	,	,	PUNCT
ejpam-4793	753	3	for	for	ADP
ejpam-4793	753	4	all	all	PRON
ejpam-4793	753	5	α	α	PRON
ejpam-4793	753	6	∈	∈	PROPN
ejpam-4793	753	7	γ	γ	X
ejpam-4793	753	8	,	,	PUNCT
ejpam-4793	753	9	(	(	PUNCT
ejpam-4793	753	10	αx∗s)∩(αh∗s	αx∗s)∩(αh∗s	PROPN
ejpam-4793	753	11	)	)	PUNCT
ejpam-4793	753	12	̸=	̸=	PROPN
ejpam-4793	753	13	∅.	∅.	ADP
ejpam-4793	753	14	this	this	PRON
ejpam-4793	753	15	implies	imply	VERB
ejpam-4793	753	16	that	that	SCONJ
ejpam-4793	753	17	there	there	PRON
ejpam-4793	753	18	exist	exist	VERB
ejpam-4793	753	19	h1	h1	PROPN
ejpam-4793	753	20	,	,	PUNCT
ejpam-4793	753	21	h2	h2	PROPN
ejpam-4793	753	22	∈	∈	PROPN
ejpam-4793	753	23	s	s	VERB
ejpam-4793	753	24	such	such	ADJ
ejpam-4793	753	25	that	that	SCONJ
ejpam-4793	753	26	αx	αx	PROPN
ejpam-4793	753	27	∗	∗	VERB
ejpam-4793	753	28	h1	h1	NOUN
ejpam-4793	753	29	=	=	SYM
ejpam-4793	753	30	αh	αh	PROPN
ejpam-4793	753	31	∗	∗	NOUN
ejpam-4793	753	32	h2	h2	PROPN
ejpam-4793	753	33	∈	∈	PROPN
ejpam-4793	753	34	s.	s.	PROPN
ejpam-4793	753	35	since	since	SCONJ
ejpam-4793	753	36	s	s	PROPN
ejpam-4793	753	37	is	be	AUX
ejpam-4793	753	38	a	a	DET
ejpam-4793	753	39	normal	normal	ADJ
ejpam-4793	753	40	γ	γ	NOUN
ejpam-4793	753	41	-	-	ADJ
ejpam-4793	753	42	submonoid	submonoid	ADJ
ejpam-4793	753	43	and	and	CCONJ
ejpam-4793	753	44	h1	h1	PROPN
ejpam-4793	753	45	,	,	PUNCT
ejpam-4793	753	46	αx	αx	ADV
ejpam-4793	753	47	∗	∗	VERB
ejpam-4793	753	48	h1	h1	PROPN
ejpam-4793	753	49	∈	∈	PROPN
ejpam-4793	753	50	s	s	PART
ejpam-4793	753	51	,	,	PUNCT
ejpam-4793	753	52	it	it	PRON
ejpam-4793	753	53	follows	follow	VERB
ejpam-4793	753	54	that	that	SCONJ
ejpam-4793	753	55	αx	αx	PRON
ejpam-4793	753	56	∈	∈	PROPN
ejpam-4793	753	57	s	s	X
ejpam-4793	753	58	for	for	ADP
ejpam-4793	753	59	all	all	PRON
ejpam-4793	753	60	α	α	PRON
ejpam-4793	753	61	∈	∈	PROPN
ejpam-4793	753	62	γ	γ	X
ejpam-4793	753	63	.	.	PROPN
ejpam-4793	753	64	accordingly	accordingly	ADV
ejpam-4793	753	65	,	,	PUNCT
ejpam-4793	753	66	for	for	SCONJ
ejpam-4793	753	67	all	all	PRON
ejpam-4793	753	68	α	α	PRON
ejpam-4793	753	69	∈	∈	PROPN
ejpam-4793	753	70	γ	γ	X
ejpam-4793	753	71	,	,	PUNCT
ejpam-4793	753	72	αx∗α1	αx∗α1	NUM
ejpam-4793	753	73	m	m	NOUN
ejpam-4793	753	74	=	=	ADJ
ejpam-4793	753	75	α1m∗αx	α1m∗αx	NOUN
ejpam-4793	753	76	implies	imply	VERB
ejpam-4793	753	77	(	(	PUNCT
ejpam-4793	753	78	αx∗s)∩(α1m∗s	αx∗s)∩(α1m∗s	NOUN
ejpam-4793	753	79	)	)	PUNCT
ejpam-4793	753	80	̸=	̸=	PROPN
ejpam-4793	753	81	∅.	∅.	PRON
ejpam-4793	753	82	hence	hence	ADV
ejpam-4793	753	83	,	,	PUNCT
ejpam-4793	753	84	xρs1	xρs1	PROPN
ejpam-4793	753	85	m	m	PROPN
ejpam-4793	753	86	and	and	CCONJ
ejpam-4793	753	87	x	x	PROPN
ejpam-4793	753	88	∈	∈	PROPN
ejpam-4793	753	89	ρs(1	ρs(1	PROPN
ejpam-4793	753	90	m	m	PROPN
ejpam-4793	753	91	)	)	PUNCT
ejpam-4793	753	92	.	.	PUNCT
ejpam-4793	754	1	it	it	PRON
ejpam-4793	754	2	follows	follow	VERB
ejpam-4793	754	3	that	that	SCONJ
ejpam-4793	754	4	ρs(h	ρs(h	NUM
ejpam-4793	754	5	)	)	PUNCT
ejpam-4793	754	6	⊆	⊆	NUM
ejpam-4793	754	7	ρs(1	ρs(1	PROPN
ejpam-4793	754	8	m	m	PROPN
ejpam-4793	754	9	)	)	PUNCT
ejpam-4793	754	10	.	.	PUNCT
ejpam-4793	755	1	let	let	VERB
ejpam-4793	755	2	x	x	SYM
ejpam-4793	755	3	∈	∈	PROPN
ejpam-4793	755	4	ρs(1	ρs(1	PROPN
ejpam-4793	755	5	m	m	PROPN
ejpam-4793	755	6	)	)	PUNCT
ejpam-4793	755	7	.	.	PUNCT
ejpam-4793	756	1	then	then	ADV
ejpam-4793	756	2	,	,	PUNCT
ejpam-4793	756	3	(	(	PUNCT
ejpam-4793	756	4	αx	αx	ADV
ejpam-4793	756	5	∗	∗	PROPN
ejpam-4793	756	6	s	s	PART
ejpam-4793	756	7	)	)	PUNCT
ejpam-4793	756	8	∩	∩	NOUN
ejpam-4793	756	9	(	(	PUNCT
ejpam-4793	756	10	α1	α1	PROPN
ejpam-4793	756	11	m	m	PROPN
ejpam-4793	756	12	∗	∗	NOUN
ejpam-4793	756	13	s	s	PART
ejpam-4793	756	14	)	)	PUNCT
ejpam-4793	756	15	̸=	̸=	NOUN
ejpam-4793	756	16	∅	∅	NOUN
ejpam-4793	756	17	for	for	ADP
ejpam-4793	756	18	all	all	PRON
ejpam-4793	756	19	α	α	PRON
ejpam-4793	756	20	∈	∈	PROPN
ejpam-4793	756	21	γ	γ	NOUN
ejpam-4793	756	22	.	.	PUNCT
ejpam-4793	757	1	thus	thus	ADV
ejpam-4793	757	2	,	,	PUNCT
ejpam-4793	757	3	there	there	PRON
ejpam-4793	757	4	exist	exist	VERB
ejpam-4793	757	5	h1	h1	PROPN
ejpam-4793	757	6	,	,	PUNCT
ejpam-4793	757	7	h2	h2	PROPN
ejpam-4793	757	8	∈	∈	PROPN
ejpam-4793	757	9	s	s	VERB
ejpam-4793	757	10	such	such	ADJ
ejpam-4793	757	11	that	that	PRON
ejpam-4793	757	12	for	for	ADP
ejpam-4793	757	13	all	all	PRON
ejpam-4793	757	14	α	α	PRON
ejpam-4793	757	15	∈	∈	PROPN
ejpam-4793	757	16	γ	γ	X
ejpam-4793	757	17	,	,	PUNCT
ejpam-4793	757	18	αx	αx	ADV
ejpam-4793	757	19	∗	∗	NOUN
ejpam-4793	757	20	h1	h1	PROPN
ejpam-4793	757	21	=	=	SYM
ejpam-4793	757	22	α1	α1	PROPN
ejpam-4793	757	23	m	m	PROPN
ejpam-4793	757	24	∗	∗	NOUN
ejpam-4793	757	25	h2	h2	PROPN
ejpam-4793	757	26	∈	∈	PROPN
ejpam-4793	757	27	s.	s.	PROPN
ejpam-4793	757	28	since	since	SCONJ
ejpam-4793	757	29	s	s	PROPN
ejpam-4793	757	30	is	be	AUX
ejpam-4793	757	31	a	a	DET
ejpam-4793	757	32	normal	normal	ADJ
ejpam-4793	757	33	γ	γ	NOUN
ejpam-4793	757	34	-	-	ADJ
ejpam-4793	757	35	submonoid	submonoid	ADJ
ejpam-4793	757	36	and	and	CCONJ
ejpam-4793	757	37	h1	h1	PROPN
ejpam-4793	757	38	,	,	PUNCT
ejpam-4793	757	39	αx∗h1	αx∗h1	NUM
ejpam-4793	757	40	∈	∈	PROPN
ejpam-4793	757	41	s	s	PART
ejpam-4793	757	42	,	,	PUNCT
ejpam-4793	757	43	it	it	PRON
ejpam-4793	757	44	follows	follow	VERB
ejpam-4793	757	45	that	that	SCONJ
ejpam-4793	757	46	αx	αx	PROPN
ejpam-4793	757	47	∈	∈	PROPN
ejpam-4793	757	48	s.	s.	PROPN
ejpam-4793	757	49	observe	observe	VERB
ejpam-4793	757	50	that	that	SCONJ
ejpam-4793	757	51	for	for	ADP
ejpam-4793	757	52	all	all	PRON
ejpam-4793	757	53	α	α	PRON
ejpam-4793	757	54	∈	∈	PROPN
ejpam-4793	757	55	γ	γ	X
ejpam-4793	757	56	,	,	PUNCT
ejpam-4793	757	57	αh	αh	NOUN
ejpam-4793	757	58	=	=	SYM
ejpam-4793	757	59	αh∗1	αh∗1	PROPN
ejpam-4793	757	60	m	m	NOUN
ejpam-4793	757	61	∈	∈	NOUN
ejpam-4793	757	62	s	s	PRON
ejpam-4793	757	63	since	since	SCONJ
ejpam-4793	757	64	s	s	PROPN
ejpam-4793	757	65	is	be	AUX
ejpam-4793	757	66	a	a	DET
ejpam-4793	757	67	γ	γ	NOUN
ejpam-4793	757	68	-	-	PUNCT
ejpam-4793	757	69	submonoid	submonoid	ADJ
ejpam-4793	757	70	.	.	PUNCT
ejpam-4793	758	1	accordingly	accordingly	ADV
ejpam-4793	758	2	,	,	PUNCT
ejpam-4793	758	3	αx∗αh	αx∗αh	PROPN
ejpam-4793	758	4	=	=	SYM
ejpam-4793	758	5	αh∗αx	αh∗αx	PROPN
ejpam-4793	758	6	implies	imply	VERB
ejpam-4793	758	7	(	(	PUNCT
ejpam-4793	758	8	αx	αx	ADV
ejpam-4793	758	9	∗	∗	PROPN
ejpam-4793	758	10	s	s	PART
ejpam-4793	758	11	)	)	PUNCT
ejpam-4793	758	12	∩	∩	NOUN
ejpam-4793	758	13	(	(	PUNCT
ejpam-4793	758	14	αh	αh	NOUN
ejpam-4793	758	15	∗	∗	NOUN
ejpam-4793	758	16	s	s	PART
ejpam-4793	758	17	)	)	PUNCT
ejpam-4793	758	18	̸=	̸=	PROPN
ejpam-4793	758	19	∅.	∅.	PRON
ejpam-4793	758	20	hence	hence	ADV
ejpam-4793	758	21	,	,	PUNCT
ejpam-4793	758	22	xρsh	xρsh	PROPN
ejpam-4793	758	23	and	and	CCONJ
ejpam-4793	758	24	x	x	PUNCT
ejpam-4793	758	25	∈	∈	PROPN
ejpam-4793	758	26	ρs(h	ρs(h	NUM
ejpam-4793	758	27	)	)	PUNCT
ejpam-4793	758	28	.	.	PUNCT
ejpam-4793	759	1	consequently	consequently	ADV
ejpam-4793	759	2	,	,	PUNCT
ejpam-4793	759	3	ρs(1	ρs(1	PROPN
ejpam-4793	759	4	m	m	PROPN
ejpam-4793	759	5	)	)	PUNCT
ejpam-4793	759	6	⊆	⊆	NUM
ejpam-4793	759	7	ρs(h	ρs(h	NUM
ejpam-4793	759	8	)	)	PUNCT
ejpam-4793	759	9	.	.	PUNCT
ejpam-4793	760	1	therefore	therefore	ADV
ejpam-4793	760	2	,	,	PUNCT
ejpam-4793	760	3	ρs(1	ρs(1	PROPN
ejpam-4793	760	4	m	m	PROPN
ejpam-4793	760	5	)	)	PUNCT
ejpam-4793	760	6	=	=	SYM
ejpam-4793	760	7	ρs(h	ρs(h	NUM
ejpam-4793	760	8	)	)	PUNCT
ejpam-4793	760	9	.	.	PUNCT
ejpam-4793	761	1	now	now	ADV
ejpam-4793	761	2	,	,	PUNCT
ejpam-4793	761	3	suppose	suppose	VERB
ejpam-4793	761	4	ρs(1	ρs(1	NOUN
ejpam-4793	761	5	m	m	PROPN
ejpam-4793	761	6	)	)	PUNCT
ejpam-4793	762	1	=	=	SYM
ejpam-4793	762	2	ρs(h	ρs(h	NUM
ejpam-4793	762	3	)	)	PUNCT
ejpam-4793	762	4	.	.	PUNCT
ejpam-4793	763	1	then	then	ADV
ejpam-4793	763	2	,	,	PUNCT
ejpam-4793	763	3	by	by	ADP
ejpam-4793	763	4	remark	remark	NOUN
ejpam-4793	763	5	14(ii	14(ii	NUM
ejpam-4793	763	6	)	)	PUNCT
ejpam-4793	763	7	,	,	PUNCT
ejpam-4793	763	8	(	(	PUNCT
ejpam-4793	763	9	α1	α1	PROPN
ejpam-4793	763	10	m	m	PROPN
ejpam-4793	763	11	∗	∗	NOUN
ejpam-4793	763	12	s	s	NOUN
ejpam-4793	763	13	)	)	PUNCT
ejpam-4793	763	14	∩	∩	NOUN
ejpam-4793	763	15	(	(	PUNCT
ejpam-4793	763	16	αh	αh	NOUN
ejpam-4793	763	17	∗	∗	PRON
ejpam-4793	763	18	s	s	PART
ejpam-4793	763	19	)	)	PUNCT
ejpam-4793	763	20	̸=	̸=	NOUN
ejpam-4793	763	21	∅	∅	NOUN
ejpam-4793	763	22	for	for	ADP
ejpam-4793	763	23	all	all	PRON
ejpam-4793	763	24	α	α	PRON
ejpam-4793	763	25	∈	∈	PROPN
ejpam-4793	763	26	γ	γ	NOUN
ejpam-4793	763	27	.	.	PUNCT
ejpam-4793	763	28	thus	thus	ADV
ejpam-4793	763	29	,	,	PUNCT
ejpam-4793	763	30	there	there	PRON
ejpam-4793	763	31	exist	exist	VERB
ejpam-4793	763	32	h1	h1	PROPN
ejpam-4793	763	33	,	,	PUNCT
ejpam-4793	763	34	h2	h2	PROPN
ejpam-4793	763	35	∈	∈	PROPN
ejpam-4793	763	36	s	s	VERB
ejpam-4793	764	1	such	such	ADJ
ejpam-4793	764	2	that	that	SCONJ
ejpam-4793	764	3	αh	αh	NOUN
ejpam-4793	764	4	∗	∗	NOUN
ejpam-4793	764	5	h2	h2	NOUN
ejpam-4793	764	6	=	=	SYM
ejpam-4793	764	7	α1	α1	PROPN
ejpam-4793	764	8	m	m	PROPN
ejpam-4793	764	9	∗	∗	NOUN
ejpam-4793	764	10	h1	h1	PROPN
ejpam-4793	764	11	∈	∈	PROPN
ejpam-4793	764	12	s	s	NOUN
ejpam-4793	764	13	for	for	ADP
ejpam-4793	764	14	all	all	PRON
ejpam-4793	764	15	α	α	PRON
ejpam-4793	764	16	∈	∈	PROPN
ejpam-4793	764	17	γ	γ	X
ejpam-4793	764	18	.	.	PUNCT
ejpam-4793	765	1	since	since	SCONJ
ejpam-4793	765	2	s	s	PROPN
ejpam-4793	765	3	is	be	AUX
ejpam-4793	765	4	a	a	DET
ejpam-4793	765	5	normal	normal	ADJ
ejpam-4793	765	6	γ	γ	NOUN
ejpam-4793	765	7	-	-	ADJ
ejpam-4793	765	8	submonoid	submonoid	ADJ
ejpam-4793	765	9	and	and	CCONJ
ejpam-4793	765	10	h2	h2	NOUN
ejpam-4793	765	11	,	,	PUNCT
ejpam-4793	765	12	αh	αh	PROPN
ejpam-4793	765	13	∗	∗	NOUN
ejpam-4793	765	14	h2	h2	PROPN
ejpam-4793	765	15	∈	∈	PROPN
ejpam-4793	765	16	s	s	PART
ejpam-4793	765	17	,	,	PUNCT
ejpam-4793	765	18	it	it	PRON
ejpam-4793	765	19	follows	follow	VERB
ejpam-4793	765	20	that	that	SCONJ
ejpam-4793	765	21	αh	αh	PROPN
ejpam-4793	765	22	∈	∈	PROPN
ejpam-4793	765	23	s	s	NOUN
ejpam-4793	765	24	for	for	ADP
ejpam-4793	765	25	all	all	PRON
ejpam-4793	765	26	α	α	DET
ejpam-4793	765	27	∈	∈	PROPN
ejpam-4793	765	28	γ	γ	X
ejpam-4793	765	29	.	.	PROPN
ejpam-4793	765	30	therefore	therefore	ADV
ejpam-4793	765	31	,	,	PUNCT
ejpam-4793	765	32	h	h	PROPN
ejpam-4793	765	33	∈	∈	PROPN
ejpam-4793	765	34	s.	s.	PROPN
ejpam-4793	765	35	proposition	proposition	NOUN
ejpam-4793	765	36	5	5	NUM
ejpam-4793	765	37	.	.	PUNCT
ejpam-4793	766	1	let	let	VERB
ejpam-4793	766	2	s	s	PRON
ejpam-4793	766	3	be	be	AUX
ejpam-4793	766	4	a	a	DET
ejpam-4793	766	5	normal	normal	ADJ
ejpam-4793	766	6	γ	γ	NOUN
ejpam-4793	766	7	-	-	NOUN
ejpam-4793	766	8	submonoid	submonoid	NOUN
ejpam-4793	766	9	of	of	ADP
ejpam-4793	766	10	a	a	DET
ejpam-4793	766	11	commutative	commutative	ADJ
ejpam-4793	766	12	γ	γ	X
ejpam-4793	766	13	-	-	PUNCT
ejpam-4793	766	14	monoid	monoid	NOUN
ejpam-4793	766	15	m	m	PROPN
ejpam-4793	766	16	.	.	PUNCT
ejpam-4793	767	1	then	then	ADV
ejpam-4793	767	2	m	m	VERB
ejpam-4793	767	3	=	=	SYM
ejpam-4793	767	4	s	s	PRON
ejpam-4793	768	1	if	if	SCONJ
ejpam-4793	769	1	and	and	CCONJ
ejpam-4793	769	2	only	only	ADV
ejpam-4793	769	3	if	if	SCONJ
ejpam-4793	769	4	m	m	PROPN
ejpam-4793	769	5	/	/	SYM
ejpam-4793	769	6	s	s	NOUN
ejpam-4793	769	7	=	=	X
ejpam-4793	769	8	{	{	PUNCT
ejpam-4793	769	9	ρs(1	ρs(1	NOUN
ejpam-4793	769	10	m	m	PROPN
ejpam-4793	769	11	)	)	PUNCT
ejpam-4793	769	12	}	}	PUNCT
ejpam-4793	769	13	.	.	PUNCT
ejpam-4793	770	1	proof	proof	NOUN
ejpam-4793	770	2	.	.	PUNCT
ejpam-4793	771	1	suppose	suppose	VERB
ejpam-4793	771	2	m	m	VERB
ejpam-4793	771	3	=	=	PUNCT
ejpam-4793	771	4	s.	s.	PROPN
ejpam-4793	771	5	let	let	VERB
ejpam-4793	771	6	x	x	X
ejpam-4793	771	7	∈	∈	PROPN
ejpam-4793	771	8	m	m	PROPN
ejpam-4793	771	9	/	/	SYM
ejpam-4793	771	10	s	s	PART
ejpam-4793	771	11	=	=	NOUN
ejpam-4793	771	12	m	m	PROPN
ejpam-4793	771	13	/	/	SYM
ejpam-4793	771	14	m	m	PROPN
ejpam-4793	771	15	.	.	PUNCT
ejpam-4793	772	1	then	then	ADV
ejpam-4793	772	2	x	x	X
ejpam-4793	772	3	=	=	NOUN
ejpam-4793	772	4	ρm	ρm	PROPN
ejpam-4793	772	5	(	(	PUNCT
ejpam-4793	772	6	y	y	NOUN
ejpam-4793	772	7	)	)	PUNCT
ejpam-4793	772	8	for	for	ADP
ejpam-4793	772	9	some	some	DET
ejpam-4793	772	10	y	y	PROPN
ejpam-4793	772	11	∈	∈	PROPN
ejpam-4793	772	12	m	m	VERB
ejpam-4793	772	13	.	.	PUNCT
ejpam-4793	773	1	by	by	ADP
ejpam-4793	773	2	proposition	proposition	NOUN
ejpam-4793	773	3	4	4	NUM
ejpam-4793	773	4	,	,	PUNCT
ejpam-4793	773	5	we	we	PRON
ejpam-4793	773	6	have	have	VERB
ejpam-4793	773	7	ρm	ρm	NUM
ejpam-4793	773	8	(	(	PUNCT
ejpam-4793	773	9	1	1	NUM
ejpam-4793	773	10	m	m	NOUN
ejpam-4793	773	11	)	)	PUNCT
ejpam-4793	774	1	=	=	SYM
ejpam-4793	774	2	ρm	ρm	INTJ
ejpam-4793	774	3	(	(	PUNCT
ejpam-4793	774	4	y	y	NOUN
ejpam-4793	774	5	)	)	PUNCT
ejpam-4793	774	6	=	=	PUNCT
ejpam-4793	775	1	x.	x.	NOUN
ejpam-4793	775	2	hence	hence	ADV
ejpam-4793	775	3	,	,	PUNCT
ejpam-4793	775	4	m	m	PROPN
ejpam-4793	775	5	/	/	SYM
ejpam-4793	775	6	m	m	PROPN
ejpam-4793	775	7	=	=	NOUN
ejpam-4793	775	8	m	m	PROPN
ejpam-4793	775	9	/	/	SYM
ejpam-4793	775	10	s	s	NOUN
ejpam-4793	775	11	=	=	PUNCT
ejpam-4793	775	12	{	{	PUNCT
ejpam-4793	775	13	ρm	ρm	X
ejpam-4793	775	14	(	(	PUNCT
ejpam-4793	775	15	1	1	NUM
ejpam-4793	775	16	m	m	NOUN
ejpam-4793	775	17	)	)	PUNCT
ejpam-4793	775	18	}	}	PUNCT
ejpam-4793	775	19	.	.	PUNCT
ejpam-4793	776	1	conversely	conversely	ADV
ejpam-4793	776	2	,	,	PUNCT
ejpam-4793	776	3	suppose	suppose	VERB
ejpam-4793	776	4	m	m	PRON
ejpam-4793	776	5	/	/	SYM
ejpam-4793	776	6	s	s	PART
ejpam-4793	776	7	=	=	X
ejpam-4793	776	8	{	{	PUNCT
ejpam-4793	776	9	ρs(1	ρs(1	NOUN
ejpam-4793	776	10	m	m	PROPN
ejpam-4793	776	11	)	)	PUNCT
ejpam-4793	776	12	}	}	PUNCT
ejpam-4793	776	13	.	.	PUNCT
ejpam-4793	777	1	let	let	VERB
ejpam-4793	777	2	x	x	PUNCT
ejpam-4793	777	3	∈	∈	NOUN
ejpam-4793	777	4	m	m	VERB
ejpam-4793	777	5	.	.	PUNCT
ejpam-4793	778	1	then	then	ADV
ejpam-4793	778	2	ρs(x	ρs(x	NOUN
ejpam-4793	778	3	)	)	PUNCT
ejpam-4793	779	1	∈	∈	PROPN
ejpam-4793	779	2	m	m	PROPN
ejpam-4793	779	3	/	/	SYM
ejpam-4793	779	4	s.	s.	PROPN
ejpam-4793	779	5	thus	thus	ADV
ejpam-4793	779	6	,	,	PUNCT
ejpam-4793	779	7	ρs(x	ρs(x	X
ejpam-4793	779	8	)	)	PUNCT
ejpam-4793	779	9	=	=	PUNCT
ejpam-4793	779	10	ρs(1	ρs(1	PROPN
ejpam-4793	779	11	m	m	PROPN
ejpam-4793	779	12	)	)	PUNCT
ejpam-4793	779	13	.	.	PUNCT
ejpam-4793	780	1	by	by	ADP
ejpam-4793	780	2	proposition	proposition	NOUN
ejpam-4793	780	3	4	4	NUM
ejpam-4793	780	4	,	,	PUNCT
ejpam-4793	780	5	x	x	SYM
ejpam-4793	780	6	∈	∈	PROPN
ejpam-4793	780	7	s.	s.	PROPN
ejpam-4793	780	8	hence	hence	ADV
ejpam-4793	780	9	,	,	PUNCT
ejpam-4793	780	10	m	m	PROPN
ejpam-4793	780	11	⊆	⊆	NUM
ejpam-4793	780	12	s.	s.	PROPN
ejpam-4793	780	13	accordingly	accordingly	ADV
ejpam-4793	780	14	,	,	PUNCT
ejpam-4793	780	15	m	m	VERB
ejpam-4793	780	16	=	=	ADJ
ejpam-4793	780	17	s.	s.	PROPN
ejpam-4793	780	18	proposition	proposition	NOUN
ejpam-4793	780	19	6	6	NUM
ejpam-4793	780	20	.	.	PUNCT
ejpam-4793	781	1	let	let	VERB
ejpam-4793	781	2	s	s	PRON
ejpam-4793	781	3	be	be	AUX
ejpam-4793	781	4	a	a	DET
ejpam-4793	781	5	normal	normal	ADJ
ejpam-4793	781	6	γ	γ	NOUN
ejpam-4793	781	7	-	-	NOUN
ejpam-4793	781	8	submonoid	submonoid	NOUN
ejpam-4793	781	9	of	of	ADP
ejpam-4793	781	10	a	a	DET
ejpam-4793	781	11	commutative	commutative	ADJ
ejpam-4793	781	12	γ	γ	X
ejpam-4793	781	13	-	-	PUNCT
ejpam-4793	781	14	monoid	monoid	NOUN
ejpam-4793	781	15	m	m	NOUN
ejpam-4793	781	16	.	.	PUNCT
ejpam-4793	782	1	every	every	DET
ejpam-4793	782	2	γ	γ	PROPN
ejpam-4793	782	3	-	-	ADJ
ejpam-4793	782	4	submonoid	submonoid	NOUN
ejpam-4793	782	5	of	of	ADP
ejpam-4793	782	6	m	m	PROPN
ejpam-4793	782	7	/	/	SYM
ejpam-4793	782	8	s	s	PART
ejpam-4793	782	9	is	be	AUX
ejpam-4793	782	10	of	of	ADP
ejpam-4793	782	11	the	the	DET
ejpam-4793	782	12	form	form	NOUN
ejpam-4793	782	13	r	r	NOUN
ejpam-4793	782	14	/	/	SYM
ejpam-4793	782	15	s	s	NOUN
ejpam-4793	782	16	,	,	PUNCT
ejpam-4793	782	17	where	where	SCONJ
ejpam-4793	782	18	r	r	NOUN
ejpam-4793	782	19	is	be	AUX
ejpam-4793	782	20	a	a	DET
ejpam-4793	782	21	γ	γ	NOUN
ejpam-4793	782	22	-	-	ADJ
ejpam-4793	782	23	submonoid	submonoid	NOUN
ejpam-4793	782	24	of	of	ADP
ejpam-4793	782	25	m	m	AUX
ejpam-4793	782	26	containing	contain	VERB
ejpam-4793	782	27	s.	s.	PROPN
ejpam-4793	782	28	proof	proof	PROPN
ejpam-4793	782	29	.	.	PUNCT
ejpam-4793	783	1	let	let	VERB
ejpam-4793	783	2	h	h	PRON
ejpam-4793	783	3	be	be	AUX
ejpam-4793	783	4	a	a	DET
ejpam-4793	783	5	γ	γ	NOUN
ejpam-4793	783	6	-	-	ADJ
ejpam-4793	783	7	submonoid	submonoid	NOUN
ejpam-4793	783	8	of	of	ADP
ejpam-4793	783	9	m	m	PROPN
ejpam-4793	783	10	/	/	SYM
ejpam-4793	783	11	s.	s.	PROPN
ejpam-4793	783	12	then	then	ADV
ejpam-4793	783	13	h	h	VERB
ejpam-4793	783	14	⊆	⊆	NUM
ejpam-4793	783	15	m	m	PROPN
ejpam-4793	783	16	/	/	SYM
ejpam-4793	783	17	s.	s.	PROPN
ejpam-4793	783	18	let	let	VERB
ejpam-4793	784	1	r	r	NOUN
ejpam-4793	784	2	=	=	PRON
ejpam-4793	784	3	{	{	PUNCT
ejpam-4793	784	4	m	m	NOUN
ejpam-4793	784	5	∈	∈	ADJ
ejpam-4793	784	6	m	m	NOUN
ejpam-4793	784	7	:	:	PUNCT
ejpam-4793	784	8	ρs(m	ρs(m	NUM
ejpam-4793	784	9	)	)	PUNCT
ejpam-4793	784	10	∈	∈	PROPN
ejpam-4793	784	11	h	h	NOUN
ejpam-4793	784	12	}	}	PUNCT
ejpam-4793	784	13	.	.	PUNCT
ejpam-4793	785	1	we	we	PRON
ejpam-4793	785	2	show	show	VERB
ejpam-4793	785	3	that	that	SCONJ
ejpam-4793	785	4	r	r	NOUN
ejpam-4793	785	5	is	be	AUX
ejpam-4793	785	6	a	a	DET
ejpam-4793	785	7	γ	γ	NOUN
ejpam-4793	785	8	-	-	ADJ
ejpam-4793	785	9	submonoid	submonoid	NOUN
ejpam-4793	785	10	of	of	ADP
ejpam-4793	785	11	m	m	PROPN
ejpam-4793	785	12	.	.	PUNCT
ejpam-4793	786	1	note	note	VERB
ejpam-4793	786	2	that	that	SCONJ
ejpam-4793	786	3	the	the	DET
ejpam-4793	786	4	identity	identity	NOUN
ejpam-4793	786	5	in	in	ADP
ejpam-4793	786	6	m	m	PROPN
ejpam-4793	786	7	/	/	SYM
ejpam-4793	786	8	s	s	PART
ejpam-4793	786	9	is	be	AUX
ejpam-4793	786	10	ρs(1	ρs(1	PROPN
ejpam-4793	786	11	m	m	NOUN
ejpam-4793	786	12	)	)	PUNCT
ejpam-4793	786	13	∈	∈	PROPN
ejpam-4793	786	14	h	h	NOUN
ejpam-4793	786	15	and	and	CCONJ
ejpam-4793	786	16	thus	thus	ADV
ejpam-4793	786	17	,	,	PUNCT
ejpam-4793	786	18	1	1	NUM
ejpam-4793	786	19	m	m	NOUN
ejpam-4793	786	20	∈	∈	PROPN
ejpam-4793	786	21	r.	r.	PROPN
ejpam-4793	786	22	now	now	ADV
ejpam-4793	786	23	,	,	PUNCT
ejpam-4793	786	24	let	let	VERB
ejpam-4793	786	25	x	x	PRON
ejpam-4793	786	26	,	,	PUNCT
ejpam-4793	786	27	y	y	PROPN
ejpam-4793	786	28	∈	∈	PROPN
ejpam-4793	786	29	r	r	NOUN
ejpam-4793	786	30	and	and	CCONJ
ejpam-4793	786	31	α	α	NOUN
ejpam-4793	786	32	,	,	PUNCT
ejpam-4793	786	33	β	β	PROPN
ejpam-4793	786	34	∈	∈	PROPN
ejpam-4793	786	35	γ	γ	X
ejpam-4793	786	36	.	.	PROPN
ejpam-4793	786	37	then	then	ADV
ejpam-4793	786	38	ρs(x	ρs(x	NOUN
ejpam-4793	786	39	)	)	PUNCT
ejpam-4793	786	40	,	,	PUNCT
ejpam-4793	786	41	ρs(y	ρs(y	NUM
ejpam-4793	786	42	)	)	PUNCT
ejpam-4793	786	43	∈	∈	PROPN
ejpam-4793	786	44	h	h	NOUN
ejpam-4793	786	45	and	and	CCONJ
ejpam-4793	786	46	αρs(x	αρs(x	NUM
ejpam-4793	786	47	)	)	PUNCT
ejpam-4793	786	48	∗	∗	NOUN
ejpam-4793	786	49	βρs(y	βρs(y	SYM
ejpam-4793	786	50	)	)	PUNCT
ejpam-4793	786	51	∈	∈	PROPN
ejpam-4793	786	52	h	h	NOUN
ejpam-4793	786	53	since	since	SCONJ
ejpam-4793	786	54	h	h	NOUN
ejpam-4793	786	55	is	be	AUX
ejpam-4793	786	56	a	a	DET
ejpam-4793	786	57	γ	γ	NOUN
ejpam-4793	786	58	-	-	PUNCT
ejpam-4793	786	59	submonoid	submonoid	ADJ
ejpam-4793	786	60	.	.	PUNCT
ejpam-4793	787	1	accordingly	accordingly	ADV
ejpam-4793	787	2	,	,	PUNCT
ejpam-4793	787	3	we	we	PRON
ejpam-4793	787	4	have	have	AUX
ejpam-4793	787	5	ρs	ρs	ADV
ejpam-4793	787	6	(	(	PUNCT
ejpam-4793	787	7	αx	αx	ADV
ejpam-4793	787	8	∗	∗	NOUN
ejpam-4793	787	9	βy	βy	PROPN
ejpam-4793	787	10	)	)	PUNCT
ejpam-4793	787	11	=	=	SYM
ejpam-4793	787	12	ρs	ρs	PROPN
ejpam-4793	787	13	(	(	PUNCT
ejpam-4793	787	14	αx	αx	NOUN
ejpam-4793	787	15	)	)	PUNCT
ejpam-4793	787	16	◦	◦	NOUN
ejpam-4793	787	17	ρs(βy	ρs(βy	NOUN
ejpam-4793	787	18	)	)	PUNCT
ejpam-4793	787	19	=	=	PUNCT
ejpam-4793	788	1	αρs(x	αρs(x	X
ejpam-4793	788	2	)	)	PUNCT
ejpam-4793	788	3	◦	◦	NOUN
ejpam-4793	788	4	βρs(y	βρs(y	SYM
ejpam-4793	788	5	)	)	PUNCT
ejpam-4793	788	6	∈	∈	PROPN
ejpam-4793	788	7	h.	h.	NOUN
ejpam-4793	789	1	it	it	PRON
ejpam-4793	789	2	follows	follow	VERB
ejpam-4793	789	3	that	that	SCONJ
ejpam-4793	789	4	αx	αx	ADV
ejpam-4793	789	5	∗	∗	VERB
ejpam-4793	789	6	βy	βy	PRON
ejpam-4793	790	1	∈	∈	PROPN
ejpam-4793	790	2	r.	r.	PROPN
ejpam-4793	790	3	accordingly	accordingly	ADV
ejpam-4793	790	4	,	,	PUNCT
ejpam-4793	790	5	r	r	NOUN
ejpam-4793	790	6	is	be	AUX
ejpam-4793	790	7	a	a	DET
ejpam-4793	790	8	γ	γ	NOUN
ejpam-4793	790	9	-	-	ADJ
ejpam-4793	790	10	submonoid	submonoid	NOUN
ejpam-4793	790	11	of	of	ADP
ejpam-4793	790	12	m	m	PROPN
ejpam-4793	790	13	.	.	PUNCT
ejpam-4793	791	1	now	now	ADV
ejpam-4793	791	2	,	,	PUNCT
ejpam-4793	791	3	we	we	PRON
ejpam-4793	791	4	show	show	VERB
ejpam-4793	791	5	that	that	SCONJ
ejpam-4793	791	6	s	s	VERB
ejpam-4793	791	7	⊆	⊆	PROPN
ejpam-4793	791	8	r.	r.	NOUN
ejpam-4793	791	9	let	let	VERB
ejpam-4793	791	10	x	x	PROPN
ejpam-4793	791	11	∈	∈	PROPN
ejpam-4793	791	12	s.	s.	PROPN
ejpam-4793	791	13	then	then	ADV
ejpam-4793	791	14	by	by	ADP
ejpam-4793	791	15	proposition	proposition	NOUN
ejpam-4793	791	16	4	4	NUM
ejpam-4793	791	17	,	,	PUNCT
ejpam-4793	791	18	we	we	PRON
ejpam-4793	791	19	have	have	VERB
ejpam-4793	791	20	ρs(x	ρs(x	NOUN
ejpam-4793	791	21	)	)	PUNCT
ejpam-4793	791	22	=	=	PUNCT
ejpam-4793	792	1	ρs(1	ρs(1	PROPN
ejpam-4793	792	2	m	m	PROPN
ejpam-4793	792	3	)	)	PUNCT
ejpam-4793	792	4	.	.	PUNCT
ejpam-4793	793	1	since	since	SCONJ
ejpam-4793	793	2	ρs(1	ρs(1	PROPN
ejpam-4793	793	3	m	m	PROPN
ejpam-4793	793	4	)	)	PUNCT
ejpam-4793	793	5	is	be	AUX
ejpam-4793	793	6	the	the	DET
ejpam-4793	793	7	identity	identity	NOUN
ejpam-4793	793	8	in	in	ADP
ejpam-4793	793	9	m	m	PROPN
ejpam-4793	793	10	/	/	SYM
ejpam-4793	793	11	s	s	NOUN
ejpam-4793	793	12	and	and	CCONJ
ejpam-4793	793	13	h	h	NOUN
ejpam-4793	793	14	is	be	AUX
ejpam-4793	793	15	a	a	DET
ejpam-4793	793	16	γ	γ	NOUN
ejpam-4793	793	17	-	-	ADJ
ejpam-4793	793	18	submonoid	submonoid	NOUN
ejpam-4793	793	19	of	of	ADP
ejpam-4793	793	20	m	m	PROPN
ejpam-4793	793	21	/	/	SYM
ejpam-4793	793	22	s	s	PROPN
ejpam-4793	793	23	,	,	PUNCT
ejpam-4793	793	24	we	we	PRON
ejpam-4793	793	25	must	must	AUX
ejpam-4793	793	26	have	have	VERB
ejpam-4793	793	27	ρs(x	ρs(x	NOUN
ejpam-4793	793	28	)	)	PUNCT
ejpam-4793	794	1	=	=	PUNCT
ejpam-4793	794	2	ρs(1	ρs(1	PROPN
ejpam-4793	794	3	m	m	PROPN
ejpam-4793	794	4	)	)	PUNCT
ejpam-4793	795	1	∈	∈	PROPN
ejpam-4793	795	2	h.	h.	PROPN
ejpam-4793	795	3	thus	thus	ADV
ejpam-4793	795	4	,	,	PUNCT
ejpam-4793	795	5	x	x	PROPN
ejpam-4793	795	6	∈	∈	PROPN
ejpam-4793	795	7	r.	r.	PROPN
ejpam-4793	795	8	therefore	therefore	ADV
ejpam-4793	795	9	,	,	PUNCT
ejpam-4793	795	10	s	s	VERB
ejpam-4793	795	11	⊆	⊆	PROPN
ejpam-4793	795	12	r.	r.	PROPN
ejpam-4793	795	13	h.	h.	PROPN
ejpam-4793	795	14	sarapuddin	sarapuddin	PROPN
ejpam-4793	795	15	,	,	PUNCT
ejpam-4793	795	16	j.	j.	PROPN
ejpam-4793	795	17	vilela	vilela	PROPN
ejpam-4793	795	18	/	/	SYM
ejpam-4793	795	19	eur	eur	PROPN
ejpam-4793	795	20	.	.	PUNCT
ejpam-4793	796	1	j.	j.	PROPN
ejpam-4793	796	2	pure	pure	PROPN
ejpam-4793	796	3	appl	appl	PROPN
ejpam-4793	796	4	.	.	PROPN
ejpam-4793	796	5	math	math	PROPN
ejpam-4793	796	6	,	,	PUNCT
ejpam-4793	796	7	16	16	NUM
ejpam-4793	796	8	(	(	PUNCT
ejpam-4793	796	9	3	3	NUM
ejpam-4793	796	10	)	)	PUNCT
ejpam-4793	796	11	(	(	PUNCT
ejpam-4793	796	12	2023	2023	NUM
ejpam-4793	796	13	)	)	PUNCT
ejpam-4793	796	14	,	,	PUNCT
ejpam-4793	796	15	1772	1772	NUM
ejpam-4793	796	16	-	-	SYM
ejpam-4793	796	17	1793	1793	NUM
ejpam-4793	796	18	1790	1790	NUM
ejpam-4793	796	19	theorem	theorem	NOUN
ejpam-4793	796	20	12	12	NUM
ejpam-4793	796	21	.	.	PUNCT
ejpam-4793	797	1	let	let	VERB
ejpam-4793	797	2	m	m	PRON
ejpam-4793	797	3	be	be	AUX
ejpam-4793	797	4	a	a	DET
ejpam-4793	797	5	commutative	commutative	ADJ
ejpam-4793	797	6	γ	γ	X
ejpam-4793	797	7	-	-	PUNCT
ejpam-4793	797	8	monoid	monoid	NOUN
ejpam-4793	797	9	and	and	CCONJ
ejpam-4793	797	10	s	s	VERB
ejpam-4793	797	11	a	a	DET
ejpam-4793	797	12	normal	normal	ADJ
ejpam-4793	797	13	γ	γ	NOUN
ejpam-4793	797	14	-	-	NOUN
ejpam-4793	797	15	submonoid	submonoid	NOUN
ejpam-4793	797	16	of	of	ADP
ejpam-4793	797	17	m	m	PROPN
ejpam-4793	797	18	.	.	PUNCT
ejpam-4793	798	1	then	then	ADV
ejpam-4793	798	2	the	the	DET
ejpam-4793	798	3	mapping	mapping	NOUN
ejpam-4793	798	4	πs	πs	INTJ
ejpam-4793	798	5	:	:	PUNCT
ejpam-4793	798	6	m	m	VERB
ejpam-4793	798	7	−→	−→	PROPN
ejpam-4793	798	8	m	m	PROPN
ejpam-4793	798	9	/	/	SYM
ejpam-4793	798	10	s	s	AUX
ejpam-4793	798	11	given	give	VERB
ejpam-4793	798	12	by	by	ADP
ejpam-4793	798	13	πs(x	πs(x	NOUN
ejpam-4793	798	14	)	)	PUNCT
ejpam-4793	798	15	=	=	SYM
ejpam-4793	799	1	ρs(x	ρs(x	X
ejpam-4793	799	2	)	)	PUNCT
ejpam-4793	799	3	is	be	AUX
ejpam-4793	799	4	a	a	DET
ejpam-4793	799	5	γ	γ	PROPN
ejpam-4793	799	6	-	-	PUNCT
ejpam-4793	799	7	monoid	monoid	NOUN
ejpam-4793	799	8	epimorphism	epimorphism	NOUN
ejpam-4793	799	9	with	with	ADP
ejpam-4793	799	10	kernel	kernel	PROPN
ejpam-4793	799	11	s.	s.	PROPN
ejpam-4793	799	12	proof	proof	PROPN
ejpam-4793	799	13	.	.	PUNCT
ejpam-4793	800	1	let	let	VERB
ejpam-4793	800	2	x	x	PRON
ejpam-4793	800	3	,	,	PUNCT
ejpam-4793	800	4	y	y	PROPN
ejpam-4793	800	5	∈	∈	PROPN
ejpam-4793	800	6	m	m	VERB
ejpam-4793	801	1	such	such	ADJ
ejpam-4793	801	2	that	that	SCONJ
ejpam-4793	801	3	x	x	X
ejpam-4793	801	4	=	=	PUNCT
ejpam-4793	801	5	y.	y.	NOUN
ejpam-4793	801	6	then	then	ADV
ejpam-4793	801	7	,	,	PUNCT
ejpam-4793	801	8	πs(x	πs(x	NUM
ejpam-4793	801	9	)	)	PUNCT
ejpam-4793	801	10	=	=	SYM
ejpam-4793	801	11	ρs(x	ρs(x	X
ejpam-4793	801	12	)	)	PUNCT
ejpam-4793	801	13	=	=	SYM
ejpam-4793	801	14	ρs(y	ρs(y	NUM
ejpam-4793	801	15	)	)	PUNCT
ejpam-4793	801	16	=	=	SYM
ejpam-4793	801	17	π(y	π(y	PROPN
ejpam-4793	801	18	)	)	PUNCT
ejpam-4793	801	19	.	.	PUNCT
ejpam-4793	802	1	thus	thus	ADV
ejpam-4793	802	2	,	,	PUNCT
ejpam-4793	802	3	πs	πs	ADV
ejpam-4793	802	4	is	be	AUX
ejpam-4793	802	5	well	well	ADV
ejpam-4793	802	6	-	-	PUNCT
ejpam-4793	802	7	defined	define	VERB
ejpam-4793	802	8	.	.	PUNCT
ejpam-4793	803	1	now	now	ADV
ejpam-4793	803	2	,	,	PUNCT
ejpam-4793	803	3	let	let	VERB
ejpam-4793	803	4	x	x	PRON
ejpam-4793	803	5	,	,	PUNCT
ejpam-4793	803	6	y	y	PROPN
ejpam-4793	803	7	∈	∈	PROPN
ejpam-4793	803	8	m	m	VERB
ejpam-4793	803	9	.	.	PUNCT
ejpam-4793	804	1	then	then	ADV
ejpam-4793	804	2	,	,	PUNCT
ejpam-4793	804	3	we	we	PRON
ejpam-4793	804	4	have	have	VERB
ejpam-4793	804	5	πs(x	πs(x	NUM
ejpam-4793	804	6	∗	∗	NOUN
ejpam-4793	804	7	y	y	NOUN
ejpam-4793	804	8	)	)	PUNCT
ejpam-4793	805	1	=	=	SYM
ejpam-4793	805	2	ρs(x	ρs(x	X
ejpam-4793	805	3	∗	∗	X
ejpam-4793	805	4	y	y	NOUN
ejpam-4793	805	5	)	)	PUNCT
ejpam-4793	805	6	=	=	SYM
ejpam-4793	805	7	ρs(x	ρs(x	X
ejpam-4793	805	8	)	)	PUNCT
ejpam-4793	805	9	◦	◦	NOUN
ejpam-4793	805	10	ρs(y	ρs(y	NUM
ejpam-4793	805	11	)	)	PUNCT
ejpam-4793	805	12	=	=	SYM
ejpam-4793	805	13	πs(x	πs(x	X
ejpam-4793	805	14	)	)	PUNCT
ejpam-4793	805	15	◦	◦	NOUN
ejpam-4793	805	16	πs(y	πs(y	PUNCT
ejpam-4793	805	17	)	)	PUNCT
ejpam-4793	805	18	and	and	CCONJ
ejpam-4793	805	19	πs(1	πs(1	PROPN
ejpam-4793	805	20	m	m	PROPN
ejpam-4793	805	21	)	)	PUNCT
ejpam-4793	806	1	=	=	PUNCT
ejpam-4793	806	2	ρs(1	ρs(1	PROPN
ejpam-4793	806	3	m	m	PROPN
ejpam-4793	806	4	)	)	PUNCT
ejpam-4793	806	5	.	.	PUNCT
ejpam-4793	807	1	thus	thus	ADV
ejpam-4793	807	2	,	,	PUNCT
ejpam-4793	807	3	by	by	ADP
ejpam-4793	807	4	definition	definition	NOUN
ejpam-4793	807	5	4	4	NUM
ejpam-4793	807	6	,	,	PUNCT
ejpam-4793	807	7	πs	πs	ADV
ejpam-4793	807	8	is	be	AUX
ejpam-4793	807	9	a	a	DET
ejpam-4793	807	10	monoid	monoid	NOUN
ejpam-4793	807	11	homomorphism	homomorphism	NOUN
ejpam-4793	807	12	.	.	PUNCT
ejpam-4793	808	1	since	since	SCONJ
ejpam-4793	808	2	απs(x	απs(x	NUM
ejpam-4793	808	3	)	)	PUNCT
ejpam-4793	808	4	=	=	SYM
ejpam-4793	808	5	αρs(x	αρs(x	X
ejpam-4793	808	6	)	)	PUNCT
ejpam-4793	808	7	=	=	SYM
ejpam-4793	808	8	ρs	ρs	PROPN
ejpam-4793	808	9	(	(	PUNCT
ejpam-4793	808	10	αx	αx	ADV
ejpam-4793	808	11	)	)	PUNCT
ejpam-4793	808	12	=	=	SYM
ejpam-4793	808	13	πs	πs	PROPN
ejpam-4793	808	14	(	(	PUNCT
ejpam-4793	808	15	αx	αx	NOUN
ejpam-4793	808	16	)	)	PUNCT
ejpam-4793	808	17	,	,	PUNCT
ejpam-4793	808	18	by	by	ADP
ejpam-4793	808	19	definition	definition	NOUN
ejpam-4793	808	20	,	,	PUNCT
ejpam-4793	808	21	πs	πs	ADV
ejpam-4793	808	22	is	be	AUX
ejpam-4793	808	23	a	a	DET
ejpam-4793	808	24	γ	γ	NOUN
ejpam-4793	808	25	-	-	PUNCT
ejpam-4793	808	26	monoid	monoid	NOUN
ejpam-4793	808	27	homomorphism	homomorphism	NOUN
ejpam-4793	808	28	.	.	PUNCT
ejpam-4793	809	1	now	now	ADV
ejpam-4793	809	2	,	,	PUNCT
ejpam-4793	809	3	let	let	VERB
ejpam-4793	809	4	b	b	X
ejpam-4793	809	5	∈	∈	PROPN
ejpam-4793	809	6	m	m	PROPN
ejpam-4793	809	7	/	/	SYM
ejpam-4793	809	8	s.	s.	PROPN
ejpam-4793	809	9	then	then	ADV
ejpam-4793	809	10	,	,	PUNCT
ejpam-4793	809	11	b	b	X
ejpam-4793	809	12	=	=	PUNCT
ejpam-4793	809	13	ρs(a	ρs(a	NUM
ejpam-4793	809	14	)	)	PUNCT
ejpam-4793	809	15	for	for	ADP
ejpam-4793	809	16	some	some	DET
ejpam-4793	809	17	a	a	DET
ejpam-4793	809	18	∈	∈	NOUN
ejpam-4793	809	19	m	m	NOUN
ejpam-4793	809	20	.	.	PUNCT
ejpam-4793	810	1	thus	thus	ADV
ejpam-4793	810	2	,	,	PUNCT
ejpam-4793	810	3	b	b	X
ejpam-4793	810	4	=	=	PUNCT
ejpam-4793	810	5	ρs(a	ρs(a	NUM
ejpam-4793	810	6	)	)	PUNCT
ejpam-4793	810	7	=	=	PUNCT
ejpam-4793	811	1	πs(a	πs(a	X
ejpam-4793	811	2	)	)	PUNCT
ejpam-4793	811	3	and	and	CCONJ
ejpam-4793	811	4	so	so	ADV
ejpam-4793	811	5	,	,	PUNCT
ejpam-4793	811	6	π	π	PROPN
ejpam-4793	811	7	is	be	AUX
ejpam-4793	811	8	surjective	surjective	ADJ
ejpam-4793	811	9	.	.	PUNCT
ejpam-4793	812	1	therefore	therefore	ADV
ejpam-4793	812	2	,	,	PUNCT
ejpam-4793	812	3	πs	πs	ADV
ejpam-4793	812	4	is	be	AUX
ejpam-4793	812	5	an	an	DET
ejpam-4793	812	6	epimorphism	epimorphism	NOUN
ejpam-4793	812	7	.	.	PUNCT
ejpam-4793	813	1	now	now	ADV
ejpam-4793	813	2	,	,	PUNCT
ejpam-4793	813	3	since	since	SCONJ
ejpam-4793	813	4	s	s	NOUN
ejpam-4793	813	5	is	be	AUX
ejpam-4793	813	6	normal	normal	ADJ
ejpam-4793	813	7	,	,	PUNCT
ejpam-4793	813	8	by	by	ADP
ejpam-4793	813	9	proposition	proposition	NOUN
ejpam-4793	813	10	4	4	NUM
ejpam-4793	813	11	we	we	PRON
ejpam-4793	813	12	have	have	AUX
ejpam-4793	813	13	kerπs	kerπ	VERB
ejpam-4793	813	14	=	=	PUNCT
ejpam-4793	813	15	{	{	PUNCT
ejpam-4793	813	16	m	m	VERB
ejpam-4793	813	17	∈	∈	ADJ
ejpam-4793	813	18	m	m	NOUN
ejpam-4793	813	19	:	:	PUNCT
ejpam-4793	813	20	ρs(m	ρs(m	NUM
ejpam-4793	813	21	)	)	PUNCT
ejpam-4793	813	22	=	=	PUNCT
ejpam-4793	814	1	ρs(1	ρs(1	PROPN
ejpam-4793	814	2	m	m	PROPN
ejpam-4793	814	3	)	)	PUNCT
ejpam-4793	814	4	}	}	PUNCT
ejpam-4793	815	1	=	=	SYM
ejpam-4793	815	2	{	{	PUNCT
ejpam-4793	815	3	m	m	VERB
ejpam-4793	815	4	∈	∈	NOUN
ejpam-4793	815	5	m	m	VERB
ejpam-4793	815	6	:	:	PUNCT
ejpam-4793	816	1	m	m	VERB
ejpam-4793	816	2	∈	∈	PROPN
ejpam-4793	816	3	s	s	PART
ejpam-4793	816	4	}	}	PUNCT
ejpam-4793	816	5	=	=	SYM
ejpam-4793	816	6	s	s	PART
ejpam-4793	816	7	∩m	∩m	NOUN
ejpam-4793	816	8	=	=	PUNCT
ejpam-4793	816	9	s	s	NOUN
ejpam-4793	816	10	as	as	SCONJ
ejpam-4793	816	11	desired	desire	VERB
ejpam-4793	816	12	.	.	PUNCT
ejpam-4793	817	1	the	the	DET
ejpam-4793	817	2	map	map	NOUN
ejpam-4793	817	3	πs	πs	ADV
ejpam-4793	817	4	in	in	ADP
ejpam-4793	817	5	theorem	theorem	NOUN
ejpam-4793	817	6	12	12	NUM
ejpam-4793	817	7	is	be	AUX
ejpam-4793	817	8	called	call	VERB
ejpam-4793	817	9	the	the	DET
ejpam-4793	817	10	canonical	canonical	ADJ
ejpam-4793	817	11	epimorphism	epimorphism	NOUN
ejpam-4793	817	12	.	.	PUNCT
ejpam-4793	818	1	proposition	proposition	NOUN
ejpam-4793	818	2	7	7	NUM
ejpam-4793	818	3	.	.	PUNCT
ejpam-4793	819	1	let	let	VERB
ejpam-4793	819	2	m	m	PRON
ejpam-4793	819	3	be	be	AUX
ejpam-4793	819	4	a	a	DET
ejpam-4793	819	5	γ	γ	X
ejpam-4793	819	6	-	-	PUNCT
ejpam-4793	819	7	monoid	monoid	NOUN
ejpam-4793	819	8	.	.	PUNCT
ejpam-4793	820	1	then	then	ADV
ejpam-4793	820	2	for	for	ADP
ejpam-4793	820	3	any	any	DET
ejpam-4793	820	4	a	a	DET
ejpam-4793	820	5	⊆	⊆	NUM
ejpam-4793	820	6	m	m	NOUN
ejpam-4793	820	7	and	and	CCONJ
ejpam-4793	820	8	s	s	VERB
ejpam-4793	820	9	a	a	DET
ejpam-4793	820	10	commutative	commutative	ADJ
ejpam-4793	820	11	γ	γ	NOUN
ejpam-4793	820	12	-	-	ADJ
ejpam-4793	820	13	submonoid	submonoid	NOUN
ejpam-4793	820	14	of	of	ADP
ejpam-4793	820	15	m	m	PRON
ejpam-4793	820	16	,	,	PUNCT
ejpam-4793	820	17	π−1	π−1	PROPN
ejpam-4793	820	18	s	s	X
ejpam-4793	820	19	(	(	PUNCT
ejpam-4793	820	20	πs(a	πs(a	NOUN
ejpam-4793	820	21	)	)	PUNCT
ejpam-4793	820	22	)	)	PUNCT
ejpam-4793	821	1	=	=	PUNCT
ejpam-4793	821	2	⋃	⋃	NOUN
ejpam-4793	821	3	x∈a	x∈a	NOUN
ejpam-4793	821	4	ρs(x	ρs(x	NUM
ejpam-4793	821	5	)	)	PUNCT
ejpam-4793	821	6	.	.	PUNCT
ejpam-4793	822	1	proof	proof	NOUN
ejpam-4793	822	2	.	.	PUNCT
ejpam-4793	823	1	suppose	suppose	VERB
ejpam-4793	823	2	y	y	PROPN
ejpam-4793	823	3	∈	∈	PROPN
ejpam-4793	823	4	π−1	π−1	PROPN
ejpam-4793	823	5	s	s	PART
ejpam-4793	823	6	(	(	PUNCT
ejpam-4793	823	7	πs(a	πs(a	NOUN
ejpam-4793	823	8	)	)	PUNCT
ejpam-4793	823	9	)	)	PUNCT
ejpam-4793	823	10	.	.	PUNCT
ejpam-4793	824	1	then	then	ADV
ejpam-4793	824	2	ρs(y	ρs(y	NUM
ejpam-4793	824	3	)	)	PUNCT
ejpam-4793	824	4	=	=	SYM
ejpam-4793	824	5	πs(y	πs(y	X
ejpam-4793	824	6	)	)	PUNCT
ejpam-4793	824	7	∈	∈	NOUN
ejpam-4793	824	8	πs(a	πs(a	PRON
ejpam-4793	824	9	)	)	PUNCT
ejpam-4793	824	10	.	.	PUNCT
ejpam-4793	825	1	since	since	SCONJ
ejpam-4793	825	2	πs	πs	PROPN
ejpam-4793	825	3	is	be	VERB
ejpam-4793	825	4	an	an	DET
ejpam-4793	825	5	epimorphism	epimorphism	NOUN
ejpam-4793	825	6	,	,	PUNCT
ejpam-4793	825	7	there	there	PRON
ejpam-4793	825	8	exists	exist	VERB
ejpam-4793	825	9	an	an	DET
ejpam-4793	825	10	x	x	SYM
ejpam-4793	825	11	∈	∈	PROPN
ejpam-4793	825	12	a	a	DET
ejpam-4793	825	13	such	such	ADJ
ejpam-4793	825	14	that	that	PRON
ejpam-4793	825	15	πs(x	πs(x	NUM
ejpam-4793	825	16	)	)	PUNCT
ejpam-4793	825	17	=	=	SYM
ejpam-4793	826	1	ρs(y	ρs(y	NUM
ejpam-4793	826	2	)	)	PUNCT
ejpam-4793	826	3	.	.	PUNCT
ejpam-4793	827	1	hence	hence	ADV
ejpam-4793	827	2	,	,	PUNCT
ejpam-4793	827	3	ρs(x	ρs(x	ADJ
ejpam-4793	827	4	)	)	PUNCT
ejpam-4793	827	5	=	=	SYM
ejpam-4793	828	1	ρs(y	ρs(y	NUM
ejpam-4793	828	2	)	)	PUNCT
ejpam-4793	828	3	.	.	PUNCT
ejpam-4793	829	1	by	by	ADP
ejpam-4793	829	2	remark	remark	NOUN
ejpam-4793	829	3	14(ii	14(ii	NUM
ejpam-4793	829	4	)	)	PUNCT
ejpam-4793	829	5	,	,	PUNCT
ejpam-4793	829	6	(	(	PUNCT
ejpam-4793	829	7	αx	αx	ADV
ejpam-4793	829	8	∗	∗	PROPN
ejpam-4793	829	9	s	s	PART
ejpam-4793	829	10	)	)	PUNCT
ejpam-4793	829	11	∩	∩	NOUN
ejpam-4793	829	12	(	(	PUNCT
ejpam-4793	829	13	αy	αy	ADP
ejpam-4793	829	14	∗	∗	NOUN
ejpam-4793	829	15	s	s	PART
ejpam-4793	829	16	)	)	PUNCT
ejpam-4793	829	17	̸=	̸=	NOUN
ejpam-4793	829	18	∅	∅	NOUN
ejpam-4793	829	19	for	for	ADP
ejpam-4793	829	20	all	all	PRON
ejpam-4793	829	21	α	α	PRON
ejpam-4793	829	22	∈	∈	PROPN
ejpam-4793	829	23	γ	γ	X
ejpam-4793	829	24	,	,	PUNCT
ejpam-4793	829	25	that	that	ADV
ejpam-4793	829	26	is	is	ADV
ejpam-4793	829	27	,	,	PUNCT
ejpam-4793	829	28	xρsy	xρsy	PROPN
ejpam-4793	829	29	.	.	PUNCT
ejpam-4793	830	1	this	this	PRON
ejpam-4793	830	2	implies	imply	VERB
ejpam-4793	830	3	that	that	SCONJ
ejpam-4793	830	4	y	y	PROPN
ejpam-4793	830	5	∈	∈	PROPN
ejpam-4793	830	6	ρs(x	ρs(x	PROPN
ejpam-4793	830	7	)	)	PUNCT
ejpam-4793	830	8	for	for	ADP
ejpam-4793	830	9	some	some	DET
ejpam-4793	830	10	x	x	SYM
ejpam-4793	830	11	∈	∈	NOUN
ejpam-4793	830	12	a.	a.	NOUN
ejpam-4793	830	13	it	it	PRON
ejpam-4793	830	14	follows	follow	VERB
ejpam-4793	830	15	that	that	SCONJ
ejpam-4793	830	16	y	y	PROPN
ejpam-4793	830	17	∈	∈	PROPN
ejpam-4793	830	18	⋃	⋃	NOUN
ejpam-4793	830	19	x∈a	x∈a	NOUN
ejpam-4793	830	20	ρs(x	ρs(x	NOUN
ejpam-4793	830	21	)	)	PUNCT
ejpam-4793	831	1	so	so	SCONJ
ejpam-4793	831	2	that	that	SCONJ
ejpam-4793	831	3	π−1	π−1	PROPN
ejpam-4793	831	4	s	s	X
ejpam-4793	831	5	(	(	PUNCT
ejpam-4793	831	6	πs(a	πs(a	NOUN
ejpam-4793	831	7	)	)	PUNCT
ejpam-4793	831	8	)	)	PUNCT
ejpam-4793	831	9	⊆⋃	⊆⋃	PROPN
ejpam-4793	831	10	x∈a	x∈a	NOUN
ejpam-4793	831	11	ρs(x	ρs(x	PROPN
ejpam-4793	831	12	)	)	PUNCT
ejpam-4793	831	13	.	.	PUNCT
ejpam-4793	832	1	conversely	conversely	ADV
ejpam-4793	832	2	,	,	PUNCT
ejpam-4793	832	3	suppose	suppose	VERB
ejpam-4793	832	4	y	y	PROPN
ejpam-4793	832	5	∈	∈	PROPN
ejpam-4793	832	6	⋃	⋃	NOUN
ejpam-4793	832	7	x∈a	x∈a	NOUN
ejpam-4793	832	8	ρs(x	ρs(x	NUM
ejpam-4793	832	9	)	)	PUNCT
ejpam-4793	832	10	.	.	PUNCT
ejpam-4793	833	1	then	then	ADV
ejpam-4793	833	2	y	y	PROPN
ejpam-4793	833	3	∈	∈	PROPN
ejpam-4793	833	4	ρs(x	ρs(x	PROPN
ejpam-4793	833	5	)	)	PUNCT
ejpam-4793	833	6	for	for	ADP
ejpam-4793	833	7	some	some	DET
ejpam-4793	833	8	x	x	SYM
ejpam-4793	833	9	∈	∈	PROPN
ejpam-4793	833	10	a.	a.	NOUN
ejpam-4793	833	11	this	this	PRON
ejpam-4793	833	12	implies	imply	VERB
ejpam-4793	833	13	that	that	PRON
ejpam-4793	833	14	yρsx	yρsx	ADJ
ejpam-4793	833	15	,	,	PUNCT
ejpam-4793	833	16	that	that	ADV
ejpam-4793	833	17	is	is	ADV
ejpam-4793	833	18	,	,	PUNCT
ejpam-4793	833	19	(	(	PUNCT
ejpam-4793	833	20	αy	αy	ADP
ejpam-4793	833	21	∗	∗	NUM
ejpam-4793	833	22	s	s	NOUN
ejpam-4793	833	23	)	)	PUNCT
ejpam-4793	833	24	∩	∩	NOUN
ejpam-4793	833	25	(	(	PUNCT
ejpam-4793	833	26	αx	αx	ADV
ejpam-4793	833	27	∗	∗	PROPN
ejpam-4793	833	28	s	s	PART
ejpam-4793	833	29	)	)	PUNCT
ejpam-4793	833	30	̸=	̸=	NOUN
ejpam-4793	833	31	∅	∅	NOUN
ejpam-4793	833	32	for	for	ADP
ejpam-4793	833	33	all	all	PRON
ejpam-4793	833	34	α	α	PRON
ejpam-4793	833	35	∈	∈	PROPN
ejpam-4793	833	36	γ	γ	X
ejpam-4793	833	37	.	.	PUNCT
ejpam-4793	833	38	by	by	ADP
ejpam-4793	833	39	remark	remark	NOUN
ejpam-4793	833	40	14(ii	14(ii	NUM
ejpam-4793	833	41	)	)	PUNCT
ejpam-4793	833	42	,	,	PUNCT
ejpam-4793	833	43	ρs(y	ρs(y	NUM
ejpam-4793	833	44	)	)	PUNCT
ejpam-4793	833	45	=	=	SYM
ejpam-4793	833	46	ρs(x	ρs(x	X
ejpam-4793	833	47	)	)	PUNCT
ejpam-4793	833	48	.	.	PUNCT
ejpam-4793	834	1	thus	thus	ADV
ejpam-4793	834	2	,	,	PUNCT
ejpam-4793	834	3	πs(y	πs(y	PUNCT
ejpam-4793	834	4	)	)	PUNCT
ejpam-4793	834	5	=	=	SYM
ejpam-4793	834	6	πs(x	πs(x	NOUN
ejpam-4793	834	7	)	)	PUNCT
ejpam-4793	834	8	.	.	PUNCT
ejpam-4793	835	1	since	since	SCONJ
ejpam-4793	835	2	πs(x	πs(x	NOUN
ejpam-4793	835	3	)	)	PUNCT
ejpam-4793	835	4	∈	∈	NOUN
ejpam-4793	835	5	πs(a	πs(a	PRON
ejpam-4793	835	6	)	)	PUNCT
ejpam-4793	835	7	,	,	PUNCT
ejpam-4793	835	8	it	it	PRON
ejpam-4793	835	9	follows	follow	VERB
ejpam-4793	835	10	that	that	SCONJ
ejpam-4793	835	11	πs(y	πs(y	PUNCT
ejpam-4793	835	12	)	)	PUNCT
ejpam-4793	835	13	∈	∈	NOUN
ejpam-4793	835	14	πs(a	πs(a	PRON
ejpam-4793	835	15	)	)	PUNCT
ejpam-4793	835	16	implying	imply	VERB
ejpam-4793	835	17	that	that	SCONJ
ejpam-4793	835	18	y	y	PROPN
ejpam-4793	835	19	∈	∈	PROPN
ejpam-4793	835	20	π−1	π−1	PROPN
ejpam-4793	835	21	s	s	PART
ejpam-4793	835	22	(	(	PUNCT
ejpam-4793	835	23	πs(a	πs(a	NOUN
ejpam-4793	835	24	)	)	PUNCT
ejpam-4793	835	25	)	)	PUNCT
ejpam-4793	835	26	.	.	PUNCT
ejpam-4793	836	1	hence	hence	ADV
ejpam-4793	836	2	,	,	PUNCT
ejpam-4793	836	3	⋃	⋃	NOUN
ejpam-4793	836	4	x∈a	x∈a	NOUN
ejpam-4793	836	5	ρs(x	ρs(x	NOUN
ejpam-4793	836	6	)	)	PUNCT
ejpam-4793	837	1	⊆	⊆	NUM
ejpam-4793	837	2	π−1	π−1	PROPN
ejpam-4793	837	3	s	s	X
ejpam-4793	837	4	(	(	PUNCT
ejpam-4793	837	5	πs(a	πs(a	NOUN
ejpam-4793	837	6	)	)	PUNCT
ejpam-4793	837	7	)	)	PUNCT
ejpam-4793	837	8	.	.	PUNCT
ejpam-4793	838	1	therefore	therefore	ADV
ejpam-4793	838	2	,	,	PUNCT
ejpam-4793	838	3	π−1	π−1	PROPN
ejpam-4793	838	4	s	s	X
ejpam-4793	838	5	(	(	PUNCT
ejpam-4793	838	6	πs(a	πs(a	NOUN
ejpam-4793	838	7	)	)	PUNCT
ejpam-4793	838	8	)	)	PUNCT
ejpam-4793	839	1	=	=	PUNCT
ejpam-4793	839	2	⋃	⋃	NOUN
ejpam-4793	839	3	x∈a	x∈a	NOUN
ejpam-4793	839	4	ρs(x	ρs(x	NOUN
ejpam-4793	839	5	)	)	PUNCT
ejpam-4793	839	6	.	.	PUNCT
ejpam-4793	840	1	6	6	X
ejpam-4793	840	2	.	.	X
ejpam-4793	841	1	isomorphism	isomorphism	NOUN
ejpam-4793	841	2	theorems	theorem	NOUN
ejpam-4793	841	3	in	in	ADP
ejpam-4793	841	4	[	[	X
ejpam-4793	841	5	5	5	NUM
ejpam-4793	841	6	]	]	PUNCT
ejpam-4793	841	7	,	,	PUNCT
ejpam-4793	841	8	the	the	DET
ejpam-4793	841	9	isomorphism	isomorphism	NOUN
ejpam-4793	841	10	theorems	theorem	VERB
ejpam-4793	841	11	for	for	ADP
ejpam-4793	841	12	γ	γ	NOUN
ejpam-4793	841	13	-	-	PUNCT
ejpam-4793	841	14	monoids	monoid	NOUN
ejpam-4793	841	15	via	via	ADP
ejpam-4793	841	16	γ	γ	NOUN
ejpam-4793	841	17	-	-	PUNCT
ejpam-4793	841	18	order	order	NOUN
ejpam-4793	841	19	-	-	PUNCT
ejpam-4793	841	20	ideals	ideal	NOUN
ejpam-4793	841	21	are	be	AUX
ejpam-4793	841	22	established	establish	VERB
ejpam-4793	841	23	.	.	PUNCT
ejpam-4793	842	1	here	here	ADV
ejpam-4793	842	2	,	,	PUNCT
ejpam-4793	842	3	we	we	PRON
ejpam-4793	842	4	prove	prove	VERB
ejpam-4793	842	5	isomorphism	isomorphism	NOUN
ejpam-4793	842	6	theorems	theorem	NOUN
ejpam-4793	842	7	for	for	ADP
ejpam-4793	842	8	γ	γ	NOUN
ejpam-4793	842	9	-	-	PUNCT
ejpam-4793	842	10	monoids	monoid	NOUN
ejpam-4793	842	11	via	via	ADP
ejpam-4793	842	12	γ	γ	NOUN
ejpam-4793	842	13	-	-	PUNCT
ejpam-4793	842	14	submonoids	submonoid	NOUN
ejpam-4793	842	15	.	.	PUNCT
ejpam-4793	843	1	as	as	SCONJ
ejpam-4793	843	2	shown	show	VERB
ejpam-4793	843	3	already	already	ADV
ejpam-4793	843	4	in	in	ADP
ejpam-4793	843	5	example	example	NOUN
ejpam-4793	843	6	16	16	NUM
ejpam-4793	843	7	,	,	PUNCT
ejpam-4793	843	8	the	the	DET
ejpam-4793	843	9	quotient	quotient	NOUN
ejpam-4793	843	10	m	m	PROPN
ejpam-4793	843	11	/	/	SYM
ejpam-4793	843	12	s	s	PROPN
ejpam-4793	843	13	in	in	ADP
ejpam-4793	843	14	our	our	PRON
ejpam-4793	843	15	discussion	discussion	NOUN
ejpam-4793	843	16	is	be	AUX
ejpam-4793	843	17	not	not	PART
ejpam-4793	843	18	the	the	DET
ejpam-4793	843	19	same	same	ADJ
ejpam-4793	843	20	with	with	ADP
ejpam-4793	843	21	the	the	DET
ejpam-4793	843	22	quotient	quotient	NOUN
ejpam-4793	843	23	discussed	discuss	VERB
ejpam-4793	843	24	in	in	ADP
ejpam-4793	843	25	[	[	X
ejpam-4793	843	26	5	5	NUM
ejpam-4793	843	27	]	]	PUNCT
ejpam-4793	843	28	.	.	PUNCT
ejpam-4793	844	1	theorem	theorem	NOUN
ejpam-4793	844	2	13	13	NUM
ejpam-4793	844	3	.	.	PUNCT
ejpam-4793	845	1	let	let	AUX
ejpam-4793	845	2	(	(	PUNCT
ejpam-4793	845	3	m	m	NOUN
ejpam-4793	845	4	,	,	PUNCT
ejpam-4793	845	5	∗	∗	NOUN
ejpam-4793	845	6	)	)	PUNCT
ejpam-4793	845	7	and	and	CCONJ
ejpam-4793	845	8	(	(	PUNCT
ejpam-4793	845	9	n	n	CCONJ
ejpam-4793	845	10	,	,	PUNCT
ejpam-4793	845	11	·	·	PUNCT
ejpam-4793	845	12	)	)	PUNCT
ejpam-4793	845	13	be	be	AUX
ejpam-4793	845	14	commutative	commutative	ADJ
ejpam-4793	845	15	γ	γ	NOUN
ejpam-4793	845	16	-	-	PUNCT
ejpam-4793	845	17	monoids	monoid	NOUN
ejpam-4793	845	18	and	and	CCONJ
ejpam-4793	845	19	let	let	VERB
ejpam-4793	845	20	f	f	PRON
ejpam-4793	845	21	:	:	PUNCT
ejpam-4793	845	22	m	m	VERB
ejpam-4793	845	23	→	→	SYM
ejpam-4793	845	24	n	n	CCONJ
ejpam-4793	845	25	be	be	AUX
ejpam-4793	845	26	a	a	DET
ejpam-4793	845	27	γmonoid	γmonoid	NOUN
ejpam-4793	845	28	homomorphism	homomorphism	NOUN
ejpam-4793	845	29	.	.	PUNCT
ejpam-4793	846	1	there	there	PRON
ejpam-4793	846	2	exists	exist	VERB
ejpam-4793	846	3	a	a	DET
ejpam-4793	846	4	unique	unique	ADJ
ejpam-4793	846	5	γ	γ	X
ejpam-4793	846	6	-	-	PUNCT
ejpam-4793	846	7	monoid	monoid	NOUN
ejpam-4793	846	8	homomorphism	homomorphism	PROPN
ejpam-4793	846	9	φ	φ	PROPN
ejpam-4793	846	10	:	:	PUNCT
ejpam-4793	846	11	m/	m/	PROPN
ejpam-4793	846	12	ker	ker	PROPN
ejpam-4793	847	1	f	f	PROPN
ejpam-4793	847	2	→	→	PUNCT
ejpam-4793	847	3	n	n	X
ejpam-4793	847	4	such	such	ADJ
ejpam-4793	847	5	that	that	SCONJ
ejpam-4793	847	6	the	the	DET
ejpam-4793	847	7	following	follow	VERB
ejpam-4793	847	8	diagram	diagram	NOUN
ejpam-4793	847	9	is	be	AUX
ejpam-4793	847	10	commutative	commutative	ADJ
ejpam-4793	847	11	m	m	NOUN
ejpam-4793	847	12	n	n	NUM
ejpam-4793	847	13	m/	m/	NOUN
ejpam-4793	847	14	ker	ker	PROPN
ejpam-4793	848	1	f	f	PROPN
ejpam-4793	848	2	πker	πker	NOUN
ejpam-4793	848	3	f	f	PROPN
ejpam-4793	848	4	f	f	PROPN
ejpam-4793	848	5	φ	φ	PROPN
ejpam-4793	848	6	h.	h.	PROPN
ejpam-4793	848	7	sarapuddin	sarapuddin	PROPN
ejpam-4793	848	8	,	,	PUNCT
ejpam-4793	848	9	j.	j.	PROPN
ejpam-4793	848	10	vilela	vilela	PROPN
ejpam-4793	848	11	/	/	SYM
ejpam-4793	848	12	eur	eur	PROPN
ejpam-4793	848	13	.	.	PUNCT
ejpam-4793	849	1	j.	j.	PROPN
ejpam-4793	849	2	pure	pure	PROPN
ejpam-4793	849	3	appl	appl	PROPN
ejpam-4793	849	4	.	.	PROPN
ejpam-4793	849	5	math	math	PROPN
ejpam-4793	849	6	,	,	PUNCT
ejpam-4793	849	7	16	16	NUM
ejpam-4793	849	8	(	(	PUNCT
ejpam-4793	849	9	3	3	NUM
ejpam-4793	849	10	)	)	PUNCT
ejpam-4793	849	11	(	(	PUNCT
ejpam-4793	849	12	2023	2023	NUM
ejpam-4793	849	13	)	)	PUNCT
ejpam-4793	849	14	,	,	PUNCT
ejpam-4793	849	15	1772	1772	NUM
ejpam-4793	849	16	-	-	SYM
ejpam-4793	849	17	1793	1793	NUM
ejpam-4793	849	18	1791	1791	NUM
ejpam-4793	849	19	that	that	PRON
ejpam-4793	849	20	is	be	AUX
ejpam-4793	849	21	,	,	PUNCT
ejpam-4793	849	22	φ	φ	PROPN
ejpam-4793	849	23	◦	◦	PROPN
ejpam-4793	849	24	πker	πker	NOUN
ejpam-4793	849	25	f	f	PROPN
ejpam-4793	849	26	=	=	SYM
ejpam-4793	849	27	f	f	PROPN
ejpam-4793	849	28	,	,	PUNCT
ejpam-4793	849	29	where	where	SCONJ
ejpam-4793	849	30	πker	πker	NOUN
ejpam-4793	849	31	f	f	X
ejpam-4793	849	32	(	(	PUNCT
ejpam-4793	849	33	x	x	NOUN
ejpam-4793	849	34	)	)	PUNCT
ejpam-4793	849	35	:	:	PUNCT
ejpam-4793	850	1	=	=	SYM
ejpam-4793	850	2	ρker	ρker	NOUN
ejpam-4793	850	3	f	f	PROPN
ejpam-4793	850	4	(	(	PUNCT
ejpam-4793	850	5	x	x	NOUN
ejpam-4793	850	6	)	)	PUNCT
ejpam-4793	850	7	.	.	PUNCT
ejpam-4793	851	1	moreover	moreover	ADV
ejpam-4793	851	2	,	,	PUNCT
ejpam-4793	851	3	φ	φ	PROPN
ejpam-4793	851	4	is	be	AUX
ejpam-4793	851	5	onto	onto	ADP
ejpam-4793	851	6	and	and	CCONJ
ejpam-4793	851	7	it	it	PRON
ejpam-4793	851	8	has	have	VERB
ejpam-4793	851	9	a	a	DET
ejpam-4793	851	10	trivial	trivial	ADJ
ejpam-4793	851	11	kernel	kernel	NOUN
ejpam-4793	851	12	,	,	PUNCT
ejpam-4793	851	13	namely	namely	ADV
ejpam-4793	851	14	,	,	PUNCT
ejpam-4793	851	15	kerφ	kerφ	PROPN
ejpam-4793	851	16	=	=	PUNCT
ejpam-4793	851	17	{	{	PUNCT
ejpam-4793	851	18	ker	ker	NOUN
ejpam-4793	851	19	f	f	X
ejpam-4793	851	20	}	}	PUNCT
ejpam-4793	851	21	.	.	PUNCT
ejpam-4793	852	1	however	however	ADV
ejpam-4793	852	2	,	,	PUNCT
ejpam-4793	852	3	φ	φ	PROPN
ejpam-4793	852	4	is	be	AUX
ejpam-4793	852	5	a	a	DET
ejpam-4793	852	6	γ	γ	PROPN
ejpam-4793	852	7	-	-	PUNCT
ejpam-4793	852	8	monoid	monoid	NOUN
ejpam-4793	852	9	isomorphism	isomorphism	NOUN
ejpam-4793	852	10	if	if	SCONJ
ejpam-4793	852	11	and	and	CCONJ
ejpam-4793	852	12	only	only	ADV
ejpam-4793	852	13	if	if	SCONJ
ejpam-4793	852	14	ρf	ρf	PROPN
ejpam-4793	852	15	=	=	SYM
ejpam-4793	852	16	ρker	ρker	NOUN
ejpam-4793	852	17	f	f	PROPN
ejpam-4793	852	18	.	.	PUNCT
ejpam-4793	853	1	proof	proof	NOUN
ejpam-4793	853	2	.	.	PUNCT
ejpam-4793	854	1	let	let	VERB
ejpam-4793	854	2	(	(	PUNCT
ejpam-4793	854	3	m	m	NOUN
ejpam-4793	854	4	,	,	PUNCT
ejpam-4793	854	5	∗	∗	NOUN
ejpam-4793	854	6	)	)	PUNCT
ejpam-4793	854	7	and	and	CCONJ
ejpam-4793	854	8	(	(	PUNCT
ejpam-4793	854	9	n	n	CCONJ
ejpam-4793	854	10	,	,	PUNCT
ejpam-4793	854	11	·	·	PUNCT
ejpam-4793	854	12	)	)	PUNCT
ejpam-4793	854	13	be	be	AUX
ejpam-4793	854	14	commutative	commutative	ADJ
ejpam-4793	854	15	γ	γ	NOUN
ejpam-4793	854	16	-	-	PUNCT
ejpam-4793	854	17	monoids	monoid	NOUN
ejpam-4793	854	18	and	and	CCONJ
ejpam-4793	854	19	let	let	VERB
ejpam-4793	854	20	f	f	PRON
ejpam-4793	854	21	:	:	PUNCT
ejpam-4793	854	22	m	m	VERB
ejpam-4793	854	23	→	→	SYM
ejpam-4793	854	24	n	n	CCONJ
ejpam-4793	854	25	be	be	AUX
ejpam-4793	854	26	a	a	DET
ejpam-4793	854	27	γ	γ	NOUN
ejpam-4793	854	28	-	-	PUNCT
ejpam-4793	854	29	monoid	monoid	NOUN
ejpam-4793	854	30	homomorphism	homomorphism	NOUN
ejpam-4793	854	31	.	.	PUNCT
ejpam-4793	855	1	since	since	SCONJ
ejpam-4793	855	2	γ	γ	NOUN
ejpam-4793	855	3	-	-	PUNCT
ejpam-4793	855	4	monoids	monoid	NOUN
ejpam-4793	855	5	are	be	AUX
ejpam-4793	855	6	monoids	monoid	NOUN
ejpam-4793	855	7	and	and	CCONJ
ejpam-4793	855	8	γ	γ	PROPN
ejpam-4793	855	9	-	-	PUNCT
ejpam-4793	855	10	monoid	monoid	NOUN
ejpam-4793	855	11	homomorphism	homomorphism	NOUN
ejpam-4793	855	12	is	be	AUX
ejpam-4793	855	13	a	a	DET
ejpam-4793	855	14	monoid	monoid	NOUN
ejpam-4793	855	15	homomorphism	homomorphism	NOUN
ejpam-4793	855	16	,	,	PUNCT
ejpam-4793	855	17	by	by	ADP
ejpam-4793	855	18	theorem	theorem	NOUN
ejpam-4793	855	19	1	1	NUM
ejpam-4793	855	20	,	,	PUNCT
ejpam-4793	855	21	there	there	PRON
ejpam-4793	855	22	exists	exist	VERB
ejpam-4793	855	23	a	a	DET
ejpam-4793	855	24	unique	unique	ADJ
ejpam-4793	855	25	monoid	monoid	NOUN
ejpam-4793	855	26	homomorphism	homomorphism	NOUN
ejpam-4793	855	27	φ	φ	X
ejpam-4793	855	28	:	:	PUNCT
ejpam-4793	855	29	m/	m/	PROPN
ejpam-4793	855	30	ker	ker	PROPN
ejpam-4793	856	1	f	f	PROPN
ejpam-4793	856	2	→	→	PUNCT
ejpam-4793	856	3	n	n	X
ejpam-4793	856	4	such	such	ADJ
ejpam-4793	856	5	that	that	SCONJ
ejpam-4793	856	6	the	the	DET
ejpam-4793	856	7	following	follow	VERB
ejpam-4793	856	8	diagram	diagram	NOUN
ejpam-4793	856	9	is	be	AUX
ejpam-4793	856	10	commutative	commutative	ADJ
ejpam-4793	856	11	m	m	NOUN
ejpam-4793	856	12	n	n	NUM
ejpam-4793	856	13	m/	m/	NOUN
ejpam-4793	856	14	ker	ker	PROPN
ejpam-4793	857	1	f	f	PROPN
ejpam-4793	857	2	πker	πker	NOUN
ejpam-4793	857	3	f	f	PROPN
ejpam-4793	857	4	f	f	PROPN
ejpam-4793	857	5	φ	φ	PROPN
ejpam-4793	857	6	that	that	PRON
ejpam-4793	857	7	is	is	ADV
ejpam-4793	857	8	,	,	PUNCT
ejpam-4793	857	9	φ	φ	PROPN
ejpam-4793	858	1	◦	◦	NOUN
ejpam-4793	859	1	πker	πker	NOUN
ejpam-4793	859	2	f	f	PROPN
ejpam-4793	859	3	=	=	SYM
ejpam-4793	859	4	f	f	PROPN
ejpam-4793	859	5	,	,	PUNCT
ejpam-4793	859	6	where	where	SCONJ
ejpam-4793	859	7	πker	πker	NOUN
ejpam-4793	859	8	f	f	X
ejpam-4793	859	9	(	(	PUNCT
ejpam-4793	859	10	x	x	NOUN
ejpam-4793	859	11	)	)	PUNCT
ejpam-4793	859	12	:	:	PUNCT
ejpam-4793	859	13	=	=	SYM
ejpam-4793	859	14	ρker	ρker	NOUN
ejpam-4793	859	15	f	f	PROPN
ejpam-4793	859	16	(	(	PUNCT
ejpam-4793	859	17	x	x	NOUN
ejpam-4793	859	18	)	)	PUNCT
ejpam-4793	859	19	.	.	PUNCT
ejpam-4793	860	1	moreover	moreover	ADV
ejpam-4793	860	2	,	,	PUNCT
ejpam-4793	860	3	φ	φ	PROPN
ejpam-4793	860	4	is	be	AUX
ejpam-4793	860	5	onto	onto	ADP
ejpam-4793	860	6	and	and	CCONJ
ejpam-4793	860	7	it	it	PRON
ejpam-4793	860	8	has	have	VERB
ejpam-4793	860	9	a	a	DET
ejpam-4793	860	10	trivial	trivial	ADJ
ejpam-4793	860	11	kernel	kernel	NOUN
ejpam-4793	860	12	,	,	PUNCT
ejpam-4793	860	13	namely	namely	ADV
ejpam-4793	860	14	,	,	PUNCT
ejpam-4793	860	15	kerφ	kerφ	PROPN
ejpam-4793	860	16	=	=	PUNCT
ejpam-4793	860	17	{	{	PUNCT
ejpam-4793	860	18	ker	ker	NOUN
ejpam-4793	860	19	f	f	X
ejpam-4793	860	20	}	}	PUNCT
ejpam-4793	860	21	.	.	PUNCT
ejpam-4793	861	1	however	however	ADV
ejpam-4793	861	2	,	,	PUNCT
ejpam-4793	861	3	φ	φ	PROPN
ejpam-4793	861	4	is	be	AUX
ejpam-4793	861	5	an	an	DET
ejpam-4793	861	6	isomorphism	isomorphism	NOUN
ejpam-4793	861	7	if	if	SCONJ
ejpam-4793	861	8	and	and	CCONJ
ejpam-4793	861	9	only	only	ADV
ejpam-4793	861	10	if	if	SCONJ
ejpam-4793	861	11	ρf	ρf	PROPN
ejpam-4793	861	12	=	=	SYM
ejpam-4793	861	13	ρker	ρker	NOUN
ejpam-4793	861	14	f	f	PROPN
ejpam-4793	861	15	.	.	PUNCT
ejpam-4793	862	1	thus	thus	ADV
ejpam-4793	862	2	,	,	PUNCT
ejpam-4793	862	3	it	it	PRON
ejpam-4793	862	4	remains	remain	VERB
ejpam-4793	862	5	to	to	PART
ejpam-4793	862	6	show	show	VERB
ejpam-4793	862	7	that	that	SCONJ
ejpam-4793	862	8	φ	φ	PROPN
ejpam-4793	862	9	is	be	AUX
ejpam-4793	862	10	a	a	DET
ejpam-4793	862	11	γ	γ	NOUN
ejpam-4793	862	12	-	-	PUNCT
ejpam-4793	862	13	monoid	monoid	NOUN
ejpam-4793	862	14	homomorphism	homomorphism	NOUN
ejpam-4793	862	15	.	.	PUNCT
ejpam-4793	863	1	now	now	ADV
ejpam-4793	863	2	,	,	PUNCT
ejpam-4793	863	3	let	let	VERB
ejpam-4793	863	4	ρker	ρker	NOUN
ejpam-4793	863	5	f	f	PROPN
ejpam-4793	863	6	(	(	PUNCT
ejpam-4793	863	7	x	x	X
ejpam-4793	863	8	)	)	PUNCT
ejpam-4793	863	9	∈	∈	PROPN
ejpam-4793	863	10	m/	m/	NOUN
ejpam-4793	864	1	ker	ker	PROPN
ejpam-4793	865	1	f	f	PROPN
ejpam-4793	865	2	and	and	CCONJ
ejpam-4793	865	3	α	α	PROPN
ejpam-4793	865	4	∈	∈	PROPN
ejpam-4793	865	5	γ	γ	X
ejpam-4793	865	6	.	.	PROPN
ejpam-4793	866	1	since	since	SCONJ
ejpam-4793	866	2	f	f	PROPN
ejpam-4793	866	3	is	be	AUX
ejpam-4793	866	4	a	a	DET
ejpam-4793	866	5	γ	γ	PROPN
ejpam-4793	866	6	-	-	PUNCT
ejpam-4793	866	7	monoid	monoid	NOUN
ejpam-4793	866	8	homomorphism	homomorphism	NOUN
ejpam-4793	866	9	,	,	PUNCT
ejpam-4793	866	10	we	we	PRON
ejpam-4793	866	11	have	have	VERB
ejpam-4793	866	12	φ(αρker	φ(αρker	NUM
ejpam-4793	866	13	f	f	NOUN
ejpam-4793	866	14	(	(	PUNCT
ejpam-4793	866	15	x	x	NOUN
ejpam-4793	866	16	)	)	PUNCT
ejpam-4793	866	17	)	)	PUNCT
ejpam-4793	867	1	=	=	SYM
ejpam-4793	867	2	φ(ρker	φ(ρker	NOUN
ejpam-4793	867	3	f	f	PROPN
ejpam-4793	867	4	(	(	PUNCT
ejpam-4793	867	5	αx	αx	NOUN
ejpam-4793	867	6	)	)	PUNCT
ejpam-4793	867	7	)	)	PUNCT
ejpam-4793	868	1	=	=	PUNCT
ejpam-4793	868	2	f(αx	f(αx	NOUN
ejpam-4793	868	3	)	)	PUNCT
ejpam-4793	868	4	=	=	SYM
ejpam-4793	868	5	αf(x	αf(x	NUM
ejpam-4793	868	6	)	)	PUNCT
ejpam-4793	868	7	=	=	SYM
ejpam-4793	869	1	αφ(ρker	αφ(ρker	X
ejpam-4793	869	2	f	f	X
ejpam-4793	869	3	(	(	PUNCT
ejpam-4793	869	4	x	x	NOUN
ejpam-4793	869	5	)	)	PUNCT
ejpam-4793	869	6	)	)	PUNCT
ejpam-4793	869	7	.	.	PUNCT
ejpam-4793	870	1	hence	hence	ADV
ejpam-4793	870	2	,	,	PUNCT
ejpam-4793	870	3	φ	φ	PROPN
ejpam-4793	870	4	is	be	AUX
ejpam-4793	870	5	a	a	DET
ejpam-4793	870	6	γ	γ	NOUN
ejpam-4793	870	7	-	-	PUNCT
ejpam-4793	870	8	monoid	monoid	NOUN
ejpam-4793	870	9	homomorphism	homomorphism	NOUN
ejpam-4793	870	10	.	.	PUNCT
ejpam-4793	871	1	corollary	corollary	ADJ
ejpam-4793	871	2	1	1	NUM
ejpam-4793	871	3	.	.	PUNCT
ejpam-4793	872	1	let	let	AUX
ejpam-4793	872	2	m	m	PRON
ejpam-4793	872	3	and	and	CCONJ
ejpam-4793	872	4	n	n	ADV
ejpam-4793	872	5	be	be	AUX
ejpam-4793	872	6	commutative	commutative	ADJ
ejpam-4793	872	7	γ	γ	NOUN
ejpam-4793	872	8	-	-	PUNCT
ejpam-4793	872	9	monoids	monoid	NOUN
ejpam-4793	872	10	and	and	CCONJ
ejpam-4793	872	11	f	f	NOUN
ejpam-4793	872	12	:	:	PUNCT
ejpam-4793	872	13	m	m	VERB
ejpam-4793	872	14	→	→	SYM
ejpam-4793	872	15	n	n	CCONJ
ejpam-4793	872	16	be	be	AUX
ejpam-4793	872	17	a	a	DET
ejpam-4793	872	18	γ	γ	NOUN
ejpam-4793	872	19	-	-	PUNCT
ejpam-4793	872	20	monoid	monoid	NOUN
ejpam-4793	872	21	homomorphism	homomorphism	NOUN
ejpam-4793	872	22	.	.	PUNCT
ejpam-4793	873	1	then	then	ADV
ejpam-4793	873	2	f	f	PROPN
ejpam-4793	873	3	induces	induce	VERB
ejpam-4793	873	4	a	a	DET
ejpam-4793	873	5	γ	γ	PROPN
ejpam-4793	873	6	-	-	PUNCT
ejpam-4793	873	7	monoid	monoid	NOUN
ejpam-4793	873	8	isomorphism	isomorphism	NOUN
ejpam-4793	873	9	m/	m/	VERB
ejpam-4793	873	10	ker	ker	PROPN
ejpam-4793	873	11	f	f	PROPN
ejpam-4793	873	12	∼=	∼=	PROPN
ejpam-4793	873	13	imf	imf	PROPN
ejpam-4793	873	14	.	.	PUNCT
ejpam-4793	874	1	proof	proof	NOUN
ejpam-4793	874	2	.	.	PUNCT
ejpam-4793	875	1	suppose	suppose	VERB
ejpam-4793	876	1	f	f	X
ejpam-4793	876	2	:	:	PUNCT
ejpam-4793	876	3	m	m	VERB
ejpam-4793	876	4	→	→	SYM
ejpam-4793	876	5	n	n	X
ejpam-4793	876	6	is	be	AUX
ejpam-4793	876	7	a	a	DET
ejpam-4793	876	8	γ	γ	NOUN
ejpam-4793	876	9	-	-	PUNCT
ejpam-4793	876	10	monoid	monoid	NOUN
ejpam-4793	876	11	homomorphism	homomorphism	NOUN
ejpam-4793	876	12	.	.	PUNCT
ejpam-4793	877	1	then	then	ADV
ejpam-4793	877	2	,	,	PUNCT
ejpam-4793	877	3	by	by	ADP
ejpam-4793	877	4	theorem	theorem	NOUN
ejpam-4793	877	5	13	13	NUM
ejpam-4793	877	6	,	,	PUNCT
ejpam-4793	877	7	there	there	PRON
ejpam-4793	877	8	exists	exist	VERB
ejpam-4793	877	9	a	a	DET
ejpam-4793	877	10	γ	γ	PROPN
ejpam-4793	877	11	-	-	PUNCT
ejpam-4793	877	12	monoid	monoid	NOUN
ejpam-4793	877	13	homomorphism	homomorphism	PROPN
ejpam-4793	877	14	φ	φ	PROPN
ejpam-4793	877	15	:	:	PUNCT
ejpam-4793	877	16	m/	m/	PROPN
ejpam-4793	877	17	ker	ker	PROPN
ejpam-4793	878	1	f	f	PROPN
ejpam-4793	878	2	→	→	SYM
ejpam-4793	878	3	n	n	PROPN
ejpam-4793	878	4	.	.	PUNCT
ejpam-4793	879	1	if	if	SCONJ
ejpam-4793	879	2	we	we	PRON
ejpam-4793	879	3	set	set	VERB
ejpam-4793	879	4	n	n	NOUN
ejpam-4793	879	5	=	=	PROPN
ejpam-4793	879	6	imf	imf	PROPN
ejpam-4793	879	7	,	,	PUNCT
ejpam-4793	879	8	then	then	ADV
ejpam-4793	879	9	φ	φ	PROPN
ejpam-4793	879	10	:	:	PUNCT
ejpam-4793	879	11	m/	m/	PROPN
ejpam-4793	879	12	ker	ker	PROPN
ejpam-4793	880	1	f	f	PROPN
ejpam-4793	880	2	→	→	PUNCT
ejpam-4793	880	3	imf	imf	PROPN
ejpam-4793	880	4	is	be	AUX
ejpam-4793	880	5	a	a	DET
ejpam-4793	880	6	γ	γ	PROPN
ejpam-4793	880	7	-	-	PUNCT
ejpam-4793	880	8	monoid	monoid	NOUN
ejpam-4793	880	9	epimorphism	epimorphism	NOUN
ejpam-4793	880	10	.	.	PUNCT
ejpam-4793	881	1	thus	thus	ADV
ejpam-4793	881	2	,	,	PUNCT
ejpam-4793	881	3	kerφ	kerφ	PROPN
ejpam-4793	881	4	=	=	PUNCT
ejpam-4793	881	5	{	{	PUNCT
ejpam-4793	881	6	ρker	ρker	NOUN
ejpam-4793	881	7	f	f	PROPN
ejpam-4793	881	8	(	(	PUNCT
ejpam-4793	881	9	x	x	X
ejpam-4793	881	10	)	)	PUNCT
ejpam-4793	881	11	:	:	PUNCT
ejpam-4793	881	12	f(x	f(x	PROPN
ejpam-4793	881	13	)	)	PUNCT
ejpam-4793	881	14	=	=	SYM
ejpam-4793	881	15	1n	1n	NUM
ejpam-4793	881	16	}	}	PUNCT
ejpam-4793	881	17	=	=	PRON
ejpam-4793	881	18	{	{	PUNCT
ejpam-4793	881	19	ker	ker	NOUN
ejpam-4793	881	20	f	f	X
ejpam-4793	881	21	}	}	PUNCT
ejpam-4793	881	22	implies	imply	VERB
ejpam-4793	881	23	that	that	SCONJ
ejpam-4793	881	24	ρker	ρker	PROPN
ejpam-4793	881	25	f	f	PROPN
ejpam-4793	881	26	(	(	PUNCT
ejpam-4793	881	27	x	x	X
ejpam-4793	881	28	)	)	PUNCT
ejpam-4793	882	1	=	=	SYM
ejpam-4793	882	2	ker	ker	PROPN
ejpam-4793	883	1	f	f	PROPN
ejpam-4793	883	2	and	and	CCONJ
ejpam-4793	883	3	x	x	PROPN
ejpam-4793	883	4	∈	∈	PROPN
ejpam-4793	884	1	ker	ker	PROPN
ejpam-4793	885	1	f	f	X
ejpam-4793	885	2	.	.	PUNCT
ejpam-4793	886	1	hence	hence	ADV
ejpam-4793	886	2	,	,	PUNCT
ejpam-4793	886	3	by	by	ADP
ejpam-4793	886	4	proposition	proposition	NOUN
ejpam-4793	886	5	4	4	NUM
ejpam-4793	886	6	,	,	PUNCT
ejpam-4793	886	7	ρker	ρker	NOUN
ejpam-4793	886	8	f	f	PROPN
ejpam-4793	886	9	(	(	PUNCT
ejpam-4793	886	10	x	x	X
ejpam-4793	886	11	)	)	PUNCT
ejpam-4793	886	12	=	=	SYM
ejpam-4793	886	13	ρker	ρker	NOUN
ejpam-4793	886	14	f	f	PROPN
ejpam-4793	886	15	(	(	PUNCT
ejpam-4793	886	16	1	1	NUM
ejpam-4793	886	17	m	m	NOUN
ejpam-4793	886	18	)	)	PUNCT
ejpam-4793	886	19	which	which	PRON
ejpam-4793	886	20	implies	imply	VERB
ejpam-4793	886	21	that	that	SCONJ
ejpam-4793	886	22	kerφ	kerφ	PROPN
ejpam-4793	886	23	=	=	PUNCT
ejpam-4793	886	24	{	{	PUNCT
ejpam-4793	886	25	ρker	ρker	NOUN
ejpam-4793	886	26	f	f	PROPN
ejpam-4793	886	27	(	(	PUNCT
ejpam-4793	886	28	1	1	NUM
ejpam-4793	886	29	m	m	NOUN
ejpam-4793	886	30	)	)	PUNCT
ejpam-4793	886	31	}	}	PUNCT
ejpam-4793	886	32	and	and	CCONJ
ejpam-4793	886	33	φ	φ	PROPN
ejpam-4793	886	34	is	be	AUX
ejpam-4793	886	35	injective	injective	ADJ
ejpam-4793	886	36	.	.	PUNCT
ejpam-4793	887	1	accordingly	accordingly	ADV
ejpam-4793	887	2	,	,	PUNCT
ejpam-4793	887	3	m/	m/	VERB
ejpam-4793	887	4	ker	ker	PROPN
ejpam-4793	887	5	f	f	PROPN
ejpam-4793	887	6	∼=	∼=	PROPN
ejpam-4793	887	7	imf	imf	PROPN
ejpam-4793	887	8	.	.	PUNCT
ejpam-4793	888	1	corollary	corollary	ADJ
ejpam-4793	888	2	2	2	NUM
ejpam-4793	888	3	.	.	PUNCT
ejpam-4793	889	1	let	let	VERB
ejpam-4793	889	2	k	k	NOUN
ejpam-4793	889	3	and	and	CCONJ
ejpam-4793	889	4	l	l	PROPN
ejpam-4793	889	5	be	be	AUX
ejpam-4793	889	6	normal	normal	ADJ
ejpam-4793	889	7	γ	γ	NOUN
ejpam-4793	889	8	-	-	NOUN
ejpam-4793	889	9	submonoids	submonoid	NOUN
ejpam-4793	889	10	of	of	ADP
ejpam-4793	889	11	a	a	DET
ejpam-4793	889	12	commutative	commutative	ADJ
ejpam-4793	889	13	γ	γ	X
ejpam-4793	889	14	-	-	PUNCT
ejpam-4793	889	15	monoid	monoid	NOUN
ejpam-4793	889	16	m	m	PROPN
ejpam-4793	889	17	.	.	PUNCT
ejpam-4793	890	1	then	then	ADV
ejpam-4793	890	2	k/(k	k/(k	NOUN
ejpam-4793	890	3	∩	∩	ADJ
ejpam-4793	890	4	l	l	NOUN
ejpam-4793	890	5	)	)	PUNCT
ejpam-4793	890	6	∼=	∼=	PROPN
ejpam-4793	890	7	(	(	PUNCT
ejpam-4793	890	8	k	k	PROPN
ejpam-4793	890	9	∗	∗	PROPN
ejpam-4793	890	10	l)/l	l)/l	PROPN
ejpam-4793	890	11	.	.	PUNCT
ejpam-4793	891	1	proof	proof	NOUN
ejpam-4793	891	2	.	.	PUNCT
ejpam-4793	892	1	consider	consider	VERB
ejpam-4793	893	1	the	the	DET
ejpam-4793	893	2	map	map	NOUN
ejpam-4793	893	3	f	f	X
ejpam-4793	893	4	:	:	PUNCT
ejpam-4793	893	5	k	k	PROPN
ejpam-4793	893	6	→	→	SYM
ejpam-4793	893	7	k	k	PROPN
ejpam-4793	893	8	∗	∗	X
ejpam-4793	893	9	l	l	NOUN
ejpam-4793	893	10	defined	define	VERB
ejpam-4793	893	11	by	by	ADP
ejpam-4793	893	12	f(k	f(k	NOUN
ejpam-4793	893	13	)	)	PUNCT
ejpam-4793	893	14	=	=	SYM
ejpam-4793	894	1	k	k	PROPN
ejpam-4793	894	2	∗	∗	PROPN
ejpam-4793	894	3	1	1	NUM
ejpam-4793	894	4	m	m	VERB
ejpam-4793	894	5	and	and	CCONJ
ejpam-4793	894	6	πl	πl	NOUN
ejpam-4793	894	7	:	:	PUNCT
ejpam-4793	894	8	k	k	X
ejpam-4793	894	9	∗	∗	X
ejpam-4793	894	10	l	l	NOUN
ejpam-4793	894	11	→	→	PUNCT
ejpam-4793	894	12	(	(	PUNCT
ejpam-4793	894	13	k∗l)/l	k∗l)/l	NOUN
ejpam-4793	894	14	defined	define	VERB
ejpam-4793	894	15	by	by	ADP
ejpam-4793	894	16	πl(k∗l	πl(k∗l	X
ejpam-4793	894	17	)	)	PUNCT
ejpam-4793	894	18	=	=	PUNCT
ejpam-4793	894	19	ρl(k∗l	ρl(k∗l	ADV
ejpam-4793	894	20	)	)	PUNCT
ejpam-4793	894	21	.	.	PUNCT
ejpam-4793	895	1	then	then	ADV
ejpam-4793	895	2	φ	φ	X
ejpam-4793	895	3	:	:	PUNCT
ejpam-4793	896	1	k	k	X
ejpam-4793	896	2	→	→	PUNCT
ejpam-4793	896	3	(	(	PUNCT
ejpam-4793	896	4	k∗l)/l	k∗l)/l	NOUN
ejpam-4793	896	5	defined	define	VERB
ejpam-4793	896	6	by	by	ADP
ejpam-4793	896	7	φ(k	φ(k	PROPN
ejpam-4793	896	8	)	)	PUNCT
ejpam-4793	896	9	=	=	SYM
ejpam-4793	896	10	ρl(k	ρl(k	X
ejpam-4793	896	11	)	)	PUNCT
ejpam-4793	896	12	is	be	AUX
ejpam-4793	896	13	a	a	DET
ejpam-4793	896	14	γ	γ	NOUN
ejpam-4793	896	15	-	-	PUNCT
ejpam-4793	896	16	monoid	monoid	NOUN
ejpam-4793	896	17	homomorphism	homomorphism	NOUN
ejpam-4793	896	18	.	.	PUNCT
ejpam-4793	897	1	let	let	VERB
ejpam-4793	897	2	x	x	X
ejpam-4793	897	3	∈	∈	PROPN
ejpam-4793	897	4	(	(	PUNCT
ejpam-4793	897	5	k	k	PROPN
ejpam-4793	897	6	∗	∗	PROPN
ejpam-4793	897	7	l)/l	l)/l	PROPN
ejpam-4793	897	8	.	.	PUNCT
ejpam-4793	898	1	then	then	ADV
ejpam-4793	898	2	x	x	X
ejpam-4793	898	3	=	=	NOUN
ejpam-4793	898	4	ρl(k	ρl(k	NOUN
ejpam-4793	898	5	∗	∗	NOUN
ejpam-4793	898	6	l	l	NOUN
ejpam-4793	898	7	)	)	PUNCT
ejpam-4793	898	8	for	for	ADP
ejpam-4793	898	9	some	some	DET
ejpam-4793	898	10	k	k	PROPN
ejpam-4793	898	11	∈	∈	PROPN
ejpam-4793	898	12	k	k	PROPN
ejpam-4793	898	13	and	and	CCONJ
ejpam-4793	898	14	l	l	PROPN
ejpam-4793	898	15	∈	∈	PROPN
ejpam-4793	898	16	l.	l.	PROPN
ejpam-4793	898	17	observe	observe	VERB
ejpam-4793	898	18	that	that	SCONJ
ejpam-4793	898	19	x	x	NOUN
ejpam-4793	899	1	=	=	NOUN
ejpam-4793	899	2	ρl(k	ρl(k	NOUN
ejpam-4793	899	3	∗	∗	NOUN
ejpam-4793	899	4	l	l	NOUN
ejpam-4793	899	5	)	)	PUNCT
ejpam-4793	899	6	=	=	SYM
ejpam-4793	899	7	ρl(k)	ρl(k)	NOUN
ejpam-4793	899	8	◦	◦	NOUN
ejpam-4793	899	9	ρl(l	ρl(l	NUM
ejpam-4793	899	10	)	)	PUNCT
ejpam-4793	899	11	=	=	NOUN
ejpam-4793	899	12	ρl(k)	ρl(k)	NOUN
ejpam-4793	899	13	◦	◦	NOUN
ejpam-4793	899	14	ρl(1	ρl(1	NOUN
ejpam-4793	899	15	m	m	NOUN
ejpam-4793	899	16	)	)	PUNCT
ejpam-4793	899	17	=	=	SYM
ejpam-4793	899	18	ρl(k	ρl(k	NUM
ejpam-4793	899	19	)	)	PUNCT
ejpam-4793	899	20	.	.	PUNCT
ejpam-4793	900	1	so	so	ADV
ejpam-4793	900	2	,	,	PUNCT
ejpam-4793	900	3	there	there	PRON
ejpam-4793	900	4	is	be	VERB
ejpam-4793	900	5	a	a	DET
ejpam-4793	900	6	k	k	PROPN
ejpam-4793	900	7	∈	∈	PROPN
ejpam-4793	900	8	k	k	ADP
ejpam-4793	901	1	such	such	ADJ
ejpam-4793	901	2	that	that	SCONJ
ejpam-4793	901	3	φ(k	φ(k	PROPN
ejpam-4793	901	4	)	)	PUNCT
ejpam-4793	901	5	=	=	SYM
ejpam-4793	901	6	ρl(k	ρl(k	NUM
ejpam-4793	901	7	)	)	PUNCT
ejpam-4793	901	8	=	=	SYM
ejpam-4793	902	1	x	x	NOUN
ejpam-4793	902	2	and	and	CCONJ
ejpam-4793	902	3	φ	φ	PROPN
ejpam-4793	902	4	is	be	AUX
ejpam-4793	902	5	onto	onto	ADP
ejpam-4793	902	6	.	.	PUNCT
ejpam-4793	903	1	moreover	moreover	ADV
ejpam-4793	903	2	,	,	PUNCT
ejpam-4793	903	3	kerφ	kerφ	PROPN
ejpam-4793	903	4	=	=	PUNCT
ejpam-4793	903	5	{	{	PUNCT
ejpam-4793	903	6	k	k	PROPN
ejpam-4793	903	7	∈	∈	PROPN
ejpam-4793	903	8	k	k	X
ejpam-4793	903	9	:	:	PUNCT
ejpam-4793	903	10	ρl(k	ρl(k	NUM
ejpam-4793	903	11	)	)	PUNCT
ejpam-4793	904	1	=	=	PUNCT
ejpam-4793	904	2	ρl(1	ρl(1	PROPN
ejpam-4793	904	3	m	m	NOUN
ejpam-4793	904	4	)	)	PUNCT
ejpam-4793	904	5	}	}	PUNCT
ejpam-4793	905	1	=	=	PRON
ejpam-4793	905	2	{	{	PUNCT
ejpam-4793	905	3	k	k	PROPN
ejpam-4793	905	4	∈	∈	PROPN
ejpam-4793	906	1	k	k	X
ejpam-4793	906	2	:	:	PUNCT
ejpam-4793	906	3	k	k	PROPN
ejpam-4793	906	4	∈	∈	PROPN
ejpam-4793	906	5	l	l	NOUN
ejpam-4793	906	6	}	}	PUNCT
ejpam-4793	906	7	=	=	SYM
ejpam-4793	906	8	k	k	PROPN
ejpam-4793	906	9	∩	∩	PROPN
ejpam-4793	906	10	l.	l.	NOUN
ejpam-4793	906	11	by	by	ADP
ejpam-4793	906	12	corollary	corollary	ADJ
ejpam-4793	906	13	1	1	NUM
ejpam-4793	906	14	,	,	PUNCT
ejpam-4793	906	15	k/	k/	NOUN
ejpam-4793	906	16	kerφ	kerφ	PROPN
ejpam-4793	906	17	∼=	∼=	PROPN
ejpam-4793	906	18	imφ	imφ	NOUN
ejpam-4793	906	19	=	=	SYM
ejpam-4793	906	20	(	(	PUNCT
ejpam-4793	906	21	k	k	PROPN
ejpam-4793	906	22	∗	∗	PROPN
ejpam-4793	906	23	l)/l	l)/l	PROPN
ejpam-4793	906	24	.	.	PUNCT
ejpam-4793	907	1	the	the	DET
ejpam-4793	907	2	following	follow	VERB
ejpam-4793	907	3	theorem	theorem	NOUN
ejpam-4793	907	4	is	be	AUX
ejpam-4793	907	5	the	the	DET
ejpam-4793	907	6	counterpart	counterpart	NOUN
ejpam-4793	907	7	to	to	ADP
ejpam-4793	907	8	the	the	DET
ejpam-4793	907	9	third	third	ADJ
ejpam-4793	907	10	isomorphism	isomorphism	NOUN
ejpam-4793	907	11	theorem	theorem	NOUN
ejpam-4793	907	12	of	of	ADP
ejpam-4793	907	13	groups	group	NOUN
ejpam-4793	907	14	for	for	ADP
ejpam-4793	907	15	γ	γ	NOUN
ejpam-4793	907	16	-	-	PUNCT
ejpam-4793	907	17	monoids	monoid	NOUN
ejpam-4793	907	18	via	via	ADP
ejpam-4793	907	19	γ	γ	NOUN
ejpam-4793	907	20	-	-	PUNCT
ejpam-4793	907	21	submonoids	submonoid	NOUN
ejpam-4793	907	22	.	.	PUNCT
ejpam-4793	908	1	h.	h.	PROPN
ejpam-4793	908	2	sarapuddin	sarapuddin	PROPN
ejpam-4793	908	3	,	,	PUNCT
ejpam-4793	908	4	j.	j.	PROPN
ejpam-4793	908	5	vilela	vilela	PROPN
ejpam-4793	908	6	/	/	SYM
ejpam-4793	908	7	eur	eur	PROPN
ejpam-4793	908	8	.	.	PUNCT
ejpam-4793	909	1	j.	j.	PROPN
ejpam-4793	909	2	pure	pure	PROPN
ejpam-4793	909	3	appl	appl	PROPN
ejpam-4793	909	4	.	.	PROPN
ejpam-4793	909	5	math	math	PROPN
ejpam-4793	909	6	,	,	PUNCT
ejpam-4793	909	7	16	16	NUM
ejpam-4793	909	8	(	(	PUNCT
ejpam-4793	909	9	3	3	NUM
ejpam-4793	909	10	)	)	PUNCT
ejpam-4793	909	11	(	(	PUNCT
ejpam-4793	909	12	2023	2023	NUM
ejpam-4793	909	13	)	)	PUNCT
ejpam-4793	909	14	,	,	PUNCT
ejpam-4793	909	15	1772	1772	NUM
ejpam-4793	909	16	-	-	SYM
ejpam-4793	909	17	1793	1793	NUM
ejpam-4793	909	18	1792	1792	NUM
ejpam-4793	909	19	theorem	theorem	VERB
ejpam-4793	909	20	14	14	NUM
ejpam-4793	909	21	.	.	PUNCT
ejpam-4793	910	1	let	let	VERB
ejpam-4793	910	2	s	s	PRON
ejpam-4793	910	3	and	and	CCONJ
ejpam-4793	910	4	t	t	PROPN
ejpam-4793	910	5	be	be	AUX
ejpam-4793	910	6	normal	normal	ADJ
ejpam-4793	910	7	γ	γ	NOUN
ejpam-4793	910	8	-	-	NOUN
ejpam-4793	910	9	submonoids	submonoid	NOUN
ejpam-4793	910	10	of	of	ADP
ejpam-4793	910	11	a	a	DET
ejpam-4793	910	12	commutative	commutative	ADJ
ejpam-4793	910	13	γ	γ	X
ejpam-4793	910	14	-	-	PUNCT
ejpam-4793	910	15	monoid	monoid	NOUN
ejpam-4793	910	16	m	m	NOUN
ejpam-4793	910	17	with	with	ADP
ejpam-4793	910	18	s	s	PROPN
ejpam-4793	910	19	⊆	⊆	NUM
ejpam-4793	910	20	t	t	NOUN
ejpam-4793	910	21	.	.	PUNCT
ejpam-4793	911	1	then	then	ADV
ejpam-4793	911	2	(	(	PUNCT
ejpam-4793	911	3	m	m	NOUN
ejpam-4793	911	4	/	/	SYM
ejpam-4793	911	5	s)/(t	s)/(t	PROPN
ejpam-4793	911	6	/	/	SYM
ejpam-4793	911	7	s	s	NOUN
ejpam-4793	911	8	)	)	PUNCT
ejpam-4793	911	9	∼=	∼=	PROPN
ejpam-4793	911	10	m	m	NOUN
ejpam-4793	911	11	/	/	SYM
ejpam-4793	911	12	t	t	PROPN
ejpam-4793	911	13	.	.	PUNCT
ejpam-4793	912	1	proof	proof	NOUN
ejpam-4793	912	2	.	.	PUNCT
ejpam-4793	913	1	define	define	VERB
ejpam-4793	913	2	f	f	X
ejpam-4793	913	3	:	:	PUNCT
ejpam-4793	913	4	m	m	PROPN
ejpam-4793	913	5	/	/	SYM
ejpam-4793	913	6	s	s	X
ejpam-4793	913	7	→	→	SYM
ejpam-4793	913	8	m	m	PROPN
ejpam-4793	913	9	/	/	SYM
ejpam-4793	913	10	t	t	PROPN
ejpam-4793	913	11	by	by	ADP
ejpam-4793	913	12	f(ρs(h	f(ρs(h	PROPN
ejpam-4793	913	13	)	)	PUNCT
ejpam-4793	913	14	)	)	PUNCT
ejpam-4793	914	1	=	=	SYM
ejpam-4793	914	2	ρt	ρt	PROPN
ejpam-4793	914	3	(	(	PUNCT
ejpam-4793	914	4	h	h	NOUN
ejpam-4793	914	5	)	)	PUNCT
ejpam-4793	914	6	for	for	ADP
ejpam-4793	914	7	all	all	DET
ejpam-4793	914	8	ρs(h	ρs(h	NUM
ejpam-4793	914	9	)	)	PUNCT
ejpam-4793	914	10	∈	∈	PROPN
ejpam-4793	914	11	m	m	PROPN
ejpam-4793	914	12	/	/	SYM
ejpam-4793	914	13	s.	s.	PROPN
ejpam-4793	914	14	let	let	VERB
ejpam-4793	914	15	ρs(h1	ρs(h1	ADV
ejpam-4793	914	16	)	)	PUNCT
ejpam-4793	914	17	,	,	PUNCT
ejpam-4793	914	18	ρs(h2	ρs(h2	NOUN
ejpam-4793	914	19	)	)	PUNCT
ejpam-4793	914	20	∈	∈	PROPN
ejpam-4793	914	21	m	m	PROPN
ejpam-4793	914	22	/	/	SYM
ejpam-4793	914	23	s	s	X
ejpam-4793	914	24	and	and	CCONJ
ejpam-4793	914	25	suppose	suppose	VERB
ejpam-4793	914	26	that	that	SCONJ
ejpam-4793	914	27	ρs(h1	ρs(h1	NOUN
ejpam-4793	914	28	)	)	PUNCT
ejpam-4793	914	29	=	=	SYM
ejpam-4793	914	30	ρs(h2	ρs(h2	NOUN
ejpam-4793	914	31	)	)	PUNCT
ejpam-4793	914	32	.	.	PUNCT
ejpam-4793	915	1	then	then	ADV
ejpam-4793	915	2	,	,	PUNCT
ejpam-4793	915	3	(	(	PUNCT
ejpam-4793	915	4	αh1	αh1	PROPN
ejpam-4793	915	5	∗s)∩	∗s)∩	PROPN
ejpam-4793	915	6	(	(	PUNCT
ejpam-4793	915	7	αh2	αh2	NOUN
ejpam-4793	915	8	∗s	∗s	ADJ
ejpam-4793	915	9	)	)	PUNCT
ejpam-4793	915	10	̸=	̸=	NOUN
ejpam-4793	915	11	∅	∅	NOUN
ejpam-4793	915	12	for	for	ADP
ejpam-4793	915	13	all	all	PRON
ejpam-4793	915	14	α	α	PRON
ejpam-4793	915	15	∈	∈	PROPN
ejpam-4793	915	16	γ	γ	X
ejpam-4793	915	17	.	.	PUNCT
ejpam-4793	916	1	thus	thus	ADV
ejpam-4793	916	2	,	,	PUNCT
ejpam-4793	916	3	αh1	αh1	DET
ejpam-4793	916	4	∗w1	∗w1	ADJ
ejpam-4793	916	5	=	=	PUNCT
ejpam-4793	916	6	αh2	αh2	NOUN
ejpam-4793	916	7	∗w2	∗w2	NOUN
ejpam-4793	916	8	for	for	ADP
ejpam-4793	916	9	some	some	DET
ejpam-4793	916	10	w1	w1	NOUN
ejpam-4793	916	11	,	,	PUNCT
ejpam-4793	916	12	w2	w2	NOUN
ejpam-4793	916	13	∈	∈	PROPN
ejpam-4793	916	14	s	s	PART
ejpam-4793	916	15	⊆	⊆	NUM
ejpam-4793	916	16	t	t	NOUN
ejpam-4793	916	17	.	.	PUNCT
ejpam-4793	917	1	thus	thus	ADV
ejpam-4793	917	2	,	,	PUNCT
ejpam-4793	917	3	(	(	PUNCT
ejpam-4793	917	4	αh1	αh1	X
ejpam-4793	917	5	∗t	∗t	ADJ
ejpam-4793	917	6	)	)	PUNCT
ejpam-4793	917	7	∩	∩	NOUN
ejpam-4793	917	8	(	(	PUNCT
ejpam-4793	917	9	αh2	αh2	NOUN
ejpam-4793	917	10	∗	∗	NOUN
ejpam-4793	917	11	t	t	NOUN
ejpam-4793	917	12	)	)	PUNCT
ejpam-4793	917	13	̸=	̸=	PROPN
ejpam-4793	917	14	∅	∅	NOUN
ejpam-4793	917	15	for	for	ADP
ejpam-4793	917	16	all	all	PRON
ejpam-4793	917	17	α	α	PRON
ejpam-4793	917	18	∈	∈	PROPN
ejpam-4793	917	19	γ	γ	X
ejpam-4793	917	20	.	.	PUNCT
ejpam-4793	917	21	by	by	ADP
ejpam-4793	917	22	remark	remark	NOUN
ejpam-4793	917	23	14(ii	14(ii	NUM
ejpam-4793	917	24	)	)	PUNCT
ejpam-4793	917	25	,	,	PUNCT
ejpam-4793	917	26	ρt	ρt	PROPN
ejpam-4793	917	27	(	(	PUNCT
ejpam-4793	917	28	h1	h1	PROPN
ejpam-4793	917	29	)	)	PUNCT
ejpam-4793	917	30	=	=	SYM
ejpam-4793	917	31	ρt	ρt	PROPN
ejpam-4793	917	32	(	(	PUNCT
ejpam-4793	917	33	h2	h2	PROPN
ejpam-4793	917	34	)	)	PUNCT
ejpam-4793	917	35	.	.	PUNCT
ejpam-4793	918	1	thus	thus	ADV
ejpam-4793	918	2	,	,	PUNCT
ejpam-4793	918	3	f(ρs(h1	f(ρs(h1	NOUN
ejpam-4793	918	4	)	)	PUNCT
ejpam-4793	918	5	)	)	PUNCT
ejpam-4793	919	1	=	=	SYM
ejpam-4793	919	2	f(ρs(h2	f(ρs(h2	ADJ
ejpam-4793	919	3	)	)	PUNCT
ejpam-4793	919	4	)	)	PUNCT
ejpam-4793	919	5	.	.	PUNCT
ejpam-4793	920	1	hence	hence	ADV
ejpam-4793	920	2	,	,	PUNCT
ejpam-4793	920	3	f	f	PROPN
ejpam-4793	920	4	is	be	AUX
ejpam-4793	920	5	well	well	ADV
ejpam-4793	920	6	-	-	PUNCT
ejpam-4793	920	7	defined	define	VERB
ejpam-4793	920	8	.	.	PUNCT
ejpam-4793	921	1	let	let	VERB
ejpam-4793	921	2	ρs(h1	ρs(h1	NUM
ejpam-4793	921	3	)	)	PUNCT
ejpam-4793	921	4	,	,	PUNCT
ejpam-4793	921	5	ρs(h2	ρs(h2	NOUN
ejpam-4793	921	6	)	)	PUNCT
ejpam-4793	921	7	∈	∈	PROPN
ejpam-4793	921	8	m	m	PROPN
ejpam-4793	921	9	/	/	SYM
ejpam-4793	921	10	s.	s.	PROPN
ejpam-4793	921	11	then	then	ADV
ejpam-4793	921	12	f(ρs(h1	f(ρs(h1	NOUN
ejpam-4793	921	13	)	)	PUNCT
ejpam-4793	921	14	◦	◦	NOUN
ejpam-4793	921	15	ρs(h2	ρs(h2	NOUN
ejpam-4793	921	16	)	)	PUNCT
ejpam-4793	921	17	)	)	PUNCT
ejpam-4793	922	1	=	=	SYM
ejpam-4793	922	2	f(ρs(h1	f(ρs(h1	NOUN
ejpam-4793	922	3	∗	∗	NOUN
ejpam-4793	922	4	h2	h2	NOUN
ejpam-4793	922	5	)	)	PUNCT
ejpam-4793	922	6	)	)	PUNCT
ejpam-4793	923	1	=	=	SYM
ejpam-4793	924	1	ρt	ρt	PROPN
ejpam-4793	924	2	(	(	PUNCT
ejpam-4793	924	3	h1	h1	PROPN
ejpam-4793	924	4	)	)	PUNCT
ejpam-4793	924	5	◦	◦	NOUN
ejpam-4793	924	6	ρt	ρt	PROPN
ejpam-4793	924	7	(	(	PUNCT
ejpam-4793	924	8	h2	h2	NOUN
ejpam-4793	924	9	)	)	PUNCT
ejpam-4793	924	10	=	=	SYM
ejpam-4793	924	11	f(ρs(h1	f(ρs(h1	NOUN
ejpam-4793	924	12	)	)	PUNCT
ejpam-4793	924	13	)	)	PUNCT
ejpam-4793	925	1	◦	◦	NOUN
ejpam-4793	925	2	f(ρs(h2	f(ρs(h2	ADJ
ejpam-4793	925	3	)	)	PUNCT
ejpam-4793	925	4	)	)	PUNCT
ejpam-4793	925	5	.	.	PUNCT
ejpam-4793	926	1	hence	hence	ADV
ejpam-4793	926	2	,	,	PUNCT
ejpam-4793	926	3	f	f	PROPN
ejpam-4793	926	4	is	be	AUX
ejpam-4793	926	5	a	a	DET
ejpam-4793	926	6	homomorphism	homomorphism	NOUN
ejpam-4793	926	7	.	.	PUNCT
ejpam-4793	927	1	let	let	VERB
ejpam-4793	927	2	ρs(h	ρs(h	NUM
ejpam-4793	927	3	)	)	PUNCT
ejpam-4793	928	1	∈	∈	PROPN
ejpam-4793	928	2	ker	ker	PROPN
ejpam-4793	929	1	f	f	X
ejpam-4793	929	2	.	.	PUNCT
ejpam-4793	930	1	then	then	ADV
ejpam-4793	930	2	f(ρs(h	f(ρs(h	PROPN
ejpam-4793	930	3	)	)	PUNCT
ejpam-4793	930	4	)	)	PUNCT
ejpam-4793	931	1	=	=	SYM
ejpam-4793	931	2	ρt	ρt	PROPN
ejpam-4793	931	3	(	(	PUNCT
ejpam-4793	931	4	1	1	NUM
ejpam-4793	931	5	m	m	NOUN
ejpam-4793	931	6	)	)	PUNCT
ejpam-4793	931	7	,	,	PUNCT
ejpam-4793	931	8	the	the	DET
ejpam-4793	931	9	identity	identity	NOUN
ejpam-4793	931	10	in	in	ADP
ejpam-4793	931	11	m	m	PROPN
ejpam-4793	931	12	/	/	SYM
ejpam-4793	931	13	t	t	PROPN
ejpam-4793	931	14	.	.	PUNCT
ejpam-4793	932	1	thus	thus	ADV
ejpam-4793	932	2	,	,	PUNCT
ejpam-4793	932	3	ρt	ρt	PROPN
ejpam-4793	932	4	(	(	PUNCT
ejpam-4793	932	5	h	h	NOUN
ejpam-4793	932	6	)	)	PUNCT
ejpam-4793	932	7	=	=	SYM
ejpam-4793	932	8	ρt	ρt	PROPN
ejpam-4793	932	9	(	(	PUNCT
ejpam-4793	932	10	1	1	NUM
ejpam-4793	932	11	m	m	NOUN
ejpam-4793	932	12	)	)	PUNCT
ejpam-4793	932	13	.	.	PUNCT
ejpam-4793	933	1	by	by	ADP
ejpam-4793	933	2	proposition	proposition	NOUN
ejpam-4793	933	3	4	4	NUM
ejpam-4793	933	4	,	,	PUNCT
ejpam-4793	933	5	h	h	NOUN
ejpam-4793	933	6	∈	∈	PROPN
ejpam-4793	933	7	t	t	PROPN
ejpam-4793	933	8	.	.	PUNCT
ejpam-4793	934	1	hence	hence	ADV
ejpam-4793	934	2	,	,	PUNCT
ejpam-4793	934	3	ρs(h	ρs(h	NUM
ejpam-4793	934	4	)	)	PUNCT
ejpam-4793	934	5	∈	∈	PROPN
ejpam-4793	934	6	t	t	PROPN
ejpam-4793	934	7	/	/	SYM
ejpam-4793	934	8	s.	s.	PROPN
ejpam-4793	934	9	thus	thus	ADV
ejpam-4793	934	10	,	,	PUNCT
ejpam-4793	934	11	ker	ker	PROPN
ejpam-4793	934	12	f	f	PROPN
ejpam-4793	934	13	⊆	⊆	NUM
ejpam-4793	934	14	t	t	PROPN
ejpam-4793	934	15	/	/	SYM
ejpam-4793	934	16	s.	s.	PROPN
ejpam-4793	934	17	let	let	VERB
ejpam-4793	934	18	ρs(h	ρs(h	NUM
ejpam-4793	934	19	)	)	PUNCT
ejpam-4793	934	20	∈	∈	PROPN
ejpam-4793	934	21	t	t	PROPN
ejpam-4793	934	22	/	/	SYM
ejpam-4793	934	23	s.	s.	PROPN
ejpam-4793	935	1	then	then	ADV
ejpam-4793	935	2	h	h	PROPN
ejpam-4793	935	3	∈	∈	PROPN
ejpam-4793	935	4	t	t	PROPN
ejpam-4793	935	5	.	.	PUNCT
ejpam-4793	936	1	by	by	ADP
ejpam-4793	936	2	proposition	proposition	NOUN
ejpam-4793	936	3	4	4	NUM
ejpam-4793	936	4	,	,	PUNCT
ejpam-4793	936	5	ρt	ρt	PROPN
ejpam-4793	936	6	(	(	PUNCT
ejpam-4793	936	7	h	h	NOUN
ejpam-4793	936	8	)	)	PUNCT
ejpam-4793	936	9	=	=	SYM
ejpam-4793	936	10	ρt	ρt	PROPN
ejpam-4793	936	11	(	(	PUNCT
ejpam-4793	936	12	1	1	NUM
ejpam-4793	936	13	m	m	NOUN
ejpam-4793	936	14	)	)	PUNCT
ejpam-4793	936	15	.	.	PUNCT
ejpam-4793	937	1	thus	thus	ADV
ejpam-4793	937	2	,	,	PUNCT
ejpam-4793	937	3	f(ρs(h	f(ρs(h	PROPN
ejpam-4793	937	4	)	)	PUNCT
ejpam-4793	937	5	)	)	PUNCT
ejpam-4793	938	1	=	=	SYM
ejpam-4793	938	2	ρt	ρt	PROPN
ejpam-4793	938	3	(	(	PUNCT
ejpam-4793	938	4	h	h	NOUN
ejpam-4793	938	5	)	)	PUNCT
ejpam-4793	939	1	=	=	SYM
ejpam-4793	939	2	ρt	ρt	PROPN
ejpam-4793	939	3	(	(	PUNCT
ejpam-4793	939	4	1	1	NUM
ejpam-4793	939	5	m	m	NOUN
ejpam-4793	939	6	)	)	PUNCT
ejpam-4793	939	7	.	.	PUNCT
ejpam-4793	940	1	accordingly	accordingly	ADV
ejpam-4793	940	2	,	,	PUNCT
ejpam-4793	940	3	ρs(h	ρs(h	NUM
ejpam-4793	941	1	)	)	PUNCT
ejpam-4793	941	2	∈	∈	PROPN
ejpam-4793	941	3	ker	ker	PROPN
ejpam-4793	942	1	f	f	X
ejpam-4793	942	2	.	.	PUNCT
ejpam-4793	943	1	hence	hence	ADV
ejpam-4793	943	2	,	,	PUNCT
ejpam-4793	943	3	t	t	PROPN
ejpam-4793	943	4	/	/	SYM
ejpam-4793	943	5	s	s	PROPN
ejpam-4793	943	6	⊆	⊆	NUM
ejpam-4793	943	7	ker	ker	NOUN
ejpam-4793	943	8	f	f	X
ejpam-4793	943	9	.	.	PUNCT
ejpam-4793	944	1	so	so	ADV
ejpam-4793	944	2	,	,	PUNCT
ejpam-4793	944	3	t	t	PROPN
ejpam-4793	944	4	/	/	SYM
ejpam-4793	944	5	s	s	PART
ejpam-4793	945	1	=	=	NOUN
ejpam-4793	945	2	ker	ker	PROPN
ejpam-4793	945	3	f	f	PROPN
ejpam-4793	945	4	.	.	PUNCT
ejpam-4793	946	1	for	for	ADP
ejpam-4793	946	2	ρs(x	ρs(x	NUM
ejpam-4793	946	3	)	)	PUNCT
ejpam-4793	946	4	,	,	PUNCT
ejpam-4793	946	5	ρs(y	ρs(y	NUM
ejpam-4793	946	6	)	)	PUNCT
ejpam-4793	946	7	∈	∈	PROPN
ejpam-4793	946	8	m	m	PROPN
ejpam-4793	946	9	/	/	SYM
ejpam-4793	946	10	s	s	PROPN
ejpam-4793	946	11	and	and	CCONJ
ejpam-4793	946	12	α	α	PRON
ejpam-4793	946	13	∈	∈	PROPN
ejpam-4793	946	14	γ	γ	X
ejpam-4793	946	15	,	,	PUNCT
ejpam-4793	946	16	recall	recall	VERB
ejpam-4793	946	17	that	that	SCONJ
ejpam-4793	946	18	ρs(x)ρfρs(y	ρs(x)ρfρs(y	NOUN
ejpam-4793	946	19	)	)	PUNCT
ejpam-4793	947	1	if	if	SCONJ
ejpam-4793	947	2	and	and	CCONJ
ejpam-4793	947	3	only	only	ADV
ejpam-4793	947	4	if	if	SCONJ
ejpam-4793	947	5	f(αρs(x	f(αρs(x	PROPN
ejpam-4793	947	6	)	)	PUNCT
ejpam-4793	947	7	)	)	PUNCT
ejpam-4793	948	1	=	=	SYM
ejpam-4793	948	2	f(αρs(y	f(αρs(y	NOUN
ejpam-4793	948	3	)	)	PUNCT
ejpam-4793	948	4	)	)	PUNCT
ejpam-4793	948	5	.	.	PUNCT
ejpam-4793	949	1	we	we	PRON
ejpam-4793	949	2	claim	claim	VERB
ejpam-4793	949	3	that	that	SCONJ
ejpam-4793	949	4	ρf	ρf	ADP
ejpam-4793	949	5	=	=	SYM
ejpam-4793	949	6	ρker	ρker	NOUN
ejpam-4793	949	7	f	f	PROPN
ejpam-4793	949	8	.	.	PUNCT
ejpam-4793	950	1	let	let	VERB
ejpam-4793	950	2	ρs(z	ρs(z	NUM
ejpam-4793	950	3	)	)	PUNCT
ejpam-4793	951	1	∈	∈	PROPN
ejpam-4793	951	2	m	m	PROPN
ejpam-4793	951	3	/	/	SYM
ejpam-4793	951	4	s.	s.	PROPN
ejpam-4793	951	5	we	we	PRON
ejpam-4793	951	6	show	show	VERB
ejpam-4793	951	7	that	that	SCONJ
ejpam-4793	951	8	ρf	ρf	PROPN
ejpam-4793	951	9	(	(	PUNCT
ejpam-4793	951	10	ρs(z	ρs(z	NUM
ejpam-4793	951	11	)	)	PUNCT
ejpam-4793	951	12	)	)	PUNCT
ejpam-4793	952	1	=	=	SYM
ejpam-4793	952	2	ρker	ρker	NOUN
ejpam-4793	952	3	f	f	PROPN
ejpam-4793	952	4	(	(	PUNCT
ejpam-4793	952	5	ρs(z	ρs(z	NUM
ejpam-4793	952	6	)	)	PUNCT
ejpam-4793	952	7	)	)	PUNCT
ejpam-4793	952	8	.	.	PUNCT
ejpam-4793	953	1	let	let	VERB
ejpam-4793	953	2	ρs(w	ρs(w	NOUN
ejpam-4793	953	3	)	)	PUNCT
ejpam-4793	953	4	∈	∈	PROPN
ejpam-4793	953	5	ρker	ρker	NOUN
ejpam-4793	953	6	f	f	X
ejpam-4793	953	7	(	(	PUNCT
ejpam-4793	953	8	ρs(z	ρs(z	NUM
ejpam-4793	953	9	)	)	PUNCT
ejpam-4793	953	10	)	)	PUNCT
ejpam-4793	953	11	.	.	PUNCT
ejpam-4793	954	1	then	then	ADV
ejpam-4793	954	2	(	(	PUNCT
ejpam-4793	954	3	αρs(z	αρs(z	NOUN
ejpam-4793	954	4	)	)	PUNCT
ejpam-4793	954	5	◦	◦	NOUN
ejpam-4793	954	6	ker	ker	NOUN
ejpam-4793	954	7	f	f	X
ejpam-4793	954	8	)	)	PUNCT
ejpam-4793	954	9	∩	∩	NOUN
ejpam-4793	954	10	(	(	PUNCT
ejpam-4793	954	11	αρs(w	αρs(w	NUM
ejpam-4793	954	12	)	)	PUNCT
ejpam-4793	954	13	◦	◦	NOUN
ejpam-4793	954	14	ker	ker	NOUN
ejpam-4793	955	1	f	f	X
ejpam-4793	955	2	)	)	PUNCT
ejpam-4793	955	3	̸=	̸=	PROPN
ejpam-4793	955	4	∅.	∅.	ADV
ejpam-4793	955	5	thus	thus	ADV
ejpam-4793	955	6	,	,	PUNCT
ejpam-4793	955	7	there	there	PRON
ejpam-4793	955	8	exist	exist	VERB
ejpam-4793	955	9	y1	y1	NOUN
ejpam-4793	955	10	,	,	PUNCT
ejpam-4793	955	11	y2	y2	PROPN
ejpam-4793	955	12	∈	∈	PROPN
ejpam-4793	956	1	ker	ker	PROPN
ejpam-4793	956	2	f	f	X
ejpam-4793	956	3	such	such	ADJ
ejpam-4793	956	4	that	that	DET
ejpam-4793	956	5	αρs(z	αρs(z	NOUN
ejpam-4793	956	6	)	)	PUNCT
ejpam-4793	956	7	◦	◦	NOUN
ejpam-4793	956	8	y1	y1	NOUN
ejpam-4793	956	9	=	=	PUNCT
ejpam-4793	956	10	αρs(w	αρs(w	NUM
ejpam-4793	956	11	)	)	PUNCT
ejpam-4793	956	12	◦	◦	NOUN
ejpam-4793	956	13	y2	y2	NOUN
ejpam-4793	956	14	.	.	PUNCT
ejpam-4793	957	1	hence	hence	ADV
ejpam-4793	957	2	,	,	PUNCT
ejpam-4793	957	3	f(αρs(z	f(αρs(z	NOUN
ejpam-4793	957	4	)	)	PUNCT
ejpam-4793	957	5	)	)	PUNCT
ejpam-4793	958	1	=	=	PUNCT
ejpam-4793	958	2	f(αρs(z	f(αρs(z	NOUN
ejpam-4793	958	3	)	)	PUNCT
ejpam-4793	958	4	)	)	PUNCT
ejpam-4793	959	1	◦	◦	NOUN
ejpam-4793	959	2	ρt	ρt	PROPN
ejpam-4793	959	3	(	(	PUNCT
ejpam-4793	959	4	1	1	NUM
ejpam-4793	959	5	m	m	NOUN
ejpam-4793	959	6	)	)	PUNCT
ejpam-4793	960	1	=	=	PUNCT
ejpam-4793	960	2	f(αρs(z	f(αρs(z	NOUN
ejpam-4793	960	3	)	)	PUNCT
ejpam-4793	960	4	)	)	PUNCT
ejpam-4793	961	1	◦	◦	NOUN
ejpam-4793	961	2	f(y1	f(y1	NOUN
ejpam-4793	961	3	)	)	PUNCT
ejpam-4793	961	4	=	=	SYM
ejpam-4793	961	5	f(αρs(z	f(αρs(z	NOUN
ejpam-4793	961	6	)	)	PUNCT
ejpam-4793	961	7	◦	◦	NOUN
ejpam-4793	961	8	y1	y1	NOUN
ejpam-4793	961	9	)	)	PUNCT
ejpam-4793	961	10	and	and	CCONJ
ejpam-4793	961	11	f(αρs(w	f(αρs(w	NOUN
ejpam-4793	961	12	)	)	PUNCT
ejpam-4793	961	13	)	)	PUNCT
ejpam-4793	962	1	=	=	SYM
ejpam-4793	962	2	f(αρs(w	f(αρs(w	NOUN
ejpam-4793	962	3	)	)	PUNCT
ejpam-4793	962	4	)	)	PUNCT
ejpam-4793	963	1	◦	◦	NOUN
ejpam-4793	963	2	ρt	ρt	PROPN
ejpam-4793	963	3	(	(	PUNCT
ejpam-4793	963	4	1	1	NUM
ejpam-4793	963	5	m	m	NOUN
ejpam-4793	963	6	)	)	PUNCT
ejpam-4793	964	1	=	=	SYM
ejpam-4793	964	2	f(αρs(w	f(αρs(w	NOUN
ejpam-4793	964	3	)	)	PUNCT
ejpam-4793	964	4	)	)	PUNCT
ejpam-4793	965	1	◦	◦	NOUN
ejpam-4793	965	2	f(y2	f(y2	ADJ
ejpam-4793	965	3	)	)	PUNCT
ejpam-4793	966	1	=	=	SYM
ejpam-4793	966	2	f(αρs(w	f(αρs(w	ADJ
ejpam-4793	966	3	)	)	PUNCT
ejpam-4793	966	4	◦	◦	NOUN
ejpam-4793	966	5	y2	y2	NOUN
ejpam-4793	966	6	)	)	PUNCT
ejpam-4793	966	7	.	.	PUNCT
ejpam-4793	967	1	so	so	ADV
ejpam-4793	967	2	,	,	PUNCT
ejpam-4793	967	3	by	by	ADP
ejpam-4793	967	4	welldefinedness	welldefinedness	NOUN
ejpam-4793	967	5	of	of	ADP
ejpam-4793	967	6	f	f	PROPN
ejpam-4793	967	7	,	,	PUNCT
ejpam-4793	967	8	we	we	PRON
ejpam-4793	967	9	have	have	VERB
ejpam-4793	967	10	f(αρs(z	f(αρs(z	NOUN
ejpam-4793	967	11	)	)	PUNCT
ejpam-4793	967	12	)	)	PUNCT
ejpam-4793	968	1	=	=	PUNCT
ejpam-4793	968	2	f(αρs(z	f(αρs(z	NOUN
ejpam-4793	968	3	)	)	PUNCT
ejpam-4793	968	4	◦	◦	NOUN
ejpam-4793	968	5	y1	y1	NOUN
ejpam-4793	968	6	)	)	PUNCT
ejpam-4793	968	7	=	=	SYM
ejpam-4793	968	8	f(αρs(w	f(αρs(w	ADJ
ejpam-4793	968	9	)	)	PUNCT
ejpam-4793	968	10	◦	◦	NOUN
ejpam-4793	968	11	y2	y2	NOUN
ejpam-4793	968	12	)	)	PUNCT
ejpam-4793	968	13	=	=	SYM
ejpam-4793	968	14	f(αρs(w	f(αρs(w	NOUN
ejpam-4793	968	15	)	)	PUNCT
ejpam-4793	968	16	)	)	PUNCT
ejpam-4793	968	17	.	.	PUNCT
ejpam-4793	969	1	accordingly	accordingly	ADV
ejpam-4793	969	2	,	,	PUNCT
ejpam-4793	969	3	ρs(w	ρs(w	ADJ
ejpam-4793	969	4	)	)	PUNCT
ejpam-4793	969	5	∈	∈	NOUN
ejpam-4793	969	6	ρf	ρf	X
ejpam-4793	969	7	(	(	PUNCT
ejpam-4793	969	8	ρs(z	ρs(z	NUM
ejpam-4793	969	9	)	)	PUNCT
ejpam-4793	969	10	)	)	PUNCT
ejpam-4793	969	11	.	.	PUNCT
ejpam-4793	970	1	thus	thus	ADV
ejpam-4793	970	2	,	,	PUNCT
ejpam-4793	970	3	ρker	ρker	NOUN
ejpam-4793	970	4	f	f	PROPN
ejpam-4793	970	5	(	(	PUNCT
ejpam-4793	970	6	ρs(z	ρs(z	NUM
ejpam-4793	970	7	)	)	PUNCT
ejpam-4793	970	8	)	)	PUNCT
ejpam-4793	970	9	⊆	⊆	NUM
ejpam-4793	970	10	ρf	ρf	NOUN
ejpam-4793	970	11	(	(	PUNCT
ejpam-4793	970	12	ρs(z	ρs(z	NUM
ejpam-4793	970	13	)	)	PUNCT
ejpam-4793	970	14	)	)	PUNCT
ejpam-4793	970	15	.	.	PUNCT
ejpam-4793	971	1	now	now	ADV
ejpam-4793	971	2	,	,	PUNCT
ejpam-4793	971	3	let	let	VERB
ejpam-4793	971	4	ρs(w	ρs(w	NOUN
ejpam-4793	971	5	)	)	PUNCT
ejpam-4793	971	6	∈	∈	NOUN
ejpam-4793	971	7	ρf	ρf	X
ejpam-4793	971	8	(	(	PUNCT
ejpam-4793	971	9	ρs(z	ρs(z	NUM
ejpam-4793	971	10	)	)	PUNCT
ejpam-4793	971	11	)	)	PUNCT
ejpam-4793	972	1	and	and	CCONJ
ejpam-4793	972	2	α	α	PRON
ejpam-4793	972	3	∈	∈	PROPN
ejpam-4793	972	4	γ	γ	X
ejpam-4793	972	5	.	.	PROPN
ejpam-4793	972	6	then	then	ADV
ejpam-4793	972	7	f(αρs(z	f(αρs(z	NOUN
ejpam-4793	972	8	)	)	PUNCT
ejpam-4793	972	9	)	)	PUNCT
ejpam-4793	973	1	=	=	SYM
ejpam-4793	973	2	f(αρs(w	f(αρs(w	NOUN
ejpam-4793	973	3	)	)	PUNCT
ejpam-4793	973	4	)	)	PUNCT
ejpam-4793	973	5	,	,	PUNCT
ejpam-4793	973	6	that	that	ADV
ejpam-4793	973	7	is	is	ADV
ejpam-4793	973	8	,	,	PUNCT
ejpam-4793	973	9	αρt	αρt	NOUN
ejpam-4793	973	10	(	(	PUNCT
ejpam-4793	973	11	z	z	NOUN
ejpam-4793	973	12	)	)	PUNCT
ejpam-4793	973	13	=	=	SYM
ejpam-4793	974	1	αρt	αρt	NOUN
ejpam-4793	974	2	(	(	PUNCT
ejpam-4793	974	3	w	w	NOUN
ejpam-4793	974	4	)	)	PUNCT
ejpam-4793	974	5	.	.	PUNCT
ejpam-4793	975	1	thus	thus	ADV
ejpam-4793	975	2	,	,	PUNCT
ejpam-4793	975	3	ρt	ρt	PROPN
ejpam-4793	975	4	(	(	PUNCT
ejpam-4793	975	5	αz	αz	PROPN
ejpam-4793	975	6	)	)	PUNCT
ejpam-4793	975	7	=	=	SYM
ejpam-4793	975	8	ρt	ρt	PROPN
ejpam-4793	975	9	(	(	PUNCT
ejpam-4793	975	10	αw	αw	NOUN
ejpam-4793	975	11	)	)	PUNCT
ejpam-4793	975	12	implies	imply	VERB
ejpam-4793	975	13	(	(	PUNCT
ejpam-4793	975	14	αw	αw	ADP
ejpam-4793	975	15	∗	∗	NOUN
ejpam-4793	975	16	t	t	NOUN
ejpam-4793	975	17	)	)	PUNCT
ejpam-4793	975	18	∩	∩	NOUN
ejpam-4793	975	19	(	(	PUNCT
ejpam-4793	975	20	αz	αz	ADP
ejpam-4793	975	21	∗	∗	PROPN
ejpam-4793	975	22	t	t	NOUN
ejpam-4793	975	23	)	)	PUNCT
ejpam-4793	975	24	̸=	̸=	PROPN
ejpam-4793	975	25	∅.	∅.	ADP
ejpam-4793	975	26	thus	thus	ADV
ejpam-4793	975	27	,	,	PUNCT
ejpam-4793	975	28	there	there	PRON
ejpam-4793	975	29	exist	exist	VERB
ejpam-4793	975	30	h1	h1	PROPN
ejpam-4793	975	31	,	,	PUNCT
ejpam-4793	975	32	h2	h2	PROPN
ejpam-4793	975	33	∈	∈	PROPN
ejpam-4793	975	34	t	t	NOUN
ejpam-4793	975	35	such	such	ADJ
ejpam-4793	975	36	that	that	SCONJ
ejpam-4793	975	37	αw∗h1	αw∗h1	NUM
ejpam-4793	975	38	=	=	SYM
ejpam-4793	975	39	αz∗h2	αz∗h2	NUM
ejpam-4793	975	40	.	.	PUNCT
ejpam-4793	976	1	hence	hence	ADV
ejpam-4793	976	2	,	,	PUNCT
ejpam-4793	976	3	ρs(h1	ρs(h1	ADV
ejpam-4793	976	4	)	)	PUNCT
ejpam-4793	976	5	,	,	PUNCT
ejpam-4793	976	6	ρs(h2	ρs(h2	NOUN
ejpam-4793	976	7	)	)	PUNCT
ejpam-4793	976	8	∈	∈	PROPN
ejpam-4793	976	9	t	t	PROPN
ejpam-4793	976	10	/	/	SYM
ejpam-4793	976	11	s	s	PART
ejpam-4793	976	12	=	=	NOUN
ejpam-4793	976	13	ker	ker	PROPN
ejpam-4793	976	14	f	f	PROPN
ejpam-4793	976	15	.	.	PUNCT
ejpam-4793	977	1	consequently	consequently	ADV
ejpam-4793	977	2	,	,	PUNCT
ejpam-4793	977	3	ρs	ρs	PROPN
ejpam-4793	977	4	(	(	PUNCT
ejpam-4793	977	5	αw	αw	NOUN
ejpam-4793	977	6	)	)	PUNCT
ejpam-4793	977	7	◦	◦	NOUN
ejpam-4793	977	8	ρs(h1	ρs(h1	NUM
ejpam-4793	977	9	)	)	PUNCT
ejpam-4793	977	10	=	=	SYM
ejpam-4793	977	11	ρs	ρs	PROPN
ejpam-4793	977	12	(	(	PUNCT
ejpam-4793	977	13	αw	αw	ADP
ejpam-4793	977	14	∗	∗	NOUN
ejpam-4793	977	15	h1	h1	PROPN
ejpam-4793	977	16	)	)	PUNCT
ejpam-4793	977	17	=	=	SYM
ejpam-4793	978	1	ρs	ρs	PROPN
ejpam-4793	978	2	(	(	PUNCT
ejpam-4793	978	3	αz	αz	ADP
ejpam-4793	978	4	∗	∗	NOUN
ejpam-4793	978	5	h2	h2	NOUN
ejpam-4793	978	6	)	)	PUNCT
ejpam-4793	979	1	=	=	SYM
ejpam-4793	979	2	ρs	ρs	PROPN
ejpam-4793	979	3	(	(	PUNCT
ejpam-4793	979	4	αz	αz	NOUN
ejpam-4793	979	5	)	)	PUNCT
ejpam-4793	979	6	◦	◦	NOUN
ejpam-4793	979	7	ρs(h2	ρs(h2	NOUN
ejpam-4793	979	8	)	)	PUNCT
ejpam-4793	979	9	for	for	ADP
ejpam-4793	979	10	all	all	PRON
ejpam-4793	979	11	α	α	PRON
ejpam-4793	979	12	∈	∈	PROPN
ejpam-4793	979	13	γ	γ	X
ejpam-4793	979	14	.	.	PUNCT
ejpam-4793	980	1	this	this	PRON
ejpam-4793	980	2	implies	imply	VERB
ejpam-4793	980	3	that	that	SCONJ
ejpam-4793	980	4	(	(	PUNCT
ejpam-4793	980	5	αρs(w	αρs(w	NUM
ejpam-4793	980	6	)	)	PUNCT
ejpam-4793	980	7	◦	◦	NOUN
ejpam-4793	980	8	ker	ker	NOUN
ejpam-4793	981	1	f	f	X
ejpam-4793	981	2	)	)	PUNCT
ejpam-4793	981	3	∩	∩	NOUN
ejpam-4793	981	4	(	(	PUNCT
ejpam-4793	981	5	αρs(z	αρs(z	NOUN
ejpam-4793	981	6	)	)	PUNCT
ejpam-4793	981	7	◦	◦	NOUN
ejpam-4793	981	8	ker	ker	NOUN
ejpam-4793	982	1	f	f	X
ejpam-4793	982	2	)	)	PUNCT
ejpam-4793	982	3	̸=	̸=	PROPN
ejpam-4793	982	4	∅.	∅.	PRON
ejpam-4793	982	5	hence	hence	ADV
ejpam-4793	982	6	,	,	PUNCT
ejpam-4793	982	7	ρs(w	ρs(w	ADJ
ejpam-4793	982	8	)	)	PUNCT
ejpam-4793	982	9	∈	∈	PROPN
ejpam-4793	982	10	ρker	ρker	NOUN
ejpam-4793	982	11	f	f	X
ejpam-4793	982	12	(	(	PUNCT
ejpam-4793	982	13	ρs(z	ρs(z	NUM
ejpam-4793	982	14	)	)	PUNCT
ejpam-4793	982	15	)	)	PUNCT
ejpam-4793	982	16	.	.	PUNCT
ejpam-4793	983	1	accordingly	accordingly	ADV
ejpam-4793	983	2	,	,	PUNCT
ejpam-4793	983	3	ρf	ρf	X
ejpam-4793	983	4	(	(	PUNCT
ejpam-4793	983	5	ρs(z	ρs(z	NUM
ejpam-4793	983	6	)	)	PUNCT
ejpam-4793	983	7	)	)	PUNCT
ejpam-4793	984	1	⊆	⊆	NUM
ejpam-4793	984	2	ρker	ρker	NOUN
ejpam-4793	984	3	f	f	X
ejpam-4793	984	4	(	(	PUNCT
ejpam-4793	984	5	ρs(z	ρs(z	NUM
ejpam-4793	984	6	)	)	PUNCT
ejpam-4793	984	7	)	)	PUNCT
ejpam-4793	984	8	.	.	PUNCT
ejpam-4793	985	1	therefore	therefore	ADV
ejpam-4793	985	2	,	,	PUNCT
ejpam-4793	985	3	ρf	ρf	X
ejpam-4793	985	4	(	(	PUNCT
ejpam-4793	985	5	ρs(z	ρs(z	NUM
ejpam-4793	985	6	)	)	PUNCT
ejpam-4793	985	7	)	)	PUNCT
ejpam-4793	986	1	=	=	SYM
ejpam-4793	986	2	ρker	ρker	NOUN
ejpam-4793	986	3	f	f	PROPN
ejpam-4793	986	4	(	(	PUNCT
ejpam-4793	986	5	ρs(z	ρs(z	NUM
ejpam-4793	986	6	)	)	PUNCT
ejpam-4793	986	7	)	)	PUNCT
ejpam-4793	986	8	for	for	ADP
ejpam-4793	986	9	all	all	PRON
ejpam-4793	986	10	ρs(z	ρs(z	NUM
ejpam-4793	986	11	)	)	PUNCT
ejpam-4793	986	12	∈	∈	PROPN
ejpam-4793	986	13	m	m	PROPN
ejpam-4793	986	14	/	/	SYM
ejpam-4793	986	15	s	s	PROPN
ejpam-4793	986	16	,	,	PUNCT
ejpam-4793	986	17	that	that	ADV
ejpam-4793	986	18	is	is	ADV
ejpam-4793	986	19	,	,	PUNCT
ejpam-4793	986	20	ρf	ρf	ADP
ejpam-4793	986	21	=	=	PUNCT
ejpam-4793	986	22	ρker	ρker	NOUN
ejpam-4793	986	23	f	f	X
ejpam-4793	986	24	.	.	PUNCT
ejpam-4793	987	1	by	by	ADP
ejpam-4793	987	2	theorem	theorem	NOUN
ejpam-4793	987	3	13	13	NUM
ejpam-4793	987	4	,	,	PUNCT
ejpam-4793	987	5	these	these	PRON
ejpam-4793	987	6	all	all	PRON
ejpam-4793	987	7	imply	imply	VERB
ejpam-4793	987	8	that	that	SCONJ
ejpam-4793	987	9	(	(	PUNCT
ejpam-4793	987	10	m	m	NOUN
ejpam-4793	987	11	/	/	SYM
ejpam-4793	987	12	s)/(t	s)/(t	PROPN
ejpam-4793	987	13	/	/	SYM
ejpam-4793	987	14	s	s	NOUN
ejpam-4793	987	15	)	)	PUNCT
ejpam-4793	987	16	=	=	SYM
ejpam-4793	987	17	(	(	PUNCT
ejpam-4793	987	18	m	m	PROPN
ejpam-4793	987	19	/	/	SYM
ejpam-4793	987	20	s)/	s)/	PROPN
ejpam-4793	987	21	ker	ker	NOUN
ejpam-4793	988	1	f	f	PROPN
ejpam-4793	988	2	∼=	∼=	PROPN
ejpam-4793	988	3	m	m	PROPN
ejpam-4793	988	4	/	/	SYM
ejpam-4793	988	5	t	t	PROPN
ejpam-4793	988	6	.	.	PUNCT
ejpam-4793	989	1	7	7	X
ejpam-4793	989	2	.	.	X
ejpam-4793	989	3	conclusion	conclusion	NOUN
ejpam-4793	989	4	:	:	PUNCT
ejpam-4793	989	5	in	in	ADP
ejpam-4793	989	6	this	this	DET
ejpam-4793	989	7	paper	paper	NOUN
ejpam-4793	989	8	,	,	PUNCT
ejpam-4793	989	9	we	we	PRON
ejpam-4793	989	10	have	have	AUX
ejpam-4793	989	11	shown	show	VERB
ejpam-4793	989	12	that	that	SCONJ
ejpam-4793	989	13	γ	γ	NOUN
ejpam-4793	989	14	-	-	PUNCT
ejpam-4793	989	15	ideals	ideal	NOUN
ejpam-4793	989	16	and	and	CCONJ
ejpam-4793	989	17	γ	γ	NOUN
ejpam-4793	989	18	-	-	NOUN
ejpam-4793	989	19	submonoids	submonoid	NOUN
ejpam-4793	989	20	of	of	ADP
ejpam-4793	989	21	a	a	DET
ejpam-4793	989	22	γ	γ	PROPN
ejpam-4793	989	23	-	-	PUNCT
ejpam-4793	989	24	monoid	monoid	NOUN
ejpam-4793	989	25	m	m	NOUN
ejpam-4793	989	26	are	be	AUX
ejpam-4793	989	27	not	not	PART
ejpam-4793	989	28	equivalent	equivalent	ADJ
ejpam-4793	989	29	to	to	ADP
ejpam-4793	989	30	the	the	DET
ejpam-4793	989	31	existing	exist	VERB
ejpam-4793	989	32	γ	γ	NOUN
ejpam-4793	989	33	-	-	PUNCT
ejpam-4793	989	34	order	order	NOUN
ejpam-4793	989	35	-	-	PUNCT
ejpam-4793	989	36	ideals	ideal	NOUN
ejpam-4793	989	37	of	of	ADP
ejpam-4793	989	38	m	m	PROPN
ejpam-4793	989	39	.	.	PUNCT
ejpam-4793	990	1	for	for	ADP
ejpam-4793	990	2	any	any	DET
ejpam-4793	990	3	γ	γ	NOUN
ejpam-4793	990	4	-	-	PUNCT
ejpam-4793	990	5	monoids	monoid	NOUN
ejpam-4793	990	6	m	m	VERB
ejpam-4793	990	7	and	and	CCONJ
ejpam-4793	990	8	n	n	CCONJ
ejpam-4793	990	9	,	,	PUNCT
ejpam-4793	990	10	we	we	PRON
ejpam-4793	990	11	proved	prove	VERB
ejpam-4793	990	12	that	that	SCONJ
ejpam-4793	990	13	the	the	DET
ejpam-4793	990	14	kernel	kernel	NOUN
ejpam-4793	990	15	of	of	ADP
ejpam-4793	990	16	a	a	DET
ejpam-4793	990	17	γ	γ	X
ejpam-4793	990	18	-	-	PUNCT
ejpam-4793	990	19	monoid	monoid	NOUN
ejpam-4793	990	20	homomorphism	homomorphism	PROPN
ejpam-4793	990	21	φ	φ	X
ejpam-4793	990	22	:	:	PUNCT
ejpam-4793	990	23	m	m	PROPN
ejpam-4793	990	24	→	→	SYM
ejpam-4793	991	1	n	n	X
ejpam-4793	991	2	is	be	AUX
ejpam-4793	991	3	a	a	DET
ejpam-4793	991	4	γ	γ	NOUN
ejpam-4793	991	5	-	-	ADJ
ejpam-4793	991	6	submonoid	submonoid	NOUN
ejpam-4793	991	7	of	of	ADP
ejpam-4793	991	8	m	m	PROPN
ejpam-4793	991	9	.	.	PUNCT
ejpam-4793	992	1	also	also	ADV
ejpam-4793	992	2	,	,	PUNCT
ejpam-4793	992	3	for	for	ADP
ejpam-4793	992	4	any	any	DET
ejpam-4793	992	5	γ	γ	NOUN
ejpam-4793	992	6	-	-	ADJ
ejpam-4793	992	7	submonoid	submonoid	ADJ
ejpam-4793	992	8	s	s	PROPN
ejpam-4793	992	9	of	of	ADP
ejpam-4793	992	10	a	a	DET
ejpam-4793	992	11	γ	γ	X
ejpam-4793	992	12	-	-	PUNCT
ejpam-4793	992	13	monoid	monoid	NOUN
ejpam-4793	992	14	m	m	NOUN
ejpam-4793	992	15	,	,	PUNCT
ejpam-4793	992	16	ρs	ρs	ADV
ejpam-4793	992	17	is	be	AUX
ejpam-4793	992	18	a	a	DET
ejpam-4793	992	19	congruence	congruence	NOUN
ejpam-4793	992	20	relation	relation	NOUN
ejpam-4793	992	21	if	if	SCONJ
ejpam-4793	992	22	m	m	NOUN
ejpam-4793	992	23	is	be	AUX
ejpam-4793	992	24	commutative	commutative	ADJ
ejpam-4793	992	25	and	and	CCONJ
ejpam-4793	992	26	thus	thus	ADV
ejpam-4793	992	27	,	,	PUNCT
ejpam-4793	992	28	m	m	PROPN
ejpam-4793	992	29	/	/	SYM
ejpam-4793	992	30	s	s	PART
ejpam-4793	992	31	=	=	VERB
ejpam-4793	992	32	m	m	VERB
ejpam-4793	992	33	/	/	SYM
ejpam-4793	992	34	ρs	ρs	NOUN
ejpam-4793	992	35	is	be	AUX
ejpam-4793	992	36	defined	define	VERB
ejpam-4793	992	37	for	for	ADP
ejpam-4793	992	38	commutative	commutative	ADJ
ejpam-4793	992	39	γ	γ	X
ejpam-4793	992	40	-	-	PUNCT
ejpam-4793	992	41	monoid	monoid	NOUN
ejpam-4793	992	42	m	m	PROPN
ejpam-4793	992	43	.	.	PUNCT
ejpam-4793	993	1	moreover	moreover	ADV
ejpam-4793	993	2	,	,	PUNCT
ejpam-4793	993	3	isomorphism	isomorphism	NOUN
ejpam-4793	993	4	theorems	theorem	VERB
ejpam-4793	993	5	for	for	ADP
ejpam-4793	993	6	γ	γ	NOUN
ejpam-4793	993	7	-	-	PUNCT
ejpam-4793	993	8	monoids	monoid	NOUN
ejpam-4793	993	9	via	via	ADP
ejpam-4793	993	10	γ	γ	NOUN
ejpam-4793	993	11	-	-	PUNCT
ejpam-4793	993	12	submonoids	submonoid	NOUN
ejpam-4793	993	13	were	be	AUX
ejpam-4793	993	14	proved	prove	VERB
ejpam-4793	993	15	.	.	PUNCT
ejpam-4793	994	1	references	reference	NOUN
ejpam-4793	994	2	1793	1793	NUM
ejpam-4793	994	3	acknowledgements	acknowledgement	NOUN
ejpam-4793	994	4	the	the	DET
ejpam-4793	994	5	authors	author	NOUN
ejpam-4793	994	6	would	would	AUX
ejpam-4793	994	7	like	like	VERB
ejpam-4793	994	8	to	to	PART
ejpam-4793	994	9	thank	thank	VERB
ejpam-4793	994	10	the	the	DET
ejpam-4793	994	11	department	department	NOUN
ejpam-4793	994	12	of	of	ADP
ejpam-4793	994	13	science	science	NOUN
ejpam-4793	994	14	and	and	CCONJ
ejpam-4793	994	15	technology	technology	NOUN
ejpam-4793	994	16	accelerated	accelerate	VERB
ejpam-4793	994	17	science	science	NOUN
ejpam-4793	994	18	and	and	CCONJ
ejpam-4793	994	19	technology	technology	NOUN
ejpam-4793	994	20	human	human	ADJ
ejpam-4793	994	21	resource	resource	NOUN
ejpam-4793	994	22	development	development	NOUN
ejpam-4793	994	23	program	program	NOUN
ejpam-4793	994	24	(	(	PUNCT
ejpam-4793	994	25	dost	dost	NOUN
ejpam-4793	994	26	-	-	PUNCT
ejpam-4793	994	27	asthrdp)philippines	asthrdp)philippine	NOUN
ejpam-4793	994	28	,	,	PUNCT
ejpam-4793	994	29	and	and	CCONJ
ejpam-4793	994	30	msu	msu	PROPN
ejpam-4793	994	31	-	-	PUNCT
ejpam-4793	994	32	iligan	iligan	PROPN
ejpam-4793	994	33	institute	institute	PROPN
ejpam-4793	994	34	of	of	ADP
ejpam-4793	994	35	technology	technology	NOUN
ejpam-4793	994	36	for	for	ADP
ejpam-4793	994	37	funding	fund	VERB
ejpam-4793	994	38	this	this	DET
ejpam-4793	994	39	research	research	NOUN
ejpam-4793	994	40	.	.	PUNCT
ejpam-4793	995	1	references	reference	NOUN
ejpam-4793	995	2	[	[	X
ejpam-4793	995	3	1	1	NUM
ejpam-4793	995	4	]	]	PUNCT
ejpam-4793	995	5	r.	r.	PROPN
ejpam-4793	995	6	hazrat	hazrat	PROPN
ejpam-4793	995	7	and	and	CCONJ
ejpam-4793	995	8	h.	h.	PROPN
ejpam-4793	995	9	li	li	PROPN
ejpam-4793	995	10	.	.	PUNCT
ejpam-4793	996	1	the	the	DET
ejpam-4793	996	2	talented	talented	ADJ
ejpam-4793	996	3	monoid	monoid	NOUN
ejpam-4793	996	4	of	of	ADP
ejpam-4793	996	5	a	a	DET
ejpam-4793	996	6	leavitt	leavitt	ADJ
ejpam-4793	996	7	path	path	NOUN
ejpam-4793	996	8	algebra	algebra	PROPN
ejpam-4793	996	9	.	.	PUNCT
ejpam-4793	997	1	journal	journal	NOUN
ejpam-4793	997	2	of	of	ADP
ejpam-4793	997	3	algebra	algebra	PROPN
ejpam-4793	997	4	547	547	NUM
ejpam-4793	997	5	,	,	PUNCT
ejpam-4793	997	6	pages	page	NOUN
ejpam-4793	997	7	430–455	430–455	NUM
ejpam-4793	997	8	,	,	PUNCT
ejpam-4793	997	9	2020	2020	NUM
ejpam-4793	997	10	.	.	PUNCT
ejpam-4793	998	1	[	[	X
ejpam-4793	998	2	2	2	X
ejpam-4793	998	3	]	]	PUNCT
ejpam-4793	998	4	t.	t.	PROPN
ejpam-4793	998	5	w.	w.	PROPN
ejpam-4793	998	6	hungerford	hungerford	PROPN
ejpam-4793	998	7	.	.	PUNCT
ejpam-4793	999	1	algebra	algebra	PROPN
ejpam-4793	999	2	.	.	PUNCT
ejpam-4793	1000	1	1980	1980	NUM
ejpam-4793	1000	2	.	.	PUNCT
ejpam-4793	1001	1	[	[	X
ejpam-4793	1001	2	3	3	X
ejpam-4793	1001	3	]	]	X
ejpam-4793	1001	4	d.	d.	PROPN
ejpam-4793	1001	5	gonçalvez	gonçalvez	PROPN
ejpam-4793	1001	6	l.	l.	PROPN
ejpam-4793	1001	7	g.	g.	PROPN
ejpam-4793	1001	8	cordeiro	cordeiro	PROPN
ejpam-4793	1001	9	and	and	CCONJ
ejpam-4793	1001	10	r.	r.	PROPN
ejpam-4793	1001	11	hazrat	hazrat	PROPN
ejpam-4793	1001	12	.	.	PUNCT
ejpam-4793	1002	1	the	the	DET
ejpam-4793	1002	2	talented	talented	ADJ
ejpam-4793	1002	3	monoid	monoid	NOUN
ejpam-4793	1002	4	of	of	ADP
ejpam-4793	1002	5	a	a	DET
ejpam-4793	1002	6	directed	direct	VERB
ejpam-4793	1002	7	graph	graph	NOUN
ejpam-4793	1002	8	with	with	ADP
ejpam-4793	1002	9	applications	application	NOUN
ejpam-4793	1002	10	to	to	PART
ejpam-4793	1002	11	graph	graph	VERB
ejpam-4793	1002	12	algebras	algebras	PROPN
ejpam-4793	1002	13	.	.	PUNCT
ejpam-4793	1002	14	rev	rev	PROPN
ejpam-4793	1002	15	.	.	PROPN
ejpam-4793	1002	16	mat	mat	PROPN
ejpam-4793	1002	17	.	.	PROPN
ejpam-4793	1002	18	iberoam	iberoam	PROPN
ejpam-4793	1002	19	,	,	PUNCT
ejpam-4793	1002	20	38:223–256	38:223–256	PROPN
ejpam-4793	1002	21	,	,	PUNCT
ejpam-4793	1002	22	2022	2022	NUM
ejpam-4793	1002	23	.	.	PUNCT
ejpam-4793	1003	1	[	[	X
ejpam-4793	1003	2	4	4	X
ejpam-4793	1003	3	]	]	X
ejpam-4793	1003	4	y.	y.	NOUN
ejpam-4793	1003	5	give	give	VERB
ejpam-4793	1003	6	’	'	PUNCT
ejpam-4793	1003	7	on	on	ADP
ejpam-4793	1003	8	.	.	PUNCT
ejpam-4793	1004	1	normal	normal	ADJ
ejpam-4793	1004	2	monoids	monoid	NOUN
ejpam-4793	1004	3	and	and	CCONJ
ejpam-4793	1004	4	factor	factor	NOUN
ejpam-4793	1004	5	monoids	monoid	NOUN
ejpam-4793	1004	6	of	of	ADP
ejpam-4793	1004	7	commutative	commutative	ADJ
ejpam-4793	1004	8	monoids	monoid	NOUN
ejpam-4793	1004	9	.	.	PUNCT
ejpam-4793	1005	1	1963	1963	NUM
ejpam-4793	1005	2	.	.	PUNCT
ejpam-4793	1006	1	[	[	X
ejpam-4793	1006	2	5	5	NUM
ejpam-4793	1006	3	]	]	PUNCT
ejpam-4793	1006	4	a.	a.	NOUN
ejpam-4793	1006	5	sebandal	sebandal	PROPN
ejpam-4793	1006	6	and	and	CCONJ
ejpam-4793	1006	7	j.	j.	PROPN
ejpam-4793	1006	8	vilela	vilela	PROPN
ejpam-4793	1006	9	.	.	PUNCT
ejpam-4793	1007	1	the	the	DET
ejpam-4793	1007	2	jordan	jordan	PROPN
ejpam-4793	1007	3	-	-	PUNCT
ejpam-4793	1007	4	hölder	hölder	PROPN
ejpam-4793	1007	5	theorem	theorem	NOUN
ejpam-4793	1007	6	for	for	ADP
ejpam-4793	1007	7	monoids	monoid	NOUN
ejpam-4793	1007	8	with	with	ADP
ejpam-4793	1007	9	group	group	NOUN
ejpam-4793	1007	10	action	action	NOUN
ejpam-4793	1007	11	.	.	PUNCT
ejpam-4793	1008	1	journal	journal	NOUN
ejpam-4793	1008	2	of	of	ADP
ejpam-4793	1008	3	algebra	algebra	PROPN
ejpam-4793	1008	4	and	and	CCONJ
ejpam-4793	1008	5	its	its	PRON
ejpam-4793	1008	6	application	application	NOUN
ejpam-4793	1008	7	,	,	PUNCT
ejpam-4793	1008	8	22(4):1–18	22(4):1–18	NUM
ejpam-4793	1008	9	,	,	PUNCT
ejpam-4793	1008	10	2023	2023	NUM
ejpam-4793	1008	11	.	.	PUNCT
ejpam-4793	1009	1	[	[	X
ejpam-4793	1009	2	6	6	NUM
ejpam-4793	1009	3	]	]	PUNCT
ejpam-4793	1009	4	b.	b.	PROPN
ejpam-4793	1009	5	steinberg	steinberg	PROPN
ejpam-4793	1009	6	.	.	PUNCT
ejpam-4793	1009	7	representation	representation	PROPN
ejpam-4793	1009	8	theory	theory	NOUN
ejpam-4793	1009	9	of	of	ADP
ejpam-4793	1009	10	finite	finite	PROPN
ejpam-4793	1009	11	monoids	monoid	NOUN
ejpam-4793	1009	12	.	.	PUNCT
ejpam-4793	1010	1	2016	2016	NUM
ejpam-4793	1010	2	.	.	PUNCT