id	sid	tid	token	lemma	pos
ejpam-4827	1	1	european	european	PROPN
ejpam-4827	1	2	journal	journal	PROPN
ejpam-4827	1	3	of	of	ADP
ejpam-4827	1	4	pure	pure	ADJ
ejpam-4827	1	5	and	and	CCONJ
ejpam-4827	1	6	applied	apply	VERB
ejpam-4827	1	7	mathematics	mathematic	NOUN
ejpam-4827	1	8	vol	vol	NOUN
ejpam-4827	1	9	.	.	PUNCT
ejpam-4827	2	1	16	16	NUM
ejpam-4827	2	2	,	,	PUNCT
ejpam-4827	2	3	no	no	INTJ
ejpam-4827	2	4	.	.	NOUN
ejpam-4827	2	5	3	3	NUM
ejpam-4827	2	6	,	,	PUNCT
ejpam-4827	2	7	2023	2023	NUM
ejpam-4827	2	8	,	,	PUNCT
ejpam-4827	2	9	1878	1878	NUM
ejpam-4827	2	10	-	-	SYM
ejpam-4827	2	11	1893	1893	NUM
ejpam-4827	2	12	issn	issn	VERB
ejpam-4827	2	13	1307	1307	NUM
ejpam-4827	2	14	-	-	SYM
ejpam-4827	2	15	5543	5543	NUM
ejpam-4827	2	16	–	–	PUNCT
ejpam-4827	2	17	ejpam.com	ejpam.com	X
ejpam-4827	2	18	published	publish	VERB
ejpam-4827	2	19	by	by	ADP
ejpam-4827	2	20	new	new	PROPN
ejpam-4827	2	21	york	york	PROPN
ejpam-4827	2	22	business	business	PROPN
ejpam-4827	2	23	global	global	PROPN
ejpam-4827	2	24	the	the	DET
ejpam-4827	2	25	reflexive	reflexive	ADJ
ejpam-4827	2	26	condition	condition	NOUN
ejpam-4827	2	27	on	on	ADP
ejpam-4827	2	28	skew	skew	NOUN
ejpam-4827	2	29	monoid	monoid	PROPN
ejpam-4827	2	30	rings	ring	NOUN
ejpam-4827	2	31	eltiyeb	eltiyeb	PROPN
ejpam-4827	2	32	ali1,2	ali1,2	PROPN
ejpam-4827	2	33	1	1	NUM
ejpam-4827	2	34	department	department	NOUN
ejpam-4827	2	35	of	of	ADP
ejpam-4827	2	36	mathematics	mathematic	NOUN
ejpam-4827	2	37	,	,	PUNCT
ejpam-4827	2	38	faculty	faculty	NOUN
ejpam-4827	2	39	of	of	ADP
ejpam-4827	2	40	education	education	NOUN
ejpam-4827	2	41	,	,	PUNCT
ejpam-4827	2	42	university	university	NOUN
ejpam-4827	2	43	of	of	ADP
ejpam-4827	2	44	khartoum	khartoum	PROPN
ejpam-4827	2	45	,	,	PUNCT
ejpam-4827	2	46	sudan	sudan	PROPN
ejpam-4827	2	47	2	2	NUM
ejpam-4827	2	48	department	department	NOUN
ejpam-4827	2	49	of	of	ADP
ejpam-4827	2	50	mathematics	mathematic	NOUN
ejpam-4827	2	51	,	,	PUNCT
ejpam-4827	2	52	collage	collage	NOUN
ejpam-4827	2	53	of	of	ADP
ejpam-4827	2	54	science	science	NOUN
ejpam-4827	2	55	and	and	CCONJ
ejpam-4827	2	56	arts	art	NOUN
ejpam-4827	2	57	,	,	PUNCT
ejpam-4827	2	58	najran	najran	ADJ
ejpam-4827	2	59	university	university	NOUN
ejpam-4827	2	60	,	,	PUNCT
ejpam-4827	2	61	saudi	saudi	PROPN
ejpam-4827	2	62	arabia	arabia	PROPN
ejpam-4827	2	63	abstract	abstract	NOUN
ejpam-4827	2	64	.	.	PUNCT
ejpam-4827	3	1	this	this	DET
ejpam-4827	3	2	paper	paper	NOUN
ejpam-4827	3	3	is	be	AUX
ejpam-4827	3	4	devoted	devote	VERB
ejpam-4827	3	5	to	to	ADP
ejpam-4827	3	6	introducing	introduce	VERB
ejpam-4827	3	7	and	and	CCONJ
ejpam-4827	3	8	studying	study	VERB
ejpam-4827	3	9	two	two	NUM
ejpam-4827	3	10	concepts	concept	NOUN
ejpam-4827	3	11	,	,	PUNCT
ejpam-4827	3	12	σ	σ	PROPN
ejpam-4827	3	13	-	-	PUNCT
ejpam-4827	3	14	skew	skew	NOUN
ejpam-4827	3	15	strongly	strongly	ADV
ejpam-4827	3	16	m	m	VERB
ejpam-4827	3	17	reflexive	reflexive	ADJ
ejpam-4827	3	18	and	and	CCONJ
ejpam-4827	3	19	σ	σ	NOUN
ejpam-4827	3	20	-	-	PUNCT
ejpam-4827	3	21	skew	skew	NOUN
ejpam-4827	3	22	strongly	strongly	ADV
ejpam-4827	3	23	m	m	VERB
ejpam-4827	3	24	-nil	-nil	NOUN
ejpam-4827	3	25	-	-	PUNCT
ejpam-4827	3	26	reflexive	reflexive	ADJ
ejpam-4827	3	27	on	on	ADP
ejpam-4827	3	28	monoid	monoid	NOUN
ejpam-4827	3	29	rings	ring	NOUN
ejpam-4827	3	30	,	,	PUNCT
ejpam-4827	3	31	which	which	PRON
ejpam-4827	3	32	are	be	AUX
ejpam-4827	3	33	generalizations	generalization	NOUN
ejpam-4827	3	34	of	of	ADP
ejpam-4827	3	35	strongly	strongly	ADV
ejpam-4827	3	36	m	m	PRON
ejpam-4827	3	37	-reflexive	-reflexive	ADJ
ejpam-4827	3	38	and	and	CCONJ
ejpam-4827	3	39	m	m	NOUN
ejpam-4827	3	40	-compatible	-compatible	ADJ
ejpam-4827	3	41	.	.	PUNCT
ejpam-4827	4	1	the	the	DET
ejpam-4827	4	2	paper	paper	NOUN
ejpam-4827	4	3	covers	cover	VERB
ejpam-4827	4	4	the	the	DET
ejpam-4827	4	5	basic	basic	ADJ
ejpam-4827	4	6	properties	property	NOUN
ejpam-4827	4	7	of	of	ADP
ejpam-4827	4	8	skew	skew	ADJ
ejpam-4827	4	9	monoid	monoid	NOUN
ejpam-4827	4	10	rings	ring	NOUN
ejpam-4827	4	11	of	of	ADP
ejpam-4827	4	12	the	the	DET
ejpam-4827	4	13	form	form	NOUN
ejpam-4827	4	14	r∗m	r∗m	NOUN
ejpam-4827	4	15	.	.	PUNCT
ejpam-4827	5	1	it	it	PRON
ejpam-4827	5	2	is	be	AUX
ejpam-4827	5	3	shown	show	VERB
ejpam-4827	5	4	that	that	SCONJ
ejpam-4827	5	5	if	if	SCONJ
ejpam-4827	5	6	r	r	NOUN
ejpam-4827	5	7	is	be	AUX
ejpam-4827	5	8	a	a	DET
ejpam-4827	5	9	left	left	ADJ
ejpam-4827	5	10	app	app	NOUN
ejpam-4827	5	11	(	(	PUNCT
ejpam-4827	5	12	quasi	quasi	NOUN
ejpam-4827	5	13	armendariz	armendariz	NOUN
ejpam-4827	5	14	,	,	PUNCT
ejpam-4827	5	15	semiprime	semiprime	NOUN
ejpam-4827	5	16	rings	ring	NOUN
ejpam-4827	5	17	,	,	PUNCT
ejpam-4827	5	18	respectively	respectively	ADV
ejpam-4827	5	19	)	)	PUNCT
ejpam-4827	5	20	,	,	PUNCT
ejpam-4827	5	21	then	then	ADV
ejpam-4827	5	22	r	r	NOUN
ejpam-4827	5	23	is	be	AUX
ejpam-4827	5	24	σ	σ	NOUN
ejpam-4827	5	25	-	-	PUNCT
ejpam-4827	5	26	skew	skew	NOUN
ejpam-4827	5	27	strongly	strongly	ADV
ejpam-4827	5	28	m	m	VERB
ejpam-4827	5	29	-reflexive	-reflexive	ADJ
ejpam-4827	5	30	.	.	PUNCT
ejpam-4827	6	1	moreover	moreover	ADV
ejpam-4827	6	2	,	,	PUNCT
ejpam-4827	6	3	if	if	SCONJ
ejpam-4827	6	4	r	r	NOUN
ejpam-4827	6	5	is	be	AUX
ejpam-4827	6	6	a	a	DET
ejpam-4827	6	7	ni	ni	NOUN
ejpam-4827	6	8	-	-	PUNCT
ejpam-4827	6	9	ring	ring	NOUN
ejpam-4827	6	10	and	and	CCONJ
ejpam-4827	6	11	m	m	NOUN
ejpam-4827	6	12	is	be	AUX
ejpam-4827	6	13	a	a	DET
ejpam-4827	6	14	u.p	u.p	PROPN
ejpam-4827	6	15	-	-	PUNCT
ejpam-4827	6	16	monoid	monoid	PROPN
ejpam-4827	6	17	,	,	PUNCT
ejpam-4827	6	18	then	then	ADV
ejpam-4827	6	19	r	r	NOUN
ejpam-4827	6	20	is	be	AUX
ejpam-4827	6	21	σ	σ	NOUN
ejpam-4827	6	22	-	-	PUNCT
ejpam-4827	6	23	skew	skew	NOUN
ejpam-4827	6	24	strongly	strongly	ADV
ejpam-4827	6	25	m	m	VERB
ejpam-4827	6	26	-nil	-nil	NOUN
ejpam-4827	6	27	-	-	PUNCT
ejpam-4827	6	28	reflexive	reflexive	ADJ
ejpam-4827	6	29	.	.	PUNCT
ejpam-4827	7	1	additionally	additionally	ADV
ejpam-4827	7	2	,	,	PUNCT
ejpam-4827	7	3	under	under	ADP
ejpam-4827	7	4	some	some	DET
ejpam-4827	7	5	necessary	necessary	ADJ
ejpam-4827	7	6	and	and	CCONJ
ejpam-4827	7	7	sufficient	sufficient	ADJ
ejpam-4827	7	8	conditions	condition	NOUN
ejpam-4827	7	9	,	,	PUNCT
ejpam-4827	7	10	a	a	DET
ejpam-4827	7	11	skew	skew	ADJ
ejpam-4827	7	12	monoid	monoid	NOUN
ejpam-4827	7	13	ring	ring	NOUN
ejpam-4827	7	14	r	r	NOUN
ejpam-4827	7	15	∗m	∗m	NOUN
ejpam-4827	7	16	is	be	AUX
ejpam-4827	7	17	proven	prove	VERB
ejpam-4827	7	18	to	to	PART
ejpam-4827	7	19	be	be	AUX
ejpam-4827	7	20	σ	σ	NOUN
ejpam-4827	7	21	-	-	PUNCT
ejpam-4827	7	22	skew	skew	NOUN
ejpam-4827	7	23	strongly	strongly	ADV
ejpam-4827	7	24	m	m	VERB
ejpam-4827	7	25	-nil	-nil	NOUN
ejpam-4827	7	26	-	-	PUNCT
ejpam-4827	7	27	reflexive	reflexive	ADJ
ejpam-4827	7	28	when	when	SCONJ
ejpam-4827	7	29	σ	σ	NOUN
ejpam-4827	7	30	:	:	PUNCT
ejpam-4827	7	31	m	m	PROPN
ejpam-4827	7	32	→	→	SYM
ejpam-4827	7	33	aut(r	aut(r	PROPN
ejpam-4827	7	34	)	)	PUNCT
ejpam-4827	7	35	is	be	AUX
ejpam-4827	7	36	a	a	DET
ejpam-4827	7	37	monoid	monoid	NOUN
ejpam-4827	7	38	homomorphism	homomorphism	NOUN
ejpam-4827	7	39	.	.	PUNCT
ejpam-4827	8	1	furthermore	furthermore	ADV
ejpam-4827	8	2	,	,	PUNCT
ejpam-4827	8	3	if	if	SCONJ
ejpam-4827	8	4	r	r	NOUN
ejpam-4827	8	5	is	be	AUX
ejpam-4827	8	6	a	a	DET
ejpam-4827	8	7	left	left	ADJ
ejpam-4827	8	8	app	app	NOUN
ejpam-4827	8	9	,	,	PUNCT
ejpam-4827	8	10	then	then	ADV
ejpam-4827	8	11	the	the	DET
ejpam-4827	8	12	upper	upper	ADJ
ejpam-4827	8	13	triangular	triangular	NOUN
ejpam-4827	8	14	matrix	matrix	NOUN
ejpam-4827	8	15	ring	ring	NOUN
ejpam-4827	8	16	tn(r	tn(r	NOUN
ejpam-4827	8	17	)	)	PUNCT
ejpam-4827	8	18	is	be	AUX
ejpam-4827	8	19	σ̄-skew	σ̄-skew	PROPN
ejpam-4827	8	20	strongly	strongly	ADV
ejpam-4827	8	21	m	m	VERB
ejpam-4827	8	22	-nil	-nil	NOUN
ejpam-4827	8	23	-	-	PUNCT
ejpam-4827	8	24	reflexive	reflexive	ADJ
ejpam-4827	8	25	,	,	PUNCT
ejpam-4827	8	26	where	where	SCONJ
ejpam-4827	8	27	n	n	PRON
ejpam-4827	8	28	is	be	AUX
ejpam-4827	8	29	a	a	DET
ejpam-4827	8	30	positive	positive	ADJ
ejpam-4827	8	31	integer	integer	NOUN
ejpam-4827	8	32	.	.	PUNCT
ejpam-4827	9	1	finally	finally	ADV
ejpam-4827	9	2	,	,	PUNCT
ejpam-4827	9	3	the	the	DET
ejpam-4827	9	4	paper	paper	NOUN
ejpam-4827	9	5	provides	provide	VERB
ejpam-4827	9	6	some	some	DET
ejpam-4827	9	7	examples	example	NOUN
ejpam-4827	9	8	and	and	CCONJ
ejpam-4827	9	9	discusses	discuss	VERB
ejpam-4827	9	10	related	related	ADJ
ejpam-4827	9	11	results	result	NOUN
ejpam-4827	9	12	from	from	ADP
ejpam-4827	9	13	the	the	DET
ejpam-4827	9	14	subject	subject	NOUN
ejpam-4827	9	15	.	.	PUNCT
ejpam-4827	10	1	2020	2020	NUM
ejpam-4827	10	2	mathematics	mathematic	NOUN
ejpam-4827	10	3	subject	subject	NOUN
ejpam-4827	10	4	classifications	classification	NOUN
ejpam-4827	10	5	:	:	PUNCT
ejpam-4827	10	6	16s34	16s34	NUM
ejpam-4827	10	7	,	,	PUNCT
ejpam-4827	10	8	16s36	16s36	NUM
ejpam-4827	10	9	,	,	PUNCT
ejpam-4827	10	10	20m25	20m25	NUM
ejpam-4827	10	11	key	key	ADJ
ejpam-4827	10	12	words	word	NOUN
ejpam-4827	10	13	and	and	CCONJ
ejpam-4827	10	14	phrases	phrase	NOUN
ejpam-4827	10	15	:	:	PUNCT
ejpam-4827	10	16	left	leave	VERB
ejpam-4827	10	17	app	app	NOUN
ejpam-4827	10	18	-ring	-ring	PROPN
ejpam-4827	10	19	,	,	PUNCT
ejpam-4827	10	20	skew	skew	ADJ
ejpam-4827	10	21	monoid	monoid	NOUN
ejpam-4827	10	22	ring	ring	NOUN
ejpam-4827	10	23	r	r	NOUN
ejpam-4827	10	24	∗m	∗m	NOUN
ejpam-4827	10	25	,	,	PUNCT
ejpam-4827	10	26	quasi	quasi	NOUN
ejpam-4827	10	27	armendariz	armendariz	PROPN
ejpam-4827	10	28	ring	ring	NOUN
ejpam-4827	10	29	,	,	PUNCT
ejpam-4827	10	30	σ	σ	PROPN
ejpam-4827	10	31	-	-	PUNCT
ejpam-4827	10	32	skew	skew	NOUN
ejpam-4827	10	33	strongly	strongly	ADV
ejpam-4827	10	34	m	m	VERB
ejpam-4827	10	35	-nil	-nil	ADJ
ejpam-4827	10	36	-	-	PUNCT
ejpam-4827	10	37	reflexive	reflexive	ADJ
ejpam-4827	10	38	ring	ring	NOUN
ejpam-4827	10	39	1	1	NUM
ejpam-4827	10	40	.	.	PUNCT
ejpam-4827	10	41	introduction	introduction	NOUN
ejpam-4827	10	42	throughout	throughout	ADP
ejpam-4827	10	43	this	this	DET
ejpam-4827	10	44	article	article	NOUN
ejpam-4827	10	45	,	,	PUNCT
ejpam-4827	10	46	r	r	NOUN
ejpam-4827	10	47	and	and	CCONJ
ejpam-4827	10	48	m	m	PROPN
ejpam-4827	10	49	denote	denote	VERB
ejpam-4827	10	50	an	an	DET
ejpam-4827	10	51	associative	associative	ADJ
ejpam-4827	10	52	ring	ring	NOUN
ejpam-4827	10	53	with	with	ADP
ejpam-4827	10	54	identity	identity	NOUN
ejpam-4827	10	55	and	and	CCONJ
ejpam-4827	10	56	a	a	DET
ejpam-4827	10	57	monoid	monoid	NOUN
ejpam-4827	10	58	,	,	PUNCT
ejpam-4827	10	59	respectively	respectively	ADV
ejpam-4827	10	60	.	.	PUNCT
ejpam-4827	11	1	mason	mason	PROPN
ejpam-4827	11	2	introduced	introduce	VERB
ejpam-4827	11	3	the	the	DET
ejpam-4827	11	4	reflexive	reflexive	ADJ
ejpam-4827	11	5	property	property	NOUN
ejpam-4827	11	6	for	for	ADP
ejpam-4827	11	7	ideals	ideal	NOUN
ejpam-4827	11	8	and	and	CCONJ
ejpam-4827	11	9	this	this	DET
ejpam-4827	11	10	concept	concept	NOUN
ejpam-4827	11	11	was	be	AUX
ejpam-4827	11	12	generalized	generalize	VERB
ejpam-4827	11	13	by	by	ADP
ejpam-4827	11	14	some	some	DET
ejpam-4827	11	15	authors	author	NOUN
ejpam-4827	11	16	,	,	PUNCT
ejpam-4827	11	17	defining	define	VERB
ejpam-4827	11	18	idempotent	idempotent	ADJ
ejpam-4827	11	19	reflexive	reflexive	ADJ
ejpam-4827	11	20	right	right	ADJ
ejpam-4827	11	21	ideals	ideal	NOUN
ejpam-4827	11	22	and	and	CCONJ
ejpam-4827	11	23	rings	ring	NOUN
ejpam-4827	11	24	,	,	PUNCT
ejpam-4827	11	25	completely	completely	ADV
ejpam-4827	11	26	reflexive	reflexive	ADJ
ejpam-4827	11	27	,	,	PUNCT
ejpam-4827	11	28	weakly	weakly	ADV
ejpam-4827	11	29	reflexive	reflexive	ADJ
ejpam-4827	11	30	(	(	PUNCT
ejpam-4827	11	31	see	see	VERB
ejpam-4827	11	32	namely	namely	ADV
ejpam-4827	11	33	,	,	PUNCT
ejpam-4827	11	34	[	[	X
ejpam-4827	11	35	13	13	NUM
ejpam-4827	11	36	]	]	PUNCT
ejpam-4827	11	37	,	,	PUNCT
ejpam-4827	11	38	[	[	X
ejpam-4827	11	39	15	15	NUM
ejpam-4827	11	40	]	]	PUNCT
ejpam-4827	11	41	and	and	CCONJ
ejpam-4827	11	42	[	[	X
ejpam-4827	11	43	24	24	NUM
ejpam-4827	11	44	]	]	PUNCT
ejpam-4827	11	45	)	)	PUNCT
ejpam-4827	11	46	.	.	PUNCT
ejpam-4827	12	1	let	let	VERB
ejpam-4827	12	2	r	r	PRON
ejpam-4827	12	3	be	be	AUX
ejpam-4827	12	4	a	a	DET
ejpam-4827	12	5	ring	ring	NOUN
ejpam-4827	12	6	and	and	CCONJ
ejpam-4827	12	7	i	i	PRON
ejpam-4827	12	8	be	be	VERB
ejpam-4827	12	9	a	a	DET
ejpam-4827	12	10	right	right	ADJ
ejpam-4827	12	11	ideal	ideal	NOUN
ejpam-4827	12	12	of	of	ADP
ejpam-4827	12	13	r.	r.	PROPN
ejpam-4827	12	14	in	in	ADP
ejpam-4827	12	15	[	[	X
ejpam-4827	12	16	24	24	NUM
ejpam-4827	12	17	]	]	PUNCT
ejpam-4827	12	18	,	,	PUNCT
ejpam-4827	12	19	i	i	PRON
ejpam-4827	12	20	is	be	AUX
ejpam-4827	12	21	called	call	VERB
ejpam-4827	12	22	a	a	DET
ejpam-4827	12	23	reflexive	reflexive	ADJ
ejpam-4827	12	24	right	right	ADJ
ejpam-4827	12	25	ideal	ideal	NOUN
ejpam-4827	12	26	if	if	SCONJ
ejpam-4827	12	27	for	for	ADP
ejpam-4827	12	28	any	any	DET
ejpam-4827	12	29	x	x	NOUN
ejpam-4827	12	30	,	,	PUNCT
ejpam-4827	12	31	y	y	PROPN
ejpam-4827	12	32	∈	∈	PROPN
ejpam-4827	12	33	r	r	PROPN
ejpam-4827	12	34	,	,	PUNCT
ejpam-4827	12	35	xry	xry	PROPN
ejpam-4827	13	1	⊆	⊆	NUM
ejpam-4827	13	2	i	i	PRON
ejpam-4827	13	3	implies	imply	VERB
ejpam-4827	13	4	yrx	yrx	NOUN
ejpam-4827	13	5	⊆	⊆	NUM
ejpam-4827	13	6	i.	i.	NOUN
ejpam-4827	13	7	the	the	DET
ejpam-4827	13	8	reflexive	reflexive	ADJ
ejpam-4827	13	9	right	right	ADJ
ejpam-4827	13	10	ideal	ideal	ADJ
ejpam-4827	13	11	concept	concept	NOUN
ejpam-4827	13	12	is	be	AUX
ejpam-4827	13	13	also	also	ADV
ejpam-4827	13	14	specialized	specialize	VERB
ejpam-4827	13	15	to	to	ADP
ejpam-4827	13	16	the	the	DET
ejpam-4827	13	17	zero	zero	NUM
ejpam-4827	13	18	ideal	ideal	NOUN
ejpam-4827	13	19	of	of	ADP
ejpam-4827	13	20	a	a	DET
ejpam-4827	13	21	ring	ring	NOUN
ejpam-4827	13	22	,	,	PUNCT
ejpam-4827	13	23	namely	namely	ADV
ejpam-4827	13	24	,	,	PUNCT
ejpam-4827	13	25	a	a	DET
ejpam-4827	13	26	ring	ring	NOUN
ejpam-4827	13	27	r	r	NOUN
ejpam-4827	13	28	is	be	AUX
ejpam-4827	13	29	called	call	VERB
ejpam-4827	13	30	reflexive	reflexive	ADJ
ejpam-4827	13	31	[	[	X
ejpam-4827	13	32	24	24	NUM
ejpam-4827	13	33	]	]	PUNCT
ejpam-4827	13	34	,	,	PUNCT
ejpam-4827	13	35	if	if	SCONJ
ejpam-4827	13	36	its	its	PRON
ejpam-4827	13	37	zero	zero	NUM
ejpam-4827	13	38	ideal	ideal	NOUN
ejpam-4827	13	39	is	be	AUX
ejpam-4827	13	40	reflexive	reflexive	ADJ
ejpam-4827	13	41	and	and	CCONJ
ejpam-4827	13	42	a	a	DET
ejpam-4827	13	43	ring	ring	NOUN
ejpam-4827	13	44	r	r	NOUN
ejpam-4827	13	45	is	be	AUX
ejpam-4827	13	46	called	call	VERB
ejpam-4827	13	47	completely	completely	ADV
ejpam-4827	13	48	reflexive	reflexive	ADJ
ejpam-4827	13	49	if	if	SCONJ
ejpam-4827	13	50	for	for	ADP
ejpam-4827	13	51	any	any	DET
ejpam-4827	13	52	x	x	NOUN
ejpam-4827	13	53	,	,	PUNCT
ejpam-4827	13	54	y	y	PROPN
ejpam-4827	13	55	∈	∈	PROPN
ejpam-4827	13	56	r	r	NOUN
ejpam-4827	13	57	,	,	PUNCT
ejpam-4827	13	58	xy	xy	PROPN
ejpam-4827	13	59	=	=	SYM
ejpam-4827	13	60	0	0	NUM
ejpam-4827	13	61	implies	imply	VERB
ejpam-4827	13	62	yx	yx	NOUN
ejpam-4827	13	63	=	=	SYM
ejpam-4827	13	64	0	0	PROPN
ejpam-4827	13	65	.	.	PUNCT
ejpam-4827	14	1	reduced	reduce	VERB
ejpam-4827	14	2	rings	ring	NOUN
ejpam-4827	14	3	are	be	AUX
ejpam-4827	14	4	completely	completely	ADV
ejpam-4827	14	5	reflexive	reflexive	ADJ
ejpam-4827	14	6	and	and	CCONJ
ejpam-4827	14	7	every	every	DET
ejpam-4827	14	8	completely	completely	ADV
ejpam-4827	14	9	reflexive	reflexive	ADJ
ejpam-4827	14	10	ring	ring	NOUN
ejpam-4827	14	11	is	be	AUX
ejpam-4827	14	12	semicommutative	semicommutative	ADJ
ejpam-4827	14	13	.	.	PUNCT
ejpam-4827	15	1	the	the	DET
ejpam-4827	15	2	notion	notion	NOUN
ejpam-4827	15	3	of	of	ADP
ejpam-4827	15	4	armendariz	armendariz	ADJ
ejpam-4827	15	5	ring	ring	NOUN
ejpam-4827	15	6	is	be	AUX
ejpam-4827	15	7	introduced	introduce	VERB
ejpam-4827	15	8	by	by	ADP
ejpam-4827	15	9	rege	rege	PROPN
ejpam-4827	15	10	and	and	CCONJ
ejpam-4827	15	11	chhawchharia	chhawchharia	VERB
ejpam-4827	16	1	[	[	X
ejpam-4827	16	2	21	21	NUM
ejpam-4827	16	3	]	]	PUNCT
ejpam-4827	16	4	.	.	PUNCT
ejpam-4827	17	1	they	they	PRON
ejpam-4827	17	2	defined	define	VERB
ejpam-4827	17	3	a	a	DET
ejpam-4827	17	4	ring	ring	NOUN
ejpam-4827	17	5	r	r	NOUN
ejpam-4827	17	6	to	to	PART
ejpam-4827	17	7	be	be	AUX
ejpam-4827	17	8	armendariz	armendariz	ADJ
ejpam-4827	17	9	if	if	SCONJ
ejpam-4827	17	10	f(x)g(x	f(x)g(x	NOUN
ejpam-4827	17	11	)	)	PUNCT
ejpam-4827	17	12	=	=	SYM
ejpam-4827	17	13	0	0	NUM
ejpam-4827	17	14	implies	imply	VERB
ejpam-4827	17	15	aibj	aibj	NOUN
ejpam-4827	17	16	=	=	PUNCT
ejpam-4827	17	17	0	0	PROPN
ejpam-4827	17	18	for	for	ADP
ejpam-4827	17	19	all	all	DET
ejpam-4827	17	20	polynomials	polynomial	NOUN
ejpam-4827	17	21	f(x	f(x	PROPN
ejpam-4827	17	22	)	)	PUNCT
ejpam-4827	18	1	=	=	SYM
ejpam-4827	18	2	a0	a0	PROPN
ejpam-4827	18	3	+	+	CCONJ
ejpam-4827	18	4	a1x+	a1x+	NOUN
ejpam-4827	18	5	a2x	a2x	ADP
ejpam-4827	18	6	2	2	NUM
ejpam-4827	18	7	+	+	CCONJ
ejpam-4827	18	8	·	·	PUNCT
ejpam-4827	18	9	·	·	PUNCT
ejpam-4827	18	10	·	·	PUNCT
ejpam-4827	18	11	+	+	NUM
ejpam-4827	18	12	amx	amx	PROPN
ejpam-4827	18	13	m	m	PROPN
ejpam-4827	18	14	and	and	CCONJ
ejpam-4827	18	15	g(x	g(x	NOUN
ejpam-4827	18	16	)	)	PUNCT
ejpam-4827	19	1	=	=	SYM
ejpam-4827	19	2	b0	b0	NOUN
ejpam-4827	19	3	+	+	CCONJ
ejpam-4827	19	4	b1x+	b1x+	NOUN
ejpam-4827	19	5	b2x	b2x	ADP
ejpam-4827	19	6	2	2	NUM
ejpam-4827	19	7	+	+	NUM
ejpam-4827	19	8	·	·	PUNCT
ejpam-4827	19	9	·	·	PUNCT
ejpam-4827	19	10	·	·	PUNCT
ejpam-4827	19	11	+	+	NUM
ejpam-4827	19	12	bnxn	bnxn	ADJ
ejpam-4827	19	13	∈	∈	PROPN
ejpam-4827	19	14	r[x	r[x	NOUN
ejpam-4827	19	15	]	]	PUNCT
ejpam-4827	19	16	.	.	PUNCT
ejpam-4827	20	1	in	in	ADP
ejpam-4827	20	2	[	[	X
ejpam-4827	20	3	21	21	NUM
ejpam-4827	20	4	]	]	X
ejpam-4827	20	5	a	a	DET
ejpam-4827	20	6	ring	ring	NOUN
ejpam-4827	20	7	r	r	NOUN
ejpam-4827	20	8	is	be	AUX
ejpam-4827	20	9	called	call	VERB
ejpam-4827	20	10	semicommutative	semicommutative	NOUN
ejpam-4827	20	11	if	if	SCONJ
ejpam-4827	20	12	for	for	ADP
ejpam-4827	20	13	all	all	DET
ejpam-4827	20	14	x	x	NOUN
ejpam-4827	20	15	,	,	PUNCT
ejpam-4827	20	16	y	y	PROPN
ejpam-4827	20	17	∈	∈	PROPN
ejpam-4827	20	18	r	r	NOUN
ejpam-4827	20	19	,	,	PUNCT
ejpam-4827	20	20	xy	xy	PROPN
ejpam-4827	20	21	=	=	SYM
ejpam-4827	20	22	0	0	NUM
ejpam-4827	20	23	implies	imply	VERB
ejpam-4827	20	24	xry	xry	X
ejpam-4827	20	25	=	=	SYM
ejpam-4827	20	26	0	0	X
ejpam-4827	20	27	.	.	PUNCT
ejpam-4827	21	1	this	this	PRON
ejpam-4827	21	2	is	be	AUX
ejpam-4827	21	3	equivalent	equivalent	ADJ
ejpam-4827	21	4	to	to	ADP
ejpam-4827	21	5	the	the	DET
ejpam-4827	21	6	definition	definition	NOUN
ejpam-4827	21	7	that	that	SCONJ
ejpam-4827	21	8	any	any	DET
ejpam-4827	21	9	doi	doi	NOUN
ejpam-4827	21	10	:	:	PUNCT
ejpam-4827	21	11	https://doi.org/10.29020/nybg.ejpam.v16i3.4827	https://doi.org/10.29020/nybg.ejpam.v16i3.4827	PRON
ejpam-4827	21	12	email	email	NOUN
ejpam-4827	21	13	addresses	address	VERB
ejpam-4827	21	14	:	:	PUNCT
ejpam-4827	21	15	eltiyeb76@gmail.com	eltiyeb76@gmail.com	PROPN
ejpam-4827	21	16	,	,	PUNCT
ejpam-4827	21	17	emali@nu.edu.sa	emali@nu.edu.sa	PROPN
ejpam-4827	21	18	(	(	PUNCT
ejpam-4827	21	19	e.	e.	PROPN
ejpam-4827	21	20	ali	ali	PROPN
ejpam-4827	21	21	)	)	PUNCT
ejpam-4827	21	22	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4827	21	23	1878	1878	NUM
ejpam-4827	22	1	©	©	PROPN
ejpam-4827	22	2	2023	2023	NUM
ejpam-4827	22	3	ejpam	ejpam	NOUN
ejpam-4827	22	4	all	all	DET
ejpam-4827	22	5	rights	right	NOUN
ejpam-4827	22	6	reserved	reserve	VERB
ejpam-4827	22	7	.	.	PUNCT
ejpam-4827	23	1	e.	e.	PROPN
ejpam-4827	23	2	ali	ali	PROPN
ejpam-4827	23	3	/	/	SYM
ejpam-4827	23	4	eur	eur	PROPN
ejpam-4827	23	5	.	.	PUNCT
ejpam-4827	24	1	j.	j.	PROPN
ejpam-4827	24	2	pure	pure	PROPN
ejpam-4827	24	3	appl	appl	PROPN
ejpam-4827	24	4	.	.	PROPN
ejpam-4827	24	5	math	math	PROPN
ejpam-4827	24	6	,	,	PUNCT
ejpam-4827	24	7	16	16	NUM
ejpam-4827	24	8	(	(	PUNCT
ejpam-4827	24	9	3	3	NUM
ejpam-4827	24	10	)	)	PUNCT
ejpam-4827	24	11	(	(	PUNCT
ejpam-4827	24	12	2023	2023	NUM
ejpam-4827	24	13	)	)	PUNCT
ejpam-4827	24	14	,	,	PUNCT
ejpam-4827	24	15	1878	1878	NUM
ejpam-4827	24	16	-	-	SYM
ejpam-4827	24	17	1893	1893	NUM
ejpam-4827	24	18	1879	1879	NUM
ejpam-4827	24	19	left	leave	VERB
ejpam-4827	24	20	(	(	PUNCT
ejpam-4827	24	21	right	right	ADJ
ejpam-4827	24	22	)	)	PUNCT
ejpam-4827	24	23	annihilator	annihilator	NOUN
ejpam-4827	24	24	of	of	ADP
ejpam-4827	24	25	r	r	NOUN
ejpam-4827	24	26	is	be	AUX
ejpam-4827	24	27	an	an	DET
ejpam-4827	24	28	ideal	ideal	NOUN
ejpam-4827	24	29	of	of	ADP
ejpam-4827	24	30	r.	r.	PROPN
ejpam-4827	24	31	an	an	DET
ejpam-4827	24	32	ideal	ideal	ADJ
ejpam-4827	25	1	i	i	PRON
ejpam-4827	25	2	of	of	ADP
ejpam-4827	25	3	a	a	DET
ejpam-4827	25	4	ring	ring	NOUN
ejpam-4827	25	5	is	be	AUX
ejpam-4827	25	6	called	call	VERB
ejpam-4827	25	7	semiprime	semiprime	NOUN
ejpam-4827	25	8	if	if	SCONJ
ejpam-4827	25	9	xrx	xrx	PROPN
ejpam-4827	25	10	⊆	⊆	NUM
ejpam-4827	25	11	i	i	PRON
ejpam-4827	25	12	implies	imply	VERB
ejpam-4827	25	13	x	x	SYM
ejpam-4827	25	14	∈	∈	PROPN
ejpam-4827	25	15	i	i	PRON
ejpam-4827	25	16	for	for	ADP
ejpam-4827	25	17	x	x	SYM
ejpam-4827	25	18	∈	∈	PROPN
ejpam-4827	25	19	r	r	NOUN
ejpam-4827	25	20	and	and	CCONJ
ejpam-4827	25	21	r	r	NOUN
ejpam-4827	25	22	is	be	AUX
ejpam-4827	25	23	called	call	VERB
ejpam-4827	25	24	semiprime	semiprime	NOUN
ejpam-4827	25	25	if	if	SCONJ
ejpam-4827	25	26	0	0	NUM
ejpam-4827	25	27	is	be	AUX
ejpam-4827	25	28	a	a	DET
ejpam-4827	25	29	semiprime	semiprime	NOUN
ejpam-4827	25	30	ideal	ideal	NOUN
ejpam-4827	25	31	.	.	PUNCT
ejpam-4827	26	1	it	it	PRON
ejpam-4827	26	2	should	should	AUX
ejpam-4827	26	3	be	be	AUX
ejpam-4827	26	4	noted	note	VERB
ejpam-4827	26	5	that	that	SCONJ
ejpam-4827	26	6	every	every	DET
ejpam-4827	26	7	semiprime	semiprime	NOUN
ejpam-4827	26	8	ideal	ideal	NOUN
ejpam-4827	26	9	is	be	AUX
ejpam-4827	26	10	reflexive	reflexive	ADJ
ejpam-4827	26	11	,	,	PUNCT
ejpam-4827	26	12	as	as	SCONJ
ejpam-4827	26	13	can	can	AUX
ejpam-4827	26	14	be	be	AUX
ejpam-4827	26	15	easily	easily	ADV
ejpam-4827	26	16	verified	verify	VERB
ejpam-4827	26	17	and	and	CCONJ
ejpam-4827	26	18	therefore	therefore	ADV
ejpam-4827	26	19	every	every	DET
ejpam-4827	26	20	ideal	ideal	NOUN
ejpam-4827	26	21	of	of	ADP
ejpam-4827	26	22	a	a	DET
ejpam-4827	26	23	fully	fully	ADV
ejpam-4827	26	24	idempotent	idempotent	ADJ
ejpam-4827	26	25	ring	ring	NOUN
ejpam-4827	26	26	(	(	PUNCT
ejpam-4827	26	27	i.e.	i.e.	X
ejpam-4827	26	28	,	,	PUNCT
ejpam-4827	26	29	a	a	DET
ejpam-4827	26	30	ring	ring	NOUN
ejpam-4827	26	31	where	where	SCONJ
ejpam-4827	26	32	i2	i2	PROPN
ejpam-4827	26	33	=	=	PUNCT
ejpam-4827	27	1	i	i	PRON
ejpam-4827	27	2	for	for	ADP
ejpam-4827	27	3	all	all	DET
ejpam-4827	27	4	ideals	ideal	NOUN
ejpam-4827	27	5	i	i	PRON
ejpam-4827	27	6	)	)	PUNCT
ejpam-4827	27	7	is	be	AUX
ejpam-4827	27	8	reflexive	reflexive	ADJ
ejpam-4827	27	9	according	accord	VERB
ejpam-4827	27	10	to	to	ADP
ejpam-4827	27	11	[	[	X
ejpam-4827	27	12	6	6	NUM
ejpam-4827	27	13	]	]	PUNCT
ejpam-4827	27	14	.	.	PUNCT
ejpam-4827	28	1	the	the	DET
ejpam-4827	28	2	ring	ring	NOUN
ejpam-4827	28	3	r	r	NOUN
ejpam-4827	28	4	is	be	AUX
ejpam-4827	28	5	said	say	VERB
ejpam-4827	28	6	to	to	PART
ejpam-4827	28	7	be	be	AUX
ejpam-4827	28	8	weakly	weakly	ADV
ejpam-4827	28	9	reflexive	reflexive	ADJ
ejpam-4827	28	10	if	if	SCONJ
ejpam-4827	28	11	xry	xry	PROPN
ejpam-4827	28	12	=	=	SYM
ejpam-4827	28	13	0	0	NUM
ejpam-4827	28	14	implies	imply	VERB
ejpam-4827	28	15	yrx	yrx	NOUN
ejpam-4827	28	16	is	be	AUX
ejpam-4827	28	17	nilpotent	nilpotent	ADJ
ejpam-4827	28	18	for	for	ADP
ejpam-4827	28	19	x	x	X
ejpam-4827	28	20	,	,	PUNCT
ejpam-4827	28	21	y	y	PROPN
ejpam-4827	28	22	∈	∈	PROPN
ejpam-4827	28	23	r	r	NOUN
ejpam-4827	28	24	and	and	CCONJ
ejpam-4827	28	25	all	all	DET
ejpam-4827	28	26	r	r	NOUN
ejpam-4827	28	27	∈	∈	PROPN
ejpam-4827	28	28	r.	r.	NOUN
ejpam-4827	28	29	the	the	DET
ejpam-4827	28	30	rings	ring	NOUN
ejpam-4827	28	31	without	without	ADP
ejpam-4827	28	32	nonzero	nonzero	ADJ
ejpam-4827	28	33	nilpotent	nilpotent	ADJ
ejpam-4827	28	34	elements	element	NOUN
ejpam-4827	28	35	are	be	AUX
ejpam-4827	28	36	said	say	VERB
ejpam-4827	28	37	to	to	PART
ejpam-4827	28	38	be	be	AUX
ejpam-4827	28	39	reduced	reduce	VERB
ejpam-4827	28	40	rings	ring	NOUN
ejpam-4827	28	41	.	.	PUNCT
ejpam-4827	29	1	according	accord	VERB
ejpam-4827	29	2	to	to	ADP
ejpam-4827	29	3	[	[	X
ejpam-4827	29	4	9	9	NUM
ejpam-4827	29	5	]	]	X
ejpam-4827	29	6	a	a	DET
ejpam-4827	29	7	ring	ring	NOUN
ejpam-4827	29	8	r	r	NOUN
ejpam-4827	29	9	is	be	AUX
ejpam-4827	29	10	called	call	VERB
ejpam-4827	29	11	quasi	quasi	NOUN
ejpam-4827	29	12	-	-	ADJ
ejpam-4827	29	13	armendariz	armendariz	ADJ
ejpam-4827	29	14	if	if	SCONJ
ejpam-4827	29	15	whenever	whenever	SCONJ
ejpam-4827	29	16	polynomials	polynomial	VERB
ejpam-4827	29	17	f(x	f(x	PROPN
ejpam-4827	29	18	)	)	PUNCT
ejpam-4827	30	1	=	=	SYM
ejpam-4827	30	2	a0	a0	PROPN
ejpam-4827	30	3	+	+	CCONJ
ejpam-4827	30	4	a1x	a1x	NOUN
ejpam-4827	30	5	+	+	CCONJ
ejpam-4827	30	6	a2x	a2x	ADP
ejpam-4827	30	7	2	2	NUM
ejpam-4827	30	8	+	+	NOUN
ejpam-4827	30	9	·	·	PUNCT
ejpam-4827	30	10	·	·	PUNCT
ejpam-4827	30	11	·	·	PUNCT
ejpam-4827	31	1	+	+	NUM
ejpam-4827	31	2	amx	amx	PROPN
ejpam-4827	31	3	m	m	PROPN
ejpam-4827	31	4	,	,	PUNCT
ejpam-4827	31	5	g(x	g(x	NOUN
ejpam-4827	31	6	)	)	PUNCT
ejpam-4827	31	7	=	=	SYM
ejpam-4827	31	8	b0	b0	NOUN
ejpam-4827	31	9	+	+	CCONJ
ejpam-4827	31	10	b1x+	b1x+	NOUN
ejpam-4827	31	11	b2x	b2x	ADP
ejpam-4827	31	12	2	2	NUM
ejpam-4827	31	13	+	+	CCONJ
ejpam-4827	31	14	·	·	PUNCT
ejpam-4827	31	15	·	·	PUNCT
ejpam-4827	31	16	·	·	PUNCT
ejpam-4827	32	1	+	+	NUM
ejpam-4827	32	2	bnx	bnx	NOUN
ejpam-4827	32	3	n	n	PRON
ejpam-4827	32	4	∈	∈	PROPN
ejpam-4827	32	5	r[x	r[x	NOUN
ejpam-4827	32	6	]	]	X
ejpam-4827	32	7	satisfy	satisfy	NOUN
ejpam-4827	32	8	f(x)r[x]g(x	f(x)r[x]g(x	X
ejpam-4827	32	9	)	)	PUNCT
ejpam-4827	33	1	=	=	SYM
ejpam-4827	33	2	0	0	NUM
ejpam-4827	33	3	,	,	PUNCT
ejpam-4827	33	4	then	then	ADV
ejpam-4827	33	5	airbj	airbj	NOUN
ejpam-4827	33	6	=	=	NOUN
ejpam-4827	33	7	0	0	NUM
ejpam-4827	33	8	for	for	ADP
ejpam-4827	33	9	each	each	DET
ejpam-4827	33	10	i	i	PROPN
ejpam-4827	33	11	,	,	PUNCT
ejpam-4827	33	12	j.	j.	PROPN
ejpam-4827	34	1	it	it	PRON
ejpam-4827	34	2	was	be	AUX
ejpam-4827	34	3	proved	prove	VERB
ejpam-4827	34	4	in	in	ADP
ejpam-4827	34	5	[	[	X
ejpam-4827	34	6	10	10	NUM
ejpam-4827	34	7	]	]	PUNCT
ejpam-4827	34	8	,	,	PUNCT
ejpam-4827	34	9	if	if	SCONJ
ejpam-4827	34	10	r	r	NOUN
ejpam-4827	34	11	is	be	AUX
ejpam-4827	34	12	an	an	DET
ejpam-4827	34	13	armendariz	armendariz	ADJ
ejpam-4827	34	14	ring	ring	NOUN
ejpam-4827	34	15	,	,	PUNCT
ejpam-4827	34	16	then	then	ADV
ejpam-4827	34	17	r	r	NOUN
ejpam-4827	34	18	is	be	AUX
ejpam-4827	34	19	completely	completely	ADV
ejpam-4827	34	20	reflexive	reflexive	ADJ
ejpam-4827	34	21	if	if	SCONJ
ejpam-4827	34	22	and	and	CCONJ
ejpam-4827	34	23	only	only	ADV
ejpam-4827	34	24	if	if	SCONJ
ejpam-4827	34	25	r[x	r[x	PROPN
ejpam-4827	34	26	]	]	X
ejpam-4827	34	27	is	be	AUX
ejpam-4827	34	28	completely	completely	ADV
ejpam-4827	34	29	reflexive	reflexive	ADJ
ejpam-4827	34	30	.	.	PUNCT
ejpam-4827	35	1	according	accord	VERB
ejpam-4827	35	2	to	to	ADP
ejpam-4827	35	3	[	[	X
ejpam-4827	35	4	22	22	NUM
ejpam-4827	35	5	]	]	PUNCT
ejpam-4827	35	6	,	,	PUNCT
ejpam-4827	35	7	a	a	DET
ejpam-4827	35	8	ring	ring	NOUN
ejpam-4827	35	9	r	r	NOUN
ejpam-4827	35	10	is	be	AUX
ejpam-4827	35	11	σ	σ	NOUN
ejpam-4827	35	12	-	-	PUNCT
ejpam-4827	35	13	skew	skew	NOUN
ejpam-4827	35	14	nil	nil	NOUN
ejpam-4827	35	15	m	m	VERB
ejpam-4827	35	16	-mccoy	-mccoy	ADJ
ejpam-4827	35	17	if	if	SCONJ
ejpam-4827	35	18	αβ	αβ	INTJ
ejpam-4827	35	19	∈	∈	PROPN
ejpam-4827	35	20	nil(r	nil(r	NOUN
ejpam-4827	35	21	)	)	PUNCT
ejpam-4827	35	22	∗	∗	NOUN
ejpam-4827	35	23	m	m	VERB
ejpam-4827	35	24	there	there	ADV
ejpam-4827	35	25	exist	exist	VERB
ejpam-4827	35	26	a	a	DET
ejpam-4827	35	27	nonzero	nonzero	NOUN
ejpam-4827	35	28	element	element	NOUN
ejpam-4827	35	29	c	c	PROPN
ejpam-4827	35	30	∈	∈	PROPN
ejpam-4827	35	31	r	r	NOUN
ejpam-4827	35	32	such	such	ADJ
ejpam-4827	35	33	that	that	DET
ejpam-4827	35	34	aiσgi(c	aiσgi(c	NOUN
ejpam-4827	35	35	)	)	PUNCT
ejpam-4827	35	36	∈	∈	PROPN
ejpam-4827	35	37	ni(r	ni(r	PRON
ejpam-4827	35	38	)	)	PUNCT
ejpam-4827	35	39	for	for	ADP
ejpam-4827	35	40	each	each	DET
ejpam-4827	35	41	i	i	PRON
ejpam-4827	35	42	,	,	PUNCT
ejpam-4827	35	43	where	where	SCONJ
ejpam-4827	35	44	α	α	NOUN
ejpam-4827	35	45	=	=	X
ejpam-4827	36	1	a1g1	a1g1	PROPN
ejpam-4827	36	2	+	+	NUM
ejpam-4827	36	3	a2g2	a2g2	PROPN
ejpam-4827	36	4	+	+	CCONJ
ejpam-4827	36	5	·	·	PUNCT
ejpam-4827	36	6	·	·	PUNCT
ejpam-4827	36	7	·	·	PUNCT
ejpam-4827	37	1	+	+	CCONJ
ejpam-4827	37	2	amgm	amgm	ADJ
ejpam-4827	37	3	and	and	CCONJ
ejpam-4827	37	4	β	β	X
ejpam-4827	37	5	=	=	PUNCT
ejpam-4827	37	6	b1h1	b1h1	NOUN
ejpam-4827	38	1	+	+	CCONJ
ejpam-4827	38	2	b2h2	b2h2	ADP
ejpam-4827	38	3	+	+	X
ejpam-4827	38	4	·	·	PUNCT
ejpam-4827	38	5	·	·	PUNCT
ejpam-4827	38	6	·	·	PUNCT
ejpam-4827	39	1	+	+	NUM
ejpam-4827	39	2	bnhn	bnhn	NOUN
ejpam-4827	39	3	are	be	AUX
ejpam-4827	39	4	nonzero	nonzero	NOUN
ejpam-4827	39	5	element	element	NOUN
ejpam-4827	39	6	in	in	ADP
ejpam-4827	39	7	r	r	NOUN
ejpam-4827	39	8	∗m	∗m	NOUN
ejpam-4827	39	9	.	.	PUNCT
ejpam-4827	40	1	in	in	ADP
ejpam-4827	40	2	[	[	X
ejpam-4827	40	3	3	3	X
ejpam-4827	40	4	]	]	PUNCT
ejpam-4827	40	5	the	the	DET
ejpam-4827	40	6	author	author	NOUN
ejpam-4827	40	7	defined	define	VERB
ejpam-4827	40	8	“	"	PUNCT
ejpam-4827	40	9	nil	nil	ADJ
ejpam-4827	40	10	skew	skew	ADJ
ejpam-4827	40	11	generalized	generalize	VERB
ejpam-4827	40	12	power	power	NOUN
ejpam-4827	40	13	series	series	PROPN
ejpam-4827	40	14	reflexive	reflexive	ADJ
ejpam-4827	40	15	rings	ring	NOUN
ejpam-4827	40	16	"	"	PUNCT
ejpam-4827	40	17	for	for	ADP
ejpam-4827	40	18	which	which	PRON
ejpam-4827	40	19	f	f	AUX
ejpam-4827	40	20	,	,	PUNCT
ejpam-4827	40	21	g	g	PROPN
ejpam-4827	40	22	∈	∈	PROPN
ejpam-4827	41	1	[	[	X
ejpam-4827	41	2	[	[	X
ejpam-4827	41	3	rs,≤	rs,≤	NUM
ejpam-4827	41	4	,	,	PUNCT
ejpam-4827	41	5	ω	ω	NOUN
ejpam-4827	41	6	]	]	X
ejpam-4827	41	7	]	]	X
ejpam-4827	41	8	satisfying	satisfy	VERB
ejpam-4827	41	9	fhg	fhg	PROPN
ejpam-4827	41	10	∈	∈	PROPN
ejpam-4827	42	1	[	[	X
ejpam-4827	42	2	[	[	X
ejpam-4827	42	3	nil(r)s,≤	nil(r)s,≤	X
ejpam-4827	42	4	,	,	PUNCT
ejpam-4827	42	5	ω	ω	NOUN
ejpam-4827	42	6	]	]	X
ejpam-4827	42	7	]	]	PUNCT
ejpam-4827	42	8	implies	imply	VERB
ejpam-4827	42	9	that	that	SCONJ
ejpam-4827	42	10	ghf	ghf	NOUN
ejpam-4827	42	11	∈	∈	PROPN
ejpam-4827	43	1	[	[	X
ejpam-4827	43	2	[	[	X
ejpam-4827	43	3	nil(r)s,≤	nil(r)s,≤	X
ejpam-4827	43	4	,	,	PUNCT
ejpam-4827	43	5	ω	ω	NOUN
ejpam-4827	43	6	]	]	X
ejpam-4827	43	7	]	]	PUNCT
ejpam-4827	43	8	.	.	PUNCT
ejpam-4827	44	1	similarly	similarly	ADV
ejpam-4827	44	2	,	,	PUNCT
ejpam-4827	44	3	in	in	ADP
ejpam-4827	44	4	[	[	X
ejpam-4827	44	5	1	1	X
ejpam-4827	44	6	]	]	PUNCT
ejpam-4827	44	7	the	the	DET
ejpam-4827	44	8	author	author	NOUN
ejpam-4827	44	9	discussed	discuss	VERB
ejpam-4827	44	10	“	"	PUNCT
ejpam-4827	44	11	the	the	DET
ejpam-4827	44	12	nilpotent	nilpotent	ADJ
ejpam-4827	44	13	elements	element	NOUN
ejpam-4827	44	14	and	and	CCONJ
ejpam-4827	44	15	nil	nil	ADJ
ejpam-4827	44	16	-	-	PUNCT
ejpam-4827	44	17	reflexive	reflexive	ADJ
ejpam-4827	44	18	property	property	NOUN
ejpam-4827	44	19	of	of	ADP
ejpam-4827	44	20	generalized	generalized	ADJ
ejpam-4827	44	21	power	power	NOUN
ejpam-4827	44	22	series	series	PROPN
ejpam-4827	44	23	rings	rings	PROPN
ejpam-4827	44	24	"	"	PUNCT
ejpam-4827	44	25	,	,	PUNCT
ejpam-4827	44	26	where	where	SCONJ
ejpam-4827	44	27	fhg	fhg	NOUN
ejpam-4827	44	28	∈	∈	PROPN
ejpam-4827	45	1	[	[	X
ejpam-4827	45	2	[	[	X
ejpam-4827	45	3	nil(r)s,≤	nil(r)s,≤	X
ejpam-4827	45	4	]	]	X
ejpam-4827	45	5	]	]	PUNCT
ejpam-4827	45	6	implies	imply	VERB
ejpam-4827	45	7	that	that	SCONJ
ejpam-4827	45	8	ghf	ghf	NOUN
ejpam-4827	45	9	∈	∈	PROPN
ejpam-4827	46	1	[	[	X
ejpam-4827	46	2	[	[	X
ejpam-4827	46	3	nil(r)s,≤	nil(r)s,≤	X
ejpam-4827	46	4	]	]	X
ejpam-4827	46	5	]	]	PUNCT
ejpam-4827	46	6	for	for	ADP
ejpam-4827	46	7	f	f	PROPN
ejpam-4827	46	8	,	,	PUNCT
ejpam-4827	46	9	g	g	PROPN
ejpam-4827	46	10	,	,	PUNCT
ejpam-4827	46	11	h	h	NOUN
ejpam-4827	46	12	∈	∈	PROPN
ejpam-4827	47	1	[	[	X
ejpam-4827	47	2	[	[	X
ejpam-4827	47	3	rs,≤	rs,≤	X
ejpam-4827	47	4	]	]	X
ejpam-4827	47	5	]	]	PUNCT
ejpam-4827	47	6	.	.	PUNCT
ejpam-4827	48	1	the	the	DET
ejpam-4827	48	2	investigation	investigation	NOUN
ejpam-4827	48	3	of	of	ADP
ejpam-4827	48	4	the	the	DET
ejpam-4827	48	5	composition	composition	NOUN
ejpam-4827	48	6	of	of	ADP
ejpam-4827	48	7	the	the	DET
ejpam-4827	48	8	collection	collection	NOUN
ejpam-4827	48	9	of	of	ADP
ejpam-4827	48	10	nilpotent	nilpotent	ADJ
ejpam-4827	48	11	elements	element	NOUN
ejpam-4827	48	12	in	in	ADP
ejpam-4827	48	13	noncommutative	noncommutative	ADJ
ejpam-4827	48	14	ring	ring	NOUN
ejpam-4827	48	15	constructions	construction	NOUN
ejpam-4827	48	16	is	be	AUX
ejpam-4827	48	17	a	a	DET
ejpam-4827	48	18	crucial	crucial	ADJ
ejpam-4827	48	19	and	and	CCONJ
ejpam-4827	48	20	highly	highly	ADV
ejpam-4827	48	21	active	active	ADJ
ejpam-4827	48	22	field	field	NOUN
ejpam-4827	48	23	in	in	ADP
ejpam-4827	48	24	noncommutative	noncommutative	ADJ
ejpam-4827	48	25	algebra	algebra	NOUN
ejpam-4827	48	26	.	.	PUNCT
ejpam-4827	49	1	this	this	PRON
ejpam-4827	49	2	is	be	AUX
ejpam-4827	49	3	evidenced	evidence	VERB
ejpam-4827	49	4	by	by	ADP
ejpam-4827	49	5	numerous	numerous	ADJ
ejpam-4827	49	6	studies	study	NOUN
ejpam-4827	49	7	conducted	conduct	VERB
ejpam-4827	49	8	by	by	ADP
ejpam-4827	49	9	various	various	ADJ
ejpam-4827	49	10	authors	author	NOUN
ejpam-4827	49	11	see	see	VERB
ejpam-4827	49	12	[	[	X
ejpam-4827	49	13	3	3	NUM
ejpam-4827	49	14	]	]	PUNCT
ejpam-4827	49	15	,	,	PUNCT
ejpam-4827	50	1	[	[	X
ejpam-4827	50	2	1	1	NUM
ejpam-4827	50	3	]	]	PUNCT
ejpam-4827	50	4	,	,	PUNCT
ejpam-4827	50	5	[	[	X
ejpam-4827	50	6	22	22	NUM
ejpam-4827	50	7	]	]	PUNCT
ejpam-4827	50	8	,	,	PUNCT
ejpam-4827	50	9	[	[	X
ejpam-4827	50	10	15	15	NUM
ejpam-4827	50	11	]	]	PUNCT
ejpam-4827	50	12	,	,	PUNCT
ejpam-4827	50	13	[	[	X
ejpam-4827	50	14	13	13	NUM
ejpam-4827	50	15	]	]	PUNCT
ejpam-4827	50	16	,	,	PUNCT
ejpam-4827	50	17	[	[	X
ejpam-4827	50	18	4	4	NUM
ejpam-4827	50	19	]	]	PUNCT
ejpam-4827	50	20	,	,	PUNCT
ejpam-4827	50	21	[	[	X
ejpam-4827	50	22	5	5	NUM
ejpam-4827	50	23	]	]	PUNCT
ejpam-4827	50	24	,	,	PUNCT
ejpam-4827	50	25	[	[	X
ejpam-4827	50	26	2	2	NUM
ejpam-4827	50	27	]	]	PUNCT
ejpam-4827	50	28	and	and	CCONJ
ejpam-4827	50	29	[	[	X
ejpam-4827	50	30	18	18	NUM
ejpam-4827	50	31	]	]	PUNCT
ejpam-4827	50	32	.	.	PUNCT
ejpam-4827	51	1	this	this	DET
ejpam-4827	51	2	paper	paper	NOUN
ejpam-4827	51	3	is	be	AUX
ejpam-4827	51	4	devoted	devote	VERB
ejpam-4827	51	5	to	to	ADP
ejpam-4827	51	6	examining	examine	VERB
ejpam-4827	51	7	the	the	DET
ejpam-4827	51	8	nilpotent	nilpotent	ADJ
ejpam-4827	51	9	elements	element	NOUN
ejpam-4827	51	10	found	find	VERB
ejpam-4827	51	11	in	in	ADP
ejpam-4827	51	12	skew	skew	ADJ
ejpam-4827	51	13	monoid	monoid	NOUN
ejpam-4827	51	14	rings	ring	NOUN
ejpam-4827	51	15	.	.	PUNCT
ejpam-4827	52	1	in	in	ADP
ejpam-4827	52	2	this	this	DET
ejpam-4827	52	3	context	context	NOUN
ejpam-4827	52	4	,	,	PUNCT
ejpam-4827	52	5	let	let	VERB
ejpam-4827	52	6	r	r	PRON
ejpam-4827	52	7	be	be	AUX
ejpam-4827	52	8	a	a	DET
ejpam-4827	52	9	ring	ring	NOUN
ejpam-4827	52	10	and	and	CCONJ
ejpam-4827	52	11	m	m	AUX
ejpam-4827	52	12	be	be	AUX
ejpam-4827	52	13	a	a	DET
ejpam-4827	52	14	u.p.-monoid	u.p.-monoid	NOUN
ejpam-4827	52	15	.	.	PUNCT
ejpam-4827	53	1	it	it	PRON
ejpam-4827	53	2	is	be	AUX
ejpam-4827	53	3	assumed	assume	VERB
ejpam-4827	53	4	that	that	SCONJ
ejpam-4827	53	5	m	m	NOUN
ejpam-4827	53	6	operates	operate	VERB
ejpam-4827	53	7	on	on	ADP
ejpam-4827	53	8	r	r	NOUN
ejpam-4827	53	9	through	through	ADP
ejpam-4827	53	10	a	a	DET
ejpam-4827	53	11	homomorphism	homomorphism	NOUN
ejpam-4827	53	12	that	that	PRON
ejpam-4827	53	13	maps	map	VERB
ejpam-4827	53	14	to	to	ADP
ejpam-4827	53	15	the	the	DET
ejpam-4827	53	16	automorphism	automorphism	NOUN
ejpam-4827	53	17	group	group	NOUN
ejpam-4827	53	18	of	of	ADP
ejpam-4827	53	19	r.	r.	PROPN
ejpam-4827	53	20	this	this	DET
ejpam-4827	53	21	homomorphism	homomorphism	NOUN
ejpam-4827	53	22	is	be	AUX
ejpam-4827	53	23	denoted	denote	VERB
ejpam-4827	53	24	as	as	ADP
ejpam-4827	53	25	σ	σ	NOUN
ejpam-4827	53	26	:	:	PUNCT
ejpam-4827	53	27	m	m	PROPN
ejpam-4827	53	28	→	→	SYM
ejpam-4827	53	29	aut(r	aut(r	PROPN
ejpam-4827	53	30	)	)	PUNCT
ejpam-4827	53	31	,	,	PUNCT
ejpam-4827	53	32	i.e.	i.e.	X
ejpam-4827	53	33	,	,	PUNCT
ejpam-4827	53	34	a	a	DET
ejpam-4827	53	35	module	module	NOUN
ejpam-4827	53	36	homomorphism	homomorphism	NOUN
ejpam-4827	53	37	from	from	ADP
ejpam-4827	53	38	m	m	PRON
ejpam-4827	53	39	to	to	ADP
ejpam-4827	53	40	the	the	DET
ejpam-4827	53	41	group	group	NOUN
ejpam-4827	53	42	of	of	ADP
ejpam-4827	53	43	automorphisms	automorphism	NOUN
ejpam-4827	53	44	of	of	ADP
ejpam-4827	53	45	r	r	NOUN
ejpam-4827	53	46	,	,	PUNCT
ejpam-4827	53	47	here	here	ADV
ejpam-4827	53	48	,	,	PUNCT
ejpam-4827	53	49	an	an	DET
ejpam-4827	53	50	automorphism	automorphism	NOUN
ejpam-4827	53	51	of	of	ADP
ejpam-4827	53	52	r	r	NOUN
ejpam-4827	53	53	is	be	AUX
ejpam-4827	53	54	a	a	DET
ejpam-4827	53	55	ring	ring	NOUN
ejpam-4827	53	56	isomorphism	isomorphism	NOUN
ejpam-4827	53	57	from	from	ADP
ejpam-4827	53	58	r	r	NOUN
ejpam-4827	53	59	to	to	ADP
ejpam-4827	53	60	itself	itself	PRON
ejpam-4827	53	61	,	,	PUNCT
ejpam-4827	53	62	so	so	ADV
ejpam-4827	53	63	,	,	PUNCT
ejpam-4827	53	64	aut(r	aut(r	PROPN
ejpam-4827	53	65	)	)	PUNCT
ejpam-4827	53	66	is	be	AUX
ejpam-4827	53	67	the	the	DET
ejpam-4827	53	68	group	group	NOUN
ejpam-4827	53	69	of	of	ADP
ejpam-4827	53	70	all	all	DET
ejpam-4827	53	71	such	such	ADJ
ejpam-4827	53	72	isomorphisms	isomorphism	NOUN
ejpam-4827	53	73	.	.	PUNCT
ejpam-4827	54	1	for	for	ADP
ejpam-4827	54	2	any	any	DET
ejpam-4827	54	3	given	give	VERB
ejpam-4827	54	4	element	element	NOUN
ejpam-4827	54	5	g	g	PROPN
ejpam-4827	54	6	∈	∈	PROPN
ejpam-4827	54	7	m	m	NOUN
ejpam-4827	54	8	,	,	PUNCT
ejpam-4827	54	9	the	the	DET
ejpam-4827	54	10	notation	notation	NOUN
ejpam-4827	54	11	σg	σg	NOUN
ejpam-4827	54	12	denotes	denote	VERB
ejpam-4827	54	13	the	the	DET
ejpam-4827	54	14	image	image	NOUN
ejpam-4827	54	15	of	of	ADP
ejpam-4827	54	16	g	g	NOUN
ejpam-4827	54	17	under	under	ADP
ejpam-4827	54	18	the	the	DET
ejpam-4827	54	19	module	module	NOUN
ejpam-4827	54	20	homomorphism	homomorphism	PROPN
ejpam-4827	54	21	σ	σ	X
ejpam-4827	54	22	,	,	PUNCT
ejpam-4827	54	23	i.e.	i.e.	X
ejpam-4827	54	24	,	,	PUNCT
ejpam-4827	54	25	σg	σg	ADJ
ejpam-4827	54	26	=	=	SYM
ejpam-4827	54	27	σ(g	σ(g	PROPN
ejpam-4827	54	28	)	)	PUNCT
ejpam-4827	54	29	∈	∈	PROPN
ejpam-4827	54	30	aut(r	aut(r	PROPN
ejpam-4827	54	31	)	)	PUNCT
ejpam-4827	54	32	.	.	PUNCT
ejpam-4827	55	1	in	in	ADP
ejpam-4827	55	2	other	other	ADJ
ejpam-4827	55	3	words	word	NOUN
ejpam-4827	55	4	,	,	PUNCT
ejpam-4827	55	5	σg	σg	NOUN
ejpam-4827	55	6	is	be	AUX
ejpam-4827	55	7	an	an	DET
ejpam-4827	55	8	automorphism	automorphism	NOUN
ejpam-4827	55	9	of	of	ADP
ejpam-4827	55	10	r	r	NOUN
ejpam-4827	55	11	that	that	PRON
ejpam-4827	55	12	depends	depend	VERB
ejpam-4827	55	13	on	on	ADP
ejpam-4827	55	14	the	the	DET
ejpam-4827	55	15	choice	choice	NOUN
ejpam-4827	55	16	of	of	ADP
ejpam-4827	55	17	the	the	DET
ejpam-4827	55	18	element	element	NOUN
ejpam-4827	55	19	g	g	PROPN
ejpam-4827	55	20	∈	∈	PROPN
ejpam-4827	55	21	m	m	VERB
ejpam-4827	55	22	.	.	PUNCT
ejpam-4827	56	1	by	by	ADP
ejpam-4827	56	2	using	use	VERB
ejpam-4827	56	3	the	the	DET
ejpam-4827	56	4	monoid	monoid	NOUN
ejpam-4827	56	5	homomorphism	homomorphism	PROPN
ejpam-4827	56	6	σ	σ	X
ejpam-4827	56	7	,	,	PUNCT
ejpam-4827	56	8	we	we	PRON
ejpam-4827	56	9	can	can	AUX
ejpam-4827	56	10	create	create	VERB
ejpam-4827	56	11	a	a	DET
ejpam-4827	56	12	skew	skew	ADJ
ejpam-4827	56	13	monoid	monoid	NOUN
ejpam-4827	56	14	ring	ring	NOUN
ejpam-4827	56	15	denoted	denote	VERB
ejpam-4827	56	16	as	as	ADP
ejpam-4827	56	17	r∗m	r∗m	NOUN
ejpam-4827	56	18	.	.	PUNCT
ejpam-4827	57	1	this	this	DET
ejpam-4827	57	2	ring	ring	NOUN
ejpam-4827	57	3	consists	consist	VERB
ejpam-4827	57	4	of	of	ADP
ejpam-4827	57	5	finite	finite	ADJ
ejpam-4827	57	6	formal	formal	ADJ
ejpam-4827	57	7	combinations	combination	NOUN
ejpam-4827	57	8	of	of	ADP
ejpam-4827	57	9	elements	element	NOUN
ejpam-4827	57	10	in	in	ADP
ejpam-4827	57	11	m	m	PROPN
ejpam-4827	57	12	,	,	PUNCT
ejpam-4827	57	13	represented	represent	VERB
ejpam-4827	57	14	as	as	ADP
ejpam-4827	57	15	σg∈mxgg	σg∈mxgg	NOUN
ejpam-4827	57	16	.	.	PUNCT
ejpam-4827	58	1	multiplication	multiplication	NOUN
ejpam-4827	58	2	in	in	ADP
ejpam-4827	58	3	this	this	DET
ejpam-4827	58	4	ring	ring	NOUN
ejpam-4827	58	5	is	be	AUX
ejpam-4827	58	6	induced	induce	VERB
ejpam-4827	58	7	by	by	ADP
ejpam-4827	58	8	the	the	DET
ejpam-4827	58	9	formula	formula	NOUN
ejpam-4827	58	10	(	(	PUNCT
ejpam-4827	58	11	xgg)(yhh	xgg)(yhh	NOUN
ejpam-4827	58	12	)	)	PUNCT
ejpam-4827	58	13	=	=	PRON
ejpam-4827	58	14	(	(	PUNCT
ejpam-4827	58	15	xgσg(yh))(gh	xgσg(yh))(gh	PROPN
ejpam-4827	58	16	)	)	PUNCT
ejpam-4827	58	17	.	.	PUNCT
ejpam-4827	59	1	therefore	therefore	ADV
ejpam-4827	59	2	,	,	PUNCT
ejpam-4827	59	3	r	r	NOUN
ejpam-4827	59	4	∗m	∗m	NOUN
ejpam-4827	59	5	is	be	AUX
ejpam-4827	59	6	a	a	DET
ejpam-4827	59	7	ring	ring	NOUN
ejpam-4827	59	8	that	that	PRON
ejpam-4827	59	9	is	be	AUX
ejpam-4827	59	10	free	free	ADJ
ejpam-4827	59	11	as	as	ADP
ejpam-4827	59	12	a	a	DET
ejpam-4827	59	13	left	left	ADJ
ejpam-4827	59	14	r	r	NOUN
ejpam-4827	59	15	-	-	PUNCT
ejpam-4827	59	16	module	module	NOUN
ejpam-4827	59	17	with	with	ADP
ejpam-4827	59	18	basis	basis	NOUN
ejpam-4827	59	19	m	m	NOUN
ejpam-4827	59	20	.	.	PUNCT
ejpam-4827	60	1	a	a	DET
ejpam-4827	60	2	commonly	commonly	ADV
ejpam-4827	60	3	accepted	accept	VERB
ejpam-4827	60	4	fact	fact	NOUN
ejpam-4827	60	5	is	be	AUX
ejpam-4827	60	6	that	that	SCONJ
ejpam-4827	60	7	if	if	SCONJ
ejpam-4827	60	8	a	a	DET
ejpam-4827	60	9	polynomial	polynomial	ADJ
ejpam-4827	60	10	f(x	f(x	PROPN
ejpam-4827	60	11	)	)	PUNCT
ejpam-4827	60	12	is	be	AUX
ejpam-4827	60	13	defined	define	VERB
ejpam-4827	60	14	over	over	ADP
ejpam-4827	60	15	a	a	DET
ejpam-4827	60	16	commutative	commutative	ADJ
ejpam-4827	60	17	ring	ring	NOUN
ejpam-4827	60	18	,	,	PUNCT
ejpam-4827	60	19	then	then	ADV
ejpam-4827	60	20	it	it	PRON
ejpam-4827	60	21	is	be	AUX
ejpam-4827	60	22	nilpotent	nilpotent	ADJ
ejpam-4827	60	23	if	if	SCONJ
ejpam-4827	60	24	only	only	ADV
ejpam-4827	60	25	if	if	SCONJ
ejpam-4827	60	26	the	the	DET
ejpam-4827	60	27	coefficient	coefficient	NOUN
ejpam-4827	60	28	of	of	ADP
ejpam-4827	60	29	f(x	f(x	PROPN
ejpam-4827	60	30	)	)	PUNCT
ejpam-4827	60	31	is	be	AUX
ejpam-4827	60	32	too	too	ADV
ejpam-4827	60	33	.	.	PUNCT
ejpam-4827	61	1	however	however	ADV
ejpam-4827	61	2	,	,	PUNCT
ejpam-4827	61	3	it	it	PRON
ejpam-4827	61	4	should	should	AUX
ejpam-4827	61	5	be	be	AUX
ejpam-4827	61	6	noted	note	VERB
ejpam-4827	61	7	that	that	SCONJ
ejpam-4827	61	8	this	this	DET
ejpam-4827	61	9	statement	statement	NOUN
ejpam-4827	61	10	does	do	AUX
ejpam-4827	61	11	not	not	PART
ejpam-4827	61	12	hold	hold	VERB
ejpam-4827	61	13	true	true	ADJ
ejpam-4827	61	14	for	for	ADP
ejpam-4827	61	15	noncommutative	noncommutative	ADJ
ejpam-4827	61	16	rings	ring	NOUN
ejpam-4827	61	17	.	.	PUNCT
ejpam-4827	62	1	the	the	DET
ejpam-4827	62	2	researcher	researcher	NOUN
ejpam-4827	62	3	introduced	introduce	VERB
ejpam-4827	62	4	and	and	CCONJ
ejpam-4827	62	5	studied	study	VERB
ejpam-4827	62	6	two	two	NUM
ejpam-4827	62	7	concepts	concept	NOUN
ejpam-4827	62	8	,	,	PUNCT
ejpam-4827	62	9	σ	σ	PROPN
ejpam-4827	62	10	-	-	PUNCT
ejpam-4827	62	11	skew	skew	NOUN
ejpam-4827	62	12	strongly	strongly	ADV
ejpam-4827	62	13	m	m	VERB
ejpam-4827	62	14	-reflexive	-reflexive	ADJ
ejpam-4827	62	15	and	and	CCONJ
ejpam-4827	62	16	σ	σ	NOUN
ejpam-4827	62	17	-	-	PUNCT
ejpam-4827	62	18	skew	skew	NOUN
ejpam-4827	62	19	strongly	strongly	ADV
ejpam-4827	62	20	m	m	VERB
ejpam-4827	62	21	-nil	-nil	NOUN
ejpam-4827	62	22	-	-	PUNCT
ejpam-4827	62	23	reflexive	reflexive	ADJ
ejpam-4827	62	24	on	on	ADP
ejpam-4827	62	25	monoid	monoid	NOUN
ejpam-4827	62	26	rings	ring	NOUN
ejpam-4827	62	27	.	.	PUNCT
ejpam-4827	63	1	the	the	DET
ejpam-4827	63	2	paper	paper	NOUN
ejpam-4827	63	3	covers	cover	VERB
ejpam-4827	63	4	the	the	DET
ejpam-4827	63	5	basic	basic	ADJ
ejpam-4827	63	6	properties	property	NOUN
ejpam-4827	63	7	of	of	ADP
ejpam-4827	63	8	skew	skew	ADJ
ejpam-4827	63	9	monoid	monoid	NOUN
ejpam-4827	63	10	rings	ring	NOUN
ejpam-4827	63	11	of	of	ADP
ejpam-4827	63	12	the	the	DET
ejpam-4827	63	13	form	form	NOUN
ejpam-4827	63	14	r∗m	r∗m	NOUN
ejpam-4827	63	15	.	.	PUNCT
ejpam-4827	64	1	it	it	PRON
ejpam-4827	64	2	is	be	AUX
ejpam-4827	64	3	shown	show	VERB
ejpam-4827	64	4	that	that	SCONJ
ejpam-4827	64	5	if	if	SCONJ
ejpam-4827	64	6	r	r	NOUN
ejpam-4827	64	7	is	be	AUX
ejpam-4827	64	8	a	a	DET
ejpam-4827	64	9	left	left	ADJ
ejpam-4827	64	10	app	app	NOUN
ejpam-4827	64	11	(	(	PUNCT
ejpam-4827	64	12	quasi	quasi	NOUN
ejpam-4827	64	13	armendariz	armendariz	NOUN
ejpam-4827	64	14	,	,	PUNCT
ejpam-4827	64	15	semiprime	semiprime	NOUN
ejpam-4827	64	16	rings	ring	NOUN
ejpam-4827	64	17	,	,	PUNCT
ejpam-4827	64	18	respectively	respectively	ADV
ejpam-4827	64	19	)	)	PUNCT
ejpam-4827	64	20	,	,	PUNCT
ejpam-4827	64	21	then	then	ADV
ejpam-4827	64	22	r	r	NOUN
ejpam-4827	64	23	is	be	AUX
ejpam-4827	64	24	σ	σ	NOUN
ejpam-4827	64	25	-	-	PUNCT
ejpam-4827	64	26	skew	skew	NOUN
ejpam-4827	64	27	strongly	strongly	ADV
ejpam-4827	64	28	m	m	VERB
ejpam-4827	64	29	-reflexive	-reflexive	ADJ
ejpam-4827	64	30	.	.	PUNCT
ejpam-4827	65	1	moreover	moreover	ADV
ejpam-4827	65	2	,	,	PUNCT
ejpam-4827	65	3	if	if	SCONJ
ejpam-4827	65	4	r	r	NOUN
ejpam-4827	65	5	is	be	AUX
ejpam-4827	65	6	a	a	DET
ejpam-4827	65	7	ni	ni	NOUN
ejpam-4827	65	8	-	-	PUNCT
ejpam-4827	65	9	ring	ring	NOUN
ejpam-4827	65	10	and	and	CCONJ
ejpam-4827	65	11	m	m	NOUN
ejpam-4827	65	12	is	be	AUX
ejpam-4827	65	13	a	a	DET
ejpam-4827	65	14	u.p	u.p	PROPN
ejpam-4827	65	15	-	-	PUNCT
ejpam-4827	65	16	monoid	monoid	PROPN
ejpam-4827	65	17	,	,	PUNCT
ejpam-4827	65	18	then	then	ADV
ejpam-4827	65	19	r	r	NOUN
ejpam-4827	65	20	is	be	AUX
ejpam-4827	65	21	σ	σ	NOUN
ejpam-4827	65	22	-	-	PUNCT
ejpam-4827	65	23	skew	skew	NOUN
ejpam-4827	65	24	strongly	strongly	ADV
ejpam-4827	65	25	m	m	VERB
ejpam-4827	65	26	-nil	-nil	NOUN
ejpam-4827	65	27	-	-	PUNCT
ejpam-4827	65	28	reflexive	reflexive	ADJ
ejpam-4827	65	29	.	.	PUNCT
ejpam-4827	66	1	furthermore	furthermore	ADV
ejpam-4827	66	2	,	,	PUNCT
ejpam-4827	66	3	under	under	ADP
ejpam-4827	66	4	certain	certain	ADJ
ejpam-4827	66	5	conditions	condition	NOUN
ejpam-4827	66	6	,	,	PUNCT
ejpam-4827	66	7	a	a	DET
ejpam-4827	66	8	skew	skew	ADJ
ejpam-4827	66	9	monoid	monoid	NOUN
ejpam-4827	66	10	ring	ring	NOUN
ejpam-4827	66	11	r	r	NOUN
ejpam-4827	66	12	∗m	∗m	NOUN
ejpam-4827	66	13	is	be	AUX
ejpam-4827	66	14	proven	prove	VERB
ejpam-4827	66	15	to	to	PART
ejpam-4827	66	16	be	be	AUX
ejpam-4827	66	17	σ	σ	NOUN
ejpam-4827	66	18	-	-	PUNCT
ejpam-4827	66	19	skew	skew	NOUN
ejpam-4827	66	20	strongly	strongly	ADV
ejpam-4827	66	21	m	m	PROPN
ejpam-4827	66	22	e.	e.	PROPN
ejpam-4827	66	23	ali	ali	PROPN
ejpam-4827	66	24	/	/	SYM
ejpam-4827	66	25	eur	eur	PROPN
ejpam-4827	66	26	.	.	PUNCT
ejpam-4827	67	1	j.	j.	PROPN
ejpam-4827	67	2	pure	pure	PROPN
ejpam-4827	67	3	appl	appl	PROPN
ejpam-4827	67	4	.	.	PROPN
ejpam-4827	67	5	math	math	PROPN
ejpam-4827	67	6	,	,	PUNCT
ejpam-4827	67	7	16	16	NUM
ejpam-4827	67	8	(	(	PUNCT
ejpam-4827	67	9	3	3	NUM
ejpam-4827	67	10	)	)	PUNCT
ejpam-4827	67	11	(	(	PUNCT
ejpam-4827	67	12	2023	2023	NUM
ejpam-4827	67	13	)	)	PUNCT
ejpam-4827	67	14	,	,	PUNCT
ejpam-4827	67	15	1878	1878	NUM
ejpam-4827	67	16	-	-	SYM
ejpam-4827	67	17	1893	1893	NUM
ejpam-4827	67	18	1880	1880	NUM
ejpam-4827	67	19	nil	nil	ADJ
ejpam-4827	67	20	-	-	PUNCT
ejpam-4827	67	21	reflexive	reflexive	ADJ
ejpam-4827	67	22	when	when	SCONJ
ejpam-4827	67	23	σ	σ	NOUN
ejpam-4827	67	24	:	:	PUNCT
ejpam-4827	67	25	m	m	PROPN
ejpam-4827	67	26	→	→	SYM
ejpam-4827	67	27	aut(r	aut(r	PROPN
ejpam-4827	67	28	)	)	PUNCT
ejpam-4827	67	29	is	be	AUX
ejpam-4827	67	30	a	a	DET
ejpam-4827	67	31	monoid	monoid	NOUN
ejpam-4827	67	32	homomorphism	homomorphism	NOUN
ejpam-4827	67	33	.	.	PUNCT
ejpam-4827	68	1	additionally	additionally	ADV
ejpam-4827	68	2	,	,	PUNCT
ejpam-4827	68	3	it	it	PRON
ejpam-4827	68	4	is	be	AUX
ejpam-4827	68	5	proved	prove	VERB
ejpam-4827	68	6	that	that	SCONJ
ejpam-4827	68	7	a	a	DET
ejpam-4827	68	8	ring	ring	NOUN
ejpam-4827	68	9	r	r	NOUN
ejpam-4827	68	10	is	be	AUX
ejpam-4827	68	11	σ	σ	NOUN
ejpam-4827	68	12	-	-	PUNCT
ejpam-4827	68	13	skew	skew	NOUN
ejpam-4827	68	14	strongly	strongly	ADV
ejpam-4827	68	15	m	m	VERB
ejpam-4827	68	16	-nil	-nil	NOUN
ejpam-4827	68	17	-	-	PUNCT
ejpam-4827	68	18	reflexive	reflexive	ADJ
ejpam-4827	68	19	if	if	SCONJ
ejpam-4827	68	20	and	and	CCONJ
ejpam-4827	68	21	only	only	ADV
ejpam-4827	68	22	if	if	SCONJ
ejpam-4827	68	23	r	r	X
ejpam-4827	68	24	/	/	SYM
ejpam-4827	68	25	i	i	PRON
ejpam-4827	68	26	is	be	AUX
ejpam-4827	68	27	σ̄-skew	σ̄-skew	PROPN
ejpam-4827	68	28	strongly	strongly	ADV
ejpam-4827	68	29	m	m	VERB
ejpam-4827	68	30	-nil	-nil	NOUN
ejpam-4827	68	31	-	-	PUNCT
ejpam-4827	68	32	reflexive	reflexive	ADJ
ejpam-4827	68	33	.	.	PUNCT
ejpam-4827	69	1	consequently	consequently	ADV
ejpam-4827	69	2	,	,	PUNCT
ejpam-4827	69	3	if	if	SCONJ
ejpam-4827	69	4	r	r	NOUN
ejpam-4827	69	5	is	be	AUX
ejpam-4827	69	6	a	a	DET
ejpam-4827	69	7	left	left	ADJ
ejpam-4827	69	8	app	app	NOUN
ejpam-4827	69	9	,	,	PUNCT
ejpam-4827	69	10	then	then	ADV
ejpam-4827	69	11	the	the	DET
ejpam-4827	69	12	upper	upper	ADJ
ejpam-4827	69	13	triangular	triangular	NOUN
ejpam-4827	69	14	matrix	matrix	NOUN
ejpam-4827	69	15	ring	ring	NOUN
ejpam-4827	69	16	tn(r	tn(r	NOUN
ejpam-4827	69	17	)	)	PUNCT
ejpam-4827	69	18	is	be	AUX
ejpam-4827	69	19	σ̄-skew	σ̄-skew	PROPN
ejpam-4827	69	20	strongly	strongly	ADV
ejpam-4827	69	21	m	m	VERB
ejpam-4827	69	22	-nil	-nil	NOUN
ejpam-4827	69	23	-	-	PUNCT
ejpam-4827	69	24	reflexive	reflexive	ADJ
ejpam-4827	69	25	,	,	PUNCT
ejpam-4827	69	26	where	where	SCONJ
ejpam-4827	69	27	n	n	PRON
ejpam-4827	69	28	is	be	AUX
ejpam-4827	69	29	a	a	DET
ejpam-4827	69	30	positive	positive	ADJ
ejpam-4827	69	31	integer	integer	NOUN
ejpam-4827	69	32	.	.	PUNCT
ejpam-4827	70	1	finally	finally	ADV
ejpam-4827	70	2	,	,	PUNCT
ejpam-4827	70	3	the	the	DET
ejpam-4827	70	4	paper	paper	NOUN
ejpam-4827	70	5	provides	provide	VERB
ejpam-4827	70	6	some	some	DET
ejpam-4827	70	7	examples	example	NOUN
ejpam-4827	70	8	and	and	CCONJ
ejpam-4827	70	9	discusses	discuss	VERB
ejpam-4827	70	10	related	related	ADJ
ejpam-4827	70	11	results	result	NOUN
ejpam-4827	70	12	from	from	ADP
ejpam-4827	70	13	the	the	DET
ejpam-4827	70	14	subject	subject	NOUN
ejpam-4827	70	15	.	.	PUNCT
ejpam-4827	71	1	to	to	PART
ejpam-4827	71	2	recall	recall	VERB
ejpam-4827	71	3	,	,	PUNCT
ejpam-4827	71	4	a	a	DET
ejpam-4827	71	5	monoid	monoid	NOUN
ejpam-4827	71	6	m	m	NOUN
ejpam-4827	71	7	is	be	AUX
ejpam-4827	71	8	referred	refer	VERB
ejpam-4827	71	9	to	to	ADP
ejpam-4827	71	10	as	as	ADP
ejpam-4827	71	11	a	a	DET
ejpam-4827	71	12	unique	unique	ADJ
ejpam-4827	71	13	product	product	NOUN
ejpam-4827	71	14	(	(	PUNCT
ejpam-4827	71	15	u.p.)-monoid	u.p.)-monoid	PROPN
ejpam-4827	71	16	if	if	SCONJ
ejpam-4827	71	17	,	,	PUNCT
ejpam-4827	71	18	for	for	ADP
ejpam-4827	71	19	any	any	DET
ejpam-4827	71	20	two	two	NUM
ejpam-4827	71	21	non	non	ADJ
ejpam-4827	71	22	-	-	ADJ
ejpam-4827	71	23	empty	empty	ADJ
ejpam-4827	71	24	finite	finite	ADJ
ejpam-4827	71	25	subsets	subset	NOUN
ejpam-4827	71	26	x	x	PUNCT
ejpam-4827	71	27	and	and	CCONJ
ejpam-4827	71	28	y	y	PROPN
ejpam-4827	71	29	of	of	ADP
ejpam-4827	71	30	m	m	PROPN
ejpam-4827	71	31	,	,	PUNCT
ejpam-4827	71	32	there	there	PRON
ejpam-4827	71	33	exist	exist	VERB
ejpam-4827	71	34	an	an	DET
ejpam-4827	71	35	element	element	NOUN
ejpam-4827	71	36	x	x	SYM
ejpam-4827	71	37	∈	∈	PROPN
ejpam-4827	71	38	x	x	X
ejpam-4827	71	39	and	and	CCONJ
ejpam-4827	71	40	an	an	DET
ejpam-4827	71	41	element	element	NOUN
ejpam-4827	71	42	y	y	PROPN
ejpam-4827	71	43	∈	∈	PROPN
ejpam-4827	71	44	y	y	PROPN
ejpam-4827	71	45	such	such	ADJ
ejpam-4827	71	46	that	that	SCONJ
ejpam-4827	71	47	their	their	PRON
ejpam-4827	71	48	product	product	NOUN
ejpam-4827	71	49	xy	xy	INTJ
ejpam-4827	71	50	is	be	AUX
ejpam-4827	71	51	distinct	distinct	ADJ
ejpam-4827	71	52	from	from	ADP
ejpam-4827	71	53	the	the	DET
ejpam-4827	71	54	product	product	NOUN
ejpam-4827	71	55	of	of	ADP
ejpam-4827	71	56	any	any	DET
ejpam-4827	71	57	other	other	ADJ
ejpam-4827	71	58	pair	pair	NOUN
ejpam-4827	71	59	(	(	PUNCT
ejpam-4827	71	60	u	u	NOUN
ejpam-4827	71	61	,	,	PUNCT
ejpam-4827	71	62	v	v	NOUN
ejpam-4827	71	63	)	)	PUNCT
ejpam-4827	71	64	∈	∈	NOUN
ejpam-4827	71	65	x	x	SYM
ejpam-4827	71	66	×	×	PROPN
ejpam-4827	71	67	y	y	PROPN
ejpam-4827	71	68	,	,	PUNCT
ejpam-4827	71	69	i.e.	i.e.	X
ejpam-4827	71	70	,	,	PUNCT
ejpam-4827	71	71	(	(	PUNCT
ejpam-4827	71	72	u	u	NOUN
ejpam-4827	71	73	,	,	PUNCT
ejpam-4827	71	74	v	v	NOUN
ejpam-4827	71	75	)	)	PUNCT
ejpam-4827	71	76	̸=	̸=	PROPN
ejpam-4827	71	77	(	(	PUNCT
ejpam-4827	71	78	x	x	PROPN
ejpam-4827	71	79	,	,	PUNCT
ejpam-4827	71	80	y	y	NOUN
ejpam-4827	71	81	)	)	PUNCT
ejpam-4827	71	82	implies	imply	VERB
ejpam-4827	71	83	uv	uv	X
ejpam-4827	71	84	̸=	̸=	PROPN
ejpam-4827	71	85	xy	xy	NUM
ejpam-4827	71	86	.	.	PUNCT
ejpam-4827	72	1	the	the	DET
ejpam-4827	72	2	element	element	NOUN
ejpam-4827	72	3	xy	xy	PROPN
ejpam-4827	72	4	is	be	AUX
ejpam-4827	72	5	termed	term	VERB
ejpam-4827	72	6	as	as	ADP
ejpam-4827	72	7	a	a	DET
ejpam-4827	72	8	u.p.-element	u.p.-element	NOUN
ejpam-4827	72	9	of	of	ADP
ejpam-4827	72	10	the	the	DET
ejpam-4827	72	11	set	set	NOUN
ejpam-4827	72	12	xy	xy	PROPN
ejpam-4827	72	13	=	=	PUNCT
ejpam-4827	73	1	{	{	PUNCT
ejpam-4827	73	2	pq	pq	INTJ
ejpam-4827	73	3	:	:	PUNCT
ejpam-4827	73	4	p	p	X
ejpam-4827	73	5	∈	∈	PROPN
ejpam-4827	73	6	x	x	X
ejpam-4827	73	7	,	,	PUNCT
ejpam-4827	73	8	q	q	PROPN
ejpam-4827	73	9	∈	∈	PROPN
ejpam-4827	73	10	y	y	PROPN
ejpam-4827	73	11	}	}	PUNCT
ejpam-4827	73	12	.	.	PUNCT
ejpam-4827	74	1	unique	unique	ADJ
ejpam-4827	74	2	product	product	NOUN
ejpam-4827	74	3	monoids	monoid	NOUN
ejpam-4827	74	4	and	and	CCONJ
ejpam-4827	74	5	groups	group	NOUN
ejpam-4827	74	6	have	have	VERB
ejpam-4827	74	7	significant	significant	ADJ
ejpam-4827	74	8	implications	implication	NOUN
ejpam-4827	74	9	in	in	ADP
ejpam-4827	74	10	ring	ring	NOUN
ejpam-4827	74	11	theory	theory	NOUN
ejpam-4827	74	12	,	,	PUNCT
ejpam-4827	74	13	particularly	particularly	ADV
ejpam-4827	74	14	in	in	ADP
ejpam-4827	74	15	providing	provide	VERB
ejpam-4827	74	16	a	a	DET
ejpam-4827	74	17	positive	positive	ADJ
ejpam-4827	74	18	solution	solution	NOUN
ejpam-4827	74	19	to	to	ADP
ejpam-4827	74	20	the	the	DET
ejpam-4827	74	21	zero	zero	NUM
ejpam-4827	74	22	-	-	PUNCT
ejpam-4827	74	23	divisor	divisor	NOUN
ejpam-4827	74	24	problem	problem	NOUN
ejpam-4827	74	25	for	for	ADP
ejpam-4827	74	26	group	group	NOUN
ejpam-4827	74	27	rings	ring	NOUN
ejpam-4827	74	28	.	.	PUNCT
ejpam-4827	75	1	their	their	PRON
ejpam-4827	75	2	structural	structural	ADJ
ejpam-4827	75	3	properties	property	NOUN
ejpam-4827	75	4	have	have	AUX
ejpam-4827	75	5	been	be	AUX
ejpam-4827	75	6	extensively	extensively	ADV
ejpam-4827	75	7	studied	study	VERB
ejpam-4827	75	8	in	in	ADP
ejpam-4827	75	9	literature	literature	NOUN
ejpam-4827	75	10	(	(	PUNCT
ejpam-4827	75	11	see	see	VERB
ejpam-4827	75	12	references	reference	NOUN
ejpam-4827	75	13	[	[	X
ejpam-4827	75	14	17	17	NUM
ejpam-4827	75	15	,	,	PUNCT
ejpam-4827	75	16	19	19	NUM
ejpam-4827	75	17	]	]	NUM
ejpam-4827	75	18	)	)	PUNCT
ejpam-4827	75	19	.	.	PUNCT
ejpam-4827	76	1	the	the	DET
ejpam-4827	76	2	category	category	NOUN
ejpam-4827	76	3	of	of	ADP
ejpam-4827	76	4	monoids	monoid	NOUN
ejpam-4827	76	5	that	that	PRON
ejpam-4827	76	6	fall	fall	VERB
ejpam-4827	76	7	under	under	ADP
ejpam-4827	76	8	the	the	DET
ejpam-4827	76	9	classification	classification	NOUN
ejpam-4827	76	10	of	of	ADP
ejpam-4827	76	11	u.p.-monoids	u.p.-monoid	NOUN
ejpam-4827	76	12	is	be	AUX
ejpam-4827	76	13	both	both	PRON
ejpam-4827	76	14	extensive	extensive	ADJ
ejpam-4827	76	15	and	and	CCONJ
ejpam-4827	76	16	significant	significant	ADJ
ejpam-4827	76	17	.	.	PUNCT
ejpam-4827	77	1	this	this	DET
ejpam-4827	77	2	category	category	NOUN
ejpam-4827	77	3	encompasses	encompass	VERB
ejpam-4827	77	4	monoids	monoid	NOUN
ejpam-4827	77	5	that	that	PRON
ejpam-4827	77	6	are	be	AUX
ejpam-4827	77	7	either	either	ADV
ejpam-4827	77	8	right	right	ADJ
ejpam-4827	77	9	or	or	CCONJ
ejpam-4827	77	10	left	leave	VERB
ejpam-4827	77	11	totally	totally	ADV
ejpam-4827	77	12	ordered	order	VERB
ejpam-4827	77	13	,	,	PUNCT
ejpam-4827	77	14	submonoids	submonoid	NOUN
ejpam-4827	77	15	of	of	ADP
ejpam-4827	77	16	a	a	DET
ejpam-4827	77	17	free	free	ADJ
ejpam-4827	77	18	group	group	NOUN
ejpam-4827	77	19	,	,	PUNCT
ejpam-4827	77	20	and	and	CCONJ
ejpam-4827	77	21	torsion	torsion	NOUN
ejpam-4827	77	22	-	-	PUNCT
ejpam-4827	77	23	free	free	ADJ
ejpam-4827	77	24	nilpotent	nilpotent	ADJ
ejpam-4827	77	25	groups	group	NOUN
ejpam-4827	77	26	.	.	PUNCT
ejpam-4827	78	1	for	for	ADP
ejpam-4827	78	2	a	a	DET
ejpam-4827	78	3	positive	positive	ADJ
ejpam-4827	78	4	integer	integer	NOUN
ejpam-4827	78	5	n	n	CCONJ
ejpam-4827	78	6	,	,	PUNCT
ejpam-4827	78	7	let	let	AUX
ejpam-4827	78	8	matn(r	matn(r	PRON
ejpam-4827	78	9	)	)	PUNCT
ejpam-4827	78	10	denote	denote	VERB
ejpam-4827	78	11	the	the	DET
ejpam-4827	78	12	ring	ring	NOUN
ejpam-4827	78	13	of	of	ADP
ejpam-4827	78	14	all	all	DET
ejpam-4827	78	15	n×n	n×n	PROPN
ejpam-4827	78	16	matrices	matrix	NOUN
ejpam-4827	78	17	and	and	CCONJ
ejpam-4827	78	18	tn(r	tn(r	NUM
ejpam-4827	78	19	)	)	PUNCT
ejpam-4827	78	20	the	the	DET
ejpam-4827	78	21	ring	ring	NOUN
ejpam-4827	78	22	of	of	ADP
ejpam-4827	78	23	all	all	DET
ejpam-4827	78	24	n×n	n×n	PROPN
ejpam-4827	78	25	upper	upper	ADJ
ejpam-4827	78	26	triangular	triangular	NOUN
ejpam-4827	78	27	matrices	matrix	NOUN
ejpam-4827	78	28	with	with	ADP
ejpam-4827	78	29	entries	entry	NOUN
ejpam-4827	78	30	in	in	ADP
ejpam-4827	78	31	r.	r.	PROPN
ejpam-4827	78	32	we	we	PRON
ejpam-4827	78	33	write	write	VERB
ejpam-4827	78	34	r[x	r[x	NOUN
ejpam-4827	78	35	]	]	PUNCT
ejpam-4827	78	36	and	and	CCONJ
ejpam-4827	78	37	sn(r	sn(r	NUM
ejpam-4827	78	38	)	)	PUNCT
ejpam-4827	78	39	,	,	PUNCT
ejpam-4827	78	40	for	for	ADP
ejpam-4827	78	41	the	the	DET
ejpam-4827	78	42	polynomial	polynomial	ADJ
ejpam-4827	78	43	ring	ring	NOUN
ejpam-4827	78	44	over	over	ADP
ejpam-4827	78	45	a	a	DET
ejpam-4827	78	46	ring	ring	NOUN
ejpam-4827	78	47	r	r	NOUN
ejpam-4827	78	48	and	and	CCONJ
ejpam-4827	78	49	the	the	DET
ejpam-4827	78	50	subring	subring	NOUN
ejpam-4827	78	51	consisting	consist	VERB
ejpam-4827	78	52	of	of	ADP
ejpam-4827	78	53	all	all	DET
ejpam-4827	78	54	upper	upper	ADJ
ejpam-4827	78	55	triangular	triangular	NOUN
ejpam-4827	78	56	matrices	matrix	NOUN
ejpam-4827	78	57	over	over	ADP
ejpam-4827	78	58	a	a	DET
ejpam-4827	78	59	ring	ring	NOUN
ejpam-4827	78	60	r	r	NOUN
ejpam-4827	78	61	with	with	ADP
ejpam-4827	78	62	equal	equal	ADJ
ejpam-4827	78	63	main	main	ADJ
ejpam-4827	78	64	diagonal	diagonal	ADJ
ejpam-4827	78	65	entries	entry	NOUN
ejpam-4827	78	66	.	.	PUNCT
ejpam-4827	79	1	2	2	X
ejpam-4827	79	2	.	.	X
ejpam-4827	79	3	reflexive	reflexive	ADJ
ejpam-4827	79	4	-	-	PUNCT
ejpam-4827	79	5	type	type	NOUN
ejpam-4827	79	6	properties	property	NOUN
ejpam-4827	79	7	in	in	ADP
ejpam-4827	79	8	skew	skew	ADJ
ejpam-4827	79	9	monoid	monoid	NOUN
ejpam-4827	79	10	rings	ring	NOUN
ejpam-4827	79	11	in	in	ADP
ejpam-4827	79	12	this	this	DET
ejpam-4827	79	13	section	section	NOUN
ejpam-4827	79	14	,	,	PUNCT
ejpam-4827	79	15	we	we	PRON
ejpam-4827	79	16	discuss	discuss	VERB
ejpam-4827	79	17	various	various	ADJ
ejpam-4827	79	18	constructions	construction	NOUN
ejpam-4827	79	19	and	and	CCONJ
ejpam-4827	79	20	extensions	extension	NOUN
ejpam-4827	79	21	under	under	ADP
ejpam-4827	79	22	which	which	PRON
ejpam-4827	79	23	the	the	DET
ejpam-4827	79	24	class	class	NOUN
ejpam-4827	79	25	of	of	ADP
ejpam-4827	79	26	σ	σ	PROPN
ejpam-4827	79	27	-	-	PUNCT
ejpam-4827	79	28	skew	skew	NOUN
ejpam-4827	79	29	strongly	strongly	ADV
ejpam-4827	79	30	m	m	VERB
ejpam-4827	79	31	-reflexive	-reflexive	ADJ
ejpam-4827	79	32	rings	ring	NOUN
ejpam-4827	79	33	is	be	AUX
ejpam-4827	79	34	closed	close	VERB
ejpam-4827	79	35	.	.	PUNCT
ejpam-4827	80	1	by	by	ADP
ejpam-4827	80	2	definition	definition	NOUN
ejpam-4827	80	3	2.3	2.3	NUM
ejpam-4827	80	4	[	[	X
ejpam-4827	80	5	18	18	NUM
ejpam-4827	80	6	]	]	PUNCT
ejpam-4827	80	7	,	,	PUNCT
ejpam-4827	80	8	a	a	DET
ejpam-4827	80	9	monoid	monoid	NOUN
ejpam-4827	80	10	homomorphism	homomorphism	NOUN
ejpam-4827	80	11	σ	σ	X
ejpam-4827	80	12	:	:	PUNCT
ejpam-4827	80	13	m	m	PROPN
ejpam-4827	80	14	→	→	SYM
ejpam-4827	80	15	aut(r	aut(r	PROPN
ejpam-4827	80	16	)	)	PUNCT
ejpam-4827	80	17	is	be	AUX
ejpam-4827	80	18	called	call	VERB
ejpam-4827	80	19	compatible	compatible	ADJ
ejpam-4827	80	20	if	if	SCONJ
ejpam-4827	80	21	the	the	DET
ejpam-4827	80	22	ring	ring	NOUN
ejpam-4827	80	23	r	r	NOUN
ejpam-4827	80	24	is	be	AUX
ejpam-4827	80	25	σg	σg	NOUN
ejpam-4827	80	26	-	-	PUNCT
ejpam-4827	80	27	compatible	compatible	ADJ
ejpam-4827	80	28	for	for	ADP
ejpam-4827	80	29	each	each	DET
ejpam-4827	80	30	g	g	NOUN
ejpam-4827	80	31	∈m	∈m	NOUN
ejpam-4827	80	32	,	,	PUNCT
ejpam-4827	80	33	that	that	ADV
ejpam-4827	80	34	is	is	ADV
ejpam-4827	80	35	,	,	PUNCT
ejpam-4827	80	36	xy	xy	PROPN
ejpam-4827	80	37	=	=	SYM
ejpam-4827	80	38	0	0	NUM
ejpam-4827	80	39	⇔	⇔	PROPN
ejpam-4827	80	40	xσg(y	xσg(y	PROPN
ejpam-4827	80	41	)	)	PUNCT
ejpam-4827	80	42	=	=	SYM
ejpam-4827	80	43	0	0	NUM
ejpam-4827	80	44	for	for	ADP
ejpam-4827	80	45	all	all	DET
ejpam-4827	80	46	x	x	NOUN
ejpam-4827	80	47	,	,	PUNCT
ejpam-4827	80	48	y	y	PROPN
ejpam-4827	80	49	∈	∈	PROPN
ejpam-4827	80	50	r.	r.	PROPN
ejpam-4827	80	51	now	now	ADV
ejpam-4827	80	52	we	we	PRON
ejpam-4827	80	53	have	have	VERB
ejpam-4827	80	54	the	the	DET
ejpam-4827	80	55	following	following	ADJ
ejpam-4827	80	56	generalization	generalization	NOUN
ejpam-4827	80	57	of	of	ADP
ejpam-4827	80	58	reflexive	reflexive	ADJ
ejpam-4827	80	59	.	.	PUNCT
ejpam-4827	81	1	definition	definition	NOUN
ejpam-4827	81	2	1	1	NUM
ejpam-4827	81	3	.	.	PUNCT
ejpam-4827	82	1	we	we	PRON
ejpam-4827	82	2	say	say	VERB
ejpam-4827	82	3	that	that	SCONJ
ejpam-4827	82	4	a	a	DET
ejpam-4827	82	5	ring	ring	NOUN
ejpam-4827	82	6	r	r	NOUN
ejpam-4827	82	7	is	be	AUX
ejpam-4827	82	8	σ	σ	NOUN
ejpam-4827	82	9	-	-	PUNCT
ejpam-4827	82	10	skew	skew	NOUN
ejpam-4827	82	11	strongly	strongly	ADV
ejpam-4827	82	12	m	m	VERB
ejpam-4827	82	13	-reflexive	-reflexive	ADJ
ejpam-4827	82	14	(	(	PUNCT
ejpam-4827	82	15	σ	σ	NOUN
ejpam-4827	82	16	-	-	PUNCT
ejpam-4827	82	17	skew	skew	NOUN
ejpam-4827	82	18	strongly	strongly	ADV
ejpam-4827	82	19	reflexive	reflexive	ADJ
ejpam-4827	82	20	relative	relative	ADJ
ejpam-4827	82	21	to	to	ADP
ejpam-4827	82	22	a	a	DET
ejpam-4827	82	23	monoid	monoid	NOUN
ejpam-4827	82	24	m	m	PROPN
ejpam-4827	82	25	)	)	PUNCT
ejpam-4827	82	26	,	,	PUNCT
ejpam-4827	82	27	if	if	SCONJ
ejpam-4827	82	28	φ(r	φ(r	ADJ
ejpam-4827	82	29	∗m)ψ	∗m)ψ	ADJ
ejpam-4827	82	30	=	=	SYM
ejpam-4827	82	31	0	0	NUM
ejpam-4827	82	32	implies	imply	VERB
ejpam-4827	82	33	that	that	SCONJ
ejpam-4827	82	34	biσgi(rσs(aj	biσgi(rσs(aj	NOUN
ejpam-4827	82	35	)	)	PUNCT
ejpam-4827	82	36	)	)	PUNCT
ejpam-4827	83	1	=	=	SYM
ejpam-4827	83	2	0	0	NUM
ejpam-4827	83	3	,	,	PUNCT
ejpam-4827	83	4	where	where	SCONJ
ejpam-4827	83	5	φ	φ	PROPN
ejpam-4827	83	6	=	=	SYM
ejpam-4827	83	7	b1g1	b1g1	PROPN
ejpam-4827	83	8	+	+	NUM
ejpam-4827	83	9	b2g2	b2g2	PROPN
ejpam-4827	83	10	+	+	CCONJ
ejpam-4827	83	11	·	·	PUNCT
ejpam-4827	83	12	·	·	PUNCT
ejpam-4827	83	13	·	·	PUNCT
ejpam-4827	83	14	+	+	NUM
ejpam-4827	83	15	bngn	bngn	NOUN
ejpam-4827	83	16	and	and	CCONJ
ejpam-4827	83	17	ψ	ψ	X
ejpam-4827	83	18	=	=	X
ejpam-4827	83	19	a1h1	a1h1	X
ejpam-4827	83	20	+	+	NOUN
ejpam-4827	83	21	a2h2	a2h2	X
ejpam-4827	83	22	+	+	X
ejpam-4827	83	23	·	·	PUNCT
ejpam-4827	83	24	·	·	PUNCT
ejpam-4827	83	25	·	·	PUNCT
ejpam-4827	84	1	+	+	NUM
ejpam-4827	84	2	amhm	amhm	NOUN
ejpam-4827	84	3	are	be	AUX
ejpam-4827	84	4	nonzero	nonzero	NOUN
ejpam-4827	84	5	elements	element	NOUN
ejpam-4827	84	6	in	in	ADP
ejpam-4827	84	7	r	r	NOUN
ejpam-4827	84	8	∗m	∗m	NOUN
ejpam-4827	84	9	,	,	PUNCT
ejpam-4827	84	10	then	then	ADV
ejpam-4827	84	11	ψ(r	ψ(r	PROPN
ejpam-4827	84	12	∗m)φ	∗m)φ	VERB
ejpam-4827	84	13	=	=	PROPN
ejpam-4827	84	14	0	0	PROPN
ejpam-4827	84	15	for	for	ADP
ejpam-4827	84	16	all	all	DET
ejpam-4827	84	17	1	1	NUM
ejpam-4827	84	18	≤	≤	NUM
ejpam-4827	84	19	i	i	PRON
ejpam-4827	84	20	≤	≤	PROPN
ejpam-4827	84	21	n	n	CCONJ
ejpam-4827	84	22	,	,	PUNCT
ejpam-4827	84	23	1	1	NUM
ejpam-4827	84	24	≤	≤	NUM
ejpam-4827	84	25	j	j	PROPN
ejpam-4827	84	26	≤	≤	PROPN
ejpam-4827	84	27	m.	m.	NOUN
ejpam-4827	84	28	definition	definition	NOUN
ejpam-4827	84	29	2	2	NUM
ejpam-4827	84	30	.	.	PUNCT
ejpam-4827	85	1	in	in	ADP
ejpam-4827	85	2	[	[	X
ejpam-4827	85	3	20	20	NUM
ejpam-4827	85	4	]	]	PUNCT
ejpam-4827	85	5	.	.	PUNCT
ejpam-4827	86	1	a	a	DET
ejpam-4827	86	2	ring	ring	NOUN
ejpam-4827	86	3	r	r	NOUN
ejpam-4827	86	4	is	be	AUX
ejpam-4827	86	5	called	call	VERB
ejpam-4827	86	6	strongly	strongly	ADV
ejpam-4827	86	7	m	m	VERB
ejpam-4827	86	8	-reflexive	-reflexive	ADJ
ejpam-4827	86	9	,	,	PUNCT
ejpam-4827	86	10	if	if	SCONJ
ejpam-4827	86	11	whenever	whenever	SCONJ
ejpam-4827	86	12	φ	φ	X
ejpam-4827	86	13	,	,	PUNCT
ejpam-4827	86	14	ψ	ψ	X
ejpam-4827	86	15	∈	∈	PROPN
ejpam-4827	86	16	r[m	r[m	NOUN
ejpam-4827	86	17	]	]	PUNCT
ejpam-4827	86	18	with	with	ADP
ejpam-4827	86	19	φr[m	φr[m	ADV
ejpam-4827	86	20	]	]	SYM
ejpam-4827	86	21	ψ	ψ	X
ejpam-4827	86	22	=	=	SYM
ejpam-4827	86	23	0	0	NUM
ejpam-4827	86	24	,	,	PUNCT
ejpam-4827	86	25	then	then	ADV
ejpam-4827	86	26	ψr[m	ψr[m	PROPN
ejpam-4827	86	27	]	]	PUNCT
ejpam-4827	86	28	φ	φ	X
ejpam-4827	86	29	=	=	SYM
ejpam-4827	86	30	0	0	X
ejpam-4827	86	31	.	.	PUNCT
ejpam-4827	87	1	in	in	ADP
ejpam-4827	87	2	[	[	X
ejpam-4827	87	3	16	16	NUM
ejpam-4827	87	4	]	]	PUNCT
ejpam-4827	87	5	,	,	PUNCT
ejpam-4827	87	6	nasr	nasr	PROPN
ejpam-4827	87	7	-	-	PUNCT
ejpam-4827	87	8	isfahani	isfahani	PROPN
ejpam-4827	87	9	and	and	CCONJ
ejpam-4827	87	10	moussavi	moussavi	NOUN
ejpam-4827	87	11	introduced	introduce	VERB
ejpam-4827	87	12	a	a	DET
ejpam-4827	87	13	ring	ring	NOUN
ejpam-4827	87	14	r	r	NOUN
ejpam-4827	87	15	with	with	ADP
ejpam-4827	87	16	an	an	DET
ejpam-4827	87	17	endomorphism	endomorphism	PROPN
ejpam-4827	87	18	σ	σ	PROPN
ejpam-4827	87	19	and	and	CCONJ
ejpam-4827	87	20	defined	define	VERB
ejpam-4827	87	21	it	it	PRON
ejpam-4827	87	22	as	as	ADP
ejpam-4827	87	23	σ	σ	NOUN
ejpam-4827	87	24	-	-	PUNCT
ejpam-4827	87	25	weakly	weakly	ADV
ejpam-4827	87	26	rigid	rigid	ADJ
ejpam-4827	87	27	if	if	SCONJ
ejpam-4827	87	28	the	the	DET
ejpam-4827	87	29	condition	condition	NOUN
ejpam-4827	87	30	xry	xry	X
ejpam-4827	88	1	=	=	SYM
ejpam-4827	88	2	0	0	NUM
ejpam-4827	88	3	holds	hold	VERB
ejpam-4827	88	4	if	if	SCONJ
ejpam-4827	88	5	and	and	CCONJ
ejpam-4827	88	6	only	only	ADV
ejpam-4827	88	7	if	if	SCONJ
ejpam-4827	88	8	xσ(ry	xσ(ry	NOUN
ejpam-4827	88	9	)	)	PUNCT
ejpam-4827	88	10	=	=	SYM
ejpam-4827	88	11	0	0	NUM
ejpam-4827	88	12	for	for	ADP
ejpam-4827	88	13	any	any	DET
ejpam-4827	88	14	x	x	NOUN
ejpam-4827	88	15	,	,	PUNCT
ejpam-4827	88	16	y	y	PROPN
ejpam-4827	88	17	∈	∈	PROPN
ejpam-4827	88	18	r.	r.	PROPN
ejpam-4827	88	19	a	a	DET
ejpam-4827	88	20	ring	ring	NOUN
ejpam-4827	88	21	r	r	NOUN
ejpam-4827	88	22	is	be	AUX
ejpam-4827	88	23	σ	σ	NOUN
ejpam-4827	88	24	-	-	ADJ
ejpam-4827	88	25	rigid	rigid	ADJ
ejpam-4827	88	26	if	if	SCONJ
ejpam-4827	88	27	and	and	CCONJ
ejpam-4827	88	28	only	only	ADV
ejpam-4827	88	29	if	if	SCONJ
ejpam-4827	88	30	r	r	NOUN
ejpam-4827	88	31	is	be	AUX
ejpam-4827	88	32	σ	σ	NOUN
ejpam-4827	88	33	-	-	PUNCT
ejpam-4827	88	34	compatible	compatible	ADJ
ejpam-4827	88	35	and	and	CCONJ
ejpam-4827	88	36	reduced	reduce	VERB
ejpam-4827	88	37	by	by	ADP
ejpam-4827	88	38	[	[	X
ejpam-4827	88	39	8	8	NUM
ejpam-4827	88	40	]	]	PUNCT
ejpam-4827	88	41	.	.	PUNCT
ejpam-4827	89	1	according	accord	VERB
ejpam-4827	89	2	to	to	ADP
ejpam-4827	89	3	[	[	X
ejpam-4827	89	4	16	16	NUM
ejpam-4827	89	5	]	]	PUNCT
ejpam-4827	89	6	,	,	PUNCT
ejpam-4827	89	7	any	any	DET
ejpam-4827	89	8	prime	prime	ADJ
ejpam-4827	89	9	ring	ring	NOUN
ejpam-4827	89	10	that	that	PRON
ejpam-4827	89	11	has	have	VERB
ejpam-4827	89	12	an	an	DET
ejpam-4827	89	13	automorphism	automorphism	NOUN
ejpam-4827	89	14	σ	σ	NOUN
ejpam-4827	89	15	is	be	AUX
ejpam-4827	89	16	considered	consider	VERB
ejpam-4827	89	17	to	to	PART
ejpam-4827	89	18	be	be	AUX
ejpam-4827	89	19	σ	σ	NOUN
ejpam-4827	89	20	-	-	ADJ
ejpam-4827	89	21	weakly	weakly	ADV
ejpam-4827	89	22	rigid	rigid	ADJ
ejpam-4827	89	23	.	.	PUNCT
ejpam-4827	90	1	if	if	SCONJ
ejpam-4827	90	2	a	a	DET
ejpam-4827	90	3	monoid	monoid	NOUN
ejpam-4827	90	4	homomorphism	homomorphism	NOUN
ejpam-4827	90	5	σ	σ	X
ejpam-4827	90	6	:	:	PUNCT
ejpam-4827	90	7	m	m	PROPN
ejpam-4827	90	8	→	→	SYM
ejpam-4827	90	9	aut(r	aut(r	PROPN
ejpam-4827	90	10	)	)	PUNCT
ejpam-4827	90	11	is	be	AUX
ejpam-4827	90	12	weakly	weakly	ADV
ejpam-4827	90	13	-	-	PUNCT
ejpam-4827	90	14	rigid	rigid	ADJ
ejpam-4827	90	15	(	(	PUNCT
ejpam-4827	90	16	compatible	compatible	ADJ
ejpam-4827	90	17	)	)	PUNCT
ejpam-4827	90	18	,	,	PUNCT
ejpam-4827	90	19	it	it	PRON
ejpam-4827	90	20	means	mean	VERB
ejpam-4827	90	21	that	that	SCONJ
ejpam-4827	90	22	the	the	DET
ejpam-4827	90	23	ring	ring	NOUN
ejpam-4827	90	24	r	r	NOUN
ejpam-4827	90	25	is	be	AUX
ejpam-4827	90	26	also	also	ADV
ejpam-4827	90	27	weakly	weakly	ADV
ejpam-4827	90	28	rigid	rigid	ADJ
ejpam-4827	90	29	(	(	PUNCT
ejpam-4827	90	30	compatible	compatible	ADJ
ejpam-4827	90	31	)	)	PUNCT
ejpam-4827	90	32	with	with	ADP
ejpam-4827	90	33	respect	respect	NOUN
ejpam-4827	90	34	to	to	ADP
ejpam-4827	90	35	each	each	DET
ejpam-4827	90	36	g	g	PROPN
ejpam-4827	90	37	∈	∈	PROPN
ejpam-4827	90	38	m	m	VERB
ejpam-4827	90	39	under	under	ADP
ejpam-4827	90	40	the	the	DET
ejpam-4827	90	41	automorphism	automorphism	NOUN
ejpam-4827	90	42	σg	σg	NOUN
ejpam-4827	90	43	.	.	PUNCT
ejpam-4827	91	1	the	the	DET
ejpam-4827	91	2	following	follow	VERB
ejpam-4827	91	3	example	example	NOUN
ejpam-4827	91	4	illustrates	illustrate	VERB
ejpam-4827	91	5	that	that	SCONJ
ejpam-4827	91	6	the	the	DET
ejpam-4827	91	7	compatibility	compatibility	NOUN
ejpam-4827	91	8	of	of	ADP
ejpam-4827	91	9	σ	σ	PROPN
ejpam-4827	91	10	is	be	AUX
ejpam-4827	91	11	necessary	necessary	ADJ
ejpam-4827	91	12	.	.	PUNCT
ejpam-4827	92	1	e.	e.	PROPN
ejpam-4827	92	2	ali	ali	PROPN
ejpam-4827	92	3	/	/	SYM
ejpam-4827	92	4	eur	eur	PROPN
ejpam-4827	92	5	.	.	PUNCT
ejpam-4827	93	1	j.	j.	PROPN
ejpam-4827	93	2	pure	pure	PROPN
ejpam-4827	93	3	appl	appl	PROPN
ejpam-4827	93	4	.	.	PROPN
ejpam-4827	93	5	math	math	PROPN
ejpam-4827	93	6	,	,	PUNCT
ejpam-4827	93	7	16	16	NUM
ejpam-4827	93	8	(	(	PUNCT
ejpam-4827	93	9	3	3	NUM
ejpam-4827	93	10	)	)	PUNCT
ejpam-4827	93	11	(	(	PUNCT
ejpam-4827	93	12	2023	2023	NUM
ejpam-4827	93	13	)	)	PUNCT
ejpam-4827	93	14	,	,	PUNCT
ejpam-4827	93	15	1878	1878	NUM
ejpam-4827	93	16	-	-	SYM
ejpam-4827	93	17	1893	1893	NUM
ejpam-4827	93	18	1881	1881	NUM
ejpam-4827	93	19	example	example	NOUN
ejpam-4827	93	20	1	1	NUM
ejpam-4827	93	21	.	.	PUNCT
ejpam-4827	94	1	let	let	VERB
ejpam-4827	94	2	s	s	PRON
ejpam-4827	94	3	be	be	AUX
ejpam-4827	94	4	any	any	DET
ejpam-4827	94	5	nonzero	nonzero	ADJ
ejpam-4827	94	6	reversible	reversible	ADJ
ejpam-4827	94	7	ring	ring	NOUN
ejpam-4827	94	8	and	and	CCONJ
ejpam-4827	94	9	m	m	AUX
ejpam-4827	94	10	be	be	AUX
ejpam-4827	94	11	a	a	DET
ejpam-4827	94	12	monoid	monoid	NOUN
ejpam-4827	94	13	generated	generate	VERB
ejpam-4827	94	14	by	by	ADP
ejpam-4827	94	15	an	an	DET
ejpam-4827	94	16	element	element	NOUN
ejpam-4827	94	17	ρ	ρ	NOUN
ejpam-4827	94	18	such	such	ADJ
ejpam-4827	94	19	that	that	SCONJ
ejpam-4827	94	20	ρ	ρ	PROPN
ejpam-4827	94	21	has	have	VERB
ejpam-4827	94	22	infinite	infinite	ADJ
ejpam-4827	94	23	order	order	NOUN
ejpam-4827	94	24	.	.	PUNCT
ejpam-4827	95	1	suppose	suppose	VERB
ejpam-4827	95	2	r	r	NOUN
ejpam-4827	95	3	=	=	SYM
ejpam-4827	95	4	s	s	PROPN
ejpam-4827	95	5	⊕	⊕	PROPN
ejpam-4827	95	6	s	s	PART
ejpam-4827	95	7	with	with	ADP
ejpam-4827	95	8	the	the	DET
ejpam-4827	95	9	usual	usual	ADJ
ejpam-4827	95	10	addition	addition	NOUN
ejpam-4827	95	11	and	and	CCONJ
ejpam-4827	95	12	multiplication	multiplication	NOUN
ejpam-4827	95	13	,	,	PUNCT
ejpam-4827	95	14	and	and	CCONJ
ejpam-4827	95	15	define	define	VERB
ejpam-4827	95	16	σ	σ	NOUN
ejpam-4827	95	17	:	:	PUNCT
ejpam-4827	95	18	m	m	PROPN
ejpam-4827	95	19	→	→	SYM
ejpam-4827	95	20	aut(r	aut(r	PROPN
ejpam-4827	95	21	)	)	PUNCT
ejpam-4827	95	22	such	such	ADJ
ejpam-4827	95	23	that	that	DET
ejpam-4827	95	24	σρ((x	σρ((x	NOUN
ejpam-4827	95	25	,	,	PUNCT
ejpam-4827	95	26	y	y	NOUN
ejpam-4827	95	27	)	)	PUNCT
ejpam-4827	95	28	)	)	PUNCT
ejpam-4827	96	1	=	=	PRON
ejpam-4827	96	2	(	(	PUNCT
ejpam-4827	96	3	y	y	PROPN
ejpam-4827	96	4	,	,	PUNCT
ejpam-4827	96	5	x	x	NOUN
ejpam-4827	96	6	)	)	PUNCT
ejpam-4827	96	7	.	.	PUNCT
ejpam-4827	97	1	then	then	ADV
ejpam-4827	97	2	the	the	DET
ejpam-4827	97	3	ring	ring	NOUN
ejpam-4827	97	4	r	r	NOUN
ejpam-4827	97	5	is	be	AUX
ejpam-4827	97	6	reflexive	reflexive	ADJ
ejpam-4827	97	7	and	and	CCONJ
ejpam-4827	97	8	m	m	VERB
ejpam-4827	97	9	is	be	AUX
ejpam-4827	97	10	u.p.-monoid	u.p.-monoid	ADJ
ejpam-4827	97	11	,	,	PUNCT
ejpam-4827	97	12	but	but	CCONJ
ejpam-4827	97	13	σ	σ	NOUN
ejpam-4827	97	14	is	be	AUX
ejpam-4827	97	15	not	not	PART
ejpam-4827	97	16	compatible	compatible	ADJ
ejpam-4827	97	17	,	,	PUNCT
ejpam-4827	97	18	since	since	SCONJ
ejpam-4827	97	19	(	(	PUNCT
ejpam-4827	97	20	1	1	NUM
ejpam-4827	97	21	,	,	PUNCT
ejpam-4827	97	22	0)(0	0)(0	NUM
ejpam-4827	97	23	,	,	PUNCT
ejpam-4827	97	24	1	1	NUM
ejpam-4827	97	25	)	)	PUNCT
ejpam-4827	97	26	=	=	SYM
ejpam-4827	97	27	(	(	PUNCT
ejpam-4827	97	28	0	0	NUM
ejpam-4827	97	29	,	,	PUNCT
ejpam-4827	97	30	0	0	NUM
ejpam-4827	97	31	)	)	PUNCT
ejpam-4827	97	32	whereas	whereas	SCONJ
ejpam-4827	97	33	(	(	PUNCT
ejpam-4827	97	34	1	1	NUM
ejpam-4827	97	35	,	,	PUNCT
ejpam-4827	97	36	0)σρ((0	0)σρ((0	NOUN
ejpam-4827	97	37	,	,	PUNCT
ejpam-4827	97	38	1	1	NUM
ejpam-4827	97	39	)	)	PUNCT
ejpam-4827	97	40	)	)	PUNCT
ejpam-4827	98	1	=	=	PUNCT
ejpam-4827	98	2	(	(	PUNCT
ejpam-4827	98	3	1	1	NUM
ejpam-4827	98	4	,	,	PUNCT
ejpam-4827	98	5	0	0	NUM
ejpam-4827	98	6	)	)	PUNCT
ejpam-4827	98	7	.	.	PUNCT
ejpam-4827	99	1	now	now	ADV
ejpam-4827	99	2	we	we	PRON
ejpam-4827	99	3	will	will	AUX
ejpam-4827	99	4	prove	prove	VERB
ejpam-4827	99	5	that	that	SCONJ
ejpam-4827	99	6	the	the	DET
ejpam-4827	99	7	ring	ring	NOUN
ejpam-4827	99	8	r	r	NOUN
ejpam-4827	99	9	is	be	AUX
ejpam-4827	99	10	not	not	PART
ejpam-4827	99	11	σ	σ	NOUN
ejpam-4827	99	12	-	-	PUNCT
ejpam-4827	99	13	skew	skew	NOUN
ejpam-4827	99	14	strongly	strongly	ADV
ejpam-4827	99	15	m	m	VERB
ejpam-4827	99	16	-reflexive	-reflexive	ADJ
ejpam-4827	99	17	.	.	PUNCT
ejpam-4827	100	1	for	for	ADP
ejpam-4827	100	2	,	,	PUNCT
ejpam-4827	100	3	let	let	VERB
ejpam-4827	100	4	φ	φ	PROPN
ejpam-4827	100	5	=	=	SYM
ejpam-4827	100	6	(	(	PUNCT
ejpam-4827	100	7	1	1	NUM
ejpam-4827	100	8	,	,	PUNCT
ejpam-4827	100	9	0)e	0)e	NOUN
ejpam-4827	100	10	+	+	CCONJ
ejpam-4827	100	11	(	(	PUNCT
ejpam-4827	100	12	1	1	NUM
ejpam-4827	100	13	,	,	PUNCT
ejpam-4827	100	14	0)g	0)g	NOUN
ejpam-4827	100	15	and	and	CCONJ
ejpam-4827	100	16	ψ	ψ	X
ejpam-4827	100	17	=	=	SYM
ejpam-4827	100	18	(	(	PUNCT
ejpam-4827	100	19	0	0	NUM
ejpam-4827	100	20	,	,	PUNCT
ejpam-4827	100	21	1)e	1)e	NUM
ejpam-4827	100	22	−	−	PROPN
ejpam-4827	100	23	(	(	PUNCT
ejpam-4827	100	24	1	1	NUM
ejpam-4827	100	25	,	,	PUNCT
ejpam-4827	100	26	0)g	0)g	NOUN
ejpam-4827	100	27	be	be	AUX
ejpam-4827	100	28	nonzero	nonzero	ADJ
ejpam-4827	100	29	elements	element	NOUN
ejpam-4827	100	30	in	in	ADP
ejpam-4827	100	31	r	r	NOUN
ejpam-4827	100	32	∗m	∗m	NOUN
ejpam-4827	100	33	and	and	CCONJ
ejpam-4827	100	34	any	any	DET
ejpam-4827	100	35	ϕ	ϕ	PROPN
ejpam-4827	100	36	∈	∈	PROPN
ejpam-4827	100	37	r	r	NOUN
ejpam-4827	100	38	∗m	∗m	NOUN
ejpam-4827	100	39	.	.	PUNCT
ejpam-4827	101	1	then	then	ADV
ejpam-4827	101	2	we	we	PRON
ejpam-4827	101	3	can	can	AUX
ejpam-4827	101	4	easily	easily	ADV
ejpam-4827	101	5	see	see	VERB
ejpam-4827	101	6	that	that	DET
ejpam-4827	101	7	φϕψ	φϕψ	NOUN
ejpam-4827	101	8	=	=	SYM
ejpam-4827	101	9	0	0	PROPN
ejpam-4827	101	10	,	,	PUNCT
ejpam-4827	101	11	but	but	CCONJ
ejpam-4827	101	12	ψϕσg(φ	ψϕσg(φ	X
ejpam-4827	101	13	)	)	PUNCT
ejpam-4827	101	14	̸=	̸=	PROPN
ejpam-4827	101	15	0	0	NUM
ejpam-4827	101	16	.	.	PUNCT
ejpam-4827	102	1	therefore	therefore	ADV
ejpam-4827	102	2	,	,	PUNCT
ejpam-4827	102	3	r	r	NOUN
ejpam-4827	102	4	is	be	AUX
ejpam-4827	102	5	not	not	PART
ejpam-4827	102	6	σ	σ	NOUN
ejpam-4827	102	7	-	-	PUNCT
ejpam-4827	102	8	skew	skew	NOUN
ejpam-4827	102	9	strongly	strongly	ADV
ejpam-4827	102	10	m	m	VERB
ejpam-4827	102	11	-reflexive	-reflexive	ADJ
ejpam-4827	102	12	.	.	PUNCT
ejpam-4827	103	1	theorem	theorem	NOUN
ejpam-4827	103	2	1	1	NUM
ejpam-4827	103	3	.	.	PUNCT
ejpam-4827	104	1	let	let	VERB
ejpam-4827	104	2	r	r	PRON
ejpam-4827	104	3	be	be	AUX
ejpam-4827	104	4	a	a	DET
ejpam-4827	104	5	ring	ring	NOUN
ejpam-4827	104	6	,	,	PUNCT
ejpam-4827	104	7	m	m	VERB
ejpam-4827	104	8	be	be	VERB
ejpam-4827	104	9	a	a	DET
ejpam-4827	104	10	monoid	monoid	NOUN
ejpam-4827	104	11	generated	generate	VERB
ejpam-4827	104	12	by	by	ADP
ejpam-4827	104	13	an	an	DET
ejpam-4827	104	14	element	element	NOUN
ejpam-4827	104	15	ρ	ρ	NOUN
ejpam-4827	104	16	such	such	ADJ
ejpam-4827	104	17	that	that	SCONJ
ejpam-4827	104	18	ρ	ρ	PROPN
ejpam-4827	104	19	has	have	VERB
ejpam-4827	104	20	infinite	infinite	ADJ
ejpam-4827	104	21	order	order	NOUN
ejpam-4827	104	22	and	and	CCONJ
ejpam-4827	104	23	σ	σ	NOUN
ejpam-4827	104	24	:	:	PUNCT
ejpam-4827	104	25	m	m	PROPN
ejpam-4827	104	26	→	→	SYM
ejpam-4827	104	27	aut(r	aut(r	PROPN
ejpam-4827	104	28	)	)	PUNCT
ejpam-4827	104	29	a	a	DET
ejpam-4827	104	30	compatible	compatible	ADJ
ejpam-4827	104	31	monoid	monoid	NOUN
ejpam-4827	104	32	homomorphism	homomorphism	NOUN
ejpam-4827	104	33	given	give	VERB
ejpam-4827	104	34	by	by	ADP
ejpam-4827	104	35	σρ	σρ	NOUN
ejpam-4827	104	36	=	=	SYM
ejpam-4827	104	37	ψ	ψ	X
ejpam-4827	104	38	.	.	PUNCT
ejpam-4827	104	39	suppose	suppose	VERB
ejpam-4827	104	40	n	n	PRON
ejpam-4827	104	41	be	be	AUX
ejpam-4827	104	42	any	any	DET
ejpam-4827	104	43	monoid	monoid	NOUN
ejpam-4827	104	44	with	with	ADP
ejpam-4827	104	45	an	an	DET
ejpam-4827	104	46	element	element	NOUN
ejpam-4827	104	47	of	of	ADP
ejpam-4827	104	48	infinite	infinite	ADJ
ejpam-4827	104	49	order	order	NOUN
ejpam-4827	104	50	.	.	PUNCT
ejpam-4827	105	1	if	if	SCONJ
ejpam-4827	105	2	the	the	DET
ejpam-4827	105	3	skew	skew	ADJ
ejpam-4827	105	4	monoid	monoid	NOUN
ejpam-4827	105	5	ring	ring	NOUN
ejpam-4827	105	6	r∗m	r∗m	PRON
ejpam-4827	105	7	is	be	AUX
ejpam-4827	105	8	a	a	DET
ejpam-4827	105	9	strongly	strongly	ADV
ejpam-4827	105	10	n	n	ADP
ejpam-4827	105	11	-reflexive	-reflexive	NOUN
ejpam-4827	105	12	,	,	PUNCT
ejpam-4827	105	13	then	then	ADV
ejpam-4827	105	14	r	r	NOUN
ejpam-4827	105	15	is	be	AUX
ejpam-4827	105	16	σ	σ	NOUN
ejpam-4827	105	17	-	-	PUNCT
ejpam-4827	105	18	skew	skew	NOUN
ejpam-4827	105	19	strongly	strongly	ADV
ejpam-4827	105	20	m	m	VERB
ejpam-4827	105	21	-reflexive	-reflexive	ADJ
ejpam-4827	105	22	.	.	PUNCT
ejpam-4827	106	1	proof	proof	NOUN
ejpam-4827	106	2	.	.	PUNCT
ejpam-4827	107	1	let	let	VERB
ejpam-4827	107	2	φ	φ	PROPN
ejpam-4827	107	3	=	=	SYM
ejpam-4827	107	4	∑m	∑m	PROPN
ejpam-4827	107	5	i=1	i=1	PROPN
ejpam-4827	107	6	aigi	aigi	PROPN
ejpam-4827	107	7	ψ	ψ	X
ejpam-4827	107	8	=	=	SYM
ejpam-4827	107	9	∑n	∑n	PROPN
ejpam-4827	107	10	1	1	NUM
ejpam-4827	107	11	bjgj	bjgj	NOUN
ejpam-4827	107	12	be	be	VERB
ejpam-4827	107	13	nonzero	nonzero	ADJ
ejpam-4827	107	14	elements	element	NOUN
ejpam-4827	107	15	in	in	ADP
ejpam-4827	107	16	r	r	NOUN
ejpam-4827	107	17	∗	∗	NOUN
ejpam-4827	107	18	m	m	VERB
ejpam-4827	107	19	such	such	ADJ
ejpam-4827	107	20	that	that	PRON
ejpam-4827	107	21	φϕψ	φϕψ	NOUN
ejpam-4827	107	22	=	=	SYM
ejpam-4827	107	23	0	0	PROPN
ejpam-4827	107	24	for	for	ADP
ejpam-4827	107	25	any	any	DET
ejpam-4827	107	26	ϕ	ϕ	NOUN
ejpam-4827	107	27	=	=	SYM
ejpam-4827	107	28	∑v	∑v	PROPN
ejpam-4827	107	29	r=1	r=1	NOUN
ejpam-4827	107	30	ℓrgr	ℓrgr	NOUN
ejpam-4827	107	31	∈	∈	NOUN
ejpam-4827	107	32	r	r	NOUN
ejpam-4827	107	33	∗m	∗m	NOUN
ejpam-4827	107	34	.	.	PUNCT
ejpam-4827	108	1	then	then	ADV
ejpam-4827	108	2	,	,	PUNCT
ejpam-4827	108	3	for	for	ADP
ejpam-4827	108	4	each	each	DET
ejpam-4827	108	5	1	1	NUM
ejpam-4827	108	6	≤	≤	NUM
ejpam-4827	108	7	k	k	NOUN
ejpam-4827	108	8	≤	≤	NUM
ejpam-4827	108	9	m	m	VERB
ejpam-4827	108	10	+	+	X
ejpam-4827	108	11	v	v	ADJ
ejpam-4827	108	12	+	+	CCONJ
ejpam-4827	108	13	n	n	CCONJ
ejpam-4827	108	14	,	,	PUNCT
ejpam-4827	108	15	we	we	PRON
ejpam-4827	108	16	have	have	VERB
ejpam-4827	108	17	ck	ck	NOUN
ejpam-4827	108	18	=	=	NOUN
ejpam-4827	108	19	∑	∑	PUNCT
ejpam-4827	108	20	i+r+j	i+r+j	NOUN
ejpam-4827	108	21	=	=	PROPN
ejpam-4827	108	22	k	k	PROPN
ejpam-4827	108	23	aiσgi(ℓrσs(bj	aiσgi(ℓrσs(bj	PROPN
ejpam-4827	108	24	)	)	PUNCT
ejpam-4827	108	25	)	)	PUNCT
ejpam-4827	109	1	=	=	SYM
ejpam-4827	109	2	0	0	NUM
ejpam-4827	110	1	for	for	ADP
ejpam-4827	110	2	s	s	PROPN
ejpam-4827	110	3	∈	∈	PROPN
ejpam-4827	110	4	m.	m.	NOUN
ejpam-4827	110	5	now	now	ADV
ejpam-4827	110	6	,	,	PUNCT
ejpam-4827	110	7	let	let	VERB
ejpam-4827	110	8	h	h	PRON
ejpam-4827	110	9	∈	∈	PROPN
ejpam-4827	110	10	n	n	PRON
ejpam-4827	110	11	such	such	ADJ
ejpam-4827	110	12	that	that	DET
ejpam-4827	110	13	o(h	o(h	PROPN
ejpam-4827	110	14	)	)	PUNCT
ejpam-4827	111	1	=	=	SYM
ejpam-4827	111	2	∞	∞	PROPN
ejpam-4827	111	3	and	and	CCONJ
ejpam-4827	111	4	define	define	VERB
ejpam-4827	111	5	f	f	NOUN
ejpam-4827	111	6	,	,	PUNCT
ejpam-4827	111	7	g	g	PROPN
ejpam-4827	111	8	∈	∈	PROPN
ejpam-4827	111	9	(	(	PUNCT
ejpam-4827	111	10	r	r	NOUN
ejpam-4827	111	11	∗m)[n	∗m)[n	NOUN
ejpam-4827	111	12	]	]	PUNCT
ejpam-4827	111	13	as	as	ADP
ejpam-4827	111	14	in	in	ADP
ejpam-4827	111	15	the	the	DET
ejpam-4827	111	16	following	following	NOUN
ejpam-4827	111	17	:	:	PUNCT
ejpam-4827	112	1	f	f	X
ejpam-4827	112	2	=	=	PUNCT
ejpam-4827	112	3	(	(	PUNCT
ejpam-4827	112	4	a1em	a1em	X
ejpam-4827	112	5	)	)	PUNCT
ejpam-4827	112	6	en	en	X
ejpam-4827	112	7	+	+	CCONJ
ejpam-4827	112	8	(	(	PUNCT
ejpam-4827	112	9	a1g)h	a1g)h	X
ejpam-4827	112	10	+	+	X
ejpam-4827	112	11	(	(	PUNCT
ejpam-4827	112	12	a2g2)h2	a2g2)h2	NOUN
ejpam-4827	112	13	+	+	X
ejpam-4827	112	14	·	·	PUNCT
ejpam-4827	112	15	·	·	PUNCT
ejpam-4827	112	16	·	·	PUNCT
ejpam-4827	113	1	+	+	CCONJ
ejpam-4827	113	2	(	(	PUNCT
ejpam-4827	113	3	amgm)hm	amgm)hm	INTJ
ejpam-4827	113	4	and	and	CCONJ
ejpam-4827	113	5	g	g	NOUN
ejpam-4827	113	6	=	=	SYM
ejpam-4827	113	7	(	(	PUNCT
ejpam-4827	113	8	b1em	b1em	X
ejpam-4827	113	9	)	)	PUNCT
ejpam-4827	113	10	en	en	X
ejpam-4827	113	11	+	+	CCONJ
ejpam-4827	113	12	(	(	PUNCT
ejpam-4827	113	13	b1g)h	b1g)h	X
ejpam-4827	113	14	+	+	X
ejpam-4827	113	15	(	(	PUNCT
ejpam-4827	113	16	b2g2)h2	b2g2)h2	NOUN
ejpam-4827	113	17	+	+	X
ejpam-4827	113	18	·	·	PUNCT
ejpam-4827	113	19	·	·	PUNCT
ejpam-4827	113	20	·	·	PUNCT
ejpam-4827	114	1	+	+	CCONJ
ejpam-4827	114	2	(	(	PUNCT
ejpam-4827	114	3	bngn)hn	bngn)hn	NOUN
ejpam-4827	114	4	.	.	PUNCT
ejpam-4827	115	1	since	since	SCONJ
ejpam-4827	115	2	φ	φ	PROPN
ejpam-4827	115	3	,	,	PUNCT
ejpam-4827	115	4	ϕ	ϕ	PROPN
ejpam-4827	115	5	and	and	CCONJ
ejpam-4827	115	6	ψ	ψ	NOUN
ejpam-4827	115	7	are	be	AUX
ejpam-4827	115	8	nonzero	nonzero	NOUN
ejpam-4827	115	9	in	in	ADP
ejpam-4827	115	10	r	r	NOUN
ejpam-4827	115	11	∗	∗	NOUN
ejpam-4827	115	12	m	m	NOUN
ejpam-4827	115	13	,	,	PUNCT
ejpam-4827	115	14	so	so	SCONJ
ejpam-4827	115	15	f	f	PROPN
ejpam-4827	115	16	and	and	CCONJ
ejpam-4827	115	17	g	g	PROPN
ejpam-4827	115	18	are	be	AUX
ejpam-4827	115	19	nonzero	nonzero	NOUN
ejpam-4827	115	20	elements	element	NOUN
ejpam-4827	115	21	in	in	ADP
ejpam-4827	115	22	(	(	PUNCT
ejpam-4827	115	23	r	r	NOUN
ejpam-4827	115	24	∗	∗	NOUN
ejpam-4827	115	25	m)[n	m)[n	PROPN
ejpam-4827	115	26	]	]	PUNCT
ejpam-4827	115	27	.	.	PUNCT
ejpam-4827	116	1	moreover	moreover	ADV
ejpam-4827	116	2	,	,	PUNCT
ejpam-4827	116	3	from	from	ADP
ejpam-4827	116	4	φϕψ	φϕψ	NOUN
ejpam-4827	116	5	=	=	SYM
ejpam-4827	116	6	0	0	NUM
ejpam-4827	116	7	and	and	CCONJ
ejpam-4827	116	8	compatibility	compatibility	NOUN
ejpam-4827	116	9	of	of	ADP
ejpam-4827	116	10	σρ	σρ	NOUN
ejpam-4827	116	11	=	=	SYM
ejpam-4827	116	12	ψ	ψ	NOUN
ejpam-4827	116	13	,	,	PUNCT
ejpam-4827	116	14	one	one	PRON
ejpam-4827	116	15	can	can	AUX
ejpam-4827	116	16	easily	easily	ADV
ejpam-4827	116	17	obtain	obtain	VERB
ejpam-4827	116	18	that	that	DET
ejpam-4827	116	19	fhg	fhg	NOUN
ejpam-4827	116	20	=	=	NOUN
ejpam-4827	116	21	0	0	NUM
ejpam-4827	116	22	for	for	ADP
ejpam-4827	116	23	any	any	DET
ejpam-4827	116	24	h	h	NOUN
ejpam-4827	116	25	∈	∈	NOUN
ejpam-4827	116	26	(	(	PUNCT
ejpam-4827	116	27	r∗m)[n	r∗m)[n	PROPN
ejpam-4827	116	28	]	]	X
ejpam-4827	116	29	.	.	PUNCT
ejpam-4827	117	1	since	since	SCONJ
ejpam-4827	117	2	the	the	DET
ejpam-4827	117	3	skew	skew	ADJ
ejpam-4827	117	4	monoid	monoid	NOUN
ejpam-4827	117	5	ring	ring	NOUN
ejpam-4827	117	6	r∗m	r∗m	NOUN
ejpam-4827	117	7	is	be	AUX
ejpam-4827	117	8	stronglyn	stronglyn	ADJ
ejpam-4827	117	9	-reflexive	-reflexive	NOUN
ejpam-4827	117	10	.	.	PUNCT
ejpam-4827	118	1	then	then	ADV
ejpam-4827	118	2	,	,	PUNCT
ejpam-4827	118	3	aictbj	aictbj	ADJ
ejpam-4827	118	4	=	=	NOUN
ejpam-4827	118	5	0	0	NUM
ejpam-4827	118	6	for	for	ADP
ejpam-4827	118	7	each	each	DET
ejpam-4827	118	8	t	t	NOUN
ejpam-4827	118	9	and	and	CCONJ
ejpam-4827	118	10	so	so	ADV
ejpam-4827	118	11	aiσgi(rσs(bj	aiσgi(rσs(bj	NOUN
ejpam-4827	118	12	)	)	PUNCT
ejpam-4827	118	13	)	)	PUNCT
ejpam-4827	119	1	=	=	SYM
ejpam-4827	119	2	0	0	NUM
ejpam-4827	119	3	for	for	ADP
ejpam-4827	119	4	each	each	DET
ejpam-4827	119	5	1	1	NUM
ejpam-4827	119	6	≤	≤	NUM
ejpam-4827	119	7	i	i	PRON
ejpam-4827	120	1	≤	≤	NOUN
ejpam-4827	120	2	m	m	VERB
ejpam-4827	120	3	and	and	CCONJ
ejpam-4827	120	4	1	1	NUM
ejpam-4827	120	5	≤	≤	NUM
ejpam-4827	120	6	j	j	PROPN
ejpam-4827	120	7	≤	≤	NUM
ejpam-4827	120	8	n	n	CCONJ
ejpam-4827	120	9	,	,	PUNCT
ejpam-4827	120	10	s	s	PROPN
ejpam-4827	120	11	∈	∈	NOUN
ejpam-4827	120	12	m.	m.	NOUN
ejpam-4827	120	13	by	by	ADP
ejpam-4827	120	14	a	a	DET
ejpam-4827	120	15	compatible	compatible	ADJ
ejpam-4827	120	16	automorphism	automorphism	NOUN
ejpam-4827	120	17	,	,	PUNCT
ejpam-4827	120	18	we	we	PRON
ejpam-4827	120	19	have	have	VERB
ejpam-4827	120	20	bjσgj	bjσgj	NOUN
ejpam-4827	120	21	(	(	PUNCT
ejpam-4827	120	22	rσs(ai	rσs(ai	NOUN
ejpam-4827	120	23	)	)	PUNCT
ejpam-4827	120	24	)	)	PUNCT
ejpam-4827	121	1	=	=	PUNCT
ejpam-4827	121	2	0	0	X
ejpam-4827	121	3	.	.	PUNCT
ejpam-4827	122	1	therefore	therefore	ADV
ejpam-4827	122	2	,	,	PUNCT
ejpam-4827	122	3	r	r	NOUN
ejpam-4827	122	4	is	be	AUX
ejpam-4827	122	5	σ	σ	NOUN
ejpam-4827	122	6	-	-	PUNCT
ejpam-4827	122	7	skew	skew	NOUN
ejpam-4827	122	8	strongly	strongly	ADV
ejpam-4827	122	9	m	m	VERB
ejpam-4827	122	10	-reflexive	-reflexive	ADJ
ejpam-4827	122	11	.	.	PUNCT
ejpam-4827	123	1	an	an	DET
ejpam-4827	123	2	ideal	ideal	ADJ
ejpam-4827	123	3	i	i	PRON
ejpam-4827	123	4	of	of	ADP
ejpam-4827	123	5	r	r	NOUN
ejpam-4827	123	6	is	be	AUX
ejpam-4827	123	7	said	say	VERB
ejpam-4827	123	8	to	to	PART
ejpam-4827	123	9	be	be	AUX
ejpam-4827	123	10	right	right	ADJ
ejpam-4827	123	11	s	s	NOUN
ejpam-4827	123	12	-	-	NOUN
ejpam-4827	123	13	unital	unital	ADJ
ejpam-4827	123	14	if	if	SCONJ
ejpam-4827	123	15	,	,	PUNCT
ejpam-4827	123	16	for	for	ADP
ejpam-4827	123	17	each	each	DET
ejpam-4827	123	18	a	a	DET
ejpam-4827	123	19	∈	∈	NOUN
ejpam-4827	123	20	i	i	PRON
ejpam-4827	123	21	there	there	PRON
ejpam-4827	123	22	exist	exist	VERB
ejpam-4827	123	23	an	an	DET
ejpam-4827	123	24	element	element	NOUN
ejpam-4827	123	25	e	e	NOUN
ejpam-4827	123	26	∈	∈	PROPN
ejpam-4827	123	27	i	i	PRON
ejpam-4827	123	28	such	such	ADJ
ejpam-4827	123	29	that	that	SCONJ
ejpam-4827	123	30	ae	ae	PROPN
ejpam-4827	123	31	=	=	NOUN
ejpam-4827	123	32	a.	a.	NOUN
ejpam-4827	123	33	note	note	VERB
ejpam-4827	123	34	that	that	SCONJ
ejpam-4827	123	35	if	if	SCONJ
ejpam-4827	123	36	i	i	PRON
ejpam-4827	123	37	and	and	CCONJ
ejpam-4827	123	38	j	j	PROPN
ejpam-4827	123	39	are	be	AUX
ejpam-4827	123	40	right	right	ADJ
ejpam-4827	123	41	s	s	ADJ
ejpam-4827	123	42	-	-	ADJ
ejpam-4827	123	43	unital	unital	ADJ
ejpam-4827	123	44	ideals	ideal	NOUN
ejpam-4827	123	45	,	,	PUNCT
ejpam-4827	123	46	then	then	ADV
ejpam-4827	123	47	so	so	ADV
ejpam-4827	123	48	is	be	AUX
ejpam-4827	123	49	i	i	PRON
ejpam-4827	123	50	∩	∩	ADJ
ejpam-4827	123	51	j	j	PROPN
ejpam-4827	123	52	(	(	PUNCT
ejpam-4827	123	53	if	if	SCONJ
ejpam-4827	123	54	a	a	DET
ejpam-4827	123	55	∈	∈	X
ejpam-4827	123	56	i	i	PROPN
ejpam-4827	123	57	∩	∩	PROPN
ejpam-4827	123	58	j	j	PROPN
ejpam-4827	123	59	,	,	PUNCT
ejpam-4827	123	60	then	then	ADV
ejpam-4827	123	61	a	a	DET
ejpam-4827	123	62	∈	∈	NOUN
ejpam-4827	123	63	aij	aij	NOUN
ejpam-4827	123	64	⊆	⊆	NUM
ejpam-4827	123	65	a(i	a(i	NOUN
ejpam-4827	123	66	∩	∩	ADJ
ejpam-4827	123	67	j	j	NOUN
ejpam-4827	123	68	)	)	PUNCT
ejpam-4827	123	69	)	)	PUNCT
ejpam-4827	123	70	.	.	PUNCT
ejpam-4827	124	1	we	we	PRON
ejpam-4827	124	2	say	say	VERB
ejpam-4827	124	3	a	a	DET
ejpam-4827	124	4	ring	ring	NOUN
ejpam-4827	124	5	r	r	NOUN
ejpam-4827	124	6	is	be	AUX
ejpam-4827	124	7	a	a	DET
ejpam-4827	124	8	left	left	ADJ
ejpam-4827	124	9	app	app	NOUN
ejpam-4827	124	10	-ring	-ring	PROPN
ejpam-4827	124	11	if	if	SCONJ
ejpam-4827	124	12	the	the	DET
ejpam-4827	124	13	left	left	ADJ
ejpam-4827	124	14	annihilator	annihilator	PROPN
ejpam-4827	124	15	lr(ra	lr(ra	PROPN
ejpam-4827	124	16	)	)	PUNCT
ejpam-4827	124	17	is	be	AUX
ejpam-4827	124	18	right	right	ADJ
ejpam-4827	124	19	s	s	NOUN
ejpam-4827	124	20	-	-	NOUN
ejpam-4827	124	21	unital	unital	ADJ
ejpam-4827	124	22	as	as	ADP
ejpam-4827	124	23	an	an	DET
ejpam-4827	124	24	ideal	ideal	NOUN
ejpam-4827	124	25	of	of	ADP
ejpam-4827	124	26	r	r	NOUN
ejpam-4827	124	27	for	for	ADP
ejpam-4827	124	28	any	any	DET
ejpam-4827	124	29	element	element	NOUN
ejpam-4827	124	30	a	a	DET
ejpam-4827	124	31	∈	∈	PROPN
ejpam-4827	124	32	r.	r.	NOUN
ejpam-4827	124	33	the	the	DET
ejpam-4827	124	34	following	follow	VERB
ejpam-4827	124	35	result	result	NOUN
ejpam-4827	124	36	follows	follow	VERB
ejpam-4827	124	37	from	from	ADP
ejpam-4827	124	38	tominaga	tominaga	NOUN
ejpam-4827	124	39	theorem	theorem	VERB
ejpam-4827	124	40	1	1	NUM
ejpam-4827	125	1	[	[	X
ejpam-4827	125	2	23	23	NUM
ejpam-4827	125	3	]	]	PUNCT
ejpam-4827	125	4	.	.	PUNCT
ejpam-4827	126	1	lemma	lemma	PROPN
ejpam-4827	126	2	1	1	NUM
ejpam-4827	126	3	.	.	PUNCT
ejpam-4827	127	1	an	an	DET
ejpam-4827	127	2	ideal	ideal	ADJ
ejpam-4827	127	3	i	i	PRON
ejpam-4827	127	4	of	of	ADP
ejpam-4827	127	5	a	a	DET
ejpam-4827	127	6	ring	ring	NOUN
ejpam-4827	127	7	r	r	NOUN
ejpam-4827	127	8	is	be	AUX
ejpam-4827	127	9	left	leave	VERB
ejpam-4827	127	10	(	(	PUNCT
ejpam-4827	127	11	resp	resp	NOUN
ejpam-4827	127	12	.	.	PUNCT
ejpam-4827	128	1	right	right	ADJ
ejpam-4827	128	2	)	)	PUNCT
ejpam-4827	129	1	s	s	NOUN
ejpam-4827	129	2	-	-	NOUN
ejpam-4827	129	3	unital	unital	ADJ
ejpam-4827	129	4	if	if	SCONJ
ejpam-4827	129	5	and	and	CCONJ
ejpam-4827	129	6	only	only	ADV
ejpam-4827	129	7	if	if	SCONJ
ejpam-4827	129	8	for	for	ADP
ejpam-4827	129	9	any	any	DET
ejpam-4827	129	10	finitely	finitely	ADV
ejpam-4827	129	11	many	many	ADJ
ejpam-4827	129	12	elements	element	NOUN
ejpam-4827	129	13	a1	a1	NOUN
ejpam-4827	129	14	,	,	PUNCT
ejpam-4827	129	15	a2	a2	PROPN
ejpam-4827	129	16	,	,	PUNCT
ejpam-4827	129	17	.	.	PUNCT
ejpam-4827	129	18	.	.	PUNCT
ejpam-4827	130	1	.	.	PUNCT
ejpam-4827	131	1	,	,	PUNCT
ejpam-4827	131	2	an	an	DET
ejpam-4827	131	3	∈	∈	PROPN
ejpam-4827	131	4	i	i	PRON
ejpam-4827	131	5	,	,	PUNCT
ejpam-4827	131	6	there	there	PRON
ejpam-4827	131	7	exists	exist	VERB
ejpam-4827	131	8	an	an	DET
ejpam-4827	131	9	element	element	NOUN
ejpam-4827	131	10	e	e	NOUN
ejpam-4827	131	11	∈	∈	PROPN
ejpam-4827	131	12	i	i	PRON
ejpam-4827	131	13	such	such	ADJ
ejpam-4827	131	14	that	that	SCONJ
ejpam-4827	131	15	ai	ai	VERB
ejpam-4827	131	16	=	=	NOUN
ejpam-4827	131	17	eai(resp	eai(resp	NOUN
ejpam-4827	131	18	.	.	PUNCT
ejpam-4827	132	1	ai	ai	VERB
ejpam-4827	132	2	=	=	SYM
ejpam-4827	132	3	aie	aie	PROPN
ejpam-4827	132	4	)	)	PUNCT
ejpam-4827	132	5	for	for	ADP
ejpam-4827	132	6	each	each	DET
ejpam-4827	132	7	i	i	NOUN
ejpam-4827	132	8	=	=	NOUN
ejpam-4827	132	9	1	1	NUM
ejpam-4827	132	10	,	,	PUNCT
ejpam-4827	132	11	2	2	NUM
ejpam-4827	132	12	,	,	PUNCT
ejpam-4827	132	13	.	.	PUNCT
ejpam-4827	132	14	.	.	PUNCT
ejpam-4827	133	1	.	.	PUNCT
ejpam-4827	134	1	,	,	PUNCT
ejpam-4827	134	2	n.	n.	PROPN
ejpam-4827	134	3	lemma	lemma	PROPN
ejpam-4827	134	4	2	2	X
ejpam-4827	134	5	.	.	PUNCT
ejpam-4827	135	1	(	(	PUNCT
ejpam-4827	135	2	lemma	lemma	PROPN
ejpam-4827	135	3	1.13	1.13	NUM
ejpam-4827	135	4	[	[	X
ejpam-4827	135	5	14	14	NUM
ejpam-4827	135	6	]	]	PUNCT
ejpam-4827	135	7	)	)	PUNCT
ejpam-4827	135	8	.	.	PUNCT
ejpam-4827	136	1	let	let	VERB
ejpam-4827	136	2	m	m	PRON
ejpam-4827	136	3	and	and	CCONJ
ejpam-4827	136	4	n	n	ADV
ejpam-4827	136	5	be	be	VERB
ejpam-4827	136	6	u.p.-monoids	u.p.-monoid	NOUN
ejpam-4827	136	7	.	.	PUNCT
ejpam-4827	137	1	then	then	ADV
ejpam-4827	137	2	so	so	ADV
ejpam-4827	137	3	is	be	AUX
ejpam-4827	137	4	the	the	DET
ejpam-4827	137	5	monoid	monoid	PROPN
ejpam-4827	137	6	m	m	PROPN
ejpam-4827	137	7	×n	×n	PROPN
ejpam-4827	137	8	.	.	PUNCT
ejpam-4827	138	1	lemma	lemma	PROPN
ejpam-4827	138	2	3	3	NUM
ejpam-4827	138	3	.	.	PUNCT
ejpam-4827	139	1	(	(	PUNCT
ejpam-4827	139	2	example	example	NOUN
ejpam-4827	139	3	2.2	2.2	NUM
ejpam-4827	140	1	[	[	X
ejpam-4827	140	2	7	7	NUM
ejpam-4827	140	3	]	]	NUM
ejpam-4827	140	4	)	)	PUNCT
ejpam-4827	140	5	.	.	PUNCT
ejpam-4827	141	1	let	let	VERB
ejpam-4827	141	2	r	r	PRON
ejpam-4827	141	3	be	be	AUX
ejpam-4827	141	4	a	a	DET
ejpam-4827	141	5	ring	ring	NOUN
ejpam-4827	141	6	and	and	CCONJ
ejpam-4827	141	7	m	m	AUX
ejpam-4827	141	8	be	be	AUX
ejpam-4827	141	9	a	a	DET
ejpam-4827	141	10	monoid	monoid	NOUN
ejpam-4827	141	11	with	with	ADP
ejpam-4827	141	12	an	an	DET
ejpam-4827	141	13	element	element	NOUN
ejpam-4827	141	14	of	of	ADP
ejpam-4827	141	15	a	a	DET
ejpam-4827	141	16	finite	finite	ADJ
ejpam-4827	141	17	order	order	NOUN
ejpam-4827	141	18	n	n	PRON
ejpam-4827	141	19	≥	≥	NOUN
ejpam-4827	141	20	2	2	X
ejpam-4827	141	21	.	.	PUNCT
ejpam-4827	142	1	let	let	VERB
ejpam-4827	142	2	φ	φ	PROPN
ejpam-4827	142	3	=	=	PROPN
ejpam-4827	142	4	σn−1	σn−1	PROPN
ejpam-4827	142	5	i=0	i=0	PROPN
ejpam-4827	142	6	g	g	PROPN
ejpam-4827	142	7	i	i	PROPN
ejpam-4827	142	8	and	and	CCONJ
ejpam-4827	142	9	ψ	ψ	X
ejpam-4827	142	10	=	=	SYM
ejpam-4827	142	11	e−	e−	NUM
ejpam-4827	142	12	g	g	NOUN
ejpam-4827	142	13	,	,	PUNCT
ejpam-4827	142	14	where	where	SCONJ
ejpam-4827	142	15	|g|	|g|	PROPN
ejpam-4827	142	16	=	=	SYM
ejpam-4827	142	17	n.	n.	PROPN
ejpam-4827	142	18	then	then	ADV
ejpam-4827	142	19	φψ	φψ	VERB
ejpam-4827	142	20	=	=	PUNCT
ejpam-4827	142	21	0	0	PUNCT
ejpam-4827	143	1	and	and	CCONJ
ejpam-4827	143	2	so	so	ADV
ejpam-4827	143	3	r	r	NOUN
ejpam-4827	143	4	is	be	AUX
ejpam-4827	143	5	not	not	PART
ejpam-4827	143	6	m	m	NOUN
ejpam-4827	143	7	-nil	-nil	NOUN
ejpam-4827	143	8	-	-	PUNCT
ejpam-4827	143	9	armendariz	armendariz	ADJ
ejpam-4827	143	10	.	.	PUNCT
ejpam-4827	144	1	lemma	lemma	PROPN
ejpam-4827	144	2	4	4	X
ejpam-4827	144	3	.	.	PUNCT
ejpam-4827	145	1	(	(	PUNCT
ejpam-4827	145	2	lemma	lemma	PROPN
ejpam-4827	145	3	1.1	1.1	NUM
ejpam-4827	145	4	[	[	X
ejpam-4827	145	5	5	5	NUM
ejpam-4827	145	6	]	]	PUNCT
ejpam-4827	145	7	)	)	PUNCT
ejpam-4827	145	8	.	.	PUNCT
ejpam-4827	146	1	assume	assume	VERB
ejpam-4827	146	2	m	m	PROPN
ejpam-4827	146	3	is	be	AUX
ejpam-4827	146	4	a	a	DET
ejpam-4827	146	5	u.p.-monoid	u.p.-monoid	NOUN
ejpam-4827	146	6	.	.	PUNCT
ejpam-4827	147	1	then	then	ADV
ejpam-4827	147	2	m	m	PROPN
ejpam-4827	147	3	is	be	AUX
ejpam-4827	147	4	cancellative	cancellative	ADJ
ejpam-4827	147	5	(	(	PUNCT
ejpam-4827	147	6	i.e.	i.e.	X
ejpam-4827	147	7	,	,	PUNCT
ejpam-4827	147	8	for	for	ADP
ejpam-4827	147	9	g	g	PROPN
ejpam-4827	147	10	,	,	PUNCT
ejpam-4827	147	11	h	h	NOUN
ejpam-4827	147	12	,	,	PUNCT
ejpam-4827	147	13	x	x	SYM
ejpam-4827	147	14	∈m	∈m	NOUN
ejpam-4827	147	15	,	,	PUNCT
ejpam-4827	147	16	if	if	SCONJ
ejpam-4827	147	17	gx	gx	PROPN
ejpam-4827	147	18	=	=	PROPN
ejpam-4827	147	19	hx	hx	PROPN
ejpam-4827	147	20	or	or	CCONJ
ejpam-4827	147	21	xg	xg	PROPN
ejpam-4827	147	22	=	=	SYM
ejpam-4827	147	23	xh	xh	PROPN
ejpam-4827	147	24	,	,	PUNCT
ejpam-4827	147	25	then	then	ADV
ejpam-4827	147	26	g	g	PROPN
ejpam-4827	147	27	=	=	PUNCT
ejpam-4827	147	28	h	h	NOUN
ejpam-4827	147	29	)	)	PUNCT
ejpam-4827	147	30	.	.	PUNCT
ejpam-4827	148	1	e.	e.	PROPN
ejpam-4827	148	2	ali	ali	PROPN
ejpam-4827	148	3	/	/	SYM
ejpam-4827	148	4	eur	eur	PROPN
ejpam-4827	148	5	.	.	PUNCT
ejpam-4827	149	1	j.	j.	PROPN
ejpam-4827	149	2	pure	pure	PROPN
ejpam-4827	149	3	appl	appl	PROPN
ejpam-4827	149	4	.	.	PROPN
ejpam-4827	149	5	math	math	PROPN
ejpam-4827	149	6	,	,	PUNCT
ejpam-4827	149	7	16	16	NUM
ejpam-4827	149	8	(	(	PUNCT
ejpam-4827	149	9	3	3	NUM
ejpam-4827	149	10	)	)	PUNCT
ejpam-4827	149	11	(	(	PUNCT
ejpam-4827	149	12	2023	2023	NUM
ejpam-4827	149	13	)	)	PUNCT
ejpam-4827	149	14	,	,	PUNCT
ejpam-4827	149	15	1878	1878	NUM
ejpam-4827	149	16	-	-	SYM
ejpam-4827	149	17	1893	1893	NUM
ejpam-4827	149	18	1882	1882	NUM
ejpam-4827	149	19	proposition	proposition	NOUN
ejpam-4827	149	20	1	1	NUM
ejpam-4827	149	21	.	.	PUNCT
ejpam-4827	150	1	let	let	VERB
ejpam-4827	150	2	r	r	PRON
ejpam-4827	150	3	be	be	AUX
ejpam-4827	150	4	a	a	DET
ejpam-4827	150	5	ring	ring	NOUN
ejpam-4827	150	6	,	,	PUNCT
ejpam-4827	150	7	m	m	VERB
ejpam-4827	150	8	be	be	VERB
ejpam-4827	150	9	a	a	DET
ejpam-4827	150	10	u.p.-monoid	u.p.-monoid	NOUN
ejpam-4827	150	11	and	and	CCONJ
ejpam-4827	150	12	σ	σ	NOUN
ejpam-4827	150	13	:	:	PUNCT
ejpam-4827	150	14	m	m	PROPN
ejpam-4827	150	15	→	→	SYM
ejpam-4827	150	16	aut(r	aut(r	PROPN
ejpam-4827	150	17	)	)	PUNCT
ejpam-4827	150	18	a	a	DET
ejpam-4827	150	19	compatible	compatible	ADJ
ejpam-4827	150	20	monoid	monoid	NOUN
ejpam-4827	150	21	homomorphism	homomorphism	NOUN
ejpam-4827	150	22	.	.	PUNCT
ejpam-4827	151	1	if	if	SCONJ
ejpam-4827	151	2	r	r	NOUN
ejpam-4827	151	3	is	be	AUX
ejpam-4827	151	4	a	a	DET
ejpam-4827	151	5	reduced	reduce	VERB
ejpam-4827	151	6	left	leave	VERB
ejpam-4827	151	7	app	app	NOUN
ejpam-4827	151	8	-ring	-ring	PROPN
ejpam-4827	151	9	,	,	PUNCT
ejpam-4827	151	10	then	then	ADV
ejpam-4827	151	11	r	r	NOUN
ejpam-4827	151	12	is	be	AUX
ejpam-4827	151	13	σ	σ	NOUN
ejpam-4827	151	14	-	-	PUNCT
ejpam-4827	151	15	skew	skew	NOUN
ejpam-4827	151	16	strongly	strongly	ADV
ejpam-4827	151	17	m	m	VERB
ejpam-4827	151	18	reflexive	reflexive	ADJ
ejpam-4827	151	19	.	.	PUNCT
ejpam-4827	152	1	proof	proof	NOUN
ejpam-4827	152	2	.	.	PUNCT
ejpam-4827	153	1	suppose	suppose	VERB
ejpam-4827	153	2	φ	φ	X
ejpam-4827	153	3	,	,	PUNCT
ejpam-4827	153	4	ψ	ψ	ADP
ejpam-4827	153	5	∈	∈	PROPN
ejpam-4827	153	6	r∗m	r∗m	NOUN
ejpam-4827	153	7	such	such	ADJ
ejpam-4827	153	8	that	that	SCONJ
ejpam-4827	153	9	φ(r∗m)ψ	φ(r∗m)ψ	VERB
ejpam-4827	153	10	=	=	SYM
ejpam-4827	153	11	0	0	NUM
ejpam-4827	153	12	implies	imply	VERB
ejpam-4827	153	13	that	that	SCONJ
ejpam-4827	153	14	biσgi(rσs(aj	biσgi(rσs(aj	NOUN
ejpam-4827	153	15	)	)	PUNCT
ejpam-4827	153	16	)	)	PUNCT
ejpam-4827	154	1	=	=	SYM
ejpam-4827	154	2	0	0	NUM
ejpam-4827	154	3	for	for	ADP
ejpam-4827	154	4	any	any	DET
ejpam-4827	154	5	s	s	NOUN
ejpam-4827	154	6	∈m	∈m	NOUN
ejpam-4827	154	7	,	,	PUNCT
ejpam-4827	154	8	where	where	SCONJ
ejpam-4827	154	9	φ	φ	PROPN
ejpam-4827	154	10	=	=	SYM
ejpam-4827	154	11	b1g1+b2g2	b1g1+b2g2	PROPN
ejpam-4827	154	12	+	+	PROPN
ejpam-4827	154	13	·	·	PUNCT
ejpam-4827	154	14	·	·	PUNCT
ejpam-4827	154	15	·	·	PUNCT
ejpam-4827	154	16	+	+	ADJ
ejpam-4827	154	17	bngn	bngn	NOUN
ejpam-4827	154	18	and	and	CCONJ
ejpam-4827	154	19	ψ	ψ	X
ejpam-4827	154	20	=	=	PUNCT
ejpam-4827	154	21	a1h1+a2h2	a1h1+a2h2	PROPN
ejpam-4827	154	22	+	+	PROPN
ejpam-4827	154	23	·	·	PUNCT
ejpam-4827	154	24	·	·	PUNCT
ejpam-4827	154	25	·	·	PUNCT
ejpam-4827	154	26	+	+	NUM
ejpam-4827	154	27	amhm	amhm	NOUN
ejpam-4827	154	28	∈	∈	PROPN
ejpam-4827	154	29	r∗m	r∗m	NOUN
ejpam-4827	154	30	.	.	PUNCT
ejpam-4827	155	1	we	we	PRON
ejpam-4827	155	2	shall	shall	AUX
ejpam-4827	155	3	prove	prove	VERB
ejpam-4827	155	4	that	that	SCONJ
ejpam-4827	155	5	ψ(r	ψ(r	PROPN
ejpam-4827	155	6	∗m)φ	∗m)φ	VERB
ejpam-4827	155	7	=	=	SYM
ejpam-4827	155	8	0	0	PROPN
ejpam-4827	155	9	,	,	PUNCT
ejpam-4827	155	10	(	(	PUNCT
ejpam-4827	155	11	i.e.	i.e.	X
ejpam-4827	155	12	,	,	PUNCT
ejpam-4827	155	13	ajσhj	ajσhj	ADJ
ejpam-4827	155	14	(	(	PUNCT
ejpam-4827	155	15	rσs(bi	rσs(bi	NUM
ejpam-4827	155	16	)	)	PUNCT
ejpam-4827	155	17	)	)	PUNCT
ejpam-4827	156	1	=	=	SYM
ejpam-4827	156	2	0	0	NUM
ejpam-4827	156	3	for	for	ADP
ejpam-4827	156	4	all	all	DET
ejpam-4827	156	5	i	i	PROPN
ejpam-4827	156	6	,	,	PUNCT
ejpam-4827	156	7	j.	j.	PROPN
ejpam-4827	156	8	let	let	VERB
ejpam-4827	156	9	r	r	PRON
ejpam-4827	156	10	be	be	AUX
ejpam-4827	156	11	an	an	DET
ejpam-4827	156	12	arbitrary	arbitrary	ADJ
ejpam-4827	156	13	element	element	NOUN
ejpam-4827	156	14	of	of	ADP
ejpam-4827	156	15	r.	r.	PROPN
ejpam-4827	156	16	then	then	ADV
ejpam-4827	156	17	we	we	PRON
ejpam-4827	156	18	have	have	VERB
ejpam-4827	156	19	the	the	DET
ejpam-4827	156	20	following	follow	VERB
ejpam-4827	156	21	equation	equation	NOUN
ejpam-4827	156	22	:	:	PUNCT
ejpam-4827	156	23	(	(	PUNCT
ejpam-4827	156	24	b1g1	b1g1	CCONJ
ejpam-4827	156	25	+	+	CCONJ
ejpam-4827	156	26	b2g2	b2g2	PROPN
ejpam-4827	156	27	+	+	CCONJ
ejpam-4827	156	28	·	·	PUNCT
ejpam-4827	156	29	·	·	PUNCT
ejpam-4827	156	30	·	·	PUNCT
ejpam-4827	156	31	+	+	NUM
ejpam-4827	156	32	bngn)(rs)(a1h1	bngn)(rs)(a1h1	PROPN
ejpam-4827	156	33	+	+	CCONJ
ejpam-4827	156	34	a2h2	a2h2	X
ejpam-4827	156	35	+	+	X
ejpam-4827	156	36	·	·	PUNCT
ejpam-4827	156	37	·	·	PUNCT
ejpam-4827	156	38	·	·	PUNCT
ejpam-4827	157	1	+	+	NUM
ejpam-4827	157	2	amhm	amhm	NOUN
ejpam-4827	157	3	)	)	PUNCT
ejpam-4827	157	4	=	=	SYM
ejpam-4827	158	1	0	0	X
ejpam-4827	158	2	.	.	PUNCT
ejpam-4827	158	3	(	(	PUNCT
ejpam-4827	158	4	2.1	2.1	NUM
ejpam-4827	158	5	)	)	PUNCT
ejpam-4827	158	6	we	we	PRON
ejpam-4827	158	7	proceed	proceed	VERB
ejpam-4827	158	8	by	by	ADP
ejpam-4827	158	9	induction	induction	NOUN
ejpam-4827	158	10	on	on	ADP
ejpam-4827	158	11	both	both	CCONJ
ejpam-4827	158	12	n	n	PROPN
ejpam-4827	158	13	and	and	CCONJ
ejpam-4827	158	14	m	m	VERB
ejpam-4827	158	15	for	for	ADP
ejpam-4827	158	16	every	every	DET
ejpam-4827	158	17	s	s	NOUN
ejpam-4827	158	18	∈m	∈m	NOUN
ejpam-4827	158	19	and	and	CCONJ
ejpam-4827	158	20	for	for	ADP
ejpam-4827	158	21	1	1	NUM
ejpam-4827	158	22	≤	≤	NUM
ejpam-4827	158	23	i	i	PRON
ejpam-4827	158	24	≤	≤	PROPN
ejpam-4827	158	25	n	n	CCONJ
ejpam-4827	158	26	,	,	PUNCT
ejpam-4827	158	27	1	1	NUM
ejpam-4827	158	28	≤	≤	NUM
ejpam-4827	158	29	j	j	PROPN
ejpam-4827	158	30	≤	≤	PROPN
ejpam-4827	158	31	m.	m.	NOUN
ejpam-4827	158	32	if	if	SCONJ
ejpam-4827	158	33	m	m	NOUN
ejpam-4827	158	34	=	=	NOUN
ejpam-4827	158	35	1	1	NUM
ejpam-4827	158	36	,	,	PUNCT
ejpam-4827	158	37	then	then	ADV
ejpam-4827	158	38	ψ	ψ	X
ejpam-4827	158	39	=	=	PUNCT
ejpam-4827	158	40	a1h1	a1h1	PROPN
ejpam-4827	158	41	.	.	NOUN
ejpam-4827	158	42	thus	thus	ADV
ejpam-4827	158	43	0	0	X
ejpam-4827	159	1	=	=	SYM
ejpam-4827	159	2	(	(	PUNCT
ejpam-4827	159	3	b1g1	b1g1	SYM
ejpam-4827	159	4	+	+	NUM
ejpam-4827	159	5	b2g2	b2g2	NOUN
ejpam-4827	159	6	+	+	NOUN
ejpam-4827	159	7	·	·	PUNCT
ejpam-4827	159	8	·	·	PUNCT
ejpam-4827	159	9	·	·	PUNCT
ejpam-4827	159	10	+	+	NUM
ejpam-4827	159	11	bngn)(rs)(a1h1	bngn)(rs)(a1h1	NOUN
ejpam-4827	159	12	)	)	PUNCT
ejpam-4827	159	13	=	=	PUNCT
ejpam-4827	159	14	b1σg1(σs(ra1))+	b1σg1(σs(ra1))+	ADJ
ejpam-4827	159	15	b2σg2(σs(ra1	b2σg2(σs(ra1	NOUN
ejpam-4827	159	16	)	)	PUNCT
ejpam-4827	159	17	)	)	PUNCT
ejpam-4827	160	1	+	+	CCONJ
ejpam-4827	160	2	·	·	PUNCT
ejpam-4827	160	3	·	·	PUNCT
ejpam-4827	160	4	·	·	PUNCT
ejpam-4827	160	5	+	+	NUM
ejpam-4827	160	6	bnσgn(σs(ra1	bnσgn(σs(ra1	NOUN
ejpam-4827	160	7	)	)	PUNCT
ejpam-4827	160	8	)	)	PUNCT
ejpam-4827	160	9	for	for	ADP
ejpam-4827	160	10	every	every	DET
ejpam-4827	160	11	r	r	NOUN
ejpam-4827	160	12	∈	∈	PROPN
ejpam-4827	160	13	r.	r.	NOUN
ejpam-4827	160	14	by	by	ADP
ejpam-4827	160	15	lemma	lemma	PROPN
ejpam-4827	160	16	4	4	NUM
ejpam-4827	160	17	,	,	PUNCT
ejpam-4827	160	18	m	m	VERB
ejpam-4827	160	19	is	be	AUX
ejpam-4827	160	20	a	a	DET
ejpam-4827	160	21	cancellative	cancellative	ADJ
ejpam-4827	160	22	monoid	monoid	NOUN
ejpam-4827	160	23	.	.	PUNCT
ejpam-4827	161	1	thus	thus	ADV
ejpam-4827	161	2	gish1	gish1	X
ejpam-4827	161	3	̸=	̸=	PROPN
ejpam-4827	161	4	gjsh1	gjsh1	NOUN
ejpam-4827	161	5	for	for	ADP
ejpam-4827	161	6	gi	gi	NOUN
ejpam-4827	161	7	̸=	̸=	PROPN
ejpam-4827	161	8	gj	gj	NOUN
ejpam-4827	161	9	.	.	PUNCT
ejpam-4827	162	1	then	then	ADV
ejpam-4827	162	2	biσgi(σs(ra1	biσgi(σs(ra1	NUM
ejpam-4827	162	3	)	)	PUNCT
ejpam-4827	162	4	)	)	PUNCT
ejpam-4827	163	1	=	=	PUNCT
ejpam-4827	163	2	0	0	X
ejpam-4827	163	3	.	.	PUNCT
ejpam-4827	164	1	hence	hence	ADV
ejpam-4827	164	2	bi	bi	PROPN
ejpam-4827	164	3	∈	∈	PROPN
ejpam-4827	164	4	ℓr(σs(ra1	ℓr(σs(ra1	PROPN
ejpam-4827	164	5	)	)	PUNCT
ejpam-4827	164	6	)	)	PUNCT
ejpam-4827	164	7	.	.	PUNCT
ejpam-4827	165	1	by	by	ADP
ejpam-4827	165	2	hypothesis	hypothesis	NOUN
ejpam-4827	165	3	,	,	PUNCT
ejpam-4827	165	4	r	r	NOUN
ejpam-4827	165	5	is	be	AUX
ejpam-4827	165	6	left	leave	VERB
ejpam-4827	165	7	app	app	NOUN
ejpam-4827	165	8	,	,	PUNCT
ejpam-4827	165	9	ℓr(ran	ℓr(ran	PROPN
ejpam-4827	165	10	)	)	PUNCT
ejpam-4827	165	11	is	be	AUX
ejpam-4827	165	12	left	leave	VERB
ejpam-4827	165	13	s	s	PART
ejpam-4827	165	14	-	-	NOUN
ejpam-4827	165	15	unital	unital	ADJ
ejpam-4827	165	16	by	by	ADP
ejpam-4827	165	17	lemma	lemma	PROPN
ejpam-4827	165	18	1	1	NUM
ejpam-4827	165	19	.	.	PUNCT
ejpam-4827	166	1	hence	hence	ADV
ejpam-4827	166	2	,	,	PUNCT
ejpam-4827	166	3	there	there	PRON
ejpam-4827	166	4	exist	exist	VERB
ejpam-4827	166	5	em	em	PRON
ejpam-4827	166	6	∈	∈	PROPN
ejpam-4827	166	7	ℓr(ran	ℓr(ran	NOUN
ejpam-4827	166	8	)	)	PUNCT
ejpam-4827	167	1	such	such	ADJ
ejpam-4827	167	2	that	that	DET
ejpam-4827	167	3	anem	anem	PROPN
ejpam-4827	167	4	=	=	PUNCT
ejpam-4827	167	5	an	an	PRON
ejpam-4827	167	6	since	since	SCONJ
ejpam-4827	167	7	σgi	σgi	NOUN
ejpam-4827	167	8	is	be	AUX
ejpam-4827	167	9	an	an	DET
ejpam-4827	167	10	automorphism	automorphism	NOUN
ejpam-4827	167	11	,	,	PUNCT
ejpam-4827	167	12	i	i	PRON
ejpam-4827	167	13	=	=	NOUN
ejpam-4827	167	14	1	1	NUM
ejpam-4827	167	15	,	,	PUNCT
ejpam-4827	167	16	2	2	NUM
ejpam-4827	167	17	,	,	PUNCT
ejpam-4827	167	18	.	.	PUNCT
ejpam-4827	167	19	.	.	PUNCT
ejpam-4827	168	1	.	.	PUNCT
ejpam-4827	169	1	,	,	PUNCT
ejpam-4827	169	2	n.	n.	PROPN
ejpam-4827	169	3	now	now	ADV
ejpam-4827	169	4	suppose	suppose	VERB
ejpam-4827	169	5	that	that	SCONJ
ejpam-4827	169	6	m	m	PROPN
ejpam-4827	169	7	≥	≥	NOUN
ejpam-4827	169	8	2	2	NUM
ejpam-4827	169	9	.	.	PUNCT
ejpam-4827	170	1	since	since	SCONJ
ejpam-4827	170	2	m	m	PROPN
ejpam-4827	170	3	is	be	AUX
ejpam-4827	170	4	u.p.-monoid	u.p.-monoid	ADJ
ejpam-4827	170	5	,	,	PUNCT
ejpam-4827	170	6	there	there	PRON
ejpam-4827	170	7	exists	exist	VERB
ejpam-4827	170	8	p	p	PRON
ejpam-4827	170	9	,	,	PUNCT
ejpam-4827	170	10	q	q	NOUN
ejpam-4827	170	11	with	with	ADP
ejpam-4827	170	12	1	1	NUM
ejpam-4827	170	13	≤	≤	NOUN
ejpam-4827	170	14	p	p	NOUN
ejpam-4827	170	15	≤	≤	ADJ
ejpam-4827	170	16	n	n	CCONJ
ejpam-4827	170	17	and	and	CCONJ
ejpam-4827	170	18	1	1	NUM
ejpam-4827	170	19	≤	≤	NUM
ejpam-4827	170	20	q	q	PROPN
ejpam-4827	170	21	≤	≤	NUM
ejpam-4827	170	22	m	m	VERB
ejpam-4827	170	23	such	such	ADJ
ejpam-4827	170	24	that	that	DET
ejpam-4827	170	25	gpshq	gpshq	NOUN
ejpam-4827	170	26	is	be	AUX
ejpam-4827	170	27	uniquely	uniquely	ADV
ejpam-4827	170	28	presented	present	VERB
ejpam-4827	170	29	by	by	ADP
ejpam-4827	170	30	considering	consider	VERB
ejpam-4827	170	31	two	two	NUM
ejpam-4827	170	32	subsets	subset	NOUN
ejpam-4827	170	33	{	{	PUNCT
ejpam-4827	170	34	g1	g1	PROPN
ejpam-4827	170	35	,	,	PUNCT
ejpam-4827	170	36	g2	g2	PROPN
ejpam-4827	170	37	,	,	PUNCT
ejpam-4827	170	38	.	.	PUNCT
ejpam-4827	170	39	.	.	PUNCT
ejpam-4827	171	1	.	.	PUNCT
ejpam-4827	172	1	,	,	PUNCT
ejpam-4827	172	2	gn	gn	X
ejpam-4827	172	3	}	}	PUNCT
ejpam-4827	172	4	and	and	CCONJ
ejpam-4827	172	5	{	{	PUNCT
ejpam-4827	172	6	sh1	sh1	PROPN
ejpam-4827	172	7	,	,	PUNCT
ejpam-4827	172	8	sh2	sh2	PROPN
ejpam-4827	172	9	,	,	PUNCT
ejpam-4827	172	10	.	.	PUNCT
ejpam-4827	172	11	.	.	PUNCT
ejpam-4827	173	1	.	.	PUNCT
ejpam-4827	174	1	,	,	PUNCT
ejpam-4827	174	2	shm	shm	PROPN
ejpam-4827	174	3	}	}	PUNCT
ejpam-4827	174	4	of	of	ADP
ejpam-4827	174	5	m.	m.	NOUN
ejpam-4827	174	6	thus	thus	ADV
ejpam-4827	174	7	,	,	PUNCT
ejpam-4827	174	8	from	from	ADP
ejpam-4827	174	9	φ(rs)ψ	φ(rs)ψ	NOUN
ejpam-4827	174	10	=	=	SYM
ejpam-4827	174	11	0	0	NUM
ejpam-4827	175	1	it	it	PRON
ejpam-4827	175	2	follows	follow	VERB
ejpam-4827	175	3	that	that	DET
ejpam-4827	175	4	bpσgp(σs(raq	bpσgp(σs(raq	NOUN
ejpam-4827	175	5	)	)	PUNCT
ejpam-4827	175	6	)	)	PUNCT
ejpam-4827	176	1	=	=	SYM
ejpam-4827	176	2	0	0	PUNCT
ejpam-4827	177	1	and	and	CCONJ
ejpam-4827	177	2	so	so	ADV
ejpam-4827	177	3	bpσgp(σs(raq	bpσgp(σs(raq	PROPN
ejpam-4827	177	4	)	)	PUNCT
ejpam-4827	177	5	)	)	PUNCT
ejpam-4827	178	1	=	=	PUNCT
ejpam-4827	178	2	0	0	X
ejpam-4827	178	3	.	.	PUNCT
ejpam-4827	178	4	thus	thus	ADV
ejpam-4827	178	5	σgp(cpσs(raq	σgp(cpσs(raq	ADV
ejpam-4827	178	6	)	)	PUNCT
ejpam-4827	178	7	)	)	PUNCT
ejpam-4827	179	1	=	=	SYM
ejpam-4827	179	2	0	0	NUM
ejpam-4827	180	1	for	for	ADP
ejpam-4827	180	2	c	c	PROPN
ejpam-4827	180	3	∈	∈	PROPN
ejpam-4827	180	4	r	r	NOUN
ejpam-4827	180	5	,	,	PUNCT
ejpam-4827	180	6	which	which	PRON
ejpam-4827	180	7	implies	imply	VERB
ejpam-4827	180	8	that	that	DET
ejpam-4827	180	9	cpσs(raq	cpσs(raq	NOUN
ejpam-4827	180	10	)	)	PUNCT
ejpam-4827	180	11	=	=	SYM
ejpam-4827	180	12	0	0	NUM
ejpam-4827	180	13	for	for	ADP
ejpam-4827	180	14	every	every	DET
ejpam-4827	180	15	r	r	NOUN
ejpam-4827	180	16	∈	∈	NOUN
ejpam-4827	180	17	r	r	NOUN
ejpam-4827	180	18	since	since	SCONJ
ejpam-4827	180	19	σgp	σgp	PROPN
ejpam-4827	180	20	is	be	AUX
ejpam-4827	180	21	an	an	DET
ejpam-4827	180	22	automorphism	automorphism	NOUN
ejpam-4827	180	23	.	.	PUNCT
ejpam-4827	181	1	hence	hence	ADV
ejpam-4827	181	2	,	,	PUNCT
ejpam-4827	181	3	cp	cp	PROPN
ejpam-4827	181	4	∈	∈	PROPN
ejpam-4827	181	5	ℓr(σs(raq	ℓr(σs(raq	PROPN
ejpam-4827	181	6	)	)	PUNCT
ejpam-4827	181	7	)	)	PUNCT
ejpam-4827	181	8	.	.	PUNCT
ejpam-4827	182	1	since	since	SCONJ
ejpam-4827	182	2	ℓr(σs(raq	ℓr(σs(raq	PROPN
ejpam-4827	182	3	)	)	PUNCT
ejpam-4827	182	4	)	)	PUNCT
ejpam-4827	182	5	is	be	AUX
ejpam-4827	182	6	pure	pure	ADJ
ejpam-4827	182	7	as	as	ADP
ejpam-4827	182	8	a	a	DET
ejpam-4827	182	9	left	left	ADJ
ejpam-4827	182	10	ideal	ideal	NOUN
ejpam-4827	182	11	of	of	ADP
ejpam-4827	182	12	r	r	NOUN
ejpam-4827	182	13	by	by	ADP
ejpam-4827	182	14	lemma	lemma	PROPN
ejpam-4827	182	15	1	1	NUM
ejpam-4827	182	16	,	,	PUNCT
ejpam-4827	182	17	there	there	PRON
ejpam-4827	182	18	exist	exist	VERB
ejpam-4827	182	19	an	an	DET
ejpam-4827	182	20	element	element	NOUN
ejpam-4827	182	21	eq	eq	NOUN
ejpam-4827	182	22	∈	∈	PROPN
ejpam-4827	182	23	ℓr(σs(raq	ℓr(σs(raq	PROPN
ejpam-4827	182	24	)	)	PUNCT
ejpam-4827	182	25	)	)	PUNCT
ejpam-4827	183	1	such	such	ADJ
ejpam-4827	183	2	that	that	SCONJ
ejpam-4827	183	3	cp	cp	PROPN
ejpam-4827	183	4	=	=	NOUN
ejpam-4827	183	5	cpeq	cpeq	PROPN
ejpam-4827	183	6	.	.	PUNCT
ejpam-4827	184	1	thus	thus	ADV
ejpam-4827	184	2	,	,	PUNCT
ejpam-4827	184	3	for	for	ADP
ejpam-4827	184	4	every	every	DET
ejpam-4827	184	5	r	r	NOUN
ejpam-4827	184	6	∈	∈	NOUN
ejpam-4827	184	7	r	r	NOUN
ejpam-4827	184	8	,	,	PUNCT
ejpam-4827	184	9	we	we	PRON
ejpam-4827	184	10	have	have	VERB
ejpam-4827	184	11	0	0	NUM
ejpam-4827	184	12	=	=	SYM
ejpam-4827	184	13	φ(eqrs)ψ	φ(eqrs)ψ	NOUN
ejpam-4827	184	14	=	=	X
ejpam-4827	184	15	(	(	PUNCT
ejpam-4827	184	16	b1g1	b1g1	PROPN
ejpam-4827	184	17	+	+	CCONJ
ejpam-4827	184	18	b2g2	b2g2	PROPN
ejpam-4827	184	19	+	+	CCONJ
ejpam-4827	184	20	·	·	PUNCT
ejpam-4827	184	21	·	·	PUNCT
ejpam-4827	184	22	·	·	PUNCT
ejpam-4827	184	23	+	+	NUM
ejpam-4827	184	24	bngn)(eqrs)(a1h1	bngn)(eqrs)(a1h1	PROPN
ejpam-4827	184	25	+	+	CCONJ
ejpam-4827	184	26	a2h2	a2h2	X
ejpam-4827	184	27	+	+	X
ejpam-4827	184	28	·	·	PUNCT
ejpam-4827	184	29	·	·	PUNCT
ejpam-4827	184	30	·	·	PUNCT
ejpam-4827	185	1	+	+	NUM
ejpam-4827	185	2	amhm	amhm	NOUN
ejpam-4827	185	3	)	)	PUNCT
ejpam-4827	185	4	=	=	PUNCT
ejpam-4827	186	1	(	(	PUNCT
ejpam-4827	186	2	b1g1	b1g1	PROPN
ejpam-4827	186	3	+	+	CCONJ
ejpam-4827	186	4	b2g2	b2g2	PROPN
ejpam-4827	186	5	+	+	CCONJ
ejpam-4827	186	6	·	·	PUNCT
ejpam-4827	186	7	·	·	PUNCT
ejpam-4827	186	8	·	·	PUNCT
ejpam-4827	186	9	+	+	NUM
ejpam-4827	186	10	bngn)(eqrs)(a1h1	bngn)(eqrs)(a1h1	PROPN
ejpam-4827	186	11	+	+	CCONJ
ejpam-4827	186	12	a2h2	a2h2	X
ejpam-4827	186	13	+	+	X
ejpam-4827	186	14	·	·	PUNCT
ejpam-4827	186	15	·	·	PUNCT
ejpam-4827	186	16	·	·	PUNCT
ejpam-4827	187	1	+	+	PUNCT
ejpam-4827	187	2	aq−1hq−1	aq−1hq−1	CCONJ
ejpam-4827	187	3	+	+	NUM
ejpam-4827	187	4	aq+1hq+1	aq+1hq+1	ADJ
ejpam-4827	187	5	+	+	X
ejpam-4827	187	6	·	·	PUNCT
ejpam-4827	187	7	·	·	PUNCT
ejpam-4827	187	8	·	·	PUNCT
ejpam-4827	187	9	+	+	NUM
ejpam-4827	187	10	amhm	amhm	NOUN
ejpam-4827	187	11	)	)	PUNCT
ejpam-4827	187	12	=	=	PUNCT
ejpam-4827	188	1	(	(	PUNCT
ejpam-4827	188	2	b1g1	b1g1	PROPN
ejpam-4827	188	3	+	+	CCONJ
ejpam-4827	188	4	b2g2	b2g2	PROPN
ejpam-4827	188	5	+	+	CCONJ
ejpam-4827	188	6	·	·	PUNCT
ejpam-4827	188	7	·	·	PUNCT
ejpam-4827	188	8	·	·	PUNCT
ejpam-4827	188	9	+	+	NUM
ejpam-4827	188	10	bngn)((eqrσg(aq))ghq	bngn)((eqrσg(aq))ghq	NOUN
ejpam-4827	188	11	)	)	PUNCT
ejpam-4827	189	1	=	=	PUNCT
ejpam-4827	189	2	(	(	PUNCT
ejpam-4827	189	3	b1σg1(eq)g1	b1σg1(eq)g1	X
ejpam-4827	189	4	+	+	X
ejpam-4827	189	5	b2σg2(eq)g2	b2σg2(eq)g2	X
ejpam-4827	189	6	+	+	X
ejpam-4827	189	7	·	·	PUNCT
ejpam-4827	189	8	·	·	PUNCT
ejpam-4827	189	9	·	·	PUNCT
ejpam-4827	189	10	+	+	NUM
ejpam-4827	189	11	bnσgn(eq)gn)(rs	bnσgn(eq)gn)(rs	PROPN
ejpam-4827	189	12	)	)	PUNCT
ejpam-4827	189	13	·	·	PUNCT
ejpam-4827	190	1	(	(	PUNCT
ejpam-4827	190	2	a1h1	a1h1	X
ejpam-4827	190	3	+	+	X
ejpam-4827	190	4	a2h2	a2h2	X
ejpam-4827	190	5	+	+	X
ejpam-4827	190	6	·	·	PUNCT
ejpam-4827	190	7	·	·	PUNCT
ejpam-4827	190	8	·	·	PUNCT
ejpam-4827	190	9	+	+	PUNCT
ejpam-4827	190	10	aq−1hq−1	aq−1hq−1	CCONJ
ejpam-4827	190	11	+	+	NUM
ejpam-4827	190	12	aq+1hq+1	aq+1hq+1	ADJ
ejpam-4827	190	13	+	+	X
ejpam-4827	190	14	·	·	PUNCT
ejpam-4827	190	15	·	·	PUNCT
ejpam-4827	190	16	·	·	PUNCT
ejpam-4827	190	17	+	+	NUM
ejpam-4827	190	18	amhm	amhm	NOUN
ejpam-4827	190	19	)	)	PUNCT
ejpam-4827	190	20	.	.	PUNCT
ejpam-4827	191	1	(	(	PUNCT
ejpam-4827	191	2	2.2	2.2	NUM
ejpam-4827	191	3	)	)	PUNCT
ejpam-4827	191	4	moreover	moreover	ADV
ejpam-4827	191	5	,	,	PUNCT
ejpam-4827	191	6	since	since	SCONJ
ejpam-4827	191	7	biσgi(eq	biσgi(eq	NOUN
ejpam-4827	191	8	)	)	PUNCT
ejpam-4827	191	9	=	=	PUNCT
ejpam-4827	191	10	σgi(cieq	σgi(cieq	NOUN
ejpam-4827	191	11	)	)	PUNCT
ejpam-4827	191	12	by	by	ADP
ejpam-4827	191	13	induction	induction	NOUN
ejpam-4827	191	14	,	,	PUNCT
ejpam-4827	191	15	it	it	PRON
ejpam-4827	191	16	follows	follow	VERB
ejpam-4827	191	17	that	that	SCONJ
ejpam-4827	191	18	cieq	cieq	PROPN
ejpam-4827	191	19	∈	∈	PROPN
ejpam-4827	191	20	ℓr(rσs(aj	ℓr(rσs(aj	PROPN
ejpam-4827	191	21	)	)	PUNCT
ejpam-4827	191	22	)	)	PUNCT
ejpam-4827	191	23	for	for	ADP
ejpam-4827	191	24	i	i	PROPN
ejpam-4827	191	25	=	=	SYM
ejpam-4827	191	26	1	1	NUM
ejpam-4827	191	27	,	,	PUNCT
ejpam-4827	191	28	2	2	NUM
ejpam-4827	191	29	,	,	PUNCT
ejpam-4827	191	30	.	.	PUNCT
ejpam-4827	191	31	.	.	PUNCT
ejpam-4827	192	1	.	.	PUNCT
ejpam-4827	193	1	,	,	PUNCT
ejpam-4827	193	2	n	n	CCONJ
ejpam-4827	193	3	,	,	PUNCT
ejpam-4827	193	4	j	j	PROPN
ejpam-4827	193	5	=	=	SYM
ejpam-4827	193	6	1	1	NUM
ejpam-4827	193	7	,	,	PUNCT
ejpam-4827	193	8	2	2	NUM
ejpam-4827	193	9	,	,	PUNCT
ejpam-4827	193	10	.	.	PUNCT
ejpam-4827	193	11	.	.	PUNCT
ejpam-4827	194	1	.	.	PUNCT
ejpam-4827	195	1	,	,	PUNCT
ejpam-4827	195	2	q−1	q−1	PROPN
ejpam-4827	195	3	,	,	PUNCT
ejpam-4827	195	4	q+1	q+1	NOUN
ejpam-4827	195	5	,	,	PUNCT
ejpam-4827	195	6	.	.	PUNCT
ejpam-4827	195	7	.	.	PUNCT
ejpam-4827	195	8	.	.	PUNCT
ejpam-4827	196	1	,	,	PUNCT
ejpam-4827	196	2	m.	m.	NOUN
ejpam-4827	196	3	therefore	therefore	ADV
ejpam-4827	196	4	,	,	PUNCT
ejpam-4827	196	5	cp	cp	PROPN
ejpam-4827	196	6	=	=	NOUN
ejpam-4827	196	7	cpeq	cpeq	PROPN
ejpam-4827	196	8	∈	∈	PROPN
ejpam-4827	196	9	∩mj=1	∩mj=1	ADJ
ejpam-4827	196	10	ℓr(rσs(aj	ℓr(rσs(aj	PROPN
ejpam-4827	196	11	)	)	PUNCT
ejpam-4827	196	12	)	)	PUNCT
ejpam-4827	196	13	.	.	PUNCT
ejpam-4827	197	1	now	now	ADV
ejpam-4827	197	2	we	we	PRON
ejpam-4827	197	3	have	have	VERB
ejpam-4827	197	4	bpσsp(rσs(aj	bpσsp(rσs(aj	NOUN
ejpam-4827	197	5	)	)	PUNCT
ejpam-4827	197	6	)	)	PUNCT
ejpam-4827	198	1	=	=	SYM
ejpam-4827	198	2	σsp(bprσs(aj	σsp(bprσs(aj	NOUN
ejpam-4827	198	3	)	)	PUNCT
ejpam-4827	198	4	)	)	PUNCT
ejpam-4827	199	1	=	=	PUNCT
ejpam-4827	199	2	0	0	X
ejpam-4827	199	3	.	.	PUNCT
ejpam-4827	200	1	for	for	ADP
ejpam-4827	200	2	every	every	DET
ejpam-4827	200	3	gi	gi	NOUN
ejpam-4827	200	4	∈	∈	PROPN
ejpam-4827	200	5	m	m	PRON
ejpam-4827	200	6	,	,	PUNCT
ejpam-4827	200	7	since	since	SCONJ
ejpam-4827	200	8	σgi	σgi	NOUN
ejpam-4827	200	9	is	be	AUX
ejpam-4827	200	10	an	an	DET
ejpam-4827	200	11	automorphism	automorphism	NOUN
ejpam-4827	200	12	of	of	ADP
ejpam-4827	200	13	r	r	NOUN
ejpam-4827	200	14	and	and	CCONJ
ejpam-4827	200	15	σgq(r	σgq(r	PROPN
ejpam-4827	200	16	)	)	PUNCT
ejpam-4827	201	1	=	=	SYM
ejpam-4827	201	2	r	r	X
ejpam-4827	201	3	,	,	PUNCT
ejpam-4827	201	4	we	we	PRON
ejpam-4827	201	5	obtain	obtain	VERB
ejpam-4827	201	6	bpσgp(rσgsc(aj	bpσgp(rσgsc(aj	PROPN
ejpam-4827	201	7	)	)	PUNCT
ejpam-4827	201	8	)	)	PUNCT
ejpam-4827	202	1	=	=	SYM
ejpam-4827	202	2	0	0	NUM
ejpam-4827	203	1	for	for	ADP
ejpam-4827	203	2	any	any	DET
ejpam-4827	203	3	j	j	NOUN
ejpam-4827	203	4	=	=	SYM
ejpam-4827	203	5	1	1	NUM
ejpam-4827	203	6	,	,	PUNCT
ejpam-4827	203	7	2	2	NUM
ejpam-4827	203	8	,	,	PUNCT
ejpam-4827	203	9	.	.	PUNCT
ejpam-4827	203	10	.	.	PUNCT
ejpam-4827	203	11	.	.	PUNCT
ejpam-4827	204	1	,	,	PUNCT
ejpam-4827	204	2	m.	m.	NOUN
ejpam-4827	204	3	thus	thus	ADV
ejpam-4827	204	4	,	,	PUNCT
ejpam-4827	204	5	from	from	ADP
ejpam-4827	204	6	φ(rs)ψ	φ(rs)ψ	NOUN
ejpam-4827	204	7	=	=	SYM
ejpam-4827	204	8	0	0	NUM
ejpam-4827	204	9	it	it	PRON
ejpam-4827	204	10	follows	follow	VERB
ejpam-4827	204	11	that	that	PRON
ejpam-4827	204	12	0	0	X
ejpam-4827	205	1	=	=	SYM
ejpam-4827	205	2	(	(	PUNCT
ejpam-4827	205	3	b1g1	b1g1	PROPN
ejpam-4827	205	4	+	+	CCONJ
ejpam-4827	205	5	b2g2	b2g2	X
ejpam-4827	205	6	+	+	PUNCT
ejpam-4827	205	7	.	.	PUNCT
ejpam-4827	205	8	.	.	PUNCT
ejpam-4827	205	9	.	.	PUNCT
ejpam-4827	206	1	+	+	CCONJ
ejpam-4827	206	2	bp−1gp−1	bp−1gp−1	PUNCT
ejpam-4827	206	3	+	+	NUM
ejpam-4827	206	4	bp+1gp+1	bp+1gp+1	NOUN
ejpam-4827	206	5	+	+	X
ejpam-4827	206	6	·	·	PUNCT
ejpam-4827	206	7	·	·	PUNCT
ejpam-4827	206	8	·	·	PUNCT
ejpam-4827	207	1	+	+	NUM
ejpam-4827	207	2	bngn)(rs)(a1h1	bngn)(rs)(a1h1	PROPN
ejpam-4827	207	3	+	+	CCONJ
ejpam-4827	207	4	a2h2	a2h2	X
ejpam-4827	207	5	+	+	X
ejpam-4827	207	6	·	·	PUNCT
ejpam-4827	207	7	·	·	PUNCT
ejpam-4827	207	8	·	·	PUNCT
ejpam-4827	208	1	+	+	CCONJ
ejpam-4827	208	2	amhm	amhm	NOUN
ejpam-4827	208	3	)	)	PUNCT
ejpam-4827	208	4	.	.	PUNCT
ejpam-4827	209	1	by	by	ADP
ejpam-4827	209	2	using	use	VERB
ejpam-4827	209	3	the	the	DET
ejpam-4827	209	4	previous	previous	ADJ
ejpam-4827	209	5	method	method	NOUN
ejpam-4827	209	6	,	,	PUNCT
ejpam-4827	209	7	there	there	PRON
ejpam-4827	209	8	exist	exist	VERB
ejpam-4827	209	9	k	k	PROPN
ejpam-4827	209	10	∈	∈	PROPN
ejpam-4827	209	11	{	{	PUNCT
ejpam-4827	209	12	1	1	NUM
ejpam-4827	209	13	,	,	PUNCT
ejpam-4827	209	14	2	2	NUM
ejpam-4827	209	15	,	,	PUNCT
ejpam-4827	209	16	.	.	PUNCT
ejpam-4827	209	17	.	.	PUNCT
ejpam-4827	210	1	.	.	PUNCT
ejpam-4827	211	1	,	,	PUNCT
ejpam-4827	211	2	p−	p−	NOUN
ejpam-4827	211	3	1	1	NUM
ejpam-4827	211	4	,	,	PUNCT
ejpam-4827	211	5	p+1	p+1	PROPN
ejpam-4827	211	6	,	,	PUNCT
ejpam-4827	211	7	.	.	PUNCT
ejpam-4827	211	8	.	.	PUNCT
ejpam-4827	212	1	.	.	PUNCT
ejpam-4827	213	1	,	,	PUNCT
ejpam-4827	214	1	n	n	CCONJ
ejpam-4827	214	2	}	}	PUNCT
ejpam-4827	214	3	such	such	ADJ
ejpam-4827	214	4	that	that	SCONJ
ejpam-4827	214	5	ck	ck	PROPN
ejpam-4827	214	6	∈	∈	PROPN
ejpam-4827	214	7	∩mj=1ℓr(σs(raj	∩mj=1ℓr(σs(raj	NUM
ejpam-4827	214	8	)	)	PUNCT
ejpam-4827	214	9	)	)	PUNCT
ejpam-4827	214	10	.	.	PUNCT
ejpam-4827	215	1	thus	thus	ADV
ejpam-4827	215	2	,	,	PUNCT
ejpam-4827	215	3	bkσgk(σs(raj	bkσgk(σs(raj	VERB
ejpam-4827	215	4	)	)	PUNCT
ejpam-4827	215	5	)	)	PUNCT
ejpam-4827	216	1	=	=	SYM
ejpam-4827	216	2	σgk(ckσs(raj	σgk(ckσs(raj	PROPN
ejpam-4827	216	3	)	)	PUNCT
ejpam-4827	216	4	)	)	PUNCT
ejpam-4827	217	1	=	=	PUNCT
ejpam-4827	217	2	0	0	NUM
ejpam-4827	218	1	for	for	ADP
ejpam-4827	218	2	any	any	DET
ejpam-4827	218	3	j	j	NOUN
ejpam-4827	218	4	=	=	SYM
ejpam-4827	218	5	1	1	NUM
ejpam-4827	218	6	,	,	PUNCT
ejpam-4827	218	7	2	2	NUM
ejpam-4827	218	8	,	,	PUNCT
ejpam-4827	218	9	.	.	PUNCT
ejpam-4827	218	10	.	.	PUNCT
ejpam-4827	218	11	.	.	PUNCT
ejpam-4827	219	1	,	,	PUNCT
ejpam-4827	219	2	m.	m.	NOUN
ejpam-4827	219	3	hence	hence	ADV
ejpam-4827	219	4	(	(	PUNCT
ejpam-4827	219	5	b1g1+b2g2	b1g1+b2g2	X
ejpam-4827	219	6	+	+	X
ejpam-4827	219	7	·	·	PUNCT
ejpam-4827	219	8	·	·	PUNCT
ejpam-4827	219	9	·	·	PUNCT
ejpam-4827	220	1	+	+	NOUN
ejpam-4827	220	2	bp−1gp−1+bp+1gp+1	bp−1gp−1+bp+1gp+1	PROPN
ejpam-4827	220	3	+	+	X
ejpam-4827	220	4	·	·	PUNCT
ejpam-4827	220	5	·	·	PUNCT
ejpam-4827	220	6	·	·	PUNCT
ejpam-4827	220	7	+	+	CCONJ
ejpam-4827	220	8	bk−1gk−1	bk−1gk−1	NUM
ejpam-4827	220	9	+	+	NUM
ejpam-4827	220	10	bk+1gk+1	bk+1gk+1	X
ejpam-4827	220	11	+	+	X
ejpam-4827	220	12	·	·	PUNCT
ejpam-4827	220	13	·	·	PUNCT
ejpam-4827	220	14	·	·	PUNCT
ejpam-4827	220	15	+	+	NUM
ejpam-4827	220	16	bngn)(rs)(a1h1	bngn)(rs)(a1h1	PROPN
ejpam-4827	220	17	+	+	CCONJ
ejpam-4827	220	18	a2h2	a2h2	X
ejpam-4827	220	19	+	+	X
ejpam-4827	220	20	·	·	PUNCT
ejpam-4827	220	21	·	·	PUNCT
ejpam-4827	220	22	·	·	PUNCT
ejpam-4827	220	23	+	+	CCONJ
ejpam-4827	220	24	amhm	amhm	NOUN
ejpam-4827	220	25	)	)	PUNCT
ejpam-4827	220	26	=	=	SYM
ejpam-4827	221	1	0	0	X
ejpam-4827	221	2	.	.	PUNCT
ejpam-4827	221	3	continuing	continue	VERB
ejpam-4827	221	4	this	this	DET
ejpam-4827	221	5	procedure	procedure	NOUN
ejpam-4827	221	6	yields	yield	NOUN
ejpam-4827	221	7	c1	c1	PROPN
ejpam-4827	221	8	,	,	PUNCT
ejpam-4827	221	9	c2	c2	PROPN
ejpam-4827	221	10	,	,	PUNCT
ejpam-4827	221	11	.	.	PUNCT
ejpam-4827	221	12	.	.	PUNCT
ejpam-4827	222	1	.	.	PUNCT
ejpam-4827	223	1	,	,	PUNCT
ejpam-4827	223	2	cn	cn	PROPN
ejpam-4827	223	3	∈	∈	PROPN
ejpam-4827	223	4	∩mj=1ℓr(σs(raj	∩mj=1ℓr(σs(raj	NUM
ejpam-4827	223	5	)	)	PUNCT
ejpam-4827	223	6	)	)	PUNCT
ejpam-4827	223	7	for	for	ADP
ejpam-4827	223	8	every	every	DET
ejpam-4827	223	9	s	s	PROPN
ejpam-4827	223	10	∈	∈	PROPN
ejpam-4827	223	11	m.	m.	NOUN
ejpam-4827	223	12	thus	thus	ADV
ejpam-4827	223	13	,	,	PUNCT
ejpam-4827	223	14	bk	bk	PROPN
ejpam-4827	223	15	∈	∈	PROPN
ejpam-4827	223	16	ℓr(σs(raj	ℓr(σs(raj	VERB
ejpam-4827	223	17	)	)	PUNCT
ejpam-4827	223	18	)	)	PUNCT
ejpam-4827	223	19	for	for	ADP
ejpam-4827	223	20	any	any	DET
ejpam-4827	223	21	k	k	NOUN
ejpam-4827	223	22	=	=	SYM
ejpam-4827	223	23	1	1	NUM
ejpam-4827	223	24	,	,	PUNCT
ejpam-4827	223	25	2	2	NUM
ejpam-4827	223	26	,	,	PUNCT
ejpam-4827	223	27	.	.	PUNCT
ejpam-4827	223	28	.	.	PUNCT
ejpam-4827	224	1	.	.	PUNCT
ejpam-4827	225	1	,	,	PUNCT
ejpam-4827	225	2	n	n	CCONJ
ejpam-4827	225	3	,	,	PUNCT
ejpam-4827	225	4	j	j	PROPN
ejpam-4827	225	5	=	=	SYM
ejpam-4827	225	6	1	1	NUM
ejpam-4827	225	7	,	,	PUNCT
ejpam-4827	225	8	2	2	NUM
ejpam-4827	225	9	,	,	PUNCT
ejpam-4827	225	10	.	.	PUNCT
ejpam-4827	225	11	.	.	PUNCT
ejpam-4827	225	12	.	.	PUNCT
ejpam-4827	226	1	,	,	PUNCT
ejpam-4827	226	2	m.	m.	NOUN
ejpam-4827	226	3	using	use	VERB
ejpam-4827	226	4	induction	induction	NOUN
ejpam-4827	226	5	on	on	ADP
ejpam-4827	226	6	m	m	PROPN
ejpam-4827	226	7	+	+	SYM
ejpam-4827	226	8	n	n	CCONJ
ejpam-4827	226	9	,	,	PUNCT
ejpam-4827	226	10	we	we	PRON
ejpam-4827	226	11	obtain	obtain	VERB
ejpam-4827	226	12	biσgi(σs(raj	biσgi(σs(raj	NOUN
ejpam-4827	226	13	)	)	PUNCT
ejpam-4827	226	14	)	)	PUNCT
ejpam-4827	227	1	=	=	PUNCT
ejpam-4827	227	2	0	0	X
ejpam-4827	227	3	.	.	PUNCT
ejpam-4827	228	1	thus	thus	ADV
ejpam-4827	228	2	,	,	PUNCT
ejpam-4827	228	3	ajσhj	ajσhj	ADJ
ejpam-4827	228	4	(	(	PUNCT
ejpam-4827	228	5	σs(rbi	σs(rbi	PROPN
ejpam-4827	228	6	)	)	PUNCT
ejpam-4827	228	7	)	)	PUNCT
ejpam-4827	229	1	=	=	PUNCT
ejpam-4827	229	2	0	0	PUNCT
ejpam-4827	230	1	since	since	SCONJ
ejpam-4827	230	2	r	r	NOUN
ejpam-4827	230	3	is	be	AUX
ejpam-4827	230	4	reduced	reduce	VERB
ejpam-4827	230	5	.	.	PUNCT
ejpam-4827	231	1	therefore	therefore	ADV
ejpam-4827	231	2	,	,	PUNCT
ejpam-4827	231	3	r	r	NOUN
ejpam-4827	231	4	is	be	AUX
ejpam-4827	231	5	σ	σ	NOUN
ejpam-4827	231	6	-	-	PUNCT
ejpam-4827	231	7	skew	skew	NOUN
ejpam-4827	231	8	strongly	strongly	ADV
ejpam-4827	231	9	m	m	VERB
ejpam-4827	231	10	-reflexive	-reflexive	ADJ
ejpam-4827	231	11	.	.	PUNCT
ejpam-4827	232	1	e.	e.	PROPN
ejpam-4827	232	2	ali	ali	PROPN
ejpam-4827	232	3	/	/	SYM
ejpam-4827	232	4	eur	eur	PROPN
ejpam-4827	232	5	.	.	PUNCT
ejpam-4827	233	1	j.	j.	PROPN
ejpam-4827	233	2	pure	pure	PROPN
ejpam-4827	233	3	appl	appl	PROPN
ejpam-4827	233	4	.	.	PROPN
ejpam-4827	233	5	math	math	PROPN
ejpam-4827	233	6	,	,	PUNCT
ejpam-4827	233	7	16	16	NUM
ejpam-4827	233	8	(	(	PUNCT
ejpam-4827	233	9	3	3	NUM
ejpam-4827	233	10	)	)	PUNCT
ejpam-4827	233	11	(	(	PUNCT
ejpam-4827	233	12	2023	2023	NUM
ejpam-4827	233	13	)	)	PUNCT
ejpam-4827	233	14	,	,	PUNCT
ejpam-4827	233	15	1878	1878	NUM
ejpam-4827	233	16	-	-	SYM
ejpam-4827	233	17	1893	1893	NUM
ejpam-4827	233	18	1883	1883	NUM
ejpam-4827	233	19	let	let	VERB
ejpam-4827	233	20	(	(	PUNCT
ejpam-4827	233	21	m,≤	m,≤	X
ejpam-4827	233	22	)	)	PUNCT
ejpam-4827	233	23	be	be	VERB
ejpam-4827	233	24	an	an	DET
ejpam-4827	233	25	ordered	ordered	ADJ
ejpam-4827	233	26	monoid	monoid	NOUN
ejpam-4827	233	27	.	.	PUNCT
ejpam-4827	234	1	if	if	SCONJ
ejpam-4827	234	2	for	for	ADP
ejpam-4827	234	3	any	any	DET
ejpam-4827	234	4	g1	g1	NOUN
ejpam-4827	234	5	,	,	PUNCT
ejpam-4827	234	6	g2	g2	PROPN
ejpam-4827	234	7	,	,	PUNCT
ejpam-4827	234	8	h	h	NOUN
ejpam-4827	234	9	∈	∈	PROPN
ejpam-4827	234	10	m	m	PROPN
ejpam-4827	234	11	,	,	PUNCT
ejpam-4827	234	12	g1	g1	PROPN
ejpam-4827	234	13	<	<	X
ejpam-4827	234	14	g2	g2	PROPN
ejpam-4827	234	15	implies	imply	VERB
ejpam-4827	234	16	that	that	SCONJ
ejpam-4827	234	17	g1h	g1h	PROPN
ejpam-4827	234	18	<	<	X
ejpam-4827	234	19	g2h	g2h	NOUN
ejpam-4827	234	20	and	and	CCONJ
ejpam-4827	234	21	hg1	hg1	PROPN
ejpam-4827	234	22	<	<	X
ejpam-4827	234	23	hg2	hg2	PROPN
ejpam-4827	234	24	,	,	PUNCT
ejpam-4827	234	25	then	then	ADV
ejpam-4827	234	26	≤	≤	NUM
ejpam-4827	234	27	is	be	AUX
ejpam-4827	234	28	called	call	VERB
ejpam-4827	234	29	a	a	DET
ejpam-4827	234	30	strictly	strictly	ADV
ejpam-4827	234	31	ordered	order	VERB
ejpam-4827	234	32	monoid	monoid	PROPN
ejpam-4827	234	33	.	.	PUNCT
ejpam-4827	235	1	corollary	corollary	ADJ
ejpam-4827	235	2	1	1	NUM
ejpam-4827	235	3	.	.	PUNCT
ejpam-4827	236	1	let	let	AUX
ejpam-4827	236	2	r	r	PRON
ejpam-4827	236	3	be	be	AUX
ejpam-4827	236	4	an	an	DET
ejpam-4827	236	5	m	m	NOUN
ejpam-4827	236	6	-ring	-ring	ADJ
ejpam-4827	236	7	,	,	PUNCT
ejpam-4827	236	8	where	where	SCONJ
ejpam-4827	236	9	m	m	NOUN
ejpam-4827	236	10	is	be	AUX
ejpam-4827	236	11	a	a	DET
ejpam-4827	236	12	strictly	strictly	ADV
ejpam-4827	236	13	totally	totally	ADV
ejpam-4827	236	14	ordered	order	VERB
ejpam-4827	236	15	monoid	monoid	NOUN
ejpam-4827	236	16	and	and	CCONJ
ejpam-4827	236	17	σ	σ	NOUN
ejpam-4827	236	18	:	:	PUNCT
ejpam-4827	236	19	m	m	PROPN
ejpam-4827	236	20	→	→	SYM
ejpam-4827	236	21	aut(r	aut(r	PROPN
ejpam-4827	236	22	)	)	PUNCT
ejpam-4827	236	23	is	be	AUX
ejpam-4827	236	24	a	a	DET
ejpam-4827	236	25	monoid	monoid	NOUN
ejpam-4827	236	26	homomorphism	homomorphism	NOUN
ejpam-4827	236	27	.	.	PUNCT
ejpam-4827	237	1	if	if	SCONJ
ejpam-4827	237	2	r	r	NOUN
ejpam-4827	237	3	is	be	AUX
ejpam-4827	237	4	quasi	quasi	ADJ
ejpam-4827	237	5	armendariz	armendariz	NOUN
ejpam-4827	237	6	,	,	PUNCT
ejpam-4827	237	7	then	then	ADV
ejpam-4827	237	8	r	r	NOUN
ejpam-4827	237	9	is	be	AUX
ejpam-4827	237	10	σ	σ	NOUN
ejpam-4827	237	11	-	-	PUNCT
ejpam-4827	237	12	skew	skew	NOUN
ejpam-4827	237	13	strongly	strongly	ADV
ejpam-4827	237	14	m	m	VERB
ejpam-4827	237	15	-reflexive	-reflexive	ADJ
ejpam-4827	237	16	.	.	PUNCT
ejpam-4827	238	1	proof	proof	NOUN
ejpam-4827	238	2	.	.	PUNCT
ejpam-4827	239	1	let	let	VERB
ejpam-4827	239	2	ϕ	ϕ	NOUN
ejpam-4827	239	3	=	=	X
ejpam-4827	239	4	σni=1bigi	σni=1bigi	PROPN
ejpam-4827	239	5	and	and	CCONJ
ejpam-4827	239	6	ψ	ψ	X
ejpam-4827	239	7	=	=	X
ejpam-4827	239	8	σmj=1ajhj	σmj=1ajhj	NOUN
ejpam-4827	239	9	∈	∈	NOUN
ejpam-4827	239	10	r	r	NOUN
ejpam-4827	239	11	∗m	∗m	NOUN
ejpam-4827	239	12	satisfying	satisfy	VERB
ejpam-4827	239	13	ϕ(r	ϕ(r	PROPN
ejpam-4827	239	14	∗m)ψ	∗m)ψ	ADJ
ejpam-4827	239	15	=	=	SYM
ejpam-4827	239	16	0	0	NUM
ejpam-4827	239	17	implies	imply	VERB
ejpam-4827	239	18	that	that	SCONJ
ejpam-4827	239	19	biσgi(rσs(aj	biσgi(rσs(aj	NOUN
ejpam-4827	239	20	)	)	PUNCT
ejpam-4827	239	21	)	)	PUNCT
ejpam-4827	240	1	=	=	SYM
ejpam-4827	240	2	0	0	NUM
ejpam-4827	241	1	for	for	ADP
ejpam-4827	241	2	any	any	DET
ejpam-4827	241	3	s	s	NOUN
ejpam-4827	241	4	∈m	∈m	NOUN
ejpam-4827	241	5	and	and	CCONJ
ejpam-4827	241	6	any	any	DET
ejpam-4827	241	7	i	i	PROPN
ejpam-4827	241	8	,	,	PUNCT
ejpam-4827	241	9	j.	j.	PROPN
ejpam-4827	241	10	we	we	PRON
ejpam-4827	241	11	write	write	VERB
ejpam-4827	241	12	(	(	PUNCT
ejpam-4827	241	13	b1g1	b1g1	PROPN
ejpam-4827	241	14	+	+	CCONJ
ejpam-4827	241	15	b2g2	b2g2	PROPN
ejpam-4827	241	16	+	+	CCONJ
ejpam-4827	241	17	·	·	PUNCT
ejpam-4827	241	18	·	·	PUNCT
ejpam-4827	241	19	·	·	PUNCT
ejpam-4827	241	20	+	+	NUM
ejpam-4827	241	21	bngn)(rs)(a1h1	bngn)(rs)(a1h1	PROPN
ejpam-4827	241	22	+	+	CCONJ
ejpam-4827	241	23	a2h2	a2h2	X
ejpam-4827	241	24	+	+	X
ejpam-4827	241	25	·	·	PUNCT
ejpam-4827	241	26	·	·	PUNCT
ejpam-4827	241	27	·	·	PUNCT
ejpam-4827	242	1	+	+	NUM
ejpam-4827	242	2	amhm	amhm	NOUN
ejpam-4827	242	3	)	)	PUNCT
ejpam-4827	242	4	=	=	SYM
ejpam-4827	243	1	0	0	X
ejpam-4827	243	2	.	.	PUNCT
ejpam-4827	244	1	(	(	PUNCT
ejpam-4827	244	2	2.3	2.3	NUM
ejpam-4827	244	3	)	)	PUNCT
ejpam-4827	244	4	with	with	ADP
ejpam-4827	244	5	g1	g1	PROPN
ejpam-4827	244	6	<	<	X
ejpam-4827	244	7	g2	g2	PROPN
ejpam-4827	244	8	<	<	X
ejpam-4827	244	9	.	.	PUNCT
ejpam-4827	244	10	.	.	PUNCT
ejpam-4827	244	11	.	.	PUNCT
ejpam-4827	245	1	<	<	X
ejpam-4827	245	2	gn	gn	PROPN
ejpam-4827	245	3	,	,	PUNCT
ejpam-4827	245	4	h1	h1	VERB
ejpam-4827	245	5	<	<	X
ejpam-4827	245	6	h2	h2	PROPN
ejpam-4827	245	7	<	<	X
ejpam-4827	245	8	.	.	PUNCT
ejpam-4827	245	9	.	.	PUNCT
ejpam-4827	245	10	.	.	PUNCT
ejpam-4827	246	1	<	<	X
ejpam-4827	246	2	hm.we	hm.we	PROPN
ejpam-4827	246	3	will	will	AUX
ejpam-4827	246	4	use	use	VERB
ejpam-4827	246	5	transfinite	transfinite	ADJ
ejpam-4827	246	6	induction	induction	NOUN
ejpam-4827	246	7	on	on	ADP
ejpam-4827	246	8	a	a	DET
ejpam-4827	246	9	strictly	strictly	ADV
ejpam-4827	246	10	totally	totally	ADV
ejpam-4827	246	11	ordered	order	VERB
ejpam-4827	246	12	set	set	VERB
ejpam-4827	246	13	≤	≤	NOUN
ejpam-4827	246	14	to	to	PART
ejpam-4827	246	15	show	show	VERB
ejpam-4827	246	16	that	that	SCONJ
ejpam-4827	246	17	ψ(r	ψ(r	NOUN
ejpam-4827	246	18	∗m)ϕ	∗m)ϕ	NOUN
ejpam-4827	246	19	=	=	NOUN
ejpam-4827	246	20	0	0	X
ejpam-4827	246	21	.	.	PUNCT
ejpam-4827	247	1	if	if	SCONJ
ejpam-4827	247	2	we	we	PRON
ejpam-4827	247	3	take	take	VERB
ejpam-4827	247	4	m	m	VERB
ejpam-4827	247	5	=	=	NOUN
ejpam-4827	247	6	1	1	NUM
ejpam-4827	247	7	in	in	ADP
ejpam-4827	247	8	eq	eq	ADP
ejpam-4827	247	9	.	.	PUNCT
ejpam-4827	248	1	(	(	PUNCT
ejpam-4827	248	2	2.3	2.3	NUM
ejpam-4827	248	3	)	)	PUNCT
ejpam-4827	248	4	,	,	PUNCT
ejpam-4827	248	5	then	then	ADV
ejpam-4827	248	6	we	we	PRON
ejpam-4827	248	7	have	have	VERB
ejpam-4827	248	8	(	(	PUNCT
ejpam-4827	248	9	b1g1	b1g1	SYM
ejpam-4827	248	10	+	+	NUM
ejpam-4827	248	11	b2g2	b2g2	PROPN
ejpam-4827	248	12	+	+	CCONJ
ejpam-4827	248	13	·	·	PUNCT
ejpam-4827	248	14	·	·	PUNCT
ejpam-4827	248	15	·	·	PUNCT
ejpam-4827	248	16	+	+	NUM
ejpam-4827	248	17	bngn)(rs)(a1h1	bngn)(rs)(a1h1	NOUN
ejpam-4827	248	18	)	)	PUNCT
ejpam-4827	248	19	=	=	SYM
ejpam-4827	249	1	0	0	X
ejpam-4827	249	2	.	.	PUNCT
ejpam-4827	250	1	therefore	therefore	ADV
ejpam-4827	250	2	,	,	PUNCT
ejpam-4827	250	3	we	we	PRON
ejpam-4827	250	4	obtain	obtain	VERB
ejpam-4827	250	5	biσgi(rσs(a1	biσgi(rσs(a1	NOUN
ejpam-4827	250	6	)	)	PUNCT
ejpam-4827	250	7	)	)	PUNCT
ejpam-4827	251	1	=	=	SYM
ejpam-4827	251	2	0	0	NUM
ejpam-4827	251	3	for	for	ADP
ejpam-4827	251	4	each	each	DET
ejpam-4827	251	5	1	1	NUM
ejpam-4827	251	6	≤	≤	NUM
ejpam-4827	251	7	i	i	PRON
ejpam-4827	251	8	≤	≤	PROPN
ejpam-4827	251	9	i.	i.	NOUN
ejpam-4827	251	10	since	since	SCONJ
ejpam-4827	251	11	m	m	PROPN
ejpam-4827	251	12	is	be	AUX
ejpam-4827	251	13	a	a	DET
ejpam-4827	251	14	strictly	strictly	ADV
ejpam-4827	251	15	totally	totally	ADV
ejpam-4827	251	16	ordered	order	VERB
ejpam-4827	251	17	monoid	monoid	NOUN
ejpam-4827	251	18	,	,	PUNCT
ejpam-4827	251	19	we	we	PRON
ejpam-4827	251	20	have	have	VERB
ejpam-4827	251	21	g1h1	g1h1	PROPN
ejpam-4827	251	22	<	<	X
ejpam-4827	251	23	gih1	gih1	PROPN
ejpam-4827	251	24	≤	≤	PROPN
ejpam-4827	251	25	gihj	gihj	PROPN
ejpam-4827	251	26	=	=	SYM
ejpam-4827	252	1	g1h1	g1h1	PROPN
ejpam-4827	252	2	for	for	ADP
ejpam-4827	252	3	i	i	PRON
ejpam-4827	252	4	̸=	̸=	PROPN
ejpam-4827	252	5	1	1	NUM
ejpam-4827	252	6	or	or	CCONJ
ejpam-4827	252	7	j	j	PROPN
ejpam-4827	252	8	̸=	̸=	PROPN
ejpam-4827	252	9	1	1	NUM
ejpam-4827	252	10	.	.	PUNCT
ejpam-4827	253	1	thus	thus	ADV
ejpam-4827	253	2	,	,	PUNCT
ejpam-4827	253	3	eq	eq	ADJ
ejpam-4827	253	4	.	.	PUNCT
ejpam-4827	254	1	(	(	PUNCT
ejpam-4827	254	2	2.3	2.3	NUM
ejpam-4827	254	3	)	)	PUNCT
ejpam-4827	254	4	becomes	become	VERB
ejpam-4827	254	5	(	(	PUNCT
ejpam-4827	254	6	b1g1	b1g1	PROPN
ejpam-4827	254	7	+	+	CCONJ
ejpam-4827	254	8	b2g2	b2g2	PROPN
ejpam-4827	254	9	+	+	CCONJ
ejpam-4827	254	10	·	·	PUNCT
ejpam-4827	254	11	·	·	PUNCT
ejpam-4827	254	12	·	·	PUNCT
ejpam-4827	255	1	+	+	PUNCT
ejpam-4827	255	2	bngn)(rs)(a2h2	bngn)(rs)(a2h2	X
ejpam-4827	255	3	+	+	PUNCT
ejpam-4827	255	4	a3h3	a3h3	PROPN
ejpam-4827	255	5	+	+	X
ejpam-4827	255	6	·	·	PUNCT
ejpam-4827	255	7	·	·	PUNCT
ejpam-4827	255	8	·	·	PUNCT
ejpam-4827	255	9	+	+	NUM
ejpam-4827	255	10	amhm	amhm	NOUN
ejpam-4827	255	11	)	)	PUNCT
ejpam-4827	255	12	=	=	SYM
ejpam-4827	256	1	0	0	X
ejpam-4827	256	2	.	.	PUNCT
ejpam-4827	257	1	(	(	PUNCT
ejpam-4827	257	2	2.4	2.4	NUM
ejpam-4827	257	3	)	)	PUNCT
ejpam-4827	257	4	the	the	DET
ejpam-4827	257	5	case	case	NOUN
ejpam-4827	257	6	n	n	NOUN
ejpam-4827	257	7	=	=	SYM
ejpam-4827	257	8	1	1	NUM
ejpam-4827	257	9	is	be	AUX
ejpam-4827	257	10	proved	prove	VERB
ejpam-4827	257	11	by	by	ADP
ejpam-4827	257	12	similar	similar	ADJ
ejpam-4827	257	13	argument	argument	NOUN
ejpam-4827	257	14	.	.	PUNCT
ejpam-4827	258	1	by	by	ADP
ejpam-4827	258	2	applying	apply	VERB
ejpam-4827	258	3	the	the	DET
ejpam-4827	258	4	induction	induction	NOUN
ejpam-4827	258	5	hypothesis	hypothesis	NOUN
ejpam-4827	258	6	for	for	ADP
ejpam-4827	258	7	all	all	DET
ejpam-4827	258	8	2	2	NUM
ejpam-4827	258	9	≤	≤	NUM
ejpam-4827	258	10	i	i	PRON
ejpam-4827	258	11	≤	≤	ADJ
ejpam-4827	258	12	n	n	CCONJ
ejpam-4827	258	13	and	and	CCONJ
ejpam-4827	258	14	2	2	NUM
ejpam-4827	258	15	≤	≤	NUM
ejpam-4827	258	16	j	j	PROPN
ejpam-4827	258	17	≤	≤	PROPN
ejpam-4827	258	18	m.	m.	NOUN
ejpam-4827	258	19	suppose	suppose	VERB
ejpam-4827	258	20	that	that	SCONJ
ejpam-4827	258	21	biσgi(rσs(aj	biσgi(rσs(aj	NOUN
ejpam-4827	258	22	)	)	PUNCT
ejpam-4827	258	23	)	)	PUNCT
ejpam-4827	259	1	=	=	SYM
ejpam-4827	259	2	0	0	NUM
ejpam-4827	260	1	for	for	ADP
ejpam-4827	260	2	all	all	DET
ejpam-4827	260	3	1	1	NUM
ejpam-4827	260	4	≤	≤	NUM
ejpam-4827	260	5	i	i	PRON
ejpam-4827	260	6	≤	≤	PROPN
ejpam-4827	260	7	n	n	CCONJ
ejpam-4827	260	8	,	,	PUNCT
ejpam-4827	260	9	1	1	NUM
ejpam-4827	260	10	≤	≤	NUM
ejpam-4827	260	11	j	j	PROPN
ejpam-4827	260	12	≤	≤	NUM
ejpam-4827	260	13	m	m	VERB
ejpam-4827	260	14	with	with	ADP
ejpam-4827	260	15	λ	λ	NOUN
ejpam-4827	260	16	∈m	∈m	NOUN
ejpam-4827	260	17	is	be	AUX
ejpam-4827	260	18	such	such	ADJ
ejpam-4827	260	19	that	that	SCONJ
ejpam-4827	260	20	for	for	ADP
ejpam-4827	260	21	any	any	DET
ejpam-4827	260	22	gi	gi	NOUN
ejpam-4827	260	23	and	and	CCONJ
ejpam-4827	260	24	hj	hj	PROPN
ejpam-4827	260	25	,	,	PUNCT
ejpam-4827	260	26	gihj	gihj	PROPN
ejpam-4827	260	27	<	<	X
ejpam-4827	260	28	λ	λ	PROPN
ejpam-4827	260	29	.	.	PUNCT
ejpam-4827	261	1	we	we	PRON
ejpam-4827	261	2	will	will	AUX
ejpam-4827	261	3	show	show	VERB
ejpam-4827	261	4	that	that	SCONJ
ejpam-4827	261	5	biσgi(rσs(aj	biσgi(rσs(aj	NOUN
ejpam-4827	261	6	)	)	PUNCT
ejpam-4827	261	7	)	)	PUNCT
ejpam-4827	262	1	=	=	SYM
ejpam-4827	262	2	0	0	NUM
ejpam-4827	263	1	for	for	ADP
ejpam-4827	263	2	any	any	DET
ejpam-4827	263	3	gi	gi	NOUN
ejpam-4827	263	4	and	and	CCONJ
ejpam-4827	263	5	hj	hj	X
ejpam-4827	263	6	with	with	ADP
ejpam-4827	263	7	gihj	gihj	PROPN
ejpam-4827	263	8	=	=	SYM
ejpam-4827	263	9	λ	λ	PROPN
ejpam-4827	263	10	.	.	PUNCT
ejpam-4827	263	11	set	set	VERB
ejpam-4827	263	12	x	x	X
ejpam-4827	263	13	=	=	PRON
ejpam-4827	263	14	{	{	PUNCT
ejpam-4827	263	15	(	(	PUNCT
ejpam-4827	263	16	gi	gi	INTJ
ejpam-4827	263	17	,	,	PUNCT
ejpam-4827	263	18	hj)|gihj	hj)|gihj	X
ejpam-4827	263	19	=	=	SYM
ejpam-4827	263	20	λ	λ	NOUN
ejpam-4827	263	21	}	}	PUNCT
ejpam-4827	263	22	.	.	PUNCT
ejpam-4827	264	1	then	then	ADV
ejpam-4827	264	2	x	x	PRON
ejpam-4827	264	3	is	be	AUX
ejpam-4827	264	4	a	a	DET
ejpam-4827	264	5	finite	finite	ADJ
ejpam-4827	264	6	set	set	NOUN
ejpam-4827	264	7	.	.	PUNCT
ejpam-4827	265	1	we	we	PRON
ejpam-4827	265	2	write	write	VERB
ejpam-4827	265	3	x	x	PUNCT
ejpam-4827	265	4	as	as	ADP
ejpam-4827	265	5	{	{	PUNCT
ejpam-4827	265	6	(	(	PUNCT
ejpam-4827	265	7	giq	giq	NOUN
ejpam-4827	265	8	,	,	PUNCT
ejpam-4827	265	9	hjq)|q	hjq)|q	NOUN
ejpam-4827	265	10	=	=	SYM
ejpam-4827	265	11	1	1	NUM
ejpam-4827	265	12	,	,	PUNCT
ejpam-4827	265	13	2	2	NUM
ejpam-4827	265	14	,	,	PUNCT
ejpam-4827	265	15	.	.	PUNCT
ejpam-4827	265	16	.	.	PUNCT
ejpam-4827	265	17	.	.	PUNCT
ejpam-4827	266	1	,	,	PUNCT
ejpam-4827	266	2	d	d	X
ejpam-4827	266	3	}	}	PUNCT
ejpam-4827	266	4	such	such	ADJ
ejpam-4827	266	5	that	that	DET
ejpam-4827	266	6	gi1	gi1	NOUN
ejpam-4827	266	7	<	<	X
ejpam-4827	266	8	gi2	gi2	NOUN
ejpam-4827	266	9	<	<	X
ejpam-4827	266	10	.	.	PUNCT
ejpam-4827	266	11	.	.	PUNCT
ejpam-4827	266	12	.	.	PUNCT
ejpam-4827	267	1	<	<	X
ejpam-4827	267	2	gid	gid	PROPN
ejpam-4827	267	3	.	.	PUNCT
ejpam-4827	268	1	since	since	SCONJ
ejpam-4827	268	2	m	m	PROPN
ejpam-4827	268	3	is	be	AUX
ejpam-4827	268	4	cancellative	cancellative	ADJ
ejpam-4827	268	5	,	,	PUNCT
ejpam-4827	268	6	gi1	gi1	NOUN
ejpam-4827	268	7	=	=	PUNCT
ejpam-4827	268	8	gi2	gi2	NOUN
ejpam-4827	268	9	and	and	CCONJ
ejpam-4827	268	10	gi1hj1	gi1hj1	NOUN
ejpam-4827	269	1	=	=	SYM
ejpam-4827	269	2	gi2hj2	gi2hj2	NOUN
ejpam-4827	269	3	=	=	PUNCT
ejpam-4827	270	1	λ	λ	NOUN
ejpam-4827	270	2	imply	imply	VERB
ejpam-4827	270	3	hj1	hj1	NOUN
ejpam-4827	271	1	=	=	NOUN
ejpam-4827	271	2	hj2	hj2	NOUN
ejpam-4827	271	3	since	since	SCONJ
ejpam-4827	271	4	≤	≤	NUM
ejpam-4827	271	5	is	be	AUX
ejpam-4827	271	6	a	a	DET
ejpam-4827	271	7	strict	strict	ADJ
ejpam-4827	271	8	order	order	NOUN
ejpam-4827	272	1	,	,	PUNCT
ejpam-4827	272	2	gi1	gi1	NOUN
ejpam-4827	272	3	<	<	X
ejpam-4827	272	4	gi2	gi2	NOUN
ejpam-4827	272	5	and	and	CCONJ
ejpam-4827	272	6	gi1hj1	gi1hj1	NOUN
ejpam-4827	273	1	=	=	SYM
ejpam-4827	273	2	gi2hj2	gi2hj2	NOUN
ejpam-4827	273	3	=	=	NOUN
ejpam-4827	274	1	λ	λ	NOUN
ejpam-4827	274	2	imply	imply	VERB
ejpam-4827	274	3	hj2	hj2	NOUN
ejpam-4827	274	4	<	<	X
ejpam-4827	274	5	hj1	hj1	NOUN
ejpam-4827	274	6	.	.	PUNCT
ejpam-4827	275	1	thus	thus	ADV
ejpam-4827	275	2	we	we	PRON
ejpam-4827	275	3	have	have	AUX
ejpam-4827	275	4	hjd	hjd	NOUN
ejpam-4827	275	5	<	<	X
ejpam-4827	275	6	hjd−1	hjd−1	X
ejpam-4827	275	7	<	<	X
ejpam-4827	275	8	.	.	PUNCT
ejpam-4827	275	9	.	.	PUNCT
ejpam-4827	275	10	.	.	PUNCT
ejpam-4827	276	1	<	<	X
ejpam-4827	276	2	hj2	hj2	X
ejpam-4827	276	3	<	<	X
ejpam-4827	276	4	hj1	hj1	NOUN
ejpam-4827	276	5	.	.	PUNCT
ejpam-4827	277	1	now	now	ADV
ejpam-4827	277	2	∑	∑	PUNCT
ejpam-4827	277	3	(	(	PUNCT
ejpam-4827	277	4	gi	gi	INTJ
ejpam-4827	277	5	,	,	PUNCT
ejpam-4827	277	6	hj)∈x	hj)∈x	PROPN
ejpam-4827	277	7	biσgi(rσs(aj	biσgi(rσs(aj	NOUN
ejpam-4827	277	8	)	)	PUNCT
ejpam-4827	277	9	)	)	PUNCT
ejpam-4827	278	1	=	=	PUNCT
ejpam-4827	279	1	∑d	∑d	CCONJ
ejpam-4827	279	2	q=1	q=1	X
ejpam-4827	279	3	biqσgiq	biqσgiq	NOUN
ejpam-4827	279	4	(	(	PUNCT
ejpam-4827	279	5	rσs(ajq	rσs(ajq	NOUN
ejpam-4827	279	6	)	)	PUNCT
ejpam-4827	279	7	)	)	PUNCT
ejpam-4827	280	1	=	=	PUNCT
ejpam-4827	280	2	0	0	X
ejpam-4827	280	3	.	.	X
ejpam-4827	281	1	for	for	ADP
ejpam-4827	281	2	any	any	DET
ejpam-4827	281	3	q	q	ADJ
ejpam-4827	281	4	≥	≥	NUM
ejpam-4827	281	5	2	2	NUM
ejpam-4827	281	6	,	,	PUNCT
ejpam-4827	281	7	gi1hjq	gi1hjq	NOUN
ejpam-4827	281	8	<	<	X
ejpam-4827	281	9	giqhjq	giqhjq	NOUN
ejpam-4827	281	10	=	=	PUNCT
ejpam-4827	281	11	λ	λ	NOUN
ejpam-4827	281	12	,	,	PUNCT
ejpam-4827	281	13	and	and	CCONJ
ejpam-4827	281	14	so	so	ADV
ejpam-4827	281	15	bi1σgiq	bi1σgiq	PUNCT
ejpam-4827	281	16	(	(	PUNCT
ejpam-4827	281	17	rσs(ajq	rσs(ajq	NOUN
ejpam-4827	281	18	)	)	PUNCT
ejpam-4827	281	19	)	)	PUNCT
ejpam-4827	282	1	=	=	SYM
ejpam-4827	282	2	0	0	NUM
ejpam-4827	282	3	by	by	ADP
ejpam-4827	282	4	induction	induction	NOUN
ejpam-4827	282	5	hypothesis	hypothesis	NOUN
ejpam-4827	282	6	.	.	PUNCT
ejpam-4827	283	1	thus	thus	ADV
ejpam-4827	283	2	,	,	PUNCT
ejpam-4827	283	3	bi1σgiq	bi1σgiq	X
ejpam-4827	283	4	(	(	PUNCT
ejpam-4827	283	5	rajq	rajq	NOUN
ejpam-4827	283	6	)	)	PUNCT
ejpam-4827	283	7	=	=	PUNCT
ejpam-4827	283	8	0	0	PUNCT
ejpam-4827	284	1	because	because	SCONJ
ejpam-4827	284	2	r	r	NOUN
ejpam-4827	284	3	is	be	AUX
ejpam-4827	284	4	m	m	PRON
ejpam-4827	284	5	-rigidness	-rigidness	ADJ
ejpam-4827	284	6	and	and	CCONJ
ejpam-4827	284	7	σ	σ	PROPN
ejpam-4827	284	8	is	be	AUX
ejpam-4827	284	9	automorphism	automorphism	ADJ
ejpam-4827	284	10	.	.	PUNCT
ejpam-4827	285	1	for	for	ADP
ejpam-4827	285	2	the	the	DET
ejpam-4827	285	3	case	case	NOUN
ejpam-4827	285	4	where	where	SCONJ
ejpam-4827	285	5	n	n	NUM
ejpam-4827	285	6	≥	≥	X
ejpam-4827	285	7	2	2	NUM
ejpam-4827	285	8	for	for	ADP
ejpam-4827	285	9	m	m	PROPN
ejpam-4827	285	10	is	be	AUX
ejpam-4827	285	11	cancellative	cancellative	ADJ
ejpam-4827	285	12	.	.	PUNCT
ejpam-4827	286	1	we	we	PRON
ejpam-4827	286	2	can	can	AUX
ejpam-4827	286	3	repeat	repeat	VERB
ejpam-4827	286	4	this	this	DET
ejpam-4827	286	5	process	process	NOUN
ejpam-4827	286	6	to	to	PART
ejpam-4827	286	7	show	show	VERB
ejpam-4827	286	8	that	that	SCONJ
ejpam-4827	286	9	biσgi(rσs(aj	biσgi(rσs(aj	NOUN
ejpam-4827	286	10	)	)	PUNCT
ejpam-4827	286	11	)	)	PUNCT
ejpam-4827	287	1	=	=	SYM
ejpam-4827	287	2	0	0	NUM
ejpam-4827	287	3	for	for	SCONJ
ejpam-4827	287	4	all	all	PRON
ejpam-4827	287	5	s	s	PART
ejpam-4827	287	6	∈	∈	NOUN
ejpam-4827	287	7	m	m	NOUN
ejpam-4827	287	8	and	and	CCONJ
ejpam-4827	287	9	all	all	PRON
ejpam-4827	287	10	i	i	PROPN
ejpam-4827	287	11	,	,	PUNCT
ejpam-4827	287	12	j.	j.	PROPN
ejpam-4827	287	13	consequently	consequently	ADV
ejpam-4827	287	14	,	,	PUNCT
ejpam-4827	287	15	we	we	PRON
ejpam-4827	287	16	can	can	AUX
ejpam-4827	287	17	see	see	VERB
ejpam-4827	287	18	that	that	DET
ejpam-4827	287	19	ajσhj	ajσhj	ADJ
ejpam-4827	287	20	(	(	PUNCT
ejpam-4827	287	21	rσs(bi	rσs(bi	NUM
ejpam-4827	287	22	)	)	PUNCT
ejpam-4827	287	23	)	)	PUNCT
ejpam-4827	288	1	=	=	SYM
ejpam-4827	288	2	0	0	NUM
ejpam-4827	288	3	for	for	ADP
ejpam-4827	288	4	all	all	PRON
ejpam-4827	288	5	s	s	PART
ejpam-4827	288	6	∈	∈	PROPN
ejpam-4827	288	7	m	m	NOUN
ejpam-4827	288	8	,	,	PUNCT
ejpam-4827	288	9	1	1	NUM
ejpam-4827	288	10	≤	≤	NUM
ejpam-4827	288	11	j	j	PROPN
ejpam-4827	288	12	≤	≤	PROPN
ejpam-4827	288	13	m	m	PROPN
ejpam-4827	288	14	,	,	PUNCT
ejpam-4827	288	15	1	1	NUM
ejpam-4827	288	16	≤	≤	NUM
ejpam-4827	288	17	i	i	PRON
ejpam-4827	288	18	≤	≤	NOUN
ejpam-4827	288	19	n	n	X
ejpam-4827	288	20	by	by	ADP
ejpam-4827	288	21	rigidness	rigidness	NOUN
ejpam-4827	288	22	and	and	CCONJ
ejpam-4827	288	23	σ	σ	PROPN
ejpam-4827	288	24	is	be	AUX
ejpam-4827	288	25	automorphism	automorphism	NOUN
ejpam-4827	288	26	.	.	PUNCT
ejpam-4827	289	1	thus	thus	ADV
ejpam-4827	289	2	,	,	PUNCT
ejpam-4827	289	3	ψ(r	ψ(r	PROPN
ejpam-4827	289	4	∗m)ϕ	∗m)ϕ	NOUN
ejpam-4827	289	5	=	=	SYM
ejpam-4827	289	6	0	0	X
ejpam-4827	289	7	.	.	PUNCT
ejpam-4827	290	1	therefore	therefore	ADV
ejpam-4827	290	2	,	,	PUNCT
ejpam-4827	290	3	r	r	NOUN
ejpam-4827	290	4	is	be	AUX
ejpam-4827	290	5	σ	σ	NOUN
ejpam-4827	290	6	-	-	PUNCT
ejpam-4827	290	7	skew	skew	NOUN
ejpam-4827	290	8	strongly	strongly	ADV
ejpam-4827	290	9	m	m	VERB
ejpam-4827	290	10	-reflexive	-reflexive	ADJ
ejpam-4827	290	11	.	.	PUNCT
ejpam-4827	291	1	proposition	proposition	NOUN
ejpam-4827	291	2	2	2	NUM
ejpam-4827	291	3	.	.	PUNCT
ejpam-4827	292	1	let	let	VERB
ejpam-4827	292	2	r	r	PRON
ejpam-4827	292	3	be	be	AUX
ejpam-4827	292	4	a	a	DET
ejpam-4827	292	5	ring	ring	NOUN
ejpam-4827	292	6	,	,	PUNCT
ejpam-4827	292	7	m	m	AUX
ejpam-4827	292	8	be	be	AUX
ejpam-4827	292	9	a	a	DET
ejpam-4827	292	10	strictly	strictly	ADV
ejpam-4827	292	11	totally	totally	ADV
ejpam-4827	292	12	ordered	order	VERB
ejpam-4827	292	13	monoid	monoid	NOUN
ejpam-4827	292	14	and	and	CCONJ
ejpam-4827	292	15	σ	σ	NOUN
ejpam-4827	292	16	:	:	PUNCT
ejpam-4827	292	17	m	m	PROPN
ejpam-4827	292	18	→	→	SYM
ejpam-4827	292	19	aut(r	aut(r	PROPN
ejpam-4827	292	20	)	)	PUNCT
ejpam-4827	292	21	a	a	DET
ejpam-4827	292	22	compatible	compatible	ADJ
ejpam-4827	292	23	monoid	monoid	NOUN
ejpam-4827	292	24	homomorphism	homomorphism	NOUN
ejpam-4827	292	25	.	.	PUNCT
ejpam-4827	293	1	if	if	SCONJ
ejpam-4827	293	2	r	r	NOUN
ejpam-4827	293	3	is	be	AUX
ejpam-4827	293	4	semiprime	semiprime	NOUN
ejpam-4827	293	5	,	,	PUNCT
ejpam-4827	293	6	then	then	ADV
ejpam-4827	293	7	r	r	NOUN
ejpam-4827	293	8	is	be	AUX
ejpam-4827	293	9	σ	σ	NOUN
ejpam-4827	293	10	-	-	PUNCT
ejpam-4827	293	11	skew	skew	NOUN
ejpam-4827	293	12	strongly	strongly	ADV
ejpam-4827	293	13	m	m	VERB
ejpam-4827	293	14	-reflexive	-reflexive	ADJ
ejpam-4827	293	15	.	.	PUNCT
ejpam-4827	294	1	proof	proof	NOUN
ejpam-4827	294	2	.	.	PUNCT
ejpam-4827	295	1	since	since	SCONJ
ejpam-4827	295	2	a	a	DET
ejpam-4827	295	3	semiprime	semiprime	NOUN
ejpam-4827	295	4	ring	ring	NOUN
ejpam-4827	295	5	is	be	AUX
ejpam-4827	295	6	quasi	quasi	ADJ
ejpam-4827	295	7	-	-	ADJ
ejpam-4827	295	8	armendariz	armendariz	ADJ
ejpam-4827	295	9	and	and	CCONJ
ejpam-4827	295	10	so	so	ADV
ejpam-4827	295	11	reflexive	reflexive	ADJ
ejpam-4827	295	12	,	,	PUNCT
ejpam-4827	295	13	the	the	DET
ejpam-4827	295	14	proof	proof	NOUN
ejpam-4827	295	15	follows	follow	VERB
ejpam-4827	295	16	from	from	ADP
ejpam-4827	295	17	corollary	corollary	ADJ
ejpam-4827	295	18	1	1	NUM
ejpam-4827	295	19	.	.	PUNCT
ejpam-4827	296	1	for	for	ADP
ejpam-4827	296	2	a	a	DET
ejpam-4827	296	3	ring	ring	NOUN
ejpam-4827	296	4	r	r	NOUN
ejpam-4827	296	5	and	and	CCONJ
ejpam-4827	296	6	n	n	PRON
ejpam-4827	296	7	≥	≥	NOUN
ejpam-4827	296	8	2	2	NUM
ejpam-4827	296	9	,	,	PUNCT
ejpam-4827	296	10	let	let	VERB
ejpam-4827	296	11	vn(r	vn(r	PRON
ejpam-4827	296	12	)	)	PUNCT
ejpam-4827	296	13	be	be	AUX
ejpam-4827	296	14	the	the	DET
ejpam-4827	296	15	ring	ring	NOUN
ejpam-4827	296	16	of	of	ADP
ejpam-4827	296	17	all	all	DET
ejpam-4827	296	18	n×n	n×n	PROPN
ejpam-4827	296	19	upper	upper	ADJ
ejpam-4827	296	20	triangular	triangular	NOUN
ejpam-4827	296	21	matrices	matrix	NOUN
ejpam-4827	296	22	over	over	ADP
ejpam-4827	296	23	r	r	NOUN
ejpam-4827	296	24	that	that	PRON
ejpam-4827	296	25	are	be	AUX
ejpam-4827	296	26	constant	constant	ADJ
ejpam-4827	296	27	on	on	ADP
ejpam-4827	296	28	the	the	DET
ejpam-4827	296	29	diagonal	diagonal	NOUN
ejpam-4827	296	30	.	.	PUNCT
ejpam-4827	297	1	let	let	VERB
ejpam-4827	297	2	σ	σ	NOUN
ejpam-4827	297	3	:	:	PUNCT
ejpam-4827	297	4	m	m	PROPN
ejpam-4827	297	5	→	→	SYM
ejpam-4827	297	6	aut(r	aut(r	PROPN
ejpam-4827	297	7	)	)	PUNCT
ejpam-4827	297	8	be	be	AUX
ejpam-4827	297	9	a	a	DET
ejpam-4827	297	10	monoid	monoid	NOUN
ejpam-4827	297	11	homomorphism	homomorphism	NOUN
ejpam-4827	297	12	.	.	PUNCT
ejpam-4827	298	1	for	for	ADP
ejpam-4827	298	2	each	each	DET
ejpam-4827	298	3	g	g	NOUN
ejpam-4827	298	4	∈m	∈m	NOUN
ejpam-4827	298	5	,	,	PUNCT
ejpam-4827	298	6	σ	σ	PROPN
ejpam-4827	298	7	can	can	AUX
ejpam-4827	298	8	be	be	AUX
ejpam-4827	298	9	extended	extend	VERB
ejpam-4827	298	10	to	to	ADP
ejpam-4827	298	11	a	a	DET
ejpam-4827	298	12	monoid	monoid	NOUN
ejpam-4827	298	13	homomorphism	homomorphism	PROPN
ejpam-4827	298	14	σ	σ	NUM
ejpam-4827	298	15	from	from	ADP
ejpam-4827	298	16	m	m	PRON
ejpam-4827	298	17	to	to	ADP
ejpam-4827	298	18	aut(vn(r	aut(vn(r	NUM
ejpam-4827	298	19	)	)	PUNCT
ejpam-4827	298	20	)	)	PUNCT
ejpam-4827	298	21	defined	define	VERB
ejpam-4827	298	22	by	by	ADP
ejpam-4827	298	23	σ((aij	σ((aij	NOUN
ejpam-4827	298	24	)	)	PUNCT
ejpam-4827	298	25	)	)	PUNCT
ejpam-4827	299	1	=	=	SYM
ejpam-4827	299	2	(	(	PUNCT
ejpam-4827	299	3	σg(aij	σg(aij	NOUN
ejpam-4827	299	4	)	)	PUNCT
ejpam-4827	299	5	)	)	PUNCT
ejpam-4827	299	6	.	.	PUNCT
ejpam-4827	300	1	e.	e.	PROPN
ejpam-4827	300	2	ali	ali	PROPN
ejpam-4827	300	3	/	/	SYM
ejpam-4827	300	4	eur	eur	PROPN
ejpam-4827	300	5	.	.	PUNCT
ejpam-4827	301	1	j.	j.	PROPN
ejpam-4827	301	2	pure	pure	PROPN
ejpam-4827	301	3	appl	appl	PROPN
ejpam-4827	301	4	.	.	PROPN
ejpam-4827	301	5	math	math	PROPN
ejpam-4827	301	6	,	,	PUNCT
ejpam-4827	301	7	16	16	NUM
ejpam-4827	301	8	(	(	PUNCT
ejpam-4827	301	9	3	3	NUM
ejpam-4827	301	10	)	)	PUNCT
ejpam-4827	301	11	(	(	PUNCT
ejpam-4827	301	12	2023	2023	NUM
ejpam-4827	301	13	)	)	PUNCT
ejpam-4827	301	14	,	,	PUNCT
ejpam-4827	301	15	1878	1878	NUM
ejpam-4827	301	16	-	-	SYM
ejpam-4827	301	17	1893	1893	NUM
ejpam-4827	301	18	1884	1884	NUM
ejpam-4827	301	19	theorem	theorem	NOUN
ejpam-4827	301	20	2	2	NUM
ejpam-4827	301	21	.	.	PUNCT
ejpam-4827	302	1	let	let	AUX
ejpam-4827	302	2	r	r	PRON
ejpam-4827	302	3	be	be	AUX
ejpam-4827	302	4	an	an	DET
ejpam-4827	302	5	m	m	NOUN
ejpam-4827	302	6	-rigid	-rigid	ADJ
ejpam-4827	302	7	ring	ring	NOUN
ejpam-4827	302	8	,	,	PUNCT
ejpam-4827	302	9	where	where	SCONJ
ejpam-4827	302	10	m	m	NOUN
ejpam-4827	302	11	is	be	AUX
ejpam-4827	302	12	a	a	DET
ejpam-4827	302	13	monoid	monoid	NOUN
ejpam-4827	302	14	and	and	CCONJ
ejpam-4827	302	15	σ	σ	NOUN
ejpam-4827	302	16	:	:	PUNCT
ejpam-4827	302	17	m	m	PROPN
ejpam-4827	302	18	→	→	SYM
ejpam-4827	302	19	aut(r	aut(r	PROPN
ejpam-4827	302	20	)	)	PUNCT
ejpam-4827	302	21	is	be	AUX
ejpam-4827	302	22	a	a	DET
ejpam-4827	302	23	monoid	monoid	NOUN
ejpam-4827	302	24	homomorphism	homomorphism	NOUN
ejpam-4827	302	25	,	,	PUNCT
ejpam-4827	302	26	let	let	VERB
ejpam-4827	302	27	n	n	PRON
ejpam-4827	302	28	≥	≥	NOUN
ejpam-4827	302	29	2	2	NUM
ejpam-4827	302	30	.	.	PUNCT
ejpam-4827	303	1	if	if	SCONJ
ejpam-4827	303	2	r	r	NOUN
ejpam-4827	303	3	is	be	AUX
ejpam-4827	303	4	a	a	DET
ejpam-4827	303	5	left	left	ADJ
ejpam-4827	303	6	app	app	NOUN
ejpam-4827	303	7	-ring	-ring	NOUN
ejpam-4827	303	8	,	,	PUNCT
ejpam-4827	303	9	then	then	ADV
ejpam-4827	303	10	vn(r	vn(r	PUNCT
ejpam-4827	303	11	)	)	PUNCT
ejpam-4827	303	12	is	be	AUX
ejpam-4827	303	13	σ	σ	NOUN
ejpam-4827	303	14	-	-	PUNCT
ejpam-4827	303	15	skew	skew	NOUN
ejpam-4827	303	16	strongly	strongly	ADV
ejpam-4827	303	17	m	m	VERB
ejpam-4827	303	18	-reflexive	-reflexive	ADJ
ejpam-4827	303	19	.	.	PUNCT
ejpam-4827	304	1	proof	proof	NOUN
ejpam-4827	304	2	.	.	PUNCT
ejpam-4827	305	1	suppose	suppose	VERB
ejpam-4827	305	2	that	that	SCONJ
ejpam-4827	305	3	r	r	NOUN
ejpam-4827	305	4	is	be	AUX
ejpam-4827	305	5	a	a	DET
ejpam-4827	305	6	left	left	ADJ
ejpam-4827	305	7	app	app	NOUN
ejpam-4827	305	8	-ring	-ring	NOUN
ejpam-4827	305	9	and	and	CCONJ
ejpam-4827	305	10	let	let	VERB
ejpam-4827	305	11	φ	φ	NOUN
ejpam-4827	305	12	=	=	PUNCT
ejpam-4827	306	1	a1g1	a1g1	PROPN
ejpam-4827	306	2	+	+	NUM
ejpam-4827	306	3	a2g2	a2g2	PROPN
ejpam-4827	306	4	+	+	CCONJ
ejpam-4827	306	5	·	·	PUNCT
ejpam-4827	306	6	·	·	PUNCT
ejpam-4827	306	7	·	·	PUNCT
ejpam-4827	307	1	+	+	NUM
ejpam-4827	307	2	angn	angn	NOUN
ejpam-4827	307	3	and	and	CCONJ
ejpam-4827	307	4	ψ	ψ	X
ejpam-4827	307	5	=	=	X
ejpam-4827	307	6	b1h1	b1h1	NOUN
ejpam-4827	308	1	+	+	CCONJ
ejpam-4827	308	2	b2h2	b2h2	ADP
ejpam-4827	308	3	+	+	X
ejpam-4827	308	4	·	·	PUNCT
ejpam-4827	308	5	·	·	PUNCT
ejpam-4827	308	6	·	·	PUNCT
ejpam-4827	309	1	+	+	NUM
ejpam-4827	309	2	bmhm	bmhm	NOUN
ejpam-4827	309	3	∈	∈	PROPN
ejpam-4827	309	4	vn(r	vn(r	NOUN
ejpam-4827	309	5	)	)	PUNCT
ejpam-4827	309	6	∗m	∗m	NOUN
ejpam-4827	309	7	such	such	ADJ
ejpam-4827	309	8	that	that	SCONJ
ejpam-4827	309	9	φ(vn(r	φ(vn(r	NOUN
ejpam-4827	309	10	)	)	PUNCT
ejpam-4827	309	11	∗m)ψ	∗m)ψ	NOUN
ejpam-4827	309	12	=	=	PUNCT
ejpam-4827	309	13	0	0	X
ejpam-4827	309	14	.	.	PUNCT
ejpam-4827	310	1	we	we	PRON
ejpam-4827	310	2	use	use	VERB
ejpam-4827	310	3	(	(	PUNCT
ejpam-4827	310	4	a1	a1	NOUN
ejpam-4827	310	5	,	,	PUNCT
ejpam-4827	310	6	a2	a2	PROPN
ejpam-4827	310	7	,	,	PUNCT
ejpam-4827	310	8	.	.	PUNCT
ejpam-4827	310	9	.	.	PUNCT
ejpam-4827	310	10	.	.	PUNCT
ejpam-4827	311	1	,	,	PUNCT
ejpam-4827	311	2	an	an	X
ejpam-4827	311	3	)	)	PUNCT
ejpam-4827	311	4	∈	∈	PROPN
ejpam-4827	311	5	vn(r	vn(r	NOUN
ejpam-4827	311	6	)	)	PUNCT
ejpam-4827	311	7	,	,	PUNCT
ejpam-4827	311	8	where	where	SCONJ
ejpam-4827	311	9	ai	ai	VERB
ejpam-4827	311	10	=	=	ADJ
ejpam-4827	311	11			ADJ
ejpam-4827	311	12	a1	a1	NOUN
ejpam-4827	311	13	(	(	PUNCT
ejpam-4827	311	14	i	i	NOUN
ejpam-4827	311	15	)	)	PUNCT
ejpam-4827	311	16	a2	a2	PROPN
ejpam-4827	311	17	(	(	PUNCT
ejpam-4827	311	18	i	i	NOUN
ejpam-4827	311	19	)	)	PUNCT
ejpam-4827	311	20	a3	a3	NOUN
ejpam-4827	311	21	(	(	PUNCT
ejpam-4827	311	22	i	i	NOUN
ejpam-4827	311	23	)	)	PUNCT
ejpam-4827	311	24	·	·	PUNCT
ejpam-4827	311	25	·	·	PUNCT
ejpam-4827	311	26	·	·	PUNCT
ejpam-4827	312	1	an	an	PRON
ejpam-4827	312	2	(	(	PUNCT
ejpam-4827	312	3	i	i	NOUN
ejpam-4827	312	4	)	)	PUNCT
ejpam-4827	312	5	0	0	NUM
ejpam-4827	312	6	a1	a1	NOUN
ejpam-4827	312	7	(	(	PUNCT
ejpam-4827	312	8	i	i	NOUN
ejpam-4827	312	9	)	)	PUNCT
ejpam-4827	312	10	a2	a2	PROPN
ejpam-4827	312	11	(	(	PUNCT
ejpam-4827	312	12	i	i	NOUN
ejpam-4827	312	13	)	)	PUNCT
ejpam-4827	312	14	·	·	PUNCT
ejpam-4827	312	15	·	·	PUNCT
ejpam-4827	312	16	·	·	PUNCT
ejpam-4827	313	1	an−1	an−1	INTJ
ejpam-4827	313	2	(	(	PUNCT
ejpam-4827	313	3	i	i	NOUN
ejpam-4827	313	4	)	)	PUNCT
ejpam-4827	313	5	0	0	SYM
ejpam-4827	313	6	0	0	NUM
ejpam-4827	313	7	a1	a1	NOUN
ejpam-4827	313	8	(	(	PUNCT
ejpam-4827	313	9	i	i	NOUN
ejpam-4827	313	10	)	)	PUNCT
ejpam-4827	313	11	·	·	PUNCT
ejpam-4827	313	12	·	·	PUNCT
ejpam-4827	313	13	·	·	PUNCT
ejpam-4827	314	1	an−2	an−2	PROPN
ejpam-4827	314	2	(	(	PUNCT
ejpam-4827	314	3	i	i	PROPN
ejpam-4827	314	4	)	)	PUNCT
ejpam-4827	314	5	...	...	PUNCT
ejpam-4827	314	6	...	...	PUNCT
ejpam-4827	314	7	...	...	PUNCT
ejpam-4827	314	8	.	.	PUNCT
ejpam-4827	314	9	.	.	PUNCT
ejpam-4827	314	10	.	.	PUNCT
ejpam-4827	315	1	...	...	PUNCT
ejpam-4827	316	1	0	0	NUM
ejpam-4827	316	2	0	0	NUM
ejpam-4827	316	3	0	0	NUM
ejpam-4827	316	4	·	·	PUNCT
ejpam-4827	316	5	·	·	PUNCT
ejpam-4827	316	6	·	·	PUNCT
ejpam-4827	316	7	a1	a1	NOUN
ejpam-4827	316	8	(	(	PUNCT
ejpam-4827	316	9	i	i	NOUN
ejpam-4827	316	10	)	)	PUNCT
ejpam-4827	316	11			NOUN
ejpam-4827	316	12	,	,	PUNCT
ejpam-4827	316	13	bj	bj	VERB
ejpam-4827	316	14	=	=	SYM
ejpam-4827	316	15			ADJ
ejpam-4827	316	16	b1	b1	NOUN
ejpam-4827	316	17	(	(	PUNCT
ejpam-4827	316	18	j	j	PROPN
ejpam-4827	316	19	)	)	PUNCT
ejpam-4827	316	20	b2	b2	NOUN
ejpam-4827	316	21	(	(	PUNCT
ejpam-4827	316	22	j	j	PROPN
ejpam-4827	316	23	)	)	PUNCT
ejpam-4827	316	24	b3	b3	PROPN
ejpam-4827	316	25	(	(	PUNCT
ejpam-4827	316	26	j	j	PROPN
ejpam-4827	316	27	)	)	PUNCT
ejpam-4827	316	28	·	·	PUNCT
ejpam-4827	316	29	·	·	PUNCT
ejpam-4827	316	30	·	·	PUNCT
ejpam-4827	316	31	bn	bn	X
ejpam-4827	316	32	(	(	PUNCT
ejpam-4827	316	33	j	j	NOUN
ejpam-4827	316	34	)	)	PUNCT
ejpam-4827	316	35	0	0	NUM
ejpam-4827	316	36	b1	b1	PROPN
ejpam-4827	316	37	(	(	PUNCT
ejpam-4827	316	38	j	j	NOUN
ejpam-4827	316	39	)	)	PUNCT
ejpam-4827	316	40	b2	b2	NOUN
ejpam-4827	316	41	(	(	PUNCT
ejpam-4827	316	42	j	j	NOUN
ejpam-4827	316	43	)	)	PUNCT
ejpam-4827	316	44	·	·	PUNCT
ejpam-4827	316	45	·	·	PUNCT
ejpam-4827	316	46	·	·	PUNCT
ejpam-4827	317	1	bn−1	bn−1	PRON
ejpam-4827	317	2	(	(	PUNCT
ejpam-4827	317	3	j	j	NOUN
ejpam-4827	317	4	)	)	PUNCT
ejpam-4827	317	5	0	0	NUM
ejpam-4827	317	6	0	0	NUM
ejpam-4827	317	7	b1	b1	PROPN
ejpam-4827	317	8	(	(	PUNCT
ejpam-4827	317	9	j	j	NOUN
ejpam-4827	317	10	)	)	PUNCT
ejpam-4827	317	11	·	·	PUNCT
ejpam-4827	317	12	·	·	PUNCT
ejpam-4827	317	13	·	·	PUNCT
ejpam-4827	318	1	bn−2	bn−2	PROPN
ejpam-4827	318	2	(	(	PUNCT
ejpam-4827	318	3	j	j	PROPN
ejpam-4827	318	4	)	)	PUNCT
ejpam-4827	318	5	...	...	PUNCT
ejpam-4827	318	6	...	...	PUNCT
ejpam-4827	318	7	...	...	PUNCT
ejpam-4827	318	8	.	.	PUNCT
ejpam-4827	318	9	.	.	PUNCT
ejpam-4827	318	10	.	.	PUNCT
ejpam-4827	319	1	...	...	PUNCT
ejpam-4827	320	1	0	0	NUM
ejpam-4827	320	2	0	0	NUM
ejpam-4827	320	3	0	0	NUM
ejpam-4827	320	4	·	·	PUNCT
ejpam-4827	320	5	·	·	PUNCT
ejpam-4827	320	6	·	·	PUNCT
ejpam-4827	320	7	b1	b1	NOUN
ejpam-4827	320	8	(	(	PUNCT
ejpam-4827	320	9	j	j	NOUN
ejpam-4827	320	10	)	)	PUNCT
ejpam-4827	320	11			PROPN
ejpam-4827	320	12	.	.	PUNCT
ejpam-4827	321	1	we	we	PRON
ejpam-4827	321	2	note	note	VERB
ejpam-4827	321	3	that	that	SCONJ
ejpam-4827	321	4	there	there	PRON
ejpam-4827	321	5	is	be	VERB
ejpam-4827	321	6	an	an	DET
ejpam-4827	321	7	obvious	obvious	ADJ
ejpam-4827	321	8	isomorphism	isomorphism	NOUN
ejpam-4827	321	9	vn(r	vn(r	NOUN
ejpam-4827	321	10	)	)	PUNCT
ejpam-4827	321	11	∗m	∗m	PROPN
ejpam-4827	321	12	∼=	∼=	PROPN
ejpam-4827	321	13	vn(r	vn(r	NOUN
ejpam-4827	321	14	∗m	∗m	NOUN
ejpam-4827	321	15	)	)	PUNCT
ejpam-4827	321	16	.	.	PUNCT
ejpam-4827	322	1	therefore	therefore	ADV
ejpam-4827	322	2	,	,	PUNCT
ejpam-4827	322	3	we	we	PRON
ejpam-4827	322	4	can	can	AUX
ejpam-4827	322	5	rewrite	rewrite	VERB
ejpam-4827	322	6	φ	φ	PROPN
ejpam-4827	322	7	and	and	CCONJ
ejpam-4827	322	8	ψ	ψ	PROPN
ejpam-4827	322	9	as	as	ADP
ejpam-4827	322	10	φ	φ	PROPN
ejpam-4827	322	11	=	=	SYM
ejpam-4827	322	12			ADJ
ejpam-4827	322	13	φ1	φ1	PROPN
ejpam-4827	322	14	φ2	φ2	PROPN
ejpam-4827	322	15	φ3	φ3	PROPN
ejpam-4827	322	16	·	·	PUNCT
ejpam-4827	322	17	·	·	PUNCT
ejpam-4827	322	18	·	·	PUNCT
ejpam-4827	323	1	φn	φn	ADP
ejpam-4827	323	2	0	0	NUM
ejpam-4827	323	3	φ1	φ1	PROPN
ejpam-4827	323	4	φ2	φ2	PROPN
ejpam-4827	323	5	·	·	PUNCT
ejpam-4827	323	6	·	·	PUNCT
ejpam-4827	324	1	·	·	PUNCT
ejpam-4827	324	2	φn−1	φn−1	VERB
ejpam-4827	324	3	0	0	NUM
ejpam-4827	324	4	0	0	NUM
ejpam-4827	324	5	φ1	φ1	NOUN
ejpam-4827	324	6	·	·	PUNCT
ejpam-4827	324	7	·	·	PUNCT
ejpam-4827	324	8	·	·	PUNCT
ejpam-4827	325	1	φn−2	φn−2	ADV
ejpam-4827	325	2	...	...	PUNCT
ejpam-4827	325	3	...	...	PUNCT
ejpam-4827	325	4	...	...	PUNCT
ejpam-4827	325	5	.	.	PUNCT
ejpam-4827	325	6	.	.	PUNCT
ejpam-4827	325	7	.	.	PUNCT
ejpam-4827	326	1	...	...	PUNCT
ejpam-4827	327	1	0	0	NUM
ejpam-4827	327	2	0	0	NUM
ejpam-4827	327	3	0	0	NUM
ejpam-4827	327	4	·	·	PUNCT
ejpam-4827	327	5	·	·	PUNCT
ejpam-4827	327	6	·	·	PUNCT
ejpam-4827	327	7	φ1	φ1	PROPN
ejpam-4827	327	8			NOUN
ejpam-4827	327	9	,	,	PUNCT
ejpam-4827	327	10	ψ	ψ	X
ejpam-4827	327	11	=	=	PUNCT
ejpam-4827	327	12			ADJ
ejpam-4827	327	13	ψ1	ψ1	ADJ
ejpam-4827	327	14	ψ2	ψ2	NOUN
ejpam-4827	327	15	ψ3	ψ3	NOUN
ejpam-4827	327	16	·	·	PUNCT
ejpam-4827	327	17	·	·	PUNCT
ejpam-4827	327	18	·	·	PUNCT
ejpam-4827	327	19	ψn	ψn	VERB
ejpam-4827	327	20	0	0	NUM
ejpam-4827	327	21	ψ1	ψ1	ADJ
ejpam-4827	327	22	ψ2	ψ2	NOUN
ejpam-4827	327	23	·	·	PUNCT
ejpam-4827	327	24	·	·	PUNCT
ejpam-4827	327	25	·	·	PUNCT
ejpam-4827	327	26	ψn−1	ψn−1	VERB
ejpam-4827	327	27	0	0	NUM
ejpam-4827	327	28	0	0	NUM
ejpam-4827	327	29	ψ1	ψ1	NOUN
ejpam-4827	327	30	·	·	PUNCT
ejpam-4827	327	31	·	·	PUNCT
ejpam-4827	327	32	·	·	PUNCT
ejpam-4827	328	1	ψn−2	ψn−2	NOUN
ejpam-4827	328	2	...	...	PUNCT
ejpam-4827	328	3	...	...	PUNCT
ejpam-4827	328	4	...	...	PUNCT
ejpam-4827	328	5	.	.	PUNCT
ejpam-4827	328	6	.	.	PUNCT
ejpam-4827	328	7	.	.	PUNCT
ejpam-4827	329	1	...	...	PUNCT
ejpam-4827	330	1	0	0	NUM
ejpam-4827	330	2	0	0	NUM
ejpam-4827	330	3	0	0	NUM
ejpam-4827	330	4	·	·	PUNCT
ejpam-4827	330	5	·	·	PUNCT
ejpam-4827	330	6	·	·	PUNCT
ejpam-4827	330	7	ψ1	ψ1	NOUN
ejpam-4827	330	8			PROPN
ejpam-4827	330	9	.	.	PUNCT
ejpam-4827	331	1	let	let	VERB
ejpam-4827	331	2	a(vn(r))b	a(vn(r))b	NOUN
ejpam-4827	331	3	=	=	SYM
ejpam-4827	331	4	0	0	NUM
ejpam-4827	331	5	for	for	ADP
ejpam-4827	331	6	a	a	DET
ejpam-4827	331	7	=	=	SYM
ejpam-4827	331	8	(	(	PUNCT
ejpam-4827	331	9	a1	a1	PROPN
ejpam-4827	331	10	,	,	PUNCT
ejpam-4827	331	11	a2	a2	PROPN
ejpam-4827	331	12	,	,	PUNCT
ejpam-4827	331	13	.	.	PUNCT
ejpam-4827	331	14	.	.	PUNCT
ejpam-4827	332	1	.	.	PUNCT
ejpam-4827	333	1	,	,	PUNCT
ejpam-4827	333	2	an	an	X
ejpam-4827	333	3	)	)	PUNCT
ejpam-4827	333	4	,	,	PUNCT
ejpam-4827	333	5	b	b	X
ejpam-4827	333	6	=	=	SYM
ejpam-4827	333	7	(	(	PUNCT
ejpam-4827	333	8	b1	b1	PROPN
ejpam-4827	333	9	,	,	PUNCT
ejpam-4827	333	10	b2	b2	NOUN
ejpam-4827	333	11	,	,	PUNCT
ejpam-4827	333	12	.	.	PUNCT
ejpam-4827	333	13	.	.	PUNCT
ejpam-4827	334	1	.	.	PUNCT
ejpam-4827	335	1	,	,	PUNCT
ejpam-4827	335	2	bn	bn	X
ejpam-4827	335	3	)	)	PUNCT
ejpam-4827	335	4	∈	∈	PROPN
ejpam-4827	335	5	vn(r	vn(r	NOUN
ejpam-4827	335	6	)	)	PUNCT
ejpam-4827	335	7	.	.	PUNCT
ejpam-4827	336	1	for	for	ADP
ejpam-4827	336	2	any	any	DET
ejpam-4827	336	3	r	r	NOUN
ejpam-4827	336	4	∈	∈	NOUN
ejpam-4827	336	5	r	r	NOUN
ejpam-4827	336	6	,	,	PUNCT
ejpam-4827	336	7	a(r	a(r	NOUN
ejpam-4827	336	8	,	,	PUNCT
ejpam-4827	336	9	0	0	NUM
ejpam-4827	336	10	,	,	PUNCT
ejpam-4827	336	11	.	.	PUNCT
ejpam-4827	336	12	.	.	PUNCT
ejpam-4827	336	13	.	.	PUNCT
ejpam-4827	337	1	,	,	PUNCT
ejpam-4827	337	2	0)b	0)b	NUM
ejpam-4827	337	3	=	=	SYM
ejpam-4827	337	4	0	0	NUM
ejpam-4827	337	5	.	.	PUNCT
ejpam-4827	338	1	thus	thus	ADV
ejpam-4827	338	2	we	we	PRON
ejpam-4827	338	3	have	have	VERB
ejpam-4827	338	4	the	the	DET
ejpam-4827	338	5	following	follow	VERB
ejpam-4827	338	6	equations	equation	NOUN
ejpam-4827	338	7	:	:	PUNCT
ejpam-4827	338	8	a1	a1	NOUN
ejpam-4827	338	9	(	(	PUNCT
ejpam-4827	338	10	i)rb1	i)rb1	PROPN
ejpam-4827	338	11	(	(	PUNCT
ejpam-4827	338	12	j	j	NOUN
ejpam-4827	338	13	)	)	PUNCT
ejpam-4827	338	14	=	=	SYM
ejpam-4827	339	1	0	0	X
ejpam-4827	339	2	.	.	PUNCT
ejpam-4827	339	3	(	(	PUNCT
ejpam-4827	339	4	2.5	2.5	NUM
ejpam-4827	339	5	)	)	PUNCT
ejpam-4827	339	6	a1	a1	NOUN
ejpam-4827	339	7	(	(	PUNCT
ejpam-4827	339	8	i)rb2	i)rb2	PROPN
ejpam-4827	339	9	(	(	PUNCT
ejpam-4827	339	10	j	j	NOUN
ejpam-4827	339	11	)	)	PUNCT
ejpam-4827	339	12	+	+	NUM
ejpam-4827	339	13	a2	a2	PROPN
ejpam-4827	339	14	(	(	PUNCT
ejpam-4827	339	15	i)rb1	i)rb1	PROPN
ejpam-4827	339	16	(	(	PUNCT
ejpam-4827	339	17	j	j	NOUN
ejpam-4827	339	18	)	)	PUNCT
ejpam-4827	339	19	=	=	SYM
ejpam-4827	339	20	0	0	X
ejpam-4827	339	21	.	.	PUNCT
ejpam-4827	340	1	(	(	PUNCT
ejpam-4827	340	2	2.6	2.6	NUM
ejpam-4827	340	3	)	)	PUNCT
ejpam-4827	340	4	a1	a1	NOUN
ejpam-4827	340	5	(	(	PUNCT
ejpam-4827	340	6	i)rb3	i)rb3	PROPN
ejpam-4827	340	7	(	(	PUNCT
ejpam-4827	340	8	j	j	NOUN
ejpam-4827	340	9	)	)	PUNCT
ejpam-4827	340	10	+	+	NUM
ejpam-4827	340	11	a2	a2	PROPN
ejpam-4827	340	12	(	(	PUNCT
ejpam-4827	340	13	i)rb2	i)rb2	PROPN
ejpam-4827	340	14	(	(	PUNCT
ejpam-4827	340	15	j	j	NOUN
ejpam-4827	340	16	)	)	PUNCT
ejpam-4827	340	17	+	+	NUM
ejpam-4827	340	18	a3	a3	NOUN
ejpam-4827	340	19	(	(	PUNCT
ejpam-4827	340	20	i)rb1	i)rb1	PROPN
ejpam-4827	340	21	(	(	PUNCT
ejpam-4827	340	22	j	j	NOUN
ejpam-4827	340	23	)	)	PUNCT
ejpam-4827	340	24	=	=	SYM
ejpam-4827	340	25	0	0	X
ejpam-4827	340	26	.	.	PUNCT
ejpam-4827	340	27	(	(	PUNCT
ejpam-4827	340	28	2.7	2.7	NUM
ejpam-4827	340	29	)	)	PUNCT
ejpam-4827	340	30	...	...	PUNCT
ejpam-4827	341	1	a1	a1	NOUN
ejpam-4827	341	2	(	(	PUNCT
ejpam-4827	341	3	i)rbn−1	i)rbn−1	PROPN
ejpam-4827	341	4	(	(	PUNCT
ejpam-4827	341	5	j	j	NOUN
ejpam-4827	341	6	)	)	PUNCT
ejpam-4827	341	7	+	+	NUM
ejpam-4827	341	8	a2	a2	PROPN
ejpam-4827	341	9	(	(	PUNCT
ejpam-4827	341	10	i)rbn−2	i)rbn−2	PROPN
ejpam-4827	341	11	(	(	PUNCT
ejpam-4827	341	12	j	j	NOUN
ejpam-4827	341	13	)	)	PUNCT
ejpam-4827	341	14	+	+	CCONJ
ejpam-4827	341	15	·	·	PUNCT
ejpam-4827	341	16	·	·	PUNCT
ejpam-4827	341	17	·	·	PUNCT
ejpam-4827	341	18	+	+	CCONJ
ejpam-4827	341	19	an−1	an−1	PROPN
ejpam-4827	341	20	(	(	PUNCT
ejpam-4827	341	21	i)rb1	i)rb1	PROPN
ejpam-4827	341	22	(	(	PUNCT
ejpam-4827	341	23	j	j	NOUN
ejpam-4827	341	24	)	)	PUNCT
ejpam-4827	341	25	=	=	SYM
ejpam-4827	341	26	0	0	X
ejpam-4827	341	27	.	.	PUNCT
ejpam-4827	342	1	(	(	PUNCT
ejpam-4827	342	2	2.8	2.8	NUM
ejpam-4827	342	3	)	)	PUNCT
ejpam-4827	342	4	a1	a1	NOUN
ejpam-4827	342	5	(	(	PUNCT
ejpam-4827	342	6	i)rbn	i)rbn	PROPN
ejpam-4827	342	7	(	(	PUNCT
ejpam-4827	342	8	j	j	NOUN
ejpam-4827	342	9	)	)	PUNCT
ejpam-4827	342	10	+	+	NUM
ejpam-4827	342	11	a2	a2	PROPN
ejpam-4827	342	12	(	(	PUNCT
ejpam-4827	342	13	i)rbn−1	i)rbn−1	PROPN
ejpam-4827	342	14	(	(	PUNCT
ejpam-4827	342	15	j	j	NOUN
ejpam-4827	342	16	)	)	PUNCT
ejpam-4827	342	17	+	+	CCONJ
ejpam-4827	342	18	·	·	PUNCT
ejpam-4827	342	19	·	·	PUNCT
ejpam-4827	342	20	·	·	PUNCT
ejpam-4827	343	1	+	+	CCONJ
ejpam-4827	343	2	an	an	DET
ejpam-4827	343	3	(	(	PUNCT
ejpam-4827	343	4	i)rb1	i)rb1	NOUN
ejpam-4827	343	5	(	(	PUNCT
ejpam-4827	343	6	j	j	NOUN
ejpam-4827	343	7	)	)	PUNCT
ejpam-4827	343	8	=	=	SYM
ejpam-4827	343	9	0	0	X
ejpam-4827	343	10	.	.	PUNCT
ejpam-4827	343	11	(	(	PUNCT
ejpam-4827	343	12	2.9	2.9	NUM
ejpam-4827	343	13	)	)	PUNCT
ejpam-4827	343	14	now	now	ADV
ejpam-4827	343	15	for	for	ADP
ejpam-4827	343	16	a	a	DET
ejpam-4827	343	17	monoid	monoid	NOUN
ejpam-4827	343	18	m	m	NOUN
ejpam-4827	343	19	and	and	CCONJ
ejpam-4827	343	20	σ	σ	NOUN
ejpam-4827	343	21	:	:	PUNCT
ejpam-4827	343	22	m	m	PROPN
ejpam-4827	343	23	→	→	SYM
ejpam-4827	343	24	aut(r	aut(r	PROPN
ejpam-4827	343	25	)	)	PUNCT
ejpam-4827	343	26	a	a	DET
ejpam-4827	343	27	monoid	monoid	NOUN
ejpam-4827	343	28	homomorphism	homomorphism	NOUN
ejpam-4827	343	29	.	.	PUNCT
ejpam-4827	344	1	from	from	ADP
ejpam-4827	344	2	eq.(2.5	eq.(2.5	PROPN
ejpam-4827	344	3	)	)	PUNCT
ejpam-4827	344	4	we	we	PRON
ejpam-4827	344	5	see	see	VERB
ejpam-4827	344	6	a1(i)σgi(rσs(b1	a1(i)σgi(rσs(b1	NOUN
ejpam-4827	344	7	(	(	PUNCT
ejpam-4827	344	8	j	j	NOUN
ejpam-4827	344	9	)	)	PUNCT
ejpam-4827	344	10	)	)	PUNCT
ejpam-4827	344	11	)	)	PUNCT
ejpam-4827	345	1	=	=	SYM
ejpam-4827	345	2	0	0	NUM
ejpam-4827	346	1	for	for	ADP
ejpam-4827	346	2	all	all	DET
ejpam-4827	346	3	i	i	PROPN
ejpam-4827	346	4	,	,	PUNCT
ejpam-4827	346	5	j	j	PROPN
ejpam-4827	346	6	and	and	CCONJ
ejpam-4827	346	7	s	s	PROPN
ejpam-4827	346	8	∈	∈	PROPN
ejpam-4827	346	9	m.	m.	NOUN
ejpam-4827	346	10	hence	hence	ADV
ejpam-4827	346	11	a1(i	a1(i	PROPN
ejpam-4827	346	12	)	)	PUNCT
ejpam-4827	346	13	∈	∈	PROPN
ejpam-4827	346	14	ℓr(rσs(b1	ℓr(rσs(b1	PROPN
ejpam-4827	346	15	(	(	PUNCT
ejpam-4827	346	16	j	j	NOUN
ejpam-4827	346	17	)	)	PUNCT
ejpam-4827	346	18	)	)	PUNCT
ejpam-4827	346	19	)	)	PUNCT
ejpam-4827	346	20	.	.	PUNCT
ejpam-4827	347	1	by	by	ADP
ejpam-4827	347	2	hypothesis	hypothesis	NOUN
ejpam-4827	347	3	,	,	PUNCT
ejpam-4827	347	4	r	r	NOUN
ejpam-4827	347	5	is	be	AUX
ejpam-4827	347	6	a	a	DET
ejpam-4827	347	7	left	left	ADJ
ejpam-4827	347	8	app	app	NOUN
ejpam-4827	347	9	,	,	PUNCT
ejpam-4827	347	10	ℓr(rσs(b1(j	ℓr(rσs(b1(j	PROPN
ejpam-4827	347	11	)	)	PUNCT
ejpam-4827	347	12	)	)	PUNCT
ejpam-4827	347	13	)	)	PUNCT
ejpam-4827	347	14	is	be	AUX
ejpam-4827	347	15	left	leave	VERB
ejpam-4827	347	16	s	s	PART
ejpam-4827	347	17	-	-	NOUN
ejpam-4827	347	18	unital	unital	ADJ
ejpam-4827	347	19	by	by	ADP
ejpam-4827	347	20	lemma	lemma	PROPN
ejpam-4827	347	21	1	1	NUM
ejpam-4827	347	22	.	.	PUNCT
ejpam-4827	348	1	hence	hence	ADV
ejpam-4827	348	2	there	there	PRON
ejpam-4827	348	3	exist	exist	VERB
ejpam-4827	348	4	en	en	X
ejpam-4827	348	5	∈	∈	PROPN
ejpam-4827	348	6	ℓr(rσs(b1	ℓr(rσs(b1	PROPN
ejpam-4827	348	7	(	(	PUNCT
ejpam-4827	348	8	j	j	NOUN
ejpam-4827	348	9	)	)	PUNCT
ejpam-4827	348	10	)	)	PUNCT
ejpam-4827	348	11	)	)	PUNCT
ejpam-4827	348	12	such	such	ADJ
ejpam-4827	348	13	that	that	DET
ejpam-4827	348	14	a1(i)en	a1(i)en	NOUN
ejpam-4827	348	15	=	=	SYM
ejpam-4827	348	16	a1	a1	PROPN
ejpam-4827	348	17	(	(	PUNCT
ejpam-4827	348	18	i	i	NOUN
ejpam-4827	348	19	)	)	PUNCT
ejpam-4827	348	20	since	since	SCONJ
ejpam-4827	348	21	σgi	σgi	PROPN
ejpam-4827	348	22	is	be	AUX
ejpam-4827	348	23	an	an	DET
ejpam-4827	348	24	automorphism	automorphism	NOUN
ejpam-4827	348	25	,	,	PUNCT
ejpam-4827	348	26	i	i	PRON
ejpam-4827	348	27	=	=	NOUN
ejpam-4827	348	28	1	1	NUM
ejpam-4827	348	29	,	,	PUNCT
ejpam-4827	348	30	2	2	NUM
ejpam-4827	348	31	,	,	PUNCT
ejpam-4827	348	32	.	.	PUNCT
ejpam-4827	348	33	.	.	PUNCT
ejpam-4827	349	1	.	.	PUNCT
ejpam-4827	350	1	,	,	PUNCT
ejpam-4827	350	2	n.	n.	PROPN
ejpam-4827	350	3	e.	e.	PROPN
ejpam-4827	350	4	ali	ali	PROPN
ejpam-4827	350	5	/	/	SYM
ejpam-4827	350	6	eur	eur	PROPN
ejpam-4827	350	7	.	.	PUNCT
ejpam-4827	351	1	j.	j.	PROPN
ejpam-4827	351	2	pure	pure	PROPN
ejpam-4827	351	3	appl	appl	PROPN
ejpam-4827	351	4	.	.	PROPN
ejpam-4827	351	5	math	math	PROPN
ejpam-4827	351	6	,	,	PUNCT
ejpam-4827	351	7	16	16	NUM
ejpam-4827	351	8	(	(	PUNCT
ejpam-4827	351	9	3	3	NUM
ejpam-4827	351	10	)	)	PUNCT
ejpam-4827	351	11	(	(	PUNCT
ejpam-4827	351	12	2023	2023	NUM
ejpam-4827	351	13	)	)	PUNCT
ejpam-4827	351	14	,	,	PUNCT
ejpam-4827	351	15	1878	1878	NUM
ejpam-4827	351	16	-	-	SYM
ejpam-4827	351	17	1893	1893	NUM
ejpam-4827	351	18	1885	1885	NUM
ejpam-4827	351	19	this	this	PRON
ejpam-4827	351	20	implies	imply	VERB
ejpam-4827	351	21	that	that	SCONJ
ejpam-4827	351	22	,	,	PUNCT
ejpam-4827	351	23	b1(j)σhj	b1(j)σhj	PROPN
ejpam-4827	351	24	(	(	PUNCT
ejpam-4827	351	25	rσs(a1	rσs(a1	PROPN
ejpam-4827	351	26	(	(	PUNCT
ejpam-4827	351	27	i	i	NOUN
ejpam-4827	351	28	)	)	PUNCT
ejpam-4827	351	29	)	)	PUNCT
ejpam-4827	351	30	)	)	PUNCT
ejpam-4827	352	1	=	=	SYM
ejpam-4827	352	2	0	0	PUNCT
ejpam-4827	353	1	by	by	ADP
ejpam-4827	353	2	rigidness	rigidness	NOUN
ejpam-4827	353	3	and	and	CCONJ
ejpam-4827	353	4	we	we	PRON
ejpam-4827	353	5	obtain	obtain	VERB
ejpam-4827	353	6	ψ1(vn(r	ψ1(vn(r	NOUN
ejpam-4827	353	7	)	)	PUNCT
ejpam-4827	353	8	∗m)φ1	∗m)φ1	NOUN
ejpam-4827	353	9	=	=	NOUN
ejpam-4827	354	1	0	0	X
ejpam-4827	354	2	.	.	PUNCT
ejpam-4827	355	1	if	if	SCONJ
ejpam-4827	355	2	we	we	PRON
ejpam-4827	355	3	multiplying	multiply	VERB
ejpam-4827	355	4	eq.(2.6	eq.(2.6	PROPN
ejpam-4827	355	5	)	)	PUNCT
ejpam-4827	355	6	on	on	ADP
ejpam-4827	355	7	the	the	DET
ejpam-4827	355	8	right	right	ADJ
ejpam-4827	355	9	-	-	PUNCT
ejpam-4827	355	10	hand	hand	NOUN
ejpam-4827	355	11	side	side	NOUN
ejpam-4827	355	12	by	by	ADP
ejpam-4827	355	13	tb1(j	tb1(j	PROPN
ejpam-4827	355	14	)	)	PUNCT
ejpam-4827	355	15	for	for	ADP
ejpam-4827	355	16	any	any	DET
ejpam-4827	355	17	t	t	NOUN
ejpam-4827	355	18	∈	∈	PROPN
ejpam-4827	355	19	r	r	NOUN
ejpam-4827	355	20	,	,	PUNCT
ejpam-4827	355	21	then	then	ADV
ejpam-4827	355	22	a1	a1	NOUN
ejpam-4827	355	23	(	(	PUNCT
ejpam-4827	355	24	i)rb2	i)rb2	PROPN
ejpam-4827	355	25	(	(	PUNCT
ejpam-4827	355	26	j)tb1	j)tb1	PROPN
ejpam-4827	355	27	(	(	PUNCT
ejpam-4827	355	28	j	j	PROPN
ejpam-4827	355	29	)	)	PUNCT
ejpam-4827	355	30	+	+	NUM
ejpam-4827	355	31	a2	a2	PROPN
ejpam-4827	355	32	(	(	PUNCT
ejpam-4827	355	33	i)rb1	i)rb1	PROPN
ejpam-4827	355	34	(	(	PUNCT
ejpam-4827	355	35	j)tb1	j)tb1	PROPN
ejpam-4827	355	36	(	(	PUNCT
ejpam-4827	355	37	j	j	PROPN
ejpam-4827	355	38	)	)	PUNCT
ejpam-4827	355	39	=	=	SYM
ejpam-4827	356	1	0	0	X
ejpam-4827	356	2	.	.	PUNCT
ejpam-4827	357	1	(	(	PUNCT
ejpam-4827	357	2	2.10	2.10	NUM
ejpam-4827	357	3	)	)	PUNCT
ejpam-4827	357	4	hence	hence	ADV
ejpam-4827	357	5	a2	a2	PROPN
ejpam-4827	357	6	(	(	PUNCT
ejpam-4827	357	7	i)rb1	i)rb1	PROPN
ejpam-4827	357	8	(	(	PUNCT
ejpam-4827	357	9	j	j	NOUN
ejpam-4827	357	10	)	)	PUNCT
ejpam-4827	357	11	=	=	NOUN
ejpam-4827	358	1	0	0	X
ejpam-4827	358	2	.	.	PUNCT
ejpam-4827	359	1	since	since	SCONJ
ejpam-4827	359	2	r	r	NOUN
ejpam-4827	359	3	is	be	AUX
ejpam-4827	359	4	m	m	PRON
ejpam-4827	359	5	-rigid	-rigid	ADJ
ejpam-4827	359	6	we	we	PRON
ejpam-4827	359	7	have	have	VERB
ejpam-4827	359	8	a2	a2	PROPN
ejpam-4827	359	9	(	(	PUNCT
ejpam-4827	359	10	i)σgi(rσs(b1	i)σgi(rσs(b1	NOUN
ejpam-4827	359	11	(	(	PUNCT
ejpam-4827	359	12	j	j	NOUN
ejpam-4827	359	13	)	)	PUNCT
ejpam-4827	359	14	)	)	PUNCT
ejpam-4827	359	15	)	)	PUNCT
ejpam-4827	360	1	=	=	PUNCT
ejpam-4827	360	2	0	0	X
ejpam-4827	360	3	.	.	PUNCT
ejpam-4827	360	4	hence	hence	ADV
ejpam-4827	360	5	a2	a2	PROPN
ejpam-4827	360	6	(	(	PUNCT
ejpam-4827	360	7	i	i	NOUN
ejpam-4827	360	8	)	)	PUNCT
ejpam-4827	360	9	∈	∈	PROPN
ejpam-4827	360	10	ℓr(rσs(b1	ℓr(rσs(b1	PROPN
ejpam-4827	360	11	(	(	PUNCT
ejpam-4827	360	12	j	j	NOUN
ejpam-4827	360	13	)	)	PUNCT
ejpam-4827	360	14	)	)	PUNCT
ejpam-4827	360	15	)	)	PUNCT
ejpam-4827	360	16	.	.	PUNCT
ejpam-4827	361	1	by	by	ADP
ejpam-4827	361	2	hypothesis	hypothesis	NOUN
ejpam-4827	361	3	,	,	PUNCT
ejpam-4827	361	4	r	r	NOUN
ejpam-4827	361	5	is	be	AUX
ejpam-4827	361	6	a	a	DET
ejpam-4827	361	7	left	left	ADJ
ejpam-4827	361	8	app	app	NOUN
ejpam-4827	361	9	,	,	PUNCT
ejpam-4827	361	10	then	then	ADV
ejpam-4827	361	11	ℓr(rσs(b1(j	ℓr(rσs(b1(j	PROPN
ejpam-4827	361	12	)	)	PUNCT
ejpam-4827	361	13	)	)	PUNCT
ejpam-4827	361	14	)	)	PUNCT
ejpam-4827	361	15	is	be	AUX
ejpam-4827	361	16	left	leave	VERB
ejpam-4827	361	17	s	s	PART
ejpam-4827	361	18	-	-	NOUN
ejpam-4827	361	19	unital	unital	ADJ
ejpam-4827	361	20	by	by	ADP
ejpam-4827	361	21	lemma	lemma	PROPN
ejpam-4827	361	22	1	1	NUM
ejpam-4827	361	23	.	.	PUNCT
ejpam-4827	362	1	hence	hence	ADV
ejpam-4827	362	2	there	there	PRON
ejpam-4827	362	3	exist	exist	VERB
ejpam-4827	362	4	en	en	X
ejpam-4827	362	5	∈	∈	PROPN
ejpam-4827	362	6	ℓr(rσs(b1	ℓr(rσs(b1	PROPN
ejpam-4827	362	7	(	(	PUNCT
ejpam-4827	362	8	j	j	NOUN
ejpam-4827	362	9	)	)	PUNCT
ejpam-4827	362	10	)	)	PUNCT
ejpam-4827	362	11	)	)	PUNCT
ejpam-4827	362	12	such	such	ADJ
ejpam-4827	362	13	that	that	ADV
ejpam-4827	362	14	a2(i)en	a2(i)en	ADV
ejpam-4827	362	15	=	=	PROPN
ejpam-4827	362	16	a2	a2	PROPN
ejpam-4827	362	17	(	(	PUNCT
ejpam-4827	362	18	i	i	NOUN
ejpam-4827	362	19	)	)	PUNCT
ejpam-4827	362	20	since	since	SCONJ
ejpam-4827	362	21	σgi	σgi	PROPN
ejpam-4827	362	22	is	be	AUX
ejpam-4827	362	23	an	an	DET
ejpam-4827	362	24	automorphism	automorphism	NOUN
ejpam-4827	362	25	.	.	PUNCT
ejpam-4827	363	1	this	this	PRON
ejpam-4827	363	2	shows	show	VERB
ejpam-4827	363	3	that	that	SCONJ
ejpam-4827	363	4	b1(j)σhj	b1(j)σhj	PROPN
ejpam-4827	363	5	(	(	PUNCT
ejpam-4827	363	6	rσs(a2	rσs(a2	PROPN
ejpam-4827	363	7	(	(	PUNCT
ejpam-4827	363	8	i	i	NOUN
ejpam-4827	363	9	)	)	PUNCT
ejpam-4827	363	10	)	)	PUNCT
ejpam-4827	363	11	)	)	PUNCT
ejpam-4827	364	1	=	=	SYM
ejpam-4827	364	2	0	0	PUNCT
ejpam-4827	365	1	since	since	SCONJ
ejpam-4827	365	2	by	by	ADP
ejpam-4827	365	3	rigidness	rigidness	NOUN
ejpam-4827	365	4	we	we	PRON
ejpam-4827	365	5	obtain	obtain	VERB
ejpam-4827	365	6	ψ1(vn(r)∗m)φ2	ψ1(vn(r)∗m)φ2	NOUN
ejpam-4827	365	7	=	=	SYM
ejpam-4827	365	8	0	0	NUM
ejpam-4827	365	9	.	.	PUNCT
ejpam-4827	366	1	thus	thus	ADV
ejpam-4827	366	2	,	,	PUNCT
ejpam-4827	366	3	we	we	PRON
ejpam-4827	366	4	deduce	deduce	VERB
ejpam-4827	366	5	the	the	DET
ejpam-4827	366	6	other	other	ADJ
ejpam-4827	366	7	side	side	NOUN
ejpam-4827	366	8	of	of	ADP
ejpam-4827	366	9	eq	eq	PROPN
ejpam-4827	366	10	.	.	PUNCT
ejpam-4827	367	1	(	(	PUNCT
ejpam-4827	367	2	2.10	2.10	NUM
ejpam-4827	367	3	)	)	PUNCT
ejpam-4827	367	4	,	,	PUNCT
ejpam-4827	367	5	a1(i)σgi(rσs(b2	a1(i)σgi(rσs(b2	PROPN
ejpam-4827	367	6	(	(	PUNCT
ejpam-4827	367	7	j	j	NOUN
ejpam-4827	367	8	)	)	PUNCT
ejpam-4827	367	9	)	)	PUNCT
ejpam-4827	367	10	)	)	PUNCT
ejpam-4827	368	1	=	=	SYM
ejpam-4827	368	2	0	0	NUM
ejpam-4827	369	1	and	and	CCONJ
ejpam-4827	369	2	so	so	ADV
ejpam-4827	369	3	ψ2(vn(r	ψ2(vn(r	NOUN
ejpam-4827	369	4	)	)	PUNCT
ejpam-4827	369	5	∗m)φ1	∗m)φ1	NOUN
ejpam-4827	370	1	=	=	SYM
ejpam-4827	370	2	0	0	X
ejpam-4827	370	3	.	.	PUNCT
ejpam-4827	371	1	similarly	similarly	ADV
ejpam-4827	371	2	,	,	PUNCT
ejpam-4827	371	3	if	if	SCONJ
ejpam-4827	371	4	we	we	PRON
ejpam-4827	371	5	multiply	multiply	VERB
ejpam-4827	371	6	eq.(2.7	eq.(2.7	NUM
ejpam-4827	371	7	)	)	PUNCT
ejpam-4827	371	8	on	on	ADP
ejpam-4827	371	9	the	the	DET
ejpam-4827	371	10	right	right	ADJ
ejpam-4827	371	11	-	-	PUNCT
ejpam-4827	371	12	hand	hand	NOUN
ejpam-4827	371	13	side	side	NOUN
ejpam-4827	371	14	by	by	ADP
ejpam-4827	371	15	tb1	tb1	PROPN
ejpam-4827	371	16	(	(	PUNCT
ejpam-4827	371	17	j	j	PROPN
ejpam-4827	371	18	)	)	PUNCT
ejpam-4827	371	19	for	for	ADP
ejpam-4827	371	20	any	any	DET
ejpam-4827	371	21	t	t	NOUN
ejpam-4827	371	22	∈	∈	PROPN
ejpam-4827	371	23	r	r	NOUN
ejpam-4827	371	24	,	,	PUNCT
ejpam-4827	371	25	then	then	ADV
ejpam-4827	371	26	a1	a1	NOUN
ejpam-4827	371	27	(	(	PUNCT
ejpam-4827	371	28	i)rb3	i)rb3	PROPN
ejpam-4827	371	29	(	(	PUNCT
ejpam-4827	371	30	j)tb1	j)tb1	PROPN
ejpam-4827	371	31	(	(	PUNCT
ejpam-4827	371	32	j	j	PROPN
ejpam-4827	371	33	)	)	PUNCT
ejpam-4827	371	34	+	+	NUM
ejpam-4827	371	35	a2	a2	PROPN
ejpam-4827	371	36	(	(	PUNCT
ejpam-4827	371	37	i)rb2	i)rb2	PROPN
ejpam-4827	371	38	(	(	PUNCT
ejpam-4827	371	39	j)tb1	j)tb1	PROPN
ejpam-4827	371	40	(	(	PUNCT
ejpam-4827	371	41	j	j	PROPN
ejpam-4827	371	42	)	)	PUNCT
ejpam-4827	371	43	+	+	NUM
ejpam-4827	371	44	a3	a3	NOUN
ejpam-4827	371	45	(	(	PUNCT
ejpam-4827	371	46	i)rb1	i)rb1	NOUN
ejpam-4827	371	47	(	(	PUNCT
ejpam-4827	371	48	j)tb1	j)tb1	PROPN
ejpam-4827	371	49	(	(	PUNCT
ejpam-4827	371	50	j	j	PROPN
ejpam-4827	371	51	)	)	PUNCT
ejpam-4827	371	52	=	=	SYM
ejpam-4827	372	1	0	0	X
ejpam-4827	372	2	.	.	PUNCT
ejpam-4827	373	1	(	(	PUNCT
ejpam-4827	373	2	2.11	2.11	NUM
ejpam-4827	373	3	)	)	PUNCT
ejpam-4827	373	4	and	and	CCONJ
ejpam-4827	373	5	so	so	ADV
ejpam-4827	373	6	a3(i)rb1(j	a3(i)rb1(j	NOUN
ejpam-4827	373	7	)	)	PUNCT
ejpam-4827	373	8	=	=	SYM
ejpam-4827	374	1	0	0	X
ejpam-4827	374	2	.	.	PUNCT
ejpam-4827	375	1	since	since	SCONJ
ejpam-4827	375	2	r	r	NOUN
ejpam-4827	375	3	is	be	AUX
ejpam-4827	375	4	m	m	PRON
ejpam-4827	375	5	-rigid	-rigid	ADJ
ejpam-4827	375	6	and	and	CCONJ
ejpam-4827	375	7	σgi	σgi	NOUN
ejpam-4827	375	8	is	be	AUX
ejpam-4827	375	9	an	an	DET
ejpam-4827	375	10	automorphism	automorphism	NOUN
ejpam-4827	375	11	,	,	PUNCT
ejpam-4827	375	12	we	we	PRON
ejpam-4827	375	13	have	have	VERB
ejpam-4827	375	14	a3	a3	NOUN
ejpam-4827	375	15	(	(	PUNCT
ejpam-4827	375	16	i)σgi(rσs(b1	i)σgi(rσs(b1	NOUN
ejpam-4827	375	17	(	(	PUNCT
ejpam-4827	375	18	j	j	NOUN
ejpam-4827	375	19	)	)	PUNCT
ejpam-4827	375	20	)	)	PUNCT
ejpam-4827	375	21	)	)	PUNCT
ejpam-4827	376	1	=	=	PUNCT
ejpam-4827	376	2	0	0	X
ejpam-4827	376	3	.	.	PUNCT
ejpam-4827	377	1	hence	hence	ADV
ejpam-4827	377	2	,	,	PUNCT
ejpam-4827	377	3	a3(i	a3(i	PROPN
ejpam-4827	377	4	)	)	PUNCT
ejpam-4827	377	5	∈	∈	PROPN
ejpam-4827	377	6	ℓr(rσs(b1	ℓr(rσs(b1	PROPN
ejpam-4827	377	7	(	(	PUNCT
ejpam-4827	377	8	j	j	NOUN
ejpam-4827	377	9	)	)	PUNCT
ejpam-4827	377	10	)	)	PUNCT
ejpam-4827	377	11	)	)	PUNCT
ejpam-4827	377	12	.	.	PUNCT
ejpam-4827	378	1	by	by	ADP
ejpam-4827	378	2	hypothesis	hypothesis	NOUN
ejpam-4827	378	3	,	,	PUNCT
ejpam-4827	378	4	r	r	NOUN
ejpam-4827	378	5	is	be	AUX
ejpam-4827	378	6	a	a	DET
ejpam-4827	378	7	left	left	ADJ
ejpam-4827	378	8	app	app	NOUN
ejpam-4827	378	9	,	,	PUNCT
ejpam-4827	378	10	ℓr(rσs(b1	ℓr(rσs(b1	PROPN
ejpam-4827	378	11	(	(	PUNCT
ejpam-4827	378	12	j	j	NOUN
ejpam-4827	378	13	)	)	PUNCT
ejpam-4827	378	14	)	)	PUNCT
ejpam-4827	378	15	)	)	PUNCT
ejpam-4827	378	16	is	be	AUX
ejpam-4827	378	17	left	leave	VERB
ejpam-4827	378	18	s	s	PART
ejpam-4827	378	19	-	-	NOUN
ejpam-4827	378	20	unital	unital	ADJ
ejpam-4827	378	21	by	by	ADP
ejpam-4827	378	22	lemma	lemma	PROPN
ejpam-4827	378	23	1	1	NUM
ejpam-4827	378	24	.	.	PUNCT
ejpam-4827	379	1	hence	hence	ADV
ejpam-4827	379	2	,	,	PUNCT
ejpam-4827	379	3	there	there	PRON
ejpam-4827	379	4	exist	exist	VERB
ejpam-4827	379	5	en	en	X
ejpam-4827	379	6	∈	∈	PROPN
ejpam-4827	379	7	ℓr(rσs(b1	ℓr(rσs(b1	PROPN
ejpam-4827	379	8	(	(	PUNCT
ejpam-4827	379	9	j	j	NOUN
ejpam-4827	379	10	)	)	PUNCT
ejpam-4827	379	11	)	)	PUNCT
ejpam-4827	379	12	)	)	PUNCT
ejpam-4827	379	13	such	such	ADJ
ejpam-4827	379	14	that	that	DET
ejpam-4827	379	15	a3(i)en	a3(i)en	X
ejpam-4827	379	16	=	=	NOUN
ejpam-4827	379	17	a3	a3	PROPN
ejpam-4827	379	18	(	(	PUNCT
ejpam-4827	379	19	i	i	NOUN
ejpam-4827	379	20	)	)	PUNCT
ejpam-4827	379	21	since	since	SCONJ
ejpam-4827	379	22	by	by	ADP
ejpam-4827	379	23	rigidness	rigidness	NOUN
ejpam-4827	379	24	we	we	PRON
ejpam-4827	379	25	obtain	obtain	VERB
ejpam-4827	379	26	ψ1(vn(r	ψ1(vn(r	NOUN
ejpam-4827	379	27	)	)	PUNCT
ejpam-4827	379	28	∗m)φ3	∗m)φ3	ADV
ejpam-4827	379	29	=	=	SYM
ejpam-4827	380	1	0	0	X
ejpam-4827	380	2	.	.	PUNCT
ejpam-4827	381	1	then	then	ADV
ejpam-4827	381	2	,	,	PUNCT
ejpam-4827	381	3	eq	eq	NOUN
ejpam-4827	381	4	.	.	PUNCT
ejpam-4827	382	1	(	(	PUNCT
ejpam-4827	382	2	2.7	2.7	NUM
ejpam-4827	382	3	)	)	PUNCT
ejpam-4827	382	4	becomes	become	VERB
ejpam-4827	382	5	a1	a1	NOUN
ejpam-4827	382	6	(	(	PUNCT
ejpam-4827	382	7	i)rb3	i)rb3	PROPN
ejpam-4827	382	8	(	(	PUNCT
ejpam-4827	382	9	j	j	NOUN
ejpam-4827	382	10	)	)	PUNCT
ejpam-4827	383	1	+	+	NUM
ejpam-4827	383	2	a2	a2	PROPN
ejpam-4827	383	3	(	(	PUNCT
ejpam-4827	383	4	i)rb2	i)rb2	PROPN
ejpam-4827	383	5	(	(	PUNCT
ejpam-4827	383	6	j	j	NOUN
ejpam-4827	383	7	)	)	PUNCT
ejpam-4827	383	8	=	=	SYM
ejpam-4827	383	9	0	0	X
ejpam-4827	383	10	.	.	PUNCT
ejpam-4827	384	1	(	(	PUNCT
ejpam-4827	384	2	2.12	2.12	NUM
ejpam-4827	384	3	)	)	PUNCT
ejpam-4827	384	4	if	if	SCONJ
ejpam-4827	384	5	we	we	PRON
ejpam-4827	384	6	multiplying	multiply	VERB
ejpam-4827	384	7	eq.(2.12	eq.(2.12	PROPN
ejpam-4827	384	8	)	)	PUNCT
ejpam-4827	384	9	on	on	ADP
ejpam-4827	384	10	the	the	DET
ejpam-4827	384	11	right	right	ADJ
ejpam-4827	384	12	-	-	PUNCT
ejpam-4827	384	13	hand	hand	NOUN
ejpam-4827	384	14	side	side	NOUN
ejpam-4827	384	15	by	by	ADP
ejpam-4827	384	16	tb2(j	tb2(j	NOUN
ejpam-4827	384	17	)	)	PUNCT
ejpam-4827	384	18	for	for	ADP
ejpam-4827	384	19	any	any	DET
ejpam-4827	384	20	t	t	NOUN
ejpam-4827	384	21	∈	∈	PROPN
ejpam-4827	384	22	r	r	NOUN
ejpam-4827	384	23	,	,	PUNCT
ejpam-4827	384	24	then	then	ADV
ejpam-4827	384	25	a1(i)rb3(j	a1(i)rb3(j	ADV
ejpam-4827	384	26	)	)	PUNCT
ejpam-4827	385	1	=	=	SYM
ejpam-4827	385	2	0	0	NUM
ejpam-4827	385	3	and	and	CCONJ
ejpam-4827	385	4	a2(i)rb2(j	a2(i)rb2(j	NOUN
ejpam-4827	385	5	)	)	PUNCT
ejpam-4827	386	1	=	=	SYM
ejpam-4827	386	2	0	0	NUM
ejpam-4827	386	3	by	by	ADP
ejpam-4827	386	4	the	the	DET
ejpam-4827	386	5	similar	similar	ADJ
ejpam-4827	386	6	argument	argument	NOUN
ejpam-4827	386	7	to	to	ADP
ejpam-4827	386	8	above	above	ADV
ejpam-4827	386	9	.	.	PUNCT
ejpam-4827	387	1	thus	thus	ADV
ejpam-4827	387	2	,	,	PUNCT
ejpam-4827	387	3	we	we	PRON
ejpam-4827	387	4	have	have	VERB
ejpam-4827	387	5	ai(i)σgi(rσs(bj	ai(i)σgi(rσs(bj	NOUN
ejpam-4827	387	6	(	(	PUNCT
ejpam-4827	387	7	j	j	NOUN
ejpam-4827	387	8	)	)	PUNCT
ejpam-4827	387	9	)	)	PUNCT
ejpam-4827	387	10	)	)	PUNCT
ejpam-4827	388	1	=	=	SYM
ejpam-4827	388	2	0	0	NUM
ejpam-4827	388	3	and	and	CCONJ
ejpam-4827	388	4	bj(j)σhj	bj(j)σhj	NOUN
ejpam-4827	388	5	(	(	PUNCT
ejpam-4827	388	6	rσs(ai	rσs(ai	NOUN
ejpam-4827	388	7	(	(	PUNCT
ejpam-4827	388	8	i	i	NOUN
ejpam-4827	388	9	)	)	PUNCT
ejpam-4827	388	10	)	)	PUNCT
ejpam-4827	389	1	=	=	SYM
ejpam-4827	389	2	0	0	NUM
ejpam-4827	390	1	for	for	ADP
ejpam-4827	390	2	all	all	DET
ejpam-4827	390	3	2	2	NUM
ejpam-4827	390	4	≤	≤	NOUN
ejpam-4827	390	5	i+	i+	NUM
ejpam-4827	390	6	j	j	PROPN
ejpam-4827	390	7	≤	≤	ADV
ejpam-4827	390	8	4	4	NUM
ejpam-4827	390	9	.	.	PUNCT
ejpam-4827	391	1	inductively	inductively	ADV
ejpam-4827	391	2	,	,	PUNCT
ejpam-4827	391	3	we	we	PRON
ejpam-4827	391	4	assume	assume	VERB
ejpam-4827	391	5	that	that	SCONJ
ejpam-4827	391	6	ai(i)σgi(rσs(bj	ai(i)σgi(rσs(bj	PROPN
ejpam-4827	391	7	(	(	PUNCT
ejpam-4827	391	8	j	j	NOUN
ejpam-4827	391	9	)	)	PUNCT
ejpam-4827	391	10	)	)	PUNCT
ejpam-4827	391	11	)	)	PUNCT
ejpam-4827	392	1	=	=	SYM
ejpam-4827	392	2	0	0	NUM
ejpam-4827	392	3	and	and	CCONJ
ejpam-4827	392	4	bj(j)σhj	bj(j)σhj	NOUN
ejpam-4827	392	5	(	(	PUNCT
ejpam-4827	392	6	rσs(ai	rσs(ai	NOUN
ejpam-4827	392	7	(	(	PUNCT
ejpam-4827	392	8	i	i	NOUN
ejpam-4827	392	9	)	)	PUNCT
ejpam-4827	392	10	)	)	PUNCT
ejpam-4827	393	1	=	=	SYM
ejpam-4827	393	2	0	0	NUM
ejpam-4827	394	1	for	for	ADP
ejpam-4827	394	2	all	all	DET
ejpam-4827	394	3	i+	i+	NUM
ejpam-4827	394	4	j	j	PROPN
ejpam-4827	394	5	≤	≤	PROPN
ejpam-4827	394	6	n.	n.	NOUN
ejpam-4827	394	7	if	if	SCONJ
ejpam-4827	394	8	we	we	PRON
ejpam-4827	394	9	multiply	multiply	VERB
ejpam-4827	394	10	eq.(2.9	eq.(2.9	X
ejpam-4827	394	11	)	)	PUNCT
ejpam-4827	394	12	on	on	ADP
ejpam-4827	394	13	the	the	DET
ejpam-4827	394	14	right	right	ADJ
ejpam-4827	394	15	-	-	PUNCT
ejpam-4827	394	16	hand	hand	NOUN
ejpam-4827	394	17	side	side	NOUN
ejpam-4827	394	18	by	by	ADP
ejpam-4827	394	19	t1b1(j	t1b1(j	NOUN
ejpam-4827	394	20	)	)	PUNCT
ejpam-4827	394	21	,	,	PUNCT
ejpam-4827	394	22	t2b2(j	t2b2(j	PROPN
ejpam-4827	394	23	)	)	PUNCT
ejpam-4827	394	24	,	,	PUNCT
ejpam-4827	394	25	.	.	PUNCT
ejpam-4827	394	26	.	.	PUNCT
ejpam-4827	394	27	.	.	PUNCT
ejpam-4827	395	1	,	,	PUNCT
ejpam-4827	395	2	tn−1bn−1	tn−1bn−1	X
ejpam-4827	395	3	(	(	PUNCT
ejpam-4827	395	4	j	j	NOUN
ejpam-4827	395	5	)	)	PUNCT
ejpam-4827	395	6	for	for	ADP
ejpam-4827	395	7	any	any	DET
ejpam-4827	395	8	t1	t1	NOUN
ejpam-4827	395	9	,	,	PUNCT
ejpam-4827	395	10	t2	t2	NOUN
ejpam-4827	395	11	,	,	PUNCT
ejpam-4827	395	12	.	.	PUNCT
ejpam-4827	395	13	.	.	PUNCT
ejpam-4827	396	1	.	.	PUNCT
ejpam-4827	397	1	,	,	PUNCT
ejpam-4827	397	2	tn−1	tn−1	PROPN
ejpam-4827	397	3	∈	∈	PROPN
ejpam-4827	397	4	r	r	NOUN
ejpam-4827	397	5	,	,	PUNCT
ejpam-4827	397	6	in	in	ADP
ejpam-4827	397	7	turn	turn	NOUN
ejpam-4827	397	8	,	,	PUNCT
ejpam-4827	397	9	since	since	SCONJ
ejpam-4827	397	10	r	r	NOUN
ejpam-4827	397	11	is	be	AUX
ejpam-4827	397	12	m	m	PRON
ejpam-4827	397	13	-rigidness	-rigidness	ADJ
ejpam-4827	397	14	and	and	CCONJ
ejpam-4827	397	15	σgi	σgi	NOUN
ejpam-4827	397	16	is	be	AUX
ejpam-4827	397	17	an	an	DET
ejpam-4827	397	18	automorphism	automorphism	NOUN
ejpam-4827	397	19	,	,	PUNCT
ejpam-4827	397	20	we	we	PRON
ejpam-4827	397	21	have	have	VERB
ejpam-4827	397	22	an	an	DET
ejpam-4827	397	23	(	(	PUNCT
ejpam-4827	397	24	i)σgi(rσs(b1	i)σgi(rσs(b1	NOUN
ejpam-4827	397	25	(	(	PUNCT
ejpam-4827	397	26	j	j	NOUN
ejpam-4827	397	27	)	)	PUNCT
ejpam-4827	397	28	)	)	PUNCT
ejpam-4827	397	29	)	)	PUNCT
ejpam-4827	398	1	=	=	SYM
ejpam-4827	398	2	0	0	NUM
ejpam-4827	398	3	,	,	PUNCT
ejpam-4827	398	4	an−1	an−1	PROPN
ejpam-4827	398	5	(	(	PUNCT
ejpam-4827	398	6	i)σgi(rσs(b2	i)σgi(rσs(b2	PROPN
ejpam-4827	398	7	(	(	PUNCT
ejpam-4827	398	8	j	j	NOUN
ejpam-4827	398	9	)	)	PUNCT
ejpam-4827	398	10	)	)	PUNCT
ejpam-4827	398	11	)	)	PUNCT
ejpam-4827	399	1	=	=	PUNCT
ejpam-4827	399	2	0	0	NUM
ejpam-4827	399	3	,	,	PUNCT
ejpam-4827	399	4	.	.	PUNCT
ejpam-4827	399	5	.	.	PUNCT
ejpam-4827	399	6	.	.	PUNCT
ejpam-4827	400	1	,	,	PUNCT
ejpam-4827	400	2	a2	a2	PROPN
ejpam-4827	400	3	(	(	PUNCT
ejpam-4827	400	4	i)σgi(rσs(bn−1	i)σgi(rσs(bn−1	X
ejpam-4827	400	5	(	(	PUNCT
ejpam-4827	400	6	j	j	NOUN
ejpam-4827	400	7	)	)	PUNCT
ejpam-4827	400	8	)	)	PUNCT
ejpam-4827	400	9	)	)	PUNCT
ejpam-4827	401	1	=	=	SYM
ejpam-4827	401	2	0	0	NUM
ejpam-4827	401	3	and	and	CCONJ
ejpam-4827	401	4	a1(i)σgi(rσs(bn	a1(i)σgi(rσs(bn	PROPN
ejpam-4827	401	5	(	(	PUNCT
ejpam-4827	401	6	j	j	PROPN
ejpam-4827	401	7	)	)	PUNCT
ejpam-4827	401	8	)	)	PUNCT
ejpam-4827	401	9	)	)	PUNCT
ejpam-4827	402	1	=	=	PUNCT
ejpam-4827	402	2	0	0	X
ejpam-4827	402	3	.	.	PUNCT
ejpam-4827	403	1	hence	hence	ADV
ejpam-4827	403	2	,	,	PUNCT
ejpam-4827	403	3	an	an	DET
ejpam-4827	403	4	(	(	PUNCT
ejpam-4827	403	5	i	i	NOUN
ejpam-4827	403	6	)	)	PUNCT
ejpam-4827	403	7	∈	∈	PROPN
ejpam-4827	403	8	ℓr(rσs(b1	ℓr(rσs(b1	PROPN
ejpam-4827	403	9	(	(	PUNCT
ejpam-4827	403	10	j	j	NOUN
ejpam-4827	403	11	)	)	PUNCT
ejpam-4827	403	12	)	)	PUNCT
ejpam-4827	403	13	)	)	PUNCT
ejpam-4827	403	14	,	,	PUNCT
ejpam-4827	403	15	an−1	an−1	PROPN
ejpam-4827	403	16	(	(	PUNCT
ejpam-4827	403	17	i	i	NOUN
ejpam-4827	403	18	)	)	PUNCT
ejpam-4827	403	19	∈	∈	PROPN
ejpam-4827	403	20	ℓr(rσs(b2	ℓr(rσs(b2	PROPN
ejpam-4827	403	21	(	(	PUNCT
ejpam-4827	403	22	j	j	PROPN
ejpam-4827	403	23	)	)	PUNCT
ejpam-4827	403	24	)	)	PUNCT
ejpam-4827	403	25	)	)	PUNCT
ejpam-4827	403	26	,	,	PUNCT
ejpam-4827	403	27	.	.	PUNCT
ejpam-4827	403	28	.	.	PUNCT
ejpam-4827	403	29	.	.	PUNCT
ejpam-4827	404	1	,	,	PUNCT
ejpam-4827	404	2	a1	a1	NOUN
ejpam-4827	404	3	(	(	PUNCT
ejpam-4827	404	4	i	i	NOUN
ejpam-4827	404	5	)	)	PUNCT
ejpam-4827	404	6	∈	∈	PROPN
ejpam-4827	404	7	ℓr(rσs(bn	ℓr(rσs(bn	PROPN
ejpam-4827	404	8	(	(	PUNCT
ejpam-4827	404	9	j	j	NOUN
ejpam-4827	404	10	)	)	PUNCT
ejpam-4827	404	11	)	)	PUNCT
ejpam-4827	404	12	)	)	PUNCT
ejpam-4827	404	13	.	.	PUNCT
ejpam-4827	405	1	by	by	ADP
ejpam-4827	405	2	hypothesis	hypothesis	NOUN
ejpam-4827	405	3	,	,	PUNCT
ejpam-4827	405	4	r	r	NOUN
ejpam-4827	405	5	is	be	AUX
ejpam-4827	405	6	a	a	DET
ejpam-4827	405	7	left	left	ADJ
ejpam-4827	405	8	app	app	NOUN
ejpam-4827	405	9	,	,	PUNCT
ejpam-4827	405	10	ℓr(rσs(b1(j	ℓr(rσs(b1(j	PROPN
ejpam-4827	405	11	)	)	PUNCT
ejpam-4827	405	12	)	)	PUNCT
ejpam-4827	405	13	)	)	PUNCT
ejpam-4827	405	14	,	,	PUNCT
ejpam-4827	405	15	ℓr(rσs(b2(j	ℓr(rσs(b2(j	PROPN
ejpam-4827	405	16	)	)	PUNCT
ejpam-4827	405	17	)	)	PUNCT
ejpam-4827	405	18	)	)	PUNCT
ejpam-4827	405	19	,	,	PUNCT
ejpam-4827	405	20	.	.	PUNCT
ejpam-4827	405	21	.	.	PUNCT
ejpam-4827	406	1	.	.	PUNCT
ejpam-4827	407	1	,	,	PUNCT
ejpam-4827	407	2	ℓr(rσs(bn−1	ℓr(rσs(bn−1	PROPN
ejpam-4827	407	3	(	(	PUNCT
ejpam-4827	407	4	j	j	NOUN
ejpam-4827	407	5	)	)	PUNCT
ejpam-4827	407	6	)	)	PUNCT
ejpam-4827	407	7	)	)	PUNCT
ejpam-4827	408	1	and	and	CCONJ
ejpam-4827	408	2	ℓr(rσs(bn	ℓr(rσs(bn	PROPN
ejpam-4827	408	3	(	(	PUNCT
ejpam-4827	408	4	j	j	PROPN
ejpam-4827	408	5	)	)	PUNCT
ejpam-4827	408	6	)	)	PUNCT
ejpam-4827	408	7	)	)	PUNCT
ejpam-4827	408	8	,	,	PUNCT
ejpam-4827	408	9	respectively	respectively	ADV
ejpam-4827	408	10	,	,	PUNCT
ejpam-4827	408	11	is	be	AUX
ejpam-4827	408	12	left	leave	VERB
ejpam-4827	408	13	s	s	PART
ejpam-4827	408	14	-	-	NOUN
ejpam-4827	408	15	unital	unital	ADJ
ejpam-4827	408	16	by	by	ADP
ejpam-4827	408	17	lemma	lemma	PROPN
ejpam-4827	408	18	1	1	NUM
ejpam-4827	408	19	again	again	ADV
ejpam-4827	408	20	.	.	PUNCT
ejpam-4827	409	1	hence	hence	ADV
ejpam-4827	409	2	,	,	PUNCT
ejpam-4827	409	3	there	there	PRON
ejpam-4827	409	4	exist	exist	VERB
ejpam-4827	409	5	en	en	X
ejpam-4827	409	6	∈	∈	PROPN
ejpam-4827	409	7	ℓr(rσs(b1	ℓr(rσs(b1	PROPN
ejpam-4827	409	8	(	(	PUNCT
ejpam-4827	409	9	j	j	NOUN
ejpam-4827	409	10	)	)	PUNCT
ejpam-4827	409	11	)	)	PUNCT
ejpam-4827	409	12	)	)	PUNCT
ejpam-4827	409	13	,	,	PUNCT
ejpam-4827	409	14	ℓr(rσs(b1	ℓr(rσs(b1	PROPN
ejpam-4827	409	15	(	(	PUNCT
ejpam-4827	409	16	j	j	NOUN
ejpam-4827	409	17	)	)	PUNCT
ejpam-4827	409	18	)	)	PUNCT
ejpam-4827	409	19	)	)	PUNCT
ejpam-4827	409	20	,	,	PUNCT
ejpam-4827	409	21	.	.	PUNCT
ejpam-4827	409	22	.	.	PUNCT
ejpam-4827	410	1	.	.	PUNCT
ejpam-4827	411	1	,	,	PUNCT
ejpam-4827	411	2	ℓr(rσs(b1	ℓr(rσs(b1	PROPN
ejpam-4827	411	3	(	(	PUNCT
ejpam-4827	411	4	j	j	NOUN
ejpam-4827	411	5	)	)	PUNCT
ejpam-4827	411	6	)	)	PUNCT
ejpam-4827	411	7	)	)	PUNCT
ejpam-4827	412	1	such	such	ADJ
ejpam-4827	412	2	that	that	SCONJ
ejpam-4827	412	3	an(i)en	an(i)en	VERB
ejpam-4827	412	4	=	=	PUNCT
ejpam-4827	412	5	an	an	DET
ejpam-4827	412	6	(	(	PUNCT
ejpam-4827	412	7	i	i	NOUN
ejpam-4827	412	8	)	)	PUNCT
ejpam-4827	412	9	,	,	PUNCT
ejpam-4827	412	10	an−1	an−1	PROPN
ejpam-4827	412	11	(	(	PUNCT
ejpam-4827	412	12	i)en	i)en	PROPN
ejpam-4827	412	13	=	=	PRON
ejpam-4827	412	14	an−1	an−1	PROPN
ejpam-4827	412	15	(	(	PUNCT
ejpam-4827	412	16	i	i	NOUN
ejpam-4827	412	17	)	)	PUNCT
ejpam-4827	412	18	,	,	PUNCT
ejpam-4827	412	19	.	.	PUNCT
ejpam-4827	412	20	.	.	PUNCT
ejpam-4827	412	21	.	.	PUNCT
ejpam-4827	413	1	,	,	PUNCT
ejpam-4827	413	2	a1	a1	NOUN
ejpam-4827	413	3	(	(	PUNCT
ejpam-4827	413	4	i)en	i)en	PROPN
ejpam-4827	413	5	=	=	PUNCT
ejpam-4827	413	6	a1	a1	PROPN
ejpam-4827	413	7	(	(	PUNCT
ejpam-4827	413	8	i	i	NOUN
ejpam-4827	413	9	)	)	PUNCT
ejpam-4827	413	10	.	.	PUNCT
ejpam-4827	414	1	now	now	ADV
ejpam-4827	414	2	,	,	PUNCT
ejpam-4827	414	3	it	it	PRON
ejpam-4827	414	4	is	be	AUX
ejpam-4827	414	5	straightforward	straightforward	ADJ
ejpam-4827	414	6	to	to	PART
ejpam-4827	414	7	see	see	VERB
ejpam-4827	414	8	that	that	DET
ejpam-4827	414	9	aiσgi(rσs(bj	aiσgi(rσs(bj	NOUN
ejpam-4827	414	10	)	)	PUNCT
ejpam-4827	414	11	)	)	PUNCT
ejpam-4827	415	1	=	=	SYM
ejpam-4827	415	2	0	0	NUM
ejpam-4827	415	3	for	for	ADP
ejpam-4827	415	4	all	all	DET
ejpam-4827	415	5	i	i	PROPN
ejpam-4827	415	6	,	,	PUNCT
ejpam-4827	415	7	j.	j.	PROPN
ejpam-4827	415	8	since	since	SCONJ
ejpam-4827	415	9	r	r	NOUN
ejpam-4827	415	10	is	be	AUX
ejpam-4827	415	11	rigidness	rigidness	NOUN
ejpam-4827	415	12	we	we	PRON
ejpam-4827	415	13	option	option	VERB
ejpam-4827	415	14	bjσhj	bjσhj	NOUN
ejpam-4827	415	15	(	(	PUNCT
ejpam-4827	415	16	rσs(ai	rσs(ai	NOUN
ejpam-4827	415	17	)	)	PUNCT
ejpam-4827	415	18	)	)	PUNCT
ejpam-4827	416	1	=	=	PUNCT
ejpam-4827	416	2	0	0	X
ejpam-4827	416	3	.	.	PUNCT
ejpam-4827	417	1	this	this	PRON
ejpam-4827	417	2	prove	prove	VERB
ejpam-4827	417	3	that	that	SCONJ
ejpam-4827	417	4	ψ(vn(r	ψ(vn(r	NOUN
ejpam-4827	417	5	)	)	PUNCT
ejpam-4827	417	6	∗m)φ	∗m)φ	PROPN
ejpam-4827	417	7	=	=	SYM
ejpam-4827	417	8	0	0	PROPN
ejpam-4827	417	9	.	.	PUNCT
ejpam-4827	418	1	therefore	therefore	ADV
ejpam-4827	418	2	,	,	PUNCT
ejpam-4827	418	3	vn(r	vn(r	X
ejpam-4827	418	4	)	)	PUNCT
ejpam-4827	418	5	is	be	AUX
ejpam-4827	418	6	σ	σ	NOUN
ejpam-4827	418	7	-	-	PUNCT
ejpam-4827	418	8	skew	skew	NOUN
ejpam-4827	418	9	strongly	strongly	ADV
ejpam-4827	418	10	m	m	VERB
ejpam-4827	418	11	-reflexive	-reflexive	ADJ
ejpam-4827	418	12	.	.	PUNCT
ejpam-4827	419	1	e.	e.	PROPN
ejpam-4827	419	2	ali	ali	PROPN
ejpam-4827	419	3	/	/	SYM
ejpam-4827	419	4	eur	eur	PROPN
ejpam-4827	419	5	.	.	PUNCT
ejpam-4827	420	1	j.	j.	PROPN
ejpam-4827	420	2	pure	pure	PROPN
ejpam-4827	420	3	appl	appl	PROPN
ejpam-4827	420	4	.	.	PROPN
ejpam-4827	420	5	math	math	PROPN
ejpam-4827	420	6	,	,	PUNCT
ejpam-4827	420	7	16	16	NUM
ejpam-4827	420	8	(	(	PUNCT
ejpam-4827	420	9	3	3	NUM
ejpam-4827	420	10	)	)	PUNCT
ejpam-4827	420	11	(	(	PUNCT
ejpam-4827	420	12	2023	2023	NUM
ejpam-4827	420	13	)	)	PUNCT
ejpam-4827	420	14	,	,	PUNCT
ejpam-4827	420	15	1878	1878	NUM
ejpam-4827	420	16	-	-	SYM
ejpam-4827	420	17	1893	1893	NUM
ejpam-4827	420	18	1886	1886	NUM
ejpam-4827	420	19	corollary	corollary	NOUN
ejpam-4827	420	20	2	2	NUM
ejpam-4827	420	21	.	.	PUNCT
ejpam-4827	420	22	(	(	PUNCT
ejpam-4827	420	23	theorem	theorem	VERB
ejpam-4827	420	24	3.8	3.8	NUM
ejpam-4827	420	25	[	[	X
ejpam-4827	420	26	12	12	NUM
ejpam-4827	420	27	]	]	PUNCT
ejpam-4827	420	28	)	)	PUNCT
ejpam-4827	420	29	.	.	PUNCT
ejpam-4827	421	1	let	let	VERB
ejpam-4827	421	2	r	r	PRON
ejpam-4827	421	3	be	be	AUX
ejpam-4827	421	4	a	a	DET
ejpam-4827	421	5	ring	ring	NOUN
ejpam-4827	421	6	with	with	ADP
ejpam-4827	421	7	an	an	DET
ejpam-4827	421	8	endomorphism	endomorphism	PROPN
ejpam-4827	421	9	α	α	NOUN
ejpam-4827	421	10	and	and	CCONJ
ejpam-4827	421	11	n	n	PRON
ejpam-4827	421	12	≥	≥	NOUN
ejpam-4827	421	13	2	2	NUM
ejpam-4827	421	14	.	.	PUNCT
ejpam-4827	422	1	if	if	SCONJ
ejpam-4827	422	2	r	r	NOUN
ejpam-4827	422	3	is	be	AUX
ejpam-4827	422	4	a	a	DET
ejpam-4827	422	5	semiprime	semiprime	NOUN
ejpam-4827	422	6	and	and	CCONJ
ejpam-4827	422	7	right	right	ADJ
ejpam-4827	422	8	α	α	NOUN
ejpam-4827	422	9	-	-	PUNCT
ejpam-4827	422	10	skew	skew	ADJ
ejpam-4827	422	11	reflexive	reflexive	ADJ
ejpam-4827	422	12	ring	ring	NOUN
ejpam-4827	422	13	,	,	PUNCT
ejpam-4827	422	14	then	then	ADV
ejpam-4827	422	15	vn(r	vn(r	NUM
ejpam-4827	422	16	)	)	PUNCT
ejpam-4827	422	17	is	be	AUX
ejpam-4827	422	18	right	right	ADJ
ejpam-4827	422	19	α	α	NOUN
ejpam-4827	422	20	-	-	PUNCT
ejpam-4827	422	21	skew	skew	NOUN
ejpam-4827	422	22	reflexive	reflexive	NOUN
ejpam-4827	422	23	.	.	PUNCT
ejpam-4827	423	1	3	3	X
ejpam-4827	423	2	.	.	X
ejpam-4827	423	3	nilpotent	nilpotent	ADJ
ejpam-4827	423	4	elements	element	NOUN
ejpam-4827	423	5	of	of	ADP
ejpam-4827	423	6	reflexive	reflexive	ADJ
ejpam-4827	423	7	in	in	ADP
ejpam-4827	423	8	skew	skew	ADJ
ejpam-4827	423	9	monoid	monoid	NOUN
ejpam-4827	423	10	rings	ring	NOUN
ejpam-4827	423	11	in	in	ADP
ejpam-4827	423	12	this	this	DET
ejpam-4827	423	13	section	section	NOUN
ejpam-4827	423	14	,	,	PUNCT
ejpam-4827	423	15	we	we	PRON
ejpam-4827	423	16	introduce	introduce	VERB
ejpam-4827	423	17	the	the	DET
ejpam-4827	423	18	concept	concept	NOUN
ejpam-4827	423	19	of	of	ADP
ejpam-4827	423	20	σ	σ	PROPN
ejpam-4827	423	21	-	-	PUNCT
ejpam-4827	423	22	skew	skew	NOUN
ejpam-4827	423	23	strongly	strongly	ADV
ejpam-4827	423	24	m	m	VERB
ejpam-4827	423	25	-nil	-nil	ADJ
ejpam-4827	423	26	-	-	PUNCT
ejpam-4827	423	27	reflexive	reflexive	ADJ
ejpam-4827	423	28	ring	ring	NOUN
ejpam-4827	423	29	and	and	CCONJ
ejpam-4827	423	30	consider	consider	VERB
ejpam-4827	423	31	its	its	PRON
ejpam-4827	423	32	properties	property	NOUN
ejpam-4827	423	33	.	.	PUNCT
ejpam-4827	424	1	let	let	VERB
ejpam-4827	424	2	φ	φ	PROPN
ejpam-4827	424	3	=	=	SYM
ejpam-4827	424	4	b1g1	b1g1	PROPN
ejpam-4827	424	5	+	+	NUM
ejpam-4827	424	6	b2g2	b2g2	PROPN
ejpam-4827	424	7	+	+	CCONJ
ejpam-4827	424	8	·	·	PUNCT
ejpam-4827	424	9	·	·	PUNCT
ejpam-4827	424	10	·	·	PUNCT
ejpam-4827	425	1	+	+	NUM
ejpam-4827	425	2	bngn	bngn	PROPN
ejpam-4827	425	3	∈	∈	PROPN
ejpam-4827	425	4	r[m	r[m	PROPN
ejpam-4827	425	5	]	]	PUNCT
ejpam-4827	425	6	.	.	PUNCT
ejpam-4827	426	1	the	the	DET
ejpam-4827	426	2	element	element	NOUN
ejpam-4827	426	3	φ	φ	PROPN
ejpam-4827	426	4	∈	∈	PROPN
ejpam-4827	426	5	nil(r)[m	nil(r)[m	ADP
ejpam-4827	426	6	]	]	PUNCT
ejpam-4827	426	7	if	if	SCONJ
ejpam-4827	426	8	and	and	CCONJ
ejpam-4827	426	9	only	only	ADV
ejpam-4827	426	10	if	if	SCONJ
ejpam-4827	426	11	bi	bi	PROPN
ejpam-4827	426	12	∈	∈	PROPN
ejpam-4827	426	13	nil(r	nil(r	PROPN
ejpam-4827	426	14	)	)	PUNCT
ejpam-4827	426	15	for	for	ADP
ejpam-4827	426	16	all	all	DET
ejpam-4827	426	17	1	1	NUM
ejpam-4827	426	18	≤	≤	NUM
ejpam-4827	426	19	i	i	PRON
ejpam-4827	426	20	≤	≤	ADJ
ejpam-4827	427	1	n.	n.	NOUN
ejpam-4827	428	1	also	also	ADV
ejpam-4827	428	2	,	,	PUNCT
ejpam-4827	428	3	we	we	PRON
ejpam-4827	428	4	say	say	VERB
ejpam-4827	428	5	that	that	SCONJ
ejpam-4827	428	6	φ	φ	PROPN
ejpam-4827	428	7	∈	∈	PROPN
ejpam-4827	428	8	nil(r	nil(r	PROPN
ejpam-4827	428	9	∗m	∗m	NOUN
ejpam-4827	428	10	)	)	PUNCT
ejpam-4827	428	11	if	if	SCONJ
ejpam-4827	428	12	φ	φ	PROPN
ejpam-4827	428	13	is	be	AUX
ejpam-4827	428	14	a	a	DET
ejpam-4827	428	15	nilpotent	nilpotent	ADJ
ejpam-4827	428	16	element	element	NOUN
ejpam-4827	428	17	in	in	ADP
ejpam-4827	428	18	the	the	DET
ejpam-4827	428	19	skew	skew	ADJ
ejpam-4827	428	20	monoid	monoid	NOUN
ejpam-4827	428	21	ring	ring	NOUN
ejpam-4827	428	22	r	r	NOUN
ejpam-4827	428	23	∗m	∗m	NOUN
ejpam-4827	428	24	.	.	PUNCT
ejpam-4827	429	1	for	for	ADP
ejpam-4827	429	2	any	any	DET
ejpam-4827	429	3	φ	φ	PROPN
ejpam-4827	429	4	∈	∈	PROPN
ejpam-4827	429	5	r	r	NOUN
ejpam-4827	429	6	∗m	∗m	NOUN
ejpam-4827	429	7	,	,	PUNCT
ejpam-4827	429	8	we	we	PRON
ejpam-4827	429	9	denote	denote	VERB
ejpam-4827	429	10	by	by	ADP
ejpam-4827	429	11	cφ	cφ	NOUN
ejpam-4827	429	12	the	the	DET
ejpam-4827	429	13	set	set	NOUN
ejpam-4827	429	14	of	of	ADP
ejpam-4827	429	15	all	all	DET
ejpam-4827	429	16	coefficients	coefficient	NOUN
ejpam-4827	429	17	of	of	ADP
ejpam-4827	429	18	φ	φ	PROPN
ejpam-4827	429	19	.	.	PUNCT
ejpam-4827	430	1	for	for	ADP
ejpam-4827	430	2	more	more	ADJ
ejpam-4827	430	3	details	detail	NOUN
ejpam-4827	430	4	on	on	ADP
ejpam-4827	430	5	this	this	PRON
ejpam-4827	430	6	,	,	PUNCT
ejpam-4827	430	7	please	please	INTJ
ejpam-4827	430	8	refer	refer	VERB
ejpam-4827	430	9	to	to	ADP
ejpam-4827	430	10	[	[	X
ejpam-4827	430	11	18	18	NUM
ejpam-4827	430	12	]	]	PUNCT
ejpam-4827	430	13	.	.	PUNCT
ejpam-4827	431	1	definition	definition	NOUN
ejpam-4827	431	2	3	3	X
ejpam-4827	431	3	.	.	PUNCT
ejpam-4827	432	1	we	we	PRON
ejpam-4827	432	2	say	say	VERB
ejpam-4827	432	3	that	that	SCONJ
ejpam-4827	432	4	a	a	DET
ejpam-4827	432	5	ring	ring	NOUN
ejpam-4827	432	6	r	r	NOUN
ejpam-4827	432	7	is	be	AUX
ejpam-4827	432	8	σ	σ	NOUN
ejpam-4827	432	9	-	-	PUNCT
ejpam-4827	432	10	skew	skew	NOUN
ejpam-4827	432	11	strongly	strongly	ADV
ejpam-4827	432	12	m	m	VERB
ejpam-4827	432	13	-nil	-nil	NOUN
ejpam-4827	432	14	-	-	PUNCT
ejpam-4827	432	15	reflexive	reflexive	ADJ
ejpam-4827	432	16	(	(	PUNCT
ejpam-4827	432	17	σ	σ	NOUN
ejpam-4827	432	18	-	-	PUNCT
ejpam-4827	432	19	skew	skew	NOUN
ejpam-4827	432	20	strongly	strongly	ADV
ejpam-4827	432	21	nil	nil	ADJ
ejpam-4827	432	22	-	-	PUNCT
ejpam-4827	432	23	reflexive	reflexive	ADJ
ejpam-4827	432	24	relative	relative	ADJ
ejpam-4827	432	25	to	to	ADP
ejpam-4827	432	26	a	a	DET
ejpam-4827	432	27	monoid	monoid	NOUN
ejpam-4827	432	28	m	m	PROPN
ejpam-4827	432	29	)	)	PUNCT
ejpam-4827	432	30	,	,	PUNCT
ejpam-4827	432	31	if	if	SCONJ
ejpam-4827	432	32	φϕψ	φϕψ	PROPN
ejpam-4827	432	33	∈	∈	PROPN
ejpam-4827	432	34	nil(r	nil(r	PROPN
ejpam-4827	432	35	)	)	PUNCT
ejpam-4827	432	36	∗m	∗m	NOUN
ejpam-4827	432	37	implies	imply	VERB
ejpam-4827	432	38	that	that	SCONJ
ejpam-4827	432	39	biσgi(c	biσgi(c	PROPN
ejpam-4827	432	40	σs(aj	σs(aj	PROPN
ejpam-4827	432	41	)	)	PUNCT
ejpam-4827	432	42	)	)	PUNCT
ejpam-4827	433	1	∈	∈	PROPN
ejpam-4827	433	2	nil(r	nil(r	PROPN
ejpam-4827	433	3	)	)	PUNCT
ejpam-4827	433	4	,	,	PUNCT
ejpam-4827	433	5	where	where	SCONJ
ejpam-4827	433	6	φ	φ	PROPN
ejpam-4827	433	7	=	=	SYM
ejpam-4827	433	8	b1g1	b1g1	PROPN
ejpam-4827	433	9	+	+	NUM
ejpam-4827	433	10	b2g2	b2g2	PROPN
ejpam-4827	433	11	+	+	CCONJ
ejpam-4827	433	12	·	·	PUNCT
ejpam-4827	433	13	·	·	PUNCT
ejpam-4827	433	14	·	·	PUNCT
ejpam-4827	434	1	+	+	CCONJ
ejpam-4827	434	2	bngn	bngn	NOUN
ejpam-4827	434	3	,	,	PUNCT
ejpam-4827	434	4	ϕ	ϕ	X
ejpam-4827	434	5	=	=	SYM
ejpam-4827	434	6	c1l1	c1l1	X
ejpam-4827	434	7	+	+	CCONJ
ejpam-4827	434	8	c2l2	c2l2	X
ejpam-4827	434	9	+	+	NUM
ejpam-4827	434	10	·	·	PUNCT
ejpam-4827	434	11	·	·	PUNCT
ejpam-4827	434	12	·	·	PUNCT
ejpam-4827	435	1	+	+	CCONJ
ejpam-4827	435	2	cdld	cdld	ADJ
ejpam-4827	435	3	and	and	CCONJ
ejpam-4827	435	4	ψ	ψ	X
ejpam-4827	435	5	=	=	PUNCT
ejpam-4827	435	6	a1h1	a1h1	X
ejpam-4827	435	7	+	+	NOUN
ejpam-4827	435	8	a2h2	a2h2	X
ejpam-4827	435	9	+	+	X
ejpam-4827	435	10	·	·	PUNCT
ejpam-4827	435	11	·	·	PUNCT
ejpam-4827	435	12	·	·	PUNCT
ejpam-4827	435	13	+	+	NUM
ejpam-4827	435	14	amhm	amhm	NOUN
ejpam-4827	435	15	∈	∈	PROPN
ejpam-4827	435	16	r	r	NOUN
ejpam-4827	435	17	∗m	∗m	NOUN
ejpam-4827	435	18	,	,	PUNCT
ejpam-4827	435	19	then	then	ADV
ejpam-4827	435	20	ψϕφ	ψϕφ	VERB
ejpam-4827	435	21	∈	∈	PROPN
ejpam-4827	435	22	nil(r	nil(r	PROPN
ejpam-4827	435	23	)	)	PUNCT
ejpam-4827	435	24	∗m	∗m	NOUN
ejpam-4827	435	25	for	for	ADP
ejpam-4827	435	26	all	all	DET
ejpam-4827	435	27	i	i	PROPN
ejpam-4827	435	28	,	,	PUNCT
ejpam-4827	435	29	k	k	PROPN
ejpam-4827	435	30	,	,	PUNCT
ejpam-4827	435	31	j.	j.	PROPN
ejpam-4827	436	1	if	if	SCONJ
ejpam-4827	436	2	m	m	VERB
ejpam-4827	436	3	=	=	SYM
ejpam-4827	436	4	(	(	PUNCT
ejpam-4827	436	5	n	n	CCONJ
ejpam-4827	436	6	∪	∪	X
ejpam-4827	436	7	{	{	PUNCT
ejpam-4827	436	8	0},+	0},+	NUM
ejpam-4827	436	9	)	)	PUNCT
ejpam-4827	436	10	and	and	CCONJ
ejpam-4827	436	11	σg	σg	NOUN
ejpam-4827	436	12	=	=	SYM
ejpam-4827	436	13	idr	idr	PROPN
ejpam-4827	436	14	for	for	ADP
ejpam-4827	436	15	all	all	PRON
ejpam-4827	436	16	g	g	PROPN
ejpam-4827	436	17	∈	∈	PROPN
ejpam-4827	436	18	m	m	NOUN
ejpam-4827	436	19	,	,	PUNCT
ejpam-4827	436	20	then	then	ADV
ejpam-4827	436	21	a	a	DET
ejpam-4827	436	22	ring	ring	NOUN
ejpam-4827	436	23	r	r	NOUN
ejpam-4827	436	24	is	be	AUX
ejpam-4827	436	25	σ	σ	NOUN
ejpam-4827	436	26	-	-	PUNCT
ejpam-4827	436	27	skew	skew	NOUN
ejpam-4827	436	28	strongly	strongly	ADV
ejpam-4827	436	29	m	m	VERB
ejpam-4827	436	30	-nil	-nil	NOUN
ejpam-4827	436	31	-	-	PUNCT
ejpam-4827	436	32	reflexive	reflexive	ADJ
ejpam-4827	436	33	if	if	SCONJ
ejpam-4827	436	34	and	and	CCONJ
ejpam-4827	436	35	only	only	ADV
ejpam-4827	436	36	if	if	SCONJ
ejpam-4827	436	37	r	r	NOUN
ejpam-4827	436	38	is	be	AUX
ejpam-4827	436	39	strongly	strongly	ADV
ejpam-4827	436	40	nil	nil	ADJ
ejpam-4827	436	41	-	-	PUNCT
ejpam-4827	436	42	reflexive	reflexive	ADJ
ejpam-4827	436	43	.	.	PUNCT
ejpam-4827	437	1	also	also	ADV
ejpam-4827	437	2	,	,	PUNCT
ejpam-4827	437	3	if	if	SCONJ
ejpam-4827	437	4	m	m	VERB
ejpam-4827	437	5	=	=	PUNCT
ejpam-4827	437	6	{	{	PUNCT
ejpam-4827	437	7	e	e	NOUN
ejpam-4827	437	8	}	}	PUNCT
ejpam-4827	437	9	and	and	CCONJ
ejpam-4827	437	10	σg	σg	NOUN
ejpam-4827	437	11	=	=	SYM
ejpam-4827	437	12	idr	idr	PROPN
ejpam-4827	437	13	for	for	ADP
ejpam-4827	437	14	all	all	DET
ejpam-4827	437	15	g	g	NOUN
ejpam-4827	437	16	∈m	∈m	NOUN
ejpam-4827	437	17	,	,	PUNCT
ejpam-4827	437	18	then	then	ADV
ejpam-4827	437	19	any	any	DET
ejpam-4827	437	20	ring	ring	NOUN
ejpam-4827	437	21	r	r	NOUN
ejpam-4827	437	22	is	be	AUX
ejpam-4827	437	23	σ	σ	NOUN
ejpam-4827	437	24	-	-	PUNCT
ejpam-4827	437	25	skew	skew	NOUN
ejpam-4827	437	26	strongly	strongly	ADV
ejpam-4827	437	27	m	m	VERB
ejpam-4827	437	28	-nil	-nil	NOUN
ejpam-4827	437	29	-	-	PUNCT
ejpam-4827	437	30	reflexive	reflexive	ADJ
ejpam-4827	437	31	.	.	PUNCT
ejpam-4827	438	1	a	a	DET
ejpam-4827	438	2	ring	ring	NOUN
ejpam-4827	438	3	r	r	NOUN
ejpam-4827	438	4	is	be	AUX
ejpam-4827	438	5	called	call	VERB
ejpam-4827	438	6	an	an	DET
ejpam-4827	438	7	ni	ni	NOUN
ejpam-4827	438	8	ring	ring	NOUN
ejpam-4827	438	9	if	if	SCONJ
ejpam-4827	438	10	nil(r	nil(r	NOUN
ejpam-4827	438	11	)	)	PUNCT
ejpam-4827	438	12	forms	form	VERB
ejpam-4827	438	13	an	an	DET
ejpam-4827	438	14	ideal	ideal	NOUN
ejpam-4827	438	15	.	.	PUNCT
ejpam-4827	439	1	for	for	ADP
ejpam-4827	439	2	a	a	DET
ejpam-4827	439	3	unique	unique	ADJ
ejpam-4827	439	4	product	product	NOUN
ejpam-4827	439	5	monoid	monoid	NOUN
ejpam-4827	439	6	we	we	PRON
ejpam-4827	439	7	have	have	VERB
ejpam-4827	439	8	the	the	DET
ejpam-4827	439	9	following	follow	VERB
ejpam-4827	439	10	result	result	NOUN
ejpam-4827	439	11	.	.	PUNCT
ejpam-4827	440	1	theorem	theorem	NOUN
ejpam-4827	440	2	3	3	X
ejpam-4827	440	3	.	.	PUNCT
ejpam-4827	441	1	let	let	VERB
ejpam-4827	441	2	r	r	PRON
ejpam-4827	441	3	be	be	AUX
ejpam-4827	441	4	an	an	DET
ejpam-4827	441	5	ni	ni	NOUN
ejpam-4827	441	6	-	-	PUNCT
ejpam-4827	441	7	ring	ring	NOUN
ejpam-4827	441	8	,	,	PUNCT
ejpam-4827	441	9	m	m	VERB
ejpam-4827	441	10	be	be	VERB
ejpam-4827	441	11	a	a	DET
ejpam-4827	441	12	u.p.-monoid	u.p.-monoid	ADJ
ejpam-4827	441	13	,	,	PUNCT
ejpam-4827	441	14	and	and	CCONJ
ejpam-4827	441	15	σ	σ	NOUN
ejpam-4827	441	16	:	:	PUNCT
ejpam-4827	441	17	m	m	PROPN
ejpam-4827	441	18	→	→	SYM
ejpam-4827	441	19	aut(r	aut(r	PROPN
ejpam-4827	441	20	)	)	PUNCT
ejpam-4827	441	21	be	be	AUX
ejpam-4827	441	22	a	a	DET
ejpam-4827	441	23	compatible	compatible	ADJ
ejpam-4827	441	24	monoid	monoid	NOUN
ejpam-4827	441	25	homomorphism	homomorphism	NOUN
ejpam-4827	441	26	,	,	PUNCT
ejpam-4827	441	27	then	then	ADV
ejpam-4827	441	28	r	r	NOUN
ejpam-4827	441	29	is	be	AUX
ejpam-4827	441	30	σ	σ	NOUN
ejpam-4827	441	31	-	-	PUNCT
ejpam-4827	441	32	skew	skew	NOUN
ejpam-4827	441	33	strongly	strongly	ADV
ejpam-4827	441	34	m	m	VERB
ejpam-4827	441	35	-nil	-nil	ADJ
ejpam-4827	441	36	-	-	PUNCT
ejpam-4827	441	37	reflexive	reflexive	ADJ
ejpam-4827	441	38	proof	proof	NOUN
ejpam-4827	441	39	.	.	PUNCT
ejpam-4827	442	1	let	let	VERB
ejpam-4827	442	2	φ	φ	PROPN
ejpam-4827	442	3	=	=	SYM
ejpam-4827	442	4	b1g1	b1g1	PROPN
ejpam-4827	442	5	+	+	NUM
ejpam-4827	442	6	b2g2	b2g2	PROPN
ejpam-4827	442	7	+	+	CCONJ
ejpam-4827	442	8	·	·	PUNCT
ejpam-4827	442	9	·	·	PUNCT
ejpam-4827	442	10	·	·	PUNCT
ejpam-4827	443	1	+	+	CCONJ
ejpam-4827	443	2	bngn	bngn	NOUN
ejpam-4827	443	3	,	,	PUNCT
ejpam-4827	443	4	ϕ	ϕ	X
ejpam-4827	443	5	=	=	SYM
ejpam-4827	443	6	c1l1	c1l1	X
ejpam-4827	443	7	+	+	CCONJ
ejpam-4827	443	8	c2l2	c2l2	X
ejpam-4827	443	9	+	+	NUM
ejpam-4827	443	10	·	·	PUNCT
ejpam-4827	443	11	·	·	PUNCT
ejpam-4827	443	12	·	·	PUNCT
ejpam-4827	444	1	+	+	CCONJ
ejpam-4827	444	2	cdld	cdld	ADJ
ejpam-4827	444	3	and	and	CCONJ
ejpam-4827	444	4	ψ	ψ	X
ejpam-4827	444	5	=	=	PUNCT
ejpam-4827	444	6	a1h1	a1h1	X
ejpam-4827	444	7	+	+	NOUN
ejpam-4827	444	8	a2h2	a2h2	X
ejpam-4827	444	9	+	+	X
ejpam-4827	444	10	·	·	PUNCT
ejpam-4827	444	11	·	·	PUNCT
ejpam-4827	444	12	·	·	PUNCT
ejpam-4827	445	1	+	+	CCONJ
ejpam-4827	445	2	amhm	amhm	NOUN
ejpam-4827	445	3	be	be	VERB
ejpam-4827	445	4	nonzero	nonzero	NOUN
ejpam-4827	445	5	elements	element	NOUN
ejpam-4827	445	6	in	in	ADP
ejpam-4827	445	7	r	r	NOUN
ejpam-4827	445	8	∗m	∗m	NOUN
ejpam-4827	445	9	such	such	ADJ
ejpam-4827	445	10	that	that	SCONJ
ejpam-4827	445	11	φϕψ	φϕψ	PROPN
ejpam-4827	445	12	∈	∈	PROPN
ejpam-4827	445	13	nil(r	nil(r	PROPN
ejpam-4827	445	14	)	)	PUNCT
ejpam-4827	445	15	∗m	∗m	NOUN
ejpam-4827	445	16	implies	imply	VERB
ejpam-4827	445	17	that	that	SCONJ
ejpam-4827	445	18	biσgi(σs(c	biσgi(σs(c	PROPN
ejpam-4827	445	19	aj	aj	PROPN
ejpam-4827	445	20	)	)	PUNCT
ejpam-4827	445	21	)	)	PUNCT
ejpam-4827	445	22	∈	∈	PROPN
ejpam-4827	445	23	nil(r	nil(r	PROPN
ejpam-4827	445	24	)	)	PUNCT
ejpam-4827	445	25	.	.	PUNCT
ejpam-4827	446	1	we	we	PRON
ejpam-4827	446	2	will	will	AUX
ejpam-4827	446	3	show	show	VERB
ejpam-4827	446	4	that	that	DET
ejpam-4827	446	5	ajσhj	ajσhj	ADJ
ejpam-4827	446	6	(	(	PUNCT
ejpam-4827	446	7	σs(c	σs(c	NOUN
ejpam-4827	446	8	bi	bi	NOUN
ejpam-4827	446	9	)	)	PUNCT
ejpam-4827	446	10	)	)	PUNCT
ejpam-4827	447	1	∈	∈	PROPN
ejpam-4827	447	2	nil(r	nil(r	PROPN
ejpam-4827	447	3	)	)	PUNCT
ejpam-4827	447	4	for	for	ADP
ejpam-4827	447	5	each	each	DET
ejpam-4827	447	6	1	1	NUM
ejpam-4827	447	7	≤	≤	NUM
ejpam-4827	447	8	i	i	PRON
ejpam-4827	447	9	≤	≤	PROPN
ejpam-4827	447	10	n	n	CCONJ
ejpam-4827	447	11	,	,	PUNCT
ejpam-4827	447	12	1	1	NUM
ejpam-4827	447	13	≤	≤	NUM
ejpam-4827	447	14	k	k	NOUN
ejpam-4827	447	15	≤	≤	PROPN
ejpam-4827	447	16	d	d	NOUN
ejpam-4827	447	17	and	and	CCONJ
ejpam-4827	447	18	1	1	NUM
ejpam-4827	447	19	≤	≤	NUM
ejpam-4827	447	20	j	j	PROPN
ejpam-4827	447	21	≤	≤	PROPN
ejpam-4827	447	22	m.	m.	NOUN
ejpam-4827	447	23	we	we	PRON
ejpam-4827	447	24	proceed	proceed	VERB
ejpam-4827	447	25	by	by	ADP
ejpam-4827	447	26	induction	induction	NOUN
ejpam-4827	447	27	on	on	ADP
ejpam-4827	447	28	both	both	CCONJ
ejpam-4827	447	29	n	n	NOUN
ejpam-4827	447	30	and	and	CCONJ
ejpam-4827	447	31	m.	m.	NOUN
ejpam-4827	447	32	by	by	ADP
ejpam-4827	447	33	using	use	VERB
ejpam-4827	447	34	freely	freely	ADV
ejpam-4827	447	35	σ	σ	X
ejpam-4827	447	36	is	be	AUX
ejpam-4827	447	37	compatible	compatible	ADJ
ejpam-4827	447	38	monoid	monoid	NOUN
ejpam-4827	447	39	homomorphism	homomorphism	PROPN
ejpam-4827	447	40	.	.	PUNCT
ejpam-4827	448	1	let	let	VERB
ejpam-4827	448	2	n	n	NOUN
ejpam-4827	448	3	=	=	SYM
ejpam-4827	448	4	1	1	NUM
ejpam-4827	448	5	and	and	CCONJ
ejpam-4827	448	6	hence	hence	ADV
ejpam-4827	448	7	φ	φ	PROPN
ejpam-4827	448	8	=	=	SYM
ejpam-4827	449	1	b1g1	b1g1	PROPN
ejpam-4827	449	2	.	.	PUNCT
ejpam-4827	450	1	since	since	SCONJ
ejpam-4827	450	2	m	m	PROPN
ejpam-4827	450	3	is	be	AUX
ejpam-4827	450	4	u.p.monoid	u.p.monoid	ADJ
ejpam-4827	450	5	then	then	ADV
ejpam-4827	450	6	by	by	ADP
ejpam-4827	450	7	lemma	lemma	PROPN
ejpam-4827	450	8	4	4	NUM
ejpam-4827	450	9	,	,	PUNCT
ejpam-4827	450	10	m	m	VERB
ejpam-4827	450	11	is	be	AUX
ejpam-4827	450	12	cancellative	cancellative	ADJ
ejpam-4827	450	13	monoid	monoid	NOUN
ejpam-4827	450	14	,	,	PUNCT
ejpam-4827	450	15	g1hi	g1hi	PUNCT
ejpam-4827	450	16	̸=	̸=	PROPN
ejpam-4827	450	17	g1hj	g1hj	VERB
ejpam-4827	450	18	for	for	ADP
ejpam-4827	450	19	i	i	PRON
ejpam-4827	450	20	̸=	̸=	PROPN
ejpam-4827	450	21	j.	j.	PROPN
ejpam-4827	450	22	so	so	PROPN
ejpam-4827	450	23	b1σg1(c(aj	b1σg1(c(aj	PROPN
ejpam-4827	450	24	)	)	PUNCT
ejpam-4827	450	25	)	)	PUNCT
ejpam-4827	451	1	∈	∈	PROPN
ejpam-4827	451	2	nil(r	nil(r	PROPN
ejpam-4827	451	3	)	)	PUNCT
ejpam-4827	451	4	for	for	ADP
ejpam-4827	451	5	any	any	DET
ejpam-4827	451	6	c	c	PROPN
ejpam-4827	451	7	∈	∈	PROPN
ejpam-4827	451	8	r	r	NOUN
ejpam-4827	451	9	and	and	CCONJ
ejpam-4827	451	10	each	each	DET
ejpam-4827	451	11	j.	j.	PROPN
ejpam-4827	451	12	the	the	DET
ejpam-4827	451	13	proof	proof	NOUN
ejpam-4827	451	14	of	of	ADP
ejpam-4827	451	15	the	the	DET
ejpam-4827	451	16	case	case	NOUN
ejpam-4827	451	17	m	m	NOUN
ejpam-4827	451	18	=	=	SYM
ejpam-4827	451	19	1	1	NUM
ejpam-4827	451	20	is	be	AUX
ejpam-4827	451	21	similar	similar	ADJ
ejpam-4827	451	22	.	.	PUNCT
ejpam-4827	452	1	now	now	ADV
ejpam-4827	452	2	,	,	PUNCT
ejpam-4827	452	3	let	let	VERB
ejpam-4827	452	4	m	m	PRON
ejpam-4827	452	5	,	,	PUNCT
ejpam-4827	452	6	n	n	PRON
ejpam-4827	452	7	≥	≥	NOUN
ejpam-4827	452	8	2	2	NUM
ejpam-4827	452	9	.	.	PUNCT
ejpam-4827	453	1	since	since	SCONJ
ejpam-4827	453	2	m	m	PROPN
ejpam-4827	453	3	is	be	AUX
ejpam-4827	453	4	u.p.-monoid	u.p.-monoid	ADJ
ejpam-4827	453	5	,	,	PUNCT
ejpam-4827	453	6	there	there	PRON
ejpam-4827	453	7	exist	exist	VERB
ejpam-4827	453	8	i	i	PRON
ejpam-4827	453	9	,	,	PUNCT
ejpam-4827	453	10	j	j	PROPN
ejpam-4827	453	11	with	with	ADP
ejpam-4827	453	12	1	1	NUM
ejpam-4827	453	13	≤	≤	NUM
ejpam-4827	453	14	i	i	PRON
ejpam-4827	453	15	≤	≤	ADJ
ejpam-4827	453	16	n	n	CCONJ
ejpam-4827	453	17	and	and	CCONJ
ejpam-4827	453	18	1	1	NUM
ejpam-4827	453	19	≤	≤	NUM
ejpam-4827	453	20	j	j	PROPN
ejpam-4827	453	21	≤	≤	NUM
ejpam-4827	453	22	m	m	VERB
ejpam-4827	453	23	such	such	ADJ
ejpam-4827	453	24	that	that	DET
ejpam-4827	453	25	gishj	gishj	NOUN
ejpam-4827	453	26	is	be	AUX
ejpam-4827	453	27	uniquely	uniquely	ADV
ejpam-4827	453	28	presented	present	VERB
ejpam-4827	453	29	by	by	ADP
ejpam-4827	453	30	considering	consider	VERB
ejpam-4827	453	31	two	two	NUM
ejpam-4827	453	32	subsets	subset	NOUN
ejpam-4827	454	1	k	k	X
ejpam-4827	455	1	=	=	PRON
ejpam-4827	455	2	{	{	PUNCT
ejpam-4827	455	3	g1	g1	PROPN
ejpam-4827	455	4	,	,	PUNCT
ejpam-4827	455	5	g2	g2	PROPN
ejpam-4827	455	6	,	,	PUNCT
ejpam-4827	455	7	.	.	PUNCT
ejpam-4827	455	8	.	.	PUNCT
ejpam-4827	455	9	.	.	PUNCT
ejpam-4827	456	1	,	,	PUNCT
ejpam-4827	456	2	gn	gn	PROPN
ejpam-4827	456	3	}	}	PUNCT
ejpam-4827	456	4	and	and	CCONJ
ejpam-4827	456	5	h	h	NOUN
ejpam-4827	456	6	=	=	SYM
ejpam-4827	456	7	{	{	PUNCT
ejpam-4827	456	8	sh1	sh1	PROPN
ejpam-4827	456	9	,	,	PUNCT
ejpam-4827	456	10	sh2	sh2	PROPN
ejpam-4827	456	11	,	,	PUNCT
ejpam-4827	456	12	.	.	PUNCT
ejpam-4827	456	13	.	.	PUNCT
ejpam-4827	457	1	.	.	PUNCT
ejpam-4827	458	1	,	,	PUNCT
ejpam-4827	458	2	shm	shm	PROPN
ejpam-4827	458	3	}	}	PUNCT
ejpam-4827	458	4	of	of	ADP
ejpam-4827	458	5	a	a	DET
ejpam-4827	458	6	monoid	monoid	NOUN
ejpam-4827	458	7	m	m	NOUN
ejpam-4827	458	8	,	,	PUNCT
ejpam-4827	458	9	s	s	PROPN
ejpam-4827	458	10	∈	∈	PROPN
ejpam-4827	458	11	m.	m.	NOUN
ejpam-4827	458	12	we	we	PRON
ejpam-4827	458	13	may	may	AUX
ejpam-4827	458	14	assume	assume	VERB
ejpam-4827	458	15	without	without	ADP
ejpam-4827	458	16	loss	loss	NOUN
ejpam-4827	458	17	of	of	ADP
ejpam-4827	458	18	generality	generality	NOUN
ejpam-4827	458	19	,	,	PUNCT
ejpam-4827	459	1	that	that	SCONJ
ejpam-4827	459	2	i	i	PRON
ejpam-4827	459	3	=	=	SYM
ejpam-4827	459	4	n	n	PROPN
ejpam-4827	459	5	and	and	CCONJ
ejpam-4827	459	6	j	j	PROPN
ejpam-4827	459	7	=	=	SYM
ejpam-4827	459	8	m.	m.	NOUN
ejpam-4827	459	9	thus	thus	ADV
ejpam-4827	459	10	bnσgn(c	bnσgn(c	NOUN
ejpam-4827	459	11	(	(	PUNCT
ejpam-4827	459	12	am	be	AUX
ejpam-4827	459	13	)	)	PUNCT
ejpam-4827	459	14	)	)	PUNCT
ejpam-4827	459	15	∈	∈	PROPN
ejpam-4827	459	16	nil(r	nil(r	PROPN
ejpam-4827	459	17	)	)	PUNCT
ejpam-4827	459	18	.	.	PUNCT
ejpam-4827	460	1	hence	hence	ADV
ejpam-4827	460	2	for	for	ADP
ejpam-4827	460	3	any	any	DET
ejpam-4827	460	4	c	c	PROPN
ejpam-4827	460	5	∈	∈	PROPN
ejpam-4827	460	6	r	r	NOUN
ejpam-4827	460	7	,	,	PUNCT
ejpam-4827	460	8	bnσgn(ck(am	bnσgn(ck(am	NOUN
ejpam-4827	460	9	)	)	PUNCT
ejpam-4827	460	10	)	)	PUNCT
ejpam-4827	460	11	∈	∈	PROPN
ejpam-4827	460	12	nil(r	nil(r	PROPN
ejpam-4827	460	13	)	)	PUNCT
ejpam-4827	460	14	so	so	ADV
ejpam-4827	460	15	,	,	PUNCT
ejpam-4827	460	16	there	there	PRON
ejpam-4827	460	17	exist	exist	VERB
ejpam-4827	460	18	a	a	DET
ejpam-4827	460	19	positive	positive	ADJ
ejpam-4827	460	20	integer	integer	NOUN
ejpam-4827	460	21	k	k	PROPN
ejpam-4827	460	22	≥	≥	NUM
ejpam-4827	460	23	1	1	NUM
ejpam-4827	460	24	such	such	ADJ
ejpam-4827	460	25	that	that	PRON
ejpam-4827	460	26	(	(	PUNCT
ejpam-4827	460	27	bnσgn(c	bnσgn(c	NOUN
ejpam-4827	460	28	(	(	PUNCT
ejpam-4827	460	29	am	am	NOUN
ejpam-4827	460	30	)	)	PUNCT
ejpam-4827	460	31	)	)	PUNCT
ejpam-4827	460	32	)	)	PUNCT
ejpam-4827	461	1	k	k	X
ejpam-4827	462	1	=	=	PUNCT
ejpam-4827	462	2	0	0	PROPN
ejpam-4827	462	3	.	.	PUNCT
ejpam-4827	463	1	then	then	ADV
ejpam-4827	463	2	(	(	PUNCT
ejpam-4827	463	3	bnσgn(c	bnσgn(c	NOUN
ejpam-4827	463	4	(	(	PUNCT
ejpam-4827	463	5	am)))(bnσgn(c	am)))(bnσgn(c	PROPN
ejpam-4827	463	6	(	(	PUNCT
ejpam-4827	463	7	am	am	NOUN
ejpam-4827	463	8	)	)	PUNCT
ejpam-4827	463	9	)	)	PUNCT
ejpam-4827	463	10	)	)	PUNCT
ejpam-4827	463	11	·	·	PUNCT
ejpam-4827	463	12	·	·	PUNCT
ejpam-4827	463	13	·	·	PUNCT
ejpam-4827	463	14	(	(	PUNCT
ejpam-4827	463	15	bnσgn(c	bnσgn(c	NOUN
ejpam-4827	463	16	(	(	PUNCT
ejpam-4827	463	17	am	am	NOUN
ejpam-4827	463	18	)	)	PUNCT
ejpam-4827	463	19	)	)	PUNCT
ejpam-4827	463	20	)	)	PUNCT
ejpam-4827	464	1	=	=	PUNCT
ejpam-4827	464	2	0	0	X
ejpam-4827	464	3	.	.	PUNCT
ejpam-4827	465	1	now	now	ADV
ejpam-4827	465	2	,	,	PUNCT
ejpam-4827	465	3	since	since	SCONJ
ejpam-4827	465	4	by	by	ADP
ejpam-4827	465	5	hypothesis	hypothesis	NOUN
ejpam-4827	465	6	σ	σ	NOUN
ejpam-4827	465	7	:	:	PUNCT
ejpam-4827	465	8	m	m	PROPN
ejpam-4827	465	9	→	→	SYM
ejpam-4827	465	10	aut(r	aut(r	PROPN
ejpam-4827	465	11	)	)	PUNCT
ejpam-4827	465	12	is	be	AUX
ejpam-4827	465	13	a	a	DET
ejpam-4827	465	14	compatible	compatible	ADJ
ejpam-4827	465	15	monoid	monoid	NOUN
ejpam-4827	465	16	homomorphism	homomorphism	NOUN
ejpam-4827	465	17	,	,	PUNCT
ejpam-4827	465	18	we	we	PRON
ejpam-4827	465	19	have	have	VERB
ejpam-4827	465	20	that	that	PRON
ejpam-4827	465	21	(	(	PUNCT
ejpam-4827	465	22	bnσgn(c	bnσgn(c	NOUN
ejpam-4827	465	23	am	be	AUX
ejpam-4827	465	24	)	)	PUNCT
ejpam-4827	465	25	)	)	PUNCT
ejpam-4827	465	26	·	·	PUNCT
ejpam-4827	465	27	·	·	PUNCT
ejpam-4827	465	28	·	·	PUNCT
ejpam-4827	465	29	(	(	PUNCT
ejpam-4827	465	30	bnσgn(c	bnσgn(c	NOUN
ejpam-4827	465	31	am))(bnc	am))(bnc	PROPN
ejpam-4827	465	32	am	be	AUX
ejpam-4827	465	33	)	)	PUNCT
ejpam-4827	465	34	=	=	SYM
ejpam-4827	465	35	0	0	PUNCT
ejpam-4827	466	1	and	and	CCONJ
ejpam-4827	466	2	then	then	ADV
ejpam-4827	466	3	we	we	PRON
ejpam-4827	466	4	conclude	conclude	VERB
ejpam-4827	466	5	that	that	PRON
ejpam-4827	466	6	(	(	PUNCT
ejpam-4827	466	7	bnσgn(c	bnσgn(c	NOUN
ejpam-4827	466	8	am	be	AUX
ejpam-4827	466	9	)	)	PUNCT
ejpam-4827	466	10	)	)	PUNCT
ejpam-4827	466	11	·	·	PUNCT
ejpam-4827	466	12	·	·	PUNCT
ejpam-4827	466	13	·	·	PUNCT
ejpam-4827	466	14	(	(	PUNCT
ejpam-4827	466	15	bnσgn(c	bnσgn(c	NOUN
ejpam-4827	466	16	am))σgn(bnc	am))σgn(bnc	NOUN
ejpam-4827	466	17	am	be	AUX
ejpam-4827	466	18	)	)	PUNCT
ejpam-4827	466	19	=	=	SYM
ejpam-4827	467	1	0	0	X
ejpam-4827	467	2	.	.	PUNCT
ejpam-4827	468	1	so	so	ADV
ejpam-4827	468	2	(	(	PUNCT
ejpam-4827	468	3	bnσgn(c	bnσgn(c	NOUN
ejpam-4827	468	4	am	be	AUX
ejpam-4827	468	5	)	)	PUNCT
ejpam-4827	468	6	)	)	PUNCT
ejpam-4827	468	7	·	·	PUNCT
ejpam-4827	468	8	·	·	PUNCT
ejpam-4827	468	9	·	·	PUNCT
ejpam-4827	469	1	bnσgn(c	bnσgn(c	NOUN
ejpam-4827	469	2	ambnam	ambnam	NOUN
ejpam-4827	469	3	)	)	PUNCT
ejpam-4827	469	4	=	=	SYM
ejpam-4827	469	5	0	0	PUNCT
ejpam-4827	469	6	and	and	CCONJ
ejpam-4827	469	7	hence	hence	ADV
ejpam-4827	469	8	(	(	PUNCT
ejpam-4827	469	9	bnσgn(c	bnσgn(c	NOUN
ejpam-4827	469	10	am	be	AUX
ejpam-4827	469	11	)	)	PUNCT
ejpam-4827	469	12	)	)	PUNCT
ejpam-4827	469	13	·	·	PUNCT
ejpam-4827	469	14	·	·	PUNCT
ejpam-4827	469	15	·	·	PUNCT
ejpam-4827	469	16	(	(	PUNCT
ejpam-4827	469	17	bnσgn(c	bnσgn(c	NOUN
ejpam-4827	469	18	am))(bnc	am))(bnc	PROPN
ejpam-4827	469	19	am)2	am)2	NOUN
ejpam-4827	469	20	=	=	NOUN
ejpam-4827	469	21	0	0	X
ejpam-4827	469	22	.	.	PUNCT
ejpam-4827	470	1	continuing	continue	VERB
ejpam-4827	470	2	this	this	DET
ejpam-4827	470	3	procedure	procedure	NOUN
ejpam-4827	470	4	yields	yield	NOUN
ejpam-4827	470	5	that	that	PRON
ejpam-4827	470	6	(	(	PUNCT
ejpam-4827	470	7	bnc	bnc	NOUN
ejpam-4827	470	8	am	am	NOUN
ejpam-4827	470	9	)	)	PUNCT
ejpam-4827	470	10	k	k	X
ejpam-4827	471	1	=	=	PUNCT
ejpam-4827	471	2	0	0	NUM
ejpam-4827	471	3	,	,	PUNCT
ejpam-4827	471	4	hence	hence	ADV
ejpam-4827	471	5	bnσgn(c	bnσgn(c	NOUN
ejpam-4827	471	6	am	be	AUX
ejpam-4827	471	7	)	)	PUNCT
ejpam-4827	471	8	∈	∈	PROPN
ejpam-4827	471	9	nil(r	nil(r	PROPN
ejpam-4827	471	10	)	)	PUNCT
ejpam-4827	471	11	.	.	PUNCT
ejpam-4827	472	1	therefore	therefore	ADV
ejpam-4827	472	2	,	,	PUNCT
ejpam-4827	472	3	anσgn(σs(c	anσgn(σs(c	PROPN
ejpam-4827	472	4	bm	bm	PROPN
ejpam-4827	472	5	)	)	PUNCT
ejpam-4827	472	6	)	)	PUNCT
ejpam-4827	473	1	∈	∈	PROPN
ejpam-4827	473	2	nil(r	nil(r	PROPN
ejpam-4827	473	3	)	)	PUNCT
ejpam-4827	473	4	for	for	ADP
ejpam-4827	473	5	each	each	DET
ejpam-4827	473	6	c	c	PROPN
ejpam-4827	473	7	∈	∈	PROPN
ejpam-4827	473	8	r.	r.	PROPN
ejpam-4827	473	9	then	then	ADV
ejpam-4827	473	10	we	we	PRON
ejpam-4827	473	11	have	have	VERB
ejpam-4827	473	12	am(φ	am(φ	NOUN
ejpam-4827	473	13	−	−	PROPN
ejpam-4827	473	14	bngn)ϕψ	bngn)ϕψ	NOUN
ejpam-4827	474	1	=	=	SYM
ejpam-4827	474	2	amφϕψ	amφϕψ	NOUN
ejpam-4827	474	3	−	−	PROPN
ejpam-4827	474	4	ambngnϕψ	ambngnϕψ	NOUN
ejpam-4827	474	5	∈	∈	PROPN
ejpam-4827	474	6	nil(r	nil(r	PROPN
ejpam-4827	474	7	)	)	PUNCT
ejpam-4827	474	8	∗m	∗m	NOUN
ejpam-4827	474	9	,	,	PUNCT
ejpam-4827	474	10	since	since	SCONJ
ejpam-4827	474	11	nil(r	nil(r	PROPN
ejpam-4827	474	12	)	)	PUNCT
ejpam-4827	474	13	e.	e.	PROPN
ejpam-4827	474	14	ali	ali	PROPN
ejpam-4827	474	15	/	/	SYM
ejpam-4827	474	16	eur	eur	PROPN
ejpam-4827	474	17	.	.	PUNCT
ejpam-4827	475	1	j.	j.	PROPN
ejpam-4827	475	2	pure	pure	PROPN
ejpam-4827	475	3	appl	appl	PROPN
ejpam-4827	475	4	.	.	PROPN
ejpam-4827	475	5	math	math	PROPN
ejpam-4827	475	6	,	,	PUNCT
ejpam-4827	475	7	16	16	NUM
ejpam-4827	475	8	(	(	PUNCT
ejpam-4827	475	9	3	3	NUM
ejpam-4827	475	10	)	)	PUNCT
ejpam-4827	475	11	(	(	PUNCT
ejpam-4827	475	12	2023	2023	NUM
ejpam-4827	475	13	)	)	PUNCT
ejpam-4827	475	14	,	,	PUNCT
ejpam-4827	475	15	1878	1878	NUM
ejpam-4827	475	16	-	-	SYM
ejpam-4827	475	17	1893	1893	NUM
ejpam-4827	475	18	1887	1887	NUM
ejpam-4827	475	19	is	be	AUX
ejpam-4827	475	20	an	an	DET
ejpam-4827	475	21	ideal	ideal	NOUN
ejpam-4827	475	22	of	of	ADP
ejpam-4827	475	23	r.	r.	NOUN
ejpam-4827	475	24	by	by	ADP
ejpam-4827	475	25	induction	induction	NOUN
ejpam-4827	475	26	hypothesis	hypothesis	NOUN
ejpam-4827	475	27	and	and	CCONJ
ejpam-4827	475	28	by	by	ADP
ejpam-4827	475	29	compatibility	compatibility	NOUN
ejpam-4827	475	30	,	,	PUNCT
ejpam-4827	475	31	we	we	PRON
ejpam-4827	475	32	have	have	VERB
ejpam-4827	475	33	bnσgn(c	bnσgn(c	NOUN
ejpam-4827	475	34	ajbn	ajbn	NOUN
ejpam-4827	475	35	)	)	PUNCT
ejpam-4827	475	36	=	=	SYM
ejpam-4827	476	1	0	0	NUM
ejpam-4827	476	2	implies	imply	VERB
ejpam-4827	476	3	biσgn(c	biσgn(c	NOUN
ejpam-4827	476	4	am	be	AUX
ejpam-4827	476	5	)	)	PUNCT
ejpam-4827	476	6	∈	∈	PROPN
ejpam-4827	476	7	nil(r	nil(r	NOUN
ejpam-4827	476	8	)	)	PUNCT
ejpam-4827	476	9	for	for	ADP
ejpam-4827	476	10	all	all	DET
ejpam-4827	476	11	i.	i.	NOUN
ejpam-4827	476	12	applying	apply	VERB
ejpam-4827	476	13	the	the	DET
ejpam-4827	476	14	preceding	precede	VERB
ejpam-4827	476	15	method	method	NOUN
ejpam-4827	476	16	repeatedly	repeatedly	ADV
ejpam-4827	476	17	,	,	PUNCT
ejpam-4827	476	18	we	we	PRON
ejpam-4827	476	19	obtain	obtain	VERB
ejpam-4827	476	20	biσgi(σs(c	biσgi(σs(c	ADJ
ejpam-4827	476	21	aj	aj	PROPN
ejpam-4827	476	22	)	)	PUNCT
ejpam-4827	476	23	)	)	PUNCT
ejpam-4827	477	1	∈	∈	PROPN
ejpam-4827	477	2	nil(r	nil(r	PROPN
ejpam-4827	477	3	)	)	PUNCT
ejpam-4827	477	4	for	for	ADP
ejpam-4827	477	5	each	each	DET
ejpam-4827	477	6	i	i	PROPN
ejpam-4827	477	7	,	,	PUNCT
ejpam-4827	477	8	j	j	PROPN
ejpam-4827	477	9	and	and	CCONJ
ejpam-4827	477	10	each	each	DET
ejpam-4827	477	11	c	c	PROPN
ejpam-4827	477	12	∈	∈	PROPN
ejpam-4827	477	13	r.	r.	PROPN
ejpam-4827	477	14	thus	thus	ADV
ejpam-4827	477	15	,	,	PUNCT
ejpam-4827	477	16	by	by	ADP
ejpam-4827	477	17	rigidness	rigidness	NOUN
ejpam-4827	477	18	we	we	PRON
ejpam-4827	477	19	have	have	AUX
ejpam-4827	477	20	ajσhj	ajσhj	ADJ
ejpam-4827	477	21	(	(	PUNCT
ejpam-4827	477	22	σs(c	σs(c	NOUN
ejpam-4827	477	23	bi	bi	NOUN
ejpam-4827	477	24	)	)	PUNCT
ejpam-4827	477	25	)	)	PUNCT
ejpam-4827	478	1	∈	∈	PROPN
ejpam-4827	478	2	nil(r	nil(r	PROPN
ejpam-4827	478	3	)	)	PUNCT
ejpam-4827	478	4	as	as	SCONJ
ejpam-4827	478	5	desired	desire	VERB
ejpam-4827	478	6	.	.	PUNCT
ejpam-4827	479	1	corollary	corollary	ADJ
ejpam-4827	479	2	3	3	NUM
ejpam-4827	479	3	.	.	PUNCT
ejpam-4827	480	1	(	(	PUNCT
ejpam-4827	480	2	1	1	X
ejpam-4827	480	3	)	)	PUNCT
ejpam-4827	480	4	every	every	DET
ejpam-4827	480	5	σ	σ	PROPN
ejpam-4827	480	6	-	-	PUNCT
ejpam-4827	480	7	compatible	compatible	ADJ
ejpam-4827	480	8	ni	ni	NOUN
ejpam-4827	480	9	-	-	PUNCT
ejpam-4827	480	10	ring	ring	NOUN
ejpam-4827	480	11	is	be	AUX
ejpam-4827	480	12	σ	σ	NOUN
ejpam-4827	480	13	-	-	PUNCT
ejpam-4827	480	14	skew	skew	NOUN
ejpam-4827	480	15	strongly	strongly	ADV
ejpam-4827	480	16	z	z	NOUN
ejpam-4827	480	17	-	-	PUNCT
ejpam-4827	480	18	nil	nil	ADV
ejpam-4827	480	19	-	-	PUNCT
ejpam-4827	480	20	reflexive	reflexive	ADJ
ejpam-4827	480	21	.	.	PUNCT
ejpam-4827	481	1	(	(	PUNCT
ejpam-4827	481	2	2	2	X
ejpam-4827	481	3	)	)	PUNCT
ejpam-4827	481	4	every	every	DET
ejpam-4827	481	5	ni	ni	PROPN
ejpam-4827	481	6	-	-	PUNCT
ejpam-4827	481	7	ring	ring	NOUN
ejpam-4827	481	8	is	be	AUX
ejpam-4827	481	9	nil	nil	ADV
ejpam-4827	481	10	-	-	PUNCT
ejpam-4827	481	11	reflexive	reflexive	ADJ
ejpam-4827	481	12	.	.	PUNCT
ejpam-4827	482	1	proof	proof	NOUN
ejpam-4827	482	2	.	.	PUNCT
ejpam-4827	483	1	(	(	PUNCT
ejpam-4827	483	2	1	1	X
ejpam-4827	483	3	)	)	PUNCT
ejpam-4827	483	4	taking	take	VERB
ejpam-4827	483	5	m	m	NOUN
ejpam-4827	483	6	=	=	X
ejpam-4827	483	7	{	{	PUNCT
ejpam-4827	483	8	.	.	PUNCT
ejpam-4827	483	9	.	.	PUNCT
ejpam-4827	484	1	.	.	PUNCT
ejpam-4827	485	1	,	,	PUNCT
ejpam-4827	485	2	x−2	x−2	PROPN
ejpam-4827	485	3	,	,	PUNCT
ejpam-4827	485	4	x−1	x−1	PROPN
ejpam-4827	485	5	,	,	PUNCT
ejpam-4827	485	6	1	1	NUM
ejpam-4827	485	7	,	,	PUNCT
ejpam-4827	485	8	x1	x1	PROPN
ejpam-4827	485	9	,	,	PUNCT
ejpam-4827	485	10	x2	x2	PROPN
ejpam-4827	485	11	,	,	PUNCT
ejpam-4827	485	12	.	.	PUNCT
ejpam-4827	485	13	.	.	PUNCT
ejpam-4827	485	14	.	.	PUNCT
ejpam-4827	485	15	}	}	PUNCT
ejpam-4827	485	16	and	and	CCONJ
ejpam-4827	485	17	σxn(λ	σxn(λ	PROPN
ejpam-4827	485	18	)	)	PUNCT
ejpam-4827	485	19	=	=	SYM
ejpam-4827	485	20	σn(λ	σn(λ	NOUN
ejpam-4827	485	21	)	)	PUNCT
ejpam-4827	485	22	for	for	ADP
ejpam-4827	485	23	each	each	DET
ejpam-4827	485	24	n	n	PRON
ejpam-4827	485	25	∈	∈	PROPN
ejpam-4827	485	26	z	z	NOUN
ejpam-4827	485	27	and	and	CCONJ
ejpam-4827	485	28	any	any	DET
ejpam-4827	485	29	λ	λ	X
ejpam-4827	485	30	∈	∈	PROPN
ejpam-4827	485	31	r	r	NOUN
ejpam-4827	485	32	,	,	PUNCT
ejpam-4827	485	33	we	we	PRON
ejpam-4827	485	34	have	have	VERB
ejpam-4827	485	35	r	r	NOUN
ejpam-4827	485	36	∗m	∗m	NOUN
ejpam-4827	485	37	∼=	∼=	PROPN
ejpam-4827	485	38	r[x	r[x	NOUN
ejpam-4827	485	39	,	,	PUNCT
ejpam-4827	485	40	x−1	x−1	PROPN
ejpam-4827	485	41	,	,	PUNCT
ejpam-4827	485	42	σ	σ	PROPN
ejpam-4827	485	43	]	]	X
ejpam-4827	485	44	,	,	PUNCT
ejpam-4827	485	45	and	and	CCONJ
ejpam-4827	485	46	the	the	DET
ejpam-4827	485	47	result	result	NOUN
ejpam-4827	485	48	follows	follow	VERB
ejpam-4827	485	49	from	from	ADP
ejpam-4827	485	50	theorem	theorem	ADJ
ejpam-4827	485	51	3	3	NUM
ejpam-4827	485	52	.	.	PUNCT
ejpam-4827	485	53	(	(	PUNCT
ejpam-4827	485	54	2	2	X
ejpam-4827	485	55	)	)	PUNCT
ejpam-4827	485	56	taking	take	VERB
ejpam-4827	485	57	m	m	NOUN
ejpam-4827	485	58	=	=	PUNCT
ejpam-4827	485	59	{	{	PUNCT
ejpam-4827	485	60	1	1	NUM
ejpam-4827	485	61	,	,	PUNCT
ejpam-4827	485	62	x1	x1	PROPN
ejpam-4827	485	63	,	,	PUNCT
ejpam-4827	485	64	x2	x2	PROPN
ejpam-4827	485	65	,	,	PUNCT
ejpam-4827	485	66	.	.	PUNCT
ejpam-4827	485	67	.	.	PUNCT
ejpam-4827	485	68	.	.	PUNCT
ejpam-4827	485	69	}	}	PUNCT
ejpam-4827	485	70	and	and	CCONJ
ejpam-4827	485	71	σxn(λ	σxn(λ	PROPN
ejpam-4827	485	72	)	)	PUNCT
ejpam-4827	486	1	=	=	SYM
ejpam-4827	486	2	λ	λ	PROPN
ejpam-4827	486	3	for	for	ADP
ejpam-4827	486	4	each	each	DET
ejpam-4827	486	5	n	n	PRON
ejpam-4827	486	6	∈	∈	PROPN
ejpam-4827	486	7	n	n	NOUN
ejpam-4827	486	8	∪	∪	X
ejpam-4827	486	9	{	{	PUNCT
ejpam-4827	486	10	0	0	NUM
ejpam-4827	486	11	}	}	PUNCT
ejpam-4827	486	12	and	and	CCONJ
ejpam-4827	486	13	any	any	DET
ejpam-4827	486	14	r	r	NOUN
ejpam-4827	486	15	∈	∈	NOUN
ejpam-4827	486	16	r	r	NOUN
ejpam-4827	486	17	,	,	PUNCT
ejpam-4827	486	18	we	we	PRON
ejpam-4827	486	19	have	have	VERB
ejpam-4827	486	20	r	r	NOUN
ejpam-4827	486	21	∗m	∗m	NOUN
ejpam-4827	486	22	∼=	∼=	PART
ejpam-4827	486	23	r[x	r[x	NOUN
ejpam-4827	486	24	]	]	X
ejpam-4827	486	25	,	,	PUNCT
ejpam-4827	486	26	and	and	CCONJ
ejpam-4827	486	27	the	the	DET
ejpam-4827	486	28	result	result	NOUN
ejpam-4827	486	29	follows	follow	VERB
ejpam-4827	486	30	from	from	ADP
ejpam-4827	486	31	theorem	theorem	ADJ
ejpam-4827	486	32	3	3	NUM
ejpam-4827	486	33	.	.	PUNCT
ejpam-4827	486	34	corollary	corollary	ADJ
ejpam-4827	486	35	4	4	NUM
ejpam-4827	486	36	.	.	PUNCT
ejpam-4827	486	37	(	(	PUNCT
ejpam-4827	486	38	theorem	theorem	VERB
ejpam-4827	486	39	3.1	3.1	NUM
ejpam-4827	486	40	[	[	SYM
ejpam-4827	486	41	20	20	NUM
ejpam-4827	486	42	]	]	PUNCT
ejpam-4827	486	43	)	)	PUNCT
ejpam-4827	486	44	let	let	VERB
ejpam-4827	486	45	m	m	PRON
ejpam-4827	486	46	be	be	AUX
ejpam-4827	486	47	a	a	DET
ejpam-4827	486	48	u.p.-monoid	u.p.-monoid	NOUN
ejpam-4827	486	49	and	and	CCONJ
ejpam-4827	486	50	r	r	NOUN
ejpam-4827	486	51	be	be	AUX
ejpam-4827	486	52	a	a	DET
ejpam-4827	486	53	reduced	reduce	VERB
ejpam-4827	486	54	ring	ring	NOUN
ejpam-4827	486	55	.	.	PUNCT
ejpam-4827	487	1	then	then	ADV
ejpam-4827	487	2	,	,	PUNCT
ejpam-4827	487	3	r	r	NOUN
ejpam-4827	487	4	is	be	AUX
ejpam-4827	487	5	strongly	strongly	ADV
ejpam-4827	487	6	m	m	PRON
ejpam-4827	487	7	-reflexive	-reflexive	ADJ
ejpam-4827	487	8	.	.	PUNCT
ejpam-4827	488	1	corollary	corollary	ADJ
ejpam-4827	488	2	5	5	NUM
ejpam-4827	488	3	.	.	PUNCT
ejpam-4827	489	1	let	let	VERB
ejpam-4827	489	2	r	r	PRON
ejpam-4827	489	3	be	be	AUX
ejpam-4827	489	4	a	a	DET
ejpam-4827	489	5	ring	ring	NOUN
ejpam-4827	489	6	,	,	PUNCT
ejpam-4827	489	7	where	where	SCONJ
ejpam-4827	489	8	nil(r	nil(r	NOUN
ejpam-4827	489	9	)	)	PUNCT
ejpam-4827	489	10	is	be	AUX
ejpam-4827	489	11	an	an	DET
ejpam-4827	489	12	ideal	ideal	NOUN
ejpam-4827	489	13	of	of	ADP
ejpam-4827	489	14	r.	r.	PROPN
ejpam-4827	489	15	let	let	VERB
ejpam-4827	489	16	m	m	PRON
ejpam-4827	489	17	be	be	AUX
ejpam-4827	489	18	a	a	DET
ejpam-4827	489	19	u.p.-monoid	u.p.-monoid	NOUN
ejpam-4827	489	20	and	and	CCONJ
ejpam-4827	489	21	σ	σ	NOUN
ejpam-4827	489	22	:	:	PUNCT
ejpam-4827	489	23	m	m	PROPN
ejpam-4827	489	24	→	→	SYM
ejpam-4827	489	25	aut(r	aut(r	PROPN
ejpam-4827	489	26	)	)	PUNCT
ejpam-4827	489	27	be	be	AUX
ejpam-4827	489	28	a	a	DET
ejpam-4827	489	29	compatible	compatible	ADJ
ejpam-4827	489	30	monoid	monoid	NOUN
ejpam-4827	489	31	homomorphism	homomorphism	NOUN
ejpam-4827	489	32	.	.	PUNCT
ejpam-4827	490	1	then	then	ADV
ejpam-4827	490	2	,	,	PUNCT
ejpam-4827	490	3	r	r	NOUN
ejpam-4827	490	4	is	be	AUX
ejpam-4827	490	5	σ	σ	NOUN
ejpam-4827	490	6	-	-	PUNCT
ejpam-4827	490	7	skew	skew	NOUN
ejpam-4827	490	8	strongly	strongly	ADV
ejpam-4827	490	9	m	m	VERB
ejpam-4827	490	10	-nil	-nil	ADJ
ejpam-4827	490	11	-	-	PUNCT
ejpam-4827	490	12	reflexive	reflexive	ADJ
ejpam-4827	490	13	proof	proof	NOUN
ejpam-4827	490	14	.	.	PUNCT
ejpam-4827	491	1	since	since	SCONJ
ejpam-4827	491	2	nil(r	nil(r	PROPN
ejpam-4827	491	3	)	)	PUNCT
ejpam-4827	491	4	is	be	AUX
ejpam-4827	491	5	an	an	DET
ejpam-4827	491	6	ideal	ideal	NOUN
ejpam-4827	491	7	,	,	PUNCT
ejpam-4827	491	8	it	it	PRON
ejpam-4827	491	9	is	be	AUX
ejpam-4827	491	10	an	an	DET
ejpam-4827	491	11	ni	ni	NOUN
ejpam-4827	491	12	-	-	PUNCT
ejpam-4827	491	13	ring	ring	NOUN
ejpam-4827	491	14	.	.	PUNCT
ejpam-4827	492	1	therefore	therefore	ADV
ejpam-4827	492	2	,	,	PUNCT
ejpam-4827	492	3	the	the	DET
ejpam-4827	492	4	proof	proof	NOUN
ejpam-4827	492	5	follows	follow	VERB
ejpam-4827	492	6	from	from	ADP
ejpam-4827	492	7	theorem	theorem	ADJ
ejpam-4827	492	8	3	3	NUM
ejpam-4827	492	9	.	.	PUNCT
ejpam-4827	493	1	the	the	DET
ejpam-4827	493	2	following	follow	VERB
ejpam-4827	493	3	example	example	NOUN
ejpam-4827	493	4	shows	show	VERB
ejpam-4827	493	5	that	that	SCONJ
ejpam-4827	493	6	r	r	NOUN
ejpam-4827	493	7	being	be	AUX
ejpam-4827	493	8	a	a	DET
ejpam-4827	493	9	reduced	reduce	VERB
ejpam-4827	493	10	ring	ring	NOUN
ejpam-4827	493	11	in	in	ADP
ejpam-4827	493	12	theorem	theorem	ADJ
ejpam-4827	493	13	3	3	NUM
ejpam-4827	493	14	is	be	AUX
ejpam-4827	493	15	not	not	PART
ejpam-4827	493	16	superfluous	superfluous	ADJ
ejpam-4827	493	17	.	.	PUNCT
ejpam-4827	493	18	example	example	NOUN
ejpam-4827	494	1	2	2	NUM
ejpam-4827	494	2	.	.	PUNCT
ejpam-4827	495	1	let	let	VERB
ejpam-4827	495	2	m	m	PRON
ejpam-4827	495	3	be	be	AUX
ejpam-4827	495	4	a	a	DET
ejpam-4827	495	5	monoid	monoid	NOUN
ejpam-4827	495	6	with	with	ADP
ejpam-4827	495	7	|	|	ADV
ejpam-4827	495	8	m	m	VERB
ejpam-4827	495	9	|≥	|≥	ADJ
ejpam-4827	495	10	2	2	NUM
ejpam-4827	495	11	,	,	PUNCT
ejpam-4827	495	12	s	s	VERB
ejpam-4827	495	13	=	=	SYM
ejpam-4827	495	14	m2(f	m2(f	PROPN
ejpam-4827	495	15	)	)	PUNCT
ejpam-4827	495	16	and	and	CCONJ
ejpam-4827	495	17	σ	σ	NUM
ejpam-4827	495	18	:	:	PUNCT
ejpam-4827	495	19	m	m	AUX
ejpam-4827	495	20	→	→	SYM
ejpam-4827	495	21	aut(r	aut(r	PROPN
ejpam-4827	495	22	)	)	PUNCT
ejpam-4827	495	23	be	be	AUX
ejpam-4827	495	24	a	a	DET
ejpam-4827	495	25	compatible	compatible	ADJ
ejpam-4827	495	26	monoid	monoid	NOUN
ejpam-4827	495	27	homomorphism	homomorphism	NOUN
ejpam-4827	495	28	.	.	PUNCT
ejpam-4827	496	1	then	then	ADV
ejpam-4827	496	2	,	,	PUNCT
ejpam-4827	496	3	s	s	VERB
ejpam-4827	496	4	is	be	AUX
ejpam-4827	496	5	not	not	PART
ejpam-4827	496	6	σ	σ	NOUN
ejpam-4827	496	7	-	-	PUNCT
ejpam-4827	496	8	skew	skew	NOUN
ejpam-4827	496	9	strongly	strongly	ADV
ejpam-4827	496	10	m	m	VERB
ejpam-4827	496	11	-nil	-nil	NOUN
ejpam-4827	496	12	-	-	PUNCT
ejpam-4827	496	13	reflexive	reflexive	ADJ
ejpam-4827	496	14	.	.	PUNCT
ejpam-4827	497	1	proof	proof	NOUN
ejpam-4827	497	2	.	.	PUNCT
ejpam-4827	498	1	take	take	VERB
ejpam-4827	498	2	e	e	NOUN
ejpam-4827	498	3	̸=	̸=	PROPN
ejpam-4827	498	4	g	g	NOUN
ejpam-4827	498	5	∈m	∈m	NOUN
ejpam-4827	498	6	and	and	CCONJ
ejpam-4827	498	7	let	let	VERB
ejpam-4827	498	8	φ	φ	PROPN
ejpam-4827	498	9	=	=	SYM
ejpam-4827	498	10	(	(	PUNCT
ejpam-4827	498	11	0	0	NUM
ejpam-4827	498	12	1	1	NUM
ejpam-4827	498	13	0	0	NUM
ejpam-4827	498	14	0	0	NUM
ejpam-4827	498	15	)	)	PUNCT
ejpam-4827	498	16	e+	e+	PUNCT
ejpam-4827	498	17	(	(	PUNCT
ejpam-4827	498	18	1	1	NUM
ejpam-4827	498	19	0	0	NUM
ejpam-4827	498	20	0	0	NUM
ejpam-4827	498	21	0	0	NUM
ejpam-4827	498	22	)	)	PUNCT
ejpam-4827	498	23	g	g	NOUN
ejpam-4827	498	24	,	,	PUNCT
ejpam-4827	498	25	ψ	ψ	X
ejpam-4827	498	26	=	=	SYM
ejpam-4827	498	27	(	(	PUNCT
ejpam-4827	498	28	1	1	NUM
ejpam-4827	498	29	1	1	NUM
ejpam-4827	498	30	0	0	NUM
ejpam-4827	498	31	0	0	NUM
ejpam-4827	498	32	)	)	PUNCT
ejpam-4827	498	33	e+	e+	PUNCT
ejpam-4827	498	34	(	(	PUNCT
ejpam-4827	498	35	1	1	NUM
ejpam-4827	498	36	1	1	NUM
ejpam-4827	498	37	0	0	NUM
ejpam-4827	498	38	0	0	NUM
ejpam-4827	498	39	)	)	PUNCT
ejpam-4827	498	40	g	g	PROPN
ejpam-4827	498	41	∈	∈	PROPN
ejpam-4827	498	42	s	s	PART
ejpam-4827	498	43	∗m	∗m	NOUN
ejpam-4827	498	44	.	.	PUNCT
ejpam-4827	499	1	for	for	ADP
ejpam-4827	499	2	ϕ	ϕ	NOUN
ejpam-4827	499	3	=	=	PUNCT
ejpam-4827	499	4	(	(	PUNCT
ejpam-4827	499	5	1	1	NUM
ejpam-4827	499	6	0	0	NUM
ejpam-4827	499	7	0	0	NUM
ejpam-4827	499	8	1	1	NUM
ejpam-4827	499	9	)	)	PUNCT
ejpam-4827	499	10	g	g	PROPN
ejpam-4827	499	11	∈	∈	PROPN
ejpam-4827	499	12	s	s	PART
ejpam-4827	499	13	∗m	∗m	NOUN
ejpam-4827	499	14	,	,	PUNCT
ejpam-4827	499	15	it	it	PRON
ejpam-4827	499	16	is	be	AUX
ejpam-4827	499	17	easy	easy	ADJ
ejpam-4827	499	18	to	to	PART
ejpam-4827	499	19	check	check	VERB
ejpam-4827	499	20	that	that	DET
ejpam-4827	499	21	φϕψ	φϕψ	PROPN
ejpam-4827	499	22	∈	∈	PROPN
ejpam-4827	499	23	nil(s	nil(s	PROPN
ejpam-4827	499	24	∗m	∗m	NOUN
ejpam-4827	499	25	)	)	PUNCT
ejpam-4827	499	26	.	.	PUNCT
ejpam-4827	500	1	but	but	CCONJ
ejpam-4827	500	2	,	,	PUNCT
ejpam-4827	500	3	we	we	PRON
ejpam-4827	500	4	have	have	VERB
ejpam-4827	500	5	ψϕφ	ψϕφ	NUM
ejpam-4827	500	6	=	=	SYM
ejpam-4827	500	7	(	(	PUNCT
ejpam-4827	500	8	0	0	NUM
ejpam-4827	500	9	1	1	NUM
ejpam-4827	500	10	0	0	NUM
ejpam-4827	500	11	0	0	NUM
ejpam-4827	500	12	)	)	PUNCT
ejpam-4827	501	1	g	g	NOUN
ejpam-4827	501	2	+	+	CCONJ
ejpam-4827	501	3	(	(	PUNCT
ejpam-4827	501	4	1	1	NUM
ejpam-4827	501	5	0	0	NUM
ejpam-4827	501	6	0	0	NUM
ejpam-4827	501	7	−1	−1	NOUN
ejpam-4827	501	8	)	)	PUNCT
ejpam-4827	502	1	g2	g2	PROPN
ejpam-4827	502	2	+	+	CCONJ
ejpam-4827	502	3	(	(	PUNCT
ejpam-4827	502	4	0	0	NUM
ejpam-4827	502	5	0	0	NUM
ejpam-4827	502	6	−1	−1	NOUN
ejpam-4827	502	7	0	0	X
ejpam-4827	502	8	)	)	PUNCT
ejpam-4827	502	9	g3	g3	NOUN
ejpam-4827	502	10	̸=	̸=	PROPN
ejpam-4827	502	11	0	0	NUM
ejpam-4827	502	12	which	which	PRON
ejpam-4827	502	13	implies	imply	VERB
ejpam-4827	502	14	that	that	SCONJ
ejpam-4827	502	15	s	s	VERB
ejpam-4827	502	16	is	be	AUX
ejpam-4827	502	17	not	not	PART
ejpam-4827	502	18	σ	σ	NOUN
ejpam-4827	502	19	-	-	PUNCT
ejpam-4827	502	20	skew	skew	NOUN
ejpam-4827	502	21	strongly	strongly	ADV
ejpam-4827	502	22	m	m	VERB
ejpam-4827	502	23	-nil	-nil	NOUN
ejpam-4827	502	24	-	-	PUNCT
ejpam-4827	502	25	reflexive	reflexive	ADJ
ejpam-4827	502	26	.	.	PUNCT
ejpam-4827	503	1	shaban	shaban	PROPN
ejpam-4827	503	2	and	and	CCONJ
ejpam-4827	503	3	mohammed	mohammed	PROPN
ejpam-4827	504	1	[	[	X
ejpam-4827	504	2	22	22	NUM
ejpam-4827	504	3	]	]	PUNCT
ejpam-4827	504	4	proved	prove	VERB
ejpam-4827	504	5	that	that	SCONJ
ejpam-4827	504	6	nil(r	nil(r	DET
ejpam-4827	504	7	∗m	∗m	NOUN
ejpam-4827	504	8	)	)	PUNCT
ejpam-4827	504	9	̸=	̸=	PROPN
ejpam-4827	504	10	nil(r	nil(r	NUM
ejpam-4827	504	11	)	)	PUNCT
ejpam-4827	504	12	∗m	∗m	NOUN
ejpam-4827	504	13	for	for	ADP
ejpam-4827	504	14	an	an	DET
ejpam-4827	504	15	ni	ni	NOUN
ejpam-4827	504	16	-	-	PUNCT
ejpam-4827	504	17	ring	ring	NOUN
ejpam-4827	504	18	r	r	NOUN
ejpam-4827	504	19	and	and	CCONJ
ejpam-4827	504	20	a	a	DET
ejpam-4827	504	21	u.p.-monoid	u.p.-monoid	NOUN
ejpam-4827	504	22	m	m	VERB
ejpam-4827	504	23	with	with	ADP
ejpam-4827	504	24	|m	|m	NOUN
ejpam-4827	504	25	|≥	|≥	ADJ
ejpam-4827	504	26	2	2	NUM
ejpam-4827	504	27	.	.	PUNCT
ejpam-4827	504	28	based	base	VERB
ejpam-4827	504	29	on	on	ADP
ejpam-4827	504	30	their	their	PRON
ejpam-4827	504	31	work	work	NOUN
ejpam-4827	504	32	,	,	PUNCT
ejpam-4827	504	33	we	we	PRON
ejpam-4827	504	34	have	have	VERB
ejpam-4827	504	35	the	the	DET
ejpam-4827	504	36	following	follow	VERB
ejpam-4827	504	37	result	result	NOUN
ejpam-4827	504	38	.	.	PUNCT
ejpam-4827	505	1	theorem	theorem	ADJ
ejpam-4827	505	2	4	4	NUM
ejpam-4827	505	3	.	.	PUNCT
ejpam-4827	506	1	let	let	VERB
ejpam-4827	506	2	r	r	NOUN
ejpam-4827	506	3	be	be	AUX
ejpam-4827	506	4	any	any	DET
ejpam-4827	506	5	ring	ring	NOUN
ejpam-4827	506	6	,	,	PUNCT
ejpam-4827	506	7	m	m	VERB
ejpam-4827	506	8	be	be	VERB
ejpam-4827	506	9	any	any	DET
ejpam-4827	506	10	monoid	monoid	NOUN
ejpam-4827	506	11	with	with	ADP
ejpam-4827	506	12	|	|	ADV
ejpam-4827	506	13	m	m	VERB
ejpam-4827	506	14	|≥	|≥	ADJ
ejpam-4827	506	15	2	2	NUM
ejpam-4827	506	16	and	and	CCONJ
ejpam-4827	506	17	σ	σ	NOUN
ejpam-4827	506	18	:	:	PUNCT
ejpam-4827	506	19	m	m	AUX
ejpam-4827	506	20	→	→	SYM
ejpam-4827	506	21	aut(r	aut(r	PROPN
ejpam-4827	506	22	)	)	PUNCT
ejpam-4827	506	23	be	be	AUX
ejpam-4827	506	24	a	a	DET
ejpam-4827	506	25	compatible	compatible	ADJ
ejpam-4827	506	26	monoid	monoid	NOUN
ejpam-4827	506	27	homomorphism	homomorphism	NOUN
ejpam-4827	506	28	such	such	ADJ
ejpam-4827	506	29	that	that	SCONJ
ejpam-4827	506	30	nil(r	nil(r	ADJ
ejpam-4827	506	31	∗m	∗m	NOUN
ejpam-4827	506	32	)	)	PUNCT
ejpam-4827	506	33	=	=	SYM
ejpam-4827	506	34	nil(r	nil(r	ADJ
ejpam-4827	506	35	)	)	PUNCT
ejpam-4827	506	36	∗m	∗m	NOUN
ejpam-4827	506	37	.	.	PUNCT
ejpam-4827	507	1	then	then	ADV
ejpam-4827	507	2	r	r	NOUN
ejpam-4827	507	3	is	be	AUX
ejpam-4827	507	4	σ	σ	NOUN
ejpam-4827	507	5	-	-	PUNCT
ejpam-4827	507	6	skew	skew	NOUN
ejpam-4827	507	7	strongly	strongly	ADV
ejpam-4827	507	8	m	m	VERB
ejpam-4827	507	9	-nil	-nil	NOUN
ejpam-4827	507	10	-	-	PUNCT
ejpam-4827	507	11	reflexive	reflexive	ADJ
ejpam-4827	507	12	.	.	PUNCT
ejpam-4827	508	1	e.	e.	PROPN
ejpam-4827	508	2	ali	ali	PROPN
ejpam-4827	508	3	/	/	SYM
ejpam-4827	508	4	eur	eur	PROPN
ejpam-4827	508	5	.	.	PUNCT
ejpam-4827	509	1	j.	j.	PROPN
ejpam-4827	509	2	pure	pure	PROPN
ejpam-4827	509	3	appl	appl	PROPN
ejpam-4827	509	4	.	.	PROPN
ejpam-4827	509	5	math	math	PROPN
ejpam-4827	509	6	,	,	PUNCT
ejpam-4827	509	7	16	16	NUM
ejpam-4827	509	8	(	(	PUNCT
ejpam-4827	509	9	3	3	NUM
ejpam-4827	509	10	)	)	PUNCT
ejpam-4827	509	11	(	(	PUNCT
ejpam-4827	509	12	2023	2023	NUM
ejpam-4827	509	13	)	)	PUNCT
ejpam-4827	509	14	,	,	PUNCT
ejpam-4827	509	15	1878	1878	NUM
ejpam-4827	509	16	-	-	SYM
ejpam-4827	509	17	1893	1893	NUM
ejpam-4827	509	18	1888	1888	NUM
ejpam-4827	509	19	proof	proof	NOUN
ejpam-4827	509	20	.	.	PUNCT
ejpam-4827	510	1	let	let	VERB
ejpam-4827	510	2	φ	φ	PROPN
ejpam-4827	510	3	=	=	SYM
ejpam-4827	510	4	σni=1bigi	σni=1bigi	PROPN
ejpam-4827	510	5	and	and	CCONJ
ejpam-4827	510	6	ψ	ψ	X
ejpam-4827	510	7	=	=	NOUN
ejpam-4827	510	8	σmj=1ajhj	σmj=1ajhj	NOUN
ejpam-4827	510	9	be	be	VERB
ejpam-4827	510	10	nonzero	nonzero	ADJ
ejpam-4827	510	11	elements	element	NOUN
ejpam-4827	510	12	in	in	ADP
ejpam-4827	510	13	r	r	NOUN
ejpam-4827	510	14	∗m	∗m	NOUN
ejpam-4827	510	15	such	such	ADJ
ejpam-4827	510	16	that	that	SCONJ
ejpam-4827	510	17	φϕψ	φϕψ	PROPN
ejpam-4827	510	18	∈	∈	PROPN
ejpam-4827	510	19	nil(r	nil(r	PROPN
ejpam-4827	510	20	)	)	PUNCT
ejpam-4827	510	21	∗m	∗m	NOUN
ejpam-4827	510	22	for	for	ADP
ejpam-4827	510	23	any	any	DET
ejpam-4827	510	24	ϕ	ϕ	PROPN
ejpam-4827	510	25	∈	∈	PROPN
ejpam-4827	510	26	r	r	NOUN
ejpam-4827	510	27	∗m	∗m	NOUN
ejpam-4827	510	28	.	.	PUNCT
ejpam-4827	511	1	we	we	PRON
ejpam-4827	511	2	have	have	VERB
ejpam-4827	511	3	φϕψ	φϕψ	NOUN
ejpam-4827	511	4	∈	∈	PROPN
ejpam-4827	511	5	nil(r	nil(r	DET
ejpam-4827	511	6	∗m	∗m	NOUN
ejpam-4827	511	7	)	)	PUNCT
ejpam-4827	511	8	,	,	PUNCT
ejpam-4827	511	9	which	which	PRON
ejpam-4827	511	10	is	be	AUX
ejpam-4827	511	11	equivalent	equivalent	ADJ
ejpam-4827	511	12	to	to	ADP
ejpam-4827	511	13	the	the	DET
ejpam-4827	511	14	existence	existence	NOUN
ejpam-4827	511	15	of	of	ADP
ejpam-4827	511	16	a	a	DET
ejpam-4827	511	17	positive	positive	ADJ
ejpam-4827	511	18	integer	integer	NOUN
ejpam-4827	511	19	ℓ	ℓ	PROPN
ejpam-4827	512	1	such	such	ADJ
ejpam-4827	512	2	that	that	SCONJ
ejpam-4827	512	3	(	(	PUNCT
ejpam-4827	512	4	φϕψ)ℓ	φϕψ)ℓ	PROPN
ejpam-4827	512	5	=	=	SYM
ejpam-4827	512	6	(	(	PUNCT
ejpam-4827	512	7	φϕψ)(φϕψ	φϕψ)(φϕψ	NOUN
ejpam-4827	512	8	)	)	PUNCT
ejpam-4827	512	9	.	.	PUNCT
ejpam-4827	512	10	.	.	PUNCT
ejpam-4827	512	11	.	.	PUNCT
ejpam-4827	513	1	(	(	PUNCT
ejpam-4827	513	2	φϕψ	φϕψ	NOUN
ejpam-4827	513	3	)	)	PUNCT
ejpam-4827	513	4	=	=	SYM
ejpam-4827	514	1	0	0	X
ejpam-4827	514	2	.	.	PUNCT
ejpam-4827	515	1	since	since	SCONJ
ejpam-4827	515	2	σ	σ	PROPN
ejpam-4827	515	3	is	be	AUX
ejpam-4827	515	4	a	a	DET
ejpam-4827	515	5	compatible	compatible	ADJ
ejpam-4827	515	6	monoid	monoid	NOUN
ejpam-4827	515	7	homomorphism	homomorphism	NOUN
ejpam-4827	515	8	,	,	PUNCT
ejpam-4827	515	9	we	we	PRON
ejpam-4827	515	10	have	have	VERB
ejpam-4827	515	11	biσgi(c(aj	biσgi(c(aj	PROPN
ejpam-4827	515	12	)	)	PUNCT
ejpam-4827	515	13	)	)	PUNCT
ejpam-4827	516	1	∈	∈	PROPN
ejpam-4827	516	2	nil(r	nil(r	PROPN
ejpam-4827	516	3	)	)	PUNCT
ejpam-4827	516	4	.	.	PUNCT
ejpam-4827	517	1	therefore	therefore	ADV
ejpam-4827	517	2	,	,	PUNCT
ejpam-4827	517	3	ajσhj	ajσhj	PROPN
ejpam-4827	517	4	(	(	PUNCT
ejpam-4827	517	5	c(bi	c(bi	PROPN
ejpam-4827	517	6	)	)	PUNCT
ejpam-4827	517	7	)	)	PUNCT
ejpam-4827	517	8	∈	∈	PROPN
ejpam-4827	517	9	nil(r	nil(r	PROPN
ejpam-4827	517	10	)	)	PUNCT
ejpam-4827	517	11	,	,	PUNCT
ejpam-4827	517	12	and	and	CCONJ
ejpam-4827	517	13	the	the	DET
ejpam-4827	517	14	proof	proof	NOUN
ejpam-4827	517	15	is	be	AUX
ejpam-4827	517	16	complete	complete	ADJ
ejpam-4827	517	17	.	.	PUNCT
ejpam-4827	518	1	in	in	ADP
ejpam-4827	518	2	proposition	proposition	NOUN
ejpam-4827	518	3	3.4	3.4	NUM
ejpam-4827	519	1	[	[	SYM
ejpam-4827	519	2	20	20	NUM
ejpam-4827	519	3	]	]	PUNCT
ejpam-4827	519	4	,	,	PUNCT
ejpam-4827	519	5	it	it	PRON
ejpam-4827	519	6	was	be	AUX
ejpam-4827	519	7	proved	prove	VERB
ejpam-4827	519	8	that	that	SCONJ
ejpam-4827	519	9	if	if	SCONJ
ejpam-4827	519	10	i	i	PRON
ejpam-4827	519	11	is	be	AUX
ejpam-4827	519	12	a	a	DET
ejpam-4827	519	13	reduced	reduce	VERB
ejpam-4827	519	14	ideal	ideal	NOUN
ejpam-4827	519	15	and	and	CCONJ
ejpam-4827	519	16	r	r	X
ejpam-4827	519	17	/	/	SYM
ejpam-4827	519	18	i	i	PRON
ejpam-4827	519	19	is	be	AUX
ejpam-4827	519	20	strongly	strongly	ADV
ejpam-4827	519	21	m	m	VERB
ejpam-4827	519	22	-reflexive	-reflexive	ADJ
ejpam-4827	519	23	,	,	PUNCT
ejpam-4827	519	24	then	then	ADV
ejpam-4827	519	25	r	r	NOUN
ejpam-4827	519	26	is	be	AUX
ejpam-4827	519	27	strongly	strongly	ADV
ejpam-4827	519	28	m	m	VERB
ejpam-4827	519	29	-reflexive	-reflexive	ADJ
ejpam-4827	519	30	.	.	PUNCT
ejpam-4827	520	1	based	base	VERB
ejpam-4827	520	2	on	on	ADP
ejpam-4827	520	3	this	this	DET
ejpam-4827	520	4	result	result	NOUN
ejpam-4827	520	5	,	,	PUNCT
ejpam-4827	520	6	we	we	PRON
ejpam-4827	520	7	have	have	VERB
ejpam-4827	520	8	the	the	DET
ejpam-4827	520	9	following	follow	VERB
ejpam-4827	520	10	statement	statement	NOUN
ejpam-4827	520	11	.	.	PUNCT
ejpam-4827	521	1	theorem	theorem	ADJ
ejpam-4827	521	2	5	5	NUM
ejpam-4827	521	3	.	.	PUNCT
ejpam-4827	521	4	suppose	suppose	VERB
ejpam-4827	521	5	that	that	SCONJ
ejpam-4827	521	6	r	r	NOUN
ejpam-4827	521	7	is	be	AUX
ejpam-4827	521	8	a	a	DET
ejpam-4827	521	9	ring	ring	NOUN
ejpam-4827	521	10	,	,	PUNCT
ejpam-4827	521	11	m	m	VERB
ejpam-4827	521	12	is	be	AUX
ejpam-4827	521	13	a	a	DET
ejpam-4827	521	14	strictly	strictly	ADV
ejpam-4827	521	15	ordered	order	VERB
ejpam-4827	521	16	monoid	monoid	NOUN
ejpam-4827	521	17	,	,	PUNCT
ejpam-4827	521	18	and	and	CCONJ
ejpam-4827	521	19	σ	σ	NOUN
ejpam-4827	521	20	:	:	PUNCT
ejpam-4827	521	21	m	m	PROPN
ejpam-4827	521	22	→	→	SYM
ejpam-4827	521	23	aut(r	aut(r	PROPN
ejpam-4827	521	24	)	)	PUNCT
ejpam-4827	521	25	is	be	AUX
ejpam-4827	521	26	a	a	DET
ejpam-4827	521	27	compatible	compatible	ADJ
ejpam-4827	521	28	monoid	monoid	NOUN
ejpam-4827	521	29	homomorphism	homomorphism	NOUN
ejpam-4827	521	30	.	.	PUNCT
ejpam-4827	522	1	if	if	SCONJ
ejpam-4827	522	2	i	i	PRON
ejpam-4827	522	3	is	be	AUX
ejpam-4827	522	4	an	an	DET
ejpam-4827	522	5	ideal	ideal	NOUN
ejpam-4827	522	6	of	of	ADP
ejpam-4827	522	7	r	r	NOUN
ejpam-4827	522	8	contained	contain	VERB
ejpam-4827	522	9	in	in	ADP
ejpam-4827	522	10	nil(r	nil(r	PROPN
ejpam-4827	522	11	)	)	PUNCT
ejpam-4827	522	12	,	,	PUNCT
ejpam-4827	522	13	then	then	ADV
ejpam-4827	522	14	r	r	X
ejpam-4827	522	15	/	/	SYM
ejpam-4827	522	16	i	i	PRON
ejpam-4827	522	17	is	be	AUX
ejpam-4827	522	18	σ	σ	NOUN
ejpam-4827	522	19	-	-	PUNCT
ejpam-4827	522	20	skew	skew	NOUN
ejpam-4827	522	21	strongly	strongly	ADV
ejpam-4827	522	22	m	m	VERB
ejpam-4827	522	23	-nil	-nil	NOUN
ejpam-4827	522	24	-	-	PUNCT
ejpam-4827	522	25	reflexive	reflexive	ADJ
ejpam-4827	522	26	if	if	SCONJ
ejpam-4827	523	1	and	and	CCONJ
ejpam-4827	523	2	only	only	ADV
ejpam-4827	523	3	if	if	SCONJ
ejpam-4827	523	4	r	r	NOUN
ejpam-4827	523	5	is	be	AUX
ejpam-4827	523	6	σ	σ	NOUN
ejpam-4827	523	7	-	-	PUNCT
ejpam-4827	523	8	skew	skew	NOUN
ejpam-4827	523	9	strongly	strongly	ADV
ejpam-4827	523	10	m	m	VERB
ejpam-4827	523	11	-nilreflexive	-nilreflexive	ADJ
ejpam-4827	523	12	.	.	PUNCT
ejpam-4827	524	1	proof	proof	NOUN
ejpam-4827	524	2	.	.	PUNCT
ejpam-4827	525	1	“	"	PUNCT
ejpam-4827	525	2	=	=	SYM
ejpam-4827	525	3	⇒	⇒	NOUN
ejpam-4827	525	4	”	"	PUNCT
ejpam-4827	525	5	let	let	VERB
ejpam-4827	525	6	φ	φ	NUM
ejpam-4827	525	7	,	,	PUNCT
ejpam-4827	525	8	ψ	ψ	ADP
ejpam-4827	525	9	∈	∈	PROPN
ejpam-4827	525	10	r∗m	r∗m	NOUN
ejpam-4827	525	11	satisfying	satisfy	VERB
ejpam-4827	525	12	φϕψ	φϕψ	PROPN
ejpam-4827	525	13	∈	∈	PROPN
ejpam-4827	525	14	nil(r)∗m	nil(r)∗m	PROPN
ejpam-4827	525	15	for	for	ADP
ejpam-4827	525	16	all	all	DET
ejpam-4827	525	17	ϕ	ϕ	PROPN
ejpam-4827	525	18	∈	∈	PROPN
ejpam-4827	525	19	r∗m	r∗m	NOUN
ejpam-4827	525	20	.	.	PUNCT
ejpam-4827	526	1	we	we	PRON
ejpam-4827	526	2	write	write	VERB
ejpam-4827	526	3	φ	φ	PROPN
ejpam-4827	526	4	=	=	SYM
ejpam-4827	526	5	b1g1	b1g1	PROPN
ejpam-4827	526	6	+	+	NUM
ejpam-4827	526	7	b2g2	b2g2	NOUN
ejpam-4827	526	8	+	+	NOUN
ejpam-4827	526	9	·	·	PUNCT
ejpam-4827	526	10	·	·	PUNCT
ejpam-4827	526	11	·	·	PUNCT
ejpam-4827	527	1	+	+	NUM
ejpam-4827	527	2	bngn	bngn	NOUN
ejpam-4827	527	3	,	,	PUNCT
ejpam-4827	527	4	ϕ	ϕ	NOUN
ejpam-4827	527	5	=	=	SYM
ejpam-4827	527	6	c1l1	c1l1	X
ejpam-4827	527	7	+	+	CCONJ
ejpam-4827	527	8	c2l2	c2l2	ADJ
ejpam-4827	527	9	+	+	X
ejpam-4827	527	10	·	·	PUNCT
ejpam-4827	527	11	·	·	PUNCT
ejpam-4827	527	12	·	·	PUNCT
ejpam-4827	527	13	+	+	NUM
ejpam-4827	527	14	cdld	cdld	ADJ
ejpam-4827	527	15	and	and	CCONJ
ejpam-4827	527	16	ψ	ψ	X
ejpam-4827	527	17	=	=	PUNCT
ejpam-4827	527	18	a1h1	a1h1	PROPN
ejpam-4827	527	19	+	+	X
ejpam-4827	527	20	a2h2	a2h2	NOUN
ejpam-4827	527	21	+	+	X
ejpam-4827	527	22	·	·	PUNCT
ejpam-4827	527	23	·	·	PUNCT
ejpam-4827	527	24	·	·	PUNCT
ejpam-4827	527	25	+	+	NUM
ejpam-4827	527	26	amhm	amhm	NOUN
ejpam-4827	527	27	with	with	ADP
ejpam-4827	527	28	g1	g1	PROPN
ejpam-4827	527	29	<	<	X
ejpam-4827	527	30	g2	g2	PROPN
ejpam-4827	527	31	<	<	X
ejpam-4827	527	32	.	.	PUNCT
ejpam-4827	527	33	.	.	PUNCT
ejpam-4827	527	34	.	.	PUNCT
ejpam-4827	528	1	<	<	X
ejpam-4827	528	2	gn	gn	PROPN
ejpam-4827	528	3	,	,	PUNCT
ejpam-4827	528	4	h1	h1	VERB
ejpam-4827	528	5	<	<	X
ejpam-4827	528	6	h2	h2	PROPN
ejpam-4827	528	7	<	<	X
ejpam-4827	528	8	.	.	PUNCT
ejpam-4827	528	9	.	.	PUNCT
ejpam-4827	528	10	.	.	PUNCT
ejpam-4827	529	1	<	<	X
ejpam-4827	530	1	hm	hm	INTJ
ejpam-4827	530	2	.	.	PUNCT
ejpam-4827	531	1	we	we	PRON
ejpam-4827	531	2	will	will	AUX
ejpam-4827	531	3	use	use	VERB
ejpam-4827	531	4	transfinite	transfinite	ADJ
ejpam-4827	531	5	induction	induction	NOUN
ejpam-4827	531	6	on	on	ADP
ejpam-4827	531	7	the	the	DET
ejpam-4827	531	8	strictly	strictly	ADV
ejpam-4827	531	9	totally	totally	ADV
ejpam-4827	531	10	ordered	order	VERB
ejpam-4827	531	11	set	set	NOUN
ejpam-4827	531	12	(	(	PUNCT
ejpam-4827	531	13	m,≤	m,≤	NOUN
ejpam-4827	531	14	)	)	PUNCT
ejpam-4827	531	15	to	to	PART
ejpam-4827	531	16	show	show	VERB
ejpam-4827	531	17	that	that	DET
ejpam-4827	531	18	ajσhj	ajσhj	ADJ
ejpam-4827	531	19	(	(	PUNCT
ejpam-4827	531	20	σs(r	σs(r	NOUN
ejpam-4827	531	21	bi	bi	NOUN
ejpam-4827	531	22	)	)	PUNCT
ejpam-4827	531	23	)	)	PUNCT
ejpam-4827	531	24	∈	∈	PROPN
ejpam-4827	532	1	nil(r	nil(r	PROPN
ejpam-4827	532	2	)	)	PUNCT
ejpam-4827	532	3	.	.	PUNCT
ejpam-4827	533	1	since	since	SCONJ
ejpam-4827	533	2	r	r	NOUN
ejpam-4827	533	3	/	/	SYM
ejpam-4827	533	4	i	i	PRON
ejpam-4827	533	5	is	be	AUX
ejpam-4827	533	6	σ	σ	NOUN
ejpam-4827	533	7	-	-	PUNCT
ejpam-4827	533	8	skew	skew	NOUN
ejpam-4827	533	9	strongly	strongly	ADV
ejpam-4827	533	10	m	m	VERB
ejpam-4827	533	11	-nil	-nil	NOUN
ejpam-4827	533	12	-	-	PUNCT
ejpam-4827	533	13	reflexive	reflexive	ADJ
ejpam-4827	533	14	and	and	CCONJ
ejpam-4827	533	15	0̄	0̄	NUM
ejpam-4827	533	16	=	=	SYM
ejpam-4827	533	17	(	(	PUNCT
ejpam-4827	533	18	b̄1g1	b̄1g1	PROPN
ejpam-4827	533	19	+	+	CCONJ
ejpam-4827	533	20	b̄2g2	b̄2g2	NOUN
ejpam-4827	533	21	+	+	X
ejpam-4827	533	22	·	·	PUNCT
ejpam-4827	533	23	·	·	PUNCT
ejpam-4827	533	24	·	·	PUNCT
ejpam-4827	534	1	+	+	NUM
ejpam-4827	534	2	b̄ngn)(rs)(ā1h1	b̄ngn)(rs)(ā1h1	NOUN
ejpam-4827	534	3	+	+	X
ejpam-4827	534	4	ā2h2	ā2h2	PROPN
ejpam-4827	534	5	+	+	CCONJ
ejpam-4827	534	6	·	·	PUNCT
ejpam-4827	534	7	·	·	PUNCT
ejpam-4827	534	8	·	·	PUNCT
ejpam-4827	534	9	+	+	NUM
ejpam-4827	534	10	āmhm	āmhm	NOUN
ejpam-4827	534	11	)	)	PUNCT
ejpam-4827	534	12	=	=	SYM
ejpam-4827	534	13	(	(	PUNCT
ejpam-4827	534	14	b1	b1	NOUN
ejpam-4827	534	15	+	+	CCONJ
ejpam-4827	534	16	i)σ̄g1(r(a1	i)σ̄g1(r(a1	NOUN
ejpam-4827	534	17	+	+	CCONJ
ejpam-4827	534	18	i	i	NOUN
ejpam-4827	534	19	)	)	PUNCT
ejpam-4827	534	20	)	)	PUNCT
ejpam-4827	535	1	+	+	CCONJ
ejpam-4827	535	2	(	(	PUNCT
ejpam-4827	535	3	b2	b2	NOUN
ejpam-4827	535	4	+	+	CCONJ
ejpam-4827	535	5	i)σ̄g2(r(a2	i)σ̄g2(r(a2	PROPN
ejpam-4827	535	6	+	+	CCONJ
ejpam-4827	535	7	i	i	NOUN
ejpam-4827	535	8	)	)	PUNCT
ejpam-4827	535	9	)	)	PUNCT
ejpam-4827	536	1	+	+	CCONJ
ejpam-4827	536	2	·	·	PUNCT
ejpam-4827	536	3	·	·	PUNCT
ejpam-4827	536	4	·	·	PUNCT
ejpam-4827	536	5	+	+	CCONJ
ejpam-4827	536	6	(	(	PUNCT
ejpam-4827	536	7	bn	bn	INTJ
ejpam-4827	536	8	+	+	X
ejpam-4827	536	9	i)σ̄gn(r(am	i)σ̄gn(r(am	PROPN
ejpam-4827	536	10	+	+	X
ejpam-4827	536	11	i	i	NOUN
ejpam-4827	536	12	)	)	PUNCT
ejpam-4827	536	13	)	)	PUNCT
ejpam-4827	537	1	=	=	PRON
ejpam-4827	537	2	(	(	PUNCT
ejpam-4827	537	3	b1σ̄g1(r	b1σ̄g1(r	NOUN
ejpam-4827	537	4	a1	a1	NOUN
ejpam-4827	537	5	)	)	PUNCT
ejpam-4827	537	6	+	+	CCONJ
ejpam-4827	537	7	i	i	NOUN
ejpam-4827	537	8	)	)	PUNCT
ejpam-4827	538	1	+	+	CCONJ
ejpam-4827	538	2	(	(	PUNCT
ejpam-4827	538	3	b2σ̄g2(r	b2σ̄g2(r	PROPN
ejpam-4827	538	4	a2	a2	PROPN
ejpam-4827	538	5	)	)	PUNCT
ejpam-4827	539	1	+	+	CCONJ
ejpam-4827	539	2	i	i	NOUN
ejpam-4827	539	3	)	)	PUNCT
ejpam-4827	540	1	+	+	CCONJ
ejpam-4827	540	2	·	·	PUNCT
ejpam-4827	540	3	·	·	PUNCT
ejpam-4827	540	4	·	·	PUNCT
ejpam-4827	540	5	+	+	CCONJ
ejpam-4827	540	6	(	(	PUNCT
ejpam-4827	540	7	bnσ̄gn(r	bnσ̄gn(r	NOUN
ejpam-4827	540	8	am	be	AUX
ejpam-4827	540	9	)	)	PUNCT
ejpam-4827	541	1	+	+	CCONJ
ejpam-4827	541	2	i	i	X
ejpam-4827	541	3	)	)	PUNCT
ejpam-4827	541	4	∈	∈	PROPN
ejpam-4827	541	5	(	(	PUNCT
ejpam-4827	541	6	r	r	NOUN
ejpam-4827	541	7	/	/	SYM
ejpam-4827	541	8	i	i	NOUN
ejpam-4827	541	9	)	)	PUNCT
ejpam-4827	541	10	∗m	∗m	NOUN
ejpam-4827	541	11	for	for	ADP
ejpam-4827	541	12	r	r	NOUN
ejpam-4827	541	13	∈	∈	PROPN
ejpam-4827	541	14	r	r	NOUN
ejpam-4827	541	15	,	,	PUNCT
ejpam-4827	541	16	s	s	PART
ejpam-4827	541	17	∈	∈	PROPN
ejpam-4827	541	18	m	m	PRON
ejpam-4827	541	19	,	,	PUNCT
ejpam-4827	541	20	so	so	SCONJ
ejpam-4827	541	21	we	we	PRON
ejpam-4827	541	22	have	have	VERB
ejpam-4827	541	23	biσgi(r	biσgi(r	PROPN
ejpam-4827	541	24	aj	aj	PROPN
ejpam-4827	541	25	)	)	PUNCT
ejpam-4827	541	26	∈	∈	PROPN
ejpam-4827	541	27	i	i	PRON
ejpam-4827	541	28	for	for	ADP
ejpam-4827	541	29	all	all	DET
ejpam-4827	541	30	i	i	PROPN
ejpam-4827	541	31	,	,	PUNCT
ejpam-4827	541	32	j.	j.	PROPN
ejpam-4827	541	33	since	since	SCONJ
ejpam-4827	541	34	m	m	PROPN
ejpam-4827	541	35	is	be	AUX
ejpam-4827	541	36	a	a	DET
ejpam-4827	541	37	strictly	strictly	ADV
ejpam-4827	541	38	totally	totally	ADV
ejpam-4827	541	39	ordered	order	VERB
ejpam-4827	541	40	monoid	monoid	NOUN
ejpam-4827	541	41	,	,	PUNCT
ejpam-4827	541	42	we	we	PRON
ejpam-4827	541	43	have	have	VERB
ejpam-4827	541	44	g1h1	g1h1	PROPN
ejpam-4827	541	45	<	<	X
ejpam-4827	541	46	gih1	gih1	PROPN
ejpam-4827	541	47	≤	≤	PROPN
ejpam-4827	541	48	gihj	gihj	PROPN
ejpam-4827	541	49	=	=	SYM
ejpam-4827	542	1	g1h1	g1h1	PROPN
ejpam-4827	542	2	for	for	ADP
ejpam-4827	542	3	i	i	PRON
ejpam-4827	542	4	̸=	̸=	PROPN
ejpam-4827	542	5	1	1	NUM
ejpam-4827	542	6	or	or	CCONJ
ejpam-4827	542	7	j	j	PROPN
ejpam-4827	542	8	̸=	̸=	PROPN
ejpam-4827	542	9	1	1	NUM
ejpam-4827	542	10	.	.	PUNCT
ejpam-4827	543	1	it	it	PRON
ejpam-4827	543	2	follows	follow	VERB
ejpam-4827	543	3	that	that	SCONJ
ejpam-4827	543	4	b1σg1(r	b1σg1(r	ADJ
ejpam-4827	543	5	a1	a1	NOUN
ejpam-4827	543	6	)	)	PUNCT
ejpam-4827	543	7	=	=	SYM
ejpam-4827	543	8	0	0	NUM
ejpam-4827	543	9	,	,	PUNCT
ejpam-4827	543	10	i.e.	i.e.	X
ejpam-4827	543	11	,	,	PUNCT
ejpam-4827	543	12	b1σg1(σs(r	b1σg1(σs(r	ADJ
ejpam-4827	543	13	a1	a1	NOUN
ejpam-4827	543	14	)	)	PUNCT
ejpam-4827	543	15	)	)	PUNCT
ejpam-4827	544	1	∈	∈	PROPN
ejpam-4827	544	2	nil(r	nil(r	PROPN
ejpam-4827	544	3	)	)	PUNCT
ejpam-4827	544	4	since	since	SCONJ
ejpam-4827	544	5	i	i	PRON
ejpam-4827	544	6	is	be	AUX
ejpam-4827	544	7	an	an	DET
ejpam-4827	544	8	ideal	ideal	NOUN
ejpam-4827	544	9	of	of	ADP
ejpam-4827	544	10	r	r	NOUN
ejpam-4827	544	11	contained	contain	VERB
ejpam-4827	544	12	in	in	ADP
ejpam-4827	544	13	nil(r	nil(r	NOUN
ejpam-4827	544	14	)	)	PUNCT
ejpam-4827	544	15	.	.	PUNCT
ejpam-4827	545	1	now	now	ADV
ejpam-4827	545	2	suppose	suppose	VERB
ejpam-4827	545	3	that	that	SCONJ
ejpam-4827	545	4	bir	bir	PROPN
ejpam-4827	545	5	aj	aj	PROPN
ejpam-4827	545	6	=	=	PROPN
ejpam-4827	545	7	0	0	PROPN
ejpam-4827	545	8	for	for	ADP
ejpam-4827	545	9	all	all	DET
ejpam-4827	545	10	1	1	NUM
ejpam-4827	545	11	≤	≤	NUM
ejpam-4827	545	12	i	i	PRON
ejpam-4827	545	13	≤	≤	PROPN
ejpam-4827	545	14	n	n	CCONJ
ejpam-4827	545	15	,	,	PUNCT
ejpam-4827	545	16	1	1	NUM
ejpam-4827	545	17	≤	≤	NUM
ejpam-4827	545	18	j	j	PROPN
ejpam-4827	545	19	≤	≤	NUM
ejpam-4827	545	20	m	m	VERB
ejpam-4827	545	21	with	with	ADP
ejpam-4827	545	22	w	w	PROPN
ejpam-4827	545	23	∈	∈	PROPN
ejpam-4827	545	24	m	m	VERB
ejpam-4827	545	25	is	be	AUX
ejpam-4827	545	26	such	such	ADJ
ejpam-4827	545	27	that	that	SCONJ
ejpam-4827	545	28	for	for	ADP
ejpam-4827	545	29	any	any	DET
ejpam-4827	545	30	gi	gi	NOUN
ejpam-4827	545	31	and	and	CCONJ
ejpam-4827	545	32	hj	hj	PROPN
ejpam-4827	545	33	,	,	PUNCT
ejpam-4827	546	1	gihj	gihj	PROPN
ejpam-4827	546	2	<	<	X
ejpam-4827	546	3	w.	w.	PROPN
ejpam-4827	546	4	we	we	PRON
ejpam-4827	546	5	will	will	AUX
ejpam-4827	546	6	show	show	VERB
ejpam-4827	546	7	that	that	SCONJ
ejpam-4827	546	8	biσgi(σs(raj	biσgi(σs(raj	NOUN
ejpam-4827	546	9	)	)	PUNCT
ejpam-4827	546	10	)	)	PUNCT
ejpam-4827	547	1	∈	∈	PROPN
ejpam-4827	547	2	nil(r	nil(r	PROPN
ejpam-4827	547	3	)	)	PUNCT
ejpam-4827	547	4	for	for	ADP
ejpam-4827	547	5	any	any	DET
ejpam-4827	547	6	gi	gi	NOUN
ejpam-4827	547	7	and	and	CCONJ
ejpam-4827	547	8	hj	hj	X
ejpam-4827	547	9	with	with	ADP
ejpam-4827	547	10	gihj	gihj	PROPN
ejpam-4827	547	11	=	=	SYM
ejpam-4827	547	12	w.	w.	PROPN
ejpam-4827	547	13	set	set	VERB
ejpam-4827	547	14	x	x	X
ejpam-4827	548	1	=	=	PRON
ejpam-4827	548	2	{	{	PUNCT
ejpam-4827	548	3	(	(	PUNCT
ejpam-4827	548	4	gi	gi	INTJ
ejpam-4827	548	5	,	,	PUNCT
ejpam-4827	548	6	hj)|gihj	hj)|gihj	X
ejpam-4827	548	7	=	=	SYM
ejpam-4827	548	8	w	w	PROPN
ejpam-4827	548	9	}	}	PUNCT
ejpam-4827	548	10	.	.	PUNCT
ejpam-4827	549	1	then	then	ADV
ejpam-4827	549	2	x	x	PRON
ejpam-4827	549	3	is	be	AUX
ejpam-4827	549	4	a	a	DET
ejpam-4827	549	5	finite	finite	ADJ
ejpam-4827	549	6	set	set	NOUN
ejpam-4827	549	7	.	.	PUNCT
ejpam-4827	550	1	we	we	PRON
ejpam-4827	550	2	write	write	VERB
ejpam-4827	550	3	x	x	PUNCT
ejpam-4827	550	4	as	as	ADP
ejpam-4827	550	5	{	{	PUNCT
ejpam-4827	550	6	(	(	PUNCT
ejpam-4827	550	7	git	git	NOUN
ejpam-4827	550	8	,	,	PUNCT
ejpam-4827	550	9	hjt)|t	hjt)|t	PROPN
ejpam-4827	550	10	=	=	SYM
ejpam-4827	550	11	1	1	NUM
ejpam-4827	550	12	,	,	PUNCT
ejpam-4827	550	13	2	2	NUM
ejpam-4827	550	14	,	,	PUNCT
ejpam-4827	550	15	.	.	PUNCT
ejpam-4827	550	16	.	.	PUNCT
ejpam-4827	550	17	.	.	PUNCT
ejpam-4827	551	1	,	,	PUNCT
ejpam-4827	551	2	k	k	X
ejpam-4827	551	3	}	}	PUNCT
ejpam-4827	551	4	such	such	ADJ
ejpam-4827	551	5	that	that	DET
ejpam-4827	551	6	gi1	gi1	NOUN
ejpam-4827	551	7	<	<	X
ejpam-4827	551	8	gi2	gi2	NOUN
ejpam-4827	551	9	<	<	X
ejpam-4827	551	10	.	.	PUNCT
ejpam-4827	551	11	.	.	PUNCT
ejpam-4827	551	12	.	.	PUNCT
ejpam-4827	552	1	<	<	X
ejpam-4827	552	2	gik	gik	PROPN
ejpam-4827	552	3	.	.	PUNCT
ejpam-4827	553	1	since	since	SCONJ
ejpam-4827	553	2	m	m	PROPN
ejpam-4827	553	3	is	be	AUX
ejpam-4827	553	4	cancellative	cancellative	ADJ
ejpam-4827	553	5	,	,	PUNCT
ejpam-4827	553	6	gi1	gi1	NOUN
ejpam-4827	553	7	=	=	PUNCT
ejpam-4827	553	8	gi2	gi2	NOUN
ejpam-4827	553	9	and	and	CCONJ
ejpam-4827	553	10	gi1hj1	gi1hj1	NOUN
ejpam-4827	553	11	=	=	SYM
ejpam-4827	553	12	gi2hj2	gi2hj2	NOUN
ejpam-4827	553	13	=	=	NOUN
ejpam-4827	553	14	w	w	NOUN
ejpam-4827	553	15	imply	imply	VERB
ejpam-4827	553	16	hj1	hj1	NOUN
ejpam-4827	554	1	=	=	NOUN
ejpam-4827	554	2	hj2	hj2	NOUN
ejpam-4827	554	3	.	.	PUNCT
ejpam-4827	555	1	since	since	SCONJ
ejpam-4827	555	2	≤	≤	NUM
ejpam-4827	555	3	is	be	AUX
ejpam-4827	555	4	a	a	DET
ejpam-4827	555	5	strict	strict	ADJ
ejpam-4827	555	6	order	order	NOUN
ejpam-4827	555	7	,	,	PUNCT
ejpam-4827	555	8	gi1	gi1	NOUN
ejpam-4827	555	9	<	<	X
ejpam-4827	555	10	gi2	gi2	NOUN
ejpam-4827	555	11	and	and	CCONJ
ejpam-4827	555	12	gi1hj1	gi1hj1	NOUN
ejpam-4827	555	13	=	=	SYM
ejpam-4827	555	14	gi2hj2	gi2hj2	NOUN
ejpam-4827	555	15	=	=	NOUN
ejpam-4827	555	16	w	w	NOUN
ejpam-4827	555	17	imply	imply	VERB
ejpam-4827	555	18	hj2	hj2	PROPN
ejpam-4827	555	19	<	<	X
ejpam-4827	555	20	hj1	hj1	NOUN
ejpam-4827	555	21	.	.	PUNCT
ejpam-4827	556	1	thus	thus	ADV
ejpam-4827	556	2	,	,	PUNCT
ejpam-4827	556	3	we	we	PRON
ejpam-4827	556	4	have	have	VERB
ejpam-4827	556	5	hjk	hjk	PROPN
ejpam-4827	556	6	<	<	X
ejpam-4827	556	7	hjk−1	hjk−1	PROPN
ejpam-4827	556	8	<	<	X
ejpam-4827	556	9	.	.	PUNCT
ejpam-4827	556	10	.	.	PUNCT
ejpam-4827	556	11	.	.	PUNCT
ejpam-4827	557	1	<	<	X
ejpam-4827	557	2	hj2	hj2	X
ejpam-4827	557	3	<	<	X
ejpam-4827	557	4	hj1	hj1	NOUN
ejpam-4827	557	5	.	.	PUNCT
ejpam-4827	558	1	now	now	ADV
ejpam-4827	558	2	∑	∑	PUNCT
ejpam-4827	558	3	(	(	PUNCT
ejpam-4827	558	4	gi	gi	INTJ
ejpam-4827	558	5	,	,	PUNCT
ejpam-4827	558	6	hj)∈x	hj)∈x	NOUN
ejpam-4827	558	7	biσgi(σs(raj	biσgi(σs(raj	NOUN
ejpam-4827	558	8	)	)	PUNCT
ejpam-4827	558	9	)	)	PUNCT
ejpam-4827	559	1	=	=	PUNCT
ejpam-4827	559	2	k∑	k∑	PROPN
ejpam-4827	559	3	t=1	t=1	PUNCT
ejpam-4827	559	4	bitσgit	bitσgit	PROPN
ejpam-4827	559	5	(	(	PUNCT
ejpam-4827	559	6	σs(rajt	σs(rajt	PROPN
ejpam-4827	559	7	)	)	PUNCT
ejpam-4827	559	8	)	)	PUNCT
ejpam-4827	560	1	=	=	PUNCT
ejpam-4827	560	2	0	0	X
ejpam-4827	560	3	.	.	X
ejpam-4827	561	1	for	for	ADP
ejpam-4827	561	2	any	any	DET
ejpam-4827	561	3	t	t	PROPN
ejpam-4827	561	4	≥	≥	NOUN
ejpam-4827	561	5	2	2	NUM
ejpam-4827	561	6	,	,	PUNCT
ejpam-4827	561	7	gi1hjt	gi1hjt	PROPN
ejpam-4827	561	8	<	<	X
ejpam-4827	561	9	githjt	githjt	PROPN
ejpam-4827	561	10	=	=	SYM
ejpam-4827	561	11	w	w	PROPN
ejpam-4827	561	12	,	,	PUNCT
ejpam-4827	561	13	and	and	CCONJ
ejpam-4827	561	14	so	so	ADV
ejpam-4827	561	15	bi1σgit	bi1σgit	PUNCT
ejpam-4827	561	16	(	(	PUNCT
ejpam-4827	561	17	σs(rajt	σs(rajt	PROPN
ejpam-4827	561	18	)	)	PUNCT
ejpam-4827	561	19	)	)	PUNCT
ejpam-4827	562	1	=	=	SYM
ejpam-4827	562	2	0	0	NUM
ejpam-4827	562	3	by	by	ADP
ejpam-4827	562	4	induction	induction	NOUN
ejpam-4827	562	5	hypothesis	hypothesis	NOUN
ejpam-4827	562	6	.	.	PUNCT
ejpam-4827	563	1	thus	thus	ADV
ejpam-4827	563	2	,	,	PUNCT
ejpam-4827	563	3	bi1rajt	bi1rajt	ADV
ejpam-4827	563	4	=	=	SYM
ejpam-4827	563	5	0	0	PUNCT
ejpam-4827	563	6	because	because	SCONJ
ejpam-4827	563	7	r	r	NOUN
ejpam-4827	563	8	is	be	AUX
ejpam-4827	563	9	m	m	PRON
ejpam-4827	563	10	-compatible	-compatible	ADJ
ejpam-4827	563	11	.	.	PUNCT
ejpam-4827	564	1	since	since	SCONJ
ejpam-4827	564	2	i	i	PRON
ejpam-4827	564	3	is	be	AUX
ejpam-4827	564	4	reduced	reduce	VERB
ejpam-4827	564	5	and	and	CCONJ
ejpam-4827	564	6	σgit	σgit	ADV
ejpam-4827	564	7	(	(	PUNCT
ejpam-4827	564	8	ajt)i(bi1	ajt)i(bi1	PROPN
ejpam-4827	564	9	)	)	PUNCT
ejpam-4827	564	10	⊆	⊆	NUM
ejpam-4827	564	11	i	i	PRON
ejpam-4827	564	12	,	,	PUNCT
ejpam-4827	564	13	then	then	ADV
ejpam-4827	564	14	we	we	PRON
ejpam-4827	564	15	have	have	VERB
ejpam-4827	564	16	(	(	PUNCT
ejpam-4827	564	17	ajtibi1	ajtibi1	PROPN
ejpam-4827	564	18	)	)	PUNCT
ejpam-4827	564	19	2	2	NUM
ejpam-4827	565	1	=	=	SYM
ejpam-4827	565	2	0	0	PUNCT
ejpam-4827	566	1	and	and	CCONJ
ejpam-4827	566	2	i	i	PRON
ejpam-4827	566	3	is	be	AUX
ejpam-4827	566	4	reduced	reduce	VERB
ejpam-4827	566	5	.	.	PUNCT
ejpam-4827	567	1	thus	thus	ADV
ejpam-4827	567	2	,	,	PUNCT
ejpam-4827	567	3	for	for	ADP
ejpam-4827	567	4	any	any	DET
ejpam-4827	567	5	t	t	PROPN
ejpam-4827	567	6	≥	≥	NOUN
ejpam-4827	567	7	2	2	NUM
ejpam-4827	567	8	,	,	PUNCT
ejpam-4827	567	9	(	(	PUNCT
ejpam-4827	567	10	bitrajt)(bi1raj1	bitrajt)(bi1raj1	NUM
ejpam-4827	567	11	)	)	PUNCT
ejpam-4827	567	12	2	2	NUM
ejpam-4827	567	13	=	=	SYM
ejpam-4827	567	14	(	(	PUNCT
ejpam-4827	567	15	bitrajt)(bi1raj1)(bi1raj1	bitrajt)(bi1raj1)(bi1raj1	NOUN
ejpam-4827	567	16	)	)	PUNCT
ejpam-4827	567	17	∈	∈	PROPN
ejpam-4827	567	18	(	(	PUNCT
ejpam-4827	567	19	bitrajt)i(bitrajt	bitrajt)i(bitrajt	NOUN
ejpam-4827	567	20	)	)	PUNCT
ejpam-4827	567	21	=	=	SYM
ejpam-4827	567	22	(	(	PUNCT
ejpam-4827	567	23	bitrajt)i(bitrajt	bitrajt)i(bitrajt	NOUN
ejpam-4827	567	24	)	)	PUNCT
ejpam-4827	567	25	=	=	SYM
ejpam-4827	567	26	0	0	NUM
ejpam-4827	567	27	,	,	PUNCT
ejpam-4827	567	28	which	which	PRON
ejpam-4827	567	29	implies	imply	VERB
ejpam-4827	567	30	that	that	SCONJ
ejpam-4827	567	31	(	(	PUNCT
ejpam-4827	567	32	bitrajt)(bi1raj1	bitrajt)(bi1raj1	NUM
ejpam-4827	567	33	)	)	PUNCT
ejpam-4827	567	34	2	2	NUM
ejpam-4827	567	35	=	=	SYM
ejpam-4827	567	36	0	0	NUM
ejpam-4827	567	37	.	.	PUNCT
ejpam-4827	567	38	now	now	ADV
ejpam-4827	567	39	multiplying	multiply	VERB
ejpam-4827	567	40	∑k	∑k	PROPN
ejpam-4827	567	41	t=1	t=1	PROPN
ejpam-4827	567	42	bitσit(σs(r	bitσit(σs(r	PROPN
ejpam-4827	567	43	ajt	ajt	PROPN
ejpam-4827	567	44	)	)	PUNCT
ejpam-4827	567	45	)	)	PUNCT
ejpam-4827	568	1	=	=	SYM
ejpam-4827	568	2	0	0	NUM
ejpam-4827	569	1	on	on	ADP
ejpam-4827	569	2	the	the	DET
ejpam-4827	569	3	right	right	NOUN
ejpam-4827	569	4	by	by	ADP
ejpam-4827	569	5	e.	e.	PROPN
ejpam-4827	569	6	ali	ali	PROPN
ejpam-4827	569	7	/	/	SYM
ejpam-4827	569	8	eur	eur	PROPN
ejpam-4827	569	9	.	.	PUNCT
ejpam-4827	570	1	j.	j.	PROPN
ejpam-4827	570	2	pure	pure	PROPN
ejpam-4827	570	3	appl	appl	PROPN
ejpam-4827	570	4	.	.	PROPN
ejpam-4827	570	5	math	math	PROPN
ejpam-4827	570	6	,	,	PUNCT
ejpam-4827	570	7	16	16	NUM
ejpam-4827	570	8	(	(	PUNCT
ejpam-4827	570	9	3	3	NUM
ejpam-4827	570	10	)	)	PUNCT
ejpam-4827	570	11	(	(	PUNCT
ejpam-4827	570	12	2023	2023	NUM
ejpam-4827	570	13	)	)	PUNCT
ejpam-4827	570	14	,	,	PUNCT
ejpam-4827	570	15	1878	1878	NUM
ejpam-4827	570	16	-	-	SYM
ejpam-4827	570	17	1893	1893	NUM
ejpam-4827	570	18	1889	1889	NUM
ejpam-4827	570	19	(	(	PUNCT
ejpam-4827	570	20	bi1σi1(raj1	bi1σi1(raj1	NOUN
ejpam-4827	570	21	)	)	PUNCT
ejpam-4827	570	22	)	)	PUNCT
ejpam-4827	571	1	2	2	NUM
ejpam-4827	571	2	,	,	PUNCT
ejpam-4827	571	3	we	we	PRON
ejpam-4827	571	4	obtain	obtain	VERB
ejpam-4827	571	5	0	0	NUM
ejpam-4827	572	1	=	=	SYM
ejpam-4827	572	2	(	(	PUNCT
ejpam-4827	572	3	bitσit(rajt))(bi1σi1(raj1	bitσit(rajt))(bi1σi1(raj1	PROPN
ejpam-4827	572	4	)	)	PUNCT
ejpam-4827	572	5	)	)	PUNCT
ejpam-4827	573	1	2	2	NUM
ejpam-4827	573	2	=	=	SYM
ejpam-4827	573	3	(	(	PUNCT
ejpam-4827	573	4	bi1σi1(raj1	bi1σi1(raj1	NOUN
ejpam-4827	573	5	)	)	PUNCT
ejpam-4827	573	6	)	)	PUNCT
ejpam-4827	574	1	2(bi1σi1(raj1	2(bi1σi1(raj1	NUM
ejpam-4827	574	2	)	)	PUNCT
ejpam-4827	574	3	)	)	PUNCT
ejpam-4827	575	1	=	=	SYM
ejpam-4827	575	2	(	(	PUNCT
ejpam-4827	575	3	bi1σi1(raj1	bi1σi1(raj1	NOUN
ejpam-4827	575	4	)	)	PUNCT
ejpam-4827	575	5	)	)	PUNCT
ejpam-4827	576	1	3	3	X
ejpam-4827	576	2	.	.	PUNCT
ejpam-4827	576	3	since	since	SCONJ
ejpam-4827	576	4	bi1σi1(raj1	bi1σi1(raj1	NOUN
ejpam-4827	576	5	)	)	PUNCT
ejpam-4827	576	6	⊆	⊆	NUM
ejpam-4827	576	7	i	i	PRON
ejpam-4827	576	8	and	and	CCONJ
ejpam-4827	576	9	i	i	PRON
ejpam-4827	576	10	is	be	AUX
ejpam-4827	576	11	reduced	reduce	VERB
ejpam-4827	576	12	and	and	CCONJ
ejpam-4827	576	13	r	r	NOUN
ejpam-4827	576	14	is	be	AUX
ejpam-4827	576	15	m	m	PRON
ejpam-4827	576	16	-compatible	-compatible	ADJ
ejpam-4827	576	17	,	,	PUNCT
ejpam-4827	576	18	we	we	PRON
ejpam-4827	576	19	have	have	VERB
ejpam-4827	576	20	bi1σi1(raj1	bi1σi1(raj1	NOUN
ejpam-4827	576	21	)	)	PUNCT
ejpam-4827	577	1	=	=	SYM
ejpam-4827	577	2	0	0	X
ejpam-4827	577	3	.	.	PUNCT
ejpam-4827	578	1	thus	thus	ADV
ejpam-4827	578	2	,	,	PUNCT
ejpam-4827	578	3	∑k	∑k	PROPN
ejpam-4827	578	4	t=2	t=2	PUNCT
ejpam-4827	578	5	bitσit(rajt	bitσit(rajt	PROPN
ejpam-4827	578	6	)	)	PUNCT
ejpam-4827	578	7	=	=	SYM
ejpam-4827	578	8	0	0	X
ejpam-4827	578	9	.	.	PUNCT
ejpam-4827	579	1	then	then	ADV
ejpam-4827	579	2	,	,	PUNCT
ejpam-4827	579	3	bitσit(rajt	bitσit(rajt	PROPN
ejpam-4827	579	4	)	)	PUNCT
ejpam-4827	579	5	∈	∈	PROPN
ejpam-4827	579	6	nil(r	nil(r	PROPN
ejpam-4827	579	7	)	)	PUNCT
ejpam-4827	579	8	for	for	ADP
ejpam-4827	579	9	t	t	PROPN
ejpam-4827	579	10	≥	≥	X
ejpam-4827	579	11	2.multiplying	2.multiplying	NUM
ejpam-4827	579	12	(	(	PUNCT
ejpam-4827	579	13	bi2σi2(raj2	bi2σi2(raj2	PROPN
ejpam-4827	579	14	)	)	PUNCT
ejpam-4827	579	15	)	)	PUNCT
ejpam-4827	579	16	2	2	NUM
ejpam-4827	579	17	on	on	ADP
ejpam-4827	579	18	∑k	∑k	PROPN
ejpam-4827	579	19	t=2	t=2	PROPN
ejpam-4827	579	20	bitσit(rajt	bitσit(rajt	PROPN
ejpam-4827	579	21	)	)	PUNCT
ejpam-4827	579	22	=	=	SYM
ejpam-4827	579	23	0	0	NUM
ejpam-4827	580	1	from	from	ADP
ejpam-4827	580	2	the	the	DET
ejpam-4827	580	3	right	right	ADJ
ejpam-4827	580	4	-	-	PUNCT
ejpam-4827	580	5	hand	hand	NOUN
ejpam-4827	580	6	side	side	NOUN
ejpam-4827	580	7	,	,	PUNCT
ejpam-4827	580	8	we	we	PRON
ejpam-4827	580	9	obtain	obtain	VERB
ejpam-4827	580	10	bi2σi2(raj2	bi2σi2(raj2	X
ejpam-4827	580	11	)	)	PUNCT
ejpam-4827	580	12	=	=	SYM
ejpam-4827	581	1	0	0	X
ejpam-4827	581	2	.	.	PUNCT
ejpam-4827	582	1	thus	thus	ADV
ejpam-4827	582	2	,	,	PUNCT
ejpam-4827	582	3	bi2σi2(raj2	bi2σi2(raj2	X
ejpam-4827	582	4	)	)	PUNCT
ejpam-4827	582	5	∈	∈	PROPN
ejpam-4827	582	6	nil(r	nil(r	PROPN
ejpam-4827	582	7	)	)	PUNCT
ejpam-4827	582	8	by	by	ADP
ejpam-4827	582	9	the	the	DET
ejpam-4827	582	10	same	same	ADJ
ejpam-4827	582	11	way	way	NOUN
ejpam-4827	582	12	as	as	ADP
ejpam-4827	582	13	the	the	DET
ejpam-4827	582	14	above	above	NOUN
ejpam-4827	582	15	.	.	PUNCT
ejpam-4827	583	1	continuing	continue	VERB
ejpam-4827	583	2	this	this	DET
ejpam-4827	583	3	process	process	NOUN
ejpam-4827	583	4	,	,	PUNCT
ejpam-4827	583	5	we	we	PRON
ejpam-4827	583	6	can	can	AUX
ejpam-4827	583	7	prove	prove	VERB
ejpam-4827	583	8	bitσit(rajt	bitσit(rajt	PROPN
ejpam-4827	583	9	)	)	PUNCT
ejpam-4827	583	10	=	=	SYM
ejpam-4827	583	11	0	0	NUM
ejpam-4827	583	12	for	for	ADP
ejpam-4827	583	13	t	t	NOUN
ejpam-4827	583	14	=	=	SYM
ejpam-4827	583	15	1	1	NUM
ejpam-4827	583	16	,	,	PUNCT
ejpam-4827	583	17	.	.	PUNCT
ejpam-4827	583	18	.	.	PUNCT
ejpam-4827	584	1	.	.	PUNCT
ejpam-4827	585	1	,	,	PUNCT
ejpam-4827	585	2	k.	k.	PROPN
ejpam-4827	585	3	thus	thus	ADV
ejpam-4827	585	4	biσgi(σs(raj	biσgi(σs(raj	VERB
ejpam-4827	585	5	)	)	PUNCT
ejpam-4827	585	6	)	)	PUNCT
ejpam-4827	586	1	∈	∈	PROPN
ejpam-4827	586	2	nil(r	nil(r	PROPN
ejpam-4827	586	3	)	)	PUNCT
ejpam-4827	586	4	because	because	SCONJ
ejpam-4827	586	5	i	i	PRON
ejpam-4827	586	6	is	be	AUX
ejpam-4827	586	7	an	an	DET
ejpam-4827	586	8	ideal	ideal	NOUN
ejpam-4827	586	9	of	of	ADP
ejpam-4827	586	10	r	r	NOUN
ejpam-4827	586	11	contained	contain	VERB
ejpam-4827	586	12	in	in	ADP
ejpam-4827	586	13	nil(r	nil(r	NOUN
ejpam-4827	586	14	)	)	PUNCT
ejpam-4827	586	15	for	for	ADP
ejpam-4827	586	16	any	any	DET
ejpam-4827	586	17	i	i	PROPN
ejpam-4827	586	18	and	and	CCONJ
ejpam-4827	586	19	j	j	PROPN
ejpam-4827	586	20	with	with	ADP
ejpam-4827	586	21	gihj	gihj	PROPN
ejpam-4827	586	22	=	=	SYM
ejpam-4827	586	23	w.	w.	PROPN
ejpam-4827	586	24	therefore	therefore	ADV
ejpam-4827	586	25	,	,	PUNCT
ejpam-4827	586	26	by	by	ADP
ejpam-4827	586	27	transfinite	transfinite	ADJ
ejpam-4827	586	28	induction	induction	NOUN
ejpam-4827	586	29	biσgi(σs(raj	biσgi(σs(raj	NOUN
ejpam-4827	586	30	)	)	PUNCT
ejpam-4827	586	31	)	)	PUNCT
ejpam-4827	587	1	∈	∈	PROPN
ejpam-4827	587	2	nil(r	nil(r	PROPN
ejpam-4827	587	3	)	)	PUNCT
ejpam-4827	587	4	for	for	ADP
ejpam-4827	587	5	any	any	DET
ejpam-4827	587	6	i	i	PROPN
ejpam-4827	587	7	and	and	CCONJ
ejpam-4827	587	8	j.	j.	PROPN
ejpam-4827	587	9	thus	thus	ADV
ejpam-4827	587	10	,	,	PUNCT
ejpam-4827	587	11	ajσhj	ajσhj	ADJ
ejpam-4827	587	12	(	(	PUNCT
ejpam-4827	587	13	σs(rbi	σs(rbi	PROPN
ejpam-4827	587	14	)	)	PUNCT
ejpam-4827	587	15	)	)	PUNCT
ejpam-4827	588	1	∈	∈	PROPN
ejpam-4827	589	1	nil(r	nil(r	PROPN
ejpam-4827	589	2	)	)	PUNCT
ejpam-4827	589	3	.	.	PUNCT
ejpam-4827	590	1	therefore	therefore	ADV
ejpam-4827	590	2	,	,	PUNCT
ejpam-4827	590	3	r	r	NOUN
ejpam-4827	590	4	is	be	AUX
ejpam-4827	590	5	σ	σ	NOUN
ejpam-4827	590	6	-	-	PUNCT
ejpam-4827	590	7	skew	skew	NOUN
ejpam-4827	590	8	strongly	strongly	ADV
ejpam-4827	590	9	m	m	VERB
ejpam-4827	590	10	-nil	-nil	NOUN
ejpam-4827	590	11	-	-	PUNCT
ejpam-4827	590	12	reflexive	reflexive	ADJ
ejpam-4827	590	13	.	.	PUNCT
ejpam-4827	591	1	“	"	PUNCT
ejpam-4827	591	2	⇐	⇐	PROPN
ejpam-4827	591	3	=	=	SYM
ejpam-4827	591	4	”	"	PUNCT
ejpam-4827	591	5	let	let	VERB
ejpam-4827	591	6	φ̄	φ̄	SYM
ejpam-4827	591	7	,	,	PUNCT
ejpam-4827	591	8	ψ̄	ψ̄	NOUN
ejpam-4827	591	9	∈	∈	NOUN
ejpam-4827	591	10	(	(	PUNCT
ejpam-4827	591	11	r	r	NOUN
ejpam-4827	591	12	/	/	SYM
ejpam-4827	591	13	i	i	NOUN
ejpam-4827	591	14	)	)	PUNCT
ejpam-4827	591	15	∗m	∗m	NOUN
ejpam-4827	591	16	with	with	ADP
ejpam-4827	591	17	φ̄ϕ̄ψ̄	φ̄ϕ̄ψ̄	PUNCT
ejpam-4827	591	18	∈	∈	PROPN
ejpam-4827	591	19	nil(r	nil(r	PROPN
ejpam-4827	591	20	/	/	SYM
ejpam-4827	591	21	i	i	PROPN
ejpam-4827	591	22	)	)	PUNCT
ejpam-4827	591	23	∗m	∗m	NOUN
ejpam-4827	591	24	for	for	ADP
ejpam-4827	591	25	all	all	DET
ejpam-4827	591	26	ϕ̄	ϕ̄	PROPN
ejpam-4827	591	27	∈	∈	PROPN
ejpam-4827	591	28	(	(	PUNCT
ejpam-4827	591	29	r	r	NOUN
ejpam-4827	591	30	/	/	SYM
ejpam-4827	591	31	i	i	NOUN
ejpam-4827	591	32	)	)	PUNCT
ejpam-4827	591	33	∗m	∗m	NOUN
ejpam-4827	591	34	.	.	PUNCT
ejpam-4827	592	1	where	where	SCONJ
ejpam-4827	592	2	φ	φ	PROPN
ejpam-4827	592	3	=	=	SYM
ejpam-4827	592	4	b1g1+b2g2	b1g1+b2g2	PROPN
ejpam-4827	592	5	+	+	X
ejpam-4827	592	6	·	·	PUNCT
ejpam-4827	592	7	·	·	PUNCT
ejpam-4827	592	8	·	·	PUNCT
ejpam-4827	592	9	+	+	NUM
ejpam-4827	592	10	bngn	bngn	NOUN
ejpam-4827	592	11	,	,	PUNCT
ejpam-4827	592	12	ϕ	ϕ	X
ejpam-4827	592	13	=	=	SYM
ejpam-4827	592	14	c1l1+c2l2	c1l1+c2l2	PROPN
ejpam-4827	592	15	+	+	PROPN
ejpam-4827	592	16	·	·	PUNCT
ejpam-4827	592	17	·	·	PUNCT
ejpam-4827	592	18	·	·	PUNCT
ejpam-4827	592	19	+	+	ADJ
ejpam-4827	592	20	cdld	cdld	ADJ
ejpam-4827	592	21	and	and	CCONJ
ejpam-4827	592	22	ψ	ψ	X
ejpam-4827	592	23	=	=	PUNCT
ejpam-4827	592	24	a1h1+a2h2	a1h1+a2h2	PROPN
ejpam-4827	592	25	+	+	X
ejpam-4827	592	26	·	·	PUNCT
ejpam-4827	592	27	·	·	PUNCT
ejpam-4827	592	28	·	·	PUNCT
ejpam-4827	592	29	+	+	NUM
ejpam-4827	592	30	amhm	amhm	NOUN
ejpam-4827	592	31	be	be	VERB
ejpam-4827	592	32	nonzero	nonzero	NOUN
ejpam-4827	592	33	elements	element	NOUN
ejpam-4827	592	34	in	in	ADP
ejpam-4827	592	35	r∗m	r∗m	NOUN
ejpam-4827	592	36	.	.	PUNCT
ejpam-4827	593	1	since	since	SCONJ
ejpam-4827	593	2	nil(r	nil(r	PROPN
ejpam-4827	593	3	)	)	PUNCT
ejpam-4827	593	4	is	be	AUX
ejpam-4827	593	5	an	an	DET
ejpam-4827	593	6	ideal	ideal	NOUN
ejpam-4827	593	7	of	of	ADP
ejpam-4827	593	8	r	r	NOUN
ejpam-4827	593	9	,	,	PUNCT
ejpam-4827	593	10	we	we	PRON
ejpam-4827	593	11	write	write	VERB
ejpam-4827	593	12	r	r	NOUN
ejpam-4827	593	13	=	=	SYM
ejpam-4827	593	14	r	r	NOUN
ejpam-4827	593	15	/	/	SYM
ejpam-4827	593	16	nil(r	nil(r	NOUN
ejpam-4827	593	17	)	)	PUNCT
ejpam-4827	593	18	and	and	CCONJ
ejpam-4827	593	19	define	define	VERB
ejpam-4827	593	20	σ	σ	NOUN
ejpam-4827	593	21	:	:	PUNCT
ejpam-4827	593	22	m	m	PROPN
ejpam-4827	593	23	→	→	SYM
ejpam-4827	593	24	aut(r	aut(r	PROPN
ejpam-4827	593	25	)	)	PUNCT
ejpam-4827	593	26	by	by	ADP
ejpam-4827	593	27	σg(x+nil(r	σg(x+nil(r	PROPN
ejpam-4827	593	28	)	)	PUNCT
ejpam-4827	593	29	)	)	PUNCT
ejpam-4827	594	1	=	=	SYM
ejpam-4827	594	2	σg(x)+nil(r	σg(x)+nil(r	NOUN
ejpam-4827	594	3	)	)	PUNCT
ejpam-4827	594	4	,	,	PUNCT
ejpam-4827	594	5	for	for	ADP
ejpam-4827	594	6	each	each	DET
ejpam-4827	594	7	g	g	NOUN
ejpam-4827	594	8	∈m	∈m	NOUN
ejpam-4827	594	9	.	.	PUNCT
ejpam-4827	595	1	we	we	PRON
ejpam-4827	595	2	claim	claim	VERB
ejpam-4827	595	3	that	that	SCONJ
ejpam-4827	595	4	σ	σ	PROPN
ejpam-4827	595	5	is	be	AUX
ejpam-4827	595	6	a	a	DET
ejpam-4827	595	7	compatible	compatible	ADJ
ejpam-4827	595	8	monoid	monoid	NOUN
ejpam-4827	595	9	homomorphism	homomorphism	NOUN
ejpam-4827	595	10	.	.	PUNCT
ejpam-4827	596	1	for	for	ADP
ejpam-4827	596	2	each	each	DET
ejpam-4827	596	3	x	x	NOUN
ejpam-4827	596	4	,	,	PUNCT
ejpam-4827	596	5	y	y	PROPN
ejpam-4827	596	6	∈	∈	PROPN
ejpam-4827	596	7	r	r	NOUN
ejpam-4827	596	8	,	,	PUNCT
ejpam-4827	596	9	let	let	VERB
ejpam-4827	596	10	(	(	PUNCT
ejpam-4827	596	11	x+nil(r))(y+nil(r	x+nil(r))(y+nil(r	PROPN
ejpam-4827	596	12	)	)	PUNCT
ejpam-4827	596	13	)	)	PUNCT
ejpam-4827	597	1	=	=	PUNCT
ejpam-4827	598	1	0̄.	0̄.	ADP
ejpam-4827	598	2	then	then	ADV
ejpam-4827	598	3	xy	xy	PROPN
ejpam-4827	598	4	∈	∈	PROPN
ejpam-4827	598	5	nil(r	nil(r	PROPN
ejpam-4827	598	6	)	)	PUNCT
ejpam-4827	598	7	and	and	CCONJ
ejpam-4827	598	8	hence	hence	ADV
ejpam-4827	598	9	(	(	PUNCT
ejpam-4827	598	10	x	x	SYM
ejpam-4827	598	11	+	+	NUM
ejpam-4827	598	12	nil(r))σ̄(y	nil(r))σ̄(y	NOUN
ejpam-4827	599	1	+	+	CCONJ
ejpam-4827	599	2	nil(r	nil(r	ADJ
ejpam-4827	599	3	)	)	PUNCT
ejpam-4827	599	4	)	)	PUNCT
ejpam-4827	600	1	=	=	PUNCT
ejpam-4827	601	1	0̄.	0̄.	ADP
ejpam-4827	601	2	the	the	DET
ejpam-4827	601	3	converse	converse	NOUN
ejpam-4827	601	4	is	be	AUX
ejpam-4827	601	5	similar	similar	ADJ
ejpam-4827	601	6	.	.	PUNCT
ejpam-4827	602	1	therefore	therefore	ADV
ejpam-4827	602	2	,	,	PUNCT
ejpam-4827	602	3	σ	σ	PROPN
ejpam-4827	602	4	is	be	AUX
ejpam-4827	602	5	compatible	compatible	ADJ
ejpam-4827	602	6	.	.	PUNCT
ejpam-4827	603	1	for	for	ADP
ejpam-4827	603	2	any	any	DET
ejpam-4827	603	3	φ	φ	NOUN
ejpam-4827	603	4	=	=	SYM
ejpam-4827	603	5	b1g1	b1g1	PROPN
ejpam-4827	603	6	+	+	NUM
ejpam-4827	603	7	b2g2	b2g2	PROPN
ejpam-4827	603	8	+	+	CCONJ
ejpam-4827	603	9	·	·	PUNCT
ejpam-4827	603	10	·	·	PUNCT
ejpam-4827	603	11	·	·	PUNCT
ejpam-4827	604	1	+	+	NUM
ejpam-4827	604	2	bngn	bngn	NOUN
ejpam-4827	604	3	∈	∈	PROPN
ejpam-4827	604	4	r	r	NOUN
ejpam-4827	604	5	∗m	∗m	NOUN
ejpam-4827	604	6	,	,	PUNCT
ejpam-4827	604	7	we	we	PRON
ejpam-4827	604	8	denote	denote	VERB
ejpam-4827	604	9	φ̄	φ̄	NOUN
ejpam-4827	605	1	=	=	SYM
ejpam-4827	605	2	∑n	∑n	PROPN
ejpam-4827	605	3	i=1(bi+	i=1(bi+	PUNCT
ejpam-4827	605	4	nil(r))gi	nil(r))gi	NOUN
ejpam-4827	605	5	∈	∈	PROPN
ejpam-4827	605	6	r̄	r̄	NOUN
ejpam-4827	605	7	∗m	∗m	NOUN
ejpam-4827	605	8	.	.	PUNCT
ejpam-4827	606	1	it	it	PRON
ejpam-4827	606	2	is	be	AUX
ejpam-4827	606	3	easy	easy	ADJ
ejpam-4827	606	4	to	to	PART
ejpam-4827	606	5	see	see	VERB
ejpam-4827	606	6	that	that	SCONJ
ejpam-4827	606	7	the	the	DET
ejpam-4827	606	8	mapping	mapping	NOUN
ejpam-4827	606	9	σ	σ	NOUN
ejpam-4827	606	10	:	:	PUNCT
ejpam-4827	606	11	r	r	NOUN
ejpam-4827	606	12	∗m	∗m	NOUN
ejpam-4827	606	13	−→	−→	NOUN
ejpam-4827	606	14	r	r	NOUN
ejpam-4827	606	15	∗m	∗m	NOUN
ejpam-4827	606	16	defined	define	VERB
ejpam-4827	606	17	by	by	ADP
ejpam-4827	606	18	σ(φi	σ(φi	NOUN
ejpam-4827	606	19	)	)	PUNCT
ejpam-4827	606	20	=	=	PRON
ejpam-4827	607	1	φi	φi	ADV
ejpam-4827	607	2	is	be	AUX
ejpam-4827	607	3	a	a	DET
ejpam-4827	607	4	ring	ring	NOUN
ejpam-4827	607	5	homomorphism	homomorphism	NOUN
ejpam-4827	607	6	.	.	PUNCT
ejpam-4827	608	1	since	since	SCONJ
ejpam-4827	608	2	φ̄ϕ̄ψ̄	φ̄ϕ̄ψ̄	NUM
ejpam-4827	608	3	∈	∈	PROPN
ejpam-4827	608	4	nil(r	nil(r	PROPN
ejpam-4827	608	5	/	/	SYM
ejpam-4827	608	6	i	i	PROPN
ejpam-4827	608	7	)	)	PUNCT
ejpam-4827	608	8	∗m	∗m	NOUN
ejpam-4827	608	9	.	.	PUNCT
ejpam-4827	609	1	then	then	ADV
ejpam-4827	609	2	there	there	PRON
ejpam-4827	609	3	exist	exist	VERB
ejpam-4827	609	4	a	a	DET
ejpam-4827	609	5	positive	positive	ADJ
ejpam-4827	609	6	integer	integer	NOUN
ejpam-4827	609	7	ℓ	ℓ	PROPN
ejpam-4827	609	8	∈	∈	PROPN
ejpam-4827	609	9	n	n	PRON
ejpam-4827	609	10	such	such	ADJ
ejpam-4827	609	11	that	that	PRON
ejpam-4827	609	12	(	(	PUNCT
ejpam-4827	609	13	φ̄ϕ̄ψ̄)ℓ	φ̄ϕ̄ψ̄)ℓ	PUNCT
ejpam-4827	609	14	∈	∈	PROPN
ejpam-4827	609	15	nil(r	nil(r	PROPN
ejpam-4827	609	16	/	/	SYM
ejpam-4827	609	17	i	i	PROPN
ejpam-4827	609	18	)	)	PUNCT
ejpam-4827	609	19	∗m	∗m	NOUN
ejpam-4827	609	20	.	.	PUNCT
ejpam-4827	610	1	so	so	ADV
ejpam-4827	610	2	biσgi(raj)ℓ	biσgi(raj)ℓ	NOUN
ejpam-4827	610	3	∈	∈	PROPN
ejpam-4827	610	4	i	i	PRON
ejpam-4827	610	5	for	for	ADP
ejpam-4827	610	6	any	any	DET
ejpam-4827	610	7	i	i	PROPN
ejpam-4827	610	8	,	,	PUNCT
ejpam-4827	610	9	j	j	PROPN
ejpam-4827	610	10	and	and	CCONJ
ejpam-4827	610	11	r	r	PROPN
ejpam-4827	610	12	∈	∈	PROPN
ejpam-4827	610	13	r.	r.	PROPN
ejpam-4827	610	14	so	so	ADV
ejpam-4827	610	15	biσgi(σs(raj	biσgi(σs(raj	PROPN
ejpam-4827	610	16	)	)	PUNCT
ejpam-4827	610	17	)	)	PUNCT
ejpam-4827	611	1	∈	∈	PROPN
ejpam-4827	611	2	nil(r	nil(r	PROPN
ejpam-4827	611	3	)	)	PUNCT
ejpam-4827	611	4	since	since	SCONJ
ejpam-4827	611	5	i	i	PRON
ejpam-4827	611	6	⊆	⊆	NUM
ejpam-4827	611	7	nil(r	nil(r	NUM
ejpam-4827	611	8	)	)	PUNCT
ejpam-4827	611	9	and	and	CCONJ
ejpam-4827	611	10	by	by	ADP
ejpam-4827	611	11	compatibility	compatibility	NOUN
ejpam-4827	611	12	,	,	PUNCT
ejpam-4827	611	13	we	we	PRON
ejpam-4827	611	14	have	have	VERB
ejpam-4827	611	15	biσgi(raj	biσgi(raj	NOUN
ejpam-4827	611	16	)	)	PUNCT
ejpam-4827	611	17	∈	∈	PROPN
ejpam-4827	611	18	nil(r	nil(r	PROPN
ejpam-4827	611	19	)	)	PUNCT
ejpam-4827	611	20	and	and	CCONJ
ejpam-4827	611	21	ajσhj	ajσhj	ADJ
ejpam-4827	611	22	(	(	PUNCT
ejpam-4827	611	23	σs(rbi	σs(rbi	PROPN
ejpam-4827	611	24	)	)	PUNCT
ejpam-4827	611	25	)	)	PUNCT
ejpam-4827	612	1	∈	∈	PROPN
ejpam-4827	612	2	nil(r	nil(r	PROPN
ejpam-4827	612	3	)	)	PUNCT
ejpam-4827	612	4	since	since	SCONJ
ejpam-4827	612	5	r	r	NOUN
ejpam-4827	612	6	is	be	AUX
ejpam-4827	612	7	σ	σ	NOUN
ejpam-4827	612	8	-	-	PUNCT
ejpam-4827	612	9	skew	skew	NOUN
ejpam-4827	612	10	strongly	strongly	ADV
ejpam-4827	612	11	m	m	VERB
ejpam-4827	612	12	-nil	-nil	NOUN
ejpam-4827	612	13	-	-	PUNCT
ejpam-4827	612	14	reflexive	reflexive	ADJ
ejpam-4827	612	15	,	,	PUNCT
ejpam-4827	612	16	this	this	PRON
ejpam-4827	612	17	mean	mean	VERB
ejpam-4827	612	18	that	that	SCONJ
ejpam-4827	612	19	ψϕφ	ψϕφ	VERB
ejpam-4827	612	20	∈	∈	PROPN
ejpam-4827	612	21	nil(r	nil(r	NOUN
ejpam-4827	612	22	)	)	PUNCT
ejpam-4827	612	23	∗m	∗m	NOUN
ejpam-4827	612	24	.	.	PUNCT
ejpam-4827	613	1	thus	thus	ADV
ejpam-4827	613	2	ψ̄ϕ̄φ̄	ψ̄ϕ̄φ̄	PUNCT
ejpam-4827	613	3	∈	∈	PROPN
ejpam-4827	613	4	nil(r	nil(r	PROPN
ejpam-4827	613	5	/	/	SYM
ejpam-4827	613	6	i	i	PROPN
ejpam-4827	613	7	)	)	PUNCT
ejpam-4827	613	8	∗m	∗m	NOUN
ejpam-4827	613	9	.	.	PUNCT
ejpam-4827	614	1	therefore	therefore	ADV
ejpam-4827	614	2	,	,	PUNCT
ejpam-4827	614	3	r	r	X
ejpam-4827	614	4	/	/	SYM
ejpam-4827	614	5	i	i	PRON
ejpam-4827	614	6	is	be	AUX
ejpam-4827	614	7	σ̄-skew	σ̄-skew	PROPN
ejpam-4827	614	8	strongly	strongly	ADV
ejpam-4827	614	9	m	m	VERB
ejpam-4827	614	10	-nil	-nil	NOUN
ejpam-4827	614	11	-	-	PUNCT
ejpam-4827	614	12	reflexive	reflexive	ADJ
ejpam-4827	614	13	.	.	PUNCT
ejpam-4827	615	1	proposition	proposition	NOUN
ejpam-4827	615	2	3	3	NUM
ejpam-4827	615	3	.	.	PUNCT
ejpam-4827	616	1	let	let	AUX
ejpam-4827	616	2	m	m	PRON
ejpam-4827	616	3	be	be	AUX
ejpam-4827	616	4	a	a	DET
ejpam-4827	616	5	finitely	finitely	ADV
ejpam-4827	616	6	generated	generate	VERB
ejpam-4827	616	7	abelian	abelian	ADJ
ejpam-4827	616	8	group	group	NOUN
ejpam-4827	616	9	and	and	CCONJ
ejpam-4827	616	10	σ	σ	NOUN
ejpam-4827	616	11	:	:	PUNCT
ejpam-4827	616	12	m	m	PROPN
ejpam-4827	616	13	→	→	SYM
ejpam-4827	616	14	aut(r	aut(r	PROPN
ejpam-4827	616	15	)	)	PUNCT
ejpam-4827	616	16	be	be	AUX
ejpam-4827	616	17	a	a	DET
ejpam-4827	616	18	compatible	compatible	ADJ
ejpam-4827	616	19	monoid	monoid	NOUN
ejpam-4827	616	20	homomorphism	homomorphism	NOUN
ejpam-4827	616	21	.	.	PUNCT
ejpam-4827	617	1	then	then	ADV
ejpam-4827	617	2	m	m	PROPN
ejpam-4827	617	3	is	be	AUX
ejpam-4827	617	4	torsion	torsion	NOUN
ejpam-4827	617	5	free	free	ADJ
ejpam-4827	617	6	if	if	SCONJ
ejpam-4827	617	7	and	and	CCONJ
ejpam-4827	617	8	only	only	ADV
ejpam-4827	617	9	if	if	SCONJ
ejpam-4827	617	10	there	there	PRON
ejpam-4827	617	11	exist	exist	VERB
ejpam-4827	617	12	a	a	DET
ejpam-4827	617	13	non	non	ADJ
ejpam-4827	617	14	-	-	ADJ
ejpam-4827	617	15	zero	zero	ADJ
ejpam-4827	617	16	ring	ring	NOUN
ejpam-4827	617	17	r	r	NOUN
ejpam-4827	617	18	such	such	ADJ
ejpam-4827	617	19	that	that	SCONJ
ejpam-4827	617	20	r	r	NOUN
ejpam-4827	617	21	is	be	AUX
ejpam-4827	617	22	σ	σ	NOUN
ejpam-4827	617	23	-	-	PUNCT
ejpam-4827	617	24	skew	skew	NOUN
ejpam-4827	617	25	strongly	strongly	ADV
ejpam-4827	617	26	m	m	VERB
ejpam-4827	617	27	-nil	-nil	NOUN
ejpam-4827	617	28	-	-	PUNCT
ejpam-4827	617	29	reflexive	reflexive	ADJ
ejpam-4827	617	30	.	.	PUNCT
ejpam-4827	618	1	proof	proof	NOUN
ejpam-4827	618	2	.	.	PUNCT
ejpam-4827	619	1	let	let	VERB
ejpam-4827	619	2	m	m	PRON
ejpam-4827	619	3	be	be	AUX
ejpam-4827	619	4	a	a	DET
ejpam-4827	619	5	finitely	finitely	ADV
ejpam-4827	619	6	generated	generate	VERB
ejpam-4827	619	7	torsion	torsion	NOUN
ejpam-4827	619	8	-	-	PUNCT
ejpam-4827	619	9	free	free	ADJ
ejpam-4827	619	10	abelian	abelian	ADJ
ejpam-4827	619	11	group	group	NOUN
ejpam-4827	619	12	.	.	PUNCT
ejpam-4827	620	1	thus	thus	ADV
ejpam-4827	620	2	,	,	PUNCT
ejpam-4827	620	3	m	m	VERB
ejpam-4827	620	4	∼=	∼=	VERB
ejpam-4827	620	5	⨿n1	⨿n1	NOUN
ejpam-4827	620	6	z	z	NOUN
ejpam-4827	621	1	and	and	CCONJ
ejpam-4827	621	2	so	so	ADV
ejpam-4827	621	3	m	m	VERB
ejpam-4827	621	4	is	be	AUX
ejpam-4827	621	5	a	a	DET
ejpam-4827	621	6	u.p.-monoid	u.p.-monoid	ADJ
ejpam-4827	621	7	,	,	PUNCT
ejpam-4827	621	8	as	as	SCONJ
ejpam-4827	621	9	shown	show	VERB
ejpam-4827	621	10	in	in	ADP
ejpam-4827	621	11	lemma	lemma	PROPN
ejpam-4827	621	12	2	2	NUM
ejpam-4827	621	13	.	.	PUNCT
ejpam-4827	622	1	therefore	therefore	ADV
ejpam-4827	622	2	,	,	PUNCT
ejpam-4827	622	3	for	for	ADP
ejpam-4827	622	4	any	any	DET
ejpam-4827	622	5	ni	ni	PROPN
ejpam-4827	622	6	ring	ring	NOUN
ejpam-4827	622	7	r	r	NOUN
ejpam-4827	622	8	,	,	PUNCT
ejpam-4827	622	9	r	r	NOUN
ejpam-4827	622	10	is	be	AUX
ejpam-4827	622	11	σ	σ	NOUN
ejpam-4827	622	12	-	-	PUNCT
ejpam-4827	622	13	skew	skew	NOUN
ejpam-4827	622	14	strongly	strongly	ADV
ejpam-4827	622	15	m	m	VERB
ejpam-4827	622	16	-nil	-nil	NOUN
ejpam-4827	622	17	-	-	PUNCT
ejpam-4827	622	18	reflexive	reflexive	ADJ
ejpam-4827	622	19	by	by	ADP
ejpam-4827	622	20	theorem	theorem	NOUN
ejpam-4827	622	21	3	3	NUM
ejpam-4827	622	22	.	.	PUNCT
ejpam-4827	622	23	conversely	conversely	ADV
ejpam-4827	622	24	,	,	PUNCT
ejpam-4827	622	25	suppose	suppose	VERB
ejpam-4827	622	26	gℓ	gℓ	VERB
ejpam-4827	622	27	=	=	SYM
ejpam-4827	622	28	e	e	PROPN
ejpam-4827	622	29	for	for	ADP
ejpam-4827	622	30	some	some	DET
ejpam-4827	622	31	e	e	NOUN
ejpam-4827	622	32	̸=	̸=	PROPN
ejpam-4827	622	33	g	g	PROPN
ejpam-4827	622	34	∈	∈	PROPN
ejpam-4827	622	35	m	m	NOUN
ejpam-4827	622	36	and	and	CCONJ
ejpam-4827	622	37	positive	positive	ADJ
ejpam-4827	622	38	integer	integer	NOUN
ejpam-4827	622	39	ℓ.	ℓ.	NOUN
ejpam-4827	622	40	let	let	VERB
ejpam-4827	622	41	n	n	PRON
ejpam-4827	622	42	be	be	AUX
ejpam-4827	622	43	a	a	DET
ejpam-4827	622	44	cyclic	cyclic	ADJ
ejpam-4827	622	45	subgroup	subgroup	NOUN
ejpam-4827	622	46	of	of	AUX
ejpam-4827	622	47	m	m	AUX
ejpam-4827	622	48	generated	generate	VERB
ejpam-4827	622	49	by	by	ADP
ejpam-4827	622	50	{	{	PUNCT
ejpam-4827	622	51	g	g	NOUN
ejpam-4827	622	52	}	}	PUNCT
ejpam-4827	622	53	.	.	PUNCT
ejpam-4827	623	1	therefore	therefore	ADV
ejpam-4827	623	2	,	,	PUNCT
ejpam-4827	623	3	r	r	NOUN
ejpam-4827	623	4	is	be	AUX
ejpam-4827	623	5	σ	σ	NOUN
ejpam-4827	623	6	-	-	PUNCT
ejpam-4827	623	7	skew	skew	NOUN
ejpam-4827	623	8	strongly	strongly	ADV
ejpam-4827	623	9	n	n	CCONJ
ejpam-4827	623	10	-nil	-nil	ADV
ejpam-4827	623	11	-	-	PUNCT
ejpam-4827	623	12	reflexive	reflexive	ADJ
ejpam-4827	623	13	,	,	PUNCT
ejpam-4827	623	14	which	which	PRON
ejpam-4827	623	15	leads	lead	VERB
ejpam-4827	623	16	to	to	ADP
ejpam-4827	623	17	a	a	DET
ejpam-4827	623	18	contradiction	contradiction	NOUN
ejpam-4827	623	19	,	,	PUNCT
ejpam-4827	623	20	as	as	SCONJ
ejpam-4827	623	21	shown	show	VERB
ejpam-4827	623	22	in	in	ADP
ejpam-4827	623	23	lemma	lemma	PROPN
ejpam-4827	623	24	3	3	NUM
ejpam-4827	623	25	.	.	PUNCT
ejpam-4827	624	1	hence	hence	ADV
ejpam-4827	624	2	,	,	PUNCT
ejpam-4827	624	3	m	m	VERB
ejpam-4827	624	4	is	be	AUX
ejpam-4827	624	5	torsion	torsion	NOUN
ejpam-4827	624	6	-	-	PUNCT
ejpam-4827	624	7	free	free	ADJ
ejpam-4827	624	8	.	.	PUNCT
ejpam-4827	625	1	we	we	PRON
ejpam-4827	625	2	will	will	AUX
ejpam-4827	625	3	now	now	ADV
ejpam-4827	625	4	provide	provide	VERB
ejpam-4827	625	5	some	some	DET
ejpam-4827	625	6	examples	example	NOUN
ejpam-4827	625	7	of	of	ADP
ejpam-4827	625	8	σ	σ	PROPN
ejpam-4827	625	9	-	-	PUNCT
ejpam-4827	625	10	skew	skew	NOUN
ejpam-4827	625	11	strongly	strongly	ADV
ejpam-4827	625	12	m	m	VERB
ejpam-4827	625	13	-nil	-nil	ADJ
ejpam-4827	625	14	-	-	PUNCT
ejpam-4827	625	15	reflexive	reflexive	ADJ
ejpam-4827	625	16	ring	ring	NOUN
ejpam-4827	625	17	.	.	PUNCT
ejpam-4827	626	1	in	in	ADP
ejpam-4827	626	2	theorem	theorem	ADJ
ejpam-4827	626	3	2.6	2.6	NUM
ejpam-4827	626	4	[	[	SYM
ejpam-4827	626	5	13	13	NUM
ejpam-4827	626	6	]	]	PUNCT
ejpam-4827	626	7	,	,	PUNCT
ejpam-4827	626	8	kwak	kwak	PROPN
ejpam-4827	626	9	and	and	CCONJ
ejpam-4827	626	10	lee	lee	PROPN
ejpam-4827	626	11	proved	prove	VERB
ejpam-4827	626	12	that	that	SCONJ
ejpam-4827	626	13	r	r	NOUN
ejpam-4827	626	14	is	be	AUX
ejpam-4827	626	15	a	a	DET
ejpam-4827	626	16	reflexive	reflexive	ADJ
ejpam-4827	626	17	ring	ring	NOUN
ejpam-4827	626	18	if	if	SCONJ
ejpam-4827	626	19	and	and	CCONJ
ejpam-4827	626	20	only	only	ADV
ejpam-4827	626	21	if	if	SCONJ
ejpam-4827	626	22	matn(r	matn(r	NOUN
ejpam-4827	626	23	)	)	PUNCT
ejpam-4827	626	24	is	be	AUX
ejpam-4827	626	25	a	a	DET
ejpam-4827	626	26	reflexive	reflexive	ADJ
ejpam-4827	626	27	ring	ring	NOUN
ejpam-4827	626	28	for	for	ADP
ejpam-4827	626	29	all	all	DET
ejpam-4827	626	30	n	n	PRON
ejpam-4827	626	31	≥	≥	NOUN
ejpam-4827	626	32	1	1	NUM
ejpam-4827	626	33	.	.	PUNCT
ejpam-4827	627	1	however	however	ADV
ejpam-4827	627	2	,	,	PUNCT
ejpam-4827	627	3	this	this	PRON
ejpam-4827	627	4	is	be	AUX
ejpam-4827	627	5	not	not	PART
ejpam-4827	627	6	the	the	DET
ejpam-4827	627	7	case	case	NOUN
ejpam-4827	627	8	in	in	ADP
ejpam-4827	627	9	σ	σ	PROPN
ejpam-4827	627	10	-	-	PUNCT
ejpam-4827	627	11	skew	skew	NOUN
ejpam-4827	627	12	strongly	strongly	ADV
ejpam-4827	627	13	m	m	VERB
ejpam-4827	627	14	-nil	-nil	ADJ
ejpam-4827	627	15	-	-	PUNCT
ejpam-4827	627	16	reflexive	reflexive	ADJ
ejpam-4827	627	17	rings	ring	NOUN
ejpam-4827	627	18	of	of	ADP
ejpam-4827	627	19	r.	r.	PROPN
ejpam-4827	627	20	there	there	ADV
ejpam-4827	627	21	exist	exist	VERB
ejpam-4827	627	22	σ	σ	NOUN
ejpam-4827	627	23	-	-	PUNCT
ejpam-4827	627	24	skew	skew	NOUN
ejpam-4827	627	25	strongly	strongly	ADV
ejpam-4827	627	26	m	m	VERB
ejpam-4827	627	27	-nil	-nil	ADJ
ejpam-4827	627	28	-	-	PUNCT
ejpam-4827	627	29	reflexive	reflexive	ADJ
ejpam-4827	627	30	rings	ring	NOUN
ejpam-4827	627	31	over	over	ADP
ejpam-4827	627	32	which	which	PRON
ejpam-4827	627	33	matrix	matrix	NOUN
ejpam-4827	627	34	rings	ring	NOUN
ejpam-4827	627	35	need	need	AUX
ejpam-4827	627	36	not	not	PART
ejpam-4827	627	37	be	be	AUX
ejpam-4827	627	38	σ	σ	NOUN
ejpam-4827	627	39	-	-	PUNCT
ejpam-4827	627	40	skew	skew	NOUN
ejpam-4827	627	41	strongly	strongly	ADV
ejpam-4827	627	42	m	m	VERB
ejpam-4827	627	43	-reflexive	-reflexive	ADJ
ejpam-4827	627	44	,	,	PUNCT
ejpam-4827	627	45	as	as	SCONJ
ejpam-4827	627	46	shown	show	VERB
ejpam-4827	627	47	below	below	ADV
ejpam-4827	627	48	.	.	PUNCT
ejpam-4827	628	1	e.	e.	PROPN
ejpam-4827	628	2	ali	ali	PROPN
ejpam-4827	628	3	/	/	SYM
ejpam-4827	628	4	eur	eur	PROPN
ejpam-4827	628	5	.	.	PUNCT
ejpam-4827	629	1	j.	j.	PROPN
ejpam-4827	629	2	pure	pure	PROPN
ejpam-4827	629	3	appl	appl	PROPN
ejpam-4827	629	4	.	.	PROPN
ejpam-4827	629	5	math	math	PROPN
ejpam-4827	629	6	,	,	PUNCT
ejpam-4827	629	7	16	16	NUM
ejpam-4827	629	8	(	(	PUNCT
ejpam-4827	629	9	3	3	NUM
ejpam-4827	629	10	)	)	PUNCT
ejpam-4827	629	11	(	(	PUNCT
ejpam-4827	629	12	2023	2023	NUM
ejpam-4827	629	13	)	)	PUNCT
ejpam-4827	629	14	,	,	PUNCT
ejpam-4827	629	15	1878	1878	NUM
ejpam-4827	629	16	-	-	SYM
ejpam-4827	629	17	1893	1893	NUM
ejpam-4827	629	18	1890	1890	NUM
ejpam-4827	629	19	example	example	NOUN
ejpam-4827	629	20	3	3	X
ejpam-4827	629	21	.	.	PUNCT
ejpam-4827	630	1	let	let	AUX
ejpam-4827	630	2	s	s	PRON
ejpam-4827	630	3	be	be	AUX
ejpam-4827	630	4	a	a	DET
ejpam-4827	630	5	torsion	torsion	NOUN
ejpam-4827	630	6	-	-	PUNCT
ejpam-4827	630	7	free	free	ADJ
ejpam-4827	630	8	and	and	CCONJ
ejpam-4827	630	9	cancellative	cancellative	ADJ
ejpam-4827	630	10	monoid	monoid	NOUN
ejpam-4827	630	11	and	and	CCONJ
ejpam-4827	630	12	σ	σ	NOUN
ejpam-4827	630	13	:	:	PUNCT
ejpam-4827	630	14	m	m	PROPN
ejpam-4827	630	15	→	→	SYM
ejpam-4827	630	16	aut(r	aut(r	PROPN
ejpam-4827	630	17	)	)	PUNCT
ejpam-4827	630	18	be	be	AUX
ejpam-4827	630	19	a	a	DET
ejpam-4827	630	20	compatible	compatible	ADJ
ejpam-4827	630	21	monoid	monoid	NOUN
ejpam-4827	630	22	homomorphism	homomorphism	NOUN
ejpam-4827	630	23	.	.	PUNCT
ejpam-4827	631	1	(	(	PUNCT
ejpam-4827	631	2	1	1	X
ejpam-4827	631	3	)	)	PUNCT
ejpam-4827	631	4	if	if	SCONJ
ejpam-4827	631	5	r	r	NOUN
ejpam-4827	631	6	is	be	AUX
ejpam-4827	631	7	a	a	DET
ejpam-4827	631	8	ring	ring	NOUN
ejpam-4827	631	9	with	with	ADP
ejpam-4827	631	10	nil(r	nil(r	NOUN
ejpam-4827	631	11	)	)	PUNCT
ejpam-4827	631	12	an	an	DET
ejpam-4827	631	13	ideal	ideal	NOUN
ejpam-4827	631	14	of	of	ADP
ejpam-4827	631	15	r	r	NOUN
ejpam-4827	631	16	,	,	PUNCT
ejpam-4827	631	17	then	then	ADV
ejpam-4827	631	18	r	r	NOUN
ejpam-4827	631	19	is	be	AUX
ejpam-4827	631	20	σ	σ	NOUN
ejpam-4827	631	21	-	-	PUNCT
ejpam-4827	631	22	skew	skew	NOUN
ejpam-4827	631	23	strongly	strongly	ADV
ejpam-4827	631	24	m	m	VERB
ejpam-4827	631	25	-nil	-nil	NOUN
ejpam-4827	631	26	-	-	PUNCT
ejpam-4827	631	27	reflexive	reflexive	ADJ
ejpam-4827	631	28	.	.	PUNCT
ejpam-4827	632	1	(	(	PUNCT
ejpam-4827	632	2	2	2	X
ejpam-4827	632	3	)	)	PUNCT
ejpam-4827	632	4	for	for	ADP
ejpam-4827	632	5	any	any	DET
ejpam-4827	632	6	reduced	reduce	VERB
ejpam-4827	632	7	ring	ring	NOUN
ejpam-4827	632	8	r	r	NOUN
ejpam-4827	632	9	,	,	PUNCT
ejpam-4827	632	10	the	the	DET
ejpam-4827	632	11	ring	ring	NOUN
ejpam-4827	632	12	tn(r	tn(r	NOUN
ejpam-4827	632	13	)	)	PUNCT
ejpam-4827	632	14	is	be	AUX
ejpam-4827	632	15	σ	σ	NOUN
ejpam-4827	632	16	-	-	PUNCT
ejpam-4827	632	17	skew	skew	NOUN
ejpam-4827	632	18	strongly	strongly	ADV
ejpam-4827	632	19	m	m	VERB
ejpam-4827	632	20	-nil	-nil	NOUN
ejpam-4827	632	21	-	-	PUNCT
ejpam-4827	632	22	reflexive	reflexive	ADJ
ejpam-4827	632	23	.	.	PUNCT
ejpam-4827	633	1	however	however	ADV
ejpam-4827	633	2	,	,	PUNCT
ejpam-4827	633	3	the	the	DET
ejpam-4827	633	4	ring	ring	NOUN
ejpam-4827	633	5	of	of	ADP
ejpam-4827	633	6	all	all	DET
ejpam-4827	633	7	2×	2×	NUM
ejpam-4827	633	8	2	2	NUM
ejpam-4827	633	9	matrices	matrix	NOUN
ejpam-4827	633	10	over	over	ADP
ejpam-4827	633	11	any	any	DET
ejpam-4827	633	12	field	field	NOUN
ejpam-4827	633	13	is	be	AUX
ejpam-4827	633	14	not	not	PART
ejpam-4827	633	15	σ	σ	NOUN
ejpam-4827	633	16	-	-	PUNCT
ejpam-4827	633	17	skew	skew	NOUN
ejpam-4827	633	18	strongly	strongly	ADV
ejpam-4827	633	19	m	m	VERB
ejpam-4827	633	20	-nil	-nil	NOUN
ejpam-4827	633	21	-	-	PUNCT
ejpam-4827	633	22	reflexive	reflexive	ADJ
ejpam-4827	633	23	.	.	PUNCT
ejpam-4827	634	1	(	(	PUNCT
ejpam-4827	634	2	3	3	X
ejpam-4827	634	3	)	)	PUNCT
ejpam-4827	634	4	for	for	ADP
ejpam-4827	634	5	r	r	NOUN
ejpam-4827	634	6	be	be	VERB
ejpam-4827	634	7	a	a	DET
ejpam-4827	634	8	reduced	reduce	VERB
ejpam-4827	634	9	ring	ring	NOUN
ejpam-4827	634	10	.	.	PUNCT
ejpam-4827	635	1	consider	consider	VERB
ejpam-4827	635	2	the	the	DET
ejpam-4827	635	3	ring	ring	NOUN
ejpam-4827	635	4	sn(r	sn(r	PRON
ejpam-4827	635	5	)	)	PUNCT
ejpam-4827	635	6	=	=	SYM
ejpam-4827	635	7			NOUN
ejpam-4827	635	8			VERB
ejpam-4827	635	9	a	a	DET
ejpam-4827	635	10	a12	a12	NOUN
ejpam-4827	635	11	a13	a13	NOUN
ejpam-4827	635	12	·	·	PUNCT
ejpam-4827	635	13	·	·	PUNCT
ejpam-4827	635	14	·	·	PUNCT
ejpam-4827	636	1	a1n	a1n	ADP
ejpam-4827	636	2	0	0	NUM
ejpam-4827	636	3	a	a	DET
ejpam-4827	636	4	a23	a23	NOUN
ejpam-4827	636	5	·	·	PUNCT
ejpam-4827	636	6	·	·	PUNCT
ejpam-4827	636	7	·	·	PUNCT
ejpam-4827	636	8	a2n	a2n	PUNCT
ejpam-4827	636	9	0	0	NUM
ejpam-4827	636	10	0	0	NUM
ejpam-4827	636	11	a	a	PRON
ejpam-4827	636	12	·	·	PUNCT
ejpam-4827	636	13	·	·	PUNCT
ejpam-4827	636	14	·	·	PUNCT
ejpam-4827	636	15	a3n	a3n	PROPN
ejpam-4827	636	16	...	...	PUNCT
ejpam-4827	636	17	...	...	PUNCT
ejpam-4827	636	18	...	...	PUNCT
ejpam-4827	636	19	.	.	PUNCT
ejpam-4827	636	20	.	.	PUNCT
ejpam-4827	636	21	.	.	PUNCT
ejpam-4827	637	1	...	...	PUNCT
ejpam-4827	638	1	0	0	NUM
ejpam-4827	638	2	0	0	NUM
ejpam-4827	638	3	0	0	NUM
ejpam-4827	638	4	·	·	PUNCT
ejpam-4827	638	5	·	·	PUNCT
ejpam-4827	638	6	·	·	PUNCT
ejpam-4827	639	1	a	a	DET
ejpam-4827	639	2			PROPN
ejpam-4827	639	3	|	|	ADV
ejpam-4827	639	4	a	a	NOUN
ejpam-4827	639	5	,	,	PUNCT
ejpam-4827	639	6	aij	aij	PROPN
ejpam-4827	639	7	∈	∈	PROPN
ejpam-4827	639	8	r	r	NOUN
ejpam-4827	639	9	;	;	PUNCT
ejpam-4827	639	10	1	1	NUM
ejpam-4827	639	11	≤	≤	NOUN
ejpam-4827	640	1	i	i	PRON
ejpam-4827	640	2	,	,	PUNCT
ejpam-4827	640	3	j	j	PROPN
ejpam-4827	640	4	≤	≤	PROPN
ejpam-4827	640	5	n	n	PRON
ejpam-4827	640	6			NOUN
ejpam-4827	640	7	.	.	PUNCT
ejpam-4827	641	1	then	then	ADV
ejpam-4827	641	2	sn(r	sn(r	PRON
ejpam-4827	641	3	)	)	PUNCT
ejpam-4827	641	4	is	be	AUX
ejpam-4827	641	5	not	not	PART
ejpam-4827	641	6	σ	σ	NOUN
ejpam-4827	641	7	-	-	PUNCT
ejpam-4827	641	8	skew	skew	NOUN
ejpam-4827	641	9	strongly	strongly	ADV
ejpam-4827	641	10	m	m	VERB
ejpam-4827	641	11	-reflexive	-reflexive	ADJ
ejpam-4827	641	12	,	,	PUNCT
ejpam-4827	641	13	when	when	SCONJ
ejpam-4827	641	14	n	n	X
ejpam-4827	641	15	≥	≥	NOUN
ejpam-4827	641	16	4	4	NUM
ejpam-4827	641	17	,	,	PUNCT
ejpam-4827	641	18	but	but	CCONJ
ejpam-4827	641	19	sn(r	sn(r	NUM
ejpam-4827	641	20	)	)	PUNCT
ejpam-4827	641	21	and	and	CCONJ
ejpam-4827	641	22	r	r	NOUN
ejpam-4827	641	23	are	be	AUX
ejpam-4827	641	24	σ	σ	NOUN
ejpam-4827	641	25	-	-	PUNCT
ejpam-4827	641	26	skew	skew	NOUN
ejpam-4827	641	27	strongly	strongly	ADV
ejpam-4827	641	28	m	m	VERB
ejpam-4827	641	29	-nil	-nil	NOUN
ejpam-4827	641	30	-	-	PUNCT
ejpam-4827	641	31	reflexive	reflexive	ADJ
ejpam-4827	641	32	for	for	ADP
ejpam-4827	641	33	all	all	PRON
ejpam-4827	641	34	n	n	PRON
ejpam-4827	641	35	≥	≥	NUM
ejpam-4827	641	36	1	1	NUM
ejpam-4827	641	37	.	.	PUNCT
ejpam-4827	641	38	solution	solution	NOUN
ejpam-4827	641	39	(	(	PUNCT
ejpam-4827	641	40	1	1	NUM
ejpam-4827	641	41	)	)	PUNCT
ejpam-4827	641	42	.	.	PUNCT
ejpam-4827	642	1	suppose	suppose	VERB
ejpam-4827	642	2	φ	φ	X
ejpam-4827	642	3	,	,	PUNCT
ejpam-4827	642	4	ψ	ψ	PROPN
ejpam-4827	642	5	∈	∈	PROPN
ejpam-4827	642	6	r	r	NOUN
ejpam-4827	642	7	∗m	∗m	NOUN
ejpam-4827	642	8	,	,	PUNCT
ejpam-4827	642	9	with	with	ADP
ejpam-4827	642	10	φϕψ	φϕψ	NOUN
ejpam-4827	642	11	is	be	AUX
ejpam-4827	642	12	nilpotent	nilpotent	ADJ
ejpam-4827	642	13	for	for	ADP
ejpam-4827	642	14	all	all	DET
ejpam-4827	642	15	ϕ	ϕ	NOUN
ejpam-4827	642	16	∈	∈	PROPN
ejpam-4827	642	17	r	r	NOUN
ejpam-4827	642	18	∗m	∗m	NOUN
ejpam-4827	642	19	,	,	PUNCT
ejpam-4827	642	20	where	where	SCONJ
ejpam-4827	642	21	φ	φ	PROPN
ejpam-4827	642	22	=	=	SYM
ejpam-4827	642	23	b1g1	b1g1	PROPN
ejpam-4827	642	24	+	+	NUM
ejpam-4827	642	25	b2g2	b2g2	NOUN
ejpam-4827	642	26	+	+	NOUN
ejpam-4827	642	27	·	·	PUNCT
ejpam-4827	642	28	·	·	PUNCT
ejpam-4827	642	29	·	·	PUNCT
ejpam-4827	642	30	+	+	NUM
ejpam-4827	642	31	bngn	bngn	NOUN
ejpam-4827	642	32	,	,	PUNCT
ejpam-4827	642	33	ϕ	ϕ	NOUN
ejpam-4827	642	34	=	=	SYM
ejpam-4827	642	35	c1l1	c1l1	X
ejpam-4827	642	36	+	+	CCONJ
ejpam-4827	642	37	c2l2	c2l2	ADJ
ejpam-4827	642	38	+	+	X
ejpam-4827	642	39	·	·	PUNCT
ejpam-4827	642	40	·	·	PUNCT
ejpam-4827	643	1	·	·	PUNCT
ejpam-4827	643	2	+	+	NUM
ejpam-4827	643	3	cdld	cdld	ADJ
ejpam-4827	643	4	and	and	CCONJ
ejpam-4827	643	5	ψ	ψ	X
ejpam-4827	643	6	=	=	PUNCT
ejpam-4827	643	7	a1h1+a2h2	a1h1+a2h2	PROPN
ejpam-4827	643	8	+	+	X
ejpam-4827	643	9	·	·	PUNCT
ejpam-4827	643	10	·	·	PUNCT
ejpam-4827	643	11	·	·	PUNCT
ejpam-4827	643	12	+	+	NOUN
ejpam-4827	643	13	amhm	amhm	NOUN
ejpam-4827	643	14	.	.	PUNCT
ejpam-4827	644	1	so	so	ADV
ejpam-4827	644	2	there	there	PRON
ejpam-4827	644	3	exist	exist	VERB
ejpam-4827	644	4	a	a	DET
ejpam-4827	644	5	positive	positive	ADJ
ejpam-4827	644	6	integer	integer	NOUN
ejpam-4827	644	7	ℓ	ℓ	PROPN
ejpam-4827	644	8	such	such	ADJ
ejpam-4827	644	9	that	that	SCONJ
ejpam-4827	644	10	(	(	PUNCT
ejpam-4827	644	11	φϕψ)ℓ	φϕψ)ℓ	PROPN
ejpam-4827	644	12	=	=	NOUN
ejpam-4827	644	13	0	0	X
ejpam-4827	644	14	.	.	PUNCT
ejpam-4827	645	1	therefore	therefore	ADV
ejpam-4827	645	2	(	(	PUNCT
ejpam-4827	645	3	biσi(σs(caj	biσi(σs(caj	NOUN
ejpam-4827	645	4	)	)	PUNCT
ejpam-4827	645	5	)	)	PUNCT
ejpam-4827	645	6	)	)	PUNCT
ejpam-4827	646	1	ℓ	ℓ	X
ejpam-4827	646	2	=	=	SYM
ejpam-4827	646	3	0	0	NUM
ejpam-4827	646	4	,	,	PUNCT
ejpam-4827	646	5	for	for	ADP
ejpam-4827	646	6	any	any	DET
ejpam-4827	646	7	s	s	X
ejpam-4827	646	8	∈	∈	PROPN
ejpam-4827	646	9	m	m	NOUN
ejpam-4827	646	10	,	,	PUNCT
ejpam-4827	646	11	i	i	PRON
ejpam-4827	646	12	,	,	PUNCT
ejpam-4827	646	13	j.	j.	PROPN
ejpam-4827	646	14	then	then	ADV
ejpam-4827	646	15	,	,	PUNCT
ejpam-4827	646	16	biσgi(σs(c	biσgi(σs(c	PROPN
ejpam-4827	646	17	aj	aj	PROPN
ejpam-4827	646	18	)	)	PUNCT
ejpam-4827	646	19	)	)	PUNCT
ejpam-4827	647	1	∈	∈	PROPN
ejpam-4827	647	2	nil(r	nil(r	PROPN
ejpam-4827	647	3	)	)	PUNCT
ejpam-4827	647	4	and	and	CCONJ
ejpam-4827	647	5	so	so	ADV
ejpam-4827	647	6	ajσhj	ajσhj	ADJ
ejpam-4827	647	7	(	(	PUNCT
ejpam-4827	647	8	σs(c	σs(c	NOUN
ejpam-4827	647	9	bi	bi	NOUN
ejpam-4827	647	10	)	)	PUNCT
ejpam-4827	647	11	)	)	PUNCT
ejpam-4827	648	1	∈	∈	PROPN
ejpam-4827	648	2	nil(r	nil(r	PROPN
ejpam-4827	648	3	)	)	PUNCT
ejpam-4827	648	4	.	.	PUNCT
ejpam-4827	649	1	hence	hence	ADV
ejpam-4827	649	2	ψϕφ	ψϕφ	NOUN
ejpam-4827	649	3	is	be	AUX
ejpam-4827	649	4	nilpotent	nilpotent	ADJ
ejpam-4827	649	5	.	.	PUNCT
ejpam-4827	650	1	(	(	PUNCT
ejpam-4827	650	2	2	2	NUM
ejpam-4827	650	3	)	)	PUNCT
ejpam-4827	650	4	.	.	PUNCT
ejpam-4827	651	1	let	let	VERB
ejpam-4827	651	2	r	r	PRON
ejpam-4827	651	3	be	be	AUX
ejpam-4827	651	4	a	a	DET
ejpam-4827	651	5	ring	ring	NOUN
ejpam-4827	651	6	,	,	PUNCT
ejpam-4827	651	7	by	by	ADP
ejpam-4827	651	8	[	[	X
ejpam-4827	651	9	4	4	NUM
ejpam-4827	651	10	]	]	PUNCT
ejpam-4827	651	11	,	,	PUNCT
ejpam-4827	651	12	nil(tn(r	nil(tn(r	NOUN
ejpam-4827	651	13	)	)	PUNCT
ejpam-4827	651	14	)	)	PUNCT
ejpam-4827	652	1	=	=	SYM
ejpam-4827	652	2			ADJ
ejpam-4827	652	3	nil(r	nil(r	NOUN
ejpam-4827	652	4	)	)	PUNCT
ejpam-4827	652	5	r	r	NOUN
ejpam-4827	652	6	r	r	NOUN
ejpam-4827	652	7	·	·	PUNCT
ejpam-4827	652	8	·	·	PUNCT
ejpam-4827	652	9	·	·	PUNCT
ejpam-4827	653	1	r	r	NOUN
ejpam-4827	653	2	0	0	NUM
ejpam-4827	653	3	nil(r	nil(r	NOUN
ejpam-4827	653	4	)	)	PUNCT
ejpam-4827	653	5	r	r	NOUN
ejpam-4827	653	6	·	·	PUNCT
ejpam-4827	653	7	·	·	PUNCT
ejpam-4827	653	8	·	·	PUNCT
ejpam-4827	654	1	r	r	NOUN
ejpam-4827	654	2	0	0	NUM
ejpam-4827	654	3	0	0	NUM
ejpam-4827	654	4	nil(r	nil(r	NOUN
ejpam-4827	654	5	)	)	PUNCT
ejpam-4827	654	6	·	·	PUNCT
ejpam-4827	654	7	·	·	PUNCT
ejpam-4827	654	8	·	·	PUNCT
ejpam-4827	655	1	r	r	NOUN
ejpam-4827	655	2	...	...	PUNCT
ejpam-4827	655	3	...	...	PUNCT
ejpam-4827	655	4	...	...	PUNCT
ejpam-4827	655	5	.	.	PUNCT
ejpam-4827	655	6	.	.	PUNCT
ejpam-4827	655	7	.	.	PUNCT
ejpam-4827	656	1	...	...	PUNCT
ejpam-4827	657	1	0	0	NUM
ejpam-4827	657	2	0	0	NUM
ejpam-4827	657	3	0	0	NUM
ejpam-4827	657	4	·	·	PUNCT
ejpam-4827	657	5	·	·	PUNCT
ejpam-4827	657	6	·	·	PUNCT
ejpam-4827	658	1	nil(r	nil(r	X
ejpam-4827	658	2	)	)	PUNCT
ejpam-4827	658	3			NOUN
ejpam-4827	658	4	.	.	PUNCT
ejpam-4827	659	1	assuming	assume	VERB
ejpam-4827	659	2	that	that	SCONJ
ejpam-4827	659	3	r	r	NOUN
ejpam-4827	659	4	is	be	AUX
ejpam-4827	659	5	a	a	DET
ejpam-4827	659	6	reduced	reduce	VERB
ejpam-4827	659	7	,	,	PUNCT
ejpam-4827	659	8	we	we	PRON
ejpam-4827	659	9	know	know	VERB
ejpam-4827	659	10	that	that	PRON
ejpam-4827	659	11	nil(r	nil(r	PROPN
ejpam-4827	659	12	)	)	PUNCT
ejpam-4827	659	13	=	=	SYM
ejpam-4827	659	14	0	0	PUNCT
ejpam-4827	659	15	and	and	CCONJ
ejpam-4827	659	16	that	that	DET
ejpam-4827	659	17	nil(tn(r	nil(tn(r	NOUN
ejpam-4827	659	18	)	)	PUNCT
ejpam-4827	659	19	)	)	PUNCT
ejpam-4827	659	20	is	be	AUX
ejpam-4827	659	21	an	an	DET
ejpam-4827	659	22	ideal	ideal	NOUN
ejpam-4827	659	23	.	.	PUNCT
ejpam-4827	660	1	from	from	ADP
ejpam-4827	660	2	(	(	PUNCT
ejpam-4827	660	3	1	1	NUM
ejpam-4827	660	4	)	)	PUNCT
ejpam-4827	660	5	,	,	PUNCT
ejpam-4827	660	6	it	it	PRON
ejpam-4827	660	7	follows	follow	VERB
ejpam-4827	660	8	that	that	SCONJ
ejpam-4827	660	9	tn(r	tn(r	NOUN
ejpam-4827	660	10	)	)	PUNCT
ejpam-4827	660	11	is	be	AUX
ejpam-4827	660	12	σ	σ	NOUN
ejpam-4827	660	13	-	-	PUNCT
ejpam-4827	660	14	skew	skew	NOUN
ejpam-4827	660	15	strongly	strongly	ADV
ejpam-4827	660	16	m	m	VERB
ejpam-4827	660	17	-nil	-nil	NOUN
ejpam-4827	660	18	-	-	PUNCT
ejpam-4827	660	19	reflexive	reflexive	ADJ
ejpam-4827	660	20	.	.	PUNCT
ejpam-4827	661	1	however	however	ADV
ejpam-4827	661	2	,	,	PUNCT
ejpam-4827	661	3	if	if	SCONJ
ejpam-4827	661	4	we	we	PRON
ejpam-4827	661	5	take	take	VERB
ejpam-4827	661	6	a	a	DET
ejpam-4827	661	7	=	=	NOUN
ejpam-4827	661	8	e12	e12	NOUN
ejpam-4827	661	9	and	and	CCONJ
ejpam-4827	661	10	b	b	NOUN
ejpam-4827	661	11	=	=	X
ejpam-4827	661	12	e22	e22	PROPN
ejpam-4827	661	13	∈mat2(f	∈mat2(f	PRON
ejpam-4827	661	14	)	)	PUNCT
ejpam-4827	661	15	,	,	PUNCT
ejpam-4827	661	16	where	where	SCONJ
ejpam-4827	661	17	f	f	PROPN
ejpam-4827	661	18	is	be	AUX
ejpam-4827	661	19	a	a	DET
ejpam-4827	661	20	field	field	NOUN
ejpam-4827	661	21	,	,	PUNCT
ejpam-4827	661	22	and	and	CCONJ
ejpam-4827	661	23	c	c	ADP
ejpam-4827	661	24	∈mat2(f	∈mat2(f	PRON
ejpam-4827	661	25	)	)	PUNCT
ejpam-4827	661	26	,	,	PUNCT
ejpam-4827	661	27	we	we	PRON
ejpam-4827	661	28	see	see	VERB
ejpam-4827	661	29	that	that	SCONJ
ejpam-4827	661	30	acb	acb	PROPN
ejpam-4827	661	31	is	be	AUX
ejpam-4827	661	32	nilpotent	nilpotent	ADJ
ejpam-4827	661	33	,	,	PUNCT
ejpam-4827	661	34	but	but	CCONJ
ejpam-4827	661	35	bca	bca	PROPN
ejpam-4827	661	36	=	=	PUNCT
ejpam-4827	661	37	e22	e22	PROPN
ejpam-4827	661	38	is	be	AUX
ejpam-4827	661	39	not	not	PART
ejpam-4827	661	40	nilpotent	nilpotent	ADJ
ejpam-4827	661	41	.	.	PUNCT
ejpam-4827	662	1	this	this	PRON
ejpam-4827	662	2	shows	show	VERB
ejpam-4827	662	3	that	that	SCONJ
ejpam-4827	662	4	mat2(f	mat2(f	X
ejpam-4827	662	5	)	)	PUNCT
ejpam-4827	662	6	is	be	AUX
ejpam-4827	662	7	not	not	PART
ejpam-4827	662	8	σ̄-skew	σ̄-skew	PROPN
ejpam-4827	662	9	strongly	strongly	ADV
ejpam-4827	662	10	m	m	VERB
ejpam-4827	662	11	-nil	-nil	NOUN
ejpam-4827	662	12	-	-	PUNCT
ejpam-4827	662	13	reflexive	reflexive	ADJ
ejpam-4827	662	14	.	.	PUNCT
ejpam-4827	663	1	(	(	PUNCT
ejpam-4827	663	2	3	3	NUM
ejpam-4827	663	3	)	)	PUNCT
ejpam-4827	663	4	.	.	PUNCT
ejpam-4827	664	1	by	by	ADP
ejpam-4827	664	2	the	the	DET
ejpam-4827	664	3	same	same	ADJ
ejpam-4827	664	4	argument	argument	NOUN
ejpam-4827	664	5	as	as	ADP
ejpam-4827	664	6	in	in	ADP
ejpam-4827	664	7	example	example	NOUN
ejpam-4827	664	8	2.3	2.3	NUM
ejpam-4827	664	9	[	[	X
ejpam-4827	664	10	13	13	NUM
ejpam-4827	664	11	]	]	PUNCT
ejpam-4827	664	12	.	.	PUNCT
ejpam-4827	665	1	for	for	ADP
ejpam-4827	665	2	a	a	DET
ejpam-4827	665	3	nonzero	nonzero	NOUN
ejpam-4827	665	4	reduced	reduce	VERB
ejpam-4827	665	5	ring	ring	NOUN
ejpam-4827	665	6	s	s	PROPN
ejpam-4827	665	7	,	,	PUNCT
ejpam-4827	665	8	the	the	DET
ejpam-4827	665	9	ring	ring	NOUN
ejpam-4827	665	10	r	r	NOUN
ejpam-4827	665	11	=	=	PUNCT
ejpam-4827	665	12			PUNCT
ejpam-4827	665	13			PROPN
ejpam-4827	665	14	α	α	X
ejpam-4827	665	15	β	β	NOUN
ejpam-4827	665	16	δ	δ	PROPN
ejpam-4827	665	17	0	0	NUM
ejpam-4827	665	18	α	α	PROPN
ejpam-4827	665	19	γ	γ	X
ejpam-4827	665	20	0	0	NUM
ejpam-4827	665	21	0	0	NUM
ejpam-4827	665	22	α	α	PRON
ejpam-4827	665	23			PROPN
ejpam-4827	666	1	|	|	ADV
ejpam-4827	666	2	α	α	NOUN
ejpam-4827	666	3	,	,	PUNCT
ejpam-4827	666	4	β	β	X
ejpam-4827	666	5	,	,	PUNCT
ejpam-4827	666	6	γ	γ	PROPN
ejpam-4827	666	7	∈	∈	NOUN
ejpam-4827	666	8	s	s	PART
ejpam-4827	666	9			NOUN
ejpam-4827	666	10	is	be	AUX
ejpam-4827	666	11	semicommutative	semicommutative	VERB
ejpam-4827	666	12	by	by	ADP
ejpam-4827	666	13	proposition	proposition	NOUN
ejpam-4827	666	14	1.2	1.2	NUM
ejpam-4827	666	15	[	[	SYM
ejpam-4827	666	16	11	11	NUM
ejpam-4827	666	17	]	]	PUNCT
ejpam-4827	666	18	.	.	PUNCT
ejpam-4827	667	1	if	if	SCONJ
ejpam-4827	667	2	we	we	PRON
ejpam-4827	667	3	take	take	VERB
ejpam-4827	667	4	φ	φ	NOUN
ejpam-4827	667	5	=	=	PUNCT
ejpam-4827	668	1	ce23	ce23	PROPN
ejpam-4827	668	2	g	g	NOUN
ejpam-4827	668	3	and	and	CCONJ
ejpam-4827	668	4	ψ	ψ	NOUN
ejpam-4827	668	5	=	=	NOUN
ejpam-4827	668	6	ce12h	ce12h	NOUN
ejpam-4827	668	7	∈	∈	PROPN
ejpam-4827	668	8	sn(r)∗m	sn(r)∗m	NOUN
ejpam-4827	668	9	for	for	ADP
ejpam-4827	668	10	any	any	DET
ejpam-4827	668	11	g	g	NOUN
ejpam-4827	668	12	,	,	PUNCT
ejpam-4827	668	13	h	h	NOUN
ejpam-4827	668	14	∈m	∈m	NOUN
ejpam-4827	668	15	,	,	PUNCT
ejpam-4827	668	16	we	we	PRON
ejpam-4827	668	17	have	have	AUX
ejpam-4827	668	18	φ(sn(r)∗m)ψ	φ(sn(r)∗m)ψ	VERB
ejpam-4827	668	19	=	=	SYM
ejpam-4827	668	20	0	0	NUM
ejpam-4827	668	21	but	but	CCONJ
ejpam-4827	668	22	ψ(sn(r	ψ(sn(r	NUM
ejpam-4827	668	23	)	)	PUNCT
ejpam-4827	668	24	∗m)φ	∗m)φ	PROPN
ejpam-4827	668	25	̸=	̸=	PROPN
ejpam-4827	668	26	0	0	NUM
ejpam-4827	668	27	.	.	PUNCT
ejpam-4827	669	1	therefore	therefore	ADV
ejpam-4827	669	2	,	,	PUNCT
ejpam-4827	669	3	sn(r	sn(r	PRON
ejpam-4827	669	4	)	)	PUNCT
ejpam-4827	669	5	is	be	AUX
ejpam-4827	669	6	not	not	PART
ejpam-4827	669	7	σ	σ	NOUN
ejpam-4827	669	8	-	-	PUNCT
ejpam-4827	669	9	skew	skew	NOUN
ejpam-4827	669	10	strongly	strongly	ADV
ejpam-4827	669	11	m	m	VERB
ejpam-4827	669	12	-reflexive	-reflexive	ADJ
ejpam-4827	669	13	.	.	PUNCT
ejpam-4827	670	1	since	since	SCONJ
ejpam-4827	670	2	r	r	NOUN
ejpam-4827	670	3	is	be	AUX
ejpam-4827	670	4	e.	e.	PROPN
ejpam-4827	670	5	ali	ali	PROPN
ejpam-4827	670	6	/	/	SYM
ejpam-4827	670	7	eur	eur	PROPN
ejpam-4827	670	8	.	.	PUNCT
ejpam-4827	671	1	j.	j.	PROPN
ejpam-4827	671	2	pure	pure	PROPN
ejpam-4827	671	3	appl	appl	PROPN
ejpam-4827	671	4	.	.	PROPN
ejpam-4827	671	5	math	math	PROPN
ejpam-4827	671	6	,	,	PUNCT
ejpam-4827	671	7	16	16	NUM
ejpam-4827	671	8	(	(	PUNCT
ejpam-4827	671	9	3	3	NUM
ejpam-4827	671	10	)	)	PUNCT
ejpam-4827	671	11	(	(	PUNCT
ejpam-4827	671	12	2023	2023	NUM
ejpam-4827	671	13	)	)	PUNCT
ejpam-4827	671	14	,	,	PUNCT
ejpam-4827	671	15	1878	1878	NUM
ejpam-4827	671	16	-	-	SYM
ejpam-4827	671	17	1893	1893	NUM
ejpam-4827	671	18	1891	1891	NUM
ejpam-4827	671	19	reduced	reduce	VERB
ejpam-4827	671	20	,	,	PUNCT
ejpam-4827	671	21	it	it	PRON
ejpam-4827	671	22	follows	follow	VERB
ejpam-4827	671	23	that	that	SCONJ
ejpam-4827	671	24	sn(r	sn(r	PRON
ejpam-4827	671	25	)	)	PUNCT
ejpam-4827	671	26	is	be	AUX
ejpam-4827	671	27	σ	σ	NOUN
ejpam-4827	671	28	-	-	PUNCT
ejpam-4827	671	29	skew	skew	NOUN
ejpam-4827	671	30	strongly	strongly	ADV
ejpam-4827	671	31	m	m	VERB
ejpam-4827	671	32	-nil	-nil	NOUN
ejpam-4827	671	33	-	-	PUNCT
ejpam-4827	671	34	reflexive	reflexive	ADJ
ejpam-4827	671	35	.	.	PUNCT
ejpam-4827	672	1	note	note	VERB
ejpam-4827	672	2	that	that	SCONJ
ejpam-4827	672	3	nil(sn(r	nil(sn(r	NOUN
ejpam-4827	672	4	)	)	PUNCT
ejpam-4827	672	5	)	)	PUNCT
ejpam-4827	673	1	=	=	PRON
ejpam-4827	673	2			NOUN
ejpam-4827	673	3			VERB
ejpam-4827	673	4	a	a	DET
ejpam-4827	673	5	a12	a12	NOUN
ejpam-4827	673	6	a13	a13	NOUN
ejpam-4827	673	7	·	·	PUNCT
ejpam-4827	673	8	·	·	PUNCT
ejpam-4827	673	9	·	·	PUNCT
ejpam-4827	674	1	a1n	a1n	ADP
ejpam-4827	674	2	0	0	NUM
ejpam-4827	674	3	a	a	DET
ejpam-4827	674	4	a23	a23	NOUN
ejpam-4827	674	5	·	·	PUNCT
ejpam-4827	674	6	·	·	PUNCT
ejpam-4827	674	7	·	·	PUNCT
ejpam-4827	674	8	a2n	a2n	PUNCT
ejpam-4827	674	9	0	0	NUM
ejpam-4827	674	10	0	0	NUM
ejpam-4827	674	11	a	a	PRON
ejpam-4827	674	12	·	·	PUNCT
ejpam-4827	674	13	·	·	PUNCT
ejpam-4827	674	14	·	·	PUNCT
ejpam-4827	674	15	a3n	a3n	PROPN
ejpam-4827	674	16	...	...	PUNCT
ejpam-4827	674	17	...	...	PUNCT
ejpam-4827	674	18	...	...	PUNCT
ejpam-4827	674	19	.	.	PUNCT
ejpam-4827	674	20	.	.	PUNCT
ejpam-4827	674	21	.	.	PUNCT
ejpam-4827	675	1	...	...	PUNCT
ejpam-4827	676	1	0	0	NUM
ejpam-4827	676	2	0	0	NUM
ejpam-4827	676	3	0	0	NUM
ejpam-4827	676	4	·	·	PUNCT
ejpam-4827	676	5	·	·	PUNCT
ejpam-4827	676	6	·	·	PUNCT
ejpam-4827	677	1	a	a	DET
ejpam-4827	677	2			PROPN
ejpam-4827	677	3	|	|	NOUN
ejpam-4827	677	4	a	a	DET
ejpam-4827	677	5	∈	∈	PROPN
ejpam-4827	677	6	nil(r	nil(r	NOUN
ejpam-4827	677	7	)	)	PUNCT
ejpam-4827	677	8	,	,	PUNCT
ejpam-4827	677	9	aij	aij	PROPN
ejpam-4827	677	10	∈	∈	PROPN
ejpam-4827	677	11	r	r	NOUN
ejpam-4827	677	12	;	;	PUNCT
ejpam-4827	677	13	1	1	NUM
ejpam-4827	677	14	≤	≤	NOUN
ejpam-4827	677	15	i	i	PRON
ejpam-4827	677	16	,	,	PUNCT
ejpam-4827	677	17	j	j	PROPN
ejpam-4827	677	18	≤	≤	PROPN
ejpam-4827	677	19	n	n	PRON
ejpam-4827	677	20			NOUN
ejpam-4827	677	21	.	.	PUNCT
ejpam-4827	678	1	the	the	DET
ejpam-4827	678	2	ring	ring	NOUN
ejpam-4827	678	3	r	r	NOUN
ejpam-4827	678	4	being	be	AUX
ejpam-4827	678	5	reduced	reduce	VERB
ejpam-4827	678	6	implies	imply	VERB
ejpam-4827	678	7	that	that	SCONJ
ejpam-4827	678	8	nil(sn(r	nil(sn(r	NOUN
ejpam-4827	678	9	)	)	PUNCT
ejpam-4827	678	10	)	)	PUNCT
ejpam-4827	678	11	is	be	AUX
ejpam-4827	678	12	an	an	DET
ejpam-4827	678	13	ideal	ideal	NOUN
ejpam-4827	678	14	.	.	PUNCT
ejpam-4827	679	1	by	by	ADP
ejpam-4827	679	2	(	(	PUNCT
ejpam-4827	679	3	1	1	NUM
ejpam-4827	679	4	)	)	PUNCT
ejpam-4827	679	5	,	,	PUNCT
ejpam-4827	679	6	sn(r	sn(r	NUM
ejpam-4827	679	7	)	)	PUNCT
ejpam-4827	679	8	is	be	AUX
ejpam-4827	679	9	σ	σ	NOUN
ejpam-4827	679	10	-	-	PUNCT
ejpam-4827	679	11	skew	skew	NOUN
ejpam-4827	679	12	strongly	strongly	ADV
ejpam-4827	679	13	m	m	VERB
ejpam-4827	679	14	-nil	-nil	NOUN
ejpam-4827	679	15	-	-	PUNCT
ejpam-4827	679	16	reflexive	reflexive	ADJ
ejpam-4827	679	17	.	.	PUNCT
ejpam-4827	680	1	by	by	ADP
ejpam-4827	680	2	example	example	NOUN
ejpam-4827	680	3	3(2	3(2	NUM
ejpam-4827	680	4	)	)	PUNCT
ejpam-4827	680	5	for	for	ADP
ejpam-4827	680	6	n	n	X
ejpam-4827	680	7	by	by	ADP
ejpam-4827	680	8	n	n	CCONJ
ejpam-4827	680	9	upper	upper	ADJ
ejpam-4827	680	10	triangular	triangular	NOUN
ejpam-4827	680	11	matrix	matrix	NOUN
ejpam-4827	680	12	ring	ring	NOUN
ejpam-4827	680	13	over	over	ADP
ejpam-4827	680	14	r.	r.	PROPN
ejpam-4827	680	15	it	it	PRON
ejpam-4827	680	16	is	be	AUX
ejpam-4827	680	17	easy	easy	ADJ
ejpam-4827	680	18	to	to	PART
ejpam-4827	680	19	verify	verify	VERB
ejpam-4827	680	20	the	the	DET
ejpam-4827	680	21	next	next	ADJ
ejpam-4827	680	22	result	result	NOUN
ejpam-4827	680	23	.	.	PUNCT
ejpam-4827	681	1	proposition	proposition	NOUN
ejpam-4827	681	2	4	4	NUM
ejpam-4827	681	3	.	.	PUNCT
ejpam-4827	682	1	let	let	VERB
ejpam-4827	682	2	m	m	PRON
ejpam-4827	682	3	be	be	AUX
ejpam-4827	682	4	a	a	DET
ejpam-4827	682	5	torsion	torsion	NOUN
ejpam-4827	682	6	-	-	PUNCT
ejpam-4827	682	7	free	free	ADJ
ejpam-4827	682	8	and	and	CCONJ
ejpam-4827	682	9	cancellative	cancellative	ADJ
ejpam-4827	682	10	monoid	monoid	NOUN
ejpam-4827	682	11	and	and	CCONJ
ejpam-4827	682	12	σ	σ	NOUN
ejpam-4827	682	13	:	:	PUNCT
ejpam-4827	682	14	m	m	PROPN
ejpam-4827	682	15	→	→	SYM
ejpam-4827	682	16	aut(r	aut(r	PROPN
ejpam-4827	682	17	)	)	PUNCT
ejpam-4827	682	18	a	a	DET
ejpam-4827	682	19	compatible	compatible	ADJ
ejpam-4827	682	20	monoid	monoid	NOUN
ejpam-4827	682	21	homomorphism	homomorphism	NOUN
ejpam-4827	682	22	.	.	PUNCT
ejpam-4827	683	1	a	a	DET
ejpam-4827	683	2	ring	ring	NOUN
ejpam-4827	683	3	r	r	NOUN
ejpam-4827	683	4	is	be	AUX
ejpam-4827	683	5	σ	σ	NOUN
ejpam-4827	683	6	-	-	PUNCT
ejpam-4827	683	7	skew	skew	NOUN
ejpam-4827	683	8	strongly	strongly	ADV
ejpam-4827	683	9	m	m	VERB
ejpam-4827	683	10	-nil	-nil	NOUN
ejpam-4827	683	11	-	-	PUNCT
ejpam-4827	683	12	reflexive	reflexive	ADJ
ejpam-4827	683	13	if	if	SCONJ
ejpam-4827	683	14	and	and	CCONJ
ejpam-4827	683	15	only	only	ADV
ejpam-4827	683	16	if	if	SCONJ
ejpam-4827	683	17	tn(r	tn(r	NUM
ejpam-4827	683	18	)	)	PUNCT
ejpam-4827	683	19	is	be	AUX
ejpam-4827	683	20	σ	σ	NOUN
ejpam-4827	683	21	-	-	PUNCT
ejpam-4827	683	22	skew	skew	NOUN
ejpam-4827	683	23	strongly	strongly	ADV
ejpam-4827	683	24	m	m	VERB
ejpam-4827	683	25	-nil	-nil	NOUN
ejpam-4827	683	26	-	-	PUNCT
ejpam-4827	683	27	reflexive	reflexive	ADJ
ejpam-4827	683	28	,	,	PUNCT
ejpam-4827	683	29	for	for	ADP
ejpam-4827	683	30	any	any	DET
ejpam-4827	683	31	positive	positive	ADJ
ejpam-4827	683	32	integer	integer	NOUN
ejpam-4827	683	33	n.	n.	NOUN
ejpam-4827	683	34	proof	proof	NOUN
ejpam-4827	683	35	.	.	PUNCT
ejpam-4827	684	1	it	it	PRON
ejpam-4827	684	2	suffices	suffice	VERB
ejpam-4827	684	3	to	to	PART
ejpam-4827	684	4	show	show	VERB
ejpam-4827	684	5	“	"	PUNCT
ejpam-4827	684	6	⇒	⇒	NOUN
ejpam-4827	684	7	”	"	PUNCT
ejpam-4827	684	8	let	let	VERB
ejpam-4827	684	9	φ	φ	NUM
ejpam-4827	684	10	,	,	PUNCT
ejpam-4827	684	11	ψ	ψ	PROPN
ejpam-4827	684	12	∈	∈	PROPN
ejpam-4827	684	13	tn(r	tn(r	PRON
ejpam-4827	684	14	)	)	PUNCT
ejpam-4827	684	15	∗m	∗m	NOUN
ejpam-4827	684	16	such	such	ADJ
ejpam-4827	684	17	that	that	SCONJ
ejpam-4827	684	18	φϕψ	φϕψ	PROPN
ejpam-4827	684	19	∈	∈	PROPN
ejpam-4827	684	20	nil(tn(r	nil(tn(r	NOUN
ejpam-4827	684	21	)	)	PUNCT
ejpam-4827	684	22	)	)	PUNCT
ejpam-4827	685	1	∗	∗	NOUN
ejpam-4827	685	2	m	m	PROPN
ejpam-4827	685	3	,	,	PUNCT
ejpam-4827	685	4	where	where	SCONJ
ejpam-4827	685	5	φ	φ	PROPN
ejpam-4827	685	6	=	=	SYM
ejpam-4827	685	7	(	(	PUNCT
ejpam-4827	685	8	bij	bij	NOUN
ejpam-4827	685	9	)	)	PUNCT
ejpam-4827	685	10	,	,	PUNCT
ejpam-4827	685	11	ϕ	ϕ	X
ejpam-4827	685	12	=	=	SYM
ejpam-4827	685	13	(	(	PUNCT
ejpam-4827	685	14	cij	cij	PROPN
ejpam-4827	685	15	)	)	PUNCT
ejpam-4827	685	16	and	and	CCONJ
ejpam-4827	685	17	ψ	ψ	X
ejpam-4827	685	18	=	=	SYM
ejpam-4827	685	19	(	(	PUNCT
ejpam-4827	685	20	aij	aij	PROPN
ejpam-4827	685	21	)	)	PUNCT
ejpam-4827	685	22	for	for	ADP
ejpam-4827	685	23	all	all	DET
ejpam-4827	685	24	(	(	PUNCT
ejpam-4827	685	25	i	i	NOUN
ejpam-4827	685	26	,	,	PUNCT
ejpam-4827	685	27	j)th	j)th	ADJ
ejpam-4827	685	28	entry	entry	NOUN
ejpam-4827	685	29	of	of	ADP
ejpam-4827	685	30	the	the	DET
ejpam-4827	685	31	matrix	matrix	NOUN
ejpam-4827	685	32	.	.	PUNCT
ejpam-4827	686	1	since	since	SCONJ
ejpam-4827	686	2	nil(tn(r	nil(tn(r	NOUN
ejpam-4827	686	3	)	)	PUNCT
ejpam-4827	686	4	)	)	PUNCT
ejpam-4827	687	1	=	=	PRON
ejpam-4827	687	2	{	{	PUNCT
ejpam-4827	687	3	(	(	PUNCT
ejpam-4827	687	4	aij)|aij	aij)|aij	PROPN
ejpam-4827	687	5	∈	∈	PROPN
ejpam-4827	687	6	nil(r	nil(r	PROPN
ejpam-4827	687	7	)	)	PUNCT
ejpam-4827	687	8	}	}	PUNCT
ejpam-4827	687	9	,	,	PUNCT
ejpam-4827	687	10	then	then	ADV
ejpam-4827	687	11	we	we	PRON
ejpam-4827	687	12	have	have	VERB
ejpam-4827	687	13	cφiiϕiiψii	cφiiϕiiψii	PROPN
ejpam-4827	687	14	∈	∈	PROPN
ejpam-4827	687	15	nil(r	nil(r	PROPN
ejpam-4827	687	16	)	)	PUNCT
ejpam-4827	687	17	for	for	ADP
ejpam-4827	687	18	each	each	DET
ejpam-4827	687	19	1	1	NUM
ejpam-4827	687	20	≤	≤	NUM
ejpam-4827	687	21	i	i	PRON
ejpam-4827	687	22	≤	≤	PROPN
ejpam-4827	687	23	n.	n.	NOUN
ejpam-4827	687	24	since	since	SCONJ
ejpam-4827	687	25	r	r	NOUN
ejpam-4827	687	26	is	be	AUX
ejpam-4827	687	27	σ	σ	NOUN
ejpam-4827	687	28	-	-	PUNCT
ejpam-4827	687	29	skew	skew	NOUN
ejpam-4827	687	30	strongly	strongly	ADV
ejpam-4827	687	31	m	m	VERB
ejpam-4827	687	32	-nil	-nil	NOUN
ejpam-4827	687	33	-	-	PUNCT
ejpam-4827	687	34	reflexive	reflexive	ADJ
ejpam-4827	687	35	,	,	PUNCT
ejpam-4827	687	36	there	there	PRON
ejpam-4827	687	37	exist	exist	VERB
ejpam-4827	687	38	some	some	DET
ejpam-4827	687	39	positive	positive	ADJ
ejpam-4827	687	40	integer	integer	NOUN
ejpam-4827	687	41	mi	mi	PROPN
ejpam-4827	687	42	such	such	ADJ
ejpam-4827	687	43	that	that	PRON
ejpam-4827	687	44	(	(	PUNCT
ejpam-4827	687	45	biiσgii(σs(ciiaii	biiσgii(σs(ciiaii	NOUN
ejpam-4827	687	46	)	)	PUNCT
ejpam-4827	687	47	)	)	PUNCT
ejpam-4827	687	48	)	)	PUNCT
ejpam-4827	687	49	mi	mi	PROPN
ejpam-4827	688	1	=	=	PROPN
ejpam-4827	688	2	0	0	PROPN
ejpam-4827	688	3	.	.	PUNCT
ejpam-4827	689	1	then	then	ADV
ejpam-4827	689	2	,	,	PUNCT
ejpam-4827	689	3	by	by	ADP
ejpam-4827	689	4	compatibility	compatibility	NOUN
ejpam-4827	689	5	biiσgii(ciiaii	biiσgii(ciiaii	NOUN
ejpam-4827	689	6	)	)	PUNCT
ejpam-4827	689	7	is	be	AUX
ejpam-4827	689	8	nilpotent	nilpotent	ADJ
ejpam-4827	689	9	and	and	CCONJ
ejpam-4827	689	10	so	so	ADV
ejpam-4827	689	11	aiiσhii(σs(ciibii	aiiσhii(σs(ciibii	PROPN
ejpam-4827	689	12	is	be	AUX
ejpam-4827	689	13	nilpotent	nilpotent	ADJ
ejpam-4827	689	14	.	.	PUNCT
ejpam-4827	690	1	thus	thus	ADV
ejpam-4827	690	2	,	,	PUNCT
ejpam-4827	690	3	ψϕφ	ψϕφ	PROPN
ejpam-4827	690	4	∈	∈	PROPN
ejpam-4827	690	5	nil(tn(r	nil(tn(r	NOUN
ejpam-4827	690	6	)	)	PUNCT
ejpam-4827	690	7	)	)	PUNCT
ejpam-4827	690	8	∗	∗	NOUN
ejpam-4827	690	9	m.	m.	NOUN
ejpam-4827	690	10	therefore	therefore	ADV
ejpam-4827	690	11	,	,	PUNCT
ejpam-4827	690	12	tn(r	tn(r	NUM
ejpam-4827	690	13	)	)	PUNCT
ejpam-4827	690	14	is	be	AUX
ejpam-4827	690	15	σ	σ	NOUN
ejpam-4827	690	16	-	-	PUNCT
ejpam-4827	690	17	skew	skew	NOUN
ejpam-4827	690	18	strongly	strongly	ADV
ejpam-4827	690	19	m	m	VERB
ejpam-4827	690	20	-nil	-nil	NOUN
ejpam-4827	690	21	-	-	PUNCT
ejpam-4827	690	22	reflexive	reflexive	ADJ
ejpam-4827	690	23	.	.	PUNCT
ejpam-4827	691	1	proposition	proposition	NOUN
ejpam-4827	691	2	5	5	NUM
ejpam-4827	691	3	.	.	PUNCT
ejpam-4827	692	1	let	let	VERB
ejpam-4827	692	2	m	m	PRON
ejpam-4827	692	3	be	be	AUX
ejpam-4827	692	4	a	a	DET
ejpam-4827	692	5	strictly	strictly	ADV
ejpam-4827	692	6	ordered	order	VERB
ejpam-4827	692	7	monoid	monoid	NOUN
ejpam-4827	692	8	and	and	CCONJ
ejpam-4827	692	9	σ	σ	NOUN
ejpam-4827	692	10	:	:	PUNCT
ejpam-4827	692	11	m	m	PROPN
ejpam-4827	692	12	→	→	SYM
ejpam-4827	692	13	aut(r	aut(r	PROPN
ejpam-4827	692	14	)	)	PUNCT
ejpam-4827	692	15	a	a	DET
ejpam-4827	692	16	compatible	compatible	ADJ
ejpam-4827	692	17	monoid	monoid	NOUN
ejpam-4827	692	18	homomorphism	homomorphism	NOUN
ejpam-4827	692	19	.	.	PUNCT
ejpam-4827	693	1	if	if	SCONJ
ejpam-4827	693	2	r	r	NOUN
ejpam-4827	693	3	is	be	AUX
ejpam-4827	693	4	finite	finite	ADJ
ejpam-4827	693	5	subdirect	subdirect	NOUN
ejpam-4827	693	6	product	product	NOUN
ejpam-4827	693	7	of	of	ADP
ejpam-4827	693	8	σ	σ	PROPN
ejpam-4827	693	9	-	-	PUNCT
ejpam-4827	693	10	skew	skew	NOUN
ejpam-4827	693	11	strongly	strongly	ADV
ejpam-4827	693	12	m	m	VERB
ejpam-4827	693	13	-nil	-nil	ADJ
ejpam-4827	693	14	-	-	PUNCT
ejpam-4827	693	15	reflexive	reflexive	ADJ
ejpam-4827	693	16	rings	ring	NOUN
ejpam-4827	693	17	,	,	PUNCT
ejpam-4827	693	18	then	then	ADV
ejpam-4827	693	19	r	r	NOUN
ejpam-4827	693	20	is	be	AUX
ejpam-4827	693	21	σ	σ	NOUN
ejpam-4827	693	22	-	-	PUNCT
ejpam-4827	693	23	skew	skew	NOUN
ejpam-4827	693	24	strongly	strongly	ADV
ejpam-4827	693	25	m	m	VERB
ejpam-4827	693	26	-nil	-nil	NOUN
ejpam-4827	693	27	-	-	PUNCT
ejpam-4827	693	28	reflexive	reflexive	ADJ
ejpam-4827	693	29	.	.	PUNCT
ejpam-4827	694	1	proof	proof	NOUN
ejpam-4827	694	2	.	.	PUNCT
ejpam-4827	695	1	let	let	VERB
ejpam-4827	695	2	ik(k	ik(k	PRON
ejpam-4827	695	3	=	=	NOUN
ejpam-4827	695	4	1	1	NUM
ejpam-4827	695	5	,	,	PUNCT
ejpam-4827	695	6	.	.	PUNCT
ejpam-4827	695	7	.	.	PUNCT
ejpam-4827	695	8	.	.	PUNCT
ejpam-4827	696	1	,	,	PUNCT
ejpam-4827	696	2	l	l	AUX
ejpam-4827	696	3	)	)	PUNCT
ejpam-4827	696	4	be	be	AUX
ejpam-4827	696	5	ideals	ideal	NOUN
ejpam-4827	696	6	of	of	ADP
ejpam-4827	696	7	r	r	NOUN
ejpam-4827	696	8	such	such	ADJ
ejpam-4827	696	9	that	that	PRON
ejpam-4827	696	10	r	r	NOUN
ejpam-4827	696	11	/	/	SYM
ejpam-4827	696	12	ik	ik	PROPN
ejpam-4827	696	13	is	be	AUX
ejpam-4827	696	14	σ	σ	NOUN
ejpam-4827	696	15	-	-	PUNCT
ejpam-4827	696	16	skew	skew	NOUN
ejpam-4827	697	1	strongly	strongly	ADV
ejpam-4827	697	2	m	m	VERB
ejpam-4827	697	3	-nilreflexive	-nilreflexive	ADJ
ejpam-4827	697	4	and	and	CCONJ
ejpam-4827	697	5	⋂l	⋂l	PROPN
ejpam-4827	697	6	k=1	k=1	PROPN
ejpam-4827	698	1	ik	ik	PROPN
ejpam-4827	698	2	=	=	SYM
ejpam-4827	698	3	0	0	X
ejpam-4827	698	4	.	.	PUNCT
ejpam-4827	699	1	let	let	VERB
ejpam-4827	699	2	φ	φ	PROPN
ejpam-4827	699	3	and	and	CCONJ
ejpam-4827	699	4	ψ	ψ	X
ejpam-4827	699	5	be	be	AUX
ejpam-4827	699	6	elements	element	NOUN
ejpam-4827	699	7	in	in	ADP
ejpam-4827	699	8	r	r	NOUN
ejpam-4827	699	9	∗m	∗m	NOUN
ejpam-4827	699	10	such	such	ADJ
ejpam-4827	699	11	that	that	SCONJ
ejpam-4827	699	12	φϕψ	φϕψ	PROPN
ejpam-4827	699	13	∈	∈	PROPN
ejpam-4827	699	14	nil(r	nil(r	PROPN
ejpam-4827	699	15	)	)	PUNCT
ejpam-4827	699	16	∗m	∗m	NOUN
ejpam-4827	699	17	for	for	ADP
ejpam-4827	699	18	all	all	DET
ejpam-4827	699	19	ϕ	ϕ	PROPN
ejpam-4827	699	20	∈	∈	PROPN
ejpam-4827	699	21	r	r	NOUN
ejpam-4827	699	22	∗m	∗m	NOUN
ejpam-4827	699	23	.	.	PUNCT
ejpam-4827	700	1	clearly	clearly	ADV
ejpam-4827	700	2	,	,	PUNCT
ejpam-4827	700	3	φ̄ϕ̄ψ̄	φ̄ϕ̄ψ̄	X
ejpam-4827	700	4	∈	∈	PROPN
ejpam-4827	700	5	nil(r	nil(r	PROPN
ejpam-4827	700	6	/	/	SYM
ejpam-4827	700	7	ik	ik	ADJ
ejpam-4827	700	8	)	)	PUNCT
ejpam-4827	700	9	∗m	∗m	NOUN
ejpam-4827	700	10	.	.	PUNCT
ejpam-4827	701	1	since	since	SCONJ
ejpam-4827	701	2	r	r	PROPN
ejpam-4827	701	3	/	/	SYM
ejpam-4827	701	4	ik	ik	PROPN
ejpam-4827	701	5	is	be	AUX
ejpam-4827	701	6	σ	σ	NOUN
ejpam-4827	701	7	-	-	PUNCT
ejpam-4827	701	8	skew	skew	NOUN
ejpam-4827	701	9	strongly	strongly	ADV
ejpam-4827	701	10	m	m	VERB
ejpam-4827	701	11	nil	nil	ADJ
ejpam-4827	701	12	-	-	PUNCT
ejpam-4827	701	13	reflexive	reflexive	ADJ
ejpam-4827	701	14	,	,	PUNCT
ejpam-4827	701	15	we	we	PRON
ejpam-4827	701	16	have	have	VERB
ejpam-4827	701	17	(	(	PUNCT
ejpam-4827	701	18	biσgi(σs(raj	biσgi(σs(raj	NOUN
ejpam-4827	701	19	)	)	PUNCT
ejpam-4827	701	20	)	)	PUNCT
ejpam-4827	701	21	)	)	PUNCT
ejpam-4827	702	1	ℓ	ℓ	PROPN
ejpam-4827	702	2	∈	∈	PROPN
ejpam-4827	702	3	ik	ik	NOUN
ejpam-4827	702	4	,	,	PUNCT
ejpam-4827	702	5	where	where	SCONJ
ejpam-4827	702	6	σ	σ	PROPN
ejpam-4827	702	7	is	be	AUX
ejpam-4827	702	8	compatible	compatible	ADJ
ejpam-4827	702	9	and	and	CCONJ
ejpam-4827	702	10	r	r	NOUN
ejpam-4827	702	11	∈	∈	PROPN
ejpam-4827	702	12	r	r	NOUN
ejpam-4827	702	13	,	,	PUNCT
ejpam-4827	702	14	for	for	ADP
ejpam-4827	702	15	some	some	DET
ejpam-4827	702	16	positive	positive	ADJ
ejpam-4827	702	17	integer	integer	NOUN
ejpam-4827	702	18	ℓ.	ℓ.	NOUN
ejpam-4827	702	19	therefore	therefore	ADV
ejpam-4827	702	20	,	,	PUNCT
ejpam-4827	702	21	(	(	PUNCT
ejpam-4827	702	22	biσi(σs(raj)))ℓ	biσi(σs(raj)))ℓ	PROPN
ejpam-4827	702	23	∈	∈	NOUN
ejpam-4827	702	24	⋂l	⋂l	NOUN
ejpam-4827	702	25	k=1	k=1	PROPN
ejpam-4827	702	26	ik	ik	PROPN
ejpam-4827	702	27	=	=	SYM
ejpam-4827	702	28	0	0	PROPN
ejpam-4827	702	29	.	.	PUNCT
ejpam-4827	703	1	hence	hence	ADV
ejpam-4827	703	2	,	,	PUNCT
ejpam-4827	703	3	(	(	PUNCT
ejpam-4827	703	4	biσgi(σs(raj)))ℓ	biσgi(σs(raj)))ℓ	NOUN
ejpam-4827	703	5	=	=	SYM
ejpam-4827	703	6	0	0	NUM
ejpam-4827	703	7	,	,	PUNCT
ejpam-4827	703	8	and	and	CCONJ
ejpam-4827	703	9	we	we	PRON
ejpam-4827	703	10	conclude	conclude	VERB
ejpam-4827	703	11	that	that	PRON
ejpam-4827	703	12	biσgi(σs(raj	biσgi(σs(raj	NOUN
ejpam-4827	703	13	)	)	PUNCT
ejpam-4827	703	14	)	)	PUNCT
ejpam-4827	703	15	∈	∈	PROPN
ejpam-4827	703	16	nil(r	nil(r	PROPN
ejpam-4827	703	17	)	)	PUNCT
ejpam-4827	703	18	since	since	SCONJ
ejpam-4827	703	19	σ	σ	PROPN
ejpam-4827	703	20	is	be	AUX
ejpam-4827	703	21	compatible	compatible	ADJ
ejpam-4827	703	22	and	and	CCONJ
ejpam-4827	703	23	nil	nil	ADJ
ejpam-4827	703	24	-	-	PUNCT
ejpam-4827	703	25	reflexivity	reflexivity	NOUN
ejpam-4827	703	26	.	.	PUNCT
ejpam-4827	704	1	then	then	ADV
ejpam-4827	704	2	,	,	PUNCT
ejpam-4827	704	3	ajσhj	ajσhj	PROPN
ejpam-4827	704	4	(	(	PUNCT
ejpam-4827	704	5	σs(rbi	σs(rbi	PROPN
ejpam-4827	704	6	)	)	PUNCT
ejpam-4827	704	7	)	)	PUNCT
ejpam-4827	704	8	∈	∈	PROPN
ejpam-4827	704	9	nil(r	nil(r	PROPN
ejpam-4827	704	10	)	)	PUNCT
ejpam-4827	704	11	.	.	PUNCT
ejpam-4827	705	1	therefore	therefore	ADV
ejpam-4827	705	2	,	,	PUNCT
ejpam-4827	705	3	ψϕφ	ψϕφ	PROPN
ejpam-4827	705	4	∈	∈	PROPN
ejpam-4827	705	5	nil(r	nil(r	NOUN
ejpam-4827	705	6	)	)	PUNCT
ejpam-4827	705	7	∗m	∗m	NOUN
ejpam-4827	705	8	,	,	PUNCT
ejpam-4827	705	9	the	the	DET
ejpam-4827	705	10	proof	proof	NOUN
ejpam-4827	705	11	is	be	AUX
ejpam-4827	705	12	done	do	VERB
ejpam-4827	705	13	.	.	PUNCT
ejpam-4827	706	1	4	4	X
ejpam-4827	706	2	.	.	X
ejpam-4827	706	3	conclusion	conclusion	NOUN
ejpam-4827	706	4	this	this	DET
ejpam-4827	706	5	paper	paper	NOUN
ejpam-4827	706	6	introduced	introduce	VERB
ejpam-4827	706	7	and	and	CCONJ
ejpam-4827	706	8	studied	study	VERB
ejpam-4827	706	9	two	two	NUM
ejpam-4827	706	10	important	important	ADJ
ejpam-4827	706	11	concepts	concept	NOUN
ejpam-4827	706	12	,	,	PUNCT
ejpam-4827	706	13	namely	namely	ADV
ejpam-4827	706	14	σ	σ	X
ejpam-4827	706	15	-	-	PUNCT
ejpam-4827	706	16	skew	skew	NOUN
ejpam-4827	706	17	strongly	strongly	ADV
ejpam-4827	706	18	m	m	VERB
ejpam-4827	706	19	-reflexive	-reflexive	ADJ
ejpam-4827	706	20	and	and	CCONJ
ejpam-4827	706	21	σ	σ	NOUN
ejpam-4827	706	22	-	-	PUNCT
ejpam-4827	706	23	skew	skew	NOUN
ejpam-4827	706	24	strongly	strongly	ADV
ejpam-4827	706	25	m	m	VERB
ejpam-4827	706	26	-nil	-nil	NOUN
ejpam-4827	706	27	-	-	PUNCT
ejpam-4827	706	28	reflexive	reflexive	ADJ
ejpam-4827	706	29	.	.	PUNCT
ejpam-4827	707	1	the	the	DET
ejpam-4827	707	2	study	study	NOUN
ejpam-4827	707	3	covered	cover	VERB
ejpam-4827	707	4	the	the	DET
ejpam-4827	707	5	fundamental	fundamental	ADJ
ejpam-4827	707	6	properties	property	NOUN
ejpam-4827	707	7	of	of	ADP
ejpam-4827	707	8	skew	skew	ADJ
ejpam-4827	707	9	monoid	monoid	NOUN
ejpam-4827	707	10	rings	ring	NOUN
ejpam-4827	707	11	of	of	ADP
ejpam-4827	707	12	the	the	DET
ejpam-4827	707	13	form	form	NOUN
ejpam-4827	707	14	r	r	NOUN
ejpam-4827	707	15	∗	∗	NOUN
ejpam-4827	707	16	m	m	PRON
ejpam-4827	707	17	,	,	PUNCT
ejpam-4827	707	18	and	and	CCONJ
ejpam-4827	707	19	established	establish	VERB
ejpam-4827	707	20	several	several	ADJ
ejpam-4827	707	21	important	important	ADJ
ejpam-4827	707	22	results	result	NOUN
ejpam-4827	707	23	.	.	PUNCT
ejpam-4827	708	1	the	the	DET
ejpam-4827	708	2	study	study	NOUN
ejpam-4827	708	3	provided	provide	VERB
ejpam-4827	708	4	some	some	DET
ejpam-4827	708	5	examples	example	NOUN
ejpam-4827	708	6	and	and	CCONJ
ejpam-4827	708	7	discussed	discuss	VERB
ejpam-4827	708	8	related	related	ADJ
ejpam-4827	708	9	results	result	NOUN
ejpam-4827	708	10	from	from	ADP
ejpam-4827	708	11	the	the	DET
ejpam-4827	708	12	subject	subject	NOUN
ejpam-4827	708	13	.	.	PUNCT
ejpam-4827	709	1	overall	overall	ADV
ejpam-4827	709	2	,	,	PUNCT
ejpam-4827	709	3	our	our	PRON
ejpam-4827	709	4	study	study	NOUN
ejpam-4827	709	5	provides	provide	VERB
ejpam-4827	709	6	important	important	ADJ
ejpam-4827	709	7	insights	insight	NOUN
ejpam-4827	709	8	into	into	ADP
ejpam-4827	709	9	the	the	DET
ejpam-4827	709	10	properties	property	NOUN
ejpam-4827	709	11	of	of	ADP
ejpam-4827	709	12	skew	skew	ADJ
ejpam-4827	709	13	monoid	monoid	NOUN
ejpam-4827	709	14	rings	ring	NOUN
ejpam-4827	709	15	references	reference	NOUN
ejpam-4827	709	16	1892	1892	NUM
ejpam-4827	709	17	and	and	CCONJ
ejpam-4827	709	18	their	their	PRON
ejpam-4827	709	19	relationship	relationship	NOUN
ejpam-4827	709	20	to	to	ADP
ejpam-4827	709	21	nilpotent	nilpotent	ADJ
ejpam-4827	709	22	elements	element	NOUN
ejpam-4827	709	23	.	.	PUNCT
ejpam-4827	710	1	we	we	PRON
ejpam-4827	710	2	anticipate	anticipate	VERB
ejpam-4827	710	3	that	that	SCONJ
ejpam-4827	710	4	our	our	PRON
ejpam-4827	710	5	findings	finding	NOUN
ejpam-4827	710	6	will	will	AUX
ejpam-4827	710	7	have	have	VERB
ejpam-4827	710	8	significant	significant	ADJ
ejpam-4827	710	9	implications	implication	NOUN
ejpam-4827	710	10	for	for	ADP
ejpam-4827	710	11	the	the	DET
ejpam-4827	710	12	theory	theory	NOUN
ejpam-4827	710	13	of	of	ADP
ejpam-4827	710	14	noncommutative	noncommutative	ADJ
ejpam-4827	710	15	algebra	algebra	PROPN
ejpam-4827	710	16	,	,	PUNCT
ejpam-4827	710	17	and	and	CCONJ
ejpam-4827	710	18	will	will	AUX
ejpam-4827	710	19	lead	lead	VERB
ejpam-4827	710	20	to	to	ADP
ejpam-4827	710	21	further	further	ADJ
ejpam-4827	710	22	research	research	NOUN
ejpam-4827	710	23	in	in	ADP
ejpam-4827	710	24	this	this	DET
ejpam-4827	710	25	area	area	NOUN
ejpam-4827	710	26	.	.	PUNCT
ejpam-4827	711	1	acknowledgements	acknowledgement	VERB
ejpam-4827	711	2	the	the	DET
ejpam-4827	711	3	author	author	NOUN
ejpam-4827	711	4	would	would	AUX
ejpam-4827	711	5	like	like	VERB
ejpam-4827	711	6	to	to	PART
ejpam-4827	711	7	express	express	VERB
ejpam-4827	711	8	their	their	PRON
ejpam-4827	711	9	sincere	sincere	ADJ
ejpam-4827	711	10	appreciation	appreciation	NOUN
ejpam-4827	711	11	and	and	CCONJ
ejpam-4827	711	12	gratitude	gratitude	NOUN
ejpam-4827	711	13	to	to	ADP
ejpam-4827	711	14	the	the	DET
ejpam-4827	711	15	referees	referee	NOUN
ejpam-4827	711	16	and	and	CCONJ
ejpam-4827	711	17	the	the	DET
ejpam-4827	711	18	editors	editor	NOUN
ejpam-4827	711	19	for	for	ADP
ejpam-4827	711	20	their	their	PRON
ejpam-4827	711	21	valuable	valuable	ADJ
ejpam-4827	711	22	comments	comment	NOUN
ejpam-4827	711	23	and	and	CCONJ
ejpam-4827	711	24	suggestions	suggestion	NOUN
ejpam-4827	711	25	,	,	PUNCT
ejpam-4827	711	26	which	which	PRON
ejpam-4827	711	27	greatly	greatly	ADV
ejpam-4827	711	28	improved	improve	VERB
ejpam-4827	711	29	the	the	DET
ejpam-4827	711	30	quality	quality	NOUN
ejpam-4827	711	31	of	of	ADP
ejpam-4827	711	32	this	this	DET
ejpam-4827	711	33	paper	paper	NOUN
ejpam-4827	711	34	.	.	PUNCT
ejpam-4827	712	1	references	reference	NOUN
ejpam-4827	712	2	[	[	X
ejpam-4827	712	3	1	1	X
ejpam-4827	712	4	]	]	PUNCT
ejpam-4827	712	5	e.	e.	PROPN
ejpam-4827	712	6	ali	ali	PROPN
ejpam-4827	712	7	.	.	PROPN
ejpam-4827	713	1	nilpotent	nilpotent	ADJ
ejpam-4827	713	2	elements	element	NOUN
ejpam-4827	713	3	and	and	CCONJ
ejpam-4827	713	4	nil	nil	ADJ
ejpam-4827	713	5	-	-	PUNCT
ejpam-4827	713	6	reflexive	reflexive	ADJ
ejpam-4827	713	7	property	property	NOUN
ejpam-4827	713	8	of	of	ADP
ejpam-4827	713	9	generalized	generalized	ADJ
ejpam-4827	713	10	power	power	NOUN
ejpam-4827	713	11	series	series	PROPN
ejpam-4827	713	12	rings	ring	NOUN
ejpam-4827	713	13	.	.	PUNCT
ejpam-4827	714	1	adv	adv	PROPN
ejpam-4827	714	2	.	.	PUNCT
ejpam-4827	715	1	p.	p.	NOUN
ejpam-4827	715	2	math	math	PROPN
ejpam-4827	715	3	,	,	PUNCT
ejpam-4827	715	4	12(11):676–692	12(11):676–692	NUM
ejpam-4827	715	5	,	,	PUNCT
ejpam-4827	715	6	2022	2022	NUM
ejpam-4827	715	7	.	.	PUNCT
ejpam-4827	716	1	[	[	X
ejpam-4827	716	2	2	2	X
ejpam-4827	716	3	]	]	PUNCT
ejpam-4827	716	4	e.	e.	PROPN
ejpam-4827	716	5	ali	ali	PROPN
ejpam-4827	716	6	.	.	PUNCT
ejpam-4827	717	1	a	a	DET
ejpam-4827	717	2	note	note	NOUN
ejpam-4827	717	3	on	on	ADP
ejpam-4827	717	4	skew	skew	ADJ
ejpam-4827	717	5	generalized	generalized	ADJ
ejpam-4827	717	6	power	power	NOUN
ejpam-4827	717	7	serieswise	serieswise	NOUN
ejpam-4827	717	8	reversible	reversible	ADJ
ejpam-4827	717	9	property	property	NOUN
ejpam-4827	717	10	.	.	PUNCT
ejpam-4827	718	1	int	int	NOUN
ejpam-4827	718	2	.	.	PUNCT
ejpam-4827	719	1	j.	j.	PROPN
ejpam-4827	719	2	anal	anal	PROPN
ejpam-4827	719	3	.	.	PUNCT
ejpam-4827	720	1	appl	appl	PROPN
ejpam-4827	720	2	.	.	PROPN
ejpam-4827	720	3	,	,	PUNCT
ejpam-4827	720	4	21:69	21:69	NUM
ejpam-4827	720	5	,	,	PUNCT
ejpam-4827	720	6	2023	2023	NUM
ejpam-4827	720	7	.	.	PUNCT
ejpam-4827	721	1	[	[	X
ejpam-4827	721	2	3	3	X
ejpam-4827	721	3	]	]	X
ejpam-4827	721	4	e.	e.	PROPN
ejpam-4827	721	5	ali	ali	PROPN
ejpam-4827	721	6	.	.	PROPN
ejpam-4827	722	1	on	on	ADP
ejpam-4827	722	2	nil	nil	ADJ
ejpam-4827	722	3	skew	skew	ADJ
ejpam-4827	722	4	generalized	generalize	VERB
ejpam-4827	722	5	power	power	NOUN
ejpam-4827	722	6	series	series	PROPN
ejpam-4827	722	7	reflexive	reflexive	ADJ
ejpam-4827	722	8	rings	ring	NOUN
ejpam-4827	722	9	.	.	PUNCT
ejpam-4827	723	1	adv	adv	PROPN
ejpam-4827	723	2	.	.	PUNCT
ejpam-4827	723	3	math	math	PROPN
ejpam-4827	723	4	.	.	PUNCT
ejpam-4827	724	1	sci	sci	PROPN
ejpam-4827	724	2	.	.	PUNCT
ejpam-4827	725	1	j.	j.	PROPN
ejpam-4827	725	2	,	,	PUNCT
ejpam-4827	725	3	12(1):287–306	12(1):287–306	PROPN
ejpam-4827	725	4	,	,	PUNCT
ejpam-4827	725	5	2023	2023	NUM
ejpam-4827	725	6	.	.	PUNCT
ejpam-4827	726	1	[	[	X
ejpam-4827	726	2	4	4	NUM
ejpam-4827	726	3	]	]	PUNCT
ejpam-4827	726	4	r.	r.	PROPN
ejpam-4827	726	5	antoine	antoine	PROPN
ejpam-4827	726	6	.	.	PUNCT
ejpam-4827	727	1	nilpotent	nilpotent	ADJ
ejpam-4827	727	2	elements	element	NOUN
ejpam-4827	727	3	and	and	CCONJ
ejpam-4827	727	4	armendariz	armendariz	ADJ
ejpam-4827	727	5	rings	ring	NOUN
ejpam-4827	727	6	.	.	PUNCT
ejpam-4827	728	1	j.	j.	PROPN
ejpam-4827	728	2	algebra	algebra	PROPN
ejpam-4827	728	3	,	,	PUNCT
ejpam-4827	728	4	319(8):3128–3140	319(8):3128–3140	NUM
ejpam-4827	728	5	,	,	PUNCT
ejpam-4827	728	6	2008	2008	NUM
ejpam-4827	728	7	.	.	PUNCT
ejpam-4827	729	1	[	[	X
ejpam-4827	729	2	5	5	X
ejpam-4827	729	3	]	]	PUNCT
ejpam-4827	729	4	g.	g.	PROPN
ejpam-4827	729	5	f.	f.	PROPN
ejpam-4827	729	6	birkenmeier	birkenmeier	PROPN
ejpam-4827	729	7	and	and	CCONJ
ejpam-4827	729	8	j.	j.	PROPN
ejpam-4827	729	9	k.	k.	PROPN
ejpam-4827	729	10	park	park	PROPN
ejpam-4827	729	11	.	.	PUNCT
ejpam-4827	730	1	triangular	triangular	NOUN
ejpam-4827	730	2	matrix	matrix	NOUN
ejpam-4827	730	3	representation	representation	NOUN
ejpam-4827	730	4	of	of	ADP
ejpam-4827	730	5	ring	ring	NOUN
ejpam-4827	730	6	extensions	extension	NOUN
ejpam-4827	730	7	.	.	PUNCT
ejpam-4827	731	1	j.	j.	PROPN
ejpam-4827	731	2	algebra	algebra	PROPN
ejpam-4827	731	3	,	,	PUNCT
ejpam-4827	731	4	265:457–477	265:457–477	NUM
ejpam-4827	731	5	,	,	PUNCT
ejpam-4827	731	6	2003	2003	NUM
ejpam-4827	731	7	.	.	PUNCT
ejpam-4827	732	1	[	[	X
ejpam-4827	732	2	6	6	NUM
ejpam-4827	732	3	]	]	PUNCT
ejpam-4827	732	4	r.	r.	PROPN
ejpam-4827	732	5	c.	c.	PROPN
ejpam-4827	732	6	courter	courter	PROPN
ejpam-4827	732	7	.	.	PUNCT
ejpam-4827	733	1	rings	ring	NOUN
ejpam-4827	733	2	all	all	PRON
ejpam-4827	733	3	of	of	ADP
ejpam-4827	733	4	whose	whose	DET
ejpam-4827	733	5	factor	factor	NOUN
ejpam-4827	733	6	rings	ring	NOUN
ejpam-4827	733	7	are	be	AUX
ejpam-4827	733	8	semiprime	semiprime	NOUN
ejpam-4827	733	9	.	.	PUNCT
ejpam-4827	734	1	canad	canad	PROPN
ejpam-4827	734	2	.	.	PUNCT
ejpam-4827	735	1	math	math	NOUN
ejpam-4827	735	2	.	.	PUNCT
ejpam-4827	736	1	bull	bull	PROPN
ejpam-4827	736	2	.	.	PUNCT
ejpam-4827	736	3	,	,	PUNCT
ejpam-4827	736	4	12(4):417–426	12(4):417–426	NOUN
ejpam-4827	736	5	,	,	PUNCT
ejpam-4827	736	6	1969	1969	NUM
ejpam-4827	736	7	.	.	PUNCT
ejpam-4827	737	1	[	[	X
ejpam-4827	737	2	7	7	X
ejpam-4827	737	3	]	]	X
ejpam-4827	737	4	m.	m.	NOUN
ejpam-4827	737	5	habibi	habibi	PROPN
ejpam-4827	737	6	and	and	CCONJ
ejpam-4827	737	7	a.	a.	NOUN
ejpam-4827	737	8	moussavi	moussavi	PROPN
ejpam-4827	737	9	.	.	PUNCT
ejpam-4827	738	1	nilpotent	nilpotent	ADJ
ejpam-4827	738	2	elements	element	NOUN
ejpam-4827	738	3	and	and	CCONJ
ejpam-4827	738	4	nil	nil	ADJ
ejpam-4827	738	5	-	-	PUNCT
ejpam-4827	738	6	armendariz	armendariz	ADJ
ejpam-4827	738	7	property	property	NOUN
ejpam-4827	738	8	of	of	ADP
ejpam-4827	738	9	monoid	monoid	NOUN
ejpam-4827	738	10	rings	ring	NOUN
ejpam-4827	738	11	.	.	PUNCT
ejpam-4827	739	1	j.	j.	PROPN
ejpam-4827	739	2	algebra	algebra	PROPN
ejpam-4827	739	3	and	and	CCONJ
ejpam-4827	739	4	its	its	PRON
ejpam-4827	739	5	applications	application	NOUN
ejpam-4827	739	6	,	,	PUNCT
ejpam-4827	739	7	11(4):1250080	11(4):1250080	NUM
ejpam-4827	739	8	,	,	PUNCT
ejpam-4827	739	9	2012	2012	NUM
ejpam-4827	739	10	.	.	PUNCT
ejpam-4827	740	1	[	[	X
ejpam-4827	740	2	8	8	NUM
ejpam-4827	740	3	]	]	X
ejpam-4827	740	4	e.	e.	PROPN
ejpam-4827	740	5	hashemi	hashemi	PROPN
ejpam-4827	740	6	and	and	CCONJ
ejpam-4827	740	7	a.	a.	NOUN
ejpam-4827	740	8	moussavi	moussavi	PROPN
ejpam-4827	740	9	.	.	PUNCT
ejpam-4827	741	1	polynomial	polynomial	ADJ
ejpam-4827	741	2	extensions	extension	NOUN
ejpam-4827	741	3	of	of	ADP
ejpam-4827	741	4	quasi	quasi	ADJ
ejpam-4827	741	5	-	-	PROPN
ejpam-4827	741	6	baer	baer	PROPN
ejpam-4827	741	7	rings	ring	NOUN
ejpam-4827	741	8	.	.	PUNCT
ejpam-4827	742	1	acta	acta	PROPN
ejpam-4827	742	2	math	math	PROPN
ejpam-4827	742	3	.	.	PUNCT
ejpam-4827	743	1	hungar	hungar	PROPN
ejpam-4827	743	2	.	.	PUNCT
ejpam-4827	743	3	,	,	PUNCT
ejpam-4827	743	4	107(3):207–224	107(3):207–224	NUM
ejpam-4827	743	5	,	,	PUNCT
ejpam-4827	743	6	2005	2005	NUM
ejpam-4827	743	7	.	.	PUNCT
ejpam-4827	744	1	[	[	X
ejpam-4827	744	2	9	9	NUM
ejpam-4827	744	3	]	]	X
ejpam-4827	744	4	y.	y.	PROPN
ejpam-4827	744	5	hirano	hirano	PROPN
ejpam-4827	744	6	.	.	PUNCT
ejpam-4827	745	1	on	on	ADP
ejpam-4827	745	2	annihilator	annihilator	PROPN
ejpam-4827	745	3	ideals	ideal	NOUN
ejpam-4827	745	4	of	of	ADP
ejpam-4827	745	5	a	a	DET
ejpam-4827	745	6	polynomial	polynomial	ADJ
ejpam-4827	745	7	ring	ring	NOUN
ejpam-4827	745	8	over	over	ADP
ejpam-4827	745	9	a	a	DET
ejpam-4827	745	10	noncommutative	noncommutative	ADJ
ejpam-4827	745	11	ring	ring	NOUN
ejpam-4827	745	12	.	.	PUNCT
ejpam-4827	746	1	j.	j.	PROPN
ejpam-4827	746	2	pure	pure	PROPN
ejpam-4827	746	3	appl	appl	PROPN
ejpam-4827	746	4	.	.	PUNCT
ejpam-4827	747	1	algebra	algebra	PROPN
ejpam-4827	747	2	,	,	PUNCT
ejpam-4827	747	3	168:45–52	168:45–52	NUM
ejpam-4827	747	4	,	,	PUNCT
ejpam-4827	747	5	2002	2002	NUM
ejpam-4827	747	6	.	.	PUNCT
ejpam-4827	748	1	[	[	X
ejpam-4827	748	2	10	10	NUM
ejpam-4827	748	3	]	]	X
ejpam-4827	748	4	n.	n.	PROPN
ejpam-4827	748	5	k.	k.	PROPN
ejpam-4827	748	6	kim	kim	PROPN
ejpam-4827	748	7	and	and	CCONJ
ejpam-4827	748	8	y.	y.	PROPN
ejpam-4827	748	9	lee	lee	PROPN
ejpam-4827	748	10	.	.	PUNCT
ejpam-4827	749	1	extensions	extension	NOUN
ejpam-4827	749	2	of	of	ADP
ejpam-4827	749	3	reversible	reversible	ADJ
ejpam-4827	749	4	rings	ring	NOUN
ejpam-4827	749	5	.	.	PUNCT
ejpam-4827	750	1	j.	j.	PROPN
ejpam-4827	750	2	pure	pure	PROPN
ejpam-4827	750	3	appl	appl	PROPN
ejpam-4827	750	4	.	.	PUNCT
ejpam-4827	751	1	algebra	algebra	PROPN
ejpam-4827	751	2	,	,	PUNCT
ejpam-4827	751	3	185:207	185:207	NOUN
ejpam-4827	751	4	–	–	PUNCT
ejpam-4827	751	5	223	223	NUM
ejpam-4827	751	6	,	,	PUNCT
ejpam-4827	751	7	2003	2003	NUM
ejpam-4827	751	8	.	.	PUNCT
ejpam-4827	752	1	[	[	X
ejpam-4827	752	2	11	11	NUM
ejpam-4827	752	3	]	]	X
ejpam-4827	752	4	n.	n.	PROPN
ejpam-4827	752	5	k.	k.	PROPN
ejpam-4827	752	6	kim	kim	PROPN
ejpam-4827	752	7	and	and	CCONJ
ejpam-4827	752	8	y.	y.	PROPN
ejpam-4827	752	9	lee	lee	PROPN
ejpam-4827	752	10	.	.	PUNCT
ejpam-4827	752	11	extensions	extension	NOUN
ejpam-4827	752	12	of	of	ADP
ejpam-4827	752	13	reversible	reversible	ADJ
ejpam-4827	752	14	rings	ring	NOUN
ejpam-4827	752	15	.	.	PUNCT
ejpam-4827	753	1	j.	j.	PROPN
ejpam-4827	753	2	pure	pure	PROPN
ejpam-4827	753	3	appl	appl	PROPN
ejpam-4827	753	4	.	.	PUNCT
ejpam-4827	754	1	algebra	algebra	PROPN
ejpam-4827	754	2	,	,	PUNCT
ejpam-4827	754	3	185:207	185:207	NOUN
ejpam-4827	754	4	–	–	PUNCT
ejpam-4827	754	5	223	223	NUM
ejpam-4827	754	6	,	,	PUNCT
ejpam-4827	754	7	2003	2003	NUM
ejpam-4827	754	8	.	.	PUNCT
ejpam-4827	755	1	[	[	X
ejpam-4827	755	2	12	12	NUM
ejpam-4827	755	3	]	]	PUNCT
ejpam-4827	755	4	t.	t.	PROPN
ejpam-4827	755	5	kwak	kwak	PROPN
ejpam-4827	755	6	,	,	PUNCT
ejpam-4827	755	7	y.	y.	PROPN
ejpam-4827	755	8	lee	lee	PROPN
ejpam-4827	755	9	,	,	PUNCT
ejpam-4827	755	10	and	and	CCONJ
ejpam-4827	755	11	s.	s.	PROPN
ejpam-4827	755	12	yun	yun	PROPN
ejpam-4827	755	13	.	.	PUNCT
ejpam-4827	756	1	reflexive	reflexive	ADJ
ejpam-4827	756	2	property	property	NOUN
ejpam-4827	756	3	skewd	skewd	NOUN
ejpam-4827	756	4	by	by	ADP
ejpam-4827	756	5	endomorphisms	endomorphism	NOUN
ejpam-4827	756	6	.	.	PUNCT
ejpam-4827	757	1	korean	korean	PROPN
ejpam-4827	757	2	j.	j.	PROPN
ejpam-4827	757	3	math	math	PROPN
ejpam-4827	757	4	.	.	PUNCT
ejpam-4827	757	5	,	,	PUNCT
ejpam-4827	757	6	22(2):217–234	22(2):217–234	PROPN
ejpam-4827	757	7	,	,	PUNCT
ejpam-4827	757	8	2014	2014	NUM
ejpam-4827	757	9	.	.	PUNCT
ejpam-4827	758	1	references	reference	NOUN
ejpam-4827	758	2	1893	1893	NUM
ejpam-4827	759	1	[	[	X
ejpam-4827	759	2	13	13	NUM
ejpam-4827	759	3	]	]	PUNCT
ejpam-4827	759	4	t.	t.	PROPN
ejpam-4827	759	5	k.	k.	PROPN
ejpam-4827	759	6	kwak	kwak	PROPN
ejpam-4827	759	7	and	and	CCONJ
ejpam-4827	759	8	y.	y.	PROPN
ejpam-4827	759	9	lee	lee	PROPN
ejpam-4827	759	10	.	.	PUNCT
ejpam-4827	760	1	reflexive	reflexive	ADJ
ejpam-4827	760	2	property	property	NOUN
ejpam-4827	760	3	of	of	ADP
ejpam-4827	760	4	rings	ring	NOUN
ejpam-4827	760	5	.	.	PUNCT
ejpam-4827	761	1	comm	comm	NOUN
ejpam-4827	761	2	.	.	PUNCT
ejpam-4827	762	1	algebra	algebra	PROPN
ejpam-4827	762	2	,	,	PUNCT
ejpam-4827	762	3	40:1576–1594	40:1576–1594	PROPN
ejpam-4827	762	4	,	,	PUNCT
ejpam-4827	762	5	2012	2012	NUM
ejpam-4827	762	6	.	.	PUNCT
ejpam-4827	763	1	[	[	X
ejpam-4827	763	2	14	14	NUM
ejpam-4827	763	3	]	]	PUNCT
ejpam-4827	763	4	z.	z.	PROPN
ejpam-4827	763	5	k.	k.	PROPN
ejpam-4827	763	6	liu	liu	PROPN
ejpam-4827	763	7	.	.	PUNCT
ejpam-4827	764	1	armendariz	armendariz	PROPN
ejpam-4827	764	2	rings	ring	NOUN
ejpam-4827	764	3	relative	relative	ADJ
ejpam-4827	764	4	to	to	ADP
ejpam-4827	764	5	a	a	DET
ejpam-4827	764	6	monoid	monoid	NOUN
ejpam-4827	764	7	.	.	PUNCT
ejpam-4827	764	8	comm	comm	NOUN
ejpam-4827	764	9	.	.	PUNCT
ejpam-4827	765	1	algebra	algebra	NOUN
ejpam-4827	765	2	,	,	PUNCT
ejpam-4827	765	3	33(3):649–661	33(3):649–661	PROPN
ejpam-4827	765	4	,	,	PUNCT
ejpam-4827	765	5	2005	2005	NUM
ejpam-4827	765	6	.	.	PUNCT
ejpam-4827	766	1	[	[	X
ejpam-4827	766	2	15	15	NUM
ejpam-4827	766	3	]	]	X
ejpam-4827	766	4	g.	g.	PROPN
ejpam-4827	766	5	mason	mason	PROPN
ejpam-4827	766	6	.	.	PUNCT
ejpam-4827	767	1	reflexive	reflexive	ADJ
ejpam-4827	767	2	ideals	ideal	NOUN
ejpam-4827	767	3	.	.	PUNCT
ejpam-4827	768	1	comm	comm	NOUN
ejpam-4827	768	2	.	.	PUNCT
ejpam-4827	769	1	algebra	algebra	NOUN
ejpam-4827	769	2	,	,	PUNCT
ejpam-4827	769	3	9(17):1709–1724	9(17):1709–1724	NUM
ejpam-4827	769	4	,	,	PUNCT
ejpam-4827	769	5	1981	1981	NUM
ejpam-4827	769	6	.	.	PUNCT
ejpam-4827	770	1	[	[	X
ejpam-4827	770	2	16	16	NUM
ejpam-4827	770	3	]	]	PUNCT
ejpam-4827	770	4	a.	a.	PROPN
ejpam-4827	770	5	r.	r.	PROPN
ejpam-4827	770	6	nasr	nasr	PROPN
ejpam-4827	770	7	-	-	PUNCT
ejpam-4827	770	8	isfahani	isfahani	PROPN
ejpam-4827	770	9	and	and	CCONJ
ejpam-4827	770	10	a.	a.	NOUN
ejpam-4827	770	11	moussavi	moussavi	PROPN
ejpam-4827	770	12	.	.	PUNCT
ejpam-4827	771	1	on	on	ADP
ejpam-4827	771	2	weakly	weakly	ADJ
ejpam-4827	771	3	rigid	rigid	ADJ
ejpam-4827	771	4	rings	ring	NOUN
ejpam-4827	771	5	.	.	PUNCT
ejpam-4827	771	6	glasg	glasg	PROPN
ejpam-4827	771	7	.	.	PUNCT
ejpam-4827	772	1	math	math	NOUN
ejpam-4827	772	2	.	.	PUNCT
ejpam-4827	773	1	j.	j.	PROPN
ejpam-4827	773	2	,	,	PUNCT
ejpam-4827	773	3	51(3):425–440	51(3):425–440	PROPN
ejpam-4827	773	4	,	,	PUNCT
ejpam-4827	773	5	2009	2009	NUM
ejpam-4827	773	6	.	.	PUNCT
ejpam-4827	774	1	[	[	X
ejpam-4827	774	2	17	17	NUM
ejpam-4827	774	3	]	]	X
ejpam-4827	774	4	j.	j.	PROPN
ejpam-4827	774	5	okninski	okninski	PROPN
ejpam-4827	774	6	.	.	PUNCT
ejpam-4827	775	1	semigroup	semigroup	PROPN
ejpam-4827	775	2	algebra	algebra	PROPN
ejpam-4827	775	3	.	.	PUNCT
ejpam-4827	776	1	w.	w.	PROPN
ejpam-4827	776	2	a.	a.	PROPN
ejpam-4827	776	3	benjamin	benjamin	PROPN
ejpam-4827	776	4	,	,	PUNCT
ejpam-4827	776	5	new	new	PROPN
ejpam-4827	776	6	york	york	PROPN
ejpam-4827	776	7	,	,	PUNCT
ejpam-4827	776	8	1991	1991	NUM
ejpam-4827	776	9	.	.	PUNCT
ejpam-4827	777	1	[	[	X
ejpam-4827	777	2	18	18	NUM
ejpam-4827	777	3	]	]	X
ejpam-4827	777	4	l.	l.	PROPN
ejpam-4827	777	5	ouyang	ouyang	PROPN
ejpam-4827	777	6	.	.	PUNCT
ejpam-4827	778	1	on	on	ADP
ejpam-4827	778	2	weak	weak	ADJ
ejpam-4827	778	3	annihilator	annihilator	NOUN
ejpam-4827	778	4	ideals	ideal	NOUN
ejpam-4827	778	5	of	of	ADP
ejpam-4827	778	6	skew	skew	ADJ
ejpam-4827	778	7	monoid	monoid	NOUN
ejpam-4827	778	8	rings	ring	NOUN
ejpam-4827	778	9	.	.	PUNCT
ejpam-4827	779	1	comm	comm	NOUN
ejpam-4827	779	2	.	.	PUNCT
ejpam-4827	780	1	algebra	algebra	PROPN
ejpam-4827	780	2	,	,	PUNCT
ejpam-4827	780	3	39(11):4259–4272	39(11):4259–4272	NUM
ejpam-4827	780	4	,	,	PUNCT
ejpam-4827	780	5	2011	2011	NUM
ejpam-4827	780	6	.	.	PUNCT
ejpam-4827	781	1	[	[	X
ejpam-4827	781	2	19	19	NUM
ejpam-4827	781	3	]	]	X
ejpam-4827	781	4	d.	d.	PROPN
ejpam-4827	781	5	s.	s.	PROPN
ejpam-4827	781	6	passman	passman	PROPN
ejpam-4827	781	7	.	.	PUNCT
ejpam-4827	782	1	the	the	DET
ejpam-4827	782	2	algebraic	algebraic	ADJ
ejpam-4827	782	3	structure	structure	NOUN
ejpam-4827	782	4	of	of	ADP
ejpam-4827	782	5	group	group	NOUN
ejpam-4827	782	6	rings	ring	NOUN
ejpam-4827	782	7	.	.	PUNCT
ejpam-4827	783	1	wiley	wiley	PROPN
ejpam-4827	783	2	,	,	PUNCT
ejpam-4827	783	3	new	new	PROPN
ejpam-4827	783	4	york	york	PROPN
ejpam-4827	783	5	,	,	PUNCT
ejpam-4827	783	6	1977	1977	NUM
ejpam-4827	783	7	.	.	PUNCT
ejpam-4827	784	1	[	[	X
ejpam-4827	784	2	20	20	NUM
ejpam-4827	784	3	]	]	PUNCT
ejpam-4827	784	4	z.	z.	PROPN
ejpam-4827	784	5	peng	peng	PROPN
ejpam-4827	784	6	,	,	PUNCT
ejpam-4827	784	7	q.	q.	PROPN
ejpam-4827	784	8	gu	gu	PROPN
ejpam-4827	784	9	,	,	PUNCT
ejpam-4827	784	10	and	and	CCONJ
ejpam-4827	784	11	l.	l.	PROPN
ejpam-4827	784	12	zhao	zhao	PROPN
ejpam-4827	784	13	.	.	PUNCT
ejpam-4827	785	1	extensions	extension	NOUN
ejpam-4827	785	2	of	of	ADP
ejpam-4827	785	3	strongly	strongly	ADV
ejpam-4827	785	4	reflexive	reflexive	ADJ
ejpam-4827	785	5	rings	ring	NOUN
ejpam-4827	785	6	.	.	PUNCT
ejpam-4827	786	1	asian	asian	ADJ
ejpam-4827	786	2	-	-	PUNCT
ejpam-4827	786	3	european	european	ADJ
ejpam-4827	786	4	journal	journal	NOUN
ejpam-4827	786	5	of	of	ADP
ejpam-4827	786	6	mathematics	mathematic	NOUN
ejpam-4827	786	7	,	,	PUNCT
ejpam-4827	786	8	8(4):1550078	8(4):1550078	NUM
ejpam-4827	786	9	,	,	PUNCT
ejpam-4827	786	10	2015	2015	NUM
ejpam-4827	786	11	.	.	PUNCT
ejpam-4827	787	1	[	[	X
ejpam-4827	787	2	21	21	NUM
ejpam-4827	787	3	]	]	X
ejpam-4827	787	4	s.	s.	PROPN
ejpam-4827	787	5	rege	rege	PROPN
ejpam-4827	787	6	,	,	PUNCT
ejpam-4827	787	7	m.	m.	PROPN
ejpam-4827	787	8	b.	b.	PROPN
ejpam-4827	787	9	andchhawchharia	andchhawchharia	PROPN
ejpam-4827	787	10	.	.	PUNCT
ejpam-4827	788	1	armendariz	armendariz	PROPN
ejpam-4827	788	2	rings	ring	NOUN
ejpam-4827	788	3	.	.	PUNCT
ejpam-4827	789	1	proc	proc	PROPN
ejpam-4827	789	2	.	.	PUNCT
ejpam-4827	790	1	japan	japan	PROPN
ejpam-4827	790	2	acad	acad	PROPN
ejpam-4827	790	3	.	.	PUNCT
ejpam-4827	791	1	ser	ser	PROPN
ejpam-4827	791	2	.	.	PUNCT
ejpam-4827	792	1	a	a	DET
ejpam-4827	792	2	math	math	NOUN
ejpam-4827	792	3	.	.	PUNCT
ejpam-4827	793	1	sci	sci	PROPN
ejpam-4827	793	2	.	.	PROPN
ejpam-4827	793	3	,	,	PUNCT
ejpam-4827	793	4	73(1):14–17	73(1):14–17	NUM
ejpam-4827	793	5	,	,	PUNCT
ejpam-4827	793	6	1997	1997	NUM
ejpam-4827	793	7	.	.	PUNCT
ejpam-4827	794	1	[	[	X
ejpam-4827	794	2	22	22	NUM
ejpam-4827	794	3	]	]	X
ejpam-4827	794	4	s.	s.	PROPN
ejpam-4827	794	5	safarisabe	safarisabe	PROPN
ejpam-4827	794	6	and	and	CCONJ
ejpam-4827	794	7	e.	e.	PROPN
ejpam-4827	794	8	mohammed	mohammed	PROPN
ejpam-4827	794	9	.	.	PUNCT
ejpam-4827	795	1	skew	skew	PROPN
ejpam-4827	795	2	monoid	monoid	PROPN
ejpam-4827	795	3	rings	ring	NOUN
ejpam-4827	795	4	with	with	ADP
ejpam-4827	795	5	annihilator	annihilator	PROPN
ejpam-4827	795	6	conditions	condition	NOUN
ejpam-4827	795	7	.	.	PUNCT
ejpam-4827	796	1	asian	asian	ADJ
ejpam-4827	796	2	-	-	PUNCT
ejpam-4827	796	3	eur	eur	NOUN
ejpam-4827	796	4	.	.	PUNCT
ejpam-4827	797	1	j.	j.	PROPN
ejpam-4827	797	2	of	of	ADP
ejpam-4827	797	3	math	math	PROPN
ejpam-4827	797	4	,	,	PUNCT
ejpam-4827	797	5	10(1):1550033	10(1):1550033	PROPN
ejpam-4827	797	6	,	,	PUNCT
ejpam-4827	797	7	2017	2017	NUM
ejpam-4827	797	8	.	.	PUNCT
ejpam-4827	798	1	[	[	X
ejpam-4827	798	2	23	23	NUM
ejpam-4827	798	3	]	]	X
ejpam-4827	798	4	h.	h.	PROPN
ejpam-4827	798	5	tominaga	tominaga	PROPN
ejpam-4827	798	6	.	.	PUNCT
ejpam-4827	799	1	on	on	ADP
ejpam-4827	799	2	s	s	NOUN
ejpam-4827	799	3	-	-	ADJ
ejpam-4827	799	4	unital	unital	ADJ
ejpam-4827	799	5	rings	ring	NOUN
ejpam-4827	799	6	.	.	PUNCT
ejpam-4827	800	1	math	math	PROPN
ejpam-4827	800	2	.	.	PUNCT
ejpam-4827	801	1	j.	j.	PROPN
ejpam-4827	801	2	okayama	okayama	PROPN
ejpam-4827	801	3	univ	univ	PROPN
ejpam-4827	801	4	.	.	PROPN
ejpam-4827	801	5	,	,	PUNCT
ejpam-4827	801	6	18:117–134	18:117–134	PROPN
ejpam-4827	801	7	,	,	PUNCT
ejpam-4827	801	8	1976	1976	NUM
ejpam-4827	801	9	.	.	PUNCT
ejpam-4827	802	1	[	[	X
ejpam-4827	802	2	24	24	NUM
ejpam-4827	802	3	]	]	PUNCT
ejpam-4827	802	4	l.	l.	PROPN
ejpam-4827	802	5	zhao	zhao	PROPN
ejpam-4827	802	6	,	,	PUNCT
ejpam-4827	802	7	x.	x.	PROPN
ejpam-4827	802	8	zhu	zhu	PROPN
ejpam-4827	802	9	,	,	PUNCT
ejpam-4827	802	10	and	and	CCONJ
ejpam-4827	802	11	q.	q.	PROPN
ejpam-4827	802	12	gu	gu	PROPN
ejpam-4827	802	13	.	.	PROPN
ejpam-4827	802	14	reflexive	reflexive	ADJ
ejpam-4827	802	15	rings	ring	NOUN
ejpam-4827	802	16	and	and	CCONJ
ejpam-4827	802	17	their	their	PRON
ejpam-4827	802	18	extensions	extension	NOUN
ejpam-4827	802	19	.	.	PUNCT
ejpam-4827	803	1	math	math	NOUN
ejpam-4827	803	2	.	.	PUNCT
ejpam-4827	804	1	slovaca	slovaca	PROPN
ejpam-4827	804	2	,	,	PUNCT
ejpam-4827	804	3	63(3):417–430	63(3):417–430	PROPN
ejpam-4827	804	4	,	,	PUNCT
ejpam-4827	804	5	2013	2013	NUM
ejpam-4827	804	6	.	.	PUNCT