id	sid	tid	token	lemma	pos
ejpam-4596	1	1	european	european	PROPN
ejpam-4596	1	2	journal	journal	PROPN
ejpam-4596	1	3	of	of	ADP
ejpam-4596	1	4	pure	pure	ADJ
ejpam-4596	1	5	and	and	CCONJ
ejpam-4596	1	6	applied	apply	VERB
ejpam-4596	1	7	mathematics	mathematic	NOUN
ejpam-4596	1	8	vol	vol	NOUN
ejpam-4596	1	9	.	.	PROPN
ejpam-4596	2	1	15	15	NUM
ejpam-4596	2	2	,	,	PUNCT
ejpam-4596	2	3	no	no	INTJ
ejpam-4596	2	4	.	.	NOUN
ejpam-4596	2	5	4	4	NUM
ejpam-4596	2	6	,	,	PUNCT
ejpam-4596	2	7	2022	2022	NUM
ejpam-4596	2	8	,	,	PUNCT
ejpam-4596	2	9	2116	2116	NUM
ejpam-4596	2	10	-	-	SYM
ejpam-4596	2	11	2126	2126	NUM
ejpam-4596	2	12	issn	issn	PROPN
ejpam-4596	2	13	1307	1307	NUM
ejpam-4596	2	14	-	-	SYM
ejpam-4596	2	15	5543	5543	NUM
ejpam-4596	2	16	–	–	PUNCT
ejpam-4596	2	17	ejpam.com	ejpam.com	X
ejpam-4596	2	18	published	publish	VERB
ejpam-4596	2	19	by	by	ADP
ejpam-4596	2	20	new	new	PROPN
ejpam-4596	2	21	york	york	PROPN
ejpam-4596	2	22	business	business	PROPN
ejpam-4596	2	23	global	global	ADJ
ejpam-4596	2	24	regularity	regularity	NOUN
ejpam-4596	2	25	on	on	ADP
ejpam-4596	2	26	variants	variant	NOUN
ejpam-4596	2	27	of	of	ADP
ejpam-4596	2	28	transformation	transformation	NOUN
ejpam-4596	2	29	semigroups	semigroup	NOUN
ejpam-4596	2	30	that	that	PRON
ejpam-4596	2	31	preserve	preserve	VERB
ejpam-4596	2	32	an	an	DET
ejpam-4596	2	33	equivalence	equivalence	NOUN
ejpam-4596	2	34	relation	relation	NOUN
ejpam-4596	2	35	piyaporn	piyaporn	NOUN
ejpam-4596	2	36	tantong1	tantong1	NOUN
ejpam-4596	2	37	,	,	PUNCT
ejpam-4596	2	38	nares	nares	X
ejpam-4596	2	39	sawatraksa1,∗	sawatraksa1,∗	NOUN
ejpam-4596	2	40	1	1	NUM
ejpam-4596	2	41	division	division	NOUN
ejpam-4596	2	42	of	of	ADP
ejpam-4596	2	43	mathematics	mathematic	NOUN
ejpam-4596	2	44	and	and	CCONJ
ejpam-4596	2	45	statistics	statistic	NOUN
ejpam-4596	2	46	,	,	PUNCT
ejpam-4596	2	47	faculty	faculty	NOUN
ejpam-4596	2	48	of	of	ADP
ejpam-4596	2	49	science	science	NOUN
ejpam-4596	2	50	and	and	CCONJ
ejpam-4596	2	51	technology	technology	NOUN
ejpam-4596	2	52	,	,	PUNCT
ejpam-4596	2	53	nakhon	nakhon	PROPN
ejpam-4596	2	54	sawan	sawan	PROPN
ejpam-4596	2	55	rajabhat	rajabhat	PROPN
ejpam-4596	2	56	university	university	PROPN
ejpam-4596	2	57	,	,	PUNCT
ejpam-4596	2	58	nakhon	nakhon	PROPN
ejpam-4596	2	59	sawan	sawan	PROPN
ejpam-4596	2	60	,	,	PUNCT
ejpam-4596	2	61	thailand	thailand	PROPN
ejpam-4596	2	62	abstract	abstract	PROPN
ejpam-4596	2	63	.	.	PUNCT
ejpam-4596	3	1	the	the	DET
ejpam-4596	3	2	variant	variant	NOUN
ejpam-4596	3	3	of	of	ADP
ejpam-4596	3	4	a	a	DET
ejpam-4596	3	5	semigroup	semigroup	NOUN
ejpam-4596	3	6	s	s	NOUN
ejpam-4596	3	7	with	with	ADP
ejpam-4596	3	8	respect	respect	NOUN
ejpam-4596	3	9	to	to	ADP
ejpam-4596	3	10	an	an	DET
ejpam-4596	3	11	element	element	NOUN
ejpam-4596	3	12	a	a	DET
ejpam-4596	3	13	∈	∈	PROPN
ejpam-4596	3	14	s	s	NOUN
ejpam-4596	3	15	,	,	PUNCT
ejpam-4596	3	16	is	be	AUX
ejpam-4596	3	17	the	the	DET
ejpam-4596	3	18	semigroup	semigroup	NOUN
ejpam-4596	3	19	with	with	ADP
ejpam-4596	3	20	underlying	underlie	VERB
ejpam-4596	3	21	set	set	NOUN
ejpam-4596	3	22	s	s	NOUN
ejpam-4596	3	23	and	and	CCONJ
ejpam-4596	3	24	a	a	DET
ejpam-4596	3	25	new	new	ADJ
ejpam-4596	3	26	binary	binary	ADJ
ejpam-4596	3	27	operation	operation	NOUN
ejpam-4596	3	28	∗	∗	NOUN
ejpam-4596	3	29	defined	define	VERB
ejpam-4596	3	30	by	by	ADP
ejpam-4596	3	31	x	x	PROPN
ejpam-4596	3	32	∗	∗	NOUN
ejpam-4596	3	33	y	y	NOUN
ejpam-4596	3	34	=	=	SYM
ejpam-4596	3	35	xay	xay	PROPN
ejpam-4596	3	36	for	for	ADP
ejpam-4596	3	37	x	x	PROPN
ejpam-4596	3	38	,	,	PUNCT
ejpam-4596	3	39	y	y	PROPN
ejpam-4596	3	40	∈	∈	PROPN
ejpam-4596	3	41	s.	s.	PROPN
ejpam-4596	3	42	let	let	VERB
ejpam-4596	3	43	t	t	PROPN
ejpam-4596	3	44	(	(	PUNCT
ejpam-4596	3	45	x	x	X
ejpam-4596	3	46	)	)	PUNCT
ejpam-4596	3	47	be	be	VERB
ejpam-4596	3	48	the	the	DET
ejpam-4596	3	49	full	full	ADJ
ejpam-4596	3	50	transformation	transformation	NOUN
ejpam-4596	3	51	semigroup	semigroup	NOUN
ejpam-4596	3	52	on	on	ADP
ejpam-4596	3	53	a	a	DET
ejpam-4596	3	54	nonempty	nonempty	ADV
ejpam-4596	3	55	set	set	VERB
ejpam-4596	3	56	x.	x.	NOUN
ejpam-4596	3	57	for	for	ADP
ejpam-4596	3	58	an	an	DET
ejpam-4596	3	59	arbitrary	arbitrary	ADJ
ejpam-4596	3	60	equivalence	equivalence	NOUN
ejpam-4596	3	61	e	e	NOUN
ejpam-4596	3	62	on	on	ADP
ejpam-4596	3	63	x	x	PRON
ejpam-4596	3	64	,	,	PUNCT
ejpam-4596	3	65	let	let	VERB
ejpam-4596	3	66	te(x	te(x	PRON
ejpam-4596	3	67	)	)	PUNCT
ejpam-4596	4	1	=	=	PRON
ejpam-4596	4	2	{	{	PUNCT
ejpam-4596	4	3	α	α	PROPN
ejpam-4596	4	4	∈	∈	PROPN
ejpam-4596	4	5	t	t	NOUN
ejpam-4596	4	6	(	(	PUNCT
ejpam-4596	4	7	x	x	NOUN
ejpam-4596	4	8	)	)	PUNCT
ejpam-4596	4	9	:	:	PUNCT
ejpam-4596	4	10	∀a	∀a	NOUN
ejpam-4596	4	11	,	,	PUNCT
ejpam-4596	4	12	b	b	X
ejpam-4596	4	13	∈	∈	PROPN
ejpam-4596	4	14	x	x	X
ejpam-4596	4	15	,	,	PUNCT
ejpam-4596	4	16	(	(	PUNCT
ejpam-4596	4	17	a	a	PRON
ejpam-4596	4	18	,	,	PUNCT
ejpam-4596	4	19	b	b	NOUN
ejpam-4596	4	20	)	)	PUNCT
ejpam-4596	4	21	∈	∈	PROPN
ejpam-4596	4	22	e	e	NOUN
ejpam-4596	4	23	⇒	⇒	NOUN
ejpam-4596	4	24	(	(	PUNCT
ejpam-4596	4	25	aα	aα	NOUN
ejpam-4596	4	26	,	,	PUNCT
ejpam-4596	4	27	bα	bα	PROPN
ejpam-4596	4	28	)	)	PUNCT
ejpam-4596	4	29	∈	∈	PROPN
ejpam-4596	4	30	e	e	NOUN
ejpam-4596	4	31	}	}	PUNCT
ejpam-4596	4	32	.	.	PUNCT
ejpam-4596	5	1	then	then	ADV
ejpam-4596	5	2	te(x	te(x	PUNCT
ejpam-4596	5	3	)	)	PUNCT
ejpam-4596	5	4	is	be	AUX
ejpam-4596	5	5	a	a	DET
ejpam-4596	5	6	subsemigroup	subsemigroup	NOUN
ejpam-4596	5	7	of	of	ADP
ejpam-4596	5	8	t	t	PROPN
ejpam-4596	5	9	(	(	PUNCT
ejpam-4596	5	10	x	x	NOUN
ejpam-4596	5	11	)	)	PUNCT
ejpam-4596	5	12	.	.	PUNCT
ejpam-4596	6	1	in	in	ADP
ejpam-4596	6	2	this	this	DET
ejpam-4596	6	3	paper	paper	NOUN
ejpam-4596	6	4	,	,	PUNCT
ejpam-4596	6	5	we	we	PRON
ejpam-4596	6	6	investigate	investigate	VERB
ejpam-4596	6	7	regular	regular	ADV
ejpam-4596	6	8	,	,	PUNCT
ejpam-4596	6	9	left	leave	VERB
ejpam-4596	6	10	regular	regular	ADV
ejpam-4596	6	11	and	and	CCONJ
ejpam-4596	6	12	right	right	ADJ
ejpam-4596	6	13	regular	regular	ADJ
ejpam-4596	6	14	elements	element	NOUN
ejpam-4596	6	15	for	for	ADP
ejpam-4596	6	16	the	the	DET
ejpam-4596	6	17	variant	variant	NOUN
ejpam-4596	6	18	of	of	ADP
ejpam-4596	6	19	some	some	DET
ejpam-4596	6	20	subsemigroups	subsemigroup	NOUN
ejpam-4596	6	21	of	of	ADP
ejpam-4596	6	22	the	the	DET
ejpam-4596	6	23	semigroup	semigroup	PROPN
ejpam-4596	6	24	te(x	te(x	PROPN
ejpam-4596	6	25	)	)	PUNCT
ejpam-4596	6	26	.	.	PUNCT
ejpam-4596	7	1	2020	2020	NUM
ejpam-4596	7	2	mathematics	mathematic	NOUN
ejpam-4596	7	3	subject	subject	NOUN
ejpam-4596	7	4	classifications	classification	NOUN
ejpam-4596	7	5	:	:	PUNCT
ejpam-4596	7	6	20m20	20m20	NUM
ejpam-4596	7	7	,	,	PUNCT
ejpam-4596	7	8	20m17	20m17	NUM
ejpam-4596	7	9	key	key	ADJ
ejpam-4596	7	10	words	word	NOUN
ejpam-4596	7	11	and	and	CCONJ
ejpam-4596	7	12	phrases	phrase	NOUN
ejpam-4596	7	13	:	:	PUNCT
ejpam-4596	7	14	transformation	transformation	NOUN
ejpam-4596	7	15	semigroup	semigroup	PROPN
ejpam-4596	7	16	,	,	PUNCT
ejpam-4596	7	17	equivalence	equivalence	NOUN
ejpam-4596	7	18	relation	relation	NOUN
ejpam-4596	7	19	,	,	PUNCT
ejpam-4596	7	20	left	leave	VERB
ejpam-4596	7	21	regular	regular	ADV
ejpam-4596	7	22	,	,	PUNCT
ejpam-4596	7	23	right	right	ADV
ejpam-4596	7	24	regular	regular	ADJ
ejpam-4596	7	25	,	,	PUNCT
ejpam-4596	7	26	completely	completely	ADV
ejpam-4596	7	27	regular	regular	ADJ
ejpam-4596	7	28	1	1	NUM
ejpam-4596	7	29	.	.	PUNCT
ejpam-4596	8	1	introduction	introduction	NOUN
ejpam-4596	8	2	and	and	CCONJ
ejpam-4596	8	3	preliminaries	preliminary	NOUN
ejpam-4596	8	4	let	let	VERB
ejpam-4596	8	5	s	s	PRON
ejpam-4596	8	6	be	be	AUX
ejpam-4596	8	7	a	a	DET
ejpam-4596	8	8	semigroup	semigroup	NOUN
ejpam-4596	8	9	and	and	CCONJ
ejpam-4596	8	10	a	a	DET
ejpam-4596	8	11	belong	belong	NOUN
ejpam-4596	8	12	to	to	ADP
ejpam-4596	8	13	s.	s.	PROPN
ejpam-4596	8	14	we	we	PRON
ejpam-4596	8	15	define	define	VERB
ejpam-4596	8	16	a	a	DET
ejpam-4596	8	17	new	new	ADJ
ejpam-4596	8	18	binary	binary	ADJ
ejpam-4596	8	19	operation	operation	NOUN
ejpam-4596	8	20	∗	∗	NOUN
ejpam-4596	8	21	on	on	ADP
ejpam-4596	8	22	s	s	PRON
ejpam-4596	8	23	by	by	ADP
ejpam-4596	8	24	putting	put	VERB
ejpam-4596	8	25	x	x	PUNCT
ejpam-4596	8	26	∗	∗	NOUN
ejpam-4596	8	27	y	y	NOUN
ejpam-4596	8	28	=	=	SYM
ejpam-4596	8	29	xay	xay	PROPN
ejpam-4596	8	30	for	for	ADP
ejpam-4596	8	31	all	all	DET
ejpam-4596	8	32	x	x	NOUN
ejpam-4596	8	33	,	,	PUNCT
ejpam-4596	8	34	y	y	PROPN
ejpam-4596	8	35	∈	∈	PROPN
ejpam-4596	8	36	s.	s.	PROPN
ejpam-4596	8	37	the	the	DET
ejpam-4596	8	38	operation	operation	NOUN
ejpam-4596	8	39	∗	∗	NOUN
ejpam-4596	8	40	is	be	AUX
ejpam-4596	8	41	clearly	clearly	ADV
ejpam-4596	8	42	associative	associative	ADJ
ejpam-4596	8	43	.	.	PUNCT
ejpam-4596	9	1	hence	hence	ADV
ejpam-4596	9	2	(	(	PUNCT
ejpam-4596	9	3	s	s	X
ejpam-4596	9	4	,	,	PUNCT
ejpam-4596	9	5	∗	∗	NOUN
ejpam-4596	9	6	)	)	PUNCT
ejpam-4596	9	7	is	be	AUX
ejpam-4596	9	8	a	a	DET
ejpam-4596	9	9	semigroup	semigroup	NOUN
ejpam-4596	9	10	and	and	CCONJ
ejpam-4596	9	11	it	it	PRON
ejpam-4596	9	12	is	be	AUX
ejpam-4596	9	13	called	call	VERB
ejpam-4596	9	14	a	a	DET
ejpam-4596	9	15	variant	variant	NOUN
ejpam-4596	9	16	of	of	ADP
ejpam-4596	9	17	s.	s.	PROPN
ejpam-4596	9	18	we	we	PRON
ejpam-4596	9	19	usually	usually	ADV
ejpam-4596	9	20	write	write	VERB
ejpam-4596	9	21	(	(	PUNCT
ejpam-4596	9	22	s	s	PROPN
ejpam-4596	9	23	,	,	PUNCT
ejpam-4596	9	24	a	a	PRON
ejpam-4596	9	25	)	)	PUNCT
ejpam-4596	9	26	rather	rather	ADV
ejpam-4596	9	27	than	than	ADP
ejpam-4596	9	28	(	(	PUNCT
ejpam-4596	9	29	s	s	X
ejpam-4596	9	30	,	,	PUNCT
ejpam-4596	9	31	∗	∗	NOUN
ejpam-4596	9	32	)	)	PUNCT
ejpam-4596	9	33	to	to	PART
ejpam-4596	9	34	make	make	VERB
ejpam-4596	9	35	the	the	DET
ejpam-4596	9	36	element	element	NOUN
ejpam-4596	9	37	explicit	explicit	ADJ
ejpam-4596	9	38	.	.	PUNCT
ejpam-4596	10	1	variants	variant	NOUN
ejpam-4596	10	2	of	of	ADP
ejpam-4596	10	3	abstract	abstract	ADJ
ejpam-4596	10	4	semigroups	semigroup	NOUN
ejpam-4596	10	5	were	be	AUX
ejpam-4596	10	6	first	first	ADV
ejpam-4596	10	7	studied	study	VERB
ejpam-4596	10	8	by	by	ADP
ejpam-4596	10	9	hickey	hickey	PROPN
ejpam-4596	11	1	[	[	X
ejpam-4596	11	2	5	5	NUM
ejpam-4596	11	3	]	]	PUNCT
ejpam-4596	11	4	.	.	PUNCT
ejpam-4596	12	1	although	although	SCONJ
ejpam-4596	12	2	concrete	concrete	ADJ
ejpam-4596	12	3	semigroups	semigroup	NOUN
ejpam-4596	12	4	of	of	ADP
ejpam-4596	12	5	relations	relation	NOUN
ejpam-4596	12	6	had	have	AUX
ejpam-4596	12	7	earlier	early	ADV
ejpam-4596	12	8	been	be	AUX
ejpam-4596	12	9	considered	consider	VERB
ejpam-4596	12	10	by	by	ADP
ejpam-4596	12	11	magill	magill	NOUN
ejpam-4596	12	12	[	[	X
ejpam-4596	12	13	10	10	NUM
ejpam-4596	12	14	]	]	PUNCT
ejpam-4596	12	15	.	.	PUNCT
ejpam-4596	13	1	the	the	DET
ejpam-4596	13	2	study	study	NOUN
ejpam-4596	13	3	of	of	ADP
ejpam-4596	13	4	semigroup	semigroup	PROPN
ejpam-4596	13	5	variants	variant	NOUN
ejpam-4596	13	6	goes	go	VERB
ejpam-4596	13	7	back	back	ADV
ejpam-4596	13	8	to	to	ADP
ejpam-4596	13	9	the	the	DET
ejpam-4596	13	10	1960	1960	NUM
ejpam-4596	13	11	monograph	monograph	NOUN
ejpam-4596	13	12	of	of	ADP
ejpam-4596	13	13	lyapin	lyapin	PROPN
ejpam-4596	13	14	[	[	X
ejpam-4596	13	15	9	9	NUM
ejpam-4596	13	16	]	]	PUNCT
ejpam-4596	13	17	and	and	CCONJ
ejpam-4596	13	18	a	a	DET
ejpam-4596	13	19	1967	1967	NUM
ejpam-4596	13	20	paper	paper	NOUN
ejpam-4596	13	21	by	by	ADP
ejpam-4596	13	22	magill	magill	NOUN
ejpam-4596	13	23	and	and	CCONJ
ejpam-4596	13	24	subbiah	subbiah	NOUN
ejpam-4596	13	25	[	[	X
ejpam-4596	13	26	6	6	NUM
ejpam-4596	13	27	]	]	PUNCT
ejpam-4596	13	28	that	that	PRON
ejpam-4596	13	29	considers	consider	VERB
ejpam-4596	13	30	semigroups	semigroup	NOUN
ejpam-4596	13	31	of	of	ADP
ejpam-4596	13	32	functions	function	NOUN
ejpam-4596	13	33	x	x	PUNCT
ejpam-4596	13	34	→	→	SYM
ejpam-4596	13	35	y	y	PROPN
ejpam-4596	13	36	under	under	ADP
ejpam-4596	13	37	an	an	DET
ejpam-4596	13	38	operation	operation	NOUN
ejpam-4596	13	39	defined	define	VERB
ejpam-4596	13	40	by	by	ADP
ejpam-4596	13	41	f	f	PROPN
ejpam-4596	13	42	·	·	PUNCT
ejpam-4596	13	43	g	g	PROPN
ejpam-4596	13	44	=	=	SYM
ejpam-4596	13	45	f	f	PROPN
ejpam-4596	13	46	◦	◦	NOUN
ejpam-4596	13	47	θ	θ	PROPN
ejpam-4596	13	48	◦	◦	NOUN
ejpam-4596	13	49	g	g	NOUN
ejpam-4596	13	50	,	,	PUNCT
ejpam-4596	13	51	where	where	SCONJ
ejpam-4596	13	52	θ	θ	PROPN
ejpam-4596	13	53	is	be	AUX
ejpam-4596	13	54	some	some	DET
ejpam-4596	13	55	fixed	fix	VERB
ejpam-4596	13	56	function	function	NOUN
ejpam-4596	13	57	y	y	PROPN
ejpam-4596	13	58	→	→	PUNCT
ejpam-4596	13	59	x.	x.	NOUN
ejpam-4596	13	60	in	in	ADP
ejpam-4596	13	61	the	the	DET
ejpam-4596	13	62	case	case	NOUN
ejpam-4596	13	63	that	that	SCONJ
ejpam-4596	13	64	x	x	X
ejpam-4596	13	65	=	=	SYM
ejpam-4596	13	66	y	y	PROPN
ejpam-4596	13	67	,	,	PUNCT
ejpam-4596	13	68	this	this	PRON
ejpam-4596	13	69	provides	provide	VERB
ejpam-4596	13	70	an	an	DET
ejpam-4596	13	71	alternative	alternative	ADJ
ejpam-4596	13	72	product	product	NOUN
ejpam-4596	13	73	on	on	ADP
ejpam-4596	13	74	the	the	DET
ejpam-4596	13	75	full	full	ADJ
ejpam-4596	13	76	transformation	transformation	NOUN
ejpam-4596	13	77	semigroup	semigroup	PROPN
ejpam-4596	13	78	t	t	PROPN
ejpam-4596	13	79	(	(	PUNCT
ejpam-4596	13	80	x	x	X
ejpam-4596	13	81	)	)	PUNCT
ejpam-4596	13	82	(	(	PUNCT
ejpam-4596	13	83	consisting	consist	VERB
ejpam-4596	13	84	of	of	ADP
ejpam-4596	13	85	all	all	DET
ejpam-4596	13	86	functions	function	NOUN
ejpam-4596	13	87	x	x	PUNCT
ejpam-4596	13	88	→	→	SYM
ejpam-4596	13	89	x	x	X
ejpam-4596	13	90	)	)	PUNCT
ejpam-4596	13	91	.	.	PUNCT
ejpam-4596	14	1	for	for	ADP
ejpam-4596	14	2	an	an	DET
ejpam-4596	14	3	element	element	NOUN
ejpam-4596	14	4	a	a	PRON
ejpam-4596	14	5	of	of	ADP
ejpam-4596	14	6	a	a	DET
ejpam-4596	14	7	semigroup	semigroup	NOUN
ejpam-4596	14	8	s	s	PROPN
ejpam-4596	14	9	,	,	PUNCT
ejpam-4596	14	10	a	a	PRON
ejpam-4596	14	11	is	be	AUX
ejpam-4596	14	12	called	call	VERB
ejpam-4596	14	13	regular	regular	ADJ
ejpam-4596	14	14	if	if	SCONJ
ejpam-4596	14	15	there	there	PRON
ejpam-4596	14	16	exists	exist	VERB
ejpam-4596	14	17	x	x	X
ejpam-4596	14	18	∈	∈	NOUN
ejpam-4596	14	19	s	s	VERB
ejpam-4596	14	20	such	such	ADJ
ejpam-4596	14	21	that	that	SCONJ
ejpam-4596	14	22	a	a	DET
ejpam-4596	14	23	=	=	X
ejpam-4596	14	24	axa	axa	NOUN
ejpam-4596	14	25	.	.	PUNCT
ejpam-4596	15	1	a	a	DET
ejpam-4596	15	2	semigroup	semigroup	NOUN
ejpam-4596	15	3	s	s	VERB
ejpam-4596	15	4	is	be	AUX
ejpam-4596	15	5	regular	regular	ADJ
ejpam-4596	15	6	semigroup	semigroup	ADJ
ejpam-4596	15	7	if	if	SCONJ
ejpam-4596	15	8	every	every	DET
ejpam-4596	15	9	element	element	NOUN
ejpam-4596	15	10	of	of	ADP
ejpam-4596	15	11	s	s	PROPN
ejpam-4596	15	12	is	be	AUX
ejpam-4596	15	13	regular	regular	ADJ
ejpam-4596	15	14	.	.	PUNCT
ejpam-4596	16	1	regular	regular	ADJ
ejpam-4596	16	2	∗corresponding	∗corresponde	VERB
ejpam-4596	16	3	author	author	NOUN
ejpam-4596	16	4	.	.	PUNCT
ejpam-4596	17	1	doi	doi	NOUN
ejpam-4596	17	2	:	:	PUNCT
ejpam-4596	17	3	https://doi.org/10.29020/nybg.ejpam.v15i4.4596	https://doi.org/10.29020/nybg.ejpam.v15i4.4596	NUM
ejpam-4596	17	4	email	email	NOUN
ejpam-4596	17	5	addresses	address	NOUN
ejpam-4596	17	6	:	:	PUNCT
ejpam-4596	17	7	piyaporn.ta@nsru.ac.th	piyaporn.ta@nsru.ac.th	PROPN
ejpam-4596	17	8	(	(	PUNCT
ejpam-4596	17	9	p.	p.	NOUN
ejpam-4596	17	10	tantong	tantong	NOUN
ejpam-4596	17	11	)	)	PUNCT
ejpam-4596	17	12	,	,	PUNCT
ejpam-4596	17	13	nares.sa@nsru.ac.th	nares.sa@nsru.ac.th	PROPN
ejpam-4596	17	14	(	(	PUNCT
ejpam-4596	17	15	n.	n.	NOUN
ejpam-4596	17	16	sawatraksa	sawatraksa	PROPN
ejpam-4596	17	17	)	)	PUNCT
ejpam-4596	17	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4596	17	19	2116	2116	NUM
ejpam-4596	18	1	©	©	PROPN
ejpam-4596	18	2	2022	2022	NUM
ejpam-4596	18	3	ejpam	ejpam	VERB
ejpam-4596	18	4	all	all	DET
ejpam-4596	18	5	rights	right	NOUN
ejpam-4596	18	6	reserved	reserve	VERB
ejpam-4596	18	7	.	.	PUNCT
ejpam-4596	19	1	p.	p.	NOUN
ejpam-4596	19	2	tantong	tantong	NOUN
ejpam-4596	19	3	,	,	PUNCT
ejpam-4596	19	4	n.	n.	NOUN
ejpam-4596	19	5	sawatraksa	sawatraksa	PROPN
ejpam-4596	19	6	/	/	SYM
ejpam-4596	19	7	eur	eur	PROPN
ejpam-4596	19	8	.	.	PUNCT
ejpam-4596	20	1	j.	j.	PROPN
ejpam-4596	20	2	pure	pure	PROPN
ejpam-4596	20	3	appl	appl	PROPN
ejpam-4596	20	4	.	.	PROPN
ejpam-4596	20	5	math	math	PROPN
ejpam-4596	20	6	,	,	PUNCT
ejpam-4596	20	7	15	15	NUM
ejpam-4596	20	8	(	(	PUNCT
ejpam-4596	20	9	4	4	NUM
ejpam-4596	20	10	)	)	PUNCT
ejpam-4596	20	11	(	(	PUNCT
ejpam-4596	20	12	2022	2022	NUM
ejpam-4596	20	13	)	)	PUNCT
ejpam-4596	20	14	,	,	PUNCT
ejpam-4596	20	15	2116	2116	NUM
ejpam-4596	20	16	-	-	SYM
ejpam-4596	20	17	2126	2126	NUM
ejpam-4596	20	18	2117	2117	NUM
ejpam-4596	20	19	semigroups	semigroup	NOUN
ejpam-4596	20	20	were	be	AUX
ejpam-4596	20	21	introduced	introduce	VERB
ejpam-4596	20	22	by	by	ADP
ejpam-4596	20	23	green	green	ADJ
ejpam-4596	20	24	[	[	X
ejpam-4596	20	25	4	4	NUM
ejpam-4596	20	26	]	]	PUNCT
ejpam-4596	20	27	in	in	ADP
ejpam-4596	20	28	his	his	PRON
ejpam-4596	20	29	influential	influential	ADJ
ejpam-4596	20	30	1951	1951	NUM
ejpam-4596	20	31	paper	paper	NOUN
ejpam-4596	20	32	“	"	PUNCT
ejpam-4596	20	33	on	on	ADP
ejpam-4596	20	34	the	the	DET
ejpam-4596	20	35	structure	structure	NOUN
ejpam-4596	20	36	of	of	ADP
ejpam-4596	20	37	semigroups	semigroup	NOUN
ejpam-4596	20	38	”	"	PUNCT
ejpam-4596	20	39	.	.	PUNCT
ejpam-4596	21	1	the	the	DET
ejpam-4596	21	2	concept	concept	NOUN
ejpam-4596	21	3	of	of	ADP
ejpam-4596	21	4	regularity	regularity	NOUN
ejpam-4596	21	5	in	in	ADP
ejpam-4596	21	6	a	a	DET
ejpam-4596	21	7	semigroup	semigroup	NOUN
ejpam-4596	21	8	was	be	AUX
ejpam-4596	21	9	adapted	adapt	VERB
ejpam-4596	21	10	from	from	ADP
ejpam-4596	21	11	an	an	DET
ejpam-4596	21	12	analogous	analogous	ADJ
ejpam-4596	21	13	condition	condition	NOUN
ejpam-4596	21	14	for	for	ADP
ejpam-4596	21	15	rings	ring	NOUN
ejpam-4596	21	16	,	,	PUNCT
ejpam-4596	21	17	already	already	ADV
ejpam-4596	21	18	considered	consider	VERB
ejpam-4596	21	19	by	by	ADP
ejpam-4596	21	20	neumann	neumann	PROPN
ejpam-4596	22	1	[	[	X
ejpam-4596	22	2	12	12	NUM
ejpam-4596	22	3	]	]	PUNCT
ejpam-4596	22	4	.	.	PUNCT
ejpam-4596	23	1	it	it	PRON
ejpam-4596	23	2	was	be	AUX
ejpam-4596	23	3	green	green	ADJ
ejpam-4596	23	4	’s	’s	PART
ejpam-4596	23	5	study	study	NOUN
ejpam-4596	23	6	of	of	ADP
ejpam-4596	23	7	regular	regular	ADJ
ejpam-4596	23	8	semigroups	semigroup	NOUN
ejpam-4596	23	9	that	that	PRON
ejpam-4596	23	10	led	lead	VERB
ejpam-4596	23	11	him	he	PRON
ejpam-4596	23	12	to	to	PART
ejpam-4596	23	13	define	define	VERB
ejpam-4596	23	14	his	his	PRON
ejpam-4596	23	15	celebrated	celebrated	ADJ
ejpam-4596	23	16	relations	relation	NOUN
ejpam-4596	23	17	.	.	PUNCT
ejpam-4596	24	1	according	accord	VERB
ejpam-4596	24	2	to	to	ADP
ejpam-4596	24	3	a	a	DET
ejpam-4596	24	4	footnote	footnote	NOUN
ejpam-4596	24	5	in	in	ADP
ejpam-4596	24	6	green	green	ADJ
ejpam-4596	24	7	1951	1951	NUM
ejpam-4596	24	8	,	,	PUNCT
ejpam-4596	24	9	the	the	DET
ejpam-4596	24	10	suggestion	suggestion	NOUN
ejpam-4596	24	11	that	that	SCONJ
ejpam-4596	24	12	the	the	DET
ejpam-4596	24	13	notion	notion	NOUN
ejpam-4596	24	14	of	of	ADP
ejpam-4596	24	15	regularity	regularity	NOUN
ejpam-4596	24	16	be	be	AUX
ejpam-4596	24	17	applied	apply	VERB
ejpam-4596	24	18	to	to	ADP
ejpam-4596	24	19	semigroups	semigroup	NOUN
ejpam-4596	24	20	was	be	AUX
ejpam-4596	24	21	first	first	ADV
ejpam-4596	24	22	made	make	VERB
ejpam-4596	24	23	by	by	ADP
ejpam-4596	24	24	rees	ree	NOUN
ejpam-4596	24	25	[	[	X
ejpam-4596	24	26	15	15	NUM
ejpam-4596	24	27	,	,	PUNCT
ejpam-4596	24	28	16	16	NUM
ejpam-4596	24	29	]	]	PUNCT
ejpam-4596	24	30	.	.	PUNCT
ejpam-4596	25	1	this	this	DET
ejpam-4596	25	2	property	property	NOUN
ejpam-4596	25	3	of	of	ADP
ejpam-4596	25	4	regular	regular	ADJ
ejpam-4596	25	5	elements	element	NOUN
ejpam-4596	25	6	was	be	AUX
ejpam-4596	25	7	first	first	ADV
ejpam-4596	25	8	observed	observe	VERB
ejpam-4596	25	9	by	by	ADP
ejpam-4596	25	10	thierrin	thierrin	NOUN
ejpam-4596	25	11	[	[	X
ejpam-4596	25	12	17	17	NUM
ejpam-4596	25	13	]	]	PUNCT
ejpam-4596	25	14	in	in	ADP
ejpam-4596	25	15	1952	1952	NUM
ejpam-4596	25	16	.	.	PUNCT
ejpam-4596	26	1	another	another	DET
ejpam-4596	26	2	important	important	ADJ
ejpam-4596	26	3	kind	kind	NOUN
ejpam-4596	26	4	of	of	ADP
ejpam-4596	26	5	the	the	DET
ejpam-4596	26	6	regularity	regularity	NOUN
ejpam-4596	26	7	was	be	AUX
ejpam-4596	26	8	introduced	introduce	VERB
ejpam-4596	26	9	by	by	ADP
ejpam-4596	26	10	clifford	clifford	PROPN
ejpam-4596	27	1	[	[	X
ejpam-4596	27	2	1	1	X
ejpam-4596	27	3	]	]	PUNCT
ejpam-4596	27	4	in	in	ADP
ejpam-4596	27	5	1941	1941	NUM
ejpam-4596	27	6	,	,	PUNCT
ejpam-4596	27	7	who	who	PRON
ejpam-4596	27	8	studied	study	VERB
ejpam-4596	27	9	elements	element	NOUN
ejpam-4596	27	10	a	a	PRON
ejpam-4596	27	11	of	of	ADP
ejpam-4596	27	12	a	a	DET
ejpam-4596	27	13	semigroup	semigroup	NOUN
ejpam-4596	28	1	s	s	AUX
ejpam-4596	28	2	having	have	VERB
ejpam-4596	28	3	the	the	DET
ejpam-4596	28	4	property	property	NOUN
ejpam-4596	28	5	that	that	PRON
ejpam-4596	28	6	there	there	PRON
ejpam-4596	28	7	exists	exist	VERB
ejpam-4596	28	8	x	x	X
ejpam-4596	28	9	∈	∈	NOUN
ejpam-4596	28	10	s	s	VERB
ejpam-4596	28	11	such	such	ADJ
ejpam-4596	28	12	that	that	SCONJ
ejpam-4596	28	13	a	a	DET
ejpam-4596	28	14	=	=	X
ejpam-4596	28	15	axa	axa	NOUN
ejpam-4596	28	16	and	and	CCONJ
ejpam-4596	28	17	ax	ax	NOUN
ejpam-4596	28	18	=	=	SYM
ejpam-4596	28	19	xa	xa	PROPN
ejpam-4596	28	20	,	,	PUNCT
ejpam-4596	28	21	which	which	PRON
ejpam-4596	28	22	we	we	PRON
ejpam-4596	28	23	now	now	ADV
ejpam-4596	28	24	call	call	VERB
ejpam-4596	28	25	a	a	DET
ejpam-4596	28	26	completely	completely	ADV
ejpam-4596	28	27	regular	regular	ADJ
ejpam-4596	28	28	element	element	NOUN
ejpam-4596	28	29	,	,	PUNCT
ejpam-4596	28	30	and	and	CCONJ
ejpam-4596	28	31	semigroups	semigroup	VERB
ejpam-4596	28	32	whose	whose	DET
ejpam-4596	28	33	any	any	DET
ejpam-4596	28	34	element	element	NOUN
ejpam-4596	28	35	is	be	AUX
ejpam-4596	28	36	completely	completely	ADV
ejpam-4596	28	37	regular	regular	ADJ
ejpam-4596	28	38	,	,	PUNCT
ejpam-4596	28	39	are	be	AUX
ejpam-4596	28	40	called	call	VERB
ejpam-4596	28	41	completely	completely	ADV
ejpam-4596	28	42	regular	regular	ADJ
ejpam-4596	28	43	semigroups	semigroup	NOUN
ejpam-4596	28	44	.	.	PUNCT
ejpam-4596	29	1	the	the	DET
ejpam-4596	29	2	complete	complete	ADJ
ejpam-4596	29	3	regularity	regularity	NOUN
ejpam-4596	29	4	was	be	AUX
ejpam-4596	29	5	also	also	ADV
ejpam-4596	29	6	investigated	investigate	VERB
ejpam-4596	29	7	by	by	ADP
ejpam-4596	29	8	croisot	croisot	NOUN
ejpam-4596	29	9	[	[	X
ejpam-4596	29	10	2	2	X
ejpam-4596	29	11	]	]	PUNCT
ejpam-4596	29	12	in	in	ADP
ejpam-4596	29	13	1953	1953	NUM
ejpam-4596	29	14	,	,	PUNCT
ejpam-4596	29	15	who	who	PRON
ejpam-4596	29	16	also	also	ADV
ejpam-4596	29	17	studied	study	VERB
ejpam-4596	29	18	elements	element	NOUN
ejpam-4596	29	19	a	a	PRON
ejpam-4596	29	20	of	of	ADP
ejpam-4596	29	21	a	a	DET
ejpam-4596	29	22	semigroup	semigroup	NOUN
ejpam-4596	29	23	s	s	X
ejpam-4596	29	24	for	for	ADP
ejpam-4596	29	25	which	which	PRON
ejpam-4596	29	26	a	a	DET
ejpam-4596	29	27	∈	∈	PROPN
ejpam-4596	29	28	sa2	sa2	NOUN
ejpam-4596	29	29	(	(	PUNCT
ejpam-4596	29	30	resp	resp	NOUN
ejpam-4596	29	31	.	.	PUNCT
ejpam-4596	30	1	a	a	DET
ejpam-4596	30	2	∈	∈	PROPN
ejpam-4596	30	3	a2s	a2s	NOUN
ejpam-4596	30	4	)	)	PUNCT
ejpam-4596	30	5	,	,	PUNCT
ejpam-4596	30	6	called	call	VERB
ejpam-4596	30	7	left	left	ADV
ejpam-4596	30	8	regular	regular	ADJ
ejpam-4596	30	9	(	(	PUNCT
ejpam-4596	30	10	resp	resp	NOUN
ejpam-4596	30	11	.	.	PUNCT
ejpam-4596	31	1	right	right	ADV
ejpam-4596	31	2	regular	regular	ADJ
ejpam-4596	31	3	)	)	PUNCT
ejpam-4596	31	4	elements	element	NOUN
ejpam-4596	31	5	,	,	PUNCT
ejpam-4596	31	6	and	and	CCONJ
ejpam-4596	31	7	semigroups	semigroup	VERB
ejpam-4596	31	8	whose	whose	DET
ejpam-4596	31	9	every	every	DET
ejpam-4596	31	10	element	element	NOUN
ejpam-4596	31	11	is	be	AUX
ejpam-4596	31	12	left	leave	VERB
ejpam-4596	31	13	regular	regular	ADJ
ejpam-4596	31	14	(	(	PUNCT
ejpam-4596	31	15	resp	resp	NOUN
ejpam-4596	31	16	.	.	PUNCT
ejpam-4596	32	1	right	right	ADV
ejpam-4596	32	2	regular	regular	ADV
ejpam-4596	32	3	)	)	PUNCT
ejpam-4596	32	4	,	,	PUNCT
ejpam-4596	32	5	called	call	VERB
ejpam-4596	32	6	left	left	ADV
ejpam-4596	32	7	regular	regular	ADJ
ejpam-4596	32	8	(	(	PUNCT
ejpam-4596	32	9	resp	resp	NOUN
ejpam-4596	32	10	.	.	PUNCT
ejpam-4596	33	1	right	right	ADV
ejpam-4596	33	2	regular	regular	ADV
ejpam-4596	33	3	)	)	PUNCT
ejpam-4596	33	4	semigroups	semigroup	NOUN
ejpam-4596	33	5	.	.	PUNCT
ejpam-4596	34	1	in	in	ADP
ejpam-4596	34	2	[	[	X
ejpam-4596	34	3	13	13	NUM
ejpam-4596	34	4	]	]	PUNCT
ejpam-4596	34	5	,	,	PUNCT
ejpam-4596	34	6	pei	pei	PROPN
ejpam-4596	34	7	has	have	AUX
ejpam-4596	34	8	introduced	introduce	VERB
ejpam-4596	34	9	a	a	DET
ejpam-4596	34	10	family	family	NOUN
ejpam-4596	34	11	of	of	ADP
ejpam-4596	34	12	subsemigroup	subsemigroup	NOUN
ejpam-4596	34	13	of	of	ADP
ejpam-4596	34	14	t	t	PROPN
ejpam-4596	34	15	(	(	PUNCT
ejpam-4596	34	16	x	x	NOUN
ejpam-4596	34	17	)	)	PUNCT
ejpam-4596	34	18	defined	define	VERB
ejpam-4596	34	19	by	by	ADP
ejpam-4596	34	20	te(x	te(x	NOUN
ejpam-4596	34	21	)	)	PUNCT
ejpam-4596	34	22	=	=	PRON
ejpam-4596	35	1	{	{	PUNCT
ejpam-4596	35	2	α	α	PROPN
ejpam-4596	35	3	∈	∈	PROPN
ejpam-4596	35	4	t	t	NOUN
ejpam-4596	35	5	(	(	PUNCT
ejpam-4596	35	6	x	x	NOUN
ejpam-4596	35	7	)	)	PUNCT
ejpam-4596	35	8	:	:	PUNCT
ejpam-4596	35	9	∀a	∀a	NOUN
ejpam-4596	35	10	,	,	PUNCT
ejpam-4596	35	11	b	b	X
ejpam-4596	35	12	∈	∈	PROPN
ejpam-4596	35	13	x	x	X
ejpam-4596	35	14	,	,	PUNCT
ejpam-4596	35	15	(	(	PUNCT
ejpam-4596	35	16	a	a	PRON
ejpam-4596	35	17	,	,	PUNCT
ejpam-4596	35	18	b	b	NOUN
ejpam-4596	35	19	)	)	PUNCT
ejpam-4596	35	20	∈	∈	PROPN
ejpam-4596	35	21	e	e	NOUN
ejpam-4596	35	22	⇒	⇒	NOUN
ejpam-4596	35	23	(	(	PUNCT
ejpam-4596	35	24	aα	aα	NOUN
ejpam-4596	35	25	,	,	PUNCT
ejpam-4596	35	26	bα	bα	PROPN
ejpam-4596	35	27	)	)	PUNCT
ejpam-4596	35	28	∈	∈	PROPN
ejpam-4596	35	29	e	e	NOUN
ejpam-4596	35	30	}	}	PUNCT
ejpam-4596	35	31	where	where	SCONJ
ejpam-4596	35	32	e	e	NOUN
ejpam-4596	35	33	is	be	AUX
ejpam-4596	35	34	an	an	DET
ejpam-4596	35	35	arbitrary	arbitrary	ADJ
ejpam-4596	35	36	equivalence	equivalence	NOUN
ejpam-4596	35	37	relation	relation	NOUN
ejpam-4596	35	38	on	on	ADP
ejpam-4596	35	39	x.	x.	NOUN
ejpam-4596	35	40	in	in	ADP
ejpam-4596	35	41	[	[	X
ejpam-4596	35	42	13	13	NUM
ejpam-4596	35	43	]	]	PUNCT
ejpam-4596	35	44	,	,	PUNCT
ejpam-4596	35	45	the	the	DET
ejpam-4596	35	46	author	author	NOUN
ejpam-4596	35	47	investigated	investigate	VERB
ejpam-4596	35	48	regularity	regularity	NOUN
ejpam-4596	35	49	and	and	CCONJ
ejpam-4596	35	50	green	green	PROPN
ejpam-4596	35	51	’s	’s	PART
ejpam-4596	35	52	relations	relation	NOUN
ejpam-4596	35	53	for	for	ADP
ejpam-4596	35	54	te(x	te(x	NOUN
ejpam-4596	35	55	)	)	PUNCT
ejpam-4596	35	56	.	.	PUNCT
ejpam-4596	36	1	in	in	ADP
ejpam-4596	36	2	[	[	X
ejpam-4596	36	3	11	11	NUM
ejpam-4596	36	4	]	]	PUNCT
ejpam-4596	36	5	,	,	PUNCT
ejpam-4596	36	6	namnak	namnak	NOUN
ejpam-4596	36	7	and	and	CCONJ
ejpam-4596	36	8	laysirikul	laysirikul	PROPN
ejpam-4596	36	9	investigated	investigate	VERB
ejpam-4596	36	10	a	a	DET
ejpam-4596	36	11	necessary	necessary	ADJ
ejpam-4596	36	12	and	and	CCONJ
ejpam-4596	36	13	sufficient	sufficient	ADJ
ejpam-4596	36	14	condition	condition	NOUN
ejpam-4596	36	15	when	when	SCONJ
ejpam-4596	36	16	elements	element	NOUN
ejpam-4596	36	17	of	of	ADP
ejpam-4596	36	18	te(x	te(x	NOUN
ejpam-4596	36	19	)	)	PUNCT
ejpam-4596	36	20	to	to	PART
ejpam-4596	36	21	be	be	AUX
ejpam-4596	36	22	left	leave	VERB
ejpam-4596	36	23	regular	regular	ADV
ejpam-4596	36	24	,	,	PUNCT
ejpam-4596	36	25	right	right	ADV
ejpam-4596	36	26	regular	regular	ADJ
ejpam-4596	36	27	and	and	CCONJ
ejpam-4596	36	28	completely	completely	ADV
ejpam-4596	36	29	regular	regular	ADJ
ejpam-4596	36	30	.	.	PUNCT
ejpam-4596	37	1	for	for	ADP
ejpam-4596	37	2	a	a	DET
ejpam-4596	37	3	fixed	fix	VERB
ejpam-4596	37	4	element	element	NOUN
ejpam-4596	37	5	θ	θ	PROPN
ejpam-4596	37	6	in	in	ADP
ejpam-4596	37	7	te(x	te(x	PROPN
ejpam-4596	37	8	)	)	PUNCT
ejpam-4596	37	9	,	,	PUNCT
ejpam-4596	37	10	the	the	DET
ejpam-4596	37	11	variant	variant	ADJ
ejpam-4596	37	12	semigroup	semigroup	NOUN
ejpam-4596	37	13	of	of	ADP
ejpam-4596	37	14	te(x	te(x	NOUN
ejpam-4596	37	15	)	)	PUNCT
ejpam-4596	37	16	with	with	ADP
ejpam-4596	37	17	the	the	DET
ejpam-4596	37	18	sandwich	sandwich	NOUN
ejpam-4596	37	19	function	function	NOUN
ejpam-4596	37	20	θ	θ	PROPN
ejpam-4596	37	21	will	will	AUX
ejpam-4596	37	22	be	be	AUX
ejpam-4596	37	23	denoted	denote	VERB
ejpam-4596	37	24	simply	simply	ADV
ejpam-4596	37	25	by	by	ADP
ejpam-4596	37	26	te(x	te(x	NOUN
ejpam-4596	37	27	,	,	PUNCT
ejpam-4596	37	28	θ	θ	PROPN
ejpam-4596	37	29	)	)	PUNCT
ejpam-4596	37	30	.	.	PUNCT
ejpam-4596	38	1	green	green	PROPN
ejpam-4596	38	2	’s	’s	PART
ejpam-4596	38	3	equivalences	equivalence	VERB
ejpam-4596	38	4	for	for	ADP
ejpam-4596	38	5	elements	element	NOUN
ejpam-4596	38	6	in	in	ADP
ejpam-4596	38	7	the	the	DET
ejpam-4596	38	8	sandwich	sandwich	NOUN
ejpam-4596	38	9	semigroup	semigroup	PROPN
ejpam-4596	38	10	te(x	te(x	PROPN
ejpam-4596	38	11	,	,	PUNCT
ejpam-4596	38	12	θ	θ	PROPN
ejpam-4596	38	13	)	)	PUNCT
ejpam-4596	38	14	were	be	AUX
ejpam-4596	38	15	characterized	characterize	VERB
ejpam-4596	38	16	by	by	ADP
ejpam-4596	38	17	pei	pei	PROPN
ejpam-4596	38	18	alone	alone	PROPN
ejpam-4596	39	1	[	[	X
ejpam-4596	39	2	13	13	NUM
ejpam-4596	39	3	]	]	PUNCT
ejpam-4596	39	4	.	.	PUNCT
ejpam-4596	39	5	deng	deng	PROPN
ejpam-4596	39	6	,	,	PUNCT
ejpam-4596	39	7	zeng	zeng	PROPN
ejpam-4596	39	8	and	and	CCONJ
ejpam-4596	39	9	xu	xu	PROPN
ejpam-4596	40	1	[	[	X
ejpam-4596	40	2	7	7	X
ejpam-4596	40	3	]	]	PUNCT
ejpam-4596	40	4	introduced	introduce	VERB
ejpam-4596	40	5	a	a	DET
ejpam-4596	40	6	subsemigroup	subsemigroup	NOUN
ejpam-4596	40	7	of	of	ADP
ejpam-4596	40	8	t	t	PROPN
ejpam-4596	40	9	(	(	PUNCT
ejpam-4596	40	10	x	x	NOUN
ejpam-4596	40	11	)	)	PUNCT
ejpam-4596	40	12	defined	define	VERB
ejpam-4596	40	13	by	by	ADP
ejpam-4596	40	14	te∗(x	te∗(x	NOUN
ejpam-4596	40	15	)	)	PUNCT
ejpam-4596	40	16	=	=	NOUN
ejpam-4596	40	17	:	:	PUNCT
ejpam-4596	40	18	{	{	PUNCT
ejpam-4596	40	19	α	α	PROPN
ejpam-4596	40	20	∈	∈	PROPN
ejpam-4596	40	21	t	t	NOUN
ejpam-4596	40	22	(	(	PUNCT
ejpam-4596	40	23	x	x	X
ejpam-4596	40	24	)	)	PUNCT
ejpam-4596	40	25	:	:	PUNCT
ejpam-4596	40	26	∀x	∀x	NUM
ejpam-4596	40	27	,	,	PUNCT
ejpam-4596	40	28	y	y	PROPN
ejpam-4596	40	29	∈	∈	PROPN
ejpam-4596	40	30	x	x	X
ejpam-4596	40	31	,	,	PUNCT
ejpam-4596	40	32	(	(	PUNCT
ejpam-4596	40	33	x	x	NOUN
ejpam-4596	40	34	,	,	PUNCT
ejpam-4596	40	35	y	y	NOUN
ejpam-4596	40	36	)	)	PUNCT
ejpam-4596	40	37	∈	∈	PROPN
ejpam-4596	40	38	σ	σ	PUNCT
ejpam-4596	40	39	if	if	SCONJ
ejpam-4596	40	40	and	and	CCONJ
ejpam-4596	40	41	only	only	ADV
ejpam-4596	40	42	if	if	SCONJ
ejpam-4596	40	43	(	(	PUNCT
ejpam-4596	40	44	xα	xα	INTJ
ejpam-4596	40	45	,	,	PUNCT
ejpam-4596	40	46	yα	yα	NOUN
ejpam-4596	40	47	)	)	PUNCT
ejpam-4596	40	48	∈	∈	PROPN
ejpam-4596	40	49	σ	σ	PROPN
ejpam-4596	40	50	}	}	PUNCT
ejpam-4596	40	51	,	,	PUNCT
ejpam-4596	40	52	the	the	DET
ejpam-4596	40	53	so	so	ADV
ejpam-4596	40	54	-	-	PUNCT
ejpam-4596	40	55	called	call	VERB
ejpam-4596	40	56	semigroups	semigroup	NOUN
ejpam-4596	40	57	of	of	ADP
ejpam-4596	40	58	transformations	transformation	NOUN
ejpam-4596	40	59	that	that	PRON
ejpam-4596	40	60	preserve	preserve	VERB
ejpam-4596	40	61	double	double	ADJ
ejpam-4596	40	62	direction	direction	NOUN
ejpam-4596	40	63	equivalence	equivalence	NOUN
ejpam-4596	40	64	on	on	ADP
ejpam-4596	40	65	x.	x.	NOUN
ejpam-4596	40	66	they	they	PRON
ejpam-4596	40	67	investigated	investigate	VERB
ejpam-4596	40	68	the	the	DET
ejpam-4596	40	69	regularity	regularity	NOUN
ejpam-4596	40	70	and	and	CCONJ
ejpam-4596	40	71	green	green	PROPN
ejpam-4596	40	72	’s	’s	PART
ejpam-4596	40	73	relations	relation	NOUN
ejpam-4596	40	74	on	on	ADP
ejpam-4596	40	75	te∗(x	te∗(x	NOUN
ejpam-4596	40	76	)	)	PUNCT
ejpam-4596	40	77	.	.	PUNCT
ejpam-4596	41	1	later	later	ADV
ejpam-4596	41	2	,	,	PUNCT
ejpam-4596	41	3	laysilikul	laysilikul	ADJ
ejpam-4596	41	4	and	and	CCONJ
ejpam-4596	41	5	namnak	namnak	NOUN
ejpam-4596	41	6	[	[	X
ejpam-4596	41	7	8	8	NUM
ejpam-4596	41	8	]	]	PUNCT
ejpam-4596	41	9	investigated	investigate	VERB
ejpam-4596	41	10	a	a	DET
ejpam-4596	41	11	necessary	necessary	ADJ
ejpam-4596	41	12	and	and	CCONJ
ejpam-4596	41	13	sufficient	sufficient	ADJ
ejpam-4596	41	14	condition	condition	NOUN
ejpam-4596	41	15	for	for	ADP
ejpam-4596	41	16	the	the	DET
ejpam-4596	41	17	left	left	ADJ
ejpam-4596	41	18	regularity	regularity	NOUN
ejpam-4596	41	19	,	,	PUNCT
ejpam-4596	41	20	the	the	DET
ejpam-4596	41	21	right	right	ADJ
ejpam-4596	41	22	regularity	regularity	NOUN
ejpam-4596	41	23	and	and	CCONJ
ejpam-4596	41	24	the	the	DET
ejpam-4596	41	25	completely	completely	ADV
ejpam-4596	41	26	regularity	regularity	NOUN
ejpam-4596	41	27	of	of	ADP
ejpam-4596	41	28	elements	element	NOUN
ejpam-4596	41	29	in	in	ADP
ejpam-4596	41	30	te∗(x	te∗(x	NOUN
ejpam-4596	41	31	)	)	PUNCT
ejpam-4596	41	32	.	.	PUNCT
ejpam-4596	42	1	deng	deng	PROPN
ejpam-4596	43	1	[	[	X
ejpam-4596	43	2	3	3	NUM
ejpam-4596	43	3	]	]	PUNCT
ejpam-4596	43	4	discussed	discuss	VERB
ejpam-4596	43	5	the	the	DET
ejpam-4596	43	6	green	green	NOUN
ejpam-4596	43	7	’s	’s	PART
ejpam-4596	43	8	∗-relations	∗-relation	NOUN
ejpam-4596	43	9	,	,	PUNCT
ejpam-4596	43	10	certain	certain	ADJ
ejpam-4596	43	11	∗-ideal	∗-ideal	NOUN
ejpam-4596	43	12	and	and	CCONJ
ejpam-4596	43	13	certain	certain	ADJ
ejpam-4596	43	14	rees	ree	NOUN
ejpam-4596	43	15	quotient	quotient	VERB
ejpam-4596	43	16	semigroup	semigroup	ADJ
ejpam-4596	43	17	for	for	ADP
ejpam-4596	43	18	the	the	DET
ejpam-4596	43	19	semigroup	semigroup	PROPN
ejpam-4596	43	20	te∗(x	te∗(x	NOUN
ejpam-4596	43	21	)	)	PUNCT
ejpam-4596	43	22	and	and	CCONJ
ejpam-4596	43	23	proved	prove	VERB
ejpam-4596	43	24	that	that	SCONJ
ejpam-4596	43	25	regular	regular	ADJ
ejpam-4596	43	26	and	and	CCONJ
ejpam-4596	43	27	abundant	abundant	ADJ
ejpam-4596	43	28	in	in	ADP
ejpam-4596	43	29	the	the	DET
ejpam-4596	43	30	semigroup	semigroup	PROPN
ejpam-4596	43	31	te∗(x	te∗(x	NOUN
ejpam-4596	43	32	)	)	PUNCT
ejpam-4596	43	33	coincided	coincide	VERB
ejpam-4596	43	34	.	.	PUNCT
ejpam-4596	44	1	the	the	DET
ejpam-4596	44	2	variant	variant	ADJ
ejpam-4596	44	3	semigroup	semigroup	NOUN
ejpam-4596	44	4	of	of	ADP
ejpam-4596	44	5	te∗(x	te∗(x	NOUN
ejpam-4596	44	6	)	)	PUNCT
ejpam-4596	44	7	with	with	ADP
ejpam-4596	44	8	the	the	DET
ejpam-4596	44	9	sandwich	sandwich	NOUN
ejpam-4596	44	10	function	function	NOUN
ejpam-4596	44	11	θ	θ	PROPN
ejpam-4596	44	12	,	,	PUNCT
ejpam-4596	44	13	and	and	CCONJ
ejpam-4596	44	14	denoted	denote	VERB
ejpam-4596	44	15	by	by	ADP
ejpam-4596	44	16	(	(	PUNCT
ejpam-4596	44	17	te∗(x	te∗(x	NOUN
ejpam-4596	44	18	)	)	PUNCT
ejpam-4596	44	19	,	,	PUNCT
ejpam-4596	44	20	θ	θ	PROPN
ejpam-4596	44	21	)	)	PUNCT
ejpam-4596	44	22	.	.	PUNCT
ejpam-4596	45	1	yonthanthum	yonthanthum	PROPN
ejpam-4596	46	1	[	[	X
ejpam-4596	46	2	18	18	NUM
ejpam-4596	46	3	]	]	PUNCT
ejpam-4596	46	4	investigated	investigate	VERB
ejpam-4596	46	5	a	a	DET
ejpam-4596	46	6	necessary	necessary	ADJ
ejpam-4596	46	7	and	and	CCONJ
ejpam-4596	46	8	sufficient	sufficient	ADJ
ejpam-4596	46	9	condition	condition	NOUN
ejpam-4596	46	10	for	for	ADP
ejpam-4596	46	11	an	an	DET
ejpam-4596	46	12	element	element	NOUN
ejpam-4596	46	13	of	of	ADP
ejpam-4596	46	14	(	(	PUNCT
ejpam-4596	46	15	te∗(x	te∗(x	NOUN
ejpam-4596	46	16	)	)	PUNCT
ejpam-4596	46	17	,	,	PUNCT
ejpam-4596	46	18	θ	θ	PROPN
ejpam-4596	46	19	)	)	PUNCT
ejpam-4596	46	20	to	to	PART
ejpam-4596	46	21	be	be	AUX
ejpam-4596	46	22	regular	regular	ADJ
ejpam-4596	46	23	and	and	CCONJ
ejpam-4596	46	24	determined	determined	ADJ
ejpam-4596	46	25	when	when	SCONJ
ejpam-4596	46	26	(	(	PUNCT
ejpam-4596	46	27	te∗(x	te∗(x	NOUN
ejpam-4596	46	28	)	)	PUNCT
ejpam-4596	46	29	,	,	PUNCT
ejpam-4596	46	30	θ	θ	PROPN
ejpam-4596	46	31	)	)	PUNCT
ejpam-4596	46	32	is	be	AUX
ejpam-4596	46	33	a	a	DET
ejpam-4596	46	34	regular	regular	ADJ
ejpam-4596	46	35	semigroup	semigroup	NOUN
ejpam-4596	46	36	.	.	PUNCT
ejpam-4596	47	1	however	however	ADV
ejpam-4596	47	2	,	,	PUNCT
ejpam-4596	47	3	it	it	PRON
ejpam-4596	47	4	easy	easy	ADJ
ejpam-4596	47	5	to	to	PART
ejpam-4596	47	6	see	see	VERB
ejpam-4596	47	7	that	that	SCONJ
ejpam-4596	47	8	the	the	DET
ejpam-4596	47	9	notations	notation	NOUN
ejpam-4596	47	10	te(x	te(x	NOUN
ejpam-4596	47	11	)	)	PUNCT
ejpam-4596	47	12	(	(	PUNCT
ejpam-4596	47	13	te∗(x	te∗(x	NOUN
ejpam-4596	47	14	)	)	PUNCT
ejpam-4596	47	15	)	)	PUNCT
ejpam-4596	47	16	and	and	CCONJ
ejpam-4596	47	17	te(x	te(x	NOUN
ejpam-4596	47	18	,	,	PUNCT
ejpam-4596	47	19	θ	θ	PROPN
ejpam-4596	47	20	)	)	PUNCT
ejpam-4596	47	21	(	(	PUNCT
ejpam-4596	47	22	te∗(x	te∗(x	NOUN
ejpam-4596	47	23	,	,	PUNCT
ejpam-4596	47	24	θ	θ	NOUN
ejpam-4596	47	25	)	)	PUNCT
ejpam-4596	47	26	)	)	PUNCT
ejpam-4596	47	27	have	have	VERB
ejpam-4596	47	28	the	the	DET
ejpam-4596	47	29	same	same	ADJ
ejpam-4596	47	30	elements	element	NOUN
ejpam-4596	47	31	.	.	PUNCT
ejpam-4596	48	1	hence	hence	ADV
ejpam-4596	48	2	,	,	PUNCT
ejpam-4596	48	3	these	these	PRON
ejpam-4596	48	4	are	be	AUX
ejpam-4596	48	5	the	the	DET
ejpam-4596	48	6	same	same	ADJ
ejpam-4596	48	7	set	set	NOUN
ejpam-4596	48	8	but	but	CCONJ
ejpam-4596	48	9	need	need	AUX
ejpam-4596	48	10	not	not	PART
ejpam-4596	48	11	be	be	AUX
ejpam-4596	48	12	the	the	DET
ejpam-4596	48	13	same	same	ADJ
ejpam-4596	48	14	semigroup	semigroup	NOUN
ejpam-4596	48	15	.	.	PUNCT
ejpam-4596	49	1	if	if	SCONJ
ejpam-4596	49	2	θ	θ	PROPN
ejpam-4596	49	3	is	be	AUX
ejpam-4596	49	4	the	the	DET
ejpam-4596	49	5	identity	identity	NOUN
ejpam-4596	49	6	transformation	transformation	NOUN
ejpam-4596	49	7	,	,	PUNCT
ejpam-4596	49	8	then	then	ADV
ejpam-4596	49	9	these	these	DET
ejpam-4596	49	10	semigroups	semigroup	NOUN
ejpam-4596	49	11	are	be	AUX
ejpam-4596	49	12	coincided	coincide	VERB
ejpam-4596	49	13	.	.	PUNCT
ejpam-4596	50	1	therefore	therefore	ADV
ejpam-4596	50	2	semigroups	semigroup	VERB
ejpam-4596	50	3	te(x	te(x	PROPN
ejpam-4596	50	4	,	,	PUNCT
ejpam-4596	50	5	θ	θ	PROPN
ejpam-4596	50	6	)	)	PUNCT
ejpam-4596	50	7	(	(	PUNCT
ejpam-4596	50	8	te∗(x	te∗(x	NOUN
ejpam-4596	50	9	,	,	PUNCT
ejpam-4596	50	10	θ	θ	NOUN
ejpam-4596	50	11	)	)	PUNCT
ejpam-4596	50	12	)	)	PUNCT
ejpam-4596	50	13	is	be	AUX
ejpam-4596	50	14	a	a	DET
ejpam-4596	50	15	generalization	generalization	NOUN
ejpam-4596	50	16	of	of	ADP
ejpam-4596	50	17	te(x	te(x	PROPN
ejpam-4596	50	18	)	)	PUNCT
ejpam-4596	50	19	(	(	PUNCT
ejpam-4596	50	20	te∗(x	te∗(x	NOUN
ejpam-4596	50	21	)	)	PUNCT
ejpam-4596	50	22	)	)	PUNCT
ejpam-4596	50	23	and	and	CCONJ
ejpam-4596	50	24	te(x	te(x	NOUN
ejpam-4596	50	25	,	,	PUNCT
ejpam-4596	50	26	θ	θ	PROPN
ejpam-4596	50	27	)	)	PUNCT
ejpam-4596	50	28	.	.	PUNCT
ejpam-4596	51	1	p.	p.	NOUN
ejpam-4596	51	2	tantong	tantong	NOUN
ejpam-4596	51	3	,	,	PUNCT
ejpam-4596	51	4	n.	n.	NOUN
ejpam-4596	51	5	sawatraksa	sawatraksa	PROPN
ejpam-4596	51	6	/	/	SYM
ejpam-4596	51	7	eur	eur	PROPN
ejpam-4596	51	8	.	.	PUNCT
ejpam-4596	52	1	j.	j.	PROPN
ejpam-4596	52	2	pure	pure	PROPN
ejpam-4596	52	3	appl	appl	PROPN
ejpam-4596	52	4	.	.	PROPN
ejpam-4596	52	5	math	math	PROPN
ejpam-4596	52	6	,	,	PUNCT
ejpam-4596	52	7	15	15	NUM
ejpam-4596	52	8	(	(	PUNCT
ejpam-4596	52	9	4	4	NUM
ejpam-4596	52	10	)	)	PUNCT
ejpam-4596	52	11	(	(	PUNCT
ejpam-4596	52	12	2022	2022	NUM
ejpam-4596	52	13	)	)	PUNCT
ejpam-4596	52	14	,	,	PUNCT
ejpam-4596	52	15	2116	2116	NUM
ejpam-4596	52	16	-	-	SYM
ejpam-4596	52	17	2126	2126	NUM
ejpam-4596	52	18	2118	2118	NUM
ejpam-4596	52	19	in	in	ADP
ejpam-4596	52	20	this	this	DET
ejpam-4596	52	21	paper	paper	NOUN
ejpam-4596	52	22	,	,	PUNCT
ejpam-4596	52	23	for	for	ADP
ejpam-4596	52	24	a	a	DET
ejpam-4596	52	25	fixed	fix	VERB
ejpam-4596	52	26	element	element	NOUN
ejpam-4596	52	27	θ	θ	PROPN
ejpam-4596	52	28	∈	∈	PROPN
ejpam-4596	52	29	te(x	te(x	NOUN
ejpam-4596	52	30	)	)	PUNCT
ejpam-4596	52	31	(	(	PUNCT
ejpam-4596	52	32	θ	θ	PROPN
ejpam-4596	52	33	∈	∈	PROPN
ejpam-4596	52	34	te∗(x	te∗(x	NOUN
ejpam-4596	52	35	)	)	PUNCT
ejpam-4596	52	36	)	)	PUNCT
ejpam-4596	52	37	,	,	PUNCT
ejpam-4596	52	38	the	the	DET
ejpam-4596	52	39	variant	variant	NOUN
ejpam-4596	52	40	semigroup	semigroup	NOUN
ejpam-4596	52	41	of	of	ADP
ejpam-4596	52	42	te(x	te(x	NOUN
ejpam-4596	52	43	)	)	PUNCT
ejpam-4596	52	44	(	(	PUNCT
ejpam-4596	52	45	te∗(x	te∗(x	NOUN
ejpam-4596	52	46	)	)	PUNCT
ejpam-4596	52	47	)	)	PUNCT
ejpam-4596	52	48	with	with	ADP
ejpam-4596	52	49	the	the	DET
ejpam-4596	52	50	sandwich	sandwich	NOUN
ejpam-4596	52	51	function	function	NOUN
ejpam-4596	52	52	θ	θ	PROPN
ejpam-4596	52	53	will	will	AUX
ejpam-4596	52	54	be	be	AUX
ejpam-4596	52	55	denoted	denote	VERB
ejpam-4596	52	56	by	by	ADP
ejpam-4596	52	57	te(x	te(x	NOUN
ejpam-4596	52	58	,	,	PUNCT
ejpam-4596	52	59	θ	θ	PROPN
ejpam-4596	52	60	)	)	PUNCT
ejpam-4596	52	61	(	(	PUNCT
ejpam-4596	52	62	te∗(x	te∗(x	NOUN
ejpam-4596	52	63	,	,	PUNCT
ejpam-4596	52	64	θ	θ	NOUN
ejpam-4596	52	65	)	)	PUNCT
ejpam-4596	52	66	)	)	PUNCT
ejpam-4596	52	67	.	.	PUNCT
ejpam-4596	53	1	this	this	DET
ejpam-4596	53	2	paper	paper	NOUN
ejpam-4596	53	3	aims	aim	VERB
ejpam-4596	53	4	to	to	PART
ejpam-4596	53	5	characterize	characterize	VERB
ejpam-4596	53	6	the	the	DET
ejpam-4596	53	7	regular	regular	NOUN
ejpam-4596	53	8	,	,	PUNCT
ejpam-4596	53	9	the	the	DET
ejpam-4596	53	10	left	left	ADJ
ejpam-4596	53	11	regular	regular	NOUN
ejpam-4596	53	12	,	,	PUNCT
ejpam-4596	53	13	the	the	DET
ejpam-4596	53	14	right	right	ADJ
ejpam-4596	53	15	regular	regular	NOUN
ejpam-4596	53	16	and	and	CCONJ
ejpam-4596	53	17	the	the	DET
ejpam-4596	53	18	completely	completely	ADV
ejpam-4596	53	19	regular	regular	ADJ
ejpam-4596	53	20	for	for	ADP
ejpam-4596	53	21	elements	element	NOUN
ejpam-4596	53	22	of	of	ADP
ejpam-4596	53	23	te(x	te(x	NOUN
ejpam-4596	53	24	,	,	PUNCT
ejpam-4596	53	25	θ	θ	NOUN
ejpam-4596	53	26	)	)	PUNCT
ejpam-4596	53	27	and	and	CCONJ
ejpam-4596	53	28	te∗(x	te∗(x	NOUN
ejpam-4596	53	29	,	,	PUNCT
ejpam-4596	53	30	θ	θ	NOUN
ejpam-4596	53	31	)	)	PUNCT
ejpam-4596	53	32	.	.	PUNCT
ejpam-4596	54	1	moreover	moreover	ADV
ejpam-4596	54	2	,	,	PUNCT
ejpam-4596	54	3	we	we	PRON
ejpam-4596	54	4	give	give	VERB
ejpam-4596	54	5	a	a	DET
ejpam-4596	54	6	necessary	necessary	ADJ
ejpam-4596	54	7	and	and	CCONJ
ejpam-4596	54	8	sufficient	sufficient	ADJ
ejpam-4596	54	9	condition	condition	NOUN
ejpam-4596	54	10	for	for	ADP
ejpam-4596	54	11	the	the	DET
ejpam-4596	54	12	identity	identity	NOUN
ejpam-4596	54	13	transformation	transformation	NOUN
ejpam-4596	54	14	idx	idx	NOUN
ejpam-4596	54	15	of	of	ADP
ejpam-4596	54	16	the	the	DET
ejpam-4596	54	17	semigroup	semigroup	PROPN
ejpam-4596	54	18	te(x	te(x	PROPN
ejpam-4596	54	19	)	)	PUNCT
ejpam-4596	54	20	to	to	PART
ejpam-4596	54	21	be	be	AUX
ejpam-4596	54	22	regular	regular	ADJ
ejpam-4596	54	23	,	,	PUNCT
ejpam-4596	54	24	left	leave	VERB
ejpam-4596	54	25	regular	regular	ADV
ejpam-4596	54	26	,	,	PUNCT
ejpam-4596	54	27	right	right	ADV
ejpam-4596	54	28	regular	regular	ADJ
ejpam-4596	54	29	and	and	CCONJ
ejpam-4596	54	30	completely	completely	ADV
ejpam-4596	54	31	regular	regular	ADJ
ejpam-4596	54	32	elements	element	NOUN
ejpam-4596	54	33	in	in	ADP
ejpam-4596	54	34	te(x	te(x	NOUN
ejpam-4596	54	35	,	,	PUNCT
ejpam-4596	54	36	θ	θ	NOUN
ejpam-4596	54	37	)	)	PUNCT
ejpam-4596	54	38	and	and	CCONJ
ejpam-4596	54	39	te∗(x	te∗(x	NOUN
ejpam-4596	54	40	,	,	PUNCT
ejpam-4596	54	41	θ	θ	NOUN
ejpam-4596	54	42	)	)	PUNCT
ejpam-4596	54	43	.	.	PUNCT
ejpam-4596	55	1	in	in	ADP
ejpam-4596	55	2	this	this	DET
ejpam-4596	55	3	introductory	introductory	ADJ
ejpam-4596	55	4	section	section	NOUN
ejpam-4596	55	5	,	,	PUNCT
ejpam-4596	55	6	we	we	PRON
ejpam-4596	55	7	present	present	VERB
ejpam-4596	55	8	many	many	ADJ
ejpam-4596	55	9	notations	notation	NOUN
ejpam-4596	55	10	and	and	CCONJ
ejpam-4596	55	11	lemma	lemma	PROPN
ejpam-4596	55	12	most	most	ADJ
ejpam-4596	55	13	of	of	ADP
ejpam-4596	55	14	which	which	PRON
ejpam-4596	55	15	will	will	AUX
ejpam-4596	55	16	be	be	AUX
ejpam-4596	55	17	indispensable	indispensable	ADJ
ejpam-4596	55	18	for	for	ADP
ejpam-4596	55	19	our	our	PRON
ejpam-4596	55	20	research	research	NOUN
ejpam-4596	55	21	.	.	PUNCT
ejpam-4596	56	1	for	for	ADP
ejpam-4596	56	2	arbitrary	arbitrary	ADJ
ejpam-4596	56	3	semigroup	semigroup	PROPN
ejpam-4596	56	4	s	s	PROPN
ejpam-4596	56	5	,	,	PUNCT
ejpam-4596	56	6	letreg(s	letreg(s	NOUN
ejpam-4596	56	7	)	)	PUNCT
ejpam-4596	56	8	,	,	PUNCT
ejpam-4596	56	9	lreg(s	lreg(s	NOUN
ejpam-4596	56	10	)	)	PUNCT
ejpam-4596	56	11	,	,	PUNCT
ejpam-4596	56	12	rreg(s	rreg(s	NOUN
ejpam-4596	56	13	)	)	PUNCT
ejpam-4596	56	14	and	and	CCONJ
ejpam-4596	56	15	creg(s	creg(s	NOUN
ejpam-4596	56	16	)	)	PUNCT
ejpam-4596	56	17	,	,	PUNCT
ejpam-4596	56	18	denote	denote	VERB
ejpam-4596	56	19	the	the	DET
ejpam-4596	56	20	set	set	NOUN
ejpam-4596	56	21	of	of	ADP
ejpam-4596	56	22	all	all	DET
ejpam-4596	56	23	regular	regular	ADJ
ejpam-4596	56	24	elements	element	NOUN
ejpam-4596	56	25	,	,	PUNCT
ejpam-4596	56	26	the	the	DET
ejpam-4596	56	27	set	set	NOUN
ejpam-4596	56	28	of	of	ADP
ejpam-4596	56	29	all	all	PRON
ejpam-4596	56	30	left	leave	VERB
ejpam-4596	56	31	regular	regular	ADJ
ejpam-4596	56	32	elements	element	NOUN
ejpam-4596	56	33	,	,	PUNCT
ejpam-4596	56	34	the	the	DET
ejpam-4596	56	35	set	set	NOUN
ejpam-4596	56	36	of	of	ADP
ejpam-4596	56	37	all	all	DET
ejpam-4596	56	38	right	right	ADJ
ejpam-4596	56	39	regular	regular	ADJ
ejpam-4596	56	40	elements	element	NOUN
ejpam-4596	56	41	and	and	CCONJ
ejpam-4596	56	42	the	the	DET
ejpam-4596	56	43	set	set	NOUN
ejpam-4596	56	44	of	of	ADP
ejpam-4596	56	45	all	all	DET
ejpam-4596	56	46	completely	completely	ADV
ejpam-4596	56	47	regular	regular	ADJ
ejpam-4596	56	48	elements	element	NOUN
ejpam-4596	56	49	of	of	ADP
ejpam-4596	56	50	s	s	NOUN
ejpam-4596	56	51	,	,	PUNCT
ejpam-4596	56	52	respectively	respectively	ADV
ejpam-4596	56	53	.	.	PUNCT
ejpam-4596	57	1	for	for	ADP
ejpam-4596	57	2	a	a	DET
ejpam-4596	57	3	nonempty	nonempty	ADV
ejpam-4596	57	4	set	set	VERB
ejpam-4596	57	5	x	x	PUNCT
ejpam-4596	57	6	and	and	CCONJ
ejpam-4596	57	7	α	α	PROPN
ejpam-4596	57	8	∈	∈	PROPN
ejpam-4596	57	9	t	t	NOUN
ejpam-4596	57	10	(	(	PUNCT
ejpam-4596	57	11	x	x	X
ejpam-4596	57	12	)	)	PUNCT
ejpam-4596	57	13	,	,	PUNCT
ejpam-4596	57	14	we	we	PRON
ejpam-4596	57	15	denote	denote	VERB
ejpam-4596	57	16	by	by	ADP
ejpam-4596	57	17	π(α	π(α	PROPN
ejpam-4596	57	18	)	)	PUNCT
ejpam-4596	57	19	the	the	DET
ejpam-4596	57	20	partition	partition	NOUN
ejpam-4596	57	21	of	of	ADP
ejpam-4596	57	22	x	x	PUNCT
ejpam-4596	57	23	induced	induce	VERB
ejpam-4596	57	24	by	by	ADP
ejpam-4596	57	25	α	α	NOUN
ejpam-4596	57	26	,	,	PUNCT
ejpam-4596	57	27	namely	namely	ADV
ejpam-4596	57	28	,	,	PUNCT
ejpam-4596	57	29	π(α	π(α	PROPN
ejpam-4596	57	30	)	)	PUNCT
ejpam-4596	57	31	=	=	PRON
ejpam-4596	58	1	{	{	PUNCT
ejpam-4596	58	2	yα−1	yα−1	NOUN
ejpam-4596	58	3	:	:	PUNCT
ejpam-4596	58	4	y	y	PROPN
ejpam-4596	58	5	∈	∈	PROPN
ejpam-4596	58	6	xα	xα	ADP
ejpam-4596	58	7	}	}	PUNCT
ejpam-4596	58	8	.	.	PUNCT
ejpam-4596	59	1	then	then	ADV
ejpam-4596	59	2	π(α	π(α	NUM
ejpam-4596	59	3	)	)	PUNCT
ejpam-4596	60	1	=	=	SYM
ejpam-4596	60	2	x/	x/	NOUN
ejpam-4596	60	3	ker(α	ker(α	PROPN
ejpam-4596	60	4	)	)	PUNCT
ejpam-4596	60	5	where	where	SCONJ
ejpam-4596	60	6	ker(α	ker(α	X
ejpam-4596	60	7	)	)	PUNCT
ejpam-4596	60	8	=	=	PRON
ejpam-4596	60	9	{	{	PUNCT
ejpam-4596	60	10	(	(	PUNCT
ejpam-4596	60	11	x	x	NOUN
ejpam-4596	60	12	,	,	PUNCT
ejpam-4596	60	13	y	y	NOUN
ejpam-4596	60	14	)	)	PUNCT
ejpam-4596	60	15	∈	∈	PROPN
ejpam-4596	61	1	x	x	X
ejpam-4596	61	2	×	×	NOUN
ejpam-4596	61	3	x	x	X
ejpam-4596	61	4	:	:	PUNCT
ejpam-4596	61	5	xα	xα	PROPN
ejpam-4596	61	6	=	=	PUNCT
ejpam-4596	61	7	yα	yα	NOUN
ejpam-4596	61	8	}	}	PUNCT
ejpam-4596	61	9	.	.	PUNCT
ejpam-4596	62	1	for	for	ADP
ejpam-4596	62	2	a	a	DET
ejpam-4596	62	3	⊆	⊆	NUM
ejpam-4596	62	4	x	x	SYM
ejpam-4596	62	5	,	,	PUNCT
ejpam-4596	62	6	we	we	PRON
ejpam-4596	62	7	define	define	VERB
ejpam-4596	62	8	πa(α	πa(α	PUNCT
ejpam-4596	62	9	)	)	PUNCT
ejpam-4596	62	10	=	=	SYM
ejpam-4596	62	11	{	{	PUNCT
ejpam-4596	62	12	p	p	X
ejpam-4596	62	13	∈	∈	PROPN
ejpam-4596	62	14	π(α	π(α	PROPN
ejpam-4596	62	15	)	)	PUNCT
ejpam-4596	62	16	:	:	PUNCT
ejpam-4596	63	1	p	p	X
ejpam-4596	63	2	∩a	∩a	PROPN
ejpam-4596	63	3	̸=	̸=	PROPN
ejpam-4596	63	4	∅	∅	NOUN
ejpam-4596	63	5	}	}	PUNCT
ejpam-4596	63	6	.	.	PUNCT
ejpam-4596	64	1	in	in	ADP
ejpam-4596	64	2	the	the	DET
ejpam-4596	64	3	remainder	remainder	NOUN
ejpam-4596	64	4	,	,	PUNCT
ejpam-4596	64	5	let	let	VERB
ejpam-4596	64	6	e	e	PRON
ejpam-4596	64	7	be	be	AUX
ejpam-4596	64	8	an	an	DET
ejpam-4596	64	9	equivalence	equivalence	NOUN
ejpam-4596	64	10	relation	relation	NOUN
ejpam-4596	64	11	on	on	ADP
ejpam-4596	64	12	a	a	DET
ejpam-4596	64	13	nonempty	nonempty	ADV
ejpam-4596	64	14	set	set	VERB
ejpam-4596	64	15	x.	x.	NOUN
ejpam-4596	64	16	denote	denote	VERB
ejpam-4596	64	17	by	by	ADP
ejpam-4596	64	18	x	x	X
ejpam-4596	64	19	/	/	SYM
ejpam-4596	64	20	e	e	X
ejpam-4596	64	21	the	the	DET
ejpam-4596	64	22	quotient	quotient	NOUN
ejpam-4596	64	23	set	set	VERB
ejpam-4596	64	24	.	.	PUNCT
ejpam-4596	65	1	for	for	SCONJ
ejpam-4596	65	2	x	x	PROPN
ejpam-4596	65	3	∈	∈	PROPN
ejpam-4596	65	4	x	x	X
ejpam-4596	65	5	,	,	PUNCT
ejpam-4596	65	6	we	we	PRON
ejpam-4596	65	7	write	write	VERB
ejpam-4596	65	8	ex	ex	PRON
ejpam-4596	65	9	as	as	ADP
ejpam-4596	65	10	for	for	ADP
ejpam-4596	65	11	the	the	DET
ejpam-4596	65	12	set	set	NOUN
ejpam-4596	65	13	of	of	ADP
ejpam-4596	65	14	all	all	DET
ejpam-4596	65	15	elements	element	NOUN
ejpam-4596	65	16	of	of	ADP
ejpam-4596	65	17	x	x	PRON
ejpam-4596	65	18	that	that	PRON
ejpam-4596	65	19	are	be	AUX
ejpam-4596	65	20	equivalent	equivalent	ADJ
ejpam-4596	65	21	to	to	ADP
ejpam-4596	65	22	x	x	PRON
ejpam-4596	65	23	,	,	PUNCT
ejpam-4596	65	24	that	that	ADV
ejpam-4596	65	25	is	is	ADV
ejpam-4596	65	26	,	,	PUNCT
ejpam-4596	65	27	ex	ex	X
ejpam-4596	65	28	=	=	PUNCT
ejpam-4596	65	29	{	{	PUNCT
ejpam-4596	65	30	y	y	PROPN
ejpam-4596	65	31	∈	∈	PROPN
ejpam-4596	65	32	x	x	X
ejpam-4596	65	33	:	:	PUNCT
ejpam-4596	65	34	(	(	PUNCT
ejpam-4596	65	35	x	x	NOUN
ejpam-4596	65	36	,	,	PUNCT
ejpam-4596	65	37	y	y	NOUN
ejpam-4596	65	38	)	)	PUNCT
ejpam-4596	65	39	∈	∈	PROPN
ejpam-4596	65	40	e	e	NOUN
ejpam-4596	65	41	}	}	PUNCT
ejpam-4596	65	42	.	.	PUNCT
ejpam-4596	66	1	the	the	DET
ejpam-4596	66	2	following	follow	VERB
ejpam-4596	66	3	lemma	lemma	PROPN
ejpam-4596	66	4	is	be	AUX
ejpam-4596	66	5	needed	need	VERB
ejpam-4596	66	6	.	.	PUNCT
ejpam-4596	67	1	lemma	lemma	PROPN
ejpam-4596	67	2	1	1	NUM
ejpam-4596	67	3	.	.	PUNCT
ejpam-4596	68	1	[	[	X
ejpam-4596	68	2	13	13	NUM
ejpam-4596	68	3	]	]	PUNCT
ejpam-4596	68	4	let	let	VERB
ejpam-4596	68	5	α	α	PRON
ejpam-4596	68	6	∈	∈	PROPN
ejpam-4596	68	7	t	t	PROPN
ejpam-4596	68	8	(	(	PUNCT
ejpam-4596	68	9	x	x	NOUN
ejpam-4596	68	10	)	)	PUNCT
ejpam-4596	68	11	.	.	PUNCT
ejpam-4596	69	1	then	then	ADV
ejpam-4596	69	2	α	α	PROPN
ejpam-4596	69	3	∈	∈	PROPN
ejpam-4596	69	4	te(x	te(x	NOUN
ejpam-4596	69	5	)	)	PUNCT
ejpam-4596	69	6	if	if	SCONJ
ejpam-4596	69	7	and	and	CCONJ
ejpam-4596	69	8	only	only	ADV
ejpam-4596	69	9	if	if	SCONJ
ejpam-4596	69	10	for	for	ADP
ejpam-4596	69	11	each	each	DET
ejpam-4596	69	12	a	a	DET
ejpam-4596	69	13	∈	∈	PROPN
ejpam-4596	69	14	x	x	SYM
ejpam-4596	69	15	/	/	SYM
ejpam-4596	69	16	e	e	NOUN
ejpam-4596	69	17	,	,	PUNCT
ejpam-4596	69	18	there	there	PRON
ejpam-4596	69	19	exists	exist	VERB
ejpam-4596	69	20	b	b	PROPN
ejpam-4596	69	21	∈	∈	PROPN
ejpam-4596	69	22	x	x	SYM
ejpam-4596	69	23	/	/	SYM
ejpam-4596	69	24	e	e	VERB
ejpam-4596	69	25	such	such	ADJ
ejpam-4596	69	26	that	that	DET
ejpam-4596	69	27	aα	aα	PROPN
ejpam-4596	69	28	⊆	⊆	NUM
ejpam-4596	69	29	b.	b.	PROPN
ejpam-4596	69	30	2	2	NUM
ejpam-4596	69	31	.	.	PUNCT
ejpam-4596	69	32	regularity	regularity	NOUN
ejpam-4596	69	33	of	of	ADP
ejpam-4596	69	34	variants	variant	NOUN
ejpam-4596	69	35	of	of	ADP
ejpam-4596	69	36	transformation	transformation	NOUN
ejpam-4596	69	37	semigroups	semigroup	NOUN
ejpam-4596	69	38	that	that	PRON
ejpam-4596	69	39	preserve	preserve	VERB
ejpam-4596	69	40	an	an	DET
ejpam-4596	69	41	equivalence	equivalence	NOUN
ejpam-4596	69	42	relation	relation	NOUN
ejpam-4596	69	43	in	in	ADP
ejpam-4596	69	44	this	this	DET
ejpam-4596	69	45	section	section	NOUN
ejpam-4596	69	46	,	,	PUNCT
ejpam-4596	69	47	we	we	PRON
ejpam-4596	69	48	characterize	characterize	VERB
ejpam-4596	69	49	the	the	DET
ejpam-4596	69	50	regular	regular	ADJ
ejpam-4596	69	51	,	,	PUNCT
ejpam-4596	69	52	left	leave	VERB
ejpam-4596	69	53	regular	regular	ADV
ejpam-4596	69	54	,	,	PUNCT
ejpam-4596	69	55	right	right	ADV
ejpam-4596	69	56	regular	regular	ADJ
ejpam-4596	69	57	and	and	CCONJ
ejpam-4596	69	58	completely	completely	ADV
ejpam-4596	69	59	regular	regular	ADJ
ejpam-4596	69	60	elements	element	NOUN
ejpam-4596	69	61	of	of	ADP
ejpam-4596	69	62	the	the	DET
ejpam-4596	69	63	variant	variant	ADJ
ejpam-4596	69	64	semigroup	semigroup	NOUN
ejpam-4596	69	65	te(x	te(x	NOUN
ejpam-4596	69	66	,	,	PUNCT
ejpam-4596	69	67	θ	θ	PROPN
ejpam-4596	69	68	)	)	PUNCT
ejpam-4596	69	69	.	.	PUNCT
ejpam-4596	70	1	the	the	DET
ejpam-4596	70	2	identity	identity	NOUN
ejpam-4596	70	3	transformation	transformation	NOUN
ejpam-4596	70	4	on	on	ADP
ejpam-4596	70	5	x	x	PRON
ejpam-4596	70	6	,	,	PUNCT
ejpam-4596	70	7	namely	namely	ADV
ejpam-4596	70	8	,	,	PUNCT
ejpam-4596	70	9	idx	idx	PROPN
ejpam-4596	70	10	is	be	AUX
ejpam-4596	70	11	the	the	DET
ejpam-4596	70	12	identity	identity	NOUN
ejpam-4596	70	13	of	of	ADP
ejpam-4596	70	14	semigroup	semigroup	PROPN
ejpam-4596	70	15	te(x	te(x	PROPN
ejpam-4596	70	16	)	)	PUNCT
ejpam-4596	70	17	but	but	CCONJ
ejpam-4596	70	18	need	need	VERB
ejpam-4596	70	19	not	not	PART
ejpam-4596	70	20	to	to	PART
ejpam-4596	70	21	be	be	AUX
ejpam-4596	70	22	the	the	DET
ejpam-4596	70	23	identity	identity	NOUN
ejpam-4596	70	24	of	of	ADP
ejpam-4596	70	25	the	the	DET
ejpam-4596	70	26	semigroup	semigroup	PROPN
ejpam-4596	70	27	te(x	te(x	PROPN
ejpam-4596	70	28	,	,	PUNCT
ejpam-4596	70	29	θ	θ	PROPN
ejpam-4596	70	30	)	)	PUNCT
ejpam-4596	70	31	.	.	PUNCT
ejpam-4596	71	1	in	in	ADP
ejpam-4596	71	2	addition	addition	NOUN
ejpam-4596	71	3	,	,	PUNCT
ejpam-4596	71	4	we	we	PRON
ejpam-4596	71	5	give	give	VERB
ejpam-4596	71	6	a	a	DET
ejpam-4596	71	7	necessary	necessary	ADJ
ejpam-4596	71	8	and	and	CCONJ
ejpam-4596	71	9	sufficient	sufficient	ADJ
ejpam-4596	71	10	condition	condition	NOUN
ejpam-4596	71	11	for	for	ADP
ejpam-4596	71	12	idx	idx	NOUN
ejpam-4596	71	13	of	of	ADP
ejpam-4596	71	14	the	the	DET
ejpam-4596	71	15	semigroup	semigroup	PROPN
ejpam-4596	71	16	te(x	te(x	PROPN
ejpam-4596	71	17	)	)	PUNCT
ejpam-4596	71	18	to	to	PART
ejpam-4596	71	19	be	be	AUX
ejpam-4596	71	20	regular	regular	ADJ
ejpam-4596	71	21	,	,	PUNCT
ejpam-4596	71	22	left	leave	VERB
ejpam-4596	71	23	regular	regular	ADV
ejpam-4596	71	24	,	,	PUNCT
ejpam-4596	71	25	right	right	ADV
ejpam-4596	71	26	regular	regular	ADJ
ejpam-4596	71	27	and	and	CCONJ
ejpam-4596	71	28	completely	completely	ADV
ejpam-4596	71	29	regular	regular	ADJ
ejpam-4596	71	30	elements	element	NOUN
ejpam-4596	71	31	in	in	ADP
ejpam-4596	71	32	te(x	te(x	NOUN
ejpam-4596	71	33	,	,	PUNCT
ejpam-4596	71	34	θ	θ	PROPN
ejpam-4596	71	35	)	)	PUNCT
ejpam-4596	71	36	.	.	PUNCT
ejpam-4596	72	1	theorem	theorem	NOUN
ejpam-4596	72	2	1	1	NUM
ejpam-4596	72	3	.	.	PUNCT
ejpam-4596	73	1	let	let	VERB
ejpam-4596	73	2	α	α	PRON
ejpam-4596	73	3	∈	∈	PROPN
ejpam-4596	73	4	te(x	te(x	NOUN
ejpam-4596	73	5	,	,	PUNCT
ejpam-4596	73	6	θ	θ	PROPN
ejpam-4596	73	7	)	)	PUNCT
ejpam-4596	73	8	.	.	PUNCT
ejpam-4596	74	1	then	then	ADV
ejpam-4596	74	2	α	α	PROPN
ejpam-4596	74	3	∈	∈	PROPN
ejpam-4596	74	4	reg(te(x	reg(te(x	PROPN
ejpam-4596	74	5	,	,	PUNCT
ejpam-4596	74	6	θ	θ	NOUN
ejpam-4596	74	7	)	)	PUNCT
ejpam-4596	74	8	)	)	PUNCT
ejpam-4596	75	1	if	if	SCONJ
ejpam-4596	75	2	and	and	CCONJ
ejpam-4596	75	3	only	only	ADV
ejpam-4596	75	4	if	if	SCONJ
ejpam-4596	75	5	(	(	PUNCT
ejpam-4596	75	6	i	i	NOUN
ejpam-4596	75	7	)	)	PUNCT
ejpam-4596	75	8	ker(α	ker(α	PROPN
ejpam-4596	75	9	)	)	PUNCT
ejpam-4596	75	10	=	=	SYM
ejpam-4596	75	11	ker(αθ	ker(αθ	NOUN
ejpam-4596	75	12	)	)	PUNCT
ejpam-4596	75	13	and	and	CCONJ
ejpam-4596	75	14	(	(	PUNCT
ejpam-4596	75	15	ii	ii	NOUN
ejpam-4596	75	16	)	)	PUNCT
ejpam-4596	75	17	for	for	ADP
ejpam-4596	75	18	every	every	DET
ejpam-4596	75	19	a	a	DET
ejpam-4596	75	20	∈	∈	PROPN
ejpam-4596	75	21	x	x	SYM
ejpam-4596	75	22	/	/	SYM
ejpam-4596	75	23	e	e	NOUN
ejpam-4596	75	24	,	,	PUNCT
ejpam-4596	75	25	there	there	PRON
ejpam-4596	75	26	exists	exist	VERB
ejpam-4596	75	27	b	b	PROPN
ejpam-4596	75	28	∈	∈	PROPN
ejpam-4596	75	29	x	x	SYM
ejpam-4596	75	30	/	/	SYM
ejpam-4596	75	31	e	e	VERB
ejpam-4596	75	32	such	such	ADJ
ejpam-4596	75	33	that	that	SCONJ
ejpam-4596	75	34	a	a	DET
ejpam-4596	75	35	∩xαθ	∩xαθ	NOUN
ejpam-4596	75	36	⊆	⊆	NUM
ejpam-4596	75	37	bθαθ	bθαθ	NOUN
ejpam-4596	75	38	.	.	PUNCT
ejpam-4596	76	1	proof	proof	NOUN
ejpam-4596	76	2	.	.	PUNCT
ejpam-4596	77	1	assume	assume	VERB
ejpam-4596	77	2	that	that	SCONJ
ejpam-4596	77	3	α	α	PRON
ejpam-4596	77	4	is	be	AUX
ejpam-4596	77	5	regular	regular	ADJ
ejpam-4596	77	6	of	of	ADP
ejpam-4596	77	7	te(x	te(x	NOUN
ejpam-4596	77	8	,	,	PUNCT
ejpam-4596	77	9	θ	θ	PROPN
ejpam-4596	77	10	)	)	PUNCT
ejpam-4596	77	11	.	.	PUNCT
ejpam-4596	78	1	then	then	ADV
ejpam-4596	78	2	α	α	X
ejpam-4596	78	3	=	=	PUNCT
ejpam-4596	78	4	α∗β∗α	α∗β∗α	NOUN
ejpam-4596	78	5	for	for	ADP
ejpam-4596	78	6	some	some	DET
ejpam-4596	78	7	β	β	X
ejpam-4596	78	8	∈	∈	PROPN
ejpam-4596	78	9	te(x	te(x	NOUN
ejpam-4596	78	10	,	,	PUNCT
ejpam-4596	78	11	θ	θ	PROPN
ejpam-4596	78	12	)	)	PUNCT
ejpam-4596	78	13	.	.	PUNCT
ejpam-4596	79	1	thus	thus	ADV
ejpam-4596	79	2	α	α	NOUN
ejpam-4596	79	3	=	=	SYM
ejpam-4596	79	4	αθβθα	αθβθα	NOUN
ejpam-4596	79	5	.	.	PUNCT
ejpam-4596	80	1	clearly	clearly	ADV
ejpam-4596	80	2	,	,	PUNCT
ejpam-4596	80	3	ker(α	ker(α	PROPN
ejpam-4596	80	4	)	)	PUNCT
ejpam-4596	80	5	⊆	⊆	NUM
ejpam-4596	80	6	ker(αθ	ker(αθ	NOUN
ejpam-4596	80	7	)	)	PUNCT
ejpam-4596	80	8	.	.	PUNCT
ejpam-4596	81	1	for	for	ADP
ejpam-4596	81	2	the	the	DET
ejpam-4596	81	3	converse	converse	NOUN
ejpam-4596	81	4	conclusion	conclusion	NOUN
ejpam-4596	81	5	,	,	PUNCT
ejpam-4596	81	6	let	let	VERB
ejpam-4596	81	7	x	x	PRON
ejpam-4596	81	8	,	,	PUNCT
ejpam-4596	81	9	y	y	PROPN
ejpam-4596	81	10	∈	∈	PROPN
ejpam-4596	81	11	x	x	AUX
ejpam-4596	81	12	be	be	AUX
ejpam-4596	81	13	such	such	ADJ
ejpam-4596	81	14	that	that	SCONJ
ejpam-4596	81	15	(	(	PUNCT
ejpam-4596	81	16	x	x	NOUN
ejpam-4596	81	17	,	,	PUNCT
ejpam-4596	81	18	y	y	NOUN
ejpam-4596	81	19	)	)	PUNCT
ejpam-4596	81	20	∈	∈	PROPN
ejpam-4596	81	21	ker(αθ	ker(αθ	NOUN
ejpam-4596	81	22	)	)	PUNCT
ejpam-4596	81	23	.	.	PUNCT
ejpam-4596	82	1	then	then	ADV
ejpam-4596	82	2	xαθ	xαθ	PUNCT
ejpam-4596	83	1	=	=	SYM
ejpam-4596	83	2	yαθ	yαθ	PROPN
ejpam-4596	84	1	and	and	CCONJ
ejpam-4596	84	2	so	so	ADV
ejpam-4596	84	3	xα	xα	PUNCT
ejpam-4596	84	4	=	=	PUNCT
ejpam-4596	84	5	xαθ(βθα	xαθ(βθα	X
ejpam-4596	84	6	)	)	PUNCT
ejpam-4596	84	7	=	=	SYM
ejpam-4596	84	8	yαθ(βθα	yαθ(βθα	NOUN
ejpam-4596	84	9	)	)	PUNCT
ejpam-4596	85	1	=	=	SYM
ejpam-4596	85	2	yα	yα	NOUN
ejpam-4596	85	3	.	.	PUNCT
ejpam-4596	86	1	p.	p.	NOUN
ejpam-4596	86	2	tantong	tantong	NOUN
ejpam-4596	86	3	,	,	PUNCT
ejpam-4596	86	4	n.	n.	NOUN
ejpam-4596	86	5	sawatraksa	sawatraksa	PROPN
ejpam-4596	86	6	/	/	SYM
ejpam-4596	86	7	eur	eur	PROPN
ejpam-4596	86	8	.	.	PUNCT
ejpam-4596	87	1	j.	j.	PROPN
ejpam-4596	87	2	pure	pure	PROPN
ejpam-4596	87	3	appl	appl	PROPN
ejpam-4596	87	4	.	.	PROPN
ejpam-4596	87	5	math	math	PROPN
ejpam-4596	87	6	,	,	PUNCT
ejpam-4596	87	7	15	15	NUM
ejpam-4596	87	8	(	(	PUNCT
ejpam-4596	87	9	4	4	NUM
ejpam-4596	87	10	)	)	PUNCT
ejpam-4596	87	11	(	(	PUNCT
ejpam-4596	87	12	2022	2022	NUM
ejpam-4596	87	13	)	)	PUNCT
ejpam-4596	87	14	,	,	PUNCT
ejpam-4596	87	15	2116	2116	NUM
ejpam-4596	87	16	-	-	SYM
ejpam-4596	87	17	2126	2126	NUM
ejpam-4596	87	18	2119	2119	NUM
ejpam-4596	87	19	this	this	PRON
ejpam-4596	87	20	means	mean	VERB
ejpam-4596	87	21	that	that	SCONJ
ejpam-4596	87	22	(	(	PUNCT
ejpam-4596	87	23	x	x	X
ejpam-4596	87	24	,	,	PUNCT
ejpam-4596	87	25	y	y	NOUN
ejpam-4596	87	26	)	)	PUNCT
ejpam-4596	87	27	∈	∈	PROPN
ejpam-4596	87	28	ker(α	ker(α	PROPN
ejpam-4596	87	29	)	)	PUNCT
ejpam-4596	87	30	.	.	PUNCT
ejpam-4596	88	1	hence	hence	ADV
ejpam-4596	88	2	(	(	PUNCT
ejpam-4596	88	3	1	1	X
ejpam-4596	88	4	)	)	PUNCT
ejpam-4596	88	5	holds	hold	VERB
ejpam-4596	88	6	.	.	PUNCT
ejpam-4596	89	1	for	for	ADP
ejpam-4596	89	2	each	each	PRON
ejpam-4596	89	3	a	a	DET
ejpam-4596	89	4	∈	∈	PROPN
ejpam-4596	89	5	x	x	SYM
ejpam-4596	89	6	/	/	SYM
ejpam-4596	89	7	e	e	NOUN
ejpam-4596	89	8	,	,	PUNCT
ejpam-4596	89	9	by	by	ADP
ejpam-4596	89	10	lemma	lemma	PROPN
ejpam-4596	89	11	1	1	NUM
ejpam-4596	89	12	there	there	ADV
ejpam-4596	89	13	exists	exist	VERB
ejpam-4596	89	14	b	b	PROPN
ejpam-4596	89	15	∈	∈	PROPN
ejpam-4596	89	16	x	x	SYM
ejpam-4596	89	17	/	/	SYM
ejpam-4596	89	18	e	e	VERB
ejpam-4596	89	19	such	such	ADJ
ejpam-4596	89	20	that	that	SCONJ
ejpam-4596	89	21	aβ	aβ	ADP
ejpam-4596	89	22	⊆	⊆	NUM
ejpam-4596	89	23	b.	b.	NOUN
ejpam-4596	89	24	thus	thus	ADV
ejpam-4596	89	25	aβθ	aβθ	PROPN
ejpam-4596	89	26	⊆	⊆	NUM
ejpam-4596	89	27	bθ	bθ	NOUN
ejpam-4596	89	28	.	.	PUNCT
ejpam-4596	90	1	if	if	SCONJ
ejpam-4596	90	2	a	a	DET
ejpam-4596	90	3	∈	∈	PROPN
ejpam-4596	90	4	a	a	DET
ejpam-4596	90	5	∩	∩	NOUN
ejpam-4596	90	6	xαθ	xαθ	NOUN
ejpam-4596	90	7	,	,	PUNCT
ejpam-4596	90	8	then	then	ADV
ejpam-4596	90	9	a	a	DET
ejpam-4596	90	10	∈	∈	PROPN
ejpam-4596	90	11	a	a	PRON
ejpam-4596	90	12	and	and	CCONJ
ejpam-4596	90	13	a	a	DET
ejpam-4596	90	14	=	=	X
ejpam-4596	90	15	xαθ	xαθ	NOUN
ejpam-4596	90	16	for	for	ADP
ejpam-4596	90	17	some	some	DET
ejpam-4596	90	18	x	x	SYM
ejpam-4596	90	19	∈	∈	NOUN
ejpam-4596	90	20	x.	x.	NOUN
ejpam-4596	90	21	thus	thus	ADV
ejpam-4596	90	22	aβθ	aβθ	PROPN
ejpam-4596	90	23	∈	∈	PROPN
ejpam-4596	90	24	bθ	bθ	PROPN
ejpam-4596	90	25	.	.	PUNCT
ejpam-4596	91	1	this	this	PRON
ejpam-4596	91	2	implies	imply	VERB
ejpam-4596	91	3	that	that	SCONJ
ejpam-4596	91	4	a	a	DET
ejpam-4596	91	5	=	=	PUNCT
ejpam-4596	91	6	xαθ	xαθ	NOUN
ejpam-4596	91	7	=	=	SYM
ejpam-4596	91	8	xαθβθαθ	xαθβθαθ	NOUN
ejpam-4596	92	1	=	=	PUNCT
ejpam-4596	92	2	aβθαθ	aβθαθ	PROPN
ejpam-4596	92	3	∈	∈	PROPN
ejpam-4596	92	4	bθαθ	bθαθ	PROPN
ejpam-4596	92	5	.	.	PUNCT
ejpam-4596	93	1	hence	hence	ADV
ejpam-4596	93	2	a	a	DET
ejpam-4596	93	3	∩xαθ	∩xαθ	NOUN
ejpam-4596	93	4	⊆	⊆	NUM
ejpam-4596	93	5	bθαθ	bθαθ	NOUN
ejpam-4596	93	6	.	.	PUNCT
ejpam-4596	94	1	conversely	conversely	ADV
ejpam-4596	94	2	,	,	PUNCT
ejpam-4596	94	3	suppose	suppose	VERB
ejpam-4596	94	4	that	that	SCONJ
ejpam-4596	94	5	the	the	DET
ejpam-4596	94	6	conditions	condition	NOUN
ejpam-4596	94	7	(	(	PUNCT
ejpam-4596	94	8	1	1	NUM
ejpam-4596	94	9	)	)	PUNCT
ejpam-4596	94	10	and	and	CCONJ
ejpam-4596	94	11	(	(	PUNCT
ejpam-4596	94	12	2	2	X
ejpam-4596	94	13	)	)	PUNCT
ejpam-4596	94	14	hold	hold	NOUN
ejpam-4596	94	15	.	.	PUNCT
ejpam-4596	95	1	for	for	ADP
ejpam-4596	95	2	each	each	PRON
ejpam-4596	95	3	a	a	DET
ejpam-4596	95	4	∈	∈	PROPN
ejpam-4596	95	5	x	x	SYM
ejpam-4596	95	6	/	/	SYM
ejpam-4596	95	7	e	e	NOUN
ejpam-4596	95	8	,	,	PUNCT
ejpam-4596	95	9	we	we	PRON
ejpam-4596	95	10	choose	choose	VERB
ejpam-4596	95	11	a′	a′	PROPN
ejpam-4596	95	12	∈	∈	PROPN
ejpam-4596	95	13	x	x	SYM
ejpam-4596	95	14	/	/	SYM
ejpam-4596	95	15	e	e	VERB
ejpam-4596	95	16	such	such	ADJ
ejpam-4596	95	17	that	that	SCONJ
ejpam-4596	95	18	a	a	DET
ejpam-4596	95	19	∩xαθ	∩xαθ	NOUN
ejpam-4596	95	20	⊆	⊆	NUM
ejpam-4596	95	21	a′θαθ	a′θαθ	NOUN
ejpam-4596	95	22	.	.	PUNCT
ejpam-4596	96	1	let	let	VERB
ejpam-4596	96	2	x	x	SYM
ejpam-4596	96	3	∈	∈	PROPN
ejpam-4596	96	4	x.	x.	NOUN
ejpam-4596	97	1	if	if	SCONJ
ejpam-4596	97	2	x	x	SYM
ejpam-4596	97	3	∈	∈	PROPN
ejpam-4596	97	4	xαθ	xαθ	NOUN
ejpam-4596	97	5	,	,	PUNCT
ejpam-4596	97	6	then	then	ADV
ejpam-4596	97	7	by	by	ADP
ejpam-4596	97	8	(	(	PUNCT
ejpam-4596	97	9	2	2	NUM
ejpam-4596	97	10	)	)	PUNCT
ejpam-4596	97	11	,	,	PUNCT
ejpam-4596	97	12	we	we	PRON
ejpam-4596	97	13	choose	choose	VERB
ejpam-4596	97	14	and	and	CCONJ
ejpam-4596	97	15	fix	fix	VERB
ejpam-4596	97	16	an	an	DET
ejpam-4596	97	17	element	element	NOUN
ejpam-4596	97	18	x′	x′	PROPN
ejpam-4596	97	19	∈	∈	PROPN
ejpam-4596	97	20	(	(	PUNCT
ejpam-4596	97	21	ex	ex	NOUN
ejpam-4596	97	22	)	)	PUNCT
ejpam-4596	97	23	′	′	ADP
ejpam-4596	98	1	such	such	ADJ
ejpam-4596	98	2	that	that	SCONJ
ejpam-4596	98	3	x	x	X
ejpam-4596	98	4	=	=	SYM
ejpam-4596	98	5	x′θαθ	x′θαθ	PROPN
ejpam-4596	98	6	.	.	PUNCT
ejpam-4596	99	1	otherwise	otherwise	ADV
ejpam-4596	99	2	,	,	PUNCT
ejpam-4596	99	3	if	if	SCONJ
ejpam-4596	99	4	x	x	X
ejpam-4596	99	5	/∈	/∈	PUNCT
ejpam-4596	99	6	xαθ	xαθ	ADV
ejpam-4596	99	7	,	,	PUNCT
ejpam-4596	99	8	we	we	PRON
ejpam-4596	99	9	choose	choose	VERB
ejpam-4596	99	10	and	and	CCONJ
ejpam-4596	99	11	fix	fix	VERB
ejpam-4596	99	12	an	an	DET
ejpam-4596	99	13	element	element	NOUN
ejpam-4596	99	14	x′	x′	PROPN
ejpam-4596	99	15	∈	∈	PROPN
ejpam-4596	99	16	(	(	PUNCT
ejpam-4596	99	17	ex	ex	NOUN
ejpam-4596	99	18	)	)	PUNCT
ejpam-4596	99	19	′.	′.	NOUN
ejpam-4596	99	20	define	define	VERB
ejpam-4596	99	21	β	β	X
ejpam-4596	99	22	:	:	PUNCT
ejpam-4596	99	23	x	x	SYM
ejpam-4596	99	24	→	→	SYM
ejpam-4596	99	25	x	x	PUNCT
ejpam-4596	99	26	by	by	ADP
ejpam-4596	99	27	xβ	xβ	NUM
ejpam-4596	99	28	=	=	SYM
ejpam-4596	99	29	x′	x′	PROPN
ejpam-4596	99	30	for	for	ADP
ejpam-4596	99	31	all	all	DET
ejpam-4596	99	32	x	x	SYM
ejpam-4596	99	33	∈	∈	ADJ
ejpam-4596	99	34	x.	x.	NOUN
ejpam-4596	99	35	then	then	ADV
ejpam-4596	99	36	β	β	PROPN
ejpam-4596	99	37	is	be	AUX
ejpam-4596	99	38	a	a	DET
ejpam-4596	99	39	well	well	ADV
ejpam-4596	99	40	-	-	PUNCT
ejpam-4596	99	41	defined	define	VERB
ejpam-4596	99	42	mapping	mapping	NOUN
ejpam-4596	99	43	.	.	PUNCT
ejpam-4596	100	1	let	let	VERB
ejpam-4596	100	2	x	x	PRON
ejpam-4596	100	3	,	,	PUNCT
ejpam-4596	100	4	y	y	PROPN
ejpam-4596	100	5	∈	∈	PROPN
ejpam-4596	100	6	x	x	AUX
ejpam-4596	100	7	be	be	AUX
ejpam-4596	101	1	such	such	ADJ
ejpam-4596	101	2	that	that	SCONJ
ejpam-4596	101	3	(	(	PUNCT
ejpam-4596	101	4	x	x	NOUN
ejpam-4596	101	5	,	,	PUNCT
ejpam-4596	101	6	y	y	NOUN
ejpam-4596	101	7	)	)	PUNCT
ejpam-4596	101	8	∈	∈	PROPN
ejpam-4596	101	9	e.	e.	PROPN
ejpam-4596	101	10	then	then	ADV
ejpam-4596	101	11	ex	ex	X
ejpam-4596	101	12	=	=	PUNCT
ejpam-4596	101	13	ey	ey	NOUN
ejpam-4596	101	14	and	and	CCONJ
ejpam-4596	101	15	so	so	ADV
ejpam-4596	101	16	ex′	ex′	X
ejpam-4596	101	17	=	=	SYM
ejpam-4596	101	18	ey′	ey′	X
ejpam-4596	101	19	.	.	PUNCT
ejpam-4596	102	1	this	this	PRON
ejpam-4596	102	2	implies	imply	VERB
ejpam-4596	102	3	that	that	SCONJ
ejpam-4596	102	4	β	β	PROPN
ejpam-4596	102	5	∈	∈	PROPN
ejpam-4596	102	6	te(x	te(x	NOUN
ejpam-4596	102	7	,	,	PUNCT
ejpam-4596	102	8	θ	θ	PROPN
ejpam-4596	102	9	)	)	PUNCT
ejpam-4596	102	10	.	.	PUNCT
ejpam-4596	103	1	it	it	PRON
ejpam-4596	103	2	remains	remain	VERB
ejpam-4596	103	3	to	to	PART
ejpam-4596	103	4	be	be	AUX
ejpam-4596	103	5	verified	verify	VERB
ejpam-4596	104	1	that	that	SCONJ
ejpam-4596	104	2	α	α	NOUN
ejpam-4596	104	3	∗β	∗β	PROPN
ejpam-4596	104	4	∗α	∗α	PROPN
ejpam-4596	104	5	=	=	SYM
ejpam-4596	104	6	α	α	PROPN
ejpam-4596	104	7	.	.	PUNCT
ejpam-4596	105	1	if	if	SCONJ
ejpam-4596	105	2	x	x	SYM
ejpam-4596	105	3	∈	∈	PROPN
ejpam-4596	105	4	x	x	NOUN
ejpam-4596	105	5	,	,	PUNCT
ejpam-4596	105	6	then	then	ADV
ejpam-4596	105	7	xαθβθα	xαθβθα	PROPN
ejpam-4596	105	8	=	=	PROPN
ejpam-4596	106	1	(	(	PUNCT
ejpam-4596	106	2	xαθ)′θα	xαθ)′θα	ADJ
ejpam-4596	106	3	with	with	ADP
ejpam-4596	106	4	(	(	PUNCT
ejpam-4596	106	5	xαθ)′θαθ	xαθ)′θαθ	PROPN
ejpam-4596	106	6	=	=	PUNCT
ejpam-4596	106	7	xαθ	xαθ	PROPN
ejpam-4596	106	8	.	.	PUNCT
ejpam-4596	107	1	thus	thus	ADV
ejpam-4596	107	2	(	(	PUNCT
ejpam-4596	107	3	(	(	PUNCT
ejpam-4596	107	4	xαθ)′θ	xαθ)′θ	PROPN
ejpam-4596	107	5	,	,	PUNCT
ejpam-4596	107	6	x	x	X
ejpam-4596	107	7	)	)	PUNCT
ejpam-4596	107	8	∈	∈	PROPN
ejpam-4596	107	9	ker(αθ	ker(αθ	NOUN
ejpam-4596	107	10	)	)	PUNCT
ejpam-4596	107	11	=	=	SYM
ejpam-4596	107	12	ker(α	ker(α	PROPN
ejpam-4596	107	13	)	)	PUNCT
ejpam-4596	107	14	.	.	PUNCT
ejpam-4596	108	1	this	this	PRON
ejpam-4596	108	2	implies	imply	VERB
ejpam-4596	108	3	that	that	PRON
ejpam-4596	108	4	xαθβθα	xαθβθα	PROPN
ejpam-4596	109	1	=	=	PRON
ejpam-4596	110	1	(	(	PUNCT
ejpam-4596	110	2	xαθ)′θα	xαθ)′θα	PROPN
ejpam-4596	110	3	=	=	PRON
ejpam-4596	110	4	xα	xα	CCONJ
ejpam-4596	111	1	and	and	CCONJ
ejpam-4596	111	2	therefore	therefore	ADV
ejpam-4596	111	3	α	α	PROPN
ejpam-4596	111	4	=	=	SYM
ejpam-4596	111	5	α	α	PROPN
ejpam-4596	111	6	∗	∗	X
ejpam-4596	111	7	β	β	PROPN
ejpam-4596	111	8	∗	∗	NOUN
ejpam-4596	111	9	α	α	NOUN
ejpam-4596	111	10	.	.	PUNCT
ejpam-4596	112	1	hence	hence	ADV
ejpam-4596	112	2	α	α	PROPN
ejpam-4596	112	3	is	be	AUX
ejpam-4596	112	4	regular	regular	ADJ
ejpam-4596	112	5	in	in	ADP
ejpam-4596	112	6	te(x	te(x	NOUN
ejpam-4596	112	7	,	,	PUNCT
ejpam-4596	112	8	θ	θ	PROPN
ejpam-4596	112	9	)	)	PUNCT
ejpam-4596	112	10	.	.	PUNCT
ejpam-4596	113	1	let	let	VERB
ejpam-4596	113	2	e	e	PRON
ejpam-4596	113	3	be	be	AUX
ejpam-4596	113	4	an	an	DET
ejpam-4596	113	5	equivalence	equivalence	NOUN
ejpam-4596	113	6	relation	relation	NOUN
ejpam-4596	113	7	on	on	ADP
ejpam-4596	113	8	x	x	PUNCT
ejpam-4596	113	9	and	and	CCONJ
ejpam-4596	113	10	y	y	PROPN
ejpam-4596	113	11	be	be	AUX
ejpam-4596	113	12	a	a	DET
ejpam-4596	113	13	subset	subset	NOUN
ejpam-4596	113	14	of	of	ADP
ejpam-4596	113	15	x.	x.	NOUN
ejpam-4596	113	16	a	a	DET
ejpam-4596	113	17	mapping	mapping	NOUN
ejpam-4596	113	18	α	α	NOUN
ejpam-4596	113	19	:	:	PUNCT
ejpam-4596	113	20	y	y	PROPN
ejpam-4596	113	21	→	→	PUNCT
ejpam-4596	113	22	x	x	X
ejpam-4596	113	23	is	be	AUX
ejpam-4596	113	24	called	call	VERB
ejpam-4596	113	25	e	e	VERB
ejpam-4596	113	26	-	-	ADJ
ejpam-4596	113	27	preserving	preserve	VERB
ejpam-4596	113	28	if	if	SCONJ
ejpam-4596	113	29	for	for	ADP
ejpam-4596	113	30	all	all	DET
ejpam-4596	113	31	x	x	NOUN
ejpam-4596	113	32	,	,	PUNCT
ejpam-4596	113	33	y	y	PROPN
ejpam-4596	113	34	∈	∈	PROPN
ejpam-4596	113	35	y	y	PROPN
ejpam-4596	113	36	,	,	PUNCT
ejpam-4596	113	37	(	(	PUNCT
ejpam-4596	113	38	x	x	X
ejpam-4596	113	39	,	,	PUNCT
ejpam-4596	113	40	y	y	NOUN
ejpam-4596	113	41	)	)	PUNCT
ejpam-4596	113	42	∈	∈	PROPN
ejpam-4596	113	43	e	e	NOUN
ejpam-4596	113	44	implies	imply	VERB
ejpam-4596	113	45	(	(	PUNCT
ejpam-4596	113	46	xα	xα	INTJ
ejpam-4596	113	47	,	,	PUNCT
ejpam-4596	113	48	yα	yα	NOUN
ejpam-4596	113	49	)	)	PUNCT
ejpam-4596	113	50	∈	∈	PROPN
ejpam-4596	113	51	e.	e.	PROPN
ejpam-4596	114	1	if	if	SCONJ
ejpam-4596	114	2	α	α	PRON
ejpam-4596	114	3	satisfies	satisfy	VERB
ejpam-4596	114	4	the	the	DET
ejpam-4596	114	5	condition	condition	NOUN
ejpam-4596	114	6	that	that	SCONJ
ejpam-4596	114	7	(	(	PUNCT
ejpam-4596	114	8	xα	xα	INTJ
ejpam-4596	114	9	,	,	PUNCT
ejpam-4596	114	10	yα	yα	NOUN
ejpam-4596	114	11	)	)	PUNCT
ejpam-4596	114	12	∈	∈	PROPN
ejpam-4596	114	13	e	e	NOUN
ejpam-4596	114	14	if	if	SCONJ
ejpam-4596	114	15	and	and	CCONJ
ejpam-4596	114	16	only	only	ADV
ejpam-4596	114	17	if	if	SCONJ
ejpam-4596	114	18	(	(	PUNCT
ejpam-4596	114	19	x	x	NOUN
ejpam-4596	114	20	,	,	PUNCT
ejpam-4596	114	21	y	y	NOUN
ejpam-4596	114	22	)	)	PUNCT
ejpam-4596	114	23	∈	∈	PROPN
ejpam-4596	114	24	e	e	NOUN
ejpam-4596	114	25	,	,	PUNCT
ejpam-4596	114	26	then	then	ADV
ejpam-4596	114	27	α	α	PROPN
ejpam-4596	114	28	is	be	AUX
ejpam-4596	114	29	called	call	VERB
ejpam-4596	114	30	e∗-preserving	e∗-preserve	VERB
ejpam-4596	114	31	.	.	PUNCT
ejpam-4596	115	1	it	it	PRON
ejpam-4596	115	2	is	be	AUX
ejpam-4596	115	3	easy	easy	ADJ
ejpam-4596	115	4	to	to	PART
ejpam-4596	115	5	see	see	VERB
ejpam-4596	115	6	that	that	SCONJ
ejpam-4596	115	7	every	every	DET
ejpam-4596	115	8	α	α	PROPN
ejpam-4596	115	9	∈	∈	PROPN
ejpam-4596	115	10	te(x	te(x	NOUN
ejpam-4596	115	11	)	)	PUNCT
ejpam-4596	115	12	is	be	AUX
ejpam-4596	115	13	e	e	VERB
ejpam-4596	115	14	-	-	ADJ
ejpam-4596	115	15	preserving	preserve	VERB
ejpam-4596	115	16	but	but	CCONJ
ejpam-4596	115	17	need	need	AUX
ejpam-4596	115	18	not	not	PART
ejpam-4596	115	19	be	be	AUX
ejpam-4596	115	20	e∗-preserving	e∗-preserve	VERB
ejpam-4596	115	21	.	.	PUNCT
ejpam-4596	116	1	the	the	DET
ejpam-4596	116	2	following	following	ADJ
ejpam-4596	116	3	result	result	NOUN
ejpam-4596	116	4	is	be	AUX
ejpam-4596	116	5	immediate	immediate	ADJ
ejpam-4596	116	6	from	from	ADP
ejpam-4596	116	7	theorem	theorem	ADJ
ejpam-4596	116	8	1	1	NUM
ejpam-4596	116	9	.	.	PUNCT
ejpam-4596	116	10	corollary	corollary	ADJ
ejpam-4596	116	11	1	1	NUM
ejpam-4596	116	12	.	.	PUNCT
ejpam-4596	116	13	idx	idx	PROPN
ejpam-4596	116	14	∈	∈	PROPN
ejpam-4596	116	15	reg(te(x	reg(te(x	PROPN
ejpam-4596	116	16	,	,	PUNCT
ejpam-4596	116	17	θ	θ	NOUN
ejpam-4596	116	18	)	)	PUNCT
ejpam-4596	116	19	)	)	PUNCT
ejpam-4596	117	1	if	if	SCONJ
ejpam-4596	117	2	and	and	CCONJ
ejpam-4596	117	3	only	only	ADV
ejpam-4596	117	4	if	if	SCONJ
ejpam-4596	117	5	θ	θ	PROPN
ejpam-4596	117	6	is	be	AUX
ejpam-4596	117	7	an	an	DET
ejpam-4596	117	8	e∗-preserving	e∗-preserve	VERB
ejpam-4596	117	9	bijection	bijection	NOUN
ejpam-4596	117	10	.	.	PUNCT
ejpam-4596	118	1	proof	proof	NOUN
ejpam-4596	118	2	.	.	PUNCT
ejpam-4596	119	1	suppose	suppose	VERB
ejpam-4596	119	2	that	that	SCONJ
ejpam-4596	119	3	idx	idx	PROPN
ejpam-4596	119	4	∈	∈	PROPN
ejpam-4596	119	5	reg(te(x	reg(te(x	PROPN
ejpam-4596	119	6	,	,	PUNCT
ejpam-4596	119	7	θ	θ	NOUN
ejpam-4596	119	8	)	)	PUNCT
ejpam-4596	119	9	)	)	PUNCT
ejpam-4596	119	10	.	.	PUNCT
ejpam-4596	120	1	let	let	VERB
ejpam-4596	120	2	x	x	PRON
ejpam-4596	120	3	,	,	PUNCT
ejpam-4596	120	4	y	y	PROPN
ejpam-4596	120	5	∈	∈	PROPN
ejpam-4596	120	6	x	x	AUX
ejpam-4596	120	7	be	be	AUX
ejpam-4596	120	8	such	such	ADJ
ejpam-4596	121	1	that	that	SCONJ
ejpam-4596	121	2	xθ	xθ	PROPN
ejpam-4596	121	3	=	=	NOUN
ejpam-4596	121	4	yθ	yθ	PROPN
ejpam-4596	121	5	.	.	PUNCT
ejpam-4596	122	1	then	then	ADV
ejpam-4596	122	2	(	(	PUNCT
ejpam-4596	122	3	x	x	X
ejpam-4596	122	4	,	,	PUNCT
ejpam-4596	122	5	y	y	NOUN
ejpam-4596	122	6	)	)	PUNCT
ejpam-4596	122	7	∈	∈	PROPN
ejpam-4596	122	8	ker(θ	ker(θ	NOUN
ejpam-4596	122	9	)	)	PUNCT
ejpam-4596	122	10	.	.	PUNCT
ejpam-4596	123	1	by	by	ADP
ejpam-4596	123	2	theorem	theorem	NOUN
ejpam-4596	123	3	1	1	NUM
ejpam-4596	123	4	,	,	PUNCT
ejpam-4596	123	5	ker(idx	ker(idx	NOUN
ejpam-4596	123	6	)	)	PUNCT
ejpam-4596	123	7	=	=	SYM
ejpam-4596	123	8	ker(idxθ	ker(idxθ	NOUN
ejpam-4596	123	9	)	)	PUNCT
ejpam-4596	123	10	=	=	SYM
ejpam-4596	123	11	ker(θ	ker(θ	PROPN
ejpam-4596	123	12	)	)	PUNCT
ejpam-4596	123	13	,	,	PUNCT
ejpam-4596	123	14	which	which	PRON
ejpam-4596	123	15	implies	imply	VERB
ejpam-4596	123	16	that	that	SCONJ
ejpam-4596	123	17	x	x	X
ejpam-4596	124	1	=	=	PRON
ejpam-4596	124	2	xidx	xidx	NOUN
ejpam-4596	124	3	=	=	PUNCT
ejpam-4596	124	4	yidx	yidx	NOUN
ejpam-4596	124	5	=	=	NOUN
ejpam-4596	124	6	y.	y.	NOUN
ejpam-4596	124	7	hence	hence	ADV
ejpam-4596	124	8	θ	θ	PROPN
ejpam-4596	124	9	is	be	AUX
ejpam-4596	124	10	an	an	DET
ejpam-4596	124	11	injection	injection	NOUN
ejpam-4596	124	12	.	.	PUNCT
ejpam-4596	125	1	for	for	ADP
ejpam-4596	125	2	each	each	DET
ejpam-4596	125	3	x	x	SYM
ejpam-4596	125	4	∈	∈	PROPN
ejpam-4596	125	5	x	x	NOUN
ejpam-4596	125	6	,	,	PUNCT
ejpam-4596	125	7	there	there	PRON
ejpam-4596	125	8	exists	exist	VERB
ejpam-4596	125	9	b	b	PROPN
ejpam-4596	125	10	∈	∈	PROPN
ejpam-4596	125	11	x	x	SYM
ejpam-4596	125	12	/	/	SYM
ejpam-4596	125	13	e	e	VERB
ejpam-4596	125	14	such	such	ADJ
ejpam-4596	125	15	that	that	SCONJ
ejpam-4596	125	16	xθ	xθ	PROPN
ejpam-4596	125	17	∈	∈	PROPN
ejpam-4596	125	18	exθ	exθ	NOUN
ejpam-4596	125	19	∩xidxθ	∩xidxθ	NOUN
ejpam-4596	125	20	⊆	⊆	NUM
ejpam-4596	125	21	bθidxθ	bθidxθ	NOUN
ejpam-4596	125	22	.	.	PUNCT
ejpam-4596	126	1	thus	thus	ADV
ejpam-4596	126	2	xθ	xθ	PROPN
ejpam-4596	127	1	=	=	SYM
ejpam-4596	127	2	x′θ2	x′θ2	PROPN
ejpam-4596	127	3	for	for	ADP
ejpam-4596	127	4	some	some	DET
ejpam-4596	127	5	x′	x′	PROPN
ejpam-4596	127	6	∈	∈	PROPN
ejpam-4596	127	7	b.	b.	PROPN
ejpam-4596	127	8	since	since	SCONJ
ejpam-4596	127	9	θ	θ	PROPN
ejpam-4596	127	10	is	be	AUX
ejpam-4596	127	11	injective	injective	ADJ
ejpam-4596	127	12	,	,	PUNCT
ejpam-4596	127	13	we	we	PRON
ejpam-4596	127	14	deduce	deduce	VERB
ejpam-4596	127	15	that	that	PRON
ejpam-4596	127	16	x	x	X
ejpam-4596	128	1	=	=	PUNCT
ejpam-4596	128	2	x′θ	x′θ	NOUN
ejpam-4596	128	3	.	.	PUNCT
ejpam-4596	129	1	consequently	consequently	ADV
ejpam-4596	129	2	,	,	PUNCT
ejpam-4596	129	3	θ	θ	PROPN
ejpam-4596	129	4	is	be	AUX
ejpam-4596	129	5	a	a	DET
ejpam-4596	129	6	bijection	bijection	NOUN
ejpam-4596	129	7	.	.	PUNCT
ejpam-4596	130	1	finally	finally	ADV
ejpam-4596	130	2	,	,	PUNCT
ejpam-4596	130	3	for	for	ADP
ejpam-4596	130	4	any	any	DET
ejpam-4596	130	5	x	x	NOUN
ejpam-4596	130	6	,	,	PUNCT
ejpam-4596	130	7	y	y	PROPN
ejpam-4596	130	8	∈	∈	PROPN
ejpam-4596	130	9	x	x	PRON
ejpam-4596	130	10	,	,	PUNCT
ejpam-4596	130	11	if	if	SCONJ
ejpam-4596	130	12	(	(	PUNCT
ejpam-4596	130	13	xθ	xθ	PROPN
ejpam-4596	130	14	,	,	PUNCT
ejpam-4596	130	15	yθ	yθ	NOUN
ejpam-4596	130	16	)	)	PUNCT
ejpam-4596	130	17	∈	∈	PROPN
ejpam-4596	130	18	e	e	NOUN
ejpam-4596	130	19	,	,	PUNCT
ejpam-4596	130	20	then	then	ADV
ejpam-4596	130	21	there	there	PRON
ejpam-4596	130	22	exists	exist	VERB
ejpam-4596	130	23	b	b	PROPN
ejpam-4596	130	24	∈	∈	PROPN
ejpam-4596	130	25	x	x	SYM
ejpam-4596	130	26	/	/	SYM
ejpam-4596	130	27	e	e	VERB
ejpam-4596	130	28	such	such	ADJ
ejpam-4596	130	29	that	that	PRON
ejpam-4596	130	30	xθ	xθ	PROPN
ejpam-4596	130	31	,	,	PUNCT
ejpam-4596	130	32	yθ	yθ	NOUN
ejpam-4596	130	33	∈	∈	NOUN
ejpam-4596	130	34	exθ	exθ	NOUN
ejpam-4596	130	35	∩	∩	NOUN
ejpam-4596	130	36	xidxθ	xidxθ	NOUN
ejpam-4596	130	37	⊆	⊆	NUM
ejpam-4596	130	38	bθidxθ	bθidxθ	NOUN
ejpam-4596	130	39	.	.	PUNCT
ejpam-4596	131	1	therefore	therefore	ADV
ejpam-4596	131	2	xθ	xθ	PROPN
ejpam-4596	132	1	=	=	SYM
ejpam-4596	132	2	x′θ2	x′θ2	PROPN
ejpam-4596	132	3	and	and	CCONJ
ejpam-4596	132	4	yθ	yθ	PROPN
ejpam-4596	132	5	=	=	NOUN
ejpam-4596	132	6	y′θ2	y′θ2	PROPN
ejpam-4596	132	7	where	where	SCONJ
ejpam-4596	132	8	x′	x′	PROPN
ejpam-4596	132	9	,	,	PUNCT
ejpam-4596	132	10	y′	y′	NOUN
ejpam-4596	132	11	∈	∈	PROPN
ejpam-4596	132	12	b.	b.	PROPN
ejpam-4596	132	13	it	it	PRON
ejpam-4596	132	14	follows	follow	VERB
ejpam-4596	132	15	from	from	ADP
ejpam-4596	132	16	θ	θ	PROPN
ejpam-4596	132	17	is	be	AUX
ejpam-4596	132	18	injective	injective	ADJ
ejpam-4596	132	19	that	that	SCONJ
ejpam-4596	132	20	x	x	NOUN
ejpam-4596	132	21	=	=	PUNCT
ejpam-4596	132	22	x′θ	x′θ	PROPN
ejpam-4596	132	23	and	and	CCONJ
ejpam-4596	132	24	y	y	PROPN
ejpam-4596	132	25	=	=	PUNCT
ejpam-4596	132	26	y′θ	y′θ	PROPN
ejpam-4596	132	27	.	.	PUNCT
ejpam-4596	133	1	since	since	SCONJ
ejpam-4596	133	2	(	(	PUNCT
ejpam-4596	133	3	x′	x′	NUM
ejpam-4596	133	4	,	,	PUNCT
ejpam-4596	133	5	y′	y′	NUM
ejpam-4596	133	6	)	)	PUNCT
ejpam-4596	133	7	∈	∈	PROPN
ejpam-4596	133	8	e	e	NOUN
ejpam-4596	133	9	and	and	CCONJ
ejpam-4596	133	10	θ	θ	PROPN
ejpam-4596	133	11	∈	∈	PROPN
ejpam-4596	133	12	te(x	te(x	NOUN
ejpam-4596	133	13	)	)	PUNCT
ejpam-4596	133	14	,	,	PUNCT
ejpam-4596	133	15	(	(	PUNCT
ejpam-4596	133	16	x	x	X
ejpam-4596	133	17	,	,	PUNCT
ejpam-4596	133	18	y	y	NOUN
ejpam-4596	133	19	)	)	PUNCT
ejpam-4596	133	20	=	=	SYM
ejpam-4596	133	21	(	(	PUNCT
ejpam-4596	133	22	x′θ	x′θ	PROPN
ejpam-4596	133	23	,	,	PUNCT
ejpam-4596	133	24	y′θ	y′θ	NOUN
ejpam-4596	133	25	)	)	PUNCT
ejpam-4596	133	26	∈	∈	PROPN
ejpam-4596	133	27	e.	e.	NOUN
ejpam-4596	133	28	we	we	PRON
ejpam-4596	133	29	conclude	conclude	VERB
ejpam-4596	133	30	that	that	SCONJ
ejpam-4596	133	31	θ	θ	PROPN
ejpam-4596	133	32	is	be	AUX
ejpam-4596	133	33	an	an	DET
ejpam-4596	133	34	e∗-preserving	e∗-preserve	VERB
ejpam-4596	133	35	bijection	bijection	NOUN
ejpam-4596	133	36	.	.	PUNCT
ejpam-4596	134	1	conversely	conversely	ADV
ejpam-4596	134	2	,	,	PUNCT
ejpam-4596	134	3	if	if	SCONJ
ejpam-4596	134	4	θ	θ	PROPN
ejpam-4596	134	5	is	be	AUX
ejpam-4596	134	6	an	an	DET
ejpam-4596	134	7	e∗-preserving	e∗-preserving	NOUN
ejpam-4596	134	8	bijection	bijection	NOUN
ejpam-4596	134	9	,	,	PUNCT
ejpam-4596	134	10	then	then	ADV
ejpam-4596	134	11	θ−1	θ−1	PROPN
ejpam-4596	134	12	∈	∈	PROPN
ejpam-4596	134	13	te(x	te(x	NOUN
ejpam-4596	134	14	,	,	PUNCT
ejpam-4596	134	15	θ	θ	NOUN
ejpam-4596	134	16	)	)	PUNCT
ejpam-4596	134	17	and	and	CCONJ
ejpam-4596	134	18	idx	idx	NOUN
ejpam-4596	134	19	=	=	NOUN
ejpam-4596	134	20	idxθ(θ−1θ−1)θidx	idxθ(θ−1θ−1)θidx	NOUN
ejpam-4596	134	21	=	=	SYM
ejpam-4596	134	22	idx	idx	NOUN
ejpam-4596	134	23	∗	∗	NOUN
ejpam-4596	134	24	(	(	PUNCT
ejpam-4596	134	25	θ−1θ−1	θ−1θ−1	PROPN
ejpam-4596	134	26	)	)	PUNCT
ejpam-4596	134	27	∗	∗	NOUN
ejpam-4596	134	28	idx	idx	NOUN
ejpam-4596	134	29	.	.	PUNCT
ejpam-4596	135	1	hence	hence	ADV
ejpam-4596	135	2	idx	idx	PROPN
ejpam-4596	135	3	∈	∈	PROPN
ejpam-4596	135	4	reg(te(x	reg(te(x	PROPN
ejpam-4596	135	5	,	,	PUNCT
ejpam-4596	135	6	θ	θ	NOUN
ejpam-4596	135	7	)	)	PUNCT
ejpam-4596	135	8	)	)	PUNCT
ejpam-4596	135	9	.	.	PUNCT
ejpam-4596	136	1	in	in	ADP
ejpam-4596	136	2	what	what	PRON
ejpam-4596	136	3	follows	follow	VERB
ejpam-4596	136	4	we	we	PRON
ejpam-4596	136	5	investigate	investigate	VERB
ejpam-4596	136	6	when	when	SCONJ
ejpam-4596	136	7	an	an	DET
ejpam-4596	136	8	element	element	NOUN
ejpam-4596	136	9	in	in	ADP
ejpam-4596	136	10	te(x	te(x	NOUN
ejpam-4596	136	11	,	,	PUNCT
ejpam-4596	136	12	θ	θ	PROPN
ejpam-4596	136	13	)	)	PUNCT
ejpam-4596	136	14	is	be	AUX
ejpam-4596	136	15	left	leave	VERB
ejpam-4596	136	16	regular	regular	ADV
ejpam-4596	136	17	.	.	PUNCT
ejpam-4596	137	1	theorem	theorem	NOUN
ejpam-4596	137	2	2	2	NUM
ejpam-4596	137	3	.	.	PUNCT
ejpam-4596	138	1	let	let	VERB
ejpam-4596	138	2	α	α	PRON
ejpam-4596	138	3	∈	∈	PROPN
ejpam-4596	138	4	te(x	te(x	NOUN
ejpam-4596	138	5	,	,	PUNCT
ejpam-4596	138	6	θ	θ	PROPN
ejpam-4596	138	7	)	)	PUNCT
ejpam-4596	138	8	.	.	PUNCT
ejpam-4596	139	1	then	then	ADV
ejpam-4596	139	2	α	α	PROPN
ejpam-4596	139	3	∈	∈	PROPN
ejpam-4596	139	4	lreg(te(x	lreg(te(x	ADJ
ejpam-4596	139	5	,	,	PUNCT
ejpam-4596	139	6	θ	θ	NOUN
ejpam-4596	139	7	)	)	PUNCT
ejpam-4596	139	8	)	)	PUNCT
ejpam-4596	140	1	if	if	SCONJ
ejpam-4596	140	2	and	and	CCONJ
ejpam-4596	140	3	only	only	ADV
ejpam-4596	140	4	if	if	SCONJ
ejpam-4596	140	5	for	for	ADP
ejpam-4596	140	6	every	every	DET
ejpam-4596	140	7	a	a	DET
ejpam-4596	140	8	∈	∈	PROPN
ejpam-4596	140	9	x	x	SYM
ejpam-4596	140	10	/	/	SYM
ejpam-4596	140	11	e	e	NOUN
ejpam-4596	140	12	,	,	PUNCT
ejpam-4596	140	13	there	there	PRON
ejpam-4596	140	14	exists	exist	VERB
ejpam-4596	140	15	b	b	PROPN
ejpam-4596	140	16	∈	∈	PROPN
ejpam-4596	140	17	x	x	SYM
ejpam-4596	140	18	/	/	SYM
ejpam-4596	140	19	e	e	VERB
ejpam-4596	140	20	such	such	ADJ
ejpam-4596	140	21	that	that	PRON
ejpam-4596	140	22	for	for	ADP
ejpam-4596	140	23	each	each	DET
ejpam-4596	140	24	p	p	PROPN
ejpam-4596	140	25	∈	∈	PROPN
ejpam-4596	140	26	πa(α	πa(α	PUNCT
ejpam-4596	140	27	)	)	PUNCT
ejpam-4596	140	28	,	,	PUNCT
ejpam-4596	140	29	xθαθ	xθαθ	PROPN
ejpam-4596	140	30	∈	∈	PROPN
ejpam-4596	140	31	p	p	PROPN
ejpam-4596	140	32	for	for	ADP
ejpam-4596	140	33	some	some	DET
ejpam-4596	140	34	x	x	SYM
ejpam-4596	140	35	∈	∈	PROPN
ejpam-4596	140	36	b.	b.	PROPN
ejpam-4596	140	37	p.	p.	NOUN
ejpam-4596	140	38	tantong	tantong	NOUN
ejpam-4596	140	39	,	,	PUNCT
ejpam-4596	140	40	n.	n.	NOUN
ejpam-4596	140	41	sawatraksa	sawatraksa	PROPN
ejpam-4596	140	42	/	/	SYM
ejpam-4596	140	43	eur	eur	PROPN
ejpam-4596	140	44	.	.	PUNCT
ejpam-4596	141	1	j.	j.	PROPN
ejpam-4596	141	2	pure	pure	PROPN
ejpam-4596	141	3	appl	appl	PROPN
ejpam-4596	141	4	.	.	PROPN
ejpam-4596	141	5	math	math	PROPN
ejpam-4596	141	6	,	,	PUNCT
ejpam-4596	141	7	15	15	NUM
ejpam-4596	141	8	(	(	PUNCT
ejpam-4596	141	9	4	4	NUM
ejpam-4596	141	10	)	)	PUNCT
ejpam-4596	141	11	(	(	PUNCT
ejpam-4596	141	12	2022	2022	NUM
ejpam-4596	141	13	)	)	PUNCT
ejpam-4596	141	14	,	,	PUNCT
ejpam-4596	141	15	2116	2116	NUM
ejpam-4596	141	16	-	-	SYM
ejpam-4596	141	17	2126	2126	NUM
ejpam-4596	141	18	2120	2120	NUM
ejpam-4596	141	19	proof	proof	NOUN
ejpam-4596	141	20	.	.	PUNCT
ejpam-4596	142	1	assume	assume	VERB
ejpam-4596	142	2	that	that	SCONJ
ejpam-4596	142	3	α	α	PROPN
ejpam-4596	142	4	∈	∈	PROPN
ejpam-4596	142	5	lreg(te(x	lreg(te(x	NOUN
ejpam-4596	142	6	,	,	PUNCT
ejpam-4596	142	7	θ	θ	NOUN
ejpam-4596	142	8	)	)	PUNCT
ejpam-4596	142	9	)	)	PUNCT
ejpam-4596	142	10	.	.	PUNCT
ejpam-4596	143	1	then	then	ADV
ejpam-4596	143	2	α	α	X
ejpam-4596	143	3	=	=	PUNCT
ejpam-4596	143	4	β	β	X
ejpam-4596	143	5	∗	∗	X
ejpam-4596	143	6	α	α	PROPN
ejpam-4596	143	7	∗	∗	NOUN
ejpam-4596	143	8	α	α	NOUN
ejpam-4596	143	9	for	for	ADP
ejpam-4596	143	10	some	some	DET
ejpam-4596	143	11	β	β	X
ejpam-4596	143	12	∈	∈	PROPN
ejpam-4596	143	13	te(x	te(x	NOUN
ejpam-4596	143	14	,	,	PUNCT
ejpam-4596	143	15	θ	θ	NOUN
ejpam-4596	143	16	)	)	PUNCT
ejpam-4596	143	17	and	and	CCONJ
ejpam-4596	143	18	so	so	ADV
ejpam-4596	143	19	α	α	NOUN
ejpam-4596	144	1	=	=	X
ejpam-4596	144	2	βθαθα	βθαθα	ADJ
ejpam-4596	144	3	.	.	PUNCT
ejpam-4596	145	1	let	let	VERB
ejpam-4596	145	2	a	a	DET
ejpam-4596	145	3	∈	∈	ADJ
ejpam-4596	145	4	x	x	NOUN
ejpam-4596	145	5	/	/	SYM
ejpam-4596	145	6	e.	e.	PROPN
ejpam-4596	145	7	by	by	ADP
ejpam-4596	145	8	lemma	lemma	PROPN
ejpam-4596	145	9	1	1	NUM
ejpam-4596	145	10	,	,	PUNCT
ejpam-4596	145	11	there	there	PRON
ejpam-4596	145	12	exists	exist	VERB
ejpam-4596	145	13	b	b	PROPN
ejpam-4596	145	14	∈	∈	PROPN
ejpam-4596	145	15	x	x	SYM
ejpam-4596	145	16	/	/	SYM
ejpam-4596	145	17	e	e	VERB
ejpam-4596	145	18	such	such	ADJ
ejpam-4596	145	19	that	that	SCONJ
ejpam-4596	145	20	aβ	aβ	ADP
ejpam-4596	145	21	⊆	⊆	PROPN
ejpam-4596	145	22	b.	b.	NOUN
ejpam-4596	145	23	suppose	suppose	VERB
ejpam-4596	145	24	that	that	SCONJ
ejpam-4596	145	25	p	p	PROPN
ejpam-4596	145	26	∈	∈	PROPN
ejpam-4596	145	27	πa(α	πa(α	PUNCT
ejpam-4596	145	28	)	)	PUNCT
ejpam-4596	145	29	and	and	CCONJ
ejpam-4596	145	30	let	let	VERB
ejpam-4596	145	31	x	x	PUNCT
ejpam-4596	145	32	∈	∈	PROPN
ejpam-4596	145	33	p	p	NOUN
ejpam-4596	145	34	∩	∩	ADJ
ejpam-4596	145	35	a.	a.	NOUN
ejpam-4596	145	36	hence	hence	ADV
ejpam-4596	145	37	xβ	xβ	PROPN
ejpam-4596	145	38	∈	∈	PROPN
ejpam-4596	145	39	b	b	PROPN
ejpam-4596	145	40	and	and	CCONJ
ejpam-4596	145	41	xα	xα	INTJ
ejpam-4596	145	42	=	=	PUNCT
ejpam-4596	146	1	xβθαθα	xβθαθα	PROPN
ejpam-4596	146	2	which	which	PRON
ejpam-4596	146	3	means	mean	VERB
ejpam-4596	146	4	xβθαθ	xβθαθ	PROPN
ejpam-4596	146	5	∈	∈	PROPN
ejpam-4596	146	6	(	(	PUNCT
ejpam-4596	147	1	xα)α−1	xα)α−1	X
ejpam-4596	147	2	=	=	SYM
ejpam-4596	147	3	p	p	NOUN
ejpam-4596	147	4	.	.	PUNCT
ejpam-4596	148	1	conversely	conversely	ADV
ejpam-4596	148	2	,	,	PUNCT
ejpam-4596	148	3	for	for	ADP
ejpam-4596	148	4	each	each	DET
ejpam-4596	148	5	a	a	DET
ejpam-4596	148	6	∈	∈	PROPN
ejpam-4596	148	7	x	x	SYM
ejpam-4596	148	8	/	/	SYM
ejpam-4596	148	9	e	e	NOUN
ejpam-4596	148	10	,	,	PUNCT
ejpam-4596	148	11	we	we	PRON
ejpam-4596	148	12	choose	choose	VERB
ejpam-4596	148	13	a′	a′	PROPN
ejpam-4596	148	14	∈	∈	PROPN
ejpam-4596	148	15	x	x	SYM
ejpam-4596	148	16	/	/	SYM
ejpam-4596	148	17	e	e	VERB
ejpam-4596	148	18	such	such	ADJ
ejpam-4596	148	19	that	that	PRON
ejpam-4596	148	20	for	for	ADP
ejpam-4596	148	21	every	every	DET
ejpam-4596	148	22	p	p	PROPN
ejpam-4596	148	23	∈	∈	PROPN
ejpam-4596	148	24	πa(α	πa(α	PUNCT
ejpam-4596	148	25	)	)	PUNCT
ejpam-4596	148	26	,	,	PUNCT
ejpam-4596	148	27	xθαθ	xθαθ	PROPN
ejpam-4596	148	28	∈	∈	PROPN
ejpam-4596	148	29	p	p	PROPN
ejpam-4596	148	30	for	for	ADP
ejpam-4596	148	31	some	some	DET
ejpam-4596	148	32	x	x	SYM
ejpam-4596	148	33	∈	∈	NOUN
ejpam-4596	148	34	a′.	a′.	NOUN
ejpam-4596	148	35	let	let	VERB
ejpam-4596	148	36	x	x	X
ejpam-4596	148	37	∈	∈	PROPN
ejpam-4596	148	38	x.	x.	NOUN
ejpam-4596	148	39	since	since	SCONJ
ejpam-4596	148	40	x	x	PROPN
ejpam-4596	148	41	/	/	SYM
ejpam-4596	148	42	e	e	NOUN
ejpam-4596	148	43	and	and	CCONJ
ejpam-4596	148	44	π(α	π(α	NUM
ejpam-4596	148	45	)	)	PUNCT
ejpam-4596	148	46	are	be	AUX
ejpam-4596	148	47	partitions	partition	NOUN
ejpam-4596	148	48	of	of	ADP
ejpam-4596	148	49	x	x	NOUN
ejpam-4596	148	50	,	,	PUNCT
ejpam-4596	148	51	there	there	PRON
ejpam-4596	148	52	exist	exist	VERB
ejpam-4596	148	53	a	a	DET
ejpam-4596	148	54	∈	∈	PROPN
ejpam-4596	148	55	x	x	SYM
ejpam-4596	148	56	/	/	SYM
ejpam-4596	148	57	e	e	NOUN
ejpam-4596	148	58	and	and	CCONJ
ejpam-4596	148	59	p	p	PROPN
ejpam-4596	148	60	∈	∈	PROPN
ejpam-4596	148	61	π(α	π(α	PROPN
ejpam-4596	148	62	)	)	PUNCT
ejpam-4596	148	63	such	such	ADJ
ejpam-4596	148	64	that	that	SCONJ
ejpam-4596	148	65	x	x	SYM
ejpam-4596	148	66	∈	∈	PROPN
ejpam-4596	148	67	a	a	PRON
ejpam-4596	148	68	and	and	CCONJ
ejpam-4596	148	69	x	x	SYM
ejpam-4596	148	70	∈	∈	PROPN
ejpam-4596	148	71	p	p	NOUN
ejpam-4596	148	72	.	.	PUNCT
ejpam-4596	149	1	hence	hence	ADV
ejpam-4596	149	2	p	p	X
ejpam-4596	149	3	∈	∈	PROPN
ejpam-4596	149	4	πa(α	πa(α	PUNCT
ejpam-4596	149	5	)	)	PUNCT
ejpam-4596	149	6	.	.	PUNCT
ejpam-4596	150	1	by	by	ADP
ejpam-4596	150	2	assumption	assumption	NOUN
ejpam-4596	150	3	,	,	PUNCT
ejpam-4596	150	4	we	we	PRON
ejpam-4596	150	5	choose	choose	VERB
ejpam-4596	150	6	and	and	CCONJ
ejpam-4596	150	7	fix	fix	VERB
ejpam-4596	150	8	an	an	DET
ejpam-4596	150	9	element	element	NOUN
ejpam-4596	150	10	x′	x′	PROPN
ejpam-4596	150	11	∈	∈	PROPN
ejpam-4596	150	12	a′	a′	NOUN
ejpam-4596	150	13	such	such	ADJ
ejpam-4596	150	14	that	that	SCONJ
ejpam-4596	150	15	x′θαθ	x′θαθ	PROPN
ejpam-4596	150	16	∈	∈	PROPN
ejpam-4596	150	17	p	p	NOUN
ejpam-4596	150	18	and	and	CCONJ
ejpam-4596	150	19	a′	a′	NOUN
ejpam-4596	150	20	∈	∈	PROPN
ejpam-4596	150	21	x	x	X
ejpam-4596	150	22	/	/	SYM
ejpam-4596	150	23	e.	e.	PROPN
ejpam-4596	150	24	we	we	PRON
ejpam-4596	150	25	also	also	ADV
ejpam-4596	150	26	have	have	VERB
ejpam-4596	150	27	that	that	PRON
ejpam-4596	151	1	x′θαθα	x′θαθα	PROPN
ejpam-4596	151	2	=	=	PRON
ejpam-4596	151	3	xα	xα	PROPN
ejpam-4596	151	4	.	.	PUNCT
ejpam-4596	152	1	define	define	VERB
ejpam-4596	152	2	β	β	X
ejpam-4596	152	3	:	:	PUNCT
ejpam-4596	152	4	x	x	SYM
ejpam-4596	152	5	→	→	SYM
ejpam-4596	152	6	x	x	PUNCT
ejpam-4596	152	7	by	by	ADP
ejpam-4596	152	8	xβ	xβ	NUM
ejpam-4596	152	9	=	=	SYM
ejpam-4596	152	10	x′	x′	PROPN
ejpam-4596	152	11	for	for	ADP
ejpam-4596	152	12	all	all	PRON
ejpam-4596	152	13	x	x	SYM
ejpam-4596	152	14	∈	∈	NOUN
ejpam-4596	152	15	x.	x.	NOUN
ejpam-4596	152	16	let	let	VERB
ejpam-4596	152	17	x	x	PRON
ejpam-4596	152	18	,	,	PUNCT
ejpam-4596	152	19	y	y	PROPN
ejpam-4596	152	20	∈	∈	PROPN
ejpam-4596	152	21	x	x	AUX
ejpam-4596	152	22	be	be	AUX
ejpam-4596	152	23	such	such	ADJ
ejpam-4596	152	24	that	that	SCONJ
ejpam-4596	152	25	(	(	PUNCT
ejpam-4596	152	26	x	x	NOUN
ejpam-4596	152	27	,	,	PUNCT
ejpam-4596	152	28	y	y	NOUN
ejpam-4596	152	29	)	)	PUNCT
ejpam-4596	152	30	∈	∈	PROPN
ejpam-4596	152	31	e.	e.	PROPN
ejpam-4596	153	1	then	then	ADV
ejpam-4596	153	2	ex	ex	X
ejpam-4596	154	1	=	=	PUNCT
ejpam-4596	154	2	ey	ey	NOUN
ejpam-4596	154	3	and	and	CCONJ
ejpam-4596	154	4	thus	thus	ADV
ejpam-4596	154	5	ex′	ex′	X
ejpam-4596	154	6	=	=	SYM
ejpam-4596	154	7	ey′	ey′	X
ejpam-4596	154	8	.	.	PUNCT
ejpam-4596	155	1	this	this	PRON
ejpam-4596	155	2	implies	imply	VERB
ejpam-4596	155	3	that	that	SCONJ
ejpam-4596	155	4	β	β	PROPN
ejpam-4596	155	5	∈	∈	PROPN
ejpam-4596	155	6	te(x	te(x	NOUN
ejpam-4596	155	7	,	,	PUNCT
ejpam-4596	155	8	θ	θ	PROPN
ejpam-4596	155	9	)	)	PUNCT
ejpam-4596	155	10	.	.	PUNCT
ejpam-4596	156	1	if	if	SCONJ
ejpam-4596	156	2	x	x	SYM
ejpam-4596	156	3	∈	∈	PROPN
ejpam-4596	156	4	x	x	NOUN
ejpam-4596	156	5	,	,	PUNCT
ejpam-4596	156	6	then	then	ADV
ejpam-4596	156	7	xβθαθα	xβθαθα	PUNCT
ejpam-4596	156	8	=	=	PUNCT
ejpam-4596	157	1	x′θαθα	x′θαθα	PROPN
ejpam-4596	157	2	=	=	PUNCT
ejpam-4596	158	1	xα	xα	X
ejpam-4596	158	2	which	which	PRON
ejpam-4596	158	3	implies	imply	VERB
ejpam-4596	158	4	that	that	SCONJ
ejpam-4596	158	5	α	α	PRON
ejpam-4596	158	6	=	=	X
ejpam-4596	158	7	βθαθα	βθαθα	ADJ
ejpam-4596	158	8	.	.	PUNCT
ejpam-4596	159	1	therefore	therefore	ADV
ejpam-4596	159	2	α	α	X
ejpam-4596	159	3	=	=	PUNCT
ejpam-4596	159	4	β	β	X
ejpam-4596	159	5	∗	∗	X
ejpam-4596	159	6	α	α	PROPN
ejpam-4596	159	7	∗	∗	NOUN
ejpam-4596	159	8	α	α	NOUN
ejpam-4596	159	9	and	and	CCONJ
ejpam-4596	159	10	hence	hence	ADV
ejpam-4596	159	11	α	α	PRON
ejpam-4596	159	12	is	be	AUX
ejpam-4596	159	13	left	leave	VERB
ejpam-4596	159	14	regular	regular	ADV
ejpam-4596	159	15	,	,	PUNCT
ejpam-4596	159	16	as	as	SCONJ
ejpam-4596	159	17	required	require	VERB
ejpam-4596	159	18	.	.	PUNCT
ejpam-4596	160	1	corollary	corollary	ADJ
ejpam-4596	160	2	2	2	NUM
ejpam-4596	160	3	.	.	PUNCT
ejpam-4596	160	4	idx	idx	PROPN
ejpam-4596	160	5	∈	∈	PROPN
ejpam-4596	160	6	lreg(te(x	lreg(te(x	NOUN
ejpam-4596	160	7	,	,	PUNCT
ejpam-4596	160	8	θ	θ	NOUN
ejpam-4596	160	9	)	)	PUNCT
ejpam-4596	160	10	)	)	PUNCT
ejpam-4596	161	1	if	if	SCONJ
ejpam-4596	161	2	and	and	CCONJ
ejpam-4596	161	3	only	only	ADV
ejpam-4596	161	4	if	if	SCONJ
ejpam-4596	161	5	θ	θ	PROPN
ejpam-4596	161	6	is	be	AUX
ejpam-4596	161	7	a	a	DET
ejpam-4596	161	8	surjection	surjection	NOUN
ejpam-4596	161	9	.	.	PUNCT
ejpam-4596	162	1	proof	proof	NOUN
ejpam-4596	162	2	.	.	PUNCT
ejpam-4596	163	1	suppose	suppose	VERB
ejpam-4596	163	2	that	that	SCONJ
ejpam-4596	163	3	idx	idx	PROPN
ejpam-4596	163	4	∈	∈	PROPN
ejpam-4596	163	5	lreg(te(x	lreg(te(x	NOUN
ejpam-4596	163	6	,	,	PUNCT
ejpam-4596	163	7	θ	θ	NOUN
ejpam-4596	163	8	)	)	PUNCT
ejpam-4596	163	9	)	)	PUNCT
ejpam-4596	163	10	.	.	PUNCT
ejpam-4596	164	1	let	let	VERB
ejpam-4596	164	2	x	x	SYM
ejpam-4596	164	3	∈	∈	PROPN
ejpam-4596	164	4	x.	x.	NOUN
ejpam-4596	164	5	since	since	SCONJ
ejpam-4596	164	6	{	{	PUNCT
ejpam-4596	164	7	x	x	NOUN
ejpam-4596	164	8	}	}	PUNCT
ejpam-4596	164	9	∈	∈	PROPN
ejpam-4596	164	10	πex(idx	πex(idx	NOUN
ejpam-4596	164	11	)	)	PUNCT
ejpam-4596	164	12	and	and	CCONJ
ejpam-4596	164	13	by	by	ADP
ejpam-4596	164	14	theorem	theorem	NOUN
ejpam-4596	164	15	2	2	NUM
ejpam-4596	164	16	,	,	PUNCT
ejpam-4596	164	17	there	there	PRON
ejpam-4596	164	18	exists	exist	VERB
ejpam-4596	164	19	b	b	PROPN
ejpam-4596	164	20	∈	∈	PROPN
ejpam-4596	164	21	x	x	SYM
ejpam-4596	164	22	/	/	SYM
ejpam-4596	164	23	e	e	VERB
ejpam-4596	164	24	such	such	ADJ
ejpam-4596	164	25	that	that	DET
ejpam-4596	164	26	bθidxθ	bθidxθ	NOUN
ejpam-4596	164	27	∈	∈	PROPN
ejpam-4596	164	28	{	{	PUNCT
ejpam-4596	164	29	x	x	NOUN
ejpam-4596	164	30	}	}	PUNCT
ejpam-4596	164	31	for	for	ADP
ejpam-4596	164	32	some	some	DET
ejpam-4596	164	33	b	b	PROPN
ejpam-4596	164	34	∈	∈	PROPN
ejpam-4596	164	35	b.	b.	NOUN
ejpam-4596	165	1	therefore	therefore	ADV
ejpam-4596	165	2	x	x	X
ejpam-4596	165	3	=	=	SYM
ejpam-4596	165	4	bθidxθ	bθidxθ	NOUN
ejpam-4596	165	5	=	=	PROPN
ejpam-4596	165	6	bθθ	bθθ	PROPN
ejpam-4596	165	7	.	.	PUNCT
ejpam-4596	166	1	hence	hence	ADV
ejpam-4596	166	2	θ	θ	PROPN
ejpam-4596	166	3	is	be	AUX
ejpam-4596	166	4	a	a	DET
ejpam-4596	166	5	surjection	surjection	NOUN
ejpam-4596	166	6	on	on	ADP
ejpam-4596	166	7	x.	x.	NOUN
ejpam-4596	166	8	conversely	conversely	ADV
ejpam-4596	166	9	,	,	PUNCT
ejpam-4596	166	10	assume	assume	VERB
ejpam-4596	166	11	that	that	SCONJ
ejpam-4596	166	12	θ	θ	PROPN
ejpam-4596	166	13	is	be	AUX
ejpam-4596	166	14	a	a	DET
ejpam-4596	166	15	surjection	surjection	NOUN
ejpam-4596	166	16	.	.	PUNCT
ejpam-4596	167	1	thus	thus	ADV
ejpam-4596	167	2	θ2	θ2	PROPN
ejpam-4596	167	3	is	be	AUX
ejpam-4596	167	4	also	also	ADV
ejpam-4596	167	5	surjective	surjective	ADJ
ejpam-4596	167	6	.	.	PUNCT
ejpam-4596	168	1	let	let	VERB
ejpam-4596	168	2	a	a	DET
ejpam-4596	168	3	∈	∈	ADJ
ejpam-4596	168	4	x	x	SYM
ejpam-4596	168	5	/	/	SYM
ejpam-4596	168	6	e.	e.	PROPN
ejpam-4596	168	7	note	note	PROPN
ejpam-4596	168	8	that	that	SCONJ
ejpam-4596	168	9	πa(idx	πa(idx	NOUN
ejpam-4596	168	10	)	)	PUNCT
ejpam-4596	169	1	=	=	PRON
ejpam-4596	169	2	{	{	PUNCT
ejpam-4596	169	3	{	{	PUNCT
ejpam-4596	169	4	a	a	NOUN
ejpam-4596	169	5	}	}	PUNCT
ejpam-4596	169	6	:	:	PUNCT
ejpam-4596	169	7	a	a	DET
ejpam-4596	169	8	∈	∈	PROPN
ejpam-4596	169	9	a	a	PRON
ejpam-4596	169	10	}	}	PUNCT
ejpam-4596	169	11	.	.	PUNCT
ejpam-4596	170	1	let	let	VERB
ejpam-4596	170	2	p	p	PRON
ejpam-4596	170	3	∈	∈	PROPN
ejpam-4596	170	4	πa(idx	πa(idx	NOUN
ejpam-4596	170	5	)	)	PUNCT
ejpam-4596	170	6	,	,	PUNCT
ejpam-4596	170	7	then	then	ADV
ejpam-4596	170	8	p	p	NOUN
ejpam-4596	170	9	=	=	X
ejpam-4596	170	10	{	{	PUNCT
ejpam-4596	170	11	a	a	NOUN
ejpam-4596	170	12	}	}	PUNCT
ejpam-4596	170	13	where	where	SCONJ
ejpam-4596	170	14	a	a	DET
ejpam-4596	170	15	∈	∈	PROPN
ejpam-4596	170	16	a	a	PRON
ejpam-4596	171	1	and	and	CCONJ
ejpam-4596	171	2	so	so	ADV
ejpam-4596	171	3	a	a	DET
ejpam-4596	171	4	=	=	X
ejpam-4596	171	5	xθθ	xθθ	PROPN
ejpam-4596	171	6	for	for	ADP
ejpam-4596	171	7	some	some	DET
ejpam-4596	171	8	x	x	SYM
ejpam-4596	171	9	∈	∈	PROPN
ejpam-4596	171	10	x.	x.	NOUN
ejpam-4596	171	11	choose	choose	VERB
ejpam-4596	171	12	b	b	X
ejpam-4596	171	13	=	=	SYM
ejpam-4596	171	14	ex	ex	X
ejpam-4596	171	15	and	and	CCONJ
ejpam-4596	171	16	so	so	ADV
ejpam-4596	171	17	xθidxθ	xθidxθ	NOUN
ejpam-4596	171	18	=	=	SYM
ejpam-4596	171	19	xθθ	xθθ	PROPN
ejpam-4596	172	1	=	=	PUNCT
ejpam-4596	172	2	a	a	DET
ejpam-4596	172	3	∈	∈	PROPN
ejpam-4596	172	4	{	{	PUNCT
ejpam-4596	172	5	a	a	NOUN
ejpam-4596	172	6	}	}	PUNCT
ejpam-4596	172	7	=	=	SYM
ejpam-4596	172	8	p	p	NOUN
ejpam-4596	172	9	.	.	PUNCT
ejpam-4596	173	1	hence	hence	ADV
ejpam-4596	173	2	by	by	ADP
ejpam-4596	173	3	theorem	theorem	NOUN
ejpam-4596	173	4	2	2	NUM
ejpam-4596	173	5	,	,	PUNCT
ejpam-4596	173	6	we	we	PRON
ejpam-4596	173	7	conclude	conclude	VERB
ejpam-4596	173	8	that	that	SCONJ
ejpam-4596	173	9	idx	idx	PROPN
ejpam-4596	173	10	∈	∈	PROPN
ejpam-4596	173	11	lreg(te(x	lreg(te(x	NOUN
ejpam-4596	173	12	,	,	PUNCT
ejpam-4596	173	13	θ	θ	NOUN
ejpam-4596	173	14	)	)	PUNCT
ejpam-4596	173	15	)	)	PUNCT
ejpam-4596	173	16	.	.	PUNCT
ejpam-4596	174	1	theorem	theorem	NOUN
ejpam-4596	174	2	3	3	X
ejpam-4596	174	3	.	.	PUNCT
ejpam-4596	175	1	let	let	VERB
ejpam-4596	175	2	α	α	PRON
ejpam-4596	175	3	∈	∈	PROPN
ejpam-4596	175	4	te(x	te(x	NOUN
ejpam-4596	175	5	,	,	PUNCT
ejpam-4596	175	6	θ	θ	PROPN
ejpam-4596	175	7	)	)	PUNCT
ejpam-4596	175	8	.	.	PUNCT
ejpam-4596	176	1	then	then	ADV
ejpam-4596	176	2	α	α	PROPN
ejpam-4596	176	3	∈	∈	PROPN
ejpam-4596	176	4	rreg(te(x	rreg(te(x	NOUN
ejpam-4596	176	5	,	,	PUNCT
ejpam-4596	176	6	θ	θ	NOUN
ejpam-4596	176	7	)	)	PUNCT
ejpam-4596	176	8	)	)	PUNCT
ejpam-4596	177	1	if	if	SCONJ
ejpam-4596	177	2	and	and	CCONJ
ejpam-4596	177	3	only	only	ADV
ejpam-4596	177	4	if	if	SCONJ
ejpam-4596	177	5	(	(	PUNCT
ejpam-4596	177	6	θαθ)|xα	θαθ)|xα	PROPN
ejpam-4596	177	7	is	be	AUX
ejpam-4596	177	8	an	an	DET
ejpam-4596	177	9	e∗-preserving	e∗-preserving	NOUN
ejpam-4596	177	10	injection	injection	NOUN
ejpam-4596	177	11	.	.	PUNCT
ejpam-4596	178	1	proof	proof	NOUN
ejpam-4596	178	2	.	.	PUNCT
ejpam-4596	179	1	suppose	suppose	VERB
ejpam-4596	179	2	that	that	SCONJ
ejpam-4596	179	3	α	α	PROPN
ejpam-4596	179	4	∈	∈	PROPN
ejpam-4596	179	5	rreg(te(x	rreg(te(x	NOUN
ejpam-4596	179	6	,	,	PUNCT
ejpam-4596	179	7	θ	θ	NOUN
ejpam-4596	179	8	)	)	PUNCT
ejpam-4596	179	9	)	)	PUNCT
ejpam-4596	179	10	.	.	PUNCT
ejpam-4596	180	1	then	then	ADV
ejpam-4596	180	2	there	there	PRON
ejpam-4596	180	3	is	be	VERB
ejpam-4596	180	4	β	β	PROPN
ejpam-4596	180	5	∈	∈	PROPN
ejpam-4596	180	6	te(x	te(x	NOUN
ejpam-4596	180	7	,	,	PUNCT
ejpam-4596	180	8	θ	θ	NOUN
ejpam-4596	180	9	)	)	PUNCT
ejpam-4596	180	10	such	such	ADJ
ejpam-4596	180	11	that	that	SCONJ
ejpam-4596	180	12	α	α	NOUN
ejpam-4596	180	13	=	=	SYM
ejpam-4596	180	14	α	α	PROPN
ejpam-4596	180	15	∗	∗	NOUN
ejpam-4596	180	16	α	α	PROPN
ejpam-4596	180	17	∗	∗	NOUN
ejpam-4596	180	18	β	β	X
ejpam-4596	181	1	and	and	CCONJ
ejpam-4596	181	2	so	so	ADV
ejpam-4596	181	3	α	α	NOUN
ejpam-4596	181	4	=	=	SYM
ejpam-4596	181	5	αθαθβ	αθαθβ	NOUN
ejpam-4596	181	6	.	.	PUNCT
ejpam-4596	182	1	since	since	SCONJ
ejpam-4596	182	2	θαθ	θαθ	PRON
ejpam-4596	182	3	∈	∈	PROPN
ejpam-4596	182	4	te(x	te(x	NOUN
ejpam-4596	182	5	)	)	PUNCT
ejpam-4596	182	6	,	,	PUNCT
ejpam-4596	182	7	(	(	PUNCT
ejpam-4596	182	8	θαθ)|xα	θαθ)|xα	PROPN
ejpam-4596	182	9	is	be	AUX
ejpam-4596	182	10	e	e	VERB
ejpam-4596	182	11	-	-	ADJ
ejpam-4596	182	12	preserving	preserving	ADJ
ejpam-4596	182	13	.	.	PUNCT
ejpam-4596	183	1	let	let	VERB
ejpam-4596	183	2	x	x	PRON
ejpam-4596	183	3	,	,	PUNCT
ejpam-4596	183	4	y	y	PROPN
ejpam-4596	183	5	∈	∈	PROPN
ejpam-4596	183	6	xα	xα	INTJ
ejpam-4596	183	7	be	be	AUX
ejpam-4596	183	8	such	such	ADJ
ejpam-4596	183	9	that	that	SCONJ
ejpam-4596	183	10	x	x	X
ejpam-4596	183	11	=	=	PUNCT
ejpam-4596	183	12	x′α	x′α	PROPN
ejpam-4596	183	13	and	and	CCONJ
ejpam-4596	183	14	y	y	PROPN
ejpam-4596	183	15	=	=	PROPN
ejpam-4596	183	16	y′α	y′α	NOUN
ejpam-4596	183	17	where	where	SCONJ
ejpam-4596	183	18	x′	x′	PROPN
ejpam-4596	183	19	,	,	PUNCT
ejpam-4596	183	20	y′	y′	NOUN
ejpam-4596	183	21	∈	∈	PROPN
ejpam-4596	183	22	x.	x.	NOUN
ejpam-4596	184	1	if	if	SCONJ
ejpam-4596	184	2	xθαθ	xθαθ	PROPN
ejpam-4596	184	3	=	=	SYM
ejpam-4596	184	4	yθαθ	yθαθ	PROPN
ejpam-4596	184	5	,	,	PUNCT
ejpam-4596	184	6	then	then	ADV
ejpam-4596	184	7	x	x	X
ejpam-4596	184	8	=	=	PUNCT
ejpam-4596	184	9	x′α	x′α	X
ejpam-4596	185	1	=	=	PUNCT
ejpam-4596	186	1	x′αθαθβ	x′αθαθβ	PUNCT
ejpam-4596	186	2	=	=	PUNCT
ejpam-4596	187	1	xθαθβ	xθαθβ	X
ejpam-4596	188	1	=	=	SYM
ejpam-4596	188	2	yθαθβ	yθαθβ	ADJ
ejpam-4596	188	3	=	=	SYM
ejpam-4596	188	4	y′αθαθβ	y′αθαθβ	PROPN
ejpam-4596	188	5	=	=	SYM
ejpam-4596	188	6	y′α	y′α	NOUN
ejpam-4596	188	7	=	=	PUNCT
ejpam-4596	189	1	y.	y.	PROPN
ejpam-4596	189	2	it	it	PRON
ejpam-4596	189	3	follows	follow	VERB
ejpam-4596	189	4	that	that	SCONJ
ejpam-4596	189	5	(	(	PUNCT
ejpam-4596	189	6	θαθ)|xα	θαθ)|xα	PROPN
ejpam-4596	189	7	is	be	AUX
ejpam-4596	189	8	an	an	DET
ejpam-4596	189	9	injection	injection	NOUN
ejpam-4596	189	10	.	.	PUNCT
ejpam-4596	190	1	if	if	SCONJ
ejpam-4596	190	2	(	(	PUNCT
ejpam-4596	190	3	xθαθ	xθαθ	PROPN
ejpam-4596	190	4	,	,	PUNCT
ejpam-4596	190	5	yθαθ	yθαθ	PROPN
ejpam-4596	190	6	)	)	PUNCT
ejpam-4596	190	7	∈	∈	PROPN
ejpam-4596	190	8	e	e	NOUN
ejpam-4596	190	9	,	,	PUNCT
ejpam-4596	190	10	then	then	ADV
ejpam-4596	190	11	since	since	SCONJ
ejpam-4596	190	12	β	β	PROPN
ejpam-4596	190	13	∈	∈	PROPN
ejpam-4596	190	14	te(x	te(x	NOUN
ejpam-4596	190	15	,	,	PUNCT
ejpam-4596	190	16	θ	θ	PROPN
ejpam-4596	190	17	)	)	PUNCT
ejpam-4596	190	18	,	,	PUNCT
ejpam-4596	190	19	we	we	PRON
ejpam-4596	190	20	have	have	VERB
ejpam-4596	190	21	that	that	PRON
ejpam-4596	190	22	(	(	PUNCT
ejpam-4596	190	23	xθαθβ	xθαθβ	PROPN
ejpam-4596	190	24	,	,	PUNCT
ejpam-4596	190	25	yθαθβ	yθαθβ	PROPN
ejpam-4596	190	26	)	)	PUNCT
ejpam-4596	190	27	∈	∈	PROPN
ejpam-4596	190	28	e.	e.	PROPN
ejpam-4596	190	29	moreover	moreover	ADV
ejpam-4596	190	30	,	,	PUNCT
ejpam-4596	190	31	(	(	PUNCT
ejpam-4596	190	32	x	x	X
ejpam-4596	190	33	,	,	PUNCT
ejpam-4596	190	34	y	y	NOUN
ejpam-4596	190	35	)	)	PUNCT
ejpam-4596	190	36	=	=	SYM
ejpam-4596	190	37	(	(	PUNCT
ejpam-4596	190	38	x′α	x′α	PROPN
ejpam-4596	190	39	,	,	PUNCT
ejpam-4596	190	40	y′α	y′α	NOUN
ejpam-4596	190	41	)	)	PUNCT
ejpam-4596	191	1	=	=	NOUN
ejpam-4596	191	2	(	(	PUNCT
ejpam-4596	191	3	x′αθαθβ	x′αθαθβ	PROPN
ejpam-4596	191	4	,	,	PUNCT
ejpam-4596	191	5	y′αθαθβ	y′αθαθβ	PROPN
ejpam-4596	191	6	)	)	PUNCT
ejpam-4596	191	7	=	=	SYM
ejpam-4596	191	8	(	(	PUNCT
ejpam-4596	191	9	xθαθβ	xθαθβ	PROPN
ejpam-4596	191	10	,	,	PUNCT
ejpam-4596	191	11	yθαθβ	yθαθβ	PROPN
ejpam-4596	191	12	)	)	PUNCT
ejpam-4596	191	13	∈	∈	PROPN
ejpam-4596	191	14	e	e	X
ejpam-4596	191	15	which	which	PRON
ejpam-4596	191	16	implies	imply	VERB
ejpam-4596	191	17	that	that	SCONJ
ejpam-4596	191	18	(	(	PUNCT
ejpam-4596	191	19	θαθ)|xα	θαθ)|xα	PROPN
ejpam-4596	191	20	is	be	AUX
ejpam-4596	191	21	an	an	DET
ejpam-4596	191	22	e∗-preserving	e∗-preserving	NOUN
ejpam-4596	191	23	injection	injection	NOUN
ejpam-4596	191	24	.	.	PUNCT
ejpam-4596	192	1	conversely	conversely	ADV
ejpam-4596	192	2	,	,	PUNCT
ejpam-4596	192	3	assume	assume	VERB
ejpam-4596	192	4	that	that	SCONJ
ejpam-4596	192	5	(	(	PUNCT
ejpam-4596	192	6	θαθ)|xα	θαθ)|xα	PROPN
ejpam-4596	192	7	is	be	AUX
ejpam-4596	192	8	an	an	DET
ejpam-4596	192	9	e∗-preserving	e∗-preserving	NOUN
ejpam-4596	192	10	injection	injection	NOUN
ejpam-4596	192	11	.	.	PUNCT
ejpam-4596	193	1	let	let	VERB
ejpam-4596	193	2	a	a	DET
ejpam-4596	193	3	∈	∈	ADJ
ejpam-4596	193	4	x	x	SYM
ejpam-4596	193	5	/	/	SYM
ejpam-4596	193	6	e	e	VERB
ejpam-4596	193	7	be	be	VERB
ejpam-4596	193	8	such	such	ADJ
ejpam-4596	193	9	that	that	SCONJ
ejpam-4596	193	10	a	a	DET
ejpam-4596	193	11	∩	∩	NOUN
ejpam-4596	193	12	xαθαθ	xαθαθ	PROPN
ejpam-4596	193	13	̸=	̸=	PROPN
ejpam-4596	193	14	∅.	∅.	PROPN
ejpam-4596	194	1	we	we	PRON
ejpam-4596	194	2	choose	choose	VERB
ejpam-4596	194	3	and	and	CCONJ
ejpam-4596	194	4	fix	fix	VERB
ejpam-4596	194	5	an	an	DET
ejpam-4596	194	6	element	element	NOUN
ejpam-4596	194	7	xa	xa	PROPN
ejpam-4596	194	8	∈	∈	PROPN
ejpam-4596	194	9	a	a	DET
ejpam-4596	194	10	∩	∩	ADJ
ejpam-4596	194	11	xαθαθ	xαθαθ	PROPN
ejpam-4596	194	12	.	.	PUNCT
ejpam-4596	195	1	for	for	ADP
ejpam-4596	195	2	each	each	DET
ejpam-4596	195	3	x	x	SYM
ejpam-4596	195	4	∈	∈	PROPN
ejpam-4596	195	5	a	a	DET
ejpam-4596	195	6	∩	∩	ADJ
ejpam-4596	195	7	xαθαθ	xαθαθ	NOUN
ejpam-4596	195	8	,	,	PUNCT
ejpam-4596	195	9	there	there	PRON
ejpam-4596	195	10	exists	exist	VERB
ejpam-4596	195	11	a	a	DET
ejpam-4596	195	12	unique	unique	ADJ
ejpam-4596	195	13	element	element	NOUN
ejpam-4596	195	14	x′	x′	PROPN
ejpam-4596	195	15	∈	∈	PROPN
ejpam-4596	195	16	xα	xα	ADP
ejpam-4596	195	17	such	such	ADJ
ejpam-4596	195	18	that	that	SCONJ
ejpam-4596	195	19	x	x	X
ejpam-4596	195	20	=	=	SYM
ejpam-4596	195	21	x′θαθ	x′θαθ	PROPN
ejpam-4596	195	22	by	by	ADP
ejpam-4596	195	23	the	the	DET
ejpam-4596	195	24	condition	condition	NOUN
ejpam-4596	195	25	(	(	PUNCT
ejpam-4596	195	26	θαθ)|xα	θαθ)|xα	PROPN
ejpam-4596	195	27	is	be	AUX
ejpam-4596	195	28	injective	injective	ADJ
ejpam-4596	195	29	.	.	PUNCT
ejpam-4596	196	1	we	we	PRON
ejpam-4596	196	2	observe	observe	VERB
ejpam-4596	196	3	that	that	SCONJ
ejpam-4596	196	4	(	(	PUNCT
ejpam-4596	196	5	x′θαθ	x′θαθ	PROPN
ejpam-4596	196	6	,	,	PUNCT
ejpam-4596	196	7	x′aθαθ	x′aθαθ	X
ejpam-4596	196	8	)	)	PUNCT
ejpam-4596	196	9	=	=	SYM
ejpam-4596	196	10	(	(	PUNCT
ejpam-4596	197	1	x	x	X
ejpam-4596	197	2	,	,	PUNCT
ejpam-4596	197	3	xa	xa	PROPN
ejpam-4596	197	4	)	)	PUNCT
ejpam-4596	197	5	∈	∈	PROPN
ejpam-4596	197	6	e.	e.	PROPN
ejpam-4596	197	7	it	it	PRON
ejpam-4596	197	8	follows	follow	VERB
ejpam-4596	197	9	from	from	ADP
ejpam-4596	197	10	(	(	PUNCT
ejpam-4596	197	11	θαθ)|xα	θαθ)|xα	PROPN
ejpam-4596	197	12	is	be	AUX
ejpam-4596	197	13	an	an	DET
ejpam-4596	197	14	e∗-preserving	e∗-preserve	VERB
ejpam-4596	197	15	mapping	mapping	NOUN
ejpam-4596	197	16	that	that	SCONJ
ejpam-4596	197	17	(	(	PUNCT
ejpam-4596	197	18	x′	x′	NUM
ejpam-4596	197	19	,	,	PUNCT
ejpam-4596	197	20	x′a	x′a	ADV
ejpam-4596	197	21	)	)	PUNCT
ejpam-4596	197	22	∈	∈	PROPN
ejpam-4596	197	23	e.	e.	PROPN
ejpam-4596	197	24	define	define	VERB
ejpam-4596	197	25	βa	βa	INTJ
ejpam-4596	197	26	:	:	PUNCT
ejpam-4596	197	27	a	a	DET
ejpam-4596	197	28	→	→	PUNCT
ejpam-4596	197	29	ex′	ex′	X
ejpam-4596	197	30	a	a	PRON
ejpam-4596	197	31	by	by	ADP
ejpam-4596	197	32	xβa	xβa	PROPN
ejpam-4596	197	33	=	=	PRON
ejpam-4596	197	34	{	{	PUNCT
ejpam-4596	197	35	x′	x′	PROPN
ejpam-4596	198	1	if	if	SCONJ
ejpam-4596	198	2	x	x	PROPN
ejpam-4596	198	3	∈	∈	PROPN
ejpam-4596	198	4	xαθαθ	xαθαθ	PROPN
ejpam-4596	198	5	,	,	PUNCT
ejpam-4596	198	6	x′a	x′a	ADV
ejpam-4596	198	7	otherwise	otherwise	ADV
ejpam-4596	198	8	.	.	PUNCT
ejpam-4596	199	1	p.	p.	NOUN
ejpam-4596	199	2	tantong	tantong	NOUN
ejpam-4596	199	3	,	,	PUNCT
ejpam-4596	199	4	n.	n.	NOUN
ejpam-4596	199	5	sawatraksa	sawatraksa	PROPN
ejpam-4596	199	6	/	/	SYM
ejpam-4596	199	7	eur	eur	PROPN
ejpam-4596	199	8	.	.	PUNCT
ejpam-4596	200	1	j.	j.	PROPN
ejpam-4596	200	2	pure	pure	PROPN
ejpam-4596	200	3	appl	appl	PROPN
ejpam-4596	200	4	.	.	PROPN
ejpam-4596	200	5	math	math	PROPN
ejpam-4596	200	6	,	,	PUNCT
ejpam-4596	200	7	15	15	NUM
ejpam-4596	200	8	(	(	PUNCT
ejpam-4596	200	9	4	4	NUM
ejpam-4596	200	10	)	)	PUNCT
ejpam-4596	200	11	(	(	PUNCT
ejpam-4596	200	12	2022	2022	NUM
ejpam-4596	200	13	)	)	PUNCT
ejpam-4596	200	14	,	,	PUNCT
ejpam-4596	200	15	2116	2116	NUM
ejpam-4596	200	16	-	-	SYM
ejpam-4596	200	17	2126	2126	NUM
ejpam-4596	200	18	2121	2121	NUM
ejpam-4596	200	19	then	then	ADV
ejpam-4596	200	20	we	we	PRON
ejpam-4596	200	21	define	define	VERB
ejpam-4596	200	22	the	the	DET
ejpam-4596	200	23	map	map	NOUN
ejpam-4596	200	24	β	β	X
ejpam-4596	200	25	:	:	PUNCT
ejpam-4596	200	26	x	x	SYM
ejpam-4596	200	27	→	→	SYM
ejpam-4596	200	28	x	x	SYM
ejpam-4596	200	29	by	by	ADP
ejpam-4596	200	30	β|a	β|a	PUNCT
ejpam-4596	201	1	=	=	PRON
ejpam-4596	201	2	{	{	PUNCT
ejpam-4596	201	3	βa	βa	INTJ
ejpam-4596	201	4	if	if	SCONJ
ejpam-4596	201	5	a	a	DET
ejpam-4596	201	6	∩xαθαθ	∩xαθαθ	NOUN
ejpam-4596	201	7	̸=	̸=	PROPN
ejpam-4596	201	8	∅	∅	NOUN
ejpam-4596	201	9	,	,	PUNCT
ejpam-4596	201	10	ida	ida	PROPN
ejpam-4596	201	11	otherwise	otherwise	ADV
ejpam-4596	201	12	,	,	PUNCT
ejpam-4596	201	13	for	for	ADP
ejpam-4596	201	14	all	all	DET
ejpam-4596	201	15	a	a	DET
ejpam-4596	201	16	∈	∈	PROPN
ejpam-4596	201	17	x	x	X
ejpam-4596	201	18	/	/	SYM
ejpam-4596	201	19	e.	e.	PROPN
ejpam-4596	201	20	since	since	SCONJ
ejpam-4596	201	21	x	x	PROPN
ejpam-4596	201	22	/	/	SYM
ejpam-4596	201	23	e	e	NOUN
ejpam-4596	201	24	is	be	AUX
ejpam-4596	201	25	a	a	DET
ejpam-4596	201	26	partition	partition	NOUN
ejpam-4596	201	27	of	of	ADP
ejpam-4596	201	28	x	x	PRON
ejpam-4596	201	29	,	,	PUNCT
ejpam-4596	201	30	β	β	X
ejpam-4596	201	31	is	be	AUX
ejpam-4596	201	32	well	well	ADV
ejpam-4596	201	33	-	-	PUNCT
ejpam-4596	201	34	defined	define	VERB
ejpam-4596	201	35	.	.	PUNCT
ejpam-4596	202	1	let	let	VERB
ejpam-4596	202	2	x	x	PRON
ejpam-4596	202	3	,	,	PUNCT
ejpam-4596	202	4	y	y	PROPN
ejpam-4596	202	5	∈	∈	PROPN
ejpam-4596	202	6	x	x	AUX
ejpam-4596	202	7	be	be	AUX
ejpam-4596	202	8	such	such	ADJ
ejpam-4596	202	9	that	that	SCONJ
ejpam-4596	202	10	(	(	PUNCT
ejpam-4596	202	11	x	x	NOUN
ejpam-4596	202	12	,	,	PUNCT
ejpam-4596	202	13	y	y	NOUN
ejpam-4596	202	14	)	)	PUNCT
ejpam-4596	202	15	∈	∈	PROPN
ejpam-4596	202	16	e.	e.	PROPN
ejpam-4596	202	17	then	then	ADV
ejpam-4596	202	18	x	x	PRON
ejpam-4596	202	19	,	,	PUNCT
ejpam-4596	202	20	y	y	PROPN
ejpam-4596	202	21	∈	∈	PROPN
ejpam-4596	202	22	a	a	PRON
ejpam-4596	202	23	for	for	ADP
ejpam-4596	202	24	some	some	DET
ejpam-4596	202	25	a	a	DET
ejpam-4596	202	26	∈	∈	PROPN
ejpam-4596	202	27	x	x	X
ejpam-4596	202	28	/	/	SYM
ejpam-4596	202	29	e.	e.	PROPN
ejpam-4596	202	30	by	by	ADP
ejpam-4596	202	31	the	the	DET
ejpam-4596	202	32	definition	definition	NOUN
ejpam-4596	202	33	of	of	ADP
ejpam-4596	202	34	β	β	X
ejpam-4596	202	35	,	,	PUNCT
ejpam-4596	202	36	we	we	PRON
ejpam-4596	202	37	have	have	VERB
ejpam-4596	202	38	(	(	PUNCT
ejpam-4596	202	39	xβ	xβ	PROPN
ejpam-4596	202	40	,	,	PUNCT
ejpam-4596	202	41	yβ	yβ	NOUN
ejpam-4596	202	42	)	)	PUNCT
ejpam-4596	202	43	=	=	SYM
ejpam-4596	203	1	(	(	PUNCT
ejpam-4596	203	2	xβ|a	xβ|a	PROPN
ejpam-4596	203	3	,	,	PUNCT
ejpam-4596	203	4	yβ|a	yβ|a	PROPN
ejpam-4596	203	5	)	)	PUNCT
ejpam-4596	203	6	.	.	PUNCT
ejpam-4596	204	1	if	if	SCONJ
ejpam-4596	204	2	a	a	DET
ejpam-4596	204	3	∩	∩	ADJ
ejpam-4596	204	4	αβαβ	αβαβ	NOUN
ejpam-4596	204	5	=	=	NOUN
ejpam-4596	204	6	∅	∅	NOUN
ejpam-4596	204	7	,	,	PUNCT
ejpam-4596	204	8	then	then	ADV
ejpam-4596	204	9	(	(	PUNCT
ejpam-4596	204	10	xβ	xβ	PROPN
ejpam-4596	204	11	,	,	PUNCT
ejpam-4596	204	12	yβ	yβ	NOUN
ejpam-4596	204	13	)	)	PUNCT
ejpam-4596	204	14	=	=	SYM
ejpam-4596	204	15	(	(	PUNCT
ejpam-4596	204	16	xida	xida	PROPN
ejpam-4596	204	17	,	,	PUNCT
ejpam-4596	204	18	yida	yida	NOUN
ejpam-4596	204	19	)	)	PUNCT
ejpam-4596	204	20	=	=	PUNCT
ejpam-4596	204	21	(	(	PUNCT
ejpam-4596	204	22	x	x	X
ejpam-4596	204	23	,	,	PUNCT
ejpam-4596	204	24	y	y	NOUN
ejpam-4596	204	25	)	)	PUNCT
ejpam-4596	204	26	∈	∈	PROPN
ejpam-4596	204	27	e.	e.	PROPN
ejpam-4596	205	1	if	if	SCONJ
ejpam-4596	205	2	a	a	DET
ejpam-4596	205	3	∩	∩	ADJ
ejpam-4596	205	4	αβαβ	αβαβ	NOUN
ejpam-4596	205	5	̸=	̸=	PROPN
ejpam-4596	205	6	∅	∅	NOUN
ejpam-4596	205	7	,	,	PUNCT
ejpam-4596	205	8	then	then	ADV
ejpam-4596	205	9	xβ	xβ	PROPN
ejpam-4596	205	10	,	,	PUNCT
ejpam-4596	205	11	yβ	yβ	PROPN
ejpam-4596	205	12	∈	∈	NOUN
ejpam-4596	205	13	aβ	aβ	NOUN
ejpam-4596	205	14	=	=	PRON
ejpam-4596	205	15	aβa	aβa	VERB
ejpam-4596	205	16	⊆	⊆	NUM
ejpam-4596	205	17	ex′	ex′	X
ejpam-4596	205	18	a	a	PRON
ejpam-4596	205	19	.	.	PUNCT
ejpam-4596	206	1	consequently	consequently	ADV
ejpam-4596	206	2	,	,	PUNCT
ejpam-4596	206	3	β	β	PROPN
ejpam-4596	206	4	∈	∈	PROPN
ejpam-4596	206	5	te(x	te(x	NOUN
ejpam-4596	206	6	,	,	PUNCT
ejpam-4596	206	7	θ	θ	PROPN
ejpam-4596	206	8	)	)	PUNCT
ejpam-4596	206	9	.	.	PUNCT
ejpam-4596	207	1	finally	finally	ADV
ejpam-4596	207	2	,	,	PUNCT
ejpam-4596	207	3	to	to	PART
ejpam-4596	207	4	show	show	VERB
ejpam-4596	207	5	that	that	SCONJ
ejpam-4596	207	6	α	α	NOUN
ejpam-4596	207	7	=	=	SYM
ejpam-4596	207	8	αθαθβ	αθαθβ	NOUN
ejpam-4596	207	9	,	,	PUNCT
ejpam-4596	207	10	let	let	VERB
ejpam-4596	207	11	x	x	PUNCT
ejpam-4596	207	12	∈	∈	PROPN
ejpam-4596	207	13	x.	x.	NOUN
ejpam-4596	207	14	then	then	ADV
ejpam-4596	207	15	xαθαθ	xαθαθ	PROPN
ejpam-4596	207	16	∈	∈	PROPN
ejpam-4596	207	17	xαθαθ	xαθαθ	PROPN
ejpam-4596	207	18	.	.	PUNCT
ejpam-4596	208	1	then	then	ADV
ejpam-4596	208	2	there	there	PRON
ejpam-4596	208	3	exists	exist	VERB
ejpam-4596	208	4	a	a	DET
ejpam-4596	208	5	∈	∈	PROPN
ejpam-4596	208	6	x	x	SYM
ejpam-4596	208	7	/	/	SYM
ejpam-4596	208	8	e	e	VERB
ejpam-4596	208	9	such	such	ADJ
ejpam-4596	208	10	that	that	SCONJ
ejpam-4596	208	11	xαθαθ	xαθαθ	PROPN
ejpam-4596	208	12	∈	∈	PROPN
ejpam-4596	208	13	a.	a.	NOUN
ejpam-4596	208	14	by	by	ADP
ejpam-4596	208	15	the	the	DET
ejpam-4596	208	16	definition	definition	NOUN
ejpam-4596	208	17	of	of	ADP
ejpam-4596	208	18	βa	βa	PROPN
ejpam-4596	208	19	,	,	PUNCT
ejpam-4596	208	20	xαθαθβa	xαθαθβa	PROPN
ejpam-4596	208	21	=	=	SYM
ejpam-4596	209	1	(	(	PUNCT
ejpam-4596	209	2	xαθαθ)′	xαθαθ)′	PROPN
ejpam-4596	209	3	where	where	SCONJ
ejpam-4596	209	4	(	(	PUNCT
ejpam-4596	209	5	xαθαθ)′θαθ	xαθαθ)′θαθ	NOUN
ejpam-4596	209	6	=	=	PUNCT
ejpam-4596	209	7	xαθαθ	xαθαθ	PROPN
ejpam-4596	209	8	=	=	SYM
ejpam-4596	209	9	(	(	PUNCT
ejpam-4596	209	10	xα)θαθ	xα)θαθ	X
ejpam-4596	209	11	.	.	PUNCT
ejpam-4596	210	1	since	since	SCONJ
ejpam-4596	210	2	(	(	PUNCT
ejpam-4596	210	3	xαθαθ)′	xαθαθ)′	PROPN
ejpam-4596	210	4	is	be	AUX
ejpam-4596	210	5	unique	unique	ADJ
ejpam-4596	210	6	,	,	PUNCT
ejpam-4596	210	7	we	we	PRON
ejpam-4596	210	8	get	get	VERB
ejpam-4596	210	9	that	that	PRON
ejpam-4596	210	10	(	(	PUNCT
ejpam-4596	210	11	xαθαθ)′	xαθαθ)′	PROPN
ejpam-4596	210	12	=	=	PUNCT
ejpam-4596	211	1	xα	xα	PROPN
ejpam-4596	211	2	.	.	PUNCT
ejpam-4596	212	1	thus	thus	ADV
ejpam-4596	212	2	xαθαθβ	xαθαθβ	PROPN
ejpam-4596	212	3	=	=	SYM
ejpam-4596	212	4	xαθαθβa	xαθαθβa	PROPN
ejpam-4596	213	1	=	=	PUNCT
ejpam-4596	214	1	xα	xα	PROPN
ejpam-4596	214	2	.	.	PUNCT
ejpam-4596	215	1	therefore	therefore	ADV
ejpam-4596	215	2	α	α	X
ejpam-4596	215	3	=	=	PUNCT
ejpam-4596	215	4	α	α	PROPN
ejpam-4596	215	5	∗	∗	NOUN
ejpam-4596	215	6	α	α	PROPN
ejpam-4596	215	7	∗	∗	NOUN
ejpam-4596	215	8	β	β	NOUN
ejpam-4596	215	9	.	.	PUNCT
ejpam-4596	216	1	hence	hence	ADV
ejpam-4596	216	2	α	α	PROPN
ejpam-4596	216	3	∈	∈	PROPN
ejpam-4596	216	4	rreg(te(x	rreg(te(x	NOUN
ejpam-4596	216	5	,	,	PUNCT
ejpam-4596	216	6	θ	θ	NOUN
ejpam-4596	216	7	)	)	PUNCT
ejpam-4596	216	8	)	)	PUNCT
ejpam-4596	216	9	,	,	PUNCT
ejpam-4596	216	10	as	as	SCONJ
ejpam-4596	216	11	asserted	assert	VERB
ejpam-4596	216	12	.	.	PUNCT
ejpam-4596	217	1	corollary	corollary	ADJ
ejpam-4596	217	2	3	3	NUM
ejpam-4596	217	3	.	.	PUNCT
ejpam-4596	218	1	idx	idx	PROPN
ejpam-4596	218	2	∈	∈	PROPN
ejpam-4596	218	3	rreg(te(x	rreg(te(x	NOUN
ejpam-4596	218	4	,	,	PUNCT
ejpam-4596	218	5	θ	θ	NOUN
ejpam-4596	218	6	)	)	PUNCT
ejpam-4596	218	7	)	)	PUNCT
ejpam-4596	219	1	if	if	SCONJ
ejpam-4596	219	2	and	and	CCONJ
ejpam-4596	219	3	only	only	ADV
ejpam-4596	219	4	if	if	SCONJ
ejpam-4596	219	5	θ	θ	PROPN
ejpam-4596	219	6	is	be	AUX
ejpam-4596	219	7	an	an	DET
ejpam-4596	219	8	e∗-preserving	e∗-preserving	NOUN
ejpam-4596	219	9	injection	injection	NOUN
ejpam-4596	219	10	.	.	PUNCT
ejpam-4596	220	1	proof	proof	NOUN
ejpam-4596	220	2	.	.	PUNCT
ejpam-4596	221	1	assume	assume	VERB
ejpam-4596	221	2	that	that	SCONJ
ejpam-4596	221	3	idx	idx	NOUN
ejpam-4596	221	4	∈	∈	PROPN
ejpam-4596	221	5	rreg(te(x	rreg(te(x	NOUN
ejpam-4596	221	6	,	,	PUNCT
ejpam-4596	221	7	θ	θ	NOUN
ejpam-4596	221	8	)	)	PUNCT
ejpam-4596	221	9	)	)	PUNCT
ejpam-4596	221	10	.	.	PUNCT
ejpam-4596	222	1	by	by	ADP
ejpam-4596	222	2	theorem	theorem	NOUN
ejpam-4596	222	3	3	3	NUM
ejpam-4596	222	4	,	,	PUNCT
ejpam-4596	222	5	we	we	PRON
ejpam-4596	222	6	get	get	VERB
ejpam-4596	222	7	that	that	PRON
ejpam-4596	222	8	(	(	PUNCT
ejpam-4596	222	9	θidxθ)|xidx	θidxθ)|xidx	NOUN
ejpam-4596	222	10	is	be	AUX
ejpam-4596	222	11	an	an	DET
ejpam-4596	222	12	e∗-preserving	e∗-preserving	NOUN
ejpam-4596	222	13	injection	injection	NOUN
ejpam-4596	222	14	.	.	PUNCT
ejpam-4596	223	1	this	this	PRON
ejpam-4596	223	2	implies	imply	VERB
ejpam-4596	223	3	that	that	SCONJ
ejpam-4596	223	4	θθ	θθ	PRON
ejpam-4596	223	5	is	be	AUX
ejpam-4596	223	6	an	an	DET
ejpam-4596	223	7	e∗-preserving	e∗-preserving	NOUN
ejpam-4596	223	8	injection	injection	NOUN
ejpam-4596	223	9	.	.	PUNCT
ejpam-4596	224	1	hence	hence	ADV
ejpam-4596	224	2	θ	θ	PROPN
ejpam-4596	224	3	is	be	AUX
ejpam-4596	224	4	an	an	DET
ejpam-4596	224	5	e∗-preserving	e∗-preserving	NOUN
ejpam-4596	224	6	injection	injection	NOUN
ejpam-4596	224	7	.	.	PUNCT
ejpam-4596	225	1	this	this	DET
ejpam-4596	225	2	converse	converse	NOUN
ejpam-4596	225	3	of	of	ADP
ejpam-4596	225	4	corollary	corollary	NOUN
ejpam-4596	225	5	is	be	AUX
ejpam-4596	225	6	clear	clear	ADJ
ejpam-4596	225	7	.	.	PUNCT
ejpam-4596	226	1	final	final	ADJ
ejpam-4596	226	2	of	of	ADP
ejpam-4596	226	3	this	this	DET
ejpam-4596	226	4	section	section	NOUN
ejpam-4596	226	5	,	,	PUNCT
ejpam-4596	226	6	we	we	PRON
ejpam-4596	226	7	give	give	VERB
ejpam-4596	226	8	a	a	DET
ejpam-4596	226	9	characterization	characterization	NOUN
ejpam-4596	226	10	of	of	ADP
ejpam-4596	226	11	completely	completely	ADV
ejpam-4596	226	12	regular	regular	ADJ
ejpam-4596	226	13	elements	element	NOUN
ejpam-4596	226	14	in	in	ADP
ejpam-4596	226	15	te(x	te(x	NOUN
ejpam-4596	226	16	,	,	PUNCT
ejpam-4596	226	17	θ	θ	PROPN
ejpam-4596	226	18	)	)	PUNCT
ejpam-4596	226	19	.	.	PUNCT
ejpam-4596	227	1	recall	recall	VERB
ejpam-4596	227	2	that	that	PRON
ejpam-4596	227	3	,	,	PUNCT
ejpam-4596	227	4	an	an	DET
ejpam-4596	227	5	element	element	NOUN
ejpam-4596	227	6	a	a	PRON
ejpam-4596	227	7	of	of	ADP
ejpam-4596	227	8	a	a	DET
ejpam-4596	227	9	semigroup	semigroup	NOUN
ejpam-4596	227	10	s	s	VERB
ejpam-4596	227	11	is	be	AUX
ejpam-4596	227	12	completely	completely	ADV
ejpam-4596	227	13	regular	regular	ADJ
ejpam-4596	227	14	if	if	SCONJ
ejpam-4596	228	1	and	and	CCONJ
ejpam-4596	228	2	only	only	ADV
ejpam-4596	228	3	if	if	SCONJ
ejpam-4596	228	4	a	a	PRON
ejpam-4596	228	5	is	be	AUX
ejpam-4596	228	6	both	both	PRON
ejpam-4596	228	7	left	leave	VERB
ejpam-4596	228	8	and	and	CCONJ
ejpam-4596	228	9	right	right	ADV
ejpam-4596	228	10	regular	regular	ADV
ejpam-4596	229	1	[	[	X
ejpam-4596	229	2	14	14	NUM
ejpam-4596	229	3	]	]	PUNCT
ejpam-4596	229	4	.	.	PUNCT
ejpam-4596	230	1	hence	hence	ADV
ejpam-4596	230	2	,	,	PUNCT
ejpam-4596	230	3	as	as	ADP
ejpam-4596	230	4	an	an	DET
ejpam-4596	230	5	immediate	immediate	ADJ
ejpam-4596	230	6	consequence	consequence	NOUN
ejpam-4596	230	7	of	of	ADP
ejpam-4596	230	8	theorems	theorem	NOUN
ejpam-4596	230	9	2	2	NUM
ejpam-4596	230	10	and	and	CCONJ
ejpam-4596	230	11	3	3	NUM
ejpam-4596	230	12	,	,	PUNCT
ejpam-4596	230	13	we	we	PRON
ejpam-4596	230	14	have	have	VERB
ejpam-4596	230	15	the	the	DET
ejpam-4596	230	16	following	following	NOUN
ejpam-4596	230	17	.	.	PUNCT
ejpam-4596	231	1	theorem	theorem	ADJ
ejpam-4596	231	2	4	4	NUM
ejpam-4596	231	3	.	.	PUNCT
ejpam-4596	232	1	let	let	VERB
ejpam-4596	232	2	α	α	PRON
ejpam-4596	232	3	∈	∈	PROPN
ejpam-4596	232	4	te(x	te(x	NOUN
ejpam-4596	232	5	,	,	PUNCT
ejpam-4596	232	6	θ	θ	PROPN
ejpam-4596	232	7	)	)	PUNCT
ejpam-4596	232	8	.	.	PUNCT
ejpam-4596	233	1	then	then	ADV
ejpam-4596	233	2	α	α	PROPN
ejpam-4596	233	3	∈	∈	PROPN
ejpam-4596	233	4	creg(te(x	creg(te(x	NOUN
ejpam-4596	233	5	,	,	PUNCT
ejpam-4596	233	6	θ	θ	NOUN
ejpam-4596	233	7	)	)	PUNCT
ejpam-4596	233	8	)	)	PUNCT
ejpam-4596	234	1	if	if	SCONJ
ejpam-4596	234	2	and	and	CCONJ
ejpam-4596	234	3	only	only	ADV
ejpam-4596	234	4	if	if	SCONJ
ejpam-4596	234	5	(	(	PUNCT
ejpam-4596	234	6	i	i	NOUN
ejpam-4596	234	7	)	)	PUNCT
ejpam-4596	234	8	for	for	ADP
ejpam-4596	234	9	every	every	DET
ejpam-4596	234	10	a	a	DET
ejpam-4596	234	11	∈	∈	PROPN
ejpam-4596	234	12	x	x	SYM
ejpam-4596	234	13	/	/	SYM
ejpam-4596	234	14	e	e	NOUN
ejpam-4596	234	15	,	,	PUNCT
ejpam-4596	234	16	there	there	PRON
ejpam-4596	234	17	exists	exist	VERB
ejpam-4596	234	18	b	b	PROPN
ejpam-4596	234	19	∈	∈	PROPN
ejpam-4596	234	20	x	x	SYM
ejpam-4596	234	21	/	/	SYM
ejpam-4596	234	22	e	e	VERB
ejpam-4596	234	23	such	such	ADJ
ejpam-4596	234	24	that	that	PRON
ejpam-4596	234	25	for	for	ADP
ejpam-4596	234	26	each	each	DET
ejpam-4596	234	27	p	p	PROPN
ejpam-4596	234	28	∈	∈	PROPN
ejpam-4596	234	29	πa(α	πa(α	PUNCT
ejpam-4596	234	30	)	)	PUNCT
ejpam-4596	234	31	,	,	PUNCT
ejpam-4596	234	32	xθαθ	xθαθ	PROPN
ejpam-4596	234	33	∈	∈	PROPN
ejpam-4596	234	34	p	p	PROPN
ejpam-4596	234	35	for	for	ADP
ejpam-4596	234	36	some	some	DET
ejpam-4596	234	37	x	x	SYM
ejpam-4596	234	38	∈	∈	PROPN
ejpam-4596	234	39	b	b	PROPN
ejpam-4596	234	40	and	and	CCONJ
ejpam-4596	234	41	(	(	PUNCT
ejpam-4596	234	42	ii	ii	NOUN
ejpam-4596	234	43	)	)	PUNCT
ejpam-4596	234	44	(	(	PUNCT
ejpam-4596	234	45	θαθ)|xα	θαθ)|xα	PROPN
ejpam-4596	234	46	is	be	AUX
ejpam-4596	234	47	an	an	DET
ejpam-4596	234	48	e∗-preserving	e∗-preserving	NOUN
ejpam-4596	234	49	injection	injection	NOUN
ejpam-4596	234	50	.	.	PUNCT
ejpam-4596	235	1	as	as	ADP
ejpam-4596	235	2	an	an	DET
ejpam-4596	235	3	immediate	immediate	ADJ
ejpam-4596	235	4	consequence	consequence	NOUN
ejpam-4596	235	5	of	of	ADP
ejpam-4596	235	6	corollaries	corollary	NOUN
ejpam-4596	235	7	2	2	NUM
ejpam-4596	235	8	and	and	CCONJ
ejpam-4596	235	9	3	3	NUM
ejpam-4596	235	10	.	.	PUNCT
ejpam-4596	235	11	corollary	corollary	ADJ
ejpam-4596	235	12	4	4	NUM
ejpam-4596	235	13	.	.	PUNCT
ejpam-4596	235	14	idx	idx	PROPN
ejpam-4596	235	15	∈	∈	PROPN
ejpam-4596	235	16	creg(te(x	creg(te(x	NOUN
ejpam-4596	235	17	,	,	PUNCT
ejpam-4596	235	18	θ	θ	NOUN
ejpam-4596	235	19	)	)	PUNCT
ejpam-4596	235	20	)	)	PUNCT
ejpam-4596	236	1	if	if	SCONJ
ejpam-4596	236	2	and	and	CCONJ
ejpam-4596	236	3	only	only	ADV
ejpam-4596	236	4	if	if	SCONJ
ejpam-4596	236	5	θ	θ	PROPN
ejpam-4596	236	6	is	be	AUX
ejpam-4596	236	7	an	an	DET
ejpam-4596	236	8	e∗-preserving	e∗-preserve	VERB
ejpam-4596	236	9	bijection	bijection	NOUN
ejpam-4596	236	10	.	.	PUNCT
ejpam-4596	237	1	theorem	theorem	NOUN
ejpam-4596	237	2	5	5	NUM
ejpam-4596	237	3	.	.	PUNCT
ejpam-4596	238	1	let	let	VERB
ejpam-4596	238	2	α	α	PRON
ejpam-4596	238	3	∈	∈	PROPN
ejpam-4596	238	4	te(x	te(x	NOUN
ejpam-4596	238	5	,	,	PUNCT
ejpam-4596	238	6	θ	θ	PROPN
ejpam-4596	238	7	)	)	PUNCT
ejpam-4596	238	8	.	.	PUNCT
ejpam-4596	239	1	if	if	SCONJ
ejpam-4596	239	2	α	α	PROPN
ejpam-4596	239	3	∈	∈	PROPN
ejpam-4596	239	4	creg(te(x	creg(te(x	NOUN
ejpam-4596	239	5	,	,	PUNCT
ejpam-4596	239	6	θ	θ	NOUN
ejpam-4596	239	7	)	)	PUNCT
ejpam-4596	239	8	)	)	PUNCT
ejpam-4596	239	9	,	,	PUNCT
ejpam-4596	239	10	then	then	ADV
ejpam-4596	239	11	every	every	DET
ejpam-4596	239	12	a	a	DET
ejpam-4596	239	13	∈	∈	PROPN
ejpam-4596	239	14	x	x	SYM
ejpam-4596	239	15	/	/	SYM
ejpam-4596	239	16	e	e	NOUN
ejpam-4596	239	17	,	,	PUNCT
ejpam-4596	239	18	there	there	PRON
ejpam-4596	239	19	exists	exist	VERB
ejpam-4596	239	20	b	b	PROPN
ejpam-4596	239	21	∈	∈	PROPN
ejpam-4596	239	22	x	x	SYM
ejpam-4596	239	23	/	/	SYM
ejpam-4596	239	24	e	e	VERB
ejpam-4596	239	25	such	such	ADJ
ejpam-4596	239	26	that	that	SCONJ
ejpam-4596	239	27	|p	|p	NOUN
ejpam-4596	239	28	∩bθαθ|	∩bθαθ|	NOUN
ejpam-4596	239	29	=	=	SYM
ejpam-4596	239	30	|p	|p	NOUN
ejpam-4596	239	31	∩xθαθ|	∩xθαθ|	NOUN
ejpam-4596	239	32	=	=	SYM
ejpam-4596	239	33	1	1	NUM
ejpam-4596	239	34	for	for	ADP
ejpam-4596	239	35	all	all	DET
ejpam-4596	239	36	p	p	NOUN
ejpam-4596	239	37	∈	∈	PROPN
ejpam-4596	239	38	πa(α	πa(α	PUNCT
ejpam-4596	239	39	)	)	PUNCT
ejpam-4596	239	40	.	.	PUNCT
ejpam-4596	240	1	proof	proof	NOUN
ejpam-4596	240	2	.	.	PUNCT
ejpam-4596	241	1	assume	assume	VERB
ejpam-4596	241	2	that	that	SCONJ
ejpam-4596	241	3	α	α	PROPN
ejpam-4596	241	4	∈	∈	PROPN
ejpam-4596	241	5	creg(te(x	creg(te(x	NOUN
ejpam-4596	241	6	,	,	PUNCT
ejpam-4596	241	7	θ	θ	NOUN
ejpam-4596	241	8	)	)	PUNCT
ejpam-4596	241	9	)	)	PUNCT
ejpam-4596	241	10	.	.	PUNCT
ejpam-4596	242	1	then	then	ADV
ejpam-4596	242	2	α	α	PROPN
ejpam-4596	242	3	is	be	AUX
ejpam-4596	242	4	regular	regular	ADJ
ejpam-4596	242	5	,	,	PUNCT
ejpam-4596	242	6	left	leave	VERB
ejpam-4596	242	7	regular	regular	ADV
ejpam-4596	242	8	and	and	CCONJ
ejpam-4596	242	9	right	right	ADV
ejpam-4596	242	10	regular	regular	ADV
ejpam-4596	242	11	.	.	PUNCT
ejpam-4596	243	1	let	let	VERB
ejpam-4596	243	2	a	a	DET
ejpam-4596	243	3	∈	∈	ADJ
ejpam-4596	243	4	x	x	NOUN
ejpam-4596	243	5	/	/	SYM
ejpam-4596	243	6	e.	e.	PROPN
ejpam-4596	243	7	by	by	ADP
ejpam-4596	243	8	theorem	theorem	NOUN
ejpam-4596	243	9	2	2	NUM
ejpam-4596	243	10	,	,	PUNCT
ejpam-4596	243	11	there	there	PRON
ejpam-4596	243	12	exists	exist	VERB
ejpam-4596	243	13	b	b	PROPN
ejpam-4596	243	14	∈	∈	PROPN
ejpam-4596	243	15	x	x	SYM
ejpam-4596	243	16	/	/	SYM
ejpam-4596	243	17	e	e	VERB
ejpam-4596	243	18	such	such	ADJ
ejpam-4596	243	19	that	that	PRON
ejpam-4596	243	20	for	for	ADP
ejpam-4596	243	21	each	each	DET
ejpam-4596	243	22	p	p	PROPN
ejpam-4596	243	23	∈	∈	PROPN
ejpam-4596	243	24	πa(α	πa(α	PUNCT
ejpam-4596	243	25	)	)	PUNCT
ejpam-4596	243	26	,	,	PUNCT
ejpam-4596	243	27	xθαθ	xθαθ	PROPN
ejpam-4596	243	28	∈	∈	PROPN
ejpam-4596	243	29	p	p	PROPN
ejpam-4596	243	30	for	for	ADP
ejpam-4596	243	31	some	some	DET
ejpam-4596	243	32	x	x	SYM
ejpam-4596	243	33	∈	∈	PROPN
ejpam-4596	243	34	b.	b.	PROPN
ejpam-4596	243	35	for	for	ADP
ejpam-4596	243	36	each	each	DET
ejpam-4596	243	37	p	p	PROPN
ejpam-4596	243	38	∈	∈	PROPN
ejpam-4596	243	39	πa(α	πa(α	PUNCT
ejpam-4596	243	40	)	)	PUNCT
ejpam-4596	243	41	,	,	PUNCT
ejpam-4596	243	42	we	we	PRON
ejpam-4596	243	43	have	have	VERB
ejpam-4596	243	44	xθαθ	xθαθ	PROPN
ejpam-4596	243	45	∈	∈	PROPN
ejpam-4596	243	46	p	p	NOUN
ejpam-4596	243	47	for	for	ADP
ejpam-4596	243	48	some	some	DET
ejpam-4596	243	49	x	x	SYM
ejpam-4596	243	50	∈	∈	PROPN
ejpam-4596	243	51	b.	b.	NOUN
ejpam-4596	243	52	therefore	therefore	ADV
ejpam-4596	243	53	p	p	PROPN
ejpam-4596	243	54	∩	∩	ADJ
ejpam-4596	243	55	bθαθ	bθαθ	PROPN
ejpam-4596	243	56	̸=	̸=	PROPN
ejpam-4596	243	57	∅	∅	NOUN
ejpam-4596	243	58	and	and	CCONJ
ejpam-4596	243	59	p	p	NOUN
ejpam-4596	243	60	∩	∩	ADJ
ejpam-4596	243	61	xθαθ	xθαθ	PROPN
ejpam-4596	243	62	̸=	̸=	PROPN
ejpam-4596	243	63	∅.	∅.	ADV
ejpam-4596	243	64	let	let	VERB
ejpam-4596	243	65	y	y	PROPN
ejpam-4596	243	66	∈	∈	PROPN
ejpam-4596	243	67	p	p	PROPN
ejpam-4596	243	68	∩	∩	ADJ
ejpam-4596	243	69	xθαθ	xθαθ	PROPN
ejpam-4596	243	70	.	.	PUNCT
ejpam-4596	244	1	then	then	ADV
ejpam-4596	244	2	yα	yα	VERB
ejpam-4596	244	3	=	=	PUNCT
ejpam-4596	244	4	xθαθα	xθαθα	NOUN
ejpam-4596	244	5	.	.	PUNCT
ejpam-4596	245	1	by	by	ADP
ejpam-4596	245	2	theorem	theorem	NOUN
ejpam-4596	245	3	3	3	NUM
ejpam-4596	245	4	,	,	PUNCT
ejpam-4596	245	5	we	we	PRON
ejpam-4596	245	6	obtain	obtain	VERB
ejpam-4596	245	7	that	that	SCONJ
ejpam-4596	245	8	(	(	PUNCT
ejpam-4596	245	9	θαθ)|xα	θαθ)|xα	PROPN
ejpam-4596	245	10	is	be	AUX
ejpam-4596	245	11	injective	injective	ADJ
ejpam-4596	245	12	.	.	PUNCT
ejpam-4596	246	1	claim	claim	NOUN
ejpam-4596	246	2	that	that	SCONJ
ejpam-4596	246	3	α|xθαθ	α|xθαθ	NOUN
ejpam-4596	246	4	is	be	AUX
ejpam-4596	246	5	also	also	ADV
ejpam-4596	246	6	injective	injective	ADJ
ejpam-4596	246	7	,	,	PUNCT
ejpam-4596	246	8	let	let	VERB
ejpam-4596	246	9	x1	x1	NUM
ejpam-4596	246	10	,	,	PUNCT
ejpam-4596	247	1	x2	x2	PROPN
ejpam-4596	247	2	∈	∈	PROPN
ejpam-4596	247	3	xθαθ	xθαθ	PROPN
ejpam-4596	247	4	be	be	VERB
ejpam-4596	247	5	such	such	ADJ
ejpam-4596	247	6	that	that	SCONJ
ejpam-4596	247	7	x1α	x1α	PUNCT
ejpam-4596	248	1	=	=	SYM
ejpam-4596	248	2	x2α	x2α	PROPN
ejpam-4596	248	3	.	.	PUNCT
ejpam-4596	249	1	then	then	ADV
ejpam-4596	249	2	x1	x1	PROPN
ejpam-4596	249	3	=	=	PUNCT
ejpam-4596	249	4	x′1θαθ	x′1θαθ	NOUN
ejpam-4596	249	5	and	and	CCONJ
ejpam-4596	249	6	p.	p.	NOUN
ejpam-4596	249	7	tantong	tantong	NOUN
ejpam-4596	249	8	,	,	PUNCT
ejpam-4596	249	9	n.	n.	NOUN
ejpam-4596	249	10	sawatraksa	sawatraksa	PROPN
ejpam-4596	249	11	/	/	SYM
ejpam-4596	249	12	eur	eur	PROPN
ejpam-4596	249	13	.	.	PUNCT
ejpam-4596	250	1	j.	j.	PROPN
ejpam-4596	250	2	pure	pure	PROPN
ejpam-4596	250	3	appl	appl	PROPN
ejpam-4596	250	4	.	.	PROPN
ejpam-4596	250	5	math	math	PROPN
ejpam-4596	250	6	,	,	PUNCT
ejpam-4596	250	7	15	15	NUM
ejpam-4596	250	8	(	(	PUNCT
ejpam-4596	250	9	4	4	NUM
ejpam-4596	250	10	)	)	PUNCT
ejpam-4596	250	11	(	(	PUNCT
ejpam-4596	250	12	2022	2022	NUM
ejpam-4596	250	13	)	)	PUNCT
ejpam-4596	250	14	,	,	PUNCT
ejpam-4596	250	15	2116	2116	NUM
ejpam-4596	250	16	-	-	SYM
ejpam-4596	250	17	2126	2126	NUM
ejpam-4596	250	18	2122	2122	NUM
ejpam-4596	250	19	x2	x2	NOUN
ejpam-4596	251	1	=	=	PUNCT
ejpam-4596	251	2	x′2θαθ	x′2θαθ	NOUN
ejpam-4596	251	3	for	for	ADP
ejpam-4596	251	4	some	some	DET
ejpam-4596	251	5	x′1	x′1	NOUN
ejpam-4596	251	6	,	,	PUNCT
ejpam-4596	251	7	x	x	X
ejpam-4596	251	8	′	′	NOUN
ejpam-4596	251	9	2	2	NUM
ejpam-4596	251	10	∈	∈	NOUN
ejpam-4596	251	11	x.	x.	NOUN
ejpam-4596	252	1	thus	thus	ADV
ejpam-4596	252	2	x′1θαθα	x′1θαθα	NOUN
ejpam-4596	253	1	=	=	PUNCT
ejpam-4596	253	2	x′2θαθα	x′2θαθα	PROPN
ejpam-4596	254	1	and	and	CCONJ
ejpam-4596	254	2	so	so	ADV
ejpam-4596	254	3	x′1θαθαθ	x′1θαθαθ	PROPN
ejpam-4596	254	4	=	=	SYM
ejpam-4596	254	5	x′2θαθαθ	x′2θαθαθ	PROPN
ejpam-4596	254	6	.	.	PUNCT
ejpam-4596	255	1	since	since	SCONJ
ejpam-4596	255	2	θαθ|xα	θαθ|xα	PROPN
ejpam-4596	255	3	is	be	AUX
ejpam-4596	255	4	injective	injective	ADJ
ejpam-4596	255	5	,	,	PUNCT
ejpam-4596	255	6	x′1θα	x′1θα	PROPN
ejpam-4596	255	7	=	=	SYM
ejpam-4596	255	8	x′2θα	x′2θα	PROPN
ejpam-4596	255	9	which	which	PRON
ejpam-4596	255	10	implies	imply	VERB
ejpam-4596	255	11	that	that	SCONJ
ejpam-4596	255	12	x1	x1	PROPN
ejpam-4596	255	13	=	=	PUNCT
ejpam-4596	255	14	x′1θαθ	x′1θαθ	X
ejpam-4596	256	1	=	=	PUNCT
ejpam-4596	256	2	x′2θαθ	x′2θαθ	NOUN
ejpam-4596	256	3	=	=	SYM
ejpam-4596	257	1	x2	x2	PROPN
ejpam-4596	257	2	.	.	PUNCT
ejpam-4596	258	1	so	so	ADV
ejpam-4596	258	2	,	,	PUNCT
ejpam-4596	258	3	we	we	PRON
ejpam-4596	258	4	have	have	VERB
ejpam-4596	258	5	the	the	DET
ejpam-4596	258	6	claim	claim	NOUN
ejpam-4596	258	7	.	.	PUNCT
ejpam-4596	259	1	this	this	PRON
ejpam-4596	259	2	implies	imply	VERB
ejpam-4596	259	3	that	that	SCONJ
ejpam-4596	259	4	y	y	PROPN
ejpam-4596	259	5	=	=	PUNCT
ejpam-4596	259	6	xθαθ	xθαθ	PROPN
ejpam-4596	259	7	and	and	CCONJ
ejpam-4596	259	8	hence	hence	ADV
ejpam-4596	259	9	|p	|p	X
ejpam-4596	259	10	∩xθαθ|	∩xθαθ|	NOUN
ejpam-4596	259	11	=	=	NOUN
ejpam-4596	259	12	1	1	X
ejpam-4596	259	13	.	.	PUNCT
ejpam-4596	260	1	it	it	PRON
ejpam-4596	260	2	follows	follow	VERB
ejpam-4596	260	3	from	from	ADP
ejpam-4596	260	4	p	p	PRON
ejpam-4596	260	5	∩bθαθ	∩bθαθ	PRON
ejpam-4596	260	6	⊆	⊆	NUM
ejpam-4596	260	7	p	p	ADP
ejpam-4596	260	8	∩xθαθ	∩xθαθ	NOUN
ejpam-4596	260	9	that	that	SCONJ
ejpam-4596	260	10	|p	|p	X
ejpam-4596	260	11	∩bθαθ|	∩bθαθ|	NOUN
ejpam-4596	260	12	=	=	SYM
ejpam-4596	260	13	|p	|p	NOUN
ejpam-4596	260	14	∩xθαθ|	∩xθαθ|	NOUN
ejpam-4596	260	15	=	=	NOUN
ejpam-4596	260	16	1	1	NUM
ejpam-4596	260	17	.	.	NOUN
ejpam-4596	260	18	3	3	X
ejpam-4596	260	19	.	.	X
ejpam-4596	260	20	regularity	regularity	NOUN
ejpam-4596	260	21	of	of	ADP
ejpam-4596	260	22	variants	variant	NOUN
ejpam-4596	260	23	of	of	ADP
ejpam-4596	260	24	transformation	transformation	NOUN
ejpam-4596	260	25	semigroups	semigroup	NOUN
ejpam-4596	260	26	that	that	PRON
ejpam-4596	260	27	preserve	preserve	VERB
ejpam-4596	260	28	double	double	ADJ
ejpam-4596	260	29	direction	direction	NOUN
ejpam-4596	260	30	equivalence	equivalence	NOUN
ejpam-4596	260	31	in	in	ADP
ejpam-4596	260	32	this	this	DET
ejpam-4596	260	33	section	section	NOUN
ejpam-4596	260	34	,	,	PUNCT
ejpam-4596	260	35	we	we	PRON
ejpam-4596	260	36	characterize	characterize	VERB
ejpam-4596	260	37	the	the	DET
ejpam-4596	260	38	regular	regular	ADJ
ejpam-4596	260	39	,	,	PUNCT
ejpam-4596	260	40	left	leave	VERB
ejpam-4596	260	41	regular	regular	ADV
ejpam-4596	260	42	,	,	PUNCT
ejpam-4596	260	43	right	right	ADV
ejpam-4596	260	44	regular	regular	ADJ
ejpam-4596	260	45	and	and	CCONJ
ejpam-4596	260	46	completely	completely	ADV
ejpam-4596	260	47	regular	regular	ADJ
ejpam-4596	260	48	elements	element	NOUN
ejpam-4596	260	49	of	of	ADP
ejpam-4596	260	50	the	the	DET
ejpam-4596	260	51	variant	variant	ADJ
ejpam-4596	260	52	semigroup	semigroup	NOUN
ejpam-4596	260	53	te∗(x	te∗(x	NOUN
ejpam-4596	260	54	,	,	PUNCT
ejpam-4596	260	55	θ	θ	NOUN
ejpam-4596	260	56	)	)	PUNCT
ejpam-4596	260	57	.	.	PUNCT
ejpam-4596	261	1	in	in	ADP
ejpam-4596	261	2	addition	addition	NOUN
ejpam-4596	261	3	,	,	PUNCT
ejpam-4596	261	4	we	we	PRON
ejpam-4596	261	5	give	give	VERB
ejpam-4596	261	6	a	a	DET
ejpam-4596	261	7	necessary	necessary	ADJ
ejpam-4596	261	8	and	and	CCONJ
ejpam-4596	261	9	sufficient	sufficient	ADJ
ejpam-4596	261	10	condition	condition	NOUN
ejpam-4596	261	11	for	for	ADP
ejpam-4596	261	12	the	the	DET
ejpam-4596	261	13	identity	identity	NOUN
ejpam-4596	261	14	transformation	transformation	NOUN
ejpam-4596	261	15	idx	idx	NOUN
ejpam-4596	261	16	of	of	ADP
ejpam-4596	261	17	the	the	DET
ejpam-4596	261	18	semigroup	semigroup	PROPN
ejpam-4596	261	19	te∗(x	te∗(x	NOUN
ejpam-4596	261	20	)	)	PUNCT
ejpam-4596	261	21	to	to	PART
ejpam-4596	261	22	be	be	AUX
ejpam-4596	261	23	regular	regular	ADJ
ejpam-4596	261	24	,	,	PUNCT
ejpam-4596	261	25	left	leave	VERB
ejpam-4596	261	26	regular	regular	ADV
ejpam-4596	261	27	,	,	PUNCT
ejpam-4596	261	28	right	right	ADV
ejpam-4596	261	29	regular	regular	ADJ
ejpam-4596	261	30	and	and	CCONJ
ejpam-4596	261	31	completely	completely	ADV
ejpam-4596	261	32	regular	regular	ADJ
ejpam-4596	261	33	elements	element	NOUN
ejpam-4596	261	34	in	in	ADP
ejpam-4596	261	35	te∗(x	te∗(x	NOUN
ejpam-4596	261	36	,	,	PUNCT
ejpam-4596	261	37	θ	θ	NOUN
ejpam-4596	261	38	)	)	PUNCT
ejpam-4596	261	39	.	.	PUNCT
ejpam-4596	262	1	theorem	theorem	VERB
ejpam-4596	262	2	6	6	NUM
ejpam-4596	262	3	.	.	PUNCT
ejpam-4596	263	1	let	let	VERB
ejpam-4596	263	2	α	α	PRON
ejpam-4596	263	3	∈	∈	PROPN
ejpam-4596	263	4	te∗(x	te∗(x	NOUN
ejpam-4596	263	5	,	,	PUNCT
ejpam-4596	263	6	θ	θ	NOUN
ejpam-4596	263	7	)	)	PUNCT
ejpam-4596	263	8	.	.	PUNCT
ejpam-4596	264	1	then	then	ADV
ejpam-4596	264	2	α	α	PROPN
ejpam-4596	264	3	∈	∈	PROPN
ejpam-4596	264	4	reg(te∗(x	reg(te∗(x	NOUN
ejpam-4596	264	5	,	,	PUNCT
ejpam-4596	264	6	θ	θ	NOUN
ejpam-4596	264	7	)	)	PUNCT
ejpam-4596	264	8	)	)	PUNCT
ejpam-4596	265	1	if	if	SCONJ
ejpam-4596	265	2	and	and	CCONJ
ejpam-4596	265	3	only	only	ADV
ejpam-4596	265	4	if	if	SCONJ
ejpam-4596	265	5	(	(	PUNCT
ejpam-4596	265	6	i	i	NOUN
ejpam-4596	265	7	)	)	PUNCT
ejpam-4596	265	8	ker(α	ker(α	PROPN
ejpam-4596	265	9	)	)	PUNCT
ejpam-4596	265	10	=	=	SYM
ejpam-4596	265	11	ker(αθ	ker(αθ	NOUN
ejpam-4596	265	12	)	)	PUNCT
ejpam-4596	265	13	and	and	CCONJ
ejpam-4596	265	14	(	(	PUNCT
ejpam-4596	265	15	ii	ii	NOUN
ejpam-4596	265	16	)	)	PUNCT
ejpam-4596	265	17	for	for	ADP
ejpam-4596	265	18	every	every	DET
ejpam-4596	265	19	a	a	DET
ejpam-4596	265	20	∈	∈	PROPN
ejpam-4596	265	21	x	x	SYM
ejpam-4596	265	22	/	/	SYM
ejpam-4596	265	23	e	e	NOUN
ejpam-4596	265	24	,	,	PUNCT
ejpam-4596	265	25	there	there	PRON
ejpam-4596	265	26	exists	exist	VERB
ejpam-4596	265	27	b	b	PROPN
ejpam-4596	265	28	∈	∈	PROPN
ejpam-4596	265	29	x	x	SYM
ejpam-4596	265	30	/	/	SYM
ejpam-4596	265	31	e	e	VERB
ejpam-4596	265	32	such	such	ADJ
ejpam-4596	265	33	that	that	SCONJ
ejpam-4596	265	34	a	a	DET
ejpam-4596	265	35	∩xαθ	∩xαθ	NOUN
ejpam-4596	265	36	=	=	SYM
ejpam-4596	265	37	bθαθ	bθαθ	PROPN
ejpam-4596	265	38	.	.	PUNCT
ejpam-4596	266	1	proof	proof	NOUN
ejpam-4596	266	2	.	.	PUNCT
ejpam-4596	267	1	suppose	suppose	VERB
ejpam-4596	267	2	that	that	SCONJ
ejpam-4596	267	3	α	α	PROPN
ejpam-4596	267	4	∈	∈	PROPN
ejpam-4596	267	5	reg(te∗(x	reg(te∗(x	NOUN
ejpam-4596	267	6	,	,	PUNCT
ejpam-4596	267	7	θ	θ	NOUN
ejpam-4596	267	8	)	)	PUNCT
ejpam-4596	267	9	)	)	PUNCT
ejpam-4596	267	10	.	.	PUNCT
ejpam-4596	268	1	then	then	ADV
ejpam-4596	268	2	α	α	PROPN
ejpam-4596	268	3	∈	∈	PROPN
ejpam-4596	268	4	reg(te(x	reg(te(x	PROPN
ejpam-4596	268	5	,	,	PUNCT
ejpam-4596	268	6	θ	θ	NOUN
ejpam-4596	268	7	)	)	PUNCT
ejpam-4596	268	8	)	)	PUNCT
ejpam-4596	268	9	.	.	PUNCT
ejpam-4596	269	1	by	by	ADP
ejpam-4596	269	2	theorem	theorem	NOUN
ejpam-4596	269	3	1(i	1(i	NUM
ejpam-4596	269	4	)	)	PUNCT
ejpam-4596	269	5	,	,	PUNCT
ejpam-4596	269	6	we	we	PRON
ejpam-4596	269	7	have	have	VERB
ejpam-4596	269	8	(	(	PUNCT
ejpam-4596	269	9	i	i	NOUN
ejpam-4596	269	10	)	)	PUNCT
ejpam-4596	269	11	holds	hold	VERB
ejpam-4596	269	12	.	.	PUNCT
ejpam-4596	270	1	let	let	VERB
ejpam-4596	270	2	a	a	DET
ejpam-4596	270	3	∈	∈	PROPN
ejpam-4596	270	4	x	x	X
ejpam-4596	270	5	/	/	SYM
ejpam-4596	270	6	e.	e.	PROPN
ejpam-4596	270	7	then	then	ADV
ejpam-4596	270	8	by	by	ADP
ejpam-4596	270	9	theorem	theorem	NOUN
ejpam-4596	270	10	1(ii	1(ii	NUM
ejpam-4596	270	11	)	)	PUNCT
ejpam-4596	270	12	,	,	PUNCT
ejpam-4596	270	13	there	there	PRON
ejpam-4596	270	14	exists	exist	VERB
ejpam-4596	270	15	b	b	PROPN
ejpam-4596	270	16	∈	∈	PROPN
ejpam-4596	270	17	x	x	SYM
ejpam-4596	270	18	/	/	SYM
ejpam-4596	270	19	e	e	VERB
ejpam-4596	270	20	such	such	ADJ
ejpam-4596	270	21	that	that	SCONJ
ejpam-4596	270	22	a∩xαθ	a∩xαθ	VERB
ejpam-4596	270	23	⊆	⊆	NUM
ejpam-4596	270	24	bθαθ	bθαθ	NOUN
ejpam-4596	270	25	.	.	PUNCT
ejpam-4596	271	1	since	since	SCONJ
ejpam-4596	271	2	θ	θ	PROPN
ejpam-4596	271	3	,	,	PUNCT
ejpam-4596	271	4	αθ	αθ	PROPN
ejpam-4596	271	5	∈	∈	PROPN
ejpam-4596	271	6	te(x	te(x	NOUN
ejpam-4596	271	7	)	)	PUNCT
ejpam-4596	271	8	and	and	CCONJ
ejpam-4596	271	9	by	by	ADP
ejpam-4596	271	10	lemma	lemma	PROPN
ejpam-4596	271	11	1	1	NUM
ejpam-4596	271	12	,	,	PUNCT
ejpam-4596	271	13	there	there	PRON
ejpam-4596	271	14	are	be	VERB
ejpam-4596	271	15	c	c	NOUN
ejpam-4596	271	16	,	,	PUNCT
ejpam-4596	271	17	d	d	PROPN
ejpam-4596	271	18	∈	∈	PROPN
ejpam-4596	271	19	x	x	X
ejpam-4596	271	20	/	/	SYM
ejpam-4596	271	21	e	e	VERB
ejpam-4596	271	22	such	such	ADJ
ejpam-4596	271	23	that	that	DET
ejpam-4596	271	24	bθ	bθ	PROPN
ejpam-4596	271	25	⊆	⊆	NUM
ejpam-4596	271	26	c	c	NOUN
ejpam-4596	271	27	and	and	CCONJ
ejpam-4596	271	28	cαθ	cαθ	VERB
ejpam-4596	271	29	⊆	⊆	NUM
ejpam-4596	271	30	d.	d.	PROPN
ejpam-4596	271	31	therefore	therefore	ADV
ejpam-4596	271	32	a	a	DET
ejpam-4596	271	33	∩xαθ	∩xαθ	NOUN
ejpam-4596	271	34	⊆	⊆	NUM
ejpam-4596	271	35	d.	d.	NOUN
ejpam-4596	271	36	since	since	SCONJ
ejpam-4596	271	37	a	a	PRON
ejpam-4596	271	38	and	and	CCONJ
ejpam-4596	271	39	d	d	NOUN
ejpam-4596	271	40	are	be	AUX
ejpam-4596	271	41	equivalence	equivalence	NOUN
ejpam-4596	271	42	classes	class	NOUN
ejpam-4596	271	43	of	of	ADP
ejpam-4596	271	44	x	x	PROPN
ejpam-4596	271	45	,	,	PUNCT
ejpam-4596	271	46	a	a	DET
ejpam-4596	271	47	=	=	X
ejpam-4596	271	48	d.	d.	NOUN
ejpam-4596	271	49	this	this	PRON
ejpam-4596	271	50	implies	imply	VERB
ejpam-4596	271	51	that	that	SCONJ
ejpam-4596	271	52	a	a	DET
ejpam-4596	271	53	∩	∩	NOUN
ejpam-4596	271	54	xαθ	xαθ	NOUN
ejpam-4596	271	55	⊆	⊆	NUM
ejpam-4596	271	56	bθαθ	bθαθ	PROPN
ejpam-4596	271	57	⊆	⊆	NUM
ejpam-4596	271	58	cαθ	cαθ	NOUN
ejpam-4596	271	59	=	=	PUNCT
ejpam-4596	271	60	cαθ	cαθ	PROPN
ejpam-4596	271	61	∩	∩	NOUN
ejpam-4596	271	62	xαθ	xαθ	VERB
ejpam-4596	271	63	⊆	⊆	NUM
ejpam-4596	271	64	a	a	DET
ejpam-4596	271	65	∩	∩	NOUN
ejpam-4596	271	66	xαθ	xαθ	NOUN
ejpam-4596	271	67	.	.	PUNCT
ejpam-4596	272	1	hence	hence	ADV
ejpam-4596	272	2	a	a	DET
ejpam-4596	272	3	∩xαθ	∩xαθ	NOUN
ejpam-4596	272	4	=	=	SYM
ejpam-4596	272	5	bθαθ	bθαθ	PROPN
ejpam-4596	272	6	.	.	PUNCT
ejpam-4596	273	1	conversely	conversely	ADV
ejpam-4596	273	2	,	,	PUNCT
ejpam-4596	273	3	assume	assume	VERB
ejpam-4596	273	4	that	that	SCONJ
ejpam-4596	273	5	conditions	condition	NOUN
ejpam-4596	273	6	(	(	PUNCT
ejpam-4596	273	7	i	i	NOUN
ejpam-4596	273	8	)	)	PUNCT
ejpam-4596	273	9	and	and	CCONJ
ejpam-4596	273	10	(	(	PUNCT
ejpam-4596	273	11	ii	ii	NOUN
ejpam-4596	273	12	)	)	PUNCT
ejpam-4596	273	13	hold	hold	VERB
ejpam-4596	273	14	.	.	PUNCT
ejpam-4596	274	1	define	define	VERB
ejpam-4596	274	2	β	β	NOUN
ejpam-4596	274	3	:	:	PUNCT
ejpam-4596	274	4	x	x	X
ejpam-4596	274	5	→	→	SYM
ejpam-4596	274	6	x	x	PUNCT
ejpam-4596	274	7	as	as	ADP
ejpam-4596	274	8	in	in	ADP
ejpam-4596	274	9	the	the	DET
ejpam-4596	274	10	proof	proof	NOUN
ejpam-4596	274	11	of	of	ADP
ejpam-4596	274	12	theorem	theorem	NOUN
ejpam-4596	274	13	1	1	NUM
ejpam-4596	274	14	.	.	PUNCT
ejpam-4596	274	15	then	then	ADV
ejpam-4596	274	16	β	β	PROPN
ejpam-4596	274	17	∈	∈	PROPN
ejpam-4596	274	18	te(x	te(x	NOUN
ejpam-4596	274	19	)	)	PUNCT
ejpam-4596	274	20	and	and	CCONJ
ejpam-4596	274	21	α	α	X
ejpam-4596	274	22	=	=	SYM
ejpam-4596	274	23	α	α	PROPN
ejpam-4596	274	24	∗	∗	X
ejpam-4596	274	25	β	β	PROPN
ejpam-4596	274	26	∗	∗	NOUN
ejpam-4596	274	27	α	α	NOUN
ejpam-4596	274	28	.	.	PUNCT
ejpam-4596	275	1	it	it	PRON
ejpam-4596	275	2	remains	remain	VERB
ejpam-4596	275	3	to	to	PART
ejpam-4596	275	4	show	show	VERB
ejpam-4596	275	5	that	that	SCONJ
ejpam-4596	275	6	β	β	PROPN
ejpam-4596	275	7	∈	∈	PROPN
ejpam-4596	275	8	te∗(x	te∗(x	NOUN
ejpam-4596	275	9	)	)	PUNCT
ejpam-4596	275	10	.	.	PUNCT
ejpam-4596	276	1	let	let	VERB
ejpam-4596	276	2	x	x	PRON
ejpam-4596	276	3	,	,	PUNCT
ejpam-4596	276	4	y	y	PROPN
ejpam-4596	276	5	∈	∈	PROPN
ejpam-4596	276	6	x	x	AUX
ejpam-4596	276	7	be	be	AUX
ejpam-4596	276	8	such	such	ADJ
ejpam-4596	276	9	that	that	SCONJ
ejpam-4596	276	10	(	(	PUNCT
ejpam-4596	276	11	xβ	xβ	PROPN
ejpam-4596	276	12	,	,	PUNCT
ejpam-4596	276	13	yβ	yβ	NOUN
ejpam-4596	276	14	)	)	PUNCT
ejpam-4596	276	15	∈	∈	PROPN
ejpam-4596	277	1	e.	e.	PROPN
ejpam-4596	277	2	then	then	ADV
ejpam-4596	277	3	(	(	PUNCT
ejpam-4596	277	4	x′	x′	NUM
ejpam-4596	277	5	,	,	PUNCT
ejpam-4596	277	6	y′	y′	NUM
ejpam-4596	277	7	)	)	PUNCT
ejpam-4596	278	1	∈	∈	PROPN
ejpam-4596	278	2	e	e	X
ejpam-4596	278	3	where	where	SCONJ
ejpam-4596	278	4	ex	ex	PRON
ejpam-4596	278	5	∩xαθ	∩xαθ	NOUN
ejpam-4596	278	6	=	=	SYM
ejpam-4596	278	7	ex′θαθ	ex′θαθ	PROPN
ejpam-4596	278	8	and	and	CCONJ
ejpam-4596	278	9	ey	ey	PRON
ejpam-4596	278	10	∩xαθ	∩xαθ	NOUN
ejpam-4596	278	11	=	=	SYM
ejpam-4596	278	12	ey′θαθ	ey′θαθ	NOUN
ejpam-4596	278	13	.	.	PUNCT
ejpam-4596	278	14	thus	thus	ADV
ejpam-4596	278	15	ex′	ex′	X
ejpam-4596	278	16	=	=	SYM
ejpam-4596	278	17	ey′	ey′	PROPN
ejpam-4596	278	18	and	and	CCONJ
ejpam-4596	278	19	so	so	ADV
ejpam-4596	278	20	ex	ex	ADJ
ejpam-4596	278	21	∩xαθ	∩xαθ	NOUN
ejpam-4596	278	22	=	=	SYM
ejpam-4596	278	23	ex′θαθ	ex′θαθ	PROPN
ejpam-4596	278	24	=	=	SYM
ejpam-4596	278	25	ey′θαθ	ey′θαθ	X
ejpam-4596	278	26	=	=	NOUN
ejpam-4596	278	27	ey	ey	X
ejpam-4596	278	28	∩xαθ	∩xαθ	NOUN
ejpam-4596	278	29	,	,	PUNCT
ejpam-4596	278	30	which	which	PRON
ejpam-4596	278	31	implies	imply	VERB
ejpam-4596	278	32	that	that	SCONJ
ejpam-4596	278	33	ex	ex	PRON
ejpam-4596	278	34	=	=	SYM
ejpam-4596	278	35	ey	ey	PROPN
ejpam-4596	278	36	.	.	PROPN
ejpam-4596	278	37	therefore	therefore	ADV
ejpam-4596	278	38	(	(	PUNCT
ejpam-4596	278	39	x	x	X
ejpam-4596	278	40	,	,	PUNCT
ejpam-4596	278	41	y	y	NOUN
ejpam-4596	278	42	)	)	PUNCT
ejpam-4596	278	43	∈	∈	PROPN
ejpam-4596	278	44	e	e	NOUN
ejpam-4596	278	45	and	and	CCONJ
ejpam-4596	278	46	hence	hence	ADV
ejpam-4596	278	47	β	β	NOUN
ejpam-4596	278	48	∈	∈	NOUN
ejpam-4596	278	49	te∗(x	te∗(x	NOUN
ejpam-4596	278	50	)	)	PUNCT
ejpam-4596	278	51	.	.	PUNCT
ejpam-4596	279	1	corollary	corollary	ADJ
ejpam-4596	279	2	5	5	NUM
ejpam-4596	279	3	.	.	PUNCT
ejpam-4596	279	4	idx	idx	PROPN
ejpam-4596	279	5	∈	∈	PROPN
ejpam-4596	279	6	reg(te∗(x	reg(te∗(x	NOUN
ejpam-4596	279	7	,	,	PUNCT
ejpam-4596	279	8	θ	θ	NOUN
ejpam-4596	279	9	)	)	PUNCT
ejpam-4596	279	10	)	)	PUNCT
ejpam-4596	280	1	if	if	SCONJ
ejpam-4596	280	2	and	and	CCONJ
ejpam-4596	280	3	only	only	ADV
ejpam-4596	280	4	if	if	SCONJ
ejpam-4596	280	5	θ	θ	PROPN
ejpam-4596	280	6	is	be	AUX
ejpam-4596	280	7	a	a	DET
ejpam-4596	280	8	bijection	bijection	NOUN
ejpam-4596	280	9	.	.	PUNCT
ejpam-4596	281	1	now	now	ADV
ejpam-4596	281	2	,	,	PUNCT
ejpam-4596	281	3	we	we	PRON
ejpam-4596	281	4	discuss	discuss	VERB
ejpam-4596	281	5	a	a	DET
ejpam-4596	281	6	characterization	characterization	NOUN
ejpam-4596	281	7	of	of	ADP
ejpam-4596	281	8	an	an	DET
ejpam-4596	281	9	element	element	NOUN
ejpam-4596	281	10	in	in	ADP
ejpam-4596	281	11	the	the	DET
ejpam-4596	281	12	semigroup	semigroup	PROPN
ejpam-4596	281	13	te∗(x	te∗(x	NOUN
ejpam-4596	281	14	,	,	PUNCT
ejpam-4596	281	15	θ	θ	NOUN
ejpam-4596	281	16	)	)	PUNCT
ejpam-4596	281	17	to	to	PART
ejpam-4596	281	18	be	be	AUX
ejpam-4596	281	19	left	leave	VERB
ejpam-4596	281	20	regular	regular	ADV
ejpam-4596	281	21	.	.	PUNCT
ejpam-4596	282	1	theorem	theorem	VERB
ejpam-4596	282	2	7	7	NUM
ejpam-4596	282	3	.	.	PUNCT
ejpam-4596	283	1	let	let	VERB
ejpam-4596	283	2	α	α	PRON
ejpam-4596	283	3	∈	∈	PROPN
ejpam-4596	283	4	te∗(x	te∗(x	NOUN
ejpam-4596	283	5	,	,	PUNCT
ejpam-4596	283	6	θ	θ	NOUN
ejpam-4596	283	7	)	)	PUNCT
ejpam-4596	283	8	.	.	PUNCT
ejpam-4596	284	1	then	then	ADV
ejpam-4596	284	2	α	α	PROPN
ejpam-4596	284	3	∈	∈	PROPN
ejpam-4596	284	4	lreg(te∗(x	lreg(te∗(x	PROPN
ejpam-4596	284	5	,	,	PUNCT
ejpam-4596	284	6	θ	θ	NOUN
ejpam-4596	284	7	)	)	PUNCT
ejpam-4596	284	8	)	)	PUNCT
ejpam-4596	285	1	if	if	SCONJ
ejpam-4596	285	2	and	and	CCONJ
ejpam-4596	285	3	only	only	ADV
ejpam-4596	285	4	if	if	SCONJ
ejpam-4596	285	5	for	for	ADP
ejpam-4596	285	6	every	every	DET
ejpam-4596	285	7	p	p	PROPN
ejpam-4596	285	8	∈	∈	PROPN
ejpam-4596	285	9	π(α	π(α	PROPN
ejpam-4596	285	10	)	)	PUNCT
ejpam-4596	285	11	,	,	PUNCT
ejpam-4596	285	12	p	p	X
ejpam-4596	285	13	∩xθαθ	∩xθαθ	X
ejpam-4596	285	14	̸=	̸=	PROPN
ejpam-4596	285	15	∅.	∅.	ADP
ejpam-4596	285	16	proof	proof	NOUN
ejpam-4596	285	17	.	.	PUNCT
ejpam-4596	286	1	suppose	suppose	VERB
ejpam-4596	286	2	that	that	SCONJ
ejpam-4596	286	3	α	α	PROPN
ejpam-4596	286	4	∈	∈	PROPN
ejpam-4596	286	5	lreg(te∗(x	lreg(te∗(x	PROPN
ejpam-4596	286	6	,	,	PUNCT
ejpam-4596	286	7	θ	θ	NOUN
ejpam-4596	286	8	)	)	PUNCT
ejpam-4596	286	9	)	)	PUNCT
ejpam-4596	286	10	.	.	PUNCT
ejpam-4596	287	1	then	then	ADV
ejpam-4596	287	2	α	α	PROPN
ejpam-4596	287	3	∈	∈	PROPN
ejpam-4596	287	4	lreg(te(x	lreg(te(x	ADJ
ejpam-4596	287	5	,	,	PUNCT
ejpam-4596	287	6	θ	θ	NOUN
ejpam-4596	287	7	)	)	PUNCT
ejpam-4596	287	8	)	)	PUNCT
ejpam-4596	287	9	.	.	PUNCT
ejpam-4596	288	1	let	let	VERB
ejpam-4596	288	2	p	p	X
ejpam-4596	288	3	∈	∈	PROPN
ejpam-4596	288	4	π(α	π(α	PROPN
ejpam-4596	288	5	)	)	PUNCT
ejpam-4596	288	6	and	and	CCONJ
ejpam-4596	288	7	p	p	NOUN
ejpam-4596	288	8	∈	∈	PROPN
ejpam-4596	288	9	p	p	NOUN
ejpam-4596	288	10	.	.	PUNCT
ejpam-4596	289	1	then	then	ADV
ejpam-4596	289	2	p	p	PROPN
ejpam-4596	289	3	∈	∈	PROPN
ejpam-4596	289	4	πep(α	πep(α	PROPN
ejpam-4596	289	5	)	)	PUNCT
ejpam-4596	289	6	.	.	PUNCT
ejpam-4596	290	1	by	by	ADP
ejpam-4596	290	2	theorem	theorem	NOUN
ejpam-4596	290	3	2	2	NUM
ejpam-4596	290	4	,	,	PUNCT
ejpam-4596	290	5	there	there	PRON
ejpam-4596	290	6	exists	exist	VERB
ejpam-4596	290	7	b	b	PROPN
ejpam-4596	290	8	∈	∈	PROPN
ejpam-4596	290	9	x	x	SYM
ejpam-4596	290	10	/	/	SYM
ejpam-4596	290	11	e	e	VERB
ejpam-4596	290	12	such	such	ADJ
ejpam-4596	291	1	that	that	SCONJ
ejpam-4596	291	2	xθαθ	xθαθ	PROPN
ejpam-4596	291	3	∈	∈	PROPN
ejpam-4596	291	4	p	p	NOUN
ejpam-4596	291	5	for	for	ADP
ejpam-4596	291	6	some	some	DET
ejpam-4596	291	7	x	x	SYM
ejpam-4596	291	8	∈	∈	PROPN
ejpam-4596	291	9	b.	b.	NOUN
ejpam-4596	291	10	hence	hence	ADV
ejpam-4596	291	11	p	p	X
ejpam-4596	291	12	∩xθαθ	∩xθαθ	PROPN
ejpam-4596	291	13	̸=	̸=	PROPN
ejpam-4596	291	14	∅.	∅.	ADV
ejpam-4596	291	15	conversely	conversely	ADV
ejpam-4596	291	16	,	,	PUNCT
ejpam-4596	291	17	for	for	ADP
ejpam-4596	291	18	every	every	DET
ejpam-4596	291	19	p	p	PROPN
ejpam-4596	291	20	∈	∈	PROPN
ejpam-4596	291	21	π(α	π(α	PROPN
ejpam-4596	291	22	)	)	PUNCT
ejpam-4596	291	23	,	,	PUNCT
ejpam-4596	291	24	p	p	X
ejpam-4596	291	25	∩xθαθ	∩xθαθ	PROPN
ejpam-4596	291	26	̸=	̸=	PROPN
ejpam-4596	291	27	∅.	∅.	ADV
ejpam-4596	291	28	we	we	PRON
ejpam-4596	291	29	choose	choose	VERB
ejpam-4596	291	30	and	and	CCONJ
ejpam-4596	291	31	fix	fix	VERB
ejpam-4596	291	32	an	an	DET
ejpam-4596	291	33	element	element	NOUN
ejpam-4596	291	34	xp	xp	INTJ
ejpam-4596	292	1	θαθ	θαθ	PROPN
ejpam-4596	292	2	∈	∈	PROPN
ejpam-4596	292	3	p	p	X
ejpam-4596	292	4	.	.	PUNCT
ejpam-4596	293	1	for	for	ADP
ejpam-4596	293	2	each	each	DET
ejpam-4596	293	3	x	x	SYM
ejpam-4596	293	4	∈	∈	PROPN
ejpam-4596	293	5	x	x	X
ejpam-4596	293	6	,	,	PUNCT
ejpam-4596	293	7	we	we	PRON
ejpam-4596	293	8	let	let	VERB
ejpam-4596	293	9	px	px	X
ejpam-4596	293	10	∈	∈	PROPN
ejpam-4596	293	11	π(α	π(α	PROPN
ejpam-4596	293	12	)	)	PUNCT
ejpam-4596	293	13	be	be	AUX
ejpam-4596	293	14	such	such	ADJ
ejpam-4596	293	15	that	that	SCONJ
ejpam-4596	293	16	x	x	SYM
ejpam-4596	293	17	∈	∈	PROPN
ejpam-4596	293	18	px	px	AUX
ejpam-4596	293	19	.	.	PROPN
ejpam-4596	293	20	define	define	VERB
ejpam-4596	293	21	β	β	X
ejpam-4596	293	22	:	:	PUNCT
ejpam-4596	293	23	x	x	SYM
ejpam-4596	293	24	→	→	SYM
ejpam-4596	293	25	x	x	PUNCT
ejpam-4596	293	26	by	by	ADP
ejpam-4596	293	27	xβ	xβ	PROPN
ejpam-4596	293	28	=	=	SYM
ejpam-4596	293	29	xpx	xpx	PROPN
ejpam-4596	293	30	for	for	ADP
ejpam-4596	293	31	all	all	DET
ejpam-4596	293	32	x	x	SYM
ejpam-4596	293	33	∈	∈	NOUN
ejpam-4596	293	34	x.	x.	NOUN
ejpam-4596	293	35	next	next	ADV
ejpam-4596	293	36	,	,	PUNCT
ejpam-4596	293	37	we	we	PRON
ejpam-4596	293	38	will	will	AUX
ejpam-4596	293	39	show	show	VERB
ejpam-4596	293	40	that	that	SCONJ
ejpam-4596	293	41	β	β	PROPN
ejpam-4596	293	42	∈	∈	PROPN
ejpam-4596	293	43	te∗(x	te∗(x	NOUN
ejpam-4596	293	44	,	,	PUNCT
ejpam-4596	293	45	θ	θ	NOUN
ejpam-4596	293	46	)	)	PUNCT
ejpam-4596	293	47	,	,	PUNCT
ejpam-4596	293	48	let	let	VERB
ejpam-4596	293	49	x	x	PRON
ejpam-4596	293	50	,	,	PUNCT
ejpam-4596	293	51	y	y	PROPN
ejpam-4596	293	52	∈	∈	PROPN
ejpam-4596	293	53	x.	x.	NOUN
ejpam-4596	294	1	if	if	SCONJ
ejpam-4596	294	2	(	(	PUNCT
ejpam-4596	294	3	x	x	NOUN
ejpam-4596	294	4	,	,	PUNCT
ejpam-4596	294	5	y	y	NOUN
ejpam-4596	294	6	)	)	PUNCT
ejpam-4596	294	7	∈	∈	PROPN
ejpam-4596	294	8	e	e	NOUN
ejpam-4596	294	9	,	,	PUNCT
ejpam-4596	294	10	then	then	ADV
ejpam-4596	294	11	p.	p.	PROPN
ejpam-4596	294	12	tantong	tantong	NOUN
ejpam-4596	294	13	,	,	PUNCT
ejpam-4596	294	14	n.	n.	NOUN
ejpam-4596	294	15	sawatraksa	sawatraksa	PROPN
ejpam-4596	294	16	/	/	SYM
ejpam-4596	294	17	eur	eur	PROPN
ejpam-4596	294	18	.	.	PUNCT
ejpam-4596	295	1	j.	j.	PROPN
ejpam-4596	295	2	pure	pure	PROPN
ejpam-4596	295	3	appl	appl	PROPN
ejpam-4596	295	4	.	.	PROPN
ejpam-4596	295	5	math	math	PROPN
ejpam-4596	295	6	,	,	PUNCT
ejpam-4596	295	7	15	15	NUM
ejpam-4596	295	8	(	(	PUNCT
ejpam-4596	295	9	4	4	NUM
ejpam-4596	295	10	)	)	PUNCT
ejpam-4596	295	11	(	(	PUNCT
ejpam-4596	295	12	2022	2022	NUM
ejpam-4596	295	13	)	)	PUNCT
ejpam-4596	295	14	,	,	PUNCT
ejpam-4596	295	15	2116	2116	NUM
ejpam-4596	295	16	-	-	SYM
ejpam-4596	295	17	2126	2126	NUM
ejpam-4596	295	18	2123	2123	NUM
ejpam-4596	295	19	(	(	PUNCT
ejpam-4596	295	20	xα	xα	INTJ
ejpam-4596	295	21	,	,	PUNCT
ejpam-4596	295	22	yα	yα	NOUN
ejpam-4596	295	23	)	)	PUNCT
ejpam-4596	295	24	∈	∈	PROPN
ejpam-4596	295	25	e	e	X
ejpam-4596	295	26	and	and	CCONJ
ejpam-4596	295	27	(	(	PUNCT
ejpam-4596	295	28	xβ	xβ	PROPN
ejpam-4596	295	29	,	,	PUNCT
ejpam-4596	295	30	yβ	yβ	NOUN
ejpam-4596	295	31	)	)	PUNCT
ejpam-4596	295	32	=	=	SYM
ejpam-4596	295	33	(	(	PUNCT
ejpam-4596	295	34	xpx	xpx	PROPN
ejpam-4596	295	35	,	,	PUNCT
ejpam-4596	295	36	xpy	xpy	PROPN
ejpam-4596	295	37	)	)	PUNCT
ejpam-4596	296	1	where	where	SCONJ
ejpam-4596	296	2	xpxθαθ	xpxθαθ	PROPN
ejpam-4596	296	3	∈	∈	PROPN
ejpam-4596	296	4	px	px	PROPN
ejpam-4596	296	5	and	and	CCONJ
ejpam-4596	296	6	xpyθαθ	xpyθαθ	PROPN
ejpam-4596	296	7	∈	∈	PROPN
ejpam-4596	296	8	py	py	PROPN
ejpam-4596	296	9	.	.	PUNCT
ejpam-4596	297	1	we	we	PRON
ejpam-4596	297	2	then	then	ADV
ejpam-4596	297	3	have	have	VERB
ejpam-4596	297	4	(	(	PUNCT
ejpam-4596	297	5	xpxθαθα	xpxθαθα	PROPN
ejpam-4596	297	6	,	,	PUNCT
ejpam-4596	297	7	xpyθαθα	xpyθαθα	PROPN
ejpam-4596	297	8	)	)	PUNCT
ejpam-4596	298	1	=	=	PRON
ejpam-4596	298	2	(	(	PUNCT
ejpam-4596	298	3	xα	xα	INTJ
ejpam-4596	298	4	,	,	PUNCT
ejpam-4596	298	5	yα	yα	NOUN
ejpam-4596	298	6	)	)	PUNCT
ejpam-4596	298	7	∈	∈	PROPN
ejpam-4596	298	8	e.	e.	PROPN
ejpam-4596	298	9	since	since	SCONJ
ejpam-4596	298	10	θαθα	θαθα	PROPN
ejpam-4596	298	11	∈	∈	PROPN
ejpam-4596	298	12	te∗(x	te∗(x	NOUN
ejpam-4596	298	13	)	)	PUNCT
ejpam-4596	298	14	,	,	PUNCT
ejpam-4596	298	15	(	(	PUNCT
ejpam-4596	298	16	xβ	xβ	PROPN
ejpam-4596	298	17	,	,	PUNCT
ejpam-4596	298	18	yβ	yβ	NOUN
ejpam-4596	298	19	)	)	PUNCT
ejpam-4596	298	20	=	=	SYM
ejpam-4596	299	1	(	(	PUNCT
ejpam-4596	299	2	xpx	xpx	PROPN
ejpam-4596	299	3	,	,	PUNCT
ejpam-4596	299	4	xpy	xpy	PROPN
ejpam-4596	299	5	)	)	PUNCT
ejpam-4596	299	6	∈	∈	PROPN
ejpam-4596	299	7	e.	e.	PROPN
ejpam-4596	299	8	on	on	ADP
ejpam-4596	299	9	the	the	DET
ejpam-4596	299	10	other	other	ADJ
ejpam-4596	299	11	hand	hand	NOUN
ejpam-4596	299	12	,	,	PUNCT
ejpam-4596	299	13	if	if	SCONJ
ejpam-4596	299	14	(	(	PUNCT
ejpam-4596	299	15	xβ	xβ	PROPN
ejpam-4596	299	16	,	,	PUNCT
ejpam-4596	299	17	yβ	yβ	NOUN
ejpam-4596	299	18	)	)	PUNCT
ejpam-4596	299	19	∈	∈	PROPN
ejpam-4596	299	20	e	e	NOUN
ejpam-4596	299	21	,	,	PUNCT
ejpam-4596	299	22	then	then	ADV
ejpam-4596	299	23	(	(	PUNCT
ejpam-4596	299	24	xpx	xpx	PROPN
ejpam-4596	299	25	,	,	PUNCT
ejpam-4596	299	26	xpy	xpy	PROPN
ejpam-4596	299	27	)	)	PUNCT
ejpam-4596	299	28	∈	∈	PROPN
ejpam-4596	299	29	e	e	X
ejpam-4596	299	30	where	where	SCONJ
ejpam-4596	299	31	xpxθαθ	xpxθαθ	PROPN
ejpam-4596	299	32	∈	∈	PROPN
ejpam-4596	299	33	px	px	PROPN
ejpam-4596	299	34	and	and	CCONJ
ejpam-4596	299	35	xpyθαθ	xpyθαθ	PROPN
ejpam-4596	299	36	∈	∈	PROPN
ejpam-4596	299	37	py	py	INTJ
ejpam-4596	299	38	and	and	CCONJ
ejpam-4596	299	39	so	so	ADV
ejpam-4596	299	40	(	(	PUNCT
ejpam-4596	299	41	xα	xα	INTJ
ejpam-4596	299	42	,	,	PUNCT
ejpam-4596	299	43	yα	yα	NOUN
ejpam-4596	299	44	)	)	PUNCT
ejpam-4596	299	45	=	=	SYM
ejpam-4596	300	1	(	(	PUNCT
ejpam-4596	300	2	xpxθαθα	xpxθαθα	PROPN
ejpam-4596	300	3	,	,	PUNCT
ejpam-4596	300	4	xpyθαθα	xpyθαθα	PROPN
ejpam-4596	300	5	)	)	PUNCT
ejpam-4596	300	6	∈	∈	PROPN
ejpam-4596	300	7	e.	e.	PROPN
ejpam-4596	300	8	by	by	ADP
ejpam-4596	300	9	α	α	PROPN
ejpam-4596	300	10	∈	∈	PROPN
ejpam-4596	300	11	te∗(x	te∗(x	NOUN
ejpam-4596	300	12	)	)	PUNCT
ejpam-4596	300	13	,	,	PUNCT
ejpam-4596	300	14	it	it	PRON
ejpam-4596	300	15	follows	follow	VERB
ejpam-4596	300	16	that	that	SCONJ
ejpam-4596	300	17	(	(	PUNCT
ejpam-4596	300	18	x	x	X
ejpam-4596	300	19	,	,	PUNCT
ejpam-4596	300	20	y	y	NOUN
ejpam-4596	300	21	)	)	PUNCT
ejpam-4596	300	22	∈	∈	PROPN
ejpam-4596	300	23	e.	e.	PROPN
ejpam-4596	300	24	hence	hence	ADV
ejpam-4596	300	25	β	β	PROPN
ejpam-4596	300	26	∈	∈	NOUN
ejpam-4596	300	27	te∗(x	te∗(x	NOUN
ejpam-4596	300	28	)	)	PUNCT
ejpam-4596	300	29	.	.	PUNCT
ejpam-4596	301	1	finally	finally	ADV
ejpam-4596	301	2	,	,	PUNCT
ejpam-4596	301	3	to	to	PART
ejpam-4596	301	4	show	show	VERB
ejpam-4596	301	5	that	that	SCONJ
ejpam-4596	301	6	α	α	PRON
ejpam-4596	301	7	=	=	PUNCT
ejpam-4596	301	8	β	β	X
ejpam-4596	301	9	∗	∗	X
ejpam-4596	301	10	α	α	PROPN
ejpam-4596	301	11	∗	∗	NOUN
ejpam-4596	301	12	α	α	NOUN
ejpam-4596	301	13	.	.	PUNCT
ejpam-4596	302	1	let	let	VERB
ejpam-4596	302	2	x	x	SYM
ejpam-4596	302	3	∈	∈	PROPN
ejpam-4596	302	4	x.	x.	NOUN
ejpam-4596	302	5	then	then	ADV
ejpam-4596	302	6	xpxθαθ	xpxθαθ	PROPN
ejpam-4596	302	7	∈	∈	PROPN
ejpam-4596	302	8	px	px	NOUN
ejpam-4596	302	9	and	and	CCONJ
ejpam-4596	302	10	hence	hence	ADV
ejpam-4596	302	11	xβθαθα	xβθαθα	PROPN
ejpam-4596	303	1	=	=	PUNCT
ejpam-4596	303	2	xpxθαθα	xpxθαθα	PROPN
ejpam-4596	304	1	=	=	PUNCT
ejpam-4596	304	2	xα	xα	PROPN
ejpam-4596	304	3	,	,	PUNCT
ejpam-4596	304	4	as	as	SCONJ
ejpam-4596	304	5	required	require	VERB
ejpam-4596	304	6	.	.	PUNCT
ejpam-4596	305	1	corollary	corollary	ADJ
ejpam-4596	305	2	6	6	NUM
ejpam-4596	305	3	.	.	PUNCT
ejpam-4596	305	4	idx	idx	PROPN
ejpam-4596	305	5	∈	∈	PROPN
ejpam-4596	305	6	lreg(te∗(x	lreg(te∗(x	PROPN
ejpam-4596	305	7	,	,	PUNCT
ejpam-4596	305	8	θ	θ	NOUN
ejpam-4596	305	9	)	)	PUNCT
ejpam-4596	305	10	)	)	PUNCT
ejpam-4596	306	1	if	if	SCONJ
ejpam-4596	306	2	and	and	CCONJ
ejpam-4596	306	3	only	only	ADV
ejpam-4596	306	4	if	if	SCONJ
ejpam-4596	306	5	θ	θ	PROPN
ejpam-4596	306	6	is	be	AUX
ejpam-4596	306	7	a	a	DET
ejpam-4596	306	8	surjection	surjection	NOUN
ejpam-4596	306	9	.	.	PUNCT
ejpam-4596	307	1	proof	proof	NOUN
ejpam-4596	307	2	.	.	PUNCT
ejpam-4596	308	1	the	the	DET
ejpam-4596	308	2	necessity	necessity	NOUN
ejpam-4596	308	3	is	be	AUX
ejpam-4596	308	4	clear	clear	ADJ
ejpam-4596	308	5	from	from	ADP
ejpam-4596	308	6	corollary	corollary	ADJ
ejpam-4596	308	7	2	2	NUM
ejpam-4596	308	8	.	.	PUNCT
ejpam-4596	308	9	to	to	PART
ejpam-4596	308	10	prove	prove	VERB
ejpam-4596	308	11	the	the	DET
ejpam-4596	308	12	sufficiency	sufficiency	NOUN
ejpam-4596	308	13	,	,	PUNCT
ejpam-4596	308	14	we	we	PRON
ejpam-4596	308	15	suppose	suppose	VERB
ejpam-4596	308	16	that	that	SCONJ
ejpam-4596	308	17	θ	θ	PROPN
ejpam-4596	308	18	is	be	AUX
ejpam-4596	308	19	a	a	DET
ejpam-4596	308	20	surjection	surjection	NOUN
ejpam-4596	308	21	.	.	PUNCT
ejpam-4596	309	1	then	then	ADV
ejpam-4596	309	2	θθ	θθ	NOUN
ejpam-4596	309	3	is	be	AUX
ejpam-4596	309	4	also	also	ADV
ejpam-4596	309	5	a	a	DET
ejpam-4596	309	6	surjection	surjection	NOUN
ejpam-4596	309	7	.	.	PUNCT
ejpam-4596	310	1	since	since	SCONJ
ejpam-4596	310	2	π(idx	π(idx	ADJ
ejpam-4596	310	3	)	)	PUNCT
ejpam-4596	310	4	=	=	PRON
ejpam-4596	310	5	{	{	PUNCT
ejpam-4596	310	6	{	{	PUNCT
ejpam-4596	310	7	x	x	NOUN
ejpam-4596	310	8	}	}	PUNCT
ejpam-4596	310	9	:	:	PUNCT
ejpam-4596	310	10	x	x	SYM
ejpam-4596	310	11	∈	∈	NOUN
ejpam-4596	310	12	x	x	X
ejpam-4596	310	13	}	}	PUNCT
ejpam-4596	310	14	,	,	PUNCT
ejpam-4596	310	15	p	p	NOUN
ejpam-4596	310	16	∩	∩	ADJ
ejpam-4596	310	17	xθidxθ	xθidxθ	NOUN
ejpam-4596	310	18	=	=	SYM
ejpam-4596	310	19	p	p	PROPN
ejpam-4596	310	20	∩	∩	PROPN
ejpam-4596	310	21	xθθ	xθθ	PROPN
ejpam-4596	310	22	̸=	̸=	PROPN
ejpam-4596	310	23	∅	∅	NOUN
ejpam-4596	310	24	for	for	ADP
ejpam-4596	310	25	all	all	DET
ejpam-4596	310	26	p	p	NOUN
ejpam-4596	310	27	∈	∈	PROPN
ejpam-4596	310	28	π(idx	π(idx	NOUN
ejpam-4596	310	29	)	)	PUNCT
ejpam-4596	310	30	.	.	PUNCT
ejpam-4596	311	1	hence	hence	ADV
ejpam-4596	311	2	idx	idx	NOUN
ejpam-4596	311	3	∈	∈	PROPN
ejpam-4596	311	4	lreg(te∗(x	lreg(te∗(x	PROPN
ejpam-4596	311	5	,	,	PUNCT
ejpam-4596	311	6	θ	θ	NOUN
ejpam-4596	311	7	)	)	PUNCT
ejpam-4596	311	8	)	)	PUNCT
ejpam-4596	311	9	,	,	PUNCT
ejpam-4596	311	10	by	by	ADP
ejpam-4596	311	11	theorem	theorem	NOUN
ejpam-4596	311	12	7	7	NUM
ejpam-4596	311	13	.	.	PUNCT
ejpam-4596	312	1	next	next	ADV
ejpam-4596	312	2	,	,	PUNCT
ejpam-4596	312	3	we	we	PRON
ejpam-4596	312	4	characterize	characterize	VERB
ejpam-4596	312	5	a	a	DET
ejpam-4596	312	6	right	right	ADJ
ejpam-4596	312	7	regular	regular	ADJ
ejpam-4596	312	8	element	element	NOUN
ejpam-4596	312	9	of	of	ADP
ejpam-4596	312	10	te∗(x	te∗(x	NOUN
ejpam-4596	312	11	,	,	PUNCT
ejpam-4596	312	12	θ	θ	NOUN
ejpam-4596	312	13	)	)	PUNCT
ejpam-4596	312	14	.	.	PUNCT
ejpam-4596	313	1	the	the	DET
ejpam-4596	313	2	following	follow	VERB
ejpam-4596	313	3	lemma	lemma	PROPN
ejpam-4596	313	4	is	be	AUX
ejpam-4596	313	5	needed	need	VERB
ejpam-4596	313	6	.	.	PUNCT
ejpam-4596	314	1	lemma	lemma	PROPN
ejpam-4596	314	2	2	2	NUM
ejpam-4596	314	3	.	.	PUNCT
ejpam-4596	315	1	[	[	X
ejpam-4596	315	2	8	8	NUM
ejpam-4596	315	3	]	]	PUNCT
ejpam-4596	315	4	let	let	VERB
ejpam-4596	315	5	α	α	PRON
ejpam-4596	315	6	∈	∈	PROPN
ejpam-4596	315	7	te∗(x	te∗(x	NOUN
ejpam-4596	315	8	,	,	PUNCT
ejpam-4596	315	9	θ	θ	NOUN
ejpam-4596	315	10	)	)	PUNCT
ejpam-4596	315	11	and	and	CCONJ
ejpam-4596	315	12	a	a	DET
ejpam-4596	315	13	,	,	PUNCT
ejpam-4596	315	14	b	b	X
ejpam-4596	315	15	∈	∈	PROPN
ejpam-4596	315	16	x	x	X
ejpam-4596	315	17	/	/	SYM
ejpam-4596	315	18	e.	e.	PROPN
ejpam-4596	315	19	if	if	SCONJ
ejpam-4596	315	20	aα	aα	PROPN
ejpam-4596	315	21	⊆	⊆	NUM
ejpam-4596	315	22	b	b	NOUN
ejpam-4596	315	23	,	,	PUNCT
ejpam-4596	315	24	then	then	ADV
ejpam-4596	315	25	bα−1	bα−1	NOUN
ejpam-4596	315	26	=	=	PUNCT
ejpam-4596	315	27	a.	a.	NOUN
ejpam-4596	315	28	theorem	theorem	NOUN
ejpam-4596	315	29	8	8	NUM
ejpam-4596	315	30	.	.	PUNCT
ejpam-4596	316	1	let	let	VERB
ejpam-4596	316	2	α	α	PRON
ejpam-4596	316	3	∈	∈	PROPN
ejpam-4596	316	4	te∗(x	te∗(x	NOUN
ejpam-4596	316	5	,	,	PUNCT
ejpam-4596	316	6	θ	θ	NOUN
ejpam-4596	316	7	)	)	PUNCT
ejpam-4596	316	8	.	.	PUNCT
ejpam-4596	317	1	then	then	ADV
ejpam-4596	317	2	α	α	PROPN
ejpam-4596	317	3	∈	∈	PROPN
ejpam-4596	317	4	rreg(te∗(x	rreg(te∗(x	PROPN
ejpam-4596	317	5	,	,	PUNCT
ejpam-4596	317	6	θ	θ	NOUN
ejpam-4596	317	7	)	)	PUNCT
ejpam-4596	317	8	)	)	PUNCT
ejpam-4596	318	1	if	if	SCONJ
ejpam-4596	318	2	and	and	CCONJ
ejpam-4596	318	3	only	only	ADV
ejpam-4596	318	4	if	if	SCONJ
ejpam-4596	318	5	(	(	PUNCT
ejpam-4596	318	6	i	i	NOUN
ejpam-4596	318	7	)	)	PUNCT
ejpam-4596	318	8	(	(	PUNCT
ejpam-4596	318	9	θαθ)|xα	θαθ)|xα	PROPN
ejpam-4596	318	10	is	be	AUX
ejpam-4596	318	11	an	an	DET
ejpam-4596	318	12	injection	injection	NOUN
ejpam-4596	318	13	and	and	CCONJ
ejpam-4596	318	14	(	(	PUNCT
ejpam-4596	318	15	ii	ii	NOUN
ejpam-4596	318	16	)	)	PUNCT
ejpam-4596	318	17	if	if	SCONJ
ejpam-4596	318	18	there	there	PRON
ejpam-4596	318	19	exists	exist	VERB
ejpam-4596	318	20	a	a	DET
ejpam-4596	318	21	∈	∈	PROPN
ejpam-4596	318	22	x	x	SYM
ejpam-4596	318	23	/	/	SYM
ejpam-4596	318	24	e	e	VERB
ejpam-4596	318	25	such	such	ADJ
ejpam-4596	318	26	that	that	SCONJ
ejpam-4596	318	27	a	a	DET
ejpam-4596	318	28	∩	∩	NOUN
ejpam-4596	318	29	x(αθ)2	x(αθ)2	NUM
ejpam-4596	318	30	=	=	SYM
ejpam-4596	318	31	∅	∅	NOUN
ejpam-4596	318	32	,	,	PUNCT
ejpam-4596	318	33	then	then	ADV
ejpam-4596	318	34	there	there	PRON
ejpam-4596	318	35	exists	exist	VERB
ejpam-4596	318	36	an	an	DET
ejpam-4596	318	37	injection	injection	NOUN
ejpam-4596	318	38	φ	φ	NOUN
ejpam-4596	318	39	:	:	PUNCT
ejpam-4596	318	40	{	{	PUNCT
ejpam-4596	318	41	a	a	DET
ejpam-4596	318	42	∈	∈	PROPN
ejpam-4596	318	43	x	x	SYM
ejpam-4596	318	44	/	/	SYM
ejpam-4596	318	45	e	e	NOUN
ejpam-4596	318	46	:	:	PUNCT
ejpam-4596	318	47	a	a	DET
ejpam-4596	318	48	∩x(αθ)2	∩x(αθ)2	NOUN
ejpam-4596	318	49	=	=	SYM
ejpam-4596	318	50	∅	∅	NOUN
ejpam-4596	318	51	}	}	PUNCT
ejpam-4596	318	52	→	→	SYM
ejpam-4596	318	53	{	{	PUNCT
ejpam-4596	318	54	a	a	PRON
ejpam-4596	318	55	∈	∈	PROPN
ejpam-4596	318	56	x	x	SYM
ejpam-4596	318	57	/	/	SYM
ejpam-4596	318	58	e	e	NOUN
ejpam-4596	318	59	:	:	PUNCT
ejpam-4596	318	60	a	a	DET
ejpam-4596	318	61	∩xα	∩xα	NOUN
ejpam-4596	318	62	=	=	NOUN
ejpam-4596	318	63	∅	∅	NOUN
ejpam-4596	318	64	}	}	PUNCT
ejpam-4596	318	65	.	.	PUNCT
ejpam-4596	319	1	proof	proof	NOUN
ejpam-4596	319	2	.	.	PUNCT
ejpam-4596	320	1	assume	assume	VERB
ejpam-4596	320	2	that	that	SCONJ
ejpam-4596	320	3	α	α	PRON
ejpam-4596	320	4	∈	∈	PROPN
ejpam-4596	320	5	rreg(te∗(x	rreg(te∗(x	PROPN
ejpam-4596	320	6	,	,	PUNCT
ejpam-4596	320	7	θ	θ	NOUN
ejpam-4596	320	8	)	)	PUNCT
ejpam-4596	320	9	)	)	PUNCT
ejpam-4596	320	10	.	.	PUNCT
ejpam-4596	321	1	then	then	ADV
ejpam-4596	321	2	α	α	PROPN
ejpam-4596	321	3	∈	∈	PROPN
ejpam-4596	321	4	rreg(te(x	rreg(te(x	NOUN
ejpam-4596	321	5	,	,	PUNCT
ejpam-4596	321	6	θ	θ	NOUN
ejpam-4596	321	7	)	)	PUNCT
ejpam-4596	321	8	)	)	PUNCT
ejpam-4596	321	9	.	.	PUNCT
ejpam-4596	322	1	by	by	ADP
ejpam-4596	322	2	theorem	theorem	NOUN
ejpam-4596	322	3	3	3	NUM
ejpam-4596	322	4	,	,	PUNCT
ejpam-4596	322	5	we	we	PRON
ejpam-4596	322	6	then	then	ADV
ejpam-4596	322	7	have	have	AUX
ejpam-4596	322	8	(	(	PUNCT
ejpam-4596	322	9	i	i	NOUN
ejpam-4596	322	10	)	)	PUNCT
ejpam-4596	322	11	hold	hold	VERB
ejpam-4596	322	12	.	.	PUNCT
ejpam-4596	323	1	next	next	ADV
ejpam-4596	323	2	,	,	PUNCT
ejpam-4596	323	3	we	we	PRON
ejpam-4596	323	4	prove	prove	VERB
ejpam-4596	323	5	that	that	SCONJ
ejpam-4596	323	6	(	(	PUNCT
ejpam-4596	323	7	ii	ii	NOUN
ejpam-4596	323	8	)	)	PUNCT
ejpam-4596	323	9	holds	hold	VERB
ejpam-4596	323	10	in	in	ADP
ejpam-4596	323	11	the	the	DET
ejpam-4596	323	12	following	following	NOUN
ejpam-4596	323	13	.	.	PUNCT
ejpam-4596	324	1	suppose	suppose	VERB
ejpam-4596	324	2	that	that	SCONJ
ejpam-4596	324	3	{	{	PUNCT
ejpam-4596	324	4	a	a	PRON
ejpam-4596	324	5	∈	∈	PROPN
ejpam-4596	324	6	x	x	SYM
ejpam-4596	324	7	/	/	SYM
ejpam-4596	324	8	e	e	NOUN
ejpam-4596	324	9	:	:	PUNCT
ejpam-4596	324	10	a	a	DET
ejpam-4596	324	11	∩	∩	ADJ
ejpam-4596	324	12	x(αθ)2	x(αθ)2	PUNCT
ejpam-4596	324	13	=	=	SYM
ejpam-4596	324	14	∅	∅	NOUN
ejpam-4596	324	15	}	}	PUNCT
ejpam-4596	324	16	̸=	̸=	PROPN
ejpam-4596	324	17	∅.	∅.	ADV
ejpam-4596	324	18	let	let	VERB
ejpam-4596	324	19	a	a	DET
ejpam-4596	324	20	∈	∈	NOUN
ejpam-4596	324	21	x	x	SYM
ejpam-4596	324	22	/	/	SYM
ejpam-4596	324	23	e	e	VERB
ejpam-4596	324	24	be	be	VERB
ejpam-4596	324	25	such	such	ADJ
ejpam-4596	324	26	that	that	SCONJ
ejpam-4596	324	27	a	a	DET
ejpam-4596	324	28	∩	∩	NOUN
ejpam-4596	324	29	x(αθ)2	x(αθ)2	PUNCT
ejpam-4596	324	30	=	=	PRON
ejpam-4596	324	31	∅.	∅.	NOUN
ejpam-4596	324	32	since	since	SCONJ
ejpam-4596	324	33	α	α	PROPN
ejpam-4596	324	34	∈	∈	PROPN
ejpam-4596	324	35	rreg(te∗(x	rreg(te∗(x	PROPN
ejpam-4596	324	36	,	,	PUNCT
ejpam-4596	324	37	θ	θ	NOUN
ejpam-4596	324	38	)	)	PUNCT
ejpam-4596	324	39	)	)	PUNCT
ejpam-4596	324	40	,	,	PUNCT
ejpam-4596	324	41	there	there	PRON
ejpam-4596	324	42	exists	exist	VERB
ejpam-4596	324	43	β	β	X
ejpam-4596	324	44	∈	∈	NOUN
ejpam-4596	324	45	te∗(x	te∗(x	NOUN
ejpam-4596	324	46	,	,	PUNCT
ejpam-4596	324	47	θ	θ	NOUN
ejpam-4596	324	48	)	)	PUNCT
ejpam-4596	324	49	such	such	ADJ
ejpam-4596	324	50	that	that	SCONJ
ejpam-4596	324	51	α	α	NOUN
ejpam-4596	324	52	=	=	SYM
ejpam-4596	324	53	α	α	PROPN
ejpam-4596	324	54	∗	∗	NOUN
ejpam-4596	324	55	α	α	PROPN
ejpam-4596	324	56	∗	∗	NOUN
ejpam-4596	324	57	β	β	X
ejpam-4596	325	1	and	and	CCONJ
ejpam-4596	326	1	so	so	ADV
ejpam-4596	326	2	α	α	NOUN
ejpam-4596	326	3	=	=	SYM
ejpam-4596	326	4	αθαθβ	αθαθβ	NOUN
ejpam-4596	326	5	.	.	PUNCT
ejpam-4596	327	1	by	by	ADP
ejpam-4596	327	2	lemma	lemma	PROPN
ejpam-4596	327	3	1	1	NUM
ejpam-4596	327	4	,	,	PUNCT
ejpam-4596	327	5	we	we	PRON
ejpam-4596	327	6	let	let	VERB
ejpam-4596	327	7	a′	a′	PROPN
ejpam-4596	327	8	∈	∈	PROPN
ejpam-4596	327	9	x	x	SYM
ejpam-4596	327	10	/	/	SYM
ejpam-4596	327	11	e	e	VERB
ejpam-4596	327	12	such	such	ADJ
ejpam-4596	327	13	that	that	SCONJ
ejpam-4596	327	14	aβ	aβ	DET
ejpam-4596	327	15	⊆	⊆	NUM
ejpam-4596	327	16	a′.	a′.	NOUN
ejpam-4596	327	17	claim	claim	VERB
ejpam-4596	327	18	that	that	SCONJ
ejpam-4596	327	19	a′	a′	NOUN
ejpam-4596	327	20	∩xα	∩xα	NOUN
ejpam-4596	327	21	=	=	NOUN
ejpam-4596	327	22	∅	∅	NOUN
ejpam-4596	327	23	,	,	PUNCT
ejpam-4596	327	24	suppose	suppose	VERB
ejpam-4596	327	25	not	not	PART
ejpam-4596	327	26	.	.	PUNCT
ejpam-4596	328	1	let	let	VERB
ejpam-4596	328	2	x	x	SYM
ejpam-4596	328	3	∈	∈	PROPN
ejpam-4596	328	4	x	x	AUX
ejpam-4596	328	5	be	be	AUX
ejpam-4596	328	6	such	such	ADJ
ejpam-4596	328	7	that	that	SCONJ
ejpam-4596	328	8	xα	xα	PROPN
ejpam-4596	328	9	∈	∈	PROPN
ejpam-4596	328	10	a′	a′	PROPN
ejpam-4596	328	11	and	and	CCONJ
ejpam-4596	328	12	choose	choose	VERB
ejpam-4596	328	13	a	a	DET
ejpam-4596	328	14	∈	∈	PROPN
ejpam-4596	328	15	a.	a.	NOUN
ejpam-4596	328	16	then	then	ADV
ejpam-4596	328	17	aβ	aβ	PRON
ejpam-4596	328	18	∈	∈	PROPN
ejpam-4596	328	19	a′	a′	PROPN
ejpam-4596	328	20	and	and	CCONJ
ejpam-4596	328	21	so	so	ADV
ejpam-4596	328	22	(	(	PUNCT
ejpam-4596	328	23	xαθαθβ	xαθαθβ	ADJ
ejpam-4596	328	24	,	,	PUNCT
ejpam-4596	328	25	aβ	aβ	NOUN
ejpam-4596	328	26	)	)	PUNCT
ejpam-4596	328	27	=	=	SYM
ejpam-4596	329	1	(	(	PUNCT
ejpam-4596	329	2	xα	xα	INTJ
ejpam-4596	329	3	,	,	PUNCT
ejpam-4596	329	4	aβ	aβ	NOUN
ejpam-4596	329	5	)	)	PUNCT
ejpam-4596	329	6	∈	∈	PROPN
ejpam-4596	329	7	e.	e.	PROPN
ejpam-4596	329	8	since	since	SCONJ
ejpam-4596	329	9	β	β	PROPN
ejpam-4596	329	10	∈	∈	PROPN
ejpam-4596	329	11	te∗(x	te∗(x	NOUN
ejpam-4596	329	12	)	)	PUNCT
ejpam-4596	329	13	,	,	PUNCT
ejpam-4596	329	14	we	we	PRON
ejpam-4596	329	15	get	get	VERB
ejpam-4596	329	16	(	(	PUNCT
ejpam-4596	329	17	xαθαθ	xαθαθ	PROPN
ejpam-4596	329	18	,	,	PUNCT
ejpam-4596	329	19	a	a	DET
ejpam-4596	329	20	)	)	PUNCT
ejpam-4596	329	21	∈	∈	PROPN
ejpam-4596	329	22	e.	e.	PROPN
ejpam-4596	330	1	hence	hence	ADV
ejpam-4596	330	2	xαθαθ	xαθαθ	PROPN
ejpam-4596	330	3	∈	∈	PROPN
ejpam-4596	330	4	a	a	PRON
ejpam-4596	330	5	which	which	PRON
ejpam-4596	330	6	is	be	AUX
ejpam-4596	330	7	a	a	DET
ejpam-4596	330	8	contradiction	contradiction	NOUN
ejpam-4596	330	9	.	.	PUNCT
ejpam-4596	331	1	thus	thus	ADV
ejpam-4596	331	2	a′	a′	NOUN
ejpam-4596	331	3	∩	∩	NOUN
ejpam-4596	331	4	xα	xα	PUNCT
ejpam-4596	331	5	=	=	VERB
ejpam-4596	331	6	∅.	∅.	AUX
ejpam-4596	331	7	define	define	VERB
ejpam-4596	331	8	φ	φ	PROPN
ejpam-4596	331	9	:	:	PUNCT
ejpam-4596	331	10	{	{	PUNCT
ejpam-4596	331	11	a	a	DET
ejpam-4596	331	12	∈	∈	PROPN
ejpam-4596	331	13	x	x	SYM
ejpam-4596	331	14	/	/	SYM
ejpam-4596	331	15	e	e	NOUN
ejpam-4596	331	16	:	:	PUNCT
ejpam-4596	331	17	a	a	DET
ejpam-4596	331	18	∩x(αθ)2	∩x(αθ)2	NOUN
ejpam-4596	331	19	=	=	SYM
ejpam-4596	331	20	∅	∅	NOUN
ejpam-4596	331	21	}	}	PUNCT
ejpam-4596	331	22	→	→	SYM
ejpam-4596	331	23	{	{	PUNCT
ejpam-4596	331	24	a	a	PRON
ejpam-4596	331	25	∈	∈	PROPN
ejpam-4596	331	26	x	x	SYM
ejpam-4596	331	27	/	/	SYM
ejpam-4596	331	28	e	e	NOUN
ejpam-4596	331	29	:	:	PUNCT
ejpam-4596	331	30	a	a	DET
ejpam-4596	331	31	∩xα	∩xα	NOUN
ejpam-4596	331	32	=	=	NOUN
ejpam-4596	331	33	∅	∅	NOUN
ejpam-4596	331	34	}	}	PUNCT
ejpam-4596	331	35	by	by	ADP
ejpam-4596	331	36	aφ	aφ	NOUN
ejpam-4596	331	37	=	=	PUNCT
ejpam-4596	331	38	a′	a′	PROPN
ejpam-4596	331	39	for	for	ADP
ejpam-4596	331	40	all	all	DET
ejpam-4596	331	41	a	a	DET
ejpam-4596	331	42	∈	∈	NOUN
ejpam-4596	331	43	x	x	SYM
ejpam-4596	331	44	/	/	SYM
ejpam-4596	331	45	e	e	NOUN
ejpam-4596	331	46	and	and	CCONJ
ejpam-4596	331	47	a	a	DET
ejpam-4596	331	48	∩x(αθ)2	∩x(αθ)2	NOUN
ejpam-4596	331	49	=	=	SYM
ejpam-4596	331	50	∅.	∅.	NOUN
ejpam-4596	331	51	to	to	PART
ejpam-4596	331	52	show	show	VERB
ejpam-4596	331	53	that	that	SCONJ
ejpam-4596	331	54	φ	φ	PROPN
ejpam-4596	331	55	is	be	AUX
ejpam-4596	331	56	injective	injective	ADJ
ejpam-4596	331	57	,	,	PUNCT
ejpam-4596	331	58	let	let	VERB
ejpam-4596	331	59	a	a	DET
ejpam-4596	331	60	,	,	PUNCT
ejpam-4596	331	61	b	b	X
ejpam-4596	331	62	∈	∈	PROPN
ejpam-4596	331	63	{	{	PUNCT
ejpam-4596	331	64	a	a	DET
ejpam-4596	331	65	∈	∈	PROPN
ejpam-4596	331	66	x	x	SYM
ejpam-4596	331	67	/	/	SYM
ejpam-4596	331	68	e	e	NOUN
ejpam-4596	331	69	:	:	PUNCT
ejpam-4596	331	70	a	a	DET
ejpam-4596	331	71	∩	∩	ADJ
ejpam-4596	331	72	x(αθ)2	x(αθ)2	PUNCT
ejpam-4596	331	73	=	=	SYM
ejpam-4596	331	74	∅	∅	NOUN
ejpam-4596	331	75	}	}	PUNCT
ejpam-4596	331	76	be	be	AUX
ejpam-4596	331	77	such	such	ADJ
ejpam-4596	331	78	that	that	DET
ejpam-4596	331	79	aφ	aφ	NOUN
ejpam-4596	331	80	=	=	SYM
ejpam-4596	331	81	bφ	bφ	NOUN
ejpam-4596	331	82	.	.	PUNCT
ejpam-4596	332	1	by	by	ADP
ejpam-4596	332	2	the	the	DET
ejpam-4596	332	3	definition	definition	NOUN
ejpam-4596	332	4	of	of	ADP
ejpam-4596	332	5	φ	φ	NUM
ejpam-4596	332	6	,	,	PUNCT
ejpam-4596	332	7	aφ	aφ	ADP
ejpam-4596	332	8	=	=	PUNCT
ejpam-4596	332	9	a′	a′	PROPN
ejpam-4596	332	10	and	and	CCONJ
ejpam-4596	332	11	bφ	bφ	PROPN
ejpam-4596	332	12	=	=	PUNCT
ejpam-4596	332	13	b′	b′	NUM
ejpam-4596	332	14	where	where	SCONJ
ejpam-4596	332	15	aβ	aβ	ADP
ejpam-4596	332	16	⊆	⊆	NUM
ejpam-4596	332	17	a′	a′	NOUN
ejpam-4596	332	18	and	and	CCONJ
ejpam-4596	332	19	bβ	bβ	VERB
ejpam-4596	332	20	⊆	⊆	NUM
ejpam-4596	332	21	b′	b′	NOUN
ejpam-4596	332	22	for	for	ADP
ejpam-4596	332	23	some	some	DET
ejpam-4596	332	24	a′	a′	ADJ
ejpam-4596	332	25	,	,	PUNCT
ejpam-4596	332	26	b′	b′	NUM
ejpam-4596	332	27	∈	∈	PROPN
ejpam-4596	332	28	x	x	X
ejpam-4596	332	29	/	/	SYM
ejpam-4596	332	30	e.	e.	PROPN
ejpam-4596	332	31	it	it	PRON
ejpam-4596	332	32	follows	follow	VERB
ejpam-4596	332	33	from	from	ADP
ejpam-4596	332	34	lemma	lemma	PROPN
ejpam-4596	332	35	2	2	NUM
ejpam-4596	332	36	that	that	SCONJ
ejpam-4596	332	37	a	a	DET
ejpam-4596	332	38	=	=	X
ejpam-4596	332	39	a′β−1	a′β−1	PROPN
ejpam-4596	332	40	and	and	CCONJ
ejpam-4596	332	41	b	b	X
ejpam-4596	332	42	=	=	PUNCT
ejpam-4596	332	43	b′β−1	b′β−1	PROPN
ejpam-4596	332	44	.	.	PUNCT
ejpam-4596	333	1	since	since	SCONJ
ejpam-4596	333	2	a′	a′	PROPN
ejpam-4596	333	3	=	=	SYM
ejpam-4596	333	4	b′	b′	NUM
ejpam-4596	333	5	,	,	PUNCT
ejpam-4596	333	6	we	we	PRON
ejpam-4596	333	7	deduce	deduce	VERB
ejpam-4596	333	8	that	that	SCONJ
ejpam-4596	333	9	a	a	DET
ejpam-4596	333	10	=	=	X
ejpam-4596	333	11	b.	b.	PROPN
ejpam-4596	333	12	therefore	therefore	ADV
ejpam-4596	333	13	φ	φ	PROPN
ejpam-4596	333	14	is	be	AUX
ejpam-4596	333	15	an	an	DET
ejpam-4596	333	16	injection	injection	NOUN
ejpam-4596	333	17	.	.	PUNCT
ejpam-4596	334	1	hence	hence	ADV
ejpam-4596	334	2	(	(	PUNCT
ejpam-4596	334	3	ii	ii	NOUN
ejpam-4596	334	4	)	)	PUNCT
ejpam-4596	334	5	holds	hold	VERB
ejpam-4596	334	6	.	.	PUNCT
ejpam-4596	335	1	conversely	conversely	ADV
ejpam-4596	335	2	,	,	PUNCT
ejpam-4596	335	3	suppose	suppose	VERB
ejpam-4596	335	4	that	that	SCONJ
ejpam-4596	335	5	the	the	DET
ejpam-4596	335	6	conditions	condition	NOUN
ejpam-4596	335	7	(	(	PUNCT
ejpam-4596	335	8	i	i	NOUN
ejpam-4596	335	9	)	)	PUNCT
ejpam-4596	335	10	and	and	CCONJ
ejpam-4596	335	11	(	(	PUNCT
ejpam-4596	335	12	ii	ii	NOUN
ejpam-4596	335	13	)	)	PUNCT
ejpam-4596	335	14	hold	hold	VERB
ejpam-4596	335	15	.	.	PUNCT
ejpam-4596	336	1	for	for	ADP
ejpam-4596	336	2	each	each	DET
ejpam-4596	336	3	x	x	PROPN
ejpam-4596	336	4	∈	∈	PROPN
ejpam-4596	336	5	x(αθ)2	x(αθ)2	PROPN
ejpam-4596	336	6	,	,	PUNCT
ejpam-4596	336	7	we	we	PRON
ejpam-4596	336	8	choose	choose	VERB
ejpam-4596	336	9	and	and	CCONJ
ejpam-4596	336	10	fix	fix	VERB
ejpam-4596	336	11	an	an	DET
ejpam-4596	336	12	element	element	NOUN
ejpam-4596	336	13	x′	x′	PROPN
ejpam-4596	336	14	∈	∈	PROPN
ejpam-4596	336	15	xα	xα	ADP
ejpam-4596	337	1	such	such	ADJ
ejpam-4596	337	2	that	that	SCONJ
ejpam-4596	337	3	x	x	X
ejpam-4596	337	4	=	=	SYM
ejpam-4596	337	5	x′θαθ	x′θαθ	PROPN
ejpam-4596	337	6	.	.	PUNCT
ejpam-4596	338	1	let	let	VERB
ejpam-4596	338	2	a	a	DET
ejpam-4596	338	3	∈	∈	ADJ
ejpam-4596	338	4	x	x	SYM
ejpam-4596	338	5	/	/	SYM
ejpam-4596	338	6	e	e	VERB
ejpam-4596	338	7	be	be	VERB
ejpam-4596	338	8	such	such	ADJ
ejpam-4596	338	9	that	that	SCONJ
ejpam-4596	338	10	a	a	DET
ejpam-4596	338	11	∩x(αθ)2	∩x(αθ)2	NOUN
ejpam-4596	338	12	̸=	̸=	PROPN
ejpam-4596	338	13	∅.	∅.	NOUN
ejpam-4596	338	14	then	then	ADV
ejpam-4596	338	15	we	we	PRON
ejpam-4596	338	16	fix	fix	VERB
ejpam-4596	338	17	xa	xa	PROPN
ejpam-4596	338	18	∈	∈	PROPN
ejpam-4596	338	19	a	a	DET
ejpam-4596	338	20	∩xα	∩xα	NOUN
ejpam-4596	338	21	and	and	CCONJ
ejpam-4596	338	22	define	define	VERB
ejpam-4596	338	23	βa	βa	INTJ
ejpam-4596	338	24	:	:	PUNCT
ejpam-4596	338	25	a	a	DET
ejpam-4596	338	26	→	→	SYM
ejpam-4596	338	27	x	x	X
ejpam-4596	338	28	by	by	ADP
ejpam-4596	338	29	xβa	xβa	PROPN
ejpam-4596	339	1	=	=	PRON
ejpam-4596	339	2	{	{	PUNCT
ejpam-4596	339	3	x′	x′	PROPN
ejpam-4596	339	4	if	if	SCONJ
ejpam-4596	339	5	x	x	PROPN
ejpam-4596	339	6	∈	∈	PROPN
ejpam-4596	339	7	x(αθ)2	x(αθ)2	PROPN
ejpam-4596	339	8	,	,	PUNCT
ejpam-4596	339	9	x′a	x′a	ADV
ejpam-4596	339	10	otherwise	otherwise	ADV
ejpam-4596	339	11	.	.	PUNCT
ejpam-4596	340	1	p.	p.	NOUN
ejpam-4596	340	2	tantong	tantong	NOUN
ejpam-4596	340	3	,	,	PUNCT
ejpam-4596	340	4	n.	n.	NOUN
ejpam-4596	340	5	sawatraksa	sawatraksa	PROPN
ejpam-4596	340	6	/	/	SYM
ejpam-4596	340	7	eur	eur	PROPN
ejpam-4596	340	8	.	.	PUNCT
ejpam-4596	341	1	j.	j.	PROPN
ejpam-4596	341	2	pure	pure	PROPN
ejpam-4596	341	3	appl	appl	PROPN
ejpam-4596	341	4	.	.	PROPN
ejpam-4596	341	5	math	math	PROPN
ejpam-4596	341	6	,	,	PUNCT
ejpam-4596	341	7	15	15	NUM
ejpam-4596	341	8	(	(	PUNCT
ejpam-4596	341	9	4	4	NUM
ejpam-4596	341	10	)	)	PUNCT
ejpam-4596	341	11	(	(	PUNCT
ejpam-4596	341	12	2022	2022	NUM
ejpam-4596	341	13	)	)	PUNCT
ejpam-4596	341	14	,	,	PUNCT
ejpam-4596	341	15	2116	2116	NUM
ejpam-4596	341	16	-	-	SYM
ejpam-4596	341	17	2126	2126	NUM
ejpam-4596	341	18	2124	2124	NUM
ejpam-4596	341	19	let	let	VERB
ejpam-4596	341	20	a	a	DET
ejpam-4596	341	21	∈	∈	NOUN
ejpam-4596	341	22	x	x	SYM
ejpam-4596	341	23	/	/	SYM
ejpam-4596	341	24	e	e	VERB
ejpam-4596	341	25	be	be	VERB
ejpam-4596	341	26	such	such	ADJ
ejpam-4596	341	27	that	that	SCONJ
ejpam-4596	341	28	a	a	DET
ejpam-4596	341	29	∩x(αθ)2	∩x(αθ)2	NOUN
ejpam-4596	341	30	=	=	SYM
ejpam-4596	341	31	∅	∅	NOUN
ejpam-4596	341	32	and	and	CCONJ
ejpam-4596	341	33	x	x	PUNCT
ejpam-4596	341	34	∈	∈	NOUN
ejpam-4596	341	35	a.	a.	NOUN
ejpam-4596	341	36	by	by	ADP
ejpam-4596	341	37	(	(	PUNCT
ejpam-4596	341	38	ii	ii	NOUN
ejpam-4596	341	39	)	)	PUNCT
ejpam-4596	341	40	,	,	PUNCT
ejpam-4596	341	41	we	we	PRON
ejpam-4596	341	42	fix	fix	VERB
ejpam-4596	341	43	x̃	x̃	PROPN
ejpam-4596	341	44	∈	∈	PROPN
ejpam-4596	341	45	aφ	aφ	NOUN
ejpam-4596	341	46	and	and	CCONJ
ejpam-4596	341	47	define	define	VERB
ejpam-4596	341	48	βa	βa	INTJ
ejpam-4596	341	49	:	:	PUNCT
ejpam-4596	341	50	a	a	PRON
ejpam-4596	341	51	→	→	SYM
ejpam-4596	341	52	x	x	X
ejpam-4596	341	53	by	by	ADP
ejpam-4596	341	54	xβa	xβa	PROPN
ejpam-4596	342	1	=	=	PROPN
ejpam-4596	342	2	x̃	x̃	PROPN
ejpam-4596	342	3	for	for	ADP
ejpam-4596	342	4	all	all	DET
ejpam-4596	342	5	x	x	SYM
ejpam-4596	342	6	∈	∈	NOUN
ejpam-4596	342	7	a.	a.	NOUN
ejpam-4596	342	8	let	let	VERB
ejpam-4596	342	9	β	β	X
ejpam-4596	342	10	:	:	PUNCT
ejpam-4596	342	11	x	x	SYM
ejpam-4596	342	12	→	→	SYM
ejpam-4596	342	13	x	x	SYM
ejpam-4596	342	14	by	by	ADP
ejpam-4596	342	15	β|a	β|a	PUNCT
ejpam-4596	343	1	=	=	NOUN
ejpam-4596	344	1	βa	βa	PROPN
ejpam-4596	345	1	for	for	ADP
ejpam-4596	345	2	all	all	DET
ejpam-4596	345	3	a	a	DET
ejpam-4596	345	4	∈	∈	PROPN
ejpam-4596	345	5	x	x	X
ejpam-4596	345	6	/	/	SYM
ejpam-4596	345	7	e.	e.	PROPN
ejpam-4596	345	8	since	since	SCONJ
ejpam-4596	345	9	x	x	PROPN
ejpam-4596	345	10	/	/	SYM
ejpam-4596	345	11	e	e	NOUN
ejpam-4596	345	12	is	be	AUX
ejpam-4596	345	13	a	a	DET
ejpam-4596	345	14	partition	partition	NOUN
ejpam-4596	345	15	of	of	ADP
ejpam-4596	345	16	x	x	PRON
ejpam-4596	345	17	,	,	PUNCT
ejpam-4596	345	18	β	β	X
ejpam-4596	345	19	is	be	AUX
ejpam-4596	345	20	well	well	ADV
ejpam-4596	345	21	-	-	PUNCT
ejpam-4596	345	22	defined	define	VERB
ejpam-4596	345	23	.	.	PUNCT
ejpam-4596	346	1	let	let	VERB
ejpam-4596	346	2	x	x	PRON
ejpam-4596	346	3	,	,	PUNCT
ejpam-4596	346	4	y	y	PROPN
ejpam-4596	346	5	∈	∈	PROPN
ejpam-4596	346	6	x	x	AUX
ejpam-4596	346	7	be	be	AUX
ejpam-4596	346	8	such	such	ADJ
ejpam-4596	346	9	that	that	SCONJ
ejpam-4596	346	10	(	(	PUNCT
ejpam-4596	346	11	x	x	NOUN
ejpam-4596	346	12	,	,	PUNCT
ejpam-4596	346	13	y	y	NOUN
ejpam-4596	346	14	)	)	PUNCT
ejpam-4596	346	15	∈	∈	PROPN
ejpam-4596	346	16	e.	e.	PROPN
ejpam-4596	346	17	then	then	ADV
ejpam-4596	346	18	x	x	PRON
ejpam-4596	346	19	,	,	PUNCT
ejpam-4596	346	20	y	y	PROPN
ejpam-4596	346	21	∈	∈	PROPN
ejpam-4596	346	22	a	a	PRON
ejpam-4596	346	23	for	for	ADP
ejpam-4596	346	24	some	some	PRON
ejpam-4596	346	25	a	a	DET
ejpam-4596	346	26	∈	∈	PROPN
ejpam-4596	346	27	x	x	X
ejpam-4596	346	28	/	/	SYM
ejpam-4596	346	29	e.	e.	PROPN
ejpam-4596	346	30	there	there	PRON
ejpam-4596	346	31	are	be	VERB
ejpam-4596	346	32	two	two	NUM
ejpam-4596	346	33	cases	case	NOUN
ejpam-4596	346	34	to	to	PART
ejpam-4596	346	35	consider	consider	VERB
ejpam-4596	346	36	.	.	PUNCT
ejpam-4596	347	1	case	case	NOUN
ejpam-4596	347	2	1	1	NUM
ejpam-4596	347	3	.	.	PUNCT
ejpam-4596	347	4	a	a	DET
ejpam-4596	347	5	∩x(αθ)2	∩x(αθ)2	NOUN
ejpam-4596	347	6	=	=	SYM
ejpam-4596	347	7	∅.	∅.	NOUN
ejpam-4596	347	8	then	then	ADV
ejpam-4596	347	9	(	(	PUNCT
ejpam-4596	347	10	xβ	xβ	PROPN
ejpam-4596	347	11	,	,	PUNCT
ejpam-4596	347	12	yβ	yβ	NOUN
ejpam-4596	347	13	)	)	PUNCT
ejpam-4596	347	14	=	=	SYM
ejpam-4596	348	1	(	(	PUNCT
ejpam-4596	348	2	x̃	x̃	PROPN
ejpam-4596	348	3	,	,	PUNCT
ejpam-4596	348	4	ỹ	ỹ	PROPN
ejpam-4596	348	5	)	)	PUNCT
ejpam-4596	348	6	∈	∈	PROPN
ejpam-4596	348	7	e.	e.	PROPN
ejpam-4596	348	8	case	case	NOUN
ejpam-4596	348	9	2	2	X
ejpam-4596	348	10	.	.	PUNCT
ejpam-4596	348	11	a	a	DET
ejpam-4596	348	12	∩x(αθ)2	∩x(αθ)2	NOUN
ejpam-4596	348	13	̸=	̸=	PROPN
ejpam-4596	348	14	∅.	∅.	NOUN
ejpam-4596	348	15	without	without	ADP
ejpam-4596	348	16	loss	loss	NOUN
ejpam-4596	348	17	of	of	ADP
ejpam-4596	348	18	generality	generality	NOUN
ejpam-4596	348	19	,	,	PUNCT
ejpam-4596	348	20	we	we	PRON
ejpam-4596	348	21	assume	assume	VERB
ejpam-4596	349	1	that	that	SCONJ
ejpam-4596	349	2	x	x	X
ejpam-4596	349	3	,	,	PUNCT
ejpam-4596	349	4	y	y	PROPN
ejpam-4596	349	5	∈	∈	PROPN
ejpam-4596	349	6	x(αθ)2	x(αθ)2	PROPN
ejpam-4596	349	7	.	.	PUNCT
ejpam-4596	350	1	hence	hence	ADV
ejpam-4596	350	2	xβ	xβ	PROPN
ejpam-4596	350	3	=	=	SYM
ejpam-4596	350	4	x′	x′	PROPN
ejpam-4596	350	5	and	and	CCONJ
ejpam-4596	350	6	yβ	yβ	NOUN
ejpam-4596	351	1	=	=	PUNCT
ejpam-4596	351	2	y′	y′	ADV
ejpam-4596	352	1	where	where	SCONJ
ejpam-4596	352	2	x	x	ADP
ejpam-4596	352	3	=	=	SYM
ejpam-4596	352	4	x′θαθ	x′θαθ	PROPN
ejpam-4596	352	5	and	and	CCONJ
ejpam-4596	352	6	y	y	PROPN
ejpam-4596	352	7	=	=	SYM
ejpam-4596	352	8	y′θαθ	y′θαθ	PROPN
ejpam-4596	352	9	,	,	PUNCT
ejpam-4596	352	10	respectively	respectively	ADV
ejpam-4596	352	11	.	.	PUNCT
ejpam-4596	353	1	since	since	SCONJ
ejpam-4596	353	2	θαθ	θαθ	PRON
ejpam-4596	353	3	∈	∈	PROPN
ejpam-4596	353	4	te∗(x	te∗(x	NOUN
ejpam-4596	353	5	,	,	PUNCT
ejpam-4596	353	6	θ	θ	NOUN
ejpam-4596	353	7	)	)	PUNCT
ejpam-4596	353	8	and	and	CCONJ
ejpam-4596	353	9	(	(	PUNCT
ejpam-4596	353	10	x′θαθ	x′θαθ	PROPN
ejpam-4596	353	11	,	,	PUNCT
ejpam-4596	353	12	y′θαθ	y′θαθ	NOUN
ejpam-4596	353	13	)	)	PUNCT
ejpam-4596	353	14	∈	∈	PROPN
ejpam-4596	353	15	e	e	NOUN
ejpam-4596	353	16	,	,	PUNCT
ejpam-4596	353	17	we	we	PRON
ejpam-4596	353	18	conclude	conclude	VERB
ejpam-4596	353	19	that	that	PRON
ejpam-4596	353	20	(	(	PUNCT
ejpam-4596	353	21	xβ	xβ	PROPN
ejpam-4596	353	22	,	,	PUNCT
ejpam-4596	353	23	yβ	yβ	NOUN
ejpam-4596	353	24	)	)	PUNCT
ejpam-4596	353	25	=	=	PUNCT
ejpam-4596	353	26	(	(	PUNCT
ejpam-4596	353	27	x′	x′	NUM
ejpam-4596	353	28	,	,	PUNCT
ejpam-4596	353	29	y′	y′	NUM
ejpam-4596	353	30	)	)	PUNCT
ejpam-4596	353	31	∈	∈	PROPN
ejpam-4596	353	32	e.	e.	PROPN
ejpam-4596	353	33	on	on	ADP
ejpam-4596	353	34	the	the	DET
ejpam-4596	353	35	other	other	ADJ
ejpam-4596	353	36	hand	hand	NOUN
ejpam-4596	353	37	,	,	PUNCT
ejpam-4596	353	38	let	let	VERB
ejpam-4596	353	39	x	x	PRON
ejpam-4596	353	40	,	,	PUNCT
ejpam-4596	353	41	y	y	PROPN
ejpam-4596	353	42	∈	∈	PROPN
ejpam-4596	353	43	x	x	AUX
ejpam-4596	353	44	be	be	AUX
ejpam-4596	353	45	such	such	ADJ
ejpam-4596	353	46	that	that	SCONJ
ejpam-4596	353	47	(	(	PUNCT
ejpam-4596	353	48	xβ	xβ	PROPN
ejpam-4596	353	49	,	,	PUNCT
ejpam-4596	353	50	yβ	yβ	NOUN
ejpam-4596	353	51	)	)	PUNCT
ejpam-4596	353	52	∈	∈	PROPN
ejpam-4596	353	53	e.	e.	PROPN
ejpam-4596	353	54	thus	thus	ADV
ejpam-4596	353	55	xβ	xβ	PROPN
ejpam-4596	353	56	,	,	PUNCT
ejpam-4596	353	57	yβ	yβ	PROPN
ejpam-4596	353	58	∈	∈	PROPN
ejpam-4596	353	59	b	b	PROPN
ejpam-4596	353	60	for	for	ADP
ejpam-4596	353	61	some	some	DET
ejpam-4596	353	62	b	b	NOUN
ejpam-4596	353	63	∈	∈	PROPN
ejpam-4596	353	64	x	x	PROPN
ejpam-4596	353	65	/	/	SYM
ejpam-4596	353	66	e.	e.	PROPN
ejpam-4596	354	1	if	if	SCONJ
ejpam-4596	354	2	b	b	PROPN
ejpam-4596	354	3	∩xα	∩xα	NOUN
ejpam-4596	354	4	=	=	NOUN
ejpam-4596	354	5	∅	∅	NOUN
ejpam-4596	354	6	,	,	PUNCT
ejpam-4596	354	7	then	then	ADV
ejpam-4596	354	8	by	by	ADP
ejpam-4596	354	9	the	the	DET
ejpam-4596	354	10	definition	definition	NOUN
ejpam-4596	354	11	of	of	ADP
ejpam-4596	354	12	β	β	X
ejpam-4596	354	13	,	,	PUNCT
ejpam-4596	354	14	xβ	xβ	PROPN
ejpam-4596	354	15	,	,	PUNCT
ejpam-4596	354	16	yβ	yβ	PROPN
ejpam-4596	354	17	∈	∈	PROPN
ejpam-4596	354	18	b	b	X
ejpam-4596	354	19	=	=	PUNCT
ejpam-4596	354	20	exφ	exφ	PROPN
ejpam-4596	354	21	=	=	SYM
ejpam-4596	354	22	eyφ	eyφ	PROPN
ejpam-4596	354	23	.	.	PUNCT
ejpam-4596	355	1	since	since	SCONJ
ejpam-4596	355	2	φ	φ	PROPN
ejpam-4596	355	3	is	be	AUX
ejpam-4596	355	4	an	an	DET
ejpam-4596	355	5	injection	injection	NOUN
ejpam-4596	355	6	,	,	PUNCT
ejpam-4596	355	7	ex	ex	X
ejpam-4596	355	8	=	=	NOUN
ejpam-4596	355	9	ey	ey	NOUN
ejpam-4596	355	10	and	and	CCONJ
ejpam-4596	355	11	hence	hence	ADV
ejpam-4596	355	12	(	(	PUNCT
ejpam-4596	355	13	x	x	X
ejpam-4596	355	14	,	,	PUNCT
ejpam-4596	355	15	y	y	NOUN
ejpam-4596	355	16	)	)	PUNCT
ejpam-4596	355	17	∈	∈	PROPN
ejpam-4596	355	18	e.	e.	PROPN
ejpam-4596	356	1	if	if	SCONJ
ejpam-4596	356	2	b	b	PROPN
ejpam-4596	356	3	∩xα	∩xα	NOUN
ejpam-4596	356	4	̸=	̸=	PROPN
ejpam-4596	356	5	∅	∅	NOUN
ejpam-4596	356	6	,	,	PUNCT
ejpam-4596	356	7	then	then	ADV
ejpam-4596	356	8	by	by	ADP
ejpam-4596	356	9	the	the	DET
ejpam-4596	356	10	definition	definition	NOUN
ejpam-4596	356	11	of	of	ADP
ejpam-4596	356	12	β	β	X
ejpam-4596	356	13	,	,	PUNCT
ejpam-4596	356	14	we	we	PRON
ejpam-4596	356	15	may	may	AUX
ejpam-4596	356	16	assume	assume	VERB
ejpam-4596	356	17	that	that	SCONJ
ejpam-4596	356	18	xβ	xβ	PROPN
ejpam-4596	356	19	=	=	SYM
ejpam-4596	356	20	x′	x′	PROPN
ejpam-4596	356	21	,	,	PUNCT
ejpam-4596	356	22	yβ	yβ	NOUN
ejpam-4596	356	23	=	=	NOUN
ejpam-4596	356	24	y′	y′	NOUN
ejpam-4596	356	25	for	for	ADP
ejpam-4596	356	26	some	some	DET
ejpam-4596	356	27	x′	x′	NUM
ejpam-4596	356	28	,	,	PUNCT
ejpam-4596	356	29	y′	y′	NOUN
ejpam-4596	356	30	∈	∈	PROPN
ejpam-4596	356	31	xα	xα	ADJ
ejpam-4596	356	32	with	with	ADP
ejpam-4596	356	33	x	x	X
ejpam-4596	356	34	=	=	SYM
ejpam-4596	356	35	x′θαθ	x′θαθ	PROPN
ejpam-4596	356	36	and	and	CCONJ
ejpam-4596	356	37	y	y	PROPN
ejpam-4596	356	38	=	=	SYM
ejpam-4596	356	39	y′θαθ	y′θαθ	PROPN
ejpam-4596	356	40	.	.	PUNCT
ejpam-4596	357	1	since	since	SCONJ
ejpam-4596	357	2	(	(	PUNCT
ejpam-4596	357	3	x′	x′	NUM
ejpam-4596	357	4	,	,	PUNCT
ejpam-4596	357	5	y′	y′	NUM
ejpam-4596	357	6	)	)	PUNCT
ejpam-4596	357	7	=	=	SYM
ejpam-4596	357	8	(	(	PUNCT
ejpam-4596	357	9	xβ	xβ	PROPN
ejpam-4596	357	10	,	,	PUNCT
ejpam-4596	357	11	yβ	yβ	NOUN
ejpam-4596	357	12	)	)	PUNCT
ejpam-4596	357	13	∈	∈	PROPN
ejpam-4596	357	14	e	e	NOUN
ejpam-4596	357	15	and	and	CCONJ
ejpam-4596	357	16	θαθ	θαθ	DET
ejpam-4596	357	17	∈	∈	PROPN
ejpam-4596	357	18	te∗(x	te∗(x	NOUN
ejpam-4596	357	19	)	)	PUNCT
ejpam-4596	357	20	,	,	PUNCT
ejpam-4596	357	21	we	we	PRON
ejpam-4596	357	22	deduce	deduce	VERB
ejpam-4596	357	23	that	that	SCONJ
ejpam-4596	357	24	(	(	PUNCT
ejpam-4596	357	25	x	x	X
ejpam-4596	357	26	,	,	PUNCT
ejpam-4596	357	27	y	y	NOUN
ejpam-4596	357	28	)	)	PUNCT
ejpam-4596	357	29	=	=	SYM
ejpam-4596	357	30	(	(	PUNCT
ejpam-4596	357	31	x′θαθ	x′θαθ	PROPN
ejpam-4596	357	32	,	,	PUNCT
ejpam-4596	357	33	y′θαθ	y′θαθ	NOUN
ejpam-4596	357	34	)	)	PUNCT
ejpam-4596	357	35	∈	∈	PROPN
ejpam-4596	357	36	e.	e.	PROPN
ejpam-4596	358	1	it	it	PRON
ejpam-4596	358	2	follows	follow	VERB
ejpam-4596	358	3	that	that	SCONJ
ejpam-4596	358	4	β	β	PROPN
ejpam-4596	358	5	∈	∈	PROPN
ejpam-4596	358	6	te∗(x	te∗(x	NOUN
ejpam-4596	358	7	)	)	PUNCT
ejpam-4596	358	8	.	.	PUNCT
ejpam-4596	359	1	let	let	VERB
ejpam-4596	359	2	x	x	SYM
ejpam-4596	359	3	∈	∈	PROPN
ejpam-4596	359	4	x	x	NOUN
ejpam-4596	359	5	,	,	PUNCT
ejpam-4596	359	6	then	then	ADV
ejpam-4596	359	7	x(αθ)2	x(αθ)2	NUM
ejpam-4596	359	8	∈	∈	PROPN
ejpam-4596	359	9	x(αθ)2	x(αθ)2	PUNCT
ejpam-4596	359	10	and	and	CCONJ
ejpam-4596	359	11	there	there	PRON
ejpam-4596	359	12	exists	exist	VERB
ejpam-4596	359	13	(	(	PUNCT
ejpam-4596	359	14	x(αθ)2)′	x(αθ)2)′	NOUN
ejpam-4596	359	15	∈	∈	PROPN
ejpam-4596	360	1	xα	xα	ADP
ejpam-4596	360	2	such	such	ADJ
ejpam-4596	360	3	that	that	PRON
ejpam-4596	360	4	(	(	PUNCT
ejpam-4596	360	5	x(αθ)2)′θαθ	x(αθ)2)′θαθ	PROPN
ejpam-4596	360	6	=	=	SYM
ejpam-4596	360	7	x(αθ)2	x(αθ)2	PROPN
ejpam-4596	360	8	=	=	SYM
ejpam-4596	360	9	(	(	PUNCT
ejpam-4596	360	10	xα)θαθ	xα)θαθ	X
ejpam-4596	360	11	.	.	PUNCT
ejpam-4596	361	1	we	we	PRON
ejpam-4596	361	2	note	note	VERB
ejpam-4596	361	3	by	by	ADP
ejpam-4596	361	4	(	(	PUNCT
ejpam-4596	361	5	1	1	NUM
ejpam-4596	361	6	)	)	PUNCT
ejpam-4596	361	7	that	that	SCONJ
ejpam-4596	361	8	(	(	PUNCT
ejpam-4596	361	9	x(αθ)2)′	x(αθ)2)′	NOUN
ejpam-4596	361	10	=	=	SYM
ejpam-4596	362	1	xα	xα	PROPN
ejpam-4596	362	2	.	.	PUNCT
ejpam-4596	363	1	therefore	therefore	ADV
ejpam-4596	363	2	x(α	x(α	PROPN
ejpam-4596	363	3	∗	∗	PROPN
ejpam-4596	363	4	α	α	PROPN
ejpam-4596	363	5	∗	∗	NOUN
ejpam-4596	363	6	β	β	NOUN
ejpam-4596	363	7	)	)	PUNCT
ejpam-4596	364	1	=	=	SYM
ejpam-4596	364	2	xαθαθβ	xαθαθβ	NOUN
ejpam-4596	364	3	=	=	SYM
ejpam-4596	365	1	x(αθ)2β	x(αθ)2β	PROPN
ejpam-4596	365	2	=	=	SYM
ejpam-4596	365	3	(	(	PUNCT
ejpam-4596	365	4	x(αθ)2)′	x(αθ)2)′	NOUN
ejpam-4596	365	5	=	=	PUNCT
ejpam-4596	366	1	xα	xα	PROPN
ejpam-4596	366	2	.	.	PUNCT
ejpam-4596	367	1	hence	hence	ADV
ejpam-4596	367	2	α	α	PROPN
ejpam-4596	367	3	is	be	AUX
ejpam-4596	367	4	right	right	ADV
ejpam-4596	367	5	regular	regular	ADJ
ejpam-4596	367	6	,	,	PUNCT
ejpam-4596	367	7	as	as	SCONJ
ejpam-4596	367	8	required	require	VERB
ejpam-4596	367	9	.	.	PUNCT
ejpam-4596	368	1	corollary	corollary	ADJ
ejpam-4596	368	2	7	7	NUM
ejpam-4596	368	3	.	.	PUNCT
ejpam-4596	368	4	idx	idx	PROPN
ejpam-4596	368	5	∈	∈	PROPN
ejpam-4596	368	6	rreg(te∗(x	rreg(te∗(x	PROPN
ejpam-4596	368	7	,	,	PUNCT
ejpam-4596	368	8	θ	θ	NOUN
ejpam-4596	368	9	)	)	PUNCT
ejpam-4596	368	10	)	)	PUNCT
ejpam-4596	369	1	if	if	SCONJ
ejpam-4596	369	2	and	and	CCONJ
ejpam-4596	369	3	only	only	ADV
ejpam-4596	369	4	if	if	SCONJ
ejpam-4596	369	5	(	(	PUNCT
ejpam-4596	369	6	i	i	NOUN
ejpam-4596	369	7	)	)	PUNCT
ejpam-4596	369	8	θ	θ	PROPN
ejpam-4596	369	9	is	be	AUX
ejpam-4596	369	10	an	an	DET
ejpam-4596	369	11	injection	injection	NOUN
ejpam-4596	369	12	and	and	CCONJ
ejpam-4596	369	13	(	(	PUNCT
ejpam-4596	369	14	ii	ii	NOUN
ejpam-4596	369	15	)	)	PUNCT
ejpam-4596	369	16	a	a	DET
ejpam-4596	369	17	∩xθ	∩xθ	ADJ
ejpam-4596	369	18	̸=	̸=	PROPN
ejpam-4596	369	19	∅	∅	NOUN
ejpam-4596	369	20	for	for	ADP
ejpam-4596	369	21	all	all	DET
ejpam-4596	369	22	a	a	DET
ejpam-4596	369	23	∈	∈	PROPN
ejpam-4596	369	24	x	x	SYM
ejpam-4596	369	25	/	/	SYM
ejpam-4596	369	26	e.	e.	PROPN
ejpam-4596	369	27	proof	proof	PROPN
ejpam-4596	369	28	.	.	PUNCT
ejpam-4596	370	1	assume	assume	VERB
ejpam-4596	370	2	that	that	SCONJ
ejpam-4596	370	3	idx	idx	NOUN
ejpam-4596	370	4	∈	∈	PROPN
ejpam-4596	370	5	rreg(te∗(x	rreg(te∗(x	PROPN
ejpam-4596	370	6	,	,	PUNCT
ejpam-4596	370	7	θ	θ	NOUN
ejpam-4596	370	8	)	)	PUNCT
ejpam-4596	370	9	)	)	PUNCT
ejpam-4596	370	10	.	.	PUNCT
ejpam-4596	371	1	by	by	ADP
ejpam-4596	371	2	corollary	corollary	ADJ
ejpam-4596	371	3	3	3	NUM
ejpam-4596	371	4	,	,	PUNCT
ejpam-4596	371	5	we	we	PRON
ejpam-4596	371	6	get	get	VERB
ejpam-4596	371	7	that	that	SCONJ
ejpam-4596	371	8	θ	θ	PROPN
ejpam-4596	371	9	is	be	AUX
ejpam-4596	371	10	injective	injective	ADJ
ejpam-4596	371	11	.	.	PUNCT
ejpam-4596	372	1	suppose	suppose	VERB
ejpam-4596	372	2	that	that	SCONJ
ejpam-4596	372	3	a	a	DET
ejpam-4596	372	4	∩	∩	ADJ
ejpam-4596	372	5	xθ2	xθ2	NOUN
ejpam-4596	372	6	=	=	NOUN
ejpam-4596	372	7	∅	∅	NOUN
ejpam-4596	372	8	for	for	ADP
ejpam-4596	372	9	some	some	DET
ejpam-4596	372	10	a	a	DET
ejpam-4596	372	11	∈	∈	PROPN
ejpam-4596	372	12	x	x	X
ejpam-4596	372	13	/	/	SYM
ejpam-4596	372	14	e.	e.	PROPN
ejpam-4596	372	15	by	by	ADP
ejpam-4596	372	16	(	(	PUNCT
ejpam-4596	372	17	2	2	NUM
ejpam-4596	372	18	)	)	PUNCT
ejpam-4596	372	19	of	of	ADP
ejpam-4596	372	20	theorem	theorem	NOUN
ejpam-4596	372	21	8	8	NUM
ejpam-4596	372	22	,	,	PUNCT
ejpam-4596	372	23	there	there	PRON
ejpam-4596	372	24	exists	exist	VERB
ejpam-4596	372	25	an	an	DET
ejpam-4596	372	26	injection	injection	NOUN
ejpam-4596	372	27	φ	φ	NOUN
ejpam-4596	372	28	:	:	PUNCT
ejpam-4596	372	29	{	{	PUNCT
ejpam-4596	372	30	a	a	DET
ejpam-4596	372	31	∈	∈	PROPN
ejpam-4596	372	32	x	x	SYM
ejpam-4596	372	33	/	/	SYM
ejpam-4596	372	34	e	e	NOUN
ejpam-4596	372	35	:	:	PUNCT
ejpam-4596	372	36	a	a	DET
ejpam-4596	372	37	∩x(idxθ)2	∩x(idxθ)2	PROPN
ejpam-4596	372	38	=	=	SYM
ejpam-4596	372	39	∅	∅	NOUN
ejpam-4596	372	40	}	}	PUNCT
ejpam-4596	372	41	→	→	SYM
ejpam-4596	372	42	{	{	PUNCT
ejpam-4596	372	43	a	a	DET
ejpam-4596	372	44	∈	∈	PROPN
ejpam-4596	372	45	x	x	SYM
ejpam-4596	372	46	/	/	SYM
ejpam-4596	372	47	e	e	NOUN
ejpam-4596	372	48	:	:	PUNCT
ejpam-4596	372	49	a	a	DET
ejpam-4596	372	50	∩xidx	∩xidx	NOUN
ejpam-4596	372	51	=	=	SYM
ejpam-4596	372	52	∅	∅	NOUN
ejpam-4596	372	53	}	}	PUNCT
ejpam-4596	372	54	.	.	PUNCT
ejpam-4596	373	1	this	this	PRON
ejpam-4596	373	2	is	be	AUX
ejpam-4596	373	3	a	a	DET
ejpam-4596	373	4	contradiction	contradiction	NOUN
ejpam-4596	373	5	with	with	ADP
ejpam-4596	373	6	{	{	PUNCT
ejpam-4596	373	7	a	a	DET
ejpam-4596	373	8	∈	∈	NOUN
ejpam-4596	373	9	x	x	SYM
ejpam-4596	373	10	/	/	SYM
ejpam-4596	373	11	e	e	NOUN
ejpam-4596	373	12	:	:	PUNCT
ejpam-4596	373	13	a	a	DET
ejpam-4596	373	14	∩	∩	ADJ
ejpam-4596	373	15	xidx	xidx	NOUN
ejpam-4596	373	16	=	=	VERB
ejpam-4596	373	17	∅	∅	NOUN
ejpam-4596	373	18	}	}	PUNCT
ejpam-4596	373	19	=	=	SYM
ejpam-4596	373	20	{	{	PUNCT
ejpam-4596	373	21	a	a	DET
ejpam-4596	373	22	∈	∈	PROPN
ejpam-4596	373	23	x	x	SYM
ejpam-4596	373	24	/	/	SYM
ejpam-4596	373	25	e	e	NOUN
ejpam-4596	373	26	:	:	PUNCT
ejpam-4596	373	27	a	a	DET
ejpam-4596	373	28	∩	∩	X
ejpam-4596	373	29	x	x	SYM
ejpam-4596	373	30	=	=	VERB
ejpam-4596	373	31	∅	∅	NOUN
ejpam-4596	373	32	}	}	PUNCT
ejpam-4596	373	33	=	=	PUNCT
ejpam-4596	373	34	∅.	∅.	AUX
ejpam-4596	373	35	thus	thus	ADV
ejpam-4596	373	36	a∩xθ2	a∩xθ2	VERB
ejpam-4596	373	37	̸=	̸=	PROPN
ejpam-4596	373	38	∅	∅	NOUN
ejpam-4596	373	39	for	for	ADP
ejpam-4596	373	40	all	all	DET
ejpam-4596	373	41	a	a	DET
ejpam-4596	373	42	∈	∈	NOUN
ejpam-4596	373	43	x	x	X
ejpam-4596	373	44	/	/	SYM
ejpam-4596	373	45	e.	e.	PROPN
ejpam-4596	373	46	it	it	PRON
ejpam-4596	373	47	follows	follow	VERB
ejpam-4596	373	48	from	from	ADP
ejpam-4596	373	49	xθ2	xθ2	PROPN
ejpam-4596	373	50	⊆	⊆	NUM
ejpam-4596	373	51	xθ	xθ	PROPN
ejpam-4596	373	52	that	that	SCONJ
ejpam-4596	373	53	a∩xθ	a∩xθ	PROPN
ejpam-4596	373	54	̸=	̸=	PROPN
ejpam-4596	373	55	∅	∅	NOUN
ejpam-4596	373	56	for	for	ADP
ejpam-4596	373	57	all	all	DET
ejpam-4596	373	58	a	a	DET
ejpam-4596	373	59	∈	∈	NOUN
ejpam-4596	373	60	x	x	X
ejpam-4596	373	61	/	/	SYM
ejpam-4596	373	62	e.	e.	PROPN
ejpam-4596	373	63	the	the	DET
ejpam-4596	373	64	converse	converse	NOUN
ejpam-4596	373	65	of	of	ADP
ejpam-4596	373	66	corollary	corollary	NOUN
ejpam-4596	373	67	follows	follow	VERB
ejpam-4596	373	68	from	from	ADP
ejpam-4596	373	69	theorem	theorem	ADJ
ejpam-4596	373	70	8	8	NUM
ejpam-4596	373	71	.	.	PUNCT
ejpam-4596	374	1	the	the	DET
ejpam-4596	374	2	following	following	ADJ
ejpam-4596	374	3	result	result	NOUN
ejpam-4596	374	4	is	be	AUX
ejpam-4596	374	5	obtained	obtain	VERB
ejpam-4596	374	6	directly	directly	ADV
ejpam-4596	374	7	from	from	ADP
ejpam-4596	374	8	theorem	theorem	ADJ
ejpam-4596	374	9	7	7	NUM
ejpam-4596	374	10	and	and	CCONJ
ejpam-4596	374	11	theorem	theorem	VERB
ejpam-4596	374	12	8	8	NUM
ejpam-4596	374	13	.	.	PUNCT
ejpam-4596	374	14	theorem	theorem	NOUN
ejpam-4596	374	15	9	9	NUM
ejpam-4596	374	16	.	.	PUNCT
ejpam-4596	375	1	let	let	VERB
ejpam-4596	375	2	α	α	PRON
ejpam-4596	375	3	∈	∈	PROPN
ejpam-4596	375	4	te∗(x	te∗(x	NOUN
ejpam-4596	375	5	,	,	PUNCT
ejpam-4596	375	6	θ	θ	NOUN
ejpam-4596	375	7	)	)	PUNCT
ejpam-4596	375	8	.	.	PUNCT
ejpam-4596	376	1	then	then	ADV
ejpam-4596	376	2	α	α	PROPN
ejpam-4596	376	3	∈	∈	PROPN
ejpam-4596	376	4	creg(te∗(x	creg(te∗(x	PROPN
ejpam-4596	376	5	,	,	PUNCT
ejpam-4596	376	6	θ	θ	NOUN
ejpam-4596	376	7	)	)	PUNCT
ejpam-4596	376	8	)	)	PUNCT
ejpam-4596	377	1	if	if	SCONJ
ejpam-4596	377	2	and	and	CCONJ
ejpam-4596	377	3	only	only	ADV
ejpam-4596	377	4	if	if	SCONJ
ejpam-4596	377	5	(	(	PUNCT
ejpam-4596	377	6	i	i	NOUN
ejpam-4596	377	7	)	)	PUNCT
ejpam-4596	377	8	for	for	ADP
ejpam-4596	377	9	every	every	DET
ejpam-4596	377	10	p	p	PROPN
ejpam-4596	377	11	∈	∈	PROPN
ejpam-4596	377	12	π(α	π(α	PROPN
ejpam-4596	377	13	)	)	PUNCT
ejpam-4596	377	14	,	,	PUNCT
ejpam-4596	377	15	p	p	X
ejpam-4596	377	16	∩xθαθ	∩xθαθ	NUM
ejpam-4596	377	17	̸=	̸=	PROPN
ejpam-4596	377	18	∅	∅	NOUN
ejpam-4596	377	19	,	,	PUNCT
ejpam-4596	377	20	(	(	PUNCT
ejpam-4596	377	21	ii	ii	NOUN
ejpam-4596	377	22	)	)	PUNCT
ejpam-4596	377	23	(	(	PUNCT
ejpam-4596	377	24	θαθ)|xα	θαθ)|xα	PROPN
ejpam-4596	377	25	is	be	AUX
ejpam-4596	377	26	an	an	DET
ejpam-4596	377	27	injection	injection	NOUN
ejpam-4596	377	28	and	and	CCONJ
ejpam-4596	377	29	(	(	PUNCT
ejpam-4596	377	30	iii	iii	X
ejpam-4596	377	31	)	)	PUNCT
ejpam-4596	377	32	if	if	SCONJ
ejpam-4596	377	33	there	there	PRON
ejpam-4596	377	34	exists	exist	VERB
ejpam-4596	377	35	a	a	DET
ejpam-4596	377	36	∈	∈	PROPN
ejpam-4596	377	37	x	x	SYM
ejpam-4596	377	38	/	/	SYM
ejpam-4596	377	39	e	e	VERB
ejpam-4596	377	40	such	such	ADJ
ejpam-4596	377	41	that	that	SCONJ
ejpam-4596	377	42	a	a	DET
ejpam-4596	377	43	∩	∩	NOUN
ejpam-4596	377	44	x(αθ)2	x(αθ)2	NUM
ejpam-4596	377	45	=	=	SYM
ejpam-4596	377	46	∅	∅	NOUN
ejpam-4596	377	47	,	,	PUNCT
ejpam-4596	377	48	then	then	ADV
ejpam-4596	377	49	there	there	PRON
ejpam-4596	377	50	exists	exist	VERB
ejpam-4596	377	51	an	an	DET
ejpam-4596	377	52	injection	injection	NOUN
ejpam-4596	377	53	φ	φ	NOUN
ejpam-4596	377	54	:	:	PUNCT
ejpam-4596	377	55	{	{	PUNCT
ejpam-4596	377	56	a	a	DET
ejpam-4596	377	57	∈	∈	PROPN
ejpam-4596	377	58	x	x	SYM
ejpam-4596	377	59	/	/	SYM
ejpam-4596	377	60	e	e	NOUN
ejpam-4596	377	61	:	:	PUNCT
ejpam-4596	377	62	a	a	DET
ejpam-4596	377	63	∩x(αθ)2	∩x(αθ)2	NOUN
ejpam-4596	377	64	=	=	SYM
ejpam-4596	377	65	∅	∅	NOUN
ejpam-4596	377	66	}	}	PUNCT
ejpam-4596	377	67	→	→	SYM
ejpam-4596	377	68	{	{	PUNCT
ejpam-4596	377	69	a	a	PRON
ejpam-4596	377	70	∈	∈	PROPN
ejpam-4596	377	71	x	x	SYM
ejpam-4596	377	72	/	/	SYM
ejpam-4596	377	73	e	e	NOUN
ejpam-4596	377	74	:	:	PUNCT
ejpam-4596	377	75	a	a	DET
ejpam-4596	377	76	∩xα	∩xα	NOUN
ejpam-4596	377	77	=	=	NOUN
ejpam-4596	377	78	∅	∅	NOUN
ejpam-4596	377	79	}	}	PUNCT
ejpam-4596	377	80	.	.	PUNCT
ejpam-4596	378	1	corollary	corollary	ADJ
ejpam-4596	378	2	8	8	NUM
ejpam-4596	378	3	.	.	PUNCT
ejpam-4596	379	1	idx	idx	PROPN
ejpam-4596	379	2	∈	∈	PROPN
ejpam-4596	379	3	creg(te∗(x	creg(te∗(x	PROPN
ejpam-4596	379	4	,	,	PUNCT
ejpam-4596	379	5	θ	θ	NOUN
ejpam-4596	379	6	)	)	PUNCT
ejpam-4596	379	7	)	)	PUNCT
ejpam-4596	380	1	if	if	SCONJ
ejpam-4596	380	2	and	and	CCONJ
ejpam-4596	380	3	only	only	ADV
ejpam-4596	380	4	if	if	SCONJ
ejpam-4596	380	5	θ	θ	PROPN
ejpam-4596	380	6	is	be	AUX
ejpam-4596	380	7	a	a	DET
ejpam-4596	380	8	bijection	bijection	NOUN
ejpam-4596	380	9	.	.	PUNCT
ejpam-4596	381	1	references	reference	NOUN
ejpam-4596	381	2	2125	2125	NUM
ejpam-4596	381	3	4	4	NUM
ejpam-4596	381	4	.	.	PUNCT
ejpam-4596	381	5	conclusion	conclusion	NOUN
ejpam-4596	381	6	and	and	CCONJ
ejpam-4596	381	7	discussion	discussion	NOUN
ejpam-4596	381	8	in	in	ADP
ejpam-4596	381	9	this	this	DET
ejpam-4596	381	10	work	work	NOUN
ejpam-4596	381	11	,	,	PUNCT
ejpam-4596	381	12	we	we	PRON
ejpam-4596	381	13	presented	present	VERB
ejpam-4596	381	14	necessary	necessary	ADJ
ejpam-4596	381	15	and	and	CCONJ
ejpam-4596	381	16	sufficient	sufficient	ADJ
ejpam-4596	381	17	conditions	condition	NOUN
ejpam-4596	381	18	when	when	SCONJ
ejpam-4596	381	19	elements	element	NOUN
ejpam-4596	381	20	of	of	ADP
ejpam-4596	381	21	the	the	DET
ejpam-4596	381	22	semigroups	semigroup	NOUN
ejpam-4596	381	23	te(x	te(x	PROPN
ejpam-4596	381	24	,	,	PUNCT
ejpam-4596	381	25	θ	θ	NOUN
ejpam-4596	381	26	)	)	PUNCT
ejpam-4596	381	27	and	and	CCONJ
ejpam-4596	381	28	te∗(x	te∗(x	NOUN
ejpam-4596	381	29	,	,	PUNCT
ejpam-4596	381	30	θ	θ	NOUN
ejpam-4596	381	31	)	)	PUNCT
ejpam-4596	381	32	to	to	PART
ejpam-4596	381	33	be	be	AUX
ejpam-4596	381	34	regular	regular	ADJ
ejpam-4596	381	35	,	,	PUNCT
ejpam-4596	381	36	left	leave	VERB
ejpam-4596	381	37	regular	regular	ADV
ejpam-4596	381	38	,	,	PUNCT
ejpam-4596	381	39	right	right	ADV
ejpam-4596	381	40	regular	regular	ADJ
ejpam-4596	381	41	and	and	CCONJ
ejpam-4596	381	42	completely	completely	ADV
ejpam-4596	381	43	regular	regular	ADJ
ejpam-4596	381	44	.	.	PUNCT
ejpam-4596	382	1	the	the	DET
ejpam-4596	382	2	identity	identity	NOUN
ejpam-4596	382	3	transformation	transformation	NOUN
ejpam-4596	382	4	is	be	AUX
ejpam-4596	382	5	a	a	DET
ejpam-4596	382	6	regular	regular	ADJ
ejpam-4596	382	7	,	,	PUNCT
ejpam-4596	382	8	a	a	DET
ejpam-4596	382	9	left	left	ADJ
ejpam-4596	382	10	regular	regular	NOUN
ejpam-4596	382	11	,	,	PUNCT
ejpam-4596	382	12	a	a	DET
ejpam-4596	382	13	right	right	NOUN
ejpam-4596	382	14	regular	regular	ADJ
ejpam-4596	382	15	and	and	CCONJ
ejpam-4596	382	16	a	a	DET
ejpam-4596	382	17	completely	completely	ADV
ejpam-4596	382	18	regular	regular	ADJ
ejpam-4596	382	19	element	element	NOUN
ejpam-4596	382	20	for	for	ADP
ejpam-4596	382	21	the	the	DET
ejpam-4596	382	22	semigroups	semigroup	NOUN
ejpam-4596	382	23	te(x	te(x	PROPN
ejpam-4596	382	24	)	)	PUNCT
ejpam-4596	382	25	and	and	CCONJ
ejpam-4596	382	26	te∗(x	te∗(x	NOUN
ejpam-4596	382	27	)	)	PUNCT
ejpam-4596	382	28	but	but	CCONJ
ejpam-4596	382	29	need	need	VERB
ejpam-4596	382	30	not	not	PART
ejpam-4596	382	31	for	for	ADP
ejpam-4596	382	32	te(x	te(x	NOUN
ejpam-4596	382	33	,	,	PUNCT
ejpam-4596	382	34	θ	θ	NOUN
ejpam-4596	382	35	)	)	PUNCT
ejpam-4596	382	36	and	and	CCONJ
ejpam-4596	382	37	te∗(x	te∗(x	NOUN
ejpam-4596	382	38	,	,	PUNCT
ejpam-4596	382	39	θ	θ	NOUN
ejpam-4596	382	40	)	)	PUNCT
ejpam-4596	382	41	.	.	PUNCT
ejpam-4596	383	1	hence	hence	ADV
ejpam-4596	383	2	,	,	PUNCT
ejpam-4596	383	3	we	we	PRON
ejpam-4596	383	4	presented	present	VERB
ejpam-4596	383	5	properties	property	NOUN
ejpam-4596	383	6	for	for	ADP
ejpam-4596	383	7	θ	θ	PROPN
ejpam-4596	383	8	that	that	SCONJ
ejpam-4596	383	9	the	the	DET
ejpam-4596	383	10	identity	identity	NOUN
ejpam-4596	383	11	transformation	transformation	NOUN
ejpam-4596	383	12	is	be	AUX
ejpam-4596	383	13	regular	regular	ADJ
ejpam-4596	383	14	,	,	PUNCT
ejpam-4596	383	15	left	leave	VERB
ejpam-4596	383	16	regular	regular	ADV
ejpam-4596	383	17	,	,	PUNCT
ejpam-4596	383	18	right	right	ADV
ejpam-4596	383	19	regular	regular	ADJ
ejpam-4596	383	20	and	and	CCONJ
ejpam-4596	383	21	completely	completely	ADV
ejpam-4596	383	22	regular	regular	ADJ
ejpam-4596	383	23	.	.	PUNCT
ejpam-4596	384	1	the	the	DET
ejpam-4596	384	2	important	important	ADJ
ejpam-4596	384	3	theory	theory	NOUN
ejpam-4596	384	4	in	in	ADP
ejpam-4596	384	5	algebraic	algebraic	ADJ
ejpam-4596	384	6	semigroups	semigroup	NOUN
ejpam-4596	384	7	theory	theory	NOUN
ejpam-4596	384	8	is	be	AUX
ejpam-4596	384	9	cayley	cayley	NOUN
ejpam-4596	384	10	’s	’s	PART
ejpam-4596	384	11	theorem	theorem	NOUN
ejpam-4596	384	12	for	for	ADP
ejpam-4596	384	13	semigroups	semigroup	NOUN
ejpam-4596	384	14	that	that	SCONJ
ejpam-4596	384	15	every	every	DET
ejpam-4596	384	16	semigroup	semigroup	NOUN
ejpam-4596	384	17	is	be	AUX
ejpam-4596	384	18	isomorphic	isomorphic	ADJ
ejpam-4596	384	19	to	to	ADP
ejpam-4596	384	20	a	a	DET
ejpam-4596	384	21	subsemigroup	subsemigroup	NOUN
ejpam-4596	384	22	of	of	ADP
ejpam-4596	384	23	some	some	DET
ejpam-4596	384	24	full	full	ADJ
ejpam-4596	384	25	transformation	transformation	NOUN
ejpam-4596	384	26	semigroup	semigroup	NOUN
ejpam-4596	384	27	.	.	PUNCT
ejpam-4596	385	1	hence	hence	ADV
ejpam-4596	385	2	in	in	ADP
ejpam-4596	385	3	order	order	NOUN
ejpam-4596	385	4	to	to	PART
ejpam-4596	385	5	study	study	VERB
ejpam-4596	385	6	structure	structure	NOUN
ejpam-4596	385	7	of	of	ADP
ejpam-4596	385	8	semigroups	semigroup	NOUN
ejpam-4596	385	9	,	,	PUNCT
ejpam-4596	385	10	it	it	PRON
ejpam-4596	385	11	suffices	suffice	VERB
ejpam-4596	385	12	to	to	PART
ejpam-4596	385	13	consider	consider	VERB
ejpam-4596	385	14	subsemigroups	subsemigroup	NOUN
ejpam-4596	385	15	of	of	ADP
ejpam-4596	385	16	the	the	DET
ejpam-4596	385	17	full	full	ADJ
ejpam-4596	385	18	transformation	transformation	NOUN
ejpam-4596	385	19	semigroup	semigroup	NOUN
ejpam-4596	385	20	.	.	PUNCT
ejpam-4596	386	1	therefore	therefore	ADV
ejpam-4596	386	2	,	,	PUNCT
ejpam-4596	386	3	several	several	ADJ
ejpam-4596	386	4	researchers	researcher	NOUN
ejpam-4596	386	5	are	be	AUX
ejpam-4596	386	6	interested	interested	ADJ
ejpam-4596	386	7	in	in	ADP
ejpam-4596	386	8	the	the	DET
ejpam-4596	386	9	characterizations	characterization	NOUN
ejpam-4596	386	10	of	of	ADP
ejpam-4596	386	11	subsemigroups	subsemigroup	NOUN
ejpam-4596	386	12	of	of	ADP
ejpam-4596	386	13	the	the	DET
ejpam-4596	386	14	full	full	ADJ
ejpam-4596	386	15	transformation	transformation	NOUN
ejpam-4596	386	16	semigroup	semigroup	NOUN
ejpam-4596	386	17	.	.	PUNCT
ejpam-4596	387	1	in	in	ADP
ejpam-4596	387	2	future	future	ADJ
ejpam-4596	387	3	work	work	NOUN
ejpam-4596	387	4	,	,	PUNCT
ejpam-4596	387	5	we	we	PRON
ejpam-4596	387	6	intend	intend	VERB
ejpam-4596	387	7	to	to	PART
ejpam-4596	387	8	expand	expand	VERB
ejpam-4596	387	9	other	other	ADJ
ejpam-4596	387	10	algebraic	algebraic	ADJ
ejpam-4596	387	11	structures	structure	NOUN
ejpam-4596	387	12	for	for	ADP
ejpam-4596	387	13	the	the	DET
ejpam-4596	387	14	variants	variant	NOUN
ejpam-4596	387	15	of	of	ADP
ejpam-4596	387	16	transformation	transformation	NOUN
ejpam-4596	387	17	semigroups	semigroup	NOUN
ejpam-4596	387	18	that	that	PRON
ejpam-4596	387	19	preserve	preserve	VERB
ejpam-4596	387	20	an	an	DET
ejpam-4596	387	21	equivalence	equivalence	NOUN
ejpam-4596	387	22	relation	relation	NOUN
ejpam-4596	387	23	and	and	CCONJ
ejpam-4596	387	24	other	other	ADJ
ejpam-4596	387	25	transformation	transformation	NOUN
ejpam-4596	387	26	semigroups	semigroup	NOUN
ejpam-4596	387	27	.	.	PUNCT
ejpam-4596	388	1	references	reference	NOUN
ejpam-4596	388	2	[	[	X
ejpam-4596	388	3	1	1	X
ejpam-4596	388	4	]	]	PUNCT
ejpam-4596	388	5	ah	ah	INTJ
ejpam-4596	388	6	clifford	clifford	PROPN
ejpam-4596	388	7	.	.	PUNCT
ejpam-4596	389	1	semigroups	semigroup	NOUN
ejpam-4596	389	2	admitting	admit	VERB
ejpam-4596	389	3	relative	relative	ADJ
ejpam-4596	389	4	inverses	inverse	NOUN
ejpam-4596	389	5	.	.	PUNCT
ejpam-4596	390	1	annals	annal	NOUN
ejpam-4596	390	2	of	of	ADP
ejpam-4596	390	3	mathematics	mathematic	NOUN
ejpam-4596	390	4	,	,	PUNCT
ejpam-4596	390	5	42:1037	42:1037	NUM
ejpam-4596	390	6	–	–	PUNCT
ejpam-4596	390	7	1049	1049	NUM
ejpam-4596	390	8	,	,	PUNCT
ejpam-4596	390	9	1941	1941	NUM
ejpam-4596	390	10	.	.	PUNCT
ejpam-4596	391	1	[	[	X
ejpam-4596	391	2	2	2	NUM
ejpam-4596	391	3	]	]	X
ejpam-4596	391	4	r	r	NOUN
ejpam-4596	391	5	croisot	croisot	NOUN
ejpam-4596	391	6	.	.	PUNCT
ejpam-4596	392	1	demi	demi	PROPN
ejpam-4596	392	2	-	-	PUNCT
ejpam-4596	392	3	groupes	groupe	NOUN
ejpam-4596	392	4	inversifs	inversifs	PROPN
ejpam-4596	392	5	et	et	PROPN
ejpam-4596	392	6	demi	demi	PROPN
ejpam-4596	392	7	-	-	PUNCT
ejpam-4596	392	8	groupes	groupes	PROPN
ejpam-4596	392	9	reunions	reunion	NOUN
ejpam-4596	392	10	de	de	X
ejpam-4596	392	11	demi	demi	X
ejpam-4596	392	12	-	-	PUNCT
ejpam-4596	392	13	groupes	groupe	NOUN
ejpam-4596	392	14	simples	simple	NOUN
ejpam-4596	392	15	.	.	PUNCT
ejpam-4596	393	1	annales	annale	NOUN
ejpam-4596	393	2	scientifiques	scientifique	NOUN
ejpam-4596	393	3	de	de	ADP
ejpam-4596	393	4	l’école	l’école	ADJ
ejpam-4596	393	5	normale	normale	PROPN
ejpam-4596	393	6	supérieure	supérieure	PROPN
ejpam-4596	393	7	,	,	PUNCT
ejpam-4596	393	8	70(4):361–379	70(4):361–379	PROPN
ejpam-4596	393	9	,	,	PUNCT
ejpam-4596	393	10	1953	1953	NUM
ejpam-4596	393	11	.	.	PUNCT
ejpam-4596	394	1	[	[	X
ejpam-4596	394	2	3	3	NUM
ejpam-4596	394	3	]	]	X
ejpam-4596	394	4	l	l	PROPN
ejpam-4596	394	5	deng	deng	PROPN
ejpam-4596	394	6	.	.	PUNCT
ejpam-4596	395	1	on	on	ADP
ejpam-4596	395	2	certain	certain	ADJ
ejpam-4596	395	3	semigroups	semigroup	NOUN
ejpam-4596	395	4	of	of	ADP
ejpam-4596	395	5	transformations	transformation	NOUN
ejpam-4596	395	6	that	that	PRON
ejpam-4596	395	7	preserve	preserve	VERB
ejpam-4596	395	8	double	double	ADJ
ejpam-4596	395	9	direction	direction	NOUN
ejpam-4596	395	10	equivalence	equivalence	NOUN
ejpam-4596	395	11	.	.	PUNCT
ejpam-4596	396	1	bulletin	bulletin	NOUN
ejpam-4596	396	2	of	of	ADP
ejpam-4596	396	3	the	the	DET
ejpam-4596	396	4	iranian	iranian	PROPN
ejpam-4596	396	5	mathematical	mathematical	ADJ
ejpam-4596	396	6	society	society	NOUN
ejpam-4596	396	7	,	,	PUNCT
ejpam-4596	396	8	42:1015–1024	42:1015–1024	NUM
ejpam-4596	396	9	,	,	PUNCT
ejpam-4596	396	10	2016	2016	NUM
ejpam-4596	396	11	.	.	PUNCT
ejpam-4596	397	1	[	[	X
ejpam-4596	397	2	4	4	X
ejpam-4596	397	3	]	]	PUNCT
ejpam-4596	397	4	ja	ja	PROPN
ejpam-4596	397	5	green	green	PROPN
ejpam-4596	397	6	.	.	PUNCT
ejpam-4596	398	1	on	on	ADP
ejpam-4596	398	2	the	the	DET
ejpam-4596	398	3	structure	structure	NOUN
ejpam-4596	398	4	of	of	ADP
ejpam-4596	398	5	semigroups	semigroup	NOUN
ejpam-4596	398	6	.	.	PUNCT
ejpam-4596	399	1	annals	annal	NOUN
ejpam-4596	399	2	of	of	ADP
ejpam-4596	399	3	mathematics	mathematic	NOUN
ejpam-4596	399	4	,	,	PUNCT
ejpam-4596	399	5	54:163–172	54:163–172	PROPN
ejpam-4596	399	6	,	,	PUNCT
ejpam-4596	399	7	1951	1951	NUM
ejpam-4596	399	8	.	.	PUNCT
ejpam-4596	400	1	[	[	X
ejpam-4596	400	2	5	5	NUM
ejpam-4596	400	3	]	]	X
ejpam-4596	400	4	jb	jb	PROPN
ejpam-4596	400	5	hickey	hickey	PROPN
ejpam-4596	400	6	.	.	PUNCT
ejpam-4596	401	1	semigroups	semigroups	X
ejpam-4596	401	2	under	under	ADP
ejpam-4596	401	3	a	a	DET
ejpam-4596	401	4	sandwich	sandwich	NOUN
ejpam-4596	401	5	operation	operation	NOUN
ejpam-4596	401	6	.	.	PUNCT
ejpam-4596	402	1	proceeding	proceed	VERB
ejpam-4596	402	2	of	of	ADP
ejpam-4596	402	3	the	the	DET
ejpam-4596	402	4	edinburgh	edinburgh	PROPN
ejpam-4596	402	5	mathematical	mathematical	PROPN
ejpam-4596	402	6	society	society	NOUN
ejpam-4596	402	7	,	,	PUNCT
ejpam-4596	402	8	26:371–382	26:371–382	NUM
ejpam-4596	402	9	,	,	PUNCT
ejpam-4596	402	10	1983	1983	NUM
ejpam-4596	402	11	.	.	PUNCT
ejpam-4596	403	1	[	[	X
ejpam-4596	403	2	6	6	NUM
ejpam-4596	403	3	]	]	X
ejpam-4596	403	4	jr	jr	PROPN
ejpam-4596	403	5	kd	kd	PROPN
ejpam-4596	403	6	magill	magill	PROPN
ejpam-4596	403	7	and	and	CCONJ
ejpam-4596	403	8	s	s	VERB
ejpam-4596	403	9	subbiah	subbiah	NOUN
ejpam-4596	403	10	.	.	PUNCT
ejpam-4596	404	1	green	green	PROPN
ejpam-4596	404	2	’s	’s	PART
ejpam-4596	404	3	relations	relation	NOUN
ejpam-4596	404	4	for	for	ADP
ejpam-4596	404	5	regular	regular	ADJ
ejpam-4596	404	6	elements	element	NOUN
ejpam-4596	404	7	of	of	ADP
ejpam-4596	404	8	sandwich	sandwich	NOUN
ejpam-4596	404	9	semigroups	semigroup	NOUN
ejpam-4596	404	10	.	.	PUNCT
ejpam-4596	405	1	ii	ii	PROPN
ejpam-4596	405	2	.	.	PUNCT
ejpam-4596	406	1	semigroups	semigroup	NOUN
ejpam-4596	406	2	of	of	ADP
ejpam-4596	406	3	continuous	continuous	ADJ
ejpam-4596	406	4	functions	function	NOUN
ejpam-4596	406	5	.	.	PUNCT
ejpam-4596	407	1	journal	journal	NOUN
ejpam-4596	407	2	of	of	ADP
ejpam-4596	407	3	the	the	DET
ejpam-4596	407	4	australian	australian	ADJ
ejpam-4596	407	5	mathematical	mathematical	ADJ
ejpam-4596	407	6	society	society	NOUN
ejpam-4596	407	7	.	.	PUNCT
ejpam-4596	408	1	series	series	PROPN
ejpam-4596	408	2	a	a	PROPN
ejpam-4596	408	3	,	,	PUNCT
ejpam-4596	408	4	25:45–65	25:45–65	PROPN
ejpam-4596	408	5	,	,	PUNCT
ejpam-4596	408	6	1978	1978	NUM
ejpam-4596	408	7	.	.	PUNCT
ejpam-4596	409	1	[	[	X
ejpam-4596	409	2	7	7	X
ejpam-4596	409	3	]	]	X
ejpam-4596	409	4	j	j	PROPN
ejpam-4596	409	5	zeng	zeng	PROPN
ejpam-4596	409	6	l	l	PROPN
ejpam-4596	409	7	deng	deng	PROPN
ejpam-4596	409	8	and	and	CCONJ
ejpam-4596	409	9	b	b	PROPN
ejpam-4596	409	10	xu	xu	PROPN
ejpam-4596	409	11	.	.	PUNCT
ejpam-4596	410	1	green	green	PROPN
ejpam-4596	410	2	’s	’s	PART
ejpam-4596	410	3	relations	relation	NOUN
ejpam-4596	410	4	and	and	CCONJ
ejpam-4596	410	5	regularity	regularity	NOUN
ejpam-4596	410	6	for	for	ADP
ejpam-4596	410	7	semigroups	semigroup	NOUN
ejpam-4596	410	8	of	of	ADP
ejpam-4596	410	9	transformations	transformation	NOUN
ejpam-4596	410	10	that	that	PRON
ejpam-4596	410	11	preserve	preserve	VERB
ejpam-4596	410	12	double	double	ADJ
ejpam-4596	410	13	direction	direction	NOUN
ejpam-4596	410	14	equivalence	equivalence	NOUN
ejpam-4596	410	15	.	.	PUNCT
ejpam-4596	411	1	semigroup	semigroup	PROPN
ejpam-4596	411	2	forum	forum	PROPN
ejpam-4596	411	3	,	,	PUNCT
ejpam-4596	411	4	80:416–425	80:416–425	PROPN
ejpam-4596	411	5	,	,	PUNCT
ejpam-4596	411	6	2010	2010	NUM
ejpam-4596	411	7	.	.	PUNCT
ejpam-4596	412	1	[	[	X
ejpam-4596	412	2	8	8	NUM
ejpam-4596	412	3	]	]	X
ejpam-4596	412	4	e	e	NOUN
ejpam-4596	412	5	laysirikul	laysirikul	NOUN
ejpam-4596	412	6	and	and	CCONJ
ejpam-4596	412	7	c	c	PROPN
ejpam-4596	412	8	namnak	namnak	PROPN
ejpam-4596	412	9	.	.	PUNCT
ejpam-4596	413	1	regularity	regularity	NOUN
ejpam-4596	413	2	conditions	condition	NOUN
ejpam-4596	413	3	for	for	ADP
ejpam-4596	413	4	semigroups	semigroup	NOUN
ejpam-4596	413	5	of	of	ADP
ejpam-4596	413	6	transformations	transformation	NOUN
ejpam-4596	413	7	that	that	PRON
ejpam-4596	413	8	preserving	preserve	VERB
ejpam-4596	413	9	double	double	ADJ
ejpam-4596	413	10	direction	direction	NOUN
ejpam-4596	413	11	equivalence	equivalence	NOUN
ejpam-4596	413	12	.	.	PUNCT
ejpam-4596	414	1	proceedings	proceeding	NOUN
ejpam-4596	414	2	of	of	ADP
ejpam-4596	414	3	the	the	DET
ejpam-4596	414	4	17th	17th	ADJ
ejpam-4596	414	5	annual	annual	ADJ
ejpam-4596	414	6	meeting	meeting	NOUN
ejpam-4596	414	7	in	in	ADP
ejpam-4596	414	8	mathematics	mathematic	NOUN
ejpam-4596	414	9	.	.	PUNCT
ejpam-4596	414	10	,	,	PUNCT
ejpam-4596	414	11	pages	page	NOUN
ejpam-4596	414	12	155–160	155–160	NUM
ejpam-4596	414	13	,	,	PUNCT
ejpam-4596	414	14	2012	2012	NUM
ejpam-4596	414	15	.	.	PUNCT
ejpam-4596	415	1	[	[	X
ejpam-4596	415	2	9	9	NUM
ejpam-4596	415	3	]	]	SYM
ejpam-4596	415	4	es	es	X
ejpam-4596	415	5	lyapin	lyapin	ADJ
ejpam-4596	415	6	.	.	PUNCT
ejpam-4596	416	1	semigroups	semigroup	NOUN
ejpam-4596	416	2	(	(	PUNCT
ejpam-4596	416	3	in	in	ADP
ejpam-4596	416	4	russian	russian	NOUN
ejpam-4596	416	5	)	)	PUNCT
ejpam-4596	416	6	.	.	PUNCT
ejpam-4596	417	1	gosudarstv	gosudarstv	NOUN
ejpam-4596	417	2	.	.	PUNCT
ejpam-4596	418	1	izdat	izdat	PROPN
ejpam-4596	418	2	.	.	PUNCT
ejpam-4596	419	1	fiz	fiz	PROPN
ejpam-4596	419	2	-	-	NOUN
ejpam-4596	419	3	mat	mat	NOUN
ejpam-4596	419	4	.	.	NOUN
ejpam-4596	419	5	lit	light	VERB
ejpam-4596	419	6	.	.	PUNCT
ejpam-4596	420	1	moscow	moscow	PROPN
ejpam-4596	420	2	,	,	PUNCT
ejpam-4596	420	3	1960	1960	NUM
ejpam-4596	420	4	.	.	PUNCT
ejpam-4596	421	1	references	reference	NOUN
ejpam-4596	421	2	2126	2126	NUM
ejpam-4596	421	3	[	[	X
ejpam-4596	421	4	10	10	NUM
ejpam-4596	421	5	]	]	X
ejpam-4596	421	6	kd	kd	PROPN
ejpam-4596	421	7	magill	magill	PROPN
ejpam-4596	421	8	.	.	PUNCT
ejpam-4596	422	1	semigroup	semigroup	PROPN
ejpam-4596	422	2	structures	structure	NOUN
ejpam-4596	422	3	for	for	ADP
ejpam-4596	422	4	families	family	NOUN
ejpam-4596	422	5	of	of	ADP
ejpam-4596	422	6	functions	function	NOUN
ejpam-4596	422	7	i.	i.	PROPN
ejpam-4596	422	8	some	some	DET
ejpam-4596	422	9	homomorphism	homomorphism	NOUN
ejpam-4596	422	10	theorems	theorem	NOUN
ejpam-4596	422	11	.	.	PROPN
ejpam-4596	422	12	journal	journal	PROPN
ejpam-4596	422	13	of	of	ADP
ejpam-4596	422	14	the	the	DET
ejpam-4596	422	15	australian	australian	ADJ
ejpam-4596	422	16	mathematical	mathematical	ADJ
ejpam-4596	422	17	society	society	NOUN
ejpam-4596	422	18	,	,	PUNCT
ejpam-4596	422	19	7:81–94	7:81–94	NUM
ejpam-4596	422	20	,	,	PUNCT
ejpam-4596	422	21	1967	1967	NUM
ejpam-4596	422	22	.	.	PUNCT
ejpam-4596	423	1	[	[	X
ejpam-4596	423	2	11	11	NUM
ejpam-4596	423	3	]	]	X
ejpam-4596	423	4	c	c	X
ejpam-4596	423	5	namnak	namnak	NOUN
ejpam-4596	423	6	and	and	CCONJ
ejpam-4596	423	7	e	e	PROPN
ejpam-4596	423	8	laysirikul	laysirikul	NOUN
ejpam-4596	423	9	.	.	PUNCT
ejpam-4596	424	1	regularity	regularity	NOUN
ejpam-4596	424	2	for	for	ADP
ejpam-4596	424	3	semigroups	semigroup	NOUN
ejpam-4596	424	4	of	of	ADP
ejpam-4596	424	5	transformations	transformation	NOUN
ejpam-4596	424	6	that	that	PRON
ejpam-4596	424	7	preserve	preserve	VERB
ejpam-4596	424	8	equivalence	equivalence	NOUN
ejpam-4596	424	9	.	.	PUNCT
ejpam-4596	425	1	jp	jp	PROPN
ejpam-4596	425	2	journal	journal	PROPN
ejpam-4596	425	3	of	of	ADP
ejpam-4596	425	4	algebra	algebra	PROPN
ejpam-4596	425	5	,	,	PUNCT
ejpam-4596	425	6	number	number	NOUN
ejpam-4596	425	7	theory	theory	NOUN
ejpam-4596	425	8	and	and	CCONJ
ejpam-4596	425	9	applications	application	NOUN
ejpam-4596	425	10	,	,	PUNCT
ejpam-4596	425	11	28:97	28:97	NUM
ejpam-4596	425	12	–	–	PUNCT
ejpam-4596	425	13	105	105	NUM
ejpam-4596	425	14	,	,	PUNCT
ejpam-4596	425	15	2013	2013	NUM
ejpam-4596	425	16	.	.	PUNCT
ejpam-4596	426	1	[	[	X
ejpam-4596	426	2	12	12	NUM
ejpam-4596	426	3	]	]	X
ejpam-4596	426	4	j	j	PROPN
ejpam-4596	426	5	neumann	neumann	PROPN
ejpam-4596	426	6	.	.	PUNCT
ejpam-4596	427	1	on	on	ADP
ejpam-4596	427	2	regular	regular	ADJ
ejpam-4596	427	3	rings	ring	NOUN
ejpam-4596	427	4	.	.	PUNCT
ejpam-4596	428	1	proceedings	proceeding	NOUN
ejpam-4596	428	2	of	of	ADP
ejpam-4596	428	3	the	the	DET
ejpam-4596	428	4	national	national	PROPN
ejpam-4596	428	5	academy	academy	PROPN
ejpam-4596	428	6	of	of	ADP
ejpam-4596	428	7	sciences	sciences	PROPN
ejpam-4596	428	8	of	of	ADP
ejpam-4596	428	9	the	the	DET
ejpam-4596	428	10	united	united	PROPN
ejpam-4596	428	11	states	states	PROPN
ejpam-4596	428	12	of	of	ADP
ejpam-4596	428	13	america	america	PROPN
ejpam-4596	428	14	,	,	PUNCT
ejpam-4596	428	15	22:707–713	22:707–713	NUM
ejpam-4596	428	16	,	,	PUNCT
ejpam-4596	428	17	1936	1936	NUM
ejpam-4596	428	18	.	.	PUNCT
ejpam-4596	429	1	[	[	X
ejpam-4596	429	2	13	13	NUM
ejpam-4596	429	3	]	]	X
ejpam-4596	429	4	h	h	PROPN
ejpam-4596	429	5	pei	pei	PROPN
ejpam-4596	429	6	.	.	PUNCT
ejpam-4596	430	1	regularity	regularity	NOUN
ejpam-4596	430	2	and	and	CCONJ
ejpam-4596	430	3	green	green	PROPN
ejpam-4596	430	4	’s	’s	PART
ejpam-4596	430	5	relations	relation	NOUN
ejpam-4596	430	6	for	for	ADP
ejpam-4596	430	7	semigroup	semigroup	NOUN
ejpam-4596	430	8	of	of	ADP
ejpam-4596	430	9	transformations	transformation	NOUN
ejpam-4596	430	10	that	that	PRON
ejpam-4596	430	11	preserve	preserve	VERB
ejpam-4596	430	12	an	an	DET
ejpam-4596	430	13	equivalence	equivalence	NOUN
ejpam-4596	430	14	.	.	PUNCT
ejpam-4596	431	1	communications	communication	NOUN
ejpam-4596	431	2	in	in	ADP
ejpam-4596	431	3	algebra	algebra	NOUN
ejpam-4596	431	4	,	,	PUNCT
ejpam-4596	431	5	14:497–503	14:497–503	PROPN
ejpam-4596	431	6	,	,	PUNCT
ejpam-4596	431	7	2005	2005	NUM
ejpam-4596	431	8	.	.	PUNCT
ejpam-4596	432	1	[	[	X
ejpam-4596	432	2	14	14	NUM
ejpam-4596	432	3	]	]	X
ejpam-4596	432	4	m	m	VERB
ejpam-4596	432	5	petrich	petrich	NOUN
ejpam-4596	432	6	and	and	CCONJ
ejpam-4596	432	7	nr	nr	PRON
ejpam-4596	432	8	reilly	reilly	ADV
ejpam-4596	432	9	.	.	PUNCT
ejpam-4596	433	1	completely	completely	ADV
ejpam-4596	433	2	regular	regular	ADJ
ejpam-4596	433	3	semigroups	semigroup	NOUN
ejpam-4596	433	4	.	.	PUNCT
ejpam-4596	434	1	wiley	wiley	PROPN
ejpam-4596	434	2	,	,	PUNCT
ejpam-4596	434	3	new	new	PROPN
ejpam-4596	434	4	york	york	PROPN
ejpam-4596	434	5	,	,	PUNCT
ejpam-4596	434	6	1999	1999	NUM
ejpam-4596	434	7	.	.	PUNCT
ejpam-4596	435	1	[	[	X
ejpam-4596	435	2	15	15	NUM
ejpam-4596	435	3	]	]	X
ejpam-4596	435	4	d	d	X
ejpam-4596	435	5	rees	rees	PROPN
ejpam-4596	435	6	.	.	PUNCT
ejpam-4596	436	1	on	on	ADP
ejpam-4596	436	2	semigroups	semigroup	NOUN
ejpam-4596	436	3	.	.	PUNCT
ejpam-4596	437	1	mathematical	mathematical	ADJ
ejpam-4596	437	2	proceedings	proceeding	NOUN
ejpam-4596	437	3	of	of	ADP
ejpam-4596	437	4	the	the	DET
ejpam-4596	437	5	cambridge	cambridge	PROPN
ejpam-4596	437	6	philosophical	philosophical	ADJ
ejpam-4596	437	7	society	society	NOUN
ejpam-4596	437	8	,	,	PUNCT
ejpam-4596	437	9	36:387–400	36:387–400	NUM
ejpam-4596	437	10	,	,	PUNCT
ejpam-4596	437	11	1940	1940	NUM
ejpam-4596	437	12	.	.	PUNCT
ejpam-4596	438	1	[	[	X
ejpam-4596	438	2	16	16	NUM
ejpam-4596	438	3	]	]	X
ejpam-4596	438	4	d	d	X
ejpam-4596	438	5	rees	rees	PROPN
ejpam-4596	438	6	.	.	PUNCT
ejpam-4596	439	1	note	note	VERB
ejpam-4596	439	2	on	on	ADP
ejpam-4596	439	3	semigroups	semigroup	NOUN
ejpam-4596	439	4	.	.	PUNCT
ejpam-4596	440	1	mathematical	mathematical	ADJ
ejpam-4596	440	2	proceedings	proceeding	NOUN
ejpam-4596	440	3	of	of	ADP
ejpam-4596	440	4	the	the	DET
ejpam-4596	440	5	cambridge	cambridge	PROPN
ejpam-4596	440	6	philosophical	philosophical	ADJ
ejpam-4596	440	7	society	society	NOUN
ejpam-4596	440	8	,	,	PUNCT
ejpam-4596	440	9	37(4	37(4	PROPN
ejpam-4596	440	10	)	)	PUNCT
ejpam-4596	440	11	,	,	PUNCT
ejpam-4596	440	12	1941	1941	NUM
ejpam-4596	440	13	.	.	PUNCT
ejpam-4596	441	1	[	[	X
ejpam-4596	441	2	17	17	NUM
ejpam-4596	441	3	]	]	X
ejpam-4596	441	4	g	g	PROPN
ejpam-4596	441	5	thierrin	thierrin	NOUN
ejpam-4596	441	6	.	.	PUNCT
ejpam-4596	442	1	sur	sur	PROPN
ejpam-4596	442	2	les	les	PROPN
ejpam-4596	442	3	elements	elements	PROPN
ejpam-4596	442	4	inversifs	inversifs	PROPN
ejpam-4596	442	5	et	et	PROPN
ejpam-4596	442	6	les	les	X
ejpam-4596	442	7	elements	element	NOUN
ejpam-4596	442	8	unitaires	unitaires	PROPN
ejpam-4596	442	9	dun	dun	PROPN
ejpam-4596	442	10	demi	demi	PROPN
ejpam-4596	442	11	-	-	PUNCT
ejpam-4596	442	12	groupe	groupe	ADJ
ejpam-4596	442	13	inversif	inversif	NOUN
ejpam-4596	442	14	.	.	PUNCT
ejpam-4596	443	1	proceedings	proceeding	NOUN
ejpam-4596	443	2	of	of	ADP
ejpam-4596	443	3	the	the	DET
ejpam-4596	443	4	academy	academy	NOUN
ejpam-4596	443	5	of	of	ADP
ejpam-4596	443	6	sciences	sciences	PROPN
ejpam-4596	443	7	,	,	PUNCT
ejpam-4596	443	8	234:33–34	234:33–34	NUM
ejpam-4596	443	9	,	,	PUNCT
ejpam-4596	443	10	1952	1952	NUM
ejpam-4596	443	11	.	.	PUNCT
ejpam-4596	444	1	[	[	X
ejpam-4596	444	2	18	18	NUM
ejpam-4596	444	3	]	]	SYM
ejpam-4596	444	4	w	w	NOUN
ejpam-4596	444	5	yonthanthum	yonthanthum	NOUN
ejpam-4596	444	6	.	.	PUNCT
ejpam-4596	445	1	regular	regular	ADJ
ejpam-4596	445	2	elements	element	NOUN
ejpam-4596	445	3	of	of	ADP
ejpam-4596	445	4	the	the	DET
ejpam-4596	445	5	variant	variant	ADJ
ejpam-4596	445	6	semigroups	semigroup	NOUN
ejpam-4596	445	7	of	of	ADP
ejpam-4596	445	8	transformations	transformation	NOUN
ejpam-4596	445	9	preserving	preserve	VERB
ejpam-4596	445	10	double	double	ADJ
ejpam-4596	445	11	direction	direction	NOUN
ejpam-4596	445	12	equivalences	equivalence	NOUN
ejpam-4596	445	13	.	.	PUNCT
ejpam-4596	446	1	thai	thai	PROPN
ejpam-4596	446	2	journal	journal	PROPN
ejpam-4596	446	3	of	of	ADP
ejpam-4596	446	4	mathematics	mathematic	NOUN
ejpam-4596	446	5	,	,	PUNCT
ejpam-4596	446	6	16:165–171	16:165–171	NUM
ejpam-4596	446	7	,	,	PUNCT
ejpam-4596	446	8	2018	2018	NUM
ejpam-4596	446	9	.	.	PUNCT