id	sid	tid	token	lemma	pos
ejpam-5623	1	1	european	european	PROPN
ejpam-5623	1	2	journal	journal	PROPN
ejpam-5623	1	3	of	of	ADP
ejpam-5623	1	4	pure	pure	ADJ
ejpam-5623	1	5	and	and	CCONJ
ejpam-5623	1	6	applied	applied	ADJ
ejpam-5623	1	7	mathematics	mathematic	NOUN
ejpam-5623	1	8	2025	2025	NUM
ejpam-5623	1	9	,	,	PUNCT
ejpam-5623	1	10	vol	vol	NOUN
ejpam-5623	1	11	.	.	PROPN
ejpam-5623	1	12	18	18	NUM
ejpam-5623	1	13	,	,	PUNCT
ejpam-5623	1	14	issue	issue	NOUN
ejpam-5623	1	15	1	1	NUM
ejpam-5623	1	16	,	,	PUNCT
ejpam-5623	1	17	article	article	NOUN
ejpam-5623	1	18	number	number	NOUN
ejpam-5623	1	19	5623	5623	NUM
ejpam-5623	1	20	issn	issn	PROPN
ejpam-5623	1	21	1307	1307	NUM
ejpam-5623	1	22	-	-	SYM
ejpam-5623	1	23	5543	5543	NUM
ejpam-5623	1	24	–	–	PUNCT
ejpam-5623	1	25	ejpam.com	ejpam.com	X
ejpam-5623	1	26	published	publish	VERB
ejpam-5623	1	27	by	by	ADP
ejpam-5623	1	28	new	new	PROPN
ejpam-5623	1	29	york	york	PROPN
ejpam-5623	1	30	business	business	PROPN
ejpam-5623	1	31	global	global	PROPN
ejpam-5623	1	32	left	leave	VERB
ejpam-5623	1	33	and	and	CCONJ
ejpam-5623	1	34	right	right	ADJ
ejpam-5623	1	35	regular	regular	ADJ
ejpam-5623	1	36	elements	element	NOUN
ejpam-5623	1	37	of	of	ADP
ejpam-5623	1	38	some	some	DET
ejpam-5623	1	39	subsemigroups	subsemigroup	NOUN
ejpam-5623	1	40	of	of	ADP
ejpam-5623	1	41	the	the	DET
ejpam-5623	1	42	linear	linear	NOUN
ejpam-5623	1	43	transformations	transformation	NOUN
ejpam-5623	1	44	semigroups	semigroup	NOUN
ejpam-5623	1	45	with	with	ADP
ejpam-5623	1	46	invariant	invariant	ADJ
ejpam-5623	1	47	subspace	subspace	NOUN
ejpam-5623	1	48	nares	nares	PROPN
ejpam-5623	1	49	sawatraksa1	sawatraksa1	PROPN
ejpam-5623	1	50	,	,	PUNCT
ejpam-5623	1	51	piyaporn	piyaporn	ADJ
ejpam-5623	1	52	tantong1,∗	tantong1,∗	NOUN
ejpam-5623	1	53	1	1	NUM
ejpam-5623	1	54	division	division	NOUN
ejpam-5623	1	55	of	of	ADP
ejpam-5623	1	56	mathematics	mathematic	NOUN
ejpam-5623	1	57	and	and	CCONJ
ejpam-5623	1	58	statistics	statistic	NOUN
ejpam-5623	1	59	,	,	PUNCT
ejpam-5623	1	60	faculty	faculty	NOUN
ejpam-5623	1	61	of	of	ADP
ejpam-5623	1	62	science	science	NOUN
ejpam-5623	1	63	and	and	CCONJ
ejpam-5623	1	64	technology	technology	NOUN
ejpam-5623	1	65	,	,	PUNCT
ejpam-5623	1	66	nakhon	nakhon	PROPN
ejpam-5623	1	67	sawan	sawan	PROPN
ejpam-5623	1	68	rajabhat	rajabhat	PROPN
ejpam-5623	1	69	university	university	PROPN
ejpam-5623	1	70	,	,	PUNCT
ejpam-5623	1	71	nakhon	nakhon	PROPN
ejpam-5623	1	72	sawan	sawan	PROPN
ejpam-5623	1	73	,	,	PUNCT
ejpam-5623	1	74	thailand	thailand	PROPN
ejpam-5623	1	75	abstract	abstract	PROPN
ejpam-5623	1	76	.	.	PUNCT
ejpam-5623	2	1	in	in	ADP
ejpam-5623	2	2	this	this	DET
ejpam-5623	2	3	paper	paper	NOUN
ejpam-5623	2	4	,	,	PUNCT
ejpam-5623	2	5	we	we	PRON
ejpam-5623	2	6	investigate	investigate	VERB
ejpam-5623	2	7	the	the	DET
ejpam-5623	2	8	left	left	ADJ
ejpam-5623	2	9	regularity	regularity	NOUN
ejpam-5623	2	10	,	,	PUNCT
ejpam-5623	2	11	right	right	ADJ
ejpam-5623	2	12	regularity	regularity	NOUN
ejpam-5623	2	13	,	,	PUNCT
ejpam-5623	2	14	and	and	CCONJ
ejpam-5623	2	15	complete	complete	ADJ
ejpam-5623	2	16	regularity	regularity	NOUN
ejpam-5623	2	17	of	of	ADP
ejpam-5623	2	18	elements	element	NOUN
ejpam-5623	2	19	in	in	ADP
ejpam-5623	2	20	subsemigroups	subsemigroup	NOUN
ejpam-5623	2	21	of	of	ADP
ejpam-5623	2	22	the	the	DET
ejpam-5623	2	23	semigroups	semigroup	NOUN
ejpam-5623	2	24	of	of	ADP
ejpam-5623	2	25	linear	linear	ADJ
ejpam-5623	2	26	transformations	transformation	NOUN
ejpam-5623	2	27	with	with	ADP
ejpam-5623	2	28	invariant	invariant	ADJ
ejpam-5623	2	29	subspaces	subspace	NOUN
ejpam-5623	2	30	.	.	PUNCT
ejpam-5623	3	1	we	we	PRON
ejpam-5623	3	2	provide	provide	VERB
ejpam-5623	3	3	necessary	necessary	ADJ
ejpam-5623	3	4	and	and	CCONJ
ejpam-5623	3	5	sufficient	sufficient	ADJ
ejpam-5623	3	6	conditions	condition	NOUN
ejpam-5623	3	7	for	for	SCONJ
ejpam-5623	3	8	these	these	DET
ejpam-5623	3	9	subsemigroups	subsemigroup	NOUN
ejpam-5623	3	10	to	to	PART
ejpam-5623	3	11	be	be	AUX
ejpam-5623	3	12	left	leave	VERB
ejpam-5623	3	13	regular	regular	ADV
ejpam-5623	3	14	,	,	PUNCT
ejpam-5623	3	15	right	right	ADV
ejpam-5623	3	16	regular	regular	ADJ
ejpam-5623	3	17	,	,	PUNCT
ejpam-5623	3	18	and	and	CCONJ
ejpam-5623	3	19	completely	completely	ADV
ejpam-5623	3	20	regular	regular	ADJ
ejpam-5623	3	21	.	.	PUNCT
ejpam-5623	4	1	specifically	specifically	ADV
ejpam-5623	4	2	,	,	PUNCT
ejpam-5623	4	3	we	we	PRON
ejpam-5623	4	4	examine	examine	VERB
ejpam-5623	4	5	semigroups	semigroup	NOUN
ejpam-5623	4	6	of	of	ADP
ejpam-5623	4	7	linear	linear	ADJ
ejpam-5623	4	8	transformations	transformation	NOUN
ejpam-5623	4	9	with	with	ADP
ejpam-5623	4	10	restricted	restricted	ADJ
ejpam-5623	4	11	range	range	NOUN
ejpam-5623	4	12	,	,	PUNCT
ejpam-5623	4	13	invariant	invariant	ADJ
ejpam-5623	4	14	subspaces	subspace	NOUN
ejpam-5623	4	15	,	,	PUNCT
ejpam-5623	4	16	and	and	CCONJ
ejpam-5623	4	17	fixed	fix	VERB
ejpam-5623	4	18	subspaces	subspace	NOUN
ejpam-5623	4	19	.	.	PUNCT
ejpam-5623	5	1	the	the	DET
ejpam-5623	5	2	results	result	NOUN
ejpam-5623	5	3	offer	offer	VERB
ejpam-5623	5	4	a	a	DET
ejpam-5623	5	5	comprehensive	comprehensive	ADJ
ejpam-5623	5	6	characterization	characterization	NOUN
ejpam-5623	5	7	of	of	ADP
ejpam-5623	5	8	regular	regular	ADJ
ejpam-5623	5	9	elements	element	NOUN
ejpam-5623	5	10	within	within	ADP
ejpam-5623	5	11	these	these	DET
ejpam-5623	5	12	algebraic	algebraic	ADJ
ejpam-5623	5	13	structures	structure	NOUN
ejpam-5623	5	14	and	and	CCONJ
ejpam-5623	5	15	extend	extend	VERB
ejpam-5623	5	16	existing	exist	VERB
ejpam-5623	5	17	work	work	NOUN
ejpam-5623	5	18	in	in	ADP
ejpam-5623	5	19	this	this	DET
ejpam-5623	5	20	field	field	NOUN
ejpam-5623	5	21	.	.	PUNCT
ejpam-5623	6	1	our	our	PRON
ejpam-5623	6	2	findings	finding	NOUN
ejpam-5623	6	3	have	have	VERB
ejpam-5623	6	4	potential	potential	ADJ
ejpam-5623	6	5	applications	application	NOUN
ejpam-5623	6	6	in	in	ADP
ejpam-5623	6	7	algebraic	algebraic	ADJ
ejpam-5623	6	8	theory	theory	NOUN
ejpam-5623	6	9	,	,	PUNCT
ejpam-5623	6	10	particularly	particularly	ADV
ejpam-5623	6	11	in	in	ADP
ejpam-5623	6	12	the	the	DET
ejpam-5623	6	13	study	study	NOUN
ejpam-5623	6	14	of	of	ADP
ejpam-5623	6	15	transformation	transformation	NOUN
ejpam-5623	6	16	semigroups	semigroup	NOUN
ejpam-5623	6	17	and	and	CCONJ
ejpam-5623	6	18	their	their	PRON
ejpam-5623	6	19	subsemigroups	subsemigroup	NOUN
ejpam-5623	6	20	.	.	PUNCT
ejpam-5623	7	1	2020	2020	NUM
ejpam-5623	7	2	mathematics	mathematic	NOUN
ejpam-5623	7	3	subject	subject	NOUN
ejpam-5623	7	4	classifications	classification	NOUN
ejpam-5623	7	5	:	:	PUNCT
ejpam-5623	7	6	20m20	20m20	NUM
ejpam-5623	7	7	,	,	PUNCT
ejpam-5623	7	8	15a04	15a04	NUM
ejpam-5623	7	9	,	,	PUNCT
ejpam-5623	7	10	47a15	47a15	NUM
ejpam-5623	7	11	,	,	PUNCT
ejpam-5623	7	12	54d15	54d15	NUM
ejpam-5623	7	13	key	key	ADJ
ejpam-5623	7	14	words	word	NOUN
ejpam-5623	7	15	and	and	CCONJ
ejpam-5623	7	16	phrases	phrase	NOUN
ejpam-5623	7	17	:	:	PUNCT
ejpam-5623	7	18	left	leave	VERB
ejpam-5623	7	19	regular	regular	ADJ
ejpam-5623	7	20	,	,	PUNCT
ejpam-5623	7	21	right	right	ADV
ejpam-5623	7	22	regular	regular	ADJ
ejpam-5623	7	23	,	,	PUNCT
ejpam-5623	7	24	linear	linear	ADJ
ejpam-5623	7	25	transformation	transformation	NOUN
ejpam-5623	7	26	,	,	PUNCT
ejpam-5623	7	27	invariant	invariant	ADJ
ejpam-5623	7	28	subspace	subspace	NOUN
ejpam-5623	7	29	1	1	NUM
ejpam-5623	7	30	.	.	PUNCT
ejpam-5623	7	31	introduction	introduction	NOUN
ejpam-5623	7	32	and	and	CCONJ
ejpam-5623	7	33	preliminaries	preliminary	NOUN
ejpam-5623	7	34	an	an	DET
ejpam-5623	7	35	element	element	NOUN
ejpam-5623	7	36	a	a	PRON
ejpam-5623	7	37	of	of	ADP
ejpam-5623	7	38	a	a	DET
ejpam-5623	7	39	semigroup	semigroup	NOUN
ejpam-5623	7	40	s	s	PART
ejpam-5623	7	41	is	be	AUX
ejpam-5623	7	42	called	call	VERB
ejpam-5623	7	43	left	leave	VERB
ejpam-5623	7	44	regular	regular	ADV
ejpam-5623	7	45	if	if	SCONJ
ejpam-5623	7	46	a	a	DET
ejpam-5623	7	47	=	=	X
ejpam-5623	7	48	xa2	xa2	PROPN
ejpam-5623	7	49	for	for	ADP
ejpam-5623	7	50	some	some	DET
ejpam-5623	7	51	x	x	SYM
ejpam-5623	7	52	∈	∈	PROPN
ejpam-5623	7	53	s	s	NOUN
ejpam-5623	7	54	,	,	PUNCT
ejpam-5623	7	55	right	right	ADV
ejpam-5623	7	56	regular	regular	ADV
ejpam-5623	7	57	if	if	SCONJ
ejpam-5623	7	58	a	a	DET
ejpam-5623	7	59	=	=	PUNCT
ejpam-5623	7	60	a2x	a2x	NOUN
ejpam-5623	7	61	for	for	ADP
ejpam-5623	7	62	some	some	DET
ejpam-5623	7	63	x	x	SYM
ejpam-5623	7	64	∈	∈	PROPN
ejpam-5623	7	65	s	s	NOUN
ejpam-5623	7	66	,	,	PUNCT
ejpam-5623	7	67	and	and	CCONJ
ejpam-5623	7	68	completely	completely	ADV
ejpam-5623	7	69	regular	regular	ADJ
ejpam-5623	7	70	if	if	SCONJ
ejpam-5623	7	71	a	a	DET
ejpam-5623	7	72	=	=	X
ejpam-5623	7	73	axa	axa	NOUN
ejpam-5623	7	74	and	and	CCONJ
ejpam-5623	7	75	ax	ax	NOUN
ejpam-5623	7	76	=	=	PUNCT
ejpam-5623	7	77	xa	xa	PROPN
ejpam-5623	7	78	for	for	ADP
ejpam-5623	7	79	some	some	DET
ejpam-5623	7	80	x	x	SYM
ejpam-5623	7	81	∈	∈	PROPN
ejpam-5623	7	82	s.	s.	PROPN
ejpam-5623	7	83	for	for	ADP
ejpam-5623	7	84	a	a	DET
ejpam-5623	7	85	semigroup	semigroup	PROPN
ejpam-5623	7	86	s	s	NOUN
ejpam-5623	7	87	,	,	PUNCT
ejpam-5623	7	88	let	let	VERB
ejpam-5623	7	89	lreg(s	lreg(s	VERB
ejpam-5623	7	90	)	)	PUNCT
ejpam-5623	7	91	,	,	PUNCT
ejpam-5623	7	92	rreg(s	rreg(s	NOUN
ejpam-5623	7	93	)	)	PUNCT
ejpam-5623	7	94	and	and	CCONJ
ejpam-5623	7	95	creg(s	creg(s	NOUN
ejpam-5623	7	96	)	)	PUNCT
ejpam-5623	7	97	denote	denote	VERB
ejpam-5623	7	98	the	the	DET
ejpam-5623	7	99	set	set	NOUN
ejpam-5623	7	100	of	of	ADP
ejpam-5623	7	101	all	all	PRON
ejpam-5623	7	102	left	leave	VERB
ejpam-5623	7	103	regular	regular	ADJ
ejpam-5623	7	104	elements	element	NOUN
ejpam-5623	7	105	,	,	PUNCT
ejpam-5623	7	106	right	right	ADJ
ejpam-5623	7	107	regular	regular	ADJ
ejpam-5623	7	108	elements	element	NOUN
ejpam-5623	7	109	,	,	PUNCT
ejpam-5623	7	110	and	and	CCONJ
ejpam-5623	7	111	completely	completely	ADV
ejpam-5623	7	112	regular	regular	ADJ
ejpam-5623	7	113	elements	element	NOUN
ejpam-5623	7	114	of	of	ADP
ejpam-5623	7	115	s	s	NOUN
ejpam-5623	7	116	,	,	PUNCT
ejpam-5623	7	117	respectively	respectively	ADV
ejpam-5623	7	118	.	.	PUNCT
ejpam-5623	8	1	it	it	PRON
ejpam-5623	8	2	is	be	AUX
ejpam-5623	8	3	important	important	ADJ
ejpam-5623	8	4	to	to	PART
ejpam-5623	8	5	note	note	VERB
ejpam-5623	8	6	that	that	SCONJ
ejpam-5623	8	7	every	every	DET
ejpam-5623	8	8	completely	completely	ADV
ejpam-5623	8	9	regular	regular	ADJ
ejpam-5623	8	10	element	element	NOUN
ejpam-5623	8	11	is	be	AUX
ejpam-5623	8	12	also	also	ADV
ejpam-5623	8	13	left	leave	VERB
ejpam-5623	8	14	and	and	CCONJ
ejpam-5623	8	15	right	right	ADV
ejpam-5623	8	16	regular	regular	ADV
ejpam-5623	8	17	.	.	PUNCT
ejpam-5623	9	1	additionally	additionally	ADV
ejpam-5623	9	2	,	,	PUNCT
ejpam-5623	9	3	petrich	petrich	NOUN
ejpam-5623	9	4	and	and	CCONJ
ejpam-5623	9	5	reilly	reilly	ADV
ejpam-5623	9	6	[	[	X
ejpam-5623	9	7	10	10	NUM
ejpam-5623	9	8	,	,	PUNCT
ejpam-5623	9	9	proposition	proposition	NOUN
ejpam-5623	9	10	2.1.3	2.1.3	NUM
ejpam-5623	9	11	]	]	PUNCT
ejpam-5623	9	12	proved	prove	VERB
ejpam-5623	9	13	that	that	SCONJ
ejpam-5623	9	14	an	an	DET
ejpam-5623	9	15	element	element	NOUN
ejpam-5623	9	16	a	a	PRON
ejpam-5623	9	17	of	of	ADP
ejpam-5623	9	18	a	a	DET
ejpam-5623	9	19	semigroup	semigroup	NOUN
ejpam-5623	9	20	s	s	VERB
ejpam-5623	9	21	is	be	AUX
ejpam-5623	9	22	completely	completely	ADV
ejpam-5623	9	23	regular	regular	ADJ
ejpam-5623	9	24	if	if	SCONJ
ejpam-5623	9	25	and	and	CCONJ
ejpam-5623	9	26	only	only	ADV
ejpam-5623	9	27	if	if	SCONJ
ejpam-5623	9	28	a	a	PRON
ejpam-5623	9	29	is	be	AUX
ejpam-5623	9	30	both	both	PRON
ejpam-5623	9	31	a	a	DET
ejpam-5623	9	32	left	left	ADJ
ejpam-5623	9	33	and	and	CCONJ
ejpam-5623	9	34	right	right	ADJ
ejpam-5623	9	35	regular	regular	ADJ
ejpam-5623	9	36	element	element	NOUN
ejpam-5623	9	37	of	of	ADP
ejpam-5623	9	38	s.	s.	PROPN
ejpam-5623	9	39	a	a	DET
ejpam-5623	9	40	semigroup	semigroup	PROPN
ejpam-5623	9	41	s	s	PART
ejpam-5623	9	42	is	be	AUX
ejpam-5623	9	43	called	call	VERB
ejpam-5623	9	44	left	left	ADJ
ejpam-5623	9	45	(	(	PUNCT
ejpam-5623	9	46	right	right	ADJ
ejpam-5623	9	47	,	,	PUNCT
ejpam-5623	9	48	completely	completely	ADV
ejpam-5623	9	49	)	)	PUNCT
ejpam-5623	9	50	regular	regular	ADJ
ejpam-5623	9	51	if	if	SCONJ
ejpam-5623	9	52	all	all	DET
ejpam-5623	9	53	its	its	PRON
ejpam-5623	9	54	elements	element	NOUN
ejpam-5623	9	55	are	be	AUX
ejpam-5623	9	56	left	leave	VERB
ejpam-5623	9	57	(	(	PUNCT
ejpam-5623	9	58	right	right	ADJ
ejpam-5623	9	59	,	,	PUNCT
ejpam-5623	9	60	completely	completely	ADV
ejpam-5623	9	61	)	)	PUNCT
ejpam-5623	9	62	regular	regular	ADJ
ejpam-5623	9	63	,	,	PUNCT
ejpam-5623	9	64	that	that	ADV
ejpam-5623	9	65	is	is	ADV
ejpam-5623	9	66	,	,	PUNCT
ejpam-5623	9	67	lreg(s	lreg(s	NOUN
ejpam-5623	9	68	)	)	PUNCT
ejpam-5623	10	1	=	=	SYM
ejpam-5623	10	2	s	s	X
ejpam-5623	10	3	(	(	PUNCT
ejpam-5623	10	4	rreg(s	rreg(s	NOUN
ejpam-5623	10	5	)	)	PUNCT
ejpam-5623	10	6	=	=	SYM
ejpam-5623	10	7	s	s	PROPN
ejpam-5623	10	8	,	,	PUNCT
ejpam-5623	10	9	creg(s	creg(s	NOUN
ejpam-5623	10	10	)	)	PUNCT
ejpam-5623	10	11	=	=	SYM
ejpam-5623	10	12	s	s	X
ejpam-5623	10	13	)	)	PUNCT
ejpam-5623	10	14	.	.	PUNCT
ejpam-5623	11	1	the	the	DET
ejpam-5623	11	2	characterizations	characterization	NOUN
ejpam-5623	11	3	of	of	ADP
ejpam-5623	11	4	left	left	ADJ
ejpam-5623	11	5	regularity	regularity	NOUN
ejpam-5623	11	6	,	,	PUNCT
ejpam-5623	11	7	right	right	ADJ
ejpam-5623	11	8	regularity	regularity	NOUN
ejpam-5623	11	9	,	,	PUNCT
ejpam-5623	11	10	and	and	CCONJ
ejpam-5623	11	11	complete	complete	ADJ
ejpam-5623	11	12	regularity	regularity	NOUN
ejpam-5623	11	13	for	for	ADP
ejpam-5623	11	14	semigroups	semigroup	NOUN
ejpam-5623	11	15	have	have	AUX
ejpam-5623	11	16	been	be	AUX
ejpam-5623	11	17	studied	study	VERB
ejpam-5623	11	18	in	in	ADP
ejpam-5623	11	19	detail	detail	NOUN
ejpam-5623	11	20	,	,	PUNCT
ejpam-5623	11	21	as	as	SCONJ
ejpam-5623	11	22	seen	see	VERB
ejpam-5623	11	23	in	in	ADP
ejpam-5623	11	24	[	[	X
ejpam-5623	11	25	2	2	NUM
ejpam-5623	11	26	,	,	PUNCT
ejpam-5623	11	27	6–9	6–9	PROPN
ejpam-5623	11	28	,	,	PUNCT
ejpam-5623	11	29	12	12	NUM
ejpam-5623	11	30	,	,	PUNCT
ejpam-5623	11	31	15	15	NUM
ejpam-5623	11	32	]	]	PUNCT
ejpam-5623	11	33	.	.	PUNCT
ejpam-5623	12	1	∗corresponding	∗corresponde	VERB
ejpam-5623	12	2	author	author	NOUN
ejpam-5623	12	3	.	.	PUNCT
ejpam-5623	13	1	doi	doi	NOUN
ejpam-5623	13	2	:	:	PUNCT
ejpam-5623	13	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5623	https://doi.org/10.29020/nybg.ejpam.v18i1.5623	SYM
ejpam-5623	13	4	email	email	NOUN
ejpam-5623	13	5	addresses	address	VERB
ejpam-5623	13	6	:	:	PUNCT
ejpam-5623	13	7	nares.sa@nsru.ac.th	nares.sa@nsru.ac.th	PROPN
ejpam-5623	13	8	(	(	PUNCT
ejpam-5623	13	9	n.	n.	NOUN
ejpam-5623	13	10	sawatraksa	sawatraksa	PROPN
ejpam-5623	13	11	)	)	PUNCT
ejpam-5623	13	12	,	,	PUNCT
ejpam-5623	13	13	piyaporn.ta@nsru.ac.th	piyaporn.ta@nsru.ac.th	PRON
ejpam-5623	13	14	(	(	PUNCT
ejpam-5623	13	15	p.	p.	NOUN
ejpam-5623	13	16	tantong	tantong	NOUN
ejpam-5623	13	17	)	)	PUNCT
ejpam-5623	13	18	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5623	13	19	1	1	NUM
ejpam-5623	14	1	copyright	copyright	NOUN
ejpam-5623	14	2	:	:	PUNCT
ejpam-5623	14	3	©	©	PROPN
ejpam-5623	14	4	2025	2025	NUM
ejpam-5623	14	5	the	the	DET
ejpam-5623	14	6	author(s	author(s	NOUN
ejpam-5623	14	7	)	)	PUNCT
ejpam-5623	14	8	.	.	PUNCT
ejpam-5623	15	1	(	(	PUNCT
ejpam-5623	15	2	cc	cc	NOUN
ejpam-5623	15	3	by	by	ADP
ejpam-5623	15	4	-	-	PUNCT
ejpam-5623	15	5	nc	nc	PROPN
ejpam-5623	15	6	4.0	4.0	NUM
ejpam-5623	15	7	)	)	PUNCT
ejpam-5623	15	8	n.	n.	NOUN
ejpam-5623	15	9	sawatraksa	sawatraksa	NOUN
ejpam-5623	15	10	,	,	PUNCT
ejpam-5623	15	11	p.	p.	NOUN
ejpam-5623	15	12	tantong	tantong	NOUN
ejpam-5623	15	13	/	/	SYM
ejpam-5623	15	14	eur	eur	PROPN
ejpam-5623	15	15	.	.	PUNCT
ejpam-5623	16	1	j.	j.	PROPN
ejpam-5623	16	2	pure	pure	PROPN
ejpam-5623	16	3	appl	appl	PROPN
ejpam-5623	16	4	.	.	PROPN
ejpam-5623	16	5	math	math	PROPN
ejpam-5623	16	6	,	,	PUNCT
ejpam-5623	16	7	18	18	NUM
ejpam-5623	16	8	(	(	PUNCT
ejpam-5623	16	9	1	1	NUM
ejpam-5623	16	10	)	)	PUNCT
ejpam-5623	16	11	(	(	PUNCT
ejpam-5623	16	12	2025	2025	NUM
ejpam-5623	16	13	)	)	PUNCT
ejpam-5623	16	14	,	,	PUNCT
ejpam-5623	16	15	5623	5623	NUM
ejpam-5623	16	16	2	2	NUM
ejpam-5623	16	17	of	of	ADP
ejpam-5623	16	18	15	15	NUM
ejpam-5623	16	19	let	let	VERB
ejpam-5623	16	20	v	v	PART
ejpam-5623	16	21	be	be	AUX
ejpam-5623	16	22	a	a	DET
ejpam-5623	16	23	vector	vector	NOUN
ejpam-5623	16	24	space	space	NOUN
ejpam-5623	16	25	over	over	ADP
ejpam-5623	16	26	a	a	DET
ejpam-5623	16	27	field	field	NOUN
ejpam-5623	16	28	,	,	PUNCT
ejpam-5623	16	29	and	and	CCONJ
ejpam-5623	16	30	let	let	VERB
ejpam-5623	16	31	l(v	l(v	NOUN
ejpam-5623	16	32	)	)	PUNCT
ejpam-5623	16	33	be	be	AUX
ejpam-5623	16	34	the	the	DET
ejpam-5623	16	35	semigroup	semigroup	NOUN
ejpam-5623	16	36	(	(	PUNCT
ejpam-5623	16	37	under	under	ADP
ejpam-5623	16	38	composition	composition	NOUN
ejpam-5623	16	39	)	)	PUNCT
ejpam-5623	16	40	consisting	consist	VERB
ejpam-5623	16	41	of	of	ADP
ejpam-5623	16	42	all	all	DET
ejpam-5623	16	43	linear	linear	ADJ
ejpam-5623	16	44	operators	operator	NOUN
ejpam-5623	16	45	on	on	ADP
ejpam-5623	16	46	v	v	NUM
ejpam-5623	16	47	.	.	PUNCT
ejpam-5623	17	1	it	it	PRON
ejpam-5623	17	2	is	be	AUX
ejpam-5623	17	3	well	well	ADV
ejpam-5623	17	4	known	know	VERB
ejpam-5623	17	5	that	that	SCONJ
ejpam-5623	17	6	l(v	l(v	NOUN
ejpam-5623	17	7	)	)	PUNCT
ejpam-5623	17	8	is	be	AUX
ejpam-5623	17	9	a	a	DET
ejpam-5623	17	10	regular	regular	ADJ
ejpam-5623	17	11	semigroup	semigroup	NOUN
ejpam-5623	17	12	[	[	X
ejpam-5623	17	13	4	4	NUM
ejpam-5623	17	14	,	,	PUNCT
ejpam-5623	17	15	page	page	NOUN
ejpam-5623	17	16	63	63	NUM
ejpam-5623	17	17	]	]	PUNCT
ejpam-5623	17	18	.	.	PUNCT
ejpam-5623	18	1	moreover	moreover	ADV
ejpam-5623	18	2	,	,	PUNCT
ejpam-5623	18	3	tantong	tantong	NOUN
ejpam-5623	18	4	[	[	X
ejpam-5623	18	5	15	15	NUM
ejpam-5623	18	6	]	]	X
ejpam-5623	18	7	characterized	characterize	VERB
ejpam-5623	18	8	left	leave	VERB
ejpam-5623	18	9	,	,	PUNCT
ejpam-5623	18	10	right	right	ADJ
ejpam-5623	18	11	and	and	CCONJ
ejpam-5623	18	12	completely	completely	ADV
ejpam-5623	18	13	regularity	regularity	NOUN
ejpam-5623	18	14	for	for	ADP
ejpam-5623	18	15	elements	element	NOUN
ejpam-5623	18	16	of	of	ADP
ejpam-5623	18	17	l(v	l(v	NOUN
ejpam-5623	18	18	)	)	PUNCT
ejpam-5623	18	19	and	and	CCONJ
ejpam-5623	18	20	provided	provide	VERB
ejpam-5623	18	21	necessary	necessary	ADJ
ejpam-5623	18	22	and	and	CCONJ
ejpam-5623	18	23	sufficient	sufficient	ADJ
ejpam-5623	18	24	conditions	condition	NOUN
ejpam-5623	18	25	for	for	ADP
ejpam-5623	18	26	l(v	l(v	NOUN
ejpam-5623	18	27	)	)	PUNCT
ejpam-5623	18	28	to	to	PART
ejpam-5623	18	29	be	be	AUX
ejpam-5623	18	30	left	leave	VERB
ejpam-5623	18	31	regular	regular	ADV
ejpam-5623	18	32	,	,	PUNCT
ejpam-5623	18	33	right	right	ADV
ejpam-5623	18	34	regular	regular	ADJ
ejpam-5623	18	35	and	and	CCONJ
ejpam-5623	18	36	completely	completely	ADV
ejpam-5623	18	37	regular	regular	ADJ
ejpam-5623	18	38	in	in	ADP
ejpam-5623	18	39	terms	term	NOUN
ejpam-5623	18	40	of	of	ADP
ejpam-5623	18	41	the	the	DET
ejpam-5623	18	42	dimension	dimension	NOUN
ejpam-5623	18	43	of	of	ADP
ejpam-5623	18	44	v	v	NOUN
ejpam-5623	18	45	.	.	PUNCT
ejpam-5623	19	1	let	let	VERB
ejpam-5623	19	2	w	w	NOUN
ejpam-5623	19	3	be	be	AUX
ejpam-5623	19	4	a	a	DET
ejpam-5623	19	5	fixed	fix	VERB
ejpam-5623	19	6	subspace	subspace	NOUN
ejpam-5623	19	7	of	of	ADP
ejpam-5623	19	8	v	v	NOUN
ejpam-5623	19	9	.	.	PUNCT
ejpam-5623	20	1	in	in	ADP
ejpam-5623	20	2	2008	2008	NUM
ejpam-5623	20	3	,	,	PUNCT
ejpam-5623	20	4	sullivan	sullivan	PROPN
ejpam-5623	20	5	[	[	X
ejpam-5623	20	6	14	14	NUM
ejpam-5623	20	7	]	]	PUNCT
ejpam-5623	20	8	defined	define	VERB
ejpam-5623	20	9	a	a	DET
ejpam-5623	20	10	subsemigroup	subsemigroup	NOUN
ejpam-5623	20	11	of	of	ADP
ejpam-5623	20	12	the	the	DET
ejpam-5623	20	13	linear	linear	PROPN
ejpam-5623	20	14	transformation	transformation	NOUN
ejpam-5623	20	15	semigroup	semigroup	NOUN
ejpam-5623	20	16	as	as	ADP
ejpam-5623	20	17	:	:	PUNCT
ejpam-5623	20	18	l(v	l(v	NOUN
ejpam-5623	20	19	,	,	PUNCT
ejpam-5623	20	20	w	w	NOUN
ejpam-5623	20	21	)	)	PUNCT
ejpam-5623	21	1	=	=	PRON
ejpam-5623	21	2	{	{	PUNCT
ejpam-5623	21	3	α	α	NOUN
ejpam-5623	21	4	∈	∈	PROPN
ejpam-5623	21	5	l(v	l(v	NOUN
ejpam-5623	21	6	)	)	PUNCT
ejpam-5623	21	7	:	:	PUNCT
ejpam-5623	22	1	v	v	X
ejpam-5623	22	2	α	α	NOUN
ejpam-5623	22	3	⊆	⊆	NUM
ejpam-5623	22	4	w	w	NOUN
ejpam-5623	22	5	}	}	PUNCT
ejpam-5623	22	6	.	.	PUNCT
ejpam-5623	23	1	this	this	DET
ejpam-5623	23	2	semigroup	semigroup	PROPN
ejpam-5623	23	3	is	be	AUX
ejpam-5623	23	4	called	call	VERB
ejpam-5623	23	5	the	the	DET
ejpam-5623	23	6	semigroup	semigroup	NOUN
ejpam-5623	23	7	of	of	ADP
ejpam-5623	23	8	linear	linear	ADJ
ejpam-5623	23	9	transformations	transformation	NOUN
ejpam-5623	23	10	with	with	ADP
ejpam-5623	23	11	restricted	restricted	ADJ
ejpam-5623	23	12	range	range	NOUN
ejpam-5623	23	13	.	.	PUNCT
ejpam-5623	24	1	the	the	DET
ejpam-5623	24	2	author	author	NOUN
ejpam-5623	24	3	examined	examine	VERB
ejpam-5623	24	4	green	green	PROPN
ejpam-5623	24	5	’s	’s	PART
ejpam-5623	24	6	relations	relation	NOUN
ejpam-5623	24	7	and	and	CCONJ
ejpam-5623	24	8	ideals	ideal	NOUN
ejpam-5623	24	9	for	for	ADP
ejpam-5623	24	10	the	the	DET
ejpam-5623	24	11	semigroup	semigroup	PROPN
ejpam-5623	24	12	l(v	l(v	PROPN
ejpam-5623	24	13	,	,	PUNCT
ejpam-5623	24	14	w	w	PROPN
ejpam-5623	24	15	)	)	PUNCT
ejpam-5623	24	16	.	.	PUNCT
ejpam-5623	25	1	later	later	ADV
ejpam-5623	25	2	,	,	PUNCT
ejpam-5623	25	3	in	in	ADP
ejpam-5623	25	4	2019	2019	NUM
ejpam-5623	25	5	,	,	PUNCT
ejpam-5623	25	6	sangkhanan	sangkhanan	NOUN
ejpam-5623	25	7	and	and	CCONJ
ejpam-5623	25	8	sanwong	sanwong	NOUN
ejpam-5623	26	1	[	[	X
ejpam-5623	26	2	11	11	NUM
ejpam-5623	26	3	]	]	PUNCT
ejpam-5623	26	4	proved	prove	VERB
ejpam-5623	26	5	certain	certain	ADJ
ejpam-5623	26	6	isomorphism	isomorphism	NOUN
ejpam-5623	26	7	theorems	theorem	NOUN
ejpam-5623	26	8	and	and	CCONJ
ejpam-5623	26	9	calculated	calculate	VERB
ejpam-5623	26	10	the	the	DET
ejpam-5623	26	11	ranks	rank	NOUN
ejpam-5623	26	12	of	of	ADP
ejpam-5623	26	13	these	these	DET
ejpam-5623	26	14	semigroups	semigroup	NOUN
ejpam-5623	26	15	for	for	ADP
ejpam-5623	26	16	any	any	DET
ejpam-5623	26	17	proper	proper	ADJ
ejpam-5623	26	18	subspace	subspace	NOUN
ejpam-5623	26	19	w	w	PROPN
ejpam-5623	26	20	of	of	ADP
ejpam-5623	26	21	a	a	DET
ejpam-5623	26	22	finite	finite	ADJ
ejpam-5623	26	23	dimensional	dimensional	ADJ
ejpam-5623	26	24	vector	vector	NOUN
ejpam-5623	26	25	space	space	NOUN
ejpam-5623	26	26	v	v	NOUN
ejpam-5623	26	27	over	over	ADP
ejpam-5623	26	28	a	a	DET
ejpam-5623	26	29	finite	finite	ADJ
ejpam-5623	26	30	field	field	NOUN
ejpam-5623	26	31	.	.	PUNCT
ejpam-5623	27	1	additionally	additionally	ADV
ejpam-5623	27	2	,	,	PUNCT
ejpam-5623	27	3	sullivan	sullivan	PROPN
ejpam-5623	28	1	[	[	X
ejpam-5623	28	2	14	14	NUM
ejpam-5623	28	3	]	]	PUNCT
ejpam-5623	28	4	showed	show	VERB
ejpam-5623	28	5	that	that	SCONJ
ejpam-5623	28	6	:	:	PUNCT
ejpam-5623	28	7	q	q	X
ejpam-5623	28	8	=	=	SYM
ejpam-5623	28	9	{	{	PUNCT
ejpam-5623	28	10	α	α	NOUN
ejpam-5623	28	11	∈	∈	PROPN
ejpam-5623	28	12	l(v	l(v	PROPN
ejpam-5623	28	13	,	,	PUNCT
ejpam-5623	28	14	w	w	PROPN
ejpam-5623	28	15	)	)	PUNCT
ejpam-5623	28	16	:	:	PUNCT
ejpam-5623	28	17	v	v	X
ejpam-5623	28	18	α	α	NOUN
ejpam-5623	28	19	⊆	⊆	NUM
ejpam-5623	28	20	wα	wα	NOUN
ejpam-5623	28	21	}	}	PUNCT
ejpam-5623	28	22	is	be	AUX
ejpam-5623	28	23	the	the	DET
ejpam-5623	28	24	largest	large	ADJ
ejpam-5623	28	25	regular	regular	ADJ
ejpam-5623	28	26	subsemigroup	subsemigroup	NOUN
ejpam-5623	28	27	of	of	ADP
ejpam-5623	28	28	l(v	l(v	PROPN
ejpam-5623	28	29	,	,	PUNCT
ejpam-5623	28	30	w	w	NOUN
ejpam-5623	28	31	)	)	PUNCT
ejpam-5623	28	32	.	.	PUNCT
ejpam-5623	29	1	in	in	ADP
ejpam-5623	29	2	2015	2015	NUM
ejpam-5623	29	3	,	,	PUNCT
ejpam-5623	29	4	sangkhanan	sangkhanan	NOUN
ejpam-5623	29	5	and	and	CCONJ
ejpam-5623	29	6	sommanee	sommanee	NOUN
ejpam-5623	29	7	[	[	X
ejpam-5623	29	8	13	13	NUM
ejpam-5623	29	9	]	]	PUNCT
ejpam-5623	29	10	described	describe	VERB
ejpam-5623	29	11	all	all	DET
ejpam-5623	29	12	the	the	DET
ejpam-5623	29	13	maximal	maximal	ADJ
ejpam-5623	29	14	regular	regular	ADJ
ejpam-5623	29	15	subsemigroups	subsemigroup	NOUN
ejpam-5623	29	16	of	of	ADP
ejpam-5623	29	17	q	q	NOUN
ejpam-5623	29	18	when	when	SCONJ
ejpam-5623	29	19	w	w	NOUN
ejpam-5623	29	20	is	be	AUX
ejpam-5623	29	21	a	a	DET
ejpam-5623	29	22	finite	finite	ADJ
ejpam-5623	29	23	dimensional	dimensional	ADJ
ejpam-5623	29	24	subspace	subspace	NOUN
ejpam-5623	29	25	of	of	ADP
ejpam-5623	29	26	v	v	NOUN
ejpam-5623	29	27	over	over	ADP
ejpam-5623	29	28	a	a	DET
ejpam-5623	29	29	finite	finite	ADJ
ejpam-5623	29	30	field	field	NOUN
ejpam-5623	29	31	.	.	PUNCT
ejpam-5623	30	1	furthermore	furthermore	ADV
ejpam-5623	30	2	,	,	PUNCT
ejpam-5623	30	3	they	they	PRON
ejpam-5623	30	4	computed	compute	VERB
ejpam-5623	30	5	the	the	DET
ejpam-5623	30	6	rank	rank	NOUN
ejpam-5623	30	7	and	and	CCONJ
ejpam-5623	30	8	idempotent	idempotent	ADJ
ejpam-5623	30	9	rank	rank	NOUN
ejpam-5623	30	10	of	of	ADP
ejpam-5623	30	11	q	q	PROPN
ejpam-5623	30	12	where	where	SCONJ
ejpam-5623	30	13	w	w	NOUN
ejpam-5623	30	14	is	be	AUX
ejpam-5623	30	15	an	an	DET
ejpam-5623	30	16	n	n	ADV
ejpam-5623	30	17	-	-	PUNCT
ejpam-5623	30	18	dimensional	dimensional	ADJ
ejpam-5623	30	19	subspace	subspace	NOUN
ejpam-5623	30	20	of	of	ADP
ejpam-5623	30	21	an	an	DET
ejpam-5623	30	22	m	m	ADV
ejpam-5623	30	23	-	-	ADJ
ejpam-5623	30	24	dimensional	dimensional	ADJ
ejpam-5623	30	25	vector	vector	NOUN
ejpam-5623	30	26	space	space	NOUN
ejpam-5623	30	27	v	v	NOUN
ejpam-5623	30	28	over	over	ADP
ejpam-5623	30	29	a	a	DET
ejpam-5623	30	30	finite	finite	ADJ
ejpam-5623	30	31	field	field	NOUN
ejpam-5623	30	32	.	.	PUNCT
ejpam-5623	31	1	let	let	VERB
ejpam-5623	31	2	w	w	NOUN
ejpam-5623	31	3	be	be	AUX
ejpam-5623	31	4	a	a	DET
ejpam-5623	31	5	subspace	subspace	NOUN
ejpam-5623	31	6	of	of	ADP
ejpam-5623	31	7	a	a	DET
ejpam-5623	31	8	vector	vector	NOUN
ejpam-5623	31	9	space	space	NOUN
ejpam-5623	31	10	v	v	NOUN
ejpam-5623	31	11	.	.	PUNCT
ejpam-5623	32	1	the	the	DET
ejpam-5623	32	2	semigroup	semigroup	NOUN
ejpam-5623	32	3	of	of	ADP
ejpam-5623	32	4	linear	linear	ADJ
ejpam-5623	32	5	transformations	transformation	NOUN
ejpam-5623	32	6	with	with	ADP
ejpam-5623	32	7	invariant	invariant	ADJ
ejpam-5623	32	8	subspace	subspace	NOUN
ejpam-5623	32	9	are	be	AUX
ejpam-5623	32	10	defined	define	VERB
ejpam-5623	32	11	by	by	ADP
ejpam-5623	32	12	:	:	PUNCT
ejpam-5623	32	13	s(v	s(v	PROPN
ejpam-5623	32	14	,	,	PUNCT
ejpam-5623	32	15	w	w	NOUN
ejpam-5623	32	16	)	)	PUNCT
ejpam-5623	33	1	=	=	PRON
ejpam-5623	33	2	{	{	PUNCT
ejpam-5623	33	3	α	α	NOUN
ejpam-5623	33	4	∈	∈	PROPN
ejpam-5623	33	5	l(v	l(v	NOUN
ejpam-5623	33	6	)	)	PUNCT
ejpam-5623	33	7	:	:	PUNCT
ejpam-5623	33	8	wα	wα	NOUN
ejpam-5623	33	9	⊆	⊆	NUM
ejpam-5623	33	10	w	w	NOUN
ejpam-5623	33	11	}	}	PUNCT
ejpam-5623	33	12	.	.	PUNCT
ejpam-5623	34	1	in	in	ADP
ejpam-5623	34	2	2012	2012	NUM
ejpam-5623	34	3	,	,	PUNCT
ejpam-5623	34	4	huisheng	huisheng	NOUN
ejpam-5623	35	1	[	[	X
ejpam-5623	35	2	5	5	NUM
ejpam-5623	35	3	]	]	PUNCT
ejpam-5623	35	4	described	describe	VERB
ejpam-5623	35	5	the	the	DET
ejpam-5623	35	6	relations	relation	NOUN
ejpam-5623	35	7	l∗	l∗	PROPN
ejpam-5623	35	8	and	and	CCONJ
ejpam-5623	35	9	r∗	r∗	VERB
ejpam-5623	35	10	on	on	ADP
ejpam-5623	35	11	s(v	s(v	PROPN
ejpam-5623	35	12	,	,	PUNCT
ejpam-5623	35	13	w	w	NOUN
ejpam-5623	35	14	)	)	PUNCT
ejpam-5623	35	15	.	.	PUNCT
ejpam-5623	36	1	in	in	ADP
ejpam-5623	36	2	the	the	DET
ejpam-5623	36	3	same	same	ADJ
ejpam-5623	36	4	year	year	NOUN
ejpam-5623	36	5	,	,	PUNCT
ejpam-5623	36	6	honyam	honyam	NOUN
ejpam-5623	36	7	and	and	CCONJ
ejpam-5623	36	8	sanwong	sanwong	ADJ
ejpam-5623	37	1	[	[	X
ejpam-5623	37	2	3	3	NUM
ejpam-5623	37	3	]	]	PUNCT
ejpam-5623	37	4	presented	present	VERB
ejpam-5623	37	5	the	the	DET
ejpam-5623	37	6	relations	relation	NOUN
ejpam-5623	37	7	of	of	ADP
ejpam-5623	37	8	green	green	ADJ
ejpam-5623	37	9	and	and	CCONJ
ejpam-5623	37	10	ideals	ideal	NOUN
ejpam-5623	37	11	of	of	ADP
ejpam-5623	37	12	the	the	DET
ejpam-5623	37	13	semigroup	semigroup	NOUN
ejpam-5623	37	14	and	and	CCONJ
ejpam-5623	37	15	proved	prove	VERB
ejpam-5623	37	16	that	that	SCONJ
ejpam-5623	37	17	it	it	PRON
ejpam-5623	37	18	is	be	AUX
ejpam-5623	37	19	never	never	ADV
ejpam-5623	37	20	isomorphic	isomorphic	ADJ
ejpam-5623	37	21	to	to	ADP
ejpam-5623	37	22	t	t	PROPN
ejpam-5623	37	23	(	(	PUNCT
ejpam-5623	37	24	u	u	NOUN
ejpam-5623	37	25	)	)	PUNCT
ejpam-5623	37	26	for	for	ADP
ejpam-5623	37	27	any	any	DET
ejpam-5623	37	28	vector	vector	NOUN
ejpam-5623	37	29	space	space	NOUN
ejpam-5623	37	30	u	u	NOUN
ejpam-5623	37	31	when	when	SCONJ
ejpam-5623	37	32	w	w	NOUN
ejpam-5623	37	33	is	be	AUX
ejpam-5623	37	34	a	a	DET
ejpam-5623	37	35	nonzero	nonzero	ADJ
ejpam-5623	37	36	proper	proper	ADJ
ejpam-5623	37	37	subspace	subspace	NOUN
ejpam-5623	37	38	of	of	ADP
ejpam-5623	37	39	v	v	NOUN
ejpam-5623	37	40	.	.	PUNCT
ejpam-5623	38	1	in	in	ADP
ejpam-5623	38	2	2019	2019	NUM
ejpam-5623	38	3	,	,	PUNCT
ejpam-5623	38	4	chaiya	chaiya	NOUN
ejpam-5623	39	1	[	[	X
ejpam-5623	39	2	1	1	X
ejpam-5623	39	3	]	]	PUNCT
ejpam-5623	39	4	characterized	characterize	VERB
ejpam-5623	39	5	the	the	DET
ejpam-5623	39	6	natural	natural	ADJ
ejpam-5623	39	7	partial	partial	ADJ
ejpam-5623	39	8	order	order	NOUN
ejpam-5623	39	9	on	on	ADP
ejpam-5623	39	10	s(v	s(v	PROPN
ejpam-5623	39	11	,	,	PUNCT
ejpam-5623	39	12	w	w	NOUN
ejpam-5623	39	13	)	)	PUNCT
ejpam-5623	39	14	and	and	CCONJ
ejpam-5623	39	15	determined	determine	VERB
ejpam-5623	39	16	the	the	DET
ejpam-5623	39	17	compatibility	compatibility	NOUN
ejpam-5623	39	18	of	of	ADP
ejpam-5623	39	19	their	their	PRON
ejpam-5623	39	20	elements	element	NOUN
ejpam-5623	39	21	and	and	CCONJ
ejpam-5623	39	22	found	find	VERB
ejpam-5623	39	23	all	all	DET
ejpam-5623	39	24	maximal	maximal	ADJ
ejpam-5623	39	25	and	and	CCONJ
ejpam-5623	39	26	minimal	minimal	ADJ
ejpam-5623	39	27	elements	element	NOUN
ejpam-5623	39	28	.	.	PUNCT
ejpam-5623	40	1	furthermore	furthermore	ADV
ejpam-5623	40	2	,	,	PUNCT
ejpam-5623	40	3	she	she	PRON
ejpam-5623	40	4	presented	present	VERB
ejpam-5623	40	5	necessary	necessary	ADJ
ejpam-5623	40	6	and	and	CCONJ
ejpam-5623	40	7	sufficient	sufficient	ADJ
ejpam-5623	40	8	conditions	condition	NOUN
ejpam-5623	40	9	for	for	ADP
ejpam-5623	40	10	s(v	s(v	PROPN
ejpam-5623	40	11	,	,	PUNCT
ejpam-5623	40	12	w	w	NOUN
ejpam-5623	40	13	)	)	PUNCT
ejpam-5623	40	14	to	to	PART
ejpam-5623	40	15	be	be	AUX
ejpam-5623	40	16	factorizable	factorizable	ADJ
ejpam-5623	40	17	,	,	PUNCT
ejpam-5623	40	18	unit	unit	NOUN
ejpam-5623	40	19	-	-	PUNCT
ejpam-5623	40	20	regular	regular	ADJ
ejpam-5623	40	21	,	,	PUNCT
ejpam-5623	40	22	and	and	CCONJ
ejpam-5623	40	23	directly	directly	ADV
ejpam-5623	40	24	finite	finite	VERB
ejpam-5623	40	25	.	.	PUNCT
ejpam-5623	41	1	for	for	ADP
ejpam-5623	41	2	a	a	DET
ejpam-5623	41	3	fixed	fix	VERB
ejpam-5623	41	4	subspace	subspace	NOUN
ejpam-5623	41	5	w	w	PROPN
ejpam-5623	41	6	of	of	ADP
ejpam-5623	41	7	a	a	DET
ejpam-5623	41	8	vector	vector	NOUN
ejpam-5623	41	9	space	space	NOUN
ejpam-5623	41	10	v	v	NOUN
ejpam-5623	41	11	,	,	PUNCT
ejpam-5623	41	12	let	let	VERB
ejpam-5623	41	13	fix(v	fix(v	PROPN
ejpam-5623	41	14	,	,	PUNCT
ejpam-5623	41	15	w	w	NOUN
ejpam-5623	41	16	)	)	PUNCT
ejpam-5623	42	1	=	=	PRON
ejpam-5623	42	2	{	{	PUNCT
ejpam-5623	42	3	α	α	NOUN
ejpam-5623	42	4	∈	∈	PROPN
ejpam-5623	42	5	l(v	l(v	NOUN
ejpam-5623	42	6	)	)	PUNCT
ejpam-5623	42	7	:	:	PUNCT
ejpam-5623	42	8	wα	wα	NOUN
ejpam-5623	42	9	=	=	PUNCT
ejpam-5623	42	10	w	w	PROPN
ejpam-5623	42	11	for	for	ADP
ejpam-5623	42	12	all	all	DET
ejpam-5623	42	13	w	w	PROPN
ejpam-5623	42	14	∈	∈	PROPN
ejpam-5623	42	15	w	w	PROPN
ejpam-5623	42	16	}	}	PUNCT
ejpam-5623	42	17	.	.	PUNCT
ejpam-5623	43	1	then	then	ADV
ejpam-5623	43	2	,	,	PUNCT
ejpam-5623	43	3	fix(v	fix(v	PROPN
ejpam-5623	43	4	,	,	PUNCT
ejpam-5623	43	5	w	w	PROPN
ejpam-5623	43	6	)	)	PUNCT
ejpam-5623	43	7	is	be	AUX
ejpam-5623	43	8	a	a	DET
ejpam-5623	43	9	subsemigroup	subsemigroup	NOUN
ejpam-5623	43	10	of	of	ADP
ejpam-5623	43	11	s(v	s(v	PROPN
ejpam-5623	43	12	,	,	PUNCT
ejpam-5623	43	13	w	w	NOUN
ejpam-5623	43	14	)	)	PUNCT
ejpam-5623	43	15	and	and	CCONJ
ejpam-5623	43	16	we	we	PRON
ejpam-5623	43	17	call	call	VERB
ejpam-5623	43	18	it	it	PRON
ejpam-5623	43	19	the	the	DET
ejpam-5623	43	20	semigroup	semigroup	NOUN
ejpam-5623	43	21	of	of	ADP
ejpam-5623	43	22	linear	linear	ADJ
ejpam-5623	43	23	transformations	transformation	NOUN
ejpam-5623	43	24	with	with	ADP
ejpam-5623	43	25	fixed	fix	VERB
ejpam-5623	43	26	subspaces	subspace	NOUN
ejpam-5623	43	27	.	.	PUNCT
ejpam-5623	44	1	in	in	ADP
ejpam-5623	44	2	2018	2018	NUM
ejpam-5623	44	3	,	,	PUNCT
ejpam-5623	44	4	chaiya	chaiya	PROPN
ejpam-5623	44	5	et	et	PROPN
ejpam-5623	44	6	.	.	PUNCT
ejpam-5623	45	1	al	al	PROPN
ejpam-5623	45	2	.	.	PUNCT
ejpam-5623	46	1	[	[	X
ejpam-5623	46	2	16	16	NUM
ejpam-5623	46	3	]	]	PUNCT
ejpam-5623	46	4	discussed	discuss	VERB
ejpam-5623	46	5	the	the	DET
ejpam-5623	46	6	green	green	PROPN
ejpam-5623	46	7	’s	’s	PART
ejpam-5623	46	8	relations	relation	NOUN
ejpam-5623	46	9	,	,	PUNCT
ejpam-5623	46	10	regularity	regularity	NOUN
ejpam-5623	46	11	,	,	PUNCT
ejpam-5623	46	12	and	and	CCONJ
ejpam-5623	46	13	ideals	ideal	NOUN
ejpam-5623	46	14	of	of	ADP
ejpam-5623	46	15	fix(v	fix(v	PROPN
ejpam-5623	46	16	,	,	PUNCT
ejpam-5623	46	17	w	w	NOUN
ejpam-5623	46	18	)	)	PUNCT
ejpam-5623	46	19	,	,	PUNCT
ejpam-5623	46	20	and	and	CCONJ
ejpam-5623	46	21	characterized	characterize	VERB
ejpam-5623	46	22	when	when	SCONJ
ejpam-5623	46	23	fix(v	fix(v	PROPN
ejpam-5623	46	24	,	,	PUNCT
ejpam-5623	46	25	w	w	NOUN
ejpam-5623	46	26	)	)	PUNCT
ejpam-5623	46	27	is	be	AUX
ejpam-5623	46	28	factorisable	factorisable	ADJ
ejpam-5623	46	29	,	,	PUNCT
ejpam-5623	46	30	unit	unit	NOUN
ejpam-5623	46	31	-	-	PUNCT
ejpam-5623	46	32	regular	regular	ADJ
ejpam-5623	46	33	,	,	PUNCT
ejpam-5623	46	34	and	and	CCONJ
ejpam-5623	46	35	directly	directly	ADV
ejpam-5623	46	36	finite	finite	VERB
ejpam-5623	46	37	.	.	PUNCT
ejpam-5623	47	1	the	the	DET
ejpam-5623	47	2	objective	objective	NOUN
ejpam-5623	47	3	of	of	ADP
ejpam-5623	47	4	this	this	DET
ejpam-5623	47	5	paper	paper	NOUN
ejpam-5623	47	6	is	be	AUX
ejpam-5623	47	7	to	to	PART
ejpam-5623	47	8	characterize	characterize	VERB
ejpam-5623	47	9	the	the	DET
ejpam-5623	47	10	left	left	ADJ
ejpam-5623	47	11	,	,	PUNCT
ejpam-5623	47	12	right	right	ADJ
ejpam-5623	47	13	,	,	PUNCT
ejpam-5623	47	14	and	and	CCONJ
ejpam-5623	47	15	complete	complete	ADJ
ejpam-5623	47	16	regularity	regularity	NOUN
ejpam-5623	47	17	of	of	ADP
ejpam-5623	47	18	elements	element	NOUN
ejpam-5623	47	19	within	within	ADP
ejpam-5623	47	20	the	the	DET
ejpam-5623	47	21	semigroups	semigroup	NOUN
ejpam-5623	47	22	l(v	l(v	PROPN
ejpam-5623	47	23	,	,	PUNCT
ejpam-5623	47	24	w	w	PROPN
ejpam-5623	47	25	)	)	PUNCT
ejpam-5623	47	26	,	,	PUNCT
ejpam-5623	47	27	s(v	s(v	PROPN
ejpam-5623	47	28	,	,	PUNCT
ejpam-5623	47	29	w	w	NOUN
ejpam-5623	47	30	)	)	PUNCT
ejpam-5623	47	31	,	,	PUNCT
ejpam-5623	47	32	q	q	NOUN
ejpam-5623	47	33	,	,	PUNCT
ejpam-5623	47	34	and	and	CCONJ
ejpam-5623	47	35	fix(v	fix(v	PROPN
ejpam-5623	47	36	,	,	PUNCT
ejpam-5623	47	37	w	w	NOUN
ejpam-5623	47	38	)	)	PUNCT
ejpam-5623	47	39	.	.	PUNCT
ejpam-5623	48	1	we	we	PRON
ejpam-5623	48	2	also	also	ADV
ejpam-5623	48	3	present	present	VERB
ejpam-5623	48	4	a	a	DET
ejpam-5623	48	5	method	method	NOUN
ejpam-5623	48	6	to	to	PART
ejpam-5623	48	7	construct	construct	VERB
ejpam-5623	48	8	an	an	DET
ejpam-5623	48	9	element	element	NOUN
ejpam-5623	48	10	β	β	NOUN
ejpam-5623	48	11	in	in	ADP
ejpam-5623	48	12	these	these	DET
ejpam-5623	48	13	semigroups	semigroup	NOUN
ejpam-5623	48	14	such	such	ADJ
ejpam-5623	48	15	that	that	SCONJ
ejpam-5623	48	16	it	it	PRON
ejpam-5623	48	17	is	be	AUX
ejpam-5623	48	18	left	leave	VERB
ejpam-5623	48	19	and	and	CCONJ
ejpam-5623	48	20	right	right	ADV
ejpam-5623	48	21	regular	regular	ADV
ejpam-5623	48	22	.	.	PUNCT
ejpam-5623	49	1	furthermore	furthermore	ADV
ejpam-5623	49	2	,	,	PUNCT
ejpam-5623	49	3	we	we	PRON
ejpam-5623	49	4	establish	establish	VERB
ejpam-5623	49	5	necessary	necessary	ADJ
ejpam-5623	49	6	and	and	CCONJ
ejpam-5623	49	7	sufficient	sufficient	ADJ
ejpam-5623	49	8	conditions	condition	NOUN
ejpam-5623	49	9	for	for	ADP
ejpam-5623	49	10	the	the	DET
ejpam-5623	49	11	semigroups	semigroup	NOUN
ejpam-5623	49	12	l(v	l(v	PROPN
ejpam-5623	49	13	,	,	PUNCT
ejpam-5623	49	14	w	w	PROPN
ejpam-5623	49	15	)	)	PUNCT
ejpam-5623	49	16	,	,	PUNCT
ejpam-5623	49	17	s(v	s(v	PROPN
ejpam-5623	49	18	,	,	PUNCT
ejpam-5623	49	19	w	w	NOUN
ejpam-5623	49	20	)	)	PUNCT
ejpam-5623	49	21	,	,	PUNCT
ejpam-5623	49	22	q	q	NOUN
ejpam-5623	49	23	,	,	PUNCT
ejpam-5623	49	24	and	and	CCONJ
ejpam-5623	49	25	fix(v	fix(v	PROPN
ejpam-5623	49	26	,	,	PUNCT
ejpam-5623	49	27	w	w	NOUN
ejpam-5623	49	28	)	)	PUNCT
ejpam-5623	49	29	to	to	PART
ejpam-5623	49	30	be	be	AUX
ejpam-5623	49	31	left	leave	VERB
ejpam-5623	49	32	regular	regular	ADV
ejpam-5623	49	33	,	,	PUNCT
ejpam-5623	49	34	right	right	ADV
ejpam-5623	49	35	regular	regular	ADJ
ejpam-5623	49	36	,	,	PUNCT
ejpam-5623	49	37	and	and	CCONJ
ejpam-5623	49	38	completely	completely	ADV
ejpam-5623	49	39	regular	regular	ADJ
ejpam-5623	49	40	.	.	PUNCT
ejpam-5623	50	1	n.	n.	PROPN
ejpam-5623	50	2	sawatraksa	sawatraksa	PROPN
ejpam-5623	50	3	,	,	PUNCT
ejpam-5623	50	4	p.	p.	NOUN
ejpam-5623	50	5	tantong	tantong	NOUN
ejpam-5623	50	6	/	/	SYM
ejpam-5623	50	7	eur	eur	PROPN
ejpam-5623	50	8	.	.	PUNCT
ejpam-5623	51	1	j.	j.	PROPN
ejpam-5623	51	2	pure	pure	PROPN
ejpam-5623	51	3	appl	appl	PROPN
ejpam-5623	51	4	.	.	PROPN
ejpam-5623	51	5	math	math	PROPN
ejpam-5623	51	6	,	,	PUNCT
ejpam-5623	51	7	18	18	NUM
ejpam-5623	51	8	(	(	PUNCT
ejpam-5623	51	9	1	1	NUM
ejpam-5623	51	10	)	)	PUNCT
ejpam-5623	51	11	(	(	PUNCT
ejpam-5623	51	12	2025	2025	NUM
ejpam-5623	51	13	)	)	PUNCT
ejpam-5623	51	14	,	,	PUNCT
ejpam-5623	51	15	5623	5623	NUM
ejpam-5623	51	16	3	3	NUM
ejpam-5623	51	17	of	of	ADP
ejpam-5623	51	18	15	15	NUM
ejpam-5623	51	19	2	2	NUM
ejpam-5623	51	20	.	.	PUNCT
ejpam-5623	52	1	semigroups	semigroup	NOUN
ejpam-5623	52	2	of	of	ADP
ejpam-5623	52	3	linear	linear	ADJ
ejpam-5623	52	4	transformations	transformation	NOUN
ejpam-5623	52	5	with	with	ADP
ejpam-5623	52	6	invariant	invariant	ADJ
ejpam-5623	52	7	subspaces	subspace	NOUN
ejpam-5623	52	8	in	in	ADP
ejpam-5623	52	9	this	this	DET
ejpam-5623	52	10	section	section	NOUN
ejpam-5623	52	11	,	,	PUNCT
ejpam-5623	52	12	we	we	PRON
ejpam-5623	52	13	assume	assume	VERB
ejpam-5623	52	14	that	that	SCONJ
ejpam-5623	52	15	v	v	NOUN
ejpam-5623	52	16	is	be	AUX
ejpam-5623	52	17	a	a	DET
ejpam-5623	52	18	vector	vector	NOUN
ejpam-5623	52	19	space	space	NOUN
ejpam-5623	52	20	over	over	ADP
ejpam-5623	52	21	a	a	DET
ejpam-5623	52	22	field	field	NOUN
ejpam-5623	52	23	f	f	NOUN
ejpam-5623	52	24	,	,	PUNCT
ejpam-5623	52	25	andw	andw	NOUN
ejpam-5623	52	26	is	be	AUX
ejpam-5623	52	27	a	a	DET
ejpam-5623	52	28	subspace	subspace	NOUN
ejpam-5623	52	29	of	of	ADP
ejpam-5623	52	30	v	v	NOUN
ejpam-5623	52	31	.	.	PUNCT
ejpam-5623	53	1	let	let	VERB
ejpam-5623	53	2	s	s	PRON
ejpam-5623	53	3	represent	represent	VERB
ejpam-5623	53	4	one	one	NUM
ejpam-5623	53	5	of	of	ADP
ejpam-5623	53	6	the	the	DET
ejpam-5623	53	7	semigroups	semigroup	NOUN
ejpam-5623	53	8	s(v	s(v	PROPN
ejpam-5623	53	9	,	,	PUNCT
ejpam-5623	53	10	w	w	NOUN
ejpam-5623	53	11	)	)	PUNCT
ejpam-5623	53	12	,	,	PUNCT
ejpam-5623	53	13	l(v	l(v	PROPN
ejpam-5623	53	14	,	,	PUNCT
ejpam-5623	53	15	w	w	NOUN
ejpam-5623	53	16	)	)	PUNCT
ejpam-5623	53	17	,	,	PUNCT
ejpam-5623	53	18	and	and	CCONJ
ejpam-5623	53	19	q.	q.	NOUN
ejpam-5623	53	20	we	we	PRON
ejpam-5623	53	21	first	first	ADV
ejpam-5623	53	22	characterize	characterize	VERB
ejpam-5623	53	23	right	right	ADJ
ejpam-5623	53	24	regularity	regularity	NOUN
ejpam-5623	53	25	for	for	ADP
ejpam-5623	53	26	elements	element	NOUN
ejpam-5623	53	27	of	of	ADP
ejpam-5623	53	28	s.	s.	PROPN
ejpam-5623	53	29	theorem	theorem	VERB
ejpam-5623	53	30	1	1	X
ejpam-5623	53	31	.	.	PUNCT
ejpam-5623	54	1	let	let	VERB
ejpam-5623	54	2	α	α	PRON
ejpam-5623	54	3	∈	∈	PROPN
ejpam-5623	54	4	s.	s.	PROPN
ejpam-5623	54	5	then	then	ADV
ejpam-5623	54	6	,	,	PUNCT
ejpam-5623	54	7	α	α	PROPN
ejpam-5623	54	8	∈	∈	PROPN
ejpam-5623	54	9	rreg(s	rreg(s	PROPN
ejpam-5623	54	10	)	)	PUNCT
ejpam-5623	55	1	if	if	SCONJ
ejpam-5623	55	2	and	and	CCONJ
ejpam-5623	55	3	only	only	ADV
ejpam-5623	55	4	if	if	SCONJ
ejpam-5623	55	5	α|v	α|v	ADV
ejpam-5623	55	6	α	α	PROPN
ejpam-5623	55	7	is	be	AUX
ejpam-5623	55	8	a	a	DET
ejpam-5623	55	9	one	one	NUM
ejpam-5623	55	10	-	-	PUNCT
ejpam-5623	55	11	to	to	ADP
ejpam-5623	55	12	-	-	PUNCT
ejpam-5623	55	13	one	one	NUM
ejpam-5623	55	14	transformations	transformation	NOUN
ejpam-5623	55	15	on	on	ADP
ejpam-5623	55	16	v	v	NUM
ejpam-5623	55	17	α	α	NOUN
ejpam-5623	55	18	.	.	PUNCT
ejpam-5623	56	1	proof	proof	NOUN
ejpam-5623	56	2	.	.	PUNCT
ejpam-5623	57	1	assume	assume	VERB
ejpam-5623	57	2	that	that	SCONJ
ejpam-5623	57	3	α	α	PRON
ejpam-5623	57	4	is	be	AUX
ejpam-5623	57	5	a	a	DET
ejpam-5623	57	6	right	right	ADJ
ejpam-5623	57	7	regular	regular	ADJ
ejpam-5623	57	8	element	element	NOUN
ejpam-5623	57	9	of	of	ADP
ejpam-5623	57	10	s.	s.	PROPN
ejpam-5623	57	11	thus	thus	ADV
ejpam-5623	57	12	,	,	PUNCT
ejpam-5623	57	13	there	there	PRON
ejpam-5623	57	14	exists	exist	VERB
ejpam-5623	57	15	an	an	DET
ejpam-5623	57	16	element	element	NOUN
ejpam-5623	57	17	β	β	NOUN
ejpam-5623	57	18	of	of	ADP
ejpam-5623	57	19	s	s	PRON
ejpam-5623	58	1	such	such	ADJ
ejpam-5623	58	2	that	that	SCONJ
ejpam-5623	58	3	α	α	NOUN
ejpam-5623	58	4	=	=	SYM
ejpam-5623	58	5	α2β	α2β	NOUN
ejpam-5623	58	6	.	.	PUNCT
ejpam-5623	58	7	let	let	VERB
ejpam-5623	58	8	v1	v1	NOUN
ejpam-5623	58	9	,	,	PUNCT
ejpam-5623	58	10	v2	v2	PROPN
ejpam-5623	58	11	∈	∈	PROPN
ejpam-5623	58	12	v	v	ADP
ejpam-5623	58	13	α	α	NOUN
ejpam-5623	58	14	,	,	PUNCT
ejpam-5623	58	15	and	and	CCONJ
ejpam-5623	58	16	suppose	suppose	VERB
ejpam-5623	58	17	v1α	v1α	PUNCT
ejpam-5623	58	18	=	=	SYM
ejpam-5623	58	19	v2α	v2α	PROPN
ejpam-5623	58	20	.	.	PUNCT
ejpam-5623	59	1	then	then	ADV
ejpam-5623	59	2	,	,	PUNCT
ejpam-5623	59	3	there	there	PRON
ejpam-5623	59	4	exist	exist	VERB
ejpam-5623	59	5	v′1	v′1	NOUN
ejpam-5623	59	6	,	,	PUNCT
ejpam-5623	59	7	v	v	ADJ
ejpam-5623	59	8	′	′	NUM
ejpam-5623	59	9	2	2	NUM
ejpam-5623	59	10	∈	∈	NOUN
ejpam-5623	59	11	v	v	ADP
ejpam-5623	59	12	such	such	ADJ
ejpam-5623	59	13	that	that	DET
ejpam-5623	59	14	v′1α	v′1α	X
ejpam-5623	59	15	=	=	SYM
ejpam-5623	59	16	v1	v1	NOUN
ejpam-5623	59	17	and	and	CCONJ
ejpam-5623	59	18	v′2α	v′2α	NUM
ejpam-5623	59	19	=	=	SYM
ejpam-5623	59	20	v2	v2	PROPN
ejpam-5623	59	21	.	.	PUNCT
ejpam-5623	60	1	therefore	therefore	ADV
ejpam-5623	60	2	,	,	PUNCT
ejpam-5623	60	3	v1	v1	PROPN
ejpam-5623	60	4	=	=	SYM
ejpam-5623	60	5	v′1α	v′1α	PUNCT
ejpam-5623	60	6	=	=	SYM
ejpam-5623	61	1	v′1α	v′1α	PUNCT
ejpam-5623	61	2	2β	2β	NOUN
ejpam-5623	61	3	=	=	SYM
ejpam-5623	61	4	v1αβ	v1αβ	PUNCT
ejpam-5623	61	5	=	=	SYM
ejpam-5623	61	6	v2αβ	v2αβ	PUNCT
ejpam-5623	61	7	=	=	PUNCT
ejpam-5623	62	1	v′2α	v′2α	NUM
ejpam-5623	62	2	2β	2β	NOUN
ejpam-5623	62	3	=	=	SYM
ejpam-5623	62	4	v′2α	v′2α	NUM
ejpam-5623	62	5	=	=	SYM
ejpam-5623	62	6	v2	v2	PROPN
ejpam-5623	62	7	.	.	PUNCT
ejpam-5623	63	1	hence	hence	ADV
ejpam-5623	63	2	,	,	PUNCT
ejpam-5623	63	3	α|v	α|v	PROPN
ejpam-5623	63	4	α	α	PROPN
ejpam-5623	63	5	is	be	AUX
ejpam-5623	63	6	one	one	NUM
ejpam-5623	63	7	-	-	PUNCT
ejpam-5623	63	8	to	to	ADP
ejpam-5623	63	9	-	-	PUNCT
ejpam-5623	63	10	one	one	NUM
ejpam-5623	63	11	.	.	PUNCT
ejpam-5623	64	1	conversely	conversely	ADV
ejpam-5623	64	2	,	,	PUNCT
ejpam-5623	64	3	assume	assume	VERB
ejpam-5623	64	4	that	that	SCONJ
ejpam-5623	64	5	α|v	α|v	PROPN
ejpam-5623	64	6	α	α	PROPN
ejpam-5623	64	7	is	be	AUX
ejpam-5623	64	8	one	one	NUM
ejpam-5623	64	9	-	-	PUNCT
ejpam-5623	64	10	to	to	ADP
ejpam-5623	64	11	-	-	PUNCT
ejpam-5623	64	12	one	one	NUM
ejpam-5623	64	13	.	.	PUNCT
ejpam-5623	65	1	we	we	PRON
ejpam-5623	65	2	will	will	AUX
ejpam-5623	65	3	construct	construct	VERB
ejpam-5623	65	4	β	β	PROPN
ejpam-5623	65	5	∈	∈	PROPN
ejpam-5623	65	6	s	s	VERB
ejpam-5623	65	7	such	such	ADJ
ejpam-5623	65	8	that	that	SCONJ
ejpam-5623	65	9	α	α	NOUN
ejpam-5623	65	10	=	=	SYM
ejpam-5623	65	11	α2β	α2β	NOUN
ejpam-5623	65	12	.	.	PUNCT
ejpam-5623	66	1	let	let	VERB
ejpam-5623	66	2	b	b	X
ejpam-5623	66	3	be	be	AUX
ejpam-5623	66	4	a	a	DET
ejpam-5623	66	5	basis	basis	NOUN
ejpam-5623	66	6	for	for	ADP
ejpam-5623	66	7	v	v	NOUN
ejpam-5623	66	8	α	α	NOUN
ejpam-5623	66	9	,	,	PUNCT
ejpam-5623	66	10	and	and	CCONJ
ejpam-5623	66	11	let	let	VERB
ejpam-5623	66	12	b′	b′	NOUN
ejpam-5623	66	13	=	=	PUNCT
ejpam-5623	66	14	{	{	PUNCT
ejpam-5623	66	15	vα	vα	X
ejpam-5623	66	16	:	:	PUNCT
ejpam-5623	66	17	v	v	NUM
ejpam-5623	66	18	∈	∈	PROPN
ejpam-5623	66	19	b	b	NOUN
ejpam-5623	66	20	}	}	PUNCT
ejpam-5623	66	21	.	.	PUNCT
ejpam-5623	67	1	to	to	PART
ejpam-5623	67	2	show	show	VERB
ejpam-5623	67	3	that	that	SCONJ
ejpam-5623	67	4	b′	b′	NOUN
ejpam-5623	67	5	is	be	AUX
ejpam-5623	67	6	a	a	DET
ejpam-5623	67	7	linearly	linearly	ADV
ejpam-5623	67	8	independent	independent	ADJ
ejpam-5623	67	9	subset	subset	NOUN
ejpam-5623	67	10	of	of	ADP
ejpam-5623	67	11	v	v	NOUN
ejpam-5623	67	12	,	,	PUNCT
ejpam-5623	67	13	let	let	VERB
ejpam-5623	67	14	a1	a1	NOUN
ejpam-5623	67	15	,	,	PUNCT
ejpam-5623	67	16	a2	a2	PROPN
ejpam-5623	67	17	,	,	PUNCT
ejpam-5623	67	18	.	.	PUNCT
ejpam-5623	67	19	.	.	PUNCT
ejpam-5623	68	1	.	.	PUNCT
ejpam-5623	69	1	,	,	PUNCT
ejpam-5623	69	2	an	an	DET
ejpam-5623	69	3	∈	∈	PROPN
ejpam-5623	69	4	f	f	X
ejpam-5623	69	5	,	,	PUNCT
ejpam-5623	69	6	and	and	CCONJ
ejpam-5623	69	7	v1	v1	NOUN
ejpam-5623	69	8	,	,	PUNCT
ejpam-5623	69	9	v2	v2	NOUN
ejpam-5623	69	10	,	,	PUNCT
ejpam-5623	69	11	.	.	PUNCT
ejpam-5623	69	12	.	.	PUNCT
ejpam-5623	70	1	.	.	PUNCT
ejpam-5623	71	1	,	,	PUNCT
ejpam-5623	71	2	vn	vn	PROPN
ejpam-5623	71	3	∈	∈	PROPN
ejpam-5623	72	1	b	b	PROPN
ejpam-5623	72	2	be	be	AUX
ejpam-5623	72	3	such	such	ADJ
ejpam-5623	72	4	that	that	SCONJ
ejpam-5623	72	5	a1(v1α	a1(v1α	NOUN
ejpam-5623	72	6	)	)	PUNCT
ejpam-5623	72	7	+	+	PUNCT
ejpam-5623	72	8	a2(v2α	a2(v2α	X
ejpam-5623	72	9	)	)	PUNCT
ejpam-5623	73	1	+	+	CCONJ
ejpam-5623	73	2	.	.	PUNCT
ejpam-5623	73	3	.	.	PUNCT
ejpam-5623	74	1	.+	.+	NOUN
ejpam-5623	74	2	an(vnα	an(vnα	X
ejpam-5623	74	3	)	)	PUNCT
ejpam-5623	75	1	=	=	SYM
ejpam-5623	75	2	0	0	X
ejpam-5623	75	3	.	.	PUNCT
ejpam-5623	76	1	thus	thus	ADV
ejpam-5623	76	2	,	,	PUNCT
ejpam-5623	76	3	(	(	PUNCT
ejpam-5623	76	4	a1v1+a2v2	a1v1+a2v2	VERB
ejpam-5623	76	5	+	+	PROPN
ejpam-5623	76	6	.	.	PUNCT
ejpam-5623	76	7	.	.	PUNCT
ejpam-5623	77	1	.+anvn)α	.+anvn)α	PUNCT
ejpam-5623	78	1	=	=	PUNCT
ejpam-5623	78	2	0	0	X
ejpam-5623	78	3	.	.	PUNCT
ejpam-5623	78	4	since	since	SCONJ
ejpam-5623	78	5	a1v1+a2v2	a1v1+a2v2	PROPN
ejpam-5623	78	6	+	+	PROPN
ejpam-5623	78	7	.	.	PUNCT
ejpam-5623	78	8	.	.	PUNCT
ejpam-5623	78	9	.+anvn	.+anvn	PUNCT
ejpam-5623	79	1	∈	∈	PROPN
ejpam-5623	79	2	v	v	ADP
ejpam-5623	79	3	α	α	NOUN
ejpam-5623	79	4	and	and	CCONJ
ejpam-5623	79	5	α|v	α|v	PROPN
ejpam-5623	80	1	α	α	PROPN
ejpam-5623	80	2	is	be	AUX
ejpam-5623	80	3	one	one	NUM
ejpam-5623	80	4	-	-	PUNCT
ejpam-5623	80	5	toone	toone	NOUN
ejpam-5623	80	6	,	,	PUNCT
ejpam-5623	80	7	it	it	PRON
ejpam-5623	80	8	follows	follow	VERB
ejpam-5623	80	9	that	that	SCONJ
ejpam-5623	80	10	a1v1+a2v2	a1v1+a2v2	PROPN
ejpam-5623	80	11	+	+	PROPN
ejpam-5623	80	12	.	.	PUNCT
ejpam-5623	80	13	.	.	PUNCT
ejpam-5623	80	14	.+anvn	.+anvn	X
ejpam-5623	81	1	=	=	PUNCT
ejpam-5623	81	2	0	0	X
ejpam-5623	81	3	.	.	PUNCT
ejpam-5623	82	1	since	since	SCONJ
ejpam-5623	82	2	v1	v1	NOUN
ejpam-5623	82	3	,	,	PUNCT
ejpam-5623	82	4	v2	v2	PROPN
ejpam-5623	82	5	,	,	PUNCT
ejpam-5623	82	6	.	.	PUNCT
ejpam-5623	82	7	.	.	PUNCT
ejpam-5623	83	1	.	.	PUNCT
ejpam-5623	84	1	,	,	PUNCT
ejpam-5623	84	2	vn	vn	PROPN
ejpam-5623	84	3	are	be	AUX
ejpam-5623	84	4	linearly	linearly	ADV
ejpam-5623	84	5	independent	independent	ADJ
ejpam-5623	84	6	,	,	PUNCT
ejpam-5623	84	7	we	we	PRON
ejpam-5623	84	8	conclude	conclude	VERB
ejpam-5623	84	9	that	that	PRON
ejpam-5623	84	10	ai	ai	VERB
ejpam-5623	84	11	=	=	NOUN
ejpam-5623	84	12	0	0	NUM
ejpam-5623	84	13	for	for	ADP
ejpam-5623	84	14	all	all	DET
ejpam-5623	84	15	i	i	PRON
ejpam-5623	84	16	=	=	NOUN
ejpam-5623	84	17	1	1	NUM
ejpam-5623	84	18	,	,	PUNCT
ejpam-5623	84	19	2	2	NUM
ejpam-5623	84	20	,	,	PUNCT
ejpam-5623	84	21	.	.	PUNCT
ejpam-5623	84	22	.	.	PUNCT
ejpam-5623	85	1	.	.	PUNCT
ejpam-5623	86	1	,	,	PUNCT
ejpam-5623	86	2	n.	n.	PROPN
ejpam-5623	86	3	hence	hence	ADV
ejpam-5623	86	4	,	,	PUNCT
ejpam-5623	86	5	b′	b′	NUM
ejpam-5623	86	6	is	be	AUX
ejpam-5623	86	7	linearly	linearly	ADV
ejpam-5623	86	8	independent	independent	ADJ
ejpam-5623	86	9	.	.	PUNCT
ejpam-5623	87	1	we	we	PRON
ejpam-5623	87	2	then	then	ADV
ejpam-5623	87	3	construct	construct	VERB
ejpam-5623	87	4	a	a	DET
ejpam-5623	87	5	basis	basis	NOUN
ejpam-5623	87	6	b′′	b′′	VERB
ejpam-5623	87	7	for	for	ADP
ejpam-5623	87	8	v	v	ADP
ejpam-5623	87	9	such	such	ADJ
ejpam-5623	87	10	that	that	DET
ejpam-5623	87	11	b′	b′	NUM
ejpam-5623	87	12	⊆	⊆	NUM
ejpam-5623	87	13	b′′.	b′′.	NOUN
ejpam-5623	87	14	for	for	ADP
ejpam-5623	87	15	each	each	DET
ejpam-5623	87	16	u	u	PROPN
ejpam-5623	87	17	∈	∈	PROPN
ejpam-5623	87	18	b′	b′	NOUN
ejpam-5623	87	19	,	,	PUNCT
ejpam-5623	87	20	there	there	PRON
ejpam-5623	87	21	exists	exist	VERB
ejpam-5623	87	22	a	a	DET
ejpam-5623	87	23	unique	unique	ADJ
ejpam-5623	87	24	u′	u′	PROPN
ejpam-5623	87	25	∈	∈	PROPN
ejpam-5623	87	26	b	b	NOUN
ejpam-5623	87	27	such	such	ADJ
ejpam-5623	87	28	that	that	DET
ejpam-5623	87	29	u′α	u′α	PROPN
ejpam-5623	87	30	=	=	SYM
ejpam-5623	87	31	u	u	NOUN
ejpam-5623	87	32	by	by	ADP
ejpam-5623	87	33	assumption	assumption	NOUN
ejpam-5623	87	34	.	.	PUNCT
ejpam-5623	88	1	define	define	VERB
ejpam-5623	88	2	β	β	NOUN
ejpam-5623	88	3	:	:	PUNCT
ejpam-5623	88	4	b′′	b′′	PROPN
ejpam-5623	88	5	→	→	SYM
ejpam-5623	88	6	v	v	NOUN
ejpam-5623	88	7	by	by	ADP
ejpam-5623	88	8	vβ	vβ	X
ejpam-5623	88	9	=	=	PUNCT
ejpam-5623	88	10	{	{	PUNCT
ejpam-5623	88	11	v′	v′	NOUN
ejpam-5623	88	12	if	if	SCONJ
ejpam-5623	88	13	v	v	NUM
ejpam-5623	88	14	∈	∈	PROPN
ejpam-5623	88	15	b′	b′	NOUN
ejpam-5623	88	16	,	,	PUNCT
ejpam-5623	88	17	0	0	NUM
ejpam-5623	88	18	otherwise	otherwise	ADV
ejpam-5623	88	19	.	.	PUNCT
ejpam-5623	89	1	thus	thus	ADV
ejpam-5623	89	2	,	,	PUNCT
ejpam-5623	89	3	β	β	X
ejpam-5623	89	4	is	be	AUX
ejpam-5623	89	5	well	well	ADV
ejpam-5623	89	6	-	-	PUNCT
ejpam-5623	89	7	defined	define	VERB
ejpam-5623	89	8	and	and	CCONJ
ejpam-5623	89	9	can	can	AUX
ejpam-5623	89	10	be	be	AUX
ejpam-5623	89	11	extended	extend	VERB
ejpam-5623	89	12	to	to	ADP
ejpam-5623	89	13	a	a	DET
ejpam-5623	89	14	linear	linear	ADJ
ejpam-5623	89	15	transformation	transformation	NOUN
ejpam-5623	89	16	on	on	ADP
ejpam-5623	89	17	v	v	NOUN
ejpam-5623	89	18	.	.	PUNCT
ejpam-5623	90	1	let	let	VERB
ejpam-5623	90	2	v	v	NUM
ejpam-5623	90	3	∈	∈	NOUN
ejpam-5623	90	4	v	v	NOUN
ejpam-5623	90	5	.	.	PUNCT
ejpam-5623	91	1	since	since	SCONJ
ejpam-5623	91	2	b′	b′	NOUN
ejpam-5623	91	3	⊆	⊆	NUM
ejpam-5623	91	4	b′′	b′′	PROPN
ejpam-5623	91	5	,	,	PUNCT
ejpam-5623	91	6	there	there	PRON
ejpam-5623	91	7	exist	exist	VERB
ejpam-5623	91	8	positive	positive	ADJ
ejpam-5623	91	9	integers	integer	NOUN
ejpam-5623	91	10	k	k	PROPN
ejpam-5623	91	11	and	and	CCONJ
ejpam-5623	91	12	n	n	CCONJ
ejpam-5623	91	13	such	such	ADJ
ejpam-5623	91	14	that	that	DET
ejpam-5623	91	15	v1	v1	NOUN
ejpam-5623	91	16	,	,	PUNCT
ejpam-5623	91	17	v2	v2	NOUN
ejpam-5623	91	18	,	,	PUNCT
ejpam-5623	91	19	.	.	PUNCT
ejpam-5623	91	20	.	.	PUNCT
ejpam-5623	92	1	.	.	PUNCT
ejpam-5623	93	1	,	,	PUNCT
ejpam-5623	93	2	vk	vk	ADP
ejpam-5623	93	3	∈	∈	PROPN
ejpam-5623	93	4	b′	b′	NUM
ejpam-5623	93	5	and	and	CCONJ
ejpam-5623	93	6	vk+1	vk+1	NOUN
ejpam-5623	93	7	,	,	PUNCT
ejpam-5623	93	8	vk+2	vk+2	NOUN
ejpam-5623	93	9	,	,	PUNCT
ejpam-5623	93	10	.	.	PUNCT
ejpam-5623	93	11	.	.	PUNCT
ejpam-5623	94	1	.	.	PUNCT
ejpam-5623	95	1	,	,	PUNCT
ejpam-5623	95	2	vn	vn	PROPN
ejpam-5623	95	3	∈	∈	PROPN
ejpam-5623	95	4	b′′	b′′	PROPN
ejpam-5623	95	5	\b′	\b′	PROPN
ejpam-5623	95	6	and	and	CCONJ
ejpam-5623	95	7	a1	a1	PROPN
ejpam-5623	95	8	,	,	PUNCT
ejpam-5623	95	9	a2	a2	PROPN
ejpam-5623	95	10	,	,	PUNCT
ejpam-5623	95	11	.	.	PUNCT
ejpam-5623	95	12	.	.	PUNCT
ejpam-5623	96	1	.	.	PUNCT
ejpam-5623	97	1	,	,	PUNCT
ejpam-5623	97	2	an	an	DET
ejpam-5623	97	3	∈	∈	PROPN
ejpam-5623	97	4	f	f	X
ejpam-5623	97	5	with	with	ADP
ejpam-5623	97	6	v	v	NOUN
ejpam-5623	97	7	=	=	SYM
ejpam-5623	97	8	a1v1	a1v1	NOUN
ejpam-5623	97	9	+	+	CCONJ
ejpam-5623	97	10	a2v2	a2v2	PROPN
ejpam-5623	97	11	+	+	NUM
ejpam-5623	97	12	.	.	PUNCT
ejpam-5623	97	13	.	.	PUNCT
ejpam-5623	98	1	.+	.+	NOUN
ejpam-5623	98	2	anvn	anvn	PROPN
ejpam-5623	98	3	.	.	PUNCT
ejpam-5623	99	1	this	this	PRON
ejpam-5623	99	2	implies	imply	VERB
ejpam-5623	99	3	that	that	SCONJ
ejpam-5623	99	4	vβ	vβ	X
ejpam-5623	99	5	=	=	PUNCT
ejpam-5623	99	6	(	(	PUNCT
ejpam-5623	99	7	a1v1	a1v1	PROPN
ejpam-5623	99	8	+	+	SYM
ejpam-5623	99	9	a2v2	a2v2	PUNCT
ejpam-5623	99	10	+	+	CCONJ
ejpam-5623	99	11	.	.	PUNCT
ejpam-5623	99	12	.	.	PUNCT
ejpam-5623	100	1	.+	.+	NOUN
ejpam-5623	100	2	akvk	akvk	VERB
ejpam-5623	100	3	+	+	CCONJ
ejpam-5623	100	4	ak+1vk+1	ak+1vk+1	ADJ
ejpam-5623	100	5	+	+	CCONJ
ejpam-5623	100	6	ak+2vk+2	ak+2vk+2	NOUN
ejpam-5623	100	7	+	+	X
ejpam-5623	100	8	.	.	PUNCT
ejpam-5623	100	9	.	.	PUNCT
ejpam-5623	101	1	.+	.+	NOUN
ejpam-5623	101	2	anvn)β	anvn)β	PUNCT
ejpam-5623	101	3	=	=	SYM
ejpam-5623	101	4	a1(v1β	a1(v1β	NOUN
ejpam-5623	101	5	)	)	PUNCT
ejpam-5623	102	1	+	+	CCONJ
ejpam-5623	102	2	a2(v2β	a2(v2β	NOUN
ejpam-5623	102	3	)	)	PUNCT
ejpam-5623	103	1	+	+	CCONJ
ejpam-5623	103	2	.	.	PUNCT
ejpam-5623	103	3	.	.	PUNCT
ejpam-5623	104	1	.+	.+	NOUN
ejpam-5623	104	2	ak(vkβ	ak(vkβ	PROPN
ejpam-5623	104	3	)	)	PUNCT
ejpam-5623	104	4	+	+	CCONJ
ejpam-5623	104	5	ak+1(vk+1β	ak+1(vk+1β	NOUN
ejpam-5623	104	6	)	)	PUNCT
ejpam-5623	105	1	+	+	NUM
ejpam-5623	105	2	ak+2(vk+2β	ak+2(vk+2β	NOUN
ejpam-5623	105	3	)	)	PUNCT
ejpam-5623	106	1	+	+	CCONJ
ejpam-5623	106	2	.	.	PUNCT
ejpam-5623	106	3	.	.	PUNCT
ejpam-5623	107	1	.+	.+	NOUN
ejpam-5623	107	2	an(vnβ	an(vnβ	PART
ejpam-5623	107	3	)	)	PUNCT
ejpam-5623	107	4	=	=	PUNCT
ejpam-5623	108	1	a1v	a1v	ADJ
ejpam-5623	108	2	′	′	NOUN
ejpam-5623	108	3	1	1	NUM
ejpam-5623	109	1	+	+	CCONJ
ejpam-5623	109	2	a2v	a2v	INTJ
ejpam-5623	109	3	′	′	NUM
ejpam-5623	109	4	2	2	NUM
ejpam-5623	109	5	+	+	CCONJ
ejpam-5623	109	6	.	.	PUNCT
ejpam-5623	109	7	.	.	PUNCT
ejpam-5623	110	1	.+	.+	NOUN
ejpam-5623	110	2	akv	akv	VERB
ejpam-5623	110	3	′	′	NUM
ejpam-5623	110	4	k	k	PROPN
ejpam-5623	111	1	+	+	PUNCT
ejpam-5623	111	2	ak+1(0	ak+1(0	X
ejpam-5623	111	3	)	)	PUNCT
ejpam-5623	112	1	+	+	CCONJ
ejpam-5623	112	2	ak+2(0	ak+2(0	NUM
ejpam-5623	112	3	)	)	PUNCT
ejpam-5623	113	1	+	+	CCONJ
ejpam-5623	113	2	.	.	PUNCT
ejpam-5623	113	3	.	.	PUNCT
ejpam-5623	114	1	.+	.+	NOUN
ejpam-5623	114	2	an(0	an(0	PROPN
ejpam-5623	114	3	)	)	PUNCT
ejpam-5623	114	4	=	=	PUNCT
ejpam-5623	115	1	a1v	a1v	ADJ
ejpam-5623	115	2	′	′	NOUN
ejpam-5623	115	3	1	1	NUM
ejpam-5623	116	1	+	+	CCONJ
ejpam-5623	116	2	a2v	a2v	INTJ
ejpam-5623	116	3	′	′	NUM
ejpam-5623	116	4	2	2	NUM
ejpam-5623	116	5	+	+	CCONJ
ejpam-5623	116	6	.	.	PUNCT
ejpam-5623	116	7	.	.	PUNCT
ejpam-5623	117	1	.+	.+	NOUN
ejpam-5623	117	2	akv	akv	VERB
ejpam-5623	117	3	′	′	PROPN
ejpam-5623	117	4	k.	k.	PROPN
ejpam-5623	118	1	since	since	SCONJ
ejpam-5623	118	2	v′1	v′1	PROPN
ejpam-5623	118	3	,	,	PUNCT
ejpam-5623	118	4	v	v	ADJ
ejpam-5623	118	5	′	′	NUM
ejpam-5623	118	6	2	2	NUM
ejpam-5623	118	7	,	,	PUNCT
ejpam-5623	118	8	.	.	PUNCT
ejpam-5623	118	9	.	.	PUNCT
ejpam-5623	119	1	.	.	PUNCT
ejpam-5623	120	1	,	,	PUNCT
ejpam-5623	120	2	v	v	X
ejpam-5623	120	3	′	′	NUM
ejpam-5623	121	1	k	k	NOUN
ejpam-5623	121	2	are	be	AUX
ejpam-5623	121	3	all	all	PRON
ejpam-5623	121	4	elements	element	NOUN
ejpam-5623	121	5	in	in	ADP
ejpam-5623	121	6	a	a	DET
ejpam-5623	121	7	basis	basis	NOUN
ejpam-5623	121	8	b	b	NOUN
ejpam-5623	121	9	of	of	ADP
ejpam-5623	121	10	v	v	NUM
ejpam-5623	121	11	α	α	NOUN
ejpam-5623	121	12	,	,	PUNCT
ejpam-5623	121	13	we	we	PRON
ejpam-5623	121	14	conclude	conclude	VERB
ejpam-5623	121	15	that	that	SCONJ
ejpam-5623	121	16	a1v	a1v	PROPN
ejpam-5623	121	17	′	′	NOUN
ejpam-5623	121	18	1	1	NUM
ejpam-5623	121	19	+	+	CCONJ
ejpam-5623	121	20	a2v	a2v	INTJ
ejpam-5623	121	21	′	′	NUM
ejpam-5623	121	22	2	2	NUM
ejpam-5623	121	23	+	+	CCONJ
ejpam-5623	121	24	.	.	PUNCT
ejpam-5623	121	25	.	.	PUNCT
ejpam-5623	121	26	.	.	PUNCT
ejpam-5623	122	1	+	+	CCONJ
ejpam-5623	122	2	akv	akv	PROPN
ejpam-5623	122	3	′	′	NUM
ejpam-5623	122	4	k	k	PROPN
ejpam-5623	122	5	∈	∈	PROPN
ejpam-5623	122	6	v	v	ADP
ejpam-5623	122	7	α	α	NOUN
ejpam-5623	122	8	.	.	PUNCT
ejpam-5623	123	1	therefore	therefore	ADV
ejpam-5623	123	2	,	,	PUNCT
ejpam-5623	123	3	β	β	X
ejpam-5623	123	4	∈	∈	PROPN
ejpam-5623	123	5	l(v	l(v	PROPN
ejpam-5623	123	6	,	,	PUNCT
ejpam-5623	123	7	w	w	PROPN
ejpam-5623	123	8	)	)	PUNCT
ejpam-5623	123	9	.	.	PUNCT
ejpam-5623	124	1	this	this	PRON
ejpam-5623	124	2	implies	imply	VERB
ejpam-5623	124	3	that	that	SCONJ
ejpam-5623	124	4	wβ	wβ	ADP
ejpam-5623	124	5	⊆	⊆	NUM
ejpam-5623	124	6	v	v	NOUN
ejpam-5623	124	7	β	β	NOUN
ejpam-5623	124	8	⊆	⊆	NUM
ejpam-5623	124	9	w	w	NOUN
ejpam-5623	124	10	,	,	PUNCT
ejpam-5623	124	11	and	and	CCONJ
ejpam-5623	124	12	so	so	ADV
ejpam-5623	124	13	n.	n.	PROPN
ejpam-5623	124	14	sawatraksa	sawatraksa	PROPN
ejpam-5623	124	15	,	,	PUNCT
ejpam-5623	124	16	p.	p.	NOUN
ejpam-5623	124	17	tantong	tantong	NOUN
ejpam-5623	124	18	/	/	SYM
ejpam-5623	124	19	eur	eur	PROPN
ejpam-5623	124	20	.	.	PUNCT
ejpam-5623	125	1	j.	j.	PROPN
ejpam-5623	125	2	pure	pure	PROPN
ejpam-5623	125	3	appl	appl	PROPN
ejpam-5623	125	4	.	.	PROPN
ejpam-5623	125	5	math	math	PROPN
ejpam-5623	125	6	,	,	PUNCT
ejpam-5623	125	7	18	18	NUM
ejpam-5623	125	8	(	(	PUNCT
ejpam-5623	125	9	1	1	NUM
ejpam-5623	125	10	)	)	PUNCT
ejpam-5623	125	11	(	(	PUNCT
ejpam-5623	125	12	2025	2025	NUM
ejpam-5623	125	13	)	)	PUNCT
ejpam-5623	125	14	,	,	PUNCT
ejpam-5623	125	15	5623	5623	NUM
ejpam-5623	125	16	4	4	NUM
ejpam-5623	125	17	of	of	ADP
ejpam-5623	125	18	15	15	NUM
ejpam-5623	125	19	β	β	NOUN
ejpam-5623	125	20	∈	∈	PROPN
ejpam-5623	125	21	s(v	s(v	PROPN
ejpam-5623	125	22	,	,	PUNCT
ejpam-5623	125	23	w	w	NOUN
ejpam-5623	125	24	)	)	PUNCT
ejpam-5623	125	25	.	.	PUNCT
ejpam-5623	126	1	moreover	moreover	ADV
ejpam-5623	126	2	,	,	PUNCT
ejpam-5623	126	3	we	we	PRON
ejpam-5623	126	4	will	will	AUX
ejpam-5623	126	5	now	now	ADV
ejpam-5623	126	6	show	show	VERB
ejpam-5623	126	7	that	that	SCONJ
ejpam-5623	126	8	β	β	PROPN
ejpam-5623	126	9	∈	∈	PROPN
ejpam-5623	126	10	q.	q.	PROPN
ejpam-5623	126	11	let	let	VERB
ejpam-5623	126	12	w	w	NOUN
ejpam-5623	126	13	=	=	PUNCT
ejpam-5623	126	14	a1v1	a1v1	PROPN
ejpam-5623	126	15	+	+	CCONJ
ejpam-5623	126	16	a2v2	a2v2	PROPN
ejpam-5623	126	17	+	+	CCONJ
ejpam-5623	126	18	.	.	PUNCT
ejpam-5623	126	19	.	.	PUNCT
ejpam-5623	127	1	.+	.+	NOUN
ejpam-5623	127	2	akvk	akvk	VERB
ejpam-5623	127	3	.	.	PUNCT
ejpam-5623	128	1	then	then	ADV
ejpam-5623	128	2	,	,	PUNCT
ejpam-5623	128	3	w	w	PROPN
ejpam-5623	128	4	∈	∈	PROPN
ejpam-5623	128	5	w	w	NOUN
ejpam-5623	128	6	,	,	PUNCT
ejpam-5623	128	7	and	and	CCONJ
ejpam-5623	128	8	wβ	wβ	ADP
ejpam-5623	128	9	=	=	PUNCT
ejpam-5623	128	10	(	(	PUNCT
ejpam-5623	128	11	a1v1	a1v1	PROPN
ejpam-5623	128	12	+	+	SYM
ejpam-5623	128	13	a2v2	a2v2	PUNCT
ejpam-5623	128	14	+	+	CCONJ
ejpam-5623	128	15	.	.	PUNCT
ejpam-5623	128	16	.	.	PUNCT
ejpam-5623	129	1	.+	.+	NOUN
ejpam-5623	129	2	akvk)β	akvk)β	PUNCT
ejpam-5623	129	3	=	=	SYM
ejpam-5623	129	4	a1(v1β	a1(v1β	NOUN
ejpam-5623	129	5	)	)	PUNCT
ejpam-5623	130	1	+	+	CCONJ
ejpam-5623	130	2	a2(v2β	a2(v2β	NOUN
ejpam-5623	130	3	)	)	PUNCT
ejpam-5623	131	1	+	+	CCONJ
ejpam-5623	131	2	.	.	PUNCT
ejpam-5623	131	3	.	.	PUNCT
ejpam-5623	132	1	.+	.+	NOUN
ejpam-5623	132	2	ak(vkβ	ak(vkβ	PROPN
ejpam-5623	132	3	)	)	PUNCT
ejpam-5623	132	4	=	=	PUNCT
ejpam-5623	133	1	a1v	a1v	ADJ
ejpam-5623	133	2	′	′	NOUN
ejpam-5623	133	3	1	1	NUM
ejpam-5623	134	1	+	+	CCONJ
ejpam-5623	134	2	a2v	a2v	INTJ
ejpam-5623	134	3	′	′	NUM
ejpam-5623	134	4	2	2	NUM
ejpam-5623	134	5	+	+	CCONJ
ejpam-5623	134	6	.	.	PUNCT
ejpam-5623	134	7	.	.	PUNCT
ejpam-5623	135	1	.+	.+	NOUN
ejpam-5623	135	2	akv	akv	VERB
ejpam-5623	135	3	′	′	NOUN
ejpam-5623	135	4	k	k	PROPN
ejpam-5623	136	1	=	=	PUNCT
ejpam-5623	136	2	vβ	vβ	X
ejpam-5623	136	3	.	.	PUNCT
ejpam-5623	137	1	we	we	PRON
ejpam-5623	137	2	can	can	AUX
ejpam-5623	137	3	conclude	conclude	VERB
ejpam-5623	137	4	that	that	SCONJ
ejpam-5623	137	5	β	β	PROPN
ejpam-5623	137	6	∈	∈	PROPN
ejpam-5623	137	7	q.	q.	NOUN
ejpam-5623	137	8	this	this	PRON
ejpam-5623	137	9	shows	show	VERB
ejpam-5623	137	10	that	that	SCONJ
ejpam-5623	137	11	β	β	PROPN
ejpam-5623	137	12	∈	∈	PROPN
ejpam-5623	137	13	s.	s.	PROPN
ejpam-5623	137	14	finally	finally	ADV
ejpam-5623	137	15	,	,	PUNCT
ejpam-5623	137	16	we	we	PRON
ejpam-5623	137	17	show	show	VERB
ejpam-5623	137	18	that	that	SCONJ
ejpam-5623	137	19	α	α	NOUN
ejpam-5623	137	20	=	=	SYM
ejpam-5623	137	21	α2β	α2β	NOUN
ejpam-5623	137	22	.	.	PUNCT
ejpam-5623	138	1	let	let	VERB
ejpam-5623	138	2	v	v	NUM
ejpam-5623	138	3	∈	∈	NOUN
ejpam-5623	138	4	v	v	NOUN
ejpam-5623	138	5	.	.	PUNCT
ejpam-5623	139	1	since	since	SCONJ
ejpam-5623	139	2	vα	vα	INTJ
ejpam-5623	139	3	∈	∈	PROPN
ejpam-5623	139	4	v	v	ADP
ejpam-5623	139	5	α	α	NOUN
ejpam-5623	139	6	,	,	PUNCT
ejpam-5623	139	7	we	we	PRON
ejpam-5623	139	8	can	can	AUX
ejpam-5623	139	9	express	express	VERB
ejpam-5623	139	10	vα	vα	ADP
ejpam-5623	139	11	=	=	PUNCT
ejpam-5623	139	12	a1u	a1u	PROPN
ejpam-5623	139	13	′	′	NOUN
ejpam-5623	139	14	1	1	NUM
ejpam-5623	140	1	+	+	CCONJ
ejpam-5623	140	2	a2u	a2u	ADP
ejpam-5623	140	3	′	′	NUM
ejpam-5623	140	4	2	2	NUM
ejpam-5623	140	5	+	+	CCONJ
ejpam-5623	140	6	.	.	PUNCT
ejpam-5623	140	7	.	.	PUNCT
ejpam-5623	141	1	.+	.+	NOUN
ejpam-5623	141	2	anu	anu	VERB
ejpam-5623	142	1	′	′	NUM
ejpam-5623	142	2	n	n	CCONJ
ejpam-5623	142	3	where	where	SCONJ
ejpam-5623	142	4	u′1	u′1	NOUN
ejpam-5623	142	5	,	,	PUNCT
ejpam-5623	142	6	u	u	NOUN
ejpam-5623	142	7	′	′	NOUN
ejpam-5623	142	8	2	2	NUM
ejpam-5623	142	9	,	,	PUNCT
ejpam-5623	142	10	.	.	PUNCT
ejpam-5623	142	11	.	.	PUNCT
ejpam-5623	143	1	.	.	PUNCT
ejpam-5623	144	1	,	,	PUNCT
ejpam-5623	144	2	u	u	NOUN
ejpam-5623	144	3	′	′	NOUN
ejpam-5623	144	4	n	n	CCONJ
ejpam-5623	144	5	∈	∈	PROPN
ejpam-5623	144	6	b	b	PROPN
ejpam-5623	144	7	with	with	ADP
ejpam-5623	144	8	u′iα	u′iα	NOUN
ejpam-5623	144	9	=	=	SYM
ejpam-5623	144	10	ui	ui	PROPN
ejpam-5623	144	11	for	for	ADP
ejpam-5623	144	12	all	all	PRON
ejpam-5623	144	13	i	i	PRON
ejpam-5623	144	14	∈	∈	PROPN
ejpam-5623	144	15	{	{	PUNCT
ejpam-5623	144	16	1	1	NUM
ejpam-5623	144	17	,	,	PUNCT
ejpam-5623	144	18	2	2	NUM
ejpam-5623	144	19	,	,	PUNCT
ejpam-5623	144	20	.	.	PUNCT
ejpam-5623	144	21	.	.	PUNCT
ejpam-5623	145	1	.	.	PUNCT
ejpam-5623	145	2	,	,	PUNCT
ejpam-5623	146	1	n	n	CCONJ
ejpam-5623	146	2	}	}	PUNCT
ejpam-5623	146	3	,	,	PUNCT
ejpam-5623	146	4	and	and	CCONJ
ejpam-5623	146	5	a1	a1	NOUN
ejpam-5623	146	6	,	,	PUNCT
ejpam-5623	146	7	a2	a2	PROPN
ejpam-5623	146	8	,	,	PUNCT
ejpam-5623	146	9	.	.	PUNCT
ejpam-5623	146	10	.	.	PUNCT
ejpam-5623	147	1	.	.	PUNCT
ejpam-5623	148	1	,	,	PUNCT
ejpam-5623	148	2	an	an	DET
ejpam-5623	148	3	∈	∈	PROPN
ejpam-5623	148	4	f.	f.	PROPN
ejpam-5623	148	5	therefore	therefore	ADV
ejpam-5623	148	6	,	,	PUNCT
ejpam-5623	148	7	vα2β	vα2β	PROPN
ejpam-5623	148	8	=	=	PUNCT
ejpam-5623	148	9	(	(	PUNCT
ejpam-5623	148	10	a1u	a1u	PROPN
ejpam-5623	148	11	′	′	NOUN
ejpam-5623	148	12	1	1	NUM
ejpam-5623	149	1	+	+	CCONJ
ejpam-5623	149	2	a2u	a2u	ADP
ejpam-5623	149	3	′	′	NUM
ejpam-5623	149	4	2	2	NUM
ejpam-5623	149	5	+	+	CCONJ
ejpam-5623	149	6	.	.	PUNCT
ejpam-5623	149	7	.	.	PUNCT
ejpam-5623	150	1	.+	.+	NOUN
ejpam-5623	150	2	anu	anu	INTJ
ejpam-5623	151	1	′	′	NUM
ejpam-5623	152	1	n)αβ	n)αβ	PROPN
ejpam-5623	153	1	=	=	PRON
ejpam-5623	154	1	(	(	PUNCT
ejpam-5623	154	2	a1u	a1u	PROPN
ejpam-5623	154	3	′	′	NUM
ejpam-5623	154	4	1α+	1α+	NUM
ejpam-5623	154	5	a2u	a2u	ADP
ejpam-5623	155	1	′	′	NUM
ejpam-5623	155	2	2α+	2α+	NUM
ejpam-5623	155	3	.	.	PUNCT
ejpam-5623	155	4	.	.	PUNCT
ejpam-5623	156	1	.+	.+	NOUN
ejpam-5623	156	2	anu	anu	VERB
ejpam-5623	156	3	′	′	NUM
ejpam-5623	156	4	nα)β	nα)β	PROPN
ejpam-5623	156	5	=	=	PUNCT
ejpam-5623	156	6	(	(	PUNCT
ejpam-5623	156	7	a1u1	a1u1	PUNCT
ejpam-5623	156	8	+	+	X
ejpam-5623	156	9	a2u2	a2u2	PROPN
ejpam-5623	156	10	+	+	X
ejpam-5623	156	11	.	.	PUNCT
ejpam-5623	156	12	.	.	PUNCT
ejpam-5623	157	1	.+	.+	NOUN
ejpam-5623	157	2	anun)β	anun)β	PROPN
ejpam-5623	158	1	=	=	SYM
ejpam-5623	158	2	a1u	a1u	PROPN
ejpam-5623	159	1	′	′	NOUN
ejpam-5623	159	2	1	1	NUM
ejpam-5623	160	1	+	+	CCONJ
ejpam-5623	160	2	a2u	a2u	ADP
ejpam-5623	160	3	′	′	NUM
ejpam-5623	160	4	2	2	NUM
ejpam-5623	160	5	+	+	CCONJ
ejpam-5623	160	6	.	.	PUNCT
ejpam-5623	160	7	.	.	PUNCT
ejpam-5623	161	1	.+	.+	NOUN
ejpam-5623	161	2	anu	anu	VERB
ejpam-5623	161	3	′	′	NUM
ejpam-5623	162	1	n	n	PROPN
ejpam-5623	162	2	=	=	SYM
ejpam-5623	162	3	vα	vα	PROPN
ejpam-5623	162	4	.	.	PUNCT
ejpam-5623	163	1	hence	hence	ADV
ejpam-5623	163	2	,	,	PUNCT
ejpam-5623	163	3	α	α	PROPN
ejpam-5623	163	4	is	be	AUX
ejpam-5623	163	5	right	right	ADV
ejpam-5623	163	6	regular	regular	ADJ
ejpam-5623	163	7	,	,	PUNCT
ejpam-5623	163	8	as	as	SCONJ
ejpam-5623	163	9	required	require	VERB
ejpam-5623	163	10	.	.	PUNCT
ejpam-5623	164	1	the	the	DET
ejpam-5623	164	2	following	following	ADJ
ejpam-5623	164	3	next	next	ADJ
ejpam-5623	164	4	theorem	theorem	NOUN
ejpam-5623	164	5	characterizes	characterize	VERB
ejpam-5623	164	6	when	when	SCONJ
ejpam-5623	164	7	s	s	NOUN
ejpam-5623	164	8	is	be	AUX
ejpam-5623	164	9	a	a	DET
ejpam-5623	164	10	right	right	ADJ
ejpam-5623	164	11	regular	regular	ADJ
ejpam-5623	164	12	semigroup	semigroup	NOUN
ejpam-5623	164	13	.	.	PUNCT
ejpam-5623	165	1	theorem	theorem	NOUN
ejpam-5623	165	2	2	2	NUM
ejpam-5623	165	3	.	.	PUNCT
ejpam-5623	166	1	the	the	DET
ejpam-5623	166	2	following	follow	VERB
ejpam-5623	166	3	statements	statement	NOUN
ejpam-5623	166	4	are	be	AUX
ejpam-5623	166	5	equivalent	equivalent	ADJ
ejpam-5623	166	6	:	:	PUNCT
ejpam-5623	166	7	(	(	PUNCT
ejpam-5623	166	8	i	i	NOUN
ejpam-5623	166	9	)	)	PUNCT
ejpam-5623	166	10	s	s	VERB
ejpam-5623	166	11	is	be	AUX
ejpam-5623	166	12	a	a	DET
ejpam-5623	166	13	right	right	ADJ
ejpam-5623	166	14	regular	regular	ADJ
ejpam-5623	166	15	semigroup	semigroup	NOUN
ejpam-5623	166	16	.	.	PUNCT
ejpam-5623	167	1	(	(	PUNCT
ejpam-5623	167	2	ii	ii	NOUN
ejpam-5623	167	3	)	)	PUNCT
ejpam-5623	167	4	rreg(s	rreg(s	NOUN
ejpam-5623	167	5	)	)	PUNCT
ejpam-5623	167	6	is	be	AUX
ejpam-5623	167	7	a	a	DET
ejpam-5623	167	8	subsemigroup	subsemigroup	NOUN
ejpam-5623	167	9	of	of	ADP
ejpam-5623	167	10	s.	s.	PROPN
ejpam-5623	167	11	(	(	PUNCT
ejpam-5623	167	12	iii	iii	NOUN
ejpam-5623	167	13	)	)	PUNCT
ejpam-5623	167	14	dim(w	dim(w	NOUN
ejpam-5623	167	15	)	)	PUNCT
ejpam-5623	167	16	≤	≤	NUM
ejpam-5623	167	17	1	1	NUM
ejpam-5623	167	18	.	.	PUNCT
ejpam-5623	168	1	proof	proof	NOUN
ejpam-5623	168	2	.	.	PUNCT
ejpam-5623	169	1	(	(	PUNCT
ejpam-5623	169	2	i	i	NOUN
ejpam-5623	169	3	)	)	PUNCT
ejpam-5623	169	4	⇒	⇒	PROPN
ejpam-5623	169	5	(	(	PUNCT
ejpam-5623	169	6	ii	ii	NOUN
ejpam-5623	169	7	)	)	PUNCT
ejpam-5623	169	8	this	this	PRON
ejpam-5623	169	9	is	be	AUX
ejpam-5623	169	10	clear	clear	ADJ
ejpam-5623	169	11	by	by	ADP
ejpam-5623	169	12	definition	definition	NOUN
ejpam-5623	169	13	.	.	PUNCT
ejpam-5623	170	1	(	(	PUNCT
ejpam-5623	170	2	ii	ii	NOUN
ejpam-5623	170	3	)	)	PUNCT
ejpam-5623	170	4	⇒	⇒	NOUN
ejpam-5623	170	5	(	(	PUNCT
ejpam-5623	170	6	iii	iii	X
ejpam-5623	170	7	)	)	PUNCT
ejpam-5623	170	8	we	we	PRON
ejpam-5623	170	9	will	will	AUX
ejpam-5623	170	10	prove	prove	VERB
ejpam-5623	170	11	the	the	DET
ejpam-5623	170	12	contrapositive	contrapositive	NOUN
ejpam-5623	170	13	.	.	PUNCT
ejpam-5623	171	1	assume	assume	VERB
ejpam-5623	171	2	that	that	SCONJ
ejpam-5623	171	3	dim(w	dim(w	ADJ
ejpam-5623	171	4	)	)	PUNCT
ejpam-5623	171	5	≥	≥	NOUN
ejpam-5623	171	6	2	2	NUM
ejpam-5623	171	7	.	.	PUNCT
ejpam-5623	172	1	then	then	ADV
ejpam-5623	172	2	,	,	PUNCT
ejpam-5623	172	3	there	there	PRON
ejpam-5623	172	4	exists	exist	VERB
ejpam-5623	172	5	a	a	DET
ejpam-5623	172	6	basis	basis	NOUN
ejpam-5623	172	7	bw	bw	NOUN
ejpam-5623	172	8	of	of	ADP
ejpam-5623	172	9	w	w	ADP
ejpam-5623	172	10	such	such	ADJ
ejpam-5623	172	11	that	that	SCONJ
ejpam-5623	172	12	|bw	|bw	ADJ
ejpam-5623	172	13	|	|	NOUN
ejpam-5623	172	14	≥	≥	NOUN
ejpam-5623	172	15	2	2	NUM
ejpam-5623	172	16	.	.	PUNCT
ejpam-5623	173	1	now	now	ADV
ejpam-5623	173	2	,	,	PUNCT
ejpam-5623	173	3	let	let	VERB
ejpam-5623	173	4	b	b	X
ejpam-5623	173	5	be	be	AUX
ejpam-5623	173	6	a	a	DET
ejpam-5623	173	7	basis	basis	NOUN
ejpam-5623	173	8	for	for	ADP
ejpam-5623	173	9	v	v	NOUN
ejpam-5623	173	10	such	such	ADJ
ejpam-5623	173	11	that	that	PRON
ejpam-5623	173	12	bw	bw	PROPN
ejpam-5623	173	13	⊆	⊆	NUM
ejpam-5623	173	14	b.	b.	PROPN
ejpam-5623	173	15	let	let	VERB
ejpam-5623	173	16	a	a	PRON
ejpam-5623	173	17	and	and	CCONJ
ejpam-5623	173	18	b	b	NOUN
ejpam-5623	173	19	be	be	AUX
ejpam-5623	173	20	distinct	distinct	ADJ
ejpam-5623	173	21	elements	element	NOUN
ejpam-5623	173	22	of	of	ADP
ejpam-5623	173	23	bw	bw	PROPN
ejpam-5623	173	24	.	.	PUNCT
ejpam-5623	174	1	define	define	VERB
ejpam-5623	174	2	two	two	NUM
ejpam-5623	174	3	mappings	mapping	NOUN
ejpam-5623	174	4	α	α	NOUN
ejpam-5623	174	5	and	and	CCONJ
ejpam-5623	174	6	β	β	NOUN
ejpam-5623	174	7	from	from	ADP
ejpam-5623	174	8	b	b	PROPN
ejpam-5623	174	9	into	into	ADP
ejpam-5623	174	10	v	v	NOUN
ejpam-5623	174	11	as	as	SCONJ
ejpam-5623	174	12	follows	follow	VERB
ejpam-5623	174	13	:	:	PUNCT
ejpam-5623	174	14	xα	xα	PROPN
ejpam-5623	175	1	=	=	PUNCT
ejpam-5623	176	1			PUNCT
ejpam-5623	176	2	a	a	PRON
ejpam-5623	176	3	if	if	NOUN
ejpam-5623	176	4	x	x	PROPN
ejpam-5623	176	5	=	=	SYM
ejpam-5623	176	6	b	b	PROPN
ejpam-5623	176	7	,	,	PUNCT
ejpam-5623	176	8	b	b	NOUN
ejpam-5623	177	1	if	if	SCONJ
ejpam-5623	177	2	x	x	PROPN
ejpam-5623	177	3	=	=	PUNCT
ejpam-5623	177	4	a	a	X
ejpam-5623	177	5	,	,	PUNCT
ejpam-5623	177	6	0	0	NUM
ejpam-5623	177	7	otherwise	otherwise	ADV
ejpam-5623	177	8	,	,	PUNCT
ejpam-5623	177	9	and	and	CCONJ
ejpam-5623	177	10	xβ	xβ	ADV
ejpam-5623	177	11	=	=	SYM
ejpam-5623	177	12	{	{	PUNCT
ejpam-5623	177	13	b	b	NOUN
ejpam-5623	177	14	if	if	SCONJ
ejpam-5623	177	15	x	x	PROPN
ejpam-5623	177	16	=	=	SYM
ejpam-5623	177	17	b	b	PROPN
ejpam-5623	177	18	,	,	PUNCT
ejpam-5623	177	19	0	0	NUM
ejpam-5623	177	20	otherwise	otherwise	ADV
ejpam-5623	177	21	.	.	PUNCT
ejpam-5623	178	1	it	it	PRON
ejpam-5623	178	2	is	be	AUX
ejpam-5623	178	3	easy	easy	ADJ
ejpam-5623	178	4	to	to	PART
ejpam-5623	178	5	verify	verify	VERB
ejpam-5623	178	6	that	that	SCONJ
ejpam-5623	178	7	α	α	PROPN
ejpam-5623	178	8	and	and	CCONJ
ejpam-5623	178	9	β	β	X
ejpam-5623	178	10	are	be	AUX
ejpam-5623	178	11	well	well	ADV
ejpam-5623	178	12	-	-	PUNCT
ejpam-5623	178	13	defined	define	VERB
ejpam-5623	178	14	and	and	CCONJ
ejpam-5623	178	15	can	can	AUX
ejpam-5623	178	16	be	be	AUX
ejpam-5623	178	17	extended	extend	VERB
ejpam-5623	178	18	to	to	ADP
ejpam-5623	178	19	linear	linear	ADJ
ejpam-5623	178	20	transformations	transformation	NOUN
ejpam-5623	178	21	on	on	ADP
ejpam-5623	178	22	v	v	NUM
ejpam-5623	178	23	.	.	PUNCT
ejpam-5623	179	1	from	from	ADP
ejpam-5623	179	2	their	their	PRON
ejpam-5623	179	3	definitions	definition	NOUN
ejpam-5623	179	4	,	,	PUNCT
ejpam-5623	179	5	α	α	X
ejpam-5623	179	6	,	,	PUNCT
ejpam-5623	179	7	β	β	X
ejpam-5623	179	8	∈	∈	PROPN
ejpam-5623	179	9	s	s	NOUN
ejpam-5623	179	10	,	,	PUNCT
ejpam-5623	179	11	and	and	CCONJ
ejpam-5623	179	12	we	we	PRON
ejpam-5623	179	13	will	will	AUX
ejpam-5623	179	14	show	show	VERB
ejpam-5623	179	15	that	that	SCONJ
ejpam-5623	179	16	both	both	PRON
ejpam-5623	179	17	are	be	AUX
ejpam-5623	179	18	right	right	ADJ
ejpam-5623	179	19	regular	regular	ADJ
ejpam-5623	179	20	n.	n.	NOUN
ejpam-5623	179	21	sawatraksa	sawatraksa	NOUN
ejpam-5623	179	22	,	,	PUNCT
ejpam-5623	179	23	p.	p.	NOUN
ejpam-5623	179	24	tantong	tantong	NOUN
ejpam-5623	179	25	/	/	SYM
ejpam-5623	179	26	eur	eur	PROPN
ejpam-5623	179	27	.	.	PUNCT
ejpam-5623	180	1	j.	j.	PROPN
ejpam-5623	180	2	pure	pure	PROPN
ejpam-5623	180	3	appl	appl	PROPN
ejpam-5623	180	4	.	.	PROPN
ejpam-5623	180	5	math	math	PROPN
ejpam-5623	180	6	,	,	PUNCT
ejpam-5623	180	7	18	18	NUM
ejpam-5623	180	8	(	(	PUNCT
ejpam-5623	180	9	1	1	NUM
ejpam-5623	180	10	)	)	PUNCT
ejpam-5623	180	11	(	(	PUNCT
ejpam-5623	180	12	2025	2025	NUM
ejpam-5623	180	13	)	)	PUNCT
ejpam-5623	180	14	,	,	PUNCT
ejpam-5623	180	15	5623	5623	NUM
ejpam-5623	180	16	5	5	NUM
ejpam-5623	180	17	of	of	ADP
ejpam-5623	180	18	15	15	NUM
ejpam-5623	180	19	elements	element	NOUN
ejpam-5623	180	20	of	of	ADP
ejpam-5623	180	21	s	s	NOUN
ejpam-5623	180	22	,	,	PUNCT
ejpam-5623	180	23	but	but	CCONJ
ejpam-5623	180	24	their	their	PRON
ejpam-5623	180	25	product	product	NOUN
ejpam-5623	181	1	αβ	αβ	INTJ
ejpam-5623	181	2	is	be	AUX
ejpam-5623	181	3	not	not	PART
ejpam-5623	181	4	right	right	ADV
ejpam-5623	181	5	regular	regular	ADJ
ejpam-5623	181	6	.	.	PUNCT
ejpam-5623	182	1	to	to	PART
ejpam-5623	182	2	check	check	VERB
ejpam-5623	182	3	that	that	SCONJ
ejpam-5623	182	4	α	α	PRON
ejpam-5623	182	5	is	be	AUX
ejpam-5623	182	6	right	right	ADV
ejpam-5623	182	7	regular	regular	ADV
ejpam-5623	182	8	,	,	PUNCT
ejpam-5623	182	9	we	we	PRON
ejpam-5623	182	10	use	use	VERB
ejpam-5623	182	11	theorem	theorem	NOUN
ejpam-5623	182	12	1	1	X
ejpam-5623	182	13	.	.	PUNCT
ejpam-5623	182	14	suppose	suppose	VERB
ejpam-5623	182	15	u	u	PRON
ejpam-5623	182	16	∈	∈	PROPN
ejpam-5623	182	17	v	v	ADP
ejpam-5623	182	18	α	α	NOUN
ejpam-5623	182	19	and	and	CCONJ
ejpam-5623	182	20	uα	uα	NOUN
ejpam-5623	183	1	=	=	NOUN
ejpam-5623	183	2	0	0	PROPN
ejpam-5623	183	3	.	.	PUNCT
ejpam-5623	184	1	from	from	ADP
ejpam-5623	184	2	the	the	DET
ejpam-5623	184	3	definition	definition	NOUN
ejpam-5623	184	4	of	of	ADP
ejpam-5623	184	5	α	α	NOUN
ejpam-5623	184	6	,	,	PUNCT
ejpam-5623	184	7	we	we	PRON
ejpam-5623	184	8	know	know	VERB
ejpam-5623	184	9	that	that	SCONJ
ejpam-5623	184	10	v	v	ADP
ejpam-5623	184	11	α	α	NOUN
ejpam-5623	184	12	=	=	PUNCT
ejpam-5623	184	13	{	{	PUNCT
ejpam-5623	184	14	k1a+	k1a+	PROPN
ejpam-5623	184	15	k2b	k2b	PROPN
ejpam-5623	184	16	:	:	PUNCT
ejpam-5623	184	17	k1	k1	PROPN
ejpam-5623	184	18	,	,	PUNCT
ejpam-5623	184	19	k2	k2	PROPN
ejpam-5623	184	20	∈	∈	PROPN
ejpam-5623	185	1	f	f	X
ejpam-5623	185	2	}	}	PUNCT
ejpam-5623	185	3	.	.	PUNCT
ejpam-5623	186	1	then	then	ADV
ejpam-5623	186	2	,	,	PUNCT
ejpam-5623	186	3	u	u	NOUN
ejpam-5623	186	4	=	=	PUNCT
ejpam-5623	186	5	k1a	k1a	PROPN
ejpam-5623	187	1	+	+	CCONJ
ejpam-5623	187	2	k2b	k2b	PROPN
ejpam-5623	187	3	where	where	SCONJ
ejpam-5623	187	4	k1	k1	NOUN
ejpam-5623	187	5	,	,	PUNCT
ejpam-5623	187	6	k2	k2	PROPN
ejpam-5623	187	7	∈	∈	PROPN
ejpam-5623	187	8	f.	f.	PROPN
ejpam-5623	187	9	thus	thus	ADV
ejpam-5623	187	10	0	0	X
ejpam-5623	188	1	=	=	SYM
ejpam-5623	188	2	uα	uα	PROPN
ejpam-5623	188	3	=	=	PUNCT
ejpam-5623	189	1	k1b	k1b	PROPN
ejpam-5623	189	2	+	+	CCONJ
ejpam-5623	189	3	k2a	k2a	PROPN
ejpam-5623	189	4	.	.	PUNCT
ejpam-5623	190	1	since	since	SCONJ
ejpam-5623	190	2	a	a	DET
ejpam-5623	190	3	,	,	PUNCT
ejpam-5623	190	4	b	b	NOUN
ejpam-5623	190	5	are	be	AUX
ejpam-5623	190	6	linearly	linearly	ADV
ejpam-5623	190	7	independent	independent	ADJ
ejpam-5623	190	8	,	,	PUNCT
ejpam-5623	190	9	we	we	PRON
ejpam-5623	190	10	must	must	AUX
ejpam-5623	190	11	have	have	VERB
ejpam-5623	190	12	k1	k1	NOUN
ejpam-5623	190	13	=	=	SYM
ejpam-5623	190	14	k2	k2	X
ejpam-5623	190	15	=	=	SYM
ejpam-5623	190	16	0	0	PROPN
ejpam-5623	190	17	,	,	PUNCT
ejpam-5623	190	18	implying	imply	VERB
ejpam-5623	190	19	that	that	SCONJ
ejpam-5623	190	20	u	u	NOUN
ejpam-5623	190	21	=	=	NOUN
ejpam-5623	190	22	0	0	NUM
ejpam-5623	190	23	.	.	PUNCT
ejpam-5623	191	1	thus	thus	ADV
ejpam-5623	191	2	ker(α|v	ker(α|v	X
ejpam-5623	191	3	α	α	X
ejpam-5623	191	4	)	)	PUNCT
ejpam-5623	191	5	=	=	PUNCT
ejpam-5623	191	6	{	{	PUNCT
ejpam-5623	191	7	0	0	NUM
ejpam-5623	191	8	}	}	PUNCT
ejpam-5623	191	9	,	,	PUNCT
ejpam-5623	191	10	and	and	CCONJ
ejpam-5623	191	11	hence	hence	ADV
ejpam-5623	191	12	,	,	PUNCT
ejpam-5623	191	13	α|v	α|v	PROPN
ejpam-5623	191	14	α	α	PROPN
ejpam-5623	191	15	is	be	AUX
ejpam-5623	191	16	one	one	NUM
ejpam-5623	191	17	-	-	PUNCT
ejpam-5623	191	18	to	to	ADP
ejpam-5623	191	19	-	-	PUNCT
ejpam-5623	191	20	one	one	NUM
ejpam-5623	191	21	.	.	PUNCT
ejpam-5623	192	1	by	by	ADP
ejpam-5623	192	2	theorem	theorem	NOUN
ejpam-5623	192	3	1	1	NUM
ejpam-5623	192	4	,	,	PUNCT
ejpam-5623	192	5	α	α	PROPN
ejpam-5623	192	6	is	be	AUX
ejpam-5623	192	7	right	right	ADV
ejpam-5623	192	8	regular	regular	ADV
ejpam-5623	192	9	.	.	PUNCT
ejpam-5623	193	1	similarly	similarly	ADV
ejpam-5623	193	2	,	,	PUNCT
ejpam-5623	193	3	we	we	PRON
ejpam-5623	193	4	can	can	AUX
ejpam-5623	193	5	show	show	VERB
ejpam-5623	193	6	that	that	SCONJ
ejpam-5623	193	7	β	β	NOUN
ejpam-5623	193	8	is	be	AUX
ejpam-5623	193	9	right	right	ADV
ejpam-5623	193	10	regular	regular	ADJ
ejpam-5623	193	11	by	by	ADP
ejpam-5623	193	12	applying	apply	VERB
ejpam-5623	193	13	theorem	theorem	NOUN
ejpam-5623	193	14	1	1	NUM
ejpam-5623	193	15	to	to	PART
ejpam-5623	193	16	β|v	β|v	PUNCT
ejpam-5623	194	1	β	β	X
ejpam-5623	194	2	.	.	PUNCT
ejpam-5623	195	1	finally	finally	ADV
ejpam-5623	195	2	,	,	PUNCT
ejpam-5623	195	3	we	we	PRON
ejpam-5623	195	4	will	will	AUX
ejpam-5623	195	5	show	show	VERB
ejpam-5623	195	6	that	that	SCONJ
ejpam-5623	195	7	αβ	αβ	PRON
ejpam-5623	195	8	is	be	AUX
ejpam-5623	195	9	not	not	PART
ejpam-5623	195	10	right	right	ADV
ejpam-5623	195	11	regular	regular	ADV
ejpam-5623	195	12	.	.	PUNCT
ejpam-5623	196	1	note	note	VERB
ejpam-5623	196	2	that	that	DET
ejpam-5623	196	3	0αβ	0αβ	NOUN
ejpam-5623	197	1	=	=	X
ejpam-5623	197	2	0	0	PUNCT
ejpam-5623	197	3	=	=	SYM
ejpam-5623	197	4	aβ	aβ	PROPN
ejpam-5623	197	5	=	=	SYM
ejpam-5623	197	6	bαβ	bαβ	PROPN
ejpam-5623	197	7	.	.	PUNCT
ejpam-5623	198	1	since	since	SCONJ
ejpam-5623	198	2	0	0	NUM
ejpam-5623	198	3	,	,	PUNCT
ejpam-5623	198	4	b	b	X
ejpam-5623	198	5	∈	∈	PROPN
ejpam-5623	198	6	v	v	ADP
ejpam-5623	198	7	αβ	αβ	NOUN
ejpam-5623	198	8	and	and	CCONJ
ejpam-5623	198	9	0	0	NUM
ejpam-5623	198	10	̸=	̸=	PROPN
ejpam-5623	198	11	b	b	NUM
ejpam-5623	198	12	,	,	PUNCT
ejpam-5623	198	13	it	it	PRON
ejpam-5623	198	14	follows	follow	VERB
ejpam-5623	198	15	that	that	SCONJ
ejpam-5623	198	16	αβ|v	αβ|v	ADV
ejpam-5623	199	1	αβ	αβ	PRON
ejpam-5623	199	2	is	be	AUX
ejpam-5623	199	3	not	not	PART
ejpam-5623	199	4	one	one	NUM
ejpam-5623	199	5	-	-	PUNCT
ejpam-5623	199	6	to	to	ADP
ejpam-5623	199	7	-	-	PUNCT
ejpam-5623	199	8	one	one	NUM
ejpam-5623	199	9	.	.	PUNCT
ejpam-5623	200	1	from	from	ADP
ejpam-5623	200	2	theorem	theorem	ADJ
ejpam-5623	200	3	1	1	NUM
ejpam-5623	200	4	,	,	PUNCT
ejpam-5623	200	5	αβ	αβ	PRON
ejpam-5623	200	6	is	be	AUX
ejpam-5623	200	7	not	not	PART
ejpam-5623	200	8	right	right	ADV
ejpam-5623	200	9	regular	regular	ADV
ejpam-5623	200	10	.	.	PUNCT
ejpam-5623	201	1	hence	hence	ADV
ejpam-5623	201	2	,	,	PUNCT
ejpam-5623	201	3	rreg(s	rreg(s	NOUN
ejpam-5623	201	4	)	)	PUNCT
ejpam-5623	201	5	is	be	AUX
ejpam-5623	201	6	not	not	PART
ejpam-5623	201	7	a	a	DET
ejpam-5623	201	8	subsemigroup	subsemigroup	NOUN
ejpam-5623	201	9	of	of	ADP
ejpam-5623	201	10	s.	s.	PROPN
ejpam-5623	201	11	(	(	PUNCT
ejpam-5623	201	12	iii	iii	PROPN
ejpam-5623	201	13	)	)	PUNCT
ejpam-5623	201	14	⇒	⇒	NOUN
ejpam-5623	201	15	(	(	PUNCT
ejpam-5623	201	16	i	i	NOUN
ejpam-5623	201	17	)	)	PUNCT
ejpam-5623	201	18	suppose	suppose	VERB
ejpam-5623	201	19	that	that	SCONJ
ejpam-5623	201	20	dim(w	dim(w	X
ejpam-5623	201	21	)	)	PUNCT
ejpam-5623	201	22	≤	≤	NUM
ejpam-5623	201	23	1	1	NUM
ejpam-5623	201	24	.	.	PUNCT
ejpam-5623	202	1	let	let	VERB
ejpam-5623	202	2	α	α	PRON
ejpam-5623	202	3	∈	∈	PROPN
ejpam-5623	202	4	s.	s.	PROPN
ejpam-5623	202	5	if	if	SCONJ
ejpam-5623	202	6	α	α	PRON
ejpam-5623	202	7	is	be	AUX
ejpam-5623	202	8	the	the	DET
ejpam-5623	202	9	zero	zero	NUM
ejpam-5623	202	10	transformation	transformation	NOUN
ejpam-5623	202	11	,	,	PUNCT
ejpam-5623	202	12	then	then	ADV
ejpam-5623	202	13	it	it	PRON
ejpam-5623	202	14	is	be	AUX
ejpam-5623	202	15	trivially	trivially	ADV
ejpam-5623	202	16	right	right	ADV
ejpam-5623	202	17	regular	regular	ADJ
ejpam-5623	202	18	.	.	PUNCT
ejpam-5623	203	1	suppose	suppose	VERB
ejpam-5623	203	2	that	that	SCONJ
ejpam-5623	203	3	α	α	PROPN
ejpam-5623	203	4	is	be	AUX
ejpam-5623	203	5	not	not	PART
ejpam-5623	203	6	the	the	DET
ejpam-5623	203	7	zero	zero	NUM
ejpam-5623	203	8	transformation	transformation	NOUN
ejpam-5623	203	9	.	.	PUNCT
ejpam-5623	204	1	then	then	ADV
ejpam-5623	204	2	,	,	PUNCT
ejpam-5623	204	3	dim(v	dim(v	PROPN
ejpam-5623	204	4	α	α	NUM
ejpam-5623	204	5	)	)	PUNCT
ejpam-5623	204	6	≥	≥	NOUN
ejpam-5623	204	7	1	1	NUM
ejpam-5623	204	8	.	.	PUNCT
ejpam-5623	205	1	since	since	SCONJ
ejpam-5623	205	2	dim(w	dim(w	X
ejpam-5623	205	3	)	)	PUNCT
ejpam-5623	205	4	≤	≤	NUM
ejpam-5623	205	5	1	1	NUM
ejpam-5623	205	6	,	,	PUNCT
ejpam-5623	205	7	by	by	ADP
ejpam-5623	205	8	the	the	DET
ejpam-5623	205	9	dimension	dimension	NOUN
ejpam-5623	205	10	theorem	theorem	PROPN
ejpam-5623	205	11	,	,	PUNCT
ejpam-5623	205	12	dim(kerα	dim(kerα	PROPN
ejpam-5623	205	13	)	)	PUNCT
ejpam-5623	205	14	=	=	SYM
ejpam-5623	205	15	0	0	NUM
ejpam-5623	205	16	,	,	PUNCT
ejpam-5623	205	17	which	which	PRON
ejpam-5623	205	18	implies	imply	VERB
ejpam-5623	205	19	that	that	SCONJ
ejpam-5623	205	20	ker(α|v	ker(α|v	PROPN
ejpam-5623	205	21	α	α	X
ejpam-5623	205	22	)	)	PUNCT
ejpam-5623	205	23	=	=	PUNCT
ejpam-5623	205	24	{	{	PUNCT
ejpam-5623	205	25	0	0	NUM
ejpam-5623	205	26	}	}	PUNCT
ejpam-5623	205	27	.	.	PUNCT
ejpam-5623	206	1	therefore	therefore	ADV
ejpam-5623	206	2	,	,	PUNCT
ejpam-5623	206	3	α|v	α|v	PROPN
ejpam-5623	206	4	α	α	NOUN
ejpam-5623	206	5	must	must	AUX
ejpam-5623	206	6	be	be	AUX
ejpam-5623	206	7	one	one	NUM
ejpam-5623	206	8	-	-	PUNCT
ejpam-5623	206	9	to	to	ADP
ejpam-5623	206	10	-	-	PUNCT
ejpam-5623	206	11	one	one	NUM
ejpam-5623	206	12	,	,	PUNCT
ejpam-5623	206	13	and	and	CCONJ
ejpam-5623	206	14	by	by	ADP
ejpam-5623	206	15	theorem	theorem	NOUN
ejpam-5623	206	16	1	1	NUM
ejpam-5623	206	17	,	,	PUNCT
ejpam-5623	206	18	α	α	PROPN
ejpam-5623	206	19	is	be	AUX
ejpam-5623	206	20	right	right	ADV
ejpam-5623	206	21	regular	regular	ADV
ejpam-5623	206	22	.	.	PUNCT
ejpam-5623	207	1	consequently	consequently	ADV
ejpam-5623	207	2	,	,	PUNCT
ejpam-5623	207	3	s	s	NOUN
ejpam-5623	207	4	=	=	SYM
ejpam-5623	207	5	rreg(s	rreg(s	PROPN
ejpam-5623	207	6	)	)	PUNCT
ejpam-5623	207	7	,	,	PUNCT
ejpam-5623	207	8	meaning	mean	VERB
ejpam-5623	207	9	that	that	SCONJ
ejpam-5623	207	10	s	s	VERB
ejpam-5623	207	11	is	be	AUX
ejpam-5623	207	12	a	a	DET
ejpam-5623	207	13	right	right	ADJ
ejpam-5623	207	14	regular	regular	ADJ
ejpam-5623	207	15	semigroup	semigroup	NOUN
ejpam-5623	207	16	.	.	PUNCT
ejpam-5623	208	1	next	next	ADV
ejpam-5623	208	2	,	,	PUNCT
ejpam-5623	208	3	we	we	PRON
ejpam-5623	208	4	give	give	VERB
ejpam-5623	208	5	a	a	DET
ejpam-5623	208	6	characterization	characterization	NOUN
ejpam-5623	208	7	for	for	ADP
ejpam-5623	208	8	left	leave	VERB
ejpam-5623	208	9	regular	regular	ADJ
ejpam-5623	208	10	elements	element	NOUN
ejpam-5623	208	11	in	in	ADP
ejpam-5623	208	12	s	s	NOUN
ejpam-5623	208	13	\q	\q	NOUN
ejpam-5623	208	14	and	and	CCONJ
ejpam-5623	208	15	q	q	NOUN
ejpam-5623	208	16	,	,	PUNCT
ejpam-5623	208	17	respectively	respectively	ADV
ejpam-5623	208	18	.	.	PUNCT
ejpam-5623	209	1	let	let	VERB
ejpam-5623	209	2	s∗	s∗	PROPN
ejpam-5623	209	3	be	be	AUX
ejpam-5623	209	4	either	either	CCONJ
ejpam-5623	209	5	l(v	l(v	PROPN
ejpam-5623	209	6	,	,	PUNCT
ejpam-5623	209	7	w	w	NOUN
ejpam-5623	209	8	)	)	PUNCT
ejpam-5623	209	9	or	or	CCONJ
ejpam-5623	209	10	s(v	s(v	PROPN
ejpam-5623	209	11	,	,	PUNCT
ejpam-5623	209	12	w	w	NOUN
ejpam-5623	209	13	)	)	PUNCT
ejpam-5623	209	14	.	.	PUNCT
ejpam-5623	210	1	theorem	theorem	NOUN
ejpam-5623	210	2	3	3	X
ejpam-5623	210	3	.	.	PUNCT
ejpam-5623	211	1	let	let	VERB
ejpam-5623	211	2	α	α	PRON
ejpam-5623	211	3	∈	∈	PROPN
ejpam-5623	211	4	s∗.	s∗.	VERB
ejpam-5623	211	5	the	the	DET
ejpam-5623	211	6	following	following	ADJ
ejpam-5623	211	7	statements	statement	NOUN
ejpam-5623	211	8	are	be	AUX
ejpam-5623	211	9	equivalent	equivalent	ADJ
ejpam-5623	211	10	:	:	PUNCT
ejpam-5623	211	11	(	(	PUNCT
ejpam-5623	211	12	i	i	NOUN
ejpam-5623	211	13	)	)	PUNCT
ejpam-5623	211	14	α	α	PROPN
ejpam-5623	211	15	∈	∈	PROPN
ejpam-5623	211	16	lreg(s∗	lreg(s∗	PROPN
ejpam-5623	211	17	)	)	PUNCT
ejpam-5623	211	18	.	.	PUNCT
ejpam-5623	212	1	(	(	PUNCT
ejpam-5623	212	2	ii	ii	NOUN
ejpam-5623	212	3	)	)	PUNCT
ejpam-5623	212	4	α|wα	α|wα	NOUN
ejpam-5623	212	5	is	be	AUX
ejpam-5623	212	6	an	an	PRON
ejpam-5623	212	7	onto	onto	ADP
ejpam-5623	212	8	transformation	transformation	NOUN
ejpam-5623	212	9	on	on	ADP
ejpam-5623	212	10	v	v	NUM
ejpam-5623	212	11	α	α	NOUN
ejpam-5623	212	12	.	.	PUNCT
ejpam-5623	213	1	(	(	PUNCT
ejpam-5623	213	2	iii	iii	X
ejpam-5623	213	3	)	)	PUNCT
ejpam-5623	213	4	v	v	NOUN
ejpam-5623	213	5	α	α	NOUN
ejpam-5623	213	6	⊆	⊆	NUM
ejpam-5623	213	7	wα2	wα2	NOUN
ejpam-5623	213	8	.	.	PUNCT
ejpam-5623	214	1	(	(	PUNCT
ejpam-5623	214	2	iv	iv	X
ejpam-5623	214	3	)	)	PUNCT
ejpam-5623	214	4	for	for	ADP
ejpam-5623	214	5	every	every	DET
ejpam-5623	214	6	basis	basis	NOUN
ejpam-5623	214	7	b	b	NOUN
ejpam-5623	214	8	for	for	ADP
ejpam-5623	214	9	v	v	NUM
ejpam-5623	214	10	,	,	PUNCT
ejpam-5623	214	11	we	we	PRON
ejpam-5623	214	12	have	have	VERB
ejpam-5623	214	13	bα	bα	NOUN
ejpam-5623	214	14	⊆	⊆	NUM
ejpam-5623	214	15	wα2	wα2	NOUN
ejpam-5623	214	16	.	.	PUNCT
ejpam-5623	215	1	(	(	PUNCT
ejpam-5623	215	2	v	v	X
ejpam-5623	215	3	)	)	PUNCT
ejpam-5623	215	4	there	there	PRON
ejpam-5623	215	5	exists	exist	VERB
ejpam-5623	215	6	a	a	DET
ejpam-5623	215	7	basis	basis	NOUN
ejpam-5623	215	8	b	b	NOUN
ejpam-5623	215	9	for	for	ADP
ejpam-5623	215	10	v	v	ADP
ejpam-5623	215	11	such	such	ADJ
ejpam-5623	215	12	that	that	DET
ejpam-5623	215	13	bα	bα	PROPN
ejpam-5623	215	14	⊆	⊆	NUM
ejpam-5623	215	15	wα2	wα2	NOUN
ejpam-5623	215	16	.	.	PUNCT
ejpam-5623	216	1	proof	proof	NOUN
ejpam-5623	216	2	.	.	PUNCT
ejpam-5623	217	1	(	(	PUNCT
ejpam-5623	217	2	i	i	NOUN
ejpam-5623	217	3	)	)	PUNCT
ejpam-5623	217	4	⇒	⇒	PROPN
ejpam-5623	217	5	(	(	PUNCT
ejpam-5623	217	6	ii	ii	NOUN
ejpam-5623	217	7	)	)	PUNCT
ejpam-5623	217	8	assume	assume	VERB
ejpam-5623	217	9	that	that	SCONJ
ejpam-5623	217	10	α	α	PRON
ejpam-5623	217	11	is	be	AUX
ejpam-5623	217	12	left	leave	VERB
ejpam-5623	217	13	regular	regular	ADV
ejpam-5623	217	14	in	in	ADP
ejpam-5623	217	15	s∗.	s∗.	ADJ
ejpam-5623	217	16	then	then	ADV
ejpam-5623	217	17	,	,	PUNCT
ejpam-5623	217	18	α	α	X
ejpam-5623	217	19	=	=	PUNCT
ejpam-5623	217	20	βα2	βα2	ADJ
ejpam-5623	217	21	for	for	ADP
ejpam-5623	217	22	some	some	DET
ejpam-5623	217	23	β	β	X
ejpam-5623	217	24	∈	∈	PROPN
ejpam-5623	217	25	s∗.	s∗.	ADJ
ejpam-5623	217	26	for	for	ADP
ejpam-5623	217	27	any	any	DET
ejpam-5623	217	28	y	y	PROPN
ejpam-5623	217	29	∈	∈	PROPN
ejpam-5623	217	30	v	v	ADP
ejpam-5623	217	31	α	α	NOUN
ejpam-5623	217	32	,	,	PUNCT
ejpam-5623	217	33	then	then	ADV
ejpam-5623	217	34	there	there	PRON
ejpam-5623	217	35	exists	exist	VERB
ejpam-5623	217	36	x	x	X
ejpam-5623	217	37	∈	∈	NOUN
ejpam-5623	217	38	v	v	ADP
ejpam-5623	217	39	such	such	ADJ
ejpam-5623	217	40	that	that	PRON
ejpam-5623	217	41	y	y	PROPN
ejpam-5623	217	42	=	=	PUNCT
ejpam-5623	218	1	xα	xα	PROPN
ejpam-5623	218	2	.	.	PUNCT
ejpam-5623	219	1	thus	thus	ADV
ejpam-5623	219	2	,	,	PUNCT
ejpam-5623	219	3	xβ	xβ	PROPN
ejpam-5623	219	4	∈	∈	PROPN
ejpam-5623	219	5	w	w	PROPN
ejpam-5623	219	6	,	,	PUNCT
ejpam-5623	219	7	and	and	CCONJ
ejpam-5623	219	8	y	y	PROPN
ejpam-5623	219	9	=	=	PUNCT
ejpam-5623	219	10	xα	xα	PROPN
ejpam-5623	220	1	=	=	PUNCT
ejpam-5623	220	2	xβα2	xβα2	NOUN
ejpam-5623	221	1	=	=	SYM
ejpam-5623	222	1	(	(	PUNCT
ejpam-5623	222	2	xβα)α	xβα)α	PROPN
ejpam-5623	222	3	,	,	PUNCT
ejpam-5623	222	4	which	which	PRON
ejpam-5623	222	5	proves	prove	VERB
ejpam-5623	222	6	that	that	SCONJ
ejpam-5623	222	7	α|wα	α|wα	NOUN
ejpam-5623	222	8	is	be	AUX
ejpam-5623	222	9	onto	onto	ADP
ejpam-5623	222	10	.	.	PUNCT
ejpam-5623	223	1	(	(	PUNCT
ejpam-5623	223	2	ii	ii	NOUN
ejpam-5623	223	3	)	)	PUNCT
ejpam-5623	223	4	⇒	⇒	NOUN
ejpam-5623	223	5	(	(	PUNCT
ejpam-5623	223	6	iii	iii	X
ejpam-5623	223	7	)	)	PUNCT
ejpam-5623	223	8	if	if	SCONJ
ejpam-5623	223	9	α|wα	α|wα	NUM
ejpam-5623	223	10	:	:	PUNCT
ejpam-5623	223	11	wα	wα	NOUN
ejpam-5623	223	12	→	→	SYM
ejpam-5623	223	13	v	v	NUM
ejpam-5623	223	14	α	α	NOUN
ejpam-5623	223	15	is	be	AUX
ejpam-5623	223	16	onto	onto	ADP
ejpam-5623	223	17	,	,	PUNCT
ejpam-5623	223	18	then	then	ADV
ejpam-5623	223	19	wα2	wα2	NOUN
ejpam-5623	223	20	=	=	SYM
ejpam-5623	223	21	(	(	PUNCT
ejpam-5623	223	22	wα)α	wα)α	PROPN
ejpam-5623	223	23	=	=	SYM
ejpam-5623	223	24	(	(	PUNCT
ejpam-5623	223	25	wα)α|wα	wα)α|wα	NOUN
ejpam-5623	223	26	=	=	X
ejpam-5623	223	27	v	v	NUM
ejpam-5623	223	28	α	α	NOUN
ejpam-5623	223	29	,	,	PUNCT
ejpam-5623	223	30	which	which	PRON
ejpam-5623	223	31	proves	prove	VERB
ejpam-5623	223	32	that	that	SCONJ
ejpam-5623	223	33	v	v	ADP
ejpam-5623	223	34	α	α	NOUN
ejpam-5623	223	35	⊆	⊆	NUM
ejpam-5623	223	36	wα2	wα2	NOUN
ejpam-5623	223	37	.	.	PUNCT
ejpam-5623	224	1	(	(	PUNCT
ejpam-5623	224	2	iii	iii	X
ejpam-5623	224	3	)	)	PUNCT
ejpam-5623	224	4	⇒	⇒	NOUN
ejpam-5623	224	5	(	(	PUNCT
ejpam-5623	224	6	iv	iv	X
ejpam-5623	224	7	)	)	PUNCT
ejpam-5623	224	8	and	and	CCONJ
ejpam-5623	224	9	(	(	PUNCT
ejpam-5623	224	10	iv	iv	X
ejpam-5623	224	11	)	)	PUNCT
ejpam-5623	224	12	⇒	⇒	NOUN
ejpam-5623	224	13	(	(	PUNCT
ejpam-5623	224	14	v	v	NOUN
ejpam-5623	224	15	)	)	PUNCT
ejpam-5623	224	16	are	be	AUX
ejpam-5623	224	17	clear	clear	ADJ
ejpam-5623	224	18	by	by	ADP
ejpam-5623	224	19	definition	definition	NOUN
ejpam-5623	224	20	.	.	PUNCT
ejpam-5623	225	1	(	(	PUNCT
ejpam-5623	225	2	v	v	NOUN
ejpam-5623	225	3	)	)	PUNCT
ejpam-5623	225	4	⇒	⇒	NOUN
ejpam-5623	225	5	(	(	PUNCT
ejpam-5623	225	6	i	i	NOUN
ejpam-5623	225	7	)	)	PUNCT
ejpam-5623	225	8	suppose	suppose	VERB
ejpam-5623	225	9	there	there	PRON
ejpam-5623	225	10	is	be	VERB
ejpam-5623	225	11	a	a	DET
ejpam-5623	225	12	basis	basis	NOUN
ejpam-5623	225	13	b	b	NOUN
ejpam-5623	225	14	for	for	ADP
ejpam-5623	225	15	v	v	ADP
ejpam-5623	225	16	such	such	ADJ
ejpam-5623	225	17	that	that	DET
ejpam-5623	225	18	bα	bα	PROPN
ejpam-5623	225	19	⊆	⊆	NUM
ejpam-5623	225	20	wα2	wα2	NOUN
ejpam-5623	225	21	.	.	PUNCT
ejpam-5623	226	1	for	for	ADP
ejpam-5623	226	2	each	each	DET
ejpam-5623	226	3	v	v	NUM
ejpam-5623	226	4	∈	∈	PROPN
ejpam-5623	226	5	b	b	NOUN
ejpam-5623	226	6	,	,	PUNCT
ejpam-5623	226	7	we	we	PRON
ejpam-5623	226	8	choose	choose	VERB
ejpam-5623	226	9	and	and	CCONJ
ejpam-5623	226	10	fix	fix	VERB
ejpam-5623	226	11	v′	v′	NOUN
ejpam-5623	226	12	∈	∈	PROPN
ejpam-5623	226	13	w	w	ADP
ejpam-5623	226	14	such	such	ADJ
ejpam-5623	226	15	that	that	DET
ejpam-5623	226	16	vα	vα	PROPN
ejpam-5623	226	17	=	=	SYM
ejpam-5623	226	18	v′α2	v′α2	PROPN
ejpam-5623	226	19	.	.	PUNCT
ejpam-5623	227	1	define	define	VERB
ejpam-5623	227	2	β	β	X
ejpam-5623	227	3	:	:	PUNCT
ejpam-5623	227	4	b	b	X
ejpam-5623	227	5	→	→	SYM
ejpam-5623	227	6	v	v	NOUN
ejpam-5623	227	7	by	by	ADP
ejpam-5623	227	8	vβ	vβ	X
ejpam-5623	227	9	=	=	PUNCT
ejpam-5623	227	10	v′	v′	NOUN
ejpam-5623	227	11	for	for	ADP
ejpam-5623	227	12	all	all	DET
ejpam-5623	227	13	v	v	PROPN
ejpam-5623	227	14	∈	∈	PROPN
ejpam-5623	227	15	b.	b.	PROPN
ejpam-5623	227	16	n.	n.	PROPN
ejpam-5623	227	17	sawatraksa	sawatraksa	PROPN
ejpam-5623	227	18	,	,	PUNCT
ejpam-5623	227	19	p.	p.	NOUN
ejpam-5623	227	20	tantong	tantong	NOUN
ejpam-5623	227	21	/	/	SYM
ejpam-5623	227	22	eur	eur	PROPN
ejpam-5623	227	23	.	.	PUNCT
ejpam-5623	228	1	j.	j.	PROPN
ejpam-5623	228	2	pure	pure	PROPN
ejpam-5623	228	3	appl	appl	PROPN
ejpam-5623	228	4	.	.	PROPN
ejpam-5623	228	5	math	math	PROPN
ejpam-5623	228	6	,	,	PUNCT
ejpam-5623	228	7	18	18	NUM
ejpam-5623	228	8	(	(	PUNCT
ejpam-5623	228	9	1	1	NUM
ejpam-5623	228	10	)	)	PUNCT
ejpam-5623	228	11	(	(	PUNCT
ejpam-5623	228	12	2025	2025	NUM
ejpam-5623	228	13	)	)	PUNCT
ejpam-5623	228	14	,	,	PUNCT
ejpam-5623	228	15	5623	5623	NUM
ejpam-5623	228	16	6	6	NUM
ejpam-5623	228	17	of	of	ADP
ejpam-5623	228	18	15	15	NUM
ejpam-5623	228	19	by	by	ADP
ejpam-5623	228	20	the	the	DET
ejpam-5623	228	21	uniqueness	uniqueness	NOUN
ejpam-5623	228	22	condition	condition	NOUN
ejpam-5623	228	23	,	,	PUNCT
ejpam-5623	228	24	β	β	X
ejpam-5623	228	25	is	be	AUX
ejpam-5623	228	26	well	well	ADV
ejpam-5623	228	27	-	-	PUNCT
ejpam-5623	228	28	defined	define	VERB
ejpam-5623	228	29	and	and	CCONJ
ejpam-5623	228	30	can	can	AUX
ejpam-5623	228	31	be	be	AUX
ejpam-5623	228	32	extended	extend	VERB
ejpam-5623	228	33	to	to	ADP
ejpam-5623	228	34	a	a	DET
ejpam-5623	228	35	linear	linear	ADJ
ejpam-5623	228	36	transformation	transformation	NOUN
ejpam-5623	228	37	on	on	ADP
ejpam-5623	228	38	v	v	NUM
ejpam-5623	228	39	.	.	PUNCT
ejpam-5623	229	1	we	we	PRON
ejpam-5623	229	2	will	will	AUX
ejpam-5623	229	3	now	now	ADV
ejpam-5623	229	4	show	show	VERB
ejpam-5623	229	5	that	that	SCONJ
ejpam-5623	229	6	β	β	PROPN
ejpam-5623	229	7	∈	∈	PROPN
ejpam-5623	229	8	s∗	s∗	PROPN
ejpam-5623	229	9	,	,	PUNCT
ejpam-5623	229	10	and	and	CCONJ
ejpam-5623	229	11	that	that	SCONJ
ejpam-5623	229	12	α	α	X
ejpam-5623	229	13	=	=	PUNCT
ejpam-5623	229	14	βα2	βα2	ADJ
ejpam-5623	229	15	,	,	PUNCT
ejpam-5623	229	16	proving	prove	VERB
ejpam-5623	229	17	that	that	SCONJ
ejpam-5623	229	18	α	α	NOUN
ejpam-5623	229	19	is	be	AUX
ejpam-5623	229	20	left	leave	VERB
ejpam-5623	229	21	regular	regular	ADV
ejpam-5623	229	22	.	.	PUNCT
ejpam-5623	230	1	for	for	ADP
ejpam-5623	230	2	any	any	DET
ejpam-5623	230	3	v	v	NUM
ejpam-5623	230	4	∈	∈	NOUN
ejpam-5623	230	5	v	v	NOUN
ejpam-5623	230	6	,	,	PUNCT
ejpam-5623	230	7	there	there	PRON
ejpam-5623	230	8	exist	exist	VERB
ejpam-5623	230	9	v1	v1	NOUN
ejpam-5623	230	10	,	,	PUNCT
ejpam-5623	230	11	v2	v2	NOUN
ejpam-5623	230	12	,	,	PUNCT
ejpam-5623	230	13	.	.	PUNCT
ejpam-5623	230	14	.	.	PUNCT
ejpam-5623	231	1	.	.	PUNCT
ejpam-5623	232	1	,	,	PUNCT
ejpam-5623	232	2	vn	vn	PROPN
ejpam-5623	232	3	∈	∈	PROPN
ejpam-5623	232	4	b	b	PROPN
ejpam-5623	232	5	and	and	CCONJ
ejpam-5623	232	6	a1	a1	PROPN
ejpam-5623	232	7	,	,	PUNCT
ejpam-5623	232	8	a2	a2	PROPN
ejpam-5623	232	9	,	,	PUNCT
ejpam-5623	232	10	.	.	PUNCT
ejpam-5623	232	11	.	.	PUNCT
ejpam-5623	233	1	.	.	PUNCT
ejpam-5623	234	1	,	,	PUNCT
ejpam-5623	234	2	an	an	DET
ejpam-5623	234	3	∈	∈	PROPN
ejpam-5623	234	4	f	f	NOUN
ejpam-5623	234	5	such	such	ADJ
ejpam-5623	234	6	that	that	DET
ejpam-5623	234	7	v	v	NOUN
ejpam-5623	234	8	=	=	SYM
ejpam-5623	234	9	a1v1	a1v1	NOUN
ejpam-5623	234	10	+	+	CCONJ
ejpam-5623	234	11	a2v2	a2v2	PROPN
ejpam-5623	234	12	+	+	CCONJ
ejpam-5623	234	13	.	.	PUNCT
ejpam-5623	234	14	.	.	PUNCT
ejpam-5623	235	1	.+	.+	NOUN
ejpam-5623	235	2	anvn	anvn	PROPN
ejpam-5623	235	3	.	.	PUNCT
ejpam-5623	236	1	thus	thus	ADV
ejpam-5623	236	2	,	,	PUNCT
ejpam-5623	236	3	vβ	vβ	X
ejpam-5623	236	4	=	=	PRON
ejpam-5623	236	5	(	(	PUNCT
ejpam-5623	236	6	a1v1	a1v1	PROPN
ejpam-5623	236	7	+	+	SYM
ejpam-5623	236	8	a2v2	a2v2	PUNCT
ejpam-5623	236	9	+	+	CCONJ
ejpam-5623	236	10	.	.	PUNCT
ejpam-5623	236	11	.	.	PUNCT
ejpam-5623	237	1	.+	.+	NOUN
ejpam-5623	237	2	anvn)β	anvn)β	PUNCT
ejpam-5623	237	3	=	=	SYM
ejpam-5623	237	4	a1(v1β	a1(v1β	NOUN
ejpam-5623	237	5	)	)	PUNCT
ejpam-5623	238	1	+	+	CCONJ
ejpam-5623	238	2	a2(v2β	a2(v2β	NOUN
ejpam-5623	238	3	)	)	PUNCT
ejpam-5623	239	1	+	+	CCONJ
ejpam-5623	239	2	.	.	PUNCT
ejpam-5623	239	3	.	.	PUNCT
ejpam-5623	240	1	.+	.+	NOUN
ejpam-5623	240	2	an(vnβ	an(vnβ	PART
ejpam-5623	240	3	)	)	PUNCT
ejpam-5623	240	4	=	=	PUNCT
ejpam-5623	241	1	a1v	a1v	ADJ
ejpam-5623	241	2	′	′	NOUN
ejpam-5623	241	3	1	1	NUM
ejpam-5623	242	1	+	+	CCONJ
ejpam-5623	242	2	a2v	a2v	INTJ
ejpam-5623	242	3	′	′	NUM
ejpam-5623	242	4	2	2	NUM
ejpam-5623	242	5	+	+	CCONJ
ejpam-5623	242	6	.	.	PUNCT
ejpam-5623	242	7	.	.	PUNCT
ejpam-5623	243	1	.+	.+	NOUN
ejpam-5623	243	2	anv	anv	VERB
ejpam-5623	243	3	′	′	NUM
ejpam-5623	243	4	n	n	CCONJ
ejpam-5623	243	5	∈	∈	PROPN
ejpam-5623	243	6	w.	w.	NOUN
ejpam-5623	243	7	hence	hence	ADV
ejpam-5623	243	8	β	β	PROPN
ejpam-5623	243	9	∈	∈	PROPN
ejpam-5623	243	10	l(v	l(v	PROPN
ejpam-5623	243	11	,	,	PUNCT
ejpam-5623	243	12	w	w	NOUN
ejpam-5623	243	13	)	)	PUNCT
ejpam-5623	243	14	or	or	CCONJ
ejpam-5623	243	15	β	β	X
ejpam-5623	243	16	∈	∈	PROPN
ejpam-5623	243	17	s(v	s(v	PROPN
ejpam-5623	243	18	,	,	PUNCT
ejpam-5623	243	19	w	w	NOUN
ejpam-5623	243	20	)	)	PUNCT
ejpam-5623	243	21	,	,	PUNCT
ejpam-5623	243	22	depending	depend	VERB
ejpam-5623	243	23	on	on	ADP
ejpam-5623	243	24	the	the	DET
ejpam-5623	243	25	semigroup	semigroup	NOUN
ejpam-5623	243	26	.	.	PUNCT
ejpam-5623	244	1	finally	finally	ADV
ejpam-5623	244	2	,	,	PUNCT
ejpam-5623	244	3	for	for	ADP
ejpam-5623	244	4	each	each	DET
ejpam-5623	244	5	v	v	ADP
ejpam-5623	244	6	∈	∈	PROPN
ejpam-5623	244	7	b	b	NOUN
ejpam-5623	244	8	,	,	PUNCT
ejpam-5623	244	9	we	we	PRON
ejpam-5623	244	10	have	have	VERB
ejpam-5623	244	11	vβα2	vβα2	NOUN
ejpam-5623	244	12	=	=	SYM
ejpam-5623	244	13	vβαα	vβαα	NOUN
ejpam-5623	244	14	=	=	PUNCT
ejpam-5623	244	15	v′αα	v′αα	NOUN
ejpam-5623	244	16	=	=	SYM
ejpam-5623	244	17	vα	vα	PROPN
ejpam-5623	244	18	,	,	PUNCT
ejpam-5623	244	19	showing	show	VERB
ejpam-5623	244	20	that	that	PRON
ejpam-5623	244	21	βα2	βα2	NOUN
ejpam-5623	244	22	=	=	SYM
ejpam-5623	244	23	α	α	NOUN
ejpam-5623	244	24	,	,	PUNCT
ejpam-5623	244	25	as	as	SCONJ
ejpam-5623	244	26	required	require	VERB
ejpam-5623	244	27	.	.	PUNCT
ejpam-5623	245	1	the	the	DET
ejpam-5623	245	2	following	following	ADJ
ejpam-5623	245	3	result	result	NOUN
ejpam-5623	245	4	follows	follow	VERB
ejpam-5623	245	5	from	from	ADP
ejpam-5623	245	6	theorems	theorem	NOUN
ejpam-5623	245	7	1	1	NUM
ejpam-5623	245	8	and	and	CCONJ
ejpam-5623	245	9	3	3	NUM
ejpam-5623	245	10	.	.	PUNCT
ejpam-5623	245	11	corollary	corollary	ADJ
ejpam-5623	245	12	1	1	NUM
ejpam-5623	245	13	.	.	PUNCT
ejpam-5623	246	1	let	let	VERB
ejpam-5623	246	2	α	α	PRON
ejpam-5623	246	3	∈	∈	PROPN
ejpam-5623	246	4	s∗.	s∗.	VERB
ejpam-5623	246	5	the	the	DET
ejpam-5623	246	6	following	following	ADJ
ejpam-5623	246	7	statements	statement	NOUN
ejpam-5623	246	8	are	be	AUX
ejpam-5623	246	9	equivalent	equivalent	ADJ
ejpam-5623	246	10	:	:	PUNCT
ejpam-5623	246	11	(	(	PUNCT
ejpam-5623	246	12	i	i	NOUN
ejpam-5623	246	13	)	)	PUNCT
ejpam-5623	246	14	α	α	PROPN
ejpam-5623	246	15	∈	∈	PROPN
ejpam-5623	246	16	creg(s∗	creg(s∗	NOUN
ejpam-5623	246	17	)	)	PUNCT
ejpam-5623	246	18	.	.	PUNCT
ejpam-5623	247	1	(	(	PUNCT
ejpam-5623	247	2	ii	ii	NOUN
ejpam-5623	247	3	)	)	PUNCT
ejpam-5623	247	4	α|v	α|v	VERB
ejpam-5623	248	1	α	α	NOUN
ejpam-5623	248	2	:	:	PUNCT
ejpam-5623	248	3	v	v	ADP
ejpam-5623	248	4	α	α	PROPN
ejpam-5623	248	5	→	→	SYM
ejpam-5623	248	6	v	v	NUM
ejpam-5623	248	7	α	α	NOUN
ejpam-5623	248	8	is	be	AUX
ejpam-5623	248	9	one	one	NUM
ejpam-5623	248	10	-	-	PUNCT
ejpam-5623	248	11	to	to	ADP
ejpam-5623	248	12	-	-	PUNCT
ejpam-5623	248	13	one	one	NUM
ejpam-5623	248	14	,	,	PUNCT
ejpam-5623	248	15	and	and	CCONJ
ejpam-5623	248	16	α|wα	α|wα	NUM
ejpam-5623	248	17	:	:	PUNCT
ejpam-5623	248	18	wα	wα	NOUN
ejpam-5623	248	19	→	→	SYM
ejpam-5623	248	20	v	v	NUM
ejpam-5623	248	21	α	α	NOUN
ejpam-5623	248	22	is	be	AUX
ejpam-5623	248	23	onto	onto	ADP
ejpam-5623	248	24	.	.	PUNCT
ejpam-5623	249	1	(	(	PUNCT
ejpam-5623	249	2	iii	iii	NOUN
ejpam-5623	249	3	)	)	PUNCT
ejpam-5623	249	4	for	for	ADP
ejpam-5623	249	5	every	every	DET
ejpam-5623	249	6	v	v	NUM
ejpam-5623	249	7	∈	∈	NOUN
ejpam-5623	249	8	v	v	NOUN
ejpam-5623	249	9	,	,	PUNCT
ejpam-5623	249	10	there	there	PRON
ejpam-5623	249	11	exists	exist	VERB
ejpam-5623	249	12	a	a	DET
ejpam-5623	249	13	unique	unique	ADJ
ejpam-5623	249	14	v′	v′	NOUN
ejpam-5623	249	15	∈	∈	NOUN
ejpam-5623	249	16	wα	wα	NOUN
ejpam-5623	249	17	such	such	ADJ
ejpam-5623	250	1	that	that	DET
ejpam-5623	250	2	vα	vα	ADP
ejpam-5623	250	3	=	=	PUNCT
ejpam-5623	250	4	v′α	v′α	ADJ
ejpam-5623	250	5	.	.	PUNCT
ejpam-5623	251	1	(	(	PUNCT
ejpam-5623	251	2	iv	iv	X
ejpam-5623	251	3	)	)	PUNCT
ejpam-5623	251	4	for	for	ADP
ejpam-5623	251	5	every	every	DET
ejpam-5623	251	6	basis	basis	NOUN
ejpam-5623	251	7	b	b	NOUN
ejpam-5623	251	8	of	of	ADP
ejpam-5623	251	9	v	v	NOUN
ejpam-5623	251	10	,	,	PUNCT
ejpam-5623	251	11	and	and	CCONJ
ejpam-5623	251	12	for	for	ADP
ejpam-5623	251	13	every	every	DET
ejpam-5623	251	14	v	v	NUM
ejpam-5623	251	15	∈	∈	PROPN
ejpam-5623	251	16	b	b	NOUN
ejpam-5623	251	17	,	,	PUNCT
ejpam-5623	251	18	there	there	PRON
ejpam-5623	251	19	exists	exist	VERB
ejpam-5623	251	20	a	a	DET
ejpam-5623	251	21	unique	unique	ADJ
ejpam-5623	251	22	v′	v′	NOUN
ejpam-5623	251	23	∈	∈	NOUN
ejpam-5623	251	24	wα	wα	NOUN
ejpam-5623	251	25	such	such	ADJ
ejpam-5623	252	1	that	that	DET
ejpam-5623	252	2	vα	vα	ADP
ejpam-5623	252	3	=	=	PUNCT
ejpam-5623	252	4	v′α	v′α	ADJ
ejpam-5623	252	5	.	.	PUNCT
ejpam-5623	253	1	(	(	PUNCT
ejpam-5623	253	2	v	v	X
ejpam-5623	253	3	)	)	PUNCT
ejpam-5623	253	4	there	there	PRON
ejpam-5623	253	5	exists	exist	VERB
ejpam-5623	253	6	a	a	DET
ejpam-5623	253	7	basis	basis	NOUN
ejpam-5623	253	8	b	b	NOUN
ejpam-5623	253	9	of	of	ADP
ejpam-5623	253	10	v	v	NOUN
ejpam-5623	253	11	,	,	PUNCT
ejpam-5623	253	12	and	and	CCONJ
ejpam-5623	253	13	for	for	ADP
ejpam-5623	253	14	every	every	DET
ejpam-5623	253	15	v	v	NUM
ejpam-5623	253	16	∈	∈	PROPN
ejpam-5623	253	17	b	b	NOUN
ejpam-5623	253	18	,	,	PUNCT
ejpam-5623	253	19	there	there	PRON
ejpam-5623	253	20	exists	exist	VERB
ejpam-5623	253	21	a	a	DET
ejpam-5623	253	22	unique	unique	ADJ
ejpam-5623	253	23	v′	v′	NOUN
ejpam-5623	253	24	∈	∈	NOUN
ejpam-5623	253	25	wα	wα	NOUN
ejpam-5623	253	26	such	such	ADJ
ejpam-5623	253	27	that	that	DET
ejpam-5623	253	28	vα	vα	ADP
ejpam-5623	253	29	=	=	PUNCT
ejpam-5623	253	30	v′α	v′α	ADJ
ejpam-5623	253	31	.	.	PUNCT
ejpam-5623	254	1	next	next	ADV
ejpam-5623	254	2	,	,	PUNCT
ejpam-5623	254	3	we	we	PRON
ejpam-5623	254	4	give	give	VERB
ejpam-5623	254	5	a	a	DET
ejpam-5623	254	6	necessary	necessary	ADJ
ejpam-5623	254	7	and	and	CCONJ
ejpam-5623	254	8	sufficient	sufficient	ADJ
ejpam-5623	254	9	condition	condition	NOUN
ejpam-5623	254	10	when	when	SCONJ
ejpam-5623	254	11	the	the	DET
ejpam-5623	254	12	semigroup	semigroup	NOUN
ejpam-5623	254	13	s∗	s∗	VERB
ejpam-5623	254	14	to	to	PART
ejpam-5623	254	15	be	be	AUX
ejpam-5623	254	16	left	leave	VERB
ejpam-5623	254	17	regular	regular	ADV
ejpam-5623	254	18	.	.	PUNCT
ejpam-5623	255	1	theorem	theorem	VERB
ejpam-5623	255	2	4	4	NUM
ejpam-5623	255	3	.	.	PUNCT
ejpam-5623	256	1	the	the	DET
ejpam-5623	256	2	following	follow	VERB
ejpam-5623	256	3	statements	statement	NOUN
ejpam-5623	256	4	are	be	AUX
ejpam-5623	256	5	equivalent	equivalent	ADJ
ejpam-5623	256	6	:	:	PUNCT
ejpam-5623	256	7	(	(	PUNCT
ejpam-5623	256	8	i	i	NOUN
ejpam-5623	256	9	)	)	PUNCT
ejpam-5623	256	10	s∗	s∗	PROPN
ejpam-5623	256	11	is	be	AUX
ejpam-5623	256	12	a	a	DET
ejpam-5623	256	13	left	left	ADJ
ejpam-5623	256	14	regular	regular	ADJ
ejpam-5623	256	15	semigroup	semigroup	NOUN
ejpam-5623	256	16	.	.	PUNCT
ejpam-5623	257	1	(	(	PUNCT
ejpam-5623	257	2	ii	ii	NOUN
ejpam-5623	257	3	)	)	PUNCT
ejpam-5623	257	4	lreg(s∗	lreg(s∗	PROPN
ejpam-5623	257	5	)	)	PUNCT
ejpam-5623	257	6	is	be	AUX
ejpam-5623	257	7	a	a	DET
ejpam-5623	257	8	subsemigroup	subsemigroup	NOUN
ejpam-5623	257	9	of	of	ADP
ejpam-5623	257	10	s∗.	s∗.	PROPN
ejpam-5623	257	11	(	(	PUNCT
ejpam-5623	257	12	iii	iii	NOUN
ejpam-5623	257	13	)	)	PUNCT
ejpam-5623	257	14	dim(w	dim(w	NOUN
ejpam-5623	257	15	)	)	PUNCT
ejpam-5623	257	16	≤	≤	NUM
ejpam-5623	257	17	1	1	NUM
ejpam-5623	257	18	.	.	PUNCT
ejpam-5623	258	1	proof	proof	NOUN
ejpam-5623	258	2	.	.	PUNCT
ejpam-5623	259	1	(	(	PUNCT
ejpam-5623	259	2	i	i	NOUN
ejpam-5623	259	3	)	)	PUNCT
ejpam-5623	259	4	⇒	⇒	PROPN
ejpam-5623	259	5	(	(	PUNCT
ejpam-5623	259	6	ii	ii	NOUN
ejpam-5623	259	7	)	)	PUNCT
ejpam-5623	259	8	this	this	PRON
ejpam-5623	259	9	is	be	AUX
ejpam-5623	259	10	clear	clear	ADJ
ejpam-5623	259	11	by	by	ADP
ejpam-5623	259	12	definition	definition	NOUN
ejpam-5623	259	13	.	.	PUNCT
ejpam-5623	260	1	(	(	PUNCT
ejpam-5623	260	2	ii	ii	NOUN
ejpam-5623	260	3	)	)	PUNCT
ejpam-5623	260	4	⇒	⇒	NOUN
ejpam-5623	260	5	(	(	PUNCT
ejpam-5623	260	6	iii	iii	X
ejpam-5623	260	7	)	)	PUNCT
ejpam-5623	260	8	we	we	PRON
ejpam-5623	260	9	will	will	AUX
ejpam-5623	260	10	prove	prove	VERB
ejpam-5623	260	11	by	by	ADP
ejpam-5623	260	12	contrapositive	contrapositive	PROPN
ejpam-5623	260	13	.	.	PUNCT
ejpam-5623	260	14	assume	assume	VERB
ejpam-5623	260	15	that	that	SCONJ
ejpam-5623	260	16	dim(w	dim(w	ADJ
ejpam-5623	260	17	)	)	PUNCT
ejpam-5623	260	18	≥	≥	NOUN
ejpam-5623	260	19	2	2	NUM
ejpam-5623	260	20	.	.	PUNCT
ejpam-5623	261	1	then	then	ADV
ejpam-5623	261	2	,	,	PUNCT
ejpam-5623	261	3	there	there	PRON
ejpam-5623	261	4	exists	exist	VERB
ejpam-5623	261	5	a	a	DET
ejpam-5623	261	6	basis	basis	NOUN
ejpam-5623	261	7	bw	bw	NOUN
ejpam-5623	261	8	of	of	ADP
ejpam-5623	261	9	w	w	ADP
ejpam-5623	261	10	such	such	ADJ
ejpam-5623	261	11	that	that	SCONJ
ejpam-5623	261	12	|bw	|bw	ADJ
ejpam-5623	261	13	|	|	NOUN
ejpam-5623	261	14	≥	≥	NOUN
ejpam-5623	261	15	2	2	NUM
ejpam-5623	261	16	.	.	PUNCT
ejpam-5623	262	1	let	let	VERB
ejpam-5623	262	2	b	b	X
ejpam-5623	262	3	be	be	AUX
ejpam-5623	262	4	a	a	DET
ejpam-5623	262	5	basis	basis	NOUN
ejpam-5623	262	6	for	for	ADP
ejpam-5623	262	7	v	v	NOUN
ejpam-5623	262	8	such	such	ADJ
ejpam-5623	262	9	that	that	DET
ejpam-5623	262	10	bw	bw	PROPN
ejpam-5623	262	11	⊆	⊆	NUM
ejpam-5623	262	12	b.	b.	PROPN
ejpam-5623	262	13	n.	n.	PROPN
ejpam-5623	262	14	sawatraksa	sawatraksa	PROPN
ejpam-5623	262	15	,	,	PUNCT
ejpam-5623	262	16	p.	p.	NOUN
ejpam-5623	262	17	tantong	tantong	NOUN
ejpam-5623	262	18	/	/	SYM
ejpam-5623	262	19	eur	eur	PROPN
ejpam-5623	262	20	.	.	PUNCT
ejpam-5623	263	1	j.	j.	PROPN
ejpam-5623	263	2	pure	pure	PROPN
ejpam-5623	263	3	appl	appl	PROPN
ejpam-5623	263	4	.	.	PROPN
ejpam-5623	263	5	math	math	PROPN
ejpam-5623	263	6	,	,	PUNCT
ejpam-5623	263	7	18	18	NUM
ejpam-5623	263	8	(	(	PUNCT
ejpam-5623	263	9	1	1	NUM
ejpam-5623	263	10	)	)	PUNCT
ejpam-5623	263	11	(	(	PUNCT
ejpam-5623	263	12	2025	2025	NUM
ejpam-5623	263	13	)	)	PUNCT
ejpam-5623	263	14	,	,	PUNCT
ejpam-5623	263	15	5623	5623	NUM
ejpam-5623	263	16	7	7	NUM
ejpam-5623	263	17	of	of	ADP
ejpam-5623	263	18	15	15	NUM
ejpam-5623	263	19	let	let	VERB
ejpam-5623	263	20	a	a	PRON
ejpam-5623	263	21	and	and	CCONJ
ejpam-5623	263	22	b	b	NOUN
ejpam-5623	263	23	be	be	AUX
ejpam-5623	263	24	distinct	distinct	ADJ
ejpam-5623	263	25	elements	element	NOUN
ejpam-5623	263	26	of	of	ADP
ejpam-5623	263	27	bw	bw	PROPN
ejpam-5623	263	28	.	.	PUNCT
ejpam-5623	264	1	define	define	VERB
ejpam-5623	264	2	two	two	NUM
ejpam-5623	264	3	transformations	transformation	NOUN
ejpam-5623	264	4	α	α	NOUN
ejpam-5623	264	5	and	and	CCONJ
ejpam-5623	264	6	β	β	X
ejpam-5623	264	7	on	on	ADP
ejpam-5623	264	8	b	b	NOUN
ejpam-5623	264	9	as	as	SCONJ
ejpam-5623	264	10	follows	follow	VERB
ejpam-5623	264	11	:	:	PUNCT
ejpam-5623	264	12	xα	xα	PROPN
ejpam-5623	265	1	=	=	PUNCT
ejpam-5623	265	2			PUNCT
ejpam-5623	265	3	a	a	PRON
ejpam-5623	265	4	if	if	NOUN
ejpam-5623	265	5	x	x	PROPN
ejpam-5623	265	6	=	=	SYM
ejpam-5623	265	7	b	b	PROPN
ejpam-5623	265	8	,	,	PUNCT
ejpam-5623	265	9	b	b	NOUN
ejpam-5623	265	10	if	if	SCONJ
ejpam-5623	265	11	x	x	PROPN
ejpam-5623	265	12	=	=	PUNCT
ejpam-5623	265	13	a	a	X
ejpam-5623	265	14	,	,	PUNCT
ejpam-5623	265	15	0	0	NUM
ejpam-5623	265	16	otherwise	otherwise	ADV
ejpam-5623	265	17	,	,	PUNCT
ejpam-5623	265	18	and	and	CCONJ
ejpam-5623	265	19	xβ	xβ	ADV
ejpam-5623	265	20	=	=	SYM
ejpam-5623	265	21	{	{	PUNCT
ejpam-5623	265	22	b	b	NOUN
ejpam-5623	265	23	if	if	SCONJ
ejpam-5623	265	24	x	x	PROPN
ejpam-5623	265	25	=	=	SYM
ejpam-5623	265	26	b	b	PROPN
ejpam-5623	265	27	,	,	PUNCT
ejpam-5623	265	28	0	0	NUM
ejpam-5623	265	29	otherwise	otherwise	ADV
ejpam-5623	265	30	.	.	PUNCT
ejpam-5623	266	1	both	both	DET
ejpam-5623	266	2	α	α	NOUN
ejpam-5623	266	3	and	and	CCONJ
ejpam-5623	266	4	β	β	X
ejpam-5623	266	5	are	be	AUX
ejpam-5623	266	6	well	well	ADV
ejpam-5623	266	7	-	-	PUNCT
ejpam-5623	266	8	defined	define	VERB
ejpam-5623	266	9	and	and	CCONJ
ejpam-5623	266	10	can	can	AUX
ejpam-5623	266	11	be	be	AUX
ejpam-5623	266	12	extended	extend	VERB
ejpam-5623	266	13	to	to	ADP
ejpam-5623	266	14	the	the	DET
ejpam-5623	266	15	linear	linear	ADJ
ejpam-5623	266	16	transformations	transformation	NOUN
ejpam-5623	266	17	on	on	ADP
ejpam-5623	266	18	v	v	NOUN
ejpam-5623	266	19	.	.	PUNCT
ejpam-5623	267	1	since	since	SCONJ
ejpam-5623	267	2	a	a	DET
ejpam-5623	267	3	,	,	PUNCT
ejpam-5623	267	4	b	b	NOUN
ejpam-5623	267	5	,	,	PUNCT
ejpam-5623	267	6	0	0	NUM
ejpam-5623	267	7	∈	∈	PROPN
ejpam-5623	267	8	w	w	NOUN
ejpam-5623	267	9	,	,	PUNCT
ejpam-5623	267	10	it	it	PRON
ejpam-5623	267	11	follows	follow	VERB
ejpam-5623	267	12	that	that	SCONJ
ejpam-5623	267	13	α	α	X
ejpam-5623	267	14	,	,	PUNCT
ejpam-5623	267	15	β	β	X
ejpam-5623	267	16	∈	∈	PROPN
ejpam-5623	267	17	s∗.	s∗.	ADJ
ejpam-5623	267	18	from	from	ADP
ejpam-5623	267	19	the	the	DET
ejpam-5623	267	20	previous	previous	ADJ
ejpam-5623	267	21	theorem	theorem	NOUN
ejpam-5623	267	22	(	(	PUNCT
ejpam-5623	267	23	theorem	theorem	NOUN
ejpam-5623	267	24	3	3	NUM
ejpam-5623	267	25	)	)	PUNCT
ejpam-5623	267	26	,	,	PUNCT
ejpam-5623	267	27	we	we	PRON
ejpam-5623	267	28	know	know	VERB
ejpam-5623	267	29	that	that	SCONJ
ejpam-5623	267	30	α	α	PROPN
ejpam-5623	267	31	and	and	CCONJ
ejpam-5623	267	32	β	β	X
ejpam-5623	267	33	are	be	AUX
ejpam-5623	267	34	left	leave	VERB
ejpam-5623	267	35	regular	regular	ADV
ejpam-5623	267	36	because	because	SCONJ
ejpam-5623	267	37	bα	bα	PROPN
ejpam-5623	267	38	⊆	⊆	NUM
ejpam-5623	267	39	wα2	wα2	NOUN
ejpam-5623	267	40	and	and	CCONJ
ejpam-5623	267	41	bβ	bβ	VERB
ejpam-5623	267	42	⊆	⊆	NUM
ejpam-5623	267	43	wβ2	wβ2	NOUN
ejpam-5623	267	44	.	.	PUNCT
ejpam-5623	268	1	however	however	ADV
ejpam-5623	268	2	,	,	PUNCT
ejpam-5623	268	3	their	their	PRON
ejpam-5623	268	4	product	product	NOUN
ejpam-5623	268	5	αβ	αβ	INTJ
ejpam-5623	268	6	is	be	AUX
ejpam-5623	268	7	not	not	PART
ejpam-5623	268	8	left	leave	VERB
ejpam-5623	268	9	regular	regular	ADV
ejpam-5623	268	10	.	.	PUNCT
ejpam-5623	269	1	to	to	PART
ejpam-5623	269	2	see	see	VERB
ejpam-5623	269	3	this	this	PRON
ejpam-5623	269	4	,	,	PUNCT
ejpam-5623	269	5	note	note	VERB
ejpam-5623	269	6	that	that	SCONJ
ejpam-5623	269	7	:	:	PUNCT
ejpam-5623	269	8	aα	aα	NOUN
ejpam-5623	269	9	=	=	SYM
ejpam-5623	269	10	b	b	PROPN
ejpam-5623	269	11	,	,	PUNCT
ejpam-5623	269	12	bα	bα	PROPN
ejpam-5623	269	13	=	=	SYM
ejpam-5623	269	14	a	a	NOUN
ejpam-5623	269	15	,	,	PUNCT
ejpam-5623	269	16	aβ	aβ	NOUN
ejpam-5623	269	17	=	=	SYM
ejpam-5623	269	18	0	0	NUM
ejpam-5623	269	19	,	,	PUNCT
ejpam-5623	269	20	bβ	bβ	NOUN
ejpam-5623	269	21	=	=	SYM
ejpam-5623	269	22	b.	b.	PROPN
ejpam-5623	269	23	thus	thus	ADV
ejpam-5623	269	24	,	,	PUNCT
ejpam-5623	269	25	aαβ	aαβ	PROPN
ejpam-5623	269	26	=	=	SYM
ejpam-5623	269	27	b	b	PROPN
ejpam-5623	269	28	and	and	CCONJ
ejpam-5623	269	29	bαβ	bαβ	ADJ
ejpam-5623	269	30	=	=	NOUN
ejpam-5623	269	31	0	0	NUM
ejpam-5623	270	1	=	=	PUNCT
ejpam-5623	270	2	0αβ	0αβ	NOUN
ejpam-5623	270	3	,	,	PUNCT
ejpam-5623	270	4	showing	show	VERB
ejpam-5623	270	5	that	that	SCONJ
ejpam-5623	270	6	αβ	αβ	PRON
ejpam-5623	270	7	is	be	AUX
ejpam-5623	270	8	not	not	PART
ejpam-5623	270	9	one	one	NUM
ejpam-5623	270	10	-	-	PUNCT
ejpam-5623	270	11	to	to	ADP
ejpam-5623	270	12	-	-	PUNCT
ejpam-5623	270	13	one	one	NUM
ejpam-5623	270	14	on	on	ADP
ejpam-5623	270	15	its	its	PRON
ejpam-5623	270	16	image	image	NOUN
ejpam-5623	270	17	.	.	PUNCT
ejpam-5623	271	1	therefore	therefore	ADV
ejpam-5623	271	2	,	,	PUNCT
ejpam-5623	271	3	αβ	αβ	PRON
ejpam-5623	271	4	is	be	AUX
ejpam-5623	271	5	not	not	PART
ejpam-5623	271	6	left	leave	VERB
ejpam-5623	271	7	regular	regular	ADV
ejpam-5623	271	8	.	.	PUNCT
ejpam-5623	272	1	this	this	PRON
ejpam-5623	272	2	implies	imply	VERB
ejpam-5623	272	3	that	that	SCONJ
ejpam-5623	272	4	lreg(s∗	lreg(s∗	PROPN
ejpam-5623	272	5	)	)	PUNCT
ejpam-5623	272	6	is	be	AUX
ejpam-5623	272	7	not	not	PART
ejpam-5623	272	8	a	a	DET
ejpam-5623	272	9	subsemigroup	subsemigroup	NOUN
ejpam-5623	272	10	of	of	ADP
ejpam-5623	272	11	s∗.	s∗.	PROPN
ejpam-5623	272	12	(	(	PUNCT
ejpam-5623	272	13	iii	iii	NOUN
ejpam-5623	272	14	)	)	PUNCT
ejpam-5623	272	15	⇒	⇒	NOUN
ejpam-5623	272	16	(	(	PUNCT
ejpam-5623	272	17	i	i	NOUN
ejpam-5623	272	18	)	)	PUNCT
ejpam-5623	272	19	suppose	suppose	VERB
ejpam-5623	272	20	that	that	SCONJ
ejpam-5623	272	21	dim(w	dim(w	X
ejpam-5623	272	22	)	)	PUNCT
ejpam-5623	272	23	≤	≤	NUM
ejpam-5623	272	24	1	1	NUM
ejpam-5623	272	25	.	.	PUNCT
ejpam-5623	273	1	let	let	VERB
ejpam-5623	273	2	α	α	PRON
ejpam-5623	273	3	∈	∈	ADJ
ejpam-5623	273	4	s∗.	s∗.	ADJ
ejpam-5623	273	5	if	if	SCONJ
ejpam-5623	273	6	α	α	PRON
ejpam-5623	273	7	is	be	AUX
ejpam-5623	273	8	the	the	DET
ejpam-5623	273	9	zero	zero	NUM
ejpam-5623	273	10	transformation	transformation	NOUN
ejpam-5623	273	11	,	,	PUNCT
ejpam-5623	273	12	it	it	PRON
ejpam-5623	273	13	is	be	AUX
ejpam-5623	273	14	trivially	trivially	ADV
ejpam-5623	273	15	left	leave	VERB
ejpam-5623	273	16	regular	regular	ADV
ejpam-5623	273	17	.	.	PUNCT
ejpam-5623	274	1	now	now	ADV
ejpam-5623	274	2	,	,	PUNCT
ejpam-5623	274	3	assume	assume	VERB
ejpam-5623	274	4	that	that	SCONJ
ejpam-5623	274	5	α	α	PRON
ejpam-5623	274	6	is	be	AUX
ejpam-5623	274	7	not	not	PART
ejpam-5623	274	8	the	the	DET
ejpam-5623	274	9	zero	zero	NUM
ejpam-5623	274	10	transformation	transformation	NOUN
ejpam-5623	274	11	.	.	PUNCT
ejpam-5623	275	1	then	then	ADV
ejpam-5623	275	2	,	,	PUNCT
ejpam-5623	275	3	dim(v	dim(v	PROPN
ejpam-5623	275	4	α	α	X
ejpam-5623	275	5	)	)	PUNCT
ejpam-5623	275	6	̸=	̸=	PROPN
ejpam-5623	275	7	0	0	NUM
ejpam-5623	275	8	.	.	PUNCT
ejpam-5623	276	1	since	since	SCONJ
ejpam-5623	276	2	dim(w	dim(w	X
ejpam-5623	276	3	)	)	PUNCT
ejpam-5623	276	4	≤	≤	NUM
ejpam-5623	276	5	1	1	NUM
ejpam-5623	276	6	,	,	PUNCT
ejpam-5623	276	7	v	v	ADP
ejpam-5623	276	8	α	α	NOUN
ejpam-5623	276	9	⊆	⊆	NUM
ejpam-5623	276	10	w	w	NOUN
ejpam-5623	276	11	,	,	PUNCT
ejpam-5623	276	12	and	and	CCONJ
ejpam-5623	276	13	thus	thus	ADV
ejpam-5623	276	14	α	α	PRON
ejpam-5623	276	15	is	be	AUX
ejpam-5623	276	16	surjective	surjective	ADJ
ejpam-5623	276	17	onw	onw	NOUN
ejpam-5623	276	18	.	.	PUNCT
ejpam-5623	277	1	by	by	ADP
ejpam-5623	277	2	assumption	assumption	NOUN
ejpam-5623	277	3	,	,	PUNCT
ejpam-5623	277	4	we	we	PRON
ejpam-5623	277	5	have	have	VERB
ejpam-5623	277	6	dim(v	dim(v	PROPN
ejpam-5623	277	7	α	α	NOUN
ejpam-5623	277	8	)	)	PUNCT
ejpam-5623	277	9	=	=	SYM
ejpam-5623	278	1	1	1	X
ejpam-5623	278	2	.	.	PUNCT
ejpam-5623	278	3	hence	hence	ADV
ejpam-5623	278	4	,	,	PUNCT
ejpam-5623	278	5	lreg(s∗	lreg(s∗	PROPN
ejpam-5623	278	6	)	)	PUNCT
ejpam-5623	278	7	=	=	SYM
ejpam-5623	278	8	s∗	s∗	PROPN
ejpam-5623	278	9	,	,	PUNCT
ejpam-5623	278	10	meaning	mean	VERB
ejpam-5623	278	11	that	that	SCONJ
ejpam-5623	278	12	s∗	s∗	PROPN
ejpam-5623	278	13	is	be	AUX
ejpam-5623	278	14	a	a	DET
ejpam-5623	278	15	left	left	ADJ
ejpam-5623	278	16	regular	regular	ADJ
ejpam-5623	278	17	semigroup	semigroup	NOUN
ejpam-5623	278	18	.	.	PUNCT
ejpam-5623	279	1	corollary	corollary	ADJ
ejpam-5623	279	2	2	2	NUM
ejpam-5623	279	3	.	.	PUNCT
ejpam-5623	280	1	the	the	DET
ejpam-5623	280	2	following	follow	VERB
ejpam-5623	280	3	statements	statement	NOUN
ejpam-5623	280	4	are	be	AUX
ejpam-5623	280	5	equivalent	equivalent	ADJ
ejpam-5623	280	6	:	:	PUNCT
ejpam-5623	280	7	(	(	PUNCT
ejpam-5623	280	8	i	i	NOUN
ejpam-5623	280	9	)	)	PUNCT
ejpam-5623	280	10	s∗	s∗	PROPN
ejpam-5623	280	11	is	be	AUX
ejpam-5623	280	12	a	a	DET
ejpam-5623	280	13	left	left	ADJ
ejpam-5623	280	14	regular	regular	ADJ
ejpam-5623	280	15	semigroup	semigroup	NOUN
ejpam-5623	280	16	.	.	PUNCT
ejpam-5623	281	1	(	(	PUNCT
ejpam-5623	281	2	ii	ii	NOUN
ejpam-5623	281	3	)	)	PUNCT
ejpam-5623	281	4	creg(s∗	creg(s∗	NOUN
ejpam-5623	281	5	)	)	PUNCT
ejpam-5623	281	6	is	be	AUX
ejpam-5623	281	7	a	a	DET
ejpam-5623	281	8	subsemigroup	subsemigroup	NOUN
ejpam-5623	281	9	of	of	ADP
ejpam-5623	281	10	s∗.	s∗.	PROPN
ejpam-5623	281	11	(	(	PUNCT
ejpam-5623	281	12	iii	iii	NOUN
ejpam-5623	281	13	)	)	PUNCT
ejpam-5623	281	14	dim(w	dim(w	NOUN
ejpam-5623	281	15	)	)	PUNCT
ejpam-5623	281	16	≤	≤	NUM
ejpam-5623	281	17	1	1	NUM
ejpam-5623	281	18	.	.	PUNCT
ejpam-5623	282	1	the	the	DET
ejpam-5623	282	2	following	follow	VERB
ejpam-5623	282	3	theorem	theorem	NOUN
ejpam-5623	282	4	gives	give	VERB
ejpam-5623	282	5	a	a	DET
ejpam-5623	282	6	necessary	necessary	ADJ
ejpam-5623	282	7	and	and	CCONJ
ejpam-5623	282	8	sufficient	sufficient	ADJ
ejpam-5623	282	9	condition	condition	NOUN
ejpam-5623	282	10	for	for	ADP
ejpam-5623	282	11	an	an	DET
ejpam-5623	282	12	element	element	NOUN
ejpam-5623	282	13	of	of	ADP
ejpam-5623	282	14	q	q	NOUN
ejpam-5623	282	15	to	to	PART
ejpam-5623	282	16	be	be	AUX
ejpam-5623	282	17	left	leave	VERB
ejpam-5623	282	18	regular	regular	ADV
ejpam-5623	282	19	.	.	PUNCT
ejpam-5623	283	1	theorem	theorem	NOUN
ejpam-5623	283	2	5	5	NUM
ejpam-5623	283	3	.	.	PUNCT
ejpam-5623	284	1	let	let	VERB
ejpam-5623	284	2	α	α	PRON
ejpam-5623	284	3	∈	∈	PROPN
ejpam-5623	284	4	q.	q.	NOUN
ejpam-5623	284	5	the	the	DET
ejpam-5623	284	6	following	follow	VERB
ejpam-5623	284	7	statements	statement	NOUN
ejpam-5623	284	8	are	be	AUX
ejpam-5623	284	9	equivalent	equivalent	ADJ
ejpam-5623	284	10	:	:	PUNCT
ejpam-5623	284	11	(	(	PUNCT
ejpam-5623	284	12	i	i	NOUN
ejpam-5623	284	13	)	)	PUNCT
ejpam-5623	284	14	α	α	PROPN
ejpam-5623	284	15	∈	∈	PROPN
ejpam-5623	284	16	lreg(q	lreg(q	NOUN
ejpam-5623	284	17	)	)	PUNCT
ejpam-5623	284	18	.	.	PUNCT
ejpam-5623	285	1	(	(	PUNCT
ejpam-5623	285	2	ii	ii	NOUN
ejpam-5623	285	3	)	)	PUNCT
ejpam-5623	285	4	α|wα	α|wα	NOUN
ejpam-5623	285	5	is	be	AUX
ejpam-5623	285	6	an	an	PRON
ejpam-5623	285	7	onto	onto	ADP
ejpam-5623	285	8	transformation	transformation	NOUN
ejpam-5623	285	9	on	on	ADP
ejpam-5623	285	10	v	v	NUM
ejpam-5623	285	11	α	α	NOUN
ejpam-5623	285	12	.	.	PUNCT
ejpam-5623	286	1	(	(	PUNCT
ejpam-5623	286	2	iii	iii	X
ejpam-5623	286	3	)	)	PUNCT
ejpam-5623	286	4	v	v	NOUN
ejpam-5623	286	5	α	α	NOUN
ejpam-5623	286	6	⊆	⊆	NUM
ejpam-5623	286	7	wα2	wα2	NOUN
ejpam-5623	286	8	.	.	PUNCT
ejpam-5623	287	1	proof	proof	NOUN
ejpam-5623	287	2	.	.	PUNCT
ejpam-5623	288	1	(	(	PUNCT
ejpam-5623	288	2	i	i	NOUN
ejpam-5623	288	3	)	)	PUNCT
ejpam-5623	288	4	⇒	⇒	PROPN
ejpam-5623	288	5	(	(	PUNCT
ejpam-5623	288	6	ii	ii	NOUN
ejpam-5623	288	7	)	)	PUNCT
ejpam-5623	288	8	and	and	CCONJ
ejpam-5623	288	9	(	(	PUNCT
ejpam-5623	288	10	ii	ii	NOUN
ejpam-5623	288	11	)	)	PUNCT
ejpam-5623	288	12	⇒	⇒	NOUN
ejpam-5623	288	13	(	(	PUNCT
ejpam-5623	288	14	iii	iii	X
ejpam-5623	288	15	)	)	PUNCT
ejpam-5623	288	16	these	these	PRON
ejpam-5623	288	17	follow	follow	VERB
ejpam-5623	288	18	directly	directly	ADV
ejpam-5623	288	19	from	from	ADP
ejpam-5623	288	20	theorem	theorem	ADJ
ejpam-5623	288	21	3	3	NUM
ejpam-5623	288	22	.	.	PUNCT
ejpam-5623	288	23	(	(	PUNCT
ejpam-5623	288	24	iii	iii	NOUN
ejpam-5623	288	25	)	)	PUNCT
ejpam-5623	288	26	⇒	⇒	NOUN
ejpam-5623	288	27	(	(	PUNCT
ejpam-5623	288	28	i	i	NOUN
ejpam-5623	288	29	)	)	PUNCT
ejpam-5623	288	30	assume	assume	VERB
ejpam-5623	288	31	that	that	SCONJ
ejpam-5623	289	1	v	v	X
ejpam-5623	289	2	α	α	NOUN
ejpam-5623	289	3	⊆	⊆	NUM
ejpam-5623	289	4	wα2	wα2	NOUN
ejpam-5623	289	5	.	.	PUNCT
ejpam-5623	290	1	let	let	VERB
ejpam-5623	290	2	bw	bw	PART
ejpam-5623	290	3	be	be	AUX
ejpam-5623	290	4	a	a	DET
ejpam-5623	290	5	basis	basis	NOUN
ejpam-5623	290	6	for	for	ADP
ejpam-5623	290	7	w	w	NOUN
ejpam-5623	290	8	,	,	PUNCT
ejpam-5623	290	9	and	and	CCONJ
ejpam-5623	290	10	let	let	VERB
ejpam-5623	290	11	b	b	X
ejpam-5623	290	12	be	be	AUX
ejpam-5623	290	13	a	a	DET
ejpam-5623	290	14	basis	basis	NOUN
ejpam-5623	290	15	for	for	ADP
ejpam-5623	290	16	v	v	NOUN
ejpam-5623	290	17	such	such	ADJ
ejpam-5623	290	18	that	that	PRON
ejpam-5623	290	19	bw	bw	PROPN
ejpam-5623	290	20	⊆	⊆	NUM
ejpam-5623	290	21	b.	b.	NOUN
ejpam-5623	290	22	for	for	ADP
ejpam-5623	290	23	each	each	DET
ejpam-5623	290	24	v	v	ADP
ejpam-5623	290	25	∈	∈	PROPN
ejpam-5623	290	26	b	b	NOUN
ejpam-5623	290	27	\	\	PROPN
ejpam-5623	290	28	bw	bw	NOUN
ejpam-5623	290	29	,	,	PUNCT
ejpam-5623	290	30	since	since	SCONJ
ejpam-5623	290	31	v	v	NUM
ejpam-5623	290	32	α	α	PRON
ejpam-5623	290	33	⊆	⊆	NUM
ejpam-5623	290	34	wα	wα	NOUN
ejpam-5623	290	35	,	,	PUNCT
ejpam-5623	290	36	we	we	PRON
ejpam-5623	290	37	choose	choose	VERB
ejpam-5623	290	38	and	and	CCONJ
ejpam-5623	290	39	fix	fix	VERB
ejpam-5623	290	40	n.	n.	NOUN
ejpam-5623	290	41	sawatraksa	sawatraksa	NOUN
ejpam-5623	290	42	,	,	PUNCT
ejpam-5623	290	43	p.	p.	NOUN
ejpam-5623	290	44	tantong	tantong	NOUN
ejpam-5623	290	45	/	/	SYM
ejpam-5623	290	46	eur	eur	PROPN
ejpam-5623	290	47	.	.	PUNCT
ejpam-5623	291	1	j.	j.	PROPN
ejpam-5623	291	2	pure	pure	PROPN
ejpam-5623	291	3	appl	appl	PROPN
ejpam-5623	291	4	.	.	PROPN
ejpam-5623	291	5	math	math	PROPN
ejpam-5623	291	6	,	,	PUNCT
ejpam-5623	291	7	18	18	NUM
ejpam-5623	291	8	(	(	PUNCT
ejpam-5623	291	9	1	1	NUM
ejpam-5623	291	10	)	)	PUNCT
ejpam-5623	291	11	(	(	PUNCT
ejpam-5623	291	12	2025	2025	NUM
ejpam-5623	291	13	)	)	PUNCT
ejpam-5623	291	14	,	,	PUNCT
ejpam-5623	291	15	5623	5623	NUM
ejpam-5623	291	16	8	8	NUM
ejpam-5623	291	17	of	of	ADP
ejpam-5623	291	18	15	15	NUM
ejpam-5623	291	19	v′	v′	NOUN
ejpam-5623	291	20	∈	∈	PROPN
ejpam-5623	292	1	w	w	ADP
ejpam-5623	292	2	such	such	ADJ
ejpam-5623	292	3	that	that	DET
ejpam-5623	292	4	vα	vα	ADP
ejpam-5623	292	5	=	=	PUNCT
ejpam-5623	292	6	v′α	v′α	ADJ
ejpam-5623	292	7	.	.	PUNCT
ejpam-5623	293	1	for	for	ADP
ejpam-5623	293	2	each	each	DET
ejpam-5623	293	3	w	w	PROPN
ejpam-5623	293	4	∈	∈	PROPN
ejpam-5623	293	5	w	w	NOUN
ejpam-5623	293	6	,	,	PUNCT
ejpam-5623	293	7	we	we	PRON
ejpam-5623	293	8	let	let	VERB
ejpam-5623	293	9	w′	w′	PROPN
ejpam-5623	293	10	=	=	PUNCT
ejpam-5623	293	11	w.	w.	PROPN
ejpam-5623	293	12	for	for	ADP
ejpam-5623	293	13	each	each	DET
ejpam-5623	293	14	w′	w′	PROPN
ejpam-5623	293	15	∈	∈	PROPN
ejpam-5623	293	16	w	w	PROPN
ejpam-5623	293	17	and	and	CCONJ
ejpam-5623	293	18	by	by	ADP
ejpam-5623	293	19	assumption	assumption	NOUN
ejpam-5623	293	20	,	,	PUNCT
ejpam-5623	293	21	we	we	PRON
ejpam-5623	293	22	choose	choose	VERB
ejpam-5623	293	23	and	and	CCONJ
ejpam-5623	293	24	fix	fix	VERB
ejpam-5623	293	25	w′′	w′′	NOUN
ejpam-5623	293	26	∈	∈	PROPN
ejpam-5623	293	27	w	w	ADP
ejpam-5623	293	28	such	such	ADJ
ejpam-5623	293	29	that	that	DET
ejpam-5623	293	30	w′α	w′α	NOUN
ejpam-5623	293	31	=	=	SYM
ejpam-5623	293	32	w′′α2	w′′α2	PROPN
ejpam-5623	293	33	.	.	PUNCT
ejpam-5623	294	1	define	define	VERB
ejpam-5623	294	2	β	β	X
ejpam-5623	294	3	:	:	PUNCT
ejpam-5623	294	4	b	b	X
ejpam-5623	294	5	→	→	SYM
ejpam-5623	294	6	v	v	NOUN
ejpam-5623	294	7	by	by	ADP
ejpam-5623	294	8	vβ	vβ	DET
ejpam-5623	294	9	=	=	NOUN
ejpam-5623	294	10	v′′	v′′	NOUN
ejpam-5623	294	11	for	for	ADP
ejpam-5623	295	1	all	all	DET
ejpam-5623	295	2	v	v	ADP
ejpam-5623	295	3	∈	∈	NOUN
ejpam-5623	295	4	b	b	NOUN
ejpam-5623	296	1	it	it	PRON
ejpam-5623	296	2	is	be	AUX
ejpam-5623	296	3	easy	easy	ADJ
ejpam-5623	296	4	to	to	PART
ejpam-5623	296	5	verify	verify	VERB
ejpam-5623	296	6	that	that	SCONJ
ejpam-5623	296	7	β	β	NOUN
ejpam-5623	296	8	is	be	AUX
ejpam-5623	296	9	well	well	ADV
ejpam-5623	296	10	-	-	PUNCT
ejpam-5623	296	11	defined	define	VERB
ejpam-5623	296	12	,	,	PUNCT
ejpam-5623	296	13	and	and	CCONJ
ejpam-5623	296	14	it	it	PRON
ejpam-5623	296	15	can	can	AUX
ejpam-5623	296	16	be	be	AUX
ejpam-5623	296	17	extended	extend	VERB
ejpam-5623	296	18	to	to	ADP
ejpam-5623	296	19	a	a	DET
ejpam-5623	296	20	linear	linear	ADJ
ejpam-5623	296	21	transformation	transformation	NOUN
ejpam-5623	296	22	on	on	ADP
ejpam-5623	296	23	v	v	NUM
ejpam-5623	296	24	.	.	PUNCT
ejpam-5623	297	1	by	by	ADP
ejpam-5623	297	2	the	the	DET
ejpam-5623	297	3	method	method	NOUN
ejpam-5623	297	4	of	of	ADP
ejpam-5623	297	5	constructing	construct	VERB
ejpam-5623	297	6	β	β	X
ejpam-5623	297	7	,	,	PUNCT
ejpam-5623	297	8	it	it	PRON
ejpam-5623	297	9	is	be	AUX
ejpam-5623	297	10	easy	easy	ADJ
ejpam-5623	297	11	to	to	PART
ejpam-5623	297	12	see	see	VERB
ejpam-5623	297	13	that	that	SCONJ
ejpam-5623	297	14	β	β	PROPN
ejpam-5623	297	15	∈	∈	PROPN
ejpam-5623	297	16	l(v	l(v	PROPN
ejpam-5623	297	17	,	,	PUNCT
ejpam-5623	297	18	w	w	PROPN
ejpam-5623	297	19	)	)	PUNCT
ejpam-5623	297	20	.	.	PUNCT
ejpam-5623	298	1	we	we	PRON
ejpam-5623	298	2	now	now	ADV
ejpam-5623	298	3	show	show	VERB
ejpam-5623	298	4	that	that	SCONJ
ejpam-5623	298	5	v	v	ADP
ejpam-5623	298	6	β	β	NOUN
ejpam-5623	298	7	⊆	⊆	NUM
ejpam-5623	298	8	wβ	wβ	ADP
ejpam-5623	298	9	.	.	PUNCT
ejpam-5623	299	1	let	let	VERB
ejpam-5623	299	2	v	v	NUM
ejpam-5623	299	3	∈	∈	PROPN
ejpam-5623	299	4	v	v	NOUN
ejpam-5623	299	5	.	.	PUNCT
ejpam-5623	300	1	then	then	ADV
ejpam-5623	300	2	,	,	PUNCT
ejpam-5623	300	3	v	v	NOUN
ejpam-5623	300	4	=	=	SYM
ejpam-5623	300	5	a1v1	a1v1	PROPN
ejpam-5623	300	6	+	+	CCONJ
ejpam-5623	300	7	a2v2	a2v2	PROPN
ejpam-5623	300	8	+	+	CCONJ
ejpam-5623	300	9	.	.	PUNCT
ejpam-5623	300	10	.	.	PUNCT
ejpam-5623	301	1	.+	.+	NOUN
ejpam-5623	301	2	anvn	anvn	VERB
ejpam-5623	301	3	where	where	SCONJ
ejpam-5623	301	4	v1	v1	NOUN
ejpam-5623	301	5	,	,	PUNCT
ejpam-5623	301	6	v2	v2	NOUN
ejpam-5623	301	7	,	,	PUNCT
ejpam-5623	301	8	.	.	PUNCT
ejpam-5623	301	9	.	.	PUNCT
ejpam-5623	302	1	.	.	PUNCT
ejpam-5623	303	1	,	,	PUNCT
ejpam-5623	303	2	vk	vk	ADP
ejpam-5623	303	3	∈	∈	PROPN
ejpam-5623	303	4	b	b	PROPN
ejpam-5623	303	5	,	,	PUNCT
ejpam-5623	303	6	and	and	CCONJ
ejpam-5623	303	7	a1	a1	NOUN
ejpam-5623	303	8	,	,	PUNCT
ejpam-5623	303	9	a2	a2	PROPN
ejpam-5623	303	10	,	,	PUNCT
ejpam-5623	303	11	.	.	PUNCT
ejpam-5623	303	12	.	.	PUNCT
ejpam-5623	304	1	.	.	PUNCT
ejpam-5623	305	1	,	,	PUNCT
ejpam-5623	305	2	ak	ak	PROPN
ejpam-5623	305	3	∈	∈	PROPN
ejpam-5623	305	4	f.	f.	PROPN
ejpam-5623	305	5	this	this	PRON
ejpam-5623	305	6	implies	imply	VERB
ejpam-5623	305	7	that	that	SCONJ
ejpam-5623	305	8	vβ	vβ	X
ejpam-5623	305	9	=	=	PUNCT
ejpam-5623	305	10	(	(	PUNCT
ejpam-5623	305	11	a1v1	a1v1	PROPN
ejpam-5623	305	12	+	+	SYM
ejpam-5623	305	13	a2v2	a2v2	PUNCT
ejpam-5623	305	14	+	+	CCONJ
ejpam-5623	305	15	.	.	PUNCT
ejpam-5623	305	16	.	.	PUNCT
ejpam-5623	306	1	.+	.+	NOUN
ejpam-5623	306	2	akvk)β	akvk)β	PUNCT
ejpam-5623	306	3	=	=	SYM
ejpam-5623	306	4	a1(v1β	a1(v1β	NOUN
ejpam-5623	306	5	)	)	PUNCT
ejpam-5623	307	1	+	+	CCONJ
ejpam-5623	307	2	a2(v2β	a2(v2β	NOUN
ejpam-5623	307	3	)	)	PUNCT
ejpam-5623	308	1	+	+	CCONJ
ejpam-5623	308	2	.	.	PUNCT
ejpam-5623	308	3	.	.	PUNCT
ejpam-5623	309	1	.+	.+	NOUN
ejpam-5623	309	2	ak(vkβ	ak(vkβ	PROPN
ejpam-5623	309	3	)	)	PUNCT
ejpam-5623	309	4	=	=	PUNCT
ejpam-5623	310	1	a1v	a1v	PROPN
ejpam-5623	310	2	′′	′′	PROPN
ejpam-5623	310	3	1	1	NUM
ejpam-5623	310	4	+	+	CCONJ
ejpam-5623	310	5	a2v	a2v	ADV
ejpam-5623	310	6	′′	′′	PROPN
ejpam-5623	310	7	2	2	NUM
ejpam-5623	310	8	+	+	CCONJ
ejpam-5623	310	9	.	.	PUNCT
ejpam-5623	310	10	.	.	PUNCT
ejpam-5623	311	1	.+	.+	NOUN
ejpam-5623	311	2	akv	akv	VERB
ejpam-5623	311	3	′′	′′	PROPN
ejpam-5623	311	4	k	k	PROPN
ejpam-5623	311	5	=	=	PUNCT
ejpam-5623	311	6	a1v	a1v	PROPN
ejpam-5623	312	1	′	′	NUM
ejpam-5623	312	2	1β	1β	NUM
ejpam-5623	313	1	+	+	CCONJ
ejpam-5623	314	1	a2v	a2v	INTJ
ejpam-5623	314	2	′	′	NUM
ejpam-5623	314	3	2β	2β	NOUN
ejpam-5623	314	4	+	+	CCONJ
ejpam-5623	314	5	.	.	PUNCT
ejpam-5623	314	6	.	.	PUNCT
ejpam-5623	315	1	.+	.+	NOUN
ejpam-5623	315	2	akv	akv	VERB
ejpam-5623	315	3	′	′	NUM
ejpam-5623	315	4	kβ	kβ	PROPN
ejpam-5623	315	5	=	=	PUNCT
ejpam-5623	315	6	(	(	PUNCT
ejpam-5623	315	7	a1v	a1v	PROPN
ejpam-5623	315	8	′	′	NOUN
ejpam-5623	315	9	1	1	NUM
ejpam-5623	316	1	+	+	CCONJ
ejpam-5623	316	2	a2v	a2v	INTJ
ejpam-5623	316	3	′	′	NUM
ejpam-5623	316	4	2	2	NUM
ejpam-5623	316	5	+	+	CCONJ
ejpam-5623	316	6	.	.	PUNCT
ejpam-5623	316	7	.	.	PUNCT
ejpam-5623	317	1	.+	.+	NOUN
ejpam-5623	317	2	akv	akv	VERB
ejpam-5623	317	3	′	′	NUM
ejpam-5623	317	4	k)β	k)β	NOUN
ejpam-5623	317	5	.	.	PUNCT
ejpam-5623	318	1	since	since	SCONJ
ejpam-5623	318	2	w	w	PROPN
ejpam-5623	318	3	is	be	AUX
ejpam-5623	318	4	a	a	DET
ejpam-5623	318	5	subspace	subspace	NOUN
ejpam-5623	318	6	of	of	ADP
ejpam-5623	318	7	v	v	NOUN
ejpam-5623	318	8	and	and	CCONJ
ejpam-5623	318	9	v′1	v′1	VERB
ejpam-5623	318	10	,	,	PUNCT
ejpam-5623	318	11	v	v	ADJ
ejpam-5623	318	12	′	′	NUM
ejpam-5623	318	13	2	2	NUM
ejpam-5623	318	14	,	,	PUNCT
ejpam-5623	318	15	.	.	PUNCT
ejpam-5623	318	16	.	.	PUNCT
ejpam-5623	319	1	.	.	PUNCT
ejpam-5623	320	1	,	,	PUNCT
ejpam-5623	320	2	v	v	X
ejpam-5623	320	3	′	′	NUM
ejpam-5623	321	1	k	k	PROPN
ejpam-5623	321	2	∈	∈	PROPN
ejpam-5623	321	3	w	w	NOUN
ejpam-5623	321	4	,	,	PUNCT
ejpam-5623	321	5	we	we	PRON
ejpam-5623	321	6	have	have	VERB
ejpam-5623	321	7	that	that	PRON
ejpam-5623	321	8	a1v	a1v	PROPN
ejpam-5623	321	9	′	′	NUM
ejpam-5623	322	1	1+a2v	1+a2v	NUM
ejpam-5623	322	2	′	′	NUM
ejpam-5623	322	3	2	2	NUM
ejpam-5623	322	4	+	+	NUM
ejpam-5623	322	5	.	.	PUNCT
ejpam-5623	322	6	.	.	PUNCT
ejpam-5623	322	7	.+akv	.+akv	PUNCT
ejpam-5623	323	1	′	′	NUM
ejpam-5623	324	1	k	k	X
ejpam-5623	324	2	∈	∈	PROPN
ejpam-5623	324	3	w	w	X
ejpam-5623	324	4	.	.	PUNCT
ejpam-5623	325	1	this	this	PRON
ejpam-5623	325	2	implies	imply	VERB
ejpam-5623	325	3	that	that	SCONJ
ejpam-5623	325	4	β	β	PROPN
ejpam-5623	325	5	∈	∈	PROPN
ejpam-5623	325	6	q.	q.	NOUN
ejpam-5623	325	7	moreover	moreover	ADV
ejpam-5623	325	8	,	,	PUNCT
ejpam-5623	325	9	we	we	PRON
ejpam-5623	325	10	will	will	AUX
ejpam-5623	325	11	now	now	ADV
ejpam-5623	325	12	show	show	VERB
ejpam-5623	325	13	that	that	SCONJ
ejpam-5623	325	14	α	α	PRON
ejpam-5623	325	15	=	=	X
ejpam-5623	325	16	βα2	βα2	ADJ
ejpam-5623	325	17	.	.	PUNCT
ejpam-5623	326	1	for	for	ADP
ejpam-5623	326	2	each	each	DET
ejpam-5623	326	3	v	v	NUM
ejpam-5623	326	4	∈	∈	PROPN
ejpam-5623	326	5	b	b	NOUN
ejpam-5623	326	6	,	,	PUNCT
ejpam-5623	326	7	we	we	PRON
ejpam-5623	326	8	have	have	VERB
ejpam-5623	326	9	vβα2	vβα2	NOUN
ejpam-5623	326	10	=	=	PUNCT
ejpam-5623	327	1	v′′αα	v′′αα	PUNCT
ejpam-5623	327	2	=	=	PROPN
ejpam-5623	327	3	v′α	v′α	NOUN
ejpam-5623	328	1	=	=	SYM
ejpam-5623	328	2	vα	vα	PROPN
ejpam-5623	328	3	.	.	PUNCT
ejpam-5623	328	4	hence	hence	ADV
ejpam-5623	328	5	,	,	PUNCT
ejpam-5623	328	6	βα2	βα2	NOUN
ejpam-5623	328	7	=	=	SYM
ejpam-5623	328	8	α	α	PROPN
ejpam-5623	328	9	,	,	PUNCT
ejpam-5623	328	10	proving	prove	VERB
ejpam-5623	328	11	that	that	SCONJ
ejpam-5623	328	12	α	α	NOUN
ejpam-5623	328	13	is	be	AUX
ejpam-5623	328	14	left	leave	VERB
ejpam-5623	328	15	regular	regular	ADV
ejpam-5623	328	16	.	.	PUNCT
ejpam-5623	329	1	the	the	DET
ejpam-5623	329	2	following	follow	VERB
ejpam-5623	329	3	corollary	corollary	NOUN
ejpam-5623	329	4	follows	follow	VERB
ejpam-5623	329	5	directly	directly	ADV
ejpam-5623	329	6	from	from	ADP
ejpam-5623	329	7	theorems	theorem	NOUN
ejpam-5623	329	8	1	1	NUM
ejpam-5623	329	9	and	and	CCONJ
ejpam-5623	329	10	5	5	NUM
ejpam-5623	329	11	.	.	PUNCT
ejpam-5623	329	12	corollary	corollary	ADJ
ejpam-5623	329	13	3	3	X
ejpam-5623	329	14	.	.	PUNCT
ejpam-5623	330	1	let	let	VERB
ejpam-5623	330	2	α	α	PRON
ejpam-5623	330	3	∈	∈	PROPN
ejpam-5623	330	4	q.	q.	NOUN
ejpam-5623	330	5	the	the	DET
ejpam-5623	330	6	following	follow	VERB
ejpam-5623	330	7	statements	statement	NOUN
ejpam-5623	330	8	are	be	AUX
ejpam-5623	330	9	equivalent	equivalent	ADJ
ejpam-5623	330	10	:	:	PUNCT
ejpam-5623	330	11	(	(	PUNCT
ejpam-5623	330	12	i	i	NOUN
ejpam-5623	330	13	)	)	PUNCT
ejpam-5623	330	14	α	α	PROPN
ejpam-5623	330	15	∈	∈	PROPN
ejpam-5623	330	16	creg(q	creg(q	NOUN
ejpam-5623	330	17	)	)	PUNCT
ejpam-5623	330	18	.	.	PUNCT
ejpam-5623	331	1	(	(	PUNCT
ejpam-5623	331	2	ii	ii	NOUN
ejpam-5623	331	3	)	)	PUNCT
ejpam-5623	331	4	α|v	α|v	VERB
ejpam-5623	332	1	α	α	NOUN
ejpam-5623	332	2	:	:	PUNCT
ejpam-5623	332	3	v	v	ADP
ejpam-5623	332	4	α	α	PROPN
ejpam-5623	332	5	→	→	SYM
ejpam-5623	332	6	v	v	NUM
ejpam-5623	332	7	α	α	NOUN
ejpam-5623	332	8	is	be	AUX
ejpam-5623	332	9	one	one	NUM
ejpam-5623	332	10	-	-	PUNCT
ejpam-5623	332	11	to	to	ADP
ejpam-5623	332	12	-	-	PUNCT
ejpam-5623	332	13	one	one	NUM
ejpam-5623	332	14	,	,	PUNCT
ejpam-5623	332	15	and	and	CCONJ
ejpam-5623	332	16	α|wα	α|wα	NUM
ejpam-5623	332	17	:	:	PUNCT
ejpam-5623	332	18	wα	wα	NOUN
ejpam-5623	332	19	→	→	SYM
ejpam-5623	332	20	v	v	NUM
ejpam-5623	332	21	α	α	NOUN
ejpam-5623	332	22	is	be	AUX
ejpam-5623	332	23	onto	onto	ADP
ejpam-5623	332	24	.	.	PUNCT
ejpam-5623	333	1	(	(	PUNCT
ejpam-5623	333	2	iii	iii	NOUN
ejpam-5623	333	3	)	)	PUNCT
ejpam-5623	333	4	for	for	ADP
ejpam-5623	333	5	every	every	DET
ejpam-5623	333	6	w	w	PROPN
ejpam-5623	333	7	∈	∈	PROPN
ejpam-5623	333	8	w	w	NOUN
ejpam-5623	333	9	,	,	PUNCT
ejpam-5623	333	10	there	there	PRON
ejpam-5623	333	11	exists	exist	VERB
ejpam-5623	333	12	a	a	DET
ejpam-5623	333	13	unique	unique	ADJ
ejpam-5623	333	14	w′	w′	PROPN
ejpam-5623	333	15	∈	∈	NOUN
ejpam-5623	333	16	wα	wα	NOUN
ejpam-5623	333	17	such	such	ADJ
ejpam-5623	333	18	that	that	DET
ejpam-5623	333	19	wα	wα	NOUN
ejpam-5623	333	20	=	=	SYM
ejpam-5623	333	21	w′α	w′α	NOUN
ejpam-5623	333	22	.	.	PUNCT
ejpam-5623	334	1	lastly	lastly	ADV
ejpam-5623	334	2	,	,	PUNCT
ejpam-5623	334	3	we	we	PRON
ejpam-5623	334	4	characterize	characterize	VERB
ejpam-5623	334	5	the	the	DET
ejpam-5623	334	6	left	left	ADJ
ejpam-5623	334	7	regular	regular	ADJ
ejpam-5623	334	8	semigroup	semigroup	ADJ
ejpam-5623	334	9	structure	structure	NOUN
ejpam-5623	334	10	of	of	ADP
ejpam-5623	334	11	the	the	DET
ejpam-5623	334	12	semigroup	semigroup	PROPN
ejpam-5623	334	13	q.	q.	PROPN
ejpam-5623	334	14	theorem	theorem	VERB
ejpam-5623	334	15	6	6	NUM
ejpam-5623	334	16	.	.	PUNCT
ejpam-5623	335	1	the	the	DET
ejpam-5623	335	2	following	follow	VERB
ejpam-5623	335	3	statements	statement	NOUN
ejpam-5623	335	4	are	be	AUX
ejpam-5623	335	5	equivalent	equivalent	ADJ
ejpam-5623	335	6	:	:	PUNCT
ejpam-5623	335	7	(	(	PUNCT
ejpam-5623	335	8	i	i	NOUN
ejpam-5623	335	9	)	)	PUNCT
ejpam-5623	335	10	q	q	X
ejpam-5623	335	11	is	be	AUX
ejpam-5623	335	12	a	a	DET
ejpam-5623	335	13	left	left	ADJ
ejpam-5623	335	14	regular	regular	ADJ
ejpam-5623	335	15	semigroup	semigroup	NOUN
ejpam-5623	335	16	.	.	PUNCT
ejpam-5623	336	1	(	(	PUNCT
ejpam-5623	336	2	ii	ii	NOUN
ejpam-5623	336	3	)	)	PUNCT
ejpam-5623	336	4	creg(q	creg(q	NOUN
ejpam-5623	336	5	)	)	PUNCT
ejpam-5623	336	6	is	be	AUX
ejpam-5623	336	7	a	a	DET
ejpam-5623	336	8	subsemigroup	subsemigroup	NOUN
ejpam-5623	336	9	of	of	ADP
ejpam-5623	336	10	s∗.	s∗.	PROPN
ejpam-5623	336	11	(	(	PUNCT
ejpam-5623	336	12	iii	iii	NOUN
ejpam-5623	336	13	)	)	PUNCT
ejpam-5623	336	14	dim(w	dim(w	NOUN
ejpam-5623	336	15	)	)	PUNCT
ejpam-5623	336	16	≤	≤	NUM
ejpam-5623	336	17	1	1	NUM
ejpam-5623	336	18	.	.	PUNCT
ejpam-5623	337	1	proof	proof	NOUN
ejpam-5623	337	2	.	.	PUNCT
ejpam-5623	338	1	the	the	DET
ejpam-5623	338	2	proof	proof	NOUN
ejpam-5623	338	3	can	can	AUX
ejpam-5623	338	4	be	be	AUX
ejpam-5623	338	5	established	establish	VERB
ejpam-5623	338	6	in	in	ADP
ejpam-5623	338	7	the	the	DET
ejpam-5623	338	8	same	same	ADJ
ejpam-5623	338	9	way	way	NOUN
ejpam-5623	338	10	as	as	SCONJ
ejpam-5623	338	11	theorem	theorem	ADJ
ejpam-5623	338	12	4	4	NUM
ejpam-5623	338	13	.	.	PUNCT
ejpam-5623	339	1	the	the	DET
ejpam-5623	339	2	following	follow	VERB
ejpam-5623	339	3	corollary	corollary	NOUN
ejpam-5623	339	4	is	be	AUX
ejpam-5623	339	5	an	an	DET
ejpam-5623	339	6	immediate	immediate	ADJ
ejpam-5623	339	7	consequence	consequence	NOUN
ejpam-5623	339	8	of	of	ADP
ejpam-5623	339	9	theorems	theorem	NOUN
ejpam-5623	339	10	2	2	NUM
ejpam-5623	339	11	and	and	CCONJ
ejpam-5623	339	12	6	6	NUM
ejpam-5623	339	13	.	.	PUNCT
ejpam-5623	339	14	corollary	corollary	ADJ
ejpam-5623	339	15	4	4	NUM
ejpam-5623	339	16	.	.	PUNCT
ejpam-5623	340	1	the	the	DET
ejpam-5623	340	2	following	follow	VERB
ejpam-5623	340	3	statements	statement	NOUN
ejpam-5623	340	4	are	be	AUX
ejpam-5623	340	5	equivalent	equivalent	ADJ
ejpam-5623	340	6	:	:	PUNCT
ejpam-5623	340	7	(	(	PUNCT
ejpam-5623	340	8	i	i	NOUN
ejpam-5623	340	9	)	)	PUNCT
ejpam-5623	340	10	q	q	X
ejpam-5623	340	11	is	be	AUX
ejpam-5623	340	12	a	a	DET
ejpam-5623	340	13	left	left	ADJ
ejpam-5623	340	14	regular	regular	ADJ
ejpam-5623	340	15	semigroup	semigroup	NOUN
ejpam-5623	340	16	.	.	PUNCT
ejpam-5623	341	1	(	(	PUNCT
ejpam-5623	341	2	ii	ii	NOUN
ejpam-5623	341	3	)	)	PUNCT
ejpam-5623	341	4	creg(q	creg(q	NOUN
ejpam-5623	341	5	)	)	PUNCT
ejpam-5623	341	6	is	be	AUX
ejpam-5623	341	7	a	a	DET
ejpam-5623	341	8	subsemigroup	subsemigroup	NOUN
ejpam-5623	341	9	of	of	ADP
ejpam-5623	341	10	s∗.	s∗.	PROPN
ejpam-5623	341	11	(	(	PUNCT
ejpam-5623	341	12	iii	iii	NOUN
ejpam-5623	341	13	)	)	PUNCT
ejpam-5623	341	14	dim(w	dim(w	NOUN
ejpam-5623	341	15	)	)	PUNCT
ejpam-5623	341	16	≤	≤	NUM
ejpam-5623	341	17	1	1	NUM
ejpam-5623	341	18	.	.	PUNCT
ejpam-5623	341	19	n.	n.	PROPN
ejpam-5623	341	20	sawatraksa	sawatraksa	PROPN
ejpam-5623	341	21	,	,	PUNCT
ejpam-5623	341	22	p.	p.	NOUN
ejpam-5623	341	23	tantong	tantong	NOUN
ejpam-5623	341	24	/	/	SYM
ejpam-5623	341	25	eur	eur	PROPN
ejpam-5623	341	26	.	.	PUNCT
ejpam-5623	342	1	j.	j.	PROPN
ejpam-5623	342	2	pure	pure	PROPN
ejpam-5623	342	3	appl	appl	PROPN
ejpam-5623	342	4	.	.	PROPN
ejpam-5623	342	5	math	math	PROPN
ejpam-5623	342	6	,	,	PUNCT
ejpam-5623	342	7	18	18	NUM
ejpam-5623	342	8	(	(	PUNCT
ejpam-5623	342	9	1	1	NUM
ejpam-5623	342	10	)	)	PUNCT
ejpam-5623	342	11	(	(	PUNCT
ejpam-5623	342	12	2025	2025	NUM
ejpam-5623	342	13	)	)	PUNCT
ejpam-5623	342	14	,	,	PUNCT
ejpam-5623	342	15	5623	5623	NUM
ejpam-5623	342	16	9	9	NUM
ejpam-5623	342	17	of	of	ADP
ejpam-5623	342	18	15	15	NUM
ejpam-5623	342	19	3	3	NUM
ejpam-5623	342	20	.	.	PUNCT
ejpam-5623	343	1	semigroups	semigroup	NOUN
ejpam-5623	343	2	of	of	ADP
ejpam-5623	343	3	linear	linear	ADJ
ejpam-5623	343	4	transformations	transformation	NOUN
ejpam-5623	343	5	with	with	ADP
ejpam-5623	343	6	fixed	fix	VERB
ejpam-5623	343	7	subspaces	subspace	NOUN
ejpam-5623	343	8	in	in	ADP
ejpam-5623	343	9	this	this	DET
ejpam-5623	343	10	section	section	NOUN
ejpam-5623	343	11	,	,	PUNCT
ejpam-5623	343	12	we	we	PRON
ejpam-5623	343	13	now	now	ADV
ejpam-5623	343	14	explore	explore	VERB
ejpam-5623	343	15	the	the	DET
ejpam-5623	343	16	necessary	necessary	ADJ
ejpam-5623	343	17	and	and	CCONJ
ejpam-5623	343	18	sufficient	sufficient	ADJ
ejpam-5623	343	19	conditions	condition	NOUN
ejpam-5623	343	20	for	for	ADP
ejpam-5623	343	21	an	an	DET
ejpam-5623	343	22	element	element	NOUN
ejpam-5623	343	23	of	of	ADP
ejpam-5623	343	24	fix(v	fix(v	PROPN
ejpam-5623	343	25	,	,	PUNCT
ejpam-5623	343	26	w	w	NOUN
ejpam-5623	343	27	)	)	PUNCT
ejpam-5623	343	28	to	to	PART
ejpam-5623	343	29	be	be	AUX
ejpam-5623	343	30	right	right	ADV
ejpam-5623	343	31	regular	regular	ADJ
ejpam-5623	343	32	.	.	PUNCT
ejpam-5623	344	1	theorem	theorem	VERB
ejpam-5623	344	2	7	7	NUM
ejpam-5623	344	3	.	.	PUNCT
ejpam-5623	345	1	let	let	VERB
ejpam-5623	345	2	α	α	PRON
ejpam-5623	345	3	∈	∈	PROPN
ejpam-5623	345	4	fix(v	fix(v	PROPN
ejpam-5623	345	5	,	,	PUNCT
ejpam-5623	345	6	w	w	PROPN
ejpam-5623	345	7	)	)	PUNCT
ejpam-5623	345	8	.	.	PUNCT
ejpam-5623	346	1	then	then	ADV
ejpam-5623	346	2	,	,	PUNCT
ejpam-5623	346	3	α	α	PROPN
ejpam-5623	346	4	∈	∈	PROPN
ejpam-5623	346	5	rreg(fix(v	rreg(fix(v	NOUN
ejpam-5623	346	6	,	,	PUNCT
ejpam-5623	346	7	w	w	NOUN
ejpam-5623	346	8	)	)	PUNCT
ejpam-5623	346	9	)	)	PUNCT
ejpam-5623	347	1	if	if	SCONJ
ejpam-5623	347	2	and	and	CCONJ
ejpam-5623	347	3	only	only	ADV
ejpam-5623	347	4	if	if	SCONJ
ejpam-5623	347	5	α|v	α|v	ADV
ejpam-5623	347	6	α	α	PROPN
ejpam-5623	347	7	is	be	AUX
ejpam-5623	347	8	a	a	DET
ejpam-5623	347	9	one	one	NUM
ejpam-5623	347	10	-	-	PUNCT
ejpam-5623	347	11	to	to	ADP
ejpam-5623	347	12	-	-	PUNCT
ejpam-5623	347	13	one	one	NUM
ejpam-5623	347	14	transformation	transformation	NOUN
ejpam-5623	347	15	on	on	ADP
ejpam-5623	347	16	v	v	NUM
ejpam-5623	347	17	α	α	NOUN
ejpam-5623	347	18	.	.	PUNCT
ejpam-5623	348	1	proof	proof	NOUN
ejpam-5623	348	2	.	.	PUNCT
ejpam-5623	349	1	the	the	DET
ejpam-5623	349	2	necessity	necessity	NOUN
ejpam-5623	349	3	follows	follow	VERB
ejpam-5623	349	4	directly	directly	ADV
ejpam-5623	349	5	from	from	ADP
ejpam-5623	349	6	theorem	theorem	ADJ
ejpam-5623	349	7	1	1	NUM
ejpam-5623	349	8	.	.	PUNCT
ejpam-5623	349	9	to	to	PART
ejpam-5623	349	10	prove	prove	VERB
ejpam-5623	349	11	the	the	DET
ejpam-5623	349	12	sufficiency	sufficiency	NOUN
ejpam-5623	349	13	,	,	PUNCT
ejpam-5623	349	14	we	we	PRON
ejpam-5623	349	15	suppose	suppose	VERB
ejpam-5623	349	16	that	that	SCONJ
ejpam-5623	349	17	α|v	α|v	PROPN
ejpam-5623	349	18	α	α	PROPN
ejpam-5623	349	19	is	be	AUX
ejpam-5623	349	20	one	one	NUM
ejpam-5623	349	21	-	-	PUNCT
ejpam-5623	349	22	to	to	ADP
ejpam-5623	349	23	-	-	PUNCT
ejpam-5623	349	24	one	one	NUM
ejpam-5623	349	25	.	.	PUNCT
ejpam-5623	350	1	since	since	SCONJ
ejpam-5623	350	2	α	α	PROPN
ejpam-5623	350	3	∈	∈	PROPN
ejpam-5623	350	4	fix(v	fix(v	PROPN
ejpam-5623	350	5	,	,	PUNCT
ejpam-5623	350	6	w	w	PROPN
ejpam-5623	350	7	)	)	PUNCT
ejpam-5623	350	8	,	,	PUNCT
ejpam-5623	350	9	we	we	PRON
ejpam-5623	350	10	get	get	VERB
ejpam-5623	350	11	that	that	PRON
ejpam-5623	350	12	w	w	ADP
ejpam-5623	350	13	⊆	⊆	NUM
ejpam-5623	350	14	v	v	ADP
ejpam-5623	350	15	α	α	NOUN
ejpam-5623	350	16	.	.	PUNCT
ejpam-5623	351	1	let	let	VERB
ejpam-5623	351	2	bw	bw	PART
ejpam-5623	351	3	be	be	AUX
ejpam-5623	351	4	a	a	DET
ejpam-5623	351	5	basis	basis	NOUN
ejpam-5623	351	6	for	for	ADP
ejpam-5623	351	7	w	w	PROPN
ejpam-5623	351	8	and	and	CCONJ
ejpam-5623	351	9	b	b	NOUN
ejpam-5623	351	10	be	be	AUX
ejpam-5623	351	11	a	a	DET
ejpam-5623	351	12	basis	basis	NOUN
ejpam-5623	351	13	for	for	ADP
ejpam-5623	351	14	v	v	NOUN
ejpam-5623	351	15	α	α	NOUN
ejpam-5623	351	16	such	such	ADJ
ejpam-5623	351	17	that	that	DET
ejpam-5623	351	18	bw	bw	PROPN
ejpam-5623	351	19	⊆	⊆	NUM
ejpam-5623	351	20	b.	b.	NOUN
ejpam-5623	351	21	define	define	VERB
ejpam-5623	351	22	b′	b′	NUM
ejpam-5623	351	23	=	=	PUNCT
ejpam-5623	351	24	{	{	PUNCT
ejpam-5623	351	25	uα	uα	PROPN
ejpam-5623	351	26	|u	|u	PROPN
ejpam-5623	351	27	∈	∈	PROPN
ejpam-5623	351	28	b	b	NOUN
ejpam-5623	351	29	}	}	PUNCT
ejpam-5623	351	30	.	.	PUNCT
ejpam-5623	352	1	then	then	ADV
ejpam-5623	352	2	,	,	PUNCT
ejpam-5623	352	3	bw	bw	PROPN
ejpam-5623	352	4	⊆	⊆	NUM
ejpam-5623	352	5	b′.	b′.	NOUN
ejpam-5623	352	6	by	by	ADP
ejpam-5623	352	7	the	the	DET
ejpam-5623	352	8	same	same	ADJ
ejpam-5623	352	9	reasoning	reasoning	NOUN
ejpam-5623	352	10	used	use	VERB
ejpam-5623	352	11	in	in	ADP
ejpam-5623	352	12	theorem	theorem	NOUN
ejpam-5623	352	13	1	1	NUM
ejpam-5623	352	14	,	,	PUNCT
ejpam-5623	352	15	b′	b′	NUM
ejpam-5623	352	16	is	be	AUX
ejpam-5623	352	17	linearly	linearly	ADV
ejpam-5623	352	18	independent	independent	ADJ
ejpam-5623	352	19	,	,	PUNCT
ejpam-5623	352	20	and	and	CCONJ
ejpam-5623	352	21	there	there	PRON
ejpam-5623	352	22	exists	exist	VERB
ejpam-5623	352	23	a	a	DET
ejpam-5623	352	24	basis	basis	NOUN
ejpam-5623	352	25	b′′	b′′	VERB
ejpam-5623	352	26	for	for	ADP
ejpam-5623	352	27	v	v	ADP
ejpam-5623	352	28	such	such	ADJ
ejpam-5623	352	29	that	that	DET
ejpam-5623	352	30	b′	b′	NUM
ejpam-5623	352	31	⊆	⊆	NUM
ejpam-5623	352	32	b′′.	b′′.	NOUN
ejpam-5623	352	33	for	for	ADP
ejpam-5623	352	34	each	each	DET
ejpam-5623	352	35	u	u	PROPN
ejpam-5623	352	36	∈	∈	PROPN
ejpam-5623	352	37	b′	b′	NOUN
ejpam-5623	352	38	,	,	PUNCT
ejpam-5623	352	39	there	there	PRON
ejpam-5623	352	40	exists	exist	VERB
ejpam-5623	352	41	a	a	DET
ejpam-5623	352	42	unique	unique	ADJ
ejpam-5623	352	43	u′	u′	PROPN
ejpam-5623	352	44	∈	∈	PROPN
ejpam-5623	352	45	b	b	NOUN
ejpam-5623	352	46	such	such	ADJ
ejpam-5623	352	47	that	that	DET
ejpam-5623	352	48	u′α	u′α	PROPN
ejpam-5623	352	49	=	=	SYM
ejpam-5623	352	50	u	u	NOUN
ejpam-5623	352	51	by	by	ADP
ejpam-5623	352	52	assumption	assumption	NOUN
ejpam-5623	352	53	.	.	PUNCT
ejpam-5623	353	1	define	define	VERB
ejpam-5623	353	2	β	β	NOUN
ejpam-5623	353	3	:	:	PUNCT
ejpam-5623	353	4	b′′	b′′	PROPN
ejpam-5623	353	5	→	→	SYM
ejpam-5623	353	6	v	v	NOUN
ejpam-5623	353	7	by	by	ADP
ejpam-5623	353	8	vβ	vβ	X
ejpam-5623	353	9	=	=	PUNCT
ejpam-5623	353	10	{	{	PUNCT
ejpam-5623	353	11	v′	v′	NOUN
ejpam-5623	353	12	if	if	SCONJ
ejpam-5623	353	13	v	v	NUM
ejpam-5623	353	14	∈	∈	PROPN
ejpam-5623	353	15	b′	b′	NOUN
ejpam-5623	353	16	,	,	PUNCT
ejpam-5623	353	17	0	0	NUM
ejpam-5623	353	18	otherwise	otherwise	ADV
ejpam-5623	353	19	.	.	PUNCT
ejpam-5623	354	1	by	by	ADP
ejpam-5623	354	2	the	the	DET
ejpam-5623	354	3	uniqueness	uniqueness	NOUN
ejpam-5623	354	4	,	,	PUNCT
ejpam-5623	354	5	β	β	X
ejpam-5623	354	6	is	be	AUX
ejpam-5623	354	7	well	well	ADV
ejpam-5623	354	8	-	-	PUNCT
ejpam-5623	354	9	defined	define	VERB
ejpam-5623	354	10	.	.	PUNCT
ejpam-5623	355	1	hence	hence	ADV
ejpam-5623	355	2	,	,	PUNCT
ejpam-5623	355	3	β	β	PROPN
ejpam-5623	355	4	can	can	AUX
ejpam-5623	355	5	be	be	AUX
ejpam-5623	355	6	extended	extend	VERB
ejpam-5623	355	7	to	to	ADP
ejpam-5623	355	8	a	a	DET
ejpam-5623	355	9	linear	linear	ADJ
ejpam-5623	355	10	transformation	transformation	NOUN
ejpam-5623	355	11	on	on	ADP
ejpam-5623	355	12	v	v	NOUN
ejpam-5623	355	13	.	.	PUNCT
ejpam-5623	356	1	let	let	VERB
ejpam-5623	356	2	w	w	PROPN
ejpam-5623	356	3	∈	∈	PROPN
ejpam-5623	356	4	w	w	PROPN
ejpam-5623	356	5	.	.	PUNCT
ejpam-5623	357	1	then	then	ADV
ejpam-5623	357	2	,	,	PUNCT
ejpam-5623	357	3	there	there	PRON
ejpam-5623	357	4	are	be	VERB
ejpam-5623	357	5	w1	w1	NOUN
ejpam-5623	357	6	,	,	PUNCT
ejpam-5623	357	7	w2	w2	NOUN
ejpam-5623	357	8	,	,	PUNCT
ejpam-5623	357	9	.	.	PUNCT
ejpam-5623	357	10	.	.	PUNCT
ejpam-5623	358	1	.	.	PUNCT
ejpam-5623	359	1	,	,	PUNCT
ejpam-5623	359	2	wk	wk	X
ejpam-5623	359	3	∈	∈	PROPN
ejpam-5623	359	4	bw	bw	NOUN
ejpam-5623	359	5	,	,	PUNCT
ejpam-5623	359	6	and	and	CCONJ
ejpam-5623	359	7	a1	a1	NOUN
ejpam-5623	359	8	,	,	PUNCT
ejpam-5623	359	9	a2	a2	PROPN
ejpam-5623	359	10	,	,	PUNCT
ejpam-5623	359	11	.	.	PUNCT
ejpam-5623	359	12	.	.	PUNCT
ejpam-5623	359	13	.	.	PUNCT
ejpam-5623	360	1	,	,	PUNCT
ejpam-5623	360	2	ak	ak	PROPN
ejpam-5623	360	3	∈	∈	PROPN
ejpam-5623	360	4	f	f	PROPN
ejpam-5623	360	5	such	such	ADJ
ejpam-5623	360	6	that	that	PRON
ejpam-5623	360	7	w	w	NOUN
ejpam-5623	360	8	=	=	PUNCT
ejpam-5623	360	9	a1w1+a2w2	a1w1+a2w2	NOUN
ejpam-5623	360	10	+	+	NUM
ejpam-5623	360	11	.	.	PUNCT
ejpam-5623	360	12	.	.	PUNCT
ejpam-5623	361	1	.+akwk	.+akwk	PROPN
ejpam-5623	361	2	.	.	PUNCT
ejpam-5623	362	1	since	since	SCONJ
ejpam-5623	362	2	bw	bw	PROPN
ejpam-5623	362	3	⊆	⊆	NUM
ejpam-5623	362	4	b	b	NOUN
ejpam-5623	362	5	,	,	PUNCT
ejpam-5623	362	6	and	and	CCONJ
ejpam-5623	362	7	by	by	ADP
ejpam-5623	362	8	the	the	DET
ejpam-5623	362	9	definition	definition	NOUN
ejpam-5623	362	10	of	of	ADP
ejpam-5623	362	11	α	α	NOUN
ejpam-5623	362	12	,	,	PUNCT
ejpam-5623	362	13	we	we	PRON
ejpam-5623	362	14	observe	observe	VERB
ejpam-5623	362	15	that	that	SCONJ
ejpam-5623	362	16	w′	w′	PROPN
ejpam-5623	362	17	iα	iα	PROPN
ejpam-5623	362	18	=	=	PROPN
ejpam-5623	362	19	wi	wi	PROPN
ejpam-5623	362	20	=	=	PUNCT
ejpam-5623	362	21	wiα	wiα	PROPN
ejpam-5623	362	22	.	.	PUNCT
ejpam-5623	363	1	it	it	PRON
ejpam-5623	363	2	follows	follow	VERB
ejpam-5623	363	3	from	from	ADP
ejpam-5623	363	4	assumption	assumption	NOUN
ejpam-5623	363	5	that	that	SCONJ
ejpam-5623	363	6	wi	wi	PROPN
ejpam-5623	363	7	=	=	SYM
ejpam-5623	363	8	w′	w′	PROPN
ejpam-5623	363	9	i.	i.	PROPN
ejpam-5623	363	10	this	this	PRON
ejpam-5623	363	11	implies	imply	VERB
ejpam-5623	363	12	that	that	SCONJ
ejpam-5623	363	13	wβ	wβ	ADP
ejpam-5623	363	14	=	=	SYM
ejpam-5623	363	15	(	(	PUNCT
ejpam-5623	363	16	a1w1	a1w1	X
ejpam-5623	363	17	+	+	X
ejpam-5623	363	18	a2w2	a2w2	X
ejpam-5623	363	19	+	+	X
ejpam-5623	363	20	.	.	PUNCT
ejpam-5623	363	21	.	.	PUNCT
ejpam-5623	364	1	.+	.+	NOUN
ejpam-5623	364	2	akwk)β	akwk)β	PUNCT
ejpam-5623	365	1	=	=	PUNCT
ejpam-5623	365	2	a1(w1β	a1(w1β	PROPN
ejpam-5623	365	3	)	)	PUNCT
ejpam-5623	365	4	+	+	NUM
ejpam-5623	365	5	a2(w2β	a2(w2β	ADV
ejpam-5623	365	6	)	)	PUNCT
ejpam-5623	366	1	+	+	CCONJ
ejpam-5623	366	2	.	.	PUNCT
ejpam-5623	366	3	.	.	PUNCT
ejpam-5623	367	1	.+	.+	NOUN
ejpam-5623	367	2	ak(wkβ	ak(wkβ	PROPN
ejpam-5623	367	3	)	)	PUNCT
ejpam-5623	367	4	=	=	PUNCT
ejpam-5623	368	1	a1w1	a1w1	PROPN
ejpam-5623	368	2	+	+	NOUN
ejpam-5623	368	3	a2w2	a2w2	X
ejpam-5623	368	4	+	+	X
ejpam-5623	368	5	.	.	PUNCT
ejpam-5623	368	6	.	.	PUNCT
ejpam-5623	369	1	.+	.+	NOUN
ejpam-5623	369	2	akwk	akwk	NOUN
ejpam-5623	370	1	=	=	PUNCT
ejpam-5623	370	2	w.	w.	PROPN
ejpam-5623	370	3	it	it	PRON
ejpam-5623	370	4	implies	imply	VERB
ejpam-5623	370	5	that	that	SCONJ
ejpam-5623	370	6	β	β	NOUN
ejpam-5623	370	7	belong	belong	VERB
ejpam-5623	370	8	to	to	ADP
ejpam-5623	370	9	fix(v	fix(v	PROPN
ejpam-5623	370	10	,	,	PUNCT
ejpam-5623	370	11	w	w	PROPN
ejpam-5623	370	12	)	)	PUNCT
ejpam-5623	370	13	.	.	PUNCT
ejpam-5623	371	1	we	we	PRON
ejpam-5623	371	2	now	now	ADV
ejpam-5623	371	3	demonstrate	demonstrate	VERB
ejpam-5623	371	4	that	that	SCONJ
ejpam-5623	371	5	α	α	NOUN
ejpam-5623	371	6	=	=	SYM
ejpam-5623	371	7	α2β	α2β	NOUN
ejpam-5623	371	8	.	.	PUNCT
ejpam-5623	372	1	if	if	SCONJ
ejpam-5623	372	2	v	v	NUM
ejpam-5623	372	3	∈	∈	PROPN
ejpam-5623	372	4	v	v	NOUN
ejpam-5623	372	5	,	,	PUNCT
ejpam-5623	372	6	then	then	ADV
ejpam-5623	372	7	we	we	PRON
ejpam-5623	372	8	have	have	VERB
ejpam-5623	372	9	vα	vα	INTJ
ejpam-5623	372	10	∈	∈	PROPN
ejpam-5623	372	11	v	v	ADP
ejpam-5623	372	12	α	α	NOUN
ejpam-5623	372	13	,	,	PUNCT
ejpam-5623	372	14	and	and	CCONJ
ejpam-5623	372	15	we	we	PRON
ejpam-5623	372	16	can	can	AUX
ejpam-5623	372	17	write	write	VERB
ejpam-5623	372	18	vα	vα	X
ejpam-5623	372	19	=	=	PUNCT
ejpam-5623	372	20	a1u	a1u	PROPN
ejpam-5623	373	1	′	′	NUM
ejpam-5623	374	1	1+a2u	1+a2u	NUM
ejpam-5623	374	2	′	′	NUM
ejpam-5623	374	3	2	2	NUM
ejpam-5623	374	4	+	+	NOUN
ejpam-5623	374	5	.	.	PUNCT
ejpam-5623	374	6	.	.	PUNCT
ejpam-5623	374	7	.+anu	.+anu	PROPN
ejpam-5623	375	1	′	′	NUM
ejpam-5623	376	1	n	n	CCONJ
ejpam-5623	376	2	where	where	SCONJ
ejpam-5623	376	3	u′1	u′1	NOUN
ejpam-5623	376	4	,	,	PUNCT
ejpam-5623	376	5	u	u	NOUN
ejpam-5623	376	6	′	′	NOUN
ejpam-5623	376	7	2	2	NUM
ejpam-5623	376	8	,	,	PUNCT
ejpam-5623	376	9	.	.	PUNCT
ejpam-5623	376	10	.	.	PUNCT
ejpam-5623	377	1	.	.	PUNCT
ejpam-5623	378	1	,	,	PUNCT
ejpam-5623	378	2	u	u	NOUN
ejpam-5623	378	3	′	′	NOUN
ejpam-5623	378	4	n	n	CCONJ
ejpam-5623	378	5	∈	∈	PROPN
ejpam-5623	378	6	b	b	PROPN
ejpam-5623	378	7	with	with	ADP
ejpam-5623	378	8	u′iα	u′iα	NOUN
ejpam-5623	378	9	=	=	SYM
ejpam-5623	378	10	ui	ui	PROPN
ejpam-5623	378	11	for	for	ADP
ejpam-5623	378	12	all	all	PRON
ejpam-5623	378	13	i	i	PRON
ejpam-5623	378	14	∈	∈	PROPN
ejpam-5623	378	15	{	{	PUNCT
ejpam-5623	378	16	1	1	NUM
ejpam-5623	378	17	,	,	PUNCT
ejpam-5623	378	18	2	2	NUM
ejpam-5623	378	19	,	,	PUNCT
ejpam-5623	378	20	.	.	PUNCT
ejpam-5623	378	21	.	.	PUNCT
ejpam-5623	379	1	.	.	PUNCT
ejpam-5623	379	2	,	,	PUNCT
ejpam-5623	380	1	n	n	CCONJ
ejpam-5623	380	2	}	}	PUNCT
ejpam-5623	380	3	and	and	CCONJ
ejpam-5623	380	4	a1	a1	NOUN
ejpam-5623	380	5	,	,	PUNCT
ejpam-5623	380	6	a2	a2	PROPN
ejpam-5623	380	7	,	,	PUNCT
ejpam-5623	380	8	.	.	PUNCT
ejpam-5623	380	9	.	.	PUNCT
ejpam-5623	381	1	.	.	PUNCT
ejpam-5623	382	1	,	,	PUNCT
ejpam-5623	382	2	an	an	DET
ejpam-5623	382	3	∈	∈	PROPN
ejpam-5623	382	4	f.	f.	PROPN
ejpam-5623	382	5	therefore	therefore	ADV
ejpam-5623	382	6	,	,	PUNCT
ejpam-5623	382	7	vα2β	vα2β	PROPN
ejpam-5623	382	8	=	=	PUNCT
ejpam-5623	382	9	(	(	PUNCT
ejpam-5623	382	10	a1u	a1u	PROPN
ejpam-5623	382	11	′	′	NOUN
ejpam-5623	382	12	1	1	NUM
ejpam-5623	383	1	+	+	CCONJ
ejpam-5623	383	2	a2u	a2u	ADP
ejpam-5623	383	3	′	′	NUM
ejpam-5623	383	4	2	2	NUM
ejpam-5623	383	5	+	+	CCONJ
ejpam-5623	383	6	.	.	PUNCT
ejpam-5623	383	7	.	.	PUNCT
ejpam-5623	384	1	.+	.+	NOUN
ejpam-5623	384	2	anu	anu	INTJ
ejpam-5623	385	1	′	′	NUM
ejpam-5623	386	1	n)αβ	n)αβ	PROPN
ejpam-5623	387	1	=	=	PRON
ejpam-5623	388	1	(	(	PUNCT
ejpam-5623	388	2	a1u	a1u	PROPN
ejpam-5623	388	3	′	′	NUM
ejpam-5623	388	4	1α+	1α+	NUM
ejpam-5623	388	5	a2u	a2u	ADP
ejpam-5623	389	1	′	′	NUM
ejpam-5623	389	2	2α+	2α+	NUM
ejpam-5623	389	3	.	.	PUNCT
ejpam-5623	389	4	.	.	PUNCT
ejpam-5623	390	1	.+	.+	NOUN
ejpam-5623	390	2	anu	anu	VERB
ejpam-5623	390	3	′	′	NUM
ejpam-5623	390	4	nα)β	nα)β	PROPN
ejpam-5623	390	5	=	=	PUNCT
ejpam-5623	390	6	(	(	PUNCT
ejpam-5623	390	7	a1u1	a1u1	PUNCT
ejpam-5623	390	8	+	+	X
ejpam-5623	390	9	a2u2	a2u2	PROPN
ejpam-5623	390	10	+	+	X
ejpam-5623	390	11	.	.	PUNCT
ejpam-5623	390	12	.	.	PUNCT
ejpam-5623	391	1	.+	.+	NOUN
ejpam-5623	391	2	anun)β	anun)β	PROPN
ejpam-5623	392	1	=	=	SYM
ejpam-5623	392	2	a1u	a1u	PROPN
ejpam-5623	393	1	′	′	NOUN
ejpam-5623	393	2	1	1	NUM
ejpam-5623	394	1	+	+	CCONJ
ejpam-5623	394	2	a2u	a2u	ADP
ejpam-5623	394	3	′	′	NUM
ejpam-5623	394	4	2	2	NUM
ejpam-5623	394	5	+	+	CCONJ
ejpam-5623	394	6	.	.	PUNCT
ejpam-5623	394	7	.	.	PUNCT
ejpam-5623	395	1	.+	.+	NOUN
ejpam-5623	395	2	anu	anu	VERB
ejpam-5623	395	3	′	′	NUM
ejpam-5623	396	1	n	n	PROPN
ejpam-5623	396	2	=	=	SYM
ejpam-5623	396	3	vα	vα	PROPN
ejpam-5623	396	4	.	.	PUNCT
ejpam-5623	397	1	hence	hence	ADV
ejpam-5623	397	2	,	,	PUNCT
ejpam-5623	397	3	α	α	PROPN
ejpam-5623	397	4	is	be	AUX
ejpam-5623	397	5	right	right	ADV
ejpam-5623	397	6	regular	regular	ADV
ejpam-5623	397	7	.	.	PUNCT
ejpam-5623	398	1	this	this	PRON
ejpam-5623	398	2	completes	complete	VERB
ejpam-5623	398	3	the	the	DET
ejpam-5623	398	4	proof	proof	NOUN
ejpam-5623	398	5	of	of	ADP
ejpam-5623	398	6	theorem	theorem	PROPN
ejpam-5623	398	7	.	.	PUNCT
ejpam-5623	399	1	next	next	ADV
ejpam-5623	399	2	,	,	PUNCT
ejpam-5623	399	3	we	we	PRON
ejpam-5623	399	4	give	give	VERB
ejpam-5623	399	5	a	a	DET
ejpam-5623	399	6	necessary	necessary	ADJ
ejpam-5623	399	7	and	and	CCONJ
ejpam-5623	399	8	sufficient	sufficient	ADJ
ejpam-5623	399	9	condition	condition	NOUN
ejpam-5623	399	10	when	when	SCONJ
ejpam-5623	399	11	the	the	DET
ejpam-5623	399	12	semigroup	semigroup	PROPN
ejpam-5623	399	13	fix(v	fix(v	PROPN
ejpam-5623	399	14	,	,	PUNCT
ejpam-5623	399	15	w	w	NOUN
ejpam-5623	399	16	)	)	PUNCT
ejpam-5623	399	17	to	to	PART
ejpam-5623	399	18	be	be	AUX
ejpam-5623	399	19	left	leave	VERB
ejpam-5623	399	20	regular	regular	ADV
ejpam-5623	399	21	.	.	PUNCT
ejpam-5623	400	1	tantong	tantong	PROPN
ejpam-5623	401	1	[	[	X
ejpam-5623	401	2	15	15	NUM
ejpam-5623	401	3	]	]	PUNCT
ejpam-5623	401	4	,	,	PUNCT
ejpam-5623	401	5	prove	prove	VERB
ejpam-5623	401	6	that	that	SCONJ
ejpam-5623	401	7	l(v	l(v	NOUN
ejpam-5623	401	8	)	)	PUNCT
ejpam-5623	401	9	is	be	AUX
ejpam-5623	401	10	a	a	DET
ejpam-5623	401	11	right	right	ADJ
ejpam-5623	401	12	regular	regular	ADJ
ejpam-5623	401	13	semigroup	semigroup	NOUN
ejpam-5623	402	1	if	if	SCONJ
ejpam-5623	402	2	and	and	CCONJ
ejpam-5623	402	3	only	only	ADV
ejpam-5623	402	4	if	if	SCONJ
ejpam-5623	402	5	dim(v	dim(v	PROPN
ejpam-5623	402	6	)	)	PUNCT
ejpam-5623	402	7	≤	≤	NUM
ejpam-5623	402	8	1	1	NUM
ejpam-5623	402	9	.	.	PUNCT
ejpam-5623	403	1	we	we	PRON
ejpam-5623	403	2	will	will	AUX
ejpam-5623	403	3	use	use	VERB
ejpam-5623	403	4	this	this	DET
ejpam-5623	403	5	result	result	NOUN
ejpam-5623	403	6	in	in	ADP
ejpam-5623	403	7	the	the	DET
ejpam-5623	403	8	proof	proof	NOUN
ejpam-5623	403	9	of	of	ADP
ejpam-5623	403	10	the	the	DET
ejpam-5623	403	11	next	next	ADJ
ejpam-5623	403	12	theorem	theorem	PROPN
ejpam-5623	403	13	.	.	PUNCT
ejpam-5623	403	14	theorem	theorem	VERB
ejpam-5623	403	15	8	8	NUM
ejpam-5623	403	16	.	.	PUNCT
ejpam-5623	404	1	the	the	DET
ejpam-5623	404	2	following	follow	VERB
ejpam-5623	404	3	statements	statement	NOUN
ejpam-5623	404	4	are	be	AUX
ejpam-5623	404	5	equivalent	equivalent	ADJ
ejpam-5623	404	6	:	:	PUNCT
ejpam-5623	404	7	n.	n.	NOUN
ejpam-5623	404	8	sawatraksa	sawatraksa	NOUN
ejpam-5623	404	9	,	,	PUNCT
ejpam-5623	404	10	p.	p.	NOUN
ejpam-5623	404	11	tantong	tantong	NOUN
ejpam-5623	404	12	/	/	SYM
ejpam-5623	404	13	eur	eur	PROPN
ejpam-5623	404	14	.	.	PUNCT
ejpam-5623	405	1	j.	j.	PROPN
ejpam-5623	405	2	pure	pure	PROPN
ejpam-5623	405	3	appl	appl	PROPN
ejpam-5623	405	4	.	.	PROPN
ejpam-5623	405	5	math	math	PROPN
ejpam-5623	405	6	,	,	PUNCT
ejpam-5623	405	7	18	18	NUM
ejpam-5623	405	8	(	(	PUNCT
ejpam-5623	405	9	1	1	NUM
ejpam-5623	405	10	)	)	PUNCT
ejpam-5623	405	11	(	(	PUNCT
ejpam-5623	405	12	2025	2025	NUM
ejpam-5623	405	13	)	)	PUNCT
ejpam-5623	405	14	,	,	PUNCT
ejpam-5623	405	15	5623	5623	NUM
ejpam-5623	405	16	10	10	NUM
ejpam-5623	405	17	of	of	ADP
ejpam-5623	405	18	15	15	NUM
ejpam-5623	405	19	(	(	PUNCT
ejpam-5623	405	20	i	i	NOUN
ejpam-5623	405	21	)	)	PUNCT
ejpam-5623	405	22	fix(v	fix(v	PROPN
ejpam-5623	405	23	,	,	PUNCT
ejpam-5623	405	24	w	w	PROPN
ejpam-5623	405	25	)	)	PUNCT
ejpam-5623	405	26	is	be	AUX
ejpam-5623	405	27	a	a	DET
ejpam-5623	405	28	right	right	ADJ
ejpam-5623	405	29	regular	regular	ADJ
ejpam-5623	405	30	semigroup	semigroup	NOUN
ejpam-5623	405	31	.	.	PUNCT
ejpam-5623	406	1	(	(	PUNCT
ejpam-5623	406	2	ii	ii	NOUN
ejpam-5623	406	3	)	)	PUNCT
ejpam-5623	406	4	rreg(fix(v	rreg(fix(v	PROPN
ejpam-5623	406	5	,	,	PUNCT
ejpam-5623	406	6	w	w	NOUN
ejpam-5623	406	7	)	)	PUNCT
ejpam-5623	406	8	)	)	PUNCT
ejpam-5623	406	9	is	be	AUX
ejpam-5623	406	10	a	a	DET
ejpam-5623	406	11	subsemigroup	subsemigroup	NOUN
ejpam-5623	406	12	of	of	ADP
ejpam-5623	406	13	fix(v	fix(v	PROPN
ejpam-5623	406	14	,	,	PUNCT
ejpam-5623	406	15	w	w	PROPN
ejpam-5623	406	16	)	)	PUNCT
ejpam-5623	406	17	.	.	PUNCT
ejpam-5623	407	1	(	(	PUNCT
ejpam-5623	407	2	iii	iii	X
ejpam-5623	407	3	)	)	PUNCT
ejpam-5623	407	4	v	v	NOUN
ejpam-5623	407	5	=	=	SYM
ejpam-5623	407	6	w	w	PROPN
ejpam-5623	407	7	or	or	CCONJ
ejpam-5623	407	8	dim(v	dim(v	PROPN
ejpam-5623	407	9	)	)	PUNCT
ejpam-5623	407	10	≤	≤	NUM
ejpam-5623	407	11	1	1	NUM
ejpam-5623	407	12	.	.	PUNCT
ejpam-5623	408	1	proof	proof	NOUN
ejpam-5623	408	2	.	.	PUNCT
ejpam-5623	409	1	(	(	PUNCT
ejpam-5623	409	2	i	i	NOUN
ejpam-5623	409	3	)	)	PUNCT
ejpam-5623	409	4	⇒	⇒	PROPN
ejpam-5623	409	5	(	(	PUNCT
ejpam-5623	409	6	ii	ii	NOUN
ejpam-5623	409	7	)	)	PUNCT
ejpam-5623	409	8	this	this	PRON
ejpam-5623	409	9	is	be	AUX
ejpam-5623	409	10	clear	clear	ADJ
ejpam-5623	409	11	by	by	ADP
ejpam-5623	409	12	definition	definition	NOUN
ejpam-5623	409	13	.	.	PUNCT
ejpam-5623	410	1	(	(	PUNCT
ejpam-5623	410	2	ii	ii	NOUN
ejpam-5623	410	3	)	)	PUNCT
ejpam-5623	410	4	⇒	⇒	NOUN
ejpam-5623	410	5	(	(	PUNCT
ejpam-5623	410	6	iii	iii	X
ejpam-5623	410	7	)	)	PUNCT
ejpam-5623	410	8	we	we	PRON
ejpam-5623	410	9	will	will	AUX
ejpam-5623	410	10	prove	prove	VERB
ejpam-5623	410	11	the	the	DET
ejpam-5623	410	12	contrapositive	contrapositive	NOUN
ejpam-5623	410	13	.	.	PUNCT
ejpam-5623	411	1	assume	assume	VERB
ejpam-5623	411	2	that	that	SCONJ
ejpam-5623	411	3	w	w	PROPN
ejpam-5623	411	4	̸=	̸=	PROPN
ejpam-5623	411	5	v	v	NOUN
ejpam-5623	411	6	and	and	CCONJ
ejpam-5623	411	7	dim(v	dim(v	PROPN
ejpam-5623	411	8	)	)	PUNCT
ejpam-5623	411	9	>	>	X
ejpam-5623	412	1	1	1	X
ejpam-5623	412	2	.	.	PUNCT
ejpam-5623	413	1	if	if	SCONJ
ejpam-5623	413	2	dim(w	dim(w	X
ejpam-5623	413	3	)	)	PUNCT
ejpam-5623	413	4	=	=	SYM
ejpam-5623	413	5	0	0	NUM
ejpam-5623	413	6	,	,	PUNCT
ejpam-5623	413	7	then	then	ADV
ejpam-5623	413	8	fix(v	fix(v	PROPN
ejpam-5623	413	9	,	,	PUNCT
ejpam-5623	413	10	w	w	NOUN
ejpam-5623	413	11	)	)	PUNCT
ejpam-5623	413	12	=	=	SYM
ejpam-5623	413	13	l(v	l(v	NOUN
ejpam-5623	413	14	)	)	PUNCT
ejpam-5623	413	15	and	and	CCONJ
ejpam-5623	413	16	hence	hence	ADV
ejpam-5623	413	17	rreg(fix(v	rreg(fix(v	NOUN
ejpam-5623	413	18	,	,	PUNCT
ejpam-5623	413	19	w	w	NOUN
ejpam-5623	413	20	)	)	PUNCT
ejpam-5623	413	21	)	)	PUNCT
ejpam-5623	413	22	is	be	AUX
ejpam-5623	413	23	not	not	PART
ejpam-5623	413	24	a	a	DET
ejpam-5623	413	25	subsemigroup	subsemigroup	NOUN
ejpam-5623	413	26	of	of	ADP
ejpam-5623	413	27	fix(v	fix(v	PROPN
ejpam-5623	413	28	,	,	PUNCT
ejpam-5623	413	29	w	w	PROPN
ejpam-5623	413	30	)	)	PUNCT
ejpam-5623	413	31	.	.	PUNCT
ejpam-5623	414	1	suppose	suppose	VERB
ejpam-5623	414	2	that	that	SCONJ
ejpam-5623	414	3	dim(w	dim(w	NOUN
ejpam-5623	414	4	)	)	PUNCT
ejpam-5623	414	5	>	>	X
ejpam-5623	414	6	0	0	X
ejpam-5623	414	7	.	.	PUNCT
ejpam-5623	415	1	let	let	VERB
ejpam-5623	415	2	a	a	PRON
ejpam-5623	415	3	be	be	AUX
ejpam-5623	415	4	a	a	DET
ejpam-5623	415	5	non	non	ADJ
ejpam-5623	415	6	-	-	ADJ
ejpam-5623	415	7	zero	zero	NUM
ejpam-5623	415	8	element	element	NOUN
ejpam-5623	415	9	of	of	ADP
ejpam-5623	415	10	w	w	PROPN
ejpam-5623	415	11	.	.	PUNCT
ejpam-5623	416	1	then	then	ADV
ejpam-5623	416	2	,	,	PUNCT
ejpam-5623	416	3	there	there	PRON
ejpam-5623	416	4	is	be	VERB
ejpam-5623	416	5	a	a	DET
ejpam-5623	416	6	basis	basis	NOUN
ejpam-5623	416	7	bw	bw	NOUN
ejpam-5623	416	8	for	for	ADP
ejpam-5623	416	9	w	w	PROPN
ejpam-5623	416	10	such	such	ADJ
ejpam-5623	416	11	that	that	SCONJ
ejpam-5623	416	12	a	a	DET
ejpam-5623	416	13	∈	∈	PROPN
ejpam-5623	416	14	bw	bw	NOUN
ejpam-5623	416	15	.	.	PUNCT
ejpam-5623	417	1	let	let	VERB
ejpam-5623	417	2	b	b	X
ejpam-5623	417	3	be	be	AUX
ejpam-5623	417	4	a	a	DET
ejpam-5623	417	5	basis	basis	NOUN
ejpam-5623	417	6	for	for	ADP
ejpam-5623	417	7	v	v	NOUN
ejpam-5623	417	8	with	with	ADP
ejpam-5623	417	9	bw	bw	PROPN
ejpam-5623	417	10	⊆	⊆	NUM
ejpam-5623	417	11	b.	b.	NOUN
ejpam-5623	417	12	by	by	ADP
ejpam-5623	417	13	assumption	assumption	NOUN
ejpam-5623	417	14	,	,	PUNCT
ejpam-5623	417	15	let	let	VERB
ejpam-5623	417	16	b	b	X
ejpam-5623	417	17	∈	∈	PROPN
ejpam-5623	417	18	b	b	PROPN
ejpam-5623	417	19	\bw	\bw	PROPN
ejpam-5623	417	20	.	.	PUNCT
ejpam-5623	418	1	define	define	VERB
ejpam-5623	418	2	transformations	transformation	NOUN
ejpam-5623	418	3	α	α	NOUN
ejpam-5623	418	4	and	and	CCONJ
ejpam-5623	418	5	β	β	PROPN
ejpam-5623	418	6	as	as	SCONJ
ejpam-5623	418	7	follows	follow	VERB
ejpam-5623	418	8	:	:	PUNCT
ejpam-5623	418	9	xα	xα	ADP
ejpam-5623	418	10	=	=	PUNCT
ejpam-5623	419	1			PUNCT
ejpam-5623	419	2	x	x	INTJ
ejpam-5623	419	3	if	if	SCONJ
ejpam-5623	419	4	x	x	SYM
ejpam-5623	419	5	∈	∈	PROPN
ejpam-5623	419	6	bw	bw	NOUN
ejpam-5623	419	7	,	,	PUNCT
ejpam-5623	419	8	b	b	PROPN
ejpam-5623	419	9	if	if	SCONJ
ejpam-5623	419	10	x	x	PROPN
ejpam-5623	419	11	=	=	SYM
ejpam-5623	419	12	b	b	PROPN
ejpam-5623	419	13	,	,	PUNCT
ejpam-5623	419	14	0	0	NUM
ejpam-5623	419	15	otherwise	otherwise	ADV
ejpam-5623	419	16	,	,	PUNCT
ejpam-5623	419	17	and	and	CCONJ
ejpam-5623	419	18	xβ	xβ	ADV
ejpam-5623	419	19	=	=	PUNCT
ejpam-5623	420	1			PUNCT
ejpam-5623	420	2	x	x	INTJ
ejpam-5623	420	3	if	if	SCONJ
ejpam-5623	420	4	x	x	SYM
ejpam-5623	420	5	∈	∈	PROPN
ejpam-5623	420	6	bw	bw	NOUN
ejpam-5623	420	7	,	,	PUNCT
ejpam-5623	420	8	a	a	DET
ejpam-5623	420	9	if	if	NOUN
ejpam-5623	420	10	x	x	PROPN
ejpam-5623	420	11	=	=	SYM
ejpam-5623	420	12	b	b	PROPN
ejpam-5623	420	13	,	,	PUNCT
ejpam-5623	420	14	0	0	NUM
ejpam-5623	420	15	otherwise	otherwise	ADV
ejpam-5623	420	16	.	.	PUNCT
ejpam-5623	421	1	it	it	PRON
ejpam-5623	421	2	is	be	AUX
ejpam-5623	421	3	easy	easy	ADJ
ejpam-5623	421	4	to	to	PART
ejpam-5623	421	5	verify	verify	VERB
ejpam-5623	421	6	that	that	SCONJ
ejpam-5623	421	7	α	α	PROPN
ejpam-5623	421	8	and	and	CCONJ
ejpam-5623	421	9	β	β	X
ejpam-5623	421	10	are	be	AUX
ejpam-5623	421	11	well	well	ADV
ejpam-5623	421	12	-	-	PUNCT
ejpam-5623	421	13	defined	define	VERB
ejpam-5623	421	14	and	and	CCONJ
ejpam-5623	421	15	can	can	AUX
ejpam-5623	421	16	be	be	AUX
ejpam-5623	421	17	extended	extend	VERB
ejpam-5623	421	18	to	to	ADP
ejpam-5623	421	19	the	the	DET
ejpam-5623	421	20	linear	linear	ADJ
ejpam-5623	421	21	transformations	transformation	NOUN
ejpam-5623	421	22	on	on	ADP
ejpam-5623	421	23	v	v	NUM
ejpam-5623	421	24	.	.	PUNCT
ejpam-5623	422	1	by	by	ADP
ejpam-5623	422	2	the	the	DET
ejpam-5623	422	3	definitions	definition	NOUN
ejpam-5623	422	4	of	of	ADP
ejpam-5623	422	5	α	α	NOUN
ejpam-5623	422	6	and	and	CCONJ
ejpam-5623	422	7	β	β	X
ejpam-5623	422	8	,	,	PUNCT
ejpam-5623	422	9	we	we	PRON
ejpam-5623	422	10	see	see	VERB
ejpam-5623	422	11	that	that	SCONJ
ejpam-5623	422	12	both	both	CCONJ
ejpam-5623	422	13	α|w	α|w	PROPN
ejpam-5623	422	14	and	and	CCONJ
ejpam-5623	422	15	β|w	β|w	ADV
ejpam-5623	422	16	are	be	AUX
ejpam-5623	422	17	the	the	DET
ejpam-5623	422	18	identity	identity	NOUN
ejpam-5623	422	19	transformation	transformation	NOUN
ejpam-5623	422	20	on	on	ADP
ejpam-5623	422	21	w	w	PROPN
ejpam-5623	422	22	,	,	PUNCT
ejpam-5623	422	23	so	so	SCONJ
ejpam-5623	422	24	that	that	SCONJ
ejpam-5623	422	25	α	α	X
ejpam-5623	422	26	,	,	PUNCT
ejpam-5623	422	27	β	β	PROPN
ejpam-5623	422	28	∈	∈	PROPN
ejpam-5623	422	29	fix(v	fix(v	PROPN
ejpam-5623	422	30	,	,	PUNCT
ejpam-5623	422	31	w	w	PROPN
ejpam-5623	422	32	)	)	PUNCT
ejpam-5623	422	33	.	.	PUNCT
ejpam-5623	423	1	next	next	ADV
ejpam-5623	423	2	,	,	PUNCT
ejpam-5623	423	3	we	we	PRON
ejpam-5623	423	4	check	check	VERB
ejpam-5623	423	5	that	that	SCONJ
ejpam-5623	423	6	α	α	PROPN
ejpam-5623	423	7	and	and	CCONJ
ejpam-5623	423	8	β	β	X
ejpam-5623	423	9	are	be	AUX
ejpam-5623	423	10	right	right	ADV
ejpam-5623	423	11	regular	regular	ADJ
ejpam-5623	423	12	elements	element	NOUN
ejpam-5623	423	13	of	of	ADP
ejpam-5623	423	14	fix(v	fix(v	PROPN
ejpam-5623	423	15	,	,	PUNCT
ejpam-5623	423	16	w	w	NOUN
ejpam-5623	423	17	)	)	PUNCT
ejpam-5623	423	18	by	by	ADP
ejpam-5623	423	19	using	use	VERB
ejpam-5623	423	20	theorem	theorem	NOUN
ejpam-5623	423	21	7	7	NUM
ejpam-5623	423	22	.	.	PUNCT
ejpam-5623	424	1	let	let	VERB
ejpam-5623	424	2	u	u	PRON
ejpam-5623	424	3	∈	∈	PROPN
ejpam-5623	424	4	v	v	ADP
ejpam-5623	424	5	α	α	NOUN
ejpam-5623	424	6	be	be	AUX
ejpam-5623	424	7	such	such	ADJ
ejpam-5623	424	8	that	that	SCONJ
ejpam-5623	424	9	uα	uα	PROPN
ejpam-5623	424	10	=	=	NOUN
ejpam-5623	424	11	0	0	PROPN
ejpam-5623	424	12	.	.	PUNCT
ejpam-5623	425	1	by	by	ADP
ejpam-5623	425	2	the	the	DET
ejpam-5623	425	3	definition	definition	NOUN
ejpam-5623	425	4	of	of	ADP
ejpam-5623	425	5	α	α	NOUN
ejpam-5623	425	6	,	,	PUNCT
ejpam-5623	425	7	we	we	PRON
ejpam-5623	425	8	know	know	VERB
ejpam-5623	425	9	that	that	SCONJ
ejpam-5623	425	10	v	v	ADP
ejpam-5623	425	11	α	α	NOUN
ejpam-5623	425	12	=	=	SYM
ejpam-5623	425	13	{	{	PUNCT
ejpam-5623	425	14	w	w	PROPN
ejpam-5623	425	15	+	+	X
ejpam-5623	425	16	kb	kb	PROPN
ejpam-5623	425	17	:	:	PUNCT
ejpam-5623	425	18	w	w	PROPN
ejpam-5623	425	19	∈	∈	PROPN
ejpam-5623	425	20	w	w	NOUN
ejpam-5623	425	21	and	and	CCONJ
ejpam-5623	425	22	k	k	PROPN
ejpam-5623	425	23	∈	∈	PROPN
ejpam-5623	425	24	f	f	X
ejpam-5623	425	25	}	}	PUNCT
ejpam-5623	425	26	.	.	PUNCT
ejpam-5623	426	1	then	then	ADV
ejpam-5623	426	2	,	,	PUNCT
ejpam-5623	426	3	u	u	NOUN
ejpam-5623	426	4	=	=	PROPN
ejpam-5623	426	5	w	w	PROPN
ejpam-5623	427	1	+	+	PROPN
ejpam-5623	427	2	kb	kb	PROPN
ejpam-5623	427	3	where	where	SCONJ
ejpam-5623	427	4	k	k	PROPN
ejpam-5623	427	5	∈	∈	PROPN
ejpam-5623	427	6	f	f	PROPN
ejpam-5623	427	7	and	and	CCONJ
ejpam-5623	427	8	w	w	PROPN
ejpam-5623	427	9	∈	∈	PROPN
ejpam-5623	427	10	w	w	NOUN
ejpam-5623	427	11	.	.	PUNCT
ejpam-5623	428	1	thus	thus	ADV
ejpam-5623	428	2	0	0	X
ejpam-5623	428	3	=	=	SYM
ejpam-5623	428	4	uα	uα	NOUN
ejpam-5623	429	1	=	=	PUNCT
ejpam-5623	430	1	(	(	PUNCT
ejpam-5623	430	2	w	w	PROPN
ejpam-5623	430	3	+	+	CCONJ
ejpam-5623	430	4	kb)α	kb)α	NOUN
ejpam-5623	430	5	=	=	SYM
ejpam-5623	430	6	wα+	wα+	ADJ
ejpam-5623	430	7	bα	bα	NOUN
ejpam-5623	431	1	=	=	PUNCT
ejpam-5623	431	2	w	w	PROPN
ejpam-5623	431	3	+	+	NUM
ejpam-5623	431	4	kb	kb	PROPN
ejpam-5623	431	5	=	=	SYM
ejpam-5623	431	6	u	u	PROPN
ejpam-5623	431	7	hence	hence	ADV
ejpam-5623	431	8	,	,	PUNCT
ejpam-5623	431	9	ker(α|v	ker(α|v	PROPN
ejpam-5623	431	10	α	α	X
ejpam-5623	431	11	)	)	PUNCT
ejpam-5623	431	12	=	=	PRON
ejpam-5623	431	13	{	{	PUNCT
ejpam-5623	431	14	0	0	NUM
ejpam-5623	431	15	}	}	PUNCT
ejpam-5623	431	16	and	and	CCONJ
ejpam-5623	431	17	so	so	ADV
ejpam-5623	431	18	α|v	α|v	ADV
ejpam-5623	431	19	α	α	PROPN
ejpam-5623	431	20	is	be	AUX
ejpam-5623	431	21	one	one	NUM
ejpam-5623	431	22	-	-	PUNCT
ejpam-5623	431	23	to	to	ADP
ejpam-5623	431	24	-	-	PUNCT
ejpam-5623	431	25	one	one	NUM
ejpam-5623	431	26	.	.	PUNCT
ejpam-5623	432	1	it	it	PRON
ejpam-5623	432	2	follows	follow	VERB
ejpam-5623	432	3	from	from	ADP
ejpam-5623	432	4	theorem	theorem	NOUN
ejpam-5623	432	5	7	7	NUM
ejpam-5623	432	6	that	that	SCONJ
ejpam-5623	432	7	α	α	PRON
ejpam-5623	432	8	is	be	AUX
ejpam-5623	432	9	right	right	ADV
ejpam-5623	432	10	regular	regular	ADJ
ejpam-5623	432	11	of	of	ADP
ejpam-5623	432	12	fix(v	fix(v	PROPN
ejpam-5623	432	13	,	,	PUNCT
ejpam-5623	432	14	w	w	NOUN
ejpam-5623	432	15	)	)	PUNCT
ejpam-5623	432	16	.	.	PUNCT
ejpam-5623	433	1	similarly	similarly	ADV
ejpam-5623	433	2	,	,	PUNCT
ejpam-5623	433	3	we	we	PRON
ejpam-5623	433	4	can	can	AUX
ejpam-5623	433	5	show	show	VERB
ejpam-5623	433	6	that	that	PRON
ejpam-5623	433	7	β|v	β|v	PUNCT
ejpam-5623	434	1	β	β	X
ejpam-5623	434	2	is	be	AUX
ejpam-5623	434	3	one	one	NUM
ejpam-5623	434	4	-	-	PUNCT
ejpam-5623	434	5	to	to	ADP
ejpam-5623	434	6	-	-	PUNCT
ejpam-5623	434	7	one	one	NUM
ejpam-5623	434	8	.	.	PUNCT
ejpam-5623	435	1	it	it	PRON
ejpam-5623	435	2	follows	follow	VERB
ejpam-5623	435	3	from	from	ADP
ejpam-5623	435	4	theorem	theorem	ADJ
ejpam-5623	435	5	7	7	NUM
ejpam-5623	435	6	that	that	SCONJ
ejpam-5623	435	7	β	β	PROPN
ejpam-5623	435	8	is	be	AUX
ejpam-5623	435	9	right	right	ADV
ejpam-5623	435	10	regular	regular	ADV
ejpam-5623	435	11	.	.	PUNCT
ejpam-5623	436	1	finally	finally	ADV
ejpam-5623	436	2	,	,	PUNCT
ejpam-5623	436	3	we	we	PRON
ejpam-5623	436	4	will	will	AUX
ejpam-5623	436	5	show	show	VERB
ejpam-5623	436	6	that	that	SCONJ
ejpam-5623	436	7	αβ	αβ	PRON
ejpam-5623	436	8	is	be	AUX
ejpam-5623	436	9	not	not	PART
ejpam-5623	436	10	right	right	ADV
ejpam-5623	436	11	regular	regular	ADV
ejpam-5623	436	12	.	.	PUNCT
ejpam-5623	437	1	note	note	VERB
ejpam-5623	437	2	that	that	SCONJ
ejpam-5623	437	3	bαβ	bαβ	ADJ
ejpam-5623	437	4	=	=	SYM
ejpam-5623	437	5	bβ	bβ	NOUN
ejpam-5623	437	6	=	=	PUNCT
ejpam-5623	437	7	a	a	NOUN
ejpam-5623	437	8	=	=	X
ejpam-5623	437	9	aβ	aβ	NOUN
ejpam-5623	437	10	=	=	NOUN
ejpam-5623	437	11	aαβ	aαβ	NOUN
ejpam-5623	437	12	.	.	PUNCT
ejpam-5623	438	1	since	since	SCONJ
ejpam-5623	438	2	a	a	DET
ejpam-5623	438	3	,	,	PUNCT
ejpam-5623	438	4	b	b	X
ejpam-5623	438	5	∈	∈	PROPN
ejpam-5623	438	6	v	v	ADP
ejpam-5623	438	7	αβ	αβ	NOUN
ejpam-5623	438	8	and	and	CCONJ
ejpam-5623	438	9	a	a	DET
ejpam-5623	438	10	̸=	̸=	PROPN
ejpam-5623	438	11	b	b	NUM
ejpam-5623	438	12	,	,	PUNCT
ejpam-5623	438	13	it	it	PRON
ejpam-5623	438	14	follows	follow	VERB
ejpam-5623	438	15	that	that	SCONJ
ejpam-5623	438	16	αβ|v	αβ|v	ADV
ejpam-5623	439	1	αβ	αβ	PRON
ejpam-5623	439	2	is	be	AUX
ejpam-5623	439	3	not	not	PART
ejpam-5623	439	4	one	one	NUM
ejpam-5623	439	5	-	-	PUNCT
ejpam-5623	439	6	to	to	ADP
ejpam-5623	439	7	-	-	PUNCT
ejpam-5623	439	8	one	one	NUM
ejpam-5623	439	9	.	.	PUNCT
ejpam-5623	440	1	from	from	ADP
ejpam-5623	440	2	theorem	theorem	ADJ
ejpam-5623	440	3	7	7	NUM
ejpam-5623	440	4	,	,	PUNCT
ejpam-5623	440	5	αβ	αβ	PRON
ejpam-5623	440	6	is	be	AUX
ejpam-5623	440	7	not	not	PART
ejpam-5623	440	8	right	right	ADV
ejpam-5623	440	9	regular	regular	ADV
ejpam-5623	440	10	.	.	PUNCT
ejpam-5623	441	1	hence	hence	ADV
ejpam-5623	441	2	,	,	PUNCT
ejpam-5623	441	3	rreg(fix(v	rreg(fix(v	NOUN
ejpam-5623	441	4	,	,	PUNCT
ejpam-5623	441	5	w	w	NOUN
ejpam-5623	441	6	)	)	PUNCT
ejpam-5623	441	7	)	)	PUNCT
ejpam-5623	441	8	is	be	AUX
ejpam-5623	441	9	not	not	PART
ejpam-5623	441	10	a	a	DET
ejpam-5623	441	11	subsemigroup	subsemigroup	NOUN
ejpam-5623	441	12	of	of	ADP
ejpam-5623	441	13	fix(v	fix(v	PROPN
ejpam-5623	441	14	,	,	PUNCT
ejpam-5623	441	15	w	w	PROPN
ejpam-5623	441	16	)	)	PUNCT
ejpam-5623	441	17	.	.	PUNCT
ejpam-5623	442	1	(	(	PUNCT
ejpam-5623	442	2	iii	iii	X
ejpam-5623	442	3	)	)	PUNCT
ejpam-5623	442	4	⇒	⇒	NOUN
ejpam-5623	442	5	(	(	PUNCT
ejpam-5623	442	6	i	i	NOUN
ejpam-5623	442	7	)	)	PUNCT
ejpam-5623	442	8	assume	assume	VERB
ejpam-5623	442	9	that	that	SCONJ
ejpam-5623	442	10	w	w	PROPN
ejpam-5623	442	11	=	=	SYM
ejpam-5623	442	12	v	v	NOUN
ejpam-5623	442	13	or	or	CCONJ
ejpam-5623	442	14	dim(v	dim(v	PROPN
ejpam-5623	442	15	)	)	PUNCT
ejpam-5623	442	16	≤	≤	NUM
ejpam-5623	443	1	1	1	NUM
ejpam-5623	443	2	.	.	PUNCT
ejpam-5623	444	1	if	if	SCONJ
ejpam-5623	444	2	w	w	PROPN
ejpam-5623	444	3	=	=	SYM
ejpam-5623	444	4	v	v	NOUN
ejpam-5623	444	5	,	,	PUNCT
ejpam-5623	444	6	then	then	ADV
ejpam-5623	444	7	fix(v	fix(v	PROPN
ejpam-5623	444	8	,	,	PUNCT
ejpam-5623	444	9	w	w	NOUN
ejpam-5623	444	10	)	)	PUNCT
ejpam-5623	444	11	consists	consist	VERB
ejpam-5623	444	12	only	only	ADV
ejpam-5623	444	13	of	of	ADP
ejpam-5623	444	14	the	the	DET
ejpam-5623	444	15	identity	identity	NOUN
ejpam-5623	444	16	transformation	transformation	NOUN
ejpam-5623	444	17	,	,	PUNCT
ejpam-5623	444	18	and	and	CCONJ
ejpam-5623	444	19	hence	hence	ADV
ejpam-5623	444	20	,	,	PUNCT
ejpam-5623	444	21	fix(v	fix(v	PROPN
ejpam-5623	444	22	,	,	PUNCT
ejpam-5623	444	23	w	w	PROPN
ejpam-5623	444	24	)	)	PUNCT
ejpam-5623	444	25	is	be	AUX
ejpam-5623	444	26	trivially	trivially	ADV
ejpam-5623	444	27	a	a	DET
ejpam-5623	444	28	right	right	ADJ
ejpam-5623	444	29	regular	regular	ADJ
ejpam-5623	444	30	n.	n.	NOUN
ejpam-5623	444	31	sawatraksa	sawatraksa	NOUN
ejpam-5623	444	32	,	,	PUNCT
ejpam-5623	444	33	p.	p.	NOUN
ejpam-5623	444	34	tantong	tantong	NOUN
ejpam-5623	444	35	/	/	SYM
ejpam-5623	444	36	eur	eur	PROPN
ejpam-5623	444	37	.	.	PUNCT
ejpam-5623	445	1	j.	j.	PROPN
ejpam-5623	445	2	pure	pure	PROPN
ejpam-5623	445	3	appl	appl	PROPN
ejpam-5623	445	4	.	.	PROPN
ejpam-5623	445	5	math	math	PROPN
ejpam-5623	445	6	,	,	PUNCT
ejpam-5623	445	7	18	18	NUM
ejpam-5623	445	8	(	(	PUNCT
ejpam-5623	445	9	1	1	NUM
ejpam-5623	445	10	)	)	PUNCT
ejpam-5623	445	11	(	(	PUNCT
ejpam-5623	445	12	2025	2025	NUM
ejpam-5623	445	13	)	)	PUNCT
ejpam-5623	445	14	,	,	PUNCT
ejpam-5623	445	15	5623	5623	NUM
ejpam-5623	445	16	11	11	NUM
ejpam-5623	445	17	of	of	ADP
ejpam-5623	445	18	15	15	NUM
ejpam-5623	445	19	semigroup	semigroup	NOUN
ejpam-5623	445	20	.	.	PUNCT
ejpam-5623	446	1	if	if	SCONJ
ejpam-5623	446	2	dim(v	dim(v	PROPN
ejpam-5623	446	3	)	)	PUNCT
ejpam-5623	446	4	≤	≤	NUM
ejpam-5623	446	5	1	1	NUM
ejpam-5623	446	6	,	,	PUNCT
ejpam-5623	446	7	then	then	ADV
ejpam-5623	446	8	fix(v	fix(v	PROPN
ejpam-5623	446	9	,	,	PUNCT
ejpam-5623	446	10	w	w	NOUN
ejpam-5623	446	11	)	)	PUNCT
ejpam-5623	446	12	=	=	SYM
ejpam-5623	446	13	l(v	l(v	NOUN
ejpam-5623	446	14	)	)	PUNCT
ejpam-5623	446	15	and	and	CCONJ
ejpam-5623	446	16	hence	hence	ADV
ejpam-5623	446	17	,	,	PUNCT
ejpam-5623	446	18	it	it	PRON
ejpam-5623	446	19	is	be	AUX
ejpam-5623	446	20	a	a	DET
ejpam-5623	446	21	right	right	ADJ
ejpam-5623	446	22	regular	regular	ADJ
ejpam-5623	446	23	semigroup	semigroup	NOUN
ejpam-5623	446	24	.	.	PUNCT
ejpam-5623	447	1	secondly	secondly	ADV
ejpam-5623	447	2	,	,	PUNCT
ejpam-5623	447	3	we	we	PRON
ejpam-5623	447	4	investigate	investigate	VERB
ejpam-5623	447	5	the	the	DET
ejpam-5623	447	6	condition	condition	NOUN
ejpam-5623	447	7	under	under	ADP
ejpam-5623	447	8	which	which	PRON
ejpam-5623	447	9	an	an	DET
ejpam-5623	447	10	element	element	NOUN
ejpam-5623	447	11	of	of	ADP
ejpam-5623	447	12	fix(v	fix(v	PROPN
ejpam-5623	447	13	,	,	PUNCT
ejpam-5623	447	14	w	w	PROPN
ejpam-5623	447	15	)	)	PUNCT
ejpam-5623	447	16	is	be	AUX
ejpam-5623	447	17	left	leave	VERB
ejpam-5623	447	18	regular	regular	ADV
ejpam-5623	447	19	.	.	PUNCT
ejpam-5623	448	1	theorem	theorem	NOUN
ejpam-5623	448	2	9	9	NUM
ejpam-5623	448	3	.	.	PUNCT
ejpam-5623	449	1	let	let	VERB
ejpam-5623	449	2	α	α	PRON
ejpam-5623	449	3	∈	∈	PROPN
ejpam-5623	449	4	fix(v	fix(v	PROPN
ejpam-5623	449	5	,	,	PUNCT
ejpam-5623	449	6	w	w	PROPN
ejpam-5623	449	7	)	)	PUNCT
ejpam-5623	449	8	.	.	PUNCT
ejpam-5623	450	1	the	the	DET
ejpam-5623	450	2	following	follow	VERB
ejpam-5623	450	3	statements	statement	NOUN
ejpam-5623	450	4	are	be	AUX
ejpam-5623	450	5	equivalent	equivalent	ADJ
ejpam-5623	450	6	:	:	PUNCT
ejpam-5623	450	7	(	(	PUNCT
ejpam-5623	450	8	i	i	NOUN
ejpam-5623	450	9	)	)	PUNCT
ejpam-5623	450	10	α	α	PROPN
ejpam-5623	450	11	∈	∈	PROPN
ejpam-5623	450	12	lreg(fix(v	lreg(fix(v	PROPN
ejpam-5623	450	13	,	,	PUNCT
ejpam-5623	450	14	w	w	NOUN
ejpam-5623	450	15	)	)	PUNCT
ejpam-5623	450	16	)	)	PUNCT
ejpam-5623	450	17	.	.	PUNCT
ejpam-5623	451	1	(	(	PUNCT
ejpam-5623	451	2	ii	ii	NOUN
ejpam-5623	451	3	)	)	PUNCT
ejpam-5623	451	4	α|v	α|v	VERB
ejpam-5623	452	1	α	α	PROPN
ejpam-5623	452	2	is	be	AUX
ejpam-5623	452	3	an	an	PRON
ejpam-5623	452	4	onto	onto	ADP
ejpam-5623	452	5	transformation	transformation	NOUN
ejpam-5623	452	6	on	on	ADP
ejpam-5623	452	7	v	v	NUM
ejpam-5623	452	8	α	α	NOUN
ejpam-5623	452	9	.	.	PUNCT
ejpam-5623	453	1	(	(	PUNCT
ejpam-5623	453	2	iii	iii	X
ejpam-5623	453	3	)	)	PUNCT
ejpam-5623	453	4	v	v	NOUN
ejpam-5623	453	5	α	α	NOUN
ejpam-5623	453	6	⊆	⊆	PROPN
ejpam-5623	453	7	v	v	NOUN
ejpam-5623	453	8	α2	α2	NOUN
ejpam-5623	453	9	.	.	PUNCT
ejpam-5623	454	1	(	(	PUNCT
ejpam-5623	454	2	iv	iv	X
ejpam-5623	454	3	)	)	PUNCT
ejpam-5623	454	4	for	for	ADP
ejpam-5623	454	5	every	every	DET
ejpam-5623	454	6	basis	basis	NOUN
ejpam-5623	454	7	b	b	NOUN
ejpam-5623	454	8	for	for	ADP
ejpam-5623	454	9	v	v	NUM
ejpam-5623	454	10	,	,	PUNCT
ejpam-5623	454	11	we	we	PRON
ejpam-5623	454	12	have	have	VERB
ejpam-5623	454	13	bα	bα	PROPN
ejpam-5623	454	14	⊆	⊆	NUM
ejpam-5623	454	15	v	v	NOUN
ejpam-5623	454	16	α2	α2	NOUN
ejpam-5623	454	17	.	.	PUNCT
ejpam-5623	455	1	(	(	PUNCT
ejpam-5623	455	2	v	v	X
ejpam-5623	455	3	)	)	PUNCT
ejpam-5623	455	4	there	there	PRON
ejpam-5623	455	5	exist	exist	VERB
ejpam-5623	455	6	two	two	NUM
ejpam-5623	455	7	basses	bass	NOUN
ejpam-5623	455	8	b	b	NOUN
ejpam-5623	455	9	for	for	ADP
ejpam-5623	455	10	v	v	NOUN
ejpam-5623	455	11	and	and	CCONJ
ejpam-5623	455	12	bw	bw	NOUN
ejpam-5623	455	13	for	for	ADP
ejpam-5623	455	14	w	w	ADP
ejpam-5623	455	15	such	such	ADJ
ejpam-5623	455	16	that	that	DET
ejpam-5623	455	17	bw	bw	PROPN
ejpam-5623	455	18	⊆	⊆	NUM
ejpam-5623	455	19	b	b	NOUN
ejpam-5623	455	20	and	and	CCONJ
ejpam-5623	455	21	bα	bα	PROPN
ejpam-5623	455	22	⊆	⊆	NUM
ejpam-5623	455	23	v	v	NOUN
ejpam-5623	455	24	α2	α2	ADJ
ejpam-5623	455	25	.	.	PUNCT
ejpam-5623	456	1	proof	proof	NOUN
ejpam-5623	456	2	.	.	PUNCT
ejpam-5623	457	1	(	(	PUNCT
ejpam-5623	457	2	i	i	NOUN
ejpam-5623	457	3	)	)	PUNCT
ejpam-5623	457	4	⇒	⇒	PROPN
ejpam-5623	457	5	(	(	PUNCT
ejpam-5623	457	6	ii	ii	NOUN
ejpam-5623	457	7	)	)	PUNCT
ejpam-5623	457	8	assume	assume	VERB
ejpam-5623	457	9	that	that	SCONJ
ejpam-5623	457	10	α	α	PRON
ejpam-5623	457	11	is	be	AUX
ejpam-5623	457	12	left	leave	VERB
ejpam-5623	457	13	regular	regular	ADV
ejpam-5623	457	14	of	of	ADP
ejpam-5623	457	15	fix(v	fix(v	PROPN
ejpam-5623	457	16	,	,	PUNCT
ejpam-5623	457	17	w	w	NOUN
ejpam-5623	457	18	)	)	PUNCT
ejpam-5623	457	19	.	.	PUNCT
ejpam-5623	458	1	then	then	ADV
ejpam-5623	458	2	,	,	PUNCT
ejpam-5623	458	3	α	α	PROPN
ejpam-5623	458	4	=	=	PUNCT
ejpam-5623	458	5	βα2	βα2	ADJ
ejpam-5623	458	6	for	for	ADP
ejpam-5623	458	7	some	some	DET
ejpam-5623	458	8	β	β	PROPN
ejpam-5623	458	9	∈	∈	PROPN
ejpam-5623	458	10	fix(v	fix(v	PROPN
ejpam-5623	458	11	,	,	PUNCT
ejpam-5623	458	12	w	w	NOUN
ejpam-5623	458	13	)	)	PUNCT
ejpam-5623	458	14	.	.	PUNCT
ejpam-5623	459	1	if	if	SCONJ
ejpam-5623	459	2	y	y	PROPN
ejpam-5623	459	3	∈	∈	PROPN
ejpam-5623	459	4	v	v	ADP
ejpam-5623	459	5	α	α	NOUN
ejpam-5623	459	6	,	,	PUNCT
ejpam-5623	459	7	then	then	ADV
ejpam-5623	459	8	y	y	PROPN
ejpam-5623	459	9	=	=	PUNCT
ejpam-5623	460	1	xα	xα	INTJ
ejpam-5623	460	2	for	for	ADP
ejpam-5623	460	3	some	some	DET
ejpam-5623	460	4	x	x	SYM
ejpam-5623	460	5	∈	∈	PROPN
ejpam-5623	460	6	v	v	NOUN
ejpam-5623	460	7	.	.	PUNCT
ejpam-5623	461	1	thus	thus	ADV
ejpam-5623	461	2	,	,	PUNCT
ejpam-5623	461	3	y	y	PROPN
ejpam-5623	461	4	=	=	PUNCT
ejpam-5623	461	5	xα	xα	PROPN
ejpam-5623	461	6	=	=	PUNCT
ejpam-5623	461	7	xβα2	xβα2	NOUN
ejpam-5623	462	1	=	=	SYM
ejpam-5623	463	1	(	(	PUNCT
ejpam-5623	463	2	xβα)α	xβα)α	PROPN
ejpam-5623	463	3	.	.	PUNCT
ejpam-5623	464	1	this	this	PRON
ejpam-5623	464	2	prove	prove	VERB
ejpam-5623	464	3	that	that	SCONJ
ejpam-5623	464	4	α|v	α|v	VERB
ejpam-5623	465	1	α	α	X
ejpam-5623	465	2	:	:	PUNCT
ejpam-5623	465	3	v	v	ADP
ejpam-5623	465	4	α	α	PROPN
ejpam-5623	465	5	→	→	SYM
ejpam-5623	465	6	v	v	NUM
ejpam-5623	465	7	α	α	NOUN
ejpam-5623	465	8	is	be	AUX
ejpam-5623	465	9	onto	onto	ADP
ejpam-5623	465	10	.	.	PUNCT
ejpam-5623	466	1	(	(	PUNCT
ejpam-5623	466	2	ii	ii	NOUN
ejpam-5623	466	3	)	)	PUNCT
ejpam-5623	466	4	⇒	⇒	NOUN
ejpam-5623	466	5	(	(	PUNCT
ejpam-5623	466	6	iii	iii	NOUN
ejpam-5623	466	7	)	)	PUNCT
ejpam-5623	466	8	assume	assume	VERB
ejpam-5623	466	9	that	that	SCONJ
ejpam-5623	466	10	α|v	α|v	VERB
ejpam-5623	466	11	α	α	X
ejpam-5623	466	12	:	:	PUNCT
ejpam-5623	466	13	v	v	ADP
ejpam-5623	466	14	α	α	PROPN
ejpam-5623	466	15	→	→	SYM
ejpam-5623	466	16	v	v	NUM
ejpam-5623	466	17	α	α	NOUN
ejpam-5623	466	18	is	be	AUX
ejpam-5623	466	19	onto	onto	ADP
ejpam-5623	466	20	.	.	PUNCT
ejpam-5623	467	1	then	then	ADV
ejpam-5623	467	2	,	,	PUNCT
ejpam-5623	467	3	v	v	ADP
ejpam-5623	467	4	α2	α2	NOUN
ejpam-5623	467	5	=	=	SYM
ejpam-5623	467	6	(	(	PUNCT
ejpam-5623	467	7	v	v	NOUN
ejpam-5623	467	8	α)α	α)α	NOUN
ejpam-5623	467	9	=	=	SYM
ejpam-5623	467	10	(	(	PUNCT
ejpam-5623	467	11	v	v	NUM
ejpam-5623	467	12	α)α|v	α)α|v	NOUN
ejpam-5623	468	1	α	α	NOUN
ejpam-5623	468	2	=	=	PROPN
ejpam-5623	468	3	v	v	NUM
ejpam-5623	468	4	α	α	NOUN
ejpam-5623	468	5	,	,	PUNCT
ejpam-5623	468	6	which	which	PRON
ejpam-5623	468	7	proves	prove	VERB
ejpam-5623	468	8	that	that	SCONJ
ejpam-5623	468	9	v	v	ADP
ejpam-5623	468	10	α	α	NOUN
ejpam-5623	468	11	⊆	⊆	PROPN
ejpam-5623	468	12	v	v	NOUN
ejpam-5623	468	13	α2	α2	NOUN
ejpam-5623	468	14	.	.	PUNCT
ejpam-5623	469	1	(	(	PUNCT
ejpam-5623	469	2	iii	iii	X
ejpam-5623	469	3	)	)	PUNCT
ejpam-5623	469	4	⇒	⇒	NOUN
ejpam-5623	469	5	(	(	PUNCT
ejpam-5623	469	6	iv	iv	X
ejpam-5623	469	7	)	)	PUNCT
ejpam-5623	469	8	this	this	DET
ejpam-5623	469	9	implication	implication	NOUN
ejpam-5623	469	10	is	be	AUX
ejpam-5623	469	11	clear	clear	ADJ
ejpam-5623	469	12	by	by	ADP
ejpam-5623	469	13	definition	definition	NOUN
ejpam-5623	469	14	.	.	PUNCT
ejpam-5623	470	1	if	if	SCONJ
ejpam-5623	470	2	v	v	NOUN
ejpam-5623	470	3	α	α	NOUN
ejpam-5623	470	4	⊆	⊆	PROPN
ejpam-5623	470	5	v	v	ADP
ejpam-5623	470	6	α2	α2	NOUN
ejpam-5623	470	7	,	,	PUNCT
ejpam-5623	470	8	then	then	ADV
ejpam-5623	470	9	for	for	ADP
ejpam-5623	470	10	every	every	DET
ejpam-5623	470	11	basis	basis	NOUN
ejpam-5623	470	12	b	b	NOUN
ejpam-5623	470	13	of	of	ADP
ejpam-5623	470	14	v	v	NUM
ejpam-5623	470	15	,	,	PUNCT
ejpam-5623	470	16	we	we	PRON
ejpam-5623	470	17	will	will	AUX
ejpam-5623	470	18	have	have	VERB
ejpam-5623	470	19	bα	bα	PROPN
ejpam-5623	470	20	⊆	⊆	NUM
ejpam-5623	470	21	v	v	X
ejpam-5623	470	22	α2	α2	NOUN
ejpam-5623	470	23	(	(	PUNCT
ejpam-5623	470	24	iv	iv	NOUN
ejpam-5623	470	25	)	)	PUNCT
ejpam-5623	470	26	⇒	⇒	NOUN
ejpam-5623	470	27	(	(	PUNCT
ejpam-5623	470	28	v	v	NOUN
ejpam-5623	470	29	)	)	PUNCT
ejpam-5623	470	30	suppose	suppose	VERB
ejpam-5623	470	31	that	that	SCONJ
ejpam-5623	470	32	the	the	DET
ejpam-5623	470	33	condition	condition	NOUN
ejpam-5623	470	34	(	(	PUNCT
ejpam-5623	470	35	iv	iv	X
ejpam-5623	470	36	)	)	PUNCT
ejpam-5623	470	37	holds	hold	NOUN
ejpam-5623	470	38	.	.	PUNCT
ejpam-5623	471	1	let	let	VERB
ejpam-5623	471	2	bw	bw	PART
ejpam-5623	471	3	be	be	AUX
ejpam-5623	471	4	a	a	DET
ejpam-5623	471	5	basis	basis	NOUN
ejpam-5623	471	6	for	for	ADP
ejpam-5623	471	7	w	w	PROPN
ejpam-5623	471	8	.	.	PUNCT
ejpam-5623	472	1	then	then	ADV
ejpam-5623	472	2	,	,	PUNCT
ejpam-5623	472	3	there	there	PRON
ejpam-5623	472	4	exists	exist	VERB
ejpam-5623	472	5	a	a	DET
ejpam-5623	472	6	basis	basis	NOUN
ejpam-5623	472	7	b	b	NOUN
ejpam-5623	472	8	for	for	ADP
ejpam-5623	472	9	v	v	ADP
ejpam-5623	472	10	such	such	ADJ
ejpam-5623	472	11	that	that	PRON
ejpam-5623	472	12	bw	bw	PROPN
ejpam-5623	472	13	⊆	⊆	NUM
ejpam-5623	472	14	b.	b.	NOUN
ejpam-5623	472	15	by	by	ADP
ejpam-5623	472	16	the	the	DET
ejpam-5623	472	17	assumption	assumption	NOUN
ejpam-5623	472	18	,	,	PUNCT
ejpam-5623	472	19	we	we	PRON
ejpam-5623	472	20	have	have	VERB
ejpam-5623	472	21	bα	bα	PROPN
ejpam-5623	472	22	⊆	⊆	NUM
ejpam-5623	472	23	v	v	NOUN
ejpam-5623	472	24	α2	α2	NOUN
ejpam-5623	472	25	,	,	PUNCT
ejpam-5623	472	26	which	which	PRON
ejpam-5623	472	27	implies	imply	VERB
ejpam-5623	472	28	that	that	SCONJ
ejpam-5623	472	29	condition	condition	NOUN
ejpam-5623	472	30	(	(	PUNCT
ejpam-5623	472	31	v	v	NOUN
ejpam-5623	472	32	)	)	PUNCT
ejpam-5623	472	33	holds	hold	VERB
ejpam-5623	472	34	.	.	PUNCT
ejpam-5623	473	1	(	(	PUNCT
ejpam-5623	473	2	v	v	NOUN
ejpam-5623	473	3	)	)	PUNCT
ejpam-5623	473	4	⇒	⇒	NOUN
ejpam-5623	473	5	(	(	PUNCT
ejpam-5623	473	6	i	i	NOUN
ejpam-5623	473	7	)	)	PUNCT
ejpam-5623	473	8	suppose	suppose	VERB
ejpam-5623	473	9	that	that	SCONJ
ejpam-5623	473	10	there	there	PRON
ejpam-5623	473	11	is	be	VERB
ejpam-5623	473	12	a	a	DET
ejpam-5623	473	13	basis	basis	NOUN
ejpam-5623	473	14	b	b	NOUN
ejpam-5623	473	15	for	for	ADP
ejpam-5623	473	16	v	v	NOUN
ejpam-5623	473	17	and	and	CCONJ
ejpam-5623	473	18	a	a	DET
ejpam-5623	473	19	basis	basis	NOUN
ejpam-5623	473	20	bw	bw	NOUN
ejpam-5623	473	21	for	for	ADP
ejpam-5623	473	22	w	w	PROPN
ejpam-5623	473	23	such	such	ADJ
ejpam-5623	473	24	that	that	DET
ejpam-5623	473	25	bw	bw	PROPN
ejpam-5623	473	26	⊆	⊆	NUM
ejpam-5623	473	27	b	b	NOUN
ejpam-5623	473	28	and	and	CCONJ
ejpam-5623	473	29	bα	bα	PROPN
ejpam-5623	473	30	⊆	⊆	NUM
ejpam-5623	473	31	v	v	NOUN
ejpam-5623	473	32	α2	α2	NOUN
ejpam-5623	473	33	.	.	PUNCT
ejpam-5623	474	1	for	for	ADP
ejpam-5623	474	2	each	each	DET
ejpam-5623	474	3	u	u	PROPN
ejpam-5623	474	4	∈	∈	PROPN
ejpam-5623	474	5	b	b	NOUN
ejpam-5623	474	6	\	\	PROPN
ejpam-5623	474	7	bw	bw	NOUN
ejpam-5623	474	8	,	,	PUNCT
ejpam-5623	474	9	we	we	PRON
ejpam-5623	474	10	choose	choose	VERB
ejpam-5623	474	11	and	and	CCONJ
ejpam-5623	474	12	fix	fix	VERB
ejpam-5623	474	13	u′	u′	PROPN
ejpam-5623	474	14	∈	∈	NOUN
ejpam-5623	474	15	v	v	ADP
ejpam-5623	474	16	such	such	ADJ
ejpam-5623	474	17	that	that	PRON
ejpam-5623	474	18	uα	uα	PROPN
ejpam-5623	474	19	=	=	SYM
ejpam-5623	474	20	u′α2	u′α2	PROPN
ejpam-5623	474	21	.	.	PUNCT
ejpam-5623	475	1	for	for	ADP
ejpam-5623	475	2	each	each	DET
ejpam-5623	475	3	u	u	PROPN
ejpam-5623	475	4	∈	∈	PROPN
ejpam-5623	475	5	bw	bw	NOUN
ejpam-5623	475	6	,	,	PUNCT
ejpam-5623	475	7	we	we	PRON
ejpam-5623	475	8	set	set	VERB
ejpam-5623	475	9	u′	u′	PROPN
ejpam-5623	475	10	=	=	SYM
ejpam-5623	475	11	u.	u.	PROPN
ejpam-5623	475	12	then	then	ADV
ejpam-5623	475	13	,	,	PUNCT
ejpam-5623	475	14	uαα	uαα	VERB
ejpam-5623	475	15	=	=	PUNCT
ejpam-5623	475	16	uα	uα	PROPN
ejpam-5623	475	17	=	=	SYM
ejpam-5623	475	18	u′α	u′α	PROPN
ejpam-5623	475	19	.	.	PUNCT
ejpam-5623	475	20	define	define	VERB
ejpam-5623	475	21	β	β	X
ejpam-5623	475	22	:	:	PUNCT
ejpam-5623	475	23	b	b	X
ejpam-5623	475	24	→	→	SYM
ejpam-5623	475	25	v	v	NOUN
ejpam-5623	475	26	by	by	ADP
ejpam-5623	475	27	vβ	vβ	X
ejpam-5623	475	28	=	=	PUNCT
ejpam-5623	475	29	v′	v′	NOUN
ejpam-5623	475	30	for	for	ADP
ejpam-5623	475	31	all	all	DET
ejpam-5623	475	32	v	v	PROPN
ejpam-5623	475	33	∈	∈	PROPN
ejpam-5623	475	34	b.	b.	NOUN
ejpam-5623	476	1	it	it	PRON
ejpam-5623	476	2	is	be	AUX
ejpam-5623	476	3	easy	easy	ADJ
ejpam-5623	476	4	to	to	PART
ejpam-5623	476	5	verify	verify	VERB
ejpam-5623	476	6	that	that	SCONJ
ejpam-5623	476	7	β	β	NOUN
ejpam-5623	476	8	is	be	AUX
ejpam-5623	476	9	well	well	ADV
ejpam-5623	476	10	-	-	PUNCT
ejpam-5623	476	11	defined	define	VERB
ejpam-5623	476	12	and	and	CCONJ
ejpam-5623	476	13	can	can	AUX
ejpam-5623	476	14	be	be	AUX
ejpam-5623	476	15	extended	extend	VERB
ejpam-5623	476	16	to	to	ADP
ejpam-5623	476	17	a	a	DET
ejpam-5623	476	18	linear	linear	ADJ
ejpam-5623	476	19	transformation	transformation	NOUN
ejpam-5623	476	20	on	on	ADP
ejpam-5623	476	21	v	v	NUM
ejpam-5623	476	22	.	.	PUNCT
ejpam-5623	477	1	we	we	PRON
ejpam-5623	477	2	will	will	AUX
ejpam-5623	477	3	now	now	ADV
ejpam-5623	477	4	show	show	VERB
ejpam-5623	477	5	that	that	SCONJ
ejpam-5623	477	6	β	β	PROPN
ejpam-5623	477	7	∈	∈	PROPN
ejpam-5623	477	8	fix(v	fix(v	PROPN
ejpam-5623	477	9	,	,	PUNCT
ejpam-5623	477	10	w	w	NOUN
ejpam-5623	477	11	)	)	PUNCT
ejpam-5623	477	12	,	,	PUNCT
ejpam-5623	477	13	and	and	CCONJ
ejpam-5623	477	14	that	that	SCONJ
ejpam-5623	477	15	α	α	X
ejpam-5623	477	16	=	=	PUNCT
ejpam-5623	477	17	βα2	βα2	PROPN
ejpam-5623	477	18	.	.	PUNCT
ejpam-5623	478	1	for	for	ADP
ejpam-5623	478	2	each	each	DET
ejpam-5623	478	3	w	w	PROPN
ejpam-5623	478	4	∈	∈	PROPN
ejpam-5623	478	5	w	w	NOUN
ejpam-5623	478	6	,	,	PUNCT
ejpam-5623	478	7	we	we	PRON
ejpam-5623	478	8	have	have	VERB
ejpam-5623	478	9	w	w	NOUN
ejpam-5623	478	10	=	=	PUNCT
ejpam-5623	478	11	a1w1+a2w2	a1w1+a2w2	NOUN
ejpam-5623	478	12	+	+	PROPN
ejpam-5623	478	13	.	.	PUNCT
ejpam-5623	478	14	.	.	PUNCT
ejpam-5623	479	1	.+anwn	.+anwn	PUNCT
ejpam-5623	480	1	for	for	ADP
ejpam-5623	480	2	some	some	DET
ejpam-5623	480	3	w1	w1	NOUN
ejpam-5623	480	4	,	,	PUNCT
ejpam-5623	480	5	w2	w2	NOUN
ejpam-5623	480	6	,	,	PUNCT
ejpam-5623	480	7	.	.	PUNCT
ejpam-5623	480	8	.	.	PUNCT
ejpam-5623	480	9	.	.	PUNCT
ejpam-5623	481	1	,	,	PUNCT
ejpam-5623	481	2	wn	wn	PROPN
ejpam-5623	481	3	∈	∈	PROPN
ejpam-5623	481	4	bw	bw	PROPN
ejpam-5623	481	5	,	,	PUNCT
ejpam-5623	481	6	and	and	CCONJ
ejpam-5623	481	7	scalars	scalar	VERB
ejpam-5623	481	8	a1	a1	PROPN
ejpam-5623	481	9	,	,	PUNCT
ejpam-5623	481	10	a2	a2	PROPN
ejpam-5623	481	11	,	,	PUNCT
ejpam-5623	481	12	.	.	PUNCT
ejpam-5623	481	13	.	.	PUNCT
ejpam-5623	482	1	.	.	PUNCT
ejpam-5623	483	1	,	,	PUNCT
ejpam-5623	483	2	an	an	DET
ejpam-5623	483	3	∈	∈	PROPN
ejpam-5623	483	4	f.	f.	PROPN
ejpam-5623	483	5	therefore	therefore	ADV
ejpam-5623	483	6	,	,	PUNCT
ejpam-5623	483	7	wβ	wβ	ADP
ejpam-5623	483	8	=	=	SYM
ejpam-5623	483	9	(	(	PUNCT
ejpam-5623	483	10	a1w1	a1w1	X
ejpam-5623	483	11	+	+	X
ejpam-5623	483	12	a2w2	a2w2	X
ejpam-5623	483	13	+	+	X
ejpam-5623	483	14	.	.	PUNCT
ejpam-5623	483	15	.	.	PUNCT
ejpam-5623	484	1	.+	.+	NOUN
ejpam-5623	484	2	anwn)β	anwn)β	PROPN
ejpam-5623	484	3	=	=	SYM
ejpam-5623	484	4	a1(w1β	a1(w1β	PROPN
ejpam-5623	484	5	)	)	PUNCT
ejpam-5623	484	6	+	+	NUM
ejpam-5623	484	7	a2(w2β	a2(w2β	ADV
ejpam-5623	484	8	)	)	PUNCT
ejpam-5623	484	9	+	+	CCONJ
ejpam-5623	484	10	.	.	PUNCT
ejpam-5623	484	11	.	.	PUNCT
ejpam-5623	485	1	.+	.+	NOUN
ejpam-5623	485	2	an(wnβ	an(wnβ	PROPN
ejpam-5623	485	3	)	)	PUNCT
ejpam-5623	485	4	n.	n.	NOUN
ejpam-5623	485	5	sawatraksa	sawatraksa	NOUN
ejpam-5623	485	6	,	,	PUNCT
ejpam-5623	485	7	p.	p.	NOUN
ejpam-5623	485	8	tantong	tantong	NOUN
ejpam-5623	485	9	/	/	SYM
ejpam-5623	485	10	eur	eur	PROPN
ejpam-5623	485	11	.	.	PUNCT
ejpam-5623	486	1	j.	j.	PROPN
ejpam-5623	486	2	pure	pure	PROPN
ejpam-5623	486	3	appl	appl	PROPN
ejpam-5623	486	4	.	.	PROPN
ejpam-5623	486	5	math	math	PROPN
ejpam-5623	486	6	,	,	PUNCT
ejpam-5623	486	7	18	18	NUM
ejpam-5623	486	8	(	(	PUNCT
ejpam-5623	486	9	1	1	NUM
ejpam-5623	486	10	)	)	PUNCT
ejpam-5623	486	11	(	(	PUNCT
ejpam-5623	486	12	2025	2025	NUM
ejpam-5623	486	13	)	)	PUNCT
ejpam-5623	486	14	,	,	PUNCT
ejpam-5623	486	15	5623	5623	NUM
ejpam-5623	486	16	12	12	NUM
ejpam-5623	486	17	of	of	ADP
ejpam-5623	486	18	15	15	NUM
ejpam-5623	486	19	=	=	SYM
ejpam-5623	486	20	a1w	a1w	PROPN
ejpam-5623	487	1	′	′	NUM
ejpam-5623	487	2	1	1	NUM
ejpam-5623	488	1	+	+	CCONJ
ejpam-5623	488	2	a2w	a2w	PROPN
ejpam-5623	488	3	′	′	NOUN
ejpam-5623	488	4	2	2	NUM
ejpam-5623	488	5	+	+	CCONJ
ejpam-5623	488	6	.	.	PUNCT
ejpam-5623	488	7	.	.	PUNCT
ejpam-5623	489	1	.+	.+	NOUN
ejpam-5623	489	2	anw	anw	VERB
ejpam-5623	490	1	′	′	NUM
ejpam-5623	491	1	n	n	NOUN
ejpam-5623	491	2	=	=	PUNCT
ejpam-5623	491	3	a1w1	a1w1	PROPN
ejpam-5623	491	4	+	+	NOUN
ejpam-5623	491	5	a2w2	a2w2	X
ejpam-5623	491	6	+	+	X
ejpam-5623	491	7	.	.	PUNCT
ejpam-5623	491	8	.	.	PUNCT
ejpam-5623	492	1	.+	.+	NOUN
ejpam-5623	492	2	anwn	anwn	VERB
ejpam-5623	492	3	=	=	SYM
ejpam-5623	492	4	w.	w.	PROPN
ejpam-5623	492	5	thus	thus	ADV
ejpam-5623	492	6	,	,	PUNCT
ejpam-5623	492	7	α	α	PROPN
ejpam-5623	492	8	∈	∈	PROPN
ejpam-5623	492	9	fix(v	fix(v	PROPN
ejpam-5623	492	10	,	,	PUNCT
ejpam-5623	492	11	w	w	PROPN
ejpam-5623	492	12	)	)	PUNCT
ejpam-5623	492	13	.	.	PUNCT
ejpam-5623	493	1	finally	finally	ADV
ejpam-5623	493	2	,	,	PUNCT
ejpam-5623	493	3	for	for	ADP
ejpam-5623	493	4	every	every	DET
ejpam-5623	493	5	v	v	PROPN
ejpam-5623	493	6	∈	∈	PROPN
ejpam-5623	493	7	b	b	NOUN
ejpam-5623	493	8	,	,	PUNCT
ejpam-5623	493	9	we	we	PRON
ejpam-5623	493	10	have	have	VERB
ejpam-5623	493	11	vβα2	vβα2	NOUN
ejpam-5623	493	12	=	=	SYM
ejpam-5623	493	13	vβαα	vβαα	NOUN
ejpam-5623	493	14	=	=	PUNCT
ejpam-5623	494	1	v′αα	v′αα	NOUN
ejpam-5623	494	2	=	=	SYM
ejpam-5623	494	3	vα	vα	PROPN
ejpam-5623	494	4	.	.	PUNCT
ejpam-5623	494	5	therefore	therefore	ADV
ejpam-5623	494	6	βα2	βα2	ADV
ejpam-5623	495	1	=	=	SYM
ejpam-5623	495	2	α	α	PROPN
ejpam-5623	495	3	,	,	PUNCT
ejpam-5623	495	4	which	which	PRON
ejpam-5623	495	5	shows	show	VERB
ejpam-5623	495	6	that	that	SCONJ
ejpam-5623	495	7	α	α	PRON
ejpam-5623	495	8	is	be	AUX
ejpam-5623	495	9	left	leave	VERB
ejpam-5623	495	10	regular	regular	ADV
ejpam-5623	495	11	,	,	PUNCT
ejpam-5623	495	12	as	as	SCONJ
ejpam-5623	495	13	required	require	VERB
ejpam-5623	495	14	.	.	PUNCT
ejpam-5623	496	1	next	next	ADJ
ejpam-5623	496	2	corollary	corollary	NOUN
ejpam-5623	496	3	is	be	AUX
ejpam-5623	496	4	result	result	NOUN
ejpam-5623	496	5	from	from	ADP
ejpam-5623	496	6	theorems	theorem	NOUN
ejpam-5623	496	7	7	7	NUM
ejpam-5623	496	8	and	and	CCONJ
ejpam-5623	496	9	9	9	NUM
ejpam-5623	496	10	.	.	PUNCT
ejpam-5623	496	11	corollary	corollary	ADJ
ejpam-5623	496	12	5	5	NUM
ejpam-5623	496	13	.	.	PUNCT
ejpam-5623	497	1	let	let	VERB
ejpam-5623	497	2	α	α	PRON
ejpam-5623	497	3	∈	∈	PROPN
ejpam-5623	497	4	fix(v	fix(v	PROPN
ejpam-5623	497	5	,	,	PUNCT
ejpam-5623	497	6	w	w	PROPN
ejpam-5623	497	7	)	)	PUNCT
ejpam-5623	497	8	.	.	PUNCT
ejpam-5623	498	1	the	the	DET
ejpam-5623	498	2	following	follow	VERB
ejpam-5623	498	3	statements	statement	NOUN
ejpam-5623	498	4	are	be	AUX
ejpam-5623	498	5	equivalent	equivalent	ADJ
ejpam-5623	498	6	:	:	PUNCT
ejpam-5623	498	7	(	(	PUNCT
ejpam-5623	498	8	i	i	NOUN
ejpam-5623	498	9	)	)	PUNCT
ejpam-5623	498	10	α	α	PROPN
ejpam-5623	498	11	∈	∈	PROPN
ejpam-5623	498	12	creg(fix(v	creg(fix(v	NOUN
ejpam-5623	498	13	,	,	PUNCT
ejpam-5623	498	14	w	w	NOUN
ejpam-5623	498	15	)	)	PUNCT
ejpam-5623	498	16	)	)	PUNCT
ejpam-5623	498	17	.	.	PUNCT
ejpam-5623	499	1	(	(	PUNCT
ejpam-5623	499	2	ii	ii	NOUN
ejpam-5623	499	3	)	)	PUNCT
ejpam-5623	499	4	α|v	α|v	VERB
ejpam-5623	500	1	α	α	NOUN
ejpam-5623	500	2	:	:	PUNCT
ejpam-5623	500	3	v	v	ADP
ejpam-5623	500	4	α	α	PROPN
ejpam-5623	500	5	→	→	SYM
ejpam-5623	500	6	v	v	NUM
ejpam-5623	500	7	α	α	NOUN
ejpam-5623	500	8	is	be	AUX
ejpam-5623	500	9	a	a	DET
ejpam-5623	500	10	bijective	bijective	ADJ
ejpam-5623	500	11	tranformation	tranformation	NOUN
ejpam-5623	500	12	.	.	PUNCT
ejpam-5623	501	1	(	(	PUNCT
ejpam-5623	501	2	iii	iii	NOUN
ejpam-5623	501	3	)	)	PUNCT
ejpam-5623	501	4	for	for	ADP
ejpam-5623	501	5	every	every	DET
ejpam-5623	501	6	v	v	NUM
ejpam-5623	501	7	∈	∈	NOUN
ejpam-5623	501	8	v	v	NOUN
ejpam-5623	501	9	,	,	PUNCT
ejpam-5623	501	10	there	there	PRON
ejpam-5623	501	11	exists	exist	VERB
ejpam-5623	501	12	a	a	DET
ejpam-5623	501	13	unique	unique	ADJ
ejpam-5623	501	14	v′	v′	NOUN
ejpam-5623	501	15	∈	∈	NOUN
ejpam-5623	501	16	v	v	ADP
ejpam-5623	501	17	α	α	PRON
ejpam-5623	501	18	such	such	ADJ
ejpam-5623	501	19	that	that	DET
ejpam-5623	501	20	vα	vα	ADP
ejpam-5623	501	21	=	=	PUNCT
ejpam-5623	502	1	v′α	v′α	ADJ
ejpam-5623	502	2	.	.	PUNCT
ejpam-5623	503	1	(	(	PUNCT
ejpam-5623	503	2	iv	iv	X
ejpam-5623	503	3	)	)	PUNCT
ejpam-5623	503	4	for	for	ADP
ejpam-5623	503	5	every	every	DET
ejpam-5623	503	6	basis	basis	NOUN
ejpam-5623	503	7	b	b	NOUN
ejpam-5623	503	8	for	for	ADP
ejpam-5623	503	9	v	v	NOUN
ejpam-5623	503	10	,	,	PUNCT
ejpam-5623	503	11	and	and	CCONJ
ejpam-5623	503	12	for	for	ADP
ejpam-5623	503	13	every	every	DET
ejpam-5623	503	14	v	v	NUM
ejpam-5623	503	15	∈	∈	PROPN
ejpam-5623	503	16	b	b	NOUN
ejpam-5623	503	17	,	,	PUNCT
ejpam-5623	503	18	there	there	PRON
ejpam-5623	503	19	exists	exist	VERB
ejpam-5623	503	20	a	a	DET
ejpam-5623	503	21	unique	unique	ADJ
ejpam-5623	503	22	v′	v′	NOUN
ejpam-5623	503	23	∈	∈	NOUN
ejpam-5623	503	24	v	v	ADP
ejpam-5623	503	25	α	α	PRON
ejpam-5623	504	1	such	such	ADJ
ejpam-5623	504	2	that	that	DET
ejpam-5623	504	3	vα	vα	ADP
ejpam-5623	504	4	=	=	PUNCT
ejpam-5623	504	5	v′α	v′α	ADJ
ejpam-5623	504	6	.	.	PUNCT
ejpam-5623	505	1	(	(	PUNCT
ejpam-5623	505	2	v	v	X
ejpam-5623	505	3	)	)	PUNCT
ejpam-5623	505	4	there	there	PRON
ejpam-5623	505	5	exist	exist	VERB
ejpam-5623	505	6	two	two	NUM
ejpam-5623	505	7	basses	bass	NOUN
ejpam-5623	505	8	b	b	NOUN
ejpam-5623	505	9	for	for	ADP
ejpam-5623	505	10	v	v	NOUN
ejpam-5623	505	11	,	,	PUNCT
ejpam-5623	505	12	and	and	CCONJ
ejpam-5623	505	13	bw	bw	X
ejpam-5623	505	14	for	for	ADP
ejpam-5623	505	15	w	w	ADP
ejpam-5623	505	16	such	such	ADJ
ejpam-5623	505	17	that	that	DET
ejpam-5623	505	18	bw	bw	PROPN
ejpam-5623	505	19	⊆	⊆	NUM
ejpam-5623	505	20	b	b	NOUN
ejpam-5623	505	21	,	,	PUNCT
ejpam-5623	505	22	and	and	CCONJ
ejpam-5623	505	23	for	for	ADP
ejpam-5623	505	24	every	every	DET
ejpam-5623	505	25	v	v	NUM
ejpam-5623	505	26	∈	∈	PROPN
ejpam-5623	505	27	b	b	NOUN
ejpam-5623	505	28	,	,	PUNCT
ejpam-5623	505	29	there	there	PRON
ejpam-5623	505	30	exists	exist	VERB
ejpam-5623	505	31	a	a	DET
ejpam-5623	505	32	unique	unique	ADJ
ejpam-5623	505	33	v′	v′	NOUN
ejpam-5623	505	34	∈	∈	NOUN
ejpam-5623	505	35	v	v	ADP
ejpam-5623	505	36	α	α	PRON
ejpam-5623	506	1	such	such	ADJ
ejpam-5623	506	2	that	that	DET
ejpam-5623	506	3	vα	vα	ADP
ejpam-5623	506	4	=	=	PUNCT
ejpam-5623	506	5	v′α	v′α	ADJ
ejpam-5623	506	6	.	.	PUNCT
ejpam-5623	507	1	finally	finally	ADV
ejpam-5623	507	2	,	,	PUNCT
ejpam-5623	507	3	we	we	PRON
ejpam-5623	507	4	show	show	VERB
ejpam-5623	507	5	that	that	SCONJ
ejpam-5623	507	6	for	for	SCONJ
ejpam-5623	507	7	the	the	DET
ejpam-5623	507	8	semigroup	semigroup	PROPN
ejpam-5623	507	9	fix(v	fix(v	PROPN
ejpam-5623	507	10	,	,	PUNCT
ejpam-5623	507	11	w	w	NOUN
ejpam-5623	507	12	)	)	PUNCT
ejpam-5623	507	13	to	to	PART
ejpam-5623	507	14	be	be	AUX
ejpam-5623	507	15	left	leave	VERB
ejpam-5623	507	16	regular	regular	ADV
ejpam-5623	507	17	whenever	whenever	SCONJ
ejpam-5623	507	18	v	v	NOUN
ejpam-5623	507	19	is	be	AUX
ejpam-5623	507	20	a	a	DET
ejpam-5623	507	21	finite	finite	ADJ
ejpam-5623	507	22	dimensional	dimensional	ADJ
ejpam-5623	507	23	vector	vector	NOUN
ejpam-5623	507	24	space	space	NOUN
ejpam-5623	507	25	.	.	PUNCT
ejpam-5623	508	1	theorem	theorem	ADJ
ejpam-5623	508	2	10	10	NUM
ejpam-5623	508	3	.	.	PUNCT
ejpam-5623	509	1	let	let	VERB
ejpam-5623	509	2	v	v	PART
ejpam-5623	509	3	be	be	AUX
ejpam-5623	509	4	a	a	DET
ejpam-5623	509	5	finite	finite	ADJ
ejpam-5623	509	6	dimensional	dimensional	ADJ
ejpam-5623	509	7	vector	vector	NOUN
ejpam-5623	509	8	space	space	NOUN
ejpam-5623	509	9	.	.	PUNCT
ejpam-5623	510	1	the	the	DET
ejpam-5623	510	2	following	follow	VERB
ejpam-5623	510	3	statements	statement	NOUN
ejpam-5623	510	4	are	be	AUX
ejpam-5623	510	5	equivalent	equivalent	ADJ
ejpam-5623	510	6	:	:	PUNCT
ejpam-5623	510	7	(	(	PUNCT
ejpam-5623	510	8	i	i	NOUN
ejpam-5623	510	9	)	)	PUNCT
ejpam-5623	510	10	fix(v	fix(v	PROPN
ejpam-5623	510	11	,	,	PUNCT
ejpam-5623	510	12	w	w	PROPN
ejpam-5623	510	13	)	)	PUNCT
ejpam-5623	510	14	is	be	AUX
ejpam-5623	510	15	a	a	DET
ejpam-5623	510	16	left	left	ADJ
ejpam-5623	510	17	regular	regular	ADJ
ejpam-5623	510	18	semigroup	semigroup	NOUN
ejpam-5623	510	19	.	.	PUNCT
ejpam-5623	511	1	(	(	PUNCT
ejpam-5623	511	2	ii	ii	X
ejpam-5623	511	3	)	)	PUNCT
ejpam-5623	511	4	lreg(fix(v	lreg(fix(v	PROPN
ejpam-5623	511	5	,	,	PUNCT
ejpam-5623	511	6	w	w	NOUN
ejpam-5623	511	7	)	)	PUNCT
ejpam-5623	511	8	)	)	PUNCT
ejpam-5623	511	9	is	be	AUX
ejpam-5623	511	10	a	a	DET
ejpam-5623	511	11	subsemigroup	subsemigroup	NOUN
ejpam-5623	511	12	of	of	ADP
ejpam-5623	511	13	fix(v	fix(v	PROPN
ejpam-5623	511	14	,	,	PUNCT
ejpam-5623	511	15	w	w	PROPN
ejpam-5623	511	16	)	)	PUNCT
ejpam-5623	511	17	.	.	PUNCT
ejpam-5623	512	1	(	(	PUNCT
ejpam-5623	512	2	iii	iii	X
ejpam-5623	512	3	)	)	PUNCT
ejpam-5623	512	4	v	v	NOUN
ejpam-5623	512	5	=	=	SYM
ejpam-5623	512	6	w	w	PROPN
ejpam-5623	512	7	or	or	CCONJ
ejpam-5623	512	8	dim(v	dim(v	ADJ
ejpam-5623	512	9	)	)	PUNCT
ejpam-5623	512	10	=	=	PUNCT
ejpam-5623	512	11	dim(w	dim(w	X
ejpam-5623	512	12	)	)	PUNCT
ejpam-5623	513	1	+	+	CCONJ
ejpam-5623	513	2	1	1	X
ejpam-5623	513	3	.	.	X
ejpam-5623	513	4	proof	proof	NOUN
ejpam-5623	513	5	.	.	PUNCT
ejpam-5623	514	1	(	(	PUNCT
ejpam-5623	514	2	i	i	NOUN
ejpam-5623	514	3	)	)	PUNCT
ejpam-5623	514	4	⇒	⇒	PROPN
ejpam-5623	514	5	(	(	PUNCT
ejpam-5623	514	6	ii	ii	NOUN
ejpam-5623	514	7	)	)	PUNCT
ejpam-5623	514	8	this	this	PRON
ejpam-5623	514	9	is	be	AUX
ejpam-5623	514	10	clear	clear	ADJ
ejpam-5623	514	11	by	by	ADP
ejpam-5623	514	12	definition	definition	NOUN
ejpam-5623	514	13	.	.	PUNCT
ejpam-5623	515	1	(	(	PUNCT
ejpam-5623	515	2	ii	ii	NOUN
ejpam-5623	515	3	)	)	PUNCT
ejpam-5623	515	4	⇒	⇒	NOUN
ejpam-5623	515	5	(	(	PUNCT
ejpam-5623	515	6	iii	iii	X
ejpam-5623	515	7	)	)	PUNCT
ejpam-5623	515	8	suppose	suppose	VERB
ejpam-5623	515	9	that	that	SCONJ
ejpam-5623	515	10	w	w	PROPN
ejpam-5623	515	11	̸=	̸=	PROPN
ejpam-5623	515	12	v	v	NOUN
ejpam-5623	515	13	and	and	CCONJ
ejpam-5623	515	14	dim(v	dim(v	PROPN
ejpam-5623	515	15	)	)	PUNCT
ejpam-5623	515	16	̸=	̸=	PROPN
ejpam-5623	515	17	dim(w	dim(w	PROPN
ejpam-5623	515	18	)	)	PUNCT
ejpam-5623	516	1	+	+	CCONJ
ejpam-5623	516	2	1	1	X
ejpam-5623	516	3	.	.	PUNCT
ejpam-5623	516	4	then	then	ADV
ejpam-5623	516	5	,	,	PUNCT
ejpam-5623	516	6	dim(v	dim(v	PROPN
ejpam-5623	516	7	)	)	PUNCT
ejpam-5623	516	8	−	−	PROPN
ejpam-5623	516	9	dim(w	dim(w	X
ejpam-5623	516	10	)	)	PUNCT
ejpam-5623	516	11	>	>	X
ejpam-5623	517	1	1	1	X
ejpam-5623	517	2	.	.	PUNCT
ejpam-5623	517	3	let	let	VERB
ejpam-5623	517	4	bw	bw	PART
ejpam-5623	517	5	be	be	AUX
ejpam-5623	517	6	a	a	DET
ejpam-5623	517	7	basis	basis	NOUN
ejpam-5623	517	8	for	for	ADP
ejpam-5623	517	9	w	w	NOUN
ejpam-5623	517	10	and	and	CCONJ
ejpam-5623	517	11	extend	extend	VERB
ejpam-5623	517	12	it	it	PRON
ejpam-5623	517	13	to	to	ADP
ejpam-5623	517	14	a	a	DET
ejpam-5623	517	15	basis	basis	NOUN
ejpam-5623	517	16	b	b	NOUN
ejpam-5623	517	17	for	for	ADP
ejpam-5623	517	18	v	v	NOUN
ejpam-5623	517	19	.	.	PUNCT
ejpam-5623	518	1	let	let	VERB
ejpam-5623	518	2	a	a	PRON
ejpam-5623	518	3	and	and	CCONJ
ejpam-5623	518	4	b	b	NOUN
ejpam-5623	518	5	be	be	AUX
ejpam-5623	518	6	distinct	distinct	ADJ
ejpam-5623	518	7	elements	element	NOUN
ejpam-5623	518	8	of	of	ADP
ejpam-5623	518	9	b	b	PROPN
ejpam-5623	518	10	\bw	\bw	PROPN
ejpam-5623	518	11	.	.	PUNCT
ejpam-5623	519	1	define	define	VERB
ejpam-5623	519	2	two	two	NUM
ejpam-5623	519	3	mappings	mapping	NOUN
ejpam-5623	519	4	α	α	NOUN
ejpam-5623	519	5	and	and	CCONJ
ejpam-5623	519	6	β	β	NOUN
ejpam-5623	519	7	from	from	ADP
ejpam-5623	519	8	b	b	PROPN
ejpam-5623	519	9	into	into	ADP
ejpam-5623	519	10	v	v	NOUN
ejpam-5623	519	11	as	as	SCONJ
ejpam-5623	519	12	follows	follow	VERB
ejpam-5623	519	13	:	:	PUNCT
ejpam-5623	519	14	xα	xα	PROPN
ejpam-5623	520	1	=	=	PUNCT
ejpam-5623	521	1			PUNCT
ejpam-5623	521	2	a	a	PRON
ejpam-5623	521	3	if	if	NOUN
ejpam-5623	521	4	x	x	PROPN
ejpam-5623	521	5	=	=	SYM
ejpam-5623	521	6	b	b	PROPN
ejpam-5623	521	7	,	,	PUNCT
ejpam-5623	521	8	b	b	NOUN
ejpam-5623	522	1	if	if	SCONJ
ejpam-5623	522	2	x	x	PROPN
ejpam-5623	522	3	=	=	SYM
ejpam-5623	522	4	a	a	DET
ejpam-5623	522	5	,	,	PUNCT
ejpam-5623	522	6	x	x	X
ejpam-5623	522	7	otherwise	otherwise	ADV
ejpam-5623	522	8	,	,	PUNCT
ejpam-5623	522	9	and	and	CCONJ
ejpam-5623	522	10	xβ	xβ	ADV
ejpam-5623	522	11	=	=	PRON
ejpam-5623	522	12	{	{	PUNCT
ejpam-5623	522	13	x	x	X
ejpam-5623	522	14	if	if	SCONJ
ejpam-5623	522	15	x	x	SYM
ejpam-5623	522	16	∈	∈	PROPN
ejpam-5623	522	17	bw	bw	NOUN
ejpam-5623	522	18	∪	∪	PROPN
ejpam-5623	522	19	{	{	PUNCT
ejpam-5623	522	20	a	a	NOUN
ejpam-5623	522	21	}	}	PUNCT
ejpam-5623	522	22	,	,	PUNCT
ejpam-5623	522	23	0	0	NUM
ejpam-5623	522	24	otherwise	otherwise	ADV
ejpam-5623	522	25	.	.	PUNCT
ejpam-5623	523	1	n.	n.	PROPN
ejpam-5623	523	2	sawatraksa	sawatraksa	PROPN
ejpam-5623	523	3	,	,	PUNCT
ejpam-5623	523	4	p.	p.	NOUN
ejpam-5623	523	5	tantong	tantong	NOUN
ejpam-5623	523	6	/	/	SYM
ejpam-5623	523	7	eur	eur	PROPN
ejpam-5623	523	8	.	.	PUNCT
ejpam-5623	524	1	j.	j.	PROPN
ejpam-5623	524	2	pure	pure	PROPN
ejpam-5623	524	3	appl	appl	PROPN
ejpam-5623	524	4	.	.	PROPN
ejpam-5623	524	5	math	math	PROPN
ejpam-5623	524	6	,	,	PUNCT
ejpam-5623	524	7	18	18	NUM
ejpam-5623	524	8	(	(	PUNCT
ejpam-5623	524	9	1	1	NUM
ejpam-5623	524	10	)	)	PUNCT
ejpam-5623	524	11	(	(	PUNCT
ejpam-5623	524	12	2025	2025	NUM
ejpam-5623	524	13	)	)	PUNCT
ejpam-5623	524	14	,	,	PUNCT
ejpam-5623	524	15	5623	5623	NUM
ejpam-5623	524	16	13	13	NUM
ejpam-5623	524	17	of	of	ADP
ejpam-5623	524	18	15	15	NUM
ejpam-5623	524	19	clearly	clearly	ADV
ejpam-5623	524	20	,	,	PUNCT
ejpam-5623	524	21	both	both	CCONJ
ejpam-5623	524	22	α	α	NOUN
ejpam-5623	524	23	and	and	CCONJ
ejpam-5623	524	24	β	β	X
ejpam-5623	524	25	are	be	AUX
ejpam-5623	524	26	well	well	ADV
ejpam-5623	524	27	-	-	PUNCT
ejpam-5623	524	28	defined	define	VERB
ejpam-5623	524	29	and	and	CCONJ
ejpam-5623	524	30	can	can	AUX
ejpam-5623	524	31	be	be	AUX
ejpam-5623	524	32	extended	extend	VERB
ejpam-5623	524	33	to	to	ADP
ejpam-5623	524	34	the	the	DET
ejpam-5623	524	35	linear	linear	ADJ
ejpam-5623	524	36	transformation	transformation	NOUN
ejpam-5623	524	37	on	on	ADP
ejpam-5623	524	38	v	v	NOUN
ejpam-5623	524	39	.	.	PUNCT
ejpam-5623	525	1	let	let	VERB
ejpam-5623	525	2	w	w	PROPN
ejpam-5623	525	3	∈	∈	PROPN
ejpam-5623	525	4	w	w	PROPN
ejpam-5623	525	5	.	.	PUNCT
ejpam-5623	526	1	then	then	ADV
ejpam-5623	526	2	,	,	PUNCT
ejpam-5623	526	3	w	w	NOUN
ejpam-5623	526	4	=	=	PUNCT
ejpam-5623	526	5	k1w1	k1w1	PROPN
ejpam-5623	526	6	+	+	CCONJ
ejpam-5623	526	7	k2w2	k2w2	X
ejpam-5623	526	8	+	+	X
ejpam-5623	526	9	.	.	PUNCT
ejpam-5623	526	10	.	.	PUNCT
ejpam-5623	527	1	.+	.+	NOUN
ejpam-5623	527	2	knwn	knwn	VERB
ejpam-5623	527	3	for	for	ADP
ejpam-5623	527	4	some	some	DET
ejpam-5623	527	5	k1	k1	NOUN
ejpam-5623	527	6	,	,	PUNCT
ejpam-5623	527	7	k2	k2	NOUN
ejpam-5623	527	8	,	,	PUNCT
ejpam-5623	527	9	.	.	PUNCT
ejpam-5623	527	10	.	.	PUNCT
ejpam-5623	527	11	.	.	PUNCT
ejpam-5623	528	1	,	,	PUNCT
ejpam-5623	528	2	kn	kn	PROPN
ejpam-5623	528	3	∈	∈	PROPN
ejpam-5623	528	4	f	f	PROPN
ejpam-5623	528	5	and	and	CCONJ
ejpam-5623	528	6	w1	w1	NOUN
ejpam-5623	528	7	,	,	PUNCT
ejpam-5623	528	8	w2	w2	NOUN
ejpam-5623	528	9	,	,	PUNCT
ejpam-5623	528	10	.	.	PUNCT
ejpam-5623	528	11	.	.	PUNCT
ejpam-5623	528	12	.	.	PUNCT
ejpam-5623	529	1	,	,	PUNCT
ejpam-5623	529	2	wn	wn	PROPN
ejpam-5623	529	3	∈	∈	PROPN
ejpam-5623	529	4	b.	b.	PROPN
ejpam-5623	529	5	therefore	therefore	ADV
ejpam-5623	529	6	,	,	PUNCT
ejpam-5623	529	7	wα	wα	NOUN
ejpam-5623	529	8	=	=	PUNCT
ejpam-5623	529	9	(	(	PUNCT
ejpam-5623	529	10	k1w1	k1w1	X
ejpam-5623	529	11	+	+	X
ejpam-5623	529	12	k2w2	k2w2	X
ejpam-5623	529	13	+	+	X
ejpam-5623	529	14	.	.	PUNCT
ejpam-5623	529	15	.	.	PUNCT
ejpam-5623	530	1	.+	.+	NOUN
ejpam-5623	530	2	knwn)α	knwn)α	PROPN
ejpam-5623	531	1	=	=	PRON
ejpam-5623	531	2	(	(	PUNCT
ejpam-5623	531	3	k1w1)α+	k1w1)α+	NOUN
ejpam-5623	531	4	(	(	PUNCT
ejpam-5623	531	5	k2w2	k2w2	X
ejpam-5623	531	6	)	)	PUNCT
ejpam-5623	531	7	+	+	CCONJ
ejpam-5623	531	8	.	.	PUNCT
ejpam-5623	531	9	.	.	PUNCT
ejpam-5623	532	1	.+	.+	NOUN
ejpam-5623	532	2	(	(	PUNCT
ejpam-5623	532	3	knwn	knwn	NOUN
ejpam-5623	532	4	)	)	PUNCT
ejpam-5623	532	5	=	=	PUNCT
ejpam-5623	533	1	k1w1α+	k1w1α+	PROPN
ejpam-5623	533	2	k2w2α+	k2w2α+	INTJ
ejpam-5623	533	3	.	.	PUNCT
ejpam-5623	533	4	.	.	PUNCT
ejpam-5623	534	1	.+	.+	NOUN
ejpam-5623	534	2	knwnα	knwnα	NOUN
ejpam-5623	534	3	=	=	PUNCT
ejpam-5623	535	1	k1w1	k1w1	X
ejpam-5623	535	2	+	+	CCONJ
ejpam-5623	535	3	k2w2	k2w2	X
ejpam-5623	535	4	+	+	X
ejpam-5623	535	5	.	.	PUNCT
ejpam-5623	535	6	.	.	PUNCT
ejpam-5623	536	1	.+	.+	NOUN
ejpam-5623	536	2	knwn	knwn	NOUN
ejpam-5623	536	3	=	=	SYM
ejpam-5623	536	4	w.	w.	PROPN
ejpam-5623	536	5	hence	hence	ADV
ejpam-5623	536	6	,	,	PUNCT
ejpam-5623	536	7	α	α	PROPN
ejpam-5623	536	8	∈	∈	PROPN
ejpam-5623	536	9	fix(v	fix(v	PROPN
ejpam-5623	536	10	,	,	PUNCT
ejpam-5623	536	11	w	w	NOUN
ejpam-5623	536	12	)	)	PUNCT
ejpam-5623	536	13	.	.	PUNCT
ejpam-5623	537	1	by	by	ADP
ejpam-5623	537	2	the	the	DET
ejpam-5623	537	3	symmetry	symmetry	NOUN
ejpam-5623	537	4	,	,	PUNCT
ejpam-5623	537	5	we	we	PRON
ejpam-5623	537	6	can	can	AUX
ejpam-5623	537	7	show	show	VERB
ejpam-5623	537	8	that	that	SCONJ
ejpam-5623	537	9	β	β	PROPN
ejpam-5623	537	10	∈	∈	PROPN
ejpam-5623	537	11	fix(v	fix(v	PROPN
ejpam-5623	537	12	,	,	PUNCT
ejpam-5623	537	13	w	w	PROPN
ejpam-5623	537	14	)	)	PUNCT
ejpam-5623	537	15	.	.	PUNCT
ejpam-5623	538	1	it	it	PRON
ejpam-5623	538	2	follows	follow	VERB
ejpam-5623	538	3	from	from	ADP
ejpam-5623	538	4	b	b	PROPN
ejpam-5623	538	5	⊆	⊆	NUM
ejpam-5623	538	6	v	v	ADP
ejpam-5623	538	7	α	α	NOUN
ejpam-5623	538	8	and	and	CCONJ
ejpam-5623	538	9	theorem	theorem	VERB
ejpam-5623	538	10	9	9	NUM
ejpam-5623	538	11	that	that	SCONJ
ejpam-5623	538	12	α	α	NOUN
ejpam-5623	538	13	is	be	AUX
ejpam-5623	538	14	left	leave	VERB
ejpam-5623	538	15	regular	regular	ADV
ejpam-5623	538	16	.	.	PUNCT
ejpam-5623	539	1	let	let	VERB
ejpam-5623	539	2	v	v	NUM
ejpam-5623	539	3	∈	∈	PROPN
ejpam-5623	539	4	b.	b.	NOUN
ejpam-5623	540	1	if	if	SCONJ
ejpam-5623	540	2	v	v	NUM
ejpam-5623	540	3	∈	∈	PROPN
ejpam-5623	540	4	bw	bw	NOUN
ejpam-5623	540	5	∪	∪	X
ejpam-5623	540	6	{	{	PUNCT
ejpam-5623	540	7	a	a	NOUN
ejpam-5623	540	8	}	}	PUNCT
ejpam-5623	540	9	,	,	PUNCT
ejpam-5623	540	10	then	then	ADV
ejpam-5623	540	11	vβ	vβ	ADP
ejpam-5623	540	12	=	=	SYM
ejpam-5623	540	13	v	v	PROPN
ejpam-5623	540	14	=	=	SYM
ejpam-5623	540	15	vβ2	vβ2	NOUN
ejpam-5623	540	16	.	.	PUNCT
ejpam-5623	541	1	otherwise	otherwise	ADV
ejpam-5623	541	2	,	,	PUNCT
ejpam-5623	541	3	vβ	vβ	ADP
ejpam-5623	541	4	=	=	NOUN
ejpam-5623	541	5	0	0	PUNCT
ejpam-5623	542	1	=	=	SYM
ejpam-5623	542	2	(	(	PUNCT
ejpam-5623	542	3	vβ)β	vβ)β	PROPN
ejpam-5623	542	4	.	.	PUNCT
ejpam-5623	543	1	this	this	PRON
ejpam-5623	543	2	implies	imply	VERB
ejpam-5623	543	3	that	that	SCONJ
ejpam-5623	543	4	β	β	PROPN
ejpam-5623	543	5	is	be	AUX
ejpam-5623	543	6	a	a	DET
ejpam-5623	543	7	left	left	ADJ
ejpam-5623	543	8	regular	regular	ADJ
ejpam-5623	543	9	element	element	NOUN
ejpam-5623	543	10	of	of	ADP
ejpam-5623	543	11	fix(v	fix(v	PROPN
ejpam-5623	543	12	,	,	PUNCT
ejpam-5623	543	13	w	w	NOUN
ejpam-5623	543	14	)	)	PUNCT
ejpam-5623	543	15	by	by	ADP
ejpam-5623	543	16	theorem	theorem	NOUN
ejpam-5623	543	17	9	9	NUM
ejpam-5623	543	18	.	.	PUNCT
ejpam-5623	544	1	finally	finally	ADV
ejpam-5623	544	2	,	,	PUNCT
ejpam-5623	544	3	we	we	PRON
ejpam-5623	544	4	will	will	AUX
ejpam-5623	544	5	show	show	VERB
ejpam-5623	544	6	that	that	SCONJ
ejpam-5623	544	7	βα	βα	PRON
ejpam-5623	544	8	is	be	AUX
ejpam-5623	544	9	not	not	PART
ejpam-5623	544	10	left	leave	VERB
ejpam-5623	544	11	regular	regular	ADV
ejpam-5623	544	12	.	.	PUNCT
ejpam-5623	545	1	note	note	VERB
ejpam-5623	545	2	that	that	SCONJ
ejpam-5623	545	3	aβα	aβα	NOUN
ejpam-5623	545	4	=	=	SYM
ejpam-5623	545	5	aα	aα	NOUN
ejpam-5623	545	6	=	=	PROPN
ejpam-5623	545	7	b.	b.	PROPN
ejpam-5623	546	1	thus	thus	ADV
ejpam-5623	546	2	,	,	PUNCT
ejpam-5623	546	3	b	b	PROPN
ejpam-5623	546	4	∈	∈	PROPN
ejpam-5623	546	5	v	v	ADP
ejpam-5623	546	6	βα	βα	PROPN
ejpam-5623	546	7	.	.	PUNCT
ejpam-5623	546	8	claim	claim	VERB
ejpam-5623	546	9	that	that	SCONJ
ejpam-5623	546	10	b	b	X
ejpam-5623	546	11	̸=	̸=	PROPN
ejpam-5623	546	12	vβα	vβα	PROPN
ejpam-5623	546	13	for	for	ADP
ejpam-5623	546	14	all	all	DET
ejpam-5623	546	15	v	v	ADP
ejpam-5623	546	16	∈	∈	PRON
ejpam-5623	546	17	v	v	NOUN
ejpam-5623	546	18	βα	βα	PROPN
ejpam-5623	546	19	.	.	PUNCT
ejpam-5623	546	20	suppose	suppose	VERB
ejpam-5623	546	21	that	that	SCONJ
ejpam-5623	546	22	vβα	vβα	PROPN
ejpam-5623	546	23	=	=	SYM
ejpam-5623	546	24	b	b	PROPN
ejpam-5623	546	25	for	for	ADP
ejpam-5623	546	26	some	some	DET
ejpam-5623	546	27	v	v	NOUN
ejpam-5623	546	28	∈	∈	NUM
ejpam-5623	546	29	v	v	ADP
ejpam-5623	546	30	βα	βα	PROPN
ejpam-5623	546	31	.	.	PUNCT
ejpam-5623	547	1	then	then	ADV
ejpam-5623	547	2	,	,	PUNCT
ejpam-5623	547	3	v	v	NOUN
ejpam-5623	547	4	=	=	PUNCT
ejpam-5623	547	5	v′βα	v′βα	NOUN
ejpam-5623	547	6	for	for	ADP
ejpam-5623	547	7	some	some	DET
ejpam-5623	547	8	v′	v′	NOUN
ejpam-5623	547	9	∈	∈	PROPN
ejpam-5623	547	10	v	v	NOUN
ejpam-5623	547	11	.	.	PUNCT
ejpam-5623	548	1	thus	thus	ADV
ejpam-5623	548	2	,	,	PUNCT
ejpam-5623	548	3	v′	v′	X
ejpam-5623	548	4	=	=	SYM
ejpam-5623	548	5	a1v1+a2v2	a1v1+a2v2	PROPN
ejpam-5623	548	6	+	+	PROPN
ejpam-5623	548	7	.	.	PUNCT
ejpam-5623	548	8	.	.	PUNCT
ejpam-5623	549	1	.+anvn	.+anvn	PUNCT
ejpam-5623	550	1	where	where	SCONJ
ejpam-5623	550	2	v1	v1	NOUN
ejpam-5623	550	3	,	,	PUNCT
ejpam-5623	550	4	v2	v2	NOUN
ejpam-5623	550	5	,	,	PUNCT
ejpam-5623	550	6	.	.	PUNCT
ejpam-5623	550	7	.	.	PUNCT
ejpam-5623	551	1	.	.	PUNCT
ejpam-5623	552	1	,	,	PUNCT
ejpam-5623	552	2	vn	vn	PROPN
ejpam-5623	552	3	∈	∈	PROPN
ejpam-5623	552	4	b	b	PROPN
ejpam-5623	552	5	and	and	CCONJ
ejpam-5623	552	6	a1	a1	PROPN
ejpam-5623	552	7	,	,	PUNCT
ejpam-5623	552	8	a2	a2	PROPN
ejpam-5623	552	9	,	,	PUNCT
ejpam-5623	552	10	.	.	PUNCT
ejpam-5623	552	11	.	.	PUNCT
ejpam-5623	553	1	.	.	PUNCT
ejpam-5623	554	1	,	,	PUNCT
ejpam-5623	554	2	an	an	DET
ejpam-5623	554	3	∈	∈	PROPN
ejpam-5623	554	4	f.	f.	NOUN
ejpam-5623	555	1	if	if	SCONJ
ejpam-5623	555	2	v′	v′	PROPN
ejpam-5623	555	3	∈	∈	PROPN
ejpam-5623	555	4	w	w	NOUN
ejpam-5623	555	5	,	,	PUNCT
ejpam-5623	555	6	then	then	ADV
ejpam-5623	555	7	v	v	NOUN
ejpam-5623	555	8	=	=	NOUN
ejpam-5623	555	9	v′βα	v′βα	NOUN
ejpam-5623	555	10	=	=	SYM
ejpam-5623	555	11	v′	v′	NOUN
ejpam-5623	555	12	,	,	PUNCT
ejpam-5623	555	13	and	and	CCONJ
ejpam-5623	555	14	so	so	ADV
ejpam-5623	555	15	b	b	X
ejpam-5623	555	16	=	=	SYM
ejpam-5623	555	17	vβα	vβα	PROPN
ejpam-5623	555	18	=	=	SYM
ejpam-5623	555	19	v	v	ADP
ejpam-5623	555	20	∈	∈	PROPN
ejpam-5623	555	21	w	w	NOUN
ejpam-5623	555	22	,	,	PUNCT
ejpam-5623	555	23	which	which	PRON
ejpam-5623	555	24	is	be	AUX
ejpam-5623	555	25	a	a	DET
ejpam-5623	555	26	contradiction	contradiction	NOUN
ejpam-5623	555	27	.	.	PUNCT
ejpam-5623	556	1	hence	hence	ADV
ejpam-5623	556	2	,	,	PUNCT
ejpam-5623	556	3	v′	v′	PROPN
ejpam-5623	556	4	/∈	/∈	PUNCT
ejpam-5623	557	1	w	w	INTJ
ejpam-5623	557	2	.	.	PUNCT
ejpam-5623	558	1	since	since	SCONJ
ejpam-5623	558	2	b	b	PROPN
ejpam-5623	558	3	̸=	̸=	PROPN
ejpam-5623	558	4	0	0	NUM
ejpam-5623	558	5	and	and	CCONJ
ejpam-5623	558	6	vβα	vβα	PROPN
ejpam-5623	558	7	=	=	SYM
ejpam-5623	558	8	b	b	PROPN
ejpam-5623	558	9	,	,	PUNCT
ejpam-5623	558	10	we	we	PRON
ejpam-5623	558	11	get	get	VERB
ejpam-5623	558	12	that	that	PRON
ejpam-5623	558	13	v	v	ADP
ejpam-5623	558	14	̸=	̸=	PROPN
ejpam-5623	558	15	0	0	NUM
ejpam-5623	558	16	,	,	PUNCT
ejpam-5623	558	17	so	so	SCONJ
ejpam-5623	558	18	that	that	SCONJ
ejpam-5623	558	19	0	0	NUM
ejpam-5623	558	20	̸=	̸=	NOUN
ejpam-5623	558	21	v′βα	v′βα	NOUN
ejpam-5623	558	22	=	=	PUNCT
ejpam-5623	558	23	a1v1βα	a1v1βα	NOUN
ejpam-5623	558	24	+	+	CCONJ
ejpam-5623	558	25	a2v2βα	a2v2βα	NOUN
ejpam-5623	558	26	+	+	X
ejpam-5623	558	27	.	.	PUNCT
ejpam-5623	558	28	.	.	PUNCT
ejpam-5623	558	29	.	.	PUNCT
ejpam-5623	559	1	+	+	CCONJ
ejpam-5623	560	1	anvnβα	anvnβα	VERB
ejpam-5623	560	2	.	.	PUNCT
ejpam-5623	561	1	by	by	ADP
ejpam-5623	561	2	the	the	DET
ejpam-5623	561	3	definition	definition	NOUN
ejpam-5623	561	4	of	of	ADP
ejpam-5623	561	5	β	β	X
ejpam-5623	561	6	,	,	PUNCT
ejpam-5623	561	7	there	there	PRON
ejpam-5623	561	8	exists	exist	VERB
ejpam-5623	561	9	k	k	PROPN
ejpam-5623	561	10	∈	∈	PROPN
ejpam-5623	561	11	{	{	PUNCT
ejpam-5623	561	12	1	1	NUM
ejpam-5623	561	13	,	,	PUNCT
ejpam-5623	561	14	2	2	NUM
ejpam-5623	561	15	,	,	PUNCT
ejpam-5623	561	16	.	.	PUNCT
ejpam-5623	561	17	.	.	PUNCT
ejpam-5623	562	1	.	.	PUNCT
ejpam-5623	563	1	,	,	PUNCT
ejpam-5623	563	2	n	n	CCONJ
ejpam-5623	563	3	}	}	PUNCT
ejpam-5623	563	4	such	such	ADJ
ejpam-5623	563	5	that	that	DET
ejpam-5623	563	6	vkβ	vkβ	NOUN
ejpam-5623	563	7	=	=	PUNCT
ejpam-5623	563	8	a	a	PRON
ejpam-5623	563	9	and	and	CCONJ
ejpam-5623	563	10	so	so	ADV
ejpam-5623	563	11	v	v	ADJ
ejpam-5623	563	12	=	=	NOUN
ejpam-5623	563	13	akvkβα	akvkβα	NOUN
ejpam-5623	563	14	=	=	PUNCT
ejpam-5623	563	15	akaα	akaα	ADJ
ejpam-5623	563	16	=	=	PUNCT
ejpam-5623	563	17	akb	akb	PROPN
ejpam-5623	563	18	.	.	PUNCT
ejpam-5623	564	1	therefore	therefore	ADV
ejpam-5623	564	2	,	,	PUNCT
ejpam-5623	564	3	b	b	X
ejpam-5623	564	4	=	=	SYM
ejpam-5623	564	5	vβα	vβα	PROPN
ejpam-5623	565	1	=	=	PUNCT
ejpam-5623	565	2	akbβα	akbβα	NOUN
ejpam-5623	565	3	=	=	SYM
ejpam-5623	565	4	ak(0α	ak(0α	NOUN
ejpam-5623	565	5	)	)	PUNCT
ejpam-5623	566	1	=	=	SYM
ejpam-5623	566	2	0	0	NUM
ejpam-5623	566	3	,	,	PUNCT
ejpam-5623	566	4	which	which	PRON
ejpam-5623	566	5	is	be	AUX
ejpam-5623	566	6	a	a	DET
ejpam-5623	566	7	contradiction	contradiction	NOUN
ejpam-5623	566	8	.	.	PUNCT
ejpam-5623	567	1	so	so	ADV
ejpam-5623	567	2	we	we	PRON
ejpam-5623	567	3	have	have	VERB
ejpam-5623	567	4	the	the	DET
ejpam-5623	567	5	claim	claim	NOUN
ejpam-5623	567	6	.	.	PUNCT
ejpam-5623	568	1	hence	hence	ADV
ejpam-5623	568	2	,	,	PUNCT
ejpam-5623	568	3	βα|v	βα|v	X
ejpam-5623	568	4	βα	βα	X
ejpam-5623	568	5	is	be	AUX
ejpam-5623	568	6	not	not	PART
ejpam-5623	568	7	onto	onto	ADP
ejpam-5623	568	8	.	.	PUNCT
ejpam-5623	569	1	from	from	ADP
ejpam-5623	569	2	theorem	theorem	NOUN
ejpam-5623	569	3	9	9	NUM
ejpam-5623	569	4	,	,	PUNCT
ejpam-5623	569	5	βα	βα	PRON
ejpam-5623	569	6	is	be	AUX
ejpam-5623	569	7	not	not	PART
ejpam-5623	569	8	left	leave	VERB
ejpam-5623	569	9	regular	regular	ADV
ejpam-5623	569	10	.	.	PUNCT
ejpam-5623	570	1	hence	hence	ADV
ejpam-5623	570	2	,	,	PUNCT
ejpam-5623	570	3	lreg(fix(v	lreg(fix(v	PROPN
ejpam-5623	570	4	,	,	PUNCT
ejpam-5623	570	5	w	w	NOUN
ejpam-5623	570	6	)	)	PUNCT
ejpam-5623	570	7	)	)	PUNCT
ejpam-5623	570	8	is	be	AUX
ejpam-5623	570	9	not	not	PART
ejpam-5623	570	10	a	a	DET
ejpam-5623	570	11	subsemigroup	subsemigroup	NOUN
ejpam-5623	570	12	of	of	ADP
ejpam-5623	570	13	fix(v	fix(v	PROPN
ejpam-5623	570	14	,	,	PUNCT
ejpam-5623	570	15	w	w	PROPN
ejpam-5623	570	16	)	)	PUNCT
ejpam-5623	570	17	.	.	PUNCT
ejpam-5623	571	1	(	(	PUNCT
ejpam-5623	571	2	iii	iii	X
ejpam-5623	571	3	)	)	PUNCT
ejpam-5623	571	4	⇒	⇒	NOUN
ejpam-5623	571	5	(	(	PUNCT
ejpam-5623	571	6	i	i	NOUN
ejpam-5623	571	7	)	)	PUNCT
ejpam-5623	571	8	assume	assume	VERB
ejpam-5623	571	9	that	that	SCONJ
ejpam-5623	571	10	w	w	PROPN
ejpam-5623	571	11	=	=	SYM
ejpam-5623	571	12	v	v	NOUN
ejpam-5623	571	13	or	or	CCONJ
ejpam-5623	571	14	dim(w	dim(w	ADJ
ejpam-5623	571	15	)	)	PUNCT
ejpam-5623	572	1	+	+	CCONJ
ejpam-5623	572	2	1	1	NUM
ejpam-5623	572	3	=	=	SYM
ejpam-5623	572	4	dim(v	dim(v	PROPN
ejpam-5623	572	5	)	)	PUNCT
ejpam-5623	572	6	.	.	PUNCT
ejpam-5623	573	1	if	if	SCONJ
ejpam-5623	573	2	w	w	PROPN
ejpam-5623	573	3	=	=	SYM
ejpam-5623	573	4	v	v	NOUN
ejpam-5623	573	5	,	,	PUNCT
ejpam-5623	573	6	then	then	ADV
ejpam-5623	573	7	fix(v	fix(v	PROPN
ejpam-5623	573	8	,	,	PUNCT
ejpam-5623	573	9	w	w	PROPN
ejpam-5623	573	10	)	)	PUNCT
ejpam-5623	573	11	contains	contain	VERB
ejpam-5623	573	12	only	only	ADV
ejpam-5623	573	13	the	the	DET
ejpam-5623	573	14	identity	identity	NOUN
ejpam-5623	573	15	transformation	transformation	NOUN
ejpam-5623	573	16	and	and	CCONJ
ejpam-5623	573	17	is	be	AUX
ejpam-5623	573	18	trivially	trivially	ADV
ejpam-5623	573	19	left	leave	VERB
ejpam-5623	573	20	regular	regular	ADV
ejpam-5623	573	21	.	.	PUNCT
ejpam-5623	574	1	suppose	suppose	VERB
ejpam-5623	574	2	that	that	SCONJ
ejpam-5623	574	3	dim(w	dim(w	NOUN
ejpam-5623	574	4	)	)	PUNCT
ejpam-5623	575	1	+	+	CCONJ
ejpam-5623	575	2	1	1	NUM
ejpam-5623	575	3	=	=	SYM
ejpam-5623	575	4	dim(v	dim(v	PROPN
ejpam-5623	575	5	)	)	PUNCT
ejpam-5623	575	6	.	.	PUNCT
ejpam-5623	576	1	let	let	VERB
ejpam-5623	576	2	α	α	PRON
ejpam-5623	576	3	∈	∈	PROPN
ejpam-5623	576	4	fix(v	fix(v	PROPN
ejpam-5623	576	5	,	,	PUNCT
ejpam-5623	576	6	w	w	PROPN
ejpam-5623	576	7	)	)	PUNCT
ejpam-5623	576	8	.	.	PUNCT
ejpam-5623	577	1	then	then	ADV
ejpam-5623	577	2	,	,	PUNCT
ejpam-5623	577	3	by	by	ADP
ejpam-5623	577	4	theorem	theorem	NOUN
ejpam-5623	577	5	9	9	NUM
ejpam-5623	577	6	,	,	PUNCT
ejpam-5623	577	7	we	we	PRON
ejpam-5623	577	8	get	get	VERB
ejpam-5623	577	9	that	that	DET
ejpam-5623	577	10	α|v	α|v	NOUN
ejpam-5623	578	1	α	α	NOUN
ejpam-5623	578	2	:	:	PUNCT
ejpam-5623	578	3	v	v	ADP
ejpam-5623	578	4	α	α	PROPN
ejpam-5623	578	5	→	→	SYM
ejpam-5623	578	6	v	v	NUM
ejpam-5623	578	7	α	α	NOUN
ejpam-5623	578	8	is	be	AUX
ejpam-5623	578	9	onto	onto	ADP
ejpam-5623	578	10	.	.	PUNCT
ejpam-5623	579	1	by	by	ADP
ejpam-5623	579	2	the	the	DET
ejpam-5623	579	3	dimension	dimension	NOUN
ejpam-5623	579	4	theorem	theorem	NOUN
ejpam-5623	579	5	and	and	CCONJ
ejpam-5623	579	6	assumptions	assumption	NOUN
ejpam-5623	579	7	,	,	PUNCT
ejpam-5623	579	8	we	we	PRON
ejpam-5623	579	9	have	have	VERB
ejpam-5623	579	10	dim(w	dim(w	NOUN
ejpam-5623	579	11	)	)	PUNCT
ejpam-5623	580	1	+	+	CCONJ
ejpam-5623	580	2	1	1	NUM
ejpam-5623	580	3	=	=	SYM
ejpam-5623	580	4	dim(kerα	dim(kerα	PROPN
ejpam-5623	580	5	)	)	PUNCT
ejpam-5623	581	1	+	+	NUM
ejpam-5623	581	2	dim(v	dim(v	PROPN
ejpam-5623	581	3	α	α	NOUN
ejpam-5623	581	4	)	)	PUNCT
ejpam-5623	581	5	.	.	PUNCT
ejpam-5623	582	1	since	since	SCONJ
ejpam-5623	582	2	v	v	NOUN
ejpam-5623	582	3	α	α	PROPN
ejpam-5623	582	4	contains	contain	VERB
ejpam-5623	582	5	the	the	DET
ejpam-5623	582	6	subspace	subspace	NOUN
ejpam-5623	582	7	w	w	PROPN
ejpam-5623	582	8	,	,	PUNCT
ejpam-5623	582	9	clarifying	clarify	VERB
ejpam-5623	582	10	this	this	PRON
ejpam-5623	582	11	will	will	AUX
ejpam-5623	582	12	help	help	VERB
ejpam-5623	582	13	in	in	ADP
ejpam-5623	582	14	explaining	explain	VERB
ejpam-5623	582	15	the	the	DET
ejpam-5623	582	16	two	two	NUM
ejpam-5623	582	17	cases	case	NOUN
ejpam-5623	582	18	.	.	PUNCT
ejpam-5623	583	1	case	case	NOUN
ejpam-5623	583	2	1	1	NUM
ejpam-5623	583	3	.	.	X
ejpam-5623	584	1	dim(w	dim(w	NOUN
ejpam-5623	584	2	)	)	PUNCT
ejpam-5623	585	1	=	=	PUNCT
ejpam-5623	585	2	dim(v	dim(v	PROPN
ejpam-5623	585	3	α	α	NOUN
ejpam-5623	585	4	)	)	PUNCT
ejpam-5623	585	5	and	and	CCONJ
ejpam-5623	585	6	dim(kerα	dim(kerα	NOUN
ejpam-5623	585	7	)	)	PUNCT
ejpam-5623	585	8	=	=	SYM
ejpam-5623	585	9	1	1	X
ejpam-5623	585	10	.	.	PUNCT
ejpam-5623	585	11	then	then	ADV
ejpam-5623	585	12	,	,	PUNCT
ejpam-5623	585	13	v	v	ADP
ejpam-5623	585	14	α	α	NOUN
ejpam-5623	585	15	=	=	SYM
ejpam-5623	585	16	w	w	PROPN
ejpam-5623	585	17	.	.	PUNCT
ejpam-5623	586	1	since	since	SCONJ
ejpam-5623	586	2	α|w	α|w	PROPN
ejpam-5623	586	3	is	be	AUX
ejpam-5623	586	4	the	the	DET
ejpam-5623	586	5	identity	identity	NOUN
ejpam-5623	586	6	transformation	transformation	NOUN
ejpam-5623	586	7	on	on	ADP
ejpam-5623	586	8	w	w	PROPN
ejpam-5623	586	9	,	,	PUNCT
ejpam-5623	586	10	by	by	ADP
ejpam-5623	586	11	theorem	theorem	NOUN
ejpam-5623	586	12	9	9	NUM
ejpam-5623	586	13	,	,	PUNCT
ejpam-5623	586	14	we	we	PRON
ejpam-5623	586	15	have	have	VERB
ejpam-5623	586	16	α	α	PROPN
ejpam-5623	586	17	∈	∈	PROPN
ejpam-5623	586	18	lreg(fix(v	lreg(fix(v	PROPN
ejpam-5623	586	19	,	,	PUNCT
ejpam-5623	586	20	w	w	NOUN
ejpam-5623	586	21	)	)	PUNCT
ejpam-5623	586	22	)	)	PUNCT
ejpam-5623	586	23	.	.	PUNCT
ejpam-5623	587	1	case	case	NOUN
ejpam-5623	587	2	2	2	NUM
ejpam-5623	587	3	.	.	X
ejpam-5623	587	4	dim(w	dim(w	NOUN
ejpam-5623	587	5	)	)	PUNCT
ejpam-5623	588	1	+	+	CCONJ
ejpam-5623	588	2	1	1	NUM
ejpam-5623	588	3	=	=	SYM
ejpam-5623	588	4	dim(v	dim(v	PROPN
ejpam-5623	588	5	α	α	NOUN
ejpam-5623	588	6	)	)	PUNCT
ejpam-5623	588	7	and	and	CCONJ
ejpam-5623	588	8	dim(kerα	dim(kerα	NOUN
ejpam-5623	588	9	)	)	PUNCT
ejpam-5623	588	10	=	=	SYM
ejpam-5623	589	1	0	0	X
ejpam-5623	589	2	.	.	PUNCT
ejpam-5623	590	1	then	then	ADV
ejpam-5623	590	2	,	,	PUNCT
ejpam-5623	590	3	v	v	ADP
ejpam-5623	590	4	α	α	NOUN
ejpam-5623	590	5	=	=	NOUN
ejpam-5623	590	6	v	v	NOUN
ejpam-5623	590	7	.	.	PUNCT
ejpam-5623	591	1	therefore	therefore	ADV
ejpam-5623	591	2	,	,	PUNCT
ejpam-5623	591	3	α	α	PROPN
ejpam-5623	591	4	is	be	AUX
ejpam-5623	591	5	onto	onto	ADP
ejpam-5623	591	6	.	.	PUNCT
ejpam-5623	592	1	hence	hence	ADV
ejpam-5623	592	2	,	,	PUNCT
ejpam-5623	592	3	by	by	ADP
ejpam-5623	592	4	theorem	theorem	NOUN
ejpam-5623	592	5	9	9	NUM
ejpam-5623	592	6	,	,	PUNCT
ejpam-5623	592	7	we	we	PRON
ejpam-5623	592	8	have	have	VERB
ejpam-5623	592	9	α	α	PROPN
ejpam-5623	592	10	∈	∈	PROPN
ejpam-5623	592	11	lreg(fix(v	lreg(fix(v	PROPN
ejpam-5623	592	12	,	,	PUNCT
ejpam-5623	592	13	w	w	NOUN
ejpam-5623	592	14	)	)	PUNCT
ejpam-5623	592	15	)	)	PUNCT
ejpam-5623	592	16	.	.	PUNCT
ejpam-5623	593	1	hence	hence	ADV
ejpam-5623	593	2	,	,	PUNCT
ejpam-5623	593	3	fix(v	fix(v	PROPN
ejpam-5623	593	4	,	,	PUNCT
ejpam-5623	593	5	w	w	PROPN
ejpam-5623	593	6	)	)	PUNCT
ejpam-5623	593	7	is	be	AUX
ejpam-5623	593	8	a	a	DET
ejpam-5623	593	9	left	left	ADJ
ejpam-5623	593	10	regular	regular	ADJ
ejpam-5623	593	11	semigroup	semigroup	NOUN
ejpam-5623	593	12	,	,	PUNCT
ejpam-5623	593	13	as	as	SCONJ
ejpam-5623	593	14	required	require	VERB
ejpam-5623	593	15	.	.	PUNCT
ejpam-5623	594	1	the	the	DET
ejpam-5623	594	2	next	next	ADJ
ejpam-5623	594	3	corollary	corollary	NOUN
ejpam-5623	594	4	is	be	AUX
ejpam-5623	594	5	a	a	DET
ejpam-5623	594	6	result	result	NOUN
ejpam-5623	594	7	from	from	ADP
ejpam-5623	594	8	theorems	theorem	NOUN
ejpam-5623	594	9	8	8	NUM
ejpam-5623	594	10	and	and	CCONJ
ejpam-5623	594	11	10	10	NUM
ejpam-5623	594	12	.	.	PUNCT
ejpam-5623	595	1	corollary	corollary	ADJ
ejpam-5623	595	2	6	6	NUM
ejpam-5623	595	3	.	.	PUNCT
ejpam-5623	596	1	the	the	DET
ejpam-5623	596	2	following	follow	VERB
ejpam-5623	596	3	statements	statement	NOUN
ejpam-5623	596	4	are	be	AUX
ejpam-5623	596	5	equivalent	equivalent	ADJ
ejpam-5623	596	6	:	:	PUNCT
ejpam-5623	596	7	n.	n.	NOUN
ejpam-5623	596	8	sawatraksa	sawatraksa	NOUN
ejpam-5623	596	9	,	,	PUNCT
ejpam-5623	596	10	p.	p.	NOUN
ejpam-5623	596	11	tantong	tantong	NOUN
ejpam-5623	596	12	/	/	SYM
ejpam-5623	596	13	eur	eur	PROPN
ejpam-5623	596	14	.	.	PUNCT
ejpam-5623	597	1	j.	j.	PROPN
ejpam-5623	597	2	pure	pure	PROPN
ejpam-5623	597	3	appl	appl	PROPN
ejpam-5623	597	4	.	.	PROPN
ejpam-5623	597	5	math	math	PROPN
ejpam-5623	597	6	,	,	PUNCT
ejpam-5623	597	7	18	18	NUM
ejpam-5623	597	8	(	(	PUNCT
ejpam-5623	597	9	1	1	NUM
ejpam-5623	597	10	)	)	PUNCT
ejpam-5623	597	11	(	(	PUNCT
ejpam-5623	597	12	2025	2025	NUM
ejpam-5623	597	13	)	)	PUNCT
ejpam-5623	597	14	,	,	PUNCT
ejpam-5623	597	15	5623	5623	NUM
ejpam-5623	597	16	14	14	NUM
ejpam-5623	597	17	of	of	ADP
ejpam-5623	597	18	15	15	NUM
ejpam-5623	597	19	(	(	PUNCT
ejpam-5623	597	20	i	i	NOUN
ejpam-5623	597	21	)	)	PUNCT
ejpam-5623	597	22	fix(v	fix(v	PROPN
ejpam-5623	597	23	,	,	PUNCT
ejpam-5623	597	24	w	w	PROPN
ejpam-5623	597	25	)	)	PUNCT
ejpam-5623	597	26	is	be	AUX
ejpam-5623	597	27	a	a	DET
ejpam-5623	597	28	completely	completely	ADV
ejpam-5623	597	29	regular	regular	ADJ
ejpam-5623	597	30	semigroup	semigroup	NOUN
ejpam-5623	597	31	.	.	PUNCT
ejpam-5623	598	1	(	(	PUNCT
ejpam-5623	598	2	ii	ii	NOUN
ejpam-5623	598	3	)	)	PUNCT
ejpam-5623	598	4	creg(fix(v	creg(fix(v	NOUN
ejpam-5623	598	5	,	,	PUNCT
ejpam-5623	598	6	w	w	NOUN
ejpam-5623	598	7	)	)	PUNCT
ejpam-5623	598	8	)	)	PUNCT
ejpam-5623	598	9	is	be	AUX
ejpam-5623	598	10	a	a	DET
ejpam-5623	598	11	subsemigroup	subsemigroup	NOUN
ejpam-5623	598	12	of	of	ADP
ejpam-5623	598	13	fix(v	fix(v	PROPN
ejpam-5623	598	14	,	,	PUNCT
ejpam-5623	598	15	w	w	PROPN
ejpam-5623	598	16	)	)	PUNCT
ejpam-5623	598	17	.	.	PUNCT
ejpam-5623	599	1	(	(	PUNCT
ejpam-5623	599	2	iii	iii	X
ejpam-5623	599	3	)	)	PUNCT
ejpam-5623	599	4	v	v	NOUN
ejpam-5623	599	5	=	=	SYM
ejpam-5623	599	6	w	w	PROPN
ejpam-5623	599	7	or	or	CCONJ
ejpam-5623	599	8	dim(v	dim(v	PROPN
ejpam-5623	599	9	)	)	PUNCT
ejpam-5623	599	10	≤	≤	NUM
ejpam-5623	599	11	1	1	NUM
ejpam-5623	599	12	.	.	X
ejpam-5623	599	13	4	4	NUM
ejpam-5623	599	14	.	.	X
ejpam-5623	599	15	conclusions	conclusion	NOUN
ejpam-5623	599	16	in	in	ADP
ejpam-5623	599	17	this	this	DET
ejpam-5623	599	18	paper	paper	NOUN
ejpam-5623	599	19	,	,	PUNCT
ejpam-5623	599	20	we	we	PRON
ejpam-5623	599	21	have	have	AUX
ejpam-5623	599	22	investigated	investigate	VERB
ejpam-5623	599	23	the	the	DET
ejpam-5623	599	24	left	left	ADJ
ejpam-5623	599	25	,	,	PUNCT
ejpam-5623	599	26	right	right	ADJ
ejpam-5623	599	27	,	,	PUNCT
ejpam-5623	599	28	and	and	CCONJ
ejpam-5623	599	29	complete	complete	ADJ
ejpam-5623	599	30	regularity	regularity	NOUN
ejpam-5623	599	31	of	of	ADP
ejpam-5623	599	32	elements	element	NOUN
ejpam-5623	599	33	in	in	ADP
ejpam-5623	599	34	the	the	DET
ejpam-5623	599	35	semigroups	semigroup	NOUN
ejpam-5623	599	36	of	of	ADP
ejpam-5623	599	37	linear	linear	ADJ
ejpam-5623	599	38	transformations	transformation	NOUN
ejpam-5623	599	39	with	with	ADP
ejpam-5623	599	40	invariant	invariant	ADJ
ejpam-5623	599	41	subspaces	subspace	NOUN
ejpam-5623	599	42	.	.	PUNCT
ejpam-5623	600	1	specifically	specifically	ADV
ejpam-5623	600	2	,	,	PUNCT
ejpam-5623	600	3	we	we	PRON
ejpam-5623	600	4	focused	focus	VERB
ejpam-5623	600	5	on	on	ADP
ejpam-5623	600	6	the	the	DET
ejpam-5623	600	7	subsemigroups	subsemigroup	NOUN
ejpam-5623	600	8	l(v	l(v	PROPN
ejpam-5623	600	9	,	,	PUNCT
ejpam-5623	600	10	w	w	PROPN
ejpam-5623	600	11	)	)	PUNCT
ejpam-5623	600	12	,	,	PUNCT
ejpam-5623	600	13	s(v	s(v	PROPN
ejpam-5623	600	14	,	,	PUNCT
ejpam-5623	600	15	w	w	NOUN
ejpam-5623	600	16	)	)	PUNCT
ejpam-5623	600	17	,	,	PUNCT
ejpam-5623	600	18	q	q	NOUN
ejpam-5623	600	19	,	,	PUNCT
ejpam-5623	600	20	and	and	CCONJ
ejpam-5623	600	21	fix(v	fix(v	PROPN
ejpam-5623	600	22	,	,	PUNCT
ejpam-5623	600	23	w	w	NOUN
ejpam-5623	600	24	)	)	PUNCT
ejpam-5623	600	25	.	.	PUNCT
ejpam-5623	601	1	we	we	PRON
ejpam-5623	601	2	provided	provide	VERB
ejpam-5623	601	3	necessary	necessary	ADJ
ejpam-5623	601	4	and	and	CCONJ
ejpam-5623	601	5	sufficient	sufficient	ADJ
ejpam-5623	601	6	conditions	condition	NOUN
ejpam-5623	601	7	for	for	ADP
ejpam-5623	601	8	these	these	DET
ejpam-5623	601	9	semigroups	semigroup	NOUN
ejpam-5623	601	10	to	to	PART
ejpam-5623	601	11	exhibit	exhibit	VERB
ejpam-5623	601	12	left	leave	VERB
ejpam-5623	601	13	,	,	PUNCT
ejpam-5623	601	14	right	right	ADJ
ejpam-5623	601	15	,	,	PUNCT
ejpam-5623	601	16	and	and	CCONJ
ejpam-5623	601	17	complete	complete	ADJ
ejpam-5623	601	18	regularity	regularity	NOUN
ejpam-5623	601	19	.	.	PUNCT
ejpam-5623	602	1	our	our	PRON
ejpam-5623	602	2	results	result	NOUN
ejpam-5623	602	3	contribute	contribute	VERB
ejpam-5623	602	4	to	to	ADP
ejpam-5623	602	5	the	the	DET
ejpam-5623	602	6	characterization	characterization	NOUN
ejpam-5623	602	7	of	of	ADP
ejpam-5623	602	8	regular	regular	ADJ
ejpam-5623	602	9	elements	element	NOUN
ejpam-5623	602	10	within	within	ADP
ejpam-5623	602	11	these	these	DET
ejpam-5623	602	12	mathematical	mathematical	ADJ
ejpam-5623	602	13	structures	structure	NOUN
ejpam-5623	602	14	.	.	PUNCT
ejpam-5623	603	1	one	one	NUM
ejpam-5623	603	2	of	of	ADP
ejpam-5623	603	3	the	the	DET
ejpam-5623	603	4	key	key	ADJ
ejpam-5623	603	5	findings	finding	NOUN
ejpam-5623	603	6	is	be	AUX
ejpam-5623	603	7	the	the	DET
ejpam-5623	603	8	relationship	relationship	NOUN
ejpam-5623	603	9	between	between	ADP
ejpam-5623	603	10	the	the	DET
ejpam-5623	603	11	subspaces	subspace	NOUN
ejpam-5623	603	12	v	v	NOUN
ejpam-5623	603	13	and	and	CCONJ
ejpam-5623	603	14	w	w	NOUN
ejpam-5623	603	15	in	in	ADP
ejpam-5623	603	16	determining	determine	VERB
ejpam-5623	603	17	the	the	DET
ejpam-5623	603	18	regularity	regularity	NOUN
ejpam-5623	603	19	of	of	ADP
ejpam-5623	603	20	the	the	DET
ejpam-5623	603	21	transformations	transformation	NOUN
ejpam-5623	603	22	.	.	PUNCT
ejpam-5623	604	1	for	for	ADP
ejpam-5623	604	2	instance	instance	NOUN
ejpam-5623	604	3	,	,	PUNCT
ejpam-5623	604	4	we	we	PRON
ejpam-5623	604	5	have	have	AUX
ejpam-5623	604	6	shown	show	VERB
ejpam-5623	604	7	that	that	SCONJ
ejpam-5623	604	8	an	an	DET
ejpam-5623	604	9	element	element	NOUN
ejpam-5623	604	10	of	of	ADP
ejpam-5623	604	11	l(v	l(v	PROPN
ejpam-5623	604	12	,	,	PUNCT
ejpam-5623	604	13	w	w	PROPN
ejpam-5623	604	14	)	)	PUNCT
ejpam-5623	604	15	is	be	AUX
ejpam-5623	604	16	right	right	ADV
ejpam-5623	604	17	regular	regular	ADV
ejpam-5623	604	18	if	if	SCONJ
ejpam-5623	604	19	and	and	CCONJ
ejpam-5623	604	20	only	only	ADV
ejpam-5623	604	21	if	if	SCONJ
ejpam-5623	604	22	its	its	PRON
ejpam-5623	604	23	restriction	restriction	NOUN
ejpam-5623	604	24	to	to	ADP
ejpam-5623	604	25	its	its	PRON
ejpam-5623	604	26	image	image	NOUN
ejpam-5623	604	27	is	be	AUX
ejpam-5623	604	28	a	a	DET
ejpam-5623	604	29	one	one	NUM
ejpam-5623	604	30	-	-	PUNCT
ejpam-5623	604	31	to	to	ADP
ejpam-5623	604	32	-	-	PUNCT
ejpam-5623	604	33	one	one	NUM
ejpam-5623	604	34	transformation	transformation	NOUN
ejpam-5623	604	35	.	.	PUNCT
ejpam-5623	605	1	similarly	similarly	ADV
ejpam-5623	605	2	,	,	PUNCT
ejpam-5623	605	3	left	leave	VERB
ejpam-5623	605	4	regularity	regularity	NOUN
ejpam-5623	605	5	is	be	AUX
ejpam-5623	605	6	characterized	characterize	VERB
ejpam-5623	605	7	by	by	ADP
ejpam-5623	605	8	onto	onto	ADP
ejpam-5623	605	9	mappings	mapping	NOUN
ejpam-5623	605	10	in	in	ADP
ejpam-5623	605	11	certain	certain	ADJ
ejpam-5623	605	12	conditions	condition	NOUN
ejpam-5623	605	13	.	.	PUNCT
ejpam-5623	606	1	these	these	DET
ejpam-5623	606	2	results	result	NOUN
ejpam-5623	606	3	align	align	VERB
ejpam-5623	606	4	with	with	ADP
ejpam-5623	606	5	and	and	CCONJ
ejpam-5623	606	6	extend	extend	VERB
ejpam-5623	606	7	previous	previous	ADJ
ejpam-5623	606	8	work	work	NOUN
ejpam-5623	606	9	,	,	PUNCT
ejpam-5623	606	10	offering	offer	VERB
ejpam-5623	606	11	a	a	DET
ejpam-5623	606	12	broader	broad	ADJ
ejpam-5623	606	13	understanding	understanding	NOUN
ejpam-5623	606	14	of	of	ADP
ejpam-5623	606	15	regular	regular	ADJ
ejpam-5623	606	16	semigroups	semigroup	NOUN
ejpam-5623	606	17	and	and	CCONJ
ejpam-5623	606	18	their	their	PRON
ejpam-5623	606	19	elements	element	NOUN
ejpam-5623	606	20	.	.	PUNCT
ejpam-5623	607	1	the	the	DET
ejpam-5623	607	2	significance	significance	NOUN
ejpam-5623	607	3	of	of	ADP
ejpam-5623	607	4	this	this	DET
ejpam-5623	607	5	study	study	NOUN
ejpam-5623	607	6	lies	lie	VERB
ejpam-5623	607	7	in	in	ADP
ejpam-5623	607	8	its	its	PRON
ejpam-5623	607	9	application	application	NOUN
ejpam-5623	607	10	to	to	ADP
ejpam-5623	607	11	algebraic	algebraic	ADJ
ejpam-5623	607	12	structures	structure	NOUN
ejpam-5623	607	13	such	such	ADJ
ejpam-5623	607	14	as	as	ADP
ejpam-5623	607	15	semigroups	semigroup	NOUN
ejpam-5623	607	16	,	,	PUNCT
ejpam-5623	607	17	transformation	transformation	NOUN
ejpam-5623	607	18	semigroups	semigroup	NOUN
ejpam-5623	607	19	,	,	PUNCT
ejpam-5623	607	20	and	and	CCONJ
ejpam-5623	607	21	subspaces	subspace	NOUN
ejpam-5623	607	22	.	.	PUNCT
ejpam-5623	608	1	specifically	specifically	ADV
ejpam-5623	608	2	,	,	PUNCT
ejpam-5623	608	3	cayley	cayley	PROPN
ejpam-5623	608	4	’s	’s	PART
ejpam-5623	608	5	theorem	theorem	ADJ
ejpam-5623	608	6	states	state	NOUN
ejpam-5623	608	7	that	that	SCONJ
ejpam-5623	608	8	every	every	DET
ejpam-5623	608	9	semigroup	semigroup	NOUN
ejpam-5623	608	10	can	can	AUX
ejpam-5623	608	11	be	be	AUX
ejpam-5623	608	12	embedded	embed	VERB
ejpam-5623	608	13	in	in	ADP
ejpam-5623	608	14	a	a	DET
ejpam-5623	608	15	full	full	ADJ
ejpam-5623	608	16	transformation	transformation	NOUN
ejpam-5623	608	17	semigroup	semigroup	NOUN
ejpam-5623	608	18	.	.	PUNCT
ejpam-5623	609	1	our	our	PRON
ejpam-5623	609	2	characterization	characterization	NOUN
ejpam-5623	609	3	of	of	ADP
ejpam-5623	609	4	subsemigroups	subsemigroup	NOUN
ejpam-5623	609	5	of	of	ADP
ejpam-5623	609	6	transformation	transformation	NOUN
ejpam-5623	609	7	semigroups	semigroup	NOUN
ejpam-5623	609	8	provides	provide	VERB
ejpam-5623	609	9	deeper	deep	ADJ
ejpam-5623	609	10	insights	insight	NOUN
ejpam-5623	609	11	into	into	ADP
ejpam-5623	609	12	their	their	PRON
ejpam-5623	609	13	structure	structure	NOUN
ejpam-5623	609	14	,	,	PUNCT
ejpam-5623	609	15	allowing	allow	VERB
ejpam-5623	609	16	for	for	ADP
ejpam-5623	609	17	further	further	ADJ
ejpam-5623	609	18	exploration	exploration	NOUN
ejpam-5623	609	19	of	of	ADP
ejpam-5623	609	20	their	their	PRON
ejpam-5623	609	21	properties	property	NOUN
ejpam-5623	609	22	and	and	CCONJ
ejpam-5623	609	23	potential	potential	ADJ
ejpam-5623	609	24	applications	application	NOUN
ejpam-5623	609	25	.	.	PUNCT
ejpam-5623	610	1	in	in	ADP
ejpam-5623	610	2	future	future	ADJ
ejpam-5623	610	3	research	research	NOUN
ejpam-5623	610	4	,	,	PUNCT
ejpam-5623	610	5	this	this	DET
ejpam-5623	610	6	framework	framework	NOUN
ejpam-5623	610	7	can	can	AUX
ejpam-5623	610	8	be	be	AUX
ejpam-5623	610	9	expanded	expand	VERB
ejpam-5623	610	10	to	to	PART
ejpam-5623	610	11	explore	explore	VERB
ejpam-5623	610	12	other	other	ADJ
ejpam-5623	610	13	algebraic	algebraic	ADJ
ejpam-5623	610	14	structures	structure	NOUN
ejpam-5623	610	15	,	,	PUNCT
ejpam-5623	610	16	including	include	VERB
ejpam-5623	610	17	transformation	transformation	NOUN
ejpam-5623	610	18	semigroups	semigroup	NOUN
ejpam-5623	610	19	that	that	PRON
ejpam-5623	610	20	preserve	preserve	VERB
ejpam-5623	610	21	different	different	ADJ
ejpam-5623	610	22	types	type	NOUN
ejpam-5623	610	23	of	of	ADP
ejpam-5623	610	24	equivalence	equivalence	NOUN
ejpam-5623	610	25	relations	relation	NOUN
ejpam-5623	610	26	.	.	PUNCT
ejpam-5623	611	1	additionally	additionally	ADV
ejpam-5623	611	2	,	,	PUNCT
ejpam-5623	611	3	studying	study	VERB
ejpam-5623	611	4	the	the	DET
ejpam-5623	611	5	interaction	interaction	NOUN
ejpam-5623	611	6	between	between	ADP
ejpam-5623	611	7	these	these	DET
ejpam-5623	611	8	regular	regular	ADJ
ejpam-5623	611	9	elements	element	NOUN
ejpam-5623	611	10	and	and	CCONJ
ejpam-5623	611	11	more	more	ADV
ejpam-5623	611	12	complex	complex	ADJ
ejpam-5623	611	13	algebraic	algebraic	ADJ
ejpam-5623	611	14	operations	operation	NOUN
ejpam-5623	611	15	could	could	AUX
ejpam-5623	611	16	lead	lead	VERB
ejpam-5623	611	17	to	to	ADP
ejpam-5623	611	18	new	new	ADJ
ejpam-5623	611	19	theoretical	theoretical	ADJ
ejpam-5623	611	20	developments	development	NOUN
ejpam-5623	611	21	and	and	CCONJ
ejpam-5623	611	22	applications	application	NOUN
ejpam-5623	611	23	in	in	ADP
ejpam-5623	611	24	fields	field	NOUN
ejpam-5623	611	25	such	such	ADJ
ejpam-5623	611	26	as	as	ADP
ejpam-5623	611	27	linear	linear	PROPN
ejpam-5623	611	28	algebra	algebra	NOUN
ejpam-5623	611	29	,	,	PUNCT
ejpam-5623	611	30	automata	automata	NOUN
ejpam-5623	611	31	theory	theory	NOUN
ejpam-5623	611	32	,	,	PUNCT
ejpam-5623	611	33	and	and	CCONJ
ejpam-5623	611	34	representation	representation	NOUN
ejpam-5623	611	35	theory	theory	NOUN
ejpam-5623	611	36	.	.	PUNCT
ejpam-5623	612	1	acknowledgements	acknowledgement	NOUN
ejpam-5623	612	2	the	the	DET
ejpam-5623	612	3	authors	author	NOUN
ejpam-5623	612	4	would	would	AUX
ejpam-5623	612	5	like	like	VERB
ejpam-5623	612	6	to	to	PART
ejpam-5623	612	7	express	express	VERB
ejpam-5623	612	8	their	their	PRON
ejpam-5623	612	9	sincere	sincere	ADJ
ejpam-5623	612	10	gratitude	gratitude	NOUN
ejpam-5623	612	11	to	to	ADP
ejpam-5623	612	12	the	the	DET
ejpam-5623	612	13	research	research	NOUN
ejpam-5623	612	14	and	and	CCONJ
ejpam-5623	612	15	development	development	PROPN
ejpam-5623	612	16	institute	institute	PROPN
ejpam-5623	612	17	,	,	PUNCT
ejpam-5623	612	18	nakhon	nakhon	PROPN
ejpam-5623	612	19	sawan	sawan	PROPN
ejpam-5623	612	20	rajabhat	rajabhat	PROPN
ejpam-5623	612	21	university	university	PROPN
ejpam-5623	612	22	,	,	PUNCT
ejpam-5623	612	23	for	for	ADP
ejpam-5623	612	24	their	their	PRON
ejpam-5623	612	25	financial	financial	ADJ
ejpam-5623	612	26	support	support	NOUN
ejpam-5623	612	27	in	in	ADP
ejpam-5623	612	28	this	this	DET
ejpam-5623	612	29	research	research	NOUN
ejpam-5623	612	30	project	project	NOUN
ejpam-5623	612	31	.	.	PUNCT
ejpam-5623	613	1	their	their	PRON
ejpam-5623	613	2	generous	generous	ADJ
ejpam-5623	613	3	contribution	contribution	NOUN
ejpam-5623	613	4	has	have	AUX
ejpam-5623	613	5	been	be	AUX
ejpam-5623	613	6	essential	essential	ADJ
ejpam-5623	613	7	to	to	ADP
ejpam-5623	613	8	the	the	DET
ejpam-5623	613	9	successful	successful	ADJ
ejpam-5623	613	10	completion	completion	NOUN
ejpam-5623	613	11	of	of	ADP
ejpam-5623	613	12	this	this	DET
ejpam-5623	613	13	study	study	NOUN
ejpam-5623	613	14	.	.	PUNCT
ejpam-5623	614	1	references	reference	NOUN
ejpam-5623	614	2	[	[	X
ejpam-5623	614	3	1	1	NUM
ejpam-5623	614	4	]	]	X
ejpam-5623	614	5	y	y	PROPN
ejpam-5623	614	6	chaiya	chaiya	PROPN
ejpam-5623	614	7	.	.	PUNCT
ejpam-5623	615	1	natural	natural	ADJ
ejpam-5623	615	2	partial	partial	ADJ
ejpam-5623	615	3	order	order	NOUN
ejpam-5623	615	4	and	and	CCONJ
ejpam-5623	615	5	finiteness	finiteness	NOUN
ejpam-5623	615	6	conditions	condition	NOUN
ejpam-5623	615	7	on	on	ADP
ejpam-5623	615	8	semigroups	semigroup	NOUN
ejpam-5623	615	9	of	of	ADP
ejpam-5623	615	10	linear	linear	ADJ
ejpam-5623	615	11	transformations	transformation	NOUN
ejpam-5623	615	12	with	with	ADP
ejpam-5623	615	13	invariant	invariant	ADJ
ejpam-5623	615	14	subspaces	subspace	NOUN
ejpam-5623	615	15	.	.	PUNCT
ejpam-5623	616	1	semigroup	semigroup	PROPN
ejpam-5623	616	2	forum	forum	PROPN
ejpam-5623	616	3	,	,	PUNCT
ejpam-5623	616	4	99:579–590	99:579–590	NUM
ejpam-5623	616	5	,	,	PUNCT
ejpam-5623	616	6	2019	2019	NUM
ejpam-5623	616	7	.	.	PUNCT
ejpam-5623	617	1	n.	n.	PROPN
ejpam-5623	617	2	sawatraksa	sawatraksa	PROPN
ejpam-5623	617	3	,	,	PUNCT
ejpam-5623	617	4	p.	p.	NOUN
ejpam-5623	617	5	tantong	tantong	NOUN
ejpam-5623	617	6	/	/	SYM
ejpam-5623	617	7	eur	eur	PROPN
ejpam-5623	617	8	.	.	PUNCT
ejpam-5623	618	1	j.	j.	PROPN
ejpam-5623	618	2	pure	pure	PROPN
ejpam-5623	618	3	appl	appl	PROPN
ejpam-5623	618	4	.	.	PROPN
ejpam-5623	618	5	math	math	PROPN
ejpam-5623	618	6	,	,	PUNCT
ejpam-5623	618	7	18	18	NUM
ejpam-5623	618	8	(	(	PUNCT
ejpam-5623	618	9	1	1	NUM
ejpam-5623	618	10	)	)	PUNCT
ejpam-5623	618	11	(	(	PUNCT
ejpam-5623	618	12	2025	2025	NUM
ejpam-5623	618	13	)	)	PUNCT
ejpam-5623	618	14	,	,	PUNCT
ejpam-5623	618	15	5623	5623	NUM
ejpam-5623	618	16	15	15	NUM
ejpam-5623	618	17	of	of	ADP
ejpam-5623	618	18	15	15	NUM
ejpam-5623	618	19	[	[	X
ejpam-5623	618	20	2	2	NUM
ejpam-5623	618	21	]	]	SYM
ejpam-5623	618	22	r	r	NOUN
ejpam-5623	618	23	chinram	chinram	NOUN
ejpam-5623	618	24	and	and	CCONJ
ejpam-5623	618	25	w	w	PROPN
ejpam-5623	618	26	yonthanthum	yonthanthum	NOUN
ejpam-5623	618	27	.	.	PUNCT
ejpam-5623	619	1	regularity	regularity	NOUN
ejpam-5623	619	2	of	of	ADP
ejpam-5623	619	3	the	the	DET
ejpam-5623	619	4	semigroups	semigroup	NOUN
ejpam-5623	619	5	of	of	ADP
ejpam-5623	619	6	transformations	transformation	NOUN
ejpam-5623	619	7	with	with	ADP
ejpam-5623	619	8	a	a	DET
ejpam-5623	619	9	fixed	fix	VERB
ejpam-5623	619	10	point	point	NOUN
ejpam-5623	619	11	set	set	VERB
ejpam-5623	619	12	.	.	PUNCT
ejpam-5623	620	1	thai	thai	PROPN
ejpam-5623	620	2	journal	journal	PROPN
ejpam-5623	620	3	of	of	ADP
ejpam-5623	620	4	mathematics	mathematic	NOUN
ejpam-5623	620	5	,	,	PUNCT
ejpam-5623	620	6	18(3):1261–1268	18(3):1261–1268	NUM
ejpam-5623	620	7	,	,	PUNCT
ejpam-5623	620	8	2020	2020	NUM
ejpam-5623	620	9	.	.	PUNCT
ejpam-5623	621	1	[	[	X
ejpam-5623	621	2	3	3	X
ejpam-5623	621	3	]	]	X
ejpam-5623	621	4	p	p	X
ejpam-5623	621	5	honyam	honyam	PROPN
ejpam-5623	621	6	and	and	CCONJ
ejpam-5623	621	7	j	j	PROPN
ejpam-5623	621	8	sanwong	sanwong	PROPN
ejpam-5623	621	9	.	.	PUNCT
ejpam-5623	622	1	semigroups	semigroup	NOUN
ejpam-5623	622	2	of	of	ADP
ejpam-5623	622	3	linear	linear	ADJ
ejpam-5623	622	4	transformations	transformation	NOUN
ejpam-5623	622	5	with	with	ADP
ejpam-5623	622	6	invariant	invariant	ADJ
ejpam-5623	622	7	subspace	subspace	NOUN
ejpam-5623	622	8	.	.	PUNCT
ejpam-5623	623	1	international	international	ADJ
ejpam-5623	623	2	journal	journal	PROPN
ejpam-5623	623	3	of	of	ADP
ejpam-5623	623	4	algebra	algebra	PROPN
ejpam-5623	623	5	,	,	PUNCT
ejpam-5623	623	6	6(8):375–386	6(8):375–386	NUM
ejpam-5623	623	7	,	,	PUNCT
ejpam-5623	623	8	2012	2012	NUM
ejpam-5623	623	9	.	.	PUNCT
ejpam-5623	624	1	[	[	X
ejpam-5623	624	2	4	4	X
ejpam-5623	624	3	]	]	PUNCT
ejpam-5623	624	4	jm	jm	PROPN
ejpam-5623	624	5	howie	howie	PROPN
ejpam-5623	624	6	.	.	PUNCT
ejpam-5623	625	1	fundamentals	fundamental	NOUN
ejpam-5623	625	2	of	of	ADP
ejpam-5623	625	3	semigroup	semigroup	PROPN
ejpam-5623	625	4	theory	theory	NOUN
ejpam-5623	625	5	.	.	PUNCT
ejpam-5623	626	1	oxford	oxford	PROPN
ejpam-5623	626	2	university	university	PROPN
ejpam-5623	626	3	press	press	NOUN
ejpam-5623	626	4	,	,	PUNCT
ejpam-5623	626	5	new	new	PROPN
ejpam-5623	626	6	york	york	PROPN
ejpam-5623	626	7	,	,	PUNCT
ejpam-5623	626	8	1995	1995	NUM
ejpam-5623	626	9	.	.	PUNCT
ejpam-5623	627	1	[	[	X
ejpam-5623	627	2	5	5	X
ejpam-5623	627	3	]	]	PUNCT
ejpam-5623	627	4	p	p	X
ejpam-5623	627	5	huisheng	huisheng	NOUN
ejpam-5623	627	6	.	.	PUNCT
ejpam-5623	628	1	a	a	DET
ejpam-5623	628	2	note	note	NOUN
ejpam-5623	628	3	on	on	ADP
ejpam-5623	628	4	semigroups	semigroup	NOUN
ejpam-5623	628	5	of	of	ADP
ejpam-5623	628	6	linear	linear	ADJ
ejpam-5623	628	7	transformations	transformation	NOUN
ejpam-5623	628	8	with	with	ADP
ejpam-5623	628	9	invariant	invariant	ADJ
ejpam-5623	628	10	subspace	subspace	NOUN
ejpam-5623	628	11	.	.	PUNCT
ejpam-5623	629	1	international	international	ADJ
ejpam-5623	629	2	journal	journal	PROPN
ejpam-5623	629	3	of	of	ADP
ejpam-5623	629	4	algebra	algebra	PROPN
ejpam-5623	629	5	,	,	PUNCT
ejpam-5623	629	6	6(27):1319–1324	6(27):1319–1324	NUM
ejpam-5623	629	7	,	,	PUNCT
ejpam-5623	629	8	2012	2012	NUM
ejpam-5623	629	9	.	.	PUNCT
ejpam-5623	630	1	[	[	X
ejpam-5623	630	2	6	6	NUM
ejpam-5623	630	3	]	]	PUNCT
ejpam-5623	630	4	e	e	X
ejpam-5623	630	5	laysirikul	laysirikul	PROPN
ejpam-5623	630	6	k	k	PROPN
ejpam-5623	630	7	sripon	sripon	ADV
ejpam-5623	630	8	and	and	CCONJ
ejpam-5623	630	9	w	w	NOUN
ejpam-5623	630	10	sommanee	sommanee	NOUN
ejpam-5623	630	11	.	.	PUNCT
ejpam-5623	631	1	left	leave	VERB
ejpam-5623	631	2	(	(	PUNCT
ejpam-5623	631	3	right	right	ADJ
ejpam-5623	631	4	)	)	PUNCT
ejpam-5623	631	5	regular	regular	ADJ
ejpam-5623	631	6	elements	element	NOUN
ejpam-5623	631	7	of	of	ADP
ejpam-5623	631	8	some	some	DET
ejpam-5623	631	9	transformation	transformation	NOUN
ejpam-5623	631	10	semigroups	semigroup	NOUN
ejpam-5623	631	11	.	.	PUNCT
ejpam-5623	632	1	mathematics	mathematic	NOUN
ejpam-5623	632	2	,	,	PUNCT
ejpam-5623	632	3	11(10):2230	11(10):2230	NUM
ejpam-5623	632	4	,	,	PUNCT
ejpam-5623	632	5	2023	2023	NUM
ejpam-5623	632	6	.	.	PUNCT
ejpam-5623	633	1	[	[	X
ejpam-5623	633	2	7	7	NUM
ejpam-5623	633	3	]	]	X
ejpam-5623	633	4	e	e	NOUN
ejpam-5623	633	5	laysirikul	laysirikul	NOUN
ejpam-5623	633	6	.	.	PUNCT
ejpam-5623	634	1	semigroups	semigroup	NOUN
ejpam-5623	634	2	of	of	ADP
ejpam-5623	634	3	full	full	ADJ
ejpam-5623	634	4	transformations	transformation	NOUN
ejpam-5623	634	5	with	with	ADP
ejpam-5623	634	6	restriction	restriction	NOUN
ejpam-5623	634	7	on	on	ADP
ejpam-5623	634	8	the	the	DET
ejpam-5623	634	9	fixed	fix	VERB
ejpam-5623	634	10	set	set	NOUN
ejpam-5623	634	11	is	be	AUX
ejpam-5623	634	12	bijective	bijective	ADJ
ejpam-5623	634	13	.	.	PUNCT
ejpam-5623	635	1	thai	thai	PROPN
ejpam-5623	635	2	journal	journal	PROPN
ejpam-5623	635	3	of	of	ADP
ejpam-5623	635	4	mathematics	mathematics	PROPN
ejpam-5623	635	5	.	.	PUNCT
ejpam-5623	635	6	,	,	PUNCT
ejpam-5623	635	7	33:109–118	33:109–118	NUM
ejpam-5623	635	8	,	,	PUNCT
ejpam-5623	635	9	2016	2016	NUM
ejpam-5623	635	10	.	.	PUNCT
ejpam-5623	636	1	[	[	X
ejpam-5623	636	2	8	8	NUM
ejpam-5623	636	3	]	]	X
ejpam-5623	636	4	c	c	X
ejpam-5623	636	5	namnak	namnak	NOUN
ejpam-5623	636	6	and	and	CCONJ
ejpam-5623	636	7	e	e	PROPN
ejpam-5623	636	8	laysirikul	laysirikul	NOUN
ejpam-5623	636	9	.	.	PUNCT
ejpam-5623	637	1	regularity	regularity	NOUN
ejpam-5623	637	2	for	for	ADP
ejpam-5623	637	3	semigroups	semigroup	NOUN
ejpam-5623	637	4	of	of	ADP
ejpam-5623	637	5	transformations	transformation	NOUN
ejpam-5623	637	6	that	that	PRON
ejpam-5623	637	7	preserve	preserve	VERB
ejpam-5623	637	8	equivalence	equivalence	NOUN
ejpam-5623	637	9	.	.	PUNCT
ejpam-5623	638	1	jp	jp	PROPN
ejpam-5623	638	2	journal	journal	PROPN
ejpam-5623	638	3	of	of	ADP
ejpam-5623	638	4	algebra	algebra	PROPN
ejpam-5623	638	5	,	,	PUNCT
ejpam-5623	638	6	number	number	NOUN
ejpam-5623	638	7	theory	theory	NOUN
ejpam-5623	638	8	and	and	CCONJ
ejpam-5623	638	9	applications	application	NOUN
ejpam-5623	638	10	,	,	PUNCT
ejpam-5623	638	11	28:97	28:97	NUM
ejpam-5623	638	12	–	–	PUNCT
ejpam-5623	638	13	105	105	NUM
ejpam-5623	638	14	,	,	PUNCT
ejpam-5623	638	15	2013	2013	NUM
ejpam-5623	638	16	.	.	PUNCT
ejpam-5623	639	1	[	[	X
ejpam-5623	639	2	9	9	NUM
ejpam-5623	639	3	]	]	X
ejpam-5623	639	4	c	c	NOUN
ejpam-5623	639	5	namnak	namnak	NOUN
ejpam-5623	639	6	and	and	CCONJ
ejpam-5623	639	7	e	e	PROPN
ejpam-5623	639	8	laysirikul	laysirikul	NOUN
ejpam-5623	639	9	.	.	PUNCT
ejpam-5623	640	1	right	right	ADV
ejpam-5623	640	2	regular	regular	ADV
ejpam-5623	640	3	and	and	CCONJ
ejpam-5623	640	4	left	leave	VERB
ejpam-5623	640	5	regular	regular	ADJ
ejpam-5623	640	6	elements	element	NOUN
ejpam-5623	640	7	of	of	ADP
ejpam-5623	640	8	e	e	NOUN
ejpam-5623	640	9	-	-	VERB
ejpam-5623	640	10	orderpreserving	orderpreserve	VERB
ejpam-5623	640	11	transformation	transformation	NOUN
ejpam-5623	640	12	semigroups	semigroup	NOUN
ejpam-5623	640	13	.	.	PUNCT
ejpam-5623	641	1	international	international	ADJ
ejpam-5623	641	2	journal	journal	NOUN
ejpam-5623	641	3	of	of	ADP
ejpam-5623	641	4	algebra	algebra	PROPN
ejpam-5623	641	5	,	,	PUNCT
ejpam-5623	641	6	7:289–296	7:289–296	NUM
ejpam-5623	641	7	,	,	PUNCT
ejpam-5623	641	8	2013	2013	NUM
ejpam-5623	641	9	.	.	PUNCT
ejpam-5623	642	1	[	[	X
ejpam-5623	642	2	10	10	NUM
ejpam-5623	642	3	]	]	X
ejpam-5623	642	4	m	m	VERB
ejpam-5623	642	5	petrich	petrich	NOUN
ejpam-5623	642	6	and	and	CCONJ
ejpam-5623	642	7	nr	nr	PRON
ejpam-5623	642	8	reilly	reilly	ADV
ejpam-5623	642	9	.	.	PUNCT
ejpam-5623	643	1	completely	completely	ADV
ejpam-5623	643	2	regular	regular	ADJ
ejpam-5623	643	3	semigroups	semigroup	NOUN
ejpam-5623	643	4	.	.	PUNCT
ejpam-5623	644	1	wiley	wiley	PROPN
ejpam-5623	644	2	,	,	PUNCT
ejpam-5623	644	3	new	new	PROPN
ejpam-5623	644	4	york	york	PROPN
ejpam-5623	644	5	,	,	PUNCT
ejpam-5623	644	6	1999	1999	NUM
ejpam-5623	644	7	.	.	PUNCT
ejpam-5623	645	1	[	[	X
ejpam-5623	645	2	11	11	NUM
ejpam-5623	645	3	]	]	X
ejpam-5623	645	4	k	k	PROPN
ejpam-5623	645	5	sangkhanan	sangkhanan	PROPN
ejpam-5623	645	6	and	and	CCONJ
ejpam-5623	645	7	j	j	PROPN
ejpam-5623	645	8	sanwong	sanwong	PROPN
ejpam-5623	645	9	.	.	PUNCT
ejpam-5623	646	1	ranks	rank	VERB
ejpam-5623	646	2	and	and	CCONJ
ejpam-5623	646	3	isomorphism	isomorphism	NOUN
ejpam-5623	646	4	theorems	theorem	NOUN
ejpam-5623	646	5	of	of	ADP
ejpam-5623	646	6	semigroups	semigroup	NOUN
ejpam-5623	646	7	of	of	ADP
ejpam-5623	646	8	linear	linear	ADJ
ejpam-5623	646	9	transformations	transformation	NOUN
ejpam-5623	646	10	with	with	ADP
ejpam-5623	646	11	restricted	restricted	ADJ
ejpam-5623	646	12	range	range	NOUN
ejpam-5623	646	13	.	.	PUNCT
ejpam-5623	647	1	semigroup	semigroup	PROPN
ejpam-5623	647	2	forum	forum	PROPN
ejpam-5623	647	3	,	,	PUNCT
ejpam-5623	647	4	98:456–471	98:456–471	PROPN
ejpam-5623	647	5	,	,	PUNCT
ejpam-5623	647	6	2019	2019	NUM
ejpam-5623	647	7	.	.	PUNCT
ejpam-5623	648	1	[	[	X
ejpam-5623	648	2	12	12	NUM
ejpam-5623	648	3	]	]	PUNCT
ejpam-5623	648	4	n	n	PRON
ejpam-5623	648	5	sirasuntorn	sirasuntorn	NOUN
ejpam-5623	648	6	and	and	CCONJ
ejpam-5623	648	7	y	y	PROPN
ejpam-5623	648	8	kemprasit	kemprasit	PROPN
ejpam-5623	648	9	.	.	PUNCT
ejpam-5623	649	1	left	leave	VERB
ejpam-5623	649	2	regular	regular	ADV
ejpam-5623	649	3	and	and	CCONJ
ejpam-5623	649	4	right	right	ADJ
ejpam-5623	649	5	regular	regular	ADJ
ejpam-5623	649	6	elements	element	NOUN
ejpam-5623	649	7	of	of	ADP
ejpam-5623	649	8	semigroups	semigroup	NOUN
ejpam-5623	649	9	of	of	ADP
ejpam-5623	649	10	1	1	NUM
ejpam-5623	649	11	-	-	SYM
ejpam-5623	649	12	1	1	NUM
ejpam-5623	649	13	transformations	transformation	NOUN
ejpam-5623	649	14	and	and	CCONJ
ejpam-5623	649	15	1	1	NUM
ejpam-5623	649	16	-	-	SYM
ejpam-5623	649	17	1	1	NUM
ejpam-5623	649	18	linear	linear	ADJ
ejpam-5623	649	19	transformations	transformation	NOUN
ejpam-5623	649	20	.	.	PUNCT
ejpam-5623	650	1	international	international	ADJ
ejpam-5623	650	2	journal	journal	NOUN
ejpam-5623	650	3	of	of	ADP
ejpam-5623	650	4	algebra	algebra	PROPN
ejpam-5623	650	5	,	,	PUNCT
ejpam-5623	650	6	4:1399–1406	4:1399–1406	NUM
ejpam-5623	650	7	,	,	PUNCT
ejpam-5623	650	8	2010	2010	NUM
ejpam-5623	650	9	.	.	PUNCT
ejpam-5623	651	1	[	[	X
ejpam-5623	651	2	13	13	NUM
ejpam-5623	651	3	]	]	SYM
ejpam-5623	651	4	w	w	NOUN
ejpam-5623	651	5	sommanee	sommanee	NOUN
ejpam-5623	651	6	and	and	CCONJ
ejpam-5623	651	7	k	k	PROPN
ejpam-5623	651	8	sangkhanan	sangkhanan	PROPN
ejpam-5623	651	9	.	.	PUNCT
ejpam-5623	652	1	the	the	DET
ejpam-5623	652	2	regular	regular	ADJ
ejpam-5623	652	3	part	part	NOUN
ejpam-5623	652	4	of	of	ADP
ejpam-5623	652	5	a	a	DET
ejpam-5623	652	6	semigroup	semigroup	NOUN
ejpam-5623	652	7	of	of	ADP
ejpam-5623	652	8	linear	linear	ADJ
ejpam-5623	652	9	transformations	transformation	NOUN
ejpam-5623	652	10	with	with	ADP
ejpam-5623	652	11	restricted	restricted	ADJ
ejpam-5623	652	12	range	range	NOUN
ejpam-5623	652	13	.	.	PUNCT
ejpam-5623	653	1	journal	journal	NOUN
ejpam-5623	653	2	of	of	ADP
ejpam-5623	653	3	the	the	DET
ejpam-5623	653	4	australian	australian	ADJ
ejpam-5623	653	5	mathematical	mathematical	ADJ
ejpam-5623	653	6	society	society	NOUN
ejpam-5623	653	7	,	,	PUNCT
ejpam-5623	653	8	103(3):402–419	103(3):402–419	NUM
ejpam-5623	653	9	,	,	PUNCT
ejpam-5623	653	10	1941	1941	NUM
ejpam-5623	653	11	.	.	PUNCT
ejpam-5623	654	1	[	[	X
ejpam-5623	654	2	14	14	NUM
ejpam-5623	654	3	]	]	X
ejpam-5623	654	4	rp	rp	PROPN
ejpam-5623	654	5	sullivan	sullivan	PROPN
ejpam-5623	654	6	.	.	PUNCT
ejpam-5623	655	1	semigroups	semigroup	NOUN
ejpam-5623	655	2	of	of	ADP
ejpam-5623	655	3	linear	linear	ADJ
ejpam-5623	655	4	transformations	transformation	NOUN
ejpam-5623	655	5	with	with	ADP
ejpam-5623	655	6	restricted	restricted	ADJ
ejpam-5623	655	7	range	range	NOUN
ejpam-5623	655	8	.	.	PUNCT
ejpam-5623	656	1	bulletin	bulletin	NOUN
ejpam-5623	656	2	of	of	ADP
ejpam-5623	656	3	the	the	DET
ejpam-5623	656	4	australian	australian	ADJ
ejpam-5623	656	5	mathematical	mathematical	ADJ
ejpam-5623	656	6	society	society	NOUN
ejpam-5623	656	7	,	,	PUNCT
ejpam-5623	656	8	77:441–453	77:441–453	NUM
ejpam-5623	656	9	,	,	PUNCT
ejpam-5623	656	10	2008	2008	NUM
ejpam-5623	656	11	.	.	PUNCT
ejpam-5623	657	1	[	[	X
ejpam-5623	657	2	15	15	NUM
ejpam-5623	657	3	]	]	X
ejpam-5623	657	4	p	p	X
ejpam-5623	657	5	tantong	tantong	NOUN
ejpam-5623	657	6	.	.	PUNCT
ejpam-5623	658	1	on	on	ADP
ejpam-5623	658	2	regularity	regularity	NOUN
ejpam-5623	658	3	of	of	ADP
ejpam-5623	658	4	linear	linear	ADJ
ejpam-5623	658	5	transformation	transformation	NOUN
ejpam-5623	658	6	semigroups	semigroup	NOUN
ejpam-5623	658	7	.	.	PUNCT
ejpam-5623	659	1	far	far	PROPN
ejpam-5623	659	2	east	east	PROPN
ejpam-5623	659	3	journal	journal	PROPN
ejpam-5623	659	4	of	of	ADP
ejpam-5623	659	5	mathematical	mathematical	ADJ
ejpam-5623	659	6	sciences	sciences	PROPN
ejpam-5623	659	7	,	,	PUNCT
ejpam-5623	659	8	108(2):401–411	108(2):401–411	PROPN
ejpam-5623	659	9	,	,	PUNCT
ejpam-5623	659	10	2018	2018	NUM
ejpam-5623	659	11	.	.	PUNCT
ejpam-5623	660	1	[	[	X
ejpam-5623	660	2	16	16	NUM
ejpam-5623	660	3	]	]	X
ejpam-5623	660	4	c	c	NOUN
ejpam-5623	660	5	pookpienlert	pookpienlert	PROPN
ejpam-5623	660	6	y	y	PROPN
ejpam-5623	660	7	chaiya	chaiya	PROPN
ejpam-5623	660	8	and	and	CCONJ
ejpam-5623	660	9	j	j	PROPN
ejpam-5623	660	10	sanwong	sanwong	PROPN
ejpam-5623	660	11	.	.	PUNCT
ejpam-5623	661	1	semigroups	semigroup	NOUN
ejpam-5623	661	2	of	of	ADP
ejpam-5623	661	3	linear	linear	ADJ
ejpam-5623	661	4	transformations	transformation	NOUN
ejpam-5623	661	5	with	with	ADP
ejpam-5623	661	6	fixed	fix	VERB
ejpam-5623	661	7	subspaces	subspace	NOUN
ejpam-5623	661	8	:	:	PUNCT
ejpam-5623	661	9	green	green	PROPN
ejpam-5623	661	10	’s	’s	PART
ejpam-5623	661	11	relations	relation	NOUN
ejpam-5623	661	12	,	,	PUNCT
ejpam-5623	661	13	ideals	ideal	NOUN
ejpam-5623	661	14	and	and	CCONJ
ejpam-5623	661	15	finiteness	finiteness	NOUN
ejpam-5623	661	16	conditions	condition	NOUN
ejpam-5623	661	17	.	.	PUNCT
ejpam-5623	662	1	asian	asian	ADJ
ejpam-5623	662	2	-	-	PUNCT
ejpam-5623	662	3	european	european	ADJ
ejpam-5623	662	4	journal	journal	NOUN
ejpam-5623	662	5	of	of	ADP
ejpam-5623	662	6	mathematics	mathematic	NOUN
ejpam-5623	662	7	,	,	PUNCT
ejpam-5623	662	8	12(4):579–590	12(4):579–590	PROPN
ejpam-5623	662	9	,	,	PUNCT
ejpam-5623	662	10	2019	2019	NUM
ejpam-5623	662	11	.	.	PUNCT