id	sid	tid	token	lemma	pos
ejpam-6501	1	1	european	european	PROPN
ejpam-6501	1	2	journal	journal	PROPN
ejpam-6501	1	3	of	of	ADP
ejpam-6501	1	4	pure	pure	ADJ
ejpam-6501	1	5	and	and	CCONJ
ejpam-6501	1	6	applied	applied	ADJ
ejpam-6501	1	7	mathematics	mathematic	NOUN
ejpam-6501	1	8	2025	2025	NUM
ejpam-6501	1	9	,	,	PUNCT
ejpam-6501	1	10	vol	vol	NOUN
ejpam-6501	1	11	.	.	PROPN
ejpam-6501	1	12	18	18	NUM
ejpam-6501	1	13	,	,	PUNCT
ejpam-6501	1	14	issue	issue	NOUN
ejpam-6501	1	15	4	4	NUM
ejpam-6501	1	16	,	,	PUNCT
ejpam-6501	1	17	article	article	NOUN
ejpam-6501	1	18	number	number	NOUN
ejpam-6501	1	19	6501	6501	NUM
ejpam-6501	1	20	issn	issn	PROPN
ejpam-6501	1	21	1307	1307	NUM
ejpam-6501	1	22	-	-	SYM
ejpam-6501	1	23	5543	5543	NUM
ejpam-6501	1	24	–	–	PUNCT
ejpam-6501	1	25	ejpam.com	ejpam.com	X
ejpam-6501	1	26	published	publish	VERB
ejpam-6501	1	27	by	by	ADP
ejpam-6501	1	28	new	new	PROPN
ejpam-6501	1	29	york	york	PROPN
ejpam-6501	1	30	business	business	PROPN
ejpam-6501	1	31	global	global	PROPN
ejpam-6501	1	32	on	on	ADP
ejpam-6501	1	33	quotient	quotient	NOUN
ejpam-6501	1	34	ink	ink	NOUN
ejpam-6501	1	35	-	-	PUNCT
ejpam-6501	1	36	algebras	algebras	PROPN
ejpam-6501	1	37	maliwan	maliwan	PROPN
ejpam-6501	1	38	phattarachaleekul	phattarachaleekul	PROPN
ejpam-6501	1	39	department	department	PROPN
ejpam-6501	1	40	of	of	ADP
ejpam-6501	1	41	mathematics	mathematic	NOUN
ejpam-6501	1	42	,	,	PUNCT
ejpam-6501	1	43	faculty	faculty	NOUN
ejpam-6501	1	44	of	of	ADP
ejpam-6501	1	45	science	science	NOUN
ejpam-6501	1	46	,	,	PUNCT
ejpam-6501	1	47	mahasarakham	mahasarakham	PROPN
ejpam-6501	1	48	university	university	PROPN
ejpam-6501	1	49	,	,	PUNCT
ejpam-6501	1	50	maha	maha	PROPN
ejpam-6501	1	51	sarakham	sarakham	PROPN
ejpam-6501	1	52	44150	44150	NUM
ejpam-6501	1	53	,	,	PUNCT
ejpam-6501	1	54	thailand	thailand	PROPN
ejpam-6501	1	55	abstract	abstract	PROPN
ejpam-6501	1	56	.	.	PUNCT
ejpam-6501	2	1	this	this	DET
ejpam-6501	2	2	paper	paper	NOUN
ejpam-6501	2	3	introduces	introduce	VERB
ejpam-6501	2	4	the	the	DET
ejpam-6501	2	5	notion	notion	NOUN
ejpam-6501	2	6	of	of	ADP
ejpam-6501	2	7	quotient	quotient	NOUN
ejpam-6501	2	8	ink	ink	NOUN
ejpam-6501	2	9	-	-	PUNCT
ejpam-6501	2	10	algebra	algebra	NOUN
ejpam-6501	2	11	(	(	PUNCT
ejpam-6501	2	12	x	x	X
ejpam-6501	2	13	,	,	PUNCT
ejpam-6501	2	14	∗	∗	NOUN
ejpam-6501	2	15	,	,	PUNCT
ejpam-6501	2	16	0	0	NUM
ejpam-6501	2	17	)	)	PUNCT
ejpam-6501	2	18	.	.	PUNCT
ejpam-6501	3	1	we	we	PRON
ejpam-6501	3	2	formalize	formalize	VERB
ejpam-6501	3	3	the	the	DET
ejpam-6501	3	4	notion	notion	NOUN
ejpam-6501	3	5	of	of	ADP
ejpam-6501	3	6	congruence	congruence	PROPN
ejpam-6501	3	7	relations	relation	NOUN
ejpam-6501	3	8	in	in	ADP
ejpam-6501	3	9	ink	ink	NOUN
ejpam-6501	3	10	-	-	PUNCT
ejpam-6501	3	11	algebras	algebras	X
ejpam-6501	3	12	,	,	PUNCT
ejpam-6501	3	13	proving	prove	VERB
ejpam-6501	3	14	that	that	SCONJ
ejpam-6501	3	15	the	the	DET
ejpam-6501	3	16	set	set	NOUN
ejpam-6501	3	17	of	of	ADP
ejpam-6501	3	18	equivalence	equivalence	NOUN
ejpam-6501	3	19	classes	class	NOUN
ejpam-6501	3	20	forms	form	VERB
ejpam-6501	3	21	a	a	DET
ejpam-6501	3	22	partition	partition	NOUN
ejpam-6501	3	23	.	.	PUNCT
ejpam-6501	4	1	additionally	additionally	ADV
ejpam-6501	4	2	,	,	PUNCT
ejpam-6501	4	3	we	we	PRON
ejpam-6501	4	4	define	define	VERB
ejpam-6501	4	5	a	a	DET
ejpam-6501	4	6	normal	normal	ADJ
ejpam-6501	4	7	subset	subset	NOUN
ejpam-6501	4	8	n	n	PROPN
ejpam-6501	4	9	of	of	ADP
ejpam-6501	4	10	x	x	PUNCT
ejpam-6501	4	11	and	and	CCONJ
ejpam-6501	4	12	show	show	VERB
ejpam-6501	4	13	that	that	SCONJ
ejpam-6501	4	14	the	the	DET
ejpam-6501	4	15	quotient	quotient	NOUN
ejpam-6501	4	16	set	set	VERB
ejpam-6501	4	17	x	x	X
ejpam-6501	4	18	/	/	SYM
ejpam-6501	4	19	n	n	NOUN
ejpam-6501	4	20	=	=	PRON
ejpam-6501	4	21	{	{	PUNCT
ejpam-6501	4	22	[	[	X
ejpam-6501	4	23	x]n	x]n	PROPN
ejpam-6501	4	24	|x	|x	X
ejpam-6501	4	25	∈	∈	PROPN
ejpam-6501	4	26	x	x	X
ejpam-6501	4	27	}	}	PUNCT
ejpam-6501	4	28	where	where	SCONJ
ejpam-6501	4	29	[	[	X
ejpam-6501	4	30	x]n	x]n	NOUN
ejpam-6501	4	31	=	=	SYM
ejpam-6501	4	32	{	{	PUNCT
ejpam-6501	4	33	y	y	PROPN
ejpam-6501	4	34	∈	∈	PROPN
ejpam-6501	4	35	x|x	x|x	PUNCT
ejpam-6501	5	1	∼	∼	NOUN
ejpam-6501	5	2	n	n	PRON
ejpam-6501	5	3	y	y	NOUN
ejpam-6501	5	4	}	}	PUNCT
ejpam-6501	5	5	and	and	CCONJ
ejpam-6501	5	6	a	a	DET
ejpam-6501	5	7	binary	binary	ADJ
ejpam-6501	5	8	operation	operation	NOUN
ejpam-6501	5	9	⊙	⊙	NOUN
ejpam-6501	5	10	on	on	ADP
ejpam-6501	5	11	x	x	PROPN
ejpam-6501	5	12	/	/	SYM
ejpam-6501	5	13	n	n	PRON
ejpam-6501	5	14	such	such	ADJ
ejpam-6501	5	15	that	that	SCONJ
ejpam-6501	6	1	[	[	X
ejpam-6501	6	2	x]n	x]n	PROPN
ejpam-6501	6	3	⊙	⊙	VERB
ejpam-6501	7	1	[	[	X
ejpam-6501	7	2	y]n	y]n	NOUN
ejpam-6501	7	3	=	=	X
ejpam-6501	8	1	[	[	X
ejpam-6501	8	2	x	x	X
ejpam-6501	8	3	∗	∗	X
ejpam-6501	8	4	y]n	y]n	NOUN
ejpam-6501	8	5	for	for	ADP
ejpam-6501	8	6	all	all	PRON
ejpam-6501	8	7	[	[	X
ejpam-6501	8	8	x]n	x]n	X
ejpam-6501	8	9	,	,	PUNCT
ejpam-6501	8	10	[	[	X
ejpam-6501	8	11	y]n	y]n	X
ejpam-6501	8	12	∈	∈	ADJ
ejpam-6501	8	13	x	x	NOUN
ejpam-6501	8	14	/	/	SYM
ejpam-6501	8	15	n	n	PRON
ejpam-6501	8	16	forms	form	VERB
ejpam-6501	8	17	a	a	DET
ejpam-6501	8	18	quotient	quotient	NOUN
ejpam-6501	8	19	ink	ink	NOUN
ejpam-6501	8	20	-	-	PUNCT
ejpam-6501	8	21	algebra	algebra	NOUN
ejpam-6501	8	22	.	.	PUNCT
ejpam-6501	9	1	2020	2020	NUM
ejpam-6501	9	2	mathematics	mathematic	NOUN
ejpam-6501	9	3	subject	subject	NOUN
ejpam-6501	9	4	classifications	classification	NOUN
ejpam-6501	9	5	:	:	PUNCT
ejpam-6501	9	6	06f35	06f35	NUM
ejpam-6501	9	7	,	,	PUNCT
ejpam-6501	9	8	03g25	03g25	NOUN
ejpam-6501	9	9	key	key	ADJ
ejpam-6501	9	10	words	word	NOUN
ejpam-6501	9	11	and	and	CCONJ
ejpam-6501	9	12	phrases	phrase	NOUN
ejpam-6501	9	13	:	:	PUNCT
ejpam-6501	9	14	ink	ink	NOUN
ejpam-6501	9	15	-	-	PUNCT
ejpam-6501	9	16	algebras	algebra	NOUN
ejpam-6501	9	17	,	,	PUNCT
ejpam-6501	9	18	ideal	ideal	ADJ
ejpam-6501	9	19	,	,	PUNCT
ejpam-6501	9	20	normal	normal	ADJ
ejpam-6501	9	21	subset	subset	NOUN
ejpam-6501	9	22	,	,	PUNCT
ejpam-6501	9	23	partitions	partition	NOUN
ejpam-6501	9	24	,	,	PUNCT
ejpam-6501	9	25	quotient	quotient	VERB
ejpam-6501	9	26	ink	ink	NOUN
ejpam-6501	9	27	-	-	PUNCT
ejpam-6501	9	28	algebras	algebras	PROPN
ejpam-6501	9	29	1	1	NUM
ejpam-6501	9	30	.	.	X
ejpam-6501	9	31	introduction	introduction	NOUN
ejpam-6501	9	32	algebraic	algebraic	ADJ
ejpam-6501	9	33	structures	structure	NOUN
ejpam-6501	9	34	play	play	VERB
ejpam-6501	9	35	a	a	DET
ejpam-6501	9	36	crucial	crucial	ADJ
ejpam-6501	9	37	role	role	NOUN
ejpam-6501	9	38	in	in	ADP
ejpam-6501	9	39	various	various	ADJ
ejpam-6501	9	40	branches	branch	NOUN
ejpam-6501	9	41	of	of	ADP
ejpam-6501	9	42	mathematics	mathematic	NOUN
ejpam-6501	9	43	,	,	PUNCT
ejpam-6501	9	44	including	include	VERB
ejpam-6501	9	45	logic	logic	NOUN
ejpam-6501	9	46	,	,	PUNCT
ejpam-6501	9	47	topology	topology	NOUN
ejpam-6501	9	48	,	,	PUNCT
ejpam-6501	9	49	and	and	CCONJ
ejpam-6501	9	50	theoretical	theoretical	ADJ
ejpam-6501	9	51	computer	computer	NOUN
ejpam-6501	9	52	science	science	NOUN
ejpam-6501	9	53	.	.	PUNCT
ejpam-6501	10	1	among	among	ADP
ejpam-6501	10	2	these	these	PRON
ejpam-6501	10	3	,	,	PUNCT
ejpam-6501	10	4	be	be	AUX
ejpam-6501	10	5	-	-	PUNCT
ejpam-6501	10	6	algebras	algebra	NOUN
ejpam-6501	10	7	and	and	CCONJ
ejpam-6501	10	8	inkalgebras	inkalgebra	NOUN
ejpam-6501	10	9	have	have	AUX
ejpam-6501	10	10	been	be	AUX
ejpam-6501	10	11	extensively	extensively	ADV
ejpam-6501	10	12	studied	study	VERB
ejpam-6501	10	13	due	due	ADP
ejpam-6501	10	14	to	to	ADP
ejpam-6501	10	15	their	their	PRON
ejpam-6501	10	16	applicability	applicability	NOUN
ejpam-6501	10	17	in	in	ADP
ejpam-6501	10	18	non	non	ADJ
ejpam-6501	10	19	-	-	ADJ
ejpam-6501	10	20	classical	classical	ADJ
ejpam-6501	10	21	logics	logic	NOUN
ejpam-6501	10	22	and	and	CCONJ
ejpam-6501	10	23	algebraic	algebraic	ADJ
ejpam-6501	10	24	systems	system	NOUN
ejpam-6501	10	25	.	.	PUNCT
ejpam-6501	11	1	be	be	AUX
ejpam-6501	11	2	-	-	PUNCT
ejpam-6501	11	3	algebras	algebras	X
ejpam-6501	11	4	,	,	PUNCT
ejpam-6501	11	5	introduced	introduce	VERB
ejpam-6501	11	6	as	as	ADP
ejpam-6501	11	7	a	a	DET
ejpam-6501	11	8	generalization	generalization	NOUN
ejpam-6501	11	9	of	of	ADP
ejpam-6501	11	10	bck	bck	PROPN
ejpam-6501	11	11	/	/	SYM
ejpam-6501	11	12	bci	bci	NOUN
ejpam-6501	11	13	-	-	PUNCT
ejpam-6501	11	14	algebras	algebras	X
ejpam-6501	11	15	,	,	PUNCT
ejpam-6501	11	16	provide	provide	VERB
ejpam-6501	11	17	a	a	DET
ejpam-6501	11	18	framework	framework	NOUN
ejpam-6501	11	19	for	for	ADP
ejpam-6501	11	20	handling	handle	VERB
ejpam-6501	11	21	abstract	abstract	ADJ
ejpam-6501	11	22	algebraic	algebraic	ADJ
ejpam-6501	11	23	operations	operation	NOUN
ejpam-6501	11	24	with	with	ADP
ejpam-6501	11	25	specific	specific	ADJ
ejpam-6501	11	26	axiomatic	axiomatic	ADJ
ejpam-6501	11	27	structures	structure	NOUN
ejpam-6501	11	28	.	.	PUNCT
ejpam-6501	12	1	meanwhile	meanwhile	ADV
ejpam-6501	12	2	,	,	PUNCT
ejpam-6501	12	3	ink	ink	NOUN
ejpam-6501	12	4	-	-	PUNCT
ejpam-6501	12	5	algebras	algebra	NOUN
ejpam-6501	12	6	have	have	AUX
ejpam-6501	12	7	been	be	AUX
ejpam-6501	12	8	recently	recently	ADV
ejpam-6501	12	9	developed	develop	VERB
ejpam-6501	12	10	as	as	ADP
ejpam-6501	12	11	an	an	DET
ejpam-6501	12	12	extension	extension	NOUN
ejpam-6501	12	13	of	of	ADP
ejpam-6501	12	14	be	be	NOUN
ejpam-6501	12	15	-	-	PUNCT
ejpam-6501	12	16	algebras	algebras	ADJ
ejpam-6501	12	17	with	with	ADP
ejpam-6501	12	18	additional	additional	ADJ
ejpam-6501	12	19	constraints	constraint	NOUN
ejpam-6501	12	20	and	and	CCONJ
ejpam-6501	12	21	properties	property	NOUN
ejpam-6501	12	22	.	.	PUNCT
ejpam-6501	13	1	the	the	DET
ejpam-6501	13	2	study	study	NOUN
ejpam-6501	13	3	of	of	ADP
ejpam-6501	13	4	ink	ink	NOUN
ejpam-6501	13	5	-	-	PUNCT
ejpam-6501	13	6	algebras	algebras	PROPN
ejpam-6501	13	7	has	have	AUX
ejpam-6501	13	8	garnered	garner	VERB
ejpam-6501	13	9	interest	interest	NOUN
ejpam-6501	13	10	due	due	ADP
ejpam-6501	13	11	to	to	ADP
ejpam-6501	13	12	their	their	PRON
ejpam-6501	13	13	unique	unique	ADJ
ejpam-6501	13	14	structural	structural	ADJ
ejpam-6501	13	15	properties	property	NOUN
ejpam-6501	13	16	and	and	CCONJ
ejpam-6501	13	17	potential	potential	ADJ
ejpam-6501	13	18	applications	application	NOUN
ejpam-6501	13	19	in	in	ADP
ejpam-6501	13	20	algebraic	algebraic	ADJ
ejpam-6501	13	21	logic	logic	NOUN
ejpam-6501	13	22	.	.	PUNCT
ejpam-6501	14	1	understanding	understand	VERB
ejpam-6501	14	2	congruence	congruence	PROPN
ejpam-6501	14	3	relations	relation	NOUN
ejpam-6501	14	4	,	,	PUNCT
ejpam-6501	14	5	ideals	ideal	NOUN
ejpam-6501	14	6	,	,	PUNCT
ejpam-6501	14	7	and	and	CCONJ
ejpam-6501	14	8	quotient	quotient	VERB
ejpam-6501	14	9	structures	structure	NOUN
ejpam-6501	14	10	within	within	ADP
ejpam-6501	14	11	these	these	DET
ejpam-6501	14	12	algebras	algebra	NOUN
ejpam-6501	14	13	is	be	AUX
ejpam-6501	14	14	fundamental	fundamental	ADJ
ejpam-6501	14	15	in	in	ADP
ejpam-6501	14	16	advancing	advance	VERB
ejpam-6501	14	17	the	the	DET
ejpam-6501	14	18	theoretical	theoretical	ADJ
ejpam-6501	14	19	framework	framework	NOUN
ejpam-6501	14	20	of	of	ADP
ejpam-6501	14	21	algebraic	algebraic	ADJ
ejpam-6501	14	22	systems	system	NOUN
ejpam-6501	14	23	.	.	PUNCT
ejpam-6501	15	1	the	the	DET
ejpam-6501	15	2	notion	notion	NOUN
ejpam-6501	15	3	of	of	ADP
ejpam-6501	15	4	congruences	congruence	NOUN
ejpam-6501	15	5	provides	provide	VERB
ejpam-6501	15	6	insight	insight	NOUN
ejpam-6501	15	7	into	into	ADP
ejpam-6501	15	8	the	the	DET
ejpam-6501	15	9	partitioning	partitioning	NOUN
ejpam-6501	15	10	of	of	ADP
ejpam-6501	15	11	algebraic	algebraic	ADJ
ejpam-6501	15	12	structures	structure	NOUN
ejpam-6501	15	13	,	,	PUNCT
ejpam-6501	15	14	while	while	SCONJ
ejpam-6501	15	15	ideals	ideal	NOUN
ejpam-6501	15	16	and	and	CCONJ
ejpam-6501	15	17	quotient	quotient	NOUN
ejpam-6501	15	18	algebras	algebras	PROPN
ejpam-6501	15	19	help	help	VERB
ejpam-6501	15	20	in	in	ADP
ejpam-6501	15	21	forming	form	VERB
ejpam-6501	15	22	new	new	ADJ
ejpam-6501	15	23	algebraic	algebraic	ADJ
ejpam-6501	15	24	systems	system	NOUN
ejpam-6501	15	25	from	from	ADP
ejpam-6501	15	26	existing	exist	VERB
ejpam-6501	15	27	ones	one	NOUN
ejpam-6501	15	28	.	.	PUNCT
ejpam-6501	16	1	in	in	ADP
ejpam-6501	16	2	2002	2002	NUM
ejpam-6501	16	3	,	,	PUNCT
ejpam-6501	16	4	j.	j.	PROPN
ejpam-6501	16	5	neggers	neggers	PROPN
ejpam-6501	16	6	and	and	CCONJ
ejpam-6501	16	7	h.	h.	PROPN
ejpam-6501	16	8	s.	s.	PROPN
ejpam-6501	16	9	kim	kim	PROPN
ejpam-6501	17	1	(	(	PUNCT
ejpam-6501	17	2	[	[	X
ejpam-6501	17	3	1	1	NUM
ejpam-6501	17	4	]	]	PUNCT
ejpam-6501	17	5	)	)	PUNCT
ejpam-6501	17	6	introduced	introduce	VERB
ejpam-6501	17	7	the	the	DET
ejpam-6501	17	8	notion	notion	NOUN
ejpam-6501	17	9	of	of	ADP
ejpam-6501	17	10	b	b	NOUN
ejpam-6501	17	11	-	-	PUNCT
ejpam-6501	17	12	algebras	algebra	NOUN
ejpam-6501	17	13	and	and	CCONJ
ejpam-6501	17	14	some	some	DET
ejpam-6501	17	15	properties	property	NOUN
ejpam-6501	17	16	of	of	ADP
ejpam-6501	17	17	exponents	exponent	NOUN
ejpam-6501	17	18	on	on	ADP
ejpam-6501	17	19	them	they	PRON
ejpam-6501	17	20	.	.	PUNCT
ejpam-6501	18	1	one	one	NUM
ejpam-6501	18	2	such	such	ADJ
ejpam-6501	18	3	structure	structure	NOUN
ejpam-6501	18	4	is	be	AUX
ejpam-6501	18	5	the	the	DET
ejpam-6501	18	6	ink	ink	NOUN
ejpam-6501	18	7	-	-	PUNCT
ejpam-6501	18	8	algebras	algebra	NOUN
ejpam-6501	18	9	,	,	PUNCT
ejpam-6501	18	10	introduced	introduce	VERB
ejpam-6501	18	11	in	in	ADP
ejpam-6501	18	12	2017	2017	NUM
ejpam-6501	18	13	(	(	PUNCT
ejpam-6501	18	14	[	[	X
ejpam-6501	18	15	2	2	NUM
ejpam-6501	18	16	]	]	PUNCT
ejpam-6501	18	17	)	)	PUNCT
ejpam-6501	18	18	by	by	ADP
ejpam-6501	18	19	kaviyarasu	kaviyarasu	PROPN
ejpam-6501	18	20	and	and	CCONJ
ejpam-6501	18	21	indhira	indhira	PROPN
ejpam-6501	18	22	,	,	PUNCT
ejpam-6501	18	23	who	who	PRON
ejpam-6501	18	24	introduced	introduce	VERB
ejpam-6501	18	25	a	a	DET
ejpam-6501	18	26	new	new	ADJ
ejpam-6501	18	27	notion	notion	NOUN
ejpam-6501	18	28	called	call	VERB
ejpam-6501	18	29	ink	ink	NOUN
ejpam-6501	18	30	-	-	PUNCT
ejpam-6501	18	31	algebras	algebras	PROPN
ejpam-6501	18	32	to	to	PART
ejpam-6501	18	33	generalize	generalize	VERB
ejpam-6501	18	34	and	and	CCONJ
ejpam-6501	18	35	extend	extend	VERB
ejpam-6501	18	36	the	the	DET
ejpam-6501	18	37	properties	property	NOUN
ejpam-6501	18	38	of	of	ADP
ejpam-6501	18	39	b	b	NOUN
ejpam-6501	18	40	-	-	PUNCT
ejpam-6501	18	41	algebras	algebras	PROPN
ejpam-6501	18	42	.	.	PUNCT
ejpam-6501	19	1	ink	ink	NOUN
ejpam-6501	19	2	-	-	PUNCT
ejpam-6501	19	3	algebras	algebra	NOUN
ejpam-6501	19	4	have	have	AUX
ejpam-6501	19	5	been	be	AUX
ejpam-6501	19	6	widely	widely	ADV
ejpam-6501	19	7	studied	study	VERB
ejpam-6501	19	8	in	in	ADP
ejpam-6501	19	9	relation	relation	NOUN
ejpam-6501	19	10	to	to	ADP
ejpam-6501	19	11	their	their	PRON
ejpam-6501	19	12	ideal	ideal	ADJ
ejpam-6501	19	13	structures	structure	NOUN
ejpam-6501	19	14	,	,	PUNCT
ejpam-6501	19	15	congruences	congruence	NOUN
ejpam-6501	19	16	,	,	PUNCT
ejpam-6501	19	17	and	and	CCONJ
ejpam-6501	19	18	algebraic	algebraic	ADJ
ejpam-6501	19	19	operations	operation	NOUN
ejpam-6501	19	20	.	.	PUNCT
ejpam-6501	20	1	in	in	ADP
ejpam-6501	20	2	doi	doi	NOUN
ejpam-6501	20	3	:	:	PUNCT
ejpam-6501	20	4	https://doi.org/10.29020/nybg.ejpam.v18i4.6501	https://doi.org/10.29020/nybg.ejpam.v18i4.6501	PRON
ejpam-6501	20	5	email	email	NOUN
ejpam-6501	20	6	address	address	NOUN
ejpam-6501	20	7	:	:	PUNCT
ejpam-6501	20	8	maliwan.t@msu.ac.th	maliwan.t@msu.ac.th	PROPN
ejpam-6501	20	9	(	(	PUNCT
ejpam-6501	20	10	m.	m.	NOUN
ejpam-6501	20	11	phattarachaleekul	phattarachaleekul	PROPN
ejpam-6501	20	12	)	)	PUNCT
ejpam-6501	20	13	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6501	20	14	1	1	NUM
ejpam-6501	20	15	copyright	copyright	NOUN
ejpam-6501	20	16	:	:	PUNCT
ejpam-6501	20	17	©	©	PROPN
ejpam-6501	20	18	2025	2025	NUM
ejpam-6501	20	19	the	the	DET
ejpam-6501	20	20	author(s	author(s	NOUN
ejpam-6501	20	21	)	)	PUNCT
ejpam-6501	20	22	.	.	PUNCT
ejpam-6501	21	1	(	(	PUNCT
ejpam-6501	21	2	cc	cc	NOUN
ejpam-6501	21	3	by	by	ADP
ejpam-6501	21	4	-	-	PUNCT
ejpam-6501	21	5	nc	nc	PROPN
ejpam-6501	21	6	4.0	4.0	NUM
ejpam-6501	21	7	)	)	PUNCT
ejpam-6501	21	8	m.	m.	NOUN
ejpam-6501	21	9	phattarachaleekul	phattarachaleekul	PROPN
ejpam-6501	21	10	/	/	SYM
ejpam-6501	21	11	eur	eur	PROPN
ejpam-6501	21	12	.	.	PUNCT
ejpam-6501	22	1	j.	j.	PROPN
ejpam-6501	22	2	pure	pure	PROPN
ejpam-6501	22	3	appl	appl	PROPN
ejpam-6501	22	4	.	.	PROPN
ejpam-6501	22	5	math	math	PROPN
ejpam-6501	22	6	,	,	PUNCT
ejpam-6501	22	7	18	18	NUM
ejpam-6501	22	8	(	(	PUNCT
ejpam-6501	22	9	4	4	NUM
ejpam-6501	22	10	)	)	PUNCT
ejpam-6501	22	11	(	(	PUNCT
ejpam-6501	22	12	2025	2025	NUM
ejpam-6501	22	13	)	)	PUNCT
ejpam-6501	22	14	,	,	PUNCT
ejpam-6501	22	15	6501	6501	NUM
ejpam-6501	22	16	2	2	NUM
ejpam-6501	22	17	of	of	ADP
ejpam-6501	22	18	8	8	NUM
ejpam-6501	22	19	particular	particular	ADJ
ejpam-6501	22	20	,	,	PUNCT
ejpam-6501	22	21	the	the	DET
ejpam-6501	22	22	concept	concept	NOUN
ejpam-6501	22	23	of	of	ADP
ejpam-6501	22	24	quotient	quotient	NOUN
ejpam-6501	22	25	ink	ink	NOUN
ejpam-6501	22	26	-	-	PUNCT
ejpam-6501	22	27	algebras	algebras	PROPN
ejpam-6501	22	28	arises	arise	VERB
ejpam-6501	22	29	naturally	naturally	ADV
ejpam-6501	22	30	when	when	SCONJ
ejpam-6501	22	31	considering	consider	VERB
ejpam-6501	22	32	the	the	DET
ejpam-6501	22	33	role	role	NOUN
ejpam-6501	22	34	of	of	ADP
ejpam-6501	22	35	congruences	congruence	NOUN
ejpam-6501	22	36	and	and	CCONJ
ejpam-6501	22	37	normal	normal	ADJ
ejpam-6501	22	38	subsets	subset	NOUN
ejpam-6501	22	39	in	in	ADP
ejpam-6501	22	40	these	these	DET
ejpam-6501	22	41	algebras	algebra	NOUN
ejpam-6501	22	42	.	.	PUNCT
ejpam-6501	23	1	a	a	DET
ejpam-6501	23	2	quotient	quotient	NOUN
ejpam-6501	23	3	ink	ink	NOUN
ejpam-6501	23	4	-	-	PUNCT
ejpam-6501	23	5	algebra	algebra	NOUN
ejpam-6501	23	6	is	be	AUX
ejpam-6501	23	7	constructed	construct	VERB
ejpam-6501	23	8	by	by	ADP
ejpam-6501	23	9	partitioning	partition	VERB
ejpam-6501	23	10	an	an	DET
ejpam-6501	23	11	ink	ink	NOUN
ejpam-6501	23	12	-	-	PUNCT
ejpam-6501	23	13	algebra	algebra	NOUN
ejpam-6501	23	14	using	use	VERB
ejpam-6501	23	15	a	a	DET
ejpam-6501	23	16	congruence	congruence	NOUN
ejpam-6501	23	17	relation	relation	NOUN
ejpam-6501	23	18	induced	induce	VERB
ejpam-6501	23	19	by	by	ADP
ejpam-6501	23	20	an	an	DET
ejpam-6501	23	21	ideal	ideal	NOUN
ejpam-6501	23	22	.	.	PUNCT
ejpam-6501	24	1	this	this	DET
ejpam-6501	24	2	process	process	NOUN
ejpam-6501	24	3	mirrors	mirror	VERB
ejpam-6501	24	4	the	the	DET
ejpam-6501	24	5	formation	formation	NOUN
ejpam-6501	24	6	of	of	ADP
ejpam-6501	24	7	quotient	quotient	NOUN
ejpam-6501	24	8	groups	group	NOUN
ejpam-6501	24	9	in	in	ADP
ejpam-6501	24	10	group	group	NOUN
ejpam-6501	24	11	theory	theory	NOUN
ejpam-6501	24	12	and	and	CCONJ
ejpam-6501	24	13	quotient	quotient	NOUN
ejpam-6501	24	14	rings	ring	NOUN
ejpam-6501	24	15	in	in	ADP
ejpam-6501	24	16	ring	ring	NOUN
ejpam-6501	24	17	theory	theory	NOUN
ejpam-6501	24	18	.	.	PUNCT
ejpam-6501	25	1	the	the	DET
ejpam-6501	25	2	primary	primary	ADJ
ejpam-6501	25	3	objective	objective	NOUN
ejpam-6501	25	4	of	of	ADP
ejpam-6501	25	5	forming	form	VERB
ejpam-6501	25	6	a	a	DET
ejpam-6501	25	7	quotient	quotient	NOUN
ejpam-6501	25	8	ink	ink	NOUN
ejpam-6501	25	9	-	-	PUNCT
ejpam-6501	25	10	algebra	algebra	NOUN
ejpam-6501	25	11	is	be	AUX
ejpam-6501	25	12	to	to	PART
ejpam-6501	25	13	simplify	simplify	VERB
ejpam-6501	25	14	the	the	DET
ejpam-6501	25	15	structure	structure	NOUN
ejpam-6501	25	16	while	while	SCONJ
ejpam-6501	25	17	preserving	preserve	VERB
ejpam-6501	25	18	essential	essential	ADJ
ejpam-6501	25	19	algebraic	algebraic	ADJ
ejpam-6501	25	20	properties	property	NOUN
ejpam-6501	25	21	.	.	PUNCT
ejpam-6501	26	1	given	give	VERB
ejpam-6501	26	2	an	an	DET
ejpam-6501	26	3	inkalgebra	inkalgebra	NOUN
ejpam-6501	26	4	(	(	PUNCT
ejpam-6501	26	5	x	x	X
ejpam-6501	26	6	,	,	PUNCT
ejpam-6501	26	7	∗	∗	NOUN
ejpam-6501	26	8	,	,	PUNCT
ejpam-6501	26	9	0	0	NUM
ejpam-6501	26	10	)	)	PUNCT
ejpam-6501	26	11	,	,	PUNCT
ejpam-6501	26	12	an	an	DET
ejpam-6501	26	13	ideal	ideal	NOUN
ejpam-6501	26	14	i	i	PRON
ejpam-6501	26	15	ofx	ofx	PROPN
ejpam-6501	26	16	induces	induce	VERB
ejpam-6501	26	17	a	a	DET
ejpam-6501	26	18	congruence	congruence	NOUN
ejpam-6501	26	19	relation∼	relation∼	NOUN
ejpam-6501	26	20	i	i	PRON
ejpam-6501	26	21	onx	onx	PROPN
ejpam-6501	26	22	as	as	ADP
ejpam-6501	26	23	x	x	PUNCT
ejpam-6501	26	24	∼	∼	NOUN
ejpam-6501	26	25	i	i	NOUN
ejpam-6501	26	26	y	y	PROPN
ejpam-6501	26	27	iff	iff	VERB
ejpam-6501	26	28	x∗y	x∗y	PROPN
ejpam-6501	26	29	∈	∈	PROPN
ejpam-6501	27	1	i	i	PRON
ejpam-6501	27	2	and	and	CCONJ
ejpam-6501	27	3	y	y	PROPN
ejpam-6501	27	4	∗	∗	NOUN
ejpam-6501	27	5	x	x	PUNCT
ejpam-6501	27	6	∈	∈	NOUN
ejpam-6501	27	7	i	i	PRON
ejpam-6501	27	8	for	for	ADP
ejpam-6501	27	9	any	any	DET
ejpam-6501	27	10	x	x	NOUN
ejpam-6501	27	11	,	,	PUNCT
ejpam-6501	27	12	y	y	PROPN
ejpam-6501	27	13	∈	∈	PROPN
ejpam-6501	27	14	x.	x.	AUX
ejpam-6501	27	15	define	define	VERB
ejpam-6501	27	16	an	an	DET
ejpam-6501	27	17	equivalence	equivalence	NOUN
ejpam-6501	27	18	classes	class	NOUN
ejpam-6501	27	19	[	[	X
ejpam-6501	27	20	x]i	x]i	X
ejpam-6501	27	21	=	=	SYM
ejpam-6501	27	22	{	{	PUNCT
ejpam-6501	27	23	y	y	PROPN
ejpam-6501	27	24	∈	∈	PROPN
ejpam-6501	27	25	x|x	x|x	PUNCT
ejpam-6501	28	1	∼	∼	NOUN
ejpam-6501	28	2	i	i	PRON
ejpam-6501	28	3	y	y	NOUN
ejpam-6501	28	4	}	}	PUNCT
ejpam-6501	28	5	and	and	CCONJ
ejpam-6501	28	6	hence	hence	ADV
ejpam-6501	28	7	the	the	DET
ejpam-6501	28	8	set	set	NOUN
ejpam-6501	28	9	of	of	ADP
ejpam-6501	28	10	all	all	DET
ejpam-6501	28	11	equivalence	equivalence	NOUN
ejpam-6501	28	12	classes	class	NOUN
ejpam-6501	28	13	forms	form	VERB
ejpam-6501	28	14	a	a	DET
ejpam-6501	28	15	partition	partition	NOUN
ejpam-6501	28	16	of	of	ADP
ejpam-6501	28	17	x.	x.	NOUN
ejpam-6501	28	18	moreover	moreover	ADV
ejpam-6501	28	19	,	,	PUNCT
ejpam-6501	28	20	this	this	DET
ejpam-6501	28	21	paper	paper	NOUN
ejpam-6501	28	22	aims	aim	VERB
ejpam-6501	28	23	to	to	PART
ejpam-6501	28	24	explore	explore	VERB
ejpam-6501	28	25	the	the	DET
ejpam-6501	28	26	construction	construction	NOUN
ejpam-6501	28	27	and	and	CCONJ
ejpam-6501	28	28	properties	property	NOUN
ejpam-6501	28	29	of	of	ADP
ejpam-6501	28	30	quotient	quotient	NOUN
ejpam-6501	28	31	ink	ink	NOUN
ejpam-6501	28	32	-	-	PUNCT
ejpam-6501	28	33	algebras	algebras	X
ejpam-6501	28	34	(	(	PUNCT
ejpam-6501	28	35	x	x	SYM
ejpam-6501	28	36	/	/	SYM
ejpam-6501	28	37	n,⊙	n,⊙	ADJ
ejpam-6501	28	38	,	,	PUNCT
ejpam-6501	28	39	[	[	X
ejpam-6501	28	40	0]i	0]i	NUM
ejpam-6501	28	41	)	)	PUNCT
ejpam-6501	28	42	,	,	PUNCT
ejpam-6501	28	43	including	include	VERB
ejpam-6501	28	44	the	the	DET
ejpam-6501	28	45	role	role	NOUN
ejpam-6501	28	46	of	of	ADP
ejpam-6501	28	47	normal	normal	ADJ
ejpam-6501	28	48	subsets	subset	NOUN
ejpam-6501	28	49	defining	define	VERB
ejpam-6501	28	50	these	these	DET
ejpam-6501	28	51	structures	structure	NOUN
ejpam-6501	28	52	.	.	PUNCT
ejpam-6501	29	1	we	we	PRON
ejpam-6501	29	2	establish	establish	VERB
ejpam-6501	29	3	key	key	ADJ
ejpam-6501	29	4	theorems	theorem	NOUN
ejpam-6501	29	5	that	that	PRON
ejpam-6501	29	6	demonstrate	demonstrate	VERB
ejpam-6501	29	7	how	how	SCONJ
ejpam-6501	29	8	the	the	DET
ejpam-6501	29	9	quotient	quotient	NOUN
ejpam-6501	29	10	operation	operation	NOUN
ejpam-6501	29	11	preserves	preserve	VERB
ejpam-6501	29	12	the	the	DET
ejpam-6501	29	13	axioms	axiom	NOUN
ejpam-6501	29	14	of	of	ADP
ejpam-6501	29	15	an	an	DET
ejpam-6501	29	16	ink	ink	NOUN
ejpam-6501	29	17	-	-	PUNCT
ejpam-6501	29	18	algebra	algebra	NOUN
ejpam-6501	29	19	.	.	PUNCT
ejpam-6501	30	1	furthermore	furthermore	ADV
ejpam-6501	30	2	,	,	PUNCT
ejpam-6501	30	3	we	we	PRON
ejpam-6501	30	4	provide	provide	VERB
ejpam-6501	30	5	illustrative	illustrative	ADJ
ejpam-6501	30	6	examples	example	NOUN
ejpam-6501	30	7	to	to	PART
ejpam-6501	30	8	highlight	highlight	VERB
ejpam-6501	30	9	the	the	DET
ejpam-6501	30	10	behavior	behavior	NOUN
ejpam-6501	30	11	of	of	ADP
ejpam-6501	30	12	quotient	quotient	NOUN
ejpam-6501	30	13	inkalgebras	inkalgebra	NOUN
ejpam-6501	30	14	and	and	CCONJ
ejpam-6501	30	15	their	their	PRON
ejpam-6501	30	16	applications	application	NOUN
ejpam-6501	30	17	within	within	ADP
ejpam-6501	30	18	algebraic	algebraic	ADJ
ejpam-6501	30	19	systems	system	NOUN
ejpam-6501	30	20	.	.	PUNCT
ejpam-6501	31	1	this	this	DET
ejpam-6501	31	2	paper	paper	NOUN
ejpam-6501	31	3	aims	aim	VERB
ejpam-6501	31	4	to	to	PART
ejpam-6501	31	5	explore	explore	VERB
ejpam-6501	31	6	fundamental	fundamental	ADJ
ejpam-6501	31	7	properties	property	NOUN
ejpam-6501	31	8	of	of	ADP
ejpam-6501	31	9	ink	ink	NOUN
ejpam-6501	31	10	-	-	PUNCT
ejpam-6501	31	11	algebras	algebra	NOUN
ejpam-6501	31	12	,	,	PUNCT
ejpam-6501	31	13	particularly	particularly	ADV
ejpam-6501	31	14	focusing	focus	VERB
ejpam-6501	31	15	on	on	ADP
ejpam-6501	31	16	congruence	congruence	NOUN
ejpam-6501	31	17	relations	relation	NOUN
ejpam-6501	31	18	and	and	CCONJ
ejpam-6501	31	19	their	their	PRON
ejpam-6501	31	20	impact	impact	NOUN
ejpam-6501	31	21	on	on	ADP
ejpam-6501	31	22	the	the	DET
ejpam-6501	31	23	structure	structure	NOUN
ejpam-6501	31	24	of	of	ADP
ejpam-6501	31	25	ink	ink	NOUN
ejpam-6501	31	26	-	-	PUNCT
ejpam-6501	31	27	algebras	algebras	PROPN
ejpam-6501	31	28	.	.	PUNCT
ejpam-6501	32	1	we	we	PRON
ejpam-6501	32	2	establish	establish	VERB
ejpam-6501	32	3	necessary	necessary	ADJ
ejpam-6501	32	4	definitions	definition	NOUN
ejpam-6501	32	5	,	,	PUNCT
ejpam-6501	32	6	investigate	investigate	VERB
ejpam-6501	32	7	the	the	DET
ejpam-6501	32	8	conditions	condition	NOUN
ejpam-6501	32	9	for	for	ADP
ejpam-6501	32	10	congruences	congruence	NOUN
ejpam-6501	32	11	,	,	PUNCT
ejpam-6501	32	12	and	and	CCONJ
ejpam-6501	32	13	present	present	ADJ
ejpam-6501	32	14	results	result	NOUN
ejpam-6501	32	15	related	relate	VERB
ejpam-6501	32	16	to	to	ADP
ejpam-6501	32	17	the	the	DET
ejpam-6501	32	18	formation	formation	NOUN
ejpam-6501	32	19	of	of	ADP
ejpam-6501	32	20	quotient	quotient	NOUN
ejpam-6501	32	21	ink	ink	NOUN
ejpam-6501	32	22	-	-	PUNCT
ejpam-6501	32	23	algebras	algebras	PROPN
ejpam-6501	32	24	.	.	PUNCT
ejpam-6501	33	1	the	the	DET
ejpam-6501	33	2	organization	organization	NOUN
ejpam-6501	33	3	of	of	ADP
ejpam-6501	33	4	the	the	DET
ejpam-6501	33	5	paper	paper	NOUN
ejpam-6501	33	6	is	be	AUX
ejpam-6501	33	7	as	as	SCONJ
ejpam-6501	33	8	follows	follow	VERB
ejpam-6501	33	9	:	:	PUNCT
ejpam-6501	33	10	section	section	NOUN
ejpam-6501	33	11	2	2	NUM
ejpam-6501	33	12	introduces	introduce	NOUN
ejpam-6501	33	13	preliminaries	preliminary	NOUN
ejpam-6501	33	14	,	,	PUNCT
ejpam-6501	33	15	including	include	VERB
ejpam-6501	33	16	essential	essential	ADJ
ejpam-6501	33	17	definitions	definition	NOUN
ejpam-6501	33	18	and	and	CCONJ
ejpam-6501	33	19	fundamental	fundamental	ADJ
ejpam-6501	33	20	results	result	NOUN
ejpam-6501	33	21	on	on	ADP
ejpam-6501	33	22	ink	ink	NOUN
ejpam-6501	33	23	-	-	PUNCT
ejpam-6501	33	24	algebras	algebras	PROPN
ejpam-6501	33	25	.	.	PUNCT
ejpam-6501	34	1	section	section	NOUN
ejpam-6501	34	2	3	3	NUM
ejpam-6501	34	3	discusses	discuss	VERB
ejpam-6501	34	4	congruence	congruence	NOUN
ejpam-6501	34	5	relations	relation	NOUN
ejpam-6501	34	6	and	and	CCONJ
ejpam-6501	34	7	their	their	PRON
ejpam-6501	34	8	characterization	characterization	NOUN
ejpam-6501	34	9	in	in	ADP
ejpam-6501	34	10	ink	ink	NOUN
ejpam-6501	34	11	-	-	PUNCT
ejpam-6501	34	12	algebras	algebras	PROPN
ejpam-6501	34	13	.	.	PUNCT
ejpam-6501	35	1	finally	finally	ADV
ejpam-6501	35	2	,	,	PUNCT
ejpam-6501	35	3	section	section	NOUN
ejpam-6501	35	4	4	4	NUM
ejpam-6501	35	5	explores	explore	VERB
ejpam-6501	35	6	the	the	DET
ejpam-6501	35	7	construction	construction	NOUN
ejpam-6501	35	8	of	of	ADP
ejpam-6501	35	9	quotient	quotient	NOUN
ejpam-6501	35	10	ink	ink	NOUN
ejpam-6501	35	11	-	-	PUNCT
ejpam-6501	35	12	algebras	algebra	NOUN
ejpam-6501	35	13	and	and	CCONJ
ejpam-6501	35	14	their	their	PRON
ejpam-6501	35	15	properties	property	NOUN
ejpam-6501	35	16	.	.	PUNCT
ejpam-6501	36	1	2	2	X
ejpam-6501	36	2	.	.	X
ejpam-6501	36	3	preliminaries	preliminary	NOUN
ejpam-6501	36	4	first	first	ADV
ejpam-6501	36	5	,	,	PUNCT
ejpam-6501	36	6	we	we	PRON
ejpam-6501	36	7	will	will	AUX
ejpam-6501	36	8	review	review	VERB
ejpam-6501	36	9	some	some	DET
ejpam-6501	36	10	essential	essential	ADJ
ejpam-6501	36	11	notations	notation	NOUN
ejpam-6501	36	12	and	and	CCONJ
ejpam-6501	36	13	definitions	definition	NOUN
ejpam-6501	36	14	of	of	ADP
ejpam-6501	36	15	ink	ink	NOUN
ejpam-6501	36	16	-	-	PUNCT
ejpam-6501	36	17	algebras	algebra	NOUN
ejpam-6501	36	18	and	and	CCONJ
ejpam-6501	36	19	ordinary	ordinary	ADJ
ejpam-6501	36	20	senses	sense	NOUN
ejpam-6501	36	21	that	that	PRON
ejpam-6501	36	22	are	be	AUX
ejpam-6501	36	23	needed	need	VERB
ejpam-6501	36	24	for	for	ADP
ejpam-6501	36	25	this	this	DET
ejpam-6501	36	26	study	study	NOUN
ejpam-6501	36	27	in	in	ADP
ejpam-6501	36	28	this	this	DET
ejpam-6501	36	29	section	section	NOUN
ejpam-6501	36	30	.	.	PUNCT
ejpam-6501	37	1	throughout	throughout	ADP
ejpam-6501	37	2	this	this	DET
ejpam-6501	37	3	paper	paper	NOUN
ejpam-6501	37	4	,	,	PUNCT
ejpam-6501	37	5	x	x	PRON
ejpam-6501	37	6	will	will	AUX
ejpam-6501	37	7	denote	denote	VERB
ejpam-6501	37	8	the	the	DET
ejpam-6501	37	9	ink	ink	NOUN
ejpam-6501	37	10	-	-	PUNCT
ejpam-6501	37	11	algebra	algebra	NOUN
ejpam-6501	37	12	(	(	PUNCT
ejpam-6501	37	13	x	x	X
ejpam-6501	37	14	,	,	PUNCT
ejpam-6501	37	15	∗	∗	NOUN
ejpam-6501	37	16	,	,	PUNCT
ejpam-6501	37	17	0	0	NUM
ejpam-6501	37	18	)	)	PUNCT
ejpam-6501	37	19	unless	unless	SCONJ
ejpam-6501	37	20	otherwise	otherwise	ADV
ejpam-6501	37	21	specified	specify	VERB
ejpam-6501	37	22	.	.	PUNCT
ejpam-6501	38	1	definition	definition	NOUN
ejpam-6501	38	2	1	1	NUM
ejpam-6501	38	3	.	.	PUNCT
ejpam-6501	39	1	[	[	X
ejpam-6501	39	2	2	2	X
ejpam-6501	39	3	]	]	PUNCT
ejpam-6501	39	4	a	a	DET
ejpam-6501	39	5	be	be	NOUN
ejpam-6501	39	6	-	-	PUNCT
ejpam-6501	39	7	algebras	algebras	X
ejpam-6501	39	8	(	(	PUNCT
ejpam-6501	39	9	x	x	X
ejpam-6501	39	10	,	,	PUNCT
ejpam-6501	39	11	∗	∗	NOUN
ejpam-6501	39	12	,	,	PUNCT
ejpam-6501	39	13	0	0	NUM
ejpam-6501	39	14	)	)	PUNCT
ejpam-6501	39	15	is	be	AUX
ejpam-6501	39	16	a	a	DET
ejpam-6501	39	17	non	non	ADJ
ejpam-6501	39	18	-	-	ADJ
ejpam-6501	39	19	empty	empty	ADJ
ejpam-6501	39	20	set	set	NOUN
ejpam-6501	39	21	x	x	PUNCT
ejpam-6501	39	22	with	with	ADP
ejpam-6501	39	23	a	a	DET
ejpam-6501	39	24	constant	constant	ADJ
ejpam-6501	39	25	0	0	NUM
ejpam-6501	39	26	and	and	CCONJ
ejpam-6501	39	27	a	a	DET
ejpam-6501	39	28	binary	binary	ADJ
ejpam-6501	39	29	operation	operation	NOUN
ejpam-6501	39	30	∗	∗	NOUN
ejpam-6501	39	31	satisfying	satisfy	VERB
ejpam-6501	39	32	the	the	DET
ejpam-6501	39	33	following	follow	VERB
ejpam-6501	39	34	axioms	axiom	NOUN
ejpam-6501	39	35	:	:	PUNCT
ejpam-6501	39	36	(	(	PUNCT
ejpam-6501	39	37	be1	be1	NOUN
ejpam-6501	39	38	)	)	PUNCT
ejpam-6501	39	39	x	x	SYM
ejpam-6501	39	40	∗	∗	NOUN
ejpam-6501	39	41	x	x	SYM
ejpam-6501	40	1	=	=	SYM
ejpam-6501	40	2	0	0	NUM
ejpam-6501	40	3	,	,	PUNCT
ejpam-6501	40	4	(	(	PUNCT
ejpam-6501	40	5	be2	be2	PROPN
ejpam-6501	40	6	)	)	PUNCT
ejpam-6501	41	1	x	x	SYM
ejpam-6501	41	2	∗	∗	NOUN
ejpam-6501	41	3	0	0	NUM
ejpam-6501	42	1	=	=	SYM
ejpam-6501	42	2	0	0	NUM
ejpam-6501	42	3	,	,	PUNCT
ejpam-6501	42	4	(	(	PUNCT
ejpam-6501	42	5	be3	be3	PROPN
ejpam-6501	42	6	)	)	PUNCT
ejpam-6501	42	7	0	0	NUM
ejpam-6501	42	8	∗	∗	NOUN
ejpam-6501	42	9	x	x	X
ejpam-6501	43	1	=	=	SYM
ejpam-6501	43	2	x	x	X
ejpam-6501	43	3	,	,	PUNCT
ejpam-6501	43	4	(	(	PUNCT
ejpam-6501	43	5	be4	be4	NOUN
ejpam-6501	43	6	)	)	PUNCT
ejpam-6501	44	1	x	x	SYM
ejpam-6501	44	2	∗	∗	NOUN
ejpam-6501	44	3	(	(	PUNCT
ejpam-6501	44	4	y	y	PROPN
ejpam-6501	44	5	∗	∗	PROPN
ejpam-6501	44	6	z	z	NOUN
ejpam-6501	44	7	)	)	PUNCT
ejpam-6501	45	1	=	=	SYM
ejpam-6501	45	2	y	y	PROPN
ejpam-6501	45	3	∗	∗	NOUN
ejpam-6501	45	4	(	(	PUNCT
ejpam-6501	45	5	x	x	X
ejpam-6501	45	6	∗	∗	PROPN
ejpam-6501	45	7	z	z	NOUN
ejpam-6501	45	8	)	)	PUNCT
ejpam-6501	45	9	for	for	ADP
ejpam-6501	45	10	all	all	DET
ejpam-6501	45	11	x	x	NOUN
ejpam-6501	45	12	,	,	PUNCT
ejpam-6501	45	13	y	y	PROPN
ejpam-6501	45	14	,	,	PUNCT
ejpam-6501	45	15	z	z	PROPN
ejpam-6501	45	16	∈	∈	PROPN
ejpam-6501	45	17	x.	x.	NOUN
ejpam-6501	45	18	definition	definition	NOUN
ejpam-6501	45	19	2	2	NUM
ejpam-6501	45	20	.	.	PUNCT
ejpam-6501	46	1	[	[	X
ejpam-6501	46	2	3	3	X
ejpam-6501	46	3	]	]	PUNCT
ejpam-6501	46	4	a	a	DET
ejpam-6501	46	5	ink	ink	NOUN
ejpam-6501	46	6	-	-	PUNCT
ejpam-6501	46	7	algebras	algebras	NOUN
ejpam-6501	46	8	(	(	PUNCT
ejpam-6501	46	9	x	x	X
ejpam-6501	46	10	,	,	PUNCT
ejpam-6501	46	11	∗	∗	NOUN
ejpam-6501	46	12	,	,	PUNCT
ejpam-6501	46	13	0	0	NUM
ejpam-6501	46	14	)	)	PUNCT
ejpam-6501	46	15	is	be	AUX
ejpam-6501	46	16	a	a	DET
ejpam-6501	46	17	non	non	ADJ
ejpam-6501	46	18	-	-	ADJ
ejpam-6501	46	19	empty	empty	ADJ
ejpam-6501	46	20	set	set	NOUN
ejpam-6501	46	21	x	x	PUNCT
ejpam-6501	46	22	with	with	ADP
ejpam-6501	46	23	a	a	DET
ejpam-6501	46	24	constant	constant	ADJ
ejpam-6501	46	25	0	0	NUM
ejpam-6501	46	26	and	and	CCONJ
ejpam-6501	46	27	a	a	DET
ejpam-6501	46	28	binary	binary	ADJ
ejpam-6501	46	29	operation	operation	NOUN
ejpam-6501	46	30	∗	∗	NOUN
ejpam-6501	46	31	satisfies	satisfy	VERB
ejpam-6501	46	32	the	the	DET
ejpam-6501	46	33	following	following	ADJ
ejpam-6501	46	34	axioms	axiom	NOUN
ejpam-6501	46	35	for	for	ADP
ejpam-6501	46	36	all	all	DET
ejpam-6501	46	37	x	x	NOUN
ejpam-6501	46	38	,	,	PUNCT
ejpam-6501	46	39	y	y	PROPN
ejpam-6501	46	40	,	,	PUNCT
ejpam-6501	46	41	z	z	PROPN
ejpam-6501	46	42	∈	∈	PROPN
ejpam-6501	46	43	x.	x.	NOUN
ejpam-6501	46	44	(	(	PUNCT
ejpam-6501	46	45	ink1	ink1	PROPN
ejpam-6501	46	46	)	)	PUNCT
ejpam-6501	46	47	(	(	PUNCT
ejpam-6501	46	48	(	(	PUNCT
ejpam-6501	46	49	x	x	SYM
ejpam-6501	46	50	∗	∗	PROPN
ejpam-6501	46	51	y	y	NOUN
ejpam-6501	46	52	)	)	PUNCT
ejpam-6501	46	53	∗	∗	NOUN
ejpam-6501	46	54	(	(	PUNCT
ejpam-6501	46	55	x	x	X
ejpam-6501	46	56	∗	∗	PROPN
ejpam-6501	46	57	z	z	NOUN
ejpam-6501	46	58	)	)	PUNCT
ejpam-6501	46	59	)	)	PUNCT
ejpam-6501	47	1	∗	∗	NOUN
ejpam-6501	47	2	(	(	PUNCT
ejpam-6501	47	3	z	z	NOUN
ejpam-6501	47	4	∗	∗	NOUN
ejpam-6501	47	5	y	y	NOUN
ejpam-6501	47	6	)	)	PUNCT
ejpam-6501	47	7	=	=	SYM
ejpam-6501	47	8	0	0	NUM
ejpam-6501	47	9	,	,	PUNCT
ejpam-6501	47	10	(	(	PUNCT
ejpam-6501	47	11	ink2	ink2	PROPN
ejpam-6501	47	12	)	)	PUNCT
ejpam-6501	47	13	(	(	PUNCT
ejpam-6501	47	14	(	(	PUNCT
ejpam-6501	47	15	x	x	SYM
ejpam-6501	47	16	∗	∗	PROPN
ejpam-6501	47	17	z	z	NOUN
ejpam-6501	47	18	)	)	PUNCT
ejpam-6501	47	19	∗	∗	NOUN
ejpam-6501	47	20	(	(	PUNCT
ejpam-6501	47	21	y	y	PROPN
ejpam-6501	47	22	∗	∗	PROPN
ejpam-6501	47	23	z	z	NOUN
ejpam-6501	47	24	)	)	PUNCT
ejpam-6501	47	25	)	)	PUNCT
ejpam-6501	47	26	∗	∗	NOUN
ejpam-6501	47	27	(	(	PUNCT
ejpam-6501	47	28	x	x	X
ejpam-6501	47	29	∗	∗	NOUN
ejpam-6501	47	30	y	y	NOUN
ejpam-6501	47	31	)	)	PUNCT
ejpam-6501	47	32	=	=	SYM
ejpam-6501	47	33	0	0	NUM
ejpam-6501	47	34	,	,	PUNCT
ejpam-6501	47	35	(	(	PUNCT
ejpam-6501	47	36	ink3	ink3	PROPN
ejpam-6501	47	37	)	)	PUNCT
ejpam-6501	47	38	x	x	SYM
ejpam-6501	47	39	∗	∗	NOUN
ejpam-6501	47	40	0	0	NUM
ejpam-6501	48	1	=	=	SYM
ejpam-6501	48	2	x	x	NOUN
ejpam-6501	48	3	,	,	PUNCT
ejpam-6501	48	4	m.	m.	NOUN
ejpam-6501	48	5	phattarachaleekul	phattarachaleekul	PROPN
ejpam-6501	48	6	/	/	SYM
ejpam-6501	48	7	eur	eur	PROPN
ejpam-6501	48	8	.	.	PUNCT
ejpam-6501	49	1	j.	j.	PROPN
ejpam-6501	49	2	pure	pure	PROPN
ejpam-6501	49	3	appl	appl	PROPN
ejpam-6501	49	4	.	.	PROPN
ejpam-6501	49	5	math	math	PROPN
ejpam-6501	49	6	,	,	PUNCT
ejpam-6501	49	7	18	18	NUM
ejpam-6501	49	8	(	(	PUNCT
ejpam-6501	49	9	4	4	NUM
ejpam-6501	49	10	)	)	PUNCT
ejpam-6501	49	11	(	(	PUNCT
ejpam-6501	49	12	2025	2025	NUM
ejpam-6501	49	13	)	)	PUNCT
ejpam-6501	49	14	,	,	PUNCT
ejpam-6501	49	15	6501	6501	NUM
ejpam-6501	49	16	3	3	NUM
ejpam-6501	49	17	of	of	ADP
ejpam-6501	49	18	8	8	NUM
ejpam-6501	49	19	(	(	PUNCT
ejpam-6501	49	20	ink4	ink4	PROPN
ejpam-6501	49	21	)	)	PUNCT
ejpam-6501	49	22	x	x	PROPN
ejpam-6501	50	1	∗	∗	NOUN
ejpam-6501	50	2	y	y	NOUN
ejpam-6501	50	3	=	=	SYM
ejpam-6501	50	4	0	0	PROPN
ejpam-6501	51	1	and	and	CCONJ
ejpam-6501	51	2	y	y	PROPN
ejpam-6501	51	3	∗	∗	NOUN
ejpam-6501	51	4	x	x	PUNCT
ejpam-6501	51	5	=	=	SYM
ejpam-6501	51	6	0	0	NUM
ejpam-6501	51	7	implies	imply	VERB
ejpam-6501	51	8	x	x	PUNCT
ejpam-6501	51	9	=	=	SYM
ejpam-6501	51	10	y.	y.	PROPN
ejpam-6501	51	11	a	a	DET
ejpam-6501	51	12	non	non	ADJ
ejpam-6501	51	13	-	-	ADJ
ejpam-6501	51	14	empty	empty	ADJ
ejpam-6501	51	15	subset	subset	NOUN
ejpam-6501	51	16	s	s	NOUN
ejpam-6501	51	17	of	of	ADP
ejpam-6501	51	18	x	x	VERB
ejpam-6501	51	19	is	be	AUX
ejpam-6501	51	20	said	say	VERB
ejpam-6501	51	21	to	to	PART
ejpam-6501	51	22	be	be	AUX
ejpam-6501	51	23	a	a	DET
ejpam-6501	51	24	subalgebra	subalgebra	NOUN
ejpam-6501	51	25	of	of	ADP
ejpam-6501	51	26	x	x	PRON
ejpam-6501	51	27	if	if	SCONJ
ejpam-6501	51	28	x	x	PROPN
ejpam-6501	51	29	∗	∗	VERB
ejpam-6501	51	30	y	y	PROPN
ejpam-6501	51	31	∈	∈	PROPN
ejpam-6501	51	32	s	s	PROPN
ejpam-6501	51	33	for	for	ADP
ejpam-6501	51	34	all	all	DET
ejpam-6501	51	35	x	x	NOUN
ejpam-6501	51	36	,	,	PUNCT
ejpam-6501	51	37	y	y	PROPN
ejpam-6501	51	38	∈	∈	PROPN
ejpam-6501	51	39	s.	s.	PROPN
ejpam-6501	51	40	example	example	NOUN
ejpam-6501	52	1	1	1	NUM
ejpam-6501	52	2	.	.	PUNCT
ejpam-6501	53	1	[	[	X
ejpam-6501	53	2	4	4	NUM
ejpam-6501	53	3	]	]	PUNCT
ejpam-6501	53	4	:	:	PUNCT
ejpam-6501	53	5	let	let	VERB
ejpam-6501	53	6	x	x	PUNCT
ejpam-6501	53	7	=	=	PUNCT
ejpam-6501	53	8	{	{	PUNCT
ejpam-6501	53	9	0	0	NUM
ejpam-6501	53	10	,	,	PUNCT
ejpam-6501	53	11	1	1	NUM
ejpam-6501	53	12	,	,	PUNCT
ejpam-6501	53	13	2	2	NUM
ejpam-6501	53	14	,	,	PUNCT
ejpam-6501	53	15	3	3	NUM
ejpam-6501	53	16	}	}	PUNCT
ejpam-6501	53	17	and	and	CCONJ
ejpam-6501	53	18	a	a	DET
ejpam-6501	53	19	binary	binary	ADJ
ejpam-6501	53	20	operations	operation	NOUN
ejpam-6501	53	21	∗	∗	NOUN
ejpam-6501	53	22	on	on	ADP
ejpam-6501	53	23	x	x	PUNCT
ejpam-6501	53	24	defined	define	VERB
ejpam-6501	53	25	by	by	ADP
ejpam-6501	53	26	the	the	DET
ejpam-6501	53	27	following	follow	VERB
ejpam-6501	53	28	table	table	NOUN
ejpam-6501	53	29	:	:	PUNCT
ejpam-6501	53	30	∗	∗	NOUN
ejpam-6501	53	31	0	0	NUM
ejpam-6501	54	1	1	1	NUM
ejpam-6501	54	2	2	2	NUM
ejpam-6501	54	3	3	3	NUM
ejpam-6501	54	4	0	0	NUM
ejpam-6501	54	5	0	0	NUM
ejpam-6501	54	6	0	0	NUM
ejpam-6501	54	7	2	2	NUM
ejpam-6501	54	8	2	2	NUM
ejpam-6501	54	9	1	1	NUM
ejpam-6501	54	10	1	1	NUM
ejpam-6501	54	11	0	0	NUM
ejpam-6501	54	12	2	2	NUM
ejpam-6501	54	13	2	2	NUM
ejpam-6501	54	14	2	2	NUM
ejpam-6501	54	15	2	2	NUM
ejpam-6501	54	16	2	2	NUM
ejpam-6501	54	17	0	0	NUM
ejpam-6501	54	18	0	0	NUM
ejpam-6501	54	19	3	3	NUM
ejpam-6501	54	20	3	3	NUM
ejpam-6501	54	21	2	2	NUM
ejpam-6501	54	22	1	1	NUM
ejpam-6501	54	23	0	0	NUM
ejpam-6501	54	24	then	then	ADV
ejpam-6501	54	25	(	(	PUNCT
ejpam-6501	54	26	x	x	X
ejpam-6501	54	27	,	,	PUNCT
ejpam-6501	54	28	∗	∗	NOUN
ejpam-6501	54	29	,	,	PUNCT
ejpam-6501	54	30	0	0	NUM
ejpam-6501	54	31	)	)	PUNCT
ejpam-6501	54	32	is	be	AUX
ejpam-6501	54	33	a	a	DET
ejpam-6501	54	34	ink	ink	NOUN
ejpam-6501	54	35	-	-	PUNCT
ejpam-6501	54	36	algebra	algebra	NOUN
ejpam-6501	54	37	but	but	CCONJ
ejpam-6501	54	38	not	not	PART
ejpam-6501	54	39	a	a	DET
ejpam-6501	54	40	be	be	NOUN
ejpam-6501	54	41	-	-	PUNCT
ejpam-6501	54	42	algebra	algebra	NOUN
ejpam-6501	54	43	,	,	PUNCT
ejpam-6501	54	44	since	since	SCONJ
ejpam-6501	54	45	0	0	NUM
ejpam-6501	54	46	∗	∗	NOUN
ejpam-6501	54	47	1	1	NUM
ejpam-6501	54	48	=	=	SYM
ejpam-6501	54	49	0	0	NUM
ejpam-6501	54	50	̸=	̸=	PROPN
ejpam-6501	54	51	1	1	NUM
ejpam-6501	54	52	and	and	CCONJ
ejpam-6501	54	53	0	0	NUM
ejpam-6501	54	54	∗	∗	NOUN
ejpam-6501	54	55	3	3	NUM
ejpam-6501	54	56	=	=	SYM
ejpam-6501	54	57	2	2	NUM
ejpam-6501	54	58	̸=	̸=	PROPN
ejpam-6501	54	59	3	3	NUM
ejpam-6501	54	60	.	.	PUNCT
ejpam-6501	54	61	example	example	NOUN
ejpam-6501	55	1	2	2	NUM
ejpam-6501	55	2	.	.	PUNCT
ejpam-6501	56	1	[	[	X
ejpam-6501	56	2	5	5	X
ejpam-6501	56	3	]	]	PUNCT
ejpam-6501	56	4	let	let	VERB
ejpam-6501	56	5	x	x	PUNCT
ejpam-6501	56	6	=	=	PRON
ejpam-6501	56	7	{	{	PUNCT
ejpam-6501	56	8	1	1	NUM
ejpam-6501	56	9	,	,	PUNCT
ejpam-6501	56	10	a	a	DET
ejpam-6501	56	11	,	,	PUNCT
ejpam-6501	56	12	b	b	NOUN
ejpam-6501	56	13	,	,	PUNCT
ejpam-6501	56	14	c	c	NOUN
ejpam-6501	56	15	,	,	PUNCT
ejpam-6501	56	16	d	d	NOUN
ejpam-6501	56	17	}	}	PUNCT
ejpam-6501	56	18	with	with	ADP
ejpam-6501	56	19	a	a	DET
ejpam-6501	56	20	binary	binary	ADJ
ejpam-6501	56	21	operations	operation	NOUN
ejpam-6501	56	22	∗	∗	NOUN
ejpam-6501	56	23	on	on	ADP
ejpam-6501	56	24	x	x	PUNCT
ejpam-6501	56	25	defined	define	VERB
ejpam-6501	56	26	by	by	ADP
ejpam-6501	56	27	the	the	DET
ejpam-6501	56	28	following	follow	VERB
ejpam-6501	56	29	table	table	NOUN
ejpam-6501	56	30	:	:	PUNCT
ejpam-6501	56	31	∗	∗	NOUN
ejpam-6501	56	32	1	1	NUM
ejpam-6501	56	33	a	a	DET
ejpam-6501	56	34	b	b	NOUN
ejpam-6501	56	35	c	c	NOUN
ejpam-6501	57	1	d	d	SYM
ejpam-6501	57	2	1	1	NUM
ejpam-6501	57	3	1	1	NUM
ejpam-6501	57	4	a	a	DET
ejpam-6501	57	5	b	b	NOUN
ejpam-6501	57	6	c	c	NOUN
ejpam-6501	57	7	d	d	NOUN
ejpam-6501	57	8	a	a	PRON
ejpam-6501	57	9	1	1	NUM
ejpam-6501	57	10	1	1	NUM
ejpam-6501	57	11	b	b	NOUN
ejpam-6501	57	12	c	c	NOUN
ejpam-6501	57	13	d	d	PROPN
ejpam-6501	57	14	b	b	PROPN
ejpam-6501	57	15	1	1	NUM
ejpam-6501	57	16	a	a	DET
ejpam-6501	57	17	1	1	NUM
ejpam-6501	57	18	c	c	NOUN
ejpam-6501	57	19	c	c	NOUN
ejpam-6501	57	20	c	c	NOUN
ejpam-6501	57	21	1	1	NUM
ejpam-6501	57	22	1	1	NUM
ejpam-6501	57	23	b	b	SYM
ejpam-6501	57	24	1	1	NUM
ejpam-6501	57	25	b	b	SYM
ejpam-6501	57	26	d	d	SYM
ejpam-6501	57	27	1	1	NUM
ejpam-6501	57	28	1	1	NUM
ejpam-6501	57	29	1	1	NUM
ejpam-6501	57	30	1	1	NUM
ejpam-6501	57	31	1	1	NUM
ejpam-6501	57	32	then	then	ADV
ejpam-6501	57	33	(	(	PUNCT
ejpam-6501	57	34	x	x	X
ejpam-6501	57	35	,	,	PUNCT
ejpam-6501	57	36	∗	∗	NOUN
ejpam-6501	57	37	,	,	PUNCT
ejpam-6501	57	38	0	0	NUM
ejpam-6501	57	39	)	)	PUNCT
ejpam-6501	57	40	is	be	AUX
ejpam-6501	57	41	a	a	DET
ejpam-6501	57	42	be	be	NOUN
ejpam-6501	57	43	-	-	PUNCT
ejpam-6501	57	44	algebra	algebra	NOUN
ejpam-6501	57	45	but	but	CCONJ
ejpam-6501	57	46	not	not	PART
ejpam-6501	57	47	an	an	DET
ejpam-6501	57	48	ink	ink	NOUN
ejpam-6501	57	49	-	-	PUNCT
ejpam-6501	57	50	algebra	algebra	NOUN
ejpam-6501	57	51	,	,	PUNCT
ejpam-6501	57	52	since	since	SCONJ
ejpam-6501	57	53	a	a	DET
ejpam-6501	57	54	∗	∗	NOUN
ejpam-6501	57	55	1	1	NUM
ejpam-6501	57	56	=	=	SYM
ejpam-6501	57	57	1	1	NUM
ejpam-6501	57	58	̸=	̸=	PROPN
ejpam-6501	57	59	a.	a.	NOUN
ejpam-6501	57	60	definition	definition	NOUN
ejpam-6501	57	61	3	3	NUM
ejpam-6501	57	62	.	.	PUNCT
ejpam-6501	58	1	[	[	X
ejpam-6501	58	2	3	3	X
ejpam-6501	58	3	]	]	X
ejpam-6501	58	4	let	let	VERB
ejpam-6501	58	5	(	(	PUNCT
ejpam-6501	58	6	x	x	NOUN
ejpam-6501	58	7	,	,	PUNCT
ejpam-6501	58	8	∗	∗	NOUN
ejpam-6501	58	9	,	,	PUNCT
ejpam-6501	58	10	0	0	NUM
ejpam-6501	58	11	)	)	PUNCT
ejpam-6501	58	12	be	be	AUX
ejpam-6501	58	13	a	a	DET
ejpam-6501	58	14	ink	ink	NOUN
ejpam-6501	58	15	-	-	PUNCT
ejpam-6501	58	16	algebra	algebra	NOUN
ejpam-6501	58	17	.	.	PUNCT
ejpam-6501	59	1	a	a	DET
ejpam-6501	59	2	nonempty	nonempty	NOUN
ejpam-6501	59	3	subset	subset	VERB
ejpam-6501	59	4	i	i	PRON
ejpam-6501	59	5	of	of	ADP
ejpam-6501	59	6	x	x	PRON
ejpam-6501	59	7	is	be	AUX
ejpam-6501	59	8	called	call	VERB
ejpam-6501	59	9	an	an	DET
ejpam-6501	59	10	ideal	ideal	NOUN
ejpam-6501	59	11	of	of	ADP
ejpam-6501	59	12	x	x	PRON
ejpam-6501	59	13	if	if	SCONJ
ejpam-6501	59	14	it	it	PRON
ejpam-6501	59	15	satisfies	satisfy	VERB
ejpam-6501	59	16	the	the	DET
ejpam-6501	59	17	following	follow	VERB
ejpam-6501	59	18	conditions	condition	NOUN
ejpam-6501	59	19	:	:	PUNCT
ejpam-6501	59	20	(	(	PUNCT
ejpam-6501	59	21	i	i	NOUN
ejpam-6501	59	22	)	)	PUNCT
ejpam-6501	59	23	0	0	PUNCT
ejpam-6501	60	1	∈	∈	PROPN
ejpam-6501	61	1	i	i	PRON
ejpam-6501	61	2	,	,	PUNCT
ejpam-6501	61	3	(	(	PUNCT
ejpam-6501	61	4	ii	ii	NOUN
ejpam-6501	61	5	)	)	PUNCT
ejpam-6501	61	6	x	x	SYM
ejpam-6501	62	1	∗	∗	NOUN
ejpam-6501	62	2	y	y	NOUN
ejpam-6501	62	3	∈	∈	PROPN
ejpam-6501	63	1	i	i	PRON
ejpam-6501	63	2	and	and	CCONJ
ejpam-6501	63	3	y	y	PROPN
ejpam-6501	63	4	∈	∈	PROPN
ejpam-6501	64	1	i	i	PRON
ejpam-6501	64	2	imply	imply	VERB
ejpam-6501	64	3	that	that	SCONJ
ejpam-6501	64	4	x	x	X
ejpam-6501	64	5	∈	∈	NOUN
ejpam-6501	64	6	i	i	PRON
ejpam-6501	64	7	for	for	ADP
ejpam-6501	64	8	all	all	DET
ejpam-6501	64	9	x	x	NOUN
ejpam-6501	64	10	,	,	PUNCT
ejpam-6501	64	11	y	y	PROPN
ejpam-6501	64	12	∈	∈	PROPN
ejpam-6501	64	13	x.	x.	NOUN
ejpam-6501	64	14	theorem	theorem	VERB
ejpam-6501	64	15	1	1	NUM
ejpam-6501	64	16	.	.	PUNCT
ejpam-6501	65	1	[	[	X
ejpam-6501	65	2	2	2	NUM
ejpam-6501	65	3	]	]	X
ejpam-6501	65	4	let	let	VERB
ejpam-6501	65	5	(	(	PUNCT
ejpam-6501	65	6	x	x	NOUN
ejpam-6501	65	7	,	,	PUNCT
ejpam-6501	65	8	∗	∗	NOUN
ejpam-6501	65	9	,	,	PUNCT
ejpam-6501	65	10	0	0	NUM
ejpam-6501	65	11	)	)	PUNCT
ejpam-6501	65	12	be	be	AUX
ejpam-6501	65	13	an	an	DET
ejpam-6501	65	14	ink	ink	NOUN
ejpam-6501	65	15	-	-	PUNCT
ejpam-6501	65	16	algebra	algebra	NOUN
ejpam-6501	65	17	.	.	PUNCT
ejpam-6501	66	1	then	then	ADV
ejpam-6501	66	2	the	the	DET
ejpam-6501	66	3	following	follow	VERB
ejpam-6501	66	4	conditions	condition	NOUN
ejpam-6501	66	5	hold	hold	VERB
ejpam-6501	66	6	for	for	ADP
ejpam-6501	66	7	all	all	DET
ejpam-6501	66	8	x	x	NOUN
ejpam-6501	66	9	,	,	PUNCT
ejpam-6501	66	10	y	y	PROPN
ejpam-6501	66	11	,	,	PUNCT
ejpam-6501	66	12	z	z	PROPN
ejpam-6501	66	13	∈	∈	PROPN
ejpam-6501	66	14	x	x	X
ejpam-6501	66	15	;	;	PUNCT
ejpam-6501	66	16	(	(	PUNCT
ejpam-6501	66	17	i	i	NOUN
ejpam-6501	66	18	)	)	PUNCT
ejpam-6501	66	19	x	x	SYM
ejpam-6501	66	20	∗	∗	NOUN
ejpam-6501	66	21	0	0	NUM
ejpam-6501	67	1	=	=	SYM
ejpam-6501	67	2	0	0	NUM
ejpam-6501	67	3	implies	imply	VERB
ejpam-6501	67	4	x	x	PUNCT
ejpam-6501	67	5	=	=	SYM
ejpam-6501	67	6	0	0	NUM
ejpam-6501	67	7	,	,	PUNCT
ejpam-6501	67	8	(	(	PUNCT
ejpam-6501	67	9	ii	ii	NOUN
ejpam-6501	67	10	)	)	PUNCT
ejpam-6501	67	11	(	(	PUNCT
ejpam-6501	67	12	x	x	SYM
ejpam-6501	67	13	∗	∗	NOUN
ejpam-6501	67	14	(	(	PUNCT
ejpam-6501	67	15	x	x	X
ejpam-6501	67	16	∗	∗	PROPN
ejpam-6501	67	17	y	y	NOUN
ejpam-6501	67	18	)	)	PUNCT
ejpam-6501	67	19	∗	∗	PROPN
ejpam-6501	67	20	y	y	NOUN
ejpam-6501	67	21	)	)	PUNCT
ejpam-6501	67	22	=	=	SYM
ejpam-6501	67	23	0	0	NUM
ejpam-6501	67	24	,	,	PUNCT
ejpam-6501	67	25	(	(	PUNCT
ejpam-6501	67	26	iii	iii	NOUN
ejpam-6501	67	27	)	)	PUNCT
ejpam-6501	67	28	0	0	NUM
ejpam-6501	68	1	∗	∗	NOUN
ejpam-6501	68	2	(	(	PUNCT
ejpam-6501	68	3	x	x	X
ejpam-6501	68	4	∗	∗	PROPN
ejpam-6501	68	5	y	y	NOUN
ejpam-6501	68	6	)	)	PUNCT
ejpam-6501	68	7	=	=	SYM
ejpam-6501	68	8	(	(	PUNCT
ejpam-6501	68	9	0	0	NUM
ejpam-6501	68	10	∗	∗	NOUN
ejpam-6501	68	11	x	x	NOUN
ejpam-6501	68	12	)	)	PUNCT
ejpam-6501	68	13	∗	∗	NOUN
ejpam-6501	68	14	(	(	PUNCT
ejpam-6501	68	15	0	0	NUM
ejpam-6501	68	16	∗	∗	PROPN
ejpam-6501	68	17	y	y	PROPN
ejpam-6501	68	18	)	)	PUNCT
ejpam-6501	68	19	,	,	PUNCT
ejpam-6501	68	20	m.	m.	NOUN
ejpam-6501	68	21	phattarachaleekul	phattarachaleekul	PROPN
ejpam-6501	68	22	/	/	SYM
ejpam-6501	68	23	eur	eur	PROPN
ejpam-6501	68	24	.	.	PUNCT
ejpam-6501	69	1	j.	j.	PROPN
ejpam-6501	69	2	pure	pure	PROPN
ejpam-6501	69	3	appl	appl	PROPN
ejpam-6501	69	4	.	.	PROPN
ejpam-6501	69	5	math	math	PROPN
ejpam-6501	69	6	,	,	PUNCT
ejpam-6501	69	7	18	18	NUM
ejpam-6501	69	8	(	(	PUNCT
ejpam-6501	69	9	4	4	NUM
ejpam-6501	69	10	)	)	PUNCT
ejpam-6501	69	11	(	(	PUNCT
ejpam-6501	69	12	2025	2025	NUM
ejpam-6501	69	13	)	)	PUNCT
ejpam-6501	69	14	,	,	PUNCT
ejpam-6501	69	15	6501	6501	NUM
ejpam-6501	69	16	4	4	NUM
ejpam-6501	69	17	of	of	ADP
ejpam-6501	69	18	8	8	NUM
ejpam-6501	69	19	(	(	PUNCT
ejpam-6501	69	20	iv	iv	X
ejpam-6501	69	21	)	)	PUNCT
ejpam-6501	69	22	x	x	PROPN
ejpam-6501	69	23	∗	∗	NOUN
ejpam-6501	69	24	y	y	NOUN
ejpam-6501	69	25	=	=	SYM
ejpam-6501	69	26	0	0	PROPN
ejpam-6501	70	1	and	and	CCONJ
ejpam-6501	70	2	y	y	PROPN
ejpam-6501	70	3	∗	∗	NOUN
ejpam-6501	70	4	x	x	PUNCT
ejpam-6501	70	5	=	=	SYM
ejpam-6501	70	6	0	0	NUM
ejpam-6501	70	7	implies	imply	VERB
ejpam-6501	70	8	x	x	PUNCT
ejpam-6501	70	9	=	=	SYM
ejpam-6501	70	10	y	y	PROPN
ejpam-6501	70	11	,	,	PUNCT
ejpam-6501	70	12	(	(	PUNCT
ejpam-6501	70	13	v	v	NOUN
ejpam-6501	70	14	)	)	PUNCT
ejpam-6501	70	15	(	(	PUNCT
ejpam-6501	70	16	x	x	SYM
ejpam-6501	70	17	∗	∗	PROPN
ejpam-6501	70	18	y	y	NOUN
ejpam-6501	70	19	)	)	PUNCT
ejpam-6501	70	20	∗	∗	NOUN
ejpam-6501	70	21	z	z	NOUN
ejpam-6501	70	22	=	=	SYM
ejpam-6501	70	23	(	(	PUNCT
ejpam-6501	70	24	x	x	X
ejpam-6501	70	25	∗	∗	PROPN
ejpam-6501	70	26	z	z	NOUN
ejpam-6501	70	27	)	)	PUNCT
ejpam-6501	70	28	∗	∗	PROPN
ejpam-6501	70	29	y	y	PROPN
ejpam-6501	70	30	,	,	PUNCT
ejpam-6501	70	31	3	3	X
ejpam-6501	70	32	.	.	PUNCT
ejpam-6501	70	33	congruence	congruence	PROPN
ejpam-6501	70	34	relations	relation	NOUN
ejpam-6501	70	35	and	and	CCONJ
ejpam-6501	70	36	partitions	partition	NOUN
ejpam-6501	70	37	on	on	ADP
ejpam-6501	70	38	ink	ink	NOUN
ejpam-6501	70	39	-	-	PUNCT
ejpam-6501	70	40	algebras	algebras	NOUN
ejpam-6501	70	41	in	in	ADP
ejpam-6501	70	42	the	the	DET
ejpam-6501	70	43	study	study	NOUN
ejpam-6501	70	44	of	of	ADP
ejpam-6501	70	45	algebraic	algebraic	ADJ
ejpam-6501	70	46	structures	structure	NOUN
ejpam-6501	70	47	,	,	PUNCT
ejpam-6501	70	48	congruence	congruence	PROPN
ejpam-6501	70	49	relations	relation	NOUN
ejpam-6501	70	50	play	play	VERB
ejpam-6501	70	51	a	a	DET
ejpam-6501	70	52	fundamental	fundamental	ADJ
ejpam-6501	70	53	role	role	NOUN
ejpam-6501	70	54	in	in	ADP
ejpam-6501	70	55	constructing	construct	VERB
ejpam-6501	70	56	quotient	quotient	NOUN
ejpam-6501	70	57	structures	structure	NOUN
ejpam-6501	70	58	and	and	CCONJ
ejpam-6501	70	59	analyzing	analyze	VERB
ejpam-6501	70	60	their	their	PRON
ejpam-6501	70	61	properties	property	NOUN
ejpam-6501	70	62	.	.	PUNCT
ejpam-6501	71	1	for	for	ADP
ejpam-6501	71	2	ink	ink	NOUN
ejpam-6501	71	3	-	-	PUNCT
ejpam-6501	71	4	algebras	algebra	NOUN
ejpam-6501	71	5	,	,	PUNCT
ejpam-6501	71	6	the	the	DET
ejpam-6501	71	7	characterization	characterization	NOUN
ejpam-6501	71	8	of	of	ADP
ejpam-6501	71	9	congruence	congruence	PROPN
ejpam-6501	71	10	relations	relation	NOUN
ejpam-6501	71	11	provides	provide	VERB
ejpam-6501	71	12	a	a	DET
ejpam-6501	71	13	pathway	pathway	NOUN
ejpam-6501	71	14	to	to	ADP
ejpam-6501	71	15	understanding	understand	VERB
ejpam-6501	71	16	the	the	DET
ejpam-6501	71	17	internal	internal	ADJ
ejpam-6501	71	18	symmetry	symmetry	NOUN
ejpam-6501	71	19	and	and	CCONJ
ejpam-6501	71	20	decomposability	decomposability	NOUN
ejpam-6501	71	21	of	of	ADP
ejpam-6501	71	22	such	such	ADJ
ejpam-6501	71	23	algebras	algebra	NOUN
ejpam-6501	71	24	via	via	ADP
ejpam-6501	71	25	partitions	partition	NOUN
ejpam-6501	71	26	induced	induce	VERB
ejpam-6501	71	27	by	by	ADP
ejpam-6501	71	28	these	these	DET
ejpam-6501	71	29	relations	relation	NOUN
ejpam-6501	71	30	.	.	PUNCT
ejpam-6501	72	1	definition	definition	NOUN
ejpam-6501	72	2	4	4	NUM
ejpam-6501	72	3	.	.	PUNCT
ejpam-6501	73	1	let	let	VERB
ejpam-6501	73	2	∼	∼	NOUN
ejpam-6501	73	3	be	be	AUX
ejpam-6501	73	4	a	a	DET
ejpam-6501	73	5	binary	binary	ADJ
ejpam-6501	73	6	relation	relation	NOUN
ejpam-6501	73	7	on	on	ADP
ejpam-6501	73	8	a	a	DET
ejpam-6501	73	9	set	set	NOUN
ejpam-6501	73	10	x.	x.	NOUN
ejpam-6501	73	11	then	then	ADV
ejpam-6501	73	12	∼	∼	NOUN
ejpam-6501	73	13	is	be	AUX
ejpam-6501	73	14	called	call	VERB
ejpam-6501	73	15	:	:	PUNCT
ejpam-6501	73	16	(	(	PUNCT
ejpam-6501	73	17	i	i	NOUN
ejpam-6501	73	18	)	)	PUNCT
ejpam-6501	73	19	reflexive	reflexive	VERB
ejpam-6501	73	20	if	if	SCONJ
ejpam-6501	73	21	x	x	PUNCT
ejpam-6501	73	22	∼	∼	NOUN
ejpam-6501	73	23	x	x	PUNCT
ejpam-6501	73	24	for	for	ADP
ejpam-6501	73	25	all	all	DET
ejpam-6501	73	26	x	x	SYM
ejpam-6501	73	27	∈	∈	PROPN
ejpam-6501	73	28	x	x	X
ejpam-6501	73	29	;	;	PUNCT
ejpam-6501	73	30	(	(	PUNCT
ejpam-6501	73	31	ii	ii	NOUN
ejpam-6501	73	32	)	)	PUNCT
ejpam-6501	73	33	symmetric	symmetric	NOUN
ejpam-6501	73	34	if	if	SCONJ
ejpam-6501	73	35	x	x	PRON
ejpam-6501	73	36	∼	∼	NOUN
ejpam-6501	73	37	y	y	PROPN
ejpam-6501	73	38	implies	imply	VERB
ejpam-6501	73	39	y	y	PROPN
ejpam-6501	73	40	∼	∼	VERB
ejpam-6501	73	41	x	x	PUNCT
ejpam-6501	73	42	for	for	ADP
ejpam-6501	73	43	all	all	DET
ejpam-6501	73	44	x	x	NOUN
ejpam-6501	73	45	,	,	PUNCT
ejpam-6501	73	46	y	y	PROPN
ejpam-6501	73	47	∈	∈	PROPN
ejpam-6501	74	1	x	x	X
ejpam-6501	74	2	;	;	PUNCT
ejpam-6501	74	3	(	(	PUNCT
ejpam-6501	74	4	iii	iii	X
ejpam-6501	74	5	)	)	PUNCT
ejpam-6501	74	6	transitive	transitive	ADJ
ejpam-6501	74	7	if	if	SCONJ
ejpam-6501	74	8	x	x	ADP
ejpam-6501	74	9	∼	∼	NOUN
ejpam-6501	74	10	y	y	NOUN
ejpam-6501	74	11	and	and	CCONJ
ejpam-6501	74	12	y	y	PROPN
ejpam-6501	74	13	∼	∼	NOUN
ejpam-6501	74	14	z	z	PROPN
ejpam-6501	74	15	implies	imply	VERB
ejpam-6501	74	16	x	x	PUNCT
ejpam-6501	74	17	∼	∼	NOUN
ejpam-6501	74	18	z	z	NOUN
ejpam-6501	74	19	for	for	ADP
ejpam-6501	74	20	all	all	DET
ejpam-6501	74	21	x	x	NOUN
ejpam-6501	74	22	,	,	PUNCT
ejpam-6501	74	23	y	y	PROPN
ejpam-6501	74	24	,	,	PUNCT
ejpam-6501	74	25	z	z	PROPN
ejpam-6501	74	26	∈	∈	PROPN
ejpam-6501	75	1	x	x	X
ejpam-6501	75	2	;	;	PUNCT
ejpam-6501	75	3	(	(	PUNCT
ejpam-6501	75	4	iv	iv	X
ejpam-6501	75	5	)	)	PUNCT
ejpam-6501	75	6	compatible	compatible	ADJ
ejpam-6501	75	7	if	if	SCONJ
ejpam-6501	75	8	a	a	DET
ejpam-6501	75	9	∼	∼	NOUN
ejpam-6501	75	10	b	b	NOUN
ejpam-6501	75	11	and	and	CCONJ
ejpam-6501	75	12	x	x	ADJ
ejpam-6501	75	13	∼	∼	NOUN
ejpam-6501	75	14	y	y	PROPN
ejpam-6501	75	15	implies	imply	VERB
ejpam-6501	75	16	a	a	DET
ejpam-6501	75	17	∗	∗	NOUN
ejpam-6501	75	18	x	x	PUNCT
ejpam-6501	75	19	∼	∼	NOUN
ejpam-6501	75	20	b	b	NOUN
ejpam-6501	75	21	∗	∗	X
ejpam-6501	75	22	y	y	NOUN
ejpam-6501	75	23	for	for	ADP
ejpam-6501	75	24	all	all	DET
ejpam-6501	75	25	a	a	DET
ejpam-6501	75	26	,	,	PUNCT
ejpam-6501	75	27	b	b	NOUN
ejpam-6501	75	28	,	,	PUNCT
ejpam-6501	75	29	x	x	X
ejpam-6501	75	30	,	,	PUNCT
ejpam-6501	75	31	y	y	PROPN
ejpam-6501	75	32	∈	∈	PROPN
ejpam-6501	75	33	x.	x.	NOUN
ejpam-6501	76	1	a	a	DET
ejpam-6501	76	2	relation	relation	NOUN
ejpam-6501	76	3	∼	∼	NOUN
ejpam-6501	76	4	is	be	AUX
ejpam-6501	76	5	said	say	VERB
ejpam-6501	76	6	to	to	PART
ejpam-6501	76	7	be	be	AUX
ejpam-6501	76	8	an	an	DET
ejpam-6501	76	9	equivalence	equivalence	NOUN
ejpam-6501	76	10	relation	relation	NOUN
ejpam-6501	76	11	if	if	SCONJ
ejpam-6501	76	12	∼	∼	NOUN
ejpam-6501	76	13	is	be	AUX
ejpam-6501	76	14	reflexive	reflexive	ADJ
ejpam-6501	76	15	,	,	PUNCT
ejpam-6501	76	16	symmetric	symmetric	ADJ
ejpam-6501	76	17	and	and	CCONJ
ejpam-6501	76	18	transitive	transitive	ADJ
ejpam-6501	76	19	.	.	PUNCT
ejpam-6501	77	1	an	an	DET
ejpam-6501	77	2	equivalence	equivalence	NOUN
ejpam-6501	77	3	relation	relation	NOUN
ejpam-6501	77	4	on	on	ADP
ejpam-6501	77	5	x	x	SYM
ejpam-6501	77	6	that	that	PRON
ejpam-6501	77	7	is	be	AUX
ejpam-6501	77	8	compatible	compatible	ADJ
ejpam-6501	77	9	with	with	ADP
ejpam-6501	77	10	the	the	DET
ejpam-6501	77	11	operation	operation	NOUN
ejpam-6501	77	12	∗	∗	NOUN
ejpam-6501	77	13	is	be	AUX
ejpam-6501	77	14	called	call	VERB
ejpam-6501	77	15	a	a	DET
ejpam-6501	77	16	congruence	congruence	NOUN
ejpam-6501	77	17	on	on	ADP
ejpam-6501	77	18	x.	x.	PROPN
ejpam-6501	77	19	example	example	NOUN
ejpam-6501	77	20	3	3	NUM
ejpam-6501	77	21	.	.	PUNCT
ejpam-6501	78	1	[	[	X
ejpam-6501	78	2	3	3	X
ejpam-6501	78	3	]	]	X
ejpam-6501	78	4	let	let	VERB
ejpam-6501	78	5	x	x	PUNCT
ejpam-6501	78	6	=	=	PUNCT
ejpam-6501	78	7	{	{	PUNCT
ejpam-6501	78	8	0	0	NUM
ejpam-6501	78	9	,	,	PUNCT
ejpam-6501	78	10	1	1	NUM
ejpam-6501	78	11	,	,	PUNCT
ejpam-6501	78	12	a	a	DET
ejpam-6501	78	13	,	,	PUNCT
ejpam-6501	78	14	b	b	NOUN
ejpam-6501	78	15	}	}	PUNCT
ejpam-6501	78	16	with	with	ADP
ejpam-6501	78	17	a	a	DET
ejpam-6501	78	18	binary	binary	ADJ
ejpam-6501	78	19	operations	operation	NOUN
ejpam-6501	78	20	∗	∗	NOUN
ejpam-6501	78	21	on	on	ADP
ejpam-6501	78	22	x	x	PUNCT
ejpam-6501	78	23	defined	define	VERB
ejpam-6501	78	24	by	by	ADP
ejpam-6501	78	25	the	the	DET
ejpam-6501	78	26	following	follow	VERB
ejpam-6501	78	27	table	table	NOUN
ejpam-6501	78	28	:	:	PUNCT
ejpam-6501	78	29	∗	∗	NOUN
ejpam-6501	78	30	0	0	NUM
ejpam-6501	78	31	1	1	NUM
ejpam-6501	78	32	a	a	DET
ejpam-6501	78	33	b	b	NOUN
ejpam-6501	78	34	0	0	NUM
ejpam-6501	78	35	0	0	NUM
ejpam-6501	78	36	0	0	NUM
ejpam-6501	78	37	a	a	DET
ejpam-6501	78	38	a	a	DET
ejpam-6501	78	39	1	1	NUM
ejpam-6501	78	40	1	1	NUM
ejpam-6501	78	41	0	0	NUM
ejpam-6501	78	42	b	b	NOUN
ejpam-6501	78	43	a	a	DET
ejpam-6501	78	44	a	a	PRON
ejpam-6501	78	45	a	a	PRON
ejpam-6501	78	46	a	a	DET
ejpam-6501	78	47	0	0	NUM
ejpam-6501	78	48	0	0	NUM
ejpam-6501	79	1	b	b	X
ejpam-6501	79	2	b	b	PROPN
ejpam-6501	79	3	a	a	DET
ejpam-6501	79	4	1	1	NUM
ejpam-6501	79	5	0	0	NUM
ejpam-6501	79	6	then	then	ADV
ejpam-6501	79	7	(	(	PUNCT
ejpam-6501	79	8	x	x	X
ejpam-6501	79	9	,	,	PUNCT
ejpam-6501	79	10	∗	∗	NOUN
ejpam-6501	79	11	,	,	PUNCT
ejpam-6501	79	12	0	0	NUM
ejpam-6501	79	13	)	)	PUNCT
ejpam-6501	79	14	is	be	AUX
ejpam-6501	79	15	a	a	DET
ejpam-6501	79	16	ink	ink	NOUN
ejpam-6501	79	17	-	-	PUNCT
ejpam-6501	79	18	algebra	algebra	NOUN
ejpam-6501	79	19	.	.	PUNCT
ejpam-6501	80	1	the	the	DET
ejpam-6501	80	2	set	set	NOUN
ejpam-6501	80	3	i	i	NOUN
ejpam-6501	80	4	=	=	PUNCT
ejpam-6501	80	5	{	{	PUNCT
ejpam-6501	80	6	0	0	NUM
ejpam-6501	80	7	,	,	PUNCT
ejpam-6501	80	8	a	a	PRON
ejpam-6501	80	9	}	}	PUNCT
ejpam-6501	80	10	is	be	AUX
ejpam-6501	80	11	an	an	DET
ejpam-6501	80	12	ideal	ideal	NOUN
ejpam-6501	80	13	of	of	ADP
ejpam-6501	80	14	x	x	PUNCT
ejpam-6501	80	15	and	and	CCONJ
ejpam-6501	80	16	the	the	DET
ejpam-6501	80	17	relation	relation	NOUN
ejpam-6501	80	18	∼	∼	NOUN
ejpam-6501	80	19	i	i	PRON
ejpam-6501	80	20	=	=	SYM
ejpam-6501	80	21	{	{	PUNCT
ejpam-6501	80	22	(	(	PUNCT
ejpam-6501	80	23	0,0),(1,1),(a	0,0),(1,1),(a	PROPN
ejpam-6501	80	24	,	,	PUNCT
ejpam-6501	80	25	a),(b	a),(b	PROPN
ejpam-6501	80	26	,	,	PUNCT
ejpam-6501	80	27	b),(0,a),(a,0),(1,b),(b,1	b),(0,a),(a,0),(1,b),(b,1	PROPN
ejpam-6501	80	28	)	)	PUNCT
ejpam-6501	80	29	}	}	PUNCT
ejpam-6501	80	30	is	be	AUX
ejpam-6501	80	31	a	a	DET
ejpam-6501	80	32	congruence	congruence	NOUN
ejpam-6501	80	33	on	on	ADP
ejpam-6501	80	34	x.	x.	NOUN
ejpam-6501	80	35	theorem	theorem	VERB
ejpam-6501	80	36	2	2	X
ejpam-6501	80	37	.	.	PUNCT
ejpam-6501	81	1	let	let	VERB
ejpam-6501	81	2	(	(	PUNCT
ejpam-6501	81	3	x	x	X
ejpam-6501	81	4	,	,	PUNCT
ejpam-6501	81	5	∗	∗	NOUN
ejpam-6501	81	6	,	,	PUNCT
ejpam-6501	81	7	0	0	NUM
ejpam-6501	81	8	)	)	PUNCT
ejpam-6501	81	9	be	be	AUX
ejpam-6501	81	10	a	a	DET
ejpam-6501	81	11	ink	ink	NOUN
ejpam-6501	81	12	-	-	PUNCT
ejpam-6501	81	13	algebra	algebra	NOUN
ejpam-6501	82	1	and	and	CCONJ
ejpam-6501	82	2	i	i	PRON
ejpam-6501	82	3	is	be	AUX
ejpam-6501	82	4	an	an	DET
ejpam-6501	82	5	ideal	ideal	NOUN
ejpam-6501	82	6	of	of	ADP
ejpam-6501	82	7	x.	x.	NOUN
ejpam-6501	82	8	then	then	ADV
ejpam-6501	82	9	the	the	DET
ejpam-6501	82	10	relation	relation	NOUN
ejpam-6501	82	11	∼	∼	NOUN
ejpam-6501	82	12	i	i	PRON
ejpam-6501	82	13	=	=	PUNCT
ejpam-6501	82	14	{	{	PUNCT
ejpam-6501	82	15	(	(	PUNCT
ejpam-6501	82	16	x	x	NOUN
ejpam-6501	82	17	,	,	PUNCT
ejpam-6501	82	18	y	y	NOUN
ejpam-6501	82	19	)	)	PUNCT
ejpam-6501	82	20	∈	∈	PROPN
ejpam-6501	83	1	x	x	PUNCT
ejpam-6501	83	2	×x|	×x|	PROPN
ejpam-6501	83	3	x	x	SYM
ejpam-6501	83	4	∗	∗	NOUN
ejpam-6501	83	5	y	y	NOUN
ejpam-6501	83	6	∈	∈	PROPN
ejpam-6501	84	1	i	i	PRON
ejpam-6501	84	2	and	and	CCONJ
ejpam-6501	84	3	y	y	PROPN
ejpam-6501	84	4	∗	∗	NOUN
ejpam-6501	84	5	x	x	PUNCT
ejpam-6501	84	6	∈	∈	PROPN
ejpam-6501	84	7	i	i	PRON
ejpam-6501	84	8	}	}	PUNCT
ejpam-6501	84	9	is	be	AUX
ejpam-6501	84	10	a	a	DET
ejpam-6501	84	11	congruence	congruence	NOUN
ejpam-6501	84	12	on	on	ADP
ejpam-6501	84	13	x.	x.	NOUN
ejpam-6501	84	14	proof	proof	NOUN
ejpam-6501	84	15	.	.	PUNCT
ejpam-6501	85	1	since	since	SCONJ
ejpam-6501	85	2	x	x	X
ejpam-6501	85	3	∗	∗	NOUN
ejpam-6501	85	4	x	x	X
ejpam-6501	85	5	=	=	SYM
ejpam-6501	85	6	0	0	NUM
ejpam-6501	85	7	∈	∈	PROPN
ejpam-6501	85	8	i	i	PRON
ejpam-6501	85	9	for	for	ADP
ejpam-6501	85	10	all	all	DET
ejpam-6501	85	11	x	x	SYM
ejpam-6501	85	12	∈	∈	PROPN
ejpam-6501	85	13	x	x	NOUN
ejpam-6501	85	14	,	,	PUNCT
ejpam-6501	85	15	that	that	PRON
ejpam-6501	85	16	is	is	ADV
ejpam-6501	85	17	x	x	INTJ
ejpam-6501	85	18	∼	∼	NOUN
ejpam-6501	85	19	i	i	PRON
ejpam-6501	85	20	x.	x.	NOUN
ejpam-6501	85	21	if	if	SCONJ
ejpam-6501	85	22	x	x	X
ejpam-6501	85	23	,	,	PUNCT
ejpam-6501	85	24	y	y	PROPN
ejpam-6501	85	25	∈	∈	PROPN
ejpam-6501	85	26	x	x	X
ejpam-6501	85	27	and	and	CCONJ
ejpam-6501	85	28	x	x	ADJ
ejpam-6501	85	29	∼	∼	NOUN
ejpam-6501	85	30	i	i	NOUN
ejpam-6501	85	31	y	y	NOUN
ejpam-6501	85	32	,	,	PUNCT
ejpam-6501	85	33	clearly	clearly	ADV
ejpam-6501	85	34	y	y	VERB
ejpam-6501	85	35	∼	∼	NOUN
ejpam-6501	85	36	i	i	PRON
ejpam-6501	85	37	x.	x.	VERB
ejpam-6501	86	1	next	next	ADJ
ejpam-6501	86	2	if	if	SCONJ
ejpam-6501	86	3	x	x	ADP
ejpam-6501	86	4	∼	∼	VERB
ejpam-6501	86	5	i	i	PRON
ejpam-6501	86	6	y	y	PROPN
ejpam-6501	86	7	and	and	CCONJ
ejpam-6501	86	8	y	y	PROPN
ejpam-6501	86	9	∼	∼	NOUN
ejpam-6501	87	1	i	i	NOUN
ejpam-6501	87	2	z	z	NOUN
ejpam-6501	87	3	,	,	PUNCT
ejpam-6501	87	4	then	then	ADV
ejpam-6501	87	5	x	x	X
ejpam-6501	87	6	∗	∗	VERB
ejpam-6501	87	7	y	y	PROPN
ejpam-6501	87	8	∈	∈	PROPN
ejpam-6501	88	1	i	i	PRON
ejpam-6501	88	2	,	,	PUNCT
ejpam-6501	88	3	y	y	PROPN
ejpam-6501	88	4	∗	∗	NOUN
ejpam-6501	88	5	x	x	PUNCT
ejpam-6501	88	6	∈	∈	PROPN
ejpam-6501	89	1	i	i	PRON
ejpam-6501	89	2	,	,	PUNCT
ejpam-6501	89	3	y	y	PROPN
ejpam-6501	89	4	∗	∗	NOUN
ejpam-6501	89	5	z	z	PROPN
ejpam-6501	89	6	∈	∈	PROPN
ejpam-6501	90	1	i	i	PRON
ejpam-6501	90	2	and	and	CCONJ
ejpam-6501	90	3	z	z	PROPN
ejpam-6501	90	4	∗	∗	NOUN
ejpam-6501	90	5	y	y	PROPN
ejpam-6501	90	6	∈	∈	PROPN
ejpam-6501	90	7	i.	i.	NOUN
ejpam-6501	90	8	by	by	ADP
ejpam-6501	90	9	m.	m.	PROPN
ejpam-6501	90	10	phattarachaleekul	phattarachaleekul	PROPN
ejpam-6501	90	11	/	/	SYM
ejpam-6501	90	12	eur	eur	PROPN
ejpam-6501	90	13	.	.	PUNCT
ejpam-6501	91	1	j.	j.	PROPN
ejpam-6501	91	2	pure	pure	PROPN
ejpam-6501	91	3	appl	appl	PROPN
ejpam-6501	91	4	.	.	PROPN
ejpam-6501	91	5	math	math	PROPN
ejpam-6501	91	6	,	,	PUNCT
ejpam-6501	91	7	18	18	NUM
ejpam-6501	91	8	(	(	PUNCT
ejpam-6501	91	9	4	4	NUM
ejpam-6501	91	10	)	)	PUNCT
ejpam-6501	91	11	(	(	PUNCT
ejpam-6501	91	12	2025	2025	NUM
ejpam-6501	91	13	)	)	PUNCT
ejpam-6501	91	14	,	,	PUNCT
ejpam-6501	91	15	6501	6501	NUM
ejpam-6501	91	16	5	5	NUM
ejpam-6501	91	17	of	of	ADP
ejpam-6501	91	18	8	8	NUM
ejpam-6501	91	19	(	(	PUNCT
ejpam-6501	91	20	ink2	ink2	PROPN
ejpam-6501	91	21	)	)	PUNCT
ejpam-6501	91	22	,	,	PUNCT
ejpam-6501	91	23	(	(	PUNCT
ejpam-6501	91	24	(	(	PUNCT
ejpam-6501	91	25	x	x	SYM
ejpam-6501	91	26	∗	∗	PROPN
ejpam-6501	91	27	z	z	NOUN
ejpam-6501	91	28	)	)	PUNCT
ejpam-6501	91	29	∗	∗	NOUN
ejpam-6501	91	30	(	(	PUNCT
ejpam-6501	91	31	y	y	PROPN
ejpam-6501	91	32	∗	∗	PROPN
ejpam-6501	91	33	z	z	NOUN
ejpam-6501	91	34	)	)	PUNCT
ejpam-6501	91	35	)	)	PUNCT
ejpam-6501	91	36	∗	∗	NOUN
ejpam-6501	91	37	(	(	PUNCT
ejpam-6501	91	38	x	x	X
ejpam-6501	91	39	∗	∗	NOUN
ejpam-6501	91	40	y	y	NOUN
ejpam-6501	91	41	)	)	PUNCT
ejpam-6501	91	42	=	=	SYM
ejpam-6501	92	1	0	0	NUM
ejpam-6501	92	2	∈	∈	PROPN
ejpam-6501	93	1	i	i	PRON
ejpam-6501	93	2	and	and	CCONJ
ejpam-6501	93	3	i	i	PRON
ejpam-6501	93	4	is	be	AUX
ejpam-6501	93	5	ideal	ideal	ADJ
ejpam-6501	93	6	,	,	PUNCT
ejpam-6501	93	7	implies	imply	VERB
ejpam-6501	93	8	that	that	SCONJ
ejpam-6501	93	9	(	(	PUNCT
ejpam-6501	93	10	x	x	X
ejpam-6501	93	11	∗	∗	PROPN
ejpam-6501	93	12	z	z	NOUN
ejpam-6501	93	13	)	)	PUNCT
ejpam-6501	93	14	∗	∗	NOUN
ejpam-6501	93	15	(	(	PUNCT
ejpam-6501	93	16	y	y	PROPN
ejpam-6501	93	17	∗	∗	PROPN
ejpam-6501	93	18	z	z	NOUN
ejpam-6501	93	19	)	)	PUNCT
ejpam-6501	93	20	∈	∈	PROPN
ejpam-6501	93	21	i	i	PRON
ejpam-6501	93	22	,	,	PUNCT
ejpam-6501	93	23	and	and	CCONJ
ejpam-6501	93	24	hence	hence	ADV
ejpam-6501	93	25	x	x	ADP
ejpam-6501	93	26	∗	∗	NOUN
ejpam-6501	93	27	z	z	PROPN
ejpam-6501	93	28	∈	∈	PROPN
ejpam-6501	93	29	i.	i.	NOUN
ejpam-6501	93	30	similarly	similarly	ADV
ejpam-6501	93	31	,	,	PUNCT
ejpam-6501	93	32	we	we	PRON
ejpam-6501	93	33	can	can	AUX
ejpam-6501	93	34	prove	prove	VERB
ejpam-6501	93	35	that	that	SCONJ
ejpam-6501	93	36	(	(	PUNCT
ejpam-6501	93	37	(	(	PUNCT
ejpam-6501	93	38	z	z	NOUN
ejpam-6501	93	39	∗	∗	NOUN
ejpam-6501	93	40	x	x	NOUN
ejpam-6501	93	41	)	)	PUNCT
ejpam-6501	93	42	∗	∗	NOUN
ejpam-6501	93	43	(	(	PUNCT
ejpam-6501	93	44	y	y	PROPN
ejpam-6501	93	45	∗	∗	NOUN
ejpam-6501	93	46	x	x	NOUN
ejpam-6501	93	47	)	)	PUNCT
ejpam-6501	93	48	)	)	PUNCT
ejpam-6501	93	49	∗	∗	NOUN
ejpam-6501	93	50	(	(	PUNCT
ejpam-6501	93	51	z	z	NOUN
ejpam-6501	93	52	∗	∗	NOUN
ejpam-6501	93	53	y	y	NOUN
ejpam-6501	93	54	)	)	PUNCT
ejpam-6501	93	55	=	=	SYM
ejpam-6501	94	1	0	0	NUM
ejpam-6501	94	2	∈	∈	PROPN
ejpam-6501	94	3	i	i	PRON
ejpam-6501	94	4	,	,	PUNCT
ejpam-6501	94	5	then	then	ADV
ejpam-6501	94	6	(	(	PUNCT
ejpam-6501	94	7	z	z	NOUN
ejpam-6501	94	8	∗	∗	X
ejpam-6501	94	9	x	x	NOUN
ejpam-6501	94	10	)	)	PUNCT
ejpam-6501	94	11	∗	∗	NOUN
ejpam-6501	94	12	(	(	PUNCT
ejpam-6501	94	13	y	y	PROPN
ejpam-6501	94	14	∗	∗	X
ejpam-6501	94	15	x	x	NOUN
ejpam-6501	94	16	)	)	PUNCT
ejpam-6501	94	17	∈	∈	PROPN
ejpam-6501	94	18	i	i	PRON
ejpam-6501	94	19	,	,	PUNCT
ejpam-6501	94	20	thus	thus	ADV
ejpam-6501	94	21	z	z	NOUN
ejpam-6501	94	22	∗	∗	NOUN
ejpam-6501	94	23	x	x	PUNCT
ejpam-6501	94	24	∈	∈	NOUN
ejpam-6501	94	25	i	i	PRON
ejpam-6501	94	26	and	and	CCONJ
ejpam-6501	94	27	hence	hence	ADV
ejpam-6501	94	28	x	x	VERB
ejpam-6501	94	29	∼	∼	NOUN
ejpam-6501	95	1	i	i	PRON
ejpam-6501	95	2	z.	z.	VERB
ejpam-6501	95	3	next	next	ADJ
ejpam-6501	95	4	to	to	PART
ejpam-6501	95	5	show	show	VERB
ejpam-6501	95	6	∼	∼	NOUN
ejpam-6501	95	7	i	i	PRON
ejpam-6501	95	8	is	be	AUX
ejpam-6501	95	9	compatible	compatible	ADJ
ejpam-6501	95	10	,	,	PUNCT
ejpam-6501	95	11	let	let	VERB
ejpam-6501	95	12	w	w	NOUN
ejpam-6501	95	13	,	,	PUNCT
ejpam-6501	95	14	x	x	NOUN
ejpam-6501	95	15	,	,	PUNCT
ejpam-6501	95	16	y	y	PROPN
ejpam-6501	95	17	,	,	PUNCT
ejpam-6501	95	18	z	z	NOUN
ejpam-6501	95	19	∈	∈	PROPN
ejpam-6501	95	20	x	x	PUNCT
ejpam-6501	95	21	such	such	ADJ
ejpam-6501	95	22	that	that	DET
ejpam-6501	95	23	w	w	ADJ
ejpam-6501	95	24	∼	∼	NOUN
ejpam-6501	95	25	i	i	PRON
ejpam-6501	95	26	y	y	PROPN
ejpam-6501	95	27	and	and	CCONJ
ejpam-6501	95	28	x	x	ADJ
ejpam-6501	95	29	∼	∼	NOUN
ejpam-6501	96	1	i	i	NOUN
ejpam-6501	96	2	z	z	NOUN
ejpam-6501	96	3	,	,	PUNCT
ejpam-6501	96	4	then	then	ADV
ejpam-6501	96	5	w∗y	w∗y	PUNCT
ejpam-6501	96	6	∈	∈	PROPN
ejpam-6501	96	7	i	i	PRON
ejpam-6501	96	8	,	,	PUNCT
ejpam-6501	96	9	y∗w	y∗w	PROPN
ejpam-6501	96	10	∈	∈	PROPN
ejpam-6501	97	1	i	i	PRON
ejpam-6501	97	2	,	,	PUNCT
ejpam-6501	97	3	x∗z	x∗z	PROPN
ejpam-6501	97	4	∈	∈	PROPN
ejpam-6501	97	5	i	i	PROPN
ejpam-6501	97	6	and	and	CCONJ
ejpam-6501	97	7	z∗x	z∗x	NUM
ejpam-6501	97	8	∈	∈	PROPN
ejpam-6501	97	9	i.	i.	NOUN
ejpam-6501	97	10	since	since	SCONJ
ejpam-6501	97	11	(	(	PUNCT
ejpam-6501	97	12	ink1	ink1	ADV
ejpam-6501	97	13	)	)	PUNCT
ejpam-6501	97	14	satisfying	satisfy	VERB
ejpam-6501	97	15	the	the	DET
ejpam-6501	97	16	identity	identity	NOUN
ejpam-6501	97	17	(	(	PUNCT
ejpam-6501	97	18	(	(	PUNCT
ejpam-6501	97	19	w∗x)∗(w∗z))∗(z∗x	w∗x)∗(w∗z))∗(z∗x	PROPN
ejpam-6501	97	20	)	)	PUNCT
ejpam-6501	97	21	=	=	SYM
ejpam-6501	97	22	0	0	PUNCT
ejpam-6501	98	1	∈	∈	PROPN
ejpam-6501	98	2	i	i	PRON
ejpam-6501	98	3	,	,	PUNCT
ejpam-6501	98	4	so	so	CCONJ
ejpam-6501	98	5	(	(	PUNCT
ejpam-6501	98	6	w∗x)∗(w∗z	w∗x)∗(w∗z	NOUN
ejpam-6501	98	7	)	)	PUNCT
ejpam-6501	98	8	∈	∈	PROPN
ejpam-6501	99	1	i	i	PRON
ejpam-6501	99	2	and	and	CCONJ
ejpam-6501	99	3	consider	consider	VERB
ejpam-6501	99	4	(	(	PUNCT
ejpam-6501	99	5	(	(	PUNCT
ejpam-6501	99	6	w	w	NOUN
ejpam-6501	99	7	∗	∗	PROPN
ejpam-6501	99	8	z	z	NOUN
ejpam-6501	99	9	)	)	PUNCT
ejpam-6501	99	10	∗	∗	NOUN
ejpam-6501	99	11	(	(	PUNCT
ejpam-6501	99	12	w	w	NOUN
ejpam-6501	99	13	∗	∗	NOUN
ejpam-6501	99	14	x	x	NOUN
ejpam-6501	99	15	)	)	PUNCT
ejpam-6501	99	16	)	)	PUNCT
ejpam-6501	99	17	∗	∗	NOUN
ejpam-6501	99	18	(	(	PUNCT
ejpam-6501	99	19	x	x	X
ejpam-6501	99	20	∗	∗	PROPN
ejpam-6501	99	21	z	z	NOUN
ejpam-6501	99	22	)	)	PUNCT
ejpam-6501	99	23	=	=	SYM
ejpam-6501	100	1	0	0	PUNCT
ejpam-6501	101	1	∈	∈	PROPN
ejpam-6501	102	1	i	i	PRON
ejpam-6501	102	2	,	,	PUNCT
ejpam-6501	102	3	we	we	PRON
ejpam-6501	102	4	have	have	VERB
ejpam-6501	102	5	(	(	PUNCT
ejpam-6501	102	6	w	w	NOUN
ejpam-6501	102	7	∗	∗	PROPN
ejpam-6501	102	8	z	z	NOUN
ejpam-6501	102	9	)	)	PUNCT
ejpam-6501	102	10	∗	∗	NOUN
ejpam-6501	102	11	(	(	PUNCT
ejpam-6501	102	12	w	w	NOUN
ejpam-6501	102	13	∗	∗	NOUN
ejpam-6501	102	14	x	x	NOUN
ejpam-6501	102	15	)	)	PUNCT
ejpam-6501	102	16	∈	∈	PROPN
ejpam-6501	103	1	i	i	PROPN
ejpam-6501	103	2	,	,	PUNCT
ejpam-6501	103	3	implies	imply	VERB
ejpam-6501	103	4	that	that	SCONJ
ejpam-6501	103	5	(	(	PUNCT
ejpam-6501	103	6	w	w	NOUN
ejpam-6501	103	7	∗	∗	NOUN
ejpam-6501	103	8	x	x	NOUN
ejpam-6501	103	9	)	)	PUNCT
ejpam-6501	103	10	∼	∼	NOUN
ejpam-6501	103	11	i	i	PRON
ejpam-6501	103	12	(	(	PUNCT
ejpam-6501	103	13	w	w	NOUN
ejpam-6501	103	14	∗	∗	NOUN
ejpam-6501	103	15	z	z	NOUN
ejpam-6501	103	16	)	)	PUNCT
ejpam-6501	103	17	.	.	PUNCT
ejpam-6501	104	1	because	because	SCONJ
ejpam-6501	104	2	(	(	PUNCT
ejpam-6501	104	3	(	(	PUNCT
ejpam-6501	104	4	w	w	NOUN
ejpam-6501	104	5	∗	∗	PROPN
ejpam-6501	104	6	z	z	NOUN
ejpam-6501	104	7	)	)	PUNCT
ejpam-6501	104	8	∗	∗	NOUN
ejpam-6501	104	9	(	(	PUNCT
ejpam-6501	104	10	w	w	NOUN
ejpam-6501	104	11	∗	∗	PROPN
ejpam-6501	104	12	y	y	PROPN
ejpam-6501	104	13	)	)	PUNCT
ejpam-6501	104	14	)	)	PUNCT
ejpam-6501	104	15	∗	∗	NOUN
ejpam-6501	104	16	(	(	PUNCT
ejpam-6501	104	17	y	y	PROPN
ejpam-6501	104	18	∗	∗	PROPN
ejpam-6501	104	19	z	z	NOUN
ejpam-6501	104	20	)	)	PUNCT
ejpam-6501	104	21	=	=	SYM
ejpam-6501	104	22	(	(	PUNCT
ejpam-6501	104	23	(	(	PUNCT
ejpam-6501	104	24	w	w	NOUN
ejpam-6501	104	25	∗	∗	PROPN
ejpam-6501	104	26	z	z	NOUN
ejpam-6501	104	27	)	)	PUNCT
ejpam-6501	104	28	∗	∗	NOUN
ejpam-6501	104	29	(	(	PUNCT
ejpam-6501	104	30	y	y	PROPN
ejpam-6501	104	31	∗	∗	PROPN
ejpam-6501	104	32	z	z	NOUN
ejpam-6501	104	33	)	)	PUNCT
ejpam-6501	104	34	)	)	PUNCT
ejpam-6501	104	35	∗	∗	NOUN
ejpam-6501	104	36	(	(	PUNCT
ejpam-6501	104	37	w	w	NOUN
ejpam-6501	104	38	∗	∗	PROPN
ejpam-6501	104	39	y	y	NOUN
ejpam-6501	104	40	)	)	PUNCT
ejpam-6501	104	41	=	=	SYM
ejpam-6501	104	42	0	0	NUM
ejpam-6501	105	1	∈	∈	PROPN
ejpam-6501	105	2	i	i	PRON
ejpam-6501	105	3	by	by	ADP
ejpam-6501	105	4	theorem	theorem	NOUN
ejpam-6501	105	5	1(iv	1(iv	NUM
ejpam-6501	105	6	)	)	PUNCT
ejpam-6501	105	7	and	and	CCONJ
ejpam-6501	105	8	(	(	PUNCT
ejpam-6501	105	9	ink1	ink1	PROPN
ejpam-6501	105	10	)	)	PUNCT
ejpam-6501	105	11	,	,	PUNCT
ejpam-6501	105	12	implies	imply	VERB
ejpam-6501	105	13	that	that	SCONJ
ejpam-6501	105	14	(	(	PUNCT
ejpam-6501	105	15	y	y	PROPN
ejpam-6501	105	16	∗	∗	PROPN
ejpam-6501	105	17	z	z	PROPN
ejpam-6501	105	18	)	)	PUNCT
ejpam-6501	105	19	∗	∗	NOUN
ejpam-6501	105	20	(	(	PUNCT
ejpam-6501	105	21	w	w	NOUN
ejpam-6501	105	22	∗	∗	PROPN
ejpam-6501	105	23	z	z	NOUN
ejpam-6501	105	24	)	)	PUNCT
ejpam-6501	105	25	∈	∈	PROPN
ejpam-6501	105	26	i	i	PRON
ejpam-6501	105	27	and	and	CCONJ
ejpam-6501	105	28	hence	hence	ADV
ejpam-6501	105	29	(	(	PUNCT
ejpam-6501	105	30	w	w	NOUN
ejpam-6501	105	31	∗	∗	NOUN
ejpam-6501	105	32	z	z	NOUN
ejpam-6501	105	33	)	)	PUNCT
ejpam-6501	105	34	∼	∼	NOUN
ejpam-6501	105	35	i	i	PRON
ejpam-6501	105	36	(	(	PUNCT
ejpam-6501	105	37	y	y	PROPN
ejpam-6501	105	38	∗	∗	PROPN
ejpam-6501	105	39	z	z	PROPN
ejpam-6501	105	40	)	)	PUNCT
ejpam-6501	105	41	.	.	PUNCT
ejpam-6501	106	1	since	since	SCONJ
ejpam-6501	106	2	∼	∼	NOUN
ejpam-6501	106	3	i	i	PRON
ejpam-6501	106	4	is	be	AUX
ejpam-6501	106	5	transitive	transitive	ADJ
ejpam-6501	106	6	,	,	PUNCT
ejpam-6501	106	7	so	so	CCONJ
ejpam-6501	106	8	(	(	PUNCT
ejpam-6501	106	9	w	w	NOUN
ejpam-6501	106	10	∗	∗	NOUN
ejpam-6501	106	11	x	x	NOUN
ejpam-6501	106	12	)	)	PUNCT
ejpam-6501	106	13	∼	∼	NOUN
ejpam-6501	106	14	i	i	PRON
ejpam-6501	106	15	(	(	PUNCT
ejpam-6501	106	16	y	y	PROPN
ejpam-6501	106	17	∗	∗	PROPN
ejpam-6501	106	18	z	z	PROPN
ejpam-6501	106	19	)	)	PUNCT
ejpam-6501	106	20	.	.	PUNCT
ejpam-6501	107	1	therefore	therefore	ADV
ejpam-6501	107	2	∼	∼	NOUN
ejpam-6501	107	3	i	i	PRON
ejpam-6501	107	4	is	be	AUX
ejpam-6501	107	5	a	a	DET
ejpam-6501	107	6	congruence	congruence	NOUN
ejpam-6501	107	7	relation	relation	NOUN
ejpam-6501	107	8	on	on	ADP
ejpam-6501	107	9	x.	x.	PROPN
ejpam-6501	107	10	lemma	lemma	PROPN
ejpam-6501	107	11	1	1	X
ejpam-6501	107	12	.	.	PUNCT
ejpam-6501	108	1	let	let	VERB
ejpam-6501	108	2	i	i	PRON
ejpam-6501	108	3	be	be	AUX
ejpam-6501	108	4	an	an	DET
ejpam-6501	108	5	ideal	ideal	NOUN
ejpam-6501	108	6	of	of	ADP
ejpam-6501	108	7	a	a	DET
ejpam-6501	108	8	ink	ink	NOUN
ejpam-6501	108	9	-	-	PUNCT
ejpam-6501	108	10	algebra	algebra	NOUN
ejpam-6501	108	11	(	(	PUNCT
ejpam-6501	108	12	x	x	X
ejpam-6501	108	13	,	,	PUNCT
ejpam-6501	108	14	∗	∗	NOUN
ejpam-6501	108	15	,	,	PUNCT
ejpam-6501	108	16	0	0	NUM
ejpam-6501	108	17	)	)	PUNCT
ejpam-6501	108	18	and	and	CCONJ
ejpam-6501	108	19	∼	∼	NOUN
ejpam-6501	108	20	i	i	PRON
ejpam-6501	108	21	is	be	AUX
ejpam-6501	108	22	a	a	DET
ejpam-6501	108	23	congruence	congruence	NOUN
ejpam-6501	108	24	relation	relation	NOUN
ejpam-6501	108	25	on	on	ADP
ejpam-6501	108	26	x.	x.	PROPN
ejpam-6501	108	27	then	then	ADV
ejpam-6501	108	28	the	the	DET
ejpam-6501	108	29	following	follow	VERB
ejpam-6501	108	30	conditions	condition	NOUN
ejpam-6501	108	31	hold	hold	VERB
ejpam-6501	108	32	;	;	PUNCT
ejpam-6501	108	33	(	(	PUNCT
ejpam-6501	108	34	i	i	NOUN
ejpam-6501	108	35	)	)	PUNCT
ejpam-6501	109	1	x	x	SYM
ejpam-6501	109	2	∈	∈	PROPN
ejpam-6501	110	1	[	[	X
ejpam-6501	110	2	x]i	x]i	NOUN
ejpam-6501	110	3	for	for	ADP
ejpam-6501	110	4	all	all	DET
ejpam-6501	110	5	x	x	SYM
ejpam-6501	110	6	∈	∈	PROPN
ejpam-6501	110	7	x	x	X
ejpam-6501	110	8	,	,	PUNCT
ejpam-6501	110	9	(	(	PUNCT
ejpam-6501	110	10	ii	ii	NOUN
ejpam-6501	110	11	)	)	PUNCT
ejpam-6501	110	12	x	x	PUNCT
ejpam-6501	110	13	∼	∼	NOUN
ejpam-6501	110	14	i	i	PRON
ejpam-6501	110	15	y	y	NOUN
ejpam-6501	110	16	if	if	SCONJ
ejpam-6501	111	1	and	and	CCONJ
ejpam-6501	111	2	only	only	ADV
ejpam-6501	111	3	if	if	SCONJ
ejpam-6501	111	4	[	[	X
ejpam-6501	111	5	x]i	x]i	X
ejpam-6501	111	6	=	=	SYM
ejpam-6501	112	1	[	[	X
ejpam-6501	112	2	y]i	y]i	ADJ
ejpam-6501	112	3	for	for	ADP
ejpam-6501	112	4	all	all	DET
ejpam-6501	112	5	x	x	NOUN
ejpam-6501	112	6	,	,	PUNCT
ejpam-6501	112	7	y	y	PROPN
ejpam-6501	112	8	∈	∈	PROPN
ejpam-6501	112	9	x.	x.	NOUN
ejpam-6501	112	10	proof	proof	NOUN
ejpam-6501	112	11	.	.	PUNCT
ejpam-6501	113	1	(	(	PUNCT
ejpam-6501	113	2	i	i	NOUN
ejpam-6501	113	3	)	)	PUNCT
ejpam-6501	113	4	clearly	clearly	ADV
ejpam-6501	113	5	,	,	PUNCT
ejpam-6501	113	6	for	for	ADP
ejpam-6501	113	7	any	any	DET
ejpam-6501	113	8	x	x	SYM
ejpam-6501	113	9	∈	∈	PROPN
ejpam-6501	113	10	x	x	NOUN
ejpam-6501	113	11	,	,	PUNCT
ejpam-6501	113	12	x	x	X
ejpam-6501	113	13	∗	∗	NOUN
ejpam-6501	113	14	x	x	X
ejpam-6501	113	15	=	=	SYM
ejpam-6501	113	16	0	0	NUM
ejpam-6501	113	17	∈	∈	PROPN
ejpam-6501	114	1	i	i	PRON
ejpam-6501	114	2	that	that	PRON
ejpam-6501	114	3	is	be	AUX
ejpam-6501	114	4	x	x	PUNCT
ejpam-6501	114	5	∼	∼	NOUN
ejpam-6501	114	6	i	i	NOUN
ejpam-6501	114	7	x	x	NOUN
ejpam-6501	114	8	,	,	PUNCT
ejpam-6501	114	9	and	and	CCONJ
ejpam-6501	114	10	hence	hence	ADV
ejpam-6501	114	11	x	x	X
ejpam-6501	114	12	∈	∈	PROPN
ejpam-6501	115	1	[	[	X
ejpam-6501	115	2	x]i	x]i	X
ejpam-6501	115	3	.	.	PUNCT
ejpam-6501	116	1	(	(	PUNCT
ejpam-6501	116	2	ii	ii	NOUN
ejpam-6501	116	3	)	)	PUNCT
ejpam-6501	116	4	assume	assume	VERB
ejpam-6501	116	5	that	that	SCONJ
ejpam-6501	116	6	x	x	PUNCT
ejpam-6501	116	7	∼	∼	NOUN
ejpam-6501	116	8	i	i	NOUN
ejpam-6501	116	9	y	y	PROPN
ejpam-6501	116	10	,	,	PUNCT
ejpam-6501	116	11	then	then	ADV
ejpam-6501	116	12	y	y	PROPN
ejpam-6501	116	13	∼	∼	NOUN
ejpam-6501	116	14	i	i	PRON
ejpam-6501	116	15	x.	x.	VERB
ejpam-6501	116	16	let	let	VERB
ejpam-6501	116	17	a	a	DET
ejpam-6501	116	18	∈	∈	NOUN
ejpam-6501	117	1	[	[	X
ejpam-6501	117	2	x]i	x]i	X
ejpam-6501	117	3	,	,	PUNCT
ejpam-6501	117	4	then	then	ADV
ejpam-6501	117	5	x	x	PUNCT
ejpam-6501	117	6	∼	∼	NOUN
ejpam-6501	117	7	i	i	PRON
ejpam-6501	117	8	a	a	NOUN
ejpam-6501	117	9	and	and	CCONJ
ejpam-6501	117	10	hence	hence	ADV
ejpam-6501	117	11	a	a	DET
ejpam-6501	117	12	∼	∼	NOUN
ejpam-6501	117	13	i	i	NOUN
ejpam-6501	117	14	x	x	NOUN
ejpam-6501	117	15	,	,	PUNCT
ejpam-6501	117	16	thus	thus	ADV
ejpam-6501	117	17	a	a	DET
ejpam-6501	117	18	∼	∼	NOUN
ejpam-6501	117	19	i	i	NOUN
ejpam-6501	117	20	y	y	NOUN
ejpam-6501	117	21	,	,	PUNCT
ejpam-6501	117	22	so	so	SCONJ
ejpam-6501	117	23	a	a	DET
ejpam-6501	117	24	∈	∈	PROPN
ejpam-6501	118	1	[	[	X
ejpam-6501	118	2	y]i	y]i	NOUN
ejpam-6501	118	3	.	.	PUNCT
ejpam-6501	119	1	therefore	therefore	ADV
ejpam-6501	119	2	[	[	X
ejpam-6501	119	3	x]i	x]i	NOUN
ejpam-6501	119	4	⊆	⊆	NUM
ejpam-6501	119	5	[	[	X
ejpam-6501	119	6	y]i	y]i	ADJ
ejpam-6501	119	7	.	.	PUNCT
ejpam-6501	120	1	simmilary	simmilary	PROPN
ejpam-6501	120	2	,	,	PUNCT
ejpam-6501	120	3	to	to	PART
ejpam-6501	120	4	prove	prove	VERB
ejpam-6501	120	5	[	[	X
ejpam-6501	120	6	y]i	y]i	ADJ
ejpam-6501	120	7	⊆	⊆	NUM
ejpam-6501	120	8	[	[	X
ejpam-6501	120	9	x]i	x]i	X
ejpam-6501	120	10	.	.	PUNCT
ejpam-6501	121	1	conversely	conversely	ADV
ejpam-6501	121	2	,	,	PUNCT
ejpam-6501	121	3	suppose	suppose	VERB
ejpam-6501	121	4	that	that	SCONJ
ejpam-6501	122	1	[	[	X
ejpam-6501	122	2	x]i	x]i	X
ejpam-6501	122	3	=	=	SYM
ejpam-6501	123	1	[	[	X
ejpam-6501	123	2	y]i	y]i	ADJ
ejpam-6501	123	3	,	,	PUNCT
ejpam-6501	123	4	so	so	CCONJ
ejpam-6501	123	5	x	x	SYM
ejpam-6501	123	6	∈	∈	PROPN
ejpam-6501	124	1	[	[	X
ejpam-6501	124	2	x]i	x]i	X
ejpam-6501	124	3	=	=	SYM
ejpam-6501	125	1	[	[	X
ejpam-6501	125	2	y]i	y]i	ADJ
ejpam-6501	125	3	by	by	ADP
ejpam-6501	125	4	(	(	PUNCT
ejpam-6501	125	5	i	i	NOUN
ejpam-6501	125	6	)	)	PUNCT
ejpam-6501	125	7	,	,	PUNCT
ejpam-6501	125	8	and	and	CCONJ
ejpam-6501	125	9	hence	hence	ADV
ejpam-6501	125	10	y	y	VERB
ejpam-6501	125	11	∼	∼	NOUN
ejpam-6501	125	12	i	i	NOUN
ejpam-6501	125	13	x	x	NOUN
ejpam-6501	125	14	,	,	PUNCT
ejpam-6501	125	15	thus	thus	ADV
ejpam-6501	125	16	x	x	ADP
ejpam-6501	125	17	∼	∼	NOUN
ejpam-6501	125	18	i	i	PRON
ejpam-6501	125	19	y.	y.	NOUN
ejpam-6501	125	20	theorem	theorem	VERB
ejpam-6501	125	21	3	3	X
ejpam-6501	125	22	.	.	PUNCT
ejpam-6501	125	23	let	let	VERB
ejpam-6501	125	24	(	(	PUNCT
ejpam-6501	125	25	x	x	X
ejpam-6501	125	26	,	,	PUNCT
ejpam-6501	125	27	∗	∗	NOUN
ejpam-6501	125	28	,	,	PUNCT
ejpam-6501	125	29	0	0	NUM
ejpam-6501	125	30	)	)	PUNCT
ejpam-6501	125	31	be	be	AUX
ejpam-6501	125	32	a	a	DET
ejpam-6501	125	33	ink	ink	NOUN
ejpam-6501	125	34	-	-	PUNCT
ejpam-6501	125	35	algebra	algebra	NOUN
ejpam-6501	125	36	and	and	CCONJ
ejpam-6501	125	37	i	i	PRON
ejpam-6501	125	38	is	be	AUX
ejpam-6501	125	39	an	an	DET
ejpam-6501	125	40	ideal	ideal	NOUN
ejpam-6501	125	41	of	of	ADP
ejpam-6501	125	42	x.	x.	NOUN
ejpam-6501	125	43	then	then	ADV
ejpam-6501	125	44	the	the	DET
ejpam-6501	125	45	set	set	NOUN
ejpam-6501	125	46	p	p	X
ejpam-6501	125	47	=	=	PUNCT
ejpam-6501	125	48	{	{	PUNCT
ejpam-6501	125	49	[	[	X
ejpam-6501	125	50	x]i	x]i	NOUN
ejpam-6501	125	51	|x	|x	NOUN
ejpam-6501	125	52	∈	∈	PROPN
ejpam-6501	125	53	x	x	PRON
ejpam-6501	125	54	}	}	PUNCT
ejpam-6501	125	55	forms	form	VERB
ejpam-6501	125	56	a	a	DET
ejpam-6501	125	57	partition	partition	NOUN
ejpam-6501	125	58	of	of	ADP
ejpam-6501	125	59	x.	x.	NOUN
ejpam-6501	125	60	proof	proof	NOUN
ejpam-6501	125	61	.	.	PUNCT
ejpam-6501	126	1	let	let	VERB
ejpam-6501	126	2	[	[	X
ejpam-6501	126	3	x]i	x]i	X
ejpam-6501	126	4	,	,	PUNCT
ejpam-6501	127	1	[	[	X
ejpam-6501	127	2	y]i	y]i	NOUN
ejpam-6501	127	3	∈	∈	X
ejpam-6501	127	4	p	p	NOUN
ejpam-6501	127	5	such	such	ADJ
ejpam-6501	127	6	that	that	SCONJ
ejpam-6501	128	1	[	[	X
ejpam-6501	128	2	x]i	x]i	X
ejpam-6501	128	3	̸=	̸=	PROPN
ejpam-6501	129	1	[	[	PUNCT
ejpam-6501	129	2	y]i	y]i	ADJ
ejpam-6501	129	3	.	.	PUNCT
ejpam-6501	130	1	suppose	suppose	VERB
ejpam-6501	130	2	that	that	SCONJ
ejpam-6501	130	3	[	[	X
ejpam-6501	130	4	x]i	x]i	X
ejpam-6501	130	5	∩	∩	NOUN
ejpam-6501	130	6	[	[	X
ejpam-6501	130	7	y]i	y]i	ADJ
ejpam-6501	130	8	̸=	̸=	NOUN
ejpam-6501	130	9	∅	∅	NOUN
ejpam-6501	130	10	,	,	PUNCT
ejpam-6501	130	11	there	there	PRON
ejpam-6501	130	12	is	be	VERB
ejpam-6501	130	13	a	a	DET
ejpam-6501	130	14	b	b	PROPN
ejpam-6501	130	15	∈	∈	PROPN
ejpam-6501	131	1	[	[	X
ejpam-6501	131	2	x]i	x]i	X
ejpam-6501	131	3	∩	∩	NOUN
ejpam-6501	131	4	[	[	X
ejpam-6501	131	5	y]i	y]i	ADJ
ejpam-6501	131	6	,	,	PUNCT
ejpam-6501	131	7	then	then	ADV
ejpam-6501	131	8	x	x	PUNCT
ejpam-6501	131	9	∼	∼	NOUN
ejpam-6501	131	10	i	i	NOUN
ejpam-6501	131	11	b	b	NOUN
ejpam-6501	131	12	,	,	PUNCT
ejpam-6501	131	13	y	y	PROPN
ejpam-6501	131	14	∼	∼	NOUN
ejpam-6501	131	15	i	i	NOUN
ejpam-6501	131	16	b	b	PROPN
ejpam-6501	131	17	and	and	CCONJ
ejpam-6501	131	18	hence	hence	ADV
ejpam-6501	131	19	b	b	NOUN
ejpam-6501	131	20	∼	∼	NOUN
ejpam-6501	131	21	i	i	NOUN
ejpam-6501	131	22	x	x	NOUN
ejpam-6501	131	23	,	,	PUNCT
ejpam-6501	131	24	so	so	SCONJ
ejpam-6501	131	25	we	we	PRON
ejpam-6501	131	26	have	have	VERB
ejpam-6501	131	27	y	y	NOUN
ejpam-6501	131	28	∼	∼	NOUN
ejpam-6501	131	29	i	i	PRON
ejpam-6501	131	30	x	x	PROPN
ejpam-6501	131	31	,	,	PUNCT
ejpam-6501	131	32	implies	imply	VERB
ejpam-6501	131	33	that	that	SCONJ
ejpam-6501	132	1	[	[	X
ejpam-6501	132	2	x]i	x]i	X
ejpam-6501	132	3	=	=	SYM
ejpam-6501	133	1	[	[	X
ejpam-6501	133	2	y]i	y]i	ADJ
ejpam-6501	133	3	that	that	PRON
ejpam-6501	133	4	’s	’	VERB
ejpam-6501	133	5	a	a	DET
ejpam-6501	133	6	contradiction	contradiction	NOUN
ejpam-6501	133	7	,	,	PUNCT
ejpam-6501	133	8	thus	thus	ADV
ejpam-6501	133	9	[	[	X
ejpam-6501	133	10	x]i	x]i	X
ejpam-6501	133	11	∩	∩	NOUN
ejpam-6501	133	12	[	[	X
ejpam-6501	133	13	y]i	y]i	ADJ
ejpam-6501	133	14	=	=	X
ejpam-6501	133	15	∅.	∅.	VERB
ejpam-6501	133	16	next	next	ADV
ejpam-6501	133	17	to	to	PART
ejpam-6501	133	18	show	show	VERB
ejpam-6501	133	19	⋃	⋃	PROPN
ejpam-6501	133	20	x∈x	x∈x	NOUN
ejpam-6501	134	1	[	[	X
ejpam-6501	134	2	x]i	x]i	X
ejpam-6501	134	3	=	=	SYM
ejpam-6501	134	4	x.	x.	NOUN
ejpam-6501	134	5	clearly	clearly	ADV
ejpam-6501	134	6	for	for	ADP
ejpam-6501	134	7	any	any	DET
ejpam-6501	134	8	x	x	SYM
ejpam-6501	134	9	∈	∈	PROPN
ejpam-6501	134	10	x	x	X
ejpam-6501	134	11	,	,	PUNCT
ejpam-6501	134	12	[	[	X
ejpam-6501	134	13	x]i	x]i	NOUN
ejpam-6501	134	14	⊆	⊆	NUM
ejpam-6501	134	15	x	x	NOUN
ejpam-6501	134	16	and	and	CCONJ
ejpam-6501	134	17	hence	hence	ADV
ejpam-6501	134	18	⋃	⋃	NOUN
ejpam-6501	134	19	x∈x	x∈x	NOUN
ejpam-6501	135	1	[	[	X
ejpam-6501	135	2	x]i	x]i	NOUN
ejpam-6501	135	3	⊆	⊆	NUM
ejpam-6501	135	4	x.	x.	NOUN
ejpam-6501	135	5	and	and	CCONJ
ejpam-6501	135	6	for	for	ADP
ejpam-6501	135	7	any	any	DET
ejpam-6501	135	8	x	x	SYM
ejpam-6501	135	9	∈	∈	PROPN
ejpam-6501	135	10	x	x	NOUN
ejpam-6501	135	11	,	,	PUNCT
ejpam-6501	135	12	x	x	SYM
ejpam-6501	135	13	∈	∈	PROPN
ejpam-6501	136	1	[	[	X
ejpam-6501	136	2	x]i	x]i	NOUN
ejpam-6501	136	3	⊆	⊆	NUM
ejpam-6501	136	4	⋃	⋃	NOUN
ejpam-6501	136	5	x∈x	x∈x	NOUN
ejpam-6501	137	1	[	[	X
ejpam-6501	137	2	x]i	x]i	X
ejpam-6501	137	3	.	.	PUNCT
ejpam-6501	138	1	thus	thus	ADV
ejpam-6501	138	2	p	p	X
ejpam-6501	138	3	=	=	PUNCT
ejpam-6501	138	4	{	{	PUNCT
ejpam-6501	138	5	[	[	X
ejpam-6501	138	6	x]i	x]i	NOUN
ejpam-6501	138	7	|x	|x	NOUN
ejpam-6501	138	8	∈	∈	PROPN
ejpam-6501	138	9	x	x	PRON
ejpam-6501	138	10	}	}	PUNCT
ejpam-6501	138	11	is	be	AUX
ejpam-6501	138	12	a	a	DET
ejpam-6501	138	13	partition	partition	NOUN
ejpam-6501	138	14	of	of	ADP
ejpam-6501	138	15	x.	x.	NOUN
ejpam-6501	138	16	example	example	NOUN
ejpam-6501	139	1	4	4	NUM
ejpam-6501	139	2	.	.	PUNCT
ejpam-6501	139	3	from	from	ADP
ejpam-6501	139	4	example	example	NOUN
ejpam-6501	139	5	3	3	NUM
ejpam-6501	139	6	,	,	PUNCT
ejpam-6501	139	7	we	we	PRON
ejpam-6501	139	8	see	see	VERB
ejpam-6501	139	9	that	that	SCONJ
ejpam-6501	139	10	i	i	PRON
ejpam-6501	139	11	=	=	PUNCT
ejpam-6501	139	12	{	{	PUNCT
ejpam-6501	139	13	0	0	NUM
ejpam-6501	139	14	,	,	PUNCT
ejpam-6501	139	15	a	a	PRON
ejpam-6501	139	16	}	}	PUNCT
ejpam-6501	139	17	is	be	AUX
ejpam-6501	139	18	an	an	DET
ejpam-6501	139	19	ideal	ideal	NOUN
ejpam-6501	139	20	of	of	ADP
ejpam-6501	139	21	x	x	PUNCT
ejpam-6501	139	22	and	and	CCONJ
ejpam-6501	139	23	relation	relation	NOUN
ejpam-6501	139	24	∼	∼	NOUN
ejpam-6501	139	25	i	i	PRON
ejpam-6501	139	26	=	=	PUNCT
ejpam-6501	139	27	{	{	PUNCT
ejpam-6501	139	28	(	(	PUNCT
ejpam-6501	139	29	0	0	NUM
ejpam-6501	139	30	,	,	PUNCT
ejpam-6501	139	31	0	0	NUM
ejpam-6501	139	32	)	)	PUNCT
ejpam-6501	139	33	,	,	PUNCT
ejpam-6501	139	34	(	(	PUNCT
ejpam-6501	139	35	1	1	NUM
ejpam-6501	139	36	,	,	PUNCT
ejpam-6501	139	37	1	1	NUM
ejpam-6501	139	38	)	)	PUNCT
ejpam-6501	139	39	,	,	PUNCT
ejpam-6501	139	40	(	(	PUNCT
ejpam-6501	139	41	a	a	X
ejpam-6501	139	42	,	,	PUNCT
ejpam-6501	139	43	a	a	NOUN
ejpam-6501	139	44	)	)	PUNCT
ejpam-6501	139	45	,	,	PUNCT
ejpam-6501	139	46	(	(	PUNCT
ejpam-6501	139	47	b	b	X
ejpam-6501	139	48	,	,	PUNCT
ejpam-6501	139	49	b	b	NOUN
ejpam-6501	139	50	)	)	PUNCT
ejpam-6501	139	51	,	,	PUNCT
ejpam-6501	139	52	(	(	PUNCT
ejpam-6501	139	53	0	0	NUM
ejpam-6501	139	54	,	,	PUNCT
ejpam-6501	139	55	a	a	PRON
ejpam-6501	139	56	)	)	PUNCT
ejpam-6501	139	57	,	,	PUNCT
ejpam-6501	139	58	(	(	PUNCT
ejpam-6501	139	59	a	a	PRON
ejpam-6501	139	60	,	,	PUNCT
ejpam-6501	139	61	0	0	NUM
ejpam-6501	139	62	)	)	PUNCT
ejpam-6501	139	63	,	,	PUNCT
ejpam-6501	139	64	(	(	PUNCT
ejpam-6501	139	65	1	1	NUM
ejpam-6501	139	66	,	,	PUNCT
ejpam-6501	139	67	b	b	NOUN
ejpam-6501	139	68	)	)	PUNCT
ejpam-6501	139	69	,	,	PUNCT
ejpam-6501	139	70	(	(	PUNCT
ejpam-6501	139	71	b	b	X
ejpam-6501	139	72	,	,	PUNCT
ejpam-6501	139	73	1	1	NUM
ejpam-6501	139	74	)	)	PUNCT
ejpam-6501	139	75	}	}	PUNCT
ejpam-6501	139	76	,	,	PUNCT
ejpam-6501	139	77	is	be	AUX
ejpam-6501	139	78	a	a	DET
ejpam-6501	139	79	congruence	congruence	NOUN
ejpam-6501	139	80	.	.	PUNCT
ejpam-6501	140	1	consider	consider	VERB
ejpam-6501	140	2	[	[	X
ejpam-6501	140	3	0]i	0]i	NOUN
ejpam-6501	140	4	=	=	SYM
ejpam-6501	140	5	{	{	PUNCT
ejpam-6501	140	6	0	0	NUM
ejpam-6501	140	7	,	,	PUNCT
ejpam-6501	140	8	a	a	PRON
ejpam-6501	140	9	}	}	PUNCT
ejpam-6501	140	10	=	=	SYM
ejpam-6501	141	1	[	[	X
ejpam-6501	141	2	a]i	a]i	NOUN
ejpam-6501	141	3	and	and	CCONJ
ejpam-6501	141	4	[	[	X
ejpam-6501	141	5	1]i	1]i	NUM
ejpam-6501	141	6	=	=	SYM
ejpam-6501	141	7	{	{	PUNCT
ejpam-6501	141	8	1	1	NUM
ejpam-6501	141	9	,	,	PUNCT
ejpam-6501	141	10	b	b	NOUN
ejpam-6501	141	11	}	}	PUNCT
ejpam-6501	141	12	=	=	PUNCT
ejpam-6501	142	1	[	[	X
ejpam-6501	142	2	b]i	b]i	NOUN
ejpam-6501	142	3	,	,	PUNCT
ejpam-6501	142	4	and	and	CCONJ
ejpam-6501	142	5	hence	hence	ADV
ejpam-6501	142	6	the	the	DET
ejpam-6501	142	7	set	set	NOUN
ejpam-6501	142	8	p	p	X
ejpam-6501	142	9	=	=	PUNCT
ejpam-6501	142	10	{	{	PUNCT
ejpam-6501	142	11	[	[	X
ejpam-6501	142	12	0]i	0]i	X
ejpam-6501	142	13	,	,	PUNCT
ejpam-6501	142	14	[	[	X
ejpam-6501	142	15	1]i	1]i	NUM
ejpam-6501	142	16	}	}	PUNCT
ejpam-6501	142	17	is	be	AUX
ejpam-6501	142	18	a	a	DET
ejpam-6501	142	19	partition	partition	NOUN
ejpam-6501	142	20	of	of	ADP
ejpam-6501	142	21	x.	x.	PROPN
ejpam-6501	142	22	m.	m.	PROPN
ejpam-6501	142	23	phattarachaleekul	phattarachaleekul	PROPN
ejpam-6501	142	24	/	/	SYM
ejpam-6501	142	25	eur	eur	PROPN
ejpam-6501	142	26	.	.	PUNCT
ejpam-6501	143	1	j.	j.	PROPN
ejpam-6501	143	2	pure	pure	PROPN
ejpam-6501	143	3	appl	appl	PROPN
ejpam-6501	143	4	.	.	PROPN
ejpam-6501	143	5	math	math	PROPN
ejpam-6501	143	6	,	,	PUNCT
ejpam-6501	143	7	18	18	NUM
ejpam-6501	143	8	(	(	PUNCT
ejpam-6501	143	9	4	4	NUM
ejpam-6501	143	10	)	)	PUNCT
ejpam-6501	143	11	(	(	PUNCT
ejpam-6501	143	12	2025	2025	NUM
ejpam-6501	143	13	)	)	PUNCT
ejpam-6501	143	14	,	,	PUNCT
ejpam-6501	143	15	6501	6501	NUM
ejpam-6501	143	16	6	6	NUM
ejpam-6501	143	17	of	of	ADP
ejpam-6501	143	18	8	8	NUM
ejpam-6501	143	19	4	4	NUM
ejpam-6501	143	20	.	.	PUNCT
ejpam-6501	143	21	quotient	quotient	VERB
ejpam-6501	143	22	ink	ink	NOUN
ejpam-6501	143	23	-	-	PUNCT
ejpam-6501	143	24	algebras	algebra	VERB
ejpam-6501	143	25	the	the	DET
ejpam-6501	143	26	notion	notion	NOUN
ejpam-6501	143	27	of	of	ADP
ejpam-6501	143	28	quotient	quotient	NOUN
ejpam-6501	143	29	structures	structure	NOUN
ejpam-6501	143	30	is	be	AUX
ejpam-6501	143	31	fundamental	fundamental	ADJ
ejpam-6501	143	32	in	in	ADP
ejpam-6501	143	33	universal	universal	ADJ
ejpam-6501	143	34	algebra	algebra	NOUN
ejpam-6501	143	35	,	,	PUNCT
ejpam-6501	143	36	serving	serve	VERB
ejpam-6501	143	37	as	as	ADP
ejpam-6501	143	38	a	a	DET
ejpam-6501	143	39	tool	tool	NOUN
ejpam-6501	143	40	for	for	ADP
ejpam-6501	143	41	classifying	classify	VERB
ejpam-6501	143	42	and	and	CCONJ
ejpam-6501	143	43	analyzing	analyze	VERB
ejpam-6501	143	44	algebraic	algebraic	ADJ
ejpam-6501	143	45	systems	system	NOUN
ejpam-6501	143	46	via	via	ADP
ejpam-6501	143	47	congruence	congruence	PROPN
ejpam-6501	143	48	relations	relation	NOUN
ejpam-6501	143	49	.	.	PUNCT
ejpam-6501	144	1	in	in	ADP
ejpam-6501	144	2	this	this	DET
ejpam-6501	144	3	section	section	NOUN
ejpam-6501	144	4	,	,	PUNCT
ejpam-6501	144	5	we	we	PRON
ejpam-6501	144	6	introduce	introduce	VERB
ejpam-6501	144	7	the	the	DET
ejpam-6501	144	8	concept	concept	NOUN
ejpam-6501	144	9	of	of	ADP
ejpam-6501	144	10	quotient	quotient	NOUN
ejpam-6501	144	11	ink	ink	NOUN
ejpam-6501	144	12	-	-	PUNCT
ejpam-6501	144	13	algebras	algebras	ADV
ejpam-6501	144	14	,	,	PUNCT
ejpam-6501	144	15	establish	establish	VERB
ejpam-6501	144	16	their	their	PRON
ejpam-6501	144	17	basic	basic	ADJ
ejpam-6501	144	18	properties	property	NOUN
ejpam-6501	144	19	,	,	PUNCT
ejpam-6501	144	20	and	and	CCONJ
ejpam-6501	144	21	explore	explore	VERB
ejpam-6501	144	22	structural	structural	ADJ
ejpam-6501	144	23	implications	implication	NOUN
ejpam-6501	144	24	arising	arise	VERB
ejpam-6501	144	25	from	from	ADP
ejpam-6501	144	26	the	the	DET
ejpam-6501	144	27	induced	induced	ADJ
ejpam-6501	144	28	congruences	congruence	NOUN
ejpam-6501	144	29	.	.	PUNCT
ejpam-6501	145	1	definition	definition	NOUN
ejpam-6501	145	2	5	5	NUM
ejpam-6501	145	3	.	.	PUNCT
ejpam-6501	146	1	let	let	VERB
ejpam-6501	146	2	(	(	PUNCT
ejpam-6501	146	3	x	x	X
ejpam-6501	146	4	,	,	PUNCT
ejpam-6501	146	5	∗	∗	NOUN
ejpam-6501	146	6	,	,	PUNCT
ejpam-6501	146	7	0	0	NUM
ejpam-6501	146	8	)	)	PUNCT
ejpam-6501	146	9	be	be	AUX
ejpam-6501	146	10	a	a	DET
ejpam-6501	146	11	ink	ink	NOUN
ejpam-6501	146	12	-	-	PUNCT
ejpam-6501	146	13	algebra	algebra	NOUN
ejpam-6501	146	14	.	.	PUNCT
ejpam-6501	147	1	a	a	DET
ejpam-6501	147	2	nonempty	nonempty	NOUN
ejpam-6501	147	3	subset	subset	VERB
ejpam-6501	147	4	n	n	PROPN
ejpam-6501	147	5	of	of	ADP
ejpam-6501	147	6	x	x	VERB
ejpam-6501	147	7	is	be	AUX
ejpam-6501	147	8	said	say	VERB
ejpam-6501	147	9	to	to	PART
ejpam-6501	147	10	be	be	AUX
ejpam-6501	147	11	normal	normal	ADJ
ejpam-6501	147	12	of	of	ADP
ejpam-6501	147	13	x	x	SYM
ejpam-6501	147	14	if	if	SCONJ
ejpam-6501	147	15	(	(	PUNCT
ejpam-6501	147	16	x	x	NOUN
ejpam-6501	147	17	∗	∗	NOUN
ejpam-6501	147	18	a	a	NOUN
ejpam-6501	147	19	)	)	PUNCT
ejpam-6501	147	20	∗	∗	NOUN
ejpam-6501	147	21	(	(	PUNCT
ejpam-6501	147	22	y	y	PROPN
ejpam-6501	147	23	∗	∗	X
ejpam-6501	147	24	b	b	NOUN
ejpam-6501	147	25	)	)	PUNCT
ejpam-6501	147	26	∈	∈	PROPN
ejpam-6501	147	27	n	n	NOUN
ejpam-6501	147	28	for	for	ADP
ejpam-6501	147	29	any	any	DET
ejpam-6501	147	30	x	x	PROPN
ejpam-6501	147	31	∗	∗	PROPN
ejpam-6501	147	32	y	y	PROPN
ejpam-6501	147	33	,	,	PUNCT
ejpam-6501	147	34	a	a	DET
ejpam-6501	147	35	∗	∗	NOUN
ejpam-6501	147	36	b	b	NOUN
ejpam-6501	147	37	∈	∈	PROPN
ejpam-6501	147	38	n.	n.	NOUN
ejpam-6501	147	39	theorem	theorem	VERB
ejpam-6501	147	40	4	4	NUM
ejpam-6501	147	41	.	.	PUNCT
ejpam-6501	148	1	every	every	DET
ejpam-6501	148	2	normal	normal	ADJ
ejpam-6501	148	3	subset	subset	NOUN
ejpam-6501	148	4	of	of	ADP
ejpam-6501	148	5	an	an	DET
ejpam-6501	148	6	ink	ink	NOUN
ejpam-6501	148	7	-	-	PUNCT
ejpam-6501	148	8	algebra	algebra	NOUN
ejpam-6501	148	9	is	be	AUX
ejpam-6501	148	10	a	a	DET
ejpam-6501	148	11	subalgebra	subalgebra	NOUN
ejpam-6501	148	12	.	.	PUNCT
ejpam-6501	149	1	proof	proof	NOUN
ejpam-6501	149	2	.	.	PUNCT
ejpam-6501	150	1	assume	assume	VERB
ejpam-6501	150	2	that	that	SCONJ
ejpam-6501	150	3	n	n	PRON
ejpam-6501	150	4	is	be	AUX
ejpam-6501	150	5	a	a	DET
ejpam-6501	150	6	normal	normal	ADJ
ejpam-6501	150	7	subset	subset	NOUN
ejpam-6501	150	8	of	of	ADP
ejpam-6501	150	9	a	a	DET
ejpam-6501	150	10	ink	ink	NOUN
ejpam-6501	150	11	-	-	PUNCT
ejpam-6501	150	12	algebra	algebra	NOUN
ejpam-6501	150	13	(	(	PUNCT
ejpam-6501	150	14	x	x	X
ejpam-6501	150	15	,	,	PUNCT
ejpam-6501	150	16	∗	∗	NOUN
ejpam-6501	150	17	,	,	PUNCT
ejpam-6501	150	18	0	0	NUM
ejpam-6501	150	19	)	)	PUNCT
ejpam-6501	150	20	and	and	CCONJ
ejpam-6501	150	21	x	x	X
ejpam-6501	150	22	,	,	PUNCT
ejpam-6501	150	23	y	y	PROPN
ejpam-6501	150	24	∈	∈	PROPN
ejpam-6501	150	25	n	n	ADV
ejpam-6501	150	26	.	.	PUNCT
ejpam-6501	151	1	by	by	ADP
ejpam-6501	151	2	(	(	PUNCT
ejpam-6501	151	3	ink3	ink3	PROPN
ejpam-6501	151	4	)	)	PUNCT
ejpam-6501	151	5	,	,	PUNCT
ejpam-6501	151	6	x	x	X
ejpam-6501	151	7	∗	∗	NOUN
ejpam-6501	151	8	0	0	NUM
ejpam-6501	152	1	=	=	SYM
ejpam-6501	152	2	x	x	SYM
ejpam-6501	152	3	∈	∈	PROPN
ejpam-6501	152	4	n	n	NOUN
ejpam-6501	152	5	and	and	CCONJ
ejpam-6501	152	6	y	y	PROPN
ejpam-6501	152	7	∗	∗	NOUN
ejpam-6501	152	8	0	0	NUM
ejpam-6501	153	1	=	=	SYM
ejpam-6501	153	2	y	y	PROPN
ejpam-6501	153	3	∈	∈	PROPN
ejpam-6501	153	4	n	n	CCONJ
ejpam-6501	153	5	,	,	PUNCT
ejpam-6501	153	6	hence	hence	ADV
ejpam-6501	153	7	x	x	INTJ
ejpam-6501	153	8	∗	∗	NOUN
ejpam-6501	153	9	y	y	NOUN
ejpam-6501	153	10	=	=	SYM
ejpam-6501	153	11	(	(	PUNCT
ejpam-6501	153	12	x	x	X
ejpam-6501	153	13	∗	∗	PROPN
ejpam-6501	153	14	y	y	NOUN
ejpam-6501	153	15	)	)	PUNCT
ejpam-6501	153	16	∗	∗	NOUN
ejpam-6501	153	17	0	0	NUM
ejpam-6501	154	1	=	=	SYM
ejpam-6501	154	2	(	(	PUNCT
ejpam-6501	154	3	x	x	X
ejpam-6501	154	4	∗	∗	PROPN
ejpam-6501	154	5	y	y	NOUN
ejpam-6501	154	6	)	)	PUNCT
ejpam-6501	154	7	∗	∗	NOUN
ejpam-6501	154	8	(	(	PUNCT
ejpam-6501	154	9	0	0	NUM
ejpam-6501	154	10	∗	∗	NOUN
ejpam-6501	154	11	0	0	NUM
ejpam-6501	154	12	)	)	PUNCT
ejpam-6501	154	13	∈	∈	PROPN
ejpam-6501	154	14	n	n	NOUN
ejpam-6501	154	15	.	.	PUNCT
ejpam-6501	155	1	the	the	DET
ejpam-6501	155	2	next	next	ADJ
ejpam-6501	155	3	example	example	NOUN
ejpam-6501	155	4	is	be	AUX
ejpam-6501	155	5	shown	show	VERB
ejpam-6501	155	6	that	that	SCONJ
ejpam-6501	155	7	the	the	DET
ejpam-6501	155	8	converse	converse	NOUN
ejpam-6501	155	9	of	of	ADP
ejpam-6501	155	10	theorem	theorem	NOUN
ejpam-6501	155	11	4	4	NUM
ejpam-6501	155	12	is	be	AUX
ejpam-6501	155	13	not	not	PART
ejpam-6501	155	14	true	true	ADJ
ejpam-6501	155	15	.	.	PUNCT
ejpam-6501	156	1	example	example	NOUN
ejpam-6501	156	2	5	5	NUM
ejpam-6501	156	3	.	.	X
ejpam-6501	157	1	consider	consider	VERB
ejpam-6501	157	2	the	the	DET
ejpam-6501	157	3	example	example	NOUN
ejpam-6501	157	4	3	3	NUM
ejpam-6501	157	5	,	,	PUNCT
ejpam-6501	157	6	i	i	PRON
ejpam-6501	157	7	=	=	PUNCT
ejpam-6501	157	8	{	{	PUNCT
ejpam-6501	157	9	0	0	NUM
ejpam-6501	157	10	,	,	PUNCT
ejpam-6501	157	11	a	a	PRON
ejpam-6501	157	12	}	}	PUNCT
ejpam-6501	157	13	is	be	AUX
ejpam-6501	157	14	a	a	DET
ejpam-6501	157	15	subalgebra	subalgebra	NOUN
ejpam-6501	157	16	of	of	ADP
ejpam-6501	157	17	x	x	PUNCT
ejpam-6501	157	18	and	and	CCONJ
ejpam-6501	157	19	so	so	ADV
ejpam-6501	157	20	is	be	AUX
ejpam-6501	157	21	ideal	ideal	ADJ
ejpam-6501	157	22	but	but	CCONJ
ejpam-6501	157	23	not	not	PART
ejpam-6501	157	24	normal	normal	ADJ
ejpam-6501	157	25	,	,	PUNCT
ejpam-6501	157	26	because	because	SCONJ
ejpam-6501	157	27	1	1	NUM
ejpam-6501	157	28	∗	∗	NOUN
ejpam-6501	157	29	b	b	NOUN
ejpam-6501	157	30	=	=	PUNCT
ejpam-6501	157	31	a	a	DET
ejpam-6501	157	32	∈	∈	PROPN
ejpam-6501	158	1	i	i	PRON
ejpam-6501	158	2	and	and	CCONJ
ejpam-6501	158	3	a	a	DET
ejpam-6501	158	4	∗	∗	NOUN
ejpam-6501	158	5	b	b	NOUN
ejpam-6501	158	6	=	=	SYM
ejpam-6501	158	7	0	0	SYM
ejpam-6501	158	8	∈	∈	PROPN
ejpam-6501	159	1	i	i	PRON
ejpam-6501	159	2	,	,	PUNCT
ejpam-6501	159	3	while	while	SCONJ
ejpam-6501	159	4	(	(	PUNCT
ejpam-6501	159	5	1	1	NUM
ejpam-6501	159	6	∗	∗	NOUN
ejpam-6501	159	7	a	a	NOUN
ejpam-6501	159	8	)	)	PUNCT
ejpam-6501	159	9	∗	∗	NOUN
ejpam-6501	159	10	(	(	PUNCT
ejpam-6501	159	11	b	b	NOUN
ejpam-6501	159	12	∗	∗	X
ejpam-6501	159	13	b	b	NOUN
ejpam-6501	159	14	)	)	PUNCT
ejpam-6501	160	1	=	=	SYM
ejpam-6501	160	2	b	b	NOUN
ejpam-6501	160	3	∗	∗	NOUN
ejpam-6501	160	4	0	0	NUM
ejpam-6501	161	1	=	=	SYM
ejpam-6501	161	2	b	b	PROPN
ejpam-6501	161	3	/∈	/∈	NOUN
ejpam-6501	161	4	i.	i.	NOUN
ejpam-6501	161	5	n	n	PROPN
ejpam-6501	161	6	=	=	PUNCT
ejpam-6501	161	7	{	{	PUNCT
ejpam-6501	161	8	0	0	NUM
ejpam-6501	161	9	,	,	PUNCT
ejpam-6501	161	10	1	1	NUM
ejpam-6501	161	11	}	}	PUNCT
ejpam-6501	161	12	is	be	AUX
ejpam-6501	161	13	a	a	DET
ejpam-6501	161	14	normal	normal	ADJ
ejpam-6501	161	15	subset	subset	NOUN
ejpam-6501	161	16	of	of	ADP
ejpam-6501	161	17	x	x	PUNCT
ejpam-6501	161	18	and	and	CCONJ
ejpam-6501	161	19	so	so	ADV
ejpam-6501	161	20	is	be	AUX
ejpam-6501	161	21	a	a	DET
ejpam-6501	161	22	subalgebra	subalgebra	NOUN
ejpam-6501	161	23	.	.	PUNCT
ejpam-6501	162	1	lemma	lemma	PROPN
ejpam-6501	162	2	2	2	X
ejpam-6501	162	3	.	.	PUNCT
ejpam-6501	163	1	if	if	SCONJ
ejpam-6501	163	2	s	s	PROPN
ejpam-6501	163	3	is	be	AUX
ejpam-6501	163	4	a	a	DET
ejpam-6501	163	5	subalgebra	subalgebra	NOUN
ejpam-6501	163	6	of	of	ADP
ejpam-6501	163	7	ink	ink	NOUN
ejpam-6501	163	8	-	-	PUNCT
ejpam-6501	163	9	algebra	algebra	NOUN
ejpam-6501	163	10	(	(	PUNCT
ejpam-6501	163	11	x	x	X
ejpam-6501	163	12	,	,	PUNCT
ejpam-6501	163	13	∗	∗	NOUN
ejpam-6501	163	14	,	,	PUNCT
ejpam-6501	163	15	0	0	NUM
ejpam-6501	163	16	)	)	PUNCT
ejpam-6501	163	17	,	,	PUNCT
ejpam-6501	163	18	then	then	ADV
ejpam-6501	163	19	0	0	NUM
ejpam-6501	163	20	∈	∈	PROPN
ejpam-6501	163	21	s.	s.	PROPN
ejpam-6501	163	22	proof	proof	PROPN
ejpam-6501	163	23	.	.	PUNCT
ejpam-6501	164	1	let	let	VERB
ejpam-6501	164	2	x	x	PUNCT
ejpam-6501	164	3	∈	∈	PROPN
ejpam-6501	164	4	s	s	NOUN
ejpam-6501	164	5	,	,	PUNCT
ejpam-6501	164	6	by	by	ADP
ejpam-6501	164	7	ink-3	ink-3	X
ejpam-6501	164	8	,	,	PUNCT
ejpam-6501	164	9	x	x	X
ejpam-6501	164	10	∗	∗	NOUN
ejpam-6501	164	11	0	0	NUM
ejpam-6501	165	1	=	=	SYM
ejpam-6501	165	2	x	x	PUNCT
ejpam-6501	165	3	∈	∈	NOUN
ejpam-6501	165	4	s	s	X
ejpam-6501	165	5	and	and	CCONJ
ejpam-6501	165	6	hence	hence	ADV
ejpam-6501	165	7	by	by	ADP
ejpam-6501	165	8	ink-2	ink-2	PRON
ejpam-6501	165	9	,	,	PUNCT
ejpam-6501	165	10	0	0	NUM
ejpam-6501	166	1	=	=	SYM
ejpam-6501	166	2	(	(	PUNCT
ejpam-6501	166	3	(	(	PUNCT
ejpam-6501	166	4	0	0	NUM
ejpam-6501	166	5	∗	∗	NOUN
ejpam-6501	166	6	x	x	NOUN
ejpam-6501	166	7	)	)	PUNCT
ejpam-6501	166	8	∗	∗	NOUN
ejpam-6501	166	9	(	(	PUNCT
ejpam-6501	166	10	0	0	NUM
ejpam-6501	166	11	∗	∗	NOUN
ejpam-6501	166	12	x	x	NOUN
ejpam-6501	166	13	)	)	PUNCT
ejpam-6501	166	14	∗	∗	NOUN
ejpam-6501	166	15	(	(	PUNCT
ejpam-6501	166	16	x	x	X
ejpam-6501	166	17	∗	∗	NOUN
ejpam-6501	166	18	x	x	NOUN
ejpam-6501	166	19	)	)	PUNCT
ejpam-6501	166	20	)	)	PUNCT
ejpam-6501	167	1	∈	∈	PROPN
ejpam-6501	167	2	s.	s.	PROPN
ejpam-6501	167	3	lemma	lemma	PROPN
ejpam-6501	167	4	3	3	X
ejpam-6501	167	5	.	.	PUNCT
ejpam-6501	168	1	let	let	VERB
ejpam-6501	168	2	(	(	PUNCT
ejpam-6501	168	3	x	x	X
ejpam-6501	168	4	,	,	PUNCT
ejpam-6501	168	5	∗	∗	NOUN
ejpam-6501	168	6	,	,	PUNCT
ejpam-6501	168	7	0	0	NUM
ejpam-6501	168	8	)	)	PUNCT
ejpam-6501	168	9	be	be	AUX
ejpam-6501	168	10	a	a	DET
ejpam-6501	168	11	ink	ink	NOUN
ejpam-6501	168	12	-	-	PUNCT
ejpam-6501	168	13	algebra	algebra	NOUN
ejpam-6501	169	1	and	and	CCONJ
ejpam-6501	169	2	i	i	PRON
ejpam-6501	169	3	is	be	AUX
ejpam-6501	169	4	an	an	DET
ejpam-6501	169	5	ideal	ideal	NOUN
ejpam-6501	169	6	of	of	ADP
ejpam-6501	169	7	x	x	SYM
ejpam-6501	169	8	such	such	ADJ
ejpam-6501	169	9	that	that	SCONJ
ejpam-6501	169	10	0	0	NUM
ejpam-6501	169	11	∗	∗	NOUN
ejpam-6501	169	12	a	a	DET
ejpam-6501	169	13	∈	∈	NOUN
ejpam-6501	169	14	i	i	PRON
ejpam-6501	169	15	for	for	ADP
ejpam-6501	169	16	all	all	DET
ejpam-6501	169	17	a	a	DET
ejpam-6501	169	18	∈	∈	NOUN
ejpam-6501	170	1	i	i	PRON
ejpam-6501	170	2	,	,	PUNCT
ejpam-6501	170	3	then	then	ADV
ejpam-6501	170	4	i	i	PRON
ejpam-6501	170	5	is	be	AUX
ejpam-6501	170	6	a	a	DET
ejpam-6501	170	7	subalgebra	subalgebra	NOUN
ejpam-6501	170	8	of	of	ADP
ejpam-6501	170	9	x.	x.	NOUN
ejpam-6501	170	10	proof	proof	NOUN
ejpam-6501	170	11	.	.	PUNCT
ejpam-6501	171	1	let	let	VERB
ejpam-6501	171	2	x	x	PRON
ejpam-6501	171	3	,	,	PUNCT
ejpam-6501	171	4	y	y	PROPN
ejpam-6501	171	5	∈	∈	PROPN
ejpam-6501	172	1	i	i	PRON
ejpam-6501	172	2	,	,	PUNCT
ejpam-6501	172	3	then	then	ADV
ejpam-6501	172	4	0	0	NUM
ejpam-6501	172	5	∗	∗	NOUN
ejpam-6501	172	6	y	y	NOUN
ejpam-6501	172	7	=	=	SYM
ejpam-6501	172	8	(	(	PUNCT
ejpam-6501	172	9	x	x	X
ejpam-6501	172	10	∗	∗	NOUN
ejpam-6501	172	11	x	x	NOUN
ejpam-6501	172	12	)	)	PUNCT
ejpam-6501	172	13	∗	∗	NOUN
ejpam-6501	172	14	y	y	NOUN
ejpam-6501	172	15	=	=	SYM
ejpam-6501	172	16	(	(	PUNCT
ejpam-6501	172	17	x	x	X
ejpam-6501	172	18	∗	∗	PROPN
ejpam-6501	172	19	y	y	NOUN
ejpam-6501	172	20	)	)	PUNCT
ejpam-6501	172	21	∗	∗	NOUN
ejpam-6501	172	22	x	x	X
ejpam-6501	172	23	∈	∈	NOUN
ejpam-6501	172	24	i	i	PRON
ejpam-6501	172	25	implies	imply	VERB
ejpam-6501	172	26	that	that	SCONJ
ejpam-6501	172	27	x	x	SYM
ejpam-6501	172	28	∗	∗	NOUN
ejpam-6501	172	29	y	y	PROPN
ejpam-6501	172	30	∈	∈	PROPN
ejpam-6501	172	31	i.	i.	NOUN
ejpam-6501	172	32	definition	definition	NOUN
ejpam-6501	172	33	6	6	NUM
ejpam-6501	172	34	.	.	PUNCT
ejpam-6501	173	1	let	let	VERB
ejpam-6501	173	2	(	(	PUNCT
ejpam-6501	173	3	x	x	X
ejpam-6501	173	4	,	,	PUNCT
ejpam-6501	173	5	∗	∗	NOUN
ejpam-6501	173	6	,	,	PUNCT
ejpam-6501	173	7	0	0	NUM
ejpam-6501	173	8	)	)	PUNCT
ejpam-6501	173	9	be	be	AUX
ejpam-6501	173	10	an	an	DET
ejpam-6501	173	11	ink	ink	NOUN
ejpam-6501	173	12	-	-	PUNCT
ejpam-6501	173	13	algebra	algebra	NOUN
ejpam-6501	173	14	and	and	CCONJ
ejpam-6501	173	15	let	let	VERB
ejpam-6501	173	16	i	i	PRON
ejpam-6501	173	17	be	be	AUX
ejpam-6501	173	18	a	a	DET
ejpam-6501	173	19	subset	subset	NOUN
ejpam-6501	173	20	of	of	ADP
ejpam-6501	173	21	x	x	PUNCT
ejpam-6501	173	22	with	with	ADP
ejpam-6501	173	23	an	an	DET
ejpam-6501	173	24	associated	associated	ADJ
ejpam-6501	173	25	congruence	congruence	NOUN
ejpam-6501	173	26	relation	relation	NOUN
ejpam-6501	173	27	∼	∼	NOUN
ejpam-6501	173	28	i	i	PRON
ejpam-6501	173	29	.	.	PUNCT
ejpam-6501	174	1	define	define	VERB
ejpam-6501	174	2	the	the	DET
ejpam-6501	174	3	set	set	NOUN
ejpam-6501	174	4	x	x	NOUN
ejpam-6501	174	5	/	/	SYM
ejpam-6501	174	6	i	i	PRON
ejpam-6501	174	7	=	=	PUNCT
ejpam-6501	174	8	{	{	PUNCT
ejpam-6501	175	1	[	[	X
ejpam-6501	175	2	x]i	x]i	X
ejpam-6501	175	3	|	|	ADV
ejpam-6501	175	4	x	x	SYM
ejpam-6501	175	5	∈	∈	NOUN
ejpam-6501	175	6	x	x	X
ejpam-6501	175	7	}	}	PUNCT
ejpam-6501	175	8	,	,	PUNCT
ejpam-6501	175	9	where	where	SCONJ
ejpam-6501	175	10	[	[	X
ejpam-6501	175	11	x]i	x]i	X
ejpam-6501	175	12	=	=	SYM
ejpam-6501	175	13	{	{	PUNCT
ejpam-6501	175	14	y	y	PROPN
ejpam-6501	175	15	∈	∈	PROPN
ejpam-6501	175	16	x|x	x|x	PUNCT
ejpam-6501	176	1	∼	∼	NOUN
ejpam-6501	176	2	i	i	PRON
ejpam-6501	176	3	y	y	NOUN
ejpam-6501	176	4	}	}	PUNCT
ejpam-6501	176	5	,	,	PUNCT
ejpam-6501	176	6	and	and	CCONJ
ejpam-6501	176	7	a	a	DET
ejpam-6501	176	8	binary	binary	ADJ
ejpam-6501	176	9	operation	operation	NOUN
ejpam-6501	176	10	⊙	⊙	NOUN
ejpam-6501	176	11	on	on	ADP
ejpam-6501	176	12	x	x	PROPN
ejpam-6501	176	13	/	/	SYM
ejpam-6501	176	14	i	i	PRON
ejpam-6501	176	15	defined	define	VERB
ejpam-6501	176	16	by	by	ADP
ejpam-6501	176	17	[	[	X
ejpam-6501	176	18	x]i	x]i	PROPN
ejpam-6501	176	19	⊙	⊙	PROPN
ejpam-6501	177	1	[	[	X
ejpam-6501	177	2	y]i	y]i	NOUN
ejpam-6501	177	3	=	=	PUNCT
ejpam-6501	177	4	[	[	X
ejpam-6501	177	5	x	x	X
ejpam-6501	177	6	∗	∗	NOUN
ejpam-6501	177	7	y]i	y]i	ADJ
ejpam-6501	177	8	for	for	ADP
ejpam-6501	177	9	all	all	DET
ejpam-6501	177	10	[	[	X
ejpam-6501	177	11	x]i	x]i	X
ejpam-6501	177	12	,	,	PUNCT
ejpam-6501	177	13	[	[	X
ejpam-6501	177	14	y]i	y]i	ADJ
ejpam-6501	177	15	∈	∈	ADJ
ejpam-6501	177	16	x	x	X
ejpam-6501	177	17	/	/	SYM
ejpam-6501	177	18	i.	i.	NOUN
ejpam-6501	177	19	we	we	PRON
ejpam-6501	177	20	say	say	VERB
ejpam-6501	177	21	that	that	SCONJ
ejpam-6501	177	22	(	(	PUNCT
ejpam-6501	177	23	x	x	X
ejpam-6501	177	24	/	/	SYM
ejpam-6501	177	25	n,⊙	n,⊙	ADJ
ejpam-6501	177	26	,	,	PUNCT
ejpam-6501	177	27	[	[	X
ejpam-6501	177	28	0]n	0]n	NUM
ejpam-6501	177	29	)	)	PUNCT
ejpam-6501	177	30	is	be	AUX
ejpam-6501	177	31	a	a	DET
ejpam-6501	177	32	quotient	quotient	NOUN
ejpam-6501	177	33	ink	ink	NOUN
ejpam-6501	177	34	-	-	PUNCT
ejpam-6501	177	35	algebra	algebra	NOUN
ejpam-6501	177	36	if	if	SCONJ
ejpam-6501	177	37	it	it	PRON
ejpam-6501	177	38	satisfies	satisfy	VERB
ejpam-6501	177	39	definition	definition	NOUN
ejpam-6501	177	40	2	2	NUM
ejpam-6501	177	41	.	.	PUNCT
ejpam-6501	177	42	theorem	theorem	NOUN
ejpam-6501	177	43	5	5	NUM
ejpam-6501	177	44	.	.	PUNCT
ejpam-6501	178	1	let	let	AUX
ejpam-6501	178	2	(	(	PUNCT
ejpam-6501	178	3	x	x	X
ejpam-6501	178	4	,	,	PUNCT
ejpam-6501	178	5	∗	∗	NOUN
ejpam-6501	178	6	,	,	PUNCT
ejpam-6501	178	7	0	0	NUM
ejpam-6501	178	8	)	)	PUNCT
ejpam-6501	178	9	be	be	AUX
ejpam-6501	178	10	a	a	DET
ejpam-6501	178	11	ink	ink	NOUN
ejpam-6501	178	12	-	-	PUNCT
ejpam-6501	178	13	algebra	algebra	NOUN
ejpam-6501	178	14	and	and	CCONJ
ejpam-6501	178	15	n	n	NOUN
ejpam-6501	178	16	is	be	AUX
ejpam-6501	178	17	a	a	DET
ejpam-6501	178	18	normal	normal	ADJ
ejpam-6501	178	19	subset	subset	NOUN
ejpam-6501	178	20	of	of	ADP
ejpam-6501	178	21	x	x	PRON
ejpam-6501	178	22	,	,	PUNCT
ejpam-6501	178	23	then	then	ADV
ejpam-6501	178	24	(	(	PUNCT
ejpam-6501	178	25	x	x	X
ejpam-6501	178	26	/	/	SYM
ejpam-6501	178	27	n,⊙	n,⊙	ADJ
ejpam-6501	178	28	,	,	PUNCT
ejpam-6501	178	29	[	[	X
ejpam-6501	178	30	0]i	0]i	NUM
ejpam-6501	178	31	)	)	PUNCT
ejpam-6501	178	32	is	be	AUX
ejpam-6501	178	33	a	a	DET
ejpam-6501	178	34	quotient	quotient	NOUN
ejpam-6501	178	35	ink	ink	NOUN
ejpam-6501	178	36	-	-	PUNCT
ejpam-6501	178	37	algebra	algebra	NOUN
ejpam-6501	178	38	.	.	PUNCT
ejpam-6501	179	1	proof	proof	NOUN
ejpam-6501	179	2	.	.	PUNCT
ejpam-6501	180	1	clearly	clearly	ADV
ejpam-6501	180	2	the	the	DET
ejpam-6501	180	3	operation	operation	NOUN
ejpam-6501	180	4	⊙	⊙	PROPN
ejpam-6501	180	5	is	be	AUX
ejpam-6501	180	6	well	well	ADV
ejpam-6501	180	7	-	-	PUNCT
ejpam-6501	180	8	defined	define	VERB
ejpam-6501	180	9	.	.	PUNCT
ejpam-6501	181	1	that	that	PRON
ejpam-6501	181	2	is	be	AUX
ejpam-6501	181	3	,	,	PUNCT
ejpam-6501	181	4	for	for	ADP
ejpam-6501	181	5	any	any	DET
ejpam-6501	181	6	[	[	X
ejpam-6501	181	7	a]n	a]n	NOUN
ejpam-6501	181	8	,	,	PUNCT
ejpam-6501	182	1	[	[	X
ejpam-6501	182	2	b]n	b]n	X
ejpam-6501	182	3	,	,	PUNCT
ejpam-6501	182	4	[	[	X
ejpam-6501	182	5	x]n	x]n	X
ejpam-6501	182	6	,	,	PUNCT
ejpam-6501	183	1	[	[	X
ejpam-6501	183	2	y]n	y]n	X
ejpam-6501	183	3	∈	∈	ADJ
ejpam-6501	183	4	x	x	NOUN
ejpam-6501	183	5	/	/	SYM
ejpam-6501	183	6	n	n	CCONJ
ejpam-6501	183	7	,	,	PUNCT
ejpam-6501	183	8	if	if	SCONJ
ejpam-6501	183	9	(	(	PUNCT
ejpam-6501	183	10	[	[	X
ejpam-6501	183	11	a]n	a]n	NOUN
ejpam-6501	183	12	,	,	PUNCT
ejpam-6501	183	13	[	[	X
ejpam-6501	183	14	b]n	b]n	X
ejpam-6501	183	15	)	)	PUNCT
ejpam-6501	183	16	=	=	SYM
ejpam-6501	183	17	(	(	PUNCT
ejpam-6501	184	1	[	[	X
ejpam-6501	184	2	x]n	x]n	X
ejpam-6501	184	3	,	,	PUNCT
ejpam-6501	185	1	[	[	X
ejpam-6501	185	2	y]n	y]n	NOUN
ejpam-6501	185	3	)	)	PUNCT
ejpam-6501	185	4	,	,	PUNCT
ejpam-6501	185	5	then	then	ADV
ejpam-6501	186	1	[	[	X
ejpam-6501	186	2	a]n	a]n	NOUN
ejpam-6501	186	3	=	=	PUNCT
ejpam-6501	187	1	[	[	X
ejpam-6501	187	2	x]n	x]n	PROPN
ejpam-6501	187	3	and	and	CCONJ
ejpam-6501	187	4	[	[	X
ejpam-6501	187	5	b]n	b]n	X
ejpam-6501	187	6	=	=	X
ejpam-6501	188	1	[	[	X
ejpam-6501	188	2	y]n	y]n	X
ejpam-6501	188	3	,	,	PUNCT
ejpam-6501	188	4	which	which	PRON
ejpam-6501	188	5	implies	imply	VERB
ejpam-6501	188	6	that	that	DET
ejpam-6501	188	7	m.	m.	NOUN
ejpam-6501	188	8	phattarachaleekul	phattarachaleekul	PROPN
ejpam-6501	188	9	/	/	SYM
ejpam-6501	188	10	eur	eur	PROPN
ejpam-6501	188	11	.	.	PUNCT
ejpam-6501	189	1	j.	j.	PROPN
ejpam-6501	189	2	pure	pure	PROPN
ejpam-6501	189	3	appl	appl	PROPN
ejpam-6501	189	4	.	.	PROPN
ejpam-6501	189	5	math	math	PROPN
ejpam-6501	189	6	,	,	PUNCT
ejpam-6501	189	7	18	18	NUM
ejpam-6501	189	8	(	(	PUNCT
ejpam-6501	189	9	4	4	NUM
ejpam-6501	189	10	)	)	PUNCT
ejpam-6501	189	11	(	(	PUNCT
ejpam-6501	189	12	2025	2025	NUM
ejpam-6501	189	13	)	)	PUNCT
ejpam-6501	189	14	,	,	PUNCT
ejpam-6501	189	15	6501	6501	NUM
ejpam-6501	189	16	7	7	NUM
ejpam-6501	189	17	of	of	ADP
ejpam-6501	189	18	8	8	NUM
ejpam-6501	189	19	a	a	DET
ejpam-6501	189	20	∼	∼	NOUN
ejpam-6501	189	21	n	n	NOUN
ejpam-6501	189	22	x	x	NOUN
ejpam-6501	189	23	and	and	CCONJ
ejpam-6501	189	24	b	b	NOUN
ejpam-6501	189	25	∼	∼	NOUN
ejpam-6501	189	26	n	n	PRON
ejpam-6501	189	27	y	y	NOUN
ejpam-6501	189	28	,	,	PUNCT
ejpam-6501	189	29	thus	thus	ADV
ejpam-6501	189	30	(	(	PUNCT
ejpam-6501	189	31	a	a	DET
ejpam-6501	189	32	∗	∗	NOUN
ejpam-6501	189	33	b	b	NOUN
ejpam-6501	189	34	)	)	PUNCT
ejpam-6501	189	35	∼	∼	NOUN
ejpam-6501	189	36	n	n	NOUN
ejpam-6501	189	37	(	(	PUNCT
ejpam-6501	189	38	x	x	X
ejpam-6501	189	39	∗	∗	PROPN
ejpam-6501	189	40	y	y	PROPN
ejpam-6501	189	41	)	)	PUNCT
ejpam-6501	189	42	,	,	PUNCT
ejpam-6501	189	43	and	and	CCONJ
ejpam-6501	189	44	hence	hence	ADV
ejpam-6501	189	45	[	[	X
ejpam-6501	189	46	a	a	DET
ejpam-6501	189	47	∗	∗	NOUN
ejpam-6501	189	48	b]n	b]n	NOUN
ejpam-6501	189	49	=	=	PUNCT
ejpam-6501	190	1	[	[	X
ejpam-6501	190	2	x	x	X
ejpam-6501	190	3	∗	∗	PROPN
ejpam-6501	190	4	y]n	y]n	INTJ
ejpam-6501	190	5	,	,	PUNCT
ejpam-6501	190	6	implies	imply	VERB
ejpam-6501	190	7	that	that	SCONJ
ejpam-6501	190	8	[	[	X
ejpam-6501	190	9	a]n	a]n	NOUN
ejpam-6501	190	10	⊙	⊙	VERB
ejpam-6501	191	1	[	[	X
ejpam-6501	191	2	b]n	b]n	X
ejpam-6501	191	3	=	=	PUNCT
ejpam-6501	192	1	[	[	X
ejpam-6501	192	2	x]n	x]n	PROPN
ejpam-6501	192	3	⊙	⊙	PROPN
ejpam-6501	193	1	[	[	X
ejpam-6501	193	2	y]n	y]n	INTJ
ejpam-6501	193	3	,	,	PUNCT
ejpam-6501	193	4	and	and	CCONJ
ejpam-6501	193	5	satisfies	satisfy	VERB
ejpam-6501	193	6	the	the	DET
ejpam-6501	193	7	following	following	NOUN
ejpam-6501	193	8	;	;	PUNCT
ejpam-6501	193	9	1	1	X
ejpam-6501	193	10	)	)	PUNCT
ejpam-6501	194	1	[	[	X
ejpam-6501	194	2	(	(	PUNCT
ejpam-6501	194	3	[	[	X
ejpam-6501	194	4	x]n	x]n	PROPN
ejpam-6501	194	5	⊙	⊙	VERB
ejpam-6501	195	1	[	[	X
ejpam-6501	195	2	y]n	y]n	NOUN
ejpam-6501	195	3	)	)	PUNCT
ejpam-6501	195	4	⊙	⊙	NOUN
ejpam-6501	195	5	(	(	PUNCT
ejpam-6501	195	6	[	[	X
ejpam-6501	195	7	x]n	x]n	PROPN
ejpam-6501	195	8	⊙	⊙	VERB
ejpam-6501	196	1	[	[	X
ejpam-6501	196	2	z]n	z]n	PROPN
ejpam-6501	196	3	)	)	PUNCT
ejpam-6501	196	4	]	]	PUNCT
ejpam-6501	196	5	⊙	⊙	X
ejpam-6501	196	6	(	(	PUNCT
ejpam-6501	196	7	[	[	X
ejpam-6501	196	8	z]n	z]n	X
ejpam-6501	196	9	⊙	⊙	VERB
ejpam-6501	197	1	[	[	X
ejpam-6501	197	2	y]n	y]n	NOUN
ejpam-6501	197	3	)	)	PUNCT
ejpam-6501	197	4	=	=	PUNCT
ejpam-6501	198	1	[	[	X
ejpam-6501	198	2	(	(	PUNCT
ejpam-6501	198	3	[	[	X
ejpam-6501	198	4	x	x	X
ejpam-6501	198	5	∗	∗	PROPN
ejpam-6501	198	6	y]n	y]n	NOUN
ejpam-6501	198	7	)	)	PUNCT
ejpam-6501	198	8	⊙	⊙	NOUN
ejpam-6501	198	9	(	(	PUNCT
ejpam-6501	198	10	[	[	X
ejpam-6501	198	11	x	x	X
ejpam-6501	198	12	∗	∗	NOUN
ejpam-6501	198	13	z]n	z]n	PROPN
ejpam-6501	198	14	)	)	PUNCT
ejpam-6501	199	1	]	]	PUNCT
ejpam-6501	199	2	⊙	⊙	X
ejpam-6501	199	3	(	(	PUNCT
ejpam-6501	199	4	[	[	X
ejpam-6501	199	5	z	z	X
ejpam-6501	199	6	∗	∗	NOUN
ejpam-6501	199	7	y]n	y]n	NOUN
ejpam-6501	199	8	)	)	PUNCT
ejpam-6501	200	1	=	=	PUNCT
ejpam-6501	201	1	[	[	X
ejpam-6501	201	2	(	(	PUNCT
ejpam-6501	201	3	x	x	X
ejpam-6501	201	4	∗	∗	PROPN
ejpam-6501	201	5	y	y	NOUN
ejpam-6501	201	6	)	)	PUNCT
ejpam-6501	201	7	∗	∗	NOUN
ejpam-6501	201	8	(	(	PUNCT
ejpam-6501	201	9	x	x	SYM
ejpam-6501	201	10	∗	∗	PROPN
ejpam-6501	201	11	z)]n	z)]n	NOUN
ejpam-6501	201	12	⊙	⊙	NOUN
ejpam-6501	202	1	[	[	X
ejpam-6501	202	2	z	z	X
ejpam-6501	202	3	∗	∗	X
ejpam-6501	202	4	y]n	y]n	NOUN
ejpam-6501	203	1	=	=	X
ejpam-6501	204	1	[	[	X
ejpam-6501	204	2	(	(	PUNCT
ejpam-6501	204	3	(	(	PUNCT
ejpam-6501	204	4	x	x	SYM
ejpam-6501	204	5	∗	∗	PROPN
ejpam-6501	204	6	y	y	NOUN
ejpam-6501	204	7	)	)	PUNCT
ejpam-6501	204	8	∗	∗	NOUN
ejpam-6501	204	9	(	(	PUNCT
ejpam-6501	204	10	x	x	X
ejpam-6501	204	11	∗	∗	PROPN
ejpam-6501	204	12	z	z	NOUN
ejpam-6501	204	13	)	)	PUNCT
ejpam-6501	204	14	)	)	PUNCT
ejpam-6501	204	15	∗	∗	NOUN
ejpam-6501	204	16	(	(	PUNCT
ejpam-6501	204	17	z	z	NOUN
ejpam-6501	204	18	∗	∗	NOUN
ejpam-6501	204	19	y)]n	y)]n	NOUN
ejpam-6501	204	20	=	=	PUNCT
ejpam-6501	205	1	[	[	X
ejpam-6501	205	2	0]n	0]n	NUM
ejpam-6501	205	3	,	,	PUNCT
ejpam-6501	205	4	2	2	X
ejpam-6501	205	5	)	)	PUNCT
ejpam-6501	206	1	[	[	X
ejpam-6501	206	2	(	(	PUNCT
ejpam-6501	206	3	[	[	X
ejpam-6501	206	4	x]n	x]n	PROPN
ejpam-6501	206	5	⊙	⊙	PROPN
ejpam-6501	207	1	[	[	X
ejpam-6501	207	2	z]n	z]n	PROPN
ejpam-6501	207	3	)	)	PUNCT
ejpam-6501	207	4	⊙	⊙	NOUN
ejpam-6501	207	5	(	(	PUNCT
ejpam-6501	208	1	[	[	X
ejpam-6501	208	2	y]n	y]n	X
ejpam-6501	208	3	⊙	⊙	NOUN
ejpam-6501	209	1	[	[	X
ejpam-6501	209	2	z]n	z]n	PROPN
ejpam-6501	209	3	)	)	PUNCT
ejpam-6501	209	4	]	]	PUNCT
ejpam-6501	209	5	⊙	⊙	X
ejpam-6501	209	6	(	(	PUNCT
ejpam-6501	209	7	[	[	X
ejpam-6501	209	8	z]n	z]n	X
ejpam-6501	209	9	⊙	⊙	VERB
ejpam-6501	210	1	[	[	X
ejpam-6501	210	2	y]n	y]n	NOUN
ejpam-6501	210	3	)	)	PUNCT
ejpam-6501	210	4	=	=	PUNCT
ejpam-6501	211	1	[	[	X
ejpam-6501	211	2	(	(	PUNCT
ejpam-6501	211	3	[	[	X
ejpam-6501	211	4	x	x	X
ejpam-6501	211	5	∗	∗	PROPN
ejpam-6501	211	6	y]n	y]n	NOUN
ejpam-6501	211	7	)	)	PUNCT
ejpam-6501	211	8	⊙	⊙	NOUN
ejpam-6501	211	9	(	(	PUNCT
ejpam-6501	211	10	[	[	X
ejpam-6501	211	11	x	x	X
ejpam-6501	211	12	∗	∗	NOUN
ejpam-6501	211	13	z]n	z]n	PROPN
ejpam-6501	211	14	)	)	PUNCT
ejpam-6501	212	1	]	]	PUNCT
ejpam-6501	212	2	⊙	⊙	X
ejpam-6501	212	3	(	(	PUNCT
ejpam-6501	212	4	[	[	X
ejpam-6501	212	5	x	x	X
ejpam-6501	212	6	∗	∗	NOUN
ejpam-6501	212	7	y]n	y]n	NOUN
ejpam-6501	212	8	)	)	PUNCT
ejpam-6501	213	1	=	=	PUNCT
ejpam-6501	214	1	[	[	X
ejpam-6501	214	2	(	(	PUNCT
ejpam-6501	214	3	x	x	X
ejpam-6501	214	4	∗	∗	PROPN
ejpam-6501	214	5	z	z	NOUN
ejpam-6501	214	6	)	)	PUNCT
ejpam-6501	214	7	∗	∗	NOUN
ejpam-6501	214	8	(	(	PUNCT
ejpam-6501	214	9	y	y	PROPN
ejpam-6501	214	10	∗	∗	PROPN
ejpam-6501	214	11	z)]n	z)]n	PROPN
ejpam-6501	214	12	⊙	⊙	NOUN
ejpam-6501	215	1	[	[	X
ejpam-6501	215	2	x	x	X
ejpam-6501	215	3	∗	∗	X
ejpam-6501	215	4	y]n	y]n	NOUN
ejpam-6501	216	1	=	=	X
ejpam-6501	217	1	[	[	X
ejpam-6501	217	2	(	(	PUNCT
ejpam-6501	217	3	(	(	PUNCT
ejpam-6501	217	4	x	x	NOUN
ejpam-6501	217	5	∗	∗	PROPN
ejpam-6501	217	6	z	z	NOUN
ejpam-6501	217	7	)	)	PUNCT
ejpam-6501	217	8	∗	∗	NOUN
ejpam-6501	217	9	(	(	PUNCT
ejpam-6501	217	10	y	y	PROPN
ejpam-6501	217	11	∗	∗	NOUN
ejpam-6501	217	12	z))n	z))n	NOUN
ejpam-6501	217	13	∗	∗	NOUN
ejpam-6501	217	14	(	(	PUNCT
ejpam-6501	217	15	x	x	SYM
ejpam-6501	217	16	∗	∗	VERB
ejpam-6501	217	17	y)]n	y)]n	NOUN
ejpam-6501	217	18	=	=	PUNCT
ejpam-6501	218	1	[	[	X
ejpam-6501	218	2	0]n	0]n	NUM
ejpam-6501	218	3	,	,	PUNCT
ejpam-6501	218	4	3	3	X
ejpam-6501	218	5	)	)	PUNCT
ejpam-6501	219	1	[	[	X
ejpam-6501	219	2	x]n	x]n	PROPN
ejpam-6501	219	3	⊙	⊙	VERB
ejpam-6501	220	1	[	[	X
ejpam-6501	220	2	0]n	0]n	NUM
ejpam-6501	220	3	=	=	PUNCT
ejpam-6501	221	1	[	[	X
ejpam-6501	221	2	x	x	X
ejpam-6501	221	3	∗	∗	NOUN
ejpam-6501	221	4	0]n	0]n	NUM
ejpam-6501	221	5	=	=	PUNCT
ejpam-6501	222	1	[	[	X
ejpam-6501	222	2	x]n	x]n	NOUN
ejpam-6501	222	3	4	4	NUM
ejpam-6501	222	4	)	)	PUNCT
ejpam-6501	222	5	assume	assume	VERB
ejpam-6501	222	6	that	that	SCONJ
ejpam-6501	222	7	[	[	X
ejpam-6501	222	8	x]n	x]n	PROPN
ejpam-6501	222	9	⊙	⊙	VERB
ejpam-6501	223	1	[	[	X
ejpam-6501	223	2	y]n	y]n	NOUN
ejpam-6501	223	3	=	=	X
ejpam-6501	224	1	[	[	X
ejpam-6501	224	2	0]n	0]n	NOUN
ejpam-6501	224	3	,	,	PUNCT
ejpam-6501	224	4	then	then	ADV
ejpam-6501	224	5	[	[	X
ejpam-6501	224	6	x∗y]n	x∗y]n	X
ejpam-6501	224	7	=	=	PUNCT
ejpam-6501	225	1	[	[	X
ejpam-6501	225	2	0]n	0]n	NOUN
ejpam-6501	225	3	,	,	PUNCT
ejpam-6501	225	4	implies	imply	VERB
ejpam-6501	225	5	that	that	SCONJ
ejpam-6501	225	6	x∗y	x∗y	PUNCT
ejpam-6501	225	7	=	=	SYM
ejpam-6501	225	8	0	0	NUM
ejpam-6501	225	9	,	,	PUNCT
ejpam-6501	225	10	so	so	ADV
ejpam-6501	225	11	x	x	SYM
ejpam-6501	225	12	=	=	PUNCT
ejpam-6501	225	13	y.	y.	NOUN
ejpam-6501	225	14	therefore	therefore	ADV
ejpam-6501	225	15	(	(	PUNCT
ejpam-6501	225	16	x	x	X
ejpam-6501	225	17	/	/	SYM
ejpam-6501	225	18	n,⊙	n,⊙	ADJ
ejpam-6501	225	19	,	,	PUNCT
ejpam-6501	225	20	[	[	X
ejpam-6501	225	21	0]n	0]n	NUM
ejpam-6501	225	22	)	)	PUNCT
ejpam-6501	225	23	is	be	AUX
ejpam-6501	225	24	a	a	DET
ejpam-6501	225	25	quotient	quotient	NOUN
ejpam-6501	225	26	ink	ink	NOUN
ejpam-6501	225	27	-	-	PUNCT
ejpam-6501	225	28	algebra	algebra	NOUN
ejpam-6501	225	29	.	.	PUNCT
ejpam-6501	226	1	5	5	X
ejpam-6501	226	2	.	.	X
ejpam-6501	226	3	conclusion	conclusion	NOUN
ejpam-6501	226	4	in	in	ADP
ejpam-6501	226	5	this	this	DET
ejpam-6501	226	6	paper	paper	NOUN
ejpam-6501	226	7	,	,	PUNCT
ejpam-6501	226	8	we	we	PRON
ejpam-6501	226	9	investigated	investigate	VERB
ejpam-6501	226	10	the	the	DET
ejpam-6501	226	11	structure	structure	NOUN
ejpam-6501	226	12	of	of	ADP
ejpam-6501	226	13	congruence	congruence	NOUN
ejpam-6501	226	14	relations	relation	NOUN
ejpam-6501	226	15	induced	induce	VERB
ejpam-6501	226	16	by	by	ADP
ejpam-6501	226	17	ideals	ideal	NOUN
ejpam-6501	226	18	in	in	ADP
ejpam-6501	226	19	ink	ink	NOUN
ejpam-6501	226	20	-	-	PUNCT
ejpam-6501	226	21	algebras	algebras	PROPN
ejpam-6501	226	22	.	.	PUNCT
ejpam-6501	227	1	we	we	PRON
ejpam-6501	227	2	began	begin	VERB
ejpam-6501	227	3	by	by	ADP
ejpam-6501	227	4	reviewing	review	VERB
ejpam-6501	227	5	the	the	DET
ejpam-6501	227	6	foundational	foundational	ADJ
ejpam-6501	227	7	definitions	definition	NOUN
ejpam-6501	227	8	and	and	CCONJ
ejpam-6501	227	9	properties	property	NOUN
ejpam-6501	227	10	of	of	ADP
ejpam-6501	227	11	ink	ink	NOUN
ejpam-6501	227	12	-	-	PUNCT
ejpam-6501	227	13	algebras	algebras	PROPN
ejpam-6501	227	14	,	,	PUNCT
ejpam-6501	227	15	a	a	DET
ejpam-6501	227	16	class	class	NOUN
ejpam-6501	227	17	of	of	ADP
ejpam-6501	227	18	non	non	ADJ
ejpam-6501	227	19	-	-	ADJ
ejpam-6501	227	20	classical	classical	ADJ
ejpam-6501	227	21	algebras	algebra	NOUN
ejpam-6501	227	22	characterized	characterize	VERB
ejpam-6501	227	23	by	by	ADP
ejpam-6501	227	24	specific	specific	ADJ
ejpam-6501	227	25	axioms	axiom	NOUN
ejpam-6501	227	26	on	on	ADP
ejpam-6501	227	27	binary	binary	ADJ
ejpam-6501	227	28	operation	operation	NOUN
ejpam-6501	227	29	axioms	axiom	NOUN
ejpam-6501	227	30	.	.	PUNCT
ejpam-6501	228	1	notably	notably	ADV
ejpam-6501	228	2	,	,	PUNCT
ejpam-6501	228	3	we	we	PRON
ejpam-6501	228	4	established	establish	VERB
ejpam-6501	228	5	the	the	DET
ejpam-6501	228	6	necessary	necessary	ADJ
ejpam-6501	228	7	and	and	CCONJ
ejpam-6501	228	8	sufficient	sufficient	ADJ
ejpam-6501	228	9	conditions	condition	NOUN
ejpam-6501	228	10	under	under	ADP
ejpam-6501	228	11	which	which	PRON
ejpam-6501	228	12	a	a	DET
ejpam-6501	228	13	binary	binary	ADJ
ejpam-6501	228	14	relation	relation	NOUN
ejpam-6501	228	15	derived	derive	VERB
ejpam-6501	228	16	from	from	ADP
ejpam-6501	228	17	an	an	DET
ejpam-6501	228	18	ideal	ideal	NOUN
ejpam-6501	228	19	constitutes	constitute	VERB
ejpam-6501	228	20	a	a	DET
ejpam-6501	228	21	congruence	congruence	NOUN
ejpam-6501	228	22	relation	relation	NOUN
ejpam-6501	228	23	.	.	PUNCT
ejpam-6501	229	1	the	the	DET
ejpam-6501	229	2	main	main	ADJ
ejpam-6501	229	3	results	result	NOUN
ejpam-6501	229	4	demonstrated	demonstrate	VERB
ejpam-6501	229	5	that	that	SCONJ
ejpam-6501	229	6	for	for	ADP
ejpam-6501	229	7	any	any	DET
ejpam-6501	229	8	ideal	ideal	NOUN
ejpam-6501	229	9	i	i	PRON
ejpam-6501	229	10	of	of	ADP
ejpam-6501	229	11	an	an	DET
ejpam-6501	229	12	ink	ink	NOUN
ejpam-6501	229	13	-	-	PUNCT
ejpam-6501	229	14	algebra	algebra	NOUN
ejpam-6501	229	15	(	(	PUNCT
ejpam-6501	229	16	x	x	X
ejpam-6501	229	17	,	,	PUNCT
ejpam-6501	229	18	∗	∗	NOUN
ejpam-6501	229	19	,	,	PUNCT
ejpam-6501	229	20	0	0	NUM
ejpam-6501	229	21	)	)	PUNCT
ejpam-6501	229	22	,	,	PUNCT
ejpam-6501	229	23	the	the	DET
ejpam-6501	229	24	relation	relation	NOUN
ejpam-6501	229	25	∼	∼	VERB
ejpam-6501	229	26	i	i	PRON
ejpam-6501	229	27	,	,	PUNCT
ejpam-6501	229	28	defined	define	VERB
ejpam-6501	229	29	by	by	ADP
ejpam-6501	229	30	x	x	PUNCT
ejpam-6501	229	31	∼	∼	NOUN
ejpam-6501	230	1	i	i	PRON
ejpam-6501	230	2	y	y	NOUN
ejpam-6501	230	3	if	if	SCONJ
ejpam-6501	230	4	and	and	CCONJ
ejpam-6501	230	5	only	only	ADV
ejpam-6501	230	6	if	if	SCONJ
ejpam-6501	230	7	x	x	X
ejpam-6501	230	8	∗	∗	VERB
ejpam-6501	230	9	y	y	NOUN
ejpam-6501	230	10	∈	∈	PROPN
ejpam-6501	231	1	i	i	PRON
ejpam-6501	231	2	and	and	CCONJ
ejpam-6501	231	3	y	y	PROPN
ejpam-6501	231	4	∗	∗	NOUN
ejpam-6501	231	5	x	x	PUNCT
ejpam-6501	231	6	∈	∈	PROPN
ejpam-6501	231	7	i	i	PRON
ejpam-6501	231	8	,	,	PUNCT
ejpam-6501	231	9	is	be	AUX
ejpam-6501	231	10	a	a	DET
ejpam-6501	231	11	congruence	congruence	NOUN
ejpam-6501	231	12	relation	relation	NOUN
ejpam-6501	231	13	on	on	ADP
ejpam-6501	231	14	x.	x.	NOUN
ejpam-6501	231	15	furthermore	furthermore	ADV
ejpam-6501	231	16	,	,	PUNCT
ejpam-6501	231	17	it	it	PRON
ejpam-6501	231	18	was	be	AUX
ejpam-6501	231	19	shown	show	VERB
ejpam-6501	231	20	that	that	SCONJ
ejpam-6501	231	21	the	the	DET
ejpam-6501	231	22	equivalence	equivalence	NOUN
ejpam-6501	231	23	classes	class	NOUN
ejpam-6501	231	24	under	under	ADP
ejpam-6501	231	25	this	this	DET
ejpam-6501	231	26	congruence	congruence	NOUN
ejpam-6501	231	27	relation	relation	NOUN
ejpam-6501	231	28	form	form	VERB
ejpam-6501	231	29	a	a	DET
ejpam-6501	231	30	partition	partition	NOUN
ejpam-6501	231	31	of	of	ADP
ejpam-6501	231	32	the	the	DET
ejpam-6501	231	33	underlying	underlie	VERB
ejpam-6501	231	34	set	set	NOUN
ejpam-6501	231	35	x	x	NOUN
ejpam-6501	231	36	,	,	PUNCT
ejpam-6501	231	37	enabling	enable	VERB
ejpam-6501	231	38	a	a	DET
ejpam-6501	231	39	quotient	quotient	NOUN
ejpam-6501	231	40	-	-	PUNCT
ejpam-6501	231	41	like	like	ADJ
ejpam-6501	231	42	decomposition	decomposition	NOUN
ejpam-6501	231	43	of	of	ADP
ejpam-6501	231	44	the	the	DET
ejpam-6501	231	45	algebraic	algebraic	ADJ
ejpam-6501	231	46	structure	structure	NOUN
ejpam-6501	231	47	.	.	PUNCT
ejpam-6501	232	1	in	in	ADP
ejpam-6501	232	2	addition	addition	NOUN
ejpam-6501	232	3	,	,	PUNCT
ejpam-6501	232	4	we	we	PRON
ejpam-6501	232	5	introduced	introduce	VERB
ejpam-6501	232	6	the	the	DET
ejpam-6501	232	7	notion	notion	NOUN
ejpam-6501	232	8	of	of	ADP
ejpam-6501	232	9	normal	normal	ADJ
ejpam-6501	232	10	subsets	subset	NOUN
ejpam-6501	232	11	and	and	CCONJ
ejpam-6501	232	12	demonstrated	demonstrate	VERB
ejpam-6501	232	13	how	how	SCONJ
ejpam-6501	232	14	they	they	PRON
ejpam-6501	232	15	play	play	VERB
ejpam-6501	232	16	a	a	DET
ejpam-6501	232	17	central	central	ADJ
ejpam-6501	232	18	role	role	NOUN
ejpam-6501	232	19	in	in	ADP
ejpam-6501	232	20	defining	define	VERB
ejpam-6501	232	21	well	well	ADV
ejpam-6501	232	22	-	-	PUNCT
ejpam-6501	232	23	structured	structure	VERB
ejpam-6501	232	24	quotient	quotient	NOUN
ejpam-6501	232	25	ink	ink	NOUN
ejpam-6501	232	26	-	-	PUNCT
ejpam-6501	232	27	algebras	algebras	PROPN
ejpam-6501	232	28	.	.	PUNCT
ejpam-6501	233	1	our	our	PRON
ejpam-6501	233	2	construction	construction	NOUN
ejpam-6501	233	3	ensures	ensure	VERB
ejpam-6501	233	4	that	that	SCONJ
ejpam-6501	233	5	the	the	DET
ejpam-6501	233	6	quotient	quotient	NOUN
ejpam-6501	233	7	algebra	algebra	VERB
ejpam-6501	233	8	x	x	PRON
ejpam-6501	233	9	/	/	SYM
ejpam-6501	233	10	n	n	PRON
ejpam-6501	233	11	inherits	inherit	VERB
ejpam-6501	233	12	the	the	DET
ejpam-6501	233	13	fundamental	fundamental	ADJ
ejpam-6501	233	14	axioms	axiom	NOUN
ejpam-6501	233	15	of	of	ADP
ejpam-6501	233	16	ink	ink	NOUN
ejpam-6501	233	17	-	-	PUNCT
ejpam-6501	233	18	algebras	algebras	PROPN
ejpam-6501	233	19	.	.	PUNCT
ejpam-6501	234	1	the	the	DET
ejpam-6501	234	2	provided	provide	VERB
ejpam-6501	234	3	theorems	theorem	NOUN
ejpam-6501	234	4	and	and	CCONJ
ejpam-6501	234	5	examples	example	NOUN
ejpam-6501	234	6	confirm	confirm	VERB
ejpam-6501	234	7	that	that	SCONJ
ejpam-6501	234	8	the	the	DET
ejpam-6501	234	9	operations	operation	NOUN
ejpam-6501	234	10	on	on	ADP
ejpam-6501	234	11	the	the	DET
ejpam-6501	234	12	quotient	quotient	NOUN
ejpam-6501	234	13	set	set	NOUN
ejpam-6501	234	14	are	be	AUX
ejpam-6501	234	15	well	well	ADV
ejpam-6501	234	16	-	-	PUNCT
ejpam-6501	234	17	defined	define	VERB
ejpam-6501	234	18	and	and	CCONJ
ejpam-6501	234	19	consistent	consistent	ADJ
ejpam-6501	234	20	with	with	ADP
ejpam-6501	234	21	the	the	DET
ejpam-6501	234	22	original	original	ADJ
ejpam-6501	234	23	algebra	algebra	NOUN
ejpam-6501	234	24	.	.	PUNCT
ejpam-6501	235	1	these	these	DET
ejpam-6501	235	2	results	result	VERB
ejpam-6501	235	3	not	not	PART
ejpam-6501	235	4	only	only	ADV
ejpam-6501	235	5	extend	extend	VERB
ejpam-6501	235	6	the	the	DET
ejpam-6501	235	7	theory	theory	NOUN
ejpam-6501	235	8	of	of	ADP
ejpam-6501	235	9	ink	ink	NOUN
ejpam-6501	235	10	-	-	PUNCT
ejpam-6501	235	11	algebras	algebras	PROPN
ejpam-6501	235	12	but	but	CCONJ
ejpam-6501	235	13	also	also	ADV
ejpam-6501	235	14	pave	pave	VERB
ejpam-6501	235	15	the	the	DET
ejpam-6501	235	16	way	way	NOUN
ejpam-6501	235	17	for	for	ADP
ejpam-6501	235	18	further	further	ADJ
ejpam-6501	235	19	exploration	exploration	NOUN
ejpam-6501	235	20	of	of	ADP
ejpam-6501	235	21	categorical	categorical	ADJ
ejpam-6501	235	22	,	,	PUNCT
ejpam-6501	235	23	topological	topological	ADJ
ejpam-6501	235	24	,	,	PUNCT
ejpam-6501	235	25	or	or	CCONJ
ejpam-6501	235	26	fuzzy	fuzzy	ADJ
ejpam-6501	235	27	extensions	extension	NOUN
ejpam-6501	235	28	of	of	ADP
ejpam-6501	235	29	quotient	quotient	NOUN
ejpam-6501	235	30	structures	structure	NOUN
ejpam-6501	235	31	in	in	ADP
ejpam-6501	235	32	algebraic	algebraic	ADJ
ejpam-6501	235	33	logic	logic	NOUN
ejpam-6501	235	34	and	and	CCONJ
ejpam-6501	235	35	non	non	ADJ
ejpam-6501	235	36	-	-	ADJ
ejpam-6501	235	37	classical	classical	ADJ
ejpam-6501	235	38	algebraic	algebraic	ADJ
ejpam-6501	235	39	systems	system	NOUN
ejpam-6501	235	40	.	.	PUNCT
ejpam-6501	236	1	acknowledgements	acknowledgement	NOUN
ejpam-6501	236	2	this	this	DET
ejpam-6501	236	3	research	research	NOUN
ejpam-6501	236	4	project	project	NOUN
ejpam-6501	236	5	was	be	AUX
ejpam-6501	236	6	financially	financially	ADV
ejpam-6501	236	7	supported	support	VERB
ejpam-6501	236	8	by	by	ADP
ejpam-6501	236	9	thailand	thailand	PROPN
ejpam-6501	236	10	science	science	PROPN
ejpam-6501	236	11	research	research	PROPN
ejpam-6501	236	12	and	and	CCONJ
ejpam-6501	236	13	innovation	innovation	NOUN
ejpam-6501	236	14	(	(	PUNCT
ejpam-6501	236	15	tsri	tsri	PROPN
ejpam-6501	236	16	)	)	PUNCT
ejpam-6501	236	17	.	.	PUNCT
ejpam-6501	237	1	the	the	DET
ejpam-6501	237	2	author	author	NOUN
ejpam-6501	237	3	would	would	AUX
ejpam-6501	237	4	also	also	ADV
ejpam-6501	237	5	like	like	VERB
ejpam-6501	237	6	to	to	PART
ejpam-6501	237	7	express	express	VERB
ejpam-6501	237	8	sincere	sincere	ADJ
ejpam-6501	237	9	gratitude	gratitude	NOUN
ejpam-6501	237	10	to	to	ADP
ejpam-6501	237	11	mahasarakham	mahasarakham	PROPN
ejpam-6501	237	12	university	university	PROPN
ejpam-6501	237	13	for	for	ADP
ejpam-6501	237	14	its	its	PRON
ejpam-6501	237	15	continuous	continuous	ADJ
ejpam-6501	237	16	academic	academic	ADJ
ejpam-6501	237	17	and	and	CCONJ
ejpam-6501	237	18	institutional	institutional	ADJ
ejpam-6501	237	19	support	support	NOUN
ejpam-6501	237	20	throughout	throughout	ADP
ejpam-6501	237	21	the	the	DET
ejpam-6501	237	22	development	development	NOUN
ejpam-6501	237	23	of	of	ADP
ejpam-6501	237	24	this	this	DET
ejpam-6501	237	25	work	work	NOUN
ejpam-6501	237	26	.	.	PUNCT
ejpam-6501	238	1	m.	m.	NOUN
ejpam-6501	238	2	phattarachaleekul	phattarachaleekul	PROPN
ejpam-6501	238	3	/	/	SYM
ejpam-6501	238	4	eur	eur	PROPN
ejpam-6501	238	5	.	.	PUNCT
ejpam-6501	239	1	j.	j.	PROPN
ejpam-6501	239	2	pure	pure	PROPN
ejpam-6501	239	3	appl	appl	PROPN
ejpam-6501	239	4	.	.	PROPN
ejpam-6501	239	5	math	math	PROPN
ejpam-6501	239	6	,	,	PUNCT
ejpam-6501	239	7	18	18	NUM
ejpam-6501	239	8	(	(	PUNCT
ejpam-6501	239	9	4	4	NUM
ejpam-6501	239	10	)	)	PUNCT
ejpam-6501	239	11	(	(	PUNCT
ejpam-6501	239	12	2025	2025	NUM
ejpam-6501	239	13	)	)	PUNCT
ejpam-6501	239	14	,	,	PUNCT
ejpam-6501	239	15	6501	6501	NUM
ejpam-6501	239	16	8	8	NUM
ejpam-6501	239	17	of	of	ADP
ejpam-6501	239	18	8	8	NUM
ejpam-6501	239	19	references	reference	NOUN
ejpam-6501	239	20	[	[	X
ejpam-6501	239	21	1	1	NUM
ejpam-6501	239	22	]	]	PUNCT
ejpam-6501	239	23	j.	j.	PROPN
ejpam-6501	239	24	neggers	neggers	PROPN
ejpam-6501	239	25	and	and	CCONJ
ejpam-6501	239	26	h.	h.	PROPN
ejpam-6501	239	27	s.	s.	PROPN
ejpam-6501	239	28	kim	kim	PROPN
ejpam-6501	239	29	.	.	PUNCT
ejpam-6501	240	1	on	on	ADP
ejpam-6501	240	2	b	b	NOUN
ejpam-6501	240	3	-	-	PUNCT
ejpam-6501	240	4	algebras	algebras	X
ejpam-6501	240	5	.	.	PUNCT
ejpam-6501	241	1	matematički	matematički	PROPN
ejpam-6501	241	2	vesnik	vesnik	PROPN
ejpam-6501	241	3	,	,	PUNCT
ejpam-6501	241	4	54:21–29	54:21–29	NUM
ejpam-6501	241	5	,	,	PUNCT
ejpam-6501	241	6	2002	2002	NUM
ejpam-6501	241	7	.	.	PUNCT
ejpam-6501	242	1	[	[	X
ejpam-6501	242	2	2	2	NUM
ejpam-6501	242	3	]	]	PUNCT
ejpam-6501	242	4	m.	m.	NOUN
ejpam-6501	242	5	kaviyarasu	kaviyarasu	PROPN
ejpam-6501	242	6	,	,	PUNCT
ejpam-6501	242	7	k.	k.	PROPN
ejpam-6501	242	8	indhira	indhira	PROPN
ejpam-6501	242	9	,	,	PUNCT
ejpam-6501	242	10	and	and	CCONJ
ejpam-6501	242	11	v.	v.	ADP
ejpam-6501	242	12	m.	m.	NOUN
ejpam-6501	242	13	chandrasekaran	chandrasekaran	VERB
ejpam-6501	242	14	.	.	PUNCT
ejpam-6501	243	1	introduction	introduction	NOUN
ejpam-6501	243	2	on	on	ADP
ejpam-6501	243	3	ink	ink	NOUN
ejpam-6501	243	4	-	-	PUNCT
ejpam-6501	243	5	algebras	algebras	PROPN
ejpam-6501	243	6	.	.	PUNCT
ejpam-6501	244	1	international	international	ADJ
ejpam-6501	244	2	journal	journal	PROPN
ejpam-6501	244	3	of	of	ADP
ejpam-6501	244	4	pure	pure	ADJ
ejpam-6501	244	5	and	and	CCONJ
ejpam-6501	244	6	applied	applied	ADJ
ejpam-6501	244	7	mathematics	mathematic	NOUN
ejpam-6501	244	8	,	,	PUNCT
ejpam-6501	244	9	115(9):1–10	115(9):1–10	NUM
ejpam-6501	244	10	,	,	PUNCT
ejpam-6501	244	11	2017	2017	NUM
ejpam-6501	244	12	.	.	PUNCT
ejpam-6501	245	1	[	[	X
ejpam-6501	245	2	3	3	X
ejpam-6501	245	3	]	]	X
ejpam-6501	245	4	m.	m.	NOUN
ejpam-6501	245	5	kaviyarasu	kaviyarasu	PROPN
ejpam-6501	245	6	,	,	PUNCT
ejpam-6501	245	7	k.	k.	PROPN
ejpam-6501	245	8	indhira	indhira	PROPN
ejpam-6501	245	9	,	,	PUNCT
ejpam-6501	245	10	and	and	CCONJ
ejpam-6501	245	11	v.	v.	ADP
ejpam-6501	245	12	m.	m.	NOUN
ejpam-6501	245	13	chandrasekaran	chandrasekaran	VERB
ejpam-6501	245	14	.	.	PUNCT
ejpam-6501	246	1	fuzzy	fuzzy	ADJ
ejpam-6501	246	2	p	p	NOUN
ejpam-6501	246	3	-	-	PUNCT
ejpam-6501	246	4	ideal	ideal	NOUN
ejpam-6501	246	5	in	in	ADP
ejpam-6501	246	6	ink	ink	NOUN
ejpam-6501	246	7	-	-	PUNCT
ejpam-6501	246	8	algebra	algebra	NOUN
ejpam-6501	246	9	.	.	PUNCT
ejpam-6501	247	1	journal	journal	PROPN
ejpam-6501	247	2	of	of	ADP
ejpam-6501	247	3	xi’an	xi’an	PROPN
ejpam-6501	247	4	university	university	PROPN
ejpam-6501	247	5	of	of	ADP
ejpam-6501	247	6	architecture	architecture	NOUN
ejpam-6501	247	7	and	and	CCONJ
ejpam-6501	247	8	technology	technology	NOUN
ejpam-6501	247	9	,	,	PUNCT
ejpam-6501	247	10	12(3):4746–4752	12(3):4746–4752	NUM
ejpam-6501	247	11	,	,	PUNCT
ejpam-6501	247	12	2020	2020	NUM
ejpam-6501	247	13	.	.	PUNCT
ejpam-6501	248	1	[	[	X
ejpam-6501	248	2	4	4	NUM
ejpam-6501	248	3	]	]	X
ejpam-6501	248	4	m.	m.	NOUN
ejpam-6501	248	5	kaviyarasu	kaviyarasu	PROPN
ejpam-6501	248	6	,	,	PUNCT
ejpam-6501	248	7	k.	k.	PROPN
ejpam-6501	248	8	indhira	indhira	PROPN
ejpam-6501	248	9	,	,	PUNCT
ejpam-6501	248	10	and	and	CCONJ
ejpam-6501	248	11	v.	v.	ADP
ejpam-6501	248	12	m.	m.	NOUN
ejpam-6501	248	13	chandrasekaran	chandrasekaran	VERB
ejpam-6501	248	14	.	.	PUNCT
ejpam-6501	249	1	neutrosophic	neutrosophic	PROPN
ejpam-6501	249	2	set	set	VERB
ejpam-6501	249	3	in	in	ADP
ejpam-6501	249	4	inkalgebra	inkalgebra	NOUN
ejpam-6501	249	5	.	.	PUNCT
ejpam-6501	250	1	advances	advance	NOUN
ejpam-6501	250	2	in	in	ADP
ejpam-6501	250	3	mathematics	mathematic	NOUN
ejpam-6501	250	4	:	:	PUNCT
ejpam-6501	250	5	scientific	scientific	ADJ
ejpam-6501	250	6	journal	journal	NOUN
ejpam-6501	250	7	,	,	PUNCT
ejpam-6501	250	8	9(7):4345–4352	9(7):4345–4352	NUM
ejpam-6501	250	9	,	,	PUNCT
ejpam-6501	250	10	2020	2020	NUM
ejpam-6501	250	11	.	.	PUNCT
ejpam-6501	251	1	[	[	X
ejpam-6501	251	2	5	5	X
ejpam-6501	251	3	]	]	X
ejpam-6501	251	4	y.	y.	PROPN
ejpam-6501	251	5	h.	h.	PROPN
ejpam-6501	251	6	yon	yon	PROPN
ejpam-6501	251	7	,	,	PUNCT
ejpam-6501	251	8	s.	s.	PROPN
ejpam-6501	251	9	m.	m.	PROPN
ejpam-6501	251	10	lee	lee	PROPN
ejpam-6501	251	11	,	,	PUNCT
ejpam-6501	251	12	and	and	CCONJ
ejpam-6501	251	13	k.	k.	PROPN
ejpam-6501	251	14	h.	h.	PROPN
ejpam-6501	251	15	kim	kim	PROPN
ejpam-6501	251	16	.	.	PUNCT
ejpam-6501	252	1	on	on	ADP
ejpam-6501	252	2	congruences	congruence	NOUN
ejpam-6501	252	3	and	and	CCONJ
ejpam-6501	252	4	be	be	NOUN
ejpam-6501	252	5	-	-	PUNCT
ejpam-6501	252	6	relation	relation	NOUN
ejpam-6501	252	7	in	in	ADP
ejpam-6501	252	8	be	be	NOUN
ejpam-6501	252	9	-	-	PUNCT
ejpam-6501	252	10	algebra	algebra	NOUN
ejpam-6501	252	11	.	.	PUNCT
ejpam-6501	253	1	international	international	ADJ
ejpam-6501	253	2	mathematical	mathematical	PROPN
ejpam-6501	253	3	forum	forum	PROPN
ejpam-6501	253	4	,	,	PUNCT
ejpam-6501	253	5	46(5):2263–2270	46(5):2263–2270	NUM
ejpam-6501	253	6	,	,	PUNCT
ejpam-6501	253	7	2010	2010	NUM
ejpam-6501	253	8	.	.	PUNCT