id	sid	tid	token	lemma	pos
ejpam-6493	1	1	european	european	PROPN
ejpam-6493	1	2	journal	journal	PROPN
ejpam-6493	1	3	of	of	ADP
ejpam-6493	1	4	pure	pure	ADJ
ejpam-6493	1	5	and	and	CCONJ
ejpam-6493	1	6	applied	applied	ADJ
ejpam-6493	1	7	mathematics	mathematic	NOUN
ejpam-6493	1	8	2025	2025	NUM
ejpam-6493	1	9	,	,	PUNCT
ejpam-6493	1	10	vol	vol	NOUN
ejpam-6493	1	11	.	.	PROPN
ejpam-6493	1	12	18	18	NUM
ejpam-6493	1	13	,	,	PUNCT
ejpam-6493	1	14	issue	issue	NOUN
ejpam-6493	1	15	3	3	NUM
ejpam-6493	1	16	,	,	PUNCT
ejpam-6493	1	17	article	article	NOUN
ejpam-6493	1	18	number	number	NOUN
ejpam-6493	1	19	6493	6493	NUM
ejpam-6493	1	20	issn	issn	PROPN
ejpam-6493	1	21	1307	1307	NUM
ejpam-6493	1	22	-	-	SYM
ejpam-6493	1	23	5543	5543	NUM
ejpam-6493	1	24	–	–	PUNCT
ejpam-6493	1	25	ejpam.com	ejpam.com	X
ejpam-6493	1	26	published	publish	VERB
ejpam-6493	1	27	by	by	ADP
ejpam-6493	1	28	new	new	PROPN
ejpam-6493	1	29	york	york	PROPN
ejpam-6493	1	30	business	business	PROPN
ejpam-6493	1	31	global	global	PROPN
ejpam-6493	1	32	a	a	DET
ejpam-6493	1	33	note	note	NOUN
ejpam-6493	1	34	on	on	ADP
ejpam-6493	1	35	quotient	quotient	NOUN
ejpam-6493	1	36	ternary	ternary	ADJ
ejpam-6493	1	37	semirings	semiring	NOUN
ejpam-6493	1	38	and	and	CCONJ
ejpam-6493	1	39	isomorphism	isomorphism	NOUN
ejpam-6493	1	40	theorems	theorem	NOUN
ejpam-6493	1	41	montakarn	montakarn	PROPN
ejpam-6493	1	42	petapirak1	petapirak1	PROPN
ejpam-6493	1	43	,	,	PUNCT
ejpam-6493	1	44	arbaz	arbaz	PROPN
ejpam-6493	1	45	jehan	jehan	PROPN
ejpam-6493	1	46	khan1	khan1	PROPN
ejpam-6493	1	47	,	,	PUNCT
ejpam-6493	1	48	ronnason	ronnason	NOUN
ejpam-6493	1	49	chinram1,∗	chinram1,∗	NOUN
ejpam-6493	1	50	1	1	NUM
ejpam-6493	1	51	division	division	NOUN
ejpam-6493	1	52	of	of	ADP
ejpam-6493	1	53	computational	computational	ADJ
ejpam-6493	1	54	science	science	NOUN
ejpam-6493	1	55	,	,	PUNCT
ejpam-6493	1	56	faculty	faculty	NOUN
ejpam-6493	1	57	of	of	ADP
ejpam-6493	1	58	science	science	NOUN
ejpam-6493	1	59	,	,	PUNCT
ejpam-6493	1	60	prince	prince	NOUN
ejpam-6493	1	61	of	of	ADP
ejpam-6493	1	62	songkla	songkla	PROPN
ejpam-6493	1	63	university	university	PROPN
ejpam-6493	1	64	,	,	PUNCT
ejpam-6493	1	65	hat	hat	PROPN
ejpam-6493	1	66	yai	yai	PROPN
ejpam-6493	1	67	,	,	PUNCT
ejpam-6493	1	68	songkhla	songkhla	VERB
ejpam-6493	1	69	90110	90110	NUM
ejpam-6493	1	70	,	,	PUNCT
ejpam-6493	1	71	thailand	thailand	PROPN
ejpam-6493	1	72	abstract	abstract	NOUN
ejpam-6493	1	73	.	.	PUNCT
ejpam-6493	2	1	in	in	ADP
ejpam-6493	2	2	this	this	DET
ejpam-6493	2	3	paper	paper	NOUN
ejpam-6493	2	4	,	,	PUNCT
ejpam-6493	2	5	we	we	PRON
ejpam-6493	2	6	present	present	VERB
ejpam-6493	2	7	a	a	DET
ejpam-6493	2	8	variety	variety	NOUN
ejpam-6493	2	9	of	of	ADP
ejpam-6493	2	10	notions	notion	NOUN
ejpam-6493	2	11	of	of	ADP
ejpam-6493	2	12	quotient	quotient	NOUN
ejpam-6493	2	13	ternary	ternary	ADJ
ejpam-6493	2	14	semirings	semiring	NOUN
ejpam-6493	2	15	,	,	PUNCT
ejpam-6493	2	16	along	along	ADP
ejpam-6493	2	17	with	with	ADP
ejpam-6493	2	18	examples	example	NOUN
ejpam-6493	2	19	and	and	CCONJ
ejpam-6493	2	20	remarks	remark	NOUN
ejpam-6493	2	21	.	.	PUNCT
ejpam-6493	3	1	moreover	moreover	ADV
ejpam-6493	3	2	,	,	PUNCT
ejpam-6493	3	3	we	we	PRON
ejpam-6493	3	4	establish	establish	VERB
ejpam-6493	3	5	various	various	ADJ
ejpam-6493	3	6	isomorphism	isomorphism	NOUN
ejpam-6493	3	7	theorems	theorem	NOUN
ejpam-6493	3	8	for	for	ADP
ejpam-6493	3	9	ternary	ternary	ADJ
ejpam-6493	3	10	semirings	semiring	NOUN
ejpam-6493	3	11	.	.	PUNCT
ejpam-6493	4	1	2020	2020	NUM
ejpam-6493	4	2	mathematics	mathematics	PROPN
ejpam-6493	4	3	subject	subject	NOUN
ejpam-6493	4	4	classifications	classification	NOUN
ejpam-6493	4	5	:	:	PUNCT
ejpam-6493	4	6	16y60	16y60	NUM
ejpam-6493	4	7	,	,	PUNCT
ejpam-6493	4	8	16y99	16y99	NUM
ejpam-6493	4	9	key	key	ADJ
ejpam-6493	4	10	words	word	NOUN
ejpam-6493	4	11	and	and	CCONJ
ejpam-6493	4	12	phrases	phrase	NOUN
ejpam-6493	4	13	:	:	PUNCT
ejpam-6493	4	14	q	q	NOUN
ejpam-6493	4	15	-	-	PUNCT
ejpam-6493	4	16	ideals	ideal	NOUN
ejpam-6493	4	17	,	,	PUNCT
ejpam-6493	4	18	congruences	congruence	NOUN
ejpam-6493	4	19	,	,	PUNCT
ejpam-6493	4	20	k	k	NOUN
ejpam-6493	4	21	-	-	NOUN
ejpam-6493	4	22	ideals	ideal	NOUN
ejpam-6493	4	23	,	,	PUNCT
ejpam-6493	4	24	quotient	quotient	VERB
ejpam-6493	4	25	ternary	ternary	ADJ
ejpam-6493	4	26	semirings	semiring	NOUN
ejpam-6493	4	27	,	,	PUNCT
ejpam-6493	4	28	isomorphism	isomorphism	NOUN
ejpam-6493	4	29	theorems	theorem	NOUN
ejpam-6493	4	30	1	1	NUM
ejpam-6493	4	31	.	.	X
ejpam-6493	4	32	introduction	introduction	NOUN
ejpam-6493	4	33	the	the	DET
ejpam-6493	4	34	notion	notion	NOUN
ejpam-6493	4	35	of	of	ADP
ejpam-6493	4	36	ideals	ideal	NOUN
ejpam-6493	4	37	is	be	AUX
ejpam-6493	4	38	fundamental	fundamental	ADJ
ejpam-6493	4	39	in	in	ADP
ejpam-6493	4	40	ring	ring	NOUN
ejpam-6493	4	41	theory	theory	NOUN
ejpam-6493	4	42	,	,	PUNCT
ejpam-6493	4	43	as	as	SCONJ
ejpam-6493	4	44	it	it	PRON
ejpam-6493	4	45	plays	play	VERB
ejpam-6493	4	46	an	an	DET
ejpam-6493	4	47	important	important	ADJ
ejpam-6493	4	48	role	role	NOUN
ejpam-6493	4	49	in	in	ADP
ejpam-6493	4	50	defining	define	VERB
ejpam-6493	4	51	quotient	quotient	NOUN
ejpam-6493	4	52	rings	ring	NOUN
ejpam-6493	4	53	.	.	PUNCT
ejpam-6493	5	1	therefore	therefore	ADV
ejpam-6493	5	2	ideal	ideal	ADJ
ejpam-6493	5	3	theory	theory	NOUN
ejpam-6493	5	4	has	have	AUX
ejpam-6493	5	5	been	be	AUX
ejpam-6493	5	6	a	a	DET
ejpam-6493	5	7	major	major	ADJ
ejpam-6493	5	8	area	area	NOUN
ejpam-6493	5	9	of	of	ADP
ejpam-6493	5	10	research	research	NOUN
ejpam-6493	5	11	in	in	ADP
ejpam-6493	5	12	the	the	DET
ejpam-6493	5	13	study	study	NOUN
ejpam-6493	5	14	of	of	ADP
ejpam-6493	5	15	rings	ring	NOUN
ejpam-6493	5	16	.	.	PUNCT
ejpam-6493	6	1	meanwhile	meanwhile	ADV
ejpam-6493	6	2	,	,	PUNCT
ejpam-6493	6	3	the	the	DET
ejpam-6493	6	4	concept	concept	NOUN
ejpam-6493	6	5	of	of	ADP
ejpam-6493	6	6	congruences	congruence	NOUN
ejpam-6493	6	7	is	be	AUX
ejpam-6493	6	8	crucial	crucial	ADJ
ejpam-6493	6	9	for	for	ADP
ejpam-6493	6	10	defining	define	VERB
ejpam-6493	6	11	quotient	quotient	NOUN
ejpam-6493	6	12	semigroups	semigroup	NOUN
ejpam-6493	6	13	.	.	PUNCT
ejpam-6493	7	1	semirings	semiring	NOUN
ejpam-6493	7	2	,	,	PUNCT
ejpam-6493	7	3	as	as	ADP
ejpam-6493	7	4	algebraic	algebraic	ADJ
ejpam-6493	7	5	structures	structure	NOUN
ejpam-6493	7	6	,	,	PUNCT
ejpam-6493	7	7	were	be	AUX
ejpam-6493	7	8	definitely	definitely	ADV
ejpam-6493	7	9	one	one	NUM
ejpam-6493	7	10	of	of	ADP
ejpam-6493	7	11	natural	natural	ADJ
ejpam-6493	7	12	choices	choice	NOUN
ejpam-6493	7	13	of	of	ADP
ejpam-6493	7	14	generalizations	generalization	NOUN
ejpam-6493	7	15	of	of	ADP
ejpam-6493	7	16	rings	ring	NOUN
ejpam-6493	7	17	.	.	PUNCT
ejpam-6493	8	1	many	many	ADJ
ejpam-6493	8	2	properties	property	NOUN
ejpam-6493	8	3	that	that	PRON
ejpam-6493	8	4	hold	hold	VERB
ejpam-6493	8	5	for	for	ADP
ejpam-6493	8	6	rings	ring	NOUN
ejpam-6493	8	7	can	can	AUX
ejpam-6493	8	8	be	be	AUX
ejpam-6493	8	9	extended	extend	VERB
ejpam-6493	8	10	to	to	ADP
ejpam-6493	8	11	semirings	semiring	NOUN
ejpam-6493	8	12	.	.	PUNCT
ejpam-6493	9	1	quotient	quotient	PROPN
ejpam-6493	9	2	semirings	semiring	NOUN
ejpam-6493	9	3	were	be	AUX
ejpam-6493	9	4	previously	previously	ADV
ejpam-6493	9	5	studied	study	VERB
ejpam-6493	9	6	in	in	ADP
ejpam-6493	9	7	[	[	X
ejpam-6493	9	8	1–3	1–3	NOUN
ejpam-6493	9	9	]	]	X
ejpam-6493	9	10	.	.	PUNCT
ejpam-6493	10	1	the	the	DET
ejpam-6493	10	2	concept	concept	NOUN
ejpam-6493	10	3	of	of	ADP
ejpam-6493	10	4	ternary	ternary	ADJ
ejpam-6493	10	5	semirings	semiring	NOUN
ejpam-6493	10	6	was	be	AUX
ejpam-6493	10	7	first	first	ADV
ejpam-6493	10	8	introduced	introduce	VERB
ejpam-6493	10	9	in	in	ADP
ejpam-6493	10	10	2003	2003	NUM
ejpam-6493	10	11	[	[	X
ejpam-6493	10	12	4	4	NUM
ejpam-6493	10	13	]	]	PUNCT
ejpam-6493	10	14	.	.	PUNCT
ejpam-6493	11	1	every	every	DET
ejpam-6493	11	2	semiring	semiring	NOUN
ejpam-6493	11	3	can	can	AUX
ejpam-6493	11	4	always	always	ADV
ejpam-6493	11	5	be	be	AUX
ejpam-6493	11	6	turned	turn	VERB
ejpam-6493	11	7	to	to	ADP
ejpam-6493	11	8	a	a	DET
ejpam-6493	11	9	ternary	ternary	ADJ
ejpam-6493	11	10	semiring	semiring	NOUN
ejpam-6493	11	11	,	,	PUNCT
ejpam-6493	11	12	though	though	SCONJ
ejpam-6493	11	13	a	a	DET
ejpam-6493	11	14	ternary	ternary	ADJ
ejpam-6493	11	15	semiring	semiring	NOUN
ejpam-6493	11	16	does	do	AUX
ejpam-6493	11	17	not	not	PART
ejpam-6493	11	18	always	always	ADV
ejpam-6493	11	19	reduce	reduce	VERB
ejpam-6493	11	20	to	to	ADP
ejpam-6493	11	21	a	a	DET
ejpam-6493	11	22	semiring	semiring	NOUN
ejpam-6493	11	23	.	.	PUNCT
ejpam-6493	12	1	the	the	DET
ejpam-6493	12	2	notion	notion	NOUN
ejpam-6493	12	3	of	of	ADP
ejpam-6493	12	4	ternary	ternary	ADJ
ejpam-6493	12	5	semirings	semiring	NOUN
ejpam-6493	12	6	can	can	AUX
ejpam-6493	12	7	,	,	PUNCT
ejpam-6493	12	8	in	in	ADP
ejpam-6493	12	9	some	some	DET
ejpam-6493	12	10	sense	sense	NOUN
ejpam-6493	12	11	,	,	PUNCT
ejpam-6493	12	12	be	be	AUX
ejpam-6493	12	13	regarded	regard	VERB
ejpam-6493	12	14	as	as	ADP
ejpam-6493	12	15	a	a	DET
ejpam-6493	12	16	generalization	generalization	NOUN
ejpam-6493	12	17	of	of	ADP
ejpam-6493	12	18	semirings	semiring	NOUN
ejpam-6493	12	19	,	,	PUNCT
ejpam-6493	12	20	but	but	CCONJ
ejpam-6493	12	21	it	it	PRON
ejpam-6493	12	22	goes	go	VERB
ejpam-6493	12	23	beyond	beyond	ADP
ejpam-6493	12	24	a	a	DET
ejpam-6493	12	25	simple	simple	ADJ
ejpam-6493	12	26	generalization	generalization	NOUN
ejpam-6493	12	27	,	,	PUNCT
ejpam-6493	12	28	because	because	SCONJ
ejpam-6493	12	29	certain	certain	ADJ
ejpam-6493	12	30	concepts	concept	NOUN
ejpam-6493	12	31	,	,	PUNCT
ejpam-6493	12	32	such	such	ADJ
ejpam-6493	12	33	as	as	ADP
ejpam-6493	12	34	lateral	lateral	ADJ
ejpam-6493	12	35	ideals	ideal	NOUN
ejpam-6493	12	36	,	,	PUNCT
ejpam-6493	12	37	have	have	VERB
ejpam-6493	12	38	no	no	DET
ejpam-6493	12	39	analog	analog	NOUN
ejpam-6493	12	40	in	in	ADP
ejpam-6493	12	41	semirings	semiring	NOUN
ejpam-6493	12	42	.	.	PUNCT
ejpam-6493	13	1	in	in	ADP
ejpam-6493	13	2	2011	2011	NUM
ejpam-6493	13	3	,	,	PUNCT
ejpam-6493	13	4	chaudhari	chaudhari	NOUN
ejpam-6493	13	5	and	and	CCONJ
ejpam-6493	13	6	ingale	ingale	VERB
ejpam-6493	13	7	[	[	X
ejpam-6493	13	8	5	5	NUM
ejpam-6493	13	9	]	]	PUNCT
ejpam-6493	13	10	introduced	introduce	VERB
ejpam-6493	13	11	a	a	DET
ejpam-6493	13	12	partitioning	partition	VERB
ejpam-6493	13	13	ideal	ideal	NOUN
ejpam-6493	13	14	(	(	PUNCT
ejpam-6493	13	15	shortly	shortly	ADV
ejpam-6493	13	16	,	,	PUNCT
ejpam-6493	13	17	q	q	NOUN
ejpam-6493	13	18	-	-	PUNCT
ejpam-6493	13	19	ideal	ideal	NOUN
ejpam-6493	13	20	)	)	PUNCT
ejpam-6493	13	21	of	of	ADP
ejpam-6493	13	22	a	a	DET
ejpam-6493	13	23	ternary	ternary	ADJ
ejpam-6493	13	24	semiring	semiring	NOUN
ejpam-6493	13	25	.	.	PUNCT
ejpam-6493	14	1	q	q	X
ejpam-6493	14	2	-	-	PUNCT
ejpam-6493	14	3	ideals	ideal	NOUN
ejpam-6493	14	4	are	be	AUX
ejpam-6493	14	5	useful	useful	ADJ
ejpam-6493	14	6	to	to	PART
ejpam-6493	14	7	develop	develop	VERB
ejpam-6493	14	8	the	the	DET
ejpam-6493	14	9	quotient	quotient	NOUN
ejpam-6493	14	10	structures	structure	NOUN
ejpam-6493	14	11	of	of	ADP
ejpam-6493	14	12	ternary	ternary	ADJ
ejpam-6493	14	13	semirings	semiring	NOUN
ejpam-6493	14	14	.	.	PUNCT
ejpam-6493	15	1	in	in	ADP
ejpam-6493	15	2	2021	2021	NUM
ejpam-6493	15	3	,	,	PUNCT
ejpam-6493	15	4	sunitha	sunitha	PROPN
ejpam-6493	15	5	et	et	PROPN
ejpam-6493	15	6	al	al	PROPN
ejpam-6493	15	7	.	.	PUNCT
ejpam-6493	16	1	[	[	X
ejpam-6493	16	2	6	6	NUM
ejpam-6493	16	3	]	]	PUNCT
ejpam-6493	16	4	provided	provide	VERB
ejpam-6493	16	5	the	the	DET
ejpam-6493	16	6	characterization	characterization	NOUN
ejpam-6493	16	7	of	of	ADP
ejpam-6493	16	8	full	full	ADJ
ejpam-6493	16	9	k	k	NOUN
ejpam-6493	16	10	-	-	NOUN
ejpam-6493	16	11	ideals	ideal	NOUN
ejpam-6493	16	12	in	in	ADP
ejpam-6493	16	13	ternary	ternary	ADJ
ejpam-6493	16	14	semirings	semiring	NOUN
ejpam-6493	16	15	.	.	PUNCT
ejpam-6493	17	1	in	in	ADP
ejpam-6493	17	2	2024	2024	NUM
ejpam-6493	17	3	,	,	PUNCT
ejpam-6493	17	4	sanborisoot	sanborisoot	NOUN
ejpam-6493	17	5	and	and	CCONJ
ejpam-6493	17	6	ayutthaya	ayutthaya	PROPN
ejpam-6493	18	1	[	[	X
ejpam-6493	18	2	7	7	NUM
ejpam-6493	18	3	]	]	PUNCT
ejpam-6493	18	4	constructed	construct	VERB
ejpam-6493	18	5	a	a	DET
ejpam-6493	18	6	congruence	congruence	NOUN
ejpam-6493	18	7	relation	relation	NOUN
ejpam-6493	18	8	with	with	ADP
ejpam-6493	18	9	respect	respect	NOUN
ejpam-6493	18	10	to	to	ADP
ejpam-6493	18	11	a	a	DET
ejpam-6493	18	12	full	full	ADJ
ejpam-6493	18	13	k	k	NOUN
ejpam-6493	18	14	-	-	NOUN
ejpam-6493	18	15	ideal	ideal	NOUN
ejpam-6493	18	16	on	on	ADP
ejpam-6493	18	17	a	a	DET
ejpam-6493	18	18	ternary	ternary	ADJ
ejpam-6493	18	19	semiring	semiring	NOUN
ejpam-6493	18	20	for	for	ADP
ejpam-6493	18	21	the	the	DET
ejpam-6493	18	22	purpose	purpose	NOUN
ejpam-6493	18	23	of	of	ADP
ejpam-6493	18	24	forming	form	VERB
ejpam-6493	18	25	a	a	DET
ejpam-6493	18	26	ternary	ternary	ADJ
ejpam-6493	18	27	ring	ring	NOUN
ejpam-6493	18	28	from	from	ADP
ejpam-6493	18	29	the	the	DET
ejpam-6493	18	30	quotient	quotient	NOUN
ejpam-6493	18	31	ternary	ternary	ADJ
ejpam-6493	18	32	semiring	semiring	NOUN
ejpam-6493	18	33	.	.	PUNCT
ejpam-6493	19	1	quotient	quotient	VERB
ejpam-6493	19	2	ternary	ternary	ADJ
ejpam-6493	19	3	semirings	semiring	NOUN
ejpam-6493	19	4	∗corresponding	∗corresponde	VERB
ejpam-6493	19	5	author	author	NOUN
ejpam-6493	19	6	.	.	PUNCT
ejpam-6493	20	1	doi	doi	NOUN
ejpam-6493	20	2	:	:	PUNCT
ejpam-6493	20	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6493	https://doi.org/10.29020/nybg.ejpam.v18i3.6493	NOUN
ejpam-6493	20	4	email	email	NOUN
ejpam-6493	20	5	addresses	address	NOUN
ejpam-6493	20	6	:	:	PUNCT
ejpam-6493	20	7	montakarn.p@psu.ac.th	montakarn.p@psu.ac.th	PROPN
ejpam-6493	20	8	(	(	PUNCT
ejpam-6493	20	9	m.	m.	NOUN
ejpam-6493	20	10	petapirak	petapirak	PROPN
ejpam-6493	20	11	)	)	PUNCT
ejpam-6493	20	12	,	,	PUNCT
ejpam-6493	20	13	arbazjehankhan@gmail.com	arbazjehankhan@gmail.com	X
ejpam-6493	20	14	(	(	PUNCT
ejpam-6493	20	15	a.	a.	PROPN
ejpam-6493	20	16	j.	j.	PROPN
ejpam-6493	20	17	khan	khan	PROPN
ejpam-6493	20	18	)	)	PUNCT
ejpam-6493	20	19	,	,	PUNCT
ejpam-6493	20	20	ronnason.c@psu.ac.th	ronnason.c@psu.ac.th	PROPN
ejpam-6493	20	21	(	(	PUNCT
ejpam-6493	20	22	r.	r.	PROPN
ejpam-6493	20	23	chinram	chinram	PROPN
ejpam-6493	20	24	)	)	PUNCT
ejpam-6493	20	25	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6493	21	1	1	1	NUM
ejpam-6493	21	2	copyright	copyright	NOUN
ejpam-6493	21	3	:	:	PUNCT
ejpam-6493	21	4	©	©	PROPN
ejpam-6493	21	5	2025	2025	NUM
ejpam-6493	21	6	the	the	DET
ejpam-6493	21	7	author(s	author(s	NOUN
ejpam-6493	21	8	)	)	PUNCT
ejpam-6493	21	9	.	.	PUNCT
ejpam-6493	22	1	(	(	PUNCT
ejpam-6493	22	2	cc	cc	NOUN
ejpam-6493	22	3	by	by	ADP
ejpam-6493	22	4	-	-	PUNCT
ejpam-6493	22	5	nc	nc	PROPN
ejpam-6493	22	6	4.0	4.0	NUM
ejpam-6493	22	7	)	)	PUNCT
ejpam-6493	22	8	m.	m.	NOUN
ejpam-6493	22	9	petapirak	petapirak	PROPN
ejpam-6493	22	10	,	,	PUNCT
ejpam-6493	22	11	a.	a.	PROPN
ejpam-6493	22	12	j.	j.	PROPN
ejpam-6493	22	13	khan	khan	PROPN
ejpam-6493	22	14	,	,	PUNCT
ejpam-6493	22	15	r.	r.	PROPN
ejpam-6493	22	16	chinram	chinram	PROPN
ejpam-6493	22	17	/	/	SYM
ejpam-6493	22	18	eur	eur	PROPN
ejpam-6493	22	19	.	.	PUNCT
ejpam-6493	23	1	j.	j.	PROPN
ejpam-6493	23	2	pure	pure	PROPN
ejpam-6493	23	3	appl	appl	PROPN
ejpam-6493	23	4	.	.	PROPN
ejpam-6493	23	5	math	math	PROPN
ejpam-6493	23	6	,	,	PUNCT
ejpam-6493	23	7	18	18	NUM
ejpam-6493	23	8	(	(	PUNCT
ejpam-6493	23	9	3	3	NUM
ejpam-6493	23	10	)	)	PUNCT
ejpam-6493	23	11	(	(	PUNCT
ejpam-6493	23	12	2025	2025	NUM
ejpam-6493	23	13	)	)	PUNCT
ejpam-6493	23	14	,	,	PUNCT
ejpam-6493	23	15	6493	6493	NUM
ejpam-6493	23	16	2	2	NUM
ejpam-6493	23	17	of	of	ADP
ejpam-6493	23	18	9	9	NUM
ejpam-6493	23	19	can	can	AUX
ejpam-6493	23	20	be	be	AUX
ejpam-6493	23	21	defined	define	VERB
ejpam-6493	23	22	by	by	ADP
ejpam-6493	23	23	congruences	congruence	NOUN
ejpam-6493	23	24	and/or	and/or	CCONJ
ejpam-6493	23	25	ideals	ideal	NOUN
ejpam-6493	23	26	.	.	PUNCT
ejpam-6493	24	1	moreover	moreover	ADV
ejpam-6493	24	2	,	,	PUNCT
ejpam-6493	24	3	additional	additional	ADJ
ejpam-6493	24	4	research	research	NOUN
ejpam-6493	24	5	papers	paper	NOUN
ejpam-6493	24	6	related	relate	VERB
ejpam-6493	24	7	to	to	ADP
ejpam-6493	24	8	ternary	ternary	ADJ
ejpam-6493	24	9	semirings	semiring	NOUN
ejpam-6493	24	10	,	,	PUNCT
ejpam-6493	24	11	published	publish	VERB
ejpam-6493	24	12	during	during	ADP
ejpam-6493	24	13	2023	2023	NUM
ejpam-6493	24	14	-	-	SYM
ejpam-6493	24	15	2025	2025	NUM
ejpam-6493	24	16	,	,	PUNCT
ejpam-6493	24	17	can	can	AUX
ejpam-6493	24	18	be	be	AUX
ejpam-6493	24	19	found	find	VERB
ejpam-6493	24	20	in	in	ADP
ejpam-6493	24	21	[	[	X
ejpam-6493	24	22	8–11	8–11	NOUN
ejpam-6493	24	23	]	]	PUNCT
ejpam-6493	24	24	.	.	PUNCT
ejpam-6493	25	1	in	in	ADP
ejpam-6493	25	2	this	this	DET
ejpam-6493	25	3	paper	paper	NOUN
ejpam-6493	25	4	,	,	PUNCT
ejpam-6493	25	5	we	we	PRON
ejpam-6493	25	6	aim	aim	VERB
ejpam-6493	25	7	to	to	PART
ejpam-6493	25	8	present	present	VERB
ejpam-6493	25	9	various	various	ADJ
ejpam-6493	25	10	notions	notion	NOUN
ejpam-6493	25	11	of	of	ADP
ejpam-6493	25	12	quotient	quotient	NOUN
ejpam-6493	25	13	ternary	ternary	ADJ
ejpam-6493	25	14	semirings	semiring	NOUN
ejpam-6493	25	15	,	,	PUNCT
ejpam-6493	25	16	examine	examine	VERB
ejpam-6493	25	17	their	their	PRON
ejpam-6493	25	18	properties	property	NOUN
ejpam-6493	25	19	,	,	PUNCT
ejpam-6493	25	20	and	and	CCONJ
ejpam-6493	25	21	provide	provide	VERB
ejpam-6493	25	22	a	a	DET
ejpam-6493	25	23	discussion	discussion	NOUN
ejpam-6493	25	24	on	on	ADP
ejpam-6493	25	25	isomorphism	isomorphism	NOUN
ejpam-6493	25	26	theorems	theorem	NOUN
ejpam-6493	25	27	concerning	concern	VERB
ejpam-6493	25	28	these	these	DET
ejpam-6493	25	29	quotient	quotient	NOUN
ejpam-6493	25	30	ternary	ternary	ADJ
ejpam-6493	25	31	semirings	semiring	NOUN
ejpam-6493	25	32	.	.	PUNCT
ejpam-6493	26	1	2	2	X
ejpam-6493	26	2	.	.	X
ejpam-6493	26	3	preliminaries	preliminary	NOUN
ejpam-6493	26	4	in	in	ADP
ejpam-6493	26	5	this	this	DET
ejpam-6493	26	6	section	section	NOUN
ejpam-6493	26	7	,	,	PUNCT
ejpam-6493	26	8	we	we	PRON
ejpam-6493	26	9	will	will	AUX
ejpam-6493	26	10	recall	recall	VERB
ejpam-6493	26	11	some	some	DET
ejpam-6493	26	12	basic	basic	ADJ
ejpam-6493	26	13	definitions	definition	NOUN
ejpam-6493	26	14	of	of	ADP
ejpam-6493	26	15	ternary	ternary	ADJ
ejpam-6493	26	16	semirings	semiring	NOUN
ejpam-6493	26	17	.	.	PUNCT
ejpam-6493	27	1	definition	definition	NOUN
ejpam-6493	27	2	1	1	NUM
ejpam-6493	27	3	(	(	PUNCT
ejpam-6493	27	4	[	[	X
ejpam-6493	27	5	4	4	NUM
ejpam-6493	27	6	]	]	NUM
ejpam-6493	27	7	)	)	PUNCT
ejpam-6493	27	8	.	.	PUNCT
ejpam-6493	28	1	a	a	DET
ejpam-6493	28	2	non	non	ADJ
ejpam-6493	28	3	-	-	ADJ
ejpam-6493	28	4	empty	empty	ADJ
ejpam-6493	28	5	set	set	VERB
ejpam-6493	28	6	r	r	NOUN
ejpam-6493	28	7	together	together	ADV
ejpam-6493	28	8	with	with	ADP
ejpam-6493	28	9	a	a	DET
ejpam-6493	28	10	binary	binary	ADJ
ejpam-6493	28	11	operation	operation	NOUN
ejpam-6493	28	12	,	,	PUNCT
ejpam-6493	28	13	called	call	VERB
ejpam-6493	28	14	the	the	DET
ejpam-6493	28	15	addition	addition	NOUN
ejpam-6493	28	16	,	,	PUNCT
ejpam-6493	28	17	and	and	CCONJ
ejpam-6493	28	18	the	the	DET
ejpam-6493	28	19	ternary	ternary	ADJ
ejpam-6493	28	20	multiplication	multiplication	NOUN
ejpam-6493	28	21	,	,	PUNCT
ejpam-6493	28	22	denoted	denote	VERB
ejpam-6493	28	23	by	by	ADP
ejpam-6493	28	24	juxtaposition	juxtaposition	NOUN
ejpam-6493	28	25	,	,	PUNCT
ejpam-6493	28	26	is	be	AUX
ejpam-6493	28	27	said	say	VERB
ejpam-6493	28	28	to	to	PART
ejpam-6493	28	29	be	be	AUX
ejpam-6493	28	30	a	a	DET
ejpam-6493	28	31	ternary	ternary	ADJ
ejpam-6493	28	32	semiring	semiring	NOUN
ejpam-6493	28	33	if	if	SCONJ
ejpam-6493	28	34	r	r	NOUN
ejpam-6493	28	35	is	be	AUX
ejpam-6493	28	36	an	an	DET
ejpam-6493	28	37	additive	additive	ADJ
ejpam-6493	28	38	commutative	commutative	ADJ
ejpam-6493	28	39	semigroup	semigroup	NOUN
ejpam-6493	28	40	satisfying	satisfy	VERB
ejpam-6493	28	41	the	the	DET
ejpam-6493	28	42	following	follow	VERB
ejpam-6493	28	43	conditions	condition	NOUN
ejpam-6493	28	44	:	:	PUNCT
ejpam-6493	28	45	(	(	PUNCT
ejpam-6493	28	46	i	i	NOUN
ejpam-6493	28	47	)	)	PUNCT
ejpam-6493	28	48	(	(	PUNCT
ejpam-6493	29	1	abc)de	abc)de	NOUN
ejpam-6493	29	2	=	=	SYM
ejpam-6493	29	3	a(bcd)e	a(bcd)e	NOUN
ejpam-6493	29	4	=	=	PUNCT
ejpam-6493	29	5	ab(cde	ab(cde	PROPN
ejpam-6493	29	6	)	)	PUNCT
ejpam-6493	29	7	,	,	PUNCT
ejpam-6493	29	8	(	(	PUNCT
ejpam-6493	29	9	ii	ii	NOUN
ejpam-6493	29	10	)	)	PUNCT
ejpam-6493	29	11	(	(	PUNCT
ejpam-6493	29	12	a+	a+	PUNCT
ejpam-6493	29	13	b)cd	b)cd	PROPN
ejpam-6493	29	14	=	=	SYM
ejpam-6493	29	15	acd+	acd+	NOUN
ejpam-6493	29	16	bcd	bcd	PROPN
ejpam-6493	29	17	,	,	PUNCT
ejpam-6493	29	18	(	(	PUNCT
ejpam-6493	29	19	iii	iii	NOUN
ejpam-6493	29	20	)	)	PUNCT
ejpam-6493	29	21	a(b+	a(b+	ADV
ejpam-6493	29	22	c)d	c)d	NOUN
ejpam-6493	29	23	=	=	SYM
ejpam-6493	29	24	abd+	abd+	ADV
ejpam-6493	29	25	acd	acd	PROPN
ejpam-6493	29	26	,	,	PUNCT
ejpam-6493	29	27	(	(	PUNCT
ejpam-6493	29	28	iv	iv	X
ejpam-6493	29	29	)	)	PUNCT
ejpam-6493	29	30	ab(c+	ab(c+	PROPN
ejpam-6493	30	1	d	d	X
ejpam-6493	30	2	)	)	PUNCT
ejpam-6493	30	3	=	=	PUNCT
ejpam-6493	30	4	abc+	abc+	PROPN
ejpam-6493	30	5	abd	abd	NOUN
ejpam-6493	30	6	for	for	ADP
ejpam-6493	30	7	all	all	DET
ejpam-6493	30	8	a	a	DET
ejpam-6493	30	9	,	,	PUNCT
ejpam-6493	30	10	b	b	NOUN
ejpam-6493	30	11	,	,	PUNCT
ejpam-6493	30	12	c	c	NOUN
ejpam-6493	30	13	,	,	PUNCT
ejpam-6493	30	14	d	d	NOUN
ejpam-6493	30	15	,	,	PUNCT
ejpam-6493	30	16	e	e	PROPN
ejpam-6493	30	17	∈	∈	PROPN
ejpam-6493	30	18	r.	r.	NOUN
ejpam-6493	30	19	definition	definition	NOUN
ejpam-6493	30	20	2	2	NUM
ejpam-6493	30	21	(	(	PUNCT
ejpam-6493	30	22	[	[	X
ejpam-6493	30	23	4	4	NUM
ejpam-6493	30	24	]	]	NUM
ejpam-6493	30	25	)	)	PUNCT
ejpam-6493	30	26	.	.	PUNCT
ejpam-6493	31	1	let	let	VERB
ejpam-6493	31	2	r	r	PRON
ejpam-6493	31	3	be	be	AUX
ejpam-6493	31	4	a	a	DET
ejpam-6493	31	5	ternary	ternary	ADJ
ejpam-6493	31	6	semiring	semiring	NOUN
ejpam-6493	31	7	.	.	PUNCT
ejpam-6493	32	1	if	if	SCONJ
ejpam-6493	32	2	0	0	NUM
ejpam-6493	32	3	∈	∈	NOUN
ejpam-6493	32	4	r	r	NOUN
ejpam-6493	32	5	such	such	ADJ
ejpam-6493	32	6	that	that	PRON
ejpam-6493	32	7	0	0	NUM
ejpam-6493	33	1	+	+	NUM
ejpam-6493	33	2	x	x	SYM
ejpam-6493	33	3	=	=	SYM
ejpam-6493	33	4	x	x	X
ejpam-6493	33	5	and	and	CCONJ
ejpam-6493	33	6	0xy	0xy	NOUN
ejpam-6493	33	7	=	=	PUNCT
ejpam-6493	33	8	x0y	x0y	PUNCT
ejpam-6493	34	1	=	=	PUNCT
ejpam-6493	34	2	xy0	xy0	X
ejpam-6493	35	1	=	=	SYM
ejpam-6493	35	2	0	0	NUM
ejpam-6493	35	3	for	for	ADP
ejpam-6493	35	4	all	all	DET
ejpam-6493	35	5	x	x	NOUN
ejpam-6493	35	6	,	,	PUNCT
ejpam-6493	35	7	y	y	PROPN
ejpam-6493	35	8	∈	∈	PROPN
ejpam-6493	35	9	r	r	NOUN
ejpam-6493	35	10	,	,	PUNCT
ejpam-6493	35	11	then	then	ADV
ejpam-6493	35	12	0	0	NUM
ejpam-6493	35	13	is	be	AUX
ejpam-6493	35	14	called	call	VERB
ejpam-6493	35	15	a	a	DET
ejpam-6493	35	16	zero	zero	NUM
ejpam-6493	35	17	element	element	NOUN
ejpam-6493	35	18	.	.	PUNCT
ejpam-6493	36	1	in	in	ADP
ejpam-6493	36	2	this	this	DET
ejpam-6493	36	3	case	case	NOUN
ejpam-6493	36	4	,	,	PUNCT
ejpam-6493	36	5	r	r	NOUN
ejpam-6493	36	6	is	be	AUX
ejpam-6493	36	7	called	call	VERB
ejpam-6493	36	8	a	a	DET
ejpam-6493	36	9	ternary	ternary	ADJ
ejpam-6493	36	10	semiring	semiring	NOUN
ejpam-6493	36	11	with	with	ADP
ejpam-6493	36	12	zero	zero	NUM
ejpam-6493	36	13	.	.	PUNCT
ejpam-6493	37	1	definition	definition	NOUN
ejpam-6493	37	2	3	3	NUM
ejpam-6493	37	3	(	(	PUNCT
ejpam-6493	37	4	[	[	X
ejpam-6493	37	5	4	4	NUM
ejpam-6493	37	6	]	]	NUM
ejpam-6493	37	7	)	)	PUNCT
ejpam-6493	37	8	.	.	PUNCT
ejpam-6493	38	1	an	an	DET
ejpam-6493	38	2	additive	additive	ADJ
ejpam-6493	38	3	semigroup	semigroup	NOUN
ejpam-6493	38	4	s	s	PROPN
ejpam-6493	38	5	of	of	ADP
ejpam-6493	38	6	a	a	DET
ejpam-6493	38	7	ternary	ternary	ADJ
ejpam-6493	38	8	semiring	semiring	NOUN
ejpam-6493	38	9	r	r	NOUN
ejpam-6493	38	10	is	be	AUX
ejpam-6493	38	11	called	call	VERB
ejpam-6493	38	12	a	a	DET
ejpam-6493	38	13	ternary	ternary	ADJ
ejpam-6493	38	14	subsemiring	subsemiring	NOUN
ejpam-6493	38	15	of	of	ADP
ejpam-6493	38	16	r	r	NOUN
ejpam-6493	38	17	if	if	SCONJ
ejpam-6493	38	18	s1s2s3	s1s2s3	PROPN
ejpam-6493	38	19	∈	∈	PROPN
ejpam-6493	38	20	s	s	X
ejpam-6493	38	21	for	for	ADP
ejpam-6493	38	22	all	all	DET
ejpam-6493	38	23	s1	s1	NOUN
ejpam-6493	38	24	,	,	PUNCT
ejpam-6493	38	25	s2	s2	PROPN
ejpam-6493	38	26	,	,	PUNCT
ejpam-6493	38	27	s3	s3	PROPN
ejpam-6493	38	28	∈	∈	PROPN
ejpam-6493	38	29	s.	s.	PROPN
ejpam-6493	38	30	definition	definition	NOUN
ejpam-6493	38	31	4	4	NUM
ejpam-6493	38	32	(	(	PUNCT
ejpam-6493	38	33	[	[	X
ejpam-6493	38	34	4	4	NUM
ejpam-6493	38	35	]	]	NUM
ejpam-6493	38	36	)	)	PUNCT
ejpam-6493	38	37	.	.	PUNCT
ejpam-6493	39	1	an	an	DET
ejpam-6493	39	2	additive	additive	ADJ
ejpam-6493	39	3	subsemigroup	subsemigroup	NOUN
ejpam-6493	39	4	i	i	PRON
ejpam-6493	39	5	of	of	ADP
ejpam-6493	39	6	a	a	DET
ejpam-6493	39	7	ternary	ternary	ADJ
ejpam-6493	39	8	semiring	semiring	NOUN
ejpam-6493	39	9	r	r	NOUN
ejpam-6493	39	10	is	be	AUX
ejpam-6493	39	11	called	call	VERB
ejpam-6493	39	12	(	(	PUNCT
ejpam-6493	39	13	1	1	NUM
ejpam-6493	39	14	)	)	PUNCT
ejpam-6493	39	15	a	a	DET
ejpam-6493	39	16	left	left	ADJ
ejpam-6493	39	17	ideal	ideal	NOUN
ejpam-6493	39	18	of	of	ADP
ejpam-6493	39	19	r	r	NOUN
ejpam-6493	39	20	if	if	SCONJ
ejpam-6493	39	21	r1r2a	r1r2a	PUNCT
ejpam-6493	39	22	∈	∈	PROPN
ejpam-6493	39	23	i	i	PRON
ejpam-6493	39	24	for	for	ADP
ejpam-6493	39	25	all	all	DET
ejpam-6493	39	26	r1	r1	NOUN
ejpam-6493	39	27	,	,	PUNCT
ejpam-6493	39	28	r2	r2	PROPN
ejpam-6493	39	29	∈	∈	PROPN
ejpam-6493	39	30	r	r	NOUN
ejpam-6493	39	31	and	and	CCONJ
ejpam-6493	39	32	a	a	DET
ejpam-6493	39	33	∈	∈	NOUN
ejpam-6493	39	34	i	i	PRON
ejpam-6493	39	35	,	,	PUNCT
ejpam-6493	39	36	(	(	PUNCT
ejpam-6493	39	37	2	2	X
ejpam-6493	39	38	)	)	PUNCT
ejpam-6493	39	39	a	a	DET
ejpam-6493	39	40	right	right	ADJ
ejpam-6493	39	41	ideal	ideal	NOUN
ejpam-6493	39	42	of	of	ADP
ejpam-6493	39	43	r	r	NOUN
ejpam-6493	39	44	if	if	SCONJ
ejpam-6493	39	45	ar1r2	ar1r2	PROPN
ejpam-6493	39	46	∈	∈	PROPN
ejpam-6493	39	47	i	i	PRON
ejpam-6493	39	48	for	for	ADP
ejpam-6493	39	49	all	all	DET
ejpam-6493	39	50	r1	r1	NOUN
ejpam-6493	39	51	,	,	PUNCT
ejpam-6493	39	52	r2	r2	PROPN
ejpam-6493	39	53	∈	∈	PROPN
ejpam-6493	39	54	r	r	NOUN
ejpam-6493	39	55	and	and	CCONJ
ejpam-6493	39	56	a	a	DET
ejpam-6493	39	57	∈	∈	NOUN
ejpam-6493	39	58	i	i	PRON
ejpam-6493	39	59	,	,	PUNCT
ejpam-6493	39	60	(	(	PUNCT
ejpam-6493	39	61	3	3	X
ejpam-6493	39	62	)	)	PUNCT
ejpam-6493	39	63	a	a	DET
ejpam-6493	39	64	lateral	lateral	ADJ
ejpam-6493	39	65	ideal	ideal	NOUN
ejpam-6493	39	66	of	of	ADP
ejpam-6493	39	67	r	r	NOUN
ejpam-6493	39	68	if	if	SCONJ
ejpam-6493	39	69	r1ar2	r1ar2	VERB
ejpam-6493	39	70	∈	∈	PROPN
ejpam-6493	39	71	i	i	PRON
ejpam-6493	39	72	for	for	ADP
ejpam-6493	39	73	all	all	DET
ejpam-6493	39	74	r1	r1	NOUN
ejpam-6493	39	75	,	,	PUNCT
ejpam-6493	39	76	r2	r2	PROPN
ejpam-6493	39	77	∈	∈	PROPN
ejpam-6493	39	78	r	r	NOUN
ejpam-6493	39	79	and	and	CCONJ
ejpam-6493	39	80	a	a	DET
ejpam-6493	39	81	∈	∈	NOUN
ejpam-6493	39	82	i	i	PRON
ejpam-6493	39	83	,	,	PUNCT
ejpam-6493	39	84	(	(	PUNCT
ejpam-6493	39	85	4	4	X
ejpam-6493	39	86	)	)	PUNCT
ejpam-6493	39	87	an	an	DET
ejpam-6493	39	88	ideal	ideal	NOUN
ejpam-6493	39	89	of	of	ADP
ejpam-6493	39	90	r	r	NOUN
ejpam-6493	39	91	if	if	SCONJ
ejpam-6493	39	92	i	i	PRON
ejpam-6493	39	93	is	be	AUX
ejpam-6493	39	94	a	a	DET
ejpam-6493	39	95	left	left	ADJ
ejpam-6493	39	96	ideal	ideal	NOUN
ejpam-6493	39	97	,	,	PUNCT
ejpam-6493	39	98	a	a	DET
ejpam-6493	39	99	right	right	ADJ
ejpam-6493	39	100	ideal	ideal	NOUN
ejpam-6493	39	101	,	,	PUNCT
ejpam-6493	39	102	and	and	CCONJ
ejpam-6493	39	103	a	a	DET
ejpam-6493	39	104	lateral	lateral	ADJ
ejpam-6493	39	105	ideal	ideal	NOUN
ejpam-6493	39	106	of	of	ADP
ejpam-6493	39	107	r.	r.	PROPN
ejpam-6493	39	108	an	an	DET
ejpam-6493	39	109	ideal	ideal	NOUN
ejpam-6493	39	110	i	i	PRON
ejpam-6493	39	111	of	of	ADP
ejpam-6493	39	112	r	r	NOUN
ejpam-6493	39	113	is	be	AUX
ejpam-6493	39	114	called	call	VERB
ejpam-6493	39	115	a	a	DET
ejpam-6493	39	116	proper	proper	ADJ
ejpam-6493	39	117	ideal	ideal	NOUN
ejpam-6493	39	118	if	if	SCONJ
ejpam-6493	39	119	i	i	PRON
ejpam-6493	39	120	̸=	̸=	PROPN
ejpam-6493	39	121	r.	r.	PROPN
ejpam-6493	39	122	definition	definition	NOUN
ejpam-6493	39	123	5	5	NUM
ejpam-6493	39	124	.	.	PUNCT
ejpam-6493	40	1	an	an	DET
ejpam-6493	40	2	ideal	ideal	ADJ
ejpam-6493	40	3	i	i	PRON
ejpam-6493	40	4	of	of	ADP
ejpam-6493	40	5	a	a	DET
ejpam-6493	40	6	semiring	semiring	NOUN
ejpam-6493	40	7	r	r	NOUN
ejpam-6493	40	8	is	be	AUX
ejpam-6493	40	9	called	call	VERB
ejpam-6493	40	10	a	a	DET
ejpam-6493	40	11	k	k	NOUN
ejpam-6493	40	12	-	-	NOUN
ejpam-6493	40	13	ideal	ideal	NOUN
ejpam-6493	40	14	of	of	ADP
ejpam-6493	40	15	r	r	NOUN
ejpam-6493	40	16	if	if	SCONJ
ejpam-6493	40	17	,	,	PUNCT
ejpam-6493	40	18	for	for	ADP
ejpam-6493	40	19	any	any	DET
ejpam-6493	40	20	x	x	NOUN
ejpam-6493	40	21	,	,	PUNCT
ejpam-6493	40	22	y	y	PROPN
ejpam-6493	40	23	∈	∈	PROPN
ejpam-6493	40	24	r	r	NOUN
ejpam-6493	40	25	,	,	PUNCT
ejpam-6493	40	26	x	x	SYM
ejpam-6493	40	27	∈	∈	NOUN
ejpam-6493	40	28	i	i	PRON
ejpam-6493	40	29	and	and	CCONJ
ejpam-6493	40	30	x+	x+	ADJ
ejpam-6493	40	31	y	y	PROPN
ejpam-6493	40	32	∈	∈	PROPN
ejpam-6493	41	1	i	i	PRON
ejpam-6493	41	2	,	,	PUNCT
ejpam-6493	41	3	it	it	PRON
ejpam-6493	41	4	follows	follow	VERB
ejpam-6493	41	5	y	y	PROPN
ejpam-6493	41	6	∈	∈	PROPN
ejpam-6493	41	7	i.	i.	NOUN
ejpam-6493	41	8	definition	definition	NOUN
ejpam-6493	41	9	6	6	NUM
ejpam-6493	41	10	.	.	PUNCT
ejpam-6493	42	1	let	let	VERB
ejpam-6493	42	2	r	r	NOUN
ejpam-6493	42	3	and	and	CCONJ
ejpam-6493	42	4	t	t	PROPN
ejpam-6493	42	5	be	be	AUX
ejpam-6493	42	6	two	two	NUM
ejpam-6493	42	7	ternary	ternary	ADJ
ejpam-6493	42	8	semirings	semiring	NOUN
ejpam-6493	42	9	and	and	CCONJ
ejpam-6493	42	10	φ	φ	PROPN
ejpam-6493	42	11	be	be	VERB
ejpam-6493	42	12	a	a	DET
ejpam-6493	42	13	mapping	mapping	NOUN
ejpam-6493	42	14	which	which	PRON
ejpam-6493	42	15	maps	map	VERB
ejpam-6493	42	16	r	r	NOUN
ejpam-6493	42	17	into	into	ADP
ejpam-6493	42	18	t	t	PROPN
ejpam-6493	42	19	.	.	PUNCT
ejpam-6493	43	1	then	then	ADV
ejpam-6493	43	2	the	the	DET
ejpam-6493	43	3	mapping	mapping	NOUN
ejpam-6493	43	4	φ	φ	NOUN
ejpam-6493	43	5	:	:	PUNCT
ejpam-6493	43	6	r	r	NOUN
ejpam-6493	43	7	→	→	SYM
ejpam-6493	43	8	t	t	PROPN
ejpam-6493	43	9	is	be	AUX
ejpam-6493	43	10	called	call	VERB
ejpam-6493	43	11	a	a	DET
ejpam-6493	43	12	homomorphism	homomorphism	NOUN
ejpam-6493	43	13	of	of	ADP
ejpam-6493	43	14	r	r	NOUN
ejpam-6493	43	15	into	into	ADP
ejpam-6493	43	16	t	t	PROPN
ejpam-6493	43	17	if	if	SCONJ
ejpam-6493	43	18	the	the	DET
ejpam-6493	43	19	following	follow	VERB
ejpam-6493	43	20	conditions	condition	NOUN
ejpam-6493	43	21	hold	hold	VERB
ejpam-6493	43	22	:	:	PUNCT
ejpam-6493	43	23	m.	m.	NOUN
ejpam-6493	43	24	petapirak	petapirak	PROPN
ejpam-6493	43	25	,	,	PUNCT
ejpam-6493	43	26	a.	a.	PROPN
ejpam-6493	43	27	j.	j.	PROPN
ejpam-6493	43	28	khan	khan	PROPN
ejpam-6493	43	29	,	,	PUNCT
ejpam-6493	43	30	r.	r.	PROPN
ejpam-6493	43	31	chinram	chinram	PROPN
ejpam-6493	43	32	/	/	SYM
ejpam-6493	43	33	eur	eur	PROPN
ejpam-6493	43	34	.	.	PUNCT
ejpam-6493	44	1	j.	j.	PROPN
ejpam-6493	44	2	pure	pure	PROPN
ejpam-6493	44	3	appl	appl	PROPN
ejpam-6493	44	4	.	.	PROPN
ejpam-6493	44	5	math	math	PROPN
ejpam-6493	44	6	,	,	PUNCT
ejpam-6493	44	7	18	18	NUM
ejpam-6493	44	8	(	(	PUNCT
ejpam-6493	44	9	3	3	NUM
ejpam-6493	44	10	)	)	PUNCT
ejpam-6493	44	11	(	(	PUNCT
ejpam-6493	44	12	2025	2025	NUM
ejpam-6493	44	13	)	)	PUNCT
ejpam-6493	44	14	,	,	PUNCT
ejpam-6493	44	15	6493	6493	NUM
ejpam-6493	44	16	3	3	NUM
ejpam-6493	44	17	of	of	ADP
ejpam-6493	44	18	9	9	NUM
ejpam-6493	44	19	(	(	PUNCT
ejpam-6493	44	20	i	i	NOUN
ejpam-6493	44	21	)	)	PUNCT
ejpam-6493	44	22	φ(a+	φ(a+	PROPN
ejpam-6493	44	23	b	b	X
ejpam-6493	44	24	)	)	PUNCT
ejpam-6493	44	25	=	=	SYM
ejpam-6493	44	26	φ(a	φ(a	ADJ
ejpam-6493	44	27	)	)	PUNCT
ejpam-6493	45	1	+	+	ADJ
ejpam-6493	45	2	φ(b	φ(b	NOUN
ejpam-6493	45	3	)	)	PUNCT
ejpam-6493	45	4	,	,	PUNCT
ejpam-6493	45	5	(	(	PUNCT
ejpam-6493	45	6	ii	ii	NOUN
ejpam-6493	45	7	)	)	PUNCT
ejpam-6493	45	8	φ(abc	φ(abc	PROPN
ejpam-6493	45	9	)	)	PUNCT
ejpam-6493	45	10	=	=	SYM
ejpam-6493	46	1	φ(a)φ(b)φ(c	φ(a)φ(b)φ(c	NOUN
ejpam-6493	46	2	)	)	PUNCT
ejpam-6493	46	3	for	for	ADP
ejpam-6493	46	4	all	all	DET
ejpam-6493	46	5	a	a	DET
ejpam-6493	46	6	,	,	PUNCT
ejpam-6493	46	7	b	b	NOUN
ejpam-6493	46	8	,	,	PUNCT
ejpam-6493	46	9	c	c	PROPN
ejpam-6493	46	10	∈	∈	PROPN
ejpam-6493	46	11	r.	r.	PROPN
ejpam-6493	46	12	definition	definition	NOUN
ejpam-6493	46	13	7	7	NUM
ejpam-6493	46	14	.	.	PUNCT
ejpam-6493	47	1	let	let	VERB
ejpam-6493	47	2	r	r	NOUN
ejpam-6493	47	3	and	and	CCONJ
ejpam-6493	47	4	t	t	PROPN
ejpam-6493	47	5	be	be	AUX
ejpam-6493	47	6	two	two	NUM
ejpam-6493	47	7	ternary	ternary	ADJ
ejpam-6493	47	8	semirings	semiring	NOUN
ejpam-6493	47	9	.	.	PUNCT
ejpam-6493	48	1	the	the	DET
ejpam-6493	48	2	homomorphism	homomorphism	PROPN
ejpam-6493	48	3	φ	φ	X
ejpam-6493	48	4	:	:	PUNCT
ejpam-6493	48	5	r	r	NOUN
ejpam-6493	48	6	→	→	SYM
ejpam-6493	48	7	t	t	PROPN
ejpam-6493	48	8	is	be	AUX
ejpam-6493	48	9	called	call	VERB
ejpam-6493	48	10	an	an	DET
ejpam-6493	48	11	isomorphism	isomorphism	NOUN
ejpam-6493	48	12	of	of	ADP
ejpam-6493	48	13	r	r	NOUN
ejpam-6493	48	14	onto	onto	ADP
ejpam-6493	48	15	t	t	PROPN
ejpam-6493	48	16	if	if	SCONJ
ejpam-6493	48	17	φ	φ	PROPN
ejpam-6493	48	18	is	be	AUX
ejpam-6493	48	19	a	a	DET
ejpam-6493	48	20	bijection	bijection	NOUN
ejpam-6493	48	21	.	.	PUNCT
ejpam-6493	49	1	if	if	SCONJ
ejpam-6493	49	2	φ	φ	PROPN
ejpam-6493	49	3	is	be	AUX
ejpam-6493	49	4	an	an	DET
ejpam-6493	49	5	isomorphism	isomorphism	NOUN
ejpam-6493	49	6	,	,	PUNCT
ejpam-6493	49	7	we	we	PRON
ejpam-6493	49	8	say	say	VERB
ejpam-6493	49	9	that	that	SCONJ
ejpam-6493	49	10	r	r	NOUN
ejpam-6493	49	11	and	and	CCONJ
ejpam-6493	49	12	t	t	NOUN
ejpam-6493	49	13	are	be	AUX
ejpam-6493	49	14	isomorphic	isomorphic	ADJ
ejpam-6493	49	15	and	and	CCONJ
ejpam-6493	49	16	use	use	VERB
ejpam-6493	49	17	the	the	DET
ejpam-6493	49	18	notation	notation	NOUN
ejpam-6493	49	19	r	r	NOUN
ejpam-6493	49	20	∼=	∼=	PROPN
ejpam-6493	49	21	t	t	NOUN
ejpam-6493	49	22	.	.	PUNCT
ejpam-6493	50	1	3	3	X
ejpam-6493	50	2	.	.	X
ejpam-6493	50	3	quotient	quotient	VERB
ejpam-6493	50	4	ternary	ternary	ADJ
ejpam-6493	50	5	semirings	semiring	NOUN
ejpam-6493	50	6	in	in	ADP
ejpam-6493	50	7	this	this	DET
ejpam-6493	50	8	section	section	NOUN
ejpam-6493	50	9	,	,	PUNCT
ejpam-6493	50	10	we	we	PRON
ejpam-6493	50	11	will	will	AUX
ejpam-6493	50	12	present	present	VERB
ejpam-6493	50	13	the	the	DET
ejpam-6493	50	14	various	various	ADJ
ejpam-6493	50	15	kinds	kind	NOUN
ejpam-6493	50	16	of	of	ADP
ejpam-6493	50	17	quotient	quotient	NOUN
ejpam-6493	50	18	ternary	ternary	ADJ
ejpam-6493	50	19	semirings	semiring	NOUN
ejpam-6493	50	20	constructed	construct	VERB
ejpam-6493	50	21	by	by	ADP
ejpam-6493	50	22	using	use	VERB
ejpam-6493	50	23	different	different	ADJ
ejpam-6493	50	24	concepts	concept	NOUN
ejpam-6493	50	25	.	.	PUNCT
ejpam-6493	51	1	3.1	3.1	NUM
ejpam-6493	51	2	.	.	PUNCT
ejpam-6493	52	1	quotient	quotient	AUX
ejpam-6493	52	2	ternary	ternary	ADJ
ejpam-6493	52	3	semirings	semiring	NOUN
ejpam-6493	52	4	modulo	modulo	VERB
ejpam-6493	52	5	q	q	X
ejpam-6493	52	6	-	-	PUNCT
ejpam-6493	52	7	ideals	ideal	NOUN
ejpam-6493	52	8	firstly	firstly	ADV
ejpam-6493	52	9	,	,	PUNCT
ejpam-6493	52	10	we	we	PRON
ejpam-6493	52	11	recall	recall	VERB
ejpam-6493	52	12	some	some	DET
ejpam-6493	52	13	results	result	NOUN
ejpam-6493	52	14	from	from	ADP
ejpam-6493	52	15	[	[	X
ejpam-6493	52	16	5	5	NUM
ejpam-6493	52	17	]	]	PUNCT
ejpam-6493	52	18	.	.	PUNCT
ejpam-6493	53	1	definition	definition	NOUN
ejpam-6493	53	2	8	8	NUM
ejpam-6493	53	3	(	(	PUNCT
ejpam-6493	53	4	[	[	X
ejpam-6493	53	5	5	5	NUM
ejpam-6493	53	6	]	]	NUM
ejpam-6493	53	7	)	)	PUNCT
ejpam-6493	53	8	.	.	PUNCT
ejpam-6493	54	1	an	an	DET
ejpam-6493	54	2	ideal	ideal	ADJ
ejpam-6493	54	3	i	i	PRON
ejpam-6493	54	4	of	of	ADP
ejpam-6493	54	5	a	a	DET
ejpam-6493	54	6	ternary	ternary	ADJ
ejpam-6493	54	7	semiring	semiring	NOUN
ejpam-6493	54	8	r	r	NOUN
ejpam-6493	54	9	is	be	AUX
ejpam-6493	54	10	called	call	VERB
ejpam-6493	54	11	a	a	DET
ejpam-6493	54	12	partitioning	partition	VERB
ejpam-6493	54	13	ideal	ideal	NOUN
ejpam-6493	54	14	(	(	PUNCT
ejpam-6493	54	15	shortly	shortly	ADV
ejpam-6493	54	16	,	,	PUNCT
ejpam-6493	54	17	q	q	NOUN
ejpam-6493	54	18	-	-	PUNCT
ejpam-6493	54	19	ideal	ideal	NOUN
ejpam-6493	54	20	)	)	PUNCT
ejpam-6493	54	21	if	if	SCONJ
ejpam-6493	54	22	there	there	PRON
ejpam-6493	54	23	exists	exist	VERB
ejpam-6493	54	24	a	a	DET
ejpam-6493	54	25	subset	subset	NOUN
ejpam-6493	54	26	q	q	NOUN
ejpam-6493	54	27	of	of	ADP
ejpam-6493	54	28	r	r	NOUN
ejpam-6493	54	29	such	such	ADJ
ejpam-6493	54	30	that	that	SCONJ
ejpam-6493	54	31	(	(	PUNCT
ejpam-6493	54	32	1	1	X
ejpam-6493	54	33	)	)	PUNCT
ejpam-6493	54	34	r	r	NOUN
ejpam-6493	54	35	=	=	SYM
ejpam-6493	54	36	∪{q	∪{q	NOUN
ejpam-6493	54	37	+	+	X
ejpam-6493	54	38	i	i	PRON
ejpam-6493	54	39	|	|	ADV
ejpam-6493	54	40	q	q	X
ejpam-6493	54	41	∈	∈	PROPN
ejpam-6493	54	42	q	q	X
ejpam-6493	54	43	}	}	PUNCT
ejpam-6493	54	44	,	,	PUNCT
ejpam-6493	54	45	(	(	PUNCT
ejpam-6493	54	46	2	2	X
ejpam-6493	54	47	)	)	PUNCT
ejpam-6493	54	48	for	for	ADP
ejpam-6493	54	49	any	any	DET
ejpam-6493	54	50	q1	q1	NOUN
ejpam-6493	54	51	,	,	PUNCT
ejpam-6493	54	52	q2	q2	PROPN
ejpam-6493	54	53	∈	∈	PROPN
ejpam-6493	54	54	q	q	NOUN
ejpam-6493	54	55	,	,	PUNCT
ejpam-6493	54	56	(	(	PUNCT
ejpam-6493	54	57	q1	q1	NOUN
ejpam-6493	54	58	+	+	CCONJ
ejpam-6493	54	59	i	i	NOUN
ejpam-6493	54	60	)	)	PUNCT
ejpam-6493	54	61	∩	∩	NOUN
ejpam-6493	54	62	(	(	PUNCT
ejpam-6493	54	63	q2	q2	NOUN
ejpam-6493	54	64	+	+	CCONJ
ejpam-6493	54	65	i	i	NOUN
ejpam-6493	54	66	)	)	PUNCT
ejpam-6493	54	67	̸=	̸=	PROPN
ejpam-6493	54	68	∅	∅	VERB
ejpam-6493	54	69	⇔	⇔	PROPN
ejpam-6493	54	70	q1	q1	PROPN
ejpam-6493	54	71	=	=	PROPN
ejpam-6493	54	72	q2	q2	PROPN
ejpam-6493	54	73	.	.	PUNCT
ejpam-6493	55	1	let	let	VERB
ejpam-6493	55	2	i	i	PRON
ejpam-6493	55	3	be	be	AUX
ejpam-6493	55	4	a	a	DET
ejpam-6493	55	5	q	q	NOUN
ejpam-6493	55	6	-	-	PUNCT
ejpam-6493	55	7	ideal	ideal	NOUN
ejpam-6493	55	8	of	of	ADP
ejpam-6493	55	9	a	a	DET
ejpam-6493	55	10	ternary	ternary	ADJ
ejpam-6493	55	11	semiring	semire	VERB
ejpam-6493	55	12	r.	r.	NOUN
ejpam-6493	55	13	we	we	PRON
ejpam-6493	55	14	let	let	VERB
ejpam-6493	55	15	r	r	VERB
ejpam-6493	55	16	/	/	SYM
ejpam-6493	55	17	i(q	i(q	NOUN
ejpam-6493	55	18	)	)	PUNCT
ejpam-6493	55	19	=	=	SYM
ejpam-6493	55	20	{	{	PUNCT
ejpam-6493	56	1	q+	q+	NOUN
ejpam-6493	56	2	i	i	PRON
ejpam-6493	56	3	|	|	ADV
ejpam-6493	56	4	q	q	X
ejpam-6493	56	5	∈	∈	PROPN
ejpam-6493	56	6	q	q	X
ejpam-6493	56	7	}	}	PUNCT
ejpam-6493	56	8	and	and	CCONJ
ejpam-6493	56	9	define	define	VERB
ejpam-6493	56	10	the	the	DET
ejpam-6493	56	11	addition	addition	NOUN
ejpam-6493	56	12	⊕	⊕	PROPN
ejpam-6493	56	13	and	and	CCONJ
ejpam-6493	56	14	ternary	ternary	ADJ
ejpam-6493	56	15	multiplication	multiplication	NOUN
ejpam-6493	56	16	,	,	PUNCT
ejpam-6493	56	17	for	for	ADP
ejpam-6493	56	18	q1	q1	PROPN
ejpam-6493	56	19	,	,	PUNCT
ejpam-6493	56	20	q2	q2	NOUN
ejpam-6493	56	21	,	,	PUNCT
ejpam-6493	56	22	q3	q3	PROPN
ejpam-6493	56	23	∈	∈	PROPN
ejpam-6493	56	24	q	q	X
ejpam-6493	56	25	,	,	PUNCT
ejpam-6493	56	26	by	by	ADP
ejpam-6493	56	27	(	(	PUNCT
ejpam-6493	56	28	q1	q1	PROPN
ejpam-6493	56	29	+	+	CCONJ
ejpam-6493	56	30	i)⊕	i)⊕	PROPN
ejpam-6493	56	31	(	(	PUNCT
ejpam-6493	56	32	q2	q2	NOUN
ejpam-6493	56	33	+	+	CCONJ
ejpam-6493	56	34	i	i	NOUN
ejpam-6493	56	35	)	)	PUNCT
ejpam-6493	56	36	=	=	PUNCT
ejpam-6493	56	37	q∗	q∗	NOUN
ejpam-6493	56	38	+	+	CCONJ
ejpam-6493	56	39	i	i	PRON
ejpam-6493	56	40	and	and	CCONJ
ejpam-6493	56	41	(	(	PUNCT
ejpam-6493	56	42	q1	q1	PROPN
ejpam-6493	56	43	+	+	CCONJ
ejpam-6493	56	44	i)(q2	i)(q2	PROPN
ejpam-6493	56	45	+	+	CCONJ
ejpam-6493	56	46	i)(q3	i)(q3	PROPN
ejpam-6493	57	1	+	+	CCONJ
ejpam-6493	57	2	i	i	NOUN
ejpam-6493	57	3	)	)	PUNCT
ejpam-6493	57	4	=	=	PUNCT
ejpam-6493	58	1	q′	q′	NOUN
ejpam-6493	59	1	+	+	CCONJ
ejpam-6493	59	2	i	i	PRON
ejpam-6493	59	3	where	where	SCONJ
ejpam-6493	59	4	q∗	q∗	PROPN
ejpam-6493	59	5	∈	∈	PROPN
ejpam-6493	59	6	q	q	NOUN
ejpam-6493	59	7	is	be	AUX
ejpam-6493	59	8	a	a	DET
ejpam-6493	59	9	unique	unique	ADJ
ejpam-6493	59	10	element	element	NOUN
ejpam-6493	59	11	such	such	ADJ
ejpam-6493	59	12	that	that	DET
ejpam-6493	59	13	q1	q1	PROPN
ejpam-6493	59	14	+	+	CCONJ
ejpam-6493	59	15	q2	q2	NOUN
ejpam-6493	60	1	+	+	CCONJ
ejpam-6493	60	2	i	i	PROPN
ejpam-6493	60	3	⊆	⊆	NUM
ejpam-6493	60	4	q∗	q∗	NOUN
ejpam-6493	61	1	+	+	CCONJ
ejpam-6493	61	2	i	i	PRON
ejpam-6493	61	3	and	and	CCONJ
ejpam-6493	61	4	q′	q′	NOUN
ejpam-6493	61	5	∈	∈	PROPN
ejpam-6493	61	6	q	q	NOUN
ejpam-6493	61	7	is	be	AUX
ejpam-6493	61	8	a	a	DET
ejpam-6493	61	9	unique	unique	ADJ
ejpam-6493	61	10	element	element	NOUN
ejpam-6493	61	11	such	such	ADJ
ejpam-6493	61	12	that	that	SCONJ
ejpam-6493	61	13	q1q2q3	q1q2q3	NOUN
ejpam-6493	61	14	+	+	PROPN
ejpam-6493	61	15	i	i	PRON
ejpam-6493	61	16	⊆	⊆	NUM
ejpam-6493	61	17	q′	q′	NOUN
ejpam-6493	61	18	+	+	CCONJ
ejpam-6493	61	19	i.	i.	NOUN
ejpam-6493	61	20	then	then	ADV
ejpam-6493	61	21	r	r	NOUN
ejpam-6493	61	22	/	/	SYM
ejpam-6493	61	23	i(q	i(q	NOUN
ejpam-6493	61	24	)	)	PUNCT
ejpam-6493	61	25	forms	form	VERB
ejpam-6493	61	26	a	a	DET
ejpam-6493	61	27	ternary	ternary	ADJ
ejpam-6493	61	28	semiring	semiring	NOUN
ejpam-6493	61	29	under	under	ADP
ejpam-6493	61	30	this	this	DET
ejpam-6493	61	31	addition	addition	NOUN
ejpam-6493	61	32	and	and	CCONJ
ejpam-6493	61	33	ternary	ternary	ADJ
ejpam-6493	61	34	multiplication	multiplication	NOUN
ejpam-6493	61	35	.	.	PUNCT
ejpam-6493	62	1	this	this	DET
ejpam-6493	62	2	ternary	ternary	ADJ
ejpam-6493	62	3	semiring	semiring	NOUN
ejpam-6493	62	4	will	will	AUX
ejpam-6493	62	5	be	be	AUX
ejpam-6493	62	6	called	call	VERB
ejpam-6493	62	7	a	a	DET
ejpam-6493	62	8	quotient	quotient	NOUN
ejpam-6493	62	9	ternary	ternary	ADJ
ejpam-6493	62	10	semiring	semiring	NOUN
ejpam-6493	62	11	of	of	ADP
ejpam-6493	62	12	r	r	NOUN
ejpam-6493	62	13	by	by	ADP
ejpam-6493	62	14	a	a	DET
ejpam-6493	62	15	q	q	NOUN
ejpam-6493	62	16	-	-	PUNCT
ejpam-6493	62	17	ideal	ideal	NOUN
ejpam-6493	62	18	i	i	PRON
ejpam-6493	62	19	[	[	X
ejpam-6493	62	20	5	5	NUM
ejpam-6493	62	21	]	]	PUNCT
ejpam-6493	62	22	.	.	PUNCT
ejpam-6493	63	1	example	example	NOUN
ejpam-6493	64	1	1	1	X
ejpam-6493	64	2	.	.	X
ejpam-6493	64	3	we	we	PRON
ejpam-6493	64	4	consider	consider	VERB
ejpam-6493	64	5	a	a	DET
ejpam-6493	64	6	ternary	ternary	ADJ
ejpam-6493	64	7	semiring	semiring	NOUN
ejpam-6493	64	8	r	r	NOUN
ejpam-6493	64	9	=	=	PUNCT
ejpam-6493	64	10	z−	z−	X
ejpam-6493	64	11	0	0	PUNCT
ejpam-6493	65	1	under	under	ADP
ejpam-6493	65	2	the	the	DET
ejpam-6493	65	3	usual	usual	ADJ
ejpam-6493	65	4	addition	addition	NOUN
ejpam-6493	65	5	and	and	CCONJ
ejpam-6493	65	6	ternary	ternary	ADJ
ejpam-6493	65	7	multiplication	multiplication	NOUN
ejpam-6493	65	8	of	of	ADP
ejpam-6493	65	9	integers	integer	NOUN
ejpam-6493	65	10	.	.	PUNCT
ejpam-6493	66	1	let	let	VERB
ejpam-6493	66	2	i	i	PRON
ejpam-6493	66	3	=	=	PUNCT
ejpam-6493	67	1	5z−	5z−	NUM
ejpam-6493	67	2	0	0	NUM
ejpam-6493	67	3	=	=	SYM
ejpam-6493	67	4	{	{	PUNCT
ejpam-6493	67	5	0,−5,−10,−15	0,−5,−10,−15	PROPN
ejpam-6493	67	6	,	,	PUNCT
ejpam-6493	67	7	.	.	PUNCT
ejpam-6493	67	8	.	.	PUNCT
ejpam-6493	67	9	.	.	PUNCT
ejpam-6493	67	10	}	}	PUNCT
ejpam-6493	67	11	.	.	PUNCT
ejpam-6493	68	1	it	it	PRON
ejpam-6493	68	2	is	be	AUX
ejpam-6493	68	3	easy	easy	ADJ
ejpam-6493	68	4	to	to	PART
ejpam-6493	68	5	show	show	VERB
ejpam-6493	68	6	that	that	SCONJ
ejpam-6493	68	7	i	i	PRON
ejpam-6493	68	8	is	be	AUX
ejpam-6493	68	9	a	a	DET
ejpam-6493	68	10	q	q	NOUN
ejpam-6493	68	11	-	-	PUNCT
ejpam-6493	68	12	ideal	ideal	NOUN
ejpam-6493	68	13	of	of	ADP
ejpam-6493	68	14	z−	z−	PROPN
ejpam-6493	68	15	0	0	NUM
ejpam-6493	69	1	where	where	SCONJ
ejpam-6493	69	2	q	q	NOUN
ejpam-6493	69	3	=	=	PUNCT
ejpam-6493	69	4	{	{	PUNCT
ejpam-6493	69	5	0,−1,−2,−3,−4	0,−1,−2,−3,−4	NUM
ejpam-6493	69	6	}	}	PUNCT
ejpam-6493	69	7	and	and	CCONJ
ejpam-6493	69	8	z−	z−	X
ejpam-6493	69	9	0	0	PUNCT
ejpam-6493	70	1	/i(q	/i(q	X
ejpam-6493	70	2	)	)	PUNCT
ejpam-6493	70	3	=	=	PRON
ejpam-6493	70	4	{	{	PUNCT
ejpam-6493	70	5	0	0	NUM
ejpam-6493	70	6	+	+	CCONJ
ejpam-6493	70	7	i,−1	i,−1	PROPN
ejpam-6493	70	8	+	+	CCONJ
ejpam-6493	70	9	i,−2	i,−2	VERB
ejpam-6493	70	10	+	+	CCONJ
ejpam-6493	70	11	i,−3	i,−3	PROPN
ejpam-6493	71	1	+	+	X
ejpam-6493	71	2	i,−4	i,−4	ADV
ejpam-6493	71	3	+	+	CCONJ
ejpam-6493	71	4	i	i	X
ejpam-6493	71	5	}	}	PUNCT
ejpam-6493	71	6	where	where	SCONJ
ejpam-6493	71	7	0	0	PUNCT
ejpam-6493	71	8	+	+	CCONJ
ejpam-6493	71	9	i	i	PRON
ejpam-6493	71	10	=	=	PUNCT
ejpam-6493	71	11	{	{	PUNCT
ejpam-6493	71	12	0,−5,−10,−15	0,−5,−10,−15	PROPN
ejpam-6493	71	13	,	,	PUNCT
ejpam-6493	71	14	.	.	PUNCT
ejpam-6493	71	15	.	.	PUNCT
ejpam-6493	72	1	.	.	PUNCT
ejpam-6493	73	1	}	}	PUNCT
ejpam-6493	73	2	,	,	PUNCT
ejpam-6493	73	3	−1	−1	NOUN
ejpam-6493	74	1	+	+	CCONJ
ejpam-6493	74	2	i	i	PRON
ejpam-6493	74	3	=	=	SYM
ejpam-6493	74	4	{	{	PUNCT
ejpam-6493	74	5	−1,−6,−11,−16	−1,−6,−11,−16	PROPN
ejpam-6493	74	6	,	,	PUNCT
ejpam-6493	74	7	.	.	PUNCT
ejpam-6493	74	8	.	.	PUNCT
ejpam-6493	74	9	.	.	PUNCT
ejpam-6493	75	1	}	}	PUNCT
ejpam-6493	75	2	,	,	PUNCT
ejpam-6493	75	3	−2	−2	NOUN
ejpam-6493	76	1	+	+	CCONJ
ejpam-6493	76	2	i	i	NOUN
ejpam-6493	76	3	=	=	PUNCT
ejpam-6493	76	4	{	{	PUNCT
ejpam-6493	76	5	−2,−7,−12,−17	−2,−7,−12,−17	NOUN
ejpam-6493	76	6	,	,	PUNCT
ejpam-6493	76	7	.	.	PUNCT
ejpam-6493	76	8	.	.	PUNCT
ejpam-6493	76	9	.	.	PUNCT
ejpam-6493	77	1	}	}	PUNCT
ejpam-6493	77	2	,	,	PUNCT
ejpam-6493	77	3	−3	−3	PROPN
ejpam-6493	78	1	+	+	CCONJ
ejpam-6493	78	2	i	i	NOUN
ejpam-6493	78	3	=	=	SYM
ejpam-6493	78	4	{	{	PUNCT
ejpam-6493	78	5	−3,−8,−13,−18	−3,−8,−13,−18	NOUN
ejpam-6493	78	6	,	,	PUNCT
ejpam-6493	78	7	.	.	PUNCT
ejpam-6493	78	8	.	.	PUNCT
ejpam-6493	79	1	.	.	PUNCT
ejpam-6493	79	2	}	}	PUNCT
ejpam-6493	79	3	,	,	PUNCT
ejpam-6493	79	4	−4	−4	X
ejpam-6493	80	1	+	+	CCONJ
ejpam-6493	80	2	i	i	NOUN
ejpam-6493	80	3	=	=	SYM
ejpam-6493	80	4	{	{	PUNCT
ejpam-6493	80	5	−4,−9,−14,−19	−4,−9,−14,−19	PROPN
ejpam-6493	80	6	,	,	PUNCT
ejpam-6493	80	7	.	.	PUNCT
ejpam-6493	80	8	.	.	PUNCT
ejpam-6493	80	9	.	.	PUNCT
ejpam-6493	80	10	}	}	PUNCT
ejpam-6493	80	11	.	.	PUNCT
ejpam-6493	81	1	later	later	ADV
ejpam-6493	81	2	,	,	PUNCT
ejpam-6493	81	3	let	let	VERB
ejpam-6493	81	4	j	j	PROPN
ejpam-6493	81	5	=	=	SYM
ejpam-6493	81	6	2z−	2z−	PROPN
ejpam-6493	81	7	0	0	NUM
ejpam-6493	81	8	\	\	NOUN
ejpam-6493	81	9	{	{	PUNCT
ejpam-6493	81	10	−2	−2	NOUN
ejpam-6493	81	11	}	}	PUNCT
ejpam-6493	81	12	=	=	SYM
ejpam-6493	81	13	{	{	PUNCT
ejpam-6493	81	14	0,−4,−6,−8	0,−4,−6,−8	ADJ
ejpam-6493	81	15	,	,	PUNCT
ejpam-6493	81	16	.	.	PUNCT
ejpam-6493	81	17	.	.	PUNCT
ejpam-6493	81	18	.	.	PUNCT
ejpam-6493	81	19	}	}	PUNCT
ejpam-6493	81	20	.	.	PUNCT
ejpam-6493	82	1	clearly	clearly	ADV
ejpam-6493	82	2	,	,	PUNCT
ejpam-6493	82	3	j	j	PROPN
ejpam-6493	82	4	is	be	AUX
ejpam-6493	82	5	an	an	DET
ejpam-6493	82	6	ideal	ideal	NOUN
ejpam-6493	82	7	.	.	PUNCT
ejpam-6493	83	1	the	the	DET
ejpam-6493	83	2	quotient	quotient	NOUN
ejpam-6493	83	3	ternary	ternary	NOUN
ejpam-6493	83	4	semiring	semire	VERB
ejpam-6493	83	5	z−	z−	PROPN
ejpam-6493	83	6	0	0	PUNCT
ejpam-6493	84	1	/j(q	/j(q	X
ejpam-6493	84	2	)	)	PUNCT
ejpam-6493	84	3	does	do	AUX
ejpam-6493	84	4	not	not	PART
ejpam-6493	84	5	easily	easily	ADV
ejpam-6493	84	6	hold	hold	VERB
ejpam-6493	84	7	according	accord	VERB
ejpam-6493	84	8	to	to	ADP
ejpam-6493	84	9	this	this	DET
ejpam-6493	84	10	concept	concept	NOUN
ejpam-6493	84	11	.	.	PUNCT
ejpam-6493	85	1	m.	m.	NOUN
ejpam-6493	85	2	petapirak	petapirak	PROPN
ejpam-6493	85	3	,	,	PUNCT
ejpam-6493	85	4	a.	a.	PROPN
ejpam-6493	85	5	j.	j.	PROPN
ejpam-6493	85	6	khan	khan	PROPN
ejpam-6493	85	7	,	,	PUNCT
ejpam-6493	85	8	r.	r.	PROPN
ejpam-6493	85	9	chinram	chinram	PROPN
ejpam-6493	85	10	/	/	SYM
ejpam-6493	85	11	eur	eur	PROPN
ejpam-6493	85	12	.	.	PUNCT
ejpam-6493	86	1	j.	j.	PROPN
ejpam-6493	86	2	pure	pure	PROPN
ejpam-6493	86	3	appl	appl	PROPN
ejpam-6493	86	4	.	.	PROPN
ejpam-6493	86	5	math	math	PROPN
ejpam-6493	86	6	,	,	PUNCT
ejpam-6493	86	7	18	18	NUM
ejpam-6493	86	8	(	(	PUNCT
ejpam-6493	86	9	3	3	NUM
ejpam-6493	86	10	)	)	PUNCT
ejpam-6493	86	11	(	(	PUNCT
ejpam-6493	86	12	2025	2025	NUM
ejpam-6493	86	13	)	)	PUNCT
ejpam-6493	86	14	,	,	PUNCT
ejpam-6493	86	15	6493	6493	NUM
ejpam-6493	86	16	4	4	NUM
ejpam-6493	86	17	of	of	ADP
ejpam-6493	86	18	9	9	NUM
ejpam-6493	86	19	definition	definition	NOUN
ejpam-6493	86	20	9	9	NUM
ejpam-6493	86	21	(	(	PUNCT
ejpam-6493	86	22	[	[	X
ejpam-6493	86	23	5	5	NUM
ejpam-6493	86	24	]	]	PUNCT
ejpam-6493	86	25	)	)	PUNCT
ejpam-6493	86	26	.	.	PUNCT
ejpam-6493	87	1	let	let	VERB
ejpam-6493	87	2	r	r	NOUN
ejpam-6493	87	3	and	and	CCONJ
ejpam-6493	87	4	t	t	PROPN
ejpam-6493	87	5	be	be	AUX
ejpam-6493	87	6	two	two	NUM
ejpam-6493	87	7	ternary	ternary	ADJ
ejpam-6493	87	8	semirings	semiring	NOUN
ejpam-6493	87	9	such	such	ADJ
ejpam-6493	87	10	that	that	SCONJ
ejpam-6493	87	11	t	t	PROPN
ejpam-6493	87	12	has	have	VERB
ejpam-6493	87	13	a	a	DET
ejpam-6493	87	14	zero	zero	NUM
ejpam-6493	87	15	0	0	NUM
ejpam-6493	87	16	t	t	NOUN
ejpam-6493	87	17	.	.	PUNCT
ejpam-6493	88	1	an	an	DET
ejpam-6493	88	2	onto	onto	ADP
ejpam-6493	88	3	homomorphism	homomorphism	PROPN
ejpam-6493	88	4	φ	φ	X
ejpam-6493	88	5	:	:	PUNCT
ejpam-6493	88	6	r	r	NOUN
ejpam-6493	88	7	→	→	SYM
ejpam-6493	88	8	t	t	PROPN
ejpam-6493	88	9	is	be	AUX
ejpam-6493	88	10	called	call	VERB
ejpam-6493	88	11	maximal	maximal	ADJ
ejpam-6493	88	12	if	if	SCONJ
ejpam-6493	88	13	,	,	PUNCT
ejpam-6493	88	14	for	for	ADP
ejpam-6493	88	15	each	each	PRON
ejpam-6493	88	16	a	a	DET
ejpam-6493	88	17	∈	∈	PROPN
ejpam-6493	88	18	t	t	NOUN
ejpam-6493	88	19	,	,	PUNCT
ejpam-6493	88	20	there	there	PRON
ejpam-6493	88	21	exists	exist	VERB
ejpam-6493	88	22	a	a	DET
ejpam-6493	88	23	unique	unique	ADJ
ejpam-6493	88	24	qa	qa	PROPN
ejpam-6493	88	25	∈	∈	PROPN
ejpam-6493	88	26	φ−1({a	φ−1({a	NOUN
ejpam-6493	88	27	}	}	PUNCT
ejpam-6493	88	28	)	)	PUNCT
ejpam-6493	88	29	such	such	ADJ
ejpam-6493	88	30	that	that	SCONJ
ejpam-6493	88	31	x	x	X
ejpam-6493	89	1	+	+	CCONJ
ejpam-6493	89	2	ker(φ	ker(φ	X
ejpam-6493	89	3	)	)	PUNCT
ejpam-6493	89	4	⊆	⊆	NUM
ejpam-6493	89	5	qa	qa	NOUN
ejpam-6493	89	6	+	+	CCONJ
ejpam-6493	89	7	ker(φ	ker(φ	X
ejpam-6493	89	8	)	)	PUNCT
ejpam-6493	89	9	for	for	ADP
ejpam-6493	89	10	each	each	DET
ejpam-6493	89	11	x	x	SYM
ejpam-6493	89	12	∈	∈	PROPN
ejpam-6493	89	13	φ−1({a	φ−1({a	NOUN
ejpam-6493	89	14	}	}	PUNCT
ejpam-6493	89	15	)	)	PUNCT
ejpam-6493	89	16	where	where	SCONJ
ejpam-6493	89	17	ker(φ	ker(φ	X
ejpam-6493	89	18	)	)	PUNCT
ejpam-6493	89	19	=	=	PRON
ejpam-6493	89	20	{	{	PUNCT
ejpam-6493	89	21	x	x	PUNCT
ejpam-6493	89	22	∈	∈	NOUN
ejpam-6493	89	23	r	r	NOUN
ejpam-6493	89	24	|	|	NOUN
ejpam-6493	89	25	φ(x	φ(x	NOUN
ejpam-6493	89	26	)	)	PUNCT
ejpam-6493	89	27	=	=	PUNCT
ejpam-6493	89	28	0	0	NUM
ejpam-6493	89	29	t	t	NOUN
ejpam-6493	89	30	}	}	PUNCT
ejpam-6493	89	31	.	.	PUNCT
ejpam-6493	90	1	example	example	NOUN
ejpam-6493	91	1	2	2	NUM
ejpam-6493	91	2	.	.	X
ejpam-6493	91	3	consider	consider	VERB
ejpam-6493	91	4	the	the	DET
ejpam-6493	91	5	ternary	ternary	ADJ
ejpam-6493	91	6	semiring	semire	VERB
ejpam-6493	91	7	z−	z−	PROPN
ejpam-6493	91	8	0	0	PUNCT
ejpam-6493	91	9	under	under	ADP
ejpam-6493	91	10	the	the	DET
ejpam-6493	91	11	usual	usual	ADJ
ejpam-6493	91	12	addition	addition	NOUN
ejpam-6493	91	13	and	and	CCONJ
ejpam-6493	91	14	ternary	ternary	ADJ
ejpam-6493	91	15	multiplication	multiplication	NOUN
ejpam-6493	91	16	of	of	ADP
ejpam-6493	91	17	integers	integer	NOUN
ejpam-6493	91	18	,	,	PUNCT
ejpam-6493	91	19	and	and	CCONJ
ejpam-6493	91	20	the	the	DET
ejpam-6493	91	21	ternary	ternary	ADJ
ejpam-6493	91	22	semiring	semire	VERB
ejpam-6493	91	23	z3	z3	PROPN
ejpam-6493	91	24	under	under	ADP
ejpam-6493	91	25	the	the	DET
ejpam-6493	91	26	usual	usual	ADJ
ejpam-6493	91	27	addition	addition	NOUN
ejpam-6493	91	28	and	and	CCONJ
ejpam-6493	91	29	ternary	ternary	ADJ
ejpam-6493	91	30	multiplication	multiplication	NOUN
ejpam-6493	91	31	of	of	ADP
ejpam-6493	91	32	integers	integer	NOUN
ejpam-6493	91	33	modulo	modulo	VERB
ejpam-6493	91	34	3	3	X
ejpam-6493	91	35	.	.	PUNCT
ejpam-6493	92	1	let	let	VERB
ejpam-6493	92	2	φ	φ	NOUN
ejpam-6493	92	3	:	:	PUNCT
ejpam-6493	92	4	z−	z−	X
ejpam-6493	92	5	0	0	PUNCT
ejpam-6493	93	1	→	→	SYM
ejpam-6493	93	2	z3	z3	PROPN
ejpam-6493	93	3	be	be	AUX
ejpam-6493	93	4	defined	define	VERB
ejpam-6493	93	5	by	by	ADP
ejpam-6493	93	6	φ(n	φ(n	NOUN
ejpam-6493	93	7	)	)	PUNCT
ejpam-6493	93	8	=	=	SYM
ejpam-6493	93	9	n	n	CCONJ
ejpam-6493	93	10	for	for	ADP
ejpam-6493	93	11	all	all	DET
ejpam-6493	93	12	n	n	PRON
ejpam-6493	93	13	∈	∈	NOUN
ejpam-6493	93	14	z−	z−	X
ejpam-6493	93	15	0	0	NUM
ejpam-6493	93	16	.	.	PUNCT
ejpam-6493	94	1	then	then	ADV
ejpam-6493	94	2	φ	φ	PROPN
ejpam-6493	94	3	is	be	AUX
ejpam-6493	94	4	an	an	PRON
ejpam-6493	94	5	onto	onto	ADP
ejpam-6493	94	6	homomorphism	homomorphism	NOUN
ejpam-6493	94	7	and	and	CCONJ
ejpam-6493	94	8	ker(φ	ker(φ	NOUN
ejpam-6493	94	9	)	)	PUNCT
ejpam-6493	94	10	=	=	SYM
ejpam-6493	95	1	3z−	3z−	NUM
ejpam-6493	95	2	0	0	NUM
ejpam-6493	95	3	=	=	SYM
ejpam-6493	95	4	{	{	PUNCT
ejpam-6493	95	5	0,−3,−6,−9	0,−3,−6,−9	NOUN
ejpam-6493	95	6	,	,	PUNCT
ejpam-6493	95	7	.	.	PUNCT
ejpam-6493	95	8	.	.	PUNCT
ejpam-6493	95	9	.	.	PUNCT
ejpam-6493	95	10	}	}	PUNCT
ejpam-6493	95	11	.	.	PUNCT
ejpam-6493	96	1	let	let	VERB
ejpam-6493	96	2	a	a	DET
ejpam-6493	96	3	∈	∈	PROPN
ejpam-6493	96	4	z3	z3	NOUN
ejpam-6493	96	5	and	and	CCONJ
ejpam-6493	96	6	qa	qa	PROPN
ejpam-6493	96	7	∈	∈	PROPN
ejpam-6493	96	8	φ−1({a	φ−1({a	NOUN
ejpam-6493	96	9	}	}	PUNCT
ejpam-6493	96	10	)	)	PUNCT
ejpam-6493	96	11	.	.	PUNCT
ejpam-6493	97	1	assume	assume	VERB
ejpam-6493	97	2	that	that	SCONJ
ejpam-6493	97	3	a	a	DET
ejpam-6493	97	4	=	=	SYM
ejpam-6493	97	5	0	0	PUNCT
ejpam-6493	97	6	and	and	CCONJ
ejpam-6493	97	7	suppose	suppose	VERB
ejpam-6493	97	8	that	that	SCONJ
ejpam-6493	97	9	qa	qa	PROPN
ejpam-6493	97	10	̸=	̸=	PROPN
ejpam-6493	97	11	0	0	NUM
ejpam-6493	97	12	.	.	PUNCT
ejpam-6493	98	1	so	so	ADV
ejpam-6493	98	2	0	0	NUM
ejpam-6493	98	3	/∈	/∈	PUNCT
ejpam-6493	99	1	qa	qa	PROPN
ejpam-6493	99	2	+	+	CCONJ
ejpam-6493	99	3	ker(φ	ker(φ	NOUN
ejpam-6493	99	4	)	)	PUNCT
ejpam-6493	99	5	.	.	PUNCT
ejpam-6493	100	1	we	we	PRON
ejpam-6493	100	2	have	have	VERB
ejpam-6493	100	3	that	that	DET
ejpam-6493	100	4	0	0	NUM
ejpam-6493	100	5	∈	∈	PROPN
ejpam-6493	100	6	φ−1({a	φ−1({a	NOUN
ejpam-6493	100	7	}	}	PUNCT
ejpam-6493	100	8	)	)	PUNCT
ejpam-6493	100	9	.	.	PUNCT
ejpam-6493	101	1	then	then	ADV
ejpam-6493	101	2	,	,	PUNCT
ejpam-6493	101	3	by	by	ADP
ejpam-6493	101	4	definition	definition	NOUN
ejpam-6493	101	5	,	,	PUNCT
ejpam-6493	101	6	we	we	PRON
ejpam-6493	101	7	obtain	obtain	VERB
ejpam-6493	101	8	that	that	DET
ejpam-6493	101	9	0	0	NUM
ejpam-6493	101	10	∈	∈	NOUN
ejpam-6493	101	11	0	0	PUNCT
ejpam-6493	102	1	+	+	CCONJ
ejpam-6493	102	2	ker(φ	ker(φ	X
ejpam-6493	102	3	)	)	PUNCT
ejpam-6493	102	4	⊆	⊆	NUM
ejpam-6493	102	5	qa	qa	NOUN
ejpam-6493	102	6	+	+	CCONJ
ejpam-6493	102	7	ker(φ	ker(φ	NOUN
ejpam-6493	102	8	)	)	PUNCT
ejpam-6493	102	9	,	,	PUNCT
ejpam-6493	102	10	a	a	DET
ejpam-6493	102	11	contradiction	contradiction	NOUN
ejpam-6493	102	12	.	.	PUNCT
ejpam-6493	103	1	we	we	PRON
ejpam-6493	103	2	can	can	AUX
ejpam-6493	103	3	conclude	conclude	VERB
ejpam-6493	103	4	that	that	SCONJ
ejpam-6493	103	5	if	if	SCONJ
ejpam-6493	103	6	a	a	DET
ejpam-6493	103	7	=	=	NOUN
ejpam-6493	103	8	0	0	NUM
ejpam-6493	103	9	,	,	PUNCT
ejpam-6493	103	10	then	then	ADV
ejpam-6493	103	11	qa	qa	PROPN
ejpam-6493	103	12	must	must	AUX
ejpam-6493	103	13	be	be	AUX
ejpam-6493	103	14	0	0	NUM
ejpam-6493	103	15	.	.	PUNCT
ejpam-6493	104	1	similarly	similarly	ADV
ejpam-6493	104	2	,	,	PUNCT
ejpam-6493	104	3	if	if	SCONJ
ejpam-6493	104	4	a	a	DET
ejpam-6493	104	5	=	=	NOUN
ejpam-6493	104	6	1	1	NUM
ejpam-6493	104	7	,	,	PUNCT
ejpam-6493	104	8	then	then	ADV
ejpam-6493	104	9	qa	qa	PROPN
ejpam-6493	104	10	must	must	AUX
ejpam-6493	104	11	be	be	AUX
ejpam-6493	104	12	−2	−2	NOUN
ejpam-6493	104	13	,	,	PUNCT
ejpam-6493	104	14	and	and	CCONJ
ejpam-6493	104	15	if	if	SCONJ
ejpam-6493	104	16	a	a	DET
ejpam-6493	104	17	=	=	SYM
ejpam-6493	104	18	2	2	NUM
ejpam-6493	104	19	,	,	PUNCT
ejpam-6493	104	20	then	then	ADV
ejpam-6493	104	21	qa	qa	PROPN
ejpam-6493	104	22	must	must	AUX
ejpam-6493	104	23	be	be	AUX
ejpam-6493	104	24	−1	−1	NOUN
ejpam-6493	104	25	.	.	PUNCT
ejpam-6493	105	1	therefore	therefore	ADV
ejpam-6493	105	2	,	,	PUNCT
ejpam-6493	105	3	φ	φ	PROPN
ejpam-6493	105	4	is	be	AUX
ejpam-6493	105	5	maximal	maximal	ADJ
ejpam-6493	105	6	.	.	PUNCT
ejpam-6493	106	1	in	in	ADP
ejpam-6493	106	2	addition	addition	NOUN
ejpam-6493	106	3	,	,	PUNCT
ejpam-6493	106	4	taking	take	VERB
ejpam-6493	106	5	q	q	NOUN
ejpam-6493	106	6	=	=	PRON
ejpam-6493	106	7	{	{	PUNCT
ejpam-6493	106	8	0,−1,−2	0,−1,−2	NUM
ejpam-6493	106	9	}	}	PUNCT
ejpam-6493	106	10	,	,	PUNCT
ejpam-6493	106	11	it	it	PRON
ejpam-6493	106	12	follows	follow	VERB
ejpam-6493	106	13	that	that	SCONJ
ejpam-6493	106	14	,	,	PUNCT
ejpam-6493	106	15	z−	z−	PROPN
ejpam-6493	106	16	0	0	PUNCT
ejpam-6493	106	17	/ker(φ)(q	/ker(φ)(q	PUNCT
ejpam-6493	106	18	)	)	PUNCT
ejpam-6493	106	19	=	=	PRON
ejpam-6493	106	20	{	{	PUNCT
ejpam-6493	107	1	0	0	NUM
ejpam-6493	107	2	+	+	CCONJ
ejpam-6493	107	3	ker(φ),−1	ker(φ),−1	PROPN
ejpam-6493	107	4	+	+	CCONJ
ejpam-6493	107	5	ker(φ),−2	ker(φ),−2	PROPN
ejpam-6493	107	6	+	+	CCONJ
ejpam-6493	107	7	ker(φ	ker(φ	NOUN
ejpam-6493	107	8	)	)	PUNCT
ejpam-6493	107	9	}	}	PUNCT
ejpam-6493	107	10	where	where	SCONJ
ejpam-6493	107	11	0	0	PUNCT
ejpam-6493	107	12	+	+	CCONJ
ejpam-6493	107	13	ker(φ	ker(φ	X
ejpam-6493	107	14	)	)	PUNCT
ejpam-6493	107	15	=	=	SYM
ejpam-6493	107	16	{	{	PUNCT
ejpam-6493	107	17	0,−3,−6,−9	0,−3,−6,−9	NOUN
ejpam-6493	107	18	,	,	PUNCT
ejpam-6493	107	19	.	.	PUNCT
ejpam-6493	107	20	.	.	PUNCT
ejpam-6493	108	1	.	.	PUNCT
ejpam-6493	109	1	}	}	PUNCT
ejpam-6493	109	2	,	,	PUNCT
ejpam-6493	109	3	−1	−1	NOUN
ejpam-6493	109	4	+	+	CCONJ
ejpam-6493	110	1	ker(φ	ker(φ	X
ejpam-6493	110	2	)	)	PUNCT
ejpam-6493	110	3	=	=	PRON
ejpam-6493	110	4	{	{	PUNCT
ejpam-6493	110	5	−1,−4,−7,−10	−1,−4,−7,−10	NOUN
ejpam-6493	110	6	,	,	PUNCT
ejpam-6493	110	7	.	.	PUNCT
ejpam-6493	110	8	.	.	PUNCT
ejpam-6493	110	9	.	.	PUNCT
ejpam-6493	111	1	}	}	PUNCT
ejpam-6493	111	2	,	,	PUNCT
ejpam-6493	111	3	−2	−2	NOUN
ejpam-6493	111	4	+	+	CCONJ
ejpam-6493	111	5	ker(φ	ker(φ	NOUN
ejpam-6493	111	6	)	)	PUNCT
ejpam-6493	111	7	=	=	SYM
ejpam-6493	111	8	{	{	PUNCT
ejpam-6493	111	9	−2,−5,−8,−11	−2,−5,−8,−11	NOUN
ejpam-6493	111	10	,	,	PUNCT
ejpam-6493	111	11	.	.	PUNCT
ejpam-6493	111	12	.	.	PUNCT
ejpam-6493	111	13	.	.	PUNCT
ejpam-6493	111	14	}	}	PUNCT
ejpam-6493	111	15	.	.	PUNCT
ejpam-6493	112	1	theorem	theorem	ADJ
ejpam-6493	112	2	1	1	NUM
ejpam-6493	112	3	(	(	PUNCT
ejpam-6493	112	4	[	[	X
ejpam-6493	112	5	5	5	NUM
ejpam-6493	112	6	]	]	PUNCT
ejpam-6493	112	7	)	)	PUNCT
ejpam-6493	112	8	.	.	PUNCT
ejpam-6493	113	1	let	let	VERB
ejpam-6493	113	2	r	r	NOUN
ejpam-6493	113	3	and	and	CCONJ
ejpam-6493	113	4	t	t	PROPN
ejpam-6493	113	5	be	be	AUX
ejpam-6493	113	6	two	two	NUM
ejpam-6493	113	7	ternary	ternary	ADJ
ejpam-6493	113	8	semirings	semiring	NOUN
ejpam-6493	113	9	such	such	ADJ
ejpam-6493	113	10	that	that	SCONJ
ejpam-6493	113	11	t	t	PROPN
ejpam-6493	113	12	has	have	VERB
ejpam-6493	113	13	a	a	DET
ejpam-6493	113	14	zero	zero	NUM
ejpam-6493	113	15	0	0	NUM
ejpam-6493	113	16	t	t	NOUN
ejpam-6493	113	17	.	.	PUNCT
ejpam-6493	114	1	if	if	SCONJ
ejpam-6493	114	2	φ	φ	PROPN
ejpam-6493	114	3	:	:	PUNCT
ejpam-6493	114	4	r	r	NOUN
ejpam-6493	114	5	→	→	SYM
ejpam-6493	114	6	t	t	PROPN
ejpam-6493	114	7	is	be	AUX
ejpam-6493	114	8	a	a	DET
ejpam-6493	114	9	maximal	maximal	ADJ
ejpam-6493	114	10	homomorphism	homomorphism	NOUN
ejpam-6493	114	11	,	,	PUNCT
ejpam-6493	114	12	then	then	ADV
ejpam-6493	114	13	there	there	PRON
ejpam-6493	114	14	exists	exist	VERB
ejpam-6493	114	15	a	a	DET
ejpam-6493	114	16	subset	subset	NOUN
ejpam-6493	114	17	q	q	NOUN
ejpam-6493	114	18	of	of	ADP
ejpam-6493	114	19	r	r	NOUN
ejpam-6493	114	20	such	such	ADJ
ejpam-6493	114	21	that	that	PRON
ejpam-6493	114	22	ker(φ	ker(φ	NOUN
ejpam-6493	114	23	)	)	PUNCT
ejpam-6493	114	24	is	be	AUX
ejpam-6493	114	25	a	a	DET
ejpam-6493	114	26	q	q	NOUN
ejpam-6493	114	27	-	-	PUNCT
ejpam-6493	114	28	ideal	ideal	NOUN
ejpam-6493	114	29	of	of	ADP
ejpam-6493	114	30	r	r	NOUN
ejpam-6493	114	31	and	and	CCONJ
ejpam-6493	114	32	r	r	NOUN
ejpam-6493	114	33	/	/	SYM
ejpam-6493	114	34	ker(φ)(q	ker(φ)(q	NOUN
ejpam-6493	114	35	)	)	PUNCT
ejpam-6493	114	36	∼=	∼=	PROPN
ejpam-6493	114	37	t	t	NOUN
ejpam-6493	114	38	.	.	PUNCT
ejpam-6493	115	1	3.2	3.2	NUM
ejpam-6493	115	2	.	.	PUNCT
ejpam-6493	116	1	quotient	quotient	AUX
ejpam-6493	116	2	ternary	ternary	ADJ
ejpam-6493	116	3	semirings	semiring	NOUN
ejpam-6493	116	4	modulo	modulo	PROPN
ejpam-6493	116	5	congruences	congruence	VERB
ejpam-6493	116	6	definition	definition	NOUN
ejpam-6493	116	7	10	10	NUM
ejpam-6493	116	8	.	.	PUNCT
ejpam-6493	117	1	let	let	VERB
ejpam-6493	117	2	r	r	PRON
ejpam-6493	117	3	be	be	AUX
ejpam-6493	117	4	a	a	DET
ejpam-6493	117	5	ternary	ternary	ADJ
ejpam-6493	117	6	semiring	semiring	NOUN
ejpam-6493	117	7	.	.	PUNCT
ejpam-6493	118	1	an	an	DET
ejpam-6493	118	2	equivalence	equivalence	NOUN
ejpam-6493	118	3	relation	relation	NOUN
ejpam-6493	118	4	ρ	ρ	NOUN
ejpam-6493	118	5	on	on	ADP
ejpam-6493	118	6	r	r	NOUN
ejpam-6493	118	7	is	be	AUX
ejpam-6493	118	8	called	call	VERB
ejpam-6493	118	9	a	a	DET
ejpam-6493	118	10	congruence	congruence	NOUN
ejpam-6493	118	11	on	on	ADP
ejpam-6493	118	12	r	r	NOUN
ejpam-6493	118	13	if	if	SCONJ
ejpam-6493	118	14	,	,	PUNCT
ejpam-6493	118	15	for	for	ADP
ejpam-6493	118	16	all	all	DET
ejpam-6493	118	17	a1	a1	NOUN
ejpam-6493	118	18	,	,	PUNCT
ejpam-6493	118	19	a2	a2	PROPN
ejpam-6493	118	20	,	,	PUNCT
ejpam-6493	118	21	b1	b1	NOUN
ejpam-6493	118	22	,	,	PUNCT
ejpam-6493	118	23	b2	b2	NOUN
ejpam-6493	118	24	,	,	PUNCT
ejpam-6493	118	25	c1	c1	NOUN
ejpam-6493	118	26	,	,	PUNCT
ejpam-6493	118	27	c2	c2	PROPN
ejpam-6493	118	28	∈	∈	PROPN
ejpam-6493	118	29	r	r	PROPN
ejpam-6493	118	30	,	,	PUNCT
ejpam-6493	118	31	the	the	DET
ejpam-6493	118	32	following	follow	VERB
ejpam-6493	118	33	conditions	condition	NOUN
ejpam-6493	118	34	hold	hold	VERB
ejpam-6493	118	35	:	:	PUNCT
ejpam-6493	118	36	(	(	PUNCT
ejpam-6493	118	37	1	1	X
ejpam-6493	118	38	)	)	PUNCT
ejpam-6493	118	39	if	if	SCONJ
ejpam-6493	118	40	(	(	PUNCT
ejpam-6493	118	41	a1	a1	NOUN
ejpam-6493	118	42	,	,	PUNCT
ejpam-6493	118	43	a2	a2	NOUN
ejpam-6493	118	44	)	)	PUNCT
ejpam-6493	118	45	∈	∈	PROPN
ejpam-6493	118	46	ρ	ρ	PROPN
ejpam-6493	118	47	and	and	CCONJ
ejpam-6493	118	48	(	(	PUNCT
ejpam-6493	118	49	b1	b1	NOUN
ejpam-6493	118	50	,	,	PUNCT
ejpam-6493	118	51	b2	b2	NOUN
ejpam-6493	118	52	)	)	PUNCT
ejpam-6493	118	53	∈	∈	PROPN
ejpam-6493	118	54	ρ	ρ	PROPN
ejpam-6493	118	55	,	,	PUNCT
ejpam-6493	118	56	then	then	ADV
ejpam-6493	118	57	(	(	PUNCT
ejpam-6493	118	58	a1	a1	NOUN
ejpam-6493	118	59	+	+	CCONJ
ejpam-6493	118	60	b1	b1	NOUN
ejpam-6493	118	61	,	,	PUNCT
ejpam-6493	118	62	a2	a2	PROPN
ejpam-6493	118	63	+	+	CCONJ
ejpam-6493	118	64	b2	b2	NOUN
ejpam-6493	118	65	)	)	PUNCT
ejpam-6493	118	66	∈	∈	PROPN
ejpam-6493	118	67	ρ	ρ	PROPN
ejpam-6493	118	68	,	,	PUNCT
ejpam-6493	118	69	(	(	PUNCT
ejpam-6493	118	70	2	2	X
ejpam-6493	118	71	)	)	PUNCT
ejpam-6493	118	72	if	if	SCONJ
ejpam-6493	118	73	(	(	PUNCT
ejpam-6493	118	74	a1	a1	NOUN
ejpam-6493	118	75	,	,	PUNCT
ejpam-6493	118	76	a2	a2	PROPN
ejpam-6493	118	77	)	)	PUNCT
ejpam-6493	118	78	∈	∈	PROPN
ejpam-6493	118	79	ρ	ρ	PROPN
ejpam-6493	118	80	,	,	PUNCT
ejpam-6493	118	81	(	(	PUNCT
ejpam-6493	118	82	b1	b1	NOUN
ejpam-6493	118	83	,	,	PUNCT
ejpam-6493	118	84	b2	b2	NOUN
ejpam-6493	118	85	)	)	PUNCT
ejpam-6493	118	86	∈	∈	PROPN
ejpam-6493	118	87	ρ	ρ	PROPN
ejpam-6493	118	88	and	and	CCONJ
ejpam-6493	118	89	(	(	PUNCT
ejpam-6493	118	90	c1	c1	PROPN
ejpam-6493	118	91	,	,	PUNCT
ejpam-6493	118	92	c2	c2	PROPN
ejpam-6493	118	93	)	)	PUNCT
ejpam-6493	118	94	∈	∈	PROPN
ejpam-6493	118	95	ρ	ρ	PROPN
ejpam-6493	118	96	,	,	PUNCT
ejpam-6493	118	97	then	then	ADV
ejpam-6493	118	98	(	(	PUNCT
ejpam-6493	118	99	a1b1c1	a1b1c1	NOUN
ejpam-6493	118	100	,	,	PUNCT
ejpam-6493	118	101	a2b2c2	a2b2c2	NOUN
ejpam-6493	118	102	)	)	PUNCT
ejpam-6493	118	103	∈	∈	PROPN
ejpam-6493	118	104	ρ	ρ	NOUN
ejpam-6493	118	105	.	.	PUNCT
ejpam-6493	119	1	we	we	PRON
ejpam-6493	119	2	denote	denote	VERB
ejpam-6493	119	3	the	the	DET
ejpam-6493	119	4	congruence	congruence	ADJ
ejpam-6493	119	5	class	class	NOUN
ejpam-6493	119	6	of	of	ADP
ejpam-6493	119	7	a	a	DET
ejpam-6493	119	8	∈	∈	NOUN
ejpam-6493	119	9	r	r	NOUN
ejpam-6493	119	10	by	by	ADP
ejpam-6493	119	11	a	a	DET
ejpam-6493	119	12	+	+	NOUN
ejpam-6493	119	13	ρ	ρ	NOUN
ejpam-6493	119	14	and	and	CCONJ
ejpam-6493	119	15	let	let	VERB
ejpam-6493	119	16	r	r	VERB
ejpam-6493	119	17	/	/	SYM
ejpam-6493	119	18	ρ	ρ	NOUN
ejpam-6493	119	19	=	=	PUNCT
ejpam-6493	119	20	{	{	PUNCT
ejpam-6493	119	21	a	a	DET
ejpam-6493	119	22	+	+	NOUN
ejpam-6493	119	23	ρ	ρ	NOUN
ejpam-6493	119	24	|	|	NOUN
ejpam-6493	119	25	a	a	DET
ejpam-6493	119	26	∈	∈	NOUN
ejpam-6493	119	27	r	r	NOUN
ejpam-6493	119	28	}	}	PUNCT
ejpam-6493	119	29	.	.	PUNCT
ejpam-6493	120	1	furthermore	furthermore	ADV
ejpam-6493	120	2	,	,	PUNCT
ejpam-6493	120	3	define	define	VERB
ejpam-6493	120	4	a	a	DET
ejpam-6493	120	5	binary	binary	ADJ
ejpam-6493	120	6	addition	addition	NOUN
ejpam-6493	120	7	⊕	⊕	PROPN
ejpam-6493	120	8	and	and	CCONJ
ejpam-6493	120	9	ternary	ternary	ADJ
ejpam-6493	120	10	multiplication	multiplication	NOUN
ejpam-6493	120	11	on	on	ADP
ejpam-6493	120	12	r	r	PROPN
ejpam-6493	120	13	/	/	SYM
ejpam-6493	120	14	ρ	ρ	NOUN
ejpam-6493	120	15	,	,	PUNCT
ejpam-6493	120	16	for	for	ADP
ejpam-6493	120	17	all	all	DET
ejpam-6493	120	18	a+	a+	PRON
ejpam-6493	120	19	ρ	ρ	PROPN
ejpam-6493	120	20	,	,	PUNCT
ejpam-6493	120	21	b+	b+	X
ejpam-6493	120	22	ρ	ρ	NOUN
ejpam-6493	120	23	,	,	PUNCT
ejpam-6493	120	24	c+	c+	VERB
ejpam-6493	120	25	ρ	ρ	NUM
ejpam-6493	120	26	∈	∈	PROPN
ejpam-6493	120	27	r	r	PROPN
ejpam-6493	120	28	/	/	SYM
ejpam-6493	120	29	ρ	ρ	PROPN
ejpam-6493	120	30	,	,	PUNCT
ejpam-6493	120	31	by	by	ADP
ejpam-6493	120	32	(	(	PUNCT
ejpam-6493	120	33	a+	a+	PUNCT
ejpam-6493	120	34	ρ)⊕	ρ)⊕	PROPN
ejpam-6493	120	35	(	(	PUNCT
ejpam-6493	120	36	b+	b+	X
ejpam-6493	120	37	ρ	ρ	NOUN
ejpam-6493	120	38	)	)	PUNCT
ejpam-6493	120	39	=	=	SYM
ejpam-6493	120	40	(	(	PUNCT
ejpam-6493	120	41	a+	a+	NOUN
ejpam-6493	120	42	b	b	NOUN
ejpam-6493	120	43	)	)	PUNCT
ejpam-6493	120	44	+	+	CCONJ
ejpam-6493	120	45	ρ	ρ	PROPN
ejpam-6493	120	46	and	and	CCONJ
ejpam-6493	120	47	(	(	PUNCT
ejpam-6493	120	48	a+	a+	X
ejpam-6493	120	49	ρ)(b+	ρ)(b+	PROPN
ejpam-6493	120	50	ρ)(c+	ρ)(c+	PROPN
ejpam-6493	120	51	ρ	ρ	PROPN
ejpam-6493	120	52	)	)	PUNCT
ejpam-6493	120	53	=	=	SYM
ejpam-6493	120	54	abc+	abc+	PROPN
ejpam-6493	120	55	ρ	ρ	PROPN
ejpam-6493	120	56	.	.	PUNCT
ejpam-6493	121	1	then	then	ADV
ejpam-6493	121	2	r	r	PROPN
ejpam-6493	121	3	/	/	SYM
ejpam-6493	121	4	ρ	ρ	PROPN
ejpam-6493	121	5	is	be	AUX
ejpam-6493	121	6	a	a	DET
ejpam-6493	121	7	ternary	ternary	ADJ
ejpam-6493	121	8	semiring	semiring	NOUN
ejpam-6493	121	9	under	under	ADP
ejpam-6493	121	10	this	this	DET
ejpam-6493	121	11	binary	binary	ADJ
ejpam-6493	121	12	addition	addition	NOUN
ejpam-6493	121	13	and	and	CCONJ
ejpam-6493	121	14	ternary	ternary	ADJ
ejpam-6493	121	15	multiplication	multiplication	NOUN
ejpam-6493	121	16	.	.	PUNCT
ejpam-6493	122	1	proposition	proposition	NOUN
ejpam-6493	122	2	1	1	NUM
ejpam-6493	122	3	.	.	PUNCT
ejpam-6493	123	1	let	let	VERB
ejpam-6493	123	2	r	r	NOUN
ejpam-6493	123	3	and	and	CCONJ
ejpam-6493	123	4	t	t	PROPN
ejpam-6493	123	5	be	be	AUX
ejpam-6493	123	6	ternary	ternary	ADJ
ejpam-6493	123	7	semirings	semiring	NOUN
ejpam-6493	123	8	and	and	CCONJ
ejpam-6493	123	9	φ	φ	NOUN
ejpam-6493	123	10	:	:	PUNCT
ejpam-6493	124	1	r→	r→	PROPN
ejpam-6493	124	2	t	t	PROPN
ejpam-6493	124	3	be	be	AUX
ejpam-6493	124	4	a	a	DET
ejpam-6493	124	5	ternary	ternary	ADJ
ejpam-6493	124	6	semiring	semire	VERB
ejpam-6493	124	7	homomorphism	homomorphism	NOUN
ejpam-6493	124	8	.	.	PUNCT
ejpam-6493	125	1	define	define	VERB
ejpam-6493	125	2	a	a	DET
ejpam-6493	125	3	relation	relation	NOUN
ejpam-6493	125	4	k(φ	k(φ	PROPN
ejpam-6493	125	5	)	)	PUNCT
ejpam-6493	125	6	on	on	ADP
ejpam-6493	125	7	r	r	NOUN
ejpam-6493	125	8	by	by	ADP
ejpam-6493	125	9	k(φ	k(φ	PROPN
ejpam-6493	125	10	)	)	PUNCT
ejpam-6493	126	1	=	=	PRON
ejpam-6493	126	2	{	{	PUNCT
ejpam-6493	126	3	(	(	PUNCT
ejpam-6493	126	4	a	a	PRON
ejpam-6493	126	5	,	,	PUNCT
ejpam-6493	126	6	b	b	NOUN
ejpam-6493	126	7	)	)	PUNCT
ejpam-6493	126	8	∈	∈	PROPN
ejpam-6493	126	9	r×r	r×r	PROPN
ejpam-6493	126	10	|	|	ADV
ejpam-6493	126	11	φ(a	φ(a	ADJ
ejpam-6493	126	12	)	)	PUNCT
ejpam-6493	126	13	=	=	SYM
ejpam-6493	126	14	φ(b	φ(b	NOUN
ejpam-6493	126	15	)	)	PUNCT
ejpam-6493	126	16	}	}	PUNCT
ejpam-6493	126	17	.	.	PUNCT
ejpam-6493	127	1	then	then	ADV
ejpam-6493	127	2	k(φ	k(φ	PROPN
ejpam-6493	127	3	)	)	PUNCT
ejpam-6493	127	4	is	be	AUX
ejpam-6493	127	5	a	a	DET
ejpam-6493	127	6	congruence	congruence	NOUN
ejpam-6493	127	7	on	on	ADP
ejpam-6493	127	8	r.	r.	PROPN
ejpam-6493	127	9	m.	m.	PROPN
ejpam-6493	127	10	petapirak	petapirak	PROPN
ejpam-6493	127	11	,	,	PUNCT
ejpam-6493	127	12	a.	a.	PROPN
ejpam-6493	127	13	j.	j.	PROPN
ejpam-6493	127	14	khan	khan	PROPN
ejpam-6493	127	15	,	,	PUNCT
ejpam-6493	127	16	r.	r.	PROPN
ejpam-6493	127	17	chinram	chinram	PROPN
ejpam-6493	127	18	/	/	SYM
ejpam-6493	127	19	eur	eur	PROPN
ejpam-6493	127	20	.	.	PUNCT
ejpam-6493	128	1	j.	j.	PROPN
ejpam-6493	128	2	pure	pure	PROPN
ejpam-6493	128	3	appl	appl	PROPN
ejpam-6493	128	4	.	.	PROPN
ejpam-6493	128	5	math	math	PROPN
ejpam-6493	128	6	,	,	PUNCT
ejpam-6493	128	7	18	18	NUM
ejpam-6493	128	8	(	(	PUNCT
ejpam-6493	128	9	3	3	NUM
ejpam-6493	128	10	)	)	PUNCT
ejpam-6493	128	11	(	(	PUNCT
ejpam-6493	128	12	2025	2025	NUM
ejpam-6493	128	13	)	)	PUNCT
ejpam-6493	128	14	,	,	PUNCT
ejpam-6493	128	15	6493	6493	NUM
ejpam-6493	128	16	5	5	NUM
ejpam-6493	128	17	of	of	ADP
ejpam-6493	128	18	9	9	NUM
ejpam-6493	128	19	proof	proof	NOUN
ejpam-6493	128	20	.	.	PUNCT
ejpam-6493	129	1	straightforward	straightforward	ADJ
ejpam-6493	129	2	.	.	PUNCT
ejpam-6493	130	1	example	example	NOUN
ejpam-6493	131	1	3	3	X
ejpam-6493	131	2	.	.	PUNCT
ejpam-6493	132	1	let	let	AUX
ejpam-6493	132	2	r	r	NOUN
ejpam-6493	132	3	=	=	SYM
ejpam-6493	132	4	z−	z−	NOUN
ejpam-6493	132	5	be	be	AUX
ejpam-6493	132	6	a	a	DET
ejpam-6493	132	7	ternary	ternary	ADJ
ejpam-6493	132	8	semiring	semiring	NOUN
ejpam-6493	132	9	under	under	ADP
ejpam-6493	132	10	the	the	DET
ejpam-6493	132	11	usual	usual	ADJ
ejpam-6493	132	12	addition	addition	NOUN
ejpam-6493	132	13	and	and	CCONJ
ejpam-6493	132	14	ternary	ternary	ADJ
ejpam-6493	132	15	multiplication	multiplication	NOUN
ejpam-6493	132	16	of	of	ADP
ejpam-6493	132	17	integers	integer	NOUN
ejpam-6493	132	18	,	,	PUNCT
ejpam-6493	132	19	and	and	CCONJ
ejpam-6493	132	20	t	t	X
ejpam-6493	132	21	=	=	PUNCT
ejpam-6493	132	22	z4	z4	NOUN
ejpam-6493	132	23	be	be	AUX
ejpam-6493	132	24	a	a	DET
ejpam-6493	132	25	ternary	ternary	ADJ
ejpam-6493	132	26	semiring	semiring	NOUN
ejpam-6493	132	27	under	under	ADP
ejpam-6493	132	28	the	the	DET
ejpam-6493	132	29	usual	usual	ADJ
ejpam-6493	132	30	addition	addition	NOUN
ejpam-6493	132	31	and	and	CCONJ
ejpam-6493	132	32	ternary	ternary	ADJ
ejpam-6493	132	33	multiplication	multiplication	NOUN
ejpam-6493	132	34	of	of	ADP
ejpam-6493	132	35	integers	integer	NOUN
ejpam-6493	132	36	modulo	modulo	VERB
ejpam-6493	132	37	4	4	NUM
ejpam-6493	132	38	.	.	PUNCT
ejpam-6493	132	39	define	define	VERB
ejpam-6493	132	40	a	a	DET
ejpam-6493	132	41	function	function	NOUN
ejpam-6493	132	42	φ	φ	NOUN
ejpam-6493	132	43	:	:	PUNCT
ejpam-6493	132	44	r→	r→	PROPN
ejpam-6493	132	45	t	t	PROPN
ejpam-6493	132	46	by	by	ADP
ejpam-6493	132	47	φ(a	φ(a	ADJ
ejpam-6493	132	48	)	)	PUNCT
ejpam-6493	132	49	=	=	PUNCT
ejpam-6493	133	1	a	a	PRON
ejpam-6493	133	2	for	for	ADP
ejpam-6493	133	3	all	all	DET
ejpam-6493	133	4	a	a	DET
ejpam-6493	133	5	∈	∈	PROPN
ejpam-6493	133	6	r.	r.	NOUN
ejpam-6493	133	7	then	then	ADV
ejpam-6493	133	8	φ	φ	PROPN
ejpam-6493	133	9	is	be	AUX
ejpam-6493	133	10	a	a	DET
ejpam-6493	133	11	homomorphism	homomorphism	NOUN
ejpam-6493	133	12	.	.	PUNCT
ejpam-6493	134	1	by	by	ADP
ejpam-6493	134	2	proposition	proposition	NOUN
ejpam-6493	134	3	1	1	NUM
ejpam-6493	134	4	,	,	PUNCT
ejpam-6493	134	5	k(φ	k(φ	PROPN
ejpam-6493	134	6	)	)	PUNCT
ejpam-6493	134	7	is	be	AUX
ejpam-6493	134	8	a	a	DET
ejpam-6493	134	9	congruence	congruence	NOUN
ejpam-6493	134	10	on	on	ADP
ejpam-6493	134	11	r.	r.	PROPN
ejpam-6493	134	12	we	we	PRON
ejpam-6493	134	13	have	have	VERB
ejpam-6493	134	14	that	that	PRON
ejpam-6493	134	15	r	r	PROPN
ejpam-6493	134	16	/	/	SYM
ejpam-6493	134	17	k(φ	k(φ	PROPN
ejpam-6493	134	18	)	)	PUNCT
ejpam-6493	135	1	=	=	PRON
ejpam-6493	135	2	{	{	PUNCT
ejpam-6493	135	3	−1	−1	NOUN
ejpam-6493	135	4	+	+	NOUN
ejpam-6493	135	5	k(φ),−2	k(φ),−2	X
ejpam-6493	135	6	+	+	NOUN
ejpam-6493	135	7	k(φ),−3	k(φ),−3	X
ejpam-6493	135	8	+	+	NOUN
ejpam-6493	135	9	k(φ),−4	k(φ),−4	NOUN
ejpam-6493	135	10	+	+	ADJ
ejpam-6493	135	11	k(φ	k(φ	X
ejpam-6493	135	12	)	)	PUNCT
ejpam-6493	135	13	}	}	PUNCT
ejpam-6493	135	14	where	where	SCONJ
ejpam-6493	135	15	−1	−1	NOUN
ejpam-6493	135	16	+	+	NOUN
ejpam-6493	135	17	k(φ	k(φ	X
ejpam-6493	135	18	)	)	PUNCT
ejpam-6493	136	1	=	=	PRON
ejpam-6493	136	2	{	{	PUNCT
ejpam-6493	136	3	−1,−5,−9,−13	−1,−5,−9,−13	X
ejpam-6493	136	4	,	,	PUNCT
ejpam-6493	136	5	.	.	PUNCT
ejpam-6493	136	6	.	.	PUNCT
ejpam-6493	137	1	.	.	PUNCT
ejpam-6493	138	1	}	}	PUNCT
ejpam-6493	138	2	,	,	PUNCT
ejpam-6493	138	3	−2	−2	PROPN
ejpam-6493	138	4	+	+	ADJ
ejpam-6493	138	5	k(φ	k(φ	X
ejpam-6493	138	6	)	)	PUNCT
ejpam-6493	139	1	=	=	PRON
ejpam-6493	139	2	{	{	PUNCT
ejpam-6493	139	3	−2,−6,−10,−14	−2,−6,−10,−14	PROPN
ejpam-6493	139	4	,	,	PUNCT
ejpam-6493	139	5	.	.	PUNCT
ejpam-6493	139	6	.	.	PUNCT
ejpam-6493	140	1	.	.	PUNCT
ejpam-6493	141	1	}	}	PUNCT
ejpam-6493	141	2	,	,	PUNCT
ejpam-6493	142	1	−3	−3	PROPN
ejpam-6493	143	1	+	+	PROPN
ejpam-6493	143	2	k(φ	k(φ	X
ejpam-6493	143	3	)	)	PUNCT
ejpam-6493	144	1	=	=	PRON
ejpam-6493	144	2	{	{	PUNCT
ejpam-6493	144	3	−3,−7,−11,−15	−3,−7,−11,−15	PROPN
ejpam-6493	144	4	,	,	PUNCT
ejpam-6493	144	5	.	.	PUNCT
ejpam-6493	144	6	.	.	PUNCT
ejpam-6493	145	1	.	.	PUNCT
ejpam-6493	145	2	}	}	PUNCT
ejpam-6493	145	3	,	,	PUNCT
ejpam-6493	145	4	−4	−4	X
ejpam-6493	146	1	+	+	VERB
ejpam-6493	147	1	k(φ	k(φ	X
ejpam-6493	147	2	)	)	PUNCT
ejpam-6493	148	1	=	=	PRON
ejpam-6493	148	2	{	{	PUNCT
ejpam-6493	148	3	−4,−8,−12,−16	−4,−8,−12,−16	PROPN
ejpam-6493	148	4	,	,	PUNCT
ejpam-6493	148	5	.	.	PUNCT
ejpam-6493	148	6	.	.	PUNCT
ejpam-6493	149	1	.	.	PUNCT
ejpam-6493	149	2	}	}	PUNCT
ejpam-6493	149	3	.	.	PUNCT
ejpam-6493	150	1	3.3	3.3	NUM
ejpam-6493	150	2	.	.	PUNCT
ejpam-6493	151	1	quotient	quotient	VERB
ejpam-6493	151	2	ternary	ternary	ADJ
ejpam-6493	151	3	semirings	semiring	NOUN
ejpam-6493	151	4	modulo	modulo	PROPN
ejpam-6493	151	5	ideals	ideal	NOUN
ejpam-6493	151	6	let	let	VERB
ejpam-6493	151	7	i	i	PRON
ejpam-6493	151	8	be	be	AUX
ejpam-6493	151	9	an	an	DET
ejpam-6493	151	10	ideal	ideal	NOUN
ejpam-6493	151	11	of	of	ADP
ejpam-6493	151	12	a	a	DET
ejpam-6493	151	13	ternary	ternary	ADJ
ejpam-6493	151	14	semiring	semire	VERB
ejpam-6493	151	15	r.	r.	NOUN
ejpam-6493	151	16	we	we	PRON
ejpam-6493	151	17	define	define	VERB
ejpam-6493	151	18	a	a	DET
ejpam-6493	151	19	relation	relation	NOUN
ejpam-6493	151	20	ρi	ρi	NOUN
ejpam-6493	151	21	on	on	ADP
ejpam-6493	151	22	r	r	NOUN
ejpam-6493	151	23	as	as	ADP
ejpam-6493	151	24	ρi	ρi	NUM
ejpam-6493	151	25	=	=	PUNCT
ejpam-6493	152	1	{	{	PUNCT
ejpam-6493	152	2	(	(	PUNCT
ejpam-6493	152	3	x	x	NOUN
ejpam-6493	152	4	,	,	PUNCT
ejpam-6493	152	5	y	y	NOUN
ejpam-6493	152	6	)	)	PUNCT
ejpam-6493	152	7	∈	∈	PROPN
ejpam-6493	152	8	r×r	r×r	NOUN
ejpam-6493	153	1	|	|	ADV
ejpam-6493	153	2	x+	x+	PUNCT
ejpam-6493	153	3	a	a	DET
ejpam-6493	153	4	=	=	SYM
ejpam-6493	153	5	y	y	PROPN
ejpam-6493	153	6	+	+	CCONJ
ejpam-6493	153	7	b	b	NOUN
ejpam-6493	153	8	for	for	ADP
ejpam-6493	153	9	some	some	DET
ejpam-6493	153	10	a	a	PRON
ejpam-6493	153	11	,	,	PUNCT
ejpam-6493	153	12	b	b	X
ejpam-6493	153	13	∈	∈	PROPN
ejpam-6493	153	14	i	i	X
ejpam-6493	153	15	}	}	PUNCT
ejpam-6493	153	16	.	.	PUNCT
ejpam-6493	154	1	this	this	PRON
ejpam-6493	154	2	means	mean	VERB
ejpam-6493	154	3	that	that	SCONJ
ejpam-6493	154	4	(	(	PUNCT
ejpam-6493	154	5	x	x	X
ejpam-6493	154	6	,	,	PUNCT
ejpam-6493	154	7	y	y	NOUN
ejpam-6493	154	8	)	)	PUNCT
ejpam-6493	154	9	∈	∈	PROPN
ejpam-6493	154	10	ρi	ρi	NOUN
ejpam-6493	154	11	if	if	SCONJ
ejpam-6493	154	12	and	and	CCONJ
ejpam-6493	154	13	only	only	ADV
ejpam-6493	154	14	if	if	SCONJ
ejpam-6493	154	15	there	there	PRON
ejpam-6493	154	16	exist	exist	VERB
ejpam-6493	154	17	a1	a1	NOUN
ejpam-6493	154	18	,	,	PUNCT
ejpam-6493	154	19	a2	a2	PROPN
ejpam-6493	154	20	∈	∈	PROPN
ejpam-6493	154	21	i	i	PRON
ejpam-6493	154	22	satisfying	satisfy	VERB
ejpam-6493	154	23	x+	x+	ADJ
ejpam-6493	154	24	a1	a1	NOUN
ejpam-6493	154	25	=	=	SYM
ejpam-6493	154	26	y	y	PROPN
ejpam-6493	154	27	+	+	PROPN
ejpam-6493	154	28	a2	a2	PROPN
ejpam-6493	154	29	.	.	PUNCT
ejpam-6493	155	1	then	then	ADV
ejpam-6493	155	2	ρi	ρi	PROPN
ejpam-6493	155	3	is	be	AUX
ejpam-6493	155	4	a	a	DET
ejpam-6493	155	5	congruence	congruence	NOUN
ejpam-6493	155	6	relation	relation	NOUN
ejpam-6493	155	7	on	on	ADP
ejpam-6493	155	8	r	r	NOUN
ejpam-6493	155	9	,	,	PUNCT
ejpam-6493	155	10	and	and	CCONJ
ejpam-6493	155	11	we	we	PRON
ejpam-6493	155	12	denote	denote	VERB
ejpam-6493	155	13	the	the	DET
ejpam-6493	155	14	congruence	congruence	ADJ
ejpam-6493	155	15	class	class	NOUN
ejpam-6493	155	16	of	of	ADP
ejpam-6493	155	17	x	x	PUNCT
ejpam-6493	155	18	by	by	ADP
ejpam-6493	155	19	a	a	DET
ejpam-6493	155	20	coset	coset	NOUN
ejpam-6493	156	1	x+	x+	X
ejpam-6493	156	2	i.	i.	NOUN
ejpam-6493	156	3	the	the	DET
ejpam-6493	156	4	collection	collection	NOUN
ejpam-6493	156	5	of	of	ADP
ejpam-6493	156	6	all	all	DET
ejpam-6493	156	7	congruence	congruence	NOUN
ejpam-6493	156	8	classes	class	NOUN
ejpam-6493	156	9	is	be	AUX
ejpam-6493	156	10	denoted	denote	VERB
ejpam-6493	156	11	by	by	ADP
ejpam-6493	156	12	r	r	NOUN
ejpam-6493	156	13	/	/	SYM
ejpam-6493	156	14	i.	i.	NOUN
ejpam-6493	156	15	example	example	NOUN
ejpam-6493	156	16	4	4	X
ejpam-6493	156	17	.	.	PUNCT
ejpam-6493	157	1	we	we	PRON
ejpam-6493	157	2	consider	consider	VERB
ejpam-6493	157	3	a	a	DET
ejpam-6493	157	4	ternary	ternary	ADJ
ejpam-6493	157	5	semiring	semire	VERB
ejpam-6493	157	6	z−	z−	PROPN
ejpam-6493	157	7	0	0	PUNCT
ejpam-6493	158	1	under	under	ADP
ejpam-6493	158	2	the	the	DET
ejpam-6493	158	3	usual	usual	ADJ
ejpam-6493	158	4	addition	addition	NOUN
ejpam-6493	158	5	and	and	CCONJ
ejpam-6493	158	6	ternary	ternary	ADJ
ejpam-6493	158	7	multiplication	multiplication	NOUN
ejpam-6493	158	8	of	of	ADP
ejpam-6493	158	9	integers	integer	NOUN
ejpam-6493	158	10	.	.	PUNCT
ejpam-6493	159	1	(	(	PUNCT
ejpam-6493	159	2	1	1	X
ejpam-6493	159	3	)	)	PUNCT
ejpam-6493	159	4	let	let	VERB
ejpam-6493	159	5	i	i	PRON
ejpam-6493	159	6	=	=	PUNCT
ejpam-6493	160	1	5z−	5z−	NUM
ejpam-6493	160	2	0	0	NUM
ejpam-6493	160	3	=	=	SYM
ejpam-6493	160	4	{	{	PUNCT
ejpam-6493	160	5	0,−5,−10,−15	0,−5,−10,−15	PROPN
ejpam-6493	160	6	,	,	PUNCT
ejpam-6493	160	7	.	.	PUNCT
ejpam-6493	160	8	.	.	PUNCT
ejpam-6493	160	9	.	.	PUNCT
ejpam-6493	160	10	}	}	PUNCT
ejpam-6493	160	11	.	.	PUNCT
ejpam-6493	161	1	it	it	PRON
ejpam-6493	161	2	is	be	AUX
ejpam-6493	161	3	easy	easy	ADJ
ejpam-6493	161	4	to	to	PART
ejpam-6493	161	5	show	show	VERB
ejpam-6493	161	6	that	that	SCONJ
ejpam-6493	161	7	i	i	PRON
ejpam-6493	161	8	is	be	AUX
ejpam-6493	161	9	a	a	DET
ejpam-6493	161	10	k	k	NOUN
ejpam-6493	161	11	-	-	NOUN
ejpam-6493	161	12	ideal	ideal	NOUN
ejpam-6493	161	13	of	of	ADP
ejpam-6493	161	14	z−	z−	PROPN
ejpam-6493	161	15	0	0	NUM
ejpam-6493	161	16	.	.	PUNCT
ejpam-6493	162	1	then	then	ADV
ejpam-6493	162	2	z−	z−	PROPN
ejpam-6493	162	3	0	0	PUNCT
ejpam-6493	162	4	/i	/i	PUNCT
ejpam-6493	162	5	=	=	X
ejpam-6493	162	6	{	{	PUNCT
ejpam-6493	162	7	0	0	NUM
ejpam-6493	162	8	+	+	CCONJ
ejpam-6493	162	9	i,−1	i,−1	PROPN
ejpam-6493	162	10	+	+	CCONJ
ejpam-6493	162	11	i,−2	i,−2	VERB
ejpam-6493	162	12	+	+	CCONJ
ejpam-6493	162	13	i,−3	i,−3	PROPN
ejpam-6493	163	1	+	+	X
ejpam-6493	163	2	i,−4	i,−4	ADV
ejpam-6493	163	3	+	+	CCONJ
ejpam-6493	163	4	i	i	X
ejpam-6493	163	5	}	}	PUNCT
ejpam-6493	163	6	where	where	SCONJ
ejpam-6493	163	7	0	0	PUNCT
ejpam-6493	163	8	+	+	CCONJ
ejpam-6493	163	9	i	i	PRON
ejpam-6493	163	10	=	=	PUNCT
ejpam-6493	163	11	{	{	PUNCT
ejpam-6493	163	12	0,−5,−10,−15	0,−5,−10,−15	PROPN
ejpam-6493	163	13	,	,	PUNCT
ejpam-6493	163	14	.	.	PUNCT
ejpam-6493	163	15	.	.	PUNCT
ejpam-6493	164	1	.	.	PUNCT
ejpam-6493	165	1	}	}	PUNCT
ejpam-6493	165	2	,	,	PUNCT
ejpam-6493	165	3	−1	−1	NOUN
ejpam-6493	166	1	+	+	CCONJ
ejpam-6493	166	2	i	i	PRON
ejpam-6493	166	3	=	=	SYM
ejpam-6493	166	4	{	{	PUNCT
ejpam-6493	166	5	−1,−6,−11,−16	−1,−6,−11,−16	PROPN
ejpam-6493	166	6	,	,	PUNCT
ejpam-6493	166	7	.	.	PUNCT
ejpam-6493	166	8	.	.	PUNCT
ejpam-6493	166	9	.	.	PUNCT
ejpam-6493	167	1	}	}	PUNCT
ejpam-6493	167	2	,	,	PUNCT
ejpam-6493	167	3	−2	−2	NOUN
ejpam-6493	168	1	+	+	CCONJ
ejpam-6493	168	2	i	i	NOUN
ejpam-6493	168	3	=	=	PUNCT
ejpam-6493	168	4	{	{	PUNCT
ejpam-6493	168	5	−2,−7,−12,−17	−2,−7,−12,−17	NOUN
ejpam-6493	168	6	,	,	PUNCT
ejpam-6493	168	7	.	.	PUNCT
ejpam-6493	168	8	.	.	PUNCT
ejpam-6493	168	9	.	.	PUNCT
ejpam-6493	169	1	}	}	PUNCT
ejpam-6493	169	2	,	,	PUNCT
ejpam-6493	169	3	−3	−3	PROPN
ejpam-6493	170	1	+	+	CCONJ
ejpam-6493	170	2	i	i	NOUN
ejpam-6493	170	3	=	=	SYM
ejpam-6493	170	4	{	{	PUNCT
ejpam-6493	170	5	−3,−8,−13,−18	−3,−8,−13,−18	NOUN
ejpam-6493	170	6	,	,	PUNCT
ejpam-6493	170	7	.	.	PUNCT
ejpam-6493	170	8	.	.	PUNCT
ejpam-6493	171	1	.	.	PUNCT
ejpam-6493	171	2	}	}	PUNCT
ejpam-6493	171	3	,	,	PUNCT
ejpam-6493	171	4	−4	−4	X
ejpam-6493	172	1	+	+	CCONJ
ejpam-6493	172	2	i	i	NOUN
ejpam-6493	172	3	=	=	SYM
ejpam-6493	172	4	{	{	PUNCT
ejpam-6493	172	5	−4,−9,−14,−19	−4,−9,−14,−19	PROPN
ejpam-6493	172	6	,	,	PUNCT
ejpam-6493	172	7	.	.	PUNCT
ejpam-6493	172	8	.	.	PUNCT
ejpam-6493	172	9	.	.	PUNCT
ejpam-6493	172	10	}	}	PUNCT
ejpam-6493	172	11	.	.	PUNCT
ejpam-6493	173	1	this	this	DET
ejpam-6493	173	2	quotient	quotient	NOUN
ejpam-6493	173	3	ternary	ternary	ADJ
ejpam-6493	173	4	semiring	semiring	NOUN
ejpam-6493	173	5	obtained	obtain	VERB
ejpam-6493	173	6	here	here	ADV
ejpam-6493	173	7	is	be	AUX
ejpam-6493	173	8	similar	similar	ADJ
ejpam-6493	173	9	to	to	ADP
ejpam-6493	173	10	z−	z−	PROPN
ejpam-6493	173	11	0	0	PUNCT
ejpam-6493	174	1	/i(q	/i(q	PROPN
ejpam-6493	174	2	)	)	PUNCT
ejpam-6493	174	3	considered	consider	VERB
ejpam-6493	174	4	in	in	ADP
ejpam-6493	174	5	example	example	NOUN
ejpam-6493	174	6	1	1	X
ejpam-6493	174	7	.	.	X
ejpam-6493	174	8	note	note	VERB
ejpam-6493	174	9	that	that	SCONJ
ejpam-6493	174	10	every	every	DET
ejpam-6493	174	11	k	k	NOUN
ejpam-6493	174	12	-	-	NOUN
ejpam-6493	174	13	ideal	ideal	NOUN
ejpam-6493	174	14	of	of	ADP
ejpam-6493	174	15	a	a	DET
ejpam-6493	174	16	ternary	ternary	ADJ
ejpam-6493	174	17	semiring	semiring	NOUN
ejpam-6493	174	18	r	r	NOUN
ejpam-6493	174	19	is	be	AUX
ejpam-6493	174	20	a	a	DET
ejpam-6493	174	21	q	q	NOUN
ejpam-6493	174	22	-	-	PUNCT
ejpam-6493	174	23	ideal	ideal	NOUN
ejpam-6493	174	24	of	of	ADP
ejpam-6493	174	25	r.	r.	PROPN
ejpam-6493	174	26	m.	m.	PROPN
ejpam-6493	174	27	petapirak	petapirak	PROPN
ejpam-6493	174	28	,	,	PUNCT
ejpam-6493	174	29	a.	a.	PROPN
ejpam-6493	174	30	j.	j.	PROPN
ejpam-6493	174	31	khan	khan	PROPN
ejpam-6493	174	32	,	,	PUNCT
ejpam-6493	174	33	r.	r.	PROPN
ejpam-6493	174	34	chinram	chinram	PROPN
ejpam-6493	174	35	/	/	SYM
ejpam-6493	174	36	eur	eur	PROPN
ejpam-6493	174	37	.	.	PUNCT
ejpam-6493	175	1	j.	j.	PROPN
ejpam-6493	175	2	pure	pure	PROPN
ejpam-6493	175	3	appl	appl	PROPN
ejpam-6493	175	4	.	.	PROPN
ejpam-6493	175	5	math	math	PROPN
ejpam-6493	175	6	,	,	PUNCT
ejpam-6493	175	7	18	18	NUM
ejpam-6493	175	8	(	(	PUNCT
ejpam-6493	175	9	3	3	NUM
ejpam-6493	175	10	)	)	PUNCT
ejpam-6493	175	11	(	(	PUNCT
ejpam-6493	175	12	2025	2025	NUM
ejpam-6493	175	13	)	)	PUNCT
ejpam-6493	175	14	,	,	PUNCT
ejpam-6493	175	15	6493	6493	NUM
ejpam-6493	175	16	6	6	NUM
ejpam-6493	175	17	of	of	ADP
ejpam-6493	175	18	9	9	NUM
ejpam-6493	175	19	(	(	PUNCT
ejpam-6493	175	20	2	2	NUM
ejpam-6493	175	21	)	)	PUNCT
ejpam-6493	175	22	let	let	VERB
ejpam-6493	175	23	j	j	PROPN
ejpam-6493	175	24	=	=	SYM
ejpam-6493	175	25	2z−	2z−	PROPN
ejpam-6493	175	26	0	0	NUM
ejpam-6493	175	27	∖	∖	X
ejpam-6493	175	28	{	{	PUNCT
ejpam-6493	175	29	−2	−2	NOUN
ejpam-6493	175	30	}	}	PUNCT
ejpam-6493	175	31	=	=	SYM
ejpam-6493	175	32	{	{	PUNCT
ejpam-6493	175	33	0,−4,−6,−8	0,−4,−6,−8	ADJ
ejpam-6493	175	34	,	,	PUNCT
ejpam-6493	175	35	.	.	PUNCT
ejpam-6493	175	36	.	.	PUNCT
ejpam-6493	175	37	.	.	PUNCT
ejpam-6493	175	38	}	}	PUNCT
ejpam-6493	175	39	.	.	PUNCT
ejpam-6493	176	1	it	it	PRON
ejpam-6493	176	2	is	be	AUX
ejpam-6493	176	3	clear	clear	ADJ
ejpam-6493	176	4	that	that	SCONJ
ejpam-6493	176	5	j	j	PROPN
ejpam-6493	176	6	is	be	AUX
ejpam-6493	176	7	an	an	DET
ejpam-6493	176	8	ideal	ideal	NOUN
ejpam-6493	176	9	of	of	ADP
ejpam-6493	176	10	z−	z−	PROPN
ejpam-6493	176	11	0	0	PUNCT
ejpam-6493	177	1	but	but	CCONJ
ejpam-6493	177	2	not	not	PART
ejpam-6493	177	3	a	a	DET
ejpam-6493	177	4	k	k	NOUN
ejpam-6493	177	5	-	-	NOUN
ejpam-6493	177	6	ideal	ideal	ADJ
ejpam-6493	177	7	.	.	PUNCT
ejpam-6493	178	1	we	we	PRON
ejpam-6493	178	2	have	have	VERB
ejpam-6493	178	3	that	that	DET
ejpam-6493	178	4	z−	z−	PROPN
ejpam-6493	178	5	0	0	NUM
ejpam-6493	179	1	/j	/j	PUNCT
ejpam-6493	180	1	=	=	PUNCT
ejpam-6493	180	2	{	{	PUNCT
ejpam-6493	180	3	0	0	NUM
ejpam-6493	181	1	+	+	NUM
ejpam-6493	181	2	j,−1	j,−1	PROPN
ejpam-6493	181	3	+	+	CCONJ
ejpam-6493	181	4	j	j	NOUN
ejpam-6493	181	5	}	}	PUNCT
ejpam-6493	181	6	where	where	SCONJ
ejpam-6493	181	7	0	0	NUM
ejpam-6493	182	1	+	+	NUM
ejpam-6493	182	2	j	j	NOUN
ejpam-6493	182	3	=	=	SYM
ejpam-6493	182	4	{	{	PUNCT
ejpam-6493	182	5	0,−2,−4,−6	0,−2,−4,−6	NOUN
ejpam-6493	182	6	,	,	PUNCT
ejpam-6493	182	7	.	.	PUNCT
ejpam-6493	182	8	.	.	PUNCT
ejpam-6493	182	9	.	.	PUNCT
ejpam-6493	182	10	}	}	PUNCT
ejpam-6493	182	11	,	,	PUNCT
ejpam-6493	182	12	−1	−1	NOUN
ejpam-6493	182	13	+	+	CCONJ
ejpam-6493	182	14	j	j	NOUN
ejpam-6493	182	15	=	=	PUNCT
ejpam-6493	182	16	{	{	PUNCT
ejpam-6493	182	17	−1,−3,−5,−7	−1,−3,−5,−7	PROPN
ejpam-6493	182	18	,	,	PUNCT
ejpam-6493	182	19	.	.	PUNCT
ejpam-6493	182	20	.	.	PUNCT
ejpam-6493	182	21	.	.	PUNCT
ejpam-6493	182	22	}	}	PUNCT
ejpam-6493	182	23	.	.	PUNCT
ejpam-6493	183	1	it	it	PRON
ejpam-6493	183	2	is	be	AUX
ejpam-6493	183	3	also	also	ADV
ejpam-6493	183	4	worth	worth	ADJ
ejpam-6493	183	5	noting	note	VERB
ejpam-6493	183	6	that	that	SCONJ
ejpam-6493	183	7	,	,	PUNCT
ejpam-6493	183	8	as	as	ADP
ejpam-6493	183	9	in	in	ADP
ejpam-6493	183	10	example	example	NOUN
ejpam-6493	183	11	1	1	NUM
ejpam-6493	183	12	,	,	PUNCT
ejpam-6493	183	13	considering	consider	VERB
ejpam-6493	183	14	a	a	DET
ejpam-6493	183	15	quotient	quotient	NOUN
ejpam-6493	183	16	ternary	ternary	ADJ
ejpam-6493	183	17	semiring	semire	VERB
ejpam-6493	183	18	z−	z−	PROPN
ejpam-6493	183	19	0	0	PUNCT
ejpam-6493	184	1	/j(q	/j(q	X
ejpam-6493	184	2	)	)	PUNCT
ejpam-6493	184	3	is	be	AUX
ejpam-6493	184	4	not	not	PART
ejpam-6493	184	5	straightforward	straightforward	ADJ
ejpam-6493	184	6	.	.	PUNCT
ejpam-6493	185	1	4	4	X
ejpam-6493	185	2	.	.	X
ejpam-6493	185	3	main	main	ADJ
ejpam-6493	185	4	results	result	NOUN
ejpam-6493	185	5	in	in	ADP
ejpam-6493	185	6	this	this	DET
ejpam-6493	185	7	section	section	NOUN
ejpam-6493	185	8	,	,	PUNCT
ejpam-6493	185	9	we	we	PRON
ejpam-6493	185	10	will	will	AUX
ejpam-6493	185	11	consider	consider	VERB
ejpam-6493	185	12	quotient	quotient	NOUN
ejpam-6493	185	13	ternary	ternary	ADJ
ejpam-6493	185	14	semirings	semiring	NOUN
ejpam-6493	185	15	via	via	ADP
ejpam-6493	185	16	congruences	congruence	NOUN
ejpam-6493	185	17	and	and	CCONJ
ejpam-6493	185	18	ideals	ideal	NOUN
ejpam-6493	185	19	from	from	ADP
ejpam-6493	185	20	subsections	subsection	NOUN
ejpam-6493	185	21	3.2	3.2	NUM
ejpam-6493	185	22	and	and	CCONJ
ejpam-6493	185	23	3.3	3.3	NUM
ejpam-6493	185	24	,	,	PUNCT
ejpam-6493	185	25	respectively	respectively	ADV
ejpam-6493	185	26	.	.	PUNCT
ejpam-6493	186	1	firstly	firstly	ADV
ejpam-6493	186	2	,	,	PUNCT
ejpam-6493	186	3	as	as	SCONJ
ejpam-6493	186	4	shown	show	VERB
ejpam-6493	186	5	in	in	ADP
ejpam-6493	186	6	example	example	NOUN
ejpam-6493	186	7	4	4	NUM
ejpam-6493	186	8	(	(	PUNCT
ejpam-6493	186	9	2	2	NUM
ejpam-6493	186	10	)	)	PUNCT
ejpam-6493	186	11	,	,	PUNCT
ejpam-6493	186	12	we	we	PRON
ejpam-6493	186	13	see	see	VERB
ejpam-6493	186	14	that	that	PRON
ejpam-6493	186	15	−2+j	−2+j	PUNCT
ejpam-6493	186	16	=	=	PRON
ejpam-6493	186	17	0+j	0+j	NUM
ejpam-6493	186	18	but	but	CCONJ
ejpam-6493	186	19	−2	−2	NOUN
ejpam-6493	186	20	/∈	/∈	PUNCT
ejpam-6493	187	1	j	j	PROPN
ejpam-6493	187	2	.	.	PUNCT
ejpam-6493	188	1	moreover	moreover	ADV
ejpam-6493	188	2	,	,	PUNCT
ejpam-6493	188	3	j	j	PROPN
ejpam-6493	188	4	⊆	⊆	NUM
ejpam-6493	188	5	0	0	NUM
ejpam-6493	189	1	+	+	CCONJ
ejpam-6493	189	2	j	j	PROPN
ejpam-6493	189	3	and	and	CCONJ
ejpam-6493	189	4	0	0	NUM
ejpam-6493	190	1	+	+	CCONJ
ejpam-6493	190	2	j	j	NOUN
ejpam-6493	190	3	need	need	AUX
ejpam-6493	190	4	not	not	PART
ejpam-6493	190	5	be	be	AUX
ejpam-6493	190	6	equal	equal	ADJ
ejpam-6493	190	7	to	to	ADP
ejpam-6493	190	8	j	j	PROPN
ejpam-6493	190	9	.	.	PUNCT
ejpam-6493	191	1	the	the	DET
ejpam-6493	191	2	following	follow	VERB
ejpam-6493	191	3	proposition	proposition	NOUN
ejpam-6493	191	4	shows	show	VERB
ejpam-6493	191	5	some	some	DET
ejpam-6493	191	6	properties	property	NOUN
ejpam-6493	191	7	of	of	ADP
ejpam-6493	191	8	cosets	coset	NOUN
ejpam-6493	191	9	in	in	ADP
ejpam-6493	191	10	a	a	DET
ejpam-6493	191	11	ternary	ternary	ADJ
ejpam-6493	191	12	semiring	semire	VERB
ejpam-6493	191	13	r	r	NOUN
ejpam-6493	191	14	/	/	SYM
ejpam-6493	191	15	i.	i.	NOUN
ejpam-6493	191	16	proposition	proposition	NOUN
ejpam-6493	191	17	2	2	X
ejpam-6493	191	18	.	.	PUNCT
ejpam-6493	192	1	let	let	VERB
ejpam-6493	192	2	r	r	PRON
ejpam-6493	192	3	be	be	AUX
ejpam-6493	192	4	a	a	DET
ejpam-6493	192	5	ternary	ternary	ADJ
ejpam-6493	192	6	semiring	semiring	NOUN
ejpam-6493	192	7	,	,	PUNCT
ejpam-6493	192	8	i	i	PRON
ejpam-6493	192	9	an	an	DET
ejpam-6493	192	10	ideal	ideal	NOUN
ejpam-6493	192	11	of	of	ADP
ejpam-6493	192	12	r	r	NOUN
ejpam-6493	192	13	,	,	PUNCT
ejpam-6493	192	14	and	and	CCONJ
ejpam-6493	192	15	a	a	DET
ejpam-6493	192	16	,	,	PUNCT
ejpam-6493	192	17	b	b	X
ejpam-6493	192	18	∈	∈	PROPN
ejpam-6493	192	19	r.	r.	NOUN
ejpam-6493	192	20	then	then	ADV
ejpam-6493	192	21	the	the	DET
ejpam-6493	192	22	following	follow	VERB
ejpam-6493	192	23	statements	statement	NOUN
ejpam-6493	192	24	hold	hold	VERB
ejpam-6493	192	25	:	:	PUNCT
ejpam-6493	192	26	(	(	PUNCT
ejpam-6493	192	27	1	1	X
ejpam-6493	192	28	)	)	PUNCT
ejpam-6493	192	29	if	if	SCONJ
ejpam-6493	192	30	r	r	NOUN
ejpam-6493	192	31	has	have	VERB
ejpam-6493	192	32	a	a	DET
ejpam-6493	192	33	zero	zero	NUM
ejpam-6493	192	34	and	and	CCONJ
ejpam-6493	192	35	i	i	PRON
ejpam-6493	192	36	is	be	AUX
ejpam-6493	192	37	a	a	DET
ejpam-6493	192	38	k	k	NOUN
ejpam-6493	192	39	-	-	NOUN
ejpam-6493	192	40	ideal	ideal	NOUN
ejpam-6493	192	41	of	of	ADP
ejpam-6493	192	42	r	r	NOUN
ejpam-6493	192	43	,	,	PUNCT
ejpam-6493	192	44	then	then	ADV
ejpam-6493	192	45	0	0	NUM
ejpam-6493	193	1	+	+	CCONJ
ejpam-6493	193	2	i	i	NOUN
ejpam-6493	193	3	=	=	SYM
ejpam-6493	193	4	i.	i.	NOUN
ejpam-6493	193	5	(	(	PUNCT
ejpam-6493	193	6	2	2	NUM
ejpam-6493	193	7	)	)	PUNCT
ejpam-6493	193	8	if	if	SCONJ
ejpam-6493	193	9	r	r	NOUN
ejpam-6493	193	10	has	have	VERB
ejpam-6493	193	11	a	a	DET
ejpam-6493	193	12	zero	zero	NUM
ejpam-6493	193	13	and	and	CCONJ
ejpam-6493	193	14	a	a	DET
ejpam-6493	193	15	∈	∈	PROPN
ejpam-6493	194	1	i	i	PRON
ejpam-6493	194	2	,	,	PUNCT
ejpam-6493	194	3	then	then	ADV
ejpam-6493	194	4	a+	a+	PUNCT
ejpam-6493	194	5	i	i	NOUN
ejpam-6493	194	6	=	=	PUNCT
ejpam-6493	194	7	0	0	PUNCT
ejpam-6493	195	1	+	+	NUM
ejpam-6493	195	2	i.	i.	NOUN
ejpam-6493	195	3	(	(	PUNCT
ejpam-6493	195	4	3	3	NUM
ejpam-6493	195	5	)	)	PUNCT
ejpam-6493	195	6	if	if	SCONJ
ejpam-6493	195	7	i	i	PRON
ejpam-6493	195	8	is	be	AUX
ejpam-6493	195	9	a	a	DET
ejpam-6493	195	10	k	k	NOUN
ejpam-6493	195	11	-	-	NOUN
ejpam-6493	195	12	ideal	ideal	NOUN
ejpam-6493	195	13	of	of	ADP
ejpam-6493	195	14	r	r	NOUN
ejpam-6493	195	15	and	and	CCONJ
ejpam-6493	195	16	a	a	DET
ejpam-6493	195	17	∈	∈	NOUN
ejpam-6493	196	1	i	i	PRON
ejpam-6493	196	2	,	,	PUNCT
ejpam-6493	196	3	then	then	ADV
ejpam-6493	196	4	a+	a+	PUNCT
ejpam-6493	196	5	i	i	PRON
ejpam-6493	196	6	=	=	PUNCT
ejpam-6493	196	7	b+	b+	PUNCT
ejpam-6493	196	8	i	i	PRON
ejpam-6493	196	9	if	if	SCONJ
ejpam-6493	196	10	and	and	CCONJ
ejpam-6493	196	11	only	only	ADV
ejpam-6493	196	12	if	if	SCONJ
ejpam-6493	196	13	b	b	PROPN
ejpam-6493	196	14	∈	∈	PROPN
ejpam-6493	196	15	i.	i.	NOUN
ejpam-6493	196	16	(	(	PUNCT
ejpam-6493	196	17	4	4	NUM
ejpam-6493	196	18	)	)	PUNCT
ejpam-6493	196	19	if	if	SCONJ
ejpam-6493	196	20	r	r	NOUN
ejpam-6493	196	21	has	have	VERB
ejpam-6493	196	22	a	a	DET
ejpam-6493	196	23	zero	zero	NUM
ejpam-6493	196	24	and	and	CCONJ
ejpam-6493	196	25	i	i	PRON
ejpam-6493	196	26	is	be	AUX
ejpam-6493	196	27	a	a	DET
ejpam-6493	196	28	k	k	NOUN
ejpam-6493	196	29	-	-	NOUN
ejpam-6493	196	30	ideal	ideal	NOUN
ejpam-6493	196	31	of	of	ADP
ejpam-6493	196	32	r	r	NOUN
ejpam-6493	196	33	,	,	PUNCT
ejpam-6493	196	34	then	then	ADV
ejpam-6493	196	35	a+	a+	PUNCT
ejpam-6493	196	36	i	i	NOUN
ejpam-6493	196	37	=	=	NOUN
ejpam-6493	196	38	0	0	PUNCT
ejpam-6493	197	1	+	+	CCONJ
ejpam-6493	197	2	i	i	PRON
ejpam-6493	197	3	if	if	SCONJ
ejpam-6493	197	4	and	and	CCONJ
ejpam-6493	197	5	only	only	ADV
ejpam-6493	197	6	if	if	SCONJ
ejpam-6493	197	7	a	a	DET
ejpam-6493	197	8	∈	∈	PROPN
ejpam-6493	197	9	i.	i.	NOUN
ejpam-6493	197	10	proof	proof	NOUN
ejpam-6493	197	11	.	.	PUNCT
ejpam-6493	198	1	(	(	PUNCT
ejpam-6493	198	2	1	1	X
ejpam-6493	198	3	)	)	PUNCT
ejpam-6493	198	4	assume	assume	VERB
ejpam-6493	198	5	0	0	NUM
ejpam-6493	198	6	∈	∈	NOUN
ejpam-6493	198	7	r	r	NOUN
ejpam-6493	199	1	and	and	CCONJ
ejpam-6493	199	2	i	i	PRON
ejpam-6493	199	3	is	be	AUX
ejpam-6493	199	4	a	a	DET
ejpam-6493	199	5	k	k	NOUN
ejpam-6493	199	6	-	-	NOUN
ejpam-6493	199	7	ideal	ideal	NOUN
ejpam-6493	199	8	of	of	ADP
ejpam-6493	199	9	r.	r.	PROPN
ejpam-6493	199	10	clearly	clearly	ADV
ejpam-6493	199	11	,	,	PUNCT
ejpam-6493	199	12	i	i	PRON
ejpam-6493	199	13	⊆	⊆	NUM
ejpam-6493	199	14	0	0	NUM
ejpam-6493	200	1	+	+	CCONJ
ejpam-6493	200	2	i.	i.	NOUN
ejpam-6493	200	3	let	let	VERB
ejpam-6493	200	4	x	x	SYM
ejpam-6493	200	5	∈	∈	PROPN
ejpam-6493	200	6	0	0	PUNCT
ejpam-6493	201	1	+	+	CCONJ
ejpam-6493	201	2	i.	i.	NOUN
ejpam-6493	201	3	then	then	ADV
ejpam-6493	201	4	(	(	PUNCT
ejpam-6493	201	5	0	0	NUM
ejpam-6493	201	6	,	,	PUNCT
ejpam-6493	201	7	x	x	X
ejpam-6493	201	8	)	)	PUNCT
ejpam-6493	201	9	∈	∈	PROPN
ejpam-6493	201	10	ρi	ρi	NOUN
ejpam-6493	201	11	.	.	PUNCT
ejpam-6493	202	1	thus	thus	ADV
ejpam-6493	202	2	there	there	PRON
ejpam-6493	202	3	exist	exist	VERB
ejpam-6493	202	4	a	a	DET
ejpam-6493	202	5	,	,	PUNCT
ejpam-6493	202	6	b	b	X
ejpam-6493	202	7	∈	∈	NOUN
ejpam-6493	202	8	i	i	PRON
ejpam-6493	202	9	such	such	VERB
ejpam-6493	202	10	that	that	SCONJ
ejpam-6493	202	11	x	x	X
ejpam-6493	203	1	+	+	CCONJ
ejpam-6493	203	2	a	a	DET
ejpam-6493	203	3	=	=	SYM
ejpam-6493	203	4	0	0	PUNCT
ejpam-6493	203	5	+	+	NUM
ejpam-6493	203	6	b.	b.	NOUN
ejpam-6493	203	7	given	give	VERB
ejpam-6493	203	8	that	that	SCONJ
ejpam-6493	203	9	i	i	PRON
ejpam-6493	203	10	is	be	AUX
ejpam-6493	203	11	a	a	DET
ejpam-6493	203	12	k	k	NOUN
ejpam-6493	203	13	-	-	NOUN
ejpam-6493	203	14	ideal	ideal	ADJ
ejpam-6493	203	15	,	,	PUNCT
ejpam-6493	203	16	x+	x+	PUNCT
ejpam-6493	203	17	a	a	DET
ejpam-6493	203	18	∈	∈	PROPN
ejpam-6493	204	1	i	i	PRON
ejpam-6493	204	2	and	and	CCONJ
ejpam-6493	204	3	a	a	DET
ejpam-6493	204	4	∈	∈	NOUN
ejpam-6493	204	5	i	i	PRON
ejpam-6493	204	6	imply	imply	VERB
ejpam-6493	204	7	that	that	SCONJ
ejpam-6493	204	8	x	x	X
ejpam-6493	204	9	∈	∈	NOUN
ejpam-6493	204	10	i.	i.	NOUN
ejpam-6493	204	11	hence	hence	ADV
ejpam-6493	204	12	0	0	NUM
ejpam-6493	205	1	+	+	CCONJ
ejpam-6493	205	2	i	i	NOUN
ejpam-6493	205	3	=	=	SYM
ejpam-6493	205	4	i.	i.	NOUN
ejpam-6493	205	5	(	(	PUNCT
ejpam-6493	205	6	2	2	X
ejpam-6493	205	7	)	)	PUNCT
ejpam-6493	205	8	assume	assume	VERB
ejpam-6493	205	9	0	0	NUM
ejpam-6493	205	10	∈	∈	NOUN
ejpam-6493	205	11	r	r	NOUN
ejpam-6493	205	12	and	and	CCONJ
ejpam-6493	205	13	a	a	DET
ejpam-6493	205	14	∈	∈	PROPN
ejpam-6493	205	15	i.	i.	NOUN
ejpam-6493	205	16	since	since	SCONJ
ejpam-6493	205	17	0	0	NUM
ejpam-6493	206	1	+	+	CCONJ
ejpam-6493	206	2	a	a	DET
ejpam-6493	206	3	=	=	X
ejpam-6493	206	4	a	a	DET
ejpam-6493	206	5	+	+	NOUN
ejpam-6493	206	6	0	0	NUM
ejpam-6493	206	7	=	=	SYM
ejpam-6493	206	8	a	a	PRON
ejpam-6493	206	9	where	where	SCONJ
ejpam-6493	206	10	a	a	DET
ejpam-6493	206	11	∈	∈	PROPN
ejpam-6493	206	12	i	i	X
ejpam-6493	206	13	,	,	PUNCT
ejpam-6493	206	14	by	by	ADP
ejpam-6493	206	15	the	the	DET
ejpam-6493	206	16	definition	definition	NOUN
ejpam-6493	206	17	of	of	ADP
ejpam-6493	206	18	a	a	DET
ejpam-6493	206	19	congruence	congruence	NOUN
ejpam-6493	206	20	relation	relation	NOUN
ejpam-6493	206	21	ρi	ρi	NOUN
ejpam-6493	206	22	,	,	PUNCT
ejpam-6493	206	23	we	we	PRON
ejpam-6493	206	24	have	have	VERB
ejpam-6493	206	25	(	(	PUNCT
ejpam-6493	206	26	0	0	NUM
ejpam-6493	206	27	,	,	PUNCT
ejpam-6493	206	28	a	a	PRON
ejpam-6493	206	29	)	)	PUNCT
ejpam-6493	206	30	∈	∈	PROPN
ejpam-6493	206	31	ρi	ρi	NOUN
ejpam-6493	206	32	.	.	PUNCT
ejpam-6493	207	1	hence	hence	ADV
ejpam-6493	207	2	0	0	PUNCT
ejpam-6493	208	1	+	+	CCONJ
ejpam-6493	208	2	i	i	PRON
ejpam-6493	208	3	=	=	SYM
ejpam-6493	208	4	a+	a+	PUNCT
ejpam-6493	208	5	i.	i.	NOUN
ejpam-6493	208	6	(	(	PUNCT
ejpam-6493	208	7	3	3	X
ejpam-6493	208	8	)	)	PUNCT
ejpam-6493	208	9	let	let	VERB
ejpam-6493	208	10	i	i	PRON
ejpam-6493	208	11	be	be	AUX
ejpam-6493	208	12	a	a	DET
ejpam-6493	208	13	k	k	NOUN
ejpam-6493	208	14	-	-	NOUN
ejpam-6493	208	15	ideal	ideal	NOUN
ejpam-6493	208	16	of	of	ADP
ejpam-6493	208	17	r	r	NOUN
ejpam-6493	208	18	and	and	CCONJ
ejpam-6493	208	19	a	a	DET
ejpam-6493	208	20	∈	∈	PROPN
ejpam-6493	208	21	i.	i.	NOUN
ejpam-6493	208	22	assume	assume	VERB
ejpam-6493	208	23	b	b	X
ejpam-6493	208	24	∈	∈	NOUN
ejpam-6493	208	25	r	r	NOUN
ejpam-6493	209	1	such	such	DET
ejpam-6493	209	2	that	that	SCONJ
ejpam-6493	209	3	a	a	DET
ejpam-6493	209	4	+	+	NOUN
ejpam-6493	209	5	i	i	NOUN
ejpam-6493	209	6	=	=	SYM
ejpam-6493	209	7	b	b	PROPN
ejpam-6493	209	8	+	+	NUM
ejpam-6493	209	9	i.	i.	NOUN
ejpam-6493	209	10	then	then	ADV
ejpam-6493	209	11	(	(	PUNCT
ejpam-6493	209	12	a	a	PRON
ejpam-6493	209	13	,	,	PUNCT
ejpam-6493	209	14	b	b	NOUN
ejpam-6493	209	15	)	)	PUNCT
ejpam-6493	209	16	∈	∈	PROPN
ejpam-6493	209	17	ρi	ρi	NOUN
ejpam-6493	209	18	which	which	PRON
ejpam-6493	209	19	implies	imply	VERB
ejpam-6493	209	20	a+	a+	PUNCT
ejpam-6493	209	21	u	u	NOUN
ejpam-6493	209	22	=	=	PUNCT
ejpam-6493	209	23	b+	b+	X
ejpam-6493	209	24	v	v	NOUN
ejpam-6493	209	25	for	for	ADP
ejpam-6493	209	26	some	some	DET
ejpam-6493	209	27	u	u	NOUN
ejpam-6493	209	28	,	,	PUNCT
ejpam-6493	209	29	v	v	PROPN
ejpam-6493	209	30	∈	∈	PROPN
ejpam-6493	209	31	i.	i.	NOUN
ejpam-6493	209	32	then	then	ADV
ejpam-6493	209	33	b+	b+	VERB
ejpam-6493	209	34	v	v	NOUN
ejpam-6493	209	35	=	=	SYM
ejpam-6493	209	36	a+	a+	PUNCT
ejpam-6493	209	37	u	u	PROPN
ejpam-6493	209	38	∈	∈	PROPN
ejpam-6493	209	39	i	i	PRON
ejpam-6493	209	40	because	because	SCONJ
ejpam-6493	209	41	a	a	DET
ejpam-6493	209	42	,	,	PUNCT
ejpam-6493	209	43	u	u	PROPN
ejpam-6493	209	44	∈	∈	PROPN
ejpam-6493	209	45	i.	i.	NOUN
ejpam-6493	209	46	since	since	SCONJ
ejpam-6493	209	47	i	i	PRON
ejpam-6493	209	48	is	be	AUX
ejpam-6493	209	49	a	a	DET
ejpam-6493	209	50	k	k	NOUN
ejpam-6493	209	51	-	-	NOUN
ejpam-6493	209	52	ideal	ideal	ADJ
ejpam-6493	209	53	,	,	PUNCT
ejpam-6493	209	54	v	v	NOUN
ejpam-6493	209	55	∈	∈	X
ejpam-6493	210	1	i	i	PRON
ejpam-6493	210	2	and	and	CCONJ
ejpam-6493	210	3	b+	b+	VERB
ejpam-6493	210	4	v	v	X
ejpam-6493	210	5	∈	∈	NOUN
ejpam-6493	210	6	i	i	PRON
ejpam-6493	210	7	imply	imply	VERB
ejpam-6493	210	8	that	that	SCONJ
ejpam-6493	210	9	b	b	X
ejpam-6493	210	10	∈	∈	PROPN
ejpam-6493	210	11	i.	i.	NOUN
ejpam-6493	210	12	conversely	conversely	ADV
ejpam-6493	210	13	,	,	PUNCT
ejpam-6493	210	14	we	we	PRON
ejpam-6493	210	15	assume	assume	VERB
ejpam-6493	210	16	b	b	X
ejpam-6493	210	17	∈	∈	PROPN
ejpam-6493	210	18	i.	i.	NOUN
ejpam-6493	210	19	then	then	ADV
ejpam-6493	210	20	a	a	PRON
ejpam-6493	210	21	,	,	PUNCT
ejpam-6493	210	22	b	b	X
ejpam-6493	210	23	∈	∈	PROPN
ejpam-6493	211	1	i	i	PRON
ejpam-6493	211	2	and	and	CCONJ
ejpam-6493	211	3	a+	a+	PRON
ejpam-6493	211	4	b	b	X
ejpam-6493	211	5	=	=	X
ejpam-6493	211	6	b+	b+	X
ejpam-6493	211	7	a	a	DET
ejpam-6493	211	8	give	give	NOUN
ejpam-6493	211	9	(	(	PUNCT
ejpam-6493	211	10	a	a	DET
ejpam-6493	211	11	,	,	PUNCT
ejpam-6493	211	12	b	b	NOUN
ejpam-6493	211	13	)	)	PUNCT
ejpam-6493	211	14	∈	∈	PROPN
ejpam-6493	211	15	ρi	ρi	NOUN
ejpam-6493	211	16	and	and	CCONJ
ejpam-6493	211	17	consequently	consequently	ADV
ejpam-6493	211	18	,	,	PUNCT
ejpam-6493	211	19	a+	a+	PUNCT
ejpam-6493	212	1	i	i	PROPN
ejpam-6493	212	2	=	=	PUNCT
ejpam-6493	212	3	b+	b+	PROPN
ejpam-6493	212	4	i.	i.	PROPN
ejpam-6493	212	5	(	(	PUNCT
ejpam-6493	212	6	4	4	X
ejpam-6493	212	7	)	)	PUNCT
ejpam-6493	212	8	assume	assume	VERB
ejpam-6493	212	9	that	that	SCONJ
ejpam-6493	212	10	a+	a+	PUNCT
ejpam-6493	212	11	i	i	PRON
ejpam-6493	212	12	=	=	NOUN
ejpam-6493	212	13	0	0	NUM
ejpam-6493	212	14	+	+	NUM
ejpam-6493	212	15	i.	i.	NOUN
ejpam-6493	212	16	since	since	SCONJ
ejpam-6493	212	17	0	0	NUM
ejpam-6493	212	18	∈	∈	PROPN
ejpam-6493	212	19	i	i	PRON
ejpam-6493	212	20	,	,	PUNCT
ejpam-6493	212	21	by	by	ADP
ejpam-6493	212	22	(	(	PUNCT
ejpam-6493	212	23	3	3	NUM
ejpam-6493	212	24	)	)	PUNCT
ejpam-6493	212	25	,	,	PUNCT
ejpam-6493	212	26	we	we	PRON
ejpam-6493	212	27	have	have	VERB
ejpam-6493	212	28	that	that	SCONJ
ejpam-6493	212	29	a	a	DET
ejpam-6493	212	30	∈	∈	PROPN
ejpam-6493	212	31	i.	i.	NOUN
ejpam-6493	212	32	conversely	conversely	ADV
ejpam-6493	212	33	,	,	PUNCT
ejpam-6493	212	34	it	it	PRON
ejpam-6493	212	35	is	be	AUX
ejpam-6493	212	36	clear	clear	ADJ
ejpam-6493	212	37	by	by	ADP
ejpam-6493	212	38	(	(	PUNCT
ejpam-6493	212	39	1	1	NUM
ejpam-6493	212	40	)	)	PUNCT
ejpam-6493	212	41	and	and	CCONJ
ejpam-6493	212	42	(	(	PUNCT
ejpam-6493	212	43	2	2	NUM
ejpam-6493	212	44	)	)	PUNCT
ejpam-6493	212	45	.	.	PUNCT
ejpam-6493	213	1	let	let	VERB
ejpam-6493	213	2	r	r	NOUN
ejpam-6493	213	3	be	be	AUX
ejpam-6493	213	4	any	any	DET
ejpam-6493	213	5	ternary	ternary	ADJ
ejpam-6493	213	6	semiring	semiring	NOUN
ejpam-6493	213	7	and	and	CCONJ
ejpam-6493	213	8	i	i	PRON
ejpam-6493	213	9	be	be	VERB
ejpam-6493	213	10	an	an	DET
ejpam-6493	213	11	ideal	ideal	NOUN
ejpam-6493	213	12	of	of	ADP
ejpam-6493	213	13	r.	r.	PROPN
ejpam-6493	213	14	the	the	DET
ejpam-6493	213	15	k	k	NOUN
ejpam-6493	213	16	-	-	NOUN
ejpam-6493	213	17	closure	closure	NOUN
ejpam-6493	213	18	of	of	ADP
ejpam-6493	213	19	i	i	PRON
ejpam-6493	213	20	is	be	AUX
ejpam-6493	213	21	defined	define	VERB
ejpam-6493	213	22	by	by	ADP
ejpam-6493	213	23	ck(i	ck(i	NOUN
ejpam-6493	213	24	)	)	PUNCT
ejpam-6493	213	25	=	=	PRON
ejpam-6493	214	1	{	{	PUNCT
ejpam-6493	214	2	r	r	NOUN
ejpam-6493	214	3	∈	∈	NOUN
ejpam-6493	214	4	r	r	NOUN
ejpam-6493	214	5	|	|	NOUN
ejpam-6493	215	1	r	r	NOUN
ejpam-6493	216	1	+	+	NOUN
ejpam-6493	217	1	x	x	SYM
ejpam-6493	217	2	∈	∈	NOUN
ejpam-6493	217	3	i	i	PRON
ejpam-6493	217	4	for	for	ADP
ejpam-6493	217	5	some	some	DET
ejpam-6493	217	6	x	x	SYM
ejpam-6493	217	7	∈	∈	PROPN
ejpam-6493	217	8	i	i	X
ejpam-6493	217	9	}	}	PUNCT
ejpam-6493	217	10	.	.	PUNCT
ejpam-6493	218	1	then	then	ADV
ejpam-6493	218	2	ck(i	ck(i	PUNCT
ejpam-6493	218	3	)	)	PUNCT
ejpam-6493	218	4	is	be	AUX
ejpam-6493	218	5	the	the	DET
ejpam-6493	218	6	smallest	small	ADJ
ejpam-6493	218	7	k	k	ADJ
ejpam-6493	218	8	-	-	NOUN
ejpam-6493	218	9	ideal	ideal	NOUN
ejpam-6493	218	10	of	of	ADP
ejpam-6493	218	11	r	r	NOUN
ejpam-6493	218	12	containing	contain	VERB
ejpam-6493	218	13	i	i	PRON
ejpam-6493	218	14	,	,	PUNCT
ejpam-6493	218	15	as	as	SCONJ
ejpam-6493	218	16	shown	show	VERB
ejpam-6493	218	17	in	in	ADP
ejpam-6493	218	18	[	[	X
ejpam-6493	218	19	11	11	NUM
ejpam-6493	218	20	]	]	PUNCT
ejpam-6493	218	21	.	.	PUNCT
ejpam-6493	219	1	theorem	theorem	NOUN
ejpam-6493	219	2	2	2	NUM
ejpam-6493	219	3	.	.	PUNCT
ejpam-6493	220	1	let	let	VERB
ejpam-6493	220	2	i	i	PRON
ejpam-6493	220	3	be	be	AUX
ejpam-6493	220	4	an	an	DET
ejpam-6493	220	5	ideal	ideal	NOUN
ejpam-6493	220	6	of	of	ADP
ejpam-6493	220	7	a	a	DET
ejpam-6493	220	8	ternary	ternary	ADJ
ejpam-6493	220	9	semiring	semire	VERB
ejpam-6493	220	10	r.	r.	PROPN
ejpam-6493	220	11	then	then	ADV
ejpam-6493	220	12	r	r	X
ejpam-6493	220	13	/	/	SYM
ejpam-6493	220	14	i	i	PRON
ejpam-6493	220	15	∼=	∼=	ADV
ejpam-6493	220	16	r	r	NOUN
ejpam-6493	220	17	/	/	SYM
ejpam-6493	220	18	ck(i	ck(i	NUM
ejpam-6493	220	19	)	)	PUNCT
ejpam-6493	220	20	.	.	PUNCT
ejpam-6493	221	1	m.	m.	NOUN
ejpam-6493	221	2	petapirak	petapirak	PROPN
ejpam-6493	221	3	,	,	PUNCT
ejpam-6493	221	4	a.	a.	PROPN
ejpam-6493	221	5	j.	j.	PROPN
ejpam-6493	221	6	khan	khan	PROPN
ejpam-6493	221	7	,	,	PUNCT
ejpam-6493	221	8	r.	r.	PROPN
ejpam-6493	221	9	chinram	chinram	PROPN
ejpam-6493	221	10	/	/	SYM
ejpam-6493	221	11	eur	eur	PROPN
ejpam-6493	221	12	.	.	PUNCT
ejpam-6493	222	1	j.	j.	PROPN
ejpam-6493	222	2	pure	pure	PROPN
ejpam-6493	222	3	appl	appl	PROPN
ejpam-6493	222	4	.	.	PROPN
ejpam-6493	222	5	math	math	PROPN
ejpam-6493	222	6	,	,	PUNCT
ejpam-6493	222	7	18	18	NUM
ejpam-6493	222	8	(	(	PUNCT
ejpam-6493	222	9	3	3	NUM
ejpam-6493	222	10	)	)	PUNCT
ejpam-6493	222	11	(	(	PUNCT
ejpam-6493	222	12	2025	2025	NUM
ejpam-6493	222	13	)	)	PUNCT
ejpam-6493	222	14	,	,	PUNCT
ejpam-6493	222	15	6493	6493	NUM
ejpam-6493	222	16	7	7	NUM
ejpam-6493	222	17	of	of	ADP
ejpam-6493	222	18	9	9	NUM
ejpam-6493	222	19	proof	proof	NOUN
ejpam-6493	222	20	.	.	PUNCT
ejpam-6493	223	1	define	define	VERB
ejpam-6493	223	2	φ	φ	NOUN
ejpam-6493	223	3	:	:	PUNCT
ejpam-6493	223	4	r	r	X
ejpam-6493	223	5	/	/	SYM
ejpam-6493	223	6	i	i	NOUN
ejpam-6493	223	7	→	→	SYM
ejpam-6493	223	8	r	r	NOUN
ejpam-6493	223	9	/	/	SYM
ejpam-6493	223	10	ck(i	ck(i	NUM
ejpam-6493	223	11	)	)	PUNCT
ejpam-6493	223	12	by	by	ADP
ejpam-6493	223	13	φ(a+	φ(a+	ADJ
ejpam-6493	223	14	i	i	NOUN
ejpam-6493	223	15	)	)	PUNCT
ejpam-6493	223	16	=	=	SYM
ejpam-6493	223	17	a+	a+	PUNCT
ejpam-6493	223	18	ck(i	ck(i	PUNCT
ejpam-6493	223	19	)	)	PUNCT
ejpam-6493	223	20	for	for	ADP
ejpam-6493	223	21	all	all	DET
ejpam-6493	223	22	a	a	DET
ejpam-6493	223	23	∈	∈	PROPN
ejpam-6493	223	24	r.	r.	NOUN
ejpam-6493	223	25	first	first	ADV
ejpam-6493	223	26	,	,	PUNCT
ejpam-6493	223	27	we	we	PRON
ejpam-6493	223	28	let	let	VERB
ejpam-6493	223	29	a	a	DET
ejpam-6493	223	30	,	,	PUNCT
ejpam-6493	223	31	b	b	X
ejpam-6493	223	32	∈	∈	NOUN
ejpam-6493	223	33	r	r	NOUN
ejpam-6493	224	1	such	such	DET
ejpam-6493	224	2	that	that	SCONJ
ejpam-6493	224	3	a	a	DET
ejpam-6493	224	4	+	+	NOUN
ejpam-6493	224	5	i	i	NOUN
ejpam-6493	224	6	=	=	SYM
ejpam-6493	224	7	b	b	PROPN
ejpam-6493	224	8	+	+	NUM
ejpam-6493	224	9	i.	i.	NOUN
ejpam-6493	224	10	so	so	SCONJ
ejpam-6493	224	11	there	there	PRON
ejpam-6493	224	12	exist	exist	VERB
ejpam-6493	224	13	x	x	NOUN
ejpam-6493	224	14	,	,	PUNCT
ejpam-6493	224	15	y	y	PROPN
ejpam-6493	224	16	∈	∈	PROPN
ejpam-6493	225	1	i	i	PRON
ejpam-6493	225	2	such	such	ADJ
ejpam-6493	225	3	that	that	SCONJ
ejpam-6493	225	4	a	a	DET
ejpam-6493	225	5	+	+	NOUN
ejpam-6493	225	6	x	x	SYM
ejpam-6493	225	7	=	=	SYM
ejpam-6493	225	8	b	b	PROPN
ejpam-6493	225	9	+	+	CCONJ
ejpam-6493	225	10	y.	y.	NOUN
ejpam-6493	225	11	since	since	SCONJ
ejpam-6493	225	12	x	x	X
ejpam-6493	225	13	,	,	PUNCT
ejpam-6493	225	14	y	y	PROPN
ejpam-6493	225	15	∈	∈	PROPN
ejpam-6493	225	16	i	i	PROPN
ejpam-6493	225	17	⊆	⊆	NUM
ejpam-6493	225	18	ck(i	ck(i	NUM
ejpam-6493	225	19	)	)	PUNCT
ejpam-6493	225	20	,	,	PUNCT
ejpam-6493	225	21	a+x	a+x	X
ejpam-6493	225	22	=	=	SYM
ejpam-6493	225	23	a+y	a+y	NUM
ejpam-6493	225	24	yields	yield	NOUN
ejpam-6493	225	25	a+ck(i	a+ck(i	PROPN
ejpam-6493	225	26	)	)	PUNCT
ejpam-6493	225	27	=	=	PUNCT
ejpam-6493	225	28	b+ck(i	b+ck(i	NUM
ejpam-6493	225	29	)	)	PUNCT
ejpam-6493	225	30	.	.	PUNCT
ejpam-6493	226	1	then	then	ADV
ejpam-6493	226	2	φ	φ	PROPN
ejpam-6493	226	3	is	be	AUX
ejpam-6493	226	4	well	well	ADV
ejpam-6493	226	5	-	-	PUNCT
ejpam-6493	226	6	defined	define	VERB
ejpam-6493	226	7	.	.	PUNCT
ejpam-6493	227	1	clearly	clearly	ADV
ejpam-6493	227	2	,	,	PUNCT
ejpam-6493	227	3	φ	φ	PROPN
ejpam-6493	227	4	is	be	AUX
ejpam-6493	227	5	an	an	PRON
ejpam-6493	227	6	onto	onto	ADP
ejpam-6493	227	7	homomorphism	homomorphism	NOUN
ejpam-6493	227	8	.	.	PUNCT
ejpam-6493	228	1	let	let	VERB
ejpam-6493	228	2	a+	a+	PUNCT
ejpam-6493	228	3	i	i	PRON
ejpam-6493	228	4	,	,	PUNCT
ejpam-6493	228	5	b+	b+	X
ejpam-6493	228	6	i	i	PROPN
ejpam-6493	228	7	∈	∈	PROPN
ejpam-6493	228	8	r	r	X
ejpam-6493	228	9	/	/	SYM
ejpam-6493	228	10	i	i	PRON
ejpam-6493	228	11	such	such	ADJ
ejpam-6493	228	12	that	that	SCONJ
ejpam-6493	228	13	φ(a+	φ(a+	ADJ
ejpam-6493	228	14	i	i	NOUN
ejpam-6493	228	15	)	)	PUNCT
ejpam-6493	228	16	=	=	SYM
ejpam-6493	229	1	φ(b+	φ(b+	PUNCT
ejpam-6493	230	1	i	i	NOUN
ejpam-6493	230	2	)	)	PUNCT
ejpam-6493	230	3	.	.	PUNCT
ejpam-6493	231	1	thus	thus	ADV
ejpam-6493	231	2	a+ck(i	a+ck(i	NUM
ejpam-6493	231	3	)	)	PUNCT
ejpam-6493	231	4	=	=	PUNCT
ejpam-6493	231	5	b+ck(i	b+ck(i	NUM
ejpam-6493	231	6	)	)	PUNCT
ejpam-6493	231	7	.	.	PUNCT
ejpam-6493	232	1	so	so	ADV
ejpam-6493	232	2	there	there	PRON
ejpam-6493	232	3	exist	exist	VERB
ejpam-6493	232	4	x	x	NOUN
ejpam-6493	232	5	,	,	PUNCT
ejpam-6493	232	6	y	y	PROPN
ejpam-6493	232	7	∈	∈	PROPN
ejpam-6493	232	8	ck(i	ck(i	PUNCT
ejpam-6493	232	9	)	)	PUNCT
ejpam-6493	233	1	such	such	ADJ
ejpam-6493	233	2	that	that	SCONJ
ejpam-6493	233	3	a+x	a+x	ADP
ejpam-6493	233	4	=	=	SYM
ejpam-6493	233	5	b+y	b+y	PROPN
ejpam-6493	233	6	.	.	PUNCT
ejpam-6493	234	1	since	since	SCONJ
ejpam-6493	234	2	x	x	X
ejpam-6493	234	3	,	,	PUNCT
ejpam-6493	234	4	y	y	PROPN
ejpam-6493	234	5	∈	∈	PROPN
ejpam-6493	234	6	ck(i	ck(i	PUNCT
ejpam-6493	234	7	)	)	PUNCT
ejpam-6493	234	8	,	,	PUNCT
ejpam-6493	234	9	there	there	PRON
ejpam-6493	234	10	exist	exist	VERB
ejpam-6493	234	11	s	s	PROPN
ejpam-6493	234	12	,	,	PUNCT
ejpam-6493	234	13	r	r	NOUN
ejpam-6493	234	14	∈	∈	PROPN
ejpam-6493	235	1	i	i	PRON
ejpam-6493	235	2	such	such	VERB
ejpam-6493	235	3	that	that	SCONJ
ejpam-6493	235	4	x+	x+	PROPN
ejpam-6493	235	5	s	s	X
ejpam-6493	235	6	∈	∈	X
ejpam-6493	235	7	i	i	PRON
ejpam-6493	235	8	and	and	CCONJ
ejpam-6493	235	9	y	y	PROPN
ejpam-6493	236	1	+	+	CCONJ
ejpam-6493	236	2	r	r	NOUN
ejpam-6493	236	3	∈	∈	PROPN
ejpam-6493	236	4	i.	i.	NOUN
ejpam-6493	236	5	thus	thus	ADV
ejpam-6493	236	6	a+	a+	PUNCT
ejpam-6493	237	1	x+	x+	PUNCT
ejpam-6493	237	2	r	r	NOUN
ejpam-6493	237	3	+	+	SYM
ejpam-6493	237	4	s	s	NOUN
ejpam-6493	237	5	=	=	PUNCT
ejpam-6493	237	6	b+	b+	X
ejpam-6493	237	7	y	y	PROPN
ejpam-6493	237	8	+	+	NOUN
ejpam-6493	237	9	r	r	NOUN
ejpam-6493	237	10	+	+	SYM
ejpam-6493	237	11	s	s	NOUN
ejpam-6493	237	12	and	and	CCONJ
ejpam-6493	237	13	x	x	PUNCT
ejpam-6493	238	1	+	+	CCONJ
ejpam-6493	238	2	r	r	NOUN
ejpam-6493	238	3	+	+	SYM
ejpam-6493	238	4	s	s	PROPN
ejpam-6493	238	5	,	,	PUNCT
ejpam-6493	238	6	y	y	PROPN
ejpam-6493	239	1	+	+	NOUN
ejpam-6493	239	2	r	r	NOUN
ejpam-6493	239	3	+	+	SYM
ejpam-6493	239	4	s	s	PART
ejpam-6493	239	5	∈	∈	NOUN
ejpam-6493	239	6	i.	i.	NOUN
ejpam-6493	239	7	hence	hence	ADV
ejpam-6493	239	8	a	a	PRON
ejpam-6493	240	1	+	+	X
ejpam-6493	240	2	i	i	NOUN
ejpam-6493	240	3	=	=	SYM
ejpam-6493	240	4	b	b	PROPN
ejpam-6493	240	5	+	+	NUM
ejpam-6493	240	6	i.	i.	NOUN
ejpam-6493	240	7	this	this	PRON
ejpam-6493	240	8	shows	show	VERB
ejpam-6493	240	9	that	that	SCONJ
ejpam-6493	240	10	φ	φ	PROPN
ejpam-6493	240	11	is	be	AUX
ejpam-6493	240	12	one	one	NUM
ejpam-6493	240	13	-	-	PUNCT
ejpam-6493	240	14	to	to	ADP
ejpam-6493	240	15	-	-	PUNCT
ejpam-6493	240	16	one	one	NUM
ejpam-6493	240	17	and	and	CCONJ
ejpam-6493	240	18	r	r	NOUN
ejpam-6493	240	19	/	/	SYM
ejpam-6493	240	20	i	i	PRON
ejpam-6493	240	21	∼=	∼=	ADV
ejpam-6493	240	22	r	r	NOUN
ejpam-6493	240	23	/	/	SYM
ejpam-6493	240	24	ck(i	ck(i	NUM
ejpam-6493	240	25	)	)	PUNCT
ejpam-6493	240	26	.	.	PUNCT
ejpam-6493	241	1	let	let	VERB
ejpam-6493	241	2	r	r	NOUN
ejpam-6493	241	3	and	and	CCONJ
ejpam-6493	241	4	t	t	PROPN
ejpam-6493	241	5	be	be	AUX
ejpam-6493	241	6	ternary	ternary	ADJ
ejpam-6493	241	7	semirings	semiring	NOUN
ejpam-6493	241	8	and	and	CCONJ
ejpam-6493	241	9	φ	φ	NOUN
ejpam-6493	241	10	:	:	PUNCT
ejpam-6493	241	11	r	r	NOUN
ejpam-6493	241	12	→	→	SYM
ejpam-6493	241	13	t	t	PROPN
ejpam-6493	241	14	be	be	AUX
ejpam-6493	241	15	a	a	DET
ejpam-6493	241	16	homomorphism	homomorphism	NOUN
ejpam-6493	241	17	.	.	PUNCT
ejpam-6493	242	1	recall	recall	NOUN
ejpam-6493	242	2	from	from	ADP
ejpam-6493	242	3	proposition	proposition	NOUN
ejpam-6493	242	4	1	1	NUM
ejpam-6493	242	5	that	that	SCONJ
ejpam-6493	242	6	k(φ	k(φ	PROPN
ejpam-6493	242	7	)	)	PUNCT
ejpam-6493	243	1	=	=	PRON
ejpam-6493	243	2	{	{	PUNCT
ejpam-6493	243	3	(	(	PUNCT
ejpam-6493	243	4	a	a	PRON
ejpam-6493	243	5	,	,	PUNCT
ejpam-6493	243	6	b	b	NOUN
ejpam-6493	243	7	)	)	PUNCT
ejpam-6493	243	8	∈	∈	PROPN
ejpam-6493	243	9	r	r	NOUN
ejpam-6493	243	10	×	×	NOUN
ejpam-6493	243	11	r	r	NOUN
ejpam-6493	243	12	|	|	ADV
ejpam-6493	243	13	φ(a	φ(a	ADJ
ejpam-6493	243	14	)	)	PUNCT
ejpam-6493	243	15	=	=	SYM
ejpam-6493	244	1	φ(b	φ(b	NOUN
ejpam-6493	244	2	)	)	PUNCT
ejpam-6493	244	3	}	}	PUNCT
ejpam-6493	244	4	is	be	AUX
ejpam-6493	244	5	a	a	DET
ejpam-6493	244	6	congruence	congruence	NOUN
ejpam-6493	244	7	on	on	ADP
ejpam-6493	244	8	r.	r.	PROPN
ejpam-6493	244	9	let	let	VERB
ejpam-6493	244	10	im(φ	im(φ	PRON
ejpam-6493	244	11	)	)	PUNCT
ejpam-6493	245	1	denote	denote	VERB
ejpam-6493	245	2	the	the	DET
ejpam-6493	245	3	sets	set	NOUN
ejpam-6493	245	4	of	of	ADP
ejpam-6493	245	5	all	all	DET
ejpam-6493	245	6	images	image	NOUN
ejpam-6493	245	7	of	of	ADP
ejpam-6493	245	8	φ	φ	NUM
ejpam-6493	245	9	,	,	PUNCT
ejpam-6493	245	10	that	that	ADV
ejpam-6493	245	11	is	is	ADV
ejpam-6493	245	12	,	,	PUNCT
ejpam-6493	245	13	im(φ	im(φ	PRON
ejpam-6493	245	14	)	)	PUNCT
ejpam-6493	246	1	=	=	SYM
ejpam-6493	246	2	{	{	PUNCT
ejpam-6493	246	3	φ(x	φ(x	NOUN
ejpam-6493	246	4	)	)	PUNCT
ejpam-6493	246	5	|	|	ADV
ejpam-6493	246	6	x	x	SYM
ejpam-6493	246	7	∈	∈	NOUN
ejpam-6493	246	8	r	r	NOUN
ejpam-6493	246	9	}	}	PUNCT
ejpam-6493	246	10	.	.	PUNCT
ejpam-6493	247	1	proposition	proposition	NOUN
ejpam-6493	247	2	3	3	X
ejpam-6493	247	3	.	.	PUNCT
ejpam-6493	248	1	let	let	VERB
ejpam-6493	248	2	r	r	NOUN
ejpam-6493	248	3	and	and	CCONJ
ejpam-6493	248	4	t	t	PROPN
ejpam-6493	248	5	be	be	AUX
ejpam-6493	248	6	ternary	ternary	ADJ
ejpam-6493	248	7	semirings	semiring	NOUN
ejpam-6493	248	8	and	and	CCONJ
ejpam-6493	248	9	φ	φ	NOUN
ejpam-6493	248	10	:	:	PUNCT
ejpam-6493	249	1	r→	r→	PROPN
ejpam-6493	249	2	t	t	PROPN
ejpam-6493	249	3	be	be	AUX
ejpam-6493	249	4	a	a	DET
ejpam-6493	249	5	ternary	ternary	ADJ
ejpam-6493	249	6	semiring	semiring	NOUN
ejpam-6493	249	7	homomorphism	homomorphism	NOUN
ejpam-6493	249	8	.	.	PUNCT
ejpam-6493	250	1	then	then	ADV
ejpam-6493	250	2	r	r	X
ejpam-6493	250	3	/	/	SYM
ejpam-6493	250	4	k(φ	k(φ	PROPN
ejpam-6493	250	5	)	)	PUNCT
ejpam-6493	250	6	∼=	∼=	PROPN
ejpam-6493	250	7	im(φ	im(φ	NUM
ejpam-6493	250	8	)	)	PUNCT
ejpam-6493	250	9	.	.	PUNCT
ejpam-6493	251	1	proof	proof	NOUN
ejpam-6493	251	2	.	.	PUNCT
ejpam-6493	252	1	define	define	VERB
ejpam-6493	252	2	ψ	ψ	X
ejpam-6493	252	3	:	:	PUNCT
ejpam-6493	252	4	r	r	X
ejpam-6493	252	5	/	/	SYM
ejpam-6493	252	6	k(φ	k(φ	PROPN
ejpam-6493	252	7	)	)	PUNCT
ejpam-6493	252	8	→	→	SYM
ejpam-6493	252	9	im(φ	im(φ	NUM
ejpam-6493	252	10	)	)	PUNCT
ejpam-6493	252	11	by	by	ADP
ejpam-6493	252	12	ψ(a	ψ(a	PROPN
ejpam-6493	252	13	+	+	CCONJ
ejpam-6493	252	14	k(φ	k(φ	PROPN
ejpam-6493	252	15	)	)	PUNCT
ejpam-6493	252	16	)	)	PUNCT
ejpam-6493	253	1	=	=	PUNCT
ejpam-6493	253	2	φ(a	φ(a	ADJ
ejpam-6493	253	3	)	)	PUNCT
ejpam-6493	253	4	for	for	SCONJ
ejpam-6493	253	5	all	all	DET
ejpam-6493	253	6	a	a	DET
ejpam-6493	253	7	∈	∈	PROPN
ejpam-6493	253	8	r.	r.	NOUN
ejpam-6493	253	9	let	let	VERB
ejpam-6493	253	10	a	a	DET
ejpam-6493	253	11	+	+	PROPN
ejpam-6493	253	12	k(φ	k(φ	X
ejpam-6493	253	13	)	)	PUNCT
ejpam-6493	254	1	=	=	SYM
ejpam-6493	254	2	b	b	X
ejpam-6493	255	1	+	+	PROPN
ejpam-6493	256	1	k(φ	k(φ	X
ejpam-6493	256	2	)	)	PUNCT
ejpam-6493	256	3	.	.	PUNCT
ejpam-6493	257	1	hence	hence	ADV
ejpam-6493	257	2	φ(a	φ(a	ADJ
ejpam-6493	257	3	)	)	PUNCT
ejpam-6493	258	1	=	=	PUNCT
ejpam-6493	258	2	φ(b	φ(b	NOUN
ejpam-6493	258	3	)	)	PUNCT
ejpam-6493	258	4	.	.	PUNCT
ejpam-6493	259	1	thus	thus	ADV
ejpam-6493	259	2	ψ	ψ	X
ejpam-6493	259	3	is	be	AUX
ejpam-6493	259	4	well	well	ADV
ejpam-6493	259	5	-	-	PUNCT
ejpam-6493	259	6	defined	define	VERB
ejpam-6493	259	7	.	.	PUNCT
ejpam-6493	260	1	clearly	clearly	ADV
ejpam-6493	260	2	,	,	PUNCT
ejpam-6493	260	3	ψ	ψ	X
ejpam-6493	260	4	is	be	AUX
ejpam-6493	260	5	an	an	PRON
ejpam-6493	260	6	onto	onto	ADP
ejpam-6493	260	7	homomorphism	homomorphism	NOUN
ejpam-6493	260	8	.	.	PUNCT
ejpam-6493	261	1	next	next	ADV
ejpam-6493	261	2	,	,	PUNCT
ejpam-6493	261	3	let	let	VERB
ejpam-6493	261	4	a	a	DET
ejpam-6493	261	5	+	+	X
ejpam-6493	261	6	k(φ	k(φ	PROPN
ejpam-6493	261	7	)	)	PUNCT
ejpam-6493	261	8	,	,	PUNCT
ejpam-6493	261	9	b	b	X
ejpam-6493	261	10	+	+	CCONJ
ejpam-6493	262	1	k(φ	k(φ	PROPN
ejpam-6493	262	2	)	)	PUNCT
ejpam-6493	262	3	∈	∈	PROPN
ejpam-6493	262	4	r	r	PROPN
ejpam-6493	262	5	/	/	SYM
ejpam-6493	262	6	k(φ	k(φ	PROPN
ejpam-6493	262	7	)	)	PUNCT
ejpam-6493	262	8	be	be	VERB
ejpam-6493	262	9	such	such	ADJ
ejpam-6493	262	10	that	that	SCONJ
ejpam-6493	262	11	ψ(a	ψ(a	PROPN
ejpam-6493	262	12	+	+	CCONJ
ejpam-6493	262	13	k(φ	k(φ	PROPN
ejpam-6493	262	14	)	)	PUNCT
ejpam-6493	262	15	)	)	PUNCT
ejpam-6493	263	1	=	=	SYM
ejpam-6493	263	2	ψ(b+k(φ	ψ(b+k(φ	NOUN
ejpam-6493	263	3	)	)	PUNCT
ejpam-6493	263	4	)	)	PUNCT
ejpam-6493	263	5	.	.	PUNCT
ejpam-6493	264	1	then	then	ADV
ejpam-6493	264	2	φ(a	φ(a	ADJ
ejpam-6493	264	3	)	)	PUNCT
ejpam-6493	264	4	=	=	SYM
ejpam-6493	264	5	φ(b	φ(b	NOUN
ejpam-6493	264	6	)	)	PUNCT
ejpam-6493	264	7	which	which	PRON
ejpam-6493	264	8	implies	imply	VERB
ejpam-6493	264	9	(	(	PUNCT
ejpam-6493	264	10	a	a	PRON
ejpam-6493	264	11	,	,	PUNCT
ejpam-6493	264	12	b	b	NOUN
ejpam-6493	264	13	)	)	PUNCT
ejpam-6493	264	14	∈	∈	PROPN
ejpam-6493	264	15	k(φ	k(φ	PROPN
ejpam-6493	264	16	)	)	PUNCT
ejpam-6493	264	17	.	.	PUNCT
ejpam-6493	265	1	hence	hence	ADV
ejpam-6493	265	2	a+k(φ	a+k(φ	ADV
ejpam-6493	265	3	)	)	PUNCT
ejpam-6493	265	4	=	=	SYM
ejpam-6493	265	5	b+k(φ	b+k(φ	NOUN
ejpam-6493	265	6	)	)	PUNCT
ejpam-6493	265	7	.	.	PUNCT
ejpam-6493	266	1	proposition	proposition	NOUN
ejpam-6493	266	2	4	4	NUM
ejpam-6493	266	3	.	.	PUNCT
ejpam-6493	267	1	let	let	VERB
ejpam-6493	267	2	r	r	NOUN
ejpam-6493	267	3	and	and	CCONJ
ejpam-6493	267	4	t	t	PROPN
ejpam-6493	267	5	be	be	AUX
ejpam-6493	267	6	ternary	ternary	ADJ
ejpam-6493	267	7	semirings	semiring	NOUN
ejpam-6493	267	8	with	with	ADP
ejpam-6493	267	9	zeroes	zero	NOUN
ejpam-6493	267	10	0r	0r	PROPN
ejpam-6493	267	11	and	and	CCONJ
ejpam-6493	267	12	0	0	NUM
ejpam-6493	267	13	t	t	NOUN
ejpam-6493	267	14	,	,	PUNCT
ejpam-6493	267	15	respectively	respectively	ADV
ejpam-6493	267	16	,	,	PUNCT
ejpam-6493	267	17	and	and	CCONJ
ejpam-6493	268	1	φ	φ	NOUN
ejpam-6493	268	2	:	:	PUNCT
ejpam-6493	269	1	r	r	NOUN
ejpam-6493	269	2	→	→	SYM
ejpam-6493	269	3	t	t	PROPN
ejpam-6493	269	4	be	be	AUX
ejpam-6493	269	5	a	a	DET
ejpam-6493	269	6	homomorphism	homomorphism	NOUN
ejpam-6493	269	7	such	such	ADJ
ejpam-6493	269	8	that	that	DET
ejpam-6493	269	9	φ(0r	φ(0r	NOUN
ejpam-6493	269	10	)	)	PUNCT
ejpam-6493	269	11	=	=	SYM
ejpam-6493	269	12	0	0	NUM
ejpam-6493	269	13	t	t	NOUN
ejpam-6493	269	14	.	.	PUNCT
ejpam-6493	270	1	then	then	ADV
ejpam-6493	270	2	ker(φ	ker(φ	X
ejpam-6493	270	3	)	)	PUNCT
ejpam-6493	270	4	=	=	PRON
ejpam-6493	271	1	{	{	PUNCT
ejpam-6493	271	2	x	x	PUNCT
ejpam-6493	271	3	∈	∈	NOUN
ejpam-6493	271	4	r	r	NOUN
ejpam-6493	271	5	|	|	NOUN
ejpam-6493	271	6	φ(x	φ(x	NOUN
ejpam-6493	271	7	)	)	PUNCT
ejpam-6493	271	8	=	=	PUNCT
ejpam-6493	271	9	0	0	NUM
ejpam-6493	271	10	t	t	NOUN
ejpam-6493	271	11	}	}	PUNCT
ejpam-6493	271	12	is	be	AUX
ejpam-6493	271	13	a	a	DET
ejpam-6493	271	14	k	k	NOUN
ejpam-6493	271	15	-	-	NOUN
ejpam-6493	271	16	ideal	ideal	NOUN
ejpam-6493	271	17	of	of	ADP
ejpam-6493	271	18	r.	r.	PROPN
ejpam-6493	271	19	proof	proof	NOUN
ejpam-6493	271	20	.	.	PUNCT
ejpam-6493	272	1	let	let	VERB
ejpam-6493	272	2	φ	φ	NOUN
ejpam-6493	272	3	:	:	PUNCT
ejpam-6493	272	4	r	r	X
ejpam-6493	272	5	→	→	SYM
ejpam-6493	272	6	t	t	PROPN
ejpam-6493	272	7	be	be	AUX
ejpam-6493	272	8	a	a	DET
ejpam-6493	272	9	homomorphism	homomorphism	NOUN
ejpam-6493	272	10	where	where	SCONJ
ejpam-6493	272	11	r	r	NOUN
ejpam-6493	272	12	and	and	CCONJ
ejpam-6493	272	13	t	t	PROPN
ejpam-6493	272	14	are	be	AUX
ejpam-6493	272	15	ternary	ternary	ADJ
ejpam-6493	272	16	semirings	semiring	NOUN
ejpam-6493	272	17	.	.	PUNCT
ejpam-6493	273	1	since	since	SCONJ
ejpam-6493	273	2	0r	0r	PROPN
ejpam-6493	273	3	∈	∈	PROPN
ejpam-6493	273	4	ker(φ	ker(φ	PROPN
ejpam-6493	273	5	)	)	PUNCT
ejpam-6493	273	6	,	,	PUNCT
ejpam-6493	273	7	we	we	PRON
ejpam-6493	273	8	see	see	VERB
ejpam-6493	273	9	that	that	SCONJ
ejpam-6493	273	10	ker(φ	ker(φ	NOUN
ejpam-6493	273	11	)	)	PUNCT
ejpam-6493	273	12	̸=	̸=	PROPN
ejpam-6493	273	13	∅.	∅.	ADV
ejpam-6493	273	14	let	let	VERB
ejpam-6493	273	15	a	a	DET
ejpam-6493	273	16	,	,	PUNCT
ejpam-6493	273	17	b	b	X
ejpam-6493	273	18	∈	∈	PROPN
ejpam-6493	273	19	ker(φ	ker(φ	NOUN
ejpam-6493	273	20	)	)	PUNCT
ejpam-6493	273	21	.	.	PUNCT
ejpam-6493	274	1	therefore	therefore	ADV
ejpam-6493	274	2	φ(a	φ(a	ADJ
ejpam-6493	274	3	)	)	PUNCT
ejpam-6493	274	4	=	=	SYM
ejpam-6493	274	5	0	0	NUM
ejpam-6493	274	6	t	t	NOUN
ejpam-6493	274	7	and	and	CCONJ
ejpam-6493	274	8	φ(b	φ(b	PROPN
ejpam-6493	274	9	)	)	PUNCT
ejpam-6493	274	10	=	=	SYM
ejpam-6493	274	11	0	0	NUM
ejpam-6493	274	12	t	t	NOUN
ejpam-6493	274	13	.	.	PUNCT
ejpam-6493	275	1	since	since	SCONJ
ejpam-6493	275	2	φ	φ	PROPN
ejpam-6493	275	3	is	be	AUX
ejpam-6493	275	4	a	a	DET
ejpam-6493	275	5	homomorphism	homomorphism	NOUN
ejpam-6493	275	6	,	,	PUNCT
ejpam-6493	275	7	we	we	PRON
ejpam-6493	275	8	have	have	AUX
ejpam-6493	275	9	φ(a	φ(a	ADJ
ejpam-6493	276	1	+	+	CCONJ
ejpam-6493	277	1	b	b	X
ejpam-6493	277	2	)	)	PUNCT
ejpam-6493	277	3	=	=	SYM
ejpam-6493	277	4	φ(a	φ(a	ADJ
ejpam-6493	277	5	)	)	PUNCT
ejpam-6493	278	1	+	+	ADJ
ejpam-6493	278	2	φ(b	φ(b	NOUN
ejpam-6493	278	3	)	)	PUNCT
ejpam-6493	278	4	=	=	SYM
ejpam-6493	278	5	0	0	NUM
ejpam-6493	278	6	t	t	NOUN
ejpam-6493	278	7	.	.	PUNCT
ejpam-6493	279	1	this	this	PRON
ejpam-6493	279	2	implies	imply	VERB
ejpam-6493	279	3	that	that	SCONJ
ejpam-6493	279	4	a+	a+	PUNCT
ejpam-6493	279	5	b	b	X
ejpam-6493	279	6	∈	∈	PROPN
ejpam-6493	279	7	ker(φ	ker(φ	NOUN
ejpam-6493	279	8	)	)	PUNCT
ejpam-6493	279	9	.	.	PUNCT
ejpam-6493	280	1	next	next	ADV
ejpam-6493	280	2	,	,	PUNCT
ejpam-6493	280	3	we	we	PRON
ejpam-6493	280	4	let	let	VERB
ejpam-6493	280	5	r	r	NOUN
ejpam-6493	280	6	∈	∈	PROPN
ejpam-6493	280	7	ker(φ	ker(φ	NOUN
ejpam-6493	280	8	)	)	PUNCT
ejpam-6493	280	9	and	and	CCONJ
ejpam-6493	280	10	a	a	DET
ejpam-6493	280	11	,	,	PUNCT
ejpam-6493	280	12	b	b	X
ejpam-6493	280	13	∈	∈	PROPN
ejpam-6493	280	14	r.	r.	NOUN
ejpam-6493	280	15	we	we	PRON
ejpam-6493	280	16	have	have	VERB
ejpam-6493	280	17	that	that	DET
ejpam-6493	280	18	φ(rab	φ(rab	NOUN
ejpam-6493	280	19	)	)	PUNCT
ejpam-6493	280	20	=	=	SYM
ejpam-6493	280	21	φ(r)φ(a)φ(b	φ(r)φ(a)φ(b	PROPN
ejpam-6493	280	22	)	)	PUNCT
ejpam-6493	280	23	=	=	SYM
ejpam-6493	280	24	0tφ(a)φ(b	0tφ(a)φ(b	NUM
ejpam-6493	280	25	)	)	PUNCT
ejpam-6493	280	26	=	=	SYM
ejpam-6493	280	27	0	0	NUM
ejpam-6493	280	28	t	t	NOUN
ejpam-6493	280	29	.	.	PUNCT
ejpam-6493	281	1	hence	hence	ADV
ejpam-6493	281	2	rab	rab	PROPN
ejpam-6493	281	3	∈	∈	PROPN
ejpam-6493	281	4	ker(φ	ker(φ	PROPN
ejpam-6493	281	5	)	)	PUNCT
ejpam-6493	281	6	.	.	PUNCT
ejpam-6493	282	1	similarly	similarly	ADV
ejpam-6493	282	2	,	,	PUNCT
ejpam-6493	282	3	arb	arb	PROPN
ejpam-6493	282	4	,	,	PUNCT
ejpam-6493	282	5	abr	abr	NOUN
ejpam-6493	282	6	∈	∈	PROPN
ejpam-6493	282	7	ker(φ	ker(φ	PROPN
ejpam-6493	282	8	)	)	PUNCT
ejpam-6493	282	9	.	.	PUNCT
ejpam-6493	283	1	then	then	ADV
ejpam-6493	283	2	ker(φ	ker(φ	X
ejpam-6493	283	3	)	)	PUNCT
ejpam-6493	283	4	is	be	AUX
ejpam-6493	283	5	an	an	DET
ejpam-6493	283	6	ideal	ideal	NOUN
ejpam-6493	283	7	of	of	ADP
ejpam-6493	283	8	r.	r.	PROPN
ejpam-6493	283	9	furthermore	furthermore	ADV
ejpam-6493	283	10	,	,	PUNCT
ejpam-6493	283	11	let	let	VERB
ejpam-6493	283	12	a	a	PRON
ejpam-6493	283	13	,	,	PUNCT
ejpam-6493	283	14	b	b	X
ejpam-6493	283	15	∈	∈	NOUN
ejpam-6493	283	16	r	r	NOUN
ejpam-6493	283	17	be	be	VERB
ejpam-6493	283	18	such	such	ADJ
ejpam-6493	283	19	that	that	SCONJ
ejpam-6493	283	20	a	a	DET
ejpam-6493	283	21	+	+	NOUN
ejpam-6493	283	22	b	b	NOUN
ejpam-6493	283	23	∈	∈	ADJ
ejpam-6493	283	24	ker(φ	ker(φ	NOUN
ejpam-6493	283	25	)	)	PUNCT
ejpam-6493	283	26	and	and	CCONJ
ejpam-6493	283	27	a	a	DET
ejpam-6493	283	28	∈	∈	PROPN
ejpam-6493	283	29	ker(φ	ker(φ	NOUN
ejpam-6493	283	30	)	)	PUNCT
ejpam-6493	283	31	.	.	PUNCT
ejpam-6493	284	1	then	then	ADV
ejpam-6493	284	2	0	0	NUM
ejpam-6493	284	3	t	t	NOUN
ejpam-6493	284	4	=	=	SYM
ejpam-6493	284	5	φ(a	φ(a	PROPN
ejpam-6493	284	6	+	+	NUM
ejpam-6493	284	7	b	b	X
ejpam-6493	284	8	)	)	PUNCT
ejpam-6493	284	9	=	=	SYM
ejpam-6493	284	10	φ(a	φ(a	ADJ
ejpam-6493	284	11	)	)	PUNCT
ejpam-6493	285	1	+	+	ADJ
ejpam-6493	285	2	φ(b	φ(b	NOUN
ejpam-6493	285	3	)	)	PUNCT
ejpam-6493	285	4	=	=	SYM
ejpam-6493	285	5	0	0	NUM
ejpam-6493	285	6	t	t	NOUN
ejpam-6493	285	7	+	+	X
ejpam-6493	285	8	φ(b	φ(b	NOUN
ejpam-6493	285	9	)	)	PUNCT
ejpam-6493	285	10	=	=	SYM
ejpam-6493	286	1	φ(b	φ(b	NOUN
ejpam-6493	286	2	)	)	PUNCT
ejpam-6493	286	3	,	,	PUNCT
ejpam-6493	286	4	so	so	ADV
ejpam-6493	286	5	b	b	X
ejpam-6493	286	6	∈	∈	PROPN
ejpam-6493	286	7	ker(φ	ker(φ	NOUN
ejpam-6493	286	8	)	)	PUNCT
ejpam-6493	286	9	.	.	PUNCT
ejpam-6493	287	1	therefore	therefore	ADV
ejpam-6493	287	2	,	,	PUNCT
ejpam-6493	287	3	ker(φ	ker(φ	X
ejpam-6493	287	4	)	)	PUNCT
ejpam-6493	287	5	is	be	AUX
ejpam-6493	287	6	a	a	DET
ejpam-6493	287	7	k	k	NOUN
ejpam-6493	287	8	-	-	NOUN
ejpam-6493	287	9	ideal	ideal	NOUN
ejpam-6493	287	10	of	of	ADP
ejpam-6493	287	11	r.	r.	PROPN
ejpam-6493	287	12	let	let	VERB
ejpam-6493	287	13	r	r	NOUN
ejpam-6493	287	14	and	and	CCONJ
ejpam-6493	287	15	t	t	PROPN
ejpam-6493	287	16	be	be	AUX
ejpam-6493	287	17	ternary	ternary	ADJ
ejpam-6493	287	18	semirings	semiring	NOUN
ejpam-6493	287	19	with	with	ADP
ejpam-6493	287	20	zeroes	zero	NOUN
ejpam-6493	287	21	0r	0r	PROPN
ejpam-6493	287	22	and	and	CCONJ
ejpam-6493	287	23	0	0	NUM
ejpam-6493	287	24	t	t	NOUN
ejpam-6493	287	25	,	,	PUNCT
ejpam-6493	287	26	respectively	respectively	ADV
ejpam-6493	287	27	,	,	PUNCT
ejpam-6493	287	28	and	and	CCONJ
ejpam-6493	287	29	φ	φ	NUM
ejpam-6493	287	30	:	:	PUNCT
ejpam-6493	288	1	r→	r→	PROPN
ejpam-6493	288	2	t	t	PROPN
ejpam-6493	288	3	be	be	AUX
ejpam-6493	288	4	a	a	DET
ejpam-6493	288	5	homomorphism	homomorphism	NOUN
ejpam-6493	288	6	such	such	ADJ
ejpam-6493	288	7	that	that	DET
ejpam-6493	288	8	φ(0r	φ(0r	NOUN
ejpam-6493	288	9	)	)	PUNCT
ejpam-6493	288	10	=	=	SYM
ejpam-6493	288	11	0	0	NUM
ejpam-6493	288	12	t	t	NOUN
ejpam-6493	288	13	.	.	PUNCT
ejpam-6493	289	1	a	a	DET
ejpam-6493	289	2	homomorphism	homomorphism	NOUN
ejpam-6493	289	3	φ	φ	PROPN
ejpam-6493	289	4	is	be	AUX
ejpam-6493	289	5	called	call	VERB
ejpam-6493	289	6	a	a	DET
ejpam-6493	289	7	k	k	ADJ
ejpam-6493	289	8	-	-	PUNCT
ejpam-6493	289	9	kernel	kernel	NOUN
ejpam-6493	289	10	homomorphism	homomorphism	NOUN
ejpam-6493	289	11	if	if	SCONJ
ejpam-6493	289	12	for	for	ADP
ejpam-6493	289	13	all	all	DET
ejpam-6493	289	14	s1	s1	NOUN
ejpam-6493	289	15	,	,	PUNCT
ejpam-6493	289	16	s2	s2	NOUN
ejpam-6493	289	17	∈	∈	PROPN
ejpam-6493	289	18	r	r	NOUN
ejpam-6493	289	19	,	,	PUNCT
ejpam-6493	289	20	φ(s1	φ(s1	NOUN
ejpam-6493	289	21	)	)	PUNCT
ejpam-6493	290	1	=	=	SYM
ejpam-6493	290	2	φ(s2	φ(s2	NOUN
ejpam-6493	290	3	)	)	PUNCT
ejpam-6493	290	4	implies	imply	VERB
ejpam-6493	290	5	s1	s1	NOUN
ejpam-6493	290	6	+	+	CCONJ
ejpam-6493	290	7	a	a	DET
ejpam-6493	290	8	=	=	X
ejpam-6493	290	9	s2	s2	NOUN
ejpam-6493	290	10	+	+	CCONJ
ejpam-6493	290	11	b	b	NOUN
ejpam-6493	290	12	for	for	ADP
ejpam-6493	290	13	some	some	DET
ejpam-6493	290	14	a	a	PRON
ejpam-6493	290	15	,	,	PUNCT
ejpam-6493	290	16	b	b	X
ejpam-6493	290	17	∈	∈	PROPN
ejpam-6493	290	18	ker(φ	ker(φ	NOUN
ejpam-6493	290	19	)	)	PUNCT
ejpam-6493	290	20	.	.	PUNCT
ejpam-6493	291	1	next	next	ADV
ejpam-6493	291	2	,	,	PUNCT
ejpam-6493	291	3	we	we	PRON
ejpam-6493	291	4	provide	provide	VERB
ejpam-6493	291	5	an	an	DET
ejpam-6493	291	6	analogue	analogue	NOUN
ejpam-6493	291	7	of	of	ADP
ejpam-6493	291	8	the	the	DET
ejpam-6493	291	9	first	first	ADJ
ejpam-6493	291	10	isomorphism	isomorphism	NOUN
ejpam-6493	291	11	theorem	theorem	NOUN
ejpam-6493	291	12	.	.	PUNCT
ejpam-6493	291	13	theorem	theorem	NOUN
ejpam-6493	291	14	3	3	X
ejpam-6493	291	15	.	.	PUNCT
ejpam-6493	292	1	let	let	VERB
ejpam-6493	292	2	r	r	NOUN
ejpam-6493	292	3	and	and	CCONJ
ejpam-6493	292	4	t	t	PROPN
ejpam-6493	292	5	be	be	AUX
ejpam-6493	292	6	ternary	ternary	ADJ
ejpam-6493	292	7	semirings	semiring	NOUN
ejpam-6493	292	8	and	and	CCONJ
ejpam-6493	292	9	φ	φ	NOUN
ejpam-6493	292	10	:	:	PUNCT
ejpam-6493	293	1	r→	r→	PROPN
ejpam-6493	293	2	t	t	PROPN
ejpam-6493	293	3	be	be	AUX
ejpam-6493	293	4	a	a	DET
ejpam-6493	293	5	homomorphism	homomorphism	NOUN
ejpam-6493	293	6	with	with	ADP
ejpam-6493	293	7	ker(φ	ker(φ	NOUN
ejpam-6493	293	8	)	)	PUNCT
ejpam-6493	293	9	.	.	PUNCT
ejpam-6493	294	1	if	if	SCONJ
ejpam-6493	294	2	φ	φ	PROPN
ejpam-6493	294	3	is	be	AUX
ejpam-6493	294	4	a	a	DET
ejpam-6493	294	5	k	k	ADJ
ejpam-6493	294	6	-	-	PUNCT
ejpam-6493	294	7	kernel	kernel	NOUN
ejpam-6493	294	8	homomorphism	homomorphism	NOUN
ejpam-6493	294	9	,	,	PUNCT
ejpam-6493	294	10	then	then	ADV
ejpam-6493	294	11	r	r	X
ejpam-6493	294	12	/	/	SYM
ejpam-6493	294	13	ker(φ	ker(φ	NOUN
ejpam-6493	294	14	)	)	PUNCT
ejpam-6493	294	15	∼=	∼=	PROPN
ejpam-6493	294	16	im(φ	im(φ	NUM
ejpam-6493	294	17	)	)	PUNCT
ejpam-6493	294	18	.	.	PUNCT
ejpam-6493	295	1	proof	proof	NOUN
ejpam-6493	295	2	.	.	PUNCT
ejpam-6493	296	1	assume	assume	VERB
ejpam-6493	296	2	that	that	SCONJ
ejpam-6493	296	3	φ	φ	PROPN
ejpam-6493	296	4	is	be	AUX
ejpam-6493	296	5	a	a	DET
ejpam-6493	296	6	k	k	ADJ
ejpam-6493	296	7	-	-	PUNCT
ejpam-6493	296	8	kernel	kernel	NOUN
ejpam-6493	296	9	homomorphism	homomorphism	NOUN
ejpam-6493	296	10	.	.	PUNCT
ejpam-6493	297	1	let	let	AUX
ejpam-6493	297	2	ψ	ψ	X
ejpam-6493	297	3	:	:	PUNCT
ejpam-6493	297	4	r	r	X
ejpam-6493	297	5	/	/	SYM
ejpam-6493	297	6	ker(φ	ker(φ	NOUN
ejpam-6493	297	7	)	)	PUNCT
ejpam-6493	297	8	→	→	SYM
ejpam-6493	297	9	im(φ	im(φ	NUM
ejpam-6493	297	10	)	)	PUNCT
ejpam-6493	297	11	be	be	AUX
ejpam-6493	297	12	defined	define	VERB
ejpam-6493	297	13	by	by	ADP
ejpam-6493	297	14	ψ(s+ker(φ	ψ(s+ker(φ	ADJ
ejpam-6493	297	15	)	)	PUNCT
ejpam-6493	297	16	)	)	PUNCT
ejpam-6493	298	1	=	=	SYM
ejpam-6493	298	2	φ(s	φ(s	NOUN
ejpam-6493	298	3	)	)	PUNCT
ejpam-6493	298	4	for	for	ADP
ejpam-6493	298	5	all	all	PRON
ejpam-6493	298	6	s	s	PROPN
ejpam-6493	298	7	∈	∈	PROPN
ejpam-6493	298	8	r.	r.	NOUN
ejpam-6493	298	9	let	let	VERB
ejpam-6493	298	10	s1	s1	NOUN
ejpam-6493	298	11	,	,	PUNCT
ejpam-6493	298	12	s2	s2	PROPN
ejpam-6493	298	13	,	,	PUNCT
ejpam-6493	298	14	s3	s3	PROPN
ejpam-6493	298	15	be	be	VERB
ejpam-6493	298	16	any	any	DET
ejpam-6493	298	17	three	three	NUM
ejpam-6493	298	18	elements	element	NOUN
ejpam-6493	298	19	in	in	ADP
ejpam-6493	298	20	r.	r.	PROPN
ejpam-6493	298	21	to	to	PART
ejpam-6493	298	22	prove	prove	VERB
ejpam-6493	298	23	that	that	SCONJ
ejpam-6493	298	24	ψ	ψ	NOUN
ejpam-6493	298	25	is	be	AUX
ejpam-6493	298	26	well	well	ADV
ejpam-6493	298	27	-	-	PUNCT
ejpam-6493	298	28	defined	define	VERB
ejpam-6493	298	29	,	,	PUNCT
ejpam-6493	298	30	assume	assume	VERB
ejpam-6493	298	31	that	that	SCONJ
ejpam-6493	298	32	s1	s1	NOUN
ejpam-6493	298	33	+	+	CCONJ
ejpam-6493	298	34	ker(φ	ker(φ	NOUN
ejpam-6493	298	35	)	)	PUNCT
ejpam-6493	298	36	=	=	SYM
ejpam-6493	298	37	s2	s2	NOUN
ejpam-6493	298	38	+	+	CCONJ
ejpam-6493	298	39	ker(φ	ker(φ	NOUN
ejpam-6493	298	40	)	)	PUNCT
ejpam-6493	298	41	.	.	PUNCT
ejpam-6493	299	1	then	then	ADV
ejpam-6493	299	2	s1	s1	PROPN
ejpam-6493	299	3	+	+	CCONJ
ejpam-6493	299	4	a	a	DET
ejpam-6493	299	5	=	=	X
ejpam-6493	299	6	s2	s2	NOUN
ejpam-6493	299	7	+	+	CCONJ
ejpam-6493	299	8	b	b	NOUN
ejpam-6493	299	9	for	for	ADP
ejpam-6493	299	10	some	some	DET
ejpam-6493	299	11	a	a	PRON
ejpam-6493	299	12	,	,	PUNCT
ejpam-6493	299	13	b	b	X
ejpam-6493	299	14	∈	∈	PROPN
ejpam-6493	299	15	ker(φ	ker(φ	NOUN
ejpam-6493	299	16	)	)	PUNCT
ejpam-6493	299	17	.	.	PUNCT
ejpam-6493	300	1	so	so	ADV
ejpam-6493	300	2	φ(a	φ(a	ADJ
ejpam-6493	300	3	)	)	PUNCT
ejpam-6493	301	1	=	=	PUNCT
ejpam-6493	301	2	φ(b	φ(b	X
ejpam-6493	301	3	)	)	PUNCT
ejpam-6493	301	4	=	=	SYM
ejpam-6493	301	5	0	0	NUM
ejpam-6493	301	6	t	t	NOUN
ejpam-6493	301	7	and	and	CCONJ
ejpam-6493	301	8	φ(s1	φ(s1	NOUN
ejpam-6493	301	9	)	)	PUNCT
ejpam-6493	301	10	=	=	SYM
ejpam-6493	301	11	φ(s1	φ(s1	NOUN
ejpam-6493	301	12	)	)	PUNCT
ejpam-6493	301	13	+	+	CCONJ
ejpam-6493	301	14	φ(a	φ(a	ADJ
ejpam-6493	301	15	)	)	PUNCT
ejpam-6493	301	16	=	=	SYM
ejpam-6493	301	17	φ(s1	φ(s1	NOUN
ejpam-6493	301	18	+	+	CCONJ
ejpam-6493	301	19	a	a	X
ejpam-6493	301	20	)	)	PUNCT
ejpam-6493	301	21	=	=	SYM
ejpam-6493	301	22	φ(s2	φ(s2	NOUN
ejpam-6493	302	1	+	+	CCONJ
ejpam-6493	302	2	b	b	X
ejpam-6493	302	3	)	)	PUNCT
ejpam-6493	302	4	=	=	SYM
ejpam-6493	302	5	φ(s2	φ(s2	NOUN
ejpam-6493	302	6	)	)	PUNCT
ejpam-6493	303	1	+	+	ADJ
ejpam-6493	303	2	φ(b	φ(b	NOUN
ejpam-6493	303	3	)	)	PUNCT
ejpam-6493	303	4	=	=	SYM
ejpam-6493	303	5	φ(s2	φ(s2	NOUN
ejpam-6493	303	6	)	)	PUNCT
ejpam-6493	303	7	.	.	PUNCT
ejpam-6493	304	1	m.	m.	NOUN
ejpam-6493	304	2	petapirak	petapirak	PROPN
ejpam-6493	304	3	,	,	PUNCT
ejpam-6493	304	4	a.	a.	PROPN
ejpam-6493	304	5	j.	j.	PROPN
ejpam-6493	304	6	khan	khan	PROPN
ejpam-6493	304	7	,	,	PUNCT
ejpam-6493	304	8	r.	r.	PROPN
ejpam-6493	304	9	chinram	chinram	PROPN
ejpam-6493	304	10	/	/	SYM
ejpam-6493	304	11	eur	eur	PROPN
ejpam-6493	304	12	.	.	PUNCT
ejpam-6493	305	1	j.	j.	PROPN
ejpam-6493	305	2	pure	pure	PROPN
ejpam-6493	305	3	appl	appl	PROPN
ejpam-6493	305	4	.	.	PROPN
ejpam-6493	305	5	math	math	PROPN
ejpam-6493	305	6	,	,	PUNCT
ejpam-6493	305	7	18	18	NUM
ejpam-6493	305	8	(	(	PUNCT
ejpam-6493	305	9	3	3	NUM
ejpam-6493	305	10	)	)	PUNCT
ejpam-6493	305	11	(	(	PUNCT
ejpam-6493	305	12	2025	2025	NUM
ejpam-6493	305	13	)	)	PUNCT
ejpam-6493	305	14	,	,	PUNCT
ejpam-6493	305	15	6493	6493	NUM
ejpam-6493	305	16	8	8	NUM
ejpam-6493	305	17	of	of	ADP
ejpam-6493	305	18	9	9	NUM
ejpam-6493	305	19	in	in	ADP
ejpam-6493	305	20	addition	addition	NOUN
ejpam-6493	305	21	,	,	PUNCT
ejpam-6493	305	22	as	as	SCONJ
ejpam-6493	305	23	φ	φ	PROPN
ejpam-6493	305	24	is	be	AUX
ejpam-6493	305	25	a	a	DET
ejpam-6493	305	26	homomorphism	homomorphism	NOUN
ejpam-6493	305	27	,	,	PUNCT
ejpam-6493	305	28	we	we	PRON
ejpam-6493	305	29	have	have	VERB
ejpam-6493	305	30	ψ((s1	ψ((s1	NOUN
ejpam-6493	305	31	+	+	CCONJ
ejpam-6493	305	32	ker(φ	ker(φ	NOUN
ejpam-6493	305	33	)	)	PUNCT
ejpam-6493	305	34	)	)	PUNCT
ejpam-6493	306	1	+	+	CCONJ
ejpam-6493	306	2	(	(	PUNCT
ejpam-6493	306	3	s2	s2	NOUN
ejpam-6493	306	4	+	+	CCONJ
ejpam-6493	306	5	ker(φ	ker(φ	NOUN
ejpam-6493	306	6	)	)	PUNCT
ejpam-6493	306	7	)	)	PUNCT
ejpam-6493	306	8	)	)	PUNCT
ejpam-6493	307	1	=	=	PRON
ejpam-6493	307	2	ψ(s1	ψ(s1	NOUN
ejpam-6493	307	3	+	+	CCONJ
ejpam-6493	307	4	s2	s2	NOUN
ejpam-6493	307	5	+	+	CCONJ
ejpam-6493	307	6	ker(φ	ker(φ	NOUN
ejpam-6493	307	7	)	)	PUNCT
ejpam-6493	307	8	)	)	PUNCT
ejpam-6493	308	1	=	=	SYM
ejpam-6493	308	2	φ(s1	φ(s1	NOUN
ejpam-6493	308	3	+	+	CCONJ
ejpam-6493	308	4	s2	s2	PROPN
ejpam-6493	308	5	)	)	PUNCT
ejpam-6493	308	6	=	=	SYM
ejpam-6493	308	7	φ(s1	φ(s1	NOUN
ejpam-6493	308	8	)	)	PUNCT
ejpam-6493	308	9	+	+	CCONJ
ejpam-6493	308	10	φ(s2	φ(s2	NOUN
ejpam-6493	308	11	)	)	PUNCT
ejpam-6493	308	12	=	=	SYM
ejpam-6493	308	13	ψ(s1	ψ(s1	NOUN
ejpam-6493	308	14	+	+	X
ejpam-6493	308	15	ker(φ	ker(φ	NOUN
ejpam-6493	308	16	)	)	PUNCT
ejpam-6493	308	17	)	)	PUNCT
ejpam-6493	309	1	+	+	CCONJ
ejpam-6493	309	2	ψ(s2	ψ(s2	NOUN
ejpam-6493	309	3	+	+	X
ejpam-6493	309	4	ker(φ	ker(φ	NOUN
ejpam-6493	309	5	)	)	PUNCT
ejpam-6493	309	6	)	)	PUNCT
ejpam-6493	309	7	and	and	CCONJ
ejpam-6493	309	8	ψ((s1	ψ((s1	X
ejpam-6493	309	9	+	+	CCONJ
ejpam-6493	309	10	ker(φ))(s2	ker(φ))(s2	PROPN
ejpam-6493	309	11	+	+	X
ejpam-6493	309	12	ker(φ))(s3	ker(φ))(s3	NOUN
ejpam-6493	309	13	+	+	CCONJ
ejpam-6493	309	14	ker(φ	ker(φ	NOUN
ejpam-6493	309	15	)	)	PUNCT
ejpam-6493	309	16	)	)	PUNCT
ejpam-6493	309	17	)	)	PUNCT
ejpam-6493	310	1	=	=	SYM
ejpam-6493	310	2	ψ(s1s2s3	ψ(s1s2s3	PROPN
ejpam-6493	310	3	+	+	CCONJ
ejpam-6493	310	4	ker(φ	ker(φ	NOUN
ejpam-6493	310	5	)	)	PUNCT
ejpam-6493	310	6	)	)	PUNCT
ejpam-6493	311	1	=	=	SYM
ejpam-6493	311	2	φ(s1s2s3	φ(s1s2s3	PROPN
ejpam-6493	311	3	)	)	PUNCT
ejpam-6493	311	4	=	=	SYM
ejpam-6493	311	5	φ(s1)φ(s2)φ(s3	φ(s1)φ(s2)φ(s3	NOUN
ejpam-6493	311	6	)	)	PUNCT
ejpam-6493	311	7	=	=	SYM
ejpam-6493	311	8	ψ(s1	ψ(s1	NOUN
ejpam-6493	311	9	+	+	CCONJ
ejpam-6493	311	10	ker(φ))ψ(s2	ker(φ))ψ(s2	X
ejpam-6493	311	11	+	+	CCONJ
ejpam-6493	311	12	ker(φ))ψ(s3	ker(φ))ψ(s3	NOUN
ejpam-6493	311	13	+	+	X
ejpam-6493	311	14	ker(φ	ker(φ	NOUN
ejpam-6493	311	15	)	)	PUNCT
ejpam-6493	311	16	)	)	PUNCT
ejpam-6493	311	17	.	.	PUNCT
ejpam-6493	312	1	hence	hence	ADV
ejpam-6493	312	2	,	,	PUNCT
ejpam-6493	312	3	ψ	ψ	X
ejpam-6493	312	4	is	be	AUX
ejpam-6493	312	5	a	a	DET
ejpam-6493	312	6	homomorphism	homomorphism	NOUN
ejpam-6493	312	7	.	.	PUNCT
ejpam-6493	313	1	clearly	clearly	ADV
ejpam-6493	313	2	,	,	PUNCT
ejpam-6493	313	3	ψ	ψ	X
ejpam-6493	313	4	is	be	AUX
ejpam-6493	313	5	onto	onto	ADP
ejpam-6493	313	6	.	.	PUNCT
ejpam-6493	314	1	next	next	ADV
ejpam-6493	314	2	,	,	PUNCT
ejpam-6493	314	3	we	we	PRON
ejpam-6493	314	4	will	will	AUX
ejpam-6493	314	5	show	show	VERB
ejpam-6493	314	6	that	that	SCONJ
ejpam-6493	314	7	ψ	ψ	NOUN
ejpam-6493	314	8	is	be	AUX
ejpam-6493	314	9	oneto	oneto	NOUN
ejpam-6493	314	10	-	-	PUNCT
ejpam-6493	314	11	one	one	NUM
ejpam-6493	314	12	.	.	PUNCT
ejpam-6493	315	1	assume	assume	VERB
ejpam-6493	315	2	that	that	SCONJ
ejpam-6493	315	3	s1	s1	NOUN
ejpam-6493	315	4	,	,	PUNCT
ejpam-6493	315	5	s2	s2	NOUN
ejpam-6493	315	6	∈	∈	PROPN
ejpam-6493	315	7	r	r	NOUN
ejpam-6493	315	8	such	such	ADJ
ejpam-6493	315	9	that	that	DET
ejpam-6493	315	10	ψ(s1	ψ(s1	NOUN
ejpam-6493	315	11	+	+	X
ejpam-6493	315	12	ker(φ	ker(φ	NOUN
ejpam-6493	315	13	)	)	PUNCT
ejpam-6493	315	14	)	)	PUNCT
ejpam-6493	315	15	=	=	PRON
ejpam-6493	316	1	ψ(s2	ψ(s2	NOUN
ejpam-6493	317	1	+	+	X
ejpam-6493	317	2	ker(φ	ker(φ	NOUN
ejpam-6493	317	3	)	)	PUNCT
ejpam-6493	317	4	)	)	PUNCT
ejpam-6493	317	5	.	.	PUNCT
ejpam-6493	318	1	then	then	ADV
ejpam-6493	318	2	φ(s1	φ(s1	NOUN
ejpam-6493	318	3	)	)	PUNCT
ejpam-6493	318	4	=	=	SYM
ejpam-6493	318	5	φ(s2	φ(s2	NOUN
ejpam-6493	318	6	)	)	PUNCT
ejpam-6493	318	7	.	.	PUNCT
ejpam-6493	319	1	by	by	ADP
ejpam-6493	319	2	assumption	assumption	NOUN
ejpam-6493	319	3	,	,	PUNCT
ejpam-6493	319	4	we	we	PRON
ejpam-6493	319	5	obtain	obtain	VERB
ejpam-6493	319	6	s1+a	s1+a	ADP
ejpam-6493	319	7	=	=	NOUN
ejpam-6493	319	8	s2+b	s2+b	VERB
ejpam-6493	319	9	for	for	ADP
ejpam-6493	319	10	some	some	DET
ejpam-6493	319	11	a	a	DET
ejpam-6493	319	12	,	,	PUNCT
ejpam-6493	319	13	b	b	X
ejpam-6493	319	14	∈	∈	PROPN
ejpam-6493	319	15	ker(φ	ker(φ	NOUN
ejpam-6493	319	16	)	)	PUNCT
ejpam-6493	319	17	.	.	PUNCT
ejpam-6493	320	1	therefore	therefore	ADV
ejpam-6493	320	2	,	,	PUNCT
ejpam-6493	320	3	s1	s1	PROPN
ejpam-6493	320	4	+	+	CCONJ
ejpam-6493	320	5	ker(φ	ker(φ	NOUN
ejpam-6493	320	6	)	)	PUNCT
ejpam-6493	320	7	=	=	SYM
ejpam-6493	320	8	s2	s2	NOUN
ejpam-6493	320	9	+	+	CCONJ
ejpam-6493	320	10	ker(φ	ker(φ	X
ejpam-6493	320	11	)	)	PUNCT
ejpam-6493	320	12	and	and	CCONJ
ejpam-6493	320	13	this	this	PRON
ejpam-6493	320	14	leads	lead	VERB
ejpam-6493	320	15	to	to	ADP
ejpam-6493	320	16	the	the	DET
ejpam-6493	320	17	result	result	NOUN
ejpam-6493	320	18	r	r	X
ejpam-6493	320	19	/	/	SYM
ejpam-6493	320	20	ker(φ	ker(φ	NOUN
ejpam-6493	320	21	)	)	PUNCT
ejpam-6493	320	22	∼=	∼=	PROPN
ejpam-6493	320	23	im(φ	im(φ	NUM
ejpam-6493	320	24	)	)	PUNCT
ejpam-6493	320	25	.	.	PUNCT
ejpam-6493	321	1	example	example	NOUN
ejpam-6493	322	1	5	5	NUM
ejpam-6493	322	2	.	.	PUNCT
ejpam-6493	323	1	let	let	VERB
ejpam-6493	323	2	r	r	NOUN
ejpam-6493	323	3	=	=	SYM
ejpam-6493	323	4	z−	z−	NOUN
ejpam-6493	323	5	be	be	AUX
ejpam-6493	323	6	a	a	DET
ejpam-6493	323	7	ternary	ternary	ADJ
ejpam-6493	323	8	semiring	semiring	NOUN
ejpam-6493	323	9	under	under	ADP
ejpam-6493	323	10	the	the	DET
ejpam-6493	323	11	usual	usual	ADJ
ejpam-6493	323	12	addition	addition	NOUN
ejpam-6493	323	13	and	and	CCONJ
ejpam-6493	323	14	ternary	ternary	ADJ
ejpam-6493	323	15	multiplication	multiplication	NOUN
ejpam-6493	323	16	of	of	ADP
ejpam-6493	323	17	integers	integer	NOUN
ejpam-6493	323	18	and	and	CCONJ
ejpam-6493	323	19	t	t	NOUN
ejpam-6493	323	20	=	=	PUNCT
ejpam-6493	323	21	z4	z4	AUX
ejpam-6493	323	22	be	be	AUX
ejpam-6493	323	23	a	a	DET
ejpam-6493	323	24	ternary	ternary	ADJ
ejpam-6493	323	25	semiring	semiring	NOUN
ejpam-6493	323	26	under	under	ADP
ejpam-6493	323	27	the	the	DET
ejpam-6493	323	28	usual	usual	ADJ
ejpam-6493	323	29	addition	addition	NOUN
ejpam-6493	323	30	and	and	CCONJ
ejpam-6493	323	31	ternary	ternary	ADJ
ejpam-6493	323	32	multiplication	multiplication	NOUN
ejpam-6493	323	33	of	of	ADP
ejpam-6493	323	34	integers	integer	NOUN
ejpam-6493	323	35	modulo	modulo	VERB
ejpam-6493	323	36	4	4	NUM
ejpam-6493	323	37	.	.	PUNCT
ejpam-6493	323	38	define	define	VERB
ejpam-6493	323	39	a	a	DET
ejpam-6493	323	40	function	function	NOUN
ejpam-6493	323	41	φ	φ	NOUN
ejpam-6493	323	42	:	:	PUNCT
ejpam-6493	323	43	r→	r→	PROPN
ejpam-6493	323	44	t	t	PROPN
ejpam-6493	323	45	by	by	ADP
ejpam-6493	323	46	φ(a	φ(a	ADJ
ejpam-6493	323	47	)	)	PUNCT
ejpam-6493	323	48	=	=	PUNCT
ejpam-6493	324	1	a	a	PRON
ejpam-6493	324	2	for	for	ADP
ejpam-6493	324	3	all	all	DET
ejpam-6493	324	4	a	a	DET
ejpam-6493	324	5	∈	∈	PROPN
ejpam-6493	324	6	r.	r.	NOUN
ejpam-6493	324	7	then	then	ADV
ejpam-6493	324	8	φ	φ	PROPN
ejpam-6493	324	9	is	be	AUX
ejpam-6493	324	10	a	a	DET
ejpam-6493	324	11	homomorphism	homomorphism	NOUN
ejpam-6493	324	12	.	.	PUNCT
ejpam-6493	325	1	we	we	PRON
ejpam-6493	325	2	have	have	VERB
ejpam-6493	325	3	that	that	PRON
ejpam-6493	325	4	ker(φ	ker(φ	NOUN
ejpam-6493	325	5	)	)	PUNCT
ejpam-6493	325	6	=	=	SYM
ejpam-6493	325	7	{	{	PUNCT
ejpam-6493	325	8	−4,−8,−12,−16	−4,−8,−12,−16	PROPN
ejpam-6493	325	9	,	,	PUNCT
ejpam-6493	325	10	.	.	PUNCT
ejpam-6493	325	11	.	.	PUNCT
ejpam-6493	326	1	.	.	PUNCT
ejpam-6493	326	2	}	}	PUNCT
ejpam-6493	326	3	is	be	AUX
ejpam-6493	326	4	a	a	DET
ejpam-6493	326	5	k	k	NOUN
ejpam-6493	326	6	-	-	NOUN
ejpam-6493	326	7	ideal	ideal	NOUN
ejpam-6493	326	8	of	of	ADP
ejpam-6493	326	9	r.	r.	PROPN
ejpam-6493	326	10	we	we	PRON
ejpam-6493	326	11	have	have	VERB
ejpam-6493	326	12	that	that	DET
ejpam-6493	326	13	r	r	NOUN
ejpam-6493	326	14	/	/	SYM
ejpam-6493	326	15	ker(φ	ker(φ	NOUN
ejpam-6493	326	16	)	)	PUNCT
ejpam-6493	327	1	=	=	PRON
ejpam-6493	327	2	{	{	PUNCT
ejpam-6493	327	3	−1	−1	NOUN
ejpam-6493	327	4	+	+	CCONJ
ejpam-6493	328	1	ker(φ),−2	ker(φ),−2	PROPN
ejpam-6493	328	2	+	+	CCONJ
ejpam-6493	328	3	ker(φ),−3	ker(φ),−3	PROPN
ejpam-6493	328	4	+	+	CCONJ
ejpam-6493	328	5	ker(φ),−4	ker(φ),−4	PROPN
ejpam-6493	328	6	+	+	CCONJ
ejpam-6493	328	7	ker(φ	ker(φ	NOUN
ejpam-6493	328	8	)	)	PUNCT
ejpam-6493	328	9	}	}	PUNCT
ejpam-6493	328	10	where	where	SCONJ
ejpam-6493	328	11	−1	−1	NOUN
ejpam-6493	328	12	+	+	CCONJ
ejpam-6493	328	13	ker(φ	ker(φ	X
ejpam-6493	328	14	)	)	PUNCT
ejpam-6493	328	15	=	=	NOUN
ejpam-6493	328	16	{	{	PUNCT
ejpam-6493	328	17	−1,−5,−9,−13	−1,−5,−9,−13	X
ejpam-6493	328	18	,	,	PUNCT
ejpam-6493	328	19	.	.	PUNCT
ejpam-6493	328	20	.	.	PUNCT
ejpam-6493	328	21	.	.	PUNCT
ejpam-6493	329	1	}	}	PUNCT
ejpam-6493	329	2	,	,	PUNCT
ejpam-6493	329	3	−2	−2	NOUN
ejpam-6493	329	4	+	+	CCONJ
ejpam-6493	329	5	ker(φ	ker(φ	NOUN
ejpam-6493	329	6	)	)	PUNCT
ejpam-6493	329	7	=	=	SYM
ejpam-6493	329	8	{	{	PUNCT
ejpam-6493	329	9	−2,−6,−10,−14	−2,−6,−10,−14	PROPN
ejpam-6493	329	10	,	,	PUNCT
ejpam-6493	329	11	.	.	PUNCT
ejpam-6493	329	12	.	.	PUNCT
ejpam-6493	329	13	.	.	PUNCT
ejpam-6493	330	1	}	}	PUNCT
ejpam-6493	330	2	,	,	PUNCT
ejpam-6493	330	3	−3	−3	PROPN
ejpam-6493	330	4	+	+	CCONJ
ejpam-6493	330	5	ker(φ	ker(φ	X
ejpam-6493	330	6	)	)	PUNCT
ejpam-6493	330	7	=	=	PRON
ejpam-6493	330	8	{	{	PUNCT
ejpam-6493	330	9	−3,−7,−11,−15	−3,−7,−11,−15	PROPN
ejpam-6493	330	10	,	,	PUNCT
ejpam-6493	330	11	.	.	PUNCT
ejpam-6493	330	12	.	.	PUNCT
ejpam-6493	331	1	.	.	PUNCT
ejpam-6493	331	2	}	}	PUNCT
ejpam-6493	331	3	,	,	PUNCT
ejpam-6493	331	4	−4	−4	X
ejpam-6493	332	1	+	+	CCONJ
ejpam-6493	332	2	ker(φ	ker(φ	X
ejpam-6493	332	3	)	)	PUNCT
ejpam-6493	332	4	=	=	SYM
ejpam-6493	332	5	{	{	PUNCT
ejpam-6493	332	6	−4,−8,−12,−16	−4,−8,−12,−16	PROPN
ejpam-6493	332	7	,	,	PUNCT
ejpam-6493	332	8	.	.	PUNCT
ejpam-6493	332	9	.	.	PUNCT
ejpam-6493	333	1	.	.	PUNCT
ejpam-6493	333	2	}	}	PUNCT
ejpam-6493	333	3	.	.	PUNCT
ejpam-6493	334	1	we	we	PRON
ejpam-6493	334	2	end	end	VERB
ejpam-6493	334	3	this	this	DET
ejpam-6493	334	4	section	section	NOUN
ejpam-6493	334	5	with	with	ADP
ejpam-6493	334	6	a	a	DET
ejpam-6493	334	7	summary	summary	NOUN
ejpam-6493	334	8	of	of	ADP
ejpam-6493	334	9	the	the	DET
ejpam-6493	334	10	relationship	relationship	NOUN
ejpam-6493	334	11	between	between	ADP
ejpam-6493	334	12	the	the	DET
ejpam-6493	334	13	isomorphisms	isomorphism	NOUN
ejpam-6493	334	14	in	in	ADP
ejpam-6493	334	15	the	the	DET
ejpam-6493	334	16	following	follow	VERB
ejpam-6493	334	17	corollary	corollary	NOUN
ejpam-6493	334	18	.	.	PUNCT
ejpam-6493	335	1	corollary	corollary	ADJ
ejpam-6493	335	2	1	1	NUM
ejpam-6493	335	3	.	.	PUNCT
ejpam-6493	336	1	let	let	VERB
ejpam-6493	336	2	r	r	NOUN
ejpam-6493	336	3	and	and	CCONJ
ejpam-6493	336	4	t	t	PROPN
ejpam-6493	336	5	be	be	AUX
ejpam-6493	336	6	ternary	ternary	ADJ
ejpam-6493	336	7	semirings	semiring	NOUN
ejpam-6493	336	8	with	with	ADP
ejpam-6493	336	9	zeroes	zero	NOUN
ejpam-6493	336	10	0r	0r	PROPN
ejpam-6493	336	11	and	and	CCONJ
ejpam-6493	336	12	0	0	NUM
ejpam-6493	336	13	t	t	NOUN
ejpam-6493	336	14	,	,	PUNCT
ejpam-6493	336	15	respectively	respectively	ADV
ejpam-6493	336	16	,	,	PUNCT
ejpam-6493	336	17	and	and	CCONJ
ejpam-6493	336	18	φ	φ	NUM
ejpam-6493	336	19	:	:	PUNCT
ejpam-6493	337	1	r→	r→	PROPN
ejpam-6493	337	2	t	t	PROPN
ejpam-6493	337	3	be	be	AUX
ejpam-6493	337	4	a	a	DET
ejpam-6493	337	5	homomorphism	homomorphism	NOUN
ejpam-6493	337	6	such	such	ADJ
ejpam-6493	337	7	that	that	DET
ejpam-6493	337	8	φ(0r	φ(0r	NOUN
ejpam-6493	337	9	)	)	PUNCT
ejpam-6493	337	10	=	=	SYM
ejpam-6493	337	11	0	0	NUM
ejpam-6493	337	12	t	t	NOUN
ejpam-6493	337	13	.	.	PUNCT
ejpam-6493	338	1	if	if	SCONJ
ejpam-6493	338	2	φ	φ	PROPN
ejpam-6493	338	3	is	be	AUX
ejpam-6493	338	4	a	a	DET
ejpam-6493	338	5	k	k	ADJ
ejpam-6493	338	6	-	-	PUNCT
ejpam-6493	338	7	kernel	kernel	NOUN
ejpam-6493	338	8	homomorphism	homomorphism	NOUN
ejpam-6493	338	9	,	,	PUNCT
ejpam-6493	338	10	then	then	ADV
ejpam-6493	338	11	r	r	X
ejpam-6493	338	12	/	/	SYM
ejpam-6493	338	13	ker(φ	ker(φ	NOUN
ejpam-6493	338	14	)	)	PUNCT
ejpam-6493	338	15	∼=	∼=	ADP
ejpam-6493	338	16	r	r	NOUN
ejpam-6493	338	17	/	/	SYM
ejpam-6493	338	18	k(φ	k(φ	PROPN
ejpam-6493	338	19	)	)	PUNCT
ejpam-6493	338	20	.	.	PUNCT
ejpam-6493	339	1	5	5	X
ejpam-6493	339	2	.	.	X
ejpam-6493	339	3	conclusion	conclusion	NOUN
ejpam-6493	339	4	in	in	ADP
ejpam-6493	339	5	this	this	DET
ejpam-6493	339	6	paper	paper	NOUN
ejpam-6493	339	7	,	,	PUNCT
ejpam-6493	339	8	we	we	PRON
ejpam-6493	339	9	present	present	VERB
ejpam-6493	339	10	multiple	multiple	ADJ
ejpam-6493	339	11	approaches	approach	NOUN
ejpam-6493	339	12	for	for	ADP
ejpam-6493	339	13	constructions	construction	NOUN
ejpam-6493	339	14	of	of	ADP
ejpam-6493	339	15	quotient	quotient	NOUN
ejpam-6493	339	16	ternary	ternary	ADJ
ejpam-6493	339	17	semirings	semiring	NOUN
ejpam-6493	339	18	,	,	PUNCT
ejpam-6493	339	19	namely	namely	ADV
ejpam-6493	339	20	quotient	quotient	VERB
ejpam-6493	339	21	ternary	ternary	ADJ
ejpam-6493	339	22	semirings	semiring	NOUN
ejpam-6493	339	23	modulo	modulo	NOUN
ejpam-6493	339	24	congruences	congruence	NOUN
ejpam-6493	339	25	and	and	CCONJ
ejpam-6493	339	26	those	those	DET
ejpam-6493	339	27	modulo	modulo	NOUN
ejpam-6493	339	28	ideals	ideal	NOUN
ejpam-6493	339	29	.	.	PUNCT
ejpam-6493	340	1	m.	m.	NOUN
ejpam-6493	340	2	petapirak	petapirak	PROPN
ejpam-6493	340	3	,	,	PUNCT
ejpam-6493	340	4	a.	a.	PROPN
ejpam-6493	340	5	j.	j.	PROPN
ejpam-6493	340	6	khan	khan	PROPN
ejpam-6493	340	7	,	,	PUNCT
ejpam-6493	340	8	r.	r.	PROPN
ejpam-6493	340	9	chinram	chinram	PROPN
ejpam-6493	340	10	/	/	SYM
ejpam-6493	340	11	eur	eur	PROPN
ejpam-6493	340	12	.	.	PUNCT
ejpam-6493	341	1	j.	j.	PROPN
ejpam-6493	341	2	pure	pure	PROPN
ejpam-6493	341	3	appl	appl	PROPN
ejpam-6493	341	4	.	.	PROPN
ejpam-6493	341	5	math	math	PROPN
ejpam-6493	341	6	,	,	PUNCT
ejpam-6493	341	7	18	18	NUM
ejpam-6493	341	8	(	(	PUNCT
ejpam-6493	341	9	3	3	NUM
ejpam-6493	341	10	)	)	PUNCT
ejpam-6493	341	11	(	(	PUNCT
ejpam-6493	341	12	2025	2025	NUM
ejpam-6493	341	13	)	)	PUNCT
ejpam-6493	341	14	,	,	PUNCT
ejpam-6493	341	15	6493	6493	NUM
ejpam-6493	341	16	9	9	NUM
ejpam-6493	341	17	of	of	ADP
ejpam-6493	341	18	9	9	NUM
ejpam-6493	341	19	we	we	PRON
ejpam-6493	341	20	investigate	investigate	VERB
ejpam-6493	341	21	some	some	DET
ejpam-6493	341	22	relationships	relationship	NOUN
ejpam-6493	341	23	between	between	ADP
ejpam-6493	341	24	ideals	ideal	NOUN
ejpam-6493	341	25	and	and	CCONJ
ejpam-6493	341	26	quotient	quotient	NOUN
ejpam-6493	341	27	structures	structure	NOUN
ejpam-6493	341	28	in	in	ADP
ejpam-6493	341	29	ternary	ternary	ADJ
ejpam-6493	341	30	semirings	semiring	NOUN
ejpam-6493	341	31	.	.	PUNCT
ejpam-6493	342	1	a	a	DET
ejpam-6493	342	2	key	key	ADJ
ejpam-6493	342	3	contribution	contribution	NOUN
ejpam-6493	342	4	of	of	ADP
ejpam-6493	342	5	this	this	DET
ejpam-6493	342	6	work	work	NOUN
ejpam-6493	342	7	is	be	AUX
ejpam-6493	342	8	the	the	DET
ejpam-6493	342	9	establishment	establishment	NOUN
ejpam-6493	342	10	of	of	ADP
ejpam-6493	342	11	various	various	ADJ
ejpam-6493	342	12	isomorphism	isomorphism	NOUN
ejpam-6493	342	13	theorems	theorem	NOUN
ejpam-6493	342	14	,	,	PUNCT
ejpam-6493	342	15	including	include	VERB
ejpam-6493	342	16	the	the	DET
ejpam-6493	342	17	fundamental	fundamental	ADJ
ejpam-6493	342	18	results	result	NOUN
ejpam-6493	342	19	that	that	PRON
ejpam-6493	342	20	relate	relate	VERB
ejpam-6493	342	21	quotient	quotient	NOUN
ejpam-6493	342	22	structures	structure	NOUN
ejpam-6493	342	23	such	such	ADJ
ejpam-6493	342	24	as	as	ADP
ejpam-6493	342	25	r	r	NOUN
ejpam-6493	342	26	/	/	SYM
ejpam-6493	342	27	i	i	PRON
ejpam-6493	342	28	∼=	∼=	ADV
ejpam-6493	342	29	r	r	NOUN
ejpam-6493	342	30	/	/	SYM
ejpam-6493	342	31	ck(i	ck(i	NUM
ejpam-6493	342	32	)	)	PUNCT
ejpam-6493	342	33	and	and	CCONJ
ejpam-6493	342	34	r	r	X
ejpam-6493	342	35	/	/	SYM
ejpam-6493	342	36	ker(φ	ker(φ	NOUN
ejpam-6493	342	37	)	)	PUNCT
ejpam-6493	342	38	∼=	∼=	PROPN
ejpam-6493	342	39	im(φ	im(φ	NUM
ejpam-6493	342	40	)	)	PUNCT
ejpam-6493	342	41	,	,	PUNCT
ejpam-6493	342	42	under	under	ADP
ejpam-6493	342	43	suitable	suitable	ADJ
ejpam-6493	342	44	conditions	condition	NOUN
ejpam-6493	342	45	.	.	PUNCT
ejpam-6493	343	1	these	these	DET
ejpam-6493	343	2	results	result	NOUN
ejpam-6493	343	3	help	help	AUX
ejpam-6493	343	4	clarify	clarify	VERB
ejpam-6493	343	5	the	the	DET
ejpam-6493	343	6	connections	connection	NOUN
ejpam-6493	343	7	between	between	ADP
ejpam-6493	343	8	homomorphisms	homomorphism	NOUN
ejpam-6493	343	9	,	,	PUNCT
ejpam-6493	343	10	ideals	ideal	NOUN
ejpam-6493	343	11	,	,	PUNCT
ejpam-6493	343	12	and	and	CCONJ
ejpam-6493	343	13	congruences	congruence	NOUN
ejpam-6493	343	14	,	,	PUNCT
ejpam-6493	343	15	and	and	CCONJ
ejpam-6493	343	16	provide	provide	VERB
ejpam-6493	343	17	a	a	DET
ejpam-6493	343	18	basis	basis	NOUN
ejpam-6493	343	19	for	for	ADP
ejpam-6493	343	20	understanding	understand	VERB
ejpam-6493	343	21	the	the	DET
ejpam-6493	343	22	structure	structure	NOUN
ejpam-6493	343	23	of	of	ADP
ejpam-6493	343	24	ternary	ternary	ADJ
ejpam-6493	343	25	semirings	semiring	NOUN
ejpam-6493	343	26	.	.	PUNCT
ejpam-6493	344	1	acknowledgements	acknowledgement	NOUN
ejpam-6493	344	2	the	the	DET
ejpam-6493	344	3	authors	author	NOUN
ejpam-6493	344	4	would	would	AUX
ejpam-6493	344	5	like	like	VERB
ejpam-6493	344	6	to	to	PART
ejpam-6493	344	7	thank	thank	VERB
ejpam-6493	344	8	the	the	DET
ejpam-6493	344	9	referees	referee	NOUN
ejpam-6493	344	10	for	for	ADP
ejpam-6493	344	11	his	his	PRON
ejpam-6493	344	12	/	/	SYM
ejpam-6493	344	13	her	her	PRON
ejpam-6493	344	14	careful	careful	ADJ
ejpam-6493	344	15	reading	reading	NOUN
ejpam-6493	344	16	and	and	CCONJ
ejpam-6493	344	17	many	many	ADJ
ejpam-6493	344	18	helpful	helpful	ADJ
ejpam-6493	344	19	comments	comment	NOUN
ejpam-6493	344	20	on	on	ADP
ejpam-6493	344	21	the	the	DET
ejpam-6493	344	22	manuscript	manuscript	NOUN
ejpam-6493	344	23	of	of	ADP
ejpam-6493	344	24	this	this	DET
ejpam-6493	344	25	paper	paper	NOUN
ejpam-6493	344	26	.	.	PUNCT
ejpam-6493	345	1	this	this	DET
ejpam-6493	345	2	work	work	NOUN
ejpam-6493	345	3	was	be	AUX
ejpam-6493	345	4	supported	support	VERB
ejpam-6493	345	5	in	in	ADP
ejpam-6493	345	6	part	part	NOUN
ejpam-6493	345	7	by	by	ADP
ejpam-6493	345	8	the	the	DET
ejpam-6493	345	9	psu	psu	NOUN
ejpam-6493	345	10	-	-	ADJ
ejpam-6493	345	11	tuyf	tuyf	ADJ
ejpam-6493	345	12	charitable	charitable	ADJ
ejpam-6493	345	13	trust	trust	NOUN
ejpam-6493	345	14	fund	fund	NOUN
ejpam-6493	345	15	,	,	PUNCT
ejpam-6493	345	16	prince	prince	NOUN
ejpam-6493	345	17	of	of	ADP
ejpam-6493	345	18	songkla	songkla	PROPN
ejpam-6493	345	19	university	university	PROPN
ejpam-6493	345	20	,	,	PUNCT
ejpam-6493	345	21	contract	contract	NOUN
ejpam-6493	345	22	no.1	no.1	NOUN
ejpam-6493	345	23	-	-	SYM
ejpam-6493	345	24	2567	2567	NUM
ejpam-6493	345	25	-	-	SYM
ejpam-6493	345	26	01	01	NUM
ejpam-6493	345	27	.	.	PUNCT
ejpam-6493	346	1	references	reference	NOUN
ejpam-6493	346	2	[	[	X
ejpam-6493	346	3	1	1	NUM
ejpam-6493	346	4	]	]	PUNCT
ejpam-6493	346	5	p.	p.	PROPN
ejpam-6493	346	6	j.	j.	PROPN
ejpam-6493	346	7	allen	allen	PROPN
ejpam-6493	346	8	.	.	PUNCT
ejpam-6493	347	1	a	a	DET
ejpam-6493	347	2	fundamental	fundamental	ADJ
ejpam-6493	347	3	theorem	theorem	NOUN
ejpam-6493	347	4	of	of	ADP
ejpam-6493	347	5	homomorphisms	homomorphism	NOUN
ejpam-6493	347	6	for	for	ADP
ejpam-6493	347	7	semirings	semiring	NOUN
ejpam-6493	347	8	.	.	PUNCT
ejpam-6493	348	1	proceedings	proceeding	NOUN
ejpam-6493	348	2	of	of	ADP
ejpam-6493	348	3	the	the	DET
ejpam-6493	348	4	american	american	PROPN
ejpam-6493	348	5	mathematical	mathematical	PROPN
ejpam-6493	348	6	society	society	NOUN
ejpam-6493	348	7	,	,	PUNCT
ejpam-6493	348	8	24:463–465	24:463–465	NUM
ejpam-6493	348	9	,	,	PUNCT
ejpam-6493	348	10	1969	1969	NUM
ejpam-6493	348	11	.	.	PUNCT
ejpam-6493	349	1	[	[	X
ejpam-6493	349	2	2	2	X
ejpam-6493	349	3	]	]	PUNCT
ejpam-6493	349	4	s.	s.	PROPN
ejpam-6493	349	5	e.	e.	PROPN
ejpam-6493	349	6	atani	atani	PROPN
ejpam-6493	349	7	and	and	CCONJ
ejpam-6493	349	8	a.	a.	NOUN
ejpam-6493	349	9	g.	g.	PROPN
ejpam-6493	349	10	garfami	garfami	PROPN
ejpam-6493	349	11	.	.	PUNCT
ejpam-6493	350	1	ideals	ideal	NOUN
ejpam-6493	350	2	in	in	ADP
ejpam-6493	350	3	quotient	quotient	NOUN
ejpam-6493	350	4	semirings	semirings	PROPN
ejpam-6493	350	5	.	.	PUNCT
ejpam-6493	351	1	chiang	chiang	PROPN
ejpam-6493	351	2	mai	mai	PROPN
ejpam-6493	351	3	journal	journal	PROPN
ejpam-6493	351	4	of	of	ADP
ejpam-6493	351	5	science	science	NOUN
ejpam-6493	351	6	,	,	PUNCT
ejpam-6493	351	7	40(1):77–82	40(1):77–82	NUM
ejpam-6493	351	8	,	,	PUNCT
ejpam-6493	351	9	2013	2013	NUM
ejpam-6493	351	10	.	.	PUNCT
ejpam-6493	352	1	[	[	X
ejpam-6493	352	2	3	3	X
ejpam-6493	352	3	]	]	X
ejpam-6493	352	4	d.	d.	PROPN
ejpam-6493	352	5	r.	r.	PROPN
ejpam-6493	352	6	latorre	latorre	PROPN
ejpam-6493	352	7	.	.	PUNCT
ejpam-6493	353	1	a	a	DET
ejpam-6493	353	2	note	note	NOUN
ejpam-6493	353	3	on	on	ADP
ejpam-6493	353	4	quotient	quotient	NOUN
ejpam-6493	353	5	semirings	semiring	NOUN
ejpam-6493	353	6	.	.	PUNCT
ejpam-6493	354	1	proceedings	proceeding	NOUN
ejpam-6493	354	2	of	of	ADP
ejpam-6493	354	3	the	the	DET
ejpam-6493	354	4	american	american	PROPN
ejpam-6493	354	5	mathematical	mathematical	PROPN
ejpam-6493	354	6	society	society	NOUN
ejpam-6493	354	7	,	,	PUNCT
ejpam-6493	354	8	24:463–465	24:463–465	NUM
ejpam-6493	354	9	,	,	PUNCT
ejpam-6493	354	10	1970	1970	NUM
ejpam-6493	354	11	.	.	PUNCT
ejpam-6493	355	1	[	[	X
ejpam-6493	355	2	4	4	X
ejpam-6493	355	3	]	]	PUNCT
ejpam-6493	355	4	t.	t.	PROPN
ejpam-6493	355	5	k.	k.	PROPN
ejpam-6493	355	6	dutta	dutta	PROPN
ejpam-6493	355	7	and	and	CCONJ
ejpam-6493	355	8	s.	s.	PROPN
ejpam-6493	355	9	kar	kar	PROPN
ejpam-6493	355	10	.	.	PUNCT
ejpam-6493	356	1	on	on	ADP
ejpam-6493	356	2	regular	regular	ADJ
ejpam-6493	356	3	ternary	ternary	ADJ
ejpam-6493	356	4	semirings	semiring	NOUN
ejpam-6493	356	5	.	.	PUNCT
ejpam-6493	357	1	advances	advance	NOUN
ejpam-6493	357	2	in	in	ADP
ejpam-6493	357	3	algebra	algebra	NOUN
ejpam-6493	357	4	,	,	PUNCT
ejpam-6493	357	5	proceedings	proceeding	NOUN
ejpam-6493	357	6	of	of	ADP
ejpam-6493	357	7	the	the	DET
ejpam-6493	357	8	icm	icm	PROPN
ejpam-6493	357	9	satellite	satellite	PROPN
ejpam-6493	357	10	conference	conference	NOUN
ejpam-6493	357	11	in	in	ADP
ejpam-6493	357	12	algebra	algebra	PROPN
ejpam-6493	357	13	and	and	CCONJ
ejpam-6493	357	14	related	related	ADJ
ejpam-6493	357	15	topics	topic	NOUN
ejpam-6493	357	16	,	,	PUNCT
ejpam-6493	357	17	world	world	NOUN
ejpam-6493	357	18	scientific	scientific	ADJ
ejpam-6493	357	19	,	,	PUNCT
ejpam-6493	357	20	pages	page	NOUN
ejpam-6493	357	21	343–355	343–355	NUM
ejpam-6493	357	22	,	,	PUNCT
ejpam-6493	357	23	2003	2003	NUM
ejpam-6493	357	24	.	.	PUNCT
ejpam-6493	358	1	[	[	X
ejpam-6493	358	2	5	5	X
ejpam-6493	358	3	]	]	PUNCT
ejpam-6493	358	4	j.	j.	PROPN
ejpam-6493	358	5	n.	n.	PROPN
ejpam-6493	358	6	chaudhari	chaudhari	PROPN
ejpam-6493	358	7	and	and	CCONJ
ejpam-6493	358	8	k.	k.	PROPN
ejpam-6493	358	9	j.	j.	PROPN
ejpam-6493	358	10	ingale	ingale	PROPN
ejpam-6493	358	11	.	.	PUNCT
ejpam-6493	359	1	on	on	ADP
ejpam-6493	359	2	partitioning	partition	VERB
ejpam-6493	359	3	and	and	CCONJ
ejpam-6493	359	4	subtractive	subtractive	ADJ
ejpam-6493	359	5	ideals	ideal	NOUN
ejpam-6493	359	6	of	of	ADP
ejpam-6493	359	7	ternary	ternary	ADJ
ejpam-6493	359	8	semirings	semiring	NOUN
ejpam-6493	359	9	.	.	PUNCT
ejpam-6493	360	1	kyungpook	kyungpook	PROPN
ejpam-6493	360	2	mathematical	mathematical	PROPN
ejpam-6493	360	3	journal	journal	NOUN
ejpam-6493	360	4	,	,	PUNCT
ejpam-6493	360	5	51(1):69–76	51(1):69–76	NUM
ejpam-6493	360	6	,	,	PUNCT
ejpam-6493	360	7	2011	2011	NUM
ejpam-6493	360	8	.	.	PUNCT
ejpam-6493	361	1	[	[	X
ejpam-6493	361	2	6	6	NUM
ejpam-6493	361	3	]	]	PUNCT
ejpam-6493	361	4	t.	t.	NOUN
ejpam-6493	361	5	sunitha	sunitha	PROPN
ejpam-6493	361	6	;	;	PUNCT
ejpam-6493	361	7	u.	u.	PROPN
ejpam-6493	361	8	nagi	nagi	PROPN
ejpam-6493	361	9	reddy	reddy	PROPN
ejpam-6493	361	10	and	and	CCONJ
ejpam-6493	361	11	g.	g.	PROPN
ejpam-6493	361	12	shobhalatha	shobhalatha	PROPN
ejpam-6493	361	13	.	.	PUNCT
ejpam-6493	362	1	a	a	DET
ejpam-6493	362	2	note	note	NOUN
ejpam-6493	362	3	on	on	ADP
ejpam-6493	362	4	full	full	ADJ
ejpam-6493	362	5	k	k	NOUN
ejpam-6493	362	6	-	-	NOUN
ejpam-6493	362	7	ideals	ideal	NOUN
ejpam-6493	362	8	in	in	ADP
ejpam-6493	362	9	ternary	ternary	ADJ
ejpam-6493	362	10	semirings	semiring	NOUN
ejpam-6493	362	11	.	.	PUNCT
ejpam-6493	363	1	indian	indian	PROPN
ejpam-6493	363	2	journal	journal	PROPN
ejpam-6493	363	3	of	of	ADP
ejpam-6493	363	4	science	science	NOUN
ejpam-6493	363	5	and	and	CCONJ
ejpam-6493	363	6	technology	technology	NOUN
ejpam-6493	363	7	,	,	PUNCT
ejpam-6493	363	8	14(21):1786–1790	14(21):1786–1790	NUM
ejpam-6493	363	9	,	,	PUNCT
ejpam-6493	363	10	2021	2021	NUM
ejpam-6493	363	11	.	.	PUNCT
ejpam-6493	364	1	[	[	X
ejpam-6493	364	2	7	7	X
ejpam-6493	364	3	]	]	X
ejpam-6493	364	4	j.	j.	PROPN
ejpam-6493	364	5	sanborisoot	sanborisoot	PROPN
ejpam-6493	364	6	and	and	CCONJ
ejpam-6493	364	7	p.	p.	PROPN
ejpam-6493	364	8	p.	p.	PROPN
ejpam-6493	364	9	n.	n.	PROPN
ejpam-6493	364	10	ayutthaya	ayutthaya	PROPN
ejpam-6493	364	11	.	.	PUNCT
ejpam-6493	365	1	on	on	ADP
ejpam-6493	365	2	ternary	ternary	ADJ
ejpam-6493	365	3	ring	ring	NOUN
ejpam-6493	365	4	congruences	congruence	NOUN
ejpam-6493	365	5	of	of	ADP
ejpam-6493	365	6	ternary	ternary	ADJ
ejpam-6493	365	7	semirings	semiring	NOUN
ejpam-6493	365	8	.	.	PUNCT
ejpam-6493	366	1	discussiones	discussione	NOUN
ejpam-6493	366	2	mathematicae	mathematicae	VERB
ejpam-6493	366	3	general	general	ADJ
ejpam-6493	366	4	algebra	algebra	PROPN
ejpam-6493	366	5	and	and	CCONJ
ejpam-6493	366	6	applications	application	NOUN
ejpam-6493	366	7	,	,	PUNCT
ejpam-6493	366	8	44:43–55	44:43–55	NUM
ejpam-6493	366	9	,	,	PUNCT
ejpam-6493	366	10	2024	2024	NUM
ejpam-6493	366	11	.	.	PUNCT
ejpam-6493	367	1	[	[	X
ejpam-6493	367	2	8	8	NUM
ejpam-6493	367	3	]	]	X
ejpam-6493	367	4	n.	n.	NOUN
ejpam-6493	367	5	sarasit	sarasit	PROPN
ejpam-6493	367	6	;	;	PUNCT
ejpam-6493	367	7	r.	r.	PROPN
ejpam-6493	367	8	chinram	chinram	PROPN
ejpam-6493	367	9	and	and	CCONJ
ejpam-6493	367	10	a.	a.	NOUN
ejpam-6493	367	11	rattana	rattana	PROPN
ejpam-6493	367	12	.	.	PUNCT
ejpam-6493	368	1	applications	application	NOUN
ejpam-6493	368	2	of	of	ADP
ejpam-6493	368	3	fuzzy	fuzzy	ADJ
ejpam-6493	368	4	sets	set	NOUN
ejpam-6493	368	5	for	for	ADP
ejpam-6493	368	6	almostity	almostity	NOUN
ejpam-6493	368	7	of	of	ADP
ejpam-6493	368	8	ternary	ternary	ADJ
ejpam-6493	368	9	semirings	semiring	NOUN
ejpam-6493	368	10	.	.	PUNCT
ejpam-6493	369	1	international	international	ADJ
ejpam-6493	369	2	journal	journal	PROPN
ejpam-6493	369	3	of	of	ADP
ejpam-6493	369	4	applied	apply	VERB
ejpam-6493	369	5	mathematics	mathematic	NOUN
ejpam-6493	369	6	,	,	PUNCT
ejpam-6493	369	7	36(4):497–508	36(4):497–508	PROPN
ejpam-6493	369	8	,	,	PUNCT
ejpam-6493	369	9	2023	2023	NUM
ejpam-6493	369	10	.	.	PUNCT
ejpam-6493	370	1	[	[	X
ejpam-6493	370	2	9	9	NUM
ejpam-6493	370	3	]	]	PUNCT
ejpam-6493	370	4	p.	p.	NOUN
ejpam-6493	370	5	luangchaisri	luangchaisri	VERB
ejpam-6493	370	6	and	and	CCONJ
ejpam-6493	370	7	t.	t.	PROPN
ejpam-6493	370	8	changphas	changphas	PROPN
ejpam-6493	370	9	.	.	PUNCT
ejpam-6493	371	1	prime	prime	ADJ
ejpam-6493	371	2	one	one	NUM
ejpam-6493	371	3	-	-	PUNCT
ejpam-6493	371	4	sided	sided	ADJ
ejpam-6493	371	5	ideals	ideal	NOUN
ejpam-6493	371	6	in	in	ADP
ejpam-6493	371	7	ternary	ternary	ADJ
ejpam-6493	371	8	semirings	semiring	NOUN
ejpam-6493	371	9	.	.	PUNCT
ejpam-6493	372	1	international	international	ADJ
ejpam-6493	372	2	journal	journal	PROPN
ejpam-6493	372	3	of	of	ADP
ejpam-6493	372	4	mathematics	mathematic	NOUN
ejpam-6493	372	5	and	and	CCONJ
ejpam-6493	372	6	computer	computer	NOUN
ejpam-6493	372	7	science	science	NOUN
ejpam-6493	372	8	,	,	PUNCT
ejpam-6493	372	9	19(2):403–409	19(2):403–409	NUM
ejpam-6493	372	10	,	,	PUNCT
ejpam-6493	372	11	2024	2024	NUM
ejpam-6493	372	12	.	.	PUNCT
ejpam-6493	373	1	[	[	X
ejpam-6493	373	2	10	10	NUM
ejpam-6493	373	3	]	]	PUNCT
ejpam-6493	373	4	t.	t.	NOUN
ejpam-6493	373	5	panityakul	panityakul	NOUN
ejpam-6493	373	6	and	and	CCONJ
ejpam-6493	373	7	r.	r.	PROPN
ejpam-6493	373	8	chinram	chinram	PROPN
ejpam-6493	373	9	.	.	PUNCT
ejpam-6493	374	1	applications	application	NOUN
ejpam-6493	374	2	of	of	ADP
ejpam-6493	374	3	neutrosophic	neutrosophic	ADJ
ejpam-6493	374	4	n	n	CCONJ
ejpam-6493	374	5	-	-	PUNCT
ejpam-6493	374	6	structures	structure	NOUN
ejpam-6493	374	7	in	in	ADP
ejpam-6493	374	8	ternary	ternary	ADJ
ejpam-6493	374	9	semirings	semiring	NOUN
ejpam-6493	374	10	:	:	PUNCT
ejpam-6493	374	11	a	a	DET
ejpam-6493	374	12	study	study	NOUN
ejpam-6493	374	13	on	on	ADP
ejpam-6493	374	14	neutrosophic	neutrosophic	ADJ
ejpam-6493	374	15	ternary	ternary	ADJ
ejpam-6493	374	16	n	n	CCONJ
ejpam-6493	374	17	-	-	PUNCT
ejpam-6493	374	18	subsemirings	subsemiring	NOUN
ejpam-6493	374	19	.	.	PUNCT
ejpam-6493	375	1	international	international	ADJ
ejpam-6493	375	2	journal	journal	PROPN
ejpam-6493	375	3	of	of	ADP
ejpam-6493	375	4	neutrosophic	neutrosophic	ADJ
ejpam-6493	375	5	science	science	NOUN
ejpam-6493	375	6	,	,	PUNCT
ejpam-6493	375	7	26(2):120–131	26(2):120–131	NUM
ejpam-6493	375	8	,	,	PUNCT
ejpam-6493	375	9	2025	2025	NUM
ejpam-6493	375	10	.	.	PUNCT
ejpam-6493	376	1	[	[	X
ejpam-6493	376	2	11	11	NUM
ejpam-6493	376	3	]	]	PUNCT
ejpam-6493	376	4	a.	a.	NOUN
ejpam-6493	376	5	j.	j.	PROPN
ejpam-6493	376	6	khan	khan	PROPN
ejpam-6493	376	7	;	;	PUNCT
ejpam-6493	376	8	m.	m.	NOUN
ejpam-6493	376	9	petapirak	petapirak	PROPN
ejpam-6493	376	10	and	and	CCONJ
ejpam-6493	376	11	r.	r.	PROPN
ejpam-6493	376	12	chinram	chinram	PROPN
ejpam-6493	376	13	.	.	PUNCT
ejpam-6493	377	1	a	a	DET
ejpam-6493	377	2	note	note	NOUN
ejpam-6493	377	3	on	on	ADP
ejpam-6493	377	4	k	k	NOUN
ejpam-6493	377	5	-	-	NOUN
ejpam-6493	377	6	ideals	ideal	NOUN
ejpam-6493	377	7	in	in	ADP
ejpam-6493	377	8	ternary	ternary	ADJ
ejpam-6493	377	9	semirings	semiring	NOUN
ejpam-6493	377	10	.	.	PUNCT
ejpam-6493	378	1	european	european	PROPN
ejpam-6493	378	2	journal	journal	PROPN
ejpam-6493	378	3	of	of	ADP
ejpam-6493	378	4	pure	pure	ADJ
ejpam-6493	378	5	and	and	CCONJ
ejpam-6493	378	6	applied	applied	ADJ
ejpam-6493	378	7	mathematics	mathematic	NOUN
ejpam-6493	378	8	,	,	PUNCT
ejpam-6493	378	9	18(2):6096	18(2):6096	NUM
ejpam-6493	378	10	,	,	PUNCT
ejpam-6493	378	11	2025	2025	NUM
ejpam-6493	378	12	.	.	PUNCT