id	sid	tid	token	lemma	pos
ejpam-6327	1	1	european	european	PROPN
ejpam-6327	1	2	journal	journal	PROPN
ejpam-6327	1	3	of	of	ADP
ejpam-6327	1	4	pure	pure	ADJ
ejpam-6327	1	5	and	and	CCONJ
ejpam-6327	1	6	applied	applied	ADJ
ejpam-6327	1	7	mathematics	mathematic	NOUN
ejpam-6327	1	8	2025	2025	NUM
ejpam-6327	1	9	,	,	PUNCT
ejpam-6327	1	10	vol	vol	NOUN
ejpam-6327	1	11	.	.	PROPN
ejpam-6327	1	12	18	18	NUM
ejpam-6327	1	13	,	,	PUNCT
ejpam-6327	1	14	issue	issue	NOUN
ejpam-6327	1	15	3	3	NUM
ejpam-6327	1	16	,	,	PUNCT
ejpam-6327	1	17	article	article	NOUN
ejpam-6327	1	18	number	number	NOUN
ejpam-6327	1	19	6327	6327	NUM
ejpam-6327	1	20	issn	issn	PROPN
ejpam-6327	1	21	1307	1307	NUM
ejpam-6327	1	22	-	-	SYM
ejpam-6327	1	23	5543	5543	NUM
ejpam-6327	1	24	–	–	PUNCT
ejpam-6327	1	25	ejpam.com	ejpam.com	X
ejpam-6327	1	26	published	publish	VERB
ejpam-6327	1	27	by	by	ADP
ejpam-6327	1	28	new	new	PROPN
ejpam-6327	1	29	york	york	PROPN
ejpam-6327	1	30	business	business	PROPN
ejpam-6327	1	31	global	global	PROPN
ejpam-6327	1	32	on	on	ADP
ejpam-6327	1	33	quotient	quotient	NOUN
ejpam-6327	1	34	ks	k	NOUN
ejpam-6327	1	35	-	-	PUNCT
ejpam-6327	1	36	semigroups	semigroup	NOUN
ejpam-6327	1	37	induced	induce	VERB
ejpam-6327	1	38	by	by	ADP
ejpam-6327	1	39	fuzzy	fuzzy	ADJ
ejpam-6327	1	40	ks	ks	NOUN
ejpam-6327	1	41	-	-	PUNCT
ejpam-6327	1	42	ideals	ideal	NOUN
ejpam-6327	1	43	hulsen	hulsen	ADJ
ejpam-6327	1	44	t.	t.	PROPN
ejpam-6327	1	45	sarapuddin1,∗	sarapuddin1,∗	PROPN
ejpam-6327	1	46	,	,	PUNCT
ejpam-6327	1	47	jocelyn	jocelyn	PROPN
ejpam-6327	1	48	p.	p.	PROPN
ejpam-6327	1	49	vilela1	vilela1	NOUN
ejpam-6327	2	1	1	1	NUM
ejpam-6327	2	2	department	department	NOUN
ejpam-6327	2	3	of	of	ADP
ejpam-6327	2	4	mathematics	mathematic	NOUN
ejpam-6327	2	5	and	and	CCONJ
ejpam-6327	2	6	statistics	statistic	NOUN
ejpam-6327	2	7	,	,	PUNCT
ejpam-6327	2	8	college	college	NOUN
ejpam-6327	2	9	of	of	ADP
ejpam-6327	2	10	science	science	NOUN
ejpam-6327	2	11	and	and	CCONJ
ejpam-6327	2	12	mathematics	mathematic	NOUN
ejpam-6327	2	13	,	,	PUNCT
ejpam-6327	2	14	center	center	NOUN
ejpam-6327	2	15	of	of	ADP
ejpam-6327	2	16	mathematical	mathematical	ADJ
ejpam-6327	2	17	and	and	CCONJ
ejpam-6327	2	18	theoretical	theoretical	ADJ
ejpam-6327	2	19	physical	physical	ADJ
ejpam-6327	2	20	sciences	science	NOUN
ejpam-6327	2	21	-	-	PUNCT
ejpam-6327	2	22	prism	prism	NOUN
ejpam-6327	2	23	,	,	PUNCT
ejpam-6327	2	24	msu	msu	PROPN
ejpam-6327	2	25	-	-	PUNCT
ejpam-6327	2	26	iligan	iligan	PROPN
ejpam-6327	2	27	institute	institute	PROPN
ejpam-6327	2	28	of	of	ADP
ejpam-6327	2	29	technology	technology	PROPN
ejpam-6327	2	30	,	,	PUNCT
ejpam-6327	2	31	9200	9200	NUM
ejpam-6327	2	32	iligan	iligan	ADJ
ejpam-6327	2	33	city	city	NOUN
ejpam-6327	2	34	,	,	PUNCT
ejpam-6327	2	35	philippines	philippine	NOUN
ejpam-6327	2	36	abstract	abstract	ADJ
ejpam-6327	2	37	.	.	PUNCT
ejpam-6327	3	1	in	in	ADP
ejpam-6327	3	2	this	this	DET
ejpam-6327	3	3	paper	paper	NOUN
ejpam-6327	3	4	,	,	PUNCT
ejpam-6327	3	5	we	we	PRON
ejpam-6327	3	6	introduce	introduce	VERB
ejpam-6327	3	7	new	new	ADJ
ejpam-6327	3	8	types	type	NOUN
ejpam-6327	3	9	of	of	ADP
ejpam-6327	3	10	fuzzy	fuzzy	ADJ
ejpam-6327	3	11	ideals	ideal	NOUN
ejpam-6327	3	12	on	on	ADP
ejpam-6327	3	13	ks	ks	NOUN
ejpam-6327	3	14	-	-	PUNCT
ejpam-6327	3	15	semigroups	semigroup	NOUN
ejpam-6327	3	16	,	,	PUNCT
ejpam-6327	3	17	compare	compare	VERB
ejpam-6327	3	18	them	they	PRON
ejpam-6327	3	19	with	with	ADP
ejpam-6327	3	20	existing	exist	VERB
ejpam-6327	3	21	fuzzy	fuzzy	ADJ
ejpam-6327	3	22	ks	ks	NOUN
ejpam-6327	3	23	-	-	PUNCT
ejpam-6327	3	24	ideals	ideal	NOUN
ejpam-6327	3	25	and	and	CCONJ
ejpam-6327	3	26	investigate	investigate	VERB
ejpam-6327	3	27	their	their	PRON
ejpam-6327	3	28	properties	property	NOUN
ejpam-6327	3	29	.	.	PUNCT
ejpam-6327	4	1	we	we	PRON
ejpam-6327	4	2	discuss	discuss	VERB
ejpam-6327	4	3	the	the	DET
ejpam-6327	4	4	construction	construction	NOUN
ejpam-6327	4	5	of	of	ADP
ejpam-6327	4	6	quotient	quotient	NOUN
ejpam-6327	4	7	ks	k	NOUN
ejpam-6327	4	8	-	-	PUNCT
ejpam-6327	4	9	semigroups	semigroup	NOUN
ejpam-6327	4	10	induced	induce	VERB
ejpam-6327	4	11	by	by	ADP
ejpam-6327	4	12	fuzzy	fuzzy	ADJ
ejpam-6327	4	13	ks	ks	NOUN
ejpam-6327	4	14	-	-	PUNCT
ejpam-6327	4	15	ideals	ideal	NOUN
ejpam-6327	4	16	,	,	PUNCT
ejpam-6327	4	17	show	show	VERB
ejpam-6327	4	18	that	that	SCONJ
ejpam-6327	4	19	this	this	DET
ejpam-6327	4	20	quotient	quotient	NOUN
ejpam-6327	4	21	structure	structure	NOUN
ejpam-6327	4	22	is	be	AUX
ejpam-6327	4	23	a	a	DET
ejpam-6327	4	24	generalization	generalization	NOUN
ejpam-6327	4	25	of	of	ADP
ejpam-6327	4	26	the	the	DET
ejpam-6327	4	27	existing	exist	VERB
ejpam-6327	4	28	quotient	quotient	NOUN
ejpam-6327	4	29	structure	structure	NOUN
ejpam-6327	4	30	and	and	CCONJ
ejpam-6327	4	31	prove	prove	VERB
ejpam-6327	4	32	properties	property	NOUN
ejpam-6327	4	33	of	of	ADP
ejpam-6327	4	34	this	this	DET
ejpam-6327	4	35	generalized	generalized	ADJ
ejpam-6327	4	36	structure	structure	NOUN
ejpam-6327	4	37	.	.	PUNCT
ejpam-6327	5	1	moreover	moreover	ADV
ejpam-6327	5	2	,	,	PUNCT
ejpam-6327	5	3	we	we	PRON
ejpam-6327	5	4	investigate	investigate	VERB
ejpam-6327	5	5	the	the	DET
ejpam-6327	5	6	quotient	quotient	NOUN
ejpam-6327	5	7	structure	structure	NOUN
ejpam-6327	5	8	of	of	ADP
ejpam-6327	5	9	product	product	NOUN
ejpam-6327	5	10	ks	ks	NOUN
ejpam-6327	5	11	-	-	PUNCT
ejpam-6327	5	12	semigroups	semigroup	NOUN
ejpam-6327	5	13	induced	induce	VERB
ejpam-6327	5	14	by	by	ADP
ejpam-6327	5	15	fuzzy	fuzzy	ADJ
ejpam-6327	5	16	ksideals	ksideal	NOUN
ejpam-6327	5	17	.	.	PUNCT
ejpam-6327	6	1	2020	2020	NUM
ejpam-6327	6	2	mathematics	mathematic	NOUN
ejpam-6327	6	3	subject	subject	NOUN
ejpam-6327	6	4	classifications	classification	NOUN
ejpam-6327	6	5	:	:	PUNCT
ejpam-6327	6	6	08a30	08a30	NOUN
ejpam-6327	6	7	,	,	PUNCT
ejpam-6327	6	8	08a72	08a72	NOUN
ejpam-6327	6	9	key	key	ADJ
ejpam-6327	6	10	words	word	NOUN
ejpam-6327	6	11	and	and	CCONJ
ejpam-6327	6	12	phrases	phrase	NOUN
ejpam-6327	6	13	:	:	PUNCT
ejpam-6327	6	14	quotient	quotient	NOUN
ejpam-6327	6	15	ks	k	NOUN
ejpam-6327	6	16	-	-	PUNCT
ejpam-6327	6	17	semigroups	semigroup	NOUN
ejpam-6327	6	18	,	,	PUNCT
ejpam-6327	6	19	congruence	congruence	PROPN
ejpam-6327	6	20	relation	relation	NOUN
ejpam-6327	6	21	,	,	PUNCT
ejpam-6327	6	22	fuzzy	fuzzy	ADJ
ejpam-6327	6	23	ks	k	NOUN
ejpam-6327	6	24	-	-	PUNCT
ejpam-6327	6	25	ideals	ideal	NOUN
ejpam-6327	6	26	,	,	PUNCT
ejpam-6327	6	27	fuzzy	fuzzy	ADJ
ejpam-6327	6	28	ks	ks	NOUN
ejpam-6327	6	29	-	-	ADJ
ejpam-6327	6	30	p	p	NOUN
ejpam-6327	6	31	-	-	PUNCT
ejpam-6327	6	32	ideals	ideal	NOUN
ejpam-6327	6	33	,	,	PUNCT
ejpam-6327	6	34	fuzzy	fuzzy	ADJ
ejpam-6327	6	35	(	(	PUNCT
ejpam-6327	6	36	commutative	commutative	ADJ
ejpam-6327	6	37	,	,	PUNCT
ejpam-6327	6	38	implicative	implicative	ADJ
ejpam-6327	6	39	)	)	PUNCT
ejpam-6327	6	40	ks	k	NOUN
ejpam-6327	6	41	-	-	PUNCT
ejpam-6327	6	42	ideals	ideal	NOUN
ejpam-6327	6	43	1	1	NUM
ejpam-6327	6	44	.	.	PUNCT
ejpam-6327	7	1	introduction	introduction	NOUN
ejpam-6327	7	2	the	the	DET
ejpam-6327	7	3	class	class	NOUN
ejpam-6327	7	4	of	of	ADP
ejpam-6327	7	5	bck	bck	PROPN
ejpam-6327	7	6	-	-	PUNCT
ejpam-6327	7	7	algebras	algebras	PROPN
ejpam-6327	7	8	was	be	AUX
ejpam-6327	7	9	introduced	introduce	VERB
ejpam-6327	7	10	by	by	ADP
ejpam-6327	7	11	imai	imai	PROPN
ejpam-6327	7	12	and	and	CCONJ
ejpam-6327	7	13	iseki	iseki	PROPN
ejpam-6327	8	1	[	[	X
ejpam-6327	8	2	1	1	X
ejpam-6327	8	3	]	]	PUNCT
ejpam-6327	8	4	in	in	ADP
ejpam-6327	8	5	1966	1966	NUM
ejpam-6327	8	6	that	that	PRON
ejpam-6327	8	7	describes	describe	VERB
ejpam-6327	8	8	fragments	fragment	NOUN
ejpam-6327	8	9	of	of	ADP
ejpam-6327	8	10	propositional	propositional	ADJ
ejpam-6327	8	11	calculus	calculus	NOUN
ejpam-6327	8	12	involving	involve	VERB
ejpam-6327	8	13	implication	implication	NOUN
ejpam-6327	8	14	,	,	PUNCT
ejpam-6327	8	15	known	know	VERB
ejpam-6327	8	16	as	as	ADP
ejpam-6327	8	17	bck	bck	PROPN
ejpam-6327	8	18	logic	logic	NOUN
ejpam-6327	8	19	.	.	PUNCT
ejpam-6327	9	1	it	it	PRON
ejpam-6327	9	2	was	be	AUX
ejpam-6327	9	3	also	also	ADV
ejpam-6327	9	4	developed	develop	VERB
ejpam-6327	9	5	as	as	ADP
ejpam-6327	9	6	a	a	DET
ejpam-6327	9	7	generalization	generalization	NOUN
ejpam-6327	9	8	of	of	ADP
ejpam-6327	9	9	set	set	ADJ
ejpam-6327	9	10	difference	difference	NOUN
ejpam-6327	9	11	in	in	ADP
ejpam-6327	9	12	set	set	NOUN
ejpam-6327	9	13	theory	theory	NOUN
ejpam-6327	9	14	.	.	PUNCT
ejpam-6327	10	1	since	since	SCONJ
ejpam-6327	10	2	then	then	ADV
ejpam-6327	10	3	,	,	PUNCT
ejpam-6327	10	4	a	a	DET
ejpam-6327	10	5	great	great	ADJ
ejpam-6327	10	6	deal	deal	NOUN
ejpam-6327	10	7	of	of	ADP
ejpam-6327	10	8	literature	literature	NOUN
ejpam-6327	10	9	has	have	AUX
ejpam-6327	10	10	been	be	AUX
ejpam-6327	10	11	produced	produce	VERB
ejpam-6327	10	12	in	in	ADP
ejpam-6327	10	13	the	the	DET
ejpam-6327	10	14	theory	theory	NOUN
ejpam-6327	10	15	of	of	ADP
ejpam-6327	10	16	bck	bck	PROPN
ejpam-6327	10	17	-	-	PUNCT
ejpam-6327	10	18	algebras	algebras	PROPN
ejpam-6327	10	19	.	.	PUNCT
ejpam-6327	11	1	the	the	DET
ejpam-6327	11	2	initial	initial	ADJ
ejpam-6327	11	3	paper	paper	NOUN
ejpam-6327	11	4	on	on	ADP
ejpam-6327	11	5	the	the	DET
ejpam-6327	11	6	class	class	NOUN
ejpam-6327	11	7	of	of	ADP
ejpam-6327	11	8	semigroups	semigroup	NOUN
ejpam-6327	11	9	emerged	emerge	VERB
ejpam-6327	11	10	in	in	ADP
ejpam-6327	11	11	1905	1905	NUM
ejpam-6327	11	12	as	as	ADP
ejpam-6327	11	13	a	a	DET
ejpam-6327	11	14	concise	concise	ADJ
ejpam-6327	11	15	work	work	NOUN
ejpam-6327	11	16	by	by	ADP
ejpam-6327	11	17	dickson	dickson	PROPN
ejpam-6327	11	18	.	.	PUNCT
ejpam-6327	12	1	however	however	ADV
ejpam-6327	12	2	,	,	PUNCT
ejpam-6327	12	3	the	the	DET
ejpam-6327	12	4	true	true	ADJ
ejpam-6327	12	5	inception	inception	NOUN
ejpam-6327	12	6	of	of	ADP
ejpam-6327	12	7	the	the	DET
ejpam-6327	12	8	theory	theory	NOUN
ejpam-6327	12	9	occurred	occur	VERB
ejpam-6327	12	10	in	in	ADP
ejpam-6327	12	11	1928	1928	NUM
ejpam-6327	12	12	when	when	SCONJ
ejpam-6327	12	13	suschkewitsch	suschkewitsch	NOUN
ejpam-6327	12	14	[	[	X
ejpam-6327	12	15	2	2	NUM
ejpam-6327	12	16	]	]	PUNCT
ejpam-6327	12	17	published	publish	VERB
ejpam-6327	12	18	a	a	DET
ejpam-6327	12	19	paper	paper	NOUN
ejpam-6327	12	20	of	of	ADP
ejpam-6327	12	21	paramount	paramount	ADJ
ejpam-6327	12	22	significance	significance	NOUN
ejpam-6327	12	23	.	.	PUNCT
ejpam-6327	13	1	in	in	ADP
ejpam-6327	13	2	contemporary	contemporary	ADJ
ejpam-6327	13	3	language	language	NOUN
ejpam-6327	13	4	,	,	PUNCT
ejpam-6327	13	5	he	he	PRON
ejpam-6327	13	6	demonstrated	demonstrate	VERB
ejpam-6327	13	7	that	that	SCONJ
ejpam-6327	13	8	within	within	ADP
ejpam-6327	13	9	any	any	DET
ejpam-6327	13	10	finite	finite	ADJ
ejpam-6327	13	11	semigroup	semigroup	NOUN
ejpam-6327	13	12	,	,	PUNCT
ejpam-6327	13	13	there	there	PRON
ejpam-6327	13	14	exists	exist	VERB
ejpam-6327	13	15	a	a	DET
ejpam-6327	13	16	“	"	PUNCT
ejpam-6327	13	17	kernel	kernel	NOUN
ejpam-6327	13	18	”	"	PUNCT
ejpam-6327	13	19	(	(	PUNCT
ejpam-6327	13	20	referred	refer	VERB
ejpam-6327	13	21	to	to	ADP
ejpam-6327	13	22	as	as	ADP
ejpam-6327	13	23	a	a	DET
ejpam-6327	13	24	simple	simple	ADJ
ejpam-6327	13	25	ideal	ideal	NOUN
ejpam-6327	13	26	)	)	PUNCT
ejpam-6327	13	27	,	,	PUNCT
ejpam-6327	13	28	and	and	CCONJ
ejpam-6327	13	29	he	he	PRON
ejpam-6327	13	30	comprehensively	comprehensively	ADV
ejpam-6327	13	31	characterized	characterize	VERB
ejpam-6327	13	32	the	the	DET
ejpam-6327	13	33	structure	structure	NOUN
ejpam-6327	13	34	of	of	ADP
ejpam-6327	13	35	finite	finite	ADJ
ejpam-6327	13	36	simple	simple	ADJ
ejpam-6327	13	37	semigroups	semigroup	NOUN
ejpam-6327	13	38	.	.	PUNCT
ejpam-6327	14	1	semigroups	semigroup	NOUN
ejpam-6327	14	2	provide	provide	VERB
ejpam-6327	14	3	a	a	DET
ejpam-6327	14	4	foundational	foundational	ADJ
ejpam-6327	14	5	framework	framework	NOUN
ejpam-6327	14	6	for	for	ADP
ejpam-6327	14	7	understanding	understand	VERB
ejpam-6327	14	8	how	how	SCONJ
ejpam-6327	14	9	elements	element	NOUN
ejpam-6327	14	10	combine	combine	VERB
ejpam-6327	14	11	under	under	ADP
ejpam-6327	14	12	certain	certain	ADJ
ejpam-6327	14	13	operations	operation	NOUN
ejpam-6327	14	14	,	,	PUNCT
ejpam-6327	14	15	and	and	CCONJ
ejpam-6327	14	16	their	their	PRON
ejpam-6327	14	17	applications	application	NOUN
ejpam-6327	14	18	span	span	VERB
ejpam-6327	14	19	across	across	ADP
ejpam-6327	14	20	multiple	multiple	ADJ
ejpam-6327	14	21	branches	branch	NOUN
ejpam-6327	14	22	of	of	ADP
ejpam-6327	14	23	mathematics	mathematic	NOUN
ejpam-6327	14	24	and	and	CCONJ
ejpam-6327	14	25	various	various	ADJ
ejpam-6327	14	26	interdisciplinary	interdisciplinary	ADJ
ejpam-6327	14	27	fields	field	NOUN
ejpam-6327	14	28	such	such	ADJ
ejpam-6327	14	29	as	as	ADP
ejpam-6327	14	30	coding	code	VERB
ejpam-6327	14	31	theory	theory	NOUN
ejpam-6327	14	32	,	,	PUNCT
ejpam-6327	14	33	automata	automata	NOUN
ejpam-6327	14	34	,	,	PUNCT
ejpam-6327	14	35	etc	etc	X
ejpam-6327	14	36	.	.	X
ejpam-6327	14	37	for	for	ADP
ejpam-6327	14	38	more	more	ADJ
ejpam-6327	14	39	notions	notion	NOUN
ejpam-6327	14	40	about	about	ADP
ejpam-6327	14	41	semigroups	semigroup	NOUN
ejpam-6327	14	42	and	and	CCONJ
ejpam-6327	14	43	history	history	NOUN
ejpam-6327	14	44	,	,	PUNCT
ejpam-6327	14	45	we	we	PRON
ejpam-6327	14	46	refer	refer	VERB
ejpam-6327	14	47	to	to	ADP
ejpam-6327	14	48	[	[	X
ejpam-6327	14	49	3	3	NUM
ejpam-6327	14	50	]	]	PUNCT
ejpam-6327	14	51	.	.	PUNCT
ejpam-6327	15	1	∗corresponding	∗corresponde	VERB
ejpam-6327	15	2	author	author	NOUN
ejpam-6327	15	3	.	.	PUNCT
ejpam-6327	16	1	doi	doi	NOUN
ejpam-6327	16	2	:	:	PUNCT
ejpam-6327	16	3	https://doi.org/10.29020/nybg.ejpam.v18i3.6327	https://doi.org/10.29020/nybg.ejpam.v18i3.6327	PROPN
ejpam-6327	16	4	email	email	NOUN
ejpam-6327	16	5	addresses	address	NOUN
ejpam-6327	16	6	:	:	PUNCT
ejpam-6327	16	7	hulsen.sarapuddin@g.msuiit.edu.ph	hulsen.sarapuddin@g.msuiit.edu.ph	PROPN
ejpam-6327	16	8	(	(	PUNCT
ejpam-6327	16	9	h.	h.	PROPN
ejpam-6327	16	10	t.	t.	PROPN
ejpam-6327	16	11	sarapuddin	sarapuddin	PROPN
ejpam-6327	16	12	)	)	PUNCT
ejpam-6327	16	13	,	,	PUNCT
ejpam-6327	16	14	jocelyn.vilela@g.msuiit.edu.ph	jocelyn.vilela@g.msuiit.edu.ph	PROPN
ejpam-6327	16	15	(	(	PUNCT
ejpam-6327	16	16	j.	j.	PROPN
ejpam-6327	16	17	p.	p.	PROPN
ejpam-6327	16	18	vilela	vilela	PROPN
ejpam-6327	16	19	)	)	PUNCT
ejpam-6327	16	20	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6327	17	1	1	1	NUM
ejpam-6327	17	2	copyright	copyright	NOUN
ejpam-6327	17	3	:	:	PUNCT
ejpam-6327	17	4	©	©	PROPN
ejpam-6327	17	5	2025	2025	NUM
ejpam-6327	17	6	the	the	DET
ejpam-6327	17	7	author(s	author(s	NOUN
ejpam-6327	17	8	)	)	PUNCT
ejpam-6327	17	9	.	.	PUNCT
ejpam-6327	18	1	(	(	PUNCT
ejpam-6327	18	2	cc	cc	NOUN
ejpam-6327	18	3	by	by	ADP
ejpam-6327	18	4	-	-	PUNCT
ejpam-6327	18	5	nc	nc	PROPN
ejpam-6327	18	6	4.0	4.0	NUM
ejpam-6327	18	7	)	)	PUNCT
ejpam-6327	18	8	h.	h.	NOUN
ejpam-6327	18	9	sarapuddin	sarapuddin	PROPN
ejpam-6327	18	10	,	,	PUNCT
ejpam-6327	18	11	j.	j.	PROPN
ejpam-6327	18	12	vilela	vilela	PROPN
ejpam-6327	18	13	/	/	SYM
ejpam-6327	18	14	eur	eur	PROPN
ejpam-6327	18	15	.	.	PUNCT
ejpam-6327	19	1	j.	j.	PROPN
ejpam-6327	19	2	pure	pure	PROPN
ejpam-6327	19	3	appl	appl	PROPN
ejpam-6327	19	4	.	.	PROPN
ejpam-6327	19	5	math	math	PROPN
ejpam-6327	19	6	,	,	PUNCT
ejpam-6327	19	7	18	18	NUM
ejpam-6327	19	8	(	(	PUNCT
ejpam-6327	19	9	3	3	NUM
ejpam-6327	19	10	)	)	PUNCT
ejpam-6327	19	11	(	(	PUNCT
ejpam-6327	19	12	2025	2025	NUM
ejpam-6327	19	13	)	)	PUNCT
ejpam-6327	19	14	,	,	PUNCT
ejpam-6327	19	15	6327	6327	NUM
ejpam-6327	19	16	2	2	NUM
ejpam-6327	19	17	of	of	ADP
ejpam-6327	19	18	23	23	NUM
ejpam-6327	19	19	the	the	DET
ejpam-6327	19	20	notion	notion	NOUN
ejpam-6327	19	21	of	of	ADP
ejpam-6327	19	22	fuzzy	fuzzy	ADJ
ejpam-6327	19	23	sets	set	NOUN
ejpam-6327	19	24	was	be	AUX
ejpam-6327	19	25	introduced	introduce	VERB
ejpam-6327	19	26	by	by	ADP
ejpam-6327	19	27	zadeh	zadeh	PROPN
ejpam-6327	19	28	[	[	X
ejpam-6327	19	29	4	4	X
ejpam-6327	19	30	]	]	PUNCT
ejpam-6327	19	31	in	in	ADP
ejpam-6327	19	32	1965	1965	NUM
ejpam-6327	19	33	to	to	PART
ejpam-6327	19	34	provide	provide	VERB
ejpam-6327	19	35	a	a	DET
ejpam-6327	19	36	mathematical	mathematical	ADJ
ejpam-6327	19	37	framework	framework	NOUN
ejpam-6327	19	38	for	for	ADP
ejpam-6327	19	39	dealing	deal	VERB
ejpam-6327	19	40	with	with	ADP
ejpam-6327	19	41	vagueness	vagueness	NOUN
ejpam-6327	19	42	and	and	CCONJ
ejpam-6327	19	43	imprecision	imprecision	NOUN
ejpam-6327	19	44	.	.	PUNCT
ejpam-6327	20	1	it	it	PRON
ejpam-6327	20	2	departs	depart	VERB
ejpam-6327	20	3	from	from	ADP
ejpam-6327	20	4	classical	classical	ADJ
ejpam-6327	20	5	set	set	NOUN
ejpam-6327	20	6	theory	theory	NOUN
ejpam-6327	20	7	by	by	ADP
ejpam-6327	20	8	allowing	allow	VERB
ejpam-6327	20	9	elements	element	NOUN
ejpam-6327	20	10	to	to	PART
ejpam-6327	20	11	have	have	VERB
ejpam-6327	20	12	degrees	degree	NOUN
ejpam-6327	20	13	of	of	ADP
ejpam-6327	20	14	membership	membership	NOUN
ejpam-6327	20	15	in	in	ADP
ejpam-6327	20	16	a	a	DET
ejpam-6327	20	17	set	set	NOUN
ejpam-6327	20	18	,	,	PUNCT
ejpam-6327	20	19	rather	rather	ADV
ejpam-6327	20	20	than	than	ADP
ejpam-6327	20	21	requiring	require	VERB
ejpam-6327	20	22	absolute	absolute	ADJ
ejpam-6327	20	23	membership	membership	NOUN
ejpam-6327	20	24	or	or	CCONJ
ejpam-6327	20	25	non	non	ADJ
ejpam-6327	20	26	-	-	NOUN
ejpam-6327	20	27	membership	membership	NOUN
ejpam-6327	20	28	.	.	PUNCT
ejpam-6327	21	1	fuzzy	fuzzy	ADJ
ejpam-6327	21	2	sets	set	NOUN
ejpam-6327	21	3	quickly	quickly	ADV
ejpam-6327	21	4	garnered	garner	VERB
ejpam-6327	21	5	attention	attention	NOUN
ejpam-6327	21	6	across	across	ADP
ejpam-6327	21	7	various	various	ADJ
ejpam-6327	21	8	disciplines	discipline	NOUN
ejpam-6327	21	9	,	,	PUNCT
ejpam-6327	21	10	including	include	VERB
ejpam-6327	21	11	algebraic	algebraic	ADJ
ejpam-6327	21	12	structures	structure	NOUN
ejpam-6327	21	13	.	.	PUNCT
ejpam-6327	22	1	the	the	DET
ejpam-6327	22	2	combination	combination	NOUN
ejpam-6327	22	3	of	of	ADP
ejpam-6327	22	4	the	the	DET
ejpam-6327	22	5	two	two	NUM
ejpam-6327	22	6	concepts	concept	NOUN
ejpam-6327	22	7	gave	give	VERB
ejpam-6327	22	8	rise	rise	NOUN
ejpam-6327	22	9	to	to	ADP
ejpam-6327	22	10	fuzzy	fuzzy	ADJ
ejpam-6327	22	11	algebraic	algebraic	ADJ
ejpam-6327	22	12	structures	structure	NOUN
ejpam-6327	22	13	.	.	PUNCT
ejpam-6327	23	1	the	the	DET
ejpam-6327	23	2	latter	latter	ADJ
ejpam-6327	23	3	was	be	AUX
ejpam-6327	23	4	established	establish	VERB
ejpam-6327	23	5	by	by	ADP
ejpam-6327	23	6	rosenfeld	rosenfeld	PROPN
ejpam-6327	23	7	in	in	ADP
ejpam-6327	23	8	1971	1971	NUM
ejpam-6327	23	9	when	when	SCONJ
ejpam-6327	23	10	he	he	PRON
ejpam-6327	23	11	applied	apply	VERB
ejpam-6327	23	12	the	the	DET
ejpam-6327	23	13	concept	concept	NOUN
ejpam-6327	23	14	of	of	ADP
ejpam-6327	23	15	fuzzy	fuzzy	ADJ
ejpam-6327	23	16	sets	set	NOUN
ejpam-6327	23	17	to	to	ADP
ejpam-6327	23	18	groups	group	NOUN
ejpam-6327	23	19	.	.	PUNCT
ejpam-6327	24	1	in	in	ADP
ejpam-6327	24	2	2006	2006	NUM
ejpam-6327	24	3	,	,	PUNCT
ejpam-6327	24	4	kim	kim	PROPN
ejpam-6327	24	5	[	[	X
ejpam-6327	24	6	5	5	NUM
ejpam-6327	24	7	]	]	PUNCT
ejpam-6327	24	8	introduced	introduce	VERB
ejpam-6327	24	9	the	the	DET
ejpam-6327	24	10	class	class	NOUN
ejpam-6327	24	11	of	of	ADP
ejpam-6327	24	12	ks	ks	NOUN
ejpam-6327	24	13	-	-	PUNCT
ejpam-6327	24	14	semigroups	semigroup	NOUN
ejpam-6327	24	15	which	which	PRON
ejpam-6327	24	16	is	be	AUX
ejpam-6327	24	17	both	both	PRON
ejpam-6327	24	18	a	a	DET
ejpam-6327	24	19	bck	bck	NOUN
ejpam-6327	24	20	-	-	PUNCT
ejpam-6327	24	21	algebra	algebra	NOUN
ejpam-6327	24	22	and	and	CCONJ
ejpam-6327	24	23	a	a	DET
ejpam-6327	24	24	semigroup	semigroup	NOUN
ejpam-6327	24	25	.	.	PUNCT
ejpam-6327	25	1	the	the	DET
ejpam-6327	25	2	quotient	quotient	NOUN
ejpam-6327	25	3	ks	ks	NOUN
ejpam-6327	25	4	-	-	PUNCT
ejpam-6327	25	5	semigroups	semigroup	NOUN
ejpam-6327	25	6	via	via	ADP
ejpam-6327	25	7	ideals	ideal	NOUN
ejpam-6327	25	8	was	be	AUX
ejpam-6327	25	9	established	establish	VERB
ejpam-6327	25	10	and	and	CCONJ
ejpam-6327	25	11	the	the	DET
ejpam-6327	25	12	isomorphism	isomorphism	NOUN
ejpam-6327	25	13	theorems	theorem	NOUN
ejpam-6327	25	14	were	be	AUX
ejpam-6327	25	15	proved	prove	VERB
ejpam-6327	25	16	in	in	ADP
ejpam-6327	25	17	2009	2009	NUM
ejpam-6327	25	18	by	by	ADP
ejpam-6327	25	19	cawi	cawi	NOUN
ejpam-6327	25	20	and	and	CCONJ
ejpam-6327	25	21	vilela	vilela	NOUN
ejpam-6327	26	1	[	[	X
ejpam-6327	26	2	6	6	NUM
ejpam-6327	26	3	]	]	PUNCT
ejpam-6327	26	4	.	.	PUNCT
ejpam-6327	27	1	in	in	ADP
ejpam-6327	27	2	2007	2007	NUM
ejpam-6327	27	3	,	,	PUNCT
ejpam-6327	27	4	prince	prince	PROPN
ejpam-6327	27	5	williams	williams	PROPN
ejpam-6327	27	6	and	and	CCONJ
ejpam-6327	27	7	husain	husain	PROPN
ejpam-6327	27	8	[	[	X
ejpam-6327	27	9	7	7	NUM
ejpam-6327	27	10	]	]	PUNCT
ejpam-6327	27	11	applied	apply	VERB
ejpam-6327	27	12	the	the	DET
ejpam-6327	27	13	concept	concept	NOUN
ejpam-6327	27	14	of	of	ADP
ejpam-6327	27	15	fuzzy	fuzzy	ADJ
ejpam-6327	27	16	sets	set	NOUN
ejpam-6327	27	17	to	to	ADP
ejpam-6327	27	18	ks	ks	NOUN
ejpam-6327	27	19	-	-	PUNCT
ejpam-6327	27	20	semigroup	semigroup	NOUN
ejpam-6327	27	21	,	,	PUNCT
ejpam-6327	27	22	and	and	CCONJ
ejpam-6327	27	23	referred	refer	VERB
ejpam-6327	27	24	to	to	ADP
ejpam-6327	27	25	it	it	PRON
ejpam-6327	27	26	as	as	ADP
ejpam-6327	27	27	a	a	DET
ejpam-6327	27	28	fuzzy	fuzzy	ADJ
ejpam-6327	27	29	ks	ks	NOUN
ejpam-6327	27	30	-	-	PUNCT
ejpam-6327	27	31	semigroup	semigroup	NOUN
ejpam-6327	27	32	.	.	PUNCT
ejpam-6327	28	1	the	the	DET
ejpam-6327	28	2	notions	notion	NOUN
ejpam-6327	28	3	of	of	ADP
ejpam-6327	28	4	fuzzy	fuzzy	ADJ
ejpam-6327	28	5	ks	ks	NOUN
ejpam-6327	28	6	-	-	PUNCT
ejpam-6327	28	7	ideals	ideal	NOUN
ejpam-6327	28	8	and	and	CCONJ
ejpam-6327	28	9	fuzzy	fuzzy	ADJ
ejpam-6327	28	10	ks	ks	NOUN
ejpam-6327	28	11	-	-	ADJ
ejpam-6327	28	12	p	p	NOUN
ejpam-6327	28	13	-	-	PUNCT
ejpam-6327	28	14	ideals	ideal	NOUN
ejpam-6327	28	15	were	be	AUX
ejpam-6327	28	16	introduced	introduce	VERB
ejpam-6327	28	17	and	and	CCONJ
ejpam-6327	28	18	some	some	PRON
ejpam-6327	28	19	of	of	ADP
ejpam-6327	28	20	their	their	PRON
ejpam-6327	28	21	properties	property	NOUN
ejpam-6327	28	22	were	be	AUX
ejpam-6327	28	23	investigated	investigate	VERB
ejpam-6327	28	24	.	.	PUNCT
ejpam-6327	29	1	in	in	ADP
ejpam-6327	29	2	2011	2011	NUM
ejpam-6327	29	3	,	,	PUNCT
ejpam-6327	29	4	bautista	bautista	PROPN
ejpam-6327	29	5	and	and	CCONJ
ejpam-6327	29	6	vilela	vilela	VERB
ejpam-6327	30	1	[	[	X
ejpam-6327	30	2	8	8	NUM
ejpam-6327	30	3	]	]	PUNCT
ejpam-6327	30	4	introduced	introduce	VERB
ejpam-6327	30	5	and	and	CCONJ
ejpam-6327	30	6	investigated	investigate	VERB
ejpam-6327	30	7	fuzzy	fuzzy	ADJ
ejpam-6327	30	8	topology	topology	NOUN
ejpam-6327	30	9	on	on	ADP
ejpam-6327	30	10	a	a	DET
ejpam-6327	30	11	ks	ks	NOUN
ejpam-6327	30	12	-	-	PUNCT
ejpam-6327	30	13	semigroup	semigroup	NOUN
ejpam-6327	30	14	and	and	CCONJ
ejpam-6327	30	15	fuzzy	fuzzy	ADJ
ejpam-6327	30	16	topological	topological	ADJ
ejpam-6327	30	17	ks	ks	PROPN
ejpam-6327	30	18	-	-	PUNCT
ejpam-6327	30	19	semigroup	semigroup	NOUN
ejpam-6327	30	20	.	.	PUNCT
ejpam-6327	31	1	there	there	PRON
ejpam-6327	31	2	are	be	VERB
ejpam-6327	31	3	many	many	ADJ
ejpam-6327	31	4	research	research	NOUN
ejpam-6327	31	5	items	item	NOUN
ejpam-6327	31	6	in	in	ADP
ejpam-6327	31	7	the	the	DET
ejpam-6327	31	8	literature	literature	NOUN
ejpam-6327	31	9	characterizing	characterize	VERB
ejpam-6327	31	10	the	the	DET
ejpam-6327	31	11	class	class	NOUN
ejpam-6327	31	12	of	of	ADP
ejpam-6327	31	13	semigroups	semigroup	NOUN
ejpam-6327	31	14	through	through	ADP
ejpam-6327	31	15	its	its	PRON
ejpam-6327	31	16	(	(	PUNCT
ejpam-6327	31	17	fuzzy	fuzzy	ADJ
ejpam-6327	31	18	)	)	PUNCT
ejpam-6327	31	19	sets	set	NOUN
ejpam-6327	31	20	,	,	PUNCT
ejpam-6327	31	21	particularly	particularly	ADV
ejpam-6327	31	22	,	,	PUNCT
ejpam-6327	31	23	through	through	ADP
ejpam-6327	31	24	generalization	generalization	NOUN
ejpam-6327	31	25	and	and	CCONJ
ejpam-6327	31	26	fuzzification	fuzzification	NOUN
ejpam-6327	31	27	of	of	ADP
ejpam-6327	31	28	ideals	ideal	NOUN
ejpam-6327	31	29	.	.	PUNCT
ejpam-6327	32	1	for	for	ADP
ejpam-6327	32	2	example	example	NOUN
ejpam-6327	32	3	,	,	PUNCT
ejpam-6327	32	4	generalization	generalization	NOUN
ejpam-6327	32	5	of	of	ADP
ejpam-6327	32	6	bi	bi	NOUN
ejpam-6327	32	7	-	-	NOUN
ejpam-6327	32	8	antiideals	antiideal	NOUN
ejpam-6327	32	9	in	in	ADP
ejpam-6327	32	10	semigroups	semigroup	NOUN
ejpam-6327	32	11	and	and	CCONJ
ejpam-6327	32	12	their	their	PRON
ejpam-6327	32	13	fuzzification	fuzzification	NOUN
ejpam-6327	32	14	were	be	AUX
ejpam-6327	32	15	studied	study	VERB
ejpam-6327	32	16	in	in	ADP
ejpam-6327	32	17	[	[	X
ejpam-6327	32	18	9	9	NUM
ejpam-6327	32	19	]	]	PUNCT
ejpam-6327	32	20	and	and	CCONJ
ejpam-6327	32	21	(	(	PUNCT
ejpam-6327	32	22	fuzzy	fuzzy	ADJ
ejpam-6327	32	23	)	)	PUNCT
ejpam-6327	32	24	ks	ks	NOUN
ejpam-6327	32	25	-	-	PUNCT
ejpam-6327	32	26	h	h	NOUN
ejpam-6327	32	27	-	-	PUNCT
ejpam-6327	32	28	ideals	ideal	NOUN
ejpam-6327	32	29	in	in	ADP
ejpam-6327	32	30	ks	ks	NOUN
ejpam-6327	32	31	-	-	PUNCT
ejpam-6327	32	32	semigroups	semigroup	NOUN
ejpam-6327	32	33	and	and	CCONJ
ejpam-6327	32	34	their	their	PRON
ejpam-6327	32	35	properties	property	NOUN
ejpam-6327	32	36	were	be	AUX
ejpam-6327	32	37	studied	study	VERB
ejpam-6327	32	38	in	in	ADP
ejpam-6327	32	39	[	[	X
ejpam-6327	32	40	10	10	NUM
ejpam-6327	32	41	]	]	PUNCT
ejpam-6327	32	42	.	.	PUNCT
ejpam-6327	33	1	moreover	moreover	ADV
ejpam-6327	33	2	,	,	PUNCT
ejpam-6327	33	3	the	the	DET
ejpam-6327	33	4	construction	construction	NOUN
ejpam-6327	33	5	of	of	ADP
ejpam-6327	33	6	quotient	quotient	NOUN
ejpam-6327	33	7	semigroups	semigroup	NOUN
ejpam-6327	33	8	induced	induce	VERB
ejpam-6327	33	9	by	by	ADP
ejpam-6327	33	10	fuzzy	fuzzy	ADJ
ejpam-6327	33	11	ideals	ideal	NOUN
ejpam-6327	33	12	and	and	CCONJ
ejpam-6327	33	13	their	their	PRON
ejpam-6327	33	14	properties	property	NOUN
ejpam-6327	33	15	were	be	AUX
ejpam-6327	33	16	studied	study	VERB
ejpam-6327	33	17	in	in	ADP
ejpam-6327	33	18	[	[	X
ejpam-6327	33	19	11	11	NUM
ejpam-6327	33	20	]	]	PUNCT
ejpam-6327	33	21	.	.	PUNCT
ejpam-6327	34	1	in	in	ADP
ejpam-6327	34	2	this	this	DET
ejpam-6327	34	3	paper	paper	NOUN
ejpam-6327	34	4	,	,	PUNCT
ejpam-6327	34	5	we	we	PRON
ejpam-6327	34	6	introduce	introduce	VERB
ejpam-6327	34	7	new	new	ADJ
ejpam-6327	34	8	types	type	NOUN
ejpam-6327	34	9	of	of	ADP
ejpam-6327	34	10	fuzzy	fuzzy	ADJ
ejpam-6327	34	11	ideals	ideal	NOUN
ejpam-6327	34	12	on	on	ADP
ejpam-6327	34	13	ks	ks	NOUN
ejpam-6327	34	14	-	-	PUNCT
ejpam-6327	34	15	semigroups	semigroup	NOUN
ejpam-6327	34	16	,	,	PUNCT
ejpam-6327	34	17	compare	compare	VERB
ejpam-6327	34	18	them	they	PRON
ejpam-6327	34	19	with	with	ADP
ejpam-6327	34	20	existing	exist	VERB
ejpam-6327	34	21	fuzzy	fuzzy	ADJ
ejpam-6327	34	22	ks	ks	NOUN
ejpam-6327	34	23	-	-	PUNCT
ejpam-6327	34	24	ideals	ideal	NOUN
ejpam-6327	34	25	and	and	CCONJ
ejpam-6327	34	26	investigate	investigate	VERB
ejpam-6327	34	27	their	their	PRON
ejpam-6327	34	28	properties	property	NOUN
ejpam-6327	34	29	.	.	PUNCT
ejpam-6327	35	1	we	we	PRON
ejpam-6327	35	2	construct	construct	VERB
ejpam-6327	35	3	quotient	quotient	NOUN
ejpam-6327	35	4	ks	k	NOUN
ejpam-6327	35	5	-	-	PUNCT
ejpam-6327	35	6	semigroups	semigroup	NOUN
ejpam-6327	35	7	induced	induce	VERB
ejpam-6327	35	8	by	by	ADP
ejpam-6327	35	9	fuzzy	fuzzy	ADJ
ejpam-6327	35	10	ks	ks	NOUN
ejpam-6327	35	11	-	-	PUNCT
ejpam-6327	35	12	ideals	ideal	NOUN
ejpam-6327	35	13	and	and	CCONJ
ejpam-6327	35	14	investigate	investigate	VERB
ejpam-6327	35	15	their	their	PRON
ejpam-6327	35	16	properties	property	NOUN
ejpam-6327	35	17	.	.	PUNCT
ejpam-6327	36	1	furthermore	furthermore	ADV
ejpam-6327	36	2	,	,	PUNCT
ejpam-6327	36	3	we	we	PRON
ejpam-6327	36	4	investigate	investigate	VERB
ejpam-6327	36	5	the	the	DET
ejpam-6327	36	6	quotient	quotient	NOUN
ejpam-6327	36	7	structure	structure	NOUN
ejpam-6327	36	8	of	of	ADP
ejpam-6327	36	9	product	product	NOUN
ejpam-6327	36	10	ks	ks	NOUN
ejpam-6327	36	11	-	-	PUNCT
ejpam-6327	36	12	semigroups	semigroup	NOUN
ejpam-6327	36	13	induced	induce	VERB
ejpam-6327	36	14	by	by	ADP
ejpam-6327	36	15	fuzzy	fuzzy	ADJ
ejpam-6327	36	16	ks	ks	NOUN
ejpam-6327	36	17	-	-	NOUN
ejpam-6327	36	18	ideals	ideal	NOUN
ejpam-6327	36	19	.	.	PUNCT
ejpam-6327	37	1	2	2	X
ejpam-6327	37	2	.	.	X
ejpam-6327	37	3	preliminaries	preliminary	NOUN
ejpam-6327	37	4	in	in	ADP
ejpam-6327	37	5	this	this	DET
ejpam-6327	37	6	section	section	NOUN
ejpam-6327	37	7	,	,	PUNCT
ejpam-6327	37	8	we	we	PRON
ejpam-6327	37	9	present	present	VERB
ejpam-6327	37	10	some	some	DET
ejpam-6327	37	11	basic	basic	ADJ
ejpam-6327	37	12	concepts	concept	NOUN
ejpam-6327	37	13	and	and	CCONJ
ejpam-6327	37	14	known	know	VERB
ejpam-6327	37	15	results	result	NOUN
ejpam-6327	37	16	that	that	PRON
ejpam-6327	37	17	are	be	AUX
ejpam-6327	37	18	useful	useful	ADJ
ejpam-6327	37	19	in	in	ADP
ejpam-6327	37	20	this	this	DET
ejpam-6327	37	21	study	study	NOUN
ejpam-6327	37	22	.	.	PUNCT
ejpam-6327	38	1	definition	definition	NOUN
ejpam-6327	38	2	1	1	NUM
ejpam-6327	38	3	.	.	PUNCT
ejpam-6327	39	1	[	[	X
ejpam-6327	39	2	5	5	NUM
ejpam-6327	39	3	]	]	PUNCT
ejpam-6327	39	4	an	an	DET
ejpam-6327	39	5	algebraic	algebraic	ADJ
ejpam-6327	39	6	system	system	NOUN
ejpam-6327	39	7	(	(	PUNCT
ejpam-6327	39	8	x	x	X
ejpam-6327	39	9	,	,	PUNCT
ejpam-6327	39	10	∗	∗	NOUN
ejpam-6327	39	11	,	,	PUNCT
ejpam-6327	39	12	0	0	NUM
ejpam-6327	39	13	)	)	PUNCT
ejpam-6327	39	14	is	be	AUX
ejpam-6327	39	15	called	call	VERB
ejpam-6327	39	16	a	a	DET
ejpam-6327	39	17	bck	bck	NOUN
ejpam-6327	39	18	-	-	PUNCT
ejpam-6327	39	19	algebra	algebra	NOUN
ejpam-6327	39	20	if	if	SCONJ
ejpam-6327	39	21	it	it	PRON
ejpam-6327	39	22	satisfies	satisfy	VERB
ejpam-6327	39	23	the	the	DET
ejpam-6327	39	24	following	follow	VERB
ejpam-6327	39	25	conditions	condition	NOUN
ejpam-6327	39	26	:	:	PUNCT
ejpam-6327	39	27	for	for	ADP
ejpam-6327	39	28	all	all	DET
ejpam-6327	39	29	x	x	NOUN
ejpam-6327	39	30	,	,	PUNCT
ejpam-6327	39	31	y	y	PROPN
ejpam-6327	39	32	,	,	PUNCT
ejpam-6327	39	33	z	z	PROPN
ejpam-6327	39	34	∈	∈	PROPN
ejpam-6327	39	35	x	x	X
ejpam-6327	39	36	,	,	PUNCT
ejpam-6327	39	37	(	(	PUNCT
ejpam-6327	39	38	i	i	NOUN
ejpam-6327	39	39	)	)	PUNCT
ejpam-6327	39	40	(	(	PUNCT
ejpam-6327	39	41	(	(	PUNCT
ejpam-6327	39	42	x	x	SYM
ejpam-6327	39	43	∗	∗	PROPN
ejpam-6327	39	44	y	y	NOUN
ejpam-6327	39	45	)	)	PUNCT
ejpam-6327	39	46	∗	∗	NOUN
ejpam-6327	39	47	(	(	PUNCT
ejpam-6327	39	48	x	x	X
ejpam-6327	39	49	∗	∗	PROPN
ejpam-6327	39	50	z	z	NOUN
ejpam-6327	39	51	)	)	PUNCT
ejpam-6327	39	52	)	)	PUNCT
ejpam-6327	39	53	∗	∗	NOUN
ejpam-6327	39	54	(	(	PUNCT
ejpam-6327	39	55	z	z	NOUN
ejpam-6327	39	56	∗	∗	NOUN
ejpam-6327	39	57	y	y	NOUN
ejpam-6327	39	58	)	)	PUNCT
ejpam-6327	40	1	=	=	SYM
ejpam-6327	40	2	0	0	NUM
ejpam-6327	40	3	,	,	PUNCT
ejpam-6327	40	4	(	(	PUNCT
ejpam-6327	40	5	iv	iv	X
ejpam-6327	40	6	)	)	PUNCT
ejpam-6327	40	7	0	0	NUM
ejpam-6327	40	8	∗	∗	NOUN
ejpam-6327	40	9	x	x	X
ejpam-6327	41	1	=	=	SYM
ejpam-6327	41	2	0	0	NUM
ejpam-6327	41	3	,	,	PUNCT
ejpam-6327	41	4	(	(	PUNCT
ejpam-6327	41	5	ii	ii	NOUN
ejpam-6327	41	6	)	)	PUNCT
ejpam-6327	41	7	(	(	PUNCT
ejpam-6327	41	8	x	x	SYM
ejpam-6327	41	9	∗	∗	NOUN
ejpam-6327	41	10	(	(	PUNCT
ejpam-6327	41	11	x	x	X
ejpam-6327	41	12	∗	∗	PROPN
ejpam-6327	41	13	y	y	NOUN
ejpam-6327	41	14	)	)	PUNCT
ejpam-6327	41	15	)	)	PUNCT
ejpam-6327	42	1	∗	∗	NOUN
ejpam-6327	42	2	y	y	NOUN
ejpam-6327	42	3	=	=	SYM
ejpam-6327	42	4	0	0	PROPN
ejpam-6327	42	5	,	,	PUNCT
ejpam-6327	42	6	(	(	PUNCT
ejpam-6327	42	7	v	v	NOUN
ejpam-6327	42	8	)	)	PUNCT
ejpam-6327	42	9	x	x	PUNCT
ejpam-6327	42	10	∗	∗	NOUN
ejpam-6327	42	11	y	y	NOUN
ejpam-6327	42	12	=	=	SYM
ejpam-6327	42	13	0	0	PROPN
ejpam-6327	43	1	and	and	CCONJ
ejpam-6327	43	2	y	y	PROPN
ejpam-6327	43	3	∗	∗	NOUN
ejpam-6327	43	4	x	x	PUNCT
ejpam-6327	44	1	=	=	SYM
ejpam-6327	44	2	0	0	NUM
ejpam-6327	44	3	imply	imply	VERB
ejpam-6327	44	4	x	x	X
ejpam-6327	44	5	=	=	SYM
ejpam-6327	44	6	y.	y.	NOUN
ejpam-6327	44	7	(	(	PUNCT
ejpam-6327	44	8	iii	iii	NOUN
ejpam-6327	44	9	)	)	PUNCT
ejpam-6327	44	10	x	x	SYM
ejpam-6327	44	11	∗	∗	NOUN
ejpam-6327	44	12	x	x	SYM
ejpam-6327	44	13	=	=	SYM
ejpam-6327	44	14	0	0	NUM
ejpam-6327	44	15	,	,	PUNCT
ejpam-6327	44	16	definition	definition	NOUN
ejpam-6327	44	17	2	2	NUM
ejpam-6327	44	18	.	.	PUNCT
ejpam-6327	45	1	[	[	X
ejpam-6327	45	2	5	5	X
ejpam-6327	45	3	]	]	PUNCT
ejpam-6327	45	4	if	if	SCONJ
ejpam-6327	45	5	x	x	PRON
ejpam-6327	45	6	is	be	AUX
ejpam-6327	45	7	a	a	DET
ejpam-6327	45	8	bck	bck	NOUN
ejpam-6327	45	9	-	-	PUNCT
ejpam-6327	45	10	algebra	algebra	NOUN
ejpam-6327	45	11	,	,	PUNCT
ejpam-6327	45	12	then	then	ADV
ejpam-6327	45	13	the	the	DET
ejpam-6327	45	14	relation	relation	NOUN
ejpam-6327	45	15	x	x	PUNCT
ejpam-6327	45	16	≤	≤	ADJ
ejpam-6327	45	17	y	y	NOUN
ejpam-6327	45	18	if	if	SCONJ
ejpam-6327	46	1	and	and	CCONJ
ejpam-6327	46	2	only	only	ADV
ejpam-6327	46	3	if	if	SCONJ
ejpam-6327	46	4	x	x	X
ejpam-6327	46	5	∗	∗	VERB
ejpam-6327	46	6	y	y	NOUN
ejpam-6327	46	7	=	=	SYM
ejpam-6327	46	8	0	0	NUM
ejpam-6327	46	9	is	be	AUX
ejpam-6327	46	10	a	a	DET
ejpam-6327	46	11	partial	partial	ADJ
ejpam-6327	46	12	order	order	NOUN
ejpam-6327	46	13	on	on	ADP
ejpam-6327	46	14	x	x	NOUN
ejpam-6327	46	15	,	,	PUNCT
ejpam-6327	46	16	which	which	PRON
ejpam-6327	46	17	will	will	AUX
ejpam-6327	46	18	be	be	AUX
ejpam-6327	46	19	called	call	VERB
ejpam-6327	46	20	the	the	DET
ejpam-6327	46	21	natural	natural	ADJ
ejpam-6327	46	22	ordering	ordering	NOUN
ejpam-6327	46	23	on	on	ADP
ejpam-6327	46	24	x.	x.	PROPN
ejpam-6327	46	25	remark	remark	PROPN
ejpam-6327	46	26	1	1	NUM
ejpam-6327	46	27	.	.	PUNCT
ejpam-6327	47	1	[	[	X
ejpam-6327	47	2	5	5	NUM
ejpam-6327	47	3	]	]	PUNCT
ejpam-6327	47	4	a	a	DET
ejpam-6327	47	5	bck	bck	NOUN
ejpam-6327	47	6	-	-	PUNCT
ejpam-6327	47	7	algebra	algebra	NOUN
ejpam-6327	47	8	x	x	PUNCT
ejpam-6327	47	9	has	have	VERB
ejpam-6327	47	10	the	the	DET
ejpam-6327	47	11	following	follow	VERB
ejpam-6327	47	12	properties	property	NOUN
ejpam-6327	47	13	for	for	ADP
ejpam-6327	47	14	any	any	DET
ejpam-6327	47	15	x	x	NOUN
ejpam-6327	47	16	,	,	PUNCT
ejpam-6327	47	17	y	y	PROPN
ejpam-6327	47	18	,	,	PUNCT
ejpam-6327	47	19	z	z	PROPN
ejpam-6327	47	20	∈	∈	PROPN
ejpam-6327	48	1	x	x	X
ejpam-6327	48	2	:	:	PUNCT
ejpam-6327	48	3	(	(	PUNCT
ejpam-6327	48	4	i	i	NOUN
ejpam-6327	48	5	)	)	PUNCT
ejpam-6327	48	6	x	x	SYM
ejpam-6327	48	7	∗	∗	NOUN
ejpam-6327	48	8	0	0	NUM
ejpam-6327	49	1	=	=	SYM
ejpam-6327	49	2	x	x	NOUN
ejpam-6327	49	3	,	,	PUNCT
ejpam-6327	49	4	(	(	PUNCT
ejpam-6327	49	5	iv	iv	X
ejpam-6327	49	6	)	)	PUNCT
ejpam-6327	49	7	x	x	X
ejpam-6327	49	8	≤	≤	NUM
ejpam-6327	49	9	y	y	NOUN
ejpam-6327	49	10	implies	imply	VERB
ejpam-6327	49	11	that	that	SCONJ
ejpam-6327	49	12	x	x	SYM
ejpam-6327	49	13	∗	∗	NOUN
ejpam-6327	49	14	z	z	NOUN
ejpam-6327	49	15	≤	≤	NOUN
ejpam-6327	49	16	y	y	PROPN
ejpam-6327	49	17	∗	∗	NOUN
ejpam-6327	49	18	z	z	PROPN
ejpam-6327	49	19	and	and	CCONJ
ejpam-6327	49	20	z	z	PROPN
ejpam-6327	49	21	∗	∗	NOUN
ejpam-6327	49	22	y	y	PROPN
ejpam-6327	49	23	≤	≤	PROPN
ejpam-6327	49	24	z	z	NOUN
ejpam-6327	49	25	∗	∗	NOUN
ejpam-6327	49	26	x	x	PROPN
ejpam-6327	49	27	,	,	PUNCT
ejpam-6327	49	28	h.	h.	PROPN
ejpam-6327	49	29	sarapuddin	sarapuddin	PROPN
ejpam-6327	49	30	,	,	PUNCT
ejpam-6327	49	31	j.	j.	PROPN
ejpam-6327	49	32	vilela	vilela	PROPN
ejpam-6327	49	33	/	/	SYM
ejpam-6327	49	34	eur	eur	PROPN
ejpam-6327	49	35	.	.	PUNCT
ejpam-6327	50	1	j.	j.	PROPN
ejpam-6327	50	2	pure	pure	PROPN
ejpam-6327	50	3	appl	appl	PROPN
ejpam-6327	50	4	.	.	PROPN
ejpam-6327	50	5	math	math	PROPN
ejpam-6327	50	6	,	,	PUNCT
ejpam-6327	50	7	18	18	NUM
ejpam-6327	50	8	(	(	PUNCT
ejpam-6327	50	9	3	3	NUM
ejpam-6327	50	10	)	)	PUNCT
ejpam-6327	50	11	(	(	PUNCT
ejpam-6327	50	12	2025	2025	NUM
ejpam-6327	50	13	)	)	PUNCT
ejpam-6327	50	14	,	,	PUNCT
ejpam-6327	50	15	6327	6327	NUM
ejpam-6327	50	16	3	3	NUM
ejpam-6327	50	17	of	of	ADP
ejpam-6327	50	18	23	23	NUM
ejpam-6327	50	19	(	(	PUNCT
ejpam-6327	50	20	ii	ii	NOUN
ejpam-6327	50	21	)	)	PUNCT
ejpam-6327	50	22	x	x	PROPN
ejpam-6327	51	1	∗	∗	NOUN
ejpam-6327	51	2	y	y	NOUN
ejpam-6327	51	3	≤	≤	NUM
ejpam-6327	51	4	x	x	X
ejpam-6327	51	5	,	,	PUNCT
ejpam-6327	51	6	(	(	PUNCT
ejpam-6327	51	7	v	v	NOUN
ejpam-6327	51	8	)	)	PUNCT
ejpam-6327	51	9	(	(	PUNCT
ejpam-6327	51	10	x	x	SYM
ejpam-6327	51	11	∗	∗	PROPN
ejpam-6327	51	12	z	z	NOUN
ejpam-6327	51	13	)	)	PUNCT
ejpam-6327	51	14	∗	∗	NOUN
ejpam-6327	51	15	(	(	PUNCT
ejpam-6327	51	16	y	y	PROPN
ejpam-6327	51	17	∗	∗	PROPN
ejpam-6327	51	18	z	z	NOUN
ejpam-6327	51	19	)	)	PUNCT
ejpam-6327	51	20	≤	≤	NUM
ejpam-6327	51	21	x	x	PUNCT
ejpam-6327	51	22	∗	∗	NOUN
ejpam-6327	51	23	y.	y.	PROPN
ejpam-6327	51	24	(	(	PUNCT
ejpam-6327	51	25	iii	iii	PROPN
ejpam-6327	51	26	)	)	PUNCT
ejpam-6327	51	27	(	(	PUNCT
ejpam-6327	51	28	x	x	SYM
ejpam-6327	51	29	∗	∗	PROPN
ejpam-6327	51	30	y	y	NOUN
ejpam-6327	51	31	)	)	PUNCT
ejpam-6327	51	32	∗	∗	NOUN
ejpam-6327	51	33	z	z	NOUN
ejpam-6327	51	34	=	=	SYM
ejpam-6327	51	35	(	(	PUNCT
ejpam-6327	51	36	x	x	X
ejpam-6327	51	37	∗	∗	PROPN
ejpam-6327	51	38	z	z	NOUN
ejpam-6327	51	39	)	)	PUNCT
ejpam-6327	51	40	∗	∗	PROPN
ejpam-6327	51	41	y	y	PROPN
ejpam-6327	51	42	,	,	PUNCT
ejpam-6327	51	43	proposition	proposition	NOUN
ejpam-6327	51	44	1	1	NUM
ejpam-6327	51	45	.	.	PUNCT
ejpam-6327	52	1	[	[	X
ejpam-6327	52	2	12	12	NUM
ejpam-6327	52	3	]	]	PUNCT
ejpam-6327	52	4	in	in	ADP
ejpam-6327	52	5	a	a	DET
ejpam-6327	52	6	bck	bck	NOUN
ejpam-6327	52	7	-	-	PUNCT
ejpam-6327	52	8	algebra	algebra	NOUN
ejpam-6327	52	9	x	x	NOUN
ejpam-6327	52	10	,	,	PUNCT
ejpam-6327	52	11	the	the	DET
ejpam-6327	52	12	following	follow	VERB
ejpam-6327	52	13	hold	hold	NOUN
ejpam-6327	52	14	:	:	PUNCT
ejpam-6327	52	15	for	for	ADP
ejpam-6327	52	16	all	all	DET
ejpam-6327	52	17	x	x	NOUN
ejpam-6327	52	18	,	,	PUNCT
ejpam-6327	52	19	y	y	PROPN
ejpam-6327	52	20	,	,	PUNCT
ejpam-6327	52	21	z	z	PROPN
ejpam-6327	52	22	∈	∈	PROPN
ejpam-6327	52	23	x	x	X
ejpam-6327	52	24	,	,	PUNCT
ejpam-6327	52	25	(	(	PUNCT
ejpam-6327	52	26	i	i	NOUN
ejpam-6327	52	27	)	)	PUNCT
ejpam-6327	52	28	(	(	PUNCT
ejpam-6327	52	29	(	(	PUNCT
ejpam-6327	52	30	x	x	SYM
ejpam-6327	52	31	∗	∗	PROPN
ejpam-6327	52	32	z	z	NOUN
ejpam-6327	52	33	)	)	PUNCT
ejpam-6327	52	34	∗	∗	PROPN
ejpam-6327	52	35	z	z	NOUN
ejpam-6327	52	36	)	)	PUNCT
ejpam-6327	52	37	∗	∗	NOUN
ejpam-6327	52	38	(	(	PUNCT
ejpam-6327	52	39	y	y	PROPN
ejpam-6327	52	40	∗	∗	PROPN
ejpam-6327	52	41	z	z	PROPN
ejpam-6327	52	42	)	)	PUNCT
ejpam-6327	52	43	≤	≤	NOUN
ejpam-6327	52	44	(	(	PUNCT
ejpam-6327	52	45	x	x	X
ejpam-6327	52	46	∗	∗	PROPN
ejpam-6327	52	47	y	y	NOUN
ejpam-6327	52	48	)	)	PUNCT
ejpam-6327	52	49	∗	∗	NOUN
ejpam-6327	52	50	z	z	PROPN
ejpam-6327	52	51	,	,	PUNCT
ejpam-6327	52	52	(	(	PUNCT
ejpam-6327	52	53	ii	ii	NOUN
ejpam-6327	52	54	)	)	PUNCT
ejpam-6327	52	55	(	(	PUNCT
ejpam-6327	52	56	x	x	SYM
ejpam-6327	52	57	∗	∗	PROPN
ejpam-6327	52	58	z	z	NOUN
ejpam-6327	52	59	)	)	PUNCT
ejpam-6327	52	60	∗	∗	NOUN
ejpam-6327	52	61	(	(	PUNCT
ejpam-6327	52	62	x	x	X
ejpam-6327	52	63	∗	∗	NOUN
ejpam-6327	52	64	(	(	PUNCT
ejpam-6327	52	65	x	x	X
ejpam-6327	52	66	∗	∗	PROPN
ejpam-6327	52	67	z	z	NOUN
ejpam-6327	52	68	)	)	PUNCT
ejpam-6327	52	69	)	)	PUNCT
ejpam-6327	53	1	=	=	PRON
ejpam-6327	53	2	(	(	PUNCT
ejpam-6327	53	3	x	x	X
ejpam-6327	53	4	∗	∗	PROPN
ejpam-6327	53	5	z	z	NOUN
ejpam-6327	53	6	)	)	PUNCT
ejpam-6327	53	7	∗	∗	PROPN
ejpam-6327	53	8	z	z	PROPN
ejpam-6327	53	9	,	,	PUNCT
ejpam-6327	53	10	(	(	PUNCT
ejpam-6327	53	11	iii	iii	NOUN
ejpam-6327	53	12	)	)	PUNCT
ejpam-6327	53	13	(	(	PUNCT
ejpam-6327	53	14	x	x	SYM
ejpam-6327	53	15	∗	∗	NOUN
ejpam-6327	53	16	(	(	PUNCT
ejpam-6327	53	17	y	y	PROPN
ejpam-6327	53	18	∗	∗	NOUN
ejpam-6327	53	19	(	(	PUNCT
ejpam-6327	53	20	y	y	PROPN
ejpam-6327	53	21	∗	∗	NOUN
ejpam-6327	53	22	x	x	NOUN
ejpam-6327	53	23	)	)	PUNCT
ejpam-6327	53	24	)	)	PUNCT
ejpam-6327	53	25	)	)	PUNCT
ejpam-6327	54	1	∗	∗	NOUN
ejpam-6327	54	2	(	(	PUNCT
ejpam-6327	54	3	y	y	PROPN
ejpam-6327	54	4	∗	∗	NOUN
ejpam-6327	54	5	(	(	PUNCT
ejpam-6327	54	6	x	x	SYM
ejpam-6327	54	7	∗	∗	NOUN
ejpam-6327	54	8	(	(	PUNCT
ejpam-6327	54	9	y	y	PROPN
ejpam-6327	54	10	∗	∗	NOUN
ejpam-6327	54	11	(	(	PUNCT
ejpam-6327	54	12	y	y	PROPN
ejpam-6327	54	13	∗	∗	NOUN
ejpam-6327	54	14	x	x	NOUN
ejpam-6327	54	15	)	)	PUNCT
ejpam-6327	54	16	)	)	PUNCT
ejpam-6327	54	17	)	)	PUNCT
ejpam-6327	54	18	)	)	PUNCT
ejpam-6327	54	19	≤	≤	NUM
ejpam-6327	55	1	x	x	X
ejpam-6327	55	2	∗	∗	NOUN
ejpam-6327	55	3	y.	y.	PROPN
ejpam-6327	55	4	definition	definition	NOUN
ejpam-6327	55	5	3	3	NUM
ejpam-6327	55	6	.	.	PUNCT
ejpam-6327	56	1	[	[	X
ejpam-6327	56	2	13	13	NUM
ejpam-6327	56	3	]	]	PUNCT
ejpam-6327	56	4	let	let	VERB
ejpam-6327	56	5	x	x	PRON
ejpam-6327	56	6	be	be	AUX
ejpam-6327	56	7	a	a	DET
ejpam-6327	56	8	non	non	ADJ
ejpam-6327	56	9	-	-	ADJ
ejpam-6327	56	10	empty	empty	ADJ
ejpam-6327	56	11	set	set	NOUN
ejpam-6327	56	12	.	.	PUNCT
ejpam-6327	57	1	the	the	DET
ejpam-6327	57	2	system	system	NOUN
ejpam-6327	57	3	(	(	PUNCT
ejpam-6327	57	4	x	x	X
ejpam-6327	57	5	,	,	PUNCT
ejpam-6327	57	6	·	·	PUNCT
ejpam-6327	57	7	)	)	PUNCT
ejpam-6327	57	8	is	be	AUX
ejpam-6327	57	9	called	call	VERB
ejpam-6327	57	10	a	a	DET
ejpam-6327	57	11	semigroup	semigroup	NOUN
ejpam-6327	57	12	if	if	SCONJ
ejpam-6327	57	13	“	"	PUNCT
ejpam-6327	57	14	·	·	PUNCT
ejpam-6327	57	15	”	"	PUNCT
ejpam-6327	57	16	is	be	AUX
ejpam-6327	57	17	an	an	DET
ejpam-6327	57	18	associative	associative	ADJ
ejpam-6327	57	19	binary	binary	ADJ
ejpam-6327	57	20	operation	operation	NOUN
ejpam-6327	57	21	.	.	PUNCT
ejpam-6327	58	1	for	for	ADP
ejpam-6327	58	2	convenience	convenience	NOUN
ejpam-6327	58	3	,	,	PUNCT
ejpam-6327	58	4	we	we	PRON
ejpam-6327	58	5	write	write	VERB
ejpam-6327	58	6	x	x	X
ejpam-6327	58	7	·	·	PUNCT
ejpam-6327	58	8	y	y	X
ejpam-6327	58	9	by	by	ADP
ejpam-6327	58	10	xy	xy	PROPN
ejpam-6327	58	11	.	.	PUNCT
ejpam-6327	59	1	definition	definition	NOUN
ejpam-6327	59	2	4	4	NUM
ejpam-6327	59	3	.	.	PUNCT
ejpam-6327	60	1	[	[	X
ejpam-6327	60	2	5	5	NUM
ejpam-6327	60	3	]	]	PUNCT
ejpam-6327	60	4	an	an	DET
ejpam-6327	60	5	algebraic	algebraic	ADJ
ejpam-6327	60	6	system	system	NOUN
ejpam-6327	60	7	(	(	PUNCT
ejpam-6327	60	8	x	x	X
ejpam-6327	60	9	,	,	PUNCT
ejpam-6327	60	10	∗	∗	NOUN
ejpam-6327	60	11	,	,	PUNCT
ejpam-6327	60	12	·	·	PUNCT
ejpam-6327	60	13	,	,	PUNCT
ejpam-6327	60	14	0	0	NUM
ejpam-6327	60	15	)	)	PUNCT
ejpam-6327	60	16	is	be	AUX
ejpam-6327	60	17	called	call	VERB
ejpam-6327	60	18	a	a	DET
ejpam-6327	60	19	ks	ks	NOUN
ejpam-6327	60	20	-	-	PUNCT
ejpam-6327	60	21	semigroup	semigroup	NOUN
ejpam-6327	60	22	if	if	SCONJ
ejpam-6327	60	23	it	it	PRON
ejpam-6327	60	24	satisfies	satisfy	VERB
ejpam-6327	60	25	the	the	DET
ejpam-6327	60	26	following	follow	VERB
ejpam-6327	60	27	conditions	condition	NOUN
ejpam-6327	60	28	:	:	PUNCT
ejpam-6327	60	29	(	(	PUNCT
ejpam-6327	60	30	i	i	NOUN
ejpam-6327	60	31	)	)	PUNCT
ejpam-6327	60	32	(	(	PUNCT
ejpam-6327	60	33	x	x	X
ejpam-6327	60	34	,	,	PUNCT
ejpam-6327	60	35	∗	∗	NOUN
ejpam-6327	60	36	,	,	PUNCT
ejpam-6327	60	37	0	0	NUM
ejpam-6327	60	38	)	)	PUNCT
ejpam-6327	60	39	is	be	AUX
ejpam-6327	60	40	a	a	DET
ejpam-6327	60	41	bck	bck	NOUN
ejpam-6327	60	42	-	-	PUNCT
ejpam-6327	60	43	algebra	algebra	NOUN
ejpam-6327	60	44	(	(	PUNCT
ejpam-6327	60	45	ii	ii	NOUN
ejpam-6327	60	46	)	)	PUNCT
ejpam-6327	60	47	(	(	PUNCT
ejpam-6327	60	48	x	x	X
ejpam-6327	60	49	,	,	PUNCT
ejpam-6327	60	50	·	·	PUNCT
ejpam-6327	60	51	)	)	PUNCT
ejpam-6327	60	52	is	be	AUX
ejpam-6327	60	53	a	a	DET
ejpam-6327	60	54	semigroup	semigroup	ADJ
ejpam-6327	60	55	(	(	PUNCT
ejpam-6327	60	56	iii	iii	NOUN
ejpam-6327	60	57	)	)	PUNCT
ejpam-6327	60	58	the	the	DET
ejpam-6327	60	59	operation	operation	NOUN
ejpam-6327	60	60	·	·	PUNCT
ejpam-6327	60	61	is	be	AUX
ejpam-6327	60	62	distributive	distributive	ADJ
ejpam-6327	60	63	over	over	ADP
ejpam-6327	60	64	the	the	DET
ejpam-6327	60	65	operation	operation	NOUN
ejpam-6327	60	66	∗	∗	NOUN
ejpam-6327	60	67	on	on	ADP
ejpam-6327	60	68	both	both	DET
ejpam-6327	60	69	sides	side	NOUN
ejpam-6327	60	70	,	,	PUNCT
ejpam-6327	60	71	that	that	ADV
ejpam-6327	60	72	is	is	ADV
ejpam-6327	60	73	,	,	PUNCT
ejpam-6327	60	74	for	for	ADP
ejpam-6327	60	75	all	all	DET
ejpam-6327	60	76	x	x	NOUN
ejpam-6327	60	77	,	,	PUNCT
ejpam-6327	60	78	y	y	PROPN
ejpam-6327	60	79	,	,	PUNCT
ejpam-6327	60	80	z	z	PROPN
ejpam-6327	60	81	∈	∈	PROPN
ejpam-6327	60	82	x	x	X
ejpam-6327	60	83	,	,	PUNCT
ejpam-6327	60	84	x	x	X
ejpam-6327	60	85	·	·	PUNCT
ejpam-6327	60	86	(	(	PUNCT
ejpam-6327	60	87	y	y	PROPN
ejpam-6327	60	88	∗	∗	PROPN
ejpam-6327	60	89	z	z	NOUN
ejpam-6327	60	90	)	)	PUNCT
ejpam-6327	60	91	=	=	SYM
ejpam-6327	60	92	(	(	PUNCT
ejpam-6327	60	93	x	x	X
ejpam-6327	60	94	·	·	PUNCT
ejpam-6327	60	95	y	y	X
ejpam-6327	60	96	)	)	PUNCT
ejpam-6327	60	97	∗	∗	NOUN
ejpam-6327	60	98	(	(	PUNCT
ejpam-6327	60	99	x	x	SYM
ejpam-6327	60	100	·	·	PUNCT
ejpam-6327	60	101	z	z	X
ejpam-6327	60	102	)	)	PUNCT
ejpam-6327	60	103	and	and	CCONJ
ejpam-6327	60	104	(	(	PUNCT
ejpam-6327	60	105	x	x	PROPN
ejpam-6327	60	106	∗	∗	PROPN
ejpam-6327	60	107	y	y	PROPN
ejpam-6327	60	108	)	)	PUNCT
ejpam-6327	60	109	·	·	PUNCT
ejpam-6327	61	1	z	z	X
ejpam-6327	62	1	=	=	SYM
ejpam-6327	63	1	(	(	PUNCT
ejpam-6327	63	2	x	x	SYM
ejpam-6327	63	3	·	·	PUNCT
ejpam-6327	63	4	z	z	X
ejpam-6327	63	5	)	)	PUNCT
ejpam-6327	63	6	∗	∗	NOUN
ejpam-6327	63	7	(	(	PUNCT
ejpam-6327	63	8	y	y	PROPN
ejpam-6327	63	9	·	·	PUNCT
ejpam-6327	63	10	z	z	X
ejpam-6327	63	11	)	)	PUNCT
ejpam-6327	63	12	.	.	PUNCT
ejpam-6327	64	1	example	example	NOUN
ejpam-6327	65	1	1	1	NUM
ejpam-6327	65	2	.	.	PUNCT
ejpam-6327	66	1	[	[	X
ejpam-6327	66	2	5	5	X
ejpam-6327	66	3	]	]	PUNCT
ejpam-6327	66	4	let	let	VERB
ejpam-6327	66	5	x	x	PUNCT
ejpam-6327	66	6	=	=	PUNCT
ejpam-6327	66	7	{	{	PUNCT
ejpam-6327	66	8	0	0	NUM
ejpam-6327	66	9	,	,	PUNCT
ejpam-6327	66	10	a	a	DET
ejpam-6327	66	11	,	,	PUNCT
ejpam-6327	66	12	b	b	NOUN
ejpam-6327	66	13	,	,	PUNCT
ejpam-6327	66	14	c	c	NOUN
ejpam-6327	66	15	}	}	PUNCT
ejpam-6327	66	16	.	.	PUNCT
ejpam-6327	67	1	define	define	VERB
ejpam-6327	67	2	∗	∗	NOUN
ejpam-6327	67	3	and	and	CCONJ
ejpam-6327	67	4	·	·	PUNCT
ejpam-6327	67	5	by	by	ADP
ejpam-6327	67	6	the	the	DET
ejpam-6327	67	7	following	following	ADJ
ejpam-6327	67	8	tables	table	NOUN
ejpam-6327	67	9	:	:	PUNCT
ejpam-6327	67	10	∗	∗	NOUN
ejpam-6327	67	11	0	0	NUM
ejpam-6327	68	1	a	a	DET
ejpam-6327	68	2	b	b	NOUN
ejpam-6327	68	3	c	c	NOUN
ejpam-6327	68	4	0	0	NUM
ejpam-6327	68	5	0	0	NUM
ejpam-6327	68	6	0	0	NUM
ejpam-6327	68	7	0	0	NUM
ejpam-6327	68	8	0	0	NUM
ejpam-6327	68	9	a	a	DET
ejpam-6327	68	10	a	a	DET
ejpam-6327	68	11	0	0	PUNCT
ejpam-6327	68	12	a	a	DET
ejpam-6327	68	13	a	a	DET
ejpam-6327	68	14	b	b	PROPN
ejpam-6327	68	15	b	b	PROPN
ejpam-6327	68	16	b	b	PROPN
ejpam-6327	68	17	0	0	NUM
ejpam-6327	68	18	b	b	PROPN
ejpam-6327	69	1	c	c	NOUN
ejpam-6327	69	2	c	c	NOUN
ejpam-6327	69	3	c	c	NOUN
ejpam-6327	69	4	c	c	NOUN
ejpam-6327	69	5	0	0	PUNCT
ejpam-6327	69	6	·	·	SYM
ejpam-6327	69	7	0	0	PUNCT
ejpam-6327	70	1	a	a	DET
ejpam-6327	70	2	b	b	X
ejpam-6327	70	3	c	c	NOUN
ejpam-6327	70	4	0	0	NUM
ejpam-6327	70	5	0	0	NUM
ejpam-6327	70	6	0	0	NUM
ejpam-6327	70	7	0	0	NUM
ejpam-6327	70	8	0	0	NUM
ejpam-6327	70	9	a	a	DET
ejpam-6327	70	10	0	0	NUM
ejpam-6327	70	11	a	a	DET
ejpam-6327	70	12	0	0	NUM
ejpam-6327	70	13	0	0	NUM
ejpam-6327	70	14	b	b	NOUN
ejpam-6327	70	15	0	0	NUM
ejpam-6327	70	16	0	0	NUM
ejpam-6327	70	17	b	b	NOUN
ejpam-6327	70	18	0	0	NUM
ejpam-6327	70	19	c	c	NOUN
ejpam-6327	70	20	0	0	PUNCT
ejpam-6327	70	21	c	c	NOUN
ejpam-6327	70	22	0	0	NUM
ejpam-6327	70	23	0	0	NUM
ejpam-6327	71	1	then	then	ADV
ejpam-6327	71	2	x	x	X
ejpam-6327	71	3	is	be	AUX
ejpam-6327	71	4	a	a	DET
ejpam-6327	71	5	ks	ks	NOUN
ejpam-6327	71	6	-	-	PUNCT
ejpam-6327	71	7	semigroup	semigroup	NOUN
ejpam-6327	71	8	.	.	PUNCT
ejpam-6327	72	1	definition	definition	NOUN
ejpam-6327	72	2	5	5	NUM
ejpam-6327	72	3	.	.	PUNCT
ejpam-6327	73	1	[	[	X
ejpam-6327	73	2	5	5	NUM
ejpam-6327	73	3	]	]	PUNCT
ejpam-6327	73	4	a	a	DET
ejpam-6327	73	5	ks	ks	NOUN
ejpam-6327	73	6	-	-	NOUN
ejpam-6327	73	7	semigroup	semigroup	NOUN
ejpam-6327	73	8	x	x	VERB
ejpam-6327	73	9	is	be	AUX
ejpam-6327	73	10	said	say	VERB
ejpam-6327	73	11	to	to	PART
ejpam-6327	73	12	be	be	AUX
ejpam-6327	73	13	(	(	PUNCT
ejpam-6327	73	14	i	i	NOUN
ejpam-6327	73	15	)	)	PUNCT
ejpam-6327	73	16	commutative	commutative	ADJ
ejpam-6327	73	17	if	if	SCONJ
ejpam-6327	73	18	x	x	PROPN
ejpam-6327	73	19	∗	∗	NOUN
ejpam-6327	73	20	(	(	PUNCT
ejpam-6327	73	21	x	x	X
ejpam-6327	73	22	∗	∗	NOUN
ejpam-6327	73	23	y	y	NOUN
ejpam-6327	73	24	)	)	PUNCT
ejpam-6327	74	1	=	=	SYM
ejpam-6327	74	2	y	y	PROPN
ejpam-6327	74	3	∗	∗	NOUN
ejpam-6327	74	4	(	(	PUNCT
ejpam-6327	74	5	y	y	PROPN
ejpam-6327	74	6	∗	∗	X
ejpam-6327	74	7	x	x	NOUN
ejpam-6327	74	8	)	)	PUNCT
ejpam-6327	74	9	for	for	ADP
ejpam-6327	74	10	all	all	DET
ejpam-6327	74	11	x	x	NOUN
ejpam-6327	74	12	,	,	PUNCT
ejpam-6327	74	13	y	y	PROPN
ejpam-6327	74	14	∈	∈	PROPN
ejpam-6327	74	15	x	x	X
ejpam-6327	74	16	,	,	PUNCT
ejpam-6327	74	17	(	(	PUNCT
ejpam-6327	74	18	ii	ii	NOUN
ejpam-6327	74	19	)	)	PUNCT
ejpam-6327	74	20	positive	positive	ADJ
ejpam-6327	74	21	implicative	implicative	NOUN
ejpam-6327	74	22	if	if	SCONJ
ejpam-6327	74	23	(	(	PUNCT
ejpam-6327	74	24	x	x	PROPN
ejpam-6327	74	25	∗	∗	PROPN
ejpam-6327	74	26	y	y	NOUN
ejpam-6327	74	27	)	)	PUNCT
ejpam-6327	74	28	∗	∗	NOUN
ejpam-6327	74	29	z	z	NOUN
ejpam-6327	74	30	=	=	SYM
ejpam-6327	74	31	(	(	PUNCT
ejpam-6327	74	32	x	x	X
ejpam-6327	74	33	∗	∗	PROPN
ejpam-6327	74	34	z	z	NOUN
ejpam-6327	74	35	)	)	PUNCT
ejpam-6327	74	36	∗	∗	NOUN
ejpam-6327	74	37	(	(	PUNCT
ejpam-6327	74	38	y	y	PROPN
ejpam-6327	74	39	∗	∗	PROPN
ejpam-6327	74	40	z	z	PROPN
ejpam-6327	74	41	)	)	PUNCT
ejpam-6327	74	42	,	,	PUNCT
ejpam-6327	74	43	or	or	CCONJ
ejpam-6327	74	44	equivalently	equivalently	ADV
ejpam-6327	74	45	,	,	PUNCT
ejpam-6327	74	46	x	x	X
ejpam-6327	74	47	∗	∗	NOUN
ejpam-6327	74	48	y	y	NOUN
ejpam-6327	74	49	=	=	SYM
ejpam-6327	74	50	(	(	PUNCT
ejpam-6327	74	51	x	x	X
ejpam-6327	74	52	∗	∗	PROPN
ejpam-6327	74	53	y	y	NOUN
ejpam-6327	74	54	)	)	PUNCT
ejpam-6327	74	55	∗	∗	VERB
ejpam-6327	74	56	y	y	PROPN
ejpam-6327	74	57	for	for	ADP
ejpam-6327	74	58	all	all	DET
ejpam-6327	74	59	x	x	NOUN
ejpam-6327	74	60	,	,	PUNCT
ejpam-6327	74	61	y	y	PROPN
ejpam-6327	74	62	,	,	PUNCT
ejpam-6327	74	63	z	z	PROPN
ejpam-6327	74	64	∈	∈	PROPN
ejpam-6327	74	65	x	x	X
ejpam-6327	74	66	,	,	PUNCT
ejpam-6327	74	67	(	(	PUNCT
ejpam-6327	74	68	iii	iii	NOUN
ejpam-6327	74	69	)	)	PUNCT
ejpam-6327	74	70	implicative	implicative	NOUN
ejpam-6327	74	71	if	if	SCONJ
ejpam-6327	74	72	x∗	x∗	PROPN
ejpam-6327	74	73	(	(	PUNCT
ejpam-6327	74	74	y	y	NOUN
ejpam-6327	74	75	∗x	∗x	ADV
ejpam-6327	74	76	)	)	PUNCT
ejpam-6327	74	77	=	=	SYM
ejpam-6327	74	78	x	x	PUNCT
ejpam-6327	74	79	for	for	ADP
ejpam-6327	74	80	all	all	DET
ejpam-6327	74	81	x	x	NOUN
ejpam-6327	74	82	,	,	PUNCT
ejpam-6327	74	83	y	y	PROPN
ejpam-6327	74	84	∈	∈	PROPN
ejpam-6327	74	85	x	x	X
ejpam-6327	74	86	,	,	PUNCT
ejpam-6327	74	87	or	or	CCONJ
ejpam-6327	74	88	if	if	SCONJ
ejpam-6327	74	89	it	it	PRON
ejpam-6327	74	90	is	be	AUX
ejpam-6327	74	91	both	both	CCONJ
ejpam-6327	74	92	commutative	commutative	ADJ
ejpam-6327	74	93	and	and	CCONJ
ejpam-6327	74	94	positive	positive	ADJ
ejpam-6327	74	95	implicative	implicative	NOUN
ejpam-6327	74	96	.	.	PUNCT
ejpam-6327	75	1	proposition	proposition	NOUN
ejpam-6327	75	2	2	2	NUM
ejpam-6327	75	3	.	.	PUNCT
ejpam-6327	76	1	[	[	X
ejpam-6327	76	2	5	5	NUM
ejpam-6327	76	3	]	]	PUNCT
ejpam-6327	76	4	a	a	DET
ejpam-6327	76	5	ks	ks	NOUN
ejpam-6327	76	6	-	-	PUNCT
ejpam-6327	76	7	semigroup	semigroup	NOUN
ejpam-6327	76	8	x	x	PUNCT
ejpam-6327	76	9	is	be	AUX
ejpam-6327	76	10	commutative	commutative	ADJ
ejpam-6327	76	11	if	if	SCONJ
ejpam-6327	76	12	and	and	CCONJ
ejpam-6327	76	13	only	only	ADV
ejpam-6327	76	14	if	if	SCONJ
ejpam-6327	76	15	for	for	ADP
ejpam-6327	76	16	all	all	DET
ejpam-6327	76	17	x	x	NOUN
ejpam-6327	76	18	,	,	PUNCT
ejpam-6327	76	19	y	y	PROPN
ejpam-6327	76	20	∈	∈	PROPN
ejpam-6327	76	21	x	x	X
ejpam-6327	76	22	,	,	PUNCT
ejpam-6327	76	23	x	x	SYM
ejpam-6327	76	24	≤	≤	ADJ
ejpam-6327	76	25	y	y	PROPN
ejpam-6327	76	26	implies	imply	VERB
ejpam-6327	76	27	x	x	PUNCT
ejpam-6327	76	28	=	=	SYM
ejpam-6327	76	29	y	y	PROPN
ejpam-6327	76	30	∗	∗	NOUN
ejpam-6327	76	31	(	(	PUNCT
ejpam-6327	76	32	y	y	PROPN
ejpam-6327	76	33	∗	∗	NOUN
ejpam-6327	76	34	x	x	NOUN
ejpam-6327	76	35	)	)	PUNCT
ejpam-6327	76	36	.	.	PUNCT
ejpam-6327	77	1	definition	definition	NOUN
ejpam-6327	77	2	6	6	NUM
ejpam-6327	77	3	.	.	PUNCT
ejpam-6327	78	1	[	[	X
ejpam-6327	78	2	5	5	NUM
ejpam-6327	78	3	]	]	PUNCT
ejpam-6327	78	4	a	a	DET
ejpam-6327	78	5	non	non	ADJ
ejpam-6327	78	6	-	-	ADJ
ejpam-6327	78	7	empty	empty	ADJ
ejpam-6327	78	8	subset	subset	NOUN
ejpam-6327	78	9	a	a	PRON
ejpam-6327	78	10	of	of	ADP
ejpam-6327	78	11	a	a	DET
ejpam-6327	78	12	semigroup	semigroup	NOUN
ejpam-6327	78	13	(	(	PUNCT
ejpam-6327	78	14	x	x	X
ejpam-6327	78	15	,	,	PUNCT
ejpam-6327	78	16	·	·	PUNCT
ejpam-6327	78	17	)	)	PUNCT
ejpam-6327	78	18	is	be	AUX
ejpam-6327	78	19	said	say	VERB
ejpam-6327	78	20	to	to	PART
ejpam-6327	78	21	be	be	AUX
ejpam-6327	78	22	left	leave	VERB
ejpam-6327	78	23	(	(	PUNCT
ejpam-6327	78	24	resp	resp	NOUN
ejpam-6327	78	25	.	.	PUNCT
ejpam-6327	79	1	right	right	ADJ
ejpam-6327	79	2	)	)	PUNCT
ejpam-6327	79	3	stable	stable	ADJ
ejpam-6327	79	4	if	if	SCONJ
ejpam-6327	79	5	xa	xa	PROPN
ejpam-6327	79	6	∈	∈	PROPN
ejpam-6327	79	7	a	a	DET
ejpam-6327	79	8	(	(	PUNCT
ejpam-6327	79	9	resp	resp	NOUN
ejpam-6327	79	10	.	.	PUNCT
ejpam-6327	80	1	ax	ax	NOUN
ejpam-6327	80	2	∈	∈	PROPN
ejpam-6327	80	3	a	a	X
ejpam-6327	80	4	)	)	PUNCT
ejpam-6327	80	5	whenever	whenever	SCONJ
ejpam-6327	80	6	x	x	SYM
ejpam-6327	80	7	∈	∈	PROPN
ejpam-6327	80	8	x	x	X
ejpam-6327	80	9	and	and	CCONJ
ejpam-6327	80	10	a	a	DET
ejpam-6327	80	11	∈	∈	PROPN
ejpam-6327	80	12	a.	a.	NOUN
ejpam-6327	80	13	a	a	DET
ejpam-6327	80	14	non	non	ADJ
ejpam-6327	80	15	-	-	ADJ
ejpam-6327	80	16	empty	empty	ADJ
ejpam-6327	80	17	subset	subset	NOUN
ejpam-6327	80	18	of	of	ADP
ejpam-6327	80	19	x	x	PRON
ejpam-6327	80	20	which	which	PRON
ejpam-6327	80	21	is	be	AUX
ejpam-6327	80	22	both	both	PRON
ejpam-6327	80	23	left	leave	VERB
ejpam-6327	80	24	and	and	CCONJ
ejpam-6327	80	25	right	right	ADJ
ejpam-6327	80	26	stable	stable	ADJ
ejpam-6327	80	27	is	be	AUX
ejpam-6327	80	28	called	call	VERB
ejpam-6327	80	29	two	two	NUM
ejpam-6327	80	30	-	-	PUNCT
ejpam-6327	80	31	sided	sided	ADJ
ejpam-6327	80	32	stable	stable	ADJ
ejpam-6327	80	33	or	or	CCONJ
ejpam-6327	80	34	simply	simply	ADV
ejpam-6327	80	35	stable	stable	ADJ
ejpam-6327	80	36	.	.	PUNCT
ejpam-6327	81	1	h.	h.	PROPN
ejpam-6327	81	2	sarapuddin	sarapuddin	PROPN
ejpam-6327	81	3	,	,	PUNCT
ejpam-6327	81	4	j.	j.	PROPN
ejpam-6327	81	5	vilela	vilela	PROPN
ejpam-6327	81	6	/	/	SYM
ejpam-6327	81	7	eur	eur	PROPN
ejpam-6327	81	8	.	.	PUNCT
ejpam-6327	82	1	j.	j.	PROPN
ejpam-6327	82	2	pure	pure	PROPN
ejpam-6327	82	3	appl	appl	PROPN
ejpam-6327	82	4	.	.	PROPN
ejpam-6327	82	5	math	math	PROPN
ejpam-6327	82	6	,	,	PUNCT
ejpam-6327	82	7	18	18	NUM
ejpam-6327	82	8	(	(	PUNCT
ejpam-6327	82	9	3	3	NUM
ejpam-6327	82	10	)	)	PUNCT
ejpam-6327	82	11	(	(	PUNCT
ejpam-6327	82	12	2025	2025	NUM
ejpam-6327	82	13	)	)	PUNCT
ejpam-6327	82	14	,	,	PUNCT
ejpam-6327	82	15	6327	6327	NUM
ejpam-6327	82	16	4	4	NUM
ejpam-6327	82	17	of	of	ADP
ejpam-6327	82	18	23	23	NUM
ejpam-6327	82	19	definition	definition	NOUN
ejpam-6327	82	20	7	7	NUM
ejpam-6327	82	21	.	.	PUNCT
ejpam-6327	83	1	[	[	X
ejpam-6327	83	2	5	5	NUM
ejpam-6327	83	3	]	]	PUNCT
ejpam-6327	83	4	a	a	DET
ejpam-6327	83	5	non	non	ADJ
ejpam-6327	83	6	-	-	ADJ
ejpam-6327	83	7	empty	empty	ADJ
ejpam-6327	83	8	subset	subset	NOUN
ejpam-6327	83	9	a	a	PRON
ejpam-6327	83	10	of	of	ADP
ejpam-6327	83	11	a	a	DET
ejpam-6327	83	12	ks	ks	NOUN
ejpam-6327	83	13	-	-	PUNCT
ejpam-6327	83	14	semigroup	semigroup	NOUN
ejpam-6327	83	15	x	x	PUNCT
ejpam-6327	83	16	is	be	AUX
ejpam-6327	83	17	called	call	VERB
ejpam-6327	83	18	a	a	DET
ejpam-6327	83	19	left	left	ADJ
ejpam-6327	83	20	(	(	PUNCT
ejpam-6327	83	21	resp	resp	NOUN
ejpam-6327	83	22	.	.	PUNCT
ejpam-6327	84	1	right	right	ADJ
ejpam-6327	84	2	)	)	PUNCT
ejpam-6327	84	3	ideal	ideal	NOUN
ejpam-6327	84	4	of	of	ADP
ejpam-6327	84	5	x	x	PRON
ejpam-6327	84	6	if	if	SCONJ
ejpam-6327	84	7	(	(	PUNCT
ejpam-6327	84	8	i	i	NOUN
ejpam-6327	84	9	)	)	PUNCT
ejpam-6327	84	10	a	a	PRON
ejpam-6327	84	11	is	be	AUX
ejpam-6327	84	12	a	a	DET
ejpam-6327	84	13	left	left	ADJ
ejpam-6327	84	14	(	(	PUNCT
ejpam-6327	84	15	resp	resp	NOUN
ejpam-6327	84	16	.	.	PUNCT
ejpam-6327	85	1	right	right	ADJ
ejpam-6327	85	2	)	)	PUNCT
ejpam-6327	85	3	stable	stable	ADJ
ejpam-6327	85	4	subset	subset	NOUN
ejpam-6327	85	5	of	of	ADP
ejpam-6327	85	6	(	(	PUNCT
ejpam-6327	85	7	x	x	NOUN
ejpam-6327	85	8	,	,	PUNCT
ejpam-6327	85	9	·	·	PUNCT
ejpam-6327	85	10	)	)	PUNCT
ejpam-6327	85	11	,	,	PUNCT
ejpam-6327	85	12	(	(	PUNCT
ejpam-6327	85	13	ii	ii	NOUN
ejpam-6327	85	14	)	)	PUNCT
ejpam-6327	85	15	for	for	ADP
ejpam-6327	85	16	any	any	DET
ejpam-6327	85	17	x	x	NOUN
ejpam-6327	85	18	,	,	PUNCT
ejpam-6327	85	19	y	y	PROPN
ejpam-6327	85	20	∈	∈	PROPN
ejpam-6327	85	21	x	x	X
ejpam-6327	85	22	,	,	PUNCT
ejpam-6327	85	23	x	x	SYM
ejpam-6327	85	24	∗	∗	NOUN
ejpam-6327	85	25	y	y	PROPN
ejpam-6327	85	26	∈	∈	PROPN
ejpam-6327	86	1	a	a	PRON
ejpam-6327	86	2	and	and	CCONJ
ejpam-6327	86	3	y	y	PROPN
ejpam-6327	86	4	∈	∈	PROPN
ejpam-6327	86	5	a	a	DET
ejpam-6327	86	6	imply	imply	NOUN
ejpam-6327	86	7	that	that	SCONJ
ejpam-6327	86	8	x	x	PUNCT
ejpam-6327	86	9	∈	∈	NOUN
ejpam-6327	86	10	a.	a.	NOUN
ejpam-6327	86	11	if	if	SCONJ
ejpam-6327	86	12	a	a	PRON
ejpam-6327	86	13	is	be	AUX
ejpam-6327	86	14	both	both	PRON
ejpam-6327	86	15	a	a	DET
ejpam-6327	86	16	left	left	NOUN
ejpam-6327	86	17	and	and	CCONJ
ejpam-6327	86	18	a	a	DET
ejpam-6327	86	19	right	right	ADJ
ejpam-6327	86	20	ideal	ideal	NOUN
ejpam-6327	86	21	,	,	PUNCT
ejpam-6327	86	22	then	then	ADV
ejpam-6327	86	23	it	it	PRON
ejpam-6327	86	24	is	be	AUX
ejpam-6327	86	25	called	call	VERB
ejpam-6327	86	26	a	a	DET
ejpam-6327	86	27	two	two	NUM
ejpam-6327	86	28	-	-	PUNCT
ejpam-6327	86	29	sided	sided	ADJ
ejpam-6327	86	30	ideal	ideal	NOUN
ejpam-6327	86	31	or	or	CCONJ
ejpam-6327	86	32	simply	simply	ADV
ejpam-6327	86	33	an	an	DET
ejpam-6327	86	34	ideal	ideal	NOUN
ejpam-6327	86	35	.	.	PUNCT
ejpam-6327	87	1	remark	remark	NOUN
ejpam-6327	87	2	2	2	NUM
ejpam-6327	87	3	.	.	PUNCT
ejpam-6327	88	1	[	[	X
ejpam-6327	88	2	5	5	X
ejpam-6327	88	3	]	]	PUNCT
ejpam-6327	88	4	let	let	VERB
ejpam-6327	88	5	x	x	PRON
ejpam-6327	88	6	be	be	AUX
ejpam-6327	88	7	a	a	DET
ejpam-6327	88	8	ks	ks	NOUN
ejpam-6327	88	9	-	-	PUNCT
ejpam-6327	88	10	semigroup	semigroup	NOUN
ejpam-6327	88	11	.	.	PUNCT
ejpam-6327	89	1	then	then	ADV
ejpam-6327	89	2	(	(	PUNCT
ejpam-6327	89	3	i	i	NOUN
ejpam-6327	89	4	)	)	PUNCT
ejpam-6327	89	5	{	{	PUNCT
ejpam-6327	89	6	0	0	NUM
ejpam-6327	89	7	}	}	PUNCT
ejpam-6327	89	8	and	and	CCONJ
ejpam-6327	89	9	x	x	PRON
ejpam-6327	89	10	are	be	AUX
ejpam-6327	89	11	ideals	ideal	NOUN
ejpam-6327	89	12	of	of	ADP
ejpam-6327	89	13	x.	x.	PROPN
ejpam-6327	89	14	(	(	PUNCT
ejpam-6327	89	15	ii	ii	PROPN
ejpam-6327	89	16	)	)	PUNCT
ejpam-6327	89	17	if	if	SCONJ
ejpam-6327	89	18	a	a	PRON
ejpam-6327	89	19	is	be	AUX
ejpam-6327	89	20	a	a	DET
ejpam-6327	89	21	left	left	ADJ
ejpam-6327	89	22	(	(	PUNCT
ejpam-6327	89	23	resp	resp	NOUN
ejpam-6327	89	24	.	.	PUNCT
ejpam-6327	90	1	right	right	ADJ
ejpam-6327	90	2	)	)	PUNCT
ejpam-6327	90	3	ideal	ideal	NOUN
ejpam-6327	90	4	of	of	ADP
ejpam-6327	90	5	x	x	PRON
ejpam-6327	90	6	,	,	PUNCT
ejpam-6327	90	7	then	then	ADV
ejpam-6327	90	8	0	0	NUM
ejpam-6327	90	9	∈	∈	NOUN
ejpam-6327	90	10	a.	a.	NOUN
ejpam-6327	90	11	definition	definition	NOUN
ejpam-6327	90	12	8	8	NUM
ejpam-6327	90	13	.	.	PUNCT
ejpam-6327	91	1	[	[	X
ejpam-6327	91	2	5	5	X
ejpam-6327	91	3	]	]	PUNCT
ejpam-6327	91	4	let	let	VERB
ejpam-6327	91	5	x	x	PRON
ejpam-6327	91	6	be	be	AUX
ejpam-6327	91	7	a	a	DET
ejpam-6327	91	8	ks	ks	NOUN
ejpam-6327	91	9	-	-	PUNCT
ejpam-6327	91	10	semigroup	semigroup	NOUN
ejpam-6327	91	11	and	and	CCONJ
ejpam-6327	91	12	let	let	VERB
ejpam-6327	91	13	∼	∼	NOUN
ejpam-6327	91	14	be	be	AUX
ejpam-6327	91	15	a	a	DET
ejpam-6327	91	16	binary	binary	ADJ
ejpam-6327	91	17	relation	relation	NOUN
ejpam-6327	91	18	on	on	ADP
ejpam-6327	91	19	x.	x.	NOUN
ejpam-6327	91	20	then	then	ADV
ejpam-6327	91	21	(	(	PUNCT
ejpam-6327	91	22	i	i	NOUN
ejpam-6327	91	23	)	)	PUNCT
ejpam-6327	91	24	∼	∼	NOUN
ejpam-6327	91	25	is	be	AUX
ejpam-6327	91	26	said	say	VERB
ejpam-6327	91	27	to	to	PART
ejpam-6327	91	28	be	be	AUX
ejpam-6327	91	29	right	right	ADJ
ejpam-6327	91	30	(	(	PUNCT
ejpam-6327	91	31	resp	resp	NOUN
ejpam-6327	91	32	.	.	PUNCT
ejpam-6327	92	1	left	left	ADJ
ejpam-6327	92	2	)	)	PUNCT
ejpam-6327	92	3	compatible	compatible	ADJ
ejpam-6327	92	4	if	if	SCONJ
ejpam-6327	92	5	whenever	whenever	SCONJ
ejpam-6327	92	6	x	x	PRON
ejpam-6327	92	7	∼	∼	NOUN
ejpam-6327	92	8	y	y	NOUN
ejpam-6327	92	9	then	then	ADV
ejpam-6327	92	10	x	x	X
ejpam-6327	92	11	∗	∗	NOUN
ejpam-6327	92	12	z	z	NOUN
ejpam-6327	92	13	∼	∼	NOUN
ejpam-6327	92	14	y	y	PROPN
ejpam-6327	92	15	∗	∗	NOUN
ejpam-6327	92	16	z	z	PROPN
ejpam-6327	92	17	(	(	PUNCT
ejpam-6327	92	18	resp	resp	NOUN
ejpam-6327	92	19	.	.	PUNCT
ejpam-6327	93	1	z	z	NOUN
ejpam-6327	93	2	∗	∗	NOUN
ejpam-6327	93	3	x	x	PUNCT
ejpam-6327	93	4	∼	∼	NOUN
ejpam-6327	93	5	z	z	NOUN
ejpam-6327	93	6	∗	∗	NOUN
ejpam-6327	93	7	y	y	PROPN
ejpam-6327	93	8	)	)	PUNCT
ejpam-6327	93	9	and	and	CCONJ
ejpam-6327	93	10	xz	xz	PROPN
ejpam-6327	93	11	∼	∼	NOUN
ejpam-6327	93	12	yz	yz	PROPN
ejpam-6327	93	13	(	(	PUNCT
ejpam-6327	93	14	resp	resp	NOUN
ejpam-6327	93	15	.	.	PUNCT
ejpam-6327	94	1	zx	zx	NOUN
ejpam-6327	94	2	∼	∼	X
ejpam-6327	94	3	zy	zy	NOUN
ejpam-6327	94	4	)	)	PUNCT
ejpam-6327	94	5	for	for	ADP
ejpam-6327	94	6	all	all	DET
ejpam-6327	94	7	x	x	NOUN
ejpam-6327	94	8	,	,	PUNCT
ejpam-6327	94	9	y	y	PROPN
ejpam-6327	94	10	,	,	PUNCT
ejpam-6327	94	11	z	z	PROPN
ejpam-6327	94	12	∈	∈	PROPN
ejpam-6327	94	13	x	x	X
ejpam-6327	94	14	;	;	PUNCT
ejpam-6327	94	15	(	(	PUNCT
ejpam-6327	94	16	ii	ii	NOUN
ejpam-6327	94	17	)	)	PUNCT
ejpam-6327	94	18	∼	∼	NOUN
ejpam-6327	94	19	is	be	AUX
ejpam-6327	94	20	said	say	VERB
ejpam-6327	94	21	to	to	PART
ejpam-6327	94	22	be	be	AUX
ejpam-6327	94	23	compatible	compatible	ADJ
ejpam-6327	94	24	if	if	SCONJ
ejpam-6327	94	25	x	x	PUNCT
ejpam-6327	94	26	∼	∼	NOUN
ejpam-6327	94	27	y	y	NOUN
ejpam-6327	94	28	and	and	CCONJ
ejpam-6327	94	29	u	u	NOUN
ejpam-6327	94	30	∼	∼	NOUN
ejpam-6327	94	31	v	v	NOUN
ejpam-6327	94	32	imply	imply	VERB
ejpam-6327	94	33	x	x	X
ejpam-6327	94	34	∗	∗	NOUN
ejpam-6327	94	35	u	u	NOUN
ejpam-6327	94	36	∼	∼	NOUN
ejpam-6327	94	37	y	y	PROPN
ejpam-6327	94	38	∗	∗	NOUN
ejpam-6327	94	39	v	v	NOUN
ejpam-6327	94	40	and	and	CCONJ
ejpam-6327	94	41	xu	xu	PROPN
ejpam-6327	94	42	∼	∼	NOUN
ejpam-6327	94	43	yv	yv	PROPN
ejpam-6327	94	44	for	for	ADP
ejpam-6327	94	45	all	all	DET
ejpam-6327	94	46	x	x	NOUN
ejpam-6327	94	47	,	,	PUNCT
ejpam-6327	94	48	y	y	PROPN
ejpam-6327	94	49	,	,	PUNCT
ejpam-6327	94	50	u	u	NOUN
ejpam-6327	94	51	,	,	PUNCT
ejpam-6327	94	52	v	v	NOUN
ejpam-6327	94	53	∈	∈	PROPN
ejpam-6327	94	54	x	x	X
ejpam-6327	94	55	;	;	PUNCT
ejpam-6327	94	56	(	(	PUNCT
ejpam-6327	94	57	iii	iii	X
ejpam-6327	94	58	)	)	PUNCT
ejpam-6327	94	59	a	a	DET
ejpam-6327	94	60	compatible	compatible	ADJ
ejpam-6327	94	61	equivalence	equivalence	NOUN
ejpam-6327	94	62	relation	relation	NOUN
ejpam-6327	94	63	is	be	AUX
ejpam-6327	94	64	called	call	VERB
ejpam-6327	94	65	a	a	DET
ejpam-6327	94	66	congruence	congruence	NOUN
ejpam-6327	94	67	relation	relation	NOUN
ejpam-6327	94	68	.	.	PUNCT
ejpam-6327	95	1	theorem	theorem	NOUN
ejpam-6327	95	2	1	1	NUM
ejpam-6327	95	3	.	.	PUNCT
ejpam-6327	96	1	[	[	X
ejpam-6327	96	2	5	5	X
ejpam-6327	96	3	]	]	PUNCT
ejpam-6327	96	4	let	let	VERB
ejpam-6327	96	5	x	x	PRON
ejpam-6327	96	6	be	be	AUX
ejpam-6327	96	7	a	a	DET
ejpam-6327	96	8	ks	ks	NOUN
ejpam-6327	96	9	-	-	PUNCT
ejpam-6327	96	10	semigroup	semigroup	NOUN
ejpam-6327	96	11	.	.	PUNCT
ejpam-6327	97	1	an	an	DET
ejpam-6327	97	2	equivalence	equivalence	NOUN
ejpam-6327	97	3	relation	relation	NOUN
ejpam-6327	97	4	∼	∼	NOUN
ejpam-6327	97	5	on	on	ADP
ejpam-6327	97	6	x	x	SYM
ejpam-6327	97	7	is	be	AUX
ejpam-6327	97	8	a	a	DET
ejpam-6327	97	9	congruence	congruence	NOUN
ejpam-6327	97	10	relation	relation	NOUN
ejpam-6327	97	11	if	if	SCONJ
ejpam-6327	97	12	and	and	CCONJ
ejpam-6327	97	13	only	only	ADV
ejpam-6327	97	14	if	if	SCONJ
ejpam-6327	97	15	it	it	PRON
ejpam-6327	97	16	is	be	AUX
ejpam-6327	97	17	both	both	PRON
ejpam-6327	97	18	left	leave	VERB
ejpam-6327	97	19	and	and	CCONJ
ejpam-6327	97	20	right	right	ADV
ejpam-6327	97	21	compatible	compatible	ADJ
ejpam-6327	97	22	.	.	PUNCT
ejpam-6327	98	1	let	let	VERB
ejpam-6327	98	2	x	x	PRON
ejpam-6327	98	3	be	be	AUX
ejpam-6327	98	4	a	a	DET
ejpam-6327	98	5	non	non	ADJ
ejpam-6327	98	6	-	-	ADJ
ejpam-6327	98	7	empty	empty	ADJ
ejpam-6327	98	8	set	set	NOUN
ejpam-6327	98	9	.	.	PUNCT
ejpam-6327	99	1	a	a	DET
ejpam-6327	99	2	fuzzy	fuzzy	ADJ
ejpam-6327	99	3	set	set	NOUN
ejpam-6327	99	4	on	on	ADP
ejpam-6327	99	5	x	x	SYM
ejpam-6327	99	6	is	be	AUX
ejpam-6327	99	7	a	a	DET
ejpam-6327	99	8	function	function	NOUN
ejpam-6327	99	9	µ	µ	NOUN
ejpam-6327	99	10	:	:	PUNCT
ejpam-6327	99	11	x	x	SYM
ejpam-6327	99	12	→	→	SYM
ejpam-6327	99	13	[	[	X
ejpam-6327	99	14	0	0	NUM
ejpam-6327	99	15	,	,	PUNCT
ejpam-6327	99	16	1	1	NUM
ejpam-6327	99	17	]	]	PUNCT
ejpam-6327	99	18	.	.	PUNCT
ejpam-6327	100	1	the	the	DET
ejpam-6327	100	2	characteristic	characteristic	ADJ
ejpam-6327	100	3	function	function	NOUN
ejpam-6327	100	4	of	of	ADP
ejpam-6327	100	5	a	a	DET
ejpam-6327	100	6	set	set	NOUN
ejpam-6327	100	7	i	i	PRON
ejpam-6327	100	8	is	be	AUX
ejpam-6327	100	9	a	a	DET
ejpam-6327	100	10	fuzzy	fuzzy	ADJ
ejpam-6327	100	11	set	set	NOUN
ejpam-6327	100	12	denoted	denote	VERB
ejpam-6327	100	13	by	by	ADP
ejpam-6327	100	14	χi	χi	X
ejpam-6327	100	15	.	.	PUNCT
ejpam-6327	101	1	definition	definition	NOUN
ejpam-6327	101	2	9	9	NUM
ejpam-6327	101	3	.	.	PUNCT
ejpam-6327	102	1	[	[	X
ejpam-6327	102	2	7	7	X
ejpam-6327	102	3	]	]	X
ejpam-6327	102	4	a	a	DET
ejpam-6327	102	5	fuzzy	fuzzy	ADJ
ejpam-6327	102	6	set	set	VERB
ejpam-6327	102	7	µ	µ	NOUN
ejpam-6327	102	8	on	on	ADP
ejpam-6327	102	9	a	a	DET
ejpam-6327	102	10	ks	ks	NOUN
ejpam-6327	102	11	-	-	PUNCT
ejpam-6327	102	12	semigroup	semigroup	NOUN
ejpam-6327	102	13	x	x	PUNCT
ejpam-6327	102	14	is	be	AUX
ejpam-6327	102	15	called	call	VERB
ejpam-6327	102	16	a	a	DET
ejpam-6327	102	17	left	left	ADJ
ejpam-6327	102	18	(	(	PUNCT
ejpam-6327	102	19	resp	resp	NOUN
ejpam-6327	102	20	.	.	PUNCT
ejpam-6327	103	1	right	right	ADJ
ejpam-6327	103	2	)	)	PUNCT
ejpam-6327	103	3	fuzzy	fuzzy	ADJ
ejpam-6327	103	4	ks	ks	NOUN
ejpam-6327	103	5	-	-	PUNCT
ejpam-6327	103	6	ideal	ideal	NOUN
ejpam-6327	103	7	of	of	ADP
ejpam-6327	103	8	x	x	PRON
ejpam-6327	103	9	if	if	SCONJ
ejpam-6327	103	10	for	for	ADP
ejpam-6327	103	11	all	all	DET
ejpam-6327	103	12	x	x	NOUN
ejpam-6327	103	13	,	,	PUNCT
ejpam-6327	103	14	y	y	PROPN
ejpam-6327	103	15	,	,	PUNCT
ejpam-6327	103	16	a	a	DET
ejpam-6327	103	17	∈	∈	PROPN
ejpam-6327	103	18	x	x	X
ejpam-6327	103	19	,	,	PUNCT
ejpam-6327	103	20	(	(	PUNCT
ejpam-6327	103	21	f1	f1	NOUN
ejpam-6327	103	22	)	)	PUNCT
ejpam-6327	103	23	µ(0	µ(0	PROPN
ejpam-6327	103	24	)	)	PUNCT
ejpam-6327	103	25	≥	≥	NOUN
ejpam-6327	103	26	µ(x	µ(x	VERB
ejpam-6327	103	27	)	)	PUNCT
ejpam-6327	103	28	,	,	PUNCT
ejpam-6327	103	29	(	(	PUNCT
ejpam-6327	103	30	f2	f2	X
ejpam-6327	103	31	)	)	PUNCT
ejpam-6327	103	32	µ(x	µ(x	PROPN
ejpam-6327	103	33	)	)	PUNCT
ejpam-6327	103	34	≥	≥	NOUN
ejpam-6327	104	1	min{µ(x	min{µ(x	PROPN
ejpam-6327	104	2	∗	∗	X
ejpam-6327	104	3	y	y	NOUN
ejpam-6327	104	4	)	)	PUNCT
ejpam-6327	104	5	,	,	PUNCT
ejpam-6327	104	6	µ(y	µ(y	PROPN
ejpam-6327	104	7	)	)	PUNCT
ejpam-6327	104	8	}	}	PUNCT
ejpam-6327	104	9	,	,	PUNCT
ejpam-6327	104	10	(	(	PUNCT
ejpam-6327	104	11	f3	f3	ADJ
ejpam-6327	104	12	)	)	PUNCT
ejpam-6327	104	13	µ(xa	µ(xa	PROPN
ejpam-6327	104	14	)	)	PUNCT
ejpam-6327	104	15	≥	≥	NOUN
ejpam-6327	104	16	min{µ(x	min{µ(x	NOUN
ejpam-6327	104	17	)	)	PUNCT
ejpam-6327	104	18	,	,	PUNCT
ejpam-6327	104	19	µ(a	µ(a	PROPN
ejpam-6327	104	20	)	)	PUNCT
ejpam-6327	104	21	}	}	PUNCT
ejpam-6327	104	22	(	(	PUNCT
ejpam-6327	104	23	resp	resp	NOUN
ejpam-6327	104	24	.	.	PUNCT
ejpam-6327	105	1	µ(ax	µ(ax	NOUN
ejpam-6327	105	2	)	)	PUNCT
ejpam-6327	105	3	≥	≥	NOUN
ejpam-6327	105	4	min{µ(x	min{µ(x	NOUN
ejpam-6327	105	5	)	)	PUNCT
ejpam-6327	105	6	,	,	PUNCT
ejpam-6327	105	7	µ(a	µ(a	PROPN
ejpam-6327	105	8	)	)	PUNCT
ejpam-6327	105	9	}	}	PUNCT
ejpam-6327	105	10	)	)	PUNCT
ejpam-6327	105	11	.	.	PUNCT
ejpam-6327	106	1	a	a	DET
ejpam-6327	106	2	fuzzy	fuzzy	ADJ
ejpam-6327	106	3	set	set	VERB
ejpam-6327	106	4	µ	µ	NOUN
ejpam-6327	106	5	on	on	ADP
ejpam-6327	106	6	a	a	DET
ejpam-6327	106	7	ks	ks	NOUN
ejpam-6327	106	8	-	-	PUNCT
ejpam-6327	106	9	semigroup	semigroup	NOUN
ejpam-6327	106	10	x	x	PUNCT
ejpam-6327	106	11	is	be	AUX
ejpam-6327	106	12	called	call	VERB
ejpam-6327	106	13	a	a	DET
ejpam-6327	106	14	fuzzy	fuzzy	ADJ
ejpam-6327	106	15	ks	ks	NOUN
ejpam-6327	106	16	-	-	NOUN
ejpam-6327	106	17	ideal	ideal	NOUN
ejpam-6327	106	18	of	of	ADP
ejpam-6327	106	19	x	x	PRON
ejpam-6327	106	20	if	if	SCONJ
ejpam-6327	106	21	it	it	PRON
ejpam-6327	106	22	is	be	AUX
ejpam-6327	106	23	both	both	CCONJ
ejpam-6327	106	24	a	a	DET
ejpam-6327	106	25	left	left	NOUN
ejpam-6327	106	26	and	and	CCONJ
ejpam-6327	106	27	a	a	DET
ejpam-6327	106	28	right	right	ADJ
ejpam-6327	106	29	fuzzy	fuzzy	ADJ
ejpam-6327	106	30	ks	ks	NOUN
ejpam-6327	106	31	-	-	PUNCT
ejpam-6327	106	32	ideal	ideal	NOUN
ejpam-6327	106	33	of	of	ADP
ejpam-6327	106	34	x.	x.	NOUN
ejpam-6327	106	35	definition	definition	NOUN
ejpam-6327	106	36	10	10	NUM
ejpam-6327	106	37	.	.	PUNCT
ejpam-6327	107	1	[	[	X
ejpam-6327	107	2	7	7	X
ejpam-6327	107	3	]	]	X
ejpam-6327	107	4	let	let	VERB
ejpam-6327	107	5	x	x	PRON
ejpam-6327	107	6	be	be	AUX
ejpam-6327	107	7	a	a	DET
ejpam-6327	107	8	ks	ks	NOUN
ejpam-6327	107	9	-	-	PUNCT
ejpam-6327	107	10	semigroup	semigroup	NOUN
ejpam-6327	107	11	.	.	PUNCT
ejpam-6327	108	1	a	a	DET
ejpam-6327	108	2	fuzzy	fuzzy	ADJ
ejpam-6327	108	3	set	set	VERB
ejpam-6327	108	4	µ	µ	NOUN
ejpam-6327	108	5	on	on	ADP
ejpam-6327	108	6	x	x	VERB
ejpam-6327	108	7	is	be	AUX
ejpam-6327	108	8	called	call	VERB
ejpam-6327	108	9	a	a	DET
ejpam-6327	108	10	left	left	ADJ
ejpam-6327	108	11	(	(	PUNCT
ejpam-6327	108	12	resp	resp	NOUN
ejpam-6327	108	13	.	.	PUNCT
ejpam-6327	109	1	right	right	ADJ
ejpam-6327	109	2	)	)	PUNCT
ejpam-6327	109	3	fuzzy	fuzzy	ADJ
ejpam-6327	109	4	ks	ks	NOUN
ejpam-6327	109	5	-	-	ADJ
ejpam-6327	109	6	p	p	NOUN
ejpam-6327	109	7	-	-	PUNCT
ejpam-6327	109	8	ideal	ideal	NOUN
ejpam-6327	109	9	of	of	ADP
ejpam-6327	109	10	x	x	PRON
ejpam-6327	109	11	if	if	SCONJ
ejpam-6327	109	12	it	it	PRON
ejpam-6327	109	13	satisfies	satisfy	VERB
ejpam-6327	109	14	(	(	PUNCT
ejpam-6327	109	15	f1	f1	NOUN
ejpam-6327	109	16	)	)	PUNCT
ejpam-6327	109	17	,	,	PUNCT
ejpam-6327	109	18	(	(	PUNCT
ejpam-6327	109	19	f3	f3	ADJ
ejpam-6327	109	20	)	)	PUNCT
ejpam-6327	109	21	and	and	CCONJ
ejpam-6327	109	22	(	(	PUNCT
ejpam-6327	109	23	f4	f4	PROPN
ejpam-6327	109	24	)	)	PUNCT
ejpam-6327	109	25	:	:	PUNCT
ejpam-6327	110	1	µ(x	µ(x	ADJ
ejpam-6327	110	2	∗	∗	NOUN
ejpam-6327	110	3	z	z	NOUN
ejpam-6327	110	4	)	)	PUNCT
ejpam-6327	110	5	≥	≥	NOUN
ejpam-6327	110	6	min{µ((x	min{µ((x	NOUN
ejpam-6327	110	7	∗	∗	X
ejpam-6327	110	8	y	y	NOUN
ejpam-6327	110	9	)	)	PUNCT
ejpam-6327	110	10	∗	∗	NOUN
ejpam-6327	110	11	z	z	PROPN
ejpam-6327	110	12	)	)	PUNCT
ejpam-6327	110	13	,	,	PUNCT
ejpam-6327	110	14	µ(y	µ(y	PROPN
ejpam-6327	110	15	∗	∗	PROPN
ejpam-6327	110	16	z	z	PROPN
ejpam-6327	110	17	)	)	PUNCT
ejpam-6327	110	18	}	}	PUNCT
ejpam-6327	110	19	for	for	ADP
ejpam-6327	110	20	all	all	DET
ejpam-6327	110	21	x	x	NOUN
ejpam-6327	110	22	,	,	PUNCT
ejpam-6327	110	23	y	y	PROPN
ejpam-6327	110	24	,	,	PUNCT
ejpam-6327	110	25	z	z	PROPN
ejpam-6327	110	26	∈	∈	PROPN
ejpam-6327	110	27	x.	x.	NOUN
ejpam-6327	110	28	a	a	DET
ejpam-6327	110	29	fuzzy	fuzzy	ADJ
ejpam-6327	110	30	set	set	VERB
ejpam-6327	110	31	µ	µ	NOUN
ejpam-6327	110	32	on	on	ADP
ejpam-6327	110	33	a	a	DET
ejpam-6327	110	34	ks	ks	NOUN
ejpam-6327	110	35	-	-	PUNCT
ejpam-6327	110	36	semigroup	semigroup	NOUN
ejpam-6327	110	37	x	x	PUNCT
ejpam-6327	110	38	is	be	AUX
ejpam-6327	110	39	called	call	VERB
ejpam-6327	110	40	a	a	DET
ejpam-6327	110	41	fuzzy	fuzzy	ADJ
ejpam-6327	110	42	ks	ks	NOUN
ejpam-6327	110	43	-	-	ADJ
ejpam-6327	110	44	p	p	NOUN
ejpam-6327	110	45	-	-	PUNCT
ejpam-6327	110	46	ideal	ideal	NOUN
ejpam-6327	110	47	of	of	ADP
ejpam-6327	110	48	x	x	PRON
ejpam-6327	110	49	if	if	SCONJ
ejpam-6327	110	50	it	it	PRON
ejpam-6327	110	51	is	be	AUX
ejpam-6327	110	52	both	both	CCONJ
ejpam-6327	110	53	a	a	DET
ejpam-6327	110	54	left	left	NOUN
ejpam-6327	110	55	and	and	CCONJ
ejpam-6327	110	56	a	a	DET
ejpam-6327	110	57	right	right	ADJ
ejpam-6327	110	58	fuzzy	fuzzy	ADJ
ejpam-6327	110	59	ks	ks	NOUN
ejpam-6327	110	60	-	-	ADJ
ejpam-6327	110	61	p	p	NOUN
ejpam-6327	110	62	-	-	PUNCT
ejpam-6327	110	63	ideal	ideal	NOUN
ejpam-6327	110	64	of	of	ADP
ejpam-6327	110	65	x.	x.	PROPN
ejpam-6327	110	66	theorem	theorem	NOUN
ejpam-6327	110	67	2	2	NUM
ejpam-6327	110	68	.	.	PUNCT
ejpam-6327	111	1	[	[	X
ejpam-6327	111	2	7	7	X
ejpam-6327	111	3	]	]	X
ejpam-6327	111	4	let	let	VERB
ejpam-6327	111	5	x	x	PRON
ejpam-6327	111	6	be	be	AUX
ejpam-6327	111	7	a	a	DET
ejpam-6327	111	8	ks	ks	NOUN
ejpam-6327	111	9	-	-	PUNCT
ejpam-6327	111	10	semigroup	semigroup	NOUN
ejpam-6327	111	11	.	.	PUNCT
ejpam-6327	112	1	then	then	ADV
ejpam-6327	112	2	any	any	DET
ejpam-6327	112	3	fuzzy	fuzzy	ADJ
ejpam-6327	112	4	ks	ks	NOUN
ejpam-6327	112	5	-	-	ADJ
ejpam-6327	112	6	p	p	NOUN
ejpam-6327	112	7	-	-	PUNCT
ejpam-6327	112	8	ideal	ideal	NOUN
ejpam-6327	112	9	of	of	ADP
ejpam-6327	112	10	x	x	PUNCT
ejpam-6327	112	11	is	be	AUX
ejpam-6327	112	12	a	a	DET
ejpam-6327	112	13	fuzzy	fuzzy	ADJ
ejpam-6327	112	14	ks	ks	NOUN
ejpam-6327	112	15	-	-	NOUN
ejpam-6327	112	16	ideal	ideal	NOUN
ejpam-6327	112	17	of	of	ADP
ejpam-6327	112	18	x.	x.	NOUN
ejpam-6327	112	19	however	however	ADV
ejpam-6327	112	20	,	,	PUNCT
ejpam-6327	112	21	the	the	DET
ejpam-6327	112	22	converse	converse	NOUN
ejpam-6327	112	23	is	be	AUX
ejpam-6327	112	24	not	not	PART
ejpam-6327	112	25	true	true	ADJ
ejpam-6327	112	26	.	.	PUNCT
ejpam-6327	113	1	h.	h.	PROPN
ejpam-6327	113	2	sarapuddin	sarapuddin	PROPN
ejpam-6327	113	3	,	,	PUNCT
ejpam-6327	113	4	j.	j.	PROPN
ejpam-6327	113	5	vilela	vilela	PROPN
ejpam-6327	113	6	/	/	SYM
ejpam-6327	113	7	eur	eur	PROPN
ejpam-6327	113	8	.	.	PUNCT
ejpam-6327	114	1	j.	j.	PROPN
ejpam-6327	114	2	pure	pure	PROPN
ejpam-6327	114	3	appl	appl	PROPN
ejpam-6327	114	4	.	.	PROPN
ejpam-6327	114	5	math	math	PROPN
ejpam-6327	114	6	,	,	PUNCT
ejpam-6327	114	7	18	18	NUM
ejpam-6327	114	8	(	(	PUNCT
ejpam-6327	114	9	3	3	NUM
ejpam-6327	114	10	)	)	PUNCT
ejpam-6327	114	11	(	(	PUNCT
ejpam-6327	114	12	2025	2025	NUM
ejpam-6327	114	13	)	)	PUNCT
ejpam-6327	114	14	,	,	PUNCT
ejpam-6327	114	15	6327	6327	NUM
ejpam-6327	114	16	5	5	NUM
ejpam-6327	114	17	of	of	ADP
ejpam-6327	114	18	23	23	NUM
ejpam-6327	114	19	3	3	NUM
ejpam-6327	114	20	.	.	PUNCT
ejpam-6327	115	1	some	some	DET
ejpam-6327	115	2	fuzzy	fuzzy	ADJ
ejpam-6327	115	3	ideals	ideal	NOUN
ejpam-6327	115	4	on	on	ADP
ejpam-6327	115	5	ks	ks	NOUN
ejpam-6327	115	6	-	-	PUNCT
ejpam-6327	115	7	semigroup	semigroup	NOUN
ejpam-6327	115	8	in	in	ADP
ejpam-6327	115	9	this	this	DET
ejpam-6327	115	10	section	section	NOUN
ejpam-6327	115	11	,	,	PUNCT
ejpam-6327	115	12	we	we	PRON
ejpam-6327	115	13	investigate	investigate	VERB
ejpam-6327	115	14	useful	useful	ADJ
ejpam-6327	115	15	properties	property	NOUN
ejpam-6327	115	16	of	of	ADP
ejpam-6327	115	17	fuzzy	fuzzy	ADJ
ejpam-6327	115	18	ks	ks	NOUN
ejpam-6327	115	19	-	-	PUNCT
ejpam-6327	115	20	ideals	ideal	NOUN
ejpam-6327	115	21	and	and	CCONJ
ejpam-6327	115	22	give	give	VERB
ejpam-6327	115	23	criterion	criterion	NOUN
ejpam-6327	115	24	for	for	ADP
ejpam-6327	115	25	a	a	DET
ejpam-6327	115	26	fuzzy	fuzzy	ADJ
ejpam-6327	115	27	ks	ks	NOUN
ejpam-6327	115	28	-	-	PUNCT
ejpam-6327	115	29	ideal	ideal	NOUN
ejpam-6327	115	30	to	to	PART
ejpam-6327	115	31	be	be	AUX
ejpam-6327	115	32	a	a	DET
ejpam-6327	115	33	fuzzy	fuzzy	ADJ
ejpam-6327	115	34	ks	ks	NOUN
ejpam-6327	115	35	-	-	ADJ
ejpam-6327	115	36	p	p	NOUN
ejpam-6327	115	37	-	-	PUNCT
ejpam-6327	115	38	ideal	ideal	NOUN
ejpam-6327	115	39	.	.	PUNCT
ejpam-6327	116	1	moreover	moreover	ADV
ejpam-6327	116	2	,	,	PUNCT
ejpam-6327	116	3	we	we	PRON
ejpam-6327	116	4	introduce	introduce	VERB
ejpam-6327	116	5	new	new	ADJ
ejpam-6327	116	6	types	type	NOUN
ejpam-6327	116	7	of	of	ADP
ejpam-6327	116	8	fuzzy	fuzzy	ADJ
ejpam-6327	116	9	ideals	ideal	NOUN
ejpam-6327	116	10	on	on	ADP
ejpam-6327	116	11	ks	ks	NOUN
ejpam-6327	116	12	-	-	PUNCT
ejpam-6327	116	13	semigroups	semigroup	NOUN
ejpam-6327	116	14	,	,	PUNCT
ejpam-6327	116	15	compare	compare	VERB
ejpam-6327	116	16	them	they	PRON
ejpam-6327	116	17	with	with	ADP
ejpam-6327	116	18	existing	exist	VERB
ejpam-6327	116	19	fuzzy	fuzzy	ADJ
ejpam-6327	116	20	ks	ks	NOUN
ejpam-6327	116	21	-	-	PUNCT
ejpam-6327	116	22	ideals	ideal	NOUN
ejpam-6327	116	23	and	and	CCONJ
ejpam-6327	116	24	investigate	investigate	VERB
ejpam-6327	116	25	their	their	PRON
ejpam-6327	116	26	properties	property	NOUN
ejpam-6327	116	27	.	.	PUNCT
ejpam-6327	117	1	the	the	DET
ejpam-6327	117	2	following	follow	VERB
ejpam-6327	117	3	lemma	lemma	PROPN
ejpam-6327	117	4	gives	give	VERB
ejpam-6327	117	5	some	some	DET
ejpam-6327	117	6	useful	useful	ADJ
ejpam-6327	117	7	properties	property	NOUN
ejpam-6327	117	8	of	of	ADP
ejpam-6327	117	9	fuzzy	fuzzy	ADJ
ejpam-6327	117	10	ks	ks	NOUN
ejpam-6327	117	11	-	-	NOUN
ejpam-6327	117	12	ideals	ideal	NOUN
ejpam-6327	117	13	.	.	PUNCT
ejpam-6327	118	1	lemma	lemma	PROPN
ejpam-6327	118	2	1	1	X
ejpam-6327	118	3	.	.	PUNCT
ejpam-6327	119	1	let	let	VERB
ejpam-6327	119	2	µ	µ	X
ejpam-6327	119	3	be	be	AUX
ejpam-6327	119	4	a	a	DET
ejpam-6327	119	5	fuzzy	fuzzy	ADJ
ejpam-6327	119	6	ks	ks	NOUN
ejpam-6327	119	7	-	-	NOUN
ejpam-6327	119	8	ideal	ideal	NOUN
ejpam-6327	119	9	of	of	ADP
ejpam-6327	119	10	a	a	DET
ejpam-6327	119	11	ks	ks	NOUN
ejpam-6327	119	12	-	-	PUNCT
ejpam-6327	119	13	semigroup	semigroup	NOUN
ejpam-6327	119	14	x.	x.	NOUN
ejpam-6327	119	15	(	(	PUNCT
ejpam-6327	119	16	i	i	NOUN
ejpam-6327	119	17	)	)	PUNCT
ejpam-6327	119	18	if	if	SCONJ
ejpam-6327	119	19	x	x	X
ejpam-6327	119	20	,	,	PUNCT
ejpam-6327	119	21	y	y	PROPN
ejpam-6327	119	22	∈	∈	PROPN
ejpam-6327	119	23	x	x	PUNCT
ejpam-6327	119	24	such	such	ADJ
ejpam-6327	119	25	that	that	SCONJ
ejpam-6327	119	26	x	x	X
ejpam-6327	119	27	≤	≤	NUM
ejpam-6327	119	28	y	y	PROPN
ejpam-6327	119	29	,	,	PUNCT
ejpam-6327	119	30	then	then	ADV
ejpam-6327	119	31	µ(y	µ(y	PROPN
ejpam-6327	119	32	)	)	PUNCT
ejpam-6327	119	33	≤	≤	NOUN
ejpam-6327	119	34	µ(x	µ(x	NOUN
ejpam-6327	119	35	)	)	PUNCT
ejpam-6327	119	36	.	.	PUNCT
ejpam-6327	120	1	(	(	PUNCT
ejpam-6327	120	2	ii	ii	NOUN
ejpam-6327	120	3	)	)	PUNCT
ejpam-6327	120	4	if	if	SCONJ
ejpam-6327	120	5	x	x	X
ejpam-6327	120	6	,	,	PUNCT
ejpam-6327	120	7	y	y	PROPN
ejpam-6327	120	8	,	,	PUNCT
ejpam-6327	120	9	z	z	NOUN
ejpam-6327	120	10	∈	∈	PROPN
ejpam-6327	120	11	x	x	PUNCT
ejpam-6327	120	12	such	such	ADJ
ejpam-6327	120	13	that	that	SCONJ
ejpam-6327	120	14	x	x	PROPN
ejpam-6327	120	15	∗	∗	NOUN
ejpam-6327	120	16	y	y	NOUN
ejpam-6327	120	17	≤	≤	PROPN
ejpam-6327	120	18	z	z	PROPN
ejpam-6327	120	19	,	,	PUNCT
ejpam-6327	120	20	then	then	ADV
ejpam-6327	120	21	µ(x	µ(x	NOUN
ejpam-6327	120	22	)	)	PUNCT
ejpam-6327	120	23	≥	≥	NOUN
ejpam-6327	120	24	min{µ(y	min{µ(y	PROPN
ejpam-6327	120	25	)	)	PUNCT
ejpam-6327	120	26	,	,	PUNCT
ejpam-6327	120	27	µ(z	µ(z	PROPN
ejpam-6327	120	28	)	)	PUNCT
ejpam-6327	120	29	}	}	PUNCT
ejpam-6327	120	30	.	.	PUNCT
ejpam-6327	121	1	proof	proof	NOUN
ejpam-6327	121	2	.	.	PUNCT
ejpam-6327	122	1	let	let	VERB
ejpam-6327	122	2	µ	µ	X
ejpam-6327	122	3	be	be	AUX
ejpam-6327	122	4	a	a	DET
ejpam-6327	122	5	fuzzy	fuzzy	ADJ
ejpam-6327	122	6	ks	ks	NOUN
ejpam-6327	122	7	-	-	NOUN
ejpam-6327	122	8	ideal	ideal	NOUN
ejpam-6327	122	9	of	of	ADP
ejpam-6327	122	10	a	a	DET
ejpam-6327	122	11	ks	ks	NOUN
ejpam-6327	122	12	-	-	PUNCT
ejpam-6327	122	13	semigroup	semigroup	NOUN
ejpam-6327	122	14	x.	x.	NOUN
ejpam-6327	123	1	(	(	PUNCT
ejpam-6327	123	2	i	i	NOUN
ejpam-6327	123	3	)	)	PUNCT
ejpam-6327	123	4	suppose	suppose	VERB
ejpam-6327	123	5	x	x	PRON
ejpam-6327	123	6	,	,	PUNCT
ejpam-6327	123	7	y	y	PROPN
ejpam-6327	123	8	∈	∈	PROPN
ejpam-6327	123	9	x	x	PUNCT
ejpam-6327	123	10	such	such	ADJ
ejpam-6327	123	11	that	that	SCONJ
ejpam-6327	123	12	x	x	X
ejpam-6327	123	13	≤	≤	ADJ
ejpam-6327	123	14	y.	y.	NOUN
ejpam-6327	123	15	then	then	ADV
ejpam-6327	123	16	by	by	ADP
ejpam-6327	123	17	(	(	PUNCT
ejpam-6327	123	18	f2	f2	PROPN
ejpam-6327	123	19	)	)	PUNCT
ejpam-6327	123	20	,	,	PUNCT
ejpam-6327	123	21	µ(x	µ(x	X
ejpam-6327	123	22	)	)	PUNCT
ejpam-6327	123	23	≥	≥	NOUN
ejpam-6327	123	24	min{µ(x	min{µ(x	PROPN
ejpam-6327	123	25	∗	∗	X
ejpam-6327	123	26	y	y	NOUN
ejpam-6327	123	27	)	)	PUNCT
ejpam-6327	123	28	,	,	PUNCT
ejpam-6327	123	29	µ(y	µ(y	PROPN
ejpam-6327	123	30	)	)	PUNCT
ejpam-6327	123	31	}	}	PUNCT
ejpam-6327	123	32	and	and	CCONJ
ejpam-6327	123	33	x	x	SYM
ejpam-6327	123	34	∗	∗	NOUN
ejpam-6327	123	35	y	y	NOUN
ejpam-6327	123	36	=	=	SYM
ejpam-6327	123	37	0	0	PROPN
ejpam-6327	123	38	.	.	PUNCT
ejpam-6327	124	1	thus	thus	ADV
ejpam-6327	124	2	,	,	PUNCT
ejpam-6327	124	3	µ(x	µ(x	NOUN
ejpam-6327	124	4	)	)	PUNCT
ejpam-6327	124	5	≥	≥	NOUN
ejpam-6327	124	6	min{µ(x	min{µ(x	PROPN
ejpam-6327	124	7	∗	∗	X
ejpam-6327	124	8	y	y	NOUN
ejpam-6327	124	9	)	)	PUNCT
ejpam-6327	124	10	,	,	PUNCT
ejpam-6327	124	11	µ(y	µ(y	PROPN
ejpam-6327	124	12	)	)	PUNCT
ejpam-6327	124	13	}	}	PUNCT
ejpam-6327	124	14	=	=	SYM
ejpam-6327	124	15	min{µ(0	min{µ(0	NOUN
ejpam-6327	124	16	)	)	PUNCT
ejpam-6327	124	17	,	,	PUNCT
ejpam-6327	124	18	µ(y	µ(y	PROPN
ejpam-6327	124	19	)	)	PUNCT
ejpam-6327	124	20	}	}	PUNCT
ejpam-6327	124	21	=	=	SYM
ejpam-6327	124	22	µ(y	µ(y	PROPN
ejpam-6327	124	23	)	)	PUNCT
ejpam-6327	124	24	.	.	PUNCT
ejpam-6327	125	1	(	(	PUNCT
ejpam-6327	125	2	ii	ii	NOUN
ejpam-6327	125	3	)	)	PUNCT
ejpam-6327	125	4	suppose	suppose	VERB
ejpam-6327	125	5	x	x	SYM
ejpam-6327	125	6	,	,	PUNCT
ejpam-6327	125	7	y	y	PROPN
ejpam-6327	125	8	,	,	PUNCT
ejpam-6327	125	9	z	z	NOUN
ejpam-6327	125	10	∈	∈	PROPN
ejpam-6327	125	11	x	x	PUNCT
ejpam-6327	125	12	such	such	ADJ
ejpam-6327	125	13	that	that	SCONJ
ejpam-6327	125	14	x	x	PROPN
ejpam-6327	125	15	∗	∗	NOUN
ejpam-6327	125	16	y	y	PROPN
ejpam-6327	125	17	≤	≤	PROPN
ejpam-6327	125	18	z.	z.	PROPN
ejpam-6327	125	19	then	then	ADV
ejpam-6327	125	20	by	by	ADP
ejpam-6327	125	21	(	(	PUNCT
ejpam-6327	125	22	i	i	NOUN
ejpam-6327	125	23	)	)	PUNCT
ejpam-6327	125	24	,	,	PUNCT
ejpam-6327	125	25	µ(z	µ(z	PROPN
ejpam-6327	125	26	)	)	PUNCT
ejpam-6327	125	27	≤	≤	NOUN
ejpam-6327	125	28	µ(x	µ(x	ADJ
ejpam-6327	125	29	∗	∗	NOUN
ejpam-6327	125	30	y	y	NOUN
ejpam-6327	125	31	)	)	PUNCT
ejpam-6327	125	32	.	.	PUNCT
ejpam-6327	126	1	by	by	ADP
ejpam-6327	126	2	(	(	PUNCT
ejpam-6327	126	3	f2	f2	PROPN
ejpam-6327	126	4	)	)	PUNCT
ejpam-6327	126	5	,	,	PUNCT
ejpam-6327	126	6	µ(x	µ(x	X
ejpam-6327	126	7	)	)	PUNCT
ejpam-6327	126	8	≥	≥	NOUN
ejpam-6327	126	9	min{µ(x	min{µ(x	PROPN
ejpam-6327	126	10	∗	∗	X
ejpam-6327	126	11	y	y	NOUN
ejpam-6327	126	12	)	)	PUNCT
ejpam-6327	126	13	,	,	PUNCT
ejpam-6327	126	14	µ(y	µ(y	PROPN
ejpam-6327	126	15	)	)	PUNCT
ejpam-6327	126	16	}	}	PUNCT
ejpam-6327	126	17	≥	≥	NOUN
ejpam-6327	126	18	min{µ(z	min{µ(z	PROPN
ejpam-6327	126	19	)	)	PUNCT
ejpam-6327	126	20	,	,	PUNCT
ejpam-6327	126	21	µ(y	µ(y	PROPN
ejpam-6327	126	22	)	)	PUNCT
ejpam-6327	126	23	}	}	PUNCT
ejpam-6327	126	24	.	.	PUNCT
ejpam-6327	127	1	in	in	ADP
ejpam-6327	127	2	theorem	theorem	NOUN
ejpam-6327	127	3	2	2	NUM
ejpam-6327	127	4	,	,	PUNCT
ejpam-6327	127	5	a	a	DET
ejpam-6327	127	6	fuzzy	fuzzy	ADJ
ejpam-6327	127	7	ks	ks	NOUN
ejpam-6327	127	8	-	-	PUNCT
ejpam-6327	127	9	ideal	ideal	NOUN
ejpam-6327	127	10	may	may	AUX
ejpam-6327	127	11	not	not	PART
ejpam-6327	127	12	be	be	AUX
ejpam-6327	127	13	a	a	DET
ejpam-6327	127	14	fuzzy	fuzzy	ADJ
ejpam-6327	127	15	ks	ks	NOUN
ejpam-6327	127	16	-	-	ADJ
ejpam-6327	127	17	p	p	NOUN
ejpam-6327	127	18	-	-	PUNCT
ejpam-6327	127	19	ideal	ideal	NOUN
ejpam-6327	127	20	.	.	PUNCT
ejpam-6327	128	1	the	the	DET
ejpam-6327	128	2	following	follow	VERB
ejpam-6327	128	3	theorem	theorem	NOUN
ejpam-6327	128	4	gives	give	VERB
ejpam-6327	128	5	a	a	DET
ejpam-6327	128	6	criterion	criterion	NOUN
ejpam-6327	128	7	for	for	ADP
ejpam-6327	128	8	a	a	DET
ejpam-6327	128	9	fuzzy	fuzzy	ADJ
ejpam-6327	128	10	ks	ks	NOUN
ejpam-6327	128	11	-	-	PUNCT
ejpam-6327	128	12	ideal	ideal	NOUN
ejpam-6327	128	13	to	to	PART
ejpam-6327	128	14	be	be	AUX
ejpam-6327	128	15	a	a	DET
ejpam-6327	128	16	fuzzy	fuzzy	ADJ
ejpam-6327	128	17	ks	ks	NOUN
ejpam-6327	128	18	-	-	ADJ
ejpam-6327	128	19	p	p	NOUN
ejpam-6327	128	20	-	-	PUNCT
ejpam-6327	128	21	ideal	ideal	NOUN
ejpam-6327	128	22	.	.	PUNCT
ejpam-6327	129	1	theorem	theorem	NOUN
ejpam-6327	129	2	3	3	NUM
ejpam-6327	129	3	.	.	PUNCT
ejpam-6327	130	1	a	a	DET
ejpam-6327	130	2	fuzzy	fuzzy	ADJ
ejpam-6327	130	3	set	set	VERB
ejpam-6327	130	4	µ	µ	NOUN
ejpam-6327	130	5	on	on	ADP
ejpam-6327	130	6	a	a	DET
ejpam-6327	130	7	ks	ks	NOUN
ejpam-6327	130	8	-	-	PUNCT
ejpam-6327	130	9	semigroup	semigroup	NOUN
ejpam-6327	130	10	x	x	PUNCT
ejpam-6327	130	11	is	be	AUX
ejpam-6327	130	12	a	a	DET
ejpam-6327	130	13	fuzzy	fuzzy	ADJ
ejpam-6327	130	14	ks	ks	NOUN
ejpam-6327	130	15	-	-	ADJ
ejpam-6327	130	16	p	p	NOUN
ejpam-6327	130	17	-	-	PUNCT
ejpam-6327	130	18	ideal	ideal	NOUN
ejpam-6327	130	19	if	if	SCONJ
ejpam-6327	130	20	and	and	CCONJ
ejpam-6327	130	21	only	only	ADV
ejpam-6327	130	22	if	if	SCONJ
ejpam-6327	130	23	it	it	PRON
ejpam-6327	130	24	is	be	AUX
ejpam-6327	130	25	a	a	DET
ejpam-6327	130	26	fuzzy	fuzzy	ADJ
ejpam-6327	130	27	ks	ks	NOUN
ejpam-6327	130	28	-	-	PUNCT
ejpam-6327	130	29	ideal	ideal	ADJ
ejpam-6327	130	30	satisfying	satisfying	ADJ
ejpam-6327	130	31	µ(x	µ(x	ADJ
ejpam-6327	130	32	∗	∗	NOUN
ejpam-6327	130	33	y	y	NOUN
ejpam-6327	130	34	)	)	PUNCT
ejpam-6327	131	1	=	=	SYM
ejpam-6327	131	2	µ((x	µ((x	NOUN
ejpam-6327	131	3	∗	∗	PROPN
ejpam-6327	131	4	y	y	NOUN
ejpam-6327	131	5	)	)	PUNCT
ejpam-6327	131	6	∗	∗	PROPN
ejpam-6327	131	7	y	y	PROPN
ejpam-6327	131	8	)	)	PUNCT
ejpam-6327	131	9	for	for	ADP
ejpam-6327	131	10	all	all	DET
ejpam-6327	131	11	x	x	NOUN
ejpam-6327	131	12	,	,	PUNCT
ejpam-6327	131	13	y	y	PROPN
ejpam-6327	131	14	∈	∈	PROPN
ejpam-6327	131	15	x.	x.	NOUN
ejpam-6327	131	16	proof	proof	NOUN
ejpam-6327	131	17	.	.	PUNCT
ejpam-6327	132	1	let	let	VERB
ejpam-6327	132	2	x	x	PRON
ejpam-6327	132	3	be	be	AUX
ejpam-6327	132	4	a	a	DET
ejpam-6327	132	5	ks	ks	NOUN
ejpam-6327	132	6	-	-	PUNCT
ejpam-6327	132	7	semigroup	semigroup	NOUN
ejpam-6327	132	8	and	and	CCONJ
ejpam-6327	132	9	µ	µ	PRON
ejpam-6327	132	10	be	be	AUX
ejpam-6327	132	11	a	a	DET
ejpam-6327	132	12	fuzzy	fuzzy	ADJ
ejpam-6327	132	13	set	set	NOUN
ejpam-6327	132	14	on	on	ADP
ejpam-6327	132	15	x.	x.	NOUN
ejpam-6327	132	16	suppose	suppose	VERB
ejpam-6327	132	17	µ	µ	PRON
ejpam-6327	132	18	is	be	AUX
ejpam-6327	132	19	a	a	DET
ejpam-6327	132	20	fuzzy	fuzzy	ADJ
ejpam-6327	132	21	ks	ks	NOUN
ejpam-6327	132	22	-	-	ADJ
ejpam-6327	132	23	p	p	NOUN
ejpam-6327	132	24	-	-	PUNCT
ejpam-6327	132	25	ideal	ideal	NOUN
ejpam-6327	132	26	of	of	ADP
ejpam-6327	132	27	x.	x.	NOUN
ejpam-6327	132	28	then	then	ADV
ejpam-6327	132	29	by	by	ADP
ejpam-6327	132	30	theorem	theorem	NOUN
ejpam-6327	132	31	2	2	NUM
ejpam-6327	132	32	,	,	PUNCT
ejpam-6327	132	33	µ	µ	X
ejpam-6327	132	34	is	be	AUX
ejpam-6327	132	35	a	a	DET
ejpam-6327	132	36	fuzzy	fuzzy	ADJ
ejpam-6327	132	37	ks	ks	NOUN
ejpam-6327	132	38	-	-	PUNCT
ejpam-6327	132	39	ideal	ideal	NOUN
ejpam-6327	132	40	.	.	PUNCT
ejpam-6327	133	1	by	by	ADP
ejpam-6327	133	2	(	(	PUNCT
ejpam-6327	133	3	f4	f4	PROPN
ejpam-6327	133	4	)	)	PUNCT
ejpam-6327	133	5	,	,	PUNCT
ejpam-6327	133	6	(	(	PUNCT
ejpam-6327	133	7	f1	f1	NOUN
ejpam-6327	133	8	)	)	PUNCT
ejpam-6327	133	9	and	and	CCONJ
ejpam-6327	133	10	definition	definition	NOUN
ejpam-6327	133	11	1	1	NUM
ejpam-6327	133	12	,	,	PUNCT
ejpam-6327	133	13	for	for	ADP
ejpam-6327	133	14	all	all	DET
ejpam-6327	133	15	x	x	NOUN
ejpam-6327	133	16	,	,	PUNCT
ejpam-6327	133	17	y	y	PROPN
ejpam-6327	133	18	∈	∈	PROPN
ejpam-6327	133	19	x	x	PROPN
ejpam-6327	133	20	,	,	PUNCT
ejpam-6327	133	21	µ(x	µ(x	X
ejpam-6327	133	22	∗	∗	NOUN
ejpam-6327	133	23	y	y	NOUN
ejpam-6327	133	24	)	)	PUNCT
ejpam-6327	133	25	≥	≥	NOUN
ejpam-6327	133	26	min{µ((x	min{µ((x	PRON
ejpam-6327	133	27	∗	∗	X
ejpam-6327	133	28	y	y	NOUN
ejpam-6327	133	29	)	)	PUNCT
ejpam-6327	133	30	∗	∗	PROPN
ejpam-6327	133	31	y	y	PROPN
ejpam-6327	133	32	)	)	PUNCT
ejpam-6327	133	33	,	,	PUNCT
ejpam-6327	133	34	µ(y	µ(y	PROPN
ejpam-6327	133	35	∗	∗	PROPN
ejpam-6327	133	36	y	y	PROPN
ejpam-6327	133	37	)	)	PUNCT
ejpam-6327	133	38	}	}	PUNCT
ejpam-6327	134	1	=	=	SYM
ejpam-6327	134	2	min{µ((x	min{µ((x	PRON
ejpam-6327	134	3	∗	∗	X
ejpam-6327	134	4	y	y	NOUN
ejpam-6327	134	5	)	)	PUNCT
ejpam-6327	134	6	∗	∗	PROPN
ejpam-6327	134	7	y	y	PROPN
ejpam-6327	134	8	)	)	PUNCT
ejpam-6327	134	9	,	,	PUNCT
ejpam-6327	134	10	µ(0	µ(0	NOUN
ejpam-6327	134	11	)	)	PUNCT
ejpam-6327	134	12	}	}	PUNCT
ejpam-6327	135	1	=	=	SYM
ejpam-6327	135	2	µ((x	µ((x	NOUN
ejpam-6327	135	3	∗	∗	X
ejpam-6327	135	4	y	y	NOUN
ejpam-6327	135	5	)	)	PUNCT
ejpam-6327	135	6	∗	∗	PROPN
ejpam-6327	135	7	y	y	PROPN
ejpam-6327	135	8	)	)	PUNCT
ejpam-6327	135	9	.	.	PUNCT
ejpam-6327	136	1	moreover	moreover	ADV
ejpam-6327	136	2	,	,	PUNCT
ejpam-6327	136	3	observe	observe	VERB
ejpam-6327	136	4	that	that	SCONJ
ejpam-6327	136	5	by	by	ADP
ejpam-6327	136	6	remark	remark	NOUN
ejpam-6327	136	7	1(ii	1(ii	NUM
ejpam-6327	136	8	)	)	PUNCT
ejpam-6327	136	9	,	,	PUNCT
ejpam-6327	136	10	(	(	PUNCT
ejpam-6327	136	11	x	x	X
ejpam-6327	136	12	∗	∗	PROPN
ejpam-6327	136	13	y	y	NOUN
ejpam-6327	136	14	)	)	PUNCT
ejpam-6327	136	15	∗	∗	NOUN
ejpam-6327	136	16	y	y	PROPN
ejpam-6327	136	17	≤	≤	NUM
ejpam-6327	136	18	x	x	PUNCT
ejpam-6327	136	19	∗	∗	NOUN
ejpam-6327	136	20	y.	y.	PROPN
ejpam-6327	136	21	thus	thus	ADV
ejpam-6327	136	22	,	,	PUNCT
ejpam-6327	136	23	by	by	ADP
ejpam-6327	136	24	lemma	lemma	PROPN
ejpam-6327	136	25	1(i	1(i	NUM
ejpam-6327	136	26	)	)	PUNCT
ejpam-6327	136	27	,	,	PUNCT
ejpam-6327	136	28	µ((x	µ((x	NOUN
ejpam-6327	136	29	∗	∗	NOUN
ejpam-6327	136	30	y	y	NOUN
ejpam-6327	136	31	)	)	PUNCT
ejpam-6327	136	32	∗	∗	PROPN
ejpam-6327	136	33	y	y	PROPN
ejpam-6327	136	34	)	)	PUNCT
ejpam-6327	136	35	≥	≥	NOUN
ejpam-6327	136	36	µ(x	µ(x	X
ejpam-6327	136	37	∗	∗	NOUN
ejpam-6327	136	38	y	y	NOUN
ejpam-6327	136	39	)	)	PUNCT
ejpam-6327	136	40	.	.	PUNCT
ejpam-6327	137	1	hence	hence	ADV
ejpam-6327	137	2	,	,	PUNCT
ejpam-6327	137	3	µ(x	µ(x	ADJ
ejpam-6327	137	4	∗	∗	NOUN
ejpam-6327	137	5	y	y	NOUN
ejpam-6327	137	6	)	)	PUNCT
ejpam-6327	138	1	=	=	SYM
ejpam-6327	138	2	µ((x	µ((x	NOUN
ejpam-6327	138	3	∗	∗	PROPN
ejpam-6327	138	4	y	y	NOUN
ejpam-6327	138	5	)	)	PUNCT
ejpam-6327	138	6	∗	∗	PROPN
ejpam-6327	138	7	y	y	PROPN
ejpam-6327	138	8	)	)	PUNCT
ejpam-6327	138	9	for	for	ADP
ejpam-6327	138	10	all	all	DET
ejpam-6327	138	11	x	x	NOUN
ejpam-6327	138	12	,	,	PUNCT
ejpam-6327	138	13	y	y	PROPN
ejpam-6327	138	14	∈	∈	PROPN
ejpam-6327	138	15	x.	x.	NOUN
ejpam-6327	138	16	conversely	conversely	ADV
ejpam-6327	138	17	,	,	PUNCT
ejpam-6327	138	18	suppose	suppose	VERB
ejpam-6327	138	19	µ	µ	PRON
ejpam-6327	138	20	is	be	AUX
ejpam-6327	138	21	a	a	DET
ejpam-6327	138	22	fuzzy	fuzzy	ADJ
ejpam-6327	138	23	ks	ks	NOUN
ejpam-6327	138	24	-	-	NOUN
ejpam-6327	138	25	ideal	ideal	NOUN
ejpam-6327	138	26	of	of	ADP
ejpam-6327	138	27	x	x	SYM
ejpam-6327	138	28	satisfying	satisfy	VERB
ejpam-6327	138	29	µ(x	µ(x	ADJ
ejpam-6327	138	30	∗	∗	NOUN
ejpam-6327	138	31	y	y	NOUN
ejpam-6327	138	32	)	)	PUNCT
ejpam-6327	138	33	=	=	SYM
ejpam-6327	139	1	µ((x	µ((x	NOUN
ejpam-6327	139	2	∗	∗	PROPN
ejpam-6327	139	3	y	y	NOUN
ejpam-6327	139	4	)	)	PUNCT
ejpam-6327	139	5	∗	∗	PROPN
ejpam-6327	139	6	y	y	PROPN
ejpam-6327	139	7	)	)	PUNCT
ejpam-6327	139	8	for	for	ADP
ejpam-6327	139	9	all	all	DET
ejpam-6327	139	10	x	x	NOUN
ejpam-6327	139	11	,	,	PUNCT
ejpam-6327	139	12	y	y	PROPN
ejpam-6327	139	13	∈	∈	PROPN
ejpam-6327	139	14	x.	x.	NOUN
ejpam-6327	139	15	note	note	VERB
ejpam-6327	139	16	that	that	SCONJ
ejpam-6327	139	17	by	by	ADP
ejpam-6327	139	18	proposition	proposition	NOUN
ejpam-6327	139	19	1(i	1(i	NUM
ejpam-6327	139	20	)	)	PUNCT
ejpam-6327	139	21	,	,	PUNCT
ejpam-6327	139	22	(	(	PUNCT
ejpam-6327	139	23	(	(	PUNCT
ejpam-6327	139	24	x	x	SYM
ejpam-6327	139	25	∗	∗	PROPN
ejpam-6327	139	26	z	z	NOUN
ejpam-6327	139	27	)	)	PUNCT
ejpam-6327	139	28	∗	∗	PROPN
ejpam-6327	139	29	z	z	NOUN
ejpam-6327	139	30	)	)	PUNCT
ejpam-6327	139	31	∗	∗	NOUN
ejpam-6327	139	32	(	(	PUNCT
ejpam-6327	139	33	y	y	PROPN
ejpam-6327	139	34	∗	∗	PROPN
ejpam-6327	139	35	z	z	PROPN
ejpam-6327	139	36	)	)	PUNCT
ejpam-6327	139	37	≤	≤	NOUN
ejpam-6327	139	38	(	(	PUNCT
ejpam-6327	139	39	x	x	X
ejpam-6327	139	40	∗	∗	PROPN
ejpam-6327	139	41	y	y	NOUN
ejpam-6327	139	42	)	)	PUNCT
ejpam-6327	139	43	∗	∗	NOUN
ejpam-6327	139	44	z.	z.	PROPN
ejpam-6327	140	1	thus	thus	ADV
ejpam-6327	140	2	,	,	PUNCT
ejpam-6327	140	3	by	by	ADP
ejpam-6327	140	4	lemma	lemma	PROPN
ejpam-6327	140	5	1(ii	1(ii	NUM
ejpam-6327	140	6	)	)	PUNCT
ejpam-6327	140	7	,	,	PUNCT
ejpam-6327	140	8	we	we	PRON
ejpam-6327	140	9	have	have	VERB
ejpam-6327	140	10	µ(x	µ(x	ADJ
ejpam-6327	140	11	∗	∗	NOUN
ejpam-6327	140	12	z	z	NOUN
ejpam-6327	140	13	)	)	PUNCT
ejpam-6327	140	14	=	=	SYM
ejpam-6327	140	15	µ((x	µ((x	NOUN
ejpam-6327	140	16	∗	∗	NOUN
ejpam-6327	140	17	z	z	NOUN
ejpam-6327	140	18	)	)	PUNCT
ejpam-6327	140	19	∗	∗	NOUN
ejpam-6327	140	20	z	z	NOUN
ejpam-6327	140	21	)	)	PUNCT
ejpam-6327	140	22	≥	≥	NOUN
ejpam-6327	140	23	min{µ((x	min{µ((x	NOUN
ejpam-6327	140	24	∗	∗	X
ejpam-6327	140	25	y	y	NOUN
ejpam-6327	140	26	)	)	PUNCT
ejpam-6327	140	27	∗	∗	NOUN
ejpam-6327	140	28	z	z	PROPN
ejpam-6327	140	29	)	)	PUNCT
ejpam-6327	140	30	,	,	PUNCT
ejpam-6327	140	31	µ(y	µ(y	PROPN
ejpam-6327	140	32	∗	∗	PROPN
ejpam-6327	140	33	z	z	PROPN
ejpam-6327	140	34	)	)	PUNCT
ejpam-6327	140	35	}	}	PUNCT
ejpam-6327	140	36	.	.	PUNCT
ejpam-6327	141	1	therefore	therefore	ADV
ejpam-6327	141	2	,	,	PUNCT
ejpam-6327	141	3	µ	µ	X
ejpam-6327	141	4	is	be	AUX
ejpam-6327	141	5	a	a	DET
ejpam-6327	141	6	fuzzy	fuzzy	ADJ
ejpam-6327	141	7	ks	ks	NOUN
ejpam-6327	141	8	-	-	ADJ
ejpam-6327	141	9	p	p	NOUN
ejpam-6327	141	10	-	-	PUNCT
ejpam-6327	141	11	ideal	ideal	NOUN
ejpam-6327	141	12	of	of	ADP
ejpam-6327	141	13	x.	x.	NOUN
ejpam-6327	141	14	now	now	ADV
ejpam-6327	141	15	,	,	PUNCT
ejpam-6327	141	16	we	we	PRON
ejpam-6327	141	17	introduce	introduce	VERB
ejpam-6327	141	18	some	some	DET
ejpam-6327	141	19	fuzzy	fuzzy	ADJ
ejpam-6327	141	20	ideals	ideal	NOUN
ejpam-6327	141	21	on	on	ADP
ejpam-6327	141	22	ks	ks	NOUN
ejpam-6327	141	23	-	-	PUNCT
ejpam-6327	141	24	semigroups	semigroup	NOUN
ejpam-6327	141	25	,	,	PUNCT
ejpam-6327	141	26	investigate	investigate	VERB
ejpam-6327	141	27	their	their	PRON
ejpam-6327	141	28	relationships	relationship	NOUN
ejpam-6327	141	29	with	with	ADP
ejpam-6327	141	30	fuzzy	fuzzy	ADJ
ejpam-6327	141	31	ks	ks	NOUN
ejpam-6327	141	32	-	-	PUNCT
ejpam-6327	141	33	ideals	ideal	NOUN
ejpam-6327	141	34	and	and	CCONJ
ejpam-6327	141	35	discuss	discuss	VERB
ejpam-6327	141	36	some	some	PRON
ejpam-6327	141	37	of	of	ADP
ejpam-6327	141	38	their	their	PRON
ejpam-6327	141	39	properties	property	NOUN
ejpam-6327	141	40	.	.	PUNCT
ejpam-6327	142	1	h.	h.	PROPN
ejpam-6327	142	2	sarapuddin	sarapuddin	PROPN
ejpam-6327	142	3	,	,	PUNCT
ejpam-6327	142	4	j.	j.	PROPN
ejpam-6327	142	5	vilela	vilela	PROPN
ejpam-6327	142	6	/	/	SYM
ejpam-6327	142	7	eur	eur	PROPN
ejpam-6327	142	8	.	.	PUNCT
ejpam-6327	143	1	j.	j.	PROPN
ejpam-6327	143	2	pure	pure	PROPN
ejpam-6327	143	3	appl	appl	PROPN
ejpam-6327	143	4	.	.	PROPN
ejpam-6327	143	5	math	math	PROPN
ejpam-6327	143	6	,	,	PUNCT
ejpam-6327	143	7	18	18	NUM
ejpam-6327	143	8	(	(	PUNCT
ejpam-6327	143	9	3	3	NUM
ejpam-6327	143	10	)	)	PUNCT
ejpam-6327	143	11	(	(	PUNCT
ejpam-6327	143	12	2025	2025	NUM
ejpam-6327	143	13	)	)	PUNCT
ejpam-6327	143	14	,	,	PUNCT
ejpam-6327	143	15	6327	6327	NUM
ejpam-6327	143	16	6	6	NUM
ejpam-6327	143	17	of	of	ADP
ejpam-6327	143	18	23	23	NUM
ejpam-6327	143	19	definition	definition	NOUN
ejpam-6327	143	20	11	11	NUM
ejpam-6327	143	21	.	.	PUNCT
ejpam-6327	144	1	let	let	VERB
ejpam-6327	144	2	x	x	PRON
ejpam-6327	144	3	be	be	AUX
ejpam-6327	144	4	a	a	DET
ejpam-6327	144	5	ks	ks	NOUN
ejpam-6327	144	6	-	-	PUNCT
ejpam-6327	144	7	semigroup	semigroup	NOUN
ejpam-6327	144	8	.	.	PUNCT
ejpam-6327	145	1	a	a	DET
ejpam-6327	145	2	fuzzy	fuzzy	ADJ
ejpam-6327	145	3	set	set	VERB
ejpam-6327	145	4	µ	µ	NOUN
ejpam-6327	145	5	on	on	ADP
ejpam-6327	145	6	x	x	VERB
ejpam-6327	145	7	is	be	AUX
ejpam-6327	145	8	called	call	VERB
ejpam-6327	145	9	a	a	DET
ejpam-6327	145	10	left	left	ADJ
ejpam-6327	145	11	(	(	PUNCT
ejpam-6327	145	12	resp	resp	NOUN
ejpam-6327	145	13	.	.	PUNCT
ejpam-6327	146	1	right	right	ADJ
ejpam-6327	146	2	)	)	PUNCT
ejpam-6327	146	3	fuzzy	fuzzy	ADJ
ejpam-6327	146	4	commutative	commutative	ADJ
ejpam-6327	146	5	ks	ks	NOUN
ejpam-6327	146	6	-	-	PUNCT
ejpam-6327	146	7	ideal	ideal	NOUN
ejpam-6327	146	8	of	of	ADP
ejpam-6327	146	9	x	x	PRON
ejpam-6327	146	10	if	if	SCONJ
ejpam-6327	146	11	it	it	PRON
ejpam-6327	146	12	satisfies	satisfy	VERB
ejpam-6327	146	13	(	(	PUNCT
ejpam-6327	146	14	f1	f1	NOUN
ejpam-6327	146	15	)	)	PUNCT
ejpam-6327	146	16	,	,	PUNCT
ejpam-6327	146	17	(	(	PUNCT
ejpam-6327	146	18	f3	f3	ADJ
ejpam-6327	146	19	)	)	PUNCT
ejpam-6327	146	20	and	and	CCONJ
ejpam-6327	146	21	(	(	PUNCT
ejpam-6327	146	22	f5	f5	PROPN
ejpam-6327	146	23	)	)	PUNCT
ejpam-6327	146	24	:	:	PUNCT
ejpam-6327	147	1	µ(x	µ(x	ADJ
ejpam-6327	147	2	∗	∗	NOUN
ejpam-6327	147	3	(	(	PUNCT
ejpam-6327	147	4	y	y	PROPN
ejpam-6327	147	5	∗	∗	NOUN
ejpam-6327	147	6	(	(	PUNCT
ejpam-6327	147	7	y	y	PROPN
ejpam-6327	147	8	∗	∗	NOUN
ejpam-6327	147	9	x	x	NOUN
ejpam-6327	147	10	)	)	PUNCT
ejpam-6327	147	11	)	)	PUNCT
ejpam-6327	147	12	)	)	PUNCT
ejpam-6327	147	13	≥	≥	X
ejpam-6327	147	14	min{µ((x	min{µ((x	DET
ejpam-6327	147	15	∗	∗	X
ejpam-6327	147	16	y	y	NOUN
ejpam-6327	147	17	)	)	PUNCT
ejpam-6327	147	18	∗	∗	NOUN
ejpam-6327	147	19	z	z	NOUN
ejpam-6327	147	20	)	)	PUNCT
ejpam-6327	147	21	,	,	PUNCT
ejpam-6327	147	22	µ(z	µ(z	PROPN
ejpam-6327	147	23	)	)	PUNCT
ejpam-6327	147	24	}	}	PUNCT
ejpam-6327	147	25	for	for	ADP
ejpam-6327	147	26	all	all	DET
ejpam-6327	147	27	x	x	NOUN
ejpam-6327	147	28	,	,	PUNCT
ejpam-6327	147	29	y	y	PROPN
ejpam-6327	147	30	,	,	PUNCT
ejpam-6327	147	31	z	z	PROPN
ejpam-6327	147	32	∈	∈	PROPN
ejpam-6327	147	33	x.	x.	NOUN
ejpam-6327	148	1	a	a	DET
ejpam-6327	148	2	fuzzy	fuzzy	ADJ
ejpam-6327	148	3	set	set	VERB
ejpam-6327	148	4	µ	µ	NOUN
ejpam-6327	148	5	on	on	ADP
ejpam-6327	148	6	a	a	DET
ejpam-6327	148	7	ks	ks	NOUN
ejpam-6327	148	8	-	-	PUNCT
ejpam-6327	148	9	semigroup	semigroup	NOUN
ejpam-6327	148	10	x	x	PUNCT
ejpam-6327	148	11	is	be	AUX
ejpam-6327	148	12	called	call	VERB
ejpam-6327	148	13	a	a	DET
ejpam-6327	148	14	fuzzy	fuzzy	ADJ
ejpam-6327	148	15	commutative	commutative	ADJ
ejpam-6327	148	16	ks	ks	NOUN
ejpam-6327	148	17	-	-	PUNCT
ejpam-6327	148	18	ideal	ideal	NOUN
ejpam-6327	148	19	of	of	ADP
ejpam-6327	148	20	x	x	PRON
ejpam-6327	148	21	if	if	SCONJ
ejpam-6327	148	22	it	it	PRON
ejpam-6327	148	23	is	be	AUX
ejpam-6327	148	24	both	both	CCONJ
ejpam-6327	148	25	a	a	DET
ejpam-6327	148	26	left	left	NOUN
ejpam-6327	148	27	and	and	CCONJ
ejpam-6327	148	28	a	a	DET
ejpam-6327	148	29	right	right	ADJ
ejpam-6327	148	30	fuzzy	fuzzy	ADJ
ejpam-6327	148	31	commutative	commutative	ADJ
ejpam-6327	148	32	ks	ks	NOUN
ejpam-6327	148	33	-	-	PUNCT
ejpam-6327	148	34	ideal	ideal	NOUN
ejpam-6327	148	35	of	of	ADP
ejpam-6327	148	36	x.	x.	PROPN
ejpam-6327	148	37	example	example	NOUN
ejpam-6327	149	1	2	2	X
ejpam-6327	149	2	.	.	PUNCT
ejpam-6327	149	3	let	let	VERB
ejpam-6327	149	4	x	x	PUNCT
ejpam-6327	149	5	=	=	PUNCT
ejpam-6327	149	6	{	{	PUNCT
ejpam-6327	149	7	0	0	NUM
ejpam-6327	149	8	,	,	PUNCT
ejpam-6327	149	9	a	a	DET
ejpam-6327	149	10	,	,	PUNCT
ejpam-6327	149	11	b	b	NOUN
ejpam-6327	149	12	,	,	PUNCT
ejpam-6327	149	13	c	c	NOUN
ejpam-6327	149	14	}	}	PUNCT
ejpam-6327	149	15	.	.	PUNCT
ejpam-6327	150	1	define	define	VERB
ejpam-6327	150	2	the	the	DET
ejpam-6327	150	3	operations	operation	NOUN
ejpam-6327	150	4	∗	∗	NOUN
ejpam-6327	150	5	and	and	CCONJ
ejpam-6327	150	6	·	·	PUNCT
ejpam-6327	150	7	by	by	ADP
ejpam-6327	150	8	the	the	DET
ejpam-6327	150	9	following	follow	VERB
ejpam-6327	150	10	tables	table	NOUN
ejpam-6327	150	11	.	.	PUNCT
ejpam-6327	151	1	∗	∗	NOUN
ejpam-6327	151	2	0	0	NUM
ejpam-6327	152	1	a	a	DET
ejpam-6327	152	2	b	b	NOUN
ejpam-6327	152	3	c	c	NOUN
ejpam-6327	152	4	0	0	NUM
ejpam-6327	152	5	0	0	NUM
ejpam-6327	152	6	0	0	NUM
ejpam-6327	152	7	0	0	NUM
ejpam-6327	152	8	0	0	NUM
ejpam-6327	152	9	a	a	DET
ejpam-6327	152	10	a	a	DET
ejpam-6327	152	11	0	0	NUM
ejpam-6327	152	12	0	0	NUM
ejpam-6327	152	13	a	a	DET
ejpam-6327	152	14	b	b	PROPN
ejpam-6327	152	15	b	b	PROPN
ejpam-6327	152	16	a	a	PRON
ejpam-6327	152	17	0	0	NUM
ejpam-6327	152	18	b	b	NOUN
ejpam-6327	152	19	c	c	NOUN
ejpam-6327	152	20	c	c	NOUN
ejpam-6327	152	21	c	c	NOUN
ejpam-6327	152	22	c	c	NOUN
ejpam-6327	152	23	0	0	PUNCT
ejpam-6327	152	24	·	·	SYM
ejpam-6327	152	25	0	0	PUNCT
ejpam-6327	153	1	a	a	DET
ejpam-6327	153	2	b	b	X
ejpam-6327	153	3	c	c	NOUN
ejpam-6327	153	4	0	0	NUM
ejpam-6327	153	5	0	0	NUM
ejpam-6327	153	6	0	0	NUM
ejpam-6327	153	7	0	0	NUM
ejpam-6327	153	8	0	0	NUM
ejpam-6327	153	9	a	a	DET
ejpam-6327	153	10	0	0	NUM
ejpam-6327	153	11	0	0	NUM
ejpam-6327	153	12	0	0	NUM
ejpam-6327	153	13	0	0	NUM
ejpam-6327	153	14	b	b	X
ejpam-6327	153	15	0	0	NUM
ejpam-6327	153	16	0	0	NUM
ejpam-6327	153	17	0	0	NUM
ejpam-6327	153	18	0	0	NUM
ejpam-6327	154	1	c	c	NOUN
ejpam-6327	154	2	0	0	NUM
ejpam-6327	155	1	a	a	DET
ejpam-6327	155	2	b	b	NOUN
ejpam-6327	155	3	c	c	NOUN
ejpam-6327	155	4	by	by	ADP
ejpam-6327	155	5	routine	routine	ADJ
ejpam-6327	155	6	calculations	calculation	NOUN
ejpam-6327	155	7	,	,	PUNCT
ejpam-6327	155	8	we	we	PRON
ejpam-6327	155	9	can	can	AUX
ejpam-6327	155	10	see	see	VERB
ejpam-6327	155	11	that	that	PRON
ejpam-6327	155	12	x	x	PRON
ejpam-6327	155	13	is	be	AUX
ejpam-6327	155	14	a	a	DET
ejpam-6327	155	15	ks	ks	NOUN
ejpam-6327	155	16	-	-	PUNCT
ejpam-6327	155	17	semigroup	semigroup	NOUN
ejpam-6327	155	18	.	.	PUNCT
ejpam-6327	156	1	let	let	VERB
ejpam-6327	156	2	t0	t0	NOUN
ejpam-6327	156	3	,	,	PUNCT
ejpam-6327	156	4	t1	t1	PROPN
ejpam-6327	156	5	,	,	PUNCT
ejpam-6327	156	6	t2	t2	PROPN
ejpam-6327	156	7	∈	∈	PROPN
ejpam-6327	157	1	[	[	X
ejpam-6327	157	2	0	0	NUM
ejpam-6327	157	3	,	,	PUNCT
ejpam-6327	157	4	1	1	NUM
ejpam-6327	157	5	]	]	PUNCT
ejpam-6327	158	1	such	such	ADJ
ejpam-6327	158	2	that	that	SCONJ
ejpam-6327	158	3	t0	t0	PROPN
ejpam-6327	158	4	>	>	X
ejpam-6327	158	5	t1	t1	PROPN
ejpam-6327	158	6	>	>	X
ejpam-6327	158	7	t2	t2	PROPN
ejpam-6327	158	8	.	.	PUNCT
ejpam-6327	159	1	define	define	VERB
ejpam-6327	159	2	a	a	DET
ejpam-6327	159	3	fuzzy	fuzzy	ADJ
ejpam-6327	159	4	set	set	VERB
ejpam-6327	159	5	µ	µ	NOUN
ejpam-6327	159	6	on	on	ADP
ejpam-6327	159	7	x	x	PUNCT
ejpam-6327	159	8	by	by	ADP
ejpam-6327	159	9	µ(0	µ(0	NOUN
ejpam-6327	159	10	)	)	PUNCT
ejpam-6327	159	11	=	=	SYM
ejpam-6327	159	12	t0	t0	PROPN
ejpam-6327	159	13	,	,	PUNCT
ejpam-6327	159	14	µ(a	µ(a	PROPN
ejpam-6327	159	15	)	)	PUNCT
ejpam-6327	160	1	=	=	SYM
ejpam-6327	160	2	t1	t1	NOUN
ejpam-6327	160	3	and	and	CCONJ
ejpam-6327	160	4	µ(b	µ(b	PROPN
ejpam-6327	160	5	)	)	PUNCT
ejpam-6327	160	6	=	=	SYM
ejpam-6327	160	7	µ(c	µ(c	PROPN
ejpam-6327	160	8	)	)	PUNCT
ejpam-6327	160	9	=	=	SYM
ejpam-6327	160	10	t2	t2	NOUN
ejpam-6327	160	11	.	.	PUNCT
ejpam-6327	161	1	then	then	ADV
ejpam-6327	161	2	µ	µ	X
ejpam-6327	161	3	is	be	AUX
ejpam-6327	161	4	a	a	DET
ejpam-6327	161	5	fuzzy	fuzzy	ADJ
ejpam-6327	161	6	commutative	commutative	ADJ
ejpam-6327	161	7	ks	ks	NOUN
ejpam-6327	161	8	-	-	PUNCT
ejpam-6327	161	9	ideal	ideal	NOUN
ejpam-6327	161	10	of	of	ADP
ejpam-6327	161	11	x.	x.	NOUN
ejpam-6327	161	12	the	the	DET
ejpam-6327	161	13	following	follow	VERB
ejpam-6327	161	14	theorem	theorem	NOUN
ejpam-6327	161	15	tells	tell	VERB
ejpam-6327	161	16	us	we	PRON
ejpam-6327	161	17	that	that	SCONJ
ejpam-6327	161	18	every	every	DET
ejpam-6327	161	19	fuzzy	fuzzy	ADJ
ejpam-6327	161	20	commutative	commutative	ADJ
ejpam-6327	161	21	ks	ks	NOUN
ejpam-6327	161	22	-	-	PUNCT
ejpam-6327	161	23	ideal	ideal	NOUN
ejpam-6327	161	24	is	be	AUX
ejpam-6327	161	25	a	a	DET
ejpam-6327	161	26	fuzzy	fuzzy	ADJ
ejpam-6327	161	27	ksideal	ksideal	NOUN
ejpam-6327	161	28	.	.	PUNCT
ejpam-6327	162	1	theorem	theorem	ADJ
ejpam-6327	162	2	4	4	NUM
ejpam-6327	162	3	.	.	PUNCT
ejpam-6327	163	1	let	let	VERB
ejpam-6327	163	2	x	x	PRON
ejpam-6327	163	3	be	be	AUX
ejpam-6327	163	4	a	a	DET
ejpam-6327	163	5	ks	ks	NOUN
ejpam-6327	163	6	-	-	PUNCT
ejpam-6327	163	7	semigroup	semigroup	NOUN
ejpam-6327	163	8	.	.	PUNCT
ejpam-6327	164	1	then	then	ADV
ejpam-6327	164	2	any	any	DET
ejpam-6327	164	3	fuzzy	fuzzy	ADJ
ejpam-6327	164	4	commutative	commutative	ADJ
ejpam-6327	164	5	ks	ks	NOUN
ejpam-6327	164	6	-	-	PUNCT
ejpam-6327	164	7	ideal	ideal	NOUN
ejpam-6327	164	8	of	of	ADP
ejpam-6327	164	9	x	x	PUNCT
ejpam-6327	164	10	is	be	AUX
ejpam-6327	164	11	a	a	DET
ejpam-6327	164	12	fuzzy	fuzzy	ADJ
ejpam-6327	164	13	ks	ks	NOUN
ejpam-6327	164	14	-	-	NOUN
ejpam-6327	164	15	ideal	ideal	NOUN
ejpam-6327	164	16	of	of	ADP
ejpam-6327	164	17	x.	x.	NOUN
ejpam-6327	164	18	proof	proof	NOUN
ejpam-6327	164	19	.	.	PUNCT
ejpam-6327	165	1	let	let	VERB
ejpam-6327	165	2	µ	µ	X
ejpam-6327	165	3	be	be	AUX
ejpam-6327	165	4	a	a	DET
ejpam-6327	165	5	fuzzy	fuzzy	ADJ
ejpam-6327	165	6	commutative	commutative	ADJ
ejpam-6327	165	7	ks	ks	NOUN
ejpam-6327	165	8	-	-	PUNCT
ejpam-6327	165	9	ideal	ideal	NOUN
ejpam-6327	165	10	of	of	ADP
ejpam-6327	165	11	x.	x.	NOUN
ejpam-6327	165	12	then	then	ADV
ejpam-6327	165	13	by	by	ADP
ejpam-6327	165	14	remark	remark	NOUN
ejpam-6327	165	15	1(i	1(i	NUM
ejpam-6327	165	16	)	)	PUNCT
ejpam-6327	165	17	and	and	CCONJ
ejpam-6327	165	18	definition	definition	NOUN
ejpam-6327	165	19	1	1	NUM
ejpam-6327	165	20	,	,	PUNCT
ejpam-6327	165	21	for	for	ADP
ejpam-6327	165	22	any	any	DET
ejpam-6327	165	23	x	x	NOUN
ejpam-6327	165	24	,	,	PUNCT
ejpam-6327	165	25	y	y	PROPN
ejpam-6327	165	26	∈	∈	PROPN
ejpam-6327	165	27	x	x	NOUN
ejpam-6327	165	28	,	,	PUNCT
ejpam-6327	165	29	µ(x	µ(x	ADJ
ejpam-6327	165	30	)	)	PUNCT
ejpam-6327	165	31	=	=	NOUN
ejpam-6327	165	32	µ(x	µ(x	ADJ
ejpam-6327	165	33	∗	∗	NOUN
ejpam-6327	165	34	0	0	NUM
ejpam-6327	165	35	)	)	PUNCT
ejpam-6327	166	1	=	=	NOUN
ejpam-6327	166	2	µ(x	µ(x	ADJ
ejpam-6327	166	3	∗	∗	NOUN
ejpam-6327	166	4	(	(	PUNCT
ejpam-6327	166	5	0	0	NUM
ejpam-6327	166	6	∗	∗	NOUN
ejpam-6327	166	7	0	0	NUM
ejpam-6327	166	8	)	)	PUNCT
ejpam-6327	166	9	)	)	PUNCT
ejpam-6327	167	1	=	=	SYM
ejpam-6327	167	2	µ(x	µ(x	ADJ
ejpam-6327	167	3	∗	∗	NOUN
ejpam-6327	167	4	(	(	PUNCT
ejpam-6327	167	5	0	0	NUM
ejpam-6327	167	6	∗	∗	NOUN
ejpam-6327	167	7	(	(	PUNCT
ejpam-6327	167	8	0	0	NUM
ejpam-6327	167	9	∗	∗	NOUN
ejpam-6327	167	10	x	x	NOUN
ejpam-6327	167	11	)	)	PUNCT
ejpam-6327	167	12	)	)	PUNCT
ejpam-6327	167	13	)	)	PUNCT
ejpam-6327	167	14	≥	≥	X
ejpam-6327	167	15	min{µ((x	min{µ((x	NOUN
ejpam-6327	167	16	∗	∗	NOUN
ejpam-6327	167	17	0	0	NUM
ejpam-6327	167	18	)	)	PUNCT
ejpam-6327	167	19	∗	∗	NOUN
ejpam-6327	167	20	y	y	PROPN
ejpam-6327	167	21	)	)	PUNCT
ejpam-6327	167	22	,	,	PUNCT
ejpam-6327	167	23	µ(y	µ(y	PROPN
ejpam-6327	167	24	)	)	PUNCT
ejpam-6327	167	25	}	}	PUNCT
ejpam-6327	168	1	=	=	SYM
ejpam-6327	168	2	min{µ(x	min{µ(x	PROPN
ejpam-6327	168	3	∗	∗	X
ejpam-6327	168	4	y	y	NOUN
ejpam-6327	168	5	)	)	PUNCT
ejpam-6327	168	6	,	,	PUNCT
ejpam-6327	168	7	µ(y	µ(y	PROPN
ejpam-6327	168	8	)	)	PUNCT
ejpam-6327	168	9	}	}	PUNCT
ejpam-6327	168	10	.	.	PUNCT
ejpam-6327	169	1	therefore	therefore	ADV
ejpam-6327	169	2	,	,	PUNCT
ejpam-6327	169	3	µ	µ	X
ejpam-6327	169	4	is	be	AUX
ejpam-6327	169	5	a	a	DET
ejpam-6327	169	6	fuzzy	fuzzy	ADJ
ejpam-6327	169	7	ks	ks	NOUN
ejpam-6327	169	8	-	-	NOUN
ejpam-6327	169	9	ideal	ideal	NOUN
ejpam-6327	169	10	of	of	ADP
ejpam-6327	169	11	x.	x.	NOUN
ejpam-6327	169	12	a	a	DET
ejpam-6327	169	13	fuzzy	fuzzy	ADJ
ejpam-6327	169	14	ks	ks	NOUN
ejpam-6327	169	15	-	-	NOUN
ejpam-6327	169	16	ideal	ideal	NOUN
ejpam-6327	169	17	of	of	ADP
ejpam-6327	169	18	a	a	DET
ejpam-6327	169	19	ks	ks	NOUN
ejpam-6327	169	20	-	-	PUNCT
ejpam-6327	169	21	semigroup	semigroup	NOUN
ejpam-6327	169	22	x	x	NOUN
ejpam-6327	169	23	may	may	AUX
ejpam-6327	169	24	not	not	PART
ejpam-6327	169	25	be	be	AUX
ejpam-6327	169	26	a	a	DET
ejpam-6327	169	27	fuzzy	fuzzy	ADJ
ejpam-6327	169	28	commutative	commutative	ADJ
ejpam-6327	169	29	ks	ks	NOUN
ejpam-6327	169	30	-	-	PUNCT
ejpam-6327	169	31	ideal	ideal	NOUN
ejpam-6327	169	32	of	of	ADP
ejpam-6327	169	33	x	x	PART
ejpam-6327	169	34	as	as	SCONJ
ejpam-6327	169	35	shown	show	VERB
ejpam-6327	169	36	in	in	ADP
ejpam-6327	169	37	the	the	DET
ejpam-6327	169	38	following	follow	VERB
ejpam-6327	169	39	example	example	NOUN
ejpam-6327	169	40	.	.	PUNCT
ejpam-6327	170	1	example	example	NOUN
ejpam-6327	171	1	3	3	X
ejpam-6327	171	2	.	.	PUNCT
ejpam-6327	171	3	let	let	VERB
ejpam-6327	171	4	x	x	PUNCT
ejpam-6327	171	5	=	=	PUNCT
ejpam-6327	171	6	{	{	PUNCT
ejpam-6327	171	7	0	0	NUM
ejpam-6327	171	8	,	,	PUNCT
ejpam-6327	171	9	1	1	NUM
ejpam-6327	171	10	,	,	PUNCT
ejpam-6327	171	11	2	2	NUM
ejpam-6327	171	12	,	,	PUNCT
ejpam-6327	171	13	3	3	NUM
ejpam-6327	171	14	,	,	PUNCT
ejpam-6327	171	15	4	4	NUM
ejpam-6327	171	16	}	}	PUNCT
ejpam-6327	171	17	.	.	PUNCT
ejpam-6327	172	1	define	define	VERB
ejpam-6327	172	2	the	the	DET
ejpam-6327	172	3	operations	operation	NOUN
ejpam-6327	172	4	∗	∗	NOUN
ejpam-6327	172	5	and	and	CCONJ
ejpam-6327	172	6	·	·	PUNCT
ejpam-6327	172	7	by	by	ADP
ejpam-6327	172	8	the	the	DET
ejpam-6327	172	9	following	follow	VERB
ejpam-6327	172	10	tables	table	NOUN
ejpam-6327	172	11	.	.	PUNCT
ejpam-6327	173	1	∗	∗	NOUN
ejpam-6327	173	2	0	0	NUM
ejpam-6327	174	1	1	1	NUM
ejpam-6327	174	2	2	2	NUM
ejpam-6327	174	3	3	3	NUM
ejpam-6327	174	4	4	4	NUM
ejpam-6327	174	5	0	0	NUM
ejpam-6327	174	6	0	0	NUM
ejpam-6327	174	7	0	0	NUM
ejpam-6327	174	8	0	0	NUM
ejpam-6327	174	9	0	0	NUM
ejpam-6327	174	10	0	0	NUM
ejpam-6327	174	11	1	1	NUM
ejpam-6327	174	12	1	1	NUM
ejpam-6327	174	13	0	0	NUM
ejpam-6327	174	14	1	1	NUM
ejpam-6327	174	15	0	0	NUM
ejpam-6327	174	16	0	0	NUM
ejpam-6327	174	17	2	2	NUM
ejpam-6327	174	18	2	2	NUM
ejpam-6327	174	19	2	2	NUM
ejpam-6327	174	20	0	0	NUM
ejpam-6327	174	21	0	0	NUM
ejpam-6327	174	22	0	0	NUM
ejpam-6327	174	23	3	3	NUM
ejpam-6327	174	24	3	3	NUM
ejpam-6327	174	25	3	3	NUM
ejpam-6327	174	26	3	3	NUM
ejpam-6327	174	27	0	0	NUM
ejpam-6327	174	28	0	0	NUM
ejpam-6327	174	29	4	4	NUM
ejpam-6327	174	30	4	4	NUM
ejpam-6327	174	31	4	4	NUM
ejpam-6327	174	32	4	4	NUM
ejpam-6327	174	33	3	3	NUM
ejpam-6327	174	34	0	0	NUM
ejpam-6327	174	35	·	·	SYM
ejpam-6327	174	36	0	0	NUM
ejpam-6327	174	37	1	1	NUM
ejpam-6327	174	38	2	2	NUM
ejpam-6327	174	39	3	3	NUM
ejpam-6327	174	40	4	4	NUM
ejpam-6327	174	41	0	0	NUM
ejpam-6327	174	42	0	0	NUM
ejpam-6327	174	43	0	0	NUM
ejpam-6327	174	44	0	0	NUM
ejpam-6327	174	45	0	0	NUM
ejpam-6327	174	46	0	0	NUM
ejpam-6327	174	47	1	1	NUM
ejpam-6327	174	48	0	0	NUM
ejpam-6327	174	49	0	0	NUM
ejpam-6327	174	50	0	0	NUM
ejpam-6327	174	51	0	0	NUM
ejpam-6327	174	52	0	0	NUM
ejpam-6327	174	53	2	2	NUM
ejpam-6327	174	54	0	0	NUM
ejpam-6327	174	55	0	0	NUM
ejpam-6327	174	56	0	0	NUM
ejpam-6327	174	57	0	0	NUM
ejpam-6327	174	58	0	0	NUM
ejpam-6327	174	59	3	3	NUM
ejpam-6327	174	60	0	0	NUM
ejpam-6327	174	61	0	0	NUM
ejpam-6327	174	62	0	0	NUM
ejpam-6327	174	63	0	0	NUM
ejpam-6327	174	64	3	3	NUM
ejpam-6327	174	65	4	4	NUM
ejpam-6327	174	66	0	0	NUM
ejpam-6327	174	67	1	1	NUM
ejpam-6327	174	68	2	2	NUM
ejpam-6327	174	69	3	3	NUM
ejpam-6327	174	70	4	4	NUM
ejpam-6327	174	71	h.	h.	NOUN
ejpam-6327	174	72	sarapuddin	sarapuddin	PROPN
ejpam-6327	174	73	,	,	PUNCT
ejpam-6327	174	74	j.	j.	PROPN
ejpam-6327	174	75	vilela	vilela	PROPN
ejpam-6327	174	76	/	/	SYM
ejpam-6327	174	77	eur	eur	PROPN
ejpam-6327	174	78	.	.	PUNCT
ejpam-6327	175	1	j.	j.	PROPN
ejpam-6327	175	2	pure	pure	PROPN
ejpam-6327	175	3	appl	appl	PROPN
ejpam-6327	175	4	.	.	PROPN
ejpam-6327	175	5	math	math	PROPN
ejpam-6327	175	6	,	,	PUNCT
ejpam-6327	175	7	18	18	NUM
ejpam-6327	175	8	(	(	PUNCT
ejpam-6327	175	9	3	3	NUM
ejpam-6327	175	10	)	)	PUNCT
ejpam-6327	175	11	(	(	PUNCT
ejpam-6327	175	12	2025	2025	NUM
ejpam-6327	175	13	)	)	PUNCT
ejpam-6327	175	14	,	,	PUNCT
ejpam-6327	175	15	6327	6327	NUM
ejpam-6327	175	16	7	7	NUM
ejpam-6327	175	17	of	of	ADP
ejpam-6327	175	18	23	23	NUM
ejpam-6327	175	19	then	then	ADV
ejpam-6327	175	20	x	x	X
ejpam-6327	175	21	is	be	AUX
ejpam-6327	175	22	a	a	DET
ejpam-6327	175	23	ks	ks	NOUN
ejpam-6327	175	24	-	-	PUNCT
ejpam-6327	175	25	semigroup	semigroup	NOUN
ejpam-6327	175	26	.	.	PUNCT
ejpam-6327	176	1	let	let	VERB
ejpam-6327	176	2	t0	t0	NOUN
ejpam-6327	176	3	,	,	PUNCT
ejpam-6327	176	4	t1	t1	PROPN
ejpam-6327	176	5	,	,	PUNCT
ejpam-6327	176	6	t2	t2	PROPN
ejpam-6327	176	7	∈	∈	PROPN
ejpam-6327	177	1	[	[	X
ejpam-6327	177	2	0	0	NUM
ejpam-6327	177	3	,	,	PUNCT
ejpam-6327	177	4	1	1	NUM
ejpam-6327	177	5	]	]	PUNCT
ejpam-6327	178	1	such	such	ADJ
ejpam-6327	178	2	that	that	SCONJ
ejpam-6327	178	3	t0	t0	PROPN
ejpam-6327	178	4	>	>	X
ejpam-6327	178	5	t1	t1	PROPN
ejpam-6327	178	6	>	>	X
ejpam-6327	178	7	t2	t2	PROPN
ejpam-6327	178	8	.	.	PUNCT
ejpam-6327	179	1	define	define	VERB
ejpam-6327	179	2	a	a	DET
ejpam-6327	179	3	fuzzy	fuzzy	ADJ
ejpam-6327	179	4	set	set	VERB
ejpam-6327	179	5	µ	µ	NOUN
ejpam-6327	179	6	on	on	ADP
ejpam-6327	179	7	x	x	PUNCT
ejpam-6327	179	8	by	by	ADP
ejpam-6327	179	9	µ(0	µ(0	NOUN
ejpam-6327	179	10	)	)	PUNCT
ejpam-6327	179	11	=	=	SYM
ejpam-6327	179	12	t0	t0	PROPN
ejpam-6327	179	13	,	,	PUNCT
ejpam-6327	179	14	µ(1	µ(1	PROPN
ejpam-6327	179	15	)	)	PUNCT
ejpam-6327	180	1	=	=	SYM
ejpam-6327	180	2	t1	t1	NOUN
ejpam-6327	180	3	and	and	CCONJ
ejpam-6327	180	4	µ(2	µ(2	PROPN
ejpam-6327	180	5	)	)	PUNCT
ejpam-6327	180	6	=	=	SYM
ejpam-6327	180	7	µ(3	µ(3	PROPN
ejpam-6327	180	8	)	)	PUNCT
ejpam-6327	180	9	=	=	SYM
ejpam-6327	180	10	µ(4	µ(4	PROPN
ejpam-6327	180	11	)	)	PUNCT
ejpam-6327	180	12	=	=	SYM
ejpam-6327	180	13	t2	t2	NOUN
ejpam-6327	180	14	.	.	PUNCT
ejpam-6327	181	1	by	by	ADP
ejpam-6327	181	2	routine	routine	ADJ
ejpam-6327	181	3	calculations	calculation	NOUN
ejpam-6327	181	4	,	,	PUNCT
ejpam-6327	181	5	we	we	PRON
ejpam-6327	181	6	can	can	AUX
ejpam-6327	181	7	see	see	VERB
ejpam-6327	181	8	that	that	SCONJ
ejpam-6327	181	9	µ	µ	NOUN
ejpam-6327	181	10	is	be	AUX
ejpam-6327	181	11	a	a	DET
ejpam-6327	181	12	fuzzy	fuzzy	ADJ
ejpam-6327	181	13	ks	ks	NOUN
ejpam-6327	181	14	-	-	NOUN
ejpam-6327	181	15	ideal	ideal	NOUN
ejpam-6327	181	16	of	of	ADP
ejpam-6327	181	17	x.	x.	NOUN
ejpam-6327	181	18	however	however	ADV
ejpam-6327	181	19	,	,	PUNCT
ejpam-6327	181	20	µ	µ	X
ejpam-6327	181	21	is	be	AUX
ejpam-6327	181	22	not	not	PART
ejpam-6327	181	23	a	a	DET
ejpam-6327	181	24	fuzzy	fuzzy	ADJ
ejpam-6327	181	25	commutative	commutative	ADJ
ejpam-6327	181	26	ks	ks	NOUN
ejpam-6327	181	27	-	-	PUNCT
ejpam-6327	181	28	ideal	ideal	NOUN
ejpam-6327	181	29	of	of	ADP
ejpam-6327	181	30	x	x	PRON
ejpam-6327	181	31	since	since	SCONJ
ejpam-6327	181	32	(	(	PUNCT
ejpam-6327	181	33	f5	f5	NOUN
ejpam-6327	181	34	)	)	PUNCT
ejpam-6327	181	35	is	be	AUX
ejpam-6327	181	36	not	not	PART
ejpam-6327	181	37	satisfied	satisfied	ADJ
ejpam-6327	181	38	:	:	PUNCT
ejpam-6327	181	39	µ(2	µ(2	PROPN
ejpam-6327	181	40	∗	∗	NOUN
ejpam-6327	181	41	(	(	PUNCT
ejpam-6327	181	42	3	3	NUM
ejpam-6327	181	43	∗	∗	NOUN
ejpam-6327	181	44	(	(	PUNCT
ejpam-6327	181	45	3	3	NUM
ejpam-6327	181	46	∗	∗	NOUN
ejpam-6327	181	47	2	2	NUM
ejpam-6327	181	48	)	)	PUNCT
ejpam-6327	181	49	)	)	PUNCT
ejpam-6327	181	50	)	)	PUNCT
ejpam-6327	182	1	=	=	PUNCT
ejpam-6327	182	2	µ(2	µ(2	PROPN
ejpam-6327	182	3	)	)	PUNCT
ejpam-6327	182	4	=	=	SYM
ejpam-6327	182	5	t2	t2	PROPN
ejpam-6327	182	6	<	<	X
ejpam-6327	182	7	t0	t0	PROPN
ejpam-6327	182	8	=	=	PUNCT
ejpam-6327	182	9	µ(0	µ(0	NOUN
ejpam-6327	182	10	)	)	PUNCT
ejpam-6327	182	11	=	=	SYM
ejpam-6327	182	12	min{µ((2	min{µ((2	PROPN
ejpam-6327	182	13	∗	∗	NOUN
ejpam-6327	182	14	3	3	NUM
ejpam-6327	182	15	)	)	PUNCT
ejpam-6327	182	16	∗	∗	NOUN
ejpam-6327	182	17	0	0	NUM
ejpam-6327	182	18	)	)	PUNCT
ejpam-6327	182	19	,	,	PUNCT
ejpam-6327	182	20	µ(0	µ(0	NOUN
ejpam-6327	182	21	)	)	PUNCT
ejpam-6327	182	22	}	}	PUNCT
ejpam-6327	182	23	.	.	PUNCT
ejpam-6327	183	1	the	the	DET
ejpam-6327	183	2	following	follow	VERB
ejpam-6327	183	3	theorem	theorem	NOUN
ejpam-6327	183	4	gives	give	VERB
ejpam-6327	183	5	a	a	DET
ejpam-6327	183	6	criterion	criterion	NOUN
ejpam-6327	183	7	for	for	ADP
ejpam-6327	183	8	a	a	DET
ejpam-6327	183	9	fuzzy	fuzzy	ADJ
ejpam-6327	183	10	ks	ks	NOUN
ejpam-6327	183	11	-	-	PUNCT
ejpam-6327	183	12	ideal	ideal	NOUN
ejpam-6327	183	13	to	to	PART
ejpam-6327	183	14	be	be	AUX
ejpam-6327	183	15	a	a	DET
ejpam-6327	183	16	fuzzy	fuzzy	ADJ
ejpam-6327	183	17	commutative	commutative	ADJ
ejpam-6327	183	18	ks	ks	NOUN
ejpam-6327	183	19	-	-	PUNCT
ejpam-6327	183	20	ideal	ideal	NOUN
ejpam-6327	183	21	.	.	PUNCT
ejpam-6327	184	1	theorem	theorem	NOUN
ejpam-6327	184	2	5	5	NUM
ejpam-6327	184	3	.	.	PUNCT
ejpam-6327	185	1	a	a	DET
ejpam-6327	185	2	fuzzy	fuzzy	ADJ
ejpam-6327	185	3	set	set	VERB
ejpam-6327	185	4	µ	µ	NOUN
ejpam-6327	185	5	on	on	ADP
ejpam-6327	185	6	a	a	DET
ejpam-6327	185	7	ks	ks	NOUN
ejpam-6327	185	8	-	-	PUNCT
ejpam-6327	185	9	semigroup	semigroup	NOUN
ejpam-6327	185	10	x	x	PUNCT
ejpam-6327	185	11	is	be	AUX
ejpam-6327	185	12	a	a	DET
ejpam-6327	185	13	fuzzy	fuzzy	ADJ
ejpam-6327	185	14	commutative	commutative	ADJ
ejpam-6327	185	15	ks	ks	NOUN
ejpam-6327	185	16	-	-	PUNCT
ejpam-6327	185	17	ideal	ideal	NOUN
ejpam-6327	185	18	if	if	SCONJ
ejpam-6327	185	19	and	and	CCONJ
ejpam-6327	185	20	only	only	ADV
ejpam-6327	185	21	if	if	SCONJ
ejpam-6327	185	22	it	it	PRON
ejpam-6327	185	23	is	be	AUX
ejpam-6327	185	24	a	a	DET
ejpam-6327	185	25	fuzzy	fuzzy	ADJ
ejpam-6327	185	26	ks	ks	NOUN
ejpam-6327	185	27	-	-	PUNCT
ejpam-6327	185	28	ideal	ideal	ADJ
ejpam-6327	185	29	satisfying	satisfying	ADJ
ejpam-6327	185	30	µ(x	µ(x	ADJ
ejpam-6327	185	31	∗	∗	NOUN
ejpam-6327	185	32	(	(	PUNCT
ejpam-6327	185	33	y	y	PROPN
ejpam-6327	185	34	∗	∗	NOUN
ejpam-6327	185	35	(	(	PUNCT
ejpam-6327	185	36	y	y	PROPN
ejpam-6327	185	37	∗	∗	NOUN
ejpam-6327	185	38	x	x	NOUN
ejpam-6327	185	39	)	)	PUNCT
ejpam-6327	185	40	)	)	PUNCT
ejpam-6327	185	41	)	)	PUNCT
ejpam-6327	186	1	=	=	NOUN
ejpam-6327	186	2	µ(x	µ(x	VERB
ejpam-6327	186	3	∗	∗	NOUN
ejpam-6327	186	4	y	y	NOUN
ejpam-6327	186	5	)	)	PUNCT
ejpam-6327	186	6	for	for	ADP
ejpam-6327	186	7	all	all	DET
ejpam-6327	186	8	x	x	NOUN
ejpam-6327	186	9	,	,	PUNCT
ejpam-6327	186	10	y	y	PROPN
ejpam-6327	186	11	∈	∈	PROPN
ejpam-6327	186	12	x.	x.	NOUN
ejpam-6327	186	13	proof	proof	NOUN
ejpam-6327	186	14	.	.	PUNCT
ejpam-6327	187	1	let	let	VERB
ejpam-6327	187	2	x	x	PRON
ejpam-6327	187	3	be	be	AUX
ejpam-6327	187	4	a	a	DET
ejpam-6327	187	5	ks	ks	NOUN
ejpam-6327	187	6	-	-	PUNCT
ejpam-6327	187	7	semigroup	semigroup	NOUN
ejpam-6327	187	8	and	and	CCONJ
ejpam-6327	187	9	µ	µ	PRON
ejpam-6327	187	10	be	be	AUX
ejpam-6327	187	11	a	a	DET
ejpam-6327	187	12	fuzzy	fuzzy	ADJ
ejpam-6327	187	13	set	set	NOUN
ejpam-6327	187	14	on	on	ADP
ejpam-6327	187	15	x.	x.	NOUN
ejpam-6327	187	16	suppose	suppose	VERB
ejpam-6327	187	17	µ	µ	PRON
ejpam-6327	187	18	is	be	AUX
ejpam-6327	187	19	a	a	DET
ejpam-6327	187	20	fuzzy	fuzzy	ADJ
ejpam-6327	187	21	commutative	commutative	ADJ
ejpam-6327	187	22	ks	ks	NOUN
ejpam-6327	187	23	-	-	PUNCT
ejpam-6327	187	24	ideal	ideal	NOUN
ejpam-6327	187	25	of	of	ADP
ejpam-6327	187	26	x.	x.	NOUN
ejpam-6327	187	27	then	then	ADV
ejpam-6327	187	28	by	by	ADP
ejpam-6327	187	29	theorem	theorem	NOUN
ejpam-6327	187	30	4	4	NUM
ejpam-6327	187	31	,	,	PUNCT
ejpam-6327	187	32	µ	µ	PRON
ejpam-6327	187	33	is	be	AUX
ejpam-6327	187	34	a	a	DET
ejpam-6327	187	35	fuzzy	fuzzy	ADJ
ejpam-6327	187	36	ks	ks	NOUN
ejpam-6327	187	37	-	-	PUNCT
ejpam-6327	187	38	ideal	ideal	NOUN
ejpam-6327	187	39	.	.	PUNCT
ejpam-6327	188	1	by	by	ADP
ejpam-6327	188	2	(	(	PUNCT
ejpam-6327	188	3	f5	f5	NOUN
ejpam-6327	188	4	)	)	PUNCT
ejpam-6327	188	5	,	,	PUNCT
ejpam-6327	188	6	for	for	ADP
ejpam-6327	188	7	all	all	DET
ejpam-6327	188	8	x	x	NOUN
ejpam-6327	188	9	,	,	PUNCT
ejpam-6327	188	10	y	y	PROPN
ejpam-6327	188	11	∈	∈	PROPN
ejpam-6327	188	12	x	x	PROPN
ejpam-6327	188	13	,	,	PUNCT
ejpam-6327	188	14	µ(x	µ(x	ADJ
ejpam-6327	188	15	∗	∗	NOUN
ejpam-6327	188	16	(	(	PUNCT
ejpam-6327	188	17	y	y	PROPN
ejpam-6327	188	18	∗	∗	NOUN
ejpam-6327	188	19	(	(	PUNCT
ejpam-6327	188	20	y	y	PROPN
ejpam-6327	188	21	∗	∗	NOUN
ejpam-6327	188	22	x	x	NOUN
ejpam-6327	188	23	)	)	PUNCT
ejpam-6327	188	24	)	)	PUNCT
ejpam-6327	188	25	)	)	PUNCT
ejpam-6327	188	26	≥	≥	X
ejpam-6327	188	27	min{µ((x	min{µ((x	PRON
ejpam-6327	188	28	∗	∗	X
ejpam-6327	188	29	y	y	NOUN
ejpam-6327	188	30	)	)	PUNCT
ejpam-6327	188	31	∗	∗	NOUN
ejpam-6327	188	32	0	0	NUM
ejpam-6327	188	33	)	)	PUNCT
ejpam-6327	188	34	,	,	PUNCT
ejpam-6327	188	35	µ(0	µ(0	NOUN
ejpam-6327	188	36	)	)	PUNCT
ejpam-6327	188	37	}	}	PUNCT
ejpam-6327	189	1	=	=	SYM
ejpam-6327	189	2	min{µ((x	min{µ((x	PRON
ejpam-6327	189	3	∗	∗	X
ejpam-6327	189	4	y	y	NOUN
ejpam-6327	189	5	)	)	PUNCT
ejpam-6327	189	6	)	)	PUNCT
ejpam-6327	189	7	,	,	PUNCT
ejpam-6327	189	8	µ(0	µ(0	NOUN
ejpam-6327	189	9	)	)	PUNCT
ejpam-6327	189	10	}	}	PUNCT
ejpam-6327	189	11	=	=	SYM
ejpam-6327	189	12	µ(x	µ(x	ADJ
ejpam-6327	189	13	∗	∗	NOUN
ejpam-6327	189	14	y	y	NOUN
ejpam-6327	189	15	)	)	PUNCT
ejpam-6327	189	16	.	.	PUNCT
ejpam-6327	190	1	moreover	moreover	ADV
ejpam-6327	190	2	,	,	PUNCT
ejpam-6327	190	3	note	note	VERB
ejpam-6327	190	4	that	that	SCONJ
ejpam-6327	190	5	by	by	ADP
ejpam-6327	190	6	remark	remark	NOUN
ejpam-6327	190	7	1(ii	1(ii	NUM
ejpam-6327	190	8	)	)	PUNCT
ejpam-6327	190	9	,	,	PUNCT
ejpam-6327	190	10	y	y	PROPN
ejpam-6327	190	11	∗	∗	NOUN
ejpam-6327	190	12	(	(	PUNCT
ejpam-6327	190	13	y	y	PROPN
ejpam-6327	190	14	∗	∗	X
ejpam-6327	190	15	x	x	NOUN
ejpam-6327	190	16	)	)	PUNCT
ejpam-6327	190	17	≤	≤	PUNCT
ejpam-6327	191	1	y.	y.	PROPN
ejpam-6327	191	2	thus	thus	ADV
ejpam-6327	191	3	,	,	PUNCT
ejpam-6327	191	4	by	by	ADP
ejpam-6327	191	5	remark	remark	NOUN
ejpam-6327	191	6	1(iv	1(iv	NUM
ejpam-6327	191	7	)	)	PUNCT
ejpam-6327	191	8	,	,	PUNCT
ejpam-6327	191	9	x	x	X
ejpam-6327	191	10	∗	∗	NOUN
ejpam-6327	191	11	y	y	NOUN
ejpam-6327	191	12	≤	≤	NUM
ejpam-6327	191	13	x	x	PUNCT
ejpam-6327	191	14	∗	∗	NOUN
ejpam-6327	191	15	(	(	PUNCT
ejpam-6327	191	16	y	y	PROPN
ejpam-6327	191	17	∗	∗	NOUN
ejpam-6327	191	18	(	(	PUNCT
ejpam-6327	191	19	y	y	PROPN
ejpam-6327	191	20	∗	∗	NOUN
ejpam-6327	191	21	x	x	NOUN
ejpam-6327	191	22	)	)	PUNCT
ejpam-6327	191	23	)	)	PUNCT
ejpam-6327	191	24	.	.	PUNCT
ejpam-6327	192	1	hence	hence	ADV
ejpam-6327	192	2	,	,	PUNCT
ejpam-6327	192	3	by	by	ADP
ejpam-6327	192	4	lemma	lemma	PROPN
ejpam-6327	192	5	1(i	1(i	NUM
ejpam-6327	192	6	)	)	PUNCT
ejpam-6327	192	7	,	,	PUNCT
ejpam-6327	192	8	µ(x	µ(x	VERB
ejpam-6327	192	9	∗	∗	NOUN
ejpam-6327	192	10	y	y	NOUN
ejpam-6327	192	11	)	)	PUNCT
ejpam-6327	192	12	≥	≥	NOUN
ejpam-6327	192	13	µ(x	µ(x	ADJ
ejpam-6327	192	14	∗	∗	NOUN
ejpam-6327	192	15	(	(	PUNCT
ejpam-6327	192	16	y	y	PROPN
ejpam-6327	192	17	∗	∗	NOUN
ejpam-6327	192	18	(	(	PUNCT
ejpam-6327	192	19	y	y	PROPN
ejpam-6327	192	20	∗	∗	NOUN
ejpam-6327	192	21	x	x	NOUN
ejpam-6327	192	22	)	)	PUNCT
ejpam-6327	192	23	)	)	PUNCT
ejpam-6327	192	24	)	)	PUNCT
ejpam-6327	192	25	.	.	PUNCT
ejpam-6327	193	1	therefore	therefore	ADV
ejpam-6327	193	2	,	,	PUNCT
ejpam-6327	193	3	µ(x	µ(x	ADJ
ejpam-6327	193	4	∗	∗	NOUN
ejpam-6327	193	5	(	(	PUNCT
ejpam-6327	193	6	y	y	PROPN
ejpam-6327	193	7	∗	∗	NOUN
ejpam-6327	193	8	(	(	PUNCT
ejpam-6327	193	9	y	y	PROPN
ejpam-6327	193	10	∗	∗	NOUN
ejpam-6327	193	11	x	x	NOUN
ejpam-6327	193	12	)	)	PUNCT
ejpam-6327	193	13	)	)	PUNCT
ejpam-6327	193	14	)	)	PUNCT
ejpam-6327	194	1	=	=	NOUN
ejpam-6327	194	2	µ(x	µ(x	VERB
ejpam-6327	194	3	∗	∗	NOUN
ejpam-6327	194	4	y	y	NOUN
ejpam-6327	194	5	)	)	PUNCT
ejpam-6327	194	6	.	.	PUNCT
ejpam-6327	195	1	conversely	conversely	ADV
ejpam-6327	195	2	,	,	PUNCT
ejpam-6327	195	3	suppose	suppose	VERB
ejpam-6327	195	4	µ	µ	PRON
ejpam-6327	195	5	is	be	AUX
ejpam-6327	195	6	a	a	DET
ejpam-6327	195	7	fuzzy	fuzzy	ADJ
ejpam-6327	195	8	ks	ks	NOUN
ejpam-6327	195	9	-	-	NOUN
ejpam-6327	195	10	ideal	ideal	NOUN
ejpam-6327	195	11	of	of	ADP
ejpam-6327	195	12	x	x	SYM
ejpam-6327	195	13	satisfying	satisfy	VERB
ejpam-6327	195	14	µ(x	µ(x	ADJ
ejpam-6327	195	15	∗	∗	NOUN
ejpam-6327	195	16	(	(	PUNCT
ejpam-6327	195	17	y	y	PROPN
ejpam-6327	195	18	∗	∗	NOUN
ejpam-6327	195	19	(	(	PUNCT
ejpam-6327	195	20	y	y	PROPN
ejpam-6327	195	21	∗	∗	NOUN
ejpam-6327	195	22	x	x	NOUN
ejpam-6327	195	23	)	)	PUNCT
ejpam-6327	195	24	)	)	PUNCT
ejpam-6327	195	25	)	)	PUNCT
ejpam-6327	196	1	=	=	NOUN
ejpam-6327	196	2	µ(x	µ(x	VERB
ejpam-6327	196	3	∗	∗	NOUN
ejpam-6327	196	4	y	y	NOUN
ejpam-6327	196	5	)	)	PUNCT
ejpam-6327	196	6	for	for	ADP
ejpam-6327	196	7	all	all	DET
ejpam-6327	196	8	x	x	NOUN
ejpam-6327	196	9	,	,	PUNCT
ejpam-6327	196	10	y	y	PROPN
ejpam-6327	196	11	∈	∈	PROPN
ejpam-6327	196	12	x.	x.	NOUN
ejpam-6327	196	13	by	by	ADP
ejpam-6327	196	14	(	(	PUNCT
ejpam-6327	196	15	f2	f2	PROPN
ejpam-6327	196	16	)	)	PUNCT
ejpam-6327	196	17	,	,	PUNCT
ejpam-6327	196	18	µ(x	µ(x	ADJ
ejpam-6327	196	19	∗	∗	NOUN
ejpam-6327	196	20	(	(	PUNCT
ejpam-6327	196	21	y	y	PROPN
ejpam-6327	196	22	∗	∗	NOUN
ejpam-6327	196	23	(	(	PUNCT
ejpam-6327	196	24	y	y	PROPN
ejpam-6327	196	25	∗	∗	NOUN
ejpam-6327	196	26	x	x	NOUN
ejpam-6327	196	27	)	)	PUNCT
ejpam-6327	196	28	)	)	PUNCT
ejpam-6327	196	29	)	)	PUNCT
ejpam-6327	197	1	=	=	NOUN
ejpam-6327	197	2	µ(x	µ(x	VERB
ejpam-6327	197	3	∗	∗	NOUN
ejpam-6327	197	4	y	y	NOUN
ejpam-6327	197	5	)	)	PUNCT
ejpam-6327	197	6	≥	≥	NOUN
ejpam-6327	197	7	min{µ((x	min{µ((x	PRON
ejpam-6327	197	8	∗	∗	X
ejpam-6327	197	9	y	y	NOUN
ejpam-6327	197	10	)	)	PUNCT
ejpam-6327	197	11	∗	∗	NOUN
ejpam-6327	197	12	z	z	NOUN
ejpam-6327	197	13	)	)	PUNCT
ejpam-6327	197	14	,	,	PUNCT
ejpam-6327	197	15	µ(z	µ(z	PROPN
ejpam-6327	197	16	)	)	PUNCT
ejpam-6327	197	17	}	}	PUNCT
ejpam-6327	197	18	for	for	ADP
ejpam-6327	197	19	all	all	DET
ejpam-6327	197	20	x	x	NOUN
ejpam-6327	197	21	,	,	PUNCT
ejpam-6327	197	22	y	y	PROPN
ejpam-6327	197	23	,	,	PUNCT
ejpam-6327	197	24	z	z	PROPN
ejpam-6327	197	25	∈	∈	PROPN
ejpam-6327	197	26	x.	x.	NOUN
ejpam-6327	197	27	therefore	therefore	ADV
ejpam-6327	197	28	,	,	PUNCT
ejpam-6327	197	29	µ	µ	X
ejpam-6327	197	30	is	be	AUX
ejpam-6327	197	31	a	a	DET
ejpam-6327	197	32	fuzzy	fuzzy	ADJ
ejpam-6327	197	33	commutative	commutative	ADJ
ejpam-6327	197	34	ks	ks	NOUN
ejpam-6327	197	35	-	-	PUNCT
ejpam-6327	197	36	ideal	ideal	NOUN
ejpam-6327	197	37	of	of	ADP
ejpam-6327	197	38	x.	x.	NOUN
ejpam-6327	197	39	the	the	DET
ejpam-6327	197	40	following	follow	VERB
ejpam-6327	197	41	theorem	theorem	NOUN
ejpam-6327	197	42	provides	provide	VERB
ejpam-6327	197	43	a	a	DET
ejpam-6327	197	44	necessary	necessary	ADJ
ejpam-6327	197	45	condition	condition	NOUN
ejpam-6327	197	46	for	for	ADP
ejpam-6327	197	47	a	a	DET
ejpam-6327	197	48	fuzzy	fuzzy	ADJ
ejpam-6327	197	49	ks	ks	NOUN
ejpam-6327	197	50	-	-	PUNCT
ejpam-6327	197	51	ideal	ideal	NOUN
ejpam-6327	197	52	to	to	PART
ejpam-6327	197	53	be	be	AUX
ejpam-6327	197	54	a	a	DET
ejpam-6327	197	55	fuzzy	fuzzy	ADJ
ejpam-6327	197	56	commutative	commutative	ADJ
ejpam-6327	197	57	ks	ks	NOUN
ejpam-6327	197	58	-	-	PUNCT
ejpam-6327	197	59	ideal	ideal	NOUN
ejpam-6327	197	60	.	.	PUNCT
ejpam-6327	198	1	theorem	theorem	NOUN
ejpam-6327	198	2	6	6	NUM
ejpam-6327	198	3	.	.	PUNCT
ejpam-6327	199	1	let	let	VERB
ejpam-6327	199	2	x	x	PRON
ejpam-6327	199	3	be	be	AUX
ejpam-6327	199	4	a	a	DET
ejpam-6327	199	5	commutative	commutative	ADJ
ejpam-6327	199	6	ks	ks	NOUN
ejpam-6327	199	7	-	-	PUNCT
ejpam-6327	199	8	semigroup	semigroup	NOUN
ejpam-6327	199	9	.	.	PUNCT
ejpam-6327	200	1	then	then	ADV
ejpam-6327	200	2	every	every	DET
ejpam-6327	200	3	fuzzy	fuzzy	ADJ
ejpam-6327	200	4	ks	ks	NOUN
ejpam-6327	200	5	-	-	PUNCT
ejpam-6327	200	6	ideal	ideal	NOUN
ejpam-6327	200	7	is	be	AUX
ejpam-6327	200	8	a	a	DET
ejpam-6327	200	9	fuzzy	fuzzy	ADJ
ejpam-6327	200	10	commutative	commutative	ADJ
ejpam-6327	200	11	ks	ks	NOUN
ejpam-6327	200	12	-	-	PUNCT
ejpam-6327	200	13	ideal	ideal	NOUN
ejpam-6327	200	14	.	.	PUNCT
ejpam-6327	201	1	proof	proof	NOUN
ejpam-6327	201	2	.	.	PUNCT
ejpam-6327	202	1	let	let	VERB
ejpam-6327	202	2	µ	µ	X
ejpam-6327	202	3	be	be	AUX
ejpam-6327	202	4	a	a	DET
ejpam-6327	202	5	fuzzy	fuzzy	ADJ
ejpam-6327	202	6	ks	ks	NOUN
ejpam-6327	202	7	-	-	NOUN
ejpam-6327	202	8	ideal	ideal	NOUN
ejpam-6327	202	9	of	of	ADP
ejpam-6327	202	10	a	a	DET
ejpam-6327	202	11	commutative	commutative	ADJ
ejpam-6327	202	12	ks	ks	NOUN
ejpam-6327	202	13	-	-	PUNCT
ejpam-6327	202	14	semigroup	semigroup	NOUN
ejpam-6327	202	15	x.	x.	NOUN
ejpam-6327	203	1	it	it	PRON
ejpam-6327	203	2	is	be	AUX
ejpam-6327	203	3	sufficient	sufficient	ADJ
ejpam-6327	203	4	to	to	PART
ejpam-6327	203	5	show	show	VERB
ejpam-6327	203	6	that	that	SCONJ
ejpam-6327	203	7	µ	µ	ADJ
ejpam-6327	203	8	satisfies	satisfie	NOUN
ejpam-6327	203	9	(	(	PUNCT
ejpam-6327	203	10	f5	f5	NOUN
ejpam-6327	203	11	)	)	PUNCT
ejpam-6327	203	12	.	.	PUNCT
ejpam-6327	204	1	let	let	VERB
ejpam-6327	204	2	x	x	PRON
ejpam-6327	204	3	,	,	PUNCT
ejpam-6327	204	4	y	y	PROPN
ejpam-6327	204	5	,	,	PUNCT
ejpam-6327	204	6	z	z	PROPN
ejpam-6327	204	7	∈	∈	PROPN
ejpam-6327	204	8	x.	x.	NOUN
ejpam-6327	204	9	then	then	ADV
ejpam-6327	204	10	by	by	ADP
ejpam-6327	204	11	remark	remark	NOUN
ejpam-6327	204	12	1	1	NUM
ejpam-6327	204	13	and	and	CCONJ
ejpam-6327	204	14	sincex	sincex	PROPN
ejpam-6327	204	15	is	be	AUX
ejpam-6327	204	16	commutative	commutative	ADJ
ejpam-6327	204	17	,	,	PUNCT
ejpam-6327	204	18	(	(	PUNCT
ejpam-6327	204	19	(	(	PUNCT
ejpam-6327	204	20	x	x	SYM
ejpam-6327	204	21	∗	∗	NOUN
ejpam-6327	204	22	(	(	PUNCT
ejpam-6327	204	23	y	y	PROPN
ejpam-6327	204	24	∗	∗	NOUN
ejpam-6327	204	25	(	(	PUNCT
ejpam-6327	204	26	y	y	PROPN
ejpam-6327	204	27	∗	∗	NOUN
ejpam-6327	204	28	x	x	NOUN
ejpam-6327	204	29	)	)	PUNCT
ejpam-6327	204	30	)	)	PUNCT
ejpam-6327	204	31	)	)	PUNCT
ejpam-6327	205	1	∗	∗	NOUN
ejpam-6327	205	2	(	(	PUNCT
ejpam-6327	205	3	(	(	PUNCT
ejpam-6327	205	4	x	x	SYM
ejpam-6327	205	5	∗	∗	PROPN
ejpam-6327	205	6	y	y	NOUN
ejpam-6327	205	7	)	)	PUNCT
ejpam-6327	205	8	∗	∗	NOUN
ejpam-6327	205	9	z	z	NOUN
ejpam-6327	205	10	)	)	PUNCT
ejpam-6327	205	11	)	)	PUNCT
ejpam-6327	205	12	∗	∗	NOUN
ejpam-6327	205	13	z	z	NOUN
ejpam-6327	205	14	=	=	SYM
ejpam-6327	205	15	(	(	PUNCT
ejpam-6327	205	16	(	(	PUNCT
ejpam-6327	205	17	x	x	SYM
ejpam-6327	205	18	∗	∗	NOUN
ejpam-6327	205	19	(	(	PUNCT
ejpam-6327	205	20	y	y	PROPN
ejpam-6327	205	21	∗	∗	NOUN
ejpam-6327	205	22	(	(	PUNCT
ejpam-6327	205	23	y	y	PROPN
ejpam-6327	205	24	∗	∗	NOUN
ejpam-6327	205	25	x	x	NOUN
ejpam-6327	205	26	)	)	PUNCT
ejpam-6327	205	27	)	)	PUNCT
ejpam-6327	205	28	)	)	PUNCT
ejpam-6327	205	29	∗	∗	PROPN
ejpam-6327	205	30	z	z	NOUN
ejpam-6327	205	31	)	)	PUNCT
ejpam-6327	205	32	∗	∗	NOUN
ejpam-6327	205	33	(	(	PUNCT
ejpam-6327	205	34	(	(	PUNCT
ejpam-6327	205	35	x	x	SYM
ejpam-6327	205	36	∗	∗	PROPN
ejpam-6327	205	37	y	y	NOUN
ejpam-6327	205	38	)	)	PUNCT
ejpam-6327	205	39	∗	∗	NOUN
ejpam-6327	205	40	z	z	NOUN
ejpam-6327	205	41	)	)	PUNCT
ejpam-6327	205	42	≤	≤	NOUN
ejpam-6327	205	43	(	(	PUNCT
ejpam-6327	205	44	x	x	SYM
ejpam-6327	205	45	∗	∗	NOUN
ejpam-6327	205	46	(	(	PUNCT
ejpam-6327	205	47	y	y	PROPN
ejpam-6327	205	48	∗	∗	NOUN
ejpam-6327	205	49	(	(	PUNCT
ejpam-6327	205	50	y	y	PROPN
ejpam-6327	205	51	∗	∗	NOUN
ejpam-6327	205	52	x	x	NOUN
ejpam-6327	205	53	)	)	PUNCT
ejpam-6327	205	54	)	)	PUNCT
ejpam-6327	205	55	)	)	PUNCT
ejpam-6327	205	56	∗	∗	NOUN
ejpam-6327	205	57	(	(	PUNCT
ejpam-6327	205	58	x	x	X
ejpam-6327	205	59	∗	∗	PROPN
ejpam-6327	205	60	y	y	NOUN
ejpam-6327	205	61	)	)	PUNCT
ejpam-6327	205	62	=	=	PRON
ejpam-6327	206	1	(	(	PUNCT
ejpam-6327	206	2	x	x	X
ejpam-6327	206	3	∗	∗	NOUN
ejpam-6327	206	4	(	(	PUNCT
ejpam-6327	206	5	x	x	X
ejpam-6327	206	6	∗	∗	PROPN
ejpam-6327	206	7	y	y	PROPN
ejpam-6327	206	8	)	)	PUNCT
ejpam-6327	206	9	)	)	PUNCT
ejpam-6327	206	10	∗	∗	NOUN
ejpam-6327	206	11	(	(	PUNCT
ejpam-6327	206	12	y	y	PROPN
ejpam-6327	206	13	∗	∗	NOUN
ejpam-6327	206	14	(	(	PUNCT
ejpam-6327	206	15	y	y	PROPN
ejpam-6327	206	16	∗	∗	NOUN
ejpam-6327	206	17	x	x	NOUN
ejpam-6327	206	18	)	)	PUNCT
ejpam-6327	206	19	)	)	PUNCT
ejpam-6327	207	1	=	=	PUNCT
ejpam-6327	207	2	0	0	X
ejpam-6327	207	3	.	.	PUNCT
ejpam-6327	208	1	by	by	ADP
ejpam-6327	208	2	definition	definition	NOUN
ejpam-6327	208	3	2	2	NUM
ejpam-6327	208	4	,	,	PUNCT
ejpam-6327	208	5	(	(	PUNCT
ejpam-6327	208	6	x	x	SYM
ejpam-6327	208	7	∗	∗	NOUN
ejpam-6327	208	8	(	(	PUNCT
ejpam-6327	208	9	y	y	PROPN
ejpam-6327	208	10	∗	∗	NOUN
ejpam-6327	208	11	(	(	PUNCT
ejpam-6327	208	12	y	y	PROPN
ejpam-6327	208	13	∗	∗	NOUN
ejpam-6327	208	14	x	x	NOUN
ejpam-6327	208	15	)	)	PUNCT
ejpam-6327	208	16	)	)	PUNCT
ejpam-6327	208	17	)	)	PUNCT
ejpam-6327	208	18	∗	∗	NOUN
ejpam-6327	208	19	(	(	PUNCT
ejpam-6327	208	20	(	(	PUNCT
ejpam-6327	208	21	x	x	SYM
ejpam-6327	208	22	∗	∗	PROPN
ejpam-6327	208	23	y	y	NOUN
ejpam-6327	208	24	)	)	PUNCT
ejpam-6327	208	25	∗	∗	NOUN
ejpam-6327	208	26	z	z	NOUN
ejpam-6327	208	27	)	)	PUNCT
ejpam-6327	208	28	≤	≤	NOUN
ejpam-6327	208	29	z.	z.	PROPN
ejpam-6327	209	1	thus	thus	ADV
ejpam-6327	209	2	,	,	PUNCT
ejpam-6327	209	3	by	by	ADP
ejpam-6327	209	4	lemma	lemma	PROPN
ejpam-6327	209	5	1(ii	1(ii	NUM
ejpam-6327	209	6	)	)	PUNCT
ejpam-6327	209	7	,	,	PUNCT
ejpam-6327	209	8	µ(x	µ(x	ADJ
ejpam-6327	209	9	∗	∗	NOUN
ejpam-6327	209	10	(	(	PUNCT
ejpam-6327	209	11	y	y	PROPN
ejpam-6327	209	12	∗	∗	NOUN
ejpam-6327	209	13	(	(	PUNCT
ejpam-6327	209	14	y	y	PROPN
ejpam-6327	209	15	∗	∗	NOUN
ejpam-6327	209	16	x	x	NOUN
ejpam-6327	209	17	)	)	PUNCT
ejpam-6327	209	18	)	)	PUNCT
ejpam-6327	209	19	)	)	PUNCT
ejpam-6327	209	20	≥	≥	X
ejpam-6327	209	21	min{µ((x	min{µ((x	DET
ejpam-6327	209	22	∗	∗	X
ejpam-6327	209	23	y	y	NOUN
ejpam-6327	209	24	)	)	PUNCT
ejpam-6327	209	25	∗	∗	NOUN
ejpam-6327	209	26	z	z	NOUN
ejpam-6327	209	27	)	)	PUNCT
ejpam-6327	209	28	,	,	PUNCT
ejpam-6327	209	29	µ(z	µ(z	PROPN
ejpam-6327	209	30	)	)	PUNCT
ejpam-6327	209	31	}	}	PUNCT
ejpam-6327	209	32	.	.	PUNCT
ejpam-6327	210	1	h.	h.	PROPN
ejpam-6327	210	2	sarapuddin	sarapuddin	PROPN
ejpam-6327	210	3	,	,	PUNCT
ejpam-6327	210	4	j.	j.	PROPN
ejpam-6327	210	5	vilela	vilela	PROPN
ejpam-6327	210	6	/	/	SYM
ejpam-6327	210	7	eur	eur	PROPN
ejpam-6327	210	8	.	.	PUNCT
ejpam-6327	211	1	j.	j.	PROPN
ejpam-6327	211	2	pure	pure	PROPN
ejpam-6327	211	3	appl	appl	PROPN
ejpam-6327	211	4	.	.	PROPN
ejpam-6327	211	5	math	math	PROPN
ejpam-6327	211	6	,	,	PUNCT
ejpam-6327	211	7	18	18	NUM
ejpam-6327	211	8	(	(	PUNCT
ejpam-6327	211	9	3	3	NUM
ejpam-6327	211	10	)	)	PUNCT
ejpam-6327	211	11	(	(	PUNCT
ejpam-6327	211	12	2025	2025	NUM
ejpam-6327	211	13	)	)	PUNCT
ejpam-6327	211	14	,	,	PUNCT
ejpam-6327	211	15	6327	6327	NUM
ejpam-6327	211	16	8	8	NUM
ejpam-6327	211	17	of	of	ADP
ejpam-6327	211	18	23	23	NUM
ejpam-6327	211	19	hence	hence	ADV
ejpam-6327	211	20	,	,	PUNCT
ejpam-6327	211	21	(	(	PUNCT
ejpam-6327	211	22	f5	f5	NOUN
ejpam-6327	211	23	)	)	PUNCT
ejpam-6327	211	24	holds	hold	NOUN
ejpam-6327	211	25	.	.	PUNCT
ejpam-6327	212	1	therefore	therefore	ADV
ejpam-6327	212	2	,	,	PUNCT
ejpam-6327	212	3	µ	µ	X
ejpam-6327	212	4	is	be	AUX
ejpam-6327	212	5	a	a	DET
ejpam-6327	212	6	fuzzy	fuzzy	ADJ
ejpam-6327	212	7	commutative	commutative	ADJ
ejpam-6327	212	8	ks	ks	NOUN
ejpam-6327	212	9	-	-	PUNCT
ejpam-6327	212	10	ideal	ideal	NOUN
ejpam-6327	212	11	of	of	ADP
ejpam-6327	212	12	x.	x.	NOUN
ejpam-6327	212	13	definition	definition	NOUN
ejpam-6327	212	14	12	12	NUM
ejpam-6327	212	15	.	.	PUNCT
ejpam-6327	213	1	let	let	VERB
ejpam-6327	213	2	x	x	PRON
ejpam-6327	213	3	be	be	AUX
ejpam-6327	213	4	a	a	DET
ejpam-6327	213	5	ks	ks	NOUN
ejpam-6327	213	6	-	-	PUNCT
ejpam-6327	213	7	semigroup	semigroup	NOUN
ejpam-6327	213	8	.	.	PUNCT
ejpam-6327	214	1	a	a	DET
ejpam-6327	214	2	fuzzy	fuzzy	ADJ
ejpam-6327	214	3	set	set	VERB
ejpam-6327	214	4	µ	µ	NOUN
ejpam-6327	214	5	on	on	ADP
ejpam-6327	214	6	x	x	VERB
ejpam-6327	214	7	is	be	AUX
ejpam-6327	214	8	called	call	VERB
ejpam-6327	214	9	a	a	DET
ejpam-6327	214	10	left	left	ADJ
ejpam-6327	214	11	(	(	PUNCT
ejpam-6327	214	12	resp	resp	NOUN
ejpam-6327	214	13	.	.	PUNCT
ejpam-6327	215	1	right	right	ADJ
ejpam-6327	215	2	)	)	PUNCT
ejpam-6327	215	3	fuzzy	fuzzy	ADJ
ejpam-6327	215	4	implicative	implicative	ADJ
ejpam-6327	215	5	ks	ks	NOUN
ejpam-6327	215	6	-	-	PUNCT
ejpam-6327	215	7	ideal	ideal	NOUN
ejpam-6327	215	8	of	of	ADP
ejpam-6327	215	9	x	x	PRON
ejpam-6327	215	10	if	if	SCONJ
ejpam-6327	215	11	it	it	PRON
ejpam-6327	215	12	satisfies	satisfy	VERB
ejpam-6327	215	13	(	(	PUNCT
ejpam-6327	215	14	f1	f1	NOUN
ejpam-6327	215	15	)	)	PUNCT
ejpam-6327	215	16	,	,	PUNCT
ejpam-6327	215	17	(	(	PUNCT
ejpam-6327	215	18	f3	f3	ADJ
ejpam-6327	215	19	)	)	PUNCT
ejpam-6327	215	20	and	and	CCONJ
ejpam-6327	215	21	(	(	PUNCT
ejpam-6327	215	22	f6	f6	PROPN
ejpam-6327	215	23	)	)	PUNCT
ejpam-6327	215	24	:	:	PUNCT
ejpam-6327	216	1	µ(x	µ(x	X
ejpam-6327	216	2	)	)	PUNCT
ejpam-6327	216	3	≥	≥	NOUN
ejpam-6327	216	4	min{µ((x	min{µ((x	NOUN
ejpam-6327	216	5	∗	∗	NOUN
ejpam-6327	216	6	(	(	PUNCT
ejpam-6327	216	7	y	y	PROPN
ejpam-6327	216	8	∗	∗	NOUN
ejpam-6327	216	9	x	x	NOUN
ejpam-6327	216	10	)	)	PUNCT
ejpam-6327	216	11	)	)	PUNCT
ejpam-6327	216	12	∗	∗	PROPN
ejpam-6327	216	13	z	z	NOUN
ejpam-6327	216	14	)	)	PUNCT
ejpam-6327	216	15	,	,	PUNCT
ejpam-6327	216	16	µ(z	µ(z	PROPN
ejpam-6327	216	17	)	)	PUNCT
ejpam-6327	216	18	}	}	PUNCT
ejpam-6327	216	19	for	for	ADP
ejpam-6327	216	20	all	all	DET
ejpam-6327	216	21	x	x	NOUN
ejpam-6327	216	22	,	,	PUNCT
ejpam-6327	216	23	y	y	PROPN
ejpam-6327	216	24	,	,	PUNCT
ejpam-6327	216	25	z	z	PROPN
ejpam-6327	216	26	∈	∈	PROPN
ejpam-6327	216	27	x.	x.	NOUN
ejpam-6327	216	28	a	a	DET
ejpam-6327	216	29	fuzzy	fuzzy	ADJ
ejpam-6327	216	30	set	set	VERB
ejpam-6327	216	31	µ	µ	NOUN
ejpam-6327	216	32	on	on	ADP
ejpam-6327	216	33	a	a	DET
ejpam-6327	216	34	ks	ks	NOUN
ejpam-6327	216	35	-	-	PUNCT
ejpam-6327	216	36	semigroup	semigroup	NOUN
ejpam-6327	216	37	x	x	PUNCT
ejpam-6327	216	38	is	be	AUX
ejpam-6327	216	39	called	call	VERB
ejpam-6327	216	40	a	a	DET
ejpam-6327	216	41	fuzzy	fuzzy	ADJ
ejpam-6327	216	42	implicative	implicative	ADJ
ejpam-6327	216	43	ks	ks	NOUN
ejpam-6327	216	44	-	-	PUNCT
ejpam-6327	216	45	ideal	ideal	NOUN
ejpam-6327	216	46	of	of	ADP
ejpam-6327	216	47	x	x	PRON
ejpam-6327	216	48	if	if	SCONJ
ejpam-6327	216	49	it	it	PRON
ejpam-6327	216	50	is	be	AUX
ejpam-6327	216	51	both	both	CCONJ
ejpam-6327	216	52	a	a	DET
ejpam-6327	216	53	left	left	NOUN
ejpam-6327	216	54	and	and	CCONJ
ejpam-6327	216	55	a	a	DET
ejpam-6327	216	56	right	right	ADJ
ejpam-6327	216	57	fuzzy	fuzzy	ADJ
ejpam-6327	216	58	implicative	implicative	ADJ
ejpam-6327	216	59	ks	ks	NOUN
ejpam-6327	216	60	-	-	PUNCT
ejpam-6327	216	61	ideal	ideal	NOUN
ejpam-6327	216	62	of	of	ADP
ejpam-6327	216	63	x.	x.	PROPN
ejpam-6327	216	64	example	example	NOUN
ejpam-6327	216	65	4	4	X
ejpam-6327	216	66	.	.	PUNCT
ejpam-6327	217	1	let	let	VERB
ejpam-6327	217	2	x	x	PUNCT
ejpam-6327	217	3	=	=	PUNCT
ejpam-6327	217	4	{	{	PUNCT
ejpam-6327	217	5	0	0	NUM
ejpam-6327	217	6	,	,	PUNCT
ejpam-6327	217	7	1	1	NUM
ejpam-6327	217	8	,	,	PUNCT
ejpam-6327	217	9	2	2	NUM
ejpam-6327	217	10	,	,	PUNCT
ejpam-6327	217	11	3	3	NUM
ejpam-6327	217	12	,	,	PUNCT
ejpam-6327	217	13	4	4	NUM
ejpam-6327	217	14	}	}	PUNCT
ejpam-6327	217	15	.	.	PUNCT
ejpam-6327	218	1	define	define	VERB
ejpam-6327	218	2	the	the	DET
ejpam-6327	218	3	operations	operation	NOUN
ejpam-6327	218	4	∗	∗	NOUN
ejpam-6327	218	5	and	and	CCONJ
ejpam-6327	218	6	·	·	PUNCT
ejpam-6327	218	7	by	by	ADP
ejpam-6327	218	8	the	the	DET
ejpam-6327	218	9	following	follow	VERB
ejpam-6327	218	10	tables	table	NOUN
ejpam-6327	218	11	.	.	PUNCT
ejpam-6327	219	1	∗	∗	NOUN
ejpam-6327	219	2	0	0	NUM
ejpam-6327	220	1	1	1	NUM
ejpam-6327	220	2	2	2	NUM
ejpam-6327	220	3	3	3	NUM
ejpam-6327	220	4	4	4	NUM
ejpam-6327	220	5	0	0	NUM
ejpam-6327	220	6	0	0	NUM
ejpam-6327	220	7	0	0	NUM
ejpam-6327	220	8	0	0	NUM
ejpam-6327	220	9	0	0	NUM
ejpam-6327	220	10	0	0	NUM
ejpam-6327	220	11	1	1	NUM
ejpam-6327	220	12	1	1	NUM
ejpam-6327	220	13	0	0	NUM
ejpam-6327	220	14	1	1	NUM
ejpam-6327	220	15	0	0	NUM
ejpam-6327	220	16	0	0	NUM
ejpam-6327	220	17	2	2	NUM
ejpam-6327	220	18	2	2	NUM
ejpam-6327	220	19	2	2	NUM
ejpam-6327	220	20	0	0	NUM
ejpam-6327	220	21	0	0	NUM
ejpam-6327	220	22	0	0	NUM
ejpam-6327	220	23	3	3	NUM
ejpam-6327	220	24	3	3	NUM
ejpam-6327	220	25	3	3	NUM
ejpam-6327	220	26	3	3	NUM
ejpam-6327	220	27	0	0	NUM
ejpam-6327	220	28	0	0	NUM
ejpam-6327	220	29	4	4	NUM
ejpam-6327	220	30	4	4	NUM
ejpam-6327	220	31	3	3	NUM
ejpam-6327	220	32	4	4	NUM
ejpam-6327	220	33	1	1	NUM
ejpam-6327	220	34	0	0	NUM
ejpam-6327	220	35	·	·	SYM
ejpam-6327	220	36	0	0	NUM
ejpam-6327	220	37	1	1	NUM
ejpam-6327	220	38	2	2	NUM
ejpam-6327	220	39	3	3	NUM
ejpam-6327	220	40	4	4	NUM
ejpam-6327	220	41	0	0	NUM
ejpam-6327	220	42	0	0	NUM
ejpam-6327	220	43	0	0	NUM
ejpam-6327	220	44	0	0	NUM
ejpam-6327	220	45	0	0	NUM
ejpam-6327	220	46	0	0	NUM
ejpam-6327	220	47	1	1	NUM
ejpam-6327	220	48	0	0	NUM
ejpam-6327	220	49	0	0	NUM
ejpam-6327	220	50	0	0	NUM
ejpam-6327	220	51	0	0	NUM
ejpam-6327	220	52	0	0	NUM
ejpam-6327	220	53	2	2	NUM
ejpam-6327	220	54	0	0	NUM
ejpam-6327	220	55	0	0	NUM
ejpam-6327	220	56	0	0	NUM
ejpam-6327	220	57	0	0	NUM
ejpam-6327	220	58	0	0	NUM
ejpam-6327	220	59	3	3	NUM
ejpam-6327	220	60	0	0	NUM
ejpam-6327	220	61	0	0	NUM
ejpam-6327	220	62	0	0	NUM
ejpam-6327	220	63	0	0	NUM
ejpam-6327	220	64	0	0	NUM
ejpam-6327	220	65	4	4	NUM
ejpam-6327	220	66	0	0	NUM
ejpam-6327	220	67	0	0	NUM
ejpam-6327	220	68	0	0	NUM
ejpam-6327	220	69	0	0	NUM
ejpam-6327	220	70	4	4	NUM
ejpam-6327	220	71	then	then	ADV
ejpam-6327	220	72	x	x	VERB
ejpam-6327	220	73	is	be	AUX
ejpam-6327	220	74	a	a	DET
ejpam-6327	220	75	ks	ks	NOUN
ejpam-6327	220	76	-	-	PUNCT
ejpam-6327	220	77	semigroup	semigroup	NOUN
ejpam-6327	220	78	.	.	PUNCT
ejpam-6327	221	1	let	let	VERB
ejpam-6327	221	2	s	s	NOUN
ejpam-6327	221	3	,	,	PUNCT
ejpam-6327	221	4	t	t	PROPN
ejpam-6327	221	5	∈	∈	PROPN
ejpam-6327	222	1	[	[	X
ejpam-6327	222	2	0	0	NUM
ejpam-6327	222	3	,	,	PUNCT
ejpam-6327	222	4	1	1	NUM
ejpam-6327	222	5	]	]	PUNCT
ejpam-6327	222	6	such	such	ADJ
ejpam-6327	222	7	that	that	DET
ejpam-6327	222	8	s	s	VERB
ejpam-6327	222	9	<	<	X
ejpam-6327	222	10	t.	t.	X
ejpam-6327	222	11	define	define	VERB
ejpam-6327	222	12	a	a	DET
ejpam-6327	222	13	fuzzy	fuzzy	ADJ
ejpam-6327	222	14	set	set	VERB
ejpam-6327	222	15	µ	µ	NOUN
ejpam-6327	222	16	on	on	ADP
ejpam-6327	222	17	x	x	PUNCT
ejpam-6327	222	18	by	by	ADP
ejpam-6327	222	19	µ(0	µ(0	NOUN
ejpam-6327	222	20	)	)	PUNCT
ejpam-6327	222	21	=	=	SYM
ejpam-6327	222	22	µ(1	µ(1	PROPN
ejpam-6327	222	23	)	)	PUNCT
ejpam-6327	222	24	=	=	PUNCT
ejpam-6327	223	1	µ(2	µ(2	PROPN
ejpam-6327	223	2	)	)	PUNCT
ejpam-6327	223	3	=	=	SYM
ejpam-6327	223	4	t	t	PROPN
ejpam-6327	223	5	and	and	CCONJ
ejpam-6327	223	6	µ(3	µ(3	PROPN
ejpam-6327	223	7	)	)	PUNCT
ejpam-6327	223	8	=	=	SYM
ejpam-6327	223	9	µ(4	µ(4	PROPN
ejpam-6327	223	10	)	)	PUNCT
ejpam-6327	223	11	=	=	VERB
ejpam-6327	224	1	s.	s.	PROPN
ejpam-6327	224	2	by	by	ADP
ejpam-6327	224	3	routine	routine	ADJ
ejpam-6327	224	4	calculations	calculation	NOUN
ejpam-6327	224	5	,	,	PUNCT
ejpam-6327	224	6	µ	µ	X
ejpam-6327	224	7	is	be	AUX
ejpam-6327	224	8	a	a	DET
ejpam-6327	224	9	fuzzy	fuzzy	ADJ
ejpam-6327	224	10	implicative	implicative	ADJ
ejpam-6327	224	11	ks	ks	NOUN
ejpam-6327	224	12	-	-	PUNCT
ejpam-6327	224	13	ideal	ideal	NOUN
ejpam-6327	224	14	of	of	ADP
ejpam-6327	224	15	x.	x.	NOUN
ejpam-6327	224	16	we	we	PRON
ejpam-6327	224	17	now	now	ADV
ejpam-6327	224	18	give	give	VERB
ejpam-6327	224	19	a	a	DET
ejpam-6327	224	20	relationship	relationship	NOUN
ejpam-6327	224	21	between	between	ADP
ejpam-6327	224	22	a	a	DET
ejpam-6327	224	23	fuzzy	fuzzy	ADJ
ejpam-6327	224	24	ks	ks	NOUN
ejpam-6327	224	25	-	-	PUNCT
ejpam-6327	224	26	ideal	ideal	NOUN
ejpam-6327	224	27	and	and	CCONJ
ejpam-6327	224	28	a	a	DET
ejpam-6327	224	29	fuzzy	fuzzy	ADJ
ejpam-6327	224	30	implicative	implicative	ADJ
ejpam-6327	224	31	ks	ks	NOUN
ejpam-6327	224	32	-	-	PUNCT
ejpam-6327	224	33	ideal	ideal	NOUN
ejpam-6327	224	34	.	.	PUNCT
ejpam-6327	225	1	theorem	theorem	ADJ
ejpam-6327	225	2	7	7	NUM
ejpam-6327	225	3	.	.	PUNCT
ejpam-6327	226	1	let	let	VERB
ejpam-6327	226	2	x	x	PRON
ejpam-6327	226	3	be	be	AUX
ejpam-6327	226	4	a	a	DET
ejpam-6327	226	5	ks	ks	NOUN
ejpam-6327	226	6	-	-	PUNCT
ejpam-6327	226	7	semigroup	semigroup	NOUN
ejpam-6327	226	8	.	.	PUNCT
ejpam-6327	227	1	then	then	ADV
ejpam-6327	227	2	any	any	DET
ejpam-6327	227	3	fuzzy	fuzzy	ADJ
ejpam-6327	227	4	implicative	implicative	ADJ
ejpam-6327	227	5	ks	ks	NOUN
ejpam-6327	227	6	-	-	PUNCT
ejpam-6327	227	7	ideal	ideal	NOUN
ejpam-6327	227	8	of	of	ADP
ejpam-6327	227	9	x	x	PUNCT
ejpam-6327	227	10	is	be	AUX
ejpam-6327	227	11	a	a	DET
ejpam-6327	227	12	fuzzy	fuzzy	ADJ
ejpam-6327	227	13	ks	ks	NOUN
ejpam-6327	227	14	-	-	NOUN
ejpam-6327	227	15	ideal	ideal	NOUN
ejpam-6327	227	16	of	of	ADP
ejpam-6327	227	17	x.	x.	NOUN
ejpam-6327	227	18	proof	proof	NOUN
ejpam-6327	227	19	.	.	PUNCT
ejpam-6327	228	1	let	let	VERB
ejpam-6327	228	2	µ	µ	X
ejpam-6327	228	3	be	be	AUX
ejpam-6327	228	4	a	a	DET
ejpam-6327	228	5	fuzzy	fuzzy	ADJ
ejpam-6327	228	6	implicative	implicative	ADJ
ejpam-6327	228	7	ks	ks	NOUN
ejpam-6327	228	8	-	-	PUNCT
ejpam-6327	228	9	ideal	ideal	NOUN
ejpam-6327	228	10	of	of	ADP
ejpam-6327	228	11	a	a	DET
ejpam-6327	228	12	ks	ks	NOUN
ejpam-6327	228	13	-	-	PUNCT
ejpam-6327	228	14	semigroup	semigroup	NOUN
ejpam-6327	228	15	x.	x.	NOUN
ejpam-6327	228	16	then	then	ADV
ejpam-6327	228	17	for	for	ADP
ejpam-6327	228	18	all	all	DET
ejpam-6327	228	19	x	x	NOUN
ejpam-6327	228	20	,	,	PUNCT
ejpam-6327	228	21	y	y	PROPN
ejpam-6327	228	22	∈	∈	PROPN
ejpam-6327	228	23	x	x	NOUN
ejpam-6327	228	24	,	,	PUNCT
ejpam-6327	228	25	µ(x	µ(x	NOUN
ejpam-6327	228	26	)	)	PUNCT
ejpam-6327	228	27	≥	≥	NOUN
ejpam-6327	228	28	min{µ((x	min{µ((x	NOUN
ejpam-6327	228	29	∗	∗	NOUN
ejpam-6327	228	30	(	(	PUNCT
ejpam-6327	228	31	x	x	X
ejpam-6327	228	32	∗	∗	NOUN
ejpam-6327	228	33	x	x	NOUN
ejpam-6327	228	34	)	)	PUNCT
ejpam-6327	228	35	)	)	PUNCT
ejpam-6327	228	36	∗	∗	PROPN
ejpam-6327	228	37	y	y	PROPN
ejpam-6327	228	38	)	)	PUNCT
ejpam-6327	228	39	,	,	PUNCT
ejpam-6327	228	40	µ(y	µ(y	PROPN
ejpam-6327	228	41	)	)	PUNCT
ejpam-6327	228	42	}	}	PUNCT
ejpam-6327	229	1	=	=	SYM
ejpam-6327	229	2	min{µ((x	min{µ((x	NOUN
ejpam-6327	229	3	∗	∗	NOUN
ejpam-6327	229	4	0	0	NUM
ejpam-6327	229	5	)	)	PUNCT
ejpam-6327	229	6	∗	∗	NOUN
ejpam-6327	229	7	y	y	PROPN
ejpam-6327	229	8	)	)	PUNCT
ejpam-6327	229	9	,	,	PUNCT
ejpam-6327	229	10	µ(y	µ(y	PROPN
ejpam-6327	229	11	)	)	PUNCT
ejpam-6327	229	12	}	}	PUNCT
ejpam-6327	230	1	=	=	SYM
ejpam-6327	230	2	min{µ(x	min{µ(x	PROPN
ejpam-6327	230	3	∗	∗	X
ejpam-6327	230	4	y	y	NOUN
ejpam-6327	230	5	)	)	PUNCT
ejpam-6327	230	6	,	,	PUNCT
ejpam-6327	230	7	µ(y	µ(y	PROPN
ejpam-6327	230	8	)	)	PUNCT
ejpam-6327	230	9	}	}	PUNCT
ejpam-6327	230	10	.	.	PUNCT
ejpam-6327	231	1	therefore	therefore	ADV
ejpam-6327	231	2	,	,	PUNCT
ejpam-6327	231	3	µ	µ	X
ejpam-6327	231	4	is	be	AUX
ejpam-6327	231	5	a	a	DET
ejpam-6327	231	6	fuzzy	fuzzy	ADJ
ejpam-6327	231	7	ks	ks	NOUN
ejpam-6327	231	8	-	-	NOUN
ejpam-6327	231	9	ideal	ideal	NOUN
ejpam-6327	231	10	of	of	ADP
ejpam-6327	231	11	x.	x.	NOUN
ejpam-6327	231	12	the	the	DET
ejpam-6327	231	13	converse	converse	NOUN
ejpam-6327	231	14	of	of	ADP
ejpam-6327	231	15	theorem	theorem	NOUN
ejpam-6327	231	16	7	7	NUM
ejpam-6327	231	17	may	may	AUX
ejpam-6327	231	18	not	not	PART
ejpam-6327	231	19	be	be	AUX
ejpam-6327	231	20	true	true	ADJ
ejpam-6327	231	21	as	as	SCONJ
ejpam-6327	231	22	shown	show	VERB
ejpam-6327	231	23	in	in	ADP
ejpam-6327	231	24	the	the	DET
ejpam-6327	231	25	following	follow	VERB
ejpam-6327	231	26	example	example	NOUN
ejpam-6327	231	27	.	.	PUNCT
ejpam-6327	232	1	example	example	NOUN
ejpam-6327	232	2	5	5	NUM
ejpam-6327	232	3	.	.	X
ejpam-6327	233	1	consider	consider	VERB
ejpam-6327	233	2	the	the	DET
ejpam-6327	233	3	ks	ks	NOUN
ejpam-6327	233	4	-	-	PUNCT
ejpam-6327	233	5	semigroup	semigroup	NOUN
ejpam-6327	233	6	in	in	ADP
ejpam-6327	233	7	example	example	NOUN
ejpam-6327	233	8	4	4	NUM
ejpam-6327	233	9	.	.	PUNCT
ejpam-6327	233	10	define	define	VERB
ejpam-6327	233	11	a	a	DET
ejpam-6327	233	12	fuzzy	fuzzy	ADJ
ejpam-6327	233	13	set	set	VERB
ejpam-6327	233	14	µ	µ	NOUN
ejpam-6327	233	15	on	on	ADP
ejpam-6327	233	16	x	x	PUNCT
ejpam-6327	233	17	by	by	ADP
ejpam-6327	233	18	µ(0	µ(0	NOUN
ejpam-6327	233	19	)	)	PUNCT
ejpam-6327	233	20	=	=	SYM
ejpam-6327	233	21	µ(2	µ(2	PROPN
ejpam-6327	233	22	)	)	PUNCT
ejpam-6327	233	23	=	=	NUM
ejpam-6327	233	24	0.8	0.8	NUM
ejpam-6327	233	25	and	and	CCONJ
ejpam-6327	233	26	µ(1	µ(1	PROPN
ejpam-6327	233	27	)	)	PUNCT
ejpam-6327	233	28	=	=	SYM
ejpam-6327	233	29	µ(3	µ(3	PROPN
ejpam-6327	233	30	)	)	PUNCT
ejpam-6327	233	31	=	=	SYM
ejpam-6327	233	32	µ(4	µ(4	PROPN
ejpam-6327	233	33	)	)	PUNCT
ejpam-6327	233	34	=	=	NUM
ejpam-6327	233	35	0.5	0.5	NUM
ejpam-6327	233	36	.	.	PUNCT
ejpam-6327	233	37	by	by	ADP
ejpam-6327	233	38	routine	routine	ADJ
ejpam-6327	233	39	calculations	calculation	NOUN
ejpam-6327	233	40	,	,	PUNCT
ejpam-6327	233	41	µ	µ	X
ejpam-6327	233	42	is	be	AUX
ejpam-6327	233	43	a	a	DET
ejpam-6327	233	44	fuzzy	fuzzy	ADJ
ejpam-6327	233	45	ks	ks	NOUN
ejpam-6327	233	46	-	-	NOUN
ejpam-6327	233	47	ideal	ideal	NOUN
ejpam-6327	233	48	of	of	ADP
ejpam-6327	233	49	x.	x.	NOUN
ejpam-6327	233	50	however	however	ADV
ejpam-6327	233	51	,	,	PUNCT
ejpam-6327	233	52	µ	µ	X
ejpam-6327	233	53	is	be	AUX
ejpam-6327	233	54	not	not	PART
ejpam-6327	233	55	a	a	PRON
ejpam-6327	233	56	fuzzy	fuzzy	ADJ
ejpam-6327	233	57	implicative	implicative	ADJ
ejpam-6327	233	58	ks	ks	NOUN
ejpam-6327	233	59	-	-	PUNCT
ejpam-6327	233	60	ideal	ideal	NOUN
ejpam-6327	233	61	of	of	ADP
ejpam-6327	233	62	x	x	PRON
ejpam-6327	233	63	since	since	SCONJ
ejpam-6327	233	64	µ(1	µ(1	PROPN
ejpam-6327	233	65	)	)	PUNCT
ejpam-6327	234	1	=	=	PUNCT
ejpam-6327	234	2	0.5	0.5	NUM
ejpam-6327	234	3	<	<	SYM
ejpam-6327	234	4	0.8	0.8	NUM
ejpam-6327	234	5	=	=	SYM
ejpam-6327	234	6	min{µ((1	min{µ((1	PROPN
ejpam-6327	234	7	∗	∗	NOUN
ejpam-6327	234	8	(	(	PUNCT
ejpam-6327	234	9	3	3	NUM
ejpam-6327	234	10	∗	∗	NOUN
ejpam-6327	234	11	1	1	NUM
ejpam-6327	234	12	)	)	PUNCT
ejpam-6327	234	13	)	)	PUNCT
ejpam-6327	234	14	∗	∗	NOUN
ejpam-6327	234	15	2	2	NUM
ejpam-6327	234	16	)	)	PUNCT
ejpam-6327	234	17	,	,	PUNCT
ejpam-6327	234	18	µ(2	µ(2	PROPN
ejpam-6327	234	19	)	)	PUNCT
ejpam-6327	234	20	}	}	PUNCT
ejpam-6327	234	21	.	.	PUNCT
ejpam-6327	235	1	the	the	DET
ejpam-6327	235	2	following	follow	VERB
ejpam-6327	235	3	result	result	NOUN
ejpam-6327	235	4	provides	provide	VERB
ejpam-6327	235	5	a	a	DET
ejpam-6327	235	6	criterion	criterion	NOUN
ejpam-6327	235	7	for	for	ADP
ejpam-6327	235	8	a	a	DET
ejpam-6327	235	9	fuzzy	fuzzy	ADJ
ejpam-6327	235	10	ks	ks	NOUN
ejpam-6327	235	11	-	-	PUNCT
ejpam-6327	235	12	ideal	ideal	NOUN
ejpam-6327	235	13	to	to	PART
ejpam-6327	235	14	be	be	AUX
ejpam-6327	235	15	a	a	DET
ejpam-6327	235	16	fuzzy	fuzzy	ADJ
ejpam-6327	235	17	implicative	implicative	ADJ
ejpam-6327	235	18	ks	ks	NOUN
ejpam-6327	235	19	-	-	PUNCT
ejpam-6327	235	20	ideal	ideal	NOUN
ejpam-6327	235	21	.	.	PUNCT
ejpam-6327	236	1	h.	h.	PROPN
ejpam-6327	236	2	sarapuddin	sarapuddin	PROPN
ejpam-6327	236	3	,	,	PUNCT
ejpam-6327	236	4	j.	j.	PROPN
ejpam-6327	236	5	vilela	vilela	PROPN
ejpam-6327	236	6	/	/	SYM
ejpam-6327	236	7	eur	eur	PROPN
ejpam-6327	236	8	.	.	PUNCT
ejpam-6327	237	1	j.	j.	PROPN
ejpam-6327	237	2	pure	pure	PROPN
ejpam-6327	237	3	appl	appl	PROPN
ejpam-6327	237	4	.	.	PROPN
ejpam-6327	237	5	math	math	PROPN
ejpam-6327	237	6	,	,	PUNCT
ejpam-6327	237	7	18	18	NUM
ejpam-6327	237	8	(	(	PUNCT
ejpam-6327	237	9	3	3	NUM
ejpam-6327	237	10	)	)	PUNCT
ejpam-6327	237	11	(	(	PUNCT
ejpam-6327	237	12	2025	2025	NUM
ejpam-6327	237	13	)	)	PUNCT
ejpam-6327	237	14	,	,	PUNCT
ejpam-6327	237	15	6327	6327	NUM
ejpam-6327	237	16	9	9	NUM
ejpam-6327	237	17	of	of	ADP
ejpam-6327	237	18	23	23	NUM
ejpam-6327	237	19	theorem	theorem	NOUN
ejpam-6327	237	20	8	8	NUM
ejpam-6327	237	21	.	.	PUNCT
ejpam-6327	238	1	a	a	DET
ejpam-6327	238	2	fuzzy	fuzzy	ADJ
ejpam-6327	238	3	set	set	VERB
ejpam-6327	238	4	µ	µ	NOUN
ejpam-6327	238	5	on	on	ADP
ejpam-6327	238	6	a	a	DET
ejpam-6327	238	7	ks	ks	NOUN
ejpam-6327	238	8	-	-	PUNCT
ejpam-6327	238	9	semigroup	semigroup	NOUN
ejpam-6327	238	10	x	x	PUNCT
ejpam-6327	238	11	is	be	AUX
ejpam-6327	238	12	a	a	DET
ejpam-6327	238	13	fuzzy	fuzzy	ADJ
ejpam-6327	238	14	implicative	implicative	ADJ
ejpam-6327	238	15	ks	ks	NOUN
ejpam-6327	238	16	-	-	PUNCT
ejpam-6327	238	17	ideal	ideal	NOUN
ejpam-6327	238	18	if	if	SCONJ
ejpam-6327	238	19	and	and	CCONJ
ejpam-6327	238	20	only	only	ADV
ejpam-6327	238	21	if	if	SCONJ
ejpam-6327	238	22	it	it	PRON
ejpam-6327	238	23	is	be	AUX
ejpam-6327	238	24	a	a	DET
ejpam-6327	238	25	fuzzy	fuzzy	ADJ
ejpam-6327	238	26	ks	ks	NOUN
ejpam-6327	238	27	-	-	PUNCT
ejpam-6327	238	28	ideal	ideal	ADJ
ejpam-6327	238	29	satisfying	satisfy	VERB
ejpam-6327	238	30	µ(x	µ(x	NOUN
ejpam-6327	238	31	)	)	PUNCT
ejpam-6327	238	32	=	=	NOUN
ejpam-6327	238	33	µ(x	µ(x	ADJ
ejpam-6327	238	34	∗	∗	NOUN
ejpam-6327	238	35	(	(	PUNCT
ejpam-6327	238	36	y	y	PROPN
ejpam-6327	238	37	∗	∗	NOUN
ejpam-6327	238	38	x	x	NOUN
ejpam-6327	238	39	)	)	PUNCT
ejpam-6327	238	40	)	)	PUNCT
ejpam-6327	238	41	for	for	ADP
ejpam-6327	238	42	all	all	DET
ejpam-6327	238	43	x	x	NOUN
ejpam-6327	238	44	,	,	PUNCT
ejpam-6327	238	45	y	y	PROPN
ejpam-6327	238	46	∈	∈	PROPN
ejpam-6327	238	47	x.	x.	NOUN
ejpam-6327	238	48	proof	proof	NOUN
ejpam-6327	238	49	.	.	PUNCT
ejpam-6327	239	1	let	let	VERB
ejpam-6327	239	2	x	x	PRON
ejpam-6327	239	3	be	be	AUX
ejpam-6327	239	4	a	a	DET
ejpam-6327	239	5	ks	ks	NOUN
ejpam-6327	239	6	-	-	PUNCT
ejpam-6327	239	7	semigroup	semigroup	NOUN
ejpam-6327	239	8	and	and	CCONJ
ejpam-6327	239	9	µ	µ	PRON
ejpam-6327	239	10	be	be	AUX
ejpam-6327	239	11	a	a	DET
ejpam-6327	239	12	fuzzy	fuzzy	ADJ
ejpam-6327	239	13	set	set	NOUN
ejpam-6327	239	14	on	on	ADP
ejpam-6327	239	15	x.	x.	NOUN
ejpam-6327	239	16	suppose	suppose	VERB
ejpam-6327	239	17	µ	µ	PRON
ejpam-6327	239	18	is	be	AUX
ejpam-6327	239	19	a	a	DET
ejpam-6327	239	20	fuzzy	fuzzy	ADJ
ejpam-6327	239	21	implicative	implicative	ADJ
ejpam-6327	239	22	ks	ks	NOUN
ejpam-6327	239	23	-	-	PUNCT
ejpam-6327	239	24	ideal	ideal	NOUN
ejpam-6327	239	25	of	of	ADP
ejpam-6327	239	26	x.	x.	NOUN
ejpam-6327	239	27	then	then	ADV
ejpam-6327	239	28	by	by	ADP
ejpam-6327	239	29	theorem	theorem	NOUN
ejpam-6327	239	30	7	7	NUM
ejpam-6327	239	31	,	,	PUNCT
ejpam-6327	239	32	µ	µ	PRON
ejpam-6327	239	33	is	be	AUX
ejpam-6327	239	34	a	a	DET
ejpam-6327	239	35	fuzzy	fuzzy	ADJ
ejpam-6327	239	36	ks	ks	NOUN
ejpam-6327	239	37	-	-	NOUN
ejpam-6327	239	38	ideal	ideal	NOUN
ejpam-6327	239	39	of	of	ADP
ejpam-6327	239	40	x.	x.	NOUN
ejpam-6327	239	41	by	by	ADP
ejpam-6327	239	42	(	(	PUNCT
ejpam-6327	239	43	f6	f6	PROPN
ejpam-6327	239	44	)	)	PUNCT
ejpam-6327	239	45	,	,	PUNCT
ejpam-6327	239	46	for	for	ADP
ejpam-6327	239	47	all	all	DET
ejpam-6327	239	48	x	x	NOUN
ejpam-6327	239	49	,	,	PUNCT
ejpam-6327	239	50	y	y	PROPN
ejpam-6327	239	51	∈	∈	PROPN
ejpam-6327	239	52	x	x	NOUN
ejpam-6327	239	53	,	,	PUNCT
ejpam-6327	239	54	µ(x	µ(x	NOUN
ejpam-6327	239	55	)	)	PUNCT
ejpam-6327	239	56	≥	≥	NOUN
ejpam-6327	239	57	min{µ(x	min{µ(x	NOUN
ejpam-6327	239	58	∗	∗	NOUN
ejpam-6327	239	59	(	(	PUNCT
ejpam-6327	239	60	y	y	PROPN
ejpam-6327	239	61	∗	∗	X
ejpam-6327	239	62	x	x	NOUN
ejpam-6327	239	63	)	)	PUNCT
ejpam-6327	239	64	∗	∗	NOUN
ejpam-6327	239	65	0	0	NUM
ejpam-6327	239	66	)	)	PUNCT
ejpam-6327	239	67	,	,	PUNCT
ejpam-6327	239	68	µ(0	µ(0	NOUN
ejpam-6327	239	69	)	)	PUNCT
ejpam-6327	239	70	}	}	PUNCT
ejpam-6327	240	1	=	=	SYM
ejpam-6327	240	2	min{µ(x	min{µ(x	NOUN
ejpam-6327	240	3	∗	∗	NOUN
ejpam-6327	240	4	(	(	PUNCT
ejpam-6327	240	5	y	y	PROPN
ejpam-6327	240	6	∗	∗	NOUN
ejpam-6327	240	7	x	x	NOUN
ejpam-6327	240	8	)	)	PUNCT
ejpam-6327	240	9	)	)	PUNCT
ejpam-6327	240	10	,	,	PUNCT
ejpam-6327	240	11	µ(0	µ(0	NOUN
ejpam-6327	240	12	)	)	PUNCT
ejpam-6327	240	13	}	}	PUNCT
ejpam-6327	240	14	=	=	SYM
ejpam-6327	240	15	µ(x	µ(x	ADJ
ejpam-6327	240	16	∗	∗	NOUN
ejpam-6327	240	17	(	(	PUNCT
ejpam-6327	240	18	y	y	PROPN
ejpam-6327	240	19	∗	∗	NOUN
ejpam-6327	240	20	x	x	NOUN
ejpam-6327	240	21	)	)	PUNCT
ejpam-6327	240	22	)	)	PUNCT
ejpam-6327	240	23	.	.	PUNCT
ejpam-6327	241	1	moreover	moreover	ADV
ejpam-6327	241	2	,	,	PUNCT
ejpam-6327	241	3	observe	observe	VERB
ejpam-6327	241	4	that	that	SCONJ
ejpam-6327	241	5	x∗(y∗x	x∗(y∗x	NOUN
ejpam-6327	241	6	)	)	PUNCT
ejpam-6327	241	7	≤	≤	NUM
ejpam-6327	241	8	x	x	PUNCT
ejpam-6327	241	9	by	by	ADP
ejpam-6327	241	10	remark	remark	NOUN
ejpam-6327	241	11	1(ii	1(ii	NUM
ejpam-6327	241	12	)	)	PUNCT
ejpam-6327	241	13	.	.	PUNCT
ejpam-6327	242	1	thus	thus	ADV
ejpam-6327	242	2	,	,	PUNCT
ejpam-6327	242	3	by	by	ADP
ejpam-6327	242	4	lemma	lemma	PROPN
ejpam-6327	242	5	1(i	1(i	NUM
ejpam-6327	242	6	)	)	PUNCT
ejpam-6327	242	7	,	,	PUNCT
ejpam-6327	242	8	µ(x∗(y∗x	µ(x∗(y∗x	NUM
ejpam-6327	242	9	)	)	PUNCT
ejpam-6327	242	10	)	)	PUNCT
ejpam-6327	242	11	≥	≥	NOUN
ejpam-6327	242	12	µ(x	µ(x	VERB
ejpam-6327	242	13	)	)	PUNCT
ejpam-6327	242	14	.	.	PUNCT
ejpam-6327	243	1	hence	hence	ADV
ejpam-6327	243	2	,	,	PUNCT
ejpam-6327	243	3	µ(x	µ(x	X
ejpam-6327	243	4	)	)	PUNCT
ejpam-6327	243	5	=	=	NOUN
ejpam-6327	243	6	µ(x	µ(x	ADJ
ejpam-6327	243	7	∗	∗	NOUN
ejpam-6327	243	8	(	(	PUNCT
ejpam-6327	243	9	y	y	PROPN
ejpam-6327	243	10	∗	∗	NOUN
ejpam-6327	243	11	x	x	NOUN
ejpam-6327	243	12	)	)	PUNCT
ejpam-6327	243	13	)	)	PUNCT
ejpam-6327	243	14	.	.	PUNCT
ejpam-6327	244	1	conversely	conversely	ADV
ejpam-6327	244	2	,	,	PUNCT
ejpam-6327	244	3	suppose	suppose	VERB
ejpam-6327	244	4	µ	µ	PRON
ejpam-6327	244	5	is	be	AUX
ejpam-6327	244	6	a	a	DET
ejpam-6327	244	7	fuzzy	fuzzy	ADJ
ejpam-6327	244	8	ks	ks	NOUN
ejpam-6327	244	9	-	-	NOUN
ejpam-6327	244	10	ideal	ideal	NOUN
ejpam-6327	244	11	of	of	ADP
ejpam-6327	244	12	x	x	PUNCT
ejpam-6327	244	13	satisfying	satisfy	VERB
ejpam-6327	244	14	µ(x	µ(x	NOUN
ejpam-6327	244	15	)	)	PUNCT
ejpam-6327	244	16	=	=	NOUN
ejpam-6327	244	17	µ(x	µ(x	ADJ
ejpam-6327	244	18	∗	∗	NOUN
ejpam-6327	244	19	(	(	PUNCT
ejpam-6327	244	20	y	y	PROPN
ejpam-6327	244	21	∗	∗	NOUN
ejpam-6327	244	22	x	x	NOUN
ejpam-6327	244	23	)	)	PUNCT
ejpam-6327	244	24	)	)	PUNCT
ejpam-6327	244	25	for	for	ADP
ejpam-6327	244	26	all	all	DET
ejpam-6327	244	27	x	x	NOUN
ejpam-6327	244	28	,	,	PUNCT
ejpam-6327	244	29	y	y	PROPN
ejpam-6327	244	30	∈	∈	PROPN
ejpam-6327	244	31	x.	x.	NOUN
ejpam-6327	244	32	by	by	ADP
ejpam-6327	244	33	(	(	PUNCT
ejpam-6327	244	34	f2	f2	PROPN
ejpam-6327	244	35	)	)	PUNCT
ejpam-6327	244	36	,	,	PUNCT
ejpam-6327	244	37	µ(x	µ(x	X
ejpam-6327	244	38	)	)	PUNCT
ejpam-6327	244	39	=	=	SYM
ejpam-6327	244	40	µ(x∗	µ(x∗	NOUN
ejpam-6327	244	41	(	(	PUNCT
ejpam-6327	244	42	y	y	NOUN
ejpam-6327	244	43	∗x	∗x	PROPN
ejpam-6327	244	44	)	)	PUNCT
ejpam-6327	244	45	)	)	PUNCT
ejpam-6327	244	46	≥	≥	AUX
ejpam-6327	245	1	min{µ((x∗	min{µ((x∗	NOUN
ejpam-6327	245	2	(	(	PUNCT
ejpam-6327	245	3	y	y	PROPN
ejpam-6327	245	4	∗x))∗	∗x))∗	NUM
ejpam-6327	245	5	z	z	NOUN
ejpam-6327	245	6	)	)	PUNCT
ejpam-6327	245	7	,	,	PUNCT
ejpam-6327	245	8	µ(z	µ(z	PROPN
ejpam-6327	245	9	)	)	PUNCT
ejpam-6327	245	10	}	}	PUNCT
ejpam-6327	245	11	for	for	ADP
ejpam-6327	245	12	all	all	DET
ejpam-6327	245	13	x	x	NOUN
ejpam-6327	245	14	,	,	PUNCT
ejpam-6327	245	15	y	y	PROPN
ejpam-6327	245	16	,	,	PUNCT
ejpam-6327	245	17	z	z	PROPN
ejpam-6327	245	18	∈	∈	PROPN
ejpam-6327	245	19	x.	x.	NOUN
ejpam-6327	245	20	thus	thus	ADV
ejpam-6327	245	21	,	,	PUNCT
ejpam-6327	245	22	µ	µ	X
ejpam-6327	245	23	is	be	AUX
ejpam-6327	245	24	a	a	DET
ejpam-6327	245	25	fuzzy	fuzzy	ADJ
ejpam-6327	245	26	implicative	implicative	ADJ
ejpam-6327	245	27	ks	ks	NOUN
ejpam-6327	245	28	-	-	PUNCT
ejpam-6327	245	29	ideal	ideal	NOUN
ejpam-6327	245	30	of	of	ADP
ejpam-6327	245	31	x.	x.	NOUN
ejpam-6327	245	32	the	the	DET
ejpam-6327	245	33	following	follow	VERB
ejpam-6327	245	34	theorem	theorem	NOUN
ejpam-6327	245	35	gives	give	VERB
ejpam-6327	245	36	a	a	DET
ejpam-6327	245	37	necessary	necessary	ADJ
ejpam-6327	245	38	condition	condition	NOUN
ejpam-6327	245	39	for	for	ADP
ejpam-6327	245	40	a	a	DET
ejpam-6327	245	41	fuzzy	fuzzy	ADJ
ejpam-6327	245	42	ks	ks	NOUN
ejpam-6327	245	43	-	-	PUNCT
ejpam-6327	245	44	ideal	ideal	NOUN
ejpam-6327	245	45	to	to	PART
ejpam-6327	245	46	be	be	AUX
ejpam-6327	245	47	a	a	DET
ejpam-6327	245	48	fuzzy	fuzzy	ADJ
ejpam-6327	245	49	implicative	implicative	ADJ
ejpam-6327	245	50	ks	ks	NOUN
ejpam-6327	245	51	-	-	PUNCT
ejpam-6327	245	52	ideal	ideal	NOUN
ejpam-6327	245	53	.	.	PUNCT
ejpam-6327	246	1	theorem	theorem	NOUN
ejpam-6327	246	2	9	9	NUM
ejpam-6327	246	3	.	.	PUNCT
ejpam-6327	247	1	if	if	SCONJ
ejpam-6327	247	2	x	x	PRON
ejpam-6327	247	3	is	be	AUX
ejpam-6327	247	4	an	an	DET
ejpam-6327	247	5	implicative	implicative	ADJ
ejpam-6327	247	6	ks	ks	NOUN
ejpam-6327	247	7	-	-	PUNCT
ejpam-6327	247	8	semigroup	semigroup	PROPN
ejpam-6327	247	9	,	,	PUNCT
ejpam-6327	247	10	then	then	ADV
ejpam-6327	247	11	every	every	DET
ejpam-6327	247	12	fuzzy	fuzzy	ADJ
ejpam-6327	247	13	ks	ks	NOUN
ejpam-6327	247	14	-	-	PUNCT
ejpam-6327	247	15	ideal	ideal	NOUN
ejpam-6327	247	16	of	of	ADP
ejpam-6327	247	17	x	x	PUNCT
ejpam-6327	247	18	is	be	AUX
ejpam-6327	247	19	fuzzy	fuzzy	ADJ
ejpam-6327	247	20	implicative	implicative	ADJ
ejpam-6327	247	21	ks	ks	NOUN
ejpam-6327	247	22	-	-	PUNCT
ejpam-6327	247	23	ideal	ideal	NOUN
ejpam-6327	247	24	.	.	PUNCT
ejpam-6327	248	1	proof	proof	NOUN
ejpam-6327	248	2	.	.	PUNCT
ejpam-6327	249	1	let	let	VERB
ejpam-6327	249	2	x	x	PRON
ejpam-6327	249	3	be	be	AUX
ejpam-6327	249	4	an	an	DET
ejpam-6327	249	5	implicative	implicative	ADJ
ejpam-6327	249	6	ks	ks	NOUN
ejpam-6327	249	7	-	-	PUNCT
ejpam-6327	249	8	semigroup	semigroup	NOUN
ejpam-6327	249	9	and	and	CCONJ
ejpam-6327	249	10	µ	µ	DET
ejpam-6327	249	11	a	a	DET
ejpam-6327	249	12	fuzzy	fuzzy	ADJ
ejpam-6327	249	13	ks	ks	NOUN
ejpam-6327	249	14	-	-	NOUN
ejpam-6327	249	15	ideal	ideal	NOUN
ejpam-6327	249	16	of	of	ADP
ejpam-6327	249	17	x.	x.	NOUN
ejpam-6327	249	18	then	then	ADV
ejpam-6327	249	19	by	by	ADP
ejpam-6327	249	20	definition	definition	NOUN
ejpam-6327	249	21	5(ii	5(ii	NUM
ejpam-6327	249	22	)	)	PUNCT
ejpam-6327	249	23	,	,	PUNCT
ejpam-6327	249	24	x	x	PUNCT
ejpam-6327	249	25	=	=	PUNCT
ejpam-6327	249	26	x	x	SYM
ejpam-6327	249	27	∗	∗	NOUN
ejpam-6327	249	28	(	(	PUNCT
ejpam-6327	249	29	y	y	PROPN
ejpam-6327	249	30	∗	∗	X
ejpam-6327	249	31	x	x	NOUN
ejpam-6327	249	32	)	)	PUNCT
ejpam-6327	249	33	for	for	ADP
ejpam-6327	249	34	all	all	DET
ejpam-6327	249	35	x	x	NOUN
ejpam-6327	249	36	,	,	PUNCT
ejpam-6327	249	37	y	y	PROPN
ejpam-6327	249	38	∈	∈	PROPN
ejpam-6327	249	39	x.	x.	NOUN
ejpam-6327	249	40	since	since	SCONJ
ejpam-6327	249	41	µ	µ	NOUN
ejpam-6327	249	42	is	be	AUX
ejpam-6327	249	43	a	a	DET
ejpam-6327	249	44	fuzzy	fuzzy	ADJ
ejpam-6327	249	45	ks	ks	NOUN
ejpam-6327	249	46	-	-	NOUN
ejpam-6327	249	47	ideal	ideal	NOUN
ejpam-6327	249	48	of	of	ADP
ejpam-6327	249	49	x	x	PRON
ejpam-6327	249	50	,	,	PUNCT
ejpam-6327	249	51	by	by	ADP
ejpam-6327	249	52	(	(	PUNCT
ejpam-6327	249	53	f2	f2	PROPN
ejpam-6327	249	54	)	)	PUNCT
ejpam-6327	249	55	,	,	PUNCT
ejpam-6327	249	56	µ(x	µ(x	X
ejpam-6327	249	57	)	)	PUNCT
ejpam-6327	249	58	≥	≥	NOUN
ejpam-6327	250	1	min{µ(x	min{µ(x	NOUN
ejpam-6327	250	2	∗	∗	NOUN
ejpam-6327	250	3	z	z	NOUN
ejpam-6327	250	4	)	)	PUNCT
ejpam-6327	250	5	,	,	PUNCT
ejpam-6327	250	6	µ(z	µ(z	PROPN
ejpam-6327	250	7	)	)	PUNCT
ejpam-6327	250	8	}	}	PUNCT
ejpam-6327	251	1	=	=	SYM
ejpam-6327	251	2	min{µ((x	min{µ((x	NOUN
ejpam-6327	251	3	∗	∗	NOUN
ejpam-6327	251	4	(	(	PUNCT
ejpam-6327	251	5	y	y	PROPN
ejpam-6327	251	6	∗	∗	NOUN
ejpam-6327	251	7	x	x	NOUN
ejpam-6327	251	8	)	)	PUNCT
ejpam-6327	251	9	)	)	PUNCT
ejpam-6327	251	10	∗	∗	PROPN
ejpam-6327	251	11	z	z	NOUN
ejpam-6327	251	12	)	)	PUNCT
ejpam-6327	251	13	,	,	PUNCT
ejpam-6327	251	14	µ(z	µ(z	PROPN
ejpam-6327	251	15	)	)	PUNCT
ejpam-6327	251	16	}	}	PUNCT
ejpam-6327	251	17	for	for	ADP
ejpam-6327	251	18	all	all	DET
ejpam-6327	251	19	z	z	NOUN
ejpam-6327	251	20	∈	∈	NOUN
ejpam-6327	251	21	x.	x.	NOUN
ejpam-6327	251	22	thus	thus	ADV
ejpam-6327	251	23	,	,	PUNCT
ejpam-6327	251	24	(	(	PUNCT
ejpam-6327	251	25	f6	f6	PROPN
ejpam-6327	251	26	)	)	PUNCT
ejpam-6327	251	27	holds	hold	VERB
ejpam-6327	251	28	.	.	PUNCT
ejpam-6327	252	1	hence	hence	ADV
ejpam-6327	252	2	,	,	PUNCT
ejpam-6327	252	3	µ	µ	X
ejpam-6327	252	4	is	be	AUX
ejpam-6327	252	5	a	a	DET
ejpam-6327	252	6	fuzzy	fuzzy	ADJ
ejpam-6327	252	7	implicative	implicative	ADJ
ejpam-6327	252	8	ks	ks	NOUN
ejpam-6327	252	9	-	-	PUNCT
ejpam-6327	252	10	ideal	ideal	NOUN
ejpam-6327	252	11	of	of	ADP
ejpam-6327	252	12	x.	x.	PROPN
ejpam-6327	252	13	theorem	theorem	VERB
ejpam-6327	252	14	10	10	NUM
ejpam-6327	252	15	.	.	PUNCT
ejpam-6327	253	1	a	a	DET
ejpam-6327	253	2	fuzzy	fuzzy	ADJ
ejpam-6327	253	3	set	set	VERB
ejpam-6327	253	4	µ	µ	NOUN
ejpam-6327	253	5	on	on	ADP
ejpam-6327	253	6	a	a	DET
ejpam-6327	253	7	ks	ks	NOUN
ejpam-6327	253	8	-	-	PUNCT
ejpam-6327	253	9	semigroup	semigroup	NOUN
ejpam-6327	253	10	x	x	PUNCT
ejpam-6327	253	11	is	be	AUX
ejpam-6327	253	12	a	a	DET
ejpam-6327	253	13	fuzzy	fuzzy	ADJ
ejpam-6327	253	14	implicative	implicative	ADJ
ejpam-6327	253	15	ks	ks	NOUN
ejpam-6327	253	16	-	-	PUNCT
ejpam-6327	253	17	ideal	ideal	NOUN
ejpam-6327	253	18	if	if	SCONJ
ejpam-6327	253	19	and	and	CCONJ
ejpam-6327	253	20	only	only	ADV
ejpam-6327	253	21	if	if	SCONJ
ejpam-6327	253	22	it	it	PRON
ejpam-6327	253	23	is	be	AUX
ejpam-6327	253	24	both	both	CCONJ
ejpam-6327	253	25	a	a	DET
ejpam-6327	253	26	fuzzy	fuzzy	ADJ
ejpam-6327	253	27	ks	ks	NOUN
ejpam-6327	253	28	-	-	ADJ
ejpam-6327	253	29	p	p	ADJ
ejpam-6327	253	30	-	-	PUNCT
ejpam-6327	253	31	ideal	ideal	ADJ
ejpam-6327	253	32	and	and	CCONJ
ejpam-6327	253	33	fuzzy	fuzzy	ADJ
ejpam-6327	253	34	commutative	commutative	ADJ
ejpam-6327	253	35	ks	ks	NOUN
ejpam-6327	253	36	-	-	PUNCT
ejpam-6327	253	37	ideal	ideal	NOUN
ejpam-6327	253	38	.	.	PUNCT
ejpam-6327	254	1	proof	proof	NOUN
ejpam-6327	254	2	.	.	PUNCT
ejpam-6327	255	1	let	let	VERB
ejpam-6327	255	2	x	x	PRON
ejpam-6327	255	3	be	be	AUX
ejpam-6327	255	4	a	a	DET
ejpam-6327	255	5	ks	ks	NOUN
ejpam-6327	255	6	-	-	PUNCT
ejpam-6327	255	7	semigroup	semigroup	NOUN
ejpam-6327	255	8	and	and	CCONJ
ejpam-6327	255	9	µ	µ	PRON
ejpam-6327	255	10	be	be	AUX
ejpam-6327	255	11	a	a	DET
ejpam-6327	255	12	fuzzy	fuzzy	ADJ
ejpam-6327	255	13	set	set	NOUN
ejpam-6327	255	14	on	on	ADP
ejpam-6327	255	15	x.	x.	NOUN
ejpam-6327	255	16	suppose	suppose	VERB
ejpam-6327	255	17	µ	µ	PRON
ejpam-6327	255	18	is	be	AUX
ejpam-6327	255	19	a	a	DET
ejpam-6327	255	20	fuzzy	fuzzy	ADJ
ejpam-6327	255	21	implicative	implicative	ADJ
ejpam-6327	255	22	ks	ks	NOUN
ejpam-6327	255	23	-	-	PUNCT
ejpam-6327	255	24	ideal	ideal	NOUN
ejpam-6327	255	25	of	of	ADP
ejpam-6327	255	26	x.	x.	NOUN
ejpam-6327	255	27	by	by	ADP
ejpam-6327	255	28	theorem	theorem	ADJ
ejpam-6327	255	29	8	8	NUM
ejpam-6327	255	30	,	,	PUNCT
ejpam-6327	255	31	proposition	proposition	NOUN
ejpam-6327	255	32	1	1	NUM
ejpam-6327	255	33	and	and	CCONJ
ejpam-6327	255	34	lemma	lemma	PROPN
ejpam-6327	255	35	1(ii	1(ii	NUM
ejpam-6327	255	36	)	)	PUNCT
ejpam-6327	255	37	,	,	PUNCT
ejpam-6327	255	38	for	for	ADP
ejpam-6327	255	39	all	all	DET
ejpam-6327	255	40	x	x	NOUN
ejpam-6327	255	41	,	,	PUNCT
ejpam-6327	255	42	y	y	PROPN
ejpam-6327	255	43	,	,	PUNCT
ejpam-6327	255	44	z	z	PROPN
ejpam-6327	255	45	∈	∈	PROPN
ejpam-6327	255	46	x	x	SYM
ejpam-6327	255	47	,	,	PUNCT
ejpam-6327	255	48	µ(x	µ(x	X
ejpam-6327	255	49	∗	∗	NOUN
ejpam-6327	255	50	z	z	NOUN
ejpam-6327	255	51	)	)	PUNCT
ejpam-6327	255	52	=	=	SYM
ejpam-6327	255	53	µ((x	µ((x	NOUN
ejpam-6327	255	54	∗	∗	NOUN
ejpam-6327	255	55	z	z	NOUN
ejpam-6327	255	56	)	)	PUNCT
ejpam-6327	255	57	∗	∗	NOUN
ejpam-6327	255	58	(	(	PUNCT
ejpam-6327	255	59	x	x	X
ejpam-6327	255	60	∗	∗	NOUN
ejpam-6327	255	61	(	(	PUNCT
ejpam-6327	255	62	x	x	X
ejpam-6327	255	63	∗	∗	PROPN
ejpam-6327	255	64	z	z	NOUN
ejpam-6327	255	65	)	)	PUNCT
ejpam-6327	255	66	)	)	PUNCT
ejpam-6327	255	67	)	)	PUNCT
ejpam-6327	256	1	=	=	PUNCT
ejpam-6327	256	2	µ((x	µ((x	NOUN
ejpam-6327	256	3	∗	∗	NOUN
ejpam-6327	256	4	z	z	NOUN
ejpam-6327	256	5	)	)	PUNCT
ejpam-6327	256	6	∗	∗	NOUN
ejpam-6327	256	7	z	z	NOUN
ejpam-6327	256	8	)	)	PUNCT
ejpam-6327	256	9	≥	≥	NOUN
ejpam-6327	256	10	min{µ((x	min{µ((x	NOUN
ejpam-6327	256	11	∗	∗	X
ejpam-6327	256	12	y	y	NOUN
ejpam-6327	256	13	)	)	PUNCT
ejpam-6327	256	14	∗	∗	NOUN
ejpam-6327	256	15	z	z	PROPN
ejpam-6327	256	16	)	)	PUNCT
ejpam-6327	256	17	,	,	PUNCT
ejpam-6327	256	18	µ(y	µ(y	PROPN
ejpam-6327	256	19	∗	∗	PROPN
ejpam-6327	256	20	z	z	PROPN
ejpam-6327	256	21	)	)	PUNCT
ejpam-6327	256	22	}	}	PUNCT
ejpam-6327	256	23	.	.	PUNCT
ejpam-6327	257	1	thus	thus	ADV
ejpam-6327	257	2	,	,	PUNCT
ejpam-6327	257	3	(	(	PUNCT
ejpam-6327	257	4	f4	f4	NOUN
ejpam-6327	257	5	)	)	PUNCT
ejpam-6327	257	6	holds	hold	NOUN
ejpam-6327	257	7	.	.	PUNCT
ejpam-6327	258	1	hence	hence	ADV
ejpam-6327	258	2	,	,	PUNCT
ejpam-6327	258	3	µ	µ	X
ejpam-6327	258	4	is	be	AUX
ejpam-6327	258	5	a	a	DET
ejpam-6327	258	6	fuzzy	fuzzy	ADJ
ejpam-6327	258	7	ks	ks	NOUN
ejpam-6327	258	8	-	-	ADJ
ejpam-6327	258	9	p	p	NOUN
ejpam-6327	258	10	-	-	PUNCT
ejpam-6327	258	11	ideal	ideal	NOUN
ejpam-6327	258	12	.	.	PUNCT
ejpam-6327	259	1	moreover	moreover	ADV
ejpam-6327	259	2	,	,	PUNCT
ejpam-6327	259	3	by	by	ADP
ejpam-6327	259	4	proposition	proposition	NOUN
ejpam-6327	259	5	1	1	NUM
ejpam-6327	259	6	,	,	PUNCT
ejpam-6327	259	7	lemma	lemma	PROPN
ejpam-6327	259	8	1(i	1(i	NUM
ejpam-6327	259	9	)	)	PUNCT
ejpam-6327	259	10	and	and	CCONJ
ejpam-6327	259	11	theorem	theorem	VERB
ejpam-6327	259	12	8	8	NUM
ejpam-6327	259	13	,	,	PUNCT
ejpam-6327	259	14	µ(x	µ(x	ADJ
ejpam-6327	259	15	∗	∗	NOUN
ejpam-6327	259	16	y	y	NOUN
ejpam-6327	259	17	)	)	PUNCT
ejpam-6327	259	18	≤	≤	NOUN
ejpam-6327	259	19	µ((x	µ((x	NOUN
ejpam-6327	259	20	∗	∗	NOUN
ejpam-6327	259	21	(	(	PUNCT
ejpam-6327	259	22	y	y	PROPN
ejpam-6327	259	23	∗	∗	NOUN
ejpam-6327	259	24	(	(	PUNCT
ejpam-6327	259	25	y	y	PROPN
ejpam-6327	259	26	∗	∗	NOUN
ejpam-6327	259	27	x	x	NOUN
ejpam-6327	259	28	)	)	PUNCT
ejpam-6327	259	29	)	)	PUNCT
ejpam-6327	259	30	)	)	PUNCT
ejpam-6327	260	1	∗	∗	NOUN
ejpam-6327	260	2	(	(	PUNCT
ejpam-6327	260	3	y	y	PROPN
ejpam-6327	260	4	∗	∗	NOUN
ejpam-6327	260	5	(	(	PUNCT
ejpam-6327	260	6	x	x	SYM
ejpam-6327	260	7	∗	∗	NOUN
ejpam-6327	260	8	(	(	PUNCT
ejpam-6327	260	9	y	y	PROPN
ejpam-6327	260	10	∗	∗	NOUN
ejpam-6327	260	11	(	(	PUNCT
ejpam-6327	260	12	y	y	PROPN
ejpam-6327	260	13	∗	∗	NOUN
ejpam-6327	260	14	x	x	NOUN
ejpam-6327	260	15	)	)	PUNCT
ejpam-6327	260	16	)	)	PUNCT
ejpam-6327	260	17	)	)	PUNCT
ejpam-6327	260	18	)	)	PUNCT
ejpam-6327	260	19	)	)	PUNCT
ejpam-6327	261	1	=	=	PUNCT
ejpam-6327	261	2	µ(x	µ(x	ADJ
ejpam-6327	261	3	∗	∗	NOUN
ejpam-6327	261	4	(	(	PUNCT
ejpam-6327	261	5	y	y	PROPN
ejpam-6327	261	6	∗	∗	NOUN
ejpam-6327	261	7	(	(	PUNCT
ejpam-6327	261	8	y	y	PROPN
ejpam-6327	261	9	∗	∗	NOUN
ejpam-6327	261	10	x	x	NOUN
ejpam-6327	261	11	)	)	PUNCT
ejpam-6327	261	12	)	)	PUNCT
ejpam-6327	261	13	)	)	PUNCT
ejpam-6327	261	14	for	for	ADP
ejpam-6327	261	15	all	all	DET
ejpam-6327	261	16	x	x	NOUN
ejpam-6327	261	17	,	,	PUNCT
ejpam-6327	261	18	y	y	PROPN
ejpam-6327	261	19	∈	∈	PROPN
ejpam-6327	261	20	x.	x.	NOUN
ejpam-6327	261	21	by	by	ADP
ejpam-6327	261	22	remark	remark	NOUN
ejpam-6327	261	23	1	1	NUM
ejpam-6327	261	24	,	,	PUNCT
ejpam-6327	261	25	y	y	PROPN
ejpam-6327	261	26	∗	∗	NOUN
ejpam-6327	261	27	(	(	PUNCT
ejpam-6327	261	28	y	y	PROPN
ejpam-6327	261	29	∗	∗	X
ejpam-6327	261	30	x	x	NOUN
ejpam-6327	261	31	)	)	PUNCT
ejpam-6327	261	32	≤	≤	PUNCT
ejpam-6327	262	1	y	y	PROPN
ejpam-6327	262	2	implies	imply	VERB
ejpam-6327	262	3	that	that	SCONJ
ejpam-6327	262	4	x	x	SYM
ejpam-6327	262	5	∗	∗	VERB
ejpam-6327	262	6	y	y	NOUN
ejpam-6327	262	7	≤	≤	NUM
ejpam-6327	262	8	x	x	PUNCT
ejpam-6327	262	9	∗	∗	NOUN
ejpam-6327	262	10	(	(	PUNCT
ejpam-6327	262	11	y	y	PROPN
ejpam-6327	262	12	∗	∗	NOUN
ejpam-6327	262	13	(	(	PUNCT
ejpam-6327	262	14	y	y	PROPN
ejpam-6327	262	15	∗	∗	NOUN
ejpam-6327	262	16	x	x	NOUN
ejpam-6327	262	17	)	)	PUNCT
ejpam-6327	262	18	)	)	PUNCT
ejpam-6327	262	19	.	.	PUNCT
ejpam-6327	263	1	thus	thus	ADV
ejpam-6327	263	2	,	,	PUNCT
ejpam-6327	263	3	by	by	ADP
ejpam-6327	263	4	lemma	lemma	PROPN
ejpam-6327	263	5	1(i	1(i	NUM
ejpam-6327	263	6	)	)	PUNCT
ejpam-6327	263	7	,	,	PUNCT
ejpam-6327	263	8	µ(x∗y	µ(x∗y	ADJ
ejpam-6327	263	9	)	)	PUNCT
ejpam-6327	263	10	≤	≤	NUM
ejpam-6327	263	11	µ(x∗	µ(x∗	NOUN
ejpam-6327	263	12	(	(	PUNCT
ejpam-6327	263	13	y	y	PROPN
ejpam-6327	263	14	∗	∗	NOUN
ejpam-6327	263	15	(	(	PUNCT
ejpam-6327	263	16	y	y	NOUN
ejpam-6327	263	17	∗x	∗x	PROPN
ejpam-6327	263	18	)	)	PUNCT
ejpam-6327	263	19	)	)	PUNCT
ejpam-6327	263	20	)	)	PUNCT
ejpam-6327	263	21	.	.	PUNCT
ejpam-6327	264	1	hence	hence	ADV
ejpam-6327	264	2	,	,	PUNCT
ejpam-6327	264	3	µ(x∗y	µ(x∗y	ADJ
ejpam-6327	264	4	)	)	PUNCT
ejpam-6327	265	1	=	=	SYM
ejpam-6327	265	2	µ(x∗	µ(x∗	NOUN
ejpam-6327	265	3	(	(	PUNCT
ejpam-6327	265	4	y	y	PROPN
ejpam-6327	265	5	∗	∗	NOUN
ejpam-6327	265	6	(	(	PUNCT
ejpam-6327	265	7	y	y	NOUN
ejpam-6327	265	8	∗x	∗x	PROPN
ejpam-6327	265	9	)	)	PUNCT
ejpam-6327	265	10	)	)	PUNCT
ejpam-6327	265	11	)	)	PUNCT
ejpam-6327	265	12	.	.	PUNCT
ejpam-6327	266	1	therefore	therefore	ADV
ejpam-6327	266	2	,	,	PUNCT
ejpam-6327	266	3	by	by	ADP
ejpam-6327	266	4	theorem	theorem	NOUN
ejpam-6327	266	5	5	5	NUM
ejpam-6327	266	6	,	,	PUNCT
ejpam-6327	266	7	µ	µ	PRON
ejpam-6327	266	8	is	be	AUX
ejpam-6327	266	9	a	a	DET
ejpam-6327	266	10	fuzzy	fuzzy	ADJ
ejpam-6327	266	11	commutative	commutative	ADJ
ejpam-6327	266	12	ks	ks	NOUN
ejpam-6327	266	13	-	-	PUNCT
ejpam-6327	266	14	ideal	ideal	NOUN
ejpam-6327	266	15	of	of	ADP
ejpam-6327	266	16	x.	x.	NOUN
ejpam-6327	266	17	conversely	conversely	ADV
ejpam-6327	266	18	,	,	PUNCT
ejpam-6327	266	19	suppose	suppose	VERB
ejpam-6327	266	20	µ	µ	X
ejpam-6327	266	21	is	be	AUX
ejpam-6327	266	22	both	both	PRON
ejpam-6327	266	23	fuzzy	fuzzy	ADJ
ejpam-6327	266	24	ks	ks	NOUN
ejpam-6327	266	25	-	-	ADJ
ejpam-6327	266	26	p	p	NOUN
ejpam-6327	266	27	-	-	PUNCT
ejpam-6327	266	28	ideal	ideal	ADJ
ejpam-6327	266	29	and	and	CCONJ
ejpam-6327	266	30	fuzzy	fuzzy	ADJ
ejpam-6327	266	31	commutative	commutative	ADJ
ejpam-6327	266	32	ks	ks	NOUN
ejpam-6327	266	33	-	-	PUNCT
ejpam-6327	266	34	ideal	ideal	NOUN
ejpam-6327	266	35	of	of	ADP
ejpam-6327	266	36	x.	x.	NOUN
ejpam-6327	266	37	by	by	ADP
ejpam-6327	266	38	definition	definition	NOUN
ejpam-6327	266	39	1(i	1(i	NUM
ejpam-6327	266	40	)	)	PUNCT
ejpam-6327	266	41	,	,	PUNCT
ejpam-6327	266	42	(	(	PUNCT
ejpam-6327	266	43	y	y	NOUN
ejpam-6327	266	44	∗	∗	X
ejpam-6327	266	45	(	(	PUNCT
ejpam-6327	266	46	y	y	NOUN
ejpam-6327	266	47	∗x	∗x	NOUN
ejpam-6327	266	48	)	)	PUNCT
ejpam-6327	266	49	)	)	PUNCT
ejpam-6327	266	50	∗	∗	NOUN
ejpam-6327	266	51	(	(	PUNCT
ejpam-6327	266	52	y	y	NOUN
ejpam-6327	266	53	∗x	∗x	NOUN
ejpam-6327	266	54	)	)	PUNCT
ejpam-6327	266	55	≤	≤	NUM
ejpam-6327	266	56	x	x	X
ejpam-6327	266	57	∗	∗	NOUN
ejpam-6327	266	58	(	(	PUNCT
ejpam-6327	266	59	y	y	NOUN
ejpam-6327	266	60	∗x	∗x	PROPN
ejpam-6327	266	61	)	)	PUNCT
ejpam-6327	266	62	for	for	ADP
ejpam-6327	266	63	all	all	DET
ejpam-6327	266	64	x	x	NOUN
ejpam-6327	266	65	,	,	PUNCT
ejpam-6327	266	66	y	y	PROPN
ejpam-6327	266	67	∈	∈	PROPN
ejpam-6327	266	68	x.	x.	NOUN
ejpam-6327	266	69	thus	thus	ADV
ejpam-6327	266	70	,	,	PUNCT
ejpam-6327	266	71	by	by	ADP
ejpam-6327	266	72	lemma	lemma	PROPN
ejpam-6327	266	73	1(i	1(i	NUM
ejpam-6327	266	74	)	)	PUNCT
ejpam-6327	266	75	,	,	PUNCT
ejpam-6327	266	76	µ(x∗(y∗x	µ(x∗(y∗x	NUM
ejpam-6327	266	77	)	)	PUNCT
ejpam-6327	266	78	)	)	PUNCT
ejpam-6327	266	79	≤	≤	NUM
ejpam-6327	266	80	µ((y∗(y∗x))∗(y∗x	µ((y∗(y∗x))∗(y∗x	NOUN
ejpam-6327	266	81	)	)	PUNCT
ejpam-6327	266	82	)	)	PUNCT
ejpam-6327	266	83	.	.	PUNCT
ejpam-6327	267	1	by	by	ADP
ejpam-6327	267	2	theorem	theorem	NOUN
ejpam-6327	267	3	3	3	NUM
ejpam-6327	267	4	,	,	PUNCT
ejpam-6327	267	5	µ((y∗(y∗x))∗(y∗x	µ((y∗(y∗x))∗(y∗x	NUM
ejpam-6327	267	6	)	)	PUNCT
ejpam-6327	267	7	)	)	PUNCT
ejpam-6327	267	8	=	=	SYM
ejpam-6327	267	9	µ(y∗(y∗x	µ(y∗(y∗x	X
ejpam-6327	267	10	)	)	PUNCT
ejpam-6327	267	11	)	)	PUNCT
ejpam-6327	267	12	.	.	PUNCT
ejpam-6327	268	1	thus	thus	ADV
ejpam-6327	268	2	,	,	PUNCT
ejpam-6327	268	3	we	we	PRON
ejpam-6327	268	4	have	have	VERB
ejpam-6327	268	5	µ(x	µ(x	ADJ
ejpam-6327	268	6	∗	∗	NOUN
ejpam-6327	268	7	(	(	PUNCT
ejpam-6327	268	8	y	y	PROPN
ejpam-6327	268	9	∗	∗	NOUN
ejpam-6327	268	10	x	x	NOUN
ejpam-6327	268	11	)	)	PUNCT
ejpam-6327	268	12	)	)	PUNCT
ejpam-6327	268	13	≤	≤	PROPN
ejpam-6327	268	14	µ(y	µ(y	X
ejpam-6327	268	15	∗	∗	NOUN
ejpam-6327	268	16	(	(	PUNCT
ejpam-6327	268	17	y	y	PROPN
ejpam-6327	268	18	∗	∗	NOUN
ejpam-6327	268	19	x	x	NOUN
ejpam-6327	268	20	)	)	PUNCT
ejpam-6327	268	21	)	)	PUNCT
ejpam-6327	268	22	.	.	PUNCT
ejpam-6327	269	1	(	(	PUNCT
ejpam-6327	269	2	1	1	X
ejpam-6327	269	3	)	)	PUNCT
ejpam-6327	269	4	h.	h.	NOUN
ejpam-6327	269	5	sarapuddin	sarapuddin	PROPN
ejpam-6327	269	6	,	,	PUNCT
ejpam-6327	269	7	j.	j.	PROPN
ejpam-6327	269	8	vilela	vilela	PROPN
ejpam-6327	269	9	/	/	SYM
ejpam-6327	269	10	eur	eur	PROPN
ejpam-6327	269	11	.	.	PUNCT
ejpam-6327	270	1	j.	j.	PROPN
ejpam-6327	270	2	pure	pure	PROPN
ejpam-6327	270	3	appl	appl	PROPN
ejpam-6327	270	4	.	.	PROPN
ejpam-6327	270	5	math	math	PROPN
ejpam-6327	270	6	,	,	PUNCT
ejpam-6327	270	7	18	18	NUM
ejpam-6327	270	8	(	(	PUNCT
ejpam-6327	270	9	3	3	NUM
ejpam-6327	270	10	)	)	PUNCT
ejpam-6327	270	11	(	(	PUNCT
ejpam-6327	270	12	2025	2025	NUM
ejpam-6327	270	13	)	)	PUNCT
ejpam-6327	270	14	,	,	PUNCT
ejpam-6327	270	15	6327	6327	NUM
ejpam-6327	270	16	10	10	NUM
ejpam-6327	270	17	of	of	ADP
ejpam-6327	270	18	23	23	NUM
ejpam-6327	270	19	now	now	ADV
ejpam-6327	270	20	,	,	PUNCT
ejpam-6327	270	21	by	by	ADP
ejpam-6327	270	22	remark	remark	NOUN
ejpam-6327	270	23	1	1	NUM
ejpam-6327	270	24	,	,	PUNCT
ejpam-6327	270	25	y	y	PROPN
ejpam-6327	270	26	∗	∗	NOUN
ejpam-6327	270	27	x	x	X
ejpam-6327	270	28	≤	≤	NUM
ejpam-6327	270	29	y	y	PROPN
ejpam-6327	270	30	implies	imply	VERB
ejpam-6327	270	31	that	that	SCONJ
ejpam-6327	270	32	x	x	SYM
ejpam-6327	270	33	∗	∗	VERB
ejpam-6327	270	34	y	y	NOUN
ejpam-6327	270	35	≤	≤	NUM
ejpam-6327	270	36	x	x	PUNCT
ejpam-6327	270	37	∗	∗	NOUN
ejpam-6327	270	38	(	(	PUNCT
ejpam-6327	270	39	y	y	PROPN
ejpam-6327	270	40	∗	∗	X
ejpam-6327	270	41	x	x	NOUN
ejpam-6327	270	42	)	)	PUNCT
ejpam-6327	270	43	for	for	ADP
ejpam-6327	270	44	all	all	DET
ejpam-6327	270	45	x	x	NOUN
ejpam-6327	270	46	,	,	PUNCT
ejpam-6327	270	47	y	y	PROPN
ejpam-6327	270	48	∈	∈	PROPN
ejpam-6327	270	49	x.	x.	NOUN
ejpam-6327	271	1	thus	thus	ADV
ejpam-6327	271	2	,	,	PUNCT
ejpam-6327	271	3	by	by	ADP
ejpam-6327	271	4	lemma	lemma	PROPN
ejpam-6327	271	5	1(i	1(i	NUM
ejpam-6327	271	6	)	)	PUNCT
ejpam-6327	271	7	,	,	PUNCT
ejpam-6327	271	8	µ(x	µ(x	ADJ
ejpam-6327	271	9	∗	∗	NOUN
ejpam-6327	271	10	(	(	PUNCT
ejpam-6327	271	11	y	y	PROPN
ejpam-6327	271	12	∗	∗	NOUN
ejpam-6327	271	13	x	x	NOUN
ejpam-6327	271	14	)	)	PUNCT
ejpam-6327	271	15	)	)	PUNCT
ejpam-6327	271	16	≤	≤	NOUN
ejpam-6327	271	17	µ(x	µ(x	ADJ
ejpam-6327	271	18	∗	∗	NOUN
ejpam-6327	271	19	y	y	NOUN
ejpam-6327	271	20	)	)	PUNCT
ejpam-6327	271	21	.	.	PUNCT
ejpam-6327	272	1	since	since	SCONJ
ejpam-6327	272	2	µ	µ	NOUN
ejpam-6327	272	3	is	be	AUX
ejpam-6327	272	4	a	a	DET
ejpam-6327	272	5	fuzzy	fuzzy	ADJ
ejpam-6327	272	6	commutative	commutative	ADJ
ejpam-6327	272	7	ks	ks	NOUN
ejpam-6327	272	8	-	-	PUNCT
ejpam-6327	272	9	ideal	ideal	NOUN
ejpam-6327	272	10	of	of	ADP
ejpam-6327	272	11	x	x	PRON
ejpam-6327	272	12	,	,	PUNCT
ejpam-6327	272	13	by	by	ADP
ejpam-6327	272	14	theorem	theorem	NOUN
ejpam-6327	272	15	5	5	NUM
ejpam-6327	272	16	,	,	PUNCT
ejpam-6327	272	17	µ(x	µ(x	ADJ
ejpam-6327	272	18	∗	∗	NOUN
ejpam-6327	272	19	y	y	NOUN
ejpam-6327	272	20	)	)	PUNCT
ejpam-6327	273	1	=	=	PUNCT
ejpam-6327	273	2	µ(x	µ(x	ADJ
ejpam-6327	273	3	∗	∗	NOUN
ejpam-6327	273	4	(	(	PUNCT
ejpam-6327	273	5	y	y	PROPN
ejpam-6327	273	6	∗	∗	NOUN
ejpam-6327	273	7	(	(	PUNCT
ejpam-6327	273	8	y	y	PROPN
ejpam-6327	273	9	∗	∗	NOUN
ejpam-6327	273	10	x	x	NOUN
ejpam-6327	273	11	)	)	PUNCT
ejpam-6327	273	12	)	)	PUNCT
ejpam-6327	273	13	)	)	PUNCT
ejpam-6327	273	14	.	.	PUNCT
ejpam-6327	274	1	hence	hence	ADV
ejpam-6327	274	2	,	,	PUNCT
ejpam-6327	274	3	for	for	ADP
ejpam-6327	274	4	all	all	DET
ejpam-6327	274	5	x	x	NOUN
ejpam-6327	274	6	,	,	PUNCT
ejpam-6327	274	7	y	y	PROPN
ejpam-6327	274	8	∈	∈	PROPN
ejpam-6327	274	9	x	x	PROPN
ejpam-6327	274	10	,	,	PUNCT
ejpam-6327	274	11	µ(x	µ(x	ADJ
ejpam-6327	274	12	∗	∗	NOUN
ejpam-6327	274	13	(	(	PUNCT
ejpam-6327	274	14	y	y	PROPN
ejpam-6327	274	15	∗	∗	NOUN
ejpam-6327	274	16	x	x	NOUN
ejpam-6327	274	17	)	)	PUNCT
ejpam-6327	274	18	)	)	PUNCT
ejpam-6327	274	19	≤	≤	NOUN
ejpam-6327	274	20	µ(x	µ(x	ADJ
ejpam-6327	274	21	∗	∗	NOUN
ejpam-6327	274	22	(	(	PUNCT
ejpam-6327	274	23	y	y	PROPN
ejpam-6327	274	24	∗	∗	NOUN
ejpam-6327	274	25	(	(	PUNCT
ejpam-6327	274	26	y	y	PROPN
ejpam-6327	274	27	∗	∗	NOUN
ejpam-6327	274	28	x	x	NOUN
ejpam-6327	274	29	)	)	PUNCT
ejpam-6327	274	30	)	)	PUNCT
ejpam-6327	274	31	)	)	PUNCT
ejpam-6327	274	32	.	.	PUNCT
ejpam-6327	275	1	(	(	PUNCT
ejpam-6327	275	2	2	2	X
ejpam-6327	275	3	)	)	PUNCT
ejpam-6327	275	4	note	note	NOUN
ejpam-6327	275	5	that	that	SCONJ
ejpam-6327	275	6	by	by	ADP
ejpam-6327	275	7	definition	definition	NOUN
ejpam-6327	275	8	1(ii	1(ii	NUM
ejpam-6327	275	9	)	)	PUNCT
ejpam-6327	275	10	,	,	PUNCT
ejpam-6327	275	11	x	x	X
ejpam-6327	275	12	∗	∗	NOUN
ejpam-6327	275	13	(	(	PUNCT
ejpam-6327	275	14	x	x	SYM
ejpam-6327	275	15	∗	∗	NOUN
ejpam-6327	275	16	(	(	PUNCT
ejpam-6327	275	17	y	y	PROPN
ejpam-6327	275	18	∗	∗	NOUN
ejpam-6327	275	19	(	(	PUNCT
ejpam-6327	275	20	y	y	PROPN
ejpam-6327	275	21	∗	∗	NOUN
ejpam-6327	275	22	x	x	NOUN
ejpam-6327	275	23	)	)	PUNCT
ejpam-6327	275	24	)	)	PUNCT
ejpam-6327	275	25	)	)	PUNCT
ejpam-6327	276	1	≤	≤	NUM
ejpam-6327	277	1	y	y	PROPN
ejpam-6327	277	2	∗	∗	NOUN
ejpam-6327	277	3	(	(	PUNCT
ejpam-6327	277	4	y	y	PROPN
ejpam-6327	277	5	∗	∗	NOUN
ejpam-6327	277	6	x	x	NOUN
ejpam-6327	277	7	)	)	PUNCT
ejpam-6327	277	8	.	.	PUNCT
ejpam-6327	278	1	thus	thus	ADV
ejpam-6327	278	2	,	,	PUNCT
ejpam-6327	278	3	by	by	ADP
ejpam-6327	278	4	lemma	lemma	PROPN
ejpam-6327	278	5	1(ii	1(ii	NUM
ejpam-6327	278	6	)	)	PUNCT
ejpam-6327	278	7	,	,	PUNCT
ejpam-6327	278	8	µ(x	µ(x	X
ejpam-6327	278	9	)	)	PUNCT
ejpam-6327	278	10	≥	≥	NOUN
ejpam-6327	278	11	min{µ(x	min{µ(x	NOUN
ejpam-6327	278	12	∗	∗	NOUN
ejpam-6327	278	13	(	(	PUNCT
ejpam-6327	278	14	y	y	PROPN
ejpam-6327	278	15	∗	∗	NOUN
ejpam-6327	278	16	(	(	PUNCT
ejpam-6327	278	17	y	y	PROPN
ejpam-6327	278	18	∗	∗	NOUN
ejpam-6327	278	19	x	x	NOUN
ejpam-6327	278	20	)	)	PUNCT
ejpam-6327	278	21	)	)	PUNCT
ejpam-6327	278	22	)	)	PUNCT
ejpam-6327	278	23	,	,	PUNCT
ejpam-6327	278	24	µ(y	µ(y	PROPN
ejpam-6327	278	25	∗	∗	NOUN
ejpam-6327	278	26	(	(	PUNCT
ejpam-6327	278	27	y	y	PROPN
ejpam-6327	278	28	∗	∗	NOUN
ejpam-6327	278	29	x	x	NOUN
ejpam-6327	278	30	)	)	PUNCT
ejpam-6327	278	31	)	)	PUNCT
ejpam-6327	278	32	}	}	PUNCT
ejpam-6327	278	33	.	.	PUNCT
ejpam-6327	279	1	combining	combine	VERB
ejpam-6327	279	2	(	(	PUNCT
ejpam-6327	279	3	1	1	NUM
ejpam-6327	279	4	)	)	PUNCT
ejpam-6327	279	5	and	and	CCONJ
ejpam-6327	279	6	(	(	PUNCT
ejpam-6327	279	7	2	2	NUM
ejpam-6327	279	8	)	)	PUNCT
ejpam-6327	279	9	,	,	PUNCT
ejpam-6327	279	10	we	we	PRON
ejpam-6327	279	11	get	get	VERB
ejpam-6327	279	12	µ(x	µ(x	NOUN
ejpam-6327	279	13	)	)	PUNCT
ejpam-6327	279	14	≥	≥	NOUN
ejpam-6327	280	1	min{µ(x	min{µ(x	NOUN
ejpam-6327	280	2	∗	∗	NOUN
ejpam-6327	280	3	(	(	PUNCT
ejpam-6327	280	4	y	y	PROPN
ejpam-6327	280	5	∗	∗	NOUN
ejpam-6327	280	6	(	(	PUNCT
ejpam-6327	280	7	y	y	PROPN
ejpam-6327	280	8	∗	∗	NOUN
ejpam-6327	280	9	x	x	NOUN
ejpam-6327	280	10	)	)	PUNCT
ejpam-6327	280	11	)	)	PUNCT
ejpam-6327	280	12	)	)	PUNCT
ejpam-6327	280	13	,	,	PUNCT
ejpam-6327	280	14	µ(y	µ(y	PROPN
ejpam-6327	280	15	∗	∗	NOUN
ejpam-6327	280	16	(	(	PUNCT
ejpam-6327	280	17	y	y	PROPN
ejpam-6327	280	18	∗	∗	NOUN
ejpam-6327	280	19	x	x	NOUN
ejpam-6327	280	20	)	)	PUNCT
ejpam-6327	280	21	)	)	PUNCT
ejpam-6327	280	22	}	}	PUNCT
ejpam-6327	280	23	≥	≥	AUX
ejpam-6327	280	24	µ(x	µ(x	ADJ
ejpam-6327	280	25	∗	∗	NOUN
ejpam-6327	280	26	(	(	PUNCT
ejpam-6327	280	27	y	y	PROPN
ejpam-6327	280	28	∗	∗	NOUN
ejpam-6327	280	29	x	x	NOUN
ejpam-6327	280	30	)	)	PUNCT
ejpam-6327	280	31	)	)	PUNCT
ejpam-6327	280	32	.	.	PUNCT
ejpam-6327	281	1	since	since	SCONJ
ejpam-6327	281	2	x	x	PROPN
ejpam-6327	281	3	∗	∗	NOUN
ejpam-6327	281	4	(	(	PUNCT
ejpam-6327	281	5	y	y	PROPN
ejpam-6327	281	6	∗	∗	X
ejpam-6327	281	7	x	x	NOUN
ejpam-6327	281	8	)	)	PUNCT
ejpam-6327	281	9	≤	≤	NUM
ejpam-6327	281	10	x	x	X
ejpam-6327	281	11	,	,	PUNCT
ejpam-6327	281	12	it	it	PRON
ejpam-6327	281	13	follows	follow	VERB
ejpam-6327	281	14	that	that	SCONJ
ejpam-6327	281	15	by	by	ADP
ejpam-6327	281	16	lemma	lemma	PROPN
ejpam-6327	281	17	1(i	1(i	NUM
ejpam-6327	281	18	)	)	PUNCT
ejpam-6327	281	19	,	,	PUNCT
ejpam-6327	281	20	µ(x	µ(x	NOUN
ejpam-6327	281	21	)	)	PUNCT
ejpam-6327	281	22	≤	≤	NOUN
ejpam-6327	281	23	µ(x	µ(x	ADJ
ejpam-6327	281	24	∗	∗	NOUN
ejpam-6327	281	25	(	(	PUNCT
ejpam-6327	281	26	y	y	PROPN
ejpam-6327	281	27	∗	∗	NOUN
ejpam-6327	281	28	x	x	NOUN
ejpam-6327	281	29	)	)	PUNCT
ejpam-6327	281	30	)	)	PUNCT
ejpam-6327	281	31	.	.	PUNCT
ejpam-6327	282	1	thus	thus	ADV
ejpam-6327	282	2	,	,	PUNCT
ejpam-6327	282	3	µ(x	µ(x	X
ejpam-6327	282	4	)	)	PUNCT
ejpam-6327	282	5	=	=	NOUN
ejpam-6327	282	6	µ(x	µ(x	ADJ
ejpam-6327	282	7	∗	∗	NOUN
ejpam-6327	282	8	(	(	PUNCT
ejpam-6327	282	9	y	y	PROPN
ejpam-6327	282	10	∗	∗	NOUN
ejpam-6327	282	11	x	x	NOUN
ejpam-6327	282	12	)	)	PUNCT
ejpam-6327	282	13	)	)	PUNCT
ejpam-6327	282	14	for	for	ADP
ejpam-6327	282	15	all	all	DET
ejpam-6327	282	16	x	x	NOUN
ejpam-6327	282	17	,	,	PUNCT
ejpam-6327	282	18	y	y	PROPN
ejpam-6327	282	19	∈	∈	PROPN
ejpam-6327	282	20	x.	x.	NOUN
ejpam-6327	282	21	therefore	therefore	ADV
ejpam-6327	282	22	,	,	PUNCT
ejpam-6327	282	23	by	by	ADP
ejpam-6327	282	24	theorem	theorem	NOUN
ejpam-6327	282	25	8	8	NUM
ejpam-6327	282	26	,	,	PUNCT
ejpam-6327	282	27	µ	µ	PRON
ejpam-6327	282	28	is	be	AUX
ejpam-6327	282	29	a	a	DET
ejpam-6327	282	30	fuzzy	fuzzy	ADJ
ejpam-6327	282	31	implicative	implicative	ADJ
ejpam-6327	282	32	ks	ks	NOUN
ejpam-6327	282	33	-	-	PUNCT
ejpam-6327	282	34	ideal	ideal	NOUN
ejpam-6327	282	35	of	of	ADP
ejpam-6327	282	36	x.	x.	NOUN
ejpam-6327	282	37	4	4	NUM
ejpam-6327	282	38	.	.	PUNCT
ejpam-6327	283	1	quotient	quotient	NOUN
ejpam-6327	283	2	ks	k	NOUN
ejpam-6327	283	3	-	-	PUNCT
ejpam-6327	283	4	semigroups	semigroup	NOUN
ejpam-6327	283	5	induced	induce	VERB
ejpam-6327	283	6	by	by	ADP
ejpam-6327	283	7	fuzzy	fuzzy	ADJ
ejpam-6327	283	8	ks	ks	NOUN
ejpam-6327	283	9	-	-	PUNCT
ejpam-6327	283	10	ideals	ideal	NOUN
ejpam-6327	283	11	recall	recall	VERB
ejpam-6327	283	12	that	that	SCONJ
ejpam-6327	283	13	in	in	ADP
ejpam-6327	283	14	[	[	X
ejpam-6327	283	15	5	5	NUM
ejpam-6327	283	16	]	]	PUNCT
ejpam-6327	283	17	,	,	PUNCT
ejpam-6327	283	18	if	if	SCONJ
ejpam-6327	283	19	a	a	PRON
ejpam-6327	283	20	is	be	AUX
ejpam-6327	283	21	an	an	DET
ejpam-6327	283	22	ideal	ideal	NOUN
ejpam-6327	283	23	of	of	ADP
ejpam-6327	283	24	a	a	DET
ejpam-6327	283	25	ks	ks	NOUN
ejpam-6327	283	26	-	-	PUNCT
ejpam-6327	283	27	semigroup	semigroup	NOUN
ejpam-6327	283	28	x	x	NOUN
ejpam-6327	283	29	,	,	PUNCT
ejpam-6327	283	30	then	then	ADV
ejpam-6327	283	31	the	the	DET
ejpam-6327	283	32	relation	relation	NOUN
ejpam-6327	283	33	“	"	PUNCT
ejpam-6327	283	34	∼a	∼a	PROPN
ejpam-6327	283	35	”	"	PUNCT
ejpam-6327	283	36	on	on	ADP
ejpam-6327	283	37	x	x	PUNCT
ejpam-6327	283	38	defined	define	VERB
ejpam-6327	283	39	by	by	ADP
ejpam-6327	283	40	x	x	PROPN
ejpam-6327	283	41	∼a	∼a	PROPN
ejpam-6327	283	42	y	y	PROPN
ejpam-6327	284	1	if	if	SCONJ
ejpam-6327	284	2	and	and	CCONJ
ejpam-6327	284	3	only	only	ADV
ejpam-6327	284	4	if	if	SCONJ
ejpam-6327	284	5	x	x	X
ejpam-6327	284	6	∗	∗	VERB
ejpam-6327	284	7	y	y	PROPN
ejpam-6327	284	8	∈	∈	PROPN
ejpam-6327	284	9	a	a	DET
ejpam-6327	284	10	and	and	CCONJ
ejpam-6327	284	11	y	y	PROPN
ejpam-6327	284	12	∗	∗	NOUN
ejpam-6327	285	1	x	x	PUNCT
ejpam-6327	285	2	∈	∈	PROPN
ejpam-6327	285	3	a	a	PRON
ejpam-6327	285	4	is	be	AUX
ejpam-6327	285	5	a	a	DET
ejpam-6327	285	6	congruence	congruence	NOUN
ejpam-6327	285	7	relation	relation	NOUN
ejpam-6327	285	8	on	on	ADP
ejpam-6327	285	9	x.	x.	NOUN
ejpam-6327	285	10	denote	denote	NOUN
ejpam-6327	285	11	ax	ax	NOUN
ejpam-6327	285	12	as	as	ADP
ejpam-6327	285	13	the	the	DET
ejpam-6327	285	14	equivalence	equivalence	NOUN
ejpam-6327	285	15	class	class	NOUN
ejpam-6327	285	16	containing	contain	VERB
ejpam-6327	285	17	x	x	SYM
ejpam-6327	285	18	∈	∈	PROPN
ejpam-6327	285	19	x	x	X
ejpam-6327	285	20	and	and	CCONJ
ejpam-6327	285	21	x	x	X
ejpam-6327	285	22	/	/	SYM
ejpam-6327	285	23	a	a	PRON
ejpam-6327	285	24	as	as	ADP
ejpam-6327	285	25	the	the	DET
ejpam-6327	285	26	set	set	NOUN
ejpam-6327	285	27	of	of	ADP
ejpam-6327	285	28	all	all	DET
ejpam-6327	285	29	equivalence	equivalence	NOUN
ejpam-6327	285	30	classes	class	NOUN
ejpam-6327	285	31	of	of	ADP
ejpam-6327	285	32	x	x	PUNCT
ejpam-6327	285	33	with	with	ADP
ejpam-6327	285	34	respect	respect	NOUN
ejpam-6327	285	35	to	to	ADP
ejpam-6327	285	36	“	"	PUNCT
ejpam-6327	285	37	∼a	∼a	PROPN
ejpam-6327	285	38	”	"	PUNCT
ejpam-6327	285	39	,	,	PUNCT
ejpam-6327	285	40	that	that	ADV
ejpam-6327	285	41	is	is	ADV
ejpam-6327	285	42	,	,	PUNCT
ejpam-6327	285	43	ax	ax	NOUN
ejpam-6327	285	44	=	=	PUNCT
ejpam-6327	285	45	{	{	PUNCT
ejpam-6327	285	46	y	y	PROPN
ejpam-6327	285	47	∈	∈	PROPN
ejpam-6327	285	48	x	x	X
ejpam-6327	285	49	:	:	PUNCT
ejpam-6327	285	50	x	x	X
ejpam-6327	285	51	∼a	∼a	PROPN
ejpam-6327	285	52	y	y	NOUN
ejpam-6327	285	53	}	}	PUNCT
ejpam-6327	285	54	and	and	CCONJ
ejpam-6327	285	55	x	x	X
ejpam-6327	285	56	/	/	SYM
ejpam-6327	285	57	a	a	NOUN
ejpam-6327	285	58	=	=	X
ejpam-6327	285	59	{	{	PUNCT
ejpam-6327	285	60	ax	ax	NOUN
ejpam-6327	285	61	:	:	PUNCT
ejpam-6327	285	62	x	x	PUNCT
ejpam-6327	285	63	∈	∈	NOUN
ejpam-6327	285	64	x	x	X
ejpam-6327	285	65	}	}	PUNCT
ejpam-6327	285	66	.	.	PUNCT
ejpam-6327	286	1	furthermore	furthermore	ADV
ejpam-6327	286	2	,	,	PUNCT
ejpam-6327	286	3	(	(	PUNCT
ejpam-6327	286	4	x	x	X
ejpam-6327	286	5	/	/	SYM
ejpam-6327	286	6	a,⊛,⊙	a,⊛,⊙	ADJ
ejpam-6327	286	7	,	,	PUNCT
ejpam-6327	286	8	a0	a0	PROPN
ejpam-6327	286	9	)	)	PUNCT
ejpam-6327	286	10	is	be	AUX
ejpam-6327	286	11	a	a	DET
ejpam-6327	286	12	ks	ks	NOUN
ejpam-6327	286	13	-	-	PUNCT
ejpam-6327	286	14	semigroup	semigroup	NOUN
ejpam-6327	286	15	under	under	ADP
ejpam-6327	286	16	the	the	DET
ejpam-6327	286	17	binary	binary	ADJ
ejpam-6327	286	18	operation	operation	NOUN
ejpam-6327	286	19	ax	ax	NOUN
ejpam-6327	286	20	⊛ay	⊛ay	PROPN
ejpam-6327	287	1	=	=	PUNCT
ejpam-6327	287	2	ax∗y	ax∗y	X
ejpam-6327	287	3	and	and	CCONJ
ejpam-6327	287	4	ax	ax	NOUN
ejpam-6327	287	5	⊙ay	⊙ay	NOUN
ejpam-6327	287	6	=	=	PROPN
ejpam-6327	287	7	axy	axy	PROPN
ejpam-6327	287	8	for	for	ADP
ejpam-6327	287	9	all	all	DET
ejpam-6327	287	10	ax	ax	NOUN
ejpam-6327	287	11	,	,	PUNCT
ejpam-6327	287	12	ay	ay	NOUN
ejpam-6327	287	13	∈	∈	PROPN
ejpam-6327	287	14	x	x	X
ejpam-6327	287	15	/	/	SYM
ejpam-6327	287	16	a.	a.	NOUN
ejpam-6327	287	17	in	in	ADP
ejpam-6327	287	18	this	this	DET
ejpam-6327	287	19	section	section	NOUN
ejpam-6327	287	20	,	,	PUNCT
ejpam-6327	287	21	we	we	PRON
ejpam-6327	287	22	present	present	VERB
ejpam-6327	287	23	the	the	DET
ejpam-6327	287	24	construction	construction	NOUN
ejpam-6327	287	25	of	of	ADP
ejpam-6327	287	26	quotient	quotient	NOUN
ejpam-6327	287	27	ks	k	NOUN
ejpam-6327	287	28	-	-	PUNCT
ejpam-6327	287	29	semigroups	semigroup	NOUN
ejpam-6327	287	30	via	via	ADP
ejpam-6327	287	31	fuzzy	fuzzy	ADJ
ejpam-6327	287	32	ksideals	ksideal	NOUN
ejpam-6327	287	33	.	.	PUNCT
ejpam-6327	288	1	definition	definition	NOUN
ejpam-6327	288	2	13	13	NUM
ejpam-6327	288	3	.	.	PUNCT
ejpam-6327	289	1	let	let	VERB
ejpam-6327	289	2	µ	µ	X
ejpam-6327	289	3	be	be	AUX
ejpam-6327	289	4	a	a	DET
ejpam-6327	289	5	non	non	ADJ
ejpam-6327	289	6	-	-	ADJ
ejpam-6327	289	7	zero	zero	ADJ
ejpam-6327	289	8	fuzzy	fuzzy	ADJ
ejpam-6327	289	9	ks	ks	NOUN
ejpam-6327	289	10	-	-	NOUN
ejpam-6327	289	11	ideal	ideal	NOUN
ejpam-6327	289	12	of	of	ADP
ejpam-6327	289	13	a	a	DET
ejpam-6327	289	14	ks	ks	NOUN
ejpam-6327	289	15	-	-	PUNCT
ejpam-6327	289	16	semigroup	semigroup	PROPN
ejpam-6327	289	17	x.	x.	NOUN
ejpam-6327	289	18	define	define	VERB
ejpam-6327	289	19	a	a	DET
ejpam-6327	289	20	binary	binary	ADJ
ejpam-6327	289	21	relation	relation	NOUN
ejpam-6327	289	22	∼µ	∼µ	NOUN
ejpam-6327	289	23	on	on	ADP
ejpam-6327	289	24	x	x	PUNCT
ejpam-6327	289	25	by	by	ADP
ejpam-6327	289	26	x	x	PROPN
ejpam-6327	289	27	∼µ	∼µ	PROPN
ejpam-6327	289	28	y	y	NOUN
ejpam-6327	289	29	if	if	SCONJ
ejpam-6327	290	1	and	and	CCONJ
ejpam-6327	290	2	only	only	ADV
ejpam-6327	290	3	if	if	SCONJ
ejpam-6327	290	4	µ(x	µ(x	NOUN
ejpam-6327	290	5	∗	∗	NOUN
ejpam-6327	290	6	y	y	NOUN
ejpam-6327	290	7	)	)	PUNCT
ejpam-6327	290	8	>	>	X
ejpam-6327	290	9	0	0	PUNCT
ejpam-6327	290	10	and	and	CCONJ
ejpam-6327	290	11	µ(y	µ(y	PROPN
ejpam-6327	290	12	∗	∗	NOUN
ejpam-6327	290	13	x	x	NOUN
ejpam-6327	290	14	)	)	PUNCT
ejpam-6327	290	15	>	>	X
ejpam-6327	291	1	0	0	X
ejpam-6327	291	2	.	.	PUNCT
ejpam-6327	292	1	we	we	PRON
ejpam-6327	292	2	will	will	AUX
ejpam-6327	292	3	prove	prove	VERB
ejpam-6327	292	4	that	that	SCONJ
ejpam-6327	292	5	∼µ	∼µ	PROPN
ejpam-6327	292	6	is	be	AUX
ejpam-6327	292	7	a	a	DET
ejpam-6327	292	8	congruence	congruence	NOUN
ejpam-6327	292	9	relation	relation	NOUN
ejpam-6327	292	10	on	on	ADP
ejpam-6327	292	11	a	a	DET
ejpam-6327	292	12	ks	ks	NOUN
ejpam-6327	292	13	-	-	PUNCT
ejpam-6327	292	14	semigroup	semigroup	NOUN
ejpam-6327	292	15	x.	x.	NOUN
ejpam-6327	292	16	proposition	proposition	NOUN
ejpam-6327	292	17	3	3	NUM
ejpam-6327	292	18	.	.	PUNCT
ejpam-6327	293	1	∼µ	∼µ	PROPN
ejpam-6327	293	2	is	be	AUX
ejpam-6327	293	3	an	an	DET
ejpam-6327	293	4	equivalence	equivalence	NOUN
ejpam-6327	293	5	relation	relation	NOUN
ejpam-6327	293	6	on	on	ADP
ejpam-6327	293	7	a	a	DET
ejpam-6327	293	8	ks	ks	NOUN
ejpam-6327	293	9	-	-	PUNCT
ejpam-6327	293	10	semigroup	semigroup	ADJ
ejpam-6327	293	11	x.	x.	NOUN
ejpam-6327	293	12	proof	proof	NOUN
ejpam-6327	293	13	.	.	PUNCT
ejpam-6327	294	1	let	let	VERB
ejpam-6327	294	2	µ	µ	X
ejpam-6327	294	3	be	be	AUX
ejpam-6327	294	4	a	a	DET
ejpam-6327	294	5	non	non	ADJ
ejpam-6327	294	6	-	-	ADJ
ejpam-6327	294	7	zero	zero	ADJ
ejpam-6327	294	8	fuzzy	fuzzy	ADJ
ejpam-6327	294	9	ks	ks	NOUN
ejpam-6327	294	10	-	-	NOUN
ejpam-6327	294	11	ideal	ideal	NOUN
ejpam-6327	294	12	of	of	ADP
ejpam-6327	294	13	a	a	DET
ejpam-6327	294	14	ks	ks	NOUN
ejpam-6327	294	15	-	-	PUNCT
ejpam-6327	294	16	semigroup	semigroup	NOUN
ejpam-6327	294	17	x.	x.	NOUN
ejpam-6327	294	18	let	let	VERB
ejpam-6327	295	1	x	x	X
ejpam-6327	295	2	∈	∈	PROPN
ejpam-6327	295	3	x.	x.	NOUN
ejpam-6327	295	4	since	since	SCONJ
ejpam-6327	295	5	µ	µ	PROPN
ejpam-6327	295	6	is	be	AUX
ejpam-6327	295	7	non	non	ADJ
ejpam-6327	295	8	-	-	ADJ
ejpam-6327	295	9	zero	zero	NUM
ejpam-6327	295	10	,	,	PUNCT
ejpam-6327	295	11	there	there	PRON
ejpam-6327	295	12	exists	exist	VERB
ejpam-6327	295	13	z	z	NOUN
ejpam-6327	295	14	∈	∈	PROPN
ejpam-6327	295	15	x	x	PUNCT
ejpam-6327	296	1	such	such	ADJ
ejpam-6327	296	2	that	that	SCONJ
ejpam-6327	296	3	µ(z	µ(z	PROPN
ejpam-6327	296	4	)	)	PUNCT
ejpam-6327	296	5	̸=	̸=	PROPN
ejpam-6327	296	6	0	0	NUM
ejpam-6327	296	7	,	,	PUNCT
ejpam-6327	296	8	that	that	ADV
ejpam-6327	296	9	is	is	ADV
ejpam-6327	296	10	,	,	PUNCT
ejpam-6327	296	11	µ(z	µ(z	PROPN
ejpam-6327	296	12	)	)	PUNCT
ejpam-6327	296	13	>	>	X
ejpam-6327	296	14	0	0	X
ejpam-6327	296	15	.	.	PUNCT
ejpam-6327	297	1	then	then	ADV
ejpam-6327	297	2	by	by	ADP
ejpam-6327	297	3	(	(	PUNCT
ejpam-6327	297	4	f1	f1	NOUN
ejpam-6327	297	5	)	)	PUNCT
ejpam-6327	297	6	,	,	PUNCT
ejpam-6327	297	7	µ(0	µ(0	NOUN
ejpam-6327	297	8	)	)	PUNCT
ejpam-6327	297	9	≥	≥	NOUN
ejpam-6327	297	10	µ(z	µ(z	PROPN
ejpam-6327	297	11	)	)	PUNCT
ejpam-6327	297	12	>	>	X
ejpam-6327	298	1	0	0	X
ejpam-6327	298	2	.	.	PUNCT
ejpam-6327	298	3	thus	thus	ADV
ejpam-6327	298	4	by	by	ADP
ejpam-6327	298	5	definition	definition	NOUN
ejpam-6327	298	6	1	1	NUM
ejpam-6327	298	7	,	,	PUNCT
ejpam-6327	298	8	µ(x	µ(x	ADJ
ejpam-6327	298	9	∗	∗	NOUN
ejpam-6327	298	10	x	x	NOUN
ejpam-6327	298	11	)	)	PUNCT
ejpam-6327	298	12	=	=	SYM
ejpam-6327	298	13	µ(0	µ(0	NOUN
ejpam-6327	298	14	)	)	PUNCT
ejpam-6327	298	15	>	>	X
ejpam-6327	298	16	0	0	X
ejpam-6327	298	17	.	.	PUNCT
ejpam-6327	299	1	hence	hence	ADV
ejpam-6327	299	2	,	,	PUNCT
ejpam-6327	299	3	x	x	PROPN
ejpam-6327	299	4	∼µ	∼µ	NOUN
ejpam-6327	299	5	x	x	NOUN
ejpam-6327	299	6	,	,	PUNCT
ejpam-6327	299	7	which	which	PRON
ejpam-6327	299	8	means	mean	VERB
ejpam-6327	299	9	∼µ	∼µ	PROPN
ejpam-6327	299	10	is	be	AUX
ejpam-6327	299	11	reflexive	reflexive	ADJ
ejpam-6327	299	12	.	.	PUNCT
ejpam-6327	300	1	suppose	suppose	VERB
ejpam-6327	300	2	x	x	X
ejpam-6327	301	1	∼µ	∼µ	ADP
ejpam-6327	301	2	y.	y.	NOUN
ejpam-6327	301	3	then	then	ADV
ejpam-6327	301	4	µ(x	µ(x	VERB
ejpam-6327	301	5	∗	∗	NOUN
ejpam-6327	301	6	y	y	NOUN
ejpam-6327	301	7	)	)	PUNCT
ejpam-6327	301	8	>	>	X
ejpam-6327	301	9	0	0	PUNCT
ejpam-6327	302	1	and	and	CCONJ
ejpam-6327	302	2	µ(y	µ(y	PROPN
ejpam-6327	302	3	∗	∗	NOUN
ejpam-6327	302	4	x	x	NOUN
ejpam-6327	302	5	)	)	PUNCT
ejpam-6327	302	6	>	>	X
ejpam-6327	302	7	0	0	NUM
ejpam-6327	302	8	,	,	PUNCT
ejpam-6327	302	9	or	or	CCONJ
ejpam-6327	302	10	equivalently	equivalently	ADV
ejpam-6327	302	11	,	,	PUNCT
ejpam-6327	302	12	µ(y	µ(y	PROPN
ejpam-6327	302	13	∗	∗	NOUN
ejpam-6327	302	14	x	x	NOUN
ejpam-6327	302	15	)	)	PUNCT
ejpam-6327	302	16	>	>	X
ejpam-6327	302	17	0	0	PUNCT
ejpam-6327	303	1	and	and	CCONJ
ejpam-6327	303	2	µ(x	µ(x	ADJ
ejpam-6327	303	3	∗	∗	NOUN
ejpam-6327	303	4	y	y	NOUN
ejpam-6327	303	5	)	)	PUNCT
ejpam-6327	303	6	>	>	X
ejpam-6327	304	1	0	0	X
ejpam-6327	304	2	.	.	PUNCT
ejpam-6327	305	1	thus	thus	ADV
ejpam-6327	305	2	,	,	PUNCT
ejpam-6327	305	3	y	y	PROPN
ejpam-6327	305	4	∼µ	∼µ	PROPN
ejpam-6327	305	5	x.	x.	NOUN
ejpam-6327	305	6	hence	hence	ADV
ejpam-6327	305	7	,	,	PUNCT
ejpam-6327	305	8	∼µ	∼µ	PROPN
ejpam-6327	305	9	is	be	AUX
ejpam-6327	305	10	symmetric	symmetric	ADJ
ejpam-6327	305	11	.	.	PUNCT
ejpam-6327	306	1	now	now	ADV
ejpam-6327	306	2	,	,	PUNCT
ejpam-6327	306	3	suppose	suppose	VERB
ejpam-6327	306	4	x	x	PUNCT
ejpam-6327	306	5	∼µ	∼µ	ADP
ejpam-6327	306	6	y	y	PROPN
ejpam-6327	306	7	and	and	CCONJ
ejpam-6327	306	8	y	y	PROPN
ejpam-6327	306	9	∼µ	∼µ	PROPN
ejpam-6327	306	10	z.	z.	PROPN
ejpam-6327	307	1	then	then	ADV
ejpam-6327	307	2	µ(x	µ(x	VERB
ejpam-6327	307	3	∗	∗	NOUN
ejpam-6327	307	4	y	y	NOUN
ejpam-6327	307	5	)	)	PUNCT
ejpam-6327	307	6	>	>	X
ejpam-6327	308	1	0	0	NUM
ejpam-6327	308	2	,	,	PUNCT
ejpam-6327	308	3	µ(y	µ(y	PROPN
ejpam-6327	308	4	∗	∗	NOUN
ejpam-6327	308	5	x	x	NOUN
ejpam-6327	308	6	)	)	PUNCT
ejpam-6327	308	7	>	>	X
ejpam-6327	308	8	0	0	NUM
ejpam-6327	308	9	,	,	PUNCT
ejpam-6327	308	10	µ(y	µ(y	PROPN
ejpam-6327	308	11	∗	∗	PROPN
ejpam-6327	308	12	z	z	PROPN
ejpam-6327	308	13	)	)	PUNCT
ejpam-6327	308	14	>	>	X
ejpam-6327	308	15	0	0	PUNCT
ejpam-6327	308	16	and	and	CCONJ
ejpam-6327	308	17	µ(z	µ(z	PROPN
ejpam-6327	308	18	∗	∗	X
ejpam-6327	308	19	y	y	PROPN
ejpam-6327	308	20	)	)	PUNCT
ejpam-6327	308	21	>	>	X
ejpam-6327	309	1	0	0	X
ejpam-6327	309	2	.	.	PUNCT
ejpam-6327	310	1	since	since	SCONJ
ejpam-6327	310	2	x	x	PRON
ejpam-6327	310	3	is	be	AUX
ejpam-6327	310	4	a	a	DET
ejpam-6327	310	5	ks	ks	NOUN
ejpam-6327	310	6	-	-	PUNCT
ejpam-6327	310	7	semigroup	semigroup	NOUN
ejpam-6327	310	8	,	,	PUNCT
ejpam-6327	310	9	(	(	PUNCT
ejpam-6327	310	10	x	x	X
ejpam-6327	310	11	∗	∗	PROPN
ejpam-6327	310	12	z	z	NOUN
ejpam-6327	310	13	)	)	PUNCT
ejpam-6327	310	14	∗	∗	NOUN
ejpam-6327	310	15	(	(	PUNCT
ejpam-6327	310	16	x	x	X
ejpam-6327	310	17	∗	∗	PROPN
ejpam-6327	310	18	y	y	NOUN
ejpam-6327	310	19	)	)	PUNCT
ejpam-6327	310	20	≤	≤	NOUN
ejpam-6327	311	1	y	y	PROPN
ejpam-6327	311	2	∗	∗	NOUN
ejpam-6327	311	3	z	z	PROPN
ejpam-6327	311	4	and	and	CCONJ
ejpam-6327	311	5	(	(	PUNCT
ejpam-6327	311	6	z	z	NOUN
ejpam-6327	311	7	∗	∗	X
ejpam-6327	311	8	x	x	NOUN
ejpam-6327	311	9	)	)	PUNCT
ejpam-6327	311	10	∗	∗	NOUN
ejpam-6327	311	11	(	(	PUNCT
ejpam-6327	311	12	z	z	NOUN
ejpam-6327	311	13	∗	∗	PROPN
ejpam-6327	311	14	y	y	NOUN
ejpam-6327	311	15	)	)	PUNCT
ejpam-6327	311	16	≤	≤	NOUN
ejpam-6327	312	1	y	y	PROPN
ejpam-6327	312	2	∗	∗	X
ejpam-6327	312	3	x	x	PUNCT
ejpam-6327	312	4	for	for	ADP
ejpam-6327	312	5	all	all	DET
ejpam-6327	312	6	x	x	NOUN
ejpam-6327	312	7	,	,	PUNCT
ejpam-6327	312	8	y	y	PROPN
ejpam-6327	312	9	,	,	PUNCT
ejpam-6327	312	10	z	z	PROPN
ejpam-6327	312	11	∈	∈	PROPN
ejpam-6327	312	12	x.	x.	NOUN
ejpam-6327	312	13	thus	thus	ADV
ejpam-6327	312	14	,	,	PUNCT
ejpam-6327	312	15	by	by	ADP
ejpam-6327	312	16	lemma	lemma	PROPN
ejpam-6327	312	17	1(ii	1(ii	NUM
ejpam-6327	312	18	)	)	PUNCT
ejpam-6327	312	19	,	,	PUNCT
ejpam-6327	312	20	we	we	PRON
ejpam-6327	312	21	have	have	VERB
ejpam-6327	312	22	µ(x	µ(x	ADJ
ejpam-6327	312	23	∗	∗	NOUN
ejpam-6327	312	24	z	z	NOUN
ejpam-6327	312	25	)	)	PUNCT
ejpam-6327	312	26	≥	≥	NOUN
ejpam-6327	313	1	min{µ(x	min{µ(x	PROPN
ejpam-6327	313	2	∗	∗	X
ejpam-6327	313	3	y	y	PROPN
ejpam-6327	313	4	)	)	PUNCT
ejpam-6327	313	5	,	,	PUNCT
ejpam-6327	314	1	µ(y	µ(y	PROPN
ejpam-6327	314	2	∗	∗	PROPN
ejpam-6327	314	3	z	z	PROPN
ejpam-6327	314	4	)	)	PUNCT
ejpam-6327	314	5	}	}	PUNCT
ejpam-6327	314	6	>	>	X
ejpam-6327	314	7	0	0	PUNCT
ejpam-6327	314	8	and	and	CCONJ
ejpam-6327	314	9	µ(z	µ(z	PROPN
ejpam-6327	314	10	∗	∗	X
ejpam-6327	314	11	x	x	NOUN
ejpam-6327	314	12	)	)	PUNCT
ejpam-6327	314	13	≥	≥	NOUN
ejpam-6327	314	14	min{µ(y	min{µ(y	PROPN
ejpam-6327	314	15	∗	∗	PROPN
ejpam-6327	314	16	x	x	NOUN
ejpam-6327	314	17	)	)	PUNCT
ejpam-6327	314	18	,	,	PUNCT
ejpam-6327	314	19	µ(z	µ(z	PROPN
ejpam-6327	314	20	∗	∗	PROPN
ejpam-6327	314	21	y	y	PROPN
ejpam-6327	314	22	)	)	PUNCT
ejpam-6327	314	23	}	}	PUNCT
ejpam-6327	314	24	>	>	X
ejpam-6327	315	1	0	0	X
ejpam-6327	315	2	.	.	PUNCT
ejpam-6327	316	1	thus	thus	ADV
ejpam-6327	316	2	,	,	PUNCT
ejpam-6327	316	3	x	x	PUNCT
ejpam-6327	316	4	∼µ	∼µ	PROPN
ejpam-6327	316	5	z.	z.	PROPN
ejpam-6327	316	6	hence	hence	ADV
ejpam-6327	316	7	,	,	PUNCT
ejpam-6327	316	8	∼µ	∼µ	PROPN
ejpam-6327	316	9	is	be	AUX
ejpam-6327	316	10	transitive	transitive	ADJ
ejpam-6327	316	11	.	.	PUNCT
ejpam-6327	317	1	therefore	therefore	ADV
ejpam-6327	317	2	,	,	PUNCT
ejpam-6327	317	3	∼µ	∼µ	PROPN
ejpam-6327	317	4	is	be	AUX
ejpam-6327	317	5	an	an	DET
ejpam-6327	317	6	equivalence	equivalence	NOUN
ejpam-6327	317	7	relation	relation	NOUN
ejpam-6327	317	8	on	on	ADP
ejpam-6327	317	9	x.	x.	NOUN
ejpam-6327	317	10	the	the	DET
ejpam-6327	317	11	following	follow	VERB
ejpam-6327	317	12	example	example	NOUN
ejpam-6327	317	13	shows	show	VERB
ejpam-6327	317	14	that	that	SCONJ
ejpam-6327	317	15	if	if	SCONJ
ejpam-6327	317	16	a	a	DET
ejpam-6327	317	17	non	non	ADJ
ejpam-6327	317	18	-	-	ADJ
ejpam-6327	317	19	zero	zero	ADJ
ejpam-6327	317	20	fuzzy	fuzzy	ADJ
ejpam-6327	317	21	ks	ks	NOUN
ejpam-6327	317	22	-	-	PUNCT
ejpam-6327	317	23	ideal	ideal	NOUN
ejpam-6327	317	24	µ	µ	NOUN
ejpam-6327	317	25	of	of	ADP
ejpam-6327	317	26	a	a	DET
ejpam-6327	317	27	ks	ks	NOUN
ejpam-6327	317	28	-	-	PUNCT
ejpam-6327	317	29	semigroup	semigroup	NOUN
ejpam-6327	317	30	x	x	NOUN
ejpam-6327	317	31	does	do	AUX
ejpam-6327	317	32	not	not	PART
ejpam-6327	317	33	satisfy	satisfy	VERB
ejpam-6327	317	34	condition	condition	NOUN
ejpam-6327	317	35	(	(	PUNCT
ejpam-6327	317	36	c	c	NOUN
ejpam-6327	317	37	):	):	PUNCT
ejpam-6327	317	38	µ(xy	µ(xy	PROPN
ejpam-6327	317	39	)	)	PUNCT
ejpam-6327	317	40	≥	≥	NOUN
ejpam-6327	317	41	µ(x	µ(x	VERB
ejpam-6327	317	42	)	)	PUNCT
ejpam-6327	317	43	and	and	CCONJ
ejpam-6327	317	44	µ(xy	µ(xy	PROPN
ejpam-6327	317	45	)	)	PUNCT
ejpam-6327	317	46	≥	≥	NOUN
ejpam-6327	317	47	µ(y	µ(y	PROPN
ejpam-6327	317	48	)	)	PUNCT
ejpam-6327	317	49	for	for	ADP
ejpam-6327	317	50	all	all	DET
ejpam-6327	317	51	x	x	NOUN
ejpam-6327	317	52	,	,	PUNCT
ejpam-6327	317	53	y	y	PROPN
ejpam-6327	317	54	∈	∈	PROPN
ejpam-6327	317	55	x	x	NOUN
ejpam-6327	317	56	,	,	PUNCT
ejpam-6327	317	57	then	then	ADV
ejpam-6327	317	58	∼µ	∼µ	PROPN
ejpam-6327	317	59	is	be	AUX
ejpam-6327	317	60	not	not	PART
ejpam-6327	317	61	compatible	compatible	ADJ
ejpam-6327	317	62	.	.	PUNCT
ejpam-6327	318	1	h.	h.	PROPN
ejpam-6327	318	2	sarapuddin	sarapuddin	PROPN
ejpam-6327	318	3	,	,	PUNCT
ejpam-6327	318	4	j.	j.	PROPN
ejpam-6327	318	5	vilela	vilela	PROPN
ejpam-6327	318	6	/	/	SYM
ejpam-6327	318	7	eur	eur	PROPN
ejpam-6327	318	8	.	.	PUNCT
ejpam-6327	319	1	j.	j.	PROPN
ejpam-6327	319	2	pure	pure	PROPN
ejpam-6327	319	3	appl	appl	PROPN
ejpam-6327	319	4	.	.	PROPN
ejpam-6327	319	5	math	math	PROPN
ejpam-6327	319	6	,	,	PUNCT
ejpam-6327	319	7	18	18	NUM
ejpam-6327	319	8	(	(	PUNCT
ejpam-6327	319	9	3	3	NUM
ejpam-6327	319	10	)	)	PUNCT
ejpam-6327	319	11	(	(	PUNCT
ejpam-6327	319	12	2025	2025	NUM
ejpam-6327	319	13	)	)	PUNCT
ejpam-6327	319	14	,	,	PUNCT
ejpam-6327	319	15	6327	6327	NUM
ejpam-6327	319	16	11	11	NUM
ejpam-6327	319	17	of	of	ADP
ejpam-6327	319	18	23	23	NUM
ejpam-6327	319	19	example	example	NOUN
ejpam-6327	319	20	6	6	NUM
ejpam-6327	319	21	.	.	PUNCT
ejpam-6327	319	22	consider	consider	VERB
ejpam-6327	319	23	the	the	DET
ejpam-6327	319	24	ks	ks	NOUN
ejpam-6327	319	25	-	-	NOUN
ejpam-6327	319	26	semigroup	semigroup	NOUN
ejpam-6327	319	27	x	x	X
ejpam-6327	319	28	=	=	PUNCT
ejpam-6327	319	29	{	{	PUNCT
ejpam-6327	319	30	0	0	NUM
ejpam-6327	319	31	,	,	PUNCT
ejpam-6327	319	32	a	a	DET
ejpam-6327	319	33	,	,	PUNCT
ejpam-6327	319	34	b	b	NOUN
ejpam-6327	319	35	,	,	PUNCT
ejpam-6327	319	36	c	c	NOUN
ejpam-6327	319	37	}	}	PUNCT
ejpam-6327	319	38	in	in	ADP
ejpam-6327	319	39	example	example	NOUN
ejpam-6327	319	40	1	1	X
ejpam-6327	319	41	.	.	X
ejpam-6327	319	42	define	define	VERB
ejpam-6327	319	43	a	a	DET
ejpam-6327	319	44	fuzzy	fuzzy	ADJ
ejpam-6327	319	45	set	set	VERB
ejpam-6327	319	46	µ	µ	NOUN
ejpam-6327	319	47	in	in	ADP
ejpam-6327	319	48	x	x	PUNCT
ejpam-6327	319	49	by	by	ADP
ejpam-6327	319	50	µ(0	µ(0	NOUN
ejpam-6327	319	51	)	)	PUNCT
ejpam-6327	319	52	=	=	SYM
ejpam-6327	319	53	µ(a	µ(a	PROPN
ejpam-6327	319	54	)	)	PUNCT
ejpam-6327	319	55	=	=	SYM
ejpam-6327	319	56	µ(b	µ(b	PROPN
ejpam-6327	319	57	)	)	PUNCT
ejpam-6327	319	58	=	=	SYM
ejpam-6327	319	59	0.8	0.8	NUM
ejpam-6327	319	60	and	and	CCONJ
ejpam-6327	319	61	µ(c	µ(c	PROPN
ejpam-6327	319	62	)	)	PUNCT
ejpam-6327	319	63	=	=	SYM
ejpam-6327	320	1	0	0	X
ejpam-6327	320	2	.	.	PUNCT
ejpam-6327	321	1	then	then	ADV
ejpam-6327	321	2	µ	µ	X
ejpam-6327	321	3	is	be	AUX
ejpam-6327	321	4	a	a	DET
ejpam-6327	321	5	fuzzy	fuzzy	ADJ
ejpam-6327	321	6	ks	ks	NOUN
ejpam-6327	321	7	-	-	NOUN
ejpam-6327	321	8	ideal	ideal	NOUN
ejpam-6327	321	9	of	of	ADP
ejpam-6327	321	10	x.	x.	NOUN
ejpam-6327	321	11	note	note	VERB
ejpam-6327	321	12	that	that	SCONJ
ejpam-6327	321	13	µ(ca	µ(ca	ADP
ejpam-6327	321	14	)	)	PUNCT
ejpam-6327	321	15	=	=	SYM
ejpam-6327	321	16	µ(c	µ(c	PROPN
ejpam-6327	321	17	)	)	PUNCT
ejpam-6327	321	18	=	=	SYM
ejpam-6327	321	19	min{µ(c	min{µ(c	PROPN
ejpam-6327	321	20	)	)	PUNCT
ejpam-6327	321	21	,	,	PUNCT
ejpam-6327	321	22	µ(a	µ(a	PROPN
ejpam-6327	321	23	)	)	PUNCT
ejpam-6327	321	24	}	}	PUNCT
ejpam-6327	321	25	<	<	X
ejpam-6327	321	26	µ(a	µ(a	PROPN
ejpam-6327	321	27	)	)	PUNCT
ejpam-6327	321	28	.	.	PUNCT
ejpam-6327	322	1	thus	thus	ADV
ejpam-6327	322	2	,	,	PUNCT
ejpam-6327	322	3	µ	µ	PRON
ejpam-6327	322	4	does	do	AUX
ejpam-6327	322	5	not	not	PART
ejpam-6327	322	6	satisfy	satisfy	VERB
ejpam-6327	322	7	condition	condition	NOUN
ejpam-6327	322	8	(	(	PUNCT
ejpam-6327	322	9	c	c	NOUN
ejpam-6327	322	10	)	)	PUNCT
ejpam-6327	322	11	.	.	PUNCT
ejpam-6327	323	1	observe	observe	VERB
ejpam-6327	323	2	that	that	SCONJ
ejpam-6327	323	3	µ(a	µ(a	PROPN
ejpam-6327	323	4	∗	∗	NOUN
ejpam-6327	323	5	b	b	NOUN
ejpam-6327	323	6	)	)	PUNCT
ejpam-6327	323	7	=	=	SYM
ejpam-6327	323	8	µ(a	µ(a	PROPN
ejpam-6327	323	9	)	)	PUNCT
ejpam-6327	323	10	=	=	PUNCT
ejpam-6327	323	11	0.8	0.8	NUM
ejpam-6327	323	12	>	>	SYM
ejpam-6327	323	13	0	0	PUNCT
ejpam-6327	323	14	and	and	CCONJ
ejpam-6327	323	15	µ(b	µ(b	PROPN
ejpam-6327	323	16	∗	∗	X
ejpam-6327	323	17	a	a	X
ejpam-6327	323	18	)	)	PUNCT
ejpam-6327	323	19	=	=	SYM
ejpam-6327	323	20	µ(b	µ(b	PROPN
ejpam-6327	323	21	)	)	PUNCT
ejpam-6327	323	22	=	=	PUNCT
ejpam-6327	323	23	0.8	0.8	NUM
ejpam-6327	323	24	>	>	X
ejpam-6327	323	25	0	0	NUM
ejpam-6327	323	26	.	.	PUNCT
ejpam-6327	324	1	thus	thus	ADV
ejpam-6327	324	2	,	,	PUNCT
ejpam-6327	324	3	a	a	DET
ejpam-6327	324	4	∼µ	∼µ	PROPN
ejpam-6327	324	5	b.	b.	PROPN
ejpam-6327	324	6	however	however	ADV
ejpam-6327	324	7	,	,	PUNCT
ejpam-6327	324	8	∼µ	∼µ	PROPN
ejpam-6327	324	9	is	be	AUX
ejpam-6327	324	10	not	not	PART
ejpam-6327	324	11	compatible	compatible	ADJ
ejpam-6327	324	12	since	since	SCONJ
ejpam-6327	324	13	µ(ca	µ(ca	ADP
ejpam-6327	324	14	∗	∗	NOUN
ejpam-6327	324	15	cb	cb	NOUN
ejpam-6327	324	16	)	)	PUNCT
ejpam-6327	324	17	=	=	SYM
ejpam-6327	324	18	µ(c	µ(c	PROPN
ejpam-6327	324	19	∗	∗	NOUN
ejpam-6327	324	20	0	0	NUM
ejpam-6327	324	21	)	)	PUNCT
ejpam-6327	324	22	=	=	SYM
ejpam-6327	324	23	µ(c	µ(c	PROPN
ejpam-6327	324	24	)	)	PUNCT
ejpam-6327	324	25	=	=	SYM
ejpam-6327	324	26	0	0	NUM
ejpam-6327	324	27	,	,	PUNCT
ejpam-6327	324	28	that	that	ADV
ejpam-6327	324	29	is	is	ADV
ejpam-6327	324	30	,	,	PUNCT
ejpam-6327	324	31	ca	can	AUX
ejpam-6327	324	32	≁µ	≁µ	NOUN
ejpam-6327	324	33	cb	cb	PROPN
ejpam-6327	324	34	.	.	PROPN
ejpam-6327	324	35	proposition	proposition	PROPN
ejpam-6327	324	36	4	4	NUM
ejpam-6327	324	37	.	.	PUNCT
ejpam-6327	325	1	let	let	VERB
ejpam-6327	325	2	x	x	PRON
ejpam-6327	325	3	be	be	AUX
ejpam-6327	325	4	a	a	DET
ejpam-6327	325	5	ks	ks	NOUN
ejpam-6327	325	6	-	-	PUNCT
ejpam-6327	325	7	semigroup	semigroup	NOUN
ejpam-6327	325	8	and	and	CCONJ
ejpam-6327	325	9	µ	µ	DET
ejpam-6327	325	10	a	a	DET
ejpam-6327	325	11	non	non	ADJ
ejpam-6327	325	12	-	-	ADJ
ejpam-6327	325	13	zero	zero	ADJ
ejpam-6327	325	14	fuzzy	fuzzy	ADJ
ejpam-6327	325	15	ks	ks	NOUN
ejpam-6327	325	16	-	-	NOUN
ejpam-6327	325	17	ideal	ideal	NOUN
ejpam-6327	325	18	of	of	ADP
ejpam-6327	325	19	x	x	SYM
ejpam-6327	325	20	satisfying	satisfy	VERB
ejpam-6327	325	21	condition	condition	NOUN
ejpam-6327	325	22	(	(	PUNCT
ejpam-6327	325	23	c	c	NOUN
ejpam-6327	325	24	)	)	PUNCT
ejpam-6327	325	25	.	.	PUNCT
ejpam-6327	326	1	then	then	ADV
ejpam-6327	326	2	∼µ	∼µ	PROPN
ejpam-6327	326	3	is	be	AUX
ejpam-6327	326	4	both	both	PRON
ejpam-6327	326	5	left	leave	VERB
ejpam-6327	326	6	and	and	CCONJ
ejpam-6327	326	7	right	right	ADV
ejpam-6327	326	8	compatible	compatible	ADJ
ejpam-6327	326	9	.	.	PUNCT
ejpam-6327	327	1	proof	proof	NOUN
ejpam-6327	327	2	.	.	PUNCT
ejpam-6327	328	1	let	let	VERB
ejpam-6327	328	2	x	x	PRON
ejpam-6327	328	3	be	be	AUX
ejpam-6327	328	4	a	a	DET
ejpam-6327	328	5	ks	ks	NOUN
ejpam-6327	328	6	-	-	PUNCT
ejpam-6327	328	7	semigroup	semigroup	NOUN
ejpam-6327	328	8	and	and	CCONJ
ejpam-6327	328	9	µ	µ	DET
ejpam-6327	328	10	a	a	DET
ejpam-6327	328	11	non	non	ADJ
ejpam-6327	328	12	-	-	ADJ
ejpam-6327	328	13	zero	zero	ADJ
ejpam-6327	328	14	fuzzy	fuzzy	ADJ
ejpam-6327	328	15	ks	ks	NOUN
ejpam-6327	328	16	-	-	NOUN
ejpam-6327	328	17	ideal	ideal	NOUN
ejpam-6327	328	18	of	of	ADP
ejpam-6327	328	19	x	x	SYM
ejpam-6327	328	20	satisfying	satisfy	VERB
ejpam-6327	328	21	condition	condition	NOUN
ejpam-6327	328	22	(	(	PUNCT
ejpam-6327	328	23	c	c	NOUN
ejpam-6327	328	24	)	)	PUNCT
ejpam-6327	328	25	.	.	PUNCT
ejpam-6327	329	1	let	let	VERB
ejpam-6327	329	2	x	x	PRON
ejpam-6327	329	3	,	,	PUNCT
ejpam-6327	329	4	y	y	PROPN
ejpam-6327	329	5	,	,	PUNCT
ejpam-6327	329	6	z	z	NOUN
ejpam-6327	329	7	∈	∈	PROPN
ejpam-6327	329	8	x	x	PUNCT
ejpam-6327	329	9	such	such	ADJ
ejpam-6327	329	10	that	that	SCONJ
ejpam-6327	329	11	x	x	PROPN
ejpam-6327	329	12	∼µ	∼µ	ADP
ejpam-6327	329	13	y.	y.	NOUN
ejpam-6327	329	14	then	then	ADV
ejpam-6327	329	15	µ(x∗y	µ(x∗y	VERB
ejpam-6327	329	16	)	)	PUNCT
ejpam-6327	329	17	>	>	X
ejpam-6327	329	18	0	0	PUNCT
ejpam-6327	330	1	and	and	CCONJ
ejpam-6327	330	2	µ(y	µ(y	PROPN
ejpam-6327	330	3	∗x	∗x	PROPN
ejpam-6327	330	4	)	)	PUNCT
ejpam-6327	330	5	>	>	X
ejpam-6327	331	1	0	0	X
ejpam-6327	331	2	.	.	PUNCT
ejpam-6327	332	1	since	since	SCONJ
ejpam-6327	332	2	(	(	PUNCT
ejpam-6327	332	3	z	z	NOUN
ejpam-6327	332	4	∗	∗	NOUN
ejpam-6327	332	5	x	x	NOUN
ejpam-6327	332	6	)	)	PUNCT
ejpam-6327	332	7	∗	∗	NOUN
ejpam-6327	332	8	(	(	PUNCT
ejpam-6327	332	9	z	z	NOUN
ejpam-6327	332	10	∗	∗	PROPN
ejpam-6327	332	11	y	y	NOUN
ejpam-6327	332	12	)	)	PUNCT
ejpam-6327	332	13	≤	≤	NOUN
ejpam-6327	332	14	y	y	PROPN
ejpam-6327	332	15	∗	∗	NOUN
ejpam-6327	332	16	x	x	PUNCT
ejpam-6327	332	17	and	and	CCONJ
ejpam-6327	332	18	(	(	PUNCT
ejpam-6327	332	19	z	z	NOUN
ejpam-6327	332	20	∗	∗	PROPN
ejpam-6327	332	21	y	y	PROPN
ejpam-6327	332	22	)	)	PUNCT
ejpam-6327	332	23	∗	∗	NOUN
ejpam-6327	332	24	(	(	PUNCT
ejpam-6327	332	25	z	z	NOUN
ejpam-6327	332	26	∗	∗	X
ejpam-6327	332	27	x	x	NOUN
ejpam-6327	332	28	)	)	PUNCT
ejpam-6327	332	29	≤	≤	NUM
ejpam-6327	332	30	x	x	PUNCT
ejpam-6327	332	31	∗	∗	PROPN
ejpam-6327	332	32	y	y	PROPN
ejpam-6327	332	33	,	,	PUNCT
ejpam-6327	332	34	we	we	PRON
ejpam-6327	332	35	have	have	VERB
ejpam-6327	332	36	µ((z	µ((z	NOUN
ejpam-6327	332	37	∗	∗	NOUN
ejpam-6327	332	38	x	x	NOUN
ejpam-6327	332	39	)	)	PUNCT
ejpam-6327	332	40	∗	∗	NOUN
ejpam-6327	332	41	(	(	PUNCT
ejpam-6327	332	42	z	z	NOUN
ejpam-6327	332	43	∗	∗	PROPN
ejpam-6327	332	44	y	y	PROPN
ejpam-6327	332	45	)	)	PUNCT
ejpam-6327	332	46	)	)	PUNCT
ejpam-6327	332	47	≥	≥	PROPN
ejpam-6327	332	48	µ(y	µ(y	PROPN
ejpam-6327	332	49	∗	∗	PROPN
ejpam-6327	332	50	x	x	NOUN
ejpam-6327	332	51	)	)	PUNCT
ejpam-6327	332	52	>	>	X
ejpam-6327	332	53	0	0	PUNCT
ejpam-6327	333	1	and	and	CCONJ
ejpam-6327	333	2	µ((z	µ((z	NOUN
ejpam-6327	333	3	∗	∗	PROPN
ejpam-6327	333	4	y	y	PROPN
ejpam-6327	333	5	)	)	PUNCT
ejpam-6327	333	6	∗	∗	NOUN
ejpam-6327	333	7	(	(	PUNCT
ejpam-6327	333	8	z	z	NOUN
ejpam-6327	333	9	∗	∗	NOUN
ejpam-6327	333	10	x	x	NOUN
ejpam-6327	333	11	)	)	PUNCT
ejpam-6327	333	12	)	)	PUNCT
ejpam-6327	333	13	≥	≥	NOUN
ejpam-6327	333	14	µ(x	µ(x	X
ejpam-6327	333	15	∗	∗	NOUN
ejpam-6327	333	16	y	y	NOUN
ejpam-6327	333	17	)	)	PUNCT
ejpam-6327	333	18	>	>	X
ejpam-6327	333	19	0	0	PUNCT
ejpam-6327	333	20	by	by	ADP
ejpam-6327	333	21	lemma	lemma	PROPN
ejpam-6327	333	22	1(i	1(i	NUM
ejpam-6327	333	23	)	)	PUNCT
ejpam-6327	333	24	.	.	PUNCT
ejpam-6327	334	1	thus	thus	ADV
ejpam-6327	334	2	,	,	PUNCT
ejpam-6327	334	3	z	z	NOUN
ejpam-6327	334	4	∗	∗	NOUN
ejpam-6327	334	5	x	x	SYM
ejpam-6327	334	6	∼µ	∼µ	PROPN
ejpam-6327	334	7	z	z	NOUN
ejpam-6327	334	8	∗	∗	NOUN
ejpam-6327	334	9	y.	y.	NOUN
ejpam-6327	334	10	now	now	ADV
ejpam-6327	334	11	,	,	PUNCT
ejpam-6327	334	12	since	since	SCONJ
ejpam-6327	334	13	zx	zx	NUM
ejpam-6327	334	14	∗	∗	NOUN
ejpam-6327	334	15	zy	zy	X
ejpam-6327	334	16	=	=	PUNCT
ejpam-6327	334	17	z(x	z(x	X
ejpam-6327	334	18	∗	∗	NOUN
ejpam-6327	334	19	y	y	NOUN
ejpam-6327	334	20	)	)	PUNCT
ejpam-6327	334	21	,	,	PUNCT
ejpam-6327	334	22	we	we	PRON
ejpam-6327	334	23	have	have	VERB
ejpam-6327	334	24	µ(zx	µ(zx	PROPN
ejpam-6327	334	25	∗	∗	NOUN
ejpam-6327	334	26	zy	zy	NOUN
ejpam-6327	334	27	)	)	PUNCT
ejpam-6327	334	28	=	=	SYM
ejpam-6327	335	1	µ(z(x	µ(z(x	PROPN
ejpam-6327	335	2	∗	∗	NOUN
ejpam-6327	335	3	y	y	NOUN
ejpam-6327	335	4	)	)	PUNCT
ejpam-6327	335	5	)	)	PUNCT
ejpam-6327	335	6	.	.	PUNCT
ejpam-6327	336	1	thus	thus	ADV
ejpam-6327	336	2	by	by	ADP
ejpam-6327	336	3	condition	condition	NOUN
ejpam-6327	336	4	(	(	PUNCT
ejpam-6327	336	5	c	c	NOUN
ejpam-6327	336	6	)	)	PUNCT
ejpam-6327	336	7	,	,	PUNCT
ejpam-6327	336	8	µ(zx	µ(zx	PROPN
ejpam-6327	336	9	∗	∗	X
ejpam-6327	336	10	zy	zy	NOUN
ejpam-6327	336	11	)	)	PUNCT
ejpam-6327	336	12	=	=	SYM
ejpam-6327	337	1	µ(z(x	µ(z(x	PROPN
ejpam-6327	337	2	∗	∗	NOUN
ejpam-6327	337	3	y	y	NOUN
ejpam-6327	337	4	)	)	PUNCT
ejpam-6327	337	5	)	)	PUNCT
ejpam-6327	337	6	≥	≥	NOUN
ejpam-6327	337	7	µ(x	µ(x	X
ejpam-6327	337	8	∗	∗	NOUN
ejpam-6327	337	9	y	y	NOUN
ejpam-6327	337	10	)	)	PUNCT
ejpam-6327	337	11	>	>	X
ejpam-6327	338	1	0	0	X
ejpam-6327	338	2	.	.	PUNCT
ejpam-6327	338	3	similarly	similarly	ADV
ejpam-6327	338	4	,	,	PUNCT
ejpam-6327	338	5	µ(zy	µ(zy	ADP
ejpam-6327	338	6	∗	∗	NOUN
ejpam-6327	338	7	zx	zx	NUM
ejpam-6327	338	8	)	)	PUNCT
ejpam-6327	338	9	>	>	X
ejpam-6327	339	1	0	0	X
ejpam-6327	339	2	.	.	PUNCT
ejpam-6327	340	1	hence	hence	ADV
ejpam-6327	340	2	,	,	PUNCT
ejpam-6327	340	3	zx	zx	PROPN
ejpam-6327	340	4	∼µ	∼µ	PROPN
ejpam-6327	340	5	zy	zy	PROPN
ejpam-6327	340	6	.	.	PUNCT
ejpam-6327	341	1	by	by	ADP
ejpam-6327	341	2	definition	definition	NOUN
ejpam-6327	341	3	8(i	8(i	NUM
ejpam-6327	341	4	)	)	PUNCT
ejpam-6327	341	5	,	,	PUNCT
ejpam-6327	341	6	∼µ	∼µ	PROPN
ejpam-6327	341	7	is	be	AUX
ejpam-6327	341	8	left	leave	VERB
ejpam-6327	341	9	compatible	compatible	ADJ
ejpam-6327	341	10	.	.	PUNCT
ejpam-6327	342	1	now	now	ADV
ejpam-6327	342	2	,	,	PUNCT
ejpam-6327	342	3	since	since	SCONJ
ejpam-6327	342	4	(	(	PUNCT
ejpam-6327	342	5	x	x	PROPN
ejpam-6327	342	6	∗	∗	PROPN
ejpam-6327	342	7	z	z	NOUN
ejpam-6327	342	8	)	)	PUNCT
ejpam-6327	342	9	∗	∗	NOUN
ejpam-6327	342	10	(	(	PUNCT
ejpam-6327	342	11	y	y	PROPN
ejpam-6327	342	12	∗	∗	PROPN
ejpam-6327	342	13	z	z	NOUN
ejpam-6327	342	14	)	)	PUNCT
ejpam-6327	342	15	≤	≤	NUM
ejpam-6327	342	16	x	x	PUNCT
ejpam-6327	342	17	∗	∗	NOUN
ejpam-6327	342	18	y	y	PROPN
ejpam-6327	342	19	and	and	CCONJ
ejpam-6327	342	20	(	(	PUNCT
ejpam-6327	342	21	y	y	PROPN
ejpam-6327	342	22	∗	∗	PROPN
ejpam-6327	342	23	z	z	PROPN
ejpam-6327	342	24	)	)	PUNCT
ejpam-6327	342	25	∗	∗	NOUN
ejpam-6327	342	26	(	(	PUNCT
ejpam-6327	342	27	x	x	X
ejpam-6327	342	28	∗	∗	PROPN
ejpam-6327	342	29	z	z	NOUN
ejpam-6327	342	30	)	)	PUNCT
ejpam-6327	342	31	≤	≤	NOUN
ejpam-6327	342	32	y	y	PROPN
ejpam-6327	342	33	∗	∗	NOUN
ejpam-6327	342	34	x	x	SYM
ejpam-6327	342	35	,	,	PUNCT
ejpam-6327	342	36	we	we	PRON
ejpam-6327	342	37	have	have	VERB
ejpam-6327	342	38	µ((x	µ((x	NOUN
ejpam-6327	342	39	∗	∗	NOUN
ejpam-6327	342	40	z	z	NOUN
ejpam-6327	342	41	)	)	PUNCT
ejpam-6327	342	42	∗	∗	NOUN
ejpam-6327	342	43	(	(	PUNCT
ejpam-6327	342	44	y	y	PROPN
ejpam-6327	342	45	∗	∗	PROPN
ejpam-6327	342	46	z	z	NOUN
ejpam-6327	342	47	)	)	PUNCT
ejpam-6327	342	48	)	)	PUNCT
ejpam-6327	342	49	≥	≥	NOUN
ejpam-6327	342	50	µ(x	µ(x	X
ejpam-6327	342	51	∗	∗	NOUN
ejpam-6327	342	52	y	y	NOUN
ejpam-6327	342	53	)	)	PUNCT
ejpam-6327	342	54	>	>	X
ejpam-6327	342	55	0	0	NUM
ejpam-6327	342	56	and	and	CCONJ
ejpam-6327	342	57	µ((y	µ((y	NOUN
ejpam-6327	342	58	∗	∗	NOUN
ejpam-6327	342	59	z	z	NOUN
ejpam-6327	342	60	)	)	PUNCT
ejpam-6327	342	61	∗	∗	NOUN
ejpam-6327	342	62	(	(	PUNCT
ejpam-6327	342	63	x	x	X
ejpam-6327	342	64	∗	∗	PROPN
ejpam-6327	342	65	z	z	NOUN
ejpam-6327	342	66	)	)	PUNCT
ejpam-6327	342	67	)	)	PUNCT
ejpam-6327	342	68	>	>	X
ejpam-6327	343	1	µ(y	µ(y	PROPN
ejpam-6327	343	2	∗	∗	NOUN
ejpam-6327	343	3	x	x	NOUN
ejpam-6327	343	4	)	)	PUNCT
ejpam-6327	343	5	>	>	X
ejpam-6327	343	6	0	0	PUNCT
ejpam-6327	344	1	by	by	ADP
ejpam-6327	344	2	lemma	lemma	PROPN
ejpam-6327	344	3	1(i	1(i	NUM
ejpam-6327	344	4	)	)	PUNCT
ejpam-6327	344	5	.	.	PUNCT
ejpam-6327	345	1	thus	thus	ADV
ejpam-6327	345	2	,	,	PUNCT
ejpam-6327	345	3	x	x	X
ejpam-6327	345	4	∗	∗	NOUN
ejpam-6327	345	5	z	z	NOUN
ejpam-6327	345	6	∼µ	∼µ	PROPN
ejpam-6327	345	7	y	y	PROPN
ejpam-6327	345	8	∗	∗	NOUN
ejpam-6327	345	9	z.	z.	PROPN
ejpam-6327	346	1	moreover	moreover	ADV
ejpam-6327	346	2	,	,	PUNCT
ejpam-6327	346	3	since	since	SCONJ
ejpam-6327	346	4	xz	xz	PROPN
ejpam-6327	346	5	∗	∗	X
ejpam-6327	346	6	yz	yz	PROPN
ejpam-6327	346	7	=	=	PUNCT
ejpam-6327	347	1	(	(	PUNCT
ejpam-6327	347	2	x	x	X
ejpam-6327	347	3	∗	∗	NOUN
ejpam-6327	347	4	y)z	y)z	NOUN
ejpam-6327	347	5	,	,	PUNCT
ejpam-6327	347	6	we	we	PRON
ejpam-6327	347	7	have	have	VERB
ejpam-6327	347	8	µ(xz	µ(xz	NOUN
ejpam-6327	347	9	∗	∗	NOUN
ejpam-6327	347	10	yz	yz	NOUN
ejpam-6327	347	11	)	)	PUNCT
ejpam-6327	347	12	=	=	SYM
ejpam-6327	348	1	µ((x	µ((x	NOUN
ejpam-6327	348	2	∗	∗	NOUN
ejpam-6327	348	3	y)z	y)z	NOUN
ejpam-6327	348	4	)	)	PUNCT
ejpam-6327	348	5	.	.	PUNCT
ejpam-6327	349	1	thus	thus	ADV
ejpam-6327	349	2	by	by	ADP
ejpam-6327	349	3	condition	condition	NOUN
ejpam-6327	349	4	(	(	PUNCT
ejpam-6327	349	5	c	c	NOUN
ejpam-6327	349	6	)	)	PUNCT
ejpam-6327	349	7	,	,	PUNCT
ejpam-6327	349	8	µ(xz	µ(xz	X
ejpam-6327	349	9	∗	∗	X
ejpam-6327	349	10	yz	yz	NOUN
ejpam-6327	349	11	)	)	PUNCT
ejpam-6327	349	12	=	=	SYM
ejpam-6327	349	13	µ((x	µ((x	NOUN
ejpam-6327	349	14	∗	∗	NOUN
ejpam-6327	349	15	y)z	y)z	NOUN
ejpam-6327	349	16	)	)	PUNCT
ejpam-6327	349	17	≥	≥	NOUN
ejpam-6327	349	18	µ(x	µ(x	X
ejpam-6327	349	19	∗	∗	NOUN
ejpam-6327	349	20	y	y	NOUN
ejpam-6327	349	21	)	)	PUNCT
ejpam-6327	349	22	>	>	X
ejpam-6327	350	1	0	0	X
ejpam-6327	350	2	.	.	PUNCT
ejpam-6327	351	1	similarly	similarly	ADV
ejpam-6327	351	2	,	,	PUNCT
ejpam-6327	351	3	µ(yz	µ(yz	PROPN
ejpam-6327	351	4	∗xz	∗xz	NOUN
ejpam-6327	351	5	)	)	PUNCT
ejpam-6327	351	6	>	>	X
ejpam-6327	352	1	0	0	X
ejpam-6327	352	2	.	.	PUNCT
ejpam-6327	353	1	hence	hence	ADV
ejpam-6327	353	2	,	,	PUNCT
ejpam-6327	353	3	xz	xz	PROPN
ejpam-6327	353	4	∼µ	∼µ	PROPN
ejpam-6327	353	5	yz	yz	PROPN
ejpam-6327	353	6	.	.	PUNCT
ejpam-6327	354	1	by	by	ADP
ejpam-6327	354	2	definition	definition	NOUN
ejpam-6327	354	3	8(i	8(i	NUM
ejpam-6327	354	4	)	)	PUNCT
ejpam-6327	354	5	,	,	PUNCT
ejpam-6327	354	6	∼µ	∼µ	PROPN
ejpam-6327	354	7	is	be	AUX
ejpam-6327	354	8	right	right	ADV
ejpam-6327	354	9	compatible	compatible	ADJ
ejpam-6327	354	10	.	.	PUNCT
ejpam-6327	355	1	the	the	DET
ejpam-6327	355	2	following	follow	VERB
ejpam-6327	355	3	corollary	corollary	NOUN
ejpam-6327	355	4	follows	follow	VERB
ejpam-6327	355	5	from	from	ADP
ejpam-6327	355	6	propositions	proposition	NOUN
ejpam-6327	355	7	3	3	NUM
ejpam-6327	355	8	and	and	CCONJ
ejpam-6327	355	9	4	4	NUM
ejpam-6327	355	10	and	and	CCONJ
ejpam-6327	355	11	theorem	theorem	VERB
ejpam-6327	355	12	1	1	NUM
ejpam-6327	355	13	.	.	PUNCT
ejpam-6327	355	14	corollary	corollary	ADJ
ejpam-6327	355	15	1	1	NUM
ejpam-6327	355	16	.	.	PUNCT
ejpam-6327	356	1	let	let	VERB
ejpam-6327	356	2	x	x	PRON
ejpam-6327	356	3	be	be	AUX
ejpam-6327	356	4	a	a	DET
ejpam-6327	356	5	ks	ks	NOUN
ejpam-6327	356	6	-	-	PUNCT
ejpam-6327	356	7	semigroup	semigroup	NOUN
ejpam-6327	356	8	and	and	CCONJ
ejpam-6327	356	9	µ	µ	DET
ejpam-6327	356	10	a	a	DET
ejpam-6327	356	11	non	non	ADJ
ejpam-6327	356	12	-	-	ADJ
ejpam-6327	356	13	zero	zero	ADJ
ejpam-6327	356	14	fuzzy	fuzzy	ADJ
ejpam-6327	356	15	ks	ks	NOUN
ejpam-6327	356	16	-	-	NOUN
ejpam-6327	356	17	ideal	ideal	NOUN
ejpam-6327	356	18	of	of	ADP
ejpam-6327	356	19	x	x	SYM
ejpam-6327	356	20	satisfying	satisfy	VERB
ejpam-6327	356	21	condition	condition	NOUN
ejpam-6327	356	22	(	(	PUNCT
ejpam-6327	356	23	c	c	NOUN
ejpam-6327	356	24	)	)	PUNCT
ejpam-6327	356	25	.	.	PUNCT
ejpam-6327	357	1	then	then	ADV
ejpam-6327	357	2	∼µ	∼µ	PROPN
ejpam-6327	357	3	is	be	AUX
ejpam-6327	357	4	a	a	DET
ejpam-6327	357	5	congruence	congruence	NOUN
ejpam-6327	357	6	relation	relation	NOUN
ejpam-6327	357	7	on	on	ADP
ejpam-6327	357	8	x.	x.	NOUN
ejpam-6327	357	9	let	let	VERB
ejpam-6327	357	10	x	x	PRON
ejpam-6327	357	11	be	be	AUX
ejpam-6327	357	12	a	a	DET
ejpam-6327	357	13	ks	ks	NOUN
ejpam-6327	357	14	-	-	PUNCT
ejpam-6327	357	15	semigroup	semigroup	NOUN
ejpam-6327	357	16	and	and	CCONJ
ejpam-6327	357	17	µ	µ	DET
ejpam-6327	357	18	a	a	DET
ejpam-6327	357	19	non	non	ADJ
ejpam-6327	357	20	-	-	ADJ
ejpam-6327	357	21	zero	zero	ADJ
ejpam-6327	357	22	fuzzy	fuzzy	ADJ
ejpam-6327	357	23	ks	ks	NOUN
ejpam-6327	357	24	-	-	NOUN
ejpam-6327	357	25	ideal	ideal	NOUN
ejpam-6327	357	26	of	of	ADP
ejpam-6327	357	27	x	x	SYM
ejpam-6327	357	28	satisfying	satisfy	VERB
ejpam-6327	357	29	condition	condition	NOUN
ejpam-6327	357	30	(	(	PUNCT
ejpam-6327	357	31	c	c	NOUN
ejpam-6327	357	32	)	)	PUNCT
ejpam-6327	357	33	.	.	PUNCT
ejpam-6327	358	1	denote	denote	VERB
ejpam-6327	358	2	µx	µx	VERB
ejpam-6327	358	3	as	as	ADP
ejpam-6327	358	4	the	the	DET
ejpam-6327	358	5	equivalence	equivalence	NOUN
ejpam-6327	358	6	class	class	NOUN
ejpam-6327	358	7	containing	contain	VERB
ejpam-6327	358	8	x	x	SYM
ejpam-6327	358	9	∈	∈	PROPN
ejpam-6327	358	10	x	x	X
ejpam-6327	358	11	and	and	CCONJ
ejpam-6327	358	12	x/µ	x/µ	PROPN
ejpam-6327	358	13	as	as	ADP
ejpam-6327	358	14	the	the	DET
ejpam-6327	358	15	set	set	NOUN
ejpam-6327	358	16	of	of	ADP
ejpam-6327	358	17	all	all	DET
ejpam-6327	358	18	equivalence	equivalence	NOUN
ejpam-6327	358	19	classes	class	NOUN
ejpam-6327	358	20	of	of	ADP
ejpam-6327	358	21	x	x	PUNCT
ejpam-6327	358	22	with	with	ADP
ejpam-6327	358	23	respect	respect	NOUN
ejpam-6327	358	24	to	to	ADP
ejpam-6327	358	25	“	"	PUNCT
ejpam-6327	358	26	∼µ	∼µ	PROPN
ejpam-6327	358	27	”	"	PUNCT
ejpam-6327	358	28	,	,	PUNCT
ejpam-6327	358	29	that	that	ADV
ejpam-6327	358	30	is	is	ADV
ejpam-6327	358	31	,	,	PUNCT
ejpam-6327	358	32	µx	µx	ADP
ejpam-6327	358	33	=	=	SYM
ejpam-6327	358	34	{	{	PUNCT
ejpam-6327	358	35	y	y	PROPN
ejpam-6327	358	36	∈	∈	PROPN
ejpam-6327	358	37	x	x	X
ejpam-6327	358	38	:	:	PUNCT
ejpam-6327	358	39	y	y	PROPN
ejpam-6327	358	40	∼µ	∼µ	ADP
ejpam-6327	358	41	x	x	PUNCT
ejpam-6327	358	42	}	}	PUNCT
ejpam-6327	358	43	and	and	CCONJ
ejpam-6327	358	44	x/µ	x/µ	PROPN
ejpam-6327	359	1	=	=	PRON
ejpam-6327	359	2	{	{	PUNCT
ejpam-6327	359	3	µx	µx	NOUN
ejpam-6327	359	4	:	:	PUNCT
ejpam-6327	359	5	x	x	SYM
ejpam-6327	359	6	∈	∈	NOUN
ejpam-6327	359	7	x	x	X
ejpam-6327	359	8	}	}	PUNCT
ejpam-6327	359	9	.	.	PUNCT
ejpam-6327	360	1	remark	remark	NOUN
ejpam-6327	360	2	3	3	NUM
ejpam-6327	360	3	.	.	PUNCT
ejpam-6327	360	4	µx	µx	PROPN
ejpam-6327	361	1	=	=	SYM
ejpam-6327	361	2	µy	µy	VERB
ejpam-6327	361	3	if	if	SCONJ
ejpam-6327	361	4	and	and	CCONJ
ejpam-6327	361	5	only	only	ADV
ejpam-6327	361	6	if	if	SCONJ
ejpam-6327	361	7	x	x	SYM
ejpam-6327	361	8	∼µ	∼µ	PROPN
ejpam-6327	361	9	y.	y.	NOUN
ejpam-6327	361	10	theorem	theorem	VERB
ejpam-6327	361	11	11	11	NUM
ejpam-6327	361	12	.	.	PUNCT
ejpam-6327	362	1	let	let	VERB
ejpam-6327	362	2	x	x	PRON
ejpam-6327	362	3	be	be	AUX
ejpam-6327	362	4	a	a	DET
ejpam-6327	362	5	ks	ks	NOUN
ejpam-6327	362	6	-	-	PUNCT
ejpam-6327	362	7	semigroup	semigroup	NOUN
ejpam-6327	362	8	and	and	CCONJ
ejpam-6327	362	9	µ	µ	DET
ejpam-6327	362	10	a	a	DET
ejpam-6327	362	11	non	non	ADJ
ejpam-6327	362	12	-	-	ADJ
ejpam-6327	362	13	zero	zero	ADJ
ejpam-6327	362	14	fuzzy	fuzzy	ADJ
ejpam-6327	362	15	ks	ks	NOUN
ejpam-6327	362	16	-	-	NOUN
ejpam-6327	362	17	ideal	ideal	NOUN
ejpam-6327	362	18	of	of	ADP
ejpam-6327	362	19	x	x	SYM
ejpam-6327	362	20	satisfying	satisfy	VERB
ejpam-6327	362	21	condition	condition	NOUN
ejpam-6327	362	22	(	(	PUNCT
ejpam-6327	362	23	c	c	NOUN
ejpam-6327	362	24	)	)	PUNCT
ejpam-6327	362	25	.	.	PUNCT
ejpam-6327	363	1	then	then	ADV
ejpam-6327	363	2	(	(	PUNCT
ejpam-6327	363	3	x/µ,⊛,⊙	x/µ,⊛,⊙	PROPN
ejpam-6327	363	4	,	,	PUNCT
ejpam-6327	363	5	µ0	µ0	NOUN
ejpam-6327	363	6	)	)	PUNCT
ejpam-6327	363	7	is	be	AUX
ejpam-6327	363	8	a	a	DET
ejpam-6327	363	9	ks	ks	NOUN
ejpam-6327	363	10	-	-	PUNCT
ejpam-6327	363	11	semigroup	semigroup	NOUN
ejpam-6327	363	12	under	under	ADP
ejpam-6327	363	13	the	the	DET
ejpam-6327	363	14	binary	binary	ADJ
ejpam-6327	363	15	operations	operation	NOUN
ejpam-6327	363	16	µx	µx	VERB
ejpam-6327	363	17	⊛	⊛	NOUN
ejpam-6327	363	18	µy	µy	ADV
ejpam-6327	363	19	=	=	PUNCT
ejpam-6327	363	20	µx∗y	µx∗y	X
ejpam-6327	363	21	and	and	CCONJ
ejpam-6327	363	22	µx	µx	VERB
ejpam-6327	363	23	⊙	⊙	PROPN
ejpam-6327	363	24	µy	µy	X
ejpam-6327	363	25	=	=	PUNCT
ejpam-6327	363	26	µxy	µxy	PROPN
ejpam-6327	363	27	for	for	ADP
ejpam-6327	363	28	all	all	PRON
ejpam-6327	363	29	µx	µx	VERB
ejpam-6327	363	30	,	,	PUNCT
ejpam-6327	363	31	µy	µy	PROPN
ejpam-6327	363	32	∈	∈	NOUN
ejpam-6327	363	33	x/µ.	x/µ.	NOUN
ejpam-6327	363	34	proof	proof	NOUN
ejpam-6327	363	35	.	.	PUNCT
ejpam-6327	364	1	let	let	VERB
ejpam-6327	364	2	x	x	PRON
ejpam-6327	364	3	be	be	AUX
ejpam-6327	364	4	a	a	DET
ejpam-6327	364	5	ks	ks	NOUN
ejpam-6327	364	6	-	-	PUNCT
ejpam-6327	364	7	semigroup	semigroup	NOUN
ejpam-6327	364	8	and	and	CCONJ
ejpam-6327	364	9	µ	µ	DET
ejpam-6327	364	10	a	a	DET
ejpam-6327	364	11	non	non	ADJ
ejpam-6327	364	12	-	-	ADJ
ejpam-6327	364	13	zero	zero	ADJ
ejpam-6327	364	14	fuzzy	fuzzy	ADJ
ejpam-6327	364	15	ks	ks	NOUN
ejpam-6327	364	16	-	-	NOUN
ejpam-6327	364	17	ideal	ideal	NOUN
ejpam-6327	364	18	of	of	ADP
ejpam-6327	364	19	x	x	SYM
ejpam-6327	364	20	satisfying	satisfy	VERB
ejpam-6327	364	21	condition	condition	NOUN
ejpam-6327	364	22	(	(	PUNCT
ejpam-6327	364	23	c	c	NOUN
ejpam-6327	364	24	)	)	PUNCT
ejpam-6327	364	25	.	.	PUNCT
ejpam-6327	365	1	by	by	ADP
ejpam-6327	365	2	corollary	corollary	ADJ
ejpam-6327	365	3	1	1	NUM
ejpam-6327	365	4	,	,	PUNCT
ejpam-6327	365	5	the	the	DET
ejpam-6327	365	6	operations	operation	NOUN
ejpam-6327	365	7	⊛	⊛	NUM
ejpam-6327	365	8	and	and	CCONJ
ejpam-6327	365	9	⊙	⊙	NOUN
ejpam-6327	365	10	are	be	AUX
ejpam-6327	365	11	well	well	ADV
ejpam-6327	365	12	-	-	PUNCT
ejpam-6327	365	13	defined	define	VERB
ejpam-6327	365	14	.	.	PUNCT
ejpam-6327	366	1	for	for	ADP
ejpam-6327	366	2	every	every	DET
ejpam-6327	366	3	µx	µx	ADJ
ejpam-6327	366	4	,	,	PUNCT
ejpam-6327	366	5	µy	µy	INTJ
ejpam-6327	366	6	,	,	PUNCT
ejpam-6327	366	7	µz	µz	PROPN
ejpam-6327	366	8	∈	∈	PROPN
ejpam-6327	366	9	x/µ	x/µ	PROPN
ejpam-6327	366	10	,	,	PUNCT
ejpam-6327	366	11	h.	h.	PROPN
ejpam-6327	366	12	sarapuddin	sarapuddin	PROPN
ejpam-6327	366	13	,	,	PUNCT
ejpam-6327	366	14	j.	j.	PROPN
ejpam-6327	366	15	vilela	vilela	PROPN
ejpam-6327	366	16	/	/	SYM
ejpam-6327	366	17	eur	eur	PROPN
ejpam-6327	366	18	.	.	PUNCT
ejpam-6327	367	1	j.	j.	PROPN
ejpam-6327	367	2	pure	pure	PROPN
ejpam-6327	367	3	appl	appl	PROPN
ejpam-6327	367	4	.	.	PROPN
ejpam-6327	367	5	math	math	PROPN
ejpam-6327	367	6	,	,	PUNCT
ejpam-6327	367	7	18	18	NUM
ejpam-6327	367	8	(	(	PUNCT
ejpam-6327	367	9	3	3	NUM
ejpam-6327	367	10	)	)	PUNCT
ejpam-6327	367	11	(	(	PUNCT
ejpam-6327	367	12	2025	2025	NUM
ejpam-6327	367	13	)	)	PUNCT
ejpam-6327	367	14	,	,	PUNCT
ejpam-6327	367	15	6327	6327	NUM
ejpam-6327	367	16	12	12	NUM
ejpam-6327	367	17	of	of	ADP
ejpam-6327	367	18	23	23	NUM
ejpam-6327	367	19	(	(	PUNCT
ejpam-6327	367	20	i	i	NOUN
ejpam-6327	367	21	)	)	PUNCT
ejpam-6327	367	22	(	(	PUNCT
ejpam-6327	367	23	(	(	PUNCT
ejpam-6327	367	24	µx	µx	X
ejpam-6327	367	25	⊛	⊛	ADJ
ejpam-6327	367	26	µy)⊛	µy)⊛	PROPN
ejpam-6327	367	27	(	(	PUNCT
ejpam-6327	367	28	µx	µx	ADJ
ejpam-6327	367	29	⊛	⊛	ADJ
ejpam-6327	367	30	µz))⊛	µz))⊛	X
ejpam-6327	367	31	(	(	PUNCT
ejpam-6327	367	32	µz	µz	PROPN
ejpam-6327	367	33	⊛	⊛	NUM
ejpam-6327	367	34	µy	µy	X
ejpam-6327	367	35	)	)	PUNCT
ejpam-6327	367	36	=	=	SYM
ejpam-6327	367	37	µ((x∗y)∗(x∗z))∗(z∗y	µ((x∗y)∗(x∗z))∗(z∗y	PROPN
ejpam-6327	367	38	)	)	PUNCT
ejpam-6327	367	39	=	=	SYM
ejpam-6327	367	40	µ0	µ0	NOUN
ejpam-6327	367	41	;	;	PUNCT
ejpam-6327	367	42	(	(	PUNCT
ejpam-6327	367	43	ii	ii	NOUN
ejpam-6327	367	44	)	)	PUNCT
ejpam-6327	367	45	(	(	PUNCT
ejpam-6327	368	1	µx	µx	ADP
ejpam-6327	368	2	⊛	⊛	NOUN
ejpam-6327	368	3	(	(	PUNCT
ejpam-6327	368	4	µx	µx	VERB
ejpam-6327	368	5	⊛	⊛	NUM
ejpam-6327	368	6	µy))⊛	µy))⊛	VERB
ejpam-6327	368	7	µy	µy	X
ejpam-6327	368	8	=	=	PUNCT
ejpam-6327	368	9	µ(x∗(x∗y))∗y	µ(x∗(x∗y))∗y	PROPN
ejpam-6327	368	10	=	=	SYM
ejpam-6327	368	11	µ0	µ0	NOUN
ejpam-6327	368	12	;	;	PUNCT
ejpam-6327	368	13	(	(	PUNCT
ejpam-6327	368	14	iii	iii	NOUN
ejpam-6327	368	15	)	)	PUNCT
ejpam-6327	368	16	µx	µx	VERB
ejpam-6327	368	17	⊛	⊛	NOUN
ejpam-6327	368	18	µx	µx	VERB
ejpam-6327	368	19	=	=	SYM
ejpam-6327	368	20	µx∗x	µx∗x	ADV
ejpam-6327	368	21	=	=	SYM
ejpam-6327	368	22	µ0	µ0	NOUN
ejpam-6327	368	23	;	;	PUNCT
ejpam-6327	368	24	(	(	PUNCT
ejpam-6327	368	25	iv	iv	X
ejpam-6327	368	26	)	)	PUNCT
ejpam-6327	368	27	µ0	µ0	NOUN
ejpam-6327	368	28	⊛	⊛	ADJ
ejpam-6327	368	29	µx	µx	VERB
ejpam-6327	369	1	=	=	SYM
ejpam-6327	369	2	µ0∗x	µ0∗x	PROPN
ejpam-6327	369	3	=	=	SYM
ejpam-6327	369	4	µ0	µ0	PROPN
ejpam-6327	369	5	.	.	PUNCT
ejpam-6327	370	1	(	(	PUNCT
ejpam-6327	370	2	v	v	NOUN
ejpam-6327	370	3	)	)	PUNCT
ejpam-6327	370	4	suppose	suppose	VERB
ejpam-6327	370	5	µx	µx	ADP
ejpam-6327	370	6	⊛	⊛	ADP
ejpam-6327	370	7	µy	µy	VERB
ejpam-6327	370	8	=	=	SYM
ejpam-6327	370	9	µ0	µ0	PROPN
ejpam-6327	370	10	and	and	CCONJ
ejpam-6327	370	11	µy	µy	X
ejpam-6327	370	12	⊛	⊛	NUM
ejpam-6327	370	13	µx	µx	VERB
ejpam-6327	370	14	=	=	NOUN
ejpam-6327	370	15	µ0	µ0	NOUN
ejpam-6327	370	16	.	.	PUNCT
ejpam-6327	371	1	then	then	ADV
ejpam-6327	371	2	µx∗y	µx∗y	PRON
ejpam-6327	371	3	=	=	PUNCT
ejpam-6327	371	4	µy∗x	µy∗x	SYM
ejpam-6327	371	5	=	=	SYM
ejpam-6327	371	6	µ0	µ0	NOUN
ejpam-6327	371	7	.	.	PUNCT
ejpam-6327	372	1	this	this	PRON
ejpam-6327	372	2	implies	imply	VERB
ejpam-6327	372	3	that	that	SCONJ
ejpam-6327	372	4	x	x	SYM
ejpam-6327	372	5	∗	∗	VERB
ejpam-6327	372	6	y	y	NOUN
ejpam-6327	372	7	∼µ	∼µ	PROPN
ejpam-6327	372	8	0	0	NUM
ejpam-6327	373	1	and	and	CCONJ
ejpam-6327	373	2	y	y	PROPN
ejpam-6327	373	3	∗	∗	NOUN
ejpam-6327	373	4	x	x	NOUN
ejpam-6327	373	5	∼µ	∼µ	NOUN
ejpam-6327	373	6	0	0	NUM
ejpam-6327	373	7	.	.	PUNCT
ejpam-6327	374	1	thus	thus	ADV
ejpam-6327	374	2	,	,	PUNCT
ejpam-6327	374	3	µ(x	µ(x	VERB
ejpam-6327	374	4	∗	∗	NOUN
ejpam-6327	374	5	y	y	NOUN
ejpam-6327	374	6	)	)	PUNCT
ejpam-6327	375	1	=	=	SYM
ejpam-6327	375	2	µ((x	µ((x	NOUN
ejpam-6327	375	3	∗	∗	PROPN
ejpam-6327	375	4	y	y	NOUN
ejpam-6327	375	5	)	)	PUNCT
ejpam-6327	375	6	∗	∗	NOUN
ejpam-6327	375	7	0	0	NUM
ejpam-6327	375	8	)	)	PUNCT
ejpam-6327	375	9	>	>	X
ejpam-6327	375	10	0	0	PUNCT
ejpam-6327	375	11	and	and	CCONJ
ejpam-6327	375	12	µ(y	µ(y	PROPN
ejpam-6327	375	13	∗	∗	NOUN
ejpam-6327	375	14	x	x	NOUN
ejpam-6327	375	15	)	)	PUNCT
ejpam-6327	375	16	=	=	PUNCT
ejpam-6327	375	17	µ((y	µ((y	NOUN
ejpam-6327	375	18	∗	∗	NOUN
ejpam-6327	375	19	x	x	NOUN
ejpam-6327	375	20	)	)	PUNCT
ejpam-6327	375	21	∗	∗	NOUN
ejpam-6327	375	22	0	0	NUM
ejpam-6327	375	23	)	)	PUNCT
ejpam-6327	375	24	>	>	X
ejpam-6327	375	25	0	0	NUM
ejpam-6327	375	26	,	,	PUNCT
ejpam-6327	375	27	that	that	ADV
ejpam-6327	375	28	is	is	ADV
ejpam-6327	375	29	,	,	PUNCT
ejpam-6327	375	30	x	x	PROPN
ejpam-6327	375	31	∼µ	∼µ	PROPN
ejpam-6327	375	32	y.	y.	NOUN
ejpam-6327	375	33	hence	hence	ADV
ejpam-6327	375	34	,	,	PUNCT
ejpam-6327	375	35	µx	µx	ADP
ejpam-6327	375	36	=	=	SYM
ejpam-6327	375	37	µy	µy	X
ejpam-6327	375	38	.	.	PUNCT
ejpam-6327	376	1	therefore	therefore	ADV
ejpam-6327	376	2	,	,	PUNCT
ejpam-6327	376	3	(	(	PUNCT
ejpam-6327	376	4	x/µ,⊛	x/µ,⊛	PROPN
ejpam-6327	376	5	,	,	PUNCT
ejpam-6327	376	6	µ0	µ0	NOUN
ejpam-6327	376	7	)	)	PUNCT
ejpam-6327	376	8	is	be	AUX
ejpam-6327	376	9	a	a	DET
ejpam-6327	376	10	bck	bck	NOUN
ejpam-6327	376	11	-	-	PUNCT
ejpam-6327	376	12	algebra	algebra	NOUN
ejpam-6327	376	13	.	.	PUNCT
ejpam-6327	377	1	moreover	moreover	ADV
ejpam-6327	377	2	,	,	PUNCT
ejpam-6327	377	3	(	(	PUNCT
ejpam-6327	377	4	µx	µx	INTJ
ejpam-6327	377	5	⊙	⊙	PROPN
ejpam-6327	377	6	µy)⊙	µy)⊙	PROPN
ejpam-6327	377	7	µz	µz	PROPN
ejpam-6327	377	8	=	=	SYM
ejpam-6327	377	9	µxy	µxy	PROPN
ejpam-6327	377	10	⊙	⊙	PROPN
ejpam-6327	377	11	µz	µz	PROPN
ejpam-6327	377	12	=	=	SYM
ejpam-6327	377	13	µ(xy)z	µ(xy)z	PROPN
ejpam-6327	377	14	=	=	SYM
ejpam-6327	377	15	µx(yz	µx(yz	PROPN
ejpam-6327	377	16	)	)	PUNCT
ejpam-6327	377	17	=	=	PRON
ejpam-6327	377	18	µx	µx	VERB
ejpam-6327	377	19	⊙	⊙	PROPN
ejpam-6327	377	20	µyz	µyz	PROPN
ejpam-6327	378	1	=	=	SYM
ejpam-6327	378	2	µx	µx	VERB
ejpam-6327	378	3	⊙	⊙	PROPN
ejpam-6327	378	4	(	(	PUNCT
ejpam-6327	378	5	µy	µy	PROPN
ejpam-6327	378	6	⊙	⊙	PROPN
ejpam-6327	378	7	µz	µz	PROPN
ejpam-6327	378	8	)	)	PUNCT
ejpam-6327	378	9	.	.	PUNCT
ejpam-6327	379	1	thus	thus	ADV
ejpam-6327	379	2	,	,	PUNCT
ejpam-6327	379	3	(	(	PUNCT
ejpam-6327	379	4	x/µ,⊙	x/µ,⊙	NOUN
ejpam-6327	379	5	)	)	PUNCT
ejpam-6327	379	6	is	be	AUX
ejpam-6327	379	7	a	a	DET
ejpam-6327	379	8	semigroup	semigroup	NOUN
ejpam-6327	379	9	.	.	PUNCT
ejpam-6327	380	1	now	now	ADV
ejpam-6327	380	2	,	,	PUNCT
ejpam-6327	380	3	observe	observe	VERB
ejpam-6327	380	4	that	that	SCONJ
ejpam-6327	380	5	µx	µx	ADV
ejpam-6327	380	6	⊙	⊙	PROPN
ejpam-6327	380	7	(	(	PUNCT
ejpam-6327	380	8	µy	µy	PROPN
ejpam-6327	380	9	⊛	⊛	NUM
ejpam-6327	380	10	µz	µz	PROPN
ejpam-6327	380	11	)	)	PUNCT
ejpam-6327	380	12	=	=	VERB
ejpam-6327	380	13	µx	µx	VERB
ejpam-6327	380	14	⊙	⊙	NOUN
ejpam-6327	380	15	µy∗z	µy∗z	PROPN
ejpam-6327	380	16	=	=	PUNCT
ejpam-6327	380	17	µx(y∗z	µx(y∗z	X
ejpam-6327	380	18	)	)	PUNCT
ejpam-6327	381	1	=	=	SYM
ejpam-6327	381	2	µxy∗xz	µxy∗xz	X
ejpam-6327	381	3	=	=	PUNCT
ejpam-6327	381	4	µxy	µxy	PROPN
ejpam-6327	381	5	⊛	⊛	ADJ
ejpam-6327	381	6	µxz	µxz	VERB
ejpam-6327	381	7	=	=	PUNCT
ejpam-6327	381	8	(	(	PUNCT
ejpam-6327	381	9	µx	µx	INTJ
ejpam-6327	381	10	⊙	⊙	PROPN
ejpam-6327	381	11	µy)⊛	µy)⊛	PROPN
ejpam-6327	381	12	(	(	PUNCT
ejpam-6327	381	13	µx	µx	PROPN
ejpam-6327	381	14	⊙	⊙	PROPN
ejpam-6327	381	15	µz	µz	PROPN
ejpam-6327	381	16	)	)	PUNCT
ejpam-6327	381	17	and	and	CCONJ
ejpam-6327	381	18	(	(	PUNCT
ejpam-6327	381	19	µx	µx	VERB
ejpam-6327	381	20	⊛	⊛	ADJ
ejpam-6327	381	21	µy)⊙	µy)⊙	NOUN
ejpam-6327	381	22	µz	µz	NOUN
ejpam-6327	381	23	=	=	PUNCT
ejpam-6327	381	24	µx∗y	µx∗y	X
ejpam-6327	381	25	⊙	⊙	VERB
ejpam-6327	381	26	µz	µz	NOUN
ejpam-6327	381	27	=	=	PUNCT
ejpam-6327	381	28	µ(x∗y)z	µ(x∗y)z	X
ejpam-6327	381	29	=	=	PUNCT
ejpam-6327	381	30	µxz∗yz	µxz∗yz	ADJ
ejpam-6327	381	31	=	=	PUNCT
ejpam-6327	381	32	µxz	µxz	VERB
ejpam-6327	381	33	⊛	⊛	ADJ
ejpam-6327	381	34	µyz	µyz	NOUN
ejpam-6327	381	35	=	=	PUNCT
ejpam-6327	381	36	(	(	PUNCT
ejpam-6327	381	37	µx	µx	INTJ
ejpam-6327	381	38	⊙	⊙	PROPN
ejpam-6327	381	39	µz)⊛	µz)⊛	PROPN
ejpam-6327	381	40	(	(	PUNCT
ejpam-6327	381	41	µy	µy	PROPN
ejpam-6327	381	42	⊙	⊙	PROPN
ejpam-6327	381	43	µz	µz	PROPN
ejpam-6327	381	44	)	)	PUNCT
ejpam-6327	381	45	.	.	PUNCT
ejpam-6327	382	1	therefore	therefore	ADV
ejpam-6327	382	2	,	,	PUNCT
ejpam-6327	382	3	(	(	PUNCT
ejpam-6327	382	4	x/µ,⊛,⊙	x/µ,⊛,⊙	NUM
ejpam-6327	382	5	,	,	PUNCT
ejpam-6327	382	6	µ0	µ0	NOUN
ejpam-6327	382	7	)	)	PUNCT
ejpam-6327	382	8	is	be	AUX
ejpam-6327	382	9	a	a	DET
ejpam-6327	382	10	ks	ks	NOUN
ejpam-6327	382	11	-	-	PUNCT
ejpam-6327	382	12	semigroup	semigroup	NOUN
ejpam-6327	382	13	.	.	PUNCT
ejpam-6327	383	1	the	the	DET
ejpam-6327	383	2	algebraic	algebraic	ADJ
ejpam-6327	383	3	system	system	NOUN
ejpam-6327	383	4	(	(	PUNCT
ejpam-6327	383	5	x/µ,⊛,⊙	x/µ,⊛,⊙	NUM
ejpam-6327	383	6	,	,	PUNCT
ejpam-6327	383	7	µ0	µ0	NOUN
ejpam-6327	383	8	)	)	PUNCT
ejpam-6327	383	9	is	be	AUX
ejpam-6327	383	10	called	call	VERB
ejpam-6327	383	11	the	the	DET
ejpam-6327	383	12	quotient	quotient	NOUN
ejpam-6327	383	13	ks	ks	NOUN
ejpam-6327	383	14	-	-	PUNCT
ejpam-6327	383	15	semigroup	semigroup	NOUN
ejpam-6327	383	16	induced	induce	VERB
ejpam-6327	383	17	by	by	ADP
ejpam-6327	383	18	a	a	DET
ejpam-6327	383	19	fuzzy	fuzzy	ADJ
ejpam-6327	383	20	ks	ks	NOUN
ejpam-6327	383	21	-	-	PUNCT
ejpam-6327	383	22	ideal	ideal	ADJ
ejpam-6327	383	23	µ.	µ.	NOUN
ejpam-6327	383	24	the	the	DET
ejpam-6327	383	25	following	follow	VERB
ejpam-6327	383	26	theorem	theorem	NOUN
ejpam-6327	383	27	tells	tell	VERB
ejpam-6327	383	28	us	we	PRON
ejpam-6327	383	29	that	that	SCONJ
ejpam-6327	383	30	the	the	DET
ejpam-6327	383	31	condition	condition	NOUN
ejpam-6327	383	32	(	(	PUNCT
ejpam-6327	383	33	c	c	X
ejpam-6327	383	34	)	)	PUNCT
ejpam-6327	383	35	is	be	AUX
ejpam-6327	383	36	necessary	necessary	ADJ
ejpam-6327	383	37	to	to	PART
ejpam-6327	383	38	extend	extend	VERB
ejpam-6327	383	39	the	the	DET
ejpam-6327	383	40	notion	notion	NOUN
ejpam-6327	383	41	of	of	ADP
ejpam-6327	383	42	ideal	ideal	NOUN
ejpam-6327	383	43	of	of	ADP
ejpam-6327	383	44	ks	ks	NOUN
ejpam-6327	383	45	-	-	PUNCT
ejpam-6327	383	46	semigroup	semigroup	NOUN
ejpam-6327	383	47	.	.	PUNCT
ejpam-6327	384	1	theorem	theorem	NOUN
ejpam-6327	384	2	12	12	NUM
ejpam-6327	384	3	.	.	PUNCT
ejpam-6327	385	1	let	let	VERB
ejpam-6327	385	2	i	i	PRON
ejpam-6327	385	3	be	be	AUX
ejpam-6327	385	4	a	a	DET
ejpam-6327	385	5	nonempty	nonempty	ADJ
ejpam-6327	385	6	subset	subset	NOUN
ejpam-6327	385	7	of	of	ADP
ejpam-6327	385	8	a	a	DET
ejpam-6327	385	9	ks	ks	NOUN
ejpam-6327	385	10	-	-	PUNCT
ejpam-6327	385	11	semigroup	semigroup	NOUN
ejpam-6327	385	12	x.	x.	NOUN
ejpam-6327	385	13	then	then	ADV
ejpam-6327	385	14	χi	χi	PROPN
ejpam-6327	385	15	is	be	AUX
ejpam-6327	385	16	a	a	DET
ejpam-6327	385	17	fuzzy	fuzzy	ADJ
ejpam-6327	385	18	ks	ks	NOUN
ejpam-6327	385	19	-	-	NOUN
ejpam-6327	385	20	ideal	ideal	NOUN
ejpam-6327	385	21	of	of	ADP
ejpam-6327	385	22	x	x	SYM
ejpam-6327	385	23	satisfying	satisfy	VERB
ejpam-6327	385	24	condition	condition	NOUN
ejpam-6327	385	25	(	(	PUNCT
ejpam-6327	385	26	c	c	NOUN
ejpam-6327	385	27	)	)	PUNCT
ejpam-6327	385	28	if	if	SCONJ
ejpam-6327	386	1	and	and	CCONJ
ejpam-6327	386	2	only	only	ADV
ejpam-6327	386	3	if	if	SCONJ
ejpam-6327	386	4	i	i	PRON
ejpam-6327	386	5	is	be	AUX
ejpam-6327	386	6	an	an	DET
ejpam-6327	386	7	ideal	ideal	NOUN
ejpam-6327	386	8	of	of	ADP
ejpam-6327	386	9	x.	x.	NOUN
ejpam-6327	386	10	proof	proof	NOUN
ejpam-6327	386	11	.	.	PUNCT
ejpam-6327	387	1	let	let	VERB
ejpam-6327	387	2	i	i	PRON
ejpam-6327	387	3	be	be	AUX
ejpam-6327	387	4	a	a	DET
ejpam-6327	387	5	nonempty	nonempty	ADJ
ejpam-6327	387	6	subset	subset	NOUN
ejpam-6327	387	7	of	of	ADP
ejpam-6327	387	8	a	a	DET
ejpam-6327	387	9	ks	ks	NOUN
ejpam-6327	387	10	-	-	PUNCT
ejpam-6327	387	11	semigroup	semigroup	PROPN
ejpam-6327	387	12	x.	x.	NOUN
ejpam-6327	387	13	suppose	suppose	VERB
ejpam-6327	387	14	χi	χi	NOUN
ejpam-6327	387	15	is	be	AUX
ejpam-6327	387	16	a	a	DET
ejpam-6327	387	17	fuzzy	fuzzy	ADJ
ejpam-6327	387	18	ks	ks	NOUN
ejpam-6327	387	19	-	-	NOUN
ejpam-6327	387	20	ideal	ideal	NOUN
ejpam-6327	387	21	of	of	ADP
ejpam-6327	387	22	x	x	SYM
ejpam-6327	387	23	satisfying	satisfy	VERB
ejpam-6327	387	24	condition	condition	NOUN
ejpam-6327	387	25	(	(	PUNCT
ejpam-6327	387	26	c	c	NOUN
ejpam-6327	387	27	)	)	PUNCT
ejpam-6327	387	28	.	.	PUNCT
ejpam-6327	388	1	let	let	VERB
ejpam-6327	388	2	x	x	PUNCT
ejpam-6327	388	3	∈	∈	PROPN
ejpam-6327	388	4	x	x	X
ejpam-6327	388	5	and	and	CCONJ
ejpam-6327	388	6	a	a	DET
ejpam-6327	388	7	∈	∈	PROPN
ejpam-6327	388	8	i.	i.	NOUN
ejpam-6327	388	9	then	then	ADV
ejpam-6327	388	10	by	by	ADP
ejpam-6327	388	11	condition	condition	NOUN
ejpam-6327	388	12	(	(	PUNCT
ejpam-6327	388	13	c	c	NOUN
ejpam-6327	388	14	)	)	PUNCT
ejpam-6327	388	15	,	,	PUNCT
ejpam-6327	388	16	χi(xa	χi(xa	PROPN
ejpam-6327	388	17	)	)	PUNCT
ejpam-6327	388	18	≥	≥	NOUN
ejpam-6327	388	19	χi(a	χi(a	PUNCT
ejpam-6327	388	20	)	)	PUNCT
ejpam-6327	389	1	=	=	SYM
ejpam-6327	390	1	1	1	X
ejpam-6327	390	2	.	.	PUNCT
ejpam-6327	390	3	thus	thus	ADV
ejpam-6327	390	4	,	,	PUNCT
ejpam-6327	390	5	xa	xa	PROPN
ejpam-6327	390	6	∈	∈	PROPN
ejpam-6327	390	7	i.	i.	NOUN
ejpam-6327	390	8	similarly	similarly	ADV
ejpam-6327	390	9	,	,	PUNCT
ejpam-6327	390	10	ax	ax	NOUN
ejpam-6327	390	11	∈	∈	PROPN
ejpam-6327	390	12	i.	i.	NOUN
ejpam-6327	390	13	hence	hence	ADV
ejpam-6327	390	14	,	,	PUNCT
ejpam-6327	390	15	i	i	PRON
ejpam-6327	390	16	is	be	AUX
ejpam-6327	390	17	stable	stable	ADJ
ejpam-6327	390	18	.	.	PUNCT
ejpam-6327	391	1	now	now	ADV
ejpam-6327	391	2	,	,	PUNCT
ejpam-6327	391	3	let	let	VERB
ejpam-6327	391	4	x	x	PRON
ejpam-6327	391	5	,	,	PUNCT
ejpam-6327	391	6	y	y	PROPN
ejpam-6327	391	7	∈	∈	PROPN
ejpam-6327	391	8	x	x	PUNCT
ejpam-6327	391	9	such	such	ADJ
ejpam-6327	391	10	that	that	SCONJ
ejpam-6327	391	11	x	x	PROPN
ejpam-6327	391	12	∗	∗	NOUN
ejpam-6327	391	13	y	y	NOUN
ejpam-6327	391	14	∈	∈	PROPN
ejpam-6327	392	1	i	i	PRON
ejpam-6327	392	2	and	and	CCONJ
ejpam-6327	392	3	y	y	PROPN
ejpam-6327	392	4	∈	∈	PROPN
ejpam-6327	392	5	i.	i.	NOUN
ejpam-6327	392	6	then	then	ADV
ejpam-6327	392	7	χi(x	χi(x	VERB
ejpam-6327	392	8	∗	∗	PROPN
ejpam-6327	392	9	y	y	NOUN
ejpam-6327	392	10	)	)	PUNCT
ejpam-6327	392	11	=	=	PRON
ejpam-6327	392	12	χi(y	χi(y	X
ejpam-6327	392	13	)	)	PUNCT
ejpam-6327	392	14	=	=	SYM
ejpam-6327	393	1	1	1	X
ejpam-6327	393	2	.	.	PUNCT
ejpam-6327	393	3	since	since	SCONJ
ejpam-6327	393	4	χi	χi	PROPN
ejpam-6327	393	5	is	be	AUX
ejpam-6327	393	6	a	a	DET
ejpam-6327	393	7	fuzzy	fuzzy	ADJ
ejpam-6327	393	8	ks	ks	NOUN
ejpam-6327	393	9	-	-	NOUN
ejpam-6327	393	10	ideal	ideal	NOUN
ejpam-6327	393	11	in	in	ADP
ejpam-6327	393	12	x	x	X
ejpam-6327	393	13	,	,	PUNCT
ejpam-6327	393	14	χi(x	χi(x	NUM
ejpam-6327	393	15	)	)	PUNCT
ejpam-6327	393	16	≥	≥	NOUN
ejpam-6327	393	17	min{χi(x	min{χi(x	NOUN
ejpam-6327	393	18	∗	∗	X
ejpam-6327	393	19	y	y	PROPN
ejpam-6327	393	20	)	)	PUNCT
ejpam-6327	393	21	,	,	PUNCT
ejpam-6327	393	22	χi(y	χi(y	NOUN
ejpam-6327	393	23	)	)	PUNCT
ejpam-6327	393	24	}	}	PUNCT
ejpam-6327	393	25	=	=	SYM
ejpam-6327	393	26	1	1	X
ejpam-6327	393	27	.	.	PUNCT
ejpam-6327	393	28	thus	thus	ADV
ejpam-6327	393	29	,	,	PUNCT
ejpam-6327	393	30	χi(x	χi(x	NUM
ejpam-6327	393	31	)	)	PUNCT
ejpam-6327	393	32	=	=	SYM
ejpam-6327	394	1	1	1	NUM
ejpam-6327	394	2	,	,	PUNCT
ejpam-6327	394	3	that	that	ADV
ejpam-6327	394	4	is	is	ADV
ejpam-6327	394	5	,	,	PUNCT
ejpam-6327	394	6	x	x	SYM
ejpam-6327	394	7	∈	∈	PROPN
ejpam-6327	394	8	i.	i.	NOUN
ejpam-6327	394	9	hence	hence	ADV
ejpam-6327	394	10	,	,	PUNCT
ejpam-6327	394	11	i	i	PRON
ejpam-6327	394	12	is	be	AUX
ejpam-6327	394	13	an	an	DET
ejpam-6327	394	14	ideal	ideal	NOUN
ejpam-6327	394	15	of	of	ADP
ejpam-6327	394	16	x.	x.	NOUN
ejpam-6327	394	17	conversely	conversely	ADV
ejpam-6327	394	18	,	,	PUNCT
ejpam-6327	394	19	suppose	suppose	VERB
ejpam-6327	394	20	i	i	PRON
ejpam-6327	394	21	is	be	AUX
ejpam-6327	394	22	an	an	DET
ejpam-6327	394	23	ideal	ideal	NOUN
ejpam-6327	394	24	of	of	ADP
ejpam-6327	394	25	x.	x.	NOUN
ejpam-6327	394	26	then	then	ADV
ejpam-6327	394	27	by	by	ADP
ejpam-6327	394	28	remark	remark	NOUN
ejpam-6327	394	29	2(ii	2(ii	NUM
ejpam-6327	394	30	)	)	PUNCT
ejpam-6327	394	31	,	,	PUNCT
ejpam-6327	394	32	0	0	NUM
ejpam-6327	394	33	∈	∈	PROPN
ejpam-6327	394	34	i.	i.	NOUN
ejpam-6327	394	35	thus	thus	ADV
ejpam-6327	394	36	,	,	PUNCT
ejpam-6327	394	37	χi(0	χi(0	NOUN
ejpam-6327	394	38	)	)	PUNCT
ejpam-6327	394	39	=	=	SYM
ejpam-6327	394	40	1	1	NUM
ejpam-6327	394	41	≥	≥	NOUN
ejpam-6327	394	42	χi(x	χi(x	NUM
ejpam-6327	394	43	)	)	PUNCT
ejpam-6327	394	44	for	for	ADP
ejpam-6327	394	45	all	all	PRON
ejpam-6327	394	46	x	x	SYM
ejpam-6327	394	47	∈	∈	NOUN
ejpam-6327	394	48	x.	x.	NOUN
ejpam-6327	394	49	now	now	ADV
ejpam-6327	394	50	,	,	PUNCT
ejpam-6327	394	51	let	let	VERB
ejpam-6327	394	52	x	x	PRON
ejpam-6327	394	53	,	,	PUNCT
ejpam-6327	394	54	y	y	PROPN
ejpam-6327	394	55	∈	∈	PROPN
ejpam-6327	394	56	x.	x.	NOUN
ejpam-6327	395	1	we	we	PRON
ejpam-6327	395	2	consider	consider	VERB
ejpam-6327	395	3	the	the	DET
ejpam-6327	395	4	following	follow	VERB
ejpam-6327	395	5	cases	case	NOUN
ejpam-6327	395	6	.	.	PUNCT
ejpam-6327	396	1	case	case	NOUN
ejpam-6327	396	2	1	1	NUM
ejpam-6327	396	3	:	:	PUNCT
ejpam-6327	396	4	x	x	SYM
ejpam-6327	396	5	∗	∗	NOUN
ejpam-6327	396	6	y	y	NOUN
ejpam-6327	396	7	∈	∈	PROPN
ejpam-6327	397	1	i	i	PRON
ejpam-6327	397	2	and	and	CCONJ
ejpam-6327	397	3	y	y	PROPN
ejpam-6327	397	4	∈	∈	PROPN
ejpam-6327	397	5	i.	i.	NOUN
ejpam-6327	397	6	since	since	SCONJ
ejpam-6327	397	7	i	i	PRON
ejpam-6327	397	8	is	be	AUX
ejpam-6327	397	9	an	an	DET
ejpam-6327	397	10	ideal	ideal	NOUN
ejpam-6327	397	11	of	of	ADP
ejpam-6327	397	12	x	x	X
ejpam-6327	397	13	,	,	PUNCT
ejpam-6327	397	14	x	x	PROPN
ejpam-6327	397	15	∈	∈	PROPN
ejpam-6327	397	16	i.	i.	NOUN
ejpam-6327	397	17	thus	thus	ADV
ejpam-6327	397	18	,	,	PUNCT
ejpam-6327	397	19	χi(x	χi(x	NUM
ejpam-6327	397	20	)	)	PUNCT
ejpam-6327	397	21	=	=	SYM
ejpam-6327	397	22	1	1	NUM
ejpam-6327	397	23	≥	≥	NOUN
ejpam-6327	397	24	min{χi(x	min{χi(x	NOUN
ejpam-6327	397	25	∗	∗	X
ejpam-6327	397	26	y	y	PROPN
ejpam-6327	397	27	)	)	PUNCT
ejpam-6327	397	28	,	,	PUNCT
ejpam-6327	397	29	χi(y	χi(y	NOUN
ejpam-6327	397	30	)	)	PUNCT
ejpam-6327	397	31	}	}	PUNCT
ejpam-6327	397	32	.	.	PUNCT
ejpam-6327	398	1	h.	h.	PROPN
ejpam-6327	398	2	sarapuddin	sarapuddin	PROPN
ejpam-6327	398	3	,	,	PUNCT
ejpam-6327	398	4	j.	j.	PROPN
ejpam-6327	398	5	vilela	vilela	PROPN
ejpam-6327	398	6	/	/	SYM
ejpam-6327	398	7	eur	eur	PROPN
ejpam-6327	398	8	.	.	PUNCT
ejpam-6327	399	1	j.	j.	PROPN
ejpam-6327	399	2	pure	pure	PROPN
ejpam-6327	399	3	appl	appl	PROPN
ejpam-6327	399	4	.	.	PROPN
ejpam-6327	399	5	math	math	PROPN
ejpam-6327	399	6	,	,	PUNCT
ejpam-6327	399	7	18	18	NUM
ejpam-6327	399	8	(	(	PUNCT
ejpam-6327	399	9	3	3	NUM
ejpam-6327	399	10	)	)	PUNCT
ejpam-6327	399	11	(	(	PUNCT
ejpam-6327	399	12	2025	2025	NUM
ejpam-6327	399	13	)	)	PUNCT
ejpam-6327	399	14	,	,	PUNCT
ejpam-6327	399	15	6327	6327	NUM
ejpam-6327	399	16	13	13	NUM
ejpam-6327	399	17	of	of	ADP
ejpam-6327	399	18	23	23	NUM
ejpam-6327	399	19	case	case	NOUN
ejpam-6327	399	20	2	2	NUM
ejpam-6327	399	21	:	:	PUNCT
ejpam-6327	399	22	x	x	SYM
ejpam-6327	399	23	∗	∗	NOUN
ejpam-6327	399	24	y	y	NOUN
ejpam-6327	399	25	∈	∈	PROPN
ejpam-6327	400	1	i	i	PRON
ejpam-6327	400	2	and	and	CCONJ
ejpam-6327	400	3	y	y	PROPN
ejpam-6327	400	4	/∈	/∈	PUNCT
ejpam-6327	400	5	i.	i.	PROPN
ejpam-6327	400	6	then	then	ADV
ejpam-6327	400	7	χi(y	χi(y	VERB
ejpam-6327	400	8	)	)	PUNCT
ejpam-6327	400	9	=	=	SYM
ejpam-6327	400	10	0	0	X
ejpam-6327	400	11	.	.	PUNCT
ejpam-6327	400	12	thus	thus	ADV
ejpam-6327	400	13	,	,	PUNCT
ejpam-6327	400	14	χi(x	χi(x	NUM
ejpam-6327	400	15	)	)	PUNCT
ejpam-6327	400	16	≥	≥	NOUN
ejpam-6327	400	17	0	0	NUM
ejpam-6327	400	18	=	=	SYM
ejpam-6327	400	19	min{χi(x	min{χi(x	NOUN
ejpam-6327	400	20	∗	∗	NOUN
ejpam-6327	400	21	y	y	PROPN
ejpam-6327	400	22	)	)	PUNCT
ejpam-6327	400	23	,	,	PUNCT
ejpam-6327	400	24	χi(y	χi(y	NOUN
ejpam-6327	400	25	)	)	PUNCT
ejpam-6327	400	26	}	}	PUNCT
ejpam-6327	400	27	.	.	PUNCT
ejpam-6327	401	1	case	case	NOUN
ejpam-6327	401	2	3	3	NUM
ejpam-6327	401	3	:	:	PUNCT
ejpam-6327	401	4	x	x	SYM
ejpam-6327	401	5	∗	∗	PUNCT
ejpam-6327	401	6	y	y	NOUN
ejpam-6327	401	7	/∈	/∈	PUNCT
ejpam-6327	402	1	i	i	PRON
ejpam-6327	402	2	and	and	CCONJ
ejpam-6327	402	3	y	y	PROPN
ejpam-6327	402	4	∈	∈	PROPN
ejpam-6327	402	5	i.	i.	NOUN
ejpam-6327	402	6	then	then	ADV
ejpam-6327	402	7	χi(x	χi(x	VERB
ejpam-6327	402	8	∗	∗	PROPN
ejpam-6327	402	9	y	y	NOUN
ejpam-6327	402	10	)	)	PUNCT
ejpam-6327	402	11	=	=	SYM
ejpam-6327	403	1	0	0	X
ejpam-6327	403	2	.	.	PUNCT
ejpam-6327	403	3	thus	thus	ADV
ejpam-6327	403	4	,	,	PUNCT
ejpam-6327	403	5	χi(x	χi(x	NUM
ejpam-6327	403	6	)	)	PUNCT
ejpam-6327	403	7	≥	≥	NOUN
ejpam-6327	403	8	0	0	NUM
ejpam-6327	403	9	=	=	SYM
ejpam-6327	403	10	min{χi(x	min{χi(x	NOUN
ejpam-6327	403	11	∗	∗	NOUN
ejpam-6327	403	12	y	y	PROPN
ejpam-6327	403	13	)	)	PUNCT
ejpam-6327	403	14	,	,	PUNCT
ejpam-6327	403	15	χi(y	χi(y	NOUN
ejpam-6327	403	16	)	)	PUNCT
ejpam-6327	403	17	}	}	PUNCT
ejpam-6327	403	18	.	.	PUNCT
ejpam-6327	404	1	case	case	NOUN
ejpam-6327	404	2	4	4	NUM
ejpam-6327	404	3	:	:	PUNCT
ejpam-6327	404	4	x	x	SYM
ejpam-6327	404	5	∗	∗	PUNCT
ejpam-6327	404	6	y	y	NOUN
ejpam-6327	404	7	/∈	/∈	PUNCT
ejpam-6327	405	1	i	i	PRON
ejpam-6327	405	2	and	and	CCONJ
ejpam-6327	405	3	y	y	PROPN
ejpam-6327	405	4	/∈	/∈	PUNCT
ejpam-6327	405	5	i.	i.	PROPN
ejpam-6327	405	6	then	then	ADV
ejpam-6327	405	7	χi(x	χi(x	VERB
ejpam-6327	405	8	∗	∗	PROPN
ejpam-6327	405	9	y	y	NOUN
ejpam-6327	405	10	)	)	PUNCT
ejpam-6327	405	11	=	=	PRON
ejpam-6327	405	12	χi(y	χi(y	X
ejpam-6327	405	13	)	)	PUNCT
ejpam-6327	405	14	=	=	SYM
ejpam-6327	406	1	0	0	X
ejpam-6327	406	2	.	.	PUNCT
ejpam-6327	406	3	thus	thus	ADV
ejpam-6327	406	4	,	,	PUNCT
ejpam-6327	406	5	χi(x	χi(x	NUM
ejpam-6327	406	6	)	)	PUNCT
ejpam-6327	406	7	≥	≥	NOUN
ejpam-6327	406	8	0	0	NUM
ejpam-6327	406	9	=	=	SYM
ejpam-6327	406	10	min{χi(x	min{χi(x	NOUN
ejpam-6327	406	11	∗	∗	NOUN
ejpam-6327	406	12	y	y	PROPN
ejpam-6327	406	13	)	)	PUNCT
ejpam-6327	406	14	,	,	PUNCT
ejpam-6327	406	15	χi(y	χi(y	NOUN
ejpam-6327	406	16	)	)	PUNCT
ejpam-6327	406	17	}	}	PUNCT
ejpam-6327	406	18	.	.	PUNCT
ejpam-6327	407	1	in	in	ADP
ejpam-6327	407	2	either	either	DET
ejpam-6327	407	3	case	case	NOUN
ejpam-6327	407	4	,	,	PUNCT
ejpam-6327	407	5	χi(x	χi(x	NUM
ejpam-6327	407	6	)	)	PUNCT
ejpam-6327	407	7	≥	≥	X
ejpam-6327	407	8	min{χi(x∗y	min{χi(x∗y	X
ejpam-6327	407	9	)	)	PUNCT
ejpam-6327	407	10	,	,	PUNCT
ejpam-6327	407	11	χi(y	χi(y	NOUN
ejpam-6327	407	12	)	)	PUNCT
ejpam-6327	407	13	}	}	PUNCT
ejpam-6327	407	14	.	.	PUNCT
ejpam-6327	408	1	now	now	ADV
ejpam-6327	408	2	,	,	PUNCT
ejpam-6327	408	3	let	let	VERB
ejpam-6327	408	4	x	x	PRON
ejpam-6327	408	5	,	,	PUNCT
ejpam-6327	408	6	a	a	DET
ejpam-6327	408	7	∈	∈	NOUN
ejpam-6327	408	8	x.	x.	NOUN
ejpam-6327	408	9	we	we	PRON
ejpam-6327	408	10	consider	consider	VERB
ejpam-6327	408	11	the	the	DET
ejpam-6327	408	12	following	follow	VERB
ejpam-6327	408	13	cases	case	NOUN
ejpam-6327	408	14	.	.	PUNCT
ejpam-6327	409	1	case	case	NOUN
ejpam-6327	409	2	1	1	NUM
ejpam-6327	409	3	:	:	SYM
ejpam-6327	409	4	x	x	X
ejpam-6327	409	5	,	,	PUNCT
ejpam-6327	409	6	a	a	DET
ejpam-6327	409	7	∈	∈	PROPN
ejpam-6327	409	8	i.	i.	NOUN
ejpam-6327	409	9	since	since	SCONJ
ejpam-6327	409	10	i	i	PRON
ejpam-6327	409	11	is	be	AUX
ejpam-6327	409	12	an	an	DET
ejpam-6327	409	13	ideal	ideal	NOUN
ejpam-6327	409	14	of	of	ADP
ejpam-6327	409	15	x	x	PROPN
ejpam-6327	409	16	,	,	PUNCT
ejpam-6327	409	17	xa	xa	PROPN
ejpam-6327	409	18	,	,	PUNCT
ejpam-6327	409	19	ax	ax	NOUN
ejpam-6327	409	20	∈	∈	PROPN
ejpam-6327	409	21	i.	i.	NOUN
ejpam-6327	409	22	thus	thus	ADV
ejpam-6327	409	23	,	,	PUNCT
ejpam-6327	409	24	χi(xa	χi(xa	PROPN
ejpam-6327	409	25	)	)	PUNCT
ejpam-6327	409	26	=	=	SYM
ejpam-6327	409	27	1	1	NUM
ejpam-6327	409	28	≥	≥	NOUN
ejpam-6327	409	29	min{χi(x	min{χi(x	PROPN
ejpam-6327	409	30	)	)	PUNCT
ejpam-6327	409	31	,	,	PUNCT
ejpam-6327	409	32	χi(a	χi(a	X
ejpam-6327	409	33	)	)	PUNCT
ejpam-6327	409	34	}	}	PUNCT
ejpam-6327	409	35	and	and	CCONJ
ejpam-6327	409	36	χi(ax	χi(ax	PROPN
ejpam-6327	409	37	)	)	PUNCT
ejpam-6327	409	38	=	=	SYM
ejpam-6327	409	39	1	1	NUM
ejpam-6327	409	40	≥	≥	NOUN
ejpam-6327	409	41	min{χi(x	min{χi(x	PROPN
ejpam-6327	409	42	)	)	PUNCT
ejpam-6327	409	43	,	,	PUNCT
ejpam-6327	409	44	χi(a	χi(a	X
ejpam-6327	409	45	)	)	PUNCT
ejpam-6327	409	46	}	}	PUNCT
ejpam-6327	409	47	.	.	PUNCT
ejpam-6327	410	1	case	case	NOUN
ejpam-6327	410	2	2	2	NUM
ejpam-6327	410	3	:	:	PUNCT
ejpam-6327	410	4	x	x	X
ejpam-6327	410	5	/∈	/∈	INTJ
ejpam-6327	411	1	i	i	PRON
ejpam-6327	411	2	or	or	CCONJ
ejpam-6327	411	3	a	a	DET
ejpam-6327	411	4	/∈	/∈	NOUN
ejpam-6327	411	5	i.	i.	NOUN
ejpam-6327	412	1	if	if	SCONJ
ejpam-6327	412	2	xa	xa	PROPN
ejpam-6327	412	3	∈	∈	PROPN
ejpam-6327	412	4	i	i	PRON
ejpam-6327	412	5	,	,	PUNCT
ejpam-6327	412	6	then	then	ADV
ejpam-6327	412	7	χi(xa	χi(xa	PROPN
ejpam-6327	412	8	)	)	PUNCT
ejpam-6327	412	9	=	=	SYM
ejpam-6327	412	10	1	1	X
ejpam-6327	412	11	.	.	PUNCT
ejpam-6327	412	12	thus	thus	ADV
ejpam-6327	412	13	,	,	PUNCT
ejpam-6327	412	14	χi(xa	χi(xa	PROPN
ejpam-6327	412	15	)	)	PUNCT
ejpam-6327	412	16	=	=	SYM
ejpam-6327	412	17	1	1	NUM
ejpam-6327	412	18	≥	≥	NOUN
ejpam-6327	412	19	min{χi(x	min{χi(x	PROPN
ejpam-6327	412	20	)	)	PUNCT
ejpam-6327	412	21	,	,	PUNCT
ejpam-6327	412	22	χi(a	χi(a	X
ejpam-6327	412	23	)	)	PUNCT
ejpam-6327	412	24	}	}	PUNCT
ejpam-6327	412	25	.	.	PUNCT
ejpam-6327	413	1	if	if	SCONJ
ejpam-6327	413	2	xa	xa	PROPN
ejpam-6327	413	3	/∈	/∈	VERB
ejpam-6327	414	1	i	i	PRON
ejpam-6327	414	2	,	,	PUNCT
ejpam-6327	414	3	then	then	ADV
ejpam-6327	414	4	x	x	PRON
ejpam-6327	414	5	,	,	PUNCT
ejpam-6327	414	6	a	a	DET
ejpam-6327	414	7	/∈	/∈	NOUN
ejpam-6327	414	8	i.	i.	NOUN
ejpam-6327	414	9	thus	thus	ADV
ejpam-6327	414	10	,	,	PUNCT
ejpam-6327	414	11	χi(xa	χi(xa	PROPN
ejpam-6327	414	12	)	)	PUNCT
ejpam-6327	414	13	=	=	SYM
ejpam-6327	414	14	0	0	NUM
ejpam-6327	414	15	and	and	CCONJ
ejpam-6327	414	16	χi(x	χi(x	NUM
ejpam-6327	414	17	)	)	PUNCT
ejpam-6327	415	1	=	=	SYM
ejpam-6327	415	2	0	0	NUM
ejpam-6327	415	3	=	=	SYM
ejpam-6327	415	4	χi(a	χi(a	X
ejpam-6327	415	5	)	)	PUNCT
ejpam-6327	415	6	.	.	PUNCT
ejpam-6327	416	1	hence	hence	ADV
ejpam-6327	416	2	,	,	PUNCT
ejpam-6327	416	3	χi(xa	χi(xa	PROPN
ejpam-6327	416	4	)	)	PUNCT
ejpam-6327	416	5	=	=	SYM
ejpam-6327	416	6	0	0	NUM
ejpam-6327	416	7	≥	≥	NOUN
ejpam-6327	416	8	0	0	NUM
ejpam-6327	416	9	=	=	SYM
ejpam-6327	416	10	min{χi(x	min{χi(x	NOUN
ejpam-6327	416	11	)	)	PUNCT
ejpam-6327	416	12	,	,	PUNCT
ejpam-6327	416	13	χi(a	χi(a	X
ejpam-6327	416	14	)	)	PUNCT
ejpam-6327	416	15	}	}	PUNCT
ejpam-6327	416	16	.	.	PUNCT
ejpam-6327	417	1	similarly	similarly	ADV
ejpam-6327	417	2	,	,	PUNCT
ejpam-6327	417	3	χi(ax	χi(ax	PROPN
ejpam-6327	417	4	)	)	PUNCT
ejpam-6327	417	5	≥	≥	NOUN
ejpam-6327	417	6	min{χi(x	min{χi(x	PROPN
ejpam-6327	417	7	)	)	PUNCT
ejpam-6327	417	8	,	,	PUNCT
ejpam-6327	417	9	χi(a	χi(a	X
ejpam-6327	417	10	)	)	PUNCT
ejpam-6327	417	11	}	}	PUNCT
ejpam-6327	417	12	.	.	PUNCT
ejpam-6327	418	1	thus	thus	ADV
ejpam-6327	418	2	,	,	PUNCT
ejpam-6327	418	3	χi	χi	PROPN
ejpam-6327	418	4	is	be	AUX
ejpam-6327	418	5	a	a	DET
ejpam-6327	418	6	fuzzy	fuzzy	ADJ
ejpam-6327	418	7	ks	ks	NOUN
ejpam-6327	418	8	-	-	NOUN
ejpam-6327	418	9	ideal	ideal	NOUN
ejpam-6327	418	10	of	of	ADP
ejpam-6327	418	11	x	x	X
ejpam-6327	418	12	.	.	PUNCT
ejpam-6327	419	1	moreover	moreover	ADV
ejpam-6327	419	2	,	,	PUNCT
ejpam-6327	419	3	by	by	ADP
ejpam-6327	419	4	observing	observe	VERB
ejpam-6327	419	5	case	case	NOUN
ejpam-6327	419	6	1	1	NUM
ejpam-6327	419	7	and	and	CCONJ
ejpam-6327	419	8	case	case	NOUN
ejpam-6327	419	9	2	2	NUM
ejpam-6327	419	10	above	above	ADV
ejpam-6327	419	11	,	,	PUNCT
ejpam-6327	419	12	χi	χi	NOUN
ejpam-6327	419	13	satisfies	satisfy	VERB
ejpam-6327	419	14	condition	condition	NOUN
ejpam-6327	419	15	(	(	PUNCT
ejpam-6327	419	16	c	c	NOUN
ejpam-6327	419	17	)	)	PUNCT
ejpam-6327	419	18	.	.	PUNCT
ejpam-6327	420	1	the	the	DET
ejpam-6327	420	2	following	follow	VERB
ejpam-6327	420	3	proposition	proposition	NOUN
ejpam-6327	420	4	tells	tell	VERB
ejpam-6327	420	5	us	we	PRON
ejpam-6327	420	6	that	that	SCONJ
ejpam-6327	420	7	the	the	DET
ejpam-6327	420	8	equivalence	equivalence	NOUN
ejpam-6327	420	9	class	class	NOUN
ejpam-6327	420	10	containing	contain	VERB
ejpam-6327	420	11	any	any	DET
ejpam-6327	420	12	element	element	NOUN
ejpam-6327	420	13	of	of	ADP
ejpam-6327	420	14	a	a	DET
ejpam-6327	420	15	ks	ks	NOUN
ejpam-6327	420	16	-	-	PUNCT
ejpam-6327	420	17	semigroup	semigroup	NOUN
ejpam-6327	420	18	x	x	NOUN
ejpam-6327	420	19	are	be	AUX
ejpam-6327	420	20	equal	equal	ADJ
ejpam-6327	420	21	with	with	ADP
ejpam-6327	420	22	respect	respect	NOUN
ejpam-6327	420	23	to	to	ADP
ejpam-6327	420	24	∼i	∼i	PROPN
ejpam-6327	420	25	and	and	CCONJ
ejpam-6327	420	26	∼χi	∼χi	PROPN
ejpam-6327	420	27	for	for	ADP
ejpam-6327	420	28	any	any	DET
ejpam-6327	420	29	ideal	ideal	NOUN
ejpam-6327	420	30	i	i	PRON
ejpam-6327	420	31	of	of	ADP
ejpam-6327	420	32	x.	x.	NOUN
ejpam-6327	420	33	proposition	proposition	NOUN
ejpam-6327	420	34	5	5	NUM
ejpam-6327	420	35	.	.	PUNCT
ejpam-6327	421	1	let	let	VERB
ejpam-6327	421	2	i	i	PRON
ejpam-6327	421	3	be	be	AUX
ejpam-6327	421	4	an	an	DET
ejpam-6327	421	5	ideal	ideal	NOUN
ejpam-6327	421	6	of	of	ADP
ejpam-6327	421	7	a	a	DET
ejpam-6327	421	8	ks	ks	NOUN
ejpam-6327	421	9	-	-	PUNCT
ejpam-6327	421	10	semigroup	semigroup	NOUN
ejpam-6327	421	11	x.	x.	NOUN
ejpam-6327	422	1	then	then	ADV
ejpam-6327	422	2	x	x	X
ejpam-6327	422	3	∼i	∼i	PROPN
ejpam-6327	422	4	y	y	PROPN
ejpam-6327	422	5	if	if	SCONJ
ejpam-6327	422	6	and	and	CCONJ
ejpam-6327	422	7	only	only	ADV
ejpam-6327	422	8	if	if	SCONJ
ejpam-6327	422	9	x	x	PRON
ejpam-6327	422	10	∼χi	∼χi	PROPN
ejpam-6327	422	11	y.	y.	NOUN
ejpam-6327	422	12	proof	proof	NOUN
ejpam-6327	422	13	.	.	PUNCT
ejpam-6327	423	1	let	let	VERB
ejpam-6327	423	2	i	i	PRON
ejpam-6327	423	3	be	be	AUX
ejpam-6327	423	4	an	an	DET
ejpam-6327	423	5	ideal	ideal	NOUN
ejpam-6327	423	6	of	of	ADP
ejpam-6327	423	7	a	a	DET
ejpam-6327	423	8	ks	ks	NOUN
ejpam-6327	423	9	-	-	PUNCT
ejpam-6327	423	10	semigroup	semigroup	NOUN
ejpam-6327	423	11	x.	x.	NOUN
ejpam-6327	423	12	then	then	ADV
ejpam-6327	423	13	for	for	ADP
ejpam-6327	423	14	all	all	DET
ejpam-6327	423	15	x	x	NOUN
ejpam-6327	423	16	,	,	PUNCT
ejpam-6327	423	17	y	y	PROPN
ejpam-6327	423	18	∈	∈	PROPN
ejpam-6327	423	19	x	x	X
ejpam-6327	423	20	,	,	PUNCT
ejpam-6327	423	21	x	x	PROPN
ejpam-6327	423	22	∼i	∼i	PROPN
ejpam-6327	423	23	y	y	PROPN
ejpam-6327	423	24	⇔	⇔	PROPN
ejpam-6327	423	25	x	x	PROPN
ejpam-6327	423	26	∗	∗	VERB
ejpam-6327	423	27	y	y	NOUN
ejpam-6327	423	28	∈	∈	PROPN
ejpam-6327	424	1	i	i	PRON
ejpam-6327	424	2	and	and	CCONJ
ejpam-6327	424	3	y	y	PROPN
ejpam-6327	424	4	∗	∗	NOUN
ejpam-6327	424	5	x	x	X
ejpam-6327	424	6	∈	∈	PROPN
ejpam-6327	424	7	i	i	PRON
ejpam-6327	424	8	⇔	⇔	X
ejpam-6327	424	9	χi(x	χi(x	X
ejpam-6327	424	10	∗	∗	PROPN
ejpam-6327	424	11	y	y	NOUN
ejpam-6327	424	12	)	)	PUNCT
ejpam-6327	424	13	=	=	SYM
ejpam-6327	424	14	1	1	NUM
ejpam-6327	424	15	and	and	CCONJ
ejpam-6327	424	16	χi(y	χi(y	VERB
ejpam-6327	424	17	∗	∗	NOUN
ejpam-6327	424	18	x	x	NOUN
ejpam-6327	424	19	)	)	PUNCT
ejpam-6327	424	20	=	=	SYM
ejpam-6327	424	21	1	1	NUM
ejpam-6327	424	22	⇔	⇔	X
ejpam-6327	424	23	χi(x	χi(x	X
ejpam-6327	424	24	∗	∗	PROPN
ejpam-6327	424	25	y	y	PROPN
ejpam-6327	424	26	)	)	PUNCT
ejpam-6327	424	27	>	>	X
ejpam-6327	424	28	0	0	PUNCT
ejpam-6327	424	29	and	and	CCONJ
ejpam-6327	424	30	χi(y	χi(y	VERB
ejpam-6327	424	31	∗	∗	NOUN
ejpam-6327	424	32	x	x	X
ejpam-6327	424	33	)	)	PUNCT
ejpam-6327	424	34	>	>	X
ejpam-6327	424	35	0	0	NUM
ejpam-6327	425	1	⇔	⇔	X
ejpam-6327	425	2	x	x	X
ejpam-6327	425	3	∼χi	∼χi	PROPN
ejpam-6327	425	4	y.	y.	NOUN
ejpam-6327	425	5	let	let	VERB
ejpam-6327	425	6	i	i	PRON
ejpam-6327	425	7	be	be	AUX
ejpam-6327	425	8	an	an	DET
ejpam-6327	425	9	ideal	ideal	NOUN
ejpam-6327	425	10	of	of	ADP
ejpam-6327	425	11	a	a	DET
ejpam-6327	425	12	ks	ks	NOUN
ejpam-6327	425	13	-	-	PUNCT
ejpam-6327	425	14	semigroup	semigroup	NOUN
ejpam-6327	425	15	x.	x.	NOUN
ejpam-6327	425	16	then	then	ADV
ejpam-6327	425	17	for	for	ADP
ejpam-6327	425	18	all	all	DET
ejpam-6327	425	19	x	x	SYM
ejpam-6327	425	20	∈	∈	PROPN
ejpam-6327	425	21	x	x	NOUN
ejpam-6327	425	22	,	,	PUNCT
ejpam-6327	425	23	ix	ix	ADV
ejpam-6327	425	24	=	=	PUNCT
ejpam-6327	425	25	(	(	PUNCT
ejpam-6327	425	26	χi)x	χi)x	PROPN
ejpam-6327	425	27	.	.	PUNCT
ejpam-6327	426	1	hence	hence	ADV
ejpam-6327	426	2	,	,	PUNCT
ejpam-6327	426	3	x	x	X
ejpam-6327	426	4	/	/	SYM
ejpam-6327	426	5	i	i	NOUN
ejpam-6327	426	6	=	=	SYM
ejpam-6327	426	7	x	x	X
ejpam-6327	426	8	/	/	SYM
ejpam-6327	426	9	χi	χi	PROPN
ejpam-6327	426	10	.	.	PUNCT
ejpam-6327	427	1	therefore	therefore	ADV
ejpam-6327	427	2	,	,	PUNCT
ejpam-6327	427	3	if	if	SCONJ
ejpam-6327	427	4	µ	µ	NOUN
ejpam-6327	427	5	is	be	AUX
ejpam-6327	427	6	a	a	DET
ejpam-6327	427	7	non	non	ADJ
ejpam-6327	427	8	-	-	ADJ
ejpam-6327	427	9	zero	zero	ADJ
ejpam-6327	427	10	fuzzy	fuzzy	ADJ
ejpam-6327	427	11	ks	ks	NOUN
ejpam-6327	427	12	-	-	NOUN
ejpam-6327	427	13	ideal	ideal	NOUN
ejpam-6327	427	14	of	of	ADP
ejpam-6327	427	15	x	x	SYM
ejpam-6327	427	16	satisfying	satisfy	VERB
ejpam-6327	427	17	condition	condition	NOUN
ejpam-6327	427	18	(	(	PUNCT
ejpam-6327	427	19	c	c	NOUN
ejpam-6327	427	20	)	)	PUNCT
ejpam-6327	427	21	,	,	PUNCT
ejpam-6327	427	22	the	the	DET
ejpam-6327	427	23	quotient	quotient	NOUN
ejpam-6327	427	24	ks	ks	NOUN
ejpam-6327	427	25	-	-	PUNCT
ejpam-6327	427	26	semigroup	semigroup	PROPN
ejpam-6327	427	27	x/µ	x/µ	PROPN
ejpam-6327	427	28	is	be	AUX
ejpam-6327	427	29	a	a	DET
ejpam-6327	427	30	generalization	generalization	NOUN
ejpam-6327	427	31	of	of	ADP
ejpam-6327	427	32	the	the	DET
ejpam-6327	427	33	quotient	quotient	NOUN
ejpam-6327	427	34	ks	ks	NOUN
ejpam-6327	427	35	-	-	PUNCT
ejpam-6327	427	36	semigroup	semigroup	ADJ
ejpam-6327	427	37	x	x	NOUN
ejpam-6327	427	38	/	/	SYM
ejpam-6327	427	39	i.	i.	NOUN
ejpam-6327	427	40	theorem	theorem	VERB
ejpam-6327	427	41	13	13	NUM
ejpam-6327	427	42	.	.	PUNCT
ejpam-6327	428	1	let	let	VERB
ejpam-6327	428	2	x	x	PRON
ejpam-6327	428	3	be	be	AUX
ejpam-6327	428	4	a	a	DET
ejpam-6327	428	5	ks	ks	NOUN
ejpam-6327	428	6	-	-	PUNCT
ejpam-6327	428	7	semigroup	semigroup	NOUN
ejpam-6327	428	8	and	and	CCONJ
ejpam-6327	428	9	µ	µ	DET
ejpam-6327	428	10	a	a	DET
ejpam-6327	428	11	non	non	ADJ
ejpam-6327	428	12	-	-	ADJ
ejpam-6327	428	13	zero	zero	ADJ
ejpam-6327	428	14	fuzzy	fuzzy	ADJ
ejpam-6327	428	15	ks	ks	NOUN
ejpam-6327	428	16	-	-	NOUN
ejpam-6327	428	17	ideal	ideal	NOUN
ejpam-6327	428	18	of	of	ADP
ejpam-6327	428	19	x	x	SYM
ejpam-6327	428	20	satisfying	satisfy	VERB
ejpam-6327	428	21	condition	condition	NOUN
ejpam-6327	428	22	(	(	PUNCT
ejpam-6327	428	23	c	c	NOUN
ejpam-6327	428	24	)	)	PUNCT
ejpam-6327	428	25	.	.	PUNCT
ejpam-6327	429	1	if	if	SCONJ
ejpam-6327	429	2	j	j	PROPN
ejpam-6327	429	3	is	be	AUX
ejpam-6327	429	4	an	an	DET
ejpam-6327	429	5	ideal	ideal	NOUN
ejpam-6327	429	6	of	of	ADP
ejpam-6327	429	7	x	x	NOUN
ejpam-6327	429	8	,	,	PUNCT
ejpam-6327	429	9	then	then	ADV
ejpam-6327	429	10	j/µ	j/µ	NOUN
ejpam-6327	429	11	=	=	PUNCT
ejpam-6327	429	12	{	{	PUNCT
ejpam-6327	429	13	µj	µj	X
ejpam-6327	429	14	:	:	PUNCT
ejpam-6327	429	15	j	j	PROPN
ejpam-6327	429	16	∈	∈	PROPN
ejpam-6327	429	17	j	j	PROPN
ejpam-6327	429	18	}	}	PUNCT
ejpam-6327	429	19	is	be	AUX
ejpam-6327	429	20	an	an	DET
ejpam-6327	429	21	ideal	ideal	NOUN
ejpam-6327	429	22	of	of	ADP
ejpam-6327	429	23	x/µ.	x/µ.	PROPN
ejpam-6327	429	24	h.	h.	PROPN
ejpam-6327	429	25	sarapuddin	sarapuddin	PROPN
ejpam-6327	429	26	,	,	PUNCT
ejpam-6327	429	27	j.	j.	PROPN
ejpam-6327	429	28	vilela	vilela	PROPN
ejpam-6327	429	29	/	/	SYM
ejpam-6327	429	30	eur	eur	PROPN
ejpam-6327	429	31	.	.	PUNCT
ejpam-6327	430	1	j.	j.	PROPN
ejpam-6327	430	2	pure	pure	PROPN
ejpam-6327	430	3	appl	appl	PROPN
ejpam-6327	430	4	.	.	PROPN
ejpam-6327	430	5	math	math	PROPN
ejpam-6327	430	6	,	,	PUNCT
ejpam-6327	430	7	18	18	NUM
ejpam-6327	430	8	(	(	PUNCT
ejpam-6327	430	9	3	3	NUM
ejpam-6327	430	10	)	)	PUNCT
ejpam-6327	430	11	(	(	PUNCT
ejpam-6327	430	12	2025	2025	NUM
ejpam-6327	430	13	)	)	PUNCT
ejpam-6327	430	14	,	,	PUNCT
ejpam-6327	430	15	6327	6327	NUM
ejpam-6327	430	16	14	14	NUM
ejpam-6327	430	17	of	of	ADP
ejpam-6327	430	18	23	23	NUM
ejpam-6327	430	19	proof	proof	NOUN
ejpam-6327	430	20	.	.	PUNCT
ejpam-6327	431	1	let	let	VERB
ejpam-6327	431	2	µ	µ	X
ejpam-6327	431	3	be	be	AUX
ejpam-6327	431	4	a	a	DET
ejpam-6327	431	5	non	non	ADJ
ejpam-6327	431	6	-	-	ADJ
ejpam-6327	431	7	zero	zero	ADJ
ejpam-6327	431	8	fuzzy	fuzzy	ADJ
ejpam-6327	431	9	ks	ks	NOUN
ejpam-6327	431	10	-	-	NOUN
ejpam-6327	431	11	ideal	ideal	NOUN
ejpam-6327	431	12	in	in	ADP
ejpam-6327	431	13	a	a	DET
ejpam-6327	431	14	ks	ks	NOUN
ejpam-6327	431	15	-	-	PUNCT
ejpam-6327	431	16	semigroup	semigroup	NOUN
ejpam-6327	431	17	x	x	PUNCT
ejpam-6327	431	18	satisfying	satisfy	VERB
ejpam-6327	431	19	condition	condition	NOUN
ejpam-6327	431	20	(	(	PUNCT
ejpam-6327	431	21	c	c	NOUN
ejpam-6327	431	22	)	)	PUNCT
ejpam-6327	431	23	and	and	CCONJ
ejpam-6327	431	24	j	j	PROPN
ejpam-6327	431	25	an	an	DET
ejpam-6327	431	26	ideal	ideal	NOUN
ejpam-6327	431	27	of	of	ADP
ejpam-6327	431	28	x.	x.	NOUN
ejpam-6327	431	29	since	since	SCONJ
ejpam-6327	431	30	j	j	PROPN
ejpam-6327	431	31	⊆	⊆	NUM
ejpam-6327	431	32	x	x	NUM
ejpam-6327	431	33	,	,	PUNCT
ejpam-6327	431	34	it	it	PRON
ejpam-6327	431	35	follows	follow	VERB
ejpam-6327	431	36	that	that	SCONJ
ejpam-6327	431	37	j/µ	j/µ	NOUN
ejpam-6327	431	38	⊆	⊆	NUM
ejpam-6327	431	39	x/µ.	x/µ.	NOUN
ejpam-6327	432	1	now	now	ADV
ejpam-6327	432	2	,	,	PUNCT
ejpam-6327	432	3	let	let	VERB
ejpam-6327	432	4	µa	µa	PRON
ejpam-6327	432	5	∈	∈	VERB
ejpam-6327	432	6	j/µ	j/µ	NOUN
ejpam-6327	432	7	and	and	CCONJ
ejpam-6327	432	8	µx	µx	INTJ
ejpam-6327	432	9	∈	∈	PROPN
ejpam-6327	432	10	x/µ.	x/µ.	PROPN
ejpam-6327	433	1	then	then	ADV
ejpam-6327	433	2	a	a	DET
ejpam-6327	433	3	∈	∈	PROPN
ejpam-6327	433	4	j	j	NOUN
ejpam-6327	433	5	and	and	CCONJ
ejpam-6327	433	6	x	x	SYM
ejpam-6327	433	7	∈	∈	PROPN
ejpam-6327	433	8	x.	x.	NOUN
ejpam-6327	433	9	since	since	SCONJ
ejpam-6327	433	10	j	j	PROPN
ejpam-6327	433	11	is	be	AUX
ejpam-6327	433	12	an	an	DET
ejpam-6327	433	13	ideal	ideal	NOUN
ejpam-6327	433	14	of	of	ADP
ejpam-6327	433	15	x	x	PRON
ejpam-6327	433	16	,	,	PUNCT
ejpam-6327	433	17	ax	ax	NOUN
ejpam-6327	433	18	∈	∈	PROPN
ejpam-6327	433	19	j	j	PROPN
ejpam-6327	433	20	.	.	PUNCT
ejpam-6327	434	1	thus	thus	ADV
ejpam-6327	434	2	,	,	PUNCT
ejpam-6327	434	3	µaµx	µaµx	NOUN
ejpam-6327	434	4	=	=	PUNCT
ejpam-6327	434	5	µax	µax	PRON
ejpam-6327	434	6	∈	∈	PROPN
ejpam-6327	434	7	j/µ.	j/µ.	AUX
ejpam-6327	434	8	similarly	similarly	ADV
ejpam-6327	434	9	,	,	PUNCT
ejpam-6327	434	10	µxµa	µxµa	PROPN
ejpam-6327	434	11	∈	∈	PROPN
ejpam-6327	434	12	j/µ.	j/µ.	X
ejpam-6327	434	13	for	for	ADP
ejpam-6327	434	14	any	any	DET
ejpam-6327	434	15	µx	µx	ADJ
ejpam-6327	434	16	,	,	PUNCT
ejpam-6327	434	17	µy	µy	ADP
ejpam-6327	434	18	∈	∈	PROPN
ejpam-6327	434	19	x/µ	x/µ	PROPN
ejpam-6327	434	20	,	,	PUNCT
ejpam-6327	434	21	suppose	suppose	VERB
ejpam-6327	434	22	µx∗y	µx∗y	X
ejpam-6327	434	23	=	=	PRON
ejpam-6327	434	24	µx	µx	ADP
ejpam-6327	434	25	∗	∗	NOUN
ejpam-6327	434	26	µy	µy	ADP
ejpam-6327	434	27	∈	∈	PROPN
ejpam-6327	434	28	j/µ	j/µ	NOUN
ejpam-6327	434	29	and	and	CCONJ
ejpam-6327	434	30	µy	µy	ADP
ejpam-6327	434	31	∈	∈	PROPN
ejpam-6327	435	1	j/µ.	j/µ.	CCONJ
ejpam-6327	435	2	then	then	ADV
ejpam-6327	435	3	x	x	X
ejpam-6327	435	4	∗	∗	VERB
ejpam-6327	435	5	y	y	PROPN
ejpam-6327	435	6	∈	∈	PROPN
ejpam-6327	435	7	j	j	PROPN
ejpam-6327	435	8	and	and	CCONJ
ejpam-6327	435	9	y	y	PROPN
ejpam-6327	435	10	∈	∈	PROPN
ejpam-6327	435	11	j	j	PROPN
ejpam-6327	435	12	.	.	PUNCT
ejpam-6327	436	1	since	since	SCONJ
ejpam-6327	436	2	j	j	PROPN
ejpam-6327	436	3	is	be	AUX
ejpam-6327	436	4	an	an	DET
ejpam-6327	436	5	ideal	ideal	NOUN
ejpam-6327	436	6	of	of	ADP
ejpam-6327	436	7	x	x	X
ejpam-6327	436	8	,	,	PUNCT
ejpam-6327	436	9	x	x	PROPN
ejpam-6327	436	10	∈	∈	PROPN
ejpam-6327	436	11	j	j	PROPN
ejpam-6327	436	12	.	.	PUNCT
ejpam-6327	437	1	thus	thus	ADV
ejpam-6327	437	2	,	,	PUNCT
ejpam-6327	437	3	µx	µx	PUNCT
ejpam-6327	437	4	∈	∈	PROPN
ejpam-6327	437	5	j/µ.	j/µ.	AUX
ejpam-6327	437	6	therefore	therefore	ADV
ejpam-6327	437	7	,	,	PUNCT
ejpam-6327	437	8	j/µ	j/µ	NOUN
ejpam-6327	437	9	is	be	AUX
ejpam-6327	437	10	an	an	DET
ejpam-6327	437	11	ideal	ideal	NOUN
ejpam-6327	437	12	of	of	ADP
ejpam-6327	437	13	x/µ.	x/µ.	PROPN
ejpam-6327	437	14	theorem	theorem	ADJ
ejpam-6327	437	15	14	14	NUM
ejpam-6327	437	16	.	.	PUNCT
ejpam-6327	438	1	let	let	VERB
ejpam-6327	438	2	x	x	PRON
ejpam-6327	438	3	be	be	AUX
ejpam-6327	438	4	a	a	DET
ejpam-6327	438	5	ks	ks	NOUN
ejpam-6327	438	6	-	-	PUNCT
ejpam-6327	438	7	semigroup	semigroup	NOUN
ejpam-6327	438	8	and	and	CCONJ
ejpam-6327	438	9	µ	µ	DET
ejpam-6327	438	10	a	a	DET
ejpam-6327	438	11	non	non	ADJ
ejpam-6327	438	12	-	-	ADJ
ejpam-6327	438	13	zero	zero	ADJ
ejpam-6327	438	14	fuzzy	fuzzy	ADJ
ejpam-6327	438	15	ks	ks	NOUN
ejpam-6327	438	16	-	-	NOUN
ejpam-6327	438	17	ideal	ideal	NOUN
ejpam-6327	438	18	of	of	ADP
ejpam-6327	438	19	x	x	SYM
ejpam-6327	438	20	satisfying	satisfy	VERB
ejpam-6327	438	21	condition	condition	NOUN
ejpam-6327	438	22	(	(	PUNCT
ejpam-6327	438	23	c	c	NOUN
ejpam-6327	438	24	)	)	PUNCT
ejpam-6327	438	25	.	.	PUNCT
ejpam-6327	439	1	if	if	SCONJ
ejpam-6327	439	2	j∗	j∗	PROPN
ejpam-6327	439	3	is	be	AUX
ejpam-6327	439	4	an	an	DET
ejpam-6327	439	5	ideal	ideal	NOUN
ejpam-6327	439	6	of	of	ADP
ejpam-6327	439	7	x/µ	x/µ	PROPN
ejpam-6327	439	8	,	,	PUNCT
ejpam-6327	439	9	then	then	ADV
ejpam-6327	439	10	there	there	PRON
ejpam-6327	439	11	exists	exist	VERB
ejpam-6327	439	12	an	an	DET
ejpam-6327	439	13	ideal	ideal	ADJ
ejpam-6327	439	14	j	j	NOUN
ejpam-6327	439	15	=	=	PUNCT
ejpam-6327	439	16	⋃	⋃	NOUN
ejpam-6327	439	17	{	{	PUNCT
ejpam-6327	439	18	x	x	SYM
ejpam-6327	439	19	∈	∈	PROPN
ejpam-6327	439	20	x	x	X
ejpam-6327	439	21	:	:	PUNCT
ejpam-6327	439	22	µx	µx	PROPN
ejpam-6327	439	23	∈	∈	PROPN
ejpam-6327	439	24	j∗	j∗	NOUN
ejpam-6327	439	25	}	}	PUNCT
ejpam-6327	439	26	of	of	ADP
ejpam-6327	439	27	x	x	SYM
ejpam-6327	439	28	such	such	ADJ
ejpam-6327	439	29	that	that	DET
ejpam-6327	439	30	j/µ	j/µ	NOUN
ejpam-6327	439	31	=	=	SYM
ejpam-6327	439	32	j∗.	j∗.	NUM
ejpam-6327	439	33	definition	definition	NOUN
ejpam-6327	439	34	14	14	NUM
ejpam-6327	439	35	.	.	PUNCT
ejpam-6327	440	1	[	[	X
ejpam-6327	440	2	5	5	X
ejpam-6327	440	3	]	]	PUNCT
ejpam-6327	440	4	let	let	VERB
ejpam-6327	440	5	x	x	PRON
ejpam-6327	440	6	and	and	CCONJ
ejpam-6327	440	7	y	y	PROPN
ejpam-6327	440	8	be	be	AUX
ejpam-6327	440	9	ks	k	NOUN
ejpam-6327	440	10	-	-	PUNCT
ejpam-6327	440	11	semigroups	semigroup	NOUN
ejpam-6327	440	12	and	and	CCONJ
ejpam-6327	440	13	f	f	X
ejpam-6327	440	14	:	:	PUNCT
ejpam-6327	440	15	x	x	X
ejpam-6327	440	16	→	→	SYM
ejpam-6327	440	17	y	y	X
ejpam-6327	440	18	be	be	AUX
ejpam-6327	440	19	a	a	DET
ejpam-6327	440	20	mapping	mapping	NOUN
ejpam-6327	440	21	.	.	PUNCT
ejpam-6327	441	1	then	then	ADV
ejpam-6327	441	2	f	f	PROPN
ejpam-6327	441	3	is	be	AUX
ejpam-6327	441	4	called	call	VERB
ejpam-6327	441	5	a	a	DET
ejpam-6327	441	6	ks	ks	NOUN
ejpam-6327	441	7	-	-	PUNCT
ejpam-6327	441	8	semigroup	semigroup	ADJ
ejpam-6327	441	9	homomorphism	homomorphism	NOUN
ejpam-6327	441	10	(	(	PUNCT
ejpam-6327	441	11	briefly	briefly	ADV
ejpam-6327	441	12	,	,	PUNCT
ejpam-6327	441	13	homomorphism	homomorphism	PROPN
ejpam-6327	441	14	)	)	PUNCT
ejpam-6327	441	15	if	if	SCONJ
ejpam-6327	441	16	for	for	ADP
ejpam-6327	441	17	all	all	DET
ejpam-6327	441	18	x	x	NOUN
ejpam-6327	441	19	,	,	PUNCT
ejpam-6327	441	20	y	y	PROPN
ejpam-6327	441	21	∈	∈	PROPN
ejpam-6327	441	22	x	x	PROPN
ejpam-6327	441	23	,	,	PUNCT
ejpam-6327	441	24	f(x	f(x	PROPN
ejpam-6327	441	25	∗	∗	NOUN
ejpam-6327	441	26	y	y	NOUN
ejpam-6327	441	27	)	)	PUNCT
ejpam-6327	441	28	=	=	SYM
ejpam-6327	441	29	f(x	f(x	PROPN
ejpam-6327	441	30	)	)	PUNCT
ejpam-6327	441	31	∗	∗	NOUN
ejpam-6327	441	32	f(y	f(y	NOUN
ejpam-6327	441	33	)	)	PUNCT
ejpam-6327	441	34	and	and	CCONJ
ejpam-6327	441	35	f(xy	f(xy	NUM
ejpam-6327	441	36	)	)	PUNCT
ejpam-6327	441	37	=	=	SYM
ejpam-6327	441	38	f(x)f(y	f(x)f(y	NOUN
ejpam-6327	441	39	)	)	PUNCT
ejpam-6327	441	40	.	.	PUNCT
ejpam-6327	442	1	if	if	SCONJ
ejpam-6327	442	2	f	f	PROPN
ejpam-6327	442	3	is	be	AUX
ejpam-6327	442	4	a	a	DET
ejpam-6327	442	5	one	one	NUM
ejpam-6327	442	6	-	-	PUNCT
ejpam-6327	442	7	to	to	ADP
ejpam-6327	442	8	-	-	PUNCT
ejpam-6327	442	9	one	one	NUM
ejpam-6327	442	10	homomorphism	homomorphism	NOUN
ejpam-6327	442	11	,	,	PUNCT
ejpam-6327	442	12	f	f	PROPN
ejpam-6327	442	13	is	be	AUX
ejpam-6327	442	14	said	say	VERB
ejpam-6327	442	15	to	to	PART
ejpam-6327	442	16	be	be	AUX
ejpam-6327	442	17	a	a	DET
ejpam-6327	442	18	monomorphism	monomorphism	NOUN
ejpam-6327	442	19	.	.	PUNCT
ejpam-6327	443	1	if	if	SCONJ
ejpam-6327	443	2	f	f	PROPN
ejpam-6327	443	3	is	be	AUX
ejpam-6327	443	4	an	an	PRON
ejpam-6327	443	5	onto	onto	ADP
ejpam-6327	443	6	homomorphism	homomorphism	NOUN
ejpam-6327	443	7	,	,	PUNCT
ejpam-6327	443	8	f	f	PROPN
ejpam-6327	443	9	is	be	AUX
ejpam-6327	443	10	called	call	VERB
ejpam-6327	443	11	an	an	DET
ejpam-6327	443	12	epimorphism	epimorphism	NOUN
ejpam-6327	443	13	.	.	PUNCT
ejpam-6327	444	1	if	if	SCONJ
ejpam-6327	444	2	f	f	PROPN
ejpam-6327	444	3	is	be	AUX
ejpam-6327	444	4	a	a	DET
ejpam-6327	444	5	bijective	bijective	ADJ
ejpam-6327	444	6	homomorphism	homomorphism	NOUN
ejpam-6327	444	7	,	,	PUNCT
ejpam-6327	444	8	f	f	PROPN
ejpam-6327	444	9	is	be	AUX
ejpam-6327	444	10	called	call	VERB
ejpam-6327	444	11	an	an	DET
ejpam-6327	444	12	isomorphism	isomorphism	NOUN
ejpam-6327	444	13	.	.	PUNCT
ejpam-6327	445	1	in	in	ADP
ejpam-6327	445	2	this	this	DET
ejpam-6327	445	3	case	case	NOUN
ejpam-6327	445	4	,	,	PUNCT
ejpam-6327	445	5	x	x	PUNCT
ejpam-6327	445	6	and	and	CCONJ
ejpam-6327	445	7	y	y	PROPN
ejpam-6327	445	8	are	be	AUX
ejpam-6327	445	9	said	say	VERB
ejpam-6327	445	10	to	to	PART
ejpam-6327	445	11	be	be	AUX
ejpam-6327	445	12	isomorphic	isomorphic	ADJ
ejpam-6327	445	13	(	(	PUNCT
ejpam-6327	445	14	written	write	VERB
ejpam-6327	445	15	x	x	PUNCT
ejpam-6327	445	16	∼=	∼=	PROPN
ejpam-6327	445	17	y	y	PROPN
ejpam-6327	445	18	)	)	PUNCT
ejpam-6327	445	19	.	.	PUNCT
ejpam-6327	446	1	theorem	theorem	NOUN
ejpam-6327	446	2	15	15	NUM
ejpam-6327	446	3	.	.	PUNCT
ejpam-6327	447	1	let	let	VERB
ejpam-6327	447	2	µ	µ	X
ejpam-6327	447	3	be	be	AUX
ejpam-6327	447	4	a	a	DET
ejpam-6327	447	5	non	non	ADJ
ejpam-6327	447	6	-	-	ADJ
ejpam-6327	447	7	zero	zero	ADJ
ejpam-6327	447	8	fuzzy	fuzzy	ADJ
ejpam-6327	447	9	ks	ks	NOUN
ejpam-6327	447	10	-	-	NOUN
ejpam-6327	447	11	ideal	ideal	NOUN
ejpam-6327	447	12	of	of	ADP
ejpam-6327	447	13	a	a	DET
ejpam-6327	447	14	ks	ks	NOUN
ejpam-6327	447	15	-	-	PUNCT
ejpam-6327	447	16	semigroup	semigroup	NOUN
ejpam-6327	447	17	x	x	PUNCT
ejpam-6327	447	18	satisfying	satisfy	VERB
ejpam-6327	447	19	condition	condition	NOUN
ejpam-6327	447	20	(	(	PUNCT
ejpam-6327	447	21	c	c	NOUN
ejpam-6327	447	22	)	)	PUNCT
ejpam-6327	447	23	.	.	PUNCT
ejpam-6327	448	1	if	if	SCONJ
ejpam-6327	448	2	j	j	PROPN
ejpam-6327	448	3	is	be	AUX
ejpam-6327	448	4	an	an	DET
ejpam-6327	448	5	ideal	ideal	NOUN
ejpam-6327	448	6	of	of	ADP
ejpam-6327	448	7	x	x	PRON
ejpam-6327	448	8	,	,	PUNCT
ejpam-6327	448	9	then	then	ADV
ejpam-6327	448	10	x/µ	x/µ	PROPN
ejpam-6327	448	11	j/µ	j/µ	NOUN
ejpam-6327	448	12	∼=	∼=	PROPN
ejpam-6327	448	13	x	x	PROPN
ejpam-6327	448	14	/	/	SYM
ejpam-6327	448	15	j	j	PROPN
ejpam-6327	448	16	.	.	PUNCT
ejpam-6327	449	1	proof	proof	NOUN
ejpam-6327	449	2	.	.	PUNCT
ejpam-6327	450	1	let	let	VERB
ejpam-6327	450	2	µ	µ	X
ejpam-6327	450	3	be	be	AUX
ejpam-6327	450	4	a	a	DET
ejpam-6327	450	5	non	non	ADJ
ejpam-6327	450	6	-	-	ADJ
ejpam-6327	450	7	zero	zero	ADJ
ejpam-6327	450	8	fuzzy	fuzzy	ADJ
ejpam-6327	450	9	ks	ks	NOUN
ejpam-6327	450	10	-	-	NOUN
ejpam-6327	450	11	ideal	ideal	NOUN
ejpam-6327	450	12	of	of	ADP
ejpam-6327	450	13	a	a	DET
ejpam-6327	450	14	ks	ks	NOUN
ejpam-6327	450	15	-	-	PUNCT
ejpam-6327	450	16	semigroupx	semigroupx	NOUN
ejpam-6327	450	17	satisfying	satisfy	VERB
ejpam-6327	450	18	condition	condition	NOUN
ejpam-6327	450	19	(	(	PUNCT
ejpam-6327	450	20	c	c	NOUN
ejpam-6327	450	21	)	)	PUNCT
ejpam-6327	450	22	and	and	CCONJ
ejpam-6327	450	23	j	j	X
ejpam-6327	450	24	an	an	DET
ejpam-6327	450	25	ideal	ideal	ADJ
ejpam-6327	450	26	ofx	ofx	NOUN
ejpam-6327	450	27	.	.	PUNCT
ejpam-6327	451	1	by	by	ADP
ejpam-6327	451	2	theorem	theorem	NOUN
ejpam-6327	451	3	13	13	NUM
ejpam-6327	451	4	,	,	PUNCT
ejpam-6327	451	5	j/µ	j/µ	NOUN
ejpam-6327	451	6	is	be	AUX
ejpam-6327	451	7	an	an	DET
ejpam-6327	451	8	ideal	ideal	ADJ
ejpam-6327	451	9	ofx/µ	ofx/µ	NOUN
ejpam-6327	451	10	,	,	PUNCT
ejpam-6327	451	11	x/µ	x/µ	PROPN
ejpam-6327	452	1	j/µ	j/µ	NOUN
ejpam-6327	452	2	=	=	PUNCT
ejpam-6327	452	3	{	{	PUNCT
ejpam-6327	452	4	(	(	PUNCT
ejpam-6327	452	5	j/µ)µx	j/µ)µx	NOUN
ejpam-6327	452	6	:	:	PUNCT
ejpam-6327	452	7	µx	µx	VERB
ejpam-6327	452	8	∈	∈	PROPN
ejpam-6327	452	9	x/µ	x/µ	PROPN
ejpam-6327	452	10	}	}	PUNCT
ejpam-6327	452	11	and	and	CCONJ
ejpam-6327	452	12	x	x	X
ejpam-6327	452	13	/	/	SYM
ejpam-6327	452	14	j	j	PROPN
ejpam-6327	452	15	=	=	PRON
ejpam-6327	452	16	{	{	PUNCT
ejpam-6327	452	17	jx	jx	NOUN
ejpam-6327	452	18	:	:	PUNCT
ejpam-6327	452	19	x	x	SYM
ejpam-6327	452	20	∈	∈	NOUN
ejpam-6327	452	21	x	x	X
ejpam-6327	452	22	}	}	PUNCT
ejpam-6327	452	23	.	.	PUNCT
ejpam-6327	453	1	consider	consider	VERB
ejpam-6327	453	2	a	a	DET
ejpam-6327	453	3	map	map	NOUN
ejpam-6327	453	4	φ	φ	X
ejpam-6327	453	5	:	:	PUNCT
ejpam-6327	454	1	x/µ	x/µ	PROPN
ejpam-6327	454	2	j/µ	j/µ	NOUN
ejpam-6327	454	3	→	→	SYM
ejpam-6327	454	4	x	x	X
ejpam-6327	454	5	/	/	SYM
ejpam-6327	454	6	j	j	PROPN
ejpam-6327	454	7	defined	define	VERB
ejpam-6327	454	8	by	by	ADP
ejpam-6327	454	9	φ((j/µ)µx	φ((j/µ)µx	NOUN
ejpam-6327	454	10	)	)	PUNCT
ejpam-6327	455	1	=	=	SYM
ejpam-6327	455	2	jx	jx	PROPN
ejpam-6327	455	3	.	.	PUNCT
ejpam-6327	456	1	let	let	VERB
ejpam-6327	456	2	(	(	PUNCT
ejpam-6327	456	3	j/µ)µx	j/µ)µx	NOUN
ejpam-6327	456	4	,	,	PUNCT
ejpam-6327	456	5	(	(	PUNCT
ejpam-6327	456	6	j/µ)µy	j/µ)µy	NOUN
ejpam-6327	456	7	∈	∈	PROPN
ejpam-6327	456	8	x/µ	x/µ	PUNCT
ejpam-6327	457	1	j/µ	j/µ	NOUN
ejpam-6327	457	2	such	such	ADJ
ejpam-6327	457	3	that	that	SCONJ
ejpam-6327	457	4	(	(	PUNCT
ejpam-6327	457	5	j/µ)µx	j/µ)µx	NOUN
ejpam-6327	457	6	=	=	PUNCT
ejpam-6327	457	7	(	(	PUNCT
ejpam-6327	457	8	j/µ)µy	j/µ)µy	NOUN
ejpam-6327	457	9	.	.	PUNCT
ejpam-6327	458	1	then	then	ADV
ejpam-6327	458	2	µx	µx	VERB
ejpam-6327	458	3	∼j/µ	∼j/µ	PROPN
ejpam-6327	458	4	µy	µy	NOUN
ejpam-6327	458	5	.	.	PUNCT
ejpam-6327	459	1	thus	thus	ADV
ejpam-6327	459	2	,	,	PUNCT
ejpam-6327	459	3	µx∗y	µx∗y	X
ejpam-6327	459	4	=	=	PUNCT
ejpam-6327	459	5	µx	µx	ADP
ejpam-6327	459	6	∗	∗	NOUN
ejpam-6327	459	7	µy	µy	ADP
ejpam-6327	459	8	∈	∈	PROPN
ejpam-6327	459	9	j/µ	j/µ	NOUN
ejpam-6327	459	10	,	,	PUNCT
ejpam-6327	459	11	µy∗x	µy∗x	X
ejpam-6327	459	12	=	=	SYM
ejpam-6327	460	1	µy	µy	ADP
ejpam-6327	460	2	∗	∗	NOUN
ejpam-6327	460	3	µx	µx	AUX
ejpam-6327	460	4	∈	∈	PROPN
ejpam-6327	460	5	j/µ.	j/µ.	AUX
ejpam-6327	461	1	this	this	PRON
ejpam-6327	461	2	means	mean	VERB
ejpam-6327	461	3	that	that	SCONJ
ejpam-6327	461	4	x	x	PUNCT
ejpam-6327	461	5	∗	∗	VERB
ejpam-6327	461	6	y	y	PROPN
ejpam-6327	461	7	∈	∈	PROPN
ejpam-6327	461	8	j	j	PROPN
ejpam-6327	461	9	,	,	PUNCT
ejpam-6327	461	10	y	y	PROPN
ejpam-6327	461	11	∗	∗	NOUN
ejpam-6327	461	12	x	x	PUNCT
ejpam-6327	461	13	∈	∈	PROPN
ejpam-6327	461	14	j	j	PROPN
ejpam-6327	461	15	,	,	PUNCT
ejpam-6327	461	16	that	that	ADV
ejpam-6327	461	17	is	is	ADV
ejpam-6327	461	18	,	,	PUNCT
ejpam-6327	461	19	x	x	SYM
ejpam-6327	461	20	∼j	∼j	PROPN
ejpam-6327	461	21	y	y	PROPN
ejpam-6327	461	22	and	and	CCONJ
ejpam-6327	461	23	jx	jx	PROPN
ejpam-6327	461	24	=	=	PROPN
ejpam-6327	461	25	jy	jy	PROPN
ejpam-6327	461	26	.	.	PUNCT
ejpam-6327	462	1	thus	thus	ADV
ejpam-6327	462	2	,	,	PUNCT
ejpam-6327	462	3	φ((j/µ)µx	φ((j/µ)µx	NOUN
ejpam-6327	462	4	)	)	PUNCT
ejpam-6327	463	1	=	=	SYM
ejpam-6327	463	2	jx	jx	PROPN
ejpam-6327	463	3	=	=	SYM
ejpam-6327	463	4	jy	jy	PROPN
ejpam-6327	463	5	=	=	SYM
ejpam-6327	463	6	φ((j/µ)µy	φ((j/µ)µy	PROPN
ejpam-6327	463	7	)	)	PUNCT
ejpam-6327	463	8	.	.	PUNCT
ejpam-6327	464	1	hence	hence	ADV
ejpam-6327	464	2	,	,	PUNCT
ejpam-6327	464	3	φ	φ	PROPN
ejpam-6327	464	4	is	be	AUX
ejpam-6327	464	5	well	well	ADV
ejpam-6327	464	6	-	-	PUNCT
ejpam-6327	464	7	defined	define	VERB
ejpam-6327	464	8	.	.	PUNCT
ejpam-6327	465	1	now	now	ADV
ejpam-6327	465	2	,	,	PUNCT
ejpam-6327	465	3	let	let	VERB
ejpam-6327	465	4	(	(	PUNCT
ejpam-6327	465	5	j/µ)µx	j/µ)µx	NOUN
ejpam-6327	465	6	,	,	PUNCT
ejpam-6327	465	7	(	(	PUNCT
ejpam-6327	465	8	j/µ)µy	j/µ)µy	NOUN
ejpam-6327	465	9	∈	∈	PROPN
ejpam-6327	466	1	x/µ	x/µ	PROPN
ejpam-6327	467	1	j/µ	j/µ	NOUN
ejpam-6327	467	2	.	.	PUNCT
ejpam-6327	468	1	then	then	ADV
ejpam-6327	468	2	φ((j/µ)µx	φ((j/µ)µx	NOUN
ejpam-6327	468	3	∗	∗	NOUN
ejpam-6327	468	4	(	(	PUNCT
ejpam-6327	468	5	j/µ)µy	j/µ)µy	NOUN
ejpam-6327	468	6	)	)	PUNCT
ejpam-6327	468	7	=	=	SYM
ejpam-6327	468	8	φ((j/µ)µx∗µy	φ((j/µ)µx∗µy	X
ejpam-6327	468	9	)	)	PUNCT
ejpam-6327	468	10	=	=	SYM
ejpam-6327	468	11	φ((j/µ)µx∗y	φ((j/µ)µx∗y	X
ejpam-6327	468	12	)	)	PUNCT
ejpam-6327	468	13	=	=	SYM
ejpam-6327	469	1	jx∗y	jx∗y	X
ejpam-6327	469	2	=	=	SYM
ejpam-6327	469	3	jx	jx	PROPN
ejpam-6327	469	4	∗	∗	VERB
ejpam-6327	469	5	jy	jy	PROPN
ejpam-6327	469	6	=	=	PROPN
ejpam-6327	469	7	φ((j/µ)µx	φ((j/µ)µx	PROPN
ejpam-6327	469	8	)	)	PUNCT
ejpam-6327	469	9	∗	∗	NOUN
ejpam-6327	469	10	φ((j/µ)µy	φ((j/µ)µy	NOUN
ejpam-6327	469	11	)	)	PUNCT
ejpam-6327	469	12	and	and	CCONJ
ejpam-6327	469	13	φ((j/µ)µx(j/µ)µy	φ((j/µ)µx(j/µ)µy	NOUN
ejpam-6327	469	14	)	)	PUNCT
ejpam-6327	469	15	=	=	SYM
ejpam-6327	469	16	φ((j/µ)µxµy	φ((j/µ)µxµy	NOUN
ejpam-6327	469	17	)	)	PUNCT
ejpam-6327	469	18	h.	h.	NOUN
ejpam-6327	469	19	sarapuddin	sarapuddin	PROPN
ejpam-6327	469	20	,	,	PUNCT
ejpam-6327	469	21	j.	j.	PROPN
ejpam-6327	469	22	vilela	vilela	PROPN
ejpam-6327	469	23	/	/	SYM
ejpam-6327	469	24	eur	eur	PROPN
ejpam-6327	469	25	.	.	PUNCT
ejpam-6327	470	1	j.	j.	PROPN
ejpam-6327	470	2	pure	pure	PROPN
ejpam-6327	470	3	appl	appl	PROPN
ejpam-6327	470	4	.	.	PROPN
ejpam-6327	470	5	math	math	PROPN
ejpam-6327	470	6	,	,	PUNCT
ejpam-6327	470	7	18	18	NUM
ejpam-6327	470	8	(	(	PUNCT
ejpam-6327	470	9	3	3	NUM
ejpam-6327	470	10	)	)	PUNCT
ejpam-6327	470	11	(	(	PUNCT
ejpam-6327	470	12	2025	2025	NUM
ejpam-6327	470	13	)	)	PUNCT
ejpam-6327	470	14	,	,	PUNCT
ejpam-6327	470	15	6327	6327	NUM
ejpam-6327	470	16	15	15	NUM
ejpam-6327	470	17	of	of	ADP
ejpam-6327	470	18	23	23	NUM
ejpam-6327	470	19	=	=	SYM
ejpam-6327	470	20	φ((j/µ)µxy	φ((j/µ)µxy	PROPN
ejpam-6327	470	21	)	)	PUNCT
ejpam-6327	471	1	=	=	NOUN
ejpam-6327	471	2	jxy	jxy	NOUN
ejpam-6327	472	1	=	=	PUNCT
ejpam-6327	472	2	jxjy	jxjy	PROPN
ejpam-6327	472	3	=	=	SYM
ejpam-6327	472	4	φ((j/µ)µx)φ((j/µ)µy	φ((j/µ)µx)φ((j/µ)µy	PROPN
ejpam-6327	472	5	)	)	PUNCT
ejpam-6327	472	6	.	.	PUNCT
ejpam-6327	473	1	thus	thus	ADV
ejpam-6327	473	2	,	,	PUNCT
ejpam-6327	473	3	φ	φ	PROPN
ejpam-6327	473	4	is	be	AUX
ejpam-6327	473	5	a	a	DET
ejpam-6327	473	6	ks	ks	NOUN
ejpam-6327	473	7	-	-	PUNCT
ejpam-6327	473	8	semigroup	semigroup	ADJ
ejpam-6327	473	9	homomorphism	homomorphism	NOUN
ejpam-6327	473	10	.	.	PUNCT
ejpam-6327	474	1	for	for	ADP
ejpam-6327	474	2	each	each	DET
ejpam-6327	474	3	jx	jx	PROPN
ejpam-6327	474	4	∈	∈	PROPN
ejpam-6327	474	5	x	x	PROPN
ejpam-6327	474	6	/	/	SYM
ejpam-6327	474	7	j	j	PROPN
ejpam-6327	474	8	,	,	PUNCT
ejpam-6327	474	9	there	there	PRON
ejpam-6327	474	10	exists	exist	VERB
ejpam-6327	474	11	(	(	PUNCT
ejpam-6327	474	12	j/µ)µx	j/µ)µx	NOUN
ejpam-6327	474	13	such	such	ADJ
ejpam-6327	474	14	that	that	DET
ejpam-6327	474	15	φ((j/µ)µx	φ((j/µ)µx	NOUN
ejpam-6327	474	16	)	)	PUNCT
ejpam-6327	474	17	=	=	SYM
ejpam-6327	474	18	jx	jx	PROPN
ejpam-6327	474	19	.	.	PUNCT
ejpam-6327	475	1	thus	thus	ADV
ejpam-6327	475	2	,	,	PUNCT
ejpam-6327	475	3	φ	φ	PROPN
ejpam-6327	475	4	is	be	AUX
ejpam-6327	475	5	onto	onto	ADP
ejpam-6327	475	6	.	.	PUNCT
ejpam-6327	476	1	now	now	ADV
ejpam-6327	476	2	,	,	PUNCT
ejpam-6327	476	3	suppose	suppose	VERB
ejpam-6327	476	4	φ((j/µ)µx	φ((j/µ)µx	NOUN
ejpam-6327	476	5	)	)	PUNCT
ejpam-6327	476	6	=	=	SYM
ejpam-6327	476	7	φ((j/µ)µy	φ((j/µ)µy	NOUN
ejpam-6327	476	8	)	)	PUNCT
ejpam-6327	476	9	.	.	PUNCT
ejpam-6327	477	1	then	then	ADV
ejpam-6327	477	2	jx	jx	PROPN
ejpam-6327	477	3	=	=	PROPN
ejpam-6327	477	4	jy	jy	PROPN
ejpam-6327	477	5	.	.	PUNCT
ejpam-6327	478	1	thus	thus	ADV
ejpam-6327	478	2	,	,	PUNCT
ejpam-6327	478	3	x	x	SYM
ejpam-6327	478	4	∼j	∼j	PROPN
ejpam-6327	478	5	y.	y.	NOUN
ejpam-6327	478	6	let	let	VERB
ejpam-6327	478	7	µα	µα	ADP
ejpam-6327	478	8	∈	∈	NOUN
ejpam-6327	478	9	(	(	PUNCT
ejpam-6327	478	10	j/µ)µx	j/µ)µx	NOUN
ejpam-6327	478	11	.	.	PUNCT
ejpam-6327	479	1	then	then	ADV
ejpam-6327	479	2	µα	µα	ADP
ejpam-6327	479	3	∼j/µ	∼j/µ	PRON
ejpam-6327	479	4	µx	µx	VERB
ejpam-6327	479	5	.	.	PUNCT
ejpam-6327	480	1	it	it	PRON
ejpam-6327	480	2	follows	follow	VERB
ejpam-6327	480	3	that	that	PRON
ejpam-6327	480	4	µα∗x	µα∗x	PROPN
ejpam-6327	480	5	=	=	PUNCT
ejpam-6327	480	6	µα∗µx	µα∗µx	PROPN
ejpam-6327	480	7	∈	∈	PROPN
ejpam-6327	480	8	j/µ	j/µ	NOUN
ejpam-6327	480	9	and	and	CCONJ
ejpam-6327	480	10	µx∗α	µx∗α	PROPN
ejpam-6327	480	11	=	=	PUNCT
ejpam-6327	480	12	µx∗µα	µx∗µα	PROPN
ejpam-6327	480	13	∈	∈	PROPN
ejpam-6327	480	14	j/µ.	j/µ.	AUX
ejpam-6327	480	15	thus	thus	ADV
ejpam-6327	480	16	,	,	PUNCT
ejpam-6327	480	17	α∗x	α∗x	PROPN
ejpam-6327	480	18	∈	∈	PROPN
ejpam-6327	480	19	j	j	PROPN
ejpam-6327	480	20	and	and	CCONJ
ejpam-6327	480	21	x∗α	x∗α	PRON
ejpam-6327	480	22	∈	∈	PROPN
ejpam-6327	480	23	j	j	PROPN
ejpam-6327	480	24	,	,	PUNCT
ejpam-6327	480	25	that	that	ADV
ejpam-6327	480	26	is	is	ADV
ejpam-6327	480	27	,	,	PUNCT
ejpam-6327	480	28	α	α	PRON
ejpam-6327	480	29	∼j	∼j	PROPN
ejpam-6327	480	30	x.	x.	NOUN
ejpam-6327	480	31	since	since	SCONJ
ejpam-6327	480	32	∼j	∼j	PROPN
ejpam-6327	480	33	is	be	AUX
ejpam-6327	480	34	an	an	DET
ejpam-6327	480	35	equivalence	equivalence	NOUN
ejpam-6327	480	36	relation	relation	NOUN
ejpam-6327	480	37	,	,	PUNCT
ejpam-6327	480	38	α	α	NOUN
ejpam-6327	480	39	∼j	∼j	PROPN
ejpam-6327	480	40	y.	y.	PROPN
ejpam-6327	480	41	thus	thus	ADV
ejpam-6327	480	42	,	,	PUNCT
ejpam-6327	480	43	µα	µα	ADP
ejpam-6327	480	44	∈	∈	PROPN
ejpam-6327	480	45	(	(	PUNCT
ejpam-6327	480	46	j/µ)µy	j/µ)µy	NOUN
ejpam-6327	480	47	.	.	PUNCT
ejpam-6327	481	1	hence	hence	ADV
ejpam-6327	481	2	,	,	PUNCT
ejpam-6327	481	3	(	(	PUNCT
ejpam-6327	481	4	j/µ)µx	j/µ)µx	NOUN
ejpam-6327	481	5	⊆	⊆	NUM
ejpam-6327	481	6	(	(	PUNCT
ejpam-6327	481	7	j/µ)µy	j/µ)µy	NOUN
ejpam-6327	481	8	.	.	PUNCT
ejpam-6327	482	1	similarly	similarly	ADV
ejpam-6327	482	2	,	,	PUNCT
ejpam-6327	482	3	(	(	PUNCT
ejpam-6327	482	4	j/µ)µy	j/µ)µy	NOUN
ejpam-6327	482	5	⊆	⊆	NUM
ejpam-6327	482	6	(	(	PUNCT
ejpam-6327	482	7	j/µ)µx	j/µ)µx	NOUN
ejpam-6327	482	8	.	.	PUNCT
ejpam-6327	483	1	thus	thus	ADV
ejpam-6327	483	2	,	,	PUNCT
ejpam-6327	483	3	(	(	PUNCT
ejpam-6327	483	4	j/µ)µx	j/µ)µx	NOUN
ejpam-6327	483	5	=	=	PUNCT
ejpam-6327	483	6	(	(	PUNCT
ejpam-6327	483	7	j/µ)µy	j/µ)µy	NOUN
ejpam-6327	483	8	.	.	PUNCT
ejpam-6327	484	1	hence	hence	ADV
ejpam-6327	484	2	,	,	PUNCT
ejpam-6327	484	3	φ	φ	PROPN
ejpam-6327	484	4	is	be	AUX
ejpam-6327	484	5	one	one	NUM
ejpam-6327	484	6	-	-	PUNCT
ejpam-6327	484	7	to	to	ADP
ejpam-6327	484	8	-	-	PUNCT
ejpam-6327	484	9	one	one	NUM
ejpam-6327	484	10	.	.	PUNCT
ejpam-6327	485	1	therefore	therefore	ADV
ejpam-6327	485	2	,	,	PUNCT
ejpam-6327	485	3	φ	φ	PROPN
ejpam-6327	485	4	is	be	AUX
ejpam-6327	485	5	an	an	DET
ejpam-6327	485	6	isomorphism	isomorphism	NOUN
ejpam-6327	485	7	,	,	PUNCT
ejpam-6327	485	8	that	that	ADV
ejpam-6327	485	9	is	is	ADV
ejpam-6327	485	10	,	,	PUNCT
ejpam-6327	485	11	x/µ	x/µ	PROPN
ejpam-6327	485	12	j/µ	j/µ	NOUN
ejpam-6327	485	13	∼=	∼=	PROPN
ejpam-6327	485	14	x	x	SYM
ejpam-6327	485	15	/	/	SYM
ejpam-6327	485	16	j	j	PROPN
ejpam-6327	485	17	.	.	PUNCT
ejpam-6327	486	1	the	the	DET
ejpam-6327	486	2	following	follow	VERB
ejpam-6327	486	3	are	be	AUX
ejpam-6327	486	4	the	the	DET
ejpam-6327	486	5	properties	property	NOUN
ejpam-6327	486	6	of	of	ADP
ejpam-6327	486	7	the	the	DET
ejpam-6327	486	8	generalized	generalized	ADJ
ejpam-6327	486	9	quotient	quotient	NOUN
ejpam-6327	486	10	via	via	ADP
ejpam-6327	486	11	fuzzy	fuzzy	ADJ
ejpam-6327	486	12	ks	ks	NOUN
ejpam-6327	486	13	-	-	PUNCT
ejpam-6327	486	14	ideals	ideal	NOUN
ejpam-6327	486	15	with	with	ADP
ejpam-6327	486	16	respect	respect	NOUN
ejpam-6327	486	17	to	to	ADP
ejpam-6327	486	18	fuzzy	fuzzy	ADJ
ejpam-6327	486	19	ks	ks	NOUN
ejpam-6327	486	20	-	-	ADJ
ejpam-6327	486	21	p	p	NOUN
ejpam-6327	486	22	-	-	PUNCT
ejpam-6327	486	23	ideal	ideal	ADJ
ejpam-6327	486	24	,	,	PUNCT
ejpam-6327	486	25	fuzzy	fuzzy	ADJ
ejpam-6327	486	26	commutative	commutative	ADJ
ejpam-6327	486	27	ks	ks	NOUN
ejpam-6327	486	28	-	-	PUNCT
ejpam-6327	486	29	ideal	ideal	ADJ
ejpam-6327	486	30	and	and	CCONJ
ejpam-6327	486	31	fuzzy	fuzzy	ADJ
ejpam-6327	486	32	implicative	implicative	ADJ
ejpam-6327	486	33	ks	ks	NOUN
ejpam-6327	486	34	-	-	PUNCT
ejpam-6327	486	35	ideal	ideal	NOUN
ejpam-6327	486	36	.	.	PUNCT
ejpam-6327	487	1	theorem	theorem	VERB
ejpam-6327	487	2	16	16	NUM
ejpam-6327	487	3	.	.	PUNCT
ejpam-6327	488	1	let	let	VERB
ejpam-6327	488	2	µ	µ	X
ejpam-6327	488	3	be	be	AUX
ejpam-6327	488	4	a	a	DET
ejpam-6327	488	5	non	non	ADJ
ejpam-6327	488	6	-	-	ADJ
ejpam-6327	488	7	zero	zero	ADJ
ejpam-6327	488	8	fuzzy	fuzzy	ADJ
ejpam-6327	488	9	commutative	commutative	ADJ
ejpam-6327	488	10	ks	ks	NOUN
ejpam-6327	488	11	-	-	PUNCT
ejpam-6327	488	12	ideal	ideal	NOUN
ejpam-6327	488	13	of	of	ADP
ejpam-6327	488	14	a	a	DET
ejpam-6327	488	15	ks	ks	NOUN
ejpam-6327	488	16	-	-	PUNCT
ejpam-6327	488	17	semigroup	semigroup	NOUN
ejpam-6327	488	18	x	x	PUNCT
ejpam-6327	488	19	satisfying	satisfy	VERB
ejpam-6327	488	20	condition	condition	NOUN
ejpam-6327	488	21	(	(	PUNCT
ejpam-6327	488	22	c	c	NOUN
ejpam-6327	488	23	)	)	PUNCT
ejpam-6327	488	24	.	.	PUNCT
ejpam-6327	489	1	then	then	ADV
ejpam-6327	489	2	x/µ	x/µ	PROPN
ejpam-6327	489	3	is	be	AUX
ejpam-6327	489	4	a	a	DET
ejpam-6327	489	5	commutative	commutative	ADJ
ejpam-6327	489	6	ks	ks	NOUN
ejpam-6327	489	7	-	-	PUNCT
ejpam-6327	489	8	semigroup	semigroup	NOUN
ejpam-6327	489	9	.	.	PUNCT
ejpam-6327	490	1	proof	proof	NOUN
ejpam-6327	490	2	.	.	PUNCT
ejpam-6327	491	1	let	let	VERB
ejpam-6327	491	2	µ	µ	X
ejpam-6327	491	3	be	be	AUX
ejpam-6327	491	4	a	a	DET
ejpam-6327	491	5	non	non	ADJ
ejpam-6327	491	6	-	-	ADJ
ejpam-6327	491	7	zero	zero	ADJ
ejpam-6327	491	8	fuzzy	fuzzy	ADJ
ejpam-6327	491	9	commutative	commutative	ADJ
ejpam-6327	491	10	ks	ks	NOUN
ejpam-6327	491	11	-	-	PUNCT
ejpam-6327	491	12	ideal	ideal	NOUN
ejpam-6327	491	13	of	of	ADP
ejpam-6327	491	14	a	a	DET
ejpam-6327	491	15	ks	ks	NOUN
ejpam-6327	491	16	-	-	PUNCT
ejpam-6327	491	17	semigroupx	semigroupx	NOUN
ejpam-6327	491	18	satisfying	satisfy	VERB
ejpam-6327	491	19	condition	condition	NOUN
ejpam-6327	491	20	(	(	PUNCT
ejpam-6327	491	21	c	c	NOUN
ejpam-6327	491	22	)	)	PUNCT
ejpam-6327	491	23	.	.	PUNCT
ejpam-6327	492	1	for	for	ADP
ejpam-6327	492	2	all	all	DET
ejpam-6327	492	3	x	x	NOUN
ejpam-6327	492	4	,	,	PUNCT
ejpam-6327	492	5	y	y	PROPN
ejpam-6327	492	6	∈	∈	PROPN
ejpam-6327	492	7	x	x	X
ejpam-6327	492	8	,	,	PUNCT
ejpam-6327	492	9	denote	denote	VERB
ejpam-6327	492	10	u	u	NOUN
ejpam-6327	492	11	=	=	PROPN
ejpam-6327	492	12	y	y	PROPN
ejpam-6327	492	13	∗	∗	NOUN
ejpam-6327	492	14	(	(	PUNCT
ejpam-6327	492	15	y	y	PROPN
ejpam-6327	492	16	∗	∗	NOUN
ejpam-6327	492	17	x	x	NOUN
ejpam-6327	492	18	)	)	PUNCT
ejpam-6327	492	19	.	.	PUNCT
ejpam-6327	493	1	then	then	ADV
ejpam-6327	493	2	u	u	PRON
ejpam-6327	493	3	∗	∗	NOUN
ejpam-6327	493	4	x	x	PUNCT
ejpam-6327	493	5	=	=	SYM
ejpam-6327	493	6	(	(	PUNCT
ejpam-6327	493	7	y	y	PROPN
ejpam-6327	493	8	∗	∗	NOUN
ejpam-6327	493	9	(	(	PUNCT
ejpam-6327	493	10	y	y	PROPN
ejpam-6327	493	11	∗	∗	NOUN
ejpam-6327	493	12	x	x	NOUN
ejpam-6327	493	13	)	)	PUNCT
ejpam-6327	493	14	)	)	PUNCT
ejpam-6327	493	15	∗	∗	NOUN
ejpam-6327	493	16	x	x	X
ejpam-6327	493	17	=	=	SYM
ejpam-6327	493	18	(	(	PUNCT
ejpam-6327	493	19	y∗x)∗(y∗x	y∗x)∗(y∗x	PROPN
ejpam-6327	493	20	)	)	PUNCT
ejpam-6327	494	1	=	=	PUNCT
ejpam-6327	494	2	0	0	X
ejpam-6327	494	3	.	.	PUNCT
ejpam-6327	494	4	thus	thus	ADV
ejpam-6327	494	5	,	,	PUNCT
ejpam-6327	494	6	by	by	ADP
ejpam-6327	494	7	theorem	theorem	NOUN
ejpam-6327	494	8	5	5	NUM
ejpam-6327	494	9	,	,	PUNCT
ejpam-6327	494	10	µ(u∗(x∗(x∗u	µ(u∗(x∗(x∗u	NOUN
ejpam-6327	494	11	)	)	PUNCT
ejpam-6327	494	12	)	)	PUNCT
ejpam-6327	494	13	)	)	PUNCT
ejpam-6327	494	14	=	=	SYM
ejpam-6327	494	15	µ(u∗x	µ(u∗x	X
ejpam-6327	494	16	)	)	PUNCT
ejpam-6327	494	17	=	=	SYM
ejpam-6327	494	18	µ(0	µ(0	NOUN
ejpam-6327	494	19	)	)	PUNCT
ejpam-6327	494	20	>	>	X
ejpam-6327	494	21	0	0	X
ejpam-6327	494	22	.	.	PUNCT
ejpam-6327	495	1	moreover	moreover	ADV
ejpam-6327	495	2	,	,	PUNCT
ejpam-6327	495	3	since	since	SCONJ
ejpam-6327	495	4	(	(	PUNCT
ejpam-6327	495	5	x	x	SYM
ejpam-6327	495	6	∗	∗	NOUN
ejpam-6327	495	7	(	(	PUNCT
ejpam-6327	495	8	x	x	X
ejpam-6327	495	9	∗	∗	NOUN
ejpam-6327	495	10	u	u	NOUN
ejpam-6327	495	11	)	)	PUNCT
ejpam-6327	495	12	)	)	PUNCT
ejpam-6327	495	13	∗	∗	NOUN
ejpam-6327	495	14	u	u	NOUN
ejpam-6327	495	15	=	=	PUNCT
ejpam-6327	495	16	(	(	PUNCT
ejpam-6327	495	17	x	x	X
ejpam-6327	495	18	∗	∗	X
ejpam-6327	495	19	u	u	NOUN
ejpam-6327	495	20	)	)	PUNCT
ejpam-6327	495	21	∗	∗	NOUN
ejpam-6327	495	22	(	(	PUNCT
ejpam-6327	495	23	x	x	X
ejpam-6327	495	24	∗	∗	X
ejpam-6327	495	25	u	u	NOUN
ejpam-6327	495	26	)	)	PUNCT
ejpam-6327	495	27	=	=	SYM
ejpam-6327	495	28	0	0	NUM
ejpam-6327	495	29	,	,	PUNCT
ejpam-6327	495	30	it	it	PRON
ejpam-6327	495	31	follows	follow	VERB
ejpam-6327	495	32	that	that	SCONJ
ejpam-6327	495	33	µ((x	µ((x	NOUN
ejpam-6327	495	34	∗	∗	NOUN
ejpam-6327	495	35	(	(	PUNCT
ejpam-6327	495	36	x	x	X
ejpam-6327	495	37	∗	∗	NOUN
ejpam-6327	495	38	u	u	NOUN
ejpam-6327	495	39	)	)	PUNCT
ejpam-6327	495	40	)	)	PUNCT
ejpam-6327	495	41	∗	∗	NOUN
ejpam-6327	495	42	u	u	NOUN
ejpam-6327	495	43	)	)	PUNCT
ejpam-6327	495	44	=	=	SYM
ejpam-6327	495	45	µ(0	µ(0	NOUN
ejpam-6327	495	46	)	)	PUNCT
ejpam-6327	495	47	>	>	X
ejpam-6327	495	48	0	0	X
ejpam-6327	495	49	.	.	PUNCT
ejpam-6327	496	1	thus	thus	ADV
ejpam-6327	496	2	,	,	PUNCT
ejpam-6327	496	3	y	y	PROPN
ejpam-6327	496	4	∗	∗	NOUN
ejpam-6327	496	5	(	(	PUNCT
ejpam-6327	496	6	y	y	PROPN
ejpam-6327	496	7	∗	∗	NOUN
ejpam-6327	496	8	x	x	NOUN
ejpam-6327	496	9	)	)	PUNCT
ejpam-6327	496	10	∼µ	∼µ	NOUN
ejpam-6327	496	11	x	x	NOUN
ejpam-6327	496	12	∗	∗	NOUN
ejpam-6327	496	13	(	(	PUNCT
ejpam-6327	496	14	x	x	SYM
ejpam-6327	496	15	∗	∗	NOUN
ejpam-6327	496	16	(	(	PUNCT
ejpam-6327	496	17	y	y	PROPN
ejpam-6327	496	18	∗	∗	NOUN
ejpam-6327	496	19	(	(	PUNCT
ejpam-6327	496	20	y	y	PROPN
ejpam-6327	496	21	∗	∗	NOUN
ejpam-6327	496	22	x	x	NOUN
ejpam-6327	496	23	)	)	PUNCT
ejpam-6327	496	24	)	)	PUNCT
ejpam-6327	496	25	)	)	PUNCT
ejpam-6327	496	26	.	.	PUNCT
ejpam-6327	497	1	hence	hence	ADV
ejpam-6327	497	2	,	,	PUNCT
ejpam-6327	497	3	µy	µy	X
ejpam-6327	497	4	⊛	⊛	NUM
ejpam-6327	497	5	(	(	PUNCT
ejpam-6327	497	6	µy	µy	NUM
ejpam-6327	497	7	⊛	⊛	NUM
ejpam-6327	497	8	µx	µx	VERB
ejpam-6327	497	9	)	)	PUNCT
ejpam-6327	497	10	=	=	PRON
ejpam-6327	497	11	µx	µx	X
ejpam-6327	497	12	⊛	⊛	PROPN
ejpam-6327	497	13	(	(	PUNCT
ejpam-6327	497	14	µx	µx	X
ejpam-6327	497	15	⊛	⊛	PROPN
ejpam-6327	497	16	(	(	PUNCT
ejpam-6327	497	17	µy	µy	ADP
ejpam-6327	497	18	⊛	⊛	NUM
ejpam-6327	497	19	(	(	PUNCT
ejpam-6327	497	20	µy	µy	X
ejpam-6327	497	21	⊛	⊛	NUM
ejpam-6327	497	22	µx	µx	VERB
ejpam-6327	497	23	)	)	PUNCT
ejpam-6327	497	24	)	)	PUNCT
ejpam-6327	497	25	)	)	PUNCT
ejpam-6327	497	26	.	.	PUNCT
ejpam-6327	498	1	therefore	therefore	ADV
ejpam-6327	498	2	,	,	PUNCT
ejpam-6327	498	3	by	by	ADP
ejpam-6327	498	4	proposition	proposition	NOUN
ejpam-6327	498	5	2	2	NUM
ejpam-6327	498	6	,	,	PUNCT
ejpam-6327	498	7	x/µ	x/µ	PROPN
ejpam-6327	498	8	is	be	AUX
ejpam-6327	498	9	a	a	DET
ejpam-6327	498	10	commutative	commutative	ADJ
ejpam-6327	498	11	ks	ks	NOUN
ejpam-6327	498	12	-	-	PUNCT
ejpam-6327	498	13	semigroup	semigroup	NOUN
ejpam-6327	498	14	.	.	PUNCT
ejpam-6327	499	1	theorem	theorem	PROPN
ejpam-6327	499	2	17	17	NUM
ejpam-6327	499	3	.	.	PUNCT
ejpam-6327	500	1	let	let	VERB
ejpam-6327	500	2	µ	µ	X
ejpam-6327	500	3	be	be	AUX
ejpam-6327	500	4	a	a	DET
ejpam-6327	500	5	non	non	ADJ
ejpam-6327	500	6	-	-	ADJ
ejpam-6327	500	7	zero	zero	ADJ
ejpam-6327	500	8	fuzzy	fuzzy	ADJ
ejpam-6327	500	9	ks	ks	NOUN
ejpam-6327	500	10	-	-	ADJ
ejpam-6327	500	11	p	p	NOUN
ejpam-6327	500	12	-	-	PUNCT
ejpam-6327	500	13	ideal	ideal	NOUN
ejpam-6327	500	14	of	of	ADP
ejpam-6327	500	15	a	a	DET
ejpam-6327	500	16	ks	ks	NOUN
ejpam-6327	500	17	-	-	PUNCT
ejpam-6327	500	18	semigroup	semigroup	NOUN
ejpam-6327	500	19	x	x	PUNCT
ejpam-6327	500	20	satisfying	satisfy	VERB
ejpam-6327	500	21	condition	condition	NOUN
ejpam-6327	500	22	(	(	PUNCT
ejpam-6327	500	23	c	c	NOUN
ejpam-6327	500	24	)	)	PUNCT
ejpam-6327	500	25	.	.	PUNCT
ejpam-6327	501	1	then	then	ADV
ejpam-6327	501	2	x/µ	x/µ	PROPN
ejpam-6327	501	3	is	be	AUX
ejpam-6327	501	4	a	a	DET
ejpam-6327	501	5	positive	positive	ADJ
ejpam-6327	501	6	implicative	implicative	ADJ
ejpam-6327	501	7	ks	ks	NOUN
ejpam-6327	501	8	-	-	PUNCT
ejpam-6327	501	9	semigroup	semigroup	NOUN
ejpam-6327	501	10	.	.	PUNCT
ejpam-6327	502	1	proof	proof	NOUN
ejpam-6327	502	2	.	.	PUNCT
ejpam-6327	503	1	let	let	VERB
ejpam-6327	503	2	µ	µ	X
ejpam-6327	503	3	be	be	AUX
ejpam-6327	503	4	a	a	DET
ejpam-6327	503	5	non	non	ADJ
ejpam-6327	503	6	-	-	ADJ
ejpam-6327	503	7	zero	zero	ADJ
ejpam-6327	503	8	fuzzy	fuzzy	ADJ
ejpam-6327	503	9	ks	ks	NOUN
ejpam-6327	503	10	-	-	ADJ
ejpam-6327	503	11	p	p	NOUN
ejpam-6327	503	12	-	-	PUNCT
ejpam-6327	503	13	ideal	ideal	NOUN
ejpam-6327	503	14	of	of	ADP
ejpam-6327	503	15	a	a	DET
ejpam-6327	503	16	ks	ks	NOUN
ejpam-6327	503	17	-	-	PUNCT
ejpam-6327	503	18	semigroup	semigroup	NOUN
ejpam-6327	503	19	x	x	PUNCT
ejpam-6327	503	20	satisfying	satisfy	VERB
ejpam-6327	503	21	condition	condition	NOUN
ejpam-6327	503	22	(	(	PUNCT
ejpam-6327	503	23	c	c	NOUN
ejpam-6327	503	24	)	)	PUNCT
ejpam-6327	503	25	.	.	PUNCT
ejpam-6327	504	1	let	let	VERB
ejpam-6327	504	2	x	x	PRON
ejpam-6327	504	3	,	,	PUNCT
ejpam-6327	504	4	y	y	PROPN
ejpam-6327	504	5	∈	∈	PROPN
ejpam-6327	504	6	x	x	X
ejpam-6327	504	7	and	and	CCONJ
ejpam-6327	504	8	denote	denote	VERB
ejpam-6327	504	9	u	u	NOUN
ejpam-6327	504	10	=	=	NOUN
ejpam-6327	504	11	x	x	SYM
ejpam-6327	504	12	∗	∗	NOUN
ejpam-6327	504	13	(	(	PUNCT
ejpam-6327	504	14	(	(	PUNCT
ejpam-6327	504	15	x	x	SYM
ejpam-6327	504	16	∗	∗	PROPN
ejpam-6327	504	17	y	y	NOUN
ejpam-6327	504	18	)	)	PUNCT
ejpam-6327	504	19	∗	∗	PROPN
ejpam-6327	504	20	y	y	PROPN
ejpam-6327	504	21	)	)	PUNCT
ejpam-6327	504	22	.	.	PUNCT
ejpam-6327	505	1	then	then	ADV
ejpam-6327	505	2	(	(	PUNCT
ejpam-6327	505	3	u	u	NOUN
ejpam-6327	505	4	∗	∗	PROPN
ejpam-6327	505	5	y	y	NOUN
ejpam-6327	505	6	)	)	PUNCT
ejpam-6327	505	7	∗	∗	NOUN
ejpam-6327	505	8	y	y	NOUN
ejpam-6327	505	9	=	=	SYM
ejpam-6327	505	10	0	0	PROPN
ejpam-6327	505	11	.	.	PUNCT
ejpam-6327	506	1	thus	thus	ADV
ejpam-6327	506	2	,	,	PUNCT
ejpam-6327	506	3	µ((u∗y)∗y	µ((u∗y)∗y	ADV
ejpam-6327	506	4	)	)	PUNCT
ejpam-6327	506	5	=	=	SYM
ejpam-6327	506	6	µ(0	µ(0	NOUN
ejpam-6327	506	7	)	)	PUNCT
ejpam-6327	506	8	>	>	X
ejpam-6327	506	9	0	0	PUNCT
ejpam-6327	506	10	since	since	SCONJ
ejpam-6327	506	11	µ	µ	NOUN
ejpam-6327	506	12	is	be	AUX
ejpam-6327	506	13	non	non	ADJ
ejpam-6327	506	14	-	-	ADJ
ejpam-6327	506	15	constant	constant	ADJ
ejpam-6327	506	16	.	.	PUNCT
ejpam-6327	507	1	by	by	ADP
ejpam-6327	507	2	theorem	theorem	ADJ
ejpam-6327	507	3	3	3	NUM
ejpam-6327	507	4	,	,	PUNCT
ejpam-6327	507	5	µ(u∗y	µ(u∗y	NUM
ejpam-6327	507	6	)	)	PUNCT
ejpam-6327	507	7	=	=	PUNCT
ejpam-6327	507	8	µ((u∗y)∗y	µ((u∗y)∗y	X
ejpam-6327	507	9	)	)	PUNCT
ejpam-6327	507	10	>	>	X
ejpam-6327	507	11	0	0	X
ejpam-6327	507	12	.	.	PUNCT
ejpam-6327	508	1	thus	thus	ADV
ejpam-6327	508	2	,	,	PUNCT
ejpam-6327	508	3	µ((x	µ((x	NOUN
ejpam-6327	508	4	∗	∗	NOUN
ejpam-6327	508	5	y	y	NOUN
ejpam-6327	508	6	)	)	PUNCT
ejpam-6327	508	7	∗	∗	NOUN
ejpam-6327	508	8	(	(	PUNCT
ejpam-6327	508	9	(	(	PUNCT
ejpam-6327	508	10	x	x	SYM
ejpam-6327	508	11	∗	∗	PROPN
ejpam-6327	508	12	y	y	NOUN
ejpam-6327	508	13	)	)	PUNCT
ejpam-6327	508	14	∗	∗	PROPN
ejpam-6327	508	15	y	y	PROPN
ejpam-6327	508	16	)	)	PUNCT
ejpam-6327	508	17	)	)	PUNCT
ejpam-6327	509	1	=	=	PUNCT
ejpam-6327	509	2	µ(u	µ(u	NOUN
ejpam-6327	509	3	∗	∗	PROPN
ejpam-6327	509	4	y	y	PROPN
ejpam-6327	509	5	)	)	PUNCT
ejpam-6327	509	6	>	>	X
ejpam-6327	510	1	0	0	X
ejpam-6327	510	2	.	.	PUNCT
ejpam-6327	511	1	moreover	moreover	ADV
ejpam-6327	511	2	,	,	PUNCT
ejpam-6327	511	3	since	since	SCONJ
ejpam-6327	511	4	(	(	PUNCT
ejpam-6327	511	5	(	(	PUNCT
ejpam-6327	511	6	x	x	SYM
ejpam-6327	511	7	∗	∗	PROPN
ejpam-6327	511	8	y	y	NOUN
ejpam-6327	511	9	)	)	PUNCT
ejpam-6327	511	10	∗	∗	PROPN
ejpam-6327	511	11	y	y	NOUN
ejpam-6327	511	12	)	)	PUNCT
ejpam-6327	511	13	∗	∗	NOUN
ejpam-6327	511	14	(	(	PUNCT
ejpam-6327	511	15	x	x	X
ejpam-6327	511	16	∗	∗	NOUN
ejpam-6327	511	17	y	y	NOUN
ejpam-6327	511	18	)	)	PUNCT
ejpam-6327	511	19	=	=	SYM
ejpam-6327	511	20	0	0	NUM
ejpam-6327	511	21	,	,	PUNCT
ejpam-6327	511	22	µ(((x	µ(((x	PROPN
ejpam-6327	511	23	∗	∗	NOUN
ejpam-6327	511	24	y	y	NOUN
ejpam-6327	511	25	)	)	PUNCT
ejpam-6327	511	26	∗	∗	PROPN
ejpam-6327	511	27	y	y	NOUN
ejpam-6327	511	28	)	)	PUNCT
ejpam-6327	511	29	∗	∗	NOUN
ejpam-6327	511	30	(	(	PUNCT
ejpam-6327	511	31	x	x	X
ejpam-6327	511	32	∗	∗	PROPN
ejpam-6327	511	33	y	y	NOUN
ejpam-6327	511	34	)	)	PUNCT
ejpam-6327	511	35	)	)	PUNCT
ejpam-6327	512	1	=	=	PUNCT
ejpam-6327	512	2	µ(0	µ(0	NOUN
ejpam-6327	512	3	)	)	PUNCT
ejpam-6327	512	4	>	>	X
ejpam-6327	513	1	0	0	X
ejpam-6327	513	2	.	.	PUNCT
ejpam-6327	514	1	this	this	PRON
ejpam-6327	514	2	implies	imply	VERB
ejpam-6327	514	3	that	that	SCONJ
ejpam-6327	514	4	x	x	SYM
ejpam-6327	514	5	∗	∗	NOUN
ejpam-6327	514	6	y	y	NOUN
ejpam-6327	514	7	∼µ	∼µ	PROPN
ejpam-6327	514	8	(	(	PUNCT
ejpam-6327	514	9	x	x	PROPN
ejpam-6327	514	10	∗	∗	PROPN
ejpam-6327	514	11	y	y	NOUN
ejpam-6327	514	12	)	)	PUNCT
ejpam-6327	514	13	∗	∗	NOUN
ejpam-6327	514	14	y.	y.	PROPN
ejpam-6327	514	15	hence	hence	ADV
ejpam-6327	514	16	,	,	PUNCT
ejpam-6327	514	17	µx	µx	VERB
ejpam-6327	514	18	⊛	⊛	ADV
ejpam-6327	514	19	µy	µy	ADV
ejpam-6327	514	20	=	=	PUNCT
ejpam-6327	514	21	µx∗y	µx∗y	PUNCT
ejpam-6327	514	22	=	=	PUNCT
ejpam-6327	514	23	µ(x∗y)∗y	µ(x∗y)∗y	NOUN
ejpam-6327	514	24	=	=	PUNCT
ejpam-6327	514	25	(	(	PUNCT
ejpam-6327	514	26	µx	µx	X
ejpam-6327	514	27	⊛	⊛	ADJ
ejpam-6327	514	28	µy)⊛	µy)⊛	PROPN
ejpam-6327	514	29	µy	µy	PROPN
ejpam-6327	514	30	.	.	PUNCT
ejpam-6327	515	1	therefore	therefore	ADV
ejpam-6327	515	2	,	,	PUNCT
ejpam-6327	515	3	x/µ	x/µ	PROPN
ejpam-6327	515	4	is	be	AUX
ejpam-6327	515	5	a	a	DET
ejpam-6327	515	6	positive	positive	ADJ
ejpam-6327	515	7	implicative	implicative	ADJ
ejpam-6327	515	8	ks	ks	NOUN
ejpam-6327	515	9	-	-	PUNCT
ejpam-6327	515	10	semigroup	semigroup	NOUN
ejpam-6327	515	11	.	.	PUNCT
ejpam-6327	516	1	h.	h.	PROPN
ejpam-6327	516	2	sarapuddin	sarapuddin	PROPN
ejpam-6327	516	3	,	,	PUNCT
ejpam-6327	516	4	j.	j.	PROPN
ejpam-6327	516	5	vilela	vilela	PROPN
ejpam-6327	516	6	/	/	SYM
ejpam-6327	516	7	eur	eur	PROPN
ejpam-6327	516	8	.	.	PUNCT
ejpam-6327	517	1	j.	j.	PROPN
ejpam-6327	517	2	pure	pure	PROPN
ejpam-6327	517	3	appl	appl	PROPN
ejpam-6327	517	4	.	.	PROPN
ejpam-6327	517	5	math	math	PROPN
ejpam-6327	517	6	,	,	PUNCT
ejpam-6327	517	7	18	18	NUM
ejpam-6327	517	8	(	(	PUNCT
ejpam-6327	517	9	3	3	NUM
ejpam-6327	517	10	)	)	PUNCT
ejpam-6327	517	11	(	(	PUNCT
ejpam-6327	517	12	2025	2025	NUM
ejpam-6327	517	13	)	)	PUNCT
ejpam-6327	517	14	,	,	PUNCT
ejpam-6327	517	15	6327	6327	NUM
ejpam-6327	517	16	16	16	NUM
ejpam-6327	517	17	of	of	ADP
ejpam-6327	517	18	23	23	NUM
ejpam-6327	517	19	theorem	theorem	NOUN
ejpam-6327	517	20	18	18	NUM
ejpam-6327	517	21	.	.	PUNCT
ejpam-6327	518	1	let	let	VERB
ejpam-6327	518	2	µ	µ	X
ejpam-6327	518	3	be	be	AUX
ejpam-6327	518	4	a	a	DET
ejpam-6327	518	5	non	non	ADJ
ejpam-6327	518	6	-	-	ADJ
ejpam-6327	518	7	zero	zero	ADJ
ejpam-6327	518	8	fuzzy	fuzzy	ADJ
ejpam-6327	518	9	implicative	implicative	ADJ
ejpam-6327	518	10	ks	ks	NOUN
ejpam-6327	518	11	-	-	PUNCT
ejpam-6327	518	12	ideal	ideal	NOUN
ejpam-6327	518	13	of	of	ADP
ejpam-6327	518	14	a	a	DET
ejpam-6327	518	15	ks	ks	NOUN
ejpam-6327	518	16	-	-	PUNCT
ejpam-6327	518	17	semigroup	semigroup	NOUN
ejpam-6327	518	18	x	x	PUNCT
ejpam-6327	518	19	satisfying	satisfy	VERB
ejpam-6327	518	20	condition	condition	NOUN
ejpam-6327	518	21	(	(	PUNCT
ejpam-6327	518	22	c	c	NOUN
ejpam-6327	518	23	)	)	PUNCT
ejpam-6327	518	24	.	.	PUNCT
ejpam-6327	519	1	then	then	ADV
ejpam-6327	519	2	x/µ	x/µ	PROPN
ejpam-6327	519	3	is	be	AUX
ejpam-6327	519	4	a	a	DET
ejpam-6327	519	5	implicative	implicative	ADJ
ejpam-6327	519	6	ks	ks	NOUN
ejpam-6327	519	7	-	-	PUNCT
ejpam-6327	519	8	semigroup	semigroup	NOUN
ejpam-6327	519	9	.	.	PUNCT
ejpam-6327	520	1	proof	proof	NOUN
ejpam-6327	520	2	.	.	PUNCT
ejpam-6327	521	1	let	let	VERB
ejpam-6327	521	2	µ	µ	X
ejpam-6327	521	3	be	be	AUX
ejpam-6327	521	4	a	a	DET
ejpam-6327	521	5	non	non	ADJ
ejpam-6327	521	6	-	-	ADJ
ejpam-6327	521	7	zero	zero	ADJ
ejpam-6327	521	8	fuzzy	fuzzy	ADJ
ejpam-6327	521	9	implicative	implicative	ADJ
ejpam-6327	521	10	ks	ks	NOUN
ejpam-6327	521	11	-	-	PUNCT
ejpam-6327	521	12	ideal	ideal	NOUN
ejpam-6327	521	13	of	of	ADP
ejpam-6327	521	14	a	a	DET
ejpam-6327	521	15	ks	ks	NOUN
ejpam-6327	521	16	-	-	PUNCT
ejpam-6327	521	17	semigroup	semigroup	NOUN
ejpam-6327	521	18	x	x	PUNCT
ejpam-6327	521	19	satisfying	satisfy	VERB
ejpam-6327	521	20	condition	condition	NOUN
ejpam-6327	521	21	(	(	PUNCT
ejpam-6327	521	22	c	c	NOUN
ejpam-6327	521	23	)	)	PUNCT
ejpam-6327	521	24	.	.	PUNCT
ejpam-6327	522	1	then	then	ADV
ejpam-6327	522	2	by	by	ADP
ejpam-6327	522	3	theorem	theorem	NOUN
ejpam-6327	522	4	10	10	NUM
ejpam-6327	522	5	,	,	PUNCT
ejpam-6327	522	6	µ	µ	PRON
ejpam-6327	522	7	is	be	AUX
ejpam-6327	522	8	both	both	PRON
ejpam-6327	522	9	a	a	DET
ejpam-6327	522	10	non	non	ADJ
ejpam-6327	522	11	-	-	ADJ
ejpam-6327	522	12	zero	zero	ADJ
ejpam-6327	522	13	fuzzy	fuzzy	ADJ
ejpam-6327	522	14	commutative	commutative	ADJ
ejpam-6327	522	15	ks	ks	NOUN
ejpam-6327	522	16	-	-	PUNCT
ejpam-6327	522	17	ideal	ideal	ADJ
ejpam-6327	522	18	and	and	CCONJ
ejpam-6327	522	19	fuzzy	fuzzy	ADJ
ejpam-6327	522	20	ks	ks	NOUN
ejpam-6327	522	21	-	-	ADJ
ejpam-6327	522	22	p	p	ADJ
ejpam-6327	522	23	-	-	PUNCT
ejpam-6327	522	24	ideal	ideal	NOUN
ejpam-6327	522	25	satisfying	satisfy	VERB
ejpam-6327	522	26	condition	condition	NOUN
ejpam-6327	522	27	(	(	PUNCT
ejpam-6327	522	28	c	c	NOUN
ejpam-6327	522	29	)	)	PUNCT
ejpam-6327	522	30	.	.	PUNCT
ejpam-6327	523	1	thus	thus	ADV
ejpam-6327	523	2	,	,	PUNCT
ejpam-6327	523	3	by	by	ADP
ejpam-6327	523	4	theorems	theorem	NOUN
ejpam-6327	523	5	16	16	NUM
ejpam-6327	523	6	and	and	CCONJ
ejpam-6327	523	7	17	17	NUM
ejpam-6327	523	8	,	,	PUNCT
ejpam-6327	523	9	x/µ	x/µ	PROPN
ejpam-6327	523	10	is	be	AUX
ejpam-6327	523	11	both	both	CCONJ
ejpam-6327	523	12	commutative	commutative	ADJ
ejpam-6327	523	13	and	and	CCONJ
ejpam-6327	523	14	positive	positive	ADJ
ejpam-6327	523	15	implicative	implicative	ADJ
ejpam-6327	523	16	ks	ks	NOUN
ejpam-6327	523	17	-	-	PUNCT
ejpam-6327	523	18	semigroup	semigroup	NOUN
ejpam-6327	523	19	.	.	PUNCT
ejpam-6327	524	1	therefore	therefore	ADV
ejpam-6327	524	2	,	,	PUNCT
ejpam-6327	524	3	by	by	ADP
ejpam-6327	524	4	definition	definition	NOUN
ejpam-6327	524	5	5(iii	5(iii	NUM
ejpam-6327	524	6	)	)	PUNCT
ejpam-6327	524	7	,	,	PUNCT
ejpam-6327	524	8	x/µ	x/µ	PROPN
ejpam-6327	524	9	is	be	AUX
ejpam-6327	524	10	implicative	implicative	ADJ
ejpam-6327	524	11	.	.	PUNCT
ejpam-6327	525	1	5	5	X
ejpam-6327	525	2	.	.	NOUN
ejpam-6327	525	3	homomorphic	homomorphic	ADJ
ejpam-6327	525	4	properties	property	NOUN
ejpam-6327	525	5	of	of	ADP
ejpam-6327	525	6	the	the	DET
ejpam-6327	525	7	generalized	generalized	ADJ
ejpam-6327	525	8	quotient	quotient	NOUN
ejpam-6327	525	9	via	via	ADP
ejpam-6327	525	10	fuzzy	fuzzy	ADJ
ejpam-6327	525	11	ks	ks	NOUN
ejpam-6327	525	12	-	-	NOUN
ejpam-6327	525	13	ideals	ideal	NOUN
ejpam-6327	525	14	in	in	ADP
ejpam-6327	525	15	this	this	DET
ejpam-6327	525	16	section	section	NOUN
ejpam-6327	525	17	,	,	PUNCT
ejpam-6327	525	18	we	we	PRON
ejpam-6327	525	19	investigate	investigate	VERB
ejpam-6327	525	20	some	some	DET
ejpam-6327	525	21	results	result	NOUN
ejpam-6327	525	22	on	on	ADP
ejpam-6327	525	23	homomorphism	homomorphism	NOUN
ejpam-6327	525	24	of	of	ADP
ejpam-6327	525	25	ks	ks	NOUN
ejpam-6327	525	26	-	-	PUNCT
ejpam-6327	525	27	semigroup	semigroup	NOUN
ejpam-6327	525	28	.	.	PUNCT
ejpam-6327	526	1	lemma	lemma	PROPN
ejpam-6327	526	2	2	2	X
ejpam-6327	526	3	.	.	PUNCT
ejpam-6327	527	1	if	if	SCONJ
ejpam-6327	527	2	µ	µ	NOUN
ejpam-6327	527	3	is	be	AUX
ejpam-6327	527	4	a	a	DET
ejpam-6327	527	5	non	non	ADJ
ejpam-6327	527	6	-	-	ADJ
ejpam-6327	527	7	zero	zero	ADJ
ejpam-6327	527	8	fuzzy	fuzzy	ADJ
ejpam-6327	527	9	ks	ks	NOUN
ejpam-6327	527	10	-	-	NOUN
ejpam-6327	527	11	ideal	ideal	NOUN
ejpam-6327	527	12	of	of	ADP
ejpam-6327	527	13	a	a	DET
ejpam-6327	527	14	ks	ks	NOUN
ejpam-6327	527	15	-	-	PUNCT
ejpam-6327	527	16	semigroup	semigroup	NOUN
ejpam-6327	527	17	x	x	PUNCT
ejpam-6327	527	18	satisfying	satisfy	VERB
ejpam-6327	527	19	condition	condition	NOUN
ejpam-6327	527	20	(	(	PUNCT
ejpam-6327	527	21	c	c	NOUN
ejpam-6327	527	22	)	)	PUNCT
ejpam-6327	527	23	,	,	PUNCT
ejpam-6327	527	24	then	then	ADV
ejpam-6327	527	25	the	the	DET
ejpam-6327	527	26	set	set	NOUN
ejpam-6327	527	27	xµ	xµ	X
ejpam-6327	528	1	=	=	SYM
ejpam-6327	529	1	{	{	PUNCT
ejpam-6327	529	2	x	x	PUNCT
ejpam-6327	529	3	∈	∈	NOUN
ejpam-6327	529	4	x	x	X
ejpam-6327	529	5	:	:	PUNCT
ejpam-6327	529	6	µ(x	µ(x	X
ejpam-6327	529	7	)	)	PUNCT
ejpam-6327	529	8	>	>	X
ejpam-6327	529	9	0	0	NUM
ejpam-6327	529	10	}	}	PUNCT
ejpam-6327	529	11	is	be	AUX
ejpam-6327	529	12	an	an	DET
ejpam-6327	529	13	ideal	ideal	NOUN
ejpam-6327	529	14	of	of	ADP
ejpam-6327	529	15	x.	x.	NOUN
ejpam-6327	529	16	proof	proof	NOUN
ejpam-6327	529	17	.	.	PUNCT
ejpam-6327	530	1	let	let	VERB
ejpam-6327	530	2	µ	µ	X
ejpam-6327	530	3	be	be	AUX
ejpam-6327	530	4	a	a	DET
ejpam-6327	530	5	non	non	ADJ
ejpam-6327	530	6	-	-	ADJ
ejpam-6327	530	7	zero	zero	ADJ
ejpam-6327	530	8	fuzzy	fuzzy	ADJ
ejpam-6327	530	9	ks	ks	NOUN
ejpam-6327	530	10	-	-	NOUN
ejpam-6327	530	11	ideal	ideal	NOUN
ejpam-6327	530	12	of	of	ADP
ejpam-6327	530	13	a	a	DET
ejpam-6327	530	14	ks	ks	NOUN
ejpam-6327	530	15	-	-	PUNCT
ejpam-6327	530	16	semigroup	semigroup	NOUN
ejpam-6327	530	17	x	x	PUNCT
ejpam-6327	530	18	satisfying	satisfy	VERB
ejpam-6327	530	19	condition	condition	NOUN
ejpam-6327	530	20	(	(	PUNCT
ejpam-6327	530	21	c	c	NOUN
ejpam-6327	530	22	)	)	PUNCT
ejpam-6327	530	23	and	and	CCONJ
ejpam-6327	530	24	let	let	VERB
ejpam-6327	530	25	xµ	xµ	PROPN
ejpam-6327	530	26	=	=	PRON
ejpam-6327	530	27	{	{	PUNCT
ejpam-6327	530	28	x	x	PUNCT
ejpam-6327	530	29	∈	∈	NOUN
ejpam-6327	530	30	x	x	X
ejpam-6327	530	31	:	:	PUNCT
ejpam-6327	530	32	µ(x	µ(x	X
ejpam-6327	530	33	)	)	PUNCT
ejpam-6327	530	34	>	>	X
ejpam-6327	530	35	0	0	NUM
ejpam-6327	530	36	}	}	PUNCT
ejpam-6327	530	37	.	.	PUNCT
ejpam-6327	531	1	let	let	VERB
ejpam-6327	531	2	x	x	PUNCT
ejpam-6327	531	3	∈	∈	PROPN
ejpam-6327	531	4	x	x	X
ejpam-6327	531	5	and	and	CCONJ
ejpam-6327	531	6	a	a	DET
ejpam-6327	531	7	∈	∈	NOUN
ejpam-6327	531	8	xµ.	xµ.	NOUN
ejpam-6327	531	9	then	then	ADV
ejpam-6327	531	10	µ(a	µ(a	PROPN
ejpam-6327	531	11	)	)	PUNCT
ejpam-6327	531	12	>	>	X
ejpam-6327	532	1	0	0	X
ejpam-6327	532	2	.	.	PUNCT
ejpam-6327	533	1	thus	thus	ADV
ejpam-6327	533	2	,	,	PUNCT
ejpam-6327	533	3	µ(xa	µ(xa	PROPN
ejpam-6327	533	4	)	)	PUNCT
ejpam-6327	533	5	≥	≥	NOUN
ejpam-6327	533	6	µ(a	µ(a	NOUN
ejpam-6327	533	7	)	)	PUNCT
ejpam-6327	534	1	>	>	SYM
ejpam-6327	534	2	0	0	NUM
ejpam-6327	534	3	and	and	CCONJ
ejpam-6327	534	4	µ(ax	µ(ax	NOUN
ejpam-6327	534	5	)	)	PUNCT
ejpam-6327	534	6	≥	≥	NOUN
ejpam-6327	534	7	µ(a	µ(a	NOUN
ejpam-6327	534	8	)	)	PUNCT
ejpam-6327	535	1	>	>	X
ejpam-6327	536	1	0	0	X
ejpam-6327	536	2	.	.	PUNCT
ejpam-6327	537	1	this	this	PRON
ejpam-6327	537	2	implies	imply	VERB
ejpam-6327	537	3	that	that	SCONJ
ejpam-6327	537	4	xa	xa	PROPN
ejpam-6327	537	5	,	,	PUNCT
ejpam-6327	537	6	ax	ax	NOUN
ejpam-6327	537	7	∈	∈	PROPN
ejpam-6327	537	8	xµ.	xµ.	VERB
ejpam-6327	537	9	thus	thus	ADV
ejpam-6327	537	10	,	,	PUNCT
ejpam-6327	537	11	xµ	xµ	PROPN
ejpam-6327	537	12	is	be	AUX
ejpam-6327	537	13	stable	stable	ADJ
ejpam-6327	537	14	.	.	PUNCT
ejpam-6327	538	1	now	now	ADV
ejpam-6327	538	2	,	,	PUNCT
ejpam-6327	538	3	suppose	suppose	VERB
ejpam-6327	538	4	x	x	PUNCT
ejpam-6327	538	5	∗	∗	VERB
ejpam-6327	538	6	y	y	PROPN
ejpam-6327	538	7	∈	∈	PROPN
ejpam-6327	538	8	xµ	xµ	X
ejpam-6327	538	9	and	and	CCONJ
ejpam-6327	538	10	y	y	PROPN
ejpam-6327	538	11	∈	∈	PROPN
ejpam-6327	538	12	xµ.	xµ.	PROPN
ejpam-6327	538	13	then	then	ADV
ejpam-6327	538	14	µ(x	µ(x	ADJ
ejpam-6327	538	15	∗	∗	NOUN
ejpam-6327	538	16	y	y	NOUN
ejpam-6327	538	17	)	)	PUNCT
ejpam-6327	538	18	=	=	SYM
ejpam-6327	538	19	µ(y	µ(y	PROPN
ejpam-6327	538	20	)	)	PUNCT
ejpam-6327	538	21	>	>	X
ejpam-6327	539	1	0	0	X
ejpam-6327	539	2	.	.	PUNCT
ejpam-6327	540	1	since	since	SCONJ
ejpam-6327	540	2	µ	µ	NOUN
ejpam-6327	540	3	is	be	AUX
ejpam-6327	540	4	a	a	DET
ejpam-6327	540	5	fuzzy	fuzzy	ADJ
ejpam-6327	540	6	ks	ks	NOUN
ejpam-6327	540	7	-	-	NOUN
ejpam-6327	540	8	ideal	ideal	NOUN
ejpam-6327	540	9	of	of	ADP
ejpam-6327	540	10	x	x	PRON
ejpam-6327	540	11	,	,	PUNCT
ejpam-6327	540	12	µ(x	µ(x	NOUN
ejpam-6327	540	13	)	)	PUNCT
ejpam-6327	540	14	≥	≥	NOUN
ejpam-6327	540	15	min{µ(x	min{µ(x	PROPN
ejpam-6327	540	16	∗	∗	X
ejpam-6327	540	17	y	y	NOUN
ejpam-6327	540	18	)	)	PUNCT
ejpam-6327	540	19	,	,	PUNCT
ejpam-6327	540	20	µ(y	µ(y	PROPN
ejpam-6327	540	21	)	)	PUNCT
ejpam-6327	540	22	}	}	PUNCT
ejpam-6327	540	23	>	>	X
ejpam-6327	540	24	0	0	X
ejpam-6327	540	25	.	.	PUNCT
ejpam-6327	541	1	thus	thus	ADV
ejpam-6327	541	2	,	,	PUNCT
ejpam-6327	541	3	x	x	SYM
ejpam-6327	541	4	∈	∈	PROPN
ejpam-6327	541	5	xµ.	xµ.	X
ejpam-6327	541	6	therefore	therefore	ADV
ejpam-6327	541	7	,	,	PUNCT
ejpam-6327	541	8	xµ	xµ	PROPN
ejpam-6327	541	9	is	be	AUX
ejpam-6327	541	10	an	an	DET
ejpam-6327	541	11	ideal	ideal	NOUN
ejpam-6327	541	12	of	of	ADP
ejpam-6327	541	13	x.	x.	NOUN
ejpam-6327	541	14	definition	definition	NOUN
ejpam-6327	541	15	15	15	NUM
ejpam-6327	541	16	.	.	PUNCT
ejpam-6327	542	1	[	[	X
ejpam-6327	542	2	5	5	X
ejpam-6327	542	3	]	]	PUNCT
ejpam-6327	542	4	let	let	VERB
ejpam-6327	542	5	f	f	PRON
ejpam-6327	542	6	:	:	PUNCT
ejpam-6327	542	7	x	x	X
ejpam-6327	542	8	→	→	SYM
ejpam-6327	542	9	y	y	X
ejpam-6327	542	10	be	be	AUX
ejpam-6327	542	11	a	a	DET
ejpam-6327	542	12	homomorphism	homomorphism	NOUN
ejpam-6327	542	13	of	of	ADP
ejpam-6327	542	14	ks	ks	NOUN
ejpam-6327	542	15	-	-	PUNCT
ejpam-6327	542	16	semigroup	semigroup	NOUN
ejpam-6327	542	17	.	.	PUNCT
ejpam-6327	543	1	then	then	ADV
ejpam-6327	543	2	the	the	DET
ejpam-6327	543	3	kernel	kernel	NOUN
ejpam-6327	543	4	of	of	ADP
ejpam-6327	543	5	f	f	PROPN
ejpam-6327	543	6	is	be	AUX
ejpam-6327	543	7	denoted	denote	VERB
ejpam-6327	543	8	and	and	CCONJ
ejpam-6327	543	9	defined	define	VERB
ejpam-6327	543	10	as	as	ADP
ejpam-6327	543	11	ker	ker	PROPN
ejpam-6327	543	12	f	f	PROPN
ejpam-6327	544	1	=	=	PRON
ejpam-6327	544	2	{	{	PUNCT
ejpam-6327	544	3	x	x	SYM
ejpam-6327	544	4	∈	∈	PROPN
ejpam-6327	544	5	x	x	X
ejpam-6327	544	6	:	:	PUNCT
ejpam-6327	544	7	f(x	f(x	PROPN
ejpam-6327	544	8	)	)	PUNCT
ejpam-6327	544	9	=	=	PUNCT
ejpam-6327	545	1	0	0	NUM
ejpam-6327	545	2	}	}	PUNCT
ejpam-6327	545	3	.	.	PUNCT
ejpam-6327	546	1	theorem	theorem	NOUN
ejpam-6327	546	2	19	19	NUM
ejpam-6327	546	3	.	.	PUNCT
ejpam-6327	547	1	let	let	VERB
ejpam-6327	547	2	µ	µ	X
ejpam-6327	547	3	be	be	AUX
ejpam-6327	547	4	a	a	DET
ejpam-6327	547	5	non	non	ADJ
ejpam-6327	547	6	-	-	ADJ
ejpam-6327	547	7	zero	zero	ADJ
ejpam-6327	547	8	fuzzy	fuzzy	ADJ
ejpam-6327	547	9	ks	ks	NOUN
ejpam-6327	547	10	-	-	NOUN
ejpam-6327	547	11	ideal	ideal	NOUN
ejpam-6327	547	12	of	of	ADP
ejpam-6327	547	13	a	a	DET
ejpam-6327	547	14	ks	ks	NOUN
ejpam-6327	547	15	-	-	PUNCT
ejpam-6327	547	16	semigroup	semigroup	NOUN
ejpam-6327	547	17	x	x	PUNCT
ejpam-6327	547	18	satisfying	satisfy	VERB
ejpam-6327	547	19	condition	condition	NOUN
ejpam-6327	547	20	(	(	PUNCT
ejpam-6327	547	21	c	c	NOUN
ejpam-6327	547	22	)	)	PUNCT
ejpam-6327	547	23	.	.	PUNCT
ejpam-6327	548	1	then	then	ADV
ejpam-6327	548	2	the	the	DET
ejpam-6327	548	3	mapping	mapping	NOUN
ejpam-6327	548	4	γ	γ	X
ejpam-6327	548	5	:	:	PUNCT
ejpam-6327	548	6	x	x	SYM
ejpam-6327	548	7	→	→	SYM
ejpam-6327	548	8	x/µ	x/µ	PROPN
ejpam-6327	548	9	given	give	VERB
ejpam-6327	548	10	by	by	ADP
ejpam-6327	548	11	γ(x	γ(x	NOUN
ejpam-6327	548	12	)	)	PUNCT
ejpam-6327	548	13	=	=	PRON
ejpam-6327	549	1	µx	µx	X
ejpam-6327	549	2	is	be	AUX
ejpam-6327	549	3	an	an	DET
ejpam-6327	549	4	epimorphism	epimorphism	NOUN
ejpam-6327	549	5	with	with	ADP
ejpam-6327	549	6	ker	ker	PROPN
ejpam-6327	549	7	γ	γ	X
ejpam-6327	549	8	=	=	SYM
ejpam-6327	549	9	xµ.	xµ.	PROPN
ejpam-6327	549	10	recall	recall	VERB
ejpam-6327	549	11	that	that	SCONJ
ejpam-6327	549	12	in	in	ADP
ejpam-6327	549	13	[	[	X
ejpam-6327	549	14	7	7	NUM
ejpam-6327	549	15	]	]	PUNCT
ejpam-6327	549	16	,	,	PUNCT
ejpam-6327	549	17	if	if	SCONJ
ejpam-6327	549	18	f	f	X
ejpam-6327	549	19	:	:	PUNCT
ejpam-6327	549	20	x	x	X
ejpam-6327	549	21	→	→	SYM
ejpam-6327	549	22	y	y	PROPN
ejpam-6327	549	23	is	be	AUX
ejpam-6327	549	24	a	a	DET
ejpam-6327	549	25	mapping	mapping	NOUN
ejpam-6327	549	26	of	of	ADP
ejpam-6327	549	27	ks	ks	NOUN
ejpam-6327	549	28	-	-	PUNCT
ejpam-6327	549	29	semigroup	semigroup	PROPN
ejpam-6327	549	30	and	and	CCONJ
ejpam-6327	549	31	ν	ν	PROPN
ejpam-6327	549	32	is	be	AUX
ejpam-6327	549	33	a	a	DET
ejpam-6327	549	34	fuzzy	fuzzy	ADJ
ejpam-6327	549	35	set	set	NOUN
ejpam-6327	549	36	on	on	ADP
ejpam-6327	549	37	y	y	PROPN
ejpam-6327	549	38	,	,	PUNCT
ejpam-6327	549	39	then	then	ADV
ejpam-6327	549	40	the	the	DET
ejpam-6327	549	41	fuzzy	fuzzy	ADJ
ejpam-6327	549	42	set	set	NOUN
ejpam-6327	549	43	νf	νf	NOUN
ejpam-6327	549	44	=	=	PUNCT
ejpam-6327	550	1	ν	ν	NOUN
ejpam-6327	550	2	◦	◦	NOUN
ejpam-6327	550	3	f	f	X
ejpam-6327	550	4	on	on	ADP
ejpam-6327	550	5	x	x	VERB
ejpam-6327	550	6	is	be	AUX
ejpam-6327	550	7	said	say	VERB
ejpam-6327	550	8	to	to	PART
ejpam-6327	550	9	be	be	AUX
ejpam-6327	550	10	the	the	DET
ejpam-6327	550	11	preimage	preimage	NOUN
ejpam-6327	550	12	of	of	ADP
ejpam-6327	550	13	ν	ν	NOUN
ejpam-6327	550	14	under	under	ADP
ejpam-6327	550	15	f	f	PROPN
ejpam-6327	550	16	.	.	PUNCT
ejpam-6327	551	1	if	if	SCONJ
ejpam-6327	551	2	f	f	PROPN
ejpam-6327	551	3	is	be	AUX
ejpam-6327	551	4	a	a	DET
ejpam-6327	551	5	homomorphism	homomorphism	NOUN
ejpam-6327	551	6	and	and	CCONJ
ejpam-6327	551	7	ν	ν	NOUN
ejpam-6327	551	8	is	be	AUX
ejpam-6327	551	9	a	a	DET
ejpam-6327	551	10	fuzzy	fuzzy	ADJ
ejpam-6327	551	11	ks	ks	NOUN
ejpam-6327	551	12	-	-	NOUN
ejpam-6327	551	13	ideal	ideal	NOUN
ejpam-6327	551	14	of	of	ADP
ejpam-6327	551	15	y	y	PROPN
ejpam-6327	551	16	,	,	PUNCT
ejpam-6327	551	17	then	then	ADV
ejpam-6327	551	18	νf	νf	NOUN
ejpam-6327	551	19	is	be	AUX
ejpam-6327	551	20	a	a	DET
ejpam-6327	551	21	fuzzy	fuzzy	ADJ
ejpam-6327	551	22	ks	ks	NOUN
ejpam-6327	551	23	-	-	NOUN
ejpam-6327	551	24	ideal	ideal	NOUN
ejpam-6327	551	25	of	of	ADP
ejpam-6327	551	26	x.	x.	NOUN
ejpam-6327	551	27	furthermore	furthermore	ADV
ejpam-6327	551	28	,	,	PUNCT
ejpam-6327	551	29	if	if	SCONJ
ejpam-6327	551	30	f	f	PROPN
ejpam-6327	551	31	is	be	AUX
ejpam-6327	551	32	an	an	DET
ejpam-6327	551	33	epimorphism	epimorphism	NOUN
ejpam-6327	551	34	and	and	CCONJ
ejpam-6327	551	35	νf	νf	NOUN
ejpam-6327	551	36	is	be	AUX
ejpam-6327	551	37	a	a	DET
ejpam-6327	551	38	fuzzy	fuzzy	ADJ
ejpam-6327	551	39	ks	ks	NOUN
ejpam-6327	551	40	-	-	NOUN
ejpam-6327	551	41	ideal	ideal	NOUN
ejpam-6327	551	42	of	of	ADP
ejpam-6327	551	43	x	x	PRON
ejpam-6327	551	44	,	,	PUNCT
ejpam-6327	551	45	then	then	ADV
ejpam-6327	551	46	ν	ν	PROPN
ejpam-6327	551	47	is	be	AUX
ejpam-6327	551	48	a	a	DET
ejpam-6327	551	49	fuzzy	fuzzy	ADJ
ejpam-6327	551	50	ks	ks	NOUN
ejpam-6327	551	51	-	-	NOUN
ejpam-6327	551	52	ideal	ideal	NOUN
ejpam-6327	551	53	of	of	ADP
ejpam-6327	551	54	y	y	PROPN
ejpam-6327	551	55	.	.	PUNCT
ejpam-6327	552	1	lemma	lemma	PROPN
ejpam-6327	552	2	3	3	X
ejpam-6327	552	3	.	.	PUNCT
ejpam-6327	553	1	let	let	VERB
ejpam-6327	553	2	f	f	NOUN
ejpam-6327	553	3	:	:	PUNCT
ejpam-6327	553	4	x	x	X
ejpam-6327	553	5	→	→	SYM
ejpam-6327	553	6	y	y	PROPN
ejpam-6327	553	7	be	be	AUX
ejpam-6327	553	8	a	a	DET
ejpam-6327	553	9	ks	ks	NOUN
ejpam-6327	553	10	-	-	PUNCT
ejpam-6327	553	11	semigroup	semigroup	ADJ
ejpam-6327	553	12	homomorphism	homomorphism	NOUN
ejpam-6327	553	13	.	.	PUNCT
ejpam-6327	554	1	if	if	SCONJ
ejpam-6327	554	2	µ	µ	NOUN
ejpam-6327	554	3	is	be	AUX
ejpam-6327	554	4	a	a	DET
ejpam-6327	554	5	non	non	ADJ
ejpam-6327	554	6	-	-	ADJ
ejpam-6327	554	7	zero	zero	ADJ
ejpam-6327	554	8	fuzzy	fuzzy	ADJ
ejpam-6327	554	9	ks	ks	NOUN
ejpam-6327	554	10	-	-	NOUN
ejpam-6327	554	11	ideal	ideal	NOUN
ejpam-6327	554	12	of	of	ADP
ejpam-6327	554	13	y	y	PROPN
ejpam-6327	554	14	satisfying	satisfy	VERB
ejpam-6327	554	15	condition	condition	NOUN
ejpam-6327	554	16	(	(	PUNCT
ejpam-6327	554	17	c	c	NOUN
ejpam-6327	554	18	)	)	PUNCT
ejpam-6327	554	19	,	,	PUNCT
ejpam-6327	554	20	then	then	ADV
ejpam-6327	554	21	µf	µf	PROPN
ejpam-6327	554	22	is	be	AUX
ejpam-6327	554	23	a	a	DET
ejpam-6327	554	24	non	non	ADJ
ejpam-6327	554	25	-	-	ADJ
ejpam-6327	554	26	zero	zero	ADJ
ejpam-6327	554	27	fuzzy	fuzzy	ADJ
ejpam-6327	554	28	ks	ks	NOUN
ejpam-6327	554	29	-	-	NOUN
ejpam-6327	554	30	ideal	ideal	NOUN
ejpam-6327	554	31	of	of	ADP
ejpam-6327	554	32	x	x	SYM
ejpam-6327	554	33	satisfying	satisfy	VERB
ejpam-6327	554	34	condition	condition	NOUN
ejpam-6327	554	35	(	(	PUNCT
ejpam-6327	554	36	c	c	NOUN
ejpam-6327	554	37	)	)	PUNCT
ejpam-6327	554	38	.	.	PUNCT
ejpam-6327	555	1	proof	proof	NOUN
ejpam-6327	555	2	.	.	PUNCT
ejpam-6327	556	1	let	let	VERB
ejpam-6327	556	2	f	f	NOUN
ejpam-6327	556	3	:	:	PUNCT
ejpam-6327	556	4	x	x	X
ejpam-6327	556	5	→	→	SYM
ejpam-6327	556	6	y	y	PROPN
ejpam-6327	556	7	be	be	AUX
ejpam-6327	556	8	a	a	DET
ejpam-6327	556	9	ks	ks	NOUN
ejpam-6327	556	10	-	-	PUNCT
ejpam-6327	556	11	semigroup	semigroup	NOUN
ejpam-6327	556	12	homomorphism	homomorphism	NOUN
ejpam-6327	556	13	and	and	CCONJ
ejpam-6327	556	14	µ	µ	DET
ejpam-6327	556	15	a	a	DET
ejpam-6327	556	16	non	non	ADJ
ejpam-6327	556	17	-	-	ADJ
ejpam-6327	556	18	zero	zero	ADJ
ejpam-6327	556	19	fuzzy	fuzzy	ADJ
ejpam-6327	556	20	ksideal	ksideal	NOUN
ejpam-6327	556	21	of	of	ADP
ejpam-6327	556	22	y	y	NOUN
ejpam-6327	556	23	satisfying	satisfy	VERB
ejpam-6327	556	24	condition	condition	NOUN
ejpam-6327	556	25	(	(	PUNCT
ejpam-6327	556	26	c	c	NOUN
ejpam-6327	556	27	)	)	PUNCT
ejpam-6327	556	28	.	.	PUNCT
ejpam-6327	557	1	then	then	ADV
ejpam-6327	557	2	µf	µf	PROPN
ejpam-6327	557	3	is	be	AUX
ejpam-6327	557	4	a	a	DET
ejpam-6327	557	5	fuzzy	fuzzy	ADJ
ejpam-6327	557	6	ks	ks	NOUN
ejpam-6327	557	7	-	-	NOUN
ejpam-6327	557	8	ideal	ideal	NOUN
ejpam-6327	557	9	of	of	ADP
ejpam-6327	557	10	x.	x.	NOUN
ejpam-6327	557	11	now	now	ADV
ejpam-6327	557	12	,	,	PUNCT
ejpam-6327	557	13	let	let	VERB
ejpam-6327	557	14	x	x	PRON
ejpam-6327	557	15	,	,	PUNCT
ejpam-6327	557	16	y	y	PROPN
ejpam-6327	557	17	∈	∈	PROPN
ejpam-6327	557	18	x.	x.	NOUN
ejpam-6327	557	19	then	then	ADV
ejpam-6327	557	20	,	,	PUNCT
ejpam-6327	557	21	µf	µf	X
ejpam-6327	557	22	(	(	PUNCT
ejpam-6327	557	23	xy	xy	NOUN
ejpam-6327	557	24	)	)	PUNCT
ejpam-6327	557	25	=	=	SYM
ejpam-6327	557	26	µ(f(xy	µ(f(xy	NOUN
ejpam-6327	557	27	)	)	PUNCT
ejpam-6327	557	28	)	)	PUNCT
ejpam-6327	557	29	=	=	SYM
ejpam-6327	557	30	µ(f(x)f(y	µ(f(x)f(y	ADJ
ejpam-6327	557	31	)	)	PUNCT
ejpam-6327	557	32	)	)	PUNCT
ejpam-6327	557	33	≥	≥	PROPN
ejpam-6327	557	34	µ(f(x	µ(f(x	PROPN
ejpam-6327	557	35	)	)	PUNCT
ejpam-6327	557	36	)	)	PUNCT
ejpam-6327	558	1	=	=	SYM
ejpam-6327	558	2	µf	µf	X
ejpam-6327	558	3	(	(	PUNCT
ejpam-6327	558	4	x	x	X
ejpam-6327	558	5	)	)	PUNCT
ejpam-6327	558	6	h.	h.	PROPN
ejpam-6327	558	7	sarapuddin	sarapuddin	PROPN
ejpam-6327	558	8	,	,	PUNCT
ejpam-6327	558	9	j.	j.	PROPN
ejpam-6327	558	10	vilela	vilela	PROPN
ejpam-6327	558	11	/	/	SYM
ejpam-6327	558	12	eur	eur	PROPN
ejpam-6327	558	13	.	.	PUNCT
ejpam-6327	559	1	j.	j.	PROPN
ejpam-6327	559	2	pure	pure	PROPN
ejpam-6327	559	3	appl	appl	PROPN
ejpam-6327	559	4	.	.	PROPN
ejpam-6327	559	5	math	math	PROPN
ejpam-6327	559	6	,	,	PUNCT
ejpam-6327	559	7	18	18	NUM
ejpam-6327	559	8	(	(	PUNCT
ejpam-6327	559	9	3	3	NUM
ejpam-6327	559	10	)	)	PUNCT
ejpam-6327	559	11	(	(	PUNCT
ejpam-6327	559	12	2025	2025	NUM
ejpam-6327	559	13	)	)	PUNCT
ejpam-6327	559	14	,	,	PUNCT
ejpam-6327	559	15	6327	6327	NUM
ejpam-6327	559	16	17	17	NUM
ejpam-6327	559	17	of	of	ADP
ejpam-6327	559	18	23	23	NUM
ejpam-6327	559	19	and	and	CCONJ
ejpam-6327	559	20	µf	µf	X
ejpam-6327	559	21	(	(	PUNCT
ejpam-6327	559	22	xy	xy	NOUN
ejpam-6327	559	23	)	)	PUNCT
ejpam-6327	559	24	=	=	SYM
ejpam-6327	559	25	µ(f(xy	µ(f(xy	NOUN
ejpam-6327	559	26	)	)	PUNCT
ejpam-6327	559	27	)	)	PUNCT
ejpam-6327	560	1	=	=	SYM
ejpam-6327	560	2	µ(f(x)f(y	µ(f(x)f(y	ADJ
ejpam-6327	560	3	)	)	PUNCT
ejpam-6327	560	4	)	)	PUNCT
ejpam-6327	560	5	≥	≥	NOUN
ejpam-6327	560	6	µ(f(y	µ(f(y	NUM
ejpam-6327	560	7	)	)	PUNCT
ejpam-6327	560	8	)	)	PUNCT
ejpam-6327	561	1	=	=	SYM
ejpam-6327	561	2	µf	µf	X
ejpam-6327	561	3	(	(	PUNCT
ejpam-6327	561	4	y	y	NOUN
ejpam-6327	561	5	)	)	PUNCT
ejpam-6327	561	6	.	.	PUNCT
ejpam-6327	562	1	theorem	theorem	ADJ
ejpam-6327	562	2	20	20	NUM
ejpam-6327	562	3	.	.	PUNCT
ejpam-6327	563	1	let	let	VERB
ejpam-6327	563	2	f	f	NOUN
ejpam-6327	563	3	:	:	PUNCT
ejpam-6327	563	4	x	x	X
ejpam-6327	563	5	→	→	SYM
ejpam-6327	563	6	y	y	X
ejpam-6327	563	7	be	be	AUX
ejpam-6327	563	8	an	an	DET
ejpam-6327	563	9	epimorphism	epimorphism	NOUN
ejpam-6327	563	10	and	and	CCONJ
ejpam-6327	563	11	µ	µ	PRON
ejpam-6327	563	12	be	be	AUX
ejpam-6327	563	13	a	a	DET
ejpam-6327	563	14	non	non	ADJ
ejpam-6327	563	15	-	-	ADJ
ejpam-6327	563	16	zero	zero	ADJ
ejpam-6327	563	17	fuzzy	fuzzy	ADJ
ejpam-6327	563	18	ks	ks	NOUN
ejpam-6327	563	19	-	-	NOUN
ejpam-6327	563	20	ideal	ideal	NOUN
ejpam-6327	563	21	of	of	ADP
ejpam-6327	563	22	y	y	PROPN
ejpam-6327	563	23	satisfying	satisfy	VERB
ejpam-6327	563	24	condition	condition	NOUN
ejpam-6327	563	25	(	(	PUNCT
ejpam-6327	563	26	c	c	NOUN
ejpam-6327	563	27	)	)	PUNCT
ejpam-6327	563	28	.	.	PUNCT
ejpam-6327	564	1	then	then	ADV
ejpam-6327	564	2	x/µf	x/µf	PROPN
ejpam-6327	564	3	∼=	∼=	PROPN
ejpam-6327	564	4	y/µ.	y/µ.	NOUN
ejpam-6327	564	5	proof	proof	NOUN
ejpam-6327	564	6	.	.	PUNCT
ejpam-6327	565	1	let	let	VERB
ejpam-6327	565	2	f	f	NOUN
ejpam-6327	565	3	:	:	PUNCT
ejpam-6327	565	4	x	x	X
ejpam-6327	565	5	→	→	SYM
ejpam-6327	565	6	y	y	X
ejpam-6327	565	7	be	be	AUX
ejpam-6327	565	8	an	an	DET
ejpam-6327	565	9	epimorphism	epimorphism	NOUN
ejpam-6327	565	10	and	and	CCONJ
ejpam-6327	565	11	µ	µ	PRON
ejpam-6327	565	12	be	be	AUX
ejpam-6327	565	13	a	a	DET
ejpam-6327	565	14	non	non	ADJ
ejpam-6327	565	15	-	-	ADJ
ejpam-6327	565	16	zero	zero	ADJ
ejpam-6327	565	17	fuzzy	fuzzy	ADJ
ejpam-6327	565	18	ks	ks	NOUN
ejpam-6327	565	19	-	-	NOUN
ejpam-6327	565	20	ideal	ideal	NOUN
ejpam-6327	565	21	of	of	ADP
ejpam-6327	565	22	y	y	PROPN
ejpam-6327	565	23	satisfying	satisfy	VERB
ejpam-6327	565	24	condition	condition	NOUN
ejpam-6327	565	25	(	(	PUNCT
ejpam-6327	565	26	c	c	NOUN
ejpam-6327	565	27	)	)	PUNCT
ejpam-6327	565	28	.	.	PUNCT
ejpam-6327	566	1	then	then	ADV
ejpam-6327	566	2	by	by	ADP
ejpam-6327	566	3	lemma	lemma	PROPN
ejpam-6327	566	4	3	3	NUM
ejpam-6327	566	5	,	,	PUNCT
ejpam-6327	566	6	µf	µf	X
ejpam-6327	566	7	is	be	AUX
ejpam-6327	566	8	a	a	DET
ejpam-6327	566	9	non	non	ADJ
ejpam-6327	566	10	-	-	ADJ
ejpam-6327	566	11	zero	zero	ADJ
ejpam-6327	566	12	fuzzy	fuzzy	ADJ
ejpam-6327	566	13	ks	ks	NOUN
ejpam-6327	566	14	-	-	NOUN
ejpam-6327	566	15	ideal	ideal	NOUN
ejpam-6327	566	16	of	of	ADP
ejpam-6327	566	17	x	x	SYM
ejpam-6327	566	18	satisfying	satisfy	VERB
ejpam-6327	566	19	condition	condition	NOUN
ejpam-6327	566	20	(	(	PUNCT
ejpam-6327	566	21	c	c	NOUN
ejpam-6327	566	22	)	)	PUNCT
ejpam-6327	566	23	.	.	PUNCT
ejpam-6327	567	1	by	by	ADP
ejpam-6327	567	2	theorem	theorem	NOUN
ejpam-6327	567	3	11	11	NUM
ejpam-6327	567	4	,	,	PUNCT
ejpam-6327	567	5	x/µf	x/µf	PROPN
ejpam-6327	567	6	and	and	CCONJ
ejpam-6327	567	7	y/µ	y/µ	PROPN
ejpam-6327	567	8	are	be	AUX
ejpam-6327	567	9	ks	ks	NOUN
ejpam-6327	567	10	-	-	PUNCT
ejpam-6327	567	11	semigroups	semigroup	NOUN
ejpam-6327	567	12	.	.	PUNCT
ejpam-6327	568	1	consider	consider	VERB
ejpam-6327	568	2	a	a	DET
ejpam-6327	568	3	mapping	mapping	NOUN
ejpam-6327	568	4	φ	φ	NOUN
ejpam-6327	568	5	:	:	PUNCT
ejpam-6327	569	1	x/µf	x/µf	PROPN
ejpam-6327	569	2	→	→	PUNCT
ejpam-6327	569	3	y/µ	y/µ	NOUN
ejpam-6327	569	4	defined	define	VERB
ejpam-6327	569	5	by	by	ADP
ejpam-6327	569	6	φ(µfx	φ(µfx	PROPN
ejpam-6327	569	7	)	)	PUNCT
ejpam-6327	569	8	=	=	SYM
ejpam-6327	569	9	µf(x	µf(x	PRON
ejpam-6327	569	10	)	)	PUNCT
ejpam-6327	569	11	for	for	ADP
ejpam-6327	569	12	each	each	DET
ejpam-6327	569	13	x	x	SYM
ejpam-6327	569	14	∈	∈	PROPN
ejpam-6327	569	15	x.	x.	NOUN
ejpam-6327	569	16	let	let	VERB
ejpam-6327	569	17	µfx	µfx	ADJ
ejpam-6327	569	18	,	,	PUNCT
ejpam-6327	569	19	µ	µ	X
ejpam-6327	569	20	f	f	X
ejpam-6327	569	21	y	y	PROPN
ejpam-6327	569	22	∈	∈	PROPN
ejpam-6327	570	1	x/µf	x/µf	PROPN
ejpam-6327	570	2	such	such	ADJ
ejpam-6327	570	3	that	that	DET
ejpam-6327	570	4	µfx	µfx	NOUN
ejpam-6327	570	5	=	=	PUNCT
ejpam-6327	570	6	µfy	µfy	NOUN
ejpam-6327	570	7	.	.	PUNCT
ejpam-6327	571	1	then	then	ADV
ejpam-6327	571	2	x	x	PROPN
ejpam-6327	571	3	∼µf	∼µf	PROPN
ejpam-6327	571	4	y	y	PROPN
ejpam-6327	571	5	,	,	PUNCT
ejpam-6327	571	6	that	that	ADV
ejpam-6327	571	7	is	is	ADV
ejpam-6327	571	8	,	,	PUNCT
ejpam-6327	571	9	µf	µf	X
ejpam-6327	571	10	(	(	PUNCT
ejpam-6327	571	11	x	x	X
ejpam-6327	571	12	∗	∗	PROPN
ejpam-6327	571	13	y	y	NOUN
ejpam-6327	571	14	)	)	PUNCT
ejpam-6327	571	15	=	=	SYM
ejpam-6327	571	16	µf	µf	X
ejpam-6327	571	17	(	(	PUNCT
ejpam-6327	571	18	0	0	NUM
ejpam-6327	571	19	)	)	PUNCT
ejpam-6327	571	20	and	and	CCONJ
ejpam-6327	571	21	µf	µf	X
ejpam-6327	571	22	(	(	PUNCT
ejpam-6327	571	23	y	y	PROPN
ejpam-6327	571	24	∗	∗	NOUN
ejpam-6327	571	25	x	x	NOUN
ejpam-6327	571	26	)	)	PUNCT
ejpam-6327	571	27	=	=	SYM
ejpam-6327	571	28	µf	µf	X
ejpam-6327	571	29	(	(	PUNCT
ejpam-6327	571	30	0	0	NUM
ejpam-6327	571	31	)	)	PUNCT
ejpam-6327	571	32	.	.	PUNCT
ejpam-6327	572	1	since	since	SCONJ
ejpam-6327	572	2	f	f	PROPN
ejpam-6327	572	3	is	be	AUX
ejpam-6327	572	4	a	a	DET
ejpam-6327	572	5	homomorphism	homomorphism	NOUN
ejpam-6327	572	6	,	,	PUNCT
ejpam-6327	572	7	µ(f(x	µ(f(x	PROPN
ejpam-6327	572	8	)	)	PUNCT
ejpam-6327	572	9	∗	∗	NOUN
ejpam-6327	572	10	f(y	f(y	NOUN
ejpam-6327	572	11	)	)	PUNCT
ejpam-6327	572	12	)	)	PUNCT
ejpam-6327	573	1	=	=	SYM
ejpam-6327	573	2	µ(f(x	µ(f(x	PROPN
ejpam-6327	573	3	∗	∗	NOUN
ejpam-6327	573	4	y	y	PROPN
ejpam-6327	573	5	)	)	PUNCT
ejpam-6327	573	6	)	)	PUNCT
ejpam-6327	574	1	=	=	PRON
ejpam-6327	574	2	µf	µf	X
ejpam-6327	574	3	(	(	PUNCT
ejpam-6327	574	4	x	x	X
ejpam-6327	574	5	∗	∗	PROPN
ejpam-6327	574	6	y	y	NOUN
ejpam-6327	574	7	)	)	PUNCT
ejpam-6327	574	8	=	=	SYM
ejpam-6327	574	9	µf	µf	X
ejpam-6327	574	10	(	(	PUNCT
ejpam-6327	574	11	0	0	NUM
ejpam-6327	574	12	)	)	PUNCT
ejpam-6327	574	13	and	and	CCONJ
ejpam-6327	574	14	µ(f(y)∗f(x	µ(f(y)∗f(x	NUM
ejpam-6327	574	15	)	)	PUNCT
ejpam-6327	574	16	)	)	PUNCT
ejpam-6327	575	1	=	=	SYM
ejpam-6327	575	2	µ(f(y∗x	µ(f(y∗x	NOUN
ejpam-6327	575	3	)	)	PUNCT
ejpam-6327	575	4	)	)	PUNCT
ejpam-6327	576	1	=	=	PRON
ejpam-6327	576	2	µf	µf	X
ejpam-6327	576	3	(	(	PUNCT
ejpam-6327	576	4	y∗x	y∗x	X
ejpam-6327	576	5	)	)	PUNCT
ejpam-6327	576	6	=	=	SYM
ejpam-6327	576	7	µf	µf	X
ejpam-6327	576	8	(	(	PUNCT
ejpam-6327	576	9	0	0	NUM
ejpam-6327	576	10	)	)	PUNCT
ejpam-6327	576	11	.	.	PUNCT
ejpam-6327	577	1	thus	thus	ADV
ejpam-6327	577	2	,	,	PUNCT
ejpam-6327	577	3	f(x	f(x	PROPN
ejpam-6327	577	4	)	)	PUNCT
ejpam-6327	577	5	∼µ	∼µ	NOUN
ejpam-6327	577	6	f(y	f(y	NOUN
ejpam-6327	577	7	)	)	PUNCT
ejpam-6327	577	8	,	,	PUNCT
ejpam-6327	577	9	that	that	ADV
ejpam-6327	577	10	is	is	ADV
ejpam-6327	577	11	,	,	PUNCT
ejpam-6327	577	12	µf(x	µf(x	ADV
ejpam-6327	577	13	)	)	PUNCT
ejpam-6327	577	14	=	=	SYM
ejpam-6327	577	15	µf(y	µf(y	PRON
ejpam-6327	577	16	)	)	PUNCT
ejpam-6327	577	17	.	.	PUNCT
ejpam-6327	578	1	hence	hence	ADV
ejpam-6327	578	2	,	,	PUNCT
ejpam-6327	578	3	φ	φ	PROPN
ejpam-6327	578	4	is	be	AUX
ejpam-6327	578	5	well	well	ADV
ejpam-6327	578	6	-	-	PUNCT
ejpam-6327	578	7	defined	define	VERB
ejpam-6327	578	8	.	.	PUNCT
ejpam-6327	579	1	now	now	ADV
ejpam-6327	579	2	,	,	PUNCT
ejpam-6327	579	3	let	let	VERB
ejpam-6327	579	4	µfx	µfx	VERB
ejpam-6327	579	5	,	,	PUNCT
ejpam-6327	579	6	µ	µ	X
ejpam-6327	579	7	f	f	X
ejpam-6327	579	8	y	y	PROPN
ejpam-6327	579	9	∈	∈	PROPN
ejpam-6327	580	1	x/µf	x/µf	PROPN
ejpam-6327	580	2	.	.	PUNCT
ejpam-6327	581	1	then	then	ADV
ejpam-6327	581	2	φ(µfx	φ(µfx	NOUN
ejpam-6327	581	3	∗	∗	NOUN
ejpam-6327	581	4	µfy	µfy	PROPN
ejpam-6327	581	5	)	)	PUNCT
ejpam-6327	581	6	=	=	SYM
ejpam-6327	582	1	φ(µfx∗y	φ(µfx∗y	X
ejpam-6327	582	2	)	)	PUNCT
ejpam-6327	582	3	=	=	SYM
ejpam-6327	582	4	µf(x∗y	µf(x∗y	X
ejpam-6327	582	5	)	)	PUNCT
ejpam-6327	582	6	=	=	SYM
ejpam-6327	582	7	µf(x)∗f(y	µf(x)∗f(y	PROPN
ejpam-6327	582	8	)	)	PUNCT
ejpam-6327	582	9	=	=	SYM
ejpam-6327	582	10	µf(x	µf(x	PRON
ejpam-6327	582	11	)	)	PUNCT
ejpam-6327	582	12	∗	∗	NOUN
ejpam-6327	582	13	µf(y	µf(y	NUM
ejpam-6327	582	14	)	)	PUNCT
ejpam-6327	582	15	=	=	SYM
ejpam-6327	582	16	φ(µfx	φ(µfx	ADJ
ejpam-6327	582	17	)	)	PUNCT
ejpam-6327	582	18	∗	∗	NOUN
ejpam-6327	582	19	φ(µfy	φ(µfy	NUM
ejpam-6327	582	20	)	)	PUNCT
ejpam-6327	582	21	and	and	CCONJ
ejpam-6327	582	22	φ(µfxµ	φ(µfxµ	PROPN
ejpam-6327	582	23	f	f	PROPN
ejpam-6327	582	24	y	y	PROPN
ejpam-6327	582	25	)	)	PUNCT
ejpam-6327	582	26	=	=	SYM
ejpam-6327	582	27	φ(µfxy	φ(µfxy	NOUN
ejpam-6327	582	28	)	)	PUNCT
ejpam-6327	582	29	=	=	SYM
ejpam-6327	582	30	µf(xy	µf(xy	NOUN
ejpam-6327	582	31	)	)	PUNCT
ejpam-6327	582	32	=	=	SYM
ejpam-6327	582	33	µf(x)f(y	µf(x)f(y	NOUN
ejpam-6327	582	34	)	)	PUNCT
ejpam-6327	582	35	=	=	SYM
ejpam-6327	582	36	µf(x)µf(y	µf(x)µf(y	NOUN
ejpam-6327	582	37	)	)	PUNCT
ejpam-6327	582	38	=	=	PUNCT
ejpam-6327	583	1	φ(µfx)φ(µ	φ(µfx)φ(µ	PROPN
ejpam-6327	583	2	f	f	PROPN
ejpam-6327	583	3	y	y	PROPN
ejpam-6327	583	4	)	)	PUNCT
ejpam-6327	583	5	.	.	PUNCT
ejpam-6327	584	1	thus	thus	ADV
ejpam-6327	584	2	,	,	PUNCT
ejpam-6327	584	3	by	by	ADP
ejpam-6327	584	4	definition	definition	NOUN
ejpam-6327	584	5	14(i	14(i	NUM
ejpam-6327	584	6	)	)	PUNCT
ejpam-6327	584	7	,	,	PUNCT
ejpam-6327	584	8	φ	φ	PROPN
ejpam-6327	584	9	is	be	AUX
ejpam-6327	584	10	a	a	DET
ejpam-6327	584	11	homomorphism	homomorphism	NOUN
ejpam-6327	584	12	.	.	PUNCT
ejpam-6327	585	1	now	now	ADV
ejpam-6327	585	2	,	,	PUNCT
ejpam-6327	585	3	since	since	SCONJ
ejpam-6327	585	4	f	f	PROPN
ejpam-6327	585	5	is	be	AUX
ejpam-6327	585	6	an	an	DET
ejpam-6327	585	7	epimorphism	epimorphism	NOUN
ejpam-6327	585	8	,	,	PUNCT
ejpam-6327	585	9	for	for	ADP
ejpam-6327	585	10	each	each	DET
ejpam-6327	585	11	y	y	PROPN
ejpam-6327	585	12	∈	∈	PROPN
ejpam-6327	585	13	y	y	PROPN
ejpam-6327	585	14	,	,	PUNCT
ejpam-6327	585	15	there	there	PRON
ejpam-6327	585	16	exists	exist	VERB
ejpam-6327	585	17	x	x	X
ejpam-6327	585	18	∈	∈	PROPN
ejpam-6327	585	19	x	x	PUNCT
ejpam-6327	585	20	such	such	ADJ
ejpam-6327	585	21	that	that	SCONJ
ejpam-6327	585	22	y	y	PROPN
ejpam-6327	585	23	=	=	SYM
ejpam-6327	585	24	f(x	f(x	PROPN
ejpam-6327	585	25	)	)	PUNCT
ejpam-6327	585	26	.	.	PUNCT
ejpam-6327	586	1	thus	thus	ADV
ejpam-6327	586	2	,	,	PUNCT
ejpam-6327	586	3	φ(µfx	φ(µfx	PROPN
ejpam-6327	586	4	)	)	PUNCT
ejpam-6327	586	5	=	=	SYM
ejpam-6327	586	6	µf(x	µf(x	PRON
ejpam-6327	586	7	)	)	PUNCT
ejpam-6327	586	8	=	=	SYM
ejpam-6327	587	1	µy	µy	X
ejpam-6327	587	2	.	.	PUNCT
ejpam-6327	588	1	hence	hence	ADV
ejpam-6327	588	2	φ	φ	PROPN
ejpam-6327	588	3	is	be	AUX
ejpam-6327	588	4	onto	onto	ADP
ejpam-6327	588	5	.	.	PUNCT
ejpam-6327	589	1	suppose	suppose	VERB
ejpam-6327	589	2	φ(µfx	φ(µfx	NOUN
ejpam-6327	589	3	)	)	PUNCT
ejpam-6327	589	4	=	=	SYM
ejpam-6327	589	5	φ(µfy	φ(µfy	PROPN
ejpam-6327	589	6	)	)	PUNCT
ejpam-6327	589	7	for	for	ADP
ejpam-6327	589	8	µ	µ	PROPN
ejpam-6327	589	9	f	f	X
ejpam-6327	589	10	x	x	PROPN
ejpam-6327	589	11	,	,	PUNCT
ejpam-6327	589	12	µ	µ	PROPN
ejpam-6327	589	13	f	f	X
ejpam-6327	589	14	y	y	PROPN
ejpam-6327	589	15	∈	∈	PROPN
ejpam-6327	590	1	x/µf	x/µf	PROPN
ejpam-6327	590	2	.	.	PUNCT
ejpam-6327	591	1	then	then	ADV
ejpam-6327	591	2	µf(x	µf(x	PRON
ejpam-6327	591	3	)	)	PUNCT
ejpam-6327	591	4	=	=	SYM
ejpam-6327	591	5	µf(y	µf(y	PRON
ejpam-6327	591	6	)	)	PUNCT
ejpam-6327	591	7	⇒	⇒	VERB
ejpam-6327	591	8	f(x	f(x	PROPN
ejpam-6327	591	9	)	)	PUNCT
ejpam-6327	591	10	∼µ	∼µ	PROPN
ejpam-6327	591	11	f(y	f(y	NOUN
ejpam-6327	591	12	)	)	PUNCT
ejpam-6327	591	13	⇒	⇒	PROPN
ejpam-6327	591	14	µ(f(x	µ(f(x	PROPN
ejpam-6327	591	15	)	)	PUNCT
ejpam-6327	591	16	∗	∗	NOUN
ejpam-6327	591	17	f(y	f(y	NOUN
ejpam-6327	591	18	)	)	PUNCT
ejpam-6327	591	19	)	)	PUNCT
ejpam-6327	591	20	>	>	PUNCT
ejpam-6327	591	21	0	0	NUM
ejpam-6327	591	22	and	and	CCONJ
ejpam-6327	591	23	µ(f(y	µ(f(y	NUM
ejpam-6327	591	24	)	)	PUNCT
ejpam-6327	591	25	∗	∗	NOUN
ejpam-6327	591	26	f(x	f(x	PROPN
ejpam-6327	591	27	)	)	PUNCT
ejpam-6327	591	28	)	)	PUNCT
ejpam-6327	592	1	>	>	SYM
ejpam-6327	592	2	0	0	NUM
ejpam-6327	593	1	⇒	⇒	PROPN
ejpam-6327	593	2	µ(f(x	µ(f(x	PROPN
ejpam-6327	593	3	∗	∗	PROPN
ejpam-6327	593	4	y	y	PROPN
ejpam-6327	593	5	)	)	PUNCT
ejpam-6327	593	6	)	)	PUNCT
ejpam-6327	594	1	>	>	X
ejpam-6327	594	2	0	0	PUNCT
ejpam-6327	595	1	and	and	CCONJ
ejpam-6327	595	2	µ(f(y	µ(f(y	PROPN
ejpam-6327	595	3	∗	∗	NOUN
ejpam-6327	595	4	x	x	NOUN
ejpam-6327	595	5	)	)	PUNCT
ejpam-6327	595	6	)	)	PUNCT
ejpam-6327	596	1	>	>	SYM
ejpam-6327	596	2	0	0	PUNCT
ejpam-6327	597	1	⇒	⇒	NOUN
ejpam-6327	597	2	µf	µf	VERB
ejpam-6327	597	3	(	(	PUNCT
ejpam-6327	597	4	x	x	X
ejpam-6327	597	5	∗	∗	PROPN
ejpam-6327	597	6	y	y	PROPN
ejpam-6327	597	7	)	)	PUNCT
ejpam-6327	597	8	>	>	X
ejpam-6327	597	9	0	0	PUNCT
ejpam-6327	597	10	and	and	CCONJ
ejpam-6327	597	11	µf	µf	X
ejpam-6327	597	12	(	(	PUNCT
ejpam-6327	597	13	y	y	PROPN
ejpam-6327	597	14	∗	∗	NOUN
ejpam-6327	597	15	x	x	NOUN
ejpam-6327	597	16	)	)	PUNCT
ejpam-6327	597	17	>	>	SYM
ejpam-6327	597	18	0	0	NUM
ejpam-6327	598	1	⇒	⇒	NOUN
ejpam-6327	598	2	x	x	PUNCT
ejpam-6327	598	3	∼µf	∼µf	ADP
ejpam-6327	598	4	y	y	PROPN
ejpam-6327	598	5	⇒	⇒	PROPN
ejpam-6327	598	6	µfx	µfx	VERB
ejpam-6327	599	1	=	=	PUNCT
ejpam-6327	599	2	µfy	µfy	NOUN
ejpam-6327	599	3	.	.	PUNCT
ejpam-6327	600	1	thus	thus	ADV
ejpam-6327	600	2	,	,	PUNCT
ejpam-6327	600	3	φ	φ	PROPN
ejpam-6327	600	4	is	be	AUX
ejpam-6327	600	5	one	one	NUM
ejpam-6327	600	6	-	-	PUNCT
ejpam-6327	600	7	to	to	ADP
ejpam-6327	600	8	-	-	PUNCT
ejpam-6327	600	9	one	one	NUM
ejpam-6327	600	10	.	.	PUNCT
ejpam-6327	601	1	hence	hence	ADV
ejpam-6327	601	2	,	,	PUNCT
ejpam-6327	601	3	φ	φ	PROPN
ejpam-6327	601	4	is	be	AUX
ejpam-6327	601	5	an	an	DET
ejpam-6327	601	6	isomorphism	isomorphism	NOUN
ejpam-6327	601	7	.	.	PUNCT
ejpam-6327	602	1	therefore	therefore	ADV
ejpam-6327	602	2	,	,	PUNCT
ejpam-6327	602	3	x/µf	x/µf	PROPN
ejpam-6327	602	4	∼=	∼=	PROPN
ejpam-6327	602	5	y/µ.	y/µ.	NOUN
ejpam-6327	602	6	6	6	NUM
ejpam-6327	602	7	.	.	PUNCT
ejpam-6327	603	1	quotient	quotient	NOUN
ejpam-6327	603	2	structure	structure	NOUN
ejpam-6327	603	3	of	of	ADP
ejpam-6327	603	4	product	product	NOUN
ejpam-6327	603	5	ks	ks	NOUN
ejpam-6327	603	6	-	-	PUNCT
ejpam-6327	603	7	semigroup	semigroup	NOUN
ejpam-6327	603	8	via	via	ADP
ejpam-6327	603	9	fuzzy	fuzzy	ADJ
ejpam-6327	603	10	ks	ks	NOUN
ejpam-6327	603	11	-	-	NOUN
ejpam-6327	603	12	ideals	ideal	NOUN
ejpam-6327	603	13	in	in	ADP
ejpam-6327	603	14	this	this	DET
ejpam-6327	603	15	section	section	NOUN
ejpam-6327	604	1	,	,	PUNCT
ejpam-6327	604	2	we	we	PRON
ejpam-6327	604	3	investigate	investigate	VERB
ejpam-6327	604	4	the	the	DET
ejpam-6327	604	5	quotient	quotient	NOUN
ejpam-6327	604	6	structure	structure	NOUN
ejpam-6327	604	7	of	of	ADP
ejpam-6327	604	8	product	product	NOUN
ejpam-6327	604	9	ks	ks	NOUN
ejpam-6327	604	10	-	-	PUNCT
ejpam-6327	604	11	semigroup	semigroup	NOUN
ejpam-6327	604	12	induced	induce	VERB
ejpam-6327	604	13	by	by	ADP
ejpam-6327	604	14	fuzzy	fuzzy	ADJ
ejpam-6327	604	15	ks	ks	NOUN
ejpam-6327	604	16	-	-	NOUN
ejpam-6327	604	17	ideals	ideal	NOUN
ejpam-6327	604	18	.	.	PUNCT
ejpam-6327	605	1	h.	h.	PROPN
ejpam-6327	605	2	sarapuddin	sarapuddin	PROPN
ejpam-6327	605	3	,	,	PUNCT
ejpam-6327	605	4	j.	j.	PROPN
ejpam-6327	605	5	vilela	vilela	PROPN
ejpam-6327	605	6	/	/	SYM
ejpam-6327	605	7	eur	eur	PROPN
ejpam-6327	605	8	.	.	PUNCT
ejpam-6327	606	1	j.	j.	PROPN
ejpam-6327	606	2	pure	pure	PROPN
ejpam-6327	606	3	appl	appl	PROPN
ejpam-6327	606	4	.	.	PROPN
ejpam-6327	606	5	math	math	PROPN
ejpam-6327	606	6	,	,	PUNCT
ejpam-6327	606	7	18	18	NUM
ejpam-6327	606	8	(	(	PUNCT
ejpam-6327	606	9	3	3	NUM
ejpam-6327	606	10	)	)	PUNCT
ejpam-6327	606	11	(	(	PUNCT
ejpam-6327	606	12	2025	2025	NUM
ejpam-6327	606	13	)	)	PUNCT
ejpam-6327	606	14	,	,	PUNCT
ejpam-6327	606	15	6327	6327	NUM
ejpam-6327	606	16	18	18	NUM
ejpam-6327	606	17	of	of	ADP
ejpam-6327	606	18	23	23	NUM
ejpam-6327	606	19	definition	definition	NOUN
ejpam-6327	606	20	16	16	NUM
ejpam-6327	606	21	.	.	PUNCT
ejpam-6327	607	1	let	let	VERB
ejpam-6327	607	2	(	(	PUNCT
ejpam-6327	607	3	x	x	NOUN
ejpam-6327	607	4	,	,	PUNCT
ejpam-6327	607	5	∗x	∗x	NOUN
ejpam-6327	607	6	,	,	PUNCT
ejpam-6327	607	7	·	·	PUNCT
ejpam-6327	607	8	x	x	SYM
ejpam-6327	607	9	,	,	PUNCT
ejpam-6327	607	10	0x	0x	NOUN
ejpam-6327	607	11	)	)	PUNCT
ejpam-6327	607	12	and	and	CCONJ
ejpam-6327	607	13	(	(	PUNCT
ejpam-6327	607	14	y	y	PROPN
ejpam-6327	607	15	,	,	PUNCT
ejpam-6327	607	16	∗y	∗y	PROPN
ejpam-6327	607	17	,	,	PUNCT
ejpam-6327	607	18	·	·	PUNCT
ejpam-6327	607	19	y	y	NOUN
ejpam-6327	607	20	,	,	PUNCT
ejpam-6327	607	21	0y	0y	NUM
ejpam-6327	607	22	)	)	PUNCT
ejpam-6327	607	23	be	be	VERB
ejpam-6327	607	24	ks	k	NOUN
ejpam-6327	607	25	-	-	PUNCT
ejpam-6327	607	26	semigroups	semigroup	NOUN
ejpam-6327	607	27	.	.	PUNCT
ejpam-6327	608	1	define	define	VERB
ejpam-6327	608	2	the	the	DET
ejpam-6327	608	3	operations	operation	NOUN
ejpam-6327	608	4	∗	∗	NOUN
ejpam-6327	608	5	and	and	CCONJ
ejpam-6327	608	6	·	·	PUNCT
ejpam-6327	608	7	in	in	ADP
ejpam-6327	608	8	x	x	SYM
ejpam-6327	608	9	×	×	PROPN
ejpam-6327	608	10	y	y	PROPN
ejpam-6327	608	11	by	by	ADP
ejpam-6327	608	12	(	(	PUNCT
ejpam-6327	608	13	x1	x1	PROPN
ejpam-6327	608	14	,	,	PUNCT
ejpam-6327	608	15	y1	y1	NOUN
ejpam-6327	608	16	)	)	PUNCT
ejpam-6327	608	17	∗	∗	NOUN
ejpam-6327	608	18	(	(	PUNCT
ejpam-6327	608	19	x2	x2	NOUN
ejpam-6327	608	20	,	,	PUNCT
ejpam-6327	608	21	y2	y2	NOUN
ejpam-6327	608	22	)	)	PUNCT
ejpam-6327	608	23	=	=	SYM
ejpam-6327	609	1	(	(	PUNCT
ejpam-6327	609	2	x1	x1	ADJ
ejpam-6327	609	3	∗x	∗x	PROPN
ejpam-6327	609	4	x2	x2	PROPN
ejpam-6327	609	5	,	,	PUNCT
ejpam-6327	609	6	y1	y1	PROPN
ejpam-6327	609	7	∗y	∗y	PROPN
ejpam-6327	609	8	y2	y2	PROPN
ejpam-6327	609	9	)	)	PUNCT
ejpam-6327	609	10	and	and	CCONJ
ejpam-6327	609	11	(	(	PUNCT
ejpam-6327	609	12	x1	x1	PROPN
ejpam-6327	609	13	,	,	PUNCT
ejpam-6327	609	14	y1	y1	PROPN
ejpam-6327	609	15	)	)	PUNCT
ejpam-6327	609	16	·	·	PUNCT
ejpam-6327	609	17	(	(	PUNCT
ejpam-6327	609	18	x2	x2	NOUN
ejpam-6327	609	19	,	,	PUNCT
ejpam-6327	609	20	y2	y2	NOUN
ejpam-6327	609	21	)	)	PUNCT
ejpam-6327	609	22	=	=	SYM
ejpam-6327	609	23	(	(	PUNCT
ejpam-6327	609	24	x1	x1	PROPN
ejpam-6327	609	25	·	·	PUNCT
ejpam-6327	609	26	x	x	SYM
ejpam-6327	609	27	x2	x2	PROPN
ejpam-6327	609	28	,	,	PUNCT
ejpam-6327	609	29	y1	y1	PROPN
ejpam-6327	609	30	·	·	PUNCT
ejpam-6327	609	31	y	y	PROPN
ejpam-6327	609	32	y2	y2	PROPN
ejpam-6327	609	33	)	)	PUNCT
ejpam-6327	609	34	.	.	PUNCT
ejpam-6327	610	1	then	then	ADV
ejpam-6327	610	2	(	(	PUNCT
ejpam-6327	610	3	x	x	SYM
ejpam-6327	610	4	×	×	PROPN
ejpam-6327	610	5	y	y	PROPN
ejpam-6327	610	6	,	,	PUNCT
ejpam-6327	610	7	∗	∗	NOUN
ejpam-6327	610	8	,	,	PUNCT
ejpam-6327	610	9	·	·	PUNCT
ejpam-6327	610	10	,	,	PUNCT
ejpam-6327	610	11	0	0	NUM
ejpam-6327	610	12	)	)	PUNCT
ejpam-6327	610	13	is	be	AUX
ejpam-6327	610	14	also	also	ADV
ejpam-6327	610	15	ks	ks	NOUN
ejpam-6327	610	16	-	-	PUNCT
ejpam-6327	610	17	semigroup	semigroup	NOUN
ejpam-6327	610	18	,	,	PUNCT
ejpam-6327	610	19	where	where	SCONJ
ejpam-6327	610	20	0	0	X
ejpam-6327	610	21	=	=	SYM
ejpam-6327	610	22	(	(	PUNCT
ejpam-6327	610	23	0x	0x	NOUN
ejpam-6327	610	24	,	,	PUNCT
ejpam-6327	610	25	0y	0y	NUM
ejpam-6327	610	26	)	)	PUNCT
ejpam-6327	610	27	and	and	CCONJ
ejpam-6327	610	28	it	it	PRON
ejpam-6327	610	29	is	be	AUX
ejpam-6327	610	30	called	call	VERB
ejpam-6327	610	31	product	product	NOUN
ejpam-6327	610	32	ks	ks	NOUN
ejpam-6327	610	33	-	-	PUNCT
ejpam-6327	610	34	semigroup	semigroup	NOUN
ejpam-6327	610	35	.	.	PUNCT
ejpam-6327	611	1	definition	definition	NOUN
ejpam-6327	611	2	17	17	NUM
ejpam-6327	611	3	.	.	PUNCT
ejpam-6327	612	1	[	[	X
ejpam-6327	612	2	4	4	X
ejpam-6327	612	3	]	]	X
ejpam-6327	612	4	let	let	VERB
ejpam-6327	612	5	µ	µ	NOUN
ejpam-6327	612	6	and	and	CCONJ
ejpam-6327	612	7	ν	ν	PROPN
ejpam-6327	612	8	be	be	AUX
ejpam-6327	612	9	fuzzy	fuzzy	ADJ
ejpam-6327	612	10	sets	set	NOUN
ejpam-6327	612	11	on	on	ADP
ejpam-6327	612	12	the	the	DET
ejpam-6327	612	13	sets	set	NOUN
ejpam-6327	612	14	x	x	PUNCT
ejpam-6327	612	15	and	and	CCONJ
ejpam-6327	612	16	y	y	PROPN
ejpam-6327	612	17	,	,	PUNCT
ejpam-6327	612	18	respectively	respectively	ADV
ejpam-6327	612	19	.	.	PUNCT
ejpam-6327	613	1	the	the	DET
ejpam-6327	613	2	cartesian	cartesian	ADJ
ejpam-6327	613	3	product	product	NOUN
ejpam-6327	613	4	µ	µ	X
ejpam-6327	613	5	×	×	NOUN
ejpam-6327	613	6	ν	ν	NOUN
ejpam-6327	613	7	:	:	PUNCT
ejpam-6327	613	8	x	x	SYM
ejpam-6327	613	9	×	×	NOUN
ejpam-6327	613	10	y	y	X
ejpam-6327	613	11	→	→	PUNCT
ejpam-6327	613	12	[	[	X
ejpam-6327	613	13	0	0	NUM
ejpam-6327	613	14	,	,	PUNCT
ejpam-6327	613	15	1	1	NUM
ejpam-6327	613	16	]	]	PUNCT
ejpam-6327	613	17	is	be	AUX
ejpam-6327	613	18	defined	define	VERB
ejpam-6327	613	19	by	by	ADP
ejpam-6327	613	20	µ	µ	NUM
ejpam-6327	613	21	×	×	NOUN
ejpam-6327	613	22	ν(x	ν(x	PROPN
ejpam-6327	613	23	,	,	PUNCT
ejpam-6327	613	24	y	y	NOUN
ejpam-6327	613	25	)	)	PUNCT
ejpam-6327	613	26	=	=	SYM
ejpam-6327	614	1	min{µ(x	min{µ(x	NOUN
ejpam-6327	614	2	)	)	PUNCT
ejpam-6327	614	3	,	,	PUNCT
ejpam-6327	614	4	ν(y	ν(y	PROPN
ejpam-6327	614	5	)	)	PUNCT
ejpam-6327	614	6	}	}	PUNCT
ejpam-6327	614	7	for	for	ADP
ejpam-6327	614	8	all	all	DET
ejpam-6327	614	9	(	(	PUNCT
ejpam-6327	614	10	x	x	NOUN
ejpam-6327	614	11	,	,	PUNCT
ejpam-6327	614	12	y	y	NOUN
ejpam-6327	614	13	)	)	PUNCT
ejpam-6327	614	14	∈	∈	PROPN
ejpam-6327	614	15	x	x	PUNCT
ejpam-6327	614	16	×	×	NOUN
ejpam-6327	614	17	y	y	PROPN
ejpam-6327	614	18	.	.	PUNCT
ejpam-6327	615	1	in	in	ADP
ejpam-6327	615	2	[	[	X
ejpam-6327	615	3	7	7	NUM
ejpam-6327	615	4	]	]	PUNCT
ejpam-6327	615	5	,	,	PUNCT
ejpam-6327	615	6	if	if	SCONJ
ejpam-6327	615	7	µ	µ	NOUN
ejpam-6327	615	8	and	and	CCONJ
ejpam-6327	615	9	ν	ν	NOUN
ejpam-6327	615	10	are	be	AUX
ejpam-6327	615	11	fuzzy	fuzzy	ADJ
ejpam-6327	615	12	ks	ks	NOUN
ejpam-6327	615	13	-	-	PUNCT
ejpam-6327	615	14	ideals	ideal	NOUN
ejpam-6327	615	15	of	of	ADP
ejpam-6327	615	16	a	a	DET
ejpam-6327	615	17	ks	ks	NOUN
ejpam-6327	615	18	-	-	PUNCT
ejpam-6327	615	19	semigroup	semigroup	NOUN
ejpam-6327	615	20	x	x	NOUN
ejpam-6327	615	21	,	,	PUNCT
ejpam-6327	615	22	then	then	ADV
ejpam-6327	615	23	µ×	µ×	X
ejpam-6327	615	24	ν	ν	NOUN
ejpam-6327	615	25	is	be	AUX
ejpam-6327	615	26	also	also	ADV
ejpam-6327	615	27	a	a	DET
ejpam-6327	615	28	fuzzy	fuzzy	ADJ
ejpam-6327	615	29	ks	ks	NOUN
ejpam-6327	615	30	-	-	NOUN
ejpam-6327	615	31	ideal	ideal	NOUN
ejpam-6327	615	32	in	in	ADP
ejpam-6327	615	33	x	x	PROPN
ejpam-6327	615	34	×x	×x	X
ejpam-6327	615	35	.	.	PUNCT
ejpam-6327	616	1	the	the	DET
ejpam-6327	616	2	following	following	ADJ
ejpam-6327	616	3	result	result	NOUN
ejpam-6327	616	4	is	be	AUX
ejpam-6327	616	5	a	a	DET
ejpam-6327	616	6	general	general	ADJ
ejpam-6327	616	7	case	case	NOUN
ejpam-6327	616	8	.	.	PUNCT
ejpam-6327	617	1	proposition	proposition	NOUN
ejpam-6327	617	2	6	6	NUM
ejpam-6327	617	3	.	.	PUNCT
ejpam-6327	618	1	let	let	VERB
ejpam-6327	618	2	µ	µ	NOUN
ejpam-6327	618	3	and	and	CCONJ
ejpam-6327	618	4	ν	ν	PROPN
ejpam-6327	618	5	be	be	AUX
ejpam-6327	618	6	fuzzy	fuzzy	ADJ
ejpam-6327	618	7	ks	ks	NOUN
ejpam-6327	618	8	-	-	PUNCT
ejpam-6327	618	9	ideals	ideal	NOUN
ejpam-6327	618	10	of	of	ADP
ejpam-6327	618	11	ks	ks	NOUN
ejpam-6327	618	12	-	-	PUNCT
ejpam-6327	618	13	semigroups	semigroups	X
ejpam-6327	618	14	x	x	X
ejpam-6327	618	15	and	and	CCONJ
ejpam-6327	618	16	y	y	PROPN
ejpam-6327	618	17	,	,	PUNCT
ejpam-6327	618	18	respectively	respectively	ADV
ejpam-6327	618	19	.	.	PUNCT
ejpam-6327	619	1	then	then	ADV
ejpam-6327	619	2	µ×	µ×	PRON
ejpam-6327	619	3	ν	ν	NOUN
ejpam-6327	619	4	is	be	AUX
ejpam-6327	619	5	also	also	ADV
ejpam-6327	619	6	a	a	DET
ejpam-6327	619	7	fuzzy	fuzzy	ADJ
ejpam-6327	619	8	ks	ks	NOUN
ejpam-6327	619	9	-	-	NOUN
ejpam-6327	619	10	ideal	ideal	NOUN
ejpam-6327	619	11	of	of	ADP
ejpam-6327	619	12	x	x	SYM
ejpam-6327	619	13	×	×	PROPN
ejpam-6327	619	14	y	y	PROPN
ejpam-6327	619	15	.	.	PUNCT
ejpam-6327	620	1	proof	proof	NOUN
ejpam-6327	620	2	.	.	PUNCT
ejpam-6327	621	1	let	let	VERB
ejpam-6327	621	2	µ	µ	NOUN
ejpam-6327	621	3	and	and	CCONJ
ejpam-6327	621	4	ν	ν	PROPN
ejpam-6327	621	5	be	be	AUX
ejpam-6327	621	6	fuzzy	fuzzy	ADJ
ejpam-6327	621	7	ks	ks	NOUN
ejpam-6327	621	8	-	-	PUNCT
ejpam-6327	621	9	ideals	ideal	NOUN
ejpam-6327	621	10	of	of	ADP
ejpam-6327	621	11	ks	ks	NOUN
ejpam-6327	621	12	-	-	PUNCT
ejpam-6327	621	13	semigroups	semigroups	X
ejpam-6327	621	14	x	x	X
ejpam-6327	621	15	and	and	CCONJ
ejpam-6327	621	16	y	y	PROPN
ejpam-6327	621	17	,	,	PUNCT
ejpam-6327	621	18	respectively	respectively	ADV
ejpam-6327	621	19	.	.	PUNCT
ejpam-6327	622	1	(	(	PUNCT
ejpam-6327	622	2	i	i	NOUN
ejpam-6327	622	3	)	)	PUNCT
ejpam-6327	622	4	for	for	ADP
ejpam-6327	622	5	any	any	DET
ejpam-6327	622	6	(	(	PUNCT
ejpam-6327	622	7	x	x	NOUN
ejpam-6327	622	8	,	,	PUNCT
ejpam-6327	622	9	y	y	NOUN
ejpam-6327	622	10	)	)	PUNCT
ejpam-6327	622	11	∈	∈	PROPN
ejpam-6327	622	12	x	x	SYM
ejpam-6327	622	13	×	×	PROPN
ejpam-6327	622	14	y	y	PROPN
ejpam-6327	622	15	,	,	PUNCT
ejpam-6327	622	16	µ×	µ×	X
ejpam-6327	622	17	ν(0x	ν(0x	NOUN
ejpam-6327	622	18	,	,	PUNCT
ejpam-6327	622	19	0y	0y	NUM
ejpam-6327	622	20	)	)	PUNCT
ejpam-6327	622	21	=	=	SYM
ejpam-6327	622	22	min{µ(0x	min{µ(0x	NUM
ejpam-6327	622	23	)	)	PUNCT
ejpam-6327	622	24	,	,	PUNCT
ejpam-6327	622	25	ν(0y	ν(0y	PROPN
ejpam-6327	622	26	)	)	PUNCT
ejpam-6327	622	27	}	}	PUNCT
ejpam-6327	622	28	≥	≥	NOUN
ejpam-6327	622	29	min{µ(x	min{µ(x	NOUN
ejpam-6327	622	30	)	)	PUNCT
ejpam-6327	622	31	,	,	PUNCT
ejpam-6327	622	32	µ(y	µ(y	PROPN
ejpam-6327	622	33	)	)	PUNCT
ejpam-6327	622	34	}	}	PUNCT
ejpam-6327	622	35	=	=	PUNCT
ejpam-6327	623	1	µ×	µ×	X
ejpam-6327	623	2	ν(x	ν(x	PROPN
ejpam-6327	623	3	,	,	PUNCT
ejpam-6327	623	4	y	y	NOUN
ejpam-6327	623	5	)	)	PUNCT
ejpam-6327	623	6	.	.	PUNCT
ejpam-6327	624	1	(	(	PUNCT
ejpam-6327	624	2	ii	ii	NOUN
ejpam-6327	624	3	)	)	PUNCT
ejpam-6327	624	4	let	let	VERB
ejpam-6327	624	5	(	(	PUNCT
ejpam-6327	624	6	x1	x1	ADJ
ejpam-6327	624	7	,	,	PUNCT
ejpam-6327	624	8	x2	x2	PROPN
ejpam-6327	624	9	)	)	PUNCT
ejpam-6327	624	10	,	,	PUNCT
ejpam-6327	624	11	(	(	PUNCT
ejpam-6327	624	12	y1	y1	INTJ
ejpam-6327	624	13	,	,	PUNCT
ejpam-6327	624	14	y2	y2	NOUN
ejpam-6327	624	15	)	)	PUNCT
ejpam-6327	624	16	∈	∈	PROPN
ejpam-6327	625	1	x	x	PUNCT
ejpam-6327	625	2	×	×	NOUN
ejpam-6327	625	3	y	y	PROPN
ejpam-6327	625	4	.	.	PUNCT
ejpam-6327	626	1	then	then	ADV
ejpam-6327	626	2	µ×	µ×	X
ejpam-6327	626	3	ν(x1	ν(x1	PROPN
ejpam-6327	626	4	,	,	PUNCT
ejpam-6327	626	5	x2	x2	PROPN
ejpam-6327	626	6	)	)	PUNCT
ejpam-6327	626	7	=	=	SYM
ejpam-6327	626	8	min{µ(x1	min{µ(x1	NOUN
ejpam-6327	626	9	)	)	PUNCT
ejpam-6327	626	10	,	,	PUNCT
ejpam-6327	626	11	ν(x2	ν(x2	NOUN
ejpam-6327	626	12	)	)	PUNCT
ejpam-6327	626	13	}	}	PUNCT
ejpam-6327	626	14	≥	≥	PROPN
ejpam-6327	626	15	min{min{µ(x1	min{min{µ(x1	NOUN
ejpam-6327	626	16	∗x	∗x	PROPN
ejpam-6327	626	17	y1	y1	NOUN
ejpam-6327	626	18	)	)	PUNCT
ejpam-6327	626	19	,	,	PUNCT
ejpam-6327	626	20	µ(y1)},min{ν(x2	µ(y1)},min{ν(x2	X
ejpam-6327	626	21	∗y	∗y	PROPN
ejpam-6327	626	22	y2	y2	PROPN
ejpam-6327	626	23	)	)	PUNCT
ejpam-6327	626	24	,	,	PUNCT
ejpam-6327	626	25	ν(y2	ν(y2	NOUN
ejpam-6327	626	26	)	)	PUNCT
ejpam-6327	626	27	}	}	PUNCT
ejpam-6327	626	28	}	}	PUNCT
ejpam-6327	626	29	=	=	SYM
ejpam-6327	626	30	min{min{µ(x1	min{min{µ(x1	X
ejpam-6327	626	31	∗x	∗x	PROPN
ejpam-6327	626	32	y1	y1	NOUN
ejpam-6327	626	33	)	)	PUNCT
ejpam-6327	626	34	,	,	PUNCT
ejpam-6327	626	35	ν(x2	ν(x2	NOUN
ejpam-6327	626	36	∗y	∗y	PROPN
ejpam-6327	626	37	y2)},min{µ(y1	y2)},min{µ(y1	PROPN
ejpam-6327	626	38	)	)	PUNCT
ejpam-6327	626	39	,	,	PUNCT
ejpam-6327	626	40	ν(y2	ν(y2	NOUN
ejpam-6327	626	41	)	)	PUNCT
ejpam-6327	626	42	}	}	PUNCT
ejpam-6327	626	43	}	}	PUNCT
ejpam-6327	626	44	=	=	SYM
ejpam-6327	626	45	min{µ×	min{µ×	PROPN
ejpam-6327	626	46	ν(x1	ν(x1	NOUN
ejpam-6327	626	47	∗x	∗x	PROPN
ejpam-6327	626	48	y1	y1	PROPN
ejpam-6327	626	49	,	,	PUNCT
ejpam-6327	626	50	x2	x2	PROPN
ejpam-6327	626	51	∗y	∗y	PROPN
ejpam-6327	626	52	y2	y2	PROPN
ejpam-6327	626	53	)	)	PUNCT
ejpam-6327	626	54	,	,	PUNCT
ejpam-6327	626	55	µ×	µ×	X
ejpam-6327	626	56	ν(y1	ν(y1	NOUN
ejpam-6327	626	57	,	,	PUNCT
ejpam-6327	626	58	y2	y2	PROPN
ejpam-6327	626	59	)	)	PUNCT
ejpam-6327	626	60	}	}	PUNCT
ejpam-6327	626	61	=	=	SYM
ejpam-6327	626	62	min{µ×	min{µ×	PROPN
ejpam-6327	626	63	ν((x1	ν((x1	PROPN
ejpam-6327	626	64	,	,	PUNCT
ejpam-6327	626	65	x2	x2	NUM
ejpam-6327	626	66	)	)	PUNCT
ejpam-6327	626	67	∗	∗	NOUN
ejpam-6327	626	68	(	(	PUNCT
ejpam-6327	626	69	y1	y1	NOUN
ejpam-6327	626	70	,	,	PUNCT
ejpam-6327	626	71	y2	y2	PROPN
ejpam-6327	626	72	)	)	PUNCT
ejpam-6327	626	73	)	)	PUNCT
ejpam-6327	626	74	,	,	PUNCT
ejpam-6327	626	75	µ×	µ×	X
ejpam-6327	626	76	ν(y1	ν(y1	NOUN
ejpam-6327	626	77	,	,	PUNCT
ejpam-6327	626	78	y2	y2	PROPN
ejpam-6327	626	79	)	)	PUNCT
ejpam-6327	626	80	}	}	PUNCT
ejpam-6327	626	81	.	.	PUNCT
ejpam-6327	627	1	(	(	PUNCT
ejpam-6327	627	2	iii	iii	X
ejpam-6327	627	3	)	)	PUNCT
ejpam-6327	627	4	let	let	VERB
ejpam-6327	627	5	(	(	PUNCT
ejpam-6327	627	6	x1	x1	ADJ
ejpam-6327	627	7	,	,	PUNCT
ejpam-6327	627	8	x2	x2	PROPN
ejpam-6327	627	9	)	)	PUNCT
ejpam-6327	627	10	,	,	PUNCT
ejpam-6327	627	11	(	(	PUNCT
ejpam-6327	627	12	y1	y1	INTJ
ejpam-6327	627	13	,	,	PUNCT
ejpam-6327	627	14	y2	y2	NOUN
ejpam-6327	627	15	)	)	PUNCT
ejpam-6327	627	16	∈	∈	PROPN
ejpam-6327	628	1	x	x	PUNCT
ejpam-6327	628	2	×	×	NOUN
ejpam-6327	628	3	y	y	PROPN
ejpam-6327	628	4	.	.	PUNCT
ejpam-6327	629	1	then	then	ADV
ejpam-6327	629	2	µ×	µ×	X
ejpam-6327	629	3	ν((x1	ν((x1	ADJ
ejpam-6327	629	4	,	,	PUNCT
ejpam-6327	629	5	x2)(y1	x2)(y1	NUM
ejpam-6327	629	6	,	,	PUNCT
ejpam-6327	629	7	y2	y2	PROPN
ejpam-6327	629	8	)	)	PUNCT
ejpam-6327	629	9	)	)	PUNCT
ejpam-6327	630	1	=	=	PRON
ejpam-6327	630	2	µ×	µ×	X
ejpam-6327	630	3	ν(x1	ν(x1	NOUN
ejpam-6327	630	4	·	·	PUNCT
ejpam-6327	630	5	x	x	SYM
ejpam-6327	630	6	y1	y1	PROPN
ejpam-6327	630	7	,	,	PUNCT
ejpam-6327	630	8	x2	x2	PROPN
ejpam-6327	630	9	·	·	PUNCT
ejpam-6327	630	10	y	y	PROPN
ejpam-6327	630	11	y2	y2	PROPN
ejpam-6327	630	12	)	)	PUNCT
ejpam-6327	630	13	=	=	SYM
ejpam-6327	631	1	min{µ(x1	min{µ(x1	NOUN
ejpam-6327	631	2	·	·	PUNCT
ejpam-6327	631	3	x	x	SYM
ejpam-6327	631	4	y1	y1	PROPN
ejpam-6327	631	5	)	)	PUNCT
ejpam-6327	631	6	,	,	PUNCT
ejpam-6327	631	7	ν(x2	ν(x2	NOUN
ejpam-6327	631	8	·	·	PUNCT
ejpam-6327	631	9	y	y	PROPN
ejpam-6327	631	10	y2	y2	PROPN
ejpam-6327	631	11	)	)	PUNCT
ejpam-6327	631	12	}	}	PUNCT
ejpam-6327	631	13	≥	≥	NOUN
ejpam-6327	631	14	min{min{µ(x1	min{min{µ(x1	NOUN
ejpam-6327	631	15	)	)	PUNCT
ejpam-6327	631	16	,	,	PUNCT
ejpam-6327	631	17	µ(y1)},min{ν(x2	µ(y1)},min{ν(x2	NOUN
ejpam-6327	631	18	)	)	PUNCT
ejpam-6327	631	19	,	,	PUNCT
ejpam-6327	631	20	ν(y2	ν(y2	NOUN
ejpam-6327	631	21	)	)	PUNCT
ejpam-6327	631	22	}	}	PUNCT
ejpam-6327	631	23	}	}	PUNCT
ejpam-6327	631	24	=	=	SYM
ejpam-6327	631	25	min{min{µ(x1	min{min{µ(x1	X
ejpam-6327	631	26	)	)	PUNCT
ejpam-6327	631	27	,	,	PUNCT
ejpam-6327	631	28	ν(x2)},min{µ(y1	ν(x2)},min{µ(y1	PROPN
ejpam-6327	631	29	)	)	PUNCT
ejpam-6327	631	30	,	,	PUNCT
ejpam-6327	631	31	ν(y2	ν(y2	NOUN
ejpam-6327	631	32	)	)	PUNCT
ejpam-6327	631	33	}	}	PUNCT
ejpam-6327	631	34	}	}	PUNCT
ejpam-6327	632	1	=	=	SYM
ejpam-6327	632	2	min{µ×	min{µ×	PROPN
ejpam-6327	632	3	ν(x1	ν(x1	NOUN
ejpam-6327	632	4	,	,	PUNCT
ejpam-6327	632	5	x2	x2	PROPN
ejpam-6327	632	6	)	)	PUNCT
ejpam-6327	632	7	,	,	PUNCT
ejpam-6327	632	8	µ×	µ×	X
ejpam-6327	632	9	ν(y1	ν(y1	NOUN
ejpam-6327	632	10	,	,	PUNCT
ejpam-6327	632	11	y2	y2	PROPN
ejpam-6327	632	12	)	)	PUNCT
ejpam-6327	632	13	}	}	PUNCT
ejpam-6327	632	14	.	.	PUNCT
ejpam-6327	633	1	therefore	therefore	ADV
ejpam-6327	633	2	,	,	PUNCT
ejpam-6327	633	3	µ×	µ×	PRON
ejpam-6327	633	4	ν	ν	NOUN
ejpam-6327	633	5	is	be	AUX
ejpam-6327	633	6	a	a	DET
ejpam-6327	633	7	fuzzy	fuzzy	ADJ
ejpam-6327	633	8	ks	ks	NOUN
ejpam-6327	633	9	-	-	NOUN
ejpam-6327	633	10	ideal	ideal	NOUN
ejpam-6327	633	11	of	of	ADP
ejpam-6327	633	12	x	x	SYM
ejpam-6327	633	13	×	×	PROPN
ejpam-6327	633	14	y	y	PROPN
ejpam-6327	633	15	.	.	PUNCT
ejpam-6327	634	1	h.	h.	PROPN
ejpam-6327	634	2	sarapuddin	sarapuddin	PROPN
ejpam-6327	634	3	,	,	PUNCT
ejpam-6327	634	4	j.	j.	PROPN
ejpam-6327	634	5	vilela	vilela	PROPN
ejpam-6327	634	6	/	/	SYM
ejpam-6327	634	7	eur	eur	PROPN
ejpam-6327	634	8	.	.	PUNCT
ejpam-6327	635	1	j.	j.	PROPN
ejpam-6327	635	2	pure	pure	PROPN
ejpam-6327	635	3	appl	appl	PROPN
ejpam-6327	635	4	.	.	PROPN
ejpam-6327	635	5	math	math	PROPN
ejpam-6327	635	6	,	,	PUNCT
ejpam-6327	635	7	18	18	NUM
ejpam-6327	635	8	(	(	PUNCT
ejpam-6327	635	9	3	3	NUM
ejpam-6327	635	10	)	)	PUNCT
ejpam-6327	635	11	(	(	PUNCT
ejpam-6327	635	12	2025	2025	NUM
ejpam-6327	635	13	)	)	PUNCT
ejpam-6327	635	14	,	,	PUNCT
ejpam-6327	635	15	6327	6327	NUM
ejpam-6327	635	16	19	19	NUM
ejpam-6327	635	17	of	of	ADP
ejpam-6327	635	18	23	23	NUM
ejpam-6327	635	19	corollary	corollary	ADJ
ejpam-6327	635	20	2	2	NUM
ejpam-6327	635	21	.	.	PUNCT
ejpam-6327	636	1	let	let	VERB
ejpam-6327	636	2	µ	µ	NOUN
ejpam-6327	636	3	and	and	CCONJ
ejpam-6327	636	4	ν	ν	PROPN
ejpam-6327	636	5	be	be	AUX
ejpam-6327	636	6	non	non	ADJ
ejpam-6327	636	7	-	-	ADJ
ejpam-6327	636	8	zero	zero	ADJ
ejpam-6327	636	9	fuzzy	fuzzy	ADJ
ejpam-6327	636	10	ks	ks	NOUN
ejpam-6327	636	11	-	-	PUNCT
ejpam-6327	636	12	ideals	ideal	NOUN
ejpam-6327	636	13	of	of	ADP
ejpam-6327	636	14	ks	ks	NOUN
ejpam-6327	636	15	-	-	PUNCT
ejpam-6327	636	16	semigroups	semigroups	X
ejpam-6327	636	17	x	x	X
ejpam-6327	636	18	and	and	CCONJ
ejpam-6327	636	19	y	y	PROPN
ejpam-6327	636	20	,	,	PUNCT
ejpam-6327	636	21	respectively	respectively	ADV
ejpam-6327	636	22	,	,	PUNCT
ejpam-6327	636	23	both	both	PRON
ejpam-6327	636	24	satisfying	satisfy	VERB
ejpam-6327	636	25	condition	condition	NOUN
ejpam-6327	636	26	(	(	PUNCT
ejpam-6327	636	27	c	c	NOUN
ejpam-6327	636	28	)	)	PUNCT
ejpam-6327	636	29	.	.	PUNCT
ejpam-6327	637	1	then	then	ADV
ejpam-6327	637	2	µ×ν	µ×ν	PROPN
ejpam-6327	637	3	is	be	AUX
ejpam-6327	637	4	a	a	DET
ejpam-6327	637	5	non	non	ADJ
ejpam-6327	637	6	-	-	ADJ
ejpam-6327	637	7	zero	zero	ADJ
ejpam-6327	637	8	fuzzy	fuzzy	ADJ
ejpam-6327	637	9	ks	ks	NOUN
ejpam-6327	637	10	-	-	NOUN
ejpam-6327	637	11	ideal	ideal	NOUN
ejpam-6327	637	12	of	of	ADP
ejpam-6327	637	13	x×y	x×y	PUNCT
ejpam-6327	637	14	satisfying	satisfy	VERB
ejpam-6327	637	15	condition	condition	NOUN
ejpam-6327	637	16	(	(	PUNCT
ejpam-6327	637	17	c	c	NOUN
ejpam-6327	637	18	)	)	PUNCT
ejpam-6327	637	19	.	.	PUNCT
ejpam-6327	638	1	proof	proof	NOUN
ejpam-6327	638	2	.	.	PUNCT
ejpam-6327	639	1	let	let	VERB
ejpam-6327	639	2	µ	µ	NOUN
ejpam-6327	639	3	and	and	CCONJ
ejpam-6327	639	4	ν	ν	PROPN
ejpam-6327	639	5	be	be	AUX
ejpam-6327	639	6	non	non	ADJ
ejpam-6327	639	7	-	-	ADJ
ejpam-6327	639	8	zero	zero	ADJ
ejpam-6327	639	9	fuzzy	fuzzy	ADJ
ejpam-6327	639	10	ks	ks	NOUN
ejpam-6327	639	11	-	-	PUNCT
ejpam-6327	639	12	ideals	ideal	NOUN
ejpam-6327	639	13	of	of	ADP
ejpam-6327	639	14	ks	ks	NOUN
ejpam-6327	639	15	-	-	PUNCT
ejpam-6327	639	16	semigroups	semigroups	X
ejpam-6327	639	17	x	x	X
ejpam-6327	639	18	and	and	CCONJ
ejpam-6327	639	19	y	y	PROPN
ejpam-6327	639	20	,	,	PUNCT
ejpam-6327	639	21	respectively	respectively	ADV
ejpam-6327	639	22	,	,	PUNCT
ejpam-6327	639	23	both	both	PRON
ejpam-6327	639	24	satisfying	satisfy	VERB
ejpam-6327	639	25	condition	condition	NOUN
ejpam-6327	639	26	(	(	PUNCT
ejpam-6327	639	27	c	c	NOUN
ejpam-6327	639	28	)	)	PUNCT
ejpam-6327	639	29	.	.	PUNCT
ejpam-6327	640	1	then	then	ADV
ejpam-6327	640	2	there	there	PRON
ejpam-6327	640	3	exist	exist	VERB
ejpam-6327	640	4	x	x	X
ejpam-6327	640	5	∈	∈	PROPN
ejpam-6327	640	6	x	x	X
ejpam-6327	640	7	and	and	CCONJ
ejpam-6327	640	8	y	y	PROPN
ejpam-6327	640	9	∈	∈	PROPN
ejpam-6327	640	10	y	y	PROPN
ejpam-6327	640	11	such	such	ADJ
ejpam-6327	640	12	that	that	SCONJ
ejpam-6327	640	13	µ(x	µ(x	VERB
ejpam-6327	640	14	)	)	PUNCT
ejpam-6327	640	15	̸=	̸=	PROPN
ejpam-6327	640	16	0	0	NUM
ejpam-6327	640	17	and	and	CCONJ
ejpam-6327	640	18	ν(y	ν(y	PROPN
ejpam-6327	640	19	)	)	PUNCT
ejpam-6327	640	20	̸=	̸=	PROPN
ejpam-6327	640	21	0	0	NUM
ejpam-6327	640	22	.	.	PUNCT
ejpam-6327	641	1	thus	thus	ADV
ejpam-6327	641	2	,	,	PUNCT
ejpam-6327	641	3	µ	µ	PRON
ejpam-6327	641	4	×	×	NOUN
ejpam-6327	641	5	ν(x	ν(x	PROPN
ejpam-6327	641	6	,	,	PUNCT
ejpam-6327	641	7	y	y	NOUN
ejpam-6327	641	8	)	)	PUNCT
ejpam-6327	641	9	=	=	SYM
ejpam-6327	641	10	min{µ(x	min{µ(x	NOUN
ejpam-6327	641	11	)	)	PUNCT
ejpam-6327	641	12	,	,	PUNCT
ejpam-6327	641	13	ν(y	ν(y	PROPN
ejpam-6327	641	14	)	)	PUNCT
ejpam-6327	641	15	}	}	PUNCT
ejpam-6327	642	1	=	=	SYM
ejpam-6327	642	2	̸	̸	NUM
ejpam-6327	642	3	0	0	NUM
ejpam-6327	642	4	.	.	PUNCT
ejpam-6327	643	1	thus	thus	ADV
ejpam-6327	643	2	,	,	PUNCT
ejpam-6327	643	3	µ	µ	DET
ejpam-6327	643	4	×	×	NOUN
ejpam-6327	643	5	ν	ν	NOUN
ejpam-6327	643	6	is	be	AUX
ejpam-6327	643	7	non	non	ADJ
ejpam-6327	643	8	-	-	ADJ
ejpam-6327	643	9	zero	zero	NUM
ejpam-6327	643	10	.	.	PUNCT
ejpam-6327	644	1	by	by	ADP
ejpam-6327	644	2	proposition	proposition	NOUN
ejpam-6327	644	3	6	6	NUM
ejpam-6327	644	4	,	,	PUNCT
ejpam-6327	644	5	µ×ν	µ×ν	PROPN
ejpam-6327	644	6	is	be	AUX
ejpam-6327	644	7	a	a	DET
ejpam-6327	644	8	fuzzy	fuzzy	ADJ
ejpam-6327	644	9	ks	ks	NOUN
ejpam-6327	644	10	-	-	NOUN
ejpam-6327	644	11	ideal	ideal	NOUN
ejpam-6327	644	12	of	of	ADP
ejpam-6327	644	13	x×y	x×y	PROPN
ejpam-6327	644	14	.	.	PUNCT
ejpam-6327	645	1	moreover	moreover	ADV
ejpam-6327	645	2	,	,	PUNCT
ejpam-6327	645	3	since	since	SCONJ
ejpam-6327	645	4	µ	µ	NOUN
ejpam-6327	645	5	and	and	CCONJ
ejpam-6327	645	6	ν	ν	NOUN
ejpam-6327	645	7	satisfy	satisfy	VERB
ejpam-6327	645	8	condition	condition	NOUN
ejpam-6327	645	9	(	(	PUNCT
ejpam-6327	645	10	c	c	NOUN
ejpam-6327	645	11	)	)	PUNCT
ejpam-6327	645	12	,	,	PUNCT
ejpam-6327	645	13	for	for	ADP
ejpam-6327	645	14	all	all	PRON
ejpam-6327	645	15	(	(	PUNCT
ejpam-6327	645	16	x1	x1	PROPN
ejpam-6327	645	17	,	,	PUNCT
ejpam-6327	645	18	x2	x2	PROPN
ejpam-6327	645	19	)	)	PUNCT
ejpam-6327	645	20	,	,	PUNCT
ejpam-6327	645	21	(	(	PUNCT
ejpam-6327	645	22	y1	y1	INTJ
ejpam-6327	645	23	,	,	PUNCT
ejpam-6327	645	24	y2	y2	NOUN
ejpam-6327	645	25	)	)	PUNCT
ejpam-6327	645	26	∈	∈	PROPN
ejpam-6327	645	27	x	x	SYM
ejpam-6327	645	28	×	×	PROPN
ejpam-6327	645	29	y	y	PROPN
ejpam-6327	645	30	,	,	PUNCT
ejpam-6327	645	31	µ×	µ×	X
ejpam-6327	645	32	ν((x1	ν((x1	ADJ
ejpam-6327	645	33	,	,	PUNCT
ejpam-6327	645	34	x2)(y1	x2)(y1	NUM
ejpam-6327	645	35	,	,	PUNCT
ejpam-6327	645	36	y2	y2	PROPN
ejpam-6327	645	37	)	)	PUNCT
ejpam-6327	645	38	)	)	PUNCT
ejpam-6327	646	1	=	=	PRON
ejpam-6327	646	2	µ×	µ×	X
ejpam-6327	646	3	ν(x1	ν(x1	NOUN
ejpam-6327	646	4	·	·	PUNCT
ejpam-6327	646	5	x	x	SYM
ejpam-6327	646	6	y1	y1	PROPN
ejpam-6327	646	7	,	,	PUNCT
ejpam-6327	646	8	x2	x2	PROPN
ejpam-6327	646	9	·	·	PUNCT
ejpam-6327	646	10	y	y	PROPN
ejpam-6327	646	11	y2	y2	PROPN
ejpam-6327	646	12	)	)	PUNCT
ejpam-6327	646	13	=	=	SYM
ejpam-6327	647	1	min{µ(x1	min{µ(x1	NOUN
ejpam-6327	647	2	·	·	PUNCT
ejpam-6327	647	3	x	x	SYM
ejpam-6327	647	4	y1	y1	PROPN
ejpam-6327	647	5	)	)	PUNCT
ejpam-6327	647	6	,	,	PUNCT
ejpam-6327	647	7	ν(x2	ν(x2	NOUN
ejpam-6327	647	8	·	·	PUNCT
ejpam-6327	647	9	y	y	PROPN
ejpam-6327	647	10	y2	y2	PROPN
ejpam-6327	647	11	)	)	PUNCT
ejpam-6327	647	12	}	}	PUNCT
ejpam-6327	647	13	≥	≥	NOUN
ejpam-6327	647	14	min{µ(x1	min{µ(x1	NOUN
ejpam-6327	647	15	)	)	PUNCT
ejpam-6327	647	16	,	,	PUNCT
ejpam-6327	647	17	ν(x2	ν(x2	NOUN
ejpam-6327	647	18	)	)	PUNCT
ejpam-6327	647	19	}	}	PUNCT
ejpam-6327	648	1	=	=	PUNCT
ejpam-6327	648	2	µ×	µ×	X
ejpam-6327	648	3	ν(x1	ν(x1	NOUN
ejpam-6327	648	4	,	,	PUNCT
ejpam-6327	648	5	x2	x2	PROPN
ejpam-6327	648	6	)	)	PUNCT
ejpam-6327	648	7	and	and	CCONJ
ejpam-6327	648	8	µ×	µ×	X
ejpam-6327	648	9	ν((x1	ν((x1	ADJ
ejpam-6327	648	10	,	,	PUNCT
ejpam-6327	648	11	x2)(y1	x2)(y1	NUM
ejpam-6327	648	12	,	,	PUNCT
ejpam-6327	648	13	y2	y2	PROPN
ejpam-6327	648	14	)	)	PUNCT
ejpam-6327	648	15	)	)	PUNCT
ejpam-6327	648	16	=	=	PRON
ejpam-6327	649	1	µ×	µ×	X
ejpam-6327	649	2	ν(x1	ν(x1	NOUN
ejpam-6327	649	3	·	·	PUNCT
ejpam-6327	649	4	x	x	SYM
ejpam-6327	649	5	y1	y1	PROPN
ejpam-6327	649	6	,	,	PUNCT
ejpam-6327	649	7	x2	x2	PROPN
ejpam-6327	649	8	·	·	PUNCT
ejpam-6327	649	9	y	y	PROPN
ejpam-6327	649	10	y2	y2	PROPN
ejpam-6327	649	11	)	)	PUNCT
ejpam-6327	649	12	=	=	SYM
ejpam-6327	650	1	min{µ(x1	min{µ(x1	NOUN
ejpam-6327	650	2	·	·	PUNCT
ejpam-6327	650	3	x	x	SYM
ejpam-6327	650	4	y1	y1	PROPN
ejpam-6327	650	5	)	)	PUNCT
ejpam-6327	650	6	,	,	PUNCT
ejpam-6327	650	7	ν(x2	ν(x2	NOUN
ejpam-6327	650	8	·	·	PUNCT
ejpam-6327	650	9	y	y	PROPN
ejpam-6327	650	10	y2	y2	PROPN
ejpam-6327	650	11	)	)	PUNCT
ejpam-6327	650	12	}	}	PUNCT
ejpam-6327	650	13	≥	≥	NOUN
ejpam-6327	650	14	min{µ(y1	min{µ(y1	NOUN
ejpam-6327	650	15	)	)	PUNCT
ejpam-6327	650	16	,	,	PUNCT
ejpam-6327	650	17	ν(y2	ν(y2	NOUN
ejpam-6327	650	18	)	)	PUNCT
ejpam-6327	650	19	}	}	PUNCT
ejpam-6327	651	1	=	=	SYM
ejpam-6327	651	2	µ×	µ×	X
ejpam-6327	651	3	ν(y1	ν(y1	NOUN
ejpam-6327	651	4	,	,	PUNCT
ejpam-6327	651	5	y2	y2	PROPN
ejpam-6327	651	6	)	)	PUNCT
ejpam-6327	651	7	.	.	PUNCT
ejpam-6327	652	1	therefore	therefore	ADV
ejpam-6327	652	2	,	,	PUNCT
ejpam-6327	652	3	µ×	µ×	PRON
ejpam-6327	652	4	ν	ν	ADP
ejpam-6327	652	5	satisfies	satisfie	NOUN
ejpam-6327	652	6	condition	condition	NOUN
ejpam-6327	652	7	(	(	PUNCT
ejpam-6327	652	8	c	c	NOUN
ejpam-6327	652	9	)	)	PUNCT
ejpam-6327	652	10	.	.	PUNCT
ejpam-6327	653	1	the	the	DET
ejpam-6327	653	2	following	follow	VERB
ejpam-6327	653	3	corollary	corollary	NOUN
ejpam-6327	653	4	follows	follow	VERB
ejpam-6327	653	5	from	from	ADP
ejpam-6327	653	6	theorems	theorem	NOUN
ejpam-6327	653	7	16	16	NUM
ejpam-6327	653	8	,	,	PUNCT
ejpam-6327	653	9	17	17	NUM
ejpam-6327	653	10	and	and	CCONJ
ejpam-6327	653	11	18	18	NUM
ejpam-6327	653	12	.	.	PUNCT
ejpam-6327	654	1	corollary	corollary	ADJ
ejpam-6327	654	2	3	3	NUM
ejpam-6327	654	3	.	.	PUNCT
ejpam-6327	655	1	let	let	VERB
ejpam-6327	655	2	µ	µ	NOUN
ejpam-6327	655	3	and	and	CCONJ
ejpam-6327	655	4	ν	ν	PROPN
ejpam-6327	655	5	be	be	AUX
ejpam-6327	655	6	fuzzy	fuzzy	ADJ
ejpam-6327	655	7	ks	ks	NOUN
ejpam-6327	655	8	-	-	PUNCT
ejpam-6327	655	9	ideals	ideal	NOUN
ejpam-6327	655	10	of	of	ADP
ejpam-6327	655	11	ks	ks	NOUN
ejpam-6327	655	12	-	-	PUNCT
ejpam-6327	655	13	semigroups	semigroups	X
ejpam-6327	655	14	x	x	X
ejpam-6327	655	15	and	and	CCONJ
ejpam-6327	655	16	y	y	PROPN
ejpam-6327	655	17	,	,	PUNCT
ejpam-6327	655	18	respectively	respectively	ADV
ejpam-6327	655	19	.	.	PUNCT
ejpam-6327	656	1	(	(	PUNCT
ejpam-6327	656	2	i	i	NOUN
ejpam-6327	656	3	)	)	PUNCT
ejpam-6327	656	4	if	if	SCONJ
ejpam-6327	656	5	µ×ν	µ×ν	PROPN
ejpam-6327	656	6	is	be	AUX
ejpam-6327	656	7	a	a	DET
ejpam-6327	656	8	non	non	ADJ
ejpam-6327	656	9	-	-	ADJ
ejpam-6327	656	10	zero	zero	ADJ
ejpam-6327	656	11	fuzzy	fuzzy	ADJ
ejpam-6327	656	12	commutative	commutative	ADJ
ejpam-6327	656	13	ks	ks	NOUN
ejpam-6327	656	14	-	-	PUNCT
ejpam-6327	656	15	ideal	ideal	NOUN
ejpam-6327	656	16	of	of	ADP
ejpam-6327	656	17	x×y	x×y	PUNCT
ejpam-6327	656	18	satisfying	satisfy	VERB
ejpam-6327	656	19	condition	condition	NOUN
ejpam-6327	656	20	(	(	PUNCT
ejpam-6327	656	21	c	c	NOUN
ejpam-6327	656	22	)	)	PUNCT
ejpam-6327	656	23	,	,	PUNCT
ejpam-6327	656	24	then	then	ADV
ejpam-6327	656	25	x	x	SYM
ejpam-6327	656	26	×	×	NOUN
ejpam-6327	656	27	y/µ×	y/µ×	NOUN
ejpam-6327	656	28	ν	ν	PROPN
ejpam-6327	656	29	is	be	AUX
ejpam-6327	656	30	a	a	DET
ejpam-6327	656	31	commutative	commutative	ADJ
ejpam-6327	656	32	ks	ks	NOUN
ejpam-6327	656	33	-	-	PUNCT
ejpam-6327	656	34	semigroup	semigroup	NOUN
ejpam-6327	656	35	.	.	PUNCT
ejpam-6327	657	1	(	(	PUNCT
ejpam-6327	657	2	ii	ii	NOUN
ejpam-6327	657	3	)	)	PUNCT
ejpam-6327	657	4	if	if	SCONJ
ejpam-6327	657	5	µ	µ	PRON
ejpam-6327	657	6	×	×	NOUN
ejpam-6327	657	7	ν	ν	NOUN
ejpam-6327	657	8	is	be	AUX
ejpam-6327	657	9	a	a	DET
ejpam-6327	657	10	non	non	ADJ
ejpam-6327	657	11	-	-	ADJ
ejpam-6327	657	12	zero	zero	ADJ
ejpam-6327	657	13	fuzzy	fuzzy	ADJ
ejpam-6327	657	14	ks	ks	NOUN
ejpam-6327	657	15	-	-	ADJ
ejpam-6327	657	16	p	p	NOUN
ejpam-6327	657	17	-	-	PUNCT
ejpam-6327	657	18	ideal	ideal	NOUN
ejpam-6327	657	19	of	of	ADP
ejpam-6327	657	20	x	x	SYM
ejpam-6327	657	21	×	×	NOUN
ejpam-6327	657	22	y	y	NOUN
ejpam-6327	657	23	satisfying	satisfy	VERB
ejpam-6327	657	24	condition	condition	NOUN
ejpam-6327	657	25	(	(	PUNCT
ejpam-6327	657	26	c	c	NOUN
ejpam-6327	657	27	)	)	PUNCT
ejpam-6327	657	28	,	,	PUNCT
ejpam-6327	657	29	then	then	ADV
ejpam-6327	657	30	x	x	SYM
ejpam-6327	657	31	×	×	NOUN
ejpam-6327	657	32	y/µ×	y/µ×	NOUN
ejpam-6327	657	33	ν	ν	PROPN
ejpam-6327	657	34	is	be	AUX
ejpam-6327	657	35	a	a	DET
ejpam-6327	657	36	positive	positive	ADJ
ejpam-6327	657	37	implicative	implicative	ADJ
ejpam-6327	657	38	ks	ks	NOUN
ejpam-6327	657	39	-	-	PUNCT
ejpam-6327	657	40	semigroup	semigroup	NOUN
ejpam-6327	657	41	.	.	PUNCT
ejpam-6327	658	1	(	(	PUNCT
ejpam-6327	658	2	iii	iii	X
ejpam-6327	658	3	)	)	PUNCT
ejpam-6327	658	4	if	if	SCONJ
ejpam-6327	658	5	µ×	µ×	PRON
ejpam-6327	658	6	ν	ν	NOUN
ejpam-6327	658	7	is	be	AUX
ejpam-6327	658	8	a	a	DET
ejpam-6327	658	9	non	non	ADJ
ejpam-6327	658	10	-	-	ADJ
ejpam-6327	658	11	zero	zero	ADJ
ejpam-6327	658	12	fuzzy	fuzzy	ADJ
ejpam-6327	658	13	implicative	implicative	ADJ
ejpam-6327	658	14	ks	ks	NOUN
ejpam-6327	658	15	-	-	PUNCT
ejpam-6327	658	16	ideal	ideal	NOUN
ejpam-6327	658	17	of	of	ADP
ejpam-6327	658	18	x	x	SYM
ejpam-6327	658	19	×	×	NOUN
ejpam-6327	658	20	y	y	NOUN
ejpam-6327	658	21	satisfying	satisfy	VERB
ejpam-6327	658	22	condition	condition	NOUN
ejpam-6327	658	23	(	(	PUNCT
ejpam-6327	658	24	c	c	NOUN
ejpam-6327	658	25	)	)	PUNCT
ejpam-6327	658	26	,	,	PUNCT
ejpam-6327	658	27	then	then	ADV
ejpam-6327	658	28	x	x	SYM
ejpam-6327	658	29	×	×	NOUN
ejpam-6327	658	30	y/µ×	y/µ×	NOUN
ejpam-6327	658	31	ν	ν	PROPN
ejpam-6327	658	32	is	be	AUX
ejpam-6327	658	33	an	an	DET
ejpam-6327	658	34	implicative	implicative	ADJ
ejpam-6327	658	35	ks	ks	NOUN
ejpam-6327	658	36	-	-	PUNCT
ejpam-6327	658	37	semigroup	semigroup	PROPN
ejpam-6327	658	38	.	.	PUNCT
ejpam-6327	659	1	recall	recall	VERB
ejpam-6327	659	2	that	that	PRON
ejpam-6327	659	3	in	in	ADP
ejpam-6327	659	4	[	[	X
ejpam-6327	659	5	5	5	NUM
ejpam-6327	659	6	]	]	PUNCT
ejpam-6327	659	7	,	,	PUNCT
ejpam-6327	659	8	if	if	SCONJ
ejpam-6327	659	9	f	f	X
ejpam-6327	659	10	:	:	PUNCT
ejpam-6327	659	11	x	x	X
ejpam-6327	659	12	→	→	SYM
ejpam-6327	659	13	y	y	PROPN
ejpam-6327	659	14	is	be	AUX
ejpam-6327	659	15	an	an	DET
ejpam-6327	659	16	epimorphism	epimorphism	NOUN
ejpam-6327	659	17	of	of	ADP
ejpam-6327	659	18	ks	ks	PROPN
ejpam-6327	659	19	-	-	PUNCT
ejpam-6327	659	20	semigroup	semigroup	NOUN
ejpam-6327	659	21	,	,	PUNCT
ejpam-6327	659	22	thenx/	thenx/	NUM
ejpam-6327	659	23	ker	ker	NOUN
ejpam-6327	659	24	f	f	X
ejpam-6327	659	25	∼=	∼=	PROPN
ejpam-6327	659	26	y	y	PROPN
ejpam-6327	659	27	.	.	PUNCT
ejpam-6327	660	1	theorem	theorem	PROPN
ejpam-6327	660	2	21	21	NUM
ejpam-6327	660	3	.	.	PUNCT
ejpam-6327	661	1	let	let	VERB
ejpam-6327	661	2	µ	µ	NOUN
ejpam-6327	661	3	and	and	CCONJ
ejpam-6327	661	4	ν	ν	PROPN
ejpam-6327	661	5	be	be	AUX
ejpam-6327	661	6	non	non	ADJ
ejpam-6327	661	7	-	-	ADJ
ejpam-6327	661	8	zero	zero	ADJ
ejpam-6327	661	9	fuzzy	fuzzy	ADJ
ejpam-6327	661	10	ks	ks	NOUN
ejpam-6327	661	11	-	-	PUNCT
ejpam-6327	661	12	ideals	ideal	NOUN
ejpam-6327	661	13	of	of	ADP
ejpam-6327	661	14	ks	ks	NOUN
ejpam-6327	661	15	-	-	PUNCT
ejpam-6327	661	16	semigroups	semigroups	X
ejpam-6327	661	17	x	x	X
ejpam-6327	661	18	and	and	CCONJ
ejpam-6327	661	19	y	y	PROPN
ejpam-6327	661	20	,	,	PUNCT
ejpam-6327	661	21	respectively	respectively	ADV
ejpam-6327	661	22	,	,	PUNCT
ejpam-6327	661	23	both	both	PRON
ejpam-6327	661	24	satisfying	satisfy	VERB
ejpam-6327	661	25	condition	condition	NOUN
ejpam-6327	661	26	(	(	PUNCT
ejpam-6327	661	27	c	c	NOUN
ejpam-6327	661	28	)	)	PUNCT
ejpam-6327	661	29	.	.	PUNCT
ejpam-6327	662	1	then	then	ADV
ejpam-6327	662	2	x	x	X
ejpam-6327	662	3	×	×	PROPN
ejpam-6327	662	4	y	y	PROPN
ejpam-6327	662	5	µ×	µ×	X
ejpam-6327	662	6	ν	ν	PRON
ejpam-6327	662	7	∼=	∼=	PROPN
ejpam-6327	662	8	x/µ×	x/µ×	NOUN
ejpam-6327	662	9	y	y	PROPN
ejpam-6327	662	10	/	/	SYM
ejpam-6327	662	11	ν	ν	PROPN
ejpam-6327	662	12	.	.	PUNCT
ejpam-6327	662	13	proof	proof	NOUN
ejpam-6327	662	14	.	.	PUNCT
ejpam-6327	663	1	let	let	VERB
ejpam-6327	663	2	µ	µ	NOUN
ejpam-6327	663	3	and	and	CCONJ
ejpam-6327	663	4	ν	ν	PROPN
ejpam-6327	663	5	be	be	AUX
ejpam-6327	663	6	non	non	ADJ
ejpam-6327	663	7	-	-	ADJ
ejpam-6327	663	8	zero	zero	ADJ
ejpam-6327	663	9	fuzzy	fuzzy	ADJ
ejpam-6327	663	10	ks	ks	NOUN
ejpam-6327	663	11	-	-	PUNCT
ejpam-6327	663	12	ideals	ideal	NOUN
ejpam-6327	663	13	of	of	ADP
ejpam-6327	663	14	ks	ks	NOUN
ejpam-6327	663	15	-	-	PUNCT
ejpam-6327	663	16	semigroups	semigroups	X
ejpam-6327	663	17	x	x	X
ejpam-6327	663	18	and	and	CCONJ
ejpam-6327	663	19	y	y	PROPN
ejpam-6327	663	20	,	,	PUNCT
ejpam-6327	663	21	respectively	respectively	ADV
ejpam-6327	663	22	,	,	PUNCT
ejpam-6327	663	23	both	both	PRON
ejpam-6327	663	24	satisfying	satisfy	VERB
ejpam-6327	663	25	condition	condition	NOUN
ejpam-6327	663	26	(	(	PUNCT
ejpam-6327	663	27	c	c	NOUN
ejpam-6327	663	28	)	)	PUNCT
ejpam-6327	663	29	.	.	PUNCT
ejpam-6327	664	1	define	define	VERB
ejpam-6327	664	2	a	a	DET
ejpam-6327	664	3	map	map	NOUN
ejpam-6327	664	4	ψ	ψ	X
ejpam-6327	664	5	:	:	PUNCT
ejpam-6327	664	6	x	x	SYM
ejpam-6327	664	7	×	×	NOUN
ejpam-6327	664	8	y	y	PROPN
ejpam-6327	664	9	→	→	SYM
ejpam-6327	664	10	x/µ	x/µ	PROPN
ejpam-6327	664	11	×	×	PROPN
ejpam-6327	664	12	y	y	PROPN
ejpam-6327	664	13	/	/	SYM
ejpam-6327	664	14	ν	ν	NOUN
ejpam-6327	664	15	by	by	ADP
ejpam-6327	664	16	h.	h.	PROPN
ejpam-6327	664	17	sarapuddin	sarapuddin	PROPN
ejpam-6327	664	18	,	,	PUNCT
ejpam-6327	664	19	j.	j.	PROPN
ejpam-6327	664	20	vilela	vilela	PROPN
ejpam-6327	664	21	/	/	SYM
ejpam-6327	664	22	eur	eur	PROPN
ejpam-6327	664	23	.	.	PUNCT
ejpam-6327	665	1	j.	j.	PROPN
ejpam-6327	665	2	pure	pure	PROPN
ejpam-6327	665	3	appl	appl	PROPN
ejpam-6327	665	4	.	.	PROPN
ejpam-6327	665	5	math	math	PROPN
ejpam-6327	665	6	,	,	PUNCT
ejpam-6327	665	7	18	18	NUM
ejpam-6327	665	8	(	(	PUNCT
ejpam-6327	665	9	3	3	NUM
ejpam-6327	665	10	)	)	PUNCT
ejpam-6327	665	11	(	(	PUNCT
ejpam-6327	665	12	2025	2025	NUM
ejpam-6327	665	13	)	)	PUNCT
ejpam-6327	665	14	,	,	PUNCT
ejpam-6327	665	15	6327	6327	NUM
ejpam-6327	665	16	20	20	NUM
ejpam-6327	665	17	of	of	ADP
ejpam-6327	665	18	23	23	NUM
ejpam-6327	665	19	ψ(x	ψ(x	NOUN
ejpam-6327	665	20	,	,	PUNCT
ejpam-6327	665	21	y	y	NOUN
ejpam-6327	665	22	)	)	PUNCT
ejpam-6327	665	23	=	=	SYM
ejpam-6327	665	24	(	(	PUNCT
ejpam-6327	665	25	µx	µx	ADJ
ejpam-6327	665	26	,	,	PUNCT
ejpam-6327	665	27	νy	νy	NOUN
ejpam-6327	665	28	)	)	PUNCT
ejpam-6327	665	29	.	.	PUNCT
ejpam-6327	666	1	let	let	VERB
ejpam-6327	666	2	(	(	PUNCT
ejpam-6327	666	3	x1	x1	PROPN
ejpam-6327	666	4	,	,	PUNCT
ejpam-6327	666	5	y1	y1	PROPN
ejpam-6327	666	6	)	)	PUNCT
ejpam-6327	666	7	,	,	PUNCT
ejpam-6327	667	1	(	(	PUNCT
ejpam-6327	667	2	x2	x2	PROPN
ejpam-6327	667	3	,	,	PUNCT
ejpam-6327	667	4	y2	y2	NOUN
ejpam-6327	667	5	)	)	PUNCT
ejpam-6327	667	6	∈	∈	PROPN
ejpam-6327	667	7	x	x	PUNCT
ejpam-6327	667	8	×	×	NOUN
ejpam-6327	667	9	y	y	PROPN
ejpam-6327	667	10	such	such	ADJ
ejpam-6327	667	11	that	that	PRON
ejpam-6327	667	12	(	(	PUNCT
ejpam-6327	667	13	x1	x1	PROPN
ejpam-6327	667	14	,	,	PUNCT
ejpam-6327	667	15	y1	y1	NOUN
ejpam-6327	667	16	)	)	PUNCT
ejpam-6327	667	17	=	=	SYM
ejpam-6327	668	1	(	(	PUNCT
ejpam-6327	668	2	x2	x2	PROPN
ejpam-6327	668	3	,	,	PUNCT
ejpam-6327	668	4	y2	y2	PROPN
ejpam-6327	668	5	)	)	PUNCT
ejpam-6327	668	6	.	.	PUNCT
ejpam-6327	669	1	then	then	ADV
ejpam-6327	669	2	x1	x1	PROPN
ejpam-6327	669	3	=	=	SYM
ejpam-6327	669	4	x2	x2	PROPN
ejpam-6327	669	5	and	and	CCONJ
ejpam-6327	669	6	y1	y1	NOUN
ejpam-6327	669	7	=	=	PUNCT
ejpam-6327	669	8	y2	y2	PROPN
ejpam-6327	669	9	.	.	PUNCT
ejpam-6327	670	1	thus	thus	ADV
ejpam-6327	670	2	,	,	PUNCT
ejpam-6327	670	3	ψ(x1	ψ(x1	ADJ
ejpam-6327	670	4	,	,	PUNCT
ejpam-6327	670	5	y1	y1	NOUN
ejpam-6327	670	6	)	)	PUNCT
ejpam-6327	670	7	=	=	SYM
ejpam-6327	670	8	(	(	PUNCT
ejpam-6327	670	9	µx1	µx1	NOUN
ejpam-6327	670	10	,	,	PUNCT
ejpam-6327	670	11	νy1	νy1	NOUN
ejpam-6327	670	12	)	)	PUNCT
ejpam-6327	671	1	=	=	PRON
ejpam-6327	671	2	(	(	PUNCT
ejpam-6327	671	3	µx2	µx2	PROPN
ejpam-6327	671	4	,	,	PUNCT
ejpam-6327	671	5	νy2	νy2	NOUN
ejpam-6327	671	6	)	)	PUNCT
ejpam-6327	671	7	=	=	SYM
ejpam-6327	671	8	ψ(x2	ψ(x2	NOUN
ejpam-6327	671	9	,	,	PUNCT
ejpam-6327	671	10	y2	y2	PROPN
ejpam-6327	671	11	)	)	PUNCT
ejpam-6327	671	12	.	.	PUNCT
ejpam-6327	672	1	thus	thus	ADV
ejpam-6327	672	2	,	,	PUNCT
ejpam-6327	672	3	ψ	ψ	X
ejpam-6327	672	4	is	be	AUX
ejpam-6327	672	5	well	well	ADV
ejpam-6327	672	6	-	-	PUNCT
ejpam-6327	672	7	defined	define	VERB
ejpam-6327	672	8	.	.	PUNCT
ejpam-6327	673	1	now	now	ADV
ejpam-6327	673	2	,	,	PUNCT
ejpam-6327	673	3	let	let	VERB
ejpam-6327	673	4	(	(	PUNCT
ejpam-6327	673	5	x1	x1	ADJ
ejpam-6327	673	6	,	,	PUNCT
ejpam-6327	673	7	y1	y1	PROPN
ejpam-6327	673	8	)	)	PUNCT
ejpam-6327	673	9	,	,	PUNCT
ejpam-6327	674	1	(	(	PUNCT
ejpam-6327	674	2	x2	x2	PROPN
ejpam-6327	674	3	,	,	PUNCT
ejpam-6327	674	4	y2	y2	NOUN
ejpam-6327	674	5	)	)	PUNCT
ejpam-6327	674	6	∈	∈	PROPN
ejpam-6327	674	7	x	x	PUNCT
ejpam-6327	674	8	×	×	NOUN
ejpam-6327	674	9	y	y	PROPN
ejpam-6327	674	10	.	.	PUNCT
ejpam-6327	675	1	then	then	ADV
ejpam-6327	675	2	ψ((x1	ψ((x1	NOUN
ejpam-6327	675	3	,	,	PUNCT
ejpam-6327	675	4	y1	y1	NOUN
ejpam-6327	675	5	)	)	PUNCT
ejpam-6327	675	6	∗	∗	NOUN
ejpam-6327	675	7	(	(	PUNCT
ejpam-6327	675	8	x2	x2	PROPN
ejpam-6327	675	9	,	,	PUNCT
ejpam-6327	675	10	y2	y2	PROPN
ejpam-6327	675	11	)	)	PUNCT
ejpam-6327	675	12	)	)	PUNCT
ejpam-6327	676	1	=	=	SYM
ejpam-6327	676	2	ψ(x1	ψ(x1	VERB
ejpam-6327	676	3	∗x	∗x	PROPN
ejpam-6327	676	4	x2	x2	PROPN
ejpam-6327	676	5	,	,	PUNCT
ejpam-6327	676	6	y1	y1	PROPN
ejpam-6327	676	7	∗y	∗y	PROPN
ejpam-6327	676	8	y2	y2	PROPN
ejpam-6327	676	9	)	)	PUNCT
ejpam-6327	676	10	=	=	PRON
ejpam-6327	676	11	(	(	PUNCT
ejpam-6327	676	12	µx1∗xx2	µx1∗xx2	NOUN
ejpam-6327	676	13	,	,	PUNCT
ejpam-6327	676	14	νy1∗y	νy1∗y	PROPN
ejpam-6327	676	15	y2	y2	PROPN
ejpam-6327	676	16	)	)	PUNCT
ejpam-6327	676	17	=	=	PRON
ejpam-6327	676	18	(	(	PUNCT
ejpam-6327	676	19	µx1	µx1	NOUN
ejpam-6327	676	20	∗x	∗x	PROPN
ejpam-6327	676	21	µx2	µx2	PROPN
ejpam-6327	676	22	,	,	PUNCT
ejpam-6327	676	23	νy1	νy1	VERB
ejpam-6327	676	24	∗y	∗y	PROPN
ejpam-6327	676	25	νy2	νy2	NOUN
ejpam-6327	676	26	)	)	PUNCT
ejpam-6327	677	1	=	=	SYM
ejpam-6327	677	2	(	(	PUNCT
ejpam-6327	677	3	µx1	µx1	NOUN
ejpam-6327	677	4	,	,	PUNCT
ejpam-6327	677	5	νy1	νy1	NOUN
ejpam-6327	677	6	)	)	PUNCT
ejpam-6327	677	7	∗	∗	NOUN
ejpam-6327	677	8	(	(	PUNCT
ejpam-6327	677	9	µx2	µx2	PROPN
ejpam-6327	677	10	,	,	PUNCT
ejpam-6327	677	11	νy2	νy2	NOUN
ejpam-6327	677	12	)	)	PUNCT
ejpam-6327	677	13	=	=	SYM
ejpam-6327	677	14	ψ(x1	ψ(x1	NOUN
ejpam-6327	677	15	,	,	PUNCT
ejpam-6327	677	16	y1	y1	NOUN
ejpam-6327	677	17	)	)	PUNCT
ejpam-6327	677	18	∗	∗	NOUN
ejpam-6327	677	19	ψ(x2	ψ(x2	NOUN
ejpam-6327	677	20	,	,	PUNCT
ejpam-6327	677	21	y2	y2	PROPN
ejpam-6327	677	22	)	)	PUNCT
ejpam-6327	677	23	and	and	CCONJ
ejpam-6327	677	24	ψ((x1	ψ((x1	NOUN
ejpam-6327	677	25	,	,	PUNCT
ejpam-6327	677	26	y1)(x2	y1)(x2	NOUN
ejpam-6327	677	27	,	,	PUNCT
ejpam-6327	677	28	y2	y2	PROPN
ejpam-6327	677	29	)	)	PUNCT
ejpam-6327	677	30	)	)	PUNCT
ejpam-6327	678	1	=	=	SYM
ejpam-6327	678	2	ψ(x1	ψ(x1	VERB
ejpam-6327	678	3	·	·	SYM
ejpam-6327	678	4	x	x	SYM
ejpam-6327	678	5	x2	x2	PROPN
ejpam-6327	678	6	,	,	PUNCT
ejpam-6327	678	7	y1	y1	PROPN
ejpam-6327	678	8	·	·	PUNCT
ejpam-6327	678	9	y	y	PROPN
ejpam-6327	678	10	y2	y2	PROPN
ejpam-6327	678	11	)	)	PUNCT
ejpam-6327	678	12	=	=	PRON
ejpam-6327	678	13	(	(	PUNCT
ejpam-6327	678	14	µx1·xx2	µx1·xx2	NOUN
ejpam-6327	678	15	,	,	PUNCT
ejpam-6327	678	16	νy1·y	νy1·y	PROPN
ejpam-6327	678	17	y2	y2	PROPN
ejpam-6327	678	18	)	)	PUNCT
ejpam-6327	678	19	=	=	PRON
ejpam-6327	678	20	(	(	PUNCT
ejpam-6327	678	21	µx1	µx1	NOUN
ejpam-6327	678	22	·	·	PUNCT
ejpam-6327	678	23	x	x	X
ejpam-6327	678	24	µx2	µx2	PROPN
ejpam-6327	678	25	,	,	PUNCT
ejpam-6327	678	26	νy1	νy1	NOUN
ejpam-6327	678	27	·	·	SYM
ejpam-6327	678	28	y	y	PROPN
ejpam-6327	678	29	νy2	νy2	NOUN
ejpam-6327	678	30	)	)	PUNCT
ejpam-6327	678	31	=	=	PRON
ejpam-6327	678	32	(	(	PUNCT
ejpam-6327	678	33	µx1	µx1	NOUN
ejpam-6327	678	34	,	,	PUNCT
ejpam-6327	678	35	νy1)(µx2	νy1)(µx2	NOUN
ejpam-6327	678	36	,	,	PUNCT
ejpam-6327	678	37	νy2	νy2	NOUN
ejpam-6327	678	38	)	)	PUNCT
ejpam-6327	678	39	=	=	SYM
ejpam-6327	678	40	ψ(x1	ψ(x1	NOUN
ejpam-6327	678	41	,	,	PUNCT
ejpam-6327	678	42	y1)ψ(x2	y1)ψ(x2	NOUN
ejpam-6327	678	43	,	,	PUNCT
ejpam-6327	678	44	y2	y2	PROPN
ejpam-6327	678	45	)	)	PUNCT
ejpam-6327	678	46	.	.	PUNCT
ejpam-6327	679	1	thus	thus	ADV
ejpam-6327	679	2	,	,	PUNCT
ejpam-6327	679	3	ψ	ψ	X
ejpam-6327	679	4	is	be	AUX
ejpam-6327	679	5	a	a	DET
ejpam-6327	679	6	homomorphism	homomorphism	NOUN
ejpam-6327	679	7	.	.	PUNCT
ejpam-6327	680	1	moreover	moreover	ADV
ejpam-6327	680	2	,	,	PUNCT
ejpam-6327	680	3	for	for	ADP
ejpam-6327	680	4	each	each	PRON
ejpam-6327	680	5	(	(	PUNCT
ejpam-6327	680	6	µx	µx	ADJ
ejpam-6327	680	7	,	,	PUNCT
ejpam-6327	680	8	νy	νy	NOUN
ejpam-6327	680	9	)	)	PUNCT
ejpam-6327	680	10	∈	∈	PROPN
ejpam-6327	681	1	x/µ	x/µ	X
ejpam-6327	681	2	×	×	PROPN
ejpam-6327	681	3	y	y	PROPN
ejpam-6327	681	4	/	/	SYM
ejpam-6327	681	5	ν	ν	NOUN
ejpam-6327	681	6	,	,	PUNCT
ejpam-6327	681	7	there	there	PRON
ejpam-6327	681	8	exists	exist	VERB
ejpam-6327	681	9	(	(	PUNCT
ejpam-6327	681	10	x	x	X
ejpam-6327	681	11	,	,	PUNCT
ejpam-6327	681	12	y	y	NOUN
ejpam-6327	681	13	)	)	PUNCT
ejpam-6327	681	14	∈	∈	PROPN
ejpam-6327	682	1	x	x	PUNCT
ejpam-6327	682	2	×	×	NOUN
ejpam-6327	682	3	y	y	PROPN
ejpam-6327	682	4	such	such	ADJ
ejpam-6327	682	5	that	that	DET
ejpam-6327	682	6	ψ(x	ψ(x	PROPN
ejpam-6327	682	7	,	,	PUNCT
ejpam-6327	682	8	y	y	NOUN
ejpam-6327	682	9	)	)	PUNCT
ejpam-6327	682	10	=	=	SYM
ejpam-6327	682	11	(	(	PUNCT
ejpam-6327	682	12	µx	µx	ADJ
ejpam-6327	682	13	,	,	PUNCT
ejpam-6327	682	14	νy	νy	NOUN
ejpam-6327	682	15	)	)	PUNCT
ejpam-6327	682	16	.	.	PUNCT
ejpam-6327	683	1	thus	thus	ADV
ejpam-6327	683	2	,	,	PUNCT
ejpam-6327	683	3	ψ	ψ	X
ejpam-6327	683	4	is	be	AUX
ejpam-6327	683	5	an	an	DET
ejpam-6327	683	6	epimorphism	epimorphism	NOUN
ejpam-6327	683	7	.	.	PUNCT
ejpam-6327	684	1	by	by	ADP
ejpam-6327	684	2	first	first	PROPN
ejpam-6327	684	3	isomorphism	isomorphism	PROPN
ejpam-6327	684	4	theorem	theorem	VERB
ejpam-6327	684	5	,	,	PUNCT
ejpam-6327	684	6	we	we	PRON
ejpam-6327	684	7	have	have	VERB
ejpam-6327	684	8	x	x	SYM
ejpam-6327	684	9	×	×	NOUN
ejpam-6327	684	10	y	y	PROPN
ejpam-6327	684	11	kerψ	kerψ	NOUN
ejpam-6327	684	12	∼=	∼=	PROPN
ejpam-6327	684	13	x/µ×	x/µ×	ADV
ejpam-6327	684	14	y	y	PROPN
ejpam-6327	684	15	/	/	SYM
ejpam-6327	684	16	ν	ν	PROPN
ejpam-6327	684	17	.	.	PUNCT
ejpam-6327	685	1	let	let	VERB
ejpam-6327	685	2	k	k	NOUN
ejpam-6327	685	3	=	=	PUNCT
ejpam-6327	685	4	kerψ	kerψ	ADJ
ejpam-6327	685	5	.	.	PUNCT
ejpam-6327	686	1	then	then	ADV
ejpam-6327	686	2	x	x	SYM
ejpam-6327	686	3	×	×	NOUN
ejpam-6327	686	4	y	y	NOUN
ejpam-6327	686	5	kerψ	kerψ	NOUN
ejpam-6327	686	6	=	=	PUNCT
ejpam-6327	687	1	x	x	SYM
ejpam-6327	687	2	×	×	NOUN
ejpam-6327	687	3	y	y	PROPN
ejpam-6327	687	4	k	k	PROPN
ejpam-6327	688	1	=	=	PRON
ejpam-6327	688	2	{	{	PUNCT
ejpam-6327	688	3	k(x	k(x	PROPN
ejpam-6327	688	4	,	,	PUNCT
ejpam-6327	688	5	y	y	PROPN
ejpam-6327	688	6	)	)	PUNCT
ejpam-6327	688	7	:	:	PUNCT
ejpam-6327	688	8	(	(	PUNCT
ejpam-6327	688	9	x	x	X
ejpam-6327	688	10	,	,	PUNCT
ejpam-6327	688	11	y	y	NOUN
ejpam-6327	688	12	)	)	PUNCT
ejpam-6327	688	13	∈	∈	PROPN
ejpam-6327	688	14	x	x	SYM
ejpam-6327	688	15	×	×	NOUN
ejpam-6327	688	16	y	y	PROPN
ejpam-6327	688	17	}	}	PUNCT
ejpam-6327	688	18	.	.	PUNCT
ejpam-6327	689	1	similarly	similarly	ADV
ejpam-6327	689	2	,	,	PUNCT
ejpam-6327	689	3	x	x	X
ejpam-6327	689	4	×	×	VERB
ejpam-6327	689	5	y	y	PROPN
ejpam-6327	689	6	µ×	µ×	X
ejpam-6327	689	7	ν	ν	X
ejpam-6327	689	8	=	=	PRON
ejpam-6327	689	9	{	{	PUNCT
ejpam-6327	689	10	µ×	µ×	X
ejpam-6327	689	11	ν(x	ν(x	PROPN
ejpam-6327	689	12	,	,	PUNCT
ejpam-6327	689	13	y	y	NOUN
ejpam-6327	689	14	)	)	PUNCT
ejpam-6327	689	15	:	:	PUNCT
ejpam-6327	689	16	(	(	PUNCT
ejpam-6327	689	17	x	x	X
ejpam-6327	689	18	,	,	PUNCT
ejpam-6327	689	19	y	y	NOUN
ejpam-6327	689	20	)	)	PUNCT
ejpam-6327	689	21	∈	∈	PROPN
ejpam-6327	689	22	x	x	SYM
ejpam-6327	689	23	×	×	NOUN
ejpam-6327	689	24	y	y	PROPN
ejpam-6327	689	25	}	}	PUNCT
ejpam-6327	689	26	.	.	PUNCT
ejpam-6327	690	1	claim	claim	NOUN
ejpam-6327	690	2	:	:	PUNCT
ejpam-6327	690	3	k(x	k(x	PROPN
ejpam-6327	690	4	,	,	PUNCT
ejpam-6327	690	5	y	y	NOUN
ejpam-6327	690	6	)	)	PUNCT
ejpam-6327	690	7	∼=	∼=	PROPN
ejpam-6327	690	8	µ×	µ×	X
ejpam-6327	690	9	ν(x	ν(x	PROPN
ejpam-6327	690	10	,	,	PUNCT
ejpam-6327	690	11	y	y	NOUN
ejpam-6327	690	12	)	)	PUNCT
ejpam-6327	690	13	.	.	PUNCT
ejpam-6327	691	1	for	for	ADP
ejpam-6327	691	2	(	(	PUNCT
ejpam-6327	691	3	α	α	X
ejpam-6327	691	4	,	,	PUNCT
ejpam-6327	691	5	β	β	NOUN
ejpam-6327	691	6	)	)	PUNCT
ejpam-6327	691	7	∈	∈	PROPN
ejpam-6327	691	8	k(x	k(x	PROPN
ejpam-6327	691	9	,	,	PUNCT
ejpam-6327	691	10	y	y	PROPN
ejpam-6327	691	11	)	)	PUNCT
ejpam-6327	691	12	,	,	PUNCT
ejpam-6327	691	13	(	(	PUNCT
ejpam-6327	691	14	α	α	X
ejpam-6327	691	15	,	,	PUNCT
ejpam-6327	691	16	β	β	NOUN
ejpam-6327	691	17	)	)	PUNCT
ejpam-6327	691	18	∈	∈	PROPN
ejpam-6327	691	19	k(x	k(x	PROPN
ejpam-6327	691	20	,	,	PUNCT
ejpam-6327	691	21	y	y	PROPN
ejpam-6327	691	22	)	)	PUNCT
ejpam-6327	691	23	⇔	⇔	X
ejpam-6327	691	24	(	(	PUNCT
ejpam-6327	691	25	α	α	NOUN
ejpam-6327	691	26	,	,	PUNCT
ejpam-6327	691	27	β	β	NOUN
ejpam-6327	691	28	)	)	PUNCT
ejpam-6327	691	29	∼k	∼k	PROPN
ejpam-6327	691	30	(	(	PUNCT
ejpam-6327	691	31	x	x	NOUN
ejpam-6327	691	32	,	,	PUNCT
ejpam-6327	691	33	y	y	PROPN
ejpam-6327	691	34	)	)	PUNCT
ejpam-6327	691	35	⇔	⇔	X
ejpam-6327	691	36	(	(	PUNCT
ejpam-6327	691	37	α	α	NOUN
ejpam-6327	691	38	,	,	PUNCT
ejpam-6327	691	39	β	β	NOUN
ejpam-6327	691	40	)	)	PUNCT
ejpam-6327	691	41	∗	∗	NOUN
ejpam-6327	691	42	(	(	PUNCT
ejpam-6327	691	43	x	x	X
ejpam-6327	691	44	,	,	PUNCT
ejpam-6327	691	45	y	y	NOUN
ejpam-6327	691	46	)	)	PUNCT
ejpam-6327	691	47	∈	∈	PROPN
ejpam-6327	691	48	k	k	NOUN
ejpam-6327	691	49	,	,	PUNCT
ejpam-6327	691	50	(	(	PUNCT
ejpam-6327	691	51	x	x	X
ejpam-6327	691	52	,	,	PUNCT
ejpam-6327	691	53	y	y	NOUN
ejpam-6327	691	54	)	)	PUNCT
ejpam-6327	691	55	∗	∗	NOUN
ejpam-6327	691	56	(	(	PUNCT
ejpam-6327	691	57	α	α	NOUN
ejpam-6327	691	58	,	,	PUNCT
ejpam-6327	691	59	β	β	NOUN
ejpam-6327	691	60	)	)	PUNCT
ejpam-6327	691	61	∈	∈	PROPN
ejpam-6327	691	62	k	k	PROPN
ejpam-6327	691	63	⇔	⇔	PROPN
ejpam-6327	691	64	ψ((α	ψ((α	PROPN
ejpam-6327	691	65	,	,	PUNCT
ejpam-6327	691	66	β	β	X
ejpam-6327	691	67	)	)	PUNCT
ejpam-6327	691	68	∗	∗	NOUN
ejpam-6327	691	69	(	(	PUNCT
ejpam-6327	691	70	x	x	X
ejpam-6327	691	71	,	,	PUNCT
ejpam-6327	691	72	y	y	NOUN
ejpam-6327	691	73	)	)	PUNCT
ejpam-6327	691	74	)	)	PUNCT
ejpam-6327	692	1	=	=	SYM
ejpam-6327	692	2	(	(	PUNCT
ejpam-6327	692	3	µ0	µ0	NOUN
ejpam-6327	692	4	,	,	PUNCT
ejpam-6327	692	5	ν0	ν0	PROPN
ejpam-6327	692	6	)	)	PUNCT
ejpam-6327	692	7	=	=	PUNCT
ejpam-6327	692	8	ψ((x	ψ((x	NOUN
ejpam-6327	692	9	,	,	PUNCT
ejpam-6327	692	10	y	y	NOUN
ejpam-6327	692	11	)	)	PUNCT
ejpam-6327	692	12	∗	∗	NOUN
ejpam-6327	692	13	(	(	PUNCT
ejpam-6327	692	14	α	α	NOUN
ejpam-6327	692	15	,	,	PUNCT
ejpam-6327	692	16	β	β	NOUN
ejpam-6327	692	17	)	)	PUNCT
ejpam-6327	692	18	)	)	PUNCT
ejpam-6327	692	19	,	,	PUNCT
ejpam-6327	692	20	where	where	SCONJ
ejpam-6327	692	21	µ0	µ0	NOUN
ejpam-6327	692	22	and	and	CCONJ
ejpam-6327	692	23	ν0	ν0	PROPN
ejpam-6327	692	24	are	be	AUX
ejpam-6327	692	25	fixed	fix	VERB
ejpam-6327	692	26	elements	element	NOUN
ejpam-6327	692	27	in	in	ADP
ejpam-6327	692	28	x/µ	x/µ	PROPN
ejpam-6327	692	29	and	and	CCONJ
ejpam-6327	692	30	y	y	PROPN
ejpam-6327	692	31	/	/	SYM
ejpam-6327	692	32	ν	ν	PROPN
ejpam-6327	692	33	,	,	PUNCT
ejpam-6327	692	34	respectively	respectively	ADV
ejpam-6327	692	35	⇔	⇔	PROPN
ejpam-6327	692	36	ψ(α	ψ(α	PROPN
ejpam-6327	692	37	∗	∗	X
ejpam-6327	692	38	x	x	NOUN
ejpam-6327	692	39	,	,	PUNCT
ejpam-6327	692	40	β	β	X
ejpam-6327	692	41	∗	∗	X
ejpam-6327	692	42	y	y	PROPN
ejpam-6327	692	43	)	)	PUNCT
ejpam-6327	692	44	=	=	SYM
ejpam-6327	692	45	(	(	PUNCT
ejpam-6327	692	46	µ0	µ0	NOUN
ejpam-6327	692	47	,	,	PUNCT
ejpam-6327	692	48	ν0	ν0	PROPN
ejpam-6327	692	49	)	)	PUNCT
ejpam-6327	692	50	=	=	SYM
ejpam-6327	692	51	ψ(x	ψ(x	NOUN
ejpam-6327	692	52	∗	∗	NOUN
ejpam-6327	692	53	α	α	NOUN
ejpam-6327	692	54	,	,	PUNCT
ejpam-6327	692	55	y	y	PROPN
ejpam-6327	692	56	∗	∗	NOUN
ejpam-6327	692	57	β	β	NOUN
ejpam-6327	692	58	)	)	PUNCT
ejpam-6327	692	59	⇔	⇔	PROPN
ejpam-6327	692	60	(	(	PUNCT
ejpam-6327	692	61	µα∗x	µα∗x	PROPN
ejpam-6327	692	62	,	,	PUNCT
ejpam-6327	692	63	νβ∗y	νβ∗y	PROPN
ejpam-6327	692	64	)	)	PUNCT
ejpam-6327	692	65	=	=	PUNCT
ejpam-6327	692	66	(	(	PUNCT
ejpam-6327	692	67	µ0	µ0	NOUN
ejpam-6327	692	68	,	,	PUNCT
ejpam-6327	692	69	ν0	ν0	PROPN
ejpam-6327	692	70	)	)	PUNCT
ejpam-6327	692	71	=	=	NOUN
ejpam-6327	692	72	(	(	PUNCT
ejpam-6327	692	73	µx∗α	µx∗α	X
ejpam-6327	692	74	,	,	PUNCT
ejpam-6327	692	75	νy∗β	νy∗β	NOUN
ejpam-6327	692	76	)	)	PUNCT
ejpam-6327	692	77	h.	h.	PROPN
ejpam-6327	692	78	sarapuddin	sarapuddin	PROPN
ejpam-6327	692	79	,	,	PUNCT
ejpam-6327	692	80	j.	j.	PROPN
ejpam-6327	692	81	vilela	vilela	PROPN
ejpam-6327	692	82	/	/	SYM
ejpam-6327	692	83	eur	eur	PROPN
ejpam-6327	692	84	.	.	PUNCT
ejpam-6327	693	1	j.	j.	PROPN
ejpam-6327	693	2	pure	pure	PROPN
ejpam-6327	693	3	appl	appl	PROPN
ejpam-6327	693	4	.	.	PROPN
ejpam-6327	693	5	math	math	PROPN
ejpam-6327	693	6	,	,	PUNCT
ejpam-6327	693	7	18	18	NUM
ejpam-6327	693	8	(	(	PUNCT
ejpam-6327	693	9	3	3	NUM
ejpam-6327	693	10	)	)	PUNCT
ejpam-6327	693	11	(	(	PUNCT
ejpam-6327	693	12	2025	2025	NUM
ejpam-6327	693	13	)	)	PUNCT
ejpam-6327	693	14	,	,	PUNCT
ejpam-6327	693	15	6327	6327	NUM
ejpam-6327	693	16	21	21	NUM
ejpam-6327	693	17	of	of	ADP
ejpam-6327	693	18	23	23	NUM
ejpam-6327	693	19	⇔	⇔	X
ejpam-6327	693	20	µα∗x	µα∗x	PROPN
ejpam-6327	693	21	=	=	PROPN
ejpam-6327	693	22	µx∗α	µx∗α	PROPN
ejpam-6327	693	23	=	=	SYM
ejpam-6327	693	24	µ0	µ0	PROPN
ejpam-6327	693	25	,	,	PUNCT
ejpam-6327	693	26	νβ∗y	νβ∗y	NOUN
ejpam-6327	693	27	=	=	SYM
ejpam-6327	693	28	νy∗β	νy∗β	PROPN
ejpam-6327	693	29	=	=	PUNCT
ejpam-6327	693	30	ν0	ν0	PROPN
ejpam-6327	693	31	⇔	⇔	NOUN
ejpam-6327	693	32	µα	µα	ADP
ejpam-6327	693	33	=	=	SYM
ejpam-6327	693	34	µx	µx	NOUN
ejpam-6327	693	35	,	,	PUNCT
ejpam-6327	693	36	νβ	νβ	PROPN
ejpam-6327	693	37	=	=	PUNCT
ejpam-6327	693	38	νy	νy	PROPN
ejpam-6327	693	39	.	.	PUNCT
ejpam-6327	694	1	also	also	ADV
ejpam-6327	694	2	,	,	PUNCT
ejpam-6327	694	3	for	for	ADP
ejpam-6327	694	4	(	(	PUNCT
ejpam-6327	694	5	α	α	X
ejpam-6327	694	6	,	,	PUNCT
ejpam-6327	694	7	β	β	NOUN
ejpam-6327	694	8	)	)	PUNCT
ejpam-6327	694	9	∈	∈	PROPN
ejpam-6327	694	10	µ×	µ×	X
ejpam-6327	694	11	ν(x	ν(x	PROPN
ejpam-6327	694	12	,	,	PUNCT
ejpam-6327	694	13	y	y	NOUN
ejpam-6327	694	14	)	)	PUNCT
ejpam-6327	694	15	,	,	PUNCT
ejpam-6327	694	16	(	(	PUNCT
ejpam-6327	694	17	α	α	X
ejpam-6327	694	18	,	,	PUNCT
ejpam-6327	694	19	β	β	NOUN
ejpam-6327	694	20	)	)	PUNCT
ejpam-6327	694	21	∈	∈	PROPN
ejpam-6327	694	22	µ×	µ×	X
ejpam-6327	694	23	ν(x	ν(x	PROPN
ejpam-6327	694	24	,	,	PUNCT
ejpam-6327	694	25	y	y	NOUN
ejpam-6327	694	26	)	)	PUNCT
ejpam-6327	694	27	⇔	⇔	X
ejpam-6327	694	28	(	(	PUNCT
ejpam-6327	694	29	α	α	NOUN
ejpam-6327	694	30	,	,	PUNCT
ejpam-6327	694	31	β	β	NOUN
ejpam-6327	694	32	)	)	PUNCT
ejpam-6327	694	33	∼µ×ν	∼µ×ν	NOUN
ejpam-6327	694	34	(	(	PUNCT
ejpam-6327	694	35	x	x	NOUN
ejpam-6327	694	36	,	,	PUNCT
ejpam-6327	694	37	y	y	PROPN
ejpam-6327	694	38	)	)	PUNCT
ejpam-6327	694	39	⇔	⇔	PROPN
ejpam-6327	694	40	µ×	µ×	X
ejpam-6327	694	41	ν((α	ν((α	PROPN
ejpam-6327	694	42	,	,	PUNCT
ejpam-6327	694	43	β	β	NOUN
ejpam-6327	694	44	)	)	PUNCT
ejpam-6327	694	45	∗	∗	NOUN
ejpam-6327	694	46	(	(	PUNCT
ejpam-6327	694	47	x	x	X
ejpam-6327	694	48	,	,	PUNCT
ejpam-6327	694	49	y	y	NOUN
ejpam-6327	694	50	)	)	PUNCT
ejpam-6327	694	51	)	)	PUNCT
ejpam-6327	694	52	>	>	X
ejpam-6327	694	53	0	0	PUNCT
ejpam-6327	695	1	and	and	CCONJ
ejpam-6327	695	2	µ×	µ×	PRON
ejpam-6327	695	3	ν((x	ν((x	NOUN
ejpam-6327	695	4	,	,	PUNCT
ejpam-6327	695	5	y	y	NOUN
ejpam-6327	695	6	)	)	PUNCT
ejpam-6327	695	7	∗	∗	NOUN
ejpam-6327	695	8	(	(	PUNCT
ejpam-6327	695	9	α	α	NOUN
ejpam-6327	695	10	,	,	PUNCT
ejpam-6327	695	11	β	β	NOUN
ejpam-6327	695	12	)	)	PUNCT
ejpam-6327	695	13	)	)	PUNCT
ejpam-6327	695	14	>	>	X
ejpam-6327	695	15	0	0	NUM
ejpam-6327	696	1	⇔	⇔	X
ejpam-6327	696	2	µ×	µ×	X
ejpam-6327	696	3	ν(α	ν(α	PROPN
ejpam-6327	696	4	∗	∗	VERB
ejpam-6327	696	5	x	x	PROPN
ejpam-6327	696	6	,	,	PUNCT
ejpam-6327	696	7	β	β	X
ejpam-6327	696	8	∗	∗	X
ejpam-6327	696	9	y	y	PROPN
ejpam-6327	696	10	)	)	PUNCT
ejpam-6327	696	11	>	>	X
ejpam-6327	696	12	0	0	PUNCT
ejpam-6327	697	1	and	and	CCONJ
ejpam-6327	697	2	µ×	µ×	PRON
ejpam-6327	697	3	ν(x	ν(x	PROPN
ejpam-6327	697	4	∗	∗	NOUN
ejpam-6327	697	5	α	α	PROPN
ejpam-6327	697	6	,	,	PUNCT
ejpam-6327	697	7	y	y	PROPN
ejpam-6327	697	8	∗	∗	NOUN
ejpam-6327	697	9	β	β	NOUN
ejpam-6327	697	10	)	)	PUNCT
ejpam-6327	697	11	>	>	PUNCT
ejpam-6327	697	12	0	0	NUM
ejpam-6327	697	13	⇔	⇔	PROPN
ejpam-6327	697	14	min{µ(α	min{µ(α	PROPN
ejpam-6327	697	15	∗	∗	NOUN
ejpam-6327	697	16	x	x	NOUN
ejpam-6327	697	17	)	)	PUNCT
ejpam-6327	697	18	,	,	PUNCT
ejpam-6327	697	19	ν(β	ν(β	PROPN
ejpam-6327	697	20	∗	∗	PROPN
ejpam-6327	697	21	y	y	PROPN
ejpam-6327	697	22	)	)	PUNCT
ejpam-6327	697	23	}	}	PUNCT
ejpam-6327	697	24	>	>	X
ejpam-6327	697	25	0	0	PUNCT
ejpam-6327	697	26	and	and	CCONJ
ejpam-6327	697	27	min{µ(x	min{µ(x	PROPN
ejpam-6327	697	28	∗	∗	NOUN
ejpam-6327	697	29	α	α	NOUN
ejpam-6327	697	30	)	)	PUNCT
ejpam-6327	697	31	,	,	PUNCT
ejpam-6327	697	32	ν(y	ν(y	PROPN
ejpam-6327	697	33	∗	∗	NOUN
ejpam-6327	697	34	β	β	NOUN
ejpam-6327	697	35	)	)	PUNCT
ejpam-6327	697	36	}	}	PUNCT
ejpam-6327	697	37	>	>	X
ejpam-6327	697	38	0	0	NUM
ejpam-6327	698	1	⇔	⇔	X
ejpam-6327	698	2	µ(α	µ(α	PROPN
ejpam-6327	698	3	∗	∗	NOUN
ejpam-6327	698	4	x	x	NOUN
ejpam-6327	698	5	)	)	PUNCT
ejpam-6327	698	6	>	>	X
ejpam-6327	698	7	0	0	NUM
ejpam-6327	698	8	,	,	PUNCT
ejpam-6327	698	9	ν(β	ν(β	PROPN
ejpam-6327	698	10	∗	∗	PROPN
ejpam-6327	698	11	y	y	PROPN
ejpam-6327	698	12	)	)	PUNCT
ejpam-6327	698	13	>	>	X
ejpam-6327	698	14	0	0	PUNCT
ejpam-6327	698	15	and	and	CCONJ
ejpam-6327	698	16	µ(x	µ(x	ADJ
ejpam-6327	698	17	∗	∗	NOUN
ejpam-6327	698	18	α	α	NOUN
ejpam-6327	698	19	)	)	PUNCT
ejpam-6327	698	20	>	>	X
ejpam-6327	698	21	0	0	PROPN
ejpam-6327	698	22	,	,	PUNCT
ejpam-6327	698	23	ν(y	ν(y	PROPN
ejpam-6327	698	24	∗	∗	NOUN
ejpam-6327	698	25	β	β	NOUN
ejpam-6327	698	26	)	)	PUNCT
ejpam-6327	698	27	>	>	PUNCT
ejpam-6327	698	28	0	0	NUM
ejpam-6327	698	29	⇔	⇔	X
ejpam-6327	698	30	µ(α	µ(α	PROPN
ejpam-6327	698	31	∗	∗	NOUN
ejpam-6327	698	32	x	x	NOUN
ejpam-6327	698	33	)	)	PUNCT
ejpam-6327	698	34	>	>	X
ejpam-6327	698	35	0	0	PUNCT
ejpam-6327	698	36	and	and	CCONJ
ejpam-6327	698	37	µ(x	µ(x	ADJ
ejpam-6327	698	38	∗	∗	NOUN
ejpam-6327	698	39	α	α	NOUN
ejpam-6327	698	40	)	)	PUNCT
ejpam-6327	698	41	>	>	X
ejpam-6327	698	42	0	0	NUM
ejpam-6327	698	43	,	,	PUNCT
ejpam-6327	698	44	ν(β	ν(β	PROPN
ejpam-6327	698	45	∗	∗	PROPN
ejpam-6327	698	46	y	y	PROPN
ejpam-6327	698	47	)	)	PUNCT
ejpam-6327	698	48	>	>	X
ejpam-6327	698	49	0	0	PUNCT
ejpam-6327	698	50	and	and	CCONJ
ejpam-6327	698	51	ν(y	ν(y	PROPN
ejpam-6327	698	52	∗	∗	X
ejpam-6327	698	53	β	β	NOUN
ejpam-6327	698	54	)	)	PUNCT
ejpam-6327	698	55	>	>	X
ejpam-6327	698	56	0	0	NUM
ejpam-6327	699	1	⇔	⇔	PROPN
ejpam-6327	699	2	α	α	PROPN
ejpam-6327	699	3	∼µ	∼µ	PROPN
ejpam-6327	699	4	x	x	PROPN
ejpam-6327	699	5	,	,	PUNCT
ejpam-6327	699	6	β	β	PROPN
ejpam-6327	699	7	∼ν	∼ν	PROPN
ejpam-6327	699	8	y	y	PROPN
ejpam-6327	699	9	⇔	⇔	PROPN
ejpam-6327	699	10	µα	µα	ADP
ejpam-6327	699	11	=	=	X
ejpam-6327	699	12	µβ	µβ	PROPN
ejpam-6327	699	13	,	,	PUNCT
ejpam-6327	699	14	νβ	νβ	PROPN
ejpam-6327	699	15	=	=	PUNCT
ejpam-6327	699	16	νy	νy	PROPN
ejpam-6327	699	17	.	.	PUNCT
ejpam-6327	700	1	thus	thus	ADV
ejpam-6327	700	2	,	,	PUNCT
ejpam-6327	700	3	(	(	PUNCT
ejpam-6327	700	4	α	α	X
ejpam-6327	700	5	,	,	PUNCT
ejpam-6327	700	6	β	β	NOUN
ejpam-6327	700	7	)	)	PUNCT
ejpam-6327	700	8	∈	∈	PROPN
ejpam-6327	700	9	k(x	k(x	PROPN
ejpam-6327	700	10	,	,	PUNCT
ejpam-6327	700	11	y	y	PROPN
ejpam-6327	700	12	)	)	PUNCT
ejpam-6327	700	13	⇔	⇔	X
ejpam-6327	700	14	(	(	PUNCT
ejpam-6327	700	15	α	α	NOUN
ejpam-6327	700	16	,	,	PUNCT
ejpam-6327	700	17	β	β	NOUN
ejpam-6327	700	18	)	)	PUNCT
ejpam-6327	700	19	∈	∈	PROPN
ejpam-6327	700	20	µ×	µ×	X
ejpam-6327	700	21	ν(x	ν(x	PROPN
ejpam-6327	700	22	,	,	PUNCT
ejpam-6327	700	23	y	y	NOUN
ejpam-6327	700	24	)	)	PUNCT
ejpam-6327	700	25	.	.	PUNCT
ejpam-6327	701	1	this	this	PRON
ejpam-6327	701	2	proves	prove	VERB
ejpam-6327	701	3	the	the	DET
ejpam-6327	701	4	claim	claim	NOUN
ejpam-6327	701	5	.	.	PUNCT
ejpam-6327	702	1	hence	hence	ADV
ejpam-6327	702	2	,	,	PUNCT
ejpam-6327	702	3	x	x	X
ejpam-6327	702	4	×	×	VERB
ejpam-6327	702	5	y	y	PROPN
ejpam-6327	702	6	µ×	µ×	X
ejpam-6327	702	7	ν	ν	NOUN
ejpam-6327	702	8	∼=	∼=	PROPN
ejpam-6327	702	9	x	x	SYM
ejpam-6327	702	10	×	×	PROPN
ejpam-6327	702	11	y	y	NOUN
ejpam-6327	702	12	kerψ	kerψ	NOUN
ejpam-6327	702	13	∼=	∼=	PROPN
ejpam-6327	702	14	x/µ×	x/µ×	ADV
ejpam-6327	702	15	y	y	PROPN
ejpam-6327	702	16	/	/	SYM
ejpam-6327	702	17	ν	ν	PROPN
ejpam-6327	702	18	.	.	PUNCT
ejpam-6327	703	1	lemma	lemma	PROPN
ejpam-6327	703	2	4	4	X
ejpam-6327	703	3	.	.	PUNCT
ejpam-6327	704	1	let	let	VERB
ejpam-6327	704	2	j	j	PROPN
ejpam-6327	704	3	and	and	CCONJ
ejpam-6327	704	4	j	j	PROPN
ejpam-6327	704	5	be	be	VERB
ejpam-6327	704	6	ideals	ideal	NOUN
ejpam-6327	704	7	of	of	ADP
ejpam-6327	704	8	ks	ks	NOUN
ejpam-6327	704	9	-	-	PUNCT
ejpam-6327	704	10	semigroups	semigroups	X
ejpam-6327	704	11	x	x	X
ejpam-6327	704	12	and	and	CCONJ
ejpam-6327	704	13	y	y	PROPN
ejpam-6327	704	14	,	,	PUNCT
ejpam-6327	704	15	respectively	respectively	ADV
ejpam-6327	704	16	.	.	PUNCT
ejpam-6327	705	1	then	then	ADV
ejpam-6327	705	2	j	j	PROPN
ejpam-6327	705	3	×	×	NOUN
ejpam-6327	705	4	i	i	PRON
ejpam-6327	705	5	is	be	AUX
ejpam-6327	705	6	an	an	DET
ejpam-6327	705	7	ideal	ideal	NOUN
ejpam-6327	705	8	of	of	ADP
ejpam-6327	705	9	x	x	SYM
ejpam-6327	705	10	×	×	PROPN
ejpam-6327	705	11	y	y	PROPN
ejpam-6327	705	12	.	.	PUNCT
ejpam-6327	706	1	moreover	moreover	ADV
ejpam-6327	706	2	,	,	PUNCT
ejpam-6327	706	3	x	x	PROPN
ejpam-6327	706	4	×	×	PROPN
ejpam-6327	706	5	y	y	PROPN
ejpam-6327	706	6	/	/	SYM
ejpam-6327	706	7	j	j	PROPN
ejpam-6327	706	8	×	×	NOUN
ejpam-6327	706	9	i	i	PRON
ejpam-6327	706	10	∼=	∼=	PROPN
ejpam-6327	706	11	x	x	SYM
ejpam-6327	706	12	/	/	SYM
ejpam-6327	706	13	j	j	PROPN
ejpam-6327	706	14	×	×	PROPN
ejpam-6327	706	15	y	y	PROPN
ejpam-6327	706	16	/	/	SYM
ejpam-6327	706	17	i.	i.	NOUN
ejpam-6327	706	18	proof	proof	NOUN
ejpam-6327	706	19	.	.	PUNCT
ejpam-6327	707	1	let	let	VERB
ejpam-6327	707	2	j	j	PROPN
ejpam-6327	707	3	and	and	CCONJ
ejpam-6327	707	4	j	j	PROPN
ejpam-6327	707	5	be	be	VERB
ejpam-6327	707	6	ideals	ideal	NOUN
ejpam-6327	707	7	of	of	ADP
ejpam-6327	707	8	ks	ks	NOUN
ejpam-6327	707	9	-	-	PUNCT
ejpam-6327	707	10	semigroups	semigroups	X
ejpam-6327	707	11	x	x	X
ejpam-6327	707	12	and	and	CCONJ
ejpam-6327	707	13	y	y	PROPN
ejpam-6327	707	14	,	,	PUNCT
ejpam-6327	707	15	respectively	respectively	ADV
ejpam-6327	707	16	.	.	PUNCT
ejpam-6327	708	1	let	let	VERB
ejpam-6327	708	2	(	(	PUNCT
ejpam-6327	708	3	x	x	NOUN
ejpam-6327	708	4	,	,	PUNCT
ejpam-6327	708	5	y	y	NOUN
ejpam-6327	708	6	)	)	PUNCT
ejpam-6327	708	7	∈	∈	PROPN
ejpam-6327	708	8	x	x	SYM
ejpam-6327	708	9	×	×	PROPN
ejpam-6327	708	10	y	y	PROPN
ejpam-6327	708	11	and	and	CCONJ
ejpam-6327	708	12	(	(	PUNCT
ejpam-6327	708	13	a	a	PRON
ejpam-6327	708	14	,	,	PUNCT
ejpam-6327	708	15	b	b	NOUN
ejpam-6327	708	16	)	)	PUNCT
ejpam-6327	708	17	∈	∈	PROPN
ejpam-6327	709	1	j	j	PROPN
ejpam-6327	709	2	×	×	PROPN
ejpam-6327	709	3	i.	i.	NOUN
ejpam-6327	709	4	since	since	SCONJ
ejpam-6327	709	5	j	j	PROPN
ejpam-6327	709	6	and	and	CCONJ
ejpam-6327	709	7	i	i	PRON
ejpam-6327	709	8	are	be	AUX
ejpam-6327	709	9	ideals	ideal	NOUN
ejpam-6327	709	10	,	,	PUNCT
ejpam-6327	709	11	(	(	PUNCT
ejpam-6327	709	12	x	x	NOUN
ejpam-6327	709	13	,	,	PUNCT
ejpam-6327	709	14	y	y	PROPN
ejpam-6327	709	15	)	)	PUNCT
ejpam-6327	709	16	·	·	PUNCT
ejpam-6327	709	17	(	(	PUNCT
ejpam-6327	709	18	a	a	DET
ejpam-6327	709	19	,	,	PUNCT
ejpam-6327	709	20	b	b	NOUN
ejpam-6327	709	21	)	)	PUNCT
ejpam-6327	709	22	=	=	SYM
ejpam-6327	709	23	(	(	PUNCT
ejpam-6327	709	24	x	x	X
ejpam-6327	709	25	·	·	PUNCT
ejpam-6327	709	26	x	x	SYM
ejpam-6327	709	27	a	a	X
ejpam-6327	709	28	,	,	PUNCT
ejpam-6327	709	29	y	y	PROPN
ejpam-6327	709	30	·	·	PUNCT
ejpam-6327	709	31	y	y	PROPN
ejpam-6327	709	32	b	b	PROPN
ejpam-6327	709	33	)	)	PUNCT
ejpam-6327	709	34	∈	∈	PROPN
ejpam-6327	710	1	j	j	PROPN
ejpam-6327	710	2	×	×	PROPN
ejpam-6327	710	3	i.	i.	NOUN
ejpam-6327	710	4	similarly	similarly	ADV
ejpam-6327	710	5	,	,	PUNCT
ejpam-6327	710	6	(	(	PUNCT
ejpam-6327	710	7	a	a	DET
ejpam-6327	710	8	,	,	PUNCT
ejpam-6327	710	9	b	b	NOUN
ejpam-6327	710	10	)	)	PUNCT
ejpam-6327	710	11	·	·	PUNCT
ejpam-6327	710	12	(	(	PUNCT
ejpam-6327	710	13	x	x	X
ejpam-6327	710	14	,	,	PUNCT
ejpam-6327	710	15	y	y	PROPN
ejpam-6327	710	16	)	)	PUNCT
ejpam-6327	710	17	∈	∈	PROPN
ejpam-6327	711	1	j	j	PROPN
ejpam-6327	711	2	×	×	PROPN
ejpam-6327	711	3	i.	i.	PROPN
ejpam-6327	711	4	now	now	ADV
ejpam-6327	711	5	,	,	PUNCT
ejpam-6327	711	6	let	let	VERB
ejpam-6327	711	7	(	(	PUNCT
ejpam-6327	711	8	x1	x1	ADJ
ejpam-6327	711	9	,	,	PUNCT
ejpam-6327	711	10	y1	y1	PROPN
ejpam-6327	711	11	)	)	PUNCT
ejpam-6327	711	12	,	,	PUNCT
ejpam-6327	711	13	(	(	PUNCT
ejpam-6327	711	14	x2	x2	PROPN
ejpam-6327	711	15	,	,	PUNCT
ejpam-6327	711	16	y2	y2	NOUN
ejpam-6327	711	17	)	)	PUNCT
ejpam-6327	711	18	∈	∈	PROPN
ejpam-6327	712	1	x	x	PUNCT
ejpam-6327	712	2	×	×	NOUN
ejpam-6327	712	3	y	y	PROPN
ejpam-6327	712	4	such	such	ADJ
ejpam-6327	712	5	that	that	PRON
ejpam-6327	712	6	(	(	PUNCT
ejpam-6327	712	7	x1	x1	PROPN
ejpam-6327	712	8	∗x	∗x	PROPN
ejpam-6327	712	9	x2	x2	PROPN
ejpam-6327	712	10	,	,	PUNCT
ejpam-6327	712	11	y1	y1	PROPN
ejpam-6327	712	12	∗y	∗y	PROPN
ejpam-6327	712	13	y2	y2	PROPN
ejpam-6327	712	14	)	)	PUNCT
ejpam-6327	712	15	=	=	PRON
ejpam-6327	712	16	(	(	PUNCT
ejpam-6327	712	17	x1	x1	PROPN
ejpam-6327	712	18	,	,	PUNCT
ejpam-6327	712	19	y1	y1	NOUN
ejpam-6327	712	20	)	)	PUNCT
ejpam-6327	712	21	∗	∗	NOUN
ejpam-6327	712	22	(	(	PUNCT
ejpam-6327	712	23	x2	x2	PROPN
ejpam-6327	712	24	,	,	PUNCT
ejpam-6327	712	25	y2	y2	PROPN
ejpam-6327	712	26	)	)	PUNCT
ejpam-6327	712	27	∈	∈	PROPN
ejpam-6327	713	1	j	j	X
ejpam-6327	713	2	×	×	NOUN
ejpam-6327	713	3	i	i	PROPN
ejpam-6327	713	4	and	and	CCONJ
ejpam-6327	713	5	(	(	PUNCT
ejpam-6327	713	6	x2	x2	PROPN
ejpam-6327	713	7	,	,	PUNCT
ejpam-6327	713	8	y2	y2	PROPN
ejpam-6327	713	9	)	)	PUNCT
ejpam-6327	713	10	∈	∈	PROPN
ejpam-6327	714	1	j	j	PROPN
ejpam-6327	714	2	×	×	PROPN
ejpam-6327	714	3	i.	i.	NOUN
ejpam-6327	714	4	since	since	SCONJ
ejpam-6327	714	5	j	j	PROPN
ejpam-6327	714	6	and	and	CCONJ
ejpam-6327	714	7	i	i	PRON
ejpam-6327	714	8	are	be	AUX
ejpam-6327	714	9	ideals	ideal	NOUN
ejpam-6327	714	10	,	,	PUNCT
ejpam-6327	714	11	(	(	PUNCT
ejpam-6327	714	12	x1	x1	PROPN
ejpam-6327	714	13	,	,	PUNCT
ejpam-6327	714	14	y1	y1	ADJ
ejpam-6327	714	15	)	)	PUNCT
ejpam-6327	714	16	∈	∈	PROPN
ejpam-6327	715	1	j	j	PROPN
ejpam-6327	715	2	×	×	PROPN
ejpam-6327	715	3	i.	i.	PROPN
ejpam-6327	715	4	thus	thus	ADV
ejpam-6327	715	5	,	,	PUNCT
ejpam-6327	715	6	j	j	PROPN
ejpam-6327	715	7	×	×	NOUN
ejpam-6327	715	8	i	i	PRON
ejpam-6327	715	9	is	be	AUX
ejpam-6327	715	10	an	an	DET
ejpam-6327	715	11	ideal	ideal	NOUN
ejpam-6327	715	12	of	of	ADP
ejpam-6327	715	13	x	x	SYM
ejpam-6327	715	14	×	×	PROPN
ejpam-6327	715	15	y	y	PROPN
ejpam-6327	715	16	.	.	PUNCT
ejpam-6327	716	1	define	define	VERB
ejpam-6327	716	2	φ	φ	NOUN
ejpam-6327	716	3	:	:	PUNCT
ejpam-6327	716	4	x×y	x×y	PUNCT
ejpam-6327	716	5	→	→	SYM
ejpam-6327	716	6	x	x	X
ejpam-6327	716	7	/	/	SYM
ejpam-6327	716	8	j×y	j×y	PROPN
ejpam-6327	716	9	/	/	SYM
ejpam-6327	716	10	i	i	PROPN
ejpam-6327	716	11	by	by	ADP
ejpam-6327	716	12	φ(x	φ(x	PROPN
ejpam-6327	716	13	,	,	PUNCT
ejpam-6327	716	14	y	y	NOUN
ejpam-6327	716	15	)	)	PUNCT
ejpam-6327	716	16	=	=	SYM
ejpam-6327	716	17	(	(	PUNCT
ejpam-6327	716	18	jx	jx	PROPN
ejpam-6327	716	19	,	,	PUNCT
ejpam-6327	716	20	iy	iy	PROPN
ejpam-6327	716	21	)	)	PUNCT
ejpam-6327	716	22	.	.	PUNCT
ejpam-6327	717	1	let	let	VERB
ejpam-6327	717	2	(	(	PUNCT
ejpam-6327	717	3	x1	x1	PROPN
ejpam-6327	717	4	,	,	PUNCT
ejpam-6327	717	5	y1	y1	PROPN
ejpam-6327	717	6	)	)	PUNCT
ejpam-6327	717	7	,	,	PUNCT
ejpam-6327	717	8	(	(	PUNCT
ejpam-6327	717	9	x2	x2	PROPN
ejpam-6327	717	10	,	,	PUNCT
ejpam-6327	717	11	y2	y2	NOUN
ejpam-6327	717	12	)	)	PUNCT
ejpam-6327	717	13	∈	∈	PROPN
ejpam-6327	717	14	x×y	x×y	PUNCT
ejpam-6327	717	15	such	such	ADJ
ejpam-6327	717	16	that	that	SCONJ
ejpam-6327	717	17	(	(	PUNCT
ejpam-6327	717	18	x1	x1	PROPN
ejpam-6327	717	19	,	,	PUNCT
ejpam-6327	717	20	y1	y1	NOUN
ejpam-6327	717	21	)	)	PUNCT
ejpam-6327	717	22	=	=	SYM
ejpam-6327	717	23	(	(	PUNCT
ejpam-6327	717	24	x2	x2	PROPN
ejpam-6327	717	25	,	,	PUNCT
ejpam-6327	717	26	y2	y2	PROPN
ejpam-6327	717	27	)	)	PUNCT
ejpam-6327	717	28	.	.	PUNCT
ejpam-6327	718	1	then	then	ADV
ejpam-6327	718	2	x1	x1	PROPN
ejpam-6327	718	3	=	=	SYM
ejpam-6327	718	4	x2	x2	PROPN
ejpam-6327	718	5	and	and	CCONJ
ejpam-6327	718	6	y1	y1	NOUN
ejpam-6327	718	7	=	=	PUNCT
ejpam-6327	718	8	y2	y2	NOUN
ejpam-6327	719	1	so	so	SCONJ
ejpam-6327	719	2	that	that	SCONJ
ejpam-6327	719	3	x1	x1	PROPN
ejpam-6327	719	4	∼j	∼j	PROPN
ejpam-6327	719	5	x2	x2	NOUN
ejpam-6327	719	6	and	and	CCONJ
ejpam-6327	719	7	y1	y1	INTJ
ejpam-6327	719	8	∼i	∼i	PROPN
ejpam-6327	719	9	y2	y2	PROPN
ejpam-6327	719	10	.	.	PUNCT
ejpam-6327	720	1	thus	thus	ADV
ejpam-6327	720	2	,	,	PUNCT
ejpam-6327	720	3	jx1	jx1	NOUN
ejpam-6327	720	4	=	=	SYM
ejpam-6327	720	5	jx2	jx2	NOUN
ejpam-6327	720	6	and	and	CCONJ
ejpam-6327	720	7	iy1	iy1	VERB
ejpam-6327	720	8	=	=	SYM
ejpam-6327	720	9	iy2	iy2	NOUN
ejpam-6327	720	10	,	,	PUNCT
ejpam-6327	720	11	that	that	ADV
ejpam-6327	720	12	is	is	ADV
ejpam-6327	720	13	,	,	PUNCT
ejpam-6327	720	14	(	(	PUNCT
ejpam-6327	720	15	jx1	jx1	NOUN
ejpam-6327	720	16	,	,	PUNCT
ejpam-6327	720	17	iy1	iy1	NOUN
ejpam-6327	720	18	)	)	PUNCT
ejpam-6327	720	19	=	=	PUNCT
ejpam-6327	721	1	(	(	PUNCT
ejpam-6327	721	2	jx2	jx2	NOUN
ejpam-6327	721	3	,	,	PUNCT
ejpam-6327	721	4	iy2	iy2	PROPN
ejpam-6327	721	5	)	)	PUNCT
ejpam-6327	721	6	.	.	PUNCT
ejpam-6327	722	1	this	this	PRON
ejpam-6327	722	2	shows	show	VERB
ejpam-6327	722	3	that	that	SCONJ
ejpam-6327	722	4	φ	φ	PROPN
ejpam-6327	722	5	is	be	AUX
ejpam-6327	722	6	well	well	ADV
ejpam-6327	722	7	-	-	PUNCT
ejpam-6327	722	8	defined	define	VERB
ejpam-6327	722	9	.	.	PUNCT
ejpam-6327	723	1	now	now	ADV
ejpam-6327	723	2	,	,	PUNCT
ejpam-6327	723	3	let	let	VERB
ejpam-6327	723	4	(	(	PUNCT
ejpam-6327	723	5	x1	x1	ADJ
ejpam-6327	723	6	,	,	PUNCT
ejpam-6327	723	7	y1	y1	PROPN
ejpam-6327	723	8	)	)	PUNCT
ejpam-6327	723	9	,	,	PUNCT
ejpam-6327	724	1	(	(	PUNCT
ejpam-6327	724	2	x2	x2	PROPN
ejpam-6327	724	3	,	,	PUNCT
ejpam-6327	724	4	y2	y2	NOUN
ejpam-6327	724	5	)	)	PUNCT
ejpam-6327	724	6	∈	∈	PROPN
ejpam-6327	724	7	x	x	PUNCT
ejpam-6327	724	8	×	×	NOUN
ejpam-6327	724	9	y	y	PROPN
ejpam-6327	724	10	.	.	PUNCT
ejpam-6327	725	1	then	then	ADV
ejpam-6327	725	2	φ((x1	φ((x1	PROPN
ejpam-6327	725	3	,	,	PUNCT
ejpam-6327	725	4	y1	y1	NOUN
ejpam-6327	725	5	)	)	PUNCT
ejpam-6327	725	6	∗	∗	NOUN
ejpam-6327	725	7	(	(	PUNCT
ejpam-6327	725	8	x2	x2	PROPN
ejpam-6327	725	9	,	,	PUNCT
ejpam-6327	725	10	y2	y2	PROPN
ejpam-6327	725	11	)	)	PUNCT
ejpam-6327	725	12	)	)	PUNCT
ejpam-6327	726	1	=	=	PRON
ejpam-6327	726	2	φ(x1	φ(x1	NOUN
ejpam-6327	726	3	∗x	∗x	PROPN
ejpam-6327	726	4	x2	x2	PROPN
ejpam-6327	726	5	,	,	PUNCT
ejpam-6327	726	6	y1	y1	PROPN
ejpam-6327	726	7	∗y	∗y	PROPN
ejpam-6327	726	8	y2	y2	PROPN
ejpam-6327	726	9	)	)	PUNCT
ejpam-6327	726	10	=	=	PRON
ejpam-6327	726	11	(	(	PUNCT
ejpam-6327	726	12	jx1∗xx2	jx1∗xx2	PROPN
ejpam-6327	726	13	,	,	PUNCT
ejpam-6327	726	14	iy1∗y	iy1∗y	PROPN
ejpam-6327	726	15	y2	y2	NOUN
ejpam-6327	726	16	)	)	PUNCT
ejpam-6327	726	17	=	=	PUNCT
ejpam-6327	726	18	(	(	PUNCT
ejpam-6327	726	19	jx1	jx1	NOUN
ejpam-6327	726	20	∗x	∗x	ADV
ejpam-6327	726	21	/	/	SYM
ejpam-6327	726	22	j	j	NOUN
ejpam-6327	726	23	jx2	jx2	NOUN
ejpam-6327	726	24	,	,	PUNCT
ejpam-6327	726	25	iy1	iy1	VERB
ejpam-6327	726	26	∗y	∗y	PROPN
ejpam-6327	726	27	/	/	SYM
ejpam-6327	726	28	i	i	PROPN
ejpam-6327	726	29	iy2	iy2	PROPN
ejpam-6327	726	30	)	)	PUNCT
ejpam-6327	726	31	=	=	PRON
ejpam-6327	726	32	(	(	PUNCT
ejpam-6327	726	33	jx1	jx1	PROPN
ejpam-6327	726	34	,	,	PUNCT
ejpam-6327	726	35	iy1	iy1	NOUN
ejpam-6327	726	36	)	)	PUNCT
ejpam-6327	726	37	∗	∗	NOUN
ejpam-6327	726	38	(	(	PUNCT
ejpam-6327	726	39	jx2	jx2	NOUN
ejpam-6327	726	40	,	,	PUNCT
ejpam-6327	726	41	iy2	iy2	PROPN
ejpam-6327	726	42	)	)	PUNCT
ejpam-6327	726	43	=	=	SYM
ejpam-6327	726	44	φ(x1	φ(x1	NOUN
ejpam-6327	726	45	,	,	PUNCT
ejpam-6327	726	46	y1	y1	NOUN
ejpam-6327	726	47	)	)	PUNCT
ejpam-6327	726	48	∗	∗	NOUN
ejpam-6327	726	49	φ(x2	φ(x2	NOUN
ejpam-6327	726	50	,	,	PUNCT
ejpam-6327	726	51	y2	y2	PROPN
ejpam-6327	726	52	)	)	PUNCT
ejpam-6327	726	53	and	and	CCONJ
ejpam-6327	726	54	φ((x1	φ((x1	PROPN
ejpam-6327	726	55	,	,	PUNCT
ejpam-6327	726	56	y1	y1	NOUN
ejpam-6327	726	57	)	)	PUNCT
ejpam-6327	726	58	·	·	PUNCT
ejpam-6327	726	59	(	(	PUNCT
ejpam-6327	726	60	x2	x2	PROPN
ejpam-6327	726	61	,	,	PUNCT
ejpam-6327	726	62	y2	y2	PROPN
ejpam-6327	726	63	)	)	PUNCT
ejpam-6327	726	64	)	)	PUNCT
ejpam-6327	727	1	=	=	SYM
ejpam-6327	727	2	φ(x1	φ(x1	NOUN
ejpam-6327	727	3	·	·	PUNCT
ejpam-6327	727	4	x	x	SYM
ejpam-6327	727	5	x2	x2	PROPN
ejpam-6327	727	6	,	,	PUNCT
ejpam-6327	727	7	y1	y1	PROPN
ejpam-6327	727	8	·	·	PUNCT
ejpam-6327	727	9	y	y	PROPN
ejpam-6327	727	10	y2	y2	PROPN
ejpam-6327	727	11	)	)	PUNCT
ejpam-6327	727	12	h.	h.	PROPN
ejpam-6327	727	13	sarapuddin	sarapuddin	PROPN
ejpam-6327	727	14	,	,	PUNCT
ejpam-6327	727	15	j.	j.	PROPN
ejpam-6327	727	16	vilela	vilela	PROPN
ejpam-6327	727	17	/	/	SYM
ejpam-6327	727	18	eur	eur	PROPN
ejpam-6327	727	19	.	.	PUNCT
ejpam-6327	728	1	j.	j.	PROPN
ejpam-6327	728	2	pure	pure	PROPN
ejpam-6327	728	3	appl	appl	PROPN
ejpam-6327	728	4	.	.	PROPN
ejpam-6327	728	5	math	math	PROPN
ejpam-6327	728	6	,	,	PUNCT
ejpam-6327	728	7	18	18	NUM
ejpam-6327	728	8	(	(	PUNCT
ejpam-6327	728	9	3	3	NUM
ejpam-6327	728	10	)	)	PUNCT
ejpam-6327	728	11	(	(	PUNCT
ejpam-6327	728	12	2025	2025	NUM
ejpam-6327	728	13	)	)	PUNCT
ejpam-6327	728	14	,	,	PUNCT
ejpam-6327	728	15	6327	6327	NUM
ejpam-6327	728	16	22	22	NUM
ejpam-6327	728	17	of	of	ADP
ejpam-6327	728	18	23	23	NUM
ejpam-6327	728	19	=	=	SYM
ejpam-6327	728	20	(	(	PUNCT
ejpam-6327	728	21	jx1·xx2	jx1·xx2	X
ejpam-6327	728	22	,	,	PUNCT
ejpam-6327	728	23	iy1·y	iy1·y	PROPN
ejpam-6327	728	24	y2	y2	NOUN
ejpam-6327	728	25	)	)	PUNCT
ejpam-6327	729	1	=	=	PRON
ejpam-6327	729	2	(	(	PUNCT
ejpam-6327	729	3	jx1	jx1	NOUN
ejpam-6327	729	4	·	·	SYM
ejpam-6327	729	5	x	x	SYM
ejpam-6327	729	6	/	/	SYM
ejpam-6327	729	7	j	j	PROPN
ejpam-6327	729	8	jx2	jx2	NOUN
ejpam-6327	729	9	,	,	PUNCT
ejpam-6327	729	10	iy1	iy1	VERB
ejpam-6327	729	11	·	·	SYM
ejpam-6327	729	12	y	y	X
ejpam-6327	729	13	/	/	SYM
ejpam-6327	729	14	i	i	PROPN
ejpam-6327	729	15	iy2	iy2	PROPN
ejpam-6327	729	16	)	)	PUNCT
ejpam-6327	729	17	=	=	PRON
ejpam-6327	729	18	(	(	PUNCT
ejpam-6327	729	19	jx1	jx1	PROPN
ejpam-6327	729	20	,	,	PUNCT
ejpam-6327	729	21	iy1	iy1	NOUN
ejpam-6327	729	22	)	)	PUNCT
ejpam-6327	729	23	·	·	PUNCT
ejpam-6327	730	1	(	(	PUNCT
ejpam-6327	730	2	jx2	jx2	NOUN
ejpam-6327	730	3	,	,	PUNCT
ejpam-6327	730	4	iy2	iy2	PROPN
ejpam-6327	730	5	)	)	PUNCT
ejpam-6327	730	6	=	=	SYM
ejpam-6327	730	7	φ(x1	φ(x1	NOUN
ejpam-6327	730	8	,	,	PUNCT
ejpam-6327	730	9	y1	y1	PROPN
ejpam-6327	730	10	)	)	PUNCT
ejpam-6327	730	11	·	·	PUNCT
ejpam-6327	730	12	φ(x2	φ(x2	NOUN
ejpam-6327	730	13	,	,	PUNCT
ejpam-6327	730	14	y2	y2	PROPN
ejpam-6327	730	15	)	)	PUNCT
ejpam-6327	730	16	.	.	PUNCT
ejpam-6327	731	1	thus	thus	ADV
ejpam-6327	731	2	,	,	PUNCT
ejpam-6327	731	3	φ	φ	PROPN
ejpam-6327	731	4	is	be	AUX
ejpam-6327	731	5	a	a	DET
ejpam-6327	731	6	homomorphism	homomorphism	NOUN
ejpam-6327	731	7	.	.	PUNCT
ejpam-6327	732	1	for	for	ADP
ejpam-6327	732	2	any	any	DET
ejpam-6327	732	3	(	(	PUNCT
ejpam-6327	732	4	jx	jx	PROPN
ejpam-6327	732	5	,	,	PUNCT
ejpam-6327	732	6	iy	iy	PROPN
ejpam-6327	732	7	)	)	PUNCT
ejpam-6327	732	8	∈	∈	PROPN
ejpam-6327	732	9	x	x	X
ejpam-6327	732	10	/	/	SYM
ejpam-6327	732	11	j	j	PROPN
ejpam-6327	732	12	×	×	PROPN
ejpam-6327	732	13	y	y	PROPN
ejpam-6327	732	14	/	/	SYM
ejpam-6327	732	15	i	i	PROPN
ejpam-6327	732	16	,	,	PUNCT
ejpam-6327	732	17	there	there	PRON
ejpam-6327	732	18	exists	exist	VERB
ejpam-6327	732	19	(	(	PUNCT
ejpam-6327	732	20	x	x	X
ejpam-6327	732	21	,	,	PUNCT
ejpam-6327	732	22	y	y	NOUN
ejpam-6327	732	23	)	)	PUNCT
ejpam-6327	732	24	∈	∈	PROPN
ejpam-6327	733	1	x	x	PUNCT
ejpam-6327	733	2	×	×	NOUN
ejpam-6327	733	3	y	y	PROPN
ejpam-6327	733	4	such	such	ADJ
ejpam-6327	733	5	that	that	SCONJ
ejpam-6327	733	6	φ(x	φ(x	PROPN
ejpam-6327	733	7	,	,	PUNCT
ejpam-6327	733	8	y	y	NOUN
ejpam-6327	733	9	)	)	PUNCT
ejpam-6327	733	10	=	=	SYM
ejpam-6327	733	11	(	(	PUNCT
ejpam-6327	733	12	jx	jx	PROPN
ejpam-6327	733	13	,	,	PUNCT
ejpam-6327	733	14	iy	iy	PROPN
ejpam-6327	733	15	)	)	PUNCT
ejpam-6327	733	16	.	.	PUNCT
ejpam-6327	734	1	this	this	PRON
ejpam-6327	734	2	means	mean	VERB
ejpam-6327	734	3	that	that	SCONJ
ejpam-6327	734	4	φ	φ	PROPN
ejpam-6327	734	5	is	be	AUX
ejpam-6327	734	6	an	an	DET
ejpam-6327	734	7	epimorphism	epimorphism	NOUN
ejpam-6327	734	8	.	.	PUNCT
ejpam-6327	735	1	moreover	moreover	ADV
ejpam-6327	735	2	,	,	PUNCT
ejpam-6327	735	3	kerφ	kerφ	PROPN
ejpam-6327	735	4	=	=	SYM
ejpam-6327	735	5	{	{	PUNCT
ejpam-6327	735	6	(	(	PUNCT
ejpam-6327	735	7	x	x	NOUN
ejpam-6327	735	8	,	,	PUNCT
ejpam-6327	735	9	y	y	NOUN
ejpam-6327	735	10	)	)	PUNCT
ejpam-6327	735	11	∈	∈	PROPN
ejpam-6327	735	12	x	x	SYM
ejpam-6327	735	13	×	×	NOUN
ejpam-6327	735	14	y	y	NOUN
ejpam-6327	735	15	:	:	PUNCT
ejpam-6327	735	16	φ(x	φ(x	PROPN
ejpam-6327	735	17	,	,	PUNCT
ejpam-6327	735	18	y	y	NOUN
ejpam-6327	735	19	)	)	PUNCT
ejpam-6327	735	20	=	=	SYM
ejpam-6327	735	21	(	(	PUNCT
ejpam-6327	735	22	j0	j0	PROPN
ejpam-6327	735	23	,	,	PUNCT
ejpam-6327	735	24	i0	i0	PROPN
ejpam-6327	735	25	)	)	PUNCT
ejpam-6327	735	26	}	}	PUNCT
ejpam-6327	735	27	=	=	SYM
ejpam-6327	735	28	{	{	PUNCT
ejpam-6327	735	29	(	(	PUNCT
ejpam-6327	735	30	x	x	NOUN
ejpam-6327	735	31	,	,	PUNCT
ejpam-6327	735	32	y	y	NOUN
ejpam-6327	735	33	)	)	PUNCT
ejpam-6327	735	34	∈	∈	PROPN
ejpam-6327	735	35	x	x	SYM
ejpam-6327	735	36	×	×	NOUN
ejpam-6327	735	37	y	y	PROPN
ejpam-6327	735	38	:	:	PUNCT
ejpam-6327	735	39	(	(	PUNCT
ejpam-6327	735	40	jx	jx	PROPN
ejpam-6327	735	41	,	,	PUNCT
ejpam-6327	735	42	iy	iy	PROPN
ejpam-6327	735	43	)	)	PUNCT
ejpam-6327	735	44	=	=	PRON
ejpam-6327	735	45	(	(	PUNCT
ejpam-6327	735	46	j0	j0	PROPN
ejpam-6327	735	47	,	,	PUNCT
ejpam-6327	735	48	i0	i0	PROPN
ejpam-6327	735	49	)	)	PUNCT
ejpam-6327	735	50	}	}	PUNCT
ejpam-6327	735	51	=	=	SYM
ejpam-6327	735	52	{	{	PUNCT
ejpam-6327	735	53	(	(	PUNCT
ejpam-6327	735	54	x	x	NOUN
ejpam-6327	735	55	,	,	PUNCT
ejpam-6327	735	56	y	y	NOUN
ejpam-6327	735	57	)	)	PUNCT
ejpam-6327	735	58	∈	∈	PROPN
ejpam-6327	735	59	x	x	SYM
ejpam-6327	735	60	×	×	NOUN
ejpam-6327	735	61	y	y	NOUN
ejpam-6327	735	62	:	:	PUNCT
ejpam-6327	735	63	jx	jx	PROPN
ejpam-6327	735	64	=	=	PROPN
ejpam-6327	735	65	j0	j0	PROPN
ejpam-6327	735	66	and	and	CCONJ
ejpam-6327	735	67	iy	iy	PROPN
ejpam-6327	735	68	=	=	SYM
ejpam-6327	735	69	i0	i0	PROPN
ejpam-6327	735	70	}	}	PUNCT
ejpam-6327	735	71	=	=	SYM
ejpam-6327	735	72	{	{	PUNCT
ejpam-6327	735	73	(	(	PUNCT
ejpam-6327	735	74	x	x	NOUN
ejpam-6327	735	75	,	,	PUNCT
ejpam-6327	735	76	y	y	NOUN
ejpam-6327	735	77	)	)	PUNCT
ejpam-6327	735	78	∈	∈	PROPN
ejpam-6327	735	79	x	x	SYM
ejpam-6327	735	80	×	×	NOUN
ejpam-6327	735	81	y	y	NOUN
ejpam-6327	735	82	:	:	PUNCT
ejpam-6327	735	83	x	x	SYM
ejpam-6327	735	84	∼j	∼j	X
ejpam-6327	735	85	0	0	PUNCT
ejpam-6327	735	86	and	and	CCONJ
ejpam-6327	735	87	y	y	PROPN
ejpam-6327	735	88	∼i	∼i	PROPN
ejpam-6327	735	89	0	0	NUM
ejpam-6327	735	90	}	}	PUNCT
ejpam-6327	735	91	=	=	SYM
ejpam-6327	735	92	{	{	PUNCT
ejpam-6327	735	93	(	(	PUNCT
ejpam-6327	735	94	x	x	NOUN
ejpam-6327	735	95	,	,	PUNCT
ejpam-6327	735	96	y	y	NOUN
ejpam-6327	735	97	)	)	PUNCT
ejpam-6327	735	98	∈	∈	PROPN
ejpam-6327	735	99	x	x	SYM
ejpam-6327	735	100	×	×	NOUN
ejpam-6327	735	101	y	y	NOUN
ejpam-6327	735	102	:	:	PUNCT
ejpam-6327	735	103	x	x	SYM
ejpam-6327	735	104	=	=	SYM
ejpam-6327	735	105	x	x	SYM
ejpam-6327	735	106	∗	∗	NOUN
ejpam-6327	735	107	0	0	NUM
ejpam-6327	736	1	∈	∈	PROPN
ejpam-6327	736	2	j	j	PROPN
ejpam-6327	736	3	and	and	CCONJ
ejpam-6327	736	4	y	y	PROPN
ejpam-6327	736	5	=	=	SYM
ejpam-6327	736	6	y	y	PROPN
ejpam-6327	736	7	∗	∗	NOUN
ejpam-6327	736	8	0	0	PUNCT
ejpam-6327	737	1	∈	∈	PROPN
ejpam-6327	737	2	i	i	NOUN
ejpam-6327	737	3	}	}	PUNCT
ejpam-6327	737	4	=	=	SYM
ejpam-6327	737	5	j	j	PROPN
ejpam-6327	737	6	×	×	PROPN
ejpam-6327	737	7	i.	i.	NOUN
ejpam-6327	737	8	by	by	ADP
ejpam-6327	737	9	first	first	PROPN
ejpam-6327	737	10	isomorphism	isomorphism	PROPN
ejpam-6327	737	11	theorem	theorem	VERB
ejpam-6327	737	12	,	,	PUNCT
ejpam-6327	737	13	x	x	SYM
ejpam-6327	737	14	×	×	PROPN
ejpam-6327	737	15	y	y	PROPN
ejpam-6327	737	16	/	/	SYM
ejpam-6327	737	17	j	j	PROPN
ejpam-6327	737	18	×	×	NOUN
ejpam-6327	738	1	i	i	NOUN
ejpam-6327	738	2	=	=	PUNCT
ejpam-6327	738	3	x	x	SYM
ejpam-6327	738	4	×	×	NOUN
ejpam-6327	738	5	y/	y/	NOUN
ejpam-6327	738	6	kerφ	kerφ	NOUN
ejpam-6327	738	7	∼=	∼=	PROPN
ejpam-6327	738	8	x	x	PROPN
ejpam-6327	738	9	/	/	SYM
ejpam-6327	738	10	j	j	PROPN
ejpam-6327	738	11	×	×	PROPN
ejpam-6327	738	12	y	y	PROPN
ejpam-6327	738	13	/	/	SYM
ejpam-6327	738	14	i.	i.	PROPN
ejpam-6327	738	15	theorem	theorem	VERB
ejpam-6327	738	16	22	22	NUM
ejpam-6327	738	17	.	.	PUNCT
ejpam-6327	739	1	let	let	VERB
ejpam-6327	739	2	µ	µ	NOUN
ejpam-6327	739	3	and	and	CCONJ
ejpam-6327	739	4	ν	ν	PROPN
ejpam-6327	739	5	be	be	AUX
ejpam-6327	739	6	non	non	ADJ
ejpam-6327	739	7	-	-	ADJ
ejpam-6327	739	8	zero	zero	ADJ
ejpam-6327	739	9	fuzzy	fuzzy	ADJ
ejpam-6327	739	10	ks	ks	NOUN
ejpam-6327	739	11	-	-	PUNCT
ejpam-6327	739	12	ideals	ideal	NOUN
ejpam-6327	739	13	of	of	ADP
ejpam-6327	739	14	ks	ks	NOUN
ejpam-6327	739	15	-	-	PUNCT
ejpam-6327	739	16	semigroups	semigroups	X
ejpam-6327	739	17	x	x	X
ejpam-6327	739	18	and	and	CCONJ
ejpam-6327	739	19	y	y	PROPN
ejpam-6327	739	20	,	,	PUNCT
ejpam-6327	739	21	respectively	respectively	ADV
ejpam-6327	739	22	,	,	PUNCT
ejpam-6327	739	23	both	both	PRON
ejpam-6327	739	24	satisfying	satisfy	VERB
ejpam-6327	739	25	condition	condition	NOUN
ejpam-6327	739	26	(	(	PUNCT
ejpam-6327	739	27	c	c	NOUN
ejpam-6327	739	28	)	)	PUNCT
ejpam-6327	739	29	.	.	PUNCT
ejpam-6327	740	1	if	if	SCONJ
ejpam-6327	740	2	j	j	PROPN
ejpam-6327	740	3	and	and	CCONJ
ejpam-6327	740	4	i	i	PRON
ejpam-6327	740	5	are	be	AUX
ejpam-6327	740	6	ideals	ideal	NOUN
ejpam-6327	740	7	of	of	ADP
ejpam-6327	740	8	x	x	X
ejpam-6327	740	9	and	and	CCONJ
ejpam-6327	740	10	y	y	PROPN
ejpam-6327	740	11	,	,	PUNCT
ejpam-6327	740	12	respectively	respectively	ADV
ejpam-6327	740	13	,	,	PUNCT
ejpam-6327	740	14	then	then	ADV
ejpam-6327	740	15	x	x	SYM
ejpam-6327	740	16	×	×	NOUN
ejpam-6327	740	17	y/µ×	y/µ×	NOUN
ejpam-6327	740	18	ν	ν	PROPN
ejpam-6327	740	19	j	j	PROPN
ejpam-6327	740	20	×	×	PROPN
ejpam-6327	740	21	i/µ×	i/µ×	ADV
ejpam-6327	740	22	ν	ν	PRON
ejpam-6327	740	23	∼=	∼=	PROPN
ejpam-6327	740	24	x	x	SYM
ejpam-6327	740	25	/	/	SYM
ejpam-6327	740	26	j	j	PROPN
ejpam-6327	740	27	×	×	PROPN
ejpam-6327	740	28	y	y	PROPN
ejpam-6327	740	29	/	/	SYM
ejpam-6327	740	30	i.	i.	NOUN
ejpam-6327	740	31	proof	proof	NOUN
ejpam-6327	740	32	.	.	PUNCT
ejpam-6327	741	1	let	let	VERB
ejpam-6327	741	2	µ	µ	NOUN
ejpam-6327	741	3	and	and	CCONJ
ejpam-6327	741	4	ν	ν	PROPN
ejpam-6327	741	5	be	be	AUX
ejpam-6327	741	6	non	non	ADJ
ejpam-6327	741	7	-	-	ADJ
ejpam-6327	741	8	zero	zero	ADJ
ejpam-6327	741	9	fuzzy	fuzzy	ADJ
ejpam-6327	741	10	ks	ks	NOUN
ejpam-6327	741	11	-	-	PUNCT
ejpam-6327	741	12	ideals	ideal	NOUN
ejpam-6327	741	13	of	of	ADP
ejpam-6327	741	14	ks	ks	NOUN
ejpam-6327	741	15	-	-	PUNCT
ejpam-6327	741	16	semigroups	semigroups	X
ejpam-6327	741	17	x	x	X
ejpam-6327	741	18	and	and	CCONJ
ejpam-6327	741	19	y	y	PROPN
ejpam-6327	741	20	,	,	PUNCT
ejpam-6327	741	21	respectively	respectively	ADV
ejpam-6327	741	22	,	,	PUNCT
ejpam-6327	741	23	both	both	PRON
ejpam-6327	741	24	satisfying	satisfy	VERB
ejpam-6327	741	25	condition	condition	NOUN
ejpam-6327	741	26	(	(	PUNCT
ejpam-6327	741	27	c	c	NOUN
ejpam-6327	741	28	)	)	PUNCT
ejpam-6327	741	29	.	.	PUNCT
ejpam-6327	742	1	let	let	VERB
ejpam-6327	742	2	j	j	PROPN
ejpam-6327	742	3	,	,	PUNCT
ejpam-6327	742	4	i	i	PRON
ejpam-6327	742	5	be	be	VERB
ejpam-6327	742	6	ideals	ideal	NOUN
ejpam-6327	742	7	of	of	ADP
ejpam-6327	742	8	x	x	X
ejpam-6327	742	9	and	and	CCONJ
ejpam-6327	742	10	y	y	PROPN
ejpam-6327	742	11	,	,	PUNCT
ejpam-6327	742	12	respectively	respectively	ADV
ejpam-6327	742	13	.	.	PUNCT
ejpam-6327	743	1	then	then	ADV
ejpam-6327	743	2	by	by	ADP
ejpam-6327	743	3	lemma	lemma	PROPN
ejpam-6327	743	4	4	4	NUM
ejpam-6327	743	5	,	,	PUNCT
ejpam-6327	743	6	j	j	PROPN
ejpam-6327	743	7	×	×	NOUN
ejpam-6327	743	8	i	i	PRON
ejpam-6327	743	9	is	be	AUX
ejpam-6327	743	10	an	an	DET
ejpam-6327	743	11	ideal	ideal	NOUN
ejpam-6327	743	12	of	of	ADP
ejpam-6327	743	13	x	x	SYM
ejpam-6327	743	14	×	×	PROPN
ejpam-6327	743	15	y	y	PROPN
ejpam-6327	743	16	.	.	PUNCT
ejpam-6327	744	1	by	by	ADP
ejpam-6327	744	2	proposition	proposition	NOUN
ejpam-6327	744	3	6	6	NUM
ejpam-6327	744	4	,	,	PUNCT
ejpam-6327	744	5	µ	µ	PRON
ejpam-6327	744	6	×	×	NOUN
ejpam-6327	744	7	ν	ν	NOUN
ejpam-6327	744	8	is	be	AUX
ejpam-6327	744	9	a	a	DET
ejpam-6327	744	10	fuzzy	fuzzy	ADJ
ejpam-6327	744	11	ks	ks	NOUN
ejpam-6327	744	12	-	-	NOUN
ejpam-6327	744	13	ideal	ideal	NOUN
ejpam-6327	744	14	in	in	ADP
ejpam-6327	744	15	x	x	SYM
ejpam-6327	744	16	×	×	PROPN
ejpam-6327	744	17	y	y	PROPN
ejpam-6327	744	18	,	,	PUNCT
ejpam-6327	744	19	so	so	ADV
ejpam-6327	744	20	is	be	AUX
ejpam-6327	744	21	in	in	ADP
ejpam-6327	744	22	j	j	PROPN
ejpam-6327	744	23	×	×	PROPN
ejpam-6327	744	24	i.	i.	PROPN
ejpam-6327	744	25	thus	thus	ADV
ejpam-6327	744	26	,	,	PUNCT
ejpam-6327	744	27	by	by	ADP
ejpam-6327	744	28	theorem	theorem	NOUN
ejpam-6327	744	29	21	21	NUM
ejpam-6327	744	30	,	,	PUNCT
ejpam-6327	744	31	x	x	X
ejpam-6327	744	32	×	×	VERB
ejpam-6327	744	33	y	y	PROPN
ejpam-6327	744	34	µ×	µ×	X
ejpam-6327	744	35	ν	ν	PRON
ejpam-6327	744	36	∼=	∼=	PROPN
ejpam-6327	744	37	x/µ×	x/µ×	NOUN
ejpam-6327	744	38	y	y	PROPN
ejpam-6327	744	39	/	/	SYM
ejpam-6327	744	40	ν	ν	NOUN
ejpam-6327	744	41	and	and	CCONJ
ejpam-6327	744	42	j	j	PROPN
ejpam-6327	744	43	×	×	NOUN
ejpam-6327	744	44	i	i	PRON
ejpam-6327	744	45	µ×	µ×	X
ejpam-6327	744	46	ν	ν	PRON
ejpam-6327	744	47	∼=	∼=	PROPN
ejpam-6327	744	48	j/µ×	j/µ×	NOUN
ejpam-6327	744	49	i	i	PROPN
ejpam-6327	744	50	/	/	SYM
ejpam-6327	744	51	ν	ν	NOUN
ejpam-6327	744	52	.	.	PUNCT
ejpam-6327	745	1	hence	hence	ADV
ejpam-6327	745	2	,	,	PUNCT
ejpam-6327	745	3	by	by	ADP
ejpam-6327	745	4	theorem	theorem	NOUN
ejpam-6327	745	5	15	15	NUM
ejpam-6327	745	6	and	and	CCONJ
ejpam-6327	745	7	lemma	lemma	PROPN
ejpam-6327	745	8	4	4	NUM
ejpam-6327	745	9	,	,	PUNCT
ejpam-6327	745	10	x	x	SYM
ejpam-6327	745	11	×	×	NOUN
ejpam-6327	745	12	y/µ×	y/µ×	NOUN
ejpam-6327	745	13	ν	ν	PROPN
ejpam-6327	745	14	j	j	PROPN
ejpam-6327	745	15	×	×	PROPN
ejpam-6327	745	16	i/µ×	i/µ×	ADV
ejpam-6327	745	17	ν	ν	X
ejpam-6327	745	18	∼=	∼=	PROPN
ejpam-6327	745	19	x/µ×	x/µ×	NOUN
ejpam-6327	745	20	y	y	PROPN
ejpam-6327	745	21	/	/	SYM
ejpam-6327	745	22	ν	ν	NOUN
ejpam-6327	745	23	j/µ×	j/µ×	NOUN
ejpam-6327	745	24	i	i	PROPN
ejpam-6327	745	25	/	/	SYM
ejpam-6327	745	26	ν	ν	PRON
ejpam-6327	745	27	∼=	∼=	NOUN
ejpam-6327	745	28	x/µ	x/µ	NOUN
ejpam-6327	746	1	j/µ	j/µ	NOUN
ejpam-6327	746	2	×	×	PROPN
ejpam-6327	746	3	y	y	PROPN
ejpam-6327	746	4	/	/	SYM
ejpam-6327	746	5	ν	ν	PROPN
ejpam-6327	746	6	i	i	NOUN
ejpam-6327	746	7	/	/	SYM
ejpam-6327	746	8	ν	ν	X
ejpam-6327	746	9	∼=	∼=	PROPN
ejpam-6327	746	10	x	x	SYM
ejpam-6327	746	11	/	/	SYM
ejpam-6327	746	12	j	j	PROPN
ejpam-6327	746	13	×	×	PROPN
ejpam-6327	746	14	y	y	PROPN
ejpam-6327	746	15	/	/	SYM
ejpam-6327	746	16	i.	i.	PROPN
ejpam-6327	746	17	7	7	NUM
ejpam-6327	746	18	.	.	PUNCT
ejpam-6327	746	19	conclusion	conclusion	NOUN
ejpam-6327	746	20	in	in	ADP
ejpam-6327	746	21	this	this	DET
ejpam-6327	746	22	paper	paper	NOUN
ejpam-6327	746	23	,	,	PUNCT
ejpam-6327	746	24	we	we	PRON
ejpam-6327	746	25	studied	study	VERB
ejpam-6327	746	26	new	new	ADJ
ejpam-6327	746	27	types	type	NOUN
ejpam-6327	746	28	of	of	ADP
ejpam-6327	746	29	fuzzy	fuzzy	ADJ
ejpam-6327	746	30	ideals	ideal	NOUN
ejpam-6327	746	31	,	,	PUNCT
ejpam-6327	746	32	namely	namely	ADV
ejpam-6327	746	33	,	,	PUNCT
ejpam-6327	746	34	fuzzy	fuzzy	ADJ
ejpam-6327	746	35	commutative	commutative	ADJ
ejpam-6327	746	36	ksideals	ksideal	NOUN
ejpam-6327	746	37	and	and	CCONJ
ejpam-6327	746	38	fuzzy	fuzzy	ADJ
ejpam-6327	746	39	implicative	implicative	ADJ
ejpam-6327	746	40	ks	ks	NOUN
ejpam-6327	746	41	-	-	NOUN
ejpam-6327	746	42	ideals	ideal	NOUN
ejpam-6327	746	43	.	.	PUNCT
ejpam-6327	747	1	we	we	PRON
ejpam-6327	747	2	compared	compare	VERB
ejpam-6327	747	3	them	they	PRON
ejpam-6327	747	4	with	with	ADP
ejpam-6327	747	5	existing	exist	VERB
ejpam-6327	747	6	fuzzy	fuzzy	ADJ
ejpam-6327	747	7	ks	ks	NOUN
ejpam-6327	747	8	-	-	PUNCT
ejpam-6327	747	9	ideals	ideal	NOUN
ejpam-6327	747	10	h.	h.	PROPN
ejpam-6327	747	11	sarapuddin	sarapuddin	PROPN
ejpam-6327	747	12	,	,	PUNCT
ejpam-6327	747	13	j.	j.	PROPN
ejpam-6327	747	14	vilela	vilela	PROPN
ejpam-6327	747	15	/	/	SYM
ejpam-6327	747	16	eur	eur	PROPN
ejpam-6327	747	17	.	.	PUNCT
ejpam-6327	748	1	j.	j.	PROPN
ejpam-6327	748	2	pure	pure	PROPN
ejpam-6327	748	3	appl	appl	PROPN
ejpam-6327	748	4	.	.	PROPN
ejpam-6327	748	5	math	math	PROPN
ejpam-6327	748	6	,	,	PUNCT
ejpam-6327	748	7	18	18	NUM
ejpam-6327	748	8	(	(	PUNCT
ejpam-6327	748	9	3	3	NUM
ejpam-6327	748	10	)	)	PUNCT
ejpam-6327	748	11	(	(	PUNCT
ejpam-6327	748	12	2025	2025	NUM
ejpam-6327	748	13	)	)	PUNCT
ejpam-6327	748	14	,	,	PUNCT
ejpam-6327	748	15	6327	6327	NUM
ejpam-6327	748	16	23	23	NUM
ejpam-6327	748	17	of	of	ADP
ejpam-6327	748	18	23	23	NUM
ejpam-6327	748	19	and	and	CCONJ
ejpam-6327	748	20	investigated	investigate	VERB
ejpam-6327	748	21	some	some	PRON
ejpam-6327	748	22	of	of	ADP
ejpam-6327	748	23	their	their	PRON
ejpam-6327	748	24	properties	property	NOUN
ejpam-6327	748	25	.	.	PUNCT
ejpam-6327	749	1	we	we	PRON
ejpam-6327	749	2	defined	define	VERB
ejpam-6327	749	3	binary	binary	ADJ
ejpam-6327	749	4	relation	relation	NOUN
ejpam-6327	749	5	on	on	ADP
ejpam-6327	749	6	a	a	DET
ejpam-6327	749	7	ks	ks	NOUN
ejpam-6327	749	8	-	-	PUNCT
ejpam-6327	749	9	semigroup	semigroup	NOUN
ejpam-6327	749	10	using	use	VERB
ejpam-6327	749	11	fuzzy	fuzzy	ADJ
ejpam-6327	749	12	ks	k	NOUN
ejpam-6327	749	13	-	-	NOUN
ejpam-6327	749	14	ideals	ideal	NOUN
ejpam-6327	749	15	with	with	ADP
ejpam-6327	749	16	certain	certain	ADJ
ejpam-6327	749	17	condition	condition	NOUN
ejpam-6327	749	18	and	and	CCONJ
ejpam-6327	749	19	showed	show	VERB
ejpam-6327	749	20	that	that	SCONJ
ejpam-6327	749	21	it	it	PRON
ejpam-6327	749	22	is	be	AUX
ejpam-6327	749	23	a	a	DET
ejpam-6327	749	24	congruence	congruence	NOUN
ejpam-6327	749	25	relation	relation	NOUN
ejpam-6327	749	26	on	on	ADP
ejpam-6327	749	27	a	a	DET
ejpam-6327	749	28	ks	ks	NOUN
ejpam-6327	749	29	-	-	PUNCT
ejpam-6327	749	30	semigroup	semigroup	NOUN
ejpam-6327	749	31	.	.	PUNCT
ejpam-6327	750	1	using	use	VERB
ejpam-6327	750	2	this	this	DET
ejpam-6327	750	3	congruence	congruence	NOUN
ejpam-6327	750	4	relation	relation	NOUN
ejpam-6327	750	5	,	,	PUNCT
ejpam-6327	750	6	we	we	PRON
ejpam-6327	750	7	constructed	construct	VERB
ejpam-6327	750	8	a	a	DET
ejpam-6327	750	9	quotient	quotient	NOUN
ejpam-6327	750	10	ks	ks	NOUN
ejpam-6327	750	11	-	-	PUNCT
ejpam-6327	750	12	semigroup	semigroup	NOUN
ejpam-6327	750	13	induced	induce	VERB
ejpam-6327	750	14	by	by	ADP
ejpam-6327	750	15	fuzzy	fuzzy	ADJ
ejpam-6327	750	16	ks	ks	NOUN
ejpam-6327	750	17	-	-	NOUN
ejpam-6327	750	18	ideals	ideal	NOUN
ejpam-6327	750	19	and	and	CCONJ
ejpam-6327	750	20	showed	show	VERB
ejpam-6327	750	21	that	that	SCONJ
ejpam-6327	750	22	it	it	PRON
ejpam-6327	750	23	is	be	AUX
ejpam-6327	750	24	a	a	DET
ejpam-6327	750	25	generalization	generalization	NOUN
ejpam-6327	750	26	of	of	ADP
ejpam-6327	750	27	the	the	DET
ejpam-6327	750	28	quotient	quotient	NOUN
ejpam-6327	750	29	kssemigroup	kssemigroup	NOUN
ejpam-6327	750	30	via	via	ADP
ejpam-6327	750	31	ideals	ideal	NOUN
ejpam-6327	750	32	.	.	PUNCT
ejpam-6327	751	1	we	we	PRON
ejpam-6327	751	2	investigated	investigate	VERB
ejpam-6327	751	3	the	the	DET
ejpam-6327	751	4	homomorphic	homomorphic	ADJ
ejpam-6327	751	5	properties	property	NOUN
ejpam-6327	751	6	of	of	ADP
ejpam-6327	751	7	this	this	DET
ejpam-6327	751	8	generalized	generalized	ADJ
ejpam-6327	751	9	quotient	quotient	NOUN
ejpam-6327	751	10	and	and	CCONJ
ejpam-6327	751	11	its	its	PRON
ejpam-6327	751	12	properties	property	NOUN
ejpam-6327	751	13	with	with	ADP
ejpam-6327	751	14	respect	respect	NOUN
ejpam-6327	751	15	to	to	ADP
ejpam-6327	751	16	fuzzy	fuzzy	ADJ
ejpam-6327	751	17	commutative	commutative	ADJ
ejpam-6327	751	18	ks	ks	NOUN
ejpam-6327	751	19	-	-	PUNCT
ejpam-6327	751	20	ideals	ideal	NOUN
ejpam-6327	751	21	,	,	PUNCT
ejpam-6327	751	22	fuzzy	fuzzy	ADJ
ejpam-6327	751	23	implicative	implicative	ADJ
ejpam-6327	751	24	ks	ks	NOUN
ejpam-6327	751	25	-	-	PUNCT
ejpam-6327	751	26	ideals	ideal	NOUN
ejpam-6327	751	27	and	and	CCONJ
ejpam-6327	751	28	fuzzy	fuzzy	ADJ
ejpam-6327	751	29	ks	ks	NOUN
ejpam-6327	751	30	-	-	ADJ
ejpam-6327	751	31	p	p	NOUN
ejpam-6327	751	32	-	-	PUNCT
ejpam-6327	751	33	ideals	ideal	NOUN
ejpam-6327	751	34	.	.	PUNCT
ejpam-6327	752	1	moreover	moreover	ADV
ejpam-6327	752	2	,	,	PUNCT
ejpam-6327	752	3	the	the	DET
ejpam-6327	752	4	properties	property	NOUN
ejpam-6327	752	5	of	of	ADP
ejpam-6327	752	6	this	this	DET
ejpam-6327	752	7	generalized	generalized	ADJ
ejpam-6327	752	8	structure	structure	NOUN
ejpam-6327	752	9	were	be	AUX
ejpam-6327	752	10	examined	examine	VERB
ejpam-6327	752	11	using	use	VERB
ejpam-6327	752	12	fuzzy	fuzzy	ADJ
ejpam-6327	752	13	ks	k	NOUN
ejpam-6327	752	14	-	-	PUNCT
ejpam-6327	752	15	ideals	ideal	NOUN
ejpam-6327	752	16	and	and	CCONJ
ejpam-6327	752	17	product	product	NOUN
ejpam-6327	752	18	ks	ks	NOUN
ejpam-6327	752	19	-	-	PUNCT
ejpam-6327	752	20	semigroups	semigroup	NOUN
ejpam-6327	752	21	.	.	PUNCT
ejpam-6327	753	1	acknowledgements	acknowledgement	NOUN
ejpam-6327	753	2	the	the	DET
ejpam-6327	753	3	authors	author	NOUN
ejpam-6327	753	4	would	would	AUX
ejpam-6327	753	5	like	like	VERB
ejpam-6327	753	6	to	to	PART
ejpam-6327	753	7	thank	thank	VERB
ejpam-6327	753	8	the	the	DET
ejpam-6327	753	9	department	department	NOUN
ejpam-6327	753	10	of	of	ADP
ejpam-6327	753	11	science	science	NOUN
ejpam-6327	753	12	and	and	CCONJ
ejpam-6327	753	13	technology	technology	NOUN
ejpam-6327	753	14	accelerated	accelerate	VERB
ejpam-6327	753	15	science	science	NOUN
ejpam-6327	753	16	and	and	CCONJ
ejpam-6327	753	17	technology	technology	NOUN
ejpam-6327	753	18	human	human	ADJ
ejpam-6327	753	19	resource	resource	NOUN
ejpam-6327	753	20	development	development	NOUN
ejpam-6327	753	21	program	program	NOUN
ejpam-6327	753	22	(	(	PUNCT
ejpam-6327	753	23	dost	dost	NOUN
ejpam-6327	753	24	-	-	PUNCT
ejpam-6327	753	25	asthrdp)philippines	asthrdp)philippine	NOUN
ejpam-6327	753	26	,	,	PUNCT
ejpam-6327	753	27	and	and	CCONJ
ejpam-6327	753	28	msu	msu	PROPN
ejpam-6327	753	29	-	-	PUNCT
ejpam-6327	753	30	iligan	iligan	PROPN
ejpam-6327	753	31	institute	institute	PROPN
ejpam-6327	753	32	of	of	ADP
ejpam-6327	753	33	technology	technology	NOUN
ejpam-6327	753	34	for	for	ADP
ejpam-6327	753	35	funding	fund	VERB
ejpam-6327	753	36	this	this	DET
ejpam-6327	753	37	research	research	NOUN
ejpam-6327	753	38	.	.	PUNCT
ejpam-6327	754	1	references	reference	NOUN
ejpam-6327	754	2	[	[	X
ejpam-6327	754	3	1	1	NUM
ejpam-6327	754	4	]	]	X
ejpam-6327	754	5	y.	y.	PROPN
ejpam-6327	754	6	imai	imai	PROPN
ejpam-6327	754	7	and	and	CCONJ
ejpam-6327	754	8	k.	k.	PROPN
ejpam-6327	754	9	iseki	iseki	PROPN
ejpam-6327	754	10	.	.	PUNCT
ejpam-6327	755	1	on	on	ADP
ejpam-6327	755	2	axiom	axiom	NOUN
ejpam-6327	755	3	systems	system	NOUN
ejpam-6327	755	4	of	of	ADP
ejpam-6327	755	5	propositional	propositional	ADJ
ejpam-6327	755	6	calculi	calculi	PROPN
ejpam-6327	755	7	,	,	PUNCT
ejpam-6327	755	8	xiv	xiv	PROPN
ejpam-6327	755	9	.	.	PUNCT
ejpam-6327	756	1	proc	proc	PROPN
ejpam-6327	756	2	.	.	PUNCT
ejpam-6327	757	1	japan	japan	PROPN
ejpam-6327	757	2	acad	acad	PROPN
ejpam-6327	757	3	.	.	PROPN
ejpam-6327	757	4	,	,	PUNCT
ejpam-6327	757	5	42:19–22	42:19–22	NUM
ejpam-6327	757	6	,	,	PUNCT
ejpam-6327	757	7	1966	1966	NUM
ejpam-6327	757	8	.	.	PUNCT
ejpam-6327	758	1	[	[	X
ejpam-6327	758	2	2	2	NUM
ejpam-6327	758	3	]	]	PUNCT
ejpam-6327	758	4	a.	a.	NOUN
ejpam-6327	758	5	suschkewitsch	suschkewitsch	PROPN
ejpam-6327	758	6	.	.	PUNCT
ejpam-6327	759	1	fiber	fiber	NOUN
ejpam-6327	759	2	die	die	VERB
ejpam-6327	759	3	endlichen	endlichen	PROPN
ejpam-6327	759	4	gruppen	gruppen	PROPN
ejpam-6327	759	5	ohne	ohne	NOUN
ejpam-6327	759	6	das	das	PROPN
ejpam-6327	759	7	gesetz	gesetz	VERB
ejpam-6327	759	8	der	der	ADJ
ejpam-6327	759	9	eindeutigen	eindeutigen	NOUN
ejpam-6327	759	10	umkehrbarkeit	umkehrbarkeit	NOUN
ejpam-6327	759	11	.	.	PUNCT
ejpam-6327	760	1	math	math	PROPN
ejpam-6327	760	2	.	.	PUNCT
ejpam-6327	761	1	ann	ann	PROPN
ejpam-6327	761	2	.	.	PROPN
ejpam-6327	761	3	,	,	PUNCT
ejpam-6327	761	4	99:30–50	99:30–50	NUM
ejpam-6327	761	5	,	,	PUNCT
ejpam-6327	761	6	1928	1928	NUM
ejpam-6327	761	7	.	.	PUNCT
ejpam-6327	762	1	[	[	X
ejpam-6327	762	2	3	3	NUM
ejpam-6327	762	3	]	]	PUNCT
ejpam-6327	762	4	a.	a.	NOUN
ejpam-6327	762	5	h.	h.	PROPN
ejpam-6327	762	6	clifford	clifford	PROPN
ejpam-6327	762	7	and	and	CCONJ
ejpam-6327	762	8	g.	g.	PROPN
ejpam-6327	762	9	b.	b.	PROPN
ejpam-6327	762	10	preston	preston	PROPN
ejpam-6327	762	11	.	.	PUNCT
ejpam-6327	763	1	introduction	introduction	NOUN
ejpam-6327	763	2	to	to	ADP
ejpam-6327	763	3	the	the	DET
ejpam-6327	763	4	theory	theory	NOUN
ejpam-6327	763	5	of	of	ADP
ejpam-6327	763	6	semigroups	semigroup	NOUN
ejpam-6327	763	7	.	.	PUNCT
ejpam-6327	764	1	1961	1961	NUM
ejpam-6327	764	2	.	.	PUNCT
ejpam-6327	765	1	[	[	X
ejpam-6327	765	2	4	4	X
ejpam-6327	765	3	]	]	PUNCT
ejpam-6327	765	4	l.	l.	PROPN
ejpam-6327	765	5	a.	a.	PROPN
ejpam-6327	765	6	zadeh	zadeh	PROPN
ejpam-6327	765	7	.	.	PUNCT
ejpam-6327	765	8	fuzzy	fuzzy	ADJ
ejpam-6327	765	9	sets	set	NOUN
ejpam-6327	765	10	.	.	PUNCT
ejpam-6327	766	1	information	information	NOUN
ejpam-6327	766	2	annd	annd	PROPN
ejpam-6327	766	3	control	control	PROPN
ejpam-6327	766	4	,	,	PUNCT
ejpam-6327	766	5	8:338–353	8:338–353	NUM
ejpam-6327	766	6	,	,	PUNCT
ejpam-6327	766	7	1965	1965	NUM
ejpam-6327	766	8	.	.	PUNCT
ejpam-6327	767	1	[	[	X
ejpam-6327	767	2	5	5	X
ejpam-6327	767	3	]	]	PUNCT
ejpam-6327	767	4	k.	k.	PROPN
ejpam-6327	767	5	h.	h.	PROPN
ejpam-6327	767	6	kim	kim	PROPN
ejpam-6327	767	7	.	.	PUNCT
ejpam-6327	768	1	on	on	ADP
ejpam-6327	768	2	structure	structure	NOUN
ejpam-6327	768	3	of	of	ADP
ejpam-6327	768	4	ks	ks	NOUN
ejpam-6327	768	5	-	-	PUNCT
ejpam-6327	768	6	semigroups	semigroup	NOUN
ejpam-6327	768	7	.	.	PUNCT
ejpam-6327	769	1	international	international	ADJ
ejpam-6327	769	2	mathematical	mathematical	PROPN
ejpam-6327	769	3	forum	forum	PROPN
ejpam-6327	769	4	,	,	PUNCT
ejpam-6327	769	5	4(2):67–76	4(2):67–76	NUM
ejpam-6327	769	6	,	,	PUNCT
ejpam-6327	769	7	2006	2006	NUM
ejpam-6327	769	8	.	.	PUNCT
ejpam-6327	770	1	[	[	X
ejpam-6327	770	2	6	6	NUM
ejpam-6327	770	3	]	]	PUNCT
ejpam-6327	770	4	j.	j.	PROPN
ejpam-6327	770	5	p.	p.	PROPN
ejpam-6327	770	6	vilela	vilela	PROPN
ejpam-6327	770	7	and	and	CCONJ
ejpam-6327	770	8	m.	m.	NOUN
ejpam-6327	770	9	cawi	cawi	NOUN
ejpam-6327	770	10	.	.	PUNCT
ejpam-6327	771	1	on	on	ADP
ejpam-6327	771	2	ks	ks	NOUN
ejpam-6327	771	3	-	-	PUNCT
ejpam-6327	771	4	semigroup	semigroup	ADJ
ejpam-6327	771	5	homomorphism	homomorphism	NOUN
ejpam-6327	771	6	.	.	PUNCT
ejpam-6327	772	1	international	international	ADJ
ejpam-6327	772	2	mathematical	mathematical	PROPN
ejpam-6327	772	3	forum	forum	PROPN
ejpam-6327	772	4	,	,	PUNCT
ejpam-6327	772	5	4(23):1129–1138	4(23):1129–1138	NUM
ejpam-6327	772	6	,	,	PUNCT
ejpam-6327	772	7	2009	2009	NUM
ejpam-6327	772	8	.	.	PUNCT
ejpam-6327	773	1	[	[	X
ejpam-6327	773	2	7	7	X
ejpam-6327	773	3	]	]	X
ejpam-6327	773	4	d.	d.	PROPN
ejpam-6327	773	5	r.	r.	PROPN
ejpam-6327	773	6	prince	prince	PROPN
ejpam-6327	773	7	williams	williams	PROPN
ejpam-6327	773	8	and	and	CCONJ
ejpam-6327	773	9	s.	s.	PROPN
ejpam-6327	773	10	husain	husain	PROPN
ejpam-6327	773	11	.	.	PUNCT
ejpam-6327	774	1	on	on	ADP
ejpam-6327	774	2	fuzzy	fuzzy	ADJ
ejpam-6327	774	3	ks	ks	NOUN
ejpam-6327	774	4	-	-	PUNCT
ejpam-6327	774	5	semigroups	semigroup	NOUN
ejpam-6327	774	6	.	.	PUNCT
ejpam-6327	775	1	international	international	ADJ
ejpam-6327	775	2	mathematical	mathematical	PROPN
ejpam-6327	775	3	forum	forum	PROPN
ejpam-6327	775	4	,	,	PUNCT
ejpam-6327	775	5	2(32):1577–1586	2(32):1577–1586	NUM
ejpam-6327	775	6	,	,	PUNCT
ejpam-6327	775	7	2007	2007	NUM
ejpam-6327	775	8	.	.	PUNCT
ejpam-6327	776	1	[	[	X
ejpam-6327	776	2	8	8	NUM
ejpam-6327	776	3	]	]	X
ejpam-6327	776	4	l.	l.	PROPN
ejpam-6327	776	5	r.	r.	PROPN
ejpam-6327	776	6	bautista	bautista	PROPN
ejpam-6327	776	7	and	and	CCONJ
ejpam-6327	776	8	j.	j.	PROPN
ejpam-6327	776	9	p.	p.	PROPN
ejpam-6327	776	10	vilela	vilela	PROPN
ejpam-6327	776	11	.	.	PUNCT
ejpam-6327	777	1	on	on	ADP
ejpam-6327	777	2	fuzzy	fuzzy	ADJ
ejpam-6327	777	3	topology	topology	NOUN
ejpam-6327	777	4	on	on	ADP
ejpam-6327	777	5	ks	ks	NOUN
ejpam-6327	777	6	-	-	PUNCT
ejpam-6327	777	7	semigroups	semigroup	NOUN
ejpam-6327	777	8	.	.	PUNCT
ejpam-6327	778	1	international	international	ADJ
ejpam-6327	778	2	mathematical	mathematical	PROPN
ejpam-6327	778	3	forum	forum	PROPN
ejpam-6327	778	4	,	,	PUNCT
ejpam-6327	778	5	6(39):1921–1932	6(39):1921–1932	NUM
ejpam-6327	778	6	,	,	PUNCT
ejpam-6327	778	7	2011	2011	NUM
ejpam-6327	778	8	.	.	PUNCT
ejpam-6327	779	1	[	[	X
ejpam-6327	779	2	9	9	NUM
ejpam-6327	779	3	]	]	PUNCT
ejpam-6327	779	4	m.	m.	NOUN
ejpam-6327	779	5	al	al	PROPN
ejpam-6327	779	6	tahan	tahan	PROPN
ejpam-6327	779	7	,	,	PUNCT
ejpam-6327	779	8	s.	s.	PROPN
ejpam-6327	779	9	hoskova	hoskova	PROPN
ejpam-6327	779	10	-	-	PUNCT
ejpam-6327	779	11	mayerova	mayerova	NOUN
ejpam-6327	779	12	,	,	PUNCT
ejpam-6327	779	13	and	and	CCONJ
ejpam-6327	779	14	s.	s.	PROPN
ejpam-6327	779	15	al	al	PROPN
ejpam-6327	779	16	-	-	PUNCT
ejpam-6327	779	17	kaseasbeh	kaseasbeh	PROPN
ejpam-6327	779	18	.	.	PUNCT
ejpam-6327	780	1	generalization	generalization	NOUN
ejpam-6327	780	2	of	of	ADP
ejpam-6327	780	3	biantiideals	biantiideal	NOUN
ejpam-6327	780	4	in	in	ADP
ejpam-6327	780	5	semigroups	semigroup	NOUN
ejpam-6327	780	6	.	.	PUNCT
ejpam-6327	781	1	eur	eur	PROPN
ejpam-6327	781	2	.	.	PUNCT
ejpam-6327	782	1	j.	j.	PROPN
ejpam-6327	782	2	pure	pure	PROPN
ejpam-6327	782	3	appl	appl	PROPN
ejpam-6327	782	4	.	.	PUNCT
ejpam-6327	782	5	math	math	PROPN
ejpam-6327	782	6	.	.	PUNCT
ejpam-6327	782	7	,	,	PUNCT
ejpam-6327	782	8	18(1	18(1	NUM
ejpam-6327	782	9	)	)	PUNCT
ejpam-6327	782	10	,	,	PUNCT
ejpam-6327	782	11	2025	2025	NUM
ejpam-6327	782	12	.	.	PUNCT
ejpam-6327	783	1	[	[	X
ejpam-6327	783	2	10	10	NUM
ejpam-6327	783	3	]	]	PUNCT
ejpam-6327	783	4	s.	s.	PROPN
ejpam-6327	783	5	mohammed	mohammed	PROPN
ejpam-6327	783	6	and	and	CCONJ
ejpam-6327	783	7	s.	s.	PROPN
ejpam-6327	783	8	jaber	jaber	PROPN
ejpam-6327	783	9	.	.	PUNCT
ejpam-6327	784	1	some	some	DET
ejpam-6327	784	2	types	type	NOUN
ejpam-6327	784	3	of	of	ADP
ejpam-6327	784	4	ideals	ideal	NOUN
ejpam-6327	784	5	on	on	ADP
ejpam-6327	784	6	ks	ks	NOUN
ejpam-6327	784	7	-	-	PUNCT
ejpam-6327	784	8	semigroups	semigroup	NOUN
ejpam-6327	784	9	.	.	PUNCT
ejpam-6327	785	1	mathematical	mathematical	ADJ
ejpam-6327	785	2	theory	theory	NOUN
ejpam-6327	785	3	and	and	CCONJ
ejpam-6327	785	4	modeling	modeling	NOUN
ejpam-6327	785	5	,	,	PUNCT
ejpam-6327	785	6	4(14):57–68	4(14):57–68	NUM
ejpam-6327	785	7	,	,	PUNCT
ejpam-6327	785	8	2014	2014	NUM
ejpam-6327	785	9	.	.	PUNCT
ejpam-6327	786	1	[	[	X
ejpam-6327	786	2	11	11	NUM
ejpam-6327	786	3	]	]	PUNCT
ejpam-6327	786	4	h.	h.	PROPN
ejpam-6327	786	5	k.	k.	PROPN
ejpam-6327	786	6	kim	kim	PROPN
ejpam-6327	786	7	,	,	PUNCT
ejpam-6327	786	8	d.	d.	PROPN
ejpam-6327	786	9	h.	h.	PROPN
ejpam-6327	786	10	hong	hong	PROPN
ejpam-6327	786	11	,	,	PUNCT
ejpam-6327	786	12	and	and	CCONJ
ejpam-6327	786	13	j.	j.	PROPN
ejpam-6327	786	14	y.	y.	PROPN
ejpam-6327	786	15	kim	kim	PROPN
ejpam-6327	786	16	.	.	PUNCT
ejpam-6327	787	1	on	on	ADP
ejpam-6327	787	2	fuzzy	fuzzy	ADJ
ejpam-6327	787	3	quotient	quotient	NOUN
ejpam-6327	787	4	semigroups	semigroup	NOUN
ejpam-6327	787	5	induced	induce	VERB
ejpam-6327	787	6	by	by	ADP
ejpam-6327	787	7	fuzzy	fuzzy	ADJ
ejpam-6327	787	8	ideals	ideal	NOUN
ejpam-6327	787	9	.	.	PUNCT
ejpam-6327	788	1	kyungpook	kyungpook	PROPN
ejpam-6327	788	2	math	math	PROPN
ejpam-6327	788	3	.	.	PUNCT
ejpam-6327	789	1	j.	j.	PROPN
ejpam-6327	789	2	,	,	PUNCT
ejpam-6327	789	3	35:105–112	35:105–112	NUM
ejpam-6327	789	4	,	,	PUNCT
ejpam-6327	789	5	1995	1995	NUM
ejpam-6327	789	6	.	.	PUNCT
ejpam-6327	790	1	[	[	X
ejpam-6327	790	2	12	12	NUM
ejpam-6327	790	3	]	]	PUNCT
ejpam-6327	790	4	j.	j.	PROPN
ejpam-6327	790	5	meng	meng	PROPN
ejpam-6327	790	6	.	.	PUNCT
ejpam-6327	791	1	on	on	ADP
ejpam-6327	791	2	ideals	ideal	NOUN
ejpam-6327	791	3	in	in	ADP
ejpam-6327	791	4	bck	bck	NOUN
ejpam-6327	791	5	-	-	PUNCT
ejpam-6327	791	6	algebras	algebras	PROPN
ejpam-6327	791	7	.	.	PUNCT
ejpam-6327	791	8	math	math	PROPN
ejpam-6327	791	9	.	.	PUNCT
ejpam-6327	792	1	japon	japon	PROPN
ejpam-6327	792	2	.	.	PUNCT
ejpam-6327	792	3	,	,	PUNCT
ejpam-6327	792	4	40:143–154	40:143–154	PROPN
ejpam-6327	792	5	,	,	PUNCT
ejpam-6327	792	6	1994	1994	NUM
ejpam-6327	792	7	.	.	PUNCT
ejpam-6327	793	1	[	[	X
ejpam-6327	793	2	13	13	NUM
ejpam-6327	793	3	]	]	PUNCT
ejpam-6327	793	4	j.	j.	PROPN
ejpam-6327	793	5	f.	f.	PROPN
ejpam-6327	793	6	fraleigh	fraleigh	PROPN
ejpam-6327	793	7	.	.	PUNCT
ejpam-6327	794	1	a	a	DET
ejpam-6327	794	2	first	first	ADJ
ejpam-6327	794	3	course	course	NOUN
ejpam-6327	794	4	in	in	ADP
ejpam-6327	794	5	abstract	abstract	ADJ
ejpam-6327	794	6	algebra	algebra	NOUN
ejpam-6327	794	7	,	,	PUNCT
ejpam-6327	794	8	fifth	fifth	ADJ
ejpam-6327	794	9	edition	edition	NOUN
ejpam-6327	794	10	.	.	PUNCT
ejpam-6327	795	1	addition	addition	NOUN
ejpam-6327	795	2	-	-	PUNCT
ejpam-6327	795	3	wesley	wesley	PROPN
ejpam-6327	795	4	publishing	publishing	NOUN
ejpam-6327	795	5	company	company	NOUN
ejpam-6327	795	6	,	,	PUNCT
ejpam-6327	795	7	1994	1994	NUM
ejpam-6327	795	8	.	.	PUNCT