id	sid	tid	token	lemma	pos
ejpam-3576	1	1	european	european	PROPN
ejpam-3576	1	2	journal	journal	PROPN
ejpam-3576	1	3	of	of	ADP
ejpam-3576	1	4	pure	pure	ADJ
ejpam-3576	1	5	and	and	CCONJ
ejpam-3576	1	6	applied	apply	VERB
ejpam-3576	1	7	mathematics	mathematic	NOUN
ejpam-3576	1	8	vol	vol	NOUN
ejpam-3576	1	9	.	.	PROPN
ejpam-3576	2	1	13	13	NUM
ejpam-3576	2	2	,	,	PUNCT
ejpam-3576	2	3	no	no	INTJ
ejpam-3576	2	4	.	.	NOUN
ejpam-3576	2	5	1	1	NUM
ejpam-3576	2	6	,	,	PUNCT
ejpam-3576	2	7	2020	2020	NUM
ejpam-3576	2	8	,	,	PUNCT
ejpam-3576	2	9	113	113	NUM
ejpam-3576	2	10	-	-	SYM
ejpam-3576	2	11	129	129	NUM
ejpam-3576	2	12	issn	issn	PROPN
ejpam-3576	2	13	1307	1307	NUM
ejpam-3576	2	14	-	-	SYM
ejpam-3576	2	15	5543	5543	NUM
ejpam-3576	2	16	–	–	PUNCT
ejpam-3576	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3576	2	18	published	publish	VERB
ejpam-3576	2	19	by	by	ADP
ejpam-3576	2	20	new	new	PROPN
ejpam-3576	2	21	york	york	PROPN
ejpam-3576	2	22	business	business	PROPN
ejpam-3576	2	23	global	global	ADJ
ejpam-3576	2	24	anti	anti	ADJ
ejpam-3576	2	25	fuzzy	fuzzy	ADJ
ejpam-3576	2	26	interior	interior	ADJ
ejpam-3576	2	27	ideals	ideal	NOUN
ejpam-3576	2	28	on	on	ADP
ejpam-3576	2	29	ordered	order	VERB
ejpam-3576	2	30	ag	ag	PROPN
ejpam-3576	2	31	-	-	PUNCT
ejpam-3576	2	32	groupoids	groupoid	NOUN
ejpam-3576	2	33	nasreen	nasreen	ADP
ejpam-3576	2	34	kausar1,∗	kausar1,∗	NOUN
ejpam-3576	2	35	,	,	PUNCT
ejpam-3576	2	36	meshari	meshari	PROPN
ejpam-3576	2	37	alesemi2	alesemi2	NOUN
ejpam-3576	2	38	,	,	PUNCT
ejpam-3576	2	39	salahuddin2	salahuddin2	NOUN
ejpam-3576	2	40	1	1	NUM
ejpam-3576	2	41	department	department	NOUN
ejpam-3576	2	42	of	of	ADP
ejpam-3576	2	43	mathematics	mathematic	NOUN
ejpam-3576	2	44	,	,	PUNCT
ejpam-3576	2	45	university	university	NOUN
ejpam-3576	2	46	of	of	ADP
ejpam-3576	2	47	agriculture	agriculture	PROPN
ejpam-3576	2	48	fsd	fsd	PROPN
ejpam-3576	2	49	pakistan	pakistan	PROPN
ejpam-3576	2	50	2	2	PROPN
ejpam-3576	2	51	department	department	NOUN
ejpam-3576	2	52	of	of	ADP
ejpam-3576	2	53	mathematics	mathematics	PROPN
ejpam-3576	2	54	,	,	PUNCT
ejpam-3576	2	55	jazan	jazan	PROPN
ejpam-3576	2	56	university	university	PROPN
ejpam-3576	2	57	,	,	PUNCT
ejpam-3576	2	58	jazan	jazan	NOUN
ejpam-3576	2	59	,	,	PUNCT
ejpam-3576	2	60	kingdom	kingdom	NOUN
ejpam-3576	2	61	of	of	ADP
ejpam-3576	2	62	saudi	saudi	PROPN
ejpam-3576	2	63	arabia	arabia	PROPN
ejpam-3576	2	64	abstract	abstract	NOUN
ejpam-3576	2	65	.	.	PUNCT
ejpam-3576	3	1	the	the	DET
ejpam-3576	3	2	purpose	purpose	NOUN
ejpam-3576	3	3	of	of	ADP
ejpam-3576	3	4	this	this	DET
ejpam-3576	3	5	paper	paper	NOUN
ejpam-3576	3	6	is	be	AUX
ejpam-3576	3	7	to	to	PART
ejpam-3576	3	8	investigate	investigate	VERB
ejpam-3576	3	9	,	,	PUNCT
ejpam-3576	3	10	the	the	DET
ejpam-3576	3	11	characterizations	characterization	NOUN
ejpam-3576	3	12	of	of	ADP
ejpam-3576	3	13	different	different	ADJ
ejpam-3576	3	14	classes	class	NOUN
ejpam-3576	3	15	of	of	ADP
ejpam-3576	3	16	non	non	ADJ
ejpam-3576	3	17	-	-	ADJ
ejpam-3576	3	18	associative	associative	ADJ
ejpam-3576	3	19	ordered	order	VERB
ejpam-3576	3	20	semigroups	semigroup	NOUN
ejpam-3576	3	21	by	by	ADP
ejpam-3576	3	22	using	use	VERB
ejpam-3576	3	23	anti	anti	ADJ
ejpam-3576	3	24	fuzzy	fuzzy	ADJ
ejpam-3576	3	25	left	left	NOUN
ejpam-3576	3	26	(	(	PUNCT
ejpam-3576	3	27	resp	resp	NOUN
ejpam-3576	3	28	.	.	PUNCT
ejpam-3576	4	1	right	right	ADJ
ejpam-3576	4	2	,	,	PUNCT
ejpam-3576	4	3	interior	interior	ADJ
ejpam-3576	4	4	)	)	PUNCT
ejpam-3576	4	5	ideals	ideal	NOUN
ejpam-3576	4	6	.	.	PUNCT
ejpam-3576	5	1	2020	2020	NUM
ejpam-3576	5	2	mathematics	mathematic	NOUN
ejpam-3576	5	3	subject	subject	NOUN
ejpam-3576	5	4	classifications	classification	NOUN
ejpam-3576	5	5	:	:	PUNCT
ejpam-3576	5	6	13cxx	13cxx	NUM
ejpam-3576	5	7	,	,	PUNCT
ejpam-3576	5	8	94d05	94d05	NUM
ejpam-3576	5	9	,	,	PUNCT
ejpam-3576	5	10	13axx	13axx	NUM
ejpam-3576	5	11	,	,	PUNCT
ejpam-3576	5	12	18b40	18b40	NUM
ejpam-3576	5	13	key	key	ADJ
ejpam-3576	5	14	words	word	NOUN
ejpam-3576	5	15	and	and	CCONJ
ejpam-3576	5	16	phrases	phrase	NOUN
ejpam-3576	5	17	:	:	PUNCT
ejpam-3576	5	18	fuzzy	fuzzy	ADJ
ejpam-3576	5	19	sets	set	NOUN
ejpam-3576	5	20	,	,	PUNCT
ejpam-3576	5	21	anti	anti	X
ejpam-3576	5	22	fuzzy	fuzzy	ADJ
ejpam-3576	5	23	ag	ag	PROPN
ejpam-3576	5	24	-	-	PUNCT
ejpam-3576	5	25	subgroupoids	subgroupoid	NOUN
ejpam-3576	5	26	,	,	PUNCT
ejpam-3576	5	27	anti	anti	X
ejpam-3576	5	28	fuzzy	fuzzy	ADJ
ejpam-3576	5	29	left	left	ADJ
ejpam-3576	5	30	(	(	PUNCT
ejpam-3576	5	31	resp	resp	NOUN
ejpam-3576	5	32	.	.	PUNCT
ejpam-3576	6	1	right	right	ADJ
ejpam-3576	6	2	,	,	PUNCT
ejpam-3576	6	3	interior	interior	ADJ
ejpam-3576	6	4	)	)	PUNCT
ejpam-3576	6	5	ideals	ideal	NOUN
ejpam-3576	6	6	,	,	PUNCT
ejpam-3576	6	7	left	leave	VERB
ejpam-3576	6	8	(	(	PUNCT
ejpam-3576	6	9	resp	resp	NOUN
ejpam-3576	6	10	.	.	PUNCT
ejpam-3576	7	1	right	right	ADJ
ejpam-3576	7	2	,	,	PUNCT
ejpam-3576	7	3	weakly	weakly	ADJ
ejpam-3576	7	4	,	,	PUNCT
ejpam-3576	7	5	intra-	intra-	ADJ
ejpam-3576	7	6	,	,	PUNCT
ejpam-3576	7	7	(	(	PUNCT
ejpam-3576	7	8	2	2	NUM
ejpam-3576	7	9	,	,	PUNCT
ejpam-3576	7	10	2)-	2)-	NUM
ejpam-3576	7	11	)	)	PUNCT
ejpam-3576	7	12	regular	regular	ADJ
ejpam-3576	7	13	ordered	order	VERB
ejpam-3576	7	14	ag	ag	PROPN
ejpam-3576	7	15	-	-	PUNCT
ejpam-3576	7	16	groupoids	groupoid	NOUN
ejpam-3576	7	17	.	.	PUNCT
ejpam-3576	8	1	1	1	X
ejpam-3576	8	2	.	.	X
ejpam-3576	8	3	introduction	introduction	NOUN
ejpam-3576	8	4	in	in	ADP
ejpam-3576	8	5	1972	1972	NUM
ejpam-3576	8	6	,	,	PUNCT
ejpam-3576	8	7	a	a	DET
ejpam-3576	8	8	generalization	generalization	NOUN
ejpam-3576	8	9	of	of	ADP
ejpam-3576	8	10	commutative	commutative	ADJ
ejpam-3576	8	11	semigroup	semigroup	PROPN
ejpam-3576	8	12	has	have	AUX
ejpam-3576	8	13	been	be	AUX
ejpam-3576	8	14	established	establish	VERB
ejpam-3576	8	15	by	by	ADP
ejpam-3576	8	16	naseeruddin	naseeruddin	VERB
ejpam-3576	8	17	et	et	PROPN
ejpam-3576	8	18	al	al	PROPN
ejpam-3576	8	19	.	.	PUNCT
ejpam-3576	9	1	[	[	X
ejpam-3576	9	2	14	14	NUM
ejpam-3576	9	3	]	]	PUNCT
ejpam-3576	9	4	.	.	PUNCT
ejpam-3576	10	1	in	in	ADP
ejpam-3576	10	2	ternary	ternary	ADJ
ejpam-3576	10	3	commutative	commutative	ADJ
ejpam-3576	10	4	law	law	NOUN
ejpam-3576	10	5	,	,	PUNCT
ejpam-3576	10	6	abc	abc	PROPN
ejpam-3576	10	7	=	=	SYM
ejpam-3576	10	8	cba	cba	PROPN
ejpam-3576	10	9	,	,	PUNCT
ejpam-3576	10	10	they	they	PRON
ejpam-3576	10	11	introduced	introduce	VERB
ejpam-3576	10	12	the	the	DET
ejpam-3576	10	13	braces	brace	NOUN
ejpam-3576	10	14	on	on	ADP
ejpam-3576	10	15	the	the	DET
ejpam-3576	10	16	left	left	ADJ
ejpam-3576	10	17	side	side	NOUN
ejpam-3576	10	18	of	of	ADP
ejpam-3576	10	19	this	this	DET
ejpam-3576	10	20	law	law	NOUN
ejpam-3576	10	21	and	and	CCONJ
ejpam-3576	10	22	explored	explore	VERB
ejpam-3576	10	23	a	a	DET
ejpam-3576	10	24	new	new	ADJ
ejpam-3576	10	25	pseudo	pseudo	NOUN
ejpam-3576	10	26	associative	associative	NOUN
ejpam-3576	10	27	law	law	NOUN
ejpam-3576	10	28	,	,	PUNCT
ejpam-3576	10	29	that	that	ADV
ejpam-3576	10	30	is	is	ADV
ejpam-3576	10	31	(	(	PUNCT
ejpam-3576	10	32	ab)c	ab)c	PROPN
ejpam-3576	10	33	=	=	SYM
ejpam-3576	10	34	(	(	PUNCT
ejpam-3576	10	35	cb)a	cb)a	PROPN
ejpam-3576	10	36	.	.	PUNCT
ejpam-3576	11	1	this	this	PRON
ejpam-3576	11	2	they	they	PRON
ejpam-3576	11	3	called	call	VERB
ejpam-3576	11	4	the	the	DET
ejpam-3576	11	5	left	left	ADJ
ejpam-3576	11	6	invertive	invertive	ADJ
ejpam-3576	11	7	law	law	NOUN
ejpam-3576	11	8	.	.	PUNCT
ejpam-3576	12	1	a	a	DET
ejpam-3576	12	2	groupoid	groupoid	PROPN
ejpam-3576	12	3	s	s	X
ejpam-3576	12	4	is	be	AUX
ejpam-3576	12	5	a	a	DET
ejpam-3576	12	6	left	left	NOUN
ejpam-3576	12	7	almost	almost	ADV
ejpam-3576	12	8	semigroup	semigroup	ADJ
ejpam-3576	12	9	(	(	PUNCT
ejpam-3576	12	10	abbreviated	abbreviate	VERB
ejpam-3576	12	11	as	as	ADP
ejpam-3576	12	12	la	la	NOUN
ejpam-3576	12	13	-	-	PUNCT
ejpam-3576	12	14	semigroup	semigroup	NOUN
ejpam-3576	12	15	)	)	PUNCT
ejpam-3576	12	16	,	,	PUNCT
ejpam-3576	12	17	if	if	SCONJ
ejpam-3576	12	18	it	it	PRON
ejpam-3576	12	19	satisfies	satisfy	VERB
ejpam-3576	12	20	the	the	DET
ejpam-3576	12	21	left	left	ADJ
ejpam-3576	12	22	invertive	invertive	ADJ
ejpam-3576	12	23	law	law	NOUN
ejpam-3576	12	24	:	:	PUNCT
ejpam-3576	12	25	(	(	PUNCT
ejpam-3576	12	26	ab)c	ab)c	PROPN
ejpam-3576	12	27	=	=	SYM
ejpam-3576	12	28	(	(	PUNCT
ejpam-3576	12	29	cb)a	cb)a	PROPN
ejpam-3576	12	30	.	.	PUNCT
ejpam-3576	13	1	this	this	DET
ejpam-3576	13	2	structure	structure	NOUN
ejpam-3576	13	3	is	be	AUX
ejpam-3576	13	4	also	also	ADV
ejpam-3576	13	5	known	know	VERB
ejpam-3576	13	6	as	as	ADP
ejpam-3576	13	7	abel	abel	NOUN
ejpam-3576	13	8	-	-	PUNCT
ejpam-3576	13	9	grassmann	grassmann	PROPN
ejpam-3576	13	10	’s	’s	PART
ejpam-3576	13	11	groupoid	groupoid	NOUN
ejpam-3576	13	12	(	(	PUNCT
ejpam-3576	13	13	abbreviated	abbreviate	VERB
ejpam-3576	13	14	as	as	ADP
ejpam-3576	13	15	ag	ag	PROPN
ejpam-3576	13	16	-	-	PUNCT
ejpam-3576	13	17	groupoid	groupoid	PROPN
ejpam-3576	13	18	)	)	PUNCT
ejpam-3576	13	19	by	by	ADP
ejpam-3576	13	20	protic	protic	PROPN
ejpam-3576	13	21	et	et	PROPN
ejpam-3576	13	22	al	al	PROPN
ejpam-3576	13	23	.	.	PUNCT
ejpam-3576	14	1	[	[	X
ejpam-3576	14	2	27	27	NUM
ejpam-3576	14	3	]	]	PUNCT
ejpam-3576	14	4	.	.	PUNCT
ejpam-3576	15	1	in	in	ADP
ejpam-3576	15	2	fact	fact	NOUN
ejpam-3576	15	3	an	an	DET
ejpam-3576	15	4	ag	ag	PROPN
ejpam-3576	15	5	-	-	PUNCT
ejpam-3576	15	6	groupoid	groupoid	PROPN
ejpam-3576	15	7	is	be	AUX
ejpam-3576	15	8	non	non	ADJ
ejpam-3576	15	9	-	-	ADJ
ejpam-3576	15	10	commutative	commutative	ADJ
ejpam-3576	15	11	and	and	CCONJ
ejpam-3576	15	12	non	non	ADJ
ejpam-3576	15	13	-	-	ADJ
ejpam-3576	15	14	associative	associative	ADJ
ejpam-3576	15	15	semigroup	semigroup	NOUN
ejpam-3576	15	16	.	.	PUNCT
ejpam-3576	16	1	ideals	ideal	NOUN
ejpam-3576	16	2	in	in	ADP
ejpam-3576	16	3	ag	ag	PROPN
ejpam-3576	16	4	-	-	PUNCT
ejpam-3576	16	5	groupoids	groupoid	NOUN
ejpam-3576	16	6	have	have	AUX
ejpam-3576	16	7	been	be	AUX
ejpam-3576	16	8	investigated	investigate	VERB
ejpam-3576	16	9	in	in	ADP
ejpam-3576	16	10	[	[	X
ejpam-3576	16	11	26	26	NUM
ejpam-3576	16	12	]	]	PUNCT
ejpam-3576	16	13	.	.	PUNCT
ejpam-3576	17	1	in	in	ADP
ejpam-3576	17	2	[	[	X
ejpam-3576	17	3	6	6	NUM
ejpam-3576	17	4	]	]	PUNCT
ejpam-3576	17	5	(	(	PUNCT
ejpam-3576	17	6	resp	resp	NOUN
ejpam-3576	17	7	.	.	PUNCT
ejpam-3576	18	1	[	[	X
ejpam-3576	18	2	3	3	NUM
ejpam-3576	18	3	]	]	NUM
ejpam-3576	18	4	)	)	PUNCT
ejpam-3576	18	5	,	,	PUNCT
ejpam-3576	18	6	a	a	DET
ejpam-3576	18	7	groupoid	groupoid	NOUN
ejpam-3576	18	8	s	s	NOUN
ejpam-3576	18	9	is	be	AUX
ejpam-3576	18	10	said	say	VERB
ejpam-3576	18	11	to	to	PART
ejpam-3576	18	12	be	be	AUX
ejpam-3576	18	13	medial	medial	ADJ
ejpam-3576	18	14	(	(	PUNCT
ejpam-3576	18	15	resp	resp	NOUN
ejpam-3576	18	16	.	.	PUNCT
ejpam-3576	19	1	paramedial	paramedial	PROPN
ejpam-3576	19	2	)	)	PUNCT
ejpam-3576	20	1	if	if	SCONJ
ejpam-3576	20	2	(	(	PUNCT
ejpam-3576	20	3	ab)(cd	ab)(cd	NOUN
ejpam-3576	20	4	)	)	PUNCT
ejpam-3576	20	5	=	=	SYM
ejpam-3576	20	6	(	(	PUNCT
ejpam-3576	20	7	ac)(bd	ac)(bd	PROPN
ejpam-3576	20	8	)	)	PUNCT
ejpam-3576	20	9	(	(	PUNCT
ejpam-3576	20	10	resp	resp	NOUN
ejpam-3576	20	11	.	.	PUNCT
ejpam-3576	21	1	(	(	PUNCT
ejpam-3576	21	2	ab)(cd	ab)(cd	PROPN
ejpam-3576	21	3	)	)	PUNCT
ejpam-3576	21	4	=	=	SYM
ejpam-3576	21	5	(	(	PUNCT
ejpam-3576	21	6	db)(ca	db)(ca	PROPN
ejpam-3576	21	7	)	)	PUNCT
ejpam-3576	21	8	)	)	PUNCT
ejpam-3576	21	9	.	.	PUNCT
ejpam-3576	22	1	in	in	ADP
ejpam-3576	22	2	[	[	X
ejpam-3576	22	3	14	14	NUM
ejpam-3576	22	4	]	]	X
ejpam-3576	22	5	,	,	PUNCT
ejpam-3576	22	6	an	an	DET
ejpam-3576	22	7	ag	ag	PROPN
ejpam-3576	22	8	-	-	PUNCT
ejpam-3576	22	9	groupoid	groupoid	PROPN
ejpam-3576	22	10	is	be	AUX
ejpam-3576	22	11	medial	medial	ADJ
ejpam-3576	22	12	,	,	PUNCT
ejpam-3576	22	13	but	but	CCONJ
ejpam-3576	22	14	in	in	ADP
ejpam-3576	22	15	general	general	ADJ
ejpam-3576	22	16	an	an	DET
ejpam-3576	22	17	ag	ag	PROPN
ejpam-3576	22	18	-	-	PUNCT
ejpam-3576	22	19	groupoid	groupoid	NOUN
ejpam-3576	22	20	needs	need	VERB
ejpam-3576	22	21	not	not	PART
ejpam-3576	22	22	to	to	PART
ejpam-3576	22	23	be	be	AUX
ejpam-3576	22	24	paramedial	paramedial	ADJ
ejpam-3576	22	25	.	.	PUNCT
ejpam-3576	23	1	however	however	ADV
ejpam-3576	23	2	by	by	ADP
ejpam-3576	23	3	protic	protic	PROPN
ejpam-3576	23	4	et	et	PROPN
ejpam-3576	23	5	al	al	PROPN
ejpam-3576	23	6	.	.	PUNCT
ejpam-3576	24	1	[	[	X
ejpam-3576	24	2	27	27	NUM
ejpam-3576	24	3	]	]	PUNCT
ejpam-3576	24	4	,	,	PUNCT
ejpam-3576	24	5	every	every	DET
ejpam-3576	24	6	aggroupoid	aggroupoid	ADJ
ejpam-3576	24	7	with	with	ADP
ejpam-3576	24	8	left	left	ADJ
ejpam-3576	24	9	identity	identity	NOUN
ejpam-3576	24	10	is	be	AUX
ejpam-3576	24	11	paramedial	paramedial	ADJ
ejpam-3576	24	12	and	and	CCONJ
ejpam-3576	24	13	also	also	ADV
ejpam-3576	24	14	satisfies	satisfie	NOUN
ejpam-3576	24	15	a(bc	a(bc	NOUN
ejpam-3576	24	16	)	)	PUNCT
ejpam-3576	24	17	=	=	SYM
ejpam-3576	24	18	b(ac	b(ac	PROPN
ejpam-3576	24	19	)	)	PUNCT
ejpam-3576	24	20	,	,	PUNCT
ejpam-3576	24	21	(	(	PUNCT
ejpam-3576	24	22	ab)(cd	ab)(cd	NOUN
ejpam-3576	24	23	)	)	PUNCT
ejpam-3576	24	24	=	=	SYM
ejpam-3576	24	25	(	(	PUNCT
ejpam-3576	24	26	dc)(ba	dc)(ba	PROPN
ejpam-3576	24	27	)	)	PUNCT
ejpam-3576	24	28	.	.	PUNCT
ejpam-3576	25	1	in	in	ADP
ejpam-3576	25	2	[	[	X
ejpam-3576	25	3	15	15	NUM
ejpam-3576	25	4	]	]	PUNCT
ejpam-3576	25	5	,	,	PUNCT
ejpam-3576	25	6	if	if	SCONJ
ejpam-3576	25	7	(	(	PUNCT
ejpam-3576	25	8	s	s	X
ejpam-3576	25	9	,	,	PUNCT
ejpam-3576	25	10	·	·	PUNCT
ejpam-3576	25	11	,	,	PUNCT
ejpam-3576	25	12	≤	≤	NUM
ejpam-3576	25	13	)	)	PUNCT
ejpam-3576	25	14	is	be	AUX
ejpam-3576	25	15	an	an	DET
ejpam-3576	25	16	ordered	order	VERB
ejpam-3576	25	17	semigroup	semigroup	NOUN
ejpam-3576	25	18	and	and	CCONJ
ejpam-3576	25	19	a	a	DET
ejpam-3576	25	20	⊆	⊆	NUM
ejpam-3576	25	21	s	s	NOUN
ejpam-3576	25	22	,	,	PUNCT
ejpam-3576	25	23	we	we	PRON
ejpam-3576	25	24	define	define	VERB
ejpam-3576	25	25	(	(	PUNCT
ejpam-3576	25	26	a	a	X
ejpam-3576	25	27	]	]	X
ejpam-3576	25	28	=	=	PUNCT
ejpam-3576	25	29	{	{	PUNCT
ejpam-3576	25	30	s	s	X
ejpam-3576	25	31	∈	∈	NOUN
ejpam-3576	25	32	s	s	PART
ejpam-3576	25	33	:	:	PUNCT
ejpam-3576	25	34	s	s	VERB
ejpam-3576	25	35	≤	≤	NOUN
ejpam-3576	25	36	a	a	PRON
ejpam-3576	25	37	for	for	ADP
ejpam-3576	25	38	some	some	DET
ejpam-3576	25	39	a	a	DET
ejpam-3576	25	40	∈	∈	PROPN
ejpam-3576	25	41	a	a	PRON
ejpam-3576	25	42	}	}	PUNCT
ejpam-3576	25	43	.	.	PUNCT
ejpam-3576	26	1	a	a	DET
ejpam-3576	26	2	non	non	ADJ
ejpam-3576	26	3	-	-	ADJ
ejpam-3576	26	4	empty	empty	ADJ
ejpam-3576	26	5	subset	subset	NOUN
ejpam-3576	26	6	a	a	PRON
ejpam-3576	26	7	of	of	ADP
ejpam-3576	26	8	s	s	NOUN
ejpam-3576	26	9	is	be	AUX
ejpam-3576	26	10	an	an	DET
ejpam-3576	26	11	ordered	ordered	ADJ
ejpam-3576	26	12	subsemigroup	subsemigroup	NOUN
ejpam-3576	26	13	of	of	ADP
ejpam-3576	26	14	s	s	PRON
ejpam-3576	26	15	if	if	SCONJ
ejpam-3576	26	16	a2	a2	PROPN
ejpam-3576	26	17	⊆	⊆	NUM
ejpam-3576	26	18	a.	a.	NOUN
ejpam-3576	26	19	the	the	DET
ejpam-3576	26	20	notions	notion	NOUN
ejpam-3576	26	21	of	of	ADP
ejpam-3576	26	22	ideals	ideal	NOUN
ejpam-3576	26	23	play	play	VERB
ejpam-3576	26	24	a	a	DET
ejpam-3576	26	25	crucial	crucial	ADJ
ejpam-3576	26	26	role	role	NOUN
ejpam-3576	26	27	in	in	ADP
ejpam-3576	26	28	the	the	DET
ejpam-3576	26	29	study	study	NOUN
ejpam-3576	26	30	of	of	ADP
ejpam-3576	26	31	(	(	PUNCT
ejpam-3576	26	32	ring	ring	NOUN
ejpam-3576	26	33	,	,	PUNCT
ejpam-3576	26	34	semiring	semiring	NOUN
ejpam-3576	26	35	,	,	PUNCT
ejpam-3576	26	36	near	near	ADP
ejpam-3576	26	37	-	-	PUNCT
ejpam-3576	26	38	ring	ring	NOUN
ejpam-3576	26	39	,	,	PUNCT
ejpam-3576	26	40	semigroup	semigroup	PROPN
ejpam-3576	26	41	,	,	PUNCT
ejpam-3576	26	42	ordered	order	VERB
ejpam-3576	26	43	semigroup	semigroup	PROPN
ejpam-3576	26	44	)	)	PUNCT
ejpam-3576	26	45	theory	theory	NOUN
ejpam-3576	26	46	etc	etc	X
ejpam-3576	26	47	.	.	X
ejpam-3576	26	48	∗corresponding	∗corresponde	VERB
ejpam-3576	26	49	author	author	NOUN
ejpam-3576	26	50	.	.	PUNCT
ejpam-3576	27	1	doi	doi	NOUN
ejpam-3576	27	2	:	:	PUNCT
ejpam-3576	27	3	https://doi.org/10.29020/nybg.ejpam.v13i1.3576	https://doi.org/10.29020/nybg.ejpam.v13i1.3576	PROPN
ejpam-3576	27	4	email	email	NOUN
ejpam-3576	27	5	addresses	address	NOUN
ejpam-3576	27	6	:	:	PUNCT
ejpam-3576	27	7	kausar.nasreen57@gmail.com	kausar.nasreen57@gmail.com	X
ejpam-3576	27	8	(	(	PUNCT
ejpam-3576	27	9	k.	k.	NOUN
ejpam-3576	27	10	nasreen	nasreen	PROPN
ejpam-3576	27	11	)	)	PUNCT
ejpam-3576	27	12	malesemi@jazanu.edu.sa	malesemi@jazanu.edu.sa	NOUN
ejpam-3576	27	13	(	(	PUNCT
ejpam-3576	27	14	m.	m.	PROPN
ejpam-3576	27	15	alesemi	alesemi	PROPN
ejpam-3576	27	16	)	)	PUNCT
ejpam-3576	27	17	,	,	PUNCT
ejpam-3576	27	18	drsalah12@hotmail.com	drsalah12@hotmail.com	X
ejpam-3576	27	19	(	(	PUNCT
ejpam-3576	27	20	salahuddin	salahuddin	NOUN
ejpam-3576	27	21	)	)	PUNCT
ejpam-3576	27	22	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3576	28	1	113	113	NUM
ejpam-3576	28	2	c	c	NOUN
ejpam-3576	28	3	©	©	NOUN
ejpam-3576	28	4	2020	2020	NUM
ejpam-3576	28	5	ejpam	ejpam	VERB
ejpam-3576	28	6	all	all	DET
ejpam-3576	28	7	rights	right	NOUN
ejpam-3576	28	8	reserved	reserve	VERB
ejpam-3576	28	9	.	.	PUNCT
ejpam-3576	29	1	k.	k.	PROPN
ejpam-3576	29	2	nasreen	nasreen	PROPN
ejpam-3576	29	3	,	,	PUNCT
ejpam-3576	29	4	m.	m.	NOUN
ejpam-3576	29	5	alesemi	alesemi	PROPN
ejpam-3576	29	6	,	,	PUNCT
ejpam-3576	29	7	salahuddin	salahuddin	VERB
ejpam-3576	29	8	/	/	SYM
ejpam-3576	29	9	eur	eur	PROPN
ejpam-3576	29	10	.	.	PUNCT
ejpam-3576	30	1	j.	j.	PROPN
ejpam-3576	30	2	pure	pure	PROPN
ejpam-3576	30	3	appl	appl	PROPN
ejpam-3576	30	4	.	.	PROPN
ejpam-3576	30	5	math	math	PROPN
ejpam-3576	30	6	,	,	PUNCT
ejpam-3576	30	7	13	13	NUM
ejpam-3576	30	8	(	(	PUNCT
ejpam-3576	30	9	1	1	NUM
ejpam-3576	30	10	)	)	PUNCT
ejpam-3576	30	11	(	(	PUNCT
ejpam-3576	30	12	2020	2020	NUM
ejpam-3576	30	13	)	)	PUNCT
ejpam-3576	30	14	,	,	PUNCT
ejpam-3576	30	15	113	113	NUM
ejpam-3576	30	16	-	-	SYM
ejpam-3576	30	17	129	129	NUM
ejpam-3576	30	18	114	114	NUM
ejpam-3576	30	19	a	a	DET
ejpam-3576	30	20	non	non	ADJ
ejpam-3576	30	21	-	-	ADJ
ejpam-3576	30	22	empty	empty	ADJ
ejpam-3576	30	23	subset	subset	NOUN
ejpam-3576	30	24	a	a	PRON
ejpam-3576	30	25	of	of	ADP
ejpam-3576	30	26	s	s	NOUN
ejpam-3576	30	27	is	be	AUX
ejpam-3576	30	28	a	a	DET
ejpam-3576	30	29	left	left	ADJ
ejpam-3576	30	30	(	(	PUNCT
ejpam-3576	30	31	resp	resp	NOUN
ejpam-3576	30	32	.	.	PUNCT
ejpam-3576	31	1	right	right	ADJ
ejpam-3576	31	2	)	)	PUNCT
ejpam-3576	31	3	ideal	ideal	NOUN
ejpam-3576	31	4	of	of	ADP
ejpam-3576	31	5	s	s	SYM
ejpam-3576	31	6	,	,	PUNCT
ejpam-3576	31	7	if	if	SCONJ
ejpam-3576	31	8	following	follow	VERB
ejpam-3576	31	9	hold	hold	NOUN
ejpam-3576	31	10	(	(	PUNCT
ejpam-3576	31	11	1	1	NUM
ejpam-3576	31	12	)	)	PUNCT
ejpam-3576	31	13	sa	sa	ADP
ejpam-3576	31	14	⊆	⊆	NUM
ejpam-3576	31	15	a	a	DET
ejpam-3576	31	16	(	(	PUNCT
ejpam-3576	31	17	resp	resp	NOUN
ejpam-3576	31	18	.	.	PUNCT
ejpam-3576	32	1	as	as	ADP
ejpam-3576	32	2	⊆	⊆	NUM
ejpam-3576	32	3	a	a	PRON
ejpam-3576	32	4	)	)	PUNCT
ejpam-3576	32	5	.	.	PUNCT
ejpam-3576	33	1	(	(	PUNCT
ejpam-3576	33	2	2	2	X
ejpam-3576	33	3	)	)	PUNCT
ejpam-3576	33	4	if	if	SCONJ
ejpam-3576	33	5	a	a	DET
ejpam-3576	33	6	∈	∈	PROPN
ejpam-3576	33	7	a	a	PRON
ejpam-3576	33	8	and	and	CCONJ
ejpam-3576	33	9	b	b	NOUN
ejpam-3576	33	10	∈	∈	NOUN
ejpam-3576	33	11	s	s	VERB
ejpam-3576	33	12	such	such	ADJ
ejpam-3576	33	13	that	that	DET
ejpam-3576	33	14	b	b	NOUN
ejpam-3576	33	15	≤	≤	NOUN
ejpam-3576	33	16	a	a	DET
ejpam-3576	33	17	implies	implie	NOUN
ejpam-3576	33	18	b	b	X
ejpam-3576	33	19	∈	∈	PROPN
ejpam-3576	33	20	a.	a.	NOUN
ejpam-3576	33	21	equivalent	equivalent	PROPN
ejpam-3576	33	22	definition	definition	NOUN
ejpam-3576	33	23	:	:	PUNCT
ejpam-3576	33	24	a	a	PRON
ejpam-3576	33	25	is	be	AUX
ejpam-3576	33	26	a	a	DET
ejpam-3576	33	27	left	left	ADJ
ejpam-3576	33	28	(	(	PUNCT
ejpam-3576	33	29	resp	resp	NOUN
ejpam-3576	33	30	.	.	PUNCT
ejpam-3576	34	1	right	right	ADJ
ejpam-3576	34	2	)	)	PUNCT
ejpam-3576	34	3	ideal	ideal	NOUN
ejpam-3576	34	4	of	of	ADP
ejpam-3576	34	5	s	s	PRON
ejpam-3576	34	6	if	if	SCONJ
ejpam-3576	34	7	(	(	PUNCT
ejpam-3576	34	8	a	a	X
ejpam-3576	34	9	]	]	X
ejpam-3576	34	10	⊆	⊆	NUM
ejpam-3576	34	11	a	a	PRON
ejpam-3576	34	12	and	and	CCONJ
ejpam-3576	34	13	sa	sa	ADP
ejpam-3576	34	14	⊆	⊆	NUM
ejpam-3576	34	15	a	a	DET
ejpam-3576	34	16	(	(	PUNCT
ejpam-3576	34	17	resp	resp	NOUN
ejpam-3576	34	18	.	.	PUNCT
ejpam-3576	35	1	as	as	ADP
ejpam-3576	35	2	⊆	⊆	NUM
ejpam-3576	35	3	a	a	PRON
ejpam-3576	35	4	)	)	PUNCT
ejpam-3576	35	5	.	.	PUNCT
ejpam-3576	36	1	a	a	DET
ejpam-3576	36	2	non	non	ADJ
ejpam-3576	36	3	-	-	ADJ
ejpam-3576	36	4	empty	empty	ADJ
ejpam-3576	36	5	subset	subset	NOUN
ejpam-3576	36	6	a	a	PRON
ejpam-3576	36	7	of	of	ADP
ejpam-3576	36	8	s	s	NOUN
ejpam-3576	36	9	is	be	AUX
ejpam-3576	36	10	an	an	DET
ejpam-3576	36	11	interior	interior	ADJ
ejpam-3576	36	12	ideal	ideal	NOUN
ejpam-3576	36	13	of	of	ADP
ejpam-3576	36	14	s	s	PRON
ejpam-3576	36	15	if	if	SCONJ
ejpam-3576	36	16	(	(	PUNCT
ejpam-3576	36	17	1	1	X
ejpam-3576	36	18	)	)	PUNCT
ejpam-3576	36	19	sas	sas	VERB
ejpam-3576	36	20	⊆	⊆	NUM
ejpam-3576	36	21	a.	a.	NOUN
ejpam-3576	36	22	(	(	PUNCT
ejpam-3576	36	23	2	2	NUM
ejpam-3576	36	24	)	)	PUNCT
ejpam-3576	36	25	if	if	SCONJ
ejpam-3576	36	26	a	a	DET
ejpam-3576	36	27	∈	∈	PROPN
ejpam-3576	36	28	a	a	PRON
ejpam-3576	36	29	and	and	CCONJ
ejpam-3576	36	30	b	b	NOUN
ejpam-3576	36	31	∈	∈	NOUN
ejpam-3576	36	32	s	s	VERB
ejpam-3576	36	33	such	such	ADJ
ejpam-3576	36	34	that	that	DET
ejpam-3576	36	35	b	b	NOUN
ejpam-3576	36	36	≤	≤	NOUN
ejpam-3576	36	37	a	a	DET
ejpam-3576	36	38	implies	implie	NOUN
ejpam-3576	36	39	b	b	X
ejpam-3576	36	40	∈	∈	ADJ
ejpam-3576	36	41	a.	a.	NOUN
ejpam-3576	36	42	in	in	ADP
ejpam-3576	36	43	[	[	X
ejpam-3576	36	44	17	17	NUM
ejpam-3576	36	45	,	,	PUNCT
ejpam-3576	36	46	18	18	NUM
ejpam-3576	36	47	]	]	PUNCT
ejpam-3576	36	48	,	,	PUNCT
ejpam-3576	36	49	an	an	DET
ejpam-3576	36	50	ordered	order	VERB
ejpam-3576	36	51	semigroup	semigroup	NOUN
ejpam-3576	36	52	s	s	VERB
ejpam-3576	36	53	is	be	AUX
ejpam-3576	36	54	said	say	VERB
ejpam-3576	36	55	to	to	PART
ejpam-3576	36	56	be	be	AUX
ejpam-3576	36	57	regular	regular	ADJ
ejpam-3576	36	58	,	,	PUNCT
ejpam-3576	36	59	if	if	SCONJ
ejpam-3576	36	60	for	for	ADP
ejpam-3576	36	61	every	every	DET
ejpam-3576	36	62	a	a	DET
ejpam-3576	36	63	∈	∈	ADJ
ejpam-3576	36	64	s	s	NOUN
ejpam-3576	36	65	,	,	PUNCT
ejpam-3576	36	66	there	there	PRON
ejpam-3576	36	67	exists	exist	VERB
ejpam-3576	36	68	x	x	X
ejpam-3576	36	69	∈	∈	PROPN
ejpam-3576	36	70	s	s	VERB
ejpam-3576	36	71	such	such	ADJ
ejpam-3576	36	72	that	that	SCONJ
ejpam-3576	36	73	a	a	DET
ejpam-3576	36	74	≤	≤	NUM
ejpam-3576	36	75	axa	axa	NOUN
ejpam-3576	36	76	.	.	PUNCT
ejpam-3576	37	1	equivalent	equivalent	ADJ
ejpam-3576	37	2	definitions	definition	NOUN
ejpam-3576	37	3	are	be	AUX
ejpam-3576	37	4	as	as	SCONJ
ejpam-3576	37	5	follows	follow	VERB
ejpam-3576	37	6	:	:	PUNCT
ejpam-3576	37	7	(	(	PUNCT
ejpam-3576	37	8	1	1	X
ejpam-3576	37	9	)	)	PUNCT
ejpam-3576	37	10	a	a	DET
ejpam-3576	37	11	⊆	⊆	NUM
ejpam-3576	37	12	(	(	PUNCT
ejpam-3576	37	13	asa	asa	PROPN
ejpam-3576	37	14	]	]	PUNCT
ejpam-3576	37	15	for	for	ADP
ejpam-3576	37	16	every	every	DET
ejpam-3576	37	17	a	a	DET
ejpam-3576	37	18	⊆	⊆	NUM
ejpam-3576	37	19	s.	s.	PROPN
ejpam-3576	37	20	(	(	PUNCT
ejpam-3576	37	21	2	2	NUM
ejpam-3576	37	22	)	)	PUNCT
ejpam-3576	37	23	a	a	DET
ejpam-3576	37	24	∈	∈	PROPN
ejpam-3576	37	25	(	(	PUNCT
ejpam-3576	37	26	asa	asa	PROPN
ejpam-3576	37	27	]	]	PUNCT
ejpam-3576	37	28	for	for	ADP
ejpam-3576	37	29	every	every	DET
ejpam-3576	37	30	a	a	DET
ejpam-3576	37	31	∈	∈	PROPN
ejpam-3576	37	32	s.	s.	PROPN
ejpam-3576	37	33	an	an	DET
ejpam-3576	37	34	ordered	order	VERB
ejpam-3576	37	35	semigroup	semigroup	NOUN
ejpam-3576	37	36	s	s	VERB
ejpam-3576	37	37	is	be	AUX
ejpam-3576	37	38	said	say	VERB
ejpam-3576	37	39	to	to	PART
ejpam-3576	37	40	be	be	AUX
ejpam-3576	37	41	(	(	PUNCT
ejpam-3576	37	42	2	2	NUM
ejpam-3576	37	43	,	,	PUNCT
ejpam-3576	37	44	2)-regular	2)-regular	NUM
ejpam-3576	37	45	,	,	PUNCT
ejpam-3576	37	46	if	if	SCONJ
ejpam-3576	37	47	for	for	ADP
ejpam-3576	37	48	every	every	DET
ejpam-3576	37	49	a	a	DET
ejpam-3576	37	50	∈	∈	ADJ
ejpam-3576	37	51	s	s	NOUN
ejpam-3576	37	52	,	,	PUNCT
ejpam-3576	37	53	there	there	PRON
ejpam-3576	37	54	exists	exist	VERB
ejpam-3576	37	55	x	x	X
ejpam-3576	37	56	∈	∈	PROPN
ejpam-3576	37	57	s	s	VERB
ejpam-3576	37	58	such	such	ADJ
ejpam-3576	37	59	that	that	SCONJ
ejpam-3576	37	60	a	a	DET
ejpam-3576	37	61	≤	≤	NUM
ejpam-3576	37	62	a2xa2	a2xa2	NOUN
ejpam-3576	37	63	.	.	PUNCT
ejpam-3576	38	1	equivalent	equivalent	ADJ
ejpam-3576	38	2	definitions	definition	NOUN
ejpam-3576	38	3	are	be	AUX
ejpam-3576	38	4	as	as	SCONJ
ejpam-3576	38	5	follows	follow	VERB
ejpam-3576	38	6	:	:	PUNCT
ejpam-3576	38	7	(	(	PUNCT
ejpam-3576	38	8	1	1	X
ejpam-3576	38	9	)	)	PUNCT
ejpam-3576	38	10	a	a	DET
ejpam-3576	38	11	⊆	⊆	NUM
ejpam-3576	38	12	(	(	PUNCT
ejpam-3576	38	13	a2sa2	a2sa2	NOUN
ejpam-3576	38	14	]	]	PUNCT
ejpam-3576	38	15	for	for	ADP
ejpam-3576	38	16	every	every	DET
ejpam-3576	38	17	a	a	DET
ejpam-3576	38	18	⊆	⊆	NUM
ejpam-3576	38	19	s.	s.	PROPN
ejpam-3576	38	20	(	(	PUNCT
ejpam-3576	38	21	2	2	NUM
ejpam-3576	38	22	)	)	PUNCT
ejpam-3576	38	23	a	a	DET
ejpam-3576	38	24	∈	∈	NOUN
ejpam-3576	38	25	(	(	PUNCT
ejpam-3576	38	26	a2sa2	a2sa2	NOUN
ejpam-3576	38	27	]	]	PUNCT
ejpam-3576	38	28	for	for	ADP
ejpam-3576	38	29	every	every	DET
ejpam-3576	38	30	a	a	DET
ejpam-3576	38	31	∈	∈	PROPN
ejpam-3576	38	32	s.	s.	PROPN
ejpam-3576	38	33	an	an	DET
ejpam-3576	38	34	ordered	order	VERB
ejpam-3576	38	35	semigroup	semigroup	NOUN
ejpam-3576	38	36	s	s	VERB
ejpam-3576	38	37	is	be	AUX
ejpam-3576	38	38	said	say	VERB
ejpam-3576	38	39	to	to	PART
ejpam-3576	38	40	be	be	AUX
ejpam-3576	38	41	weakly	weakly	ADV
ejpam-3576	38	42	regular	regular	ADJ
ejpam-3576	38	43	,	,	PUNCT
ejpam-3576	38	44	if	if	SCONJ
ejpam-3576	38	45	for	for	ADP
ejpam-3576	38	46	every	every	DET
ejpam-3576	38	47	a	a	DET
ejpam-3576	38	48	∈	∈	ADJ
ejpam-3576	38	49	s	s	NOUN
ejpam-3576	38	50	,	,	PUNCT
ejpam-3576	38	51	there	there	PRON
ejpam-3576	38	52	exist	exist	VERB
ejpam-3576	38	53	x	x	NOUN
ejpam-3576	38	54	,	,	PUNCT
ejpam-3576	38	55	y	y	PROPN
ejpam-3576	38	56	∈	∈	PROPN
ejpam-3576	38	57	s	s	VERB
ejpam-3576	38	58	such	such	ADJ
ejpam-3576	38	59	that	that	SCONJ
ejpam-3576	38	60	a	a	DET
ejpam-3576	38	61	≤	≤	PROPN
ejpam-3576	38	62	axay	axay	NOUN
ejpam-3576	38	63	.	.	PUNCT
ejpam-3576	39	1	equivalent	equivalent	ADJ
ejpam-3576	39	2	definitions	definition	NOUN
ejpam-3576	39	3	are	be	AUX
ejpam-3576	39	4	as	as	SCONJ
ejpam-3576	39	5	follows	follow	VERB
ejpam-3576	39	6	:	:	PUNCT
ejpam-3576	39	7	(	(	PUNCT
ejpam-3576	39	8	1	1	X
ejpam-3576	39	9	)	)	PUNCT
ejpam-3576	39	10	a	a	DET
ejpam-3576	39	11	⊆	⊆	NUM
ejpam-3576	39	12	(	(	PUNCT
ejpam-3576	39	13	(	(	PUNCT
ejpam-3576	39	14	as)2	as)2	NOUN
ejpam-3576	39	15	]	]	PUNCT
ejpam-3576	39	16	for	for	ADP
ejpam-3576	39	17	every	every	DET
ejpam-3576	39	18	a	a	DET
ejpam-3576	39	19	⊆	⊆	NUM
ejpam-3576	39	20	s.	s.	PROPN
ejpam-3576	39	21	(	(	PUNCT
ejpam-3576	39	22	2	2	NUM
ejpam-3576	39	23	)	)	PUNCT
ejpam-3576	39	24	a	a	DET
ejpam-3576	39	25	∈	∈	NOUN
ejpam-3576	39	26	(	(	PUNCT
ejpam-3576	39	27	(	(	PUNCT
ejpam-3576	39	28	as)2	as)2	NOUN
ejpam-3576	39	29	]	]	PUNCT
ejpam-3576	39	30	for	for	ADP
ejpam-3576	39	31	every	every	DET
ejpam-3576	39	32	a	a	DET
ejpam-3576	39	33	∈	∈	NOUN
ejpam-3576	39	34	s.	s.	PROPN
ejpam-3576	39	35	in	in	ADP
ejpam-3576	39	36	[	[	X
ejpam-3576	39	37	16	16	NUM
ejpam-3576	39	38	,	,	PUNCT
ejpam-3576	39	39	18	18	NUM
ejpam-3576	39	40	]	]	PUNCT
ejpam-3576	39	41	,	,	PUNCT
ejpam-3576	39	42	an	an	DET
ejpam-3576	39	43	ordered	order	VERB
ejpam-3576	39	44	semigroup	semigroup	NOUN
ejpam-3576	39	45	s	s	VERB
ejpam-3576	39	46	is	be	AUX
ejpam-3576	39	47	an	an	DET
ejpam-3576	39	48	intra	intra	ADJ
ejpam-3576	39	49	-	-	ADJ
ejpam-3576	39	50	regular	regular	ADJ
ejpam-3576	39	51	if	if	SCONJ
ejpam-3576	39	52	for	for	SCONJ
ejpam-3576	39	53	every	every	DET
ejpam-3576	39	54	a	a	DET
ejpam-3576	39	55	∈	∈	NOUN
ejpam-3576	39	56	s	s	VERB
ejpam-3576	39	57	there	there	PRON
ejpam-3576	39	58	exist	exist	VERB
ejpam-3576	39	59	x	x	NOUN
ejpam-3576	39	60	,	,	PUNCT
ejpam-3576	39	61	y	y	PROPN
ejpam-3576	39	62	∈	∈	PROPN
ejpam-3576	39	63	s	s	VERB
ejpam-3576	39	64	such	such	ADJ
ejpam-3576	39	65	that	that	SCONJ
ejpam-3576	39	66	a	a	DET
ejpam-3576	39	67	≤	≤	NOUN
ejpam-3576	39	68	xa2y	xa2y	PROPN
ejpam-3576	39	69	.	.	PUNCT
ejpam-3576	40	1	equivalent	equivalent	ADJ
ejpam-3576	40	2	definitions	definition	NOUN
ejpam-3576	40	3	are	be	AUX
ejpam-3576	40	4	as	as	SCONJ
ejpam-3576	40	5	follows	follow	VERB
ejpam-3576	40	6	:	:	PUNCT
ejpam-3576	40	7	(	(	PUNCT
ejpam-3576	40	8	1	1	X
ejpam-3576	40	9	)	)	PUNCT
ejpam-3576	40	10	a	a	DET
ejpam-3576	40	11	⊆	⊆	NUM
ejpam-3576	40	12	(	(	PUNCT
ejpam-3576	40	13	sa2s	sa2s	NOUN
ejpam-3576	40	14	]	]	PUNCT
ejpam-3576	40	15	for	for	ADP
ejpam-3576	40	16	every	every	DET
ejpam-3576	40	17	a	a	DET
ejpam-3576	40	18	⊆	⊆	NUM
ejpam-3576	40	19	s.	s.	PROPN
ejpam-3576	40	20	(	(	PUNCT
ejpam-3576	40	21	2	2	NUM
ejpam-3576	40	22	)	)	PUNCT
ejpam-3576	40	23	a	a	DET
ejpam-3576	40	24	∈	∈	PROPN
ejpam-3576	40	25	(	(	PUNCT
ejpam-3576	40	26	sa2s	sa2s	NOUN
ejpam-3576	40	27	]	]	PUNCT
ejpam-3576	40	28	for	for	ADP
ejpam-3576	40	29	every	every	DET
ejpam-3576	40	30	a	a	DET
ejpam-3576	40	31	∈	∈	NOUN
ejpam-3576	40	32	s.	s.	NOUN
ejpam-3576	40	33	we	we	PRON
ejpam-3576	40	34	define	define	VERB
ejpam-3576	40	35	anti	anti	ADJ
ejpam-3576	40	36	fuzzy	fuzzy	ADJ
ejpam-3576	40	37	left	left	NOUN
ejpam-3576	40	38	(	(	PUNCT
ejpam-3576	40	39	resp	resp	NOUN
ejpam-3576	40	40	.	.	PUNCT
ejpam-3576	41	1	right	right	ADJ
ejpam-3576	41	2	,	,	PUNCT
ejpam-3576	41	3	interior	interior	ADJ
ejpam-3576	41	4	)	)	PUNCT
ejpam-3576	41	5	ideals	ideal	NOUN
ejpam-3576	41	6	in	in	ADP
ejpam-3576	41	7	an	an	DET
ejpam-3576	41	8	ordered	order	VERB
ejpam-3576	41	9	ag	ag	PROPN
ejpam-3576	41	10	-	-	PUNCT
ejpam-3576	41	11	groupoids	groupoid	NOUN
ejpam-3576	41	12	,	,	PUNCT
ejpam-3576	41	13	basically	basically	ADV
ejpam-3576	41	14	an	an	DET
ejpam-3576	41	15	ordered	order	VERB
ejpam-3576	41	16	ag	ag	PROPN
ejpam-3576	41	17	-	-	PUNCT
ejpam-3576	41	18	groupoid	groupoid	PROPN
ejpam-3576	41	19	is	be	AUX
ejpam-3576	41	20	non	non	ADJ
ejpam-3576	41	21	-	-	ADJ
ejpam-3576	41	22	commutative	commutative	ADJ
ejpam-3576	41	23	and	and	CCONJ
ejpam-3576	41	24	non	non	ADJ
ejpam-3576	41	25	-	-	ADJ
ejpam-3576	41	26	associative	associative	ADJ
ejpam-3576	41	27	ordered	order	VERB
ejpam-3576	41	28	semigroup	semigroup	NOUN
ejpam-3576	41	29	.	.	PUNCT
ejpam-3576	42	1	in	in	ADP
ejpam-3576	42	2	this	this	DET
ejpam-3576	42	3	present	present	ADJ
ejpam-3576	42	4	paper	paper	NOUN
ejpam-3576	42	5	,	,	PUNCT
ejpam-3576	42	6	we	we	PRON
ejpam-3576	42	7	characterize	characterize	VERB
ejpam-3576	42	8	regular	regular	ADJ
ejpam-3576	42	9	(	(	PUNCT
ejpam-3576	42	10	resp	resp	NOUN
ejpam-3576	42	11	.	.	PUNCT
ejpam-3576	43	1	right	right	ADV
ejpam-3576	43	2	regular	regular	ADV
ejpam-3576	43	3	,	,	PUNCT
ejpam-3576	43	4	left	leave	VERB
ejpam-3576	43	5	regular	regular	ADV
ejpam-3576	43	6	,	,	PUNCT
ejpam-3576	43	7	(	(	PUNCT
ejpam-3576	43	8	2	2	NUM
ejpam-3576	43	9	,	,	PUNCT
ejpam-3576	43	10	2)regular	2)regular	NUM
ejpam-3576	43	11	,	,	PUNCT
ejpam-3576	43	12	weakly	weakly	ADV
ejpam-3576	43	13	regular	regular	ADJ
ejpam-3576	43	14	and	and	CCONJ
ejpam-3576	43	15	intra	intra	ADJ
ejpam-3576	43	16	-	-	ADJ
ejpam-3576	43	17	regular	regular	ADJ
ejpam-3576	43	18	)	)	PUNCT
ejpam-3576	43	19	ordered	order	VERB
ejpam-3576	43	20	ag	ag	PROPN
ejpam-3576	43	21	-	-	NOUN
ejpam-3576	43	22	groupoids	groupoid	NOUN
ejpam-3576	43	23	in	in	ADP
ejpam-3576	43	24	terms	term	NOUN
ejpam-3576	43	25	of	of	ADP
ejpam-3576	43	26	anti	anti	ADJ
ejpam-3576	43	27	fuzzy	fuzzy	ADJ
ejpam-3576	43	28	left	left	ADJ
ejpam-3576	43	29	(	(	PUNCT
ejpam-3576	43	30	resp	resp	NOUN
ejpam-3576	43	31	.	.	PUNCT
ejpam-3576	44	1	right	right	ADJ
ejpam-3576	44	2	,	,	PUNCT
ejpam-3576	44	3	interior	interior	ADJ
ejpam-3576	44	4	)	)	PUNCT
ejpam-3576	44	5	ideals	ideal	NOUN
ejpam-3576	44	6	.	.	PUNCT
ejpam-3576	45	1	in	in	ADP
ejpam-3576	45	2	this	this	DET
ejpam-3576	45	3	regard	regard	NOUN
ejpam-3576	45	4	,	,	PUNCT
ejpam-3576	45	5	we	we	PRON
ejpam-3576	45	6	prove	prove	VERB
ejpam-3576	45	7	that	that	SCONJ
ejpam-3576	45	8	in	in	ADP
ejpam-3576	45	9	(	(	PUNCT
ejpam-3576	45	10	regular	regular	ADJ
ejpam-3576	45	11	,	,	PUNCT
ejpam-3576	45	12	right	right	ADV
ejpam-3576	45	13	regular	regular	ADJ
ejpam-3576	45	14	,	,	PUNCT
ejpam-3576	45	15	weakly	weakly	ADV
ejpam-3576	45	16	regular	regular	ADJ
ejpam-3576	45	17	)	)	PUNCT
ejpam-3576	45	18	ordered	order	VERB
ejpam-3576	45	19	ag	ag	PROPN
ejpam-3576	45	20	-	-	NOUN
ejpam-3576	45	21	groupoids	groupoid	NOUN
ejpam-3576	45	22	,	,	PUNCT
ejpam-3576	45	23	the	the	DET
ejpam-3576	45	24	concept	concept	NOUN
ejpam-3576	45	25	of	of	ADP
ejpam-3576	45	26	anti	anti	ADJ
ejpam-3576	45	27	fuzzy	fuzzy	ADJ
ejpam-3576	45	28	(	(	PUNCT
ejpam-3576	45	29	interior	interior	ADJ
ejpam-3576	45	30	,	,	PUNCT
ejpam-3576	45	31	two	two	NUM
ejpam-3576	45	32	-	-	PUNCT
ejpam-3576	45	33	sided	sided	ADJ
ejpam-3576	45	34	)	)	PUNCT
ejpam-3576	45	35	ideals	ideal	NOUN
ejpam-3576	45	36	coincide	coincide	VERB
ejpam-3576	45	37	.	.	PUNCT
ejpam-3576	46	1	the	the	DET
ejpam-3576	46	2	concept	concept	NOUN
ejpam-3576	46	3	of	of	ADP
ejpam-3576	46	4	anti	anti	ADJ
ejpam-3576	46	5	fuzzy	fuzzy	ADJ
ejpam-3576	46	6	(	(	PUNCT
ejpam-3576	46	7	interior	interior	ADJ
ejpam-3576	46	8	,	,	PUNCT
ejpam-3576	46	9	two	two	NUM
ejpam-3576	46	10	-	-	PUNCT
ejpam-3576	46	11	sided	sided	ADJ
ejpam-3576	46	12	)	)	PUNCT
ejpam-3576	46	13	ideals	ideal	NOUN
ejpam-3576	46	14	coincide	coincide	VERB
ejpam-3576	46	15	in	in	ADP
ejpam-3576	46	16	(	(	PUNCT
ejpam-3576	46	17	(	(	PUNCT
ejpam-3576	46	18	2	2	NUM
ejpam-3576	46	19	,	,	PUNCT
ejpam-3576	46	20	2	2	NUM
ejpam-3576	46	21	)	)	PUNCT
ejpam-3576	46	22	,	,	PUNCT
ejpam-3576	46	23	left	leave	VERB
ejpam-3576	46	24	,	,	PUNCT
ejpam-3576	46	25	intra-	intra-	ADJ
ejpam-3576	46	26	)	)	PUNCT
ejpam-3576	46	27	regular	regular	ADJ
ejpam-3576	46	28	ordered	order	VERB
ejpam-3576	46	29	ag	ag	PROPN
ejpam-3576	46	30	-	-	NOUN
ejpam-3576	46	31	groupoids	groupoid	NOUN
ejpam-3576	46	32	with	with	ADP
ejpam-3576	46	33	left	left	ADJ
ejpam-3576	46	34	identity	identity	NOUN
ejpam-3576	46	35	.	.	PUNCT
ejpam-3576	47	1	2	2	X
ejpam-3576	47	2	.	.	X
ejpam-3576	47	3	preliminaries	preliminary	NOUN
ejpam-3576	47	4	in	in	ADP
ejpam-3576	47	5	[	[	X
ejpam-3576	47	6	31	31	NUM
ejpam-3576	47	7	]	]	PUNCT
ejpam-3576	47	8	,	,	PUNCT
ejpam-3576	47	9	an	an	DET
ejpam-3576	47	10	ordered	order	VERB
ejpam-3576	47	11	ag	ag	PROPN
ejpam-3576	47	12	-	-	PROPN
ejpam-3576	47	13	groupoid	groupoid	PROPN
ejpam-3576	47	14	s	s	PROPN
ejpam-3576	47	15	,	,	PUNCT
ejpam-3576	47	16	is	be	AUX
ejpam-3576	47	17	a	a	DET
ejpam-3576	47	18	partially	partially	ADV
ejpam-3576	47	19	ordered	order	VERB
ejpam-3576	47	20	set	set	NOUN
ejpam-3576	47	21	,	,	PUNCT
ejpam-3576	47	22	at	at	ADP
ejpam-3576	47	23	the	the	DET
ejpam-3576	47	24	same	same	ADJ
ejpam-3576	47	25	time	time	NOUN
ejpam-3576	47	26	an	an	DET
ejpam-3576	47	27	aggroupoid	aggroupoid	ADJ
ejpam-3576	47	28	such	such	ADJ
ejpam-3576	47	29	that	that	SCONJ
ejpam-3576	47	30	a	a	DET
ejpam-3576	47	31	≤	≤	PROPN
ejpam-3576	47	32	b	b	NOUN
ejpam-3576	47	33	,	,	PUNCT
ejpam-3576	47	34	implies	imply	VERB
ejpam-3576	47	35	ac	ac	PROPN
ejpam-3576	47	36	≤	≤	PUNCT
ejpam-3576	47	37	bc	bc	PROPN
ejpam-3576	47	38	and	and	CCONJ
ejpam-3576	47	39	ca	can	AUX
ejpam-3576	47	40	≤	≤	NUM
ejpam-3576	47	41	cb	cb	X
ejpam-3576	47	42	for	for	ADP
ejpam-3576	47	43	all	all	DET
ejpam-3576	47	44	a	a	DET
ejpam-3576	47	45	,	,	PUNCT
ejpam-3576	47	46	b	b	NOUN
ejpam-3576	47	47	,	,	PUNCT
ejpam-3576	47	48	c	c	PROPN
ejpam-3576	47	49	∈	∈	PROPN
ejpam-3576	47	50	s.	s.	PROPN
ejpam-3576	47	51	two	two	NUM
ejpam-3576	47	52	conditions	condition	NOUN
ejpam-3576	47	53	are	be	AUX
ejpam-3576	47	54	equivalent	equivalent	ADJ
ejpam-3576	47	55	to	to	ADP
ejpam-3576	47	56	the	the	DET
ejpam-3576	47	57	one	one	NUM
ejpam-3576	47	58	condition	condition	NOUN
ejpam-3576	47	59	(	(	PUNCT
ejpam-3576	47	60	ca)d	ca)d	PROPN
ejpam-3576	47	61	≤	≤	NOUN
ejpam-3576	47	62	(	(	PUNCT
ejpam-3576	47	63	cb)d	cb)d	PROPN
ejpam-3576	47	64	,	,	PUNCT
ejpam-3576	47	65	for	for	ADP
ejpam-3576	47	66	all	all	DET
ejpam-3576	47	67	a	a	DET
ejpam-3576	47	68	,	,	PUNCT
ejpam-3576	47	69	b	b	NOUN
ejpam-3576	47	70	,	,	PUNCT
ejpam-3576	47	71	c	c	NOUN
ejpam-3576	47	72	,	,	PUNCT
ejpam-3576	47	73	d	d	PROPN
ejpam-3576	47	74	∈	∈	PROPN
ejpam-3576	47	75	s.	s.	PROPN
ejpam-3576	47	76	an	an	DET
ejpam-3576	47	77	ordered	order	VERB
ejpam-3576	47	78	ag	ag	PROPN
ejpam-3576	47	79	-	-	PUNCT
ejpam-3576	47	80	groupoid	groupoid	PROPN
ejpam-3576	47	81	is	be	AUX
ejpam-3576	47	82	also	also	ADV
ejpam-3576	47	83	called	call	VERB
ejpam-3576	47	84	a	a	DET
ejpam-3576	47	85	po	po	NOUN
ejpam-3576	47	86	-	-	PUNCT
ejpam-3576	47	87	ag	ag	NOUN
ejpam-3576	47	88	-	-	NOUN
ejpam-3576	47	89	groupoid	groupoid	PROPN
ejpam-3576	47	90	for	for	ADP
ejpam-3576	47	91	short	short	ADJ
ejpam-3576	47	92	.	.	PUNCT
ejpam-3576	48	1	example	example	NOUN
ejpam-3576	49	1	1	1	NUM
ejpam-3576	49	2	.	.	X
ejpam-3576	49	3	consider	consider	VERB
ejpam-3576	49	4	a	a	DET
ejpam-3576	49	5	set	set	NOUN
ejpam-3576	49	6	s	s	PART
ejpam-3576	49	7	=	=	PUNCT
ejpam-3576	49	8	{	{	PUNCT
ejpam-3576	49	9	e	e	PROPN
ejpam-3576	49	10	,	,	PUNCT
ejpam-3576	49	11	f	f	PROPN
ejpam-3576	49	12	,	,	PUNCT
ejpam-3576	49	13	a	a	DET
ejpam-3576	49	14	,	,	PUNCT
ejpam-3576	49	15	b	b	NOUN
ejpam-3576	49	16	,	,	PUNCT
ejpam-3576	49	17	c	c	NOUN
ejpam-3576	49	18	}	}	PUNCT
ejpam-3576	49	19	with	with	ADP
ejpam-3576	49	20	the	the	DET
ejpam-3576	49	21	following	follow	VERB
ejpam-3576	49	22	multiplication	multiplication	NOUN
ejpam-3576	49	23	“	"	PUNCT
ejpam-3576	49	24	·	·	PUNCT
ejpam-3576	49	25	”	"	PUNCT
ejpam-3576	49	26	and	and	CCONJ
ejpam-3576	49	27	order	order	NOUN
ejpam-3576	49	28	relation	relation	NOUN
ejpam-3576	49	29	“	"	PUNCT
ejpam-3576	49	30	≤	≤	NUM
ejpam-3576	49	31	”	"	PUNCT
ejpam-3576	49	32	:	:	PUNCT
ejpam-3576	49	33	·	·	PUNCT
ejpam-3576	50	1	e	e	X
ejpam-3576	50	2	f	f	PROPN
ejpam-3576	50	3	a	a	DET
ejpam-3576	50	4	b	b	X
ejpam-3576	50	5	c	c	NOUN
ejpam-3576	50	6	e	e	X
ejpam-3576	50	7	e	e	X
ejpam-3576	50	8	f	f	PROPN
ejpam-3576	50	9	a	a	DET
ejpam-3576	50	10	b	b	X
ejpam-3576	50	11	c	c	NOUN
ejpam-3576	50	12	f	f	PROPN
ejpam-3576	50	13	f	f	PROPN
ejpam-3576	50	14	f	f	PROPN
ejpam-3576	50	15	f	f	PROPN
ejpam-3576	50	16	b	b	PROPN
ejpam-3576	50	17	c	c	PROPN
ejpam-3576	50	18	a	a	PRON
ejpam-3576	50	19	a	a	DET
ejpam-3576	50	20	f	f	NOUN
ejpam-3576	50	21	c	c	PROPN
ejpam-3576	50	22	b	b	PROPN
ejpam-3576	50	23	c	c	PROPN
ejpam-3576	50	24	b	b	PROPN
ejpam-3576	51	1	c	c	NOUN
ejpam-3576	51	2	c	c	NOUN
ejpam-3576	51	3	c	c	NOUN
ejpam-3576	51	4	f	f	PROPN
ejpam-3576	51	5	b	b	PROPN
ejpam-3576	51	6	c	c	PROPN
ejpam-3576	51	7	b	b	PROPN
ejpam-3576	51	8	b	b	PROPN
ejpam-3576	51	9	b	b	PROPN
ejpam-3576	51	10	c	c	NOUN
ejpam-3576	51	11	f	f	PROPN
ejpam-3576	51	12	≤	≤	NUM
ejpam-3576	51	13	:	:	PUNCT
ejpam-3576	52	1	=	=	SYM
ejpam-3576	52	2	{	{	PUNCT
ejpam-3576	52	3	(	(	PUNCT
ejpam-3576	52	4	e	e	NOUN
ejpam-3576	52	5	,	,	PUNCT
ejpam-3576	52	6	e	e	NOUN
ejpam-3576	52	7	)	)	PUNCT
ejpam-3576	52	8	,	,	PUNCT
ejpam-3576	52	9	(	(	PUNCT
ejpam-3576	52	10	e	e	NOUN
ejpam-3576	52	11	,	,	PUNCT
ejpam-3576	52	12	a	a	PRON
ejpam-3576	52	13	)	)	PUNCT
ejpam-3576	52	14	,	,	PUNCT
ejpam-3576	52	15	(	(	PUNCT
ejpam-3576	52	16	e	e	NOUN
ejpam-3576	52	17	,	,	PUNCT
ejpam-3576	52	18	b	b	NOUN
ejpam-3576	52	19	)	)	PUNCT
ejpam-3576	52	20	,	,	PUNCT
ejpam-3576	52	21	(	(	PUNCT
ejpam-3576	52	22	e	e	NOUN
ejpam-3576	52	23	,	,	PUNCT
ejpam-3576	52	24	c	c	NOUN
ejpam-3576	52	25	)	)	PUNCT
ejpam-3576	52	26	,	,	PUNCT
ejpam-3576	52	27	(	(	PUNCT
ejpam-3576	52	28	f	f	X
ejpam-3576	52	29	,	,	PUNCT
ejpam-3576	52	30	f	f	PROPN
ejpam-3576	52	31	)	)	PUNCT
ejpam-3576	52	32	,	,	PUNCT
ejpam-3576	52	33	(	(	PUNCT
ejpam-3576	52	34	f	f	X
ejpam-3576	52	35	,	,	PUNCT
ejpam-3576	52	36	b	b	NOUN
ejpam-3576	52	37	)	)	PUNCT
ejpam-3576	52	38	,	,	PUNCT
ejpam-3576	52	39	(	(	PUNCT
ejpam-3576	52	40	f	f	X
ejpam-3576	52	41	,	,	PUNCT
ejpam-3576	52	42	c	c	NOUN
ejpam-3576	52	43	)	)	PUNCT
ejpam-3576	52	44	,	,	PUNCT
ejpam-3576	52	45	(	(	PUNCT
ejpam-3576	52	46	a	a	X
ejpam-3576	52	47	,	,	PUNCT
ejpam-3576	52	48	a	a	NOUN
ejpam-3576	52	49	)	)	PUNCT
ejpam-3576	52	50	,	,	PUNCT
ejpam-3576	52	51	(	(	PUNCT
ejpam-3576	52	52	a	a	DET
ejpam-3576	52	53	,	,	PUNCT
ejpam-3576	52	54	c	c	NOUN
ejpam-3576	52	55	)	)	PUNCT
ejpam-3576	52	56	,	,	PUNCT
ejpam-3576	52	57	(	(	PUNCT
ejpam-3576	52	58	b	b	X
ejpam-3576	52	59	,	,	PUNCT
ejpam-3576	52	60	b	b	NOUN
ejpam-3576	52	61	)	)	PUNCT
ejpam-3576	52	62	,	,	PUNCT
ejpam-3576	52	63	(	(	PUNCT
ejpam-3576	52	64	b	b	X
ejpam-3576	52	65	,	,	PUNCT
ejpam-3576	52	66	c	c	NOUN
ejpam-3576	52	67	)	)	PUNCT
ejpam-3576	52	68	,	,	PUNCT
ejpam-3576	52	69	(	(	PUNCT
ejpam-3576	52	70	c	c	X
ejpam-3576	52	71	,	,	PUNCT
ejpam-3576	52	72	c	c	NOUN
ejpam-3576	52	73	)	)	PUNCT
ejpam-3576	52	74	}	}	PUNCT
ejpam-3576	52	75	.	.	PUNCT
ejpam-3576	53	1	then	then	ADV
ejpam-3576	53	2	(	(	PUNCT
ejpam-3576	53	3	s	s	X
ejpam-3576	53	4	,	,	PUNCT
ejpam-3576	53	5	·	·	PUNCT
ejpam-3576	53	6	,	,	PUNCT
ejpam-3576	53	7	≤	≤	NUM
ejpam-3576	53	8	)	)	PUNCT
ejpam-3576	53	9	is	be	AUX
ejpam-3576	53	10	an	an	DET
ejpam-3576	53	11	ordered	order	VERB
ejpam-3576	53	12	ag	ag	PROPN
ejpam-3576	53	13	-	-	NOUN
ejpam-3576	53	14	groupoid	groupoid	PROPN
ejpam-3576	53	15	with	with	ADP
ejpam-3576	53	16	left	left	ADJ
ejpam-3576	53	17	identity	identity	NOUN
ejpam-3576	53	18	e.	e.	PROPN
ejpam-3576	53	19	k.	k.	PROPN
ejpam-3576	53	20	nasreen	nasreen	PROPN
ejpam-3576	53	21	,	,	PUNCT
ejpam-3576	53	22	m.	m.	NOUN
ejpam-3576	53	23	alesemi	alesemi	PROPN
ejpam-3576	53	24	,	,	PUNCT
ejpam-3576	53	25	salahuddin	salahuddin	VERB
ejpam-3576	53	26	/	/	SYM
ejpam-3576	53	27	eur	eur	PROPN
ejpam-3576	53	28	.	.	PUNCT
ejpam-3576	54	1	j.	j.	PROPN
ejpam-3576	54	2	pure	pure	PROPN
ejpam-3576	54	3	appl	appl	PROPN
ejpam-3576	54	4	.	.	PROPN
ejpam-3576	54	5	math	math	PROPN
ejpam-3576	54	6	,	,	PUNCT
ejpam-3576	54	7	13	13	NUM
ejpam-3576	54	8	(	(	PUNCT
ejpam-3576	54	9	1	1	NUM
ejpam-3576	54	10	)	)	PUNCT
ejpam-3576	54	11	(	(	PUNCT
ejpam-3576	54	12	2020	2020	NUM
ejpam-3576	54	13	)	)	PUNCT
ejpam-3576	54	14	,	,	PUNCT
ejpam-3576	54	15	113	113	NUM
ejpam-3576	54	16	-	-	SYM
ejpam-3576	54	17	129	129	NUM
ejpam-3576	54	18	115	115	NUM
ejpam-3576	54	19	let	let	VERB
ejpam-3576	54	20	s	s	PRON
ejpam-3576	54	21	be	be	AUX
ejpam-3576	54	22	an	an	DET
ejpam-3576	54	23	ordered	order	VERB
ejpam-3576	54	24	ag	ag	PROPN
ejpam-3576	54	25	-	-	NOUN
ejpam-3576	54	26	groupoid	groupoid	PROPN
ejpam-3576	54	27	and	and	CCONJ
ejpam-3576	54	28	a	a	DET
ejpam-3576	54	29	⊆	⊆	NUM
ejpam-3576	54	30	s	s	NOUN
ejpam-3576	54	31	,	,	PUNCT
ejpam-3576	54	32	we	we	PRON
ejpam-3576	54	33	define	define	VERB
ejpam-3576	54	34	a	a	DET
ejpam-3576	54	35	subset	subset	NOUN
ejpam-3576	54	36	(	(	PUNCT
ejpam-3576	54	37	a	a	X
ejpam-3576	54	38	]	]	X
ejpam-3576	54	39	=	=	PUNCT
ejpam-3576	54	40	{	{	PUNCT
ejpam-3576	54	41	s	s	X
ejpam-3576	54	42	∈	∈	NOUN
ejpam-3576	54	43	s	s	PART
ejpam-3576	54	44	:	:	PUNCT
ejpam-3576	54	45	s	s	VERB
ejpam-3576	54	46	≤	≤	NOUN
ejpam-3576	54	47	a	a	PRON
ejpam-3576	54	48	for	for	ADP
ejpam-3576	54	49	some	some	DET
ejpam-3576	54	50	a	a	DET
ejpam-3576	54	51	∈	∈	PROPN
ejpam-3576	54	52	a	a	PRON
ejpam-3576	54	53	}	}	PUNCT
ejpam-3576	54	54	of	of	ADP
ejpam-3576	54	55	s	s	PRON
ejpam-3576	54	56	and	and	CCONJ
ejpam-3576	54	57	obviously	obviously	ADV
ejpam-3576	54	58	a	a	DET
ejpam-3576	54	59	⊆	⊆	NUM
ejpam-3576	54	60	(	(	PUNCT
ejpam-3576	54	61	a	a	NOUN
ejpam-3576	54	62	]	]	X
ejpam-3576	54	63	.	.	PUNCT
ejpam-3576	55	1	if	if	SCONJ
ejpam-3576	55	2	a	a	PRON
ejpam-3576	55	3	=	=	X
ejpam-3576	55	4	{	{	PUNCT
ejpam-3576	55	5	a	a	NOUN
ejpam-3576	55	6	}	}	PUNCT
ejpam-3576	55	7	,	,	PUNCT
ejpam-3576	55	8	then	then	ADV
ejpam-3576	55	9	we	we	PRON
ejpam-3576	55	10	write	write	VERB
ejpam-3576	55	11	(	(	PUNCT
ejpam-3576	55	12	a	a	PRON
ejpam-3576	55	13	]	]	X
ejpam-3576	55	14	instead	instead	ADV
ejpam-3576	55	15	of	of	ADP
ejpam-3576	55	16	(	(	PUNCT
ejpam-3576	55	17	{	{	PUNCT
ejpam-3576	55	18	a	a	NOUN
ejpam-3576	55	19	}	}	PUNCT
ejpam-3576	55	20	]	]	PUNCT
ejpam-3576	55	21	.	.	PUNCT
ejpam-3576	56	1	for	for	ADP
ejpam-3576	56	2	a	a	DET
ejpam-3576	56	3	,	,	PUNCT
ejpam-3576	56	4	b	b	PROPN
ejpam-3576	56	5	⊆	⊆	NUM
ejpam-3576	56	6	s	s	NOUN
ejpam-3576	56	7	,	,	PUNCT
ejpam-3576	56	8	then	then	ADV
ejpam-3576	56	9	ab	ab	PROPN
ejpam-3576	56	10	=	=	PUNCT
ejpam-3576	56	11	{	{	PUNCT
ejpam-3576	56	12	ab	ab	PROPN
ejpam-3576	56	13	|	|	ADV
ejpam-3576	56	14	a	a	DET
ejpam-3576	56	15	∈	∈	PROPN
ejpam-3576	56	16	a	a	PRON
ejpam-3576	56	17	,	,	PUNCT
ejpam-3576	56	18	b	b	PROPN
ejpam-3576	56	19	∈	∈	PROPN
ejpam-3576	56	20	b	b	NOUN
ejpam-3576	56	21	}	}	PUNCT
ejpam-3576	56	22	,	,	PUNCT
ejpam-3576	56	23	(	(	PUNCT
ejpam-3576	56	24	(	(	PUNCT
ejpam-3576	56	25	a	a	X
ejpam-3576	56	26	]	]	X
ejpam-3576	56	27	]	]	X
ejpam-3576	56	28	=	=	X
ejpam-3576	56	29	(	(	PUNCT
ejpam-3576	56	30	a	a	X
ejpam-3576	56	31	]	]	X
ejpam-3576	56	32	,	,	PUNCT
ejpam-3576	56	33	(	(	PUNCT
ejpam-3576	56	34	a](b	a](b	NOUN
ejpam-3576	56	35	]	]	PUNCT
ejpam-3576	56	36	⊆	⊆	NUM
ejpam-3576	56	37	(	(	PUNCT
ejpam-3576	56	38	ab	ab	X
ejpam-3576	56	39	]	]	X
ejpam-3576	56	40	,	,	PUNCT
ejpam-3576	56	41	(	(	PUNCT
ejpam-3576	56	42	(	(	PUNCT
ejpam-3576	56	43	a](b	a](b	NOUN
ejpam-3576	56	44	]	]	X
ejpam-3576	56	45	]	]	X
ejpam-3576	57	1	=	=	SYM
ejpam-3576	57	2	(	(	PUNCT
ejpam-3576	57	3	ab	ab	X
ejpam-3576	57	4	]	]	X
ejpam-3576	57	5	,	,	PUNCT
ejpam-3576	57	6	if	if	SCONJ
ejpam-3576	57	7	a	a	DET
ejpam-3576	57	8	⊆	⊆	NUM
ejpam-3576	57	9	b	b	NOUN
ejpam-3576	57	10	then	then	ADV
ejpam-3576	57	11	(	(	PUNCT
ejpam-3576	57	12	a	a	X
ejpam-3576	57	13	]	]	X
ejpam-3576	57	14	⊆	⊆	NUM
ejpam-3576	57	15	(	(	PUNCT
ejpam-3576	57	16	b	b	NOUN
ejpam-3576	57	17	]	]	X
ejpam-3576	57	18	,	,	PUNCT
ejpam-3576	57	19	(	(	PUNCT
ejpam-3576	57	20	a	a	DET
ejpam-3576	57	21	∩b	∩b	NOUN
ejpam-3576	57	22	]	]	PUNCT
ejpam-3576	57	23	6=	6=	NUM
ejpam-3576	57	24	(	(	PUNCT
ejpam-3576	57	25	a	a	PRON
ejpam-3576	57	26	]	]	X
ejpam-3576	57	27	∩	∩	NOUN
ejpam-3576	57	28	(	(	PUNCT
ejpam-3576	57	29	b	b	X
ejpam-3576	57	30	]	]	X
ejpam-3576	57	31	in	in	ADP
ejpam-3576	57	32	general	general	ADJ
ejpam-3576	57	33	.	.	PUNCT
ejpam-3576	58	1	for	for	ADP
ejpam-3576	58	2	∅	∅	NOUN
ejpam-3576	58	3	6=	6=	ADP
ejpam-3576	58	4	a	a	DET
ejpam-3576	58	5	⊆	⊆	NUM
ejpam-3576	58	6	s.	s.	PROPN
ejpam-3576	58	7	a	a	PRON
ejpam-3576	58	8	is	be	AUX
ejpam-3576	58	9	an	an	DET
ejpam-3576	58	10	ordered	order	VERB
ejpam-3576	58	11	ag	ag	NOUN
ejpam-3576	58	12	-	-	NOUN
ejpam-3576	58	13	subgroupoid	subgroupoid	NOUN
ejpam-3576	58	14	of	of	ADP
ejpam-3576	58	15	s	s	PRON
ejpam-3576	58	16	if	if	SCONJ
ejpam-3576	58	17	a2	a2	PROPN
ejpam-3576	58	18	⊆	⊆	NUM
ejpam-3576	58	19	a.	a.	NOUN
ejpam-3576	58	20	a	a	PRON
ejpam-3576	58	21	is	be	AUX
ejpam-3576	58	22	left	leave	VERB
ejpam-3576	58	23	(	(	PUNCT
ejpam-3576	58	24	resp	resp	NOUN
ejpam-3576	58	25	.	.	PUNCT
ejpam-3576	59	1	right	right	ADJ
ejpam-3576	59	2	)	)	PUNCT
ejpam-3576	59	3	ideal	ideal	NOUN
ejpam-3576	59	4	of	of	ADP
ejpam-3576	59	5	s	s	PRON
ejpam-3576	59	6	if	if	SCONJ
ejpam-3576	59	7	(	(	PUNCT
ejpam-3576	59	8	1	1	NUM
ejpam-3576	59	9	)	)	PUNCT
ejpam-3576	59	10	sa	sa	ADP
ejpam-3576	59	11	⊆	⊆	NUM
ejpam-3576	59	12	a	a	DET
ejpam-3576	59	13	(	(	PUNCT
ejpam-3576	59	14	resp	resp	NOUN
ejpam-3576	59	15	.	.	PUNCT
ejpam-3576	60	1	as	as	ADP
ejpam-3576	60	2	⊆	⊆	NUM
ejpam-3576	60	3	a	a	PRON
ejpam-3576	60	4	)	)	PUNCT
ejpam-3576	60	5	.	.	PUNCT
ejpam-3576	61	1	(	(	PUNCT
ejpam-3576	61	2	2	2	X
ejpam-3576	61	3	)	)	PUNCT
ejpam-3576	61	4	if	if	SCONJ
ejpam-3576	61	5	a	a	DET
ejpam-3576	61	6	∈	∈	PROPN
ejpam-3576	61	7	a	a	PRON
ejpam-3576	61	8	and	and	CCONJ
ejpam-3576	61	9	b	b	NOUN
ejpam-3576	61	10	∈	∈	NOUN
ejpam-3576	61	11	s	s	VERB
ejpam-3576	61	12	such	such	ADJ
ejpam-3576	61	13	that	that	DET
ejpam-3576	61	14	b	b	NOUN
ejpam-3576	61	15	≤	≤	NOUN
ejpam-3576	61	16	a	a	DET
ejpam-3576	61	17	implies	implie	NOUN
ejpam-3576	61	18	b	b	X
ejpam-3576	61	19	∈	∈	PROPN
ejpam-3576	61	20	a.	a.	NOUN
ejpam-3576	61	21	equivalent	equivalent	PROPN
ejpam-3576	61	22	definition	definition	NOUN
ejpam-3576	61	23	:	:	PUNCT
ejpam-3576	61	24	a	a	PRON
ejpam-3576	61	25	is	be	AUX
ejpam-3576	61	26	left	leave	VERB
ejpam-3576	61	27	(	(	PUNCT
ejpam-3576	61	28	resp	resp	NOUN
ejpam-3576	61	29	.	.	PUNCT
ejpam-3576	62	1	right	right	ADJ
ejpam-3576	62	2	)	)	PUNCT
ejpam-3576	62	3	ideal	ideal	NOUN
ejpam-3576	62	4	of	of	ADP
ejpam-3576	62	5	s	s	PRON
ejpam-3576	62	6	if	if	SCONJ
ejpam-3576	62	7	(	(	PUNCT
ejpam-3576	62	8	a	a	X
ejpam-3576	62	9	]	]	X
ejpam-3576	62	10	⊆	⊆	NUM
ejpam-3576	62	11	a	a	PRON
ejpam-3576	62	12	and	and	CCONJ
ejpam-3576	62	13	sa	sa	ADP
ejpam-3576	62	14	⊆	⊆	NUM
ejpam-3576	62	15	a	a	DET
ejpam-3576	62	16	(	(	PUNCT
ejpam-3576	62	17	resp	resp	NOUN
ejpam-3576	62	18	.	.	PUNCT
ejpam-3576	63	1	as	as	ADP
ejpam-3576	63	2	⊆	⊆	NUM
ejpam-3576	63	3	a	a	PRON
ejpam-3576	63	4	)	)	PUNCT
ejpam-3576	63	5	.	.	PUNCT
ejpam-3576	64	1	a	a	PRON
ejpam-3576	64	2	is	be	AUX
ejpam-3576	64	3	an	an	DET
ejpam-3576	64	4	ideal	ideal	NOUN
ejpam-3576	64	5	of	of	ADP
ejpam-3576	64	6	s	s	PRON
ejpam-3576	64	7	if	if	SCONJ
ejpam-3576	64	8	a	a	PRON
ejpam-3576	64	9	is	be	AUX
ejpam-3576	64	10	both	both	PRON
ejpam-3576	64	11	a	a	DET
ejpam-3576	64	12	left	left	NOUN
ejpam-3576	64	13	and	and	CCONJ
ejpam-3576	64	14	a	a	DET
ejpam-3576	64	15	right	right	ADJ
ejpam-3576	64	16	ideal	ideal	NOUN
ejpam-3576	64	17	of	of	ADP
ejpam-3576	64	18	s.	s.	PROPN
ejpam-3576	64	19	if	if	SCONJ
ejpam-3576	64	20	a	a	DET
ejpam-3576	64	21	,	,	PUNCT
ejpam-3576	64	22	b	b	NOUN
ejpam-3576	64	23	are	be	AUX
ejpam-3576	64	24	ideals	ideal	NOUN
ejpam-3576	64	25	of	of	ADP
ejpam-3576	64	26	s	s	NOUN
ejpam-3576	64	27	,	,	PUNCT
ejpam-3576	64	28	then	then	ADV
ejpam-3576	64	29	a	a	DET
ejpam-3576	64	30	∪b	∪b	X
ejpam-3576	64	31	and	and	CCONJ
ejpam-3576	64	32	a	a	DET
ejpam-3576	64	33	∩b	∩b	NOUN
ejpam-3576	64	34	are	be	AUX
ejpam-3576	64	35	also	also	ADV
ejpam-3576	64	36	ideals	ideal	NOUN
ejpam-3576	64	37	of	of	ADP
ejpam-3576	64	38	s.	s.	PROPN
ejpam-3576	64	39	a	a	DET
ejpam-3576	64	40	non	non	ADJ
ejpam-3576	64	41	-	-	ADJ
ejpam-3576	64	42	empty	empty	ADJ
ejpam-3576	64	43	subset	subset	NOUN
ejpam-3576	64	44	a	a	PRON
ejpam-3576	64	45	of	of	ADP
ejpam-3576	64	46	an	an	DET
ejpam-3576	64	47	ordered	order	VERB
ejpam-3576	64	48	ag	ag	PROPN
ejpam-3576	64	49	-	-	NOUN
ejpam-3576	64	50	groupoid	groupoid	PROPN
ejpam-3576	64	51	s	s	PART
ejpam-3576	64	52	is	be	AUX
ejpam-3576	64	53	an	an	DET
ejpam-3576	64	54	interior	interior	ADJ
ejpam-3576	64	55	ideal	ideal	NOUN
ejpam-3576	64	56	of	of	ADP
ejpam-3576	64	57	s	s	PRON
ejpam-3576	64	58	if	if	SCONJ
ejpam-3576	64	59	(	(	PUNCT
ejpam-3576	64	60	1	1	NUM
ejpam-3576	64	61	)	)	PUNCT
ejpam-3576	64	62	(	(	PUNCT
ejpam-3576	64	63	sa)s	sa)s	PROPN
ejpam-3576	64	64	⊆	⊆	NUM
ejpam-3576	64	65	a.	a.	NOUN
ejpam-3576	64	66	(	(	PUNCT
ejpam-3576	64	67	2	2	NUM
ejpam-3576	64	68	)	)	PUNCT
ejpam-3576	64	69	if	if	SCONJ
ejpam-3576	64	70	a	a	DET
ejpam-3576	64	71	∈	∈	PROPN
ejpam-3576	64	72	a	a	PRON
ejpam-3576	64	73	and	and	CCONJ
ejpam-3576	64	74	b	b	NOUN
ejpam-3576	64	75	∈	∈	NOUN
ejpam-3576	64	76	s	s	VERB
ejpam-3576	64	77	such	such	ADJ
ejpam-3576	64	78	that	that	DET
ejpam-3576	64	79	b	b	NOUN
ejpam-3576	64	80	≤	≤	NOUN
ejpam-3576	64	81	a	a	DET
ejpam-3576	64	82	implies	implie	NOUN
ejpam-3576	64	83	b	b	X
ejpam-3576	64	84	∈	∈	PROPN
ejpam-3576	64	85	a	a	DET
ejpam-3576	64	86	(	(	PUNCT
ejpam-3576	64	87	or	or	CCONJ
ejpam-3576	64	88	(	(	PUNCT
ejpam-3576	64	89	a	a	PRON
ejpam-3576	64	90	]	]	X
ejpam-3576	64	91	⊆	⊆	NUM
ejpam-3576	64	92	a	a	PRON
ejpam-3576	64	93	)	)	PUNCT
ejpam-3576	64	94	.	.	PUNCT
ejpam-3576	65	1	an	an	DET
ejpam-3576	65	2	ordered	order	VERB
ejpam-3576	65	3	ag	ag	PROPN
ejpam-3576	65	4	-	-	NOUN
ejpam-3576	65	5	groupoid	groupoid	PROPN
ejpam-3576	65	6	s	s	PART
ejpam-3576	65	7	is	be	AUX
ejpam-3576	65	8	left	leave	VERB
ejpam-3576	65	9	(	(	PUNCT
ejpam-3576	65	10	resp	resp	NOUN
ejpam-3576	65	11	.	.	PUNCT
ejpam-3576	66	1	right	right	ADJ
ejpam-3576	66	2	)	)	PUNCT
ejpam-3576	67	1	regular	regular	ADJ
ejpam-3576	67	2	,	,	PUNCT
ejpam-3576	67	3	if	if	SCONJ
ejpam-3576	67	4	for	for	ADP
ejpam-3576	67	5	every	every	DET
ejpam-3576	67	6	a	a	DET
ejpam-3576	67	7	∈	∈	ADJ
ejpam-3576	67	8	s	s	NOUN
ejpam-3576	67	9	,	,	PUNCT
ejpam-3576	67	10	there	there	PRON
ejpam-3576	67	11	exists	exist	VERB
ejpam-3576	67	12	x	x	X
ejpam-3576	67	13	∈	∈	PROPN
ejpam-3576	67	14	s	s	VERB
ejpam-3576	67	15	such	such	ADJ
ejpam-3576	67	16	that	that	SCONJ
ejpam-3576	67	17	a	a	DET
ejpam-3576	67	18	≤	≤	ADJ
ejpam-3576	67	19	xa2	xa2	PROPN
ejpam-3576	67	20	(	(	PUNCT
ejpam-3576	67	21	resp	resp	NOUN
ejpam-3576	67	22	.	.	PUNCT
ejpam-3576	68	1	a	a	DET
ejpam-3576	68	2	≤	≤	NUM
ejpam-3576	68	3	a2x	a2x	PROPN
ejpam-3576	68	4	)	)	PUNCT
ejpam-3576	68	5	.	.	PUNCT
ejpam-3576	69	1	equivalent	equivalent	ADJ
ejpam-3576	69	2	definitions	definition	NOUN
ejpam-3576	69	3	are	be	AUX
ejpam-3576	69	4	as	as	SCONJ
ejpam-3576	69	5	follows	follow	VERB
ejpam-3576	69	6	:	:	PUNCT
ejpam-3576	69	7	(	(	PUNCT
ejpam-3576	69	8	1	1	X
ejpam-3576	69	9	)	)	PUNCT
ejpam-3576	69	10	a	a	DET
ejpam-3576	69	11	⊆	⊆	NUM
ejpam-3576	69	12	(	(	PUNCT
ejpam-3576	69	13	sa2	sa2	PROPN
ejpam-3576	69	14	]	]	PUNCT
ejpam-3576	69	15	(	(	PUNCT
ejpam-3576	69	16	resp	resp	NOUN
ejpam-3576	69	17	.	.	PUNCT
ejpam-3576	70	1	a	a	DET
ejpam-3576	70	2	⊆	⊆	NUM
ejpam-3576	70	3	(	(	PUNCT
ejpam-3576	70	4	a2s	a2s	NOUN
ejpam-3576	70	5	]	]	X
ejpam-3576	70	6	)	)	PUNCT
ejpam-3576	70	7	for	for	ADP
ejpam-3576	70	8	every	every	DET
ejpam-3576	70	9	a	a	DET
ejpam-3576	70	10	⊆	⊆	NUM
ejpam-3576	70	11	s.	s.	PROPN
ejpam-3576	70	12	(	(	PUNCT
ejpam-3576	70	13	2	2	NUM
ejpam-3576	70	14	)	)	PUNCT
ejpam-3576	70	15	a	a	DET
ejpam-3576	70	16	∈	∈	PROPN
ejpam-3576	70	17	(	(	PUNCT
ejpam-3576	70	18	sa2	sa2	PROPN
ejpam-3576	70	19	]	]	PUNCT
ejpam-3576	70	20	(	(	PUNCT
ejpam-3576	70	21	resp	resp	NOUN
ejpam-3576	70	22	.	.	PUNCT
ejpam-3576	71	1	a	a	DET
ejpam-3576	71	2	∈	∈	PROPN
ejpam-3576	71	3	(	(	PUNCT
ejpam-3576	71	4	a2s	a2s	NOUN
ejpam-3576	71	5	]	]	X
ejpam-3576	71	6	)	)	PUNCT
ejpam-3576	71	7	for	for	ADP
ejpam-3576	71	8	every	every	DET
ejpam-3576	71	9	a	a	DET
ejpam-3576	71	10	∈	∈	PROPN
ejpam-3576	71	11	s.	s.	PROPN
ejpam-3576	71	12	an	an	DET
ejpam-3576	71	13	ordered	order	VERB
ejpam-3576	71	14	ag	ag	PROPN
ejpam-3576	71	15	-	-	PROPN
ejpam-3576	71	16	groupoid	groupoid	PROPN
ejpam-3576	71	17	s	s	PART
ejpam-3576	71	18	is	be	AUX
ejpam-3576	71	19	regular	regular	ADJ
ejpam-3576	71	20	,	,	PUNCT
ejpam-3576	71	21	if	if	SCONJ
ejpam-3576	71	22	for	for	ADP
ejpam-3576	71	23	every	every	DET
ejpam-3576	71	24	a	a	DET
ejpam-3576	71	25	∈	∈	ADJ
ejpam-3576	71	26	s	s	NOUN
ejpam-3576	71	27	,	,	PUNCT
ejpam-3576	71	28	there	there	PRON
ejpam-3576	71	29	exists	exist	VERB
ejpam-3576	71	30	x	x	X
ejpam-3576	71	31	∈	∈	PROPN
ejpam-3576	71	32	s	s	VERB
ejpam-3576	71	33	such	such	ADJ
ejpam-3576	71	34	that	that	SCONJ
ejpam-3576	71	35	a	a	DET
ejpam-3576	71	36	≤	≤	NOUN
ejpam-3576	71	37	(	(	PUNCT
ejpam-3576	71	38	ax)a	ax)a	PROPN
ejpam-3576	71	39	.	.	PUNCT
ejpam-3576	72	1	equivalent	equivalent	ADJ
ejpam-3576	72	2	definitions	definition	NOUN
ejpam-3576	72	3	:	:	PUNCT
ejpam-3576	72	4	(	(	PUNCT
ejpam-3576	72	5	1	1	X
ejpam-3576	72	6	)	)	PUNCT
ejpam-3576	72	7	a	a	DET
ejpam-3576	72	8	⊆	⊆	NUM
ejpam-3576	72	9	(	(	PUNCT
ejpam-3576	72	10	(	(	PUNCT
ejpam-3576	72	11	as)a	as)a	PROPN
ejpam-3576	72	12	]	]	X
ejpam-3576	72	13	for	for	ADP
ejpam-3576	72	14	every	every	DET
ejpam-3576	72	15	a	a	DET
ejpam-3576	72	16	⊆	⊆	NUM
ejpam-3576	72	17	s.	s.	PROPN
ejpam-3576	72	18	(	(	PUNCT
ejpam-3576	72	19	2	2	NUM
ejpam-3576	72	20	)	)	PUNCT
ejpam-3576	72	21	a	a	DET
ejpam-3576	72	22	∈	∈	NOUN
ejpam-3576	72	23	(	(	PUNCT
ejpam-3576	72	24	(	(	PUNCT
ejpam-3576	72	25	as)a	as)a	PROPN
ejpam-3576	72	26	]	]	X
ejpam-3576	72	27	for	for	ADP
ejpam-3576	72	28	every	every	DET
ejpam-3576	72	29	a	a	DET
ejpam-3576	72	30	∈	∈	PROPN
ejpam-3576	72	31	s.	s.	PROPN
ejpam-3576	72	32	an	an	DET
ejpam-3576	72	33	ordered	order	VERB
ejpam-3576	72	34	ag	ag	PROPN
ejpam-3576	72	35	-	-	PROPN
ejpam-3576	72	36	groupoid	groupoid	PROPN
ejpam-3576	72	37	s	s	PART
ejpam-3576	72	38	is	be	AUX
ejpam-3576	72	39	completely	completely	ADV
ejpam-3576	72	40	regular	regular	ADJ
ejpam-3576	72	41	,	,	PUNCT
ejpam-3576	72	42	if	if	SCONJ
ejpam-3576	72	43	it	it	PRON
ejpam-3576	72	44	is	be	AUX
ejpam-3576	72	45	regular	regular	ADJ
ejpam-3576	72	46	,	,	PUNCT
ejpam-3576	72	47	left	leave	VERB
ejpam-3576	72	48	regular	regular	ADV
ejpam-3576	72	49	,	,	PUNCT
ejpam-3576	72	50	right	right	ADV
ejpam-3576	72	51	regular	regular	ADJ
ejpam-3576	72	52	.	.	PUNCT
ejpam-3576	73	1	an	an	DET
ejpam-3576	73	2	ordered	order	VERB
ejpam-3576	73	3	ag	ag	PROPN
ejpam-3576	73	4	-	-	NOUN
ejpam-3576	73	5	groupoid	groupoid	PROPN
ejpam-3576	73	6	s	s	PART
ejpam-3576	73	7	is	be	AUX
ejpam-3576	73	8	strongly	strongly	ADV
ejpam-3576	73	9	regular	regular	ADJ
ejpam-3576	73	10	,	,	PUNCT
ejpam-3576	73	11	if	if	SCONJ
ejpam-3576	73	12	for	for	ADP
ejpam-3576	73	13	every	every	DET
ejpam-3576	73	14	a	a	DET
ejpam-3576	73	15	∈	∈	ADJ
ejpam-3576	73	16	s	s	NOUN
ejpam-3576	73	17	,	,	PUNCT
ejpam-3576	73	18	there	there	PRON
ejpam-3576	73	19	exists	exist	VERB
ejpam-3576	73	20	x	x	X
ejpam-3576	73	21	∈	∈	PROPN
ejpam-3576	73	22	s	s	VERB
ejpam-3576	73	23	such	such	ADJ
ejpam-3576	73	24	that	that	SCONJ
ejpam-3576	73	25	a	a	DET
ejpam-3576	73	26	≤	≤	NOUN
ejpam-3576	73	27	(	(	PUNCT
ejpam-3576	73	28	ax)a	ax)a	PROPN
ejpam-3576	73	29	and	and	CCONJ
ejpam-3576	73	30	ax	ax	NOUN
ejpam-3576	73	31	=	=	SYM
ejpam-3576	73	32	xa	xa	PROPN
ejpam-3576	73	33	.	.	PUNCT
ejpam-3576	74	1	every	every	DET
ejpam-3576	74	2	strongly	strongly	ADV
ejpam-3576	74	3	regular	regular	ADJ
ejpam-3576	74	4	ordered	order	VERB
ejpam-3576	74	5	ag	ag	PROPN
ejpam-3576	74	6	-	-	PUNCT
ejpam-3576	74	7	groupoid	groupoid	PROPN
ejpam-3576	74	8	is	be	AUX
ejpam-3576	74	9	right	right	ADV
ejpam-3576	74	10	regular	regular	ADV
ejpam-3576	74	11	ordered	order	VERB
ejpam-3576	74	12	ag	ag	PROPN
ejpam-3576	74	13	-	-	NOUN
ejpam-3576	74	14	groupoid	groupoid	PROPN
ejpam-3576	74	15	.	.	PUNCT
ejpam-3576	75	1	an	an	DET
ejpam-3576	75	2	ordered	order	VERB
ejpam-3576	75	3	ag	ag	PROPN
ejpam-3576	75	4	-	-	NOUN
ejpam-3576	75	5	groupoid	groupoid	PROPN
ejpam-3576	75	6	s	s	PART
ejpam-3576	75	7	is	be	AUX
ejpam-3576	75	8	said	say	VERB
ejpam-3576	75	9	to	to	PART
ejpam-3576	75	10	be	be	AUX
ejpam-3576	75	11	weakly	weakly	ADV
ejpam-3576	75	12	regular	regular	ADJ
ejpam-3576	75	13	,	,	PUNCT
ejpam-3576	75	14	if	if	SCONJ
ejpam-3576	75	15	for	for	ADP
ejpam-3576	75	16	every	every	DET
ejpam-3576	75	17	a	a	DET
ejpam-3576	75	18	∈	∈	ADJ
ejpam-3576	75	19	s	s	NOUN
ejpam-3576	75	20	,	,	PUNCT
ejpam-3576	75	21	there	there	PRON
ejpam-3576	75	22	exist	exist	VERB
ejpam-3576	75	23	x	x	NOUN
ejpam-3576	75	24	,	,	PUNCT
ejpam-3576	75	25	y	y	PROPN
ejpam-3576	75	26	∈	∈	PROPN
ejpam-3576	75	27	s	s	VERB
ejpam-3576	75	28	such	such	ADJ
ejpam-3576	75	29	that	that	SCONJ
ejpam-3576	75	30	a	a	DET
ejpam-3576	75	31	≤	≤	ADJ
ejpam-3576	75	32	(	(	PUNCT
ejpam-3576	75	33	ax)(ay	ax)(ay	NOUN
ejpam-3576	75	34	)	)	PUNCT
ejpam-3576	75	35	.	.	PUNCT
ejpam-3576	76	1	equivalent	equivalent	ADJ
ejpam-3576	76	2	definitions	definition	NOUN
ejpam-3576	76	3	are	be	AUX
ejpam-3576	76	4	as	as	SCONJ
ejpam-3576	76	5	follows	follow	VERB
ejpam-3576	76	6	:	:	PUNCT
ejpam-3576	76	7	(	(	PUNCT
ejpam-3576	76	8	1	1	X
ejpam-3576	76	9	)	)	PUNCT
ejpam-3576	76	10	a	a	DET
ejpam-3576	76	11	⊆	⊆	NUM
ejpam-3576	76	12	(	(	PUNCT
ejpam-3576	76	13	(	(	PUNCT
ejpam-3576	76	14	as)2	as)2	NOUN
ejpam-3576	76	15	]	]	PUNCT
ejpam-3576	76	16	for	for	ADP
ejpam-3576	76	17	every	every	DET
ejpam-3576	76	18	a	a	DET
ejpam-3576	76	19	⊆	⊆	NUM
ejpam-3576	76	20	s.	s.	PROPN
ejpam-3576	76	21	(	(	PUNCT
ejpam-3576	76	22	2	2	NUM
ejpam-3576	76	23	)	)	PUNCT
ejpam-3576	76	24	a	a	DET
ejpam-3576	76	25	∈	∈	NOUN
ejpam-3576	76	26	(	(	PUNCT
ejpam-3576	76	27	(	(	PUNCT
ejpam-3576	76	28	as)2	as)2	NOUN
ejpam-3576	76	29	]	]	PUNCT
ejpam-3576	76	30	for	for	ADP
ejpam-3576	76	31	every	every	DET
ejpam-3576	76	32	a	a	DET
ejpam-3576	76	33	∈	∈	PROPN
ejpam-3576	76	34	s.	s.	PROPN
ejpam-3576	76	35	an	an	DET
ejpam-3576	76	36	ordered	order	VERB
ejpam-3576	76	37	ag	ag	PROPN
ejpam-3576	76	38	-	-	PROPN
ejpam-3576	76	39	groupoid	groupoid	PROPN
ejpam-3576	76	40	s	s	PART
ejpam-3576	76	41	is	be	AUX
ejpam-3576	76	42	an	an	DET
ejpam-3576	76	43	intra	intra	ADJ
ejpam-3576	76	44	-	-	ADJ
ejpam-3576	76	45	regular	regular	ADJ
ejpam-3576	76	46	,	,	PUNCT
ejpam-3576	76	47	if	if	SCONJ
ejpam-3576	76	48	for	for	ADP
ejpam-3576	76	49	every	every	DET
ejpam-3576	76	50	a	a	DET
ejpam-3576	76	51	∈	∈	ADJ
ejpam-3576	76	52	s	s	NOUN
ejpam-3576	76	53	,	,	PUNCT
ejpam-3576	76	54	there	there	PRON
ejpam-3576	76	55	exist	exist	VERB
ejpam-3576	76	56	x	x	NOUN
ejpam-3576	76	57	,	,	PUNCT
ejpam-3576	76	58	y	y	PROPN
ejpam-3576	76	59	∈	∈	PROPN
ejpam-3576	76	60	s	s	VERB
ejpam-3576	76	61	such	such	ADJ
ejpam-3576	76	62	that	that	SCONJ
ejpam-3576	76	63	a	a	DET
ejpam-3576	76	64	≤	≤	NOUN
ejpam-3576	76	65	(	(	PUNCT
ejpam-3576	76	66	xa2)y	xa2)y	PROPN
ejpam-3576	76	67	.	.	PUNCT
ejpam-3576	77	1	equivalent	equivalent	ADJ
ejpam-3576	77	2	definitions	definition	NOUN
ejpam-3576	77	3	are	be	AUX
ejpam-3576	77	4	as	as	SCONJ
ejpam-3576	77	5	follows	follow	VERB
ejpam-3576	77	6	:	:	PUNCT
ejpam-3576	77	7	(	(	PUNCT
ejpam-3576	77	8	1	1	X
ejpam-3576	77	9	)	)	PUNCT
ejpam-3576	77	10	a	a	DET
ejpam-3576	77	11	⊆	⊆	NUM
ejpam-3576	77	12	(	(	PUNCT
ejpam-3576	77	13	(	(	PUNCT
ejpam-3576	77	14	sa2)s	sa2)s	PROPN
ejpam-3576	77	15	]	]	X
ejpam-3576	77	16	for	for	ADP
ejpam-3576	77	17	every	every	DET
ejpam-3576	77	18	a	a	DET
ejpam-3576	77	19	⊆	⊆	NUM
ejpam-3576	77	20	s.	s.	PROPN
ejpam-3576	77	21	(	(	PUNCT
ejpam-3576	77	22	2	2	NUM
ejpam-3576	77	23	)	)	PUNCT
ejpam-3576	77	24	a	a	DET
ejpam-3576	77	25	∈	∈	NOUN
ejpam-3576	77	26	(	(	PUNCT
ejpam-3576	77	27	(	(	PUNCT
ejpam-3576	77	28	sa2)s	sa2)s	PROPN
ejpam-3576	77	29	]	]	X
ejpam-3576	77	30	for	for	ADP
ejpam-3576	77	31	every	every	DET
ejpam-3576	77	32	a	a	DET
ejpam-3576	77	33	∈	∈	NOUN
ejpam-3576	77	34	s.	s.	NOUN
ejpam-3576	77	35	we	we	PRON
ejpam-3576	77	36	denote	denote	VERB
ejpam-3576	77	37	by	by	ADP
ejpam-3576	77	38	l(a	l(a	PROPN
ejpam-3576	77	39	)	)	PUNCT
ejpam-3576	77	40	,	,	PUNCT
ejpam-3576	77	41	r(a	r(a	PROPN
ejpam-3576	77	42	)	)	PUNCT
ejpam-3576	77	43	,	,	PUNCT
ejpam-3576	77	44	i(a	i(a	PROPN
ejpam-3576	77	45	)	)	PUNCT
ejpam-3576	77	46	the	the	DET
ejpam-3576	77	47	left	left	ADJ
ejpam-3576	77	48	ideal	ideal	NOUN
ejpam-3576	77	49	,	,	PUNCT
ejpam-3576	77	50	the	the	DET
ejpam-3576	77	51	right	right	ADJ
ejpam-3576	77	52	ideal	ideal	NOUN
ejpam-3576	77	53	and	and	CCONJ
ejpam-3576	77	54	the	the	DET
ejpam-3576	77	55	ideal	ideal	NOUN
ejpam-3576	77	56	of	of	ADP
ejpam-3576	77	57	s	s	PROPN
ejpam-3576	77	58	,	,	PUNCT
ejpam-3576	77	59	respectively	respectively	ADV
ejpam-3576	77	60	generated	generate	VERB
ejpam-3576	77	61	by	by	ADP
ejpam-3576	77	62	a.	a.	NOUN
ejpam-3576	77	63	we	we	PRON
ejpam-3576	77	64	have	have	VERB
ejpam-3576	77	65	l(a	l(a	PROPN
ejpam-3576	77	66	)	)	PUNCT
ejpam-3576	77	67	=	=	PRON
ejpam-3576	78	1	{	{	PUNCT
ejpam-3576	78	2	s	s	X
ejpam-3576	78	3	∈	∈	NOUN
ejpam-3576	78	4	s	s	PART
ejpam-3576	78	5	:	:	PUNCT
ejpam-3576	78	6	s	s	VERB
ejpam-3576	78	7	≤	≤	NOUN
ejpam-3576	78	8	a	a	DET
ejpam-3576	78	9	or	or	CCONJ
ejpam-3576	78	10	s	s	PROPN
ejpam-3576	78	11	≤	≤	NUM
ejpam-3576	78	12	xa	xa	PROPN
ejpam-3576	78	13	for	for	ADP
ejpam-3576	78	14	some	some	DET
ejpam-3576	78	15	x	x	SYM
ejpam-3576	78	16	∈	∈	PROPN
ejpam-3576	78	17	s	s	PART
ejpam-3576	78	18	}	}	PUNCT
ejpam-3576	78	19	=	=	SYM
ejpam-3576	78	20	(	(	PUNCT
ejpam-3576	78	21	a	a	DET
ejpam-3576	78	22	∪	∪	X
ejpam-3576	78	23	sa	sa	NOUN
ejpam-3576	78	24	]	]	PUNCT
ejpam-3576	78	25	,	,	PUNCT
ejpam-3576	78	26	r(a	r(a	PROPN
ejpam-3576	78	27	)	)	PUNCT
ejpam-3576	78	28	=	=	SYM
ejpam-3576	79	1	(	(	PUNCT
ejpam-3576	79	2	a	a	DET
ejpam-3576	79	3	∪	∪	NOUN
ejpam-3576	79	4	as	as	ADP
ejpam-3576	79	5	]	]	X
ejpam-3576	79	6	,	,	PUNCT
ejpam-3576	79	7	i(a	i(a	PROPN
ejpam-3576	79	8	)	)	PUNCT
ejpam-3576	79	9	=	=	PUNCT
ejpam-3576	80	1	(	(	PUNCT
ejpam-3576	80	2	a	a	DET
ejpam-3576	80	3	∪	∪	NOUN
ejpam-3576	80	4	sa	sa	NOUN
ejpam-3576	80	5	∪	∪	NOUN
ejpam-3576	80	6	as	as	ADP
ejpam-3576	80	7	∪	∪	ADJ
ejpam-3576	80	8	(	(	PUNCT
ejpam-3576	80	9	sa)s	sa)s	NOUN
ejpam-3576	80	10	]	]	PUNCT
ejpam-3576	80	11	.	.	PUNCT
ejpam-3576	81	1	example	example	NOUN
ejpam-3576	82	1	2	2	NUM
ejpam-3576	82	2	.	.	PUNCT
ejpam-3576	82	3	let	let	VERB
ejpam-3576	82	4	s	s	VERB
ejpam-3576	82	5	=	=	X
ejpam-3576	82	6	{	{	PUNCT
ejpam-3576	82	7	a	a	PRON
ejpam-3576	82	8	,	,	PUNCT
ejpam-3576	82	9	b	b	NOUN
ejpam-3576	82	10	,	,	PUNCT
ejpam-3576	82	11	c	c	NOUN
ejpam-3576	82	12	,	,	PUNCT
ejpam-3576	82	13	d	d	NOUN
ejpam-3576	82	14	,	,	PUNCT
ejpam-3576	82	15	e	e	NOUN
ejpam-3576	82	16	}	}	PUNCT
ejpam-3576	82	17	.	.	PUNCT
ejpam-3576	83	1	define	define	VERB
ejpam-3576	83	2	multiplication	multiplication	NOUN
ejpam-3576	83	3	“	"	PUNCT
ejpam-3576	83	4	·	·	PUNCT
ejpam-3576	83	5	”	"	PUNCT
ejpam-3576	83	6	in	in	ADP
ejpam-3576	83	7	s	s	PRON
ejpam-3576	83	8	as	as	SCONJ
ejpam-3576	83	9	follows	follow	VERB
ejpam-3576	83	10	:	:	PUNCT
ejpam-3576	83	11	·	·	PUNCT
ejpam-3576	83	12	a	a	DET
ejpam-3576	83	13	b	b	X
ejpam-3576	83	14	c	c	NOUN
ejpam-3576	83	15	d	d	PROPN
ejpam-3576	83	16	e	e	X
ejpam-3576	83	17	a	a	PRON
ejpam-3576	83	18	a	a	DET
ejpam-3576	83	19	a	a	DET
ejpam-3576	83	20	a	a	DET
ejpam-3576	83	21	a	a	DET
ejpam-3576	83	22	a	a	DET
ejpam-3576	83	23	b	b	NOUN
ejpam-3576	83	24	a	a	DET
ejpam-3576	83	25	a	a	DET
ejpam-3576	83	26	a	a	NOUN
ejpam-3576	83	27	a	a	DET
ejpam-3576	83	28	a	a	DET
ejpam-3576	83	29	c	c	NOUN
ejpam-3576	83	30	a	a	PRON
ejpam-3576	83	31	a	a	DET
ejpam-3576	83	32	e	e	NOUN
ejpam-3576	83	33	c	c	NOUN
ejpam-3576	83	34	d	d	PROPN
ejpam-3576	83	35	d	d	X
ejpam-3576	83	36	a	a	PRON
ejpam-3576	83	37	a	a	PROPN
ejpam-3576	83	38	d	d	X
ejpam-3576	83	39	e	e	X
ejpam-3576	83	40	c	c	NOUN
ejpam-3576	83	41	e	e	X
ejpam-3576	83	42	a	a	DET
ejpam-3576	83	43	a	a	DET
ejpam-3576	83	44	c	c	NOUN
ejpam-3576	83	45	d	d	X
ejpam-3576	83	46	e	e	NOUN
ejpam-3576	83	47	and	and	CCONJ
ejpam-3576	83	48	≤	≤	NUM
ejpam-3576	83	49	:	:	PUNCT
ejpam-3576	84	1	=	=	SYM
ejpam-3576	84	2	{	{	PUNCT
ejpam-3576	84	3	(	(	PUNCT
ejpam-3576	84	4	a	a	PRON
ejpam-3576	84	5	,	,	PUNCT
ejpam-3576	84	6	a	a	NOUN
ejpam-3576	84	7	)	)	PUNCT
ejpam-3576	84	8	,	,	PUNCT
ejpam-3576	84	9	(	(	PUNCT
ejpam-3576	84	10	b	b	X
ejpam-3576	84	11	,	,	PUNCT
ejpam-3576	84	12	b	b	NOUN
ejpam-3576	84	13	)	)	PUNCT
ejpam-3576	84	14	,	,	PUNCT
ejpam-3576	84	15	(	(	PUNCT
ejpam-3576	84	16	c	c	X
ejpam-3576	84	17	,	,	PUNCT
ejpam-3576	84	18	c	c	NOUN
ejpam-3576	84	19	)	)	PUNCT
ejpam-3576	84	20	,	,	PUNCT
ejpam-3576	84	21	(	(	PUNCT
ejpam-3576	84	22	d	d	X
ejpam-3576	84	23	,	,	PUNCT
ejpam-3576	84	24	d	d	NOUN
ejpam-3576	84	25	)	)	PUNCT
ejpam-3576	84	26	,	,	PUNCT
ejpam-3576	84	27	(	(	PUNCT
ejpam-3576	84	28	e	e	NOUN
ejpam-3576	84	29	,	,	PUNCT
ejpam-3576	84	30	e	e	NOUN
ejpam-3576	84	31	)	)	PUNCT
ejpam-3576	84	32	}	}	PUNCT
ejpam-3576	84	33	.	.	PUNCT
ejpam-3576	85	1	then	then	ADV
ejpam-3576	85	2	s	s	VERB
ejpam-3576	85	3	is	be	AUX
ejpam-3576	85	4	an	an	DET
ejpam-3576	85	5	ordered	order	VERB
ejpam-3576	85	6	ag	ag	PROPN
ejpam-3576	85	7	-	-	NOUN
ejpam-3576	85	8	groupoid	groupoid	PROPN
ejpam-3576	85	9	.	.	PUNCT
ejpam-3576	86	1	a	a	PRON
ejpam-3576	86	2	=	=	X
ejpam-3576	86	3	{	{	PUNCT
ejpam-3576	86	4	c	c	NOUN
ejpam-3576	86	5	,	,	PUNCT
ejpam-3576	86	6	d	d	NOUN
ejpam-3576	86	7	,	,	PUNCT
ejpam-3576	86	8	e	e	NOUN
ejpam-3576	86	9	}	}	PUNCT
ejpam-3576	86	10	is	be	AUX
ejpam-3576	86	11	an	an	DET
ejpam-3576	86	12	ag	ag	NOUN
ejpam-3576	86	13	-	-	PUNCT
ejpam-3576	86	14	subgroupoid	subgroupoid	NOUN
ejpam-3576	86	15	of	of	ADP
ejpam-3576	86	16	s	s	PRON
ejpam-3576	86	17	and	and	CCONJ
ejpam-3576	86	18	i	i	PRON
ejpam-3576	86	19	=	=	PUNCT
ejpam-3576	86	20	{	{	PUNCT
ejpam-3576	86	21	a	a	X
ejpam-3576	86	22	,	,	PUNCT
ejpam-3576	86	23	c	c	NOUN
ejpam-3576	86	24	,	,	PUNCT
ejpam-3576	86	25	d	d	NOUN
ejpam-3576	86	26	,	,	PUNCT
ejpam-3576	86	27	e	e	NOUN
ejpam-3576	86	28	}	}	PUNCT
ejpam-3576	86	29	is	be	AUX
ejpam-3576	86	30	an	an	DET
ejpam-3576	86	31	ideal	ideal	NOUN
ejpam-3576	86	32	of	of	ADP
ejpam-3576	86	33	s.	s.	PROPN
ejpam-3576	86	34	k.	k.	PROPN
ejpam-3576	86	35	nasreen	nasreen	PROPN
ejpam-3576	86	36	,	,	PUNCT
ejpam-3576	86	37	m.	m.	NOUN
ejpam-3576	86	38	alesemi	alesemi	PROPN
ejpam-3576	86	39	,	,	PUNCT
ejpam-3576	86	40	salahuddin	salahuddin	VERB
ejpam-3576	86	41	/	/	SYM
ejpam-3576	86	42	eur	eur	PROPN
ejpam-3576	86	43	.	.	PUNCT
ejpam-3576	87	1	j.	j.	PROPN
ejpam-3576	87	2	pure	pure	PROPN
ejpam-3576	87	3	appl	appl	PROPN
ejpam-3576	87	4	.	.	PROPN
ejpam-3576	87	5	math	math	PROPN
ejpam-3576	87	6	,	,	PUNCT
ejpam-3576	87	7	13	13	NUM
ejpam-3576	87	8	(	(	PUNCT
ejpam-3576	87	9	1	1	NUM
ejpam-3576	87	10	)	)	PUNCT
ejpam-3576	87	11	(	(	PUNCT
ejpam-3576	87	12	2020	2020	NUM
ejpam-3576	87	13	)	)	PUNCT
ejpam-3576	87	14	,	,	PUNCT
ejpam-3576	87	15	113	113	NUM
ejpam-3576	87	16	-	-	SYM
ejpam-3576	87	17	129	129	NUM
ejpam-3576	87	18	116	116	NUM
ejpam-3576	87	19	remark	remark	NOUN
ejpam-3576	87	20	1	1	NUM
ejpam-3576	87	21	.	.	PUNCT
ejpam-3576	88	1	every	every	DET
ejpam-3576	88	2	ideal	ideal	NOUN
ejpam-3576	88	3	(	(	PUNCT
ejpam-3576	88	4	whether	whether	SCONJ
ejpam-3576	88	5	right	right	ADJ
ejpam-3576	88	6	,	,	PUNCT
ejpam-3576	88	7	left	left	ADJ
ejpam-3576	88	8	or	or	CCONJ
ejpam-3576	88	9	two	two	NUM
ejpam-3576	88	10	-	-	PUNCT
ejpam-3576	88	11	sided	sided	ADJ
ejpam-3576	88	12	)	)	PUNCT
ejpam-3576	88	13	is	be	AUX
ejpam-3576	88	14	an	an	DET
ejpam-3576	88	15	ag	ag	NOUN
ejpam-3576	88	16	-	-	PUNCT
ejpam-3576	88	17	subgroupoid	subgroupoid	NOUN
ejpam-3576	88	18	but	but	CCONJ
ejpam-3576	88	19	the	the	DET
ejpam-3576	88	20	converse	converse	NOUN
ejpam-3576	88	21	is	be	AUX
ejpam-3576	88	22	not	not	PART
ejpam-3576	88	23	true	true	ADJ
ejpam-3576	88	24	in	in	ADP
ejpam-3576	88	25	general	general	ADJ
ejpam-3576	88	26	.	.	PUNCT
ejpam-3576	89	1	an	an	DET
ejpam-3576	89	2	ordered	order	VERB
ejpam-3576	89	3	ag	ag	PROPN
ejpam-3576	89	4	-	-	NOUN
ejpam-3576	89	5	groupoid	groupoid	PROPN
ejpam-3576	89	6	s	s	X
ejpam-3576	89	7	is	be	AUX
ejpam-3576	89	8	to	to	PART
ejpam-3576	89	9	be	be	AUX
ejpam-3576	89	10	locally	locally	ADV
ejpam-3576	89	11	associative	associative	ADJ
ejpam-3576	89	12	,	,	PUNCT
ejpam-3576	89	13	if	if	SCONJ
ejpam-3576	89	14	(	(	PUNCT
ejpam-3576	89	15	a.a).a	a.a).a	PROPN
ejpam-3576	89	16	=	=	SYM
ejpam-3576	89	17	a.(a.a	a.(a.a	PROPN
ejpam-3576	89	18	)	)	PUNCT
ejpam-3576	89	19	for	for	ADP
ejpam-3576	89	20	every	every	DET
ejpam-3576	89	21	a	a	DET
ejpam-3576	89	22	∈	∈	PROPN
ejpam-3576	89	23	s.	s.	PROPN
ejpam-3576	89	24	example	example	NOUN
ejpam-3576	90	1	3	3	X
ejpam-3576	90	2	.	.	PUNCT
ejpam-3576	91	1	let	let	VERB
ejpam-3576	91	2	s	s	VERB
ejpam-3576	91	3	=	=	X
ejpam-3576	91	4	{	{	PUNCT
ejpam-3576	91	5	a	a	PRON
ejpam-3576	91	6	,	,	PUNCT
ejpam-3576	91	7	b	b	NOUN
ejpam-3576	91	8	,	,	PUNCT
ejpam-3576	91	9	c	c	NOUN
ejpam-3576	91	10	}	}	PUNCT
ejpam-3576	91	11	.	.	PUNCT
ejpam-3576	92	1	define	define	VERB
ejpam-3576	92	2	multiplication	multiplication	NOUN
ejpam-3576	92	3	“	"	PUNCT
ejpam-3576	92	4	·	·	PUNCT
ejpam-3576	92	5	”	"	PUNCT
ejpam-3576	92	6	in	in	ADP
ejpam-3576	92	7	s	s	PRON
ejpam-3576	92	8	as	as	SCONJ
ejpam-3576	92	9	follows	follow	VERB
ejpam-3576	92	10	:	:	PUNCT
ejpam-3576	92	11	·	·	PUNCT
ejpam-3576	92	12	a	a	DET
ejpam-3576	92	13	b	b	X
ejpam-3576	92	14	c	c	NOUN
ejpam-3576	92	15	a	a	DET
ejpam-3576	92	16	c	c	NOUN
ejpam-3576	92	17	c	c	PROPN
ejpam-3576	92	18	b	b	PROPN
ejpam-3576	92	19	b	b	PROPN
ejpam-3576	92	20	b	b	PROPN
ejpam-3576	92	21	b	b	PROPN
ejpam-3576	92	22	b	b	PROPN
ejpam-3576	92	23	c	c	PROPN
ejpam-3576	92	24	b	b	PROPN
ejpam-3576	92	25	b	b	PROPN
ejpam-3576	92	26	b	b	PROPN
ejpam-3576	92	27	and	and	CCONJ
ejpam-3576	92	28	≤	≤	NUM
ejpam-3576	92	29	:	:	PUNCT
ejpam-3576	93	1	=	=	SYM
ejpam-3576	93	2	{	{	PUNCT
ejpam-3576	93	3	(	(	PUNCT
ejpam-3576	93	4	a	a	PRON
ejpam-3576	93	5	,	,	PUNCT
ejpam-3576	93	6	a	a	NOUN
ejpam-3576	93	7	)	)	PUNCT
ejpam-3576	93	8	,	,	PUNCT
ejpam-3576	93	9	(	(	PUNCT
ejpam-3576	93	10	b	b	X
ejpam-3576	93	11	,	,	PUNCT
ejpam-3576	93	12	b	b	NOUN
ejpam-3576	93	13	)	)	PUNCT
ejpam-3576	93	14	,	,	PUNCT
ejpam-3576	93	15	(	(	PUNCT
ejpam-3576	93	16	c	c	X
ejpam-3576	93	17	,	,	PUNCT
ejpam-3576	93	18	c	c	NOUN
ejpam-3576	93	19	)	)	PUNCT
ejpam-3576	93	20	}	}	PUNCT
ejpam-3576	93	21	.	.	PUNCT
ejpam-3576	94	1	then	then	ADV
ejpam-3576	94	2	(	(	PUNCT
ejpam-3576	94	3	s	s	X
ejpam-3576	94	4	,	,	PUNCT
ejpam-3576	94	5	·	·	PUNCT
ejpam-3576	94	6	,	,	PUNCT
ejpam-3576	94	7	≤	≤	NUM
ejpam-3576	94	8	)	)	PUNCT
ejpam-3576	94	9	is	be	AUX
ejpam-3576	94	10	a	a	DET
ejpam-3576	94	11	locally	locally	ADV
ejpam-3576	94	12	associative	associative	ADJ
ejpam-3576	94	13	ordered	order	VERB
ejpam-3576	94	14	ag	ag	PROPN
ejpam-3576	94	15	-	-	NOUN
ejpam-3576	94	16	groupoid	groupoid	PROPN
ejpam-3576	94	17	.	.	PUNCT
ejpam-3576	95	1	in	in	ADP
ejpam-3576	95	2	a	a	DET
ejpam-3576	95	3	locally	locally	ADV
ejpam-3576	95	4	associative	associative	NOUN
ejpam-3576	95	5	ordered	order	VERB
ejpam-3576	95	6	ag	ag	PROPN
ejpam-3576	95	7	-	-	NOUN
ejpam-3576	95	8	groupoids	groupoid	NOUN
ejpam-3576	95	9	s	s	PART
ejpam-3576	95	10	,	,	PUNCT
ejpam-3576	95	11	we	we	PRON
ejpam-3576	95	12	define	define	VERB
ejpam-3576	95	13	powers	power	NOUN
ejpam-3576	95	14	of	of	ADP
ejpam-3576	95	15	an	an	DET
ejpam-3576	95	16	element	element	NOUN
ejpam-3576	95	17	as	as	ADP
ejpam-3576	95	18	follow	follow	NOUN
ejpam-3576	95	19	:	:	PUNCT
ejpam-3576	95	20	a1	a1	NOUN
ejpam-3576	95	21	=	=	SYM
ejpam-3576	95	22	a	a	NOUN
ejpam-3576	95	23	,	,	PUNCT
ejpam-3576	95	24	an+1	an+1	NOUN
ejpam-3576	95	25	=	=	SYM
ejpam-3576	95	26	ana	ana	PROPN
ejpam-3576	95	27	.	.	PUNCT
ejpam-3576	96	1	if	if	SCONJ
ejpam-3576	96	2	s	s	PROPN
ejpam-3576	96	3	has	have	VERB
ejpam-3576	96	4	a	a	DET
ejpam-3576	96	5	left	left	ADJ
ejpam-3576	96	6	identity	identity	NOUN
ejpam-3576	96	7	e	e	NOUN
ejpam-3576	96	8	,	,	PUNCT
ejpam-3576	96	9	we	we	PRON
ejpam-3576	96	10	define	define	VERB
ejpam-3576	96	11	a0	a0	NOUN
ejpam-3576	96	12	=	=	SYM
ejpam-3576	96	13	e	e	PROPN
ejpam-3576	96	14	,	,	PUNCT
ejpam-3576	96	15	as	as	SCONJ
ejpam-3576	96	16	left	leave	VERB
ejpam-3576	96	17	identity	identity	NOUN
ejpam-3576	96	18	is	be	AUX
ejpam-3576	96	19	unique	unique	ADJ
ejpam-3576	96	20	in	in	ADP
ejpam-3576	96	21	an	an	DET
ejpam-3576	96	22	ordered	order	VERB
ejpam-3576	96	23	ag	ag	PROPN
ejpam-3576	96	24	-	-	PUNCT
ejpam-3576	96	25	groupoid.a	groupoid.a	NOUN
ejpam-3576	96	26	locally	locally	ADV
ejpam-3576	96	27	associative	associative	PROPN
ejpam-3576	96	28	ordered	order	VERB
ejpam-3576	97	1	ag	ag	PROPN
ejpam-3576	97	2	-	-	PROPN
ejpam-3576	97	3	groupoid	groupoid	PROPN
ejpam-3576	97	4	s	s	PROPN
ejpam-3576	97	5	with	with	ADP
ejpam-3576	97	6	left	left	ADJ
ejpam-3576	97	7	identity	identity	NOUN
ejpam-3576	97	8	e	e	NOUN
ejpam-3576	97	9	has	have	VERB
ejpam-3576	97	10	associative	associative	ADJ
ejpam-3576	97	11	powers	power	NOUN
ejpam-3576	97	12	.	.	PUNCT
ejpam-3576	98	1	3	3	X
ejpam-3576	98	2	.	.	X
ejpam-3576	98	3	anti	anti	X
ejpam-3576	98	4	fuzzy	fuzzy	ADJ
ejpam-3576	98	5	interior	interior	ADJ
ejpam-3576	98	6	ideals	ideal	NOUN
ejpam-3576	98	7	on	on	ADP
ejpam-3576	98	8	ordered	order	VERB
ejpam-3576	98	9	ag	ag	PROPN
ejpam-3576	98	10	-	-	NOUN
ejpam-3576	98	11	groupoids	groupoid	NOUN
ejpam-3576	98	12	a	a	DET
ejpam-3576	98	13	fuzzy	fuzzy	ADJ
ejpam-3576	98	14	set	set	VERB
ejpam-3576	98	15	µ	µ	NOUN
ejpam-3576	98	16	on	on	ADP
ejpam-3576	98	17	a	a	DET
ejpam-3576	98	18	given	give	VERB
ejpam-3576	98	19	set	set	NOUN
ejpam-3576	98	20	x	x	PUNCT
ejpam-3576	98	21	is	be	AUX
ejpam-3576	98	22	described	describe	VERB
ejpam-3576	98	23	as	as	ADP
ejpam-3576	98	24	an	an	DET
ejpam-3576	98	25	arbitrary	arbitrary	ADJ
ejpam-3576	98	26	function	function	NOUN
ejpam-3576	98	27	µ	µ	NOUN
ejpam-3576	98	28	:	:	PUNCT
ejpam-3576	98	29	x	x	SYM
ejpam-3576	98	30	→	→	SYM
ejpam-3576	99	1	[	[	X
ejpam-3576	99	2	0	0	NUM
ejpam-3576	99	3	,	,	PUNCT
ejpam-3576	99	4	1	1	NUM
ejpam-3576	99	5	]	]	PUNCT
ejpam-3576	99	6	,	,	PUNCT
ejpam-3576	99	7	where	where	SCONJ
ejpam-3576	99	8	[	[	X
ejpam-3576	99	9	0	0	NUM
ejpam-3576	99	10	,	,	PUNCT
ejpam-3576	99	11	1	1	NUM
ejpam-3576	99	12	]	]	PUNCT
ejpam-3576	99	13	is	be	AUX
ejpam-3576	99	14	the	the	DET
ejpam-3576	99	15	unit	unit	NOUN
ejpam-3576	99	16	closed	close	VERB
ejpam-3576	99	17	interval	interval	NOUN
ejpam-3576	99	18	of	of	ADP
ejpam-3576	99	19	real	real	ADJ
ejpam-3576	99	20	numbers	number	NOUN
ejpam-3576	99	21	.	.	PUNCT
ejpam-3576	100	1	the	the	DET
ejpam-3576	100	2	fundamental	fundamental	ADJ
ejpam-3576	100	3	concept	concept	NOUN
ejpam-3576	100	4	of	of	ADP
ejpam-3576	100	5	a	a	DET
ejpam-3576	100	6	fuzzy	fuzzy	ADJ
ejpam-3576	100	7	set	set	NOUN
ejpam-3576	100	8	,	,	PUNCT
ejpam-3576	100	9	introduced	introduce	VERB
ejpam-3576	100	10	by	by	ADP
ejpam-3576	100	11	zadeh	zadeh	PROPN
ejpam-3576	100	12	in	in	ADP
ejpam-3576	100	13	his	his	PRON
ejpam-3576	100	14	classic	classic	ADJ
ejpam-3576	100	15	paper	paper	NOUN
ejpam-3576	100	16	[	[	X
ejpam-3576	100	17	33	33	NUM
ejpam-3576	100	18	]	]	SYM
ejpam-3576	100	19	1965	1965	NUM
ejpam-3576	100	20	,	,	PUNCT
ejpam-3576	100	21	which	which	PRON
ejpam-3576	100	22	gives	give	VERB
ejpam-3576	100	23	a	a	DET
ejpam-3576	100	24	natural	natural	ADJ
ejpam-3576	100	25	frame	frame	NOUN
ejpam-3576	100	26	work	work	NOUN
ejpam-3576	100	27	for	for	ADP
ejpam-3576	100	28	the	the	DET
ejpam-3576	100	29	generalizations	generalization	NOUN
ejpam-3576	100	30	of	of	ADP
ejpam-3576	100	31	some	some	DET
ejpam-3576	100	32	basic	basic	ADJ
ejpam-3576	100	33	notions	notion	NOUN
ejpam-3576	100	34	of	of	ADP
ejpam-3576	100	35	algebra	algebra	NOUN
ejpam-3576	100	36	,	,	PUNCT
ejpam-3576	100	37	for	for	ADP
ejpam-3576	100	38	example	example	NOUN
ejpam-3576	100	39	set	set	NOUN
ejpam-3576	100	40	(	(	PUNCT
ejpam-3576	100	41	resp	resp	PROPN
ejpam-3576	100	42	.	.	PUNCT
ejpam-3576	101	1	semigroup	semigroup	PROPN
ejpam-3576	101	2	,	,	PUNCT
ejpam-3576	101	3	group	group	NOUN
ejpam-3576	101	4	,	,	PUNCT
ejpam-3576	101	5	ring	ring	NOUN
ejpam-3576	101	6	,	,	PUNCT
ejpam-3576	101	7	near	near	ADP
ejpam-3576	101	8	-	-	PUNCT
ejpam-3576	101	9	ring	ring	NOUN
ejpam-3576	101	10	,	,	PUNCT
ejpam-3576	101	11	semiring	semiring	NOUN
ejpam-3576	101	12	)	)	PUNCT
ejpam-3576	101	13	theory	theory	NOUN
ejpam-3576	101	14	,	,	PUNCT
ejpam-3576	101	15	groupoids	groupoid	NOUN
ejpam-3576	101	16	,	,	PUNCT
ejpam-3576	101	17	real	real	ADJ
ejpam-3576	101	18	analysis	analysis	NOUN
ejpam-3576	101	19	,	,	PUNCT
ejpam-3576	101	20	topology	topology	NOUN
ejpam-3576	101	21	,	,	PUNCT
ejpam-3576	101	22	differential	differential	ADJ
ejpam-3576	101	23	equations	equation	NOUN
ejpam-3576	101	24	and	and	CCONJ
ejpam-3576	101	25	so	so	ADV
ejpam-3576	101	26	forth	forth	ADV
ejpam-3576	101	27	.	.	PUNCT
ejpam-3576	102	1	rosenfeld	rosenfeld	PROPN
ejpam-3576	103	1	[	[	X
ejpam-3576	103	2	29	29	NUM
ejpam-3576	103	3	]	]	PUNCT
ejpam-3576	103	4	,	,	PUNCT
ejpam-3576	103	5	introduced	introduce	VERB
ejpam-3576	103	6	the	the	DET
ejpam-3576	103	7	concept	concept	NOUN
ejpam-3576	103	8	of	of	ADP
ejpam-3576	103	9	fuzzy	fuzzy	ADJ
ejpam-3576	103	10	set	set	NOUN
ejpam-3576	103	11	in	in	ADP
ejpam-3576	103	12	groups	group	NOUN
ejpam-3576	103	13	.	.	PUNCT
ejpam-3576	104	1	the	the	DET
ejpam-3576	104	2	study	study	NOUN
ejpam-3576	104	3	of	of	ADP
ejpam-3576	104	4	fuzzy	fuzzy	ADJ
ejpam-3576	104	5	set	set	VERB
ejpam-3576	104	6	in	in	ADP
ejpam-3576	104	7	semigroups	semigroup	NOUN
ejpam-3576	104	8	investigated	investigate	VERB
ejpam-3576	104	9	by	by	ADP
ejpam-3576	104	10	kuroki	kuroki	PROPN
ejpam-3576	104	11	[	[	X
ejpam-3576	104	12	21–23	21–23	PROPN
ejpam-3576	104	13	]	]	X
ejpam-3576	104	14	.	.	PUNCT
ejpam-3576	105	1	he	he	PRON
ejpam-3576	105	2	studied	study	VERB
ejpam-3576	105	3	fuzzy	fuzzy	ADJ
ejpam-3576	105	4	(	(	PUNCT
ejpam-3576	105	5	interior	interior	NOUN
ejpam-3576	105	6	,	,	PUNCT
ejpam-3576	105	7	bi-	bi-	NUM
ejpam-3576	105	8	,	,	PUNCT
ejpam-3576	105	9	quasi-	quasi-	ADJ
ejpam-3576	105	10	,	,	PUNCT
ejpam-3576	105	11	semiprime	semiprime	NOUN
ejpam-3576	105	12	quasi	quasi	NOUN
ejpam-3576	105	13	)	)	PUNCT
ejpam-3576	105	14	ideals	ideal	NOUN
ejpam-3576	105	15	in	in	ADP
ejpam-3576	105	16	semigroups	semigroup	NOUN
ejpam-3576	105	17	.	.	PUNCT
ejpam-3576	106	1	dib	dib	NOUN
ejpam-3576	106	2	and	and	CCONJ
ejpam-3576	106	3	galham	galham	NOUN
ejpam-3576	106	4	in	in	ADP
ejpam-3576	106	5	[	[	X
ejpam-3576	106	6	4	4	NUM
ejpam-3576	106	7	]	]	PUNCT
ejpam-3576	106	8	,	,	PUNCT
ejpam-3576	106	9	examined	examine	VERB
ejpam-3576	106	10	the	the	DET
ejpam-3576	106	11	definition	definition	NOUN
ejpam-3576	106	12	of	of	ADP
ejpam-3576	106	13	fuzzy	fuzzy	ADJ
ejpam-3576	106	14	groupoid	groupoid	NOUN
ejpam-3576	106	15	(	(	PUNCT
ejpam-3576	106	16	resp	resp	NOUN
ejpam-3576	106	17	.	.	PUNCT
ejpam-3576	107	1	semigroup	semigroup	PROPN
ejpam-3576	107	2	)	)	PUNCT
ejpam-3576	107	3	.	.	PUNCT
ejpam-3576	108	1	they	they	PRON
ejpam-3576	108	2	studied	study	VERB
ejpam-3576	108	3	fuzzy	fuzzy	ADJ
ejpam-3576	108	4	ideals	ideal	NOUN
ejpam-3576	108	5	and	and	CCONJ
ejpam-3576	108	6	fuzzy	fuzzy	ADJ
ejpam-3576	108	7	bi	bi	NOUN
ejpam-3576	108	8	-	-	NOUN
ejpam-3576	108	9	ideals	ideal	NOUN
ejpam-3576	108	10	of	of	ADP
ejpam-3576	108	11	fuzzy	fuzzy	ADJ
ejpam-3576	108	12	semigroups	semigroup	NOUN
ejpam-3576	108	13	.	.	PUNCT
ejpam-3576	109	1	a	a	DET
ejpam-3576	109	2	systematic	systematic	ADJ
ejpam-3576	109	3	exposition	exposition	NOUN
ejpam-3576	109	4	of	of	ADP
ejpam-3576	109	5	fuzzy	fuzzy	ADJ
ejpam-3576	109	6	semigroups	semigroup	NOUN
ejpam-3576	109	7	by	by	ADP
ejpam-3576	109	8	mordeson	mordeson	NOUN
ejpam-3576	109	9	et	et	PROPN
ejpam-3576	109	10	al	al	PROPN
ejpam-3576	109	11	.	.	PROPN
ejpam-3576	109	12	appeared	appear	VERB
ejpam-3576	109	13	in	in	ADP
ejpam-3576	109	14	[	[	X
ejpam-3576	109	15	24	24	NUM
ejpam-3576	109	16	]	]	PUNCT
ejpam-3576	109	17	,	,	PUNCT
ejpam-3576	109	18	where	where	SCONJ
ejpam-3576	109	19	one	one	PRON
ejpam-3576	109	20	can	can	AUX
ejpam-3576	109	21	find	find	VERB
ejpam-3576	109	22	theoretical	theoretical	ADJ
ejpam-3576	109	23	results	result	NOUN
ejpam-3576	109	24	on	on	ADP
ejpam-3576	109	25	fuzzy	fuzzy	ADJ
ejpam-3576	109	26	semigroups	semigroup	NOUN
ejpam-3576	109	27	and	and	CCONJ
ejpam-3576	109	28	their	their	PRON
ejpam-3576	109	29	use	use	NOUN
ejpam-3576	109	30	in	in	ADP
ejpam-3576	109	31	fuzzy	fuzzy	ADJ
ejpam-3576	109	32	finite	finite	ADJ
ejpam-3576	109	33	state	state	NOUN
ejpam-3576	109	34	machines	machine	NOUN
ejpam-3576	109	35	and	and	CCONJ
ejpam-3576	109	36	fuzzy	fuzzy	ADJ
ejpam-3576	109	37	languages	language	NOUN
ejpam-3576	109	38	.	.	PUNCT
ejpam-3576	110	1	fuzzy	fuzzy	ADJ
ejpam-3576	110	2	sets	set	NOUN
ejpam-3576	110	3	in	in	ADP
ejpam-3576	110	4	ordered	order	VERB
ejpam-3576	110	5	semigroups	semigroup	NOUN
ejpam-3576	110	6	/	/	SYM
ejpam-3576	110	7	ordered	order	VERB
ejpam-3576	110	8	groupoids	groupoid	NOUN
ejpam-3576	110	9	established	establish	VERB
ejpam-3576	110	10	by	by	ADP
ejpam-3576	110	11	kehayopulu	kehayopulu	ADJ
ejpam-3576	110	12	and	and	CCONJ
ejpam-3576	110	13	tsingelis	tsingeli	NOUN
ejpam-3576	110	14	[	[	X
ejpam-3576	110	15	19	19	NUM
ejpam-3576	110	16	]	]	PUNCT
ejpam-3576	110	17	.	.	PUNCT
ejpam-3576	111	1	they	they	PRON
ejpam-3576	111	2	also	also	ADV
ejpam-3576	111	3	studied	study	VERB
ejpam-3576	111	4	fuzzy	fuzzy	ADJ
ejpam-3576	111	5	bi	bi	NOUN
ejpam-3576	111	6	-	-	NOUN
ejpam-3576	111	7	ideals	ideal	NOUN
ejpam-3576	111	8	and	and	CCONJ
ejpam-3576	111	9	fuzzy	fuzzy	ADJ
ejpam-3576	111	10	quasi	quasi	NOUN
ejpam-3576	111	11	-	-	NOUN
ejpam-3576	111	12	ideals	ideal	NOUN
ejpam-3576	111	13	in	in	ADP
ejpam-3576	111	14	ordered	order	VERB
ejpam-3576	111	15	semigroups	semigroup	NOUN
ejpam-3576	111	16	[	[	X
ejpam-3576	111	17	19	19	NUM
ejpam-3576	111	18	,	,	PUNCT
ejpam-3576	111	19	20	20	NUM
ejpam-3576	111	20	]	]	PUNCT
ejpam-3576	111	21	.	.	PUNCT
ejpam-3576	112	1	in	in	ADP
ejpam-3576	112	2	[	[	X
ejpam-3576	112	3	2	2	NUM
ejpam-3576	112	4	]	]	PUNCT
ejpam-3576	112	5	,	,	PUNCT
ejpam-3576	112	6	biswas	biswas	PROPN
ejpam-3576	112	7	introduced	introduce	VERB
ejpam-3576	112	8	the	the	DET
ejpam-3576	112	9	concept	concept	NOUN
ejpam-3576	112	10	of	of	ADP
ejpam-3576	112	11	anti	anti	ADJ
ejpam-3576	112	12	fuzzy	fuzzy	ADJ
ejpam-3576	112	13	subgroups	subgroup	NOUN
ejpam-3576	112	14	of	of	ADP
ejpam-3576	112	15	groups	group	NOUN
ejpam-3576	112	16	and	and	CCONJ
ejpam-3576	112	17	studied	study	VERB
ejpam-3576	112	18	the	the	DET
ejpam-3576	112	19	basic	basic	ADJ
ejpam-3576	112	20	properties	property	NOUN
ejpam-3576	112	21	of	of	ADP
ejpam-3576	112	22	groups	group	NOUN
ejpam-3576	112	23	in	in	ADP
ejpam-3576	112	24	terms	term	NOUN
ejpam-3576	112	25	of	of	ADP
ejpam-3576	112	26	anti	anti	ADJ
ejpam-3576	112	27	fuzzy	fuzzy	ADJ
ejpam-3576	112	28	subgroups	subgroup	NOUN
ejpam-3576	112	29	.	.	PUNCT
ejpam-3576	113	1	hong	hong	PROPN
ejpam-3576	113	2	and	and	CCONJ
ejpam-3576	113	3	jun	jun	PROPN
ejpam-3576	114	1	[	[	X
ejpam-3576	114	2	5	5	NUM
ejpam-3576	114	3	]	]	PUNCT
ejpam-3576	114	4	modified	modify	VERB
ejpam-3576	114	5	the	the	DET
ejpam-3576	114	6	biswas	biswas	PROPN
ejpam-3576	114	7	idea	idea	NOUN
ejpam-3576	114	8	and	and	CCONJ
ejpam-3576	114	9	applied	apply	VERB
ejpam-3576	114	10	it	it	PRON
ejpam-3576	114	11	into	into	ADP
ejpam-3576	114	12	bck	bck	NOUN
ejpam-3576	114	13	-	-	PUNCT
ejpam-3576	114	14	algebra	algebra	NOUN
ejpam-3576	114	15	.	.	PUNCT
ejpam-3576	115	1	akram	akram	PROPN
ejpam-3576	115	2	and	and	CCONJ
ejpam-3576	115	3	dar	dar	PROPN
ejpam-3576	115	4	defined	define	VERB
ejpam-3576	115	5	anti	anti	X
ejpam-3576	115	6	fuzzy	fuzzy	ADJ
ejpam-3576	115	7	left	leave	VERB
ejpam-3576	115	8	h	h	NOUN
ejpam-3576	115	9	-	-	PUNCT
ejpam-3576	115	10	ideals	ideal	NOUN
ejpam-3576	115	11	of	of	ADP
ejpam-3576	115	12	hemiring	hemire	VERB
ejpam-3576	115	13	and	and	CCONJ
ejpam-3576	115	14	discussed	discuss	VERB
ejpam-3576	115	15	the	the	DET
ejpam-3576	115	16	basic	basic	ADJ
ejpam-3576	115	17	properties	property	NOUN
ejpam-3576	115	18	of	of	ADP
ejpam-3576	115	19	hemiring	hemire	VERB
ejpam-3576	115	20	[	[	X
ejpam-3576	115	21	1	1	NUM
ejpam-3576	115	22	]	]	PUNCT
ejpam-3576	115	23	.	.	PUNCT
ejpam-3576	116	1	by	by	ADP
ejpam-3576	116	2	a	a	DET
ejpam-3576	116	3	fuzzy	fuzzy	ADJ
ejpam-3576	116	4	set	set	VERB
ejpam-3576	116	5	µ	µ	NOUN
ejpam-3576	116	6	of	of	ADP
ejpam-3576	116	7	an	an	DET
ejpam-3576	116	8	ordered	order	VERB
ejpam-3576	116	9	ag	ag	PROPN
ejpam-3576	116	10	-	-	PROPN
ejpam-3576	116	11	groupoid	groupoid	PROPN
ejpam-3576	116	12	s	s	NOUN
ejpam-3576	116	13	,	,	PUNCT
ejpam-3576	116	14	we	we	PRON
ejpam-3576	116	15	mean	mean	VERB
ejpam-3576	116	16	a	a	DET
ejpam-3576	116	17	function	function	NOUN
ejpam-3576	116	18	µ	µ	NOUN
ejpam-3576	116	19	:	:	PUNCT
ejpam-3576	116	20	s	s	X
ejpam-3576	116	21	→	→	SYM
ejpam-3576	116	22	[	[	X
ejpam-3576	116	23	0	0	NUM
ejpam-3576	116	24	,	,	PUNCT
ejpam-3576	116	25	1	1	NUM
ejpam-3576	116	26	]	]	PUNCT
ejpam-3576	116	27	and	and	CCONJ
ejpam-3576	116	28	the	the	DET
ejpam-3576	116	29	complement	complement	NOUN
ejpam-3576	116	30	of	of	ADP
ejpam-3576	116	31	µ	µ	NOUN
ejpam-3576	116	32	is	be	AUX
ejpam-3576	116	33	denoted	denote	VERB
ejpam-3576	116	34	by	by	ADP
ejpam-3576	116	35	µ′	µ′	PROPN
ejpam-3576	116	36	,	,	PUNCT
ejpam-3576	116	37	is	be	AUX
ejpam-3576	116	38	a	a	DET
ejpam-3576	116	39	fuzzy	fuzzy	ADJ
ejpam-3576	116	40	set	set	NOUN
ejpam-3576	116	41	in	in	ADP
ejpam-3576	116	42	s	s	PRON
ejpam-3576	116	43	given	give	VERB
ejpam-3576	116	44	by	by	ADP
ejpam-3576	116	45	µ′(x	µ′(x	SYM
ejpam-3576	116	46	)	)	PUNCT
ejpam-3576	116	47	=	=	SYM
ejpam-3576	116	48	1−	1−	NUM
ejpam-3576	116	49	µ(x	µ(x	NOUN
ejpam-3576	116	50	)	)	PUNCT
ejpam-3576	116	51	for	for	ADP
ejpam-3576	116	52	all	all	DET
ejpam-3576	116	53	x	x	SYM
ejpam-3576	116	54	∈	∈	PROPN
ejpam-3576	116	55	s.	s.	PROPN
ejpam-3576	116	56	k.	k.	PROPN
ejpam-3576	116	57	nasreen	nasreen	PROPN
ejpam-3576	116	58	,	,	PUNCT
ejpam-3576	116	59	m.	m.	NOUN
ejpam-3576	116	60	alesemi	alesemi	PROPN
ejpam-3576	116	61	,	,	PUNCT
ejpam-3576	116	62	salahuddin	salahuddin	VERB
ejpam-3576	116	63	/	/	SYM
ejpam-3576	116	64	eur	eur	PROPN
ejpam-3576	116	65	.	.	PUNCT
ejpam-3576	117	1	j.	j.	PROPN
ejpam-3576	117	2	pure	pure	PROPN
ejpam-3576	117	3	appl	appl	PROPN
ejpam-3576	117	4	.	.	PROPN
ejpam-3576	117	5	math	math	PROPN
ejpam-3576	117	6	,	,	PUNCT
ejpam-3576	117	7	13	13	NUM
ejpam-3576	117	8	(	(	PUNCT
ejpam-3576	117	9	1	1	NUM
ejpam-3576	117	10	)	)	PUNCT
ejpam-3576	117	11	(	(	PUNCT
ejpam-3576	117	12	2020	2020	NUM
ejpam-3576	117	13	)	)	PUNCT
ejpam-3576	117	14	,	,	PUNCT
ejpam-3576	117	15	113	113	NUM
ejpam-3576	117	16	-	-	SYM
ejpam-3576	117	17	129	129	NUM
ejpam-3576	117	18	117	117	NUM
ejpam-3576	117	19	a	a	DET
ejpam-3576	117	20	fuzzy	fuzzy	ADJ
ejpam-3576	117	21	set	set	VERB
ejpam-3576	117	22	µ	µ	NOUN
ejpam-3576	117	23	of	of	ADP
ejpam-3576	117	24	an	an	DET
ejpam-3576	117	25	ordered	order	VERB
ejpam-3576	117	26	ag	ag	PROPN
ejpam-3576	117	27	-	-	NOUN
ejpam-3576	117	28	groupoid	groupoid	PROPN
ejpam-3576	117	29	s	s	PART
ejpam-3576	117	30	is	be	AUX
ejpam-3576	117	31	an	an	DET
ejpam-3576	117	32	anti	anti	ADJ
ejpam-3576	117	33	fuzzy	fuzzy	ADJ
ejpam-3576	117	34	ag	ag	PROPN
ejpam-3576	117	35	-	-	NOUN
ejpam-3576	117	36	subgroupoid	subgroupoid	NOUN
ejpam-3576	117	37	of	of	ADP
ejpam-3576	117	38	s	s	PRON
ejpam-3576	117	39	if	if	SCONJ
ejpam-3576	117	40	µ(xy	µ(xy	NUM
ejpam-3576	117	41	)	)	PUNCT
ejpam-3576	117	42	≤	≤	NOUN
ejpam-3576	117	43	max{µ(x	max{µ(x	PROPN
ejpam-3576	117	44	)	)	PUNCT
ejpam-3576	117	45	,	,	PUNCT
ejpam-3576	117	46	µ(y	µ(y	PROPN
ejpam-3576	117	47	)	)	PUNCT
ejpam-3576	117	48	}	}	PUNCT
ejpam-3576	117	49	for	for	ADP
ejpam-3576	117	50	all	all	DET
ejpam-3576	117	51	x	x	NOUN
ejpam-3576	117	52	,	,	PUNCT
ejpam-3576	117	53	y	y	PROPN
ejpam-3576	117	54	∈	∈	PROPN
ejpam-3576	117	55	s.	s.	PROPN
ejpam-3576	117	56	µ	µ	X
ejpam-3576	117	57	is	be	AUX
ejpam-3576	117	58	an	an	DET
ejpam-3576	117	59	anti	anti	ADJ
ejpam-3576	117	60	fuzzy	fuzzy	ADJ
ejpam-3576	117	61	left	left	ADJ
ejpam-3576	117	62	(	(	PUNCT
ejpam-3576	117	63	resp	resp	NOUN
ejpam-3576	117	64	.	.	PUNCT
ejpam-3576	118	1	right	right	ADJ
ejpam-3576	118	2	)	)	PUNCT
ejpam-3576	118	3	ideal	ideal	NOUN
ejpam-3576	118	4	of	of	ADP
ejpam-3576	118	5	s	s	PROPN
ejpam-3576	118	6	,	,	PUNCT
ejpam-3576	118	7	if	if	SCONJ
ejpam-3576	118	8	(	(	PUNCT
ejpam-3576	118	9	1	1	NUM
ejpam-3576	118	10	)	)	PUNCT
ejpam-3576	118	11	µ(xy	µ(xy	PROPN
ejpam-3576	118	12	)	)	PUNCT
ejpam-3576	118	13	≤	≤	NOUN
ejpam-3576	118	14	µ(y	µ(y	PROPN
ejpam-3576	118	15	)	)	PUNCT
ejpam-3576	118	16	(	(	PUNCT
ejpam-3576	118	17	resp	resp	NOUN
ejpam-3576	118	18	.	.	PUNCT
ejpam-3576	119	1	µ(xy	µ(xy	PROPN
ejpam-3576	119	2	)	)	PUNCT
ejpam-3576	119	3	≤	≤	NOUN
ejpam-3576	120	1	µ(x	µ(x	NOUN
ejpam-3576	120	2	)	)	PUNCT
ejpam-3576	120	3	)	)	PUNCT
ejpam-3576	120	4	.	.	PUNCT
ejpam-3576	121	1	(	(	PUNCT
ejpam-3576	121	2	2	2	X
ejpam-3576	121	3	)	)	PUNCT
ejpam-3576	121	4	x	x	PUNCT
ejpam-3576	121	5	≤	≤	NOUN
ejpam-3576	121	6	y	y	PROPN
ejpam-3576	121	7	,	,	PUNCT
ejpam-3576	121	8	implies	imply	VERB
ejpam-3576	121	9	µ(x	µ(x	NOUN
ejpam-3576	121	10	)	)	PUNCT
ejpam-3576	121	11	≤	≤	NOUN
ejpam-3576	121	12	µ(y	µ(y	PROPN
ejpam-3576	121	13	)	)	PUNCT
ejpam-3576	121	14	for	for	ADP
ejpam-3576	121	15	all	all	DET
ejpam-3576	121	16	x	x	NOUN
ejpam-3576	121	17	,	,	PUNCT
ejpam-3576	121	18	y	y	PROPN
ejpam-3576	121	19	∈	∈	PROPN
ejpam-3576	121	20	s.	s.	PROPN
ejpam-3576	121	21	µ	µ	X
ejpam-3576	121	22	is	be	AUX
ejpam-3576	121	23	an	an	DET
ejpam-3576	121	24	anti	anti	ADJ
ejpam-3576	121	25	fuzzy	fuzzy	ADJ
ejpam-3576	121	26	ideal	ideal	NOUN
ejpam-3576	121	27	of	of	ADP
ejpam-3576	121	28	s	s	PROPN
ejpam-3576	121	29	,	,	PUNCT
ejpam-3576	121	30	if	if	SCONJ
ejpam-3576	121	31	µ	µ	PRON
ejpam-3576	121	32	is	be	AUX
ejpam-3576	121	33	both	both	PRON
ejpam-3576	121	34	an	an	DET
ejpam-3576	121	35	anti	anti	X
ejpam-3576	121	36	fuzzy	fuzzy	ADJ
ejpam-3576	121	37	left	leave	VERB
ejpam-3576	121	38	ideal	ideal	NOUN
ejpam-3576	121	39	and	and	CCONJ
ejpam-3576	121	40	an	an	DET
ejpam-3576	121	41	anti	anti	ADJ
ejpam-3576	121	42	fuzzy	fuzzy	ADJ
ejpam-3576	121	43	right	right	ADJ
ejpam-3576	121	44	ideal	ideal	NOUN
ejpam-3576	121	45	of	of	ADP
ejpam-3576	121	46	s.	s.	PROPN
ejpam-3576	121	47	equivalently	equivalently	PROPN
ejpam-3576	121	48	,	,	PUNCT
ejpam-3576	121	49	µ	µ	PRON
ejpam-3576	121	50	is	be	AUX
ejpam-3576	121	51	an	an	DET
ejpam-3576	121	52	anti	anti	ADJ
ejpam-3576	121	53	fuzzy	fuzzy	ADJ
ejpam-3576	121	54	ideal	ideal	NOUN
ejpam-3576	121	55	of	of	ADP
ejpam-3576	121	56	s	s	PRON
ejpam-3576	121	57	if	if	SCONJ
ejpam-3576	121	58	(	(	PUNCT
ejpam-3576	121	59	1	1	NUM
ejpam-3576	121	60	)	)	PUNCT
ejpam-3576	121	61	µ(xy	µ(xy	PROPN
ejpam-3576	121	62	)	)	PUNCT
ejpam-3576	121	63	≤	≤	NOUN
ejpam-3576	121	64	max{µ(x	max{µ(x	PROPN
ejpam-3576	121	65	)	)	PUNCT
ejpam-3576	121	66	,	,	PUNCT
ejpam-3576	121	67	µ(y	µ(y	PROPN
ejpam-3576	121	68	)	)	PUNCT
ejpam-3576	121	69	}	}	PUNCT
ejpam-3576	121	70	.	.	PUNCT
ejpam-3576	122	1	(	(	PUNCT
ejpam-3576	122	2	2	2	X
ejpam-3576	122	3	)	)	PUNCT
ejpam-3576	122	4	x	x	PUNCT
ejpam-3576	122	5	≤	≤	NOUN
ejpam-3576	122	6	y	y	PROPN
ejpam-3576	122	7	,	,	PUNCT
ejpam-3576	122	8	implies	imply	VERB
ejpam-3576	122	9	µ(x	µ(x	NOUN
ejpam-3576	122	10	)	)	PUNCT
ejpam-3576	122	11	≤	≤	NOUN
ejpam-3576	122	12	µ(y	µ(y	PROPN
ejpam-3576	122	13	)	)	PUNCT
ejpam-3576	122	14	for	for	ADP
ejpam-3576	122	15	all	all	DET
ejpam-3576	122	16	x	x	NOUN
ejpam-3576	122	17	,	,	PUNCT
ejpam-3576	122	18	y	y	PROPN
ejpam-3576	122	19	∈	∈	PROPN
ejpam-3576	122	20	s.	s.	PROPN
ejpam-3576	122	21	every	every	DET
ejpam-3576	122	22	anti	anti	X
ejpam-3576	122	23	fuzzy	fuzzy	ADJ
ejpam-3576	122	24	ideal	ideal	NOUN
ejpam-3576	122	25	(	(	PUNCT
ejpam-3576	122	26	whether	whether	SCONJ
ejpam-3576	122	27	left	leave	VERB
ejpam-3576	122	28	,	,	PUNCT
ejpam-3576	122	29	right	right	INTJ
ejpam-3576	122	30	,	,	PUNCT
ejpam-3576	122	31	two	two	NUM
ejpam-3576	122	32	-	-	PUNCT
ejpam-3576	122	33	sided	sided	ADJ
ejpam-3576	122	34	)	)	PUNCT
ejpam-3576	122	35	is	be	AUX
ejpam-3576	122	36	an	an	DET
ejpam-3576	122	37	anti	anti	ADJ
ejpam-3576	122	38	fuzzy	fuzzy	ADJ
ejpam-3576	122	39	ag	ag	NOUN
ejpam-3576	122	40	-	-	PUNCT
ejpam-3576	122	41	subgroupoid	subgroupoid	NOUN
ejpam-3576	122	42	but	but	CCONJ
ejpam-3576	122	43	the	the	DET
ejpam-3576	122	44	converse	converse	NOUN
ejpam-3576	122	45	is	be	AUX
ejpam-3576	122	46	not	not	PART
ejpam-3576	122	47	true	true	ADJ
ejpam-3576	122	48	in	in	ADP
ejpam-3576	122	49	general	general	ADJ
ejpam-3576	122	50	.	.	PUNCT
ejpam-3576	123	1	a	a	DET
ejpam-3576	123	2	fuzzy	fuzzy	ADJ
ejpam-3576	123	3	set	set	VERB
ejpam-3576	123	4	µ	µ	NOUN
ejpam-3576	123	5	of	of	ADP
ejpam-3576	123	6	s	s	PROPN
ejpam-3576	123	7	is	be	AUX
ejpam-3576	123	8	an	an	DET
ejpam-3576	123	9	anti	anti	ADJ
ejpam-3576	123	10	fuzzy	fuzzy	ADJ
ejpam-3576	123	11	interior	interior	ADJ
ejpam-3576	123	12	ideal	ideal	NOUN
ejpam-3576	123	13	of	of	ADP
ejpam-3576	123	14	s	s	SYM
ejpam-3576	123	15	,	,	PUNCT
ejpam-3576	123	16	if	if	SCONJ
ejpam-3576	123	17	(	(	PUNCT
ejpam-3576	123	18	1	1	X
ejpam-3576	123	19	)	)	PUNCT
ejpam-3576	123	20	µ((xa)y	µ((xa)y	NOUN
ejpam-3576	123	21	)	)	PUNCT
ejpam-3576	123	22	≤	≤	NOUN
ejpam-3576	123	23	µ(a	µ(a	PROPN
ejpam-3576	123	24	)	)	PUNCT
ejpam-3576	123	25	.	.	PUNCT
ejpam-3576	124	1	(	(	PUNCT
ejpam-3576	124	2	2	2	X
ejpam-3576	124	3	)	)	PUNCT
ejpam-3576	124	4	x	x	PUNCT
ejpam-3576	124	5	≤	≤	NOUN
ejpam-3576	124	6	y	y	PROPN
ejpam-3576	124	7	,	,	PUNCT
ejpam-3576	124	8	implies	imply	VERB
ejpam-3576	124	9	µ(x	µ(x	NOUN
ejpam-3576	124	10	)	)	PUNCT
ejpam-3576	124	11	≤	≤	NOUN
ejpam-3576	124	12	µ(y	µ(y	PROPN
ejpam-3576	124	13	)	)	PUNCT
ejpam-3576	124	14	for	for	ADP
ejpam-3576	124	15	all	all	DET
ejpam-3576	124	16	x	x	NOUN
ejpam-3576	124	17	,	,	PUNCT
ejpam-3576	124	18	a	a	DET
ejpam-3576	124	19	,	,	PUNCT
ejpam-3576	124	20	y	y	PROPN
ejpam-3576	124	21	∈	∈	PROPN
ejpam-3576	124	22	s.	s.	PROPN
ejpam-3576	124	23	we	we	PRON
ejpam-3576	124	24	denote	denote	VERB
ejpam-3576	124	25	by	by	ADP
ejpam-3576	124	26	f	f	PROPN
ejpam-3576	124	27	(	(	PUNCT
ejpam-3576	124	28	s	s	PROPN
ejpam-3576	124	29	)	)	PUNCT
ejpam-3576	124	30	,	,	PUNCT
ejpam-3576	124	31	the	the	DET
ejpam-3576	124	32	set	set	NOUN
ejpam-3576	124	33	of	of	ADP
ejpam-3576	124	34	all	all	DET
ejpam-3576	124	35	fuzzy	fuzzy	ADJ
ejpam-3576	124	36	subsets	subset	NOUN
ejpam-3576	124	37	of	of	ADP
ejpam-3576	124	38	s.	s.	PROPN
ejpam-3576	124	39	we	we	PRON
ejpam-3576	124	40	define	define	VERB
ejpam-3576	124	41	an	an	DET
ejpam-3576	124	42	order	order	NOUN
ejpam-3576	124	43	relation	relation	NOUN
ejpam-3576	124	44	”	"	PUNCT
ejpam-3576	124	45	⊆	⊆	NOUN
ejpam-3576	124	46	”	"	PUNCT
ejpam-3576	124	47	on	on	ADP
ejpam-3576	124	48	f	f	PROPN
ejpam-3576	124	49	(	(	PUNCT
ejpam-3576	124	50	s	s	NOUN
ejpam-3576	124	51	)	)	PUNCT
ejpam-3576	125	1	such	such	ADJ
ejpam-3576	125	2	that	that	SCONJ
ejpam-3576	125	3	f	f	PROPN
ejpam-3576	125	4	⊆	⊆	NUM
ejpam-3576	125	5	g	g	NOUN
ejpam-3576	125	6	if	if	SCONJ
ejpam-3576	125	7	and	and	CCONJ
ejpam-3576	125	8	only	only	ADV
ejpam-3576	125	9	if	if	SCONJ
ejpam-3576	125	10	f(x	f(x	PROPN
ejpam-3576	125	11	)	)	PUNCT
ejpam-3576	125	12	≤	≤	PUNCT
ejpam-3576	125	13	g(x	g(x	NOUN
ejpam-3576	125	14	)	)	PUNCT
ejpam-3576	125	15	for	for	ADP
ejpam-3576	125	16	all	all	DET
ejpam-3576	125	17	x	x	SYM
ejpam-3576	125	18	∈	∈	PROPN
ejpam-3576	125	19	s.	s.	PROPN
ejpam-3576	125	20	then	then	ADV
ejpam-3576	125	21	(	(	PUNCT
ejpam-3576	125	22	f	f	X
ejpam-3576	125	23	(	(	PUNCT
ejpam-3576	125	24	s	s	NOUN
ejpam-3576	125	25	)	)	PUNCT
ejpam-3576	125	26	,	,	PUNCT
ejpam-3576	125	27	◦	◦	NOUN
ejpam-3576	125	28	,	,	PUNCT
ejpam-3576	125	29	⊆	⊆	NUM
ejpam-3576	125	30	)	)	PUNCT
ejpam-3576	125	31	is	be	AUX
ejpam-3576	125	32	an	an	DET
ejpam-3576	125	33	ordered	order	VERB
ejpam-3576	125	34	ag	ag	PROPN
ejpam-3576	125	35	-	-	NOUN
ejpam-3576	125	36	groupoid	groupoid	PROPN
ejpam-3576	125	37	.	.	PUNCT
ejpam-3576	126	1	for	for	ADP
ejpam-3576	126	2	f	f	PROPN
ejpam-3576	126	3	∧	∧	PROPN
ejpam-3576	126	4	g	g	PROPN
ejpam-3576	126	5	and	and	CCONJ
ejpam-3576	126	6	f	f	PROPN
ejpam-3576	126	7	∨	∨	NUM
ejpam-3576	126	8	g	g	PROPN
ejpam-3576	126	9	,	,	PUNCT
ejpam-3576	126	10	we	we	PRON
ejpam-3576	126	11	define	define	VERB
ejpam-3576	126	12	(	(	PUNCT
ejpam-3576	126	13	f	f	PROPN
ejpam-3576	126	14	∧	∧	PROPN
ejpam-3576	126	15	g)(x	g)(x	PROPN
ejpam-3576	126	16	)	)	PUNCT
ejpam-3576	126	17	=	=	SYM
ejpam-3576	126	18	min{f(x	min{f(x	PROPN
ejpam-3576	126	19	)	)	PUNCT
ejpam-3576	126	20	,	,	PUNCT
ejpam-3576	126	21	g(x	g(x	NOUN
ejpam-3576	126	22	)	)	PUNCT
ejpam-3576	126	23	}	}	PUNCT
ejpam-3576	126	24	and	and	CCONJ
ejpam-3576	126	25	(	(	PUNCT
ejpam-3576	126	26	f	f	PROPN
ejpam-3576	126	27	∨	∨	PROPN
ejpam-3576	126	28	g)(x	g)(x	PROPN
ejpam-3576	126	29	)	)	PUNCT
ejpam-3576	126	30	=	=	SYM
ejpam-3576	127	1	max{f(x	max{f(x	PROPN
ejpam-3576	127	2	)	)	PUNCT
ejpam-3576	127	3	,	,	PUNCT
ejpam-3576	127	4	g(x	g(x	NOUN
ejpam-3576	127	5	)	)	PUNCT
ejpam-3576	127	6	}	}	PUNCT
ejpam-3576	127	7	.	.	PUNCT
ejpam-3576	128	1	for	for	ADP
ejpam-3576	128	2	a	a	DET
ejpam-3576	128	3	∈	∈	PROPN
ejpam-3576	128	4	s	s	NOUN
ejpam-3576	128	5	,	,	PUNCT
ejpam-3576	128	6	we	we	PRON
ejpam-3576	128	7	define	define	VERB
ejpam-3576	128	8	aa	aa	NOUN
ejpam-3576	128	9	=	=	PUNCT
ejpam-3576	128	10	{	{	PUNCT
ejpam-3576	128	11	(	(	PUNCT
ejpam-3576	128	12	y	y	PROPN
ejpam-3576	128	13	,	,	PUNCT
ejpam-3576	128	14	z	z	NOUN
ejpam-3576	128	15	)	)	PUNCT
ejpam-3576	128	16	∈	∈	PROPN
ejpam-3576	128	17	s	s	PART
ejpam-3576	128	18	×	×	NOUN
ejpam-3576	128	19	s	s	X
ejpam-3576	128	20	|	|	ADV
ejpam-3576	128	21	a	a	DET
ejpam-3576	128	22	≤	≤	NUM
ejpam-3576	128	23	yz	yz	NOUN
ejpam-3576	128	24	}	}	PUNCT
ejpam-3576	128	25	.	.	PUNCT
ejpam-3576	129	1	let	let	VERB
ejpam-3576	129	2	f	f	PROPN
ejpam-3576	129	3	and	and	CCONJ
ejpam-3576	129	4	g	g	PROPN
ejpam-3576	129	5	be	be	VERB
ejpam-3576	129	6	fuzzy	fuzzy	ADJ
ejpam-3576	129	7	subsets	subset	NOUN
ejpam-3576	129	8	of	of	ADP
ejpam-3576	129	9	s	s	PROPN
ejpam-3576	129	10	,	,	PUNCT
ejpam-3576	129	11	the	the	DET
ejpam-3576	129	12	product	product	NOUN
ejpam-3576	129	13	f	f	NOUN
ejpam-3576	129	14	◦	◦	NOUN
ejpam-3576	129	15	g	g	NOUN
ejpam-3576	129	16	of	of	ADP
ejpam-3576	129	17	f	f	PROPN
ejpam-3576	129	18	and	and	CCONJ
ejpam-3576	129	19	g	g	PROPN
ejpam-3576	129	20	is	be	AUX
ejpam-3576	129	21	defined	define	VERB
ejpam-3576	129	22	by	by	ADP
ejpam-3576	129	23	:	:	PUNCT
ejpam-3576	129	24	(	(	PUNCT
ejpam-3576	129	25	f	f	X
ejpam-3576	129	26	◦	◦	NOUN
ejpam-3576	129	27	g)(a	g)(a	PROPN
ejpam-3576	129	28	)	)	PUNCT
ejpam-3576	130	1	=	=	PRON
ejpam-3576	130	2	{	{	PUNCT
ejpam-3576	130	3	∧(y	∧(y	PROPN
ejpam-3576	130	4	,	,	PUNCT
ejpam-3576	130	5	z)∈aa	z)∈aa	PROPN
ejpam-3576	130	6	max{f(y	max{f(y	PROPN
ejpam-3576	130	7	)	)	PUNCT
ejpam-3576	130	8	,	,	PUNCT
ejpam-3576	130	9	g(z	g(z	PROPN
ejpam-3576	130	10	)	)	PUNCT
ejpam-3576	130	11	}	}	PUNCT
ejpam-3576	130	12	if	if	SCONJ
ejpam-3576	130	13	aa	aa	NOUN
ejpam-3576	130	14	6=	6=	PUNCT
ejpam-3576	130	15	∅	∅	NOUN
ejpam-3576	130	16	0	0	NUM
ejpam-3576	131	1	if	if	SCONJ
ejpam-3576	131	2	aa	aa	NOUN
ejpam-3576	131	3	=	=	NOUN
ejpam-3576	131	4	∅	∅	NOUN
ejpam-3576	131	5	for	for	ADP
ejpam-3576	131	6	a	a	DET
ejpam-3576	131	7	non	non	ADJ
ejpam-3576	131	8	-	-	ADJ
ejpam-3576	131	9	empty	empty	ADJ
ejpam-3576	131	10	family	family	NOUN
ejpam-3576	131	11	of	of	ADP
ejpam-3576	131	12	fuzzy	fuzzy	ADJ
ejpam-3576	131	13	subsets	subset	NOUN
ejpam-3576	131	14	{	{	PUNCT
ejpam-3576	131	15	fi}i∈i	fi}i∈i	INTJ
ejpam-3576	131	16	,	,	PUNCT
ejpam-3576	131	17	of	of	ADP
ejpam-3576	131	18	s	s	PRON
ejpam-3576	131	19	,	,	PUNCT
ejpam-3576	131	20	the	the	DET
ejpam-3576	131	21	fuzzy	fuzzy	ADJ
ejpam-3576	131	22	subsets	subset	NOUN
ejpam-3576	131	23	∨i∈ifi	∨i∈ifi	PUNCT
ejpam-3576	131	24	and	and	CCONJ
ejpam-3576	131	25	∧i∈ifi	∧i∈ifi	PROPN
ejpam-3576	131	26	of	of	ADP
ejpam-3576	131	27	s	s	PRON
ejpam-3576	132	1	are	be	AUX
ejpam-3576	132	2	defined	define	VERB
ejpam-3576	132	3	as	as	SCONJ
ejpam-3576	132	4	follows	follow	VERB
ejpam-3576	132	5	:	:	PUNCT
ejpam-3576	132	6	(	(	PUNCT
ejpam-3576	132	7	∨i∈ifi)(a	∨i∈ifi)(a	NOUN
ejpam-3576	132	8	)	)	PUNCT
ejpam-3576	132	9	:	:	PUNCT
ejpam-3576	133	1	=	=	PUNCT
ejpam-3576	133	2	sup	sup	NOUN
ejpam-3576	133	3	i∈i	i∈i	ADJ
ejpam-3576	133	4	{	{	PUNCT
ejpam-3576	133	5	fi(a	fi(a	NUM
ejpam-3576	133	6	)	)	PUNCT
ejpam-3576	133	7	}	}	PUNCT
ejpam-3576	133	8	and	and	CCONJ
ejpam-3576	133	9	(	(	PUNCT
ejpam-3576	133	10	∧i∈ifi)(a	∧i∈ifi)(a	PROPN
ejpam-3576	133	11	)	)	PUNCT
ejpam-3576	133	12	:	:	PUNCT
ejpam-3576	134	1	=	=	PUNCT
ejpam-3576	134	2	inf	inf	PROPN
ejpam-3576	134	3	i∈i	i∈i	ADJ
ejpam-3576	134	4	{	{	PUNCT
ejpam-3576	134	5	fi(a	fi(a	NUM
ejpam-3576	134	6	)	)	PUNCT
ejpam-3576	134	7	}	}	PUNCT
ejpam-3576	134	8	.	.	PUNCT
ejpam-3576	135	1	if	if	SCONJ
ejpam-3576	135	2	i	i	PRON
ejpam-3576	135	3	is	be	AUX
ejpam-3576	135	4	a	a	DET
ejpam-3576	135	5	finite	finite	ADJ
ejpam-3576	135	6	set	set	NOUN
ejpam-3576	135	7	,	,	PUNCT
ejpam-3576	135	8	say	say	VERB
ejpam-3576	135	9	i	i	PRON
ejpam-3576	135	10	=	=	PUNCT
ejpam-3576	135	11	{	{	PUNCT
ejpam-3576	135	12	1	1	NUM
ejpam-3576	135	13	,	,	PUNCT
ejpam-3576	135	14	2	2	NUM
ejpam-3576	135	15	,	,	PUNCT
ejpam-3576	135	16	...	...	PUNCT
ejpam-3576	135	17	n	n	CCONJ
ejpam-3576	135	18	}	}	PUNCT
ejpam-3576	135	19	,	,	PUNCT
ejpam-3576	135	20	then	then	ADV
ejpam-3576	135	21	clearly	clearly	ADV
ejpam-3576	135	22	,	,	PUNCT
ejpam-3576	135	23	∨i∈ifi(a	∨i∈ifi(a	PROPN
ejpam-3576	135	24	)	)	PUNCT
ejpam-3576	135	25	=	=	SYM
ejpam-3576	135	26	max{f1(a	max{f1(a	PROPN
ejpam-3576	135	27	)	)	PUNCT
ejpam-3576	135	28	,	,	PUNCT
ejpam-3576	135	29	f2(a	f2(a	NOUN
ejpam-3576	135	30	)	)	PUNCT
ejpam-3576	135	31	,	,	PUNCT
ejpam-3576	135	32	...	...	PUNCT
ejpam-3576	135	33	,	,	PUNCT
ejpam-3576	135	34	fn(a	fn(a	NOUN
ejpam-3576	135	35	)	)	PUNCT
ejpam-3576	135	36	}	}	PUNCT
ejpam-3576	135	37	and	and	CCONJ
ejpam-3576	135	38	∧i∈i	∧i∈i	NOUN
ejpam-3576	135	39	fi(a	fi(a	NOUN
ejpam-3576	135	40	)	)	PUNCT
ejpam-3576	135	41	=	=	SYM
ejpam-3576	135	42	min{f1(a	min{f1(a	PROPN
ejpam-3576	135	43	)	)	PUNCT
ejpam-3576	135	44	,	,	PUNCT
ejpam-3576	135	45	f2(a	f2(a	NOUN
ejpam-3576	135	46	)	)	PUNCT
ejpam-3576	135	47	,	,	PUNCT
ejpam-3576	135	48	...	...	PUNCT
ejpam-3576	135	49	,	,	PUNCT
ejpam-3576	135	50	fn(a	fn(a	NOUN
ejpam-3576	135	51	)	)	PUNCT
ejpam-3576	135	52	}	}	PUNCT
ejpam-3576	135	53	.	.	PUNCT
ejpam-3576	136	1	for	for	ADP
ejpam-3576	136	2	s	s	SYM
ejpam-3576	136	3	,	,	PUNCT
ejpam-3576	136	4	the	the	DET
ejpam-3576	136	5	fuzzy	fuzzy	ADJ
ejpam-3576	136	6	subsets	subset	NOUN
ejpam-3576	136	7	“	"	PUNCT
ejpam-3576	136	8	0	0	NUM
ejpam-3576	136	9	”	"	PUNCT
ejpam-3576	136	10	and	and	CCONJ
ejpam-3576	136	11	“	"	PUNCT
ejpam-3576	136	12	1	1	NUM
ejpam-3576	136	13	”	"	PUNCT
ejpam-3576	136	14	are	be	AUX
ejpam-3576	136	15	defined	define	VERB
ejpam-3576	136	16	as	as	ADP
ejpam-3576	136	17	0(x	0(x	NOUN
ejpam-3576	136	18	)	)	PUNCT
ejpam-3576	136	19	:	:	PUNCT
ejpam-3576	137	1	=	=	SYM
ejpam-3576	137	2	0	0	NUM
ejpam-3576	137	3	and	and	CCONJ
ejpam-3576	137	4	1(x	1(x	NUM
ejpam-3576	137	5	)	)	PUNCT
ejpam-3576	137	6	:	:	PUNCT
ejpam-3576	138	1	=	=	NOUN
ejpam-3576	138	2	1	1	NUM
ejpam-3576	138	3	.	.	NUM
ejpam-3576	138	4	0	0	NUM
ejpam-3576	138	5	:	:	PUNCT
ejpam-3576	138	6	s	s	X
ejpam-3576	138	7	→	→	SYM
ejpam-3576	138	8	[	[	X
ejpam-3576	138	9	0	0	NUM
ejpam-3576	138	10	,	,	PUNCT
ejpam-3576	138	11	1	1	NUM
ejpam-3576	138	12	]	]	PUNCT
ejpam-3576	138	13	,	,	PUNCT
ejpam-3576	138	14	x	x	SYM
ejpam-3576	138	15	7→	7→	NUM
ejpam-3576	138	16	0(x	0(x	NOUN
ejpam-3576	138	17	)	)	PUNCT
ejpam-3576	138	18	:	:	PUNCT
ejpam-3576	139	1	=	=	PUNCT
ejpam-3576	139	2	0	0	X
ejpam-3576	139	3	.	.	NOUN
ejpam-3576	139	4	1	1	NUM
ejpam-3576	139	5	:	:	PUNCT
ejpam-3576	139	6	s	s	X
ejpam-3576	139	7	→	→	SYM
ejpam-3576	139	8	[	[	X
ejpam-3576	139	9	0	0	NUM
ejpam-3576	139	10	,	,	PUNCT
ejpam-3576	139	11	1	1	NUM
ejpam-3576	139	12	]	]	PUNCT
ejpam-3576	139	13	,	,	PUNCT
ejpam-3576	139	14	x	x	SYM
ejpam-3576	139	15	7→	7→	NUM
ejpam-3576	139	16	1(x	1(x	NUM
ejpam-3576	139	17	)	)	PUNCT
ejpam-3576	139	18	:	:	PUNCT
ejpam-3576	140	1	=	=	NOUN
ejpam-3576	140	2	1	1	X
ejpam-3576	140	3	.	.	PUNCT
ejpam-3576	140	4	clearly	clearly	ADV
ejpam-3576	140	5	,	,	PUNCT
ejpam-3576	140	6	the	the	DET
ejpam-3576	140	7	fuzzy	fuzzy	ADJ
ejpam-3576	140	8	subset	subset	NOUN
ejpam-3576	140	9	“	"	PUNCT
ejpam-3576	140	10	0	0	NUM
ejpam-3576	140	11	”	"	PUNCT
ejpam-3576	140	12	(	(	PUNCT
ejpam-3576	140	13	resp	resp	NOUN
ejpam-3576	140	14	.	.	PUNCT
ejpam-3576	141	1	“1	“1	NOUN
ejpam-3576	141	2	”	"	PUNCT
ejpam-3576	141	3	)	)	PUNCT
ejpam-3576	141	4	of	of	ADP
ejpam-3576	141	5	s	s	PROPN
ejpam-3576	141	6	is	be	AUX
ejpam-3576	141	7	the	the	DET
ejpam-3576	141	8	least	least	ADJ
ejpam-3576	141	9	(	(	PUNCT
ejpam-3576	141	10	resp	resp	NOUN
ejpam-3576	141	11	.	.	PUNCT
ejpam-3576	142	1	the	the	DET
ejpam-3576	142	2	greatest	great	ADJ
ejpam-3576	142	3	)	)	PUNCT
ejpam-3576	142	4	element	element	NOUN
ejpam-3576	142	5	of	of	ADP
ejpam-3576	142	6	the	the	DET
ejpam-3576	142	7	ordered	order	VERB
ejpam-3576	142	8	set	set	NOUN
ejpam-3576	142	9	(	(	PUNCT
ejpam-3576	142	10	f	f	PROPN
ejpam-3576	142	11	(	(	PUNCT
ejpam-3576	142	12	s),≤	s),≤	PROPN
ejpam-3576	142	13	)	)	PUNCT
ejpam-3576	142	14	.	.	PUNCT
ejpam-3576	143	1	the	the	DET
ejpam-3576	143	2	fuzzy	fuzzy	ADJ
ejpam-3576	143	3	subset	subset	VERB
ejpam-3576	143	4	“	"	PUNCT
ejpam-3576	143	5	0	0	NUM
ejpam-3576	143	6	”	"	PUNCT
ejpam-3576	143	7	is	be	AUX
ejpam-3576	143	8	the	the	DET
ejpam-3576	143	9	zero	zero	NUM
ejpam-3576	143	10	element	element	NOUN
ejpam-3576	143	11	of	of	ADP
ejpam-3576	143	12	(	(	PUNCT
ejpam-3576	143	13	f	f	PROPN
ejpam-3576	143	14	(	(	PUNCT
ejpam-3576	143	15	s	s	NOUN
ejpam-3576	143	16	)	)	PUNCT
ejpam-3576	143	17	,	,	PUNCT
ejpam-3576	143	18	◦	◦	NOUN
ejpam-3576	143	19	,	,	PUNCT
ejpam-3576	143	20	≤	≤	NUM
ejpam-3576	143	21	)	)	PUNCT
ejpam-3576	143	22	(	(	PUNCT
ejpam-3576	143	23	that	that	PRON
ejpam-3576	143	24	is	be	AUX
ejpam-3576	143	25	,	,	PUNCT
ejpam-3576	143	26	f	f	PROPN
ejpam-3576	143	27	◦	◦	NOUN
ejpam-3576	143	28	0	0	NUM
ejpam-3576	144	1	=	=	SYM
ejpam-3576	144	2	0	0	NUM
ejpam-3576	144	3	◦	◦	NOUN
ejpam-3576	144	4	f	f	NOUN
ejpam-3576	144	5	=	=	SYM
ejpam-3576	144	6	0	0	NUM
ejpam-3576	144	7	and	and	CCONJ
ejpam-3576	144	8	0	0	NUM
ejpam-3576	144	9	≤	≤	NUM
ejpam-3576	144	10	f	f	NOUN
ejpam-3576	145	1	for	for	ADP
ejpam-3576	145	2	every	every	DET
ejpam-3576	145	3	f	f	PROPN
ejpam-3576	145	4	∈	∈	PROPN
ejpam-3576	145	5	f	f	X
ejpam-3576	145	6	(	(	PUNCT
ejpam-3576	145	7	s	s	NOUN
ejpam-3576	145	8	)	)	PUNCT
ejpam-3576	145	9	)	)	PUNCT
ejpam-3576	145	10	.	.	PUNCT
ejpam-3576	146	1	for	for	ADP
ejpam-3576	146	2	∅	∅	NOUN
ejpam-3576	146	3	6=	6=	ADP
ejpam-3576	146	4	a	a	DET
ejpam-3576	146	5	⊆	⊆	NUM
ejpam-3576	146	6	s	s	NOUN
ejpam-3576	146	7	,	,	PUNCT
ejpam-3576	146	8	the	the	DET
ejpam-3576	146	9	anti	anti	ADJ
ejpam-3576	146	10	characteristic	characteristic	ADJ
ejpam-3576	146	11	function	function	NOUN
ejpam-3576	146	12	of	of	ADP
ejpam-3576	146	13	a	a	PRON
ejpam-3576	146	14	is	be	AUX
ejpam-3576	146	15	denoted	denote	VERB
ejpam-3576	146	16	by	by	ADP
ejpam-3576	146	17	χc	χc	PROPN
ejpam-3576	146	18	a	a	PROPN
ejpam-3576	146	19	and	and	CCONJ
ejpam-3576	146	20	defined	define	VERB
ejpam-3576	146	21	as	as	ADP
ejpam-3576	146	22	χc	χc	PRON
ejpam-3576	146	23	a(a	a(a	PROPN
ejpam-3576	146	24	)	)	PUNCT
ejpam-3576	147	1	=	=	PRON
ejpam-3576	147	2	{	{	PUNCT
ejpam-3576	147	3	0	0	NUM
ejpam-3576	147	4	if	if	SCONJ
ejpam-3576	147	5	a	a	DET
ejpam-3576	147	6	∈	∈	PROPN
ejpam-3576	147	7	a	a	DET
ejpam-3576	147	8	1	1	NUM
ejpam-3576	147	9	if	if	SCONJ
ejpam-3576	147	10	a	a	PRON
ejpam-3576	147	11	/∈	/∈	NOUN
ejpam-3576	147	12	a	a	DET
ejpam-3576	147	13	k.	k.	NOUN
ejpam-3576	147	14	nasreen	nasreen	PROPN
ejpam-3576	147	15	,	,	PUNCT
ejpam-3576	147	16	m.	m.	NOUN
ejpam-3576	147	17	alesemi	alesemi	PROPN
ejpam-3576	147	18	,	,	PUNCT
ejpam-3576	147	19	salahuddin	salahuddin	VERB
ejpam-3576	147	20	/	/	SYM
ejpam-3576	147	21	eur	eur	PROPN
ejpam-3576	147	22	.	.	PUNCT
ejpam-3576	148	1	j.	j.	PROPN
ejpam-3576	148	2	pure	pure	PROPN
ejpam-3576	148	3	appl	appl	PROPN
ejpam-3576	148	4	.	.	PROPN
ejpam-3576	148	5	math	math	PROPN
ejpam-3576	148	6	,	,	PUNCT
ejpam-3576	148	7	13	13	NUM
ejpam-3576	148	8	(	(	PUNCT
ejpam-3576	148	9	1	1	NUM
ejpam-3576	148	10	)	)	PUNCT
ejpam-3576	148	11	(	(	PUNCT
ejpam-3576	148	12	2020	2020	NUM
ejpam-3576	148	13	)	)	PUNCT
ejpam-3576	148	14	,	,	PUNCT
ejpam-3576	148	15	113	113	NUM
ejpam-3576	148	16	-	-	SYM
ejpam-3576	148	17	129	129	NUM
ejpam-3576	148	18	118	118	NUM
ejpam-3576	148	19	an	an	DET
ejpam-3576	148	20	ordered	order	VERB
ejpam-3576	148	21	ag	ag	PROPN
ejpam-3576	148	22	-	-	NOUN
ejpam-3576	148	23	groupoid	groupoid	PROPN
ejpam-3576	148	24	s	s	PART
ejpam-3576	148	25	can	can	AUX
ejpam-3576	148	26	be	be	AUX
ejpam-3576	148	27	considered	consider	VERB
ejpam-3576	148	28	a	a	DET
ejpam-3576	148	29	fuzzy	fuzzy	ADJ
ejpam-3576	148	30	subset	subset	NOUN
ejpam-3576	148	31	of	of	ADP
ejpam-3576	148	32	itself	itself	PRON
ejpam-3576	148	33	and	and	CCONJ
ejpam-3576	148	34	we	we	PRON
ejpam-3576	148	35	write	write	VERB
ejpam-3576	148	36	s	s	VERB
ejpam-3576	148	37	=	=	PUNCT
ejpam-3576	148	38	χc	χc	NOUN
ejpam-3576	148	39	s	s	X
ejpam-3576	148	40	,	,	PUNCT
ejpam-3576	148	41	i.e.	i.e.	X
ejpam-3576	148	42	,	,	PUNCT
ejpam-3576	148	43	s(x	s(x	NOUN
ejpam-3576	148	44	)	)	PUNCT
ejpam-3576	149	1	=	=	PUNCT
ejpam-3576	149	2	χc	χc	PRON
ejpam-3576	149	3	s	s	X
ejpam-3576	149	4	(	(	PUNCT
ejpam-3576	149	5	x	x	NOUN
ejpam-3576	149	6	)	)	PUNCT
ejpam-3576	149	7	=	=	SYM
ejpam-3576	149	8	0	0	NUM
ejpam-3576	149	9	for	for	ADP
ejpam-3576	149	10	all	all	DET
ejpam-3576	149	11	x	x	SYM
ejpam-3576	149	12	∈	∈	PROPN
ejpam-3576	149	13	s.	s.	PROPN
ejpam-3576	149	14	this	this	PRON
ejpam-3576	149	15	implies	imply	VERB
ejpam-3576	149	16	that	that	SCONJ
ejpam-3576	149	17	s(x	s(x	NOUN
ejpam-3576	149	18	)	)	PUNCT
ejpam-3576	149	19	=	=	SYM
ejpam-3576	149	20	0	0	NUM
ejpam-3576	149	21	for	for	ADP
ejpam-3576	149	22	all	all	DET
ejpam-3576	149	23	x	x	SYM
ejpam-3576	149	24	∈	∈	PROPN
ejpam-3576	149	25	s.	s.	PROPN
ejpam-3576	149	26	for	for	ADP
ejpam-3576	149	27	a	a	DET
ejpam-3576	149	28	,	,	PUNCT
ejpam-3576	149	29	b	b	PROPN
ejpam-3576	149	30	⊆	⊆	NUM
ejpam-3576	149	31	s	s	NOUN
ejpam-3576	149	32	,	,	PUNCT
ejpam-3576	149	33	then	then	ADV
ejpam-3576	149	34	a	a	DET
ejpam-3576	149	35	⊆	⊆	NUM
ejpam-3576	149	36	b	b	NOUN
ejpam-3576	149	37	if	if	SCONJ
ejpam-3576	149	38	and	and	CCONJ
ejpam-3576	149	39	only	only	ADV
ejpam-3576	149	40	if	if	SCONJ
ejpam-3576	149	41	χc	χc	PROPN
ejpam-3576	149	42	a	a	DET
ejpam-3576	149	43	≥	≥	NUM
ejpam-3576	149	44	χc	χc	NOUN
ejpam-3576	149	45	b	b	PROPN
ejpam-3576	149	46	,	,	PUNCT
ejpam-3576	149	47	χ	χ	PROPN
ejpam-3576	149	48	c	c	PROPN
ejpam-3576	149	49	a	a	DET
ejpam-3576	149	50	∩	∩	PROPN
ejpam-3576	149	51	χc	χc	PROPN
ejpam-3576	149	52	b	b	PROPN
ejpam-3576	149	53	=	=	PRON
ejpam-3576	149	54	χc	χc	PROPN
ejpam-3576	149	55	a∩b	a∩b	PROPN
ejpam-3576	149	56	and	and	CCONJ
ejpam-3576	149	57	χc	χc	PROPN
ejpam-3576	149	58	a	a	DET
ejpam-3576	149	59	◦	◦	NOUN
ejpam-3576	150	1	χc	χc	NOUN
ejpam-3576	150	2	b	b	PROPN
ejpam-3576	151	1	=	=	SYM
ejpam-3576	151	2	χc	χc	PROPN
ejpam-3576	151	3	(	(	PUNCT
ejpam-3576	151	4	ab	ab	PROPN
ejpam-3576	151	5	]	]	PUNCT
ejpam-3576	151	6	.	.	PUNCT
ejpam-3576	152	1	let	let	VERB
ejpam-3576	152	2	µ	µ	X
ejpam-3576	152	3	be	be	AUX
ejpam-3576	152	4	a	a	DET
ejpam-3576	152	5	fuzzy	fuzzy	ADJ
ejpam-3576	152	6	subset	subset	NOUN
ejpam-3576	152	7	of	of	ADP
ejpam-3576	152	8	s	s	PROPN
ejpam-3576	152	9	,	,	PUNCT
ejpam-3576	152	10	then	then	ADV
ejpam-3576	152	11	for	for	ADP
ejpam-3576	152	12	all	all	DET
ejpam-3576	152	13	t	t	NOUN
ejpam-3576	152	14	∈	∈	PROPN
ejpam-3576	152	15	(	(	PUNCT
ejpam-3576	152	16	0	0	NUM
ejpam-3576	152	17	,	,	PUNCT
ejpam-3576	152	18	1	1	NUM
ejpam-3576	152	19	]	]	PUNCT
ejpam-3576	152	20	,	,	PUNCT
ejpam-3576	152	21	we	we	PRON
ejpam-3576	152	22	define	define	VERB
ejpam-3576	152	23	a	a	DET
ejpam-3576	152	24	set	set	NOUN
ejpam-3576	152	25	l(µ	l(µ	NOUN
ejpam-3576	152	26	;	;	PUNCT
ejpam-3576	152	27	t	t	PROPN
ejpam-3576	152	28	)	)	PUNCT
ejpam-3576	152	29	=	=	PRON
ejpam-3576	153	1	{	{	PUNCT
ejpam-3576	153	2	x	x	PUNCT
ejpam-3576	153	3	∈	∈	PROPN
ejpam-3576	153	4	s	s	X
ejpam-3576	153	5	|	|	ADV
ejpam-3576	153	6	µ(x	µ(x	NOUN
ejpam-3576	153	7	)	)	PUNCT
ejpam-3576	153	8	≤	≤	NOUN
ejpam-3576	153	9	t	t	PROPN
ejpam-3576	153	10	}	}	PUNCT
ejpam-3576	153	11	,	,	PUNCT
ejpam-3576	153	12	which	which	PRON
ejpam-3576	153	13	is	be	AUX
ejpam-3576	153	14	called	call	VERB
ejpam-3576	153	15	lower	low	ADJ
ejpam-3576	153	16	t	t	NOUN
ejpam-3576	153	17	-	-	PUNCT
ejpam-3576	153	18	level	level	NOUN
ejpam-3576	153	19	set	set	NOUN
ejpam-3576	153	20	of	of	ADP
ejpam-3576	153	21	µ	µ	NUM
ejpam-3576	153	22	and	and	CCONJ
ejpam-3576	153	23	can	can	AUX
ejpam-3576	153	24	be	be	AUX
ejpam-3576	153	25	used	use	VERB
ejpam-3576	153	26	for	for	ADP
ejpam-3576	153	27	the	the	DET
ejpam-3576	153	28	characterization	characterization	NOUN
ejpam-3576	153	29	of	of	ADP
ejpam-3576	153	30	µ.	µ.	PROPN
ejpam-3576	153	31	example	example	NOUN
ejpam-3576	154	1	4	4	X
ejpam-3576	154	2	.	.	PUNCT
ejpam-3576	155	1	let	let	VERB
ejpam-3576	155	2	s	s	VERB
ejpam-3576	155	3	=	=	X
ejpam-3576	155	4	{	{	PUNCT
ejpam-3576	155	5	a	a	PRON
ejpam-3576	155	6	,	,	PUNCT
ejpam-3576	155	7	b	b	NOUN
ejpam-3576	155	8	,	,	PUNCT
ejpam-3576	155	9	c	c	NOUN
ejpam-3576	155	10	,	,	PUNCT
ejpam-3576	155	11	d	d	NOUN
ejpam-3576	155	12	}	}	PUNCT
ejpam-3576	155	13	.	.	PUNCT
ejpam-3576	156	1	define	define	VERB
ejpam-3576	156	2	multiplication	multiplication	NOUN
ejpam-3576	156	3	“	"	PUNCT
ejpam-3576	156	4	·	·	PUNCT
ejpam-3576	156	5	”	"	PUNCT
ejpam-3576	156	6	in	in	ADP
ejpam-3576	156	7	s	s	PRON
ejpam-3576	156	8	as	as	SCONJ
ejpam-3576	156	9	follows	follow	VERB
ejpam-3576	156	10	:	:	PUNCT
ejpam-3576	156	11	·	·	PUNCT
ejpam-3576	156	12	a	a	DET
ejpam-3576	156	13	b	b	X
ejpam-3576	156	14	c	c	NOUN
ejpam-3576	156	15	d	d	NOUN
ejpam-3576	156	16	a	a	PROPN
ejpam-3576	156	17	c	c	NOUN
ejpam-3576	156	18	d	d	NOUN
ejpam-3576	156	19	a	a	DET
ejpam-3576	156	20	b	b	PROPN
ejpam-3576	156	21	b	b	PROPN
ejpam-3576	156	22	b	b	PROPN
ejpam-3576	156	23	c	c	PROPN
ejpam-3576	156	24	d	d	NOUN
ejpam-3576	156	25	a	a	PROPN
ejpam-3576	156	26	c	c	NOUN
ejpam-3576	156	27	a	a	DET
ejpam-3576	156	28	b	b	NOUN
ejpam-3576	156	29	c	c	NOUN
ejpam-3576	157	1	d	d	PROPN
ejpam-3576	157	2	d	d	PROPN
ejpam-3576	157	3	d	d	PROPN
ejpam-3576	157	4	a	a	DET
ejpam-3576	157	5	b	b	NOUN
ejpam-3576	157	6	c	c	NOUN
ejpam-3576	157	7	and	and	CCONJ
ejpam-3576	157	8	≤	≤	NUM
ejpam-3576	157	9	:	:	PUNCT
ejpam-3576	158	1	=	=	SYM
ejpam-3576	158	2	{	{	PUNCT
ejpam-3576	158	3	(	(	PUNCT
ejpam-3576	158	4	a	a	PRON
ejpam-3576	158	5	,	,	PUNCT
ejpam-3576	158	6	a	a	NOUN
ejpam-3576	158	7	)	)	PUNCT
ejpam-3576	158	8	,	,	PUNCT
ejpam-3576	158	9	(	(	PUNCT
ejpam-3576	158	10	b	b	X
ejpam-3576	158	11	,	,	PUNCT
ejpam-3576	158	12	b	b	NOUN
ejpam-3576	158	13	)	)	PUNCT
ejpam-3576	158	14	,	,	PUNCT
ejpam-3576	158	15	(	(	PUNCT
ejpam-3576	158	16	c	c	X
ejpam-3576	158	17	,	,	PUNCT
ejpam-3576	158	18	c	c	NOUN
ejpam-3576	158	19	)	)	PUNCT
ejpam-3576	158	20	,	,	PUNCT
ejpam-3576	158	21	(	(	PUNCT
ejpam-3576	158	22	d	d	X
ejpam-3576	158	23	,	,	PUNCT
ejpam-3576	158	24	d	d	NOUN
ejpam-3576	158	25	)	)	PUNCT
ejpam-3576	158	26	}	}	PUNCT
ejpam-3576	158	27	.	.	PUNCT
ejpam-3576	159	1	then	then	ADV
ejpam-3576	159	2	s	s	VERB
ejpam-3576	159	3	is	be	AUX
ejpam-3576	159	4	an	an	DET
ejpam-3576	159	5	ordered	order	VERB
ejpam-3576	159	6	ag	ag	PROPN
ejpam-3576	159	7	-	-	NOUN
ejpam-3576	159	8	groupoid	groupoid	PROPN
ejpam-3576	159	9	.	.	PUNCT
ejpam-3576	160	1	let	let	VERB
ejpam-3576	160	2	µ	µ	X
ejpam-3576	160	3	be	be	AUX
ejpam-3576	160	4	a	a	DET
ejpam-3576	160	5	fuzzy	fuzzy	ADJ
ejpam-3576	160	6	subset	subset	NOUN
ejpam-3576	160	7	of	of	ADP
ejpam-3576	160	8	s.	s.	PROPN
ejpam-3576	160	9	we	we	PRON
ejpam-3576	160	10	define	define	VERB
ejpam-3576	160	11	µ(a	µ(a	PROPN
ejpam-3576	160	12	)	)	PUNCT
ejpam-3576	161	1	=	=	SYM
ejpam-3576	161	2	µ(c	µ(c	PROPN
ejpam-3576	161	3	)	)	PUNCT
ejpam-3576	161	4	=	=	SYM
ejpam-3576	161	5	0.7	0.7	NUM
ejpam-3576	161	6	,	,	PUNCT
ejpam-3576	161	7	µ(b	µ(b	PROPN
ejpam-3576	161	8	)	)	PUNCT
ejpam-3576	161	9	=	=	SYM
ejpam-3576	161	10	µ(d	µ(d	PROPN
ejpam-3576	161	11	)	)	PUNCT
ejpam-3576	161	12	=	=	SYM
ejpam-3576	162	1	0	0	X
ejpam-3576	162	2	.	.	X
ejpam-3576	163	1	hence	hence	ADV
ejpam-3576	163	2	µ	µ	X
ejpam-3576	163	3	is	be	AUX
ejpam-3576	163	4	an	an	DET
ejpam-3576	163	5	anti	anti	ADJ
ejpam-3576	163	6	fuzzy	fuzzy	ADJ
ejpam-3576	163	7	ag	ag	PROPN
ejpam-3576	163	8	-	-	NOUN
ejpam-3576	163	9	subgroupoid	subgroupoid	NOUN
ejpam-3576	163	10	of	of	ADP
ejpam-3576	163	11	s.	s.	PROPN
ejpam-3576	163	12	example	example	PROPN
ejpam-3576	163	13	5	5	X
ejpam-3576	163	14	.	.	PUNCT
ejpam-3576	164	1	let	let	VERB
ejpam-3576	164	2	s	s	VERB
ejpam-3576	164	3	=	=	X
ejpam-3576	164	4	{	{	PUNCT
ejpam-3576	164	5	a	a	PRON
ejpam-3576	164	6	,	,	PUNCT
ejpam-3576	164	7	b	b	NOUN
ejpam-3576	164	8	,	,	PUNCT
ejpam-3576	164	9	c	c	NOUN
ejpam-3576	164	10	,	,	PUNCT
ejpam-3576	164	11	d	d	NOUN
ejpam-3576	164	12	}	}	PUNCT
ejpam-3576	164	13	.	.	PUNCT
ejpam-3576	165	1	define	define	VERB
ejpam-3576	165	2	multiplication	multiplication	NOUN
ejpam-3576	165	3	“	"	PUNCT
ejpam-3576	165	4	·	·	PUNCT
ejpam-3576	165	5	”	"	PUNCT
ejpam-3576	165	6	in	in	ADP
ejpam-3576	165	7	s	s	PRON
ejpam-3576	165	8	as	as	SCONJ
ejpam-3576	165	9	follows	follow	VERB
ejpam-3576	165	10	:	:	PUNCT
ejpam-3576	165	11	·	·	PUNCT
ejpam-3576	165	12	a	a	DET
ejpam-3576	165	13	b	b	X
ejpam-3576	165	14	c	c	NOUN
ejpam-3576	165	15	d	d	NOUN
ejpam-3576	165	16	a	a	DET
ejpam-3576	165	17	a	a	DET
ejpam-3576	165	18	a	a	DET
ejpam-3576	165	19	a	a	DET
ejpam-3576	165	20	a	a	DET
ejpam-3576	165	21	b	b	NOUN
ejpam-3576	165	22	a	a	DET
ejpam-3576	165	23	a	a	DET
ejpam-3576	165	24	a	a	DET
ejpam-3576	165	25	a	a	DET
ejpam-3576	165	26	c	c	NOUN
ejpam-3576	165	27	a	a	DET
ejpam-3576	165	28	a	a	PROPN
ejpam-3576	165	29	d	d	NOUN
ejpam-3576	165	30	a	a	PROPN
ejpam-3576	165	31	d	d	NOUN
ejpam-3576	165	32	a	a	DET
ejpam-3576	165	33	a	a	DET
ejpam-3576	165	34	c	c	NOUN
ejpam-3576	165	35	d	d	NOUN
ejpam-3576	165	36	and	and	CCONJ
ejpam-3576	165	37	≤	≤	NUM
ejpam-3576	165	38	:	:	PUNCT
ejpam-3576	166	1	=	=	SYM
ejpam-3576	166	2	{	{	PUNCT
ejpam-3576	166	3	(	(	PUNCT
ejpam-3576	166	4	a	a	PRON
ejpam-3576	166	5	,	,	PUNCT
ejpam-3576	166	6	a	a	NOUN
ejpam-3576	166	7	)	)	PUNCT
ejpam-3576	166	8	,	,	PUNCT
ejpam-3576	166	9	(	(	PUNCT
ejpam-3576	166	10	b	b	X
ejpam-3576	166	11	,	,	PUNCT
ejpam-3576	166	12	b	b	NOUN
ejpam-3576	166	13	)	)	PUNCT
ejpam-3576	166	14	,	,	PUNCT
ejpam-3576	166	15	(	(	PUNCT
ejpam-3576	166	16	c	c	X
ejpam-3576	166	17	,	,	PUNCT
ejpam-3576	166	18	c	c	NOUN
ejpam-3576	166	19	)	)	PUNCT
ejpam-3576	166	20	,	,	PUNCT
ejpam-3576	166	21	(	(	PUNCT
ejpam-3576	166	22	d	d	X
ejpam-3576	166	23	,	,	PUNCT
ejpam-3576	166	24	d	d	NOUN
ejpam-3576	166	25	)	)	PUNCT
ejpam-3576	166	26	}	}	PUNCT
ejpam-3576	166	27	.	.	PUNCT
ejpam-3576	167	1	then	then	ADV
ejpam-3576	167	2	s	s	VERB
ejpam-3576	167	3	is	be	AUX
ejpam-3576	167	4	an	an	DET
ejpam-3576	167	5	ordered	order	VERB
ejpam-3576	167	6	ag	ag	PROPN
ejpam-3576	167	7	-	-	NOUN
ejpam-3576	167	8	groupoid	groupoid	PROPN
ejpam-3576	167	9	.	.	PUNCT
ejpam-3576	168	1	let	let	VERB
ejpam-3576	168	2	µ	µ	X
ejpam-3576	168	3	be	be	AUX
ejpam-3576	168	4	a	a	DET
ejpam-3576	168	5	fuzzy	fuzzy	ADJ
ejpam-3576	168	6	subset	subset	NOUN
ejpam-3576	168	7	of	of	ADP
ejpam-3576	168	8	s.	s.	PROPN
ejpam-3576	168	9	we	we	PRON
ejpam-3576	168	10	define	define	VERB
ejpam-3576	168	11	µ(a	µ(a	PROPN
ejpam-3576	168	12	)	)	PUNCT
ejpam-3576	169	1	=	=	SYM
ejpam-3576	169	2	µ(c	µ(c	PROPN
ejpam-3576	169	3	)	)	PUNCT
ejpam-3576	169	4	=	=	SYM
ejpam-3576	169	5	µ(d	µ(d	PROPN
ejpam-3576	169	6	)	)	PUNCT
ejpam-3576	169	7	=	=	SYM
ejpam-3576	169	8	0	0	NUM
ejpam-3576	169	9	,	,	PUNCT
ejpam-3576	169	10	µ(b	µ(b	PROPN
ejpam-3576	169	11	)	)	PUNCT
ejpam-3576	169	12	=	=	PUNCT
ejpam-3576	170	1	0.7	0.7	NUM
ejpam-3576	170	2	.	.	PUNCT
ejpam-3576	171	1	hence	hence	ADV
ejpam-3576	171	2	µ	µ	X
ejpam-3576	171	3	is	be	AUX
ejpam-3576	171	4	an	an	DET
ejpam-3576	171	5	anti	anti	ADJ
ejpam-3576	171	6	fuzzy	fuzzy	ADJ
ejpam-3576	171	7	right	right	ADJ
ejpam-3576	171	8	ideal	ideal	NOUN
ejpam-3576	171	9	of	of	ADP
ejpam-3576	171	10	s.	s.	PROPN
ejpam-3576	171	11	remark	remark	PROPN
ejpam-3576	171	12	2	2	NUM
ejpam-3576	171	13	.	.	PUNCT
ejpam-3576	171	14	example	example	NOUN
ejpam-3576	171	15	4	4	NUM
ejpam-3576	171	16	and	and	CCONJ
ejpam-3576	171	17	example	example	NOUN
ejpam-3576	171	18	5	5	NUM
ejpam-3576	171	19	show	show	VERB
ejpam-3576	171	20	that	that	SCONJ
ejpam-3576	171	21	,	,	PUNCT
ejpam-3576	171	22	every	every	DET
ejpam-3576	171	23	anti	anti	X
ejpam-3576	171	24	fuzzy	fuzzy	ADJ
ejpam-3576	171	25	ideal	ideal	NOUN
ejpam-3576	171	26	(	(	PUNCT
ejpam-3576	171	27	whether	whether	SCONJ
ejpam-3576	171	28	right	right	ADJ
ejpam-3576	171	29	,	,	PUNCT
ejpam-3576	171	30	left	leave	VERB
ejpam-3576	171	31	,	,	PUNCT
ejpam-3576	171	32	two	two	NUM
ejpam-3576	171	33	-	-	PUNCT
ejpam-3576	171	34	sided	sided	ADJ
ejpam-3576	171	35	)	)	PUNCT
ejpam-3576	171	36	is	be	AUX
ejpam-3576	171	37	an	an	DET
ejpam-3576	171	38	anti	anti	ADJ
ejpam-3576	171	39	fuzzy	fuzzy	ADJ
ejpam-3576	171	40	ag	ag	PROPN
ejpam-3576	171	41	-	-	NOUN
ejpam-3576	171	42	subgroupoid	subgroupoid	NOUN
ejpam-3576	171	43	,	,	PUNCT
ejpam-3576	171	44	but	but	CCONJ
ejpam-3576	171	45	the	the	DET
ejpam-3576	171	46	converse	converse	NOUN
ejpam-3576	171	47	is	be	AUX
ejpam-3576	171	48	not	not	PART
ejpam-3576	171	49	true	true	ADJ
ejpam-3576	171	50	.	.	PUNCT
ejpam-3576	172	1	lemma	lemma	PROPN
ejpam-3576	172	2	1	1	X
ejpam-3576	172	3	.	.	PUNCT
ejpam-3576	173	1	let	let	VERB
ejpam-3576	173	2	s	s	PRON
ejpam-3576	173	3	be	be	AUX
ejpam-3576	173	4	an	an	DET
ejpam-3576	173	5	ordered	order	VERB
ejpam-3576	173	6	ag	ag	PROPN
ejpam-3576	173	7	-	-	PUNCT
ejpam-3576	173	8	groupoid	groupoid	PROPN
ejpam-3576	173	9	and	and	CCONJ
ejpam-3576	173	10	∅	∅	NOUN
ejpam-3576	173	11	6=	6=	ADP
ejpam-3576	173	12	a	a	DET
ejpam-3576	173	13	⊆	⊆	NUM
ejpam-3576	173	14	s.	s.	PROPN
ejpam-3576	173	15	then	then	ADV
ejpam-3576	173	16	the	the	DET
ejpam-3576	173	17	anti	anti	ADJ
ejpam-3576	173	18	characteristic	characteristic	ADJ
ejpam-3576	173	19	function	function	NOUN
ejpam-3576	174	1	χc	χc	PROPN
ejpam-3576	174	2	(	(	PUNCT
ejpam-3576	174	3	a	a	PRON
ejpam-3576	174	4	]	]	X
ejpam-3576	174	5	of	of	ADP
ejpam-3576	174	6	(	(	PUNCT
ejpam-3576	174	7	a	a	PRON
ejpam-3576	174	8	]	]	X
ejpam-3576	174	9	is	be	AUX
ejpam-3576	174	10	a	a	DET
ejpam-3576	174	11	fuzzy	fuzzy	ADJ
ejpam-3576	174	12	subset	subset	NOUN
ejpam-3576	174	13	of	of	ADP
ejpam-3576	174	14	s	s	PRON
ejpam-3576	174	15	satisfying	satisfy	VERB
ejpam-3576	174	16	the	the	DET
ejpam-3576	174	17	condition	condition	NOUN
ejpam-3576	174	18	x	x	PUNCT
ejpam-3576	174	19	≤	≤	ADJ
ejpam-3576	174	20	y	y	PROPN
ejpam-3576	174	21	⇒	⇒	VERB
ejpam-3576	174	22	χc	χc	PROPN
ejpam-3576	174	23	(	(	PUNCT
ejpam-3576	174	24	a](x	a](x	X
ejpam-3576	174	25	)	)	PUNCT
ejpam-3576	174	26	≤	≤	NOUN
ejpam-3576	175	1	χc	χc	PROPN
ejpam-3576	175	2	(	(	PUNCT
ejpam-3576	175	3	a](y	a](y	PROPN
ejpam-3576	175	4	)	)	PUNCT
ejpam-3576	175	5	for	for	ADP
ejpam-3576	175	6	all	all	DET
ejpam-3576	175	7	x	x	NOUN
ejpam-3576	175	8	,	,	PUNCT
ejpam-3576	175	9	y	y	PROPN
ejpam-3576	175	10	∈	∈	PROPN
ejpam-3576	175	11	s.	s.	PROPN
ejpam-3576	175	12	proof	proof	NOUN
ejpam-3576	175	13	.	.	PUNCT
ejpam-3576	176	1	by	by	ADP
ejpam-3576	176	2	the	the	DET
ejpam-3576	176	3	definition	definition	NOUN
ejpam-3576	176	4	,	,	PUNCT
ejpam-3576	176	5	χc	χc	PROPN
ejpam-3576	176	6	(	(	PUNCT
ejpam-3576	176	7	a	a	PRON
ejpam-3576	176	8	]	]	X
ejpam-3576	176	9	is	be	AUX
ejpam-3576	176	10	a	a	DET
ejpam-3576	176	11	mapping	mapping	NOUN
ejpam-3576	176	12	of	of	ADP
ejpam-3576	176	13	s	s	NOUN
ejpam-3576	176	14	into	into	ADP
ejpam-3576	176	15	{	{	PUNCT
ejpam-3576	176	16	0	0	NUM
ejpam-3576	176	17	,	,	PUNCT
ejpam-3576	176	18	1	1	NUM
ejpam-3576	176	19	}	}	PUNCT
ejpam-3576	176	20	⊆	⊆	NUM
ejpam-3576	176	21	[	[	X
ejpam-3576	176	22	0	0	NUM
ejpam-3576	176	23	,	,	PUNCT
ejpam-3576	176	24	1	1	NUM
ejpam-3576	176	25	]	]	PUNCT
ejpam-3576	176	26	.	.	PUNCT
ejpam-3576	177	1	let	let	VERB
ejpam-3576	177	2	x	x	SYM
ejpam-3576	177	3	≤	≤	PROPN
ejpam-3576	177	4	y	y	PROPN
ejpam-3576	177	5	,	,	PUNCT
ejpam-3576	177	6	x	x	PRON
ejpam-3576	177	7	,	,	PUNCT
ejpam-3576	177	8	y	y	PROPN
ejpam-3576	177	9	∈	∈	PROPN
ejpam-3576	177	10	s.	s.	PROPN
ejpam-3576	178	1	if	if	SCONJ
ejpam-3576	178	2	y	y	PROPN
ejpam-3576	178	3	/∈	/∈	PUNCT
ejpam-3576	179	1	(	(	PUNCT
ejpam-3576	179	2	a	a	PRON
ejpam-3576	179	3	]	]	X
ejpam-3576	179	4	,	,	PUNCT
ejpam-3576	179	5	by	by	ADP
ejpam-3576	179	6	definition	definition	NOUN
ejpam-3576	179	7	χc	χc	PROPN
ejpam-3576	179	8	(	(	PUNCT
ejpam-3576	179	9	a](y	a](y	PROPN
ejpam-3576	179	10	)	)	PUNCT
ejpam-3576	179	11	=	=	SYM
ejpam-3576	179	12	1	1	NUM
ejpam-3576	179	13	,	,	PUNCT
ejpam-3576	179	14	thus	thus	ADV
ejpam-3576	179	15	χc	χc	PROPN
ejpam-3576	179	16	(	(	PUNCT
ejpam-3576	179	17	a](x	a](x	X
ejpam-3576	179	18	)	)	PUNCT
ejpam-3576	179	19	≤	≤	NOUN
ejpam-3576	179	20	χc	χc	PROPN
ejpam-3576	179	21	(	(	PUNCT
ejpam-3576	179	22	a](y	a](y	ADJ
ejpam-3576	179	23	)	)	PUNCT
ejpam-3576	179	24	.	.	PUNCT
ejpam-3576	180	1	if	if	SCONJ
ejpam-3576	180	2	y	y	PROPN
ejpam-3576	180	3	∈	∈	PROPN
ejpam-3576	180	4	(	(	PUNCT
ejpam-3576	180	5	a	a	PRON
ejpam-3576	180	6	]	]	X
ejpam-3576	180	7	,	,	PUNCT
ejpam-3576	180	8	by	by	ADP
ejpam-3576	180	9	definition	definition	NOUN
ejpam-3576	180	10	χc	χc	PROPN
ejpam-3576	180	11	(	(	PUNCT
ejpam-3576	180	12	a](y	a](y	PROPN
ejpam-3576	180	13	)	)	PUNCT
ejpam-3576	180	14	=	=	SYM
ejpam-3576	181	1	0	0	X
ejpam-3576	181	2	.	.	PUNCT
ejpam-3576	182	1	since	since	SCONJ
ejpam-3576	182	2	y	y	PROPN
ejpam-3576	182	3	∈	∈	PROPN
ejpam-3576	182	4	(	(	PUNCT
ejpam-3576	182	5	a	a	PRON
ejpam-3576	182	6	]	]	X
ejpam-3576	182	7	,	,	PUNCT
ejpam-3576	182	8	so	so	CCONJ
ejpam-3576	182	9	there	there	PRON
ejpam-3576	182	10	exists	exist	VERB
ejpam-3576	182	11	z	z	PROPN
ejpam-3576	182	12	∈	∈	PROPN
ejpam-3576	182	13	a	a	DET
ejpam-3576	182	14	such	such	ADJ
ejpam-3576	182	15	that	that	SCONJ
ejpam-3576	182	16	y	y	PROPN
ejpam-3576	182	17	≤	≤	PROPN
ejpam-3576	182	18	z.	z.	PROPN
ejpam-3576	183	1	thus	thus	ADV
ejpam-3576	183	2	x	x	X
ejpam-3576	183	3	≤	≤	PROPN
ejpam-3576	183	4	z	z	NOUN
ejpam-3576	183	5	,	,	PUNCT
ejpam-3576	183	6	i.e.	i.e.	X
ejpam-3576	183	7	,	,	PUNCT
ejpam-3576	183	8	x	x	SYM
ejpam-3576	183	9	∈	∈	PROPN
ejpam-3576	183	10	(	(	PUNCT
ejpam-3576	183	11	a	a	X
ejpam-3576	183	12	]	]	X
ejpam-3576	183	13	and	and	CCONJ
ejpam-3576	183	14	χc	χc	PROPN
ejpam-3576	183	15	(	(	PUNCT
ejpam-3576	183	16	a](x	a](x	X
ejpam-3576	183	17	)	)	PUNCT
ejpam-3576	183	18	=	=	SYM
ejpam-3576	184	1	0	0	X
ejpam-3576	184	2	.	.	PUNCT
ejpam-3576	185	1	hence	hence	ADV
ejpam-3576	185	2	χc	χc	PROPN
ejpam-3576	185	3	(	(	PUNCT
ejpam-3576	185	4	a](x	a](x	X
ejpam-3576	185	5	)	)	PUNCT
ejpam-3576	185	6	≤	≤	NOUN
ejpam-3576	185	7	χc	χc	PROPN
ejpam-3576	185	8	(	(	PUNCT
ejpam-3576	185	9	a](y	a](y	PROPN
ejpam-3576	185	10	)	)	PUNCT
ejpam-3576	185	11	.	.	PUNCT
ejpam-3576	186	1	k.	k.	PROPN
ejpam-3576	186	2	nasreen	nasreen	PROPN
ejpam-3576	186	3	,	,	PUNCT
ejpam-3576	186	4	m.	m.	NOUN
ejpam-3576	186	5	alesemi	alesemi	PROPN
ejpam-3576	186	6	,	,	PUNCT
ejpam-3576	186	7	salahuddin	salahuddin	VERB
ejpam-3576	186	8	/	/	SYM
ejpam-3576	186	9	eur	eur	PROPN
ejpam-3576	186	10	.	.	PUNCT
ejpam-3576	187	1	j.	j.	PROPN
ejpam-3576	187	2	pure	pure	PROPN
ejpam-3576	187	3	appl	appl	PROPN
ejpam-3576	187	4	.	.	PROPN
ejpam-3576	187	5	math	math	PROPN
ejpam-3576	187	6	,	,	PUNCT
ejpam-3576	187	7	13	13	NUM
ejpam-3576	187	8	(	(	PUNCT
ejpam-3576	187	9	1	1	NUM
ejpam-3576	187	10	)	)	PUNCT
ejpam-3576	187	11	(	(	PUNCT
ejpam-3576	187	12	2020	2020	NUM
ejpam-3576	187	13	)	)	PUNCT
ejpam-3576	187	14	,	,	PUNCT
ejpam-3576	187	15	113	113	NUM
ejpam-3576	187	16	-	-	SYM
ejpam-3576	187	17	129	129	NUM
ejpam-3576	187	18	119	119	NUM
ejpam-3576	187	19	proposition	proposition	NOUN
ejpam-3576	187	20	1	1	NUM
ejpam-3576	187	21	.	.	PUNCT
ejpam-3576	188	1	let	let	VERB
ejpam-3576	188	2	s	s	PRON
ejpam-3576	188	3	be	be	AUX
ejpam-3576	188	4	an	an	DET
ejpam-3576	188	5	ordered	order	VERB
ejpam-3576	188	6	ag	ag	PROPN
ejpam-3576	188	7	-	-	PUNCT
ejpam-3576	188	8	groupoid	groupoid	PROPN
ejpam-3576	188	9	and	and	CCONJ
ejpam-3576	188	10	∅	∅	NOUN
ejpam-3576	188	11	6=	6=	ADP
ejpam-3576	188	12	a	a	DET
ejpam-3576	188	13	⊆	⊆	NUM
ejpam-3576	188	14	s.	s.	PROPN
ejpam-3576	188	15	then	then	ADV
ejpam-3576	188	16	a	a	DET
ejpam-3576	188	17	=	=	X
ejpam-3576	188	18	(	(	PUNCT
ejpam-3576	188	19	a	a	X
ejpam-3576	188	20	]	]	X
ejpam-3576	188	21	if	if	SCONJ
ejpam-3576	188	22	and	and	CCONJ
ejpam-3576	188	23	only	only	ADV
ejpam-3576	188	24	if	if	SCONJ
ejpam-3576	188	25	fuzzy	fuzzy	ADJ
ejpam-3576	188	26	subset	subset	VERB
ejpam-3576	188	27	χc	χc	PRON
ejpam-3576	188	28	a	a	PRON
ejpam-3576	188	29	of	of	ADP
ejpam-3576	188	30	s	s	PROPN
ejpam-3576	188	31	has	have	VERB
ejpam-3576	188	32	the	the	DET
ejpam-3576	188	33	property	property	NOUN
ejpam-3576	188	34	x	x	SYM
ejpam-3576	188	35	≤	≤	NOUN
ejpam-3576	188	36	y	y	PROPN
ejpam-3576	188	37	⇒	⇒	NOUN
ejpam-3576	188	38	χc	χc	PROPN
ejpam-3576	188	39	a(x	a(x	PROPN
ejpam-3576	188	40	)	)	PUNCT
ejpam-3576	188	41	≤	≤	NOUN
ejpam-3576	188	42	χc	χc	ADP
ejpam-3576	188	43	a(y	a(y	PROPN
ejpam-3576	188	44	)	)	PUNCT
ejpam-3576	188	45	for	for	ADP
ejpam-3576	188	46	all	all	DET
ejpam-3576	188	47	x	x	NOUN
ejpam-3576	188	48	,	,	PUNCT
ejpam-3576	188	49	y	y	PROPN
ejpam-3576	188	50	∈	∈	PROPN
ejpam-3576	188	51	s.	s.	PROPN
ejpam-3576	188	52	proof	proof	PROPN
ejpam-3576	188	53	.	.	PUNCT
ejpam-3576	189	1	suppose	suppose	VERB
ejpam-3576	189	2	a	a	DET
ejpam-3576	189	3	=	=	X
ejpam-3576	189	4	(	(	PUNCT
ejpam-3576	189	5	a	a	X
ejpam-3576	189	6	]	]	X
ejpam-3576	189	7	,	,	PUNCT
ejpam-3576	189	8	then	then	ADV
ejpam-3576	189	9	the	the	DET
ejpam-3576	189	10	anti	anti	PROPN
ejpam-3576	189	11	characteristic	characteristic	ADJ
ejpam-3576	189	12	function	function	NOUN
ejpam-3576	190	1	χc	χc	PRON
ejpam-3576	190	2	a	a	PRON
ejpam-3576	190	3	of	of	ADP
ejpam-3576	190	4	a	a	PRON
ejpam-3576	190	5	is	be	AUX
ejpam-3576	190	6	a	a	DET
ejpam-3576	190	7	fuzzy	fuzzy	ADJ
ejpam-3576	190	8	subset	subset	NOUN
ejpam-3576	190	9	of	of	ADP
ejpam-3576	190	10	s	s	PRON
ejpam-3576	190	11	satisfying	satisfy	VERB
ejpam-3576	190	12	the	the	DET
ejpam-3576	190	13	condition	condition	NOUN
ejpam-3576	190	14	x	x	PUNCT
ejpam-3576	190	15	≤	≤	ADJ
ejpam-3576	190	16	y	y	PROPN
ejpam-3576	190	17	⇒	⇒	NOUN
ejpam-3576	190	18	χc	χc	PROPN
ejpam-3576	190	19	a(x	a(x	PROPN
ejpam-3576	190	20	)	)	PUNCT
ejpam-3576	190	21	≤	≤	NOUN
ejpam-3576	190	22	χc	χc	ADP
ejpam-3576	190	23	a(y	a(y	PROPN
ejpam-3576	190	24	)	)	PUNCT
ejpam-3576	190	25	,	,	PUNCT
ejpam-3576	190	26	by	by	ADP
ejpam-3576	190	27	the	the	DET
ejpam-3576	190	28	lemma	lemma	PROPN
ejpam-3576	190	29	1	1	NUM
ejpam-3576	190	30	.	.	PUNCT
ejpam-3576	191	1	conversely	conversely	ADV
ejpam-3576	191	2	,	,	PUNCT
ejpam-3576	191	3	let	let	VERB
ejpam-3576	191	4	x	x	X
ejpam-3576	191	5	∈	∈	PROPN
ejpam-3576	191	6	(	(	PUNCT
ejpam-3576	191	7	a	a	X
ejpam-3576	191	8	]	]	X
ejpam-3576	191	9	,	,	PUNCT
ejpam-3576	191	10	this	this	PRON
ejpam-3576	191	11	imply	imply	VERB
ejpam-3576	191	12	that	that	SCONJ
ejpam-3576	191	13	there	there	PRON
ejpam-3576	191	14	exists	exist	VERB
ejpam-3576	191	15	y	y	PROPN
ejpam-3576	191	16	∈	∈	PROPN
ejpam-3576	191	17	a	a	DET
ejpam-3576	191	18	such	such	ADJ
ejpam-3576	191	19	that	that	SCONJ
ejpam-3576	191	20	x	x	X
ejpam-3576	191	21	≤	≤	X
ejpam-3576	191	22	y.	y.	NOUN
ejpam-3576	191	23	by	by	ADP
ejpam-3576	191	24	the	the	DET
ejpam-3576	191	25	given	give	VERB
ejpam-3576	191	26	condition	condition	NOUN
ejpam-3576	191	27	,	,	PUNCT
ejpam-3576	191	28	we	we	PRON
ejpam-3576	191	29	have	have	VERB
ejpam-3576	191	30	χc	χc	PART
ejpam-3576	191	31	a(x	a(x	PROPN
ejpam-3576	191	32	)	)	PUNCT
ejpam-3576	191	33	≤	≤	NOUN
ejpam-3576	191	34	χc	χc	ADP
ejpam-3576	191	35	a(y	a(y	PROPN
ejpam-3576	191	36	)	)	PUNCT
ejpam-3576	191	37	.	.	PUNCT
ejpam-3576	192	1	since	since	SCONJ
ejpam-3576	192	2	y	y	PROPN
ejpam-3576	192	3	∈	∈	PROPN
ejpam-3576	192	4	a	a	X
ejpam-3576	192	5	,	,	PUNCT
ejpam-3576	192	6	we	we	PRON
ejpam-3576	192	7	have	have	VERB
ejpam-3576	192	8	χc	χc	PRON
ejpam-3576	192	9	a(y	a(y	PROPN
ejpam-3576	192	10	)	)	PUNCT
ejpam-3576	193	1	=	=	PUNCT
ejpam-3576	193	2	0	0	X
ejpam-3576	193	3	.	.	PUNCT
ejpam-3576	194	1	thus	thus	ADV
ejpam-3576	194	2	χc	χc	ADP
ejpam-3576	194	3	a(x	a(x	PROPN
ejpam-3576	194	4	)	)	PUNCT
ejpam-3576	194	5	=	=	SYM
ejpam-3576	194	6	0	0	NUM
ejpam-3576	194	7	,	,	PUNCT
ejpam-3576	194	8	i.e.	i.e.	X
ejpam-3576	194	9	,	,	PUNCT
ejpam-3576	194	10	x	x	SYM
ejpam-3576	194	11	∈	∈	NOUN
ejpam-3576	194	12	a.	a.	NOUN
ejpam-3576	194	13	hence	hence	ADV
ejpam-3576	194	14	a	a	PRON
ejpam-3576	194	15	=	=	X
ejpam-3576	194	16	(	(	PUNCT
ejpam-3576	194	17	a	a	X
ejpam-3576	194	18	]	]	X
ejpam-3576	194	19	.	.	PUNCT
ejpam-3576	195	1	lemma	lemma	PROPN
ejpam-3576	195	2	2	2	X
ejpam-3576	195	3	.	.	PUNCT
ejpam-3576	196	1	let	let	VERB
ejpam-3576	196	2	s	s	PRON
ejpam-3576	196	3	be	be	AUX
ejpam-3576	196	4	an	an	DET
ejpam-3576	196	5	ordered	order	VERB
ejpam-3576	196	6	ag	ag	PROPN
ejpam-3576	196	7	-	-	PUNCT
ejpam-3576	196	8	groupoid	groupoid	PROPN
ejpam-3576	196	9	and	and	CCONJ
ejpam-3576	196	10	∅	∅	NOUN
ejpam-3576	196	11	6=	6=	ADP
ejpam-3576	196	12	a	a	DET
ejpam-3576	196	13	⊆	⊆	NUM
ejpam-3576	196	14	s.	s.	PROPN
ejpam-3576	196	15	then	then	ADV
ejpam-3576	196	16	a	a	PRON
ejpam-3576	196	17	is	be	AUX
ejpam-3576	196	18	an	an	DET
ejpam-3576	196	19	agsubgroupoid	agsubgroupoid	NOUN
ejpam-3576	196	20	of	of	ADP
ejpam-3576	196	21	s	s	PRON
ejpam-3576	196	22	if	if	SCONJ
ejpam-3576	197	1	and	and	CCONJ
ejpam-3576	197	2	only	only	ADV
ejpam-3576	197	3	if	if	SCONJ
ejpam-3576	197	4	the	the	DET
ejpam-3576	197	5	anti	anti	X
ejpam-3576	197	6	characteristic	characteristic	ADJ
ejpam-3576	197	7	function	function	NOUN
ejpam-3576	197	8	χc	χc	PRON
ejpam-3576	197	9	a	a	PRON
ejpam-3576	197	10	of	of	ADP
ejpam-3576	197	11	a	a	PRON
ejpam-3576	197	12	is	be	AUX
ejpam-3576	197	13	an	an	DET
ejpam-3576	197	14	anti	anti	ADJ
ejpam-3576	197	15	fuzzy	fuzzy	ADJ
ejpam-3576	197	16	ag	ag	PROPN
ejpam-3576	197	17	-	-	NOUN
ejpam-3576	197	18	subgroupoid	subgroupoid	NOUN
ejpam-3576	197	19	of	of	ADP
ejpam-3576	197	20	s.	s.	PROPN
ejpam-3576	197	21	proof	proof	PROPN
ejpam-3576	197	22	.	.	PUNCT
ejpam-3576	198	1	suppose	suppose	VERB
ejpam-3576	198	2	a	a	PRON
ejpam-3576	198	3	is	be	AUX
ejpam-3576	198	4	an	an	DET
ejpam-3576	198	5	ag	ag	NOUN
ejpam-3576	198	6	-	-	PUNCT
ejpam-3576	198	7	subgroupoid	subgroupoid	NOUN
ejpam-3576	198	8	of	of	ADP
ejpam-3576	198	9	s	s	PROPN
ejpam-3576	198	10	and	and	CCONJ
ejpam-3576	198	11	x	x	X
ejpam-3576	198	12	,	,	PUNCT
ejpam-3576	198	13	y	y	PROPN
ejpam-3576	198	14	∈	∈	PROPN
ejpam-3576	198	15	s.	s.	PROPN
ejpam-3576	198	16	if	if	SCONJ
ejpam-3576	198	17	x	x	PROPN
ejpam-3576	198	18	,	,	PUNCT
ejpam-3576	198	19	y	y	PROPN
ejpam-3576	198	20	/∈	/∈	PUNCT
ejpam-3576	198	21	a	a	X
ejpam-3576	198	22	,	,	PUNCT
ejpam-3576	198	23	by	by	ADP
ejpam-3576	198	24	definition	definition	NOUN
ejpam-3576	198	25	χc	χc	NOUN
ejpam-3576	198	26	a(x	a(x	PROPN
ejpam-3576	198	27	)	)	PUNCT
ejpam-3576	198	28	=	=	SYM
ejpam-3576	198	29	1	1	NUM
ejpam-3576	198	30	=	=	SYM
ejpam-3576	198	31	χc	χc	PROPN
ejpam-3576	198	32	a(y	a(y	PROPN
ejpam-3576	198	33	)	)	PUNCT
ejpam-3576	198	34	.	.	PUNCT
ejpam-3576	199	1	thus	thus	ADV
ejpam-3576	199	2	χc	χc	PROPN
ejpam-3576	199	3	a(xy	a(xy	PROPN
ejpam-3576	199	4	)	)	PUNCT
ejpam-3576	199	5	≤	≤	NUM
ejpam-3576	199	6	χc	χc	ADP
ejpam-3576	199	7	a(x	a(x	PROPN
ejpam-3576	199	8	)	)	PUNCT
ejpam-3576	199	9	∨	∨	NUM
ejpam-3576	199	10	χc	χc	PROPN
ejpam-3576	199	11	a(y	a(y	PROPN
ejpam-3576	199	12	)	)	PUNCT
ejpam-3576	199	13	.	.	PUNCT
ejpam-3576	200	1	if	if	SCONJ
ejpam-3576	200	2	x	x	X
ejpam-3576	200	3	,	,	PUNCT
ejpam-3576	200	4	y	y	PROPN
ejpam-3576	200	5	∈	∈	PROPN
ejpam-3576	200	6	a	a	PRON
ejpam-3576	200	7	,	,	PUNCT
ejpam-3576	200	8	by	by	ADP
ejpam-3576	200	9	definition	definition	NOUN
ejpam-3576	200	10	χc	χc	NOUN
ejpam-3576	200	11	a(x	a(x	PROPN
ejpam-3576	200	12	)	)	PUNCT
ejpam-3576	200	13	=	=	SYM
ejpam-3576	200	14	0	0	PUNCT
ejpam-3576	201	1	=	=	PUNCT
ejpam-3576	201	2	χc	χc	PROPN
ejpam-3576	201	3	a(y	a(y	PROPN
ejpam-3576	201	4	)	)	PUNCT
ejpam-3576	201	5	.	.	PUNCT
ejpam-3576	202	1	xy	xy	PROPN
ejpam-3576	202	2	∈	∈	PROPN
ejpam-3576	203	1	a	a	PRON
ejpam-3576	203	2	,	,	PUNCT
ejpam-3576	203	3	a	a	DET
ejpam-3576	203	4	being	be	AUX
ejpam-3576	203	5	an	an	DET
ejpam-3576	203	6	ag	ag	NOUN
ejpam-3576	203	7	-	-	PUNCT
ejpam-3576	203	8	subgroupoid	subgroupoid	NOUN
ejpam-3576	203	9	of	of	ADP
ejpam-3576	203	10	s	s	PROPN
ejpam-3576	203	11	,	,	PUNCT
ejpam-3576	203	12	this	this	PRON
ejpam-3576	203	13	imply	imply	VERB
ejpam-3576	203	14	that	that	SCONJ
ejpam-3576	203	15	χc	χc	PROPN
ejpam-3576	203	16	a(xy	a(xy	PROPN
ejpam-3576	203	17	)	)	PUNCT
ejpam-3576	204	1	=	=	PUNCT
ejpam-3576	204	2	0	0	X
ejpam-3576	204	3	.	.	PUNCT
ejpam-3576	205	1	thus	thus	ADV
ejpam-3576	205	2	χc	χc	PROPN
ejpam-3576	205	3	a(xy	a(xy	PROPN
ejpam-3576	205	4	)	)	PUNCT
ejpam-3576	205	5	≤	≤	NUM
ejpam-3576	205	6	χc	χc	ADP
ejpam-3576	205	7	a(x	a(x	PROPN
ejpam-3576	205	8	)	)	PUNCT
ejpam-3576	205	9	∨	∨	NUM
ejpam-3576	205	10	χc	χc	PROPN
ejpam-3576	205	11	a(y	a(y	PROPN
ejpam-3576	205	12	)	)	PUNCT
ejpam-3576	205	13	.	.	PUNCT
ejpam-3576	206	1	hence	hence	ADV
ejpam-3576	206	2	the	the	DET
ejpam-3576	206	3	anti	anti	ADJ
ejpam-3576	206	4	characteristic	characteristic	ADJ
ejpam-3576	206	5	function	function	NOUN
ejpam-3576	206	6	χc	χc	PRON
ejpam-3576	206	7	a	a	PRON
ejpam-3576	206	8	of	of	ADP
ejpam-3576	206	9	a	a	PRON
ejpam-3576	206	10	is	be	AUX
ejpam-3576	206	11	an	an	DET
ejpam-3576	206	12	anti	anti	ADJ
ejpam-3576	206	13	fuzzy	fuzzy	ADJ
ejpam-3576	206	14	ag	ag	PROPN
ejpam-3576	206	15	-	-	NOUN
ejpam-3576	206	16	subgroupoid	subgroupoid	NOUN
ejpam-3576	206	17	of	of	ADP
ejpam-3576	206	18	s.	s.	PROPN
ejpam-3576	206	19	conversely	conversely	ADV
ejpam-3576	206	20	,	,	PUNCT
ejpam-3576	206	21	let	let	VERB
ejpam-3576	206	22	xy	xy	PROPN
ejpam-3576	206	23	∈	∈	PROPN
ejpam-3576	206	24	a2	a2	PROPN
ejpam-3576	206	25	,	,	PUNCT
ejpam-3576	206	26	x	x	PRON
ejpam-3576	206	27	,	,	PUNCT
ejpam-3576	206	28	y	y	PROPN
ejpam-3576	206	29	∈	∈	PROPN
ejpam-3576	206	30	a.	a.	NOUN
ejpam-3576	206	31	by	by	ADP
ejpam-3576	206	32	definition	definition	NOUN
ejpam-3576	206	33	of	of	ADP
ejpam-3576	206	34	anti	anti	ADJ
ejpam-3576	206	35	characteristic	characteristic	ADJ
ejpam-3576	206	36	function	function	NOUN
ejpam-3576	206	37	χc	χc	PROPN
ejpam-3576	206	38	a(x	a(x	PROPN
ejpam-3576	206	39	)	)	PUNCT
ejpam-3576	206	40	=	=	PUNCT
ejpam-3576	206	41	0	0	PUNCT
ejpam-3576	207	1	=	=	PUNCT
ejpam-3576	207	2	χc	χc	PROPN
ejpam-3576	207	3	a(y	a(y	PROPN
ejpam-3576	207	4	)	)	PUNCT
ejpam-3576	207	5	.	.	PUNCT
ejpam-3576	208	1	χc	χc	PROPN
ejpam-3576	208	2	a(xy	a(xy	PROPN
ejpam-3576	208	3	)	)	PUNCT
ejpam-3576	208	4	≤	≤	NUM
ejpam-3576	208	5	χc	χc	ADP
ejpam-3576	208	6	a(x	a(x	PROPN
ejpam-3576	208	7	)	)	PUNCT
ejpam-3576	208	8	∨	∨	NUM
ejpam-3576	208	9	χc	χc	PROPN
ejpam-3576	208	10	a(y	a(y	PROPN
ejpam-3576	208	11	)	)	PUNCT
ejpam-3576	209	1	=	=	SYM
ejpam-3576	209	2	0	0	NUM
ejpam-3576	209	3	,	,	PUNCT
ejpam-3576	209	4	χc	χc	PRON
ejpam-3576	209	5	a	a	DET
ejpam-3576	209	6	being	be	AUX
ejpam-3576	209	7	an	an	DET
ejpam-3576	209	8	anti	anti	ADJ
ejpam-3576	209	9	fuzzy	fuzzy	ADJ
ejpam-3576	209	10	ag	ag	PROPN
ejpam-3576	209	11	-	-	NOUN
ejpam-3576	209	12	subgroupoid	subgroupoid	NOUN
ejpam-3576	209	13	of	of	ADP
ejpam-3576	209	14	s.	s.	PROPN
ejpam-3576	209	15	this	this	PRON
ejpam-3576	209	16	imply	imply	VERB
ejpam-3576	209	17	that	that	SCONJ
ejpam-3576	209	18	χc	χc	PROPN
ejpam-3576	209	19	a(xy	a(xy	PROPN
ejpam-3576	209	20	)	)	PUNCT
ejpam-3576	209	21	=	=	SYM
ejpam-3576	209	22	0	0	NUM
ejpam-3576	209	23	,	,	PUNCT
ejpam-3576	209	24	i.e.	i.e.	X
ejpam-3576	209	25	,	,	PUNCT
ejpam-3576	209	26	xy	xy	PROPN
ejpam-3576	209	27	∈	∈	PROPN
ejpam-3576	209	28	a.	a.	NOUN
ejpam-3576	209	29	hence	hence	ADV
ejpam-3576	209	30	a	a	PRON
ejpam-3576	209	31	is	be	AUX
ejpam-3576	209	32	an	an	DET
ejpam-3576	209	33	ag	ag	NOUN
ejpam-3576	209	34	-	-	PUNCT
ejpam-3576	209	35	subgroupoid	subgroupoid	NOUN
ejpam-3576	209	36	of	of	ADP
ejpam-3576	209	37	s.	s.	PROPN
ejpam-3576	209	38	lemma	lemma	PROPN
ejpam-3576	210	1	3	3	X
ejpam-3576	210	2	.	.	PUNCT
ejpam-3576	210	3	let	let	VERB
ejpam-3576	210	4	s	s	PRON
ejpam-3576	210	5	be	be	AUX
ejpam-3576	210	6	an	an	DET
ejpam-3576	210	7	ordered	order	VERB
ejpam-3576	210	8	ag	ag	PROPN
ejpam-3576	210	9	-	-	PUNCT
ejpam-3576	210	10	groupoid	groupoid	PROPN
ejpam-3576	210	11	and	and	CCONJ
ejpam-3576	210	12	∅	∅	NOUN
ejpam-3576	210	13	6=	6=	ADP
ejpam-3576	210	14	a	a	DET
ejpam-3576	210	15	⊆	⊆	NUM
ejpam-3576	210	16	s.	s.	PROPN
ejpam-3576	210	17	then	then	ADV
ejpam-3576	210	18	a	a	PRON
ejpam-3576	210	19	is	be	AUX
ejpam-3576	210	20	a	a	DET
ejpam-3576	210	21	left	left	ADJ
ejpam-3576	210	22	(	(	PUNCT
ejpam-3576	210	23	resp	resp	NOUN
ejpam-3576	210	24	.	.	PUNCT
ejpam-3576	211	1	right	right	ADJ
ejpam-3576	211	2	)	)	PUNCT
ejpam-3576	211	3	ideal	ideal	NOUN
ejpam-3576	211	4	of	of	ADP
ejpam-3576	211	5	s	s	PRON
ejpam-3576	211	6	if	if	SCONJ
ejpam-3576	212	1	and	and	CCONJ
ejpam-3576	212	2	only	only	ADV
ejpam-3576	212	3	if	if	SCONJ
ejpam-3576	212	4	the	the	DET
ejpam-3576	212	5	anti	anti	X
ejpam-3576	212	6	characteristic	characteristic	ADJ
ejpam-3576	212	7	function	function	NOUN
ejpam-3576	212	8	χc	χc	PRON
ejpam-3576	212	9	a	a	PRON
ejpam-3576	212	10	of	of	ADP
ejpam-3576	212	11	a	a	PRON
ejpam-3576	212	12	is	be	AUX
ejpam-3576	212	13	an	an	DET
ejpam-3576	212	14	anti	anti	ADJ
ejpam-3576	212	15	fuzzy	fuzzy	ADJ
ejpam-3576	212	16	left	left	ADJ
ejpam-3576	212	17	(	(	PUNCT
ejpam-3576	212	18	resp	resp	NOUN
ejpam-3576	212	19	.	.	PUNCT
ejpam-3576	213	1	right	right	ADJ
ejpam-3576	213	2	)	)	PUNCT
ejpam-3576	213	3	ideal	ideal	NOUN
ejpam-3576	213	4	of	of	ADP
ejpam-3576	213	5	s.	s.	PROPN
ejpam-3576	213	6	proof	proof	PROPN
ejpam-3576	213	7	.	.	PUNCT
ejpam-3576	214	1	suppose	suppose	VERB
ejpam-3576	214	2	a	a	PRON
ejpam-3576	214	3	is	be	AUX
ejpam-3576	214	4	a	a	DET
ejpam-3576	214	5	left	left	ADJ
ejpam-3576	214	6	ideal	ideal	NOUN
ejpam-3576	214	7	of	of	ADP
ejpam-3576	214	8	s	s	PRON
ejpam-3576	214	9	and	and	CCONJ
ejpam-3576	214	10	x	x	NOUN
ejpam-3576	214	11	,	,	PUNCT
ejpam-3576	214	12	y	y	PROPN
ejpam-3576	214	13	∈	∈	PROPN
ejpam-3576	214	14	s	s	VERB
ejpam-3576	214	15	such	such	ADJ
ejpam-3576	214	16	that	that	SCONJ
ejpam-3576	214	17	x	x	X
ejpam-3576	214	18	≤	≤	X
ejpam-3576	215	1	y.	y.	NOUN
ejpam-3576	215	2	this	this	PRON
ejpam-3576	215	3	imply	imply	VERB
ejpam-3576	215	4	that	that	SCONJ
ejpam-3576	215	5	a	a	DET
ejpam-3576	215	6	=	=	X
ejpam-3576	215	7	(	(	PUNCT
ejpam-3576	215	8	a	a	X
ejpam-3576	215	9	]	]	X
ejpam-3576	215	10	,	,	PUNCT
ejpam-3576	215	11	a	a	DET
ejpam-3576	215	12	being	be	AUX
ejpam-3576	215	13	a	a	DET
ejpam-3576	215	14	left	left	ADJ
ejpam-3576	215	15	ideal	ideal	NOUN
ejpam-3576	215	16	of	of	ADP
ejpam-3576	215	17	s.	s.	PROPN
ejpam-3576	215	18	then	then	ADV
ejpam-3576	215	19	χc	χc	PROPN
ejpam-3576	215	20	a(x	a(x	PROPN
ejpam-3576	215	21	)	)	PUNCT
ejpam-3576	215	22	≤	≤	NOUN
ejpam-3576	215	23	χc	χc	ADP
ejpam-3576	215	24	a(y	a(y	PROPN
ejpam-3576	215	25	)	)	PUNCT
ejpam-3576	215	26	,	,	PUNCT
ejpam-3576	215	27	by	by	ADP
ejpam-3576	215	28	the	the	DET
ejpam-3576	215	29	proposition	proposition	NOUN
ejpam-3576	215	30	1	1	X
ejpam-3576	215	31	.	.	PUNCT
ejpam-3576	216	1	if	if	SCONJ
ejpam-3576	216	2	y	y	PROPN
ejpam-3576	216	3	/∈	/∈	VERB
ejpam-3576	216	4	a	a	X
ejpam-3576	216	5	,	,	PUNCT
ejpam-3576	216	6	by	by	ADP
ejpam-3576	216	7	definition	definition	NOUN
ejpam-3576	216	8	χc	χc	PROPN
ejpam-3576	216	9	a(y	a(y	PROPN
ejpam-3576	216	10	)	)	PUNCT
ejpam-3576	217	1	=	=	SYM
ejpam-3576	217	2	1	1	X
ejpam-3576	217	3	.	.	PUNCT
ejpam-3576	217	4	thus	thus	ADV
ejpam-3576	217	5	χc	χc	PROPN
ejpam-3576	217	6	a(xy	a(xy	PROPN
ejpam-3576	217	7	)	)	PUNCT
ejpam-3576	217	8	≤	≤	NUM
ejpam-3576	217	9	χc	χc	ADP
ejpam-3576	217	10	a(y	a(y	PROPN
ejpam-3576	217	11	)	)	PUNCT
ejpam-3576	217	12	.	.	PUNCT
ejpam-3576	218	1	if	if	SCONJ
ejpam-3576	218	2	y	y	PROPN
ejpam-3576	218	3	∈	∈	PROPN
ejpam-3576	218	4	a	a	X
ejpam-3576	218	5	,	,	PUNCT
ejpam-3576	218	6	by	by	ADP
ejpam-3576	218	7	definition	definition	NOUN
ejpam-3576	218	8	χc	χc	PROPN
ejpam-3576	218	9	a(y	a(y	PROPN
ejpam-3576	218	10	)	)	PUNCT
ejpam-3576	219	1	=	=	PUNCT
ejpam-3576	219	2	0	0	X
ejpam-3576	219	3	.	.	PUNCT
ejpam-3576	220	1	xy	xy	PROPN
ejpam-3576	220	2	∈	∈	PROPN
ejpam-3576	221	1	a	a	PRON
ejpam-3576	221	2	,	,	PUNCT
ejpam-3576	221	3	a	a	DET
ejpam-3576	221	4	being	be	AUX
ejpam-3576	221	5	a	a	DET
ejpam-3576	221	6	left	left	ADJ
ejpam-3576	221	7	ideal	ideal	NOUN
ejpam-3576	221	8	,	,	PUNCT
ejpam-3576	221	9	so	so	SCONJ
ejpam-3576	221	10	χc	χc	PROPN
ejpam-3576	221	11	a(xy	a(xy	PROPN
ejpam-3576	221	12	)	)	PUNCT
ejpam-3576	222	1	=	=	PUNCT
ejpam-3576	222	2	0	0	X
ejpam-3576	222	3	.	.	PUNCT
ejpam-3576	223	1	thus	thus	ADV
ejpam-3576	223	2	χc	χc	PROPN
ejpam-3576	223	3	a(xy	a(xy	PROPN
ejpam-3576	223	4	)	)	PUNCT
ejpam-3576	223	5	≤	≤	NUM
ejpam-3576	223	6	χc	χc	ADP
ejpam-3576	223	7	a(y	a(y	PROPN
ejpam-3576	223	8	)	)	PUNCT
ejpam-3576	223	9	.	.	PUNCT
ejpam-3576	224	1	hence	hence	ADV
ejpam-3576	224	2	the	the	DET
ejpam-3576	224	3	anti	anti	ADJ
ejpam-3576	224	4	characteristic	characteristic	ADJ
ejpam-3576	224	5	function	function	NOUN
ejpam-3576	224	6	χc	χc	PRON
ejpam-3576	224	7	a	a	PRON
ejpam-3576	224	8	of	of	ADP
ejpam-3576	224	9	a	a	PRON
ejpam-3576	224	10	is	be	AUX
ejpam-3576	224	11	an	an	DET
ejpam-3576	224	12	anti	anti	ADJ
ejpam-3576	224	13	fuzzy	fuzzy	ADJ
ejpam-3576	224	14	left	leave	VERB
ejpam-3576	224	15	ideal	ideal	NOUN
ejpam-3576	224	16	of	of	ADP
ejpam-3576	224	17	s.	s.	PROPN
ejpam-3576	224	18	conversely	conversely	ADV
ejpam-3576	224	19	,	,	PUNCT
ejpam-3576	224	20	let	let	VERB
ejpam-3576	224	21	y	y	PROPN
ejpam-3576	224	22	∈	∈	PROPN
ejpam-3576	224	23	a	a	PRON
ejpam-3576	224	24	and	and	CCONJ
ejpam-3576	224	25	x	x	SYM
ejpam-3576	224	26	∈	∈	NOUN
ejpam-3576	224	27	s	s	VERB
ejpam-3576	224	28	such	such	ADJ
ejpam-3576	224	29	that	that	SCONJ
ejpam-3576	224	30	x	x	X
ejpam-3576	224	31	≤	≤	X
ejpam-3576	224	32	y.	y.	NOUN
ejpam-3576	224	33	this	this	PRON
ejpam-3576	224	34	imply	imply	VERB
ejpam-3576	224	35	that	that	SCONJ
ejpam-3576	224	36	χc	χc	PROPN
ejpam-3576	224	37	a(x	a(x	PROPN
ejpam-3576	224	38	)	)	PUNCT
ejpam-3576	224	39	≤	≤	NOUN
ejpam-3576	224	40	χc	χc	ADP
ejpam-3576	224	41	a(y	a(y	PROPN
ejpam-3576	224	42	)	)	PUNCT
ejpam-3576	224	43	,	,	PUNCT
ejpam-3576	224	44	χc	χc	PROPN
ejpam-3576	224	45	a	a	DET
ejpam-3576	224	46	being	be	AUX
ejpam-3576	224	47	an	an	DET
ejpam-3576	224	48	anti	anti	ADJ
ejpam-3576	224	49	fuzzy	fuzzy	ADJ
ejpam-3576	224	50	left	leave	VERB
ejpam-3576	224	51	ideal	ideal	NOUN
ejpam-3576	224	52	of	of	ADP
ejpam-3576	224	53	s.	s.	PROPN
ejpam-3576	224	54	then	then	ADV
ejpam-3576	224	55	a	a	PRON
ejpam-3576	224	56	=	=	X
ejpam-3576	224	57	(	(	PUNCT
ejpam-3576	224	58	a	a	PRON
ejpam-3576	224	59	]	]	X
ejpam-3576	224	60	,	,	PUNCT
ejpam-3576	224	61	by	by	ADP
ejpam-3576	224	62	the	the	DET
ejpam-3576	224	63	proposition	proposition	NOUN
ejpam-3576	224	64	1	1	X
ejpam-3576	224	65	.	.	PUNCT
ejpam-3576	225	1	let	let	VERB
ejpam-3576	225	2	xy	xy	PROPN
ejpam-3576	225	3	∈	∈	PROPN
ejpam-3576	225	4	sa	sa	PROPN
ejpam-3576	225	5	,	,	PUNCT
ejpam-3576	225	6	where	where	SCONJ
ejpam-3576	225	7	y	y	PROPN
ejpam-3576	225	8	∈	∈	PROPN
ejpam-3576	225	9	a	a	PRON
ejpam-3576	225	10	,	,	PUNCT
ejpam-3576	225	11	x	x	SYM
ejpam-3576	225	12	∈	∈	PROPN
ejpam-3576	225	13	s.	s.	PROPN
ejpam-3576	225	14	by	by	ADP
ejpam-3576	225	15	definition	definition	NOUN
ejpam-3576	225	16	of	of	ADP
ejpam-3576	225	17	anti	anti	ADJ
ejpam-3576	225	18	characteristic	characteristic	ADJ
ejpam-3576	225	19	function	function	NOUN
ejpam-3576	226	1	χc	χc	PROPN
ejpam-3576	226	2	a(y	a(y	PROPN
ejpam-3576	226	3	)	)	PUNCT
ejpam-3576	227	1	=	=	PUNCT
ejpam-3576	227	2	0	0	X
ejpam-3576	227	3	.	.	PUNCT
ejpam-3576	227	4	χc	χc	PROPN
ejpam-3576	227	5	a(xy	a(xy	PROPN
ejpam-3576	227	6	)	)	PUNCT
ejpam-3576	227	7	≤	≤	NUM
ejpam-3576	227	8	χc	χc	ADP
ejpam-3576	228	1	a(y	a(y	PROPN
ejpam-3576	228	2	)	)	PUNCT
ejpam-3576	229	1	=	=	PUNCT
ejpam-3576	229	2	0	0	NUM
ejpam-3576	229	3	,	,	PUNCT
ejpam-3576	229	4	χc	χc	PRON
ejpam-3576	229	5	a	a	DET
ejpam-3576	229	6	being	be	AUX
ejpam-3576	229	7	an	an	DET
ejpam-3576	229	8	anti	anti	ADJ
ejpam-3576	229	9	fuzzy	fuzzy	ADJ
ejpam-3576	229	10	left	leave	VERB
ejpam-3576	229	11	ideal	ideal	NOUN
ejpam-3576	229	12	of	of	ADP
ejpam-3576	229	13	s.	s.	PROPN
ejpam-3576	229	14	thus	thus	ADV
ejpam-3576	229	15	χc	χc	PROPN
ejpam-3576	229	16	a(xy	a(xy	PROPN
ejpam-3576	229	17	)	)	PUNCT
ejpam-3576	230	1	=	=	SYM
ejpam-3576	230	2	0	0	NUM
ejpam-3576	230	3	,	,	PUNCT
ejpam-3576	230	4	i.e.	i.e.	X
ejpam-3576	230	5	,	,	PUNCT
ejpam-3576	230	6	xy	xy	PROPN
ejpam-3576	230	7	∈	∈	PROPN
ejpam-3576	230	8	a.	a.	NOUN
ejpam-3576	230	9	hence	hence	ADV
ejpam-3576	230	10	a	a	PRON
ejpam-3576	230	11	is	be	AUX
ejpam-3576	230	12	a	a	DET
ejpam-3576	230	13	left	left	ADJ
ejpam-3576	230	14	ideal	ideal	NOUN
ejpam-3576	230	15	of	of	ADP
ejpam-3576	230	16	s.	s.	PROPN
ejpam-3576	230	17	proposition	proposition	PROPN
ejpam-3576	230	18	2	2	X
ejpam-3576	230	19	.	.	PUNCT
ejpam-3576	231	1	let	let	VERB
ejpam-3576	231	2	s	s	PRON
ejpam-3576	231	3	be	be	AUX
ejpam-3576	231	4	an	an	DET
ejpam-3576	231	5	ordered	order	VERB
ejpam-3576	231	6	ag	ag	PROPN
ejpam-3576	231	7	-	-	PUNCT
ejpam-3576	231	8	groupoid	groupoid	PROPN
ejpam-3576	231	9	and	and	CCONJ
ejpam-3576	231	10	∅	∅	NOUN
ejpam-3576	231	11	6=	6=	ADP
ejpam-3576	231	12	a	a	DET
ejpam-3576	231	13	⊆	⊆	NUM
ejpam-3576	231	14	s.	s.	PROPN
ejpam-3576	231	15	then	then	ADV
ejpam-3576	231	16	a	a	PRON
ejpam-3576	231	17	is	be	AUX
ejpam-3576	231	18	an	an	DET
ejpam-3576	231	19	interior	interior	ADJ
ejpam-3576	231	20	ideal	ideal	NOUN
ejpam-3576	231	21	of	of	ADP
ejpam-3576	231	22	s	s	PRON
ejpam-3576	231	23	if	if	SCONJ
ejpam-3576	232	1	and	and	CCONJ
ejpam-3576	232	2	only	only	ADV
ejpam-3576	232	3	if	if	SCONJ
ejpam-3576	232	4	the	the	DET
ejpam-3576	232	5	anti	anti	X
ejpam-3576	232	6	characteristic	characteristic	ADJ
ejpam-3576	232	7	function	function	NOUN
ejpam-3576	232	8	χc	χc	PRON
ejpam-3576	232	9	a	a	PRON
ejpam-3576	232	10	of	of	ADP
ejpam-3576	232	11	a	a	PRON
ejpam-3576	232	12	is	be	AUX
ejpam-3576	232	13	an	an	DET
ejpam-3576	232	14	anti	anti	ADJ
ejpam-3576	232	15	fuzzy	fuzzy	ADJ
ejpam-3576	232	16	interior	interior	ADJ
ejpam-3576	232	17	ideal	ideal	NOUN
ejpam-3576	232	18	of	of	ADP
ejpam-3576	232	19	s.	s.	PROPN
ejpam-3576	232	20	proof	proof	PROPN
ejpam-3576	232	21	.	.	PUNCT
ejpam-3576	233	1	suppose	suppose	VERB
ejpam-3576	233	2	a	a	PRON
ejpam-3576	233	3	is	be	AUX
ejpam-3576	233	4	an	an	DET
ejpam-3576	233	5	interior	interior	ADJ
ejpam-3576	233	6	ideal	ideal	NOUN
ejpam-3576	233	7	of	of	ADP
ejpam-3576	233	8	s	s	PRON
ejpam-3576	233	9	and	and	CCONJ
ejpam-3576	233	10	a	a	PRON
ejpam-3576	233	11	,	,	PUNCT
ejpam-3576	233	12	x	x	X
ejpam-3576	233	13	,	,	PUNCT
ejpam-3576	233	14	y	y	PROPN
ejpam-3576	233	15	∈	∈	PROPN
ejpam-3576	233	16	s	s	VERB
ejpam-3576	233	17	such	such	ADJ
ejpam-3576	233	18	that	that	SCONJ
ejpam-3576	233	19	x	x	X
ejpam-3576	234	1	≤	≤	X
ejpam-3576	234	2	y.	y.	NOUN
ejpam-3576	234	3	this	this	PRON
ejpam-3576	234	4	imply	imply	VERB
ejpam-3576	234	5	that	that	SCONJ
ejpam-3576	234	6	a	a	PRON
ejpam-3576	234	7	=	=	X
ejpam-3576	234	8	(	(	PUNCT
ejpam-3576	234	9	a	a	X
ejpam-3576	234	10	]	]	X
ejpam-3576	234	11	,	,	PUNCT
ejpam-3576	234	12	a	a	DET
ejpam-3576	234	13	being	be	AUX
ejpam-3576	234	14	an	an	DET
ejpam-3576	234	15	interior	interior	NOUN
ejpam-3576	234	16	-	-	PUNCT
ejpam-3576	234	17	ideal	ideal	NOUN
ejpam-3576	234	18	.	.	PUNCT
ejpam-3576	235	1	then	then	ADV
ejpam-3576	235	2	χc	χc	PROPN
ejpam-3576	235	3	a(x	a(x	PROPN
ejpam-3576	235	4	)	)	PUNCT
ejpam-3576	235	5	≤	≤	NOUN
ejpam-3576	235	6	χc	χc	ADP
ejpam-3576	235	7	a(y	a(y	PROPN
ejpam-3576	235	8	)	)	PUNCT
ejpam-3576	235	9	,	,	PUNCT
ejpam-3576	235	10	by	by	ADP
ejpam-3576	235	11	the	the	DET
ejpam-3576	235	12	proposition	proposition	NOUN
ejpam-3576	235	13	1	1	NUM
ejpam-3576	235	14	.	.	PUNCT
ejpam-3576	236	1	if	if	SCONJ
ejpam-3576	236	2	a	a	DET
ejpam-3576	236	3	/∈	/∈	NOUN
ejpam-3576	236	4	a	a	NOUN
ejpam-3576	236	5	,	,	PUNCT
ejpam-3576	236	6	by	by	ADP
ejpam-3576	236	7	definition	definition	NOUN
ejpam-3576	236	8	χc	χc	PROPN
ejpam-3576	236	9	a(a	a(a	PROPN
ejpam-3576	236	10	)	)	PUNCT
ejpam-3576	237	1	=	=	PUNCT
ejpam-3576	237	2	1	1	X
ejpam-3576	237	3	.	.	PUNCT
ejpam-3576	237	4	thus	thus	ADV
ejpam-3576	237	5	χc	χc	ADP
ejpam-3576	237	6	a((xa)y	a((xa)y	NOUN
ejpam-3576	237	7	)	)	PUNCT
ejpam-3576	237	8	≤	≤	NUM
ejpam-3576	237	9	χc	χc	PRON
ejpam-3576	237	10	a(a	a(a	PROPN
ejpam-3576	237	11	)	)	PUNCT
ejpam-3576	237	12	.	.	PUNCT
ejpam-3576	238	1	if	if	SCONJ
ejpam-3576	238	2	a	a	DET
ejpam-3576	238	3	∈	∈	PROPN
ejpam-3576	238	4	a	a	X
ejpam-3576	238	5	,	,	PUNCT
ejpam-3576	238	6	by	by	ADP
ejpam-3576	238	7	definition	definition	NOUN
ejpam-3576	238	8	k.	k.	PROPN
ejpam-3576	238	9	nasreen	nasreen	PROPN
ejpam-3576	238	10	,	,	PUNCT
ejpam-3576	238	11	m.	m.	NOUN
ejpam-3576	238	12	alesemi	alesemi	PROPN
ejpam-3576	238	13	,	,	PUNCT
ejpam-3576	238	14	salahuddin	salahuddin	VERB
ejpam-3576	238	15	/	/	SYM
ejpam-3576	238	16	eur	eur	PROPN
ejpam-3576	238	17	.	.	PUNCT
ejpam-3576	239	1	j.	j.	PROPN
ejpam-3576	239	2	pure	pure	PROPN
ejpam-3576	239	3	appl	appl	PROPN
ejpam-3576	239	4	.	.	PROPN
ejpam-3576	239	5	math	math	PROPN
ejpam-3576	239	6	,	,	PUNCT
ejpam-3576	239	7	13	13	NUM
ejpam-3576	239	8	(	(	PUNCT
ejpam-3576	239	9	1	1	NUM
ejpam-3576	239	10	)	)	PUNCT
ejpam-3576	239	11	(	(	PUNCT
ejpam-3576	239	12	2020	2020	NUM
ejpam-3576	239	13	)	)	PUNCT
ejpam-3576	239	14	,	,	PUNCT
ejpam-3576	239	15	113	113	NUM
ejpam-3576	239	16	-	-	SYM
ejpam-3576	239	17	129	129	NUM
ejpam-3576	239	18	120	120	NUM
ejpam-3576	239	19	χc	χc	NOUN
ejpam-3576	239	20	a(a	a(a	PROPN
ejpam-3576	239	21	)	)	PUNCT
ejpam-3576	240	1	=	=	SYM
ejpam-3576	240	2	0	0	X
ejpam-3576	240	3	.	.	PUNCT
ejpam-3576	241	1	(	(	PUNCT
ejpam-3576	241	2	xa)y	xa)y	PROPN
ejpam-3576	241	3	∈	∈	PROPN
ejpam-3576	241	4	a	a	PRON
ejpam-3576	241	5	,	,	PUNCT
ejpam-3576	241	6	a	a	DET
ejpam-3576	241	7	being	be	AUX
ejpam-3576	241	8	an	an	DET
ejpam-3576	241	9	interior	interior	ADJ
ejpam-3576	241	10	ideal	ideal	NOUN
ejpam-3576	241	11	,	,	PUNCT
ejpam-3576	241	12	this	this	PRON
ejpam-3576	241	13	imply	imply	VERB
ejpam-3576	241	14	that	that	SCONJ
ejpam-3576	241	15	χc	χc	PROPN
ejpam-3576	241	16	a((xa)y	a((xa)y	PROPN
ejpam-3576	241	17	)	)	PUNCT
ejpam-3576	241	18	=	=	SYM
ejpam-3576	241	19	0	0	X
ejpam-3576	241	20	.	.	PUNCT
ejpam-3576	242	1	thus	thus	ADV
ejpam-3576	242	2	χc	χc	ADP
ejpam-3576	242	3	a((xa)y	a((xa)y	NOUN
ejpam-3576	242	4	)	)	PUNCT
ejpam-3576	242	5	≤	≤	NUM
ejpam-3576	242	6	χc	χc	PRON
ejpam-3576	242	7	a(a	a(a	PROPN
ejpam-3576	242	8	)	)	PUNCT
ejpam-3576	242	9	.	.	PUNCT
ejpam-3576	243	1	hence	hence	ADV
ejpam-3576	243	2	the	the	DET
ejpam-3576	243	3	anti	anti	ADJ
ejpam-3576	243	4	characteristic	characteristic	ADJ
ejpam-3576	243	5	function	function	NOUN
ejpam-3576	243	6	χc	χc	PRON
ejpam-3576	243	7	a	a	PRON
ejpam-3576	243	8	of	of	ADP
ejpam-3576	243	9	a	a	PRON
ejpam-3576	243	10	is	be	AUX
ejpam-3576	243	11	an	an	DET
ejpam-3576	243	12	anti	anti	ADJ
ejpam-3576	243	13	fuzzy	fuzzy	ADJ
ejpam-3576	243	14	interior	interior	ADJ
ejpam-3576	243	15	ideal	ideal	NOUN
ejpam-3576	243	16	of	of	ADP
ejpam-3576	243	17	s.	s.	PROPN
ejpam-3576	243	18	conversely	conversely	ADV
ejpam-3576	243	19	,	,	PUNCT
ejpam-3576	243	20	let	let	VERB
ejpam-3576	243	21	y	y	PROPN
ejpam-3576	243	22	∈	∈	PROPN
ejpam-3576	243	23	a	a	PRON
ejpam-3576	243	24	and	and	CCONJ
ejpam-3576	243	25	x	x	SYM
ejpam-3576	243	26	∈	∈	NOUN
ejpam-3576	243	27	s	s	VERB
ejpam-3576	243	28	such	such	ADJ
ejpam-3576	243	29	that	that	SCONJ
ejpam-3576	243	30	x	x	X
ejpam-3576	243	31	≤	≤	X
ejpam-3576	243	32	y.	y.	NOUN
ejpam-3576	243	33	this	this	PRON
ejpam-3576	243	34	imply	imply	VERB
ejpam-3576	243	35	that	that	SCONJ
ejpam-3576	243	36	χc	χc	PROPN
ejpam-3576	243	37	a(x	a(x	PROPN
ejpam-3576	243	38	)	)	PUNCT
ejpam-3576	243	39	≤	≤	NOUN
ejpam-3576	243	40	χc	χc	ADP
ejpam-3576	243	41	a(y	a(y	PROPN
ejpam-3576	243	42	)	)	PUNCT
ejpam-3576	243	43	,	,	PUNCT
ejpam-3576	243	44	χc	χc	PROPN
ejpam-3576	243	45	a	a	DET
ejpam-3576	243	46	being	be	AUX
ejpam-3576	243	47	an	an	DET
ejpam-3576	243	48	anti	anti	ADJ
ejpam-3576	243	49	fuzzy	fuzzy	ADJ
ejpam-3576	243	50	interior	interior	ADJ
ejpam-3576	243	51	ideal	ideal	NOUN
ejpam-3576	243	52	of	of	ADP
ejpam-3576	243	53	s.	s.	PROPN
ejpam-3576	243	54	then	then	ADV
ejpam-3576	243	55	a	a	PRON
ejpam-3576	243	56	=	=	X
ejpam-3576	243	57	(	(	PUNCT
ejpam-3576	243	58	a	a	PRON
ejpam-3576	243	59	]	]	X
ejpam-3576	243	60	,	,	PUNCT
ejpam-3576	243	61	by	by	ADP
ejpam-3576	243	62	the	the	DET
ejpam-3576	243	63	proposition	proposition	NOUN
ejpam-3576	243	64	1	1	X
ejpam-3576	243	65	.	.	PUNCT
ejpam-3576	244	1	let	let	VERB
ejpam-3576	244	2	t	t	PROPN
ejpam-3576	244	3	∈	∈	PROPN
ejpam-3576	244	4	(	(	PUNCT
ejpam-3576	244	5	sa)s	sa)s	PROPN
ejpam-3576	244	6	,	,	PUNCT
ejpam-3576	244	7	implies	imply	VERB
ejpam-3576	244	8	t	t	PROPN
ejpam-3576	244	9	=	=	SYM
ejpam-3576	244	10	(	(	PUNCT
ejpam-3576	244	11	xa)y	xa)y	PROPN
ejpam-3576	244	12	,	,	PUNCT
ejpam-3576	244	13	where	where	SCONJ
ejpam-3576	244	14	a	a	DET
ejpam-3576	244	15	∈	∈	PROPN
ejpam-3576	244	16	a	a	PRON
ejpam-3576	244	17	and	and	CCONJ
ejpam-3576	244	18	x	x	NOUN
ejpam-3576	244	19	,	,	PUNCT
ejpam-3576	244	20	y	y	PROPN
ejpam-3576	244	21	∈	∈	PROPN
ejpam-3576	244	22	s.	s.	PROPN
ejpam-3576	244	23	by	by	ADP
ejpam-3576	244	24	definition	definition	NOUN
ejpam-3576	244	25	of	of	ADP
ejpam-3576	244	26	anti	anti	ADJ
ejpam-3576	244	27	characteristic	characteristic	ADJ
ejpam-3576	244	28	function	function	NOUN
ejpam-3576	244	29	χc	χc	PROPN
ejpam-3576	244	30	a(a	a(a	PROPN
ejpam-3576	244	31	)	)	PUNCT
ejpam-3576	245	1	=	=	PUNCT
ejpam-3576	246	1	0	0	X
ejpam-3576	246	2	.	.	PUNCT
ejpam-3576	246	3	χc	χc	PROPN
ejpam-3576	246	4	a((xa)y	a((xa)y	PROPN
ejpam-3576	246	5	)	)	PUNCT
ejpam-3576	246	6	≤	≤	NUM
ejpam-3576	246	7	χc	χc	PRON
ejpam-3576	246	8	a(a	a(a	PROPN
ejpam-3576	246	9	)	)	PUNCT
ejpam-3576	247	1	=	=	PUNCT
ejpam-3576	247	2	0	0	NUM
ejpam-3576	247	3	,	,	PUNCT
ejpam-3576	247	4	χc	χc	PRON
ejpam-3576	247	5	a	a	DET
ejpam-3576	247	6	being	be	AUX
ejpam-3576	247	7	an	an	DET
ejpam-3576	247	8	anti	anti	ADJ
ejpam-3576	247	9	fuzzy	fuzzy	ADJ
ejpam-3576	247	10	interior	interior	ADJ
ejpam-3576	247	11	ideal	ideal	NOUN
ejpam-3576	247	12	of	of	ADP
ejpam-3576	247	13	s.	s.	PROPN
ejpam-3576	247	14	thus	thus	ADV
ejpam-3576	247	15	χc	χc	PROPN
ejpam-3576	247	16	a((xa)y	a((xa)y	PROPN
ejpam-3576	247	17	)	)	PUNCT
ejpam-3576	247	18	=	=	SYM
ejpam-3576	247	19	0	0	NUM
ejpam-3576	247	20	,	,	PUNCT
ejpam-3576	247	21	i.e.	i.e.	X
ejpam-3576	247	22	,	,	PUNCT
ejpam-3576	247	23	(	(	PUNCT
ejpam-3576	247	24	xa)y	xa)y	PROPN
ejpam-3576	247	25	∈	∈	PROPN
ejpam-3576	247	26	a.	a.	NOUN
ejpam-3576	247	27	hence	hence	ADV
ejpam-3576	247	28	a	a	PRON
ejpam-3576	247	29	is	be	AUX
ejpam-3576	247	30	an	an	DET
ejpam-3576	247	31	interior	interior	ADJ
ejpam-3576	247	32	ideal	ideal	NOUN
ejpam-3576	247	33	of	of	ADP
ejpam-3576	247	34	s.	s.	PROPN
ejpam-3576	247	35	lemma	lemma	PROPN
ejpam-3576	248	1	4	4	X
ejpam-3576	248	2	.	.	PUNCT
ejpam-3576	248	3	let	let	VERB
ejpam-3576	248	4	µ	µ	X
ejpam-3576	248	5	be	be	AUX
ejpam-3576	248	6	a	a	DET
ejpam-3576	248	7	fuzzy	fuzzy	ADJ
ejpam-3576	248	8	subset	subset	NOUN
ejpam-3576	248	9	of	of	ADP
ejpam-3576	248	10	an	an	DET
ejpam-3576	248	11	ordered	order	VERB
ejpam-3576	248	12	ag	ag	PROPN
ejpam-3576	248	13	-	-	PROPN
ejpam-3576	248	14	groupoid	groupoid	PROPN
ejpam-3576	248	15	s.	s.	PROPN
ejpam-3576	248	16	then	then	ADV
ejpam-3576	248	17	µ	µ	PROPN
ejpam-3576	248	18	is	be	AUX
ejpam-3576	248	19	an	an	DET
ejpam-3576	248	20	anti	anti	ADJ
ejpam-3576	248	21	fuzzy	fuzzy	ADJ
ejpam-3576	248	22	ag	ag	PROPN
ejpam-3576	248	23	-	-	NOUN
ejpam-3576	248	24	subgroupoid	subgroupoid	NOUN
ejpam-3576	248	25	of	of	ADP
ejpam-3576	248	26	s	s	PRON
ejpam-3576	248	27	if	if	SCONJ
ejpam-3576	248	28	and	and	CCONJ
ejpam-3576	248	29	only	only	ADV
ejpam-3576	248	30	if	if	SCONJ
ejpam-3576	248	31	lower	low	ADJ
ejpam-3576	248	32	t	t	NOUN
ejpam-3576	248	33	-	-	PUNCT
ejpam-3576	248	34	level	level	NOUN
ejpam-3576	248	35	l(µ	l(µ	NOUN
ejpam-3576	248	36	;	;	PUNCT
ejpam-3576	248	37	t	t	PROPN
ejpam-3576	248	38	)	)	PUNCT
ejpam-3576	248	39	of	of	ADP
ejpam-3576	248	40	µ	µ	PROPN
ejpam-3576	248	41	is	be	AUX
ejpam-3576	248	42	an	an	DET
ejpam-3576	248	43	ag	ag	NOUN
ejpam-3576	248	44	-	-	PUNCT
ejpam-3576	248	45	subgroupoid	subgroupoid	NOUN
ejpam-3576	248	46	of	of	ADP
ejpam-3576	248	47	s	s	PRON
ejpam-3576	248	48	for	for	ADP
ejpam-3576	248	49	all	all	DET
ejpam-3576	248	50	t	t	NOUN
ejpam-3576	248	51	∈	∈	PROPN
ejpam-3576	248	52	(	(	PUNCT
ejpam-3576	248	53	0	0	NUM
ejpam-3576	248	54	,	,	PUNCT
ejpam-3576	248	55	1	1	NUM
ejpam-3576	248	56	]	]	PUNCT
ejpam-3576	248	57	.	.	PUNCT
ejpam-3576	249	1	proof	proof	NOUN
ejpam-3576	249	2	.	.	PUNCT
ejpam-3576	250	1	suppose	suppose	VERB
ejpam-3576	250	2	µ	µ	PRON
ejpam-3576	250	3	is	be	AUX
ejpam-3576	250	4	an	an	DET
ejpam-3576	250	5	anti	anti	ADJ
ejpam-3576	250	6	fuzzy	fuzzy	ADJ
ejpam-3576	250	7	ag	ag	PROPN
ejpam-3576	250	8	-	-	NOUN
ejpam-3576	250	9	subgroupoid	subgroupoid	NOUN
ejpam-3576	250	10	of	of	ADP
ejpam-3576	250	11	s	s	PROPN
ejpam-3576	250	12	and	and	CCONJ
ejpam-3576	250	13	x	x	NOUN
ejpam-3576	250	14	,	,	PUNCT
ejpam-3576	250	15	y	y	PROPN
ejpam-3576	250	16	∈	∈	PROPN
ejpam-3576	250	17	l(µ	l(µ	PROPN
ejpam-3576	250	18	;	;	PUNCT
ejpam-3576	250	19	t	t	PROPN
ejpam-3576	250	20	)	)	PUNCT
ejpam-3576	250	21	,	,	PUNCT
ejpam-3576	250	22	this	this	PRON
ejpam-3576	250	23	imply	imply	VERB
ejpam-3576	250	24	that	that	SCONJ
ejpam-3576	250	25	µ(x	µ(x	VERB
ejpam-3576	250	26	)	)	PUNCT
ejpam-3576	250	27	,	,	PUNCT
ejpam-3576	250	28	µ(y	µ(y	PROPN
ejpam-3576	250	29	)	)	PUNCT
ejpam-3576	250	30	≤	≤	NOUN
ejpam-3576	250	31	t.	t.	PROPN
ejpam-3576	250	32	µ(xy	µ(xy	PROPN
ejpam-3576	250	33	)	)	PUNCT
ejpam-3576	250	34	≤	≤	NOUN
ejpam-3576	250	35	µ(x	µ(x	X
ejpam-3576	250	36	)	)	PUNCT
ejpam-3576	250	37	∨	∨	NUM
ejpam-3576	250	38	µ(y	µ(y	PROPN
ejpam-3576	250	39	)	)	PUNCT
ejpam-3576	250	40	≤	≤	NOUN
ejpam-3576	250	41	t	t	PROPN
ejpam-3576	250	42	,	,	PUNCT
ejpam-3576	250	43	µ	µ	X
ejpam-3576	250	44	being	be	AUX
ejpam-3576	250	45	an	an	DET
ejpam-3576	250	46	anti	anti	ADJ
ejpam-3576	250	47	fuzzy	fuzzy	ADJ
ejpam-3576	250	48	ag	ag	PROPN
ejpam-3576	250	49	-	-	NOUN
ejpam-3576	250	50	subgroupoid	subgroupoid	NOUN
ejpam-3576	250	51	,	,	PUNCT
ejpam-3576	250	52	i.e.	i.e.	X
ejpam-3576	250	53	,	,	PUNCT
ejpam-3576	250	54	xy	xy	PROPN
ejpam-3576	250	55	∈	∈	PROPN
ejpam-3576	251	1	l(µ	l(µ	PROPN
ejpam-3576	251	2	;	;	PUNCT
ejpam-3576	251	3	t	t	PROPN
ejpam-3576	251	4	)	)	PUNCT
ejpam-3576	251	5	.	.	PUNCT
ejpam-3576	252	1	hence	hence	ADV
ejpam-3576	252	2	l(µ	l(µ	PROPN
ejpam-3576	252	3	;	;	PUNCT
ejpam-3576	252	4	t	t	PROPN
ejpam-3576	252	5	)	)	PUNCT
ejpam-3576	252	6	is	be	AUX
ejpam-3576	252	7	an	an	DET
ejpam-3576	252	8	ag	ag	NOUN
ejpam-3576	252	9	-	-	PUNCT
ejpam-3576	252	10	subgroupoid	subgroupoid	NOUN
ejpam-3576	252	11	of	of	ADP
ejpam-3576	252	12	s.	s.	PROPN
ejpam-3576	252	13	conversely	conversely	ADV
ejpam-3576	252	14	,	,	PUNCT
ejpam-3576	252	15	we	we	PRON
ejpam-3576	252	16	have	have	VERB
ejpam-3576	252	17	to	to	PART
ejpam-3576	252	18	show	show	VERB
ejpam-3576	252	19	that	that	SCONJ
ejpam-3576	252	20	µ(xy	µ(xy	NOUN
ejpam-3576	252	21	)	)	PUNCT
ejpam-3576	252	22	≤	≤	NOUN
ejpam-3576	252	23	µ(x	µ(x	VERB
ejpam-3576	252	24	)	)	PUNCT
ejpam-3576	252	25	∨	∨	NUM
ejpam-3576	252	26	µ(y	µ(y	PROPN
ejpam-3576	252	27	)	)	PUNCT
ejpam-3576	252	28	,	,	PUNCT
ejpam-3576	252	29	x	x	X
ejpam-3576	252	30	,	,	PUNCT
ejpam-3576	252	31	y	y	PROPN
ejpam-3576	252	32	∈	∈	PROPN
ejpam-3576	252	33	s.	s.	PROPN
ejpam-3576	252	34	we	we	PRON
ejpam-3576	252	35	suppose	suppose	VERB
ejpam-3576	252	36	a	a	DET
ejpam-3576	252	37	contradiction	contradiction	NOUN
ejpam-3576	252	38	µ(xy	µ(xy	PROPN
ejpam-3576	252	39	)	)	PUNCT
ejpam-3576	252	40	>	>	X
ejpam-3576	252	41	µ(x)∧µ(y	µ(x)∧µ(y	NUM
ejpam-3576	252	42	)	)	PUNCT
ejpam-3576	252	43	.	.	PUNCT
ejpam-3576	253	1	assume	assume	VERB
ejpam-3576	253	2	µ(x	µ(x	NOUN
ejpam-3576	253	3	)	)	PUNCT
ejpam-3576	253	4	=	=	SYM
ejpam-3576	253	5	t	t	NOUN
ejpam-3576	253	6	=	=	SYM
ejpam-3576	253	7	µ(y	µ(y	PROPN
ejpam-3576	253	8	)	)	PUNCT
ejpam-3576	253	9	,	,	PUNCT
ejpam-3576	253	10	this	this	PRON
ejpam-3576	253	11	imply	imply	VERB
ejpam-3576	253	12	that	that	SCONJ
ejpam-3576	253	13	µ(x	µ(x	VERB
ejpam-3576	253	14	)	)	PUNCT
ejpam-3576	253	15	,	,	PUNCT
ejpam-3576	253	16	µ(y	µ(y	PROPN
ejpam-3576	253	17	)	)	PUNCT
ejpam-3576	253	18	≤	≤	NOUN
ejpam-3576	253	19	t	t	PROPN
ejpam-3576	253	20	,	,	PUNCT
ejpam-3576	253	21	i.e.	i.e.	X
ejpam-3576	253	22	,	,	PUNCT
ejpam-3576	253	23	x	x	PRON
ejpam-3576	253	24	,	,	PUNCT
ejpam-3576	253	25	y	y	PROPN
ejpam-3576	253	26	∈	∈	PROPN
ejpam-3576	253	27	l(µ	l(µ	PROPN
ejpam-3576	253	28	;	;	PUNCT
ejpam-3576	253	29	t	t	PROPN
ejpam-3576	253	30	)	)	PUNCT
ejpam-3576	253	31	.	.	PUNCT
ejpam-3576	254	1	but	but	CCONJ
ejpam-3576	254	2	µ(xy	µ(xy	PROPN
ejpam-3576	254	3	)	)	PUNCT
ejpam-3576	254	4	>	>	X
ejpam-3576	255	1	t	t	PROPN
ejpam-3576	255	2	,	,	PUNCT
ejpam-3576	255	3	i.e.	i.e.	X
ejpam-3576	255	4	,	,	PUNCT
ejpam-3576	255	5	xy	xy	NOUN
ejpam-3576	255	6	/∈	/∈	PUNCT
ejpam-3576	256	1	u(µ	u(µ	NOUN
ejpam-3576	256	2	;	;	PUNCT
ejpam-3576	256	3	t	t	PROPN
ejpam-3576	256	4	)	)	PUNCT
ejpam-3576	256	5	,	,	PUNCT
ejpam-3576	256	6	which	which	PRON
ejpam-3576	256	7	is	be	AUX
ejpam-3576	256	8	a	a	DET
ejpam-3576	256	9	contradiction	contradiction	NOUN
ejpam-3576	256	10	.	.	PUNCT
ejpam-3576	257	1	hence	hence	ADV
ejpam-3576	257	2	µ(xy	µ(xy	NUM
ejpam-3576	257	3	)	)	PUNCT
ejpam-3576	257	4	≤	≤	NOUN
ejpam-3576	257	5	µ(x	µ(x	X
ejpam-3576	257	6	)	)	PUNCT
ejpam-3576	257	7	∨	∨	NUM
ejpam-3576	257	8	µ(y	µ(y	PROPN
ejpam-3576	257	9	)	)	PUNCT
ejpam-3576	257	10	.	.	PUNCT
ejpam-3576	258	1	lemma	lemma	PROPN
ejpam-3576	258	2	5	5	X
ejpam-3576	258	3	.	.	PUNCT
ejpam-3576	259	1	let	let	VERB
ejpam-3576	259	2	µ	µ	X
ejpam-3576	259	3	be	be	AUX
ejpam-3576	259	4	a	a	DET
ejpam-3576	259	5	fuzzy	fuzzy	ADJ
ejpam-3576	259	6	subset	subset	NOUN
ejpam-3576	259	7	of	of	ADP
ejpam-3576	259	8	an	an	DET
ejpam-3576	259	9	ordered	order	VERB
ejpam-3576	259	10	ag	ag	PROPN
ejpam-3576	259	11	-	-	PROPN
ejpam-3576	259	12	groupoid	groupoid	PROPN
ejpam-3576	259	13	s.	s.	PROPN
ejpam-3576	259	14	then	then	ADV
ejpam-3576	259	15	µ	µ	PROPN
ejpam-3576	259	16	is	be	AUX
ejpam-3576	259	17	an	an	DET
ejpam-3576	259	18	anti	anti	ADJ
ejpam-3576	259	19	fuzzy	fuzzy	ADJ
ejpam-3576	259	20	left	left	ADJ
ejpam-3576	259	21	(	(	PUNCT
ejpam-3576	259	22	resp	resp	NOUN
ejpam-3576	259	23	.	.	PUNCT
ejpam-3576	260	1	right	right	ADJ
ejpam-3576	260	2	)	)	PUNCT
ejpam-3576	260	3	ideal	ideal	NOUN
ejpam-3576	260	4	of	of	ADP
ejpam-3576	260	5	s	s	PRON
ejpam-3576	260	6	if	if	SCONJ
ejpam-3576	261	1	and	and	CCONJ
ejpam-3576	261	2	only	only	ADV
ejpam-3576	261	3	if	if	SCONJ
ejpam-3576	261	4	lower	low	ADJ
ejpam-3576	261	5	t	t	NOUN
ejpam-3576	261	6	-	-	PUNCT
ejpam-3576	261	7	level	level	NOUN
ejpam-3576	261	8	l(µ	l(µ	NOUN
ejpam-3576	261	9	;	;	PUNCT
ejpam-3576	261	10	t	t	PROPN
ejpam-3576	261	11	)	)	PUNCT
ejpam-3576	261	12	of	of	ADP
ejpam-3576	261	13	µ	µ	PROPN
ejpam-3576	261	14	is	be	AUX
ejpam-3576	261	15	a	a	DET
ejpam-3576	261	16	left	left	ADJ
ejpam-3576	261	17	(	(	PUNCT
ejpam-3576	261	18	resp	resp	NOUN
ejpam-3576	261	19	.	.	PUNCT
ejpam-3576	262	1	right	right	ADJ
ejpam-3576	262	2	)	)	PUNCT
ejpam-3576	262	3	ideal	ideal	NOUN
ejpam-3576	262	4	of	of	ADP
ejpam-3576	262	5	s	s	PRON
ejpam-3576	262	6	for	for	ADP
ejpam-3576	262	7	all	all	DET
ejpam-3576	262	8	t	t	NOUN
ejpam-3576	262	9	∈	∈	PROPN
ejpam-3576	262	10	(	(	PUNCT
ejpam-3576	262	11	0	0	NUM
ejpam-3576	262	12	,	,	PUNCT
ejpam-3576	262	13	1	1	NUM
ejpam-3576	262	14	]	]	PUNCT
ejpam-3576	262	15	.	.	PUNCT
ejpam-3576	263	1	proof	proof	NOUN
ejpam-3576	263	2	.	.	PUNCT
ejpam-3576	264	1	suppose	suppose	VERB
ejpam-3576	264	2	µ	µ	PRON
ejpam-3576	264	3	is	be	AUX
ejpam-3576	264	4	an	an	DET
ejpam-3576	264	5	anti	anti	ADJ
ejpam-3576	264	6	fuzzy	fuzzy	ADJ
ejpam-3576	264	7	left	leave	VERB
ejpam-3576	264	8	ideal	ideal	NOUN
ejpam-3576	264	9	of	of	ADP
ejpam-3576	264	10	s.	s.	PROPN
ejpam-3576	264	11	let	let	VERB
ejpam-3576	264	12	y	y	PROPN
ejpam-3576	264	13	∈	∈	PROPN
ejpam-3576	264	14	l(µ	l(µ	PROPN
ejpam-3576	264	15	;	;	PUNCT
ejpam-3576	264	16	t	t	PROPN
ejpam-3576	264	17	)	)	PUNCT
ejpam-3576	264	18	and	and	CCONJ
ejpam-3576	264	19	x	x	PUNCT
ejpam-3576	264	20	∈	∈	NOUN
ejpam-3576	264	21	s	s	VERB
ejpam-3576	264	22	such	such	ADJ
ejpam-3576	264	23	that	that	SCONJ
ejpam-3576	264	24	x	x	X
ejpam-3576	264	25	≤	≤	NUM
ejpam-3576	264	26	y	y	PROPN
ejpam-3576	264	27	,	,	PUNCT
ejpam-3576	264	28	this	this	PRON
ejpam-3576	264	29	imply	imply	VERB
ejpam-3576	264	30	that	that	SCONJ
ejpam-3576	264	31	µ(y	µ(y	NOUN
ejpam-3576	264	32	)	)	PUNCT
ejpam-3576	264	33	≤	≤	NUM
ejpam-3576	265	1	t.	t.	NOUN
ejpam-3576	265	2	µ(x	µ(x	ADJ
ejpam-3576	265	3	)	)	PUNCT
ejpam-3576	265	4	≤	≤	NUM
ejpam-3576	265	5	µ(y	µ(y	NOUN
ejpam-3576	265	6	)	)	PUNCT
ejpam-3576	265	7	≤	≤	NOUN
ejpam-3576	265	8	t	t	NOUN
ejpam-3576	265	9	and	and	CCONJ
ejpam-3576	265	10	µ(xy	µ(xy	PROPN
ejpam-3576	265	11	)	)	PUNCT
ejpam-3576	265	12	≤	≤	NOUN
ejpam-3576	265	13	µ(y	µ(y	NOUN
ejpam-3576	265	14	)	)	PUNCT
ejpam-3576	265	15	≤	≤	NOUN
ejpam-3576	265	16	t	t	PROPN
ejpam-3576	265	17	,	,	PUNCT
ejpam-3576	265	18	µ	µ	X
ejpam-3576	265	19	being	be	AUX
ejpam-3576	265	20	an	an	DET
ejpam-3576	265	21	anti	anti	ADJ
ejpam-3576	265	22	fuzzy	fuzzy	ADJ
ejpam-3576	265	23	left	leave	VERB
ejpam-3576	265	24	ideal	ideal	NOUN
ejpam-3576	265	25	of	of	ADP
ejpam-3576	265	26	s.	s.	PROPN
ejpam-3576	265	27	thus	thus	ADV
ejpam-3576	265	28	x	x	PRON
ejpam-3576	265	29	,	,	PUNCT
ejpam-3576	265	30	xy	xy	PROPN
ejpam-3576	265	31	∈	∈	PROPN
ejpam-3576	266	1	l(µ	l(µ	PROPN
ejpam-3576	266	2	;	;	PUNCT
ejpam-3576	266	3	t	t	PROPN
ejpam-3576	266	4	)	)	PUNCT
ejpam-3576	266	5	.	.	PUNCT
ejpam-3576	267	1	hence	hence	ADV
ejpam-3576	267	2	l(µ	l(µ	PROPN
ejpam-3576	267	3	;	;	PUNCT
ejpam-3576	267	4	t	t	PROPN
ejpam-3576	267	5	)	)	PUNCT
ejpam-3576	267	6	is	be	AUX
ejpam-3576	267	7	a	a	DET
ejpam-3576	267	8	left	left	ADJ
ejpam-3576	267	9	ideal	ideal	NOUN
ejpam-3576	267	10	of	of	ADP
ejpam-3576	267	11	s.	s.	PROPN
ejpam-3576	267	12	conversely	conversely	ADV
ejpam-3576	267	13	,	,	PUNCT
ejpam-3576	267	14	suppose	suppose	VERB
ejpam-3576	267	15	l(µ	l(µ	PROPN
ejpam-3576	267	16	;	;	PUNCT
ejpam-3576	267	17	t	t	PROPN
ejpam-3576	267	18	)	)	PUNCT
ejpam-3576	267	19	is	be	AUX
ejpam-3576	267	20	a	a	DET
ejpam-3576	267	21	left	left	ADJ
ejpam-3576	267	22	ideal	ideal	NOUN
ejpam-3576	267	23	of	of	ADP
ejpam-3576	267	24	s	s	PRON
ejpam-3576	267	25	and	and	CCONJ
ejpam-3576	267	26	x	x	NOUN
ejpam-3576	267	27	,	,	PUNCT
ejpam-3576	267	28	y	y	PROPN
ejpam-3576	267	29	∈	∈	PROPN
ejpam-3576	267	30	s	s	VERB
ejpam-3576	267	31	such	such	ADJ
ejpam-3576	267	32	that	that	SCONJ
ejpam-3576	267	33	x	x	X
ejpam-3576	267	34	≤	≤	NUM
ejpam-3576	267	35	y.	y.	NOUN
ejpam-3576	267	36	we	we	PRON
ejpam-3576	267	37	have	have	VERB
ejpam-3576	267	38	to	to	PART
ejpam-3576	267	39	show	show	VERB
ejpam-3576	267	40	that	that	SCONJ
ejpam-3576	267	41	µ(x	µ(x	VERB
ejpam-3576	267	42	)	)	PUNCT
ejpam-3576	267	43	≤	≤	NUM
ejpam-3576	267	44	µ(y	µ(y	NUM
ejpam-3576	267	45	)	)	PUNCT
ejpam-3576	267	46	and	and	CCONJ
ejpam-3576	267	47	µ(xy	µ(xy	NUM
ejpam-3576	267	48	)	)	PUNCT
ejpam-3576	267	49	≤	≤	NOUN
ejpam-3576	267	50	µ(y	µ(y	NUM
ejpam-3576	267	51	)	)	PUNCT
ejpam-3576	267	52	.	.	PUNCT
ejpam-3576	268	1	we	we	PRON
ejpam-3576	268	2	suppose	suppose	VERB
ejpam-3576	268	3	a	a	DET
ejpam-3576	268	4	contradiction	contradiction	NOUN
ejpam-3576	268	5	µ(x	µ(x	VERB
ejpam-3576	268	6	)	)	PUNCT
ejpam-3576	268	7	>	>	X
ejpam-3576	268	8	µ(y	µ(y	PROPN
ejpam-3576	268	9	)	)	PUNCT
ejpam-3576	268	10	and	and	CCONJ
ejpam-3576	268	11	µ(xy	µ(xy	NUM
ejpam-3576	268	12	)	)	PUNCT
ejpam-3576	268	13	>	>	X
ejpam-3576	268	14	µ(y	µ(y	PROPN
ejpam-3576	268	15	)	)	PUNCT
ejpam-3576	268	16	.	.	PUNCT
ejpam-3576	269	1	let	let	VERB
ejpam-3576	269	2	µ(y	µ(y	NUM
ejpam-3576	269	3	)	)	PUNCT
ejpam-3576	269	4	=	=	SYM
ejpam-3576	269	5	t	t	PROPN
ejpam-3576	269	6	,	,	PUNCT
ejpam-3576	269	7	this	this	PRON
ejpam-3576	269	8	imply	imply	VERB
ejpam-3576	269	9	that	that	SCONJ
ejpam-3576	269	10	µ(y	µ(y	NOUN
ejpam-3576	269	11	)	)	PUNCT
ejpam-3576	269	12	≤	≤	NOUN
ejpam-3576	269	13	t	t	PROPN
ejpam-3576	269	14	,	,	PUNCT
ejpam-3576	269	15	i.e.	i.e.	X
ejpam-3576	269	16	,	,	PUNCT
ejpam-3576	269	17	y	y	PROPN
ejpam-3576	269	18	∈	∈	PROPN
ejpam-3576	269	19	l(µ	l(µ	PROPN
ejpam-3576	269	20	;	;	PUNCT
ejpam-3576	269	21	t	t	PROPN
ejpam-3576	269	22	)	)	PUNCT
ejpam-3576	269	23	.	.	PUNCT
ejpam-3576	270	1	but	but	CCONJ
ejpam-3576	270	2	µ(x	µ(x	NOUN
ejpam-3576	270	3	)	)	PUNCT
ejpam-3576	270	4	>	>	X
ejpam-3576	270	5	t	t	PROPN
ejpam-3576	270	6	and	and	CCONJ
ejpam-3576	270	7	µ(xy	µ(xy	PROPN
ejpam-3576	270	8	)	)	PUNCT
ejpam-3576	270	9	>	>	X
ejpam-3576	270	10	t	t	PROPN
ejpam-3576	270	11	,	,	PUNCT
ejpam-3576	270	12	i.e.	i.e.	X
ejpam-3576	270	13	,	,	PUNCT
ejpam-3576	270	14	x	x	X
ejpam-3576	270	15	,	,	PUNCT
ejpam-3576	270	16	xy	xy	PROPN
ejpam-3576	270	17	/∈	/∈	PUNCT
ejpam-3576	271	1	l(µ	l(µ	PROPN
ejpam-3576	271	2	;	;	PUNCT
ejpam-3576	271	3	t	t	PROPN
ejpam-3576	271	4	)	)	PUNCT
ejpam-3576	271	5	,	,	PUNCT
ejpam-3576	271	6	which	which	PRON
ejpam-3576	271	7	is	be	AUX
ejpam-3576	271	8	a	a	DET
ejpam-3576	271	9	contradiction	contradiction	NOUN
ejpam-3576	271	10	.	.	PUNCT
ejpam-3576	272	1	hence	hence	ADV
ejpam-3576	272	2	µ(x	µ(x	NOUN
ejpam-3576	272	3	)	)	PUNCT
ejpam-3576	272	4	≤	≤	NUM
ejpam-3576	272	5	µ(y	µ(y	NUM
ejpam-3576	272	6	)	)	PUNCT
ejpam-3576	272	7	and	and	CCONJ
ejpam-3576	272	8	µ(xy	µ(xy	NUM
ejpam-3576	272	9	)	)	PUNCT
ejpam-3576	272	10	≤	≤	NOUN
ejpam-3576	272	11	µ(y	µ(y	NUM
ejpam-3576	272	12	)	)	PUNCT
ejpam-3576	272	13	.	.	PUNCT
ejpam-3576	273	1	proposition	proposition	NOUN
ejpam-3576	273	2	3	3	X
ejpam-3576	273	3	.	.	PUNCT
ejpam-3576	274	1	let	let	VERB
ejpam-3576	274	2	µ	µ	X
ejpam-3576	274	3	be	be	AUX
ejpam-3576	274	4	a	a	DET
ejpam-3576	274	5	fuzzy	fuzzy	ADJ
ejpam-3576	274	6	subset	subset	NOUN
ejpam-3576	274	7	of	of	ADP
ejpam-3576	274	8	an	an	DET
ejpam-3576	274	9	ordered	order	VERB
ejpam-3576	274	10	ag	ag	PROPN
ejpam-3576	274	11	-	-	PROPN
ejpam-3576	274	12	groupoid	groupoid	PROPN
ejpam-3576	274	13	s.	s.	PROPN
ejpam-3576	274	14	then	then	ADV
ejpam-3576	274	15	µ	µ	PROPN
ejpam-3576	274	16	is	be	AUX
ejpam-3576	274	17	an	an	DET
ejpam-3576	274	18	anti	anti	ADJ
ejpam-3576	274	19	fuzzy	fuzzy	ADJ
ejpam-3576	274	20	interior	interior	ADJ
ejpam-3576	274	21	ideal	ideal	NOUN
ejpam-3576	274	22	of	of	ADP
ejpam-3576	274	23	s	s	PRON
ejpam-3576	274	24	if	if	SCONJ
ejpam-3576	274	25	and	and	CCONJ
ejpam-3576	274	26	only	only	ADV
ejpam-3576	274	27	if	if	SCONJ
ejpam-3576	274	28	the	the	DET
ejpam-3576	274	29	lower	low	ADJ
ejpam-3576	274	30	t	t	NOUN
ejpam-3576	274	31	-	-	PUNCT
ejpam-3576	274	32	level	level	NOUN
ejpam-3576	274	33	l(µ	l(µ	NOUN
ejpam-3576	274	34	;	;	PUNCT
ejpam-3576	274	35	t	t	PROPN
ejpam-3576	274	36	)	)	PUNCT
ejpam-3576	274	37	of	of	ADP
ejpam-3576	274	38	µ	µ	PROPN
ejpam-3576	274	39	is	be	AUX
ejpam-3576	274	40	an	an	DET
ejpam-3576	274	41	interior	interior	ADJ
ejpam-3576	274	42	ideal	ideal	NOUN
ejpam-3576	274	43	of	of	ADP
ejpam-3576	274	44	s	s	PRON
ejpam-3576	274	45	for	for	ADP
ejpam-3576	274	46	all	all	DET
ejpam-3576	274	47	t	t	NOUN
ejpam-3576	274	48	∈	∈	PROPN
ejpam-3576	274	49	(	(	PUNCT
ejpam-3576	274	50	0	0	NUM
ejpam-3576	274	51	,	,	PUNCT
ejpam-3576	274	52	1	1	NUM
ejpam-3576	274	53	]	]	PUNCT
ejpam-3576	274	54	.	.	PUNCT
ejpam-3576	275	1	proof	proof	NOUN
ejpam-3576	275	2	.	.	PUNCT
ejpam-3576	276	1	suppose	suppose	VERB
ejpam-3576	276	2	µ	µ	PRON
ejpam-3576	276	3	is	be	AUX
ejpam-3576	276	4	an	an	DET
ejpam-3576	276	5	anti	anti	ADJ
ejpam-3576	276	6	fuzzy	fuzzy	ADJ
ejpam-3576	276	7	interior	interior	ADJ
ejpam-3576	276	8	ideal	ideal	NOUN
ejpam-3576	276	9	of	of	ADP
ejpam-3576	276	10	s.	s.	PROPN
ejpam-3576	276	11	let	let	VERB
ejpam-3576	276	12	y	y	PROPN
ejpam-3576	276	13	∈	∈	PROPN
ejpam-3576	276	14	l(µ	l(µ	PROPN
ejpam-3576	276	15	;	;	PUNCT
ejpam-3576	276	16	t	t	PROPN
ejpam-3576	276	17	)	)	PUNCT
ejpam-3576	276	18	and	and	CCONJ
ejpam-3576	276	19	x	x	PUNCT
ejpam-3576	276	20	∈	∈	NOUN
ejpam-3576	276	21	s	s	VERB
ejpam-3576	276	22	such	such	ADJ
ejpam-3576	276	23	that	that	SCONJ
ejpam-3576	276	24	x	x	X
ejpam-3576	276	25	≤	≤	NUM
ejpam-3576	276	26	y	y	PROPN
ejpam-3576	276	27	,	,	PUNCT
ejpam-3576	276	28	this	this	PRON
ejpam-3576	276	29	imply	imply	VERB
ejpam-3576	276	30	that	that	SCONJ
ejpam-3576	276	31	µ(y	µ(y	NOUN
ejpam-3576	276	32	)	)	PUNCT
ejpam-3576	276	33	≤	≤	NUM
ejpam-3576	276	34	t.	t.	NOUN
ejpam-3576	276	35	µ(x	µ(x	ADJ
ejpam-3576	276	36	)	)	PUNCT
ejpam-3576	276	37	≤	≤	NUM
ejpam-3576	276	38	µ(y	µ(y	NOUN
ejpam-3576	276	39	)	)	PUNCT
ejpam-3576	276	40	≤	≤	NOUN
ejpam-3576	276	41	t	t	PROPN
ejpam-3576	276	42	,	,	PUNCT
ejpam-3576	276	43	µ	µ	X
ejpam-3576	276	44	being	be	AUX
ejpam-3576	276	45	an	an	DET
ejpam-3576	276	46	anti	anti	ADJ
ejpam-3576	276	47	fuzzy	fuzzy	ADJ
ejpam-3576	276	48	interior	interior	ADJ
ejpam-3576	276	49	ideal	ideal	NOUN
ejpam-3576	276	50	of	of	ADP
ejpam-3576	276	51	s.	s.	PROPN
ejpam-3576	276	52	thus	thus	ADV
ejpam-3576	276	53	µ(x	µ(x	NOUN
ejpam-3576	276	54	)	)	PUNCT
ejpam-3576	276	55	≤	≤	NOUN
ejpam-3576	276	56	t	t	PROPN
ejpam-3576	276	57	,	,	PUNCT
ejpam-3576	276	58	i.e.	i.e.	X
ejpam-3576	276	59	,	,	PUNCT
ejpam-3576	276	60	x	x	SYM
ejpam-3576	276	61	∈	∈	PROPN
ejpam-3576	276	62	l(µ	l(µ	PROPN
ejpam-3576	276	63	;	;	PUNCT
ejpam-3576	276	64	t	t	PROPN
ejpam-3576	276	65	)	)	PUNCT
ejpam-3576	276	66	.	.	PUNCT
ejpam-3576	277	1	let	let	VERB
ejpam-3576	277	2	a	a	DET
ejpam-3576	277	3	∈	∈	PROPN
ejpam-3576	277	4	l(µ	l(µ	PROPN
ejpam-3576	277	5	;	;	PUNCT
ejpam-3576	277	6	t	t	PROPN
ejpam-3576	277	7	)	)	PUNCT
ejpam-3576	277	8	and	and	CCONJ
ejpam-3576	277	9	x	x	X
ejpam-3576	277	10	,	,	PUNCT
ejpam-3576	277	11	y	y	PROPN
ejpam-3576	277	12	∈	∈	PROPN
ejpam-3576	277	13	s	s	AUX
ejpam-3576	277	14	,	,	PUNCT
ejpam-3576	277	15	by	by	ADP
ejpam-3576	277	16	definition	definition	NOUN
ejpam-3576	277	17	µ(a	µ(a	PROPN
ejpam-3576	277	18	)	)	PUNCT
ejpam-3576	277	19	≤	≤	NOUN
ejpam-3576	277	20	t.	t.	NOUN
ejpam-3576	277	21	µ((xa)y	µ((xa)y	NOUN
ejpam-3576	277	22	)	)	PUNCT
ejpam-3576	277	23	≤	≤	NOUN
ejpam-3576	277	24	µ(a	µ(a	NOUN
ejpam-3576	277	25	)	)	PUNCT
ejpam-3576	277	26	≤	≤	NOUN
ejpam-3576	277	27	t	t	PROPN
ejpam-3576	277	28	,	,	PUNCT
ejpam-3576	277	29	µ	µ	X
ejpam-3576	277	30	being	be	AUX
ejpam-3576	277	31	an	an	DET
ejpam-3576	277	32	anti	anti	ADJ
ejpam-3576	277	33	fuzzy	fuzzy	ADJ
ejpam-3576	277	34	interior	interior	ADJ
ejpam-3576	277	35	ideal	ideal	NOUN
ejpam-3576	277	36	of	of	ADP
ejpam-3576	277	37	s.	s.	PROPN
ejpam-3576	277	38	thus	thus	ADV
ejpam-3576	277	39	µ((xa)y	µ((xa)y	PROPN
ejpam-3576	277	40	)	)	PUNCT
ejpam-3576	277	41	≤	≤	NOUN
ejpam-3576	277	42	t	t	PROPN
ejpam-3576	277	43	,	,	PUNCT
ejpam-3576	277	44	i.e.	i.e.	X
ejpam-3576	277	45	,	,	PUNCT
ejpam-3576	277	46	(	(	PUNCT
ejpam-3576	277	47	xa)y	xa)y	PROPN
ejpam-3576	277	48	∈	∈	PROPN
ejpam-3576	277	49	l(µ	l(µ	PROPN
ejpam-3576	277	50	;	;	PUNCT
ejpam-3576	277	51	t	t	PROPN
ejpam-3576	277	52	)	)	PUNCT
ejpam-3576	277	53	.	.	PUNCT
ejpam-3576	278	1	hence	hence	ADV
ejpam-3576	278	2	l(µ	l(µ	PROPN
ejpam-3576	278	3	;	;	PUNCT
ejpam-3576	278	4	t	t	PROPN
ejpam-3576	278	5	)	)	PUNCT
ejpam-3576	278	6	is	be	AUX
ejpam-3576	278	7	an	an	DET
ejpam-3576	278	8	interior	interior	ADJ
ejpam-3576	278	9	ideal	ideal	NOUN
ejpam-3576	278	10	of	of	ADP
ejpam-3576	278	11	s.	s.	PROPN
ejpam-3576	278	12	k.	k.	PROPN
ejpam-3576	278	13	nasreen	nasreen	PROPN
ejpam-3576	278	14	,	,	PUNCT
ejpam-3576	278	15	m.	m.	NOUN
ejpam-3576	278	16	alesemi	alesemi	PROPN
ejpam-3576	278	17	,	,	PUNCT
ejpam-3576	278	18	salahuddin	salahuddin	VERB
ejpam-3576	278	19	/	/	SYM
ejpam-3576	278	20	eur	eur	PROPN
ejpam-3576	278	21	.	.	PUNCT
ejpam-3576	279	1	j.	j.	PROPN
ejpam-3576	279	2	pure	pure	PROPN
ejpam-3576	279	3	appl	appl	PROPN
ejpam-3576	279	4	.	.	PROPN
ejpam-3576	279	5	math	math	PROPN
ejpam-3576	279	6	,	,	PUNCT
ejpam-3576	279	7	13	13	NUM
ejpam-3576	279	8	(	(	PUNCT
ejpam-3576	279	9	1	1	NUM
ejpam-3576	279	10	)	)	PUNCT
ejpam-3576	279	11	(	(	PUNCT
ejpam-3576	279	12	2020	2020	NUM
ejpam-3576	279	13	)	)	PUNCT
ejpam-3576	279	14	,	,	PUNCT
ejpam-3576	279	15	113	113	NUM
ejpam-3576	279	16	-	-	SYM
ejpam-3576	279	17	129	129	NUM
ejpam-3576	279	18	121	121	NUM
ejpam-3576	279	19	conversely	conversely	ADV
ejpam-3576	279	20	,	,	PUNCT
ejpam-3576	279	21	suppose	suppose	VERB
ejpam-3576	279	22	l(µ	l(µ	PROPN
ejpam-3576	279	23	;	;	PUNCT
ejpam-3576	279	24	t	t	PROPN
ejpam-3576	279	25	)	)	PUNCT
ejpam-3576	279	26	is	be	AUX
ejpam-3576	279	27	an	an	DET
ejpam-3576	279	28	interior	interior	ADJ
ejpam-3576	279	29	ideal	ideal	NOUN
ejpam-3576	279	30	of	of	ADP
ejpam-3576	279	31	s	s	PRON
ejpam-3576	279	32	and	and	CCONJ
ejpam-3576	279	33	x	x	PROPN
ejpam-3576	279	34	,	,	PUNCT
ejpam-3576	279	35	y	y	PROPN
ejpam-3576	279	36	,	,	PUNCT
ejpam-3576	279	37	a	a	DET
ejpam-3576	279	38	∈	∈	NOUN
ejpam-3576	279	39	s	s	VERB
ejpam-3576	279	40	such	such	ADJ
ejpam-3576	279	41	that	that	SCONJ
ejpam-3576	279	42	x	x	X
ejpam-3576	279	43	≤	≤	NUM
ejpam-3576	279	44	y.	y.	NOUN
ejpam-3576	279	45	we	we	PRON
ejpam-3576	279	46	have	have	VERB
ejpam-3576	279	47	to	to	PART
ejpam-3576	279	48	show	show	VERB
ejpam-3576	279	49	that	that	SCONJ
ejpam-3576	279	50	µ(x	µ(x	VERB
ejpam-3576	279	51	)	)	PUNCT
ejpam-3576	279	52	≤	≤	NUM
ejpam-3576	279	53	µ(y	µ(y	PROPN
ejpam-3576	279	54	)	)	PUNCT
ejpam-3576	279	55	,	,	PUNCT
ejpam-3576	279	56	we	we	PRON
ejpam-3576	279	57	suppose	suppose	VERB
ejpam-3576	279	58	a	a	DET
ejpam-3576	279	59	contradiction	contradiction	NOUN
ejpam-3576	279	60	µ(x	µ(x	VERB
ejpam-3576	279	61	)	)	PUNCT
ejpam-3576	279	62	>	>	X
ejpam-3576	279	63	µ(y	µ(y	PROPN
ejpam-3576	279	64	)	)	PUNCT
ejpam-3576	279	65	.	.	PUNCT
ejpam-3576	280	1	let	let	VERB
ejpam-3576	280	2	µ(y	µ(y	NUM
ejpam-3576	280	3	)	)	PUNCT
ejpam-3576	280	4	=	=	SYM
ejpam-3576	280	5	t	t	PROPN
ejpam-3576	280	6	,	,	PUNCT
ejpam-3576	280	7	this	this	PRON
ejpam-3576	280	8	imply	imply	VERB
ejpam-3576	280	9	that	that	SCONJ
ejpam-3576	280	10	µ(y	µ(y	NOUN
ejpam-3576	280	11	)	)	PUNCT
ejpam-3576	280	12	≤	≤	NOUN
ejpam-3576	280	13	t	t	PROPN
ejpam-3576	280	14	,	,	PUNCT
ejpam-3576	280	15	i.e.	i.e.	X
ejpam-3576	280	16	,	,	PUNCT
ejpam-3576	280	17	y	y	PROPN
ejpam-3576	280	18	∈	∈	PROPN
ejpam-3576	280	19	l(µ	l(µ	PROPN
ejpam-3576	280	20	;	;	PUNCT
ejpam-3576	280	21	t	t	PROPN
ejpam-3576	280	22	)	)	PUNCT
ejpam-3576	280	23	.	.	PUNCT
ejpam-3576	281	1	but	but	CCONJ
ejpam-3576	281	2	µ(x	µ(x	NOUN
ejpam-3576	281	3	)	)	PUNCT
ejpam-3576	281	4	>	>	X
ejpam-3576	281	5	t	t	PROPN
ejpam-3576	281	6	,	,	PUNCT
ejpam-3576	281	7	i.e.	i.e.	X
ejpam-3576	281	8	,	,	PUNCT
ejpam-3576	281	9	x	x	X
ejpam-3576	281	10	/∈	/∈	PUNCT
ejpam-3576	281	11	l(µ	l(µ	PROPN
ejpam-3576	281	12	;	;	PUNCT
ejpam-3576	281	13	t	t	PROPN
ejpam-3576	281	14	)	)	PUNCT
ejpam-3576	281	15	,	,	PUNCT
ejpam-3576	281	16	which	which	PRON
ejpam-3576	281	17	is	be	AUX
ejpam-3576	281	18	a	a	DET
ejpam-3576	281	19	contradiction	contradiction	NOUN
ejpam-3576	281	20	.	.	PUNCT
ejpam-3576	282	1	hence	hence	ADV
ejpam-3576	282	2	µ(x	µ(x	NOUN
ejpam-3576	282	3	)	)	PUNCT
ejpam-3576	282	4	≤	≤	NUM
ejpam-3576	282	5	µ(y	µ(y	NUM
ejpam-3576	282	6	)	)	PUNCT
ejpam-3576	282	7	.	.	PUNCT
ejpam-3576	283	1	we	we	PRON
ejpam-3576	283	2	have	have	VERB
ejpam-3576	283	3	to	to	PART
ejpam-3576	283	4	show	show	VERB
ejpam-3576	283	5	that	that	SCONJ
ejpam-3576	283	6	µ((xa)y	µ((xa)y	NOUN
ejpam-3576	283	7	)	)	PUNCT
ejpam-3576	283	8	≤	≤	NOUN
ejpam-3576	283	9	µ(a	µ(a	PROPN
ejpam-3576	283	10	)	)	PUNCT
ejpam-3576	283	11	,	,	PUNCT
ejpam-3576	283	12	we	we	PRON
ejpam-3576	283	13	suppose	suppose	VERB
ejpam-3576	283	14	a	a	DET
ejpam-3576	283	15	contradiction	contradiction	NOUN
ejpam-3576	283	16	µ((xa)y	µ((xa)y	NUM
ejpam-3576	283	17	)	)	PUNCT
ejpam-3576	283	18	>	>	X
ejpam-3576	284	1	µ(a	µ(a	PROPN
ejpam-3576	284	2	)	)	PUNCT
ejpam-3576	284	3	.	.	PUNCT
ejpam-3576	285	1	let	let	VERB
ejpam-3576	285	2	µ(a	µ(a	PROPN
ejpam-3576	285	3	)	)	PUNCT
ejpam-3576	285	4	=	=	SYM
ejpam-3576	285	5	t	t	PROPN
ejpam-3576	285	6	,	,	PUNCT
ejpam-3576	285	7	this	this	PRON
ejpam-3576	285	8	imply	imply	VERB
ejpam-3576	285	9	that	that	SCONJ
ejpam-3576	285	10	µ(a	µ(a	PROPN
ejpam-3576	285	11	)	)	PUNCT
ejpam-3576	285	12	≤	≤	NOUN
ejpam-3576	285	13	t	t	PROPN
ejpam-3576	285	14	,	,	PUNCT
ejpam-3576	285	15	i.e.	i.e.	X
ejpam-3576	285	16	,	,	PUNCT
ejpam-3576	285	17	a	a	DET
ejpam-3576	285	18	∈	∈	PROPN
ejpam-3576	285	19	l(µ	l(µ	PROPN
ejpam-3576	285	20	;	;	PUNCT
ejpam-3576	285	21	t	t	PROPN
ejpam-3576	285	22	)	)	PUNCT
ejpam-3576	285	23	.	.	PUNCT
ejpam-3576	286	1	but	but	CCONJ
ejpam-3576	286	2	µ((xa)y	µ((xa)y	NUM
ejpam-3576	286	3	)	)	PUNCT
ejpam-3576	286	4	>	>	X
ejpam-3576	286	5	t	t	PROPN
ejpam-3576	286	6	,	,	PUNCT
ejpam-3576	286	7	i.e.	i.e.	X
ejpam-3576	286	8	,	,	PUNCT
ejpam-3576	286	9	(	(	PUNCT
ejpam-3576	286	10	xa)y	xa)y	PROPN
ejpam-3576	286	11	/∈	/∈	PUNCT
ejpam-3576	287	1	l(µ	l(µ	PROPN
ejpam-3576	287	2	;	;	PUNCT
ejpam-3576	287	3	t	t	PROPN
ejpam-3576	287	4	)	)	PUNCT
ejpam-3576	287	5	,	,	PUNCT
ejpam-3576	287	6	which	which	PRON
ejpam-3576	287	7	is	be	AUX
ejpam-3576	287	8	a	a	DET
ejpam-3576	287	9	contradiction	contradiction	NOUN
ejpam-3576	287	10	.	.	PUNCT
ejpam-3576	288	1	hence	hence	ADV
ejpam-3576	288	2	µ((xa)y	µ((xa)y	NOUN
ejpam-3576	288	3	)	)	PUNCT
ejpam-3576	288	4	≤	≤	NOUN
ejpam-3576	289	1	µ(a	µ(a	PROPN
ejpam-3576	289	2	)	)	PUNCT
ejpam-3576	289	3	.	.	PUNCT
ejpam-3576	290	1	lemma	lemma	PROPN
ejpam-3576	290	2	6	6	NUM
ejpam-3576	290	3	.	.	PUNCT
ejpam-3576	291	1	every	every	DET
ejpam-3576	291	2	anti	anti	ADJ
ejpam-3576	291	3	fuzzy	fuzzy	ADJ
ejpam-3576	291	4	right	right	ADJ
ejpam-3576	291	5	ideal	ideal	NOUN
ejpam-3576	291	6	of	of	ADP
ejpam-3576	291	7	an	an	DET
ejpam-3576	291	8	ordered	order	VERB
ejpam-3576	291	9	ag	ag	PROPN
ejpam-3576	291	10	-	-	PROPN
ejpam-3576	291	11	groupoid	groupoid	PROPN
ejpam-3576	291	12	s	s	PROPN
ejpam-3576	291	13	with	with	ADP
ejpam-3576	291	14	left	left	ADJ
ejpam-3576	291	15	identity	identity	NOUN
ejpam-3576	291	16	e	e	NOUN
ejpam-3576	291	17	,	,	PUNCT
ejpam-3576	291	18	is	be	AUX
ejpam-3576	291	19	an	an	DET
ejpam-3576	291	20	anti	anti	ADJ
ejpam-3576	291	21	fuzzy	fuzzy	ADJ
ejpam-3576	291	22	ideal	ideal	NOUN
ejpam-3576	291	23	of	of	ADP
ejpam-3576	291	24	s.	s.	PROPN
ejpam-3576	291	25	proof	proof	PROPN
ejpam-3576	291	26	.	.	PUNCT
ejpam-3576	292	1	let	let	VERB
ejpam-3576	292	2	µ	µ	X
ejpam-3576	292	3	be	be	AUX
ejpam-3576	292	4	an	an	DET
ejpam-3576	292	5	anti	anti	ADJ
ejpam-3576	292	6	fuzzy	fuzzy	ADJ
ejpam-3576	292	7	right	right	ADJ
ejpam-3576	292	8	ideal	ideal	NOUN
ejpam-3576	292	9	of	of	ADP
ejpam-3576	292	10	s	s	PRON
ejpam-3576	292	11	and	and	CCONJ
ejpam-3576	292	12	x	x	NOUN
ejpam-3576	292	13	,	,	PUNCT
ejpam-3576	292	14	y	y	PROPN
ejpam-3576	292	15	∈	∈	PROPN
ejpam-3576	292	16	s.	s.	PROPN
ejpam-3576	292	17	now	now	ADV
ejpam-3576	292	18	µ(xy	µ(xy	PROPN
ejpam-3576	292	19	)	)	PUNCT
ejpam-3576	292	20	=	=	SYM
ejpam-3576	292	21	µ((ex)y	µ((ex)y	PROPN
ejpam-3576	292	22	)	)	PUNCT
ejpam-3576	293	1	=	=	SYM
ejpam-3576	293	2	µ((yx)e	µ((yx)e	PROPN
ejpam-3576	293	3	)	)	PUNCT
ejpam-3576	293	4	≤	≤	NOUN
ejpam-3576	293	5	µ(yx	µ(yx	NOUN
ejpam-3576	293	6	)	)	PUNCT
ejpam-3576	293	7	≤	≤	NUM
ejpam-3576	293	8	µ(y	µ(y	NUM
ejpam-3576	293	9	)	)	PUNCT
ejpam-3576	293	10	.	.	PUNCT
ejpam-3576	294	1	hence	hence	ADV
ejpam-3576	294	2	µ	µ	X
ejpam-3576	294	3	is	be	AUX
ejpam-3576	294	4	an	an	DET
ejpam-3576	294	5	anti	anti	ADJ
ejpam-3576	294	6	fuzzy	fuzzy	ADJ
ejpam-3576	294	7	ideal	ideal	NOUN
ejpam-3576	294	8	of	of	ADP
ejpam-3576	294	9	s.	s.	PROPN
ejpam-3576	294	10	remark	remark	PROPN
ejpam-3576	294	11	3	3	NUM
ejpam-3576	294	12	.	.	PUNCT
ejpam-3576	295	1	the	the	DET
ejpam-3576	295	2	concept	concept	NOUN
ejpam-3576	295	3	of	of	ADP
ejpam-3576	295	4	anti	anti	ADJ
ejpam-3576	295	5	fuzzy	fuzzy	ADJ
ejpam-3576	295	6	(	(	PUNCT
ejpam-3576	295	7	right	right	ADJ
ejpam-3576	295	8	,	,	PUNCT
ejpam-3576	295	9	two	two	NUM
ejpam-3576	295	10	-	-	PUNCT
ejpam-3576	295	11	sided	sided	ADJ
ejpam-3576	295	12	)	)	PUNCT
ejpam-3576	295	13	ideals	ideal	NOUN
ejpam-3576	295	14	coincide	coincide	VERB
ejpam-3576	295	15	in	in	ADP
ejpam-3576	295	16	ordered	order	VERB
ejpam-3576	295	17	ag	ag	PROPN
ejpam-3576	295	18	-	-	NOUN
ejpam-3576	295	19	groupoids	groupoids	PROPN
ejpam-3576	295	20	s	s	VERB
ejpam-3576	295	21	with	with	ADP
ejpam-3576	295	22	left	left	ADJ
ejpam-3576	295	23	identity	identity	NOUN
ejpam-3576	295	24	.	.	PUNCT
ejpam-3576	296	1	lemma	lemma	PROPN
ejpam-3576	296	2	7	7	NUM
ejpam-3576	296	3	.	.	PUNCT
ejpam-3576	297	1	every	every	DET
ejpam-3576	297	2	anti	anti	ADJ
ejpam-3576	297	3	fuzzy	fuzzy	ADJ
ejpam-3576	297	4	ideal	ideal	NOUN
ejpam-3576	297	5	of	of	ADP
ejpam-3576	297	6	an	an	DET
ejpam-3576	297	7	ordered	order	VERB
ejpam-3576	297	8	ag	ag	PROPN
ejpam-3576	297	9	-	-	NOUN
ejpam-3576	297	10	groupoid	groupoid	PROPN
ejpam-3576	297	11	s	s	PART
ejpam-3576	297	12	is	be	AUX
ejpam-3576	297	13	an	an	DET
ejpam-3576	297	14	anti	anti	ADJ
ejpam-3576	297	15	fuzzy	fuzzy	ADJ
ejpam-3576	297	16	interior	interior	ADJ
ejpam-3576	297	17	ideal	ideal	NOUN
ejpam-3576	297	18	of	of	ADP
ejpam-3576	297	19	s.	s.	PROPN
ejpam-3576	297	20	proof	proof	PROPN
ejpam-3576	297	21	.	.	PUNCT
ejpam-3576	298	1	let	let	VERB
ejpam-3576	298	2	µ	µ	X
ejpam-3576	298	3	be	be	AUX
ejpam-3576	298	4	an	an	DET
ejpam-3576	298	5	anti	anti	ADJ
ejpam-3576	298	6	fuzzy	fuzzy	ADJ
ejpam-3576	298	7	two	two	NUM
ejpam-3576	298	8	-	-	PUNCT
ejpam-3576	298	9	sided	sided	ADJ
ejpam-3576	298	10	ideal	ideal	NOUN
ejpam-3576	298	11	of	of	ADP
ejpam-3576	298	12	s	s	PRON
ejpam-3576	298	13	and	and	CCONJ
ejpam-3576	298	14	x	x	NOUN
ejpam-3576	298	15	,	,	PUNCT
ejpam-3576	298	16	a	a	PRON
ejpam-3576	298	17	,	,	PUNCT
ejpam-3576	298	18	y	y	PROPN
ejpam-3576	298	19	∈	∈	PROPN
ejpam-3576	298	20	s.	s.	PROPN
ejpam-3576	298	21	now	now	ADV
ejpam-3576	298	22	µ((xa)y	µ((xa)y	PROPN
ejpam-3576	298	23	)	)	PUNCT
ejpam-3576	298	24	≤	≤	NUM
ejpam-3576	298	25	µ(xa	µ(xa	PROPN
ejpam-3576	298	26	)	)	PUNCT
ejpam-3576	298	27	≤	≤	NOUN
ejpam-3576	298	28	µ(a	µ(a	PROPN
ejpam-3576	298	29	)	)	PUNCT
ejpam-3576	298	30	.	.	PUNCT
ejpam-3576	299	1	hence	hence	ADV
ejpam-3576	299	2	µ	µ	X
ejpam-3576	299	3	is	be	AUX
ejpam-3576	299	4	an	an	DET
ejpam-3576	299	5	anti	anti	ADJ
ejpam-3576	299	6	fuzzy	fuzzy	ADJ
ejpam-3576	299	7	interior	interior	ADJ
ejpam-3576	299	8	ideal	ideal	NOUN
ejpam-3576	299	9	of	of	ADP
ejpam-3576	299	10	s.	s.	PROPN
ejpam-3576	299	11	proposition	proposition	PROPN
ejpam-3576	299	12	4	4	X
ejpam-3576	299	13	.	.	PUNCT
ejpam-3576	300	1	let	let	VERB
ejpam-3576	300	2	s	s	PRON
ejpam-3576	300	3	be	be	AUX
ejpam-3576	300	4	an	an	DET
ejpam-3576	300	5	ordered	order	VERB
ejpam-3576	300	6	ag	ag	PROPN
ejpam-3576	300	7	-	-	NOUN
ejpam-3576	300	8	groupoid	groupoid	PROPN
ejpam-3576	300	9	with	with	ADP
ejpam-3576	300	10	left	left	ADJ
ejpam-3576	300	11	identity	identity	NOUN
ejpam-3576	300	12	e.	e.	PROPN
ejpam-3576	300	13	then	then	ADV
ejpam-3576	300	14	µ	µ	PROPN
ejpam-3576	300	15	is	be	AUX
ejpam-3576	300	16	an	an	DET
ejpam-3576	300	17	anti	anti	ADJ
ejpam-3576	300	18	fuzzy	fuzzy	ADJ
ejpam-3576	300	19	interior	interior	ADJ
ejpam-3576	300	20	ideal	ideal	NOUN
ejpam-3576	300	21	if	if	SCONJ
ejpam-3576	300	22	and	and	CCONJ
ejpam-3576	300	23	only	only	ADV
ejpam-3576	300	24	if	if	SCONJ
ejpam-3576	300	25	µ	µ	NOUN
ejpam-3576	300	26	is	be	AUX
ejpam-3576	300	27	an	an	DET
ejpam-3576	300	28	anti	anti	ADJ
ejpam-3576	300	29	fuzzy	fuzzy	ADJ
ejpam-3576	300	30	ideal	ideal	NOUN
ejpam-3576	300	31	of	of	ADP
ejpam-3576	300	32	s.	s.	PROPN
ejpam-3576	300	33	proof	proof	PROPN
ejpam-3576	300	34	.	.	PUNCT
ejpam-3576	301	1	let	let	VERB
ejpam-3576	301	2	µ	µ	X
ejpam-3576	301	3	be	be	AUX
ejpam-3576	301	4	an	an	DET
ejpam-3576	301	5	anti	anti	ADJ
ejpam-3576	301	6	fuzzy	fuzzy	ADJ
ejpam-3576	301	7	interior	interior	ADJ
ejpam-3576	301	8	ideal	ideal	NOUN
ejpam-3576	301	9	of	of	ADP
ejpam-3576	301	10	s	s	PRON
ejpam-3576	301	11	and	and	CCONJ
ejpam-3576	301	12	x	x	NOUN
ejpam-3576	301	13	,	,	PUNCT
ejpam-3576	301	14	y	y	PROPN
ejpam-3576	301	15	∈	∈	PROPN
ejpam-3576	301	16	s.	s.	PROPN
ejpam-3576	301	17	now	now	ADV
ejpam-3576	301	18	µ(xy	µ(xy	PROPN
ejpam-3576	301	19	)	)	PUNCT
ejpam-3576	301	20	=	=	SYM
ejpam-3576	301	21	µ((ex)y	µ((ex)y	PROPN
ejpam-3576	301	22	)	)	PUNCT
ejpam-3576	301	23	≤	≤	NOUN
ejpam-3576	301	24	µ(x	µ(x	NOUN
ejpam-3576	301	25	)	)	PUNCT
ejpam-3576	301	26	.	.	PUNCT
ejpam-3576	302	1	thus	thus	ADV
ejpam-3576	302	2	µ	µ	X
ejpam-3576	302	3	is	be	AUX
ejpam-3576	302	4	an	an	DET
ejpam-3576	302	5	anti	anti	ADJ
ejpam-3576	302	6	fuzzy	fuzzy	ADJ
ejpam-3576	302	7	right	right	ADJ
ejpam-3576	302	8	ideal	ideal	NOUN
ejpam-3576	302	9	of	of	ADP
ejpam-3576	302	10	s.	s.	PROPN
ejpam-3576	302	11	hence	hence	ADV
ejpam-3576	302	12	µ	µ	PROPN
ejpam-3576	302	13	is	be	AUX
ejpam-3576	302	14	an	an	DET
ejpam-3576	302	15	anti	anti	ADJ
ejpam-3576	302	16	fuzzy	fuzzy	ADJ
ejpam-3576	302	17	ideal	ideal	NOUN
ejpam-3576	302	18	of	of	ADP
ejpam-3576	302	19	s	s	PRON
ejpam-3576	302	20	by	by	ADP
ejpam-3576	302	21	lemma	lemma	PROPN
ejpam-3576	302	22	6	6	NUM
ejpam-3576	302	23	.	.	PUNCT
ejpam-3576	303	1	converse	converse	NOUN
ejpam-3576	303	2	is	be	AUX
ejpam-3576	303	3	true	true	ADJ
ejpam-3576	303	4	by	by	ADP
ejpam-3576	303	5	lemma	lemma	PROPN
ejpam-3576	303	6	7	7	PROPN
ejpam-3576	303	7	.	.	PUNCT
ejpam-3576	304	1	lemma	lemma	PROPN
ejpam-3576	304	2	8	8	NUM
ejpam-3576	304	3	.	.	PUNCT
ejpam-3576	305	1	every	every	DET
ejpam-3576	305	2	anti	anti	ADJ
ejpam-3576	305	3	fuzzy	fuzzy	ADJ
ejpam-3576	305	4	right	right	ADJ
ejpam-3576	305	5	ideal	ideal	NOUN
ejpam-3576	305	6	of	of	ADP
ejpam-3576	305	7	a	a	DET
ejpam-3576	305	8	regular	regular	ADJ
ejpam-3576	305	9	ordered	order	VERB
ejpam-3576	305	10	ag	ag	PROPN
ejpam-3576	305	11	-	-	PROPN
ejpam-3576	305	12	groupoid	groupoid	PROPN
ejpam-3576	305	13	s	s	PROPN
ejpam-3576	305	14	,	,	PUNCT
ejpam-3576	305	15	is	be	AUX
ejpam-3576	305	16	an	an	DET
ejpam-3576	305	17	anti	anti	ADJ
ejpam-3576	305	18	fuzzy	fuzzy	ADJ
ejpam-3576	305	19	ideal	ideal	NOUN
ejpam-3576	305	20	of	of	ADP
ejpam-3576	305	21	s.	s.	PROPN
ejpam-3576	305	22	proof	proof	PROPN
ejpam-3576	305	23	.	.	PUNCT
ejpam-3576	306	1	let	let	VERB
ejpam-3576	306	2	µ	µ	X
ejpam-3576	306	3	be	be	AUX
ejpam-3576	306	4	an	an	DET
ejpam-3576	306	5	anti	anti	ADJ
ejpam-3576	306	6	fuzzy	fuzzy	ADJ
ejpam-3576	306	7	right	right	ADJ
ejpam-3576	306	8	ideal	ideal	NOUN
ejpam-3576	306	9	of	of	ADP
ejpam-3576	306	10	s	s	PRON
ejpam-3576	306	11	and	and	CCONJ
ejpam-3576	306	12	x	x	NOUN
ejpam-3576	306	13	,	,	PUNCT
ejpam-3576	306	14	y	y	PROPN
ejpam-3576	306	15	∈	∈	PROPN
ejpam-3576	306	16	s	s	PROPN
ejpam-3576	306	17	,	,	PUNCT
ejpam-3576	306	18	this	this	PRON
ejpam-3576	306	19	imply	imply	VERB
ejpam-3576	306	20	that	that	SCONJ
ejpam-3576	306	21	there	there	PRON
ejpam-3576	306	22	exists	exist	VERB
ejpam-3576	306	23	a	a	DET
ejpam-3576	306	24	∈	∈	NOUN
ejpam-3576	306	25	s	s	VERB
ejpam-3576	306	26	such	such	ADJ
ejpam-3576	306	27	that	that	SCONJ
ejpam-3576	306	28	x	x	SYM
ejpam-3576	306	29	≤	≤	X
ejpam-3576	306	30	(	(	PUNCT
ejpam-3576	306	31	xa)x	xa)x	PROPN
ejpam-3576	306	32	.	.	PUNCT
ejpam-3576	307	1	now	now	ADV
ejpam-3576	307	2	µ(xy	µ(xy	NUM
ejpam-3576	307	3	)	)	PUNCT
ejpam-3576	307	4	≤	≤	NUM
ejpam-3576	307	5	µ(((xa)x)y	µ(((xa)x)y	PROPN
ejpam-3576	307	6	)	)	PUNCT
ejpam-3576	307	7	=	=	SYM
ejpam-3576	308	1	µ((yx)(xa	µ((yx)(xa	PROPN
ejpam-3576	308	2	)	)	PUNCT
ejpam-3576	308	3	)	)	PUNCT
ejpam-3576	309	1	≤	≤	NOUN
ejpam-3576	309	2	µ(yx	µ(yx	NOUN
ejpam-3576	309	3	)	)	PUNCT
ejpam-3576	309	4	≤	≤	NUM
ejpam-3576	309	5	µ(y	µ(y	NUM
ejpam-3576	309	6	)	)	PUNCT
ejpam-3576	309	7	.	.	PUNCT
ejpam-3576	310	1	hence	hence	ADV
ejpam-3576	310	2	µ	µ	X
ejpam-3576	310	3	is	be	AUX
ejpam-3576	310	4	an	an	DET
ejpam-3576	310	5	anti	anti	ADJ
ejpam-3576	310	6	fuzzy	fuzzy	ADJ
ejpam-3576	310	7	ideal	ideal	NOUN
ejpam-3576	310	8	of	of	ADP
ejpam-3576	310	9	s.	s.	PROPN
ejpam-3576	310	10	remark	remark	PROPN
ejpam-3576	310	11	4	4	NUM
ejpam-3576	310	12	.	.	PUNCT
ejpam-3576	311	1	the	the	DET
ejpam-3576	311	2	concept	concept	NOUN
ejpam-3576	311	3	of	of	ADP
ejpam-3576	311	4	anti	anti	ADJ
ejpam-3576	311	5	fuzzy	fuzzy	ADJ
ejpam-3576	311	6	(	(	PUNCT
ejpam-3576	311	7	right	right	ADJ
ejpam-3576	311	8	,	,	PUNCT
ejpam-3576	311	9	two	two	NUM
ejpam-3576	311	10	-	-	PUNCT
ejpam-3576	311	11	sided	sided	ADJ
ejpam-3576	311	12	)	)	PUNCT
ejpam-3576	311	13	ideals	ideal	NOUN
ejpam-3576	311	14	coincide	coincide	VERB
ejpam-3576	311	15	in	in	ADP
ejpam-3576	311	16	regular	regular	ADJ
ejpam-3576	311	17	ordered	order	VERB
ejpam-3576	311	18	ag	ag	PROPN
ejpam-3576	311	19	-	-	PUNCT
ejpam-3576	311	20	groupoids	groupoid	NOUN
ejpam-3576	311	21	s.	s.	PROPN
ejpam-3576	311	22	proposition	proposition	PROPN
ejpam-3576	311	23	5	5	NUM
ejpam-3576	311	24	.	.	PUNCT
ejpam-3576	311	25	let	let	VERB
ejpam-3576	311	26	s	s	PRON
ejpam-3576	311	27	be	be	AUX
ejpam-3576	311	28	a	a	DET
ejpam-3576	311	29	regular	regular	ADJ
ejpam-3576	311	30	ordered	order	VERB
ejpam-3576	311	31	ag	ag	PROPN
ejpam-3576	311	32	-	-	NOUN
ejpam-3576	311	33	groupoid	groupoid	PROPN
ejpam-3576	311	34	.	.	PUNCT
ejpam-3576	312	1	then	then	ADV
ejpam-3576	312	2	µ	µ	X
ejpam-3576	312	3	is	be	AUX
ejpam-3576	312	4	an	an	DET
ejpam-3576	312	5	anti	anti	ADJ
ejpam-3576	312	6	fuzzy	fuzzy	ADJ
ejpam-3576	312	7	interior	interior	ADJ
ejpam-3576	312	8	ideal	ideal	NOUN
ejpam-3576	312	9	if	if	SCONJ
ejpam-3576	312	10	and	and	CCONJ
ejpam-3576	312	11	only	only	ADV
ejpam-3576	312	12	if	if	SCONJ
ejpam-3576	312	13	µ	µ	NOUN
ejpam-3576	312	14	is	be	AUX
ejpam-3576	312	15	an	an	DET
ejpam-3576	312	16	anti	anti	ADJ
ejpam-3576	312	17	fuzzy	fuzzy	ADJ
ejpam-3576	312	18	ideal	ideal	NOUN
ejpam-3576	312	19	of	of	ADP
ejpam-3576	312	20	s.	s.	PROPN
ejpam-3576	312	21	proof	proof	PROPN
ejpam-3576	312	22	.	.	PUNCT
ejpam-3576	313	1	let	let	VERB
ejpam-3576	313	2	µ	µ	X
ejpam-3576	313	3	be	be	AUX
ejpam-3576	313	4	an	an	DET
ejpam-3576	313	5	anti	anti	ADJ
ejpam-3576	313	6	fuzzy	fuzzy	ADJ
ejpam-3576	313	7	interior	interior	ADJ
ejpam-3576	313	8	ideal	ideal	NOUN
ejpam-3576	313	9	of	of	ADP
ejpam-3576	313	10	s	s	PRON
ejpam-3576	313	11	and	and	CCONJ
ejpam-3576	313	12	x	x	NOUN
ejpam-3576	313	13	,	,	PUNCT
ejpam-3576	313	14	y	y	PROPN
ejpam-3576	313	15	∈	∈	PROPN
ejpam-3576	313	16	s	s	PROPN
ejpam-3576	313	17	,	,	PUNCT
ejpam-3576	313	18	this	this	PRON
ejpam-3576	313	19	imply	imply	VERB
ejpam-3576	313	20	that	that	SCONJ
ejpam-3576	313	21	there	there	PRON
ejpam-3576	313	22	exists	exist	VERB
ejpam-3576	313	23	a	a	DET
ejpam-3576	313	24	∈	∈	NOUN
ejpam-3576	313	25	s	s	VERB
ejpam-3576	313	26	such	such	ADJ
ejpam-3576	313	27	that	that	SCONJ
ejpam-3576	313	28	x	x	SYM
ejpam-3576	313	29	≤	≤	X
ejpam-3576	313	30	(	(	PUNCT
ejpam-3576	313	31	xa)x	xa)x	PROPN
ejpam-3576	313	32	.	.	PUNCT
ejpam-3576	314	1	now	now	ADV
ejpam-3576	314	2	µ(xy	µ(xy	NUM
ejpam-3576	314	3	)	)	PUNCT
ejpam-3576	314	4	≤	≤	NUM
ejpam-3576	314	5	µ(((xa)x)y	µ(((xa)x)y	PROPN
ejpam-3576	314	6	)	)	PUNCT
ejpam-3576	314	7	=	=	SYM
ejpam-3576	315	1	µ((yx)(xa	µ((yx)(xa	NOUN
ejpam-3576	315	2	)	)	PUNCT
ejpam-3576	315	3	)	)	PUNCT
ejpam-3576	316	1	≤	≤	NOUN
ejpam-3576	316	2	µ(x	µ(x	NOUN
ejpam-3576	316	3	)	)	PUNCT
ejpam-3576	316	4	.	.	PUNCT
ejpam-3576	317	1	thus	thus	ADV
ejpam-3576	317	2	µ	µ	X
ejpam-3576	317	3	is	be	AUX
ejpam-3576	317	4	an	an	DET
ejpam-3576	317	5	anti	anti	ADJ
ejpam-3576	317	6	fuzzy	fuzzy	ADJ
ejpam-3576	317	7	right	right	ADJ
ejpam-3576	317	8	ideal	ideal	NOUN
ejpam-3576	317	9	of	of	ADP
ejpam-3576	317	10	s.	s.	PROPN
ejpam-3576	317	11	hence	hence	ADV
ejpam-3576	317	12	µ	µ	PROPN
ejpam-3576	317	13	is	be	AUX
ejpam-3576	317	14	an	an	DET
ejpam-3576	317	15	anti	anti	ADJ
ejpam-3576	317	16	fuzzy	fuzzy	ADJ
ejpam-3576	317	17	ideal	ideal	NOUN
ejpam-3576	317	18	of	of	ADP
ejpam-3576	317	19	s	s	PRON
ejpam-3576	317	20	by	by	ADP
ejpam-3576	317	21	lemma	lemma	PROPN
ejpam-3576	317	22	8	8	NUM
ejpam-3576	317	23	.	.	PUNCT
ejpam-3576	318	1	converse	converse	NOUN
ejpam-3576	318	2	is	be	AUX
ejpam-3576	318	3	true	true	ADJ
ejpam-3576	318	4	by	by	ADP
ejpam-3576	318	5	lemma	lemma	PROPN
ejpam-3576	318	6	7	7	PROPN
ejpam-3576	318	7	.	.	PUNCT
ejpam-3576	318	8	k.	k.	PROPN
ejpam-3576	318	9	nasreen	nasreen	PROPN
ejpam-3576	318	10	,	,	PUNCT
ejpam-3576	318	11	m.	m.	NOUN
ejpam-3576	318	12	alesemi	alesemi	PROPN
ejpam-3576	318	13	,	,	PUNCT
ejpam-3576	318	14	salahuddin	salahuddin	VERB
ejpam-3576	318	15	/	/	SYM
ejpam-3576	318	16	eur	eur	PROPN
ejpam-3576	318	17	.	.	PUNCT
ejpam-3576	319	1	j.	j.	PROPN
ejpam-3576	319	2	pure	pure	PROPN
ejpam-3576	319	3	appl	appl	PROPN
ejpam-3576	319	4	.	.	PROPN
ejpam-3576	319	5	math	math	PROPN
ejpam-3576	319	6	,	,	PUNCT
ejpam-3576	319	7	13	13	NUM
ejpam-3576	319	8	(	(	PUNCT
ejpam-3576	319	9	1	1	NUM
ejpam-3576	319	10	)	)	PUNCT
ejpam-3576	319	11	(	(	PUNCT
ejpam-3576	319	12	2020	2020	NUM
ejpam-3576	319	13	)	)	PUNCT
ejpam-3576	319	14	,	,	PUNCT
ejpam-3576	319	15	113	113	NUM
ejpam-3576	319	16	-	-	SYM
ejpam-3576	319	17	129	129	NUM
ejpam-3576	319	18	122	122	NUM
ejpam-3576	319	19	lemma	lemma	PROPN
ejpam-3576	319	20	9	9	NUM
ejpam-3576	319	21	.	.	PUNCT
ejpam-3576	320	1	every	every	DET
ejpam-3576	320	2	anti	anti	ADJ
ejpam-3576	320	3	fuzzy	fuzzy	ADJ
ejpam-3576	320	4	right	right	ADJ
ejpam-3576	320	5	(	(	PUNCT
ejpam-3576	320	6	resp	resp	NOUN
ejpam-3576	320	7	.	.	PUNCT
ejpam-3576	321	1	left	left	ADJ
ejpam-3576	321	2	)	)	PUNCT
ejpam-3576	321	3	ideal	ideal	NOUN
ejpam-3576	321	4	of	of	ADP
ejpam-3576	321	5	(	(	PUNCT
ejpam-3576	321	6	2	2	NUM
ejpam-3576	321	7	,	,	PUNCT
ejpam-3576	321	8	2)-regular	2)-regular	NUM
ejpam-3576	321	9	ordered	order	VERB
ejpam-3576	321	10	ag	ag	PROPN
ejpam-3576	321	11	-	-	PROPN
ejpam-3576	321	12	groupoid	groupoid	PROPN
ejpam-3576	321	13	s	s	PROPN
ejpam-3576	321	14	,	,	PUNCT
ejpam-3576	321	15	is	be	AUX
ejpam-3576	321	16	an	an	DET
ejpam-3576	321	17	anti	anti	ADJ
ejpam-3576	321	18	fuzzy	fuzzy	ADJ
ejpam-3576	321	19	ideal	ideal	NOUN
ejpam-3576	321	20	of	of	ADP
ejpam-3576	321	21	s.	s.	PROPN
ejpam-3576	321	22	proof	proof	PROPN
ejpam-3576	321	23	.	.	PUNCT
ejpam-3576	322	1	let	let	VERB
ejpam-3576	322	2	µ	µ	X
ejpam-3576	322	3	be	be	AUX
ejpam-3576	322	4	an	an	DET
ejpam-3576	322	5	anti	anti	ADJ
ejpam-3576	322	6	fuzzy	fuzzy	ADJ
ejpam-3576	322	7	right	right	ADJ
ejpam-3576	322	8	ideal	ideal	NOUN
ejpam-3576	322	9	of	of	ADP
ejpam-3576	322	10	s	s	PRON
ejpam-3576	322	11	and	and	CCONJ
ejpam-3576	322	12	x	x	NOUN
ejpam-3576	322	13	,	,	PUNCT
ejpam-3576	322	14	y	y	PROPN
ejpam-3576	322	15	∈	∈	PROPN
ejpam-3576	322	16	s	s	PROPN
ejpam-3576	322	17	,	,	PUNCT
ejpam-3576	322	18	this	this	PRON
ejpam-3576	322	19	imply	imply	VERB
ejpam-3576	322	20	that	that	SCONJ
ejpam-3576	322	21	there	there	PRON
ejpam-3576	322	22	exists	exist	VERB
ejpam-3576	322	23	a	a	DET
ejpam-3576	322	24	∈	∈	NOUN
ejpam-3576	322	25	s	s	VERB
ejpam-3576	322	26	such	such	ADJ
ejpam-3576	322	27	that	that	SCONJ
ejpam-3576	322	28	x	x	SYM
ejpam-3576	322	29	≤	≤	X
ejpam-3576	322	30	(	(	PUNCT
ejpam-3576	322	31	x2a)x2	x2a)x2	PROPN
ejpam-3576	322	32	.	.	PUNCT
ejpam-3576	323	1	now	now	ADV
ejpam-3576	323	2	µ(xy	µ(xy	NUM
ejpam-3576	323	3	)	)	PUNCT
ejpam-3576	323	4	≤	≤	NUM
ejpam-3576	323	5	µ(((x2a)x2)y	µ(((x2a)x2)y	NOUN
ejpam-3576	323	6	)	)	PUNCT
ejpam-3576	323	7	=	=	SYM
ejpam-3576	323	8	µ((yx2)(x2a	µ((yx2)(x2a	NOUN
ejpam-3576	323	9	)	)	PUNCT
ejpam-3576	323	10	)	)	PUNCT
ejpam-3576	324	1	≤	≤	NOUN
ejpam-3576	325	1	µ(yx2	µ(yx2	NOUN
ejpam-3576	325	2	)	)	PUNCT
ejpam-3576	325	3	≤	≤	NUM
ejpam-3576	325	4	µ(y	µ(y	NUM
ejpam-3576	325	5	)	)	PUNCT
ejpam-3576	325	6	.	.	PUNCT
ejpam-3576	326	1	hence	hence	ADV
ejpam-3576	326	2	µ	µ	X
ejpam-3576	326	3	is	be	AUX
ejpam-3576	326	4	an	an	DET
ejpam-3576	326	5	anti	anti	ADJ
ejpam-3576	326	6	fuzzy	fuzzy	ADJ
ejpam-3576	326	7	ideal	ideal	NOUN
ejpam-3576	326	8	of	of	ADP
ejpam-3576	326	9	s.	s.	PROPN
ejpam-3576	326	10	let	let	VERB
ejpam-3576	326	11	µ	µ	X
ejpam-3576	326	12	be	be	AUX
ejpam-3576	326	13	an	an	DET
ejpam-3576	326	14	anti	anti	ADJ
ejpam-3576	326	15	fuzzy	fuzzy	ADJ
ejpam-3576	326	16	left	leave	VERB
ejpam-3576	326	17	ideal	ideal	NOUN
ejpam-3576	326	18	of	of	ADP
ejpam-3576	326	19	s.	s.	PROPN
ejpam-3576	326	20	now	now	PROPN
ejpam-3576	326	21	µ(xy	µ(xy	PROPN
ejpam-3576	326	22	)	)	PUNCT
ejpam-3576	326	23	≤	≤	NUM
ejpam-3576	326	24	µ(((x2a)x2)y	µ(((x2a)x2)y	NOUN
ejpam-3576	326	25	)	)	PUNCT
ejpam-3576	326	26	=	=	SYM
ejpam-3576	327	1	µ((yx2)(x2a	µ((yx2)(x2a	NOUN
ejpam-3576	327	2	)	)	PUNCT
ejpam-3576	327	3	≤	≤	NUM
ejpam-3576	327	4	µ((xx)a	µ((xx)a	PROPN
ejpam-3576	327	5	)	)	PUNCT
ejpam-3576	327	6	=	=	SYM
ejpam-3576	327	7	µ((ax)x	µ((ax)x	PROPN
ejpam-3576	327	8	)	)	PUNCT
ejpam-3576	327	9	≤	≤	NOUN
ejpam-3576	327	10	µ(x	µ(x	NOUN
ejpam-3576	327	11	)	)	PUNCT
ejpam-3576	327	12	.	.	PUNCT
ejpam-3576	328	1	hence	hence	ADV
ejpam-3576	328	2	µ	µ	X
ejpam-3576	328	3	is	be	AUX
ejpam-3576	328	4	an	an	DET
ejpam-3576	328	5	anti	anti	ADJ
ejpam-3576	328	6	fuzzy	fuzzy	ADJ
ejpam-3576	328	7	ideal	ideal	NOUN
ejpam-3576	328	8	of	of	ADP
ejpam-3576	328	9	s.	s.	PROPN
ejpam-3576	328	10	remark	remark	PROPN
ejpam-3576	328	11	5	5	NUM
ejpam-3576	328	12	.	.	PUNCT
ejpam-3576	329	1	the	the	DET
ejpam-3576	329	2	concept	concept	NOUN
ejpam-3576	329	3	of	of	ADP
ejpam-3576	329	4	anti	anti	ADJ
ejpam-3576	329	5	fuzzy	fuzzy	ADJ
ejpam-3576	329	6	(	(	PUNCT
ejpam-3576	329	7	right	right	INTJ
ejpam-3576	329	8	,	,	PUNCT
ejpam-3576	329	9	left	leave	VERB
ejpam-3576	329	10	,	,	PUNCT
ejpam-3576	329	11	two	two	NUM
ejpam-3576	329	12	-	-	PUNCT
ejpam-3576	329	13	sided	sided	ADJ
ejpam-3576	329	14	)	)	PUNCT
ejpam-3576	329	15	ideals	ideal	NOUN
ejpam-3576	329	16	coincide	coincide	VERB
ejpam-3576	329	17	in	in	ADP
ejpam-3576	329	18	(	(	PUNCT
ejpam-3576	329	19	2	2	NUM
ejpam-3576	329	20	,	,	PUNCT
ejpam-3576	329	21	2)regular	2)regular	NUM
ejpam-3576	329	22	ordered	order	VERB
ejpam-3576	329	23	ag	ag	PROPN
ejpam-3576	329	24	-	-	PUNCT
ejpam-3576	329	25	groupoids	groupoid	NOUN
ejpam-3576	329	26	s.	s.	PROPN
ejpam-3576	329	27	proposition	proposition	PROPN
ejpam-3576	329	28	6	6	NUM
ejpam-3576	329	29	.	.	PUNCT
ejpam-3576	330	1	let	let	VERB
ejpam-3576	330	2	s	s	PRON
ejpam-3576	330	3	be	be	AUX
ejpam-3576	330	4	a	a	DET
ejpam-3576	330	5	(	(	PUNCT
ejpam-3576	330	6	2	2	NUM
ejpam-3576	330	7	,	,	PUNCT
ejpam-3576	330	8	2)-regular	2)-regular	NUM
ejpam-3576	330	9	ordered	order	VERB
ejpam-3576	330	10	ag	ag	PROPN
ejpam-3576	330	11	-	-	NOUN
ejpam-3576	330	12	groupoid	groupoid	PROPN
ejpam-3576	330	13	with	with	ADP
ejpam-3576	330	14	left	left	ADJ
ejpam-3576	330	15	identity	identity	NOUN
ejpam-3576	330	16	e.	e.	PROPN
ejpam-3576	330	17	then	then	ADV
ejpam-3576	330	18	µ	µ	PROPN
ejpam-3576	330	19	is	be	AUX
ejpam-3576	330	20	an	an	DET
ejpam-3576	330	21	anti	anti	ADJ
ejpam-3576	330	22	fuzzy	fuzzy	ADJ
ejpam-3576	330	23	interior	interior	ADJ
ejpam-3576	330	24	ideal	ideal	NOUN
ejpam-3576	330	25	if	if	SCONJ
ejpam-3576	330	26	and	and	CCONJ
ejpam-3576	330	27	only	only	ADV
ejpam-3576	330	28	if	if	SCONJ
ejpam-3576	330	29	µ	µ	NOUN
ejpam-3576	330	30	is	be	AUX
ejpam-3576	330	31	an	an	DET
ejpam-3576	330	32	anti	anti	ADJ
ejpam-3576	330	33	fuzzy	fuzzy	ADJ
ejpam-3576	330	34	ideal	ideal	NOUN
ejpam-3576	330	35	of	of	ADP
ejpam-3576	330	36	s.	s.	PROPN
ejpam-3576	330	37	proof	proof	PROPN
ejpam-3576	330	38	.	.	PUNCT
ejpam-3576	331	1	let	let	VERB
ejpam-3576	331	2	µ	µ	X
ejpam-3576	331	3	be	be	AUX
ejpam-3576	331	4	an	an	DET
ejpam-3576	331	5	anti	anti	ADJ
ejpam-3576	331	6	fuzzy	fuzzy	ADJ
ejpam-3576	331	7	interior	interior	ADJ
ejpam-3576	331	8	ideal	ideal	NOUN
ejpam-3576	331	9	of	of	ADP
ejpam-3576	331	10	s	s	PRON
ejpam-3576	331	11	and	and	CCONJ
ejpam-3576	331	12	x	x	NOUN
ejpam-3576	331	13	,	,	PUNCT
ejpam-3576	331	14	y	y	PROPN
ejpam-3576	331	15	∈	∈	PROPN
ejpam-3576	331	16	s	s	PROPN
ejpam-3576	331	17	,	,	PUNCT
ejpam-3576	331	18	this	this	PRON
ejpam-3576	331	19	imply	imply	VERB
ejpam-3576	331	20	that	that	SCONJ
ejpam-3576	331	21	there	there	PRON
ejpam-3576	331	22	exists	exist	VERB
ejpam-3576	331	23	a	a	DET
ejpam-3576	331	24	∈	∈	NOUN
ejpam-3576	331	25	s	s	VERB
ejpam-3576	331	26	such	such	ADJ
ejpam-3576	331	27	that	that	SCONJ
ejpam-3576	331	28	x	x	SYM
ejpam-3576	331	29	≤	≤	X
ejpam-3576	331	30	(	(	PUNCT
ejpam-3576	331	31	x2a)x2	x2a)x2	PROPN
ejpam-3576	331	32	.	.	PUNCT
ejpam-3576	332	1	now	now	ADV
ejpam-3576	332	2	µ(xy	µ(xy	NUM
ejpam-3576	332	3	)	)	PUNCT
ejpam-3576	332	4	≤	≤	NUM
ejpam-3576	332	5	µ(((x2a)x2)y	µ(((x2a)x2)y	NOUN
ejpam-3576	332	6	)	)	PUNCT
ejpam-3576	332	7	=	=	SYM
ejpam-3576	332	8	µ((yx2)(x2a	µ((yx2)(x2a	NOUN
ejpam-3576	332	9	)	)	PUNCT
ejpam-3576	332	10	)	)	PUNCT
ejpam-3576	332	11	≤	≤	NUM
ejpam-3576	332	12	µ(x2	µ(x2	NOUN
ejpam-3576	332	13	)	)	PUNCT
ejpam-3576	332	14	=	=	SYM
ejpam-3576	332	15	µ(xx	µ(xx	PROPN
ejpam-3576	332	16	)	)	PUNCT
ejpam-3576	332	17	=	=	SYM
ejpam-3576	332	18	µ((ex)x	µ((ex)x	NUM
ejpam-3576	332	19	)	)	PUNCT
ejpam-3576	332	20	≤	≤	NOUN
ejpam-3576	332	21	µ(x	µ(x	NOUN
ejpam-3576	332	22	)	)	PUNCT
ejpam-3576	332	23	.	.	PUNCT
ejpam-3576	333	1	thus	thus	ADV
ejpam-3576	333	2	µ	µ	X
ejpam-3576	333	3	is	be	AUX
ejpam-3576	333	4	an	an	DET
ejpam-3576	333	5	anti	anti	ADJ
ejpam-3576	333	6	fuzzy	fuzzy	ADJ
ejpam-3576	333	7	right	right	ADJ
ejpam-3576	333	8	ideal	ideal	NOUN
ejpam-3576	333	9	of	of	ADP
ejpam-3576	333	10	s.	s.	PROPN
ejpam-3576	333	11	hence	hence	ADV
ejpam-3576	333	12	µ	µ	PROPN
ejpam-3576	333	13	is	be	AUX
ejpam-3576	333	14	an	an	DET
ejpam-3576	333	15	anti	anti	ADJ
ejpam-3576	333	16	fuzzy	fuzzy	ADJ
ejpam-3576	333	17	ideal	ideal	NOUN
ejpam-3576	333	18	of	of	ADP
ejpam-3576	333	19	s	s	PRON
ejpam-3576	333	20	by	by	ADP
ejpam-3576	333	21	lemma	lemma	PROPN
ejpam-3576	333	22	9	9	NUM
ejpam-3576	333	23	.	.	PUNCT
ejpam-3576	334	1	converse	converse	NOUN
ejpam-3576	334	2	is	be	AUX
ejpam-3576	334	3	true	true	ADJ
ejpam-3576	334	4	by	by	ADP
ejpam-3576	334	5	lemma	lemma	PROPN
ejpam-3576	334	6	7	7	PROPN
ejpam-3576	334	7	.	.	PUNCT
ejpam-3576	334	8	lemma	lemma	PROPN
ejpam-3576	334	9	10	10	NUM
ejpam-3576	334	10	.	.	PUNCT
ejpam-3576	335	1	let	let	VERB
ejpam-3576	335	2	s	s	PRON
ejpam-3576	335	3	be	be	AUX
ejpam-3576	335	4	a	a	DET
ejpam-3576	335	5	right	right	ADJ
ejpam-3576	335	6	regular	regular	ADJ
ejpam-3576	335	7	ordered	order	VERB
ejpam-3576	335	8	ag	ag	PROPN
ejpam-3576	335	9	-	-	NOUN
ejpam-3576	335	10	groupoid	groupoid	PROPN
ejpam-3576	335	11	.	.	PUNCT
ejpam-3576	336	1	then	then	ADV
ejpam-3576	336	2	every	every	DET
ejpam-3576	336	3	anti	anti	ADJ
ejpam-3576	336	4	fuzzy	fuzzy	ADJ
ejpam-3576	336	5	right	right	ADJ
ejpam-3576	336	6	(	(	PUNCT
ejpam-3576	336	7	resp	resp	NOUN
ejpam-3576	336	8	.	.	PUNCT
ejpam-3576	337	1	left	left	ADJ
ejpam-3576	337	2	)	)	PUNCT
ejpam-3576	337	3	ideal	ideal	NOUN
ejpam-3576	337	4	of	of	ADP
ejpam-3576	337	5	s	s	PROPN
ejpam-3576	337	6	is	be	AUX
ejpam-3576	337	7	an	an	DET
ejpam-3576	337	8	anti	anti	ADJ
ejpam-3576	337	9	fuzzy	fuzzy	ADJ
ejpam-3576	337	10	ideal	ideal	NOUN
ejpam-3576	337	11	of	of	ADP
ejpam-3576	337	12	s.	s.	PROPN
ejpam-3576	337	13	proof	proof	PROPN
ejpam-3576	337	14	.	.	PUNCT
ejpam-3576	338	1	let	let	VERB
ejpam-3576	338	2	µ	µ	X
ejpam-3576	338	3	be	be	AUX
ejpam-3576	338	4	an	an	DET
ejpam-3576	338	5	anti	anti	ADJ
ejpam-3576	338	6	fuzzy	fuzzy	ADJ
ejpam-3576	338	7	right	right	ADJ
ejpam-3576	338	8	ideal	ideal	NOUN
ejpam-3576	338	9	of	of	ADP
ejpam-3576	338	10	s	s	PRON
ejpam-3576	338	11	and	and	CCONJ
ejpam-3576	338	12	x	x	NOUN
ejpam-3576	338	13	,	,	PUNCT
ejpam-3576	338	14	y	y	PROPN
ejpam-3576	338	15	∈	∈	PROPN
ejpam-3576	338	16	s	s	PROPN
ejpam-3576	338	17	,	,	PUNCT
ejpam-3576	338	18	this	this	PRON
ejpam-3576	338	19	imply	imply	VERB
ejpam-3576	338	20	that	that	SCONJ
ejpam-3576	338	21	there	there	PRON
ejpam-3576	338	22	exists	exist	VERB
ejpam-3576	338	23	a	a	DET
ejpam-3576	338	24	∈	∈	NOUN
ejpam-3576	338	25	s	s	VERB
ejpam-3576	338	26	such	such	ADJ
ejpam-3576	338	27	that	that	SCONJ
ejpam-3576	338	28	x	x	SYM
ejpam-3576	338	29	≤	≤	NUM
ejpam-3576	338	30	x2a	x2a	PROPN
ejpam-3576	338	31	.	.	PUNCT
ejpam-3576	339	1	now	now	ADV
ejpam-3576	339	2	µ(xy	µ(xy	NUM
ejpam-3576	339	3	)	)	PUNCT
ejpam-3576	339	4	≤	≤	NOUN
ejpam-3576	339	5	µ((x2a)y	µ((x2a)y	PROPN
ejpam-3576	339	6	)	)	PUNCT
ejpam-3576	339	7	=	=	SYM
ejpam-3576	339	8	µ(((xx)a)y	µ(((xx)a)y	ADJ
ejpam-3576	339	9	)	)	PUNCT
ejpam-3576	339	10	=	=	SYM
ejpam-3576	339	11	µ(((ax)x)y	µ(((ax)x)y	PROPN
ejpam-3576	339	12	)	)	PUNCT
ejpam-3576	339	13	=	=	SYM
ejpam-3576	339	14	µ((yx)(ax	µ((yx)(ax	NOUN
ejpam-3576	339	15	)	)	PUNCT
ejpam-3576	339	16	)	)	PUNCT
ejpam-3576	340	1	≤	≤	NOUN
ejpam-3576	340	2	µ(yx	µ(yx	NOUN
ejpam-3576	340	3	)	)	PUNCT
ejpam-3576	340	4	≤	≤	NUM
ejpam-3576	340	5	µ(y	µ(y	NUM
ejpam-3576	340	6	)	)	PUNCT
ejpam-3576	340	7	.	.	PUNCT
ejpam-3576	341	1	hence	hence	ADV
ejpam-3576	341	2	µ	µ	X
ejpam-3576	341	3	is	be	AUX
ejpam-3576	341	4	an	an	DET
ejpam-3576	341	5	anti	anti	ADJ
ejpam-3576	341	6	fuzzy	fuzzy	ADJ
ejpam-3576	341	7	ideal	ideal	NOUN
ejpam-3576	341	8	of	of	ADP
ejpam-3576	341	9	s.	s.	PROPN
ejpam-3576	341	10	let	let	VERB
ejpam-3576	341	11	µ	µ	X
ejpam-3576	341	12	be	be	AUX
ejpam-3576	341	13	an	an	DET
ejpam-3576	341	14	anti	anti	ADJ
ejpam-3576	341	15	fuzzy	fuzzy	ADJ
ejpam-3576	341	16	left	leave	VERB
ejpam-3576	341	17	ideal	ideal	NOUN
ejpam-3576	341	18	of	of	ADP
ejpam-3576	341	19	s.	s.	PROPN
ejpam-3576	341	20	now	now	PROPN
ejpam-3576	341	21	µ(xy	µ(xy	PROPN
ejpam-3576	341	22	)	)	PUNCT
ejpam-3576	341	23	≤	≤	NOUN
ejpam-3576	341	24	µ((x2a)y	µ((x2a)y	PROPN
ejpam-3576	341	25	)	)	PUNCT
ejpam-3576	341	26	=	=	SYM
ejpam-3576	341	27	µ(((xx)a)y	µ(((xx)a)y	ADJ
ejpam-3576	341	28	)	)	PUNCT
ejpam-3576	341	29	=	=	SYM
ejpam-3576	341	30	µ(((ax)x)y	µ(((ax)x)y	PROPN
ejpam-3576	341	31	)	)	PUNCT
ejpam-3576	341	32	=	=	SYM
ejpam-3576	341	33	µ((yx)(ax	µ((yx)(ax	NOUN
ejpam-3576	341	34	)	)	PUNCT
ejpam-3576	341	35	)	)	PUNCT
ejpam-3576	341	36	≤	≤	NUM
ejpam-3576	341	37	µ(ax	µ(ax	NOUN
ejpam-3576	341	38	)	)	PUNCT
ejpam-3576	341	39	≤	≤	NOUN
ejpam-3576	341	40	µ(x	µ(x	NOUN
ejpam-3576	341	41	)	)	PUNCT
ejpam-3576	341	42	.	.	PUNCT
ejpam-3576	342	1	hence	hence	ADV
ejpam-3576	342	2	µ	µ	X
ejpam-3576	342	3	is	be	AUX
ejpam-3576	342	4	an	an	DET
ejpam-3576	342	5	anti	anti	ADJ
ejpam-3576	342	6	fuzzy	fuzzy	ADJ
ejpam-3576	342	7	ideal	ideal	NOUN
ejpam-3576	342	8	of	of	ADP
ejpam-3576	342	9	s.	s.	PROPN
ejpam-3576	342	10	remark	remark	PROPN
ejpam-3576	342	11	6	6	NUM
ejpam-3576	342	12	.	.	PUNCT
ejpam-3576	343	1	the	the	DET
ejpam-3576	343	2	concept	concept	NOUN
ejpam-3576	343	3	of	of	ADP
ejpam-3576	343	4	anti	anti	ADJ
ejpam-3576	343	5	fuzzy	fuzzy	ADJ
ejpam-3576	343	6	(	(	PUNCT
ejpam-3576	343	7	right	right	INTJ
ejpam-3576	343	8	,	,	PUNCT
ejpam-3576	343	9	left	leave	VERB
ejpam-3576	343	10	,	,	PUNCT
ejpam-3576	343	11	two	two	NUM
ejpam-3576	343	12	-	-	PUNCT
ejpam-3576	343	13	sided	sided	ADJ
ejpam-3576	343	14	)	)	PUNCT
ejpam-3576	343	15	ideals	ideal	NOUN
ejpam-3576	343	16	coincide	coincide	VERB
ejpam-3576	343	17	in	in	ADP
ejpam-3576	343	18	right	right	ADJ
ejpam-3576	343	19	regular	regular	ADJ
ejpam-3576	343	20	ordered	order	VERB
ejpam-3576	343	21	ag	ag	PROPN
ejpam-3576	343	22	-	-	PUNCT
ejpam-3576	343	23	groupoids	groupoid	NOUN
ejpam-3576	343	24	s.	s.	PROPN
ejpam-3576	343	25	proposition	proposition	PROPN
ejpam-3576	343	26	7	7	NUM
ejpam-3576	343	27	.	.	PUNCT
ejpam-3576	343	28	let	let	VERB
ejpam-3576	343	29	s	s	PRON
ejpam-3576	343	30	be	be	AUX
ejpam-3576	343	31	a	a	DET
ejpam-3576	343	32	right	right	ADJ
ejpam-3576	343	33	regular	regular	ADJ
ejpam-3576	343	34	ordered	order	VERB
ejpam-3576	343	35	ag	ag	PROPN
ejpam-3576	343	36	-	-	NOUN
ejpam-3576	343	37	groupoid	groupoid	PROPN
ejpam-3576	343	38	.	.	PUNCT
ejpam-3576	344	1	then	then	ADV
ejpam-3576	344	2	µ	µ	X
ejpam-3576	344	3	is	be	AUX
ejpam-3576	344	4	an	an	DET
ejpam-3576	344	5	anti	anti	ADJ
ejpam-3576	344	6	fuzzy	fuzzy	ADJ
ejpam-3576	344	7	interior	interior	ADJ
ejpam-3576	344	8	ideal	ideal	NOUN
ejpam-3576	344	9	if	if	SCONJ
ejpam-3576	344	10	and	and	CCONJ
ejpam-3576	344	11	only	only	ADV
ejpam-3576	344	12	if	if	SCONJ
ejpam-3576	344	13	µ	µ	NOUN
ejpam-3576	344	14	is	be	AUX
ejpam-3576	344	15	an	an	DET
ejpam-3576	344	16	anti	anti	ADJ
ejpam-3576	344	17	fuzzy	fuzzy	ADJ
ejpam-3576	344	18	ideal	ideal	NOUN
ejpam-3576	344	19	of	of	ADP
ejpam-3576	344	20	s.	s.	PROPN
ejpam-3576	344	21	k.	k.	PROPN
ejpam-3576	344	22	nasreen	nasreen	PROPN
ejpam-3576	344	23	,	,	PUNCT
ejpam-3576	344	24	m.	m.	NOUN
ejpam-3576	344	25	alesemi	alesemi	PROPN
ejpam-3576	344	26	,	,	PUNCT
ejpam-3576	344	27	salahuddin	salahuddin	VERB
ejpam-3576	344	28	/	/	SYM
ejpam-3576	344	29	eur	eur	PROPN
ejpam-3576	344	30	.	.	PUNCT
ejpam-3576	345	1	j.	j.	PROPN
ejpam-3576	345	2	pure	pure	PROPN
ejpam-3576	345	3	appl	appl	PROPN
ejpam-3576	345	4	.	.	PROPN
ejpam-3576	345	5	math	math	PROPN
ejpam-3576	345	6	,	,	PUNCT
ejpam-3576	345	7	13	13	NUM
ejpam-3576	345	8	(	(	PUNCT
ejpam-3576	345	9	1	1	NUM
ejpam-3576	345	10	)	)	PUNCT
ejpam-3576	345	11	(	(	PUNCT
ejpam-3576	345	12	2020	2020	NUM
ejpam-3576	345	13	)	)	PUNCT
ejpam-3576	345	14	,	,	PUNCT
ejpam-3576	345	15	113	113	NUM
ejpam-3576	345	16	-	-	SYM
ejpam-3576	345	17	129	129	NUM
ejpam-3576	345	18	123	123	NUM
ejpam-3576	345	19	proof	proof	NOUN
ejpam-3576	345	20	.	.	PUNCT
ejpam-3576	346	1	let	let	VERB
ejpam-3576	346	2	µ	µ	X
ejpam-3576	346	3	be	be	AUX
ejpam-3576	346	4	an	an	DET
ejpam-3576	346	5	anti	anti	ADJ
ejpam-3576	346	6	fuzzy	fuzzy	ADJ
ejpam-3576	346	7	interior	interior	ADJ
ejpam-3576	346	8	ideal	ideal	NOUN
ejpam-3576	346	9	of	of	ADP
ejpam-3576	346	10	s	s	PRON
ejpam-3576	346	11	and	and	CCONJ
ejpam-3576	346	12	x	x	NOUN
ejpam-3576	346	13	,	,	PUNCT
ejpam-3576	346	14	y	y	PROPN
ejpam-3576	346	15	∈	∈	PROPN
ejpam-3576	346	16	s	s	PROPN
ejpam-3576	346	17	,	,	PUNCT
ejpam-3576	346	18	this	this	PRON
ejpam-3576	346	19	imply	imply	VERB
ejpam-3576	346	20	that	that	SCONJ
ejpam-3576	346	21	there	there	PRON
ejpam-3576	346	22	exists	exist	VERB
ejpam-3576	346	23	a	a	DET
ejpam-3576	346	24	∈	∈	NOUN
ejpam-3576	346	25	s	s	VERB
ejpam-3576	346	26	such	such	ADJ
ejpam-3576	346	27	that	that	SCONJ
ejpam-3576	346	28	x	x	SYM
ejpam-3576	346	29	≤	≤	NUM
ejpam-3576	346	30	x2a	x2a	PROPN
ejpam-3576	346	31	.	.	PUNCT
ejpam-3576	347	1	now	now	ADV
ejpam-3576	347	2	µ(xy	µ(xy	NUM
ejpam-3576	347	3	)	)	PUNCT
ejpam-3576	347	4	≤	≤	NOUN
ejpam-3576	347	5	µ((x2a)y	µ((x2a)y	PROPN
ejpam-3576	347	6	)	)	PUNCT
ejpam-3576	347	7	=	=	SYM
ejpam-3576	347	8	µ(((xx)a)y	µ(((xx)a)y	ADJ
ejpam-3576	347	9	)	)	PUNCT
ejpam-3576	347	10	=	=	SYM
ejpam-3576	347	11	µ(((ax)x)y	µ(((ax)x)y	PROPN
ejpam-3576	347	12	)	)	PUNCT
ejpam-3576	347	13	≤	≤	NOUN
ejpam-3576	347	14	µ(x	µ(x	NOUN
ejpam-3576	347	15	)	)	PUNCT
ejpam-3576	347	16	.	.	PUNCT
ejpam-3576	348	1	thus	thus	ADV
ejpam-3576	348	2	µ	µ	X
ejpam-3576	348	3	is	be	AUX
ejpam-3576	348	4	an	an	DET
ejpam-3576	348	5	anti	anti	ADJ
ejpam-3576	348	6	fuzzy	fuzzy	ADJ
ejpam-3576	348	7	right	right	ADJ
ejpam-3576	348	8	ideal	ideal	NOUN
ejpam-3576	348	9	of	of	ADP
ejpam-3576	348	10	s.	s.	PROPN
ejpam-3576	348	11	hence	hence	ADV
ejpam-3576	348	12	µ	µ	PROPN
ejpam-3576	348	13	is	be	AUX
ejpam-3576	348	14	an	an	DET
ejpam-3576	348	15	anti	anti	ADJ
ejpam-3576	348	16	fuzzy	fuzzy	ADJ
ejpam-3576	348	17	ideal	ideal	NOUN
ejpam-3576	348	18	of	of	ADP
ejpam-3576	348	19	s	s	PRON
ejpam-3576	348	20	by	by	ADP
ejpam-3576	348	21	lemma	lemma	PROPN
ejpam-3576	348	22	10	10	NUM
ejpam-3576	348	23	.	.	PUNCT
ejpam-3576	349	1	converse	converse	NOUN
ejpam-3576	349	2	is	be	AUX
ejpam-3576	349	3	true	true	ADJ
ejpam-3576	349	4	by	by	ADP
ejpam-3576	349	5	lemma	lemma	PROPN
ejpam-3576	349	6	7	7	PROPN
ejpam-3576	349	7	.	.	PUNCT
ejpam-3576	349	8	lemma	lemma	PROPN
ejpam-3576	349	9	11	11	NUM
ejpam-3576	349	10	.	.	PUNCT
ejpam-3576	350	1	let	let	VERB
ejpam-3576	350	2	s	s	PRON
ejpam-3576	350	3	be	be	AUX
ejpam-3576	350	4	a	a	DET
ejpam-3576	350	5	left	left	ADJ
ejpam-3576	350	6	regular	regular	ADV
ejpam-3576	350	7	ordered	order	VERB
ejpam-3576	350	8	ag	ag	PROPN
ejpam-3576	350	9	-	-	NOUN
ejpam-3576	350	10	groupoid	groupoid	PROPN
ejpam-3576	350	11	with	with	ADP
ejpam-3576	350	12	left	left	ADJ
ejpam-3576	350	13	identity	identity	NOUN
ejpam-3576	350	14	e.	e.	PROPN
ejpam-3576	350	15	then	then	ADV
ejpam-3576	350	16	every	every	DET
ejpam-3576	350	17	anti	anti	X
ejpam-3576	350	18	fuzzy	fuzzy	ADJ
ejpam-3576	350	19	right	right	ADJ
ejpam-3576	350	20	(	(	PUNCT
ejpam-3576	350	21	resp	resp	NOUN
ejpam-3576	350	22	.	.	PUNCT
ejpam-3576	351	1	left	left	ADJ
ejpam-3576	351	2	)	)	PUNCT
ejpam-3576	351	3	ideal	ideal	NOUN
ejpam-3576	351	4	of	of	ADP
ejpam-3576	351	5	s	s	PROPN
ejpam-3576	351	6	is	be	AUX
ejpam-3576	351	7	an	an	DET
ejpam-3576	351	8	anti	anti	ADJ
ejpam-3576	351	9	fuzzy	fuzzy	ADJ
ejpam-3576	351	10	ideal	ideal	NOUN
ejpam-3576	351	11	of	of	ADP
ejpam-3576	351	12	s.	s.	PROPN
ejpam-3576	351	13	proof	proof	PROPN
ejpam-3576	351	14	.	.	PUNCT
ejpam-3576	352	1	let	let	VERB
ejpam-3576	352	2	µ	µ	X
ejpam-3576	352	3	be	be	AUX
ejpam-3576	352	4	an	an	DET
ejpam-3576	352	5	anti	anti	ADJ
ejpam-3576	352	6	fuzzy	fuzzy	ADJ
ejpam-3576	352	7	right	right	ADJ
ejpam-3576	352	8	ideal	ideal	NOUN
ejpam-3576	352	9	of	of	ADP
ejpam-3576	352	10	s	s	PRON
ejpam-3576	352	11	and	and	CCONJ
ejpam-3576	352	12	x	x	NOUN
ejpam-3576	352	13	,	,	PUNCT
ejpam-3576	352	14	y	y	PROPN
ejpam-3576	352	15	∈	∈	PROPN
ejpam-3576	352	16	s	s	PROPN
ejpam-3576	352	17	,	,	PUNCT
ejpam-3576	352	18	this	this	PRON
ejpam-3576	352	19	imply	imply	VERB
ejpam-3576	352	20	that	that	SCONJ
ejpam-3576	352	21	there	there	PRON
ejpam-3576	352	22	exists	exist	VERB
ejpam-3576	352	23	a	a	DET
ejpam-3576	352	24	∈	∈	NOUN
ejpam-3576	352	25	s	s	VERB
ejpam-3576	352	26	such	such	ADJ
ejpam-3576	352	27	that	that	SCONJ
ejpam-3576	352	28	x	x	SYM
ejpam-3576	352	29	≤	≤	ADJ
ejpam-3576	352	30	ax2	ax2	NOUN
ejpam-3576	352	31	.	.	PUNCT
ejpam-3576	353	1	now	now	ADV
ejpam-3576	353	2	µ(xy	µ(xy	NUM
ejpam-3576	353	3	)	)	PUNCT
ejpam-3576	353	4	≤	≤	NOUN
ejpam-3576	353	5	µ((ax2)y	µ((ax2)y	PUNCT
ejpam-3576	353	6	)	)	PUNCT
ejpam-3576	354	1	=	=	SYM
ejpam-3576	354	2	µ((a(xx))y	µ((a(xx))y	NOUN
ejpam-3576	354	3	)	)	PUNCT
ejpam-3576	354	4	=	=	SYM
ejpam-3576	354	5	µ((x(ax))y	µ((x(ax))y	X
ejpam-3576	354	6	)	)	PUNCT
ejpam-3576	354	7	=	=	PUNCT
ejpam-3576	354	8	µ((y(ax))x	µ((y(ax))x	NOUN
ejpam-3576	354	9	)	)	PUNCT
ejpam-3576	354	10	≤	≤	NOUN
ejpam-3576	354	11	µ(y(ax	µ(y(ax	NOUN
ejpam-3576	354	12	)	)	PUNCT
ejpam-3576	354	13	)	)	PUNCT
ejpam-3576	355	1	≤	≤	NUM
ejpam-3576	355	2	µ(y	µ(y	NUM
ejpam-3576	355	3	)	)	PUNCT
ejpam-3576	355	4	.	.	PUNCT
ejpam-3576	356	1	hence	hence	ADV
ejpam-3576	356	2	µ	µ	X
ejpam-3576	356	3	is	be	AUX
ejpam-3576	356	4	an	an	DET
ejpam-3576	356	5	anti	anti	ADJ
ejpam-3576	356	6	fuzzy	fuzzy	ADJ
ejpam-3576	356	7	ideal	ideal	NOUN
ejpam-3576	356	8	of	of	ADP
ejpam-3576	356	9	s.	s.	PROPN
ejpam-3576	356	10	let	let	VERB
ejpam-3576	356	11	µ	µ	X
ejpam-3576	356	12	be	be	AUX
ejpam-3576	356	13	an	an	DET
ejpam-3576	356	14	anti	anti	ADJ
ejpam-3576	356	15	fuzzy	fuzzy	ADJ
ejpam-3576	356	16	left	leave	VERB
ejpam-3576	356	17	ideal	ideal	NOUN
ejpam-3576	356	18	of	of	ADP
ejpam-3576	356	19	s.	s.	PROPN
ejpam-3576	356	20	now	now	PROPN
ejpam-3576	356	21	µ(xy	µ(xy	PROPN
ejpam-3576	356	22	)	)	PUNCT
ejpam-3576	356	23	≤	≤	NOUN
ejpam-3576	356	24	µ((ax2)y	µ((ax2)y	PUNCT
ejpam-3576	356	25	)	)	PUNCT
ejpam-3576	356	26	=	=	SYM
ejpam-3576	356	27	µ((a(xx))y	µ((a(xx))y	NOUN
ejpam-3576	356	28	)	)	PUNCT
ejpam-3576	356	29	=	=	SYM
ejpam-3576	356	30	µ((x(ax))y	µ((x(ax))y	X
ejpam-3576	356	31	)	)	PUNCT
ejpam-3576	356	32	=	=	PUNCT
ejpam-3576	356	33	µ((y(ax))x	µ((y(ax))x	NOUN
ejpam-3576	356	34	)	)	PUNCT
ejpam-3576	356	35	≤	≤	NUM
ejpam-3576	356	36	µ((ax)x	µ((ax)x	NOUN
ejpam-3576	356	37	)	)	PUNCT
ejpam-3576	356	38	≤	≤	NOUN
ejpam-3576	356	39	µ(x	µ(x	NOUN
ejpam-3576	356	40	)	)	PUNCT
ejpam-3576	356	41	.	.	PUNCT
ejpam-3576	357	1	hence	hence	ADV
ejpam-3576	357	2	µ	µ	X
ejpam-3576	357	3	is	be	AUX
ejpam-3576	357	4	an	an	DET
ejpam-3576	357	5	anti	anti	ADJ
ejpam-3576	357	6	fuzzy	fuzzy	ADJ
ejpam-3576	357	7	ideal	ideal	NOUN
ejpam-3576	357	8	of	of	ADP
ejpam-3576	357	9	s.	s.	PROPN
ejpam-3576	357	10	remark	remark	PROPN
ejpam-3576	357	11	7	7	NUM
ejpam-3576	357	12	.	.	PUNCT
ejpam-3576	358	1	the	the	DET
ejpam-3576	358	2	concept	concept	NOUN
ejpam-3576	358	3	of	of	ADP
ejpam-3576	358	4	anti	anti	ADJ
ejpam-3576	358	5	fuzzy	fuzzy	ADJ
ejpam-3576	358	6	(	(	PUNCT
ejpam-3576	358	7	right	right	INTJ
ejpam-3576	358	8	,	,	PUNCT
ejpam-3576	358	9	left	leave	VERB
ejpam-3576	358	10	,	,	PUNCT
ejpam-3576	358	11	two	two	NUM
ejpam-3576	358	12	-	-	PUNCT
ejpam-3576	358	13	sided	sided	ADJ
ejpam-3576	358	14	)	)	PUNCT
ejpam-3576	358	15	ideals	ideal	NOUN
ejpam-3576	358	16	coincide	coincide	VERB
ejpam-3576	358	17	in	in	ADP
ejpam-3576	358	18	left	left	ADJ
ejpam-3576	358	19	regular	regular	ADJ
ejpam-3576	358	20	ordered	order	VERB
ejpam-3576	358	21	ag	ag	PROPN
ejpam-3576	358	22	-	-	NOUN
ejpam-3576	358	23	groupoids	groupoids	PROPN
ejpam-3576	358	24	s	s	VERB
ejpam-3576	358	25	with	with	ADP
ejpam-3576	358	26	left	left	ADJ
ejpam-3576	358	27	identity	identity	NOUN
ejpam-3576	358	28	.	.	PUNCT
ejpam-3576	359	1	proposition	proposition	NOUN
ejpam-3576	359	2	8	8	NUM
ejpam-3576	359	3	.	.	PUNCT
ejpam-3576	360	1	let	let	VERB
ejpam-3576	360	2	s	s	PRON
ejpam-3576	360	3	be	be	AUX
ejpam-3576	360	4	a	a	DET
ejpam-3576	360	5	left	left	ADJ
ejpam-3576	360	6	regular	regular	ADV
ejpam-3576	360	7	ordered	order	VERB
ejpam-3576	360	8	ag	ag	PROPN
ejpam-3576	360	9	-	-	NOUN
ejpam-3576	360	10	groupoid	groupoid	PROPN
ejpam-3576	360	11	with	with	ADP
ejpam-3576	360	12	left	left	ADJ
ejpam-3576	360	13	identity	identity	NOUN
ejpam-3576	360	14	e.	e.	PROPN
ejpam-3576	360	15	then	then	ADV
ejpam-3576	360	16	µ	µ	PROPN
ejpam-3576	360	17	is	be	AUX
ejpam-3576	360	18	an	an	DET
ejpam-3576	360	19	anti	anti	ADJ
ejpam-3576	360	20	fuzzy	fuzzy	ADJ
ejpam-3576	360	21	interior	interior	ADJ
ejpam-3576	360	22	ideal	ideal	NOUN
ejpam-3576	360	23	if	if	SCONJ
ejpam-3576	360	24	and	and	CCONJ
ejpam-3576	360	25	only	only	ADV
ejpam-3576	360	26	if	if	SCONJ
ejpam-3576	360	27	µ	µ	NOUN
ejpam-3576	360	28	is	be	AUX
ejpam-3576	360	29	an	an	DET
ejpam-3576	360	30	anti	anti	ADJ
ejpam-3576	360	31	fuzzy	fuzzy	ADJ
ejpam-3576	360	32	ideal	ideal	NOUN
ejpam-3576	360	33	of	of	ADP
ejpam-3576	360	34	s.	s.	PROPN
ejpam-3576	360	35	proof	proof	PROPN
ejpam-3576	360	36	.	.	PUNCT
ejpam-3576	361	1	let	let	VERB
ejpam-3576	361	2	µ	µ	X
ejpam-3576	361	3	be	be	AUX
ejpam-3576	361	4	an	an	DET
ejpam-3576	361	5	anti	anti	ADJ
ejpam-3576	361	6	fuzzy	fuzzy	ADJ
ejpam-3576	361	7	interior	interior	ADJ
ejpam-3576	361	8	ideal	ideal	NOUN
ejpam-3576	361	9	of	of	ADP
ejpam-3576	361	10	s	s	PRON
ejpam-3576	361	11	and	and	CCONJ
ejpam-3576	361	12	x	x	NOUN
ejpam-3576	361	13	,	,	PUNCT
ejpam-3576	361	14	y	y	PROPN
ejpam-3576	361	15	∈	∈	PROPN
ejpam-3576	361	16	s	s	PROPN
ejpam-3576	361	17	,	,	PUNCT
ejpam-3576	361	18	this	this	PRON
ejpam-3576	361	19	imply	imply	VERB
ejpam-3576	361	20	that	that	SCONJ
ejpam-3576	361	21	there	there	PRON
ejpam-3576	361	22	exists	exist	VERB
ejpam-3576	361	23	a	a	DET
ejpam-3576	361	24	∈	∈	NOUN
ejpam-3576	361	25	s	s	VERB
ejpam-3576	361	26	such	such	ADJ
ejpam-3576	361	27	that	that	SCONJ
ejpam-3576	361	28	x	x	SYM
ejpam-3576	361	29	≤	≤	ADJ
ejpam-3576	361	30	ax2	ax2	NOUN
ejpam-3576	361	31	.	.	PUNCT
ejpam-3576	362	1	now	now	ADV
ejpam-3576	362	2	µ(xy	µ(xy	NUM
ejpam-3576	362	3	)	)	PUNCT
ejpam-3576	362	4	≤	≤	NOUN
ejpam-3576	362	5	µ((ax2)y	µ((ax2)y	PUNCT
ejpam-3576	362	6	)	)	PUNCT
ejpam-3576	363	1	=	=	SYM
ejpam-3576	363	2	µ((a(xx))y	µ((a(xx))y	NOUN
ejpam-3576	363	3	)	)	PUNCT
ejpam-3576	363	4	=	=	SYM
ejpam-3576	363	5	µ((x(ax))y	µ((x(ax))y	X
ejpam-3576	363	6	)	)	PUNCT
ejpam-3576	363	7	=	=	SYM
ejpam-3576	363	8	µ(((ex)(ax))y	µ(((ex)(ax))y	PROPN
ejpam-3576	363	9	)	)	PUNCT
ejpam-3576	363	10	=	=	SYM
ejpam-3576	363	11	µ(((xx)(ae))y	µ(((xx)(ae))y	PROPN
ejpam-3576	363	12	)	)	PUNCT
ejpam-3576	364	1	=	=	PUNCT
ejpam-3576	364	2	µ((((ae)x)x)y	µ((((ae)x)x)y	X
ejpam-3576	364	3	)	)	PUNCT
ejpam-3576	364	4	≤	≤	NOUN
ejpam-3576	364	5	µ(x	µ(x	NOUN
ejpam-3576	364	6	)	)	PUNCT
ejpam-3576	364	7	.	.	PUNCT
ejpam-3576	365	1	thus	thus	ADV
ejpam-3576	365	2	µ	µ	X
ejpam-3576	365	3	is	be	AUX
ejpam-3576	365	4	an	an	DET
ejpam-3576	365	5	anti	anti	ADJ
ejpam-3576	365	6	fuzzy	fuzzy	ADJ
ejpam-3576	365	7	right	right	ADJ
ejpam-3576	365	8	ideal	ideal	NOUN
ejpam-3576	365	9	of	of	ADP
ejpam-3576	365	10	s.	s.	PROPN
ejpam-3576	365	11	hence	hence	ADV
ejpam-3576	365	12	µ	µ	PROPN
ejpam-3576	365	13	is	be	AUX
ejpam-3576	365	14	an	an	DET
ejpam-3576	365	15	anti	anti	ADJ
ejpam-3576	365	16	fuzzy	fuzzy	ADJ
ejpam-3576	365	17	ideal	ideal	NOUN
ejpam-3576	365	18	of	of	ADP
ejpam-3576	365	19	s	s	PRON
ejpam-3576	365	20	by	by	ADP
ejpam-3576	365	21	lemma	lemma	PROPN
ejpam-3576	365	22	11	11	NUM
ejpam-3576	365	23	.	.	PUNCT
ejpam-3576	366	1	converse	converse	NOUN
ejpam-3576	366	2	is	be	AUX
ejpam-3576	366	3	true	true	ADJ
ejpam-3576	366	4	by	by	ADP
ejpam-3576	366	5	lemma	lemma	PROPN
ejpam-3576	366	6	7	7	NUM
ejpam-3576	366	7	.	.	PUNCT
ejpam-3576	366	8	theorem	theorem	NOUN
ejpam-3576	366	9	1	1	NUM
ejpam-3576	366	10	.	.	PUNCT
ejpam-3576	367	1	let	let	VERB
ejpam-3576	367	2	s	s	PRON
ejpam-3576	367	3	be	be	AUX
ejpam-3576	367	4	a	a	DET
ejpam-3576	367	5	right	right	ADJ
ejpam-3576	367	6	regular	regular	ADJ
ejpam-3576	367	7	locally	locally	ADV
ejpam-3576	367	8	associative	associative	ADJ
ejpam-3576	368	1	ordered	order	VERB
ejpam-3576	368	2	ag	ag	PROPN
ejpam-3576	368	3	-	-	NOUN
ejpam-3576	368	4	groupoid	groupoid	PROPN
ejpam-3576	368	5	with	with	ADP
ejpam-3576	368	6	left	left	ADJ
ejpam-3576	368	7	identity	identity	NOUN
ejpam-3576	368	8	e.	e.	PROPN
ejpam-3576	368	9	then	then	ADV
ejpam-3576	368	10	for	for	ADP
ejpam-3576	368	11	every	every	DET
ejpam-3576	368	12	anti	anti	ADJ
ejpam-3576	368	13	fuzzy	fuzzy	ADJ
ejpam-3576	368	14	interior	interior	ADJ
ejpam-3576	368	15	ideal	ideal	NOUN
ejpam-3576	368	16	µ	µ	X
ejpam-3576	368	17	of	of	ADP
ejpam-3576	368	18	s	s	PROPN
ejpam-3576	368	19	,	,	PUNCT
ejpam-3576	368	20	µ(an	µ(an	ADJ
ejpam-3576	368	21	)	)	PUNCT
ejpam-3576	368	22	=	=	PUNCT
ejpam-3576	369	1	µ(a2n	µ(a2n	NOUN
ejpam-3576	369	2	)	)	PUNCT
ejpam-3576	369	3	,	,	PUNCT
ejpam-3576	369	4	where	where	SCONJ
ejpam-3576	369	5	n	n	PRON
ejpam-3576	369	6	is	be	AUX
ejpam-3576	369	7	any	any	DET
ejpam-3576	369	8	positive	positive	ADJ
ejpam-3576	369	9	integer	integer	NOUN
ejpam-3576	369	10	,	,	PUNCT
ejpam-3576	369	11	for	for	ADP
ejpam-3576	369	12	all	all	DET
ejpam-3576	369	13	a	a	DET
ejpam-3576	369	14	∈	∈	PROPN
ejpam-3576	369	15	s.	s.	PROPN
ejpam-3576	369	16	proof	proof	NOUN
ejpam-3576	369	17	.	.	PUNCT
ejpam-3576	370	1	for	for	ADP
ejpam-3576	370	2	n	n	NOUN
ejpam-3576	370	3	=	=	SYM
ejpam-3576	370	4	1	1	X
ejpam-3576	370	5	.	.	PUNCT
ejpam-3576	370	6	let	let	VERB
ejpam-3576	370	7	a	a	DET
ejpam-3576	370	8	∈	∈	ADJ
ejpam-3576	370	9	s	s	NOUN
ejpam-3576	370	10	,	,	PUNCT
ejpam-3576	370	11	this	this	PRON
ejpam-3576	370	12	imply	imply	VERB
ejpam-3576	370	13	that	that	SCONJ
ejpam-3576	370	14	there	there	PRON
ejpam-3576	370	15	exists	exist	VERB
ejpam-3576	370	16	x	x	X
ejpam-3576	370	17	∈	∈	NOUN
ejpam-3576	370	18	s	s	VERB
ejpam-3576	370	19	such	such	ADJ
ejpam-3576	370	20	that	that	SCONJ
ejpam-3576	370	21	a	a	DET
ejpam-3576	370	22	≤	≤	NOUN
ejpam-3576	370	23	a2x	a2x	PROPN
ejpam-3576	370	24	.	.	PUNCT
ejpam-3576	371	1	thus	thus	ADV
ejpam-3576	371	2	µ(a	µ(a	PROPN
ejpam-3576	371	3	)	)	PUNCT
ejpam-3576	371	4	≤	≤	NOUN
ejpam-3576	372	1	µ(a2x	µ(a2x	ADP
ejpam-3576	372	2	)	)	PUNCT
ejpam-3576	372	3	=	=	SYM
ejpam-3576	372	4	µ((ea2)x	µ((ea2)x	CCONJ
ejpam-3576	372	5	)	)	PUNCT
ejpam-3576	372	6	≤	≤	NOUN
ejpam-3576	372	7	µ(a2	µ(a2	NOUN
ejpam-3576	372	8	)	)	PUNCT
ejpam-3576	372	9	≤	≤	NUM
ejpam-3576	372	10	max{µ	max{µ	NOUN
ejpam-3576	372	11	(	(	PUNCT
ejpam-3576	372	12	a	a	NOUN
ejpam-3576	372	13	)	)	PUNCT
ejpam-3576	372	14	,	,	PUNCT
ejpam-3576	372	15	µ	µ	X
ejpam-3576	372	16	(	(	PUNCT
ejpam-3576	372	17	a	a	NOUN
ejpam-3576	372	18	)	)	PUNCT
ejpam-3576	372	19	}	}	PUNCT
ejpam-3576	372	20	=	=	SYM
ejpam-3576	372	21	µ	µ	X
ejpam-3576	372	22	(	(	PUNCT
ejpam-3576	372	23	a	a	NOUN
ejpam-3576	372	24	)	)	PUNCT
ejpam-3576	372	25	,	,	PUNCT
ejpam-3576	372	26	(	(	PUNCT
ejpam-3576	372	27	µ	µ	X
ejpam-3576	372	28	is	be	AUX
ejpam-3576	372	29	an	an	DET
ejpam-3576	372	30	anti	anti	ADJ
ejpam-3576	372	31	fuzzy	fuzzy	ADJ
ejpam-3576	372	32	ideal	ideal	NOUN
ejpam-3576	372	33	of	of	ADP
ejpam-3576	372	34	s	s	PRON
ejpam-3576	372	35	by	by	ADP
ejpam-3576	372	36	proposition	proposition	NOUN
ejpam-3576	372	37	7	7	NUM
ejpam-3576	372	38	)	)	PUNCT
ejpam-3576	372	39	.	.	PUNCT
ejpam-3576	373	1	hence	hence	ADV
ejpam-3576	373	2	µ	µ	X
ejpam-3576	373	3	(	(	PUNCT
ejpam-3576	373	4	a	a	NOUN
ejpam-3576	373	5	)	)	PUNCT
ejpam-3576	373	6	=	=	SYM
ejpam-3576	373	7	µ	µ	X
ejpam-3576	373	8	(	(	PUNCT
ejpam-3576	373	9	a2	a2	PROPN
ejpam-3576	373	10	)	)	PUNCT
ejpam-3576	373	11	.	.	PUNCT
ejpam-3576	374	1	now	now	ADV
ejpam-3576	374	2	a2	a2	PROPN
ejpam-3576	374	3	=	=	SYM
ejpam-3576	374	4	aa	aa	PROPN
ejpam-3576	374	5	≤	≤	PROPN
ejpam-3576	374	6	(	(	PUNCT
ejpam-3576	374	7	a2x)(a2x	a2x)(a2x	PROPN
ejpam-3576	374	8	)	)	PUNCT
ejpam-3576	374	9	=	=	SYM
ejpam-3576	375	1	a4x2	a4x2	PROPN
ejpam-3576	375	2	,	,	PUNCT
ejpam-3576	375	3	then	then	ADV
ejpam-3576	375	4	k.	k.	PROPN
ejpam-3576	375	5	nasreen	nasreen	PROPN
ejpam-3576	375	6	,	,	PUNCT
ejpam-3576	375	7	m.	m.	NOUN
ejpam-3576	375	8	alesemi	alesemi	PROPN
ejpam-3576	375	9	,	,	PUNCT
ejpam-3576	375	10	salahuddin	salahuddin	VERB
ejpam-3576	375	11	/	/	SYM
ejpam-3576	375	12	eur	eur	PROPN
ejpam-3576	375	13	.	.	PUNCT
ejpam-3576	376	1	j.	j.	PROPN
ejpam-3576	376	2	pure	pure	PROPN
ejpam-3576	376	3	appl	appl	PROPN
ejpam-3576	376	4	.	.	PROPN
ejpam-3576	376	5	math	math	PROPN
ejpam-3576	376	6	,	,	PUNCT
ejpam-3576	376	7	13	13	NUM
ejpam-3576	376	8	(	(	PUNCT
ejpam-3576	376	9	1	1	NUM
ejpam-3576	376	10	)	)	PUNCT
ejpam-3576	376	11	(	(	PUNCT
ejpam-3576	376	12	2020	2020	NUM
ejpam-3576	376	13	)	)	PUNCT
ejpam-3576	376	14	,	,	PUNCT
ejpam-3576	376	15	113	113	NUM
ejpam-3576	376	16	-	-	SYM
ejpam-3576	376	17	129	129	NUM
ejpam-3576	376	18	124	124	NUM
ejpam-3576	376	19	the	the	DET
ejpam-3576	376	20	result	result	NOUN
ejpam-3576	376	21	is	be	AUX
ejpam-3576	376	22	true	true	ADJ
ejpam-3576	376	23	for	for	ADP
ejpam-3576	376	24	n	n	NOUN
ejpam-3576	376	25	=	=	SYM
ejpam-3576	376	26	2	2	X
ejpam-3576	376	27	.	.	PUNCT
ejpam-3576	376	28	suppose	suppose	VERB
ejpam-3576	376	29	that	that	SCONJ
ejpam-3576	376	30	result	result	NOUN
ejpam-3576	376	31	is	be	AUX
ejpam-3576	376	32	true	true	ADJ
ejpam-3576	376	33	for	for	ADP
ejpam-3576	376	34	n	n	PROPN
ejpam-3576	376	35	=	=	SYM
ejpam-3576	376	36	k	k	NOUN
ejpam-3576	376	37	,	,	PUNCT
ejpam-3576	376	38	i.e.	i.e.	X
ejpam-3576	376	39	,	,	PUNCT
ejpam-3576	376	40	µ(ak	µ(ak	PROPN
ejpam-3576	376	41	)	)	PUNCT
ejpam-3576	376	42	=	=	PUNCT
ejpam-3576	376	43	µ(a2k	µ(a2k	X
ejpam-3576	376	44	)	)	PUNCT
ejpam-3576	376	45	.	.	PUNCT
ejpam-3576	377	1	now	now	ADV
ejpam-3576	377	2	ak+1	ak+1	VERB
ejpam-3576	377	3	=	=	PUNCT
ejpam-3576	377	4	aka	aka	ADV
ejpam-3576	377	5	≤	≤	NUM
ejpam-3576	377	6	(	(	PUNCT
ejpam-3576	377	7	a2kxk)(a2x	a2kxk)(a2x	PROPN
ejpam-3576	377	8	)	)	PUNCT
ejpam-3576	377	9	=	=	SYM
ejpam-3576	377	10	a2(k+1)x(k+1	a2(k+1)x(k+1	PROPN
ejpam-3576	377	11	)	)	PUNCT
ejpam-3576	377	12	.	.	PUNCT
ejpam-3576	378	1	thus	thus	ADV
ejpam-3576	378	2	µ(ak+1	µ(ak+1	VERB
ejpam-3576	378	3	)	)	PUNCT
ejpam-3576	378	4	≤	≤	NOUN
ejpam-3576	378	5	µ(a2(k+1)x(k+1	µ(a2(k+1)x(k+1	ADV
ejpam-3576	378	6	)	)	PUNCT
ejpam-3576	378	7	)	)	PUNCT
ejpam-3576	379	1	=	=	SYM
ejpam-3576	379	2	µ((ea2(k+1))x(k+1	µ((ea2(k+1))x(k+1	NOUN
ejpam-3576	379	3	)	)	PUNCT
ejpam-3576	379	4	)	)	PUNCT
ejpam-3576	379	5	≤	≤	NOUN
ejpam-3576	379	6	µ(a2(k+1	µ(a2(k+1	NOUN
ejpam-3576	379	7	)	)	PUNCT
ejpam-3576	379	8	)	)	PUNCT
ejpam-3576	380	1	=	=	SYM
ejpam-3576	380	2	µ(a2k+2	µ(a2k+2	NOUN
ejpam-3576	380	3	)	)	PUNCT
ejpam-3576	380	4	=	=	PUNCT
ejpam-3576	380	5	µ(ak+1ak+1	µ(ak+1ak+1	NOUN
ejpam-3576	380	6	)	)	PUNCT
ejpam-3576	380	7	≤	≤	NOUN
ejpam-3576	380	8	max{µ	max{µ	ADV
ejpam-3576	380	9	(	(	PUNCT
ejpam-3576	380	10	ak+1	ak+1	X
ejpam-3576	380	11	)	)	PUNCT
ejpam-3576	380	12	,	,	PUNCT
ejpam-3576	380	13	µ	µ	X
ejpam-3576	380	14	(	(	PUNCT
ejpam-3576	380	15	ak+1	ak+1	X
ejpam-3576	380	16	)	)	PUNCT
ejpam-3576	380	17	}	}	PUNCT
ejpam-3576	380	18	=	=	SYM
ejpam-3576	380	19	µ	µ	X
ejpam-3576	380	20	(	(	PUNCT
ejpam-3576	380	21	ak+1	ak+1	NUM
ejpam-3576	380	22	)	)	PUNCT
ejpam-3576	380	23	.	.	PUNCT
ejpam-3576	381	1	therefore	therefore	ADV
ejpam-3576	381	2	µ(ak+1	µ(ak+1	NOUN
ejpam-3576	381	3	)	)	PUNCT
ejpam-3576	381	4	=	=	SYM
ejpam-3576	381	5	µ(a2(k+1	µ(a2(k+1	ADJ
ejpam-3576	381	6	)	)	PUNCT
ejpam-3576	381	7	)	)	PUNCT
ejpam-3576	381	8	.	.	PUNCT
ejpam-3576	382	1	hence	hence	ADV
ejpam-3576	382	2	by	by	ADP
ejpam-3576	382	3	induction	induction	NOUN
ejpam-3576	382	4	method	method	NOUN
ejpam-3576	382	5	,	,	PUNCT
ejpam-3576	382	6	the	the	DET
ejpam-3576	382	7	result	result	NOUN
ejpam-3576	382	8	is	be	AUX
ejpam-3576	382	9	true	true	ADJ
ejpam-3576	382	10	for	for	ADP
ejpam-3576	382	11	all	all	DET
ejpam-3576	382	12	positive	positive	ADJ
ejpam-3576	382	13	integers	integer	NOUN
ejpam-3576	382	14	.	.	PUNCT
ejpam-3576	383	1	lemma	lemma	PROPN
ejpam-3576	383	2	12	12	NUM
ejpam-3576	383	3	.	.	PUNCT
ejpam-3576	384	1	let	let	VERB
ejpam-3576	384	2	s	s	PRON
ejpam-3576	384	3	be	be	AUX
ejpam-3576	384	4	a	a	DET
ejpam-3576	384	5	right	right	ADJ
ejpam-3576	384	6	regular	regular	ADJ
ejpam-3576	384	7	locally	locally	ADV
ejpam-3576	384	8	associative	associative	ADJ
ejpam-3576	385	1	ordered	order	VERB
ejpam-3576	385	2	ag	ag	PROPN
ejpam-3576	385	3	-	-	NOUN
ejpam-3576	385	4	groupoid	groupoid	PROPN
ejpam-3576	385	5	with	with	ADP
ejpam-3576	385	6	left	left	ADJ
ejpam-3576	385	7	identity	identity	NOUN
ejpam-3576	385	8	e.	e.	PROPN
ejpam-3576	385	9	then	then	ADV
ejpam-3576	385	10	for	for	ADP
ejpam-3576	385	11	every	every	DET
ejpam-3576	385	12	anti	anti	ADJ
ejpam-3576	385	13	fuzzy	fuzzy	ADJ
ejpam-3576	385	14	interior	interior	ADJ
ejpam-3576	385	15	ideal	ideal	NOUN
ejpam-3576	385	16	µ	µ	X
ejpam-3576	385	17	of	of	ADP
ejpam-3576	385	18	s	s	PROPN
ejpam-3576	385	19	,	,	PUNCT
ejpam-3576	385	20	µ(ab	µ(ab	PROPN
ejpam-3576	385	21	)	)	PUNCT
ejpam-3576	385	22	=	=	SYM
ejpam-3576	385	23	µ(ba	µ(ba	X
ejpam-3576	385	24	)	)	PUNCT
ejpam-3576	385	25	for	for	ADP
ejpam-3576	385	26	all	all	DET
ejpam-3576	385	27	a	a	PRON
ejpam-3576	385	28	,	,	PUNCT
ejpam-3576	385	29	b	b	X
ejpam-3576	385	30	∈	∈	PROPN
ejpam-3576	385	31	s.	s.	PROPN
ejpam-3576	385	32	proof	proof	PROPN
ejpam-3576	385	33	.	.	PUNCT
ejpam-3576	386	1	let	let	VERB
ejpam-3576	386	2	a	a	DET
ejpam-3576	386	3	,	,	PUNCT
ejpam-3576	386	4	b	b	PROPN
ejpam-3576	386	5	∈	∈	PROPN
ejpam-3576	386	6	s.	s.	PROPN
ejpam-3576	386	7	by	by	ADP
ejpam-3576	386	8	using	use	VERB
ejpam-3576	386	9	theorem	theorem	NOUN
ejpam-3576	386	10	(	(	PUNCT
ejpam-3576	386	11	for	for	ADP
ejpam-3576	386	12	n	n	NOUN
ejpam-3576	386	13	=	=	SYM
ejpam-3576	386	14	1	1	NUM
ejpam-3576	386	15	)	)	PUNCT
ejpam-3576	386	16	.	.	PUNCT
ejpam-3576	387	1	now	now	ADV
ejpam-3576	387	2	µ(ab	µ(ab	VERB
ejpam-3576	387	3	)	)	PUNCT
ejpam-3576	387	4	=	=	PUNCT
ejpam-3576	387	5	µ((ab)2	µ((ab)2	X
ejpam-3576	387	6	)	)	PUNCT
ejpam-3576	387	7	=	=	NOUN
ejpam-3576	387	8	µ((ab)(ab	µ((ab)(ab	NOUN
ejpam-3576	387	9	)	)	PUNCT
ejpam-3576	387	10	)	)	PUNCT
ejpam-3576	388	1	=	=	SYM
ejpam-3576	388	2	µ((ba)(ba	µ((ba)(ba	NOUN
ejpam-3576	388	3	)	)	PUNCT
ejpam-3576	388	4	)	)	PUNCT
ejpam-3576	389	1	=	=	PUNCT
ejpam-3576	389	2	µ((ba)2	µ((ba)2	ADP
ejpam-3576	389	3	)	)	PUNCT
ejpam-3576	389	4	=	=	PUNCT
ejpam-3576	389	5	µ(ba	µ(ba	NOUN
ejpam-3576	389	6	)	)	PUNCT
ejpam-3576	389	7	.	.	PUNCT
ejpam-3576	390	1	theorem	theorem	NOUN
ejpam-3576	390	2	2	2	NUM
ejpam-3576	390	3	.	.	PUNCT
ejpam-3576	391	1	let	let	VERB
ejpam-3576	391	2	s	s	PRON
ejpam-3576	391	3	be	be	AUX
ejpam-3576	391	4	a	a	DET
ejpam-3576	391	5	regular	regular	ADJ
ejpam-3576	391	6	and	and	CCONJ
ejpam-3576	391	7	right	right	ADV
ejpam-3576	391	8	regular	regular	ADJ
ejpam-3576	391	9	locally	locally	ADV
ejpam-3576	391	10	associative	associative	ADJ
ejpam-3576	392	1	ordered	order	VERB
ejpam-3576	392	2	ag	ag	PROPN
ejpam-3576	392	3	-	-	NOUN
ejpam-3576	392	4	groupoid	groupoid	PROPN
ejpam-3576	392	5	with	with	ADP
ejpam-3576	392	6	left	left	ADJ
ejpam-3576	392	7	identity	identity	NOUN
ejpam-3576	392	8	e.	e.	PROPN
ejpam-3576	392	9	then	then	ADV
ejpam-3576	392	10	for	for	ADP
ejpam-3576	392	11	every	every	DET
ejpam-3576	392	12	anti	anti	ADJ
ejpam-3576	392	13	fuzzy	fuzzy	ADJ
ejpam-3576	392	14	interior	interior	ADJ
ejpam-3576	392	15	ideal	ideal	NOUN
ejpam-3576	392	16	µ	µ	X
ejpam-3576	392	17	of	of	ADP
ejpam-3576	392	18	s	s	PROPN
ejpam-3576	392	19	,	,	PUNCT
ejpam-3576	392	20	µ(an	µ(an	ADJ
ejpam-3576	392	21	)	)	PUNCT
ejpam-3576	392	22	=	=	PUNCT
ejpam-3576	392	23	µ(a3n	µ(a3n	PRON
ejpam-3576	392	24	)	)	PUNCT
ejpam-3576	392	25	,	,	PUNCT
ejpam-3576	392	26	where	where	SCONJ
ejpam-3576	392	27	n	n	PRON
ejpam-3576	392	28	is	be	AUX
ejpam-3576	392	29	any	any	DET
ejpam-3576	392	30	positive	positive	ADJ
ejpam-3576	392	31	integer	integer	NOUN
ejpam-3576	392	32	,	,	PUNCT
ejpam-3576	392	33	for	for	ADP
ejpam-3576	392	34	all	all	DET
ejpam-3576	392	35	a	a	DET
ejpam-3576	392	36	∈	∈	PROPN
ejpam-3576	392	37	s.	s.	PROPN
ejpam-3576	392	38	proof	proof	NOUN
ejpam-3576	392	39	.	.	PUNCT
ejpam-3576	393	1	for	for	ADP
ejpam-3576	393	2	n	n	NOUN
ejpam-3576	393	3	=	=	SYM
ejpam-3576	393	4	1	1	X
ejpam-3576	393	5	.	.	PUNCT
ejpam-3576	393	6	let	let	VERB
ejpam-3576	393	7	a	a	DET
ejpam-3576	393	8	∈	∈	ADJ
ejpam-3576	393	9	s	s	NOUN
ejpam-3576	393	10	,	,	PUNCT
ejpam-3576	393	11	this	this	PRON
ejpam-3576	393	12	imply	imply	VERB
ejpam-3576	393	13	that	that	SCONJ
ejpam-3576	393	14	there	there	PRON
ejpam-3576	393	15	exists	exist	VERB
ejpam-3576	393	16	x	x	X
ejpam-3576	393	17	∈	∈	NOUN
ejpam-3576	393	18	s	s	VERB
ejpam-3576	393	19	such	such	ADJ
ejpam-3576	393	20	that	that	SCONJ
ejpam-3576	393	21	a	a	DET
ejpam-3576	393	22	≤	≤	NOUN
ejpam-3576	393	23	(	(	PUNCT
ejpam-3576	393	24	ax)a	ax)a	PROPN
ejpam-3576	393	25	and	and	CCONJ
ejpam-3576	393	26	a	a	DET
ejpam-3576	393	27	≤	≤	NOUN
ejpam-3576	393	28	a2x	a2x	PROPN
ejpam-3576	393	29	.	.	PUNCT
ejpam-3576	394	1	now	now	ADV
ejpam-3576	394	2	a	a	DET
ejpam-3576	394	3	≤	≤	NOUN
ejpam-3576	394	4	(	(	PUNCT
ejpam-3576	394	5	ax)a	ax)a	PROPN
ejpam-3576	394	6	≤	≤	NOUN
ejpam-3576	394	7	(	(	PUNCT
ejpam-3576	394	8	ax)(a2x	ax)(a2x	NOUN
ejpam-3576	394	9	)	)	PUNCT
ejpam-3576	394	10	=	=	PUNCT
ejpam-3576	395	1	a3x2	a3x2	PROPN
ejpam-3576	395	2	.	.	PUNCT
ejpam-3576	395	3	thus	thus	ADV
ejpam-3576	395	4	µ(a	µ(a	PROPN
ejpam-3576	395	5	)	)	PUNCT
ejpam-3576	395	6	≤	≤	NOUN
ejpam-3576	395	7	µ(a3x2	µ(a3x2	ADP
ejpam-3576	395	8	)	)	PUNCT
ejpam-3576	395	9	=	=	PUNCT
ejpam-3576	395	10	µ((ea3)x2	µ((ea3)x2	NOUN
ejpam-3576	395	11	)	)	PUNCT
ejpam-3576	395	12	≤	≤	NOUN
ejpam-3576	395	13	µ(a3	µ(a3	NOUN
ejpam-3576	395	14	)	)	PUNCT
ejpam-3576	395	15	=	=	SYM
ejpam-3576	395	16	µ(aa2	µ(aa2	PROPN
ejpam-3576	395	17	)	)	PUNCT
ejpam-3576	395	18	≤	≤	NOUN
ejpam-3576	395	19	max{µ	max{µ	NOUN
ejpam-3576	395	20	(	(	PUNCT
ejpam-3576	395	21	a	a	NOUN
ejpam-3576	395	22	)	)	PUNCT
ejpam-3576	395	23	,	,	PUNCT
ejpam-3576	395	24	µ	µ	X
ejpam-3576	395	25	(	(	PUNCT
ejpam-3576	395	26	a2	a2	PROPN
ejpam-3576	395	27	)	)	PUNCT
ejpam-3576	395	28	}	}	PUNCT
ejpam-3576	395	29	≤	≤	ADV
ejpam-3576	395	30	max{µ	max{µ	ADV
ejpam-3576	395	31	(	(	PUNCT
ejpam-3576	395	32	a	a	NOUN
ejpam-3576	395	33	)	)	PUNCT
ejpam-3576	395	34	,	,	PUNCT
ejpam-3576	395	35	µ	µ	X
ejpam-3576	395	36	(	(	PUNCT
ejpam-3576	395	37	a	a	NOUN
ejpam-3576	395	38	)	)	PUNCT
ejpam-3576	395	39	,	,	PUNCT
ejpam-3576	395	40	µ	µ	X
ejpam-3576	395	41	(	(	PUNCT
ejpam-3576	395	42	a	a	NOUN
ejpam-3576	395	43	)	)	PUNCT
ejpam-3576	395	44	}	}	PUNCT
ejpam-3576	395	45	=	=	SYM
ejpam-3576	395	46	µ	µ	X
ejpam-3576	395	47	(	(	PUNCT
ejpam-3576	395	48	a	a	NOUN
ejpam-3576	395	49	)	)	PUNCT
ejpam-3576	395	50	.	.	PUNCT
ejpam-3576	396	1	hence	hence	ADV
ejpam-3576	396	2	µ	µ	X
ejpam-3576	396	3	(	(	PUNCT
ejpam-3576	396	4	a	a	NOUN
ejpam-3576	396	5	)	)	PUNCT
ejpam-3576	396	6	=	=	SYM
ejpam-3576	396	7	µ	µ	X
ejpam-3576	396	8	(	(	PUNCT
ejpam-3576	396	9	a3	a3	PROPN
ejpam-3576	396	10	)	)	PUNCT
ejpam-3576	396	11	.	.	PUNCT
ejpam-3576	397	1	now	now	ADV
ejpam-3576	397	2	a2	a2	PROPN
ejpam-3576	397	3	=	=	SYM
ejpam-3576	397	4	aa	aa	PROPN
ejpam-3576	397	5	≤	≤	PROPN
ejpam-3576	397	6	(	(	PUNCT
ejpam-3576	397	7	a3x2)(a3x2	a3x2)(a3x2	NOUN
ejpam-3576	397	8	)	)	PUNCT
ejpam-3576	397	9	=	=	SYM
ejpam-3576	397	10	a6x4	a6x4	PROPN
ejpam-3576	397	11	,	,	PUNCT
ejpam-3576	397	12	then	then	ADV
ejpam-3576	397	13	the	the	DET
ejpam-3576	397	14	result	result	NOUN
ejpam-3576	397	15	is	be	AUX
ejpam-3576	397	16	true	true	ADJ
ejpam-3576	397	17	for	for	ADP
ejpam-3576	397	18	n	n	NOUN
ejpam-3576	397	19	=	=	SYM
ejpam-3576	397	20	2	2	X
ejpam-3576	397	21	.	.	PUNCT
ejpam-3576	397	22	suppose	suppose	VERB
ejpam-3576	397	23	that	that	SCONJ
ejpam-3576	397	24	result	result	NOUN
ejpam-3576	397	25	is	be	AUX
ejpam-3576	397	26	true	true	ADJ
ejpam-3576	397	27	for	for	ADP
ejpam-3576	397	28	n	n	PROPN
ejpam-3576	397	29	=	=	SYM
ejpam-3576	397	30	k	k	NOUN
ejpam-3576	397	31	,	,	PUNCT
ejpam-3576	397	32	i.e.	i.e.	X
ejpam-3576	397	33	,	,	PUNCT
ejpam-3576	397	34	µ(ak	µ(ak	PROPN
ejpam-3576	397	35	)	)	PUNCT
ejpam-3576	397	36	=	=	PUNCT
ejpam-3576	397	37	µ(a3k	µ(a3k	ADV
ejpam-3576	397	38	)	)	PUNCT
ejpam-3576	397	39	.	.	PUNCT
ejpam-3576	398	1	now	now	ADV
ejpam-3576	398	2	ak+1	ak+1	VERB
ejpam-3576	398	3	=	=	PUNCT
ejpam-3576	398	4	aka	aka	ADV
ejpam-3576	398	5	≤	≤	NUM
ejpam-3576	398	6	(	(	PUNCT
ejpam-3576	398	7	a3kx2k)(a3x2	a3kx2k)(a3x2	NOUN
ejpam-3576	398	8	)	)	PUNCT
ejpam-3576	398	9	=	=	PUNCT
ejpam-3576	398	10	a3(k+1)x2(k+1	a3(k+1)x2(k+1	PROPN
ejpam-3576	398	11	)	)	PUNCT
ejpam-3576	398	12	.	.	PUNCT
ejpam-3576	399	1	thus	thus	ADV
ejpam-3576	399	2	µ(ak+1	µ(ak+1	VERB
ejpam-3576	399	3	)	)	PUNCT
ejpam-3576	399	4	≤	≤	NOUN
ejpam-3576	399	5	µ(a3(k+1)x2(k+1	µ(a3(k+1)x2(k+1	PROPN
ejpam-3576	399	6	)	)	PUNCT
ejpam-3576	399	7	)	)	PUNCT
ejpam-3576	399	8	=	=	SYM
ejpam-3576	399	9	µ((ea3(k+1))x2(k+1	µ((ea3(k+1))x2(k+1	NUM
ejpam-3576	399	10	)	)	PUNCT
ejpam-3576	399	11	)	)	PUNCT
ejpam-3576	399	12	≤	≤	NOUN
ejpam-3576	399	13	µ(a3(k+1	µ(a3(k+1	VERB
ejpam-3576	399	14	)	)	PUNCT
ejpam-3576	399	15	)	)	PUNCT
ejpam-3576	400	1	=	=	SYM
ejpam-3576	400	2	µ(a3k+3	µ(a3k+3	NUM
ejpam-3576	400	3	)	)	PUNCT
ejpam-3576	400	4	=	=	PUNCT
ejpam-3576	400	5	µ(ak+1a2k+2	µ(ak+1a2k+2	VERB
ejpam-3576	400	6	)	)	PUNCT
ejpam-3576	400	7	≤	≤	NOUN
ejpam-3576	400	8	maxµ	maxµ	NOUN
ejpam-3576	400	9	(	(	PUNCT
ejpam-3576	400	10	ak+1	ak+1	X
ejpam-3576	400	11	)	)	PUNCT
ejpam-3576	400	12	,	,	PUNCT
ejpam-3576	400	13	µ	µ	X
ejpam-3576	400	14	(	(	PUNCT
ejpam-3576	400	15	a2k+2	a2k+2	ADJ
ejpam-3576	400	16	)	)	PUNCT
ejpam-3576	400	17	}	}	PUNCT
ejpam-3576	400	18	≤	≤	ADV
ejpam-3576	400	19	max{µ	max{µ	ADV
ejpam-3576	400	20	(	(	PUNCT
ejpam-3576	400	21	ak+1	ak+1	X
ejpam-3576	400	22	)	)	PUNCT
ejpam-3576	400	23	,	,	PUNCT
ejpam-3576	400	24	µ	µ	X
ejpam-3576	400	25	(	(	PUNCT
ejpam-3576	400	26	ak+1	ak+1	X
ejpam-3576	400	27	)	)	PUNCT
ejpam-3576	400	28	,	,	PUNCT
ejpam-3576	400	29	µ	µ	X
ejpam-3576	400	30	(	(	PUNCT
ejpam-3576	400	31	ak+1	ak+1	X
ejpam-3576	400	32	)	)	PUNCT
ejpam-3576	400	33	}	}	PUNCT
ejpam-3576	400	34	=	=	SYM
ejpam-3576	400	35	µ	µ	X
ejpam-3576	400	36	(	(	PUNCT
ejpam-3576	400	37	ak+1	ak+1	NUM
ejpam-3576	400	38	)	)	PUNCT
ejpam-3576	400	39	.	.	PUNCT
ejpam-3576	401	1	therefore	therefore	ADV
ejpam-3576	401	2	µ(ak+1	µ(ak+1	NOUN
ejpam-3576	401	3	)	)	PUNCT
ejpam-3576	401	4	=	=	PUNCT
ejpam-3576	401	5	µ(a3(k+1	µ(a3(k+1	X
ejpam-3576	401	6	)	)	PUNCT
ejpam-3576	401	7	)	)	PUNCT
ejpam-3576	401	8	.	.	PUNCT
ejpam-3576	402	1	hence	hence	ADV
ejpam-3576	402	2	by	by	ADP
ejpam-3576	402	3	induction	induction	NOUN
ejpam-3576	402	4	method	method	NOUN
ejpam-3576	402	5	,	,	PUNCT
ejpam-3576	402	6	the	the	DET
ejpam-3576	402	7	result	result	NOUN
ejpam-3576	402	8	is	be	AUX
ejpam-3576	402	9	true	true	ADJ
ejpam-3576	402	10	for	for	ADP
ejpam-3576	402	11	all	all	DET
ejpam-3576	402	12	positive	positive	ADJ
ejpam-3576	402	13	integers	integer	NOUN
ejpam-3576	402	14	.	.	PUNCT
ejpam-3576	403	1	lemma	lemma	PROPN
ejpam-3576	403	2	13	13	NUM
ejpam-3576	403	3	.	.	PUNCT
ejpam-3576	404	1	let	let	VERB
ejpam-3576	404	2	s	s	PRON
ejpam-3576	404	3	be	be	AUX
ejpam-3576	404	4	a	a	DET
ejpam-3576	404	5	weakly	weakly	ADJ
ejpam-3576	404	6	regular	regular	ADJ
ejpam-3576	404	7	ordered	order	VERB
ejpam-3576	404	8	ag	ag	PROPN
ejpam-3576	404	9	-	-	NOUN
ejpam-3576	404	10	groupoid	groupoid	PROPN
ejpam-3576	404	11	.	.	PUNCT
ejpam-3576	405	1	then	then	ADV
ejpam-3576	405	2	every	every	DET
ejpam-3576	405	3	anti	anti	ADJ
ejpam-3576	405	4	fuzzy	fuzzy	ADJ
ejpam-3576	405	5	right	right	ADJ
ejpam-3576	405	6	(	(	PUNCT
ejpam-3576	405	7	resp	resp	NOUN
ejpam-3576	405	8	.	.	PUNCT
ejpam-3576	405	9	left	left	ADJ
ejpam-3576	405	10	)	)	PUNCT
ejpam-3576	405	11	ideal	ideal	NOUN
ejpam-3576	405	12	is	be	AUX
ejpam-3576	405	13	an	an	DET
ejpam-3576	405	14	anti	anti	ADJ
ejpam-3576	405	15	fuzzy	fuzzy	ADJ
ejpam-3576	405	16	ideal	ideal	NOUN
ejpam-3576	405	17	of	of	ADP
ejpam-3576	405	18	s.	s.	PROPN
ejpam-3576	405	19	k.	k.	PROPN
ejpam-3576	405	20	nasreen	nasreen	PROPN
ejpam-3576	405	21	,	,	PUNCT
ejpam-3576	405	22	m.	m.	NOUN
ejpam-3576	405	23	alesemi	alesemi	PROPN
ejpam-3576	405	24	,	,	PUNCT
ejpam-3576	405	25	salahuddin	salahuddin	VERB
ejpam-3576	405	26	/	/	SYM
ejpam-3576	405	27	eur	eur	PROPN
ejpam-3576	405	28	.	.	PUNCT
ejpam-3576	406	1	j.	j.	PROPN
ejpam-3576	406	2	pure	pure	PROPN
ejpam-3576	406	3	appl	appl	PROPN
ejpam-3576	406	4	.	.	PROPN
ejpam-3576	406	5	math	math	PROPN
ejpam-3576	406	6	,	,	PUNCT
ejpam-3576	406	7	13	13	NUM
ejpam-3576	406	8	(	(	PUNCT
ejpam-3576	406	9	1	1	NUM
ejpam-3576	406	10	)	)	PUNCT
ejpam-3576	406	11	(	(	PUNCT
ejpam-3576	406	12	2020	2020	NUM
ejpam-3576	406	13	)	)	PUNCT
ejpam-3576	406	14	,	,	PUNCT
ejpam-3576	406	15	113	113	NUM
ejpam-3576	406	16	-	-	SYM
ejpam-3576	406	17	129	129	NUM
ejpam-3576	406	18	125	125	NUM
ejpam-3576	406	19	proof	proof	NOUN
ejpam-3576	406	20	.	.	PUNCT
ejpam-3576	407	1	let	let	VERB
ejpam-3576	407	2	µ	µ	X
ejpam-3576	407	3	be	be	AUX
ejpam-3576	407	4	an	an	DET
ejpam-3576	407	5	anti	anti	ADJ
ejpam-3576	407	6	fuzzy	fuzzy	ADJ
ejpam-3576	407	7	right	right	ADJ
ejpam-3576	407	8	ideal	ideal	NOUN
ejpam-3576	407	9	of	of	ADP
ejpam-3576	407	10	s	s	PRON
ejpam-3576	407	11	and	and	CCONJ
ejpam-3576	407	12	x	x	NOUN
ejpam-3576	407	13	,	,	PUNCT
ejpam-3576	407	14	y	y	PROPN
ejpam-3576	407	15	∈	∈	PROPN
ejpam-3576	407	16	s	s	PROPN
ejpam-3576	407	17	,	,	PUNCT
ejpam-3576	407	18	this	this	PRON
ejpam-3576	407	19	imply	imply	VERB
ejpam-3576	407	20	that	that	SCONJ
ejpam-3576	407	21	there	there	PRON
ejpam-3576	407	22	exist	exist	VERB
ejpam-3576	407	23	a	a	DET
ejpam-3576	407	24	,	,	PUNCT
ejpam-3576	407	25	b	b	X
ejpam-3576	407	26	∈	∈	NOUN
ejpam-3576	407	27	s	s	VERB
ejpam-3576	407	28	such	such	ADJ
ejpam-3576	407	29	that	that	SCONJ
ejpam-3576	407	30	x	x	SYM
ejpam-3576	407	31	≤	≤	X
ejpam-3576	407	32	(	(	PUNCT
ejpam-3576	407	33	xa)(xb	xa)(xb	PROPN
ejpam-3576	407	34	)	)	PUNCT
ejpam-3576	407	35	.	.	PUNCT
ejpam-3576	408	1	now	now	ADV
ejpam-3576	408	2	µ(xy	µ(xy	NUM
ejpam-3576	408	3	)	)	PUNCT
ejpam-3576	408	4	≤	≤	NUM
ejpam-3576	408	5	µ(((xa)(xb))y	µ(((xa)(xb))y	NOUN
ejpam-3576	408	6	)	)	PUNCT
ejpam-3576	408	7	=	=	SYM
ejpam-3576	409	1	µ((((xb)a)x)y	µ((((xb)a)x)y	PROPN
ejpam-3576	409	2	)	)	PUNCT
ejpam-3576	409	3	=	=	SYM
ejpam-3576	409	4	µ((((ab)x)x)y	µ((((ab)x)x)y	PROPN
ejpam-3576	409	5	)	)	PUNCT
ejpam-3576	409	6	=	=	SYM
ejpam-3576	409	7	µ((yx)((ab)x	µ((yx)((ab)x	NOUN
ejpam-3576	409	8	)	)	PUNCT
ejpam-3576	409	9	)	)	PUNCT
ejpam-3576	410	1	=	=	SYM
ejpam-3576	410	2	µ((yx)(nx	µ((yx)(nx	NOUN
ejpam-3576	410	3	)	)	PUNCT
ejpam-3576	410	4	)	)	PUNCT
ejpam-3576	410	5	say	say	VERB
ejpam-3576	410	6	ab	ab	PROPN
ejpam-3576	410	7	=	=	PUNCT
ejpam-3576	410	8	n	n	CCONJ
ejpam-3576	410	9	≤	≤	NOUN
ejpam-3576	410	10	µ(yx	µ(yx	NOUN
ejpam-3576	410	11	)	)	PUNCT
ejpam-3576	410	12	≤	≤	NUM
ejpam-3576	410	13	µ(y	µ(y	NUM
ejpam-3576	410	14	)	)	PUNCT
ejpam-3576	410	15	.	.	PUNCT
ejpam-3576	411	1	hence	hence	ADV
ejpam-3576	411	2	µ	µ	X
ejpam-3576	411	3	is	be	AUX
ejpam-3576	411	4	an	an	DET
ejpam-3576	411	5	anti	anti	ADJ
ejpam-3576	411	6	fuzzy	fuzzy	ADJ
ejpam-3576	411	7	ideal	ideal	NOUN
ejpam-3576	411	8	of	of	ADP
ejpam-3576	411	9	s.	s.	PROPN
ejpam-3576	411	10	let	let	VERB
ejpam-3576	411	11	µ	µ	X
ejpam-3576	411	12	be	be	AUX
ejpam-3576	411	13	an	an	DET
ejpam-3576	411	14	anti	anti	ADJ
ejpam-3576	411	15	fuzzy	fuzzy	ADJ
ejpam-3576	411	16	left	leave	VERB
ejpam-3576	411	17	ideal	ideal	NOUN
ejpam-3576	411	18	of	of	ADP
ejpam-3576	411	19	s.	s.	PROPN
ejpam-3576	411	20	now	now	PROPN
ejpam-3576	411	21	µ(xy	µ(xy	PROPN
ejpam-3576	411	22	)	)	PUNCT
ejpam-3576	411	23	≤	≤	NUM
ejpam-3576	411	24	µ(((xa)(xb))y	µ(((xa)(xb))y	NOUN
ejpam-3576	411	25	)	)	PUNCT
ejpam-3576	411	26	=	=	SYM
ejpam-3576	411	27	µ((((xb)a)x)y	µ((((xb)a)x)y	PROPN
ejpam-3576	411	28	)	)	PUNCT
ejpam-3576	411	29	=	=	SYM
ejpam-3576	411	30	µ((((ab)x)x)y	µ((((ab)x)x)y	PROPN
ejpam-3576	411	31	)	)	PUNCT
ejpam-3576	411	32	=	=	SYM
ejpam-3576	411	33	µ((yx)((ab)x	µ((yx)((ab)x	NOUN
ejpam-3576	411	34	)	)	PUNCT
ejpam-3576	411	35	)	)	PUNCT
ejpam-3576	411	36	=	=	SYM
ejpam-3576	412	1	µ((yx)(nx	µ((yx)(nx	NOUN
ejpam-3576	412	2	)	)	PUNCT
ejpam-3576	412	3	)	)	PUNCT
ejpam-3576	413	1	say	say	VERB
ejpam-3576	413	2	ab	ab	PROPN
ejpam-3576	413	3	=	=	PUNCT
ejpam-3576	413	4	n	n	CCONJ
ejpam-3576	413	5	≤	≤	NUM
ejpam-3576	413	6	µ(nx	µ(nx	NOUN
ejpam-3576	413	7	)	)	PUNCT
ejpam-3576	413	8	≤	≤	NOUN
ejpam-3576	413	9	µ(x	µ(x	NOUN
ejpam-3576	413	10	)	)	PUNCT
ejpam-3576	413	11	.	.	PUNCT
ejpam-3576	414	1	hence	hence	ADV
ejpam-3576	414	2	µ	µ	X
ejpam-3576	414	3	is	be	AUX
ejpam-3576	414	4	an	an	DET
ejpam-3576	414	5	anti	anti	ADJ
ejpam-3576	414	6	fuzzy	fuzzy	ADJ
ejpam-3576	414	7	ideal	ideal	NOUN
ejpam-3576	414	8	of	of	ADP
ejpam-3576	414	9	s.	s.	PROPN
ejpam-3576	414	10	remark	remark	PROPN
ejpam-3576	414	11	8	8	NUM
ejpam-3576	414	12	.	.	PUNCT
ejpam-3576	415	1	the	the	DET
ejpam-3576	415	2	concept	concept	NOUN
ejpam-3576	415	3	of	of	ADP
ejpam-3576	415	4	anti	anti	ADJ
ejpam-3576	415	5	fuzzy	fuzzy	ADJ
ejpam-3576	415	6	(	(	PUNCT
ejpam-3576	415	7	right	right	INTJ
ejpam-3576	415	8	,	,	PUNCT
ejpam-3576	415	9	left	leave	VERB
ejpam-3576	415	10	,	,	PUNCT
ejpam-3576	415	11	two	two	NUM
ejpam-3576	415	12	-	-	PUNCT
ejpam-3576	415	13	sided	sided	ADJ
ejpam-3576	415	14	)	)	PUNCT
ejpam-3576	415	15	ideals	ideal	NOUN
ejpam-3576	415	16	coincide	coincide	VERB
ejpam-3576	415	17	in	in	ADP
ejpam-3576	415	18	weakly	weakly	ADJ
ejpam-3576	415	19	regular	regular	ADJ
ejpam-3576	415	20	ordered	order	VERB
ejpam-3576	415	21	ag	ag	PROPN
ejpam-3576	415	22	-	-	PUNCT
ejpam-3576	415	23	groupoids	groupoid	NOUN
ejpam-3576	415	24	s.	s.	PROPN
ejpam-3576	415	25	proposition	proposition	PROPN
ejpam-3576	415	26	9	9	NUM
ejpam-3576	415	27	.	.	PUNCT
ejpam-3576	416	1	let	let	VERB
ejpam-3576	416	2	s	s	PRON
ejpam-3576	416	3	be	be	AUX
ejpam-3576	416	4	a	a	DET
ejpam-3576	416	5	weakly	weakly	ADJ
ejpam-3576	416	6	regular	regular	ADJ
ejpam-3576	416	7	ordered	order	VERB
ejpam-3576	416	8	ag	ag	PROPN
ejpam-3576	416	9	-	-	NOUN
ejpam-3576	416	10	groupoid	groupoid	PROPN
ejpam-3576	416	11	.	.	PUNCT
ejpam-3576	417	1	then	then	ADV
ejpam-3576	417	2	µ	µ	X
ejpam-3576	417	3	is	be	AUX
ejpam-3576	417	4	an	an	DET
ejpam-3576	417	5	anti	anti	ADJ
ejpam-3576	417	6	fuzzy	fuzzy	ADJ
ejpam-3576	417	7	interior	interior	ADJ
ejpam-3576	417	8	ideal	ideal	NOUN
ejpam-3576	417	9	if	if	SCONJ
ejpam-3576	417	10	and	and	CCONJ
ejpam-3576	417	11	only	only	ADV
ejpam-3576	417	12	if	if	SCONJ
ejpam-3576	417	13	µ	µ	NOUN
ejpam-3576	417	14	is	be	AUX
ejpam-3576	417	15	an	an	DET
ejpam-3576	417	16	anti	anti	ADJ
ejpam-3576	417	17	fuzzy	fuzzy	ADJ
ejpam-3576	417	18	ideal	ideal	NOUN
ejpam-3576	417	19	of	of	ADP
ejpam-3576	417	20	s.	s.	PROPN
ejpam-3576	417	21	proof	proof	PROPN
ejpam-3576	417	22	.	.	PUNCT
ejpam-3576	418	1	let	let	VERB
ejpam-3576	418	2	µ	µ	X
ejpam-3576	418	3	be	be	AUX
ejpam-3576	418	4	an	an	DET
ejpam-3576	418	5	anti	anti	ADJ
ejpam-3576	418	6	fuzzy	fuzzy	ADJ
ejpam-3576	418	7	interior	interior	ADJ
ejpam-3576	418	8	ideal	ideal	NOUN
ejpam-3576	418	9	of	of	ADP
ejpam-3576	418	10	s	s	PRON
ejpam-3576	418	11	and	and	CCONJ
ejpam-3576	418	12	x	x	NOUN
ejpam-3576	418	13	,	,	PUNCT
ejpam-3576	418	14	y	y	PROPN
ejpam-3576	418	15	∈	∈	PROPN
ejpam-3576	418	16	s	s	PROPN
ejpam-3576	418	17	,	,	PUNCT
ejpam-3576	418	18	this	this	PRON
ejpam-3576	418	19	imply	imply	VERB
ejpam-3576	418	20	that	that	SCONJ
ejpam-3576	418	21	there	there	PRON
ejpam-3576	418	22	exist	exist	VERB
ejpam-3576	418	23	a	a	DET
ejpam-3576	418	24	,	,	PUNCT
ejpam-3576	418	25	b	b	X
ejpam-3576	418	26	∈	∈	NOUN
ejpam-3576	418	27	s	s	VERB
ejpam-3576	418	28	such	such	ADJ
ejpam-3576	418	29	that	that	SCONJ
ejpam-3576	418	30	x	x	SYM
ejpam-3576	418	31	≤	≤	X
ejpam-3576	418	32	(	(	PUNCT
ejpam-3576	418	33	xa)(xb	xa)(xb	PROPN
ejpam-3576	418	34	)	)	PUNCT
ejpam-3576	418	35	.	.	PUNCT
ejpam-3576	419	1	now	now	ADV
ejpam-3576	419	2	µ(xy	µ(xy	NUM
ejpam-3576	419	3	)	)	PUNCT
ejpam-3576	419	4	≤	≤	NUM
ejpam-3576	419	5	µ(((xa)(xb))y	µ(((xa)(xb))y	NOUN
ejpam-3576	419	6	)	)	PUNCT
ejpam-3576	419	7	=	=	SYM
ejpam-3576	419	8	µ((((xb)a)x)y	µ((((xb)a)x)y	PROPN
ejpam-3576	419	9	)	)	PUNCT
ejpam-3576	419	10	≤	≤	NOUN
ejpam-3576	419	11	µ(x	µ(x	NOUN
ejpam-3576	419	12	)	)	PUNCT
ejpam-3576	419	13	.	.	PUNCT
ejpam-3576	420	1	thus	thus	ADV
ejpam-3576	420	2	µ	µ	X
ejpam-3576	420	3	is	be	AUX
ejpam-3576	420	4	an	an	DET
ejpam-3576	420	5	anti	anti	ADJ
ejpam-3576	420	6	fuzzy	fuzzy	ADJ
ejpam-3576	420	7	right	right	ADJ
ejpam-3576	420	8	ideal	ideal	NOUN
ejpam-3576	420	9	of	of	ADP
ejpam-3576	420	10	s.	s.	PROPN
ejpam-3576	420	11	hence	hence	ADV
ejpam-3576	420	12	µ	µ	PROPN
ejpam-3576	420	13	is	be	AUX
ejpam-3576	420	14	an	an	DET
ejpam-3576	420	15	anti	anti	ADJ
ejpam-3576	420	16	fuzzy	fuzzy	ADJ
ejpam-3576	420	17	ideal	ideal	NOUN
ejpam-3576	420	18	of	of	ADP
ejpam-3576	420	19	s	s	PRON
ejpam-3576	420	20	by	by	ADP
ejpam-3576	420	21	lemma	lemma	PROPN
ejpam-3576	420	22	13	13	NUM
ejpam-3576	420	23	.	.	PUNCT
ejpam-3576	421	1	converse	converse	NOUN
ejpam-3576	421	2	is	be	AUX
ejpam-3576	421	3	true	true	ADJ
ejpam-3576	421	4	by	by	ADP
ejpam-3576	421	5	lemma	lemma	PROPN
ejpam-3576	421	6	7	7	NUM
ejpam-3576	421	7	.	.	PUNCT
ejpam-3576	421	8	theorem	theorem	NOUN
ejpam-3576	421	9	3	3	X
ejpam-3576	421	10	.	.	PUNCT
ejpam-3576	422	1	let	let	VERB
ejpam-3576	422	2	s	s	PRON
ejpam-3576	422	3	be	be	AUX
ejpam-3576	422	4	an	an	DET
ejpam-3576	422	5	ordered	order	VERB
ejpam-3576	422	6	ag	ag	PROPN
ejpam-3576	422	7	-	-	NOUN
ejpam-3576	422	8	groupoid	groupoid	PROPN
ejpam-3576	422	9	with	with	ADP
ejpam-3576	422	10	left	left	ADJ
ejpam-3576	422	11	identity	identity	NOUN
ejpam-3576	422	12	e.	e.	PROPN
ejpam-3576	422	13	then	then	ADV
ejpam-3576	422	14	s	s	VERB
ejpam-3576	422	15	is	be	AUX
ejpam-3576	422	16	a	a	DET
ejpam-3576	422	17	weakly	weakly	ADJ
ejpam-3576	422	18	regular	regular	ADJ
ejpam-3576	422	19	if	if	SCONJ
ejpam-3576	423	1	and	and	CCONJ
ejpam-3576	423	2	only	only	ADV
ejpam-3576	423	3	if	if	SCONJ
ejpam-3576	423	4	s	s	NOUN
ejpam-3576	423	5	is	be	AUX
ejpam-3576	423	6	completely	completely	ADV
ejpam-3576	423	7	regular	regular	ADJ
ejpam-3576	423	8	.	.	PUNCT
ejpam-3576	424	1	proof	proof	NOUN
ejpam-3576	424	2	.	.	PUNCT
ejpam-3576	425	1	suppose	suppose	VERB
ejpam-3576	425	2	s	s	PRON
ejpam-3576	425	3	is	be	AUX
ejpam-3576	425	4	a	a	DET
ejpam-3576	425	5	weakly	weakly	ADJ
ejpam-3576	425	6	regular	regular	ADJ
ejpam-3576	425	7	ordered	order	VERB
ejpam-3576	425	8	ag	ag	PROPN
ejpam-3576	425	9	-	-	NOUN
ejpam-3576	425	10	groupoid	groupoid	PROPN
ejpam-3576	425	11	.	.	PUNCT
ejpam-3576	426	1	let	let	VERB
ejpam-3576	426	2	a	a	DET
ejpam-3576	426	3	∈	∈	ADJ
ejpam-3576	426	4	s	s	NOUN
ejpam-3576	426	5	,	,	PUNCT
ejpam-3576	426	6	then	then	ADV
ejpam-3576	426	7	there	there	PRON
ejpam-3576	426	8	exist	exist	VERB
ejpam-3576	426	9	x	x	NOUN
ejpam-3576	426	10	,	,	PUNCT
ejpam-3576	426	11	y	y	PROPN
ejpam-3576	426	12	∈	∈	PROPN
ejpam-3576	426	13	s	s	VERB
ejpam-3576	426	14	such	such	ADJ
ejpam-3576	426	15	that	that	SCONJ
ejpam-3576	426	16	a	a	DET
ejpam-3576	426	17	≤	≤	ADJ
ejpam-3576	426	18	(	(	PUNCT
ejpam-3576	426	19	ax)(ay	ax)(ay	NOUN
ejpam-3576	426	20	)	)	PUNCT
ejpam-3576	426	21	.	.	PUNCT
ejpam-3576	427	1	now	now	ADV
ejpam-3576	427	2	a	a	DET
ejpam-3576	427	3	≤	≤	ADJ
ejpam-3576	427	4	(	(	PUNCT
ejpam-3576	427	5	ax)(ay	ax)(ay	NOUN
ejpam-3576	427	6	)	)	PUNCT
ejpam-3576	427	7	=	=	SYM
ejpam-3576	427	8	(	(	PUNCT
ejpam-3576	427	9	aa)(xy	aa)(xy	NOUN
ejpam-3576	427	10	)	)	PUNCT
ejpam-3576	427	11	=	=	SYM
ejpam-3576	427	12	a2	a2	PROPN
ejpam-3576	427	13	t	t	PROPN
ejpam-3576	427	14	,	,	PUNCT
ejpam-3576	427	15	for	for	ADP
ejpam-3576	427	16	some	some	DET
ejpam-3576	427	17	t	t	NOUN
ejpam-3576	427	18	∈	∈	PROPN
ejpam-3576	427	19	s	s	PART
ejpam-3576	427	20	,	,	PUNCT
ejpam-3576	427	21	this	this	PRON
ejpam-3576	427	22	imply	imply	VERB
ejpam-3576	427	23	that	that	SCONJ
ejpam-3576	427	24	a	a	DET
ejpam-3576	427	25	≤	≤	NUM
ejpam-3576	427	26	a2	a2	NOUN
ejpam-3576	427	27	t.	t.	NOUN
ejpam-3576	427	28	thus	thus	ADV
ejpam-3576	427	29	s	s	PART
ejpam-3576	427	30	is	be	AUX
ejpam-3576	427	31	a	a	DET
ejpam-3576	427	32	right	right	ADJ
ejpam-3576	427	33	regular	regular	ADJ
ejpam-3576	427	34	ordered	order	VERB
ejpam-3576	427	35	ag	ag	PROPN
ejpam-3576	427	36	-	-	NOUN
ejpam-3576	427	37	groupoid	groupoid	PROPN
ejpam-3576	427	38	.	.	PUNCT
ejpam-3576	428	1	now	now	ADV
ejpam-3576	428	2	a	a	DET
ejpam-3576	428	3	≤	≤	ADJ
ejpam-3576	428	4	(	(	PUNCT
ejpam-3576	428	5	ax)(ay	ax)(ay	NOUN
ejpam-3576	428	6	)	)	PUNCT
ejpam-3576	428	7	=	=	SYM
ejpam-3576	428	8	(	(	PUNCT
ejpam-3576	428	9	yx)(aa	yx)(aa	NOUN
ejpam-3576	428	10	)	)	PUNCT
ejpam-3576	428	11	=	=	SYM
ejpam-3576	428	12	ta2	ta2	PROPN
ejpam-3576	428	13	,	,	PUNCT
ejpam-3576	428	14	for	for	ADP
ejpam-3576	428	15	some	some	DET
ejpam-3576	428	16	t	t	NOUN
ejpam-3576	428	17	∈	∈	PROPN
ejpam-3576	428	18	s	s	PART
ejpam-3576	428	19	,	,	PUNCT
ejpam-3576	428	20	this	this	PRON
ejpam-3576	428	21	imply	imply	VERB
ejpam-3576	428	22	that	that	SCONJ
ejpam-3576	428	23	a	a	DET
ejpam-3576	428	24	≤	≤	PROPN
ejpam-3576	428	25	ta2	ta2	PROPN
ejpam-3576	428	26	.	.	PUNCT
ejpam-3576	429	1	thus	thus	ADV
ejpam-3576	429	2	s	s	X
ejpam-3576	429	3	is	be	AUX
ejpam-3576	429	4	a	a	DET
ejpam-3576	429	5	left	left	ADJ
ejpam-3576	429	6	regular	regular	ADJ
ejpam-3576	429	7	ordered	order	VERB
ejpam-3576	429	8	ag	ag	PROPN
ejpam-3576	429	9	-	-	NOUN
ejpam-3576	429	10	groupoid	groupoid	PROPN
ejpam-3576	429	11	.	.	PUNCT
ejpam-3576	430	1	now	now	ADV
ejpam-3576	430	2	a	a	DET
ejpam-3576	430	3	≤	≤	ADJ
ejpam-3576	430	4	(	(	PUNCT
ejpam-3576	430	5	ax)(ay	ax)(ay	NOUN
ejpam-3576	430	6	)	)	PUNCT
ejpam-3576	430	7	=	=	SYM
ejpam-3576	430	8	(	(	PUNCT
ejpam-3576	430	9	aa)(xy	aa)(xy	NOUN
ejpam-3576	430	10	)	)	PUNCT
ejpam-3576	430	11	=	=	SYM
ejpam-3576	430	12	a2	a2	PROPN
ejpam-3576	430	13	t	t	NOUN
ejpam-3576	430	14	=	=	PUNCT
ejpam-3576	430	15	(	(	PUNCT
ejpam-3576	430	16	aa)t	aa)t	PROPN
ejpam-3576	430	17	=	=	PUNCT
ejpam-3576	430	18	(	(	PUNCT
ejpam-3576	430	19	ta)a	ta)a	ADP
ejpam-3576	430	20	≤	≤	NOUN
ejpam-3576	430	21	(	(	PUNCT
ejpam-3576	430	22	t(ta2))a	t(ta2))a	NOUN
ejpam-3576	430	23	=	=	SYM
ejpam-3576	430	24	(	(	PUNCT
ejpam-3576	430	25	t(t(aa)))a	t(t(aa)))a	NOUN
ejpam-3576	430	26	=	=	SYM
ejpam-3576	430	27	(	(	PUNCT
ejpam-3576	430	28	t(a(ta)))a	t(a(ta)))a	NOUN
ejpam-3576	430	29	=	=	SYM
ejpam-3576	430	30	(	(	PUNCT
ejpam-3576	430	31	a(t(ta)))a	a(t(ta)))a	PROPN
ejpam-3576	430	32	=	=	PUNCT
ejpam-3576	430	33	(	(	PUNCT
ejpam-3576	430	34	as)a	as)a	PROPN
ejpam-3576	430	35	,	,	PUNCT
ejpam-3576	430	36	say	say	VERB
ejpam-3576	430	37	t(ta	t(ta	PRON
ejpam-3576	430	38	)	)	PUNCT
ejpam-3576	431	1	=	=	SYM
ejpam-3576	431	2	s	s	VERB
ejpam-3576	431	3	this	this	PRON
ejpam-3576	431	4	imply	imply	VERB
ejpam-3576	431	5	that	that	SCONJ
ejpam-3576	431	6	a	a	DET
ejpam-3576	431	7	≤	≤	NOUN
ejpam-3576	431	8	(	(	PUNCT
ejpam-3576	431	9	as)a	as)a	PROPN
ejpam-3576	431	10	,	,	PUNCT
ejpam-3576	431	11	for	for	ADP
ejpam-3576	431	12	some	some	DET
ejpam-3576	431	13	s	s	ADP
ejpam-3576	431	14	∈	∈	NOUN
ejpam-3576	431	15	s.	s.	PROPN
ejpam-3576	431	16	thus	thus	ADV
ejpam-3576	431	17	s	s	VERB
ejpam-3576	431	18	is	be	AUX
ejpam-3576	431	19	a	a	DET
ejpam-3576	431	20	regular	regular	ADJ
ejpam-3576	431	21	ordered	order	VERB
ejpam-3576	431	22	ag	ag	PROPN
ejpam-3576	431	23	-	-	NOUN
ejpam-3576	431	24	groupoid	groupoid	PROPN
ejpam-3576	431	25	.	.	PUNCT
ejpam-3576	432	1	hence	hence	ADV
ejpam-3576	432	2	s	s	VERB
ejpam-3576	432	3	is	be	AUX
ejpam-3576	432	4	a	a	DET
ejpam-3576	432	5	completely	completely	ADV
ejpam-3576	432	6	regular	regular	ADJ
ejpam-3576	432	7	ordered	order	VERB
ejpam-3576	432	8	ag	ag	PROPN
ejpam-3576	432	9	-	-	NOUN
ejpam-3576	432	10	groupoid	groupoid	PROPN
ejpam-3576	432	11	.	.	PUNCT
ejpam-3576	433	1	k.	k.	PROPN
ejpam-3576	433	2	nasreen	nasreen	PROPN
ejpam-3576	433	3	,	,	PUNCT
ejpam-3576	433	4	m.	m.	NOUN
ejpam-3576	433	5	alesemi	alesemi	PROPN
ejpam-3576	433	6	,	,	PUNCT
ejpam-3576	433	7	salahuddin	salahuddin	VERB
ejpam-3576	433	8	/	/	SYM
ejpam-3576	433	9	eur	eur	PROPN
ejpam-3576	433	10	.	.	PUNCT
ejpam-3576	434	1	j.	j.	PROPN
ejpam-3576	434	2	pure	pure	PROPN
ejpam-3576	434	3	appl	appl	PROPN
ejpam-3576	434	4	.	.	PROPN
ejpam-3576	434	5	math	math	PROPN
ejpam-3576	434	6	,	,	PUNCT
ejpam-3576	434	7	13	13	NUM
ejpam-3576	434	8	(	(	PUNCT
ejpam-3576	434	9	1	1	NUM
ejpam-3576	434	10	)	)	PUNCT
ejpam-3576	434	11	(	(	PUNCT
ejpam-3576	434	12	2020	2020	NUM
ejpam-3576	434	13	)	)	PUNCT
ejpam-3576	434	14	,	,	PUNCT
ejpam-3576	434	15	113	113	NUM
ejpam-3576	434	16	-	-	SYM
ejpam-3576	434	17	129	129	NUM
ejpam-3576	434	18	126	126	NUM
ejpam-3576	434	19	conversely	conversely	ADV
ejpam-3576	434	20	,	,	PUNCT
ejpam-3576	434	21	let	let	VERB
ejpam-3576	434	22	s	s	PRON
ejpam-3576	434	23	be	be	AUX
ejpam-3576	434	24	a	a	DET
ejpam-3576	434	25	completely	completely	ADV
ejpam-3576	434	26	regular	regular	ADJ
ejpam-3576	434	27	ordered	order	VERB
ejpam-3576	434	28	ag	ag	PROPN
ejpam-3576	434	29	-	-	NOUN
ejpam-3576	434	30	groupoid	groupoid	PROPN
ejpam-3576	434	31	.	.	PUNCT
ejpam-3576	435	1	let	let	VERB
ejpam-3576	435	2	a	a	DET
ejpam-3576	435	3	∈	∈	ADJ
ejpam-3576	435	4	s	s	NOUN
ejpam-3576	435	5	,	,	PUNCT
ejpam-3576	435	6	then	then	ADV
ejpam-3576	435	7	there	there	PRON
ejpam-3576	435	8	exists	exist	VERB
ejpam-3576	435	9	x	x	X
ejpam-3576	435	10	∈	∈	NOUN
ejpam-3576	435	11	s	s	VERB
ejpam-3576	435	12	such	such	ADJ
ejpam-3576	435	13	that	that	SCONJ
ejpam-3576	435	14	a	a	DET
ejpam-3576	435	15	≤	≤	NOUN
ejpam-3576	435	16	(	(	PUNCT
ejpam-3576	435	17	ax)a	ax)a	PROPN
ejpam-3576	435	18	,	,	PUNCT
ejpam-3576	435	19	a	a	DET
ejpam-3576	435	20	≤	≤	NOUN
ejpam-3576	435	21	a2x	a2x	PROPN
ejpam-3576	435	22	and	and	CCONJ
ejpam-3576	435	23	a	a	DET
ejpam-3576	435	24	≤	≤	NUM
ejpam-3576	435	25	xa2	xa2	PROPN
ejpam-3576	435	26	.	.	PUNCT
ejpam-3576	436	1	now	now	ADV
ejpam-3576	436	2	a	a	DET
ejpam-3576	436	3	≤	≤	NOUN
ejpam-3576	436	4	(	(	PUNCT
ejpam-3576	436	5	ax)a	ax)a	PROPN
ejpam-3576	436	6	≤	≤	PROPN
ejpam-3576	436	7	(	(	PUNCT
ejpam-3576	436	8	ax)(xa2	ax)(xa2	ADV
ejpam-3576	436	9	)	)	PUNCT
ejpam-3576	436	10	=	=	PUNCT
ejpam-3576	436	11	(	(	PUNCT
ejpam-3576	436	12	ax)(x(aa	ax)(x(aa	PROPN
ejpam-3576	436	13	)	)	PUNCT
ejpam-3576	436	14	)	)	PUNCT
ejpam-3576	437	1	=	=	PUNCT
ejpam-3576	437	2	(	(	PUNCT
ejpam-3576	437	3	ax)(a(xa	ax)(a(xa	PROPN
ejpam-3576	437	4	)	)	PUNCT
ejpam-3576	437	5	)	)	PUNCT
ejpam-3576	438	1	=	=	SYM
ejpam-3576	438	2	(	(	PUNCT
ejpam-3576	438	3	ax)(ay	ax)(ay	NOUN
ejpam-3576	438	4	)	)	PUNCT
ejpam-3576	438	5	,	,	PUNCT
ejpam-3576	438	6	say	say	VERB
ejpam-3576	438	7	xa	xa	PROPN
ejpam-3576	438	8	=	=	SYM
ejpam-3576	438	9	y	y	PROPN
ejpam-3576	438	10	this	this	PRON
ejpam-3576	438	11	imply	imply	VERB
ejpam-3576	438	12	that	that	SCONJ
ejpam-3576	438	13	a	a	DET
ejpam-3576	438	14	≤	≤	ADJ
ejpam-3576	438	15	(	(	PUNCT
ejpam-3576	438	16	ax)(ay	ax)(ay	NOUN
ejpam-3576	438	17	)	)	PUNCT
ejpam-3576	438	18	,	,	PUNCT
ejpam-3576	438	19	for	for	ADP
ejpam-3576	438	20	some	some	DET
ejpam-3576	438	21	x	x	NOUN
ejpam-3576	438	22	,	,	PUNCT
ejpam-3576	438	23	y	y	PROPN
ejpam-3576	438	24	∈	∈	PROPN
ejpam-3576	438	25	s.	s.	PROPN
ejpam-3576	438	26	hence	hence	ADV
ejpam-3576	438	27	s	s	VERB
ejpam-3576	438	28	is	be	AUX
ejpam-3576	439	1	weakly	weakly	ADV
ejpam-3576	439	2	regular	regular	ADJ
ejpam-3576	439	3	ordered	order	VERB
ejpam-3576	439	4	ag	ag	PROPN
ejpam-3576	439	5	-	-	PROPN
ejpam-3576	439	6	groupoid	groupoid	PROPN
ejpam-3576	439	7	.	.	PUNCT
ejpam-3576	440	1	lemma	lemma	PROPN
ejpam-3576	440	2	14	14	NUM
ejpam-3576	440	3	.	.	PUNCT
ejpam-3576	441	1	every	every	DET
ejpam-3576	441	2	anti	anti	X
ejpam-3576	441	3	fuzzy	fuzzy	ADJ
ejpam-3576	441	4	right	right	ADJ
ejpam-3576	441	5	ideal	ideal	NOUN
ejpam-3576	441	6	of	of	ADP
ejpam-3576	441	7	an	an	DET
ejpam-3576	441	8	intra	intra	ADJ
ejpam-3576	441	9	-	-	ADJ
ejpam-3576	441	10	regular	regular	ADJ
ejpam-3576	441	11	ordered	order	VERB
ejpam-3576	441	12	ag	ag	PROPN
ejpam-3576	441	13	-	-	PROPN
ejpam-3576	441	14	groupoid	groupoid	PROPN
ejpam-3576	441	15	s	s	PART
ejpam-3576	441	16	is	be	AUX
ejpam-3576	441	17	an	an	DET
ejpam-3576	441	18	anti	anti	ADJ
ejpam-3576	441	19	fuzzy	fuzzy	ADJ
ejpam-3576	441	20	ideal	ideal	NOUN
ejpam-3576	441	21	of	of	ADP
ejpam-3576	441	22	s.	s.	PROPN
ejpam-3576	441	23	proof	proof	PROPN
ejpam-3576	441	24	.	.	PUNCT
ejpam-3576	442	1	let	let	VERB
ejpam-3576	442	2	µ	µ	X
ejpam-3576	442	3	be	be	AUX
ejpam-3576	442	4	an	an	DET
ejpam-3576	442	5	anti	anti	ADJ
ejpam-3576	442	6	fuzzy	fuzzy	ADJ
ejpam-3576	442	7	right	right	ADJ
ejpam-3576	442	8	ideal	ideal	NOUN
ejpam-3576	442	9	of	of	ADP
ejpam-3576	442	10	s	s	PRON
ejpam-3576	442	11	and	and	CCONJ
ejpam-3576	442	12	x	x	NOUN
ejpam-3576	442	13	,	,	PUNCT
ejpam-3576	442	14	y	y	PROPN
ejpam-3576	442	15	∈	∈	PROPN
ejpam-3576	442	16	s	s	PROPN
ejpam-3576	442	17	,	,	PUNCT
ejpam-3576	442	18	this	this	PRON
ejpam-3576	442	19	imply	imply	VERB
ejpam-3576	442	20	that	that	SCONJ
ejpam-3576	442	21	there	there	PRON
ejpam-3576	442	22	exist	exist	VERB
ejpam-3576	442	23	a	a	DET
ejpam-3576	442	24	,	,	PUNCT
ejpam-3576	442	25	b	b	X
ejpam-3576	442	26	∈	∈	NOUN
ejpam-3576	442	27	s	s	VERB
ejpam-3576	443	1	such	such	ADJ
ejpam-3576	443	2	that	that	SCONJ
ejpam-3576	443	3	x	x	SYM
ejpam-3576	443	4	≤	≤	X
ejpam-3576	443	5	(	(	PUNCT
ejpam-3576	443	6	ax2)b	ax2)b	PROPN
ejpam-3576	443	7	.	.	PUNCT
ejpam-3576	443	8	now	now	ADV
ejpam-3576	443	9	µ(xy	µ(xy	NUM
ejpam-3576	443	10	)	)	PUNCT
ejpam-3576	443	11	≤	≤	NUM
ejpam-3576	443	12	µ(((ax2)b)y	µ(((ax2)b)y	NOUN
ejpam-3576	443	13	)	)	PUNCT
ejpam-3576	443	14	=	=	SYM
ejpam-3576	443	15	µ((yb)(ax2	µ((yb)(ax2	PROPN
ejpam-3576	443	16	)	)	PUNCT
ejpam-3576	443	17	)	)	PUNCT
ejpam-3576	443	18	≤	≤	PUNCT
ejpam-3576	443	19	µ(yb	µ(yb	PROPN
ejpam-3576	443	20	)	)	PUNCT
ejpam-3576	443	21	≤	≤	PROPN
ejpam-3576	443	22	µ(y	µ(y	PROPN
ejpam-3576	443	23	)	)	PUNCT
ejpam-3576	443	24	.	.	PUNCT
ejpam-3576	444	1	hence	hence	ADV
ejpam-3576	444	2	µ	µ	X
ejpam-3576	444	3	is	be	AUX
ejpam-3576	444	4	an	an	DET
ejpam-3576	444	5	anti	anti	ADJ
ejpam-3576	444	6	fuzzy	fuzzy	ADJ
ejpam-3576	444	7	ideal	ideal	NOUN
ejpam-3576	444	8	of	of	ADP
ejpam-3576	444	9	s.	s.	PROPN
ejpam-3576	444	10	remark	remark	PROPN
ejpam-3576	444	11	9	9	NUM
ejpam-3576	444	12	.	.	PUNCT
ejpam-3576	445	1	the	the	DET
ejpam-3576	445	2	concept	concept	NOUN
ejpam-3576	445	3	of	of	ADP
ejpam-3576	445	4	anti	anti	ADJ
ejpam-3576	445	5	fuzzy	fuzzy	ADJ
ejpam-3576	445	6	(	(	PUNCT
ejpam-3576	445	7	right	right	ADJ
ejpam-3576	445	8	,	,	PUNCT
ejpam-3576	445	9	two	two	NUM
ejpam-3576	445	10	-	-	PUNCT
ejpam-3576	445	11	sided	sided	ADJ
ejpam-3576	445	12	)	)	PUNCT
ejpam-3576	445	13	ideals	ideal	NOUN
ejpam-3576	445	14	coincide	coincide	VERB
ejpam-3576	445	15	in	in	ADP
ejpam-3576	445	16	intraregular	intraregular	ADJ
ejpam-3576	445	17	ordered	order	VERB
ejpam-3576	445	18	ag	ag	PROPN
ejpam-3576	445	19	-	-	PUNCT
ejpam-3576	445	20	groupoids	groupoid	NOUN
ejpam-3576	445	21	s.	s.	PROPN
ejpam-3576	445	22	proposition	proposition	PROPN
ejpam-3576	445	23	10	10	NUM
ejpam-3576	445	24	.	.	PUNCT
ejpam-3576	446	1	let	let	VERB
ejpam-3576	446	2	s	s	PRON
ejpam-3576	446	3	be	be	AUX
ejpam-3576	446	4	an	an	DET
ejpam-3576	446	5	intra	intra	ADJ
ejpam-3576	446	6	-	-	ADJ
ejpam-3576	446	7	regular	regular	ADJ
ejpam-3576	446	8	ordered	order	VERB
ejpam-3576	446	9	ag	ag	PROPN
ejpam-3576	446	10	-	-	NOUN
ejpam-3576	446	11	groupoid	groupoid	PROPN
ejpam-3576	446	12	with	with	ADP
ejpam-3576	446	13	left	left	ADJ
ejpam-3576	446	14	identity	identity	NOUN
ejpam-3576	446	15	e.	e.	PROPN
ejpam-3576	446	16	then	then	ADV
ejpam-3576	446	17	µ	µ	PROPN
ejpam-3576	446	18	is	be	AUX
ejpam-3576	446	19	an	an	DET
ejpam-3576	446	20	anti	anti	ADJ
ejpam-3576	446	21	fuzzy	fuzzy	ADJ
ejpam-3576	446	22	interior	interior	ADJ
ejpam-3576	446	23	ideal	ideal	NOUN
ejpam-3576	446	24	if	if	SCONJ
ejpam-3576	446	25	and	and	CCONJ
ejpam-3576	446	26	only	only	ADV
ejpam-3576	446	27	if	if	SCONJ
ejpam-3576	446	28	µ	µ	NOUN
ejpam-3576	446	29	is	be	AUX
ejpam-3576	446	30	an	an	DET
ejpam-3576	446	31	anti	anti	ADJ
ejpam-3576	446	32	fuzzy	fuzzy	ADJ
ejpam-3576	446	33	ideal	ideal	NOUN
ejpam-3576	446	34	of	of	ADP
ejpam-3576	446	35	s.	s.	PROPN
ejpam-3576	446	36	proof	proof	PROPN
ejpam-3576	446	37	.	.	PUNCT
ejpam-3576	447	1	let	let	VERB
ejpam-3576	447	2	µ	µ	X
ejpam-3576	447	3	be	be	AUX
ejpam-3576	447	4	an	an	DET
ejpam-3576	447	5	anti	anti	ADJ
ejpam-3576	447	6	fuzzy	fuzzy	ADJ
ejpam-3576	447	7	interior	interior	ADJ
ejpam-3576	447	8	ideal	ideal	NOUN
ejpam-3576	447	9	of	of	ADP
ejpam-3576	447	10	s	s	PRON
ejpam-3576	447	11	and	and	CCONJ
ejpam-3576	447	12	x	x	NOUN
ejpam-3576	447	13	,	,	PUNCT
ejpam-3576	447	14	y	y	PROPN
ejpam-3576	447	15	∈	∈	PROPN
ejpam-3576	447	16	s	s	PROPN
ejpam-3576	447	17	,	,	PUNCT
ejpam-3576	447	18	this	this	PRON
ejpam-3576	447	19	imply	imply	VERB
ejpam-3576	447	20	that	that	SCONJ
ejpam-3576	447	21	there	there	PRON
ejpam-3576	447	22	exist	exist	VERB
ejpam-3576	447	23	a	a	DET
ejpam-3576	447	24	,	,	PUNCT
ejpam-3576	447	25	b	b	X
ejpam-3576	447	26	∈	∈	NOUN
ejpam-3576	447	27	s	s	VERB
ejpam-3576	447	28	such	such	ADJ
ejpam-3576	447	29	that	that	SCONJ
ejpam-3576	447	30	x	x	SYM
ejpam-3576	447	31	≤	≤	X
ejpam-3576	447	32	(	(	PUNCT
ejpam-3576	447	33	ax2)b	ax2)b	PROPN
ejpam-3576	447	34	.	.	PUNCT
ejpam-3576	448	1	now	now	ADV
ejpam-3576	448	2	xy	xy	X
ejpam-3576	448	3	≤	≤	PROPN
ejpam-3576	448	4	(	(	PUNCT
ejpam-3576	448	5	(	(	PUNCT
ejpam-3576	448	6	ax2)b)y	ax2)b)y	PROPN
ejpam-3576	448	7	=	=	SYM
ejpam-3576	448	8	(	(	PUNCT
ejpam-3576	448	9	yb)(ax2	yb)(ax2	NOUN
ejpam-3576	448	10	)	)	PUNCT
ejpam-3576	448	11	=	=	SYM
ejpam-3576	448	12	n(a(xx	n(a(xx	NOUN
ejpam-3576	448	13	)	)	PUNCT
ejpam-3576	448	14	)	)	PUNCT
ejpam-3576	449	1	=	=	PUNCT
ejpam-3576	449	2	n(x(ax	n(x(ax	NOUN
ejpam-3576	449	3	)	)	PUNCT
ejpam-3576	449	4	)	)	PUNCT
ejpam-3576	449	5	,	,	PUNCT
ejpam-3576	449	6	say	say	VERB
ejpam-3576	449	7	yb	yb	PROPN
ejpam-3576	449	8	=	=	SYM
ejpam-3576	449	9	n	n	PROPN
ejpam-3576	449	10	=	=	SYM
ejpam-3576	449	11	(	(	PUNCT
ejpam-3576	449	12	en)(x(ax	en)(x(ax	PROPN
ejpam-3576	449	13	)	)	PUNCT
ejpam-3576	449	14	)	)	PUNCT
ejpam-3576	450	1	=	=	SYM
ejpam-3576	450	2	(	(	PUNCT
ejpam-3576	450	3	ex)(n(ax	ex)(n(ax	PROPN
ejpam-3576	450	4	)	)	PUNCT
ejpam-3576	450	5	)	)	PUNCT
ejpam-3576	451	1	=	=	PUNCT
ejpam-3576	451	2	(	(	PUNCT
ejpam-3576	451	3	ex)m	ex)m	X
ejpam-3576	451	4	,	,	PUNCT
ejpam-3576	451	5	say	say	VERB
ejpam-3576	451	6	n(ax	n(ax	NUM
ejpam-3576	451	7	)	)	PUNCT
ejpam-3576	451	8	=	=	PUNCT
ejpam-3576	452	1	m	m	VERB
ejpam-3576	452	2	thus	thus	ADV
ejpam-3576	452	3	µ(xy	µ(xy	NUM
ejpam-3576	452	4	)	)	PUNCT
ejpam-3576	452	5	≤	≤	NOUN
ejpam-3576	452	6	µ((ex)m	µ((ex)m	NUM
ejpam-3576	452	7	)	)	PUNCT
ejpam-3576	452	8	≤	≤	NOUN
ejpam-3576	452	9	µ(x	µ(x	NOUN
ejpam-3576	452	10	)	)	PUNCT
ejpam-3576	452	11	.	.	PUNCT
ejpam-3576	453	1	hence	hence	ADV
ejpam-3576	453	2	µ	µ	X
ejpam-3576	453	3	is	be	AUX
ejpam-3576	453	4	an	an	DET
ejpam-3576	453	5	anti	anti	ADJ
ejpam-3576	453	6	fuzzy	fuzzy	ADJ
ejpam-3576	453	7	ideal	ideal	NOUN
ejpam-3576	453	8	of	of	ADP
ejpam-3576	453	9	s.	s.	PROPN
ejpam-3576	453	10	converse	converse	PROPN
ejpam-3576	453	11	is	be	AUX
ejpam-3576	453	12	true	true	ADJ
ejpam-3576	453	13	by	by	ADP
ejpam-3576	453	14	lemma	lemma	PROPN
ejpam-3576	453	15	7	7	NUM
ejpam-3576	453	16	.	.	PUNCT
ejpam-3576	453	17	theorem	theorem	NOUN
ejpam-3576	453	18	4	4	NUM
ejpam-3576	453	19	.	.	PUNCT
ejpam-3576	454	1	let	let	VERB
ejpam-3576	454	2	s	s	PRON
ejpam-3576	454	3	be	be	AUX
ejpam-3576	454	4	an	an	DET
ejpam-3576	454	5	intra	intra	ADJ
ejpam-3576	454	6	-	-	ADJ
ejpam-3576	454	7	regular	regular	ADJ
ejpam-3576	454	8	locally	locally	ADV
ejpam-3576	454	9	associative	associative	ADJ
ejpam-3576	454	10	ordered	order	VERB
ejpam-3576	454	11	ag	ag	PROPN
ejpam-3576	454	12	-	-	NOUN
ejpam-3576	454	13	groupoid	groupoid	PROPN
ejpam-3576	454	14	.	.	PUNCT
ejpam-3576	455	1	then	then	ADV
ejpam-3576	455	2	for	for	ADP
ejpam-3576	455	3	every	every	DET
ejpam-3576	455	4	anti	anti	ADJ
ejpam-3576	455	5	fuzzy	fuzzy	ADJ
ejpam-3576	455	6	interior	interior	ADJ
ejpam-3576	455	7	ideal	ideal	NOUN
ejpam-3576	455	8	µ	µ	X
ejpam-3576	455	9	of	of	ADP
ejpam-3576	455	10	s	s	PROPN
ejpam-3576	455	11	,	,	PUNCT
ejpam-3576	455	12	µ(an	µ(an	ADJ
ejpam-3576	455	13	)	)	PUNCT
ejpam-3576	455	14	=	=	PUNCT
ejpam-3576	455	15	µ(a2n	µ(a2n	NOUN
ejpam-3576	455	16	)	)	PUNCT
ejpam-3576	455	17	,	,	PUNCT
ejpam-3576	455	18	where	where	SCONJ
ejpam-3576	455	19	n	n	PRON
ejpam-3576	455	20	is	be	AUX
ejpam-3576	455	21	any	any	DET
ejpam-3576	455	22	positive	positive	ADJ
ejpam-3576	455	23	integer	integer	NOUN
ejpam-3576	455	24	,	,	PUNCT
ejpam-3576	455	25	for	for	ADP
ejpam-3576	455	26	all	all	DET
ejpam-3576	455	27	a	a	DET
ejpam-3576	455	28	∈	∈	PROPN
ejpam-3576	455	29	s.	s.	PROPN
ejpam-3576	455	30	proof	proof	NOUN
ejpam-3576	455	31	.	.	PUNCT
ejpam-3576	456	1	for	for	ADP
ejpam-3576	456	2	n	n	NOUN
ejpam-3576	456	3	=	=	SYM
ejpam-3576	456	4	1	1	X
ejpam-3576	456	5	.	.	PUNCT
ejpam-3576	456	6	let	let	VERB
ejpam-3576	456	7	a	a	DET
ejpam-3576	456	8	∈	∈	ADJ
ejpam-3576	456	9	s	s	NOUN
ejpam-3576	456	10	,	,	PUNCT
ejpam-3576	456	11	this	this	PRON
ejpam-3576	456	12	imply	imply	VERB
ejpam-3576	456	13	that	that	SCONJ
ejpam-3576	456	14	there	there	PRON
ejpam-3576	456	15	exist	exist	VERB
ejpam-3576	456	16	x	x	NOUN
ejpam-3576	456	17	,	,	PUNCT
ejpam-3576	456	18	y	y	PROPN
ejpam-3576	456	19	∈	∈	PROPN
ejpam-3576	456	20	s	s	VERB
ejpam-3576	456	21	such	such	ADJ
ejpam-3576	456	22	that	that	SCONJ
ejpam-3576	456	23	a	a	DET
ejpam-3576	456	24	≤	≤	NOUN
ejpam-3576	456	25	(	(	PUNCT
ejpam-3576	456	26	xa2)y	xa2)y	PROPN
ejpam-3576	456	27	.	.	PUNCT
ejpam-3576	457	1	thus	thus	ADV
ejpam-3576	457	2	µ	µ	X
ejpam-3576	457	3	(	(	PUNCT
ejpam-3576	457	4	a	a	PRON
ejpam-3576	457	5	)	)	PUNCT
ejpam-3576	457	6	≤	≤	NOUN
ejpam-3576	457	7	µ((xa2)y	µ((xa2)y	PUNCT
ejpam-3576	457	8	)	)	PUNCT
ejpam-3576	457	9	≤	≤	NUM
ejpam-3576	457	10	µ(a2	µ(a2	NOUN
ejpam-3576	457	11	)	)	PUNCT
ejpam-3576	457	12	=	=	SYM
ejpam-3576	457	13	µ(aa	µ(aa	PROPN
ejpam-3576	457	14	)	)	PUNCT
ejpam-3576	457	15	≤	≤	NOUN
ejpam-3576	457	16	max{µ	max{µ	NOUN
ejpam-3576	457	17	(	(	PUNCT
ejpam-3576	457	18	a	a	NOUN
ejpam-3576	457	19	)	)	PUNCT
ejpam-3576	457	20	,	,	PUNCT
ejpam-3576	457	21	µ	µ	X
ejpam-3576	457	22	(	(	PUNCT
ejpam-3576	457	23	a	a	NOUN
ejpam-3576	457	24	)	)	PUNCT
ejpam-3576	457	25	}	}	PUNCT
ejpam-3576	457	26	=	=	SYM
ejpam-3576	457	27	µ	µ	X
ejpam-3576	457	28	(	(	PUNCT
ejpam-3576	457	29	a	a	NOUN
ejpam-3576	457	30	)	)	PUNCT
ejpam-3576	457	31	,	,	PUNCT
ejpam-3576	457	32	(	(	PUNCT
ejpam-3576	457	33	µ	µ	X
ejpam-3576	457	34	is	be	AUX
ejpam-3576	457	35	an	an	DET
ejpam-3576	457	36	anti	anti	ADJ
ejpam-3576	457	37	fuzzy	fuzzy	ADJ
ejpam-3576	457	38	ideal	ideal	NOUN
ejpam-3576	457	39	of	of	ADP
ejpam-3576	457	40	s	s	PRON
ejpam-3576	457	41	by	by	ADP
ejpam-3576	457	42	proposition	proposition	NOUN
ejpam-3576	457	43	10	10	NUM
ejpam-3576	457	44	)	)	PUNCT
ejpam-3576	457	45	.	.	PUNCT
ejpam-3576	458	1	hence	hence	ADV
ejpam-3576	458	2	µ(a	µ(a	PROPN
ejpam-3576	458	3	)	)	PUNCT
ejpam-3576	458	4	=	=	SYM
ejpam-3576	458	5	µ(a2	µ(a2	PROPN
ejpam-3576	458	6	)	)	PUNCT
ejpam-3576	458	7	.	.	PUNCT
ejpam-3576	459	1	now	now	ADV
ejpam-3576	459	2	a2	a2	PROPN
ejpam-3576	459	3	=	=	SYM
ejpam-3576	459	4	aa	aa	PROPN
ejpam-3576	459	5	≤	≤	X
ejpam-3576	459	6	(	(	PUNCT
ejpam-3576	459	7	(	(	PUNCT
ejpam-3576	459	8	xa2)y)((xa2)y	xa2)y)((xa2)y	PROPN
ejpam-3576	459	9	)	)	PUNCT
ejpam-3576	459	10	=	=	PRON
ejpam-3576	460	1	(	(	PUNCT
ejpam-3576	460	2	(	(	PUNCT
ejpam-3576	460	3	xa2)(xa2))y2	xa2)(xa2))y2	NOUN
ejpam-3576	460	4	=	=	SYM
ejpam-3576	460	5	(	(	PUNCT
ejpam-3576	460	6	x2a4)y2	x2a4)y2	X
ejpam-3576	460	7	,	,	PUNCT
ejpam-3576	460	8	then	then	ADV
ejpam-3576	460	9	the	the	DET
ejpam-3576	460	10	result	result	NOUN
ejpam-3576	460	11	is	be	AUX
ejpam-3576	460	12	true	true	ADJ
ejpam-3576	460	13	for	for	ADP
ejpam-3576	460	14	n	n	NOUN
ejpam-3576	460	15	=	=	SYM
ejpam-3576	460	16	2	2	X
ejpam-3576	460	17	.	.	PUNCT
ejpam-3576	460	18	suppose	suppose	VERB
ejpam-3576	460	19	that	that	SCONJ
ejpam-3576	460	20	the	the	DET
ejpam-3576	460	21	result	result	NOUN
ejpam-3576	460	22	is	be	AUX
ejpam-3576	460	23	true	true	ADJ
ejpam-3576	460	24	for	for	ADP
ejpam-3576	460	25	n	n	PROPN
ejpam-3576	460	26	=	=	SYM
ejpam-3576	460	27	k	k	NOUN
ejpam-3576	460	28	,	,	PUNCT
ejpam-3576	460	29	i.e.	i.e.	X
ejpam-3576	460	30	,	,	PUNCT
ejpam-3576	460	31	µ(ak	µ(ak	PROPN
ejpam-3576	460	32	)	)	PUNCT
ejpam-3576	460	33	=	=	PUNCT
ejpam-3576	460	34	µ(a2k	µ(a2k	X
ejpam-3576	460	35	)	)	PUNCT
ejpam-3576	460	36	.	.	PUNCT
ejpam-3576	461	1	now	now	ADV
ejpam-3576	461	2	ak+1	ak+1	VERB
ejpam-3576	461	3	=	=	PUNCT
ejpam-3576	461	4	aka	aka	ADV
ejpam-3576	461	5	≤	≤	NUM
ejpam-3576	461	6	(	(	PUNCT
ejpam-3576	461	7	(	(	PUNCT
ejpam-3576	461	8	xka2k)yk)((xa2)y	xka2k)yk)((xa2)y	PROPN
ejpam-3576	461	9	)	)	PUNCT
ejpam-3576	461	10	=	=	PRON
ejpam-3576	461	11	(	(	PUNCT
ejpam-3576	461	12	xk+1a2(k+1))yk+1	xk+1a2(k+1))yk+1	NUM
ejpam-3576	461	13	.	.	PUNCT
ejpam-3576	462	1	thus	thus	ADV
ejpam-3576	462	2	µ	µ	X
ejpam-3576	462	3	(	(	PUNCT
ejpam-3576	462	4	ak+1	ak+1	X
ejpam-3576	462	5	)	)	PUNCT
ejpam-3576	462	6	≤	≤	NUM
ejpam-3576	462	7	µ((xk+1a2(k+1))yk+1	µ((xk+1a2(k+1))yk+1	VERB
ejpam-3576	462	8	)	)	PUNCT
ejpam-3576	462	9	≤	≤	NOUN
ejpam-3576	462	10	µ(a2(k+1	µ(a2(k+1	NOUN
ejpam-3576	462	11	)	)	PUNCT
ejpam-3576	462	12	)	)	PUNCT
ejpam-3576	463	1	=	=	SYM
ejpam-3576	463	2	µ(a(k+1)a(k+1	µ(a(k+1)a(k+1	ADJ
ejpam-3576	463	3	)	)	PUNCT
ejpam-3576	463	4	)	)	PUNCT
ejpam-3576	463	5	≤	≤	NUM
ejpam-3576	463	6	max{µ	max{µ	NOUN
ejpam-3576	463	7	(	(	PUNCT
ejpam-3576	463	8	a(k+1	a(k+1	NOUN
ejpam-3576	463	9	)	)	PUNCT
ejpam-3576	463	10	)	)	PUNCT
ejpam-3576	463	11	,	,	PUNCT
ejpam-3576	463	12	µ	µ	X
ejpam-3576	463	13	(	(	PUNCT
ejpam-3576	463	14	a(k+1	a(k+1	NOUN
ejpam-3576	463	15	)	)	PUNCT
ejpam-3576	463	16	)	)	PUNCT
ejpam-3576	463	17	}	}	PUNCT
ejpam-3576	464	1	=	=	SYM
ejpam-3576	464	2	µ	µ	X
ejpam-3576	464	3	(	(	PUNCT
ejpam-3576	464	4	a(k+1	a(k+1	NOUN
ejpam-3576	464	5	)	)	PUNCT
ejpam-3576	464	6	)	)	PUNCT
ejpam-3576	464	7	.	.	PUNCT
ejpam-3576	465	1	therefore	therefore	ADV
ejpam-3576	465	2	µ(ak+1	µ(ak+1	NOUN
ejpam-3576	465	3	)	)	PUNCT
ejpam-3576	465	4	=	=	SYM
ejpam-3576	465	5	µ(a2(k+1	µ(a2(k+1	ADJ
ejpam-3576	465	6	)	)	PUNCT
ejpam-3576	465	7	)	)	PUNCT
ejpam-3576	465	8	.	.	PUNCT
ejpam-3576	466	1	hence	hence	ADV
ejpam-3576	466	2	by	by	ADP
ejpam-3576	466	3	induction	induction	NOUN
ejpam-3576	466	4	method	method	NOUN
ejpam-3576	466	5	,	,	PUNCT
ejpam-3576	466	6	the	the	DET
ejpam-3576	466	7	result	result	NOUN
ejpam-3576	466	8	is	be	AUX
ejpam-3576	466	9	true	true	ADJ
ejpam-3576	466	10	for	for	ADP
ejpam-3576	466	11	all	all	DET
ejpam-3576	466	12	positive	positive	ADJ
ejpam-3576	466	13	integers	integer	NOUN
ejpam-3576	466	14	.	.	PUNCT
ejpam-3576	467	1	references	reference	NOUN
ejpam-3576	467	2	127	127	NUM
ejpam-3576	467	3	lemma	lemma	PROPN
ejpam-3576	467	4	15	15	NUM
ejpam-3576	467	5	.	.	PUNCT
ejpam-3576	468	1	let	let	VERB
ejpam-3576	468	2	s	s	PRON
ejpam-3576	468	3	be	be	AUX
ejpam-3576	468	4	an	an	DET
ejpam-3576	468	5	intra	intra	ADJ
ejpam-3576	468	6	-	-	ADJ
ejpam-3576	468	7	regular	regular	ADJ
ejpam-3576	468	8	locally	locally	ADV
ejpam-3576	468	9	associative	associative	ADJ
ejpam-3576	468	10	ordered	order	VERB
ejpam-3576	468	11	ag	ag	PROPN
ejpam-3576	468	12	-	-	NOUN
ejpam-3576	468	13	groupoid	groupoid	PROPN
ejpam-3576	468	14	with	with	ADP
ejpam-3576	468	15	left	left	ADJ
ejpam-3576	468	16	identity	identity	NOUN
ejpam-3576	468	17	e.	e.	PROPN
ejpam-3576	468	18	then	then	ADV
ejpam-3576	468	19	for	for	ADP
ejpam-3576	468	20	every	every	DET
ejpam-3576	468	21	anti	anti	ADJ
ejpam-3576	468	22	fuzzy	fuzzy	ADJ
ejpam-3576	468	23	interior	interior	ADJ
ejpam-3576	468	24	ideal	ideal	NOUN
ejpam-3576	468	25	µ	µ	X
ejpam-3576	468	26	of	of	ADP
ejpam-3576	468	27	s	s	PROPN
ejpam-3576	468	28	,	,	PUNCT
ejpam-3576	468	29	µ(ab	µ(ab	PROPN
ejpam-3576	468	30	)	)	PUNCT
ejpam-3576	468	31	=	=	SYM
ejpam-3576	468	32	µ(ba	µ(ba	X
ejpam-3576	468	33	)	)	PUNCT
ejpam-3576	468	34	for	for	ADP
ejpam-3576	468	35	all	all	DET
ejpam-3576	468	36	a	a	PRON
ejpam-3576	468	37	,	,	PUNCT
ejpam-3576	468	38	b	b	X
ejpam-3576	468	39	∈	∈	PROPN
ejpam-3576	468	40	s.	s.	PROPN
ejpam-3576	468	41	proof	proof	PROPN
ejpam-3576	468	42	.	.	PUNCT
ejpam-3576	469	1	same	same	ADJ
ejpam-3576	469	2	as	as	ADP
ejpam-3576	469	3	lemma	lemma	PROPN
ejpam-3576	469	4	12	12	NUM
ejpam-3576	469	5	.	.	PUNCT
ejpam-3576	470	1	references	reference	NOUN
ejpam-3576	470	2	[	[	X
ejpam-3576	470	3	1	1	NUM
ejpam-3576	470	4	]	]	PUNCT
ejpam-3576	470	5	m.	m.	NOUN
ejpam-3576	470	6	akram	akram	PROPN
ejpam-3576	470	7	and	and	CCONJ
ejpam-3576	470	8	k.	k.	PROPN
ejpam-3576	470	9	h.	h.	PROPN
ejpam-3576	470	10	dar	dar	PROPN
ejpam-3576	470	11	,	,	PUNCT
ejpam-3576	470	12	on	on	ADP
ejpam-3576	470	13	anti	anti	X
ejpam-3576	470	14	fuzzy	fuzzy	ADJ
ejpam-3576	470	15	left	leave	VERB
ejpam-3576	470	16	h	h	NOUN
ejpam-3576	470	17	-	-	PUNCT
ejpam-3576	470	18	ideals	ideal	NOUN
ejpam-3576	470	19	in	in	ADP
ejpam-3576	470	20	hemirings	hemiring	NOUN
ejpam-3576	470	21	,	,	PUNCT
ejpam-3576	470	22	int	int	NOUN
ejpam-3576	470	23	.	.	PUNCT
ejpam-3576	470	24	math	math	PROPN
ejpam-3576	470	25	.	.	PUNCT
ejpam-3576	471	1	forum	forum	PROPN
ejpam-3576	471	2	,	,	PUNCT
ejpam-3576	471	3	2(2007	2(2007	NUM
ejpam-3576	471	4	)	)	PUNCT
ejpam-3576	471	5	2295	2295	NUM
ejpam-3576	471	6	-	-	SYM
ejpam-3576	471	7	2304	2304	NUM
ejpam-3576	471	8	.	.	PUNCT
ejpam-3576	472	1	[	[	X
ejpam-3576	472	2	2	2	NUM
ejpam-3576	472	3	]	]	PUNCT
ejpam-3576	472	4	r.	r.	PROPN
ejpam-3576	472	5	biswas	biswas	PROPN
ejpam-3576	472	6	,	,	PUNCT
ejpam-3576	472	7	fuzzy	fuzzy	ADJ
ejpam-3576	472	8	subgroups	subgroup	NOUN
ejpam-3576	472	9	and	and	CCONJ
ejpam-3576	472	10	anti	anti	ADJ
ejpam-3576	472	11	fuzzy	fuzzy	ADJ
ejpam-3576	472	12	subgroups	subgroup	NOUN
ejpam-3576	472	13	,	,	PUNCT
ejpam-3576	472	14	fuzzy	fuzzy	ADJ
ejpam-3576	472	15	sets	set	NOUN
ejpam-3576	472	16	and	and	CCONJ
ejpam-3576	472	17	systems	system	NOUN
ejpam-3576	472	18	,	,	PUNCT
ejpam-3576	472	19	35(1990	35(1990	NUM
ejpam-3576	472	20	)	)	PUNCT
ejpam-3576	472	21	121	121	NUM
ejpam-3576	472	22	-	-	SYM
ejpam-3576	472	23	124	124	NUM
ejpam-3576	472	24	.	.	PUNCT
ejpam-3576	473	1	[	[	X
ejpam-3576	473	2	3	3	X
ejpam-3576	473	3	]	]	PUNCT
ejpam-3576	473	4	r.	r.	PROPN
ejpam-3576	473	5	j.	j.	PROPN
ejpam-3576	473	6	cho	cho	PROPN
ejpam-3576	473	7	,	,	PUNCT
ejpam-3576	473	8	j.	j.	PROPN
ejpam-3576	473	9	jezek	jezek	PROPN
ejpam-3576	473	10	and	and	CCONJ
ejpam-3576	473	11	t.	t.	PROPN
ejpam-3576	473	12	kepka	kepka	PROPN
ejpam-3576	473	13	praha	praha	PROPN
ejpam-3576	473	14	,	,	PUNCT
ejpam-3576	473	15	paramedial	paramedial	ADJ
ejpam-3576	473	16	groupoids	groupoid	NOUN
ejpam-3576	473	17	,	,	PUNCT
ejpam-3576	473	18	czechoslovak	czechoslovak	ADJ
ejpam-3576	473	19	math	math	NOUN
ejpam-3576	473	20	.	.	PUNCT
ejpam-3576	474	1	j.	j.	PROPN
ejpam-3576	474	2	,	,	PUNCT
ejpam-3576	474	3	49(1999	49(1999	PROPN
ejpam-3576	474	4	)	)	PUNCT
ejpam-3576	474	5	391	391	NUM
ejpam-3576	474	6	-	-	SYM
ejpam-3576	474	7	399	399	NUM
ejpam-3576	474	8	.	.	PUNCT
ejpam-3576	475	1	[	[	X
ejpam-3576	475	2	4	4	X
ejpam-3576	475	3	]	]	PUNCT
ejpam-3576	475	4	k.	k.	PROPN
ejpam-3576	475	5	a.	a.	PROPN
ejpam-3576	475	6	dib	dib	PROPN
ejpam-3576	475	7	and	and	CCONJ
ejpam-3576	475	8	n.	n.	PROPN
ejpam-3576	475	9	galham	galham	PROPN
ejpam-3576	475	10	,	,	PUNCT
ejpam-3576	475	11	fuzzy	fuzzy	ADJ
ejpam-3576	475	12	ideals	ideal	NOUN
ejpam-3576	475	13	and	and	CCONJ
ejpam-3576	475	14	fuzzy	fuzzy	ADJ
ejpam-3576	475	15	bi	bi	NOUN
ejpam-3576	475	16	-	-	NOUN
ejpam-3576	475	17	ideals	ideal	NOUN
ejpam-3576	475	18	in	in	ADP
ejpam-3576	475	19	fuzzy	fuzzy	ADJ
ejpam-3576	475	20	semigroups	semigroup	NOUN
ejpam-3576	475	21	,	,	PUNCT
ejpam-3576	475	22	fuzzy	fuzzy	ADJ
ejpam-3576	475	23	sets	set	NOUN
ejpam-3576	475	24	and	and	CCONJ
ejpam-3576	475	25	system	system	NOUN
ejpam-3576	475	26	,	,	PUNCT
ejpam-3576	475	27	92(1997	92(1997	NOUN
ejpam-3576	475	28	)	)	PUNCT
ejpam-3576	475	29	203	203	NUM
ejpam-3576	475	30	-	-	SYM
ejpam-3576	475	31	215	215	NUM
ejpam-3576	475	32	.	.	PUNCT
ejpam-3576	476	1	[	[	X
ejpam-3576	476	2	5	5	X
ejpam-3576	476	3	]	]	PUNCT
ejpam-3576	476	4	s.	s.	PROPN
ejpam-3576	476	5	m.	m.	PROPN
ejpam-3576	476	6	hong	hong	PROPN
ejpam-3576	476	7	and	and	CCONJ
ejpam-3576	476	8	y.	y.	PROPN
ejpam-3576	476	9	b.	b.	PROPN
ejpam-3576	476	10	jun	jun	PROPN
ejpam-3576	476	11	,	,	PUNCT
ejpam-3576	476	12	anti	anti	X
ejpam-3576	476	13	fuzzy	fuzzy	ADJ
ejpam-3576	476	14	ideals	ideal	NOUN
ejpam-3576	476	15	in	in	ADP
ejpam-3576	476	16	bck	bck	NOUN
ejpam-3576	476	17	-	-	PUNCT
ejpam-3576	476	18	algebra	algebra	NOUN
ejpam-3576	476	19	,	,	PUNCT
ejpam-3576	476	20	kyungpook	kyungpook	NOUN
ejpam-3576	476	21	math	math	NOUN
ejpam-3576	476	22	.	.	PUNCT
ejpam-3576	477	1	j.	j.	PROPN
ejpam-3576	477	2	,	,	PUNCT
ejpam-3576	477	3	38(1998	38(1998	NUM
ejpam-3576	477	4	)	)	PUNCT
ejpam-3576	477	5	145	145	NUM
ejpam-3576	477	6	-	-	SYM
ejpam-3576	477	7	150	150	NUM
ejpam-3576	477	8	.	.	PUNCT
ejpam-3576	478	1	[	[	X
ejpam-3576	478	2	6	6	NUM
ejpam-3576	478	3	]	]	PUNCT
ejpam-3576	478	4	j.	j.	PROPN
ejpam-3576	478	5	jezek	jezek	PROPN
ejpam-3576	478	6	and	and	CCONJ
ejpam-3576	478	7	t.	t.	PROPN
ejpam-3576	478	8	kepka	kepka	NOUN
ejpam-3576	478	9	,	,	PUNCT
ejpam-3576	478	10	medial	medial	ADJ
ejpam-3576	478	11	groupoids	groupoid	NOUN
ejpam-3576	478	12	,	,	PUNCT
ejpam-3576	478	13	rozpravy	rozpravy	PROPN
ejpam-3576	478	14	csav	csav	PROPN
ejpam-3576	478	15	rada	rada	PROPN
ejpam-3576	478	16	mat	mat	PROPN
ejpam-3576	478	17	.	.	PUNCT
ejpam-3576	479	1	a	a	DET
ejpam-3576	479	2	prir	prir	NOUN
ejpam-3576	479	3	.	.	PUNCT
ejpam-3576	480	1	ved	ve	VERB
ejpam-3576	480	2	93/2	93/2	NUM
ejpam-3576	480	3	,	,	PUNCT
ejpam-3576	480	4	1983	1983	NUM
ejpam-3576	480	5	,	,	PUNCT
ejpam-3576	480	6	93	93	NUM
ejpam-3576	480	7	pp	pp	NOUN
ejpam-3576	480	8	.	.	PUNCT
ejpam-3576	481	1	[	[	X
ejpam-3576	481	2	7	7	X
ejpam-3576	481	3	]	]	PUNCT
ejpam-3576	481	4	t.	t.	PROPN
ejpam-3576	481	5	kadir	kadir	PROPN
ejpam-3576	481	6	,	,	PUNCT
ejpam-3576	481	7	in	in	ADP
ejpam-3576	481	8	discrepancy	discrepancy	NOUN
ejpam-3576	481	9	between	between	ADP
ejpam-3576	481	10	the	the	DET
ejpam-3576	481	11	traditional	traditional	ADJ
ejpam-3576	481	12	fuzzy	fuzzy	ADJ
ejpam-3576	481	13	logic	logic	NOUN
ejpam-3576	481	14	and	and	CCONJ
ejpam-3576	481	15	inductive	inductive	ADJ
ejpam-3576	481	16	,	,	PUNCT
ejpam-3576	481	17	international	international	ADJ
ejpam-3576	481	18	journal	journal	NOUN
ejpam-3576	481	19	of	of	ADP
ejpam-3576	481	20	advanced	advanced	ADJ
ejpam-3576	481	21	and	and	CCONJ
ejpam-3576	481	22	applied	apply	VERB
ejpam-3576	481	23	sciences	science	NOUN
ejpam-3576	481	24	,	,	PUNCT
ejpam-3576	481	25	1(2014	1(2014	NUM
ejpam-3576	481	26	)	)	PUNCT
ejpam-3576	481	27	36	36	NUM
ejpam-3576	481	28	-	-	SYM
ejpam-3576	481	29	43	43	NUM
ejpam-3576	481	30	.	.	PUNCT
ejpam-3576	482	1	[	[	X
ejpam-3576	482	2	8	8	NUM
ejpam-3576	482	3	]	]	X
ejpam-3576	482	4	n.	n.	NOUN
ejpam-3576	482	5	kausar	kausar	PROPN
ejpam-3576	482	6	,	,	PUNCT
ejpam-3576	482	7	m.	m.	NOUN
ejpam-3576	482	8	waqar	waqar	PROPN
ejpam-3576	482	9	,	,	PUNCT
ejpam-3576	482	10	characterizations	characterization	NOUN
ejpam-3576	482	11	of	of	ADP
ejpam-3576	482	12	non	non	ADJ
ejpam-3576	482	13	-	-	ADJ
ejpam-3576	482	14	associative	associative	ADJ
ejpam-3576	482	15	rings	ring	NOUN
ejpam-3576	482	16	by	by	ADP
ejpam-3576	482	17	their	their	PRON
ejpam-3576	482	18	intuitionistic	intuitionistic	ADJ
ejpam-3576	482	19	fuzzy	fuzzy	ADJ
ejpam-3576	482	20	bi	bi	NOUN
ejpam-3576	482	21	-	-	NOUN
ejpam-3576	482	22	ideals	ideal	NOUN
ejpam-3576	482	23	,	,	PUNCT
ejpam-3576	482	24	european	european	ADJ
ejpam-3576	482	25	journal	journal	PROPN
ejpam-3576	482	26	of	of	ADP
ejpam-3576	482	27	pure	pure	ADJ
ejpam-3576	482	28	and	and	CCONJ
ejpam-3576	482	29	applied	applied	ADJ
ejpam-3576	482	30	mathematics	mathematic	NOUN
ejpam-3576	482	31	,	,	PUNCT
ejpam-3576	482	32	vol	vol	NOUN
ejpam-3576	482	33	.	.	PROPN
ejpam-3576	482	34	12	12	NUM
ejpam-3576	482	35	,	,	PUNCT
ejpam-3576	482	36	1(2019	1(2019	NUM
ejpam-3576	482	37	)	)	PUNCT
ejpam-3576	482	38	226	226	NUM
ejpam-3576	482	39	-	-	SYM
ejpam-3576	482	40	250	250	NUM
ejpam-3576	482	41	.	.	PUNCT
ejpam-3576	483	1	[	[	X
ejpam-3576	483	2	9	9	NUM
ejpam-3576	483	3	]	]	X
ejpam-3576	483	4	n.	n.	NOUN
ejpam-3576	483	5	kausar	kausar	PROPN
ejpam-3576	483	6	,	,	PUNCT
ejpam-3576	483	7	characterizations	characterization	NOUN
ejpam-3576	483	8	of	of	ADP
ejpam-3576	483	9	non	non	ADJ
ejpam-3576	483	10	-	-	ADJ
ejpam-3576	483	11	associative	associative	ADJ
ejpam-3576	483	12	ordered	order	VERB
ejpam-3576	483	13	semigroups	semigroup	NOUN
ejpam-3576	483	14	by	by	ADP
ejpam-3576	483	15	the	the	DET
ejpam-3576	483	16	properties	property	NOUN
ejpam-3576	483	17	of	of	ADP
ejpam-3576	483	18	their	their	PRON
ejpam-3576	483	19	fuzzy	fuzzy	ADJ
ejpam-3576	483	20	ideals	ideal	NOUN
ejpam-3576	483	21	with	with	ADP
ejpam-3576	483	22	thresholds	threshold	NOUN
ejpam-3576	483	23	(	(	PUNCT
ejpam-3576	483	24	α	α	X
ejpam-3576	483	25	,	,	PUNCT
ejpam-3576	483	26	β	β	X
ejpam-3576	483	27	]	]	X
ejpam-3576	483	28	,	,	PUNCT
ejpam-3576	483	29	prikladnaya	prikladnaya	PROPN
ejpam-3576	483	30	diskretnaya	diskretnaya	PROPN
ejpam-3576	483	31	matematika	matematika	PROPN
ejpam-3576	483	32	,	,	PUNCT
ejpam-3576	483	33	vol	vol	NOUN
ejpam-3576	483	34	.	.	PUNCT
ejpam-3576	484	1	43(2019	43(2019	NUM
ejpam-3576	484	2	)	)	PUNCT
ejpam-3576	484	3	37	37	NUM
ejpam-3576	484	4	-	-	SYM
ejpam-3576	484	5	59	59	NUM
ejpam-3576	484	6	.	.	PUNCT
ejpam-3576	485	1	[	[	X
ejpam-3576	485	2	10	10	NUM
ejpam-3576	485	3	]	]	X
ejpam-3576	485	4	n.	n.	NOUN
ejpam-3576	485	5	kausar	kausar	PROPN
ejpam-3576	485	6	,	,	PUNCT
ejpam-3576	485	7	direct	direct	ADJ
ejpam-3576	485	8	product	product	NOUN
ejpam-3576	485	9	of	of	ADP
ejpam-3576	485	10	finite	finite	PROPN
ejpam-3576	485	11	intuitionistic	intuitionistic	ADJ
ejpam-3576	485	12	fuzzy	fuzzy	ADJ
ejpam-3576	485	13	normal	normal	ADJ
ejpam-3576	485	14	subrings	subring	NOUN
ejpam-3576	485	15	over	over	ADP
ejpam-3576	485	16	nonassociative	nonassociative	ADJ
ejpam-3576	485	17	rings	ring	NOUN
ejpam-3576	485	18	,	,	PUNCT
ejpam-3576	485	19	european	european	PROPN
ejpam-3576	485	20	journal	journal	PROPN
ejpam-3576	485	21	of	of	ADP
ejpam-3576	485	22	pure	pure	ADJ
ejpam-3576	485	23	and	and	CCONJ
ejpam-3576	485	24	applied	applied	ADJ
ejpam-3576	485	25	mathematics	mathematic	NOUN
ejpam-3576	485	26	,	,	PUNCT
ejpam-3576	485	27	vol	vol	NOUN
ejpam-3576	485	28	.	.	PROPN
ejpam-3576	485	29	12	12	NUM
ejpam-3576	485	30	,	,	PUNCT
ejpam-3576	485	31	2(2019	2(2019	NUM
ejpam-3576	485	32	)	)	PUNCT
ejpam-3576	485	33	622	622	NUM
ejpam-3576	485	34	-	-	SYM
ejpam-3576	485	35	648	648	NUM
ejpam-3576	485	36	.	.	PUNCT
ejpam-3576	486	1	[	[	X
ejpam-3576	486	2	11	11	NUM
ejpam-3576	486	3	]	]	X
ejpam-3576	486	4	n.	n.	NOUN
ejpam-3576	486	5	kausar	kausar	PROPN
ejpam-3576	486	6	,	,	PUNCT
ejpam-3576	486	7	b.	b.	PROPN
ejpam-3576	486	8	islam	islam	PROPN
ejpam-3576	486	9	,	,	PUNCT
ejpam-3576	486	10	m.	m.	PROPN
ejpam-3576	486	11	javaid	javaid	PROPN
ejpam-3576	486	12	,	,	PUNCT
ejpam-3576	486	13	s	s	PROPN
ejpam-3576	486	14	,	,	PUNCT
ejpam-3576	486	15	amjad	amjad	PROPN
ejpam-3576	486	16	,	,	PUNCT
ejpam-3576	486	17	u.	u.	PROPN
ejpam-3576	486	18	ijaz	ijaz	PROPN
ejpam-3576	486	19	,	,	PUNCT
ejpam-3576	486	20	characterizations	characterization	NOUN
ejpam-3576	486	21	of	of	ADP
ejpam-3576	486	22	nonassociative	nonassociative	ADJ
ejpam-3576	486	23	rings	ring	NOUN
ejpam-3576	486	24	by	by	ADP
ejpam-3576	486	25	the	the	DET
ejpam-3576	486	26	properties	property	NOUN
ejpam-3576	486	27	of	of	ADP
ejpam-3576	486	28	their	their	PRON
ejpam-3576	486	29	fuzzy	fuzzy	ADJ
ejpam-3576	486	30	ideals	ideal	NOUN
ejpam-3576	486	31	,	,	PUNCT
ejpam-3576	486	32	journal	journal	NOUN
ejpam-3576	486	33	of	of	ADP
ejpam-3576	486	34	taibah	taibah	PROPN
ejpam-3576	486	35	university	university	PROPN
ejpam-3576	486	36	for	for	ADP
ejpam-3576	486	37	science	science	NOUN
ejpam-3576	486	38	,	,	PUNCT
ejpam-3576	486	39	vol	vol	NOUN
ejpam-3576	486	40	.	.	PROPN
ejpam-3576	486	41	13	13	NUM
ejpam-3576	486	42	,	,	PUNCT
ejpam-3576	486	43	1(2019	1(2019	NUM
ejpam-3576	486	44	)	)	PUNCT
ejpam-3576	486	45	820	820	NUM
ejpam-3576	486	46	-	-	SYM
ejpam-3576	486	47	833	833	NUM
ejpam-3576	486	48	.	.	PUNCT
ejpam-3576	487	1	[	[	X
ejpam-3576	487	2	12	12	NUM
ejpam-3576	487	3	]	]	X
ejpam-3576	487	4	n.	n.	NOUN
ejpam-3576	487	5	kausar	kausar	PROPN
ejpam-3576	487	6	,	,	PUNCT
ejpam-3576	487	7	b.	b.	PROPN
ejpam-3576	487	8	islam	islam	PROPN
ejpam-3576	487	9	,	,	PUNCT
ejpam-3576	487	10	s.	s.	PROPN
ejpam-3576	487	11	amjad	amjad	PROPN
ejpam-3576	487	12	,	,	PUNCT
ejpam-3576	487	13	m.	m.	PROPN
ejpam-3576	487	14	waqar	waqar	PROPN
ejpam-3576	487	15	,	,	PUNCT
ejpam-3576	487	16	intuitionistics	intuitionistics	NOUN
ejpam-3576	487	17	fuzzy	fuzzy	ADJ
ejpam-3576	487	18	ideals	ideal	NOUN
ejpam-3576	487	19	with	with	ADP
ejpam-3576	487	20	thresholds(α	thresholds(α	NOUN
ejpam-3576	487	21	,	,	PUNCT
ejpam-3576	487	22	β	β	X
ejpam-3576	487	23	]	]	X
ejpam-3576	487	24	in	in	ADP
ejpam-3576	487	25	la	la	NOUN
ejpam-3576	487	26	-	-	PUNCT
ejpam-3576	487	27	rings	ring	NOUN
ejpam-3576	487	28	,	,	PUNCT
ejpam-3576	487	29	european	european	PROPN
ejpam-3576	487	30	journal	journal	PROPN
ejpam-3576	487	31	of	of	ADP
ejpam-3576	487	32	pure	pure	ADJ
ejpam-3576	487	33	and	and	CCONJ
ejpam-3576	487	34	applied	applied	ADJ
ejpam-3576	487	35	mathematics	mathematic	NOUN
ejpam-3576	487	36	,	,	PUNCT
ejpam-3576	487	37	vol	vol	NOUN
ejpam-3576	487	38	.	.	PROPN
ejpam-3576	487	39	12	12	NUM
ejpam-3576	487	40	,	,	PUNCT
ejpam-3576	487	41	3(2019	3(2019	NUM
ejpam-3576	487	42	)	)	PUNCT
ejpam-3576	487	43	906	906	NUM
ejpam-3576	487	44	-	-	SYM
ejpam-3576	487	45	943	943	NUM
ejpam-3576	487	46	.	.	PUNCT
ejpam-3576	488	1	references	reference	NOUN
ejpam-3576	488	2	128	128	NUM
ejpam-3576	488	3	[	[	SYM
ejpam-3576	488	4	13	13	NUM
ejpam-3576	488	5	]	]	X
ejpam-3576	488	6	n.	n.	NOUN
ejpam-3576	488	7	kausar	kausar	PROPN
ejpam-3576	488	8	,	,	PUNCT
ejpam-3576	488	9	m.	m.	NOUN
ejpam-3576	488	10	waqar	waqar	PROPN
ejpam-3576	488	11	,	,	PUNCT
ejpam-3576	488	12	direct	direct	ADJ
ejpam-3576	488	13	product	product	NOUN
ejpam-3576	488	14	of	of	ADP
ejpam-3576	488	15	finite	finite	ADJ
ejpam-3576	488	16	fuzzy	fuzzy	ADJ
ejpam-3576	488	17	normal	normal	ADJ
ejpam-3576	488	18	subrings	subring	NOUN
ejpam-3576	488	19	over	over	ADP
ejpam-3576	488	20	nonassociative	nonassociative	ADJ
ejpam-3576	488	21	rings	ring	NOUN
ejpam-3576	488	22	,	,	PUNCT
ejpam-3576	488	23	international	international	ADJ
ejpam-3576	488	24	journal	journal	NOUN
ejpam-3576	488	25	of	of	ADP
ejpam-3576	488	26	analysis	analysis	NOUN
ejpam-3576	488	27	and	and	CCONJ
ejpam-3576	488	28	applications	application	NOUN
ejpam-3576	488	29	,	,	PUNCT
ejpam-3576	488	30	vol	vol	NOUN
ejpam-3576	488	31	.	.	PROPN
ejpam-3576	488	32	17	17	NUM
ejpam-3576	488	33	,	,	PUNCT
ejpam-3576	488	34	5(2019	5(2019	NUM
ejpam-3576	488	35	)	)	PUNCT
ejpam-3576	488	36	752	752	NUM
ejpam-3576	488	37	-	-	SYM
ejpam-3576	488	38	770	770	NUM
ejpam-3576	488	39	.	.	PUNCT
ejpam-3576	489	1	[	[	X
ejpam-3576	489	2	14	14	NUM
ejpam-3576	489	3	]	]	PUNCT
ejpam-3576	489	4	m.	m.	NOUN
ejpam-3576	489	5	a.	a.	PROPN
ejpam-3576	489	6	kazim	kazim	PROPN
ejpam-3576	489	7	and	and	CCONJ
ejpam-3576	489	8	m.	m.	PROPN
ejpam-3576	489	9	naseeruddin	naseeruddin	PROPN
ejpam-3576	489	10	,	,	PUNCT
ejpam-3576	489	11	on	on	ADP
ejpam-3576	489	12	almost	almost	ADV
ejpam-3576	489	13	semigroups	semigroup	NOUN
ejpam-3576	489	14	,	,	PUNCT
ejpam-3576	489	15	alig	alig	PROPN
ejpam-3576	489	16	.	.	PUNCT
ejpam-3576	490	1	bull	bull	PROPN
ejpam-3576	490	2	.	.	PUNCT
ejpam-3576	491	1	math	math	NOUN
ejpam-3576	491	2	.	.	PUNCT
ejpam-3576	491	3	,	,	PUNCT
ejpam-3576	491	4	2(1972	2(1972	X
ejpam-3576	491	5	)	)	PUNCT
ejpam-3576	491	6	1	1	NUM
ejpam-3576	491	7	-	-	SYM
ejpam-3576	491	8	7	7	NUM
ejpam-3576	491	9	.	.	PUNCT
ejpam-3576	492	1	[	[	X
ejpam-3576	492	2	15	15	NUM
ejpam-3576	492	3	]	]	X
ejpam-3576	492	4	n.	n.	NOUN
ejpam-3576	492	5	kehayopulu	kehayopulu	PROPN
ejpam-3576	492	6	,	,	PUNCT
ejpam-3576	492	7	on	on	ADP
ejpam-3576	492	8	weakly	weakly	ADJ
ejpam-3576	492	9	prime	prime	ADJ
ejpam-3576	492	10	ideals	ideal	NOUN
ejpam-3576	492	11	of	of	ADP
ejpam-3576	492	12	ordered	order	VERB
ejpam-3576	492	13	semigroups	semigroup	NOUN
ejpam-3576	492	14	,	,	PUNCT
ejpam-3576	492	15	math	math	NOUN
ejpam-3576	492	16	.	.	PUNCT
ejpam-3576	493	1	japon	japon	PROPN
ejpam-3576	493	2	.	.	PROPN
ejpam-3576	493	3	,	,	PUNCT
ejpam-3576	493	4	35(1990	35(1990	NUM
ejpam-3576	493	5	)	)	PUNCT
ejpam-3576	493	6	1051	1051	NUM
ejpam-3576	493	7	-	-	SYM
ejpam-3576	493	8	1056	1056	NUM
ejpam-3576	493	9	.	.	PUNCT
ejpam-3576	494	1	[	[	X
ejpam-3576	494	2	16	16	NUM
ejpam-3576	494	3	]	]	X
ejpam-3576	494	4	n.	n.	PROPN
ejpam-3576	494	5	kehayopulu	kehayopulu	PROPN
ejpam-3576	494	6	,	,	PUNCT
ejpam-3576	494	7	on	on	ADP
ejpam-3576	494	8	intra	intra	ADJ
ejpam-3576	494	9	-	-	ADJ
ejpam-3576	494	10	regular	regular	ADJ
ejpam-3576	494	11	ordered	order	VERB
ejpam-3576	494	12	semigroups	semigroup	NOUN
ejpam-3576	494	13	,	,	PUNCT
ejpam-3576	494	14	semigroup	semigroup	PROPN
ejpam-3576	494	15	forum	forum	PROPN
ejpam-3576	494	16	,	,	PUNCT
ejpam-3576	494	17	46(1993	46(1993	NUM
ejpam-3576	494	18	)	)	PUNCT
ejpam-3576	494	19	271	271	NUM
ejpam-3576	494	20	-	-	SYM
ejpam-3576	494	21	278	278	NUM
ejpam-3576	494	22	.	.	PUNCT
ejpam-3576	495	1	[	[	X
ejpam-3576	495	2	17	17	NUM
ejpam-3576	495	3	]	]	X
ejpam-3576	495	4	n.	n.	NOUN
ejpam-3576	495	5	kehayopulu	kehayopulu	PROPN
ejpam-3576	495	6	,	,	PUNCT
ejpam-3576	495	7	on	on	ADP
ejpam-3576	495	8	regular	regular	ADJ
ejpam-3576	495	9	ordered	order	VERB
ejpam-3576	495	10	semigroups	semigroup	NOUN
ejpam-3576	495	11	,	,	PUNCT
ejpam-3576	495	12	math	math	NOUN
ejpam-3576	495	13	.	.	PUNCT
ejpam-3576	496	1	japon	japon	PROPN
ejpam-3576	496	2	.	.	PROPN
ejpam-3576	496	3	,	,	PUNCT
ejpam-3576	496	4	45(1997	45(1997	NUM
ejpam-3576	496	5	)	)	PUNCT
ejpam-3576	496	6	549	549	NUM
ejpam-3576	496	7	-	-	SYM
ejpam-3576	496	8	553	553	NUM
ejpam-3576	496	9	.	.	PUNCT
ejpam-3576	497	1	[	[	X
ejpam-3576	497	2	18	18	NUM
ejpam-3576	497	3	]	]	X
ejpam-3576	497	4	n.	n.	NOUN
ejpam-3576	497	5	kehayopulu	kehayopulu	PROPN
ejpam-3576	497	6	,	,	PUNCT
ejpam-3576	497	7	on	on	ADP
ejpam-3576	497	8	completely	completely	ADV
ejpam-3576	497	9	regular	regular	ADJ
ejpam-3576	497	10	ordered	order	VERB
ejpam-3576	497	11	semigroups	semigroup	NOUN
ejpam-3576	497	12	,	,	PUNCT
ejpam-3576	497	13	sci	sci	PROPN
ejpam-3576	497	14	.	.	PROPN
ejpam-3576	497	15	math	math	PROPN
ejpam-3576	497	16	.	.	PUNCT
ejpam-3576	497	17	,	,	PUNCT
ejpam-3576	497	18	1(1998	1(1998	NUM
ejpam-3576	497	19	)	)	PUNCT
ejpam-3576	497	20	27	27	NUM
ejpam-3576	497	21	-	-	SYM
ejpam-3576	497	22	32	32	NUM
ejpam-3576	497	23	.	.	PUNCT
ejpam-3576	498	1	[	[	X
ejpam-3576	498	2	19	19	NUM
ejpam-3576	498	3	]	]	X
ejpam-3576	498	4	n.	n.	NOUN
ejpam-3576	498	5	kehayopulu	kehayopulu	ADJ
ejpam-3576	498	6	and	and	CCONJ
ejpam-3576	498	7	m.	m.	NOUN
ejpam-3576	498	8	tsingelis	tsingelis	PROPN
ejpam-3576	498	9	,	,	PUNCT
ejpam-3576	498	10	fuzzy	fuzzy	ADJ
ejpam-3576	498	11	sets	set	NOUN
ejpam-3576	498	12	in	in	ADP
ejpam-3576	498	13	ordered	order	VERB
ejpam-3576	498	14	groupoids	groupoid	NOUN
ejpam-3576	498	15	,	,	PUNCT
ejpam-3576	498	16	semigroup	semigroup	PROPN
ejpam-3576	498	17	forum	forum	PROPN
ejpam-3576	498	18	,	,	PUNCT
ejpam-3576	498	19	65(2002	65(2002	NUM
ejpam-3576	498	20	)	)	PUNCT
ejpam-3576	498	21	128	128	NUM
ejpam-3576	498	22	-	-	SYM
ejpam-3576	498	23	132	132	NUM
ejpam-3576	498	24	.	.	PUNCT
ejpam-3576	499	1	[	[	X
ejpam-3576	499	2	20	20	NUM
ejpam-3576	499	3	]	]	X
ejpam-3576	499	4	n.	n.	NOUN
ejpam-3576	499	5	kehayopulu	kehayopulu	ADJ
ejpam-3576	499	6	and	and	CCONJ
ejpam-3576	499	7	m.	m.	NOUN
ejpam-3576	499	8	tsingelis	tsingelis	PROPN
ejpam-3576	499	9	,	,	PUNCT
ejpam-3576	499	10	fuzzy	fuzzy	ADJ
ejpam-3576	499	11	bi	bi	NOUN
ejpam-3576	499	12	-	-	NOUN
ejpam-3576	499	13	ideals	ideal	NOUN
ejpam-3576	499	14	in	in	ADP
ejpam-3576	499	15	ordered	order	VERB
ejpam-3576	499	16	semigroups	semigroup	NOUN
ejpam-3576	499	17	,	,	PUNCT
ejpam-3576	499	18	inform	inform	VERB
ejpam-3576	499	19	sci	sci	PROPN
ejpam-3576	499	20	.	.	PROPN
ejpam-3576	499	21	,	,	PUNCT
ejpam-3576	499	22	171(2005	171(2005	NUM
ejpam-3576	499	23	)	)	PUNCT
ejpam-3576	499	24	13	13	NUM
ejpam-3576	499	25	-	-	SYM
ejpam-3576	499	26	28	28	NUM
ejpam-3576	499	27	.	.	PUNCT
ejpam-3576	500	1	[	[	X
ejpam-3576	500	2	21	21	NUM
ejpam-3576	500	3	]	]	X
ejpam-3576	500	4	n.	n.	PROPN
ejpam-3576	500	5	kuroki	kuroki	PROPN
ejpam-3576	500	6	,	,	PUNCT
ejpam-3576	500	7	fuzzy	fuzzy	ADJ
ejpam-3576	500	8	bi	bi	NOUN
ejpam-3576	500	9	-	-	NOUN
ejpam-3576	500	10	ideals	ideal	NOUN
ejpam-3576	500	11	in	in	ADP
ejpam-3576	500	12	semigroups	semigroup	NOUN
ejpam-3576	500	13	,	,	PUNCT
ejpam-3576	500	14	comment	comment	NOUN
ejpam-3576	500	15	.	.	PUNCT
ejpam-3576	501	1	math	math	NOUN
ejpam-3576	501	2	.	.	PUNCT
ejpam-3576	502	1	univ	univ	PROPN
ejpam-3576	502	2	.	.	PUNCT
ejpam-3576	502	3	st	st	PROPN
ejpam-3576	502	4	.	.	PROPN
ejpam-3576	502	5	pauli	pauli	PROPN
ejpam-3576	502	6	.	.	PUNCT
ejpam-3576	503	1	28(1979	28(1979	NUM
ejpam-3576	503	2	)	)	PUNCT
ejpam-3576	503	3	17	17	NUM
ejpam-3576	503	4	-	-	SYM
ejpam-3576	503	5	21	21	NUM
ejpam-3576	503	6	.	.	PUNCT
ejpam-3576	504	1	[	[	X
ejpam-3576	504	2	22	22	NUM
ejpam-3576	504	3	]	]	X
ejpam-3576	504	4	n.	n.	PROPN
ejpam-3576	504	5	kuroki	kuroki	PROPN
ejpam-3576	504	6	,	,	PUNCT
ejpam-3576	504	7	fuzzy	fuzzy	ADJ
ejpam-3576	504	8	semiprime	semiprime	NOUN
ejpam-3576	504	9	quasi	quasi	NOUN
ejpam-3576	504	10	-	-	NOUN
ejpam-3576	504	11	ideals	ideal	NOUN
ejpam-3576	504	12	in	in	ADP
ejpam-3576	504	13	semigroups	semigroup	NOUN
ejpam-3576	504	14	,	,	PUNCT
ejpam-3576	504	15	inform	inform	NOUN
ejpam-3576	504	16	.	.	PUNCT
ejpam-3576	505	1	sci	sci	PROPN
ejpam-3576	505	2	.	.	PROPN
ejpam-3576	505	3	,	,	PUNCT
ejpam-3576	505	4	75(1993	75(1993	NUM
ejpam-3576	505	5	)	)	PUNCT
ejpam-3576	505	6	201211	201211	NUM
ejpam-3576	505	7	.	.	PUNCT
ejpam-3576	506	1	[	[	X
ejpam-3576	506	2	23	23	NUM
ejpam-3576	506	3	]	]	X
ejpam-3576	506	4	n.	n.	PROPN
ejpam-3576	506	5	kuroki	kuroki	PROPN
ejpam-3576	506	6	,	,	PUNCT
ejpam-3576	506	7	fuzzy	fuzzy	ADJ
ejpam-3576	506	8	interior	interior	ADJ
ejpam-3576	506	9	ideals	ideal	NOUN
ejpam-3576	506	10	in	in	ADP
ejpam-3576	506	11	semigroups	semigroup	NOUN
ejpam-3576	506	12	,	,	PUNCT
ejpam-3576	506	13	j.	j.	PROPN
ejpam-3576	506	14	fuzzy	fuzzy	PROPN
ejpam-3576	506	15	math	math	PROPN
ejpam-3576	506	16	.	.	PUNCT
ejpam-3576	506	17	,	,	PUNCT
ejpam-3576	506	18	3(1995	3(1995	NUM
ejpam-3576	506	19	)	)	PUNCT
ejpam-3576	506	20	435	435	NUM
ejpam-3576	506	21	-	-	SYM
ejpam-3576	506	22	447	447	NUM
ejpam-3576	506	23	.	.	PUNCT
ejpam-3576	507	1	[	[	X
ejpam-3576	507	2	24	24	NUM
ejpam-3576	507	3	]	]	PUNCT
ejpam-3576	507	4	j.	j.	PROPN
ejpam-3576	507	5	n.	n.	PROPN
ejpam-3576	507	6	mordeson	mordeson	PROPN
ejpam-3576	507	7	,	,	PUNCT
ejpam-3576	507	8	d.	d.	PROPN
ejpam-3576	507	9	s.	s.	PROPN
ejpam-3576	507	10	malik	malik	PROPN
ejpam-3576	507	11	and	and	CCONJ
ejpam-3576	507	12	n.	n.	PROPN
ejpam-3576	507	13	kuroki	kuroki	PROPN
ejpam-3576	507	14	,	,	PUNCT
ejpam-3576	507	15	fuzzy	fuzzy	ADJ
ejpam-3576	507	16	semigroups	semigroup	NOUN
ejpam-3576	507	17	,	,	PUNCT
ejpam-3576	507	18	springer	springer	NOUN
ejpam-3576	507	19	berlin	berlin	PROPN
ejpam-3576	507	20	,	,	PUNCT
ejpam-3576	507	21	2003	2003	NUM
ejpam-3576	507	22	.	.	PUNCT
ejpam-3576	508	1	[	[	X
ejpam-3576	508	2	25	25	NUM
ejpam-3576	508	3	]	]	PUNCT
ejpam-3576	508	4	a.	a.	NOUN
ejpam-3576	508	5	lafi	lafi	PROPN
ejpam-3576	508	6	,	,	PUNCT
ejpam-3576	508	7	dfig	dfig	PROPN
ejpam-3576	508	8	control	control	PROPN
ejpam-3576	508	9	:	:	PUNCT
ejpam-3576	508	10	a	a	DET
ejpam-3576	508	11	fuzzy	fuzzy	ADJ
ejpam-3576	508	12	approach	approach	NOUN
ejpam-3576	508	13	,	,	PUNCT
ejpam-3576	508	14	international	international	ADJ
ejpam-3576	508	15	journal	journal	NOUN
ejpam-3576	508	16	of	of	ADP
ejpam-3576	508	17	advanced	advanced	ADJ
ejpam-3576	508	18	and	and	CCONJ
ejpam-3576	508	19	applied	apply	VERB
ejpam-3576	508	20	sciences	science	NOUN
ejpam-3576	508	21	,	,	PUNCT
ejpam-3576	508	22	6(019	6(019	NUM
ejpam-3576	508	23	107	107	NUM
ejpam-3576	508	24	-	-	SYM
ejpam-3576	508	25	116	116	NUM
ejpam-3576	508	26	.	.	PUNCT
ejpam-3576	509	1	[	[	X
ejpam-3576	509	2	26	26	NUM
ejpam-3576	509	3	]	]	X
ejpam-3576	509	4	q.	q.	PROPN
ejpam-3576	509	5	mushtaq	mushtaq	PROPN
ejpam-3576	509	6	and	and	CCONJ
ejpam-3576	509	7	s.	s.	PROPN
ejpam-3576	509	8	m.	m.	PROPN
ejpam-3576	509	9	yusuf	yusuf	PROPN
ejpam-3576	509	10	,	,	PUNCT
ejpam-3576	509	11	on	on	ADP
ejpam-3576	509	12	la	la	NOUN
ejpam-3576	509	13	-	-	PUNCT
ejpam-3576	509	14	semigroups	semigroup	NOUN
ejpam-3576	509	15	,	,	PUNCT
ejpam-3576	509	16	alig	alig	PROPN
ejpam-3576	509	17	.	.	PUNCT
ejpam-3576	510	1	bull	bull	PROPN
ejpam-3576	510	2	.	.	PUNCT
ejpam-3576	511	1	math	math	NOUN
ejpam-3576	511	2	.	.	PUNCT
ejpam-3576	511	3	,	,	PUNCT
ejpam-3576	511	4	8(1978	8(1978	NUM
ejpam-3576	511	5	)	)	PUNCT
ejpam-3576	511	6	65	65	NUM
ejpam-3576	511	7	-	-	SYM
ejpam-3576	511	8	70	70	NUM
ejpam-3576	511	9	.	.	PUNCT
ejpam-3576	512	1	[	[	X
ejpam-3576	512	2	27	27	NUM
ejpam-3576	512	3	]	]	X
ejpam-3576	512	4	p.	p.	NOUN
ejpam-3576	512	5	v.	v.	ADP
ejpam-3576	512	6	protic	protic	PROPN
ejpam-3576	512	7	and	and	CCONJ
ejpam-3576	512	8	n.	n.	PROPN
ejpam-3576	512	9	stevanovic	stevanovic	PROPN
ejpam-3576	512	10	,	,	PUNCT
ejpam-3576	512	11	ag	ag	NOUN
ejpam-3576	512	12	-	-	PUNCT
ejpam-3576	512	13	test	test	NOUN
ejpam-3576	512	14	and	and	CCONJ
ejpam-3576	512	15	some	some	DET
ejpam-3576	512	16	general	general	ADJ
ejpam-3576	512	17	properties	property	NOUN
ejpam-3576	512	18	of	of	ADP
ejpam-3576	512	19	abelgrassmann	abelgrassmann	PROPN
ejpam-3576	512	20	’s	’s	PART
ejpam-3576	512	21	groupoids	groupoid	NOUN
ejpam-3576	512	22	,	,	PUNCT
ejpam-3576	512	23	pure	pure	ADJ
ejpam-3576	512	24	math	math	NOUN
ejpam-3576	512	25	.	.	PUNCT
ejpam-3576	513	1	appl	appl	PROPN
ejpam-3576	513	2	.	.	PROPN
ejpam-3576	513	3	,	,	PUNCT
ejpam-3576	513	4	6(1995	6(1995	NUM
ejpam-3576	513	5	)	)	PUNCT
ejpam-3576	513	6	371	371	NUM
ejpam-3576	513	7	-	-	SYM
ejpam-3576	513	8	383	383	NUM
ejpam-3576	513	9	.	.	PUNCT
ejpam-3576	514	1	[	[	X
ejpam-3576	514	2	28	28	NUM
ejpam-3576	514	3	]	]	X
ejpam-3576	514	4	s.	s.	PROPN
ejpam-3576	514	5	a.	a.	PROPN
ejpam-3576	514	6	razak	razak	PROPN
ejpam-3576	514	7	,	,	PUNCT
ejpam-3576	514	8	d.	d.	PROPN
ejpam-3576	514	9	mohamad	mohamad	PROPN
ejpam-3576	514	10	,	,	PUNCT
ejpam-3576	514	11	i.	i.	PROPN
ejpam-3576	514	12	i.	i.	PROPN
ejpam-3576	514	13	abdullah	abdullah	PROPN
ejpam-3576	514	14	,	,	PUNCT
ejpam-3576	514	15	a	a	DET
ejpam-3576	514	16	group	group	NOUN
ejpam-3576	514	17	decision	decision	NOUN
ejpam-3576	514	18	making	make	VERB
ejpam-3576	514	19	problem	problem	NOUN
ejpam-3576	514	20	using	use	VERB
ejpam-3576	514	21	hierarchical	hierarchical	ADJ
ejpam-3576	514	22	based	base	VERB
ejpam-3576	514	23	fuzzy	fuzzy	ADJ
ejpam-3576	514	24	soft	soft	ADJ
ejpam-3576	514	25	matrix	matrix	NOUN
ejpam-3576	514	26	approach	approach	NOUN
ejpam-3576	514	27	,	,	PUNCT
ejpam-3576	514	28	international	international	ADJ
ejpam-3576	514	29	journal	journal	NOUN
ejpam-3576	514	30	of	of	ADP
ejpam-3576	514	31	advanced	advanced	ADJ
ejpam-3576	514	32	and	and	CCONJ
ejpam-3576	514	33	applied	apply	VERB
ejpam-3576	514	34	sciences	science	NOUN
ejpam-3576	514	35	,	,	PUNCT
ejpam-3576	514	36	4((2017	4((2017	NUM
ejpam-3576	514	37	)	)	PUNCT
ejpam-3576	514	38	26	26	NUM
ejpam-3576	514	39	-	-	SYM
ejpam-3576	514	40	32	32	NUM
ejpam-3576	514	41	.	.	PUNCT
ejpam-3576	515	1	[	[	X
ejpam-3576	515	2	29	29	NUM
ejpam-3576	515	3	]	]	PUNCT
ejpam-3576	515	4	a.	a.	NOUN
ejpam-3576	515	5	rosenfeld	rosenfeld	PROPN
ejpam-3576	515	6	,	,	PUNCT
ejpam-3576	515	7	fuzzy	fuzzy	ADJ
ejpam-3576	515	8	groups	group	NOUN
ejpam-3576	515	9	,	,	PUNCT
ejpam-3576	515	10	j.	j.	PROPN
ejpam-3576	515	11	math	math	PROPN
ejpam-3576	515	12	.	.	PUNCT
ejpam-3576	516	1	anal	anal	PROPN
ejpam-3576	516	2	.	.	PUNCT
ejpam-3576	517	1	appl	appl	PROPN
ejpam-3576	517	2	.	.	PROPN
ejpam-3576	517	3	,	,	PUNCT
ejpam-3576	517	4	35(1971	35(1971	NUM
ejpam-3576	517	5	)	)	PUNCT
ejpam-3576	517	6	512	512	NUM
ejpam-3576	517	7	-	-	SYM
ejpam-3576	517	8	517	517	NUM
ejpam-3576	517	9	.	.	PUNCT
ejpam-3576	518	1	[	[	X
ejpam-3576	518	2	30	30	NUM
ejpam-3576	518	3	]	]	PUNCT
ejpam-3576	518	4	t.	t.	NOUN
ejpam-3576	518	5	shah	shah	PROPN
ejpam-3576	518	6	,	,	PUNCT
ejpam-3576	518	7	n.	n.	PROPN
ejpam-3576	518	8	kausar	kausar	PROPN
ejpam-3576	518	9	,	,	PUNCT
ejpam-3576	518	10	i.	i.	PROPN
ejpam-3576	518	11	rehman	rehman	PROPN
ejpam-3576	518	12	,	,	PUNCT
ejpam-3576	518	13	intuitionistic	intuitionistic	ADJ
ejpam-3576	518	14	fuzzy	fuzzy	ADJ
ejpam-3576	518	15	normal	normal	ADJ
ejpam-3576	518	16	subrings	subring	NOUN
ejpam-3576	518	17	over	over	ADP
ejpam-3576	518	18	a	a	DET
ejpam-3576	518	19	nonassociative	nonassociative	ADJ
ejpam-3576	518	20	ring	ring	NOUN
ejpam-3576	518	21	,	,	PUNCT
ejpam-3576	518	22	an	an	PROPN
ejpam-3576	518	23	.	.	PUNCT
ejpam-3576	518	24	st	st	PROPN
ejpam-3576	518	25	.	.	PROPN
ejpam-3576	518	26	univ	univ	PROPN
ejpam-3576	518	27	.	.	PUNCT
ejpam-3576	519	1	ovidius	ovidius	PROPN
ejpam-3576	519	2	constanta	constanta	PROPN
ejpam-3576	519	3	,	,	PUNCT
ejpam-3576	519	4	vol	vol	NOUN
ejpam-3576	519	5	.	.	PROPN
ejpam-3576	519	6	20	20	NUM
ejpam-3576	519	7	(	(	PUNCT
ejpam-3576	519	8	2012	2012	NUM
ejpam-3576	519	9	)	)	PUNCT
ejpam-3576	519	10	369	369	NUM
ejpam-3576	519	11	-	-	SYM
ejpam-3576	519	12	386	386	NUM
ejpam-3576	519	13	.	.	PUNCT
ejpam-3576	519	14	references	reference	NOUN
ejpam-3576	519	15	129	129	NUM
ejpam-3576	520	1	[	[	X
ejpam-3576	520	2	31	31	NUM
ejpam-3576	520	3	]	]	PUNCT
ejpam-3576	520	4	t.	t.	NOUN
ejpam-3576	520	5	shah	shah	PROPN
ejpam-3576	520	6	,	,	PUNCT
ejpam-3576	520	7	n.	n.	PROPN
ejpam-3576	520	8	kausar	kausar	PROPN
ejpam-3576	520	9	,	,	PUNCT
ejpam-3576	520	10	characterizations	characterization	NOUN
ejpam-3576	520	11	of	of	ADP
ejpam-3576	520	12	non	non	ADJ
ejpam-3576	520	13	-	-	ADJ
ejpam-3576	520	14	associative	associative	ADJ
ejpam-3576	520	15	ordered	order	VERB
ejpam-3576	520	16	semigroups	semigroup	NOUN
ejpam-3576	520	17	by	by	ADP
ejpam-3576	520	18	their	their	PRON
ejpam-3576	520	19	fuzzy	fuzzy	ADJ
ejpam-3576	520	20	bi	bi	NOUN
ejpam-3576	520	21	-	-	NOUN
ejpam-3576	520	22	ideals	ideal	NOUN
ejpam-3576	520	23	,	,	PUNCT
ejpam-3576	520	24	theoretical	theoretical	ADJ
ejpam-3576	520	25	computer	computer	NOUN
ejpam-3576	520	26	science	science	NOUN
ejpam-3576	520	27	,	,	PUNCT
ejpam-3576	520	28	vol	vol	NOUN
ejpam-3576	520	29	.	.	PUNCT
ejpam-3576	521	1	529	529	NUM
ejpam-3576	521	2	(	(	PUNCT
ejpam-3576	521	3	2014	2014	NUM
ejpam-3576	521	4	)	)	PUNCT
ejpam-3576	521	5	96	96	NUM
ejpam-3576	521	6	-	-	SYM
ejpam-3576	521	7	110	110	NUM
ejpam-3576	521	8	.	.	PUNCT
ejpam-3576	522	1	[	[	X
ejpam-3576	522	2	32	32	NUM
ejpam-3576	522	3	]	]	PUNCT
ejpam-3576	522	4	o.	o.	NOUN
ejpam-3576	522	5	ozer	ozer	PROPN
ejpam-3576	522	6	,	,	PUNCT
ejpam-3576	522	7	s.	s.	PROPN
ejpam-3576	522	8	omran	omran	PROPN
ejpam-3576	522	9	,	,	PUNCT
ejpam-3576	522	10	on	on	ADP
ejpam-3576	522	11	the	the	DET
ejpam-3576	522	12	generalized	generalize	VERB
ejpam-3576	522	13	c*valued	c*value	VERB
ejpam-3576	522	14	metric	metric	ADJ
ejpam-3576	522	15	spaces	space	NOUN
ejpam-3576	522	16	related	relate	VERB
ejpam-3576	522	17	with	with	ADP
ejpam-3576	522	18	banach	banach	ADV
ejpam-3576	522	19	fixed	fix	VERB
ejpam-3576	522	20	point	point	NOUN
ejpam-3576	522	21	theory	theory	NOUN
ejpam-3576	522	22	,	,	PUNCT
ejpam-3576	522	23	international	international	ADJ
ejpam-3576	522	24	journal	journal	NOUN
ejpam-3576	522	25	of	of	ADP
ejpam-3576	522	26	advanced	advanced	ADJ
ejpam-3576	522	27	and	and	CCONJ
ejpam-3576	522	28	applied	apply	VERB
ejpam-3576	522	29	sciences	science	NOUN
ejpam-3576	522	30	,	,	PUNCT
ejpam-3576	522	31	4(2017	4(2017	NUM
ejpam-3576	522	32	)	)	PUNCT
ejpam-3576	522	33	35	35	NUM
ejpam-3576	522	34	-	-	SYM
ejpam-3576	522	35	37	37	NUM
ejpam-3576	522	36	.	.	PUNCT
ejpam-3576	523	1	[	[	X
ejpam-3576	523	2	33	33	NUM
ejpam-3576	523	3	]	]	PUNCT
ejpam-3576	523	4	l.	l.	PROPN
ejpam-3576	523	5	a.	a.	PROPN
ejpam-3576	523	6	zadeh	zadeh	PROPN
ejpam-3576	523	7	,	,	PUNCT
ejpam-3576	523	8	fuzzy	fuzzy	ADJ
ejpam-3576	523	9	sets	set	NOUN
ejpam-3576	523	10	,	,	PUNCT
ejpam-3576	523	11	inform	inform	NOUN
ejpam-3576	523	12	.	.	PUNCT
ejpam-3576	524	1	control	control	NOUN
ejpam-3576	524	2	,	,	PUNCT
ejpam-3576	524	3	8(1965	8(1965	NUM
ejpam-3576	524	4	)	)	PUNCT
ejpam-3576	524	5	338	338	NUM
ejpam-3576	524	6	-	-	SYM
ejpam-3576	524	7	353	353	NUM
ejpam-3576	524	8	.	.	PUNCT