id	sid	tid	token	lemma	pos
ejpam-3246	1	1	european	european	PROPN
ejpam-3246	1	2	journal	journal	PROPN
ejpam-3246	1	3	of	of	ADP
ejpam-3246	1	4	pure	pure	ADJ
ejpam-3246	1	5	and	and	CCONJ
ejpam-3246	1	6	applied	apply	VERB
ejpam-3246	1	7	mathematics	mathematic	NOUN
ejpam-3246	1	8	vol	vol	NOUN
ejpam-3246	1	9	.	.	PUNCT
ejpam-3246	2	1	11	11	NUM
ejpam-3246	2	2	,	,	PUNCT
ejpam-3246	2	3	no	no	INTJ
ejpam-3246	2	4	.	.	NOUN
ejpam-3246	2	5	2	2	NUM
ejpam-3246	2	6	,	,	PUNCT
ejpam-3246	2	7	2018	2018	NUM
ejpam-3246	2	8	,	,	PUNCT
ejpam-3246	2	9	417	417	NUM
ejpam-3246	2	10	-	-	SYM
ejpam-3246	2	11	430	430	NUM
ejpam-3246	2	12	issn	issn	PROPN
ejpam-3246	2	13	1307	1307	NUM
ejpam-3246	2	14	-	-	SYM
ejpam-3246	2	15	5543	5543	NUM
ejpam-3246	2	16	–	–	PUNCT
ejpam-3246	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3246	2	18	published	publish	VERB
ejpam-3246	2	19	by	by	ADP
ejpam-3246	2	20	new	new	PROPN
ejpam-3246	2	21	york	york	PROPN
ejpam-3246	2	22	business	business	PROPN
ejpam-3246	2	23	global	global	ADJ
ejpam-3246	2	24	subalgebras	subalgebra	NOUN
ejpam-3246	2	25	and	and	CCONJ
ejpam-3246	2	26	ideals	ideal	NOUN
ejpam-3246	2	27	in	in	ADP
ejpam-3246	2	28	bck	bck	PROPN
ejpam-3246	2	29	/	/	SYM
ejpam-3246	2	30	bci	bci	NOUN
ejpam-3246	2	31	-	-	PUNCT
ejpam-3246	2	32	algebras	algebras	PROPN
ejpam-3246	2	33	based	base	VERB
ejpam-3246	2	34	on	on	ADP
ejpam-3246	2	35	uni	uni	ADJ
ejpam-3246	2	36	-	-	ADJ
ejpam-3246	2	37	hesitant	hesitant	ADJ
ejpam-3246	2	38	fuzzy	fuzzy	ADJ
ejpam-3246	2	39	set	set	NOUN
ejpam-3246	2	40	theory	theory	NOUN
ejpam-3246	2	41	g.	g.	PROPN
ejpam-3246	2	42	muhiuddin1,∗	muhiuddin1,∗	PROPN
ejpam-3246	2	43	,	,	PUNCT
ejpam-3246	2	44	shuaa	shuaa	ADV
ejpam-3246	2	45	aldhafeeri2	aldhafeeri2	PROPN
ejpam-3246	3	1	1	1	NUM
ejpam-3246	3	2	department	department	NOUN
ejpam-3246	3	3	of	of	ADP
ejpam-3246	3	4	mathematics	mathematic	NOUN
ejpam-3246	3	5	,	,	PUNCT
ejpam-3246	3	6	university	university	PROPN
ejpam-3246	3	7	of	of	ADP
ejpam-3246	3	8	tabuk	tabuk	PROPN
ejpam-3246	3	9	,	,	PUNCT
ejpam-3246	3	10	tabuk	tabuk	NOUN
ejpam-3246	3	11	71491	71491	NUM
ejpam-3246	3	12	,	,	PUNCT
ejpam-3246	3	13	saudi	saudi	PROPN
ejpam-3246	3	14	arabia	arabia	PROPN
ejpam-3246	3	15	2	2	NUM
ejpam-3246	3	16	department	department	NOUN
ejpam-3246	3	17	of	of	ADP
ejpam-3246	3	18	mathematics	mathematic	NOUN
ejpam-3246	3	19	,	,	PUNCT
ejpam-3246	3	20	college	college	NOUN
ejpam-3246	3	21	of	of	ADP
ejpam-3246	3	22	basic	basic	ADJ
ejpam-3246	3	23	education	education	NOUN
ejpam-3246	3	24	,	,	PUNCT
ejpam-3246	3	25	public	public	ADJ
ejpam-3246	3	26	authority	authority	NOUN
ejpam-3246	3	27	for	for	ADP
ejpam-3246	3	28	applied	apply	VERB
ejpam-3246	3	29	education	education	NOUN
ejpam-3246	3	30	and	and	CCONJ
ejpam-3246	3	31	training	training	NOUN
ejpam-3246	3	32	,	,	PUNCT
ejpam-3246	3	33	kuwait	kuwait	PROPN
ejpam-3246	3	34	abstract	abstract	NOUN
ejpam-3246	3	35	.	.	PUNCT
ejpam-3246	4	1	in	in	ADP
ejpam-3246	4	2	the	the	DET
ejpam-3246	4	3	present	present	ADJ
ejpam-3246	4	4	paper	paper	NOUN
ejpam-3246	4	5	,	,	PUNCT
ejpam-3246	4	6	the	the	DET
ejpam-3246	4	7	notions	notion	NOUN
ejpam-3246	4	8	of	of	ADP
ejpam-3246	4	9	uni	uni	ADJ
ejpam-3246	4	10	-	-	ADJ
ejpam-3246	4	11	hesitant	hesitant	ADJ
ejpam-3246	4	12	fuzzy	fuzzy	ADJ
ejpam-3246	4	13	algebras	algebra	NOUN
ejpam-3246	4	14	and	and	CCONJ
ejpam-3246	4	15	uni	uni	ADJ
ejpam-3246	4	16	-	-	ADJ
ejpam-3246	4	17	hesitant	hesitant	ADJ
ejpam-3246	4	18	fuzzy	fuzzy	ADJ
ejpam-3246	4	19	(	(	PUNCT
ejpam-3246	4	20	closed	closed	ADJ
ejpam-3246	4	21	)	)	PUNCT
ejpam-3246	4	22	ideals	ideal	NOUN
ejpam-3246	4	23	in	in	ADP
ejpam-3246	4	24	bck	bck	NOUN
ejpam-3246	4	25	-	-	PUNCT
ejpam-3246	4	26	algebras	algebras	PROPN
ejpam-3246	4	27	and	and	CCONJ
ejpam-3246	4	28	bci	bci	NOUN
ejpam-3246	4	29	-	-	PUNCT
ejpam-3246	4	30	algebras	algebra	NOUN
ejpam-3246	4	31	are	be	AUX
ejpam-3246	4	32	introduced	introduce	VERB
ejpam-3246	4	33	,	,	PUNCT
ejpam-3246	4	34	and	and	CCONJ
ejpam-3246	4	35	several	several	ADJ
ejpam-3246	4	36	related	related	ADJ
ejpam-3246	4	37	properties	property	NOUN
ejpam-3246	4	38	are	be	AUX
ejpam-3246	4	39	investigated	investigate	VERB
ejpam-3246	4	40	.	.	PUNCT
ejpam-3246	5	1	characterizations	characterization	NOUN
ejpam-3246	5	2	of	of	ADP
ejpam-3246	5	3	uni	uni	ADJ
ejpam-3246	5	4	-	-	ADJ
ejpam-3246	5	5	hesitant	hesitant	ADJ
ejpam-3246	5	6	fuzzy	fuzzy	ADJ
ejpam-3246	5	7	algebras	algebra	NOUN
ejpam-3246	5	8	and	and	CCONJ
ejpam-3246	5	9	uni	uni	ADJ
ejpam-3246	5	10	-	-	ADJ
ejpam-3246	5	11	hesitant	hesitant	ADJ
ejpam-3246	5	12	fuzzy	fuzzy	ADJ
ejpam-3246	5	13	(	(	PUNCT
ejpam-3246	5	14	closed	closed	ADJ
ejpam-3246	5	15	)	)	PUNCT
ejpam-3246	5	16	ideals	ideal	NOUN
ejpam-3246	5	17	are	be	AUX
ejpam-3246	5	18	considered	consider	VERB
ejpam-3246	5	19	,	,	PUNCT
ejpam-3246	5	20	and	and	CCONJ
ejpam-3246	5	21	a	a	DET
ejpam-3246	5	22	new	new	ADJ
ejpam-3246	5	23	uni	uni	ADJ
ejpam-3246	5	24	-	-	ADJ
ejpam-3246	5	25	hesitant	hesitant	ADJ
ejpam-3246	5	26	fuzzy	fuzzy	ADJ
ejpam-3246	5	27	algebra	algebra	NOUN
ejpam-3246	5	28	(	(	PUNCT
ejpam-3246	5	29	resp	resp	NOUN
ejpam-3246	5	30	.	.	PUNCT
ejpam-3246	6	1	uni	uni	ADJ
ejpam-3246	6	2	-	-	ADJ
ejpam-3246	6	3	hesitant	hesitant	ADJ
ejpam-3246	6	4	fuzzy	fuzzy	ADJ
ejpam-3246	6	5	(	(	PUNCT
ejpam-3246	6	6	closed	closed	ADJ
ejpam-3246	6	7	)	)	PUNCT
ejpam-3246	6	8	ideal	ideal	NOUN
ejpam-3246	6	9	)	)	PUNCT
ejpam-3246	6	10	from	from	ADP
ejpam-3246	6	11	old	old	ADJ
ejpam-3246	6	12	one	one	NUM
ejpam-3246	6	13	is	be	AUX
ejpam-3246	6	14	established	establish	VERB
ejpam-3246	6	15	.	.	PUNCT
ejpam-3246	7	1	relations	relation	NOUN
ejpam-3246	7	2	between	between	ADP
ejpam-3246	7	3	uni	uni	ADJ
ejpam-3246	7	4	-	-	ADJ
ejpam-3246	7	5	hesitant	hesitant	ADJ
ejpam-3246	7	6	fuzzy	fuzzy	ADJ
ejpam-3246	7	7	algebras	algebra	NOUN
ejpam-3246	7	8	and	and	CCONJ
ejpam-3246	7	9	uni	uni	ADJ
ejpam-3246	7	10	-	-	ADJ
ejpam-3246	7	11	hesitant	hesitant	ADJ
ejpam-3246	7	12	fuzzy	fuzzy	ADJ
ejpam-3246	7	13	(	(	PUNCT
ejpam-3246	7	14	closed	closed	ADJ
ejpam-3246	7	15	)	)	PUNCT
ejpam-3246	7	16	ideals	ideal	NOUN
ejpam-3246	7	17	are	be	AUX
ejpam-3246	7	18	discussed	discuss	VERB
ejpam-3246	7	19	,	,	PUNCT
ejpam-3246	7	20	and	and	CCONJ
ejpam-3246	7	21	conditions	condition	NOUN
ejpam-3246	7	22	for	for	ADP
ejpam-3246	7	23	a	a	DET
ejpam-3246	7	24	uni	uni	ADJ
ejpam-3246	7	25	-	-	ADJ
ejpam-3246	7	26	hesitant	hesitant	ADJ
ejpam-3246	7	27	fuzzy	fuzzy	ADJ
ejpam-3246	7	28	ideal	ideal	NOUN
ejpam-3246	7	29	to	to	PART
ejpam-3246	7	30	be	be	AUX
ejpam-3246	7	31	hesitant	hesitant	ADJ
ejpam-3246	7	32	closed	close	VERB
ejpam-3246	7	33	are	be	AUX
ejpam-3246	7	34	provided	provide	VERB
ejpam-3246	7	35	.	.	PUNCT
ejpam-3246	8	1	2010	2010	NUM
ejpam-3246	8	2	mathematics	mathematic	NOUN
ejpam-3246	8	3	subject	subject	NOUN
ejpam-3246	8	4	classifications	classification	NOUN
ejpam-3246	8	5	:	:	PUNCT
ejpam-3246	8	6	06f35	06f35	NUM
ejpam-3246	8	7	,	,	PUNCT
ejpam-3246	8	8	03g25	03g25	NUM
ejpam-3246	8	9	,	,	PUNCT
ejpam-3246	8	10	08a72	08a72	NOUN
ejpam-3246	8	11	key	key	ADJ
ejpam-3246	8	12	words	word	NOUN
ejpam-3246	8	13	and	and	CCONJ
ejpam-3246	8	14	phrases	phrase	NOUN
ejpam-3246	8	15	:	:	PUNCT
ejpam-3246	8	16	uni	uni	ADJ
ejpam-3246	8	17	-	-	ADJ
ejpam-3246	8	18	hesitant	hesitant	ADJ
ejpam-3246	8	19	fuzzy	fuzzy	ADJ
ejpam-3246	8	20	algebra	algebra	NOUN
ejpam-3246	8	21	,	,	PUNCT
ejpam-3246	8	22	(	(	PUNCT
ejpam-3246	8	23	closed	closed	ADJ
ejpam-3246	8	24	)	)	PUNCT
ejpam-3246	8	25	uni	uni	ADJ
ejpam-3246	8	26	-	-	ADJ
ejpam-3246	8	27	hesitant	hesitant	ADJ
ejpam-3246	8	28	fuzzy	fuzzy	ADJ
ejpam-3246	8	29	ideal	ideal	NOUN
ejpam-3246	8	30	1	1	NUM
ejpam-3246	8	31	.	.	PUNCT
ejpam-3246	9	1	introduction	introduction	NOUN
ejpam-3246	9	2	the	the	DET
ejpam-3246	9	3	hesitant	hesitant	ADJ
ejpam-3246	9	4	fuzzy	fuzzy	ADJ
ejpam-3246	9	5	set	set	NOUN
ejpam-3246	9	6	which	which	PRON
ejpam-3246	9	7	is	be	AUX
ejpam-3246	9	8	introduced	introduce	VERB
ejpam-3246	9	9	by	by	ADP
ejpam-3246	9	10	torra	torra	NOUN
ejpam-3246	9	11	[	[	X
ejpam-3246	9	12	14	14	NUM
ejpam-3246	9	13	]	]	PUNCT
ejpam-3246	9	14	is	be	AUX
ejpam-3246	9	15	a	a	DET
ejpam-3246	9	16	useful	useful	ADJ
ejpam-3246	9	17	generalization	generalization	NOUN
ejpam-3246	9	18	of	of	ADP
ejpam-3246	9	19	the	the	DET
ejpam-3246	9	20	fuzzy	fuzzy	ADJ
ejpam-3246	9	21	set	set	NOUN
ejpam-3246	9	22	that	that	PRON
ejpam-3246	9	23	is	be	AUX
ejpam-3246	9	24	designed	design	VERB
ejpam-3246	9	25	for	for	ADP
ejpam-3246	9	26	situations	situation	NOUN
ejpam-3246	9	27	in	in	ADP
ejpam-3246	9	28	which	which	PRON
ejpam-3246	9	29	it	it	PRON
ejpam-3246	9	30	is	be	AUX
ejpam-3246	9	31	difficult	difficult	ADJ
ejpam-3246	9	32	to	to	PART
ejpam-3246	9	33	determine	determine	VERB
ejpam-3246	9	34	the	the	DET
ejpam-3246	9	35	membership	membership	NOUN
ejpam-3246	9	36	of	of	ADP
ejpam-3246	9	37	an	an	DET
ejpam-3246	9	38	element	element	NOUN
ejpam-3246	9	39	to	to	ADP
ejpam-3246	9	40	a	a	DET
ejpam-3246	9	41	set	set	NOUN
ejpam-3246	9	42	owing	owe	VERB
ejpam-3246	9	43	to	to	ADP
ejpam-3246	9	44	ambiguity	ambiguity	NOUN
ejpam-3246	9	45	between	between	ADP
ejpam-3246	9	46	a	a	DET
ejpam-3246	9	47	few	few	ADJ
ejpam-3246	9	48	different	different	ADJ
ejpam-3246	9	49	values	value	NOUN
ejpam-3246	9	50	.	.	PUNCT
ejpam-3246	10	1	the	the	DET
ejpam-3246	10	2	hesitant	hesitant	ADJ
ejpam-3246	10	3	fuzzy	fuzzy	ADJ
ejpam-3246	10	4	set	set	NOUN
ejpam-3246	10	5	permits	permit	VERB
ejpam-3246	10	6	the	the	DET
ejpam-3246	10	7	membership	membership	NOUN
ejpam-3246	10	8	degree	degree	NOUN
ejpam-3246	10	9	of	of	ADP
ejpam-3246	10	10	an	an	DET
ejpam-3246	10	11	element	element	NOUN
ejpam-3246	10	12	to	to	ADP
ejpam-3246	10	13	a	a	DET
ejpam-3246	10	14	set	set	NOUN
ejpam-3246	10	15	to	to	PART
ejpam-3246	10	16	be	be	AUX
ejpam-3246	10	17	represented	represent	VERB
ejpam-3246	10	18	by	by	ADP
ejpam-3246	10	19	a	a	DET
ejpam-3246	10	20	set	set	NOUN
ejpam-3246	10	21	of	of	ADP
ejpam-3246	10	22	possible	possible	ADJ
ejpam-3246	10	23	values	value	NOUN
ejpam-3246	10	24	between	between	ADP
ejpam-3246	10	25	0	0	NUM
ejpam-3246	10	26	and	and	CCONJ
ejpam-3246	10	27	1	1	NUM
ejpam-3246	10	28	(	(	PUNCT
ejpam-3246	10	29	see	see	VERB
ejpam-3246	10	30	[	[	X
ejpam-3246	10	31	14	14	NUM
ejpam-3246	10	32	]	]	PUNCT
ejpam-3246	10	33	and	and	CCONJ
ejpam-3246	10	34	[	[	X
ejpam-3246	10	35	15	15	NUM
ejpam-3246	10	36	]	]	NUM
ejpam-3246	10	37	)	)	PUNCT
ejpam-3246	10	38	.	.	PUNCT
ejpam-3246	11	1	the	the	DET
ejpam-3246	11	2	hesitant	hesitant	ADJ
ejpam-3246	11	3	fuzzy	fuzzy	ADJ
ejpam-3246	11	4	set	set	NOUN
ejpam-3246	11	5	therefore	therefore	ADV
ejpam-3246	11	6	provides	provide	VERB
ejpam-3246	11	7	a	a	DET
ejpam-3246	11	8	more	more	ADV
ejpam-3246	11	9	accurate	accurate	ADJ
ejpam-3246	11	10	representation	representation	NOUN
ejpam-3246	11	11	of	of	ADP
ejpam-3246	11	12	people	people	NOUN
ejpam-3246	11	13	’s	’s	PART
ejpam-3246	11	14	hesitancy	hesitancy	NOUN
ejpam-3246	11	15	in	in	ADP
ejpam-3246	11	16	stating	state	VERB
ejpam-3246	11	17	their	their	PRON
ejpam-3246	11	18	preferences	preference	NOUN
ejpam-3246	11	19	over	over	ADP
ejpam-3246	11	20	objects	object	NOUN
ejpam-3246	11	21	than	than	ADP
ejpam-3246	11	22	the	the	DET
ejpam-3246	11	23	fuzzy	fuzzy	ADJ
ejpam-3246	11	24	set	set	NOUN
ejpam-3246	11	25	or	or	CCONJ
ejpam-3246	11	26	its	its	PRON
ejpam-3246	11	27	classical	classical	ADJ
ejpam-3246	11	28	extensions	extension	NOUN
ejpam-3246	11	29	.	.	PUNCT
ejpam-3246	12	1	hesitant	hesitant	ADJ
ejpam-3246	12	2	fuzzy	fuzzy	ADJ
ejpam-3246	12	3	set	set	NOUN
ejpam-3246	12	4	theory	theory	NOUN
ejpam-3246	12	5	has	have	AUX
ejpam-3246	12	6	been	be	AUX
ejpam-3246	12	7	applied	apply	VERB
ejpam-3246	12	8	to	to	ADP
ejpam-3246	12	9	several	several	ADJ
ejpam-3246	12	10	practical	practical	ADJ
ejpam-3246	12	11	problems	problem	NOUN
ejpam-3246	12	12	,	,	PUNCT
ejpam-3246	12	13	primarily	primarily	ADV
ejpam-3246	12	14	in	in	ADP
ejpam-3246	12	15	the	the	DET
ejpam-3246	12	16	area	area	NOUN
ejpam-3246	12	17	of	of	ADP
ejpam-3246	12	18	decision	decision	NOUN
ejpam-3246	12	19	making	making	NOUN
ejpam-3246	12	20	(	(	PUNCT
ejpam-3246	12	21	see	see	VERB
ejpam-3246	12	22	[	[	X
ejpam-3246	12	23	13	13	NUM
ejpam-3246	12	24	]	]	PUNCT
ejpam-3246	12	25	,	,	PUNCT
ejpam-3246	13	1	[	[	X
ejpam-3246	13	2	15	15	NUM
ejpam-3246	13	3	]	]	X
ejpam-3246	14	1	[	[	X
ejpam-3246	14	2	16	16	NUM
ejpam-3246	14	3	]	]	PUNCT
ejpam-3246	14	4	,	,	PUNCT
ejpam-3246	14	5	[	[	X
ejpam-3246	14	6	17	17	NUM
ejpam-3246	14	7	]	]	PUNCT
ejpam-3246	14	8	,	,	PUNCT
ejpam-3246	14	9	[	[	X
ejpam-3246	14	10	18	18	NUM
ejpam-3246	14	11	]	]	PUNCT
ejpam-3246	14	12	,	,	PUNCT
ejpam-3246	15	1	[	[	X
ejpam-3246	15	2	19	19	NUM
ejpam-3246	15	3	]	]	PUNCT
ejpam-3246	15	4	,	,	PUNCT
ejpam-3246	15	5	[	[	X
ejpam-3246	15	6	20	20	NUM
ejpam-3246	15	7	]	]	NUM
ejpam-3246	15	8	)	)	PUNCT
ejpam-3246	15	9	.	.	PUNCT
ejpam-3246	16	1	furthermore	furthermore	ADV
ejpam-3246	16	2	,	,	PUNCT
ejpam-3246	16	3	jun	jun	PROPN
ejpam-3246	16	4	et	et	PROPN
ejpam-3246	16	5	al	al	PROPN
ejpam-3246	16	6	.	.	PROPN
ejpam-3246	16	7	applied	apply	VERB
ejpam-3246	16	8	the	the	DET
ejpam-3246	16	9	notion	notion	NOUN
ejpam-3246	16	10	of	of	ADP
ejpam-3246	16	11	hesitant	hesitant	ADJ
ejpam-3246	16	12	fuzzy	fuzzy	ADJ
ejpam-3246	16	13	sets	set	NOUN
ejpam-3246	16	14	to	to	PART
ejpam-3246	16	15	bck	bck	VERB
ejpam-3246	16	16	/	/	SYM
ejpam-3246	16	17	bci	bci	NOUN
ejpam-3246	16	18	-	-	PUNCT
ejpam-3246	16	19	algebras	algebra	NOUN
ejpam-3246	16	20	,	,	PUNCT
ejpam-3246	16	21	mtl	mtl	PROPN
ejpam-3246	16	22	-	-	PUNCT
ejpam-3246	16	23	algebras	algebras	PROPN
ejpam-3246	16	24	,	,	PUNCT
ejpam-3246	16	25	eq	eq	NOUN
ejpam-3246	16	26	-	-	PUNCT
ejpam-3246	16	27	algebras	algebras	PROPN
ejpam-3246	16	28	and	and	CCONJ
ejpam-3246	16	29	semigroups	semigroup	NOUN
ejpam-3246	16	30	(	(	PUNCT
ejpam-3246	16	31	see	see	VERB
ejpam-3246	16	32	[	[	X
ejpam-3246	16	33	2	2	NUM
ejpam-3246	16	34	]	]	PUNCT
ejpam-3246	16	35	,	,	PUNCT
ejpam-3246	16	36	[	[	X
ejpam-3246	16	37	3	3	NUM
ejpam-3246	16	38	]	]	PUNCT
ejpam-3246	16	39	,	,	PUNCT
ejpam-3246	16	40	[	[	X
ejpam-3246	16	41	4	4	X
ejpam-3246	16	42	]	]	PUNCT
ejpam-3246	16	43	and	and	CCONJ
ejpam-3246	16	44	[	[	X
ejpam-3246	16	45	5	5	NUM
ejpam-3246	16	46	]	]	NUM
ejpam-3246	16	47	)	)	PUNCT
ejpam-3246	16	48	.	.	PUNCT
ejpam-3246	17	1	recently	recently	ADV
ejpam-3246	17	2	,	,	PUNCT
ejpam-3246	17	3	muhiuddin	muhiuddin	VERB
ejpam-3246	17	4	et	et	PROPN
ejpam-3246	17	5	al	al	PROPN
ejpam-3246	17	6	.	.	PROPN
ejpam-3246	17	7	applied	apply	VERB
ejpam-3246	17	8	the	the	DET
ejpam-3246	17	9	notion	notion	NOUN
ejpam-3246	17	10	of	of	ADP
ejpam-3246	17	11	hesitant	hesitant	ADJ
ejpam-3246	17	12	fuzzy	fuzzy	ADJ
ejpam-3246	17	13	sets	set	NOUN
ejpam-3246	17	14	to	to	ADP
ejpam-3246	17	15	residuated	residuate	VERB
ejpam-3246	17	16	lattices	lattice	NOUN
ejpam-3246	17	17	,	,	PUNCT
ejpam-3246	17	18	bck	bck	PROPN
ejpam-3246	17	19	/	/	SYM
ejpam-3246	17	20	bci	bci	NOUN
ejpam-3246	17	21	-	-	PUNCT
ejpam-3246	17	22	algebras	algebra	NOUN
ejpam-3246	17	23	and	and	CCONJ
ejpam-3246	17	24	lattice	lattice	PROPN
ejpam-3246	17	25	implication	implication	NOUN
ejpam-3246	17	26	algebras	algebra	NOUN
ejpam-3246	17	27	(	(	PUNCT
ejpam-3246	17	28	see	see	VERB
ejpam-3246	17	29	[	[	X
ejpam-3246	17	30	8	8	NUM
ejpam-3246	17	31	]	]	PUNCT
ejpam-3246	17	32	,	,	PUNCT
ejpam-3246	17	33	[	[	X
ejpam-3246	17	34	9	9	NUM
ejpam-3246	17	35	]	]	PUNCT
ejpam-3246	17	36	,	,	PUNCT
ejpam-3246	17	37	∗corresponding	∗corresponde	VERB
ejpam-3246	17	38	author	author	NOUN
ejpam-3246	17	39	.	.	PUNCT
ejpam-3246	18	1	email	email	NOUN
ejpam-3246	18	2	addresses	address	NOUN
ejpam-3246	18	3	:	:	PUNCT
ejpam-3246	18	4	chishtygm@gmail.com	chishtygm@gmail.com	X
ejpam-3246	18	5	(	(	PUNCT
ejpam-3246	18	6	g.	g.	PROPN
ejpam-3246	18	7	muhiuddin	muhiuddin	PROPN
ejpam-3246	18	8	)	)	PUNCT
ejpam-3246	18	9	,	,	PUNCT
ejpam-3246	18	10	saldhafeeri@yahoo.com	saldhafeeri@yahoo.com	X
ejpam-3246	19	1	(	(	PUNCT
ejpam-3246	19	2	shuaa	shuaa	ADV
ejpam-3246	19	3	aldhafeeri	aldhafeeri	PROPN
ejpam-3246	19	4	)	)	PUNCT
ejpam-3246	19	5	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3246	20	1	417	417	NUM
ejpam-3246	20	2	c	c	X
ejpam-3246	20	3	©	©	PROPN
ejpam-3246	20	4	2018	2018	NUM
ejpam-3246	20	5	ejpam	ejpam	VERB
ejpam-3246	20	6	all	all	DET
ejpam-3246	20	7	rights	right	NOUN
ejpam-3246	20	8	reserved	reserve	VERB
ejpam-3246	20	9	.	.	PUNCT
ejpam-3246	21	1	g.	g.	PROPN
ejpam-3246	21	2	muhiuddin	muhiuddin	PROPN
ejpam-3246	21	3	,	,	PUNCT
ejpam-3246	21	4	s.	s.	PROPN
ejpam-3246	21	5	aldhafeeri	aldhafeeri	PROPN
ejpam-3246	21	6	/	/	SYM
ejpam-3246	21	7	eur	eur	PROPN
ejpam-3246	21	8	.	.	PUNCT
ejpam-3246	22	1	j.	j.	PROPN
ejpam-3246	22	2	pure	pure	PROPN
ejpam-3246	22	3	appl	appl	PROPN
ejpam-3246	22	4	.	.	PROPN
ejpam-3246	22	5	math	math	PROPN
ejpam-3246	22	6	,	,	PUNCT
ejpam-3246	22	7	11	11	NUM
ejpam-3246	22	8	(	(	PUNCT
ejpam-3246	22	9	2	2	NUM
ejpam-3246	22	10	)	)	PUNCT
ejpam-3246	22	11	(	(	PUNCT
ejpam-3246	22	12	2018	2018	NUM
ejpam-3246	22	13	)	)	PUNCT
ejpam-3246	22	14	,	,	PUNCT
ejpam-3246	22	15	417	417	NUM
ejpam-3246	22	16	-	-	SYM
ejpam-3246	22	17	430	430	NUM
ejpam-3246	22	18	418	418	NUM
ejpam-3246	23	1	[	[	X
ejpam-3246	23	2	10	10	NUM
ejpam-3246	23	3	]	]	PUNCT
ejpam-3246	23	4	,	,	PUNCT
ejpam-3246	24	1	[	[	X
ejpam-3246	24	2	11	11	NUM
ejpam-3246	24	3	]	]	PUNCT
ejpam-3246	24	4	and	and	CCONJ
ejpam-3246	24	5	[	[	X
ejpam-3246	24	6	12	12	NUM
ejpam-3246	24	7	]	]	PUNCT
ejpam-3246	24	8	)	)	PUNCT
ejpam-3246	24	9	.	.	PUNCT
ejpam-3246	25	1	in	in	ADP
ejpam-3246	25	2	this	this	DET
ejpam-3246	25	3	paper	paper	NOUN
ejpam-3246	25	4	,	,	PUNCT
ejpam-3246	25	5	we	we	PRON
ejpam-3246	25	6	introduce	introduce	VERB
ejpam-3246	25	7	the	the	DET
ejpam-3246	25	8	notions	notion	NOUN
ejpam-3246	25	9	of	of	ADP
ejpam-3246	25	10	uni	uni	ADJ
ejpam-3246	25	11	-	-	ADJ
ejpam-3246	25	12	hesitant	hesitant	ADJ
ejpam-3246	25	13	fuzzy	fuzzy	ADJ
ejpam-3246	25	14	algebras	algebra	NOUN
ejpam-3246	25	15	and	and	CCONJ
ejpam-3246	25	16	uni	uni	ADJ
ejpam-3246	25	17	-	-	ADJ
ejpam-3246	25	18	hesitant	hesitant	ADJ
ejpam-3246	25	19	fuzzy	fuzzy	ADJ
ejpam-3246	25	20	(	(	PUNCT
ejpam-3246	25	21	closed	closed	ADJ
ejpam-3246	25	22	)	)	PUNCT
ejpam-3246	25	23	ideals	ideal	NOUN
ejpam-3246	25	24	in	in	ADP
ejpam-3246	25	25	bck	bck	PROPN
ejpam-3246	25	26	/	/	SYM
ejpam-3246	25	27	bci	bci	NOUN
ejpam-3246	25	28	-	-	PUNCT
ejpam-3246	25	29	algebras	algebras	X
ejpam-3246	25	30	,	,	PUNCT
ejpam-3246	25	31	and	and	CCONJ
ejpam-3246	25	32	investigate	investigate	VERB
ejpam-3246	25	33	several	several	ADJ
ejpam-3246	25	34	related	related	ADJ
ejpam-3246	25	35	properties	property	NOUN
ejpam-3246	25	36	.	.	PUNCT
ejpam-3246	26	1	we	we	PRON
ejpam-3246	26	2	consider	consider	VERB
ejpam-3246	26	3	characterizations	characterization	NOUN
ejpam-3246	26	4	of	of	ADP
ejpam-3246	26	5	uni	uni	ADJ
ejpam-3246	26	6	-	-	ADJ
ejpam-3246	26	7	hesitant	hesitant	ADJ
ejpam-3246	26	8	fuzzy	fuzzy	ADJ
ejpam-3246	26	9	algebras	algebra	NOUN
ejpam-3246	26	10	and	and	CCONJ
ejpam-3246	26	11	uni	uni	ADJ
ejpam-3246	26	12	-	-	ADJ
ejpam-3246	26	13	hesitant	hesitant	ADJ
ejpam-3246	26	14	fuzzy	fuzzy	ADJ
ejpam-3246	26	15	(	(	PUNCT
ejpam-3246	26	16	closed	closed	ADJ
ejpam-3246	26	17	)	)	PUNCT
ejpam-3246	26	18	ideals	ideal	NOUN
ejpam-3246	26	19	.	.	PUNCT
ejpam-3246	27	1	given	give	VERB
ejpam-3246	27	2	a	a	DET
ejpam-3246	27	3	unihesitant	unihesitant	ADJ
ejpam-3246	27	4	fuzzy	fuzzy	ADJ
ejpam-3246	27	5	algebra	algebra	NOUN
ejpam-3246	27	6	(	(	PUNCT
ejpam-3246	27	7	resp	resp	NOUN
ejpam-3246	27	8	.	.	PUNCT
ejpam-3246	28	1	uni	uni	ADJ
ejpam-3246	28	2	-	-	ADJ
ejpam-3246	28	3	hesitant	hesitant	ADJ
ejpam-3246	28	4	fuzzy	fuzzy	ADJ
ejpam-3246	28	5	(	(	PUNCT
ejpam-3246	28	6	closed	closed	ADJ
ejpam-3246	28	7	)	)	PUNCT
ejpam-3246	28	8	ideal	ideal	NOUN
ejpam-3246	28	9	)	)	PUNCT
ejpam-3246	28	10	,	,	PUNCT
ejpam-3246	28	11	we	we	PRON
ejpam-3246	28	12	make	make	VERB
ejpam-3246	28	13	a	a	DET
ejpam-3246	28	14	new	new	ADJ
ejpam-3246	28	15	uni	uni	ADJ
ejpam-3246	28	16	-	-	ADJ
ejpam-3246	28	17	hesitant	hesitant	ADJ
ejpam-3246	28	18	fuzzy	fuzzy	ADJ
ejpam-3246	28	19	algebra	algebra	NOUN
ejpam-3246	28	20	(	(	PUNCT
ejpam-3246	28	21	resp	resp	NOUN
ejpam-3246	28	22	.	.	PUNCT
ejpam-3246	29	1	uni	uni	ADJ
ejpam-3246	29	2	-	-	ADJ
ejpam-3246	29	3	hesitant	hesitant	ADJ
ejpam-3246	29	4	fuzzy	fuzzy	ADJ
ejpam-3246	29	5	(	(	PUNCT
ejpam-3246	29	6	closed	closed	ADJ
ejpam-3246	29	7	)	)	PUNCT
ejpam-3246	29	8	ideal	ideal	NOUN
ejpam-3246	29	9	)	)	PUNCT
ejpam-3246	29	10	.	.	PUNCT
ejpam-3246	30	1	we	we	PRON
ejpam-3246	30	2	investigate	investigate	VERB
ejpam-3246	30	3	relations	relation	NOUN
ejpam-3246	30	4	between	between	ADP
ejpam-3246	30	5	uni	uni	ADJ
ejpam-3246	30	6	-	-	ADJ
ejpam-3246	30	7	hesitant	hesitant	ADJ
ejpam-3246	30	8	fuzzy	fuzzy	ADJ
ejpam-3246	30	9	algebras	algebra	NOUN
ejpam-3246	30	10	and	and	CCONJ
ejpam-3246	30	11	uni	uni	ADJ
ejpam-3246	30	12	-	-	ADJ
ejpam-3246	30	13	hesitant	hesitant	ADJ
ejpam-3246	30	14	fuzzy	fuzzy	ADJ
ejpam-3246	30	15	(	(	PUNCT
ejpam-3246	30	16	closed	closed	ADJ
ejpam-3246	30	17	)	)	PUNCT
ejpam-3246	30	18	ideals	ideal	NOUN
ejpam-3246	30	19	.	.	PUNCT
ejpam-3246	31	1	we	we	PRON
ejpam-3246	31	2	provide	provide	VERB
ejpam-3246	31	3	conditions	condition	NOUN
ejpam-3246	31	4	for	for	ADP
ejpam-3246	31	5	a	a	DET
ejpam-3246	31	6	uni	uni	ADJ
ejpam-3246	31	7	-	-	ADJ
ejpam-3246	31	8	hesitant	hesitant	ADJ
ejpam-3246	31	9	fuzzy	fuzzy	ADJ
ejpam-3246	31	10	ideal	ideal	NOUN
ejpam-3246	31	11	to	to	PART
ejpam-3246	31	12	be	be	AUX
ejpam-3246	31	13	hesitant	hesitant	ADJ
ejpam-3246	31	14	closed	closed	ADJ
ejpam-3246	31	15	.	.	PUNCT
ejpam-3246	32	1	2	2	X
ejpam-3246	32	2	.	.	NUM
ejpam-3246	32	3	preliminaries	preliminary	NOUN
ejpam-3246	32	4	a	a	DET
ejpam-3246	32	5	bck	bck	PROPN
ejpam-3246	32	6	/	/	SYM
ejpam-3246	32	7	bci	bci	NOUN
ejpam-3246	32	8	-	-	NOUN
ejpam-3246	32	9	algebra	algebra	NOUN
ejpam-3246	32	10	is	be	AUX
ejpam-3246	32	11	an	an	DET
ejpam-3246	32	12	important	important	ADJ
ejpam-3246	32	13	class	class	NOUN
ejpam-3246	32	14	of	of	ADP
ejpam-3246	32	15	logical	logical	ADJ
ejpam-3246	32	16	algebras	algebra	NOUN
ejpam-3246	32	17	introduced	introduce	VERB
ejpam-3246	32	18	by	by	ADP
ejpam-3246	32	19	k.	k.	PROPN
ejpam-3246	32	20	iséki	iséki	PROPN
ejpam-3246	32	21	and	and	CCONJ
ejpam-3246	32	22	was	be	AUX
ejpam-3246	32	23	extensively	extensively	ADV
ejpam-3246	32	24	investigated	investigate	VERB
ejpam-3246	32	25	by	by	ADP
ejpam-3246	32	26	several	several	ADJ
ejpam-3246	32	27	researchers	researcher	NOUN
ejpam-3246	32	28	.	.	PUNCT
ejpam-3246	33	1	an	an	DET
ejpam-3246	33	2	algebra	algebra	NOUN
ejpam-3246	33	3	(	(	PUNCT
ejpam-3246	33	4	x	x	NOUN
ejpam-3246	33	5	;	;	PUNCT
ejpam-3246	33	6	∗	∗	NOUN
ejpam-3246	33	7	,	,	PUNCT
ejpam-3246	33	8	0	0	NUM
ejpam-3246	33	9	)	)	PUNCT
ejpam-3246	33	10	of	of	ADP
ejpam-3246	33	11	type	type	NOUN
ejpam-3246	33	12	(	(	PUNCT
ejpam-3246	33	13	2	2	NUM
ejpam-3246	33	14	,	,	PUNCT
ejpam-3246	33	15	0	0	NUM
ejpam-3246	33	16	)	)	PUNCT
ejpam-3246	33	17	is	be	AUX
ejpam-3246	33	18	called	call	VERB
ejpam-3246	33	19	a	a	DET
ejpam-3246	33	20	bci	bci	NOUN
ejpam-3246	33	21	-	-	NOUN
ejpam-3246	33	22	algebra	algebra	NOUN
ejpam-3246	33	23	if	if	SCONJ
ejpam-3246	33	24	it	it	PRON
ejpam-3246	33	25	satisfies	satisfy	VERB
ejpam-3246	33	26	the	the	DET
ejpam-3246	33	27	following	follow	VERB
ejpam-3246	33	28	conditions	condition	NOUN
ejpam-3246	33	29	:	:	PUNCT
ejpam-3246	33	30	(	(	PUNCT
ejpam-3246	33	31	i	i	NOUN
ejpam-3246	33	32	)	)	PUNCT
ejpam-3246	33	33	(	(	PUNCT
ejpam-3246	33	34	∀x	∀x	X
ejpam-3246	33	35	,	,	PUNCT
ejpam-3246	33	36	y	y	PROPN
ejpam-3246	33	37	,	,	PUNCT
ejpam-3246	33	38	z	z	NOUN
ejpam-3246	33	39	∈	∈	PROPN
ejpam-3246	33	40	x	x	X
ejpam-3246	33	41	)	)	PUNCT
ejpam-3246	33	42	(	(	PUNCT
ejpam-3246	33	43	(	(	PUNCT
ejpam-3246	33	44	(	(	PUNCT
ejpam-3246	33	45	x	x	SYM
ejpam-3246	33	46	∗	∗	PROPN
ejpam-3246	33	47	y	y	NOUN
ejpam-3246	33	48	)	)	PUNCT
ejpam-3246	33	49	∗	∗	NOUN
ejpam-3246	33	50	(	(	PUNCT
ejpam-3246	33	51	x	x	X
ejpam-3246	33	52	∗	∗	PROPN
ejpam-3246	33	53	z	z	NOUN
ejpam-3246	33	54	)	)	PUNCT
ejpam-3246	33	55	)	)	PUNCT
ejpam-3246	33	56	∗	∗	NOUN
ejpam-3246	33	57	(	(	PUNCT
ejpam-3246	33	58	z	z	NOUN
ejpam-3246	33	59	∗	∗	NOUN
ejpam-3246	33	60	y	y	NOUN
ejpam-3246	33	61	)	)	PUNCT
ejpam-3246	33	62	=	=	SYM
ejpam-3246	33	63	0	0	NUM
ejpam-3246	33	64	)	)	PUNCT
ejpam-3246	33	65	,	,	PUNCT
ejpam-3246	33	66	(	(	PUNCT
ejpam-3246	33	67	ii	ii	NOUN
ejpam-3246	33	68	)	)	PUNCT
ejpam-3246	33	69	(	(	PUNCT
ejpam-3246	33	70	∀x	∀x	X
ejpam-3246	33	71	,	,	PUNCT
ejpam-3246	33	72	y	y	PROPN
ejpam-3246	33	73	∈	∈	PROPN
ejpam-3246	33	74	x	x	X
ejpam-3246	33	75	)	)	PUNCT
ejpam-3246	33	76	(	(	PUNCT
ejpam-3246	33	77	(	(	PUNCT
ejpam-3246	33	78	x	x	SYM
ejpam-3246	33	79	∗	∗	NOUN
ejpam-3246	33	80	(	(	PUNCT
ejpam-3246	33	81	x	x	X
ejpam-3246	33	82	∗	∗	PROPN
ejpam-3246	33	83	y	y	NOUN
ejpam-3246	33	84	)	)	PUNCT
ejpam-3246	33	85	)	)	PUNCT
ejpam-3246	34	1	∗	∗	NOUN
ejpam-3246	34	2	y	y	NOUN
ejpam-3246	35	1	=	=	SYM
ejpam-3246	36	1	0	0	NUM
ejpam-3246	36	2	)	)	PUNCT
ejpam-3246	37	1	,	,	PUNCT
ejpam-3246	37	2	(	(	PUNCT
ejpam-3246	37	3	iii	iii	X
ejpam-3246	37	4	)	)	PUNCT
ejpam-3246	37	5	(	(	PUNCT
ejpam-3246	37	6	∀x	∀x	X
ejpam-3246	37	7	∈	∈	PROPN
ejpam-3246	37	8	x	x	NOUN
ejpam-3246	37	9	)	)	PUNCT
ejpam-3246	37	10	(	(	PUNCT
ejpam-3246	37	11	x	x	X
ejpam-3246	37	12	∗	∗	NOUN
ejpam-3246	37	13	x	x	SYM
ejpam-3246	37	14	=	=	NOUN
ejpam-3246	37	15	0	0	NUM
ejpam-3246	37	16	)	)	PUNCT
ejpam-3246	37	17	,	,	PUNCT
ejpam-3246	37	18	(	(	PUNCT
ejpam-3246	37	19	iv	iv	X
ejpam-3246	37	20	)	)	PUNCT
ejpam-3246	37	21	(	(	PUNCT
ejpam-3246	37	22	∀x	∀x	X
ejpam-3246	37	23	,	,	PUNCT
ejpam-3246	37	24	y	y	PROPN
ejpam-3246	37	25	∈	∈	PROPN
ejpam-3246	37	26	x	x	X
ejpam-3246	37	27	)	)	PUNCT
ejpam-3246	37	28	(	(	PUNCT
ejpam-3246	37	29	x	x	SYM
ejpam-3246	37	30	∗	∗	NOUN
ejpam-3246	37	31	y	y	NOUN
ejpam-3246	37	32	=	=	SYM
ejpam-3246	37	33	0	0	PROPN
ejpam-3246	37	34	,	,	PUNCT
ejpam-3246	37	35	y	y	PROPN
ejpam-3246	37	36	∗	∗	NOUN
ejpam-3246	37	37	x	x	PUNCT
ejpam-3246	37	38	=	=	SYM
ejpam-3246	37	39	0	0	NUM
ejpam-3246	37	40	⇒	⇒	NOUN
ejpam-3246	37	41	x	x	PUNCT
ejpam-3246	38	1	=	=	SYM
ejpam-3246	38	2	y	y	PROPN
ejpam-3246	38	3	)	)	PUNCT
ejpam-3246	38	4	.	.	PUNCT
ejpam-3246	39	1	if	if	SCONJ
ejpam-3246	39	2	a	a	DET
ejpam-3246	39	3	bci	bci	NOUN
ejpam-3246	39	4	-	-	NOUN
ejpam-3246	39	5	algebra	algebra	NOUN
ejpam-3246	39	6	x	x	PRON
ejpam-3246	39	7	satisfies	satisfy	VERB
ejpam-3246	39	8	the	the	DET
ejpam-3246	39	9	following	follow	VERB
ejpam-3246	39	10	identity	identity	NOUN
ejpam-3246	39	11	:	:	PUNCT
ejpam-3246	39	12	(	(	PUNCT
ejpam-3246	39	13	v	v	NOUN
ejpam-3246	39	14	)	)	PUNCT
ejpam-3246	39	15	(	(	PUNCT
ejpam-3246	39	16	∀x	∀x	X
ejpam-3246	39	17	∈	∈	PROPN
ejpam-3246	39	18	x	x	X
ejpam-3246	39	19	)	)	PUNCT
ejpam-3246	39	20	(	(	PUNCT
ejpam-3246	39	21	0	0	NUM
ejpam-3246	39	22	∗	∗	NOUN
ejpam-3246	39	23	x	x	SYM
ejpam-3246	39	24	=	=	NOUN
ejpam-3246	39	25	0	0	NUM
ejpam-3246	39	26	)	)	PUNCT
ejpam-3246	39	27	,	,	PUNCT
ejpam-3246	39	28	then	then	ADV
ejpam-3246	39	29	x	x	PUNCT
ejpam-3246	39	30	is	be	AUX
ejpam-3246	39	31	called	call	VERB
ejpam-3246	39	32	a	a	DET
ejpam-3246	39	33	bck	bck	NOUN
ejpam-3246	39	34	-	-	PUNCT
ejpam-3246	39	35	algebra	algebra	NOUN
ejpam-3246	39	36	.	.	PUNCT
ejpam-3246	40	1	any	any	DET
ejpam-3246	40	2	bck	bck	PROPN
ejpam-3246	40	3	/	/	SYM
ejpam-3246	40	4	bci	bci	NOUN
ejpam-3246	40	5	-	-	NOUN
ejpam-3246	40	6	algebra	algebra	NOUN
ejpam-3246	40	7	x	x	PRON
ejpam-3246	40	8	satisfies	satisfy	VERB
ejpam-3246	40	9	the	the	DET
ejpam-3246	40	10	following	follow	VERB
ejpam-3246	40	11	axioms	axiom	NOUN
ejpam-3246	40	12	:	:	PUNCT
ejpam-3246	40	13	(	(	PUNCT
ejpam-3246	40	14	a1	a1	NOUN
ejpam-3246	40	15	)	)	PUNCT
ejpam-3246	40	16	(	(	PUNCT
ejpam-3246	40	17	∀x	∀x	X
ejpam-3246	40	18	∈	∈	PROPN
ejpam-3246	40	19	x	x	NOUN
ejpam-3246	40	20	)	)	PUNCT
ejpam-3246	40	21	(	(	PUNCT
ejpam-3246	40	22	x	x	NOUN
ejpam-3246	40	23	∗	∗	NOUN
ejpam-3246	40	24	0	0	NUM
ejpam-3246	41	1	=	=	SYM
ejpam-3246	41	2	x	x	NOUN
ejpam-3246	41	3	)	)	PUNCT
ejpam-3246	41	4	,	,	PUNCT
ejpam-3246	41	5	(	(	PUNCT
ejpam-3246	41	6	a2	a2	PROPN
ejpam-3246	41	7	)	)	PUNCT
ejpam-3246	41	8	(	(	PUNCT
ejpam-3246	41	9	∀x	∀x	X
ejpam-3246	41	10	,	,	PUNCT
ejpam-3246	41	11	y	y	PROPN
ejpam-3246	41	12	,	,	PUNCT
ejpam-3246	41	13	z	z	NOUN
ejpam-3246	41	14	∈	∈	PROPN
ejpam-3246	41	15	x	x	X
ejpam-3246	41	16	)	)	PUNCT
ejpam-3246	41	17	(	(	PUNCT
ejpam-3246	41	18	x	x	X
ejpam-3246	41	19	≤	≤	NOUN
ejpam-3246	41	20	y	y	PROPN
ejpam-3246	41	21	⇒	⇒	NOUN
ejpam-3246	41	22	x	x	PUNCT
ejpam-3246	42	1	∗	∗	NOUN
ejpam-3246	42	2	z	z	NOUN
ejpam-3246	42	3	≤	≤	NOUN
ejpam-3246	42	4	y	y	PROPN
ejpam-3246	42	5	∗	∗	PROPN
ejpam-3246	42	6	z	z	PROPN
ejpam-3246	42	7	,	,	PUNCT
ejpam-3246	42	8	z	z	PROPN
ejpam-3246	42	9	∗	∗	NOUN
ejpam-3246	42	10	y	y	PROPN
ejpam-3246	42	11	≤	≤	PROPN
ejpam-3246	42	12	z	z	NOUN
ejpam-3246	42	13	∗	∗	NOUN
ejpam-3246	42	14	x	x	NOUN
ejpam-3246	42	15	)	)	PUNCT
ejpam-3246	42	16	,	,	PUNCT
ejpam-3246	42	17	(	(	PUNCT
ejpam-3246	42	18	a3	a3	NOUN
ejpam-3246	42	19	)	)	PUNCT
ejpam-3246	42	20	(	(	PUNCT
ejpam-3246	42	21	∀x	∀x	X
ejpam-3246	42	22	,	,	PUNCT
ejpam-3246	42	23	y	y	PROPN
ejpam-3246	42	24	,	,	PUNCT
ejpam-3246	42	25	z	z	NOUN
ejpam-3246	42	26	∈	∈	PROPN
ejpam-3246	42	27	x	x	X
ejpam-3246	42	28	)	)	PUNCT
ejpam-3246	42	29	(	(	PUNCT
ejpam-3246	42	30	(	(	PUNCT
ejpam-3246	42	31	x	x	SYM
ejpam-3246	42	32	∗	∗	PROPN
ejpam-3246	42	33	y	y	NOUN
ejpam-3246	42	34	)	)	PUNCT
ejpam-3246	42	35	∗	∗	NOUN
ejpam-3246	42	36	z	z	NOUN
ejpam-3246	42	37	=	=	SYM
ejpam-3246	42	38	(	(	PUNCT
ejpam-3246	42	39	x	x	X
ejpam-3246	42	40	∗	∗	PROPN
ejpam-3246	42	41	z	z	NOUN
ejpam-3246	42	42	)	)	PUNCT
ejpam-3246	42	43	∗	∗	PROPN
ejpam-3246	42	44	y	y	PROPN
ejpam-3246	42	45	)	)	PUNCT
ejpam-3246	42	46	,	,	PUNCT
ejpam-3246	42	47	(	(	PUNCT
ejpam-3246	42	48	a4	a4	NOUN
ejpam-3246	42	49	)	)	PUNCT
ejpam-3246	42	50	(	(	PUNCT
ejpam-3246	42	51	∀x	∀x	X
ejpam-3246	42	52	,	,	PUNCT
ejpam-3246	42	53	y	y	PROPN
ejpam-3246	42	54	,	,	PUNCT
ejpam-3246	42	55	z	z	NOUN
ejpam-3246	42	56	∈	∈	PROPN
ejpam-3246	42	57	x	x	X
ejpam-3246	42	58	)	)	PUNCT
ejpam-3246	42	59	(	(	PUNCT
ejpam-3246	42	60	(	(	PUNCT
ejpam-3246	42	61	x	x	SYM
ejpam-3246	42	62	∗	∗	PROPN
ejpam-3246	42	63	z	z	NOUN
ejpam-3246	42	64	)	)	PUNCT
ejpam-3246	42	65	∗	∗	NOUN
ejpam-3246	42	66	(	(	PUNCT
ejpam-3246	42	67	y	y	PROPN
ejpam-3246	42	68	∗	∗	PROPN
ejpam-3246	42	69	z	z	NOUN
ejpam-3246	42	70	)	)	PUNCT
ejpam-3246	42	71	≤	≤	NUM
ejpam-3246	42	72	x	x	PUNCT
ejpam-3246	42	73	∗	∗	PROPN
ejpam-3246	42	74	y	y	PROPN
ejpam-3246	42	75	)	)	PUNCT
ejpam-3246	42	76	where	where	SCONJ
ejpam-3246	42	77	x	x	X
ejpam-3246	42	78	≤	≤	ADJ
ejpam-3246	42	79	y	y	NOUN
ejpam-3246	43	1	if	if	SCONJ
ejpam-3246	44	1	and	and	CCONJ
ejpam-3246	44	2	only	only	ADV
ejpam-3246	44	3	if	if	SCONJ
ejpam-3246	44	4	x	x	X
ejpam-3246	44	5	∗	∗	VERB
ejpam-3246	44	6	y	y	NOUN
ejpam-3246	44	7	=	=	SYM
ejpam-3246	44	8	0	0	PROPN
ejpam-3246	44	9	.	.	PUNCT
ejpam-3246	45	1	in	in	ADP
ejpam-3246	45	2	a	a	DET
ejpam-3246	45	3	bci	bci	NOUN
ejpam-3246	45	4	-	-	NOUN
ejpam-3246	45	5	algebra	algebra	NOUN
ejpam-3246	45	6	x	x	NOUN
ejpam-3246	45	7	,	,	PUNCT
ejpam-3246	45	8	the	the	DET
ejpam-3246	45	9	following	follow	VERB
ejpam-3246	45	10	hold	hold	NOUN
ejpam-3246	45	11	:	:	PUNCT
ejpam-3246	45	12	(	(	PUNCT
ejpam-3246	45	13	b1	b1	NOUN
ejpam-3246	45	14	)	)	PUNCT
ejpam-3246	45	15	(	(	PUNCT
ejpam-3246	45	16	∀x	∀x	X
ejpam-3246	45	17	,	,	PUNCT
ejpam-3246	45	18	y	y	PROPN
ejpam-3246	45	19	∈	∈	PROPN
ejpam-3246	45	20	x	x	X
ejpam-3246	45	21	)	)	PUNCT
ejpam-3246	45	22	(	(	PUNCT
ejpam-3246	45	23	x	x	SYM
ejpam-3246	45	24	∗	∗	NOUN
ejpam-3246	45	25	(	(	PUNCT
ejpam-3246	45	26	x	x	X
ejpam-3246	45	27	∗	∗	NOUN
ejpam-3246	45	28	(	(	PUNCT
ejpam-3246	45	29	x	x	X
ejpam-3246	45	30	∗	∗	PROPN
ejpam-3246	45	31	y	y	NOUN
ejpam-3246	45	32	)	)	PUNCT
ejpam-3246	45	33	)	)	PUNCT
ejpam-3246	46	1	=	=	PUNCT
ejpam-3246	47	1	x	x	X
ejpam-3246	47	2	∗	∗	PROPN
ejpam-3246	47	3	y	y	NOUN
ejpam-3246	47	4	)	)	PUNCT
ejpam-3246	47	5	,	,	PUNCT
ejpam-3246	47	6	(	(	PUNCT
ejpam-3246	47	7	b2	b2	NOUN
ejpam-3246	47	8	)	)	PUNCT
ejpam-3246	47	9	(	(	PUNCT
ejpam-3246	47	10	∀x	∀x	X
ejpam-3246	47	11	,	,	PUNCT
ejpam-3246	47	12	y	y	PROPN
ejpam-3246	47	13	∈	∈	PROPN
ejpam-3246	47	14	x	x	X
ejpam-3246	47	15	)	)	PUNCT
ejpam-3246	47	16	(	(	PUNCT
ejpam-3246	47	17	0	0	NUM
ejpam-3246	47	18	∗	∗	NOUN
ejpam-3246	47	19	(	(	PUNCT
ejpam-3246	47	20	x	x	X
ejpam-3246	47	21	∗	∗	PROPN
ejpam-3246	47	22	y	y	NOUN
ejpam-3246	47	23	)	)	PUNCT
ejpam-3246	47	24	=	=	SYM
ejpam-3246	48	1	(	(	PUNCT
ejpam-3246	48	2	0	0	NUM
ejpam-3246	48	3	∗	∗	NOUN
ejpam-3246	48	4	x	x	NOUN
ejpam-3246	48	5	)	)	PUNCT
ejpam-3246	48	6	∗	∗	NOUN
ejpam-3246	48	7	(	(	PUNCT
ejpam-3246	48	8	0	0	NUM
ejpam-3246	48	9	∗	∗	PROPN
ejpam-3246	48	10	y	y	PROPN
ejpam-3246	48	11	)	)	PUNCT
ejpam-3246	48	12	)	)	PUNCT
ejpam-3246	48	13	.	.	PUNCT
ejpam-3246	49	1	g.	g.	PROPN
ejpam-3246	49	2	muhiuddin	muhiuddin	PROPN
ejpam-3246	49	3	,	,	PUNCT
ejpam-3246	49	4	s.	s.	PROPN
ejpam-3246	49	5	aldhafeeri	aldhafeeri	PROPN
ejpam-3246	49	6	/	/	SYM
ejpam-3246	49	7	eur	eur	PROPN
ejpam-3246	49	8	.	.	PUNCT
ejpam-3246	50	1	j.	j.	PROPN
ejpam-3246	50	2	pure	pure	PROPN
ejpam-3246	50	3	appl	appl	PROPN
ejpam-3246	50	4	.	.	PROPN
ejpam-3246	50	5	math	math	PROPN
ejpam-3246	50	6	,	,	PUNCT
ejpam-3246	50	7	11	11	NUM
ejpam-3246	50	8	(	(	PUNCT
ejpam-3246	50	9	2	2	NUM
ejpam-3246	50	10	)	)	PUNCT
ejpam-3246	50	11	(	(	PUNCT
ejpam-3246	50	12	2018	2018	NUM
ejpam-3246	50	13	)	)	PUNCT
ejpam-3246	50	14	,	,	PUNCT
ejpam-3246	50	15	417	417	NUM
ejpam-3246	50	16	-	-	SYM
ejpam-3246	50	17	430	430	NUM
ejpam-3246	50	18	419	419	NUM
ejpam-3246	50	19	a	a	DET
ejpam-3246	50	20	nonempty	nonempty	ADJ
ejpam-3246	50	21	subset	subset	VERB
ejpam-3246	50	22	s	s	NOUN
ejpam-3246	50	23	of	of	ADP
ejpam-3246	50	24	a	a	DET
ejpam-3246	50	25	bck	bck	PROPN
ejpam-3246	50	26	/	/	SYM
ejpam-3246	50	27	bci	bci	NOUN
ejpam-3246	50	28	-	-	NOUN
ejpam-3246	50	29	algebra	algebra	NOUN
ejpam-3246	50	30	x	x	PUNCT
ejpam-3246	50	31	is	be	AUX
ejpam-3246	50	32	called	call	VERB
ejpam-3246	50	33	a	a	DET
ejpam-3246	50	34	subalgebra	subalgebra	NOUN
ejpam-3246	50	35	of	of	ADP
ejpam-3246	50	36	x	x	PRON
ejpam-3246	50	37	if	if	SCONJ
ejpam-3246	50	38	x∗y	x∗y	X
ejpam-3246	50	39	∈	∈	PROPN
ejpam-3246	50	40	s	s	VERB
ejpam-3246	50	41	for	for	ADP
ejpam-3246	50	42	all	all	DET
ejpam-3246	50	43	x	x	NOUN
ejpam-3246	50	44	,	,	PUNCT
ejpam-3246	50	45	y	y	PROPN
ejpam-3246	50	46	∈	∈	PROPN
ejpam-3246	50	47	s.	s.	PROPN
ejpam-3246	50	48	a	a	PRON
ejpam-3246	50	49	subset	subset	VERB
ejpam-3246	50	50	a	a	PRON
ejpam-3246	50	51	of	of	ADP
ejpam-3246	50	52	a	a	DET
ejpam-3246	50	53	bck	bck	PROPN
ejpam-3246	50	54	/	/	SYM
ejpam-3246	50	55	bci	bci	NOUN
ejpam-3246	50	56	-	-	NOUN
ejpam-3246	50	57	algebra	algebra	NOUN
ejpam-3246	50	58	x	x	PUNCT
ejpam-3246	50	59	is	be	AUX
ejpam-3246	50	60	called	call	VERB
ejpam-3246	50	61	an	an	DET
ejpam-3246	50	62	ideal	ideal	NOUN
ejpam-3246	50	63	of	of	ADP
ejpam-3246	50	64	x	x	PRON
ejpam-3246	50	65	if	if	SCONJ
ejpam-3246	50	66	it	it	PRON
ejpam-3246	50	67	satisfies	satisfy	VERB
ejpam-3246	50	68	:	:	PUNCT
ejpam-3246	50	69	0	0	NUM
ejpam-3246	50	70	∈	∈	PROPN
ejpam-3246	50	71	a	a	DET
ejpam-3246	50	72	,	,	PUNCT
ejpam-3246	50	73	(	(	PUNCT
ejpam-3246	50	74	1	1	NUM
ejpam-3246	50	75	)	)	PUNCT
ejpam-3246	50	76	(	(	PUNCT
ejpam-3246	50	77	∀x	∀x	X
ejpam-3246	50	78	∈	∈	PROPN
ejpam-3246	50	79	x	x	NOUN
ejpam-3246	50	80	)	)	PUNCT
ejpam-3246	50	81	(	(	PUNCT
ejpam-3246	50	82	∀y	∀y	PROPN
ejpam-3246	50	83	∈	∈	PROPN
ejpam-3246	50	84	a	a	NOUN
ejpam-3246	50	85	)	)	PUNCT
ejpam-3246	50	86	(	(	PUNCT
ejpam-3246	50	87	x	x	SYM
ejpam-3246	50	88	∗	∗	VERB
ejpam-3246	50	89	y	y	PROPN
ejpam-3246	50	90	∈	∈	PROPN
ejpam-3246	50	91	a	a	DET
ejpam-3246	50	92	⇒	⇒	NOUN
ejpam-3246	50	93	x	x	PUNCT
ejpam-3246	50	94	∈	∈	PROPN
ejpam-3246	50	95	a	a	PRON
ejpam-3246	50	96	)	)	PUNCT
ejpam-3246	50	97	.	.	PUNCT
ejpam-3246	51	1	(	(	PUNCT
ejpam-3246	51	2	2	2	X
ejpam-3246	51	3	)	)	PUNCT
ejpam-3246	51	4	an	an	DET
ejpam-3246	51	5	ideal	ideal	NOUN
ejpam-3246	51	6	a	a	PRON
ejpam-3246	51	7	of	of	ADP
ejpam-3246	51	8	a	a	DET
ejpam-3246	51	9	bci	bci	NOUN
ejpam-3246	51	10	-	-	NOUN
ejpam-3246	51	11	algebra	algebra	NOUN
ejpam-3246	51	12	x	x	PUNCT
ejpam-3246	51	13	is	be	AUX
ejpam-3246	51	14	said	say	VERB
ejpam-3246	51	15	to	to	PART
ejpam-3246	51	16	be	be	AUX
ejpam-3246	51	17	closed	close	VERB
ejpam-3246	51	18	if	if	SCONJ
ejpam-3246	51	19	0	0	NUM
ejpam-3246	51	20	∗	∗	NOUN
ejpam-3246	51	21	x	x	PUNCT
ejpam-3246	51	22	∈	∈	PROPN
ejpam-3246	51	23	a	a	PRON
ejpam-3246	51	24	for	for	ADP
ejpam-3246	51	25	all	all	PRON
ejpam-3246	51	26	x	x	SYM
ejpam-3246	51	27	∈	∈	NOUN
ejpam-3246	51	28	a.	a.	NOUN
ejpam-3246	51	29	we	we	PRON
ejpam-3246	51	30	refer	refer	VERB
ejpam-3246	51	31	the	the	DET
ejpam-3246	51	32	reader	reader	NOUN
ejpam-3246	51	33	to	to	ADP
ejpam-3246	51	34	the	the	DET
ejpam-3246	51	35	books	book	NOUN
ejpam-3246	52	1	[	[	X
ejpam-3246	52	2	1	1	NUM
ejpam-3246	52	3	,	,	PUNCT
ejpam-3246	52	4	6	6	NUM
ejpam-3246	52	5	]	]	PUNCT
ejpam-3246	52	6	for	for	ADP
ejpam-3246	52	7	further	further	ADJ
ejpam-3246	52	8	information	information	NOUN
ejpam-3246	52	9	regarding	regard	VERB
ejpam-3246	52	10	bck	bck	PROPN
ejpam-3246	52	11	/	/	SYM
ejpam-3246	52	12	bci	bci	NOUN
ejpam-3246	52	13	-	-	PUNCT
ejpam-3246	52	14	algebras	algebras	X
ejpam-3246	52	15	.	.	PUNCT
ejpam-3246	53	1	torra	torra	VERB
ejpam-3246	53	2	[	[	X
ejpam-3246	53	3	14	14	NUM
ejpam-3246	53	4	]	]	SYM
ejpam-3246	53	5	defined	define	VERB
ejpam-3246	53	6	hesitant	hesitant	ADJ
ejpam-3246	53	7	fuzzy	fuzzy	ADJ
ejpam-3246	53	8	sets	set	NOUN
ejpam-3246	53	9	in	in	ADP
ejpam-3246	53	10	terms	term	NOUN
ejpam-3246	53	11	of	of	ADP
ejpam-3246	53	12	a	a	DET
ejpam-3246	53	13	function	function	NOUN
ejpam-3246	53	14	that	that	PRON
ejpam-3246	53	15	returns	return	VERB
ejpam-3246	53	16	a	a	DET
ejpam-3246	53	17	set	set	NOUN
ejpam-3246	53	18	of	of	ADP
ejpam-3246	53	19	membership	membership	NOUN
ejpam-3246	53	20	values	value	NOUN
ejpam-3246	53	21	for	for	ADP
ejpam-3246	53	22	each	each	DET
ejpam-3246	53	23	element	element	NOUN
ejpam-3246	53	24	in	in	ADP
ejpam-3246	53	25	the	the	DET
ejpam-3246	53	26	domain	domain	NOUN
ejpam-3246	53	27	.	.	PUNCT
ejpam-3246	54	1	we	we	PRON
ejpam-3246	54	2	display	display	VERB
ejpam-3246	54	3	the	the	DET
ejpam-3246	54	4	basic	basic	ADJ
ejpam-3246	54	5	notions	notion	NOUN
ejpam-3246	54	6	on	on	ADP
ejpam-3246	54	7	hesitant	hesitant	ADJ
ejpam-3246	54	8	fuzzy	fuzzy	ADJ
ejpam-3246	54	9	sets	set	NOUN
ejpam-3246	54	10	.	.	PUNCT
ejpam-3246	55	1	for	for	ADP
ejpam-3246	55	2	more	more	ADJ
ejpam-3246	55	3	details	detail	NOUN
ejpam-3246	55	4	,	,	PUNCT
ejpam-3246	55	5	we	we	PRON
ejpam-3246	55	6	refer	refer	VERB
ejpam-3246	55	7	to	to	ADP
ejpam-3246	55	8	references	reference	NOUN
ejpam-3246	55	9	.	.	PUNCT
ejpam-3246	56	1	let	let	VERB
ejpam-3246	56	2	x	x	PRON
ejpam-3246	56	3	be	be	AUX
ejpam-3246	56	4	a	a	DET
ejpam-3246	56	5	reference	reference	NOUN
ejpam-3246	56	6	set	set	VERB
ejpam-3246	56	7	.	.	PUNCT
ejpam-3246	57	1	then	then	ADV
ejpam-3246	57	2	we	we	PRON
ejpam-3246	57	3	define	define	VERB
ejpam-3246	57	4	a	a	DET
ejpam-3246	57	5	hesitant	hesitant	ADJ
ejpam-3246	57	6	fuzzy	fuzzy	ADJ
ejpam-3246	57	7	set	set	NOUN
ejpam-3246	57	8	on	on	ADP
ejpam-3246	57	9	x	x	PUNCT
ejpam-3246	57	10	in	in	ADP
ejpam-3246	57	11	terms	term	NOUN
ejpam-3246	57	12	of	of	ADP
ejpam-3246	57	13	a	a	DET
ejpam-3246	57	14	function	function	NOUN
ejpam-3246	57	15	g	g	NOUN
ejpam-3246	57	16	that	that	SCONJ
ejpam-3246	57	17	when	when	SCONJ
ejpam-3246	57	18	applied	apply	VERB
ejpam-3246	57	19	to	to	ADP
ejpam-3246	57	20	x	x	PROPN
ejpam-3246	57	21	returns	returns	AUX
ejpam-3246	57	22	a	a	DET
ejpam-3246	57	23	subset	subset	NOUN
ejpam-3246	57	24	of	of	ADP
ejpam-3246	57	25	[	[	X
ejpam-3246	57	26	0	0	NUM
ejpam-3246	57	27	,	,	PUNCT
ejpam-3246	57	28	1	1	NUM
ejpam-3246	57	29	]	]	PUNCT
ejpam-3246	57	30	,	,	PUNCT
ejpam-3246	57	31	and	and	CCONJ
ejpam-3246	57	32	the	the	DET
ejpam-3246	57	33	image	image	NOUN
ejpam-3246	57	34	of	of	ADP
ejpam-3246	57	35	x	x	X
ejpam-3246	57	36	∈	∈	PROPN
ejpam-3246	57	37	x	x	PUNCT
ejpam-3246	57	38	under	under	ADP
ejpam-3246	57	39	g	g	PROPN
ejpam-3246	57	40	is	be	AUX
ejpam-3246	57	41	denoted	denote	VERB
ejpam-3246	57	42	by	by	ADP
ejpam-3246	57	43	xg	xg	PROPN
ejpam-3246	57	44	.	.	PUNCT
ejpam-3246	57	45	for	for	ADP
ejpam-3246	57	46	a	a	DET
ejpam-3246	57	47	hesitant	hesitant	ADJ
ejpam-3246	57	48	fuzzy	fuzzy	NOUN
ejpam-3246	57	49	set	set	VERB
ejpam-3246	57	50	g	g	NOUN
ejpam-3246	57	51	on	on	ADP
ejpam-3246	57	52	x	x	PUNCT
ejpam-3246	57	53	and	and	CCONJ
ejpam-3246	57	54	a	a	DET
ejpam-3246	57	55	subset	subset	ADJ
ejpam-3246	57	56	λ	λ	NOUN
ejpam-3246	57	57	of	of	ADP
ejpam-3246	57	58	[	[	X
ejpam-3246	57	59	0	0	NUM
ejpam-3246	57	60	,	,	PUNCT
ejpam-3246	57	61	1	1	NUM
ejpam-3246	57	62	]	]	PUNCT
ejpam-3246	57	63	,	,	PUNCT
ejpam-3246	57	64	the	the	DET
ejpam-3246	57	65	set	set	NOUN
ejpam-3246	57	66	l(g;λ	l(g;λ	PROPN
ejpam-3246	57	67	)	)	PUNCT
ejpam-3246	57	68	:	:	PUNCT
ejpam-3246	58	1	=	=	SYM
ejpam-3246	58	2	{	{	PUNCT
ejpam-3246	58	3	x	x	SYM
ejpam-3246	58	4	∈	∈	PROPN
ejpam-3246	58	5	x	x	INTJ
ejpam-3246	58	6	|	|	ADV
ejpam-3246	58	7	xg	xg	PROPN
ejpam-3246	58	8	⊆	⊆	NUM
ejpam-3246	58	9	λ	λ	NOUN
ejpam-3246	58	10	}	}	PUNCT
ejpam-3246	58	11	,	,	PUNCT
ejpam-3246	58	12	is	be	AUX
ejpam-3246	58	13	called	call	VERB
ejpam-3246	58	14	the	the	DET
ejpam-3246	58	15	uni	uni	ADJ
ejpam-3246	58	16	-	-	ADJ
ejpam-3246	58	17	hesitant	hesitant	ADJ
ejpam-3246	58	18	level	level	NOUN
ejpam-3246	58	19	set	set	NOUN
ejpam-3246	58	20	of	of	ADP
ejpam-3246	58	21	g.	g.	PROPN
ejpam-3246	58	22	let	let	VERB
ejpam-3246	58	23	g	g	NOUN
ejpam-3246	58	24	and	and	CCONJ
ejpam-3246	58	25	h	h	NOUN
ejpam-3246	58	26	be	be	VERB
ejpam-3246	58	27	two	two	NUM
ejpam-3246	58	28	hesitant	hesitant	ADJ
ejpam-3246	58	29	fuzzy	fuzzy	ADJ
ejpam-3246	58	30	sets	set	NOUN
ejpam-3246	58	31	on	on	ADP
ejpam-3246	58	32	x.	x.	NOUN
ejpam-3246	58	33	the	the	DET
ejpam-3246	58	34	hesitant	hesitant	PROPN
ejpam-3246	58	35	union	union	PROPN
ejpam-3246	58	36	g	g	PROPN
ejpam-3246	58	37	th	th	INTJ
ejpam-3246	58	38	and	and	CCONJ
ejpam-3246	58	39	hesitant	hesitant	ADJ
ejpam-3246	58	40	intersection	intersection	NOUN
ejpam-3246	58	41	g	g	PROPN
ejpam-3246	58	42	u	u	NOUN
ejpam-3246	58	43	h	h	NOUN
ejpam-3246	58	44	of	of	ADP
ejpam-3246	58	45	g	g	PROPN
ejpam-3246	58	46	and	and	CCONJ
ejpam-3246	58	47	h	h	NOUN
ejpam-3246	58	48	are	be	AUX
ejpam-3246	58	49	defined	define	VERB
ejpam-3246	58	50	to	to	PART
ejpam-3246	58	51	be	be	AUX
ejpam-3246	58	52	hesitant	hesitant	ADJ
ejpam-3246	58	53	fuzzy	fuzzy	ADJ
ejpam-3246	58	54	sets	set	NOUN
ejpam-3246	58	55	on	on	ADP
ejpam-3246	58	56	x	x	PUNCT
ejpam-3246	58	57	as	as	SCONJ
ejpam-3246	58	58	follows	follow	VERB
ejpam-3246	58	59	:	:	PUNCT
ejpam-3246	59	1	g	g	PROPN
ejpam-3246	59	2	t	t	PROPN
ejpam-3246	59	3	h	h	NOUN
ejpam-3246	59	4	:	:	PUNCT
ejpam-3246	59	5	x	x	X
ejpam-3246	59	6	→	→	X
ejpam-3246	59	7	p	p	X
ejpam-3246	59	8	(	(	PUNCT
ejpam-3246	59	9	[	[	X
ejpam-3246	59	10	0	0	NUM
ejpam-3246	59	11	,	,	PUNCT
ejpam-3246	59	12	1	1	NUM
ejpam-3246	59	13	]	]	NUM
ejpam-3246	59	14	)	)	PUNCT
ejpam-3246	59	15	,	,	PUNCT
ejpam-3246	59	16	x	x	X
ejpam-3246	59	17	7→	7→	NUM
ejpam-3246	59	18	xg	xg	NOUN
ejpam-3246	59	19	∪	∪	PROPN
ejpam-3246	59	20	xh	xh	PROPN
ejpam-3246	59	21	(	(	PUNCT
ejpam-3246	59	22	3	3	NUM
ejpam-3246	59	23	)	)	PUNCT
ejpam-3246	59	24	and	and	CCONJ
ejpam-3246	59	25	g	g	ADP
ejpam-3246	59	26	u	u	NOUN
ejpam-3246	59	27	h	h	NOUN
ejpam-3246	59	28	:	:	PUNCT
ejpam-3246	59	29	x	x	X
ejpam-3246	59	30	→	→	X
ejpam-3246	59	31	p	p	X
ejpam-3246	59	32	(	(	PUNCT
ejpam-3246	59	33	[	[	X
ejpam-3246	59	34	0	0	NUM
ejpam-3246	59	35	,	,	PUNCT
ejpam-3246	59	36	1	1	NUM
ejpam-3246	59	37	]	]	NUM
ejpam-3246	59	38	)	)	PUNCT
ejpam-3246	59	39	,	,	PUNCT
ejpam-3246	59	40	x	x	X
ejpam-3246	59	41	7→	7→	NUM
ejpam-3246	59	42	xg	xg	NOUN
ejpam-3246	59	43	∩	∩	PROPN
ejpam-3246	59	44	xh	xh	PROPN
ejpam-3246	59	45	,	,	PUNCT
ejpam-3246	59	46	(	(	PUNCT
ejpam-3246	59	47	4	4	X
ejpam-3246	59	48	)	)	PUNCT
ejpam-3246	59	49	respectively	respectively	ADV
ejpam-3246	59	50	.	.	PUNCT
ejpam-3246	60	1	3	3	X
ejpam-3246	60	2	.	.	X
ejpam-3246	60	3	uni	uni	ADJ
ejpam-3246	60	4	-	-	ADJ
ejpam-3246	60	5	hesitant	hesitant	ADJ
ejpam-3246	60	6	fuzzy	fuzzy	ADJ
ejpam-3246	60	7	algebras	algebra	NOUN
ejpam-3246	60	8	in	in	ADP
ejpam-3246	60	9	bck	bck	PROPN
ejpam-3246	60	10	/	/	SYM
ejpam-3246	60	11	bci	bci	NOUN
ejpam-3246	60	12	-	-	PUNCT
ejpam-3246	60	13	algebras	algebra	NOUN
ejpam-3246	60	14	in	in	ADP
ejpam-3246	60	15	what	what	PRON
ejpam-3246	60	16	follows	follow	VERB
ejpam-3246	60	17	,	,	PUNCT
ejpam-3246	60	18	let	let	VERB
ejpam-3246	60	19	x	x	PRON
ejpam-3246	60	20	denote	denote	VERB
ejpam-3246	60	21	a	a	DET
ejpam-3246	60	22	bck	bck	NOUN
ejpam-3246	60	23	/	/	SYM
ejpam-3246	60	24	bci	bci	NOUN
ejpam-3246	60	25	-	-	NOUN
ejpam-3246	60	26	algebra	algebra	NOUN
ejpam-3246	60	27	unless	unless	SCONJ
ejpam-3246	60	28	otherwise	otherwise	ADV
ejpam-3246	60	29	specified	specify	VERB
ejpam-3246	60	30	.	.	PUNCT
ejpam-3246	61	1	definition	definition	NOUN
ejpam-3246	61	2	1	1	NUM
ejpam-3246	61	3	.	.	PUNCT
ejpam-3246	62	1	a	a	DET
ejpam-3246	62	2	hesitant	hesitant	ADJ
ejpam-3246	62	3	fuzzy	fuzzy	NOUN
ejpam-3246	62	4	set	set	VERB
ejpam-3246	62	5	g	g	NOUN
ejpam-3246	62	6	on	on	ADP
ejpam-3246	62	7	x	x	PROPN
ejpam-3246	62	8	is	be	AUX
ejpam-3246	62	9	called	call	VERB
ejpam-3246	62	10	a	a	DET
ejpam-3246	62	11	uni	uni	ADJ
ejpam-3246	62	12	-	-	ADJ
ejpam-3246	62	13	hesitant	hesitant	ADJ
ejpam-3246	62	14	fuzzy	fuzzy	ADJ
ejpam-3246	62	15	algebra	algebra	NOUN
ejpam-3246	62	16	on	on	ADP
ejpam-3246	62	17	x	x	PUNCT
ejpam-3246	63	1	if	if	SCONJ
ejpam-3246	63	2	(	(	PUNCT
ejpam-3246	63	3	x	x	SYM
ejpam-3246	63	4	∗	∗	NOUN
ejpam-3246	63	5	y)g	y)g	NOUN
ejpam-3246	63	6	⊆	⊆	NUM
ejpam-3246	63	7	xg	xg	NOUN
ejpam-3246	63	8	∪	∪	ADP
ejpam-3246	63	9	yg	yg	PROPN
ejpam-3246	63	10	for	for	ADP
ejpam-3246	63	11	all	all	DET
ejpam-3246	63	12	x	x	NOUN
ejpam-3246	63	13	,	,	PUNCT
ejpam-3246	63	14	y	y	PROPN
ejpam-3246	63	15	∈	∈	PROPN
ejpam-3246	63	16	x.	x.	NOUN
ejpam-3246	63	17	example	example	NOUN
ejpam-3246	64	1	1	1	NUM
ejpam-3246	64	2	.	.	PUNCT
ejpam-3246	64	3	(	(	PUNCT
ejpam-3246	64	4	1	1	X
ejpam-3246	64	5	)	)	PUNCT
ejpam-3246	64	6	let	let	VERB
ejpam-3246	64	7	x	x	PUNCT
ejpam-3246	64	8	=	=	PUNCT
ejpam-3246	64	9	{	{	PUNCT
ejpam-3246	64	10	0	0	NUM
ejpam-3246	64	11	,	,	PUNCT
ejpam-3246	64	12	a	a	PRON
ejpam-3246	64	13	,	,	PUNCT
ejpam-3246	64	14	b	b	AUX
ejpam-3246	64	15	}	}	PUNCT
ejpam-3246	64	16	be	be	AUX
ejpam-3246	64	17	a	a	DET
ejpam-3246	64	18	bck	bck	NOUN
ejpam-3246	64	19	-	-	PUNCT
ejpam-3246	64	20	algebra	algebra	NOUN
ejpam-3246	64	21	with	with	ADP
ejpam-3246	64	22	the	the	DET
ejpam-3246	64	23	cayley	cayley	ADJ
ejpam-3246	64	24	table	table	NOUN
ejpam-3246	64	25	which	which	PRON
ejpam-3246	64	26	is	be	AUX
ejpam-3246	64	27	appeared	appear	VERB
ejpam-3246	64	28	in	in	ADP
ejpam-3246	64	29	table	table	NOUN
ejpam-3246	64	30	1	1	NUM
ejpam-3246	64	31	.	.	PUNCT
ejpam-3246	65	1	let	let	VERB
ejpam-3246	65	2	g	g	PRON
ejpam-3246	65	3	be	be	AUX
ejpam-3246	65	4	a	a	DET
ejpam-3246	65	5	hesitant	hesitant	ADJ
ejpam-3246	65	6	fuzzy	fuzzy	ADJ
ejpam-3246	65	7	set	set	NOUN
ejpam-3246	65	8	on	on	ADP
ejpam-3246	65	9	x	x	PUNCT
ejpam-3246	65	10	defined	define	VERB
ejpam-3246	65	11	as	as	SCONJ
ejpam-3246	65	12	follows	follow	VERB
ejpam-3246	65	13	:	:	PUNCT
ejpam-3246	65	14	g	g	NOUN
ejpam-3246	65	15	:	:	PUNCT
ejpam-3246	65	16	x	x	X
ejpam-3246	65	17	→	→	X
ejpam-3246	65	18	p	p	X
ejpam-3246	65	19	(	(	PUNCT
ejpam-3246	65	20	[	[	X
ejpam-3246	65	21	0	0	NUM
ejpam-3246	65	22	,	,	PUNCT
ejpam-3246	65	23	1	1	NUM
ejpam-3246	65	24	]	]	NUM
ejpam-3246	65	25	)	)	PUNCT
ejpam-3246	65	26	,	,	PUNCT
ejpam-3246	65	27	x	x	X
ejpam-3246	65	28	7→	7→	NUM
ejpam-3246	65	29			PUNCT
ejpam-3246	65	30	(	(	PUNCT
ejpam-3246	65	31	0	0	NUM
ejpam-3246	65	32	,	,	PUNCT
ejpam-3246	65	33	3	3	NUM
ejpam-3246	65	34	,	,	PUNCT
ejpam-3246	65	35	0.7	0.7	NUM
ejpam-3246	65	36	)	)	PUNCT
ejpam-3246	65	37	if	if	SCONJ
ejpam-3246	65	38	x	x	PROPN
ejpam-3246	65	39	=	=	SYM
ejpam-3246	65	40	0	0	NUM
ejpam-3246	65	41	,	,	PUNCT
ejpam-3246	65	42	(	(	PUNCT
ejpam-3246	65	43	0.2	0.2	NUM
ejpam-3246	65	44	,	,	PUNCT
ejpam-3246	65	45	0.8	0.8	NUM
ejpam-3246	65	46	]	]	PUNCT
ejpam-3246	65	47	if	if	SCONJ
ejpam-3246	65	48	x	x	X
ejpam-3246	65	49	=	=	SYM
ejpam-3246	65	50	a	a	PRON
ejpam-3246	65	51	,	,	PUNCT
ejpam-3246	65	52	(	(	PUNCT
ejpam-3246	65	53	0	0	NUM
ejpam-3246	65	54	,	,	PUNCT
ejpam-3246	65	55	1	1	NUM
ejpam-3246	65	56	)	)	PUNCT
ejpam-3246	65	57	if	if	SCONJ
ejpam-3246	65	58	x	x	PROPN
ejpam-3246	65	59	=	=	PROPN
ejpam-3246	65	60	b.	b.	PROPN
ejpam-3246	65	61	g.	g.	PROPN
ejpam-3246	65	62	muhiuddin	muhiuddin	PROPN
ejpam-3246	65	63	,	,	PUNCT
ejpam-3246	65	64	s.	s.	PROPN
ejpam-3246	65	65	aldhafeeri	aldhafeeri	PROPN
ejpam-3246	65	66	/	/	SYM
ejpam-3246	65	67	eur	eur	PROPN
ejpam-3246	65	68	.	.	PUNCT
ejpam-3246	66	1	j.	j.	PROPN
ejpam-3246	66	2	pure	pure	PROPN
ejpam-3246	66	3	appl	appl	PROPN
ejpam-3246	66	4	.	.	PROPN
ejpam-3246	66	5	math	math	PROPN
ejpam-3246	66	6	,	,	PUNCT
ejpam-3246	66	7	11	11	NUM
ejpam-3246	66	8	(	(	PUNCT
ejpam-3246	66	9	2	2	NUM
ejpam-3246	66	10	)	)	PUNCT
ejpam-3246	66	11	(	(	PUNCT
ejpam-3246	66	12	2018	2018	NUM
ejpam-3246	66	13	)	)	PUNCT
ejpam-3246	66	14	,	,	PUNCT
ejpam-3246	66	15	417	417	NUM
ejpam-3246	66	16	-	-	SYM
ejpam-3246	66	17	430	430	NUM
ejpam-3246	66	18	420	420	NUM
ejpam-3246	66	19	table	table	NOUN
ejpam-3246	66	20	1	1	NUM
ejpam-3246	66	21	:	:	PUNCT
ejpam-3246	66	22	cayley	cayley	ADJ
ejpam-3246	66	23	table	table	NOUN
ejpam-3246	66	24	for	for	ADP
ejpam-3246	66	25	the	the	DET
ejpam-3246	66	26	∗-operation	∗-operation	NOUN
ejpam-3246	66	27	∗	∗	NOUN
ejpam-3246	66	28	0	0	NUM
ejpam-3246	67	1	a	a	DET
ejpam-3246	67	2	b	b	NOUN
ejpam-3246	67	3	0	0	NUM
ejpam-3246	67	4	0	0	NUM
ejpam-3246	67	5	0	0	NUM
ejpam-3246	67	6	0	0	NUM
ejpam-3246	67	7	a	a	DET
ejpam-3246	67	8	a	a	DET
ejpam-3246	67	9	0	0	NUM
ejpam-3246	67	10	a	a	DET
ejpam-3246	67	11	b	b	PROPN
ejpam-3246	67	12	b	b	PROPN
ejpam-3246	67	13	b	b	PROPN
ejpam-3246	67	14	0	0	NUM
ejpam-3246	68	1	it	it	PRON
ejpam-3246	68	2	is	be	AUX
ejpam-3246	68	3	routine	routine	ADJ
ejpam-3246	68	4	to	to	PART
ejpam-3246	68	5	verify	verify	VERB
ejpam-3246	68	6	that	that	SCONJ
ejpam-3246	68	7	g	g	PROPN
ejpam-3246	68	8	is	be	AUX
ejpam-3246	68	9	a	a	DET
ejpam-3246	68	10	uni	uni	ADJ
ejpam-3246	68	11	-	-	ADJ
ejpam-3246	68	12	hesitant	hesitant	ADJ
ejpam-3246	68	13	fuzzy	fuzzy	ADJ
ejpam-3246	68	14	algebra	algebra	NOUN
ejpam-3246	68	15	on	on	ADP
ejpam-3246	68	16	x.	x.	NOUN
ejpam-3246	68	17	(	(	PUNCT
ejpam-3246	68	18	2	2	X
ejpam-3246	68	19	)	)	PUNCT
ejpam-3246	68	20	let	let	VERB
ejpam-3246	68	21	x	x	PUNCT
ejpam-3246	68	22	=	=	PUNCT
ejpam-3246	68	23	{	{	PUNCT
ejpam-3246	68	24	0	0	NUM
ejpam-3246	68	25	,	,	PUNCT
ejpam-3246	68	26	a	a	DET
ejpam-3246	68	27	,	,	PUNCT
ejpam-3246	68	28	b	b	NOUN
ejpam-3246	68	29	,	,	PUNCT
ejpam-3246	68	30	c	c	AUX
ejpam-3246	68	31	}	}	PUNCT
ejpam-3246	68	32	be	be	AUX
ejpam-3246	68	33	a	a	DET
ejpam-3246	68	34	bck	bck	NOUN
ejpam-3246	68	35	-	-	PUNCT
ejpam-3246	68	36	algebra	algebra	NOUN
ejpam-3246	68	37	with	with	ADP
ejpam-3246	68	38	the	the	DET
ejpam-3246	68	39	cayley	cayley	ADJ
ejpam-3246	68	40	table	table	NOUN
ejpam-3246	68	41	which	which	PRON
ejpam-3246	68	42	is	be	AUX
ejpam-3246	68	43	appeared	appear	VERB
ejpam-3246	68	44	in	in	ADP
ejpam-3246	68	45	table	table	NOUN
ejpam-3246	68	46	2	2	NUM
ejpam-3246	68	47	.	.	PUNCT
ejpam-3246	68	48	table	table	NOUN
ejpam-3246	68	49	2	2	NUM
ejpam-3246	68	50	:	:	PUNCT
ejpam-3246	68	51	cayley	cayley	ADJ
ejpam-3246	68	52	table	table	NOUN
ejpam-3246	68	53	for	for	ADP
ejpam-3246	68	54	the	the	DET
ejpam-3246	68	55	∗-operation	∗-operation	NOUN
ejpam-3246	68	56	∗	∗	NOUN
ejpam-3246	68	57	0	0	NUM
ejpam-3246	69	1	a	a	DET
ejpam-3246	69	2	b	b	NOUN
ejpam-3246	69	3	c	c	NOUN
ejpam-3246	69	4	0	0	NUM
ejpam-3246	69	5	0	0	NUM
ejpam-3246	69	6	0	0	NUM
ejpam-3246	69	7	0	0	NUM
ejpam-3246	69	8	0	0	NUM
ejpam-3246	70	1	a	a	DET
ejpam-3246	70	2	a	a	DET
ejpam-3246	70	3	0	0	NUM
ejpam-3246	70	4	0	0	NUM
ejpam-3246	70	5	a	a	DET
ejpam-3246	70	6	b	b	PROPN
ejpam-3246	70	7	b	b	PROPN
ejpam-3246	70	8	b	b	PROPN
ejpam-3246	70	9	0	0	NUM
ejpam-3246	70	10	b	b	PROPN
ejpam-3246	70	11	c	c	NOUN
ejpam-3246	70	12	c	c	NOUN
ejpam-3246	70	13	c	c	NOUN
ejpam-3246	70	14	c	c	PROPN
ejpam-3246	70	15	0	0	PUNCT
ejpam-3246	70	16	let	let	VERB
ejpam-3246	70	17	h	h	NOUN
ejpam-3246	70	18	be	be	AUX
ejpam-3246	70	19	a	a	DET
ejpam-3246	70	20	hesitant	hesitant	ADJ
ejpam-3246	70	21	fuzzy	fuzzy	ADJ
ejpam-3246	70	22	set	set	NOUN
ejpam-3246	70	23	on	on	ADP
ejpam-3246	70	24	x	x	PUNCT
ejpam-3246	70	25	defined	define	VERB
ejpam-3246	70	26	as	as	SCONJ
ejpam-3246	70	27	follows	follow	VERB
ejpam-3246	70	28	:	:	PUNCT
ejpam-3246	70	29	h	h	NOUN
ejpam-3246	70	30	:	:	PUNCT
ejpam-3246	70	31	x	x	X
ejpam-3246	71	1	→	→	X
ejpam-3246	71	2	p	p	X
ejpam-3246	71	3	(	(	PUNCT
ejpam-3246	71	4	[	[	X
ejpam-3246	71	5	0	0	NUM
ejpam-3246	71	6	,	,	PUNCT
ejpam-3246	71	7	1	1	NUM
ejpam-3246	71	8	]	]	NUM
ejpam-3246	71	9	)	)	PUNCT
ejpam-3246	71	10	,	,	PUNCT
ejpam-3246	71	11	x	x	X
ejpam-3246	71	12	7→	7→	NUM
ejpam-3246	71	13			NUM
ejpam-3246	71	14	(	(	PUNCT
ejpam-3246	71	15	0	0	NUM
ejpam-3246	71	16	,	,	PUNCT
ejpam-3246	71	17	3	3	NUM
ejpam-3246	71	18	,	,	PUNCT
ejpam-3246	71	19	0.4	0.4	NUM
ejpam-3246	71	20	)	)	PUNCT
ejpam-3246	71	21	∪	∪	NOUN
ejpam-3246	71	22	{	{	PUNCT
ejpam-3246	71	23	0.5	0.5	NUM
ejpam-3246	71	24	,	,	PUNCT
ejpam-3246	71	25	0.6	0.6	NUM
ejpam-3246	71	26	}	}	PUNCT
ejpam-3246	71	27	if	if	SCONJ
ejpam-3246	71	28	x	x	PROPN
ejpam-3246	71	29	=	=	SYM
ejpam-3246	71	30	0	0	NUM
ejpam-3246	71	31	,	,	PUNCT
ejpam-3246	71	32	(	(	PUNCT
ejpam-3246	71	33	0.2	0.2	NUM
ejpam-3246	71	34	,	,	PUNCT
ejpam-3246	71	35	0.4	0.4	NUM
ejpam-3246	71	36	]	]	PUNCT
ejpam-3246	71	37	∪	∪	X
ejpam-3246	71	38	{	{	PUNCT
ejpam-3246	71	39	0.5	0.5	NUM
ejpam-3246	71	40	,	,	PUNCT
ejpam-3246	71	41	0.6	0.6	NUM
ejpam-3246	71	42	}	}	PUNCT
ejpam-3246	71	43	if	if	SCONJ
ejpam-3246	71	44	x	x	X
ejpam-3246	71	45	=	=	PUNCT
ejpam-3246	71	46	a	a	X
ejpam-3246	71	47	,	,	PUNCT
ejpam-3246	71	48	(	(	PUNCT
ejpam-3246	71	49	0.2	0.2	NUM
ejpam-3246	71	50	,	,	PUNCT
ejpam-3246	71	51	0.7	0.7	NUM
ejpam-3246	71	52	]	]	PUNCT
ejpam-3246	71	53	if	if	SCONJ
ejpam-3246	71	54	x	x	PROPN
ejpam-3246	71	55	=	=	SYM
ejpam-3246	71	56	b	b	PROPN
ejpam-3246	71	57	,	,	PUNCT
ejpam-3246	71	58	(	(	PUNCT
ejpam-3246	71	59	0.2	0.2	NUM
ejpam-3246	71	60	,	,	PUNCT
ejpam-3246	71	61	0.5	0.5	NUM
ejpam-3246	71	62	)	)	PUNCT
ejpam-3246	71	63	∪	∪	ADP
ejpam-3246	71	64	[	[	X
ejpam-3246	71	65	0.5	0.5	NUM
ejpam-3246	71	66	,	,	PUNCT
ejpam-3246	71	67	0.6	0.6	NUM
ejpam-3246	71	68	]	]	PUNCT
ejpam-3246	71	69	if	if	SCONJ
ejpam-3246	71	70	x	x	SYM
ejpam-3246	71	71	=	=	PUNCT
ejpam-3246	71	72	c.	c.	NOUN
ejpam-3246	71	73	it	it	PRON
ejpam-3246	71	74	is	be	AUX
ejpam-3246	71	75	routine	routine	ADJ
ejpam-3246	71	76	to	to	PART
ejpam-3246	71	77	verify	verify	VERB
ejpam-3246	71	78	that	that	SCONJ
ejpam-3246	71	79	h	h	NOUN
ejpam-3246	71	80	is	be	AUX
ejpam-3246	71	81	a	a	DET
ejpam-3246	71	82	uni	uni	ADJ
ejpam-3246	71	83	-	-	ADJ
ejpam-3246	71	84	hesitant	hesitant	ADJ
ejpam-3246	71	85	fuzzy	fuzzy	ADJ
ejpam-3246	71	86	algebra	algebra	NOUN
ejpam-3246	71	87	on	on	ADP
ejpam-3246	71	88	x.	x.	NOUN
ejpam-3246	71	89	theorem	theorem	VERB
ejpam-3246	71	90	1	1	NUM
ejpam-3246	71	91	.	.	PUNCT
ejpam-3246	72	1	a	a	DET
ejpam-3246	72	2	hesitant	hesitant	ADJ
ejpam-3246	72	3	fuzzy	fuzzy	NOUN
ejpam-3246	72	4	set	set	VERB
ejpam-3246	72	5	g	g	NOUN
ejpam-3246	72	6	on	on	ADP
ejpam-3246	72	7	x	x	DET
ejpam-3246	72	8	a	a	DET
ejpam-3246	72	9	uni	uni	ADJ
ejpam-3246	72	10	-	-	ADJ
ejpam-3246	72	11	hesitant	hesitant	ADJ
ejpam-3246	72	12	fuzzy	fuzzy	ADJ
ejpam-3246	72	13	algebra	algebra	NOUN
ejpam-3246	72	14	on	on	ADP
ejpam-3246	72	15	x	x	PUNCT
ejpam-3246	72	16	if	if	SCONJ
ejpam-3246	73	1	and	and	CCONJ
ejpam-3246	73	2	only	only	ADV
ejpam-3246	73	3	if	if	SCONJ
ejpam-3246	73	4	the	the	DET
ejpam-3246	73	5	nonempty	nonempty	ADJ
ejpam-3246	73	6	uni	uni	ADJ
ejpam-3246	73	7	-	-	ADJ
ejpam-3246	73	8	hesitant	hesitant	ADJ
ejpam-3246	73	9	level	level	NOUN
ejpam-3246	73	10	set	set	VERB
ejpam-3246	73	11	l(g;λ	l(g;λ	NOUN
ejpam-3246	73	12	)	)	PUNCT
ejpam-3246	73	13	of	of	ADP
ejpam-3246	73	14	g	g	PROPN
ejpam-3246	73	15	is	be	AUX
ejpam-3246	73	16	a	a	DET
ejpam-3246	73	17	subalgebra	subalgebra	NOUN
ejpam-3246	73	18	of	of	ADP
ejpam-3246	73	19	x	x	PUNCT
ejpam-3246	73	20	for	for	ADP
ejpam-3246	73	21	all	all	DET
ejpam-3246	73	22	λ	λ	PROPN
ejpam-3246	73	23	∈	∈	PROPN
ejpam-3246	73	24	p	p	X
ejpam-3246	73	25	(	(	PUNCT
ejpam-3246	73	26	[	[	X
ejpam-3246	73	27	0	0	NUM
ejpam-3246	73	28	,	,	PUNCT
ejpam-3246	73	29	1	1	NUM
ejpam-3246	73	30	]	]	NUM
ejpam-3246	73	31	)	)	PUNCT
ejpam-3246	73	32	.	.	PUNCT
ejpam-3246	74	1	proof	proof	NOUN
ejpam-3246	74	2	.	.	PUNCT
ejpam-3246	75	1	assume	assume	VERB
ejpam-3246	75	2	that	that	SCONJ
ejpam-3246	75	3	g	g	PROPN
ejpam-3246	75	4	is	be	AUX
ejpam-3246	75	5	a	a	DET
ejpam-3246	75	6	uni	uni	ADJ
ejpam-3246	75	7	-	-	ADJ
ejpam-3246	75	8	hesitant	hesitant	ADJ
ejpam-3246	75	9	fuzzy	fuzzy	ADJ
ejpam-3246	75	10	algebra	algebra	NOUN
ejpam-3246	75	11	on	on	ADP
ejpam-3246	75	12	x.	x.	NOUN
ejpam-3246	75	13	let	let	VERB
ejpam-3246	75	14	λ	λ	X
ejpam-3246	75	15	∈	∈	VERB
ejpam-3246	75	16	p	p	X
ejpam-3246	75	17	(	(	PUNCT
ejpam-3246	75	18	[	[	X
ejpam-3246	75	19	0	0	NUM
ejpam-3246	75	20	,	,	PUNCT
ejpam-3246	75	21	1	1	NUM
ejpam-3246	75	22	]	]	PUNCT
ejpam-3246	75	23	)	)	PUNCT
ejpam-3246	75	24	and	and	CCONJ
ejpam-3246	75	25	x	x	X
ejpam-3246	75	26	,	,	PUNCT
ejpam-3246	75	27	y	y	PROPN
ejpam-3246	75	28	∈	∈	PROPN
ejpam-3246	75	29	l(g;λ	l(g;λ	PROPN
ejpam-3246	75	30	)	)	PUNCT
ejpam-3246	75	31	.	.	PUNCT
ejpam-3246	76	1	then	then	ADV
ejpam-3246	76	2	xg	xg	PROPN
ejpam-3246	76	3	⊆	⊆	NUM
ejpam-3246	76	4	λ	λ	PROPN
ejpam-3246	76	5	and	and	CCONJ
ejpam-3246	76	6	yg	yg	PROPN
ejpam-3246	76	7	⊆	⊆	NUM
ejpam-3246	76	8	λ	λ	PROPN
ejpam-3246	76	9	.	.	PUNCT
ejpam-3246	77	1	it	it	PRON
ejpam-3246	77	2	follows	follow	VERB
ejpam-3246	77	3	that	that	SCONJ
ejpam-3246	77	4	(	(	PUNCT
ejpam-3246	77	5	x	x	SYM
ejpam-3246	77	6	∗	∗	NOUN
ejpam-3246	77	7	y)g	y)g	NOUN
ejpam-3246	77	8	⊆	⊆	NUM
ejpam-3246	77	9	xg	xg	NOUN
ejpam-3246	77	10	∪	∪	ADP
ejpam-3246	77	11	yg	yg	PROPN
ejpam-3246	77	12	⊆	⊆	NUM
ejpam-3246	77	13	λ	λ	PROPN
ejpam-3246	77	14	.	.	PUNCT
ejpam-3246	78	1	hence	hence	ADV
ejpam-3246	78	2	x	x	X
ejpam-3246	78	3	∗	∗	NOUN
ejpam-3246	78	4	y	y	PROPN
ejpam-3246	78	5	∈	∈	PROPN
ejpam-3246	78	6	l(g;λ	l(g;λ	PROPN
ejpam-3246	78	7	)	)	PUNCT
ejpam-3246	78	8	.	.	PUNCT
ejpam-3246	79	1	therefore	therefore	ADV
ejpam-3246	79	2	l(g;λ	l(g;λ	X
ejpam-3246	79	3	)	)	PUNCT
ejpam-3246	79	4	is	be	AUX
ejpam-3246	79	5	a	a	DET
ejpam-3246	79	6	subalgebra	subalgebra	NOUN
ejpam-3246	79	7	of	of	ADP
ejpam-3246	79	8	a.	a.	NOUN
ejpam-3246	79	9	conversely	conversely	ADV
ejpam-3246	79	10	,	,	PUNCT
ejpam-3246	79	11	suppose	suppose	VERB
ejpam-3246	79	12	that	that	SCONJ
ejpam-3246	79	13	the	the	DET
ejpam-3246	79	14	nonempty	nonempty	ADJ
ejpam-3246	79	15	uni	uni	ADJ
ejpam-3246	79	16	-	-	ADJ
ejpam-3246	79	17	hesitant	hesitant	ADJ
ejpam-3246	79	18	level	level	NOUN
ejpam-3246	79	19	set	set	NOUN
ejpam-3246	79	20	of	of	ADP
ejpam-3246	79	21	g	g	PROPN
ejpam-3246	79	22	is	be	AUX
ejpam-3246	79	23	a	a	DET
ejpam-3246	79	24	subalgebra	subalgebra	NOUN
ejpam-3246	79	25	of	of	ADP
ejpam-3246	79	26	x	x	PUNCT
ejpam-3246	79	27	for	for	ADP
ejpam-3246	79	28	all	all	DET
ejpam-3246	79	29	λ	λ	PROPN
ejpam-3246	79	30	∈	∈	PROPN
ejpam-3246	79	31	p	p	X
ejpam-3246	79	32	(	(	PUNCT
ejpam-3246	79	33	[	[	X
ejpam-3246	79	34	0	0	NUM
ejpam-3246	79	35	,	,	PUNCT
ejpam-3246	79	36	1	1	NUM
ejpam-3246	79	37	]	]	NUM
ejpam-3246	79	38	)	)	PUNCT
ejpam-3246	79	39	.	.	PUNCT
ejpam-3246	80	1	let	let	VERB
ejpam-3246	80	2	x	x	PRON
ejpam-3246	80	3	,	,	PUNCT
ejpam-3246	80	4	y	y	PROPN
ejpam-3246	80	5	∈	∈	PROPN
ejpam-3246	80	6	x	x	AUX
ejpam-3246	80	7	be	be	AUX
ejpam-3246	80	8	such	such	ADJ
ejpam-3246	80	9	that	that	DET
ejpam-3246	80	10	xg	xg	PROPN
ejpam-3246	80	11	=	=	PROPN
ejpam-3246	80	12	λx	λx	PROPN
ejpam-3246	81	1	and	and	CCONJ
ejpam-3246	81	2	yg	yg	PROPN
ejpam-3246	81	3	=	=	PROPN
ejpam-3246	81	4	λy	λy	PROPN
ejpam-3246	81	5	.	.	PUNCT
ejpam-3246	82	1	taking	take	VERB
ejpam-3246	82	2	λ	λ	PROPN
ejpam-3246	82	3	=	=	PUNCT
ejpam-3246	82	4	λx	λx	PROPN
ejpam-3246	82	5	∪	∪	ADP
ejpam-3246	82	6	λy	λy	PROPN
ejpam-3246	82	7	implies	imply	VERB
ejpam-3246	82	8	that	that	SCONJ
ejpam-3246	82	9	x	x	X
ejpam-3246	82	10	,	,	PUNCT
ejpam-3246	82	11	y	y	PROPN
ejpam-3246	82	12	∈	∈	PROPN
ejpam-3246	82	13	l(g;λ	l(g;λ	PROPN
ejpam-3246	82	14	)	)	PUNCT
ejpam-3246	82	15	,	,	PUNCT
ejpam-3246	82	16	and	and	CCONJ
ejpam-3246	82	17	so	so	ADV
ejpam-3246	82	18	x	x	SYM
ejpam-3246	82	19	∗	∗	NOUN
ejpam-3246	82	20	y	y	PROPN
ejpam-3246	82	21	∈	∈	PROPN
ejpam-3246	82	22	l(g;λ	l(g;λ	PROPN
ejpam-3246	82	23	)	)	PUNCT
ejpam-3246	82	24	.	.	PUNCT
ejpam-3246	83	1	hence	hence	ADV
ejpam-3246	83	2	(	(	PUNCT
ejpam-3246	83	3	x	x	X
ejpam-3246	83	4	∗	∗	NOUN
ejpam-3246	83	5	y)g	y)g	NOUN
ejpam-3246	83	6	⊆	⊆	NUM
ejpam-3246	83	7	λ	λ	X
ejpam-3246	83	8	=	=	SYM
ejpam-3246	83	9	λx	λx	NOUN
ejpam-3246	83	10	∪	∪	NOUN
ejpam-3246	83	11	λy	λy	PROPN
ejpam-3246	83	12	=	=	SYM
ejpam-3246	83	13	xg	xg	PROPN
ejpam-3246	83	14	∪	∪	PROPN
ejpam-3246	83	15	yg	yg	PROPN
ejpam-3246	83	16	.	.	PUNCT
ejpam-3246	84	1	therefore	therefore	ADV
ejpam-3246	84	2	g	g	PROPN
ejpam-3246	84	3	is	be	AUX
ejpam-3246	84	4	a	a	DET
ejpam-3246	84	5	uni	uni	ADJ
ejpam-3246	84	6	-	-	ADJ
ejpam-3246	84	7	hesitant	hesitant	ADJ
ejpam-3246	84	8	fuzzy	fuzzy	ADJ
ejpam-3246	84	9	algebra	algebra	NOUN
ejpam-3246	84	10	on	on	ADP
ejpam-3246	84	11	x.	x.	PROPN
ejpam-3246	84	12	g.	g.	PROPN
ejpam-3246	84	13	muhiuddin	muhiuddin	PROPN
ejpam-3246	84	14	,	,	PUNCT
ejpam-3246	84	15	s.	s.	PROPN
ejpam-3246	84	16	aldhafeeri	aldhafeeri	PROPN
ejpam-3246	84	17	/	/	SYM
ejpam-3246	84	18	eur	eur	PROPN
ejpam-3246	84	19	.	.	PUNCT
ejpam-3246	85	1	j.	j.	PROPN
ejpam-3246	85	2	pure	pure	PROPN
ejpam-3246	85	3	appl	appl	PROPN
ejpam-3246	85	4	.	.	PROPN
ejpam-3246	85	5	math	math	PROPN
ejpam-3246	85	6	,	,	PUNCT
ejpam-3246	85	7	11	11	NUM
ejpam-3246	85	8	(	(	PUNCT
ejpam-3246	85	9	2	2	NUM
ejpam-3246	85	10	)	)	PUNCT
ejpam-3246	85	11	(	(	PUNCT
ejpam-3246	85	12	2018	2018	NUM
ejpam-3246	85	13	)	)	PUNCT
ejpam-3246	85	14	,	,	PUNCT
ejpam-3246	85	15	417	417	NUM
ejpam-3246	85	16	-	-	SYM
ejpam-3246	85	17	430	430	NUM
ejpam-3246	85	18	421	421	NUM
ejpam-3246	85	19	proposition	proposition	NOUN
ejpam-3246	85	20	1	1	NUM
ejpam-3246	85	21	.	.	PUNCT
ejpam-3246	86	1	every	every	DET
ejpam-3246	86	2	uni	uni	ADJ
ejpam-3246	86	3	-	-	ADJ
ejpam-3246	86	4	hesitant	hesitant	ADJ
ejpam-3246	86	5	fuzzy	fuzzy	ADJ
ejpam-3246	86	6	algebra	algebra	NOUN
ejpam-3246	86	7	g	g	NOUN
ejpam-3246	86	8	on	on	ADP
ejpam-3246	86	9	x	x	PUNCT
ejpam-3246	86	10	satisfies	satisfie	NOUN
ejpam-3246	86	11	0	0	NUM
ejpam-3246	86	12	g	g	PROPN
ejpam-3246	86	13	⊆	⊆	NUM
ejpam-3246	86	14	xg	xg	NOUN
ejpam-3246	86	15	for	for	ADP
ejpam-3246	86	16	all	all	DET
ejpam-3246	86	17	x	x	SYM
ejpam-3246	86	18	∈	∈	ADJ
ejpam-3246	86	19	x.	x.	NOUN
ejpam-3246	86	20	proof	proof	NOUN
ejpam-3246	86	21	.	.	PUNCT
ejpam-3246	87	1	since	since	SCONJ
ejpam-3246	87	2	x	x	X
ejpam-3246	87	3	∗	∗	NOUN
ejpam-3246	87	4	x	x	SYM
ejpam-3246	87	5	=	=	SYM
ejpam-3246	87	6	0	0	NUM
ejpam-3246	87	7	for	for	ADP
ejpam-3246	87	8	all	all	DET
ejpam-3246	87	9	x	x	SYM
ejpam-3246	87	10	∈	∈	NOUN
ejpam-3246	87	11	x	x	X
ejpam-3246	87	12	,	,	PUNCT
ejpam-3246	87	13	it	it	PRON
ejpam-3246	87	14	is	be	AUX
ejpam-3246	87	15	straightforward	straightforward	ADJ
ejpam-3246	87	16	.	.	PUNCT
ejpam-3246	88	1	proposition	proposition	NOUN
ejpam-3246	88	2	2	2	NUM
ejpam-3246	88	3	.	.	PUNCT
ejpam-3246	89	1	let	let	VERB
ejpam-3246	89	2	x	x	PRON
ejpam-3246	89	3	be	be	AUX
ejpam-3246	89	4	a	a	DET
ejpam-3246	89	5	bci	bci	NOUN
ejpam-3246	89	6	-	-	NOUN
ejpam-3246	89	7	algebra	algebra	NOUN
ejpam-3246	89	8	.	.	PUNCT
ejpam-3246	90	1	if	if	SCONJ
ejpam-3246	90	2	g	g	PROPN
ejpam-3246	90	3	is	be	AUX
ejpam-3246	90	4	a	a	DET
ejpam-3246	90	5	uni	uni	ADJ
ejpam-3246	90	6	-	-	ADJ
ejpam-3246	90	7	hesitant	hesitant	ADJ
ejpam-3246	90	8	fuzzy	fuzzy	ADJ
ejpam-3246	90	9	algebra	algebra	NOUN
ejpam-3246	90	10	on	on	ADP
ejpam-3246	90	11	x	x	NOUN
ejpam-3246	90	12	,	,	PUNCT
ejpam-3246	90	13	then	then	ADV
ejpam-3246	90	14	(	(	PUNCT
ejpam-3246	90	15	x	x	SYM
ejpam-3246	90	16	∗	∗	NOUN
ejpam-3246	90	17	(	(	PUNCT
ejpam-3246	90	18	0	0	NUM
ejpam-3246	90	19	∗	∗	NOUN
ejpam-3246	90	20	y))g	y))g	NOUN
ejpam-3246	90	21	⊆	⊆	NUM
ejpam-3246	90	22	xg	xg	NOUN
ejpam-3246	90	23	∪	∪	ADP
ejpam-3246	90	24	yg	yg	PROPN
ejpam-3246	90	25	for	for	ADP
ejpam-3246	90	26	all	all	DET
ejpam-3246	90	27	x	x	NOUN
ejpam-3246	90	28	,	,	PUNCT
ejpam-3246	90	29	y	y	PROPN
ejpam-3246	90	30	∈	∈	PROPN
ejpam-3246	90	31	x.	x.	NOUN
ejpam-3246	90	32	proof	proof	NOUN
ejpam-3246	90	33	.	.	PUNCT
ejpam-3246	91	1	using	use	VERB
ejpam-3246	91	2	proposition	proposition	NOUN
ejpam-3246	91	3	1	1	NUM
ejpam-3246	91	4	,	,	PUNCT
ejpam-3246	91	5	we	we	PRON
ejpam-3246	91	6	have	have	VERB
ejpam-3246	91	7	(	(	PUNCT
ejpam-3246	91	8	x	x	SYM
ejpam-3246	91	9	∗	∗	NOUN
ejpam-3246	91	10	(	(	PUNCT
ejpam-3246	91	11	0	0	NUM
ejpam-3246	91	12	∗	∗	NOUN
ejpam-3246	91	13	y))g	y))g	NOUN
ejpam-3246	92	1	⊆	⊆	NUM
ejpam-3246	92	2	xg	xg	NOUN
ejpam-3246	92	3	∪	∪	X
ejpam-3246	92	4	(	(	PUNCT
ejpam-3246	92	5	0	0	NUM
ejpam-3246	92	6	∗	∗	NOUN
ejpam-3246	92	7	y)g	y)g	NOUN
ejpam-3246	92	8	⊆	⊆	NUM
ejpam-3246	92	9	xg	xg	NOUN
ejpam-3246	92	10	∪	∪	ADP
ejpam-3246	92	11	0	0	NUM
ejpam-3246	92	12	g	g	NOUN
ejpam-3246	92	13	∪	∪	NOUN
ejpam-3246	92	14	yg	yg	PROPN
ejpam-3246	92	15	=	=	PROPN
ejpam-3246	92	16	xg	xg	PROPN
ejpam-3246	92	17	∪	∪	ADP
ejpam-3246	92	18	yg	yg	PROPN
ejpam-3246	92	19	for	for	ADP
ejpam-3246	92	20	all	all	DET
ejpam-3246	92	21	x	x	NOUN
ejpam-3246	92	22	,	,	PUNCT
ejpam-3246	92	23	y	y	PROPN
ejpam-3246	92	24	∈	∈	PROPN
ejpam-3246	92	25	x.	x.	NOUN
ejpam-3246	92	26	proposition	proposition	NOUN
ejpam-3246	92	27	3	3	NUM
ejpam-3246	92	28	.	.	X
ejpam-3246	93	1	for	for	ADP
ejpam-3246	93	2	any	any	DET
ejpam-3246	93	3	uni	uni	ADJ
ejpam-3246	93	4	-	-	ADJ
ejpam-3246	93	5	hesitant	hesitant	ADJ
ejpam-3246	93	6	fuzzy	fuzzy	ADJ
ejpam-3246	93	7	algebra	algebra	NOUN
ejpam-3246	93	8	g	g	NOUN
ejpam-3246	93	9	on	on	ADP
ejpam-3246	93	10	x	x	SYM
ejpam-3246	93	11	,	,	PUNCT
ejpam-3246	93	12	we	we	PRON
ejpam-3246	93	13	have	have	VERB
ejpam-3246	93	14	(	(	PUNCT
ejpam-3246	93	15	∀x	∀x	X
ejpam-3246	93	16	,	,	PUNCT
ejpam-3246	93	17	y	y	PROPN
ejpam-3246	93	18	∈	∈	PROPN
ejpam-3246	93	19	x	x	X
ejpam-3246	93	20	)	)	PUNCT
ejpam-3246	93	21	(	(	PUNCT
ejpam-3246	93	22	(	(	PUNCT
ejpam-3246	93	23	x	x	SYM
ejpam-3246	93	24	∗	∗	NOUN
ejpam-3246	93	25	y)g	y)g	NOUN
ejpam-3246	93	26	⊆	⊆	NUM
ejpam-3246	93	27	yg	yg	PROPN
ejpam-3246	93	28	⇔	⇔	PROPN
ejpam-3246	93	29	xg	xg	PROPN
ejpam-3246	94	1	=	=	NOUN
ejpam-3246	94	2	0	0	NUM
ejpam-3246	94	3	g	g	NOUN
ejpam-3246	94	4	)	)	PUNCT
ejpam-3246	94	5	.	.	PUNCT
ejpam-3246	95	1	proof	proof	NOUN
ejpam-3246	95	2	.	.	PUNCT
ejpam-3246	96	1	assume	assume	VERB
ejpam-3246	96	2	that	that	SCONJ
ejpam-3246	96	3	(	(	PUNCT
ejpam-3246	96	4	x	x	SYM
ejpam-3246	96	5	∗	∗	NOUN
ejpam-3246	96	6	y)g	y)g	NOUN
ejpam-3246	96	7	⊆	⊆	NUM
ejpam-3246	96	8	yg	yg	NOUN
ejpam-3246	96	9	for	for	ADP
ejpam-3246	96	10	all	all	DET
ejpam-3246	96	11	x	x	NOUN
ejpam-3246	96	12	,	,	PUNCT
ejpam-3246	96	13	y	y	PROPN
ejpam-3246	96	14	∈	∈	PROPN
ejpam-3246	96	15	x.	x.	NOUN
ejpam-3246	96	16	taking	take	VERB
ejpam-3246	96	17	y	y	PROPN
ejpam-3246	96	18	=	=	SYM
ejpam-3246	96	19	0	0	NUM
ejpam-3246	96	20	induces	induce	VERB
ejpam-3246	96	21	xg	xg	NOUN
ejpam-3246	96	22	=	=	SYM
ejpam-3246	97	1	(	(	PUNCT
ejpam-3246	97	2	x	x	SYM
ejpam-3246	97	3	∗	∗	VERB
ejpam-3246	97	4	0)g	0)g	NOUN
ejpam-3246	97	5	⊆	⊆	NUM
ejpam-3246	97	6	0	0	NUM
ejpam-3246	97	7	g.	g.	NOUN
ejpam-3246	97	8	it	it	PRON
ejpam-3246	97	9	follows	follow	VERB
ejpam-3246	97	10	from	from	ADP
ejpam-3246	97	11	proposition	proposition	NOUN
ejpam-3246	97	12	1	1	NUM
ejpam-3246	97	13	that	that	SCONJ
ejpam-3246	97	14	xg	xg	AUX
ejpam-3246	98	1	=	=	NOUN
ejpam-3246	98	2	0	0	NUM
ejpam-3246	98	3	g	g	NOUN
ejpam-3246	98	4	for	for	ADP
ejpam-3246	98	5	all	all	PRON
ejpam-3246	98	6	x	x	SYM
ejpam-3246	98	7	∈	∈	NOUN
ejpam-3246	98	8	x.	x.	NOUN
ejpam-3246	98	9	conversely	conversely	ADV
ejpam-3246	98	10	,	,	PUNCT
ejpam-3246	98	11	suppose	suppose	VERB
ejpam-3246	98	12	that	that	SCONJ
ejpam-3246	98	13	xg	xg	PROPN
ejpam-3246	98	14	=	=	NOUN
ejpam-3246	98	15	0	0	NUM
ejpam-3246	98	16	g	g	NOUN
ejpam-3246	98	17	for	for	ADP
ejpam-3246	98	18	all	all	DET
ejpam-3246	98	19	x	x	SYM
ejpam-3246	98	20	∈	∈	NOUN
ejpam-3246	98	21	x.	x.	NOUN
ejpam-3246	98	22	then	then	ADV
ejpam-3246	98	23	(	(	PUNCT
ejpam-3246	99	1	x	x	SYM
ejpam-3246	99	2	∗	∗	NOUN
ejpam-3246	99	3	y)g	y)g	NOUN
ejpam-3246	99	4	⊆	⊆	NUM
ejpam-3246	99	5	xg	xg	NOUN
ejpam-3246	99	6	∪	∪	ADP
ejpam-3246	99	7	yg	yg	PROPN
ejpam-3246	99	8	=	=	SYM
ejpam-3246	99	9	0	0	NUM
ejpam-3246	99	10	g	g	NOUN
ejpam-3246	99	11	∪	∪	NOUN
ejpam-3246	99	12	yg	yg	PROPN
ejpam-3246	99	13	=	=	PROPN
ejpam-3246	99	14	yg	yg	PROPN
ejpam-3246	99	15	for	for	ADP
ejpam-3246	99	16	all	all	DET
ejpam-3246	99	17	x	x	NOUN
ejpam-3246	99	18	,	,	PUNCT
ejpam-3246	99	19	y	y	PROPN
ejpam-3246	99	20	∈	∈	PROPN
ejpam-3246	99	21	x.	x.	NOUN
ejpam-3246	99	22	theorem	theorem	VERB
ejpam-3246	99	23	2	2	NUM
ejpam-3246	99	24	.	.	PUNCT
ejpam-3246	99	25	given	give	VERB
ejpam-3246	99	26	a	a	DET
ejpam-3246	99	27	uni	uni	ADJ
ejpam-3246	99	28	-	-	ADJ
ejpam-3246	99	29	hesitant	hesitant	ADJ
ejpam-3246	99	30	fuzzy	fuzzy	ADJ
ejpam-3246	99	31	algebra	algebra	NOUN
ejpam-3246	99	32	g	g	NOUN
ejpam-3246	99	33	on	on	ADP
ejpam-3246	99	34	x	x	SYM
ejpam-3246	99	35	,	,	PUNCT
ejpam-3246	99	36	the	the	DET
ejpam-3246	99	37	hesitant	hesitant	ADJ
ejpam-3246	99	38	fuzzy	fuzzy	ADJ
ejpam-3246	99	39	set	set	NOUN
ejpam-3246	99	40	g∗	g∗	NOUN
ejpam-3246	99	41	on	on	ADP
ejpam-3246	99	42	x	x	PUNCT
ejpam-3246	99	43	defined	define	VERB
ejpam-3246	99	44	by	by	ADP
ejpam-3246	99	45	g∗	g∗	PROPN
ejpam-3246	99	46	:	:	PUNCT
ejpam-3246	99	47	x	x	X
ejpam-3246	99	48	→	→	X
ejpam-3246	99	49	p	p	X
ejpam-3246	99	50	(	(	PUNCT
ejpam-3246	99	51	[	[	X
ejpam-3246	99	52	0	0	NUM
ejpam-3246	99	53	,	,	PUNCT
ejpam-3246	99	54	1	1	NUM
ejpam-3246	99	55	]	]	NUM
ejpam-3246	99	56	)	)	PUNCT
ejpam-3246	99	57	,	,	PUNCT
ejpam-3246	99	58	x	x	X
ejpam-3246	99	59	7→	7→	X
ejpam-3246	99	60	{	{	PUNCT
ejpam-3246	99	61	xg	xg	VERB
ejpam-3246	99	62	if	if	SCONJ
ejpam-3246	99	63	x	x	X
ejpam-3246	99	64	∈	∈	PROPN
ejpam-3246	99	65	l(g;λ	l(g;λ	PROPN
ejpam-3246	99	66	)	)	PUNCT
ejpam-3246	99	67	,	,	PUNCT
ejpam-3246	99	68	(	(	PUNCT
ejpam-3246	99	69	0	0	NUM
ejpam-3246	99	70	,	,	PUNCT
ejpam-3246	99	71	1	1	NUM
ejpam-3246	99	72	)	)	PUNCT
ejpam-3246	99	73	otherwise	otherwise	ADV
ejpam-3246	99	74	is	be	AUX
ejpam-3246	99	75	a	a	DET
ejpam-3246	99	76	uni	uni	ADJ
ejpam-3246	99	77	-	-	ADJ
ejpam-3246	99	78	hesitant	hesitant	ADJ
ejpam-3246	99	79	fuzzy	fuzzy	ADJ
ejpam-3246	99	80	algebra	algebra	NOUN
ejpam-3246	99	81	on	on	ADP
ejpam-3246	99	82	x.	x.	NOUN
ejpam-3246	99	83	proof	proof	NOUN
ejpam-3246	99	84	.	.	PUNCT
ejpam-3246	100	1	if	if	SCONJ
ejpam-3246	100	2	g	g	PROPN
ejpam-3246	100	3	is	be	AUX
ejpam-3246	100	4	a	a	DET
ejpam-3246	100	5	uni	uni	ADJ
ejpam-3246	100	6	-	-	ADJ
ejpam-3246	100	7	hesitant	hesitant	ADJ
ejpam-3246	100	8	fuzzy	fuzzy	ADJ
ejpam-3246	100	9	algebra	algebra	NOUN
ejpam-3246	100	10	on	on	ADP
ejpam-3246	100	11	x	x	NOUN
ejpam-3246	100	12	,	,	PUNCT
ejpam-3246	100	13	then	then	ADV
ejpam-3246	100	14	l(g;λ	l(g;λ	PROPN
ejpam-3246	100	15	)	)	PUNCT
ejpam-3246	100	16	is	be	AUX
ejpam-3246	100	17	a	a	DET
ejpam-3246	100	18	subalgebra	subalgebra	NOUN
ejpam-3246	100	19	of	of	ADP
ejpam-3246	100	20	a	a	PRON
ejpam-3246	100	21	for	for	ADP
ejpam-3246	100	22	all	all	DET
ejpam-3246	100	23	λ	λ	PROPN
ejpam-3246	100	24	∈	∈	PROPN
ejpam-3246	100	25	p	p	X
ejpam-3246	100	26	(	(	PUNCT
ejpam-3246	100	27	[	[	X
ejpam-3246	100	28	0	0	NUM
ejpam-3246	100	29	,	,	PUNCT
ejpam-3246	100	30	1	1	NUM
ejpam-3246	100	31	]	]	PUNCT
ejpam-3246	100	32	)	)	PUNCT
ejpam-3246	100	33	with	with	ADP
ejpam-3246	100	34	l(g;λ	l(g;λ	PROPN
ejpam-3246	100	35	)	)	PUNCT
ejpam-3246	100	36	6=	6=	ADP
ejpam-3246	100	37	∅	∅	NOUN
ejpam-3246	100	38	by	by	ADP
ejpam-3246	100	39	theorem	theorem	NOUN
ejpam-3246	100	40	1	1	NUM
ejpam-3246	100	41	.	.	PUNCT
ejpam-3246	101	1	let	let	VERB
ejpam-3246	101	2	x	x	PRON
ejpam-3246	101	3	,	,	PUNCT
ejpam-3246	101	4	y	y	PROPN
ejpam-3246	101	5	∈	∈	PROPN
ejpam-3246	101	6	x.	x.	NOUN
ejpam-3246	102	1	if	if	SCONJ
ejpam-3246	102	2	x	x	X
ejpam-3246	102	3	,	,	PUNCT
ejpam-3246	102	4	y	y	PROPN
ejpam-3246	102	5	∈	∈	PROPN
ejpam-3246	102	6	l(g;λ	l(g;λ	PROPN
ejpam-3246	102	7	)	)	PUNCT
ejpam-3246	102	8	,	,	PUNCT
ejpam-3246	102	9	then	then	ADV
ejpam-3246	102	10	x	x	X
ejpam-3246	102	11	∗	∗	VERB
ejpam-3246	102	12	y	y	PROPN
ejpam-3246	102	13	∈	∈	PROPN
ejpam-3246	102	14	l(g;λ	l(g;λ	X
ejpam-3246	102	15	)	)	PUNCT
ejpam-3246	102	16	and	and	CCONJ
ejpam-3246	102	17	so	so	ADV
ejpam-3246	102	18	(	(	PUNCT
ejpam-3246	102	19	x	x	X
ejpam-3246	103	1	∗	∗	NOUN
ejpam-3246	103	2	y)g∗	y)g∗	NOUN
ejpam-3246	103	3	=	=	SYM
ejpam-3246	103	4	(	(	PUNCT
ejpam-3246	103	5	x	x	X
ejpam-3246	103	6	∗	∗	NOUN
ejpam-3246	103	7	y)g	y)g	NOUN
ejpam-3246	103	8	⊆	⊆	NUM
ejpam-3246	103	9	xg	xg	NOUN
ejpam-3246	103	10	∪	∪	ADP
ejpam-3246	103	11	yg	yg	PROPN
ejpam-3246	103	12	=	=	PROPN
ejpam-3246	103	13	xg∗	xg∗	NOUN
ejpam-3246	103	14	∪	∪	VERB
ejpam-3246	103	15	yg∗.	yg∗.	PROPN
ejpam-3246	103	16	if	if	SCONJ
ejpam-3246	103	17	x	x	X
ejpam-3246	103	18	/∈	/∈	PUNCT
ejpam-3246	103	19	l(g;λ	l(g;λ	X
ejpam-3246	103	20	)	)	PUNCT
ejpam-3246	103	21	or	or	CCONJ
ejpam-3246	103	22	y	y	PROPN
ejpam-3246	103	23	/∈	/∈	PUNCT
ejpam-3246	103	24	l(g;λ	l(g;λ	PROPN
ejpam-3246	103	25	)	)	PUNCT
ejpam-3246	103	26	,	,	PUNCT
ejpam-3246	103	27	then	then	ADV
ejpam-3246	103	28	xg∗	xg∗	PUNCT
ejpam-3246	103	29	=	=	PUNCT
ejpam-3246	104	1	(	(	PUNCT
ejpam-3246	104	2	0	0	NUM
ejpam-3246	104	3	,	,	PUNCT
ejpam-3246	104	4	1	1	NUM
ejpam-3246	104	5	)	)	PUNCT
ejpam-3246	104	6	or	or	CCONJ
ejpam-3246	104	7	yg∗	yg∗	ADJ
ejpam-3246	105	1	=	=	SYM
ejpam-3246	105	2	(	(	PUNCT
ejpam-3246	105	3	0	0	NUM
ejpam-3246	105	4	,	,	PUNCT
ejpam-3246	105	5	1	1	NUM
ejpam-3246	105	6	)	)	PUNCT
ejpam-3246	105	7	.	.	PUNCT
ejpam-3246	106	1	thus	thus	ADV
ejpam-3246	106	2	(	(	PUNCT
ejpam-3246	106	3	x	x	SYM
ejpam-3246	106	4	∗	∗	NOUN
ejpam-3246	106	5	y)g∗	y)g∗	PRON
ejpam-3246	106	6	⊆	⊆	NUM
ejpam-3246	106	7	(	(	PUNCT
ejpam-3246	106	8	0	0	NUM
ejpam-3246	106	9	,	,	PUNCT
ejpam-3246	106	10	1	1	NUM
ejpam-3246	106	11	)	)	PUNCT
ejpam-3246	106	12	=	=	PUNCT
ejpam-3246	106	13	xg∗	xg∗	NOUN
ejpam-3246	106	14	∪	∪	X
ejpam-3246	106	15	yg∗.	yg∗.	PROPN
ejpam-3246	106	16	therefore	therefore	ADV
ejpam-3246	106	17	g∗	g∗	PROPN
ejpam-3246	106	18	is	be	AUX
ejpam-3246	106	19	a	a	DET
ejpam-3246	106	20	uni	uni	ADJ
ejpam-3246	106	21	-	-	ADJ
ejpam-3246	106	22	hesitant	hesitant	ADJ
ejpam-3246	106	23	fuzzy	fuzzy	ADJ
ejpam-3246	106	24	algebra	algebra	NOUN
ejpam-3246	106	25	on	on	ADP
ejpam-3246	106	26	x.	x.	PROPN
ejpam-3246	106	27	g.	g.	PROPN
ejpam-3246	106	28	muhiuddin	muhiuddin	PROPN
ejpam-3246	106	29	,	,	PUNCT
ejpam-3246	106	30	s.	s.	PROPN
ejpam-3246	106	31	aldhafeeri	aldhafeeri	PROPN
ejpam-3246	106	32	/	/	SYM
ejpam-3246	106	33	eur	eur	PROPN
ejpam-3246	106	34	.	.	PUNCT
ejpam-3246	107	1	j.	j.	PROPN
ejpam-3246	107	2	pure	pure	PROPN
ejpam-3246	107	3	appl	appl	PROPN
ejpam-3246	107	4	.	.	PROPN
ejpam-3246	107	5	math	math	PROPN
ejpam-3246	107	6	,	,	PUNCT
ejpam-3246	107	7	11	11	NUM
ejpam-3246	107	8	(	(	PUNCT
ejpam-3246	107	9	2	2	NUM
ejpam-3246	107	10	)	)	PUNCT
ejpam-3246	107	11	(	(	PUNCT
ejpam-3246	107	12	2018	2018	NUM
ejpam-3246	107	13	)	)	PUNCT
ejpam-3246	107	14	,	,	PUNCT
ejpam-3246	107	15	417	417	NUM
ejpam-3246	107	16	-	-	SYM
ejpam-3246	107	17	430	430	NUM
ejpam-3246	107	18	422	422	NUM
ejpam-3246	107	19	4	4	NUM
ejpam-3246	107	20	.	.	PUNCT
ejpam-3246	108	1	uni	uni	ADJ
ejpam-3246	108	2	-	-	ADJ
ejpam-3246	108	3	hesitant	hesitant	ADJ
ejpam-3246	108	4	fuzzy	fuzzy	ADJ
ejpam-3246	108	5	ideals	ideal	NOUN
ejpam-3246	108	6	definition	definition	NOUN
ejpam-3246	108	7	2	2	NUM
ejpam-3246	108	8	.	.	PUNCT
ejpam-3246	109	1	a	a	DET
ejpam-3246	109	2	hesitant	hesitant	ADJ
ejpam-3246	109	3	fuzzy	fuzzy	NOUN
ejpam-3246	109	4	set	set	VERB
ejpam-3246	109	5	g	g	NOUN
ejpam-3246	109	6	on	on	ADP
ejpam-3246	109	7	x	x	PROPN
ejpam-3246	109	8	is	be	AUX
ejpam-3246	109	9	called	call	VERB
ejpam-3246	109	10	a	a	DET
ejpam-3246	109	11	uni	uni	ADJ
ejpam-3246	109	12	-	-	ADJ
ejpam-3246	109	13	hesitant	hesitant	ADJ
ejpam-3246	109	14	fuzzy	fuzzy	ADJ
ejpam-3246	109	15	ideal	ideal	NOUN
ejpam-3246	109	16	on	on	ADP
ejpam-3246	109	17	x	x	SYM
ejpam-3246	109	18	if	if	SCONJ
ejpam-3246	109	19	it	it	PRON
ejpam-3246	109	20	satisfies	satisfy	VERB
ejpam-3246	109	21	0	0	NUM
ejpam-3246	109	22	g	g	PROPN
ejpam-3246	109	23	⊆	⊆	NUM
ejpam-3246	109	24	xg	xg	NOUN
ejpam-3246	109	25	for	for	ADP
ejpam-3246	109	26	all	all	DET
ejpam-3246	109	27	x	x	SYM
ejpam-3246	109	28	∈	∈	PROPN
ejpam-3246	109	29	x	x	X
ejpam-3246	109	30	and	and	CCONJ
ejpam-3246	109	31	(	(	PUNCT
ejpam-3246	109	32	∀x	∀x	X
ejpam-3246	109	33	,	,	PUNCT
ejpam-3246	109	34	y	y	PROPN
ejpam-3246	109	35	∈	∈	PROPN
ejpam-3246	109	36	x	x	X
ejpam-3246	109	37	)	)	PUNCT
ejpam-3246	109	38	(	(	PUNCT
ejpam-3246	109	39	xg	xg	PROPN
ejpam-3246	109	40	⊆	⊆	NUM
ejpam-3246	109	41	(	(	PUNCT
ejpam-3246	109	42	x	x	SYM
ejpam-3246	109	43	∗	∗	NOUN
ejpam-3246	109	44	y)g	y)g	NOUN
ejpam-3246	109	45	∪	∪	PROPN
ejpam-3246	109	46	yg	yg	PROPN
ejpam-3246	109	47	)	)	PUNCT
ejpam-3246	109	48	.	.	PUNCT
ejpam-3246	110	1	(	(	PUNCT
ejpam-3246	110	2	5	5	X
ejpam-3246	110	3	)	)	PUNCT
ejpam-3246	110	4	example	example	NOUN
ejpam-3246	111	1	2	2	NUM
ejpam-3246	111	2	.	.	PUNCT
ejpam-3246	111	3	let	let	VERB
ejpam-3246	111	4	x	x	PUNCT
ejpam-3246	111	5	=	=	PUNCT
ejpam-3246	111	6	{	{	PUNCT
ejpam-3246	111	7	0	0	NUM
ejpam-3246	111	8	,	,	PUNCT
ejpam-3246	111	9	1	1	NUM
ejpam-3246	111	10	,	,	PUNCT
ejpam-3246	111	11	2	2	NUM
ejpam-3246	111	12	,	,	PUNCT
ejpam-3246	111	13	a	a	PRON
ejpam-3246	111	14	,	,	PUNCT
ejpam-3246	111	15	b	b	X
ejpam-3246	111	16	}	}	PUNCT
ejpam-3246	111	17	be	be	AUX
ejpam-3246	111	18	a	a	DET
ejpam-3246	111	19	bci	bci	NOUN
ejpam-3246	111	20	-	-	NOUN
ejpam-3246	111	21	algebra	algebra	NOUN
ejpam-3246	111	22	with	with	ADP
ejpam-3246	111	23	the	the	DET
ejpam-3246	111	24	cayley	cayley	ADJ
ejpam-3246	111	25	table	table	NOUN
ejpam-3246	111	26	which	which	PRON
ejpam-3246	111	27	is	be	AUX
ejpam-3246	111	28	appeared	appear	VERB
ejpam-3246	111	29	in	in	ADP
ejpam-3246	111	30	table	table	NOUN
ejpam-3246	111	31	3	3	NUM
ejpam-3246	111	32	.	.	PUNCT
ejpam-3246	111	33	table	table	NOUN
ejpam-3246	111	34	3	3	NUM
ejpam-3246	111	35	:	:	PUNCT
ejpam-3246	111	36	cayley	cayley	ADJ
ejpam-3246	111	37	table	table	NOUN
ejpam-3246	111	38	for	for	ADP
ejpam-3246	111	39	the	the	DET
ejpam-3246	111	40	∗-operation	∗-operation	NOUN
ejpam-3246	111	41	∗	∗	NOUN
ejpam-3246	111	42	0	0	NUM
ejpam-3246	111	43	1	1	NUM
ejpam-3246	111	44	2	2	NUM
ejpam-3246	111	45	a	a	DET
ejpam-3246	111	46	b	b	NOUN
ejpam-3246	111	47	0	0	NUM
ejpam-3246	111	48	0	0	NUM
ejpam-3246	111	49	0	0	NUM
ejpam-3246	111	50	0	0	NUM
ejpam-3246	111	51	a	a	DET
ejpam-3246	111	52	a	a	DET
ejpam-3246	111	53	1	1	NUM
ejpam-3246	111	54	1	1	NUM
ejpam-3246	111	55	0	0	NUM
ejpam-3246	111	56	1	1	NUM
ejpam-3246	111	57	b	b	NOUN
ejpam-3246	111	58	a	a	DET
ejpam-3246	111	59	2	2	NUM
ejpam-3246	111	60	2	2	NUM
ejpam-3246	111	61	2	2	NUM
ejpam-3246	111	62	0	0	NUM
ejpam-3246	111	63	a	a	DET
ejpam-3246	111	64	a	a	DET
ejpam-3246	111	65	a	a	DET
ejpam-3246	111	66	a	a	PRON
ejpam-3246	111	67	a	a	PRON
ejpam-3246	111	68	a	a	PRON
ejpam-3246	111	69	0	0	NUM
ejpam-3246	111	70	0	0	NUM
ejpam-3246	112	1	b	b	X
ejpam-3246	112	2	b	b	PROPN
ejpam-3246	112	3	a	a	DET
ejpam-3246	112	4	b	b	NOUN
ejpam-3246	112	5	1	1	NUM
ejpam-3246	112	6	0	0	NUM
ejpam-3246	112	7	let	let	VERB
ejpam-3246	112	8	h	h	NOUN
ejpam-3246	112	9	be	be	AUX
ejpam-3246	112	10	a	a	DET
ejpam-3246	112	11	hesitant	hesitant	ADJ
ejpam-3246	112	12	fuzzy	fuzzy	ADJ
ejpam-3246	112	13	set	set	NOUN
ejpam-3246	112	14	on	on	ADP
ejpam-3246	112	15	x	x	PUNCT
ejpam-3246	112	16	defined	define	VERB
ejpam-3246	112	17	as	as	SCONJ
ejpam-3246	112	18	follows	follow	VERB
ejpam-3246	112	19	:	:	PUNCT
ejpam-3246	112	20	h	h	NOUN
ejpam-3246	112	21	:	:	PUNCT
ejpam-3246	112	22	x	x	X
ejpam-3246	113	1	→	→	X
ejpam-3246	113	2	p	p	X
ejpam-3246	113	3	(	(	PUNCT
ejpam-3246	113	4	[	[	X
ejpam-3246	113	5	0	0	NUM
ejpam-3246	113	6	,	,	PUNCT
ejpam-3246	113	7	1	1	NUM
ejpam-3246	113	8	]	]	NUM
ejpam-3246	113	9	)	)	PUNCT
ejpam-3246	113	10	,	,	PUNCT
ejpam-3246	113	11	x	x	X
ejpam-3246	113	12	7→	7→	NOUN
ejpam-3246	113	13			NOUN
ejpam-3246	113	14	{	{	PUNCT
ejpam-3246	113	15	0.2	0.2	NUM
ejpam-3246	113	16	,	,	PUNCT
ejpam-3246	113	17	0.4	0.4	NUM
ejpam-3246	113	18	,	,	PUNCT
ejpam-3246	113	19	0.6	0.6	NUM
ejpam-3246	113	20	,	,	PUNCT
ejpam-3246	113	21	0.8	0.8	NUM
ejpam-3246	113	22	}	}	PUNCT
ejpam-3246	113	23	if	if	SCONJ
ejpam-3246	113	24	x	x	PROPN
ejpam-3246	113	25	=	=	SYM
ejpam-3246	113	26	0	0	NUM
ejpam-3246	113	27	,	,	PUNCT
ejpam-3246	113	28	[	[	X
ejpam-3246	113	29	0.2	0.2	NUM
ejpam-3246	113	30	,	,	PUNCT
ejpam-3246	113	31	0.3	0.3	NUM
ejpam-3246	113	32	)	)	PUNCT
ejpam-3246	113	33	∪	∪	X
ejpam-3246	113	34	{	{	PUNCT
ejpam-3246	113	35	0.4	0.4	NUM
ejpam-3246	113	36	,	,	PUNCT
ejpam-3246	113	37	0.6	0.6	NUM
ejpam-3246	113	38	,	,	PUNCT
ejpam-3246	113	39	0.8	0.8	NUM
ejpam-3246	113	40	}	}	PUNCT
ejpam-3246	113	41	if	if	SCONJ
ejpam-3246	113	42	x	x	PROPN
ejpam-3246	113	43	=	=	SYM
ejpam-3246	113	44	1	1	NUM
ejpam-3246	113	45	,	,	PUNCT
ejpam-3246	113	46	(	(	PUNCT
ejpam-3246	113	47	0.1	0.1	NUM
ejpam-3246	113	48	,	,	PUNCT
ejpam-3246	113	49	0.2	0.2	NUM
ejpam-3246	113	50	]	]	PUNCT
ejpam-3246	113	51	∪	∪	X
ejpam-3246	113	52	{	{	PUNCT
ejpam-3246	113	53	0.4	0.4	NUM
ejpam-3246	113	54	}	}	PUNCT
ejpam-3246	113	55	∪	∪	NOUN
ejpam-3246	113	56	[	[	X
ejpam-3246	113	57	0.6	0.6	NUM
ejpam-3246	113	58	,	,	PUNCT
ejpam-3246	113	59	0.8	0.8	NUM
ejpam-3246	113	60	]	]	PUNCT
ejpam-3246	113	61	if	if	SCONJ
ejpam-3246	113	62	x	x	PROPN
ejpam-3246	113	63	=	=	SYM
ejpam-3246	113	64	2	2	NUM
ejpam-3246	113	65	,	,	PUNCT
ejpam-3246	113	66	(	(	PUNCT
ejpam-3246	113	67	0.1	0.1	NUM
ejpam-3246	113	68	,	,	PUNCT
ejpam-3246	113	69	0.4	0.4	NUM
ejpam-3246	113	70	]	]	PUNCT
ejpam-3246	113	71	∪	∪	X
ejpam-3246	113	72	[	[	X
ejpam-3246	113	73	0.6	0.6	NUM
ejpam-3246	113	74	,	,	PUNCT
ejpam-3246	113	75	0.8	0.8	NUM
ejpam-3246	113	76	]	]	PUNCT
ejpam-3246	113	77	if	if	SCONJ
ejpam-3246	113	78	x	x	X
ejpam-3246	113	79	=	=	PUNCT
ejpam-3246	113	80	a	a	X
ejpam-3246	113	81	,	,	PUNCT
ejpam-3246	113	82	[	[	X
ejpam-3246	113	83	0.1	0.1	NUM
ejpam-3246	113	84	,	,	PUNCT
ejpam-3246	113	85	0.4	0.4	NUM
ejpam-3246	113	86	]	]	PUNCT
ejpam-3246	113	87	∪	∪	X
ejpam-3246	113	88	[	[	X
ejpam-3246	113	89	0.6	0.6	NUM
ejpam-3246	113	90	,	,	PUNCT
ejpam-3246	113	91	0.9	0.9	NUM
ejpam-3246	113	92	)	)	PUNCT
ejpam-3246	113	93	if	if	SCONJ
ejpam-3246	113	94	x	x	PRON
ejpam-3246	113	95	=	=	PUNCT
ejpam-3246	113	96	b.	b.	PROPN
ejpam-3246	113	97	it	it	PRON
ejpam-3246	113	98	is	be	AUX
ejpam-3246	113	99	routine	routine	ADJ
ejpam-3246	113	100	to	to	PART
ejpam-3246	113	101	verify	verify	VERB
ejpam-3246	113	102	that	that	SCONJ
ejpam-3246	113	103	h	h	NOUN
ejpam-3246	113	104	is	be	AUX
ejpam-3246	113	105	a	a	DET
ejpam-3246	113	106	uni	uni	ADJ
ejpam-3246	113	107	-	-	ADJ
ejpam-3246	113	108	hesitant	hesitant	ADJ
ejpam-3246	113	109	fuzzy	fuzzy	ADJ
ejpam-3246	113	110	ideal	ideal	NOUN
ejpam-3246	113	111	on	on	ADP
ejpam-3246	113	112	x.	x.	PROPN
ejpam-3246	113	113	example	example	NOUN
ejpam-3246	113	114	3	3	X
ejpam-3246	113	115	.	.	X
ejpam-3246	113	116	consider	consider	VERB
ejpam-3246	113	117	a	a	DET
ejpam-3246	113	118	bci	bci	NOUN
ejpam-3246	113	119	-	-	NOUN
ejpam-3246	113	120	algebra	algebra	NOUN
ejpam-3246	113	121	x	x	SYM
ejpam-3246	113	122	=	=	SYM
ejpam-3246	113	123	{	{	PUNCT
ejpam-3246	113	124	2n	2n	NUM
ejpam-3246	113	125	|	|	CCONJ
ejpam-3246	113	126	n	n	CCONJ
ejpam-3246	113	127	∈	∈	PROPN
ejpam-3246	113	128	z	z	NOUN
ejpam-3246	113	129	}	}	PUNCT
ejpam-3246	113	130	with	with	ADP
ejpam-3246	113	131	a	a	DET
ejpam-3246	113	132	binary	binary	ADJ
ejpam-3246	113	133	operation	operation	NOUN
ejpam-3246	113	134	“	"	PUNCT
ejpam-3246	113	135	÷	÷	NOUN
ejpam-3246	113	136	”	"	PUNCT
ejpam-3246	113	137	(	(	PUNCT
ejpam-3246	113	138	usual	usual	ADJ
ejpam-3246	113	139	division	division	NOUN
ejpam-3246	113	140	)	)	PUNCT
ejpam-3246	113	141	.	.	PUNCT
ejpam-3246	114	1	let	let	VERB
ejpam-3246	114	2	g	g	PRON
ejpam-3246	114	3	be	be	AUX
ejpam-3246	114	4	a	a	DET
ejpam-3246	114	5	hesitant	hesitant	ADJ
ejpam-3246	114	6	fuzzy	fuzzy	ADJ
ejpam-3246	114	7	set	set	NOUN
ejpam-3246	114	8	on	on	ADP
ejpam-3246	114	9	x	x	PUNCT
ejpam-3246	114	10	given	give	VERB
ejpam-3246	114	11	as	as	SCONJ
ejpam-3246	114	12	follows	follow	VERB
ejpam-3246	114	13	:	:	PUNCT
ejpam-3246	114	14	g	g	NOUN
ejpam-3246	114	15	:	:	PUNCT
ejpam-3246	114	16	x	x	X
ejpam-3246	114	17	→	→	X
ejpam-3246	114	18	p	p	X
ejpam-3246	114	19	(	(	PUNCT
ejpam-3246	114	20	[	[	X
ejpam-3246	114	21	0	0	NUM
ejpam-3246	114	22	,	,	PUNCT
ejpam-3246	114	23	1	1	NUM
ejpam-3246	114	24	]	]	NUM
ejpam-3246	114	25	)	)	PUNCT
ejpam-3246	114	26	,	,	PUNCT
ejpam-3246	114	27	x	x	X
ejpam-3246	114	28	7→	7→	X
ejpam-3246	114	29	{	{	PUNCT
ejpam-3246	114	30	λ1	λ1	ADJ
ejpam-3246	114	31	if	if	SCONJ
ejpam-3246	114	32	n	n	PRON
ejpam-3246	114	33	≥	≥	NOUN
ejpam-3246	114	34	0	0	NUM
ejpam-3246	114	35	,	,	PUNCT
ejpam-3246	114	36	λ2	λ2	PRON
ejpam-3246	114	37	if	if	SCONJ
ejpam-3246	114	38	n	n	X
ejpam-3246	114	39	<	<	X
ejpam-3246	114	40	0	0	PROPN
ejpam-3246	114	41	,	,	PUNCT
ejpam-3246	114	42	where	where	SCONJ
ejpam-3246	114	43	λ1	λ1	ADJ
ejpam-3246	114	44	and	and	CCONJ
ejpam-3246	114	45	λ2	λ2	NOUN
ejpam-3246	114	46	are	be	AUX
ejpam-3246	114	47	subsets	subset	NOUN
ejpam-3246	114	48	of	of	ADP
ejpam-3246	114	49	[	[	X
ejpam-3246	114	50	0	0	NUM
ejpam-3246	114	51	,	,	PUNCT
ejpam-3246	114	52	1	1	NUM
ejpam-3246	114	53	]	]	PUNCT
ejpam-3246	114	54	with	with	ADP
ejpam-3246	114	55	λ1	λ1	PROPN
ejpam-3246	114	56	(	(	PUNCT
ejpam-3246	114	57	λ2	λ2	PROPN
ejpam-3246	114	58	.	.	PUNCT
ejpam-3246	115	1	then	then	ADV
ejpam-3246	115	2	g	g	PROPN
ejpam-3246	115	3	is	be	AUX
ejpam-3246	115	4	a	a	DET
ejpam-3246	115	5	uni	uni	ADJ
ejpam-3246	115	6	-	-	ADJ
ejpam-3246	115	7	hesitant	hesitant	ADJ
ejpam-3246	115	8	fuzzy	fuzzy	ADJ
ejpam-3246	115	9	ideal	ideal	NOUN
ejpam-3246	115	10	on	on	ADP
ejpam-3246	115	11	x	x	SYM
ejpam-3246	115	12	,	,	PUNCT
ejpam-3246	115	13	lemma	lemma	PROPN
ejpam-3246	115	14	1	1	NUM
ejpam-3246	115	15	.	.	PUNCT
ejpam-3246	116	1	every	every	DET
ejpam-3246	116	2	uni	uni	ADJ
ejpam-3246	116	3	-	-	ADJ
ejpam-3246	116	4	hesitant	hesitant	ADJ
ejpam-3246	116	5	fuzzy	fuzzy	ADJ
ejpam-3246	116	6	ideal	ideal	NOUN
ejpam-3246	116	7	g	g	NOUN
ejpam-3246	116	8	on	on	ADP
ejpam-3246	116	9	x	x	PART
ejpam-3246	116	10	satisfies	satisfie	NOUN
ejpam-3246	116	11	the	the	DET
ejpam-3246	116	12	following	follow	VERB
ejpam-3246	116	13	condition	condition	NOUN
ejpam-3246	116	14	.	.	PUNCT
ejpam-3246	117	1	(	(	PUNCT
ejpam-3246	117	2	∀x	∀x	X
ejpam-3246	117	3	,	,	PUNCT
ejpam-3246	117	4	y	y	PROPN
ejpam-3246	117	5	∈	∈	PROPN
ejpam-3246	117	6	x	x	X
ejpam-3246	117	7	)	)	PUNCT
ejpam-3246	117	8	(	(	PUNCT
ejpam-3246	117	9	x	x	SYM
ejpam-3246	117	10	∗	∗	NOUN
ejpam-3246	117	11	y	y	NOUN
ejpam-3246	117	12	=	=	SYM
ejpam-3246	117	13	0	0	NUM
ejpam-3246	117	14	⇒	⇒	PROPN
ejpam-3246	117	15	xg	xg	PROPN
ejpam-3246	117	16	⊆	⊆	NUM
ejpam-3246	117	17	yg	yg	PROPN
ejpam-3246	117	18	)	)	PUNCT
ejpam-3246	117	19	.	.	PUNCT
ejpam-3246	118	1	(	(	PUNCT
ejpam-3246	118	2	6	6	X
ejpam-3246	118	3	)	)	PUNCT
ejpam-3246	118	4	proof	proof	NOUN
ejpam-3246	118	5	.	.	PUNCT
ejpam-3246	119	1	it	it	PRON
ejpam-3246	119	2	is	be	AUX
ejpam-3246	119	3	straightforward	straightforward	ADJ
ejpam-3246	119	4	by	by	ADP
ejpam-3246	119	5	(	(	PUNCT
ejpam-3246	119	6	5	5	NUM
ejpam-3246	119	7	)	)	PUNCT
ejpam-3246	119	8	and	and	CCONJ
ejpam-3246	119	9	proposition	proposition	NOUN
ejpam-3246	119	10	1	1	NUM
ejpam-3246	119	11	.	.	X
ejpam-3246	119	12	proposition	proposition	NOUN
ejpam-3246	119	13	4	4	NUM
ejpam-3246	119	14	.	.	PUNCT
ejpam-3246	120	1	every	every	DET
ejpam-3246	120	2	uni	uni	ADJ
ejpam-3246	120	3	-	-	ADJ
ejpam-3246	120	4	hesitant	hesitant	ADJ
ejpam-3246	120	5	fuzzy	fuzzy	ADJ
ejpam-3246	120	6	ideal	ideal	NOUN
ejpam-3246	120	7	g	g	NOUN
ejpam-3246	120	8	on	on	ADP
ejpam-3246	120	9	x	x	NOUN
ejpam-3246	120	10	satisfies	satisfie	NOUN
ejpam-3246	120	11	:	:	PUNCT
ejpam-3246	120	12	g.	g.	PROPN
ejpam-3246	120	13	muhiuddin	muhiuddin	PROPN
ejpam-3246	120	14	,	,	PUNCT
ejpam-3246	120	15	s.	s.	PROPN
ejpam-3246	120	16	aldhafeeri	aldhafeeri	PROPN
ejpam-3246	120	17	/	/	SYM
ejpam-3246	120	18	eur	eur	PROPN
ejpam-3246	120	19	.	.	PUNCT
ejpam-3246	121	1	j.	j.	PROPN
ejpam-3246	121	2	pure	pure	PROPN
ejpam-3246	121	3	appl	appl	PROPN
ejpam-3246	121	4	.	.	PROPN
ejpam-3246	121	5	math	math	PROPN
ejpam-3246	121	6	,	,	PUNCT
ejpam-3246	121	7	11	11	NUM
ejpam-3246	121	8	(	(	PUNCT
ejpam-3246	121	9	2	2	NUM
ejpam-3246	121	10	)	)	PUNCT
ejpam-3246	121	11	(	(	PUNCT
ejpam-3246	121	12	2018	2018	NUM
ejpam-3246	121	13	)	)	PUNCT
ejpam-3246	121	14	,	,	PUNCT
ejpam-3246	121	15	417	417	NUM
ejpam-3246	121	16	-	-	SYM
ejpam-3246	121	17	430	430	NUM
ejpam-3246	121	18	423	423	NUM
ejpam-3246	121	19	(	(	PUNCT
ejpam-3246	121	20	1	1	NUM
ejpam-3246	121	21	)	)	PUNCT
ejpam-3246	121	22	(	(	PUNCT
ejpam-3246	121	23	∀x	∀x	X
ejpam-3246	121	24	,	,	PUNCT
ejpam-3246	121	25	y	y	PROPN
ejpam-3246	121	26	,	,	PUNCT
ejpam-3246	121	27	z	z	NOUN
ejpam-3246	121	28	∈	∈	PROPN
ejpam-3246	121	29	x	x	X
ejpam-3246	121	30	)	)	PUNCT
ejpam-3246	121	31	(	(	PUNCT
ejpam-3246	121	32	(	(	PUNCT
ejpam-3246	121	33	x	x	SYM
ejpam-3246	121	34	∗	∗	NOUN
ejpam-3246	121	35	y)g	y)g	NOUN
ejpam-3246	122	1	⊆	⊆	NUM
ejpam-3246	122	2	(	(	PUNCT
ejpam-3246	122	3	x	x	X
ejpam-3246	122	4	∗	∗	NOUN
ejpam-3246	122	5	z)g	z)g	NUM
ejpam-3246	122	6	∪	∪	X
ejpam-3246	122	7	(	(	PUNCT
ejpam-3246	122	8	z	z	NOUN
ejpam-3246	122	9	∗	∗	NOUN
ejpam-3246	122	10	y)g	y)g	NOUN
ejpam-3246	122	11	)	)	PUNCT
ejpam-3246	122	12	.	.	PUNCT
ejpam-3246	123	1	(	(	PUNCT
ejpam-3246	123	2	2	2	X
ejpam-3246	123	3	)	)	PUNCT
ejpam-3246	123	4	(	(	PUNCT
ejpam-3246	123	5	∀x	∀x	X
ejpam-3246	123	6	,	,	PUNCT
ejpam-3246	123	7	y	y	PROPN
ejpam-3246	123	8	∈	∈	PROPN
ejpam-3246	123	9	x	x	X
ejpam-3246	123	10	)	)	PUNCT
ejpam-3246	123	11	(	(	PUNCT
ejpam-3246	123	12	(	(	PUNCT
ejpam-3246	123	13	x	x	X
ejpam-3246	123	14	∗	∗	NOUN
ejpam-3246	123	15	y)g	y)g	X
ejpam-3246	124	1	=	=	SYM
ejpam-3246	124	2	0	0	NUM
ejpam-3246	124	3	g	g	NOUN
ejpam-3246	124	4	⇒	⇒	NOUN
ejpam-3246	124	5	xg	xg	PROPN
ejpam-3246	124	6	⊆	⊆	NUM
ejpam-3246	124	7	yg	yg	PROPN
ejpam-3246	124	8	)	)	PUNCT
ejpam-3246	124	9	.	.	PUNCT
ejpam-3246	125	1	proof	proof	NOUN
ejpam-3246	125	2	.	.	PUNCT
ejpam-3246	126	1	since	since	SCONJ
ejpam-3246	126	2	(	(	PUNCT
ejpam-3246	126	3	(	(	PUNCT
ejpam-3246	126	4	x	x	SYM
ejpam-3246	126	5	∗	∗	PROPN
ejpam-3246	126	6	y	y	NOUN
ejpam-3246	126	7	)	)	PUNCT
ejpam-3246	126	8	∗	∗	NOUN
ejpam-3246	126	9	(	(	PUNCT
ejpam-3246	126	10	x	x	X
ejpam-3246	126	11	∗	∗	PROPN
ejpam-3246	126	12	z	z	NOUN
ejpam-3246	126	13	)	)	PUNCT
ejpam-3246	126	14	)	)	PUNCT
ejpam-3246	126	15	∗	∗	NOUN
ejpam-3246	126	16	(	(	PUNCT
ejpam-3246	126	17	z	z	NOUN
ejpam-3246	126	18	∗	∗	NOUN
ejpam-3246	126	19	y	y	NOUN
ejpam-3246	126	20	)	)	PUNCT
ejpam-3246	126	21	=	=	SYM
ejpam-3246	126	22	0	0	NUM
ejpam-3246	126	23	,	,	PUNCT
ejpam-3246	126	24	it	it	PRON
ejpam-3246	126	25	follows	follow	VERB
ejpam-3246	126	26	from	from	ADP
ejpam-3246	126	27	lemma	lemma	PROPN
ejpam-3246	126	28	1	1	NUM
ejpam-3246	126	29	that	that	PRON
ejpam-3246	126	30	(	(	PUNCT
ejpam-3246	126	31	(	(	PUNCT
ejpam-3246	126	32	x	x	SYM
ejpam-3246	126	33	∗	∗	PROPN
ejpam-3246	126	34	y	y	NOUN
ejpam-3246	126	35	)	)	PUNCT
ejpam-3246	126	36	∗	∗	NOUN
ejpam-3246	126	37	(	(	PUNCT
ejpam-3246	126	38	x	x	NOUN
ejpam-3246	126	39	∗	∗	NOUN
ejpam-3246	126	40	z))g	z))g	NOUN
ejpam-3246	126	41	⊆	⊆	NUM
ejpam-3246	126	42	(	(	PUNCT
ejpam-3246	126	43	z	z	NOUN
ejpam-3246	126	44	∗	∗	NOUN
ejpam-3246	126	45	y)g	y)g	NOUN
ejpam-3246	126	46	.	.	PUNCT
ejpam-3246	127	1	hence	hence	ADV
ejpam-3246	127	2	(	(	PUNCT
ejpam-3246	127	3	x	x	X
ejpam-3246	127	4	∗	∗	NOUN
ejpam-3246	127	5	y)g	y)g	NOUN
ejpam-3246	127	6	⊆	⊆	NUM
ejpam-3246	127	7	(	(	PUNCT
ejpam-3246	127	8	(	(	PUNCT
ejpam-3246	127	9	x	x	SYM
ejpam-3246	127	10	∗	∗	PROPN
ejpam-3246	127	11	y	y	NOUN
ejpam-3246	127	12	)	)	PUNCT
ejpam-3246	127	13	∗	∗	NOUN
ejpam-3246	127	14	(	(	PUNCT
ejpam-3246	127	15	x	x	NOUN
ejpam-3246	127	16	∗	∗	NOUN
ejpam-3246	127	17	z))g	z))g	NOUN
ejpam-3246	127	18	∪	∪	NOUN
ejpam-3246	127	19	(	(	PUNCT
ejpam-3246	127	20	x	x	X
ejpam-3246	127	21	∗	∗	NOUN
ejpam-3246	127	22	z)g	z)g	NUM
ejpam-3246	127	23	⊆	⊆	NUM
ejpam-3246	127	24	(	(	PUNCT
ejpam-3246	127	25	x	x	X
ejpam-3246	127	26	∗	∗	NOUN
ejpam-3246	127	27	z)g	z)g	NUM
ejpam-3246	127	28	∪	∪	X
ejpam-3246	127	29	(	(	PUNCT
ejpam-3246	127	30	z	z	NOUN
ejpam-3246	127	31	∗	∗	NOUN
ejpam-3246	127	32	y)g	y)g	NOUN
ejpam-3246	127	33	for	for	ADP
ejpam-3246	127	34	all	all	DET
ejpam-3246	127	35	x	x	NOUN
ejpam-3246	127	36	,	,	PUNCT
ejpam-3246	127	37	y	y	PROPN
ejpam-3246	127	38	,	,	PUNCT
ejpam-3246	127	39	z	z	PROPN
ejpam-3246	127	40	∈	∈	PROPN
ejpam-3246	127	41	x.	x.	NOUN
ejpam-3246	127	42	(	(	PUNCT
ejpam-3246	127	43	2	2	NUM
ejpam-3246	127	44	)	)	PUNCT
ejpam-3246	127	45	if	if	SCONJ
ejpam-3246	127	46	(	(	PUNCT
ejpam-3246	127	47	x	x	X
ejpam-3246	127	48	∗	∗	NOUN
ejpam-3246	127	49	y)g	y)g	X
ejpam-3246	128	1	=	=	SYM
ejpam-3246	128	2	0	0	NUM
ejpam-3246	128	3	g	g	NOUN
ejpam-3246	128	4	,	,	PUNCT
ejpam-3246	128	5	then	then	ADV
ejpam-3246	128	6	xg	xg	PROPN
ejpam-3246	128	7	⊆	⊆	NUM
ejpam-3246	128	8	(	(	PUNCT
ejpam-3246	128	9	x	x	SYM
ejpam-3246	128	10	∗	∗	NOUN
ejpam-3246	128	11	y)g	y)g	NOUN
ejpam-3246	128	12	∪	∪	PROPN
ejpam-3246	128	13	yg	yg	PROPN
ejpam-3246	128	14	=	=	SYM
ejpam-3246	128	15	0	0	NUM
ejpam-3246	128	16	g	g	NOUN
ejpam-3246	128	17	∪	∪	NOUN
ejpam-3246	128	18	yg	yg	PROPN
ejpam-3246	128	19	=	=	PROPN
ejpam-3246	128	20	yg	yg	PROPN
ejpam-3246	128	21	for	for	ADP
ejpam-3246	128	22	all	all	DET
ejpam-3246	128	23	x	x	NOUN
ejpam-3246	128	24	,	,	PUNCT
ejpam-3246	128	25	y	y	PROPN
ejpam-3246	128	26	∈	∈	PROPN
ejpam-3246	128	27	x.	x.	NOUN
ejpam-3246	128	28	proposition	proposition	NOUN
ejpam-3246	128	29	5	5	NUM
ejpam-3246	128	30	.	.	PUNCT
ejpam-3246	129	1	for	for	ADP
ejpam-3246	129	2	any	any	DET
ejpam-3246	129	3	uni	uni	ADJ
ejpam-3246	129	4	-	-	ADJ
ejpam-3246	129	5	hesitant	hesitant	ADJ
ejpam-3246	129	6	fuzzy	fuzzy	ADJ
ejpam-3246	129	7	ideal	ideal	NOUN
ejpam-3246	129	8	g	g	NOUN
ejpam-3246	129	9	on	on	ADP
ejpam-3246	129	10	x	x	SYM
ejpam-3246	129	11	,	,	PUNCT
ejpam-3246	129	12	the	the	DET
ejpam-3246	129	13	following	follow	VERB
ejpam-3246	129	14	conditions	condition	NOUN
ejpam-3246	129	15	are	be	AUX
ejpam-3246	129	16	equivalent	equivalent	ADJ
ejpam-3246	129	17	:	:	PUNCT
ejpam-3246	129	18	(	(	PUNCT
ejpam-3246	129	19	1	1	X
ejpam-3246	129	20	)	)	PUNCT
ejpam-3246	129	21	(	(	PUNCT
ejpam-3246	129	22	∀x	∀x	X
ejpam-3246	129	23	,	,	PUNCT
ejpam-3246	129	24	y	y	PROPN
ejpam-3246	129	25	∈	∈	PROPN
ejpam-3246	129	26	x	x	X
ejpam-3246	129	27	)	)	PUNCT
ejpam-3246	129	28	(	(	PUNCT
ejpam-3246	129	29	(	(	PUNCT
ejpam-3246	129	30	x	x	SYM
ejpam-3246	129	31	∗	∗	NOUN
ejpam-3246	129	32	y)g	y)g	NOUN
ejpam-3246	129	33	⊆	⊆	NUM
ejpam-3246	129	34	(	(	PUNCT
ejpam-3246	129	35	(	(	PUNCT
ejpam-3246	129	36	x	x	SYM
ejpam-3246	129	37	∗	∗	PROPN
ejpam-3246	129	38	y	y	NOUN
ejpam-3246	129	39	)	)	PUNCT
ejpam-3246	129	40	∗	∗	NOUN
ejpam-3246	129	41	y)g	y)g	NOUN
ejpam-3246	129	42	)	)	PUNCT
ejpam-3246	129	43	.	.	PUNCT
ejpam-3246	130	1	(	(	PUNCT
ejpam-3246	130	2	2	2	X
ejpam-3246	130	3	)	)	PUNCT
ejpam-3246	130	4	(	(	PUNCT
ejpam-3246	130	5	∀x	∀x	X
ejpam-3246	130	6	,	,	PUNCT
ejpam-3246	130	7	y	y	PROPN
ejpam-3246	130	8	,	,	PUNCT
ejpam-3246	130	9	z	z	NOUN
ejpam-3246	130	10	∈	∈	PROPN
ejpam-3246	130	11	x	x	X
ejpam-3246	130	12	)	)	PUNCT
ejpam-3246	130	13	(	(	PUNCT
ejpam-3246	130	14	(	(	PUNCT
ejpam-3246	130	15	(	(	PUNCT
ejpam-3246	130	16	x	x	SYM
ejpam-3246	130	17	∗	∗	PROPN
ejpam-3246	130	18	z	z	NOUN
ejpam-3246	130	19	)	)	PUNCT
ejpam-3246	130	20	∗	∗	NOUN
ejpam-3246	130	21	(	(	PUNCT
ejpam-3246	130	22	y	y	NOUN
ejpam-3246	130	23	∗	∗	NOUN
ejpam-3246	130	24	z))g	z))g	NOUN
ejpam-3246	130	25	⊆	⊆	NUM
ejpam-3246	130	26	(	(	PUNCT
ejpam-3246	130	27	(	(	PUNCT
ejpam-3246	130	28	x	x	SYM
ejpam-3246	130	29	∗	∗	PROPN
ejpam-3246	130	30	y	y	NOUN
ejpam-3246	130	31	)	)	PUNCT
ejpam-3246	130	32	∗	∗	NOUN
ejpam-3246	130	33	z)g	z)g	NUM
ejpam-3246	130	34	)	)	PUNCT
ejpam-3246	130	35	.	.	PUNCT
ejpam-3246	131	1	proof	proof	NOUN
ejpam-3246	131	2	.	.	PUNCT
ejpam-3246	132	1	assume	assume	VERB
ejpam-3246	132	2	that	that	SCONJ
ejpam-3246	132	3	(	(	PUNCT
ejpam-3246	132	4	1	1	X
ejpam-3246	132	5	)	)	PUNCT
ejpam-3246	132	6	is	be	AUX
ejpam-3246	132	7	valid	valid	ADJ
ejpam-3246	132	8	and	and	CCONJ
ejpam-3246	132	9	let	let	VERB
ejpam-3246	132	10	x	x	PRON
ejpam-3246	132	11	,	,	PUNCT
ejpam-3246	132	12	y	y	PROPN
ejpam-3246	132	13	,	,	PUNCT
ejpam-3246	132	14	z	z	PROPN
ejpam-3246	132	15	∈	∈	PROPN
ejpam-3246	132	16	x.	x.	NOUN
ejpam-3246	133	1	since	since	SCONJ
ejpam-3246	133	2	(	(	PUNCT
ejpam-3246	133	3	(	(	PUNCT
ejpam-3246	133	4	x	x	SYM
ejpam-3246	133	5	∗	∗	NOUN
ejpam-3246	133	6	(	(	PUNCT
ejpam-3246	133	7	y	y	PROPN
ejpam-3246	133	8	∗	∗	PROPN
ejpam-3246	133	9	z	z	NOUN
ejpam-3246	133	10	)	)	PUNCT
ejpam-3246	133	11	)	)	PUNCT
ejpam-3246	133	12	∗	∗	PROPN
ejpam-3246	133	13	z	z	NOUN
ejpam-3246	133	14	)	)	PUNCT
ejpam-3246	133	15	∗	∗	NOUN
ejpam-3246	133	16	z	z	NOUN
ejpam-3246	133	17	=	=	SYM
ejpam-3246	133	18	(	(	PUNCT
ejpam-3246	133	19	(	(	PUNCT
ejpam-3246	133	20	x	x	SYM
ejpam-3246	133	21	∗	∗	PROPN
ejpam-3246	133	22	z	z	NOUN
ejpam-3246	133	23	)	)	PUNCT
ejpam-3246	133	24	∗	∗	NOUN
ejpam-3246	133	25	(	(	PUNCT
ejpam-3246	133	26	y	y	PROPN
ejpam-3246	133	27	∗	∗	PROPN
ejpam-3246	133	28	z	z	NOUN
ejpam-3246	133	29	)	)	PUNCT
ejpam-3246	133	30	)	)	PUNCT
ejpam-3246	133	31	∗	∗	NOUN
ejpam-3246	133	32	z	z	NOUN
ejpam-3246	133	33	≤	≤	NOUN
ejpam-3246	134	1	(	(	PUNCT
ejpam-3246	134	2	x	x	X
ejpam-3246	134	3	∗	∗	PROPN
ejpam-3246	134	4	y	y	NOUN
ejpam-3246	134	5	)	)	PUNCT
ejpam-3246	134	6	∗	∗	NOUN
ejpam-3246	134	7	z	z	PROPN
ejpam-3246	134	8	,	,	PUNCT
ejpam-3246	134	9	that	that	ADV
ejpam-3246	134	10	is	is	ADV
ejpam-3246	134	11	,	,	PUNCT
ejpam-3246	134	12	(	(	PUNCT
ejpam-3246	134	13	(	(	PUNCT
ejpam-3246	134	14	(	(	PUNCT
ejpam-3246	134	15	x	x	SYM
ejpam-3246	134	16	∗	∗	NOUN
ejpam-3246	134	17	(	(	PUNCT
ejpam-3246	134	18	y	y	PROPN
ejpam-3246	134	19	∗	∗	PROPN
ejpam-3246	134	20	z	z	NOUN
ejpam-3246	134	21	)	)	PUNCT
ejpam-3246	134	22	)	)	PUNCT
ejpam-3246	134	23	∗	∗	PROPN
ejpam-3246	135	1	z	z	NOUN
ejpam-3246	135	2	)	)	PUNCT
ejpam-3246	135	3	∗	∗	NOUN
ejpam-3246	135	4	z	z	NOUN
ejpam-3246	135	5	)	)	PUNCT
ejpam-3246	135	6	∗	∗	NOUN
ejpam-3246	135	7	(	(	PUNCT
ejpam-3246	135	8	(	(	PUNCT
ejpam-3246	135	9	x	x	SYM
ejpam-3246	135	10	∗	∗	PROPN
ejpam-3246	135	11	y	y	NOUN
ejpam-3246	135	12	)	)	PUNCT
ejpam-3246	135	13	∗	∗	NOUN
ejpam-3246	135	14	z	z	NOUN
ejpam-3246	135	15	)	)	PUNCT
ejpam-3246	136	1	=	=	SYM
ejpam-3246	136	2	0	0	NUM
ejpam-3246	136	3	,	,	PUNCT
ejpam-3246	136	4	it	it	PRON
ejpam-3246	136	5	follows	follow	VERB
ejpam-3246	136	6	from	from	ADP
ejpam-3246	136	7	(	(	PUNCT
ejpam-3246	136	8	a3	a3	NOUN
ejpam-3246	136	9	)	)	PUNCT
ejpam-3246	136	10	,	,	PUNCT
ejpam-3246	136	11	(	(	PUNCT
ejpam-3246	136	12	1	1	X
ejpam-3246	136	13	)	)	PUNCT
ejpam-3246	136	14	and	and	CCONJ
ejpam-3246	136	15	(	(	PUNCT
ejpam-3246	136	16	6	6	NUM
ejpam-3246	136	17	)	)	PUNCT
ejpam-3246	136	18	that	that	SCONJ
ejpam-3246	136	19	(	(	PUNCT
ejpam-3246	136	20	(	(	PUNCT
ejpam-3246	136	21	x	x	SYM
ejpam-3246	136	22	∗	∗	PROPN
ejpam-3246	136	23	z	z	NOUN
ejpam-3246	136	24	)	)	PUNCT
ejpam-3246	136	25	∗	∗	NOUN
ejpam-3246	136	26	(	(	PUNCT
ejpam-3246	136	27	y	y	NOUN
ejpam-3246	136	28	∗	∗	NOUN
ejpam-3246	136	29	z))g	z))g	NOUN
ejpam-3246	136	30	=	=	SYM
ejpam-3246	136	31	(	(	PUNCT
ejpam-3246	136	32	(	(	PUNCT
ejpam-3246	136	33	x	x	SYM
ejpam-3246	136	34	∗	∗	NOUN
ejpam-3246	136	35	(	(	PUNCT
ejpam-3246	136	36	y	y	PROPN
ejpam-3246	136	37	∗	∗	PROPN
ejpam-3246	136	38	z	z	NOUN
ejpam-3246	136	39	)	)	PUNCT
ejpam-3246	136	40	)	)	PUNCT
ejpam-3246	137	1	∗	∗	NOUN
ejpam-3246	137	2	z)g	z)g	NUM
ejpam-3246	137	3	⊆	⊆	X
ejpam-3246	137	4	(	(	PUNCT
ejpam-3246	137	5	(	(	PUNCT
ejpam-3246	137	6	(	(	PUNCT
ejpam-3246	137	7	x	x	SYM
ejpam-3246	137	8	∗	∗	NOUN
ejpam-3246	137	9	(	(	PUNCT
ejpam-3246	137	10	y	y	PROPN
ejpam-3246	137	11	∗	∗	PROPN
ejpam-3246	137	12	z	z	NOUN
ejpam-3246	137	13	)	)	PUNCT
ejpam-3246	137	14	)	)	PUNCT
ejpam-3246	137	15	∗	∗	PROPN
ejpam-3246	137	16	z	z	NOUN
ejpam-3246	137	17	)	)	PUNCT
ejpam-3246	137	18	∗	∗	NOUN
ejpam-3246	137	19	z)g	z)g	NOUN
ejpam-3246	137	20	⊆	⊆	X
ejpam-3246	137	21	(	(	PUNCT
ejpam-3246	137	22	(	(	PUNCT
ejpam-3246	137	23	x	x	SYM
ejpam-3246	137	24	∗	∗	PROPN
ejpam-3246	137	25	y	y	NOUN
ejpam-3246	137	26	)	)	PUNCT
ejpam-3246	137	27	∗	∗	NOUN
ejpam-3246	137	28	z)g	z)g	NUM
ejpam-3246	137	29	.	.	PUNCT
ejpam-3246	138	1	conversely	conversely	ADV
ejpam-3246	138	2	,	,	PUNCT
ejpam-3246	138	3	suppose	suppose	VERB
ejpam-3246	138	4	that	that	SCONJ
ejpam-3246	138	5	(	(	PUNCT
ejpam-3246	138	6	2	2	X
ejpam-3246	138	7	)	)	PUNCT
ejpam-3246	138	8	holds	hold	VERB
ejpam-3246	138	9	.	.	PUNCT
ejpam-3246	139	1	if	if	SCONJ
ejpam-3246	139	2	we	we	PRON
ejpam-3246	139	3	take	take	VERB
ejpam-3246	139	4	y	y	NOUN
ejpam-3246	139	5	=	=	NOUN
ejpam-3246	139	6	z	z	NOUN
ejpam-3246	139	7	in	in	ADP
ejpam-3246	139	8	(	(	PUNCT
ejpam-3246	139	9	2	2	NUM
ejpam-3246	139	10	)	)	PUNCT
ejpam-3246	139	11	,	,	PUNCT
ejpam-3246	139	12	then	then	ADV
ejpam-3246	139	13	(	(	PUNCT
ejpam-3246	139	14	(	(	PUNCT
ejpam-3246	139	15	x	x	SYM
ejpam-3246	139	16	∗	∗	PROPN
ejpam-3246	139	17	z	z	NOUN
ejpam-3246	139	18	)	)	PUNCT
ejpam-3246	139	19	∗	∗	NOUN
ejpam-3246	139	20	z)g	z)g	X
ejpam-3246	140	1	⊇	⊇	X
ejpam-3246	140	2	(	(	PUNCT
ejpam-3246	140	3	(	(	PUNCT
ejpam-3246	140	4	x	x	SYM
ejpam-3246	140	5	∗	∗	PROPN
ejpam-3246	140	6	z	z	NOUN
ejpam-3246	140	7	)	)	PUNCT
ejpam-3246	140	8	∗	∗	NOUN
ejpam-3246	140	9	(	(	PUNCT
ejpam-3246	140	10	z	z	NOUN
ejpam-3246	140	11	∗	∗	NOUN
ejpam-3246	140	12	z))g	z))g	NOUN
ejpam-3246	140	13	=	=	SYM
ejpam-3246	140	14	(	(	PUNCT
ejpam-3246	140	15	(	(	PUNCT
ejpam-3246	140	16	x	x	SYM
ejpam-3246	140	17	∗	∗	PROPN
ejpam-3246	140	18	z	z	NOUN
ejpam-3246	140	19	)	)	PUNCT
ejpam-3246	140	20	∗	∗	NOUN
ejpam-3246	140	21	0)g	0)g	NOUN
ejpam-3246	140	22	=	=	PUNCT
ejpam-3246	140	23	(	(	PUNCT
ejpam-3246	140	24	x	x	X
ejpam-3246	140	25	∗	∗	NOUN
ejpam-3246	140	26	z)g	z)g	NUM
ejpam-3246	140	27	by	by	ADP
ejpam-3246	140	28	(	(	PUNCT
ejpam-3246	140	29	iii	iii	NOUN
ejpam-3246	140	30	)	)	PUNCT
ejpam-3246	140	31	and	and	CCONJ
ejpam-3246	140	32	(	(	PUNCT
ejpam-3246	140	33	a1	a1	NOUN
ejpam-3246	140	34	)	)	PUNCT
ejpam-3246	140	35	.	.	PUNCT
ejpam-3246	141	1	this	this	PRON
ejpam-3246	141	2	proves	prove	VERB
ejpam-3246	141	3	(	(	PUNCT
ejpam-3246	141	4	1	1	NUM
ejpam-3246	141	5	)	)	PUNCT
ejpam-3246	141	6	.	.	PUNCT
ejpam-3246	142	1	theorem	theorem	NOUN
ejpam-3246	142	2	3	3	NUM
ejpam-3246	142	3	.	.	PUNCT
ejpam-3246	143	1	every	every	DET
ejpam-3246	143	2	uni	uni	ADJ
ejpam-3246	143	3	-	-	ADJ
ejpam-3246	143	4	hesitant	hesitant	ADJ
ejpam-3246	143	5	fuzzy	fuzzy	ADJ
ejpam-3246	143	6	ideal	ideal	NOUN
ejpam-3246	143	7	on	on	ADP
ejpam-3246	143	8	a	a	DET
ejpam-3246	143	9	bck	bck	NOUN
ejpam-3246	143	10	-	-	PUNCT
ejpam-3246	143	11	algebra	algebra	NOUN
ejpam-3246	143	12	is	be	AUX
ejpam-3246	143	13	a	a	DET
ejpam-3246	143	14	uni	uni	ADJ
ejpam-3246	143	15	-	-	ADJ
ejpam-3246	143	16	hesitant	hesitant	ADJ
ejpam-3246	143	17	fuzzy	fuzzy	ADJ
ejpam-3246	143	18	algebra	algebra	NOUN
ejpam-3246	143	19	.	.	PUNCT
ejpam-3246	144	1	proof	proof	NOUN
ejpam-3246	144	2	.	.	PUNCT
ejpam-3246	145	1	let	let	VERB
ejpam-3246	145	2	g	g	PRON
ejpam-3246	145	3	be	be	AUX
ejpam-3246	145	4	a	a	DET
ejpam-3246	145	5	uni	uni	ADJ
ejpam-3246	145	6	-	-	ADJ
ejpam-3246	145	7	hesitant	hesitant	ADJ
ejpam-3246	145	8	fuzzy	fuzzy	ADJ
ejpam-3246	145	9	ideal	ideal	NOUN
ejpam-3246	145	10	on	on	ADP
ejpam-3246	145	11	a	a	DET
ejpam-3246	145	12	bck	bck	NOUN
ejpam-3246	145	13	-	-	PUNCT
ejpam-3246	145	14	algebrax	algebrax	NOUN
ejpam-3246	145	15	.	.	PUNCT
ejpam-3246	146	1	note	note	VERB
ejpam-3246	146	2	that	that	SCONJ
ejpam-3246	146	3	(	(	PUNCT
ejpam-3246	146	4	x∗y)∗x	x∗y)∗x	PROPN
ejpam-3246	146	5	=	=	PUNCT
ejpam-3246	146	6	0	0	NUM
ejpam-3246	146	7	for	for	ADP
ejpam-3246	146	8	all	all	DET
ejpam-3246	146	9	x	x	NOUN
ejpam-3246	146	10	,	,	PUNCT
ejpam-3246	146	11	y	y	PROPN
ejpam-3246	146	12	∈	∈	PROPN
ejpam-3246	146	13	x.	x.	NOUN
ejpam-3246	146	14	using	use	VERB
ejpam-3246	146	15	lemma	lemma	PROPN
ejpam-3246	146	16	1	1	NUM
ejpam-3246	146	17	and	and	CCONJ
ejpam-3246	146	18	(	(	PUNCT
ejpam-3246	146	19	5	5	NUM
ejpam-3246	146	20	)	)	PUNCT
ejpam-3246	146	21	,	,	PUNCT
ejpam-3246	146	22	we	we	PRON
ejpam-3246	146	23	have	have	AUX
ejpam-3246	146	24	(	(	PUNCT
ejpam-3246	146	25	x	x	SYM
ejpam-3246	146	26	∗	∗	NOUN
ejpam-3246	146	27	y)g	y)g	NOUN
ejpam-3246	146	28	⊆	⊆	NUM
ejpam-3246	146	29	xg	xg	NOUN
ejpam-3246	146	30	⊆	⊆	NUM
ejpam-3246	146	31	(	(	PUNCT
ejpam-3246	146	32	x	x	SYM
ejpam-3246	146	33	∗	∗	NOUN
ejpam-3246	146	34	y)g	y)g	NOUN
ejpam-3246	146	35	∪	∪	PROPN
ejpam-3246	146	36	yg	yg	PROPN
ejpam-3246	146	37	⊆	⊆	NUM
ejpam-3246	146	38	xg	xg	PROPN
ejpam-3246	146	39	∪	∪	ADP
ejpam-3246	146	40	yg	yg	PROPN
ejpam-3246	146	41	for	for	ADP
ejpam-3246	146	42	all	all	DET
ejpam-3246	146	43	x	x	NOUN
ejpam-3246	146	44	,	,	PUNCT
ejpam-3246	146	45	y	y	PROPN
ejpam-3246	146	46	∈	∈	PROPN
ejpam-3246	146	47	x.	x.	NOUN
ejpam-3246	146	48	hence	hence	ADV
ejpam-3246	146	49	g	g	PROPN
ejpam-3246	146	50	is	be	AUX
ejpam-3246	146	51	a	a	DET
ejpam-3246	146	52	uni	uni	ADJ
ejpam-3246	146	53	-	-	ADJ
ejpam-3246	146	54	hesitant	hesitant	ADJ
ejpam-3246	146	55	fuzzy	fuzzy	ADJ
ejpam-3246	146	56	algebra	algebra	NOUN
ejpam-3246	146	57	on	on	ADP
ejpam-3246	146	58	x.	x.	NOUN
ejpam-3246	146	59	if	if	SCONJ
ejpam-3246	146	60	x	x	PRON
ejpam-3246	146	61	is	be	AUX
ejpam-3246	146	62	a	a	DET
ejpam-3246	146	63	bci	bci	NOUN
ejpam-3246	146	64	-	-	NOUN
ejpam-3246	146	65	algebra	algebra	NOUN
ejpam-3246	146	66	,	,	PUNCT
ejpam-3246	146	67	then	then	ADV
ejpam-3246	146	68	theorem	theorem	VERB
ejpam-3246	146	69	3	3	NUM
ejpam-3246	146	70	is	be	AUX
ejpam-3246	146	71	not	not	PART
ejpam-3246	146	72	true	true	ADJ
ejpam-3246	146	73	as	as	SCONJ
ejpam-3246	146	74	seen	see	VERB
ejpam-3246	146	75	in	in	ADP
ejpam-3246	146	76	the	the	DET
ejpam-3246	146	77	following	follow	VERB
ejpam-3246	146	78	example	example	NOUN
ejpam-3246	146	79	.	.	PUNCT
ejpam-3246	147	1	g.	g.	PROPN
ejpam-3246	147	2	muhiuddin	muhiuddin	PROPN
ejpam-3246	147	3	,	,	PUNCT
ejpam-3246	147	4	s.	s.	PROPN
ejpam-3246	147	5	aldhafeeri	aldhafeeri	PROPN
ejpam-3246	147	6	/	/	SYM
ejpam-3246	147	7	eur	eur	PROPN
ejpam-3246	147	8	.	.	PUNCT
ejpam-3246	148	1	j.	j.	PROPN
ejpam-3246	148	2	pure	pure	PROPN
ejpam-3246	148	3	appl	appl	PROPN
ejpam-3246	148	4	.	.	PROPN
ejpam-3246	148	5	math	math	PROPN
ejpam-3246	148	6	,	,	PUNCT
ejpam-3246	148	7	11	11	NUM
ejpam-3246	148	8	(	(	PUNCT
ejpam-3246	148	9	2	2	NUM
ejpam-3246	148	10	)	)	PUNCT
ejpam-3246	148	11	(	(	PUNCT
ejpam-3246	148	12	2018	2018	NUM
ejpam-3246	148	13	)	)	PUNCT
ejpam-3246	148	14	,	,	PUNCT
ejpam-3246	148	15	417	417	NUM
ejpam-3246	148	16	-	-	SYM
ejpam-3246	148	17	430	430	NUM
ejpam-3246	148	18	424	424	NUM
ejpam-3246	148	19	example	example	NOUN
ejpam-3246	148	20	4	4	NUM
ejpam-3246	148	21	.	.	PUNCT
ejpam-3246	149	1	let	let	VERB
ejpam-3246	149	2	(	(	PUNCT
ejpam-3246	149	3	y	y	NOUN
ejpam-3246	149	4	,	,	PUNCT
ejpam-3246	149	5	∗	∗	NOUN
ejpam-3246	149	6	,	,	PUNCT
ejpam-3246	149	7	0	0	NUM
ejpam-3246	149	8	)	)	PUNCT
ejpam-3246	149	9	be	be	AUX
ejpam-3246	149	10	a	a	DET
ejpam-3246	149	11	bci	bci	NOUN
ejpam-3246	149	12	-	-	NOUN
ejpam-3246	149	13	algebra	algebra	NOUN
ejpam-3246	149	14	and	and	CCONJ
ejpam-3246	149	15	let	let	VERB
ejpam-3246	149	16	(	(	PUNCT
ejpam-3246	149	17	z,−	z,−	PROPN
ejpam-3246	149	18	,	,	PUNCT
ejpam-3246	149	19	0	0	NUM
ejpam-3246	149	20	)	)	PUNCT
ejpam-3246	149	21	be	be	VERB
ejpam-3246	149	22	the	the	DET
ejpam-3246	149	23	adjoint	adjoint	NOUN
ejpam-3246	149	24	bci	bci	NOUN
ejpam-3246	149	25	-	-	NOUN
ejpam-3246	149	26	algebra	algebra	NOUN
ejpam-3246	149	27	of	of	ADP
ejpam-3246	149	28	the	the	DET
ejpam-3246	149	29	additive	additive	ADJ
ejpam-3246	149	30	group	group	NOUN
ejpam-3246	149	31	(	(	PUNCT
ejpam-3246	149	32	z,+	z,+	NUM
ejpam-3246	149	33	,	,	PUNCT
ejpam-3246	149	34	0	0	NUM
ejpam-3246	149	35	)	)	PUNCT
ejpam-3246	149	36	of	of	ADP
ejpam-3246	149	37	integers	integer	NOUN
ejpam-3246	149	38	.	.	PUNCT
ejpam-3246	150	1	then	then	ADV
ejpam-3246	150	2	x	x	X
ejpam-3246	150	3	:	:	PUNCT
ejpam-3246	150	4	=	=	SYM
ejpam-3246	150	5	y	y	PROPN
ejpam-3246	150	6	×z	×z	ADV
ejpam-3246	150	7	is	be	AUX
ejpam-3246	150	8	a	a	DET
ejpam-3246	150	9	bci	bci	NOUN
ejpam-3246	150	10	-	-	NOUN
ejpam-3246	150	11	algebra	algebra	NOUN
ejpam-3246	150	12	with	with	ADP
ejpam-3246	150	13	a	a	DET
ejpam-3246	150	14	binary	binary	ADJ
ejpam-3246	150	15	operation	operation	NOUN
ejpam-3246	150	16	⊗	⊗	PROPN
ejpam-3246	150	17	defined	define	VERB
ejpam-3246	150	18	as	as	SCONJ
ejpam-3246	150	19	follows	follow	VERB
ejpam-3246	150	20	:	:	PUNCT
ejpam-3246	150	21	(	(	PUNCT
ejpam-3246	150	22	∀(x	∀(x	X
ejpam-3246	150	23	,	,	PUNCT
ejpam-3246	150	24	m	m	NOUN
ejpam-3246	150	25	)	)	PUNCT
ejpam-3246	150	26	,	,	PUNCT
ejpam-3246	150	27	(	(	PUNCT
ejpam-3246	150	28	y	y	NOUN
ejpam-3246	150	29	,	,	PUNCT
ejpam-3246	150	30	n	n	CCONJ
ejpam-3246	150	31	)	)	PUNCT
ejpam-3246	150	32	∈	∈	PROPN
ejpam-3246	150	33	x	x	X
ejpam-3246	150	34	)	)	PUNCT
ejpam-3246	150	35	(	(	PUNCT
ejpam-3246	150	36	(	(	PUNCT
ejpam-3246	150	37	x	x	X
ejpam-3246	150	38	,	,	PUNCT
ejpam-3246	150	39	m)⊗	m)⊗	PROPN
ejpam-3246	150	40	(	(	PUNCT
ejpam-3246	150	41	y	y	NOUN
ejpam-3246	150	42	,	,	PUNCT
ejpam-3246	150	43	n	n	CCONJ
ejpam-3246	150	44	)	)	PUNCT
ejpam-3246	150	45	=	=	SYM
ejpam-3246	151	1	(	(	PUNCT
ejpam-3246	151	2	x	x	X
ejpam-3246	151	3	∗	∗	PROPN
ejpam-3246	151	4	y	y	PROPN
ejpam-3246	151	5	,	,	PUNCT
ejpam-3246	151	6	m−	m−	PROPN
ejpam-3246	151	7	n	n	CCONJ
ejpam-3246	151	8	)	)	PUNCT
ejpam-3246	151	9	)	)	PUNCT
ejpam-3246	151	10	.	.	PUNCT
ejpam-3246	152	1	for	for	ADP
ejpam-3246	152	2	a	a	DET
ejpam-3246	152	3	subset	subset	NOUN
ejpam-3246	152	4	a	a	PRON
ejpam-3246	152	5	=	=	X
ejpam-3246	152	6	y	y	PROPN
ejpam-3246	152	7	×	×	NOUN
ejpam-3246	152	8	(	(	PUNCT
ejpam-3246	152	9	n	n	CCONJ
ejpam-3246	152	10	∪	∪	X
ejpam-3246	152	11	{	{	PUNCT
ejpam-3246	152	12	0	0	NUM
ejpam-3246	152	13	}	}	PUNCT
ejpam-3246	152	14	)	)	PUNCT
ejpam-3246	152	15	of	of	ADP
ejpam-3246	152	16	x	x	PRON
ejpam-3246	152	17	,	,	PUNCT
ejpam-3246	152	18	let	let	VERB
ejpam-3246	152	19	g	g	PRON
ejpam-3246	152	20	be	be	AUX
ejpam-3246	152	21	a	a	DET
ejpam-3246	152	22	hesitant	hesitant	ADJ
ejpam-3246	152	23	fuzzy	fuzzy	ADJ
ejpam-3246	152	24	set	set	NOUN
ejpam-3246	152	25	on	on	ADP
ejpam-3246	152	26	x	x	PUNCT
ejpam-3246	152	27	given	give	VERB
ejpam-3246	152	28	by	by	ADP
ejpam-3246	152	29	g	g	NOUN
ejpam-3246	152	30	:	:	PUNCT
ejpam-3246	152	31	x	x	X
ejpam-3246	152	32	→	→	X
ejpam-3246	152	33	p	p	X
ejpam-3246	152	34	(	(	PUNCT
ejpam-3246	152	35	[	[	X
ejpam-3246	152	36	0	0	NUM
ejpam-3246	152	37	,	,	PUNCT
ejpam-3246	152	38	1	1	NUM
ejpam-3246	152	39	]	]	NUM
ejpam-3246	152	40	)	)	PUNCT
ejpam-3246	152	41	,	,	PUNCT
ejpam-3246	152	42	(	(	PUNCT
ejpam-3246	152	43	x	x	X
ejpam-3246	152	44	,	,	PUNCT
ejpam-3246	152	45	m	m	NOUN
ejpam-3246	152	46	)	)	PUNCT
ejpam-3246	152	47	7→	7→	NOUN
ejpam-3246	152	48	{	{	PUNCT
ejpam-3246	152	49	λ	λ	X
ejpam-3246	152	50	if	if	SCONJ
ejpam-3246	152	51	x	x	PROPN
ejpam-3246	152	52	∈	∈	PROPN
ejpam-3246	152	53	a	a	PRON
ejpam-3246	152	54	,	,	PUNCT
ejpam-3246	152	55	[	[	X
ejpam-3246	152	56	0	0	NUM
ejpam-3246	152	57	,	,	PUNCT
ejpam-3246	152	58	1	1	NUM
ejpam-3246	152	59	]	]	PUNCT
ejpam-3246	152	60	otherwise	otherwise	ADV
ejpam-3246	152	61	where	where	SCONJ
ejpam-3246	152	62	λ	λ	PROPN
ejpam-3246	152	63	∈	∈	PROPN
ejpam-3246	152	64	p	p	X
ejpam-3246	152	65	(	(	PUNCT
ejpam-3246	152	66	[	[	X
ejpam-3246	152	67	0	0	NUM
ejpam-3246	152	68	,	,	PUNCT
ejpam-3246	152	69	1	1	NUM
ejpam-3246	152	70	]	]	PUNCT
ejpam-3246	152	71	)	)	PUNCT
ejpam-3246	152	72	with	with	ADP
ejpam-3246	152	73	λ	λ	PROPN
ejpam-3246	152	74	6=	6=	PUNCT
ejpam-3246	153	1	[	[	X
ejpam-3246	153	2	0	0	NUM
ejpam-3246	153	3	,	,	PUNCT
ejpam-3246	153	4	1	1	NUM
ejpam-3246	153	5	]	]	PUNCT
ejpam-3246	153	6	.	.	PUNCT
ejpam-3246	154	1	then	then	ADV
ejpam-3246	154	2	g	g	PROPN
ejpam-3246	154	3	is	be	AUX
ejpam-3246	154	4	a	a	DET
ejpam-3246	154	5	uni	uni	ADJ
ejpam-3246	154	6	-	-	ADJ
ejpam-3246	154	7	hesitant	hesitant	ADJ
ejpam-3246	154	8	fuzzy	fuzzy	ADJ
ejpam-3246	154	9	ideal	ideal	NOUN
ejpam-3246	154	10	on	on	ADP
ejpam-3246	154	11	x.	x.	PROPN
ejpam-3246	154	12	note	note	VERB
ejpam-3246	154	13	that	that	SCONJ
ejpam-3246	154	14	(	(	PUNCT
ejpam-3246	154	15	0	0	NUM
ejpam-3246	154	16	,	,	PUNCT
ejpam-3246	154	17	2	2	X
ejpam-3246	154	18	)	)	PUNCT
ejpam-3246	154	19	∈	∈	PROPN
ejpam-3246	154	20	a	a	PRON
ejpam-3246	154	21	and	and	CCONJ
ejpam-3246	154	22	(	(	PUNCT
ejpam-3246	154	23	0	0	NUM
ejpam-3246	154	24	,	,	PUNCT
ejpam-3246	154	25	3	3	X
ejpam-3246	154	26	)	)	PUNCT
ejpam-3246	154	27	∈	∈	PROPN
ejpam-3246	154	28	a	a	PRON
ejpam-3246	154	29	,	,	PUNCT
ejpam-3246	154	30	but	but	CCONJ
ejpam-3246	154	31	(	(	PUNCT
ejpam-3246	154	32	0	0	NUM
ejpam-3246	154	33	,	,	PUNCT
ejpam-3246	154	34	2)⊗	2)⊗	NUM
ejpam-3246	154	35	(	(	PUNCT
ejpam-3246	154	36	0	0	NUM
ejpam-3246	154	37	,	,	PUNCT
ejpam-3246	154	38	3	3	NUM
ejpam-3246	154	39	)	)	PUNCT
ejpam-3246	154	40	=	=	SYM
ejpam-3246	154	41	(	(	PUNCT
ejpam-3246	154	42	0,−1	0,−1	PROPN
ejpam-3246	154	43	)	)	PUNCT
ejpam-3246	154	44	/∈	/∈	PUNCT
ejpam-3246	155	1	a.	a.	NOUN
ejpam-3246	156	1	thus	thus	ADV
ejpam-3246	156	2	(	(	PUNCT
ejpam-3246	156	3	(	(	PUNCT
ejpam-3246	156	4	0	0	NUM
ejpam-3246	156	5	,	,	PUNCT
ejpam-3246	156	6	2)⊗	2)⊗	NUM
ejpam-3246	156	7	(	(	PUNCT
ejpam-3246	156	8	0	0	NUM
ejpam-3246	156	9	,	,	PUNCT
ejpam-3246	156	10	3))g	3))g	NUM
ejpam-3246	156	11	=	=	SYM
ejpam-3246	157	1	[	[	X
ejpam-3246	157	2	0	0	NUM
ejpam-3246	157	3	,	,	PUNCT
ejpam-3246	157	4	1	1	NUM
ejpam-3246	157	5	]	]	PUNCT
ejpam-3246	157	6	*	*	PUNCT
ejpam-3246	157	7	λ	λ	SYM
ejpam-3246	157	8	=	=	SYM
ejpam-3246	157	9	(	(	PUNCT
ejpam-3246	157	10	0	0	NUM
ejpam-3246	157	11	,	,	PUNCT
ejpam-3246	157	12	2)g	2)g	NOUN
ejpam-3246	157	13	∪	∪	X
ejpam-3246	157	14	(	(	PUNCT
ejpam-3246	157	15	0	0	NUM
ejpam-3246	157	16	,	,	PUNCT
ejpam-3246	157	17	3)g	3)g	NUM
ejpam-3246	157	18	.	.	PUNCT
ejpam-3246	158	1	therefore	therefore	ADV
ejpam-3246	158	2	g	g	PROPN
ejpam-3246	158	3	is	be	AUX
ejpam-3246	158	4	not	not	PART
ejpam-3246	158	5	a	a	DET
ejpam-3246	158	6	uni	uni	ADJ
ejpam-3246	158	7	-	-	ADJ
ejpam-3246	158	8	hesitant	hesitant	ADJ
ejpam-3246	158	9	fuzzy	fuzzy	ADJ
ejpam-3246	158	10	algebra	algebra	NOUN
ejpam-3246	158	11	on	on	ADP
ejpam-3246	158	12	x.	x.	NOUN
ejpam-3246	158	13	proposition	proposition	PROPN
ejpam-3246	158	14	6	6	NUM
ejpam-3246	158	15	.	.	PUNCT
ejpam-3246	159	1	every	every	DET
ejpam-3246	159	2	uni	uni	ADJ
ejpam-3246	159	3	-	-	ADJ
ejpam-3246	159	4	hesitant	hesitant	ADJ
ejpam-3246	159	5	fuzzy	fuzzy	ADJ
ejpam-3246	159	6	ideal	ideal	NOUN
ejpam-3246	159	7	g	g	NOUN
ejpam-3246	159	8	on	on	ADP
ejpam-3246	159	9	x	x	PART
ejpam-3246	159	10	satisfies	satisfie	NOUN
ejpam-3246	159	11	the	the	DET
ejpam-3246	159	12	following	follow	VERB
ejpam-3246	159	13	condition	condition	NOUN
ejpam-3246	159	14	:	:	PUNCT
ejpam-3246	159	15	(	(	PUNCT
ejpam-3246	159	16	∀x	∀x	X
ejpam-3246	159	17	,	,	PUNCT
ejpam-3246	159	18	y	y	PROPN
ejpam-3246	159	19	,	,	PUNCT
ejpam-3246	159	20	z	z	NOUN
ejpam-3246	159	21	∈	∈	PROPN
ejpam-3246	159	22	x	x	X
ejpam-3246	159	23	)	)	PUNCT
ejpam-3246	159	24	(	(	PUNCT
ejpam-3246	159	25	(	(	PUNCT
ejpam-3246	159	26	x	x	SYM
ejpam-3246	159	27	∗	∗	PROPN
ejpam-3246	159	28	y	y	NOUN
ejpam-3246	159	29	)	)	PUNCT
ejpam-3246	159	30	∗	∗	NOUN
ejpam-3246	159	31	z	z	NOUN
ejpam-3246	159	32	=	=	SYM
ejpam-3246	159	33	0	0	NUM
ejpam-3246	159	34	⇒	⇒	NOUN
ejpam-3246	159	35	xg	xg	PROPN
ejpam-3246	159	36	⊆	⊆	NUM
ejpam-3246	159	37	yg	yg	PROPN
ejpam-3246	159	38	∪	∪	PROPN
ejpam-3246	159	39	zg	zg	PROPN
ejpam-3246	159	40	)	)	PUNCT
ejpam-3246	159	41	.	.	PUNCT
ejpam-3246	160	1	(	(	PUNCT
ejpam-3246	160	2	7	7	X
ejpam-3246	160	3	)	)	PUNCT
ejpam-3246	160	4	proof	proof	NOUN
ejpam-3246	160	5	.	.	PUNCT
ejpam-3246	161	1	let	let	VERB
ejpam-3246	161	2	x	x	PRON
ejpam-3246	161	3	,	,	PUNCT
ejpam-3246	161	4	y	y	PROPN
ejpam-3246	161	5	,	,	PUNCT
ejpam-3246	161	6	z	z	NOUN
ejpam-3246	161	7	∈	∈	PROPN
ejpam-3246	161	8	x	x	AUX
ejpam-3246	161	9	be	be	AUX
ejpam-3246	161	10	such	such	ADJ
ejpam-3246	161	11	that	that	SCONJ
ejpam-3246	161	12	(	(	PUNCT
ejpam-3246	161	13	x	x	SYM
ejpam-3246	161	14	∗	∗	PROPN
ejpam-3246	161	15	y	y	NOUN
ejpam-3246	161	16	)	)	PUNCT
ejpam-3246	161	17	∗	∗	NOUN
ejpam-3246	161	18	z	z	NOUN
ejpam-3246	162	1	=	=	SYM
ejpam-3246	162	2	0	0	X
ejpam-3246	162	3	.	.	PUNCT
ejpam-3246	163	1	then	then	ADV
ejpam-3246	163	2	(	(	PUNCT
ejpam-3246	163	3	x	x	SYM
ejpam-3246	163	4	∗	∗	NOUN
ejpam-3246	163	5	y)g	y)g	NOUN
ejpam-3246	163	6	⊆	⊆	NUM
ejpam-3246	163	7	(	(	PUNCT
ejpam-3246	163	8	(	(	PUNCT
ejpam-3246	163	9	x	x	SYM
ejpam-3246	163	10	∗	∗	PROPN
ejpam-3246	163	11	y	y	NOUN
ejpam-3246	163	12	)	)	PUNCT
ejpam-3246	163	13	∗	∗	NOUN
ejpam-3246	163	14	z)g	z)g	NUM
ejpam-3246	163	15	∪	∪	ADP
ejpam-3246	163	16	zg	zg	PROPN
ejpam-3246	163	17	=	=	SYM
ejpam-3246	163	18	0	0	NUM
ejpam-3246	163	19	g	g	NOUN
ejpam-3246	163	20	∪	∪	NOUN
ejpam-3246	163	21	zg	zg	PROPN
ejpam-3246	163	22	=	=	SYM
ejpam-3246	163	23	zg	zg	PROPN
ejpam-3246	163	24	by	by	ADP
ejpam-3246	163	25	(	(	PUNCT
ejpam-3246	163	26	5	5	NUM
ejpam-3246	163	27	)	)	PUNCT
ejpam-3246	163	28	and	and	CCONJ
ejpam-3246	163	29	proposition	proposition	NOUN
ejpam-3246	163	30	1	1	NUM
ejpam-3246	163	31	.	.	PUNCT
ejpam-3246	164	1	it	it	PRON
ejpam-3246	164	2	follows	follow	VERB
ejpam-3246	164	3	that	that	SCONJ
ejpam-3246	164	4	xg	xg	PROPN
ejpam-3246	164	5	⊆	⊆	NUM
ejpam-3246	164	6	(	(	PUNCT
ejpam-3246	164	7	x	x	SYM
ejpam-3246	164	8	∗	∗	NOUN
ejpam-3246	164	9	y)g	y)g	NOUN
ejpam-3246	164	10	∪	∪	PROPN
ejpam-3246	164	11	yg	yg	PROPN
ejpam-3246	164	12	⊆	⊆	NUM
ejpam-3246	164	13	yg	yg	PROPN
ejpam-3246	164	14	∪	∪	ADP
ejpam-3246	164	15	zg	zg	PROPN
ejpam-3246	164	16	for	for	ADP
ejpam-3246	164	17	all	all	DET
ejpam-3246	164	18	x	x	PROPN
ejpam-3246	164	19	,	,	PUNCT
ejpam-3246	164	20	y	y	PROPN
ejpam-3246	164	21	,	,	PUNCT
ejpam-3246	164	22	z	z	PROPN
ejpam-3246	164	23	∈	∈	PROPN
ejpam-3246	164	24	x.	x.	NOUN
ejpam-3246	164	25	this	this	PRON
ejpam-3246	164	26	completes	complete	VERB
ejpam-3246	164	27	the	the	DET
ejpam-3246	164	28	proof	proof	NOUN
ejpam-3246	164	29	.	.	PUNCT
ejpam-3246	165	1	we	we	PRON
ejpam-3246	165	2	provide	provide	VERB
ejpam-3246	165	3	conditions	condition	NOUN
ejpam-3246	165	4	for	for	ADP
ejpam-3246	165	5	a	a	DET
ejpam-3246	165	6	hesitant	hesitant	ADJ
ejpam-3246	165	7	fuzzy	fuzzy	ADJ
ejpam-3246	165	8	set	set	NOUN
ejpam-3246	165	9	to	to	PART
ejpam-3246	165	10	be	be	AUX
ejpam-3246	165	11	a	a	DET
ejpam-3246	165	12	hesitant	hesitant	ADJ
ejpam-3246	165	13	fuzzy	fuzzy	ADJ
ejpam-3246	165	14	ideal	ideal	NOUN
ejpam-3246	165	15	.	.	PUNCT
ejpam-3246	166	1	proposition	proposition	NOUN
ejpam-3246	166	2	7	7	NUM
ejpam-3246	166	3	.	.	PUNCT
ejpam-3246	167	1	if	if	SCONJ
ejpam-3246	167	2	a	a	DET
ejpam-3246	167	3	hesitant	hesitant	ADJ
ejpam-3246	167	4	fuzzy	fuzzy	NOUN
ejpam-3246	167	5	set	set	VERB
ejpam-3246	167	6	g	g	NOUN
ejpam-3246	167	7	on	on	ADP
ejpam-3246	167	8	x	x	X
ejpam-3246	167	9	satisfies	satisfie	NOUN
ejpam-3246	167	10	proposition	proposition	NOUN
ejpam-3246	167	11	1	1	NUM
ejpam-3246	167	12	and	and	CCONJ
ejpam-3246	167	13	(	(	PUNCT
ejpam-3246	167	14	7	7	NUM
ejpam-3246	167	15	)	)	PUNCT
ejpam-3246	167	16	,	,	PUNCT
ejpam-3246	167	17	then	then	ADV
ejpam-3246	167	18	g	g	PROPN
ejpam-3246	167	19	is	be	AUX
ejpam-3246	167	20	a	a	DET
ejpam-3246	167	21	uni	uni	ADJ
ejpam-3246	167	22	-	-	ADJ
ejpam-3246	167	23	hesitant	hesitant	ADJ
ejpam-3246	167	24	fuzzy	fuzzy	ADJ
ejpam-3246	167	25	ideal	ideal	NOUN
ejpam-3246	167	26	on	on	ADP
ejpam-3246	167	27	x.	x.	NOUN
ejpam-3246	167	28	proof	proof	NOUN
ejpam-3246	167	29	.	.	PUNCT
ejpam-3246	168	1	it	it	PRON
ejpam-3246	168	2	is	be	AUX
ejpam-3246	168	3	straightforward	straightforward	ADJ
ejpam-3246	168	4	by	by	ADP
ejpam-3246	168	5	(	(	PUNCT
ejpam-3246	168	6	ii	ii	NOUN
ejpam-3246	168	7	)	)	PUNCT
ejpam-3246	168	8	and	and	CCONJ
ejpam-3246	168	9	(	(	PUNCT
ejpam-3246	168	10	7	7	NUM
ejpam-3246	168	11	)	)	PUNCT
ejpam-3246	168	12	.	.	PUNCT
ejpam-3246	169	1	the	the	DET
ejpam-3246	169	2	following	follow	VERB
ejpam-3246	169	3	could	could	AUX
ejpam-3246	169	4	be	be	AUX
ejpam-3246	169	5	easily	easily	ADV
ejpam-3246	169	6	proved	prove	VERB
ejpam-3246	169	7	by	by	ADP
ejpam-3246	169	8	induction	induction	NOUN
ejpam-3246	169	9	.	.	PUNCT
ejpam-3246	170	1	corollary	corollary	ADJ
ejpam-3246	170	2	1	1	NUM
ejpam-3246	170	3	.	.	PUNCT
ejpam-3246	171	1	let	let	VERB
ejpam-3246	171	2	g	g	PRON
ejpam-3246	171	3	be	be	AUX
ejpam-3246	171	4	a	a	DET
ejpam-3246	171	5	hesitant	hesitant	ADJ
ejpam-3246	171	6	fuzzy	fuzzy	ADJ
ejpam-3246	171	7	set	set	NOUN
ejpam-3246	171	8	on	on	ADP
ejpam-3246	171	9	x	x	PUNCT
ejpam-3246	171	10	satisfying	satisfy	VERB
ejpam-3246	171	11	proposition	proposition	NOUN
ejpam-3246	171	12	1	1	NUM
ejpam-3246	171	13	.	.	PUNCT
ejpam-3246	172	1	then	then	ADV
ejpam-3246	172	2	g	g	PROPN
ejpam-3246	172	3	is	be	AUX
ejpam-3246	172	4	a	a	DET
ejpam-3246	172	5	uni	uni	ADJ
ejpam-3246	172	6	-	-	ADJ
ejpam-3246	172	7	hesitant	hesitant	ADJ
ejpam-3246	172	8	fuzzy	fuzzy	ADJ
ejpam-3246	172	9	ideal	ideal	NOUN
ejpam-3246	172	10	on	on	ADP
ejpam-3246	172	11	x	x	PUNCT
ejpam-3246	172	12	if	if	SCONJ
ejpam-3246	172	13	and	and	CCONJ
ejpam-3246	172	14	only	only	ADV
ejpam-3246	172	15	if	if	SCONJ
ejpam-3246	172	16	(	(	PUNCT
ejpam-3246	172	17	∀x	∀x	NUM
ejpam-3246	172	18	,	,	PUNCT
ejpam-3246	172	19	a1	a1	NOUN
ejpam-3246	172	20	,	,	PUNCT
ejpam-3246	172	21	a2	a2	PROPN
ejpam-3246	172	22	,	,	PUNCT
ejpam-3246	172	23	·	·	PUNCT
ejpam-3246	172	24	·	·	PUNCT
ejpam-3246	172	25	·	·	PUNCT
ejpam-3246	172	26	,	,	PUNCT
ejpam-3246	172	27	an	an	DET
ejpam-3246	172	28	∈	∈	PROPN
ejpam-3246	172	29	x	x	NOUN
ejpam-3246	172	30	)	)	PUNCT
ejpam-3246	172	31	x	x	X
ejpam-3246	172	32	∗	∗	PROPN
ejpam-3246	172	33	n∏	n∏	PROPN
ejpam-3246	172	34	i=1	i=1	PROPN
ejpam-3246	172	35	ai	ai	PROPN
ejpam-3246	173	1	=	=	SYM
ejpam-3246	173	2	0	0	NUM
ejpam-3246	173	3	⇒	⇒	NOUN
ejpam-3246	173	4	xg	xg	NOUN
ejpam-3246	173	5	⊆	⊆	NUM
ejpam-3246	173	6	⋃	⋃	PROPN
ejpam-3246	173	7	i=1,2	i=1,2	ADJ
ejpam-3246	173	8	,	,	PUNCT
ejpam-3246	173	9	·	·	PUNCT
ejpam-3246	173	10	·	·	PUNCT
ejpam-3246	173	11	·	·	PUNCT
ejpam-3246	173	12	,	,	PUNCT
ejpam-3246	173	13	n	n	PROPN
ejpam-3246	173	14	aig	aig	PROPN
ejpam-3246	173	15			PROPN
ejpam-3246	173	16	,	,	PUNCT
ejpam-3246	173	17	where	where	SCONJ
ejpam-3246	173	18	x	x	PUNCT
ejpam-3246	173	19	∗	∗	PROPN
ejpam-3246	173	20	n∏	n∏	PROPN
ejpam-3246	173	21	i=1	i=1	PROPN
ejpam-3246	173	22	ai	ai	PROPN
ejpam-3246	173	23	=	=	PUNCT
ejpam-3246	173	24	(	(	PUNCT
ejpam-3246	173	25	·	·	PUNCT
ejpam-3246	173	26	·	·	PUNCT
ejpam-3246	173	27	·	·	PUNCT
ejpam-3246	174	1	(	(	PUNCT
ejpam-3246	174	2	x	x	NOUN
ejpam-3246	174	3	∗	∗	NOUN
ejpam-3246	174	4	a1	a1	NOUN
ejpam-3246	174	5	)	)	PUNCT
ejpam-3246	174	6	∗	∗	NOUN
ejpam-3246	174	7	·	·	PUNCT
ejpam-3246	174	8	·	·	PUNCT
ejpam-3246	174	9	·	·	PUNCT
ejpam-3246	174	10	)	)	PUNCT
ejpam-3246	174	11	∗	∗	NOUN
ejpam-3246	175	1	an	an	PROPN
ejpam-3246	175	2	.	.	PUNCT
ejpam-3246	176	1	g.	g.	PROPN
ejpam-3246	176	2	muhiuddin	muhiuddin	PROPN
ejpam-3246	176	3	,	,	PUNCT
ejpam-3246	176	4	s.	s.	PROPN
ejpam-3246	176	5	aldhafeeri	aldhafeeri	PROPN
ejpam-3246	176	6	/	/	SYM
ejpam-3246	176	7	eur	eur	PROPN
ejpam-3246	176	8	.	.	PUNCT
ejpam-3246	177	1	j.	j.	PROPN
ejpam-3246	177	2	pure	pure	PROPN
ejpam-3246	177	3	appl	appl	PROPN
ejpam-3246	177	4	.	.	PROPN
ejpam-3246	177	5	math	math	PROPN
ejpam-3246	177	6	,	,	PUNCT
ejpam-3246	177	7	11	11	NUM
ejpam-3246	177	8	(	(	PUNCT
ejpam-3246	177	9	2	2	NUM
ejpam-3246	177	10	)	)	PUNCT
ejpam-3246	177	11	(	(	PUNCT
ejpam-3246	177	12	2018	2018	NUM
ejpam-3246	177	13	)	)	PUNCT
ejpam-3246	177	14	,	,	PUNCT
ejpam-3246	177	15	417	417	NUM
ejpam-3246	177	16	-	-	SYM
ejpam-3246	177	17	430	430	NUM
ejpam-3246	177	18	425	425	NUM
ejpam-3246	177	19	proposition	proposition	NOUN
ejpam-3246	177	20	8	8	NUM
ejpam-3246	177	21	.	.	PUNCT
ejpam-3246	178	1	let	let	VERB
ejpam-3246	178	2	x	x	PRON
ejpam-3246	178	3	be	be	AUX
ejpam-3246	178	4	a	a	DET
ejpam-3246	178	5	bck	bck	NOUN
ejpam-3246	178	6	-	-	PUNCT
ejpam-3246	178	7	algebra	algebra	NOUN
ejpam-3246	178	8	such	such	ADJ
ejpam-3246	178	9	that	that	SCONJ
ejpam-3246	178	10	(	(	PUNCT
ejpam-3246	178	11	x	x	SYM
ejpam-3246	178	12	∗	∗	NOUN
ejpam-3246	178	13	a	a	NOUN
ejpam-3246	178	14	)	)	PUNCT
ejpam-3246	178	15	∗	∗	NOUN
ejpam-3246	178	16	b	b	NOUN
ejpam-3246	178	17	=	=	SYM
ejpam-3246	178	18	0	0	NUM
ejpam-3246	178	19	,	,	PUNCT
ejpam-3246	178	20	(	(	PUNCT
ejpam-3246	178	21	8)	8)	NOUN
ejpam-3246	178	22	a	a	DET
ejpam-3246	178	23	∗	∗	NOUN
ejpam-3246	178	24	n∏	n∏	PROPN
ejpam-3246	178	25	i=1	i=1	PROPN
ejpam-3246	178	26	ai	ai	PROPN
ejpam-3246	178	27	=	=	SYM
ejpam-3246	178	28	0	0	PROPN
ejpam-3246	178	29	,	,	PUNCT
ejpam-3246	178	30	(	(	PUNCT
ejpam-3246	178	31	9	9	X
ejpam-3246	178	32	)	)	PUNCT
ejpam-3246	178	33	b	b	NOUN
ejpam-3246	178	34	∗	∗	NOUN
ejpam-3246	178	35	m∏	m∏	NOUN
ejpam-3246	178	36	j=1	j=1	NOUN
ejpam-3246	178	37	bj	bj	VERB
ejpam-3246	178	38	=	=	SYM
ejpam-3246	178	39	0	0	NUM
ejpam-3246	178	40	,	,	PUNCT
ejpam-3246	178	41	(	(	PUNCT
ejpam-3246	178	42	10	10	NUM
ejpam-3246	178	43	)	)	PUNCT
ejpam-3246	178	44	for	for	ADP
ejpam-3246	178	45	all	all	DET
ejpam-3246	178	46	x	x	NOUN
ejpam-3246	178	47	,	,	PUNCT
ejpam-3246	178	48	a	a	DET
ejpam-3246	178	49	,	,	PUNCT
ejpam-3246	178	50	b	b	NOUN
ejpam-3246	178	51	,	,	PUNCT
ejpam-3246	178	52	a1	a1	NOUN
ejpam-3246	178	53	,	,	PUNCT
ejpam-3246	178	54	a2	a2	PROPN
ejpam-3246	178	55	,	,	PUNCT
ejpam-3246	178	56	·	·	PUNCT
ejpam-3246	178	57	·	·	PUNCT
ejpam-3246	178	58	·	·	PUNCT
ejpam-3246	178	59	,	,	PUNCT
ejpam-3246	178	60	an	an	DET
ejpam-3246	178	61	,	,	PUNCT
ejpam-3246	178	62	b1	b1	NOUN
ejpam-3246	178	63	,	,	PUNCT
ejpam-3246	178	64	b2	b2	NOUN
ejpam-3246	178	65	,	,	PUNCT
ejpam-3246	178	66	·	·	PUNCT
ejpam-3246	178	67	·	·	PUNCT
ejpam-3246	178	68	·	·	PUNCT
ejpam-3246	178	69	,	,	PUNCT
ejpam-3246	178	70	bm	bm	PROPN
ejpam-3246	178	71	∈	∈	PROPN
ejpam-3246	178	72	x.	x.	NOUN
ejpam-3246	178	73	if	if	SCONJ
ejpam-3246	178	74	g	g	PROPN
ejpam-3246	178	75	is	be	AUX
ejpam-3246	178	76	a	a	DET
ejpam-3246	178	77	uni	uni	ADJ
ejpam-3246	178	78	-	-	ADJ
ejpam-3246	178	79	hesitant	hesitant	ADJ
ejpam-3246	178	80	fuzzy	fuzzy	ADJ
ejpam-3246	178	81	ideal	ideal	NOUN
ejpam-3246	178	82	on	on	ADP
ejpam-3246	178	83	x	x	NOUN
ejpam-3246	178	84	,	,	PUNCT
ejpam-3246	178	85	then	then	ADV
ejpam-3246	178	86	xg	xg	NOUN
ejpam-3246	179	1	⊆	⊆	NUM
ejpam-3246	179	2	⋃	⋃	PROPN
ejpam-3246	179	3	i=1,2	i=1,2	ADJ
ejpam-3246	179	4	,	,	PUNCT
ejpam-3246	179	5	·	·	PUNCT
ejpam-3246	179	6	·	·	PUNCT
ejpam-3246	179	7	·	·	PUNCT
ejpam-3246	179	8	,	,	PUNCT
ejpam-3246	179	9	n	n	X
ejpam-3246	179	10	j=1,2	j=1,2	ADJ
ejpam-3246	179	11	,	,	PUNCT
ejpam-3246	179	12	·	·	PUNCT
ejpam-3246	179	13	·	·	PUNCT
ejpam-3246	179	14	·	·	PUNCT
ejpam-3246	179	15	,	,	PUNCT
ejpam-3246	179	16	m	m	PROPN
ejpam-3246	179	17	(	(	PUNCT
ejpam-3246	179	18	aig	aig	PROPN
ejpam-3246	179	19	∪	∪	PROPN
ejpam-3246	179	20	bjg	bjg	PROPN
ejpam-3246	179	21	)	)	PUNCT
ejpam-3246	179	22	for	for	ADP
ejpam-3246	179	23	all	all	DET
ejpam-3246	179	24	x	x	NOUN
ejpam-3246	179	25	,	,	PUNCT
ejpam-3246	179	26	a1	a1	NOUN
ejpam-3246	179	27	,	,	PUNCT
ejpam-3246	179	28	a2	a2	PROPN
ejpam-3246	179	29	,	,	PUNCT
ejpam-3246	179	30	·	·	PUNCT
ejpam-3246	179	31	·	·	PUNCT
ejpam-3246	179	32	·	·	PUNCT
ejpam-3246	179	33	,	,	PUNCT
ejpam-3246	179	34	an	an	DET
ejpam-3246	179	35	,	,	PUNCT
ejpam-3246	179	36	b1	b1	NOUN
ejpam-3246	179	37	,	,	PUNCT
ejpam-3246	179	38	b2	b2	NOUN
ejpam-3246	179	39	,	,	PUNCT
ejpam-3246	179	40	·	·	PUNCT
ejpam-3246	179	41	·	·	PUNCT
ejpam-3246	179	42	·	·	PUNCT
ejpam-3246	179	43	,	,	PUNCT
ejpam-3246	179	44	bm	bm	PROPN
ejpam-3246	179	45	∈	∈	PROPN
ejpam-3246	179	46	x.	x.	NOUN
ejpam-3246	179	47	proof	proof	NOUN
ejpam-3246	179	48	.	.	PUNCT
ejpam-3246	180	1	using	use	VERB
ejpam-3246	180	2	(	(	PUNCT
ejpam-3246	180	3	8)	8)	NUM
ejpam-3246	180	4	and	and	CCONJ
ejpam-3246	180	5	(	(	PUNCT
ejpam-3246	180	6	a3	a3	NOUN
ejpam-3246	180	7	)	)	PUNCT
ejpam-3246	180	8	,	,	PUNCT
ejpam-3246	180	9	we	we	PRON
ejpam-3246	180	10	have	have	VERB
ejpam-3246	180	11	(	(	PUNCT
ejpam-3246	180	12	x	x	NOUN
ejpam-3246	180	13	∗	∗	NUM
ejpam-3246	180	14	b	b	NOUN
ejpam-3246	180	15	)	)	PUNCT
ejpam-3246	180	16	∗	∗	NOUN
ejpam-3246	180	17	a	a	DET
ejpam-3246	180	18	=	=	NOUN
ejpam-3246	180	19	0	0	NUM
ejpam-3246	180	20	.	.	PUNCT
ejpam-3246	181	1	it	it	PRON
ejpam-3246	181	2	follows	follow	VERB
ejpam-3246	181	3	from	from	ADP
ejpam-3246	181	4	(	(	PUNCT
ejpam-3246	181	5	a2	a2	PROPN
ejpam-3246	181	6	)	)	PUNCT
ejpam-3246	181	7	,	,	PUNCT
ejpam-3246	181	8	(	(	PUNCT
ejpam-3246	181	9	a3	a3	NOUN
ejpam-3246	181	10	)	)	PUNCT
ejpam-3246	181	11	and	and	CCONJ
ejpam-3246	181	12	(	(	PUNCT
ejpam-3246	181	13	9	9	X
ejpam-3246	181	14	)	)	PUNCT
ejpam-3246	181	15	that	that	SCONJ
ejpam-3246	181	16	(	(	PUNCT
ejpam-3246	181	17	x	x	SYM
ejpam-3246	181	18	∗	∗	PROPN
ejpam-3246	181	19	n∏	n∏	PROPN
ejpam-3246	181	20	i=1	i=1	PROPN
ejpam-3246	181	21	ai	ai	VERB
ejpam-3246	181	22	)	)	PUNCT
ejpam-3246	181	23	∗	∗	NOUN
ejpam-3246	181	24	b	b	NOUN
ejpam-3246	181	25	≤	≤	NOUN
ejpam-3246	181	26	a	a	DET
ejpam-3246	181	27	∗	∗	NOUN
ejpam-3246	181	28	n∏	n∏	PROPN
ejpam-3246	181	29	i=1	i=1	PROPN
ejpam-3246	182	1	ai	ai	VERB
ejpam-3246	182	2	=	=	ADJ
ejpam-3246	182	3	0	0	PUNCT
ejpam-3246	183	1	so	so	SCONJ
ejpam-3246	183	2	that	that	SCONJ
ejpam-3246	183	3	(	(	PUNCT
ejpam-3246	183	4	x	x	SYM
ejpam-3246	183	5	∗	∗	PROPN
ejpam-3246	183	6	n∏	n∏	PROPN
ejpam-3246	183	7	i=1	i=1	PROPN
ejpam-3246	183	8	ai	ai	VERB
ejpam-3246	183	9	)	)	PUNCT
ejpam-3246	183	10	∗	∗	NOUN
ejpam-3246	183	11	b	b	NOUN
ejpam-3246	183	12	=	=	SYM
ejpam-3246	183	13	0	0	NUM
ejpam-3246	183	14	,	,	PUNCT
ejpam-3246	183	15	i.e.	i.e.	X
ejpam-3246	183	16	,	,	PUNCT
ejpam-3246	184	1	x	x	PROPN
ejpam-3246	184	2	∗	∗	PROPN
ejpam-3246	184	3	n∏	n∏	PROPN
ejpam-3246	184	4	i=1	i=1	PROPN
ejpam-3246	185	1	ai	ai	AUX
ejpam-3246	185	2	≤	≤	PROPN
ejpam-3246	185	3	b.	b.	PROPN
ejpam-3246	185	4	using	use	VERB
ejpam-3246	185	5	(	(	PUNCT
ejpam-3246	185	6	a2	a2	PROPN
ejpam-3246	185	7	)	)	PUNCT
ejpam-3246	185	8	,	,	PUNCT
ejpam-3246	185	9	(	(	PUNCT
ejpam-3246	185	10	a3	a3	NOUN
ejpam-3246	185	11	)	)	PUNCT
ejpam-3246	185	12	and	and	CCONJ
ejpam-3246	185	13	(	(	PUNCT
ejpam-3246	185	14	10	10	NUM
ejpam-3246	185	15	)	)	PUNCT
ejpam-3246	185	16	,	,	PUNCT
ejpam-3246	185	17	we	we	PRON
ejpam-3246	185	18	have	have	AUX
ejpam-3246	185	19	(	(	PUNCT
ejpam-3246	185	20	x	x	SYM
ejpam-3246	185	21	∗	∗	PROPN
ejpam-3246	185	22	n∏	n∏	PROPN
ejpam-3246	185	23	i=1	i=1	PROPN
ejpam-3246	185	24	ai	ai	VERB
ejpam-3246	185	25	)	)	PUNCT
ejpam-3246	185	26	∗	∗	NOUN
ejpam-3246	185	27	m∏	m∏	PROPN
ejpam-3246	185	28	j=1	j=1	NOUN
ejpam-3246	185	29	bj	bj	VERB
ejpam-3246	185	30	≤	≤	NUM
ejpam-3246	185	31	b	b	NOUN
ejpam-3246	185	32	∗	∗	NOUN
ejpam-3246	185	33	m∏	m∏	NOUN
ejpam-3246	185	34	j=1	j=1	NOUN
ejpam-3246	185	35	bj	bj	VERB
ejpam-3246	185	36	=	=	SYM
ejpam-3246	185	37	0	0	NUM
ejpam-3246	185	38	,	,	PUNCT
ejpam-3246	185	39	and	and	CCONJ
ejpam-3246	185	40	so	so	ADV
ejpam-3246	185	41	(	(	PUNCT
ejpam-3246	185	42	x	x	SYM
ejpam-3246	185	43	∗	∗	PROPN
ejpam-3246	185	44	n∏	n∏	PROPN
ejpam-3246	185	45	i=1	i=1	PROPN
ejpam-3246	185	46	ai	ai	VERB
ejpam-3246	185	47	)	)	PUNCT
ejpam-3246	185	48	∗	∗	NOUN
ejpam-3246	185	49	m∏	m∏	PROPN
ejpam-3246	185	50	j=1	j=1	NOUN
ejpam-3246	185	51	bj	bj	VERB
ejpam-3246	185	52	=	=	SYM
ejpam-3246	185	53	0	0	NUM
ejpam-3246	185	54	.	.	PUNCT
ejpam-3246	186	1	thus	thus	ADV
ejpam-3246	186	2	,	,	PUNCT
ejpam-3246	186	3	by	by	ADP
ejpam-3246	186	4	corollary	corollary	ADJ
ejpam-3246	186	5	1	1	NUM
ejpam-3246	186	6	,	,	PUNCT
ejpam-3246	186	7	we	we	PRON
ejpam-3246	186	8	have	have	VERB
ejpam-3246	186	9	xg	xg	NOUN
ejpam-3246	186	10	⊆	⊆	NUM
ejpam-3246	186	11	⋃	⋃	PROPN
ejpam-3246	186	12	i=1,2	i=1,2	ADJ
ejpam-3246	186	13	,	,	PUNCT
ejpam-3246	186	14	·	·	PUNCT
ejpam-3246	186	15	·	·	PUNCT
ejpam-3246	186	16	·	·	PUNCT
ejpam-3246	186	17	,	,	PUNCT
ejpam-3246	186	18	n	n	X
ejpam-3246	186	19	j=1,2	j=1,2	ADJ
ejpam-3246	186	20	,	,	PUNCT
ejpam-3246	186	21	·	·	PUNCT
ejpam-3246	186	22	·	·	PUNCT
ejpam-3246	186	23	·	·	PUNCT
ejpam-3246	186	24	,	,	PUNCT
ejpam-3246	186	25	m	m	PROPN
ejpam-3246	186	26	(	(	PUNCT
ejpam-3246	186	27	aig	aig	PROPN
ejpam-3246	186	28	∪	∪	PROPN
ejpam-3246	186	29	bjg	bjg	PROPN
ejpam-3246	186	30	)	)	PUNCT
ejpam-3246	186	31	for	for	ADP
ejpam-3246	186	32	all	all	DET
ejpam-3246	186	33	x	x	NOUN
ejpam-3246	186	34	,	,	PUNCT
ejpam-3246	186	35	a1	a1	NOUN
ejpam-3246	186	36	,	,	PUNCT
ejpam-3246	186	37	a2	a2	PROPN
ejpam-3246	186	38	,	,	PUNCT
ejpam-3246	186	39	·	·	PUNCT
ejpam-3246	186	40	·	·	PUNCT
ejpam-3246	186	41	·	·	PUNCT
ejpam-3246	186	42	,	,	PUNCT
ejpam-3246	186	43	an	an	DET
ejpam-3246	186	44	,	,	PUNCT
ejpam-3246	186	45	b1	b1	NOUN
ejpam-3246	186	46	,	,	PUNCT
ejpam-3246	186	47	b2	b2	NOUN
ejpam-3246	186	48	,	,	PUNCT
ejpam-3246	186	49	·	·	PUNCT
ejpam-3246	186	50	·	·	PUNCT
ejpam-3246	186	51	·	·	PUNCT
ejpam-3246	186	52	,	,	PUNCT
ejpam-3246	186	53	bm	bm	PROPN
ejpam-3246	186	54	∈	∈	PROPN
ejpam-3246	186	55	x.	x.	NOUN
ejpam-3246	186	56	theorem	theorem	VERB
ejpam-3246	186	57	4	4	NUM
ejpam-3246	186	58	.	.	X
ejpam-3246	187	1	for	for	SCONJ
ejpam-3246	187	2	a	a	DET
ejpam-3246	187	3	hesitant	hesitant	ADJ
ejpam-3246	187	4	fuzzy	fuzzy	NOUN
ejpam-3246	187	5	set	set	VERB
ejpam-3246	187	6	g	g	NOUN
ejpam-3246	187	7	on	on	ADP
ejpam-3246	187	8	x	x	PRON
ejpam-3246	187	9	,	,	PUNCT
ejpam-3246	187	10	the	the	DET
ejpam-3246	187	11	following	follow	VERB
ejpam-3246	187	12	are	be	AUX
ejpam-3246	187	13	equivalent	equivalent	ADJ
ejpam-3246	187	14	.	.	PUNCT
ejpam-3246	188	1	(	(	PUNCT
ejpam-3246	188	2	i	i	NOUN
ejpam-3246	188	3	)	)	PUNCT
ejpam-3246	188	4	g	g	NOUN
ejpam-3246	188	5	is	be	AUX
ejpam-3246	188	6	a	a	DET
ejpam-3246	188	7	uni	uni	ADJ
ejpam-3246	188	8	-	-	ADJ
ejpam-3246	188	9	hesitant	hesitant	ADJ
ejpam-3246	188	10	fuzzy	fuzzy	ADJ
ejpam-3246	188	11	ideal	ideal	NOUN
ejpam-3246	188	12	on	on	ADP
ejpam-3246	188	13	x.	x.	PROPN
ejpam-3246	188	14	(	(	PUNCT
ejpam-3246	188	15	ii	ii	PROPN
ejpam-3246	188	16	)	)	PUNCT
ejpam-3246	188	17	the	the	PRON
ejpam-3246	188	18	nonempty	nonempty	ADJ
ejpam-3246	188	19	uni	uni	ADJ
ejpam-3246	188	20	-	-	ADJ
ejpam-3246	188	21	hesitant	hesitant	ADJ
ejpam-3246	188	22	level	level	NOUN
ejpam-3246	188	23	set	set	VERB
ejpam-3246	188	24	l(g;λ	l(g;λ	NOUN
ejpam-3246	188	25	)	)	PUNCT
ejpam-3246	188	26	of	of	ADP
ejpam-3246	188	27	g	g	PROPN
ejpam-3246	188	28	is	be	AUX
ejpam-3246	188	29	an	an	DET
ejpam-3246	188	30	ideal	ideal	NOUN
ejpam-3246	188	31	of	of	ADP
ejpam-3246	188	32	x	x	PUNCT
ejpam-3246	188	33	for	for	ADP
ejpam-3246	188	34	all	all	DET
ejpam-3246	188	35	λ	λ	PROPN
ejpam-3246	188	36	∈	∈	PROPN
ejpam-3246	188	37	p	p	X
ejpam-3246	188	38	(	(	PUNCT
ejpam-3246	188	39	[	[	X
ejpam-3246	188	40	0	0	NUM
ejpam-3246	188	41	,	,	PUNCT
ejpam-3246	188	42	1	1	NUM
ejpam-3246	188	43	]	]	NUM
ejpam-3246	188	44	)	)	PUNCT
ejpam-3246	188	45	.	.	PUNCT
ejpam-3246	189	1	proof	proof	NOUN
ejpam-3246	189	2	.	.	PUNCT
ejpam-3246	190	1	assume	assume	VERB
ejpam-3246	190	2	that	that	SCONJ
ejpam-3246	190	3	g	g	PROPN
ejpam-3246	190	4	is	be	AUX
ejpam-3246	190	5	a	a	DET
ejpam-3246	190	6	uni	uni	ADJ
ejpam-3246	190	7	-	-	ADJ
ejpam-3246	190	8	hesitant	hesitant	ADJ
ejpam-3246	190	9	fuzzy	fuzzy	ADJ
ejpam-3246	190	10	ideal	ideal	NOUN
ejpam-3246	190	11	on	on	ADP
ejpam-3246	190	12	x.	x.	NOUN
ejpam-3246	190	13	let	let	VERB
ejpam-3246	190	14	λ	λ	X
ejpam-3246	190	15	∈	∈	VERB
ejpam-3246	190	16	p	p	X
ejpam-3246	190	17	(	(	PUNCT
ejpam-3246	190	18	[	[	X
ejpam-3246	190	19	0	0	NUM
ejpam-3246	190	20	,	,	PUNCT
ejpam-3246	190	21	1	1	NUM
ejpam-3246	190	22	]	]	PUNCT
ejpam-3246	190	23	)	)	PUNCT
ejpam-3246	190	24	be	be	AUX
ejpam-3246	190	25	such	such	ADJ
ejpam-3246	190	26	that	that	PRON
ejpam-3246	190	27	l(g;λ	l(g;λ	PROPN
ejpam-3246	190	28	)	)	PUNCT
ejpam-3246	190	29	6=	6=	ADP
ejpam-3246	190	30	∅.	∅.	VERB
ejpam-3246	190	31	then	then	ADV
ejpam-3246	190	32	xg	xg	PROPN
ejpam-3246	190	33	⊆	⊆	NUM
ejpam-3246	190	34	λ	λ	PROPN
ejpam-3246	190	35	for	for	ADP
ejpam-3246	190	36	some	some	DET
ejpam-3246	190	37	x	x	SYM
ejpam-3246	190	38	∈	∈	PROPN
ejpam-3246	190	39	x.	x.	NOUN
ejpam-3246	191	1	it	it	PRON
ejpam-3246	191	2	follows	follow	VERB
ejpam-3246	191	3	from	from	ADP
ejpam-3246	191	4	proposition	proposition	NOUN
ejpam-3246	191	5	1	1	NUM
ejpam-3246	191	6	that	that	PRON
ejpam-3246	191	7	g.	g.	PROPN
ejpam-3246	191	8	muhiuddin	muhiuddin	PROPN
ejpam-3246	191	9	,	,	PUNCT
ejpam-3246	191	10	s.	s.	PROPN
ejpam-3246	191	11	aldhafeeri	aldhafeeri	PROPN
ejpam-3246	191	12	/	/	SYM
ejpam-3246	191	13	eur	eur	PROPN
ejpam-3246	191	14	.	.	PUNCT
ejpam-3246	192	1	j.	j.	PROPN
ejpam-3246	192	2	pure	pure	PROPN
ejpam-3246	192	3	appl	appl	PROPN
ejpam-3246	192	4	.	.	PROPN
ejpam-3246	192	5	math	math	PROPN
ejpam-3246	192	6	,	,	PUNCT
ejpam-3246	192	7	11	11	NUM
ejpam-3246	192	8	(	(	PUNCT
ejpam-3246	192	9	2	2	NUM
ejpam-3246	192	10	)	)	PUNCT
ejpam-3246	192	11	(	(	PUNCT
ejpam-3246	192	12	2018	2018	NUM
ejpam-3246	192	13	)	)	PUNCT
ejpam-3246	192	14	,	,	PUNCT
ejpam-3246	192	15	417	417	NUM
ejpam-3246	192	16	-	-	SYM
ejpam-3246	192	17	430	430	NUM
ejpam-3246	193	1	426	426	NUM
ejpam-3246	193	2	0	0	NUM
ejpam-3246	193	3	g	g	NOUN
ejpam-3246	193	4	⊆	⊆	NUM
ejpam-3246	193	5	xg	xg	NOUN
ejpam-3246	193	6	⊆	⊆	NUM
ejpam-3246	193	7	λ	λ	NOUN
ejpam-3246	193	8	.	.	PUNCT
ejpam-3246	194	1	hence	hence	ADV
ejpam-3246	194	2	0	0	NUM
ejpam-3246	194	3	∈	∈	PROPN
ejpam-3246	194	4	l(g;λ	l(g;λ	PROPN
ejpam-3246	194	5	)	)	PUNCT
ejpam-3246	194	6	.	.	PUNCT
ejpam-3246	195	1	let	let	VERB
ejpam-3246	195	2	x	x	PRON
ejpam-3246	195	3	,	,	PUNCT
ejpam-3246	195	4	y	y	PROPN
ejpam-3246	195	5	∈	∈	PROPN
ejpam-3246	195	6	x	x	AUX
ejpam-3246	195	7	be	be	AUX
ejpam-3246	195	8	such	such	ADJ
ejpam-3246	195	9	that	that	SCONJ
ejpam-3246	195	10	x	x	PUNCT
ejpam-3246	195	11	∗	∗	NOUN
ejpam-3246	195	12	y	y	PROPN
ejpam-3246	195	13	∈	∈	PROPN
ejpam-3246	195	14	l(g;λ	l(g;λ	X
ejpam-3246	195	15	)	)	PUNCT
ejpam-3246	195	16	and	and	CCONJ
ejpam-3246	195	17	y	y	PROPN
ejpam-3246	195	18	∈	∈	PROPN
ejpam-3246	195	19	l(g;λ	l(g;λ	PROPN
ejpam-3246	195	20	)	)	PUNCT
ejpam-3246	195	21	.	.	PUNCT
ejpam-3246	196	1	then	then	ADV
ejpam-3246	196	2	(	(	PUNCT
ejpam-3246	196	3	x	x	SYM
ejpam-3246	196	4	∗	∗	NOUN
ejpam-3246	196	5	y)g	y)g	NOUN
ejpam-3246	196	6	⊆	⊆	NUM
ejpam-3246	196	7	λ	λ	NOUN
ejpam-3246	196	8	and	and	CCONJ
ejpam-3246	196	9	yg	yg	PROPN
ejpam-3246	196	10	⊆	⊆	NUM
ejpam-3246	196	11	λ	λ	PROPN
ejpam-3246	196	12	.	.	PUNCT
ejpam-3246	197	1	it	it	PRON
ejpam-3246	197	2	follows	follow	VERB
ejpam-3246	197	3	from	from	ADP
ejpam-3246	197	4	(	(	PUNCT
ejpam-3246	197	5	5	5	NUM
ejpam-3246	197	6	)	)	PUNCT
ejpam-3246	197	7	that	that	PRON
ejpam-3246	197	8	xg	xg	PROPN
ejpam-3246	197	9	⊆	⊆	NUM
ejpam-3246	197	10	(	(	PUNCT
ejpam-3246	197	11	x	x	SYM
ejpam-3246	197	12	∗	∗	NOUN
ejpam-3246	197	13	y)g	y)g	NOUN
ejpam-3246	197	14	∪	∪	PROPN
ejpam-3246	197	15	yg	yg	PROPN
ejpam-3246	197	16	⊆	⊆	NUM
ejpam-3246	197	17	λ	λ	PROPN
ejpam-3246	197	18	.	.	PUNCT
ejpam-3246	198	1	thus	thus	ADV
ejpam-3246	198	2	x	x	X
ejpam-3246	198	3	∈	∈	NOUN
ejpam-3246	198	4	l(g;λ	l(g;λ	X
ejpam-3246	198	5	)	)	PUNCT
ejpam-3246	198	6	.	.	PUNCT
ejpam-3246	199	1	therefore	therefore	ADV
ejpam-3246	199	2	l(g;λ	l(g;λ	X
ejpam-3246	199	3	)	)	PUNCT
ejpam-3246	199	4	is	be	AUX
ejpam-3246	199	5	an	an	DET
ejpam-3246	199	6	ideal	ideal	NOUN
ejpam-3246	199	7	of	of	ADP
ejpam-3246	199	8	x.	x.	NOUN
ejpam-3246	199	9	conversely	conversely	ADV
ejpam-3246	199	10	suppose	suppose	VERB
ejpam-3246	199	11	that	that	SCONJ
ejpam-3246	199	12	the	the	DET
ejpam-3246	199	13	nonempty	nonempty	ADJ
ejpam-3246	199	14	uni	uni	ADJ
ejpam-3246	199	15	-	-	ADJ
ejpam-3246	199	16	hesitant	hesitant	ADJ
ejpam-3246	199	17	level	level	NOUN
ejpam-3246	199	18	set	set	NOUN
ejpam-3246	199	19	of	of	ADP
ejpam-3246	199	20	g	g	PROPN
ejpam-3246	199	21	is	be	AUX
ejpam-3246	199	22	an	an	DET
ejpam-3246	199	23	ideal	ideal	NOUN
ejpam-3246	199	24	of	of	ADP
ejpam-3246	199	25	x	x	PUNCT
ejpam-3246	199	26	for	for	ADP
ejpam-3246	199	27	all	all	DET
ejpam-3246	199	28	λ	λ	PROPN
ejpam-3246	199	29	∈	∈	PROPN
ejpam-3246	199	30	p	p	X
ejpam-3246	199	31	(	(	PUNCT
ejpam-3246	199	32	[	[	X
ejpam-3246	199	33	0	0	NUM
ejpam-3246	199	34	,	,	PUNCT
ejpam-3246	199	35	1	1	NUM
ejpam-3246	199	36	]	]	NUM
ejpam-3246	199	37	)	)	PUNCT
ejpam-3246	199	38	.	.	PUNCT
ejpam-3246	200	1	then	then	ADV
ejpam-3246	200	2	0	0	NUM
ejpam-3246	200	3	∈	∈	NOUN
ejpam-3246	200	4	l(g;λ	l(g;λ	PROPN
ejpam-3246	200	5	)	)	PUNCT
ejpam-3246	200	6	.	.	PUNCT
ejpam-3246	201	1	if	if	SCONJ
ejpam-3246	201	2	there	there	PRON
ejpam-3246	201	3	exists	exist	VERB
ejpam-3246	201	4	a	a	DET
ejpam-3246	201	5	∈	∈	NOUN
ejpam-3246	201	6	x	x	PUNCT
ejpam-3246	201	7	such	such	ADJ
ejpam-3246	201	8	that	that	SCONJ
ejpam-3246	201	9	0	0	NUM
ejpam-3246	201	10	g	g	PROPN
ejpam-3246	201	11	*	*	PUNCT
ejpam-3246	201	12	ag	ag	PROPN
ejpam-3246	201	13	,	,	PUNCT
ejpam-3246	201	14	then	then	ADV
ejpam-3246	201	15	0	0	NUM
ejpam-3246	201	16	g	g	NOUN
ejpam-3246	201	17	*	*	PUNCT
ejpam-3246	201	18	λ	λ	PROPN
ejpam-3246	201	19	for	for	ADP
ejpam-3246	201	20	λ	λ	PROPN
ejpam-3246	201	21	=	=	SYM
ejpam-3246	201	22	ag	ag	PROPN
ejpam-3246	201	23	\	\	PROPN
ejpam-3246	201	24	0	0	NUM
ejpam-3246	201	25	g.	g.	NOUN
ejpam-3246	201	26	hence	hence	ADV
ejpam-3246	201	27	0	0	NUM
ejpam-3246	201	28	/∈	/∈	PUNCT
ejpam-3246	201	29	l(g;λ	l(g;λ	PROPN
ejpam-3246	201	30	)	)	PUNCT
ejpam-3246	201	31	,	,	PUNCT
ejpam-3246	201	32	a	a	DET
ejpam-3246	201	33	contradiction	contradiction	NOUN
ejpam-3246	201	34	.	.	PUNCT
ejpam-3246	202	1	therefore	therefore	ADV
ejpam-3246	202	2	0	0	NUM
ejpam-3246	202	3	g	g	NOUN
ejpam-3246	202	4	⊆	⊆	NUM
ejpam-3246	202	5	xg	xg	NOUN
ejpam-3246	202	6	for	for	ADP
ejpam-3246	202	7	all	all	PRON
ejpam-3246	202	8	x	x	SYM
ejpam-3246	202	9	∈	∈	NOUN
ejpam-3246	202	10	x.	x.	NOUN
ejpam-3246	202	11	let	let	VERB
ejpam-3246	202	12	x	x	PRON
ejpam-3246	202	13	,	,	PUNCT
ejpam-3246	202	14	y	y	PROPN
ejpam-3246	202	15	∈	∈	PROPN
ejpam-3246	202	16	x	x	AUX
ejpam-3246	202	17	be	be	AUX
ejpam-3246	202	18	such	such	ADJ
ejpam-3246	202	19	that	that	SCONJ
ejpam-3246	202	20	(	(	PUNCT
ejpam-3246	202	21	x	x	X
ejpam-3246	202	22	∗	∗	X
ejpam-3246	202	23	y)g	y)g	NOUN
ejpam-3246	203	1	=	=	SYM
ejpam-3246	203	2	λ1	λ1	ADJ
ejpam-3246	203	3	and	and	CCONJ
ejpam-3246	203	4	yg	yg	NOUN
ejpam-3246	203	5	=	=	PROPN
ejpam-3246	203	6	λ2	λ2	PROPN
ejpam-3246	203	7	.	.	PUNCT
ejpam-3246	204	1	let	let	VERB
ejpam-3246	204	2	us	we	PRON
ejpam-3246	204	3	take	take	VERB
ejpam-3246	204	4	λ	λ	NOUN
ejpam-3246	204	5	=	=	PUNCT
ejpam-3246	204	6	λ1	λ1	ADJ
ejpam-3246	204	7	∪	∪	PROPN
ejpam-3246	204	8	λ2	λ2	PROPN
ejpam-3246	204	9	.	.	PUNCT
ejpam-3246	205	1	then	then	ADV
ejpam-3246	205	2	x	x	X
ejpam-3246	205	3	∗	∗	VERB
ejpam-3246	205	4	y	y	PROPN
ejpam-3246	205	5	∈	∈	PROPN
ejpam-3246	205	6	l(g;λ	l(g;λ	X
ejpam-3246	205	7	)	)	PUNCT
ejpam-3246	205	8	and	and	CCONJ
ejpam-3246	205	9	y	y	PROPN
ejpam-3246	205	10	∈	∈	PROPN
ejpam-3246	205	11	l(g;λ	l(g;λ	PROPN
ejpam-3246	205	12	)	)	PUNCT
ejpam-3246	205	13	.	.	PUNCT
ejpam-3246	206	1	since	since	SCONJ
ejpam-3246	206	2	l(g;λ	l(g;λ	PROPN
ejpam-3246	206	3	)	)	PUNCT
ejpam-3246	206	4	is	be	AUX
ejpam-3246	206	5	an	an	DET
ejpam-3246	206	6	ideal	ideal	NOUN
ejpam-3246	206	7	of	of	ADP
ejpam-3246	206	8	x	x	PRON
ejpam-3246	206	9	,	,	PUNCT
ejpam-3246	206	10	it	it	PRON
ejpam-3246	206	11	follows	follow	VERB
ejpam-3246	206	12	from	from	ADP
ejpam-3246	206	13	(	(	PUNCT
ejpam-3246	206	14	2	2	NUM
ejpam-3246	206	15	)	)	PUNCT
ejpam-3246	206	16	that	that	SCONJ
ejpam-3246	206	17	x	x	SYM
ejpam-3246	206	18	∈	∈	PROPN
ejpam-3246	206	19	l(g;λ	l(g;λ	PROPN
ejpam-3246	206	20	)	)	PUNCT
ejpam-3246	206	21	.	.	PUNCT
ejpam-3246	207	1	hence	hence	ADV
ejpam-3246	207	2	xg	xg	NOUN
ejpam-3246	207	3	⊆	⊆	NUM
ejpam-3246	207	4	λ	λ	X
ejpam-3246	207	5	=	=	SYM
ejpam-3246	207	6	λ1	λ1	PROPN
ejpam-3246	207	7	∪	∪	NOUN
ejpam-3246	207	8	λ2	λ2	NOUN
ejpam-3246	207	9	=	=	SYM
ejpam-3246	207	10	(	(	PUNCT
ejpam-3246	207	11	x	x	SYM
ejpam-3246	207	12	∗	∗	NOUN
ejpam-3246	207	13	y)g	y)g	NOUN
ejpam-3246	207	14	∪	∪	PROPN
ejpam-3246	207	15	yg	yg	PROPN
ejpam-3246	207	16	.	.	PUNCT
ejpam-3246	208	1	consequently	consequently	ADV
ejpam-3246	208	2	,	,	PUNCT
ejpam-3246	208	3	g	g	PROPN
ejpam-3246	208	4	is	be	AUX
ejpam-3246	208	5	a	a	DET
ejpam-3246	208	6	uni	uni	ADJ
ejpam-3246	208	7	-	-	ADJ
ejpam-3246	208	8	hesitant	hesitant	ADJ
ejpam-3246	208	9	fuzzy	fuzzy	ADJ
ejpam-3246	208	10	ideal	ideal	NOUN
ejpam-3246	208	11	on	on	ADP
ejpam-3246	208	12	x.	x.	PROPN
ejpam-3246	208	13	5	5	X
ejpam-3246	208	14	.	.	PUNCT
ejpam-3246	209	1	uni	uni	ADJ
ejpam-3246	209	2	-	-	ADJ
ejpam-3246	209	3	hesitant	hesitant	ADJ
ejpam-3246	209	4	fuzzy	fuzzy	ADJ
ejpam-3246	209	5	closed	closed	ADJ
ejpam-3246	209	6	ideals	ideal	NOUN
ejpam-3246	209	7	in	in	ADP
ejpam-3246	209	8	bci	bci	NOUN
ejpam-3246	209	9	-	-	PUNCT
ejpam-3246	209	10	algebras	algebras	ADJ
ejpam-3246	209	11	definition	definition	NOUN
ejpam-3246	209	12	3	3	NUM
ejpam-3246	209	13	.	.	PUNCT
ejpam-3246	210	1	a	a	DET
ejpam-3246	210	2	uni	uni	ADJ
ejpam-3246	210	3	-	-	ADJ
ejpam-3246	210	4	hesitant	hesitant	ADJ
ejpam-3246	210	5	fuzzy	fuzzy	ADJ
ejpam-3246	210	6	ideal	ideal	NOUN
ejpam-3246	210	7	g	g	NOUN
ejpam-3246	210	8	on	on	ADP
ejpam-3246	210	9	a	a	DET
ejpam-3246	210	10	bci	bci	NOUN
ejpam-3246	210	11	-	-	NOUN
ejpam-3246	210	12	algebra	algebra	NOUN
ejpam-3246	210	13	x	x	PUNCT
ejpam-3246	210	14	is	be	AUX
ejpam-3246	210	15	said	say	VERB
ejpam-3246	210	16	to	to	PART
ejpam-3246	210	17	be	be	AUX
ejpam-3246	210	18	hesitant	hesitant	ADJ
ejpam-3246	210	19	closed	close	VERB
ejpam-3246	210	20	if	if	SCONJ
ejpam-3246	210	21	the	the	DET
ejpam-3246	210	22	following	follow	VERB
ejpam-3246	210	23	inclusion	inclusion	NOUN
ejpam-3246	210	24	is	be	AUX
ejpam-3246	210	25	valid	valid	ADJ
ejpam-3246	210	26	.	.	PUNCT
ejpam-3246	211	1	(	(	PUNCT
ejpam-3246	211	2	∀x	∀x	X
ejpam-3246	211	3	∈	∈	PROPN
ejpam-3246	211	4	x	x	NOUN
ejpam-3246	211	5	)	)	PUNCT
ejpam-3246	211	6	(	(	PUNCT
ejpam-3246	211	7	(	(	PUNCT
ejpam-3246	211	8	0	0	NUM
ejpam-3246	211	9	∗	∗	NOUN
ejpam-3246	211	10	x)g	x)g	X
ejpam-3246	212	1	⊆	⊆	NUM
ejpam-3246	212	2	xg	xg	NUM
ejpam-3246	212	3	)	)	PUNCT
ejpam-3246	212	4	.	.	PUNCT
ejpam-3246	213	1	(	(	PUNCT
ejpam-3246	213	2	11	11	NUM
ejpam-3246	213	3	)	)	PUNCT
ejpam-3246	213	4	obviously	obviously	ADV
ejpam-3246	213	5	,	,	PUNCT
ejpam-3246	213	6	every	every	DET
ejpam-3246	213	7	uni	uni	ADJ
ejpam-3246	213	8	-	-	ADJ
ejpam-3246	213	9	hesitant	hesitant	ADJ
ejpam-3246	213	10	fuzzy	fuzzy	ADJ
ejpam-3246	213	11	closed	closed	ADJ
ejpam-3246	213	12	ideal	ideal	NOUN
ejpam-3246	213	13	is	be	AUX
ejpam-3246	213	14	a	a	DET
ejpam-3246	213	15	uni	uni	ADJ
ejpam-3246	213	16	-	-	ADJ
ejpam-3246	213	17	hesitant	hesitant	ADJ
ejpam-3246	213	18	fuzzy	fuzzy	ADJ
ejpam-3246	213	19	algebra	algebra	NOUN
ejpam-3246	213	20	.	.	PUNCT
ejpam-3246	214	1	example	example	NOUN
ejpam-3246	214	2	5	5	NUM
ejpam-3246	214	3	.	.	PUNCT
ejpam-3246	215	1	the	the	DET
ejpam-3246	215	2	uni	uni	ADJ
ejpam-3246	215	3	-	-	ADJ
ejpam-3246	215	4	hesitant	hesitant	ADJ
ejpam-3246	215	5	fuzzy	fuzzy	ADJ
ejpam-3246	215	6	ideal	ideal	ADJ
ejpam-3246	215	7	h	h	NOUN
ejpam-3246	215	8	in	in	ADP
ejpam-3246	215	9	example	example	NOUN
ejpam-3246	215	10	2	2	NUM
ejpam-3246	215	11	is	be	AUX
ejpam-3246	215	12	hesitant	hesitant	ADJ
ejpam-3246	215	13	closed	closed	ADJ
ejpam-3246	215	14	.	.	PUNCT
ejpam-3246	216	1	note	note	VERB
ejpam-3246	216	2	that	that	SCONJ
ejpam-3246	216	3	every	every	DET
ejpam-3246	216	4	uni	uni	ADJ
ejpam-3246	216	5	-	-	ADJ
ejpam-3246	216	6	hesitant	hesitant	ADJ
ejpam-3246	216	7	fuzzy	fuzzy	ADJ
ejpam-3246	216	8	ideal	ideal	NOUN
ejpam-3246	216	9	on	on	ADP
ejpam-3246	216	10	a	a	DET
ejpam-3246	216	11	bck	bck	NOUN
ejpam-3246	216	12	-	-	PUNCT
ejpam-3246	216	13	algebra	algebra	NOUN
ejpam-3246	216	14	x	x	PUNCT
ejpam-3246	216	15	is	be	AUX
ejpam-3246	216	16	hesitant	hesitant	ADJ
ejpam-3246	216	17	closed	closed	ADJ
ejpam-3246	216	18	.	.	PUNCT
ejpam-3246	217	1	the	the	DET
ejpam-3246	217	2	following	follow	VERB
ejpam-3246	217	3	example	example	NOUN
ejpam-3246	217	4	shows	show	VERB
ejpam-3246	217	5	that	that	SCONJ
ejpam-3246	217	6	there	there	PRON
ejpam-3246	217	7	exists	exist	VERB
ejpam-3246	217	8	a	a	DET
ejpam-3246	217	9	uni	uni	ADJ
ejpam-3246	217	10	-	-	ADJ
ejpam-3246	217	11	hesitant	hesitant	ADJ
ejpam-3246	217	12	fuzzy	fuzzy	ADJ
ejpam-3246	217	13	ideal	ideal	NOUN
ejpam-3246	217	14	on	on	ADP
ejpam-3246	217	15	a	a	DET
ejpam-3246	217	16	bci	bci	NOUN
ejpam-3246	217	17	-	-	NOUN
ejpam-3246	217	18	algebra	algebra	NOUN
ejpam-3246	217	19	x	x	PUNCT
ejpam-3246	217	20	which	which	PRON
ejpam-3246	217	21	is	be	AUX
ejpam-3246	217	22	not	not	PART
ejpam-3246	217	23	hesitant	hesitant	ADJ
ejpam-3246	217	24	closed	closed	ADJ
ejpam-3246	217	25	.	.	PUNCT
ejpam-3246	218	1	example	example	NOUN
ejpam-3246	219	1	6	6	NUM
ejpam-3246	219	2	.	.	PUNCT
ejpam-3246	220	1	in	in	ADP
ejpam-3246	220	2	example	example	NOUN
ejpam-3246	220	3	3	3	NUM
ejpam-3246	220	4	,	,	PUNCT
ejpam-3246	220	5	the	the	DET
ejpam-3246	220	6	uni	uni	ADJ
ejpam-3246	220	7	-	-	ADJ
ejpam-3246	220	8	hesitant	hesitant	ADJ
ejpam-3246	220	9	fuzzy	fuzzy	ADJ
ejpam-3246	220	10	ideal	ideal	NOUN
ejpam-3246	220	11	g	g	NOUN
ejpam-3246	220	12	on	on	ADP
ejpam-3246	220	13	x	x	SYM
ejpam-3246	220	14	is	be	AUX
ejpam-3246	220	15	not	not	PART
ejpam-3246	220	16	hesitant	hesitant	ADJ
ejpam-3246	220	17	closed	close	VERB
ejpam-3246	220	18	since	since	SCONJ
ejpam-3246	220	19	(	(	PUNCT
ejpam-3246	220	20	1÷	1÷	NUM
ejpam-3246	220	21	22	22	NUM
ejpam-3246	220	22	)	)	PUNCT
ejpam-3246	220	23	g	g	NOUN
ejpam-3246	220	24	=	=	NOUN
ejpam-3246	220	25	2−2	2−2	NUM
ejpam-3246	221	1	g	g	NOUN
ejpam-3246	221	2	=	=	SYM
ejpam-3246	221	3	λ2	λ2	PROPN
ejpam-3246	221	4	*	*	PUNCT
ejpam-3246	222	1	λ1	λ1	VERB
ejpam-3246	222	2	=	=	SYM
ejpam-3246	222	3	1	1	NUM
ejpam-3246	222	4	g	g	NOUN
ejpam-3246	222	5	∪	∪	ADJ
ejpam-3246	222	6	22	22	NUM
ejpam-3246	222	7	g.	g.	NOUN
ejpam-3246	222	8	we	we	PRON
ejpam-3246	222	9	provide	provide	VERB
ejpam-3246	222	10	conditions	condition	NOUN
ejpam-3246	222	11	for	for	ADP
ejpam-3246	222	12	a	a	DET
ejpam-3246	222	13	uni	uni	ADJ
ejpam-3246	222	14	-	-	ADJ
ejpam-3246	222	15	hesitant	hesitant	ADJ
ejpam-3246	222	16	fuzzy	fuzzy	ADJ
ejpam-3246	222	17	ideal	ideal	NOUN
ejpam-3246	222	18	to	to	PART
ejpam-3246	222	19	be	be	AUX
ejpam-3246	222	20	hesitant	hesitant	ADJ
ejpam-3246	222	21	closed	closed	ADJ
ejpam-3246	222	22	.	.	PUNCT
ejpam-3246	223	1	theorem	theorem	NOUN
ejpam-3246	223	2	5	5	NUM
ejpam-3246	223	3	.	.	PUNCT
ejpam-3246	224	1	let	let	VERB
ejpam-3246	224	2	g	g	PRON
ejpam-3246	224	3	be	be	AUX
ejpam-3246	224	4	a	a	DET
ejpam-3246	224	5	uni	uni	ADJ
ejpam-3246	224	6	-	-	ADJ
ejpam-3246	224	7	hesitant	hesitant	ADJ
ejpam-3246	224	8	fuzzy	fuzzy	ADJ
ejpam-3246	224	9	ideal	ideal	NOUN
ejpam-3246	224	10	on	on	ADP
ejpam-3246	224	11	a	a	DET
ejpam-3246	224	12	bci	bci	NOUN
ejpam-3246	224	13	-	-	NOUN
ejpam-3246	224	14	algebra	algebra	NOUN
ejpam-3246	224	15	x.	x.	NOUN
ejpam-3246	224	16	then	then	ADV
ejpam-3246	224	17	g	g	PROPN
ejpam-3246	224	18	is	be	AUX
ejpam-3246	224	19	hesitant	hesitant	ADJ
ejpam-3246	224	20	closed	close	VERB
ejpam-3246	224	21	if	if	SCONJ
ejpam-3246	224	22	and	and	CCONJ
ejpam-3246	224	23	only	only	ADV
ejpam-3246	224	24	if	if	SCONJ
ejpam-3246	224	25	g	g	PROPN
ejpam-3246	224	26	is	be	AUX
ejpam-3246	224	27	a	a	DET
ejpam-3246	224	28	uni	uni	ADJ
ejpam-3246	224	29	-	-	ADJ
ejpam-3246	224	30	hesitant	hesitant	ADJ
ejpam-3246	224	31	fuzzy	fuzzy	ADJ
ejpam-3246	224	32	algebra	algebra	NOUN
ejpam-3246	224	33	on	on	ADP
ejpam-3246	224	34	x.	x.	NOUN
ejpam-3246	224	35	proof	proof	PROPN
ejpam-3246	224	36	.	.	PUNCT
ejpam-3246	225	1	assume	assume	VERB
ejpam-3246	225	2	that	that	SCONJ
ejpam-3246	225	3	g	g	PROPN
ejpam-3246	225	4	is	be	AUX
ejpam-3246	225	5	hesitant	hesitant	ADJ
ejpam-3246	225	6	closed	closed	ADJ
ejpam-3246	225	7	.	.	PUNCT
ejpam-3246	226	1	then	then	ADV
ejpam-3246	226	2	(	(	PUNCT
ejpam-3246	226	3	0	0	NUM
ejpam-3246	226	4	∗	∗	NOUN
ejpam-3246	226	5	x)g	x)g	X
ejpam-3246	227	1	⊆	⊆	NUM
ejpam-3246	227	2	xg	xg	NOUN
ejpam-3246	227	3	for	for	ADP
ejpam-3246	227	4	all	all	PRON
ejpam-3246	227	5	x	x	SYM
ejpam-3246	227	6	∈	∈	ADJ
ejpam-3246	227	7	x.	x.	NOUN
ejpam-3246	227	8	it	it	PRON
ejpam-3246	227	9	follows	follow	VERB
ejpam-3246	227	10	from	from	ADP
ejpam-3246	227	11	(	(	PUNCT
ejpam-3246	227	12	5	5	NUM
ejpam-3246	227	13	)	)	PUNCT
ejpam-3246	227	14	that	that	SCONJ
ejpam-3246	227	15	(	(	PUNCT
ejpam-3246	227	16	x	x	X
ejpam-3246	227	17	∗	∗	NOUN
ejpam-3246	227	18	y)g	y)g	NOUN
ejpam-3246	227	19	⊆	⊆	NUM
ejpam-3246	227	20	(	(	PUNCT
ejpam-3246	227	21	(	(	PUNCT
ejpam-3246	227	22	x	x	SYM
ejpam-3246	227	23	∗	∗	PROPN
ejpam-3246	227	24	y	y	NOUN
ejpam-3246	227	25	)	)	PUNCT
ejpam-3246	227	26	∗	∗	NOUN
ejpam-3246	227	27	x)g	x)g	X
ejpam-3246	227	28	∪	∪	X
ejpam-3246	227	29	xg	xg	PROPN
ejpam-3246	227	30	=	=	SYM
ejpam-3246	227	31	(	(	PUNCT
ejpam-3246	227	32	0	0	NUM
ejpam-3246	227	33	∗	∗	NOUN
ejpam-3246	227	34	y)g	y)g	NOUN
ejpam-3246	227	35	∪	∪	X
ejpam-3246	227	36	xg	xg	PROPN
ejpam-3246	227	37	⊆	⊆	NUM
ejpam-3246	227	38	xg	xg	PROPN
ejpam-3246	227	39	∪	∪	ADP
ejpam-3246	227	40	yg	yg	PROPN
ejpam-3246	227	41	for	for	ADP
ejpam-3246	227	42	all	all	DET
ejpam-3246	227	43	x	x	NOUN
ejpam-3246	227	44	,	,	PUNCT
ejpam-3246	227	45	y	y	PROPN
ejpam-3246	227	46	∈	∈	PROPN
ejpam-3246	227	47	x.	x.	NOUN
ejpam-3246	227	48	hence	hence	ADV
ejpam-3246	227	49	g	g	PROPN
ejpam-3246	227	50	is	be	AUX
ejpam-3246	227	51	a	a	DET
ejpam-3246	227	52	uni	uni	ADJ
ejpam-3246	227	53	-	-	ADJ
ejpam-3246	227	54	hesitant	hesitant	ADJ
ejpam-3246	227	55	fuzzy	fuzzy	ADJ
ejpam-3246	227	56	algebra	algebra	NOUN
ejpam-3246	227	57	on	on	ADP
ejpam-3246	227	58	x.	x.	NOUN
ejpam-3246	227	59	conversely	conversely	ADV
ejpam-3246	227	60	,	,	PUNCT
ejpam-3246	227	61	suppose	suppose	VERB
ejpam-3246	227	62	that	that	SCONJ
ejpam-3246	227	63	g	g	PROPN
ejpam-3246	227	64	is	be	AUX
ejpam-3246	227	65	a	a	DET
ejpam-3246	227	66	uni	uni	ADJ
ejpam-3246	227	67	-	-	ADJ
ejpam-3246	227	68	hesitant	hesitant	ADJ
ejpam-3246	227	69	fuzzy	fuzzy	ADJ
ejpam-3246	227	70	algebra	algebra	NOUN
ejpam-3246	227	71	on	on	ADP
ejpam-3246	227	72	a	a	DET
ejpam-3246	227	73	bci	bci	NOUN
ejpam-3246	227	74	-	-	NOUN
ejpam-3246	227	75	algebra	algebra	NOUN
ejpam-3246	227	76	x.	x.	NOUN
ejpam-3246	227	77	then	then	ADV
ejpam-3246	227	78	(	(	PUNCT
ejpam-3246	227	79	0	0	NUM
ejpam-3246	227	80	∗	∗	NOUN
ejpam-3246	227	81	x)g	x)g	X
ejpam-3246	228	1	⊆	⊆	SYM
ejpam-3246	228	2	0	0	NUM
ejpam-3246	228	3	g	g	PROPN
ejpam-3246	228	4	∪	∪	X
ejpam-3246	228	5	xg	xg	PROPN
ejpam-3246	228	6	=	=	PROPN
ejpam-3246	228	7	xg	xg	PROPN
ejpam-3246	228	8	g.	g.	PROPN
ejpam-3246	228	9	muhiuddin	muhiuddin	PROPN
ejpam-3246	228	10	,	,	PUNCT
ejpam-3246	228	11	s.	s.	PROPN
ejpam-3246	228	12	aldhafeeri	aldhafeeri	PROPN
ejpam-3246	228	13	/	/	SYM
ejpam-3246	228	14	eur	eur	PROPN
ejpam-3246	228	15	.	.	PUNCT
ejpam-3246	229	1	j.	j.	PROPN
ejpam-3246	229	2	pure	pure	PROPN
ejpam-3246	229	3	appl	appl	PROPN
ejpam-3246	229	4	.	.	PROPN
ejpam-3246	229	5	math	math	PROPN
ejpam-3246	229	6	,	,	PUNCT
ejpam-3246	229	7	11	11	NUM
ejpam-3246	229	8	(	(	PUNCT
ejpam-3246	229	9	2	2	NUM
ejpam-3246	229	10	)	)	PUNCT
ejpam-3246	229	11	(	(	PUNCT
ejpam-3246	229	12	2018	2018	NUM
ejpam-3246	229	13	)	)	PUNCT
ejpam-3246	229	14	,	,	PUNCT
ejpam-3246	229	15	417	417	NUM
ejpam-3246	229	16	-	-	SYM
ejpam-3246	229	17	430	430	NUM
ejpam-3246	229	18	427	427	NUM
ejpam-3246	229	19	for	for	ADP
ejpam-3246	229	20	all	all	DET
ejpam-3246	229	21	x	x	SYM
ejpam-3246	229	22	∈	∈	NOUN
ejpam-3246	229	23	x.	x.	NOUN
ejpam-3246	229	24	therefore	therefore	ADV
ejpam-3246	229	25	g	g	PROPN
ejpam-3246	229	26	is	be	AUX
ejpam-3246	229	27	hesitant	hesitant	ADJ
ejpam-3246	229	28	closed	closed	ADJ
ejpam-3246	229	29	.	.	PUNCT
ejpam-3246	230	1	by	by	ADP
ejpam-3246	230	2	the	the	DET
ejpam-3246	230	3	similar	similar	ADJ
ejpam-3246	230	4	way	way	NOUN
ejpam-3246	230	5	to	to	PART
ejpam-3246	230	6	theorem	theorem	VERB
ejpam-3246	230	7	4	4	NUM
ejpam-3246	230	8	,	,	PUNCT
ejpam-3246	230	9	we	we	PRON
ejpam-3246	230	10	have	have	VERB
ejpam-3246	230	11	a	a	DET
ejpam-3246	230	12	characterization	characterization	NOUN
ejpam-3246	230	13	of	of	ADP
ejpam-3246	230	14	a	a	DET
ejpam-3246	230	15	uni	uni	ADJ
ejpam-3246	230	16	-	-	ADJ
ejpam-3246	230	17	hesitant	hesitant	ADJ
ejpam-3246	230	18	fuzzy	fuzzy	ADJ
ejpam-3246	230	19	closed	closed	ADJ
ejpam-3246	230	20	ideal	ideal	ADJ
ejpam-3246	230	21	.	.	PUNCT
ejpam-3246	231	1	theorem	theorem	VERB
ejpam-3246	231	2	6	6	NUM
ejpam-3246	231	3	.	.	PUNCT
ejpam-3246	231	4	for	for	ADP
ejpam-3246	231	5	a	a	DET
ejpam-3246	231	6	hesitant	hesitant	ADJ
ejpam-3246	231	7	fuzzy	fuzzy	NOUN
ejpam-3246	231	8	set	set	VERB
ejpam-3246	231	9	g	g	NOUN
ejpam-3246	231	10	on	on	ADP
ejpam-3246	231	11	a	a	DET
ejpam-3246	231	12	bci	bci	NOUN
ejpam-3246	231	13	-	-	NOUN
ejpam-3246	231	14	algebra	algebra	NOUN
ejpam-3246	231	15	x	x	NOUN
ejpam-3246	231	16	,	,	PUNCT
ejpam-3246	231	17	the	the	DET
ejpam-3246	231	18	following	follow	VERB
ejpam-3246	231	19	are	be	AUX
ejpam-3246	231	20	equivalent	equivalent	ADJ
ejpam-3246	231	21	.	.	PUNCT
ejpam-3246	232	1	(	(	PUNCT
ejpam-3246	232	2	i	i	NOUN
ejpam-3246	232	3	)	)	PUNCT
ejpam-3246	232	4	g	g	NOUN
ejpam-3246	232	5	is	be	AUX
ejpam-3246	232	6	a	a	DET
ejpam-3246	232	7	uni	uni	ADJ
ejpam-3246	232	8	-	-	ADJ
ejpam-3246	232	9	hesitant	hesitant	ADJ
ejpam-3246	232	10	fuzzy	fuzzy	ADJ
ejpam-3246	232	11	closed	close	VERB
ejpam-3246	232	12	ideal	ideal	NOUN
ejpam-3246	232	13	on	on	ADP
ejpam-3246	232	14	x.	x.	PROPN
ejpam-3246	232	15	(	(	PUNCT
ejpam-3246	232	16	ii	ii	PROPN
ejpam-3246	232	17	)	)	PUNCT
ejpam-3246	232	18	the	the	PRON
ejpam-3246	232	19	nonempty	nonempty	ADJ
ejpam-3246	232	20	uni	uni	ADJ
ejpam-3246	232	21	-	-	ADJ
ejpam-3246	232	22	hesitant	hesitant	ADJ
ejpam-3246	232	23	level	level	NOUN
ejpam-3246	232	24	set	set	VERB
ejpam-3246	232	25	l(g;λ	l(g;λ	NOUN
ejpam-3246	232	26	)	)	PUNCT
ejpam-3246	232	27	of	of	ADP
ejpam-3246	232	28	g	g	PROPN
ejpam-3246	232	29	is	be	AUX
ejpam-3246	232	30	a	a	DET
ejpam-3246	232	31	closed	closed	ADJ
ejpam-3246	232	32	ideal	ideal	NOUN
ejpam-3246	232	33	of	of	ADP
ejpam-3246	232	34	x	x	PUNCT
ejpam-3246	232	35	for	for	ADP
ejpam-3246	232	36	all	all	DET
ejpam-3246	232	37	λ	λ	PROPN
ejpam-3246	232	38	∈	∈	PROPN
ejpam-3246	232	39	p	p	X
ejpam-3246	232	40	(	(	PUNCT
ejpam-3246	232	41	[	[	X
ejpam-3246	232	42	0	0	NUM
ejpam-3246	232	43	,	,	PUNCT
ejpam-3246	232	44	1	1	NUM
ejpam-3246	232	45	]	]	NUM
ejpam-3246	232	46	)	)	PUNCT
ejpam-3246	232	47	.	.	PUNCT
ejpam-3246	233	1	theorem	theorem	VERB
ejpam-3246	233	2	7	7	NUM
ejpam-3246	233	3	.	.	PUNCT
ejpam-3246	234	1	the	the	DET
ejpam-3246	234	2	hesitant	hesitant	PROPN
ejpam-3246	234	3	union	union	NOUN
ejpam-3246	234	4	of	of	ADP
ejpam-3246	234	5	two	two	NUM
ejpam-3246	234	6	uni	uni	ADJ
ejpam-3246	234	7	-	-	ADJ
ejpam-3246	234	8	hesitant	hesitant	ADJ
ejpam-3246	234	9	fuzzy	fuzzy	ADJ
ejpam-3246	234	10	closed	closed	ADJ
ejpam-3246	234	11	ideals	ideal	NOUN
ejpam-3246	234	12	on	on	ADP
ejpam-3246	234	13	a	a	DET
ejpam-3246	234	14	bci	bci	NOUN
ejpam-3246	234	15	-	-	NOUN
ejpam-3246	234	16	algebra	algebra	NOUN
ejpam-3246	234	17	x	x	PUNCT
ejpam-3246	234	18	is	be	AUX
ejpam-3246	234	19	a	a	DET
ejpam-3246	234	20	uni	uni	ADJ
ejpam-3246	234	21	-	-	ADJ
ejpam-3246	234	22	hesitant	hesitant	ADJ
ejpam-3246	234	23	fuzzy	fuzzy	ADJ
ejpam-3246	234	24	closed	close	VERB
ejpam-3246	234	25	ideal	ideal	NOUN
ejpam-3246	234	26	on	on	ADP
ejpam-3246	234	27	x.	x.	NOUN
ejpam-3246	234	28	proof	proof	NOUN
ejpam-3246	234	29	.	.	PUNCT
ejpam-3246	235	1	let	let	VERB
ejpam-3246	235	2	g	g	NOUN
ejpam-3246	235	3	and	and	CCONJ
ejpam-3246	235	4	h	h	NOUN
ejpam-3246	235	5	be	be	AUX
ejpam-3246	235	6	uni	uni	ADJ
ejpam-3246	235	7	-	-	ADJ
ejpam-3246	235	8	hesitant	hesitant	ADJ
ejpam-3246	235	9	fuzzy	fuzzy	ADJ
ejpam-3246	235	10	closed	closed	ADJ
ejpam-3246	235	11	ideals	ideal	NOUN
ejpam-3246	235	12	on	on	ADP
ejpam-3246	235	13	a	a	DET
ejpam-3246	235	14	bci	bci	NOUN
ejpam-3246	235	15	-	-	NOUN
ejpam-3246	235	16	algebra	algebra	NOUN
ejpam-3246	235	17	x.	x.	NOUN
ejpam-3246	235	18	then	then	ADV
ejpam-3246	235	19	l(g;λ	l(g;λ	PROPN
ejpam-3246	235	20	)	)	PUNCT
ejpam-3246	235	21	and	and	CCONJ
ejpam-3246	235	22	l(h;λ	l(h;λ	PROPN
ejpam-3246	235	23	)	)	PUNCT
ejpam-3246	235	24	are	be	AUX
ejpam-3246	235	25	closed	close	VERB
ejpam-3246	235	26	ideals	ideal	NOUN
ejpam-3246	235	27	of	of	ADP
ejpam-3246	235	28	x	x	PUNCT
ejpam-3246	235	29	for	for	ADP
ejpam-3246	235	30	all	all	DET
ejpam-3246	235	31	λ	λ	PROPN
ejpam-3246	235	32	∈	∈	PROPN
ejpam-3246	235	33	p	p	X
ejpam-3246	235	34	(	(	PUNCT
ejpam-3246	235	35	[	[	X
ejpam-3246	235	36	0	0	NUM
ejpam-3246	235	37	,	,	PUNCT
ejpam-3246	235	38	1	1	NUM
ejpam-3246	235	39	]	]	PUNCT
ejpam-3246	235	40	)	)	PUNCT
ejpam-3246	235	41	whenever	whenever	SCONJ
ejpam-3246	235	42	they	they	PRON
ejpam-3246	235	43	are	be	AUX
ejpam-3246	235	44	nonempty	nonempty	ADJ
ejpam-3246	235	45	.	.	PUNCT
ejpam-3246	236	1	thus	thus	ADV
ejpam-3246	236	2	0	0	NUM
ejpam-3246	236	3	∈	∈	PROPN
ejpam-3246	236	4	l(g;λ	l(g;λ	PROPN
ejpam-3246	236	5	)	)	PUNCT
ejpam-3246	236	6	∩	∩	NOUN
ejpam-3246	236	7	l(h;λ	l(h;λ	PROPN
ejpam-3246	236	8	)	)	PUNCT
ejpam-3246	236	9	,	,	PUNCT
ejpam-3246	236	10	and	and	CCONJ
ejpam-3246	236	11	so	so	ADV
ejpam-3246	236	12	0(g	0(g	NUM
ejpam-3246	236	13	t	t	NOUN
ejpam-3246	236	14	h	h	NOUN
ejpam-3246	236	15	)	)	PUNCT
ejpam-3246	236	16	=	=	SYM
ejpam-3246	237	1	0	0	NUM
ejpam-3246	237	2	g	g	NOUN
ejpam-3246	237	3	∪	∪	X
ejpam-3246	237	4	0h	0h	X
ejpam-3246	237	5	⊆	⊆	NUM
ejpam-3246	237	6	λ	λ	NOUN
ejpam-3246	237	7	.	.	PUNCT
ejpam-3246	238	1	thus	thus	ADV
ejpam-3246	238	2	0	0	NUM
ejpam-3246	238	3	∈	∈	PROPN
ejpam-3246	238	4	l(g	l(g	PROPN
ejpam-3246	238	5	t	t	NOUN
ejpam-3246	238	6	h;λ	h;λ	NUM
ejpam-3246	238	7	)	)	PUNCT
ejpam-3246	238	8	.	.	PUNCT
ejpam-3246	239	1	let	let	VERB
ejpam-3246	239	2	x	x	PRON
ejpam-3246	239	3	,	,	PUNCT
ejpam-3246	239	4	y	y	PROPN
ejpam-3246	239	5	∈	∈	PROPN
ejpam-3246	239	6	x	x	AUX
ejpam-3246	239	7	be	be	AUX
ejpam-3246	239	8	such	such	ADJ
ejpam-3246	239	9	that	that	SCONJ
ejpam-3246	239	10	x	x	PUNCT
ejpam-3246	239	11	∗	∗	NOUN
ejpam-3246	239	12	y	y	PROPN
ejpam-3246	239	13	∈	∈	PROPN
ejpam-3246	239	14	l(g	l(g	PROPN
ejpam-3246	239	15	t	t	NOUN
ejpam-3246	239	16	h;λ	h;λ	NUM
ejpam-3246	239	17	)	)	PUNCT
ejpam-3246	239	18	and	and	CCONJ
ejpam-3246	239	19	y	y	PROPN
ejpam-3246	239	20	∈	∈	PROPN
ejpam-3246	239	21	l(g	l(g	PROPN
ejpam-3246	239	22	t	t	NOUN
ejpam-3246	239	23	h;λ	h;λ	NUM
ejpam-3246	239	24	)	)	PUNCT
ejpam-3246	239	25	.	.	PUNCT
ejpam-3246	240	1	then	then	ADV
ejpam-3246	240	2	(	(	PUNCT
ejpam-3246	240	3	x	x	SYM
ejpam-3246	240	4	∗	∗	NOUN
ejpam-3246	240	5	y)g	y)g	NOUN
ejpam-3246	240	6	∪	∪	X
ejpam-3246	240	7	(	(	PUNCT
ejpam-3246	240	8	x	x	X
ejpam-3246	240	9	∗	∗	NOUN
ejpam-3246	240	10	y)h	y)h	NOUN
ejpam-3246	241	1	=	=	SYM
ejpam-3246	242	1	(	(	PUNCT
ejpam-3246	242	2	x	x	SYM
ejpam-3246	242	3	∗	∗	NOUN
ejpam-3246	242	4	y)(g	y)(g	PROPN
ejpam-3246	242	5	t	t	PROPN
ejpam-3246	242	6	h	h	NOUN
ejpam-3246	242	7	)	)	PUNCT
ejpam-3246	242	8	⊆	⊆	NUM
ejpam-3246	242	9	λ	λ	NOUN
ejpam-3246	242	10	and	and	CCONJ
ejpam-3246	242	11	yg	yg	PROPN
ejpam-3246	242	12	∪	∪	AUX
ejpam-3246	242	13	yh	yh	PROPN
ejpam-3246	242	14	=	=	SYM
ejpam-3246	242	15	y(g	y(g	PROPN
ejpam-3246	242	16	t	t	PROPN
ejpam-3246	242	17	h	h	PROPN
ejpam-3246	242	18	)	)	PUNCT
ejpam-3246	243	1	⊆	⊆	NUM
ejpam-3246	243	2	λ	λ	NOUN
ejpam-3246	243	3	.	.	PUNCT
ejpam-3246	244	1	hence	hence	ADV
ejpam-3246	244	2	(	(	PUNCT
ejpam-3246	244	3	x	x	SYM
ejpam-3246	244	4	∗	∗	NOUN
ejpam-3246	244	5	y)g	y)g	NOUN
ejpam-3246	244	6	⊆	⊆	NUM
ejpam-3246	244	7	λ	λ	NOUN
ejpam-3246	244	8	,	,	PUNCT
ejpam-3246	244	9	(	(	PUNCT
ejpam-3246	244	10	x	x	X
ejpam-3246	244	11	∗	∗	X
ejpam-3246	244	12	y)h	y)h	NOUN
ejpam-3246	244	13	⊆	⊆	NUM
ejpam-3246	244	14	λ	λ	NOUN
ejpam-3246	244	15	,	,	PUNCT
ejpam-3246	244	16	yg	yg	PROPN
ejpam-3246	244	17	⊆	⊆	NUM
ejpam-3246	244	18	λ	λ	NOUN
ejpam-3246	244	19	and	and	CCONJ
ejpam-3246	244	20	yh	yh	NOUN
ejpam-3246	244	21	⊆	⊆	NUM
ejpam-3246	244	22	λ	λ	NOUN
ejpam-3246	244	23	.	.	PUNCT
ejpam-3246	245	1	it	it	PRON
ejpam-3246	245	2	follows	follow	VERB
ejpam-3246	245	3	from	from	ADP
ejpam-3246	245	4	(	(	PUNCT
ejpam-3246	245	5	5	5	NUM
ejpam-3246	245	6	)	)	PUNCT
ejpam-3246	245	7	that	that	PRON
ejpam-3246	245	8	xg	xg	PROPN
ejpam-3246	245	9	⊆	⊆	NUM
ejpam-3246	245	10	(	(	PUNCT
ejpam-3246	245	11	x	x	SYM
ejpam-3246	245	12	∗	∗	NOUN
ejpam-3246	245	13	y)g	y)g	NOUN
ejpam-3246	245	14	∪	∪	PROPN
ejpam-3246	245	15	yg	yg	PROPN
ejpam-3246	245	16	and	and	CCONJ
ejpam-3246	245	17	xh	xh	PROPN
ejpam-3246	245	18	⊆	⊆	NUM
ejpam-3246	245	19	(	(	PUNCT
ejpam-3246	245	20	x	x	NOUN
ejpam-3246	245	21	∗	∗	NOUN
ejpam-3246	245	22	y)h	y)h	NOUN
ejpam-3246	245	23	∪	∪	ADP
ejpam-3246	245	24	yh	yh	NOUN
ejpam-3246	245	25	.	.	PUNCT
ejpam-3246	246	1	hence	hence	ADV
ejpam-3246	246	2	x(g	x(g	PROPN
ejpam-3246	246	3	t	t	PROPN
ejpam-3246	246	4	h	h	NOUN
ejpam-3246	246	5	)	)	PUNCT
ejpam-3246	246	6	=	=	PUNCT
ejpam-3246	246	7	xg	xg	PROPN
ejpam-3246	246	8	∪	∪	PROPN
ejpam-3246	246	9	xh	xh	PROPN
ejpam-3246	246	10	⊆	⊆	NUM
ejpam-3246	246	11	(	(	PUNCT
ejpam-3246	246	12	(	(	PUNCT
ejpam-3246	246	13	x	x	SYM
ejpam-3246	246	14	∗	∗	NOUN
ejpam-3246	246	15	y)g	y)g	NOUN
ejpam-3246	246	16	∪	∪	PROPN
ejpam-3246	246	17	yg	yg	NOUN
ejpam-3246	246	18	)	)	PUNCT
ejpam-3246	246	19	∪	∪	NOUN
ejpam-3246	246	20	(	(	PUNCT
ejpam-3246	246	21	(	(	PUNCT
ejpam-3246	246	22	x	x	SYM
ejpam-3246	246	23	∗	∗	NOUN
ejpam-3246	246	24	y)h	y)h	NOUN
ejpam-3246	246	25	∪	∪	ADP
ejpam-3246	246	26	yh	yh	NOUN
ejpam-3246	246	27	)	)	PUNCT
ejpam-3246	246	28	=	=	SYM
ejpam-3246	246	29	(	(	PUNCT
ejpam-3246	246	30	(	(	PUNCT
ejpam-3246	246	31	x	x	SYM
ejpam-3246	246	32	∗	∗	NOUN
ejpam-3246	246	33	y)g	y)g	NOUN
ejpam-3246	246	34	∪	∪	X
ejpam-3246	246	35	(	(	PUNCT
ejpam-3246	246	36	x	x	NOUN
ejpam-3246	246	37	∗	∗	NOUN
ejpam-3246	246	38	y)h	y)h	NOUN
ejpam-3246	246	39	)	)	PUNCT
ejpam-3246	246	40	∪	∪	NOUN
ejpam-3246	246	41	(	(	PUNCT
ejpam-3246	246	42	yg	yg	PROPN
ejpam-3246	246	43	∪	∪	PROPN
ejpam-3246	246	44	yh	yh	NOUN
ejpam-3246	246	45	)	)	PUNCT
ejpam-3246	246	46	=	=	SYM
ejpam-3246	246	47	(	(	PUNCT
ejpam-3246	246	48	x	x	SYM
ejpam-3246	246	49	∗	∗	NOUN
ejpam-3246	246	50	y)(g	y)(g	PROPN
ejpam-3246	246	51	t	t	PROPN
ejpam-3246	246	52	h	h	NOUN
ejpam-3246	246	53	)	)	PUNCT
ejpam-3246	246	54	∪	∪	ADP
ejpam-3246	246	55	y(g	y(g	PROPN
ejpam-3246	246	56	t	t	PROPN
ejpam-3246	246	57	h	h	PROPN
ejpam-3246	246	58	)	)	PUNCT
ejpam-3246	246	59	and	and	CCONJ
ejpam-3246	246	60	(	(	PUNCT
ejpam-3246	246	61	0	0	NUM
ejpam-3246	246	62	∗	∗	NOUN
ejpam-3246	246	63	y)(g	y)(g	NUM
ejpam-3246	246	64	th	th	NOUN
ejpam-3246	246	65	)	)	PUNCT
ejpam-3246	246	66	=	=	SYM
ejpam-3246	246	67	(	(	PUNCT
ejpam-3246	246	68	0	0	NUM
ejpam-3246	246	69	∗	∗	NOUN
ejpam-3246	246	70	y)g	y)g	NOUN
ejpam-3246	246	71	∪	∪	X
ejpam-3246	246	72	(	(	PUNCT
ejpam-3246	246	73	0	0	NUM
ejpam-3246	246	74	∗	∗	NOUN
ejpam-3246	246	75	y)h	y)h	NOUN
ejpam-3246	246	76	⊆	⊆	NUM
ejpam-3246	246	77	yg	yg	NOUN
ejpam-3246	246	78	∪	∪	ADJ
ejpam-3246	246	79	yh	yh	PROPN
ejpam-3246	246	80	=	=	SYM
ejpam-3246	246	81	y(g	y(g	PROPN
ejpam-3246	246	82	th	th	NUM
ejpam-3246	246	83	)	)	PUNCT
ejpam-3246	246	84	.	.	PUNCT
ejpam-3246	247	1	thus	thus	ADV
ejpam-3246	247	2	x	x	X
ejpam-3246	247	3	∈	∈	PROPN
ejpam-3246	247	4	l(g	l(g	NOUN
ejpam-3246	247	5	th;λ	th;λ	NUM
ejpam-3246	247	6	)	)	PUNCT
ejpam-3246	247	7	and	and	CCONJ
ejpam-3246	247	8	0	0	NUM
ejpam-3246	247	9	∗	∗	NOUN
ejpam-3246	247	10	y	y	PROPN
ejpam-3246	247	11	∈	∈	PROPN
ejpam-3246	247	12	l(g	l(g	PROPN
ejpam-3246	247	13	th;λ	th;λ	NUM
ejpam-3246	247	14	)	)	PUNCT
ejpam-3246	247	15	.	.	PUNCT
ejpam-3246	248	1	therefore	therefore	ADV
ejpam-3246	248	2	l(g	l(g	NOUN
ejpam-3246	248	3	th;λ	th;λ	NUM
ejpam-3246	248	4	)	)	PUNCT
ejpam-3246	248	5	is	be	AUX
ejpam-3246	248	6	a	a	DET
ejpam-3246	248	7	closed	closed	ADJ
ejpam-3246	248	8	ideal	ideal	NOUN
ejpam-3246	248	9	of	of	ADP
ejpam-3246	248	10	x.	x.	NOUN
ejpam-3246	248	11	it	it	PRON
ejpam-3246	248	12	follows	follow	VERB
ejpam-3246	248	13	from	from	ADP
ejpam-3246	248	14	theorem	theorem	NOUN
ejpam-3246	248	15	6	6	NUM
ejpam-3246	248	16	that	that	SCONJ
ejpam-3246	248	17	g	g	PROPN
ejpam-3246	248	18	t	t	PROPN
ejpam-3246	248	19	h	h	NOUN
ejpam-3246	248	20	is	be	AUX
ejpam-3246	248	21	a	a	DET
ejpam-3246	248	22	uni	uni	ADJ
ejpam-3246	248	23	-	-	ADJ
ejpam-3246	248	24	hesitant	hesitant	ADJ
ejpam-3246	248	25	fuzzy	fuzzy	ADJ
ejpam-3246	248	26	closed	close	VERB
ejpam-3246	248	27	ideal	ideal	NOUN
ejpam-3246	248	28	on	on	ADP
ejpam-3246	248	29	x.	x.	NOUN
ejpam-3246	248	30	theorem	theorem	VERB
ejpam-3246	248	31	8	8	NUM
ejpam-3246	248	32	.	.	PUNCT
ejpam-3246	249	1	if	if	SCONJ
ejpam-3246	249	2	g	g	PROPN
ejpam-3246	249	3	is	be	AUX
ejpam-3246	249	4	a	a	DET
ejpam-3246	249	5	uni	uni	ADJ
ejpam-3246	249	6	-	-	ADJ
ejpam-3246	249	7	hesitant	hesitant	ADJ
ejpam-3246	249	8	fuzzy	fuzzy	ADJ
ejpam-3246	249	9	closed	close	VERB
ejpam-3246	249	10	ideal	ideal	NOUN
ejpam-3246	249	11	on	on	ADP
ejpam-3246	249	12	a	a	DET
ejpam-3246	249	13	bci	bci	NOUN
ejpam-3246	249	14	-	-	NOUN
ejpam-3246	249	15	algebra	algebra	NOUN
ejpam-3246	249	16	x	x	NOUN
ejpam-3246	249	17	,	,	PUNCT
ejpam-3246	249	18	then	then	ADV
ejpam-3246	249	19	the	the	DET
ejpam-3246	249	20	set	set	NOUN
ejpam-3246	249	21	a	a	PRON
ejpam-3246	249	22	:	:	PUNCT
ejpam-3246	249	23	=	=	SYM
ejpam-3246	249	24	{	{	PUNCT
ejpam-3246	249	25	x	x	SYM
ejpam-3246	249	26	∈	∈	PROPN
ejpam-3246	249	27	x	x	INTJ
ejpam-3246	249	28	|	|	ADV
ejpam-3246	249	29	xg	xg	NOUN
ejpam-3246	250	1	=	=	NOUN
ejpam-3246	250	2	0	0	NUM
ejpam-3246	250	3	g	g	NOUN
ejpam-3246	250	4	}	}	PUNCT
ejpam-3246	250	5	is	be	AUX
ejpam-3246	250	6	a	a	DET
ejpam-3246	250	7	closed	closed	ADJ
ejpam-3246	250	8	ideal	ideal	NOUN
ejpam-3246	250	9	of	of	ADP
ejpam-3246	250	10	x.	x.	NOUN
ejpam-3246	250	11	proof	proof	NOUN
ejpam-3246	250	12	.	.	PUNCT
ejpam-3246	251	1	clearly	clearly	ADV
ejpam-3246	251	2	0	0	NUM
ejpam-3246	251	3	∈	∈	NOUN
ejpam-3246	251	4	a.	a.	NOUN
ejpam-3246	251	5	let	let	VERB
ejpam-3246	251	6	x	x	PRON
ejpam-3246	251	7	,	,	PUNCT
ejpam-3246	251	8	y	y	PROPN
ejpam-3246	251	9	∈	∈	PROPN
ejpam-3246	251	10	x	x	AUX
ejpam-3246	251	11	be	be	AUX
ejpam-3246	251	12	such	such	ADJ
ejpam-3246	251	13	that	that	SCONJ
ejpam-3246	251	14	x	x	PUNCT
ejpam-3246	251	15	∗	∗	NOUN
ejpam-3246	251	16	y	y	PROPN
ejpam-3246	251	17	∈	∈	PROPN
ejpam-3246	251	18	a	a	PRON
ejpam-3246	251	19	and	and	CCONJ
ejpam-3246	251	20	y	y	PROPN
ejpam-3246	251	21	∈	∈	PROPN
ejpam-3246	251	22	a.	a.	NOUN
ejpam-3246	251	23	then	then	ADV
ejpam-3246	251	24	(	(	PUNCT
ejpam-3246	251	25	x	x	X
ejpam-3246	251	26	∗	∗	NOUN
ejpam-3246	251	27	y)g	y)g	X
ejpam-3246	252	1	=	=	SYM
ejpam-3246	252	2	0	0	NUM
ejpam-3246	252	3	g	g	NOUN
ejpam-3246	252	4	=	=	SYM
ejpam-3246	252	5	yg	yg	PROPN
ejpam-3246	252	6	.	.	PUNCT
ejpam-3246	253	1	it	it	PRON
ejpam-3246	253	2	follows	follow	VERB
ejpam-3246	253	3	from	from	ADP
ejpam-3246	253	4	(	(	PUNCT
ejpam-3246	253	5	5	5	NUM
ejpam-3246	253	6	)	)	PUNCT
ejpam-3246	253	7	and	and	CCONJ
ejpam-3246	253	8	(	(	PUNCT
ejpam-3246	253	9	11	11	NUM
ejpam-3246	253	10	)	)	PUNCT
ejpam-3246	253	11	that	that	PRON
ejpam-3246	253	12	xg	xg	PROPN
ejpam-3246	253	13	⊆	⊆	NUM
ejpam-3246	253	14	(	(	PUNCT
ejpam-3246	253	15	x	x	SYM
ejpam-3246	253	16	∗	∗	NOUN
ejpam-3246	253	17	y)g	y)g	NOUN
ejpam-3246	253	18	∪	∪	PROPN
ejpam-3246	253	19	yg	yg	NOUN
ejpam-3246	253	20	=	=	SYM
ejpam-3246	253	21	0	0	NUM
ejpam-3246	253	22	g	g	NOUN
ejpam-3246	253	23	and	and	CCONJ
ejpam-3246	253	24	(	(	PUNCT
ejpam-3246	253	25	0	0	NUM
ejpam-3246	253	26	∗	∗	NOUN
ejpam-3246	253	27	y)g	y)g	NOUN
ejpam-3246	253	28	⊆	⊆	NUM
ejpam-3246	253	29	yg	yg	PROPN
ejpam-3246	253	30	=	=	SYM
ejpam-3246	253	31	0	0	NUM
ejpam-3246	253	32	g.	g.	NOUN
ejpam-3246	253	33	since	since	SCONJ
ejpam-3246	253	34	0	0	NUM
ejpam-3246	253	35	g	g	PROPN
ejpam-3246	253	36	⊆	⊆	NUM
ejpam-3246	253	37	xg	xg	NOUN
ejpam-3246	253	38	for	for	ADP
ejpam-3246	253	39	all	all	DET
ejpam-3246	253	40	x	x	SYM
ejpam-3246	253	41	∈	∈	PROPN
ejpam-3246	253	42	x	x	NOUN
ejpam-3246	253	43	,	,	PUNCT
ejpam-3246	253	44	we	we	PRON
ejpam-3246	253	45	have	have	VERB
ejpam-3246	253	46	xg	xg	NOUN
ejpam-3246	253	47	=	=	SYM
ejpam-3246	253	48	0	0	NUM
ejpam-3246	253	49	g	g	NOUN
ejpam-3246	253	50	and	and	CCONJ
ejpam-3246	253	51	(	(	PUNCT
ejpam-3246	253	52	0	0	NUM
ejpam-3246	253	53	∗	∗	NOUN
ejpam-3246	253	54	y)g	y)g	X
ejpam-3246	254	1	=	=	SYM
ejpam-3246	254	2	0	0	NUM
ejpam-3246	254	3	g	g	NOUN
ejpam-3246	254	4	,	,	PUNCT
ejpam-3246	254	5	that	that	ADV
ejpam-3246	254	6	is	is	ADV
ejpam-3246	254	7	,	,	PUNCT
ejpam-3246	254	8	x	x	SYM
ejpam-3246	254	9	∈	∈	PROPN
ejpam-3246	254	10	a	a	PRON
ejpam-3246	254	11	and	and	CCONJ
ejpam-3246	254	12	0	0	NUM
ejpam-3246	254	13	∗	∗	NOUN
ejpam-3246	254	14	y	y	PROPN
ejpam-3246	254	15	∈	∈	PROPN
ejpam-3246	254	16	a.	a.	NOUN
ejpam-3246	254	17	therefore	therefore	ADV
ejpam-3246	254	18	a	a	PRON
ejpam-3246	254	19	is	be	AUX
ejpam-3246	254	20	a	a	DET
ejpam-3246	254	21	closed	closed	ADJ
ejpam-3246	254	22	ideal	ideal	NOUN
ejpam-3246	254	23	of	of	ADP
ejpam-3246	254	24	x.	x.	NOUN
ejpam-3246	254	25	let	let	VERB
ejpam-3246	254	26	x	x	PRON
ejpam-3246	254	27	be	be	AUX
ejpam-3246	254	28	a	a	DET
ejpam-3246	254	29	bci	bci	NOUN
ejpam-3246	254	30	-	-	NOUN
ejpam-3246	254	31	algebra	algebra	NOUN
ejpam-3246	254	32	and	and	CCONJ
ejpam-3246	254	33	b(x	b(x	NOUN
ejpam-3246	254	34	)	)	PUNCT
ejpam-3246	255	1	:	:	PUNCT
ejpam-3246	255	2	=	=	SYM
ejpam-3246	255	3	{	{	PUNCT
ejpam-3246	255	4	x	x	SYM
ejpam-3246	255	5	∈	∈	NOUN
ejpam-3246	255	6	x	x	SYM
ejpam-3246	255	7	|	|	ADV
ejpam-3246	255	8	0	0	NUM
ejpam-3246	255	9	≤	≤	NUM
ejpam-3246	255	10	x	x	X
ejpam-3246	255	11	}	}	PUNCT
ejpam-3246	255	12	.	.	PUNCT
ejpam-3246	256	1	for	for	ADP
ejpam-3246	256	2	any	any	DET
ejpam-3246	256	3	x	x	SYM
ejpam-3246	256	4	∈	∈	PROPN
ejpam-3246	256	5	x	x	X
ejpam-3246	256	6	and	and	CCONJ
ejpam-3246	256	7	n	n	CCONJ
ejpam-3246	256	8	∈	∈	PROPN
ejpam-3246	256	9	n	n	CCONJ
ejpam-3246	256	10	,	,	PUNCT
ejpam-3246	256	11	we	we	PRON
ejpam-3246	256	12	define	define	VERB
ejpam-3246	256	13	xn	xn	PUNCT
ejpam-3246	256	14	by	by	ADP
ejpam-3246	256	15	x1	x1	PROPN
ejpam-3246	256	16	=	=	SYM
ejpam-3246	256	17	x	x	NOUN
ejpam-3246	256	18	,	,	PUNCT
ejpam-3246	256	19	xn+1	xn+1	PROPN
ejpam-3246	256	20	=	=	SYM
ejpam-3246	256	21	x	x	SYM
ejpam-3246	256	22	∗	∗	NOUN
ejpam-3246	256	23	(	(	PUNCT
ejpam-3246	256	24	0	0	NUM
ejpam-3246	256	25	∗	∗	NOUN
ejpam-3246	256	26	xn	xn	NUM
ejpam-3246	256	27	)	)	PUNCT
ejpam-3246	256	28	.	.	PUNCT
ejpam-3246	257	1	g.	g.	PROPN
ejpam-3246	257	2	muhiuddin	muhiuddin	PROPN
ejpam-3246	257	3	,	,	PUNCT
ejpam-3246	257	4	s.	s.	PROPN
ejpam-3246	257	5	aldhafeeri	aldhafeeri	PROPN
ejpam-3246	257	6	/	/	SYM
ejpam-3246	257	7	eur	eur	PROPN
ejpam-3246	257	8	.	.	PUNCT
ejpam-3246	258	1	j.	j.	PROPN
ejpam-3246	258	2	pure	pure	PROPN
ejpam-3246	258	3	appl	appl	PROPN
ejpam-3246	258	4	.	.	PROPN
ejpam-3246	258	5	math	math	PROPN
ejpam-3246	258	6	,	,	PUNCT
ejpam-3246	258	7	11	11	NUM
ejpam-3246	258	8	(	(	PUNCT
ejpam-3246	258	9	2	2	NUM
ejpam-3246	258	10	)	)	PUNCT
ejpam-3246	258	11	(	(	PUNCT
ejpam-3246	258	12	2018	2018	NUM
ejpam-3246	258	13	)	)	PUNCT
ejpam-3246	258	14	,	,	PUNCT
ejpam-3246	258	15	417	417	NUM
ejpam-3246	258	16	-	-	SYM
ejpam-3246	258	17	430	430	NUM
ejpam-3246	258	18	428	428	NUM
ejpam-3246	258	19	if	if	SCONJ
ejpam-3246	258	20	there	there	PRON
ejpam-3246	258	21	is	be	VERB
ejpam-3246	258	22	an	an	DET
ejpam-3246	258	23	n	n	NUM
ejpam-3246	258	24	∈	∈	NOUN
ejpam-3246	258	25	n	n	PRON
ejpam-3246	258	26	such	such	ADJ
ejpam-3246	258	27	that	that	SCONJ
ejpam-3246	258	28	xn	xn	PROPN
ejpam-3246	258	29	∈	∈	PROPN
ejpam-3246	258	30	b(x	b(x	NOUN
ejpam-3246	258	31	)	)	PUNCT
ejpam-3246	259	1	,	,	PUNCT
ejpam-3246	259	2	then	then	ADV
ejpam-3246	259	3	we	we	PRON
ejpam-3246	259	4	say	say	VERB
ejpam-3246	259	5	that	that	SCONJ
ejpam-3246	259	6	x	x	PRON
ejpam-3246	259	7	is	be	AUX
ejpam-3246	259	8	of	of	ADP
ejpam-3246	259	9	finite	finite	ADJ
ejpam-3246	259	10	periodic	periodic	NOUN
ejpam-3246	259	11	(	(	PUNCT
ejpam-3246	259	12	see	see	VERB
ejpam-3246	259	13	[	[	X
ejpam-3246	259	14	7	7	NUM
ejpam-3246	259	15	]	]	NUM
ejpam-3246	259	16	)	)	PUNCT
ejpam-3246	259	17	,	,	PUNCT
ejpam-3246	259	18	and	and	CCONJ
ejpam-3246	259	19	we	we	PRON
ejpam-3246	259	20	denote	denote	VERB
ejpam-3246	259	21	its	its	PRON
ejpam-3246	259	22	period	period	NOUN
ejpam-3246	259	23	|x|	|x|	PROPN
ejpam-3246	259	24	by	by	ADP
ejpam-3246	259	25	|x|	|x|	PROPN
ejpam-3246	259	26	=	=	PUNCT
ejpam-3246	259	27	min{n	min{n	NOUN
ejpam-3246	259	28	∈	∈	NOUN
ejpam-3246	259	29	n	n	CCONJ
ejpam-3246	259	30	|	|	ADV
ejpam-3246	259	31	xn	xn	PROPN
ejpam-3246	259	32	∈	∈	PROPN
ejpam-3246	259	33	b(x	b(x	NOUN
ejpam-3246	259	34	)	)	PUNCT
ejpam-3246	259	35	}	}	PUNCT
ejpam-3246	259	36	.	.	PUNCT
ejpam-3246	260	1	otherwise	otherwise	ADV
ejpam-3246	260	2	,	,	PUNCT
ejpam-3246	260	3	x	x	X
ejpam-3246	260	4	is	be	AUX
ejpam-3246	260	5	of	of	ADP
ejpam-3246	260	6	infinite	infinite	ADJ
ejpam-3246	260	7	period	period	NOUN
ejpam-3246	260	8	and	and	CCONJ
ejpam-3246	260	9	denoted	denote	VERB
ejpam-3246	260	10	by	by	ADP
ejpam-3246	260	11	|x|	|x|	PROPN
ejpam-3246	260	12	=	=	SYM
ejpam-3246	260	13	∞.	∞.	PROPN
ejpam-3246	260	14	theorem	theorem	VERB
ejpam-3246	260	15	9	9	NUM
ejpam-3246	260	16	.	.	PUNCT
ejpam-3246	261	1	if	if	SCONJ
ejpam-3246	261	2	x	x	PRON
ejpam-3246	261	3	is	be	AUX
ejpam-3246	261	4	a	a	DET
ejpam-3246	261	5	bci	bci	NOUN
ejpam-3246	261	6	-	-	NOUN
ejpam-3246	261	7	algebra	algebra	NOUN
ejpam-3246	261	8	in	in	ADP
ejpam-3246	261	9	which	which	PRON
ejpam-3246	261	10	every	every	DET
ejpam-3246	261	11	element	element	NOUN
ejpam-3246	261	12	is	be	AUX
ejpam-3246	261	13	of	of	ADP
ejpam-3246	261	14	finite	finite	ADJ
ejpam-3246	261	15	period	period	NOUN
ejpam-3246	261	16	,	,	PUNCT
ejpam-3246	261	17	then	then	ADV
ejpam-3246	261	18	every	every	DET
ejpam-3246	261	19	uni	uni	ADJ
ejpam-3246	261	20	-	-	ADJ
ejpam-3246	261	21	hesitant	hesitant	ADJ
ejpam-3246	261	22	fuzzy	fuzzy	ADJ
ejpam-3246	261	23	ideal	ideal	NOUN
ejpam-3246	261	24	on	on	ADP
ejpam-3246	261	25	x	x	SYM
ejpam-3246	261	26	is	be	AUX
ejpam-3246	261	27	hesitant	hesitant	ADJ
ejpam-3246	261	28	closed	closed	ADJ
ejpam-3246	261	29	.	.	PUNCT
ejpam-3246	262	1	proof	proof	NOUN
ejpam-3246	262	2	.	.	PUNCT
ejpam-3246	263	1	let	let	VERB
ejpam-3246	263	2	g	g	PRON
ejpam-3246	263	3	be	be	AUX
ejpam-3246	263	4	a	a	DET
ejpam-3246	263	5	uni	uni	ADJ
ejpam-3246	263	6	-	-	ADJ
ejpam-3246	263	7	hesitant	hesitant	ADJ
ejpam-3246	263	8	fuzzy	fuzzy	ADJ
ejpam-3246	263	9	ideal	ideal	NOUN
ejpam-3246	263	10	on	on	ADP
ejpam-3246	263	11	x.	x.	NOUN
ejpam-3246	263	12	for	for	ADP
ejpam-3246	263	13	any	any	DET
ejpam-3246	263	14	x	x	SYM
ejpam-3246	263	15	∈	∈	PROPN
ejpam-3246	263	16	x	x	NOUN
ejpam-3246	263	17	,	,	PUNCT
ejpam-3246	263	18	assume	assume	VERB
ejpam-3246	263	19	that	that	SCONJ
ejpam-3246	263	20	|x|	|x|	PROPN
ejpam-3246	263	21	=	=	SYM
ejpam-3246	263	22	n.	n.	PROPN
ejpam-3246	263	23	then	then	ADV
ejpam-3246	263	24	xn	xn	PROPN
ejpam-3246	263	25	∈	∈	PROPN
ejpam-3246	263	26	b(x	b(x	NOUN
ejpam-3246	263	27	)	)	PUNCT
ejpam-3246	263	28	.	.	PUNCT
ejpam-3246	264	1	note	note	VERB
ejpam-3246	264	2	that	that	SCONJ
ejpam-3246	264	3	(	(	PUNCT
ejpam-3246	264	4	0	0	NUM
ejpam-3246	264	5	∗	∗	NOUN
ejpam-3246	264	6	xn−1	xn−1	PROPN
ejpam-3246	264	7	)	)	PUNCT
ejpam-3246	264	8	∗	∗	NOUN
ejpam-3246	264	9	x	x	X
ejpam-3246	264	10	=	=	SYM
ejpam-3246	264	11	(	(	PUNCT
ejpam-3246	264	12	0	0	NUM
ejpam-3246	264	13	∗	∗	NOUN
ejpam-3246	264	14	(	(	PUNCT
ejpam-3246	264	15	0	0	NUM
ejpam-3246	264	16	∗	∗	NOUN
ejpam-3246	264	17	(	(	PUNCT
ejpam-3246	264	18	0	0	NUM
ejpam-3246	264	19	∗	∗	NOUN
ejpam-3246	264	20	xn−1	xn−1	PROPN
ejpam-3246	264	21	)	)	PUNCT
ejpam-3246	264	22	)	)	PUNCT
ejpam-3246	264	23	)	)	PUNCT
ejpam-3246	265	1	∗	∗	NOUN
ejpam-3246	265	2	x	x	X
ejpam-3246	265	3	=	=	SYM
ejpam-3246	265	4	(	(	PUNCT
ejpam-3246	265	5	0	0	NUM
ejpam-3246	265	6	∗	∗	NOUN
ejpam-3246	265	7	x	x	NOUN
ejpam-3246	265	8	)	)	PUNCT
ejpam-3246	265	9	∗	∗	NOUN
ejpam-3246	265	10	(	(	PUNCT
ejpam-3246	265	11	0	0	NUM
ejpam-3246	265	12	∗	∗	NOUN
ejpam-3246	265	13	(	(	PUNCT
ejpam-3246	265	14	0	0	NUM
ejpam-3246	265	15	∗	∗	NOUN
ejpam-3246	265	16	xn−1	xn−1	PROPN
ejpam-3246	265	17	)	)	PUNCT
ejpam-3246	265	18	)	)	PUNCT
ejpam-3246	266	1	=	=	SYM
ejpam-3246	266	2	0	0	NUM
ejpam-3246	266	3	∗	∗	NOUN
ejpam-3246	266	4	(	(	PUNCT
ejpam-3246	266	5	x	x	SYM
ejpam-3246	266	6	∗	∗	NOUN
ejpam-3246	266	7	(	(	PUNCT
ejpam-3246	266	8	0	0	NUM
ejpam-3246	266	9	∗	∗	NOUN
ejpam-3246	266	10	xn−1	xn−1	PROPN
ejpam-3246	266	11	)	)	PUNCT
ejpam-3246	266	12	)	)	PUNCT
ejpam-3246	267	1	=	=	SYM
ejpam-3246	267	2	0	0	NUM
ejpam-3246	267	3	∗	∗	NOUN
ejpam-3246	267	4	xn	xn	PUNCT
ejpam-3246	268	1	=	=	SYM
ejpam-3246	268	2	0	0	NUM
ejpam-3246	268	3	,	,	PUNCT
ejpam-3246	268	4	and	and	CCONJ
ejpam-3246	268	5	so	so	ADV
ejpam-3246	268	6	(	(	PUNCT
ejpam-3246	268	7	(	(	PUNCT
ejpam-3246	268	8	0	0	NUM
ejpam-3246	268	9	∗	∗	NOUN
ejpam-3246	268	10	xn−1	xn−1	PROPN
ejpam-3246	268	11	)	)	PUNCT
ejpam-3246	268	12	∗	∗	NOUN
ejpam-3246	268	13	x	x	X
ejpam-3246	268	14	)	)	PUNCT
ejpam-3246	269	1	g	g	NOUN
ejpam-3246	269	2	=	=	SYM
ejpam-3246	269	3	0	0	NUM
ejpam-3246	269	4	g	g	NOUN
ejpam-3246	269	5	⊆	⊆	NUM
ejpam-3246	269	6	xg	xg	NOUN
ejpam-3246	269	7	by	by	ADP
ejpam-3246	269	8	proposition	proposition	NOUN
ejpam-3246	269	9	1	1	NUM
ejpam-3246	269	10	.	.	PUNCT
ejpam-3246	270	1	it	it	PRON
ejpam-3246	270	2	follows	follow	VERB
ejpam-3246	270	3	from	from	ADP
ejpam-3246	270	4	(	(	PUNCT
ejpam-3246	270	5	5	5	NUM
ejpam-3246	270	6	)	)	PUNCT
ejpam-3246	270	7	that	that	SCONJ
ejpam-3246	270	8	(	(	PUNCT
ejpam-3246	270	9	0	0	NUM
ejpam-3246	270	10	∗	∗	NOUN
ejpam-3246	270	11	xn−1	xn−1	PROPN
ejpam-3246	270	12	)	)	PUNCT
ejpam-3246	270	13	g	g	ADP
ejpam-3246	270	14	⊆	⊆	NUM
ejpam-3246	270	15	(	(	PUNCT
ejpam-3246	270	16	(	(	PUNCT
ejpam-3246	270	17	0	0	NUM
ejpam-3246	270	18	∗	∗	NOUN
ejpam-3246	270	19	xn−1	xn−1	PROPN
ejpam-3246	270	20	)	)	PUNCT
ejpam-3246	270	21	∗	∗	NOUN
ejpam-3246	270	22	x	x	X
ejpam-3246	270	23	)	)	PUNCT
ejpam-3246	271	1	g	g	PROPN
ejpam-3246	271	2	∪	∪	ADP
ejpam-3246	271	3	xg	xg	PROPN
ejpam-3246	271	4	⊆	⊆	NUM
ejpam-3246	271	5	xg	xg	NOUN
ejpam-3246	271	6	.	.	PUNCT
ejpam-3246	272	1	(	(	PUNCT
ejpam-3246	272	2	12	12	NUM
ejpam-3246	272	3	)	)	PUNCT
ejpam-3246	272	4	also	also	ADV
ejpam-3246	272	5	,	,	PUNCT
ejpam-3246	272	6	note	note	VERB
ejpam-3246	272	7	that	that	SCONJ
ejpam-3246	272	8	(	(	PUNCT
ejpam-3246	272	9	0	0	NUM
ejpam-3246	272	10	∗	∗	NOUN
ejpam-3246	272	11	xn−2	xn−2	PROPN
ejpam-3246	272	12	)	)	PUNCT
ejpam-3246	272	13	∗	∗	NOUN
ejpam-3246	272	14	x	x	X
ejpam-3246	272	15	=	=	SYM
ejpam-3246	272	16	(	(	PUNCT
ejpam-3246	272	17	0	0	NUM
ejpam-3246	272	18	∗	∗	NOUN
ejpam-3246	272	19	(	(	PUNCT
ejpam-3246	272	20	0	0	NUM
ejpam-3246	272	21	∗	∗	NOUN
ejpam-3246	272	22	(	(	PUNCT
ejpam-3246	272	23	0	0	NUM
ejpam-3246	272	24	∗	∗	NOUN
ejpam-3246	272	25	xn−2	xn−2	PROPN
ejpam-3246	272	26	)	)	PUNCT
ejpam-3246	272	27	)	)	PUNCT
ejpam-3246	272	28	)	)	PUNCT
ejpam-3246	273	1	∗	∗	NOUN
ejpam-3246	273	2	x	x	X
ejpam-3246	273	3	=	=	SYM
ejpam-3246	273	4	(	(	PUNCT
ejpam-3246	273	5	0	0	NUM
ejpam-3246	273	6	∗	∗	NOUN
ejpam-3246	273	7	x	x	NOUN
ejpam-3246	273	8	)	)	PUNCT
ejpam-3246	273	9	∗	∗	NOUN
ejpam-3246	273	10	(	(	PUNCT
ejpam-3246	273	11	0	0	NUM
ejpam-3246	273	12	∗	∗	NOUN
ejpam-3246	273	13	(	(	PUNCT
ejpam-3246	273	14	0	0	NUM
ejpam-3246	273	15	∗	∗	NOUN
ejpam-3246	273	16	xn−2	xn−2	PROPN
ejpam-3246	273	17	)	)	PUNCT
ejpam-3246	273	18	)	)	PUNCT
ejpam-3246	274	1	=	=	SYM
ejpam-3246	274	2	0	0	NUM
ejpam-3246	274	3	∗	∗	NOUN
ejpam-3246	274	4	(	(	PUNCT
ejpam-3246	274	5	x	x	SYM
ejpam-3246	274	6	∗	∗	NOUN
ejpam-3246	274	7	(	(	PUNCT
ejpam-3246	274	8	0	0	NUM
ejpam-3246	274	9	∗	∗	NOUN
ejpam-3246	274	10	xn−2	xn−2	PROPN
ejpam-3246	274	11	)	)	PUNCT
ejpam-3246	274	12	)	)	PUNCT
ejpam-3246	275	1	=	=	SYM
ejpam-3246	275	2	0	0	NUM
ejpam-3246	275	3	∗	∗	NOUN
ejpam-3246	275	4	xn−1	xn−1	PROPN
ejpam-3246	275	5	,	,	PUNCT
ejpam-3246	275	6	which	which	PRON
ejpam-3246	275	7	implies	imply	VERB
ejpam-3246	275	8	from	from	ADP
ejpam-3246	275	9	(	(	PUNCT
ejpam-3246	275	10	12	12	NUM
ejpam-3246	275	11	)	)	PUNCT
ejpam-3246	275	12	that	that	SCONJ
ejpam-3246	275	13	(	(	PUNCT
ejpam-3246	275	14	(	(	PUNCT
ejpam-3246	275	15	0	0	NUM
ejpam-3246	275	16	∗	∗	NOUN
ejpam-3246	275	17	xn−2	xn−2	PROPN
ejpam-3246	275	18	)	)	PUNCT
ejpam-3246	275	19	∗	∗	NOUN
ejpam-3246	275	20	x	x	X
ejpam-3246	275	21	)	)	PUNCT
ejpam-3246	275	22	g	g	NOUN
ejpam-3246	275	23	=	=	SYM
ejpam-3246	275	24	(	(	PUNCT
ejpam-3246	275	25	0	0	NUM
ejpam-3246	275	26	∗	∗	NOUN
ejpam-3246	275	27	xn−1	xn−1	PROPN
ejpam-3246	275	28	)	)	PUNCT
ejpam-3246	275	29	g	g	PROPN
ejpam-3246	275	30	⊆	⊆	NUM
ejpam-3246	275	31	xg	xg	NOUN
ejpam-3246	275	32	.	.	PUNCT
ejpam-3246	276	1	using	use	VERB
ejpam-3246	276	2	(	(	PUNCT
ejpam-3246	276	3	5	5	NUM
ejpam-3246	276	4	)	)	PUNCT
ejpam-3246	276	5	,	,	PUNCT
ejpam-3246	276	6	we	we	PRON
ejpam-3246	276	7	have	have	VERB
ejpam-3246	276	8	(	(	PUNCT
ejpam-3246	276	9	0	0	NUM
ejpam-3246	276	10	∗	∗	NOUN
ejpam-3246	276	11	xn−2	xn−2	PROPN
ejpam-3246	276	12	)	)	PUNCT
ejpam-3246	277	1	g	g	ADP
ejpam-3246	277	2	⊆	⊆	NUM
ejpam-3246	277	3	(	(	PUNCT
ejpam-3246	277	4	(	(	PUNCT
ejpam-3246	277	5	0	0	NUM
ejpam-3246	277	6	∗	∗	NOUN
ejpam-3246	277	7	xn−2	xn−2	PROPN
ejpam-3246	277	8	)	)	PUNCT
ejpam-3246	277	9	∗	∗	NOUN
ejpam-3246	277	10	x	x	X
ejpam-3246	277	11	)	)	PUNCT
ejpam-3246	277	12	g	g	PROPN
ejpam-3246	277	13	∪	∪	ADP
ejpam-3246	277	14	xg	xg	PROPN
ejpam-3246	277	15	⊆	⊆	NUM
ejpam-3246	277	16	xg	xg	NOUN
ejpam-3246	277	17	.	.	PUNCT
ejpam-3246	278	1	continuing	continue	VERB
ejpam-3246	278	2	this	this	DET
ejpam-3246	278	3	process	process	NOUN
ejpam-3246	278	4	,	,	PUNCT
ejpam-3246	278	5	we	we	PRON
ejpam-3246	278	6	have	have	VERB
ejpam-3246	278	7	(	(	PUNCT
ejpam-3246	278	8	0	0	NUM
ejpam-3246	278	9	∗	∗	NOUN
ejpam-3246	278	10	x)g	x)g	X
ejpam-3246	279	1	⊆	⊆	NUM
ejpam-3246	279	2	xg	xg	NOUN
ejpam-3246	279	3	for	for	ADP
ejpam-3246	279	4	all	all	DET
ejpam-3246	279	5	x	x	SYM
ejpam-3246	279	6	∈	∈	NOUN
ejpam-3246	279	7	x.	x.	NOUN
ejpam-3246	280	1	therefore	therefore	ADV
ejpam-3246	280	2	g	g	PROPN
ejpam-3246	280	3	is	be	AUX
ejpam-3246	280	4	hesitant	hesitant	ADJ
ejpam-3246	280	5	closed	closed	ADJ
ejpam-3246	280	6	.	.	PUNCT
ejpam-3246	281	1	acknowledgements	acknowledgement	NOUN
ejpam-3246	281	2	the	the	DET
ejpam-3246	281	3	authors	author	NOUN
ejpam-3246	281	4	would	would	AUX
ejpam-3246	281	5	like	like	VERB
ejpam-3246	281	6	to	to	PART
ejpam-3246	281	7	express	express	VERB
ejpam-3246	281	8	their	their	PRON
ejpam-3246	281	9	sincere	sincere	ADJ
ejpam-3246	281	10	thanks	thank	NOUN
ejpam-3246	281	11	to	to	ADP
ejpam-3246	281	12	the	the	DET
ejpam-3246	281	13	anonymous	anonymous	ADJ
ejpam-3246	281	14	referees	referee	NOUN
ejpam-3246	281	15	for	for	ADP
ejpam-3246	281	16	their	their	PRON
ejpam-3246	281	17	valuable	valuable	ADJ
ejpam-3246	281	18	comments	comment	NOUN
ejpam-3246	281	19	and	and	CCONJ
ejpam-3246	281	20	several	several	ADJ
ejpam-3246	281	21	useful	useful	ADJ
ejpam-3246	281	22	suggestions	suggestion	NOUN
ejpam-3246	281	23	.	.	PUNCT
ejpam-3246	282	1	references	reference	NOUN
ejpam-3246	282	2	429	429	NUM
ejpam-3246	282	3	references	reference	NOUN
ejpam-3246	282	4	[	[	X
ejpam-3246	282	5	1	1	NUM
ejpam-3246	282	6	]	]	X
ejpam-3246	282	7	y.	y.	PROPN
ejpam-3246	282	8	huang	huang	PROPN
ejpam-3246	282	9	,	,	PUNCT
ejpam-3246	282	10	bci	bci	PROPN
ejpam-3246	282	11	-	-	NOUN
ejpam-3246	282	12	algebra	algebra	NOUN
ejpam-3246	282	13	,	,	PUNCT
ejpam-3246	282	14	science	science	NOUN
ejpam-3246	282	15	press	press	NOUN
ejpam-3246	282	16	,	,	PUNCT
ejpam-3246	282	17	beijing	beijing	PROPN
ejpam-3246	282	18	2006	2006	NUM
ejpam-3246	282	19	.	.	PUNCT
ejpam-3246	283	1	[	[	X
ejpam-3246	283	2	2	2	X
ejpam-3246	283	3	]	]	X
ejpam-3246	283	4	y.	y.	PROPN
ejpam-3246	283	5	b.	b.	PROPN
ejpam-3246	283	6	jun	jun	PROPN
ejpam-3246	283	7	,	,	PUNCT
ejpam-3246	283	8	s.	s.	PROPN
ejpam-3246	283	9	s.	s.	PROPN
ejpam-3246	283	10	ahn	ahn	PROPN
ejpam-3246	283	11	and	and	CCONJ
ejpam-3246	283	12	g.	g.	PROPN
ejpam-3246	283	13	muhiuddin	muhiuddin	PROPN
ejpam-3246	283	14	,	,	PUNCT
ejpam-3246	283	15	hesitant	hesitant	ADJ
ejpam-3246	283	16	fuzzy	fuzzy	ADJ
ejpam-3246	283	17	soft	soft	ADJ
ejpam-3246	283	18	subalgebras	subalgebra	NOUN
ejpam-3246	283	19	and	and	CCONJ
ejpam-3246	283	20	ideals	ideal	NOUN
ejpam-3246	283	21	in	in	ADP
ejpam-3246	283	22	bck	bck	PROPN
ejpam-3246	283	23	/	/	SYM
ejpam-3246	283	24	bci	bci	NOUN
ejpam-3246	283	25	-	-	PUNCT
ejpam-3246	283	26	algebras	algebra	NOUN
ejpam-3246	283	27	,	,	PUNCT
ejpam-3246	283	28	the	the	DET
ejpam-3246	283	29	scientific	scientific	ADJ
ejpam-3246	283	30	world	world	PROPN
ejpam-3246	283	31	journal	journal	PROPN
ejpam-3246	283	32	volume	volume	NOUN
ejpam-3246	283	33	2014	2014	NUM
ejpam-3246	283	34	,	,	PUNCT
ejpam-3246	283	35	article	article	NOUN
ejpam-3246	283	36	i	i	PROPN
ejpam-3246	283	37	d	d	PROPN
ejpam-3246	283	38	763929	763929	NUM
ejpam-3246	283	39	,	,	PUNCT
ejpam-3246	283	40	(	(	PUNCT
ejpam-3246	283	41	2014	2014	NUM
ejpam-3246	283	42	)	)	PUNCT
ejpam-3246	283	43	,	,	PUNCT
ejpam-3246	283	44	7	7	NUM
ejpam-3246	283	45	pages	page	NOUN
ejpam-3246	283	46	.	.	PUNCT
ejpam-3246	284	1	[	[	X
ejpam-3246	284	2	3	3	X
ejpam-3246	284	3	]	]	X
ejpam-3246	284	4	y.	y.	PROPN
ejpam-3246	284	5	b.	b.	PROPN
ejpam-3246	284	6	jun	jun	PROPN
ejpam-3246	284	7	and	and	CCONJ
ejpam-3246	284	8	s.	s.	PROPN
ejpam-3246	284	9	z.	z.	PROPN
ejpam-3246	284	10	song	song	PROPN
ejpam-3246	284	11	,	,	PUNCT
ejpam-3246	284	12	hesitant	hesitant	ADJ
ejpam-3246	284	13	fuzzy	fuzzy	ADJ
ejpam-3246	284	14	set	set	NOUN
ejpam-3246	284	15	theory	theory	NOUN
ejpam-3246	284	16	applied	apply	VERB
ejpam-3246	284	17	to	to	ADP
ejpam-3246	284	18	filters	filter	NOUN
ejpam-3246	284	19	in	in	ADP
ejpam-3246	284	20	mtl	mtl	PROPN
ejpam-3246	284	21	-	-	PUNCT
ejpam-3246	284	22	algebras	algebras	PROPN
ejpam-3246	284	23	,	,	PUNCT
ejpam-3246	284	24	honam	honam	PROPN
ejpam-3246	284	25	math	math	NOUN
ejpam-3246	284	26	.	.	PUNCT
ejpam-3246	285	1	j.	j.	PROPN
ejpam-3246	285	2	36	36	NUM
ejpam-3246	285	3	,	,	PUNCT
ejpam-3246	285	4	(	(	PUNCT
ejpam-3246	285	5	2014	2014	NUM
ejpam-3246	285	6	)	)	PUNCT
ejpam-3246	285	7	,	,	PUNCT
ejpam-3246	285	8	no	no	INTJ
ejpam-3246	285	9	.	.	NOUN
ejpam-3246	285	10	4	4	NUM
ejpam-3246	285	11	,	,	PUNCT
ejpam-3246	285	12	813–830	813–830	NUM
ejpam-3246	285	13	.	.	PUNCT
ejpam-3246	286	1	[	[	X
ejpam-3246	286	2	4	4	X
ejpam-3246	286	3	]	]	X
ejpam-3246	286	4	y.	y.	PROPN
ejpam-3246	286	5	b.	b.	PROPN
ejpam-3246	286	6	jun	jun	PROPN
ejpam-3246	286	7	and	and	CCONJ
ejpam-3246	286	8	s.	s.	PROPN
ejpam-3246	286	9	z.	z.	PROPN
ejpam-3246	286	10	song	song	PROPN
ejpam-3246	286	11	,	,	PUNCT
ejpam-3246	286	12	hesitant	hesitant	ADJ
ejpam-3246	286	13	fuzzy	fuzzy	ADJ
ejpam-3246	286	14	prefilters	prefilter	NOUN
ejpam-3246	286	15	and	and	CCONJ
ejpam-3246	286	16	filters	filter	NOUN
ejpam-3246	286	17	of	of	ADP
ejpam-3246	286	18	eq	eq	NOUN
ejpam-3246	286	19	-	-	PUNCT
ejpam-3246	286	20	algebras	algebra	NOUN
ejpam-3246	286	21	,	,	PUNCT
ejpam-3246	286	22	appl	appl	PROPN
ejpam-3246	286	23	.	.	PROPN
ejpam-3246	286	24	math	math	PROPN
ejpam-3246	286	25	.	.	PUNCT
ejpam-3246	287	1	sci	sci	PROPN
ejpam-3246	287	2	.	.	PROPN
ejpam-3246	287	3	9	9	NUM
ejpam-3246	287	4	(	(	PUNCT
ejpam-3246	287	5	2015	2015	NUM
ejpam-3246	287	6	)	)	PUNCT
ejpam-3246	287	7	,	,	PUNCT
ejpam-3246	287	8	515–532	515–532	NUM
ejpam-3246	287	9	.	.	PUNCT
ejpam-3246	288	1	[	[	X
ejpam-3246	288	2	5	5	X
ejpam-3246	288	3	]	]	X
ejpam-3246	288	4	y.	y.	PROPN
ejpam-3246	288	5	b.	b.	PROPN
ejpam-3246	288	6	jun	jun	PROPN
ejpam-3246	288	7	and	and	CCONJ
ejpam-3246	288	8	s.	s.	PROPN
ejpam-3246	288	9	z.	z.	PROPN
ejpam-3246	288	10	song	song	PROPN
ejpam-3246	288	11	and	and	CCONJ
ejpam-3246	288	12	g.	g.	PROPN
ejpam-3246	288	13	muhiuddin	muhiuddin	PROPN
ejpam-3246	288	14	,	,	PUNCT
ejpam-3246	288	15	hesitant	hesitant	ADJ
ejpam-3246	288	16	fuzzy	fuzzy	ADJ
ejpam-3246	288	17	semigroups	semigroup	NOUN
ejpam-3246	288	18	with	with	ADP
ejpam-3246	288	19	a	a	DET
ejpam-3246	288	20	frontier	frontier	NOUN
ejpam-3246	288	21	,	,	PUNCT
ejpam-3246	288	22	journal	journal	NOUN
ejpam-3246	288	23	of	of	ADP
ejpam-3246	288	24	intelligent	intelligent	ADJ
ejpam-3246	288	25	and	and	CCONJ
ejpam-3246	288	26	fuzzy	fuzzy	ADJ
ejpam-3246	288	27	systems	system	NOUN
ejpam-3246	288	28	,	,	PUNCT
ejpam-3246	288	29	30	30	NUM
ejpam-3246	288	30	,	,	PUNCT
ejpam-3246	288	31	no	no	INTJ
ejpam-3246	288	32	.	.	NOUN
ejpam-3246	288	33	3	3	NUM
ejpam-3246	288	34	(	(	PUNCT
ejpam-3246	288	35	2016	2016	NUM
ejpam-3246	288	36	)	)	PUNCT
ejpam-3246	288	37	,	,	PUNCT
ejpam-3246	288	38	1613	1613	NUM
ejpam-3246	288	39	-	-	SYM
ejpam-3246	288	40	1618	1618	NUM
ejpam-3246	288	41	.	.	PUNCT
ejpam-3246	289	1	doi	doi	NOUN
ejpam-3246	289	2	:	:	PUNCT
ejpam-3246	289	3	10.3233	10.3233	NUM
ejpam-3246	289	4	/	/	SYM
ejpam-3246	289	5	ifs-151869	ifs-151869	NOUN
ejpam-3246	289	6	.	.	PUNCT
ejpam-3246	290	1	[	[	X
ejpam-3246	290	2	6	6	NUM
ejpam-3246	290	3	]	]	PUNCT
ejpam-3246	290	4	j.	j.	PROPN
ejpam-3246	290	5	meng	meng	PROPN
ejpam-3246	290	6	and	and	CCONJ
ejpam-3246	290	7	y.	y.	PROPN
ejpam-3246	290	8	b.	b.	PROPN
ejpam-3246	290	9	jun	jun	PROPN
ejpam-3246	290	10	,	,	PUNCT
ejpam-3246	290	11	bck	bck	PROPN
ejpam-3246	290	12	-	-	PUNCT
ejpam-3246	290	13	algebras	algebras	PROPN
ejpam-3246	290	14	,	,	PUNCT
ejpam-3246	290	15	kyungmoon	kyungmoon	PROPN
ejpam-3246	290	16	sa	sa	PROPN
ejpam-3246	290	17	co.	co.	PROPN
ejpam-3246	290	18	seoul	seoul	PROPN
ejpam-3246	290	19	1994	1994	NUM
ejpam-3246	290	20	.	.	PUNCT
ejpam-3246	291	1	[	[	X
ejpam-3246	291	2	7	7	X
ejpam-3246	291	3	]	]	PUNCT
ejpam-3246	291	4	j.	j.	PROPN
ejpam-3246	291	5	meng	meng	PROPN
ejpam-3246	291	6	and	and	CCONJ
ejpam-3246	291	7	s.	s.	PROPN
ejpam-3246	291	8	m.	m.	PROPN
ejpam-3246	291	9	wei	wei	PROPN
ejpam-3246	291	10	,	,	PUNCT
ejpam-3246	291	11	periods	period	NOUN
ejpam-3246	291	12	of	of	ADP
ejpam-3246	291	13	elements	element	NOUN
ejpam-3246	291	14	in	in	ADP
ejpam-3246	291	15	bci	bci	NOUN
ejpam-3246	291	16	-	-	PUNCT
ejpam-3246	291	17	algebras	algebra	NOUN
ejpam-3246	291	18	,	,	PUNCT
ejpam-3246	291	19	math	math	NOUN
ejpam-3246	291	20	.	.	PUNCT
ejpam-3246	292	1	japon	japon	PROPN
ejpam-3246	292	2	.	.	PUNCT
ejpam-3246	293	1	38	38	NUM
ejpam-3246	293	2	(	(	PUNCT
ejpam-3246	293	3	1993	1993	NUM
ejpam-3246	293	4	)	)	PUNCT
ejpam-3246	293	5	,	,	PUNCT
ejpam-3246	293	6	427–431	427–431	NUM
ejpam-3246	293	7	.	.	PUNCT
ejpam-3246	294	1	[	[	X
ejpam-3246	294	2	8	8	NUM
ejpam-3246	294	3	]	]	X
ejpam-3246	294	4	g.	g.	PROPN
ejpam-3246	294	5	muhiuddin	muhiuddin	PROPN
ejpam-3246	294	6	and	and	CCONJ
ejpam-3246	294	7	abdullah	abdullah	PROPN
ejpam-3246	294	8	m.	m.	PROPN
ejpam-3246	294	9	al	al	PROPN
ejpam-3246	294	10	-	-	PUNCT
ejpam-3246	294	11	roqi	roqi	ADJ
ejpam-3246	294	12	,	,	PUNCT
ejpam-3246	294	13	regular	regular	ADJ
ejpam-3246	294	14	hesitant	hesitant	ADJ
ejpam-3246	294	15	fuzzy	fuzzy	ADJ
ejpam-3246	294	16	filters	filter	NOUN
ejpam-3246	294	17	and	and	CCONJ
ejpam-3246	294	18	mv	mv	PROPN
ejpam-3246	294	19	hesitant	hesitant	ADJ
ejpam-3246	294	20	fuzzy	fuzzy	ADJ
ejpam-3246	294	21	filters	filter	NOUN
ejpam-3246	294	22	of	of	ADP
ejpam-3246	294	23	residuated	residuate	VERB
ejpam-3246	294	24	lattices	lattice	NOUN
ejpam-3246	294	25	,	,	PUNCT
ejpam-3246	294	26	j.	j.	PROPN
ejpam-3246	294	27	comput	comput	PROPN
ejpam-3246	294	28	.	.	PUNCT
ejpam-3246	295	1	anal	anal	PROPN
ejpam-3246	295	2	.	.	PUNCT
ejpam-3246	295	3	appl	appl	PROPN
ejpam-3246	295	4	.	.	PROPN
ejpam-3246	296	1	24	24	NUM
ejpam-3246	296	2	,	,	PUNCT
ejpam-3246	296	3	no	no	INTJ
ejpam-3246	296	4	.	.	NOUN
ejpam-3246	296	5	6	6	NUM
ejpam-3246	296	6	(	(	PUNCT
ejpam-3246	296	7	2018	2018	NUM
ejpam-3246	296	8	)	)	PUNCT
ejpam-3246	296	9	,	,	PUNCT
ejpam-3246	296	10	1133–1144	1133–1144	NUM
ejpam-3246	296	11	.	.	PUNCT
ejpam-3246	297	1	[	[	X
ejpam-3246	297	2	9	9	NUM
ejpam-3246	297	3	]	]	X
ejpam-3246	297	4	g.	g.	PROPN
ejpam-3246	297	5	muhiuddin	muhiuddin	PROPN
ejpam-3246	297	6	and	and	CCONJ
ejpam-3246	297	7	s.	s.	PROPN
ejpam-3246	297	8	aldhafeeri	aldhafeeri	PROPN
ejpam-3246	297	9	,	,	PUNCT
ejpam-3246	297	10	join	join	VERB
ejpam-3246	297	11	hesitant	hesitant	ADJ
ejpam-3246	297	12	fuzzy	fuzzy	ADJ
ejpam-3246	297	13	filters	filter	NOUN
ejpam-3246	297	14	of	of	ADP
ejpam-3246	297	15	residuated	residuate	VERB
ejpam-3246	297	16	lattices	lattice	NOUN
ejpam-3246	297	17	,	,	PUNCT
ejpam-3246	297	18	italian	italian	ADJ
ejpam-3246	297	19	journal	journal	NOUN
ejpam-3246	297	20	of	of	ADP
ejpam-3246	297	21	pure	pure	ADJ
ejpam-3246	297	22	and	and	CCONJ
ejpam-3246	297	23	applied	applied	ADJ
ejpam-3246	297	24	mathematics	mathematic	NOUN
ejpam-3246	297	25	,	,	PUNCT
ejpam-3246	297	26	(	(	PUNCT
ejpam-3246	297	27	accepted	accept	VERB
ejpam-3246	297	28	)	)	PUNCT
ejpam-3246	297	29	(	(	PUNCT
ejpam-3246	297	30	2017	2017	NUM
ejpam-3246	297	31	)	)	PUNCT
ejpam-3246	297	32	.	.	PUNCT
ejpam-3246	298	1	[	[	X
ejpam-3246	298	2	10	10	NUM
ejpam-3246	298	3	]	]	X
ejpam-3246	298	4	g.	g.	PROPN
ejpam-3246	298	5	muhiuddin	muhiuddin	PROPN
ejpam-3246	298	6	,	,	PUNCT
ejpam-3246	298	7	hesitant	hesitant	ADJ
ejpam-3246	298	8	fuzzy	fuzzy	ADJ
ejpam-3246	298	9	filters	filter	NOUN
ejpam-3246	298	10	and	and	CCONJ
ejpam-3246	298	11	hesitant	hesitant	ADJ
ejpam-3246	298	12	fuzzy	fuzzy	ADJ
ejpam-3246	298	13	g	g	NOUN
ejpam-3246	298	14	-	-	PUNCT
ejpam-3246	298	15	filters	filter	NOUN
ejpam-3246	298	16	in	in	ADP
ejpam-3246	298	17	residuated	residuate	VERB
ejpam-3246	298	18	lattices	lattice	NOUN
ejpam-3246	298	19	,	,	PUNCT
ejpam-3246	298	20	j.	j.	PROPN
ejpam-3246	298	21	comput	comput	PROPN
ejpam-3246	298	22	.	.	PUNCT
ejpam-3246	299	1	anal	anal	PROPN
ejpam-3246	299	2	.	.	PUNCT
ejpam-3246	299	3	appl	appl	PROPN
ejpam-3246	299	4	.	.	PROPN
ejpam-3246	300	1	21	21	NUM
ejpam-3246	300	2	,	,	PUNCT
ejpam-3246	300	3	no	no	INTJ
ejpam-3246	300	4	.	.	NOUN
ejpam-3246	300	5	2	2	NUM
ejpam-3246	300	6	(	(	PUNCT
ejpam-3246	300	7	2016	2016	NUM
ejpam-3246	300	8	)	)	PUNCT
ejpam-3246	300	9	,	,	PUNCT
ejpam-3246	300	10	394–404	394–404	NUM
ejpam-3246	300	11	.	.	PUNCT
ejpam-3246	301	1	[	[	X
ejpam-3246	301	2	11	11	NUM
ejpam-3246	301	3	]	]	X
ejpam-3246	301	4	g.	g.	PROPN
ejpam-3246	301	5	muhiuddin	muhiuddin	PROPN
ejpam-3246	301	6	,	,	PUNCT
ejpam-3246	301	7	h.	h.	PROPN
ejpam-3246	301	8	s.	s.	PROPN
ejpam-3246	301	9	kim	kim	PROPN
ejpam-3246	301	10	,	,	PUNCT
ejpam-3246	301	11	s.	s.	PROPN
ejpam-3246	301	12	z.	z.	PROPN
ejpam-3246	301	13	song	song	PROPN
ejpam-3246	301	14	and	and	CCONJ
ejpam-3246	301	15	y.	y.	PROPN
ejpam-3246	301	16	b.	b.	PROPN
ejpam-3246	301	17	jun	jun	PROPN
ejpam-3246	301	18	,	,	PUNCT
ejpam-3246	301	19	hesitant	hesitant	ADJ
ejpam-3246	301	20	fuzzy	fuzzy	ADJ
ejpam-3246	301	21	translations	translation	NOUN
ejpam-3246	301	22	and	and	CCONJ
ejpam-3246	301	23	extensions	extension	NOUN
ejpam-3246	301	24	of	of	ADP
ejpam-3246	301	25	subalgebras	subalgebra	NOUN
ejpam-3246	301	26	and	and	CCONJ
ejpam-3246	301	27	ideals	ideal	NOUN
ejpam-3246	301	28	in	in	ADP
ejpam-3246	301	29	bck	bck	PROPN
ejpam-3246	301	30	/	/	SYM
ejpam-3246	301	31	bci	bci	NOUN
ejpam-3246	301	32	-	-	PUNCT
ejpam-3246	301	33	algebras	algebra	NOUN
ejpam-3246	301	34	,	,	PUNCT
ejpam-3246	301	35	journal	journal	NOUN
ejpam-3246	301	36	of	of	ADP
ejpam-3246	301	37	intelligent	intelligent	ADJ
ejpam-3246	301	38	and	and	CCONJ
ejpam-3246	301	39	fuzzy	fuzzy	ADJ
ejpam-3246	301	40	systems	system	NOUN
ejpam-3246	301	41	,	,	PUNCT
ejpam-3246	301	42	32	32	NUM
ejpam-3246	301	43	,	,	PUNCT
ejpam-3246	301	44	no	no	INTJ
ejpam-3246	301	45	.	.	NOUN
ejpam-3246	301	46	1	1	NUM
ejpam-3246	301	47	(	(	PUNCT
ejpam-3246	301	48	2017	2017	NUM
ejpam-3246	301	49	)	)	PUNCT
ejpam-3246	301	50	,	,	PUNCT
ejpam-3246	301	51	43–48	43–48	NUM
ejpam-3246	301	52	.	.	PUNCT
ejpam-3246	302	1	[	[	X
ejpam-3246	302	2	12	12	NUM
ejpam-3246	302	3	]	]	X
ejpam-3246	302	4	g.	g.	PROPN
ejpam-3246	302	5	muhiuddin	muhiuddin	PROPN
ejpam-3246	302	6	,	,	PUNCT
ejpam-3246	302	7	e.	e.	PROPN
ejpam-3246	302	8	h.	h.	PROPN
ejpam-3246	302	9	roh	roh	PROPN
ejpam-3246	302	10	,	,	PUNCT
ejpam-3246	302	11	sun	sun	PROPN
ejpam-3246	302	12	shin	shin	PROPN
ejpam-3246	302	13	ahn	ahn	PROPN
ejpam-3246	302	14	and	and	CCONJ
ejpam-3246	302	15	y.	y.	PROPN
ejpam-3246	302	16	b.	b.	PROPN
ejpam-3246	302	17	jun	jun	PROPN
ejpam-3246	302	18	,	,	PUNCT
ejpam-3246	302	19	hesitant	hesitant	ADJ
ejpam-3246	302	20	fuzzy	fuzzy	ADJ
ejpam-3246	302	21	filters	filter	NOUN
ejpam-3246	302	22	in	in	ADP
ejpam-3246	302	23	lattice	lattice	PROPN
ejpam-3246	302	24	implication	implication	NOUN
ejpam-3246	302	25	algebras	algebra	NOUN
ejpam-3246	302	26	,	,	PUNCT
ejpam-3246	302	27	j.	j.	PROPN
ejpam-3246	302	28	comput	comput	PROPN
ejpam-3246	302	29	.	.	PUNCT
ejpam-3246	303	1	anal	anal	PROPN
ejpam-3246	303	2	.	.	PUNCT
ejpam-3246	303	3	appl	appl	PROPN
ejpam-3246	303	4	.	.	PROPN
ejpam-3246	304	1	22	22	NUM
ejpam-3246	304	2	,	,	PUNCT
ejpam-3246	304	3	no.6	no.6	PROPN
ejpam-3246	304	4	(	(	PUNCT
ejpam-3246	304	5	2017	2017	NUM
ejpam-3246	304	6	)	)	PUNCT
ejpam-3246	304	7	,	,	PUNCT
ejpam-3246	304	8	1105	1105	NUM
ejpam-3246	304	9	-	-	SYM
ejpam-3246	304	10	1113	1113	NUM
ejpam-3246	304	11	.	.	PUNCT
ejpam-3246	305	1	[	[	X
ejpam-3246	305	2	13	13	NUM
ejpam-3246	305	3	]	]	X
ejpam-3246	305	4	rosa	rosa	PROPN
ejpam-3246	305	5	m.	m.	PROPN
ejpam-3246	305	6	rodriguez	rodriguez	PROPN
ejpam-3246	305	7	,	,	PUNCT
ejpam-3246	305	8	luis	luis	PROPN
ejpam-3246	305	9	martinez	martinez	PROPN
ejpam-3246	305	10	and	and	CCONJ
ejpam-3246	305	11	francisco	francisco	PROPN
ejpam-3246	305	12	herrera	herrera	PROPN
ejpam-3246	305	13	,	,	PUNCT
ejpam-3246	305	14	hesitant	hesitant	ADJ
ejpam-3246	305	15	fuzzy	fuzzy	ADJ
ejpam-3246	305	16	linguistic	linguistic	ADJ
ejpam-3246	305	17	term	term	NOUN
ejpam-3246	305	18	sets	set	NOUN
ejpam-3246	305	19	for	for	ADP
ejpam-3246	305	20	decision	decision	NOUN
ejpam-3246	305	21	making	making	NOUN
ejpam-3246	305	22	,	,	PUNCT
ejpam-3246	305	23	ieee	ieee	NOUN
ejpam-3246	305	24	trans	tran	NOUN
ejpam-3246	305	25	.	.	PUNCT
ejpam-3246	306	1	fuzzy	fuzzy	ADJ
ejpam-3246	306	2	syst	syst	PROPN
ejpam-3246	306	3	.	.	PUNCT
ejpam-3246	307	1	20	20	NUM
ejpam-3246	307	2	(	(	PUNCT
ejpam-3246	307	3	2012	2012	NUM
ejpam-3246	307	4	)	)	PUNCT
ejpam-3246	307	5	,	,	PUNCT
ejpam-3246	307	6	no	no	INTJ
ejpam-3246	307	7	.	.	NOUN
ejpam-3246	307	8	1	1	NUM
ejpam-3246	307	9	,	,	PUNCT
ejpam-3246	307	10	109–119	109–119	NUM
ejpam-3246	307	11	.	.	PUNCT
ejpam-3246	308	1	[	[	X
ejpam-3246	308	2	14	14	NUM
ejpam-3246	308	3	]	]	X
ejpam-3246	308	4	v.	v.	CCONJ
ejpam-3246	308	5	torra	torra	ADJ
ejpam-3246	308	6	,	,	PUNCT
ejpam-3246	308	7	hesitant	hesitant	ADJ
ejpam-3246	308	8	fuzzy	fuzzy	ADJ
ejpam-3246	308	9	sets	set	NOUN
ejpam-3246	308	10	,	,	PUNCT
ejpam-3246	308	11	int	int	NOUN
ejpam-3246	308	12	.	.	PUNCT
ejpam-3246	309	1	j.	j.	PROPN
ejpam-3246	309	2	intell	intell	PROPN
ejpam-3246	309	3	.	.	PUNCT
ejpam-3246	310	1	syst	syst	PROPN
ejpam-3246	310	2	.	.	PUNCT
ejpam-3246	311	1	25	25	NUM
ejpam-3246	311	2	(	(	PUNCT
ejpam-3246	311	3	2010	2010	NUM
ejpam-3246	311	4	)	)	PUNCT
ejpam-3246	311	5	,	,	PUNCT
ejpam-3246	311	6	529–539	529–539	NUM
ejpam-3246	311	7	.	.	PUNCT
ejpam-3246	312	1	[	[	X
ejpam-3246	312	2	15	15	NUM
ejpam-3246	312	3	]	]	X
ejpam-3246	312	4	v.	v.	CCONJ
ejpam-3246	312	5	torra	torra	NOUN
ejpam-3246	312	6	and	and	CCONJ
ejpam-3246	312	7	y.	y.	PROPN
ejpam-3246	312	8	narukawa	narukawa	PROPN
ejpam-3246	312	9	,	,	PUNCT
ejpam-3246	312	10	on	on	ADP
ejpam-3246	312	11	hesitant	hesitant	ADJ
ejpam-3246	312	12	fuzzy	fuzzy	ADJ
ejpam-3246	312	13	sets	set	NOUN
ejpam-3246	312	14	and	and	CCONJ
ejpam-3246	312	15	decision	decision	NOUN
ejpam-3246	312	16	,	,	PUNCT
ejpam-3246	312	17	in	in	ADP
ejpam-3246	312	18	:	:	PUNCT
ejpam-3246	312	19	the	the	DET
ejpam-3246	312	20	18th	18th	ADJ
ejpam-3246	312	21	ieee	ieee	NOUN
ejpam-3246	312	22	international	international	ADJ
ejpam-3246	312	23	conference	conference	NOUN
ejpam-3246	312	24	on	on	ADP
ejpam-3246	312	25	fuzzy	fuzzy	ADJ
ejpam-3246	312	26	systems	system	NOUN
ejpam-3246	312	27	,	,	PUNCT
ejpam-3246	312	28	jeju	jeju	PROPN
ejpam-3246	312	29	island	island	PROPN
ejpam-3246	312	30	,	,	PUNCT
ejpam-3246	312	31	korea	korea	PROPN
ejpam-3246	312	32	,	,	PUNCT
ejpam-3246	312	33	2009	2009	NUM
ejpam-3246	312	34	,	,	PUNCT
ejpam-3246	312	35	pp	pp	ADJ
ejpam-3246	312	36	.	.	PUNCT
ejpam-3246	313	1	1378–1382	1378–1382	NUM
ejpam-3246	313	2	.	.	PUNCT
ejpam-3246	314	1	references	reference	NOUN
ejpam-3246	314	2	430	430	NUM
ejpam-3246	314	3	[	[	SYM
ejpam-3246	314	4	16	16	NUM
ejpam-3246	314	5	]	]	X
ejpam-3246	314	6	g.	g.	PROPN
ejpam-3246	314	7	wei	wei	PROPN
ejpam-3246	314	8	,	,	PUNCT
ejpam-3246	314	9	hesitant	hesitant	ADJ
ejpam-3246	314	10	fuzzy	fuzzy	ADJ
ejpam-3246	314	11	prioritized	prioritize	VERB
ejpam-3246	314	12	operators	operator	NOUN
ejpam-3246	314	13	and	and	CCONJ
ejpam-3246	314	14	their	their	PRON
ejpam-3246	314	15	application	application	NOUN
ejpam-3246	314	16	to	to	ADP
ejpam-3246	314	17	multiple	multiple	ADJ
ejpam-3246	314	18	attribute	attribute	NOUN
ejpam-3246	314	19	decision	decision	NOUN
ejpam-3246	314	20	making	making	NOUN
ejpam-3246	314	21	,	,	PUNCT
ejpam-3246	314	22	knowledge	knowledge	NOUN
ejpam-3246	314	23	-	-	PUNCT
ejpam-3246	314	24	based	base	VERB
ejpam-3246	314	25	systems	system	NOUN
ejpam-3246	314	26	31	31	NUM
ejpam-3246	314	27	(	(	PUNCT
ejpam-3246	314	28	2012	2012	NUM
ejpam-3246	314	29	)	)	PUNCT
ejpam-3246	314	30	,	,	PUNCT
ejpam-3246	314	31	176–182	176–182	NUM
ejpam-3246	314	32	.	.	PUNCT
ejpam-3246	315	1	[	[	X
ejpam-3246	315	2	17	17	NUM
ejpam-3246	315	3	]	]	PUNCT
ejpam-3246	315	4	m.	m.	NOUN
ejpam-3246	315	5	xia	xia	PROPN
ejpam-3246	315	6	and	and	CCONJ
ejpam-3246	315	7	z.	z.	PROPN
ejpam-3246	315	8	s.	s.	PROPN
ejpam-3246	315	9	xu	xu	PROPN
ejpam-3246	315	10	,	,	PUNCT
ejpam-3246	315	11	hesitant	hesitant	ADJ
ejpam-3246	315	12	fuzzy	fuzzy	ADJ
ejpam-3246	315	13	information	information	NOUN
ejpam-3246	315	14	aggregation	aggregation	NOUN
ejpam-3246	315	15	in	in	ADP
ejpam-3246	315	16	decision	decision	NOUN
ejpam-3246	315	17	making	making	NOUN
ejpam-3246	315	18	,	,	PUNCT
ejpam-3246	315	19	internat	internat	PROPN
ejpam-3246	315	20	.	.	PUNCT
ejpam-3246	316	1	j.	j.	PROPN
ejpam-3246	316	2	approx	approx	PROPN
ejpam-3246	316	3	.	.	PUNCT
ejpam-3246	317	1	reason	reason	NOUN
ejpam-3246	317	2	.	.	PUNCT
ejpam-3246	318	1	52	52	NUM
ejpam-3246	318	2	(	(	PUNCT
ejpam-3246	318	3	2011	2011	NUM
ejpam-3246	318	4	)	)	PUNCT
ejpam-3246	318	5	,	,	PUNCT
ejpam-3246	318	6	no	no	INTJ
ejpam-3246	318	7	.	.	NOUN
ejpam-3246	318	8	3	3	NUM
ejpam-3246	318	9	,	,	PUNCT
ejpam-3246	318	10	395–407	395–407	NUM
ejpam-3246	318	11	.	.	PUNCT
ejpam-3246	319	1	[	[	X
ejpam-3246	319	2	18	18	NUM
ejpam-3246	319	3	]	]	PUNCT
ejpam-3246	319	4	m.	m.	NOUN
ejpam-3246	319	5	xia	xia	PROPN
ejpam-3246	319	6	,	,	PUNCT
ejpam-3246	319	7	z.	z.	PROPN
ejpam-3246	319	8	s.	s.	PROPN
ejpam-3246	319	9	xu	xu	PROPN
ejpam-3246	319	10	and	and	CCONJ
ejpam-3246	319	11	n.	n.	PROPN
ejpam-3246	319	12	chen	chen	PROPN
ejpam-3246	319	13	,	,	PUNCT
ejpam-3246	319	14	some	some	DET
ejpam-3246	319	15	hesitant	hesitant	ADJ
ejpam-3246	319	16	fuzzy	fuzzy	ADJ
ejpam-3246	319	17	aggregation	aggregation	NOUN
ejpam-3246	319	18	operators	operator	NOUN
ejpam-3246	319	19	with	with	ADP
ejpam-3246	319	20	their	their	PRON
ejpam-3246	319	21	application	application	NOUN
ejpam-3246	319	22	in	in	ADP
ejpam-3246	319	23	group	group	NOUN
ejpam-3246	319	24	decision	decision	NOUN
ejpam-3246	319	25	making	making	NOUN
ejpam-3246	319	26	,	,	PUNCT
ejpam-3246	319	27	group	group	NOUN
ejpam-3246	319	28	decision	decision	NOUN
ejpam-3246	319	29	negotiation	negotiation	NOUN
ejpam-3246	319	30	22	22	NUM
ejpam-3246	319	31	(	(	PUNCT
ejpam-3246	319	32	2013	2013	NUM
ejpam-3246	319	33	)	)	PUNCT
ejpam-3246	319	34	,	,	PUNCT
ejpam-3246	319	35	259–279	259–279	NUM
ejpam-3246	319	36	.	.	PUNCT
ejpam-3246	320	1	[	[	X
ejpam-3246	320	2	19	19	NUM
ejpam-3246	320	3	]	]	PUNCT
ejpam-3246	320	4	z.	z.	PROPN
ejpam-3246	320	5	s.	s.	PROPN
ejpam-3246	320	6	xu	xu	PROPN
ejpam-3246	320	7	and	and	CCONJ
ejpam-3246	320	8	m.	m.	PROPN
ejpam-3246	320	9	xia	xia	PROPN
ejpam-3246	320	10	,	,	PUNCT
ejpam-3246	320	11	distance	distance	NOUN
ejpam-3246	320	12	and	and	CCONJ
ejpam-3246	320	13	similarity	similarity	NOUN
ejpam-3246	320	14	measures	measure	NOUN
ejpam-3246	320	15	for	for	ADP
ejpam-3246	320	16	hesitant	hesitant	ADJ
ejpam-3246	320	17	fuzzy	fuzzy	ADJ
ejpam-3246	320	18	sets	set	NOUN
ejpam-3246	320	19	,	,	PUNCT
ejpam-3246	320	20	inform	inform	NOUN
ejpam-3246	320	21	.	.	PUNCT
ejpam-3246	321	1	sci	sci	PROPN
ejpam-3246	321	2	.	.	PROPN
ejpam-3246	321	3	181	181	NUM
ejpam-3246	321	4	(	(	PUNCT
ejpam-3246	321	5	2011	2011	NUM
ejpam-3246	321	6	)	)	PUNCT
ejpam-3246	321	7	,	,	PUNCT
ejpam-3246	321	8	2128–2138	2128–2138	NUM
ejpam-3246	321	9	.	.	PUNCT
ejpam-3246	322	1	[	[	X
ejpam-3246	322	2	20	20	NUM
ejpam-3246	322	3	]	]	PUNCT
ejpam-3246	322	4	b.	b.	PROPN
ejpam-3246	322	5	zhu	zhu	PROPN
ejpam-3246	322	6	,	,	PUNCT
ejpam-3246	322	7	z.	z.	PROPN
ejpam-3246	322	8	xu	xu	PROPN
ejpam-3246	322	9	and	and	CCONJ
ejpam-3246	322	10	m.	m.	PROPN
ejpam-3246	322	11	xia	xia	PROPN
ejpam-3246	322	12	,	,	PUNCT
ejpam-3246	322	13	hesitant	hesitant	ADJ
ejpam-3246	322	14	fuzzy	fuzzy	ADJ
ejpam-3246	322	15	geometric	geometric	ADJ
ejpam-3246	322	16	bonferroni	bonferroni	NOUN
ejpam-3246	322	17	means	mean	VERB
ejpam-3246	322	18	,	,	PUNCT
ejpam-3246	322	19	inform	inform	NOUN
ejpam-3246	322	20	.	.	PUNCT
ejpam-3246	323	1	sci	sci	PROPN
ejpam-3246	323	2	.	.	PROPN
ejpam-3246	323	3	205	205	NUM
ejpam-3246	323	4	(	(	PUNCT
ejpam-3246	323	5	2012	2012	NUM
ejpam-3246	323	6	)	)	PUNCT
ejpam-3246	323	7	,	,	PUNCT
ejpam-3246	323	8	72–85	72–85	NUM
ejpam-3246	323	9	.	.	PUNCT