id	sid	tid	token	lemma	pos
ejpam-4933	1	1	european	european	PROPN
ejpam-4933	1	2	journal	journal	PROPN
ejpam-4933	1	3	of	of	ADP
ejpam-4933	1	4	pure	pure	ADJ
ejpam-4933	1	5	and	and	CCONJ
ejpam-4933	1	6	applied	apply	VERB
ejpam-4933	1	7	mathematics	mathematic	NOUN
ejpam-4933	1	8	vol	vol	NOUN
ejpam-4933	1	9	.	.	PUNCT
ejpam-4933	2	1	16	16	NUM
ejpam-4933	2	2	,	,	PUNCT
ejpam-4933	2	3	no	no	INTJ
ejpam-4933	2	4	.	.	NOUN
ejpam-4933	2	5	4	4	NUM
ejpam-4933	2	6	,	,	PUNCT
ejpam-4933	2	7	2023	2023	NUM
ejpam-4933	2	8	,	,	PUNCT
ejpam-4933	2	9	2009	2009	NUM
ejpam-4933	2	10	-	-	SYM
ejpam-4933	2	11	2024	2024	NUM
ejpam-4933	2	12	issn	issn	PROPN
ejpam-4933	2	13	1307	1307	NUM
ejpam-4933	2	14	-	-	SYM
ejpam-4933	2	15	5543	5543	NUM
ejpam-4933	2	16	–	–	PUNCT
ejpam-4933	3	1	ejpam.com	ejpam.com	X
ejpam-4933	3	2	published	publish	VERB
ejpam-4933	3	3	by	by	ADP
ejpam-4933	3	4	new	new	PROPN
ejpam-4933	3	5	york	york	PROPN
ejpam-4933	3	6	business	business	PROPN
ejpam-4933	3	7	global	global	ADJ
ejpam-4933	3	8	ideals	ideal	NOUN
ejpam-4933	3	9	of	of	ADP
ejpam-4933	3	10	bck	bck	NOUN
ejpam-4933	3	11	-	-	PUNCT
ejpam-4933	3	12	algebras	algebras	PROPN
ejpam-4933	3	13	and	and	CCONJ
ejpam-4933	3	14	bci	bci	NOUN
ejpam-4933	3	15	-	-	PUNCT
ejpam-4933	3	16	algebras	algebras	PROPN
ejpam-4933	3	17	based	base	VERB
ejpam-4933	3	18	on	on	ADP
ejpam-4933	3	19	a	a	DET
ejpam-4933	3	20	new	new	ADJ
ejpam-4933	3	21	form	form	NOUN
ejpam-4933	3	22	of	of	ADP
ejpam-4933	3	23	fuzzy	fuzzy	ADJ
ejpam-4933	3	24	set	set	VERB
ejpam-4933	3	25	eun	eun	PROPN
ejpam-4933	3	26	hwan	hwan	PROPN
ejpam-4933	3	27	roh1,∗	roh1,∗	PROPN
ejpam-4933	3	28	,	,	PUNCT
ejpam-4933	3	29	eunsuk	eunsuk	NOUN
ejpam-4933	3	30	yang2	yang2	NOUN
ejpam-4933	3	31	,	,	PUNCT
ejpam-4933	3	32	young	young	ADJ
ejpam-4933	3	33	bae	bae	NOUN
ejpam-4933	3	34	jun3	jun3	PROPN
ejpam-4933	3	35	1	1	NUM
ejpam-4933	3	36	department	department	NOUN
ejpam-4933	3	37	of	of	ADP
ejpam-4933	3	38	mathematics	mathematics	PROPN
ejpam-4933	3	39	education	education	NOUN
ejpam-4933	3	40	,	,	PUNCT
ejpam-4933	3	41	chinju	chinju	PROPN
ejpam-4933	3	42	national	national	PROPN
ejpam-4933	3	43	university	university	PROPN
ejpam-4933	3	44	of	of	ADP
ejpam-4933	3	45	education	education	NOUN
ejpam-4933	3	46	,	,	PUNCT
ejpam-4933	3	47	jinju	jinju	PROPN
ejpam-4933	3	48	52673	52673	NUM
ejpam-4933	3	49	,	,	PUNCT
ejpam-4933	3	50	korea	korea	PROPN
ejpam-4933	3	51	2	2	NUM
ejpam-4933	3	52	department	department	NOUN
ejpam-4933	3	53	of	of	ADP
ejpam-4933	3	54	philosophy	philosophy	NOUN
ejpam-4933	3	55	,	,	PUNCT
ejpam-4933	3	56	jeonbuk	jeonbuk	PROPN
ejpam-4933	3	57	national	national	PROPN
ejpam-4933	3	58	university	university	PROPN
ejpam-4933	3	59	,	,	PUNCT
ejpam-4933	3	60	jeonju	jeonju	NOUN
ejpam-4933	3	61	54896	54896	NUM
ejpam-4933	3	62	,	,	PUNCT
ejpam-4933	3	63	korea	korea	PROPN
ejpam-4933	3	64	3	3	NUM
ejpam-4933	3	65	department	department	PROPN
ejpam-4933	3	66	of	of	ADP
ejpam-4933	3	67	mathematics	mathematics	PROPN
ejpam-4933	3	68	education	education	NOUN
ejpam-4933	3	69	,	,	PUNCT
ejpam-4933	3	70	gyeongsang	gyeongsang	PROPN
ejpam-4933	3	71	national	national	PROPN
ejpam-4933	3	72	university	university	PROPN
ejpam-4933	3	73	,	,	PUNCT
ejpam-4933	3	74	jinju	jinju	NOUN
ejpam-4933	3	75	52828	52828	NUM
ejpam-4933	3	76	,	,	PUNCT
ejpam-4933	3	77	korea	korea	PROPN
ejpam-4933	3	78	abstract	abstract	NOUN
ejpam-4933	3	79	.	.	PUNCT
ejpam-4933	4	1	ideals	ideal	NOUN
ejpam-4933	4	2	in	in	ADP
ejpam-4933	4	3	bck	bck	PROPN
ejpam-4933	4	4	/	/	SYM
ejpam-4933	4	5	bci	bci	PROPN
ejpam-4933	4	6	algebra	algebra	NOUN
ejpam-4933	4	7	based	base	VERB
ejpam-4933	4	8	on	on	ADP
ejpam-4933	4	9	y	y	PROPN
ejpam-4933	4	10	ε	ε	PROPN
ejpam-4933	4	11	j	j	PROPN
ejpam-4933	4	12	-fuzzy	-fuzzy	PROPN
ejpam-4933	4	13	sets	set	NOUN
ejpam-4933	4	14	are	be	AUX
ejpam-4933	4	15	studied	study	VERB
ejpam-4933	4	16	.	.	PUNCT
ejpam-4933	5	1	the	the	DET
ejpam-4933	5	2	fundamental	fundamental	ADJ
ejpam-4933	5	3	properties	property	NOUN
ejpam-4933	5	4	of	of	ADP
ejpam-4933	5	5	the	the	DET
ejpam-4933	5	6	level	level	NOUN
ejpam-4933	5	7	set	set	NOUN
ejpam-4933	5	8	of	of	ADP
ejpam-4933	5	9	y	y	PROPN
ejpam-4933	5	10	ε	ε	PROPN
ejpam-4933	5	11	j	j	PROPN
ejpam-4933	5	12	-fuzzy	-fuzzy	PROPN
ejpam-4933	5	13	sets	set	NOUN
ejpam-4933	5	14	are	be	AUX
ejpam-4933	5	15	investigate	investigate	VERB
ejpam-4933	5	16	first	first	ADV
ejpam-4933	5	17	.	.	PUNCT
ejpam-4933	6	1	the	the	DET
ejpam-4933	6	2	concept	concept	NOUN
ejpam-4933	6	3	of	of	ADP
ejpam-4933	6	4	(	(	PUNCT
ejpam-4933	6	5	closed	closed	ADJ
ejpam-4933	6	6	)	)	PUNCT
ejpam-4933	6	7	y	y	PROPN
ejpam-4933	6	8	ε	ε	PROPN
ejpam-4933	6	9	j	j	PROPN
ejpam-4933	6	10	-fuzzy	-fuzzy	PROPN
ejpam-4933	6	11	ideals	ideal	NOUN
ejpam-4933	6	12	in	in	ADP
ejpam-4933	6	13	bck	bck	PROPN
ejpam-4933	6	14	/	/	SYM
ejpam-4933	6	15	bci	bci	NOUN
ejpam-4933	6	16	-	-	PUNCT
ejpam-4933	6	17	algebras	algebras	PROPN
ejpam-4933	6	18	is	be	AUX
ejpam-4933	6	19	introduces	introduce	NOUN
ejpam-4933	6	20	,	,	PUNCT
ejpam-4933	6	21	and	and	CCONJ
ejpam-4933	6	22	several	several	ADJ
ejpam-4933	6	23	properties	property	NOUN
ejpam-4933	6	24	are	be	AUX
ejpam-4933	6	25	investigated	investigate	VERB
ejpam-4933	6	26	.	.	PUNCT
ejpam-4933	7	1	the	the	DET
ejpam-4933	7	2	relationship	relationship	NOUN
ejpam-4933	7	3	between	between	ADP
ejpam-4933	7	4	y	y	PROPN
ejpam-4933	7	5	ε	ε	PROPN
ejpam-4933	7	6	j	j	PROPN
ejpam-4933	7	7	-fuzzy	-fuzzy	PROPN
ejpam-4933	7	8	ideal	ideal	ADJ
ejpam-4933	7	9	and	and	CCONJ
ejpam-4933	7	10	y	y	PROPN
ejpam-4933	7	11	ε	ε	PROPN
ejpam-4933	7	12	j	j	PROPN
ejpam-4933	7	13	-fuzzy	-fuzzy	PROPN
ejpam-4933	7	14	subalgebra	subalgebra	NOUN
ejpam-4933	7	15	are	be	AUX
ejpam-4933	7	16	discussed	discuss	VERB
ejpam-4933	7	17	,	,	PUNCT
ejpam-4933	7	18	and	and	CCONJ
ejpam-4933	7	19	also	also	ADV
ejpam-4933	7	20	the	the	DET
ejpam-4933	7	21	relationship	relationship	NOUN
ejpam-4933	7	22	between	between	ADP
ejpam-4933	7	23	y	y	PROPN
ejpam-4933	7	24	ε	ε	PROPN
ejpam-4933	7	25	j	j	PROPN
ejpam-4933	7	26	-fuzzy	-fuzzy	PROPN
ejpam-4933	7	27	ideal	ideal	ADJ
ejpam-4933	7	28	and	and	CCONJ
ejpam-4933	7	29	fuzzy	fuzzy	ADJ
ejpam-4933	7	30	ideal	ideal	NOUN
ejpam-4933	7	31	is	be	AUX
ejpam-4933	7	32	identified	identify	VERB
ejpam-4933	7	33	.	.	PUNCT
ejpam-4933	8	1	the	the	DET
ejpam-4933	8	2	characterization	characterization	NOUN
ejpam-4933	8	3	of	of	ADP
ejpam-4933	8	4	(	(	PUNCT
ejpam-4933	8	5	closed	closed	ADJ
ejpam-4933	8	6	)	)	PUNCT
ejpam-4933	8	7	y	y	PROPN
ejpam-4933	8	8	ε	ε	PROPN
ejpam-4933	8	9	j	j	PROPN
ejpam-4933	8	10	-fuzzy	-fuzzy	PROPN
ejpam-4933	8	11	ideal	ideal	NOUN
ejpam-4933	8	12	using	use	VERB
ejpam-4933	8	13	the	the	DET
ejpam-4933	8	14	y	y	NOUN
ejpam-4933	8	15	-	-	PUNCT
ejpam-4933	8	16	level	level	NOUN
ejpam-4933	8	17	set	set	NOUN
ejpam-4933	8	18	is	be	AUX
ejpam-4933	8	19	established	establish	VERB
ejpam-4933	8	20	.	.	PUNCT
ejpam-4933	9	1	the	the	DET
ejpam-4933	9	2	necessary	necessary	ADJ
ejpam-4933	9	3	and	and	CCONJ
ejpam-4933	9	4	sufficient	sufficient	ADJ
ejpam-4933	9	5	conditions	condition	NOUN
ejpam-4933	9	6	for	for	ADP
ejpam-4933	9	7	y	y	PROPN
ejpam-4933	9	8	ε	ε	PROPN
ejpam-4933	9	9	j	j	PROPN
ejpam-4933	9	10	-fuzzy	-fuzzy	PROPN
ejpam-4933	9	11	ideal	ideal	ADJ
ejpam-4933	9	12	to	to	PART
ejpam-4933	9	13	be	be	AUX
ejpam-4933	9	14	closed	close	VERB
ejpam-4933	9	15	is	be	AUX
ejpam-4933	9	16	explored	explore	VERB
ejpam-4933	9	17	,	,	PUNCT
ejpam-4933	9	18	and	and	CCONJ
ejpam-4933	9	19	conditions	condition	NOUN
ejpam-4933	9	20	for	for	ADP
ejpam-4933	9	21	y	y	PROPN
ejpam-4933	9	22	ε	ε	PROPN
ejpam-4933	9	23	j	j	PROPN
ejpam-4933	9	24	-fuzzy	-fuzzy	PROPN
ejpam-4933	9	25	subalgebra	subalgebra	NOUN
ejpam-4933	9	26	to	to	PART
ejpam-4933	9	27	be	be	AUX
ejpam-4933	9	28	y	y	PROPN
ejpam-4933	9	29	ε	ε	PROPN
ejpam-4933	9	30	j	j	PROPN
ejpam-4933	9	31	-fuzzy	-fuzzy	PROPN
ejpam-4933	9	32	ideal	ideal	NOUN
ejpam-4933	9	33	are	be	AUX
ejpam-4933	9	34	provided	provide	VERB
ejpam-4933	9	35	.	.	PUNCT
ejpam-4933	10	1	2020	2020	NUM
ejpam-4933	10	2	mathematics	mathematic	NOUN
ejpam-4933	10	3	subject	subject	NOUN
ejpam-4933	10	4	classifications	classification	NOUN
ejpam-4933	10	5	:	:	PUNCT
ejpam-4933	10	6	03g25	03g25	NUM
ejpam-4933	10	7	,	,	PUNCT
ejpam-4933	10	8	06f35	06f35	NUM
ejpam-4933	10	9	,	,	PUNCT
ejpam-4933	10	10	08a72	08a72	NOUN
ejpam-4933	10	11	key	key	ADJ
ejpam-4933	10	12	words	word	NOUN
ejpam-4933	10	13	and	and	CCONJ
ejpam-4933	10	14	phrases	phrase	NOUN
ejpam-4933	10	15	:	:	PUNCT
ejpam-4933	10	16	subalgebra	subalgebra	NOUN
ejpam-4933	10	17	,	,	PUNCT
ejpam-4933	10	18	ideal	ideal	ADJ
ejpam-4933	10	19	,	,	PUNCT
ejpam-4933	10	20	j	j	NOUN
ejpam-4933	10	21	-	-	NOUN
ejpam-4933	10	22	operator	operator	NOUN
ejpam-4933	10	23	,	,	PUNCT
ejpam-4933	10	24	nonconstant	nonconstant	ADJ
ejpam-4933	10	25	factor	factor	NOUN
ejpam-4933	10	26	,	,	PUNCT
ejpam-4933	10	27	y	y	PROPN
ejpam-4933	10	28	ε	ε	PROPN
ejpam-4933	10	29	j	j	PROPN
ejpam-4933	10	30	-fuzzy	-fuzzy	PROPN
ejpam-4933	10	31	subalgebra	subalgebra	NOUN
ejpam-4933	10	32	,	,	PUNCT
ejpam-4933	10	33	(	(	PUNCT
ejpam-4933	10	34	closed	closed	ADJ
ejpam-4933	10	35	)	)	PUNCT
ejpam-4933	10	36	y	y	PROPN
ejpam-4933	10	37	ε	ε	PROPN
ejpam-4933	10	38	j	j	PROPN
ejpam-4933	10	39	-fuzzy	-fuzzy	PROPN
ejpam-4933	10	40	ideal	ideal	ADJ
ejpam-4933	10	41	1	1	NUM
ejpam-4933	10	42	.	.	PUNCT
ejpam-4933	11	1	introduction	introduction	NOUN
ejpam-4933	11	2	fuzzy	fuzzy	ADJ
ejpam-4933	11	3	sets	set	NOUN
ejpam-4933	11	4	,	,	PUNCT
ejpam-4933	11	5	which	which	PRON
ejpam-4933	11	6	are	be	AUX
ejpam-4933	11	7	introduced	introduce	VERB
ejpam-4933	11	8	by	by	ADP
ejpam-4933	11	9	zadeh	zadeh	PROPN
ejpam-4933	12	1	[	[	X
ejpam-4933	12	2	14	14	NUM
ejpam-4933	12	3	]	]	PUNCT
ejpam-4933	12	4	,	,	PUNCT
ejpam-4933	12	5	are	be	AUX
ejpam-4933	12	6	mathematical	mathematical	ADJ
ejpam-4933	12	7	frameworks	framework	NOUN
ejpam-4933	12	8	that	that	PRON
ejpam-4933	12	9	are	be	AUX
ejpam-4933	12	10	very	very	ADV
ejpam-4933	12	11	useful	useful	ADJ
ejpam-4933	12	12	in	in	ADP
ejpam-4933	12	13	expressing	express	VERB
ejpam-4933	12	14	and	and	CCONJ
ejpam-4933	12	15	manipulating	manipulate	VERB
ejpam-4933	12	16	uncertainty	uncertainty	NOUN
ejpam-4933	12	17	and	and	CCONJ
ejpam-4933	12	18	ambiguity	ambiguity	NOUN
ejpam-4933	12	19	of	of	ADP
ejpam-4933	12	20	data	datum	NOUN
ejpam-4933	12	21	with	with	ADP
ejpam-4933	12	22	applications	application	NOUN
ejpam-4933	12	23	such	such	ADJ
ejpam-4933	12	24	as	as	ADP
ejpam-4933	12	25	pattern	pattern	NOUN
ejpam-4933	12	26	recognition	recognition	NOUN
ejpam-4933	12	27	,	,	PUNCT
ejpam-4933	12	28	decision	decision	NOUN
ejpam-4933	12	29	making	making	NOUN
ejpam-4933	12	30	,	,	PUNCT
ejpam-4933	12	31	control	control	NOUN
ejpam-4933	12	32	systems	system	NOUN
ejpam-4933	12	33	,	,	PUNCT
ejpam-4933	12	34	image	image	NOUN
ejpam-4933	12	35	processing	processing	NOUN
ejpam-4933	12	36	,	,	PUNCT
ejpam-4933	12	37	data	datum	NOUN
ejpam-4933	12	38	mining	mining	NOUN
ejpam-4933	12	39	,	,	PUNCT
ejpam-4933	12	40	expert	expert	NOUN
ejpam-4933	12	41	systems	system	NOUN
ejpam-4933	12	42	,	,	PUNCT
ejpam-4933	12	43	natural	natural	ADJ
ejpam-4933	12	44	language	language	NOUN
ejpam-4933	12	45	processing	processing	NOUN
ejpam-4933	12	46	,	,	PUNCT
ejpam-4933	12	47	risk	risk	NOUN
ejpam-4933	12	48	assessment	assessment	NOUN
ejpam-4933	12	49	and	and	CCONJ
ejpam-4933	12	50	decision	decision	NOUN
ejpam-4933	12	51	analysis	analysis	NOUN
ejpam-4933	12	52	,	,	PUNCT
ejpam-4933	12	53	etc	etc	X
ejpam-4933	12	54	.	.	X
ejpam-4933	12	55	various	various	ADJ
ejpam-4933	12	56	studies	study	NOUN
ejpam-4933	12	57	have	have	AUX
ejpam-4933	12	58	been	be	AUX
ejpam-4933	12	59	conducted	conduct	VERB
ejpam-4933	12	60	since	since	SCONJ
ejpam-4933	12	61	the	the	DET
ejpam-4933	12	62	study	study	NOUN
ejpam-4933	12	63	of	of	ADP
ejpam-4933	12	64	fuzzy	fuzzy	ADJ
ejpam-4933	12	65	sets	set	NOUN
ejpam-4933	12	66	in	in	ADP
ejpam-4933	12	67	bck	bck	NOUN
ejpam-4933	12	68	-	-	PUNCT
ejpam-4933	12	69	algebra	algebra	NOUN
ejpam-4933	12	70	began	begin	VERB
ejpam-4933	12	71	in	in	ADP
ejpam-4933	12	72	1991	1991	NUM
ejpam-4933	12	73	(	(	PUNCT
ejpam-4933	12	74	see	see	VERB
ejpam-4933	12	75	[	[	X
ejpam-4933	12	76	1	1	NUM
ejpam-4933	12	77	,	,	PUNCT
ejpam-4933	12	78	5	5	NUM
ejpam-4933	12	79	,	,	PUNCT
ejpam-4933	12	80	7–10	7–10	NOUN
ejpam-4933	12	81	]	]	PUNCT
ejpam-4933	12	82	)	)	PUNCT
ejpam-4933	12	83	.	.	PUNCT
ejpam-4933	13	1	jun	jun	PROPN
ejpam-4933	14	1	[	[	X
ejpam-4933	14	2	6	6	NUM
ejpam-4933	14	3	]	]	PUNCT
ejpam-4933	14	4	introduce	introduce	VERB
ejpam-4933	14	5	the	the	DET
ejpam-4933	14	6	notion	notion	NOUN
ejpam-4933	14	7	of	of	ADP
ejpam-4933	14	8	the	the	DET
ejpam-4933	14	9	j	j	NOUN
ejpam-4933	14	10	-	-	NOUN
ejpam-4933	14	11	operator	operator	NOUN
ejpam-4933	14	12	in	in	ADP
ejpam-4933	14	13	the	the	DET
ejpam-4933	14	14	closed	closed	ADJ
ejpam-4933	14	15	interval	interval	NOUN
ejpam-4933	14	16	[	[	X
ejpam-4933	14	17	0	0	NUM
ejpam-4933	14	18	,	,	PUNCT
ejpam-4933	14	19	1	1	NUM
ejpam-4933	14	20	]	]	PUNCT
ejpam-4933	14	21	and	and	CCONJ
ejpam-4933	14	22	investigate	investigate	VERB
ejpam-4933	14	23	several	several	ADJ
ejpam-4933	14	24	properties	property	NOUN
ejpam-4933	14	25	.	.	PUNCT
ejpam-4933	15	1	he	he	PRON
ejpam-4933	15	2	used	use	VERB
ejpam-4933	15	3	the	the	DET
ejpam-4933	15	4	joperator	joperator	NOUN
ejpam-4933	15	5	to	to	PART
ejpam-4933	15	6	create	create	VERB
ejpam-4933	15	7	a	a	DET
ejpam-4933	15	8	new	new	ADJ
ejpam-4933	15	9	fuzzy	fuzzy	ADJ
ejpam-4933	15	10	set	set	NOUN
ejpam-4933	15	11	called	call	VERB
ejpam-4933	15	12	the	the	DET
ejpam-4933	15	13	y	y	PROPN
ejpam-4933	15	14	ε	ε	PROPN
ejpam-4933	15	15	j	j	PROPN
ejpam-4933	15	16	-fuzzy	-fuzzy	PROPN
ejpam-4933	15	17	set	set	VERB
ejpam-4933	15	18	and	and	CCONJ
ejpam-4933	15	19	applied	apply	VERB
ejpam-4933	15	20	it	it	PRON
ejpam-4933	15	21	to	to	ADP
ejpam-4933	15	22	subalgebras	subalgebras	PROPN
ejpam-4933	15	23	in	in	ADP
ejpam-4933	15	24	bck	bck	PROPN
ejpam-4933	15	25	/	/	SYM
ejpam-4933	15	26	bci	bci	NOUN
ejpam-4933	15	27	-	-	PUNCT
ejpam-4933	15	28	algebras	algebras	X
ejpam-4933	15	29	.	.	PUNCT
ejpam-4933	16	1	he	he	PRON
ejpam-4933	16	2	introduced	introduce	VERB
ejpam-4933	16	3	the	the	DET
ejpam-4933	16	4	concept	concept	NOUN
ejpam-4933	16	5	of	of	ADP
ejpam-4933	16	6	the	the	DET
ejpam-4933	16	7	y	y	PROPN
ejpam-4933	16	8	ε	ε	PROPN
ejpam-4933	16	9	j	j	PROPN
ejpam-4933	16	10	-fuzzy	-fuzzy	PROPN
ejpam-4933	16	11	subalgebra	subalgebra	NOUN
ejpam-4933	16	12	and	and	CCONJ
ejpam-4933	16	13	investigated	investigate	VERB
ejpam-4933	16	14	its	its	PRON
ejpam-4933	16	15	properties	property	NOUN
ejpam-4933	16	16	.	.	PUNCT
ejpam-4933	17	1	he	he	PRON
ejpam-4933	17	2	provided	provide	VERB
ejpam-4933	17	3	conditions	condition	NOUN
ejpam-4933	17	4	for	for	ADP
ejpam-4933	17	5	a	a	DET
ejpam-4933	17	6	fuzzy	fuzzy	ADJ
ejpam-4933	17	7	set	set	NOUN
ejpam-4933	17	8	to	to	PART
ejpam-4933	17	9	be	be	AUX
ejpam-4933	17	10	a	a	DET
ejpam-4933	17	11	y	y	PROPN
ejpam-4933	17	12	ε	ε	PROPN
ejpam-4933	17	13	j	j	PROPN
ejpam-4933	17	14	-fuzzy	-fuzzy	PROPN
ejpam-4933	17	15	subalgebra	subalgebra	NOUN
ejpam-4933	17	16	,	,	PUNCT
ejpam-4933	17	17	and	and	CCONJ
ejpam-4933	17	18	discussed	discuss	VERB
ejpam-4933	17	19	the	the	DET
ejpam-4933	17	20	relationship	relationship	NOUN
ejpam-4933	17	21	between	between	ADP
ejpam-4933	17	22	the	the	DET
ejpam-4933	17	23	fuzzy	fuzzy	ADJ
ejpam-4933	17	24	subalgebra	subalgebra	NOUN
ejpam-4933	17	25	and	and	CCONJ
ejpam-4933	17	26	the	the	DET
ejpam-4933	17	27	y	y	PROPN
ejpam-4933	17	28	ε	ε	PROPN
ejpam-4933	17	29	j	j	PROPN
ejpam-4933	17	30	-fuzzy	-fuzzy	PROPN
ejpam-4933	17	31	subalgebra	subalgebra	NOUN
ejpam-4933	17	32	.	.	PUNCT
ejpam-4933	18	1	∗corresponding	∗corresponde	VERB
ejpam-4933	18	2	author	author	NOUN
ejpam-4933	18	3	.	.	PUNCT
ejpam-4933	19	1	doi	doi	NOUN
ejpam-4933	19	2	:	:	PUNCT
ejpam-4933	19	3	https://doi.org/10.29020/nybg.ejpam.v16i4.4933	https://doi.org/10.29020/nybg.ejpam.v16i4.4933	ADJ
ejpam-4933	19	4	email	email	NOUN
ejpam-4933	19	5	addresses	address	NOUN
ejpam-4933	19	6	:	:	PUNCT
ejpam-4933	19	7	ehroh9988@gmail.com	ehroh9988@gmail.com	X
ejpam-4933	19	8	(	(	PUNCT
ejpam-4933	19	9	e.	e.	PROPN
ejpam-4933	19	10	h.	h.	PROPN
ejpam-4933	19	11	roh	roh	PROPN
ejpam-4933	19	12	)	)	PUNCT
ejpam-4933	19	13	,	,	PUNCT
ejpam-4933	19	14	eunsyang@jbnu.ac.kr	eunsyang@jbnu.ac.kr	X
ejpam-4933	19	15	(	(	PUNCT
ejpam-4933	19	16	e.	e.	PROPN
ejpam-4933	19	17	yang	yang	PROPN
ejpam-4933	19	18	)	)	PUNCT
ejpam-4933	19	19	,	,	PUNCT
ejpam-4933	19	20	skywine@gmail.com	skywine@gmail.com	X
ejpam-4933	20	1	(	(	PUNCT
ejpam-4933	20	2	y.	y.	PROPN
ejpam-4933	20	3	b.	b.	PROPN
ejpam-4933	20	4	jun	jun	PROPN
ejpam-4933	20	5	)	)	PUNCT
ejpam-4933	20	6	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-4933	20	7	2009	2009	NUM
ejpam-4933	21	1	©	©	PROPN
ejpam-4933	21	2	2023	2023	NUM
ejpam-4933	21	3	ejpam	ejpam	NOUN
ejpam-4933	21	4	all	all	DET
ejpam-4933	21	5	rights	right	NOUN
ejpam-4933	21	6	reserved	reserve	VERB
ejpam-4933	21	7	.	.	PUNCT
ejpam-4933	22	1	e.	e.	PROPN
ejpam-4933	22	2	h.	h.	PROPN
ejpam-4933	22	3	roh	roh	PROPN
ejpam-4933	22	4	,	,	PUNCT
ejpam-4933	22	5	e.	e.	PROPN
ejpam-4933	22	6	yang	yang	PROPN
ejpam-4933	22	7	,	,	PUNCT
ejpam-4933	22	8	y.	y.	PROPN
ejpam-4933	22	9	b.	b.	PROPN
ejpam-4933	22	10	jun	jun	PROPN
ejpam-4933	22	11	/	/	SYM
ejpam-4933	22	12	eur	eur	PROPN
ejpam-4933	22	13	.	.	PUNCT
ejpam-4933	23	1	j.	j.	PROPN
ejpam-4933	23	2	pure	pure	PROPN
ejpam-4933	23	3	appl	appl	PROPN
ejpam-4933	23	4	.	.	PROPN
ejpam-4933	23	5	math	math	PROPN
ejpam-4933	23	6	,	,	PUNCT
ejpam-4933	23	7	16	16	NUM
ejpam-4933	23	8	(	(	PUNCT
ejpam-4933	23	9	4	4	NUM
ejpam-4933	23	10	)	)	PUNCT
ejpam-4933	23	11	(	(	PUNCT
ejpam-4933	23	12	2023	2023	NUM
ejpam-4933	23	13	)	)	PUNCT
ejpam-4933	23	14	,	,	PUNCT
ejpam-4933	23	15	2009	2009	NUM
ejpam-4933	23	16	-	-	SYM
ejpam-4933	23	17	2024	2024	NUM
ejpam-4933	23	18	2010	2010	NUM
ejpam-4933	23	19	in	in	ADP
ejpam-4933	23	20	this	this	DET
ejpam-4933	23	21	paper	paper	NOUN
ejpam-4933	24	1	,	,	PUNCT
ejpam-4933	24	2	we	we	PRON
ejpam-4933	24	3	study	study	VERB
ejpam-4933	24	4	the	the	DET
ejpam-4933	24	5	ideals	ideal	NOUN
ejpam-4933	24	6	of	of	ADP
ejpam-4933	24	7	bck	bck	PROPN
ejpam-4933	24	8	/	/	SYM
ejpam-4933	24	9	bci	bci	NOUN
ejpam-4933	24	10	-	-	PUNCT
ejpam-4933	24	11	algebras	algebras	PROPN
ejpam-4933	24	12	based	base	VERB
ejpam-4933	24	13	on	on	ADP
ejpam-4933	24	14	y	y	PROPN
ejpam-4933	24	15	ε	ε	PROPN
ejpam-4933	24	16	j	j	PROPN
ejpam-4933	24	17	-fuzzy	-fuzzy	PROPN
ejpam-4933	24	18	sets	set	NOUN
ejpam-4933	24	19	.	.	PUNCT
ejpam-4933	25	1	we	we	PRON
ejpam-4933	25	2	first	first	ADV
ejpam-4933	25	3	investigate	investigate	VERB
ejpam-4933	25	4	the	the	DET
ejpam-4933	25	5	underlying	underlie	VERB
ejpam-4933	25	6	properties	property	NOUN
ejpam-4933	25	7	of	of	ADP
ejpam-4933	25	8	the	the	DET
ejpam-4933	25	9	level	level	NOUN
ejpam-4933	25	10	sets	set	NOUN
ejpam-4933	25	11	of	of	ADP
ejpam-4933	25	12	y	y	PROPN
ejpam-4933	25	13	ε	ε	PROPN
ejpam-4933	25	14	j	j	PROPN
ejpam-4933	25	15	-fuzzy	-fuzzy	PROPN
ejpam-4933	25	16	sets	set	NOUN
ejpam-4933	25	17	.	.	PUNCT
ejpam-4933	26	1	we	we	PRON
ejpam-4933	26	2	introduce	introduce	VERB
ejpam-4933	26	3	the	the	DET
ejpam-4933	26	4	concept	concept	NOUN
ejpam-4933	26	5	of	of	ADP
ejpam-4933	26	6	y	y	PROPN
ejpam-4933	26	7	ε	ε	PROPN
ejpam-4933	26	8	j	j	PROPN
ejpam-4933	26	9	-fuzzy	-fuzzy	PROPN
ejpam-4933	26	10	ideals	ideal	NOUN
ejpam-4933	26	11	in	in	ADP
ejpam-4933	26	12	bck	bck	PROPN
ejpam-4933	26	13	/	/	SYM
ejpam-4933	26	14	bci	bci	NOUN
ejpam-4933	26	15	-	-	PUNCT
ejpam-4933	26	16	algebras	algebras	X
ejpam-4933	26	17	,	,	PUNCT
ejpam-4933	26	18	and	and	CCONJ
ejpam-4933	26	19	investigate	investigate	VERB
ejpam-4933	26	20	several	several	ADJ
ejpam-4933	26	21	properties	property	NOUN
ejpam-4933	26	22	.	.	PUNCT
ejpam-4933	27	1	we	we	PRON
ejpam-4933	27	2	discuss	discuss	VERB
ejpam-4933	27	3	the	the	DET
ejpam-4933	27	4	relationship	relationship	NOUN
ejpam-4933	27	5	between	between	ADP
ejpam-4933	27	6	y	y	PROPN
ejpam-4933	27	7	ε	ε	PROPN
ejpam-4933	27	8	j	j	PROPN
ejpam-4933	27	9	-fuzzy	-fuzzy	PROPN
ejpam-4933	27	10	ideal	ideal	ADJ
ejpam-4933	27	11	and	and	CCONJ
ejpam-4933	27	12	y	y	PROPN
ejpam-4933	27	13	ε	ε	PROPN
ejpam-4933	27	14	j	j	PROPN
ejpam-4933	27	15	-fuzzy	-fuzzy	PROPN
ejpam-4933	27	16	subalgebra	subalgebra	NOUN
ejpam-4933	27	17	,	,	PUNCT
ejpam-4933	27	18	and	and	CCONJ
ejpam-4933	27	19	also	also	ADV
ejpam-4933	27	20	identify	identify	VERB
ejpam-4933	27	21	the	the	DET
ejpam-4933	27	22	relationship	relationship	NOUN
ejpam-4933	27	23	between	between	ADP
ejpam-4933	27	24	y	y	PROPN
ejpam-4933	27	25	ε	ε	PROPN
ejpam-4933	27	26	j	j	PROPN
ejpam-4933	27	27	-fuzzy	-fuzzy	PROPN
ejpam-4933	27	28	ideal	ideal	ADJ
ejpam-4933	27	29	and	and	CCONJ
ejpam-4933	27	30	fuzzy	fuzzy	ADJ
ejpam-4933	27	31	ideal	ideal	NOUN
ejpam-4933	27	32	.	.	PUNCT
ejpam-4933	28	1	we	we	PRON
ejpam-4933	28	2	consider	consider	VERB
ejpam-4933	28	3	the	the	DET
ejpam-4933	28	4	characterization	characterization	NOUN
ejpam-4933	28	5	of	of	ADP
ejpam-4933	28	6	y	y	PROPN
ejpam-4933	28	7	ε	ε	PROPN
ejpam-4933	28	8	j	j	PROPN
ejpam-4933	28	9	-fuzzy	-fuzzy	PROPN
ejpam-4933	28	10	ideal	ideal	NOUN
ejpam-4933	28	11	using	use	VERB
ejpam-4933	28	12	the	the	DET
ejpam-4933	28	13	y	y	ADJ
ejpam-4933	28	14	-	-	PUNCT
ejpam-4933	28	15	level	level	NOUN
ejpam-4933	28	16	set	set	NOUN
ejpam-4933	28	17	.	.	PUNCT
ejpam-4933	29	1	we	we	PRON
ejpam-4933	29	2	define	define	VERB
ejpam-4933	29	3	closed	closed	ADJ
ejpam-4933	29	4	y	y	PROPN
ejpam-4933	29	5	ε	ε	PROPN
ejpam-4933	29	6	j	j	PROPN
ejpam-4933	29	7	-fuzzy	-fuzzy	NOUN
ejpam-4933	29	8	ideal	ideal	ADJ
ejpam-4933	29	9	,	,	PUNCT
ejpam-4933	29	10	and	and	CCONJ
ejpam-4933	29	11	deal	deal	VERB
ejpam-4933	29	12	with	with	ADP
ejpam-4933	29	13	its	its	PRON
ejpam-4933	29	14	properties	property	NOUN
ejpam-4933	29	15	.	.	PUNCT
ejpam-4933	30	1	we	we	PRON
ejpam-4933	30	2	explore	explore	VERB
ejpam-4933	30	3	the	the	DET
ejpam-4933	30	4	necessary	necessary	ADJ
ejpam-4933	30	5	and	and	CCONJ
ejpam-4933	30	6	sufficient	sufficient	ADJ
ejpam-4933	30	7	conditions	condition	NOUN
ejpam-4933	30	8	for	for	ADP
ejpam-4933	30	9	y	y	PROPN
ejpam-4933	30	10	ε	ε	PROPN
ejpam-4933	30	11	j	j	PROPN
ejpam-4933	30	12	-fuzzy	-fuzzy	PROPN
ejpam-4933	30	13	ideal	ideal	ADJ
ejpam-4933	30	14	to	to	PART
ejpam-4933	30	15	be	be	AUX
ejpam-4933	30	16	closed	close	VERB
ejpam-4933	30	17	.	.	PUNCT
ejpam-4933	31	1	finally	finally	ADV
ejpam-4933	31	2	,	,	PUNCT
ejpam-4933	31	3	we	we	PRON
ejpam-4933	31	4	provide	provide	VERB
ejpam-4933	31	5	conditions	condition	NOUN
ejpam-4933	31	6	for	for	ADP
ejpam-4933	31	7	y	y	PROPN
ejpam-4933	31	8	ε	ε	PROPN
ejpam-4933	31	9	j	j	PROPN
ejpam-4933	31	10	-fuzzy	-fuzzy	PROPN
ejpam-4933	31	11	subalgebra	subalgebra	NOUN
ejpam-4933	31	12	to	to	PART
ejpam-4933	31	13	be	be	AUX
ejpam-4933	31	14	y	y	PROPN
ejpam-4933	31	15	ε	ε	PROPN
ejpam-4933	31	16	j	j	PROPN
ejpam-4933	31	17	-fuzzy	-fuzzy	PROPN
ejpam-4933	31	18	ideal	ideal	ADJ
ejpam-4933	31	19	.	.	PUNCT
ejpam-4933	32	1	2	2	X
ejpam-4933	32	2	.	.	NUM
ejpam-4933	32	3	preliminaries	preliminary	NOUN
ejpam-4933	32	4	a	a	DET
ejpam-4933	32	5	bck	bck	PROPN
ejpam-4933	32	6	/	/	SYM
ejpam-4933	32	7	bci	bci	NOUN
ejpam-4933	32	8	-	-	NOUN
ejpam-4933	32	9	algebra	algebra	NOUN
ejpam-4933	32	10	is	be	AUX
ejpam-4933	32	11	an	an	DET
ejpam-4933	32	12	important	important	ADJ
ejpam-4933	32	13	class	class	NOUN
ejpam-4933	32	14	of	of	ADP
ejpam-4933	32	15	logical	logical	ADJ
ejpam-4933	32	16	algebras	algebra	NOUN
ejpam-4933	32	17	introduced	introduce	VERB
ejpam-4933	32	18	by	by	ADP
ejpam-4933	32	19	k.	k.	PROPN
ejpam-4933	32	20	iséki	iséki	PROPN
ejpam-4933	32	21	(	(	PUNCT
ejpam-4933	32	22	see	see	VERB
ejpam-4933	32	23	[	[	X
ejpam-4933	32	24	3	3	X
ejpam-4933	32	25	]	]	PUNCT
ejpam-4933	32	26	and	and	CCONJ
ejpam-4933	32	27	[	[	X
ejpam-4933	32	28	4	4	NUM
ejpam-4933	32	29	]	]	PUNCT
ejpam-4933	32	30	)	)	PUNCT
ejpam-4933	32	31	and	and	CCONJ
ejpam-4933	32	32	was	be	AUX
ejpam-4933	32	33	extensively	extensively	ADV
ejpam-4933	32	34	investigated	investigate	VERB
ejpam-4933	32	35	by	by	ADP
ejpam-4933	32	36	several	several	ADJ
ejpam-4933	32	37	researchers	researcher	NOUN
ejpam-4933	32	38	.	.	PUNCT
ejpam-4933	33	1	we	we	PRON
ejpam-4933	33	2	recall	recall	VERB
ejpam-4933	33	3	the	the	DET
ejpam-4933	33	4	definitions	definition	NOUN
ejpam-4933	33	5	and	and	CCONJ
ejpam-4933	33	6	basic	basic	ADJ
ejpam-4933	33	7	results	result	NOUN
ejpam-4933	33	8	required	require	VERB
ejpam-4933	33	9	in	in	ADP
ejpam-4933	33	10	this	this	DET
ejpam-4933	33	11	paper	paper	NOUN
ejpam-4933	33	12	.	.	PUNCT
ejpam-4933	34	1	see	see	VERB
ejpam-4933	34	2	the	the	DET
ejpam-4933	34	3	books	book	NOUN
ejpam-4933	34	4	[	[	X
ejpam-4933	34	5	2	2	NUM
ejpam-4933	34	6	,	,	PUNCT
ejpam-4933	34	7	11	11	NUM
ejpam-4933	34	8	]	]	PUNCT
ejpam-4933	34	9	for	for	ADP
ejpam-4933	34	10	further	further	ADJ
ejpam-4933	34	11	information	information	NOUN
ejpam-4933	34	12	regarding	regard	VERB
ejpam-4933	34	13	bck	bck	PROPN
ejpam-4933	34	14	/	/	SYM
ejpam-4933	34	15	bci	bci	NOUN
ejpam-4933	34	16	-	-	PUNCT
ejpam-4933	34	17	algebras	algebra	NOUN
ejpam-4933	34	18	.	.	PUNCT
ejpam-4933	35	1	by	by	ADP
ejpam-4933	35	2	a	a	DET
ejpam-4933	35	3	bci	bci	NOUN
ejpam-4933	35	4	-	-	NOUN
ejpam-4933	35	5	algebra	algebra	NOUN
ejpam-4933	35	6	,	,	PUNCT
ejpam-4933	35	7	we	we	PRON
ejpam-4933	35	8	mean	mean	VERB
ejpam-4933	35	9	a	a	DET
ejpam-4933	35	10	structure	structure	NOUN
ejpam-4933	35	11	(	(	PUNCT
ejpam-4933	35	12	x	x	X
ejpam-4933	35	13	,	,	PUNCT
ejpam-4933	35	14	∗	∗	NOUN
ejpam-4933	35	15	,	,	PUNCT
ejpam-4933	35	16	0	0	NUM
ejpam-4933	35	17	)	)	PUNCT
ejpam-4933	35	18	,	,	PUNCT
ejpam-4933	35	19	where	where	SCONJ
ejpam-4933	35	20	0	0	NUM
ejpam-4933	35	21	is	be	AUX
ejpam-4933	35	22	a	a	DET
ejpam-4933	35	23	special	special	ADJ
ejpam-4933	35	24	element	element	NOUN
ejpam-4933	35	25	and	and	CCONJ
ejpam-4933	35	26	∗	∗	NOUN
ejpam-4933	35	27	is	be	AUX
ejpam-4933	35	28	a	a	DET
ejpam-4933	35	29	binary	binary	ADJ
ejpam-4933	35	30	operation	operation	NOUN
ejpam-4933	35	31	on	on	ADP
ejpam-4933	35	32	x	x	PRON
ejpam-4933	35	33	,	,	PUNCT
ejpam-4933	35	34	that	that	PRON
ejpam-4933	35	35	satisfies	satisfy	VERB
ejpam-4933	35	36	the	the	DET
ejpam-4933	35	37	following	follow	VERB
ejpam-4933	35	38	conditions	condition	NOUN
ejpam-4933	35	39	:	:	PUNCT
ejpam-4933	35	40	(	(	PUNCT
ejpam-4933	35	41	i	i	NOUN
ejpam-4933	35	42	)	)	PUNCT
ejpam-4933	35	43	(	(	PUNCT
ejpam-4933	35	44	(	(	PUNCT
ejpam-4933	35	45	a	a	DET
ejpam-4933	35	46	∗	∗	NOUN
ejpam-4933	35	47	b	b	NOUN
ejpam-4933	35	48	)	)	PUNCT
ejpam-4933	35	49	∗	∗	NOUN
ejpam-4933	35	50	(	(	PUNCT
ejpam-4933	35	51	a	a	DET
ejpam-4933	35	52	∗	∗	NOUN
ejpam-4933	35	53	c	c	NOUN
ejpam-4933	35	54	)	)	PUNCT
ejpam-4933	35	55	)	)	PUNCT
ejpam-4933	35	56	∗	∗	NOUN
ejpam-4933	35	57	(	(	PUNCT
ejpam-4933	35	58	c	c	NOUN
ejpam-4933	35	59	∗	∗	X
ejpam-4933	35	60	b	b	NOUN
ejpam-4933	35	61	)	)	PUNCT
ejpam-4933	35	62	=	=	SYM
ejpam-4933	35	63	0	0	NUM
ejpam-4933	35	64	,	,	PUNCT
ejpam-4933	35	65	(	(	PUNCT
ejpam-4933	35	66	ii	ii	NOUN
ejpam-4933	35	67	)	)	PUNCT
ejpam-4933	35	68	(	(	PUNCT
ejpam-4933	35	69	a	a	DET
ejpam-4933	35	70	∗	∗	NOUN
ejpam-4933	35	71	(	(	PUNCT
ejpam-4933	35	72	a	a	DET
ejpam-4933	35	73	∗	∗	NOUN
ejpam-4933	35	74	b	b	NOUN
ejpam-4933	35	75	)	)	PUNCT
ejpam-4933	35	76	)	)	PUNCT
ejpam-4933	35	77	∗	∗	NOUN
ejpam-4933	35	78	b	b	X
ejpam-4933	35	79	=	=	SYM
ejpam-4933	35	80	0	0	NUM
ejpam-4933	35	81	,	,	PUNCT
ejpam-4933	35	82	(	(	PUNCT
ejpam-4933	35	83	iii	iii	X
ejpam-4933	35	84	)	)	PUNCT
ejpam-4933	35	85	a	a	DET
ejpam-4933	35	86	∗	∗	NOUN
ejpam-4933	35	87	a	a	DET
ejpam-4933	35	88	=	=	NOUN
ejpam-4933	35	89	0	0	NUM
ejpam-4933	35	90	,	,	PUNCT
ejpam-4933	35	91	(	(	PUNCT
ejpam-4933	35	92	iv	iv	X
ejpam-4933	35	93	)	)	PUNCT
ejpam-4933	35	94	a	a	DET
ejpam-4933	35	95	∗	∗	NOUN
ejpam-4933	35	96	b	b	NOUN
ejpam-4933	35	97	=	=	SYM
ejpam-4933	35	98	0	0	NUM
ejpam-4933	35	99	,	,	PUNCT
ejpam-4933	35	100	b	b	NOUN
ejpam-4933	35	101	∗	∗	NOUN
ejpam-4933	35	102	a	a	DET
ejpam-4933	35	103	=	=	SYM
ejpam-4933	35	104	0	0	PROPN
ejpam-4933	35	105	⇒	⇒	NOUN
ejpam-4933	35	106	a	a	DET
ejpam-4933	35	107	=	=	SYM
ejpam-4933	35	108	b	b	NOUN
ejpam-4933	35	109	,	,	PUNCT
ejpam-4933	35	110	for	for	ADP
ejpam-4933	35	111	all	all	DET
ejpam-4933	35	112	a	a	DET
ejpam-4933	35	113	,	,	PUNCT
ejpam-4933	35	114	b	b	NOUN
ejpam-4933	35	115	,	,	PUNCT
ejpam-4933	35	116	c	c	PROPN
ejpam-4933	35	117	∈	∈	PROPN
ejpam-4933	35	118	x.	x.	NOUN
ejpam-4933	36	1	if	if	SCONJ
ejpam-4933	36	2	a	a	DET
ejpam-4933	36	3	bci	bci	NOUN
ejpam-4933	36	4	-	-	NOUN
ejpam-4933	36	5	algebra	algebra	NOUN
ejpam-4933	36	6	(	(	PUNCT
ejpam-4933	36	7	x	x	X
ejpam-4933	36	8	,	,	PUNCT
ejpam-4933	36	9	∗	∗	NOUN
ejpam-4933	36	10	,	,	PUNCT
ejpam-4933	36	11	0	0	NUM
ejpam-4933	36	12	)	)	PUNCT
ejpam-4933	36	13	satisfies	satisfy	VERB
ejpam-4933	36	14	the	the	DET
ejpam-4933	36	15	following	follow	VERB
ejpam-4933	36	16	identity	identity	NOUN
ejpam-4933	36	17	:	:	PUNCT
ejpam-4933	36	18	(	(	PUNCT
ejpam-4933	36	19	v	v	NOUN
ejpam-4933	36	20	)	)	PUNCT
ejpam-4933	36	21	(	(	PUNCT
ejpam-4933	36	22	∀a	∀a	NOUN
ejpam-4933	36	23	∈	∈	NOUN
ejpam-4933	36	24	x	x	NOUN
ejpam-4933	36	25	)	)	PUNCT
ejpam-4933	36	26	(	(	PUNCT
ejpam-4933	36	27	0	0	NUM
ejpam-4933	36	28	∗	∗	NOUN
ejpam-4933	36	29	a	a	PRON
ejpam-4933	36	30	=	=	NOUN
ejpam-4933	36	31	0	0	NUM
ejpam-4933	36	32	)	)	PUNCT
ejpam-4933	36	33	,	,	PUNCT
ejpam-4933	36	34	then	then	ADV
ejpam-4933	36	35	(	(	PUNCT
ejpam-4933	36	36	x	x	X
ejpam-4933	36	37	,	,	PUNCT
ejpam-4933	36	38	∗	∗	NOUN
ejpam-4933	36	39	,	,	PUNCT
ejpam-4933	36	40	0	0	NUM
ejpam-4933	36	41	)	)	PUNCT
ejpam-4933	36	42	is	be	AUX
ejpam-4933	36	43	called	call	VERB
ejpam-4933	36	44	a	a	DET
ejpam-4933	36	45	bck	bck	NOUN
ejpam-4933	36	46	-	-	PUNCT
ejpam-4933	36	47	algebra	algebra	NOUN
ejpam-4933	36	48	.	.	PUNCT
ejpam-4933	37	1	the	the	DET
ejpam-4933	37	2	order	order	NOUN
ejpam-4933	37	3	relation	relation	NOUN
ejpam-4933	37	4	“	"	PUNCT
ejpam-4933	37	5	≤x	≤x	PROPN
ejpam-4933	37	6	”	"	PUNCT
ejpam-4933	37	7	in	in	ADP
ejpam-4933	37	8	a	a	DET
ejpam-4933	37	9	bck	bck	NOUN
ejpam-4933	37	10	/	/	SYM
ejpam-4933	37	11	bci	bci	NOUN
ejpam-4933	37	12	-	-	NOUN
ejpam-4933	37	13	algebra	algebra	NOUN
ejpam-4933	37	14	(	(	PUNCT
ejpam-4933	37	15	x	x	X
ejpam-4933	37	16	,	,	PUNCT
ejpam-4933	37	17	∗	∗	NOUN
ejpam-4933	37	18	,	,	PUNCT
ejpam-4933	37	19	0	0	NUM
ejpam-4933	37	20	)	)	PUNCT
ejpam-4933	37	21	is	be	AUX
ejpam-4933	37	22	defined	define	VERB
ejpam-4933	37	23	as	as	SCONJ
ejpam-4933	37	24	follows	follow	VERB
ejpam-4933	37	25	:	:	PUNCT
ejpam-4933	37	26	(	(	PUNCT
ejpam-4933	37	27	∀a	∀a	X
ejpam-4933	37	28	,	,	PUNCT
ejpam-4933	37	29	b	b	PROPN
ejpam-4933	37	30	∈	∈	PROPN
ejpam-4933	37	31	x)(a	x)(a	PUNCT
ejpam-4933	38	1	≤x	≤x	PROPN
ejpam-4933	38	2	b	b	PROPN
ejpam-4933	38	3	⇔	⇔	PROPN
ejpam-4933	38	4	a	a	DET
ejpam-4933	38	5	∗	∗	NOUN
ejpam-4933	38	6	b	b	NOUN
ejpam-4933	38	7	=	=	NOUN
ejpam-4933	38	8	0	0	NUM
ejpam-4933	38	9	)	)	PUNCT
ejpam-4933	38	10	.	.	PUNCT
ejpam-4933	39	1	(	(	PUNCT
ejpam-4933	39	2	1	1	X
ejpam-4933	39	3	)	)	PUNCT
ejpam-4933	39	4	every	every	DET
ejpam-4933	39	5	bck	bck	PROPN
ejpam-4933	39	6	/	/	SYM
ejpam-4933	39	7	bci	bci	NOUN
ejpam-4933	39	8	-	-	NOUN
ejpam-4933	39	9	algebra	algebra	NOUN
ejpam-4933	39	10	(	(	PUNCT
ejpam-4933	39	11	x	x	X
ejpam-4933	39	12	,	,	PUNCT
ejpam-4933	39	13	∗	∗	NOUN
ejpam-4933	39	14	,	,	PUNCT
ejpam-4933	39	15	0	0	NUM
ejpam-4933	39	16	)	)	PUNCT
ejpam-4933	39	17	satisfies	satisfy	VERB
ejpam-4933	39	18	the	the	DET
ejpam-4933	39	19	following	follow	VERB
ejpam-4933	39	20	conditions	condition	NOUN
ejpam-4933	39	21	:	:	PUNCT
ejpam-4933	39	22	a	a	DET
ejpam-4933	39	23	∗	∗	NOUN
ejpam-4933	39	24	0	0	NUM
ejpam-4933	40	1	=	=	SYM
ejpam-4933	40	2	a	a	PRON
ejpam-4933	40	3	,	,	PUNCT
ejpam-4933	40	4	(	(	PUNCT
ejpam-4933	40	5	2	2	X
ejpam-4933	40	6	)	)	PUNCT
ejpam-4933	40	7	a	a	DET
ejpam-4933	40	8	≤x	≤x	PROPN
ejpam-4933	40	9	b	b	PROPN
ejpam-4933	40	10	⇒	⇒	VERB
ejpam-4933	40	11	a	a	DET
ejpam-4933	40	12	∗	∗	NOUN
ejpam-4933	40	13	c	c	NOUN
ejpam-4933	40	14	≤x	≤x	PROPN
ejpam-4933	40	15	b	b	PROPN
ejpam-4933	40	16	∗	∗	NOUN
ejpam-4933	40	17	c	c	X
ejpam-4933	40	18	,	,	PUNCT
ejpam-4933	41	1	c	c	PROPN
ejpam-4933	41	2	∗	∗	NOUN
ejpam-4933	41	3	b	b	NOUN
ejpam-4933	41	4	≤x	≤x	PROPN
ejpam-4933	41	5	c	c	PROPN
ejpam-4933	41	6	∗	∗	X
ejpam-4933	41	7	a	a	PRON
ejpam-4933	41	8	,	,	PUNCT
ejpam-4933	41	9	(	(	PUNCT
ejpam-4933	41	10	3	3	NUM
ejpam-4933	41	11	)	)	PUNCT
ejpam-4933	41	12	(	(	PUNCT
ejpam-4933	41	13	a	a	DET
ejpam-4933	41	14	∗	∗	NOUN
ejpam-4933	41	15	b	b	NOUN
ejpam-4933	41	16	)	)	PUNCT
ejpam-4933	41	17	∗	∗	NOUN
ejpam-4933	41	18	c	c	NOUN
ejpam-4933	41	19	=	=	SYM
ejpam-4933	41	20	(	(	PUNCT
ejpam-4933	41	21	a	a	DET
ejpam-4933	41	22	∗	∗	NOUN
ejpam-4933	41	23	c	c	NOUN
ejpam-4933	41	24	)	)	PUNCT
ejpam-4933	41	25	∗	∗	PROPN
ejpam-4933	41	26	b	b	NOUN
ejpam-4933	41	27	,	,	PUNCT
ejpam-4933	41	28	(	(	PUNCT
ejpam-4933	41	29	4	4	NUM
ejpam-4933	41	30	)	)	PUNCT
ejpam-4933	41	31	for	for	ADP
ejpam-4933	41	32	all	all	DET
ejpam-4933	41	33	a	a	DET
ejpam-4933	41	34	,	,	PUNCT
ejpam-4933	41	35	b	b	NOUN
ejpam-4933	41	36	,	,	PUNCT
ejpam-4933	41	37	c	c	PROPN
ejpam-4933	41	38	∈	∈	PROPN
ejpam-4933	41	39	x.	x.	NOUN
ejpam-4933	42	1	every	every	DET
ejpam-4933	42	2	bck	bck	NOUN
ejpam-4933	42	3	-	-	PUNCT
ejpam-4933	42	4	algebra	algebra	NOUN
ejpam-4933	42	5	(	(	PUNCT
ejpam-4933	42	6	x	x	X
ejpam-4933	42	7	,	,	PUNCT
ejpam-4933	42	8	∗	∗	NOUN
ejpam-4933	42	9	,	,	PUNCT
ejpam-4933	42	10	0	0	NUM
ejpam-4933	42	11	)	)	PUNCT
ejpam-4933	42	12	satisfies	satisfie	NOUN
ejpam-4933	42	13	:	:	PUNCT
ejpam-4933	42	14	(	(	PUNCT
ejpam-4933	42	15	∀x	∀x	X
ejpam-4933	42	16	,	,	PUNCT
ejpam-4933	42	17	a	a	DET
ejpam-4933	42	18	∈	∈	PROPN
ejpam-4933	42	19	x)(x	x)(x	PROPN
ejpam-4933	42	20	∗	∗	VERB
ejpam-4933	42	21	a	a	DET
ejpam-4933	42	22	≤x	≤x	PROPN
ejpam-4933	42	23	x	x	NOUN
ejpam-4933	42	24	)	)	PUNCT
ejpam-4933	42	25	.	.	PUNCT
ejpam-4933	43	1	(	(	PUNCT
ejpam-4933	43	2	5	5	X
ejpam-4933	43	3	)	)	PUNCT
ejpam-4933	43	4	e.	e.	PROPN
ejpam-4933	43	5	h.	h.	PROPN
ejpam-4933	43	6	roh	roh	PROPN
ejpam-4933	43	7	,	,	PUNCT
ejpam-4933	43	8	e.	e.	PROPN
ejpam-4933	43	9	yang	yang	PROPN
ejpam-4933	43	10	,	,	PUNCT
ejpam-4933	43	11	y.	y.	PROPN
ejpam-4933	43	12	b.	b.	PROPN
ejpam-4933	43	13	jun	jun	PROPN
ejpam-4933	43	14	/	/	SYM
ejpam-4933	43	15	eur	eur	PROPN
ejpam-4933	43	16	.	.	PUNCT
ejpam-4933	44	1	j.	j.	PROPN
ejpam-4933	44	2	pure	pure	PROPN
ejpam-4933	44	3	appl	appl	PROPN
ejpam-4933	44	4	.	.	PROPN
ejpam-4933	44	5	math	math	PROPN
ejpam-4933	44	6	,	,	PUNCT
ejpam-4933	44	7	16	16	NUM
ejpam-4933	44	8	(	(	PUNCT
ejpam-4933	44	9	4	4	NUM
ejpam-4933	44	10	)	)	PUNCT
ejpam-4933	44	11	(	(	PUNCT
ejpam-4933	44	12	2023	2023	NUM
ejpam-4933	44	13	)	)	PUNCT
ejpam-4933	44	14	,	,	PUNCT
ejpam-4933	44	15	2009	2009	NUM
ejpam-4933	44	16	-	-	SYM
ejpam-4933	44	17	2024	2024	NUM
ejpam-4933	44	18	2011	2011	NUM
ejpam-4933	44	19	every	every	DET
ejpam-4933	44	20	bci	bci	NOUN
ejpam-4933	44	21	-	-	NOUN
ejpam-4933	44	22	algebra	algebra	NOUN
ejpam-4933	44	23	(	(	PUNCT
ejpam-4933	44	24	x	x	X
ejpam-4933	44	25	,	,	PUNCT
ejpam-4933	44	26	∗	∗	NOUN
ejpam-4933	44	27	,	,	PUNCT
ejpam-4933	44	28	0	0	NUM
ejpam-4933	44	29	)	)	PUNCT
ejpam-4933	44	30	satisfies	satisfie	NOUN
ejpam-4933	44	31	:	:	PUNCT
ejpam-4933	44	32	(	(	PUNCT
ejpam-4933	44	33	∀a	∀a	X
ejpam-4933	44	34	,	,	PUNCT
ejpam-4933	44	35	b	b	PROPN
ejpam-4933	44	36	∈	∈	PROPN
ejpam-4933	44	37	x)(0	x)(0	X
ejpam-4933	45	1	∗	∗	NOUN
ejpam-4933	45	2	(	(	PUNCT
ejpam-4933	45	3	a	a	DET
ejpam-4933	45	4	∗	∗	NOUN
ejpam-4933	45	5	b	b	NOUN
ejpam-4933	45	6	)	)	PUNCT
ejpam-4933	45	7	=	=	SYM
ejpam-4933	45	8	(	(	PUNCT
ejpam-4933	45	9	0	0	NUM
ejpam-4933	45	10	∗	∗	NOUN
ejpam-4933	45	11	a	a	NOUN
ejpam-4933	45	12	)	)	PUNCT
ejpam-4933	45	13	∗	∗	NOUN
ejpam-4933	45	14	(	(	PUNCT
ejpam-4933	45	15	0	0	NUM
ejpam-4933	45	16	∗	∗	NUM
ejpam-4933	45	17	b	b	NOUN
ejpam-4933	45	18	)	)	PUNCT
ejpam-4933	45	19	)	)	PUNCT
ejpam-4933	45	20	.	.	PUNCT
ejpam-4933	46	1	(	(	PUNCT
ejpam-4933	46	2	6	6	X
ejpam-4933	46	3	)	)	PUNCT
ejpam-4933	46	4	a	a	DET
ejpam-4933	46	5	bci	bci	NOUN
ejpam-4933	46	6	-	-	NOUN
ejpam-4933	46	7	algebra	algebra	NOUN
ejpam-4933	46	8	(	(	PUNCT
ejpam-4933	46	9	x	x	X
ejpam-4933	46	10	,	,	PUNCT
ejpam-4933	46	11	∗	∗	NOUN
ejpam-4933	46	12	,	,	PUNCT
ejpam-4933	46	13	0	0	NUM
ejpam-4933	46	14	)	)	PUNCT
ejpam-4933	46	15	is	be	AUX
ejpam-4933	46	16	said	say	VERB
ejpam-4933	46	17	to	to	PART
ejpam-4933	46	18	be	be	AUX
ejpam-4933	46	19	p	p	NOUN
ejpam-4933	46	20	-	-	PUNCT
ejpam-4933	46	21	semisimple	semisimple	NOUN
ejpam-4933	46	22	if	if	SCONJ
ejpam-4933	46	23	0	0	NUM
ejpam-4933	46	24	∗	∗	NOUN
ejpam-4933	46	25	(	(	PUNCT
ejpam-4933	46	26	0	0	NUM
ejpam-4933	46	27	∗	∗	NOUN
ejpam-4933	46	28	x	x	NOUN
ejpam-4933	46	29	)	)	PUNCT
ejpam-4933	46	30	=	=	PUNCT
ejpam-4933	47	1	x	x	PUNCT
ejpam-4933	47	2	for	for	ADP
ejpam-4933	47	3	all	all	DET
ejpam-4933	47	4	x	x	SYM
ejpam-4933	47	5	∈	∈	NOUN
ejpam-4933	47	6	x	x	INTJ
ejpam-4933	47	7	(	(	PUNCT
ejpam-4933	47	8	see	see	VERB
ejpam-4933	47	9	[	[	X
ejpam-4933	47	10	2	2	NUM
ejpam-4933	47	11	]	]	NUM
ejpam-4933	47	12	)	)	PUNCT
ejpam-4933	47	13	.	.	PUNCT
ejpam-4933	48	1	a	a	DET
ejpam-4933	48	2	nonempty	nonempty	NOUN
ejpam-4933	48	3	subset	subset	VERB
ejpam-4933	48	4	s	s	NOUN
ejpam-4933	48	5	of	of	ADP
ejpam-4933	48	6	x	x	PRON
ejpam-4933	48	7	is	be	AUX
ejpam-4933	48	8	called	call	VERB
ejpam-4933	48	9	a	a	DET
ejpam-4933	48	10	subalgebra	subalgebra	NOUN
ejpam-4933	48	11	of	of	ADP
ejpam-4933	48	12	a	a	DET
ejpam-4933	48	13	bck	bck	VERB
ejpam-4933	48	14	/	/	SYM
ejpam-4933	48	15	bci	bci	NOUN
ejpam-4933	48	16	-	-	NOUN
ejpam-4933	48	17	algebra	algebra	NOUN
ejpam-4933	48	18	(	(	PUNCT
ejpam-4933	48	19	x	x	X
ejpam-4933	48	20	,	,	PUNCT
ejpam-4933	48	21	∗	∗	NOUN
ejpam-4933	48	22	,	,	PUNCT
ejpam-4933	48	23	0	0	NUM
ejpam-4933	48	24	)	)	PUNCT
ejpam-4933	48	25	(	(	PUNCT
ejpam-4933	48	26	see	see	VERB
ejpam-4933	48	27	[	[	X
ejpam-4933	48	28	11	11	NUM
ejpam-4933	48	29	]	]	SYM
ejpam-4933	48	30	)	)	PUNCT
ejpam-4933	48	31	if	if	SCONJ
ejpam-4933	48	32	a	a	DET
ejpam-4933	48	33	∗	∗	X
ejpam-4933	48	34	y	y	PROPN
ejpam-4933	48	35	∈	∈	PROPN
ejpam-4933	48	36	s	s	PROPN
ejpam-4933	48	37	for	for	ADP
ejpam-4933	48	38	all	all	DET
ejpam-4933	48	39	a	a	PRON
ejpam-4933	48	40	,	,	PUNCT
ejpam-4933	48	41	y	y	PROPN
ejpam-4933	48	42	∈	∈	PROPN
ejpam-4933	48	43	s.	s.	PROPN
ejpam-4933	48	44	a	a	DET
ejpam-4933	48	45	subset	subset	VERB
ejpam-4933	48	46	a	a	PRON
ejpam-4933	48	47	of	of	ADP
ejpam-4933	48	48	x	x	PRON
ejpam-4933	48	49	is	be	AUX
ejpam-4933	48	50	called	call	VERB
ejpam-4933	48	51	an	an	DET
ejpam-4933	48	52	ideal	ideal	NOUN
ejpam-4933	48	53	of	of	ADP
ejpam-4933	48	54	a	a	DET
ejpam-4933	48	55	bck	bck	PROPN
ejpam-4933	48	56	/	/	SYM
ejpam-4933	48	57	bci	bci	NOUN
ejpam-4933	48	58	-	-	NOUN
ejpam-4933	48	59	algebra	algebra	NOUN
ejpam-4933	48	60	(	(	PUNCT
ejpam-4933	48	61	x	x	X
ejpam-4933	48	62	,	,	PUNCT
ejpam-4933	48	63	∗	∗	NOUN
ejpam-4933	48	64	,	,	PUNCT
ejpam-4933	48	65	0	0	NUM
ejpam-4933	48	66	)	)	PUNCT
ejpam-4933	48	67	(	(	PUNCT
ejpam-4933	48	68	see	see	VERB
ejpam-4933	48	69	[	[	X
ejpam-4933	48	70	11	11	NUM
ejpam-4933	48	71	]	]	SYM
ejpam-4933	48	72	)	)	PUNCT
ejpam-4933	48	73	if	if	SCONJ
ejpam-4933	48	74	it	it	PRON
ejpam-4933	48	75	satisfies	satisfy	VERB
ejpam-4933	48	76	:	:	PUNCT
ejpam-4933	48	77	0	0	NUM
ejpam-4933	48	78	∈	∈	PROPN
ejpam-4933	48	79	a	a	PRON
ejpam-4933	48	80	,	,	PUNCT
ejpam-4933	48	81	(	(	PUNCT
ejpam-4933	48	82	7	7	NUM
ejpam-4933	48	83	)	)	PUNCT
ejpam-4933	48	84	(	(	PUNCT
ejpam-4933	48	85	∀a	∀a	NOUN
ejpam-4933	48	86	∈	∈	NOUN
ejpam-4933	48	87	x	x	NOUN
ejpam-4933	48	88	)	)	PUNCT
ejpam-4933	48	89	(	(	PUNCT
ejpam-4933	48	90	∀y	∀y	PROPN
ejpam-4933	48	91	∈	∈	PROPN
ejpam-4933	48	92	a	a	NOUN
ejpam-4933	48	93	)	)	PUNCT
ejpam-4933	48	94	(	(	PUNCT
ejpam-4933	48	95	a	a	DET
ejpam-4933	48	96	∗	∗	X
ejpam-4933	48	97	y	y	PROPN
ejpam-4933	48	98	∈	∈	PROPN
ejpam-4933	48	99	a	a	DET
ejpam-4933	48	100	⇒	⇒	NOUN
ejpam-4933	48	101	a	a	DET
ejpam-4933	48	102	∈	∈	PROPN
ejpam-4933	48	103	a	a	PRON
ejpam-4933	48	104	)	)	PUNCT
ejpam-4933	48	105	.	.	PUNCT
ejpam-4933	49	1	(	(	PUNCT
ejpam-4933	49	2	8)	8)	NUM
ejpam-4933	49	3	an	an	DET
ejpam-4933	49	4	ideal	ideal	NOUN
ejpam-4933	49	5	a	a	PRON
ejpam-4933	49	6	of	of	ADP
ejpam-4933	49	7	a	a	DET
ejpam-4933	49	8	bci	bci	NOUN
ejpam-4933	49	9	-	-	NOUN
ejpam-4933	49	10	algebra	algebra	NOUN
ejpam-4933	49	11	(	(	PUNCT
ejpam-4933	49	12	x	x	X
ejpam-4933	49	13	,	,	PUNCT
ejpam-4933	49	14	∗	∗	NOUN
ejpam-4933	49	15	,	,	PUNCT
ejpam-4933	49	16	0	0	NUM
ejpam-4933	49	17	)	)	PUNCT
ejpam-4933	49	18	is	be	AUX
ejpam-4933	49	19	said	say	VERB
ejpam-4933	49	20	to	to	PART
ejpam-4933	49	21	be	be	AUX
ejpam-4933	49	22	closed	close	VERB
ejpam-4933	49	23	(	(	PUNCT
ejpam-4933	49	24	see	see	VERB
ejpam-4933	49	25	[	[	X
ejpam-4933	49	26	2	2	NUM
ejpam-4933	49	27	,	,	PUNCT
ejpam-4933	49	28	11	11	NUM
ejpam-4933	49	29	]	]	PUNCT
ejpam-4933	49	30	)	)	PUNCT
ejpam-4933	49	31	if	if	SCONJ
ejpam-4933	49	32	it	it	PRON
ejpam-4933	49	33	is	be	AUX
ejpam-4933	49	34	also	also	ADV
ejpam-4933	49	35	a	a	DET
ejpam-4933	49	36	subalgebra	subalgebra	NOUN
ejpam-4933	49	37	of	of	ADP
ejpam-4933	49	38	(	(	PUNCT
ejpam-4933	49	39	x	x	NOUN
ejpam-4933	49	40	,	,	PUNCT
ejpam-4933	49	41	∗	∗	NOUN
ejpam-4933	49	42	,	,	PUNCT
ejpam-4933	49	43	0	0	NUM
ejpam-4933	49	44	)	)	PUNCT
ejpam-4933	49	45	.	.	PUNCT
ejpam-4933	50	1	note	note	VERB
ejpam-4933	50	2	that	that	SCONJ
ejpam-4933	50	3	an	an	DET
ejpam-4933	50	4	ideal	ideal	NOUN
ejpam-4933	50	5	a	a	PRON
ejpam-4933	50	6	of	of	ADP
ejpam-4933	50	7	a	a	DET
ejpam-4933	50	8	bci	bci	NOUN
ejpam-4933	50	9	-	-	NOUN
ejpam-4933	50	10	algebra	algebra	NOUN
ejpam-4933	50	11	(	(	PUNCT
ejpam-4933	50	12	x	x	X
ejpam-4933	50	13	,	,	PUNCT
ejpam-4933	50	14	∗	∗	NOUN
ejpam-4933	50	15	,	,	PUNCT
ejpam-4933	50	16	0	0	NUM
ejpam-4933	50	17	)	)	PUNCT
ejpam-4933	50	18	is	be	AUX
ejpam-4933	50	19	closed	close	VERB
ejpam-4933	50	20	if	if	SCONJ
ejpam-4933	50	21	and	and	CCONJ
ejpam-4933	50	22	only	only	ADV
ejpam-4933	50	23	if	if	SCONJ
ejpam-4933	50	24	0	0	NUM
ejpam-4933	50	25	∗	∗	VERB
ejpam-4933	50	26	a	a	DET
ejpam-4933	50	27	∈	∈	PROPN
ejpam-4933	50	28	a	a	PRON
ejpam-4933	50	29	for	for	ADP
ejpam-4933	50	30	all	all	DET
ejpam-4933	50	31	a	a	DET
ejpam-4933	50	32	∈	∈	PROPN
ejpam-4933	50	33	a	a	DET
ejpam-4933	50	34	(	(	PUNCT
ejpam-4933	50	35	see	see	VERB
ejpam-4933	50	36	[	[	X
ejpam-4933	50	37	2	2	NUM
ejpam-4933	50	38	,	,	PUNCT
ejpam-4933	50	39	proposition	proposition	NOUN
ejpam-4933	50	40	1.4.4	1.4.4	NUM
ejpam-4933	50	41	]	]	PUNCT
ejpam-4933	50	42	)	)	PUNCT
ejpam-4933	50	43	.	.	PUNCT
ejpam-4933	51	1	every	every	DET
ejpam-4933	51	2	ideal	ideal	NOUN
ejpam-4933	51	3	a	a	PRON
ejpam-4933	51	4	of	of	ADP
ejpam-4933	51	5	a	a	DET
ejpam-4933	51	6	bck	bck	PROPN
ejpam-4933	51	7	/	/	SYM
ejpam-4933	51	8	bci	bci	NOUN
ejpam-4933	51	9	-	-	NOUN
ejpam-4933	51	10	algebra	algebra	NOUN
ejpam-4933	51	11	(	(	PUNCT
ejpam-4933	51	12	x	x	X
ejpam-4933	51	13	,	,	PUNCT
ejpam-4933	51	14	∗	∗	NOUN
ejpam-4933	51	15	,	,	PUNCT
ejpam-4933	51	16	0	0	NUM
ejpam-4933	51	17	)	)	PUNCT
ejpam-4933	51	18	satisfies	satisfy	VERB
ejpam-4933	51	19	the	the	DET
ejpam-4933	51	20	next	next	ADJ
ejpam-4933	51	21	assertion	assertion	NOUN
ejpam-4933	51	22	.	.	PUNCT
ejpam-4933	52	1	(	(	PUNCT
ejpam-4933	52	2	∀a	∀a	NOUN
ejpam-4933	52	3	,	,	PUNCT
ejpam-4933	52	4	y	y	PROPN
ejpam-4933	52	5	∈	∈	PROPN
ejpam-4933	52	6	x	x	X
ejpam-4933	52	7	)	)	PUNCT
ejpam-4933	52	8	(	(	PUNCT
ejpam-4933	52	9	a	a	DET
ejpam-4933	52	10	≤x	≤x	PROPN
ejpam-4933	52	11	y	y	PROPN
ejpam-4933	52	12	,	,	PUNCT
ejpam-4933	52	13	y	y	PROPN
ejpam-4933	52	14	∈	∈	PROPN
ejpam-4933	52	15	a	a	DET
ejpam-4933	52	16	⇒	⇒	NOUN
ejpam-4933	52	17	a	a	DET
ejpam-4933	52	18	∈	∈	PROPN
ejpam-4933	52	19	a	a	PRON
ejpam-4933	52	20	)	)	PUNCT
ejpam-4933	52	21	.	.	PUNCT
ejpam-4933	53	1	(	(	PUNCT
ejpam-4933	53	2	9	9	X
ejpam-4933	53	3	)	)	PUNCT
ejpam-4933	53	4	a	a	DET
ejpam-4933	53	5	fuzzy	fuzzy	ADJ
ejpam-4933	53	6	set	set	NOUN
ejpam-4933	53	7	in	in	ADP
ejpam-4933	53	8	a	a	DET
ejpam-4933	53	9	set	set	NOUN
ejpam-4933	53	10	x	x	PUNCT
ejpam-4933	53	11	is	be	AUX
ejpam-4933	53	12	defined	define	VERB
ejpam-4933	53	13	to	to	PART
ejpam-4933	53	14	be	be	AUX
ejpam-4933	53	15	a	a	DET
ejpam-4933	53	16	function	function	NOUN
ejpam-4933	53	17	ζ	ζ	NOUN
ejpam-4933	53	18	:	:	PUNCT
ejpam-4933	53	19	x	x	SYM
ejpam-4933	54	1	→	→	SYM
ejpam-4933	54	2	[	[	X
ejpam-4933	54	3	0	0	NUM
ejpam-4933	54	4	,	,	PUNCT
ejpam-4933	54	5	1	1	NUM
ejpam-4933	54	6	]	]	PUNCT
ejpam-4933	54	7	.	.	PUNCT
ejpam-4933	55	1	denote	denote	VERB
ejpam-4933	55	2	by	by	ADP
ejpam-4933	55	3	fs(x	fs(x	NOUN
ejpam-4933	55	4	)	)	PUNCT
ejpam-4933	55	5	the	the	DET
ejpam-4933	55	6	collection	collection	NOUN
ejpam-4933	55	7	of	of	ADP
ejpam-4933	55	8	all	all	DET
ejpam-4933	55	9	fuzzy	fuzzy	ADJ
ejpam-4933	55	10	sets	set	NOUN
ejpam-4933	55	11	in	in	ADP
ejpam-4933	55	12	x.	x.	NOUN
ejpam-4933	55	13	define	define	VERB
ejpam-4933	55	14	a	a	DET
ejpam-4933	55	15	relation	relation	NOUN
ejpam-4933	55	16	“	"	PUNCT
ejpam-4933	55	17	⊆	⊆	NUM
ejpam-4933	55	18	”	"	PUNCT
ejpam-4933	55	19	on	on	ADP
ejpam-4933	55	20	fs(x	fs(x	NOUN
ejpam-4933	55	21	)	)	PUNCT
ejpam-4933	55	22	by	by	ADP
ejpam-4933	55	23	(	(	PUNCT
ejpam-4933	55	24	∀ζ	∀ζ	PROPN
ejpam-4933	55	25	,	,	PUNCT
ejpam-4933	55	26	ξ	ξ	PROPN
ejpam-4933	55	27	∈	∈	NOUN
ejpam-4933	55	28	fs(x))(ζ	fs(x))(ζ	VERB
ejpam-4933	55	29	⊆	⊆	SYM
ejpam-4933	55	30	ξ	ξ	PROPN
ejpam-4933	55	31	⇔	⇔	X
ejpam-4933	55	32	(	(	PUNCT
ejpam-4933	55	33	∀a	∀a	NOUN
ejpam-4933	55	34	∈	∈	NOUN
ejpam-4933	55	35	x)(ζ(a	x)(ζ(a	NOUN
ejpam-4933	55	36	)	)	PUNCT
ejpam-4933	55	37	≤	≤	NOUN
ejpam-4933	55	38	ξ(a	ξ(a	NUM
ejpam-4933	55	39	)	)	PUNCT
ejpam-4933	55	40	)	)	PUNCT
ejpam-4933	55	41	)	)	PUNCT
ejpam-4933	55	42	.	.	PUNCT
ejpam-4933	56	1	the	the	DET
ejpam-4933	56	2	join	join	NOUN
ejpam-4933	56	3	(	(	PUNCT
ejpam-4933	56	4	∨	∨	NOUN
ejpam-4933	56	5	)	)	PUNCT
ejpam-4933	56	6	and	and	CCONJ
ejpam-4933	56	7	meet	meet	VERB
ejpam-4933	56	8	(	(	PUNCT
ejpam-4933	56	9	∧	∧	PROPN
ejpam-4933	56	10	)	)	PUNCT
ejpam-4933	56	11	of	of	ADP
ejpam-4933	56	12	ζ	ζ	NOUN
ejpam-4933	56	13	and	and	CCONJ
ejpam-4933	56	14	ξ	ξ	PROPN
ejpam-4933	56	15	are	be	AUX
ejpam-4933	56	16	defined	define	VERB
ejpam-4933	56	17	by	by	ADP
ejpam-4933	56	18	(	(	PUNCT
ejpam-4933	56	19	ζ	ζ	NOUN
ejpam-4933	56	20	∨	∨	NUM
ejpam-4933	56	21	ξ)(a	ξ)(a	NUM
ejpam-4933	56	22	)	)	PUNCT
ejpam-4933	56	23	=	=	SYM
ejpam-4933	56	24	max{ζ(a	max{ζ(a	PROPN
ejpam-4933	56	25	)	)	PUNCT
ejpam-4933	56	26	,	,	PUNCT
ejpam-4933	56	27	ξ(a	ξ(a	NUM
ejpam-4933	56	28	)	)	PUNCT
ejpam-4933	56	29	}	}	PUNCT
ejpam-4933	56	30	,	,	PUNCT
ejpam-4933	56	31	(	(	PUNCT
ejpam-4933	56	32	ζ	ζ	NOUN
ejpam-4933	56	33	∧	∧	PROPN
ejpam-4933	56	34	ξ)(a	ξ)(a	NUM
ejpam-4933	56	35	)	)	PUNCT
ejpam-4933	56	36	=	=	PUNCT
ejpam-4933	56	37	min{ζ(a	min{ζ(a	PROPN
ejpam-4933	56	38	)	)	PUNCT
ejpam-4933	56	39	,	,	PUNCT
ejpam-4933	56	40	ξ(a	ξ(a	NUM
ejpam-4933	56	41	)	)	PUNCT
ejpam-4933	56	42	}	}	PUNCT
ejpam-4933	56	43	,	,	PUNCT
ejpam-4933	56	44	respectively	respectively	ADV
ejpam-4933	56	45	,	,	PUNCT
ejpam-4933	56	46	for	for	ADP
ejpam-4933	56	47	all	all	DET
ejpam-4933	56	48	a	a	DET
ejpam-4933	56	49	∈	∈	NOUN
ejpam-4933	56	50	x.	x.	NOUN
ejpam-4933	57	1	the	the	DET
ejpam-4933	57	2	complement	complement	NOUN
ejpam-4933	57	3	of	of	ADP
ejpam-4933	57	4	ζ	ζ	NOUN
ejpam-4933	57	5	,	,	PUNCT
ejpam-4933	57	6	denoted	denote	VERB
ejpam-4933	57	7	by	by	ADP
ejpam-4933	57	8	ζc	ζc	NOUN
ejpam-4933	57	9	,	,	PUNCT
ejpam-4933	57	10	is	be	AUX
ejpam-4933	57	11	defined	define	VERB
ejpam-4933	57	12	by	by	ADP
ejpam-4933	57	13	(	(	PUNCT
ejpam-4933	57	14	∀a	∀a	NOUN
ejpam-4933	57	15	∈	∈	NOUN
ejpam-4933	57	16	x)(ζc(a	x)(ζc(a	PUNCT
ejpam-4933	57	17	)	)	PUNCT
ejpam-4933	57	18	=	=	SYM
ejpam-4933	58	1	1	1	NUM
ejpam-4933	58	2	−	−	NOUN
ejpam-4933	58	3	ζ(a	ζ(a	NOUN
ejpam-4933	58	4	)	)	PUNCT
ejpam-4933	58	5	)	)	PUNCT
ejpam-4933	58	6	.	.	PUNCT
ejpam-4933	59	1	a	a	DET
ejpam-4933	59	2	fuzzy	fuzzy	ADJ
ejpam-4933	59	3	set	set	VERB
ejpam-4933	59	4	ζ	ζ	NOUN
ejpam-4933	59	5	in	in	ADP
ejpam-4933	59	6	a	a	DET
ejpam-4933	59	7	set	set	NOUN
ejpam-4933	59	8	x	x	X
ejpam-4933	59	9	of	of	ADP
ejpam-4933	59	10	the	the	DET
ejpam-4933	59	11	form	form	NOUN
ejpam-4933	59	12	ζ(b	ζ(b	NOUN
ejpam-4933	59	13	)	)	PUNCT
ejpam-4933	59	14	:	:	PUNCT
ejpam-4933	60	1	=	=	X
ejpam-4933	60	2	{	{	PUNCT
ejpam-4933	60	3	t	t	PROPN
ejpam-4933	60	4	∈	∈	PROPN
ejpam-4933	60	5	(	(	PUNCT
ejpam-4933	60	6	0	0	NUM
ejpam-4933	60	7	,	,	PUNCT
ejpam-4933	60	8	1	1	NUM
ejpam-4933	60	9	]	]	PUNCT
ejpam-4933	60	10	if	if	SCONJ
ejpam-4933	60	11	b	b	X
ejpam-4933	60	12	=	=	SYM
ejpam-4933	60	13	a	a	PROPN
ejpam-4933	60	14	,	,	PUNCT
ejpam-4933	60	15	0	0	PUNCT
ejpam-4933	60	16	if	if	SCONJ
ejpam-4933	60	17	b	b	X
ejpam-4933	60	18	̸=	̸=	PROPN
ejpam-4933	60	19	a	a	PRON
ejpam-4933	60	20	,	,	PUNCT
ejpam-4933	60	21	is	be	AUX
ejpam-4933	60	22	said	say	VERB
ejpam-4933	60	23	to	to	PART
ejpam-4933	60	24	be	be	AUX
ejpam-4933	60	25	a	a	DET
ejpam-4933	60	26	fuzzy	fuzzy	ADJ
ejpam-4933	60	27	point	point	NOUN
ejpam-4933	60	28	with	with	ADP
ejpam-4933	60	29	support	support	NOUN
ejpam-4933	60	30	a	a	PRON
ejpam-4933	60	31	and	and	CCONJ
ejpam-4933	60	32	value	value	NOUN
ejpam-4933	60	33	t	t	NOUN
ejpam-4933	60	34	and	and	CCONJ
ejpam-4933	60	35	is	be	AUX
ejpam-4933	60	36	denoted	denote	VERB
ejpam-4933	60	37	by	by	ADP
ejpam-4933	60	38	⟨at⟩.	⟨at⟩.	PUNCT
ejpam-4933	60	39	for	for	ADP
ejpam-4933	60	40	a	a	DET
ejpam-4933	60	41	fuzzy	fuzzy	ADJ
ejpam-4933	60	42	set	set	VERB
ejpam-4933	60	43	ζ	ζ	NOUN
ejpam-4933	60	44	in	in	ADP
ejpam-4933	60	45	a	a	DET
ejpam-4933	60	46	set	set	NOUN
ejpam-4933	60	47	x	x	NOUN
ejpam-4933	60	48	,	,	PUNCT
ejpam-4933	60	49	we	we	PRON
ejpam-4933	60	50	say	say	VERB
ejpam-4933	60	51	that	that	SCONJ
ejpam-4933	60	52	a	a	DET
ejpam-4933	60	53	fuzzy	fuzzy	ADJ
ejpam-4933	60	54	point	point	NOUN
ejpam-4933	60	55	⟨at⟩	⟨at⟩	PROPN
ejpam-4933	60	56	is	be	AUX
ejpam-4933	60	57	(	(	PUNCT
ejpam-4933	60	58	i	i	NOUN
ejpam-4933	60	59	)	)	PUNCT
ejpam-4933	60	60	contained	contain	VERB
ejpam-4933	60	61	in	in	ADP
ejpam-4933	60	62	ζ	ζ	NOUN
ejpam-4933	60	63	,	,	PUNCT
ejpam-4933	60	64	denoted	denote	VERB
ejpam-4933	60	65	by	by	ADP
ejpam-4933	60	66	⟨at⟩	⟨at⟩	PROPN
ejpam-4933	60	67	∈	∈	PROPN
ejpam-4933	60	68	ζ	ζ	PROPN
ejpam-4933	60	69	,	,	PUNCT
ejpam-4933	60	70	(	(	PUNCT
ejpam-4933	60	71	see	see	VERB
ejpam-4933	60	72	[	[	X
ejpam-4933	60	73	12	12	NUM
ejpam-4933	60	74	]	]	SYM
ejpam-4933	60	75	)	)	PUNCT
ejpam-4933	60	76	if	if	SCONJ
ejpam-4933	60	77	ζ(a	ζ(a	PRON
ejpam-4933	60	78	)	)	PUNCT
ejpam-4933	60	79	≥	≥	PROPN
ejpam-4933	60	80	t.	t.	PROPN
ejpam-4933	60	81	(	(	PUNCT
ejpam-4933	60	82	ii	ii	NOUN
ejpam-4933	60	83	)	)	PUNCT
ejpam-4933	60	84	quasi	quasi	NOUN
ejpam-4933	60	85	-	-	VERB
ejpam-4933	60	86	coincident	coincident	ADJ
ejpam-4933	60	87	with	with	ADP
ejpam-4933	60	88	ζ	ζ	NOUN
ejpam-4933	60	89	,	,	PUNCT
ejpam-4933	60	90	denoted	denote	VERB
ejpam-4933	60	91	by	by	ADP
ejpam-4933	60	92	⟨at⟩	⟨at⟩	PROPN
ejpam-4933	60	93	q	q	PROPN
ejpam-4933	60	94	ζ	ζ	PROPN
ejpam-4933	60	95	,	,	PUNCT
ejpam-4933	60	96	(	(	PUNCT
ejpam-4933	60	97	see	see	VERB
ejpam-4933	60	98	[	[	X
ejpam-4933	60	99	12	12	NUM
ejpam-4933	60	100	]	]	SYM
ejpam-4933	60	101	)	)	PUNCT
ejpam-4933	60	102	if	if	SCONJ
ejpam-4933	60	103	ζ(a	ζ(a	PRON
ejpam-4933	60	104	)	)	PUNCT
ejpam-4933	61	1	+	+	CCONJ
ejpam-4933	61	2	t	t	X
ejpam-4933	61	3	>	>	X
ejpam-4933	61	4	1	1	X
ejpam-4933	61	5	.	.	PUNCT
ejpam-4933	61	6	e.	e.	PROPN
ejpam-4933	61	7	h.	h.	PROPN
ejpam-4933	61	8	roh	roh	PROPN
ejpam-4933	61	9	,	,	PUNCT
ejpam-4933	61	10	e.	e.	PROPN
ejpam-4933	61	11	yang	yang	PROPN
ejpam-4933	61	12	,	,	PUNCT
ejpam-4933	61	13	y.	y.	PROPN
ejpam-4933	61	14	b.	b.	PROPN
ejpam-4933	61	15	jun	jun	PROPN
ejpam-4933	61	16	/	/	SYM
ejpam-4933	61	17	eur	eur	PROPN
ejpam-4933	61	18	.	.	PUNCT
ejpam-4933	62	1	j.	j.	PROPN
ejpam-4933	62	2	pure	pure	PROPN
ejpam-4933	62	3	appl	appl	PROPN
ejpam-4933	62	4	.	.	PROPN
ejpam-4933	62	5	math	math	PROPN
ejpam-4933	62	6	,	,	PUNCT
ejpam-4933	62	7	16	16	NUM
ejpam-4933	62	8	(	(	PUNCT
ejpam-4933	62	9	4	4	NUM
ejpam-4933	62	10	)	)	PUNCT
ejpam-4933	62	11	(	(	PUNCT
ejpam-4933	62	12	2023	2023	NUM
ejpam-4933	62	13	)	)	PUNCT
ejpam-4933	62	14	,	,	PUNCT
ejpam-4933	62	15	2009	2009	NUM
ejpam-4933	62	16	-	-	SYM
ejpam-4933	62	17	2024	2024	NUM
ejpam-4933	62	18	2012	2012	NUM
ejpam-4933	62	19	if	if	SCONJ
ejpam-4933	62	20	a	a	DET
ejpam-4933	62	21	fuzzy	fuzzy	ADJ
ejpam-4933	62	22	point	point	NOUN
ejpam-4933	62	23	⟨at⟩	⟨at⟩	PROPN
ejpam-4933	62	24	is	be	AUX
ejpam-4933	62	25	contained	contain	VERB
ejpam-4933	62	26	in	in	ADP
ejpam-4933	62	27	ζ	ζ	NOUN
ejpam-4933	62	28	or	or	CCONJ
ejpam-4933	62	29	is	be	AUX
ejpam-4933	62	30	quasi	quasi	ADJ
ejpam-4933	62	31	-	-	ADJ
ejpam-4933	62	32	coincident	coincident	ADJ
ejpam-4933	62	33	with	with	ADP
ejpam-4933	62	34	ζ	ζ	NOUN
ejpam-4933	62	35	,	,	PUNCT
ejpam-4933	62	36	we	we	PRON
ejpam-4933	62	37	denote	denote	VERB
ejpam-4933	62	38	it	it	PRON
ejpam-4933	62	39	⟨at⟩	⟨at⟩	PROPN
ejpam-4933	63	1	∈∨q	∈∨q	NUM
ejpam-4933	63	2	ζ	ζ	PROPN
ejpam-4933	63	3	.	.	PUNCT
ejpam-4933	64	1	if	if	SCONJ
ejpam-4933	64	2	⟨at⟩α	⟨at⟩α	NOUN
ejpam-4933	64	3	ζ	ζ	NOUN
ejpam-4933	64	4	is	be	AUX
ejpam-4933	64	5	not	not	PART
ejpam-4933	64	6	established	establish	VERB
ejpam-4933	64	7	for	for	ADP
ejpam-4933	64	8	α	α	PRON
ejpam-4933	64	9	∈	∈	PROPN
ejpam-4933	64	10	{	{	PUNCT
ejpam-4933	64	11	∈	∈	PROPN
ejpam-4933	64	12	,	,	PUNCT
ejpam-4933	64	13	q,∈∨q	q,∈∨q	PROPN
ejpam-4933	64	14	}	}	PUNCT
ejpam-4933	64	15	,	,	PUNCT
ejpam-4933	64	16	it	it	PRON
ejpam-4933	64	17	is	be	AUX
ejpam-4933	64	18	denoted	denote	VERB
ejpam-4933	64	19	by	by	ADP
ejpam-4933	64	20	⟨at⟩α	⟨at⟩α	DET
ejpam-4933	64	21	ζ	ζ	NOUN
ejpam-4933	64	22	.	.	PUNCT
ejpam-4933	65	1	given	give	VERB
ejpam-4933	65	2	t	t	PROPN
ejpam-4933	65	3	∈	∈	PROPN
ejpam-4933	65	4	(	(	PUNCT
ejpam-4933	65	5	0	0	NUM
ejpam-4933	65	6	,	,	PUNCT
ejpam-4933	65	7	1	1	NUM
ejpam-4933	65	8	]	]	PUNCT
ejpam-4933	65	9	and	and	CCONJ
ejpam-4933	65	10	a	a	DET
ejpam-4933	65	11	fuzzy	fuzzy	ADJ
ejpam-4933	65	12	set	set	VERB
ejpam-4933	65	13	ζ	ζ	NOUN
ejpam-4933	65	14	in	in	ADP
ejpam-4933	65	15	a	a	DET
ejpam-4933	65	16	set	set	NOUN
ejpam-4933	65	17	x	x	NOUN
ejpam-4933	65	18	,	,	PUNCT
ejpam-4933	65	19	consider	consider	VERB
ejpam-4933	65	20	the	the	DET
ejpam-4933	65	21	following	follow	VERB
ejpam-4933	65	22	sets	set	NOUN
ejpam-4933	65	23	(	(	PUNCT
ejpam-4933	65	24	ζ	ζ	NOUN
ejpam-4933	65	25	,	,	PUNCT
ejpam-4933	65	26	t)∈	t)∈	NUM
ejpam-4933	65	27	:	:	PUNCT
ejpam-4933	65	28	=	=	X
ejpam-4933	65	29	{	{	PUNCT
ejpam-4933	65	30	a	a	DET
ejpam-4933	65	31	∈	∈	NOUN
ejpam-4933	65	32	x	x	PUNCT
ejpam-4933	65	33	|	|	ADV
ejpam-4933	65	34	⟨at⟩	⟨at⟩	X
ejpam-4933	65	35	∈	∈	PROPN
ejpam-4933	65	36	ζ	ζ	PROPN
ejpam-4933	65	37	}	}	PUNCT
ejpam-4933	65	38	and	and	CCONJ
ejpam-4933	65	39	(	(	PUNCT
ejpam-4933	65	40	ζ	ζ	NOUN
ejpam-4933	65	41	,	,	PUNCT
ejpam-4933	65	42	t)q	t)q	PUNCT
ejpam-4933	65	43	:	:	PUNCT
ejpam-4933	65	44	=	=	X
ejpam-4933	65	45	{	{	PUNCT
ejpam-4933	65	46	a	a	DET
ejpam-4933	65	47	∈	∈	NOUN
ejpam-4933	65	48	x	x	PUNCT
ejpam-4933	65	49	|	|	ADV
ejpam-4933	65	50	⟨at⟩	⟨at⟩	X
ejpam-4933	65	51	q	q	PROPN
ejpam-4933	65	52	ζ	ζ	PROPN
ejpam-4933	65	53	}	}	PUNCT
ejpam-4933	65	54	which	which	PRON
ejpam-4933	65	55	are	be	AUX
ejpam-4933	65	56	called	call	VERB
ejpam-4933	65	57	the	the	DET
ejpam-4933	65	58	level	level	NOUN
ejpam-4933	65	59	set	set	VERB
ejpam-4933	65	60	and	and	CCONJ
ejpam-4933	65	61	the	the	DET
ejpam-4933	65	62	q	q	NOUN
ejpam-4933	65	63	-	-	PUNCT
ejpam-4933	65	64	set	set	NOUN
ejpam-4933	65	65	of	of	ADP
ejpam-4933	65	66	ζ	ζ	NOUN
ejpam-4933	65	67	related	relate	VERB
ejpam-4933	65	68	to	to	ADP
ejpam-4933	65	69	t	t	PROPN
ejpam-4933	65	70	,	,	PUNCT
ejpam-4933	65	71	respectively	respectively	ADV
ejpam-4933	65	72	,	,	PUNCT
ejpam-4933	65	73	in	in	ADP
ejpam-4933	65	74	x.	x.	NOUN
ejpam-4933	65	75	also	also	ADV
ejpam-4933	65	76	,	,	PUNCT
ejpam-4933	65	77	we	we	PRON
ejpam-4933	65	78	consider	consider	VERB
ejpam-4933	65	79	the	the	DET
ejpam-4933	65	80	set	set	NOUN
ejpam-4933	65	81	(	(	PUNCT
ejpam-4933	65	82	ζ	ζ	NOUN
ejpam-4933	65	83	,	,	PUNCT
ejpam-4933	65	84	t)∈∨q	t)∈∨q	NOUN
ejpam-4933	65	85	:	:	PUNCT
ejpam-4933	65	86	=	=	SYM
ejpam-4933	65	87	{	{	PUNCT
ejpam-4933	65	88	a	a	DET
ejpam-4933	65	89	∈	∈	NOUN
ejpam-4933	65	90	x	x	PUNCT
ejpam-4933	65	91	|	|	ADV
ejpam-4933	65	92	⟨at⟩	⟨at⟩	X
ejpam-4933	65	93	∈∨q	∈∨q	NUM
ejpam-4933	65	94	ζ	ζ	PROPN
ejpam-4933	65	95	}	}	PUNCT
ejpam-4933	65	96	which	which	PRON
ejpam-4933	65	97	is	be	AUX
ejpam-4933	65	98	called	call	VERB
ejpam-4933	65	99	the	the	DET
ejpam-4933	65	100	∈∨q	∈∨q	NOUN
ejpam-4933	65	101	-set	-set	PUNCT
ejpam-4933	65	102	of	of	ADP
ejpam-4933	65	103	ζ	ζ	NOUN
ejpam-4933	65	104	related	relate	VERB
ejpam-4933	65	105	to	to	ADP
ejpam-4933	65	106	t.	t.	NOUN
ejpam-4933	65	107	it	it	PRON
ejpam-4933	65	108	is	be	AUX
ejpam-4933	65	109	clear	clear	ADJ
ejpam-4933	65	110	that	that	SCONJ
ejpam-4933	65	111	(	(	PUNCT
ejpam-4933	65	112	ζ	ζ	NOUN
ejpam-4933	65	113	,	,	PUNCT
ejpam-4933	65	114	t)∈∨q	t)∈∨q	NOUN
ejpam-4933	65	115	=	=	PUNCT
ejpam-4933	65	116	(	(	PUNCT
ejpam-4933	65	117	ζ	ζ	NOUN
ejpam-4933	65	118	,	,	PUNCT
ejpam-4933	65	119	t)∈	t)∈	NUM
ejpam-4933	65	120	∪	∪	X
ejpam-4933	65	121	(	(	PUNCT
ejpam-4933	65	122	ζ	ζ	NOUN
ejpam-4933	65	123	,	,	PUNCT
ejpam-4933	65	124	t)q	t)q	PUNCT
ejpam-4933	65	125	and	and	CCONJ
ejpam-4933	65	126	(	(	PUNCT
ejpam-4933	65	127	ζ	ζ	NOUN
ejpam-4933	65	128	,	,	PUNCT
ejpam-4933	65	129	t)q	t)q	PUNCT
ejpam-4933	65	130	⊆	⊆	NUM
ejpam-4933	65	131	(	(	PUNCT
ejpam-4933	65	132	ζ	ζ	NOUN
ejpam-4933	65	133	,	,	PUNCT
ejpam-4933	65	134	s)q	s)q	PUNCT
ejpam-4933	65	135	for	for	ADP
ejpam-4933	65	136	all	all	DET
ejpam-4933	65	137	t	t	PROPN
ejpam-4933	65	138	,	,	PUNCT
ejpam-4933	65	139	s	s	PART
ejpam-4933	65	140	∈	∈	PROPN
ejpam-4933	65	141	(	(	PUNCT
ejpam-4933	65	142	0	0	NUM
ejpam-4933	65	143	,	,	PUNCT
ejpam-4933	65	144	1	1	NUM
ejpam-4933	65	145	]	]	PUNCT
ejpam-4933	65	146	with	with	ADP
ejpam-4933	65	147	t	t	PROPN
ejpam-4933	65	148	≤	≤	NUM
ejpam-4933	65	149	s.	s.	PROPN
ejpam-4933	65	150	a	a	DET
ejpam-4933	65	151	fuzzy	fuzzy	ADJ
ejpam-4933	65	152	set	set	VERB
ejpam-4933	65	153	ζ	ζ	NOUN
ejpam-4933	65	154	in	in	ADP
ejpam-4933	65	155	x	x	PROPN
ejpam-4933	65	156	is	be	AUX
ejpam-4933	65	157	called	call	VERB
ejpam-4933	65	158	a	a	DET
ejpam-4933	65	159	fuzzy	fuzzy	ADJ
ejpam-4933	65	160	subalgebra	subalgebra	NOUN
ejpam-4933	65	161	of	of	ADP
ejpam-4933	65	162	a	a	DET
ejpam-4933	65	163	bck	bck	VERB
ejpam-4933	65	164	/	/	SYM
ejpam-4933	65	165	bci	bci	NOUN
ejpam-4933	65	166	-	-	NOUN
ejpam-4933	65	167	algebra	algebra	NOUN
ejpam-4933	65	168	(	(	PUNCT
ejpam-4933	65	169	x	x	X
ejpam-4933	65	170	,	,	PUNCT
ejpam-4933	65	171	∗	∗	NOUN
ejpam-4933	65	172	,	,	PUNCT
ejpam-4933	65	173	0	0	NUM
ejpam-4933	65	174	)	)	PUNCT
ejpam-4933	65	175	(	(	PUNCT
ejpam-4933	65	176	see	see	VERB
ejpam-4933	65	177	[	[	X
ejpam-4933	65	178	13	13	NUM
ejpam-4933	65	179	]	]	SYM
ejpam-4933	65	180	)	)	PUNCT
ejpam-4933	65	181	if	if	SCONJ
ejpam-4933	65	182	it	it	PRON
ejpam-4933	65	183	satisfies	satisfy	VERB
ejpam-4933	65	184	:	:	PUNCT
ejpam-4933	65	185	(	(	PUNCT
ejpam-4933	65	186	∀x	∀x	X
ejpam-4933	65	187	,	,	PUNCT
ejpam-4933	65	188	a	a	DET
ejpam-4933	65	189	∈	∈	NOUN
ejpam-4933	65	190	x)(ζ(x	x)(ζ(x	PUNCT
ejpam-4933	66	1	∗	∗	NOUN
ejpam-4933	66	2	a	a	PRON
ejpam-4933	66	3	)	)	PUNCT
ejpam-4933	66	4	≥	≥	NOUN
ejpam-4933	66	5	ζ(x	ζ(x	NOUN
ejpam-4933	66	6	)	)	PUNCT
ejpam-4933	66	7	∧	∧	PROPN
ejpam-4933	66	8	ζ(a	ζ(a	NOUN
ejpam-4933	66	9	)	)	PUNCT
ejpam-4933	66	10	)	)	PUNCT
ejpam-4933	66	11	.	.	PUNCT
ejpam-4933	67	1	(	(	PUNCT
ejpam-4933	67	2	10	10	NUM
ejpam-4933	67	3	)	)	PUNCT
ejpam-4933	67	4	a	a	DET
ejpam-4933	67	5	fuzzy	fuzzy	ADJ
ejpam-4933	67	6	set	set	VERB
ejpam-4933	67	7	ζ	ζ	NOUN
ejpam-4933	67	8	in	in	ADP
ejpam-4933	67	9	x	x	PROPN
ejpam-4933	67	10	is	be	AUX
ejpam-4933	67	11	called	call	VERB
ejpam-4933	67	12	a	a	DET
ejpam-4933	67	13	fuzzy	fuzzy	ADJ
ejpam-4933	67	14	ideal	ideal	NOUN
ejpam-4933	67	15	of	of	ADP
ejpam-4933	67	16	a	a	DET
ejpam-4933	67	17	bck	bck	PROPN
ejpam-4933	67	18	/	/	SYM
ejpam-4933	67	19	bci	bci	NOUN
ejpam-4933	67	20	-	-	NOUN
ejpam-4933	67	21	algebra	algebra	NOUN
ejpam-4933	67	22	(	(	PUNCT
ejpam-4933	67	23	x	x	X
ejpam-4933	67	24	,	,	PUNCT
ejpam-4933	67	25	∗	∗	NOUN
ejpam-4933	67	26	,	,	PUNCT
ejpam-4933	67	27	0	0	NUM
ejpam-4933	67	28	)	)	PUNCT
ejpam-4933	67	29	(	(	PUNCT
ejpam-4933	67	30	see	see	VERB
ejpam-4933	67	31	[	[	X
ejpam-4933	67	32	13	13	NUM
ejpam-4933	67	33	]	]	SYM
ejpam-4933	67	34	)	)	PUNCT
ejpam-4933	67	35	if	if	SCONJ
ejpam-4933	67	36	it	it	PRON
ejpam-4933	67	37	satisfies	satisfy	VERB
ejpam-4933	67	38	:	:	PUNCT
ejpam-4933	67	39	(	(	PUNCT
ejpam-4933	67	40	∀x	∀x	X
ejpam-4933	67	41	∈	∈	PROPN
ejpam-4933	67	42	x)(ζ(0	x)(ζ(0	NOUN
ejpam-4933	67	43	)	)	PUNCT
ejpam-4933	67	44	≥	≥	NOUN
ejpam-4933	67	45	ζ(x	ζ(x	NOUN
ejpam-4933	67	46	)	)	PUNCT
ejpam-4933	67	47	)	)	PUNCT
ejpam-4933	67	48	,	,	PUNCT
ejpam-4933	67	49	(	(	PUNCT
ejpam-4933	67	50	11	11	NUM
ejpam-4933	67	51	)	)	PUNCT
ejpam-4933	67	52	(	(	PUNCT
ejpam-4933	67	53	∀x	∀x	X
ejpam-4933	67	54	,	,	PUNCT
ejpam-4933	67	55	a	a	DET
ejpam-4933	67	56	∈	∈	PROPN
ejpam-4933	67	57	x)(ζ(x	x)(ζ(x	PUNCT
ejpam-4933	67	58	)	)	PUNCT
ejpam-4933	67	59	≥	≥	NOUN
ejpam-4933	67	60	ζ(x	ζ(x	PROPN
ejpam-4933	67	61	∗	∗	VERB
ejpam-4933	67	62	a	a	X
ejpam-4933	67	63	)	)	PUNCT
ejpam-4933	67	64	∧	∧	PROPN
ejpam-4933	67	65	ζ(a	ζ(a	NOUN
ejpam-4933	67	66	)	)	PUNCT
ejpam-4933	67	67	)	)	PUNCT
ejpam-4933	67	68	.	.	PUNCT
ejpam-4933	68	1	(	(	PUNCT
ejpam-4933	68	2	12	12	NUM
ejpam-4933	68	3	)	)	PUNCT
ejpam-4933	68	4	in	in	ADP
ejpam-4933	68	5	[	[	X
ejpam-4933	68	6	6	6	NUM
ejpam-4933	68	7	]	]	PUNCT
ejpam-4933	68	8	,	,	PUNCT
ejpam-4933	68	9	jun	jun	PROPN
ejpam-4933	68	10	introduced	introduce	VERB
ejpam-4933	68	11	the	the	DET
ejpam-4933	68	12	notion	notion	NOUN
ejpam-4933	68	13	of	of	ADP
ejpam-4933	68	14	y	y	PROPN
ejpam-4933	68	15	ε	ε	PROPN
ejpam-4933	68	16	j	j	PROPN
ejpam-4933	68	17	-fuzzy	-fuzzy	PROPN
ejpam-4933	68	18	sets	set	NOUN
ejpam-4933	68	19	based	base	VERB
ejpam-4933	68	20	on	on	ADP
ejpam-4933	68	21	the	the	DET
ejpam-4933	68	22	j	j	NOUN
ejpam-4933	68	23	-	-	NOUN
ejpam-4933	68	24	operator	operator	NOUN
ejpam-4933	68	25	in	in	ADP
ejpam-4933	68	26	the	the	DET
ejpam-4933	68	27	closed	closed	ADJ
ejpam-4933	68	28	interval	interval	NOUN
ejpam-4933	68	29	[	[	X
ejpam-4933	68	30	0	0	NUM
ejpam-4933	68	31	,	,	PUNCT
ejpam-4933	68	32	1	1	NUM
ejpam-4933	68	33	]	]	PUNCT
ejpam-4933	68	34	.	.	PUNCT
ejpam-4933	69	1	we	we	PRON
ejpam-4933	69	2	display	display	VERB
ejpam-4933	69	3	the	the	DET
ejpam-4933	69	4	basic	basic	ADJ
ejpam-4933	69	5	notions	notion	NOUN
ejpam-4933	69	6	about	about	ADP
ejpam-4933	69	7	the	the	DET
ejpam-4933	69	8	y	y	PROPN
ejpam-4933	69	9	ε	ε	PROPN
ejpam-4933	69	10	j	j	PROPN
ejpam-4933	69	11	-fuzzy	-fuzzy	PROPN
ejpam-4933	69	12	sets	set	NOUN
ejpam-4933	69	13	.	.	PUNCT
ejpam-4933	70	1	we	we	PRON
ejpam-4933	70	2	use	use	VERB
ejpam-4933	70	3	the	the	DET
ejpam-4933	70	4	notation	notation	NOUN
ejpam-4933	70	5	i	i	PRON
ejpam-4933	70	6	instead	instead	ADV
ejpam-4933	70	7	of	of	ADP
ejpam-4933	70	8	the	the	DET
ejpam-4933	70	9	closed	closed	ADJ
ejpam-4933	70	10	interval	interval	NOUN
ejpam-4933	70	11	[	[	X
ejpam-4933	70	12	0	0	NUM
ejpam-4933	70	13	,	,	PUNCT
ejpam-4933	70	14	1	1	NUM
ejpam-4933	70	15	]	]	PUNCT
ejpam-4933	70	16	.	.	PUNCT
ejpam-4933	71	1	let	let	VERB
ejpam-4933	71	2	“	"	PUNCT
ejpam-4933	71	3	≪	≪	VERB
ejpam-4933	71	4	”	"	PUNCT
ejpam-4933	71	5	be	be	AUX
ejpam-4933	71	6	the	the	DET
ejpam-4933	71	7	order	order	NOUN
ejpam-4933	71	8	relation	relation	NOUN
ejpam-4933	71	9	in	in	ADP
ejpam-4933	71	10	i2	i2	PROPN
ejpam-4933	71	11	defined	define	VERB
ejpam-4933	71	12	as	as	SCONJ
ejpam-4933	71	13	follows	follow	VERB
ejpam-4933	71	14	:	:	PUNCT
ejpam-4933	71	15	(	(	PUNCT
ejpam-4933	71	16	∀(m	∀(m	NOUN
ejpam-4933	71	17	,	,	PUNCT
ejpam-4933	71	18	n	n	CCONJ
ejpam-4933	71	19	)	)	PUNCT
ejpam-4933	71	20	,	,	PUNCT
ejpam-4933	71	21	(	(	PUNCT
ejpam-4933	71	22	j	j	NOUN
ejpam-4933	71	23	,	,	PUNCT
ejpam-4933	71	24	i	i	PROPN
ejpam-4933	71	25	)	)	PUNCT
ejpam-4933	71	26	∈	∈	PROPN
ejpam-4933	71	27	i2)((m	i2)((m	NOUN
ejpam-4933	71	28	,	,	PUNCT
ejpam-4933	71	29	n	n	CCONJ
ejpam-4933	71	30	)	)	PUNCT
ejpam-4933	71	31	≪	≪	PUNCT
ejpam-4933	71	32	(	(	PUNCT
ejpam-4933	71	33	j	j	NOUN
ejpam-4933	71	34	,	,	PUNCT
ejpam-4933	71	35	i	i	PROPN
ejpam-4933	71	36	)	)	PUNCT
ejpam-4933	71	37	⇔	⇔	PROPN
ejpam-4933	71	38	m	m	PROPN
ejpam-4933	71	39	≤	≤	PROPN
ejpam-4933	71	40	j	j	PROPN
ejpam-4933	71	41	,	,	PUNCT
ejpam-4933	71	42	n	n	CCONJ
ejpam-4933	71	43	≤	≤	NOUN
ejpam-4933	71	44	i	i	NOUN
ejpam-4933	71	45	)	)	PUNCT
ejpam-4933	71	46	for	for	ADP
ejpam-4933	71	47	every	every	DET
ejpam-4933	71	48	m	m	NOUN
ejpam-4933	71	49	,	,	PUNCT
ejpam-4933	71	50	ε	ε	PROPN
ejpam-4933	71	51	∈	∈	PROPN
ejpam-4933	72	1	i	i	PRON
ejpam-4933	72	2	,	,	PUNCT
ejpam-4933	72	3	we	we	PRON
ejpam-4933	72	4	define	define	VERB
ejpam-4933	72	5	m	m	VERB
ejpam-4933	72	6	∧	∧	PROPN
ejpam-4933	72	7	ε	ε	PROPN
ejpam-4933	72	8	:	:	PUNCT
ejpam-4933	72	9	=	=	SYM
ejpam-4933	72	10	min{m	min{m	PROPN
ejpam-4933	72	11	,	,	PUNCT
ejpam-4933	72	12	ε	ε	PROPN
ejpam-4933	72	13	}	}	PUNCT
ejpam-4933	72	14	and	and	CCONJ
ejpam-4933	72	15	m	m	PROPN
ejpam-4933	72	16	∨	∨	PROPN
ejpam-4933	72	17	ε	ε	PROPN
ejpam-4933	72	18	:	:	PUNCT
ejpam-4933	72	19	=	=	SYM
ejpam-4933	72	20	max{m	max{m	NOUN
ejpam-4933	72	21	,	,	PUNCT
ejpam-4933	72	22	ε	ε	PROPN
ejpam-4933	72	23	}	}	PUNCT
ejpam-4933	72	24	.	.	PUNCT
ejpam-4933	73	1	consider	consider	VERB
ejpam-4933	73	2	a	a	DET
ejpam-4933	73	3	binary	binary	ADJ
ejpam-4933	73	4	operation	operation	NOUN
ejpam-4933	73	5	yj	yj	PROPN
ejpam-4933	73	6	in	in	ADP
ejpam-4933	73	7	i	i	PRON
ejpam-4933	73	8	given	give	VERB
ejpam-4933	73	9	as	as	SCONJ
ejpam-4933	73	10	follows	follow	VERB
ejpam-4933	73	11	:	:	PUNCT
ejpam-4933	73	12	yj	yj	PROPN
ejpam-4933	73	13	:	:	PUNCT
ejpam-4933	73	14	i2	i2	PROPN
ejpam-4933	73	15	→	→	SYM
ejpam-4933	73	16	i	i	PROPN
ejpam-4933	73	17	,	,	PUNCT
ejpam-4933	73	18	(	(	PUNCT
ejpam-4933	73	19	m	m	PROPN
ejpam-4933	73	20	,	,	PUNCT
ejpam-4933	73	21	ε	ε	PROPN
ejpam-4933	73	22	)	)	PUNCT
ejpam-4933	73	23	7→	7→	NUM
ejpam-4933	73	24	(	(	PUNCT
ejpam-4933	73	25	1	1	NUM
ejpam-4933	73	26	−m	−m	NOUN
ejpam-4933	73	27	)	)	PUNCT
ejpam-4933	73	28	∧	∧	NOUN
ejpam-4933	73	29	(	(	PUNCT
ejpam-4933	73	30	1	1	NUM
ejpam-4933	73	31	−	−	PROPN
ejpam-4933	73	32	ε	ε	PROPN
ejpam-4933	73	33	)	)	PUNCT
ejpam-4933	73	34	.	.	PUNCT
ejpam-4933	74	1	we	we	PRON
ejpam-4933	74	2	will	will	AUX
ejpam-4933	74	3	call	call	VERB
ejpam-4933	74	4	this	this	DET
ejpam-4933	74	5	binary	binary	ADJ
ejpam-4933	74	6	operation	operation	NOUN
ejpam-4933	74	7	yj	yj	PROPN
ejpam-4933	74	8	the	the	DET
ejpam-4933	74	9	j	j	NOUN
ejpam-4933	74	10	-	-	NOUN
ejpam-4933	74	11	operator	operator	NOUN
ejpam-4933	74	12	in	in	ADP
ejpam-4933	74	13	i	i	PRON
ejpam-4933	74	14	(	(	PUNCT
ejpam-4933	74	15	see	see	VERB
ejpam-4933	74	16	[	[	X
ejpam-4933	74	17	6	6	NUM
ejpam-4933	74	18	]	]	NUM
ejpam-4933	74	19	)	)	PUNCT
ejpam-4933	74	20	.	.	PUNCT
ejpam-4933	75	1	let	let	VERB
ejpam-4933	75	2	x	x	PRON
ejpam-4933	75	3	be	be	AUX
ejpam-4933	75	4	a	a	DET
ejpam-4933	75	5	set	set	NOUN
ejpam-4933	75	6	.	.	PUNCT
ejpam-4933	76	1	given	give	VERB
ejpam-4933	76	2	a	a	DET
ejpam-4933	76	3	fuzzy	fuzzy	ADJ
ejpam-4933	76	4	set	set	VERB
ejpam-4933	76	5	ζ	ζ	NOUN
ejpam-4933	76	6	in	in	ADP
ejpam-4933	76	7	x	x	PUNCT
ejpam-4933	76	8	and	and	CCONJ
ejpam-4933	76	9	ε	ε	PROPN
ejpam-4933	76	10	∈	∈	PROPN
ejpam-4933	76	11	i	i	PRON
ejpam-4933	76	12	,	,	PUNCT
ejpam-4933	76	13	let	let	VERB
ejpam-4933	76	14	ε(ζ	ε(ζ	PROPN
ejpam-4933	76	15	)	)	PUNCT
ejpam-4933	76	16	be	be	AUX
ejpam-4933	76	17	a	a	DET
ejpam-4933	76	18	mapping	mapping	NOUN
ejpam-4933	76	19	defined	define	VERB
ejpam-4933	76	20	by	by	ADP
ejpam-4933	76	21	ε(ζ	ε(ζ	NOUN
ejpam-4933	76	22	)	)	PUNCT
ejpam-4933	76	23	:	:	PUNCT
ejpam-4933	77	1	x	x	X
ejpam-4933	77	2	→	→	PUNCT
ejpam-4933	77	3	i	i	PROPN
ejpam-4933	77	4	,	,	PUNCT
ejpam-4933	77	5	x	x	PROPN
ejpam-4933	77	6	7→	7→	NUM
ejpam-4933	77	7	yj(ε	yj(ε	NUM
ejpam-4933	77	8	,	,	PUNCT
ejpam-4933	77	9	ζ(x	ζ(x	NOUN
ejpam-4933	77	10	)	)	PUNCT
ejpam-4933	77	11	)	)	PUNCT
ejpam-4933	77	12	.	.	PUNCT
ejpam-4933	78	1	it	it	PRON
ejpam-4933	78	2	is	be	AUX
ejpam-4933	78	3	clear	clear	ADJ
ejpam-4933	78	4	that	that	SCONJ
ejpam-4933	78	5	ε(ζ	ε(ζ	NOUN
ejpam-4933	78	6	)	)	PUNCT
ejpam-4933	78	7	is	be	AUX
ejpam-4933	78	8	a	a	DET
ejpam-4933	78	9	fuzzy	fuzzy	ADJ
ejpam-4933	78	10	set	set	NOUN
ejpam-4933	78	11	in	in	ADP
ejpam-4933	78	12	x	x	PUNCT
ejpam-4933	78	13	determined	determine	VERB
ejpam-4933	78	14	by	by	ADP
ejpam-4933	78	15	the	the	DET
ejpam-4933	78	16	j	j	NOUN
ejpam-4933	78	17	-	-	NOUN
ejpam-4933	78	18	operator	operator	NOUN
ejpam-4933	78	19	and	and	CCONJ
ejpam-4933	78	20	ε	ε	PROPN
ejpam-4933	78	21	.	.	PUNCT
ejpam-4933	79	1	so	so	ADV
ejpam-4933	79	2	we	we	PRON
ejpam-4933	79	3	can	can	AUX
ejpam-4933	79	4	say	say	VERB
ejpam-4933	79	5	that	that	SCONJ
ejpam-4933	79	6	ε(ζ	ε(ζ	NOUN
ejpam-4933	79	7	)	)	PUNCT
ejpam-4933	79	8	is	be	AUX
ejpam-4933	79	9	a	a	DET
ejpam-4933	79	10	y	y	PROPN
ejpam-4933	79	11	ε	ε	PROPN
ejpam-4933	79	12	j	j	PROPN
ejpam-4933	79	13	-fuzzy	-fuzzy	PROPN
ejpam-4933	79	14	set	set	NOUN
ejpam-4933	79	15	of	of	ADP
ejpam-4933	79	16	ζ	ζ	NOUN
ejpam-4933	79	17	in	in	ADP
ejpam-4933	79	18	x	x	PART
ejpam-4933	79	19	(	(	PUNCT
ejpam-4933	79	20	see	see	VERB
ejpam-4933	79	21	[	[	X
ejpam-4933	79	22	6	6	NUM
ejpam-4933	79	23	]	]	NUM
ejpam-4933	79	24	)	)	PUNCT
ejpam-4933	79	25	.	.	PUNCT
ejpam-4933	80	1	given	give	VERB
ejpam-4933	80	2	a	a	DET
ejpam-4933	80	3	fuzzy	fuzzy	ADJ
ejpam-4933	80	4	set	set	VERB
ejpam-4933	80	5	ζ	ζ	NOUN
ejpam-4933	80	6	in	in	ADP
ejpam-4933	80	7	x	x	PUNCT
ejpam-4933	80	8	and	and	CCONJ
ejpam-4933	80	9	ε	ε	PROPN
ejpam-4933	80	10	∈	∈	PROPN
ejpam-4933	80	11	(	(	PUNCT
ejpam-4933	80	12	0	0	NUM
ejpam-4933	80	13	,	,	PUNCT
ejpam-4933	80	14	1	1	NUM
ejpam-4933	80	15	)	)	PUNCT
ejpam-4933	80	16	,	,	PUNCT
ejpam-4933	80	17	if	if	SCONJ
ejpam-4933	80	18	the	the	DET
ejpam-4933	80	19	y	y	PROPN
ejpam-4933	80	20	ε	ε	PROPN
ejpam-4933	80	21	j	j	PROPN
ejpam-4933	80	22	-fuzzy	-fuzzy	PROPN
ejpam-4933	80	23	set	set	VERB
ejpam-4933	80	24	ε(ζ	ε(ζ	NOUN
ejpam-4933	80	25	)	)	PUNCT
ejpam-4933	80	26	of	of	ADP
ejpam-4933	80	27	ζ	ζ	NOUN
ejpam-4933	80	28	is	be	AUX
ejpam-4933	80	29	not	not	PART
ejpam-4933	80	30	constant	constant	ADJ
ejpam-4933	80	31	on	on	ADP
ejpam-4933	80	32	x	x	X
ejpam-4933	80	33	,	,	PUNCT
ejpam-4933	80	34	then	then	ADV
ejpam-4933	80	35	ε	ε	PROPN
ejpam-4933	80	36	is	be	AUX
ejpam-4933	80	37	said	say	VERB
ejpam-4933	80	38	to	to	PART
ejpam-4933	80	39	be	be	AUX
ejpam-4933	80	40	a	a	DET
ejpam-4933	80	41	nonconstant	nonconstant	ADJ
ejpam-4933	80	42	factor	factor	NOUN
ejpam-4933	80	43	in	in	ADP
ejpam-4933	80	44	(	(	PUNCT
ejpam-4933	80	45	0	0	NUM
ejpam-4933	80	46	,	,	PUNCT
ejpam-4933	80	47	1	1	NUM
ejpam-4933	80	48	)	)	PUNCT
ejpam-4933	80	49	(	(	PUNCT
ejpam-4933	80	50	see	see	VERB
ejpam-4933	80	51	[	[	X
ejpam-4933	80	52	6	6	NUM
ejpam-4933	80	53	]	]	NUM
ejpam-4933	80	54	)	)	PUNCT
ejpam-4933	80	55	.	.	PUNCT
ejpam-4933	81	1	a	a	DET
ejpam-4933	81	2	fuzzy	fuzzy	ADJ
ejpam-4933	81	3	set	set	VERB
ejpam-4933	81	4	ζ	ζ	NOUN
ejpam-4933	81	5	in	in	ADP
ejpam-4933	81	6	x	x	PROPN
ejpam-4933	81	7	is	be	AUX
ejpam-4933	81	8	called	call	VERB
ejpam-4933	81	9	a	a	DET
ejpam-4933	81	10	y	y	PROPN
ejpam-4933	81	11	ε	ε	PROPN
ejpam-4933	81	12	j	j	PROPN
ejpam-4933	81	13	-fuzzy	-fuzzy	PROPN
ejpam-4933	81	14	subalgebra	subalgebra	NOUN
ejpam-4933	81	15	of	of	ADP
ejpam-4933	81	16	(	(	PUNCT
ejpam-4933	81	17	x	x	NOUN
ejpam-4933	81	18	,	,	PUNCT
ejpam-4933	81	19	∗	∗	NOUN
ejpam-4933	81	20	,	,	PUNCT
ejpam-4933	81	21	0	0	NUM
ejpam-4933	81	22	)	)	PUNCT
ejpam-4933	81	23	(	(	PUNCT
ejpam-4933	81	24	see	see	VERB
ejpam-4933	81	25	[	[	X
ejpam-4933	81	26	6	6	NUM
ejpam-4933	81	27	]	]	PUNCT
ejpam-4933	81	28	)	)	PUNCT
ejpam-4933	81	29	if	if	SCONJ
ejpam-4933	81	30	it	it	PRON
ejpam-4933	81	31	satisfies	satisfy	VERB
ejpam-4933	81	32	:	:	PUNCT
ejpam-4933	81	33	(	(	PUNCT
ejpam-4933	81	34	∀x	∀x	X
ejpam-4933	81	35	,	,	PUNCT
ejpam-4933	81	36	a	a	DET
ejpam-4933	81	37	∈	∈	PROPN
ejpam-4933	81	38	x)(yj(ε	x)(yj(ε	NOUN
ejpam-4933	81	39	,	,	PUNCT
ejpam-4933	81	40	ζ(x	ζ(x	PROPN
ejpam-4933	81	41	∗	∗	VERB
ejpam-4933	81	42	a	a	NOUN
ejpam-4933	81	43	)	)	PUNCT
ejpam-4933	81	44	)	)	PUNCT
ejpam-4933	81	45	≤	≤	NOUN
ejpam-4933	81	46	yj(ε	yj(ε	NUM
ejpam-4933	81	47	,	,	PUNCT
ejpam-4933	81	48	ζ(x	ζ(x	NOUN
ejpam-4933	81	49	)	)	PUNCT
ejpam-4933	81	50	)	)	PUNCT
ejpam-4933	82	1	∨	∨	PROPN
ejpam-4933	82	2	yj(ε	yj(ε	NUM
ejpam-4933	82	3	,	,	PUNCT
ejpam-4933	82	4	ζ(a	ζ(a	NOUN
ejpam-4933	82	5	)	)	PUNCT
ejpam-4933	82	6	)	)	PUNCT
ejpam-4933	82	7	)	)	PUNCT
ejpam-4933	82	8	.	.	PUNCT
ejpam-4933	83	1	(	(	PUNCT
ejpam-4933	83	2	13	13	NUM
ejpam-4933	83	3	)	)	PUNCT
ejpam-4933	83	4	e.	e.	PROPN
ejpam-4933	83	5	h.	h.	PROPN
ejpam-4933	83	6	roh	roh	PROPN
ejpam-4933	83	7	,	,	PUNCT
ejpam-4933	83	8	e.	e.	PROPN
ejpam-4933	83	9	yang	yang	PROPN
ejpam-4933	83	10	,	,	PUNCT
ejpam-4933	83	11	y.	y.	PROPN
ejpam-4933	83	12	b.	b.	PROPN
ejpam-4933	83	13	jun	jun	PROPN
ejpam-4933	83	14	/	/	SYM
ejpam-4933	83	15	eur	eur	PROPN
ejpam-4933	83	16	.	.	PUNCT
ejpam-4933	84	1	j.	j.	PROPN
ejpam-4933	84	2	pure	pure	PROPN
ejpam-4933	84	3	appl	appl	PROPN
ejpam-4933	84	4	.	.	PROPN
ejpam-4933	84	5	math	math	PROPN
ejpam-4933	84	6	,	,	PUNCT
ejpam-4933	84	7	16	16	NUM
ejpam-4933	84	8	(	(	PUNCT
ejpam-4933	84	9	4	4	NUM
ejpam-4933	84	10	)	)	PUNCT
ejpam-4933	84	11	(	(	PUNCT
ejpam-4933	84	12	2023	2023	NUM
ejpam-4933	84	13	)	)	PUNCT
ejpam-4933	84	14	,	,	PUNCT
ejpam-4933	84	15	2009	2009	NUM
ejpam-4933	84	16	-	-	SYM
ejpam-4933	84	17	2024	2024	NUM
ejpam-4933	84	18	2013	2013	NUM
ejpam-4933	84	19	3	3	NUM
ejpam-4933	84	20	.	.	PUNCT
ejpam-4933	84	21	level	level	NOUN
ejpam-4933	84	22	sets	set	NOUN
ejpam-4933	84	23	of	of	ADP
ejpam-4933	84	24	the	the	DET
ejpam-4933	84	25	y	y	PROPN
ejpam-4933	84	26	ε	ε	PROPN
ejpam-4933	84	27	j	j	PROPN
ejpam-4933	84	28	-fuzzy	-fuzzy	PROPN
ejpam-4933	84	29	set	set	VERB
ejpam-4933	84	30	let	let	VERB
ejpam-4933	84	31	ζ	ζ	NOUN
ejpam-4933	84	32	be	be	AUX
ejpam-4933	84	33	a	a	DET
ejpam-4933	84	34	fuzzy	fuzzy	ADJ
ejpam-4933	84	35	set	set	NOUN
ejpam-4933	84	36	in	in	ADP
ejpam-4933	84	37	x	x	PROPN
ejpam-4933	84	38	,	,	PUNCT
ejpam-4933	84	39	ε	ε	PROPN
ejpam-4933	84	40	∈	∈	PROPN
ejpam-4933	85	1	i	i	PRON
ejpam-4933	85	2	and	and	CCONJ
ejpam-4933	85	3	t	t	NOUN
ejpam-4933	85	4	∈	∈	PROPN
ejpam-4933	86	1	i	i	PRON
ejpam-4933	86	2	\	\	PROPN
ejpam-4933	86	3	{	{	PUNCT
ejpam-4933	86	4	0	0	NUM
ejpam-4933	86	5	,	,	PUNCT
ejpam-4933	86	6	1	1	NUM
ejpam-4933	86	7	}	}	PUNCT
ejpam-4933	86	8	.	.	PUNCT
ejpam-4933	87	1	given	give	VERB
ejpam-4933	87	2	a	a	DET
ejpam-4933	87	3	y	y	PROPN
ejpam-4933	87	4	ε	ε	PROPN
ejpam-4933	87	5	j	j	PROPN
ejpam-4933	87	6	-fuzzy	-fuzzy	PROPN
ejpam-4933	87	7	set	set	VERB
ejpam-4933	87	8	ε(ζ	ε(ζ	NOUN
ejpam-4933	87	9	)	)	PUNCT
ejpam-4933	87	10	,	,	PUNCT
ejpam-4933	87	11	we	we	PRON
ejpam-4933	87	12	consider	consider	VERB
ejpam-4933	87	13	the	the	DET
ejpam-4933	87	14	sets	set	NOUN
ejpam-4933	87	15	:	:	PUNCT
ejpam-4933	87	16	ε(ζ)t	ε(ζ)t	VERB
ejpam-4933	87	17	:	:	PUNCT
ejpam-4933	87	18	=	=	SYM
ejpam-4933	87	19	{	{	PUNCT
ejpam-4933	87	20	x	x	SYM
ejpam-4933	87	21	∈	∈	PROPN
ejpam-4933	87	22	x	x	NOUN
ejpam-4933	87	23	|	|	ADV
ejpam-4933	87	24	yj(ε	yj(ε	NUM
ejpam-4933	87	25	,	,	PUNCT
ejpam-4933	87	26	ζ(x	ζ(x	NOUN
ejpam-4933	87	27	)	)	PUNCT
ejpam-4933	87	28	)	)	PUNCT
ejpam-4933	88	1	≤	≤	NUM
ejpam-4933	88	2	t	t	PROPN
ejpam-4933	88	3	}	}	PUNCT
ejpam-4933	88	4	,	,	PUNCT
ejpam-4933	88	5	ε(ζ)tq	ε(ζ)tq	NUM
ejpam-4933	88	6	:	:	PUNCT
ejpam-4933	88	7	=	=	SYM
ejpam-4933	88	8	{	{	PUNCT
ejpam-4933	88	9	x	x	SYM
ejpam-4933	88	10	∈	∈	PROPN
ejpam-4933	88	11	x	x	NOUN
ejpam-4933	88	12	|	|	ADV
ejpam-4933	88	13	yj(ε	yj(ε	NUM
ejpam-4933	88	14	,	,	PUNCT
ejpam-4933	88	15	ζ(x	ζ(x	NOUN
ejpam-4933	88	16	)	)	PUNCT
ejpam-4933	88	17	)	)	PUNCT
ejpam-4933	88	18	<	<	X
ejpam-4933	88	19	1	1	NUM
ejpam-4933	88	20	−	−	PROPN
ejpam-4933	88	21	t	t	PROPN
ejpam-4933	88	22	}	}	PUNCT
ejpam-4933	88	23	,	,	PUNCT
ejpam-4933	88	24	which	which	PRON
ejpam-4933	88	25	is	be	AUX
ejpam-4933	88	26	called	call	VERB
ejpam-4933	88	27	the	the	DET
ejpam-4933	88	28	y	y	NOUN
ejpam-4933	88	29	-	-	PUNCT
ejpam-4933	88	30	level	level	NOUN
ejpam-4933	88	31	set	set	NOUN
ejpam-4933	88	32	and	and	CCONJ
ejpam-4933	88	33	yq	yq	NOUN
ejpam-4933	88	34	-	-	PUNCT
ejpam-4933	88	35	set	set	NOUN
ejpam-4933	88	36	of	of	ADP
ejpam-4933	88	37	ε(ζ	ε(ζ	NOUN
ejpam-4933	88	38	)	)	PUNCT
ejpam-4933	88	39	,	,	PUNCT
ejpam-4933	88	40	respectively	respectively	ADV
ejpam-4933	88	41	,	,	PUNCT
ejpam-4933	88	42	related	relate	VERB
ejpam-4933	88	43	to	to	ADP
ejpam-4933	88	44	t.	t.	NOUN
ejpam-4933	88	45	we	we	PRON
ejpam-4933	88	46	call	call	VERB
ejpam-4933	88	47	t	t	NOUN
ejpam-4933	88	48	the	the	DET
ejpam-4933	88	49	level	level	NOUN
ejpam-4933	88	50	degree	degree	NOUN
ejpam-4933	88	51	of	of	ADP
ejpam-4933	88	52	ε(ζ	ε(ζ	NOUN
ejpam-4933	88	53	)	)	PUNCT
ejpam-4933	88	54	.	.	PUNCT
ejpam-4933	89	1	the	the	DET
ejpam-4933	89	2	y	y	NOUN
ejpam-4933	89	3	-	-	PUNCT
ejpam-4933	89	4	level	level	NOUN
ejpam-4933	89	5	set	set	NOUN
ejpam-4933	89	6	and	and	CCONJ
ejpam-4933	89	7	the	the	DET
ejpam-4933	89	8	yq	yq	NOUN
ejpam-4933	89	9	-	-	PUNCT
ejpam-4933	89	10	set	set	NOUN
ejpam-4933	89	11	of	of	ADP
ejpam-4933	89	12	ε(ζ	ε(ζ	NOUN
ejpam-4933	89	13	)	)	PUNCT
ejpam-4933	89	14	related	relate	VERB
ejpam-4933	89	15	to	to	ADP
ejpam-4933	89	16	t	t	PROPN
ejpam-4933	89	17	are	be	AUX
ejpam-4933	89	18	calculated	calculate	VERB
ejpam-4933	89	19	as	as	SCONJ
ejpam-4933	89	20	follows	follow	VERB
ejpam-4933	89	21	:	:	PUNCT
ejpam-4933	89	22	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	89	23	=	=	PRON
ejpam-4933	89	24	{	{	PUNCT
ejpam-4933	89	25	x	x	PUNCT
ejpam-4933	89	26	∈	∈	PROPN
ejpam-4933	89	27	x	x	NOUN
ejpam-4933	89	28	|	|	ADV
ejpam-4933	89	29	yj(ε	yj(ε	NUM
ejpam-4933	89	30	,	,	PUNCT
ejpam-4933	89	31	ζ(x	ζ(x	NOUN
ejpam-4933	89	32	)	)	PUNCT
ejpam-4933	89	33	)	)	PUNCT
ejpam-4933	90	1	≤	≤	NUM
ejpam-4933	90	2	t	t	PROPN
ejpam-4933	90	3	}	}	PUNCT
ejpam-4933	90	4	=	=	SYM
ejpam-4933	90	5	{	{	PUNCT
ejpam-4933	90	6	x	x	PUNCT
ejpam-4933	90	7	∈	∈	PROPN
ejpam-4933	90	8	x	x	INTJ
ejpam-4933	90	9	|	|	ADV
ejpam-4933	90	10	(	(	PUNCT
ejpam-4933	90	11	1	1	NUM
ejpam-4933	90	12	−	−	PROPN
ejpam-4933	90	13	ε	ε	PROPN
ejpam-4933	90	14	)	)	PUNCT
ejpam-4933	90	15	∧	∧	NOUN
ejpam-4933	90	16	(	(	PUNCT
ejpam-4933	90	17	1	1	NUM
ejpam-4933	90	18	−	−	PROPN
ejpam-4933	90	19	ζ(x	ζ(x	NOUN
ejpam-4933	90	20	)	)	PUNCT
ejpam-4933	90	21	)	)	PUNCT
ejpam-4933	90	22	≤	≤	NUM
ejpam-4933	91	1	t	t	PROPN
ejpam-4933	91	2	}	}	PUNCT
ejpam-4933	91	3	=	=	SYM
ejpam-4933	91	4	{	{	PUNCT
ejpam-4933	91	5	x	x	PUNCT
ejpam-4933	91	6	∈	∈	NOUN
ejpam-4933	91	7	x	x	PUNCT
ejpam-4933	91	8	|	|	ADV
ejpam-4933	91	9	1	1	NUM
ejpam-4933	91	10	−	−	PROPN
ejpam-4933	91	11	(	(	PUNCT
ejpam-4933	91	12	ε	ε	PROPN
ejpam-4933	91	13	∨	∨	NUM
ejpam-4933	91	14	ζ(x	ζ(x	NOUN
ejpam-4933	91	15	)	)	PUNCT
ejpam-4933	91	16	)	)	PUNCT
ejpam-4933	91	17	≤	≤	NUM
ejpam-4933	92	1	t	t	PROPN
ejpam-4933	92	2	}	}	PUNCT
ejpam-4933	92	3	=	=	SYM
ejpam-4933	92	4	{	{	PUNCT
ejpam-4933	92	5	x	x	PUNCT
ejpam-4933	92	6	∈	∈	PROPN
ejpam-4933	92	7	x	x	INTJ
ejpam-4933	92	8	|	|	ADV
ejpam-4933	92	9	ε	ε	PROPN
ejpam-4933	92	10	∨	∨	NOUN
ejpam-4933	92	11	ζ(x	ζ(x	NOUN
ejpam-4933	92	12	)	)	PUNCT
ejpam-4933	92	13	≥	≥	NOUN
ejpam-4933	92	14	1	1	NUM
ejpam-4933	92	15	−	−	PROPN
ejpam-4933	92	16	t	t	PROPN
ejpam-4933	92	17	}	}	PUNCT
ejpam-4933	92	18	and	and	CCONJ
ejpam-4933	92	19	ε(ζ)tq	ε(ζ)tq	PROPN
ejpam-4933	92	20	=	=	VERB
ejpam-4933	92	21	{	{	PUNCT
ejpam-4933	92	22	x	x	PUNCT
ejpam-4933	92	23	∈	∈	PROPN
ejpam-4933	92	24	x	x	NOUN
ejpam-4933	92	25	|	|	ADV
ejpam-4933	92	26	yj(ε	yj(ε	NUM
ejpam-4933	92	27	,	,	PUNCT
ejpam-4933	92	28	ζ(x	ζ(x	NOUN
ejpam-4933	92	29	)	)	PUNCT
ejpam-4933	92	30	)	)	PUNCT
ejpam-4933	92	31	<	<	X
ejpam-4933	92	32	1	1	NUM
ejpam-4933	92	33	−	−	PROPN
ejpam-4933	92	34	t	t	PROPN
ejpam-4933	92	35	}	}	PUNCT
ejpam-4933	92	36	=	=	PUNCT
ejpam-4933	92	37	{	{	PUNCT
ejpam-4933	92	38	x	x	PUNCT
ejpam-4933	92	39	∈	∈	PROPN
ejpam-4933	92	40	x	x	INTJ
ejpam-4933	92	41	|	|	ADV
ejpam-4933	92	42	(	(	PUNCT
ejpam-4933	92	43	1	1	NUM
ejpam-4933	92	44	−	−	PROPN
ejpam-4933	92	45	ε	ε	PROPN
ejpam-4933	92	46	)	)	PUNCT
ejpam-4933	92	47	∧	∧	NOUN
ejpam-4933	92	48	(	(	PUNCT
ejpam-4933	92	49	1	1	NUM
ejpam-4933	92	50	−	−	PROPN
ejpam-4933	92	51	ζ(x	ζ(x	NOUN
ejpam-4933	92	52	)	)	PUNCT
ejpam-4933	92	53	)	)	PUNCT
ejpam-4933	93	1	<	<	X
ejpam-4933	93	2	1	1	NUM
ejpam-4933	93	3	−	−	PROPN
ejpam-4933	93	4	t	t	PROPN
ejpam-4933	93	5	}	}	PUNCT
ejpam-4933	93	6	=	=	PUNCT
ejpam-4933	93	7	{	{	PUNCT
ejpam-4933	93	8	x	x	PUNCT
ejpam-4933	93	9	∈	∈	PROPN
ejpam-4933	93	10	x	x	INTJ
ejpam-4933	93	11	|	|	ADV
ejpam-4933	93	12	ε	ε	PROPN
ejpam-4933	93	13	∨	∨	NOUN
ejpam-4933	93	14	ζ(x	ζ(x	NOUN
ejpam-4933	93	15	)	)	PUNCT
ejpam-4933	93	16	>	>	X
ejpam-4933	93	17	t	t	PROPN
ejpam-4933	93	18	}	}	PUNCT
ejpam-4933	93	19	,	,	PUNCT
ejpam-4933	93	20	respectively	respectively	ADV
ejpam-4933	93	21	.	.	PUNCT
ejpam-4933	94	1	the	the	DET
ejpam-4933	94	2	set	set	VERB
ejpam-4933	94	3	ε(ζ)t∈∨q	ε(ζ)t∈∨q	NOUN
ejpam-4933	94	4	:	:	PUNCT
ejpam-4933	94	5	=	=	SYM
ejpam-4933	94	6	{	{	PUNCT
ejpam-4933	94	7	x	x	SYM
ejpam-4933	94	8	∈	∈	PROPN
ejpam-4933	94	9	x	x	NOUN
ejpam-4933	94	10	|	|	ADV
ejpam-4933	94	11	yj(ε	yj(ε	NUM
ejpam-4933	94	12	,	,	PUNCT
ejpam-4933	94	13	ζ(x	ζ(x	NOUN
ejpam-4933	94	14	)	)	PUNCT
ejpam-4933	94	15	)	)	PUNCT
ejpam-4933	94	16	≤	≤	NUM
ejpam-4933	94	17	t	t	NOUN
ejpam-4933	94	18	or	or	CCONJ
ejpam-4933	94	19	yj(ε	yj(ε	NUM
ejpam-4933	94	20	,	,	PUNCT
ejpam-4933	94	21	ζ(x	ζ(x	NOUN
ejpam-4933	94	22	)	)	PUNCT
ejpam-4933	94	23	)	)	PUNCT
ejpam-4933	94	24	<	<	X
ejpam-4933	94	25	1	1	NUM
ejpam-4933	94	26	−	−	PROPN
ejpam-4933	94	27	t	t	PROPN
ejpam-4933	94	28	}	}	PUNCT
ejpam-4933	94	29	is	be	AUX
ejpam-4933	94	30	called	call	VERB
ejpam-4933	94	31	the	the	DET
ejpam-4933	94	32	the	the	DET
ejpam-4933	94	33	y∈∨q	y∈∨q	PROPN
ejpam-4933	94	34	-set	-set	PUNCT
ejpam-4933	94	35	of	of	ADP
ejpam-4933	94	36	ε(ζ	ε(ζ	NOUN
ejpam-4933	94	37	)	)	PUNCT
ejpam-4933	94	38	related	relate	VERB
ejpam-4933	94	39	to	to	ADP
ejpam-4933	94	40	t.	t.	NOUN
ejpam-4933	94	41	it	it	PRON
ejpam-4933	94	42	is	be	AUX
ejpam-4933	94	43	clear	clear	ADJ
ejpam-4933	94	44	that	that	SCONJ
ejpam-4933	94	45	ε(ζ)t∈∨q	ε(ζ)t∈∨q	NOUN
ejpam-4933	94	46	=	=	PUNCT
ejpam-4933	94	47	ε(ζ)t	ε(ζ)t	NOUN
ejpam-4933	94	48	∪	∪	ADP
ejpam-4933	94	49	ε(ζ)tq	ε(ζ)tq	PROPN
ejpam-4933	94	50	.	.	PUNCT
ejpam-4933	94	51	proposition	proposition	NOUN
ejpam-4933	94	52	1	1	NUM
ejpam-4933	94	53	.	.	PUNCT
ejpam-4933	95	1	let	let	VERB
ejpam-4933	95	2	ζ	ζ	NOUN
ejpam-4933	95	3	be	be	AUX
ejpam-4933	95	4	a	a	DET
ejpam-4933	95	5	fuzzy	fuzzy	ADJ
ejpam-4933	95	6	set	set	NOUN
ejpam-4933	95	7	in	in	ADP
ejpam-4933	95	8	x	x	PUNCT
ejpam-4933	95	9	and	and	CCONJ
ejpam-4933	95	10	ε	ε	PROPN
ejpam-4933	95	11	∈	∈	PROPN
ejpam-4933	96	1	i	i	PRON
ejpam-4933	96	2	that	that	PRON
ejpam-4933	96	3	satisfies	satisfy	VERB
ejpam-4933	96	4	ε	ε	PROPN
ejpam-4933	96	5	≤	≤	NOUN
ejpam-4933	96	6	ζ(x	ζ(x	NOUN
ejpam-4933	96	7	)	)	PUNCT
ejpam-4933	96	8	for	for	ADP
ejpam-4933	96	9	all	all	DET
ejpam-4933	96	10	x	x	SYM
ejpam-4933	96	11	∈	∈	ADJ
ejpam-4933	96	12	x.	x.	NOUN
ejpam-4933	96	13	then	then	ADV
ejpam-4933	96	14	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	96	15	=	=	PRON
ejpam-4933	96	16	(	(	PUNCT
ejpam-4933	96	17	ζ	ζ	NOUN
ejpam-4933	96	18	,	,	PUNCT
ejpam-4933	96	19	t)q	t)q	PUNCT
ejpam-4933	96	20	∪	∪	ADP
ejpam-4933	96	21	ζ1	ζ1	PROPN
ejpam-4933	96	22	t	t	PROPN
ejpam-4933	96	23	where	where	SCONJ
ejpam-4933	96	24	ζ1	ζ1	NOUN
ejpam-4933	96	25	t	t	NOUN
ejpam-4933	96	26	:	:	PUNCT
ejpam-4933	96	27	=	=	SYM
ejpam-4933	96	28	{	{	PUNCT
ejpam-4933	96	29	x	x	PUNCT
ejpam-4933	96	30	∈	∈	PROPN
ejpam-4933	96	31	x	x	X
ejpam-4933	96	32	|	|	ADP
ejpam-4933	96	33	ζ(x	ζ(x	NOUN
ejpam-4933	96	34	)	)	PUNCT
ejpam-4933	96	35	+	+	NUM
ejpam-4933	96	36	t	t	X
ejpam-4933	96	37	=	=	SYM
ejpam-4933	96	38	1	1	NUM
ejpam-4933	96	39	}	}	PUNCT
ejpam-4933	96	40	,	,	PUNCT
ejpam-4933	96	41	and	and	CCONJ
ejpam-4933	96	42	ε(ζ)tq	ε(ζ)tq	NUM
ejpam-4933	96	43	⊆	⊆	NUM
ejpam-4933	96	44	(	(	PUNCT
ejpam-4933	96	45	ζ	ζ	NOUN
ejpam-4933	96	46	,	,	PUNCT
ejpam-4933	96	47	t)∈.	t)∈.	PROPN
ejpam-4933	96	48	proof	proof	NOUN
ejpam-4933	96	49	.	.	PUNCT
ejpam-4933	97	1	straightforwad	straightforwad	NOUN
ejpam-4933	97	2	.	.	PUNCT
ejpam-4933	98	1	proposition	proposition	NOUN
ejpam-4933	98	2	2	2	NUM
ejpam-4933	98	3	.	.	PUNCT
ejpam-4933	98	4	let	let	VERB
ejpam-4933	98	5	ζ	ζ	NOUN
ejpam-4933	98	6	be	be	AUX
ejpam-4933	98	7	a	a	DET
ejpam-4933	98	8	fuzzy	fuzzy	ADJ
ejpam-4933	98	9	set	set	NOUN
ejpam-4933	98	10	in	in	ADP
ejpam-4933	98	11	x	x	PUNCT
ejpam-4933	98	12	and	and	CCONJ
ejpam-4933	98	13	ε	ε	PROPN
ejpam-4933	98	14	∈	∈	PROPN
ejpam-4933	98	15	i.	i.	NOUN
ejpam-4933	98	16	if	if	SCONJ
ejpam-4933	98	17	s	s	PROPN
ejpam-4933	98	18	≥	≥	PROPN
ejpam-4933	98	19	t	t	X
ejpam-4933	98	20	in	in	ADP
ejpam-4933	98	21	i\{0	i\{0	PROPN
ejpam-4933	98	22	,	,	PUNCT
ejpam-4933	98	23	1	1	NUM
ejpam-4933	98	24	}	}	PUNCT
ejpam-4933	98	25	,	,	PUNCT
ejpam-4933	98	26	then	then	ADV
ejpam-4933	98	27	ε(ζ)t	ε(ζ)t	VERB
ejpam-4933	98	28	⊆	⊆	NUM
ejpam-4933	98	29	ε(ζ)s	ε(ζ)s	PROPN
ejpam-4933	98	30	and	and	CCONJ
ejpam-4933	98	31	ε(ζ)tq	ε(ζ)tq	PROPN
ejpam-4933	98	32	⊇	⊇	PROPN
ejpam-4933	98	33	ε(ζ)sq	ε(ζ)sq	PROPN
ejpam-4933	98	34	.	.	PUNCT
ejpam-4933	98	35	proof	proof	NOUN
ejpam-4933	98	36	.	.	PUNCT
ejpam-4933	99	1	straightforward	straightforward	ADJ
ejpam-4933	99	2	.	.	PUNCT
ejpam-4933	100	1	e.	e.	PROPN
ejpam-4933	100	2	h.	h.	PROPN
ejpam-4933	100	3	roh	roh	PROPN
ejpam-4933	100	4	,	,	PUNCT
ejpam-4933	100	5	e.	e.	PROPN
ejpam-4933	100	6	yang	yang	PROPN
ejpam-4933	100	7	,	,	PUNCT
ejpam-4933	100	8	y.	y.	PROPN
ejpam-4933	100	9	b.	b.	PROPN
ejpam-4933	100	10	jun	jun	PROPN
ejpam-4933	100	11	/	/	SYM
ejpam-4933	100	12	eur	eur	PROPN
ejpam-4933	100	13	.	.	PUNCT
ejpam-4933	101	1	j.	j.	PROPN
ejpam-4933	101	2	pure	pure	PROPN
ejpam-4933	101	3	appl	appl	PROPN
ejpam-4933	101	4	.	.	PROPN
ejpam-4933	101	5	math	math	PROPN
ejpam-4933	101	6	,	,	PUNCT
ejpam-4933	101	7	16	16	NUM
ejpam-4933	101	8	(	(	PUNCT
ejpam-4933	101	9	4	4	NUM
ejpam-4933	101	10	)	)	PUNCT
ejpam-4933	101	11	(	(	PUNCT
ejpam-4933	101	12	2023	2023	NUM
ejpam-4933	101	13	)	)	PUNCT
ejpam-4933	101	14	,	,	PUNCT
ejpam-4933	101	15	2009	2009	NUM
ejpam-4933	101	16	-	-	SYM
ejpam-4933	101	17	2024	2024	NUM
ejpam-4933	101	18	2014	2014	NUM
ejpam-4933	101	19	4	4	NUM
ejpam-4933	101	20	.	.	PUNCT
ejpam-4933	102	1	y	y	PROPN
ejpam-4933	102	2	ε	ε	PROPN
ejpam-4933	102	3	j	j	PROPN
ejpam-4933	102	4	-fuzzy	-fuzzy	PROPN
ejpam-4933	102	5	ideals	ideal	NOUN
ejpam-4933	102	6	we	we	PRON
ejpam-4933	102	7	begin	begin	VERB
ejpam-4933	102	8	this	this	DET
ejpam-4933	102	9	section	section	NOUN
ejpam-4933	102	10	by	by	ADP
ejpam-4933	102	11	looking	look	VERB
ejpam-4933	102	12	at	at	ADP
ejpam-4933	102	13	a	a	DET
ejpam-4933	102	14	characterization	characterization	NOUN
ejpam-4933	102	15	of	of	ADP
ejpam-4933	102	16	y	y	PROPN
ejpam-4933	102	17	ε	ε	PROPN
ejpam-4933	102	18	j	j	PROPN
ejpam-4933	102	19	-fuzzy	-fuzzy	PROPN
ejpam-4933	102	20	subalgebra	subalgebra	NOUN
ejpam-4933	102	21	by	by	ADP
ejpam-4933	102	22	ylevel	ylevel	NOUN
ejpam-4933	102	23	set	set	NOUN
ejpam-4933	102	24	.	.	PUNCT
ejpam-4933	103	1	in	in	ADP
ejpam-4933	103	2	what	what	PRON
ejpam-4933	103	3	follows	follow	VERB
ejpam-4933	103	4	,	,	PUNCT
ejpam-4933	103	5	let	let	VERB
ejpam-4933	103	6	(	(	PUNCT
ejpam-4933	103	7	x	x	NOUN
ejpam-4933	103	8	,	,	PUNCT
ejpam-4933	103	9	∗	∗	NOUN
ejpam-4933	103	10	,	,	PUNCT
ejpam-4933	103	11	0	0	NUM
ejpam-4933	103	12	)	)	PUNCT
ejpam-4933	103	13	be	be	AUX
ejpam-4933	103	14	a	a	DET
ejpam-4933	103	15	bck	bck	NOUN
ejpam-4933	103	16	-	-	PUNCT
ejpam-4933	103	17	algebra	algebra	NOUN
ejpam-4933	103	18	or	or	CCONJ
ejpam-4933	103	19	a	a	DET
ejpam-4933	103	20	bci	bci	NOUN
ejpam-4933	103	21	-	-	NOUN
ejpam-4933	103	22	algebra	algebra	NOUN
ejpam-4933	103	23	,	,	PUNCT
ejpam-4933	103	24	and	and	CCONJ
ejpam-4933	103	25	ε	ε	PROPN
ejpam-4933	103	26	∈	∈	PROPN
ejpam-4933	103	27	(	(	PUNCT
ejpam-4933	103	28	0	0	NUM
ejpam-4933	103	29	,	,	PUNCT
ejpam-4933	103	30	1	1	NUM
ejpam-4933	103	31	)	)	PUNCT
ejpam-4933	103	32	unless	unless	SCONJ
ejpam-4933	103	33	otherwise	otherwise	ADV
ejpam-4933	103	34	specified	specify	VERB
ejpam-4933	103	35	.	.	PUNCT
ejpam-4933	104	1	theorem	theorem	NOUN
ejpam-4933	104	2	1	1	NUM
ejpam-4933	104	3	.	.	PUNCT
ejpam-4933	105	1	a	a	DET
ejpam-4933	105	2	fuzzy	fuzzy	ADJ
ejpam-4933	105	3	set	set	VERB
ejpam-4933	105	4	ζ	ζ	NOUN
ejpam-4933	105	5	in	in	ADP
ejpam-4933	105	6	x	x	SYM
ejpam-4933	105	7	is	be	AUX
ejpam-4933	105	8	a	a	DET
ejpam-4933	105	9	y	y	PROPN
ejpam-4933	105	10	ε	ε	PROPN
ejpam-4933	105	11	j	j	PROPN
ejpam-4933	105	12	-fuzzy	-fuzzy	PROPN
ejpam-4933	105	13	subalgebra	subalgebra	NOUN
ejpam-4933	105	14	of	of	ADP
ejpam-4933	105	15	(	(	PUNCT
ejpam-4933	105	16	x	x	NOUN
ejpam-4933	105	17	,	,	PUNCT
ejpam-4933	105	18	∗	∗	NOUN
ejpam-4933	105	19	,	,	PUNCT
ejpam-4933	105	20	0	0	NUM
ejpam-4933	105	21	)	)	PUNCT
ejpam-4933	105	22	if	if	SCONJ
ejpam-4933	105	23	and	and	CCONJ
ejpam-4933	105	24	only	only	ADV
ejpam-4933	105	25	if	if	SCONJ
ejpam-4933	105	26	the	the	DET
ejpam-4933	105	27	nonempty	nonempty	ADJ
ejpam-4933	105	28	y	y	NOUN
ejpam-4933	105	29	-	-	PUNCT
ejpam-4933	105	30	level	level	NOUN
ejpam-4933	105	31	set	set	NOUN
ejpam-4933	105	32	ε(ζ)t	ε(ζ)t	NOUN
ejpam-4933	105	33	of	of	ADP
ejpam-4933	105	34	ε(ζ	ε(ζ	NOUN
ejpam-4933	105	35	)	)	PUNCT
ejpam-4933	105	36	is	be	AUX
ejpam-4933	105	37	a	a	DET
ejpam-4933	105	38	subalgebra	subalgebra	NOUN
ejpam-4933	105	39	of	of	ADP
ejpam-4933	105	40	(	(	PUNCT
ejpam-4933	105	41	x	x	NOUN
ejpam-4933	105	42	,	,	PUNCT
ejpam-4933	105	43	∗	∗	NOUN
ejpam-4933	105	44	,	,	PUNCT
ejpam-4933	105	45	0	0	NUM
ejpam-4933	105	46	)	)	PUNCT
ejpam-4933	105	47	for	for	ADP
ejpam-4933	105	48	all	all	DET
ejpam-4933	105	49	t	t	NOUN
ejpam-4933	105	50	∈	∈	PROPN
ejpam-4933	106	1	i	i	PRON
ejpam-4933	106	2	\	\	PROPN
ejpam-4933	106	3	{	{	PUNCT
ejpam-4933	106	4	0	0	NUM
ejpam-4933	106	5	,	,	PUNCT
ejpam-4933	106	6	1	1	NUM
ejpam-4933	106	7	}	}	PUNCT
ejpam-4933	106	8	.	.	PUNCT
ejpam-4933	107	1	proof	proof	NOUN
ejpam-4933	107	2	.	.	PUNCT
ejpam-4933	108	1	assume	assume	VERB
ejpam-4933	108	2	that	that	SCONJ
ejpam-4933	108	3	ζ	ζ	NOUN
ejpam-4933	108	4	is	be	AUX
ejpam-4933	108	5	a	a	DET
ejpam-4933	108	6	y	y	PROPN
ejpam-4933	108	7	ε	ε	PROPN
ejpam-4933	108	8	j	j	PROPN
ejpam-4933	108	9	-fuzzy	-fuzzy	PROPN
ejpam-4933	108	10	subalgebra	subalgebra	NOUN
ejpam-4933	108	11	of	of	ADP
ejpam-4933	108	12	(	(	PUNCT
ejpam-4933	108	13	x	x	NOUN
ejpam-4933	108	14	,	,	PUNCT
ejpam-4933	108	15	∗	∗	NOUN
ejpam-4933	108	16	,	,	PUNCT
ejpam-4933	108	17	0	0	NUM
ejpam-4933	108	18	)	)	PUNCT
ejpam-4933	108	19	and	and	CCONJ
ejpam-4933	108	20	let	let	VERB
ejpam-4933	108	21	t	t	PROPN
ejpam-4933	108	22	∈	∈	PROPN
ejpam-4933	109	1	i	i	PRON
ejpam-4933	109	2	\	\	PROPN
ejpam-4933	109	3	{	{	PUNCT
ejpam-4933	109	4	0	0	NUM
ejpam-4933	109	5	,	,	PUNCT
ejpam-4933	109	6	1	1	NUM
ejpam-4933	109	7	}	}	PUNCT
ejpam-4933	109	8	be	be	AUX
ejpam-4933	109	9	such	such	ADJ
ejpam-4933	109	10	that	that	SCONJ
ejpam-4933	109	11	ε(ζ)t	ε(ζ)t	AUX
ejpam-4933	109	12	̸=	̸=	PROPN
ejpam-4933	109	13	∅.	∅.	ADV
ejpam-4933	109	14	let	let	VERB
ejpam-4933	109	15	x	x	PRON
ejpam-4933	109	16	,	,	PUNCT
ejpam-4933	109	17	y	y	PROPN
ejpam-4933	109	18	∈	∈	PROPN
ejpam-4933	109	19	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	109	20	.	.	PUNCT
ejpam-4933	110	1	then	then	ADV
ejpam-4933	110	2	yj(ε	yj(ε	NUM
ejpam-4933	110	3	,	,	PUNCT
ejpam-4933	110	4	ζ(x	ζ(x	NOUN
ejpam-4933	110	5	)	)	PUNCT
ejpam-4933	110	6	)	)	PUNCT
ejpam-4933	111	1	≤	≤	NOUN
ejpam-4933	111	2	t	t	PROPN
ejpam-4933	111	3	and	and	CCONJ
ejpam-4933	111	4	yj(ε	yj(ε	NUM
ejpam-4933	111	5	,	,	PUNCT
ejpam-4933	111	6	ζ(y	ζ(y	PROPN
ejpam-4933	111	7	)	)	PUNCT
ejpam-4933	111	8	)	)	PUNCT
ejpam-4933	111	9	≤	≤	PROPN
ejpam-4933	111	10	t	t	PROPN
ejpam-4933	111	11	,	,	PUNCT
ejpam-4933	111	12	which	which	PRON
ejpam-4933	111	13	imply	imply	VERB
ejpam-4933	111	14	from	from	ADP
ejpam-4933	111	15	(	(	PUNCT
ejpam-4933	111	16	13	13	NUM
ejpam-4933	111	17	)	)	PUNCT
ejpam-4933	111	18	that	that	PRON
ejpam-4933	111	19	yj(ε	yj(ε	ADV
ejpam-4933	111	20	,	,	PUNCT
ejpam-4933	111	21	ζ(x	ζ(x	PROPN
ejpam-4933	111	22	∗	∗	NOUN
ejpam-4933	111	23	y	y	NOUN
ejpam-4933	111	24	)	)	PUNCT
ejpam-4933	111	25	)	)	PUNCT
ejpam-4933	111	26	≤	≤	NOUN
ejpam-4933	112	1	yj(ε	yj(ε	NUM
ejpam-4933	112	2	,	,	PUNCT
ejpam-4933	112	3	ζ(x	ζ(x	NOUN
ejpam-4933	112	4	)	)	PUNCT
ejpam-4933	112	5	)	)	PUNCT
ejpam-4933	113	1	∨	∨	PROPN
ejpam-4933	113	2	yj(ε	yj(ε	NUM
ejpam-4933	113	3	,	,	PUNCT
ejpam-4933	113	4	ζ(y	ζ(y	PROPN
ejpam-4933	113	5	)	)	PUNCT
ejpam-4933	113	6	)	)	PUNCT
ejpam-4933	113	7	≤	≤	NOUN
ejpam-4933	113	8	t.	t.	NOUN
ejpam-4933	113	9	hence	hence	ADV
ejpam-4933	113	10	x	x	PROPN
ejpam-4933	113	11	∗	∗	VERB
ejpam-4933	113	12	y	y	PROPN
ejpam-4933	113	13	∈	∈	PROPN
ejpam-4933	113	14	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	113	15	,	,	PUNCT
ejpam-4933	113	16	and	and	CCONJ
ejpam-4933	113	17	therefore	therefore	ADV
ejpam-4933	113	18	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	113	19	is	be	AUX
ejpam-4933	113	20	a	a	DET
ejpam-4933	113	21	subalgebra	subalgebra	NOUN
ejpam-4933	113	22	of	of	ADP
ejpam-4933	113	23	(	(	PUNCT
ejpam-4933	113	24	x	x	NOUN
ejpam-4933	113	25	,	,	PUNCT
ejpam-4933	113	26	∗	∗	NOUN
ejpam-4933	113	27	,	,	PUNCT
ejpam-4933	113	28	0	0	NUM
ejpam-4933	113	29	)	)	PUNCT
ejpam-4933	113	30	.	.	PUNCT
ejpam-4933	114	1	conversely	conversely	ADV
ejpam-4933	114	2	,	,	PUNCT
ejpam-4933	114	3	suppose	suppose	VERB
ejpam-4933	114	4	that	that	SCONJ
ejpam-4933	114	5	the	the	DET
ejpam-4933	114	6	nonempty	nonempty	ADJ
ejpam-4933	114	7	y	y	NOUN
ejpam-4933	114	8	-	-	PUNCT
ejpam-4933	114	9	level	level	NOUN
ejpam-4933	114	10	set	set	NOUN
ejpam-4933	114	11	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	114	12	is	be	AUX
ejpam-4933	114	13	a	a	DET
ejpam-4933	114	14	subalgebra	subalgebra	NOUN
ejpam-4933	114	15	of	of	ADP
ejpam-4933	114	16	(	(	PUNCT
ejpam-4933	114	17	x	x	NOUN
ejpam-4933	114	18	,	,	PUNCT
ejpam-4933	114	19	∗	∗	NOUN
ejpam-4933	114	20	,	,	PUNCT
ejpam-4933	114	21	0	0	NUM
ejpam-4933	114	22	)	)	PUNCT
ejpam-4933	114	23	for	for	ADP
ejpam-4933	114	24	all	all	DET
ejpam-4933	114	25	t	t	NOUN
ejpam-4933	114	26	∈	∈	PROPN
ejpam-4933	114	27	i	i	PRON
ejpam-4933	114	28	\	\	PROPN
ejpam-4933	114	29	{	{	PUNCT
ejpam-4933	114	30	0	0	NUM
ejpam-4933	114	31	,	,	PUNCT
ejpam-4933	114	32	1	1	NUM
ejpam-4933	114	33	}	}	PUNCT
ejpam-4933	114	34	.	.	PUNCT
ejpam-4933	115	1	if	if	SCONJ
ejpam-4933	115	2	(	(	PUNCT
ejpam-4933	115	3	13	13	NUM
ejpam-4933	115	4	)	)	PUNCT
ejpam-4933	115	5	is	be	AUX
ejpam-4933	115	6	not	not	PART
ejpam-4933	115	7	valid	valid	ADJ
ejpam-4933	115	8	,	,	PUNCT
ejpam-4933	115	9	then	then	ADV
ejpam-4933	115	10	yj(ε	yj(ε	PRON
ejpam-4933	115	11	,	,	PUNCT
ejpam-4933	115	12	ζ(b	ζ(b	PROPN
ejpam-4933	115	13	∗	∗	NOUN
ejpam-4933	115	14	c	c	NOUN
ejpam-4933	115	15	)	)	PUNCT
ejpam-4933	115	16	)	)	PUNCT
ejpam-4933	116	1	>	>	X
ejpam-4933	116	2	t	t	PROPN
ejpam-4933	116	3	≥	≥	PROPN
ejpam-4933	116	4	yj(ε	yj(ε	NUM
ejpam-4933	116	5	,	,	PUNCT
ejpam-4933	116	6	ζ(b	ζ(b	PROPN
ejpam-4933	116	7	)	)	PUNCT
ejpam-4933	116	8	)	)	PUNCT
ejpam-4933	117	1	∨	∨	PROPN
ejpam-4933	117	2	yj(ε	yj(ε	NUM
ejpam-4933	117	3	,	,	PUNCT
ejpam-4933	117	4	ζ(c	ζ(c	NOUN
ejpam-4933	117	5	)	)	PUNCT
ejpam-4933	117	6	)	)	PUNCT
ejpam-4933	117	7	for	for	ADP
ejpam-4933	117	8	some	some	DET
ejpam-4933	117	9	b	b	NOUN
ejpam-4933	117	10	,	,	PUNCT
ejpam-4933	117	11	c	c	PROPN
ejpam-4933	117	12	∈	∈	PROPN
ejpam-4933	117	13	x	x	X
ejpam-4933	117	14	and	and	CCONJ
ejpam-4933	117	15	t	t	PROPN
ejpam-4933	117	16	∈	∈	PROPN
ejpam-4933	118	1	i	i	PRON
ejpam-4933	118	2	\	\	PROPN
ejpam-4933	118	3	{	{	PUNCT
ejpam-4933	118	4	0	0	NUM
ejpam-4933	118	5	,	,	PUNCT
ejpam-4933	118	6	1	1	NUM
ejpam-4933	118	7	}	}	PUNCT
ejpam-4933	118	8	.	.	PUNCT
ejpam-4933	119	1	hence	hence	ADV
ejpam-4933	119	2	b	b	NOUN
ejpam-4933	119	3	,	,	PUNCT
ejpam-4933	119	4	c	c	PROPN
ejpam-4933	119	5	∈	∈	PROPN
ejpam-4933	119	6	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	119	7	and	and	CCONJ
ejpam-4933	119	8	b	b	NOUN
ejpam-4933	119	9	∗	∗	NOUN
ejpam-4933	119	10	c	c	NOUN
ejpam-4933	119	11	/∈	/∈	PUNCT
ejpam-4933	120	1	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	120	2	,	,	PUNCT
ejpam-4933	120	3	which	which	PRON
ejpam-4933	120	4	is	be	AUX
ejpam-4933	120	5	a	a	DET
ejpam-4933	120	6	contradiction	contradiction	NOUN
ejpam-4933	120	7	.	.	PUNCT
ejpam-4933	121	1	therefore	therefore	ADV
ejpam-4933	121	2	yj(ε	yj(ε	NUM
ejpam-4933	121	3	,	,	PUNCT
ejpam-4933	121	4	ζ(x	ζ(x	PROPN
ejpam-4933	121	5	∗	∗	VERB
ejpam-4933	121	6	a	a	NOUN
ejpam-4933	121	7	)	)	PUNCT
ejpam-4933	121	8	)	)	PUNCT
ejpam-4933	122	1	≤	≤	NOUN
ejpam-4933	122	2	yj(ε	yj(ε	NUM
ejpam-4933	122	3	,	,	PUNCT
ejpam-4933	122	4	ζ(x	ζ(x	NOUN
ejpam-4933	122	5	)	)	PUNCT
ejpam-4933	122	6	)	)	PUNCT
ejpam-4933	123	1	∨	∨	PROPN
ejpam-4933	123	2	yj(ε	yj(ε	NUM
ejpam-4933	123	3	,	,	PUNCT
ejpam-4933	123	4	ζ(a	ζ(a	NOUN
ejpam-4933	123	5	)	)	PUNCT
ejpam-4933	123	6	)	)	PUNCT
ejpam-4933	123	7	for	for	ADP
ejpam-4933	123	8	all	all	DET
ejpam-4933	123	9	x	x	NOUN
ejpam-4933	123	10	,	,	PUNCT
ejpam-4933	123	11	a	a	DET
ejpam-4933	123	12	∈	∈	PROPN
ejpam-4933	123	13	x	x	NOUN
ejpam-4933	123	14	,	,	PUNCT
ejpam-4933	123	15	which	which	PRON
ejpam-4933	123	16	shows	show	VERB
ejpam-4933	123	17	that	that	SCONJ
ejpam-4933	123	18	ζ	ζ	NOUN
ejpam-4933	123	19	is	be	AUX
ejpam-4933	123	20	a	a	DET
ejpam-4933	123	21	y	y	PROPN
ejpam-4933	123	22	ε	ε	PROPN
ejpam-4933	123	23	j	j	PROPN
ejpam-4933	123	24	-fuzzy	-fuzzy	PROPN
ejpam-4933	123	25	subalgebra	subalgebra	NOUN
ejpam-4933	123	26	of	of	ADP
ejpam-4933	123	27	(	(	PUNCT
ejpam-4933	123	28	x	x	NOUN
ejpam-4933	123	29	,	,	PUNCT
ejpam-4933	123	30	∗	∗	NOUN
ejpam-4933	123	31	,	,	PUNCT
ejpam-4933	123	32	0	0	NUM
ejpam-4933	123	33	)	)	PUNCT
ejpam-4933	123	34	.	.	PUNCT
ejpam-4933	124	1	definition	definition	NOUN
ejpam-4933	124	2	1	1	NUM
ejpam-4933	124	3	.	.	PUNCT
ejpam-4933	125	1	a	a	DET
ejpam-4933	125	2	fuzzy	fuzzy	ADJ
ejpam-4933	125	3	set	set	VERB
ejpam-4933	125	4	ζ	ζ	NOUN
ejpam-4933	125	5	in	in	ADP
ejpam-4933	125	6	x	x	PROPN
ejpam-4933	125	7	is	be	AUX
ejpam-4933	125	8	called	call	VERB
ejpam-4933	125	9	a	a	DET
ejpam-4933	125	10	y	y	PROPN
ejpam-4933	125	11	ε	ε	PROPN
ejpam-4933	125	12	j	j	PROPN
ejpam-4933	125	13	-fuzzy	-fuzzy	PROPN
ejpam-4933	125	14	ideal	ideal	ADJ
ejpam-4933	125	15	if	if	SCONJ
ejpam-4933	125	16	it	it	PRON
ejpam-4933	125	17	satisfies	satisfy	VERB
ejpam-4933	125	18	:	:	PUNCT
ejpam-4933	125	19	(	(	PUNCT
ejpam-4933	125	20	∀x	∀x	X
ejpam-4933	125	21	∈	∈	PROPN
ejpam-4933	125	22	x)(yj(ε	x)(yj(ε	NOUN
ejpam-4933	125	23	,	,	PUNCT
ejpam-4933	125	24	ζ(0	ζ(0	NOUN
ejpam-4933	125	25	)	)	PUNCT
ejpam-4933	125	26	)	)	PUNCT
ejpam-4933	125	27	≤	≤	NOUN
ejpam-4933	125	28	yj(ε	yj(ε	NUM
ejpam-4933	125	29	,	,	PUNCT
ejpam-4933	125	30	ζ(x	ζ(x	NOUN
ejpam-4933	125	31	)	)	PUNCT
ejpam-4933	125	32	)	)	PUNCT
ejpam-4933	125	33	)	)	PUNCT
ejpam-4933	125	34	,	,	PUNCT
ejpam-4933	125	35	(	(	PUNCT
ejpam-4933	125	36	14	14	NUM
ejpam-4933	125	37	)	)	PUNCT
ejpam-4933	125	38	(	(	PUNCT
ejpam-4933	125	39	∀x	∀x	X
ejpam-4933	125	40	,	,	PUNCT
ejpam-4933	125	41	a	a	DET
ejpam-4933	125	42	∈	∈	PROPN
ejpam-4933	125	43	x)(yj(ε	x)(yj(ε	NOUN
ejpam-4933	125	44	,	,	PUNCT
ejpam-4933	125	45	ζ(x	ζ(x	NOUN
ejpam-4933	125	46	)	)	PUNCT
ejpam-4933	125	47	)	)	PUNCT
ejpam-4933	125	48	≤	≤	NOUN
ejpam-4933	125	49	yj(ε	yj(ε	NOUN
ejpam-4933	125	50	,	,	PUNCT
ejpam-4933	125	51	ζ(x	ζ(x	PROPN
ejpam-4933	125	52	∗	∗	VERB
ejpam-4933	125	53	a	a	NOUN
ejpam-4933	125	54	)	)	PUNCT
ejpam-4933	125	55	)	)	PUNCT
ejpam-4933	125	56	∨	∨	PROPN
ejpam-4933	125	57	yj(ε	yj(ε	NUM
ejpam-4933	125	58	,	,	PUNCT
ejpam-4933	125	59	ζ(a	ζ(a	NOUN
ejpam-4933	125	60	)	)	PUNCT
ejpam-4933	125	61	)	)	PUNCT
ejpam-4933	125	62	)	)	PUNCT
ejpam-4933	125	63	.	.	PUNCT
ejpam-4933	126	1	(	(	PUNCT
ejpam-4933	126	2	15	15	NUM
ejpam-4933	126	3	)	)	PUNCT
ejpam-4933	126	4	example	example	NOUN
ejpam-4933	127	1	1	1	NUM
ejpam-4933	127	2	.	.	PUNCT
ejpam-4933	127	3	let	let	VERB
ejpam-4933	127	4	x	x	PUNCT
ejpam-4933	127	5	=	=	PUNCT
ejpam-4933	127	6	{	{	PUNCT
ejpam-4933	127	7	0	0	NUM
ejpam-4933	127	8	,	,	PUNCT
ejpam-4933	127	9	1	1	NUM
ejpam-4933	127	10	,	,	PUNCT
ejpam-4933	127	11	2	2	NUM
ejpam-4933	127	12	,	,	PUNCT
ejpam-4933	127	13	a	a	PRON
ejpam-4933	127	14	,	,	PUNCT
ejpam-4933	127	15	b	b	X
ejpam-4933	127	16	}	}	PUNCT
ejpam-4933	127	17	be	be	AUX
ejpam-4933	127	18	a	a	DET
ejpam-4933	127	19	set	set	NOUN
ejpam-4933	127	20	with	with	ADP
ejpam-4933	127	21	the	the	DET
ejpam-4933	127	22	binary	binary	PROPN
ejpam-4933	127	23	operation	operation	NOUN
ejpam-4933	127	24	“	"	PUNCT
ejpam-4933	127	25	∗	∗	NOUN
ejpam-4933	127	26	”	"	PUNCT
ejpam-4933	127	27	given	give	VERB
ejpam-4933	127	28	by	by	ADP
ejpam-4933	127	29	table	table	NOUN
ejpam-4933	127	30	1	1	NUM
ejpam-4933	127	31	.	.	PUNCT
ejpam-4933	127	32	table	table	NOUN
ejpam-4933	127	33	1	1	NUM
ejpam-4933	127	34	:	:	PUNCT
ejpam-4933	127	35	cayley	cayley	ADJ
ejpam-4933	127	36	table	table	NOUN
ejpam-4933	127	37	for	for	ADP
ejpam-4933	127	38	the	the	DET
ejpam-4933	127	39	binary	binary	PROPN
ejpam-4933	127	40	operation	operation	NOUN
ejpam-4933	127	41	“	"	PUNCT
ejpam-4933	127	42	∗	∗	NOUN
ejpam-4933	127	43	”	"	PUNCT
ejpam-4933	127	44	∗	∗	NOUN
ejpam-4933	127	45	0	0	NUM
ejpam-4933	127	46	1	1	NUM
ejpam-4933	127	47	2	2	NUM
ejpam-4933	127	48	a	a	DET
ejpam-4933	127	49	b	b	NOUN
ejpam-4933	127	50	0	0	NUM
ejpam-4933	127	51	0	0	NUM
ejpam-4933	127	52	0	0	NUM
ejpam-4933	127	53	0	0	NUM
ejpam-4933	127	54	a	a	DET
ejpam-4933	127	55	a	a	DET
ejpam-4933	127	56	1	1	NUM
ejpam-4933	127	57	1	1	NUM
ejpam-4933	127	58	0	0	NUM
ejpam-4933	127	59	0	0	NUM
ejpam-4933	127	60	a	a	DET
ejpam-4933	127	61	a	a	DET
ejpam-4933	127	62	2	2	NUM
ejpam-4933	127	63	2	2	NUM
ejpam-4933	127	64	2	2	NUM
ejpam-4933	127	65	0	0	NUM
ejpam-4933	127	66	b	b	NOUN
ejpam-4933	127	67	a	a	DET
ejpam-4933	127	68	a	a	PRON
ejpam-4933	127	69	a	a	DET
ejpam-4933	127	70	a	a	DET
ejpam-4933	127	71	a	a	DET
ejpam-4933	127	72	0	0	NUM
ejpam-4933	127	73	0	0	NUM
ejpam-4933	127	74	b	b	PROPN
ejpam-4933	127	75	b	b	X
ejpam-4933	127	76	b	b	PROPN
ejpam-4933	127	77	a	a	DET
ejpam-4933	127	78	2	2	NUM
ejpam-4933	127	79	0	0	NUM
ejpam-4933	127	80	e.	e.	PROPN
ejpam-4933	127	81	h.	h.	PROPN
ejpam-4933	127	82	roh	roh	PROPN
ejpam-4933	127	83	,	,	PUNCT
ejpam-4933	127	84	e.	e.	PROPN
ejpam-4933	127	85	yang	yang	PROPN
ejpam-4933	127	86	,	,	PUNCT
ejpam-4933	127	87	y.	y.	PROPN
ejpam-4933	127	88	b.	b.	PROPN
ejpam-4933	127	89	jun	jun	PROPN
ejpam-4933	127	90	/	/	SYM
ejpam-4933	127	91	eur	eur	PROPN
ejpam-4933	127	92	.	.	PUNCT
ejpam-4933	128	1	j.	j.	PROPN
ejpam-4933	128	2	pure	pure	PROPN
ejpam-4933	128	3	appl	appl	PROPN
ejpam-4933	128	4	.	.	PROPN
ejpam-4933	128	5	math	math	PROPN
ejpam-4933	128	6	,	,	PUNCT
ejpam-4933	128	7	16	16	NUM
ejpam-4933	128	8	(	(	PUNCT
ejpam-4933	128	9	4	4	NUM
ejpam-4933	128	10	)	)	PUNCT
ejpam-4933	128	11	(	(	PUNCT
ejpam-4933	128	12	2023	2023	NUM
ejpam-4933	128	13	)	)	PUNCT
ejpam-4933	128	14	,	,	PUNCT
ejpam-4933	128	15	2009	2009	NUM
ejpam-4933	128	16	-	-	SYM
ejpam-4933	128	17	2024	2024	NUM
ejpam-4933	128	18	2015	2015	NUM
ejpam-4933	128	19	then	then	ADV
ejpam-4933	128	20	(	(	PUNCT
ejpam-4933	128	21	x	x	X
ejpam-4933	128	22	,	,	PUNCT
ejpam-4933	128	23	∗	∗	NOUN
ejpam-4933	128	24	,	,	PUNCT
ejpam-4933	128	25	0	0	NUM
ejpam-4933	128	26	)	)	PUNCT
ejpam-4933	128	27	is	be	AUX
ejpam-4933	128	28	a	a	DET
ejpam-4933	128	29	bci	bci	NOUN
ejpam-4933	128	30	-	-	NOUN
ejpam-4933	128	31	algebra	algebra	NOUN
ejpam-4933	128	32	(	(	PUNCT
ejpam-4933	128	33	see	see	VERB
ejpam-4933	128	34	[	[	X
ejpam-4933	128	35	2	2	NUM
ejpam-4933	128	36	]	]	PUNCT
ejpam-4933	128	37	)	)	PUNCT
ejpam-4933	128	38	.	.	PUNCT
ejpam-4933	129	1	define	define	VERB
ejpam-4933	129	2	a	a	DET
ejpam-4933	129	3	fuzzy	fuzzy	ADJ
ejpam-4933	129	4	set	set	VERB
ejpam-4933	129	5	ζ	ζ	NOUN
ejpam-4933	129	6	in	in	ADP
ejpam-4933	129	7	x	x	PUNCT
ejpam-4933	129	8	as	as	SCONJ
ejpam-4933	129	9	follows	follow	VERB
ejpam-4933	129	10	:	:	PUNCT
ejpam-4933	129	11	ζ	ζ	NOUN
ejpam-4933	129	12	:	:	PUNCT
ejpam-4933	129	13	x	x	X
ejpam-4933	129	14	→	→	SYM
ejpam-4933	129	15	[	[	X
ejpam-4933	129	16	0	0	NUM
ejpam-4933	129	17	,	,	PUNCT
ejpam-4933	129	18	1	1	NUM
ejpam-4933	129	19	]	]	PUNCT
ejpam-4933	129	20	,	,	PUNCT
ejpam-4933	129	21	y	y	PROPN
ejpam-4933	129	22	7→	7→	PROPN
ejpam-4933	129	23			VERB
ejpam-4933	129	24	0.68	0.68	NUM
ejpam-4933	129	25	if	if	SCONJ
ejpam-4933	129	26	y	y	PROPN
ejpam-4933	129	27	=	=	SYM
ejpam-4933	129	28	0	0	NUM
ejpam-4933	129	29	,	,	PUNCT
ejpam-4933	129	30	0.61	0.61	NUM
ejpam-4933	129	31	if	if	SCONJ
ejpam-4933	129	32	y	y	NOUN
ejpam-4933	129	33	=	=	SYM
ejpam-4933	129	34	1	1	NUM
ejpam-4933	129	35	,	,	PUNCT
ejpam-4933	129	36	0.46	0.46	NUM
ejpam-4933	129	37	if	if	SCONJ
ejpam-4933	129	38	y	y	PROPN
ejpam-4933	129	39	=	=	SYM
ejpam-4933	129	40	2	2	NUM
ejpam-4933	129	41	,	,	PUNCT
ejpam-4933	129	42	0.54	0.54	NUM
ejpam-4933	129	43	if	if	SCONJ
ejpam-4933	129	44	y	y	PROPN
ejpam-4933	129	45	=	=	PUNCT
ejpam-4933	129	46	a	a	PROPN
ejpam-4933	129	47	,	,	PUNCT
ejpam-4933	129	48	0.46	0.46	NUM
ejpam-4933	129	49	if	if	SCONJ
ejpam-4933	129	50	y	y	PROPN
ejpam-4933	129	51	=	=	SYM
ejpam-4933	129	52	b.	b.	PROPN
ejpam-4933	129	53	it	it	PRON
ejpam-4933	129	54	is	be	AUX
ejpam-4933	129	55	routine	routine	ADJ
ejpam-4933	129	56	to	to	PART
ejpam-4933	129	57	verify	verify	VERB
ejpam-4933	129	58	that	that	SCONJ
ejpam-4933	129	59	ζ	ζ	NOUN
ejpam-4933	129	60	is	be	AUX
ejpam-4933	129	61	a	a	DET
ejpam-4933	129	62	y	y	PROPN
ejpam-4933	129	63	ε	ε	PROPN
ejpam-4933	129	64	j	j	PROPN
ejpam-4933	129	65	-fuzzy	-fuzzy	PROPN
ejpam-4933	129	66	ideal	ideal	NOUN
ejpam-4933	129	67	of	of	ADP
ejpam-4933	129	68	(	(	PUNCT
ejpam-4933	129	69	x	x	NOUN
ejpam-4933	129	70	,	,	PUNCT
ejpam-4933	129	71	∗	∗	NOUN
ejpam-4933	129	72	,	,	PUNCT
ejpam-4933	129	73	0	0	NUM
ejpam-4933	129	74	)	)	PUNCT
ejpam-4933	129	75	for	for	ADP
ejpam-4933	129	76	all	all	DET
ejpam-4933	129	77	ε	ε	PROPN
ejpam-4933	129	78	∈	∈	PROPN
ejpam-4933	129	79	(	(	PUNCT
ejpam-4933	129	80	0	0	NUM
ejpam-4933	129	81	,	,	PUNCT
ejpam-4933	129	82	1	1	NUM
ejpam-4933	129	83	)	)	PUNCT
ejpam-4933	129	84	.	.	PUNCT
ejpam-4933	130	1	proposition	proposition	NOUN
ejpam-4933	130	2	3	3	NUM
ejpam-4933	130	3	.	.	PUNCT
ejpam-4933	131	1	every	every	DET
ejpam-4933	131	2	y	y	PROPN
ejpam-4933	131	3	ε	ε	PROPN
ejpam-4933	131	4	j	j	PROPN
ejpam-4933	131	5	-fuzzy	-fuzzy	PROPN
ejpam-4933	131	6	ideal	ideal	ADJ
ejpam-4933	131	7	ζ	ζ	NOUN
ejpam-4933	131	8	of	of	ADP
ejpam-4933	131	9	(	(	PUNCT
ejpam-4933	131	10	x	x	NOUN
ejpam-4933	131	11	,	,	PUNCT
ejpam-4933	131	12	∗	∗	NOUN
ejpam-4933	131	13	,	,	PUNCT
ejpam-4933	131	14	0	0	NUM
ejpam-4933	131	15	)	)	PUNCT
ejpam-4933	131	16	satisfies	satisfie	NOUN
ejpam-4933	131	17	:	:	PUNCT
ejpam-4933	131	18	(	(	PUNCT
ejpam-4933	131	19	∀x	∀x	X
ejpam-4933	131	20	,	,	PUNCT
ejpam-4933	131	21	a	a	DET
ejpam-4933	131	22	∈	∈	PROPN
ejpam-4933	131	23	x)(x	x)(x	PUNCT
ejpam-4933	131	24	≤x	≤x	VERB
ejpam-4933	131	25	a	a	DET
ejpam-4933	131	26	⇒	⇒	NOUN
ejpam-4933	131	27	yj(ε	yj(ε	NUM
ejpam-4933	131	28	,	,	PUNCT
ejpam-4933	131	29	ζ(x	ζ(x	NOUN
ejpam-4933	131	30	)	)	PUNCT
ejpam-4933	131	31	)	)	PUNCT
ejpam-4933	132	1	≤	≤	NOUN
ejpam-4933	132	2	yj(ε	yj(ε	NUM
ejpam-4933	132	3	,	,	PUNCT
ejpam-4933	132	4	ζ(a	ζ(a	NOUN
ejpam-4933	132	5	)	)	PUNCT
ejpam-4933	132	6	)	)	PUNCT
ejpam-4933	132	7	)	)	PUNCT
ejpam-4933	132	8	.	.	PUNCT
ejpam-4933	133	1	(	(	PUNCT
ejpam-4933	133	2	16	16	NUM
ejpam-4933	133	3	)	)	PUNCT
ejpam-4933	133	4	(	(	PUNCT
ejpam-4933	133	5	∀x	∀x	X
ejpam-4933	133	6	,	,	PUNCT
ejpam-4933	133	7	a	a	PRON
ejpam-4933	133	8	,	,	PUNCT
ejpam-4933	133	9	y	y	PROPN
ejpam-4933	133	10	∈	∈	PROPN
ejpam-4933	133	11	x)(x	x)(x	PROPN
ejpam-4933	133	12	∗	∗	VERB
ejpam-4933	133	13	a	a	DET
ejpam-4933	133	14	≤x	≤x	PROPN
ejpam-4933	133	15	y	y	PROPN
ejpam-4933	133	16	⇒	⇒	PROPN
ejpam-4933	133	17	yj(ε	yj(ε	SYM
ejpam-4933	133	18	,	,	PUNCT
ejpam-4933	133	19	ζ(x	ζ(x	NOUN
ejpam-4933	133	20	)	)	PUNCT
ejpam-4933	133	21	)	)	PUNCT
ejpam-4933	134	1	≤	≤	NOUN
ejpam-4933	134	2	yj(ε	yj(ε	NUM
ejpam-4933	134	3	,	,	PUNCT
ejpam-4933	134	4	ζ(a	ζ(a	NOUN
ejpam-4933	134	5	)	)	PUNCT
ejpam-4933	134	6	)	)	PUNCT
ejpam-4933	135	1	∨	∨	PROPN
ejpam-4933	135	2	yj(ε	yj(ε	NUM
ejpam-4933	135	3	,	,	PUNCT
ejpam-4933	135	4	ζ(y	ζ(y	PROPN
ejpam-4933	135	5	)	)	PUNCT
ejpam-4933	135	6	)	)	PUNCT
ejpam-4933	135	7	)	)	PUNCT
ejpam-4933	135	8	.	.	PUNCT
ejpam-4933	136	1	(	(	PUNCT
ejpam-4933	136	2	17	17	NUM
ejpam-4933	136	3	)	)	PUNCT
ejpam-4933	136	4	proof	proof	NOUN
ejpam-4933	136	5	.	.	PUNCT
ejpam-4933	137	1	let	let	VERB
ejpam-4933	137	2	ζ	ζ	NOUN
ejpam-4933	137	3	be	be	AUX
ejpam-4933	137	4	a	a	DET
ejpam-4933	137	5	y	y	PROPN
ejpam-4933	137	6	ε	ε	PROPN
ejpam-4933	137	7	j	j	PROPN
ejpam-4933	137	8	-fuzzy	-fuzzy	PROPN
ejpam-4933	137	9	ideal	ideal	NOUN
ejpam-4933	137	10	of	of	ADP
ejpam-4933	137	11	(	(	PUNCT
ejpam-4933	137	12	x	x	NOUN
ejpam-4933	137	13	,	,	PUNCT
ejpam-4933	137	14	∗	∗	NOUN
ejpam-4933	137	15	,	,	PUNCT
ejpam-4933	137	16	0	0	NUM
ejpam-4933	137	17	)	)	PUNCT
ejpam-4933	137	18	and	and	CCONJ
ejpam-4933	137	19	let	let	VERB
ejpam-4933	137	20	x	x	PRON
ejpam-4933	137	21	,	,	PUNCT
ejpam-4933	137	22	a	a	DET
ejpam-4933	137	23	∈	∈	NOUN
ejpam-4933	137	24	x	x	AUX
ejpam-4933	137	25	be	be	AUX
ejpam-4933	137	26	such	such	ADJ
ejpam-4933	137	27	that	that	SCONJ
ejpam-4933	137	28	x	x	PUNCT
ejpam-4933	137	29	≤x	≤x	NOUN
ejpam-4933	137	30	a.	a.	NOUN
ejpam-4933	137	31	then	then	ADV
ejpam-4933	137	32	x	x	X
ejpam-4933	137	33	∗	∗	VERB
ejpam-4933	137	34	a	a	PRON
ejpam-4933	137	35	=	=	NOUN
ejpam-4933	137	36	0	0	NUM
ejpam-4933	137	37	,	,	PUNCT
ejpam-4933	137	38	and	and	CCONJ
ejpam-4933	137	39	so	so	ADV
ejpam-4933	137	40	yj(ε	yj(ε	ADJ
ejpam-4933	137	41	,	,	PUNCT
ejpam-4933	137	42	ζ(x	ζ(x	NOUN
ejpam-4933	137	43	)	)	PUNCT
ejpam-4933	137	44	)	)	PUNCT
ejpam-4933	137	45	≤	≤	NOUN
ejpam-4933	137	46	yj(ε	yj(ε	NOUN
ejpam-4933	137	47	,	,	PUNCT
ejpam-4933	137	48	ζ(x	ζ(x	PROPN
ejpam-4933	137	49	∗	∗	VERB
ejpam-4933	137	50	a	a	NOUN
ejpam-4933	137	51	)	)	PUNCT
ejpam-4933	137	52	)	)	PUNCT
ejpam-4933	137	53	∨	∨	PROPN
ejpam-4933	137	54	yj(ε	yj(ε	NUM
ejpam-4933	137	55	,	,	PUNCT
ejpam-4933	137	56	ζ(a	ζ(a	NOUN
ejpam-4933	137	57	)	)	PUNCT
ejpam-4933	137	58	)	)	PUNCT
ejpam-4933	138	1	=	=	PUNCT
ejpam-4933	138	2	yj(ε	yj(ε	ADJ
ejpam-4933	138	3	,	,	PUNCT
ejpam-4933	138	4	ζ(0	ζ(0	NOUN
ejpam-4933	138	5	)	)	PUNCT
ejpam-4933	138	6	)	)	PUNCT
ejpam-4933	139	1	∨	∨	PROPN
ejpam-4933	139	2	yj(ε	yj(ε	NUM
ejpam-4933	139	3	,	,	PUNCT
ejpam-4933	139	4	ζ(a	ζ(a	NOUN
ejpam-4933	139	5	)	)	PUNCT
ejpam-4933	139	6	)	)	PUNCT
ejpam-4933	140	1	=	=	PUNCT
ejpam-4933	140	2	yj(ε	yj(ε	ADJ
ejpam-4933	140	3	,	,	PUNCT
ejpam-4933	140	4	ζ(a	ζ(a	NOUN
ejpam-4933	140	5	)	)	PUNCT
ejpam-4933	140	6	)	)	PUNCT
ejpam-4933	140	7	by	by	ADP
ejpam-4933	140	8	(	(	PUNCT
ejpam-4933	140	9	14	14	NUM
ejpam-4933	140	10	)	)	PUNCT
ejpam-4933	140	11	and	and	CCONJ
ejpam-4933	140	12	(	(	PUNCT
ejpam-4933	140	13	15	15	NUM
ejpam-4933	140	14	)	)	PUNCT
ejpam-4933	140	15	.	.	PUNCT
ejpam-4933	141	1	thus	thus	ADV
ejpam-4933	141	2	(	(	PUNCT
ejpam-4933	141	3	16	16	NUM
ejpam-4933	141	4	)	)	PUNCT
ejpam-4933	141	5	is	be	AUX
ejpam-4933	141	6	valid	valid	ADJ
ejpam-4933	141	7	.	.	PUNCT
ejpam-4933	142	1	let	let	VERB
ejpam-4933	142	2	x	x	PRON
ejpam-4933	142	3	,	,	PUNCT
ejpam-4933	142	4	a	a	PRON
ejpam-4933	142	5	,	,	PUNCT
ejpam-4933	142	6	y	y	PROPN
ejpam-4933	142	7	∈	∈	PROPN
ejpam-4933	142	8	x	x	AUX
ejpam-4933	142	9	be	be	AUX
ejpam-4933	142	10	such	such	ADJ
ejpam-4933	142	11	that	that	SCONJ
ejpam-4933	142	12	x	x	PROPN
ejpam-4933	142	13	∗	∗	VERB
ejpam-4933	142	14	a	a	DET
ejpam-4933	142	15	≤x	≤x	PROPN
ejpam-4933	142	16	y.	y.	PROPN
ejpam-4933	142	17	then	then	ADV
ejpam-4933	142	18	yj(ε	yj(ε	NUM
ejpam-4933	142	19	,	,	PUNCT
ejpam-4933	142	20	ζ(x	ζ(x	PROPN
ejpam-4933	142	21	∗	∗	VERB
ejpam-4933	142	22	a	a	NOUN
ejpam-4933	142	23	)	)	PUNCT
ejpam-4933	142	24	)	)	PUNCT
ejpam-4933	142	25	≤	≤	NOUN
ejpam-4933	142	26	yj(ε	yj(ε	X
ejpam-4933	142	27	,	,	PUNCT
ejpam-4933	142	28	ζ((x	ζ((x	ADV
ejpam-4933	142	29	∗	∗	VERB
ejpam-4933	142	30	a	a	PRON
ejpam-4933	142	31	)	)	PUNCT
ejpam-4933	142	32	∗	∗	NOUN
ejpam-4933	142	33	y	y	PROPN
ejpam-4933	142	34	)	)	PUNCT
ejpam-4933	142	35	)	)	PUNCT
ejpam-4933	143	1	∨	∨	PROPN
ejpam-4933	143	2	yj(ε	yj(ε	NUM
ejpam-4933	143	3	,	,	PUNCT
ejpam-4933	143	4	ζ(y	ζ(y	PROPN
ejpam-4933	143	5	)	)	PUNCT
ejpam-4933	143	6	)	)	PUNCT
ejpam-4933	144	1	=	=	PUNCT
ejpam-4933	144	2	yj(ε	yj(ε	ADJ
ejpam-4933	144	3	,	,	PUNCT
ejpam-4933	144	4	ζ(0	ζ(0	NOUN
ejpam-4933	144	5	)	)	PUNCT
ejpam-4933	144	6	)	)	PUNCT
ejpam-4933	145	1	∨	∨	PROPN
ejpam-4933	145	2	yj(ε	yj(ε	NUM
ejpam-4933	145	3	,	,	PUNCT
ejpam-4933	145	4	ζ(y	ζ(y	PROPN
ejpam-4933	145	5	)	)	PUNCT
ejpam-4933	145	6	)	)	PUNCT
ejpam-4933	146	1	=	=	SYM
ejpam-4933	146	2	yj(ε	yj(ε	PROPN
ejpam-4933	146	3	,	,	PUNCT
ejpam-4933	146	4	ζ(y	ζ(y	PROPN
ejpam-4933	146	5	)	)	PUNCT
ejpam-4933	146	6	)	)	PUNCT
ejpam-4933	146	7	,	,	PUNCT
ejpam-4933	146	8	and	and	CCONJ
ejpam-4933	146	9	thus	thus	ADV
ejpam-4933	146	10	yj(ε	yj(ε	NUM
ejpam-4933	146	11	,	,	PUNCT
ejpam-4933	146	12	ζ(x	ζ(x	NOUN
ejpam-4933	146	13	)	)	PUNCT
ejpam-4933	146	14	)	)	PUNCT
ejpam-4933	146	15	≤	≤	NOUN
ejpam-4933	146	16	yj(ε	yj(ε	NOUN
ejpam-4933	146	17	,	,	PUNCT
ejpam-4933	146	18	ζ(x	ζ(x	PROPN
ejpam-4933	146	19	∗	∗	VERB
ejpam-4933	146	20	a	a	NOUN
ejpam-4933	146	21	)	)	PUNCT
ejpam-4933	146	22	)	)	PUNCT
ejpam-4933	146	23	∨	∨	PROPN
ejpam-4933	146	24	yj(ε	yj(ε	NUM
ejpam-4933	146	25	,	,	PUNCT
ejpam-4933	146	26	ζ(a	ζ(a	NOUN
ejpam-4933	146	27	)	)	PUNCT
ejpam-4933	146	28	)	)	PUNCT
ejpam-4933	147	1	≤	≤	PROPN
ejpam-4933	147	2	yj(ε	yj(ε	NUM
ejpam-4933	147	3	,	,	PUNCT
ejpam-4933	147	4	ζ(y	ζ(y	PROPN
ejpam-4933	147	5	)	)	PUNCT
ejpam-4933	147	6	)	)	PUNCT
ejpam-4933	148	1	∨	∨	PROPN
ejpam-4933	148	2	yj(ε	yj(ε	NUM
ejpam-4933	148	3	,	,	PUNCT
ejpam-4933	148	4	ζ(a	ζ(a	NOUN
ejpam-4933	148	5	)	)	PUNCT
ejpam-4933	148	6	)	)	PUNCT
ejpam-4933	148	7	.	.	PUNCT
ejpam-4933	149	1	this	this	PRON
ejpam-4933	149	2	completes	complete	VERB
ejpam-4933	149	3	the	the	DET
ejpam-4933	149	4	proof	proof	NOUN
ejpam-4933	149	5	.	.	PUNCT
ejpam-4933	150	1	corollary	corollary	ADJ
ejpam-4933	150	2	1	1	NUM
ejpam-4933	150	3	.	.	PUNCT
ejpam-4933	151	1	if	if	SCONJ
ejpam-4933	151	2	ζ	ζ	NOUN
ejpam-4933	151	3	is	be	AUX
ejpam-4933	151	4	a	a	DET
ejpam-4933	151	5	fuzzy	fuzzy	ADJ
ejpam-4933	151	6	ideal	ideal	NOUN
ejpam-4933	151	7	of	of	ADP
ejpam-4933	151	8	(	(	PUNCT
ejpam-4933	151	9	x	x	NOUN
ejpam-4933	151	10	,	,	PUNCT
ejpam-4933	151	11	∗	∗	NOUN
ejpam-4933	151	12	,	,	PUNCT
ejpam-4933	151	13	0	0	NUM
ejpam-4933	151	14	)	)	PUNCT
ejpam-4933	151	15	,	,	PUNCT
ejpam-4933	151	16	then	then	ADV
ejpam-4933	151	17	its	its	PRON
ejpam-4933	151	18	y	y	PROPN
ejpam-4933	151	19	ε	ε	PROPN
ejpam-4933	151	20	j	j	PROPN
ejpam-4933	151	21	-fuzzy	-fuzzy	PROPN
ejpam-4933	151	22	set	set	VERB
ejpam-4933	151	23	ε(ζ	ε(ζ	NOUN
ejpam-4933	151	24	)	)	PUNCT
ejpam-4933	151	25	satisfies	satisfie	NOUN
ejpam-4933	151	26	:	:	PUNCT
ejpam-4933	151	27	(	(	PUNCT
ejpam-4933	151	28	∀x	∀x	X
ejpam-4933	151	29	,	,	PUNCT
ejpam-4933	151	30	a	a	DET
ejpam-4933	151	31	∈	∈	PROPN
ejpam-4933	151	32	x)(x	x)(x	PUNCT
ejpam-4933	151	33	≤x	≤x	VERB
ejpam-4933	151	34	a	a	DET
ejpam-4933	151	35	⇒	⇒	NOUN
ejpam-4933	151	36	ε(ζ)(x	ε(ζ)(x	NUM
ejpam-4933	151	37	)	)	PUNCT
ejpam-4933	151	38	≤	≤	NOUN
ejpam-4933	151	39	ε(ζ)(a	ε(ζ)(a	PROPN
ejpam-4933	151	40	)	)	PUNCT
ejpam-4933	151	41	)	)	PUNCT
ejpam-4933	151	42	.	.	PUNCT
ejpam-4933	152	1	(	(	PUNCT
ejpam-4933	152	2	∀x	∀x	X
ejpam-4933	152	3	,	,	PUNCT
ejpam-4933	152	4	a	a	PRON
ejpam-4933	152	5	,	,	PUNCT
ejpam-4933	152	6	y	y	PROPN
ejpam-4933	152	7	∈	∈	PROPN
ejpam-4933	152	8	x)(x	x)(x	PROPN
ejpam-4933	152	9	∗	∗	VERB
ejpam-4933	152	10	a	a	DET
ejpam-4933	152	11	≤x	≤x	PROPN
ejpam-4933	152	12	y	y	PROPN
ejpam-4933	152	13	⇒	⇒	PROPN
ejpam-4933	152	14	ε(ζ)(x	ε(ζ)(x	NUM
ejpam-4933	152	15	)	)	PUNCT
ejpam-4933	152	16	≤	≤	NOUN
ejpam-4933	152	17	ε(ζ)(a	ε(ζ)(a	ADJ
ejpam-4933	152	18	)	)	PUNCT
ejpam-4933	152	19	∨	∨	NUM
ejpam-4933	152	20	ε(ζ)(y	ε(ζ)(y	ADJ
ejpam-4933	152	21	)	)	PUNCT
ejpam-4933	152	22	)	)	PUNCT
ejpam-4933	152	23	.	.	PUNCT
ejpam-4933	153	1	theorem	theorem	NOUN
ejpam-4933	153	2	2	2	NUM
ejpam-4933	153	3	.	.	PUNCT
ejpam-4933	153	4	in	in	ADP
ejpam-4933	153	5	a	a	DET
ejpam-4933	153	6	bck	bck	NOUN
ejpam-4933	153	7	-	-	PUNCT
ejpam-4933	153	8	algebra	algebra	NOUN
ejpam-4933	153	9	(	(	PUNCT
ejpam-4933	153	10	x	x	X
ejpam-4933	153	11	,	,	PUNCT
ejpam-4933	153	12	∗	∗	NOUN
ejpam-4933	153	13	,	,	PUNCT
ejpam-4933	153	14	0	0	NUM
ejpam-4933	153	15	)	)	PUNCT
ejpam-4933	153	16	,	,	PUNCT
ejpam-4933	153	17	every	every	DET
ejpam-4933	153	18	y	y	PROPN
ejpam-4933	153	19	ε	ε	PROPN
ejpam-4933	153	20	j	j	PROPN
ejpam-4933	153	21	-fuzzy	-fuzzy	PROPN
ejpam-4933	153	22	ideal	ideal	NOUN
ejpam-4933	153	23	is	be	AUX
ejpam-4933	153	24	a	a	DET
ejpam-4933	153	25	y	y	PROPN
ejpam-4933	153	26	ε	ε	PROPN
ejpam-4933	153	27	j	j	PROPN
ejpam-4933	153	28	-fuzzy	-fuzzy	PROPN
ejpam-4933	153	29	subalgebra	subalgebra	NOUN
ejpam-4933	153	30	for	for	ADP
ejpam-4933	153	31	all	all	DET
ejpam-4933	153	32	ε	ε	PROPN
ejpam-4933	153	33	∈	∈	PROPN
ejpam-4933	153	34	(	(	PUNCT
ejpam-4933	153	35	0	0	NUM
ejpam-4933	153	36	,	,	PUNCT
ejpam-4933	153	37	1	1	NUM
ejpam-4933	153	38	)	)	PUNCT
ejpam-4933	153	39	.	.	PUNCT
ejpam-4933	154	1	proof	proof	NOUN
ejpam-4933	154	2	.	.	PUNCT
ejpam-4933	155	1	let	let	VERB
ejpam-4933	155	2	ζ	ζ	NOUN
ejpam-4933	155	3	be	be	AUX
ejpam-4933	155	4	a	a	DET
ejpam-4933	155	5	y	y	PROPN
ejpam-4933	155	6	ε	ε	PROPN
ejpam-4933	155	7	j	j	PROPN
ejpam-4933	155	8	-fuzzy	-fuzzy	PROPN
ejpam-4933	155	9	ideal	ideal	NOUN
ejpam-4933	155	10	of	of	ADP
ejpam-4933	155	11	a	a	DET
ejpam-4933	155	12	bck	bck	NOUN
ejpam-4933	155	13	-	-	PUNCT
ejpam-4933	155	14	algebra	algebra	NOUN
ejpam-4933	155	15	(	(	PUNCT
ejpam-4933	155	16	x	x	X
ejpam-4933	155	17	,	,	PUNCT
ejpam-4933	155	18	∗	∗	NOUN
ejpam-4933	155	19	,	,	PUNCT
ejpam-4933	155	20	0	0	NUM
ejpam-4933	155	21	)	)	PUNCT
ejpam-4933	155	22	for	for	ADP
ejpam-4933	155	23	all	all	DET
ejpam-4933	155	24	ε	ε	PROPN
ejpam-4933	155	25	∈	∈	PROPN
ejpam-4933	155	26	(	(	PUNCT
ejpam-4933	155	27	0	0	NUM
ejpam-4933	155	28	,	,	PUNCT
ejpam-4933	155	29	1	1	NUM
ejpam-4933	155	30	)	)	PUNCT
ejpam-4933	155	31	.	.	PUNCT
ejpam-4933	156	1	the	the	DET
ejpam-4933	156	2	combination	combination	NOUN
ejpam-4933	156	3	of	of	ADP
ejpam-4933	156	4	(	(	PUNCT
ejpam-4933	156	5	5	5	NUM
ejpam-4933	156	6	)	)	PUNCT
ejpam-4933	156	7	and	and	CCONJ
ejpam-4933	156	8	(	(	PUNCT
ejpam-4933	156	9	16	16	NUM
ejpam-4933	156	10	)	)	PUNCT
ejpam-4933	156	11	induces	induce	VERB
ejpam-4933	156	12	yj(ε	yj(ε	NOUN
ejpam-4933	156	13	,	,	PUNCT
ejpam-4933	156	14	ζ(x	ζ(x	PROPN
ejpam-4933	156	15	∗	∗	VERB
ejpam-4933	156	16	a	a	NOUN
ejpam-4933	156	17	)	)	PUNCT
ejpam-4933	156	18	)	)	PUNCT
ejpam-4933	156	19	≤	≤	NOUN
ejpam-4933	156	20	yj(ε	yj(ε	NUM
ejpam-4933	156	21	,	,	PUNCT
ejpam-4933	156	22	ζ(x	ζ(x	NOUN
ejpam-4933	156	23	)	)	PUNCT
ejpam-4933	156	24	)	)	PUNCT
ejpam-4933	156	25	,	,	PUNCT
ejpam-4933	156	26	and	and	CCONJ
ejpam-4933	156	27	so	so	ADV
ejpam-4933	156	28	yj(ε	yj(ε	ADJ
ejpam-4933	156	29	,	,	PUNCT
ejpam-4933	156	30	ζ(x	ζ(x	PROPN
ejpam-4933	156	31	∗	∗	VERB
ejpam-4933	156	32	a	a	NOUN
ejpam-4933	156	33	)	)	PUNCT
ejpam-4933	156	34	)	)	PUNCT
ejpam-4933	156	35	≤	≤	NOUN
ejpam-4933	156	36	yj(ε	yj(ε	NUM
ejpam-4933	156	37	,	,	PUNCT
ejpam-4933	156	38	ζ(x	ζ(x	NOUN
ejpam-4933	156	39	)	)	PUNCT
ejpam-4933	156	40	)	)	PUNCT
ejpam-4933	157	1	≤	≤	NOUN
ejpam-4933	157	2	yj(ε	yj(ε	NOUN
ejpam-4933	157	3	,	,	PUNCT
ejpam-4933	157	4	ζ(x	ζ(x	PROPN
ejpam-4933	157	5	∗	∗	VERB
ejpam-4933	157	6	a	a	NOUN
ejpam-4933	157	7	)	)	PUNCT
ejpam-4933	157	8	)	)	PUNCT
ejpam-4933	158	1	∨	∨	PROPN
ejpam-4933	158	2	yj(ε	yj(ε	NUM
ejpam-4933	158	3	,	,	PUNCT
ejpam-4933	158	4	ζ(a	ζ(a	NOUN
ejpam-4933	158	5	)	)	PUNCT
ejpam-4933	158	6	)	)	PUNCT
ejpam-4933	159	1	≤	≤	NOUN
ejpam-4933	159	2	yj(ε	yj(ε	NUM
ejpam-4933	159	3	,	,	PUNCT
ejpam-4933	159	4	ζ(x	ζ(x	NOUN
ejpam-4933	159	5	)	)	PUNCT
ejpam-4933	159	6	)	)	PUNCT
ejpam-4933	160	1	∨	∨	PROPN
ejpam-4933	160	2	yj(ε	yj(ε	NUM
ejpam-4933	160	3	,	,	PUNCT
ejpam-4933	160	4	ζ(a	ζ(a	NOUN
ejpam-4933	160	5	)	)	PUNCT
ejpam-4933	160	6	)	)	PUNCT
ejpam-4933	160	7	.	.	PUNCT
ejpam-4933	161	1	therefore	therefore	ADV
ejpam-4933	161	2	ζ	ζ	PROPN
ejpam-4933	161	3	is	be	AUX
ejpam-4933	161	4	a	a	DET
ejpam-4933	161	5	y	y	PROPN
ejpam-4933	161	6	ε	ε	PROPN
ejpam-4933	161	7	j	j	PROPN
ejpam-4933	161	8	-fuzzy	-fuzzy	PROPN
ejpam-4933	161	9	subalgebra	subalgebra	NOUN
ejpam-4933	161	10	of	of	ADP
ejpam-4933	161	11	(	(	PUNCT
ejpam-4933	161	12	x	x	NOUN
ejpam-4933	161	13	,	,	PUNCT
ejpam-4933	161	14	∗	∗	NOUN
ejpam-4933	161	15	,	,	PUNCT
ejpam-4933	161	16	0	0	NUM
ejpam-4933	161	17	)	)	PUNCT
ejpam-4933	161	18	.	.	PUNCT
ejpam-4933	162	1	in	in	ADP
ejpam-4933	162	2	a	a	DET
ejpam-4933	162	3	bci	bci	NOUN
ejpam-4933	162	4	-	-	NOUN
ejpam-4933	162	5	algebra	algebra	NOUN
ejpam-4933	162	6	,	,	PUNCT
ejpam-4933	162	7	theorem	theorem	ADJ
ejpam-4933	162	8	2	2	NUM
ejpam-4933	162	9	may	may	AUX
ejpam-4933	162	10	not	not	PART
ejpam-4933	162	11	be	be	AUX
ejpam-4933	162	12	true	true	ADJ
ejpam-4933	162	13	as	as	SCONJ
ejpam-4933	162	14	seen	see	VERB
ejpam-4933	162	15	in	in	ADP
ejpam-4933	162	16	the	the	DET
ejpam-4933	162	17	following	follow	VERB
ejpam-4933	162	18	example	example	NOUN
ejpam-4933	162	19	.	.	PUNCT
ejpam-4933	163	1	e.	e.	PROPN
ejpam-4933	163	2	h.	h.	PROPN
ejpam-4933	163	3	roh	roh	PROPN
ejpam-4933	163	4	,	,	PUNCT
ejpam-4933	163	5	e.	e.	PROPN
ejpam-4933	163	6	yang	yang	PROPN
ejpam-4933	163	7	,	,	PUNCT
ejpam-4933	163	8	y.	y.	PROPN
ejpam-4933	163	9	b.	b.	PROPN
ejpam-4933	163	10	jun	jun	PROPN
ejpam-4933	163	11	/	/	SYM
ejpam-4933	163	12	eur	eur	PROPN
ejpam-4933	163	13	.	.	PUNCT
ejpam-4933	164	1	j.	j.	PROPN
ejpam-4933	164	2	pure	pure	PROPN
ejpam-4933	164	3	appl	appl	PROPN
ejpam-4933	164	4	.	.	PROPN
ejpam-4933	164	5	math	math	PROPN
ejpam-4933	164	6	,	,	PUNCT
ejpam-4933	164	7	16	16	NUM
ejpam-4933	164	8	(	(	PUNCT
ejpam-4933	164	9	4	4	NUM
ejpam-4933	164	10	)	)	PUNCT
ejpam-4933	164	11	(	(	PUNCT
ejpam-4933	164	12	2023	2023	NUM
ejpam-4933	164	13	)	)	PUNCT
ejpam-4933	164	14	,	,	PUNCT
ejpam-4933	164	15	2009	2009	NUM
ejpam-4933	164	16	-	-	SYM
ejpam-4933	164	17	2024	2024	NUM
ejpam-4933	164	18	2016	2016	NUM
ejpam-4933	164	19	example	example	NOUN
ejpam-4933	164	20	2	2	NUM
ejpam-4933	164	21	.	.	X
ejpam-4933	165	1	let	let	VERB
ejpam-4933	165	2	(	(	PUNCT
ejpam-4933	165	3	x	x	X
ejpam-4933	165	4	,	,	PUNCT
ejpam-4933	165	5	∗	∗	NOUN
ejpam-4933	165	6	,	,	PUNCT
ejpam-4933	165	7	0	0	NUM
ejpam-4933	165	8	)	)	PUNCT
ejpam-4933	165	9	be	be	AUX
ejpam-4933	165	10	a	a	DET
ejpam-4933	165	11	bci	bci	NOUN
ejpam-4933	165	12	-	-	NOUN
ejpam-4933	165	13	algebra	algebra	NOUN
ejpam-4933	165	14	and	and	CCONJ
ejpam-4933	165	15	(	(	PUNCT
ejpam-4933	165	16	z,−	z,−	PROPN
ejpam-4933	165	17	,	,	PUNCT
ejpam-4933	165	18	0	0	NUM
ejpam-4933	165	19	)	)	PUNCT
ejpam-4933	165	20	the	the	DET
ejpam-4933	165	21	adjoint	adjoint	NOUN
ejpam-4933	165	22	bci	bci	PROPN
ejpam-4933	165	23	-	-	NOUN
ejpam-4933	165	24	algebra	algebra	NOUN
ejpam-4933	165	25	of	of	ADP
ejpam-4933	165	26	the	the	DET
ejpam-4933	165	27	additive	additive	ADJ
ejpam-4933	165	28	group	group	NOUN
ejpam-4933	165	29	(	(	PUNCT
ejpam-4933	165	30	z,+	z,+	NUM
ejpam-4933	165	31	,	,	PUNCT
ejpam-4933	165	32	0	0	NUM
ejpam-4933	165	33	)	)	PUNCT
ejpam-4933	165	34	of	of	ADP
ejpam-4933	165	35	integers	integer	NOUN
ejpam-4933	165	36	.	.	PUNCT
ejpam-4933	166	1	then	then	ADV
ejpam-4933	166	2	(	(	PUNCT
ejpam-4933	166	3	y,⊛	y,⊛	PROPN
ejpam-4933	166	4	,	,	PUNCT
ejpam-4933	166	5	(	(	PUNCT
ejpam-4933	166	6	0	0	NUM
ejpam-4933	166	7	,	,	PUNCT
ejpam-4933	166	8	0	0	NUM
ejpam-4933	166	9	)	)	PUNCT
ejpam-4933	166	10	)	)	PUNCT
ejpam-4933	166	11	is	be	AUX
ejpam-4933	166	12	a	a	DET
ejpam-4933	166	13	bci	bci	NOUN
ejpam-4933	166	14	-	-	NOUN
ejpam-4933	166	15	algebra	algebra	NOUN
ejpam-4933	166	16	(	(	PUNCT
ejpam-4933	166	17	see	see	VERB
ejpam-4933	166	18	[	[	X
ejpam-4933	166	19	2	2	NUM
ejpam-4933	166	20	]	]	PUNCT
ejpam-4933	166	21	)	)	PUNCT
ejpam-4933	166	22	where	where	SCONJ
ejpam-4933	166	23	y	y	NOUN
ejpam-4933	166	24	=	=	PUNCT
ejpam-4933	166	25	x	x	SYM
ejpam-4933	166	26	×	×	PROPN
ejpam-4933	166	27	z	z	NOUN
ejpam-4933	166	28	and	and	CCONJ
ejpam-4933	166	29	⊛	⊛	NUM
ejpam-4933	166	30	is	be	AUX
ejpam-4933	166	31	a	a	DET
ejpam-4933	166	32	binary	binary	ADJ
ejpam-4933	166	33	operation	operation	NOUN
ejpam-4933	166	34	in	in	ADP
ejpam-4933	166	35	y	y	PROPN
ejpam-4933	166	36	given	give	VERB
ejpam-4933	166	37	as	as	SCONJ
ejpam-4933	166	38	follows	follow	VERB
ejpam-4933	166	39	:	:	PUNCT
ejpam-4933	166	40	(	(	PUNCT
ejpam-4933	166	41	∀(x	∀(x	X
ejpam-4933	166	42	,	,	PUNCT
ejpam-4933	166	43	a	a	PRON
ejpam-4933	166	44	)	)	PUNCT
ejpam-4933	166	45	,	,	PUNCT
ejpam-4933	166	46	(	(	PUNCT
ejpam-4933	166	47	y	y	PROPN
ejpam-4933	166	48	,	,	PUNCT
ejpam-4933	166	49	b	b	NOUN
ejpam-4933	166	50	)	)	PUNCT
ejpam-4933	166	51	∈	∈	PROPN
ejpam-4933	166	52	y	y	PROPN
ejpam-4933	166	53	)	)	PUNCT
ejpam-4933	166	54	(	(	PUNCT
ejpam-4933	166	55	(	(	PUNCT
ejpam-4933	166	56	x	x	NOUN
ejpam-4933	166	57	,	,	PUNCT
ejpam-4933	166	58	a	a	PRON
ejpam-4933	166	59	)	)	PUNCT
ejpam-4933	166	60	⊛	⊛	NUM
ejpam-4933	166	61	(	(	PUNCT
ejpam-4933	166	62	y	y	PROPN
ejpam-4933	166	63	,	,	PUNCT
ejpam-4933	166	64	b	b	NOUN
ejpam-4933	166	65	)	)	PUNCT
ejpam-4933	166	66	=	=	SYM
ejpam-4933	166	67	(	(	PUNCT
ejpam-4933	166	68	x	x	X
ejpam-4933	166	69	∗	∗	PROPN
ejpam-4933	166	70	y	y	PROPN
ejpam-4933	166	71	,	,	PUNCT
ejpam-4933	166	72	a−	a−	PROPN
ejpam-4933	166	73	b	b	NOUN
ejpam-4933	166	74	)	)	PUNCT
ejpam-4933	166	75	)	)	PUNCT
ejpam-4933	166	76	.	.	PUNCT
ejpam-4933	167	1	define	define	VERB
ejpam-4933	167	2	a	a	DET
ejpam-4933	167	3	fuzzy	fuzzy	ADJ
ejpam-4933	167	4	set	set	VERB
ejpam-4933	167	5	ζ	ζ	NOUN
ejpam-4933	167	6	in	in	ADP
ejpam-4933	167	7	y	y	PROPN
ejpam-4933	167	8	as	as	SCONJ
ejpam-4933	167	9	follows	follow	VERB
ejpam-4933	167	10	:	:	PUNCT
ejpam-4933	167	11	ζ	ζ	NOUN
ejpam-4933	167	12	:	:	PUNCT
ejpam-4933	168	1	y	y	PROPN
ejpam-4933	168	2	→	→	PUNCT
ejpam-4933	169	1	[	[	X
ejpam-4933	169	2	0	0	NUM
ejpam-4933	169	3	,	,	PUNCT
ejpam-4933	169	4	1	1	NUM
ejpam-4933	169	5	]	]	PUNCT
ejpam-4933	169	6	,	,	PUNCT
ejpam-4933	169	7	c	c	PROPN
ejpam-4933	169	8	7→	7→	NUM
ejpam-4933	169	9			PUNCT
ejpam-4933	169	10	0.87	0.87	NUM
ejpam-4933	169	11	if	if	SCONJ
ejpam-4933	169	12	c	c	NOUN
ejpam-4933	169	13	=	=	SYM
ejpam-4933	169	14	(	(	PUNCT
ejpam-4933	169	15	0	0	NUM
ejpam-4933	169	16	,	,	PUNCT
ejpam-4933	169	17	0	0	NUM
ejpam-4933	169	18	)	)	PUNCT
ejpam-4933	169	19	,	,	PUNCT
ejpam-4933	169	20	0.73	0.73	NUM
ejpam-4933	169	21	if	if	SCONJ
ejpam-4933	169	22	c	c	PROPN
ejpam-4933	169	23	∈	∈	PROPN
ejpam-4933	169	24	x	x	SYM
ejpam-4933	169	25	×	×	PROPN
ejpam-4933	169	26	n0	n0	NUM
ejpam-4933	169	27	,	,	PUNCT
ejpam-4933	169	28	0.42	0.42	NUM
ejpam-4933	169	29	otherwise	otherwise	ADV
ejpam-4933	169	30	wher	wher	PROPN
ejpam-4933	169	31	n0	n0	PROPN
ejpam-4933	169	32	is	be	AUX
ejpam-4933	169	33	the	the	DET
ejpam-4933	169	34	set	set	NOUN
ejpam-4933	169	35	of	of	ADP
ejpam-4933	169	36	all	all	DET
ejpam-4933	169	37	nonnegative	nonnegative	ADJ
ejpam-4933	169	38	integes	intege	NOUN
ejpam-4933	169	39	.	.	PUNCT
ejpam-4933	170	1	it	it	PRON
ejpam-4933	170	2	is	be	AUX
ejpam-4933	170	3	routine	routine	ADJ
ejpam-4933	170	4	to	to	PART
ejpam-4933	170	5	verify	verify	VERB
ejpam-4933	170	6	that	that	SCONJ
ejpam-4933	170	7	ζ	ζ	NOUN
ejpam-4933	170	8	is	be	AUX
ejpam-4933	170	9	a	a	DET
ejpam-4933	170	10	y	y	PROPN
ejpam-4933	170	11	ε	ε	PROPN
ejpam-4933	170	12	j	j	PROPN
ejpam-4933	170	13	-fuzzy	-fuzzy	PROPN
ejpam-4933	170	14	ideal	ideal	NOUN
ejpam-4933	170	15	of	of	ADP
ejpam-4933	170	16	(	(	PUNCT
ejpam-4933	170	17	y,⊛	y,⊛	PROPN
ejpam-4933	170	18	,	,	PUNCT
ejpam-4933	170	19	(	(	PUNCT
ejpam-4933	170	20	0	0	NUM
ejpam-4933	170	21	,	,	PUNCT
ejpam-4933	170	22	0	0	NUM
ejpam-4933	170	23	)	)	PUNCT
ejpam-4933	170	24	)	)	PUNCT
ejpam-4933	170	25	for	for	ADP
ejpam-4933	170	26	ε	ε	PROPN
ejpam-4933	170	27	=	=	SYM
ejpam-4933	170	28	0.61	0.61	NUM
ejpam-4933	170	29	.	.	PUNCT
ejpam-4933	171	1	we	we	PRON
ejpam-4933	171	2	can	can	AUX
ejpam-4933	171	3	observe	observe	VERB
ejpam-4933	171	4	that	that	SCONJ
ejpam-4933	171	5	yj(ε	yj(ε	NOUN
ejpam-4933	171	6	,	,	PUNCT
ejpam-4933	171	7	ζ((0	ζ((0	PROPN
ejpam-4933	171	8	,	,	PUNCT
ejpam-4933	171	9	3	3	NUM
ejpam-4933	171	10	)	)	PUNCT
ejpam-4933	171	11	⊛	⊛	NUM
ejpam-4933	171	12	(	(	PUNCT
ejpam-4933	171	13	0	0	NUM
ejpam-4933	171	14	,	,	PUNCT
ejpam-4933	171	15	7	7	NUM
ejpam-4933	171	16	)	)	PUNCT
ejpam-4933	171	17	)	)	PUNCT
ejpam-4933	171	18	)	)	PUNCT
ejpam-4933	172	1	=	=	SYM
ejpam-4933	172	2	yj(0.61	yj(0.61	NUM
ejpam-4933	172	3	,	,	PUNCT
ejpam-4933	172	4	ζ(0,−4	ζ(0,−4	PROPN
ejpam-4933	172	5	)	)	PUNCT
ejpam-4933	172	6	)	)	PUNCT
ejpam-4933	172	7	=	=	PUNCT
ejpam-4933	172	8	(	(	PUNCT
ejpam-4933	172	9	1	1	NUM
ejpam-4933	172	10	−	−	NUM
ejpam-4933	172	11	0.61	0.61	NUM
ejpam-4933	172	12	)	)	PUNCT
ejpam-4933	172	13	∧	∧	NOUN
ejpam-4933	172	14	(	(	PUNCT
ejpam-4933	172	15	1	1	NUM
ejpam-4933	172	16	−	−	NOUN
ejpam-4933	172	17	0.42	0.42	NUM
ejpam-4933	172	18	)	)	PUNCT
ejpam-4933	172	19	=	=	SYM
ejpam-4933	172	20	0.39	0.39	NUM
ejpam-4933	172	21	and	and	CCONJ
ejpam-4933	172	22	yj(ε	yj(ε	NUM
ejpam-4933	172	23	,	,	PUNCT
ejpam-4933	172	24	ζ(0	ζ(0	NOUN
ejpam-4933	172	25	,	,	PUNCT
ejpam-4933	172	26	3	3	NUM
ejpam-4933	172	27	)	)	PUNCT
ejpam-4933	172	28	)	)	PUNCT
ejpam-4933	172	29	∨	∨	PROPN
ejpam-4933	172	30	yj(ε	yj(ε	NUM
ejpam-4933	172	31	,	,	PUNCT
ejpam-4933	172	32	ζ(0	ζ(0	NOUN
ejpam-4933	172	33	,	,	PUNCT
ejpam-4933	172	34	7	7	NUM
ejpam-4933	172	35	)	)	PUNCT
ejpam-4933	172	36	)	)	PUNCT
ejpam-4933	173	1	=	=	SYM
ejpam-4933	173	2	(	(	PUNCT
ejpam-4933	173	3	(	(	PUNCT
ejpam-4933	173	4	1	1	NUM
ejpam-4933	173	5	−	−	NOUN
ejpam-4933	173	6	0.61	0.61	NUM
ejpam-4933	173	7	)	)	PUNCT
ejpam-4933	173	8	∧	∧	NOUN
ejpam-4933	173	9	(	(	PUNCT
ejpam-4933	173	10	1	1	NUM
ejpam-4933	173	11	−	−	NOUN
ejpam-4933	173	12	0.73	0.73	NUM
ejpam-4933	173	13	)	)	PUNCT
ejpam-4933	173	14	)	)	PUNCT
ejpam-4933	173	15	∨	∨	NUM
ejpam-4933	173	16	(	(	PUNCT
ejpam-4933	173	17	(	(	PUNCT
ejpam-4933	173	18	1	1	NUM
ejpam-4933	173	19	−	−	NOUN
ejpam-4933	173	20	0.61	0.61	NUM
ejpam-4933	173	21	)	)	PUNCT
ejpam-4933	174	1	∧	∧	NOUN
ejpam-4933	174	2	(	(	PUNCT
ejpam-4933	174	3	1	1	NUM
ejpam-4933	174	4	−	−	NOUN
ejpam-4933	174	5	0.73	0.73	NUM
ejpam-4933	174	6	)	)	PUNCT
ejpam-4933	174	7	)	)	PUNCT
ejpam-4933	175	1	=	=	SYM
ejpam-4933	175	2	0.27	0.27	NUM
ejpam-4933	175	3	.	.	PUNCT
ejpam-4933	176	1	hence	hence	ADV
ejpam-4933	176	2	yj(ε	yj(ε	NUM
ejpam-4933	176	3	,	,	PUNCT
ejpam-4933	176	4	ζ((0	ζ((0	PROPN
ejpam-4933	176	5	,	,	PUNCT
ejpam-4933	176	6	3	3	NUM
ejpam-4933	176	7	)	)	PUNCT
ejpam-4933	176	8	⊛	⊛	NUM
ejpam-4933	176	9	(	(	PUNCT
ejpam-4933	176	10	0	0	NUM
ejpam-4933	176	11	,	,	PUNCT
ejpam-4933	176	12	7	7	NUM
ejpam-4933	176	13	)	)	PUNCT
ejpam-4933	176	14	)	)	PUNCT
ejpam-4933	176	15	)	)	PUNCT
ejpam-4933	177	1	≰	≰	PROPN
ejpam-4933	177	2	yj(ε	yj(ε	PART
ejpam-4933	177	3	,	,	PUNCT
ejpam-4933	177	4	ζ(0	ζ(0	NOUN
ejpam-4933	177	5	,	,	PUNCT
ejpam-4933	177	6	3	3	NUM
ejpam-4933	177	7	)	)	PUNCT
ejpam-4933	177	8	)	)	PUNCT
ejpam-4933	177	9	∨	∨	PROPN
ejpam-4933	177	10	yj(ε	yj(ε	NUM
ejpam-4933	177	11	,	,	PUNCT
ejpam-4933	177	12	ζ(0	ζ(0	NOUN
ejpam-4933	177	13	,	,	PUNCT
ejpam-4933	177	14	7	7	NUM
ejpam-4933	177	15	)	)	PUNCT
ejpam-4933	177	16	)	)	PUNCT
ejpam-4933	177	17	for	for	ADP
ejpam-4933	177	18	ε	ε	PROPN
ejpam-4933	177	19	=	=	SYM
ejpam-4933	177	20	0.61	0.61	NUM
ejpam-4933	177	21	,	,	PUNCT
ejpam-4933	177	22	which	which	PRON
ejpam-4933	177	23	shows	show	VERB
ejpam-4933	177	24	that	that	SCONJ
ejpam-4933	177	25	ζ	ζ	NOUN
ejpam-4933	177	26	is	be	AUX
ejpam-4933	177	27	not	not	PART
ejpam-4933	177	28	a	a	DET
ejpam-4933	177	29	y	y	PROPN
ejpam-4933	177	30	ε	ε	PROPN
ejpam-4933	177	31	j	j	PROPN
ejpam-4933	177	32	-fuzzy	-fuzzy	PROPN
ejpam-4933	177	33	subalgebra	subalgebra	NOUN
ejpam-4933	177	34	of	of	ADP
ejpam-4933	177	35	(	(	PUNCT
ejpam-4933	177	36	y,⊛	y,⊛	PROPN
ejpam-4933	177	37	,	,	PUNCT
ejpam-4933	177	38	(	(	PUNCT
ejpam-4933	177	39	0	0	NUM
ejpam-4933	177	40	,	,	PUNCT
ejpam-4933	177	41	0	0	NUM
ejpam-4933	177	42	)	)	PUNCT
ejpam-4933	177	43	)	)	PUNCT
ejpam-4933	177	44	.	.	PUNCT
ejpam-4933	178	1	the	the	DET
ejpam-4933	178	2	following	follow	VERB
ejpam-4933	178	3	example	example	NOUN
ejpam-4933	178	4	shows	show	VERB
ejpam-4933	178	5	that	that	SCONJ
ejpam-4933	178	6	there	there	PRON
ejpam-4933	178	7	exists	exist	VERB
ejpam-4933	178	8	ε	ε	PROPN
ejpam-4933	178	9	∈	∈	PROPN
ejpam-4933	178	10	(	(	PUNCT
ejpam-4933	178	11	0	0	NUM
ejpam-4933	178	12	,	,	PUNCT
ejpam-4933	178	13	1	1	NUM
ejpam-4933	178	14	)	)	PUNCT
ejpam-4933	178	15	such	such	ADJ
ejpam-4933	178	16	that	that	SCONJ
ejpam-4933	178	17	a	a	DET
ejpam-4933	178	18	y	y	PROPN
ejpam-4933	178	19	ε	ε	PROPN
ejpam-4933	178	20	j	j	PROPN
ejpam-4933	178	21	-fuzzy	-fuzzy	PROPN
ejpam-4933	178	22	subalgebra	subalgebra	NOUN
ejpam-4933	178	23	may	may	AUX
ejpam-4933	178	24	not	not	PART
ejpam-4933	178	25	be	be	AUX
ejpam-4933	178	26	a	a	DET
ejpam-4933	178	27	y	y	PROPN
ejpam-4933	178	28	ε	ε	PROPN
ejpam-4933	178	29	j	j	PROPN
ejpam-4933	178	30	-fuzzy	-fuzzy	PROPN
ejpam-4933	178	31	ideal	ideal	ADJ
ejpam-4933	178	32	.	.	PUNCT
ejpam-4933	179	1	example	example	NOUN
ejpam-4933	180	1	3	3	NUM
ejpam-4933	180	2	.	.	PUNCT
ejpam-4933	181	1	(	(	PUNCT
ejpam-4933	181	2	i	i	NOUN
ejpam-4933	181	3	)	)	PUNCT
ejpam-4933	181	4	let	let	VERB
ejpam-4933	181	5	x	x	PUNCT
ejpam-4933	181	6	=	=	PUNCT
ejpam-4933	181	7	{	{	PUNCT
ejpam-4933	181	8	0	0	NUM
ejpam-4933	181	9	,	,	PUNCT
ejpam-4933	181	10	b1	b1	NOUN
ejpam-4933	181	11	,	,	PUNCT
ejpam-4933	181	12	b2	b2	NOUN
ejpam-4933	181	13	,	,	PUNCT
ejpam-4933	181	14	b3	b3	PROPN
ejpam-4933	181	15	}	}	PUNCT
ejpam-4933	181	16	be	be	AUX
ejpam-4933	181	17	a	a	DET
ejpam-4933	181	18	set	set	NOUN
ejpam-4933	181	19	with	with	ADP
ejpam-4933	181	20	a	a	DET
ejpam-4933	181	21	binary	binary	ADJ
ejpam-4933	181	22	operation	operation	NOUN
ejpam-4933	181	23	“	"	PUNCT
ejpam-4933	181	24	∗	∗	NOUN
ejpam-4933	181	25	”	"	PUNCT
ejpam-4933	181	26	given	give	VERB
ejpam-4933	181	27	by	by	ADP
ejpam-4933	181	28	table	table	NOUN
ejpam-4933	181	29	2	2	NUM
ejpam-4933	181	30	.	.	PUNCT
ejpam-4933	181	31	table	table	NOUN
ejpam-4933	181	32	2	2	NUM
ejpam-4933	181	33	:	:	PUNCT
ejpam-4933	181	34	cayley	cayley	ADJ
ejpam-4933	181	35	table	table	NOUN
ejpam-4933	181	36	for	for	ADP
ejpam-4933	181	37	the	the	DET
ejpam-4933	181	38	binary	binary	PROPN
ejpam-4933	181	39	operation	operation	NOUN
ejpam-4933	181	40	“	"	PUNCT
ejpam-4933	181	41	∗	∗	NOUN
ejpam-4933	181	42	”	"	PUNCT
ejpam-4933	181	43	∗	∗	X
ejpam-4933	181	44	0	0	NUM
ejpam-4933	181	45	b1	b1	NOUN
ejpam-4933	181	46	b2	b2	NOUN
ejpam-4933	181	47	b3	b3	PROPN
ejpam-4933	181	48	0	0	NUM
ejpam-4933	181	49	0	0	NUM
ejpam-4933	181	50	0	0	NUM
ejpam-4933	181	51	0	0	SYM
ejpam-4933	181	52	0	0	NUM
ejpam-4933	181	53	b1	b1	NOUN
ejpam-4933	181	54	b1	b1	NOUN
ejpam-4933	181	55	0	0	NUM
ejpam-4933	181	56	0	0	NUM
ejpam-4933	181	57	b1	b1	NOUN
ejpam-4933	181	58	b2	b2	NOUN
ejpam-4933	181	59	b2	b2	NOUN
ejpam-4933	181	60	b1	b1	NOUN
ejpam-4933	181	61	0	0	NUM
ejpam-4933	181	62	b2	b2	NOUN
ejpam-4933	181	63	b3	b3	PROPN
ejpam-4933	181	64	b3	b3	PROPN
ejpam-4933	181	65	b3	b3	PROPN
ejpam-4933	181	66	b3	b3	PROPN
ejpam-4933	181	67	0	0	PUNCT
ejpam-4933	182	1	then	then	ADV
ejpam-4933	182	2	x	x	VERB
ejpam-4933	182	3	is	be	AUX
ejpam-4933	182	4	a	a	DET
ejpam-4933	182	5	bck	bck	NOUN
ejpam-4933	182	6	-	-	PUNCT
ejpam-4933	182	7	algebra	algebra	NOUN
ejpam-4933	182	8	(	(	PUNCT
ejpam-4933	182	9	see	see	VERB
ejpam-4933	182	10	[	[	X
ejpam-4933	182	11	11	11	NUM
ejpam-4933	182	12	]	]	NUM
ejpam-4933	182	13	)	)	PUNCT
ejpam-4933	182	14	.	.	PUNCT
ejpam-4933	183	1	a	a	DET
ejpam-4933	183	2	fuzzy	fuzzy	ADJ
ejpam-4933	183	3	set	set	VERB
ejpam-4933	183	4	ζ	ζ	NOUN
ejpam-4933	183	5	in	in	ADP
ejpam-4933	183	6	x	x	PUNCT
ejpam-4933	183	7	defined	define	VERB
ejpam-4933	183	8	by	by	ADP
ejpam-4933	183	9	ζ	ζ	NOUN
ejpam-4933	183	10	:	:	PUNCT
ejpam-4933	183	11	x	x	X
ejpam-4933	183	12	→	→	SYM
ejpam-4933	183	13	[	[	X
ejpam-4933	183	14	0	0	NUM
ejpam-4933	183	15	,	,	PUNCT
ejpam-4933	183	16	1	1	NUM
ejpam-4933	183	17	]	]	PUNCT
ejpam-4933	183	18	,	,	PUNCT
ejpam-4933	183	19	x	x	PROPN
ejpam-4933	183	20	7→	7→	NUM
ejpam-4933	183	21			NUM
ejpam-4933	183	22	0.63	0.63	NUM
ejpam-4933	183	23	if	if	SCONJ
ejpam-4933	183	24	x	x	X
ejpam-4933	183	25	=	=	SYM
ejpam-4933	183	26	0	0	NUM
ejpam-4933	183	27	,	,	PUNCT
ejpam-4933	183	28	0.54	0.54	NUM
ejpam-4933	183	29	if	if	SCONJ
ejpam-4933	183	30	x	x	X
ejpam-4933	183	31	=	=	SYM
ejpam-4933	183	32	b1	b1	NOUN
ejpam-4933	183	33	,	,	PUNCT
ejpam-4933	183	34	0.42	0.42	NUM
ejpam-4933	183	35	if	if	SCONJ
ejpam-4933	183	36	x	x	NOUN
ejpam-4933	183	37	=	=	SYM
ejpam-4933	183	38	b2	b2	NOUN
ejpam-4933	183	39	,	,	PUNCT
ejpam-4933	183	40	0.49	0.49	NUM
ejpam-4933	183	41	if	if	SCONJ
ejpam-4933	183	42	x	x	NOUN
ejpam-4933	183	43	=	=	PROPN
ejpam-4933	183	44	b3	b3	PROPN
ejpam-4933	183	45	e.	e.	PROPN
ejpam-4933	183	46	h.	h.	PROPN
ejpam-4933	183	47	roh	roh	PROPN
ejpam-4933	183	48	,	,	PUNCT
ejpam-4933	183	49	e.	e.	PROPN
ejpam-4933	183	50	yang	yang	PROPN
ejpam-4933	183	51	,	,	PUNCT
ejpam-4933	183	52	y.	y.	PROPN
ejpam-4933	183	53	b.	b.	PROPN
ejpam-4933	183	54	jun	jun	PROPN
ejpam-4933	183	55	/	/	SYM
ejpam-4933	183	56	eur	eur	PROPN
ejpam-4933	183	57	.	.	PUNCT
ejpam-4933	184	1	j.	j.	PROPN
ejpam-4933	184	2	pure	pure	PROPN
ejpam-4933	184	3	appl	appl	PROPN
ejpam-4933	184	4	.	.	PROPN
ejpam-4933	184	5	math	math	PROPN
ejpam-4933	184	6	,	,	PUNCT
ejpam-4933	184	7	16	16	NUM
ejpam-4933	184	8	(	(	PUNCT
ejpam-4933	184	9	4	4	NUM
ejpam-4933	184	10	)	)	PUNCT
ejpam-4933	184	11	(	(	PUNCT
ejpam-4933	184	12	2023	2023	NUM
ejpam-4933	184	13	)	)	PUNCT
ejpam-4933	184	14	,	,	PUNCT
ejpam-4933	184	15	2009	2009	NUM
ejpam-4933	184	16	-	-	SYM
ejpam-4933	184	17	2024	2024	NUM
ejpam-4933	184	18	2017	2017	NUM
ejpam-4933	184	19	is	be	AUX
ejpam-4933	184	20	a	a	DET
ejpam-4933	184	21	y	y	PROPN
ejpam-4933	184	22	ε	ε	PROPN
ejpam-4933	184	23	j	j	PROPN
ejpam-4933	184	24	-fuzzy	-fuzzy	PROPN
ejpam-4933	184	25	subalgebra	subalgebra	NOUN
ejpam-4933	184	26	of	of	ADP
ejpam-4933	184	27	(	(	PUNCT
ejpam-4933	184	28	x	x	NOUN
ejpam-4933	184	29	,	,	PUNCT
ejpam-4933	184	30	∗	∗	NOUN
ejpam-4933	184	31	,	,	PUNCT
ejpam-4933	184	32	0	0	NUM
ejpam-4933	184	33	)	)	PUNCT
ejpam-4933	184	34	for	for	ADP
ejpam-4933	184	35	ε	ε	PROPN
ejpam-4933	184	36	=	=	PROPN
ejpam-4933	184	37	0.52	0.52	NUM
ejpam-4933	184	38	.	.	PUNCT
ejpam-4933	185	1	but	but	CCONJ
ejpam-4933	185	2	it	it	PRON
ejpam-4933	185	3	is	be	AUX
ejpam-4933	185	4	not	not	PART
ejpam-4933	185	5	a	a	DET
ejpam-4933	185	6	y	y	PROPN
ejpam-4933	185	7	ε	ε	PROPN
ejpam-4933	185	8	j	j	PROPN
ejpam-4933	185	9	-fuzzy	-fuzzy	PROPN
ejpam-4933	185	10	ideal	ideal	NOUN
ejpam-4933	185	11	of	of	ADP
ejpam-4933	185	12	(	(	PUNCT
ejpam-4933	185	13	x	x	NOUN
ejpam-4933	185	14	,	,	PUNCT
ejpam-4933	185	15	∗	∗	NOUN
ejpam-4933	185	16	,	,	PUNCT
ejpam-4933	185	17	0	0	NUM
ejpam-4933	185	18	)	)	PUNCT
ejpam-4933	185	19	for	for	ADP
ejpam-4933	185	20	ε	ε	PROPN
ejpam-4933	185	21	=	=	SYM
ejpam-4933	185	22	0.52	0.52	NUM
ejpam-4933	185	23	since	since	SCONJ
ejpam-4933	185	24	yj(ε	yj(ε	NUM
ejpam-4933	185	25	,	,	PUNCT
ejpam-4933	185	26	ζ(b2	ζ(b2	NOUN
ejpam-4933	185	27	)	)	PUNCT
ejpam-4933	185	28	)	)	PUNCT
ejpam-4933	186	1	=	=	PUNCT
ejpam-4933	186	2	yj(0.52	yj(0.52	NUM
ejpam-4933	186	3	,	,	PUNCT
ejpam-4933	186	4	0.42	0.42	NUM
ejpam-4933	186	5	)	)	PUNCT
ejpam-4933	186	6	=	=	PUNCT
ejpam-4933	186	7	(	(	PUNCT
ejpam-4933	186	8	1	1	NUM
ejpam-4933	186	9	−	−	NUM
ejpam-4933	186	10	0.52	0.52	NUM
ejpam-4933	186	11	)	)	PUNCT
ejpam-4933	186	12	∧	∧	NOUN
ejpam-4933	186	13	(	(	PUNCT
ejpam-4933	186	14	1	1	NUM
ejpam-4933	186	15	−	−	NOUN
ejpam-4933	186	16	0.42	0.42	NUM
ejpam-4933	186	17	)	)	PUNCT
ejpam-4933	186	18	=	=	NOUN
ejpam-4933	186	19	0.48	0.48	NUM
ejpam-4933	186	20	≰	≰	PROPN
ejpam-4933	186	21	0.46	0.46	NUM
ejpam-4933	186	22	=	=	SYM
ejpam-4933	186	23	(	(	PUNCT
ejpam-4933	186	24	1	1	NUM
ejpam-4933	186	25	−	−	NUM
ejpam-4933	186	26	0.52	0.52	NUM
ejpam-4933	186	27	)	)	PUNCT
ejpam-4933	186	28	∧	∧	NOUN
ejpam-4933	186	29	(	(	PUNCT
ejpam-4933	186	30	1	1	NUM
ejpam-4933	186	31	−	−	NOUN
ejpam-4933	186	32	0.54	0.54	NUM
ejpam-4933	186	33	)	)	PUNCT
ejpam-4933	186	34	=	=	SYM
ejpam-4933	186	35	(	(	PUNCT
ejpam-4933	186	36	(	(	PUNCT
ejpam-4933	186	37	1	1	NUM
ejpam-4933	186	38	−	−	NOUN
ejpam-4933	186	39	0.52	0.52	NUM
ejpam-4933	186	40	)	)	PUNCT
ejpam-4933	186	41	∧	∧	NOUN
ejpam-4933	186	42	(	(	PUNCT
ejpam-4933	186	43	1	1	NUM
ejpam-4933	186	44	−	−	NOUN
ejpam-4933	186	45	0.54	0.54	NUM
ejpam-4933	186	46	)	)	PUNCT
ejpam-4933	186	47	)	)	PUNCT
ejpam-4933	186	48	∨	∨	NUM
ejpam-4933	186	49	(	(	PUNCT
ejpam-4933	186	50	(	(	PUNCT
ejpam-4933	186	51	1	1	NUM
ejpam-4933	186	52	−	−	NUM
ejpam-4933	186	53	0.52	0.52	NUM
ejpam-4933	186	54	)	)	PUNCT
ejpam-4933	186	55	∧	∧	NOUN
ejpam-4933	186	56	(	(	PUNCT
ejpam-4933	186	57	1	1	NUM
ejpam-4933	186	58	−	−	NOUN
ejpam-4933	186	59	0.54	0.54	NUM
ejpam-4933	186	60	)	)	PUNCT
ejpam-4933	186	61	)	)	PUNCT
ejpam-4933	187	1	=	=	SYM
ejpam-4933	187	2	yj(0.52	yj(0.52	ADJ
ejpam-4933	187	3	,	,	PUNCT
ejpam-4933	187	4	ζ(b1	ζ(b1	NOUN
ejpam-4933	187	5	)	)	PUNCT
ejpam-4933	187	6	)	)	PUNCT
ejpam-4933	188	1	∨	∨	ADP
ejpam-4933	188	2	yj(0.52	yj(0.52	NOUN
ejpam-4933	188	3	,	,	PUNCT
ejpam-4933	188	4	ζ(b1	ζ(b1	NOUN
ejpam-4933	188	5	)	)	PUNCT
ejpam-4933	188	6	)	)	PUNCT
ejpam-4933	189	1	=	=	SYM
ejpam-4933	189	2	yj(ε	yj(ε	PROPN
ejpam-4933	189	3	,	,	PUNCT
ejpam-4933	189	4	ζ(b2	ζ(b2	NOUN
ejpam-4933	189	5	∗	∗	NOUN
ejpam-4933	189	6	b1	b1	NOUN
ejpam-4933	189	7	)	)	PUNCT
ejpam-4933	189	8	)	)	PUNCT
ejpam-4933	189	9	∨	∨	PROPN
ejpam-4933	189	10	yj(ε	yj(ε	NUM
ejpam-4933	189	11	,	,	PUNCT
ejpam-4933	189	12	ζ(b1	ζ(b1	NOUN
ejpam-4933	189	13	)	)	PUNCT
ejpam-4933	189	14	)	)	PUNCT
ejpam-4933	189	15	.	.	PUNCT
ejpam-4933	190	1	(	(	PUNCT
ejpam-4933	190	2	ii	ii	NOUN
ejpam-4933	190	3	)	)	PUNCT
ejpam-4933	190	4	consider	consider	VERB
ejpam-4933	190	5	the	the	DET
ejpam-4933	190	6	bci	bci	NOUN
ejpam-4933	190	7	-	-	NOUN
ejpam-4933	190	8	algebra	algebra	NOUN
ejpam-4933	190	9	(	(	PUNCT
ejpam-4933	190	10	x	x	X
ejpam-4933	190	11	,	,	PUNCT
ejpam-4933	190	12	∗	∗	NOUN
ejpam-4933	190	13	,	,	PUNCT
ejpam-4933	190	14	0	0	NUM
ejpam-4933	190	15	)	)	PUNCT
ejpam-4933	190	16	in	in	ADP
ejpam-4933	190	17	example	example	NOUN
ejpam-4933	190	18	1	1	NUM
ejpam-4933	190	19	and	and	CCONJ
ejpam-4933	190	20	let	let	VERB
ejpam-4933	190	21	ζ	ζ	NOUN
ejpam-4933	190	22	be	be	AUX
ejpam-4933	190	23	a	a	DET
ejpam-4933	190	24	fuzzy	fuzzy	ADJ
ejpam-4933	190	25	set	set	NOUN
ejpam-4933	190	26	in	in	ADP
ejpam-4933	190	27	x	x	PUNCT
ejpam-4933	190	28	given	give	VERB
ejpam-4933	190	29	as	as	SCONJ
ejpam-4933	190	30	follows	follow	VERB
ejpam-4933	190	31	:	:	PUNCT
ejpam-4933	190	32	ζ	ζ	NOUN
ejpam-4933	190	33	:	:	PUNCT
ejpam-4933	190	34	x	x	X
ejpam-4933	190	35	→	→	SYM
ejpam-4933	191	1	[	[	X
ejpam-4933	191	2	0	0	NUM
ejpam-4933	191	3	,	,	PUNCT
ejpam-4933	191	4	1	1	NUM
ejpam-4933	191	5	]	]	PUNCT
ejpam-4933	191	6	,	,	PUNCT
ejpam-4933	191	7	y	y	PROPN
ejpam-4933	191	8	7→	7→	PROPN
ejpam-4933	191	9			NOUN
ejpam-4933	191	10	0.78	0.78	NUM
ejpam-4933	191	11	if	if	SCONJ
ejpam-4933	191	12	y	y	NOUN
ejpam-4933	191	13	=	=	SYM
ejpam-4933	191	14	0	0	PROPN
ejpam-4933	191	15	,	,	PUNCT
ejpam-4933	191	16	0.54	0.54	NUM
ejpam-4933	191	17	if	if	SCONJ
ejpam-4933	191	18	y	y	NOUN
ejpam-4933	191	19	=	=	SYM
ejpam-4933	191	20	1	1	NUM
ejpam-4933	191	21	,	,	PUNCT
ejpam-4933	191	22	0.37	0.37	NUM
ejpam-4933	191	23	if	if	SCONJ
ejpam-4933	191	24	y	y	PROPN
ejpam-4933	191	25	=	=	SYM
ejpam-4933	191	26	2	2	NUM
ejpam-4933	191	27	,	,	PUNCT
ejpam-4933	191	28	0.65	0.65	NUM
ejpam-4933	191	29	if	if	SCONJ
ejpam-4933	191	30	y	y	PROPN
ejpam-4933	191	31	=	=	PUNCT
ejpam-4933	191	32	a	a	PROPN
ejpam-4933	191	33	,	,	PUNCT
ejpam-4933	191	34	0.37	0.37	NUM
ejpam-4933	191	35	if	if	SCONJ
ejpam-4933	191	36	y	y	PROPN
ejpam-4933	191	37	=	=	SYM
ejpam-4933	191	38	b.	b.	PROPN
ejpam-4933	191	39	then	then	ADV
ejpam-4933	191	40	ζ	ζ	NOUN
ejpam-4933	191	41	is	be	AUX
ejpam-4933	191	42	a	a	DET
ejpam-4933	191	43	y	y	PROPN
ejpam-4933	191	44	ε	ε	PROPN
ejpam-4933	191	45	j	j	PROPN
ejpam-4933	191	46	-fuzzy	-fuzzy	PROPN
ejpam-4933	191	47	subalgebra	subalgebra	NOUN
ejpam-4933	191	48	of	of	ADP
ejpam-4933	191	49	(	(	PUNCT
ejpam-4933	191	50	x	x	NOUN
ejpam-4933	191	51	,	,	PUNCT
ejpam-4933	191	52	∗	∗	NOUN
ejpam-4933	191	53	,	,	PUNCT
ejpam-4933	191	54	0	0	NUM
ejpam-4933	191	55	)	)	PUNCT
ejpam-4933	191	56	for	for	ADP
ejpam-4933	191	57	ε	ε	PROPN
ejpam-4933	191	58	=	=	PUNCT
ejpam-4933	191	59	0.49	0.49	NUM
ejpam-4933	191	60	.	.	PUNCT
ejpam-4933	192	1	we	we	PRON
ejpam-4933	192	2	can	can	AUX
ejpam-4933	192	3	observe	observe	VERB
ejpam-4933	192	4	that	that	SCONJ
ejpam-4933	192	5	yj(ε	yj(ε	NOUN
ejpam-4933	192	6	,	,	PUNCT
ejpam-4933	192	7	ζ(1	ζ(1	PROPN
ejpam-4933	192	8	)	)	PUNCT
ejpam-4933	192	9	)	)	PUNCT
ejpam-4933	192	10	=	=	SYM
ejpam-4933	193	1	yj(0.49	yj(0.49	X
ejpam-4933	193	2	,	,	PUNCT
ejpam-4933	193	3	0.54	0.54	NUM
ejpam-4933	193	4	)	)	PUNCT
ejpam-4933	193	5	=	=	NOUN
ejpam-4933	193	6	(	(	PUNCT
ejpam-4933	193	7	1	1	NUM
ejpam-4933	193	8	−	−	NOUN
ejpam-4933	193	9	0.49	0.49	NUM
ejpam-4933	193	10	)	)	PUNCT
ejpam-4933	193	11	∧	∧	NOUN
ejpam-4933	193	12	(	(	PUNCT
ejpam-4933	193	13	1	1	NUM
ejpam-4933	193	14	−	−	NOUN
ejpam-4933	193	15	0.54	0.54	NUM
ejpam-4933	193	16	)	)	PUNCT
ejpam-4933	193	17	=	=	SYM
ejpam-4933	193	18	0.46	0.46	NUM
ejpam-4933	193	19	and	and	CCONJ
ejpam-4933	193	20	yj(ε	yj(ε	NUM
ejpam-4933	193	21	,	,	PUNCT
ejpam-4933	193	22	ζ(1	ζ(1	PROPN
ejpam-4933	193	23	∗	∗	NOUN
ejpam-4933	193	24	a	a	NOUN
ejpam-4933	193	25	)	)	PUNCT
ejpam-4933	193	26	)	)	PUNCT
ejpam-4933	193	27	∨	∨	PROPN
ejpam-4933	193	28	yj(ε	yj(ε	NUM
ejpam-4933	193	29	,	,	PUNCT
ejpam-4933	193	30	ζ(a	ζ(a	NOUN
ejpam-4933	193	31	)	)	PUNCT
ejpam-4933	193	32	)	)	PUNCT
ejpam-4933	194	1	=	=	PUNCT
ejpam-4933	194	2	yj(ε	yj(ε	ADJ
ejpam-4933	194	3	,	,	PUNCT
ejpam-4933	194	4	ζ(a	ζ(a	NOUN
ejpam-4933	194	5	)	)	PUNCT
ejpam-4933	194	6	)	)	PUNCT
ejpam-4933	195	1	∨	∨	PROPN
ejpam-4933	195	2	yj(ε	yj(ε	NUM
ejpam-4933	195	3	,	,	PUNCT
ejpam-4933	195	4	ζ(a	ζ(a	NOUN
ejpam-4933	195	5	)	)	PUNCT
ejpam-4933	195	6	)	)	PUNCT
ejpam-4933	196	1	=	=	PUNCT
ejpam-4933	196	2	yj(ε	yj(ε	ADJ
ejpam-4933	196	3	,	,	PUNCT
ejpam-4933	196	4	ζ(a	ζ(a	NOUN
ejpam-4933	196	5	)	)	PUNCT
ejpam-4933	196	6	)	)	PUNCT
ejpam-4933	197	1	=	=	SYM
ejpam-4933	197	2	yj(0.49	yj(0.49	X
ejpam-4933	197	3	,	,	PUNCT
ejpam-4933	197	4	0.65	0.65	NUM
ejpam-4933	197	5	)	)	PUNCT
ejpam-4933	197	6	=	=	PUNCT
ejpam-4933	197	7	(	(	PUNCT
ejpam-4933	197	8	1	1	NUM
ejpam-4933	197	9	−	−	NOUN
ejpam-4933	197	10	0.49	0.49	NUM
ejpam-4933	197	11	)	)	PUNCT
ejpam-4933	197	12	∧	∧	NOUN
ejpam-4933	197	13	(	(	PUNCT
ejpam-4933	197	14	1	1	NUM
ejpam-4933	197	15	−	−	NUM
ejpam-4933	197	16	0.65	0.65	NUM
ejpam-4933	197	17	)	)	PUNCT
ejpam-4933	197	18	=	=	SYM
ejpam-4933	197	19	0.35	0.35	NUM
ejpam-4933	197	20	.	.	PUNCT
ejpam-4933	198	1	hence	hence	ADV
ejpam-4933	198	2	yj(ε	yj(ε	NUM
ejpam-4933	198	3	,	,	PUNCT
ejpam-4933	198	4	ζ(1	ζ(1	PROPN
ejpam-4933	198	5	)	)	PUNCT
ejpam-4933	198	6	)	)	PUNCT
ejpam-4933	198	7	≰	≰	PROPN
ejpam-4933	198	8	yj(ε	yj(ε	PART
ejpam-4933	198	9	,	,	PUNCT
ejpam-4933	198	10	ζ(1	ζ(1	PROPN
ejpam-4933	198	11	∗	∗	NOUN
ejpam-4933	198	12	a))∨	a))∨	NOUN
ejpam-4933	198	13	yj(ε	yj(ε	PRON
ejpam-4933	198	14	,	,	PUNCT
ejpam-4933	198	15	ζ(a	ζ(a	NOUN
ejpam-4933	198	16	)	)	PUNCT
ejpam-4933	198	17	)	)	PUNCT
ejpam-4933	198	18	,	,	PUNCT
ejpam-4933	198	19	and	and	CCONJ
ejpam-4933	198	20	therefore	therefore	ADV
ejpam-4933	198	21	ζ	ζ	NOUN
ejpam-4933	198	22	is	be	AUX
ejpam-4933	198	23	not	not	PART
ejpam-4933	198	24	a	a	DET
ejpam-4933	198	25	y	y	PROPN
ejpam-4933	198	26	ε	ε	PROPN
ejpam-4933	198	27	j	j	PROPN
ejpam-4933	198	28	-fuzzy	-fuzzy	PROPN
ejpam-4933	198	29	ideal	ideal	NOUN
ejpam-4933	198	30	of	of	ADP
ejpam-4933	198	31	(	(	PUNCT
ejpam-4933	198	32	x	x	NOUN
ejpam-4933	198	33	,	,	PUNCT
ejpam-4933	198	34	∗	∗	NOUN
ejpam-4933	198	35	,	,	PUNCT
ejpam-4933	198	36	0	0	NUM
ejpam-4933	198	37	)	)	PUNCT
ejpam-4933	198	38	for	for	ADP
ejpam-4933	198	39	ε	ε	PROPN
ejpam-4933	198	40	=	=	SYM
ejpam-4933	198	41	0.49	0.49	NUM
ejpam-4933	198	42	.	.	PUNCT
ejpam-4933	199	1	theorem	theorem	NOUN
ejpam-4933	199	2	3	3	X
ejpam-4933	199	3	.	.	PUNCT
ejpam-4933	200	1	let	let	VERB
ejpam-4933	200	2	ζ	ζ	NOUN
ejpam-4933	200	3	be	be	AUX
ejpam-4933	200	4	a	a	DET
ejpam-4933	200	5	fuzzy	fuzzy	ADJ
ejpam-4933	200	6	set	set	NOUN
ejpam-4933	200	7	in	in	ADP
ejpam-4933	200	8	x.	x.	NOUN
ejpam-4933	200	9	if	if	SCONJ
ejpam-4933	200	10	ζ(x	ζ(x	NOUN
ejpam-4933	200	11	)	)	PUNCT
ejpam-4933	200	12	≤	≤	NUM
ejpam-4933	200	13	ε	ε	PROPN
ejpam-4933	200	14	for	for	ADP
ejpam-4933	200	15	all	all	DET
ejpam-4933	200	16	x	x	SYM
ejpam-4933	200	17	∈	∈	PROPN
ejpam-4933	200	18	x	x	NOUN
ejpam-4933	200	19	,	,	PUNCT
ejpam-4933	200	20	then	then	ADV
ejpam-4933	200	21	ζ	ζ	NOUN
ejpam-4933	200	22	is	be	AUX
ejpam-4933	200	23	a	a	DET
ejpam-4933	200	24	y	y	PROPN
ejpam-4933	200	25	ε	ε	PROPN
ejpam-4933	200	26	j	j	PROPN
ejpam-4933	200	27	-fuzzy	-fuzzy	PROPN
ejpam-4933	200	28	ideal	ideal	NOUN
ejpam-4933	200	29	of	of	ADP
ejpam-4933	200	30	(	(	PUNCT
ejpam-4933	200	31	x	x	NOUN
ejpam-4933	200	32	,	,	PUNCT
ejpam-4933	200	33	∗	∗	NOUN
ejpam-4933	200	34	,	,	PUNCT
ejpam-4933	200	35	0	0	NUM
ejpam-4933	200	36	)	)	PUNCT
ejpam-4933	200	37	.	.	PUNCT
ejpam-4933	201	1	proof	proof	NOUN
ejpam-4933	201	2	.	.	PUNCT
ejpam-4933	202	1	let	let	VERB
ejpam-4933	202	2	ζ	ζ	NOUN
ejpam-4933	202	3	be	be	AUX
ejpam-4933	202	4	a	a	DET
ejpam-4933	202	5	fuzzy	fuzzy	ADJ
ejpam-4933	202	6	set	set	NOUN
ejpam-4933	202	7	in	in	ADP
ejpam-4933	202	8	x	x	PUNCT
ejpam-4933	202	9	that	that	SCONJ
ejpam-4933	202	10	satisfies	satisfy	VERB
ejpam-4933	202	11	ζ(x	ζ(x	NOUN
ejpam-4933	202	12	)	)	PUNCT
ejpam-4933	202	13	≤	≤	NUM
ejpam-4933	202	14	ε	ε	PROPN
ejpam-4933	202	15	for	for	ADP
ejpam-4933	202	16	all	all	PRON
ejpam-4933	202	17	x	x	SYM
ejpam-4933	202	18	∈	∈	ADJ
ejpam-4933	202	19	x.	x.	NOUN
ejpam-4933	202	20	then	then	ADV
ejpam-4933	202	21	1	1	NUM
ejpam-4933	202	22	−	−	NOUN
ejpam-4933	202	23	ε	ε	PROPN
ejpam-4933	202	24	≤	≤	NUM
ejpam-4933	202	25	1	1	NUM
ejpam-4933	202	26	−	−	NOUN
ejpam-4933	202	27	ζ(x	ζ(x	NOUN
ejpam-4933	202	28	)	)	PUNCT
ejpam-4933	202	29	for	for	ADP
ejpam-4933	202	30	all	all	PRON
ejpam-4933	202	31	x	x	SYM
ejpam-4933	202	32	∈	∈	ADJ
ejpam-4933	202	33	x.	x.	NOUN
ejpam-4933	202	34	hence	hence	ADV
ejpam-4933	202	35	yj(ε	yj(ε	NUM
ejpam-4933	202	36	,	,	PUNCT
ejpam-4933	202	37	ζ(0	ζ(0	NOUN
ejpam-4933	202	38	)	)	PUNCT
ejpam-4933	202	39	)	)	PUNCT
ejpam-4933	203	1	=	=	PUNCT
ejpam-4933	203	2	(	(	PUNCT
ejpam-4933	203	3	1	1	NUM
ejpam-4933	203	4	−	−	PROPN
ejpam-4933	203	5	ε	ε	PROPN
ejpam-4933	203	6	)	)	PUNCT
ejpam-4933	203	7	∧	∧	NOUN
ejpam-4933	203	8	(	(	PUNCT
ejpam-4933	203	9	1	1	NUM
ejpam-4933	203	10	−	−	NOUN
ejpam-4933	203	11	ζ(0	ζ(0	NOUN
ejpam-4933	203	12	)	)	PUNCT
ejpam-4933	203	13	)	)	PUNCT
ejpam-4933	203	14	=	=	SYM
ejpam-4933	204	1	1	1	NUM
ejpam-4933	204	2	−	−	NOUN
ejpam-4933	204	3	ε	ε	PROPN
ejpam-4933	204	4	=	=	PUNCT
ejpam-4933	204	5	(	(	PUNCT
ejpam-4933	204	6	1	1	NUM
ejpam-4933	204	7	−	−	PROPN
ejpam-4933	204	8	ε	ε	PROPN
ejpam-4933	204	9	)	)	PUNCT
ejpam-4933	204	10	∧	∧	NOUN
ejpam-4933	204	11	(	(	PUNCT
ejpam-4933	204	12	1	1	NUM
ejpam-4933	204	13	−	−	PROPN
ejpam-4933	204	14	ζ(x	ζ(x	NOUN
ejpam-4933	204	15	)	)	PUNCT
ejpam-4933	204	16	)	)	PUNCT
ejpam-4933	205	1	=	=	PUNCT
ejpam-4933	205	2	yj(ε	yj(ε	ADJ
ejpam-4933	205	3	,	,	PUNCT
ejpam-4933	205	4	ζ(x	ζ(x	NOUN
ejpam-4933	205	5	)	)	PUNCT
ejpam-4933	205	6	)	)	PUNCT
ejpam-4933	205	7	for	for	ADP
ejpam-4933	205	8	all	all	PRON
ejpam-4933	205	9	x	x	SYM
ejpam-4933	205	10	∈	∈	NOUN
ejpam-4933	205	11	x.	x.	NOUN
ejpam-4933	205	12	also	also	ADV
ejpam-4933	205	13	,	,	PUNCT
ejpam-4933	205	14	we	we	PRON
ejpam-4933	205	15	have	have	VERB
ejpam-4933	205	16	yj(ε	yj(ε	NOUN
ejpam-4933	205	17	,	,	PUNCT
ejpam-4933	205	18	ζ(x	ζ(x	NOUN
ejpam-4933	205	19	)	)	PUNCT
ejpam-4933	205	20	)	)	PUNCT
ejpam-4933	206	1	=	=	SYM
ejpam-4933	206	2	1	1	NUM
ejpam-4933	206	3	−	−	NOUN
ejpam-4933	206	4	ε	ε	PROPN
ejpam-4933	206	5	=	=	SYM
ejpam-4933	206	6	yj(ε	yj(ε	PROPN
ejpam-4933	206	7	,	,	PUNCT
ejpam-4933	206	8	ζ(x	ζ(x	PROPN
ejpam-4933	206	9	∗	∗	VERB
ejpam-4933	206	10	a	a	NOUN
ejpam-4933	206	11	)	)	PUNCT
ejpam-4933	206	12	)	)	PUNCT
ejpam-4933	207	1	∨	∨	PROPN
ejpam-4933	207	2	yj(ε	yj(ε	NUM
ejpam-4933	207	3	,	,	PUNCT
ejpam-4933	207	4	ζ(a	ζ(a	NOUN
ejpam-4933	207	5	)	)	PUNCT
ejpam-4933	207	6	)	)	PUNCT
ejpam-4933	207	7	for	for	ADP
ejpam-4933	207	8	all	all	DET
ejpam-4933	207	9	x	x	NOUN
ejpam-4933	207	10	,	,	PUNCT
ejpam-4933	207	11	a	a	DET
ejpam-4933	207	12	∈	∈	NOUN
ejpam-4933	207	13	x.	x.	NOUN
ejpam-4933	207	14	therefore	therefore	ADV
ejpam-4933	207	15	ζ	ζ	PROPN
ejpam-4933	207	16	is	be	AUX
ejpam-4933	207	17	a	a	DET
ejpam-4933	207	18	y	y	PROPN
ejpam-4933	207	19	ε	ε	PROPN
ejpam-4933	207	20	j	j	PROPN
ejpam-4933	207	21	-fuzzy	-fuzzy	PROPN
ejpam-4933	207	22	ideal	ideal	NOUN
ejpam-4933	207	23	of	of	ADP
ejpam-4933	207	24	(	(	PUNCT
ejpam-4933	207	25	x	x	NOUN
ejpam-4933	207	26	,	,	PUNCT
ejpam-4933	207	27	∗	∗	NOUN
ejpam-4933	207	28	,	,	PUNCT
ejpam-4933	207	29	0	0	NUM
ejpam-4933	207	30	)	)	PUNCT
ejpam-4933	207	31	.	.	PUNCT
ejpam-4933	208	1	let	let	VERB
ejpam-4933	208	2	ζ	ζ	NOUN
ejpam-4933	208	3	be	be	AUX
ejpam-4933	208	4	a	a	DET
ejpam-4933	208	5	fuzzy	fuzzy	ADJ
ejpam-4933	208	6	set	set	NOUN
ejpam-4933	208	7	in	in	ADP
ejpam-4933	208	8	x.	x.	NOUN
ejpam-4933	208	9	if	if	SCONJ
ejpam-4933	208	10	there	there	PRON
ejpam-4933	208	11	exists	exist	VERB
ejpam-4933	208	12	z	z	NOUN
ejpam-4933	208	13	∈	∈	PROPN
ejpam-4933	208	14	x	x	PUNCT
ejpam-4933	208	15	that	that	SCONJ
ejpam-4933	208	16	satisfies	satisfy	VERB
ejpam-4933	208	17	ζ(z	ζ(z	NOUN
ejpam-4933	208	18	)	)	PUNCT
ejpam-4933	208	19	>	>	X
ejpam-4933	209	1	ε	ε	PROPN
ejpam-4933	209	2	,	,	PUNCT
ejpam-4933	209	3	then	then	ADV
ejpam-4933	209	4	ζ	ζ	NOUN
ejpam-4933	209	5	may	may	AUX
ejpam-4933	209	6	not	not	PART
ejpam-4933	209	7	be	be	AUX
ejpam-4933	209	8	a	a	DET
ejpam-4933	209	9	y	y	PROPN
ejpam-4933	209	10	ε	ε	PROPN
ejpam-4933	209	11	j	j	PROPN
ejpam-4933	209	12	-fuzzy	-fuzzy	PROPN
ejpam-4933	209	13	ideal	ideal	NOUN
ejpam-4933	209	14	of	of	ADP
ejpam-4933	209	15	(	(	PUNCT
ejpam-4933	209	16	x	x	NOUN
ejpam-4933	209	17	,	,	PUNCT
ejpam-4933	209	18	∗	∗	NOUN
ejpam-4933	209	19	,	,	PUNCT
ejpam-4933	209	20	0	0	NUM
ejpam-4933	209	21	)	)	PUNCT
ejpam-4933	209	22	as	as	SCONJ
ejpam-4933	209	23	shown	show	VERB
ejpam-4933	209	24	in	in	ADP
ejpam-4933	209	25	the	the	DET
ejpam-4933	209	26	example	example	NOUN
ejpam-4933	209	27	below	below	ADV
ejpam-4933	209	28	.	.	PUNCT
ejpam-4933	210	1	e.	e.	PROPN
ejpam-4933	210	2	h.	h.	PROPN
ejpam-4933	210	3	roh	roh	PROPN
ejpam-4933	210	4	,	,	PUNCT
ejpam-4933	210	5	e.	e.	PROPN
ejpam-4933	210	6	yang	yang	PROPN
ejpam-4933	210	7	,	,	PUNCT
ejpam-4933	210	8	y.	y.	PROPN
ejpam-4933	210	9	b.	b.	PROPN
ejpam-4933	210	10	jun	jun	PROPN
ejpam-4933	210	11	/	/	SYM
ejpam-4933	210	12	eur	eur	PROPN
ejpam-4933	210	13	.	.	PUNCT
ejpam-4933	211	1	j.	j.	PROPN
ejpam-4933	211	2	pure	pure	PROPN
ejpam-4933	211	3	appl	appl	PROPN
ejpam-4933	211	4	.	.	PROPN
ejpam-4933	211	5	math	math	PROPN
ejpam-4933	211	6	,	,	PUNCT
ejpam-4933	211	7	16	16	NUM
ejpam-4933	211	8	(	(	PUNCT
ejpam-4933	211	9	4	4	NUM
ejpam-4933	211	10	)	)	PUNCT
ejpam-4933	211	11	(	(	PUNCT
ejpam-4933	211	12	2023	2023	NUM
ejpam-4933	211	13	)	)	PUNCT
ejpam-4933	211	14	,	,	PUNCT
ejpam-4933	211	15	2009	2009	NUM
ejpam-4933	211	16	-	-	SYM
ejpam-4933	211	17	2024	2024	NUM
ejpam-4933	211	18	2018	2018	NUM
ejpam-4933	211	19	table	table	NOUN
ejpam-4933	211	20	3	3	NUM
ejpam-4933	211	21	:	:	PUNCT
ejpam-4933	211	22	cayley	cayley	ADJ
ejpam-4933	211	23	table	table	NOUN
ejpam-4933	211	24	for	for	ADP
ejpam-4933	211	25	the	the	DET
ejpam-4933	211	26	binary	binary	PROPN
ejpam-4933	211	27	operation	operation	NOUN
ejpam-4933	211	28	“	"	PUNCT
ejpam-4933	211	29	∗	∗	NOUN
ejpam-4933	211	30	”	"	PUNCT
ejpam-4933	211	31	∗	∗	X
ejpam-4933	211	32	0	0	NUM
ejpam-4933	211	33	b1	b1	PROPN
ejpam-4933	211	34	b2	b2	NOUN
ejpam-4933	211	35	b3	b3	PROPN
ejpam-4933	211	36	b4	b4	NOUN
ejpam-4933	211	37	0	0	NUM
ejpam-4933	211	38	0	0	NUM
ejpam-4933	211	39	0	0	NUM
ejpam-4933	211	40	0	0	NUM
ejpam-4933	211	41	0	0	NUM
ejpam-4933	211	42	0	0	NUM
ejpam-4933	211	43	b1	b1	NOUN
ejpam-4933	211	44	b1	b1	NOUN
ejpam-4933	211	45	0	0	NUM
ejpam-4933	211	46	0	0	SYM
ejpam-4933	211	47	0	0	NUM
ejpam-4933	211	48	b1	b1	NOUN
ejpam-4933	211	49	b2	b2	NOUN
ejpam-4933	211	50	b2	b2	NOUN
ejpam-4933	211	51	b1	b1	NOUN
ejpam-4933	211	52	0	0	NUM
ejpam-4933	211	53	0	0	NUM
ejpam-4933	211	54	b2	b2	NOUN
ejpam-4933	211	55	b3	b3	PROPN
ejpam-4933	211	56	b3	b3	PROPN
ejpam-4933	211	57	b1	b1	NOUN
ejpam-4933	211	58	b1	b1	NOUN
ejpam-4933	211	59	0	0	NUM
ejpam-4933	211	60	b3	b3	PROPN
ejpam-4933	211	61	b4	b4	NOUN
ejpam-4933	211	62	b4	b4	NOUN
ejpam-4933	211	63	b4	b4	PROPN
ejpam-4933	211	64	b4	b4	PROPN
ejpam-4933	211	65	b4	b4	PROPN
ejpam-4933	211	66	0	0	NUM
ejpam-4933	211	67	example	example	NOUN
ejpam-4933	212	1	4	4	X
ejpam-4933	212	2	.	.	PUNCT
ejpam-4933	213	1	let	let	VERB
ejpam-4933	213	2	x	x	PUNCT
ejpam-4933	213	3	=	=	PUNCT
ejpam-4933	213	4	{	{	PUNCT
ejpam-4933	213	5	0	0	NUM
ejpam-4933	213	6	,	,	PUNCT
ejpam-4933	213	7	b1	b1	NOUN
ejpam-4933	213	8	,	,	PUNCT
ejpam-4933	213	9	b2	b2	NOUN
ejpam-4933	213	10	,	,	PUNCT
ejpam-4933	213	11	b3	b3	NOUN
ejpam-4933	213	12	,	,	PUNCT
ejpam-4933	213	13	b4	b4	PROPN
ejpam-4933	213	14	}	}	PUNCT
ejpam-4933	213	15	be	be	AUX
ejpam-4933	213	16	a	a	DET
ejpam-4933	213	17	set	set	NOUN
ejpam-4933	213	18	with	with	ADP
ejpam-4933	213	19	a	a	DET
ejpam-4933	213	20	binary	binary	ADJ
ejpam-4933	213	21	operation	operation	NOUN
ejpam-4933	213	22	“	"	PUNCT
ejpam-4933	213	23	∗	∗	NOUN
ejpam-4933	213	24	”	"	PUNCT
ejpam-4933	213	25	given	give	VERB
ejpam-4933	213	26	by	by	ADP
ejpam-4933	213	27	table	table	NOUN
ejpam-4933	213	28	3	3	NUM
ejpam-4933	213	29	.	.	PUNCT
ejpam-4933	214	1	then	then	ADV
ejpam-4933	214	2	(	(	PUNCT
ejpam-4933	214	3	x	x	X
ejpam-4933	214	4	,	,	PUNCT
ejpam-4933	214	5	∗	∗	NOUN
ejpam-4933	214	6	,	,	PUNCT
ejpam-4933	214	7	0	0	NUM
ejpam-4933	214	8	)	)	PUNCT
ejpam-4933	214	9	is	be	AUX
ejpam-4933	214	10	a	a	DET
ejpam-4933	214	11	bck	bck	NOUN
ejpam-4933	214	12	-	-	PUNCT
ejpam-4933	214	13	algebra	algebra	NOUN
ejpam-4933	214	14	and	and	CCONJ
ejpam-4933	214	15	so	so	ADV
ejpam-4933	214	16	a	a	DET
ejpam-4933	214	17	bci	bci	NOUN
ejpam-4933	214	18	-	-	NOUN
ejpam-4933	214	19	algebra	algebra	NOUN
ejpam-4933	214	20	(	(	PUNCT
ejpam-4933	214	21	see	see	VERB
ejpam-4933	214	22	[	[	X
ejpam-4933	214	23	11	11	NUM
ejpam-4933	214	24	]	]	NUM
ejpam-4933	214	25	)	)	PUNCT
ejpam-4933	214	26	.	.	PUNCT
ejpam-4933	215	1	consider	consider	VERB
ejpam-4933	215	2	a	a	DET
ejpam-4933	215	3	fuzzy	fuzzy	ADJ
ejpam-4933	215	4	set	set	VERB
ejpam-4933	215	5	ζ	ζ	NOUN
ejpam-4933	215	6	in	in	ADP
ejpam-4933	215	7	x	x	PUNCT
ejpam-4933	215	8	given	give	VERB
ejpam-4933	215	9	as	as	SCONJ
ejpam-4933	215	10	follows	follow	VERB
ejpam-4933	215	11	:	:	PUNCT
ejpam-4933	215	12	ζ	ζ	NOUN
ejpam-4933	215	13	:	:	PUNCT
ejpam-4933	215	14	x	x	X
ejpam-4933	215	15	→	→	SYM
ejpam-4933	216	1	[	[	X
ejpam-4933	216	2	0	0	NUM
ejpam-4933	216	3	,	,	PUNCT
ejpam-4933	216	4	1	1	NUM
ejpam-4933	216	5	]	]	PUNCT
ejpam-4933	216	6	,	,	PUNCT
ejpam-4933	216	7	y	y	PROPN
ejpam-4933	216	8	7→	7→	PROPN
ejpam-4933	216	9			NOUN
ejpam-4933	216	10	0.93	0.93	NUM
ejpam-4933	216	11	if	if	SCONJ
ejpam-4933	216	12	y	y	PROPN
ejpam-4933	216	13	=	=	SYM
ejpam-4933	216	14	0	0	PROPN
ejpam-4933	216	15	,	,	PUNCT
ejpam-4933	216	16	0.46	0.46	NUM
ejpam-4933	216	17	if	if	SCONJ
ejpam-4933	216	18	y	y	PROPN
ejpam-4933	216	19	=	=	SYM
ejpam-4933	216	20	b1	b1	PROPN
ejpam-4933	216	21	,	,	PUNCT
ejpam-4933	216	22	0.77	0.77	NUM
ejpam-4933	216	23	if	if	SCONJ
ejpam-4933	216	24	y	y	PROPN
ejpam-4933	216	25	=	=	PUNCT
ejpam-4933	216	26	b2	b2	PROPN
ejpam-4933	216	27	,	,	PUNCT
ejpam-4933	216	28	0.58	0.58	NUM
ejpam-4933	216	29	if	if	SCONJ
ejpam-4933	216	30	y	y	PROPN
ejpam-4933	216	31	=	=	PUNCT
ejpam-4933	216	32	b3	b3	PROPN
ejpam-4933	216	33	,	,	PUNCT
ejpam-4933	216	34	0.35	0.35	NUM
ejpam-4933	216	35	if	if	SCONJ
ejpam-4933	216	36	y	y	PROPN
ejpam-4933	216	37	=	=	PUNCT
ejpam-4933	216	38	b4	b4	PROPN
ejpam-4933	216	39	.	.	PUNCT
ejpam-4933	217	1	if	if	SCONJ
ejpam-4933	217	2	ε	ε	PROPN
ejpam-4933	217	3	:	:	PUNCT
ejpam-4933	217	4	=	=	NOUN
ejpam-4933	217	5	0.53	0.53	NUM
ejpam-4933	217	6	,	,	PUNCT
ejpam-4933	217	7	then	then	ADV
ejpam-4933	217	8	yj(ε	yj(ε	NUM
ejpam-4933	217	9	,	,	PUNCT
ejpam-4933	217	10	ζ(0	ζ(0	NOUN
ejpam-4933	217	11	)	)	PUNCT
ejpam-4933	217	12	)	)	PUNCT
ejpam-4933	217	13	≤	≤	NOUN
ejpam-4933	217	14	yj(ε	yj(ε	NUM
ejpam-4933	217	15	,	,	PUNCT
ejpam-4933	217	16	ζ(x	ζ(x	NOUN
ejpam-4933	217	17	)	)	PUNCT
ejpam-4933	217	18	)	)	PUNCT
ejpam-4933	217	19	for	for	ADP
ejpam-4933	217	20	all	all	DET
ejpam-4933	217	21	x	x	SYM
ejpam-4933	217	22	∈	∈	PROPN
ejpam-4933	217	23	x.	x.	NOUN
ejpam-4933	217	24	but	but	CCONJ
ejpam-4933	217	25	yj(ε	yj(ε	NUM
ejpam-4933	217	26	,	,	PUNCT
ejpam-4933	217	27	ζ(b1	ζ(b1	VERB
ejpam-4933	217	28	∗	∗	NOUN
ejpam-4933	217	29	b3	b3	NOUN
ejpam-4933	217	30	)	)	PUNCT
ejpam-4933	217	31	)	)	PUNCT
ejpam-4933	218	1	∨	∨	PROPN
ejpam-4933	218	2	yj(ε	yj(ε	NUM
ejpam-4933	218	3	,	,	PUNCT
ejpam-4933	218	4	ζ(b3	ζ(b3	NOUN
ejpam-4933	218	5	)	)	PUNCT
ejpam-4933	218	6	)	)	PUNCT
ejpam-4933	219	1	=	=	SYM
ejpam-4933	219	2	yj(0.53	yj(0.53	NOUN
ejpam-4933	219	3	,	,	PUNCT
ejpam-4933	219	4	0.93	0.93	NUM
ejpam-4933	219	5	)	)	PUNCT
ejpam-4933	219	6	∨	∨	NUM
ejpam-4933	219	7	yj(0.53	yj(0.53	NOUN
ejpam-4933	219	8	,	,	PUNCT
ejpam-4933	219	9	0.58	0.58	NUM
ejpam-4933	219	10	)	)	PUNCT
ejpam-4933	219	11	=	=	SYM
ejpam-4933	220	1	0.07	0.07	NUM
ejpam-4933	220	2	∨	∨	NUM
ejpam-4933	220	3	0.42	0.42	NUM
ejpam-4933	220	4	=	=	SYM
ejpam-4933	220	5	0.42	0.42	NUM
ejpam-4933	220	6	<	<	X
ejpam-4933	220	7	0.47	0.47	NUM
ejpam-4933	220	8	=	=	SYM
ejpam-4933	220	9	yj(ε	yj(ε	NUM
ejpam-4933	220	10	,	,	PUNCT
ejpam-4933	220	11	ζ(b1	ζ(b1	NOUN
ejpam-4933	220	12	)	)	PUNCT
ejpam-4933	220	13	)	)	PUNCT
ejpam-4933	220	14	.	.	PUNCT
ejpam-4933	221	1	hence	hence	ADV
ejpam-4933	221	2	ζ	ζ	NOUN
ejpam-4933	221	3	is	be	AUX
ejpam-4933	221	4	not	not	PART
ejpam-4933	221	5	a	a	DET
ejpam-4933	221	6	y	y	PROPN
ejpam-4933	221	7	ε	ε	PROPN
ejpam-4933	221	8	j	j	PROPN
ejpam-4933	221	9	-fuzzy	-fuzzy	PROPN
ejpam-4933	221	10	ideal	ideal	NOUN
ejpam-4933	221	11	of	of	ADP
ejpam-4933	221	12	(	(	PUNCT
ejpam-4933	221	13	x	x	NOUN
ejpam-4933	221	14	,	,	PUNCT
ejpam-4933	221	15	∗	∗	NOUN
ejpam-4933	221	16	,	,	PUNCT
ejpam-4933	221	17	0	0	NUM
ejpam-4933	221	18	)	)	PUNCT
ejpam-4933	221	19	for	for	ADP
ejpam-4933	221	20	ε	ε	PROPN
ejpam-4933	221	21	=	=	SYM
ejpam-4933	221	22	0.53	0.53	NUM
ejpam-4933	221	23	.	.	PUNCT
ejpam-4933	222	1	theorem	theorem	VERB
ejpam-4933	222	2	4	4	NUM
ejpam-4933	222	3	.	.	PUNCT
ejpam-4933	223	1	every	every	DET
ejpam-4933	223	2	fuzzy	fuzzy	ADJ
ejpam-4933	223	3	ideal	ideal	NOUN
ejpam-4933	223	4	of	of	ADP
ejpam-4933	223	5	(	(	PUNCT
ejpam-4933	223	6	x	x	NOUN
ejpam-4933	223	7	,	,	PUNCT
ejpam-4933	223	8	∗	∗	NOUN
ejpam-4933	223	9	,	,	PUNCT
ejpam-4933	223	10	0	0	NUM
ejpam-4933	223	11	)	)	PUNCT
ejpam-4933	223	12	is	be	AUX
ejpam-4933	223	13	a	a	DET
ejpam-4933	223	14	y	y	PROPN
ejpam-4933	223	15	ε	ε	PROPN
ejpam-4933	223	16	j	j	PROPN
ejpam-4933	223	17	-fuzzy	-fuzzy	PROPN
ejpam-4933	223	18	ideal	ideal	NOUN
ejpam-4933	223	19	of	of	ADP
ejpam-4933	223	20	(	(	PUNCT
ejpam-4933	223	21	x	x	NOUN
ejpam-4933	223	22	,	,	PUNCT
ejpam-4933	223	23	∗	∗	NOUN
ejpam-4933	223	24	,	,	PUNCT
ejpam-4933	223	25	0	0	NUM
ejpam-4933	223	26	)	)	PUNCT
ejpam-4933	223	27	for	for	ADP
ejpam-4933	223	28	all	all	DET
ejpam-4933	223	29	ε	ε	PROPN
ejpam-4933	223	30	∈	∈	PROPN
ejpam-4933	223	31	(	(	PUNCT
ejpam-4933	223	32	0	0	NUM
ejpam-4933	223	33	,	,	PUNCT
ejpam-4933	223	34	1	1	NUM
ejpam-4933	223	35	)	)	PUNCT
ejpam-4933	223	36	.	.	PUNCT
ejpam-4933	224	1	proof	proof	NOUN
ejpam-4933	224	2	.	.	PUNCT
ejpam-4933	225	1	let	let	VERB
ejpam-4933	225	2	ζ	ζ	NOUN
ejpam-4933	225	3	be	be	AUX
ejpam-4933	225	4	a	a	DET
ejpam-4933	225	5	fuzzy	fuzzy	ADJ
ejpam-4933	225	6	ideal	ideal	NOUN
ejpam-4933	225	7	of	of	ADP
ejpam-4933	225	8	(	(	PUNCT
ejpam-4933	225	9	x	x	NOUN
ejpam-4933	225	10	,	,	PUNCT
ejpam-4933	225	11	∗	∗	NOUN
ejpam-4933	225	12	,	,	PUNCT
ejpam-4933	225	13	0	0	NUM
ejpam-4933	225	14	)	)	PUNCT
ejpam-4933	225	15	and	and	CCONJ
ejpam-4933	225	16	let	let	VERB
ejpam-4933	225	17	ε	ε	PROPN
ejpam-4933	225	18	∈	∈	PROPN
ejpam-4933	225	19	(	(	PUNCT
ejpam-4933	225	20	0	0	NUM
ejpam-4933	225	21	,	,	PUNCT
ejpam-4933	225	22	1	1	NUM
ejpam-4933	225	23	)	)	PUNCT
ejpam-4933	225	24	.	.	PUNCT
ejpam-4933	226	1	then	then	ADV
ejpam-4933	226	2	ζc(0	ζc(0	PROPN
ejpam-4933	226	3	)	)	PUNCT
ejpam-4933	226	4	≤	≤	NOUN
ejpam-4933	226	5	ζc(x	ζc(x	NOUN
ejpam-4933	226	6	)	)	PUNCT
ejpam-4933	226	7	and	and	CCONJ
ejpam-4933	226	8	ζc(x	ζc(x	NOUN
ejpam-4933	226	9	)	)	PUNCT
ejpam-4933	226	10	≤	≤	NOUN
ejpam-4933	227	1	ζc(x	ζc(x	NOUN
ejpam-4933	227	2	∗	∗	NOUN
ejpam-4933	227	3	a	a	PRON
ejpam-4933	227	4	)	)	PUNCT
ejpam-4933	227	5	∨	∨	NUM
ejpam-4933	227	6	ζc(a	ζc(a	NUM
ejpam-4933	227	7	)	)	PUNCT
ejpam-4933	227	8	for	for	ADP
ejpam-4933	227	9	all	all	DET
ejpam-4933	227	10	x	x	NOUN
ejpam-4933	227	11	,	,	PUNCT
ejpam-4933	227	12	a	a	DET
ejpam-4933	227	13	∈	∈	NOUN
ejpam-4933	227	14	x.	x.	NOUN
ejpam-4933	227	15	hence	hence	ADV
ejpam-4933	227	16	yj(ε	yj(ε	NUM
ejpam-4933	227	17	,	,	PUNCT
ejpam-4933	227	18	ζ(0	ζ(0	NOUN
ejpam-4933	227	19	)	)	PUNCT
ejpam-4933	227	20	)	)	PUNCT
ejpam-4933	228	1	=	=	PUNCT
ejpam-4933	228	2	(	(	PUNCT
ejpam-4933	228	3	1	1	NUM
ejpam-4933	228	4	−	−	PROPN
ejpam-4933	228	5	ε	ε	PROPN
ejpam-4933	228	6	)	)	PUNCT
ejpam-4933	228	7	∧	∧	PROPN
ejpam-4933	228	8	ζc(0	ζc(0	NOUN
ejpam-4933	228	9	)	)	PUNCT
ejpam-4933	228	10	≤	≤	NOUN
ejpam-4933	228	11	(	(	PUNCT
ejpam-4933	228	12	1	1	NUM
ejpam-4933	228	13	−	−	PROPN
ejpam-4933	228	14	ε	ε	PROPN
ejpam-4933	228	15	)	)	PUNCT
ejpam-4933	228	16	∧	∧	NOUN
ejpam-4933	228	17	ζc(x	ζc(x	NOUN
ejpam-4933	228	18	)	)	PUNCT
ejpam-4933	228	19	=	=	SYM
ejpam-4933	228	20	yj(ε	yj(ε	ADJ
ejpam-4933	228	21	,	,	PUNCT
ejpam-4933	228	22	ζ(x	ζ(x	NOUN
ejpam-4933	228	23	)	)	PUNCT
ejpam-4933	228	24	)	)	PUNCT
ejpam-4933	228	25	and	and	CCONJ
ejpam-4933	228	26	yj(ε	yj(ε	NUM
ejpam-4933	228	27	,	,	PUNCT
ejpam-4933	228	28	ζ(x	ζ(x	NOUN
ejpam-4933	228	29	)	)	PUNCT
ejpam-4933	228	30	)	)	PUNCT
ejpam-4933	229	1	=	=	PUNCT
ejpam-4933	229	2	(	(	PUNCT
ejpam-4933	229	3	1	1	NUM
ejpam-4933	229	4	−	−	PROPN
ejpam-4933	229	5	ε	ε	PROPN
ejpam-4933	229	6	)	)	PUNCT
ejpam-4933	229	7	∧	∧	NOUN
ejpam-4933	229	8	ζc(x	ζc(x	NOUN
ejpam-4933	229	9	)	)	PUNCT
ejpam-4933	229	10	≤	≤	NOUN
ejpam-4933	229	11	(	(	PUNCT
ejpam-4933	229	12	1	1	NUM
ejpam-4933	229	13	−	−	PROPN
ejpam-4933	229	14	ε	ε	PROPN
ejpam-4933	229	15	)	)	PUNCT
ejpam-4933	229	16	∧	∧	PROPN
ejpam-4933	229	17	(	(	PUNCT
ejpam-4933	229	18	ζc(x	ζc(x	NOUN
ejpam-4933	229	19	∗	∗	NOUN
ejpam-4933	229	20	a	a	PRON
ejpam-4933	229	21	)	)	PUNCT
ejpam-4933	229	22	∨	∨	NUM
ejpam-4933	229	23	ζc(a	ζc(a	NUM
ejpam-4933	229	24	)	)	PUNCT
ejpam-4933	229	25	)	)	PUNCT
ejpam-4933	230	1	=	=	PRON
ejpam-4933	231	1	(	(	PUNCT
ejpam-4933	231	2	(	(	PUNCT
ejpam-4933	231	3	1	1	NUM
ejpam-4933	231	4	−	−	PROPN
ejpam-4933	231	5	ε	ε	PROPN
ejpam-4933	231	6	)	)	PUNCT
ejpam-4933	231	7	∧	∧	PROPN
ejpam-4933	231	8	ζc(x	ζc(x	NOUN
ejpam-4933	231	9	∗	∗	NOUN
ejpam-4933	231	10	a	a	NOUN
ejpam-4933	231	11	)	)	PUNCT
ejpam-4933	231	12	)	)	PUNCT
ejpam-4933	232	1	∨	∨	NUM
ejpam-4933	232	2	(	(	PUNCT
ejpam-4933	232	3	(	(	PUNCT
ejpam-4933	232	4	1	1	NUM
ejpam-4933	232	5	−	−	NOUN
ejpam-4933	232	6	ε	ε	PROPN
ejpam-4933	232	7	)	)	PUNCT
ejpam-4933	232	8	∧	∧	NOUN
ejpam-4933	232	9	ζc(a	ζc(a	NUM
ejpam-4933	232	10	)	)	PUNCT
ejpam-4933	232	11	)	)	PUNCT
ejpam-4933	233	1	=	=	PUNCT
ejpam-4933	233	2	yj(ε	yj(ε	ADJ
ejpam-4933	233	3	,	,	PUNCT
ejpam-4933	233	4	ζ(x	ζ(x	PROPN
ejpam-4933	233	5	∗	∗	VERB
ejpam-4933	233	6	a	a	NOUN
ejpam-4933	233	7	)	)	PUNCT
ejpam-4933	233	8	)	)	PUNCT
ejpam-4933	233	9	∨	∨	PROPN
ejpam-4933	233	10	yj(ε	yj(ε	NUM
ejpam-4933	233	11	,	,	PUNCT
ejpam-4933	233	12	ζ(a	ζ(a	NOUN
ejpam-4933	233	13	)	)	PUNCT
ejpam-4933	233	14	)	)	PUNCT
ejpam-4933	233	15	for	for	ADP
ejpam-4933	233	16	all	all	DET
ejpam-4933	233	17	x	x	NOUN
ejpam-4933	233	18	,	,	PUNCT
ejpam-4933	233	19	a	a	DET
ejpam-4933	233	20	∈	∈	NOUN
ejpam-4933	233	21	x.	x.	NOUN
ejpam-4933	233	22	therefore	therefore	ADV
ejpam-4933	233	23	ζ	ζ	PROPN
ejpam-4933	233	24	is	be	AUX
ejpam-4933	233	25	a	a	DET
ejpam-4933	233	26	y	y	PROPN
ejpam-4933	233	27	ε	ε	PROPN
ejpam-4933	233	28	j	j	PROPN
ejpam-4933	233	29	-fuzzy	-fuzzy	PROPN
ejpam-4933	233	30	ideal	ideal	NOUN
ejpam-4933	233	31	of	of	ADP
ejpam-4933	233	32	(	(	PUNCT
ejpam-4933	233	33	x	x	NOUN
ejpam-4933	233	34	,	,	PUNCT
ejpam-4933	233	35	∗	∗	NOUN
ejpam-4933	233	36	,	,	PUNCT
ejpam-4933	233	37	0	0	NUM
ejpam-4933	233	38	)	)	PUNCT
ejpam-4933	233	39	for	for	ADP
ejpam-4933	233	40	all	all	DET
ejpam-4933	233	41	ε	ε	PROPN
ejpam-4933	233	42	∈	∈	PROPN
ejpam-4933	233	43	(	(	PUNCT
ejpam-4933	233	44	0	0	NUM
ejpam-4933	233	45	,	,	PUNCT
ejpam-4933	233	46	1	1	NUM
ejpam-4933	233	47	)	)	PUNCT
ejpam-4933	233	48	.	.	PUNCT
ejpam-4933	234	1	e.	e.	PROPN
ejpam-4933	234	2	h.	h.	PROPN
ejpam-4933	234	3	roh	roh	PROPN
ejpam-4933	234	4	,	,	PUNCT
ejpam-4933	234	5	e.	e.	PROPN
ejpam-4933	234	6	yang	yang	PROPN
ejpam-4933	234	7	,	,	PUNCT
ejpam-4933	234	8	y.	y.	PROPN
ejpam-4933	234	9	b.	b.	PROPN
ejpam-4933	234	10	jun	jun	PROPN
ejpam-4933	234	11	/	/	SYM
ejpam-4933	234	12	eur	eur	PROPN
ejpam-4933	234	13	.	.	PUNCT
ejpam-4933	235	1	j.	j.	PROPN
ejpam-4933	235	2	pure	pure	PROPN
ejpam-4933	235	3	appl	appl	PROPN
ejpam-4933	235	4	.	.	PROPN
ejpam-4933	235	5	math	math	PROPN
ejpam-4933	235	6	,	,	PUNCT
ejpam-4933	235	7	16	16	NUM
ejpam-4933	235	8	(	(	PUNCT
ejpam-4933	235	9	4	4	NUM
ejpam-4933	235	10	)	)	PUNCT
ejpam-4933	235	11	(	(	PUNCT
ejpam-4933	235	12	2023	2023	NUM
ejpam-4933	235	13	)	)	PUNCT
ejpam-4933	235	14	,	,	PUNCT
ejpam-4933	235	15	2009	2009	NUM
ejpam-4933	235	16	-	-	SYM
ejpam-4933	235	17	2024	2024	NUM
ejpam-4933	235	18	2019	2019	NUM
ejpam-4933	235	19	theorem	theorem	NOUN
ejpam-4933	235	20	5	5	NUM
ejpam-4933	235	21	.	.	PUNCT
ejpam-4933	236	1	if	if	SCONJ
ejpam-4933	236	2	ζ	ζ	NOUN
ejpam-4933	236	3	is	be	AUX
ejpam-4933	236	4	a	a	DET
ejpam-4933	236	5	y	y	PROPN
ejpam-4933	236	6	ε	ε	PROPN
ejpam-4933	236	7	j	j	PROPN
ejpam-4933	236	8	-fuzzy	-fuzzy	PROPN
ejpam-4933	236	9	ideal	ideal	NOUN
ejpam-4933	236	10	of	of	ADP
ejpam-4933	236	11	(	(	PUNCT
ejpam-4933	236	12	x	x	NOUN
ejpam-4933	236	13	,	,	PUNCT
ejpam-4933	236	14	∗	∗	NOUN
ejpam-4933	236	15	,	,	PUNCT
ejpam-4933	236	16	0	0	NUM
ejpam-4933	236	17	)	)	PUNCT
ejpam-4933	236	18	for	for	ADP
ejpam-4933	236	19	some	some	DET
ejpam-4933	236	20	nonconstant	nonconstant	ADJ
ejpam-4933	236	21	factor	factor	NOUN
ejpam-4933	236	22	ε	ε	PROPN
ejpam-4933	236	23	∈	∈	PROPN
ejpam-4933	236	24	(	(	PUNCT
ejpam-4933	236	25	0	0	NUM
ejpam-4933	236	26	,	,	PUNCT
ejpam-4933	236	27	1	1	NUM
ejpam-4933	236	28	)	)	PUNCT
ejpam-4933	236	29	,	,	PUNCT
ejpam-4933	236	30	then	then	ADV
ejpam-4933	236	31	it	it	PRON
ejpam-4933	236	32	is	be	AUX
ejpam-4933	236	33	a	a	DET
ejpam-4933	236	34	fuzzy	fuzzy	ADJ
ejpam-4933	236	35	ideal	ideal	NOUN
ejpam-4933	236	36	of	of	ADP
ejpam-4933	236	37	(	(	PUNCT
ejpam-4933	236	38	x	x	NOUN
ejpam-4933	236	39	,	,	PUNCT
ejpam-4933	236	40	∗	∗	NOUN
ejpam-4933	236	41	,	,	PUNCT
ejpam-4933	236	42	0	0	NUM
ejpam-4933	236	43	)	)	PUNCT
ejpam-4933	236	44	.	.	PUNCT
ejpam-4933	237	1	proof	proof	NOUN
ejpam-4933	237	2	.	.	PUNCT
ejpam-4933	238	1	assume	assume	VERB
ejpam-4933	238	2	that	that	SCONJ
ejpam-4933	238	3	ζ	ζ	NOUN
ejpam-4933	238	4	is	be	AUX
ejpam-4933	238	5	a	a	DET
ejpam-4933	238	6	y	y	PROPN
ejpam-4933	238	7	ε	ε	PROPN
ejpam-4933	238	8	j	j	PROPN
ejpam-4933	238	9	-fuzzy	-fuzzy	PROPN
ejpam-4933	238	10	ideal	ideal	NOUN
ejpam-4933	238	11	of	of	ADP
ejpam-4933	238	12	(	(	PUNCT
ejpam-4933	238	13	x	x	NOUN
ejpam-4933	238	14	,	,	PUNCT
ejpam-4933	238	15	∗	∗	NOUN
ejpam-4933	238	16	,	,	PUNCT
ejpam-4933	238	17	0	0	NUM
ejpam-4933	238	18	)	)	PUNCT
ejpam-4933	238	19	for	for	ADP
ejpam-4933	238	20	some	some	DET
ejpam-4933	238	21	nonconstant	nonconstant	ADJ
ejpam-4933	238	22	factor	factor	NOUN
ejpam-4933	238	23	ε	ε	PROPN
ejpam-4933	238	24	∈	∈	PROPN
ejpam-4933	238	25	(	(	PUNCT
ejpam-4933	238	26	0	0	NUM
ejpam-4933	238	27	,	,	PUNCT
ejpam-4933	238	28	1	1	NUM
ejpam-4933	238	29	)	)	PUNCT
ejpam-4933	238	30	.	.	PUNCT
ejpam-4933	239	1	then	then	ADV
ejpam-4933	239	2	(	(	PUNCT
ejpam-4933	239	3	1	1	NUM
ejpam-4933	239	4	−	−	PROPN
ejpam-4933	239	5	ε	ε	PROPN
ejpam-4933	239	6	)	)	PUNCT
ejpam-4933	239	7	∧	∧	NOUN
ejpam-4933	239	8	(	(	PUNCT
ejpam-4933	239	9	1	1	NUM
ejpam-4933	239	10	−	−	NOUN
ejpam-4933	239	11	ζ(0	ζ(0	NOUN
ejpam-4933	239	12	)	)	PUNCT
ejpam-4933	239	13	)	)	PUNCT
ejpam-4933	240	1	=	=	PUNCT
ejpam-4933	240	2	yj(ε	yj(ε	ADJ
ejpam-4933	240	3	,	,	PUNCT
ejpam-4933	240	4	ζ(0	ζ(0	NOUN
ejpam-4933	240	5	)	)	PUNCT
ejpam-4933	240	6	)	)	PUNCT
ejpam-4933	241	1	≤	≤	NOUN
ejpam-4933	241	2	yj(ε	yj(ε	NUM
ejpam-4933	241	3	,	,	PUNCT
ejpam-4933	241	4	ζ(x	ζ(x	NOUN
ejpam-4933	241	5	)	)	PUNCT
ejpam-4933	241	6	)	)	PUNCT
ejpam-4933	242	1	=	=	PUNCT
ejpam-4933	242	2	(	(	PUNCT
ejpam-4933	242	3	1	1	NUM
ejpam-4933	242	4	−	−	PROPN
ejpam-4933	242	5	ε	ε	PROPN
ejpam-4933	242	6	)	)	PUNCT
ejpam-4933	242	7	∧	∧	NOUN
ejpam-4933	242	8	(	(	PUNCT
ejpam-4933	242	9	1	1	NUM
ejpam-4933	242	10	−	−	PROPN
ejpam-4933	242	11	ζ(x	ζ(x	NOUN
ejpam-4933	242	12	)	)	PUNCT
ejpam-4933	242	13	)	)	PUNCT
ejpam-4933	242	14	for	for	ADP
ejpam-4933	242	15	all	all	PRON
ejpam-4933	242	16	x	x	SYM
ejpam-4933	242	17	∈	∈	ADJ
ejpam-4933	242	18	x.	x.	NOUN
ejpam-4933	242	19	hence	hence	ADV
ejpam-4933	242	20	1	1	NUM
ejpam-4933	242	21	−	−	NOUN
ejpam-4933	242	22	ζ(0	ζ(0	NOUN
ejpam-4933	242	23	)	)	PUNCT
ejpam-4933	242	24	≤	≤	NUM
ejpam-4933	242	25	1	1	NUM
ejpam-4933	242	26	−	−	NOUN
ejpam-4933	242	27	ζ(x	ζ(x	NOUN
ejpam-4933	242	28	)	)	PUNCT
ejpam-4933	242	29	,	,	PUNCT
ejpam-4933	242	30	and	and	CCONJ
ejpam-4933	242	31	so	so	ADV
ejpam-4933	242	32	ζ(0	ζ(0	PROPN
ejpam-4933	242	33	)	)	PUNCT
ejpam-4933	242	34	≥	≥	NOUN
ejpam-4933	242	35	ζ(x	ζ(x	NOUN
ejpam-4933	242	36	)	)	PUNCT
ejpam-4933	242	37	for	for	ADP
ejpam-4933	242	38	all	all	PRON
ejpam-4933	242	39	x	x	SYM
ejpam-4933	242	40	∈	∈	ADJ
ejpam-4933	242	41	x.	x.	NOUN
ejpam-4933	242	42	for	for	ADP
ejpam-4933	242	43	every	every	DET
ejpam-4933	242	44	x	x	NOUN
ejpam-4933	242	45	,	,	PUNCT
ejpam-4933	242	46	a	a	DET
ejpam-4933	242	47	∈	∈	PROPN
ejpam-4933	242	48	x	x	NOUN
ejpam-4933	242	49	,	,	PUNCT
ejpam-4933	242	50	we	we	PRON
ejpam-4933	242	51	have	have	VERB
ejpam-4933	242	52	(	(	PUNCT
ejpam-4933	242	53	1	1	NUM
ejpam-4933	242	54	−	−	PROPN
ejpam-4933	242	55	ε	ε	PROPN
ejpam-4933	242	56	)	)	PUNCT
ejpam-4933	242	57	∧	∧	NOUN
ejpam-4933	242	58	(	(	PUNCT
ejpam-4933	242	59	1	1	NUM
ejpam-4933	242	60	−	−	PROPN
ejpam-4933	242	61	ζ(x	ζ(x	NOUN
ejpam-4933	242	62	)	)	PUNCT
ejpam-4933	242	63	)	)	PUNCT
ejpam-4933	243	1	=	=	PUNCT
ejpam-4933	243	2	yj(ε	yj(ε	ADJ
ejpam-4933	243	3	,	,	PUNCT
ejpam-4933	243	4	ζ(x	ζ(x	NOUN
ejpam-4933	243	5	)	)	PUNCT
ejpam-4933	243	6	)	)	PUNCT
ejpam-4933	244	1	≤	≤	NOUN
ejpam-4933	244	2	yj(ε	yj(ε	NOUN
ejpam-4933	244	3	,	,	PUNCT
ejpam-4933	244	4	ζ(x	ζ(x	PROPN
ejpam-4933	244	5	∗	∗	VERB
ejpam-4933	244	6	a	a	NOUN
ejpam-4933	244	7	)	)	PUNCT
ejpam-4933	244	8	)	)	PUNCT
ejpam-4933	245	1	∨	∨	PROPN
ejpam-4933	245	2	yj(ε	yj(ε	NUM
ejpam-4933	245	3	,	,	PUNCT
ejpam-4933	245	4	ζ(a	ζ(a	NOUN
ejpam-4933	245	5	)	)	PUNCT
ejpam-4933	245	6	)	)	PUNCT
ejpam-4933	246	1	=	=	PRON
ejpam-4933	246	2	(	(	PUNCT
ejpam-4933	246	3	(	(	PUNCT
ejpam-4933	246	4	1	1	NUM
ejpam-4933	246	5	−	−	PROPN
ejpam-4933	246	6	ε	ε	PROPN
ejpam-4933	246	7	)	)	PUNCT
ejpam-4933	246	8	∧	∧	NOUN
ejpam-4933	246	9	(	(	PUNCT
ejpam-4933	246	10	1	1	NUM
ejpam-4933	246	11	−	−	PROPN
ejpam-4933	246	12	ζ(x	ζ(x	PROPN
ejpam-4933	246	13	∗	∗	VERB
ejpam-4933	246	14	a	a	NOUN
ejpam-4933	246	15	)	)	PUNCT
ejpam-4933	246	16	)	)	PUNCT
ejpam-4933	246	17	)	)	PUNCT
ejpam-4933	247	1	∨	∨	NUM
ejpam-4933	247	2	(	(	PUNCT
ejpam-4933	247	3	(	(	PUNCT
ejpam-4933	247	4	1	1	NUM
ejpam-4933	247	5	−	−	PROPN
ejpam-4933	247	6	ε	ε	PROPN
ejpam-4933	247	7	)	)	PUNCT
ejpam-4933	247	8	∧	∧	NOUN
ejpam-4933	247	9	(	(	PUNCT
ejpam-4933	247	10	1	1	NUM
ejpam-4933	247	11	−	−	PROPN
ejpam-4933	247	12	ζ(a	ζ(a	NOUN
ejpam-4933	247	13	)	)	PUNCT
ejpam-4933	247	14	)	)	PUNCT
ejpam-4933	247	15	)	)	PUNCT
ejpam-4933	248	1	=	=	PUNCT
ejpam-4933	249	1	(	(	PUNCT
ejpam-4933	249	2	1	1	NUM
ejpam-4933	249	3	−	−	PROPN
ejpam-4933	249	4	ε	ε	PROPN
ejpam-4933	249	5	)	)	PUNCT
ejpam-4933	249	6	∧	∧	PROPN
ejpam-4933	249	7	(	(	PUNCT
ejpam-4933	249	8	(	(	PUNCT
ejpam-4933	249	9	1	1	NUM
ejpam-4933	249	10	−	−	PROPN
ejpam-4933	249	11	ζ(x	ζ(x	PROPN
ejpam-4933	249	12	∗	∗	VERB
ejpam-4933	249	13	a	a	NOUN
ejpam-4933	249	14	)	)	PUNCT
ejpam-4933	249	15	)	)	PUNCT
ejpam-4933	249	16	∨	∨	NUM
ejpam-4933	249	17	(	(	PUNCT
ejpam-4933	249	18	1	1	NUM
ejpam-4933	249	19	−	−	NOUN
ejpam-4933	249	20	ζ(a	ζ(a	NOUN
ejpam-4933	249	21	)	)	PUNCT
ejpam-4933	249	22	)	)	PUNCT
ejpam-4933	249	23	)	)	PUNCT
ejpam-4933	249	24	.	.	PUNCT
ejpam-4933	250	1	it	it	PRON
ejpam-4933	250	2	follows	follow	VERB
ejpam-4933	250	3	that	that	SCONJ
ejpam-4933	250	4	1	1	NUM
ejpam-4933	250	5	−	−	PRON
ejpam-4933	250	6	ζ(x	ζ(x	NOUN
ejpam-4933	250	7	)	)	PUNCT
ejpam-4933	250	8	≤	≤	NOUN
ejpam-4933	250	9	(	(	PUNCT
ejpam-4933	250	10	(	(	PUNCT
ejpam-4933	250	11	1	1	NUM
ejpam-4933	250	12	−	−	PROPN
ejpam-4933	250	13	ζ(x	ζ(x	PROPN
ejpam-4933	250	14	∗	∗	VERB
ejpam-4933	250	15	a	a	NOUN
ejpam-4933	250	16	)	)	PUNCT
ejpam-4933	250	17	)	)	PUNCT
ejpam-4933	250	18	∨	∨	NUM
ejpam-4933	250	19	(	(	PUNCT
ejpam-4933	250	20	1	1	NUM
ejpam-4933	250	21	−	−	NOUN
ejpam-4933	250	22	ζ(a	ζ(a	NOUN
ejpam-4933	250	23	)	)	PUNCT
ejpam-4933	250	24	)	)	PUNCT
ejpam-4933	250	25	)	)	PUNCT
ejpam-4933	251	1	=	=	SYM
ejpam-4933	251	2	1	1	NUM
ejpam-4933	251	3	−	−	NOUN
ejpam-4933	251	4	(	(	PUNCT
ejpam-4933	251	5	ζ(x	ζ(x	PROPN
ejpam-4933	251	6	∗	∗	VERB
ejpam-4933	251	7	a	a	X
ejpam-4933	251	8	)	)	PUNCT
ejpam-4933	251	9	∧	∧	PROPN
ejpam-4933	251	10	ζ(a	ζ(a	NOUN
ejpam-4933	251	11	)	)	PUNCT
ejpam-4933	251	12	)	)	PUNCT
ejpam-4933	251	13	.	.	PUNCT
ejpam-4933	252	1	thus	thus	ADV
ejpam-4933	252	2	ζ(x	ζ(x	NOUN
ejpam-4933	252	3	)	)	PUNCT
ejpam-4933	252	4	≥	≥	NOUN
ejpam-4933	252	5	ζ(x	ζ(x	PROPN
ejpam-4933	252	6	∗	∗	VERB
ejpam-4933	252	7	a	a	X
ejpam-4933	252	8	)	)	PUNCT
ejpam-4933	252	9	∧	∧	PROPN
ejpam-4933	252	10	ζ(a	ζ(a	PROPN
ejpam-4933	252	11	)	)	PUNCT
ejpam-4933	252	12	.	.	PUNCT
ejpam-4933	253	1	therefore	therefore	ADV
ejpam-4933	253	2	ζ	ζ	PROPN
ejpam-4933	253	3	is	be	AUX
ejpam-4933	253	4	a	a	DET
ejpam-4933	253	5	fuzzy	fuzzy	ADJ
ejpam-4933	253	6	ideal	ideal	NOUN
ejpam-4933	253	7	of	of	ADP
ejpam-4933	253	8	(	(	PUNCT
ejpam-4933	253	9	x	x	NOUN
ejpam-4933	253	10	,	,	PUNCT
ejpam-4933	253	11	∗	∗	NOUN
ejpam-4933	253	12	,	,	PUNCT
ejpam-4933	253	13	0	0	NUM
ejpam-4933	253	14	)	)	PUNCT
ejpam-4933	253	15	.	.	PUNCT
ejpam-4933	254	1	theorem	theorem	VERB
ejpam-4933	254	2	6	6	NUM
ejpam-4933	254	3	.	.	PUNCT
ejpam-4933	255	1	a	a	DET
ejpam-4933	255	2	fuzzy	fuzzy	ADJ
ejpam-4933	255	3	set	set	VERB
ejpam-4933	255	4	ζ	ζ	NOUN
ejpam-4933	255	5	in	in	ADP
ejpam-4933	255	6	x	x	SYM
ejpam-4933	255	7	is	be	AUX
ejpam-4933	255	8	a	a	DET
ejpam-4933	255	9	y	y	PROPN
ejpam-4933	255	10	ε	ε	PROPN
ejpam-4933	255	11	j	j	PROPN
ejpam-4933	255	12	-fuzzy	-fuzzy	PROPN
ejpam-4933	255	13	ideal	ideal	NOUN
ejpam-4933	255	14	of	of	ADP
ejpam-4933	255	15	(	(	PUNCT
ejpam-4933	255	16	x	x	NOUN
ejpam-4933	255	17	,	,	PUNCT
ejpam-4933	255	18	∗	∗	NOUN
ejpam-4933	255	19	,	,	PUNCT
ejpam-4933	255	20	0	0	NUM
ejpam-4933	255	21	)	)	PUNCT
ejpam-4933	255	22	if	if	SCONJ
ejpam-4933	255	23	and	and	CCONJ
ejpam-4933	255	24	only	only	ADV
ejpam-4933	255	25	if	if	SCONJ
ejpam-4933	255	26	the	the	DET
ejpam-4933	255	27	nonempty	nonempty	ADJ
ejpam-4933	255	28	y	y	NOUN
ejpam-4933	255	29	-	-	PUNCT
ejpam-4933	255	30	level	level	NOUN
ejpam-4933	255	31	set	set	NOUN
ejpam-4933	255	32	ε(ζ)t	ε(ζ)t	NOUN
ejpam-4933	255	33	of	of	ADP
ejpam-4933	255	34	ε(ζ	ε(ζ	NOUN
ejpam-4933	255	35	)	)	PUNCT
ejpam-4933	255	36	is	be	AUX
ejpam-4933	255	37	an	an	DET
ejpam-4933	255	38	ideal	ideal	NOUN
ejpam-4933	255	39	of	of	ADP
ejpam-4933	255	40	(	(	PUNCT
ejpam-4933	255	41	x	x	NOUN
ejpam-4933	255	42	,	,	PUNCT
ejpam-4933	255	43	∗	∗	NOUN
ejpam-4933	255	44	,	,	PUNCT
ejpam-4933	255	45	0	0	NUM
ejpam-4933	255	46	)	)	PUNCT
ejpam-4933	255	47	for	for	ADP
ejpam-4933	255	48	all	all	DET
ejpam-4933	255	49	t	t	NOUN
ejpam-4933	255	50	∈	∈	PROPN
ejpam-4933	256	1	i	i	PRON
ejpam-4933	256	2	\	\	PROPN
ejpam-4933	256	3	{	{	PUNCT
ejpam-4933	256	4	0	0	NUM
ejpam-4933	256	5	,	,	PUNCT
ejpam-4933	256	6	1	1	NUM
ejpam-4933	256	7	}	}	PUNCT
ejpam-4933	256	8	proof	proof	NOUN
ejpam-4933	256	9	.	.	PUNCT
ejpam-4933	257	1	assume	assume	VERB
ejpam-4933	257	2	that	that	SCONJ
ejpam-4933	257	3	ζ	ζ	NOUN
ejpam-4933	257	4	is	be	AUX
ejpam-4933	257	5	a	a	DET
ejpam-4933	257	6	y	y	PROPN
ejpam-4933	257	7	ε	ε	PROPN
ejpam-4933	257	8	j	j	PROPN
ejpam-4933	257	9	-fuzzy	-fuzzy	PROPN
ejpam-4933	257	10	ideal	ideal	NOUN
ejpam-4933	257	11	of	of	ADP
ejpam-4933	257	12	(	(	PUNCT
ejpam-4933	257	13	x	x	NOUN
ejpam-4933	257	14	,	,	PUNCT
ejpam-4933	257	15	∗	∗	NOUN
ejpam-4933	257	16	,	,	PUNCT
ejpam-4933	257	17	0	0	NUM
ejpam-4933	257	18	)	)	PUNCT
ejpam-4933	257	19	and	and	CCONJ
ejpam-4933	257	20	let	let	VERB
ejpam-4933	257	21	t	t	PROPN
ejpam-4933	257	22	∈	∈	PROPN
ejpam-4933	258	1	i	i	PRON
ejpam-4933	258	2	\	\	PROPN
ejpam-4933	258	3	{	{	PUNCT
ejpam-4933	258	4	0	0	NUM
ejpam-4933	258	5	,	,	PUNCT
ejpam-4933	258	6	1	1	NUM
ejpam-4933	258	7	}	}	PUNCT
ejpam-4933	258	8	be	be	AUX
ejpam-4933	258	9	such	such	ADJ
ejpam-4933	258	10	that	that	SCONJ
ejpam-4933	258	11	ε(ζ)t	ε(ζ)t	VERB
ejpam-4933	258	12	̸=	̸=	PROPN
ejpam-4933	258	13	∅.	∅.	VERB
ejpam-4933	258	14	if	if	SCONJ
ejpam-4933	258	15	0	0	NUM
ejpam-4933	258	16	/∈	/∈	PUNCT
ejpam-4933	259	1	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	259	2	,	,	PUNCT
ejpam-4933	259	3	then	then	ADV
ejpam-4933	259	4	yj(ε	yj(ε	NUM
ejpam-4933	259	5	,	,	PUNCT
ejpam-4933	259	6	ζ(0	ζ(0	NOUN
ejpam-4933	259	7	)	)	PUNCT
ejpam-4933	259	8	)	)	PUNCT
ejpam-4933	260	1	>	>	X
ejpam-4933	260	2	t	t	PROPN
ejpam-4933	260	3	≥	≥	PROPN
ejpam-4933	260	4	yj(ε	yj(ε	NUM
ejpam-4933	260	5	,	,	PUNCT
ejpam-4933	260	6	ζ(b	ζ(b	PROPN
ejpam-4933	260	7	)	)	PUNCT
ejpam-4933	260	8	)	)	PUNCT
ejpam-4933	260	9	for	for	ADP
ejpam-4933	260	10	some	some	DET
ejpam-4933	260	11	b	b	NOUN
ejpam-4933	260	12	∈	∈	PROPN
ejpam-4933	260	13	x	x	NOUN
ejpam-4933	260	14	,	,	PUNCT
ejpam-4933	260	15	which	which	PRON
ejpam-4933	260	16	contradicts	contradict	VERB
ejpam-4933	260	17	(	(	PUNCT
ejpam-4933	260	18	14	14	NUM
ejpam-4933	260	19	)	)	PUNCT
ejpam-4933	260	20	.	.	PUNCT
ejpam-4933	261	1	hence	hence	ADV
ejpam-4933	261	2	0	0	NUM
ejpam-4933	261	3	/∈	/∈	PUNCT
ejpam-4933	262	1	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	262	2	.	.	PUNCT
ejpam-4933	263	1	let	let	VERB
ejpam-4933	263	2	x	x	PRON
ejpam-4933	263	3	,	,	PUNCT
ejpam-4933	263	4	y	y	PROPN
ejpam-4933	263	5	∈	∈	PROPN
ejpam-4933	263	6	x	x	AUX
ejpam-4933	263	7	be	be	AUX
ejpam-4933	263	8	such	such	ADJ
ejpam-4933	263	9	that	that	SCONJ
ejpam-4933	263	10	x	x	PUNCT
ejpam-4933	263	11	∗	∗	NOUN
ejpam-4933	263	12	y	y	PROPN
ejpam-4933	263	13	∈	∈	PROPN
ejpam-4933	263	14	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	263	15	and	and	CCONJ
ejpam-4933	263	16	y	y	PROPN
ejpam-4933	263	17	∈	∈	PROPN
ejpam-4933	263	18	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	263	19	.	.	PUNCT
ejpam-4933	264	1	then	then	ADV
ejpam-4933	264	2	yj(ε	yj(ε	PRON
ejpam-4933	264	3	,	,	PUNCT
ejpam-4933	264	4	ζ(x	ζ(x	PROPN
ejpam-4933	264	5	∗	∗	NOUN
ejpam-4933	264	6	y	y	NOUN
ejpam-4933	264	7	)	)	PUNCT
ejpam-4933	264	8	)	)	PUNCT
ejpam-4933	264	9	≤	≤	PROPN
ejpam-4933	264	10	t	t	PROPN
ejpam-4933	264	11	and	and	CCONJ
ejpam-4933	264	12	yj(ε	yj(ε	NUM
ejpam-4933	264	13	,	,	PUNCT
ejpam-4933	264	14	ζ(y	ζ(y	PROPN
ejpam-4933	264	15	)	)	PUNCT
ejpam-4933	264	16	)	)	PUNCT
ejpam-4933	264	17	≤	≤	NOUN
ejpam-4933	265	1	t.	t.	NOUN
ejpam-4933	265	2	it	it	PRON
ejpam-4933	265	3	follows	follow	VERB
ejpam-4933	265	4	from	from	ADP
ejpam-4933	265	5	(	(	PUNCT
ejpam-4933	265	6	15	15	NUM
ejpam-4933	265	7	)	)	PUNCT
ejpam-4933	265	8	that	that	PRON
ejpam-4933	265	9	yj(ε	yj(ε	ADJ
ejpam-4933	265	10	,	,	PUNCT
ejpam-4933	265	11	ζ(x	ζ(x	NOUN
ejpam-4933	265	12	)	)	PUNCT
ejpam-4933	265	13	)	)	PUNCT
ejpam-4933	266	1	≤	≤	NOUN
ejpam-4933	266	2	yj(ε	yj(ε	NOUN
ejpam-4933	266	3	,	,	PUNCT
ejpam-4933	266	4	ζ(x	ζ(x	PROPN
ejpam-4933	266	5	∗	∗	NOUN
ejpam-4933	266	6	y	y	NOUN
ejpam-4933	266	7	)	)	PUNCT
ejpam-4933	266	8	)	)	PUNCT
ejpam-4933	266	9	∨	∨	PROPN
ejpam-4933	266	10	yj(ε	yj(ε	NUM
ejpam-4933	266	11	,	,	PUNCT
ejpam-4933	266	12	ζ(y	ζ(y	PROPN
ejpam-4933	266	13	)	)	PUNCT
ejpam-4933	266	14	)	)	PUNCT
ejpam-4933	267	1	≤	≤	NOUN
ejpam-4933	268	1	t.	t.	NOUN
ejpam-4933	268	2	hence	hence	ADV
ejpam-4933	268	3	x	x	PROPN
ejpam-4933	268	4	∈	∈	PROPN
ejpam-4933	268	5	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	268	6	,	,	PUNCT
ejpam-4933	268	7	which	which	PRON
ejpam-4933	268	8	shows	show	VERB
ejpam-4933	268	9	that	that	SCONJ
ejpam-4933	268	10	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	268	11	is	be	AUX
ejpam-4933	268	12	an	an	DET
ejpam-4933	268	13	ideal	ideal	NOUN
ejpam-4933	268	14	of	of	ADP
ejpam-4933	268	15	(	(	PUNCT
ejpam-4933	268	16	x	x	NOUN
ejpam-4933	268	17	,	,	PUNCT
ejpam-4933	268	18	∗	∗	NOUN
ejpam-4933	268	19	,	,	PUNCT
ejpam-4933	268	20	0	0	NUM
ejpam-4933	268	21	)	)	PUNCT
ejpam-4933	268	22	.	.	PUNCT
ejpam-4933	269	1	conversely	conversely	ADV
ejpam-4933	269	2	,	,	PUNCT
ejpam-4933	269	3	suppose	suppose	VERB
ejpam-4933	269	4	that	that	SCONJ
ejpam-4933	269	5	the	the	DET
ejpam-4933	269	6	nonempty	nonempty	ADJ
ejpam-4933	269	7	y	y	NOUN
ejpam-4933	269	8	-	-	PUNCT
ejpam-4933	269	9	level	level	NOUN
ejpam-4933	269	10	set	set	NOUN
ejpam-4933	269	11	ε(ζ)t	ε(ζ)t	NOUN
ejpam-4933	269	12	of	of	ADP
ejpam-4933	269	13	ε(ζ	ε(ζ	NOUN
ejpam-4933	269	14	)	)	PUNCT
ejpam-4933	269	15	is	be	AUX
ejpam-4933	269	16	an	an	DET
ejpam-4933	269	17	ideal	ideal	NOUN
ejpam-4933	269	18	of	of	ADP
ejpam-4933	269	19	(	(	PUNCT
ejpam-4933	269	20	x	x	NOUN
ejpam-4933	269	21	,	,	PUNCT
ejpam-4933	269	22	∗	∗	NOUN
ejpam-4933	269	23	,	,	PUNCT
ejpam-4933	269	24	0	0	NUM
ejpam-4933	269	25	)	)	PUNCT
ejpam-4933	269	26	for	for	ADP
ejpam-4933	269	27	all	all	DET
ejpam-4933	269	28	t	t	NOUN
ejpam-4933	269	29	∈	∈	PROPN
ejpam-4933	270	1	i	i	PRON
ejpam-4933	270	2	\	\	PROPN
ejpam-4933	270	3	{	{	PUNCT
ejpam-4933	270	4	0	0	NUM
ejpam-4933	270	5	,	,	PUNCT
ejpam-4933	270	6	1	1	NUM
ejpam-4933	270	7	}	}	PUNCT
ejpam-4933	270	8	.	.	PUNCT
ejpam-4933	271	1	if	if	SCONJ
ejpam-4933	271	2	there	there	PRON
ejpam-4933	271	3	exists	exist	VERB
ejpam-4933	271	4	c	c	NOUN
ejpam-4933	271	5	∈	∈	PROPN
ejpam-4933	271	6	x	x	PUNCT
ejpam-4933	271	7	such	such	ADJ
ejpam-4933	271	8	that	that	SCONJ
ejpam-4933	271	9	yj(ε	yj(ε	NOUN
ejpam-4933	271	10	,	,	PUNCT
ejpam-4933	271	11	ζ(0	ζ(0	NOUN
ejpam-4933	271	12	)	)	PUNCT
ejpam-4933	271	13	)	)	PUNCT
ejpam-4933	271	14	>	>	X
ejpam-4933	272	1	yj(ε	yj(ε	SYM
ejpam-4933	272	2	,	,	PUNCT
ejpam-4933	272	3	ζ(c	ζ(c	ADJ
ejpam-4933	272	4	)	)	PUNCT
ejpam-4933	272	5	)	)	PUNCT
ejpam-4933	272	6	,	,	PUNCT
ejpam-4933	272	7	then	then	ADV
ejpam-4933	272	8	yj(ε	yj(ε	NUM
ejpam-4933	272	9	,	,	PUNCT
ejpam-4933	272	10	ζ(0	ζ(0	NOUN
ejpam-4933	272	11	)	)	PUNCT
ejpam-4933	272	12	)	)	PUNCT
ejpam-4933	273	1	>	>	X
ejpam-4933	273	2	t	t	PROPN
ejpam-4933	273	3	≥	≥	PROPN
ejpam-4933	273	4	yj(ε	yj(ε	NUM
ejpam-4933	273	5	,	,	PUNCT
ejpam-4933	273	6	ζ(c	ζ(c	NOUN
ejpam-4933	273	7	)	)	PUNCT
ejpam-4933	273	8	)	)	PUNCT
ejpam-4933	273	9	for	for	ADP
ejpam-4933	273	10	some	some	DET
ejpam-4933	273	11	t	t	NOUN
ejpam-4933	273	12	∈	∈	NOUN
ejpam-4933	274	1	i	i	PRON
ejpam-4933	274	2	\	\	PROPN
ejpam-4933	274	3	{	{	PUNCT
ejpam-4933	274	4	0	0	NUM
ejpam-4933	274	5	,	,	PUNCT
ejpam-4933	274	6	1	1	NUM
ejpam-4933	274	7	}	}	PUNCT
ejpam-4933	274	8	.	.	PUNCT
ejpam-4933	275	1	it	it	PRON
ejpam-4933	275	2	follows	follow	VERB
ejpam-4933	275	3	that	that	SCONJ
ejpam-4933	275	4	c	c	PROPN
ejpam-4933	275	5	∈	∈	PROPN
ejpam-4933	275	6	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	275	7	,	,	PUNCT
ejpam-4933	275	8	that	that	ADV
ejpam-4933	275	9	is	is	ADV
ejpam-4933	275	10	,	,	PUNCT
ejpam-4933	275	11	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	275	12	̸=	̸=	PROPN
ejpam-4933	275	13	∅.	∅.	VERB
ejpam-4933	275	14	hence	hence	ADV
ejpam-4933	275	15	0	0	NUM
ejpam-4933	275	16	∈	∈	PROPN
ejpam-4933	275	17	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	275	18	,	,	PUNCT
ejpam-4933	275	19	and	and	CCONJ
ejpam-4933	275	20	so	so	ADV
ejpam-4933	275	21	yj(ε	yj(ε	ADJ
ejpam-4933	275	22	,	,	PUNCT
ejpam-4933	275	23	ζ(0	ζ(0	NOUN
ejpam-4933	275	24	)	)	PUNCT
ejpam-4933	275	25	)	)	PUNCT
ejpam-4933	275	26	≤	≤	PROPN
ejpam-4933	275	27	t	t	PROPN
ejpam-4933	275	28	,	,	PUNCT
ejpam-4933	275	29	which	which	PRON
ejpam-4933	275	30	is	be	AUX
ejpam-4933	275	31	a	a	DET
ejpam-4933	275	32	contradiction	contradiction	NOUN
ejpam-4933	275	33	.	.	PUNCT
ejpam-4933	276	1	thus	thus	ADV
ejpam-4933	276	2	yj(ε	yj(ε	NUM
ejpam-4933	276	3	,	,	PUNCT
ejpam-4933	276	4	ζ(0	ζ(0	NOUN
ejpam-4933	276	5	)	)	PUNCT
ejpam-4933	276	6	)	)	PUNCT
ejpam-4933	277	1	≤	≤	NOUN
ejpam-4933	277	2	yj(ε	yj(ε	NUM
ejpam-4933	277	3	,	,	PUNCT
ejpam-4933	277	4	ζ(x	ζ(x	NOUN
ejpam-4933	277	5	)	)	PUNCT
ejpam-4933	277	6	)	)	PUNCT
ejpam-4933	277	7	for	for	ADP
ejpam-4933	277	8	all	all	PRON
ejpam-4933	277	9	x	x	SYM
ejpam-4933	277	10	∈	∈	PROPN
ejpam-4933	277	11	x.	x.	NOUN
ejpam-4933	277	12	suppose	suppose	VERB
ejpam-4933	277	13	that	that	SCONJ
ejpam-4933	277	14	(	(	PUNCT
ejpam-4933	277	15	15	15	NUM
ejpam-4933	277	16	)	)	PUNCT
ejpam-4933	277	17	is	be	AUX
ejpam-4933	277	18	not	not	PART
ejpam-4933	277	19	valid	valid	ADJ
ejpam-4933	277	20	.	.	PUNCT
ejpam-4933	278	1	then	then	ADV
ejpam-4933	278	2	yj(ε	yj(ε	NUM
ejpam-4933	278	3	,	,	PUNCT
ejpam-4933	278	4	ζ(x	ζ(x	NOUN
ejpam-4933	278	5	)	)	PUNCT
ejpam-4933	278	6	)	)	PUNCT
ejpam-4933	279	1	>	>	X
ejpam-4933	279	2	t	t	PROPN
ejpam-4933	279	3	≥	≥	PROPN
ejpam-4933	279	4	yj(ε	yj(ε	NUM
ejpam-4933	279	5	,	,	PUNCT
ejpam-4933	279	6	ζ(x	ζ(x	PROPN
ejpam-4933	279	7	∗	∗	VERB
ejpam-4933	279	8	a	a	NOUN
ejpam-4933	279	9	)	)	PUNCT
ejpam-4933	279	10	)	)	PUNCT
ejpam-4933	279	11	∨	∨	PROPN
ejpam-4933	279	12	yj(ε	yj(ε	NUM
ejpam-4933	279	13	,	,	PUNCT
ejpam-4933	279	14	ζ(a	ζ(a	NOUN
ejpam-4933	279	15	)	)	PUNCT
ejpam-4933	279	16	)	)	PUNCT
ejpam-4933	279	17	for	for	ADP
ejpam-4933	279	18	some	some	DET
ejpam-4933	279	19	x	x	NOUN
ejpam-4933	279	20	,	,	PUNCT
ejpam-4933	279	21	a	a	DET
ejpam-4933	279	22	∈	∈	NOUN
ejpam-4933	279	23	x	x	X
ejpam-4933	279	24	and	and	CCONJ
ejpam-4933	279	25	t	t	PROPN
ejpam-4933	279	26	∈	∈	PROPN
ejpam-4933	280	1	i	i	PRON
ejpam-4933	280	2	\	\	PROPN
ejpam-4933	280	3	{	{	PUNCT
ejpam-4933	280	4	0	0	NUM
ejpam-4933	280	5	,	,	PUNCT
ejpam-4933	280	6	1	1	NUM
ejpam-4933	280	7	}	}	PUNCT
ejpam-4933	280	8	.	.	PUNCT
ejpam-4933	281	1	it	it	PRON
ejpam-4933	281	2	follows	follow	VERB
ejpam-4933	281	3	that	that	SCONJ
ejpam-4933	281	4	x	x	PROPN
ejpam-4933	281	5	∗	∗	VERB
ejpam-4933	281	6	a	a	DET
ejpam-4933	281	7	∈	∈	NOUN
ejpam-4933	281	8	ε(ζ)t	ε(ζ)t	NOUN
ejpam-4933	281	9	and	and	CCONJ
ejpam-4933	281	10	a	a	DET
ejpam-4933	281	11	∈	∈	NOUN
ejpam-4933	281	12	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	281	13	,	,	PUNCT
ejpam-4933	281	14	but	but	CCONJ
ejpam-4933	281	15	x	x	X
ejpam-4933	281	16	/∈	/∈	PUNCT
ejpam-4933	281	17	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	281	18	.	.	PUNCT
ejpam-4933	282	1	this	this	PRON
ejpam-4933	282	2	is	be	AUX
ejpam-4933	282	3	a	a	DET
ejpam-4933	282	4	contradiction	contradiction	NOUN
ejpam-4933	282	5	,	,	PUNCT
ejpam-4933	282	6	and	and	CCONJ
ejpam-4933	282	7	thus	thus	ADV
ejpam-4933	282	8	(	(	PUNCT
ejpam-4933	282	9	15	15	NUM
ejpam-4933	282	10	)	)	PUNCT
ejpam-4933	282	11	is	be	AUX
ejpam-4933	282	12	valid	valid	ADJ
ejpam-4933	282	13	.	.	PUNCT
ejpam-4933	283	1	therefore	therefore	ADV
ejpam-4933	283	2	ζ	ζ	PROPN
ejpam-4933	283	3	is	be	AUX
ejpam-4933	283	4	a	a	DET
ejpam-4933	283	5	y	y	PROPN
ejpam-4933	283	6	ε	ε	PROPN
ejpam-4933	283	7	j	j	PROPN
ejpam-4933	283	8	-fuzzy	-fuzzy	PROPN
ejpam-4933	283	9	ideal	ideal	NOUN
ejpam-4933	283	10	of	of	ADP
ejpam-4933	283	11	(	(	PUNCT
ejpam-4933	283	12	x	x	NOUN
ejpam-4933	283	13	,	,	PUNCT
ejpam-4933	283	14	∗	∗	NOUN
ejpam-4933	283	15	,	,	PUNCT
ejpam-4933	283	16	0	0	NUM
ejpam-4933	283	17	)	)	PUNCT
ejpam-4933	283	18	.	.	PUNCT
ejpam-4933	284	1	we	we	PRON
ejpam-4933	284	2	provide	provide	VERB
ejpam-4933	284	3	conditions	condition	NOUN
ejpam-4933	284	4	for	for	ADP
ejpam-4933	284	5	y	y	PROPN
ejpam-4933	284	6	ε	ε	PROPN
ejpam-4933	284	7	j	j	PROPN
ejpam-4933	284	8	-fuzzy	-fuzzy	PROPN
ejpam-4933	284	9	subalgebra	subalgebra	NOUN
ejpam-4933	284	10	to	to	PART
ejpam-4933	284	11	be	be	AUX
ejpam-4933	284	12	y	y	PROPN
ejpam-4933	284	13	ε	ε	PROPN
ejpam-4933	284	14	j	j	PROPN
ejpam-4933	284	15	-fuzzy	-fuzzy	PROPN
ejpam-4933	284	16	ideal	ideal	ADJ
ejpam-4933	284	17	.	.	PUNCT
ejpam-4933	285	1	theorem	theorem	VERB
ejpam-4933	285	2	7	7	NUM
ejpam-4933	285	3	.	.	PUNCT
ejpam-4933	286	1	if	if	SCONJ
ejpam-4933	286	2	a	a	DET
ejpam-4933	286	3	y	y	NOUN
ejpam-4933	286	4	ε	ε	PROPN
ejpam-4933	286	5	j	j	PROPN
ejpam-4933	286	6	-fuzzy	-fuzzy	PROPN
ejpam-4933	286	7	subalgebra	subalgebra	PROPN
ejpam-4933	286	8	ζ	ζ	NOUN
ejpam-4933	286	9	of	of	ADP
ejpam-4933	286	10	(	(	PUNCT
ejpam-4933	286	11	x	x	NOUN
ejpam-4933	286	12	,	,	PUNCT
ejpam-4933	286	13	∗	∗	NOUN
ejpam-4933	286	14	,	,	PUNCT
ejpam-4933	286	15	0	0	NUM
ejpam-4933	286	16	)	)	PUNCT
ejpam-4933	286	17	satisfies	satisfy	VERB
ejpam-4933	286	18	the	the	DET
ejpam-4933	286	19	condition	condition	NOUN
ejpam-4933	286	20	(	(	PUNCT
ejpam-4933	286	21	17	17	NUM
ejpam-4933	286	22	)	)	PUNCT
ejpam-4933	286	23	,	,	PUNCT
ejpam-4933	286	24	then	then	ADV
ejpam-4933	286	25	it	it	PRON
ejpam-4933	286	26	is	be	AUX
ejpam-4933	286	27	a	a	DET
ejpam-4933	286	28	y	y	PROPN
ejpam-4933	286	29	ε	ε	PROPN
ejpam-4933	286	30	j	j	PROPN
ejpam-4933	286	31	-fuzzy	-fuzzy	PROPN
ejpam-4933	286	32	ideal	ideal	NOUN
ejpam-4933	286	33	of	of	ADP
ejpam-4933	286	34	(	(	PUNCT
ejpam-4933	286	35	x	x	NOUN
ejpam-4933	286	36	,	,	PUNCT
ejpam-4933	286	37	∗	∗	NOUN
ejpam-4933	286	38	,	,	PUNCT
ejpam-4933	286	39	0	0	NUM
ejpam-4933	286	40	)	)	PUNCT
ejpam-4933	286	41	.	.	PUNCT
ejpam-4933	287	1	proof	proof	NOUN
ejpam-4933	287	2	.	.	PUNCT
ejpam-4933	288	1	let	let	VERB
ejpam-4933	288	2	ζ	ζ	NOUN
ejpam-4933	288	3	be	be	AUX
ejpam-4933	288	4	a	a	DET
ejpam-4933	288	5	y	y	PROPN
ejpam-4933	288	6	ε	ε	PROPN
ejpam-4933	288	7	j	j	PROPN
ejpam-4933	288	8	-fuzzy	-fuzzy	PROPN
ejpam-4933	288	9	subalgebra	subalgebra	PROPN
ejpam-4933	288	10	ζ	ζ	NOUN
ejpam-4933	288	11	of	of	ADP
ejpam-4933	288	12	(	(	PUNCT
ejpam-4933	288	13	x	x	NOUN
ejpam-4933	288	14	,	,	PUNCT
ejpam-4933	288	15	∗	∗	NOUN
ejpam-4933	288	16	,	,	PUNCT
ejpam-4933	288	17	0	0	NUM
ejpam-4933	288	18	)	)	PUNCT
ejpam-4933	288	19	that	that	PRON
ejpam-4933	288	20	satisfies	satisfy	VERB
ejpam-4933	288	21	the	the	DET
ejpam-4933	288	22	condition	condition	NOUN
ejpam-4933	288	23	(	(	PUNCT
ejpam-4933	288	24	17	17	NUM
ejpam-4933	288	25	)	)	PUNCT
ejpam-4933	288	26	.	.	PUNCT
ejpam-4933	289	1	the	the	DET
ejpam-4933	289	2	combination	combination	NOUN
ejpam-4933	289	3	of	of	ADP
ejpam-4933	289	4	(	(	PUNCT
ejpam-4933	289	5	iii	iii	NOUN
ejpam-4933	289	6	)	)	PUNCT
ejpam-4933	289	7	and	and	CCONJ
ejpam-4933	289	8	(	(	PUNCT
ejpam-4933	289	9	13	13	NUM
ejpam-4933	289	10	)	)	PUNCT
ejpam-4933	289	11	induces	induce	VERB
ejpam-4933	289	12	yj(ε	yj(ε	NUM
ejpam-4933	289	13	,	,	PUNCT
ejpam-4933	289	14	ζ(0	ζ(0	NOUN
ejpam-4933	289	15	)	)	PUNCT
ejpam-4933	289	16	)	)	PUNCT
ejpam-4933	289	17	≤	≤	NOUN
ejpam-4933	289	18	yj(ε	yj(ε	NUM
ejpam-4933	289	19	,	,	PUNCT
ejpam-4933	289	20	ζ(x	ζ(x	NOUN
ejpam-4933	289	21	)	)	PUNCT
ejpam-4933	289	22	)	)	PUNCT
ejpam-4933	289	23	for	for	ADP
ejpam-4933	289	24	all	all	PRON
ejpam-4933	289	25	x	x	SYM
ejpam-4933	289	26	∈	∈	ADJ
ejpam-4933	289	27	x.	x.	NOUN
ejpam-4933	289	28	for	for	ADP
ejpam-4933	289	29	every	every	DET
ejpam-4933	289	30	x	x	PROPN
ejpam-4933	289	31	,	,	PUNCT
ejpam-4933	289	32	y	y	PROPN
ejpam-4933	289	33	∈	∈	PROPN
ejpam-4933	289	34	x	x	X
ejpam-4933	289	35	,	,	PUNCT
ejpam-4933	289	36	we	we	PRON
ejpam-4933	289	37	have	have	VERB
ejpam-4933	289	38	x	x	NOUN
ejpam-4933	289	39	∗	∗	NOUN
ejpam-4933	289	40	(	(	PUNCT
ejpam-4933	289	41	x	x	X
ejpam-4933	289	42	∗	∗	PROPN
ejpam-4933	289	43	y	y	NOUN
ejpam-4933	289	44	)	)	PUNCT
ejpam-4933	289	45	≤x	≤x	PROPN
ejpam-4933	290	1	y	y	PROPN
ejpam-4933	290	2	by	by	ADP
ejpam-4933	290	3	(	(	PUNCT
ejpam-4933	290	4	iii	iii	NOUN
ejpam-4933	290	5	)	)	PUNCT
ejpam-4933	290	6	,	,	PUNCT
ejpam-4933	290	7	(	(	PUNCT
ejpam-4933	290	8	1	1	X
ejpam-4933	290	9	)	)	PUNCT
ejpam-4933	290	10	and	and	CCONJ
ejpam-4933	290	11	(	(	PUNCT
ejpam-4933	290	12	4	4	NUM
ejpam-4933	290	13	)	)	PUNCT
ejpam-4933	290	14	.	.	PUNCT
ejpam-4933	291	1	it	it	PRON
ejpam-4933	291	2	follows	follow	VERB
ejpam-4933	291	3	from	from	ADP
ejpam-4933	291	4	(	(	PUNCT
ejpam-4933	291	5	17	17	NUM
ejpam-4933	291	6	)	)	PUNCT
ejpam-4933	291	7	that	that	PRON
ejpam-4933	291	8	yj(ε	yj(ε	ADJ
ejpam-4933	291	9	,	,	PUNCT
ejpam-4933	291	10	ζ(x	ζ(x	NOUN
ejpam-4933	291	11	)	)	PUNCT
ejpam-4933	291	12	)	)	PUNCT
ejpam-4933	291	13	≤	≤	NOUN
ejpam-4933	291	14	yj(ε	yj(ε	NOUN
ejpam-4933	291	15	,	,	PUNCT
ejpam-4933	291	16	ζ(x	ζ(x	PROPN
ejpam-4933	291	17	∗	∗	NOUN
ejpam-4933	291	18	y	y	NOUN
ejpam-4933	291	19	)	)	PUNCT
ejpam-4933	291	20	)	)	PUNCT
ejpam-4933	292	1	∨	∨	PROPN
ejpam-4933	292	2	yj(ε	yj(ε	NUM
ejpam-4933	292	3	,	,	PUNCT
ejpam-4933	292	4	ζ(y	ζ(y	PROPN
ejpam-4933	292	5	)	)	PUNCT
ejpam-4933	292	6	)	)	PUNCT
ejpam-4933	292	7	for	for	ADP
ejpam-4933	292	8	all	all	DET
ejpam-4933	292	9	x	x	NOUN
ejpam-4933	292	10	,	,	PUNCT
ejpam-4933	292	11	y	y	PROPN
ejpam-4933	292	12	∈	∈	PROPN
ejpam-4933	292	13	x.	x.	NOUN
ejpam-4933	292	14	therefore	therefore	ADV
ejpam-4933	292	15	ζ	ζ	PROPN
ejpam-4933	292	16	is	be	AUX
ejpam-4933	292	17	a	a	DET
ejpam-4933	292	18	y	y	PROPN
ejpam-4933	292	19	ε	ε	PROPN
ejpam-4933	292	20	j	j	PROPN
ejpam-4933	292	21	-fuzzy	-fuzzy	PROPN
ejpam-4933	292	22	ideal	ideal	NOUN
ejpam-4933	292	23	of	of	ADP
ejpam-4933	292	24	(	(	PUNCT
ejpam-4933	292	25	x	x	NOUN
ejpam-4933	292	26	,	,	PUNCT
ejpam-4933	292	27	∗	∗	NOUN
ejpam-4933	292	28	,	,	PUNCT
ejpam-4933	292	29	0	0	NUM
ejpam-4933	292	30	)	)	PUNCT
ejpam-4933	292	31	.	.	PUNCT
ejpam-4933	293	1	e.	e.	PROPN
ejpam-4933	293	2	h.	h.	PROPN
ejpam-4933	293	3	roh	roh	PROPN
ejpam-4933	293	4	,	,	PUNCT
ejpam-4933	293	5	e.	e.	PROPN
ejpam-4933	293	6	yang	yang	PROPN
ejpam-4933	293	7	,	,	PUNCT
ejpam-4933	293	8	y.	y.	PROPN
ejpam-4933	293	9	b.	b.	PROPN
ejpam-4933	293	10	jun	jun	PROPN
ejpam-4933	293	11	/	/	SYM
ejpam-4933	293	12	eur	eur	PROPN
ejpam-4933	293	13	.	.	PUNCT
ejpam-4933	294	1	j.	j.	PROPN
ejpam-4933	294	2	pure	pure	PROPN
ejpam-4933	294	3	appl	appl	PROPN
ejpam-4933	294	4	.	.	PROPN
ejpam-4933	294	5	math	math	PROPN
ejpam-4933	294	6	,	,	PUNCT
ejpam-4933	294	7	16	16	NUM
ejpam-4933	294	8	(	(	PUNCT
ejpam-4933	294	9	4	4	NUM
ejpam-4933	294	10	)	)	PUNCT
ejpam-4933	294	11	(	(	PUNCT
ejpam-4933	294	12	2023	2023	NUM
ejpam-4933	294	13	)	)	PUNCT
ejpam-4933	294	14	,	,	PUNCT
ejpam-4933	294	15	2009	2009	NUM
ejpam-4933	294	16	-	-	SYM
ejpam-4933	294	17	2024	2024	NUM
ejpam-4933	294	18	2020	2020	NUM
ejpam-4933	294	19	5	5	NUM
ejpam-4933	294	20	.	.	PUNCT
ejpam-4933	294	21	closed	close	VERB
ejpam-4933	294	22	y	y	PROPN
ejpam-4933	294	23	ε	ε	PROPN
ejpam-4933	294	24	j	j	PROPN
ejpam-4933	294	25	-fuzzy	-fuzzy	PROPN
ejpam-4933	294	26	ideals	ideal	NOUN
ejpam-4933	294	27	in	in	ADP
ejpam-4933	294	28	bci	bci	NOUN
ejpam-4933	294	29	-	-	PUNCT
ejpam-4933	294	30	algebras	algebra	NOUN
ejpam-4933	294	31	in	in	ADP
ejpam-4933	294	32	this	this	DET
ejpam-4933	294	33	section	section	NOUN
ejpam-4933	294	34	,	,	PUNCT
ejpam-4933	294	35	let	let	VERB
ejpam-4933	294	36	(	(	PUNCT
ejpam-4933	294	37	x	x	NOUN
ejpam-4933	294	38	,	,	PUNCT
ejpam-4933	294	39	∗	∗	NOUN
ejpam-4933	294	40	,	,	PUNCT
ejpam-4933	294	41	0	0	NUM
ejpam-4933	294	42	)	)	PUNCT
ejpam-4933	294	43	denote	denote	VERB
ejpam-4933	294	44	a	a	DET
ejpam-4933	294	45	bci	bci	NOUN
ejpam-4933	294	46	-	-	NOUN
ejpam-4933	294	47	algebra	algebra	NOUN
ejpam-4933	294	48	.	.	PUNCT
ejpam-4933	295	1	we	we	PRON
ejpam-4933	295	2	recall	recall	VERB
ejpam-4933	295	3	that	that	SCONJ
ejpam-4933	295	4	any	any	DET
ejpam-4933	295	5	y	y	PROPN
ejpam-4933	295	6	ε	ε	PROPN
ejpam-4933	295	7	j	j	PROPN
ejpam-4933	295	8	-fuzzy	-fuzzy	PROPN
ejpam-4933	295	9	ideal	ideal	NOUN
ejpam-4933	295	10	may	may	AUX
ejpam-4933	295	11	not	not	PART
ejpam-4933	295	12	be	be	AUX
ejpam-4933	295	13	a	a	DET
ejpam-4933	295	14	y	y	PROPN
ejpam-4933	295	15	ε	ε	PROPN
ejpam-4933	295	16	j	j	PROPN
ejpam-4933	295	17	-fuzzy	-fuzzy	PROPN
ejpam-4933	295	18	subalgebra	subalgebra	NOUN
ejpam-4933	295	19	in	in	ADP
ejpam-4933	295	20	bci	bci	NOUN
ejpam-4933	295	21	-	-	PUNCT
ejpam-4933	295	22	algebras	algebras	X
ejpam-4933	295	23	(	(	PUNCT
ejpam-4933	295	24	cf	cf	AUX
ejpam-4933	295	25	.	.	PUNCT
ejpam-4933	295	26	example	example	NOUN
ejpam-4933	295	27	2	2	NUM
ejpam-4933	295	28	)	)	PUNCT
ejpam-4933	295	29	.	.	PUNCT
ejpam-4933	296	1	this	this	PRON
ejpam-4933	296	2	is	be	AUX
ejpam-4933	296	3	a	a	DET
ejpam-4933	296	4	motivation	motivation	NOUN
ejpam-4933	296	5	for	for	ADP
ejpam-4933	296	6	the	the	DET
ejpam-4933	296	7	definition	definition	NOUN
ejpam-4933	296	8	below	below	ADV
ejpam-4933	296	9	.	.	PUNCT
ejpam-4933	297	1	definition	definition	NOUN
ejpam-4933	297	2	2	2	NUM
ejpam-4933	297	3	.	.	PUNCT
ejpam-4933	298	1	a	a	PRON
ejpam-4933	298	2	y	y	PROPN
ejpam-4933	298	3	ε	ε	PROPN
ejpam-4933	298	4	j	j	PROPN
ejpam-4933	298	5	-fuzzy	-fuzzy	PROPN
ejpam-4933	298	6	ideal	ideal	ADJ
ejpam-4933	298	7	ζ	ζ	NOUN
ejpam-4933	298	8	of	of	ADP
ejpam-4933	298	9	(	(	PUNCT
ejpam-4933	298	10	x	x	NOUN
ejpam-4933	298	11	,	,	PUNCT
ejpam-4933	298	12	∗	∗	NOUN
ejpam-4933	298	13	,	,	PUNCT
ejpam-4933	298	14	0	0	NUM
ejpam-4933	298	15	)	)	PUNCT
ejpam-4933	298	16	is	be	AUX
ejpam-4933	298	17	said	say	VERB
ejpam-4933	298	18	to	to	PART
ejpam-4933	298	19	be	be	AUX
ejpam-4933	298	20	closed	close	VERB
ejpam-4933	298	21	if	if	SCONJ
ejpam-4933	298	22	it	it	PRON
ejpam-4933	298	23	is	be	AUX
ejpam-4933	298	24	also	also	ADV
ejpam-4933	298	25	a	a	DET
ejpam-4933	298	26	y	y	PROPN
ejpam-4933	298	27	ε	ε	PROPN
ejpam-4933	298	28	j	j	PROPN
ejpam-4933	298	29	-fuzzy	-fuzzy	PROPN
ejpam-4933	298	30	subalgebra	subalgebra	NOUN
ejpam-4933	298	31	of	of	ADP
ejpam-4933	298	32	(	(	PUNCT
ejpam-4933	298	33	x	x	NOUN
ejpam-4933	298	34	,	,	PUNCT
ejpam-4933	298	35	∗	∗	NOUN
ejpam-4933	298	36	,	,	PUNCT
ejpam-4933	298	37	0	0	NUM
ejpam-4933	298	38	)	)	PUNCT
ejpam-4933	298	39	.	.	PUNCT
ejpam-4933	299	1	example	example	NOUN
ejpam-4933	300	1	5	5	NUM
ejpam-4933	300	2	.	.	PUNCT
ejpam-4933	300	3	let	let	VERB
ejpam-4933	300	4	x	x	PUNCT
ejpam-4933	300	5	=	=	PUNCT
ejpam-4933	300	6	{	{	PUNCT
ejpam-4933	300	7	0	0	NUM
ejpam-4933	300	8	,	,	PUNCT
ejpam-4933	300	9	b1	b1	NOUN
ejpam-4933	300	10	,	,	PUNCT
ejpam-4933	300	11	b2	b2	NOUN
ejpam-4933	300	12	,	,	PUNCT
ejpam-4933	300	13	b3	b3	NOUN
ejpam-4933	300	14	,	,	PUNCT
ejpam-4933	300	15	b4	b4	PROPN
ejpam-4933	300	16	}	}	PUNCT
ejpam-4933	300	17	be	be	AUX
ejpam-4933	300	18	a	a	DET
ejpam-4933	300	19	set	set	NOUN
ejpam-4933	300	20	with	with	ADP
ejpam-4933	300	21	a	a	DET
ejpam-4933	300	22	binary	binary	ADJ
ejpam-4933	300	23	operation	operation	NOUN
ejpam-4933	300	24	“	"	PUNCT
ejpam-4933	300	25	∗	∗	NOUN
ejpam-4933	300	26	”	"	PUNCT
ejpam-4933	300	27	given	give	VERB
ejpam-4933	300	28	by	by	ADP
ejpam-4933	300	29	table	table	NOUN
ejpam-4933	300	30	4	4	NUM
ejpam-4933	300	31	.	.	PUNCT
ejpam-4933	300	32	table	table	NOUN
ejpam-4933	300	33	4	4	NUM
ejpam-4933	300	34	:	:	PUNCT
ejpam-4933	300	35	cayley	cayley	ADJ
ejpam-4933	300	36	table	table	NOUN
ejpam-4933	300	37	for	for	ADP
ejpam-4933	300	38	the	the	DET
ejpam-4933	300	39	binary	binary	PROPN
ejpam-4933	300	40	operation	operation	NOUN
ejpam-4933	300	41	“	"	PUNCT
ejpam-4933	300	42	∗	∗	NOUN
ejpam-4933	300	43	”	"	PUNCT
ejpam-4933	300	44	∗	∗	X
ejpam-4933	300	45	0	0	NUM
ejpam-4933	300	46	b1	b1	PROPN
ejpam-4933	300	47	b2	b2	NOUN
ejpam-4933	300	48	b3	b3	PROPN
ejpam-4933	300	49	b4	b4	NOUN
ejpam-4933	300	50	0	0	NUM
ejpam-4933	300	51	0	0	NUM
ejpam-4933	300	52	0	0	NUM
ejpam-4933	300	53	0	0	NUM
ejpam-4933	300	54	b3	b3	PROPN
ejpam-4933	300	55	b3	b3	PROPN
ejpam-4933	300	56	b1	b1	NOUN
ejpam-4933	300	57	b1	b1	NOUN
ejpam-4933	300	58	0	0	NUM
ejpam-4933	300	59	0	0	NUM
ejpam-4933	300	60	b3	b3	PROPN
ejpam-4933	300	61	b3	b3	PROPN
ejpam-4933	300	62	b2	b2	NOUN
ejpam-4933	300	63	b2	b2	NOUN
ejpam-4933	300	64	b2	b2	NOUN
ejpam-4933	300	65	0	0	NUM
ejpam-4933	300	66	b4	b4	PROPN
ejpam-4933	300	67	b3	b3	PROPN
ejpam-4933	300	68	b3	b3	PROPN
ejpam-4933	300	69	b3	b3	PROPN
ejpam-4933	300	70	b3	b3	PROPN
ejpam-4933	300	71	b3	b3	PROPN
ejpam-4933	300	72	0	0	NUM
ejpam-4933	300	73	0	0	NUM
ejpam-4933	300	74	b4	b4	NOUN
ejpam-4933	300	75	b4	b4	PROPN
ejpam-4933	300	76	b4	b4	PROPN
ejpam-4933	300	77	b3	b3	PROPN
ejpam-4933	300	78	b2	b2	NOUN
ejpam-4933	300	79	0	0	PUNCT
ejpam-4933	301	1	then	then	ADV
ejpam-4933	301	2	(	(	PUNCT
ejpam-4933	301	3	x	x	X
ejpam-4933	301	4	,	,	PUNCT
ejpam-4933	301	5	∗	∗	NOUN
ejpam-4933	301	6	,	,	PUNCT
ejpam-4933	301	7	0	0	NUM
ejpam-4933	301	8	)	)	PUNCT
ejpam-4933	301	9	is	be	AUX
ejpam-4933	301	10	a	a	DET
ejpam-4933	301	11	bci	bci	NOUN
ejpam-4933	301	12	-	-	NOUN
ejpam-4933	301	13	algebra	algebra	NOUN
ejpam-4933	301	14	(	(	PUNCT
ejpam-4933	301	15	see	see	VERB
ejpam-4933	301	16	[	[	X
ejpam-4933	301	17	2	2	NUM
ejpam-4933	301	18	]	]	PUNCT
ejpam-4933	301	19	)	)	PUNCT
ejpam-4933	301	20	.	.	PUNCT
ejpam-4933	302	1	let	let	VERB
ejpam-4933	302	2	ζ	ζ	NOUN
ejpam-4933	302	3	be	be	AUX
ejpam-4933	302	4	a	a	DET
ejpam-4933	302	5	fuzzy	fuzzy	ADJ
ejpam-4933	302	6	set	set	NOUN
ejpam-4933	302	7	in	in	ADP
ejpam-4933	302	8	x	x	PUNCT
ejpam-4933	302	9	given	give	VERB
ejpam-4933	302	10	by	by	ADP
ejpam-4933	302	11	ζ	ζ	NOUN
ejpam-4933	302	12	:	:	PUNCT
ejpam-4933	302	13	x	x	X
ejpam-4933	302	14	→	→	SYM
ejpam-4933	303	1	[	[	X
ejpam-4933	303	2	0	0	NUM
ejpam-4933	303	3	,	,	PUNCT
ejpam-4933	303	4	1	1	NUM
ejpam-4933	303	5	]	]	PUNCT
ejpam-4933	303	6	,	,	PUNCT
ejpam-4933	303	7	y	y	PROPN
ejpam-4933	303	8	7→	7→	NUM
ejpam-4933	303	9			PUNCT
ejpam-4933	303	10	0.78	0.78	NUM
ejpam-4933	303	11	if	if	SCONJ
ejpam-4933	303	12	y	y	PROPN
ejpam-4933	303	13	=	=	SYM
ejpam-4933	303	14	0	0	PROPN
ejpam-4933	303	15	,	,	PUNCT
ejpam-4933	303	16	0.63	0.63	NUM
ejpam-4933	303	17	if	if	SCONJ
ejpam-4933	303	18	y	y	PROPN
ejpam-4933	303	19	∈	∈	PROPN
ejpam-4933	303	20	{	{	PUNCT
ejpam-4933	303	21	b1	b1	NOUN
ejpam-4933	303	22	,	,	PUNCT
ejpam-4933	303	23	b2	b2	NOUN
ejpam-4933	303	24	}	}	PUNCT
ejpam-4933	303	25	,	,	PUNCT
ejpam-4933	303	26	0.47	0.47	NUM
ejpam-4933	303	27	otherwise	otherwise	ADV
ejpam-4933	303	28	,	,	PUNCT
ejpam-4933	303	29	it	it	PRON
ejpam-4933	303	30	is	be	AUX
ejpam-4933	303	31	routine	routine	ADJ
ejpam-4933	303	32	to	to	PART
ejpam-4933	303	33	check	check	VERB
ejpam-4933	303	34	that	that	SCONJ
ejpam-4933	303	35	ζ	ζ	NOUN
ejpam-4933	303	36	is	be	AUX
ejpam-4933	303	37	a	a	DET
ejpam-4933	303	38	closed	closed	ADJ
ejpam-4933	303	39	y	y	PROPN
ejpam-4933	303	40	ε	ε	PROPN
ejpam-4933	303	41	j	j	PROPN
ejpam-4933	303	42	-fuzzy	-fuzzy	PROPN
ejpam-4933	303	43	ideal	ideal	NOUN
ejpam-4933	303	44	of	of	ADP
ejpam-4933	303	45	(	(	PUNCT
ejpam-4933	303	46	x	x	NOUN
ejpam-4933	303	47	,	,	PUNCT
ejpam-4933	303	48	∗	∗	NOUN
ejpam-4933	303	49	,	,	PUNCT
ejpam-4933	303	50	0	0	NUM
ejpam-4933	303	51	)	)	PUNCT
ejpam-4933	303	52	for	for	ADP
ejpam-4933	303	53	ε	ε	PROPN
ejpam-4933	303	54	:	:	PUNCT
ejpam-4933	303	55	=	=	NOUN
ejpam-4933	303	56	0.46	0.46	NUM
ejpam-4933	303	57	.	.	PUNCT
ejpam-4933	304	1	theorem	theorem	VERB
ejpam-4933	304	2	8	8	NUM
ejpam-4933	304	3	.	.	PUNCT
ejpam-4933	305	1	a	a	DET
ejpam-4933	305	2	fuzzy	fuzzy	ADJ
ejpam-4933	305	3	set	set	VERB
ejpam-4933	305	4	ζ	ζ	NOUN
ejpam-4933	305	5	in	in	ADP
ejpam-4933	305	6	x	x	PUNCT
ejpam-4933	305	7	given	give	VERB
ejpam-4933	305	8	by	by	ADP
ejpam-4933	305	9	ζ	ζ	NOUN
ejpam-4933	305	10	:	:	PUNCT
ejpam-4933	305	11	x	x	X
ejpam-4933	305	12	→	→	SYM
ejpam-4933	305	13	[	[	X
ejpam-4933	305	14	0	0	NUM
ejpam-4933	305	15	,	,	PUNCT
ejpam-4933	305	16	1	1	NUM
ejpam-4933	305	17	]	]	PUNCT
ejpam-4933	305	18	,	,	PUNCT
ejpam-4933	305	19	y	y	PROPN
ejpam-4933	305	20	7→	7→	PROPN
ejpam-4933	305	21	{	{	PUNCT
ejpam-4933	305	22	s1	s1	NOUN
ejpam-4933	305	23	if	if	SCONJ
ejpam-4933	305	24	y	y	PROPN
ejpam-4933	305	25	∈	∈	PROPN
ejpam-4933	305	26	{	{	PUNCT
ejpam-4933	305	27	x	x	SYM
ejpam-4933	305	28	∈	∈	PROPN
ejpam-4933	305	29	x	x	SYM
ejpam-4933	305	30	|	|	ADV
ejpam-4933	305	31	0	0	NUM
ejpam-4933	305	32	≤x	≤x	PROPN
ejpam-4933	305	33	x	x	SYM
ejpam-4933	305	34	}	}	PUNCT
ejpam-4933	305	35	,	,	PUNCT
ejpam-4933	305	36	s2	s2	VERB
ejpam-4933	305	37	otherwise	otherwise	ADV
ejpam-4933	305	38	,	,	PUNCT
ejpam-4933	305	39	where	where	SCONJ
ejpam-4933	305	40	s1	s1	PROPN
ejpam-4933	305	41	>	>	X
ejpam-4933	305	42	s2	s2	PROPN
ejpam-4933	305	43	in	in	ADP
ejpam-4933	305	44	(	(	PUNCT
ejpam-4933	305	45	0	0	NUM
ejpam-4933	305	46	,	,	PUNCT
ejpam-4933	305	47	1	1	NUM
ejpam-4933	305	48	)	)	PUNCT
ejpam-4933	305	49	,	,	PUNCT
ejpam-4933	305	50	is	be	AUX
ejpam-4933	305	51	a	a	DET
ejpam-4933	305	52	closed	closed	ADJ
ejpam-4933	305	53	y	y	PROPN
ejpam-4933	305	54	ε	ε	PROPN
ejpam-4933	305	55	j	j	PROPN
ejpam-4933	305	56	-fuzzy	-fuzzy	PROPN
ejpam-4933	305	57	ideal	ideal	NOUN
ejpam-4933	305	58	of	of	ADP
ejpam-4933	305	59	(	(	PUNCT
ejpam-4933	305	60	x	x	NOUN
ejpam-4933	305	61	,	,	PUNCT
ejpam-4933	305	62	∗	∗	NOUN
ejpam-4933	305	63	,	,	PUNCT
ejpam-4933	305	64	0	0	NUM
ejpam-4933	305	65	)	)	PUNCT
ejpam-4933	305	66	.	.	PUNCT
ejpam-4933	306	1	proof	proof	NOUN
ejpam-4933	306	2	.	.	PUNCT
ejpam-4933	307	1	the	the	DET
ejpam-4933	307	2	y	y	NOUN
ejpam-4933	307	3	-	-	PUNCT
ejpam-4933	307	4	level	level	NOUN
ejpam-4933	307	5	set	set	NOUN
ejpam-4933	307	6	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	307	7	is	be	AUX
ejpam-4933	307	8	calculated	calculate	VERB
ejpam-4933	307	9	as	as	SCONJ
ejpam-4933	307	10	follows	follow	VERB
ejpam-4933	307	11	:	:	PUNCT
ejpam-4933	307	12	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	307	13	=	=	PUNCT
ejpam-4933	307	14			PUNCT
ejpam-4933	307	15	∅	∅	NOUN
ejpam-4933	307	16	if	if	SCONJ
ejpam-4933	307	17	0	0	NUM
ejpam-4933	307	18	<	<	X
ejpam-4933	307	19	t	t	X
ejpam-4933	307	20	<	<	X
ejpam-4933	307	21	1	1	NUM
ejpam-4933	307	22	−	−	PROPN
ejpam-4933	307	23	s1	s1	NOUN
ejpam-4933	307	24	,	,	PUNCT
ejpam-4933	307	25	{	{	PUNCT
ejpam-4933	307	26	x	x	SYM
ejpam-4933	307	27	∈	∈	NOUN
ejpam-4933	307	28	x	x	SYM
ejpam-4933	307	29	|	|	ADV
ejpam-4933	307	30	0	0	NUM
ejpam-4933	307	31	≤x	≤x	PROPN
ejpam-4933	307	32	x	x	INTJ
ejpam-4933	307	33	}	}	PUNCT
ejpam-4933	307	34	if	if	SCONJ
ejpam-4933	307	35	1	1	NUM
ejpam-4933	307	36	−	−	PROPN
ejpam-4933	307	37	s1	s1	PROPN
ejpam-4933	307	38	≤	≤	PROPN
ejpam-4933	307	39	t	t	PROPN
ejpam-4933	307	40	<	<	X
ejpam-4933	307	41	1	1	NUM
ejpam-4933	307	42	−	−	PROPN
ejpam-4933	307	43	s2	s2	PROPN
ejpam-4933	307	44	,	,	PUNCT
ejpam-4933	307	45	x	x	SYM
ejpam-4933	307	46	if	if	SCONJ
ejpam-4933	307	47	1	1	NUM
ejpam-4933	307	48	−	−	PROPN
ejpam-4933	307	49	s2	s2	PROPN
ejpam-4933	307	50	≤	≤	NUM
ejpam-4933	307	51	t	t	PROPN
ejpam-4933	307	52	<	<	X
ejpam-4933	307	53	1	1	X
ejpam-4933	307	54	.	.	PUNCT
ejpam-4933	308	1	let	let	VERB
ejpam-4933	308	2	a	a	DET
ejpam-4933	308	3	:	:	PUNCT
ejpam-4933	308	4	=	=	SYM
ejpam-4933	308	5	{	{	PUNCT
ejpam-4933	308	6	x	x	SYM
ejpam-4933	308	7	∈	∈	NOUN
ejpam-4933	308	8	x	x	SYM
ejpam-4933	308	9	|	|	ADV
ejpam-4933	308	10	0	0	NUM
ejpam-4933	308	11	≤x	≤x	PROPN
ejpam-4933	308	12	x	x	NOUN
ejpam-4933	308	13	}	}	PUNCT
ejpam-4933	308	14	,	,	PUNCT
ejpam-4933	308	15	and	and	CCONJ
ejpam-4933	308	16	let	let	VERB
ejpam-4933	308	17	y	y	PRON
ejpam-4933	308	18	,	,	PUNCT
ejpam-4933	308	19	z	z	PROPN
ejpam-4933	308	20	∈	∈	PROPN
ejpam-4933	308	21	a.	a.	NOUN
ejpam-4933	308	22	then	then	ADV
ejpam-4933	308	23	0	0	NUM
ejpam-4933	308	24	≤x	≤x	PROPN
ejpam-4933	308	25	y	y	PROPN
ejpam-4933	308	26	and	and	CCONJ
ejpam-4933	308	27	0	0	NUM
ejpam-4933	308	28	≤x	≤x	PROPN
ejpam-4933	308	29	z	z	PROPN
ejpam-4933	308	30	,	,	PUNCT
ejpam-4933	308	31	i.e.	i.e.	X
ejpam-4933	308	32	,	,	PUNCT
ejpam-4933	308	33	0	0	NUM
ejpam-4933	308	34	∗	∗	NOUN
ejpam-4933	308	35	y	y	NOUN
ejpam-4933	308	36	=	=	SYM
ejpam-4933	308	37	0	0	NUM
ejpam-4933	308	38	and	and	CCONJ
ejpam-4933	308	39	0	0	NUM
ejpam-4933	308	40	∗	∗	NOUN
ejpam-4933	308	41	z	z	NOUN
ejpam-4933	308	42	=	=	SYM
ejpam-4933	309	1	0	0	X
ejpam-4933	309	2	.	.	PUNCT
ejpam-4933	310	1	hence	hence	ADV
ejpam-4933	310	2	0	0	NUM
ejpam-4933	310	3	∗	∗	NOUN
ejpam-4933	310	4	(	(	PUNCT
ejpam-4933	310	5	y	y	PROPN
ejpam-4933	310	6	∗	∗	PROPN
ejpam-4933	310	7	z	z	NOUN
ejpam-4933	310	8	)	)	PUNCT
ejpam-4933	310	9	=	=	SYM
ejpam-4933	310	10	(	(	PUNCT
ejpam-4933	310	11	0	0	NUM
ejpam-4933	310	12	∗	∗	PROPN
ejpam-4933	310	13	y	y	NOUN
ejpam-4933	310	14	)	)	PUNCT
ejpam-4933	310	15	∗	∗	NOUN
ejpam-4933	310	16	(	(	PUNCT
ejpam-4933	310	17	0	0	NUM
ejpam-4933	310	18	∗	∗	NOUN
ejpam-4933	310	19	z	z	NOUN
ejpam-4933	310	20	)	)	PUNCT
ejpam-4933	310	21	=	=	SYM
ejpam-4933	310	22	0	0	NUM
ejpam-4933	310	23	by	by	ADP
ejpam-4933	310	24	(	(	PUNCT
ejpam-4933	310	25	iii	iii	NOUN
ejpam-4933	310	26	)	)	PUNCT
ejpam-4933	310	27	and	and	CCONJ
ejpam-4933	310	28	(	(	PUNCT
ejpam-4933	310	29	6	6	NUM
ejpam-4933	310	30	)	)	PUNCT
ejpam-4933	310	31	,	,	PUNCT
ejpam-4933	310	32	and	and	CCONJ
ejpam-4933	310	33	so	so	ADV
ejpam-4933	310	34	0	0	NUM
ejpam-4933	310	35	≤x	≤x	PROPN
ejpam-4933	310	36	y	y	PROPN
ejpam-4933	310	37	∗	∗	PROPN
ejpam-4933	310	38	z	z	PROPN
ejpam-4933	310	39	,	,	PUNCT
ejpam-4933	310	40	e.	e.	PROPN
ejpam-4933	310	41	h.	h.	PROPN
ejpam-4933	310	42	roh	roh	PROPN
ejpam-4933	310	43	,	,	PUNCT
ejpam-4933	310	44	e.	e.	PROPN
ejpam-4933	310	45	yang	yang	PROPN
ejpam-4933	310	46	,	,	PUNCT
ejpam-4933	310	47	y.	y.	PROPN
ejpam-4933	310	48	b.	b.	PROPN
ejpam-4933	310	49	jun	jun	PROPN
ejpam-4933	310	50	/	/	SYM
ejpam-4933	310	51	eur	eur	PROPN
ejpam-4933	310	52	.	.	PUNCT
ejpam-4933	311	1	j.	j.	PROPN
ejpam-4933	311	2	pure	pure	PROPN
ejpam-4933	311	3	appl	appl	PROPN
ejpam-4933	311	4	.	.	PROPN
ejpam-4933	311	5	math	math	PROPN
ejpam-4933	311	6	,	,	PUNCT
ejpam-4933	311	7	16	16	NUM
ejpam-4933	311	8	(	(	PUNCT
ejpam-4933	311	9	4	4	NUM
ejpam-4933	311	10	)	)	PUNCT
ejpam-4933	311	11	(	(	PUNCT
ejpam-4933	311	12	2023	2023	NUM
ejpam-4933	311	13	)	)	PUNCT
ejpam-4933	311	14	,	,	PUNCT
ejpam-4933	311	15	2009	2009	NUM
ejpam-4933	311	16	-	-	SYM
ejpam-4933	311	17	2024	2024	NUM
ejpam-4933	311	18	2021	2021	NUM
ejpam-4933	311	19	i.e.	i.e.	X
ejpam-4933	311	20	,	,	PUNCT
ejpam-4933	311	21	y	y	PROPN
ejpam-4933	311	22	∗	∗	NOUN
ejpam-4933	311	23	z	z	PROPN
ejpam-4933	311	24	∈	∈	PROPN
ejpam-4933	311	25	a.	a.	NOUN
ejpam-4933	311	26	thus	thus	ADV
ejpam-4933	311	27	a	a	PRON
ejpam-4933	311	28	is	be	AUX
ejpam-4933	311	29	a	a	DET
ejpam-4933	311	30	subalgebra	subalgebra	NOUN
ejpam-4933	311	31	of	of	ADP
ejpam-4933	311	32	(	(	PUNCT
ejpam-4933	311	33	x	x	NOUN
ejpam-4933	311	34	,	,	PUNCT
ejpam-4933	311	35	∗	∗	NOUN
ejpam-4933	311	36	,	,	PUNCT
ejpam-4933	311	37	0	0	NUM
ejpam-4933	311	38	)	)	PUNCT
ejpam-4933	311	39	.	.	PUNCT
ejpam-4933	312	1	it	it	PRON
ejpam-4933	312	2	is	be	AUX
ejpam-4933	312	3	clear	clear	ADJ
ejpam-4933	312	4	that	that	SCONJ
ejpam-4933	312	5	0	0	NUM
ejpam-4933	312	6	∈	∈	NOUN
ejpam-4933	312	7	a.	a.	NOUN
ejpam-4933	312	8	let	let	VERB
ejpam-4933	312	9	y	y	NOUN
ejpam-4933	312	10	,	,	PUNCT
ejpam-4933	312	11	z	z	PROPN
ejpam-4933	312	12	∈	∈	PROPN
ejpam-4933	312	13	x	x	AUX
ejpam-4933	312	14	be	be	AUX
ejpam-4933	312	15	such	such	ADJ
ejpam-4933	312	16	that	that	SCONJ
ejpam-4933	312	17	y	y	PROPN
ejpam-4933	312	18	∗	∗	NOUN
ejpam-4933	312	19	z	z	PROPN
ejpam-4933	312	20	∈	∈	PROPN
ejpam-4933	312	21	a	a	DET
ejpam-4933	312	22	and	and	CCONJ
ejpam-4933	312	23	z	z	NOUN
ejpam-4933	312	24	∈	∈	PROPN
ejpam-4933	312	25	a.	a.	NOUN
ejpam-4933	312	26	then	then	ADV
ejpam-4933	312	27	0	0	X
ejpam-4933	313	1	=	=	SYM
ejpam-4933	313	2	0	0	NUM
ejpam-4933	313	3	∗	∗	NOUN
ejpam-4933	313	4	(	(	PUNCT
ejpam-4933	313	5	y	y	PROPN
ejpam-4933	313	6	∗	∗	PROPN
ejpam-4933	313	7	z	z	NOUN
ejpam-4933	313	8	)	)	PUNCT
ejpam-4933	313	9	=	=	SYM
ejpam-4933	313	10	(	(	PUNCT
ejpam-4933	313	11	0	0	NUM
ejpam-4933	313	12	∗	∗	PROPN
ejpam-4933	313	13	y	y	NOUN
ejpam-4933	313	14	)	)	PUNCT
ejpam-4933	313	15	∗	∗	NOUN
ejpam-4933	313	16	(	(	PUNCT
ejpam-4933	313	17	0	0	NUM
ejpam-4933	313	18	∗	∗	NOUN
ejpam-4933	313	19	z	z	NOUN
ejpam-4933	313	20	)	)	PUNCT
ejpam-4933	313	21	=	=	SYM
ejpam-4933	313	22	(	(	PUNCT
ejpam-4933	313	23	0	0	NUM
ejpam-4933	313	24	∗	∗	PROPN
ejpam-4933	313	25	y	y	NOUN
ejpam-4933	313	26	)	)	PUNCT
ejpam-4933	313	27	∗	∗	NOUN
ejpam-4933	313	28	0	0	NUM
ejpam-4933	314	1	=	=	SYM
ejpam-4933	314	2	0	0	NUM
ejpam-4933	314	3	∗	∗	NOUN
ejpam-4933	314	4	y	y	PROPN
ejpam-4933	314	5	by	by	ADP
ejpam-4933	314	6	(	(	PUNCT
ejpam-4933	314	7	2	2	NUM
ejpam-4933	314	8	)	)	PUNCT
ejpam-4933	314	9	and	and	CCONJ
ejpam-4933	314	10	(	(	PUNCT
ejpam-4933	314	11	6	6	NUM
ejpam-4933	314	12	)	)	PUNCT
ejpam-4933	314	13	.	.	PUNCT
ejpam-4933	315	1	hence	hence	ADV
ejpam-4933	315	2	y	y	PROPN
ejpam-4933	315	3	∈	∈	PROPN
ejpam-4933	315	4	a	a	PRON
ejpam-4933	315	5	,	,	PUNCT
ejpam-4933	315	6	which	which	PRON
ejpam-4933	315	7	shows	show	VERB
ejpam-4933	315	8	that	that	SCONJ
ejpam-4933	315	9	a	a	PRON
ejpam-4933	315	10	is	be	AUX
ejpam-4933	315	11	an	an	DET
ejpam-4933	315	12	ideal	ideal	NOUN
ejpam-4933	315	13	of	of	ADP
ejpam-4933	315	14	(	(	PUNCT
ejpam-4933	315	15	x	x	NOUN
ejpam-4933	315	16	,	,	PUNCT
ejpam-4933	315	17	∗	∗	NOUN
ejpam-4933	315	18	,	,	PUNCT
ejpam-4933	315	19	0	0	NUM
ejpam-4933	315	20	)	)	PUNCT
ejpam-4933	315	21	.	.	PUNCT
ejpam-4933	316	1	therefore	therefore	ADV
ejpam-4933	316	2	a	a	PRON
ejpam-4933	316	3	is	be	AUX
ejpam-4933	316	4	a	a	DET
ejpam-4933	316	5	closed	closed	ADJ
ejpam-4933	316	6	ideal	ideal	NOUN
ejpam-4933	316	7	of	of	ADP
ejpam-4933	316	8	(	(	PUNCT
ejpam-4933	316	9	x	x	NOUN
ejpam-4933	316	10	,	,	PUNCT
ejpam-4933	316	11	∗	∗	NOUN
ejpam-4933	316	12	,	,	PUNCT
ejpam-4933	316	13	0	0	NUM
ejpam-4933	316	14	)	)	PUNCT
ejpam-4933	316	15	.	.	PUNCT
ejpam-4933	317	1	by	by	ADP
ejpam-4933	317	2	the	the	DET
ejpam-4933	317	3	combination	combination	NOUN
ejpam-4933	317	4	of	of	ADP
ejpam-4933	317	5	theorems	theorem	NOUN
ejpam-4933	317	6	1	1	NUM
ejpam-4933	317	7	and	and	CCONJ
ejpam-4933	317	8	6	6	NUM
ejpam-4933	317	9	,	,	PUNCT
ejpam-4933	317	10	we	we	PRON
ejpam-4933	317	11	conclude	conclude	VERB
ejpam-4933	317	12	that	that	SCONJ
ejpam-4933	317	13	ζ	ζ	NOUN
ejpam-4933	317	14	is	be	AUX
ejpam-4933	317	15	a	a	DET
ejpam-4933	317	16	closed	closed	ADJ
ejpam-4933	317	17	y	y	PROPN
ejpam-4933	317	18	ε	ε	PROPN
ejpam-4933	317	19	j	j	PROPN
ejpam-4933	317	20	-fuzzy	-fuzzy	PROPN
ejpam-4933	317	21	ideal	ideal	NOUN
ejpam-4933	317	22	of	of	ADP
ejpam-4933	317	23	(	(	PUNCT
ejpam-4933	317	24	x	x	NOUN
ejpam-4933	317	25	,	,	PUNCT
ejpam-4933	317	26	∗	∗	NOUN
ejpam-4933	317	27	,	,	PUNCT
ejpam-4933	317	28	0	0	NUM
ejpam-4933	317	29	)	)	PUNCT
ejpam-4933	317	30	.	.	PUNCT
ejpam-4933	318	1	lemma	lemma	PROPN
ejpam-4933	318	2	1	1	NUM
ejpam-4933	318	3	.	.	PUNCT
ejpam-4933	319	1	a	a	DET
ejpam-4933	319	2	fuzzy	fuzzy	ADJ
ejpam-4933	319	3	set	set	VERB
ejpam-4933	319	4	ζ	ζ	NOUN
ejpam-4933	319	5	in	in	ADP
ejpam-4933	319	6	x	x	SYM
ejpam-4933	319	7	is	be	AUX
ejpam-4933	319	8	a	a	DET
ejpam-4933	319	9	closed	closed	ADJ
ejpam-4933	319	10	y	y	PROPN
ejpam-4933	319	11	ε	ε	PROPN
ejpam-4933	319	12	j	j	PROPN
ejpam-4933	319	13	-fuzzy	-fuzzy	PROPN
ejpam-4933	319	14	ideal	ideal	NOUN
ejpam-4933	319	15	of	of	ADP
ejpam-4933	319	16	(	(	PUNCT
ejpam-4933	319	17	x	x	NOUN
ejpam-4933	319	18	,	,	PUNCT
ejpam-4933	319	19	∗	∗	NOUN
ejpam-4933	319	20	,	,	PUNCT
ejpam-4933	319	21	0	0	NUM
ejpam-4933	319	22	)	)	PUNCT
ejpam-4933	319	23	if	if	SCONJ
ejpam-4933	319	24	and	and	CCONJ
ejpam-4933	319	25	only	only	ADV
ejpam-4933	319	26	if	if	SCONJ
ejpam-4933	319	27	the	the	DET
ejpam-4933	319	28	nonempty	nonempty	ADJ
ejpam-4933	319	29	y	y	NOUN
ejpam-4933	319	30	-	-	PUNCT
ejpam-4933	319	31	level	level	NOUN
ejpam-4933	319	32	set	set	NOUN
ejpam-4933	319	33	ε(ζ)t	ε(ζ)t	NOUN
ejpam-4933	319	34	of	of	ADP
ejpam-4933	319	35	ε(ζ	ε(ζ	NOUN
ejpam-4933	319	36	)	)	PUNCT
ejpam-4933	319	37	is	be	AUX
ejpam-4933	319	38	a	a	DET
ejpam-4933	319	39	closed	closed	ADJ
ejpam-4933	319	40	ideal	ideal	NOUN
ejpam-4933	319	41	of	of	ADP
ejpam-4933	319	42	(	(	PUNCT
ejpam-4933	319	43	x	x	NOUN
ejpam-4933	319	44	,	,	PUNCT
ejpam-4933	319	45	∗	∗	NOUN
ejpam-4933	319	46	,	,	PUNCT
ejpam-4933	319	47	0	0	NUM
ejpam-4933	319	48	)	)	PUNCT
ejpam-4933	319	49	for	for	ADP
ejpam-4933	319	50	all	all	DET
ejpam-4933	319	51	t	t	NOUN
ejpam-4933	319	52	∈	∈	PROPN
ejpam-4933	320	1	i	i	PRON
ejpam-4933	320	2	\	\	PROPN
ejpam-4933	320	3	{	{	PUNCT
ejpam-4933	320	4	0	0	NUM
ejpam-4933	320	5	,	,	PUNCT
ejpam-4933	320	6	1	1	NUM
ejpam-4933	320	7	}	}	PUNCT
ejpam-4933	320	8	.	.	PUNCT
ejpam-4933	321	1	proof	proof	NOUN
ejpam-4933	321	2	.	.	PUNCT
ejpam-4933	322	1	assume	assume	VERB
ejpam-4933	322	2	that	that	SCONJ
ejpam-4933	322	3	ζ	ζ	NOUN
ejpam-4933	322	4	is	be	AUX
ejpam-4933	322	5	a	a	DET
ejpam-4933	322	6	closed	closed	ADJ
ejpam-4933	322	7	y	y	PROPN
ejpam-4933	322	8	ε	ε	PROPN
ejpam-4933	322	9	j	j	PROPN
ejpam-4933	322	10	-fuzzy	-fuzzy	PROPN
ejpam-4933	322	11	ideal	ideal	NOUN
ejpam-4933	322	12	of	of	ADP
ejpam-4933	322	13	(	(	PUNCT
ejpam-4933	322	14	x	x	NOUN
ejpam-4933	322	15	,	,	PUNCT
ejpam-4933	322	16	∗	∗	NOUN
ejpam-4933	322	17	,	,	PUNCT
ejpam-4933	322	18	0	0	NUM
ejpam-4933	322	19	)	)	PUNCT
ejpam-4933	322	20	and	and	CCONJ
ejpam-4933	322	21	let	let	VERB
ejpam-4933	322	22	t	t	PROPN
ejpam-4933	322	23	∈	∈	PROPN
ejpam-4933	323	1	i	i	PRON
ejpam-4933	323	2	\	\	PROPN
ejpam-4933	323	3	{	{	PUNCT
ejpam-4933	323	4	0	0	NUM
ejpam-4933	323	5	,	,	PUNCT
ejpam-4933	323	6	1	1	NUM
ejpam-4933	323	7	}	}	PUNCT
ejpam-4933	323	8	be	be	AUX
ejpam-4933	323	9	such	such	ADJ
ejpam-4933	323	10	that	that	SCONJ
ejpam-4933	323	11	ε(ζ)t	ε(ζ)t	VERB
ejpam-4933	323	12	̸=	̸=	PROPN
ejpam-4933	323	13	∅.	∅.	VERB
ejpam-4933	323	14	then	then	ADV
ejpam-4933	323	15	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	323	16	is	be	AUX
ejpam-4933	323	17	an	an	DET
ejpam-4933	323	18	ideal	ideal	NOUN
ejpam-4933	323	19	of	of	ADP
ejpam-4933	323	20	(	(	PUNCT
ejpam-4933	323	21	x	x	NOUN
ejpam-4933	323	22	,	,	PUNCT
ejpam-4933	323	23	∗	∗	NOUN
ejpam-4933	323	24	,	,	PUNCT
ejpam-4933	323	25	0	0	NUM
ejpam-4933	323	26	)	)	PUNCT
ejpam-4933	323	27	by	by	ADP
ejpam-4933	323	28	theorem	theorem	NOUN
ejpam-4933	323	29	6	6	NUM
ejpam-4933	323	30	.	.	PUNCT
ejpam-4933	324	1	let	let	VERB
ejpam-4933	324	2	x	x	X
ejpam-4933	324	3	∈	∈	PROPN
ejpam-4933	324	4	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	324	5	.	.	PUNCT
ejpam-4933	325	1	then	then	ADV
ejpam-4933	325	2	yj(ε	yj(ε	NUM
ejpam-4933	325	3	,	,	PUNCT
ejpam-4933	325	4	ζ(0	ζ(0	NOUN
ejpam-4933	325	5	∗	∗	NOUN
ejpam-4933	325	6	x	x	NOUN
ejpam-4933	325	7	)	)	PUNCT
ejpam-4933	325	8	)	)	PUNCT
ejpam-4933	325	9	(	(	PUNCT
ejpam-4933	325	10	13	13	X
ejpam-4933	325	11	)	)	PUNCT
ejpam-4933	325	12	≤	≤	NOUN
ejpam-4933	325	13	yj(ε	yj(ε	NUM
ejpam-4933	325	14	,	,	PUNCT
ejpam-4933	325	15	ζ(0	ζ(0	NOUN
ejpam-4933	325	16	)	)	PUNCT
ejpam-4933	325	17	)	)	PUNCT
ejpam-4933	326	1	∨	∨	PROPN
ejpam-4933	326	2	yj(ε	yj(ε	NUM
ejpam-4933	326	3	,	,	PUNCT
ejpam-4933	326	4	ζ(x	ζ(x	NOUN
ejpam-4933	326	5	)	)	PUNCT
ejpam-4933	326	6	)	)	PUNCT
ejpam-4933	326	7	(	(	PUNCT
ejpam-4933	326	8	14	14	NUM
ejpam-4933	326	9	)	)	PUNCT
ejpam-4933	326	10	≤	≤	NOUN
ejpam-4933	326	11	yj(ε	yj(ε	NUM
ejpam-4933	326	12	,	,	PUNCT
ejpam-4933	326	13	ζ(x	ζ(x	NOUN
ejpam-4933	326	14	)	)	PUNCT
ejpam-4933	326	15	)	)	PUNCT
ejpam-4933	326	16	≤	≤	NOUN
ejpam-4933	326	17	t	t	PROPN
ejpam-4933	326	18	,	,	PUNCT
ejpam-4933	326	19	and	and	CCONJ
ejpam-4933	326	20	so	so	ADV
ejpam-4933	326	21	0	0	NUM
ejpam-4933	326	22	∗	∗	NOUN
ejpam-4933	326	23	x	x	X
ejpam-4933	326	24	∈	∈	PROPN
ejpam-4933	326	25	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	326	26	.	.	PUNCT
ejpam-4933	327	1	hence	hence	ADV
ejpam-4933	327	2	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	327	3	is	be	AUX
ejpam-4933	327	4	a	a	DET
ejpam-4933	327	5	closed	closed	ADJ
ejpam-4933	327	6	ideal	ideal	NOUN
ejpam-4933	327	7	of	of	ADP
ejpam-4933	327	8	(	(	PUNCT
ejpam-4933	327	9	x	x	NOUN
ejpam-4933	327	10	,	,	PUNCT
ejpam-4933	327	11	∗	∗	NOUN
ejpam-4933	327	12	,	,	PUNCT
ejpam-4933	327	13	0	0	NUM
ejpam-4933	327	14	)	)	PUNCT
ejpam-4933	327	15	.	.	PUNCT
ejpam-4933	328	1	conversely	conversely	ADV
ejpam-4933	328	2	,	,	PUNCT
ejpam-4933	328	3	suppose	suppose	VERB
ejpam-4933	328	4	that	that	SCONJ
ejpam-4933	328	5	the	the	DET
ejpam-4933	328	6	nonempty	nonempty	ADJ
ejpam-4933	328	7	y	y	NOUN
ejpam-4933	328	8	-	-	PUNCT
ejpam-4933	328	9	level	level	NOUN
ejpam-4933	328	10	set	set	NOUN
ejpam-4933	328	11	ε(ζ)t	ε(ζ)t	NOUN
ejpam-4933	328	12	of	of	ADP
ejpam-4933	328	13	ε(ζ	ε(ζ	NOUN
ejpam-4933	328	14	)	)	PUNCT
ejpam-4933	328	15	is	be	AUX
ejpam-4933	328	16	a	a	DET
ejpam-4933	328	17	closed	closed	ADJ
ejpam-4933	328	18	ideal	ideal	NOUN
ejpam-4933	328	19	of	of	ADP
ejpam-4933	328	20	(	(	PUNCT
ejpam-4933	328	21	x	x	NOUN
ejpam-4933	328	22	,	,	PUNCT
ejpam-4933	328	23	∗	∗	NOUN
ejpam-4933	328	24	,	,	PUNCT
ejpam-4933	328	25	0	0	NUM
ejpam-4933	328	26	)	)	PUNCT
ejpam-4933	328	27	for	for	ADP
ejpam-4933	328	28	all	all	DET
ejpam-4933	328	29	t	t	NOUN
ejpam-4933	328	30	∈	∈	PROPN
ejpam-4933	329	1	i	i	PRON
ejpam-4933	329	2	\	\	PROPN
ejpam-4933	329	3	{	{	PUNCT
ejpam-4933	329	4	0	0	NUM
ejpam-4933	329	5	,	,	PUNCT
ejpam-4933	329	6	1	1	NUM
ejpam-4933	329	7	}	}	PUNCT
ejpam-4933	329	8	.	.	PUNCT
ejpam-4933	330	1	then	then	ADV
ejpam-4933	330	2	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	330	3	is	be	AUX
ejpam-4933	330	4	an	an	DET
ejpam-4933	330	5	ideal	ideal	NOUN
ejpam-4933	330	6	of	of	ADP
ejpam-4933	330	7	(	(	PUNCT
ejpam-4933	330	8	x	x	NOUN
ejpam-4933	330	9	,	,	PUNCT
ejpam-4933	330	10	∗	∗	NOUN
ejpam-4933	330	11	,	,	PUNCT
ejpam-4933	330	12	0	0	NUM
ejpam-4933	330	13	)	)	PUNCT
ejpam-4933	330	14	,	,	PUNCT
ejpam-4933	330	15	and	and	CCONJ
ejpam-4933	330	16	thus	thus	ADV
ejpam-4933	330	17	ζ	ζ	NOUN
ejpam-4933	330	18	is	be	AUX
ejpam-4933	330	19	a	a	DET
ejpam-4933	330	20	y	y	PROPN
ejpam-4933	330	21	ε	ε	PROPN
ejpam-4933	330	22	j	j	PROPN
ejpam-4933	330	23	-fuzzy	-fuzzy	PROPN
ejpam-4933	330	24	ideal	ideal	NOUN
ejpam-4933	330	25	of	of	ADP
ejpam-4933	330	26	(	(	PUNCT
ejpam-4933	330	27	x	x	NOUN
ejpam-4933	330	28	,	,	PUNCT
ejpam-4933	330	29	∗	∗	NOUN
ejpam-4933	330	30	,	,	PUNCT
ejpam-4933	330	31	0	0	NUM
ejpam-4933	330	32	)	)	PUNCT
ejpam-4933	330	33	by	by	ADP
ejpam-4933	330	34	theorem	theorem	NOUN
ejpam-4933	330	35	6	6	NUM
ejpam-4933	330	36	.	.	PUNCT
ejpam-4933	331	1	if	if	SCONJ
ejpam-4933	331	2	ζ	ζ	NOUN
ejpam-4933	331	3	is	be	AUX
ejpam-4933	331	4	not	not	PART
ejpam-4933	331	5	a	a	DET
ejpam-4933	331	6	y	y	PROPN
ejpam-4933	331	7	ε	ε	PROPN
ejpam-4933	331	8	j	j	PROPN
ejpam-4933	331	9	-fuzzy	-fuzzy	PROPN
ejpam-4933	331	10	subalgebra	subalgebra	NOUN
ejpam-4933	331	11	of	of	ADP
ejpam-4933	331	12	(	(	PUNCT
ejpam-4933	331	13	x	x	NOUN
ejpam-4933	331	14	,	,	PUNCT
ejpam-4933	331	15	∗	∗	NOUN
ejpam-4933	331	16	,	,	PUNCT
ejpam-4933	331	17	0	0	NUM
ejpam-4933	331	18	)	)	PUNCT
ejpam-4933	331	19	,	,	PUNCT
ejpam-4933	331	20	then	then	ADV
ejpam-4933	331	21	yj(ε	yj(ε	PRON
ejpam-4933	331	22	,	,	PUNCT
ejpam-4933	331	23	ζ(x	ζ(x	PROPN
ejpam-4933	331	24	∗	∗	VERB
ejpam-4933	331	25	a	a	NOUN
ejpam-4933	331	26	)	)	PUNCT
ejpam-4933	331	27	)	)	PUNCT
ejpam-4933	332	1	>	>	X
ejpam-4933	332	2	yj(ε	yj(ε	ADJ
ejpam-4933	332	3	,	,	PUNCT
ejpam-4933	332	4	ζ(x	ζ(x	NOUN
ejpam-4933	332	5	)	)	PUNCT
ejpam-4933	332	6	)	)	PUNCT
ejpam-4933	333	1	∨	∨	PROPN
ejpam-4933	333	2	yj(ε	yj(ε	NUM
ejpam-4933	333	3	,	,	PUNCT
ejpam-4933	333	4	ζ(a	ζ(a	NOUN
ejpam-4933	333	5	)	)	PUNCT
ejpam-4933	333	6	)	)	PUNCT
ejpam-4933	333	7	for	for	ADP
ejpam-4933	333	8	some	some	DET
ejpam-4933	333	9	x	x	NOUN
ejpam-4933	333	10	,	,	PUNCT
ejpam-4933	333	11	a	a	DET
ejpam-4933	333	12	∈	∈	NOUN
ejpam-4933	333	13	x.	x.	NOUN
ejpam-4933	333	14	selecting	select	VERB
ejpam-4933	333	15	t	t	PROPN
ejpam-4933	333	16	:	:	PUNCT
ejpam-4933	333	17	=	=	NOUN
ejpam-4933	333	18	yj(ε	yj(ε	ADJ
ejpam-4933	333	19	,	,	PUNCT
ejpam-4933	333	20	ζ(x	ζ(x	NOUN
ejpam-4933	333	21	)	)	PUNCT
ejpam-4933	333	22	)	)	PUNCT
ejpam-4933	333	23	∨	∨	PROPN
ejpam-4933	333	24	yj(ε	yj(ε	NUM
ejpam-4933	333	25	,	,	PUNCT
ejpam-4933	333	26	ζ(a	ζ(a	NOUN
ejpam-4933	333	27	)	)	PUNCT
ejpam-4933	333	28	)	)	PUNCT
ejpam-4933	333	29	induces	induce	VERB
ejpam-4933	333	30	x	x	PRON
ejpam-4933	333	31	,	,	PUNCT
ejpam-4933	333	32	a	a	DET
ejpam-4933	333	33	∈	∈	NOUN
ejpam-4933	333	34	ε(ζ)t	ε(ζ)t	NOUN
ejpam-4933	333	35	and	and	CCONJ
ejpam-4933	333	36	x	x	NOUN
ejpam-4933	333	37	∗	∗	NOUN
ejpam-4933	333	38	a	a	PRON
ejpam-4933	333	39	/∈	/∈	PUNCT
ejpam-4933	334	1	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	334	2	,	,	PUNCT
ejpam-4933	334	3	which	which	PRON
ejpam-4933	334	4	is	be	AUX
ejpam-4933	334	5	a	a	DET
ejpam-4933	334	6	contradiction	contradiction	NOUN
ejpam-4933	334	7	.	.	PUNCT
ejpam-4933	335	1	hence	hence	ADV
ejpam-4933	335	2	yj(ε	yj(ε	NUM
ejpam-4933	335	3	,	,	PUNCT
ejpam-4933	335	4	ζ(x	ζ(x	PROPN
ejpam-4933	335	5	∗	∗	VERB
ejpam-4933	335	6	a	a	NOUN
ejpam-4933	335	7	)	)	PUNCT
ejpam-4933	335	8	)	)	PUNCT
ejpam-4933	336	1	≤	≤	NOUN
ejpam-4933	336	2	yj(ε	yj(ε	NUM
ejpam-4933	336	3	,	,	PUNCT
ejpam-4933	336	4	ζ(x	ζ(x	NOUN
ejpam-4933	336	5	)	)	PUNCT
ejpam-4933	336	6	)	)	PUNCT
ejpam-4933	337	1	∨	∨	PROPN
ejpam-4933	337	2	yj(ε	yj(ε	NUM
ejpam-4933	337	3	,	,	PUNCT
ejpam-4933	337	4	ζ(a	ζ(a	NOUN
ejpam-4933	337	5	)	)	PUNCT
ejpam-4933	337	6	)	)	PUNCT
ejpam-4933	337	7	for	for	ADP
ejpam-4933	337	8	all	all	DET
ejpam-4933	337	9	x	x	NOUN
ejpam-4933	337	10	,	,	PUNCT
ejpam-4933	337	11	a	a	DET
ejpam-4933	337	12	∈	∈	PROPN
ejpam-4933	337	13	x	x	NOUN
ejpam-4933	337	14	,	,	PUNCT
ejpam-4933	337	15	which	which	PRON
ejpam-4933	337	16	shows	show	VERB
ejpam-4933	337	17	that	that	SCONJ
ejpam-4933	337	18	ζ	ζ	NOUN
ejpam-4933	337	19	is	be	AUX
ejpam-4933	337	20	a	a	DET
ejpam-4933	337	21	y	y	PROPN
ejpam-4933	337	22	ε	ε	PROPN
ejpam-4933	337	23	j	j	PROPN
ejpam-4933	337	24	-fuzzy	-fuzzy	PROPN
ejpam-4933	337	25	subalgebra	subalgebra	NOUN
ejpam-4933	337	26	of	of	ADP
ejpam-4933	337	27	(	(	PUNCT
ejpam-4933	337	28	x	x	NOUN
ejpam-4933	337	29	,	,	PUNCT
ejpam-4933	337	30	∗	∗	NOUN
ejpam-4933	337	31	,	,	PUNCT
ejpam-4933	337	32	0	0	NUM
ejpam-4933	337	33	)	)	PUNCT
ejpam-4933	337	34	.	.	PUNCT
ejpam-4933	338	1	consequently	consequently	ADV
ejpam-4933	338	2	,	,	PUNCT
ejpam-4933	338	3	ζ	ζ	PROPN
ejpam-4933	338	4	is	be	AUX
ejpam-4933	338	5	a	a	DET
ejpam-4933	338	6	closed	closed	ADJ
ejpam-4933	338	7	y	y	PROPN
ejpam-4933	338	8	ε	ε	PROPN
ejpam-4933	338	9	j	j	PROPN
ejpam-4933	338	10	-fuzzy	-fuzzy	PROPN
ejpam-4933	338	11	ideal	ideal	NOUN
ejpam-4933	338	12	of	of	ADP
ejpam-4933	338	13	(	(	PUNCT
ejpam-4933	338	14	x	x	NOUN
ejpam-4933	338	15	,	,	PUNCT
ejpam-4933	338	16	∗	∗	NOUN
ejpam-4933	338	17	,	,	PUNCT
ejpam-4933	338	18	0	0	NUM
ejpam-4933	338	19	)	)	PUNCT
ejpam-4933	338	20	.	.	PUNCT
ejpam-4933	339	1	theorem	theorem	VERB
ejpam-4933	339	2	9	9	NUM
ejpam-4933	339	3	.	.	PUNCT
ejpam-4933	340	1	a	a	DET
ejpam-4933	340	2	y	y	PROPN
ejpam-4933	340	3	ε	ε	PROPN
ejpam-4933	340	4	j	j	PROPN
ejpam-4933	340	5	-fuzzy	-fuzzy	PROPN
ejpam-4933	340	6	ideal	ideal	ADJ
ejpam-4933	340	7	ζ	ζ	NOUN
ejpam-4933	340	8	of	of	ADP
ejpam-4933	340	9	(	(	PUNCT
ejpam-4933	340	10	x	x	NOUN
ejpam-4933	340	11	,	,	PUNCT
ejpam-4933	340	12	∗	∗	NOUN
ejpam-4933	340	13	,	,	PUNCT
ejpam-4933	340	14	0	0	NUM
ejpam-4933	340	15	)	)	PUNCT
ejpam-4933	340	16	is	be	AUX
ejpam-4933	340	17	closed	close	VERB
ejpam-4933	340	18	if	if	SCONJ
ejpam-4933	340	19	and	and	CCONJ
ejpam-4933	340	20	only	only	ADV
ejpam-4933	340	21	if	if	SCONJ
ejpam-4933	340	22	it	it	PRON
ejpam-4933	340	23	satisfies	satisfy	VERB
ejpam-4933	340	24	:	:	PUNCT
ejpam-4933	340	25	(	(	PUNCT
ejpam-4933	340	26	∀x	∀x	X
ejpam-4933	340	27	∈	∈	PROPN
ejpam-4933	340	28	x)(yj(ε	x)(yj(ε	NOUN
ejpam-4933	340	29	,	,	PUNCT
ejpam-4933	340	30	ζ(0	ζ(0	PROPN
ejpam-4933	340	31	∗	∗	NOUN
ejpam-4933	340	32	x	x	NOUN
ejpam-4933	340	33	)	)	PUNCT
ejpam-4933	340	34	)	)	PUNCT
ejpam-4933	340	35	≤	≤	NOUN
ejpam-4933	340	36	yj(ε	yj(ε	NUM
ejpam-4933	340	37	,	,	PUNCT
ejpam-4933	340	38	ζ(x	ζ(x	NOUN
ejpam-4933	340	39	)	)	PUNCT
ejpam-4933	340	40	)	)	PUNCT
ejpam-4933	340	41	)	)	PUNCT
ejpam-4933	340	42	.	.	PUNCT
ejpam-4933	341	1	(	(	PUNCT
ejpam-4933	341	2	18	18	NUM
ejpam-4933	341	3	)	)	PUNCT
ejpam-4933	341	4	proof	proof	NOUN
ejpam-4933	341	5	.	.	PUNCT
ejpam-4933	342	1	let	let	VERB
ejpam-4933	342	2	ζ	ζ	NOUN
ejpam-4933	342	3	be	be	AUX
ejpam-4933	342	4	a	a	DET
ejpam-4933	342	5	closed	closed	ADJ
ejpam-4933	342	6	y	y	PROPN
ejpam-4933	342	7	ε	ε	PROPN
ejpam-4933	342	8	j	j	PROPN
ejpam-4933	342	9	-fuzzy	-fuzzy	PROPN
ejpam-4933	342	10	ideal	ideal	NOUN
ejpam-4933	342	11	of	of	ADP
ejpam-4933	342	12	(	(	PUNCT
ejpam-4933	342	13	x	x	NOUN
ejpam-4933	342	14	,	,	PUNCT
ejpam-4933	342	15	∗	∗	NOUN
ejpam-4933	342	16	,	,	PUNCT
ejpam-4933	342	17	0	0	NUM
ejpam-4933	342	18	)	)	PUNCT
ejpam-4933	342	19	.	.	PUNCT
ejpam-4933	343	1	then	then	ADV
ejpam-4933	343	2	the	the	DET
ejpam-4933	343	3	nonempty	nonempty	ADJ
ejpam-4933	343	4	y	y	NOUN
ejpam-4933	343	5	-	-	PUNCT
ejpam-4933	343	6	level	level	NOUN
ejpam-4933	343	7	set	set	NOUN
ejpam-4933	343	8	ε(ζ)t	ε(ζ)t	NOUN
ejpam-4933	343	9	of	of	ADP
ejpam-4933	343	10	ε(ζ	ε(ζ	NOUN
ejpam-4933	343	11	)	)	PUNCT
ejpam-4933	343	12	is	be	AUX
ejpam-4933	343	13	a	a	DET
ejpam-4933	343	14	closed	closed	ADJ
ejpam-4933	343	15	ideal	ideal	NOUN
ejpam-4933	343	16	of	of	ADP
ejpam-4933	343	17	(	(	PUNCT
ejpam-4933	343	18	x	x	NOUN
ejpam-4933	343	19	,	,	PUNCT
ejpam-4933	343	20	∗	∗	NOUN
ejpam-4933	343	21	,	,	PUNCT
ejpam-4933	343	22	0	0	NUM
ejpam-4933	343	23	)	)	PUNCT
ejpam-4933	343	24	for	for	ADP
ejpam-4933	343	25	all	all	DET
ejpam-4933	343	26	t	t	NOUN
ejpam-4933	343	27	∈	∈	PROPN
ejpam-4933	344	1	i	i	PRON
ejpam-4933	344	2	\	\	PROPN
ejpam-4933	344	3	{	{	PUNCT
ejpam-4933	344	4	0	0	NUM
ejpam-4933	344	5	,	,	PUNCT
ejpam-4933	344	6	1	1	NUM
ejpam-4933	344	7	}	}	PUNCT
ejpam-4933	344	8	by	by	ADP
ejpam-4933	344	9	lemma	lemma	PROPN
ejpam-4933	344	10	1	1	NUM
ejpam-4933	344	11	.	.	PUNCT
ejpam-4933	345	1	then	then	ADV
ejpam-4933	345	2	yj(ε	yj(ε	NUM
ejpam-4933	345	3	,	,	PUNCT
ejpam-4933	345	4	ζ(0	ζ(0	NOUN
ejpam-4933	345	5	∗	∗	NOUN
ejpam-4933	345	6	x	x	NOUN
ejpam-4933	345	7	)	)	PUNCT
ejpam-4933	345	8	)	)	PUNCT
ejpam-4933	345	9	(	(	PUNCT
ejpam-4933	345	10	13	13	X
ejpam-4933	345	11	)	)	PUNCT
ejpam-4933	345	12	≤	≤	NOUN
ejpam-4933	345	13	yj(ε	yj(ε	NUM
ejpam-4933	345	14	,	,	PUNCT
ejpam-4933	345	15	ζ(0	ζ(0	NOUN
ejpam-4933	345	16	)	)	PUNCT
ejpam-4933	345	17	)	)	PUNCT
ejpam-4933	346	1	∨	∨	PROPN
ejpam-4933	346	2	yj(ε	yj(ε	NUM
ejpam-4933	346	3	,	,	PUNCT
ejpam-4933	346	4	ζ(x	ζ(x	NOUN
ejpam-4933	346	5	)	)	PUNCT
ejpam-4933	346	6	)	)	PUNCT
ejpam-4933	346	7	(	(	PUNCT
ejpam-4933	346	8	14	14	NUM
ejpam-4933	346	9	)	)	PUNCT
ejpam-4933	346	10	≤	≤	NOUN
ejpam-4933	346	11	yj(ε	yj(ε	NUM
ejpam-4933	346	12	,	,	PUNCT
ejpam-4933	346	13	ζ(x	ζ(x	NOUN
ejpam-4933	346	14	)	)	PUNCT
ejpam-4933	346	15	)	)	PUNCT
ejpam-4933	346	16	for	for	ADP
ejpam-4933	346	17	all	all	PRON
ejpam-4933	346	18	x	x	SYM
ejpam-4933	346	19	∈	∈	NOUN
ejpam-4933	346	20	x.	x.	NOUN
ejpam-4933	346	21	conversely	conversely	ADV
ejpam-4933	346	22	,	,	PUNCT
ejpam-4933	346	23	let	let	VERB
ejpam-4933	346	24	ζ	ζ	NOUN
ejpam-4933	346	25	be	be	AUX
ejpam-4933	346	26	a	a	DET
ejpam-4933	346	27	y	y	PROPN
ejpam-4933	346	28	ε	ε	PROPN
ejpam-4933	346	29	j	j	PROPN
ejpam-4933	346	30	-fuzzy	-fuzzy	PROPN
ejpam-4933	346	31	ideal	ideal	NOUN
ejpam-4933	346	32	of	of	ADP
ejpam-4933	346	33	(	(	PUNCT
ejpam-4933	346	34	x	x	NOUN
ejpam-4933	346	35	,	,	PUNCT
ejpam-4933	346	36	∗	∗	NOUN
ejpam-4933	346	37	,	,	PUNCT
ejpam-4933	346	38	0	0	NUM
ejpam-4933	346	39	)	)	PUNCT
ejpam-4933	346	40	that	that	PRON
ejpam-4933	346	41	satisfies	satisfy	VERB
ejpam-4933	346	42	the	the	DET
ejpam-4933	346	43	condition	condition	NOUN
ejpam-4933	346	44	(	(	PUNCT
ejpam-4933	346	45	18	18	NUM
ejpam-4933	346	46	)	)	PUNCT
ejpam-4933	346	47	.	.	PUNCT
ejpam-4933	347	1	then	then	ADV
ejpam-4933	347	2	yj(ε	yj(ε	NUM
ejpam-4933	347	3	,	,	PUNCT
ejpam-4933	347	4	ζ((x	ζ((x	ADV
ejpam-4933	347	5	∗	∗	PROPN
ejpam-4933	347	6	y	y	NOUN
ejpam-4933	347	7	)	)	PUNCT
ejpam-4933	347	8	∗	∗	NOUN
ejpam-4933	347	9	x	x	NOUN
ejpam-4933	347	10	)	)	PUNCT
ejpam-4933	347	11	)	)	PUNCT
ejpam-4933	348	1	(	(	PUNCT
ejpam-4933	348	2	4	4	X
ejpam-4933	348	3	)	)	PUNCT
ejpam-4933	348	4	=	=	SYM
ejpam-4933	348	5	yj(ε	yj(ε	ADJ
ejpam-4933	348	6	,	,	PUNCT
ejpam-4933	348	7	ζ((x	ζ((x	ADV
ejpam-4933	348	8	∗	∗	NOUN
ejpam-4933	348	9	x	x	NOUN
ejpam-4933	348	10	)	)	PUNCT
ejpam-4933	348	11	∗	∗	PROPN
ejpam-4933	348	12	y	y	PROPN
ejpam-4933	348	13	)	)	PUNCT
ejpam-4933	348	14	)	)	PUNCT
ejpam-4933	348	15	(	(	PUNCT
ejpam-4933	348	16	iii	iii	X
ejpam-4933	348	17	)	)	PUNCT
ejpam-4933	348	18	=	=	SYM
ejpam-4933	348	19	yj(ε	yj(ε	ADJ
ejpam-4933	348	20	,	,	PUNCT
ejpam-4933	348	21	ζ(0	ζ(0	PROPN
ejpam-4933	348	22	∗	∗	NOUN
ejpam-4933	348	23	y	y	PROPN
ejpam-4933	348	24	)	)	PUNCT
ejpam-4933	348	25	)	)	PUNCT
ejpam-4933	348	26	(	(	PUNCT
ejpam-4933	348	27	18	18	NUM
ejpam-4933	348	28	)	)	PUNCT
ejpam-4933	348	29	≤	≤	NOUN
ejpam-4933	348	30	yj(ε	yj(ε	NOUN
ejpam-4933	348	31	,	,	PUNCT
ejpam-4933	348	32	ζ(y	ζ(y	PROPN
ejpam-4933	348	33	)	)	PUNCT
ejpam-4933	348	34	)	)	PUNCT
ejpam-4933	349	1	e.	e.	PROPN
ejpam-4933	349	2	h.	h.	PROPN
ejpam-4933	349	3	roh	roh	PROPN
ejpam-4933	349	4	,	,	PUNCT
ejpam-4933	349	5	e.	e.	PROPN
ejpam-4933	349	6	yang	yang	PROPN
ejpam-4933	349	7	,	,	PUNCT
ejpam-4933	349	8	y.	y.	PROPN
ejpam-4933	349	9	b.	b.	PROPN
ejpam-4933	349	10	jun	jun	PROPN
ejpam-4933	349	11	/	/	SYM
ejpam-4933	349	12	eur	eur	PROPN
ejpam-4933	349	13	.	.	PUNCT
ejpam-4933	350	1	j.	j.	PROPN
ejpam-4933	350	2	pure	pure	PROPN
ejpam-4933	350	3	appl	appl	PROPN
ejpam-4933	350	4	.	.	PROPN
ejpam-4933	350	5	math	math	PROPN
ejpam-4933	350	6	,	,	PUNCT
ejpam-4933	350	7	16	16	NUM
ejpam-4933	350	8	(	(	PUNCT
ejpam-4933	350	9	4	4	NUM
ejpam-4933	350	10	)	)	PUNCT
ejpam-4933	350	11	(	(	PUNCT
ejpam-4933	350	12	2023	2023	NUM
ejpam-4933	350	13	)	)	PUNCT
ejpam-4933	350	14	,	,	PUNCT
ejpam-4933	350	15	2009	2009	NUM
ejpam-4933	350	16	-	-	SYM
ejpam-4933	350	17	2024	2024	NUM
ejpam-4933	350	18	2022	2022	NUM
ejpam-4933	350	19	for	for	ADP
ejpam-4933	350	20	all	all	DET
ejpam-4933	350	21	x	x	NOUN
ejpam-4933	350	22	,	,	PUNCT
ejpam-4933	350	23	y	y	PROPN
ejpam-4933	350	24	∈	∈	PROPN
ejpam-4933	350	25	x	x	X
ejpam-4933	350	26	,	,	PUNCT
ejpam-4933	350	27	and	and	CCONJ
ejpam-4933	350	28	so	so	ADV
ejpam-4933	350	29	yj(ε	yj(ε	ADJ
ejpam-4933	350	30	,	,	PUNCT
ejpam-4933	351	1	ζ(x	ζ(x	PROPN
ejpam-4933	351	2	∗	∗	NOUN
ejpam-4933	351	3	y	y	NOUN
ejpam-4933	351	4	)	)	PUNCT
ejpam-4933	351	5	(	(	PUNCT
ejpam-4933	351	6	15	15	NUM
ejpam-4933	351	7	)	)	PUNCT
ejpam-4933	351	8	≤	≤	NOUN
ejpam-4933	351	9	yj(ε	yj(ε	NOUN
ejpam-4933	351	10	,	,	PUNCT
ejpam-4933	351	11	ζ((x	ζ((x	ADV
ejpam-4933	351	12	∗	∗	PROPN
ejpam-4933	351	13	y	y	NOUN
ejpam-4933	351	14	)	)	PUNCT
ejpam-4933	351	15	∗	∗	NOUN
ejpam-4933	351	16	x	x	NOUN
ejpam-4933	351	17	)	)	PUNCT
ejpam-4933	351	18	)	)	PUNCT
ejpam-4933	351	19	∨	∨	PROPN
ejpam-4933	351	20	yj(ε	yj(ε	NUM
ejpam-4933	351	21	,	,	PUNCT
ejpam-4933	351	22	ζ(x	ζ(x	NOUN
ejpam-4933	351	23	)	)	PUNCT
ejpam-4933	351	24	)	)	PUNCT
ejpam-4933	352	1	≤	≤	NOUN
ejpam-4933	352	2	yj(ε	yj(ε	NUM
ejpam-4933	352	3	,	,	PUNCT
ejpam-4933	352	4	ζ(x	ζ(x	NOUN
ejpam-4933	352	5	)	)	PUNCT
ejpam-4933	352	6	)	)	PUNCT
ejpam-4933	353	1	∨	∨	PROPN
ejpam-4933	353	2	yj(ε	yj(ε	NUM
ejpam-4933	353	3	,	,	PUNCT
ejpam-4933	353	4	ζ(y	ζ(y	PROPN
ejpam-4933	353	5	)	)	PUNCT
ejpam-4933	353	6	)	)	PUNCT
ejpam-4933	353	7	for	for	ADP
ejpam-4933	353	8	all	all	DET
ejpam-4933	353	9	x	x	NOUN
ejpam-4933	353	10	,	,	PUNCT
ejpam-4933	353	11	y	y	PROPN
ejpam-4933	353	12	∈	∈	PROPN
ejpam-4933	353	13	x.	x.	NOUN
ejpam-4933	353	14	hence	hence	ADV
ejpam-4933	353	15	ζ	ζ	PROPN
ejpam-4933	353	16	is	be	AUX
ejpam-4933	353	17	a	a	DET
ejpam-4933	353	18	closed	closed	ADJ
ejpam-4933	353	19	y	y	PROPN
ejpam-4933	353	20	ε	ε	PROPN
ejpam-4933	353	21	j	j	PROPN
ejpam-4933	353	22	-fuzzy	-fuzzy	PROPN
ejpam-4933	353	23	ideal	ideal	NOUN
ejpam-4933	353	24	of	of	ADP
ejpam-4933	353	25	(	(	PUNCT
ejpam-4933	353	26	x	x	NOUN
ejpam-4933	353	27	,	,	PUNCT
ejpam-4933	353	28	∗	∗	NOUN
ejpam-4933	353	29	,	,	PUNCT
ejpam-4933	353	30	0	0	NUM
ejpam-4933	353	31	)	)	PUNCT
ejpam-4933	353	32	.	.	PUNCT
ejpam-4933	354	1	theorem	theorem	ADJ
ejpam-4933	354	2	10	10	NUM
ejpam-4933	354	3	.	.	PUNCT
ejpam-4933	355	1	given	give	VERB
ejpam-4933	355	2	an	an	DET
ejpam-4933	355	3	element	element	NOUN
ejpam-4933	355	4	a	a	DET
ejpam-4933	355	5	∈	∈	NOUN
ejpam-4933	355	6	x	x	NOUN
ejpam-4933	355	7	,	,	PUNCT
ejpam-4933	355	8	let	let	VERB
ejpam-4933	355	9	ζa	ζa	NOUN
ejpam-4933	355	10	be	be	AUX
ejpam-4933	355	11	the	the	DET
ejpam-4933	355	12	fuzzy	fuzzy	ADJ
ejpam-4933	355	13	set	set	NOUN
ejpam-4933	355	14	in	in	ADP
ejpam-4933	355	15	x	x	PUNCT
ejpam-4933	355	16	defined	define	VERB
ejpam-4933	355	17	by	by	ADP
ejpam-4933	355	18	ζa	ζa	NOUN
ejpam-4933	355	19	:	:	PUNCT
ejpam-4933	355	20	x	x	X
ejpam-4933	355	21	→	→	PUNCT
ejpam-4933	356	1	[	[	X
ejpam-4933	356	2	0	0	NUM
ejpam-4933	356	3	,	,	PUNCT
ejpam-4933	356	4	1	1	NUM
ejpam-4933	356	5	]	]	PUNCT
ejpam-4933	356	6	,	,	PUNCT
ejpam-4933	356	7	y	y	PROPN
ejpam-4933	356	8	7→	7→	PROPN
ejpam-4933	356	9	{	{	PUNCT
ejpam-4933	356	10	s1	s1	NOUN
ejpam-4933	356	11	if	if	SCONJ
ejpam-4933	356	12	y	y	PROPN
ejpam-4933	356	13	∈	∈	PROPN
ejpam-4933	356	14	xa	xa	PROPN
ejpam-4933	356	15	,	,	PUNCT
ejpam-4933	356	16	s2	s2	VERB
ejpam-4933	356	17	otherwise	otherwise	ADV
ejpam-4933	356	18	,	,	PUNCT
ejpam-4933	356	19	where	where	SCONJ
ejpam-4933	356	20	s1	s1	PROPN
ejpam-4933	356	21	>	>	X
ejpam-4933	356	22	s2	s2	PROPN
ejpam-4933	356	23	in	in	ADP
ejpam-4933	356	24	(	(	PUNCT
ejpam-4933	356	25	0	0	NUM
ejpam-4933	356	26	,	,	PUNCT
ejpam-4933	356	27	1	1	NUM
ejpam-4933	356	28	)	)	PUNCT
ejpam-4933	356	29	and	and	CCONJ
ejpam-4933	356	30	xa	xa	PROPN
ejpam-4933	356	31	:	:	PUNCT
ejpam-4933	356	32	=	=	SYM
ejpam-4933	356	33	{	{	PUNCT
ejpam-4933	356	34	x	x	SYM
ejpam-4933	356	35	∈	∈	PROPN
ejpam-4933	356	36	x	x	PUNCT
ejpam-4933	356	37	|	|	ADV
ejpam-4933	356	38	a	a	DET
ejpam-4933	356	39	∗	∗	NOUN
ejpam-4933	356	40	x	x	X
ejpam-4933	356	41	=	=	NOUN
ejpam-4933	356	42	a	a	PRON
ejpam-4933	356	43	}	}	PUNCT
ejpam-4933	356	44	.	.	PUNCT
ejpam-4933	357	1	then	then	ADV
ejpam-4933	357	2	ζa	ζa	NOUN
ejpam-4933	357	3	is	be	AUX
ejpam-4933	357	4	a	a	DET
ejpam-4933	357	5	closed	closed	ADJ
ejpam-4933	357	6	y	y	PROPN
ejpam-4933	357	7	ε	ε	PROPN
ejpam-4933	357	8	j	j	PROPN
ejpam-4933	357	9	-fuzzy	-fuzzy	PROPN
ejpam-4933	357	10	ideal	ideal	NOUN
ejpam-4933	357	11	of	of	ADP
ejpam-4933	357	12	(	(	PUNCT
ejpam-4933	357	13	x	x	NOUN
ejpam-4933	357	14	,	,	PUNCT
ejpam-4933	357	15	∗	∗	NOUN
ejpam-4933	357	16	,	,	PUNCT
ejpam-4933	357	17	0	0	NUM
ejpam-4933	357	18	)	)	PUNCT
ejpam-4933	357	19	.	.	PUNCT
ejpam-4933	358	1	proof	proof	NOUN
ejpam-4933	358	2	.	.	PUNCT
ejpam-4933	359	1	the	the	DET
ejpam-4933	359	2	y	y	NOUN
ejpam-4933	359	3	-	-	PUNCT
ejpam-4933	359	4	level	level	NOUN
ejpam-4933	359	5	set	set	NOUN
ejpam-4933	359	6	ε(ζa)t	ε(ζa)t	NOUN
ejpam-4933	359	7	is	be	AUX
ejpam-4933	359	8	calculated	calculate	VERB
ejpam-4933	359	9	as	as	SCONJ
ejpam-4933	359	10	follows	follow	VERB
ejpam-4933	359	11	:	:	PUNCT
ejpam-4933	359	12	ε(ζa)t	ε(ζa)t	PROPN
ejpam-4933	359	13	=	=	PUNCT
ejpam-4933	359	14			PUNCT
ejpam-4933	359	15	∅	∅	NOUN
ejpam-4933	359	16	if	if	SCONJ
ejpam-4933	359	17	0	0	NUM
ejpam-4933	359	18	<	<	X
ejpam-4933	359	19	t	t	X
ejpam-4933	359	20	<	<	X
ejpam-4933	359	21	1	1	NUM
ejpam-4933	359	22	−	−	PROPN
ejpam-4933	359	23	s1	s1	NOUN
ejpam-4933	359	24	,	,	PUNCT
ejpam-4933	359	25	xa	xa	PROPN
ejpam-4933	360	1	if	if	SCONJ
ejpam-4933	360	2	1	1	NUM
ejpam-4933	360	3	−	−	PROPN
ejpam-4933	360	4	s1	s1	PROPN
ejpam-4933	360	5	≤	≤	PROPN
ejpam-4933	360	6	t	t	PROPN
ejpam-4933	360	7	<	<	X
ejpam-4933	360	8	1	1	NUM
ejpam-4933	360	9	−	−	PROPN
ejpam-4933	360	10	s2	s2	PROPN
ejpam-4933	360	11	,	,	PUNCT
ejpam-4933	360	12	x	x	SYM
ejpam-4933	360	13	if	if	SCONJ
ejpam-4933	360	14	1	1	NUM
ejpam-4933	360	15	−	−	PROPN
ejpam-4933	360	16	s2	s2	PROPN
ejpam-4933	360	17	≤	≤	NUM
ejpam-4933	360	18	t	t	PROPN
ejpam-4933	360	19	<	<	X
ejpam-4933	360	20	1	1	NUM
ejpam-4933	360	21	.	.	PUNCT
ejpam-4933	361	1	it	it	PRON
ejpam-4933	361	2	is	be	AUX
ejpam-4933	361	3	clear	clear	ADJ
ejpam-4933	361	4	that	that	SCONJ
ejpam-4933	361	5	0	0	NUM
ejpam-4933	361	6	∈	∈	X
ejpam-4933	361	7	xa	xa	X
ejpam-4933	361	8	by	by	ADP
ejpam-4933	361	9	(	(	PUNCT
ejpam-4933	361	10	2	2	NUM
ejpam-4933	361	11	)	)	PUNCT
ejpam-4933	361	12	.	.	PUNCT
ejpam-4933	362	1	for	for	ADP
ejpam-4933	362	2	every	every	DET
ejpam-4933	362	3	x	x	PROPN
ejpam-4933	362	4	∈	∈	PROPN
ejpam-4933	362	5	xa	xa	PROPN
ejpam-4933	362	6	,	,	PUNCT
ejpam-4933	362	7	we	we	PRON
ejpam-4933	362	8	have	have	VERB
ejpam-4933	362	9	0	0	NUM
ejpam-4933	362	10	∗	∗	NOUN
ejpam-4933	362	11	x	x	SYM
ejpam-4933	362	12	(	(	PUNCT
ejpam-4933	362	13	iii	iii	NOUN
ejpam-4933	362	14	)	)	PUNCT
ejpam-4933	362	15	=	=	SYM
ejpam-4933	362	16	(	(	PUNCT
ejpam-4933	362	17	a	a	DET
ejpam-4933	362	18	∗	∗	X
ejpam-4933	362	19	a	a	NOUN
ejpam-4933	362	20	)	)	PUNCT
ejpam-4933	362	21	∗	∗	NOUN
ejpam-4933	362	22	x	x	SYM
ejpam-4933	362	23	(	(	PUNCT
ejpam-4933	362	24	4	4	NUM
ejpam-4933	362	25	)	)	PUNCT
ejpam-4933	362	26	=	=	SYM
ejpam-4933	362	27	(	(	PUNCT
ejpam-4933	362	28	a	a	DET
ejpam-4933	362	29	∗	∗	NOUN
ejpam-4933	362	30	x	x	NOUN
ejpam-4933	362	31	)	)	PUNCT
ejpam-4933	362	32	∗	∗	NOUN
ejpam-4933	362	33	a	a	DET
ejpam-4933	362	34	x∈xa=	x∈xa=	PROPN
ejpam-4933	362	35	a	a	DET
ejpam-4933	362	36	∗	∗	NOUN
ejpam-4933	362	37	a	a	DET
ejpam-4933	362	38	(	(	PUNCT
ejpam-4933	362	39	iii	iii	NOUN
ejpam-4933	362	40	)	)	PUNCT
ejpam-4933	362	41	=	=	SYM
ejpam-4933	362	42	0	0	NUM
ejpam-4933	362	43	∈	∈	PROPN
ejpam-4933	362	44	xa	xa	PROPN
ejpam-4933	362	45	(	(	PUNCT
ejpam-4933	362	46	19	19	NUM
ejpam-4933	362	47	)	)	PUNCT
ejpam-4933	362	48	let	let	VERB
ejpam-4933	362	49	x	x	PRON
ejpam-4933	362	50	,	,	PUNCT
ejpam-4933	362	51	y	y	PROPN
ejpam-4933	362	52	∈	∈	PROPN
ejpam-4933	362	53	x	x	AUX
ejpam-4933	362	54	be	be	AUX
ejpam-4933	362	55	such	such	ADJ
ejpam-4933	362	56	that	that	SCONJ
ejpam-4933	362	57	x	x	PUNCT
ejpam-4933	362	58	∗	∗	NOUN
ejpam-4933	362	59	y	y	PROPN
ejpam-4933	362	60	∈	∈	PROPN
ejpam-4933	362	61	xa	xa	PROPN
ejpam-4933	362	62	and	and	CCONJ
ejpam-4933	362	63	y	y	PROPN
ejpam-4933	362	64	∈	∈	PROPN
ejpam-4933	362	65	xa	xa	PROPN
ejpam-4933	362	66	.	.	PUNCT
ejpam-4933	363	1	then	then	ADV
ejpam-4933	363	2	0	0	NUM
ejpam-4933	363	3	∗	∗	NOUN
ejpam-4933	363	4	(	(	PUNCT
ejpam-4933	363	5	x	x	X
ejpam-4933	363	6	∗	∗	NOUN
ejpam-4933	363	7	y	y	NOUN
ejpam-4933	363	8	)	)	PUNCT
ejpam-4933	363	9	=	=	SYM
ejpam-4933	363	10	0	0	NUM
ejpam-4933	363	11	and	and	CCONJ
ejpam-4933	363	12	0	0	NUM
ejpam-4933	363	13	∗	∗	NOUN
ejpam-4933	363	14	y	y	NOUN
ejpam-4933	363	15	=	=	SYM
ejpam-4933	363	16	0	0	NUM
ejpam-4933	363	17	by	by	ADP
ejpam-4933	363	18	(	(	PUNCT
ejpam-4933	363	19	19	19	NUM
ejpam-4933	363	20	)	)	PUNCT
ejpam-4933	363	21	.	.	PUNCT
ejpam-4933	364	1	it	it	PRON
ejpam-4933	364	2	follows	follow	VERB
ejpam-4933	364	3	that	that	SCONJ
ejpam-4933	364	4	(	(	PUNCT
ejpam-4933	364	5	a	a	DET
ejpam-4933	364	6	∗	∗	NOUN
ejpam-4933	364	7	x	x	NOUN
ejpam-4933	364	8	)	)	PUNCT
ejpam-4933	364	9	∗	∗	NOUN
ejpam-4933	364	10	a	a	DET
ejpam-4933	364	11	(	(	PUNCT
ejpam-4933	364	12	4	4	NUM
ejpam-4933	364	13	)	)	PUNCT
ejpam-4933	364	14	=	=	SYM
ejpam-4933	364	15	(	(	PUNCT
ejpam-4933	364	16	a	a	DET
ejpam-4933	364	17	∗	∗	X
ejpam-4933	364	18	a	a	NOUN
ejpam-4933	364	19	)	)	PUNCT
ejpam-4933	364	20	∗	∗	NOUN
ejpam-4933	364	21	x	x	SYM
ejpam-4933	364	22	(	(	PUNCT
ejpam-4933	364	23	iii	iii	NOUN
ejpam-4933	364	24	)	)	PUNCT
ejpam-4933	364	25	=	=	SYM
ejpam-4933	364	26	0	0	NUM
ejpam-4933	364	27	∗	∗	NOUN
ejpam-4933	364	28	x	x	SYM
ejpam-4933	364	29	(	(	PUNCT
ejpam-4933	364	30	2	2	NUM
ejpam-4933	364	31	)	)	PUNCT
ejpam-4933	364	32	=	=	SYM
ejpam-4933	364	33	(	(	PUNCT
ejpam-4933	364	34	0	0	NUM
ejpam-4933	364	35	∗	∗	NOUN
ejpam-4933	364	36	x	x	NOUN
ejpam-4933	364	37	)	)	PUNCT
ejpam-4933	364	38	∗	∗	NOUN
ejpam-4933	364	39	(	(	PUNCT
ejpam-4933	364	40	0	0	NUM
ejpam-4933	364	41	∗	∗	PROPN
ejpam-4933	364	42	y	y	NOUN
ejpam-4933	364	43	)	)	PUNCT
ejpam-4933	364	44	(	(	PUNCT
ejpam-4933	364	45	6	6	NUM
ejpam-4933	364	46	)	)	PUNCT
ejpam-4933	364	47	=	=	SYM
ejpam-4933	364	48	0	0	NUM
ejpam-4933	364	49	∗	∗	NOUN
ejpam-4933	364	50	(	(	PUNCT
ejpam-4933	364	51	x	x	X
ejpam-4933	364	52	∗	∗	NOUN
ejpam-4933	364	53	y	y	NOUN
ejpam-4933	364	54	)	)	PUNCT
ejpam-4933	364	55	=	=	SYM
ejpam-4933	364	56	0	0	NUM
ejpam-4933	364	57	,	,	PUNCT
ejpam-4933	364	58	i.e.	i.e.	X
ejpam-4933	364	59	,	,	PUNCT
ejpam-4933	364	60	a	a	DET
ejpam-4933	364	61	∗	∗	NOUN
ejpam-4933	364	62	x	x	SYM
ejpam-4933	364	63	≤x	≤x	NOUN
ejpam-4933	364	64	a.	a.	NOUN
ejpam-4933	364	65	since	since	SCONJ
ejpam-4933	364	66	x	x	PROPN
ejpam-4933	364	67	∗	∗	VERB
ejpam-4933	364	68	y	y	PROPN
ejpam-4933	364	69	∈	∈	PROPN
ejpam-4933	364	70	xa	xa	PROPN
ejpam-4933	364	71	and	and	CCONJ
ejpam-4933	364	72	y	y	PROPN
ejpam-4933	364	73	∈	∈	PROPN
ejpam-4933	365	1	xa	xa	PROPN
ejpam-4933	365	2	,	,	PUNCT
ejpam-4933	365	3	we	we	PRON
ejpam-4933	365	4	get	get	VERB
ejpam-4933	365	5	a	a	DET
ejpam-4933	365	6	=	=	NOUN
ejpam-4933	365	7	a	a	DET
ejpam-4933	365	8	∗	∗	NOUN
ejpam-4933	365	9	(	(	PUNCT
ejpam-4933	365	10	x	x	X
ejpam-4933	365	11	∗	∗	NOUN
ejpam-4933	365	12	y	y	NOUN
ejpam-4933	365	13	)	)	PUNCT
ejpam-4933	365	14	=	=	PUNCT
ejpam-4933	365	15	(	(	PUNCT
ejpam-4933	365	16	a	a	DET
ejpam-4933	365	17	∗	∗	NOUN
ejpam-4933	365	18	y	y	NOUN
ejpam-4933	365	19	)	)	PUNCT
ejpam-4933	365	20	∗	∗	NOUN
ejpam-4933	365	21	(	(	PUNCT
ejpam-4933	365	22	x	x	X
ejpam-4933	365	23	∗	∗	PROPN
ejpam-4933	365	24	y	y	NOUN
ejpam-4933	365	25	)	)	PUNCT
ejpam-4933	365	26	≤	≤	NOUN
ejpam-4933	365	27	a	a	DET
ejpam-4933	365	28	∗	∗	NOUN
ejpam-4933	365	29	x.	x.	NOUN
ejpam-4933	365	30	hence	hence	ADV
ejpam-4933	365	31	a	a	DET
ejpam-4933	365	32	∗	∗	NOUN
ejpam-4933	365	33	x	x	X
ejpam-4933	365	34	=	=	SYM
ejpam-4933	365	35	a	a	X
ejpam-4933	365	36	,	,	PUNCT
ejpam-4933	365	37	i.e.	i.e.	X
ejpam-4933	365	38	,	,	PUNCT
ejpam-4933	365	39	x	x	SYM
ejpam-4933	365	40	∈	∈	PROPN
ejpam-4933	365	41	xa	xa	PROPN
ejpam-4933	365	42	.	.	PUNCT
ejpam-4933	366	1	this	this	PRON
ejpam-4933	366	2	shows	show	VERB
ejpam-4933	366	3	that	that	SCONJ
ejpam-4933	366	4	xa	xa	PROPN
ejpam-4933	366	5	is	be	AUX
ejpam-4933	366	6	a	a	DET
ejpam-4933	366	7	closed	closed	ADJ
ejpam-4933	366	8	ideal	ideal	NOUN
ejpam-4933	366	9	of	of	ADP
ejpam-4933	366	10	(	(	PUNCT
ejpam-4933	366	11	x	x	NOUN
ejpam-4933	366	12	,	,	PUNCT
ejpam-4933	366	13	∗	∗	NOUN
ejpam-4933	366	14	,	,	PUNCT
ejpam-4933	366	15	0	0	NUM
ejpam-4933	366	16	)	)	PUNCT
ejpam-4933	366	17	.	.	PUNCT
ejpam-4933	367	1	thus	thus	ADV
ejpam-4933	367	2	we	we	PRON
ejpam-4933	367	3	know	know	VERB
ejpam-4933	367	4	that	that	SCONJ
ejpam-4933	367	5	the	the	DET
ejpam-4933	367	6	nonempty	nonempty	ADJ
ejpam-4933	367	7	y	y	NOUN
ejpam-4933	367	8	-	-	PUNCT
ejpam-4933	367	9	level	level	NOUN
ejpam-4933	367	10	set	set	NOUN
ejpam-4933	367	11	ε(ζa)t	ε(ζa)t	NOUN
ejpam-4933	367	12	is	be	AUX
ejpam-4933	367	13	a	a	DET
ejpam-4933	367	14	closed	closed	ADJ
ejpam-4933	367	15	ideal	ideal	NOUN
ejpam-4933	367	16	of	of	ADP
ejpam-4933	367	17	(	(	PUNCT
ejpam-4933	367	18	x	x	NOUN
ejpam-4933	367	19	,	,	PUNCT
ejpam-4933	367	20	∗	∗	NOUN
ejpam-4933	367	21	,	,	PUNCT
ejpam-4933	367	22	0	0	NUM
ejpam-4933	367	23	)	)	PUNCT
ejpam-4933	367	24	for	for	ADP
ejpam-4933	367	25	all	all	DET
ejpam-4933	367	26	t	t	NOUN
ejpam-4933	367	27	∈	∈	PROPN
ejpam-4933	367	28	i	i	PRON
ejpam-4933	367	29	\{0	\{0	VERB
ejpam-4933	367	30	,	,	PUNCT
ejpam-4933	367	31	1	1	NUM
ejpam-4933	367	32	}	}	PUNCT
ejpam-4933	367	33	.	.	PUNCT
ejpam-4933	368	1	therefore	therefore	ADV
ejpam-4933	368	2	ζa	ζa	NOUN
ejpam-4933	368	3	is	be	AUX
ejpam-4933	368	4	a	a	DET
ejpam-4933	368	5	closed	closed	ADJ
ejpam-4933	368	6	y	y	PROPN
ejpam-4933	368	7	ε	ε	PROPN
ejpam-4933	368	8	j	j	PROPN
ejpam-4933	368	9	-fuzzy	-fuzzy	PROPN
ejpam-4933	368	10	ideal	ideal	NOUN
ejpam-4933	368	11	of	of	ADP
ejpam-4933	368	12	(	(	PUNCT
ejpam-4933	368	13	x	x	NOUN
ejpam-4933	368	14	,	,	PUNCT
ejpam-4933	368	15	∗	∗	NOUN
ejpam-4933	368	16	,	,	PUNCT
ejpam-4933	368	17	0	0	NUM
ejpam-4933	368	18	)	)	PUNCT
ejpam-4933	368	19	by	by	ADP
ejpam-4933	368	20	lemma	lemma	PROPN
ejpam-4933	368	21	1	1	NUM
ejpam-4933	368	22	.	.	PUNCT
ejpam-4933	369	1	we	we	PRON
ejpam-4933	369	2	explore	explore	VERB
ejpam-4933	369	3	the	the	DET
ejpam-4933	369	4	conditions	condition	NOUN
ejpam-4933	369	5	under	under	ADP
ejpam-4933	369	6	which	which	PRON
ejpam-4933	369	7	a	a	DET
ejpam-4933	369	8	y	y	PROPN
ejpam-4933	369	9	ε	ε	PROPN
ejpam-4933	369	10	j	j	PROPN
ejpam-4933	369	11	-fuzzy	-fuzzy	PROPN
ejpam-4933	369	12	subalgebra	subalgebra	NOUN
ejpam-4933	369	13	becomes	become	VERB
ejpam-4933	369	14	a	a	DET
ejpam-4933	369	15	y	y	PROPN
ejpam-4933	369	16	ε	ε	PROPN
ejpam-4933	369	17	j	j	PROPN
ejpam-4933	369	18	-fuzzy	-fuzzy	PROPN
ejpam-4933	369	19	ideal	ideal	ADJ
ejpam-4933	369	20	.	.	PUNCT
ejpam-4933	370	1	theorem	theorem	VERB
ejpam-4933	370	2	11	11	NUM
ejpam-4933	370	3	.	.	PUNCT
ejpam-4933	371	1	in	in	ADP
ejpam-4933	371	2	a	a	DET
ejpam-4933	371	3	p	p	ADJ
ejpam-4933	371	4	-	-	PUNCT
ejpam-4933	371	5	semisimple	semisimple	NOUN
ejpam-4933	371	6	bci	bci	NOUN
ejpam-4933	371	7	-	-	NOUN
ejpam-4933	371	8	algebra	algebra	NOUN
ejpam-4933	371	9	(	(	PUNCT
ejpam-4933	371	10	x	x	X
ejpam-4933	371	11	,	,	PUNCT
ejpam-4933	371	12	∗	∗	NOUN
ejpam-4933	371	13	,	,	PUNCT
ejpam-4933	371	14	0	0	NUM
ejpam-4933	371	15	)	)	PUNCT
ejpam-4933	371	16	,	,	PUNCT
ejpam-4933	371	17	every	every	DET
ejpam-4933	371	18	y	y	PROPN
ejpam-4933	371	19	ε	ε	PROPN
ejpam-4933	371	20	j	j	PROPN
ejpam-4933	371	21	-fuzzy	-fuzzy	PROPN
ejpam-4933	371	22	subalgebra	subalgebra	NOUN
ejpam-4933	371	23	is	be	AUX
ejpam-4933	371	24	a	a	DET
ejpam-4933	371	25	y	y	PROPN
ejpam-4933	371	26	ε	ε	PROPN
ejpam-4933	371	27	j	j	PROPN
ejpam-4933	371	28	-fuzzy	-fuzzy	PROPN
ejpam-4933	371	29	ideal	ideal	ADJ
ejpam-4933	371	30	.	.	PUNCT
ejpam-4933	372	1	references	reference	NOUN
ejpam-4933	372	2	2023	2023	NUM
ejpam-4933	372	3	proof	proof	NOUN
ejpam-4933	372	4	.	.	PUNCT
ejpam-4933	373	1	let	let	VERB
ejpam-4933	373	2	ζ	ζ	NOUN
ejpam-4933	373	3	be	be	AUX
ejpam-4933	373	4	a	a	DET
ejpam-4933	373	5	y	y	PROPN
ejpam-4933	373	6	ε	ε	PROPN
ejpam-4933	373	7	j	j	PROPN
ejpam-4933	373	8	-fuzzy	-fuzzy	PROPN
ejpam-4933	373	9	subalgebra	subalgebra	NOUN
ejpam-4933	373	10	of	of	ADP
ejpam-4933	373	11	a	a	DET
ejpam-4933	373	12	p	p	ADJ
ejpam-4933	373	13	-	-	PUNCT
ejpam-4933	373	14	semisimple	semisimple	NOUN
ejpam-4933	373	15	bci	bci	NOUN
ejpam-4933	373	16	-	-	NOUN
ejpam-4933	373	17	algebra	algebra	NOUN
ejpam-4933	373	18	(	(	PUNCT
ejpam-4933	373	19	x	x	X
ejpam-4933	373	20	,	,	PUNCT
ejpam-4933	373	21	∗	∗	NOUN
ejpam-4933	373	22	,	,	PUNCT
ejpam-4933	373	23	0	0	NUM
ejpam-4933	373	24	)	)	PUNCT
ejpam-4933	373	25	,	,	PUNCT
ejpam-4933	373	26	and	and	CCONJ
ejpam-4933	373	27	let	let	VERB
ejpam-4933	373	28	t	t	PROPN
ejpam-4933	373	29	∈	∈	PROPN
ejpam-4933	374	1	i	i	PRON
ejpam-4933	374	2	\	\	PROPN
ejpam-4933	374	3	{	{	PUNCT
ejpam-4933	374	4	0	0	NUM
ejpam-4933	374	5	,	,	PUNCT
ejpam-4933	374	6	1	1	NUM
ejpam-4933	374	7	}	}	PUNCT
ejpam-4933	374	8	be	be	AUX
ejpam-4933	374	9	such	such	ADJ
ejpam-4933	374	10	that	that	SCONJ
ejpam-4933	374	11	ε(ζ)t	ε(ζ)t	VERB
ejpam-4933	374	12	̸=	̸=	PROPN
ejpam-4933	374	13	∅.	∅.	VERB
ejpam-4933	374	14	then	then	ADV
ejpam-4933	374	15	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	374	16	is	be	AUX
ejpam-4933	374	17	a	a	DET
ejpam-4933	374	18	subalgebra	subalgebra	NOUN
ejpam-4933	374	19	of	of	ADP
ejpam-4933	374	20	(	(	PUNCT
ejpam-4933	374	21	x	x	NOUN
ejpam-4933	374	22	,	,	PUNCT
ejpam-4933	374	23	∗	∗	NOUN
ejpam-4933	374	24	,	,	PUNCT
ejpam-4933	374	25	0	0	NUM
ejpam-4933	374	26	)	)	PUNCT
ejpam-4933	374	27	by	by	ADP
ejpam-4933	374	28	theorem	theorem	NOUN
ejpam-4933	374	29	1	1	NUM
ejpam-4933	374	30	.	.	PUNCT
ejpam-4933	375	1	it	it	PRON
ejpam-4933	375	2	is	be	AUX
ejpam-4933	375	3	clear	clear	ADJ
ejpam-4933	375	4	that	that	SCONJ
ejpam-4933	375	5	0	0	NUM
ejpam-4933	375	6	∈	∈	PROPN
ejpam-4933	375	7	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	375	8	.	.	PUNCT
ejpam-4933	376	1	let	let	VERB
ejpam-4933	376	2	x	x	PRON
ejpam-4933	376	3	,	,	PUNCT
ejpam-4933	376	4	y	y	PROPN
ejpam-4933	376	5	∈	∈	PROPN
ejpam-4933	376	6	x	x	AUX
ejpam-4933	376	7	be	be	AUX
ejpam-4933	376	8	such	such	ADJ
ejpam-4933	376	9	that	that	SCONJ
ejpam-4933	376	10	x	x	PUNCT
ejpam-4933	376	11	∗	∗	NOUN
ejpam-4933	376	12	y	y	PROPN
ejpam-4933	376	13	∈	∈	PROPN
ejpam-4933	376	14	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	376	15	and	and	CCONJ
ejpam-4933	376	16	y	y	PROPN
ejpam-4933	376	17	∈	∈	PROPN
ejpam-4933	376	18	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	376	19	.	.	PUNCT
ejpam-4933	377	1	then	then	ADV
ejpam-4933	377	2	0	0	NUM
ejpam-4933	377	3	∗	∗	NOUN
ejpam-4933	377	4	y	y	PROPN
ejpam-4933	377	5	∈	∈	PROPN
ejpam-4933	377	6	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	378	1	and	and	CCONJ
ejpam-4933	378	2	(	(	PUNCT
ejpam-4933	378	3	x	x	PROPN
ejpam-4933	378	4	∗	∗	PROPN
ejpam-4933	378	5	y	y	NOUN
ejpam-4933	378	6	)	)	PUNCT
ejpam-4933	378	7	∗	∗	NOUN
ejpam-4933	378	8	(	(	PUNCT
ejpam-4933	378	9	0	0	NUM
ejpam-4933	378	10	∗	∗	PROPN
ejpam-4933	378	11	y	y	NOUN
ejpam-4933	378	12	)	)	PUNCT
ejpam-4933	378	13	∈	∈	PROPN
ejpam-4933	378	14	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	378	15	.	.	PUNCT
ejpam-4933	379	1	on	on	ADP
ejpam-4933	379	2	the	the	DET
ejpam-4933	379	3	other	other	ADJ
ejpam-4933	379	4	hand	hand	NOUN
ejpam-4933	379	5	,	,	PUNCT
ejpam-4933	379	6	we	we	PRON
ejpam-4933	379	7	have	have	VERB
ejpam-4933	379	8	(	(	PUNCT
ejpam-4933	379	9	(	(	PUNCT
ejpam-4933	379	10	x	x	SYM
ejpam-4933	379	11	∗	∗	PROPN
ejpam-4933	379	12	y	y	NOUN
ejpam-4933	379	13	)	)	PUNCT
ejpam-4933	379	14	∗	∗	NOUN
ejpam-4933	379	15	(	(	PUNCT
ejpam-4933	379	16	0	0	NUM
ejpam-4933	379	17	∗	∗	PROPN
ejpam-4933	379	18	y	y	PROPN
ejpam-4933	379	19	)	)	PUNCT
ejpam-4933	379	20	)	)	PUNCT
ejpam-4933	379	21	∗	∗	NOUN
ejpam-4933	379	22	x	x	SYM
ejpam-4933	379	23	(	(	PUNCT
ejpam-4933	379	24	4	4	NUM
ejpam-4933	379	25	)	)	PUNCT
ejpam-4933	379	26	=	=	SYM
ejpam-4933	379	27	(	(	PUNCT
ejpam-4933	379	28	(	(	PUNCT
ejpam-4933	379	29	x	x	SYM
ejpam-4933	379	30	∗	∗	PROPN
ejpam-4933	379	31	y	y	NOUN
ejpam-4933	379	32	)	)	PUNCT
ejpam-4933	379	33	∗	∗	NOUN
ejpam-4933	379	34	x	x	NOUN
ejpam-4933	379	35	)	)	PUNCT
ejpam-4933	379	36	∗	∗	NOUN
ejpam-4933	379	37	(	(	PUNCT
ejpam-4933	379	38	0	0	NUM
ejpam-4933	379	39	∗	∗	PROPN
ejpam-4933	379	40	y	y	NOUN
ejpam-4933	379	41	)	)	PUNCT
ejpam-4933	379	42	(	(	PUNCT
ejpam-4933	379	43	4	4	X
ejpam-4933	379	44	)	)	PUNCT
ejpam-4933	379	45	=	=	SYM
ejpam-4933	379	46	(	(	PUNCT
ejpam-4933	379	47	(	(	PUNCT
ejpam-4933	379	48	x	x	NOUN
ejpam-4933	379	49	∗	∗	NOUN
ejpam-4933	379	50	x	x	NOUN
ejpam-4933	379	51	)	)	PUNCT
ejpam-4933	379	52	∗	∗	PROPN
ejpam-4933	379	53	y	y	NOUN
ejpam-4933	379	54	)	)	PUNCT
ejpam-4933	379	55	∗	∗	NOUN
ejpam-4933	379	56	(	(	PUNCT
ejpam-4933	379	57	0	0	NUM
ejpam-4933	379	58	∗	∗	PROPN
ejpam-4933	379	59	y	y	PROPN
ejpam-4933	379	60	)	)	PUNCT
ejpam-4933	379	61	(	(	PUNCT
ejpam-4933	379	62	iii	iii	NOUN
ejpam-4933	379	63	)	)	PUNCT
ejpam-4933	379	64	=	=	SYM
ejpam-4933	380	1	(	(	PUNCT
ejpam-4933	380	2	0	0	NUM
ejpam-4933	380	3	∗	∗	PROPN
ejpam-4933	380	4	y	y	NOUN
ejpam-4933	380	5	)	)	PUNCT
ejpam-4933	380	6	∗	∗	NOUN
ejpam-4933	380	7	(	(	PUNCT
ejpam-4933	380	8	0	0	NUM
ejpam-4933	380	9	∗	∗	PROPN
ejpam-4933	380	10	y	y	PROPN
ejpam-4933	380	11	)	)	PUNCT
ejpam-4933	380	12	(	(	PUNCT
ejpam-4933	380	13	iii	iii	NOUN
ejpam-4933	380	14	)	)	PUNCT
ejpam-4933	380	15	=	=	SYM
ejpam-4933	380	16	0	0	NUM
ejpam-4933	380	17	,	,	PUNCT
ejpam-4933	380	18	that	that	ADV
ejpam-4933	380	19	is	is	ADV
ejpam-4933	380	20	,	,	PUNCT
ejpam-4933	380	21	(	(	PUNCT
ejpam-4933	380	22	x	x	X
ejpam-4933	380	23	∗	∗	PROPN
ejpam-4933	380	24	y	y	NOUN
ejpam-4933	380	25	)	)	PUNCT
ejpam-4933	380	26	∗	∗	NOUN
ejpam-4933	380	27	(	(	PUNCT
ejpam-4933	380	28	0	0	NUM
ejpam-4933	380	29	∗	∗	PROPN
ejpam-4933	380	30	y	y	NOUN
ejpam-4933	380	31	)	)	PUNCT
ejpam-4933	380	32	≤x	≤x	PROPN
ejpam-4933	380	33	x.	x.	NOUN
ejpam-4933	380	34	since	since	SCONJ
ejpam-4933	380	35	(	(	PUNCT
ejpam-4933	380	36	x	x	X
ejpam-4933	380	37	,	,	PUNCT
ejpam-4933	380	38	∗	∗	NOUN
ejpam-4933	380	39	,	,	PUNCT
ejpam-4933	380	40	0	0	NUM
ejpam-4933	380	41	)	)	PUNCT
ejpam-4933	380	42	is	be	AUX
ejpam-4933	380	43	p	p	NOUN
ejpam-4933	380	44	-	-	PUNCT
ejpam-4933	380	45	semisimple	semisimple	NOUN
ejpam-4933	380	46	,	,	PUNCT
ejpam-4933	380	47	x	x	X
ejpam-4933	380	48	is	be	AUX
ejpam-4933	380	49	a	a	DET
ejpam-4933	380	50	minimal	minimal	ADJ
ejpam-4933	380	51	element	element	NOUN
ejpam-4933	380	52	of	of	ADP
ejpam-4933	380	53	x.	x.	NOUN
ejpam-4933	380	54	it	it	PRON
ejpam-4933	380	55	follows	follow	VERB
ejpam-4933	380	56	that	that	SCONJ
ejpam-4933	380	57	x	x	PUNCT
ejpam-4933	381	1	=	=	PRON
ejpam-4933	381	2	(	(	PUNCT
ejpam-4933	381	3	x	x	X
ejpam-4933	381	4	∗	∗	PROPN
ejpam-4933	381	5	y	y	NOUN
ejpam-4933	381	6	)	)	PUNCT
ejpam-4933	381	7	∗	∗	NOUN
ejpam-4933	381	8	(	(	PUNCT
ejpam-4933	381	9	0	0	NUM
ejpam-4933	381	10	∗	∗	PROPN
ejpam-4933	381	11	y	y	NOUN
ejpam-4933	381	12	)	)	PUNCT
ejpam-4933	381	13	∈	∈	PROPN
ejpam-4933	381	14	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	381	15	.	.	PUNCT
ejpam-4933	382	1	hence	hence	ADV
ejpam-4933	382	2	ε(ζ)t	ε(ζ)t	PROPN
ejpam-4933	382	3	is	be	AUX
ejpam-4933	382	4	an	an	DET
ejpam-4933	382	5	ideal	ideal	NOUN
ejpam-4933	382	6	of	of	ADP
ejpam-4933	382	7	(	(	PUNCT
ejpam-4933	382	8	x	x	NOUN
ejpam-4933	382	9	,	,	PUNCT
ejpam-4933	382	10	∗	∗	NOUN
ejpam-4933	382	11	,	,	PUNCT
ejpam-4933	382	12	0	0	NUM
ejpam-4933	382	13	)	)	PUNCT
ejpam-4933	382	14	,	,	PUNCT
ejpam-4933	382	15	and	and	CCONJ
ejpam-4933	382	16	therefore	therefore	ADV
ejpam-4933	382	17	ζ	ζ	NOUN
ejpam-4933	382	18	is	be	AUX
ejpam-4933	382	19	a	a	DET
ejpam-4933	382	20	y	y	PROPN
ejpam-4933	382	21	ε	ε	PROPN
ejpam-4933	382	22	j	j	PROPN
ejpam-4933	382	23	-fuzzy	-fuzzy	PROPN
ejpam-4933	382	24	ideal	ideal	NOUN
ejpam-4933	382	25	of	of	ADP
ejpam-4933	382	26	(	(	PUNCT
ejpam-4933	382	27	x	x	NOUN
ejpam-4933	382	28	,	,	PUNCT
ejpam-4933	382	29	∗	∗	NOUN
ejpam-4933	382	30	,	,	PUNCT
ejpam-4933	382	31	0	0	NUM
ejpam-4933	382	32	)	)	PUNCT
ejpam-4933	382	33	by	by	ADP
ejpam-4933	382	34	theorem	theorem	ADJ
ejpam-4933	382	35	6	6	NUM
ejpam-4933	382	36	.	.	PUNCT
ejpam-4933	382	37	corollary	corollary	ADJ
ejpam-4933	382	38	2	2	NUM
ejpam-4933	382	39	.	.	PUNCT
ejpam-4933	383	1	if	if	SCONJ
ejpam-4933	383	2	a	a	DET
ejpam-4933	383	3	bci	bci	NOUN
ejpam-4933	383	4	-	-	NOUN
ejpam-4933	383	5	algebra	algebra	NOUN
ejpam-4933	383	6	(	(	PUNCT
ejpam-4933	383	7	x	x	X
ejpam-4933	383	8	,	,	PUNCT
ejpam-4933	383	9	∗	∗	NOUN
ejpam-4933	383	10	,	,	PUNCT
ejpam-4933	383	11	0	0	NUM
ejpam-4933	383	12	)	)	PUNCT
ejpam-4933	383	13	satisfies	satisfie	NOUN
ejpam-4933	383	14	:	:	PUNCT
ejpam-4933	383	15	(	(	PUNCT
ejpam-4933	383	16	∀x	∀x	X
ejpam-4933	383	17	,	,	PUNCT
ejpam-4933	383	18	y	y	PROPN
ejpam-4933	383	19	∈	∈	PROPN
ejpam-4933	383	20	x)(x	x)(x	PROPN
ejpam-4933	383	21	∗	∗	NOUN
ejpam-4933	383	22	(	(	PUNCT
ejpam-4933	383	23	0	0	NUM
ejpam-4933	383	24	∗	∗	NUM
ejpam-4933	383	25	y	y	NOUN
ejpam-4933	383	26	)	)	PUNCT
ejpam-4933	384	1	=	=	SYM
ejpam-4933	384	2	y	y	PROPN
ejpam-4933	384	3	∗	∗	NOUN
ejpam-4933	384	4	(	(	PUNCT
ejpam-4933	384	5	0	0	NUM
ejpam-4933	384	6	∗	∗	NOUN
ejpam-4933	384	7	x	x	NOUN
ejpam-4933	384	8	)	)	PUNCT
ejpam-4933	384	9	)	)	PUNCT
ejpam-4933	384	10	or	or	CCONJ
ejpam-4933	384	11	(	(	PUNCT
ejpam-4933	384	12	∀x	∀x	X
ejpam-4933	384	13	∈	∈	PROPN
ejpam-4933	384	14	x)(0	x)(0	X
ejpam-4933	385	1	∗	∗	NOUN
ejpam-4933	385	2	x	x	X
ejpam-4933	385	3	=	=	SYM
ejpam-4933	385	4	0	0	NUM
ejpam-4933	385	5	⇒	⇒	NOUN
ejpam-4933	385	6	x	x	PUNCT
ejpam-4933	386	1	=	=	NOUN
ejpam-4933	386	2	0	0	NUM
ejpam-4933	386	3	)	)	PUNCT
ejpam-4933	386	4	,	,	PUNCT
ejpam-4933	386	5	then	then	ADV
ejpam-4933	386	6	every	every	DET
ejpam-4933	386	7	y	y	PROPN
ejpam-4933	386	8	ε	ε	PROPN
ejpam-4933	386	9	j	j	PROPN
ejpam-4933	386	10	-fuzzy	-fuzzy	PROPN
ejpam-4933	386	11	subalgebra	subalgebra	NOUN
ejpam-4933	386	12	is	be	AUX
ejpam-4933	386	13	a	a	DET
ejpam-4933	386	14	y	y	PROPN
ejpam-4933	386	15	ε	ε	PROPN
ejpam-4933	386	16	j	j	PROPN
ejpam-4933	386	17	-fuzzy	-fuzzy	PROPN
ejpam-4933	386	18	ideal	ideal	ADJ
ejpam-4933	386	19	.	.	PUNCT
ejpam-4933	387	1	acknowledgements	acknowledgement	NOUN
ejpam-4933	387	2	this	this	DET
ejpam-4933	387	3	paper	paper	NOUN
ejpam-4933	387	4	was	be	AUX
ejpam-4933	387	5	supported	support	VERB
ejpam-4933	387	6	by	by	ADP
ejpam-4933	387	7	the	the	DET
ejpam-4933	387	8	research	research	NOUN
ejpam-4933	387	9	fund	fund	NOUN
ejpam-4933	387	10	in	in	ADP
ejpam-4933	387	11	chinju	chinju	PROPN
ejpam-4933	387	12	national	national	PROPN
ejpam-4933	387	13	university	university	PROPN
ejpam-4933	387	14	of	of	ADP
ejpam-4933	387	15	education	education	NOUN
ejpam-4933	387	16	,	,	PUNCT
ejpam-4933	387	17	2022	2022	NUM
ejpam-4933	387	18	.	.	PUNCT
ejpam-4933	388	1	references	reference	NOUN
ejpam-4933	388	2	[	[	X
ejpam-4933	388	3	1	1	X
ejpam-4933	388	4	]	]	PUNCT
ejpam-4933	388	5	s.	s.	PROPN
ejpam-4933	388	6	m.	m.	PROPN
ejpam-4933	388	7	hong	hong	PROPN
ejpam-4933	388	8	and	and	CCONJ
ejpam-4933	388	9	y.	y.	PROPN
ejpam-4933	388	10	b.	b.	PROPN
ejpam-4933	388	11	jun	jun	PROPN
ejpam-4933	388	12	.	.	PUNCT
ejpam-4933	389	1	anti	anti	PROPN
ejpam-4933	389	2	fuzzy	fuzzy	ADJ
ejpam-4933	389	3	ideals	ideal	NOUN
ejpam-4933	389	4	in	in	ADP
ejpam-4933	389	5	bck	bck	NOUN
ejpam-4933	389	6	-	-	PUNCT
ejpam-4933	389	7	algebras	algebras	PROPN
ejpam-4933	389	8	.	.	PUNCT
ejpam-4933	390	1	kyungpook	kyungpook	PROPN
ejpam-4933	390	2	math	math	PROPN
ejpam-4933	390	3	.	.	PUNCT
ejpam-4933	391	1	j.	j.	PROPN
ejpam-4933	391	2	,	,	PUNCT
ejpam-4933	391	3	38:145–150	38:145–150	PROPN
ejpam-4933	391	4	,	,	PUNCT
ejpam-4933	391	5	1998	1998	NUM
ejpam-4933	391	6	.	.	PUNCT
ejpam-4933	392	1	[	[	X
ejpam-4933	392	2	2	2	X
ejpam-4933	392	3	]	]	X
ejpam-4933	392	4	y.	y.	PROPN
ejpam-4933	392	5	s.	s.	PROPN
ejpam-4933	392	6	huang	huang	PROPN
ejpam-4933	392	7	.	.	PUNCT
ejpam-4933	393	1	bci	bci	PROPN
ejpam-4933	393	2	-	-	NOUN
ejpam-4933	393	3	algebra	algebra	NOUN
ejpam-4933	393	4	.	.	PUNCT
ejpam-4933	394	1	science	science	NOUN
ejpam-4933	394	2	press	press	PROPN
ejpam-4933	394	3	,	,	PUNCT
ejpam-4933	394	4	beijing	beijing	PROPN
ejpam-4933	394	5	,	,	PUNCT
ejpam-4933	394	6	china	china	PROPN
ejpam-4933	394	7	,	,	PUNCT
ejpam-4933	394	8	2006	2006	NUM
ejpam-4933	394	9	.	.	PUNCT
ejpam-4933	395	1	[	[	X
ejpam-4933	395	2	3	3	X
ejpam-4933	395	3	]	]	X
ejpam-4933	395	4	k.	k.	PROPN
ejpam-4933	395	5	iséki	iséki	PROPN
ejpam-4933	395	6	.	.	PROPN
ejpam-4933	395	7	on	on	ADP
ejpam-4933	395	8	bci	bci	NOUN
ejpam-4933	395	9	-	-	PUNCT
ejpam-4933	395	10	algebras	algebra	NOUN
ejpam-4933	395	11	.	.	PUNCT
ejpam-4933	395	12	math	math	PROPN
ejpam-4933	395	13	.	.	PUNCT
ejpam-4933	396	1	japon	japon	PROPN
ejpam-4933	396	2	.	.	PROPN
ejpam-4933	396	3	,	,	PUNCT
ejpam-4933	396	4	23:1–26	23:1–26	NUM
ejpam-4933	396	5	,	,	PUNCT
ejpam-4933	396	6	1978	1978	NUM
ejpam-4933	396	7	.	.	PUNCT
ejpam-4933	397	1	[	[	X
ejpam-4933	397	2	4	4	X
ejpam-4933	397	3	]	]	PUNCT
ejpam-4933	397	4	k.	k.	PROPN
ejpam-4933	397	5	iséki	iséki	PROPN
ejpam-4933	397	6	and	and	CCONJ
ejpam-4933	397	7	s.	s.	PROPN
ejpam-4933	397	8	tanaka	tanaka	PROPN
ejpam-4933	397	9	.	.	PUNCT
ejpam-4933	398	1	an	an	DET
ejpam-4933	398	2	introduction	introduction	NOUN
ejpam-4933	398	3	to	to	ADP
ejpam-4933	398	4	the	the	DET
ejpam-4933	398	5	theory	theory	NOUN
ejpam-4933	398	6	of	of	ADP
ejpam-4933	398	7	bck	bck	PROPN
ejpam-4933	398	8	-	-	PUNCT
ejpam-4933	398	9	algebras	algebras	PROPN
ejpam-4933	398	10	.	.	PUNCT
ejpam-4933	399	1	math	math	PROPN
ejpam-4933	399	2	.	.	PUNCT
ejpam-4933	400	1	japon	japon	PROPN
ejpam-4933	400	2	.	.	PROPN
ejpam-4933	400	3	,	,	PUNCT
ejpam-4933	400	4	23:1–26	23:1–26	NUM
ejpam-4933	400	5	,	,	PUNCT
ejpam-4933	400	6	1978	1978	NUM
ejpam-4933	400	7	.	.	PUNCT
ejpam-4933	401	1	[	[	X
ejpam-4933	401	2	5	5	NUM
ejpam-4933	401	3	]	]	PUNCT
ejpam-4933	401	4	c.	c.	PROPN
ejpam-4933	401	5	jana	jana	PROPN
ejpam-4933	401	6	,	,	PUNCT
ejpam-4933	401	7	t.	t.	PROPN
ejpam-4933	401	8	senapati	senapati	PROPN
ejpam-4933	401	9	,	,	PUNCT
ejpam-4933	401	10	and	and	CCONJ
ejpam-4933	401	11	m.	m.	NOUN
ejpam-4933	401	12	pal	pal	NOUN
ejpam-4933	401	13	.	.	PUNCT
ejpam-4933	402	1	(	(	PUNCT
ejpam-4933	402	2	∈	∈	PROPN
ejpam-4933	402	3	,	,	PUNCT
ejpam-4933	402	4	∈	∈	PROPN
ejpam-4933	402	5	∨q)-intuitionistic	∨q)-intuitionistic	ADJ
ejpam-4933	402	6	fuzzy	fuzzy	ADJ
ejpam-4933	402	7	bci	bci	NOUN
ejpam-4933	402	8	-	-	PUNCT
ejpam-4933	402	9	subalgebras	subalgebras	PROPN
ejpam-4933	402	10	of	of	ADP
ejpam-4933	402	11	a	a	DET
ejpam-4933	402	12	bci	bci	NOUN
ejpam-4933	402	13	-	-	NOUN
ejpam-4933	402	14	algebra	algebra	NOUN
ejpam-4933	402	15	.	.	PUNCT
ejpam-4933	403	1	j.	j.	PROPN
ejpam-4933	403	2	intell	intell	PROPN
ejpam-4933	403	3	.	.	PUNCT
ejpam-4933	404	1	fuzzy	fuzzy	ADJ
ejpam-4933	404	2	systems	system	NOUN
ejpam-4933	404	3	,	,	PUNCT
ejpam-4933	404	4	31:613–621	31:613–621	NUM
ejpam-4933	404	5	,	,	PUNCT
ejpam-4933	404	6	2016	2016	NUM
ejpam-4933	404	7	.	.	PUNCT
ejpam-4933	405	1	[	[	X
ejpam-4933	405	2	6	6	NUM
ejpam-4933	405	3	]	]	X
ejpam-4933	405	4	y.	y.	PROPN
ejpam-4933	405	5	b.	b.	PROPN
ejpam-4933	405	6	jun	jun	PROPN
ejpam-4933	405	7	.	.	PROPN
ejpam-4933	406	1	a	a	DET
ejpam-4933	406	2	new	new	ADJ
ejpam-4933	406	3	form	form	NOUN
ejpam-4933	406	4	of	of	ADP
ejpam-4933	406	5	fuzzy	fuzzy	ADJ
ejpam-4933	406	6	set	set	NOUN
ejpam-4933	406	7	and	and	CCONJ
ejpam-4933	406	8	its	its	PRON
ejpam-4933	406	9	application	application	NOUN
ejpam-4933	406	10	in	in	ADP
ejpam-4933	406	11	bck	bck	PROPN
ejpam-4933	406	12	-	-	PUNCT
ejpam-4933	406	13	algebras	algebra	NOUN
ejpam-4933	406	14	and	and	CCONJ
ejpam-4933	406	15	bcialgebras	bcialgebra	NOUN
ejpam-4933	406	16	.	.	PUNCT
ejpam-4933	407	1	ann	ann	PROPN
ejpam-4933	407	2	.	.	PUNCT
ejpam-4933	407	3	fuzzy	fuzzy	ADJ
ejpam-4933	407	4	math	math	NOUN
ejpam-4933	407	5	.	.	PUNCT
ejpam-4933	408	1	inform	inform	NOUN
ejpam-4933	408	2	.	.	PUNCT
ejpam-4933	409	1	,	,	PUNCT
ejpam-4933	409	2	in	in	ADP
ejpam-4933	409	3	press	press	NOUN
ejpam-4933	409	4	.	.	PUNCT
ejpam-4933	410	1	references	reference	NOUN
ejpam-4933	410	2	2024	2024	NUM
ejpam-4933	411	1	[	[	X
ejpam-4933	411	2	7	7	NUM
ejpam-4933	411	3	]	]	X
ejpam-4933	411	4	y.	y.	PROPN
ejpam-4933	411	5	b.	b.	PROPN
ejpam-4933	411	6	jun	jun	PROPN
ejpam-4933	411	7	.	.	PROPN
ejpam-4933	411	8	fuzzy	fuzzy	ADJ
ejpam-4933	411	9	subalgebras	subalgebra	NOUN
ejpam-4933	411	10	of	of	ADP
ejpam-4933	411	11	type	type	NOUN
ejpam-4933	411	12	(	(	PUNCT
ejpam-4933	411	13	α	α	NOUN
ejpam-4933	411	14	,	,	PUNCT
ejpam-4933	411	15	β	β	NOUN
ejpam-4933	411	16	)	)	PUNCT
ejpam-4933	411	17	in	in	ADP
ejpam-4933	411	18	bck	bck	PROPN
ejpam-4933	411	19	/	/	SYM
ejpam-4933	411	20	bci	bci	NOUN
ejpam-4933	411	21	-	-	PUNCT
ejpam-4933	411	22	algebras	algebra	NOUN
ejpam-4933	411	23	.	.	PUNCT
ejpam-4933	412	1	kyungpook	kyungpook	PROPN
ejpam-4933	412	2	math	math	PROPN
ejpam-4933	412	3	.	.	PUNCT
ejpam-4933	413	1	j.	j.	PROPN
ejpam-4933	413	2	,	,	PUNCT
ejpam-4933	413	3	47:403–410	47:403–410	PROPN
ejpam-4933	413	4	,	,	PUNCT
ejpam-4933	413	5	2007	2007	NUM
ejpam-4933	413	6	.	.	PUNCT
ejpam-4933	414	1	[	[	X
ejpam-4933	414	2	8	8	NUM
ejpam-4933	414	3	]	]	X
ejpam-4933	414	4	y.	y.	PROPN
ejpam-4933	414	5	b.	b.	PROPN
ejpam-4933	414	6	jun	jun	PROPN
ejpam-4933	414	7	.	.	PROPN
ejpam-4933	414	8	lukasiewicz	lukasiewicz	ADJ
ejpam-4933	414	9	fuzzy	fuzzy	ADJ
ejpam-4933	414	10	subalgebras	subalgebras	PROPN
ejpam-4933	414	11	in	in	ADP
ejpam-4933	414	12	bck	bck	PROPN
ejpam-4933	414	13	-	-	PUNCT
ejpam-4933	414	14	algebras	algebras	PROPN
ejpam-4933	414	15	and	and	CCONJ
ejpam-4933	414	16	bci	bci	NOUN
ejpam-4933	414	17	-	-	PUNCT
ejpam-4933	414	18	algebras	algebras	PROPN
ejpam-4933	414	19	.	.	PUNCT
ejpam-4933	415	1	ann	ann	PROPN
ejpam-4933	415	2	.	.	PUNCT
ejpam-4933	415	3	fuzzy	fuzzy	ADJ
ejpam-4933	415	4	math	math	NOUN
ejpam-4933	415	5	.	.	PUNCT
ejpam-4933	416	1	inform	inform	NOUN
ejpam-4933	416	2	.	.	PUNCT
ejpam-4933	416	3	,	,	PUNCT
ejpam-4933	416	4	23(2):213–223	23(2):213–223	NOUN
ejpam-4933	416	5	,	,	PUNCT
ejpam-4933	416	6	2022	2022	NUM
ejpam-4933	416	7	.	.	PUNCT
ejpam-4933	417	1	[	[	X
ejpam-4933	417	2	9	9	NUM
ejpam-4933	417	3	]	]	X
ejpam-4933	417	4	y.	y.	PROPN
ejpam-4933	417	5	b.	b.	PROPN
ejpam-4933	417	6	jun	jun	PROPN
ejpam-4933	417	7	and	and	CCONJ
ejpam-4933	417	8	s.	s.	PROPN
ejpam-4933	417	9	z.	z.	PROPN
ejpam-4933	417	10	song	song	PROPN
ejpam-4933	417	11	.	.	PUNCT
ejpam-4933	418	1	falling	fall	VERB
ejpam-4933	418	2	fuzzy	fuzzy	ADJ
ejpam-4933	418	3	quasi	quasi	ADJ
ejpam-4933	418	4	-	-	ADJ
ejpam-4933	418	5	associative	associative	ADJ
ejpam-4933	418	6	ideals	ideal	NOUN
ejpam-4933	418	7	of	of	ADP
ejpam-4933	418	8	bci	bci	NOUN
ejpam-4933	418	9	-	-	PUNCT
ejpam-4933	418	10	algebras	algebra	NOUN
ejpam-4933	418	11	.	.	PUNCT
ejpam-4933	419	1	filomat	filomat	PROPN
ejpam-4933	419	2	,	,	PUNCT
ejpam-4933	419	3	26(4):649–656	26(4):649–656	NUM
ejpam-4933	419	4	,	,	PUNCT
ejpam-4933	419	5	2012	2012	NUM
ejpam-4933	419	6	.	.	PUNCT
ejpam-4933	420	1	[	[	X
ejpam-4933	420	2	10	10	NUM
ejpam-4933	420	3	]	]	X
ejpam-4933	420	4	y.	y.	PROPN
ejpam-4933	420	5	b.	b.	PROPN
ejpam-4933	420	6	jun	jun	PROPN
ejpam-4933	420	7	and	and	CCONJ
ejpam-4933	420	8	x.	x.	PROPN
ejpam-4933	420	9	l.	l.	PROPN
ejpam-4933	420	10	xin	xin	PROPN
ejpam-4933	420	11	.	.	PUNCT
ejpam-4933	421	1	complex	complex	ADJ
ejpam-4933	421	2	fuzzy	fuzzy	ADJ
ejpam-4933	421	3	sets	set	NOUN
ejpam-4933	421	4	with	with	ADP
ejpam-4933	421	5	application	application	NOUN
ejpam-4933	421	6	in	in	ADP
ejpam-4933	421	7	bck	bck	PROPN
ejpam-4933	421	8	/	/	SYM
ejpam-4933	421	9	bci	bci	NOUN
ejpam-4933	421	10	-	-	PUNCT
ejpam-4933	421	11	algebras	algebra	NOUN
ejpam-4933	421	12	.	.	PUNCT
ejpam-4933	422	1	bulletin	bulletin	NOUN
ejpam-4933	422	2	of	of	ADP
ejpam-4933	422	3	the	the	DET
ejpam-4933	422	4	section	section	NOUN
ejpam-4933	422	5	of	of	ADP
ejpam-4933	422	6	logic	logic	NOUN
ejpam-4933	422	7	,	,	PUNCT
ejpam-4933	422	8	48(3):173–185	48(3):173–185	ADJ
ejpam-4933	422	9	,	,	PUNCT
ejpam-4933	422	10	2019	2019	NUM
ejpam-4933	422	11	.	.	PUNCT
ejpam-4933	423	1	[	[	X
ejpam-4933	423	2	11	11	NUM
ejpam-4933	423	3	]	]	PUNCT
ejpam-4933	423	4	j.	j.	PROPN
ejpam-4933	423	5	meng	meng	PROPN
ejpam-4933	423	6	and	and	CCONJ
ejpam-4933	423	7	y.	y.	PROPN
ejpam-4933	423	8	b.	b.	PROPN
ejpam-4933	424	1	jun	jun	PROPN
ejpam-4933	424	2	.	.	PUNCT
ejpam-4933	425	1	bck	bck	PROPN
ejpam-4933	425	2	-	-	PUNCT
ejpam-4933	425	3	algebra	algebra	PROPN
ejpam-4933	425	4	.	.	PUNCT
ejpam-4933	426	1	kyungmoonsa	kyungmoonsa	PROPN
ejpam-4933	426	2	co.	co.	PROPN
ejpam-4933	426	3	,	,	PUNCT
ejpam-4933	426	4	seoul	seoul	PROPN
ejpam-4933	426	5	,	,	PUNCT
ejpam-4933	426	6	korea	korea	PROPN
ejpam-4933	426	7	,	,	PUNCT
ejpam-4933	426	8	1994	1994	NUM
ejpam-4933	426	9	.	.	PUNCT
ejpam-4933	427	1	[	[	X
ejpam-4933	427	2	12	12	NUM
ejpam-4933	427	3	]	]	X
ejpam-4933	427	4	p.	p.	NOUN
ejpam-4933	427	5	m.	m.	NOUN
ejpam-4933	427	6	pu	pu	PROPN
ejpam-4933	427	7	and	and	CCONJ
ejpam-4933	427	8	y.	y.	PROPN
ejpam-4933	427	9	m.	m.	PROPN
ejpam-4933	427	10	liu	liu	PROPN
ejpam-4933	427	11	.	.	PROPN
ejpam-4933	428	1	fuzzy	fuzzy	ADJ
ejpam-4933	428	2	topology	topology	NOUN
ejpam-4933	428	3	i	i	PRON
ejpam-4933	428	4	,	,	PUNCT
ejpam-4933	428	5	neighborhood	neighborhood	NOUN
ejpam-4933	428	6	structure	structure	NOUN
ejpam-4933	428	7	of	of	ADP
ejpam-4933	428	8	a	a	DET
ejpam-4933	428	9	fuzzy	fuzzy	ADJ
ejpam-4933	428	10	pointand	pointand	NOUN
ejpam-4933	428	11	moore	moore	PROPN
ejpam-4933	428	12	-	-	PUNCT
ejpam-4933	428	13	smith	smith	PROPN
ejpam-4933	428	14	convergence	convergence	NOUN
ejpam-4933	428	15	.	.	PUNCT
ejpam-4933	429	1	j.	j.	PROPN
ejpam-4933	429	2	math	math	PROPN
ejpam-4933	429	3	.	.	PUNCT
ejpam-4933	430	1	anal	anal	PROPN
ejpam-4933	430	2	.	.	PUNCT
ejpam-4933	431	1	appl	appl	PROPN
ejpam-4933	431	2	.	.	PROPN
ejpam-4933	431	3	,	,	PUNCT
ejpam-4933	432	1	76:571–599	76:571–599	NUM
ejpam-4933	432	2	,	,	PUNCT
ejpam-4933	432	3	1980	1980	NUM
ejpam-4933	432	4	.	.	PUNCT
ejpam-4933	433	1	[	[	X
ejpam-4933	433	2	13	13	NUM
ejpam-4933	433	3	]	]	X
ejpam-4933	433	4	o.	o.	PROPN
ejpam-4933	433	5	g.	g.	PROPN
ejpam-4933	433	6	xi	xi	PROPN
ejpam-4933	433	7	.	.	PUNCT
ejpam-4933	434	1	fuzzy	fuzzy	ADJ
ejpam-4933	434	2	bck	bck	PROPN
ejpam-4933	434	3	-	-	PUNCT
ejpam-4933	434	4	algebras	algebras	PROPN
ejpam-4933	434	5	.	.	PUNCT
ejpam-4933	435	1	math	math	PROPN
ejpam-4933	435	2	.	.	PUNCT
ejpam-4933	436	1	japon	japon	PROPN
ejpam-4933	436	2	.	.	PROPN
ejpam-4933	436	3	,	,	PUNCT
ejpam-4933	436	4	36:935–942	36:935–942	NUM
ejpam-4933	436	5	,	,	PUNCT
ejpam-4933	436	6	1991	1991	NUM
ejpam-4933	436	7	.	.	PUNCT
ejpam-4933	437	1	[	[	X
ejpam-4933	437	2	14	14	NUM
ejpam-4933	437	3	]	]	X
ejpam-4933	437	4	l.	l.	PROPN
ejpam-4933	437	5	a.	a.	PROPN
ejpam-4933	437	6	zadeh	zadeh	PROPN
ejpam-4933	437	7	.	.	PUNCT
ejpam-4933	437	8	fuzzy	fuzzy	ADJ
ejpam-4933	437	9	sets	set	NOUN
ejpam-4933	437	10	.	.	PUNCT
ejpam-4933	438	1	inform	inform	NOUN
ejpam-4933	438	2	.	.	PUNCT
ejpam-4933	439	1	control	control	NOUN
ejpam-4933	439	2	,	,	PUNCT
ejpam-4933	439	3	8(3):338–353	8(3):338–353	NUM
ejpam-4933	439	4	,	,	PUNCT
ejpam-4933	439	5	1965	1965	NUM
ejpam-4933	439	6	.	.	PUNCT