id	sid	tid	token	lemma	pos
ejpam-3971	1	1	european	european	PROPN
ejpam-3971	1	2	journal	journal	PROPN
ejpam-3971	1	3	of	of	ADP
ejpam-3971	1	4	pure	pure	ADJ
ejpam-3971	1	5	and	and	CCONJ
ejpam-3971	1	6	applied	apply	VERB
ejpam-3971	1	7	mathematics	mathematic	NOUN
ejpam-3971	1	8	vol	vol	NOUN
ejpam-3971	1	9	.	.	PUNCT
ejpam-3971	2	1	14	14	NUM
ejpam-3971	2	2	,	,	PUNCT
ejpam-3971	2	3	no	no	INTJ
ejpam-3971	2	4	.	.	NOUN
ejpam-3971	2	5	3	3	NUM
ejpam-3971	2	6	,	,	PUNCT
ejpam-3971	2	7	2021	2021	NUM
ejpam-3971	2	8	,	,	PUNCT
ejpam-3971	2	9	737	737	NUM
ejpam-3971	2	10	-	-	SYM
ejpam-3971	2	11	745	745	NUM
ejpam-3971	2	12	issn	issn	PROPN
ejpam-3971	2	13	1307	1307	NUM
ejpam-3971	2	14	-	-	SYM
ejpam-3971	2	15	5543	5543	NUM
ejpam-3971	2	16	–	–	PUNCT
ejpam-3971	3	1	ejpam.com	ejpam.com	X
ejpam-3971	3	2	published	publish	VERB
ejpam-3971	3	3	by	by	ADP
ejpam-3971	3	4	new	new	PROPN
ejpam-3971	3	5	york	york	PROPN
ejpam-3971	3	6	business	business	PROPN
ejpam-3971	3	7	global	global	ADJ
ejpam-3971	3	8	fuzzy	fuzzy	ADJ
ejpam-3971	3	9	translation	translation	NOUN
ejpam-3971	3	10	and	and	CCONJ
ejpam-3971	3	11	fuzzy	fuzzy	ADJ
ejpam-3971	3	12	multiplication	multiplication	NOUN
ejpam-3971	3	13	in	in	ADP
ejpam-3971	3	14	brk	brk	PROPN
ejpam-3971	3	15	-	-	PUNCT
ejpam-3971	3	16	algebras	algebras	PROPN
ejpam-3971	3	17	halimah	halimah	PROPN
ejpam-3971	3	18	alshehri	alshehri	PROPN
ejpam-3971	3	19	department	department	PROPN
ejpam-3971	3	20	computer	computer	NOUN
ejpam-3971	3	21	science	science	NOUN
ejpam-3971	3	22	and	and	CCONJ
ejpam-3971	3	23	engineering	engineering	NOUN
ejpam-3971	3	24	,	,	PUNCT
ejpam-3971	3	25	faculty	faculty	NOUN
ejpam-3971	3	26	applied	apply	VERB
ejpam-3971	3	27	studies	study	NOUN
ejpam-3971	3	28	and	and	CCONJ
ejpam-3971	3	29	community	community	NOUN
ejpam-3971	3	30	service	service	NOUN
ejpam-3971	3	31	,	,	PUNCT
ejpam-3971	3	32	king	king	PROPN
ejpam-3971	3	33	saud	saud	PROPN
ejpam-3971	3	34	university	university	PROPN
ejpam-3971	3	35	,	,	PUNCT
ejpam-3971	3	36	riyadh	riyadh	PROPN
ejpam-3971	3	37	,	,	PUNCT
ejpam-3971	3	38	saudi	saudi	PROPN
ejpam-3971	3	39	arabia	arabia	PROPN
ejpam-3971	3	40	abstract	abstract	NOUN
ejpam-3971	3	41	.	.	PUNCT
ejpam-3971	4	1	in	in	ADP
ejpam-3971	4	2	this	this	DET
ejpam-3971	4	3	paper	paper	NOUN
ejpam-3971	4	4	,	,	PUNCT
ejpam-3971	4	5	the	the	DET
ejpam-3971	4	6	concepts	concept	NOUN
ejpam-3971	4	7	of	of	ADP
ejpam-3971	4	8	fuzzy	fuzzy	ADJ
ejpam-3971	4	9	translation	translation	NOUN
ejpam-3971	4	10	and	and	CCONJ
ejpam-3971	4	11	fuzzy	fuzzy	ADJ
ejpam-3971	4	12	multiplication	multiplication	NOUN
ejpam-3971	4	13	on	on	ADP
ejpam-3971	4	14	a	a	DET
ejpam-3971	4	15	brkalgebra	brkalgebra	NOUN
ejpam-3971	4	16	are	be	AUX
ejpam-3971	4	17	introduced	introduce	VERB
ejpam-3971	4	18	.	.	PUNCT
ejpam-3971	5	1	we	we	PRON
ejpam-3971	5	2	investigated	investigate	VERB
ejpam-3971	5	3	fuzzy	fuzzy	ADJ
ejpam-3971	5	4	translation	translation	NOUN
ejpam-3971	5	5	and	and	CCONJ
ejpam-3971	5	6	fuzzy	fuzzy	ADJ
ejpam-3971	5	7	multiplication	multiplication	NOUN
ejpam-3971	5	8	(	(	PUNCT
ejpam-3971	5	9	brk	brk	PROPN
ejpam-3971	5	10	-	-	PUNCT
ejpam-3971	5	11	subalgebras	subalgebras	PROPN
ejpam-3971	5	12	&	&	CCONJ
ejpam-3971	5	13	brk	brk	PROPN
ejpam-3971	5	14	-	-	PUNCT
ejpam-3971	5	15	ideals	ideal	NOUN
ejpam-3971	5	16	)	)	PUNCT
ejpam-3971	5	17	in	in	ADP
ejpam-3971	5	18	brk	brk	PROPN
ejpam-3971	5	19	-	-	PUNCT
ejpam-3971	5	20	algebras	algebras	PROPN
ejpam-3971	5	21	and	and	CCONJ
ejpam-3971	5	22	discussed	discuss	VERB
ejpam-3971	5	23	related	related	ADJ
ejpam-3971	5	24	properties	property	NOUN
ejpam-3971	5	25	.	.	PUNCT
ejpam-3971	6	1	finally	finally	ADV
ejpam-3971	6	2	,	,	PUNCT
ejpam-3971	6	3	we	we	PRON
ejpam-3971	6	4	presented	present	VERB
ejpam-3971	6	5	the	the	DET
ejpam-3971	6	6	nation	nation	NOUN
ejpam-3971	6	7	fuzzy	fuzzy	ADJ
ejpam-3971	6	8	magnified	magnify	VERB
ejpam-3971	6	9	-	-	PUNCT
ejpam-3971	6	10	αβ	αβ	INTJ
ejpam-3971	6	11	-	-	PUNCT
ejpam-3971	6	12	translation	translation	NOUN
ejpam-3971	6	13	on	on	ADP
ejpam-3971	6	14	brk	brk	PROPN
ejpam-3971	6	15	-	-	PUNCT
ejpam-3971	6	16	algebra	algebra	PROPN
ejpam-3971	6	17	x.	x.	NOUN
ejpam-3971	6	18	2020	2020	NUM
ejpam-3971	6	19	mathematics	mathematics	PROPN
ejpam-3971	6	20	subject	subject	NOUN
ejpam-3971	6	21	classifications	classification	NOUN
ejpam-3971	6	22	:	:	PUNCT
ejpam-3971	6	23	08a72	08a72	NUM
ejpam-3971	6	24	,	,	PUNCT
ejpam-3971	6	25	03e72	03e72	NUM
ejpam-3971	6	26	,	,	PUNCT
ejpam-3971	6	27	03b52	03b52	VERB
ejpam-3971	6	28	key	key	ADJ
ejpam-3971	6	29	words	word	NOUN
ejpam-3971	6	30	and	and	CCONJ
ejpam-3971	6	31	phrases	phrase	NOUN
ejpam-3971	6	32	:	:	PUNCT
ejpam-3971	6	33	brk	brk	PROPN
ejpam-3971	6	34	-	-	PUNCT
ejpam-3971	6	35	algebra	algebra	PROPN
ejpam-3971	6	36	,	,	PUNCT
ejpam-3971	6	37	fuzzy	fuzzy	ADJ
ejpam-3971	6	38	translation	translation	NOUN
ejpam-3971	6	39	brk	brk	PROPN
ejpam-3971	6	40	-	-	PUNCT
ejpam-3971	6	41	subalgebras	subalgebras	PROPN
ejpam-3971	6	42	,	,	PUNCT
ejpam-3971	6	43	fuzzy	fuzzy	ADJ
ejpam-3971	6	44	multiplication	multiplication	NOUN
ejpam-3971	6	45	brk	brk	PROPN
ejpam-3971	6	46	-	-	PUNCT
ejpam-3971	6	47	subalgebras	subalgebras	PROPN
ejpam-3971	6	48	,	,	PUNCT
ejpam-3971	6	49	fuzzy	fuzzy	ADJ
ejpam-3971	6	50	translation	translation	NOUN
ejpam-3971	6	51	brk	brk	PROPN
ejpam-3971	6	52	-	-	PUNCT
ejpam-3971	6	53	ideals	ideal	NOUN
ejpam-3971	6	54	,	,	PUNCT
ejpam-3971	6	55	fuzzy	fuzzy	ADJ
ejpam-3971	6	56	multiplication	multiplication	NOUN
ejpam-3971	6	57	brk	brk	PROPN
ejpam-3971	6	58	-	-	PUNCT
ejpam-3971	6	59	ideals	ideal	NOUN
ejpam-3971	6	60	,	,	PUNCT
ejpam-3971	6	61	fuzzy	fuzzy	ADJ
ejpam-3971	6	62	magnified	magnify	VERB
ejpam-3971	6	63	-	-	PUNCT
ejpam-3971	6	64	αβ	αβ	INTJ
ejpam-3971	6	65	-	-	PUNCT
ejpam-3971	6	66	translation	translation	NOUN
ejpam-3971	7	1	1	1	NUM
ejpam-3971	7	2	.	.	PUNCT
ejpam-3971	7	3	introduction	introduction	NOUN
ejpam-3971	7	4	the	the	DET
ejpam-3971	7	5	fundamental	fundamental	ADJ
ejpam-3971	7	6	concept	concept	NOUN
ejpam-3971	7	7	of	of	ADP
ejpam-3971	7	8	fuzzy	fuzzy	ADJ
ejpam-3971	7	9	set	set	NOUN
ejpam-3971	7	10	,	,	PUNCT
ejpam-3971	7	11	popularized	popularize	VERB
ejpam-3971	7	12	by	by	ADP
ejpam-3971	7	13	zadeh	zadeh	PROPN
ejpam-3971	8	1	[	[	X
ejpam-3971	8	2	10	10	NUM
ejpam-3971	8	3	]	]	PUNCT
ejpam-3971	8	4	,	,	PUNCT
ejpam-3971	8	5	was	be	AUX
ejpam-3971	8	6	used	use	VERB
ejpam-3971	8	7	to	to	PART
ejpam-3971	8	8	generalize	generalize	VERB
ejpam-3971	8	9	several	several	ADJ
ejpam-3971	8	10	basic	basic	ADJ
ejpam-3971	8	11	concepts	concept	NOUN
ejpam-3971	8	12	of	of	ADP
ejpam-3971	8	13	algebra	algebra	NOUN
ejpam-3971	8	14	.	.	PUNCT
ejpam-3971	9	1	fuzzy	fuzzy	ADJ
ejpam-3971	9	2	sets	set	NOUN
ejpam-3971	9	3	are	be	AUX
ejpam-3971	9	4	extremely	extremely	ADV
ejpam-3971	9	5	useful	useful	ADJ
ejpam-3971	9	6	to	to	PART
ejpam-3971	9	7	deal	deal	VERB
ejpam-3971	9	8	with	with	ADP
ejpam-3971	9	9	the	the	DET
ejpam-3971	9	10	many	many	ADJ
ejpam-3971	9	11	problems	problem	NOUN
ejpam-3971	9	12	in	in	ADP
ejpam-3971	9	13	applied	applied	ADJ
ejpam-3971	9	14	mathematics	mathematic	NOUN
ejpam-3971	9	15	,	,	PUNCT
ejpam-3971	9	16	control	control	NOUN
ejpam-3971	9	17	engineering	engineering	NOUN
ejpam-3971	9	18	,	,	PUNCT
ejpam-3971	9	19	information	information	NOUN
ejpam-3971	9	20	sciences	science	NOUN
ejpam-3971	9	21	,	,	PUNCT
ejpam-3971	9	22	expert	expert	NOUN
ejpam-3971	9	23	systems	system	NOUN
ejpam-3971	9	24	etc	etc	X
ejpam-3971	9	25	.	.	X
ejpam-3971	10	1	although	although	SCONJ
ejpam-3971	10	2	there	there	PRON
ejpam-3971	10	3	are	be	VERB
ejpam-3971	10	4	several	several	ADJ
ejpam-3971	10	5	generalizations	generalization	NOUN
ejpam-3971	10	6	of	of	ADP
ejpam-3971	10	7	fuzzy	fuzzy	ADJ
ejpam-3971	10	8	sets	set	NOUN
ejpam-3971	10	9	,	,	PUNCT
ejpam-3971	10	10	none	none	NOUN
ejpam-3971	10	11	of	of	ADP
ejpam-3971	10	12	them	they	PRON
ejpam-3971	10	13	address	address	VERB
ejpam-3971	10	14	the	the	DET
ejpam-3971	10	15	issues	issue	NOUN
ejpam-3971	10	16	of	of	ADP
ejpam-3971	10	17	members	member	NOUN
ejpam-3971	10	18	with	with	ADP
ejpam-3971	10	19	membership	membership	NOUN
ejpam-3971	10	20	degree	degree	NOUN
ejpam-3971	10	21	0	0	NUM
ejpam-3971	10	22	who	who	PRON
ejpam-3971	10	23	have	have	AUX
ejpam-3971	10	24	opposing	oppose	VERB
ejpam-3971	10	25	qualities	quality	NOUN
ejpam-3971	10	26	.	.	PUNCT
ejpam-3971	11	1	lee	lee	PROPN
ejpam-3971	12	1	[	[	X
ejpam-3971	12	2	7	7	X
ejpam-3971	12	3	]	]	PUNCT
ejpam-3971	12	4	handled	handle	VERB
ejpam-3971	12	5	this	this	DET
ejpam-3971	12	6	problem	problem	NOUN
ejpam-3971	12	7	by	by	ADP
ejpam-3971	12	8	introducing	introduce	VERB
ejpam-3971	12	9	the	the	DET
ejpam-3971	12	10	concept	concept	NOUN
ejpam-3971	12	11	of	of	ADP
ejpam-3971	12	12	bipolar	bipolar	ADJ
ejpam-3971	12	13	fuzzy	fuzzy	ADJ
ejpam-3971	12	14	(	(	PUNCT
ejpam-3971	12	15	bf	bf	NOUN
ejpam-3971	12	16	)	)	PUNCT
ejpam-3971	12	17	sets	set	NOUN
ejpam-3971	12	18	.	.	PUNCT
ejpam-3971	13	1	a	a	DET
ejpam-3971	13	2	bf	bf	NOUN
ejpam-3971	13	3	set	set	NOUN
ejpam-3971	13	4	is	be	AUX
ejpam-3971	13	5	a	a	DET
ejpam-3971	13	6	pair	pair	NOUN
ejpam-3971	13	7	of	of	ADP
ejpam-3971	13	8	fuzzy	fuzzy	ADJ
ejpam-3971	13	9	sets	set	NOUN
ejpam-3971	13	10	,	,	PUNCT
ejpam-3971	13	11	namely	namely	ADV
ejpam-3971	13	12	a	a	DET
ejpam-3971	13	13	membership	membership	NOUN
ejpam-3971	13	14	and	and	CCONJ
ejpam-3971	13	15	a	a	DET
ejpam-3971	13	16	non	non	ADJ
ejpam-3971	13	17	-	-	ADJ
ejpam-3971	13	18	membership	membership	ADJ
ejpam-3971	13	19	function	function	NOUN
ejpam-3971	13	20	,	,	PUNCT
ejpam-3971	13	21	which	which	PRON
ejpam-3971	13	22	represent	represent	VERB
ejpam-3971	13	23	positive	positive	ADJ
ejpam-3971	13	24	and	and	CCONJ
ejpam-3971	13	25	negative	negative	ADJ
ejpam-3971	13	26	aspects	aspect	NOUN
ejpam-3971	13	27	of	of	ADP
ejpam-3971	13	28	the	the	DET
ejpam-3971	13	29	given	give	VERB
ejpam-3971	13	30	information	information	NOUN
ejpam-3971	13	31	.	.	PUNCT
ejpam-3971	14	1	imai	imai	PROPN
ejpam-3971	14	2	and	and	CCONJ
ejpam-3971	14	3	iseki	iseki	PROPN
ejpam-3971	14	4	investigated	investigate	VERB
ejpam-3971	14	5	two	two	NUM
ejpam-3971	14	6	classes	class	NOUN
ejpam-3971	14	7	of	of	ADP
ejpam-3971	14	8	abstract	abstract	ADJ
ejpam-3971	14	9	algebras	algebra	NOUN
ejpam-3971	14	10	:	:	PUNCT
ejpam-3971	14	11	bci	bci	NOUN
ejpam-3971	14	12	-	-	PUNCT
ejpam-3971	14	13	algebras	algebra	NOUN
ejpam-3971	14	14	and	and	CCONJ
ejpam-3971	14	15	bck	bck	NOUN
ejpam-3971	14	16	-	-	PUNCT
ejpam-3971	14	17	algebras	algebras	NOUN
ejpam-3971	15	1	[	[	X
ejpam-3971	15	2	5	5	NUM
ejpam-3971	15	3	]	]	PUNCT
ejpam-3971	15	4	.	.	PUNCT
ejpam-3971	16	1	recently	recently	ADV
ejpam-3971	16	2	,	,	PUNCT
ejpam-3971	16	3	bandaru	bandaru	PROPN
ejpam-3971	17	1	[	[	X
ejpam-3971	17	2	1	1	NUM
ejpam-3971	17	3	]	]	PUNCT
ejpam-3971	17	4	investigated	investigate	VERB
ejpam-3971	17	5	brk	brk	PROPN
ejpam-3971	17	6	-	-	PUNCT
ejpam-3971	17	7	algebra	algebra	PROPN
ejpam-3971	17	8	which	which	PRON
ejpam-3971	17	9	is	be	AUX
ejpam-3971	17	10	a	a	DET
ejpam-3971	17	11	generalization	generalization	NOUN
ejpam-3971	17	12	of	of	ADP
ejpam-3971	17	13	bck	bck	PROPN
ejpam-3971	17	14	/	/	SYM
ejpam-3971	17	15	bci	bci	PROPN
ejpam-3971	17	16	/	/	SYM
ejpam-3971	17	17	bch	bch	PROPN
ejpam-3971	17	18	/	/	SYM
ejpam-3971	17	19	q	q	NOUN
ejpam-3971	17	20	/	/	SYM
ejpam-3971	17	21	qs	qs	NOUN
ejpam-3971	17	22	/	/	SYM
ejpam-3971	17	23	bm	bm	NOUN
ejpam-3971	17	24	-	-	NOUN
ejpam-3971	17	25	algebras	algebras	X
ejpam-3971	17	26	.	.	PUNCT
ejpam-3971	18	1	in	in	ADP
ejpam-3971	18	2	[	[	X
ejpam-3971	18	3	2,3	2,3	NUM
ejpam-3971	18	4	]	]	PUNCT
ejpam-3971	18	5	,	,	PUNCT
ejpam-3971	18	6	elgendy	elgendy	PROPN
ejpam-3971	18	7	introduced	introduce	VERB
ejpam-3971	18	8	fuzzy	fuzzy	ADJ
ejpam-3971	18	9	brk	brk	PROPN
ejpam-3971	18	10	-	-	PUNCT
ejpam-3971	18	11	ideal	ideal	NOUN
ejpam-3971	18	12	of	of	ADP
ejpam-3971	18	13	brk	brk	PROPN
ejpam-3971	18	14	-	-	PUNCT
ejpam-3971	18	15	algebra	algebra	PROPN
ejpam-3971	18	16	and	and	CCONJ
ejpam-3971	18	17	cubic	cubic	ADJ
ejpam-3971	18	18	brk	brk	PROPN
ejpam-3971	18	19	-	-	PUNCT
ejpam-3971	18	20	ideal	ideal	NOUN
ejpam-3971	18	21	of	of	ADP
ejpam-3971	18	22	brkalgebra	brkalgebra	NOUN
ejpam-3971	18	23	.	.	PUNCT
ejpam-3971	19	1	some	some	DET
ejpam-3971	19	2	properties	property	NOUN
ejpam-3971	19	3	of	of	ADP
ejpam-3971	19	4	n	n	ADV
ejpam-3971	19	5	-	-	PUNCT
ejpam-3971	19	6	dimensional	dimensional	ADJ
ejpam-3971	19	7	fuzzy	fuzzy	ADJ
ejpam-3971	19	8	subalgebra	subalgebra	NOUN
ejpam-3971	19	9	in	in	ADP
ejpam-3971	19	10	brk	brk	PROPN
ejpam-3971	19	11	-	-	PUNCT
ejpam-3971	19	12	algebras	algebras	PROPN
ejpam-3971	19	13	investigated	investigate	VERB
ejpam-3971	19	14	by	by	ADP
ejpam-3971	19	15	zulfiqar	zulfiqar	NOUN
ejpam-3971	19	16	[	[	X
ejpam-3971	19	17	11	11	NUM
ejpam-3971	19	18	]	]	PUNCT
ejpam-3971	19	19	.	.	PUNCT
ejpam-3971	20	1	fuzzy	fuzzy	ADJ
ejpam-3971	20	2	translations	translation	NOUN
ejpam-3971	20	3	and	and	CCONJ
ejpam-3971	20	4	fuzzy	fuzzy	ADJ
ejpam-3971	20	5	multiplications	multiplication	NOUN
ejpam-3971	20	6	of	of	ADP
ejpam-3971	20	7	bck	bck	PROPN
ejpam-3971	20	8	/	/	SYM
ejpam-3971	20	9	bci	bci	NOUN
ejpam-3971	20	10	-	-	PUNCT
ejpam-3971	20	11	algebras	algebras	PROPN
ejpam-3971	20	12	presented	present	VERB
ejpam-3971	20	13	in	in	ADP
ejpam-3971	20	14	[	[	X
ejpam-3971	20	15	8	8	NUM
ejpam-3971	20	16	]	]	PUNCT
ejpam-3971	20	17	.	.	PUNCT
ejpam-3971	21	1	the	the	DET
ejpam-3971	21	2	contents	content	NOUN
ejpam-3971	21	3	of	of	ADP
ejpam-3971	21	4	the	the	DET
ejpam-3971	21	5	current	current	ADJ
ejpam-3971	21	6	paper	paper	NOUN
ejpam-3971	21	7	are	be	AUX
ejpam-3971	21	8	structured	structure	VERB
ejpam-3971	21	9	as	as	SCONJ
ejpam-3971	21	10	follows	follow	VERB
ejpam-3971	21	11	:	:	PUNCT
ejpam-3971	21	12	in	in	ADP
ejpam-3971	21	13	sect	sect	NOUN
ejpam-3971	21	14	.	.	PUNCT
ejpam-3971	22	1	2	2	NUM
ejpam-3971	22	2	,	,	PUNCT
ejpam-3971	22	3	we	we	PRON
ejpam-3971	22	4	presented	present	VERB
ejpam-3971	22	5	some	some	DET
ejpam-3971	22	6	basic	basic	ADJ
ejpam-3971	22	7	definitions	definition	NOUN
ejpam-3971	22	8	and	and	CCONJ
ejpam-3971	22	9	preliminaries	preliminary	NOUN
ejpam-3971	22	10	.	.	PUNCT
ejpam-3971	23	1	in	in	ADP
ejpam-3971	23	2	sect	sect	NOUN
ejpam-3971	23	3	.	.	PUNCT
ejpam-3971	24	1	3	3	X
ejpam-3971	24	2	,	,	PUNCT
ejpam-3971	24	3	we	we	PRON
ejpam-3971	24	4	investigated	investigate	VERB
ejpam-3971	24	5	fuzzy	fuzzy	ADJ
ejpam-3971	24	6	doi	doi	NOUN
ejpam-3971	24	7	:	:	PUNCT
ejpam-3971	24	8	https://doi.org/10.29020/nybg.ejpam.v14i3.3971	https://doi.org/10.29020/nybg.ejpam.v14i3.3971	PROPN
ejpam-3971	24	9	email	email	NOUN
ejpam-3971	24	10	address	address	NOUN
ejpam-3971	24	11	:	:	PUNCT
ejpam-3971	24	12	haalshehri@ksu.edu.sa	haalshehri@ksu.edu.sa	PROPN
ejpam-3971	24	13	(	(	PUNCT
ejpam-3971	24	14	h.	h.	PROPN
ejpam-3971	24	15	alshehri	alshehri	PROPN
ejpam-3971	24	16	)	)	PUNCT
ejpam-3971	24	17	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3971	25	1	737	737	NUM
ejpam-3971	25	2	©	©	NOUN
ejpam-3971	25	3	2021	2021	NUM
ejpam-3971	25	4	ejpam	ejpam	VERB
ejpam-3971	25	5	all	all	DET
ejpam-3971	25	6	rights	right	NOUN
ejpam-3971	25	7	reserved	reserve	VERB
ejpam-3971	25	8	.	.	PUNCT
ejpam-3971	26	1	h.	h.	PROPN
ejpam-3971	26	2	alshehri	alshehri	PROPN
ejpam-3971	26	3	/	/	SYM
ejpam-3971	26	4	eur	eur	PROPN
ejpam-3971	26	5	.	.	PUNCT
ejpam-3971	27	1	j.	j.	PROPN
ejpam-3971	27	2	pure	pure	PROPN
ejpam-3971	27	3	appl	appl	PROPN
ejpam-3971	27	4	.	.	PROPN
ejpam-3971	27	5	math	math	PROPN
ejpam-3971	27	6	,	,	PUNCT
ejpam-3971	27	7	14	14	NUM
ejpam-3971	27	8	(	(	PUNCT
ejpam-3971	27	9	3	3	NUM
ejpam-3971	27	10	)	)	PUNCT
ejpam-3971	27	11	(	(	PUNCT
ejpam-3971	27	12	2021	2021	NUM
ejpam-3971	27	13	)	)	PUNCT
ejpam-3971	27	14	,	,	PUNCT
ejpam-3971	27	15	737	737	NUM
ejpam-3971	27	16	-	-	SYM
ejpam-3971	27	17	745	745	NUM
ejpam-3971	27	18	738	738	NUM
ejpam-3971	27	19	translation	translation	NOUN
ejpam-3971	27	20	and	and	CCONJ
ejpam-3971	27	21	fuzzy	fuzzy	ADJ
ejpam-3971	27	22	multiplication	multiplication	NOUN
ejpam-3971	27	23	of	of	ADP
ejpam-3971	27	24	brk	brk	PROPN
ejpam-3971	27	25	-	-	PUNCT
ejpam-3971	27	26	subalgebras	subalgebras	PROPN
ejpam-3971	27	27	and	and	CCONJ
ejpam-3971	27	28	discussed	discuss	VERB
ejpam-3971	27	29	related	related	ADJ
ejpam-3971	27	30	properties	property	NOUN
ejpam-3971	27	31	.	.	PUNCT
ejpam-3971	28	1	in	in	ADP
ejpam-3971	28	2	sect	sect	NOUN
ejpam-3971	28	3	.	.	PUNCT
ejpam-3971	29	1	4	4	X
ejpam-3971	29	2	,	,	PUNCT
ejpam-3971	29	3	we	we	PRON
ejpam-3971	29	4	introduced	introduce	VERB
ejpam-3971	29	5	fuzzy	fuzzy	ADJ
ejpam-3971	29	6	translation	translation	NOUN
ejpam-3971	29	7	and	and	CCONJ
ejpam-3971	29	8	fuzzy	fuzzy	ADJ
ejpam-3971	29	9	multiplication	multiplication	NOUN
ejpam-3971	29	10	of	of	ADP
ejpam-3971	29	11	brk	brk	PROPN
ejpam-3971	29	12	-	-	PUNCT
ejpam-3971	29	13	ideals	ideal	NOUN
ejpam-3971	29	14	and	and	CCONJ
ejpam-3971	29	15	discussed	discuss	VERB
ejpam-3971	29	16	related	related	ADJ
ejpam-3971	29	17	results	result	NOUN
ejpam-3971	29	18	.	.	PUNCT
ejpam-3971	30	1	in	in	ADP
ejpam-3971	30	2	sect	sect	NOUN
ejpam-3971	30	3	.	.	PUNCT
ejpam-3971	31	1	5	5	NUM
ejpam-3971	31	2	,	,	PUNCT
ejpam-3971	31	3	we	we	PRON
ejpam-3971	31	4	defined	define	VERB
ejpam-3971	31	5	concept	concept	NOUN
ejpam-3971	31	6	of	of	ADP
ejpam-3971	31	7	fuzzy	fuzzy	ADJ
ejpam-3971	31	8	magnified	magnify	VERB
ejpam-3971	31	9	-	-	PUNCT
ejpam-3971	31	10	αβ	αβ	INTJ
ejpam-3971	31	11	-	-	PUNCT
ejpam-3971	31	12	translation	translation	NOUN
ejpam-3971	31	13	of	of	ADP
ejpam-3971	31	14	brk	brk	PROPN
ejpam-3971	31	15	-	-	PUNCT
ejpam-3971	31	16	algebras	algebras	PROPN
ejpam-3971	31	17	.	.	PUNCT
ejpam-3971	32	1	at	at	ADP
ejpam-3971	32	2	last	last	ADV
ejpam-3971	32	3	,	,	PUNCT
ejpam-3971	32	4	some	some	DET
ejpam-3971	32	5	conclusions	conclusion	NOUN
ejpam-3971	32	6	and	and	CCONJ
ejpam-3971	32	7	future	future	ADJ
ejpam-3971	32	8	work	work	NOUN
ejpam-3971	32	9	were	be	AUX
ejpam-3971	32	10	presented	present	VERB
ejpam-3971	32	11	.	.	PUNCT
ejpam-3971	33	1	2	2	X
ejpam-3971	33	2	.	.	X
ejpam-3971	33	3	preliminaries	preliminary	NOUN
ejpam-3971	33	4	some	some	DET
ejpam-3971	33	5	elementary	elementary	ADJ
ejpam-3971	33	6	aspects	aspect	NOUN
ejpam-3971	33	7	that	that	PRON
ejpam-3971	33	8	are	be	AUX
ejpam-3971	33	9	important	important	ADJ
ejpam-3971	33	10	for	for	ADP
ejpam-3971	33	11	this	this	DET
ejpam-3971	33	12	paper	paper	NOUN
ejpam-3971	33	13	are	be	AUX
ejpam-3971	33	14	included	include	VERB
ejpam-3971	33	15	in	in	ADP
ejpam-3971	33	16	this	this	DET
ejpam-3971	33	17	section	section	NOUN
ejpam-3971	33	18	.	.	PUNCT
ejpam-3971	34	1	definition	definition	NOUN
ejpam-3971	34	2	1	1	NUM
ejpam-3971	34	3	(	(	PUNCT
ejpam-3971	34	4	1	1	NUM
ejpam-3971	34	5	)	)	PUNCT
ejpam-3971	34	6	.	.	PUNCT
ejpam-3971	35	1	a	a	DET
ejpam-3971	35	2	brk	brk	PROPN
ejpam-3971	35	3	-	-	PUNCT
ejpam-3971	35	4	algebra	algebra	PROPN
ejpam-3971	35	5	is	be	AUX
ejpam-3971	35	6	a	a	DET
ejpam-3971	35	7	non	non	ADJ
ejpam-3971	35	8	-	-	ADJ
ejpam-3971	35	9	empty	empty	ADJ
ejpam-3971	35	10	set	set	NOUN
ejpam-3971	35	11	x	x	PUNCT
ejpam-3971	35	12	with	with	ADP
ejpam-3971	35	13	a	a	DET
ejpam-3971	35	14	constant	constant	ADJ
ejpam-3971	35	15	0	0	NUM
ejpam-3971	35	16	and	and	CCONJ
ejpam-3971	35	17	a	a	DET
ejpam-3971	35	18	binary	binary	ADJ
ejpam-3971	35	19	operation	operation	NOUN
ejpam-3971	35	20	“	"	PUNCT
ejpam-3971	35	21	∗”satisfyingthefollowingconditions	∗”satisfyingthefollowingcondition	NOUN
ejpam-3971	35	22	:	:	PUNCT
ejpam-3971	35	23	(	(	PUNCT
ejpam-3971	35	24	brk1	brk1	PROPN
ejpam-3971	35	25	)	)	PUNCT
ejpam-3971	35	26	x	x	X
ejpam-3971	36	1	∗0	∗0	X
ejpam-3971	36	2	=	=	PUNCT
ejpam-3971	36	3	x	x	PROPN
ejpam-3971	36	4	,	,	PUNCT
ejpam-3971	36	5	(	(	PUNCT
ejpam-3971	36	6	brk2	brk2	PROPN
ejpam-3971	36	7	)	)	PUNCT
ejpam-3971	36	8	(	(	PUNCT
ejpam-3971	36	9	x	x	SYM
ejpam-3971	36	10	∗	∗	PROPN
ejpam-3971	36	11	y	y	NOUN
ejpam-3971	36	12	)	)	PUNCT
ejpam-3971	36	13	∗	∗	NOUN
ejpam-3971	36	14	x	x	PUNCT
ejpam-3971	37	1	=	=	SYM
ejpam-3971	37	2	0	0	NUM
ejpam-3971	37	3	∗	∗	NOUN
ejpam-3971	37	4	y	y	PROPN
ejpam-3971	37	5	,	,	PUNCT
ejpam-3971	37	6	for	for	ADP
ejpam-3971	37	7	all	all	DET
ejpam-3971	37	8	x	x	NOUN
ejpam-3971	37	9	,	,	PUNCT
ejpam-3971	37	10	y	y	PROPN
ejpam-3971	37	11	∈	∈	PROPN
ejpam-3971	37	12	x.	x.	NOUN
ejpam-3971	37	13	a	a	DET
ejpam-3971	37	14	partial	partial	ADJ
ejpam-3971	37	15	ordered	order	VERB
ejpam-3971	37	16	relation	relation	NOUN
ejpam-3971	37	17	≤	≤	NUM
ejpam-3971	37	18	canbedefinedbyx≤	canbedefinedbyx≤	NOUN
ejpam-3971	37	19	y	y	PROPN
ejpam-3971	38	1	if	if	SCONJ
ejpam-3971	38	2	and	and	CCONJ
ejpam-3971	38	3	only	only	ADV
ejpam-3971	38	4	if	if	SCONJ
ejpam-3971	38	5	x	x	X
ejpam-3971	38	6	∗	∗	VERB
ejpam-3971	38	7	y	y	NOUN
ejpam-3971	38	8	=	=	SYM
ejpam-3971	38	9	0	0	PROPN
ejpam-3971	38	10	.	.	PUNCT
ejpam-3971	39	1	throughout	throughout	ADP
ejpam-3971	39	2	this	this	DET
ejpam-3971	39	3	paper	paper	NOUN
ejpam-3971	39	4	,	,	PUNCT
ejpam-3971	39	5	x	x	PROPN
ejpam-3971	39	6	denotes	denotes	PROPN
ejpam-3971	39	7	brk	brk	PROPN
ejpam-3971	39	8	-	-	PUNCT
ejpam-3971	39	9	algebra	algebra	PROPN
ejpam-3971	39	10	.	.	PUNCT
ejpam-3971	40	1	definition	definition	NOUN
ejpam-3971	40	2	2	2	NUM
ejpam-3971	40	3	(	(	PUNCT
ejpam-3971	40	4	1	1	NUM
ejpam-3971	40	5	)	)	PUNCT
ejpam-3971	40	6	.	.	PUNCT
ejpam-3971	41	1	if	if	SCONJ
ejpam-3971	41	2	(	(	PUNCT
ejpam-3971	41	3	x	x	X
ejpam-3971	41	4	,	,	PUNCT
ejpam-3971	41	5	∗	∗	NOUN
ejpam-3971	41	6	,	,	PUNCT
ejpam-3971	41	7	0	0	NUM
ejpam-3971	41	8	)	)	PUNCT
ejpam-3971	41	9	is	be	AUX
ejpam-3971	41	10	a	a	DET
ejpam-3971	41	11	brk	brk	PROPN
ejpam-3971	41	12	-	-	PUNCT
ejpam-3971	41	13	algebra	algebra	PROPN
ejpam-3971	41	14	,	,	PUNCT
ejpam-3971	41	15	the	the	DET
ejpam-3971	41	16	following	follow	VERB
ejpam-3971	41	17	conditions	condition	NOUN
ejpam-3971	41	18	hold	hold	VERB
ejpam-3971	41	19	:	:	PUNCT
ejpam-3971	41	20	(	(	PUNCT
ejpam-3971	41	21	brk3	brk3	PROPN
ejpam-3971	41	22	)	)	PUNCT
ejpam-3971	41	23	x	x	SYM
ejpam-3971	42	1	∗	∗	NOUN
ejpam-3971	42	2	x	x	SYM
ejpam-3971	42	3	=	=	SYM
ejpam-3971	42	4	0	0	NUM
ejpam-3971	42	5	,	,	PUNCT
ejpam-3971	42	6	(	(	PUNCT
ejpam-3971	42	7	brk4	brk4	NOUN
ejpam-3971	42	8	)	)	PUNCT
ejpam-3971	42	9	(	(	PUNCT
ejpam-3971	42	10	x	x	SYM
ejpam-3971	42	11	∗	∗	NUM
ejpam-3971	42	12	y	y	NOUN
ejpam-3971	42	13	)	)	PUNCT
ejpam-3971	42	14	=	=	SYM
ejpam-3971	42	15	0	0	NUM
ejpam-3971	42	16	implies	imply	VERB
ejpam-3971	42	17	0	0	NUM
ejpam-3971	42	18	∗	∗	NOUN
ejpam-3971	42	19	x	x	X
ejpam-3971	42	20	=	=	SYM
ejpam-3971	42	21	0	0	NUM
ejpam-3971	42	22	∗	∗	NOUN
ejpam-3971	42	23	y	y	PROPN
ejpam-3971	42	24	for	for	ADP
ejpam-3971	42	25	all	all	DET
ejpam-3971	42	26	x	x	NOUN
ejpam-3971	42	27	,	,	PUNCT
ejpam-3971	42	28	y	y	PROPN
ejpam-3971	42	29	∈	∈	PROPN
ejpam-3971	42	30	x	x	X
ejpam-3971	42	31	,	,	PUNCT
ejpam-3971	42	32	(	(	PUNCT
ejpam-3971	42	33	brk5	brk5	PROPN
ejpam-3971	42	34	)	)	PUNCT
ejpam-3971	42	35	0	0	NUM
ejpam-3971	43	1	∗	∗	NOUN
ejpam-3971	43	2	(	(	PUNCT
ejpam-3971	43	3	x	x	X
ejpam-3971	43	4	∗	∗	PROPN
ejpam-3971	43	5	y	y	NOUN
ejpam-3971	43	6	)	)	PUNCT
ejpam-3971	43	7	=	=	SYM
ejpam-3971	43	8	(	(	PUNCT
ejpam-3971	43	9	0	0	NUM
ejpam-3971	43	10	∗	∗	NOUN
ejpam-3971	43	11	x	x	NOUN
ejpam-3971	43	12	)	)	PUNCT
ejpam-3971	43	13	∗	∗	NOUN
ejpam-3971	43	14	(	(	PUNCT
ejpam-3971	43	15	0	0	NUM
ejpam-3971	43	16	∗	∗	PROPN
ejpam-3971	43	17	y	y	PROPN
ejpam-3971	43	18	)	)	PUNCT
ejpam-3971	43	19	for	for	ADP
ejpam-3971	43	20	all	all	DET
ejpam-3971	43	21	x	x	NOUN
ejpam-3971	43	22	,	,	PUNCT
ejpam-3971	43	23	y	y	PROPN
ejpam-3971	43	24	∈	∈	PROPN
ejpam-3971	43	25	x	x	X
ejpam-3971	43	26	,	,	PUNCT
ejpam-3971	43	27	.	.	PUNCT
ejpam-3971	44	1	definition	definition	NOUN
ejpam-3971	44	2	3	3	NUM
ejpam-3971	44	3	(	(	PUNCT
ejpam-3971	44	4	1	1	NUM
ejpam-3971	44	5	)	)	PUNCT
ejpam-3971	44	6	.	.	PUNCT
ejpam-3971	45	1	a	a	DET
ejpam-3971	45	2	subset	subset	NOUN
ejpam-3971	45	3	s	s	NOUN
ejpam-3971	45	4	of	of	ADP
ejpam-3971	45	5	a	a	DET
ejpam-3971	45	6	brk	brk	PROPN
ejpam-3971	45	7	-	-	PUNCT
ejpam-3971	45	8	algebra	algebra	PROPN
ejpam-3971	45	9	x	x	PUNCT
ejpam-3971	45	10	is	be	AUX
ejpam-3971	45	11	said	say	VERB
ejpam-3971	45	12	to	to	PART
ejpam-3971	45	13	be	be	AUX
ejpam-3971	45	14	brk	brk	NOUN
ejpam-3971	45	15	-	-	PUNCT
ejpam-3971	45	16	subalgebra	subalgebra	NOUN
ejpam-3971	45	17	of	of	ADP
ejpam-3971	45	18	x	x	SYM
ejpam-3971	45	19	,	,	PUNCT
ejpam-3971	45	20	if	if	SCONJ
ejpam-3971	45	21	x	x	NOUN
ejpam-3971	45	22	,	,	PUNCT
ejpam-3971	45	23	y	y	PROPN
ejpam-3971	45	24	∈	∈	PROPN
ejpam-3971	45	25	s	s	PROPN
ejpam-3971	45	26	,	,	PUNCT
ejpam-3971	45	27	implies	imply	VERB
ejpam-3971	45	28	x	x	X
ejpam-3971	45	29	∗	∗	NOUN
ejpam-3971	45	30	y	y	PROPN
ejpam-3971	45	31	∈	∈	PROPN
ejpam-3971	45	32	s.	s.	PROPN
ejpam-3971	45	33	definition	definition	NOUN
ejpam-3971	45	34	4	4	NUM
ejpam-3971	45	35	(	(	PUNCT
ejpam-3971	45	36	1	1	NUM
ejpam-3971	45	37	)	)	PUNCT
ejpam-3971	45	38	.	.	PUNCT
ejpam-3971	46	1	a	a	DET
ejpam-3971	46	2	non	non	ADJ
ejpam-3971	46	3	-	-	ADJ
ejpam-3971	46	4	empty	empty	ADJ
ejpam-3971	46	5	subset	subset	NOUN
ejpam-3971	46	6	i	i	PRON
ejpam-3971	46	7	of	of	ADP
ejpam-3971	46	8	a	a	DET
ejpam-3971	46	9	brk	brk	PROPN
ejpam-3971	46	10	-	-	PUNCT
ejpam-3971	46	11	algebra	algebra	PROPN
ejpam-3971	46	12	x	x	PUNCT
ejpam-3971	46	13	is	be	AUX
ejpam-3971	46	14	said	say	VERB
ejpam-3971	46	15	to	to	PART
ejpam-3971	46	16	be	be	AUX
ejpam-3971	46	17	a	a	DET
ejpam-3971	46	18	brk	brk	PROPN
ejpam-3971	46	19	-	-	PUNCT
ejpam-3971	46	20	ideal	ideal	NOUN
ejpam-3971	46	21	of	of	ADP
ejpam-3971	46	22	x	x	PRON
ejpam-3971	46	23	if	if	SCONJ
ejpam-3971	46	24	it	it	PRON
ejpam-3971	46	25	satisfies	satisfy	VERB
ejpam-3971	46	26	:	:	PUNCT
ejpam-3971	46	27	(	(	PUNCT
ejpam-3971	46	28	i1	i1	PROPN
ejpam-3971	46	29	)	)	PUNCT
ejpam-3971	46	30	0	0	PUNCT
ejpam-3971	47	1	∈	∈	PROPN
ejpam-3971	47	2	i	i	PRON
ejpam-3971	47	3	,	,	PUNCT
ejpam-3971	47	4	(	(	PUNCT
ejpam-3971	47	5	i2	i2	PROPN
ejpam-3971	47	6	)	)	PUNCT
ejpam-3971	47	7	0	0	NUM
ejpam-3971	48	1	∗	∗	NOUN
ejpam-3971	48	2	(	(	PUNCT
ejpam-3971	48	3	x	x	X
ejpam-3971	48	4	∗	∗	PROPN
ejpam-3971	48	5	y	y	NOUN
ejpam-3971	48	6	)	)	PUNCT
ejpam-3971	48	7	∈	∈	PROPN
ejpam-3971	49	1	i	i	PRON
ejpam-3971	49	2	and	and	CCONJ
ejpam-3971	49	3	0	0	NUM
ejpam-3971	49	4	∗	∗	NOUN
ejpam-3971	49	5	y	y	PROPN
ejpam-3971	49	6	∈	∈	PROPN
ejpam-3971	50	1	i	i	PRON
ejpam-3971	50	2	imply	imply	VERB
ejpam-3971	50	3	0	0	NUM
ejpam-3971	50	4	∗	∗	NOUN
ejpam-3971	50	5	x	x	PUNCT
ejpam-3971	50	6	∈	∈	PROPN
ejpam-3971	50	7	i	i	PRON
ejpam-3971	50	8	,	,	PUNCT
ejpam-3971	50	9	for	for	ADP
ejpam-3971	50	10	all	all	DET
ejpam-3971	50	11	x	x	NOUN
ejpam-3971	50	12	,	,	PUNCT
ejpam-3971	50	13	y	y	PROPN
ejpam-3971	50	14	∈	∈	PROPN
ejpam-3971	50	15	x.	x.	NOUN
ejpam-3971	50	16	definition	definition	NOUN
ejpam-3971	50	17	5	5	NUM
ejpam-3971	50	18	(	(	PUNCT
ejpam-3971	50	19	10	10	NUM
ejpam-3971	50	20	)	)	PUNCT
ejpam-3971	50	21	.	.	PUNCT
ejpam-3971	51	1	a	a	DET
ejpam-3971	51	2	fuzzy	fuzzy	ADJ
ejpam-3971	51	3	subset	subset	VERB
ejpam-3971	51	4	µ	µ	X
ejpam-3971	51	5	in	in	ADP
ejpam-3971	51	6	a	a	DET
ejpam-3971	51	7	non	non	ADJ
ejpam-3971	51	8	-	-	ADJ
ejpam-3971	51	9	empty	empty	ADJ
ejpam-3971	51	10	set	set	NOUN
ejpam-3971	51	11	x	x	PUNCT
ejpam-3971	51	12	is	be	AUX
ejpam-3971	51	13	a	a	DET
ejpam-3971	51	14	function	function	NOUN
ejpam-3971	51	15	µ	µ	NOUN
ejpam-3971	51	16	:	:	PUNCT
ejpam-3971	51	17	x	x	PUNCT
ejpam-3971	51	18	−→	−→	NOUN
ejpam-3971	51	19	[	[	X
ejpam-3971	51	20	0	0	NUM
ejpam-3971	51	21	,	,	PUNCT
ejpam-3971	51	22	1	1	NUM
ejpam-3971	51	23	]	]	PUNCT
ejpam-3971	51	24	.	.	PUNCT
ejpam-3971	52	1	definition	definition	NOUN
ejpam-3971	52	2	6	6	NUM
ejpam-3971	52	3	(	(	PUNCT
ejpam-3971	52	4	2	2	NUM
ejpam-3971	52	5	)	)	PUNCT
ejpam-3971	52	6	.	.	PUNCT
ejpam-3971	53	1	a	a	DET
ejpam-3971	53	2	fuzzy	fuzzy	ADJ
ejpam-3971	53	3	subset	subset	VERB
ejpam-3971	53	4	µ	µ	X
ejpam-3971	53	5	in	in	ADP
ejpam-3971	53	6	a	a	DET
ejpam-3971	53	7	brk	brk	PROPN
ejpam-3971	53	8	-	-	PUNCT
ejpam-3971	53	9	algebra	algebra	PROPN
ejpam-3971	53	10	x	x	PUNCT
ejpam-3971	53	11	is	be	AUX
ejpam-3971	53	12	said	say	VERB
ejpam-3971	53	13	to	to	PART
ejpam-3971	53	14	be	be	AUX
ejpam-3971	53	15	a	a	DET
ejpam-3971	53	16	fuzzy	fuzzy	ADJ
ejpam-3971	53	17	brk	brk	PROPN
ejpam-3971	53	18	-subalgebra	-subalgebra	PROPN
ejpam-3971	53	19	of	of	ADP
ejpam-3971	53	20	x	x	PRON
ejpam-3971	53	21	if	if	SCONJ
ejpam-3971	53	22	µ(x	µ(x	VERB
ejpam-3971	53	23	∗	∗	NOUN
ejpam-3971	53	24	y	y	NOUN
ejpam-3971	53	25	)	)	PUNCT
ejpam-3971	53	26	≥	≥	NOUN
ejpam-3971	53	27	min{µ(x	min{µ(x	NOUN
ejpam-3971	53	28	)	)	PUNCT
ejpam-3971	53	29	,	,	PUNCT
ejpam-3971	53	30	µ(y)}∀x	µ(y)}∀x	PROPN
ejpam-3971	53	31	,	,	PUNCT
ejpam-3971	53	32	y	y	PROPN
ejpam-3971	53	33	∈	∈	PROPN
ejpam-3971	53	34	x.	x.	NOUN
ejpam-3971	53	35	definition	definition	NOUN
ejpam-3971	53	36	7	7	NUM
ejpam-3971	53	37	(	(	PUNCT
ejpam-3971	53	38	2	2	NUM
ejpam-3971	53	39	)	)	PUNCT
ejpam-3971	53	40	.	.	PUNCT
ejpam-3971	54	1	let	let	VERB
ejpam-3971	54	2	(	(	PUNCT
ejpam-3971	54	3	x	x	X
ejpam-3971	54	4	,	,	PUNCT
ejpam-3971	54	5	∗	∗	NOUN
ejpam-3971	54	6	,	,	PUNCT
ejpam-3971	54	7	0	0	NUM
ejpam-3971	54	8	)	)	PUNCT
ejpam-3971	54	9	be	be	AUX
ejpam-3971	54	10	a	a	DET
ejpam-3971	54	11	brk	brk	PROPN
ejpam-3971	54	12	-	-	PUNCT
ejpam-3971	54	13	algebra	algebra	PROPN
ejpam-3971	54	14	.	.	PUNCT
ejpam-3971	55	1	a	a	DET
ejpam-3971	55	2	fuzzy	fuzzy	ADJ
ejpam-3971	55	3	set	set	VERB
ejpam-3971	55	4	µ	µ	NOUN
ejpam-3971	55	5	in	in	ADP
ejpam-3971	55	6	x	x	AUX
ejpam-3971	55	7	is	be	AUX
ejpam-3971	55	8	called	call	VERB
ejpam-3971	55	9	a	a	DET
ejpam-3971	55	10	fuzzy	fuzzy	ADJ
ejpam-3971	55	11	brk	brk	PROPN
ejpam-3971	55	12	-	-	PUNCT
ejpam-3971	55	13	ideal	ideal	NOUN
ejpam-3971	55	14	of	of	ADP
ejpam-3971	55	15	x	x	PRON
ejpam-3971	55	16	if	if	SCONJ
ejpam-3971	55	17	it	it	PRON
ejpam-3971	55	18	satisfies	satisfy	VERB
ejpam-3971	55	19	:	:	PUNCT
ejpam-3971	55	20	(	(	PUNCT
ejpam-3971	55	21	fi1	fi1	ADJ
ejpam-3971	55	22	)	)	PUNCT
ejpam-3971	55	23	µ(0	µ(0	NOUN
ejpam-3971	55	24	)	)	PUNCT
ejpam-3971	55	25	≥	≥	NOUN
ejpam-3971	55	26	µ(x	µ(x	VERB
ejpam-3971	55	27	)	)	PUNCT
ejpam-3971	55	28	,	,	PUNCT
ejpam-3971	55	29	(	(	PUNCT
ejpam-3971	55	30	fi2	fi2	X
ejpam-3971	55	31	)	)	PUNCT
ejpam-3971	55	32	µ(0	µ(0	NOUN
ejpam-3971	55	33	∗	∗	NOUN
ejpam-3971	55	34	x	x	NOUN
ejpam-3971	55	35	)	)	PUNCT
ejpam-3971	55	36	≥	≥	NOUN
ejpam-3971	55	37	min{µ(0	min{µ(0	NOUN
ejpam-3971	55	38	∗	∗	NOUN
ejpam-3971	55	39	(	(	PUNCT
ejpam-3971	55	40	x	x	X
ejpam-3971	55	41	∗	∗	PROPN
ejpam-3971	55	42	y	y	PROPN
ejpam-3971	55	43	)	)	PUNCT
ejpam-3971	55	44	)	)	PUNCT
ejpam-3971	55	45	,	,	PUNCT
ejpam-3971	56	1	µ(0	µ(0	NOUN
ejpam-3971	56	2	∗	∗	NOUN
ejpam-3971	56	3	y)},∀x	y)},∀x	NUM
ejpam-3971	56	4	,	,	PUNCT
ejpam-3971	56	5	y	y	PROPN
ejpam-3971	56	6	∈	∈	PROPN
ejpam-3971	56	7	x.	x.	NOUN
ejpam-3971	56	8	3	3	X
ejpam-3971	56	9	.	.	NOUN
ejpam-3971	56	10	fuzzy	fuzzy	ADJ
ejpam-3971	56	11	translation	translation	NOUN
ejpam-3971	56	12	and	and	CCONJ
ejpam-3971	56	13	fuzzy	fuzzy	ADJ
ejpam-3971	56	14	multiplication	multiplication	NOUN
ejpam-3971	56	15	of	of	ADP
ejpam-3971	56	16	brk	brk	PROPN
ejpam-3971	56	17	-	-	PUNCT
ejpam-3971	56	18	subalgebras	subalgebras	PROPN
ejpam-3971	56	19	this	this	DET
ejpam-3971	56	20	section	section	NOUN
ejpam-3971	56	21	deals	deal	VERB
ejpam-3971	56	22	with	with	ADP
ejpam-3971	56	23	the	the	DET
ejpam-3971	56	24	notion	notion	NOUN
ejpam-3971	56	25	of	of	ADP
ejpam-3971	56	26	fuzzy	fuzzy	ADJ
ejpam-3971	56	27	translation	translation	NOUN
ejpam-3971	56	28	and	and	CCONJ
ejpam-3971	56	29	fuzzy	fuzzy	ADJ
ejpam-3971	56	30	multiplication	multiplication	NOUN
ejpam-3971	56	31	on	on	ADP
ejpam-3971	56	32	brk	brk	PROPN
ejpam-3971	56	33	-	-	PUNCT
ejpam-3971	56	34	algebras	algebras	PROPN
ejpam-3971	56	35	.	.	PUNCT
ejpam-3971	57	1	in	in	ADP
ejpam-3971	57	2	what	what	PRON
ejpam-3971	57	3	follows	follow	VERB
ejpam-3971	57	4	,	,	PUNCT
ejpam-3971	57	5	x	x	PRON
ejpam-3971	57	6	denotes	denote	VERB
ejpam-3971	57	7	a	a	DET
ejpam-3971	57	8	brk	brk	PROPN
ejpam-3971	57	9	-algebra	-algebra	PROPN
ejpam-3971	57	10	,	,	PUNCT
ejpam-3971	57	11	and	and	CCONJ
ejpam-3971	57	12	for	for	ADP
ejpam-3971	57	13	any	any	DET
ejpam-3971	57	14	fuzzy	fuzzy	ADJ
ejpam-3971	57	15	set	set	VERB
ejpam-3971	57	16	µ	µ	NOUN
ejpam-3971	57	17	of	of	ADP
ejpam-3971	57	18	x	x	SYM
ejpam-3971	57	19	,	,	PUNCT
ejpam-3971	57	20	we	we	PRON
ejpam-3971	57	21	denote	denote	VERB
ejpam-3971	57	22	t	t	PROPN
ejpam-3971	57	23	=	=	SYM
ejpam-3971	57	24	1−	1−	NUM
ejpam-3971	57	25	sup{µ(x)|x	sup{µ(x)|x	NOUN
ejpam-3971	57	26	∈	∈	NOUN
ejpam-3971	57	27	x	x	NOUN
ejpam-3971	57	28	}	}	PUNCT
ejpam-3971	57	29	unless	unless	SCONJ
ejpam-3971	57	30	otherwise	otherwise	ADV
ejpam-3971	57	31	specified	specify	VERB
ejpam-3971	57	32	.	.	PUNCT
ejpam-3971	58	1	we	we	PRON
ejpam-3971	58	2	start	start	VERB
ejpam-3971	58	3	with	with	ADP
ejpam-3971	58	4	,	,	PUNCT
ejpam-3971	58	5	h.	h.	PROPN
ejpam-3971	58	6	alshehri	alshehri	PROPN
ejpam-3971	58	7	/	/	SYM
ejpam-3971	58	8	eur	eur	PROPN
ejpam-3971	58	9	.	.	PUNCT
ejpam-3971	59	1	j.	j.	PROPN
ejpam-3971	59	2	pure	pure	PROPN
ejpam-3971	59	3	appl	appl	PROPN
ejpam-3971	59	4	.	.	PROPN
ejpam-3971	59	5	math	math	PROPN
ejpam-3971	59	6	,	,	PUNCT
ejpam-3971	59	7	14	14	NUM
ejpam-3971	59	8	(	(	PUNCT
ejpam-3971	59	9	3	3	NUM
ejpam-3971	59	10	)	)	PUNCT
ejpam-3971	59	11	(	(	PUNCT
ejpam-3971	59	12	2021	2021	NUM
ejpam-3971	59	13	)	)	PUNCT
ejpam-3971	59	14	,	,	PUNCT
ejpam-3971	59	15	737	737	NUM
ejpam-3971	59	16	-	-	SYM
ejpam-3971	59	17	745	745	NUM
ejpam-3971	59	18	739	739	NUM
ejpam-3971	59	19	definition	definition	NOUN
ejpam-3971	59	20	8	8	NUM
ejpam-3971	59	21	.	.	PUNCT
ejpam-3971	60	1	let	let	VERB
ejpam-3971	60	2	µ	µ	X
ejpam-3971	60	3	be	be	AUX
ejpam-3971	60	4	a	a	DET
ejpam-3971	60	5	fuzzy	fuzzy	ADJ
ejpam-3971	60	6	subset	subset	NOUN
ejpam-3971	60	7	of	of	ADP
ejpam-3971	60	8	x	x	PUNCT
ejpam-3971	60	9	and	and	CCONJ
ejpam-3971	60	10	α	α	NOUN
ejpam-3971	60	11	∈	∈	PROPN
ejpam-3971	61	1	[	[	X
ejpam-3971	61	2	0	0	NUM
ejpam-3971	61	3	,	,	PUNCT
ejpam-3971	61	4	1	1	NUM
ejpam-3971	61	5	]	]	PUNCT
ejpam-3971	61	6	.	.	PUNCT
ejpam-3971	62	1	a	a	DET
ejpam-3971	62	2	mapping	mapping	NOUN
ejpam-3971	62	3	µtα	µtα	NOUN
ejpam-3971	62	4	:	:	PUNCT
ejpam-3971	62	5	x	x	X
ejpam-3971	62	6	−→	−→	NOUN
ejpam-3971	63	1	[	[	X
ejpam-3971	63	2	0	0	NUM
ejpam-3971	63	3	,	,	PUNCT
ejpam-3971	63	4	1	1	NUM
ejpam-3971	63	5	]	]	PUNCT
ejpam-3971	63	6	is	be	AUX
ejpam-3971	63	7	said	say	VERB
ejpam-3971	63	8	to	to	PART
ejpam-3971	63	9	be	be	AUX
ejpam-3971	63	10	a	a	DET
ejpam-3971	63	11	fuzzy	fuzzy	ADJ
ejpam-3971	63	12	α−	α−	ADP
ejpam-3971	63	13	translationofµ	translationofµ	NOUN
ejpam-3971	63	14	if	if	SCONJ
ejpam-3971	63	15	it	it	PRON
ejpam-3971	63	16	satisfies	satisfy	VERB
ejpam-3971	63	17	:	:	PUNCT
ejpam-3971	63	18	µtα	µtα	PROPN
ejpam-3971	63	19	(	(	PUNCT
ejpam-3971	63	20	x	x	X
ejpam-3971	63	21	)	)	PUNCT
ejpam-3971	63	22	=	=	SYM
ejpam-3971	63	23	µ(x	µ(x	X
ejpam-3971	63	24	)	)	PUNCT
ejpam-3971	63	25	+	+	PUNCT
ejpam-3971	64	1	α,∀x	α,∀x	NUM
ejpam-3971	64	2	∈	∈	NOUN
ejpam-3971	64	3	x.	x.	NOUN
ejpam-3971	64	4	definition	definition	NOUN
ejpam-3971	64	5	9	9	NUM
ejpam-3971	64	6	.	.	PUNCT
ejpam-3971	65	1	let	let	VERB
ejpam-3971	65	2	µ	µ	X
ejpam-3971	65	3	be	be	AUX
ejpam-3971	65	4	a	a	DET
ejpam-3971	65	5	fuzzy	fuzzy	ADJ
ejpam-3971	65	6	subset	subset	NOUN
ejpam-3971	65	7	of	of	ADP
ejpam-3971	65	8	x	x	PUNCT
ejpam-3971	65	9	and	and	CCONJ
ejpam-3971	65	10	α	α	NOUN
ejpam-3971	65	11	∈	∈	PROPN
ejpam-3971	66	1	[	[	X
ejpam-3971	66	2	0	0	NUM
ejpam-3971	66	3	,	,	PUNCT
ejpam-3971	66	4	1	1	NUM
ejpam-3971	66	5	]	]	PUNCT
ejpam-3971	66	6	.	.	PUNCT
ejpam-3971	67	1	a	a	DET
ejpam-3971	67	2	mapping	mapping	NOUN
ejpam-3971	67	3	µmα	µmα	X
ejpam-3971	67	4	:	:	PUNCT
ejpam-3971	67	5	x	x	X
ejpam-3971	67	6	−→	−→	NOUN
ejpam-3971	67	7	[	[	X
ejpam-3971	67	8	0	0	NUM
ejpam-3971	67	9	,	,	PUNCT
ejpam-3971	67	10	1	1	NUM
ejpam-3971	67	11	]	]	PUNCT
ejpam-3971	67	12	is	be	AUX
ejpam-3971	67	13	said	say	VERB
ejpam-3971	67	14	to	to	PART
ejpam-3971	67	15	be	be	AUX
ejpam-3971	67	16	a	a	DET
ejpam-3971	67	17	fuzzy	fuzzy	ADJ
ejpam-3971	67	18	αmultiplication	αmultiplication	NOUN
ejpam-3971	67	19	of	of	ADP
ejpam-3971	67	20	µ	µ	NOUN
ejpam-3971	67	21	if	if	SCONJ
ejpam-3971	67	22	it	it	PRON
ejpam-3971	67	23	satisfies	satisfy	VERB
ejpam-3971	67	24	:	:	PUNCT
ejpam-3971	67	25	µmα	µmα	PROPN
ejpam-3971	67	26	(	(	PUNCT
ejpam-3971	67	27	x	x	X
ejpam-3971	67	28	)	)	PUNCT
ejpam-3971	67	29	=	=	SYM
ejpam-3971	67	30	α.µ(x	α.µ(x	X
ejpam-3971	67	31	)	)	PUNCT
ejpam-3971	67	32	,	,	PUNCT
ejpam-3971	67	33	∀x	∀x	X
ejpam-3971	67	34	∈	∈	PROPN
ejpam-3971	67	35	x.	x.	NOUN
ejpam-3971	67	36	definition	definition	NOUN
ejpam-3971	67	37	10	10	NUM
ejpam-3971	67	38	.	.	PUNCT
ejpam-3971	68	1	a	a	DET
ejpam-3971	68	2	fuzzy	fuzzy	ADJ
ejpam-3971	68	3	α	α	NOUN
ejpam-3971	68	4	-	-	NOUN
ejpam-3971	68	5	translation	translation	NOUN
ejpam-3971	68	6	set	set	VERB
ejpam-3971	68	7	µtα(x	µtα(x	PROPN
ejpam-3971	68	8	)	)	PUNCT
ejpam-3971	68	9	of	of	ADP
ejpam-3971	68	10	µ	µ	PROPN
ejpam-3971	68	11	is	be	AUX
ejpam-3971	68	12	called	call	VERB
ejpam-3971	68	13	fuzzy	fuzzy	ADJ
ejpam-3971	68	14	α	α	NOUN
ejpam-3971	68	15	-	-	PUNCT
ejpam-3971	68	16	translation	translation	NOUN
ejpam-3971	68	17	brksubalgebra	brksubalgebra	NOUN
ejpam-3971	68	18	of	of	ADP
ejpam-3971	68	19	x	x	PRON
ejpam-3971	68	20	if	if	SCONJ
ejpam-3971	68	21	it	it	PRON
ejpam-3971	68	22	satisfies	satisfy	VERB
ejpam-3971	68	23	following	follow	VERB
ejpam-3971	68	24	condition	condition	NOUN
ejpam-3971	68	25	:	:	PUNCT
ejpam-3971	68	26	µtα(x∗y	µtα(x∗y	X
ejpam-3971	68	27	)	)	PUNCT
ejpam-3971	68	28	≥	≥	PROPN
ejpam-3971	68	29	min{µtα(x	min{µtα(x	PROPN
ejpam-3971	68	30	)	)	PUNCT
ejpam-3971	68	31	,	,	PUNCT
ejpam-3971	68	32	µtα(y	µtα(y	PROPN
ejpam-3971	68	33	)	)	PUNCT
ejpam-3971	68	34	}	}	PUNCT
ejpam-3971	68	35	,	,	PUNCT
ejpam-3971	68	36	similarly	similarly	ADV
ejpam-3971	68	37	,	,	PUNCT
ejpam-3971	68	38	wesaidthatµmα	wesaidthatµmα	NOUN
ejpam-3971	68	39	(	(	PUNCT
ejpam-3971	68	40	x	x	X
ejpam-3971	68	41	)	)	PUNCT
ejpam-3971	68	42	is	be	AUX
ejpam-3971	68	43	fuzzy	fuzzy	ADJ
ejpam-3971	68	44	α	α	DET
ejpam-3971	68	45	-	-	PUNCT
ejpam-3971	68	46	multiplication	multiplication	NOUN
ejpam-3971	68	47	brk	brk	NOUN
ejpam-3971	68	48	-	-	PUNCT
ejpam-3971	68	49	subalgebra	subalgebra	PROPN
ejpam-3971	68	50	of	of	ADP
ejpam-3971	68	51	x	x	PRON
ejpam-3971	68	52	if	if	SCONJ
ejpam-3971	68	53	it	it	PRON
ejpam-3971	68	54	satisfies	satisfy	VERB
ejpam-3971	68	55	:	:	PUNCT
ejpam-3971	68	56	µmα	µmα	X
ejpam-3971	68	57	(	(	PUNCT
ejpam-3971	68	58	x∗y	x∗y	X
ejpam-3971	68	59	)	)	PUNCT
ejpam-3971	68	60	≥	≥	PROPN
ejpam-3971	68	61	min{µmα	min{µmα	NOUN
ejpam-3971	68	62	(	(	PUNCT
ejpam-3971	68	63	x	x	NOUN
ejpam-3971	68	64	)	)	PUNCT
ejpam-3971	68	65	,	,	PUNCT
ejpam-3971	68	66	µmα	µmα	X
ejpam-3971	68	67	(	(	PUNCT
ejpam-3971	68	68	y	y	NOUN
ejpam-3971	68	69	)	)	PUNCT
ejpam-3971	68	70	}	}	PUNCT
ejpam-3971	68	71	.	.	PUNCT
ejpam-3971	69	1	example	example	NOUN
ejpam-3971	70	1	1	1	NUM
ejpam-3971	70	2	.	.	X
ejpam-3971	70	3	consider	consider	VERB
ejpam-3971	70	4	a	a	DET
ejpam-3971	70	5	set	set	NOUN
ejpam-3971	70	6	x	x	X
ejpam-3971	70	7	=	=	SYM
ejpam-3971	70	8	{	{	PUNCT
ejpam-3971	70	9	0	0	NUM
ejpam-3971	70	10	,	,	PUNCT
ejpam-3971	70	11	a	a	DET
ejpam-3971	70	12	,	,	PUNCT
ejpam-3971	70	13	b	b	NOUN
ejpam-3971	70	14	,	,	PUNCT
ejpam-3971	70	15	c	c	NOUN
ejpam-3971	70	16	}	}	PUNCT
ejpam-3971	70	17	.	.	PUNCT
ejpam-3971	71	1	we	we	PRON
ejpam-3971	71	2	define	define	VERB
ejpam-3971	71	3	“	"	PUNCT
ejpam-3971	71	4	∗”onxasthefollowingtable	∗”onxasthefollowingtable	ADJ
ejpam-3971	71	5	:	:	PUNCT
ejpam-3971	71	6	∗	∗	NOUN
ejpam-3971	71	7	0	0	NUM
ejpam-3971	72	1	a	a	DET
ejpam-3971	72	2	b	b	NOUN
ejpam-3971	72	3	c	c	NOUN
ejpam-3971	72	4	0	0	NUM
ejpam-3971	72	5	0	0	NUM
ejpam-3971	72	6	a	a	DET
ejpam-3971	72	7	0	0	NUM
ejpam-3971	72	8	a	a	DET
ejpam-3971	72	9	a	a	DET
ejpam-3971	72	10	a	a	DET
ejpam-3971	72	11	0	0	NUM
ejpam-3971	72	12	a	a	DET
ejpam-3971	72	13	0	0	NUM
ejpam-3971	72	14	b	b	X
ejpam-3971	72	15	b	b	PROPN
ejpam-3971	72	16	a	a	DET
ejpam-3971	72	17	0	0	NUM
ejpam-3971	72	18	a	a	DET
ejpam-3971	72	19	c	c	NOUN
ejpam-3971	72	20	c	c	AUX
ejpam-3971	72	21	b	b	PROPN
ejpam-3971	72	22	c	c	NOUN
ejpam-3971	72	23	0	0	PUNCT
ejpam-3971	72	24	define	define	VERB
ejpam-3971	72	25	a	a	DET
ejpam-3971	72	26	fuzzy	fuzzy	ADJ
ejpam-3971	72	27	subset	subset	VERB
ejpam-3971	72	28	µ	µ	PROPN
ejpam-3971	72	29	of	of	ADP
ejpam-3971	72	30	x	x	PUNCT
ejpam-3971	72	31	by	by	ADP
ejpam-3971	72	32	µ(0	µ(0	NOUN
ejpam-3971	72	33	)	)	PUNCT
ejpam-3971	72	34	=	=	SYM
ejpam-3971	72	35	µ(a	µ(a	PROPN
ejpam-3971	72	36	)	)	PUNCT
ejpam-3971	72	37	=	=	SYM
ejpam-3971	72	38	0.6	0.6	NUM
ejpam-3971	72	39	and	and	CCONJ
ejpam-3971	72	40	µ(b	µ(b	NOUN
ejpam-3971	72	41	)	)	PUNCT
ejpam-3971	72	42	=	=	SYM
ejpam-3971	72	43	µ(c	µ(c	PROPN
ejpam-3971	72	44	)	)	PUNCT
ejpam-3971	72	45	=	=	SYM
ejpam-3971	72	46	0.1	0.1	NUM
ejpam-3971	72	47	,	,	PUNCT
ejpam-3971	72	48	routine	routine	ADJ
ejpam-3971	72	49	calculation	calculation	NOUN
ejpam-3971	72	50	gives	give	VERB
ejpam-3971	72	51	that	that	SCONJ
ejpam-3971	72	52	µ	µ	NOUN
ejpam-3971	72	53	is	be	AUX
ejpam-3971	72	54	fuzzy	fuzzy	ADJ
ejpam-3971	72	55	brk	brk	NOUN
ejpam-3971	72	56	-	-	PUNCT
ejpam-3971	72	57	subalgebra	subalgebra	PROPN
ejpam-3971	72	58	of	of	ADP
ejpam-3971	72	59	x.	x.	NOUN
ejpam-3971	72	60	here	here	ADV
ejpam-3971	72	61	t	t	PROPN
ejpam-3971	73	1	=	=	SYM
ejpam-3971	73	2	1	1	NUM
ejpam-3971	73	3	−	−	PROPN
ejpam-3971	73	4	sup{(x	sup{(x	NOUN
ejpam-3971	73	5	)	)	PUNCT
ejpam-3971	73	6	:	:	PUNCT
ejpam-3971	74	1	x	x	X
ejpam-3971	74	2	∈	∈	NOUN
ejpam-3971	74	3	x	x	X
ejpam-3971	74	4	}	}	PUNCT
ejpam-3971	74	5	=	=	SYM
ejpam-3971	74	6	1−	1−	NUM
ejpam-3971	74	7	0.6	0.6	NUM
ejpam-3971	74	8	=	=	SYM
ejpam-3971	74	9	0.4	0.4	NUM
ejpam-3971	74	10	.	.	PUNCT
ejpam-3971	75	1	choose	choose	VERB
ejpam-3971	75	2	α	α	NOUN
ejpam-3971	75	3	=	=	PUNCT
ejpam-3971	75	4	0.2	0.2	NUM
ejpam-3971	75	5	∈	∈	NOUN
ejpam-3971	76	1	[	[	X
ejpam-3971	76	2	0	0	NUM
ejpam-3971	76	3	,	,	PUNCT
ejpam-3971	76	4	t	t	NOUN
ejpam-3971	76	5	]	]	PUNCT
ejpam-3971	76	6	and	and	CCONJ
ejpam-3971	76	7	β	β	X
ejpam-3971	76	8	=	=	NOUN
ejpam-3971	76	9	0.3	0.3	NUM
ejpam-3971	76	10	∈	∈	PROPN
ejpam-3971	77	1	[	[	X
ejpam-3971	77	2	0	0	NUM
ejpam-3971	77	3	,	,	PUNCT
ejpam-3971	77	4	1	1	NUM
ejpam-3971	77	5	]	]	PUNCT
ejpam-3971	77	6	.	.	PUNCT
ejpam-3971	78	1	then	then	ADV
ejpam-3971	78	2	the	the	DET
ejpam-3971	78	3	mapping	mapping	NOUN
ejpam-3971	78	4	µt0	µt0	PROPN
ejpam-3971	78	5	.2(x	.2(x	PUNCT
ejpam-3971	78	6	):	):	PUNCT
ejpam-3971	78	7	x	x	X
ejpam-3971	78	8	−→	−→	NOUN
ejpam-3971	79	1	[	[	X
ejpam-3971	79	2	0	0	NUM
ejpam-3971	79	3	,	,	PUNCT
ejpam-3971	79	4	1	1	NUM
ejpam-3971	79	5	]	]	PUNCT
ejpam-3971	79	6	defined	define	VERB
ejpam-3971	79	7	by	by	ADP
ejpam-3971	79	8	µt0	µt0	PROPN
ejpam-3971	79	9	.2(x)=	.2(x)=	PROPN
ejpam-3971	79	10	{	{	PUNCT
ejpam-3971	80	1	0.6	0.6	NUM
ejpam-3971	80	2	+	+	CCONJ
ejpam-3971	80	3	0.2	0.2	NUM
ejpam-3971	80	4	=	=	SYM
ejpam-3971	80	5	0.8	0.8	NUM
ejpam-3971	80	6	;	;	PUNCT
ejpam-3971	80	7	x	x	SYM
ejpam-3971	80	8	=	=	SYM
ejpam-3971	80	9	0	0	NUM
ejpam-3971	80	10	,	,	PUNCT
ejpam-3971	80	11	a	a	DET
ejpam-3971	80	12	0.1	0.1	NUM
ejpam-3971	80	13	+	+	CCONJ
ejpam-3971	80	14	0.2	0.2	NUM
ejpam-3971	80	15	=	=	SYM
ejpam-3971	80	16	0.3	0.3	NUM
ejpam-3971	80	17	;	;	PUNCT
ejpam-3971	80	18	x	x	SYM
ejpam-3971	80	19	=	=	SYM
ejpam-3971	80	20	b	b	PROPN
ejpam-3971	80	21	,	,	PUNCT
ejpam-3971	80	22	c	c	PROPN
ejpam-3971	80	23	µt0	µt0	PROPN
ejpam-3971	80	24	.2(x)=µ(x	.2(x)=µ(x	NUM
ejpam-3971	80	25	)	)	PUNCT
ejpam-3971	81	1	+	+	CCONJ
ejpam-3971	81	2	0.2	0.2	NUM
ejpam-3971	81	3	,	,	PUNCT
ejpam-3971	81	4	∀x	∀x	VERB
ejpam-3971	81	5	∈	∈	PROPN
ejpam-3971	81	6	x	x	PRON
ejpam-3971	81	7	,	,	PUNCT
ejpam-3971	81	8	isafuzzy0.2−translation	isafuzzy0.2−translation	PROPN
ejpam-3971	81	9	.	.	PUNCT
ejpam-3971	82	1	µm0	µm0	PROPN
ejpam-3971	82	2	.3(x	.3(x	PROPN
ejpam-3971	82	3	):	):	PUNCT
ejpam-3971	82	4	x	x	PUNCT
ejpam-3971	82	5	−→	−→	NOUN
ejpam-3971	83	1	[	[	X
ejpam-3971	83	2	0	0	NUM
ejpam-3971	83	3	,	,	PUNCT
ejpam-3971	83	4	1	1	NUM
ejpam-3971	83	5	]	]	PUNCT
ejpam-3971	83	6	defined	define	VERB
ejpam-3971	83	7	by	by	ADP
ejpam-3971	83	8	µm0	µm0	PROPN
ejpam-3971	83	9	.3(x)=	.3(x)=	PUNCT
ejpam-3971	83	10	{	{	PUNCT
ejpam-3971	83	11	(	(	PUNCT
ejpam-3971	83	12	0.3)(0.6	0.3)(0.6	NOUN
ejpam-3971	83	13	)	)	PUNCT
ejpam-3971	83	14	=	=	SYM
ejpam-3971	83	15	0.18	0.18	NUM
ejpam-3971	83	16	;	;	PUNCT
ejpam-3971	83	17	x	x	SYM
ejpam-3971	83	18	=	=	SYM
ejpam-3971	83	19	0	0	NUM
ejpam-3971	83	20	,	,	PUNCT
ejpam-3971	83	21	a	a	DET
ejpam-3971	83	22	(	(	PUNCT
ejpam-3971	83	23	0.3)(0.1	0.3)(0.1	NUM
ejpam-3971	83	24	)	)	PUNCT
ejpam-3971	83	25	=	=	SYM
ejpam-3971	83	26	0.3	0.3	NUM
ejpam-3971	83	27	;	;	PUNCT
ejpam-3971	83	28	x	x	SYM
ejpam-3971	83	29	=	=	SYM
ejpam-3971	83	30	b	b	PROPN
ejpam-3971	83	31	,	,	PUNCT
ejpam-3971	83	32	c	c	X
ejpam-3971	83	33	µm0	µm0	PROPN
ejpam-3971	83	34	.3(x)=(0.3)µ(x	.3(x)=(0.3)µ(x	PROPN
ejpam-3971	83	35	)	)	PUNCT
ejpam-3971	83	36	,	,	PUNCT
ejpam-3971	83	37	∀x	∀x	VERB
ejpam-3971	83	38	∈	∈	PROPN
ejpam-3971	83	39	x	x	SYM
ejpam-3971	83	40	,	,	PUNCT
ejpam-3971	83	41	isafuzzy0.3−multiplication	isafuzzy0.3−multiplication	NOUN
ejpam-3971	83	42	.	.	PUNCT
ejpam-3971	84	1	theorem	theorem	NOUN
ejpam-3971	84	2	1	1	NUM
ejpam-3971	84	3	.	.	X
ejpam-3971	85	1	for	for	ADP
ejpam-3971	85	2	any	any	DET
ejpam-3971	85	3	fuzzy	fuzzy	ADJ
ejpam-3971	85	4	brk	brk	PROPN
ejpam-3971	85	5	-	-	PUNCT
ejpam-3971	85	6	subalgebra	subalgebra	PROPN
ejpam-3971	85	7	µ	µ	NOUN
ejpam-3971	85	8	of	of	ADP
ejpam-3971	85	9	x	x	X
ejpam-3971	85	10	and	and	CCONJ
ejpam-3971	85	11	α	α	NOUN
ejpam-3971	85	12	∈	∈	PROPN
ejpam-3971	86	1	[	[	X
ejpam-3971	86	2	0	0	NUM
ejpam-3971	86	3	,	,	PUNCT
ejpam-3971	86	4	t	t	X
ejpam-3971	86	5	]	]	PUNCT
ejpam-3971	86	6	,	,	PUNCT
ejpam-3971	86	7	the	the	DET
ejpam-3971	86	8	fuzzy	fuzzy	ADJ
ejpam-3971	86	9	α	α	PROPN
ejpam-3971	86	10	−	−	PROPN
ejpam-3971	86	11	translation	translation	NOUN
ejpam-3971	86	12	µtα(x	µtα(x	PROPN
ejpam-3971	86	13	)	)	PUNCT
ejpam-3971	86	14	of	of	ADP
ejpam-3971	86	15	µ	µ	PROPN
ejpam-3971	86	16	is	be	AUX
ejpam-3971	86	17	a	a	DET
ejpam-3971	86	18	fuzzy	fuzzy	ADJ
ejpam-3971	86	19	brk	brk	NOUN
ejpam-3971	86	20	-	-	PUNCT
ejpam-3971	86	21	subalgebra	subalgebra	PROPN
ejpam-3971	86	22	of	of	ADP
ejpam-3971	86	23	x.	x.	NOUN
ejpam-3971	86	24	proof	proof	NOUN
ejpam-3971	86	25	.	.	PUNCT
ejpam-3971	87	1	let	let	VERB
ejpam-3971	87	2	x	x	PRON
ejpam-3971	87	3	,	,	PUNCT
ejpam-3971	87	4	y	y	PROPN
ejpam-3971	87	5	∈	∈	PROPN
ejpam-3971	87	6	x	x	X
ejpam-3971	87	7	and	and	CCONJ
ejpam-3971	87	8	α	α	NOUN
ejpam-3971	87	9	∈	∈	PROPN
ejpam-3971	88	1	[	[	X
ejpam-3971	88	2	0	0	NUM
ejpam-3971	88	3	,	,	PUNCT
ejpam-3971	88	4	t	t	X
ejpam-3971	88	5	]	]	PUNCT
ejpam-3971	88	6	.	.	PUNCT
ejpam-3971	89	1	then	then	ADV
ejpam-3971	89	2	µ(x	µ(x	ADJ
ejpam-3971	89	3	∗	∗	NOUN
ejpam-3971	89	4	y	y	NOUN
ejpam-3971	89	5	)	)	PUNCT
ejpam-3971	89	6	≥	≥	NOUN
ejpam-3971	89	7	min{µ(x	min{µ(x	NOUN
ejpam-3971	89	8	)	)	PUNCT
ejpam-3971	89	9	,	,	PUNCT
ejpam-3971	89	10	µ(y	µ(y	PROPN
ejpam-3971	89	11	)	)	PUNCT
ejpam-3971	89	12	}	}	PUNCT
ejpam-3971	89	13	.	.	PUNCT
ejpam-3971	90	1	now	now	ADV
ejpam-3971	90	2	,	,	PUNCT
ejpam-3971	90	3	h.	h.	PROPN
ejpam-3971	90	4	alshehri	alshehri	PROPN
ejpam-3971	90	5	/	/	SYM
ejpam-3971	90	6	eur	eur	PROPN
ejpam-3971	90	7	.	.	PUNCT
ejpam-3971	91	1	j.	j.	PROPN
ejpam-3971	91	2	pure	pure	PROPN
ejpam-3971	91	3	appl	appl	PROPN
ejpam-3971	91	4	.	.	PROPN
ejpam-3971	91	5	math	math	PROPN
ejpam-3971	91	6	,	,	PUNCT
ejpam-3971	91	7	14	14	NUM
ejpam-3971	91	8	(	(	PUNCT
ejpam-3971	91	9	3	3	NUM
ejpam-3971	91	10	)	)	PUNCT
ejpam-3971	91	11	(	(	PUNCT
ejpam-3971	91	12	2021	2021	NUM
ejpam-3971	91	13	)	)	PUNCT
ejpam-3971	91	14	,	,	PUNCT
ejpam-3971	91	15	737	737	NUM
ejpam-3971	91	16	-	-	SYM
ejpam-3971	91	17	745	745	NUM
ejpam-3971	91	18	740	740	NUM
ejpam-3971	91	19	µmα	µmα	NOUN
ejpam-3971	91	20	(	(	PUNCT
ejpam-3971	91	21	x	x	SYM
ejpam-3971	91	22	∗	∗	VERB
ejpam-3971	91	23	y)=	y)=	ADJ
ejpam-3971	91	24	µ(x	µ(x	NOUN
ejpam-3971	91	25	)	)	PUNCT
ejpam-3971	91	26	+	+	CCONJ
ejpam-3971	91	27	α	α	PRON
ejpam-3971	91	28	≥	≥	NOUN
ejpam-3971	91	29	min{µ(x	min{µ(x	NOUN
ejpam-3971	91	30	)	)	PUNCT
ejpam-3971	91	31	,	,	PUNCT
ejpam-3971	91	32	µ(y)}+	µ(y)}+	NUM
ejpam-3971	91	33	α	α	NOUN
ejpam-3971	91	34	=	=	X
ejpam-3971	91	35	min{µ(x	min{µ(x	PROPN
ejpam-3971	91	36	)	)	PUNCT
ejpam-3971	91	37	+	+	CCONJ
ejpam-3971	91	38	α	α	NOUN
ejpam-3971	91	39	,	,	PUNCT
ejpam-3971	91	40	µ(y	µ(y	PROPN
ejpam-3971	91	41	)	)	PUNCT
ejpam-3971	91	42	+	+	CCONJ
ejpam-3971	91	43	α	α	X
ejpam-3971	91	44	}	}	PUNCT
ejpam-3971	91	45	=	=	ADJ
ejpam-3971	91	46	min{µmα	min{µmα	NOUN
ejpam-3971	91	47	(	(	PUNCT
ejpam-3971	91	48	x	x	NOUN
ejpam-3971	91	49	)	)	PUNCT
ejpam-3971	91	50	,	,	PUNCT
ejpam-3971	91	51	µmα	µmα	X
ejpam-3971	91	52	(	(	PUNCT
ejpam-3971	91	53	y	y	NOUN
ejpam-3971	91	54	)	)	PUNCT
ejpam-3971	91	55	}	}	PUNCT
ejpam-3971	92	1	this	this	PRON
ejpam-3971	92	2	completes	complete	VERB
ejpam-3971	92	3	the	the	DET
ejpam-3971	92	4	proof	proof	NOUN
ejpam-3971	92	5	.	.	PUNCT
ejpam-3971	93	1	the	the	DET
ejpam-3971	93	2	converse	converse	NOUN
ejpam-3971	93	3	of	of	ADP
ejpam-3971	93	4	the	the	DET
ejpam-3971	93	5	above	above	ADJ
ejpam-3971	93	6	theorem	theorem	NOUN
ejpam-3971	93	7	is	be	AUX
ejpam-3971	93	8	valid	valid	ADJ
ejpam-3971	93	9	.	.	PUNCT
ejpam-3971	94	1	theorem	theorem	NOUN
ejpam-3971	94	2	2	2	NUM
ejpam-3971	94	3	.	.	X
ejpam-3971	95	1	for	for	ADP
ejpam-3971	95	2	any	any	DET
ejpam-3971	95	3	fuzzy	fuzzy	ADJ
ejpam-3971	95	4	subset	subset	VERB
ejpam-3971	95	5	µ	µ	PROPN
ejpam-3971	95	6	of	of	ADP
ejpam-3971	95	7	x	x	X
ejpam-3971	95	8	and	and	CCONJ
ejpam-3971	95	9	α	α	NOUN
ejpam-3971	95	10	∈	∈	PROPN
ejpam-3971	95	11	[	[	X
ejpam-3971	95	12	0	0	NUM
ejpam-3971	95	13	,	,	PUNCT
ejpam-3971	95	14	t	t	X
ejpam-3971	95	15	]	]	PUNCT
ejpam-3971	95	16	,	,	PUNCT
ejpam-3971	95	17	if	if	SCONJ
ejpam-3971	95	18	the	the	DET
ejpam-3971	95	19	fuzzy	fuzzy	ADJ
ejpam-3971	95	20	α	α	NOUN
ejpam-3971	95	21	-	-	NOUN
ejpam-3971	95	22	translation	translation	NOUN
ejpam-3971	95	23	µtα(x	µtα(x	PROPN
ejpam-3971	95	24	)	)	PUNCT
ejpam-3971	95	25	of	of	ADP
ejpam-3971	95	26	µ	µ	PROPN
ejpam-3971	95	27	is	be	AUX
ejpam-3971	95	28	a	a	DET
ejpam-3971	95	29	fuzzy	fuzzy	ADJ
ejpam-3971	95	30	brk	brk	NOUN
ejpam-3971	95	31	-	-	PUNCT
ejpam-3971	95	32	subalgebra	subalgebra	PROPN
ejpam-3971	95	33	of	of	ADP
ejpam-3971	95	34	x	x	PUNCT
ejpam-3971	95	35	then	then	ADV
ejpam-3971	95	36	so	so	ADV
ejpam-3971	95	37	is	be	AUX
ejpam-3971	95	38	µ.	µ.	NOUN
ejpam-3971	95	39	proof	proof	NOUN
ejpam-3971	95	40	.	.	PUNCT
ejpam-3971	96	1	let	let	VERB
ejpam-3971	96	2	x	x	PRON
ejpam-3971	96	3	,	,	PUNCT
ejpam-3971	96	4	y	y	PROPN
ejpam-3971	96	5	∈	∈	PROPN
ejpam-3971	96	6	x.	x.	NOUN
ejpam-3971	96	7	assume	assume	VERB
ejpam-3971	96	8	that	that	SCONJ
ejpam-3971	96	9	µtα(x	µtα(x	PROPN
ejpam-3971	96	10	)	)	PUNCT
ejpam-3971	96	11	of	of	ADP
ejpam-3971	96	12	µ	µ	PROPN
ejpam-3971	96	13	is	be	AUX
ejpam-3971	96	14	a	a	DET
ejpam-3971	96	15	fuzzy	fuzzy	ADJ
ejpam-3971	96	16	brk	brk	NOUN
ejpam-3971	96	17	-	-	PUNCT
ejpam-3971	96	18	subalgebra	subalgebra	PROPN
ejpam-3971	96	19	of	of	ADP
ejpam-3971	96	20	x	x	PUNCT
ejpam-3971	96	21	for	for	ADP
ejpam-3971	96	22	α	α	PRON
ejpam-3971	96	23	∈	∈	PROPN
ejpam-3971	97	1	[	[	X
ejpam-3971	97	2	0	0	NUM
ejpam-3971	97	3	,	,	PUNCT
ejpam-3971	97	4	1	1	NUM
ejpam-3971	97	5	]	]	PUNCT
ejpam-3971	97	6	.	.	PUNCT
ejpam-3971	98	1	then	then	ADV
ejpam-3971	98	2	µ(x	µ(x	ADJ
ejpam-3971	98	3	∗	∗	NOUN
ejpam-3971	98	4	y	y	NOUN
ejpam-3971	98	5	)	)	PUNCT
ejpam-3971	99	1	+	+	CCONJ
ejpam-3971	99	2	α	α	X
ejpam-3971	99	3	=	=	SYM
ejpam-3971	99	4	µtα(x	µtα(x	PROPN
ejpam-3971	99	5	∗	∗	NOUN
ejpam-3971	99	6	y	y	PROPN
ejpam-3971	99	7	)	)	PUNCT
ejpam-3971	99	8	≥	≥	PROPN
ejpam-3971	99	9	min{µtα(x	min{µtα(x	PROPN
ejpam-3971	99	10	)	)	PUNCT
ejpam-3971	99	11	,	,	PUNCT
ejpam-3971	99	12	µtα(y	µtα(y	PROPN
ejpam-3971	99	13	)	)	PUNCT
ejpam-3971	99	14	}	}	PUNCT
ejpam-3971	99	15	=	=	SYM
ejpam-3971	99	16	min{µ(x	min{µ(x	NOUN
ejpam-3971	99	17	)	)	PUNCT
ejpam-3971	100	1	+	+	CCONJ
ejpam-3971	100	2	α	α	NOUN
ejpam-3971	100	3	,	,	PUNCT
ejpam-3971	100	4	µ(y	µ(y	PROPN
ejpam-3971	100	5	)	)	PUNCT
ejpam-3971	100	6	+	+	CCONJ
ejpam-3971	100	7	α	α	X
ejpam-3971	100	8	}	}	PUNCT
ejpam-3971	100	9	=	=	SYM
ejpam-3971	100	10	min{µ(x	min{µ(x	NOUN
ejpam-3971	100	11	)	)	PUNCT
ejpam-3971	100	12	,	,	PUNCT
ejpam-3971	100	13	µ(y)}+	µ(y)}+	NUM
ejpam-3971	100	14	α	α	NOUN
ejpam-3971	100	15	.	.	PUNCT
ejpam-3971	101	1	hence	hence	ADV
ejpam-3971	101	2	,	,	PUNCT
ejpam-3971	101	3	µ(x	µ(x	ADJ
ejpam-3971	101	4	∗	∗	NOUN
ejpam-3971	101	5	y	y	NOUN
ejpam-3971	101	6	)	)	PUNCT
ejpam-3971	101	7	≥	≥	NOUN
ejpam-3971	101	8	min{µ(x	min{µ(x	NOUN
ejpam-3971	101	9	)	)	PUNCT
ejpam-3971	101	10	,	,	PUNCT
ejpam-3971	101	11	µ(y	µ(y	PROPN
ejpam-3971	101	12	)	)	PUNCT
ejpam-3971	101	13	}	}	PUNCT
ejpam-3971	101	14	.	.	PUNCT
ejpam-3971	102	1	therefore	therefore	ADV
ejpam-3971	102	2	,	,	PUNCT
ejpam-3971	102	3	µ	µ	X
ejpam-3971	102	4	is	be	AUX
ejpam-3971	102	5	a	a	DET
ejpam-3971	102	6	fuzzy	fuzzy	ADJ
ejpam-3971	102	7	brk	brk	NOUN
ejpam-3971	102	8	-	-	PUNCT
ejpam-3971	102	9	subalgebra	subalgebra	PROPN
ejpam-3971	102	10	of	of	ADP
ejpam-3971	102	11	x.	x.	NOUN
ejpam-3971	102	12	theorem	theorem	NOUN
ejpam-3971	102	13	3	3	NUM
ejpam-3971	102	14	.	.	X
ejpam-3971	103	1	for	for	ADP
ejpam-3971	103	2	any	any	DET
ejpam-3971	103	3	fuzzy	fuzzy	ADJ
ejpam-3971	103	4	brk	brk	PROPN
ejpam-3971	103	5	-	-	PUNCT
ejpam-3971	103	6	subalgebra	subalgebra	PROPN
ejpam-3971	103	7	µ	µ	NOUN
ejpam-3971	103	8	of	of	ADP
ejpam-3971	103	9	x	x	X
ejpam-3971	103	10	and	and	CCONJ
ejpam-3971	103	11	α	α	NOUN
ejpam-3971	103	12	∈	∈	PROPN
ejpam-3971	104	1	[	[	X
ejpam-3971	104	2	0	0	NUM
ejpam-3971	104	3	,	,	PUNCT
ejpam-3971	104	4	1	1	NUM
ejpam-3971	104	5	]	]	PUNCT
ejpam-3971	104	6	,	,	PUNCT
ejpam-3971	104	7	the	the	DET
ejpam-3971	104	8	fuzzy	fuzzy	ADJ
ejpam-3971	104	9	α	α	NOUN
ejpam-3971	104	10	-	-	PUNCT
ejpam-3971	104	11	multiplication	multiplication	NOUN
ejpam-3971	104	12	µmα	µmα	NOUN
ejpam-3971	104	13	(	(	PUNCT
ejpam-3971	104	14	x	x	X
ejpam-3971	104	15	)	)	PUNCT
ejpam-3971	104	16	of	of	ADP
ejpam-3971	104	17	µ	µ	PROPN
ejpam-3971	104	18	is	be	AUX
ejpam-3971	104	19	a	a	DET
ejpam-3971	104	20	fuzzy	fuzzy	ADJ
ejpam-3971	104	21	brk	brk	NOUN
ejpam-3971	104	22	-	-	PUNCT
ejpam-3971	104	23	subalgebra	subalgebra	PROPN
ejpam-3971	104	24	of	of	ADP
ejpam-3971	104	25	x.	x.	NOUN
ejpam-3971	104	26	proof	proof	NOUN
ejpam-3971	104	27	.	.	PUNCT
ejpam-3971	105	1	let	let	VERB
ejpam-3971	105	2	x	x	PRON
ejpam-3971	105	3	,	,	PUNCT
ejpam-3971	105	4	y	y	PROPN
ejpam-3971	105	5	∈	∈	PROPN
ejpam-3971	105	6	x	x	X
ejpam-3971	105	7	and	and	CCONJ
ejpam-3971	105	8	α	α	NOUN
ejpam-3971	105	9	∈	∈	PROPN
ejpam-3971	106	1	[	[	X
ejpam-3971	106	2	0	0	NUM
ejpam-3971	106	3	,	,	PUNCT
ejpam-3971	106	4	1	1	NUM
ejpam-3971	106	5	]	]	PUNCT
ejpam-3971	106	6	.	.	PUNCT
ejpam-3971	107	1	then	then	ADV
ejpam-3971	107	2	µ(x	µ(x	ADJ
ejpam-3971	107	3	∗	∗	NOUN
ejpam-3971	107	4	y	y	NOUN
ejpam-3971	107	5	)	)	PUNCT
ejpam-3971	107	6	≥	≥	NOUN
ejpam-3971	107	7	min{µ(x	min{µ(x	NOUN
ejpam-3971	107	8	)	)	PUNCT
ejpam-3971	107	9	,	,	PUNCT
ejpam-3971	107	10	µ(y	µ(y	PROPN
ejpam-3971	107	11	)	)	PUNCT
ejpam-3971	107	12	}	}	PUNCT
ejpam-3971	107	13	.	.	PUNCT
ejpam-3971	108	1	now	now	ADV
ejpam-3971	108	2	,	,	PUNCT
ejpam-3971	108	3	µmα	µmα	X
ejpam-3971	108	4	(	(	PUNCT
ejpam-3971	108	5	x	x	X
ejpam-3971	108	6	∗	∗	PROPN
ejpam-3971	108	7	y	y	NOUN
ejpam-3971	108	8	)	)	PUNCT
ejpam-3971	108	9	=	=	PUNCT
ejpam-3971	108	10	α.µ(x	α.µ(x	NUM
ejpam-3971	108	11	∗	∗	NOUN
ejpam-3971	108	12	y	y	PROPN
ejpam-3971	108	13	)	)	PUNCT
ejpam-3971	108	14	≥	≥	NOUN
ejpam-3971	108	15	α.min{µ(x	α.min{µ(x	NOUN
ejpam-3971	108	16	)	)	PUNCT
ejpam-3971	108	17	,	,	PUNCT
ejpam-3971	108	18	µ(y	µ(y	PROPN
ejpam-3971	108	19	)	)	PUNCT
ejpam-3971	108	20	}	}	PUNCT
ejpam-3971	108	21	=	=	SYM
ejpam-3971	108	22	min{α.µ(x	min{α.µ(x	PROPN
ejpam-3971	108	23	)	)	PUNCT
ejpam-3971	108	24	,	,	PUNCT
ejpam-3971	108	25	α.µ(y	α.µ(y	NUM
ejpam-3971	108	26	)	)	PUNCT
ejpam-3971	108	27	}	}	PUNCT
ejpam-3971	109	1	=	=	PUNCT
ejpam-3971	109	2	min{µm	min{µm	INTJ
ejpam-3971	109	3	(	(	PUNCT
ejpam-3971	109	4	x	x	NOUN
ejpam-3971	109	5	)	)	PUNCT
ejpam-3971	109	6	,	,	PUNCT
ejpam-3971	109	7	µm	µm	ADP
ejpam-3971	109	8	(	(	PUNCT
ejpam-3971	109	9	y	y	NOUN
ejpam-3971	109	10	)	)	PUNCT
ejpam-3971	109	11	}	}	PUNCT
ejpam-3971	109	12	.	.	PUNCT
ejpam-3971	110	1	this	this	PRON
ejpam-3971	110	2	completes	complete	VERB
ejpam-3971	110	3	the	the	DET
ejpam-3971	110	4	proof	proof	NOUN
ejpam-3971	110	5	.	.	PUNCT
ejpam-3971	111	1	the	the	DET
ejpam-3971	111	2	following	follow	VERB
ejpam-3971	111	3	is	be	AUX
ejpam-3971	111	4	the	the	DET
ejpam-3971	111	5	converse	converse	NOUN
ejpam-3971	111	6	of	of	ADP
ejpam-3971	111	7	the	the	DET
ejpam-3971	111	8	above	above	ADJ
ejpam-3971	111	9	theorem	theorem	PROPN
ejpam-3971	111	10	.	.	PUNCT
ejpam-3971	111	11	theorem	theorem	NOUN
ejpam-3971	111	12	4	4	NUM
ejpam-3971	111	13	.	.	X
ejpam-3971	112	1	for	for	ADP
ejpam-3971	112	2	any	any	DET
ejpam-3971	112	3	fuzzy	fuzzy	ADJ
ejpam-3971	112	4	subset	subset	VERB
ejpam-3971	112	5	µ	µ	PROPN
ejpam-3971	112	6	of	of	ADP
ejpam-3971	112	7	x	x	X
ejpam-3971	112	8	and	and	CCONJ
ejpam-3971	112	9	α	α	NOUN
ejpam-3971	112	10	∈	∈	PROPN
ejpam-3971	112	11	[	[	X
ejpam-3971	112	12	0	0	NUM
ejpam-3971	112	13	,	,	PUNCT
ejpam-3971	112	14	t	t	X
ejpam-3971	112	15	]	]	PUNCT
ejpam-3971	112	16	,	,	PUNCT
ejpam-3971	112	17	if	if	SCONJ
ejpam-3971	112	18	the	the	DET
ejpam-3971	112	19	fuzzy	fuzzy	ADJ
ejpam-3971	112	20	αmultiplication	αmultiplication	NOUN
ejpam-3971	112	21	µmα	µmα	X
ejpam-3971	112	22	(	(	PUNCT
ejpam-3971	112	23	x	x	X
ejpam-3971	112	24	)	)	PUNCT
ejpam-3971	112	25	of	of	ADP
ejpam-3971	112	26	µ	µ	PROPN
ejpam-3971	112	27	is	be	AUX
ejpam-3971	112	28	a	a	DET
ejpam-3971	112	29	fuzzy	fuzzy	ADJ
ejpam-3971	112	30	brk	brk	NOUN
ejpam-3971	112	31	-	-	PUNCT
ejpam-3971	112	32	subalgebra	subalgebra	PROPN
ejpam-3971	112	33	of	of	ADP
ejpam-3971	112	34	x	x	PUNCT
ejpam-3971	112	35	then	then	ADV
ejpam-3971	112	36	so	so	ADV
ejpam-3971	112	37	is	be	AUX
ejpam-3971	112	38	µ.	µ.	NOUN
ejpam-3971	112	39	proof	proof	NOUN
ejpam-3971	112	40	.	.	PUNCT
ejpam-3971	113	1	let	let	VERB
ejpam-3971	113	2	x	x	PRON
ejpam-3971	113	3	,	,	PUNCT
ejpam-3971	113	4	y	y	PROPN
ejpam-3971	113	5	∈	∈	PROPN
ejpam-3971	113	6	x.	x.	NOUN
ejpam-3971	113	7	assume	assume	VERB
ejpam-3971	113	8	that	that	SCONJ
ejpam-3971	113	9	µmα	µmα	PROPN
ejpam-3971	113	10	(	(	PUNCT
ejpam-3971	113	11	x	x	X
ejpam-3971	113	12	)	)	PUNCT
ejpam-3971	113	13	of	of	ADP
ejpam-3971	113	14	µ	µ	PROPN
ejpam-3971	113	15	is	be	AUX
ejpam-3971	113	16	a	a	DET
ejpam-3971	113	17	fuzzy	fuzzy	ADJ
ejpam-3971	113	18	brk	brk	NOUN
ejpam-3971	113	19	-	-	PUNCT
ejpam-3971	113	20	subalgebra	subalgebra	PROPN
ejpam-3971	113	21	of	of	ADP
ejpam-3971	113	22	x	x	PUNCT
ejpam-3971	113	23	for	for	ADP
ejpam-3971	113	24	α	α	PRON
ejpam-3971	113	25	∈	∈	PROPN
ejpam-3971	114	1	[	[	X
ejpam-3971	114	2	0	0	NUM
ejpam-3971	114	3	,	,	PUNCT
ejpam-3971	114	4	1	1	NUM
ejpam-3971	114	5	]	]	PUNCT
ejpam-3971	114	6	.	.	PUNCT
ejpam-3971	115	1	then	then	ADV
ejpam-3971	115	2	α.µ(x	α.µ(x	NUM
ejpam-3971	115	3	∗	∗	NOUN
ejpam-3971	115	4	y	y	NOUN
ejpam-3971	115	5	)	)	PUNCT
ejpam-3971	116	1	=	=	SYM
ejpam-3971	116	2	µmα	µmα	NOUN
ejpam-3971	116	3	(	(	PUNCT
ejpam-3971	116	4	x	x	X
ejpam-3971	116	5	∗	∗	PROPN
ejpam-3971	116	6	y	y	PROPN
ejpam-3971	116	7	)	)	PUNCT
ejpam-3971	116	8	≥	≥	NOUN
ejpam-3971	116	9	min{µm	min{µm	NOUN
ejpam-3971	116	10	(	(	PUNCT
ejpam-3971	116	11	x	x	NOUN
ejpam-3971	116	12	)	)	PUNCT
ejpam-3971	116	13	,	,	PUNCT
ejpam-3971	116	14	µm	µm	ADP
ejpam-3971	116	15	(	(	PUNCT
ejpam-3971	116	16	y	y	NOUN
ejpam-3971	116	17	)	)	PUNCT
ejpam-3971	116	18	}	}	PUNCT
ejpam-3971	116	19	=	=	SYM
ejpam-3971	116	20	min{α.µ(x	min{α.µ(x	PROPN
ejpam-3971	116	21	)	)	PUNCT
ejpam-3971	116	22	,	,	PUNCT
ejpam-3971	116	23	α.µ(y	α.µ(y	NUM
ejpam-3971	116	24	)	)	PUNCT
ejpam-3971	116	25	}	}	PUNCT
ejpam-3971	116	26	=	=	SYM
ejpam-3971	116	27	α.min{µ(x	α.min{µ(x	NOUN
ejpam-3971	116	28	)	)	PUNCT
ejpam-3971	116	29	,	,	PUNCT
ejpam-3971	116	30	µ(y	µ(y	PROPN
ejpam-3971	116	31	)	)	PUNCT
ejpam-3971	116	32	}	}	PUNCT
ejpam-3971	116	33	hence	hence	ADV
ejpam-3971	116	34	,	,	PUNCT
ejpam-3971	116	35	µ(x	µ(x	ADJ
ejpam-3971	116	36	∗	∗	NOUN
ejpam-3971	116	37	y	y	NOUN
ejpam-3971	116	38	)	)	PUNCT
ejpam-3971	116	39	≥	≥	NOUN
ejpam-3971	116	40	min{µ(x	min{µ(x	NOUN
ejpam-3971	116	41	)	)	PUNCT
ejpam-3971	116	42	,	,	PUNCT
ejpam-3971	116	43	µ(y	µ(y	PROPN
ejpam-3971	116	44	)	)	PUNCT
ejpam-3971	116	45	}	}	PUNCT
ejpam-3971	116	46	therefore	therefore	ADV
ejpam-3971	116	47	,	,	PUNCT
ejpam-3971	116	48	µ	µ	X
ejpam-3971	116	49	is	be	AUX
ejpam-3971	116	50	a	a	DET
ejpam-3971	116	51	fuzzy	fuzzy	ADJ
ejpam-3971	116	52	brk	brk	NOUN
ejpam-3971	116	53	-	-	PUNCT
ejpam-3971	116	54	subalgebra	subalgebra	PROPN
ejpam-3971	116	55	of	of	ADP
ejpam-3971	116	56	x.	x.	NOUN
ejpam-3971	116	57	4	4	NUM
ejpam-3971	116	58	.	.	PUNCT
ejpam-3971	116	59	fuzzy	fuzzy	ADJ
ejpam-3971	116	60	translation	translation	NOUN
ejpam-3971	116	61	and	and	CCONJ
ejpam-3971	116	62	fuzzy	fuzzy	ADJ
ejpam-3971	116	63	multiplication	multiplication	NOUN
ejpam-3971	116	64	of	of	ADP
ejpam-3971	116	65	brk	brk	PROPN
ejpam-3971	116	66	-	-	PUNCT
ejpam-3971	116	67	ideals	ideal	NOUN
ejpam-3971	116	68	definition	definition	NOUN
ejpam-3971	116	69	11	11	NUM
ejpam-3971	116	70	.	.	PUNCT
ejpam-3971	117	1	a	a	DET
ejpam-3971	117	2	fuzzy	fuzzy	ADJ
ejpam-3971	117	3	α	α	NOUN
ejpam-3971	117	4	-	-	NOUN
ejpam-3971	117	5	translation	translation	NOUN
ejpam-3971	117	6	set	set	VERB
ejpam-3971	117	7	µtα(x	µtα(x	PROPN
ejpam-3971	117	8	)	)	PUNCT
ejpam-3971	117	9	of	of	ADP
ejpam-3971	117	10	µ	µ	PROPN
ejpam-3971	117	11	is	be	AUX
ejpam-3971	117	12	called	call	VERB
ejpam-3971	117	13	fuzzy	fuzzy	ADJ
ejpam-3971	117	14	α	α	NOUN
ejpam-3971	117	15	-	-	PUNCT
ejpam-3971	117	16	translation	translation	NOUN
ejpam-3971	117	17	brkideal	brkideal	NOUN
ejpam-3971	117	18	of	of	ADP
ejpam-3971	117	19	x	x	PRON
ejpam-3971	117	20	if	if	SCONJ
ejpam-3971	117	21	it	it	PRON
ejpam-3971	117	22	satisfies	satisfy	VERB
ejpam-3971	117	23	following	follow	VERB
ejpam-3971	117	24	condition	condition	NOUN
ejpam-3971	117	25	:	:	PUNCT
ejpam-3971	117	26	(	(	PUNCT
ejpam-3971	117	27	fti1	fti1	NOUN
ejpam-3971	117	28	)	)	PUNCT
ejpam-3971	117	29	µtα(0	µtα(0	NUM
ejpam-3971	117	30	)	)	PUNCT
ejpam-3971	117	31	≥	≥	PROPN
ejpam-3971	117	32	µtα(x	µtα(x	PROPN
ejpam-3971	117	33	)	)	PUNCT
ejpam-3971	117	34	,	,	PUNCT
ejpam-3971	117	35	h.	h.	PROPN
ejpam-3971	117	36	alshehri	alshehri	PROPN
ejpam-3971	117	37	/	/	SYM
ejpam-3971	117	38	eur	eur	PROPN
ejpam-3971	117	39	.	.	PUNCT
ejpam-3971	118	1	j.	j.	PROPN
ejpam-3971	118	2	pure	pure	PROPN
ejpam-3971	118	3	appl	appl	PROPN
ejpam-3971	118	4	.	.	PROPN
ejpam-3971	118	5	math	math	PROPN
ejpam-3971	118	6	,	,	PUNCT
ejpam-3971	118	7	14	14	NUM
ejpam-3971	118	8	(	(	PUNCT
ejpam-3971	118	9	3	3	NUM
ejpam-3971	118	10	)	)	PUNCT
ejpam-3971	118	11	(	(	PUNCT
ejpam-3971	118	12	2021	2021	NUM
ejpam-3971	118	13	)	)	PUNCT
ejpam-3971	118	14	,	,	PUNCT
ejpam-3971	119	1	737	737	NUM
ejpam-3971	119	2	-	-	SYM
ejpam-3971	119	3	745	745	NUM
ejpam-3971	119	4	741	741	NUM
ejpam-3971	119	5	(	(	PUNCT
ejpam-3971	119	6	fti2	fti2	PROPN
ejpam-3971	119	7	)	)	PUNCT
ejpam-3971	119	8	µtα(0	µtα(0	ADJ
ejpam-3971	119	9	∗	∗	NOUN
ejpam-3971	119	10	x	x	NOUN
ejpam-3971	119	11	)	)	PUNCT
ejpam-3971	119	12	≥	≥	NOUN
ejpam-3971	119	13	min{µtα(0	min{µtα(0	PUNCT
ejpam-3971	119	14	∗	∗	NOUN
ejpam-3971	119	15	(	(	PUNCT
ejpam-3971	119	16	x	x	X
ejpam-3971	119	17	∗	∗	PROPN
ejpam-3971	119	18	y	y	PROPN
ejpam-3971	119	19	)	)	PUNCT
ejpam-3971	119	20	)	)	PUNCT
ejpam-3971	119	21	,	,	PUNCT
ejpam-3971	119	22	µtα(0	µtα(0	PROPN
ejpam-3971	119	23	∗	∗	NOUN
ejpam-3971	119	24	y	y	PROPN
ejpam-3971	119	25	)	)	PUNCT
ejpam-3971	119	26	}	}	PUNCT
ejpam-3971	119	27	,	,	PUNCT
ejpam-3971	119	28	∀x	∀x	X
ejpam-3971	119	29	,	,	PUNCT
ejpam-3971	119	30	y	y	PROPN
ejpam-3971	119	31	∈	∈	PROPN
ejpam-3971	119	32	x.	x.	NOUN
ejpam-3971	119	33	similarly	similarly	ADV
ejpam-3971	119	34	,	,	PUNCT
ejpam-3971	119	35	we	we	PRON
ejpam-3971	119	36	said	say	VERB
ejpam-3971	119	37	that	that	SCONJ
ejpam-3971	119	38	µtα(x	µtα(x	PROPN
ejpam-3971	119	39	)	)	PUNCT
ejpam-3971	119	40	is	be	AUX
ejpam-3971	119	41	fuzzy	fuzzy	ADJ
ejpam-3971	119	42	α	α	DET
ejpam-3971	119	43	-	-	PUNCT
ejpam-3971	119	44	multiplication	multiplication	NOUN
ejpam-3971	119	45	brk	brk	PROPN
ejpam-3971	119	46	-	-	PUNCT
ejpam-3971	119	47	ideal	ideal	NOUN
ejpam-3971	119	48	of	of	ADP
ejpam-3971	119	49	x	x	PRON
ejpam-3971	119	50	if	if	SCONJ
ejpam-3971	119	51	it	it	PRON
ejpam-3971	119	52	satisfies	satisfy	VERB
ejpam-3971	119	53	:	:	PUNCT
ejpam-3971	119	54	(	(	PUNCT
ejpam-3971	119	55	fmi1	fmi1	NOUN
ejpam-3971	119	56	)	)	PUNCT
ejpam-3971	120	1	µmα	µmα	X
ejpam-3971	120	2	(	(	PUNCT
ejpam-3971	120	3	0	0	NUM
ejpam-3971	120	4	)	)	PUNCT
ejpam-3971	120	5	≥	≥	NOUN
ejpam-3971	120	6	µmα	µmα	X
ejpam-3971	120	7	(	(	PUNCT
ejpam-3971	120	8	x	x	X
ejpam-3971	120	9	)	)	PUNCT
ejpam-3971	120	10	,	,	PUNCT
ejpam-3971	120	11	(	(	PUNCT
ejpam-3971	120	12	fmi2	fmi2	PROPN
ejpam-3971	120	13	)	)	PUNCT
ejpam-3971	120	14	µmα	µmα	PROPN
ejpam-3971	120	15	(	(	PUNCT
ejpam-3971	120	16	0	0	NUM
ejpam-3971	120	17	∗	∗	NOUN
ejpam-3971	120	18	x	x	NOUN
ejpam-3971	120	19	)	)	PUNCT
ejpam-3971	120	20	≥	≥	PROPN
ejpam-3971	120	21	min{µmα	min{µmα	NOUN
ejpam-3971	120	22	(	(	PUNCT
ejpam-3971	120	23	0	0	NUM
ejpam-3971	120	24	∗	∗	NOUN
ejpam-3971	120	25	(	(	PUNCT
ejpam-3971	120	26	x	x	X
ejpam-3971	120	27	∗	∗	PROPN
ejpam-3971	120	28	y	y	PROPN
ejpam-3971	120	29	)	)	PUNCT
ejpam-3971	120	30	)	)	PUNCT
ejpam-3971	120	31	,	,	PUNCT
ejpam-3971	120	32	µmα	µmα	X
ejpam-3971	120	33	(	(	PUNCT
ejpam-3971	120	34	0	0	NUM
ejpam-3971	120	35	∗	∗	NOUN
ejpam-3971	120	36	y)},∀x	y)},∀x	NUM
ejpam-3971	120	37	,	,	PUNCT
ejpam-3971	120	38	y	y	PROPN
ejpam-3971	120	39	∈	∈	PROPN
ejpam-3971	120	40	x.	x.	NOUN
ejpam-3971	120	41	theorem	theorem	VERB
ejpam-3971	120	42	5	5	NUM
ejpam-3971	120	43	.	.	X
ejpam-3971	121	1	for	for	ADP
ejpam-3971	121	2	any	any	DET
ejpam-3971	121	3	fuzzy	fuzzy	ADJ
ejpam-3971	121	4	brk	brk	PROPN
ejpam-3971	121	5	-	-	PUNCT
ejpam-3971	121	6	ideal	ideal	PROPN
ejpam-3971	121	7	µ	µ	PROPN
ejpam-3971	121	8	of	of	ADP
ejpam-3971	121	9	x	x	X
ejpam-3971	121	10	and	and	CCONJ
ejpam-3971	121	11	α	α	NOUN
ejpam-3971	121	12	∈	∈	PROPN
ejpam-3971	122	1	[	[	X
ejpam-3971	122	2	0	0	NUM
ejpam-3971	122	3	,	,	PUNCT
ejpam-3971	122	4	t	t	X
ejpam-3971	122	5	]	]	PUNCT
ejpam-3971	122	6	,	,	PUNCT
ejpam-3971	122	7	the	the	DET
ejpam-3971	122	8	fuzzy	fuzzy	ADJ
ejpam-3971	122	9	α	α	NOUN
ejpam-3971	122	10	-	-	NOUN
ejpam-3971	122	11	translation	translation	NOUN
ejpam-3971	122	12	µtα(x	µtα(x	PROPN
ejpam-3971	122	13	)	)	PUNCT
ejpam-3971	122	14	of	of	ADP
ejpam-3971	122	15	µ	µ	PROPN
ejpam-3971	122	16	is	be	AUX
ejpam-3971	122	17	a	a	DET
ejpam-3971	122	18	fuzzy	fuzzy	ADJ
ejpam-3971	122	19	brk	brk	PROPN
ejpam-3971	122	20	-	-	PUNCT
ejpam-3971	122	21	ideal	ideal	NOUN
ejpam-3971	122	22	of	of	ADP
ejpam-3971	122	23	x.	x.	NOUN
ejpam-3971	122	24	proof	proof	NOUN
ejpam-3971	122	25	.	.	PUNCT
ejpam-3971	123	1	let	let	VERB
ejpam-3971	123	2	x	x	PRON
ejpam-3971	123	3	,	,	PUNCT
ejpam-3971	123	4	y	y	PROPN
ejpam-3971	123	5	∈	∈	PROPN
ejpam-3971	123	6	x	x	X
ejpam-3971	123	7	and	and	CCONJ
ejpam-3971	123	8	α	α	NOUN
ejpam-3971	123	9	∈	∈	PROPN
ejpam-3971	124	1	[	[	X
ejpam-3971	124	2	0	0	NUM
ejpam-3971	124	3	,	,	PUNCT
ejpam-3971	124	4	t	t	X
ejpam-3971	124	5	]	]	PUNCT
ejpam-3971	124	6	.	.	PUNCT
ejpam-3971	125	1	then	then	ADV
ejpam-3971	125	2	µ(0	µ(0	PROPN
ejpam-3971	125	3	∗	∗	NOUN
ejpam-3971	125	4	x	x	NOUN
ejpam-3971	125	5	)	)	PUNCT
ejpam-3971	125	6	≥	≥	NOUN
ejpam-3971	125	7	min{µ(0	min{µ(0	NOUN
ejpam-3971	125	8	∗	∗	NOUN
ejpam-3971	125	9	(	(	PUNCT
ejpam-3971	125	10	x	x	X
ejpam-3971	125	11	∗	∗	PROPN
ejpam-3971	125	12	y	y	PROPN
ejpam-3971	125	13	)	)	PUNCT
ejpam-3971	125	14	)	)	PUNCT
ejpam-3971	125	15	,	,	PUNCT
ejpam-3971	125	16	µ(0	µ(0	NOUN
ejpam-3971	125	17	∗	∗	PROPN
ejpam-3971	125	18	y	y	NOUN
ejpam-3971	125	19	)	)	PUNCT
ejpam-3971	125	20	}	}	PUNCT
ejpam-3971	125	21	now	now	ADV
ejpam-3971	125	22	,	,	PUNCT
ejpam-3971	125	23	µtα(0	µtα(0	ADJ
ejpam-3971	125	24	∗	∗	NOUN
ejpam-3971	125	25	x	x	NOUN
ejpam-3971	125	26	)	)	PUNCT
ejpam-3971	125	27	=	=	PUNCT
ejpam-3971	126	1	µ(0	µ(0	NOUN
ejpam-3971	126	2	∗	∗	NOUN
ejpam-3971	126	3	x	x	NOUN
ejpam-3971	126	4	)	)	PUNCT
ejpam-3971	127	1	+	+	CCONJ
ejpam-3971	127	2	α	α	PRON
ejpam-3971	127	3	≥	≥	NOUN
ejpam-3971	127	4	min{µ(0	min{µ(0	NOUN
ejpam-3971	127	5	∗	∗	NOUN
ejpam-3971	127	6	(	(	PUNCT
ejpam-3971	127	7	x	x	X
ejpam-3971	127	8	∗	∗	PROPN
ejpam-3971	127	9	y	y	PROPN
ejpam-3971	127	10	)	)	PUNCT
ejpam-3971	127	11	)	)	PUNCT
ejpam-3971	127	12	,	,	PUNCT
ejpam-3971	127	13	µ(0	µ(0	NOUN
ejpam-3971	127	14	∗	∗	VERB
ejpam-3971	127	15	y)}+	y)}+	PROPN
ejpam-3971	128	1	α	α	NOUN
ejpam-3971	128	2	=	=	NOUN
ejpam-3971	128	3	min{µ(0	min{µ(0	NOUN
ejpam-3971	128	4	∗	∗	NOUN
ejpam-3971	128	5	(	(	PUNCT
ejpam-3971	128	6	x	x	X
ejpam-3971	128	7	∗	∗	PROPN
ejpam-3971	128	8	y	y	PROPN
ejpam-3971	128	9	)	)	PUNCT
ejpam-3971	128	10	)	)	PUNCT
ejpam-3971	129	1	+	+	CCONJ
ejpam-3971	129	2	α	α	X
ejpam-3971	129	3	,	,	PUNCT
ejpam-3971	129	4	µ(0	µ(0	PROPN
ejpam-3971	129	5	∗	∗	NOUN
ejpam-3971	129	6	y	y	NOUN
ejpam-3971	129	7	)	)	PUNCT
ejpam-3971	130	1	+	+	CCONJ
ejpam-3971	130	2	α	α	X
ejpam-3971	130	3	}	}	PUNCT
ejpam-3971	130	4	=	=	ADJ
ejpam-3971	130	5	min{µmα	min{µmα	NOUN
ejpam-3971	130	6	(	(	PUNCT
ejpam-3971	130	7	0	0	NUM
ejpam-3971	130	8	∗	∗	NOUN
ejpam-3971	130	9	(	(	PUNCT
ejpam-3971	130	10	x	x	X
ejpam-3971	130	11	∗	∗	PROPN
ejpam-3971	130	12	y	y	PROPN
ejpam-3971	130	13	)	)	PUNCT
ejpam-3971	130	14	)	)	PUNCT
ejpam-3971	130	15	,	,	PUNCT
ejpam-3971	130	16	µmα	µmα	X
ejpam-3971	130	17	(	(	PUNCT
ejpam-3971	130	18	0	0	NUM
ejpam-3971	130	19	∗	∗	PROPN
ejpam-3971	130	20	y	y	PROPN
ejpam-3971	130	21	)	)	PUNCT
ejpam-3971	130	22	}	}	PUNCT
ejpam-3971	130	23	hence	hence	ADV
ejpam-3971	130	24	,	,	PUNCT
ejpam-3971	130	25	µtα(x	µtα(x	PROPN
ejpam-3971	130	26	)	)	PUNCT
ejpam-3971	130	27	is	be	AUX
ejpam-3971	130	28	a	a	DET
ejpam-3971	130	29	fuzzy	fuzzy	ADJ
ejpam-3971	130	30	brk	brk	PROPN
ejpam-3971	130	31	-	-	PUNCT
ejpam-3971	130	32	ideal	ideal	NOUN
ejpam-3971	130	33	of	of	ADP
ejpam-3971	130	34	x.	x.	NOUN
ejpam-3971	130	35	the	the	DET
ejpam-3971	130	36	following	follow	VERB
ejpam-3971	130	37	is	be	AUX
ejpam-3971	130	38	the	the	DET
ejpam-3971	130	39	converse	converse	NOUN
ejpam-3971	130	40	of	of	ADP
ejpam-3971	130	41	the	the	DET
ejpam-3971	130	42	above	above	ADJ
ejpam-3971	130	43	theorem	theorem	PROPN
ejpam-3971	130	44	.	.	PUNCT
ejpam-3971	131	1	theorem	theorem	VERB
ejpam-3971	131	2	6	6	NUM
ejpam-3971	131	3	.	.	PUNCT
ejpam-3971	132	1	for	for	ADP
ejpam-3971	132	2	any	any	DET
ejpam-3971	132	3	fuzzy	fuzzy	ADJ
ejpam-3971	132	4	subset	subset	VERB
ejpam-3971	132	5	µ	µ	PROPN
ejpam-3971	132	6	of	of	ADP
ejpam-3971	132	7	x	x	X
ejpam-3971	132	8	and	and	CCONJ
ejpam-3971	132	9	α	α	NOUN
ejpam-3971	132	10	∈	∈	PROPN
ejpam-3971	133	1	[	[	X
ejpam-3971	133	2	0	0	NUM
ejpam-3971	133	3	,	,	PUNCT
ejpam-3971	133	4	t	t	X
ejpam-3971	133	5	]	]	PUNCT
ejpam-3971	133	6	,	,	PUNCT
ejpam-3971	133	7	if	if	SCONJ
ejpam-3971	133	8	the	the	DET
ejpam-3971	133	9	fuzzy	fuzzy	ADJ
ejpam-3971	133	10	α	α	NOUN
ejpam-3971	133	11	-	-	NOUN
ejpam-3971	133	12	translation	translation	NOUN
ejpam-3971	133	13	µtα(x	µtα(x	PROPN
ejpam-3971	133	14	)	)	PUNCT
ejpam-3971	133	15	of	of	ADP
ejpam-3971	133	16	µ	µ	PROPN
ejpam-3971	133	17	is	be	AUX
ejpam-3971	133	18	a	a	DET
ejpam-3971	133	19	fuzzy	fuzzy	ADJ
ejpam-3971	133	20	brk	brk	PROPN
ejpam-3971	133	21	-	-	PUNCT
ejpam-3971	133	22	ideal	ideal	NOUN
ejpam-3971	133	23	of	of	ADP
ejpam-3971	133	24	x	x	PRON
ejpam-3971	133	25	then	then	ADV
ejpam-3971	133	26	so	so	ADV
ejpam-3971	133	27	is	be	AUX
ejpam-3971	133	28	µ.	µ.	NOUN
ejpam-3971	133	29	proof	proof	NOUN
ejpam-3971	133	30	.	.	PUNCT
ejpam-3971	134	1	let	let	VERB
ejpam-3971	134	2	x	x	PRON
ejpam-3971	134	3	,	,	PUNCT
ejpam-3971	134	4	y	y	PROPN
ejpam-3971	134	5	∈	∈	PROPN
ejpam-3971	134	6	x.	x.	NOUN
ejpam-3971	134	7	assume	assume	VERB
ejpam-3971	134	8	that	that	SCONJ
ejpam-3971	134	9	µtα(x	µtα(x	PROPN
ejpam-3971	134	10	)	)	PUNCT
ejpam-3971	134	11	of	of	ADP
ejpam-3971	134	12	µ	µ	NOUN
ejpam-3971	134	13	for	for	ADP
ejpam-3971	134	14	α	α	PRON
ejpam-3971	134	15	∈	∈	PROPN
ejpam-3971	135	1	[	[	X
ejpam-3971	135	2	0	0	NUM
ejpam-3971	135	3	,	,	PUNCT
ejpam-3971	135	4	1	1	NUM
ejpam-3971	135	5	]	]	PUNCT
ejpam-3971	135	6	.	.	PUNCT
ejpam-3971	136	1	then	then	ADV
ejpam-3971	136	2	µ(0	µ(0	PROPN
ejpam-3971	136	3	∗	∗	NOUN
ejpam-3971	136	4	x	x	NOUN
ejpam-3971	136	5	)	)	PUNCT
ejpam-3971	137	1	+	+	CCONJ
ejpam-3971	137	2	α	α	NOUN
ejpam-3971	137	3	=	=	PUNCT
ejpam-3971	137	4	µtα(0	µtα(0	ADJ
ejpam-3971	137	5	∗	∗	NOUN
ejpam-3971	137	6	x	x	NOUN
ejpam-3971	137	7	)	)	PUNCT
ejpam-3971	137	8	≥	≥	PROPN
ejpam-3971	137	9	min{µmα	min{µmα	NOUN
ejpam-3971	137	10	(	(	PUNCT
ejpam-3971	137	11	0	0	NUM
ejpam-3971	137	12	∗	∗	NOUN
ejpam-3971	137	13	(	(	PUNCT
ejpam-3971	137	14	x	x	X
ejpam-3971	137	15	∗	∗	PROPN
ejpam-3971	137	16	y	y	PROPN
ejpam-3971	137	17	)	)	PUNCT
ejpam-3971	137	18	)	)	PUNCT
ejpam-3971	137	19	,	,	PUNCT
ejpam-3971	137	20	µmα	µmα	X
ejpam-3971	137	21	(	(	PUNCT
ejpam-3971	137	22	0	0	NUM
ejpam-3971	137	23	∗	∗	PROPN
ejpam-3971	137	24	y	y	NOUN
ejpam-3971	137	25	)	)	PUNCT
ejpam-3971	137	26	}	}	PUNCT
ejpam-3971	137	27	=	=	PUNCT
ejpam-3971	137	28	min{µ(0	min{µ(0	NOUN
ejpam-3971	137	29	∗	∗	NOUN
ejpam-3971	137	30	(	(	PUNCT
ejpam-3971	137	31	x	x	X
ejpam-3971	137	32	∗	∗	PROPN
ejpam-3971	137	33	y	y	PROPN
ejpam-3971	137	34	)	)	PUNCT
ejpam-3971	137	35	)	)	PUNCT
ejpam-3971	138	1	+	+	CCONJ
ejpam-3971	138	2	α	α	X
ejpam-3971	138	3	,	,	PUNCT
ejpam-3971	138	4	µ(0	µ(0	PROPN
ejpam-3971	138	5	∗	∗	NOUN
ejpam-3971	138	6	y	y	NOUN
ejpam-3971	138	7	)	)	PUNCT
ejpam-3971	139	1	+	+	CCONJ
ejpam-3971	139	2	α	α	X
ejpam-3971	139	3	}	}	PUNCT
ejpam-3971	139	4	=	=	PUNCT
ejpam-3971	139	5	min{µ(0	min{µ(0	NOUN
ejpam-3971	139	6	∗	∗	NOUN
ejpam-3971	139	7	(	(	PUNCT
ejpam-3971	139	8	x	x	X
ejpam-3971	139	9	∗	∗	PROPN
ejpam-3971	139	10	y	y	PROPN
ejpam-3971	139	11	)	)	PUNCT
ejpam-3971	139	12	)	)	PUNCT
ejpam-3971	139	13	,	,	PUNCT
ejpam-3971	139	14	µ(0	µ(0	NOUN
ejpam-3971	139	15	∗	∗	VERB
ejpam-3971	139	16	y)}+	y)}+	NUM
ejpam-3971	139	17	α	α	NOUN
ejpam-3971	139	18	hence	hence	ADV
ejpam-3971	139	19	,	,	PUNCT
ejpam-3971	139	20	µ(0	µ(0	NOUN
ejpam-3971	139	21	∗	∗	NOUN
ejpam-3971	139	22	x	x	NOUN
ejpam-3971	139	23	)	)	PUNCT
ejpam-3971	139	24	≥	≥	NOUN
ejpam-3971	139	25	min{µ(0	min{µ(0	NOUN
ejpam-3971	139	26	∗	∗	NOUN
ejpam-3971	139	27	(	(	PUNCT
ejpam-3971	139	28	x	x	X
ejpam-3971	139	29	∗	∗	PROPN
ejpam-3971	139	30	y	y	PROPN
ejpam-3971	139	31	)	)	PUNCT
ejpam-3971	139	32	)	)	PUNCT
ejpam-3971	139	33	,	,	PUNCT
ejpam-3971	139	34	µ(0	µ(0	NOUN
ejpam-3971	139	35	∗	∗	PROPN
ejpam-3971	139	36	y	y	PROPN
ejpam-3971	139	37	)	)	PUNCT
ejpam-3971	139	38	}	}	PUNCT
ejpam-3971	139	39	.	.	PUNCT
ejpam-3971	140	1	therefore	therefore	ADV
ejpam-3971	140	2	,	,	PUNCT
ejpam-3971	140	3	µ	µ	X
ejpam-3971	140	4	is	be	AUX
ejpam-3971	140	5	a	a	DET
ejpam-3971	140	6	fuzzy	fuzzy	ADJ
ejpam-3971	140	7	brk	brk	PROPN
ejpam-3971	140	8	-	-	PUNCT
ejpam-3971	140	9	ideal	ideal	NOUN
ejpam-3971	140	10	of	of	ADP
ejpam-3971	140	11	x.	x.	PROPN
ejpam-3971	140	12	theorem	theorem	VERB
ejpam-3971	140	13	7	7	NUM
ejpam-3971	140	14	.	.	X
ejpam-3971	140	15	for	for	ADP
ejpam-3971	140	16	any	any	DET
ejpam-3971	140	17	fuzzy	fuzzy	ADJ
ejpam-3971	140	18	brk	brk	PROPN
ejpam-3971	140	19	-	-	PUNCT
ejpam-3971	140	20	ideal	ideal	PROPN
ejpam-3971	140	21	µ	µ	PROPN
ejpam-3971	140	22	of	of	ADP
ejpam-3971	140	23	x	x	X
ejpam-3971	140	24	and	and	CCONJ
ejpam-3971	140	25	α	α	NOUN
ejpam-3971	140	26	∈	∈	PROPN
ejpam-3971	141	1	[	[	X
ejpam-3971	141	2	0	0	NUM
ejpam-3971	141	3	,	,	PUNCT
ejpam-3971	141	4	t	t	X
ejpam-3971	141	5	]	]	PUNCT
ejpam-3971	141	6	,	,	PUNCT
ejpam-3971	141	7	the	the	DET
ejpam-3971	141	8	fuzzy	fuzzy	ADJ
ejpam-3971	141	9	α	α	NOUN
ejpam-3971	141	10	-	-	PUNCT
ejpam-3971	141	11	multiplication	multiplication	NOUN
ejpam-3971	141	12	µmα	µmα	NOUN
ejpam-3971	141	13	(	(	PUNCT
ejpam-3971	141	14	x	x	X
ejpam-3971	141	15	)	)	PUNCT
ejpam-3971	141	16	of	of	ADP
ejpam-3971	141	17	µ	µ	PROPN
ejpam-3971	141	18	is	be	AUX
ejpam-3971	141	19	a	a	DET
ejpam-3971	141	20	fuzzy	fuzzy	ADJ
ejpam-3971	141	21	brk	brk	PROPN
ejpam-3971	141	22	-	-	PUNCT
ejpam-3971	141	23	ideal	ideal	NOUN
ejpam-3971	141	24	of	of	ADP
ejpam-3971	141	25	x.	x.	NOUN
ejpam-3971	141	26	proof	proof	NOUN
ejpam-3971	141	27	.	.	PUNCT
ejpam-3971	142	1	let	let	VERB
ejpam-3971	142	2	x	x	PRON
ejpam-3971	142	3	,	,	PUNCT
ejpam-3971	142	4	y	y	PROPN
ejpam-3971	142	5	∈	∈	PROPN
ejpam-3971	142	6	x	x	X
ejpam-3971	142	7	and	and	CCONJ
ejpam-3971	142	8	α	α	NOUN
ejpam-3971	142	9	∈	∈	PROPN
ejpam-3971	143	1	[	[	X
ejpam-3971	143	2	0	0	NUM
ejpam-3971	143	3	,	,	PUNCT
ejpam-3971	143	4	t	t	X
ejpam-3971	143	5	]	]	PUNCT
ejpam-3971	143	6	.	.	PUNCT
ejpam-3971	144	1	then	then	ADV
ejpam-3971	144	2	µ(0	µ(0	PROPN
ejpam-3971	144	3	∗	∗	NOUN
ejpam-3971	144	4	x	x	NOUN
ejpam-3971	144	5	)	)	PUNCT
ejpam-3971	144	6	≥	≥	NOUN
ejpam-3971	144	7	min{µ(0	min{µ(0	NOUN
ejpam-3971	144	8	∗	∗	NOUN
ejpam-3971	144	9	(	(	PUNCT
ejpam-3971	144	10	x	x	X
ejpam-3971	144	11	∗	∗	PROPN
ejpam-3971	144	12	y	y	PROPN
ejpam-3971	144	13	)	)	PUNCT
ejpam-3971	144	14	)	)	PUNCT
ejpam-3971	144	15	,	,	PUNCT
ejpam-3971	144	16	µ(0	µ(0	NOUN
ejpam-3971	144	17	∗	∗	PROPN
ejpam-3971	144	18	y	y	NOUN
ejpam-3971	144	19	)	)	PUNCT
ejpam-3971	144	20	}	}	PUNCT
ejpam-3971	144	21	h.	h.	PROPN
ejpam-3971	144	22	alshehri	alshehri	PROPN
ejpam-3971	144	23	/	/	SYM
ejpam-3971	144	24	eur	eur	PROPN
ejpam-3971	144	25	.	.	PUNCT
ejpam-3971	145	1	j.	j.	PROPN
ejpam-3971	145	2	pure	pure	PROPN
ejpam-3971	145	3	appl	appl	PROPN
ejpam-3971	145	4	.	.	PROPN
ejpam-3971	145	5	math	math	PROPN
ejpam-3971	145	6	,	,	PUNCT
ejpam-3971	145	7	14	14	NUM
ejpam-3971	145	8	(	(	PUNCT
ejpam-3971	145	9	3	3	NUM
ejpam-3971	145	10	)	)	PUNCT
ejpam-3971	145	11	(	(	PUNCT
ejpam-3971	145	12	2021	2021	NUM
ejpam-3971	145	13	)	)	PUNCT
ejpam-3971	145	14	,	,	PUNCT
ejpam-3971	145	15	737	737	NUM
ejpam-3971	145	16	-	-	SYM
ejpam-3971	145	17	745	745	NUM
ejpam-3971	145	18	742	742	NUM
ejpam-3971	145	19	now	now	ADV
ejpam-3971	145	20	,	,	PUNCT
ejpam-3971	145	21	µmα	µmα	X
ejpam-3971	145	22	(	(	PUNCT
ejpam-3971	145	23	0	0	NUM
ejpam-3971	145	24	∗	∗	NOUN
ejpam-3971	145	25	x	x	NOUN
ejpam-3971	145	26	)	)	PUNCT
ejpam-3971	145	27	=	=	SYM
ejpam-3971	145	28	α.µ(0	α.µ(0	PROPN
ejpam-3971	145	29	∗	∗	NOUN
ejpam-3971	145	30	x	x	NOUN
ejpam-3971	145	31	)	)	PUNCT
ejpam-3971	145	32	≥	≥	PROPN
ejpam-3971	145	33	α.min{µ(0	α.min{µ(0	NOUN
ejpam-3971	145	34	∗	∗	NOUN
ejpam-3971	145	35	(	(	PUNCT
ejpam-3971	145	36	x	x	X
ejpam-3971	145	37	∗	∗	PROPN
ejpam-3971	145	38	y	y	PROPN
ejpam-3971	145	39	)	)	PUNCT
ejpam-3971	145	40	)	)	PUNCT
ejpam-3971	145	41	,	,	PUNCT
ejpam-3971	145	42	µ(0	µ(0	NOUN
ejpam-3971	145	43	∗	∗	PROPN
ejpam-3971	145	44	y	y	NOUN
ejpam-3971	145	45	)	)	PUNCT
ejpam-3971	145	46	}	}	PUNCT
ejpam-3971	146	1	=	=	SYM
ejpam-3971	146	2	min{α.µ(0	min{α.µ(0	NOUN
ejpam-3971	146	3	∗	∗	NOUN
ejpam-3971	146	4	(	(	PUNCT
ejpam-3971	146	5	x	x	X
ejpam-3971	146	6	∗	∗	PROPN
ejpam-3971	146	7	y	y	PROPN
ejpam-3971	146	8	)	)	PUNCT
ejpam-3971	146	9	)	)	PUNCT
ejpam-3971	146	10	,	,	PUNCT
ejpam-3971	146	11	α.µ(0	α.µ(0	PROPN
ejpam-3971	146	12	∗	∗	X
ejpam-3971	146	13	y	y	NOUN
ejpam-3971	146	14	)	)	PUNCT
ejpam-3971	146	15	}	}	PUNCT
ejpam-3971	146	16	=	=	PUNCT
ejpam-3971	146	17	min{µmα	min{µmα	NOUN
ejpam-3971	146	18	(	(	PUNCT
ejpam-3971	146	19	0	0	NUM
ejpam-3971	146	20	∗	∗	NOUN
ejpam-3971	146	21	(	(	PUNCT
ejpam-3971	146	22	x	x	X
ejpam-3971	146	23	∗	∗	PROPN
ejpam-3971	146	24	y	y	PROPN
ejpam-3971	146	25	)	)	PUNCT
ejpam-3971	146	26	)	)	PUNCT
ejpam-3971	146	27	,	,	PUNCT
ejpam-3971	146	28	µmα	µmα	X
ejpam-3971	146	29	(	(	PUNCT
ejpam-3971	146	30	0	0	NUM
ejpam-3971	146	31	∗	∗	PROPN
ejpam-3971	146	32	y	y	PROPN
ejpam-3971	146	33	)	)	PUNCT
ejpam-3971	146	34	}	}	PUNCT
ejpam-3971	146	35	hence	hence	ADV
ejpam-3971	146	36	,	,	PUNCT
ejpam-3971	146	37	µmα	µmα	X
ejpam-3971	146	38	(	(	PUNCT
ejpam-3971	146	39	x	x	X
ejpam-3971	146	40	)	)	PUNCT
ejpam-3971	146	41	is	be	AUX
ejpam-3971	146	42	a	a	DET
ejpam-3971	146	43	fuzzy	fuzzy	ADJ
ejpam-3971	146	44	brk	brk	PROPN
ejpam-3971	146	45	-	-	PUNCT
ejpam-3971	146	46	ideal	ideal	NOUN
ejpam-3971	146	47	of	of	ADP
ejpam-3971	146	48	x.	x.	NOUN
ejpam-3971	146	49	the	the	DET
ejpam-3971	146	50	following	follow	VERB
ejpam-3971	146	51	is	be	AUX
ejpam-3971	146	52	the	the	DET
ejpam-3971	146	53	converse	converse	NOUN
ejpam-3971	146	54	of	of	ADP
ejpam-3971	146	55	the	the	DET
ejpam-3971	146	56	above	above	ADJ
ejpam-3971	146	57	theorem	theorem	PROPN
ejpam-3971	146	58	.	.	PUNCT
ejpam-3971	146	59	theorem	theorem	ADJ
ejpam-3971	146	60	8	8	NUM
ejpam-3971	146	61	.	.	PUNCT
ejpam-3971	147	1	for	for	ADP
ejpam-3971	147	2	any	any	DET
ejpam-3971	147	3	fuzzy	fuzzy	ADJ
ejpam-3971	147	4	subset	subset	VERB
ejpam-3971	147	5	µ	µ	PROPN
ejpam-3971	147	6	of	of	ADP
ejpam-3971	147	7	x	x	X
ejpam-3971	147	8	and	and	CCONJ
ejpam-3971	147	9	α	α	NOUN
ejpam-3971	147	10	∈	∈	PROPN
ejpam-3971	147	11	[	[	X
ejpam-3971	147	12	0	0	NUM
ejpam-3971	147	13	,	,	PUNCT
ejpam-3971	147	14	t	t	X
ejpam-3971	147	15	]	]	PUNCT
ejpam-3971	147	16	,	,	PUNCT
ejpam-3971	147	17	if	if	SCONJ
ejpam-3971	147	18	the	the	DET
ejpam-3971	147	19	fuzzy	fuzzy	ADJ
ejpam-3971	147	20	α	α	NOUN
ejpam-3971	147	21	-	-	PUNCT
ejpam-3971	147	22	multiplication	multiplication	NOUN
ejpam-3971	147	23	µmα	µmα	NOUN
ejpam-3971	147	24	(	(	PUNCT
ejpam-3971	147	25	x	x	X
ejpam-3971	147	26	)	)	PUNCT
ejpam-3971	147	27	of	of	ADP
ejpam-3971	147	28	µ	µ	PROPN
ejpam-3971	147	29	is	be	AUX
ejpam-3971	147	30	a	a	DET
ejpam-3971	147	31	fuzzy	fuzzy	ADJ
ejpam-3971	147	32	brk	brk	PROPN
ejpam-3971	147	33	-	-	PUNCT
ejpam-3971	147	34	ideal	ideal	NOUN
ejpam-3971	147	35	of	of	ADP
ejpam-3971	147	36	x	x	PRON
ejpam-3971	147	37	then	then	ADV
ejpam-3971	147	38	so	so	ADV
ejpam-3971	147	39	is	be	AUX
ejpam-3971	147	40	µ.	µ.	NOUN
ejpam-3971	147	41	proof	proof	NOUN
ejpam-3971	147	42	.	.	PUNCT
ejpam-3971	148	1	let	let	VERB
ejpam-3971	148	2	x	x	PRON
ejpam-3971	148	3	,	,	PUNCT
ejpam-3971	148	4	y	y	PROPN
ejpam-3971	148	5	∈	∈	PROPN
ejpam-3971	148	6	x.	x.	NOUN
ejpam-3971	148	7	suppose	suppose	VERB
ejpam-3971	149	1	that	that	SCONJ
ejpam-3971	149	2	µmα	µmα	PROPN
ejpam-3971	149	3	(	(	PUNCT
ejpam-3971	149	4	x	x	X
ejpam-3971	149	5	)	)	PUNCT
ejpam-3971	149	6	of	of	ADP
ejpam-3971	149	7	µ	µ	NOUN
ejpam-3971	149	8	for	for	ADP
ejpam-3971	149	9	α	α	PRON
ejpam-3971	149	10	∈	∈	PROPN
ejpam-3971	150	1	[	[	X
ejpam-3971	150	2	0	0	NUM
ejpam-3971	150	3	,	,	PUNCT
ejpam-3971	150	4	1	1	NUM
ejpam-3971	150	5	]	]	PUNCT
ejpam-3971	150	6	.	.	PUNCT
ejpam-3971	151	1	then	then	ADV
ejpam-3971	151	2	α.µ(0	α.µ(0	PROPN
ejpam-3971	151	3	∗	∗	NOUN
ejpam-3971	151	4	x	x	NOUN
ejpam-3971	151	5	)	)	PUNCT
ejpam-3971	152	1	=	=	SYM
ejpam-3971	152	2	µmα	µmα	X
ejpam-3971	152	3	(	(	PUNCT
ejpam-3971	152	4	0	0	NUM
ejpam-3971	152	5	∗	∗	NOUN
ejpam-3971	152	6	x	x	NOUN
ejpam-3971	152	7	)	)	PUNCT
ejpam-3971	152	8	≥	≥	PROPN
ejpam-3971	152	9	min{µmα	min{µmα	NOUN
ejpam-3971	152	10	(	(	PUNCT
ejpam-3971	152	11	0	0	NUM
ejpam-3971	152	12	∗	∗	NOUN
ejpam-3971	152	13	(	(	PUNCT
ejpam-3971	152	14	x	x	X
ejpam-3971	152	15	∗	∗	PROPN
ejpam-3971	152	16	y	y	PROPN
ejpam-3971	152	17	)	)	PUNCT
ejpam-3971	152	18	)	)	PUNCT
ejpam-3971	152	19	,	,	PUNCT
ejpam-3971	152	20	µmα	µmα	X
ejpam-3971	152	21	(	(	PUNCT
ejpam-3971	152	22	0	0	NUM
ejpam-3971	152	23	∗	∗	PROPN
ejpam-3971	152	24	y	y	NOUN
ejpam-3971	152	25	)	)	PUNCT
ejpam-3971	152	26	}	}	PUNCT
ejpam-3971	152	27	=	=	SYM
ejpam-3971	152	28	min{α.µ(0	min{α.µ(0	NOUN
ejpam-3971	152	29	∗	∗	NOUN
ejpam-3971	152	30	(	(	PUNCT
ejpam-3971	152	31	x	x	X
ejpam-3971	152	32	∗	∗	PROPN
ejpam-3971	152	33	y	y	PROPN
ejpam-3971	152	34	)	)	PUNCT
ejpam-3971	152	35	)	)	PUNCT
ejpam-3971	152	36	,	,	PUNCT
ejpam-3971	152	37	α.µ(0	α.µ(0	PROPN
ejpam-3971	152	38	∗	∗	X
ejpam-3971	152	39	y	y	PROPN
ejpam-3971	152	40	)	)	PUNCT
ejpam-3971	152	41	}	}	PUNCT
ejpam-3971	153	1	=	=	PUNCT
ejpam-3971	153	2	α.min{µ(0	α.min{µ(0	NOUN
ejpam-3971	153	3	∗	∗	NOUN
ejpam-3971	153	4	(	(	PUNCT
ejpam-3971	153	5	x	x	X
ejpam-3971	153	6	∗	∗	PROPN
ejpam-3971	153	7	y	y	PROPN
ejpam-3971	153	8	)	)	PUNCT
ejpam-3971	153	9	)	)	PUNCT
ejpam-3971	153	10	,	,	PUNCT
ejpam-3971	153	11	µ(0	µ(0	NOUN
ejpam-3971	153	12	∗	∗	PROPN
ejpam-3971	153	13	y	y	NOUN
ejpam-3971	153	14	)	)	PUNCT
ejpam-3971	153	15	}	}	PUNCT
ejpam-3971	153	16	hence	hence	ADV
ejpam-3971	153	17	,	,	PUNCT
ejpam-3971	153	18	µ(0	µ(0	NOUN
ejpam-3971	153	19	∗	∗	NOUN
ejpam-3971	153	20	x	x	NOUN
ejpam-3971	153	21	)	)	PUNCT
ejpam-3971	153	22	≥	≥	NOUN
ejpam-3971	153	23	min{µ(0	min{µ(0	NOUN
ejpam-3971	153	24	∗	∗	NOUN
ejpam-3971	153	25	(	(	PUNCT
ejpam-3971	153	26	x	x	X
ejpam-3971	153	27	∗	∗	PROPN
ejpam-3971	153	28	y	y	PROPN
ejpam-3971	153	29	)	)	PUNCT
ejpam-3971	153	30	)	)	PUNCT
ejpam-3971	153	31	,	,	PUNCT
ejpam-3971	153	32	µ(0	µ(0	NOUN
ejpam-3971	153	33	∗	∗	PROPN
ejpam-3971	153	34	y	y	PROPN
ejpam-3971	153	35	)	)	PUNCT
ejpam-3971	153	36	}	}	PUNCT
ejpam-3971	153	37	.	.	PUNCT
ejpam-3971	154	1	therefore	therefore	ADV
ejpam-3971	154	2	,	,	PUNCT
ejpam-3971	154	3	µ	µ	X
ejpam-3971	154	4	is	be	AUX
ejpam-3971	154	5	a	a	DET
ejpam-3971	154	6	fuzzy	fuzzy	ADJ
ejpam-3971	154	7	brk	brk	PROPN
ejpam-3971	154	8	-	-	PUNCT
ejpam-3971	154	9	ideal	ideal	NOUN
ejpam-3971	154	10	of	of	ADP
ejpam-3971	154	11	x.	x.	PROPN
ejpam-3971	154	12	lemma	lemma	PROPN
ejpam-3971	155	1	1	1	X
ejpam-3971	155	2	.	.	PUNCT
ejpam-3971	156	1	let	let	VERB
ejpam-3971	156	2	i	i	PRON
ejpam-3971	156	3	be	be	AUX
ejpam-3971	156	4	a	a	DET
ejpam-3971	156	5	fuzzy	fuzzy	ADJ
ejpam-3971	156	6	α	α	NOUN
ejpam-3971	156	7	-	-	NOUN
ejpam-3971	156	8	translation	translation	NOUN
ejpam-3971	156	9	(	(	PUNCT
ejpam-3971	156	10	or	or	CCONJ
ejpam-3971	156	11	fuzzy	fuzzy	ADJ
ejpam-3971	156	12	α	α	NOUN
ejpam-3971	156	13	-	-	NOUN
ejpam-3971	156	14	multiplication	multiplication	NOUN
ejpam-3971	156	15	)	)	PUNCT
ejpam-3971	156	16	brk	brk	PROPN
ejpam-3971	156	17	-	-	PUNCT
ejpam-3971	156	18	ideal	ideal	NOUN
ejpam-3971	156	19	of	of	ADP
ejpam-3971	156	20	brkalgebra	brkalgebra	NOUN
ejpam-3971	156	21	x.	x.	NOUN
ejpam-3971	157	1	if	if	SCONJ
ejpam-3971	157	2	x	x	SYM
ejpam-3971	157	3	≤	≤	NOUN
ejpam-3971	157	4	y	y	NOUN
ejpam-3971	157	5	holds	hold	VERB
ejpam-3971	157	6	in	in	ADP
ejpam-3971	157	7	x	x	NOUN
ejpam-3971	157	8	,	,	PUNCT
ejpam-3971	157	9	then	then	ADV
ejpam-3971	157	10	µtα(0	µtα(0	ADJ
ejpam-3971	157	11	∗	∗	NOUN
ejpam-3971	157	12	x	x	NOUN
ejpam-3971	157	13	)	)	PUNCT
ejpam-3971	157	14	=	=	SYM
ejpam-3971	157	15	µtα(0	µtα(0	NOUN
ejpam-3971	157	16	∗	∗	PROPN
ejpam-3971	157	17	y	y	PROPN
ejpam-3971	157	18	)	)	PUNCT
ejpam-3971	157	19	(	(	PUNCT
ejpam-3971	157	20	or	or	CCONJ
ejpam-3971	157	21	µmα	µmα	X
ejpam-3971	157	22	(	(	PUNCT
ejpam-3971	157	23	0	0	NUM
ejpam-3971	157	24	∗	∗	NOUN
ejpam-3971	157	25	x	x	NOUN
ejpam-3971	157	26	)	)	PUNCT
ejpam-3971	157	27	=	=	SYM
ejpam-3971	157	28	µmα	µmα	X
ejpam-3971	157	29	(	(	PUNCT
ejpam-3971	157	30	0	0	NUM
ejpam-3971	157	31	∗	∗	PROPN
ejpam-3971	157	32	y	y	PROPN
ejpam-3971	157	33	)	)	PUNCT
ejpam-3971	157	34	)	)	PUNCT
ejpam-3971	157	35	.	.	PUNCT
ejpam-3971	158	1	proof	proof	NOUN
ejpam-3971	158	2	.	.	PUNCT
ejpam-3971	159	1	assume	assume	VERB
ejpam-3971	159	2	that	that	SCONJ
ejpam-3971	159	3	x	x	SYM
ejpam-3971	159	4	≤	≤	NOUN
ejpam-3971	159	5	y	y	NOUN
ejpam-3971	159	6	holds	hold	VERB
ejpam-3971	159	7	in	in	ADP
ejpam-3971	159	8	x.	x.	NOUN
ejpam-3971	159	9	then	then	ADV
ejpam-3971	159	10	x	x	X
ejpam-3971	159	11	∗	∗	NOUN
ejpam-3971	159	12	y	y	NOUN
ejpam-3971	159	13	=	=	SYM
ejpam-3971	159	14	0	0	X
ejpam-3971	159	15	.	.	PUNCT
ejpam-3971	160	1	by	by	ADP
ejpam-3971	160	2	(	(	PUNCT
ejpam-3971	160	3	brk4	brk4	PROPN
ejpam-3971	160	4	)	)	PUNCT
ejpam-3971	160	5	implies	imply	VERB
ejpam-3971	160	6	µtα(0	µtα(0	ADJ
ejpam-3971	160	7	∗	∗	NOUN
ejpam-3971	160	8	x	x	NOUN
ejpam-3971	160	9	)	)	PUNCT
ejpam-3971	160	10	=	=	SYM
ejpam-3971	160	11	µtα(0	µtα(0	NOUN
ejpam-3971	160	12	∗	∗	PROPN
ejpam-3971	160	13	y	y	PROPN
ejpam-3971	160	14	)	)	PUNCT
ejpam-3971	160	15	lemma	lemma	PROPN
ejpam-3971	160	16	2	2	X
ejpam-3971	160	17	.	.	PUNCT
ejpam-3971	161	1	let	let	VERB
ejpam-3971	161	2	i	i	PRON
ejpam-3971	161	3	be	be	AUX
ejpam-3971	161	4	a	a	DET
ejpam-3971	161	5	fuzzy	fuzzy	ADJ
ejpam-3971	161	6	α	α	NOUN
ejpam-3971	161	7	-	-	NOUN
ejpam-3971	161	8	translation	translation	NOUN
ejpam-3971	161	9	(	(	PUNCT
ejpam-3971	161	10	or	or	CCONJ
ejpam-3971	161	11	fuzzy	fuzzy	ADJ
ejpam-3971	161	12	α	α	NOUN
ejpam-3971	161	13	-	-	NOUN
ejpam-3971	161	14	multiplication	multiplication	NOUN
ejpam-3971	161	15	)	)	PUNCT
ejpam-3971	161	16	brk	brk	PROPN
ejpam-3971	161	17	-	-	PUNCT
ejpam-3971	161	18	ideal	ideal	NOUN
ejpam-3971	161	19	of	of	ADP
ejpam-3971	161	20	brkalgebra	brkalgebra	NOUN
ejpam-3971	161	21	x.	x.	NOUN
ejpam-3971	162	1	if	if	SCONJ
ejpam-3971	162	2	x	x	PROPN
ejpam-3971	162	3	∗	∗	VERB
ejpam-3971	162	4	y	y	NOUN
ejpam-3971	162	5	≤	≤	NUM
ejpam-3971	162	6	x	x	PUNCT
ejpam-3971	162	7	holds	hold	VERB
ejpam-3971	162	8	in	in	ADP
ejpam-3971	162	9	x	x	NOUN
ejpam-3971	162	10	,	,	PUNCT
ejpam-3971	162	11	then	then	ADV
ejpam-3971	162	12	µtα(0	µtα(0	PROPN
ejpam-3971	162	13	∗	∗	PROPN
ejpam-3971	162	14	y	y	PROPN
ejpam-3971	162	15	)	)	PUNCT
ejpam-3971	162	16	≥	≥	NOUN
ejpam-3971	162	17	µtα(0	µtα(0	PROPN
ejpam-3971	162	18	∗	∗	NOUN
ejpam-3971	162	19	x	x	NOUN
ejpam-3971	162	20	)	)	PUNCT
ejpam-3971	162	21	(	(	PUNCT
ejpam-3971	162	22	or	or	CCONJ
ejpam-3971	162	23	µmα	µmα	X
ejpam-3971	162	24	(	(	PUNCT
ejpam-3971	162	25	0	0	NUM
ejpam-3971	162	26	∗	∗	PROPN
ejpam-3971	162	27	y	y	PROPN
ejpam-3971	162	28	)	)	PUNCT
ejpam-3971	162	29	≥	≥	NOUN
ejpam-3971	162	30	µmα	µmα	X
ejpam-3971	162	31	(	(	PUNCT
ejpam-3971	162	32	0	0	NUM
ejpam-3971	162	33	∗	∗	NOUN
ejpam-3971	162	34	x	x	NOUN
ejpam-3971	162	35	)	)	PUNCT
ejpam-3971	162	36	)	)	PUNCT
ejpam-3971	162	37	.	.	PUNCT
ejpam-3971	163	1	proof	proof	NOUN
ejpam-3971	163	2	.	.	PUNCT
ejpam-3971	164	1	assume	assume	VERB
ejpam-3971	164	2	that	that	SCONJ
ejpam-3971	164	3	x	x	PROPN
ejpam-3971	164	4	∗	∗	VERB
ejpam-3971	164	5	y	y	NOUN
ejpam-3971	164	6	≤	≤	NUM
ejpam-3971	164	7	x	x	PUNCT
ejpam-3971	164	8	holds	hold	VERB
ejpam-3971	164	9	in	in	ADP
ejpam-3971	164	10	x.	x.	NOUN
ejpam-3971	164	11	then	then	ADV
ejpam-3971	164	12	(	(	PUNCT
ejpam-3971	164	13	x	x	PROPN
ejpam-3971	164	14	∗	∗	PROPN
ejpam-3971	164	15	y	y	NOUN
ejpam-3971	164	16	)	)	PUNCT
ejpam-3971	164	17	∗	∗	NOUN
ejpam-3971	164	18	x	x	PUNCT
ejpam-3971	165	1	=	=	NOUN
ejpam-3971	165	2	0	0	NUM
ejpam-3971	165	3	.	.	PUNCT
ejpam-3971	166	1	by	by	ADP
ejpam-3971	166	2	(	(	PUNCT
ejpam-3971	166	3	brk2	brk2	PROPN
ejpam-3971	166	4	)	)	PUNCT
ejpam-3971	166	5	µtα(0	µtα(0	PROPN
ejpam-3971	166	6	∗	∗	PROPN
ejpam-3971	166	7	y	y	NOUN
ejpam-3971	166	8	)	)	PUNCT
ejpam-3971	166	9	=	=	PUNCT
ejpam-3971	166	10	min{µtα(0	min{µtα(0	X
ejpam-3971	166	11	∗	∗	NOUN
ejpam-3971	166	12	(	(	PUNCT
ejpam-3971	166	13	x	x	X
ejpam-3971	166	14	∗	∗	PROPN
ejpam-3971	166	15	y	y	PROPN
ejpam-3971	166	16	)	)	PUNCT
ejpam-3971	166	17	)	)	PUNCT
ejpam-3971	166	18	,	,	PUNCT
ejpam-3971	166	19	µtα(0	µtα(0	ADJ
ejpam-3971	166	20	∗	∗	NOUN
ejpam-3971	166	21	x	x	NOUN
ejpam-3971	166	22	)	)	PUNCT
ejpam-3971	166	23	}	}	PUNCT
ejpam-3971	166	24	also	also	ADV
ejpam-3971	166	25	,	,	PUNCT
ejpam-3971	166	26	µtα(0	µtα(0	ADJ
ejpam-3971	166	27	∗	∗	NOUN
ejpam-3971	166	28	(	(	PUNCT
ejpam-3971	166	29	x	x	X
ejpam-3971	166	30	∗	∗	PROPN
ejpam-3971	166	31	y	y	PROPN
ejpam-3971	166	32	)	)	PUNCT
ejpam-3971	166	33	)	)	PUNCT
ejpam-3971	166	34	≥	≥	NOUN
ejpam-3971	166	35	min{µtα((0	min{µtα((0	PRON
ejpam-3971	166	36	∗	∗	NOUN
ejpam-3971	166	37	(	(	PUNCT
ejpam-3971	166	38	x	x	X
ejpam-3971	166	39	∗	∗	PROPN
ejpam-3971	166	40	y	y	PROPN
ejpam-3971	166	41	)	)	PUNCT
ejpam-3971	166	42	)	)	PUNCT
ejpam-3971	166	43	∗	∗	NOUN
ejpam-3971	166	44	x	x	NOUN
ejpam-3971	166	45	)	)	PUNCT
ejpam-3971	166	46	,	,	PUNCT
ejpam-3971	166	47	µtα(0	µtα(0	ADJ
ejpam-3971	166	48	∗	∗	NOUN
ejpam-3971	166	49	x	x	NOUN
ejpam-3971	166	50	)	)	PUNCT
ejpam-3971	166	51	}	}	PUNCT
ejpam-3971	166	52	=	=	SYM
ejpam-3971	166	53	min{µtα(0	min{µtα(0	NOUN
ejpam-3971	166	54	)	)	PUNCT
ejpam-3971	166	55	,	,	PUNCT
ejpam-3971	166	56	µtα(x	µtα(x	PROPN
ejpam-3971	166	57	)	)	PUNCT
ejpam-3971	166	58	}	}	PUNCT
ejpam-3971	166	59	=	=	SYM
ejpam-3971	166	60	{	{	PUNCT
ejpam-3971	166	61	µtα(0	µtα(0	ADJ
ejpam-3971	166	62	∗	∗	NOUN
ejpam-3971	166	63	x	x	NOUN
ejpam-3971	166	64	)	)	PUNCT
ejpam-3971	166	65	}	}	PUNCT
ejpam-3971	166	66	hence	hence	ADV
ejpam-3971	166	67	,	,	PUNCT
ejpam-3971	166	68	µtα(0	µtα(0	ADJ
ejpam-3971	166	69	∗	∗	PROPN
ejpam-3971	166	70	y	y	PROPN
ejpam-3971	166	71	)	)	PUNCT
ejpam-3971	166	72	≥	≥	NOUN
ejpam-3971	166	73	µtα(0	µtα(0	PROPN
ejpam-3971	166	74	∗	∗	NOUN
ejpam-3971	166	75	x	x	NOUN
ejpam-3971	166	76	)	)	PUNCT
ejpam-3971	166	77	.	.	PUNCT
ejpam-3971	167	1	h.	h.	PROPN
ejpam-3971	167	2	alshehri	alshehri	PROPN
ejpam-3971	167	3	/	/	SYM
ejpam-3971	167	4	eur	eur	PROPN
ejpam-3971	167	5	.	.	PUNCT
ejpam-3971	168	1	j.	j.	PROPN
ejpam-3971	168	2	pure	pure	PROPN
ejpam-3971	168	3	appl	appl	PROPN
ejpam-3971	168	4	.	.	PROPN
ejpam-3971	168	5	math	math	PROPN
ejpam-3971	168	6	,	,	PUNCT
ejpam-3971	168	7	14	14	NUM
ejpam-3971	168	8	(	(	PUNCT
ejpam-3971	168	9	3	3	NUM
ejpam-3971	168	10	)	)	PUNCT
ejpam-3971	168	11	(	(	PUNCT
ejpam-3971	168	12	2021	2021	NUM
ejpam-3971	168	13	)	)	PUNCT
ejpam-3971	168	14	,	,	PUNCT
ejpam-3971	168	15	737	737	NUM
ejpam-3971	168	16	-	-	SYM
ejpam-3971	168	17	745	745	NUM
ejpam-3971	168	18	743	743	NUM
ejpam-3971	168	19	5	5	NUM
ejpam-3971	168	20	.	.	PUNCT
ejpam-3971	168	21	fuzzy	fuzzy	ADJ
ejpam-3971	168	22	magnified	magnify	VERB
ejpam-3971	168	23	-	-	PUNCT
ejpam-3971	168	24	αβ	αβ	INTJ
ejpam-3971	168	25	-	-	PUNCT
ejpam-3971	168	26	translation	translation	NOUN
ejpam-3971	168	27	of	of	ADP
ejpam-3971	168	28	brk	brk	PROPN
ejpam-3971	168	29	-	-	PUNCT
ejpam-3971	168	30	algebras	algebras	PROPN
ejpam-3971	168	31	definition	definition	NOUN
ejpam-3971	168	32	12	12	NUM
ejpam-3971	168	33	.	.	PUNCT
ejpam-3971	169	1	let	let	VERB
ejpam-3971	169	2	µ	µ	X
ejpam-3971	169	3	be	be	AUX
ejpam-3971	169	4	a	a	DET
ejpam-3971	169	5	fuzzy	fuzzy	ADJ
ejpam-3971	169	6	subset	subset	NOUN
ejpam-3971	169	7	of	of	ADP
ejpam-3971	169	8	x	x	PRON
ejpam-3971	169	9	,	,	PUNCT
ejpam-3971	169	10	α	α	PROPN
ejpam-3971	169	11	∈	∈	PROPN
ejpam-3971	170	1	[	[	X
ejpam-3971	170	2	0	0	NUM
ejpam-3971	170	3	,	,	PUNCT
ejpam-3971	170	4	t	t	X
ejpam-3971	170	5	]	]	PUNCT
ejpam-3971	170	6	,	,	PUNCT
ejpam-3971	170	7	where	where	SCONJ
ejpam-3971	170	8	t	t	NOUN
ejpam-3971	170	9	=	=	SYM
ejpam-3971	170	10	1−	1−	NUM
ejpam-3971	170	11	sup{µ(x	sup{µ(x	PROPN
ejpam-3971	170	12	)	)	PUNCT
ejpam-3971	170	13	:	:	PUNCT
ejpam-3971	171	1	x	x	PUNCT
ejpam-3971	171	2	∈	∈	NOUN
ejpam-3971	171	3	x	x	X
ejpam-3971	171	4	}	}	PUNCT
ejpam-3971	171	5	and	and	CCONJ
ejpam-3971	171	6	β	β	X
ejpam-3971	171	7	∈	∈	PROPN
ejpam-3971	172	1	[	[	X
ejpam-3971	172	2	0	0	NUM
ejpam-3971	172	3	,	,	PUNCT
ejpam-3971	172	4	1	1	NUM
ejpam-3971	172	5	]	]	PUNCT
ejpam-3971	172	6	.	.	PUNCT
ejpam-3971	173	1	a	a	DET
ejpam-3971	173	2	mapping	mapping	NOUN
ejpam-3971	173	3	µmt	µmt	NOUN
ejpam-3971	173	4	(	(	PUNCT
ejpam-3971	173	5	αβ	αβ	INTJ
ejpam-3971	173	6	)	)	PUNCT
ejpam-3971	173	7	:x	:x	PROPN
ejpam-3971	174	1	−→	−→	PROPN
ejpam-3971	175	1	[	[	X
ejpam-3971	175	2	0	0	NUM
ejpam-3971	175	3	,	,	PUNCT
ejpam-3971	175	4	1	1	NUM
ejpam-3971	175	5	]	]	PUNCT
ejpam-3971	175	6	,	,	PUNCT
ejpam-3971	175	7	is	be	AUX
ejpam-3971	175	8	said	say	VERB
ejpam-3971	175	9	to	to	PART
ejpam-3971	175	10	be	be	AUX
ejpam-3971	175	11	a	a	DET
ejpam-3971	175	12	fuzzy	fuzzy	ADJ
ejpam-3971	175	13	magnified	magnify	VERB
ejpam-3971	175	14	-	-	PUNCT
ejpam-3971	175	15	αβtranslation	αβtranslation	NOUN
ejpam-3971	175	16	of	of	ADP
ejpam-3971	175	17	µ	µ	NOUN
ejpam-3971	175	18	if	if	SCONJ
ejpam-3971	175	19	it	it	PRON
ejpam-3971	175	20	satisfies	satisfy	VERB
ejpam-3971	175	21	:	:	PUNCT
ejpam-3971	175	22	µmt	µmt	NOUN
ejpam-3971	175	23	(	(	PUNCT
ejpam-3971	175	24	αβ	αβ	INTJ
ejpam-3971	175	25	)	)	PUNCT
ejpam-3971	175	26	=	=	PUNCT
ejpam-3971	175	27	α.x+	α.x+	NUM
ejpam-3971	175	28	β	β	X
ejpam-3971	175	29	.	.	PUNCT
ejpam-3971	175	30	example	example	NOUN
ejpam-3971	176	1	2	2	NUM
ejpam-3971	176	2	.	.	X
ejpam-3971	176	3	consider	consider	VERB
ejpam-3971	176	4	the	the	DET
ejpam-3971	176	5	brk	brk	PROPN
ejpam-3971	176	6	-	-	PUNCT
ejpam-3971	176	7	algebra	algebra	PROPN
ejpam-3971	176	8	x	x	X
ejpam-3971	176	9	=	=	SYM
ejpam-3971	176	10	0	0	NUM
ejpam-3971	176	11	,	,	PUNCT
ejpam-3971	176	12	a	a	DET
ejpam-3971	176	13	,	,	PUNCT
ejpam-3971	176	14	b	b	NOUN
ejpam-3971	176	15	,	,	PUNCT
ejpam-3971	176	16	c	c	PROPN
ejpam-3971	176	17	in	in	ADP
ejpam-3971	176	18	example	example	NOUN
ejpam-3971	176	19	3.4	3.4	NUM
ejpam-3971	176	20	.	.	PUNCT
ejpam-3971	177	1	define	define	VERB
ejpam-3971	177	2	a	a	DET
ejpam-3971	177	3	fuzzy	fuzzy	ADJ
ejpam-3971	177	4	subset	subset	VERB
ejpam-3971	177	5	µ	µ	PROPN
ejpam-3971	177	6	of	of	ADP
ejpam-3971	177	7	x	x	PUNCT
ejpam-3971	177	8	by	by	ADP
ejpam-3971	177	9	µ(x)=	µ(x)=	PROPN
ejpam-3971	177	10	{	{	PUNCT
ejpam-3971	177	11	0.6	0.6	NUM
ejpam-3971	177	12	;	;	PUNCT
ejpam-3971	177	13	x	x	SYM
ejpam-3971	177	14	=	=	SYM
ejpam-3971	177	15	0	0	NUM
ejpam-3971	177	16	,	,	PUNCT
ejpam-3971	177	17	a	a	DET
ejpam-3971	177	18	0.1	0.1	NUM
ejpam-3971	177	19	;	;	PUNCT
ejpam-3971	177	20	x	x	SYM
ejpam-3971	177	21	=	=	SYM
ejpam-3971	177	22	b	b	PROPN
ejpam-3971	177	23	,	,	PUNCT
ejpam-3971	177	24	c	c	PROPN
ejpam-3971	177	25	then	then	ADV
ejpam-3971	177	26	µ	µ	X
ejpam-3971	177	27	is	be	AUX
ejpam-3971	177	28	fuzzy	fuzzy	ADJ
ejpam-3971	177	29	brk	brk	NOUN
ejpam-3971	177	30	-	-	PUNCT
ejpam-3971	177	31	subalgebra	subalgebra	PROPN
ejpam-3971	177	32	of	of	ADP
ejpam-3971	177	33	x.	x.	NOUN
ejpam-3971	177	34	here	here	ADV
ejpam-3971	177	35	,	,	PUNCT
ejpam-3971	177	36	t	t	NOUN
ejpam-3971	177	37	=	=	SYM
ejpam-3971	177	38	1	1	NUM
ejpam-3971	177	39	−	−	PROPN
ejpam-3971	177	40	sup{µ(x	sup{µ(x	PROPN
ejpam-3971	177	41	)	)	PUNCT
ejpam-3971	177	42	:	:	PUNCT
ejpam-3971	178	1	x	x	PUNCT
ejpam-3971	178	2	∈	∈	NOUN
ejpam-3971	178	3	x	x	X
ejpam-3971	178	4	}	}	PUNCT
ejpam-3971	178	5	=	=	SYM
ejpam-3971	178	6	1	1	NUM
ejpam-3971	178	7	−	−	NOUN
ejpam-3971	178	8	0.6	0.6	NUM
ejpam-3971	178	9	=	=	SYM
ejpam-3971	178	10	0.4.choose	0.4.choose	NUM
ejpam-3971	178	11	α	α	NOUN
ejpam-3971	178	12	=	=	SYM
ejpam-3971	178	13	0.5	0.5	NUM
ejpam-3971	178	14	∈	∈	PROPN
ejpam-3971	179	1	[	[	X
ejpam-3971	179	2	0	0	NUM
ejpam-3971	179	3	,	,	PUNCT
ejpam-3971	179	4	t	t	NOUN
ejpam-3971	179	5	]	]	PUNCT
ejpam-3971	179	6	and	and	CCONJ
ejpam-3971	179	7	β	β	X
ejpam-3971	179	8	=	=	NOUN
ejpam-3971	179	9	0.2	0.2	NUM
ejpam-3971	179	10	∈	∈	PROPN
ejpam-3971	180	1	[	[	X
ejpam-3971	180	2	0	0	NUM
ejpam-3971	180	3	,	,	PUNCT
ejpam-3971	180	4	1	1	NUM
ejpam-3971	180	5	]	]	PUNCT
ejpam-3971	180	6	.	.	PUNCT
ejpam-3971	181	1	then	then	ADV
ejpam-3971	181	2	the	the	DET
ejpam-3971	181	3	mapping	mapping	NOUN
ejpam-3971	181	4	µmt	µmt	NOUN
ejpam-3971	181	5	(	(	PUNCT
ejpam-3971	181	6	0.5)(0.2	0.5)(0.2	NUM
ejpam-3971	181	7	)	)	PUNCT
ejpam-3971	181	8	:x	:x	PROPN
ejpam-3971	182	1	−→	−→	PROPN
ejpam-3971	182	2	[	[	X
ejpam-3971	182	3	0	0	NUM
ejpam-3971	182	4	,	,	PUNCT
ejpam-3971	182	5	1	1	NUM
ejpam-3971	182	6	]	]	PUNCT
ejpam-3971	182	7	defined	define	VERB
ejpam-3971	182	8	by	by	ADP
ejpam-3971	182	9	µmt	µmt	NOUN
ejpam-3971	182	10	(	(	PUNCT
ejpam-3971	182	11	0.5)(0.2	0.5)(0.2	NUM
ejpam-3971	182	12	)	)	PUNCT
ejpam-3971	182	13	=	=	PRON
ejpam-3971	182	14	{	{	PUNCT
ejpam-3971	182	15	(	(	PUNCT
ejpam-3971	182	16	0.5)(0.6	0.5)(0.6	NOUN
ejpam-3971	182	17	)	)	PUNCT
ejpam-3971	182	18	+	+	CCONJ
ejpam-3971	182	19	0.2	0.2	NUM
ejpam-3971	182	20	=	=	SYM
ejpam-3971	182	21	0.5	0.5	NUM
ejpam-3971	182	22	;	;	PUNCT
ejpam-3971	182	23	x	x	SYM
ejpam-3971	182	24	=	=	SYM
ejpam-3971	182	25	0	0	NUM
ejpam-3971	182	26	,	,	PUNCT
ejpam-3971	182	27	a	a	DET
ejpam-3971	182	28	(	(	PUNCT
ejpam-3971	182	29	0.5)(0.1	0.5)(0.1	NOUN
ejpam-3971	182	30	)	)	PUNCT
ejpam-3971	183	1	+	+	CCONJ
ejpam-3971	183	2	0.2	0.2	NUM
ejpam-3971	183	3	=	=	SYM
ejpam-3971	183	4	0.7	0.7	NUM
ejpam-3971	183	5	;	;	PUNCT
ejpam-3971	183	6	x	x	SYM
ejpam-3971	183	7	=	=	SYM
ejpam-3971	183	8	b	b	PROPN
ejpam-3971	183	9	,	,	PUNCT
ejpam-3971	183	10	c	c	X
ejpam-3971	183	11	which	which	DET
ejpam-3971	183	12	satisfiesµmt	satisfiesµmt	NOUN
ejpam-3971	183	13	(	(	PUNCT
ejpam-3971	183	14	0.5)(0.2	0.5)(0.2	NUM
ejpam-3971	183	15	)	)	PUNCT
ejpam-3971	183	16	=	=	NOUN
ejpam-3971	183	17	αµ(x	αµ(x	NUM
ejpam-3971	183	18	)	)	PUNCT
ejpam-3971	184	1	+	+	CCONJ
ejpam-3971	184	2	β;∀x	β;∀x	X
ejpam-3971	184	3	∈	∈	NOUN
ejpam-3971	184	4	x	x	PUNCT
ejpam-3971	184	5	is	be	AUX
ejpam-3971	184	6	fuzzy	fuzzy	ADJ
ejpam-3971	184	7	magnified-(0.5)(0.2)-translation	magnified-(0.5)(0.2)-translation	NOUN
ejpam-3971	184	8	.	.	PUNCT
ejpam-3971	185	1	theorem	theorem	NOUN
ejpam-3971	185	2	9	9	NUM
ejpam-3971	185	3	.	.	PUNCT
ejpam-3971	186	1	let	let	VERB
ejpam-3971	186	2	µ	µ	X
ejpam-3971	186	3	be	be	AUX
ejpam-3971	186	4	a	a	DET
ejpam-3971	186	5	fuzzy	fuzzy	ADJ
ejpam-3971	186	6	subset	subset	NOUN
ejpam-3971	186	7	of	of	ADP
ejpam-3971	186	8	x	x	PRON
ejpam-3971	186	9	,	,	PUNCT
ejpam-3971	186	10	α	α	PROPN
ejpam-3971	186	11	∈	∈	PROPN
ejpam-3971	187	1	[	[	X
ejpam-3971	187	2	0	0	NUM
ejpam-3971	187	3	,	,	PUNCT
ejpam-3971	187	4	t	t	NOUN
ejpam-3971	187	5	]	]	PUNCT
ejpam-3971	187	6	and	and	CCONJ
ejpam-3971	187	7	β	β	X
ejpam-3971	187	8	∈	∈	PROPN
ejpam-3971	188	1	[	[	X
ejpam-3971	188	2	0	0	NUM
ejpam-3971	188	3	,	,	PUNCT
ejpam-3971	188	4	1	1	NUM
ejpam-3971	188	5	]	]	PUNCT
ejpam-3971	188	6	.	.	PUNCT
ejpam-3971	189	1	a	a	DET
ejpam-3971	189	2	mapping	mapping	NOUN
ejpam-3971	189	3	µmt	µmt	NOUN
ejpam-3971	189	4	(	(	PUNCT
ejpam-3971	189	5	αβ	αβ	INTJ
ejpam-3971	189	6	)	)	PUNCT
ejpam-3971	189	7	:x	:x	PROPN
ejpam-3971	190	1	−→	−→	PROPN
ejpam-3971	190	2	[	[	X
ejpam-3971	190	3	0	0	NUM
ejpam-3971	190	4	,	,	PUNCT
ejpam-3971	190	5	1	1	NUM
ejpam-3971	190	6	]	]	PUNCT
ejpam-3971	190	7	is	be	AUX
ejpam-3971	190	8	said	say	VERB
ejpam-3971	190	9	to	to	PART
ejpam-3971	190	10	be	be	AUX
ejpam-3971	190	11	a	a	DET
ejpam-3971	190	12	fuzzy	fuzzy	ADJ
ejpam-3971	190	13	magnified	magnify	VERB
ejpam-3971	190	14	-	-	PUNCT
ejpam-3971	190	15	αβ	αβ	INTJ
ejpam-3971	190	16	-	-	PUNCT
ejpam-3971	190	17	translation	translation	NOUN
ejpam-3971	190	18	of	of	ADP
ejpam-3971	190	19	µ.	µ.	PROPN
ejpam-3971	190	20	then	then	ADV
ejpam-3971	190	21	µ	µ	PROPN
ejpam-3971	190	22	is	be	AUX
ejpam-3971	190	23	fuzzy	fuzzy	ADJ
ejpam-3971	190	24	brk	brk	NOUN
ejpam-3971	190	25	-	-	PUNCT
ejpam-3971	190	26	subalgebra	subalgebra	PROPN
ejpam-3971	190	27	of	of	ADP
ejpam-3971	190	28	x	x	PRON
ejpam-3971	190	29	if	if	SCONJ
ejpam-3971	190	30	and	and	CCONJ
ejpam-3971	190	31	only	only	ADV
ejpam-3971	190	32	if	if	SCONJ
ejpam-3971	190	33	µmt	µmt	NOUN
ejpam-3971	190	34	(	(	PUNCT
ejpam-3971	190	35	αβ	αβ	INTJ
ejpam-3971	190	36	)	)	PUNCT
ejpam-3971	190	37	is	be	AUX
ejpam-3971	190	38	fuzzy	fuzzy	ADJ
ejpam-3971	190	39	subalgebra	subalgebra	NOUN
ejpam-3971	190	40	of	of	ADP
ejpam-3971	190	41	x.	x.	NOUN
ejpam-3971	190	42	proof	proof	NOUN
ejpam-3971	190	43	.	.	PUNCT
ejpam-3971	191	1	let	let	VERB
ejpam-3971	191	2	µ	µ	X
ejpam-3971	191	3	be	be	AUX
ejpam-3971	191	4	a	a	DET
ejpam-3971	191	5	fuzzy	fuzzy	ADJ
ejpam-3971	191	6	subset	subset	NOUN
ejpam-3971	191	7	of	of	ADP
ejpam-3971	191	8	x	x	PRON
ejpam-3971	191	9	,	,	PUNCT
ejpam-3971	191	10	α	α	PROPN
ejpam-3971	191	11	∈	∈	PROPN
ejpam-3971	192	1	[	[	X
ejpam-3971	192	2	0	0	NUM
ejpam-3971	192	3	,	,	PUNCT
ejpam-3971	192	4	t	t	NOUN
ejpam-3971	192	5	]	]	PUNCT
ejpam-3971	192	6	and	and	CCONJ
ejpam-3971	192	7	β	β	X
ejpam-3971	192	8	∈	∈	PROPN
ejpam-3971	193	1	[	[	X
ejpam-3971	193	2	0	0	NUM
ejpam-3971	193	3	,	,	PUNCT
ejpam-3971	193	4	1	1	NUM
ejpam-3971	193	5	]	]	PUNCT
ejpam-3971	193	6	.	.	PUNCT
ejpam-3971	194	1	a	a	DET
ejpam-3971	194	2	mapping	mapping	NOUN
ejpam-3971	194	3	µmt	µmt	NOUN
ejpam-3971	194	4	(	(	PUNCT
ejpam-3971	194	5	αβ	αβ	INTJ
ejpam-3971	194	6	)	)	PUNCT
ejpam-3971	194	7	:x	:x	PROPN
ejpam-3971	195	1	−→	−→	PROPN
ejpam-3971	195	2	[	[	X
ejpam-3971	195	3	0	0	NUM
ejpam-3971	195	4	,	,	PUNCT
ejpam-3971	195	5	1	1	NUM
ejpam-3971	195	6	]	]	PUNCT
ejpam-3971	195	7	is	be	AUX
ejpam-3971	195	8	said	say	VERB
ejpam-3971	195	9	to	to	PART
ejpam-3971	195	10	be	be	AUX
ejpam-3971	195	11	a	a	DET
ejpam-3971	195	12	fuzzy	fuzzy	ADJ
ejpam-3971	195	13	magnified	magnify	VERB
ejpam-3971	195	14	-	-	PUNCT
ejpam-3971	195	15	αβ	αβ	INTJ
ejpam-3971	195	16	-	-	PUNCT
ejpam-3971	195	17	translation	translation	NOUN
ejpam-3971	195	18	of	of	ADP
ejpam-3971	195	19	µ.	µ.	PROPN
ejpam-3971	195	20	assume	assume	VERB
ejpam-3971	195	21	µ	µ	X
ejpam-3971	195	22	is	be	AUX
ejpam-3971	195	23	fuzzy	fuzzy	ADJ
ejpam-3971	195	24	brk	brk	NOUN
ejpam-3971	195	25	-	-	PUNCT
ejpam-3971	195	26	subalgebra	subalgebra	PROPN
ejpam-3971	195	27	of	of	ADP
ejpam-3971	195	28	x.	x.	NOUN
ejpam-3971	195	29	then	then	ADV
ejpam-3971	195	30	µ(x	µ(x	VERB
ejpam-3971	195	31	∗	∗	NOUN
ejpam-3971	195	32	y	y	NOUN
ejpam-3971	195	33	)	)	PUNCT
ejpam-3971	195	34	≥	≥	NOUN
ejpam-3971	195	35	min{µ(x	min{µ(x	NOUN
ejpam-3971	195	36	)	)	PUNCT
ejpam-3971	195	37	,	,	PUNCT
ejpam-3971	195	38	µ(y	µ(y	PROPN
ejpam-3971	195	39	)	)	PUNCT
ejpam-3971	195	40	}	}	PUNCT
ejpam-3971	195	41	now	now	ADV
ejpam-3971	195	42	,	,	PUNCT
ejpam-3971	195	43	µmt	µmt	NOUN
ejpam-3971	195	44	(	(	PUNCT
ejpam-3971	195	45	αβ	αβ	INTJ
ejpam-3971	195	46	)	)	PUNCT
ejpam-3971	195	47	(	(	PUNCT
ejpam-3971	195	48	x	x	SYM
ejpam-3971	195	49	∗	∗	NOUN
ejpam-3971	195	50	y)=α.µ(x	y)=α.µ(x	NOUN
ejpam-3971	195	51	∗	∗	NOUN
ejpam-3971	195	52	y	y	NOUN
ejpam-3971	195	53	)	)	PUNCT
ejpam-3971	195	54	+	+	CCONJ
ejpam-3971	195	55	β	β	X
ejpam-3971	195	56	≥	≥	NOUN
ejpam-3971	195	57	α.min{µ(x	α.min{µ(x	NUM
ejpam-3971	195	58	)	)	PUNCT
ejpam-3971	195	59	,	,	PUNCT
ejpam-3971	195	60	µ(y)}+	µ(y)}+	NUM
ejpam-3971	195	61	β	β	X
ejpam-3971	195	62	=	=	SYM
ejpam-3971	195	63	min{α.µ(x	min{α.µ(x	PROPN
ejpam-3971	195	64	)	)	PUNCT
ejpam-3971	196	1	+	+	CCONJ
ejpam-3971	196	2	β	β	NOUN
ejpam-3971	196	3	,	,	PUNCT
ejpam-3971	196	4	α.µ(y	α.µ(y	NUM
ejpam-3971	196	5	)	)	PUNCT
ejpam-3971	197	1	+	+	CCONJ
ejpam-3971	197	2	β	β	X
ejpam-3971	197	3	}	}	PUNCT
ejpam-3971	197	4	=	=	PUNCT
ejpam-3971	197	5	min{µmt	min{µmt	INTJ
ejpam-3971	197	6	(	(	PUNCT
ejpam-3971	197	7	αβ	αβ	INTJ
ejpam-3971	197	8	)	)	PUNCT
ejpam-3971	197	9	(	(	PUNCT
ejpam-3971	197	10	x),µmt	x),µmt	X
ejpam-3971	197	11	(	(	PUNCT
ejpam-3971	197	12	αβ	αβ	INTJ
ejpam-3971	197	13	)	)	PUNCT
ejpam-3971	197	14	(	(	PUNCT
ejpam-3971	197	15	y	y	NOUN
ejpam-3971	197	16	)	)	PUNCT
ejpam-3971	197	17	}	}	PUNCT
ejpam-3971	197	18	hence	hence	ADV
ejpam-3971	197	19	,	,	PUNCT
ejpam-3971	197	20	µmt	µmt	NOUN
ejpam-3971	197	21	(	(	PUNCT
ejpam-3971	197	22	αβ	αβ	INTJ
ejpam-3971	197	23	)	)	PUNCT
ejpam-3971	197	24	is	be	AUX
ejpam-3971	197	25	fuzzy	fuzzy	ADJ
ejpam-3971	197	26	subalgebra	subalgebra	NOUN
ejpam-3971	197	27	of	of	ADP
ejpam-3971	197	28	x.	x.	NOUN
ejpam-3971	197	29	conversely	conversely	ADV
ejpam-3971	197	30	assume,µmt	assume,µmt	PROPN
ejpam-3971	197	31	(	(	PUNCT
ejpam-3971	197	32	αβ	αβ	INTJ
ejpam-3971	197	33	)	)	PUNCT
ejpam-3971	197	34	is	be	AUX
ejpam-3971	197	35	fuzzy	fuzzy	ADJ
ejpam-3971	197	36	subalgebra	subalgebra	NOUN
ejpam-3971	197	37	of	of	ADP
ejpam-3971	197	38	x.	x.	NOUN
ejpam-3971	197	39	then	then	ADV
ejpam-3971	197	40	,	,	PUNCT
ejpam-3971	197	41	α.µ(x	α.µ(x	NUM
ejpam-3971	197	42	∗	∗	NOUN
ejpam-3971	197	43	y	y	PROPN
ejpam-3971	197	44	)	)	PUNCT
ejpam-3971	198	1	+	+	CCONJ
ejpam-3971	198	2	β=µmt	β=µmt	PUNCT
ejpam-3971	198	3	(	(	PUNCT
ejpam-3971	198	4	αβ	αβ	INTJ
ejpam-3971	198	5	)	)	PUNCT
ejpam-3971	198	6	(	(	PUNCT
ejpam-3971	198	7	x	x	SYM
ejpam-3971	198	8	∗	∗	NOUN
ejpam-3971	199	1	y)≥	y)≥	INTJ
ejpam-3971	199	2	min{µmt	min{µmt	INTJ
ejpam-3971	199	3	(	(	PUNCT
ejpam-3971	199	4	αβ	αβ	INTJ
ejpam-3971	199	5	)	)	PUNCT
ejpam-3971	199	6	(	(	PUNCT
ejpam-3971	199	7	x	x	NOUN
ejpam-3971	199	8	)	)	PUNCT
ejpam-3971	199	9	,	,	PUNCT
ejpam-3971	199	10	µmt	µmt	NOUN
ejpam-3971	199	11	(	(	PUNCT
ejpam-3971	199	12	αβ	αβ	INTJ
ejpam-3971	199	13	)	)	PUNCT
ejpam-3971	199	14	(	(	PUNCT
ejpam-3971	199	15	y	y	NOUN
ejpam-3971	199	16	)	)	PUNCT
ejpam-3971	199	17	}	}	PUNCT
ejpam-3971	199	18	=	=	SYM
ejpam-3971	199	19	min{α.µ(x	min{α.µ(x	PROPN
ejpam-3971	199	20	)	)	PUNCT
ejpam-3971	200	1	+	+	CCONJ
ejpam-3971	200	2	β	β	NOUN
ejpam-3971	200	3	,	,	PUNCT
ejpam-3971	200	4	α.µ(y	α.µ(y	NUM
ejpam-3971	200	5	)	)	PUNCT
ejpam-3971	200	6	+	+	CCONJ
ejpam-3971	200	7	β	β	X
ejpam-3971	200	8	}	}	PUNCT
ejpam-3971	200	9	=	=	SYM
ejpam-3971	200	10	α.min{µ(x	α.min{µ(x	NUM
ejpam-3971	200	11	)	)	PUNCT
ejpam-3971	200	12	,	,	PUNCT
ejpam-3971	200	13	µ(y)}+	µ(y)}+	NUM
ejpam-3971	200	14	β	β	NOUN
ejpam-3971	200	15	references	reference	NOUN
ejpam-3971	200	16	744	744	NUM
ejpam-3971	200	17	hence	hence	ADV
ejpam-3971	200	18	,	,	PUNCT
ejpam-3971	200	19	µ(x	µ(x	ADJ
ejpam-3971	200	20	∗	∗	NOUN
ejpam-3971	200	21	y	y	NOUN
ejpam-3971	200	22	)	)	PUNCT
ejpam-3971	200	23	≥	≥	NOUN
ejpam-3971	200	24	min{µ(x	min{µ(x	NOUN
ejpam-3971	200	25	)	)	PUNCT
ejpam-3971	200	26	,	,	PUNCT
ejpam-3971	200	27	µ(y	µ(y	PROPN
ejpam-3971	200	28	)	)	PUNCT
ejpam-3971	200	29	}	}	PUNCT
ejpam-3971	200	30	,	,	PUNCT
ejpam-3971	200	31	so	so	ADV
ejpam-3971	200	32	µ	µ	NOUN
ejpam-3971	200	33	is	be	AUX
ejpam-3971	200	34	fuzzy	fuzzy	ADJ
ejpam-3971	200	35	brk	brk	NOUN
ejpam-3971	200	36	-	-	PUNCT
ejpam-3971	200	37	subalgebra	subalgebra	PROPN
ejpam-3971	200	38	of	of	ADP
ejpam-3971	200	39	x.	x.	PROPN
ejpam-3971	200	40	theorem	theorem	VERB
ejpam-3971	200	41	10	10	NUM
ejpam-3971	200	42	.	.	PUNCT
ejpam-3971	201	1	let	let	VERB
ejpam-3971	201	2	µ	µ	X
ejpam-3971	201	3	be	be	AUX
ejpam-3971	201	4	a	a	DET
ejpam-3971	201	5	fuzzy	fuzzy	ADJ
ejpam-3971	201	6	subset	subset	NOUN
ejpam-3971	201	7	of	of	ADP
ejpam-3971	201	8	x	x	PRON
ejpam-3971	201	9	,	,	PUNCT
ejpam-3971	201	10	α	α	PROPN
ejpam-3971	201	11	∈	∈	PROPN
ejpam-3971	202	1	[	[	X
ejpam-3971	202	2	0	0	NUM
ejpam-3971	202	3	,	,	PUNCT
ejpam-3971	202	4	t	t	NOUN
ejpam-3971	202	5	]	]	PUNCT
ejpam-3971	202	6	and	and	CCONJ
ejpam-3971	202	7	β	β	X
ejpam-3971	202	8	∈	∈	PROPN
ejpam-3971	203	1	[	[	X
ejpam-3971	203	2	0	0	NUM
ejpam-3971	203	3	,	,	PUNCT
ejpam-3971	203	4	1	1	NUM
ejpam-3971	203	5	]	]	PUNCT
ejpam-3971	203	6	.	.	PUNCT
ejpam-3971	204	1	a	a	DET
ejpam-3971	204	2	mapping	mapping	NOUN
ejpam-3971	204	3	µmt	µmt	NOUN
ejpam-3971	204	4	(	(	PUNCT
ejpam-3971	204	5	αβ	αβ	INTJ
ejpam-3971	204	6	)	)	PUNCT
ejpam-3971	204	7	:x	:x	PROPN
ejpam-3971	205	1	−→	−→	PROPN
ejpam-3971	205	2	[	[	X
ejpam-3971	205	3	0	0	NUM
ejpam-3971	205	4	,	,	PUNCT
ejpam-3971	205	5	1	1	NUM
ejpam-3971	205	6	]	]	PUNCT
ejpam-3971	205	7	is	be	AUX
ejpam-3971	205	8	said	say	VERB
ejpam-3971	205	9	to	to	PART
ejpam-3971	205	10	be	be	AUX
ejpam-3971	205	11	a	a	DET
ejpam-3971	205	12	fuzzy	fuzzy	ADJ
ejpam-3971	205	13	magnified	magnify	VERB
ejpam-3971	205	14	-	-	PUNCT
ejpam-3971	205	15	αβ	αβ	INTJ
ejpam-3971	205	16	-	-	PUNCT
ejpam-3971	205	17	translation	translation	NOUN
ejpam-3971	205	18	of	of	ADP
ejpam-3971	205	19	µ.	µ.	PROPN
ejpam-3971	205	20	then	then	ADV
ejpam-3971	205	21	µ	µ	PROPN
ejpam-3971	205	22	is	be	AUX
ejpam-3971	205	23	fuzzy	fuzzy	ADJ
ejpam-3971	205	24	brk	brk	PROPN
ejpam-3971	205	25	-	-	PUNCT
ejpam-3971	205	26	ideal	ideal	NOUN
ejpam-3971	205	27	of	of	ADP
ejpam-3971	205	28	x	x	SYM
ejpam-3971	205	29	if	if	SCONJ
ejpam-3971	205	30	and	and	CCONJ
ejpam-3971	205	31	only	only	ADV
ejpam-3971	205	32	if	if	SCONJ
ejpam-3971	205	33	µmt	µmt	NOUN
ejpam-3971	205	34	(	(	PUNCT
ejpam-3971	205	35	αβ	αβ	INTJ
ejpam-3971	205	36	)	)	PUNCT
ejpam-3971	205	37	is	be	AUX
ejpam-3971	205	38	fuzzy	fuzzy	ADJ
ejpam-3971	205	39	ideal	ideal	NOUN
ejpam-3971	205	40	of	of	ADP
ejpam-3971	205	41	x.	x.	NOUN
ejpam-3971	205	42	proof	proof	NOUN
ejpam-3971	205	43	.	.	PUNCT
ejpam-3971	206	1	let	let	VERB
ejpam-3971	206	2	µ	µ	X
ejpam-3971	206	3	be	be	AUX
ejpam-3971	206	4	a	a	DET
ejpam-3971	206	5	fuzzy	fuzzy	ADJ
ejpam-3971	206	6	subset	subset	NOUN
ejpam-3971	206	7	of	of	ADP
ejpam-3971	206	8	x	x	PRON
ejpam-3971	206	9	,	,	PUNCT
ejpam-3971	206	10	α	α	PROPN
ejpam-3971	206	11	∈	∈	PROPN
ejpam-3971	207	1	[	[	X
ejpam-3971	207	2	0	0	NUM
ejpam-3971	207	3	,	,	PUNCT
ejpam-3971	207	4	t	t	NOUN
ejpam-3971	207	5	]	]	PUNCT
ejpam-3971	207	6	and	and	CCONJ
ejpam-3971	207	7	β	β	X
ejpam-3971	207	8	∈	∈	PROPN
ejpam-3971	208	1	[	[	X
ejpam-3971	208	2	0	0	NUM
ejpam-3971	208	3	,	,	PUNCT
ejpam-3971	208	4	1	1	NUM
ejpam-3971	208	5	]	]	PUNCT
ejpam-3971	208	6	.	.	PUNCT
ejpam-3971	209	1	a	a	DET
ejpam-3971	209	2	mapping	mapping	NOUN
ejpam-3971	209	3	µmt	µmt	NOUN
ejpam-3971	209	4	(	(	PUNCT
ejpam-3971	209	5	αβ	αβ	INTJ
ejpam-3971	209	6	)	)	PUNCT
ejpam-3971	209	7	:x	:x	PROPN
ejpam-3971	210	1	−→	−→	PROPN
ejpam-3971	210	2	[	[	X
ejpam-3971	210	3	0	0	NUM
ejpam-3971	210	4	,	,	PUNCT
ejpam-3971	210	5	1	1	NUM
ejpam-3971	210	6	]	]	PUNCT
ejpam-3971	210	7	is	be	AUX
ejpam-3971	210	8	said	say	VERB
ejpam-3971	210	9	to	to	PART
ejpam-3971	210	10	be	be	AUX
ejpam-3971	210	11	a	a	DET
ejpam-3971	210	12	fuzzy	fuzzy	ADJ
ejpam-3971	210	13	magnified	magnify	VERB
ejpam-3971	210	14	-	-	PUNCT
ejpam-3971	210	15	αβ	αβ	INTJ
ejpam-3971	210	16	-	-	PUNCT
ejpam-3971	210	17	translation	translation	NOUN
ejpam-3971	210	18	of	of	ADP
ejpam-3971	210	19	µ.	µ.	PROPN
ejpam-3971	210	20	assume	assume	VERB
ejpam-3971	210	21	µ	µ	X
ejpam-3971	210	22	is	be	AUX
ejpam-3971	210	23	fuzzy	fuzzy	ADJ
ejpam-3971	210	24	brk	brk	PROPN
ejpam-3971	210	25	-	-	PUNCT
ejpam-3971	210	26	ideal	ideal	NOUN
ejpam-3971	210	27	of	of	ADP
ejpam-3971	210	28	x.	x.	NOUN
ejpam-3971	210	29	then	then	ADV
ejpam-3971	210	30	µ(0	µ(0	PROPN
ejpam-3971	210	31	∗	∗	NOUN
ejpam-3971	210	32	x	x	NOUN
ejpam-3971	210	33	)	)	PUNCT
ejpam-3971	210	34	≥	≥	NOUN
ejpam-3971	210	35	min{µ(0	min{µ(0	NOUN
ejpam-3971	210	36	∗	∗	NOUN
ejpam-3971	210	37	(	(	PUNCT
ejpam-3971	210	38	x	x	X
ejpam-3971	210	39	∗	∗	PROPN
ejpam-3971	210	40	y	y	PROPN
ejpam-3971	210	41	)	)	PUNCT
ejpam-3971	210	42	)	)	PUNCT
ejpam-3971	210	43	,	,	PUNCT
ejpam-3971	210	44	µ(0	µ(0	NOUN
ejpam-3971	210	45	∗	∗	PROPN
ejpam-3971	210	46	y	y	NOUN
ejpam-3971	210	47	)	)	PUNCT
ejpam-3971	210	48	}	}	PUNCT
ejpam-3971	210	49	now	now	ADV
ejpam-3971	210	50	,	,	PUNCT
ejpam-3971	210	51	µmt	µmt	NOUN
ejpam-3971	210	52	(	(	PUNCT
ejpam-3971	210	53	αβ	αβ	INTJ
ejpam-3971	210	54	)	)	PUNCT
ejpam-3971	210	55	(	(	PUNCT
ejpam-3971	210	56	0	0	NUM
ejpam-3971	210	57	∗	∗	NOUN
ejpam-3971	210	58	x	x	NOUN
ejpam-3971	210	59	)	)	PUNCT
ejpam-3971	211	1	=	=	NOUN
ejpam-3971	211	2	α.µ(0	α.µ(0	PROPN
ejpam-3971	211	3	∗	∗	NOUN
ejpam-3971	211	4	x	x	NOUN
ejpam-3971	211	5	)	)	PUNCT
ejpam-3971	211	6	+	+	CCONJ
ejpam-3971	211	7	β	β	X
ejpam-3971	211	8	≥	≥	NOUN
ejpam-3971	211	9	α.min{µ(0	α.min{µ(0	PROPN
ejpam-3971	211	10	∗	∗	NOUN
ejpam-3971	211	11	(	(	PUNCT
ejpam-3971	211	12	x	x	X
ejpam-3971	211	13	∗	∗	PROPN
ejpam-3971	211	14	y	y	PROPN
ejpam-3971	211	15	)	)	PUNCT
ejpam-3971	211	16	)	)	PUNCT
ejpam-3971	211	17	,	,	PUNCT
ejpam-3971	211	18	µ(0	µ(0	NOUN
ejpam-3971	211	19	∗	∗	VERB
ejpam-3971	211	20	y)}+	y)}+	NUM
ejpam-3971	211	21	β	β	NOUN
ejpam-3971	211	22	=	=	PUNCT
ejpam-3971	211	23	min{α.µ(0	min{α.µ(0	NOUN
ejpam-3971	211	24	∗	∗	NOUN
ejpam-3971	211	25	(	(	PUNCT
ejpam-3971	211	26	x	x	X
ejpam-3971	211	27	∗	∗	PROPN
ejpam-3971	211	28	y	y	PROPN
ejpam-3971	211	29	)	)	PUNCT
ejpam-3971	211	30	)	)	PUNCT
ejpam-3971	212	1	+	+	CCONJ
ejpam-3971	212	2	β	β	X
ejpam-3971	212	3	,	,	PUNCT
ejpam-3971	212	4	µ(0	µ(0	PROPN
ejpam-3971	212	5	∗	∗	PROPN
ejpam-3971	212	6	y	y	NOUN
ejpam-3971	212	7	)	)	PUNCT
ejpam-3971	212	8	+	+	CCONJ
ejpam-3971	212	9	β	β	X
ejpam-3971	212	10	}	}	PUNCT
ejpam-3971	212	11	=	=	PUNCT
ejpam-3971	212	12	min{µmt	min{µmt	INTJ
ejpam-3971	212	13	(	(	PUNCT
ejpam-3971	212	14	αβ	αβ	INTJ
ejpam-3971	212	15	)	)	PUNCT
ejpam-3971	212	16	(	(	PUNCT
ejpam-3971	212	17	0	0	NUM
ejpam-3971	212	18	∗	∗	NOUN
ejpam-3971	212	19	(	(	PUNCT
ejpam-3971	212	20	x	x	X
ejpam-3971	212	21	∗	∗	PUNCT
ejpam-3971	212	22	y)),µmt	y)),µmt	NOUN
ejpam-3971	212	23	(	(	PUNCT
ejpam-3971	212	24	αβ	αβ	INTJ
ejpam-3971	212	25	)	)	PUNCT
ejpam-3971	212	26	(	(	PUNCT
ejpam-3971	212	27	0	0	NUM
ejpam-3971	212	28	∗	∗	NUM
ejpam-3971	212	29	y	y	PROPN
ejpam-3971	212	30	)	)	PUNCT
ejpam-3971	212	31	}	}	PUNCT
ejpam-3971	212	32	hence	hence	ADV
ejpam-3971	212	33	,	,	PUNCT
ejpam-3971	212	34	µmt	µmt	NOUN
ejpam-3971	212	35	(	(	PUNCT
ejpam-3971	212	36	αβ	αβ	INTJ
ejpam-3971	212	37	)	)	PUNCT
ejpam-3971	212	38	is	be	AUX
ejpam-3971	212	39	fuzzy	fuzzy	ADJ
ejpam-3971	212	40	brk	brk	PROPN
ejpam-3971	212	41	-	-	PUNCT
ejpam-3971	212	42	ideal	ideal	NOUN
ejpam-3971	212	43	of	of	ADP
ejpam-3971	212	44	x.	x.	NOUN
ejpam-3971	212	45	conversely	conversely	ADV
ejpam-3971	212	46	,	,	PUNCT
ejpam-3971	212	47	suppose	suppose	VERB
ejpam-3971	212	48	that	that	SCONJ
ejpam-3971	212	49	µmt	µmt	NOUN
ejpam-3971	212	50	(	(	PUNCT
ejpam-3971	212	51	αβ	αβ	INTJ
ejpam-3971	212	52	)	)	PUNCT
ejpam-3971	212	53	is	be	AUX
ejpam-3971	212	54	fuzzy	fuzzy	ADJ
ejpam-3971	212	55	brk	brk	PROPN
ejpam-3971	212	56	-	-	PUNCT
ejpam-3971	212	57	ideal	ideal	NOUN
ejpam-3971	212	58	of	of	ADP
ejpam-3971	212	59	x.	x.	NOUN
ejpam-3971	212	60	then	then	ADV
ejpam-3971	212	61	,	,	PUNCT
ejpam-3971	212	62	α.µ(0	α.µ(0	PROPN
ejpam-3971	212	63	∗	∗	PROPN
ejpam-3971	212	64	x)=	x)=	PROPN
ejpam-3971	212	65	µmt	µmt	PROPN
ejpam-3971	212	66	(	(	PUNCT
ejpam-3971	212	67	αβ	αβ	INTJ
ejpam-3971	212	68	)	)	PUNCT
ejpam-3971	212	69	(	(	PUNCT
ejpam-3971	212	70	0	0	NUM
ejpam-3971	212	71	∗	∗	NOUN
ejpam-3971	212	72	x)≥	x)≥	NOUN
ejpam-3971	213	1	min{µmt	min{µmt	NOUN
ejpam-3971	213	2	(	(	PUNCT
ejpam-3971	213	3	αβ	αβ	INTJ
ejpam-3971	213	4	)	)	PUNCT
ejpam-3971	213	5	(	(	PUNCT
ejpam-3971	213	6	0	0	NUM
ejpam-3971	213	7	∗	∗	NOUN
ejpam-3971	213	8	(	(	PUNCT
ejpam-3971	213	9	x	x	X
ejpam-3971	213	10	∗	∗	PUNCT
ejpam-3971	213	11	y)),µmt	y)),µmt	NOUN
ejpam-3971	213	12	(	(	PUNCT
ejpam-3971	213	13	αβ	αβ	INTJ
ejpam-3971	213	14	)	)	PUNCT
ejpam-3971	213	15	(	(	PUNCT
ejpam-3971	213	16	0	0	NUM
ejpam-3971	213	17	∗	∗	NUM
ejpam-3971	213	18	y	y	NOUN
ejpam-3971	213	19	)	)	PUNCT
ejpam-3971	213	20	}	}	PUNCT
ejpam-3971	214	1	=	=	NOUN
ejpam-3971	214	2	min{(α.µ(0	min{(α.µ(0	NOUN
ejpam-3971	214	3	∗	∗	NOUN
ejpam-3971	214	4	(	(	PUNCT
ejpam-3971	214	5	x	x	X
ejpam-3971	214	6	∗	∗	PROPN
ejpam-3971	214	7	y	y	PROPN
ejpam-3971	214	8	)	)	PUNCT
ejpam-3971	214	9	)	)	PUNCT
ejpam-3971	215	1	+	+	CCONJ
ejpam-3971	215	2	β	β	X
ejpam-3971	215	3	)	)	PUNCT
ejpam-3971	215	4	,	,	PUNCT
ejpam-3971	215	5	(	(	PUNCT
ejpam-3971	215	6	α.µ(0	α.µ(0	PROPN
ejpam-3971	215	7	∗	∗	X
ejpam-3971	215	8	y	y	PROPN
ejpam-3971	215	9	)	)	PUNCT
ejpam-3971	216	1	+	+	CCONJ
ejpam-3971	216	2	β	β	X
ejpam-3971	216	3	)	)	PUNCT
ejpam-3971	216	4	}	}	PUNCT
ejpam-3971	217	1	=	=	PUNCT
ejpam-3971	217	2	α.min{µ(0	α.min{µ(0	NOUN
ejpam-3971	217	3	∗	∗	NOUN
ejpam-3971	217	4	(	(	PUNCT
ejpam-3971	217	5	x	x	X
ejpam-3971	217	6	∗	∗	PROPN
ejpam-3971	217	7	y	y	PROPN
ejpam-3971	217	8	)	)	PUNCT
ejpam-3971	217	9	)	)	PUNCT
ejpam-3971	217	10	,	,	PUNCT
ejpam-3971	217	11	µ(0	µ(0	NOUN
ejpam-3971	217	12	∗	∗	VERB
ejpam-3971	217	13	y)}+β	y)}+β	PROPN
ejpam-3971	218	1	so	so	ADV
ejpam-3971	218	2	,	,	PUNCT
ejpam-3971	218	3	µ(0	µ(0	NOUN
ejpam-3971	218	4	∗	∗	NOUN
ejpam-3971	218	5	x	x	NOUN
ejpam-3971	218	6	)	)	PUNCT
ejpam-3971	218	7	≥	≥	NOUN
ejpam-3971	218	8	min{µ(0	min{µ(0	NOUN
ejpam-3971	218	9	∗	∗	NOUN
ejpam-3971	218	10	(	(	PUNCT
ejpam-3971	218	11	x	x	X
ejpam-3971	218	12	∗	∗	PROPN
ejpam-3971	218	13	y	y	PROPN
ejpam-3971	218	14	)	)	PUNCT
ejpam-3971	218	15	)	)	PUNCT
ejpam-3971	218	16	,	,	PUNCT
ejpam-3971	218	17	µ(0	µ(0	NOUN
ejpam-3971	218	18	∗	∗	PROPN
ejpam-3971	218	19	y	y	PROPN
ejpam-3971	218	20	)	)	PUNCT
ejpam-3971	218	21	}	}	PUNCT
ejpam-3971	218	22	.	.	PUNCT
ejpam-3971	219	1	hence,µ	hence,µ	PROPN
ejpam-3971	219	2	is	be	AUX
ejpam-3971	219	3	fuzzy	fuzzy	ADJ
ejpam-3971	219	4	brk	brk	PROPN
ejpam-3971	219	5	-	-	PUNCT
ejpam-3971	219	6	ideal	ideal	NOUN
ejpam-3971	219	7	of	of	ADP
ejpam-3971	219	8	x.	x.	NOUN
ejpam-3971	219	9	6	6	NUM
ejpam-3971	219	10	.	.	PUNCT
ejpam-3971	219	11	conclusion	conclusion	NOUN
ejpam-3971	219	12	in	in	ADP
ejpam-3971	219	13	this	this	DET
ejpam-3971	219	14	article	article	NOUN
ejpam-3971	219	15	,	,	PUNCT
ejpam-3971	219	16	we	we	PRON
ejpam-3971	219	17	investigated	investigate	VERB
ejpam-3971	219	18	the	the	DET
ejpam-3971	219	19	notion	notion	NOUN
ejpam-3971	219	20	of	of	ADP
ejpam-3971	219	21	fuzzy	fuzzy	ADJ
ejpam-3971	219	22	translation	translation	NOUN
ejpam-3971	219	23	and	and	CCONJ
ejpam-3971	219	24	fuzzy	fuzzy	ADJ
ejpam-3971	219	25	multiplication	multiplication	NOUN
ejpam-3971	219	26	brk	brk	PROPN
ejpam-3971	219	27	-	-	PUNCT
ejpam-3971	219	28	subalgebras	subalgebras	PROPN
ejpam-3971	219	29	and	and	CCONJ
ejpam-3971	219	30	discussed	discuss	VERB
ejpam-3971	219	31	related	related	ADJ
ejpam-3971	219	32	properties	property	NOUN
ejpam-3971	219	33	.	.	PUNCT
ejpam-3971	220	1	we	we	PRON
ejpam-3971	220	2	introduced	introduce	VERB
ejpam-3971	220	3	fuzzy	fuzzy	ADJ
ejpam-3971	220	4	translation	translation	NOUN
ejpam-3971	220	5	and	and	CCONJ
ejpam-3971	220	6	fuzzy	fuzzy	ADJ
ejpam-3971	220	7	multiplication	multiplication	NOUN
ejpam-3971	220	8	brk	brk	PROPN
ejpam-3971	220	9	-	-	PUNCT
ejpam-3971	220	10	ideals	ideal	NOUN
ejpam-3971	220	11	and	and	CCONJ
ejpam-3971	220	12	discussed	discuss	VERB
ejpam-3971	220	13	related	related	ADJ
ejpam-3971	220	14	results	result	NOUN
ejpam-3971	220	15	.	.	PUNCT
ejpam-3971	221	1	also	also	ADV
ejpam-3971	221	2	,	,	PUNCT
ejpam-3971	221	3	we	we	PRON
ejpam-3971	221	4	defined	define	VERB
ejpam-3971	221	5	fuzzy	fuzzy	ADV
ejpam-3971	221	6	magnified	magnify	VERB
ejpam-3971	221	7	-	-	PUNCT
ejpam-3971	221	8	αβ	αβ	INTJ
ejpam-3971	221	9	-	-	PUNCT
ejpam-3971	221	10	translation	translation	NOUN
ejpam-3971	221	11	of	of	ADP
ejpam-3971	221	12	brk	brk	PROPN
ejpam-3971	221	13	-	-	PUNCT
ejpam-3971	221	14	algebras	algebras	PROPN
ejpam-3971	221	15	.	.	PUNCT
ejpam-3971	222	1	as	as	ADP
ejpam-3971	222	2	an	an	DET
ejpam-3971	222	3	extension	extension	NOUN
ejpam-3971	222	4	of	of	ADP
ejpam-3971	222	5	above	above	ADJ
ejpam-3971	222	6	results	result	NOUN
ejpam-3971	222	7	,	,	PUNCT
ejpam-3971	222	8	one	one	PRON
ejpam-3971	222	9	could	could	AUX
ejpam-3971	222	10	study	study	VERB
ejpam-3971	222	11	anti	anti	ADJ
ejpam-3971	222	12	fuzzy	fuzzy	ADJ
ejpam-3971	222	13	translation	translation	NOUN
ejpam-3971	222	14	and	and	CCONJ
ejpam-3971	222	15	fuzzy	fuzzy	ADJ
ejpam-3971	222	16	multiplication	multiplication	NOUN
ejpam-3971	222	17	on	on	ADP
ejpam-3971	222	18	brk	brk	PROPN
ejpam-3971	222	19	-	-	PUNCT
ejpam-3971	222	20	algebras	algebras	PROPN
ejpam-3971	222	21	in	in	ADP
ejpam-3971	222	22	other	other	ADJ
ejpam-3971	222	23	algebraic	algebraic	ADJ
ejpam-3971	222	24	structures	structure	NOUN
ejpam-3971	222	25	.	.	PUNCT
ejpam-3971	223	1	method	method	NOUN
ejpam-3971	223	2	implementations	implementation	NOUN
ejpam-3971	223	3	to	to	PART
ejpam-3971	223	4	fix	fix	VERB
ejpam-3971	223	5	similar	similar	ADJ
ejpam-3971	223	6	concerns	concern	NOUN
ejpam-3971	223	7	in	in	ADP
ejpam-3971	223	8	machine	machine	NOUN
ejpam-3971	223	9	learning	learning	NOUN
ejpam-3971	223	10	,	,	PUNCT
ejpam-3971	223	11	decisionmaking	decisionmake	VERB
ejpam-3971	223	12	,	,	PUNCT
ejpam-3971	223	13	knowledge	knowledge	NOUN
ejpam-3971	223	14	science	science	NOUN
ejpam-3971	223	15	,	,	PUNCT
ejpam-3971	223	16	cognitive	cognitive	ADJ
ejpam-3971	223	17	science	science	NOUN
ejpam-3971	223	18	,	,	PUNCT
ejpam-3971	223	19	smart	smart	ADJ
ejpam-3971	223	20	decision	decision	NOUN
ejpam-3971	223	21	-	-	PUNCT
ejpam-3971	223	22	making	making	NOUN
ejpam-3971	223	23	,	,	PUNCT
ejpam-3971	223	24	etc	etc	X
ejpam-3971	223	25	.	.	X
ejpam-3971	223	26	references	reference	NOUN
ejpam-3971	223	27	[	[	X
ejpam-3971	223	28	1	1	NUM
ejpam-3971	223	29	]	]	X
ejpam-3971	223	30	r.k.bandaru	r.k.bandaru	NOUN
ejpam-3971	223	31	,	,	PUNCT
ejpam-3971	223	32	on	on	ADP
ejpam-3971	223	33	brk	brk	PROPN
ejpam-3971	223	34	-	-	PUNCT
ejpam-3971	223	35	algebras	algebras	PROPN
ejpam-3971	223	36	.	.	PUNCT
ejpam-3971	224	1	int	int	NOUN
ejpam-3971	224	2	.	.	PUNCT
ejpam-3971	225	1	j.	j.	PROPN
ejpam-3971	225	2	math	math	PROPN
ejpam-3971	225	3	.	.	PUNCT
ejpam-3971	226	1	mathematical	mathematical	PROPN
ejpam-3971	226	2	sci	sci	PROPN
ejpam-3971	226	3	.	.	PROPN
ejpam-3971	226	4	,	,	PUNCT
ejpam-3971	226	5	2012	2012	NUM
ejpam-3971	226	6	.	.	PUNCT
ejpam-3971	226	7	references	reference	NOUN
ejpam-3971	226	8	745	745	NUM
ejpam-3971	227	1	[	[	X
ejpam-3971	227	2	2	2	NUM
ejpam-3971	227	3	]	]	PUNCT
ejpam-3971	227	4	o.	o.	PROPN
ejpam-3971	227	5	r.	r.	PROPN
ejpam-3971	227	6	elgendy	elgendy	PROPN
ejpam-3971	227	7	,	,	PUNCT
ejpam-3971	227	8	fuzzy	fuzzy	ADJ
ejpam-3971	227	9	brk	brk	PROPN
ejpam-3971	227	10	-	-	PUNCT
ejpam-3971	227	11	ideal	ideal	NOUN
ejpam-3971	227	12	of	of	ADP
ejpam-3971	227	13	brk	brk	PROPN
ejpam-3971	227	14	-	-	PUNCT
ejpam-3971	227	15	algebra	algebra	PROPN
ejpam-3971	227	16	.	.	PUNCT
ejpam-3971	227	17	,	,	PUNCT
ejpam-3971	228	1	jp	jp	PROPN
ejpam-3971	228	2	j.	j.	PROPN
ejpam-3971	228	3	algebra	algebra	PROPN
ejpam-3971	228	4	number	number	NOUN
ejpam-3971	228	5	theory	theory	NOUN
ejpam-3971	228	6	appl	appl	PROPN
ejpam-3971	228	7	.	.	PUNCT
ejpam-3971	229	1	,	,	PUNCT
ejpam-3971	229	2	36	36	NUM
ejpam-3971	229	3	,	,	PUNCT
ejpam-3971	229	4	231–240	231–240	NUM
ejpam-3971	229	5	,	,	PUNCT
ejpam-3971	229	6	2015	2015	NUM
ejpam-3971	229	7	.	.	PUNCT
ejpam-3971	230	1	[	[	X
ejpam-3971	230	2	3	3	X
ejpam-3971	230	3	]	]	X
ejpam-3971	230	4	o.	o.	PROPN
ejpam-3971	230	5	r.	r.	PROPN
ejpam-3971	230	6	elgendy	elgendy	PROPN
ejpam-3971	230	7	,	,	PUNCT
ejpam-3971	230	8	cubic	cubic	PROPN
ejpam-3971	230	9	brk	brk	PROPN
ejpam-3971	230	10	-	-	PUNCT
ejpam-3971	230	11	ideal	ideal	NOUN
ejpam-3971	230	12	of	of	ADP
ejpam-3971	230	13	brk	brk	PROPN
ejpam-3971	230	14	-	-	PUNCT
ejpam-3971	230	15	algebra	algebra	PROPN
ejpam-3971	230	16	.	.	PUNCT
ejpam-3971	230	17	,	,	PUNCT
ejpam-3971	230	18	ann	ann	PROPN
ejpam-3971	230	19	.	.	PROPN
ejpam-3971	230	20	fuzzy	fuzzy	ADJ
ejpam-3971	230	21	math	math	PROPN
ejpam-3971	230	22	.	.	PUNCT
ejpam-3971	231	1	inf	inf	PROPN
ejpam-3971	231	2	.	.	PROPN
ejpam-3971	231	3	,	,	PUNCT
ejpam-3971	231	4	11	11	NUM
ejpam-3971	231	5	,	,	PUNCT
ejpam-3971	231	6	1–9	1–9	NUM
ejpam-3971	231	7	,	,	PUNCT
ejpam-3971	231	8	2016	2016	NUM
ejpam-3971	231	9	.	.	PUNCT
ejpam-3971	232	1	[	[	X
ejpam-3971	232	2	4	4	NUM
ejpam-3971	232	3	]	]	X
ejpam-3971	232	4	q.p	q.p	PROPN
ejpam-3971	232	5	.	.	PROPN
ejpam-3971	232	6	hu	hu	PROPN
ejpam-3971	232	7	and	and	CCONJ
ejpam-3971	232	8	x.	x.	PROPN
ejpam-3971	232	9	li	li	PROPN
ejpam-3971	232	10	,	,	PUNCT
ejpam-3971	232	11	on	on	ADP
ejpam-3971	232	12	bch	bch	PROPN
ejpam-3971	232	13	-	-	PUNCT
ejpam-3971	232	14	algebras	algebras	PROPN
ejpam-3971	232	15	.	.	PROPN
ejpam-3971	232	16	,	,	PUNCT
ejpam-3971	232	17	math	math	NOUN
ejpam-3971	232	18	.	.	PUNCT
ejpam-3971	233	1	sem	sem	PROPN
ejpam-3971	233	2	.	.	PUNCT
ejpam-3971	234	1	notes	notes	PROPN
ejpam-3971	234	2	kobe	kobe	PROPN
ejpam-3971	234	3	univ	univ	PROPN
ejpam-3971	234	4	.	.	PROPN
ejpam-3971	234	5	,	,	PUNCT
ejpam-3971	234	6	2	2	NUM
ejpam-3971	234	7	,	,	PUNCT
ejpam-3971	234	8	1983	1983	NUM
ejpam-3971	234	9	.	.	PUNCT
ejpam-3971	235	1	[	[	X
ejpam-3971	235	2	5	5	NUM
ejpam-3971	235	3	]	]	X
ejpam-3971	235	4	y.	y.	PROPN
ejpam-3971	235	5	imai	imai	PROPN
ejpam-3971	235	6	and	and	CCONJ
ejpam-3971	235	7	k.	k.	PROPN
ejpam-3971	235	8	is´eki	is´eki	PROPN
ejpam-3971	235	9	,	,	PUNCT
ejpam-3971	235	10	on	on	ADP
ejpam-3971	235	11	axiom	axiom	NOUN
ejpam-3971	235	12	systems	system	NOUN
ejpam-3971	235	13	of	of	ADP
ejpam-3971	235	14	propositional	propositional	ADJ
ejpam-3971	235	15	calculi	calculi	PROPN
ejpam-3971	235	16	,	,	PUNCT
ejpam-3971	235	17	xiv	xiv	PROPN
ejpam-3971	235	18	.	.	PROPN
ejpam-3971	235	19	,	,	PUNCT
ejpam-3971	235	20	proc	proc	PROPN
ejpam-3971	235	21	.	.	PUNCT
ejpam-3971	236	1	jpn	jpn	PROPN
ejpam-3971	236	2	.	.	PUNCT
ejpam-3971	236	3	acad	acad	PROPN
ejpam-3971	236	4	.	.	PROPN
ejpam-3971	236	5	,	,	PUNCT
ejpam-3971	236	6	42	42	NUM
ejpam-3971	236	7	,	,	PUNCT
ejpam-3971	236	8	19–22	19–22	NUM
ejpam-3971	236	9	1966	1966	NUM
ejpam-3971	236	10	.	.	PUNCT
ejpam-3971	237	1	[	[	X
ejpam-3971	237	2	6	6	NUM
ejpam-3971	237	3	]	]	X
ejpam-3971	237	4	h.	h.	PROPN
ejpam-3971	237	5	khizar	khizar	PROPN
ejpam-3971	237	6	,	,	PUNCT
ejpam-3971	237	7	l.	l.	PROPN
ejpam-3971	237	8	xiao	xiao	PROPN
ejpam-3971	237	9	and	and	CCONJ
ejpam-3971	237	10	c.	c.	PROPN
ejpam-3971	237	11	bing	bing	NOUN
ejpam-3971	237	12	,	,	PUNCT
ejpam-3971	237	13	bipolar	bipolar	ADJ
ejpam-3971	237	14	fuzzy	fuzzy	ADJ
ejpam-3971	237	15	brk	brk	PROPN
ejpam-3971	237	16	-	-	PUNCT
ejpam-3971	237	17	ideals	ideal	NOUN
ejpam-3971	237	18	in	in	ADP
ejpam-3971	237	19	brk	brk	PROPN
ejpam-3971	237	20	-	-	PUNCT
ejpam-3971	237	21	algebras	algebras	PROPN
ejpam-3971	237	22	.	.	PROPN
ejpam-3971	237	23	,	,	PUNCT
ejpam-3971	237	24	conference	conference	NOUN
ejpam-3971	237	25	:	:	PUNCT
ejpam-3971	237	26	international	international	ADJ
ejpam-3971	237	27	workshop	workshop	NOUN
ejpam-3971	237	28	on	on	ADP
ejpam-3971	237	29	mathematics	mathematic	NOUN
ejpam-3971	237	30	and	and	CCONJ
ejpam-3971	237	31	decision	decision	NOUN
ejpam-3971	237	32	science	science	NOUN
ejpam-3971	237	33	,	,	PUNCT
ejpam-3971	237	34	3	3	NUM
ejpam-3971	237	35	-	-	SYM
ejpam-3971	237	36	15	15	NUM
ejpam-3971	237	37	2018	2018	NUM
ejpam-3971	237	38	.	.	PUNCT
ejpam-3971	238	1	[	[	X
ejpam-3971	238	2	7	7	X
ejpam-3971	238	3	]	]	X
ejpam-3971	238	4	k.m	k.m	PROPN
ejpam-3971	238	5	.	.	PROPN
ejpam-3971	238	6	lee	lee	PROPN
ejpam-3971	238	7	,	,	PUNCT
ejpam-3971	238	8	bipolar	bipolar	ADV
ejpam-3971	238	9	-	-	PUNCT
ejpam-3971	238	10	valued	value	VERB
ejpam-3971	238	11	fuzzy	fuzzy	ADJ
ejpam-3971	238	12	sets	set	NOUN
ejpam-3971	238	13	and	and	CCONJ
ejpam-3971	238	14	their	their	PRON
ejpam-3971	238	15	operations	operation	NOUN
ejpam-3971	238	16	.	.	PUNCT
ejpam-3971	238	17	,	,	PUNCT
ejpam-3971	238	18	proceedings	proceeding	NOUN
ejpam-3971	238	19	of	of	ADP
ejpam-3971	238	20	the	the	DET
ejpam-3971	238	21	international	international	ADJ
ejpam-3971	238	22	conference	conference	NOUN
ejpam-3971	238	23	on	on	ADP
ejpam-3971	238	24	intelligent	intelligent	ADJ
ejpam-3971	238	25	technologies	technology	NOUN
ejpam-3971	238	26	,	,	PUNCT
ejpam-3971	238	27	bangkok	bangkok	PROPN
ejpam-3971	238	28	,	,	PUNCT
ejpam-3971	238	29	thailand	thailand	PROPN
ejpam-3971	238	30	,	,	PUNCT
ejpam-3971	238	31	307–312	307–312	NUM
ejpam-3971	238	32	,	,	PUNCT
ejpam-3971	238	33	2000	2000	NUM
ejpam-3971	238	34	.	.	PUNCT
ejpam-3971	239	1	[	[	X
ejpam-3971	239	2	8	8	NUM
ejpam-3971	239	3	]	]	X
ejpam-3971	239	4	k.j.lee	k.j.lee	PROPN
ejpam-3971	239	5	,	,	PUNCT
ejpam-3971	239	6	y.	y.	PROPN
ejpam-3971	239	7	b.	b.	PROPN
ejpam-3971	239	8	jun	jun	PROPN
ejpam-3971	239	9	and	and	CCONJ
ejpam-3971	239	10	m.	m.	PROPN
ejpam-3971	239	11	i.	i.	PROPN
ejpam-3971	239	12	doh	doh	PROPN
ejpam-3971	239	13	,	,	PUNCT
ejpam-3971	239	14	fuzzy	fuzzy	ADJ
ejpam-3971	239	15	translations	translation	NOUN
ejpam-3971	239	16	and	and	CCONJ
ejpam-3971	239	17	fuzzy	fuzzy	ADJ
ejpam-3971	239	18	multiplications	multiplication	NOUN
ejpam-3971	239	19	of	of	ADP
ejpam-3971	239	20	bck	bck	PROPN
ejpam-3971	239	21	/	/	SYM
ejpam-3971	239	22	bci	bci	PROPN
ejpam-3971	239	23	-algebras	-algebras	PROPN
ejpam-3971	239	24	.	.	PROPN
ejpam-3971	239	25	,	,	PUNCT
ejpam-3971	239	26	commun	commun	PROPN
ejpam-3971	239	27	.	.	PUNCT
ejpam-3971	240	1	korean	korean	ADJ
ejpam-3971	240	2	math	math	PROPN
ejpam-3971	240	3	.	.	PUNCT
ejpam-3971	241	1	soc	soc	PROPN
ejpam-3971	241	2	.	.	PROPN
ejpam-3971	241	3	,	,	PUNCT
ejpam-3971	241	4	24	24	NUM
ejpam-3971	241	5	,	,	PUNCT
ejpam-3971	241	6	353–360	353–360	NUM
ejpam-3971	241	7	,	,	PUNCT
ejpam-3971	241	8	2009	2009	NUM
ejpam-3971	241	9	.	.	PUNCT
ejpam-3971	242	1	[	[	X
ejpam-3971	242	2	9	9	NUM
ejpam-3971	242	3	]	]	X
ejpam-3971	242	4	j.	j.	PROPN
ejpam-3971	242	5	neggers	neggers	PROPN
ejpam-3971	242	6	,	,	PUNCT
ejpam-3971	242	7	s.s	s.s	PROPN
ejpam-3971	242	8	.	.	PROPN
ejpam-3971	242	9	ahn	ahn	PROPN
ejpam-3971	242	10	and	and	CCONJ
ejpam-3971	242	11	h.s.kim	h.s.kim	PROPN
ejpam-3971	242	12	,	,	PUNCT
ejpam-3971	242	13	on	on	ADP
ejpam-3971	242	14	q	q	NOUN
ejpam-3971	242	15	-	-	PUNCT
ejpam-3971	242	16	algebras	algebras	X
ejpam-3971	242	17	.	.	PROPN
ejpam-3971	242	18	,	,	PUNCT
ejpam-3971	242	19	int	int	PROPN
ejpam-3971	242	20	.	.	PUNCT
ejpam-3971	243	1	j.	j.	PROPN
ejpam-3971	243	2	math	math	PROPN
ejpam-3971	243	3	.	.	PUNCT
ejpam-3971	244	1	sci	sci	PROPN
ejpam-3971	244	2	.	.	PROPN
ejpam-3971	244	3	,	,	PUNCT
ejpam-3971	244	4	27	27	NUM
ejpam-3971	244	5	,	,	PUNCT
ejpam-3971	244	6	(	(	PUNCT
ejpam-3971	244	7	12	12	NUM
ejpam-3971	244	8	)	)	PUNCT
ejpam-3971	244	9	,	,	PUNCT
ejpam-3971	244	10	749	749	NUM
ejpam-3971	244	11	-	-	SYM
ejpam-3971	244	12	757	757	NOUN
ejpam-3971	244	13	,	,	PUNCT
ejpam-3971	244	14	2001	2001	NUM
ejpam-3971	244	15	.	.	PUNCT
ejpam-3971	245	1	[	[	X
ejpam-3971	245	2	10	10	NUM
ejpam-3971	245	3	]	]	PUNCT
ejpam-3971	245	4	l.a.zadeh	l.a.zadeh	NOUN
ejpam-3971	245	5	,	,	PUNCT
ejpam-3971	245	6	fuzzy	fuzzy	ADJ
ejpam-3971	245	7	sets	set	NOUN
ejpam-3971	245	8	.	.	PUNCT
ejpam-3971	245	9	,	,	PUNCT
ejpam-3971	245	10	inform	inform	NOUN
ejpam-3971	245	11	.	.	PUNCT
ejpam-3971	246	1	and	and	CCONJ
ejpam-3971	246	2	control	control	NOUN
ejpam-3971	246	3	,	,	PUNCT
ejpam-3971	246	4	8	8	NUM
ejpam-3971	246	5	,	,	PUNCT
ejpam-3971	246	6	338–353	338–353	NUM
ejpam-3971	246	7	,	,	PUNCT
ejpam-3971	246	8	1965	1965	NUM
ejpam-3971	246	9	.	.	PUNCT
ejpam-3971	247	1	[	[	X
ejpam-3971	247	2	11	11	NUM
ejpam-3971	247	3	]	]	PUNCT
ejpam-3971	247	4	m.zulfiqar	m.zulfiqar	NUM
ejpam-3971	247	5	,	,	PUNCT
ejpam-3971	247	6	some	some	DET
ejpam-3971	247	7	properties	property	NOUN
ejpam-3971	247	8	of	of	ADP
ejpam-3971	247	9	n	n	ADV
ejpam-3971	247	10	-	-	PUNCT
ejpam-3971	247	11	dimensional	dimensional	ADJ
ejpam-3971	247	12	fuzzy	fuzzy	ADJ
ejpam-3971	247	13	subalgebra	subalgebra	NOUN
ejpam-3971	247	14	in	in	ADP
ejpam-3971	247	15	brk	brk	PROPN
ejpam-3971	247	16	-	-	PUNCT
ejpam-3971	247	17	algebras	algebras	PROPN
ejpam-3971	247	18	.	.	PROPN
ejpam-3971	247	19	,	,	PUNCT
ejpam-3971	247	20	analele	analele	ADP
ejpam-3971	247	21	stiintifice	stiintifice	NOUN
ejpam-3971	247	22	ale	ale	PROPN
ejpam-3971	247	23	universitatii	universitatii	PROPN
ejpam-3971	247	24	ovidius	ovidius	PROPN
ejpam-3971	247	25	constanta	constanta	PROPN
ejpam-3971	247	26	,	,	PUNCT
ejpam-3971	247	27	2015	2015	NUM
ejpam-3971	247	28	.	.	PUNCT