id	sid	tid	token	lemma	pos
ejpam-4618	1	1	european	european	PROPN
ejpam-4618	1	2	journal	journal	PROPN
ejpam-4618	1	3	of	of	ADP
ejpam-4618	1	4	pure	pure	ADJ
ejpam-4618	1	5	and	and	CCONJ
ejpam-4618	1	6	applied	apply	VERB
ejpam-4618	1	7	mathematics	mathematic	NOUN
ejpam-4618	1	8	vol	vol	NOUN
ejpam-4618	1	9	.	.	PUNCT
ejpam-4618	2	1	16	16	NUM
ejpam-4618	2	2	,	,	PUNCT
ejpam-4618	2	3	no	no	INTJ
ejpam-4618	2	4	.	.	NOUN
ejpam-4618	2	5	1	1	NUM
ejpam-4618	2	6	,	,	PUNCT
ejpam-4618	2	7	2023	2023	NUM
ejpam-4618	2	8	,	,	PUNCT
ejpam-4618	2	9	253	253	NUM
ejpam-4618	2	10	-	-	SYM
ejpam-4618	2	11	260	260	NUM
ejpam-4618	2	12	issn	issn	PROPN
ejpam-4618	2	13	1307	1307	NUM
ejpam-4618	2	14	-	-	SYM
ejpam-4618	2	15	5543	5543	NUM
ejpam-4618	2	16	–	–	PUNCT
ejpam-4618	3	1	ejpam.com	ejpam.com	X
ejpam-4618	3	2	published	publish	VERB
ejpam-4618	3	3	by	by	ADP
ejpam-4618	3	4	new	new	PROPN
ejpam-4618	3	5	york	york	PROPN
ejpam-4618	3	6	business	business	PROPN
ejpam-4618	3	7	global	global	PROPN
ejpam-4618	3	8	multipolar	multipolar	ADJ
ejpam-4618	3	9	fuzzy	fuzzy	ADJ
ejpam-4618	3	10	ku	ku	NOUN
ejpam-4618	3	11	-	-	PUNCT
ejpam-4618	3	12	ideals	ideal	NOUN
ejpam-4618	3	13	in	in	ADP
ejpam-4618	3	14	ku	ku	PROPN
ejpam-4618	3	15	-	-	PUNCT
ejpam-4618	3	16	algebras	algebras	PROPN
ejpam-4618	3	17	halimah	halimah	PROPN
ejpam-4618	3	18	alshehri	alshehri	PROPN
ejpam-4618	3	19	department	department	PROPN
ejpam-4618	3	20	of	of	ADP
ejpam-4618	3	21	computer	computer	NOUN
ejpam-4618	3	22	science	science	NOUN
ejpam-4618	3	23	and	and	CCONJ
ejpam-4618	3	24	engineering	engineering	NOUN
ejpam-4618	3	25	,	,	PUNCT
ejpam-4618	3	26	faculty	faculty	NOUN
ejpam-4618	3	27	applied	apply	VERB
ejpam-4618	3	28	studies	study	NOUN
ejpam-4618	3	29	and	and	CCONJ
ejpam-4618	3	30	community	community	NOUN
ejpam-4618	3	31	service	service	NOUN
ejpam-4618	3	32	,	,	PUNCT
ejpam-4618	3	33	king	king	PROPN
ejpam-4618	3	34	saud	saud	PROPN
ejpam-4618	3	35	university	university	PROPN
ejpam-4618	3	36	,	,	PUNCT
ejpam-4618	3	37	riyadh	riyadh	PROPN
ejpam-4618	3	38	,	,	PUNCT
ejpam-4618	3	39	saudi	saudi	PROPN
ejpam-4618	3	40	arabia	arabia	PROPN
ejpam-4618	3	41	abstract	abstract	NOUN
ejpam-4618	3	42	.	.	PUNCT
ejpam-4618	4	1	this	this	DET
ejpam-4618	4	2	article	article	NOUN
ejpam-4618	4	3	presents	present	VERB
ejpam-4618	4	4	the	the	DET
ejpam-4618	4	5	idea	idea	NOUN
ejpam-4618	4	6	of	of	ADP
ejpam-4618	4	7	an	an	DET
ejpam-4618	4	8	m	m	ADJ
ejpam-4618	4	9	-	-	ADJ
ejpam-4618	4	10	polar	polar	ADJ
ejpam-4618	4	11	fuzzy	fuzzy	ADJ
ejpam-4618	4	12	ku	ku	NOUN
ejpam-4618	4	13	-	-	PUNCT
ejpam-4618	4	14	ideal	ideal	NOUN
ejpam-4618	4	15	and	and	CCONJ
ejpam-4618	4	16	investigates	investigate	VERB
ejpam-4618	4	17	its	its	PRON
ejpam-4618	4	18	characteristics	characteristic	NOUN
ejpam-4618	4	19	.	.	PUNCT
ejpam-4618	5	1	the	the	DET
ejpam-4618	5	2	relationship	relationship	NOUN
ejpam-4618	5	3	between	between	ADP
ejpam-4618	5	4	an	an	DET
ejpam-4618	5	5	m	m	ADJ
ejpam-4618	5	6	-	-	ADJ
ejpam-4618	5	7	polar	polar	ADJ
ejpam-4618	5	8	fuzzy	fuzzy	ADJ
ejpam-4618	5	9	ku	ku	NOUN
ejpam-4618	5	10	-	-	PUNCT
ejpam-4618	5	11	subalgebra	subalgebra	PROPN
ejpam-4618	5	12	and	and	CCONJ
ejpam-4618	5	13	an	an	DET
ejpam-4618	5	14	m	m	ADJ
ejpam-4618	5	15	-	-	ADJ
ejpam-4618	5	16	polar	polar	ADJ
ejpam-4618	5	17	fuzzy	fuzzy	ADJ
ejpam-4618	5	18	ku	ku	NOUN
ejpam-4618	5	19	-	-	PUNCT
ejpam-4618	5	20	ideal	ideal	NOUN
ejpam-4618	5	21	is	be	AUX
ejpam-4618	5	22	presented	present	VERB
ejpam-4618	5	23	in	in	ADP
ejpam-4618	5	24	the	the	DET
ejpam-4618	5	25	discussion	discussion	NOUN
ejpam-4618	5	26	2020	2020	NUM
ejpam-4618	5	27	mathematics	mathematic	NOUN
ejpam-4618	5	28	subject	subject	NOUN
ejpam-4618	5	29	classifications	classification	NOUN
ejpam-4618	5	30	:	:	PUNCT
ejpam-4618	5	31	03b52	03b52	NUM
ejpam-4618	5	32	,	,	PUNCT
ejpam-4618	5	33	03e72	03e72	NUM
ejpam-4618	5	34	,	,	PUNCT
ejpam-4618	5	35	08a72	08a72	NOUN
ejpam-4618	5	36	key	key	ADJ
ejpam-4618	5	37	words	word	NOUN
ejpam-4618	5	38	and	and	CCONJ
ejpam-4618	5	39	phrases	phrase	NOUN
ejpam-4618	5	40	:	:	PUNCT
ejpam-4618	5	41	ku	ku	PROPN
ejpam-4618	5	42	-	-	PUNCT
ejpam-4618	5	43	algebras	algebras	PROPN
ejpam-4618	5	44	,	,	PUNCT
ejpam-4618	5	45	m	m	ADJ
ejpam-4618	5	46	-	-	ADJ
ejpam-4618	5	47	polar	polar	ADJ
ejpam-4618	5	48	fuzzy	fuzzy	ADJ
ejpam-4618	5	49	set	set	NOUN
ejpam-4618	5	50	,	,	PUNCT
ejpam-4618	5	51	m	m	NOUN
ejpam-4618	5	52	-	-	ADJ
ejpam-4618	5	53	polar	polar	ADJ
ejpam-4618	5	54	fuzzy	fuzzy	ADJ
ejpam-4618	5	55	ku	ku	NOUN
ejpam-4618	5	56	-	-	PUNCT
ejpam-4618	5	57	subalgebra	subalgebra	PROPN
ejpam-4618	5	58	,	,	PUNCT
ejpam-4618	5	59	mpolar	mpolar	ADJ
ejpam-4618	5	60	fuzzy	fuzzy	ADJ
ejpam-4618	5	61	ku	ku	NOUN
ejpam-4618	5	62	-	-	PUNCT
ejpam-4618	5	63	ideal	ideal	ADJ
ejpam-4618	5	64	1	1	NUM
ejpam-4618	5	65	.	.	PUNCT
ejpam-4618	6	1	introduction	introduction	NOUN
ejpam-4618	6	2	a	a	DET
ejpam-4618	6	3	fuzzy	fuzzy	ADJ
ejpam-4618	6	4	set	set	NOUN
ejpam-4618	6	5	is	be	AUX
ejpam-4618	6	6	a	a	DET
ejpam-4618	6	7	useful	useful	ADJ
ejpam-4618	6	8	tool	tool	NOUN
ejpam-4618	6	9	developed	develop	VERB
ejpam-4618	6	10	by	by	ADP
ejpam-4618	6	11	zadeh	zadeh	PROPN
ejpam-4618	7	1	[	[	X
ejpam-4618	7	2	11	11	NUM
ejpam-4618	7	3	]	]	PUNCT
ejpam-4618	7	4	for	for	ADP
ejpam-4618	7	5	dealing	deal	VERB
ejpam-4618	7	6	with	with	ADP
ejpam-4618	7	7	probabilistic	probabilistic	ADJ
ejpam-4618	7	8	uncertainty	uncertainty	NOUN
ejpam-4618	7	9	related	relate	VERB
ejpam-4618	7	10	to	to	ADP
ejpam-4618	7	11	perceptions	perception	NOUN
ejpam-4618	7	12	,	,	PUNCT
ejpam-4618	7	13	state	state	NOUN
ejpam-4618	7	14	inaccuracies	inaccuracy	NOUN
ejpam-4618	7	15	,	,	PUNCT
ejpam-4618	7	16	and	and	CCONJ
ejpam-4618	7	17	preferences	preference	NOUN
ejpam-4618	7	18	.	.	PUNCT
ejpam-4618	8	1	since	since	SCONJ
ejpam-4618	8	2	that	that	DET
ejpam-4618	8	3	time	time	NOUN
ejpam-4618	8	4	,	,	PUNCT
ejpam-4618	8	5	fuzzy	fuzzy	ADJ
ejpam-4618	8	6	set	set	NOUN
ejpam-4618	8	7	theory	theory	NOUN
ejpam-4618	8	8	has	have	AUX
ejpam-4618	8	9	gained	gain	VERB
ejpam-4618	8	10	much	much	ADJ
ejpam-4618	8	11	attention	attention	NOUN
ejpam-4618	8	12	in	in	ADP
ejpam-4618	8	13	a	a	DET
ejpam-4618	8	14	variety	variety	NOUN
ejpam-4618	8	15	of	of	ADP
ejpam-4618	8	16	disciplines	discipline	NOUN
ejpam-4618	8	17	,	,	PUNCT
ejpam-4618	8	18	including	include	VERB
ejpam-4618	8	19	graph	graph	NOUN
ejpam-4618	8	20	theory	theory	NOUN
ejpam-4618	8	21	,	,	PUNCT
ejpam-4618	8	22	statistics	statistic	NOUN
ejpam-4618	8	23	,	,	PUNCT
ejpam-4618	8	24	life	life	NOUN
ejpam-4618	8	25	and	and	CCONJ
ejpam-4618	8	26	medical	medical	ADJ
ejpam-4618	8	27	sciences	science	NOUN
ejpam-4618	8	28	,	,	PUNCT
ejpam-4618	8	29	engineering	engineering	NOUN
ejpam-4618	8	30	,	,	PUNCT
ejpam-4618	8	31	social	social	ADJ
ejpam-4618	8	32	sciences	science	NOUN
ejpam-4618	8	33	,	,	PUNCT
ejpam-4618	8	34	decision	decision	NOUN
ejpam-4618	8	35	-	-	PUNCT
ejpam-4618	8	36	making	making	NOUN
ejpam-4618	8	37	,	,	PUNCT
ejpam-4618	8	38	computer	computer	NOUN
ejpam-4618	8	39	networks	network	NOUN
ejpam-4618	8	40	,	,	PUNCT
ejpam-4618	8	41	robotics	robotic	NOUN
ejpam-4618	8	42	automata	automata	NOUN
ejpam-4618	8	43	theory	theory	NOUN
ejpam-4618	8	44	,	,	PUNCT
ejpam-4618	8	45	artificial	artificial	ADJ
ejpam-4618	8	46	intelligence	intelligence	NOUN
ejpam-4618	8	47	,	,	PUNCT
ejpam-4618	8	48	pattern	pattern	NOUN
ejpam-4618	8	49	recognition	recognition	NOUN
ejpam-4618	8	50	,	,	PUNCT
ejpam-4618	8	51	and	and	CCONJ
ejpam-4618	8	52	many	many	ADJ
ejpam-4618	8	53	others	other	NOUN
ejpam-4618	8	54	.	.	PUNCT
ejpam-4618	9	1	in	in	ADP
ejpam-4618	9	2	[	[	X
ejpam-4618	9	3	8	8	NUM
ejpam-4618	9	4	]	]	PUNCT
ejpam-4618	9	5	and	and	CCONJ
ejpam-4618	9	6	[	[	X
ejpam-4618	9	7	9	9	NUM
ejpam-4618	9	8	]	]	PUNCT
ejpam-4618	9	9	,	,	PUNCT
ejpam-4618	9	10	a	a	DET
ejpam-4618	9	11	new	new	ADJ
ejpam-4618	9	12	algebraic	algebraic	ADJ
ejpam-4618	9	13	structure	structure	NOUN
ejpam-4618	9	14	called	call	VERB
ejpam-4618	9	15	ku	ku	PROPN
ejpam-4618	9	16	-	-	PUNCT
ejpam-4618	9	17	algebras	algebras	PROPN
ejpam-4618	9	18	was	be	AUX
ejpam-4618	9	19	constructed	construct	VERB
ejpam-4618	9	20	.	.	PUNCT
ejpam-4618	10	1	mostafa	mostafa	PROPN
ejpam-4618	10	2	et	et	PROPN
ejpam-4618	10	3	al	al	PROPN
ejpam-4618	10	4	.	.	PUNCT
ejpam-4618	11	1	[	[	X
ejpam-4618	11	2	7	7	X
ejpam-4618	11	3	]	]	PUNCT
ejpam-4618	11	4	introduced	introduce	VERB
ejpam-4618	11	5	the	the	DET
ejpam-4618	11	6	notion	notion	NOUN
ejpam-4618	11	7	of	of	ADP
ejpam-4618	11	8	fuzzy	fuzzy	ADJ
ejpam-4618	11	9	ku	ku	NOUN
ejpam-4618	11	10	-	-	PUNCT
ejpam-4618	11	11	ideals	ideal	NOUN
ejpam-4618	11	12	of	of	ADP
ejpam-4618	11	13	ku	ku	PROPN
ejpam-4618	11	14	-	-	PUNCT
ejpam-4618	11	15	algebras	algebras	PROPN
ejpam-4618	11	16	and	and	CCONJ
ejpam-4618	11	17	then	then	ADV
ejpam-4618	11	18	investigated	investigate	VERB
ejpam-4618	11	19	several	several	ADJ
ejpam-4618	11	20	basic	basic	ADJ
ejpam-4618	11	21	properties	property	NOUN
ejpam-4618	11	22	related	relate	VERB
ejpam-4618	11	23	to	to	ADP
ejpam-4618	11	24	fuzzy	fuzzy	ADJ
ejpam-4618	11	25	ku	ku	NOUN
ejpam-4618	11	26	-	-	PUNCT
ejpam-4618	11	27	ideals	ideal	NOUN
ejpam-4618	11	28	.	.	PUNCT
ejpam-4618	12	1	recently	recently	ADV
ejpam-4618	12	2	,	,	PUNCT
ejpam-4618	12	3	akram	akram	PROPN
ejpam-4618	12	4	and	and	CCONJ
ejpam-4618	12	5	sarwar	sarwar	NOUN
ejpam-4618	13	1	[	[	X
ejpam-4618	13	2	3	3	X
ejpam-4618	13	3	]	]	PUNCT
ejpam-4618	13	4	applied	apply	VERB
ejpam-4618	13	5	the	the	DET
ejpam-4618	13	6	notion	notion	NOUN
ejpam-4618	13	7	of	of	ADP
ejpam-4618	13	8	m	m	ADJ
ejpam-4618	13	9	-	-	ADJ
ejpam-4618	13	10	polar	polar	ADJ
ejpam-4618	13	11	fuzzy	fuzzy	ADJ
ejpam-4618	13	12	set	set	NOUN
ejpam-4618	13	13	theory	theory	NOUN
ejpam-4618	13	14	to	to	ADP
ejpam-4618	13	15	the	the	DET
ejpam-4618	13	16	graph	graph	NOUN
ejpam-4618	13	17	theory	theory	NOUN
ejpam-4618	13	18	.	.	PUNCT
ejpam-4618	14	1	also	also	ADV
ejpam-4618	14	2	,	,	PUNCT
ejpam-4618	14	3	al	al	PROPN
ejpam-4618	14	4	-	-	PROPN
ejpam-4618	14	5	masarwah	masarwah	PROPN
ejpam-4618	14	6	and	and	CCONJ
ejpam-4618	14	7	ahmad	ahmad	PROPN
ejpam-4618	15	1	[	[	X
ejpam-4618	15	2	4	4	X
ejpam-4618	15	3	]	]	PUNCT
ejpam-4618	15	4	discussed	discuss	VERB
ejpam-4618	15	5	the	the	DET
ejpam-4618	15	6	notion	notion	NOUN
ejpam-4618	15	7	of	of	ADP
ejpam-4618	15	8	m	m	ADJ
ejpam-4618	15	9	-	-	ADJ
ejpam-4618	15	10	polar	polar	ADJ
ejpam-4618	15	11	fuzzy	fuzzy	ADJ
ejpam-4618	15	12	sets	set	NOUN
ejpam-4618	15	13	with	with	ADP
ejpam-4618	15	14	an	an	DET
ejpam-4618	15	15	application	application	NOUN
ejpam-4618	15	16	to	to	PART
ejpam-4618	15	17	bck	bck	VERB
ejpam-4618	15	18	/	/	SYM
ejpam-4618	15	19	bci	bci	NOUN
ejpam-4618	15	20	-	-	PUNCT
ejpam-4618	15	21	algebras	algebras	X
ejpam-4618	15	22	.	.	PUNCT
ejpam-4618	16	1	in	in	ADP
ejpam-4618	16	2	this	this	DET
ejpam-4618	16	3	study	study	NOUN
ejpam-4618	16	4	,	,	PUNCT
ejpam-4618	16	5	we	we	PRON
ejpam-4618	16	6	will	will	AUX
ejpam-4618	16	7	introduce	introduce	VERB
ejpam-4618	16	8	the	the	DET
ejpam-4618	16	9	notions	notion	NOUN
ejpam-4618	16	10	of	of	ADP
ejpam-4618	16	11	m	m	ADJ
ejpam-4618	16	12	-	-	ADJ
ejpam-4618	16	13	polar	polar	ADJ
ejpam-4618	16	14	fuzzy	fuzzy	ADJ
ejpam-4618	16	15	subalgebras	subalgebra	NOUN
ejpam-4618	16	16	and	and	CCONJ
ejpam-4618	16	17	m	m	ADJ
ejpam-4618	16	18	-	-	ADJ
ejpam-4618	16	19	polar	polar	ADJ
ejpam-4618	16	20	fuzzy	fuzzy	ADJ
ejpam-4618	16	21	(	(	PUNCT
ejpam-4618	16	22	closed	closed	ADJ
ejpam-4618	16	23	,	,	PUNCT
ejpam-4618	16	24	commutative	commutative	ADJ
ejpam-4618	16	25	)	)	PUNCT
ejpam-4618	16	26	ideals	ideal	NOUN
ejpam-4618	16	27	and	and	CCONJ
ejpam-4618	16	28	investigate	investigate	VERB
ejpam-4618	16	29	several	several	ADJ
ejpam-4618	16	30	properties	property	NOUN
ejpam-4618	16	31	.	.	PUNCT
ejpam-4618	17	1	this	this	DET
ejpam-4618	17	2	manuscript	manuscript	NOUN
ejpam-4618	17	3	aims	aim	VERB
ejpam-4618	17	4	to	to	PART
ejpam-4618	17	5	apply	apply	VERB
ejpam-4618	17	6	the	the	DET
ejpam-4618	17	7	notion	notion	NOUN
ejpam-4618	17	8	of	of	ADP
ejpam-4618	17	9	an	an	DET
ejpam-4618	17	10	m	m	ADJ
ejpam-4618	17	11	-	-	ADJ
ejpam-4618	17	12	polar	polar	ADJ
ejpam-4618	17	13	fuzzy	fuzzy	ADJ
ejpam-4618	17	14	set	set	NOUN
ejpam-4618	17	15	to	to	ADP
ejpam-4618	17	16	fuzzy	fuzzy	ADJ
ejpam-4618	17	17	ku	ku	NOUN
ejpam-4618	17	18	-	-	PUNCT
ejpam-4618	17	19	ideal	ideal	NOUN
ejpam-4618	17	20	in	in	ADP
ejpam-4618	17	21	ku	ku	PROPN
ejpam-4618	17	22	-	-	PUNCT
ejpam-4618	17	23	algebras	algebras	PROPN
ejpam-4618	17	24	.	.	PUNCT
ejpam-4618	18	1	the	the	DET
ejpam-4618	18	2	notions	notion	NOUN
ejpam-4618	18	3	of	of	ADP
ejpam-4618	18	4	an	an	DET
ejpam-4618	18	5	m	m	ADJ
ejpam-4618	18	6	-	-	ADJ
ejpam-4618	18	7	polar	polar	ADJ
ejpam-4618	18	8	fuzzy	fuzzy	ADJ
ejpam-4618	18	9	ku	ku	NOUN
ejpam-4618	18	10	-	-	PUNCT
ejpam-4618	18	11	ideal	ideal	NOUN
ejpam-4618	18	12	were	be	AUX
ejpam-4618	18	13	introduced	introduce	VERB
ejpam-4618	18	14	and	and	CCONJ
ejpam-4618	18	15	their	their	PRON
ejpam-4618	18	16	properties	property	NOUN
ejpam-4618	18	17	were	be	AUX
ejpam-4618	18	18	investigated	investigate	VERB
ejpam-4618	18	19	.	.	PUNCT
ejpam-4618	19	1	the	the	DET
ejpam-4618	19	2	relationship	relationship	NOUN
ejpam-4618	19	3	between	between	ADP
ejpam-4618	19	4	an	an	DET
ejpam-4618	19	5	m	m	ADJ
ejpam-4618	19	6	-	-	ADJ
ejpam-4618	19	7	polar	polar	ADJ
ejpam-4618	19	8	fuzzy	fuzzy	ADJ
ejpam-4618	19	9	ku	ku	NOUN
ejpam-4618	19	10	-	-	PUNCT
ejpam-4618	19	11	subalgebra	subalgebra	PROPN
ejpam-4618	19	12	and	and	CCONJ
ejpam-4618	19	13	an	an	DET
ejpam-4618	19	14	m	m	ADJ
ejpam-4618	19	15	-	-	ADJ
ejpam-4618	19	16	polar	polar	ADJ
ejpam-4618	19	17	fuzzy	fuzzy	ADJ
ejpam-4618	19	18	ku	ku	NOUN
ejpam-4618	19	19	-	-	PUNCT
ejpam-4618	19	20	ideal	ideal	PROPN
ejpam-4618	19	21	was	be	AUX
ejpam-4618	19	22	examined	examine	VERB
ejpam-4618	19	23	.	.	PUNCT
ejpam-4618	20	1	moreover	moreover	ADV
ejpam-4618	20	2	,	,	PUNCT
ejpam-4618	20	3	the	the	DET
ejpam-4618	20	4	relationship	relationship	NOUN
ejpam-4618	20	5	between	between	ADP
ejpam-4618	20	6	an	an	DET
ejpam-4618	20	7	m	m	ADJ
ejpam-4618	20	8	-	-	ADJ
ejpam-4618	20	9	polar	polar	ADJ
ejpam-4618	20	10	fuzzy	fuzzy	ADJ
ejpam-4618	20	11	ku	ku	NOUN
ejpam-4618	20	12	-	-	PUNCT
ejpam-4618	20	13	ideal	ideal	PROPN
ejpam-4618	20	14	and	and	CCONJ
ejpam-4618	20	15	the	the	DET
ejpam-4618	20	16	ideal	ideal	NOUN
ejpam-4618	20	17	of	of	ADP
ejpam-4618	20	18	bck	bck	PROPN
ejpam-4618	20	19	/	/	SYM
ejpam-4618	20	20	bci	bci	NOUN
ejpam-4618	20	21	-	-	PUNCT
ejpam-4618	20	22	algebras	algebras	PROPN
ejpam-4618	20	23	was	be	AUX
ejpam-4618	20	24	presented	present	VERB
ejpam-4618	20	25	.	.	PUNCT
ejpam-4618	21	1	doi	doi	NOUN
ejpam-4618	21	2	:	:	PUNCT
ejpam-4618	21	3	https://doi.org/10.29020/nybg.ejpam.v16i1.4618	https://doi.org/10.29020/nybg.ejpam.v16i1.4618	PRON
ejpam-4618	21	4	email	email	NOUN
ejpam-4618	21	5	address	address	NOUN
ejpam-4618	21	6	:	:	PUNCT
ejpam-4618	21	7	haalshehri@ksu.edu.sa	haalshehri@ksu.edu.sa	PROPN
ejpam-4618	21	8	(	(	PUNCT
ejpam-4618	21	9	h.	h.	PROPN
ejpam-4618	21	10	alshehri	alshehri	PROPN
ejpam-4618	21	11	)	)	PUNCT
ejpam-4618	21	12	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4618	22	1	253	253	NUM
ejpam-4618	22	2	©	©	ADP
ejpam-4618	22	3	2023	2023	NUM
ejpam-4618	22	4	ejpam	ejpam	NOUN
ejpam-4618	22	5	all	all	DET
ejpam-4618	22	6	rights	right	NOUN
ejpam-4618	22	7	reserved	reserve	VERB
ejpam-4618	22	8	.	.	PUNCT
ejpam-4618	23	1	h.	h.	PROPN
ejpam-4618	23	2	alshehri	alshehri	PROPN
ejpam-4618	23	3	/	/	SYM
ejpam-4618	23	4	eur	eur	PROPN
ejpam-4618	23	5	.	.	PUNCT
ejpam-4618	24	1	j.	j.	PROPN
ejpam-4618	24	2	pure	pure	PROPN
ejpam-4618	24	3	appl	appl	PROPN
ejpam-4618	24	4	.	.	PROPN
ejpam-4618	24	5	math	math	PROPN
ejpam-4618	24	6	,	,	PUNCT
ejpam-4618	24	7	16	16	NUM
ejpam-4618	24	8	(	(	PUNCT
ejpam-4618	24	9	1	1	NUM
ejpam-4618	24	10	)	)	PUNCT
ejpam-4618	24	11	(	(	PUNCT
ejpam-4618	24	12	2023	2023	NUM
ejpam-4618	24	13	)	)	PUNCT
ejpam-4618	24	14	,	,	PUNCT
ejpam-4618	24	15	253	253	NUM
ejpam-4618	24	16	-	-	SYM
ejpam-4618	24	17	260	260	NUM
ejpam-4618	24	18	254	254	NUM
ejpam-4618	24	19	2	2	NUM
ejpam-4618	24	20	.	.	PUNCT
ejpam-4618	24	21	preliminaries	preliminary	NOUN
ejpam-4618	24	22	we	we	PRON
ejpam-4618	24	23	first	first	ADV
ejpam-4618	24	24	recall	recall	VERB
ejpam-4618	24	25	some	some	DET
ejpam-4618	24	26	elementary	elementary	ADJ
ejpam-4618	24	27	aspects	aspect	NOUN
ejpam-4618	24	28	which	which	PRON
ejpam-4618	24	29	are	be	AUX
ejpam-4618	24	30	used	use	VERB
ejpam-4618	24	31	in	in	ADP
ejpam-4618	24	32	the	the	DET
ejpam-4618	24	33	present	present	ADJ
ejpam-4618	24	34	paper	paper	NOUN
ejpam-4618	24	35	.	.	PUNCT
ejpam-4618	25	1	throughout	throughout	ADP
ejpam-4618	25	2	this	this	DET
ejpam-4618	25	3	paper	paper	NOUN
ejpam-4618	25	4	,	,	PUNCT
ejpam-4618	25	5	x	x	PRON
ejpam-4618	25	6	always	always	ADV
ejpam-4618	25	7	denotes	denote	VERB
ejpam-4618	25	8	a	a	DET
ejpam-4618	25	9	ku	ku	NOUN
ejpam-4618	25	10	-	-	PUNCT
ejpam-4618	25	11	algebra	algebra	NOUN
ejpam-4618	25	12	without	without	ADP
ejpam-4618	25	13	any	any	DET
ejpam-4618	25	14	specifications	specification	NOUN
ejpam-4618	25	15	.	.	PUNCT
ejpam-4618	26	1	definition	definition	NOUN
ejpam-4618	26	2	1	1	NUM
ejpam-4618	26	3	(	(	PUNCT
ejpam-4618	26	4	8)	8)	NUM
ejpam-4618	26	5	.	.	PUNCT
ejpam-4618	27	1	let	let	VERB
ejpam-4618	27	2	x	x	PRON
ejpam-4618	27	3	be	be	AUX
ejpam-4618	27	4	a	a	DET
ejpam-4618	27	5	nonempty	nonempty	NOUN
ejpam-4618	27	6	set	set	VERB
ejpam-4618	27	7	with	with	ADP
ejpam-4618	27	8	a	a	DET
ejpam-4618	27	9	binary	binary	ADJ
ejpam-4618	27	10	operation	operation	NOUN
ejpam-4618	27	11	∗	∗	NOUN
ejpam-4618	27	12	and	and	CCONJ
ejpam-4618	27	13	a	a	DET
ejpam-4618	27	14	constant	constant	ADJ
ejpam-4618	27	15	0	0	NUM
ejpam-4618	27	16	,	,	PUNCT
ejpam-4618	27	17	then	then	ADV
ejpam-4618	27	18	(	(	PUNCT
ejpam-4618	27	19	x	x	X
ejpam-4618	27	20	,	,	PUNCT
ejpam-4618	27	21	∗	∗	NOUN
ejpam-4618	27	22	,	,	PUNCT
ejpam-4618	27	23	0	0	NUM
ejpam-4618	27	24	)	)	PUNCT
ejpam-4618	27	25	is	be	AUX
ejpam-4618	27	26	called	call	VERB
ejpam-4618	27	27	a	a	DET
ejpam-4618	27	28	ku	ku	PROPN
ejpam-4618	27	29	-algebra	-algebra	PROPN
ejpam-4618	27	30	,	,	PUNCT
ejpam-4618	27	31	if	if	SCONJ
ejpam-4618	27	32	for	for	ADP
ejpam-4618	27	33	all	all	DET
ejpam-4618	27	34	u	u	NOUN
ejpam-4618	27	35	,	,	PUNCT
ejpam-4618	27	36	v	v	NOUN
ejpam-4618	27	37	,	,	PUNCT
ejpam-4618	27	38	w	w	PROPN
ejpam-4618	27	39	∈	∈	PROPN
ejpam-4618	27	40	x	x	PUNCT
ejpam-4618	27	41	the	the	DET
ejpam-4618	27	42	following	follow	VERB
ejpam-4618	27	43	axioms	axiom	NOUN
ejpam-4618	27	44	are	be	AUX
ejpam-4618	27	45	hold	hold	ADJ
ejpam-4618	27	46	:	:	PUNCT
ejpam-4618	28	1	(	(	PUNCT
ejpam-4618	28	2	ku	ku	PROPN
ejpam-4618	28	3	1	1	NUM
ejpam-4618	28	4	)	)	PUNCT
ejpam-4618	28	5	(	(	PUNCT
ejpam-4618	28	6	u	u	NOUN
ejpam-4618	28	7	∗	∗	PROPN
ejpam-4618	28	8	v	v	NOUN
ejpam-4618	28	9	)	)	PUNCT
ejpam-4618	28	10	∗	∗	NOUN
ejpam-4618	28	11	[	[	X
ejpam-4618	28	12	(	(	PUNCT
ejpam-4618	28	13	v	v	NOUN
ejpam-4618	28	14	∗	∗	NOUN
ejpam-4618	28	15	w	w	NOUN
ejpam-4618	28	16	)	)	PUNCT
ejpam-4618	28	17	)	)	PUNCT
ejpam-4618	28	18	∗	∗	NOUN
ejpam-4618	28	19	(	(	PUNCT
ejpam-4618	28	20	u	u	NOUN
ejpam-4618	28	21	∗	∗	PROPN
ejpam-4618	28	22	w	w	NOUN
ejpam-4618	28	23	)	)	PUNCT
ejpam-4618	28	24	]	]	PUNCT
ejpam-4618	29	1	=	=	SYM
ejpam-4618	29	2	0	0	NUM
ejpam-4618	29	3	,	,	PUNCT
ejpam-4618	29	4	(	(	PUNCT
ejpam-4618	29	5	ku	ku	NOUN
ejpam-4618	29	6	2	2	NUM
ejpam-4618	29	7	)	)	PUNCT
ejpam-4618	29	8	u	u	NOUN
ejpam-4618	29	9	∗	∗	NOUN
ejpam-4618	29	10	0	0	NUM
ejpam-4618	29	11	=	=	SYM
ejpam-4618	29	12	0	0	NUM
ejpam-4618	29	13	,	,	PUNCT
ejpam-4618	29	14	(	(	PUNCT
ejpam-4618	29	15	ku	ku	NOUN
ejpam-4618	29	16	3	3	NUM
ejpam-4618	29	17	)	)	PUNCT
ejpam-4618	29	18	0	0	NUM
ejpam-4618	29	19	∗	∗	NOUN
ejpam-4618	29	20	u	u	NOUN
ejpam-4618	29	21	=	=	SYM
ejpam-4618	29	22	u	u	PROPN
ejpam-4618	29	23	,	,	PUNCT
ejpam-4618	29	24	(	(	PUNCT
ejpam-4618	29	25	ku	ku	NOUN
ejpam-4618	29	26	4	4	NUM
ejpam-4618	29	27	)	)	PUNCT
ejpam-4618	29	28	u	u	NOUN
ejpam-4618	29	29	∗	∗	NOUN
ejpam-4618	29	30	v	v	NOUN
ejpam-4618	29	31	=	=	SYM
ejpam-4618	29	32	0	0	NUM
ejpam-4618	29	33	and	and	CCONJ
ejpam-4618	29	34	v	v	NOUN
ejpam-4618	29	35	∗	∗	NOUN
ejpam-4618	29	36	u	u	NOUN
ejpam-4618	29	37	=	=	SYM
ejpam-4618	29	38	0	0	NUM
ejpam-4618	29	39	implies	imply	VERB
ejpam-4618	29	40	u	u	PROPN
ejpam-4618	29	41	=	=	PROPN
ejpam-4618	29	42	v	v	PROPN
ejpam-4618	29	43	,	,	PUNCT
ejpam-4618	29	44	(	(	PUNCT
ejpam-4618	29	45	ku	ku	NOUN
ejpam-4618	29	46	5	5	NUM
ejpam-4618	29	47	)	)	PUNCT
ejpam-4618	29	48	u	u	NOUN
ejpam-4618	29	49	∗	∗	NOUN
ejpam-4618	29	50	u	u	NOUN
ejpam-4618	29	51	=	=	NOUN
ejpam-4618	29	52	0	0	NUM
ejpam-4618	29	53	on	on	ADP
ejpam-4618	29	54	a	a	DET
ejpam-4618	29	55	ku	ku	NOUN
ejpam-4618	29	56	-	-	PUNCT
ejpam-4618	29	57	algebra	algebra	PROPN
ejpam-4618	29	58	(	(	PUNCT
ejpam-4618	29	59	x	x	X
ejpam-4618	29	60	,	,	PUNCT
ejpam-4618	29	61	∗	∗	NOUN
ejpam-4618	29	62	,	,	PUNCT
ejpam-4618	29	63	0	0	NUM
ejpam-4618	29	64	)	)	PUNCT
ejpam-4618	29	65	we	we	PRON
ejpam-4618	29	66	can	can	AUX
ejpam-4618	29	67	define	define	VERB
ejpam-4618	29	68	a	a	DET
ejpam-4618	29	69	binary	binary	ADJ
ejpam-4618	29	70	relation	relation	NOUN
ejpam-4618	29	71	≤	≤	PUNCT
ejpam-4618	29	72	on	on	ADP
ejpam-4618	29	73	x	x	PUNCT
ejpam-4618	29	74	by	by	ADP
ejpam-4618	29	75	putting	put	VERB
ejpam-4618	29	76	:	:	PUNCT
ejpam-4618	29	77	u	u	PROPN
ejpam-4618	29	78	≤	≤	PROPN
ejpam-4618	29	79	v	v	ADP
ejpam-4618	29	80	⇐	⇐	ADJ
ejpam-4618	29	81	⇒	⇒	NOUN
ejpam-4618	29	82	v	v	ADP
ejpam-4618	29	83	∗	∗	NOUN
ejpam-4618	29	84	u	u	NOUN
ejpam-4618	29	85	=	=	NOUN
ejpam-4618	29	86	0	0	PROPN
ejpam-4618	29	87	.	.	PUNCT
ejpam-4618	30	1	then	then	ADV
ejpam-4618	30	2	(	(	PUNCT
ejpam-4618	30	3	x,≤	x,≤	NOUN
ejpam-4618	30	4	)	)	PUNCT
ejpam-4618	30	5	is	be	AUX
ejpam-4618	30	6	a	a	DET
ejpam-4618	30	7	partially	partially	ADV
ejpam-4618	30	8	ordered	order	VERB
ejpam-4618	30	9	set	set	NOUN
ejpam-4618	30	10	and	and	CCONJ
ejpam-4618	30	11	0	0	NUM
ejpam-4618	30	12	is	be	AUX
ejpam-4618	30	13	its	its	PRON
ejpam-4618	30	14	smallest	small	ADJ
ejpam-4618	30	15	element	element	NOUN
ejpam-4618	30	16	.	.	PUNCT
ejpam-4618	31	1	thus	thus	ADV
ejpam-4618	31	2	(	(	PUNCT
ejpam-4618	31	3	x	x	X
ejpam-4618	31	4	,	,	PUNCT
ejpam-4618	31	5	∗	∗	NOUN
ejpam-4618	31	6	,	,	PUNCT
ejpam-4618	31	7	0	0	NUM
ejpam-4618	31	8	)	)	PUNCT
ejpam-4618	31	9	satisfies	satisfy	VERB
ejpam-4618	31	10	the	the	DET
ejpam-4618	31	11	following	follow	VERB
ejpam-4618	31	12	conditions	condition	NOUN
ejpam-4618	31	13	:	:	PUNCT
ejpam-4618	31	14	for	for	ADP
ejpam-4618	31	15	all	all	DET
ejpam-4618	31	16	u	u	NOUN
ejpam-4618	31	17	,	,	PUNCT
ejpam-4618	31	18	v	v	NOUN
ejpam-4618	31	19	,	,	PUNCT
ejpam-4618	31	20	w	w	PROPN
ejpam-4618	31	21	∈	∈	PROPN
ejpam-4618	31	22	x.	x.	NOUN
ejpam-4618	31	23	(	(	PUNCT
ejpam-4618	31	24	1	1	NUM
ejpam-4618	31	25	)	)	PUNCT
ejpam-4618	31	26	:	:	PUNCT
ejpam-4618	31	27	(	(	PUNCT
ejpam-4618	31	28	v	v	X
ejpam-4618	31	29	∗	∗	NOUN
ejpam-4618	31	30	w	w	NOUN
ejpam-4618	31	31	)	)	PUNCT
ejpam-4618	31	32	∗	∗	NOUN
ejpam-4618	31	33	(	(	PUNCT
ejpam-4618	31	34	u	u	NOUN
ejpam-4618	31	35	∗	∗	PROPN
ejpam-4618	31	36	w	w	NOUN
ejpam-4618	31	37	)	)	PUNCT
ejpam-4618	31	38	≤	≤	NOUN
ejpam-4618	31	39	(	(	PUNCT
ejpam-4618	31	40	u	u	NOUN
ejpam-4618	31	41	∗	∗	NOUN
ejpam-4618	31	42	v	v	NOUN
ejpam-4618	31	43	)	)	PUNCT
ejpam-4618	31	44	(	(	PUNCT
ejpam-4618	31	45	2	2	NUM
ejpam-4618	31	46	):	):	PUNCT
ejpam-4618	31	47	0	0	NUM
ejpam-4618	31	48	≤	≤	NUM
ejpam-4618	31	49	u	u	NOUN
ejpam-4618	31	50	(	(	PUNCT
ejpam-4618	31	51	3	3	NUM
ejpam-4618	31	52	):	):	PUNCT
ejpam-4618	31	53	u	u	PROPN
ejpam-4618	31	54	≤	≤	PROPN
ejpam-4618	31	55	v	v	NOUN
ejpam-4618	31	56	,	,	PUNCT
ejpam-4618	31	57	v	v	ADJ
ejpam-4618	31	58	≤	≤	NOUN
ejpam-4618	31	59	u	u	NOUN
ejpam-4618	31	60	implies	imply	VERB
ejpam-4618	31	61	u	u	PROPN
ejpam-4618	31	62	=	=	PROPN
ejpam-4618	31	63	v	v	PROPN
ejpam-4618	31	64	,	,	PUNCT
ejpam-4618	31	65	(	(	PUNCT
ejpam-4618	31	66	4	4	NUM
ejpam-4618	31	67	):	):	PUNCT
ejpam-4618	31	68	v	v	NOUN
ejpam-4618	31	69	∗	∗	NOUN
ejpam-4618	31	70	u	u	NOUN
ejpam-4618	31	71	≤	≤	X
ejpam-4618	31	72	u.	u.	VERB
ejpam-4618	31	73	a	a	DET
ejpam-4618	31	74	subset	subset	NOUN
ejpam-4618	31	75	s	s	NOUN
ejpam-4618	31	76	of	of	ADP
ejpam-4618	31	77	a	a	DET
ejpam-4618	31	78	ku	ku	NOUN
ejpam-4618	31	79	-	-	PUNCT
ejpam-4618	31	80	algebra	algebra	PROPN
ejpam-4618	31	81	x	x	PUNCT
ejpam-4618	31	82	is	be	AUX
ejpam-4618	31	83	called	call	VERB
ejpam-4618	31	84	ku	ku	NOUN
ejpam-4618	31	85	-	-	PUNCT
ejpam-4618	31	86	subalgebra	subalgebra	NOUN
ejpam-4618	31	87	of	of	ADP
ejpam-4618	31	88	x	x	SYM
ejpam-4618	31	89	,	,	PUNCT
ejpam-4618	31	90	if	if	SCONJ
ejpam-4618	31	91	u	u	NOUN
ejpam-4618	31	92	,	,	PUNCT
ejpam-4618	31	93	v	v	ADP
ejpam-4618	31	94	∈	∈	PROPN
ejpam-4618	31	95	s	s	NOUN
ejpam-4618	31	96	,	,	PUNCT
ejpam-4618	31	97	implies	imply	VERB
ejpam-4618	31	98	(	(	PUNCT
ejpam-4618	31	99	u	u	NOUN
ejpam-4618	31	100	∗	∗	NOUN
ejpam-4618	31	101	v	v	NOUN
ejpam-4618	31	102	)	)	PUNCT
ejpam-4618	31	103	∈	∈	PROPN
ejpam-4618	31	104	s.	s.	PROPN
ejpam-4618	31	105	a	a	DET
ejpam-4618	31	106	nonempty	nonempty	NOUN
ejpam-4618	31	107	subset	subset	VERB
ejpam-4618	31	108	i	i	PRON
ejpam-4618	31	109	of	of	ADP
ejpam-4618	31	110	a	a	DET
ejpam-4618	31	111	ku	ku	NOUN
ejpam-4618	31	112	-	-	PUNCT
ejpam-4618	31	113	algebra	algebra	PROPN
ejpam-4618	31	114	x	x	PUNCT
ejpam-4618	31	115	is	be	AUX
ejpam-4618	31	116	said	say	VERB
ejpam-4618	31	117	to	to	PART
ejpam-4618	31	118	be	be	AUX
ejpam-4618	31	119	a	a	DET
ejpam-4618	31	120	ku	ku	NOUN
ejpam-4618	31	121	-	-	PUNCT
ejpam-4618	31	122	ideal	ideal	NOUN
ejpam-4618	31	123	of	of	ADP
ejpam-4618	31	124	x	x	PRON
ejpam-4618	31	125	if	if	SCONJ
ejpam-4618	31	126	it	it	PRON
ejpam-4618	31	127	satisfies	satisfy	VERB
ejpam-4618	31	128	:	:	PUNCT
ejpam-4618	31	129	(	(	PUNCT
ejpam-4618	31	130	k1	k1	NOUN
ejpam-4618	31	131	)	)	PUNCT
ejpam-4618	31	132	0	0	PUNCT
ejpam-4618	32	1	∈	∈	PROPN
ejpam-4618	32	2	i	i	PRON
ejpam-4618	32	3	,	,	PUNCT
ejpam-4618	32	4	(	(	PUNCT
ejpam-4618	32	5	k2	k2	ADJ
ejpam-4618	32	6	)	)	PUNCT
ejpam-4618	32	7	u	u	NOUN
ejpam-4618	32	8	∗	∗	NOUN
ejpam-4618	32	9	(	(	PUNCT
ejpam-4618	32	10	v	v	NOUN
ejpam-4618	32	11	∗	∗	NOUN
ejpam-4618	32	12	w	w	NOUN
ejpam-4618	32	13	)	)	PUNCT
ejpam-4618	32	14	∈	∈	PROPN
ejpam-4618	33	1	i	i	PRON
ejpam-4618	33	2	and	and	CCONJ
ejpam-4618	33	3	v	v	X
ejpam-4618	33	4	∈	∈	NOUN
ejpam-4618	34	1	i	i	PRON
ejpam-4618	34	2	imply	imply	VERB
ejpam-4618	34	3	u	u	NOUN
ejpam-4618	34	4	∗	∗	NOUN
ejpam-4618	34	5	w	w	NOUN
ejpam-4618	34	6	∈	∈	PROPN
ejpam-4618	35	1	i	i	PRON
ejpam-4618	35	2	for	for	ADP
ejpam-4618	35	3	all	all	DET
ejpam-4618	35	4	u	u	NOUN
ejpam-4618	35	5	,	,	PUNCT
ejpam-4618	35	6	v	v	NOUN
ejpam-4618	35	7	and	and	CCONJ
ejpam-4618	35	8	w	w	PROPN
ejpam-4618	35	9	∈	∈	PROPN
ejpam-4618	35	10	x.	x.	NOUN
ejpam-4618	35	11	theorem	theorem	VERB
ejpam-4618	35	12	1	1	NUM
ejpam-4618	35	13	(	(	PUNCT
ejpam-4618	35	14	7	7	NUM
ejpam-4618	35	15	)	)	PUNCT
ejpam-4618	35	16	.	.	PUNCT
ejpam-4618	36	1	in	in	ADP
ejpam-4618	36	2	a	a	DET
ejpam-4618	36	3	ku	ku	NOUN
ejpam-4618	36	4	-	-	PUNCT
ejpam-4618	36	5	algebra	algebra	PROPN
ejpam-4618	36	6	(	(	PUNCT
ejpam-4618	36	7	x	x	X
ejpam-4618	36	8	,	,	PUNCT
ejpam-4618	36	9	∗	∗	NOUN
ejpam-4618	36	10	,	,	PUNCT
ejpam-4618	36	11	0	0	NUM
ejpam-4618	36	12	)	)	PUNCT
ejpam-4618	36	13	,	,	PUNCT
ejpam-4618	36	14	the	the	DET
ejpam-4618	36	15	following	follow	VERB
ejpam-4618	36	16	axioms	axiom	NOUN
ejpam-4618	36	17	are	be	AUX
ejpam-4618	36	18	satisfied	satisfied	ADJ
ejpam-4618	36	19	:	:	PUNCT
ejpam-4618	36	20	for	for	SCONJ
ejpam-4618	36	21	all	all	DET
ejpam-4618	36	22	u	u	NOUN
ejpam-4618	36	23	,	,	PUNCT
ejpam-4618	36	24	v	v	NOUN
ejpam-4618	36	25	,	,	PUNCT
ejpam-4618	36	26	w	w	PROPN
ejpam-4618	36	27	∈	∈	PROPN
ejpam-4618	36	28	x	x	X
ejpam-4618	36	29	,	,	PUNCT
ejpam-4618	36	30	(	(	PUNCT
ejpam-4618	36	31	1	1	X
ejpam-4618	36	32	)	)	PUNCT
ejpam-4618	36	33	u	u	NOUN
ejpam-4618	36	34	≤	≤	PROPN
ejpam-4618	36	35	v	v	AUX
ejpam-4618	36	36	imply	imply	NOUN
ejpam-4618	36	37	v	v	ADP
ejpam-4618	36	38	∗	∗	NOUN
ejpam-4618	36	39	w	w	NOUN
ejpam-4618	36	40	≤	≤	NOUN
ejpam-4618	36	41	u	u	PROPN
ejpam-4618	36	42	∗	∗	NOUN
ejpam-4618	36	43	w	w	PROPN
ejpam-4618	36	44	,	,	PUNCT
ejpam-4618	36	45	(	(	PUNCT
ejpam-4618	36	46	2	2	X
ejpam-4618	36	47	)	)	PUNCT
ejpam-4618	36	48	u	u	NOUN
ejpam-4618	36	49	∗	∗	NOUN
ejpam-4618	36	50	(	(	PUNCT
ejpam-4618	36	51	v	v	NOUN
ejpam-4618	36	52	∗	∗	NOUN
ejpam-4618	36	53	w	w	NOUN
ejpam-4618	36	54	)	)	PUNCT
ejpam-4618	36	55	=	=	SYM
ejpam-4618	36	56	v	v	ADP
ejpam-4618	36	57	∗	∗	NOUN
ejpam-4618	36	58	(	(	PUNCT
ejpam-4618	36	59	u	u	NOUN
ejpam-4618	36	60	∗	∗	PROPN
ejpam-4618	36	61	w	w	PROPN
ejpam-4618	36	62	)	)	PUNCT
ejpam-4618	36	63	,	,	PUNCT
ejpam-4618	36	64	for	for	ADP
ejpam-4618	36	65	all	all	DET
ejpam-4618	36	66	u	u	NOUN
ejpam-4618	36	67	,	,	PUNCT
ejpam-4618	36	68	v	v	NOUN
ejpam-4618	36	69	,	,	PUNCT
ejpam-4618	36	70	w	w	PROPN
ejpam-4618	36	71	∈	∈	PROPN
ejpam-4618	36	72	x	x	X
ejpam-4618	36	73	,	,	PUNCT
ejpam-4618	36	74	(	(	PUNCT
ejpam-4618	36	75	3	3	NUM
ejpam-4618	36	76	)	)	PUNCT
ejpam-4618	36	77	(	(	PUNCT
ejpam-4618	36	78	(	(	PUNCT
ejpam-4618	36	79	v	v	NOUN
ejpam-4618	36	80	∗	∗	NOUN
ejpam-4618	36	81	u	u	NOUN
ejpam-4618	36	82	)	)	PUNCT
ejpam-4618	36	83	∗	∗	NOUN
ejpam-4618	36	84	u	u	NOUN
ejpam-4618	36	85	)	)	PUNCT
ejpam-4618	36	86	≤	≤	NUM
ejpam-4618	36	87	v	v	ADP
ejpam-4618	36	88	definition	definition	NOUN
ejpam-4618	36	89	2	2	NUM
ejpam-4618	36	90	(	(	PUNCT
ejpam-4618	36	91	7	7	NUM
ejpam-4618	36	92	)	)	PUNCT
ejpam-4618	36	93	.	.	PUNCT
ejpam-4618	37	1	let	let	VERB
ejpam-4618	37	2	µ	µ	X
ejpam-4618	37	3	be	be	AUX
ejpam-4618	37	4	a	a	DET
ejpam-4618	37	5	fuzzy	fuzzy	ADJ
ejpam-4618	37	6	set	set	NOUN
ejpam-4618	37	7	on	on	ADP
ejpam-4618	37	8	a	a	DET
ejpam-4618	37	9	ku	ku	NOUN
ejpam-4618	37	10	-	-	PUNCT
ejpam-4618	37	11	algebra	algebra	PROPN
ejpam-4618	37	12	x	x	NOUN
ejpam-4618	37	13	,	,	PUNCT
ejpam-4618	37	14	then	then	ADV
ejpam-4618	37	15	µ	µ	NOUN
ejpam-4618	37	16	is	be	AUX
ejpam-4618	37	17	called	call	VERB
ejpam-4618	37	18	a	a	DET
ejpam-4618	37	19	fuzzy	fuzzy	ADJ
ejpam-4618	37	20	kusubalgebra	kusubalgebra	NOUN
ejpam-4618	37	21	of	of	ADP
ejpam-4618	37	22	x	x	PRON
ejpam-4618	37	23	if	if	SCONJ
ejpam-4618	37	24	µ(u	µ(u	NOUN
ejpam-4618	37	25	∗	∗	NOUN
ejpam-4618	37	26	v	v	NOUN
ejpam-4618	37	27	)	)	PUNCT
ejpam-4618	37	28	≥	≥	NOUN
ejpam-4618	37	29	min{µ(u	min{µ(u	NOUN
ejpam-4618	37	30	)	)	PUNCT
ejpam-4618	37	31	,	,	PUNCT
ejpam-4618	37	32	µ(v	µ(v	PROPN
ejpam-4618	37	33	)	)	PUNCT
ejpam-4618	37	34	}	}	PUNCT
ejpam-4618	37	35	,	,	PUNCT
ejpam-4618	37	36	for	for	ADP
ejpam-4618	37	37	all	all	DET
ejpam-4618	37	38	u	u	NOUN
ejpam-4618	37	39	,	,	PUNCT
ejpam-4618	38	1	v	v	NOUN
ejpam-4618	38	2	∈	∈	NOUN
ejpam-4618	38	3	x.	x.	NOUN
ejpam-4618	38	4	definition	definition	NOUN
ejpam-4618	38	5	3	3	NUM
ejpam-4618	38	6	(	(	PUNCT
ejpam-4618	38	7	7	7	NUM
ejpam-4618	38	8	)	)	PUNCT
ejpam-4618	38	9	.	.	PUNCT
ejpam-4618	39	1	let	let	VERB
ejpam-4618	39	2	x	x	PRON
ejpam-4618	39	3	be	be	AUX
ejpam-4618	39	4	a	a	DET
ejpam-4618	39	5	ku	ku	NOUN
ejpam-4618	39	6	-	-	PUNCT
ejpam-4618	39	7	algebra	algebra	PROPN
ejpam-4618	39	8	.	.	PUNCT
ejpam-4618	40	1	a	a	DET
ejpam-4618	40	2	fuzzy	fuzzy	ADJ
ejpam-4618	40	3	set	set	VERB
ejpam-4618	40	4	µ	µ	NOUN
ejpam-4618	40	5	in	in	ADP
ejpam-4618	40	6	x	x	AUX
ejpam-4618	40	7	is	be	AUX
ejpam-4618	40	8	called	call	VERB
ejpam-4618	40	9	a	a	DET
ejpam-4618	40	10	fuzzy	fuzzy	ADJ
ejpam-4618	40	11	ku	ku	NOUN
ejpam-4618	40	12	-	-	PUNCT
ejpam-4618	40	13	ideal	ideal	NOUN
ejpam-4618	40	14	of	of	ADP
ejpam-4618	40	15	x	x	PRON
ejpam-4618	40	16	if	if	SCONJ
ejpam-4618	40	17	it	it	PRON
ejpam-4618	40	18	satisfies	satisfy	VERB
ejpam-4618	40	19	:	:	PUNCT
ejpam-4618	40	20	(	(	PUNCT
ejpam-4618	40	21	fk1	fk1	NOUN
ejpam-4618	40	22	)	)	PUNCT
ejpam-4618	40	23	µ(0	µ(0	NOUN
ejpam-4618	40	24	)	)	PUNCT
ejpam-4618	40	25	≥	≥	NOUN
ejpam-4618	40	26	µ(u	µ(u	NOUN
ejpam-4618	40	27	)	)	PUNCT
ejpam-4618	40	28	,	,	PUNCT
ejpam-4618	40	29	(	(	PUNCT
ejpam-4618	40	30	fk2	fk2	NOUN
ejpam-4618	40	31	)	)	PUNCT
ejpam-4618	40	32	µ(u	µ(u	PROPN
ejpam-4618	40	33	∗	∗	PROPN
ejpam-4618	40	34	w	w	NOUN
ejpam-4618	40	35	)	)	PUNCT
ejpam-4618	40	36	≥	≥	NOUN
ejpam-4618	40	37	min{µ(u	min{µ(u	PROPN
ejpam-4618	40	38	∗	∗	NOUN
ejpam-4618	40	39	(	(	PUNCT
ejpam-4618	40	40	v	v	NOUN
ejpam-4618	40	41	∗	∗	NOUN
ejpam-4618	40	42	w	w	NOUN
ejpam-4618	40	43	)	)	PUNCT
ejpam-4618	40	44	)	)	PUNCT
ejpam-4618	40	45	,	,	PUNCT
ejpam-4618	40	46	µ(v	µ(v	PROPN
ejpam-4618	40	47	)	)	PUNCT
ejpam-4618	40	48	}	}	PUNCT
ejpam-4618	40	49	,	,	PUNCT
ejpam-4618	40	50	for	for	ADP
ejpam-4618	40	51	all	all	DET
ejpam-4618	40	52	u	u	NOUN
ejpam-4618	40	53	,	,	PUNCT
ejpam-4618	40	54	v	v	NOUN
ejpam-4618	40	55	and	and	CCONJ
ejpam-4618	40	56	w	w	PROPN
ejpam-4618	40	57	∈	∈	PROPN
ejpam-4618	40	58	x.	x.	NOUN
ejpam-4618	41	1	lemma	lemma	PROPN
ejpam-4618	41	2	1	1	NUM
ejpam-4618	41	3	(	(	PUNCT
ejpam-4618	41	4	7	7	NUM
ejpam-4618	41	5	)	)	PUNCT
ejpam-4618	41	6	.	.	PUNCT
ejpam-4618	42	1	if	if	SCONJ
ejpam-4618	42	2	a	a	DET
ejpam-4618	42	3	fuzzy	fuzzy	ADJ
ejpam-4618	42	4	ku	ku	NOUN
ejpam-4618	42	5	-	-	PUNCT
ejpam-4618	42	6	subalgebra	subalgebra	NOUN
ejpam-4618	42	7	of	of	ADP
ejpam-4618	42	8	x	x	PRON
ejpam-4618	42	9	,	,	PUNCT
ejpam-4618	42	10	then	then	ADV
ejpam-4618	42	11	µ(0	µ(0	PROPN
ejpam-4618	42	12	)	)	PUNCT
ejpam-4618	42	13	≥	≥	NOUN
ejpam-4618	42	14	µ(u	µ(u	NOUN
ejpam-4618	42	15	)	)	PUNCT
ejpam-4618	42	16	,	,	PUNCT
ejpam-4618	42	17	for	for	ADP
ejpam-4618	42	18	all	all	DET
ejpam-4618	42	19	u	u	PROPN
ejpam-4618	42	20	∈	∈	NOUN
ejpam-4618	42	21	x.	x.	NOUN
ejpam-4618	42	22	proposition	proposition	NOUN
ejpam-4618	42	23	1	1	NUM
ejpam-4618	42	24	(	(	PUNCT
ejpam-4618	42	25	7	7	NUM
ejpam-4618	42	26	)	)	PUNCT
ejpam-4618	42	27	.	.	PUNCT
ejpam-4618	43	1	if	if	SCONJ
ejpam-4618	43	2	a	a	DET
ejpam-4618	43	3	fuzzy	fuzzy	ADJ
ejpam-4618	43	4	ku	ku	NOUN
ejpam-4618	43	5	-	-	PUNCT
ejpam-4618	43	6	ideal	ideal	NOUN
ejpam-4618	43	7	of	of	ADP
ejpam-4618	43	8	x	x	X
ejpam-4618	43	9	and	and	CCONJ
ejpam-4618	43	10	u	u	PROPN
ejpam-4618	43	11	≤	≤	PROPN
ejpam-4618	43	12	v	v	NOUN
ejpam-4618	43	13	,	,	PUNCT
ejpam-4618	43	14	then	then	ADV
ejpam-4618	43	15	µ(u	µ(u	NOUN
ejpam-4618	43	16	)	)	PUNCT
ejpam-4618	43	17	≥	≥	NOUN
ejpam-4618	43	18	µ(v	µ(v	PROPN
ejpam-4618	43	19	)	)	PUNCT
ejpam-4618	43	20	,	,	PUNCT
ejpam-4618	43	21	for	for	ADP
ejpam-4618	43	22	all	all	DET
ejpam-4618	43	23	u	u	NOUN
ejpam-4618	43	24	,	,	PUNCT
ejpam-4618	43	25	v	v	NOUN
ejpam-4618	43	26	∈	∈	PROPN
ejpam-4618	43	27	x.	x.	NOUN
ejpam-4618	43	28	h.	h.	PROPN
ejpam-4618	43	29	alshehri	alshehri	PROPN
ejpam-4618	43	30	/	/	SYM
ejpam-4618	43	31	eur	eur	PROPN
ejpam-4618	43	32	.	.	PUNCT
ejpam-4618	44	1	j.	j.	PROPN
ejpam-4618	44	2	pure	pure	PROPN
ejpam-4618	44	3	appl	appl	PROPN
ejpam-4618	44	4	.	.	PROPN
ejpam-4618	44	5	math	math	PROPN
ejpam-4618	44	6	,	,	PUNCT
ejpam-4618	44	7	16	16	NUM
ejpam-4618	44	8	(	(	PUNCT
ejpam-4618	44	9	1	1	NUM
ejpam-4618	44	10	)	)	PUNCT
ejpam-4618	44	11	(	(	PUNCT
ejpam-4618	44	12	2023	2023	NUM
ejpam-4618	44	13	)	)	PUNCT
ejpam-4618	44	14	,	,	PUNCT
ejpam-4618	44	15	253	253	NUM
ejpam-4618	44	16	-	-	SYM
ejpam-4618	44	17	260	260	NUM
ejpam-4618	44	18	255	255	NUM
ejpam-4618	44	19	theorem	theorem	VERB
ejpam-4618	44	20	2	2	NUM
ejpam-4618	44	21	(	(	PUNCT
ejpam-4618	44	22	7	7	NUM
ejpam-4618	44	23	)	)	PUNCT
ejpam-4618	44	24	.	.	PUNCT
ejpam-4618	45	1	a	a	DET
ejpam-4618	45	2	fuzzy	fuzzy	ADJ
ejpam-4618	45	3	ku	ku	NOUN
ejpam-4618	45	4	-	-	PUNCT
ejpam-4618	45	5	ideal	ideal	NOUN
ejpam-4618	45	6	of	of	ADP
ejpam-4618	45	7	x	x	PUNCT
ejpam-4618	45	8	is	be	AUX
ejpam-4618	45	9	a	a	DET
ejpam-4618	45	10	fuzzy	fuzzy	ADJ
ejpam-4618	45	11	ku	ku	NOUN
ejpam-4618	45	12	-	-	PUNCT
ejpam-4618	45	13	subalgebra	subalgebra	PROPN
ejpam-4618	45	14	of	of	ADP
ejpam-4618	45	15	x.	x.	NOUN
ejpam-4618	45	16	by	by	ADP
ejpam-4618	45	17	an	an	DET
ejpam-4618	45	18	m	m	ADJ
ejpam-4618	45	19	-	-	ADJ
ejpam-4618	45	20	polar	polar	ADJ
ejpam-4618	45	21	fuzzy	fuzzy	ADJ
ejpam-4618	45	22	set	set	NOUN
ejpam-4618	45	23	of	of	ADP
ejpam-4618	45	24	a	a	DET
ejpam-4618	45	25	set	set	NOUN
ejpam-4618	45	26	x	x	PUNCT
ejpam-4618	45	27	(	(	PUNCT
ejpam-4618	45	28	see	see	VERB
ejpam-4618	45	29	[	[	X
ejpam-4618	45	30	5	5	NUM
ejpam-4618	45	31	]	]	NUM
ejpam-4618	45	32	)	)	PUNCT
ejpam-4618	45	33	,	,	PUNCT
ejpam-4618	45	34	we	we	PRON
ejpam-4618	45	35	mean	mean	VERB
ejpam-4618	45	36	a	a	DET
ejpam-4618	45	37	function	function	NOUN
ejpam-4618	45	38	ô	ô	NUM
ejpam-4618	45	39	:	:	PUNCT
ejpam-4618	46	1	x	x	X
ejpam-4618	46	2	→	→	PUNCT
ejpam-4618	46	3	[	[	X
ejpam-4618	46	4	0	0	NUM
ejpam-4618	46	5	,	,	PUNCT
ejpam-4618	46	6	1]m	1]m	NUM
ejpam-4618	46	7	.	.	PUNCT
ejpam-4618	47	1	the	the	DET
ejpam-4618	47	2	membership	membership	NOUN
ejpam-4618	47	3	value	value	NOUN
ejpam-4618	47	4	of	of	ADP
ejpam-4618	47	5	every	every	DET
ejpam-4618	47	6	element	element	NOUN
ejpam-4618	47	7	u	u	NOUN
ejpam-4618	47	8	∈	∈	PROPN
ejpam-4618	47	9	x	x	AUX
ejpam-4618	47	10	is	be	AUX
ejpam-4618	47	11	denoted	denote	VERB
ejpam-4618	47	12	by	by	ADP
ejpam-4618	47	13	ô(u	ô(u	NOUN
ejpam-4618	47	14	)	)	PUNCT
ejpam-4618	47	15	:	:	PUNCT
ejpam-4618	48	1	=	=	SYM
ejpam-4618	48	2	{	{	PUNCT
ejpam-4618	48	3	(	(	PUNCT
ejpam-4618	48	4	π1	π1	NOUN
ejpam-4618	48	5	◦	◦	NOUN
ejpam-4618	48	6	ô)(u	ô)(u	PROPN
ejpam-4618	48	7	)	)	PUNCT
ejpam-4618	48	8	,	,	PUNCT
ejpam-4618	48	9	(	(	PUNCT
ejpam-4618	48	10	π2	π2	X
ejpam-4618	48	11	◦	◦	NOUN
ejpam-4618	48	12	ô)(u	ô)(u	PROPN
ejpam-4618	48	13	)	)	PUNCT
ejpam-4618	48	14	,	,	PUNCT
ejpam-4618	48	15	...	...	PUNCT
ejpam-4618	48	16	,	,	PUNCT
ejpam-4618	48	17	(	(	PUNCT
ejpam-4618	48	18	πm	πm	ADP
ejpam-4618	48	19	◦	◦	NOUN
ejpam-4618	48	20	ô)(u	ô)(u	ADJ
ejpam-4618	48	21	)	)	PUNCT
ejpam-4618	48	22	}	}	PUNCT
ejpam-4618	48	23	,	,	PUNCT
ejpam-4618	48	24	where	where	SCONJ
ejpam-4618	48	25	πi	πi	ADV
ejpam-4618	48	26	:	:	PUNCT
ejpam-4618	49	1	[	[	X
ejpam-4618	49	2	0	0	NUM
ejpam-4618	49	3	,	,	PUNCT
ejpam-4618	49	4	1	1	NUM
ejpam-4618	49	5	]	]	X
ejpam-4618	49	6	m	m	VERB
ejpam-4618	49	7	→	→	SYM
ejpam-4618	49	8	[	[	X
ejpam-4618	49	9	0	0	NUM
ejpam-4618	49	10	,	,	PUNCT
ejpam-4618	49	11	1	1	NUM
ejpam-4618	49	12	]	]	PUNCT
ejpam-4618	49	13	is	be	AUX
ejpam-4618	49	14	the	the	DET
ejpam-4618	49	15	i	i	PROPN
ejpam-4618	49	16	-	-	PUNCT
ejpam-4618	49	17	th	th	X
ejpam-4618	49	18	projection	projection	NOUN
ejpam-4618	49	19	for	for	ADP
ejpam-4618	49	20	all	all	DET
ejpam-4618	49	21	i	i	PRON
ejpam-4618	49	22	=	=	NOUN
ejpam-4618	49	23	1	1	NUM
ejpam-4618	49	24	,	,	PUNCT
ejpam-4618	49	25	2	2	NUM
ejpam-4618	49	26	,	,	PUNCT
ejpam-4618	49	27	...	...	PUNCT
ejpam-4618	49	28	,	,	PUNCT
ejpam-4618	49	29	m.	m.	NOUN
ejpam-4618	49	30	given	give	VERB
ejpam-4618	49	31	an	an	DET
ejpam-4618	49	32	m	m	ADJ
ejpam-4618	49	33	-	-	ADJ
ejpam-4618	49	34	polar	polar	ADJ
ejpam-4618	49	35	fuzzy	fuzzy	ADJ
ejpam-4618	49	36	set	set	NOUN
ejpam-4618	49	37	on	on	ADP
ejpam-4618	49	38	a	a	DET
ejpam-4618	49	39	set	set	NOUN
ejpam-4618	49	40	x	x	NOUN
ejpam-4618	49	41	,	,	PUNCT
ejpam-4618	49	42	we	we	PRON
ejpam-4618	49	43	consider	consider	VERB
ejpam-4618	49	44	the	the	DET
ejpam-4618	49	45	set	set	NOUN
ejpam-4618	49	46	u(ô	u(ô	NOUN
ejpam-4618	49	47	;	;	PUNCT
ejpam-4618	49	48	r̂	r̂	NOUN
ejpam-4618	49	49	)	)	PUNCT
ejpam-4618	49	50	:	:	PUNCT
ejpam-4618	50	1	=	=	PUNCT
ejpam-4618	50	2	{	{	PUNCT
ejpam-4618	50	3	u	u	NOUN
ejpam-4618	50	4	∈	∈	PROPN
ejpam-4618	50	5	x|ô(u	x|ô(u	PROPN
ejpam-4618	50	6	)	)	PUNCT
ejpam-4618	50	7	≥	≥	NOUN
ejpam-4618	50	8	r̂	r̂	NOUN
ejpam-4618	50	9	}	}	PUNCT
ejpam-4618	50	10	that	that	PRON
ejpam-4618	50	11	is	be	AUX
ejpam-4618	50	12	,	,	PUNCT
ejpam-4618	50	13	u(ô	u(ô	NOUN
ejpam-4618	50	14	;	;	PUNCT
ejpam-4618	50	15	r̂	r̂	NOUN
ejpam-4618	50	16	)	)	PUNCT
ejpam-4618	50	17	:	:	PUNCT
ejpam-4618	50	18	=	=	SYM
ejpam-4618	50	19	{	{	PUNCT
ejpam-4618	50	20	u	u	NOUN
ejpam-4618	50	21	∈	∈	PROPN
ejpam-4618	50	22	x|(πi	x|(πi	PROPN
ejpam-4618	50	23	◦	◦	VERB
ejpam-4618	50	24	ô)(u	ô)(u	PROPN
ejpam-4618	50	25	)	)	PUNCT
ejpam-4618	50	26	≥	≥	NUM
ejpam-4618	50	27	ri	ri	NOUN
ejpam-4618	50	28	,	,	PUNCT
ejpam-4618	50	29	i	i	PRON
ejpam-4618	50	30	=	=	NOUN
ejpam-4618	50	31	1	1	NUM
ejpam-4618	50	32	,	,	PUNCT
ejpam-4618	50	33	2	2	NUM
ejpam-4618	50	34	,	,	PUNCT
ejpam-4618	50	35	...	...	PUNCT
ejpam-4618	50	36	,	,	PUNCT
ejpam-4618	50	37	m	m	VERB
ejpam-4618	50	38	}	}	PUNCT
ejpam-4618	50	39	,	,	PUNCT
ejpam-4618	50	40	which	which	PRON
ejpam-4618	50	41	is	be	AUX
ejpam-4618	50	42	called	call	VERB
ejpam-4618	50	43	an	an	DET
ejpam-4618	50	44	m	m	ADJ
ejpam-4618	50	45	-	-	ADJ
ejpam-4618	50	46	polar	polar	ADJ
ejpam-4618	50	47	r̂-level	r̂-level	NOUN
ejpam-4618	50	48	cut	cut	NOUN
ejpam-4618	50	49	set	set	NOUN
ejpam-4618	50	50	of	of	ADP
ejpam-4618	50	51	ô.	ô.	NUM
ejpam-4618	50	52	ô(v	ô(v	X
ejpam-4618	50	53	)	)	PUNCT
ejpam-4618	50	54	=	=	PRON
ejpam-4618	50	55	{	{	PUNCT
ejpam-4618	50	56	t̂	t̂	NUM
ejpam-4618	50	57	=	=	SYM
ejpam-4618	50	58	(	(	PUNCT
ejpam-4618	50	59	t1	t1	NOUN
ejpam-4618	50	60	,	,	PUNCT
ejpam-4618	50	61	t2	t2	NOUN
ejpam-4618	50	62	,	,	PUNCT
ejpam-4618	50	63	...	...	PUNCT
ejpam-4618	50	64	,	,	PUNCT
ejpam-4618	50	65	tm	tm	PROPN
ejpam-4618	50	66	)	)	PUNCT
ejpam-4618	50	67	∈	∈	PROPN
ejpam-4618	50	68	(	(	PUNCT
ejpam-4618	50	69	0	0	NUM
ejpam-4618	50	70	,	,	PUNCT
ejpam-4618	50	71	1]m	1]m	NUM
ejpam-4618	50	72	;	;	PUNCT
ejpam-4618	50	73	u	u	PROPN
ejpam-4618	50	74	=	=	PROPN
ejpam-4618	50	75	v	v	ADP
ejpam-4618	50	76	0̂	0̂	PROPN
ejpam-4618	50	77	=	=	SYM
ejpam-4618	50	78	(	(	PUNCT
ejpam-4618	50	79	0	0	NUM
ejpam-4618	50	80	,	,	PUNCT
ejpam-4618	50	81	0	0	NUM
ejpam-4618	50	82	,	,	PUNCT
ejpam-4618	50	83	....	....	PUNCT
ejpam-4618	50	84	0	0	NUM
ejpam-4618	50	85	)	)	PUNCT
ejpam-4618	50	86	;	;	PUNCT
ejpam-4618	50	87	u	u	PROPN
ejpam-4618	50	88	̸=	̸=	PROPN
ejpam-4618	50	89	v	v	NUM
ejpam-4618	50	90	and	and	CCONJ
ejpam-4618	50	91	it	it	PRON
ejpam-4618	50	92	is	be	AUX
ejpam-4618	50	93	denoted	denote	VERB
ejpam-4618	50	94	by	by	ADP
ejpam-4618	50	95	ut̂.	ut̂.	NOUN
ejpam-4618	50	96	we	we	PRON
ejpam-4618	50	97	say	say	VERB
ejpam-4618	50	98	that	that	SCONJ
ejpam-4618	50	99	u	u	PROPN
ejpam-4618	50	100	is	be	AUX
ejpam-4618	50	101	the	the	DET
ejpam-4618	50	102	support	support	NOUN
ejpam-4618	50	103	of	of	ADP
ejpam-4618	50	104	ut̂	ut̂	PROPN
ejpam-4618	50	105	,	,	PUNCT
ejpam-4618	50	106	and	and	CCONJ
ejpam-4618	50	107	t̂	t̂	NUM
ejpam-4618	50	108	is	be	AUX
ejpam-4618	50	109	the	the	DET
ejpam-4618	50	110	value	value	NOUN
ejpam-4618	50	111	of	of	ADP
ejpam-4618	50	112	ut̂.	ut̂.	NOUN
ejpam-4618	50	113	we	we	PRON
ejpam-4618	50	114	say	say	VERB
ejpam-4618	50	115	that	that	SCONJ
ejpam-4618	50	116	an	an	DET
ejpam-4618	50	117	m	m	ADJ
ejpam-4618	50	118	-	-	ADJ
ejpam-4618	50	119	polar	polar	ADJ
ejpam-4618	50	120	fuzzy	fuzzy	ADJ
ejpam-4618	50	121	point	point	NOUN
ejpam-4618	50	122	ut̂	ut̂	PROPN
ejpam-4618	50	123	is	be	AUX
ejpam-4618	50	124	contained	contain	VERB
ejpam-4618	50	125	in	in	ADP
ejpam-4618	50	126	an	an	DET
ejpam-4618	50	127	m	m	ADJ
ejpam-4618	50	128	-	-	ADJ
ejpam-4618	50	129	polar	polar	ADJ
ejpam-4618	50	130	fuzzy	fuzzy	ADJ
ejpam-4618	50	131	set	set	NOUN
ejpam-4618	50	132	ô	ô	NUM
ejpam-4618	50	133	denoted	denote	VERB
ejpam-4618	50	134	by	by	ADP
ejpam-4618	50	135	ut̂	ut̂	PROPN
ejpam-4618	50	136	∈	∈	PROPN
ejpam-4618	50	137	ô	ô	NUM
ejpam-4618	50	138	,	,	PUNCT
ejpam-4618	50	139	if	if	SCONJ
ejpam-4618	50	140	ô(u	ô(u	NOUN
ejpam-4618	50	141	)	)	PUNCT
ejpam-4618	50	142	≥	≥	NOUN
ejpam-4618	50	143	t̂	t̂	NUM
ejpam-4618	50	144	,	,	PUNCT
ejpam-4618	50	145	that	that	ADV
ejpam-4618	50	146	is	is	ADV
ejpam-4618	50	147	,	,	PUNCT
ejpam-4618	50	148	(	(	PUNCT
ejpam-4618	50	149	πi	πi	ADP
ejpam-4618	50	150	◦	◦	VERB
ejpam-4618	50	151	ô)(u	ô)(u	ADJ
ejpam-4618	50	152	)	)	PUNCT
ejpam-4618	50	153	≥	≥	NOUN
ejpam-4618	50	154	ti	ti	NOUN
ejpam-4618	50	155	for	for	ADP
ejpam-4618	50	156	all	all	DET
ejpam-4618	50	157	i	i	PRON
ejpam-4618	51	1	=	=	NOUN
ejpam-4618	51	2	1	1	NUM
ejpam-4618	51	3	,	,	PUNCT
ejpam-4618	51	4	2	2	NUM
ejpam-4618	51	5	,	,	PUNCT
ejpam-4618	51	6	...	...	PUNCT
ejpam-4618	51	7	,	,	PUNCT
ejpam-4618	51	8	m.	m.	NOUN
ejpam-4618	51	9	definition	definition	NOUN
ejpam-4618	51	10	4	4	NUM
ejpam-4618	51	11	(	(	PUNCT
ejpam-4618	51	12	4	4	NUM
ejpam-4618	51	13	)	)	PUNCT
ejpam-4618	51	14	.	.	PUNCT
ejpam-4618	52	1	an	an	DET
ejpam-4618	52	2	m	m	ADJ
ejpam-4618	52	3	-	-	ADJ
ejpam-4618	52	4	polar	polar	ADJ
ejpam-4618	52	5	fuzzy	fuzzy	ADJ
ejpam-4618	52	6	set	set	NOUN
ejpam-4618	52	7	ô	ô	NUM
ejpam-4618	52	8	of	of	ADP
ejpam-4618	52	9	bck	bck	PROPN
ejpam-4618	52	10	/	/	SYM
ejpam-4618	52	11	bci	bci	NOUN
ejpam-4618	52	12	-	-	NOUN
ejpam-4618	52	13	algebra	algebra	NOUN
ejpam-4618	52	14	x	x	PUNCT
ejpam-4618	52	15	is	be	AUX
ejpam-4618	52	16	called	call	VERB
ejpam-4618	52	17	an	an	DET
ejpam-4618	52	18	m	m	ADJ
ejpam-4618	52	19	-	-	ADJ
ejpam-4618	52	20	polar	polar	ADJ
ejpam-4618	52	21	fuzzy	fuzzy	ADJ
ejpam-4618	52	22	subalgebra	subalgebra	NOUN
ejpam-4618	52	23	if	if	SCONJ
ejpam-4618	52	24	the	the	DET
ejpam-4618	52	25	following	follow	VERB
ejpam-4618	52	26	assertion	assertion	NOUN
ejpam-4618	52	27	is	be	AUX
ejpam-4618	52	28	valid	valid	ADJ
ejpam-4618	52	29	:	:	PUNCT
ejpam-4618	52	30	ô(u	ô(u	ADP
ejpam-4618	52	31	∗	∗	X
ejpam-4618	52	32	v	v	NOUN
ejpam-4618	52	33	)	)	PUNCT
ejpam-4618	52	34	≥	≥	NOUN
ejpam-4618	52	35	min{ô(u	min{ô(u	PROPN
ejpam-4618	52	36	)	)	PUNCT
ejpam-4618	52	37	,	,	PUNCT
ejpam-4618	52	38	ô(v	ô(v	ADV
ejpam-4618	52	39	)	)	PUNCT
ejpam-4618	52	40	}	}	PUNCT
ejpam-4618	52	41	that	that	ADV
ejpam-4618	52	42	is	is	ADV
ejpam-4618	52	43	,	,	PUNCT
ejpam-4618	52	44	(	(	PUNCT
ejpam-4618	52	45	πi	πi	ADP
ejpam-4618	52	46	◦	◦	VERB
ejpam-4618	52	47	ô)(u	ô)(u	ADJ
ejpam-4618	52	48	∗	∗	NOUN
ejpam-4618	52	49	v	v	NOUN
ejpam-4618	52	50	)	)	PUNCT
ejpam-4618	52	51	≥	≥	NOUN
ejpam-4618	52	52	min{(πi	min{(πi	AUX
ejpam-4618	52	53	◦	◦	VERB
ejpam-4618	52	54	ô)(u	ô)(u	ADJ
ejpam-4618	52	55	)	)	PUNCT
ejpam-4618	52	56	,	,	PUNCT
ejpam-4618	52	57	(	(	PUNCT
ejpam-4618	52	58	πi	πi	ADP
ejpam-4618	52	59	◦	◦	NOUN
ejpam-4618	52	60	ô)(v	ô)(v	NOUN
ejpam-4618	52	61	)	)	PUNCT
ejpam-4618	52	62	}	}	PUNCT
ejpam-4618	52	63	for	for	ADP
ejpam-4618	52	64	all	all	DET
ejpam-4618	52	65	u	u	NOUN
ejpam-4618	52	66	,	,	PUNCT
ejpam-4618	52	67	v	v	NOUN
ejpam-4618	52	68	∈	∈	PROPN
ejpam-4618	53	1	x	x	NOUN
ejpam-4618	53	2	,	,	PUNCT
ejpam-4618	53	3	i	i	NOUN
ejpam-4618	53	4	=	=	NOUN
ejpam-4618	53	5	1	1	NUM
ejpam-4618	53	6	,	,	PUNCT
ejpam-4618	53	7	2	2	NUM
ejpam-4618	53	8	,	,	PUNCT
ejpam-4618	53	9	...	...	PUNCT
ejpam-4618	53	10	,	,	PUNCT
ejpam-4618	53	11	m.	m.	NOUN
ejpam-4618	53	12	definition	definition	NOUN
ejpam-4618	53	13	5	5	NUM
ejpam-4618	53	14	(	(	PUNCT
ejpam-4618	53	15	4	4	NUM
ejpam-4618	53	16	)	)	PUNCT
ejpam-4618	53	17	.	.	PUNCT
ejpam-4618	54	1	an	an	DET
ejpam-4618	54	2	m	m	ADJ
ejpam-4618	54	3	-	-	ADJ
ejpam-4618	54	4	polar	polar	ADJ
ejpam-4618	54	5	fuzzy	fuzzy	ADJ
ejpam-4618	54	6	set	set	NOUN
ejpam-4618	54	7	ô	ô	NUM
ejpam-4618	54	8	of	of	ADP
ejpam-4618	54	9	bck	bck	PROPN
ejpam-4618	54	10	/	/	SYM
ejpam-4618	54	11	bci	bci	NOUN
ejpam-4618	54	12	-	-	NOUN
ejpam-4618	54	13	algebra	algebra	NOUN
ejpam-4618	54	14	x	x	PUNCT
ejpam-4618	54	15	is	be	AUX
ejpam-4618	54	16	called	call	VERB
ejpam-4618	54	17	an	an	DET
ejpam-4618	54	18	m	m	ADJ
ejpam-4618	54	19	-	-	ADJ
ejpam-4618	54	20	polar	polar	ADJ
ejpam-4618	54	21	fuzzy	fuzzy	ADJ
ejpam-4618	54	22	ideal	ideal	NOUN
ejpam-4618	54	23	if	if	SCONJ
ejpam-4618	54	24	the	the	DET
ejpam-4618	54	25	following	follow	VERB
ejpam-4618	54	26	assertion	assertion	NOUN
ejpam-4618	54	27	is	be	AUX
ejpam-4618	54	28	valid	valid	ADJ
ejpam-4618	54	29	:	:	PUNCT
ejpam-4618	54	30	ô(0	ô(0	NUM
ejpam-4618	54	31	)	)	PUNCT
ejpam-4618	54	32	≥	≥	NOUN
ejpam-4618	54	33	ô(u	ô(u	NOUN
ejpam-4618	54	34	)	)	PUNCT
ejpam-4618	54	35	≥	≥	NOUN
ejpam-4618	54	36	min{ô(u	min{ô(u	PROPN
ejpam-4618	54	37	∗	∗	NOUN
ejpam-4618	54	38	v	v	NOUN
ejpam-4618	54	39	)	)	PUNCT
ejpam-4618	54	40	,	,	PUNCT
ejpam-4618	54	41	ô(v	ô(v	ADV
ejpam-4618	54	42	)	)	PUNCT
ejpam-4618	54	43	}	}	PUNCT
ejpam-4618	54	44	that	that	ADV
ejpam-4618	54	45	is	is	ADV
ejpam-4618	54	46	,	,	PUNCT
ejpam-4618	54	47	(	(	PUNCT
ejpam-4618	54	48	πi	πi	ADP
ejpam-4618	54	49	◦	◦	VERB
ejpam-4618	54	50	ô)(0	ô)(0	ADJ
ejpam-4618	54	51	)	)	PUNCT
ejpam-4618	54	52	≥	≥	NOUN
ejpam-4618	54	53	(	(	PUNCT
ejpam-4618	54	54	πi	πi	ADP
ejpam-4618	54	55	◦	◦	VERB
ejpam-4618	54	56	ô)(u	ô)(u	ADJ
ejpam-4618	54	57	)	)	PUNCT
ejpam-4618	54	58	≥	≥	NOUN
ejpam-4618	54	59	min{(πi	min{(πi	NOUN
ejpam-4618	54	60	◦	◦	VERB
ejpam-4618	54	61	ô)(u	ô)(u	ADJ
ejpam-4618	54	62	∗	∗	NOUN
ejpam-4618	54	63	v	v	NOUN
ejpam-4618	54	64	)	)	PUNCT
ejpam-4618	54	65	,	,	PUNCT
ejpam-4618	54	66	(	(	PUNCT
ejpam-4618	54	67	πi	πi	ADP
ejpam-4618	54	68	◦	◦	NOUN
ejpam-4618	54	69	ô)(v	ô)(v	NOUN
ejpam-4618	54	70	)	)	PUNCT
ejpam-4618	54	71	}	}	PUNCT
ejpam-4618	54	72	for	for	ADP
ejpam-4618	54	73	all	all	DET
ejpam-4618	54	74	u	u	NOUN
ejpam-4618	54	75	,	,	PUNCT
ejpam-4618	54	76	v	v	NOUN
ejpam-4618	54	77	∈	∈	PROPN
ejpam-4618	54	78	x	x	NOUN
ejpam-4618	54	79	,	,	PUNCT
ejpam-4618	54	80	i	i	NOUN
ejpam-4618	54	81	=	=	NOUN
ejpam-4618	54	82	1	1	NUM
ejpam-4618	54	83	,	,	PUNCT
ejpam-4618	54	84	2	2	NUM
ejpam-4618	54	85	,	,	PUNCT
ejpam-4618	54	86	...	...	PUNCT
ejpam-4618	54	87	,	,	PUNCT
ejpam-4618	54	88	m.	m.	PROPN
ejpam-4618	54	89	h.	h.	PROPN
ejpam-4618	54	90	alshehri	alshehri	PROPN
ejpam-4618	54	91	/	/	SYM
ejpam-4618	54	92	eur	eur	PROPN
ejpam-4618	54	93	.	.	PUNCT
ejpam-4618	55	1	j.	j.	PROPN
ejpam-4618	55	2	pure	pure	PROPN
ejpam-4618	55	3	appl	appl	PROPN
ejpam-4618	55	4	.	.	PROPN
ejpam-4618	55	5	math	math	PROPN
ejpam-4618	55	6	,	,	PUNCT
ejpam-4618	55	7	16	16	NUM
ejpam-4618	55	8	(	(	PUNCT
ejpam-4618	55	9	1	1	NUM
ejpam-4618	55	10	)	)	PUNCT
ejpam-4618	55	11	(	(	PUNCT
ejpam-4618	55	12	2023	2023	NUM
ejpam-4618	55	13	)	)	PUNCT
ejpam-4618	55	14	,	,	PUNCT
ejpam-4618	55	15	253	253	NUM
ejpam-4618	55	16	-	-	SYM
ejpam-4618	55	17	260	260	NUM
ejpam-4618	55	18	256	256	NUM
ejpam-4618	55	19	3	3	NUM
ejpam-4618	55	20	.	.	PUNCT
ejpam-4618	56	1	m	m	ADJ
ejpam-4618	56	2	-	-	ADJ
ejpam-4618	56	3	polar	polar	ADJ
ejpam-4618	56	4	fuzzy	fuzzy	ADJ
ejpam-4618	56	5	ku	ku	PROPN
ejpam-4618	56	6	-	-	PUNCT
ejpam-4618	56	7	subalgebras	subalgebras	PROPN
ejpam-4618	56	8	and	and	CCONJ
ejpam-4618	56	9	ku	ku	PROPN
ejpam-4618	56	10	-	-	PUNCT
ejpam-4618	56	11	ideals	ideal	NOUN
ejpam-4618	56	12	in	in	ADP
ejpam-4618	56	13	this	this	DET
ejpam-4618	56	14	section	section	NOUN
ejpam-4618	56	15	,	,	PUNCT
ejpam-4618	56	16	we	we	PRON
ejpam-4618	56	17	introduce	introduce	VERB
ejpam-4618	56	18	the	the	DET
ejpam-4618	56	19	notions	notion	NOUN
ejpam-4618	56	20	of	of	ADP
ejpam-4618	56	21	an	an	DET
ejpam-4618	56	22	m	m	ADJ
ejpam-4618	56	23	-	-	ADJ
ejpam-4618	56	24	polar	polar	ADJ
ejpam-4618	56	25	fuzzy	fuzzy	ADJ
ejpam-4618	56	26	ku	ku	PROPN
ejpam-4618	56	27	-	-	PUNCT
ejpam-4618	56	28	subalgebras	subalgebras	PROPN
ejpam-4618	56	29	,	,	PUNCT
ejpam-4618	56	30	an	an	DET
ejpam-4618	56	31	mpolar	mpolar	ADJ
ejpam-4618	56	32	fuzzy	fuzzy	ADJ
ejpam-4618	56	33	ku	ku	NOUN
ejpam-4618	56	34	-	-	PUNCT
ejpam-4618	56	35	ideals	ideal	NOUN
ejpam-4618	56	36	in	in	ADP
ejpam-4618	56	37	ku	ku	PROPN
ejpam-4618	56	38	-	-	PUNCT
ejpam-4618	56	39	algebras	algebras	PROPN
ejpam-4618	56	40	and	and	CCONJ
ejpam-4618	56	41	investigate	investigate	VERB
ejpam-4618	56	42	some	some	PRON
ejpam-4618	56	43	of	of	ADP
ejpam-4618	56	44	their	their	PRON
ejpam-4618	56	45	related	related	ADJ
ejpam-4618	56	46	properties	property	NOUN
ejpam-4618	56	47	.	.	PUNCT
ejpam-4618	57	1	definition	definition	NOUN
ejpam-4618	57	2	6	6	NUM
ejpam-4618	57	3	.	.	PUNCT
ejpam-4618	58	1	an	an	DET
ejpam-4618	58	2	m	m	ADJ
ejpam-4618	58	3	-	-	ADJ
ejpam-4618	58	4	polar	polar	ADJ
ejpam-4618	58	5	fuzzy	fuzzy	ADJ
ejpam-4618	58	6	set	set	NOUN
ejpam-4618	58	7	ô	ô	NUM
ejpam-4618	58	8	of	of	ADP
ejpam-4618	58	9	x	x	PRON
ejpam-4618	58	10	is	be	AUX
ejpam-4618	58	11	called	call	VERB
ejpam-4618	58	12	an	an	DET
ejpam-4618	58	13	m	m	ADJ
ejpam-4618	58	14	-	-	ADJ
ejpam-4618	58	15	polar	polar	ADJ
ejpam-4618	58	16	fuzzy	fuzzy	ADJ
ejpam-4618	58	17	ku	ku	NOUN
ejpam-4618	58	18	-	-	PUNCT
ejpam-4618	58	19	subalgebra	subalgebra	PROPN
ejpam-4618	58	20	if	if	SCONJ
ejpam-4618	58	21	the	the	DET
ejpam-4618	58	22	following	follow	VERB
ejpam-4618	58	23	assertion	assertion	NOUN
ejpam-4618	58	24	is	be	AUX
ejpam-4618	58	25	valid	valid	ADJ
ejpam-4618	58	26	for	for	ADP
ejpam-4618	58	27	all	all	DET
ejpam-4618	58	28	u	u	NOUN
ejpam-4618	58	29	,	,	PUNCT
ejpam-4618	58	30	v	v	NOUN
ejpam-4618	58	31	∈	∈	PROPN
ejpam-4618	58	32	x.	x.	NOUN
ejpam-4618	58	33	ô(u	ô(u	PROPN
ejpam-4618	58	34	∗	∗	X
ejpam-4618	58	35	v	v	NOUN
ejpam-4618	58	36	)	)	PUNCT
ejpam-4618	58	37	≥	≥	NOUN
ejpam-4618	58	38	min{ô(u	min{ô(u	PROPN
ejpam-4618	58	39	)	)	PUNCT
ejpam-4618	58	40	,	,	PUNCT
ejpam-4618	58	41	ô(v	ô(v	ADV
ejpam-4618	58	42	)	)	PUNCT
ejpam-4618	58	43	}	}	PUNCT
ejpam-4618	58	44	(	(	PUNCT
ejpam-4618	58	45	1	1	X
ejpam-4618	58	46	)	)	PUNCT
ejpam-4618	58	47	that	that	PRON
ejpam-4618	58	48	is	be	AUX
ejpam-4618	58	49	,	,	PUNCT
ejpam-4618	58	50	(	(	PUNCT
ejpam-4618	58	51	πi	πi	ADP
ejpam-4618	58	52	◦	◦	VERB
ejpam-4618	58	53	ô)(u	ô)(u	ADJ
ejpam-4618	58	54	∗	∗	NOUN
ejpam-4618	58	55	v	v	NOUN
ejpam-4618	58	56	)	)	PUNCT
ejpam-4618	58	57	≥	≥	NOUN
ejpam-4618	58	58	min{(πi	min{(πi	AUX
ejpam-4618	58	59	◦	◦	VERB
ejpam-4618	58	60	ô)(u	ô)(u	ADJ
ejpam-4618	58	61	)	)	PUNCT
ejpam-4618	58	62	,	,	PUNCT
ejpam-4618	58	63	(	(	PUNCT
ejpam-4618	58	64	πi	πi	ADP
ejpam-4618	58	65	◦	◦	NOUN
ejpam-4618	58	66	ô)(v	ô)(v	NOUN
ejpam-4618	58	67	)	)	PUNCT
ejpam-4618	58	68	}	}	PUNCT
ejpam-4618	58	69	for	for	ADP
ejpam-4618	58	70	all	all	DET
ejpam-4618	58	71	u	u	NOUN
ejpam-4618	58	72	,	,	PUNCT
ejpam-4618	58	73	v	v	NOUN
ejpam-4618	58	74	∈	∈	PROPN
ejpam-4618	59	1	x	x	NOUN
ejpam-4618	59	2	,	,	PUNCT
ejpam-4618	59	3	i	i	NOUN
ejpam-4618	59	4	=	=	NOUN
ejpam-4618	59	5	1	1	NUM
ejpam-4618	59	6	,	,	PUNCT
ejpam-4618	59	7	2	2	NUM
ejpam-4618	59	8	,	,	PUNCT
ejpam-4618	59	9	...	...	PUNCT
ejpam-4618	59	10	,	,	PUNCT
ejpam-4618	59	11	m.	m.	NOUN
ejpam-4618	59	12	example	example	NOUN
ejpam-4618	59	13	1	1	X
ejpam-4618	59	14	.	.	PUNCT
ejpam-4618	60	1	let	let	VERB
ejpam-4618	60	2	x	x	PUNCT
ejpam-4618	60	3	=	=	PUNCT
ejpam-4618	60	4	{	{	PUNCT
ejpam-4618	60	5	0	0	NUM
ejpam-4618	60	6	,	,	PUNCT
ejpam-4618	60	7	1	1	NUM
ejpam-4618	60	8	,	,	PUNCT
ejpam-4618	60	9	2	2	NUM
ejpam-4618	60	10	,	,	PUNCT
ejpam-4618	60	11	3	3	NUM
ejpam-4618	60	12	,	,	PUNCT
ejpam-4618	60	13	4	4	NUM
ejpam-4618	60	14	}	}	PUNCT
ejpam-4618	60	15	be	be	AUX
ejpam-4618	60	16	ku	ku	NOUN
ejpam-4618	60	17	-	-	PUNCT
ejpam-4618	60	18	algebra	algebra	PROPN
ejpam-4618	60	19	with	with	ADP
ejpam-4618	60	20	a	a	DET
ejpam-4618	60	21	binary	binary	ADJ
ejpam-4618	60	22	operation	operation	NOUN
ejpam-4618	60	23	∗	∗	NOUN
ejpam-4618	60	24	defined	define	VERB
ejpam-4618	60	25	by	by	ADP
ejpam-4618	60	26	the	the	DET
ejpam-4618	60	27	following	follow	VERB
ejpam-4618	60	28	table	table	NOUN
ejpam-4618	60	29	∗	∗	NOUN
ejpam-4618	60	30	0	0	NUM
ejpam-4618	61	1	1	1	NUM
ejpam-4618	61	2	2	2	NUM
ejpam-4618	61	3	3	3	NUM
ejpam-4618	61	4	4	4	NUM
ejpam-4618	61	5	0	0	NUM
ejpam-4618	61	6	0	0	NUM
ejpam-4618	61	7	1	1	NUM
ejpam-4618	61	8	2	2	NUM
ejpam-4618	61	9	3	3	NUM
ejpam-4618	61	10	4	4	NUM
ejpam-4618	61	11	1	1	NUM
ejpam-4618	61	12	0	0	NUM
ejpam-4618	61	13	0	0	NUM
ejpam-4618	61	14	2	2	NUM
ejpam-4618	61	15	3	3	NUM
ejpam-4618	61	16	3	3	NUM
ejpam-4618	61	17	2	2	NUM
ejpam-4618	61	18	0	0	NUM
ejpam-4618	61	19	0	0	NUM
ejpam-4618	61	20	0	0	NUM
ejpam-4618	61	21	1	1	NUM
ejpam-4618	61	22	4	4	NUM
ejpam-4618	61	23	3	3	NUM
ejpam-4618	61	24	0	0	NUM
ejpam-4618	61	25	0	0	NUM
ejpam-4618	61	26	0	0	NUM
ejpam-4618	61	27	0	0	NUM
ejpam-4618	61	28	3	3	NUM
ejpam-4618	61	29	4	4	NUM
ejpam-4618	61	30	0	0	NUM
ejpam-4618	61	31	0	0	NUM
ejpam-4618	61	32	0	0	NUM
ejpam-4618	61	33	0	0	NUM
ejpam-4618	61	34	0	0	NUM
ejpam-4618	61	35	define	define	VERB
ejpam-4618	61	36	a	a	DET
ejpam-4618	61	37	3	3	NUM
ejpam-4618	61	38	-	-	PUNCT
ejpam-4618	61	39	polar	polar	ADJ
ejpam-4618	61	40	fuzzy	fuzzy	ADJ
ejpam-4618	61	41	set	set	NOUN
ejpam-4618	61	42	ô	ô	NOUN
ejpam-4618	61	43	=	=	PUNCT
ejpam-4618	61	44	x	x	PUNCT
ejpam-4618	61	45	→	→	SYM
ejpam-4618	61	46	[	[	X
ejpam-4618	61	47	0	0	NUM
ejpam-4618	61	48	,	,	PUNCT
ejpam-4618	61	49	1]3	1]3	NUM
ejpam-4618	61	50	by	by	ADP
ejpam-4618	61	51	:	:	PUNCT
ejpam-4618	61	52	ô(u)=	ô(u)=	X
ejpam-4618	61	53			X
ejpam-4618	61	54	(	(	PUNCT
ejpam-4618	61	55	0.3	0.3	NUM
ejpam-4618	61	56	,	,	PUNCT
ejpam-4618	61	57	0.4	0.4	NUM
ejpam-4618	61	58	,	,	PUNCT
ejpam-4618	61	59	0.6	0.6	NUM
ejpam-4618	61	60	)	)	PUNCT
ejpam-4618	61	61	;	;	PUNCT
ejpam-4618	61	62	u	u	NOUN
ejpam-4618	61	63	=	=	SYM
ejpam-4618	61	64	0	0	NUM
ejpam-4618	61	65	(	(	PUNCT
ejpam-4618	61	66	0.2	0.2	NUM
ejpam-4618	61	67	,	,	PUNCT
ejpam-4618	61	68	0.3	0.3	NUM
ejpam-4618	61	69	,	,	PUNCT
ejpam-4618	61	70	0.2	0.2	NUM
ejpam-4618	61	71	)	)	PUNCT
ejpam-4618	61	72	;	;	PUNCT
ejpam-4618	61	73	u	u	NOUN
ejpam-4618	61	74	=	=	SYM
ejpam-4618	61	75	1	1	NUM
ejpam-4618	61	76	(	(	PUNCT
ejpam-4618	61	77	0.1	0.1	NUM
ejpam-4618	61	78	,	,	PUNCT
ejpam-4618	61	79	0.2	0.2	NUM
ejpam-4618	61	80	,	,	PUNCT
ejpam-4618	61	81	0.3	0.3	NUM
ejpam-4618	61	82	)	)	PUNCT
ejpam-4618	61	83	;	;	PUNCT
ejpam-4618	61	84	u	u	NOUN
ejpam-4618	61	85	=	=	SYM
ejpam-4618	61	86	2	2	NUM
ejpam-4618	61	87	(	(	PUNCT
ejpam-4618	61	88	0.2	0.2	NUM
ejpam-4618	61	89	,	,	PUNCT
ejpam-4618	61	90	0.3	0.3	NUM
ejpam-4618	61	91	,	,	PUNCT
ejpam-4618	61	92	0.4	0.4	NUM
ejpam-4618	61	93	)	)	PUNCT
ejpam-4618	61	94	;	;	PUNCT
ejpam-4618	61	95	u	u	NOUN
ejpam-4618	61	96	=	=	SYM
ejpam-4618	61	97	3	3	NUM
ejpam-4618	61	98	(	(	PUNCT
ejpam-4618	61	99	0.2	0.2	NUM
ejpam-4618	61	100	,	,	PUNCT
ejpam-4618	61	101	0.3	0.3	NUM
ejpam-4618	61	102	,	,	PUNCT
ejpam-4618	61	103	0.5	0.5	NUM
ejpam-4618	61	104	)	)	PUNCT
ejpam-4618	61	105	;	;	PUNCT
ejpam-4618	61	106	u	u	NOUN
ejpam-4618	61	107	=	=	NOUN
ejpam-4618	61	108	4	4	NUM
ejpam-4618	61	109	it	it	PRON
ejpam-4618	61	110	is	be	AUX
ejpam-4618	61	111	routine	routine	ADJ
ejpam-4618	61	112	to	to	PART
ejpam-4618	61	113	verify	verify	VERB
ejpam-4618	61	114	that	that	SCONJ
ejpam-4618	61	115	ô	ô	NUM
ejpam-4618	61	116	is	be	AUX
ejpam-4618	61	117	a	a	DET
ejpam-4618	61	118	3	3	NUM
ejpam-4618	61	119	-	-	PUNCT
ejpam-4618	61	120	polar	polar	ADJ
ejpam-4618	61	121	fuzzy	fuzzy	ADJ
ejpam-4618	61	122	ku	ku	NOUN
ejpam-4618	61	123	-	-	PUNCT
ejpam-4618	61	124	subalgebra	subalgebra	PROPN
ejpam-4618	61	125	of	of	ADP
ejpam-4618	61	126	x.	x.	NOUN
ejpam-4618	61	127	theorem	theorem	NOUN
ejpam-4618	61	128	3	3	X
ejpam-4618	61	129	.	.	PUNCT
ejpam-4618	62	1	let	let	VERB
ejpam-4618	62	2	ô	ô	X
ejpam-4618	62	3	be	be	AUX
ejpam-4618	62	4	an	an	DET
ejpam-4618	62	5	m	m	ADJ
ejpam-4618	62	6	-	-	ADJ
ejpam-4618	62	7	polar	polar	ADJ
ejpam-4618	62	8	fuzzy	fuzzy	ADJ
ejpam-4618	62	9	set	set	NOUN
ejpam-4618	62	10	of	of	ADP
ejpam-4618	62	11	x.	x.	NOUN
ejpam-4618	62	12	then	then	ADV
ejpam-4618	62	13	ô	ô	NUM
ejpam-4618	62	14	is	be	AUX
ejpam-4618	62	15	an	an	DET
ejpam-4618	62	16	m	m	ADJ
ejpam-4618	62	17	-	-	ADJ
ejpam-4618	62	18	polar	polar	ADJ
ejpam-4618	62	19	fuzzy	fuzzy	ADJ
ejpam-4618	62	20	kusubalgebra	kusubalgebra	NOUN
ejpam-4618	62	21	of	of	ADP
ejpam-4618	62	22	x	x	SYM
ejpam-4618	62	23	if	if	SCONJ
ejpam-4618	62	24	and	and	CCONJ
ejpam-4618	62	25	only	only	ADV
ejpam-4618	62	26	if	if	SCONJ
ejpam-4618	62	27	u(ô	u(ô	NOUN
ejpam-4618	62	28	;	;	PUNCT
ejpam-4618	62	29	r	r	X
ejpam-4618	62	30	)	)	PUNCT
ejpam-4618	62	31	̸=	̸=	PROPN
ejpam-4618	62	32	ϕ	ϕ	NOUN
ejpam-4618	62	33	is	be	AUX
ejpam-4618	62	34	a	a	DET
ejpam-4618	62	35	ku	ku	NOUN
ejpam-4618	62	36	-	-	PUNCT
ejpam-4618	62	37	subalgebra	subalgebra	NOUN
ejpam-4618	62	38	of	of	ADP
ejpam-4618	62	39	x	x	PUNCT
ejpam-4618	62	40	for	for	ADP
ejpam-4618	62	41	all	all	DET
ejpam-4618	62	42	r̂	r̂	NOUN
ejpam-4618	62	43	=	=	SYM
ejpam-4618	62	44	(	(	PUNCT
ejpam-4618	62	45	r1	r1	PROPN
ejpam-4618	62	46	,	,	PUNCT
ejpam-4618	62	47	r2	r2	PROPN
ejpam-4618	62	48	,	,	PUNCT
ejpam-4618	62	49	...	...	PUNCT
ejpam-4618	62	50	,	,	PUNCT
ejpam-4618	62	51	rm	rm	PROPN
ejpam-4618	62	52	)	)	PUNCT
ejpam-4618	62	53	∈	∈	PROPN
ejpam-4618	63	1	[	[	X
ejpam-4618	63	2	0	0	NUM
ejpam-4618	63	3	,	,	PUNCT
ejpam-4618	63	4	1]m	1]m	NUM
ejpam-4618	63	5	.	.	PUNCT
ejpam-4618	63	6	proof	proof	NOUN
ejpam-4618	63	7	.	.	PUNCT
ejpam-4618	64	1	assume	assume	VERB
ejpam-4618	64	2	that	that	SCONJ
ejpam-4618	64	3	ô	ô	NUM
ejpam-4618	64	4	is	be	AUX
ejpam-4618	64	5	an	an	DET
ejpam-4618	64	6	m	m	ADJ
ejpam-4618	64	7	-	-	ADJ
ejpam-4618	64	8	polar	polar	ADJ
ejpam-4618	64	9	fuzzy	fuzzy	ADJ
ejpam-4618	64	10	subalgebra	subalgebra	NOUN
ejpam-4618	64	11	of	of	ADP
ejpam-4618	64	12	x	x	PUNCT
ejpam-4618	64	13	and	and	CCONJ
ejpam-4618	64	14	let	let	VERB
ejpam-4618	64	15	r̂	r̂	PRON
ejpam-4618	64	16	∈	∈	PROPN
ejpam-4618	65	1	[	[	X
ejpam-4618	65	2	0	0	NUM
ejpam-4618	65	3	,	,	PUNCT
ejpam-4618	65	4	1]m	1]m	NUM
ejpam-4618	65	5	be	be	AUX
ejpam-4618	65	6	such	such	ADJ
ejpam-4618	65	7	that	that	SCONJ
ejpam-4618	65	8	u(ô	u(ô	NOUN
ejpam-4618	65	9	;	;	PUNCT
ejpam-4618	65	10	r	r	X
ejpam-4618	65	11	)	)	PUNCT
ejpam-4618	65	12	̸=	̸=	PROPN
ejpam-4618	65	13	ϕ.	ϕ.	NOUN
ejpam-4618	65	14	let	let	VERB
ejpam-4618	65	15	u	u	NOUN
ejpam-4618	65	16	,	,	PUNCT
ejpam-4618	65	17	v	v	PROPN
ejpam-4618	65	18	∈	∈	PROPN
ejpam-4618	65	19	u(ô	u(ô	NOUN
ejpam-4618	65	20	;	;	PUNCT
ejpam-4618	65	21	r̂	r̂	NOUN
ejpam-4618	65	22	)	)	PUNCT
ejpam-4618	65	23	.	.	PUNCT
ejpam-4618	66	1	then	then	ADV
ejpam-4618	66	2	ô(u	ô(u	NOUN
ejpam-4618	66	3	)	)	PUNCT
ejpam-4618	66	4	≥	≥	NOUN
ejpam-4618	66	5	r̂	r̂	NOUN
ejpam-4618	66	6	and	and	CCONJ
ejpam-4618	66	7	ô(v	ô(v	NUM
ejpam-4618	66	8	)	)	PUNCT
ejpam-4618	66	9	≥	≥	PRON
ejpam-4618	66	10	r̂.	r̂.	NOUN
ejpam-4618	66	11	it	it	PRON
ejpam-4618	66	12	follows	follow	VERB
ejpam-4618	66	13	from	from	ADP
ejpam-4618	66	14	definition	definition	NOUN
ejpam-4618	66	15	6	6	NUM
ejpam-4618	66	16	that	that	SCONJ
ejpam-4618	66	17	ô(u	ô(u	ADP
ejpam-4618	66	18	∗	∗	X
ejpam-4618	66	19	v	v	NOUN
ejpam-4618	66	20	)	)	PUNCT
ejpam-4618	66	21	≥	≥	NOUN
ejpam-4618	66	22	min{ô(u	min{ô(u	PROPN
ejpam-4618	66	23	)	)	PUNCT
ejpam-4618	66	24	,	,	PUNCT
ejpam-4618	66	25	ô(v	ô(v	ADV
ejpam-4618	66	26	)	)	PUNCT
ejpam-4618	66	27	}	}	PUNCT
ejpam-4618	66	28	≥	≥	NOUN
ejpam-4618	66	29	r̂	r̂	NOUN
ejpam-4618	66	30	,	,	PUNCT
ejpam-4618	66	31	so	so	SCONJ
ejpam-4618	66	32	that	that	SCONJ
ejpam-4618	66	33	(	(	PUNCT
ejpam-4618	66	34	u	u	NOUN
ejpam-4618	66	35	∗	∗	NOUN
ejpam-4618	66	36	v	v	NOUN
ejpam-4618	66	37	)	)	PUNCT
ejpam-4618	66	38	∈	∈	PROPN
ejpam-4618	66	39	u(ô	u(ô	NOUN
ejpam-4618	66	40	;	;	PUNCT
ejpam-4618	66	41	r̂	r̂	NOUN
ejpam-4618	66	42	)	)	PUNCT
ejpam-4618	66	43	.	.	PUNCT
ejpam-4618	67	1	hence	hence	ADV
ejpam-4618	67	2	u(ô	u(ô	PROPN
ejpam-4618	67	3	;	;	PUNCT
ejpam-4618	67	4	r̂	r̂	NOUN
ejpam-4618	67	5	)	)	PUNCT
ejpam-4618	67	6	is	be	AUX
ejpam-4618	67	7	a	a	DET
ejpam-4618	67	8	subalgebra	subalgebra	NOUN
ejpam-4618	67	9	of	of	ADP
ejpam-4618	67	10	x.	x.	NOUN
ejpam-4618	67	11	conversely	conversely	ADV
ejpam-4618	67	12	,	,	PUNCT
ejpam-4618	67	13	assume	assume	VERB
ejpam-4618	67	14	that	that	SCONJ
ejpam-4618	67	15	u(ô	u(ô	NOUN
ejpam-4618	67	16	;	;	PUNCT
ejpam-4618	67	17	r̂	r̂	NOUN
ejpam-4618	67	18	)	)	PUNCT
ejpam-4618	67	19	is	be	AUX
ejpam-4618	67	20	a	a	DET
ejpam-4618	67	21	subalgebra	subalgebra	NOUN
ejpam-4618	67	22	of	of	ADP
ejpam-4618	67	23	x.	x.	NOUN
ejpam-4618	67	24	suppose	suppose	VERB
ejpam-4618	67	25	that	that	SCONJ
ejpam-4618	67	26	there	there	PRON
ejpam-4618	67	27	exist	exist	VERB
ejpam-4618	67	28	u	u	NOUN
ejpam-4618	67	29	,	,	PUNCT
ejpam-4618	67	30	v	v	NOUN
ejpam-4618	67	31	∈	∈	NOUN
ejpam-4618	67	32	x	x	PUNCT
ejpam-4618	67	33	such	such	ADJ
ejpam-4618	67	34	that	that	DET
ejpam-4618	67	35	ô(u∗v	ô(u∗v	NOUN
ejpam-4618	67	36	)	)	PUNCT
ejpam-4618	67	37	<	<	X
ejpam-4618	68	1	min{ô(u	min{ô(u	PROPN
ejpam-4618	68	2	)	)	PUNCT
ejpam-4618	68	3	,	,	PUNCT
ejpam-4618	68	4	ô(v	ô(v	ADV
ejpam-4618	68	5	)	)	PUNCT
ejpam-4618	68	6	}	}	PUNCT
ejpam-4618	68	7	.	.	PUNCT
ejpam-4618	69	1	then	then	ADV
ejpam-4618	69	2	there	there	PRON
ejpam-4618	69	3	exists	exist	VERB
ejpam-4618	69	4	r̂	r̂	NOUN
ejpam-4618	69	5	=	=	SYM
ejpam-4618	69	6	(	(	PUNCT
ejpam-4618	69	7	r1	r1	PROPN
ejpam-4618	69	8	,	,	PUNCT
ejpam-4618	69	9	r2	r2	PROPN
ejpam-4618	69	10	,	,	PUNCT
ejpam-4618	69	11	...	...	PUNCT
ejpam-4618	69	12	,	,	PUNCT
ejpam-4618	69	13	rm	rm	PROPN
ejpam-4618	69	14	)	)	PUNCT
ejpam-4618	69	15	∈	∈	PROPN
ejpam-4618	70	1	[	[	X
ejpam-4618	70	2	0	0	NUM
ejpam-4618	70	3	,	,	PUNCT
ejpam-4618	70	4	1]m	1]m	NUM
ejpam-4618	70	5	such	such	ADJ
ejpam-4618	70	6	taht	taht	NOUN
ejpam-4618	70	7	ô(u	ô(u	ADP
ejpam-4618	70	8	∗	∗	NOUN
ejpam-4618	70	9	v	v	NOUN
ejpam-4618	70	10	)	)	PUNCT
ejpam-4618	70	11	<	<	X
ejpam-4618	70	12	r̂	r̂	NOUN
ejpam-4618	70	13	≤	≤	ADV
ejpam-4618	70	14	min{ô(u	min{ô(u	PROPN
ejpam-4618	70	15	)	)	PUNCT
ejpam-4618	70	16	,	,	PUNCT
ejpam-4618	70	17	ô(v	ô(v	ADV
ejpam-4618	70	18	)	)	PUNCT
ejpam-4618	70	19	}	}	PUNCT
ejpam-4618	70	20	.	.	PUNCT
ejpam-4618	71	1	it	it	PRON
ejpam-4618	71	2	follows	follow	VERB
ejpam-4618	71	3	that	that	SCONJ
ejpam-4618	71	4	u	u	NOUN
ejpam-4618	71	5	,	,	PUNCT
ejpam-4618	71	6	v	v	PROPN
ejpam-4618	71	7	∈	∈	PROPN
ejpam-4618	71	8	u(ô	u(ô	NOUN
ejpam-4618	71	9	;	;	PUNCT
ejpam-4618	71	10	r̂	r̂	NOUN
ejpam-4618	71	11	)	)	PUNCT
ejpam-4618	71	12	,	,	PUNCT
ejpam-4618	71	13	but	but	CCONJ
ejpam-4618	71	14	u	u	NOUN
ejpam-4618	71	15	∗	∗	NOUN
ejpam-4618	71	16	v	v	NOUN
ejpam-4618	71	17	/∈	/∈	PUNCT
ejpam-4618	71	18	u(ô	u(ô	NOUN
ejpam-4618	71	19	;	;	PUNCT
ejpam-4618	71	20	r̂	r̂	NOUN
ejpam-4618	71	21	)	)	PUNCT
ejpam-4618	71	22	.	.	PUNCT
ejpam-4618	72	1	this	this	PRON
ejpam-4618	72	2	is	be	AUX
ejpam-4618	72	3	a	a	DET
ejpam-4618	72	4	contradiction	contradiction	NOUN
ejpam-4618	72	5	,	,	PUNCT
ejpam-4618	72	6	and	and	CCONJ
ejpam-4618	72	7	so	so	ADV
ejpam-4618	72	8	ô(u	ô(u	PROPN
ejpam-4618	72	9	∗	∗	X
ejpam-4618	72	10	v	v	NOUN
ejpam-4618	72	11	)	)	PUNCT
ejpam-4618	72	12	≥	≥	NOUN
ejpam-4618	72	13	min{ô(u	min{ô(u	PROPN
ejpam-4618	72	14	)	)	PUNCT
ejpam-4618	72	15	,	,	PUNCT
ejpam-4618	72	16	ô(v	ô(v	ADV
ejpam-4618	72	17	)	)	PUNCT
ejpam-4618	72	18	}	}	PUNCT
ejpam-4618	72	19	,	,	PUNCT
ejpam-4618	72	20	∀u	∀u	NOUN
ejpam-4618	72	21	,	,	PUNCT
ejpam-4618	72	22	v	v	PROPN
ejpam-4618	72	23	∈	∈	PROPN
ejpam-4618	72	24	x.	x.	NOUN
ejpam-4618	72	25	therefore	therefore	ADV
ejpam-4618	72	26	ô	ô	NUM
ejpam-4618	72	27	is	be	AUX
ejpam-4618	72	28	an	an	DET
ejpam-4618	72	29	m	m	ADJ
ejpam-4618	72	30	-	-	ADJ
ejpam-4618	72	31	polar	polar	ADJ
ejpam-4618	72	32	fuzzy	fuzzy	ADJ
ejpam-4618	72	33	ku	ku	NOUN
ejpam-4618	72	34	-	-	PUNCT
ejpam-4618	72	35	subalgebra	subalgebra	PROPN
ejpam-4618	72	36	of	of	ADP
ejpam-4618	72	37	x.	x.	PROPN
ejpam-4618	72	38	h.	h.	PROPN
ejpam-4618	72	39	alshehri	alshehri	PROPN
ejpam-4618	72	40	/	/	SYM
ejpam-4618	72	41	eur	eur	PROPN
ejpam-4618	72	42	.	.	PUNCT
ejpam-4618	73	1	j.	j.	PROPN
ejpam-4618	73	2	pure	pure	PROPN
ejpam-4618	73	3	appl	appl	PROPN
ejpam-4618	73	4	.	.	PROPN
ejpam-4618	73	5	math	math	PROPN
ejpam-4618	73	6	,	,	PUNCT
ejpam-4618	73	7	16	16	NUM
ejpam-4618	73	8	(	(	PUNCT
ejpam-4618	73	9	1	1	NUM
ejpam-4618	73	10	)	)	PUNCT
ejpam-4618	73	11	(	(	PUNCT
ejpam-4618	73	12	2023	2023	NUM
ejpam-4618	73	13	)	)	PUNCT
ejpam-4618	73	14	,	,	PUNCT
ejpam-4618	73	15	253	253	NUM
ejpam-4618	73	16	-	-	SYM
ejpam-4618	73	17	260	260	NUM
ejpam-4618	73	18	257	257	NUM
ejpam-4618	73	19	lemma	lemma	PROPN
ejpam-4618	73	20	2	2	NUM
ejpam-4618	73	21	.	.	PUNCT
ejpam-4618	74	1	every	every	DET
ejpam-4618	74	2	m	m	ADJ
ejpam-4618	74	3	-	-	ADJ
ejpam-4618	74	4	polar	polar	ADJ
ejpam-4618	74	5	fuzzy	fuzzy	ADJ
ejpam-4618	74	6	subalgebra	subalgebra	NOUN
ejpam-4618	74	7	ô	ô	NUM
ejpam-4618	74	8	of	of	ADP
ejpam-4618	74	9	x	x	PUNCT
ejpam-4618	74	10	satisfies	satisfy	VERB
ejpam-4618	74	11	the	the	DET
ejpam-4618	74	12	following	follow	VERB
ejpam-4618	74	13	inequality	inequality	NOUN
ejpam-4618	74	14	:	:	PUNCT
ejpam-4618	74	15	(	(	PUNCT
ejpam-4618	74	16	∀u	∀u	NOUN
ejpam-4618	74	17	∈	∈	PROPN
ejpam-4618	74	18	x)(ô(0	x)(ô(0	PROPN
ejpam-4618	74	19	)	)	PUNCT
ejpam-4618	74	20	≥	≥	NOUN
ejpam-4618	74	21	ô(u	ô(u	NOUN
ejpam-4618	74	22	)	)	PUNCT
ejpam-4618	74	23	)	)	PUNCT
ejpam-4618	74	24	(	(	PUNCT
ejpam-4618	74	25	2	2	X
ejpam-4618	74	26	)	)	PUNCT
ejpam-4618	74	27	that	that	PRON
ejpam-4618	74	28	is	be	AUX
ejpam-4618	74	29	,	,	PUNCT
ejpam-4618	74	30	(	(	PUNCT
ejpam-4618	74	31	πi	πi	ADP
ejpam-4618	74	32	◦	◦	VERB
ejpam-4618	74	33	ô)(0	ô)(0	ADJ
ejpam-4618	74	34	)	)	PUNCT
ejpam-4618	74	35	≥	≥	NOUN
ejpam-4618	74	36	(	(	PUNCT
ejpam-4618	74	37	πi	πi	ADP
ejpam-4618	74	38	◦	◦	VERB
ejpam-4618	74	39	ô)(u	ô)(u	ADJ
ejpam-4618	74	40	)	)	PUNCT
ejpam-4618	74	41	for	for	ADP
ejpam-4618	74	42	all	all	DET
ejpam-4618	74	43	u	u	NOUN
ejpam-4618	74	44	∈	∈	PROPN
ejpam-4618	74	45	x	x	NOUN
ejpam-4618	74	46	,	,	PUNCT
ejpam-4618	74	47	i	i	NOUN
ejpam-4618	74	48	=	=	NOUN
ejpam-4618	74	49	1	1	NUM
ejpam-4618	74	50	,	,	PUNCT
ejpam-4618	74	51	2	2	NUM
ejpam-4618	74	52	,	,	PUNCT
ejpam-4618	74	53	...	...	PUNCT
ejpam-4618	74	54	,	,	PUNCT
ejpam-4618	74	55	m.	m.	NOUN
ejpam-4618	74	56	proof	proof	NOUN
ejpam-4618	74	57	.	.	PUNCT
ejpam-4618	75	1	note	note	VERB
ejpam-4618	75	2	that	that	SCONJ
ejpam-4618	76	1	u	u	PRON
ejpam-4618	76	2	∗	∗	NOUN
ejpam-4618	76	3	u	u	NOUN
ejpam-4618	76	4	=	=	NOUN
ejpam-4618	76	5	0	0	NUM
ejpam-4618	76	6	for	for	ADP
ejpam-4618	76	7	all	all	DET
ejpam-4618	76	8	u	u	PROPN
ejpam-4618	76	9	∈	∈	NOUN
ejpam-4618	76	10	x.	x.	NOUN
ejpam-4618	76	11	using	use	VERB
ejpam-4618	76	12	definition	definition	NOUN
ejpam-4618	76	13	6	6	NUM
ejpam-4618	76	14	,	,	PUNCT
ejpam-4618	76	15	we	we	PRON
ejpam-4618	76	16	have	have	VERB
ejpam-4618	76	17	ô(0	ô(0	NUM
ejpam-4618	76	18	)	)	PUNCT
ejpam-4618	77	1	=	=	SYM
ejpam-4618	77	2	ô(u	ô(u	ADP
ejpam-4618	77	3	∗	∗	NOUN
ejpam-4618	77	4	u	u	NOUN
ejpam-4618	77	5	)	)	PUNCT
ejpam-4618	77	6	≥	≥	NOUN
ejpam-4618	77	7	min{ô(u	min{ô(u	PROPN
ejpam-4618	77	8	)	)	PUNCT
ejpam-4618	77	9	,	,	PUNCT
ejpam-4618	77	10	ô(u	ô(u	NOUN
ejpam-4618	77	11	)	)	PUNCT
ejpam-4618	77	12	}	}	PUNCT
ejpam-4618	77	13	=	=	SYM
ejpam-4618	77	14	ô(u	ô(u	NOUN
ejpam-4618	77	15	)	)	PUNCT
ejpam-4618	77	16	.	.	PUNCT
ejpam-4618	78	1	for	for	ADP
ejpam-4618	78	2	all	all	DET
ejpam-4618	78	3	u	u	PROPN
ejpam-4618	78	4	∈	∈	NOUN
ejpam-4618	78	5	x.	x.	NOUN
ejpam-4618	78	6	proposition	proposition	NOUN
ejpam-4618	78	7	2	2	NUM
ejpam-4618	78	8	.	.	PUNCT
ejpam-4618	79	1	if	if	SCONJ
ejpam-4618	79	2	every	every	DET
ejpam-4618	79	3	an	an	DET
ejpam-4618	79	4	m	m	ADJ
ejpam-4618	79	5	-	-	ADJ
ejpam-4618	79	6	polar	polar	ADJ
ejpam-4618	79	7	fuzzy	fuzzy	ADJ
ejpam-4618	79	8	subalgebra	subalgebra	NOUN
ejpam-4618	79	9	ô	ô	NUM
ejpam-4618	79	10	of	of	ADP
ejpam-4618	79	11	x	x	PUNCT
ejpam-4618	79	12	satisfies	satisfy	VERB
ejpam-4618	79	13	the	the	DET
ejpam-4618	79	14	following	follow	VERB
ejpam-4618	79	15	inequality	inequality	NOUN
ejpam-4618	79	16	:	:	PUNCT
ejpam-4618	79	17	(	(	PUNCT
ejpam-4618	79	18	∀u	∀u	NOUN
ejpam-4618	79	19	,	,	PUNCT
ejpam-4618	79	20	v	v	ADP
ejpam-4618	79	21	∈	∈	PROPN
ejpam-4618	79	22	x)(ô(u	x)(ô(u	PROPN
ejpam-4618	79	23	∗	∗	NOUN
ejpam-4618	79	24	v	v	NOUN
ejpam-4618	79	25	)	)	PUNCT
ejpam-4618	79	26	≥	≥	NOUN
ejpam-4618	79	27	ô(v	ô(v	NUM
ejpam-4618	79	28	)	)	PUNCT
ejpam-4618	79	29	)	)	PUNCT
ejpam-4618	80	1	(	(	PUNCT
ejpam-4618	80	2	3	3	X
ejpam-4618	80	3	)	)	PUNCT
ejpam-4618	80	4	then	then	ADV
ejpam-4618	80	5	,	,	PUNCT
ejpam-4618	80	6	ô(0	ô(0	NUM
ejpam-4618	80	7	)	)	PUNCT
ejpam-4618	80	8	=	=	SYM
ejpam-4618	80	9	ô(u	ô(u	NOUN
ejpam-4618	80	10	)	)	PUNCT
ejpam-4618	80	11	that	that	PRON
ejpam-4618	80	12	is	be	AUX
ejpam-4618	80	13	,	,	PUNCT
ejpam-4618	80	14	(	(	PUNCT
ejpam-4618	80	15	πi	πi	ADP
ejpam-4618	80	16	◦	◦	VERB
ejpam-4618	80	17	ô)(u	ô)(u	ADJ
ejpam-4618	80	18	∗	∗	NOUN
ejpam-4618	80	19	v	v	NOUN
ejpam-4618	80	20	)	)	PUNCT
ejpam-4618	80	21	≥	≥	NOUN
ejpam-4618	80	22	(	(	PUNCT
ejpam-4618	80	23	πi	πi	ADP
ejpam-4618	80	24	◦	◦	NOUN
ejpam-4618	80	25	ô)(v	ô)(v	NOUN
ejpam-4618	80	26	)	)	PUNCT
ejpam-4618	80	27	then	then	ADV
ejpam-4618	80	28	,	,	PUNCT
ejpam-4618	80	29	(	(	PUNCT
ejpam-4618	80	30	πi	πi	ADP
ejpam-4618	80	31	◦	◦	NOUN
ejpam-4618	80	32	ô)(0	ô)(0	ADJ
ejpam-4618	80	33	)	)	PUNCT
ejpam-4618	80	34	=	=	PUNCT
ejpam-4618	81	1	(	(	PUNCT
ejpam-4618	81	2	πi	πi	ADP
ejpam-4618	81	3	◦	◦	VERB
ejpam-4618	81	4	ô)(u	ô)(u	ADJ
ejpam-4618	81	5	)	)	PUNCT
ejpam-4618	81	6	,	,	PUNCT
ejpam-4618	81	7	for	for	ADP
ejpam-4618	81	8	all	all	DET
ejpam-4618	81	9	u	u	NOUN
ejpam-4618	81	10	,	,	PUNCT
ejpam-4618	81	11	v	v	NOUN
ejpam-4618	81	12	∈	∈	PROPN
ejpam-4618	81	13	x	x	NOUN
ejpam-4618	81	14	,	,	PUNCT
ejpam-4618	81	15	i	i	NOUN
ejpam-4618	81	16	=	=	NOUN
ejpam-4618	81	17	1	1	NUM
ejpam-4618	81	18	,	,	PUNCT
ejpam-4618	81	19	2	2	NUM
ejpam-4618	81	20	,	,	PUNCT
ejpam-4618	81	21	...	...	PUNCT
ejpam-4618	81	22	,	,	PUNCT
ejpam-4618	81	23	m.	m.	NOUN
ejpam-4618	81	24	proof	proof	NOUN
ejpam-4618	81	25	.	.	PUNCT
ejpam-4618	82	1	let	let	VERB
ejpam-4618	82	2	u	u	PRON
ejpam-4618	82	3	∈	∈	NOUN
ejpam-4618	82	4	x.	x.	NOUN
ejpam-4618	82	5	using	use	VERB
ejpam-4618	82	6	(	(	PUNCT
ejpam-4618	82	7	ku	ku	PROPN
ejpam-4618	82	8	2	2	NUM
ejpam-4618	82	9	)	)	PUNCT
ejpam-4618	82	10	and	and	CCONJ
ejpam-4618	82	11	(	(	PUNCT
ejpam-4618	82	12	3	3	NUM
ejpam-4618	82	13	)	)	PUNCT
ejpam-4618	82	14	,	,	PUNCT
ejpam-4618	82	15	we	we	PRON
ejpam-4618	82	16	have	have	VERB
ejpam-4618	82	17	ô(u	ô(u	NOUN
ejpam-4618	82	18	)	)	PUNCT
ejpam-4618	82	19	=	=	SYM
ejpam-4618	82	20	ô(u	ô(u	ADP
ejpam-4618	82	21	∗	∗	NOUN
ejpam-4618	82	22	0	0	NUM
ejpam-4618	82	23	)	)	PUNCT
ejpam-4618	82	24	≥	≥	NOUN
ejpam-4618	82	25	ô(0	ô(0	NUM
ejpam-4618	82	26	)	)	PUNCT
ejpam-4618	82	27	.	.	PUNCT
ejpam-4618	83	1	it	it	PRON
ejpam-4618	83	2	follows	follow	VERB
ejpam-4618	83	3	from	from	ADP
ejpam-4618	83	4	lemma	lemma	PROPN
ejpam-4618	83	5	2	2	NUM
ejpam-4618	83	6	that	that	SCONJ
ejpam-4618	83	7	ô(0	ô(0	NOUN
ejpam-4618	83	8	)	)	PUNCT
ejpam-4618	83	9	=	=	SYM
ejpam-4618	83	10	ô(u	ô(u	NOUN
ejpam-4618	83	11	)	)	PUNCT
ejpam-4618	83	12	.	.	PUNCT
ejpam-4618	84	1	definition	definition	NOUN
ejpam-4618	84	2	7	7	NUM
ejpam-4618	84	3	.	.	PUNCT
ejpam-4618	85	1	an	an	DET
ejpam-4618	85	2	m	m	ADJ
ejpam-4618	85	3	-	-	ADJ
ejpam-4618	85	4	polar	polar	ADJ
ejpam-4618	85	5	fuzzy	fuzzy	ADJ
ejpam-4618	85	6	set	set	NOUN
ejpam-4618	85	7	ô	ô	NUM
ejpam-4618	85	8	of	of	ADP
ejpam-4618	85	9	x	x	PRON
ejpam-4618	85	10	is	be	AUX
ejpam-4618	85	11	called	call	VERB
ejpam-4618	85	12	an	an	DET
ejpam-4618	85	13	m	m	ADJ
ejpam-4618	85	14	-	-	ADJ
ejpam-4618	85	15	polar	polar	ADJ
ejpam-4618	85	16	fuzzy	fuzzy	ADJ
ejpam-4618	85	17	ku	ku	NOUN
ejpam-4618	85	18	-	-	PUNCT
ejpam-4618	85	19	ideal	ideal	NOUN
ejpam-4618	85	20	if	if	SCONJ
ejpam-4618	85	21	the	the	DET
ejpam-4618	85	22	following	follow	VERB
ejpam-4618	85	23	conditions	condition	NOUN
ejpam-4618	85	24	are	be	AUX
ejpam-4618	85	25	valid	valid	ADJ
ejpam-4618	85	26	:	:	PUNCT
ejpam-4618	85	27	(	(	PUNCT
ejpam-4618	85	28	∀u	∀u	NOUN
ejpam-4618	85	29	∈	∈	PROPN
ejpam-4618	85	30	x)(ô(0	x)(ô(0	PROPN
ejpam-4618	85	31	)	)	PUNCT
ejpam-4618	85	32	≥	≥	NOUN
ejpam-4618	85	33	ô(u	ô(u	NOUN
ejpam-4618	85	34	)	)	PUNCT
ejpam-4618	85	35	)	)	PUNCT
ejpam-4618	85	36	(	(	PUNCT
ejpam-4618	85	37	∀u	∀u	NOUN
ejpam-4618	85	38	,	,	PUNCT
ejpam-4618	85	39	v	v	NOUN
ejpam-4618	85	40	,	,	PUNCT
ejpam-4618	85	41	w	w	PROPN
ejpam-4618	85	42	∈	∈	PROPN
ejpam-4618	85	43	x)(ô(u	x)(ô(u	PROPN
ejpam-4618	85	44	∗	∗	PROPN
ejpam-4618	85	45	w	w	PROPN
ejpam-4618	85	46	)	)	PUNCT
ejpam-4618	85	47	≥	≥	NOUN
ejpam-4618	85	48	min{ô(u	min{ô(u	PROPN
ejpam-4618	85	49	∗	∗	NOUN
ejpam-4618	85	50	(	(	PUNCT
ejpam-4618	85	51	v	v	NOUN
ejpam-4618	85	52	∗	∗	NOUN
ejpam-4618	85	53	w	w	NOUN
ejpam-4618	85	54	)	)	PUNCT
ejpam-4618	85	55	)	)	PUNCT
ejpam-4618	85	56	,	,	PUNCT
ejpam-4618	85	57	ô(v	ô(v	ADV
ejpam-4618	85	58	)	)	PUNCT
ejpam-4618	85	59	}	}	PUNCT
ejpam-4618	85	60	(	(	PUNCT
ejpam-4618	85	61	4	4	X
ejpam-4618	85	62	)	)	PUNCT
ejpam-4618	85	63	that	that	PRON
ejpam-4618	85	64	is	be	AUX
ejpam-4618	85	65	,	,	PUNCT
ejpam-4618	85	66	(	(	PUNCT
ejpam-4618	85	67	∀u	∀u	NOUN
ejpam-4618	85	68	∈	∈	VERB
ejpam-4618	85	69	x)((πi	x)((πi	ADV
ejpam-4618	85	70	◦	◦	VERB
ejpam-4618	85	71	ô)(0	ô)(0	ADJ
ejpam-4618	85	72	)	)	PUNCT
ejpam-4618	85	73	≥	≥	NOUN
ejpam-4618	85	74	(	(	PUNCT
ejpam-4618	85	75	πi	πi	ADP
ejpam-4618	85	76	◦	◦	VERB
ejpam-4618	85	77	ô)(u	ô)(u	ADJ
ejpam-4618	85	78	)	)	PUNCT
ejpam-4618	85	79	)	)	PUNCT
ejpam-4618	85	80	(	(	PUNCT
ejpam-4618	85	81	∀u	∀u	NOUN
ejpam-4618	85	82	,	,	PUNCT
ejpam-4618	85	83	v	v	NOUN
ejpam-4618	85	84	,	,	PUNCT
ejpam-4618	85	85	w	w	PROPN
ejpam-4618	85	86	∈	∈	PROPN
ejpam-4618	85	87	x)((πi	x)((πi	ADV
ejpam-4618	85	88	◦	◦	VERB
ejpam-4618	85	89	ô)(u	ô)(u	ADJ
ejpam-4618	85	90	∗	∗	X
ejpam-4618	85	91	w	w	NOUN
ejpam-4618	85	92	)	)	PUNCT
ejpam-4618	85	93	≥	≥	NOUN
ejpam-4618	85	94	min{(πi	min{(πi	NOUN
ejpam-4618	85	95	◦	◦	VERB
ejpam-4618	85	96	ô)(u	ô)(u	ADJ
ejpam-4618	85	97	∗	∗	NOUN
ejpam-4618	85	98	(	(	PUNCT
ejpam-4618	85	99	v	v	NOUN
ejpam-4618	85	100	∗	∗	NOUN
ejpam-4618	85	101	w	w	NOUN
ejpam-4618	85	102	)	)	PUNCT
ejpam-4618	85	103	)	)	PUNCT
ejpam-4618	85	104	,	,	PUNCT
ejpam-4618	85	105	(	(	PUNCT
ejpam-4618	85	106	πi	πi	ADP
ejpam-4618	85	107	◦	◦	NOUN
ejpam-4618	85	108	ô)(v	ô)(v	NOUN
ejpam-4618	85	109	)	)	PUNCT
ejpam-4618	85	110	}	}	PUNCT
ejpam-4618	85	111	for	for	ADP
ejpam-4618	85	112	all	all	DET
ejpam-4618	85	113	i	i	PRON
ejpam-4618	85	114	=	=	NOUN
ejpam-4618	85	115	1	1	NUM
ejpam-4618	85	116	,	,	PUNCT
ejpam-4618	85	117	2	2	NUM
ejpam-4618	85	118	,	,	PUNCT
ejpam-4618	85	119	...	...	PUNCT
ejpam-4618	85	120	,	,	PUNCT
ejpam-4618	85	121	m.	m.	NOUN
ejpam-4618	85	122	proposition	proposition	NOUN
ejpam-4618	85	123	3	3	X
ejpam-4618	85	124	.	.	PUNCT
ejpam-4618	86	1	if	if	SCONJ
ejpam-4618	86	2	ô	ô	NUM
ejpam-4618	86	3	is	be	AUX
ejpam-4618	86	4	an	an	DET
ejpam-4618	86	5	m	m	ADJ
ejpam-4618	86	6	-	-	ADJ
ejpam-4618	86	7	polar	polar	ADJ
ejpam-4618	86	8	fuzzy	fuzzy	ADJ
ejpam-4618	86	9	ku	ku	NOUN
ejpam-4618	86	10	-	-	PUNCT
ejpam-4618	86	11	ideal	ideal	NOUN
ejpam-4618	86	12	of	of	ADP
ejpam-4618	86	13	x	x	X
ejpam-4618	86	14	and	and	CCONJ
ejpam-4618	86	15	u	u	PROPN
ejpam-4618	86	16	≤	≤	PROPN
ejpam-4618	86	17	v	v	NOUN
ejpam-4618	86	18	,	,	PUNCT
ejpam-4618	86	19	then	then	ADV
ejpam-4618	86	20	(	(	PUNCT
ejpam-4618	86	21	ô(u	ô(u	NOUN
ejpam-4618	86	22	)	)	PUNCT
ejpam-4618	86	23	≥	≥	NOUN
ejpam-4618	87	1	ô(v))(∀u	ô(v))(∀u	PROPN
ejpam-4618	87	2	,	,	PUNCT
ejpam-4618	87	3	v	v	NOUN
ejpam-4618	87	4	∈	∈	PROPN
ejpam-4618	87	5	x	x	X
ejpam-4618	87	6	)	)	PUNCT
ejpam-4618	87	7	(	(	PUNCT
ejpam-4618	87	8	5	5	X
ejpam-4618	87	9	)	)	PUNCT
ejpam-4618	87	10	that	that	PRON
ejpam-4618	87	11	is	be	AUX
ejpam-4618	87	12	,	,	PUNCT
ejpam-4618	87	13	h.	h.	PROPN
ejpam-4618	87	14	alshehri	alshehri	PROPN
ejpam-4618	87	15	/	/	SYM
ejpam-4618	87	16	eur	eur	PROPN
ejpam-4618	87	17	.	.	PUNCT
ejpam-4618	88	1	j.	j.	PROPN
ejpam-4618	88	2	pure	pure	PROPN
ejpam-4618	88	3	appl	appl	PROPN
ejpam-4618	88	4	.	.	PROPN
ejpam-4618	88	5	math	math	PROPN
ejpam-4618	88	6	,	,	PUNCT
ejpam-4618	88	7	16	16	NUM
ejpam-4618	88	8	(	(	PUNCT
ejpam-4618	88	9	1	1	NUM
ejpam-4618	88	10	)	)	PUNCT
ejpam-4618	88	11	(	(	PUNCT
ejpam-4618	88	12	2023	2023	NUM
ejpam-4618	88	13	)	)	PUNCT
ejpam-4618	88	14	,	,	PUNCT
ejpam-4618	88	15	253	253	NUM
ejpam-4618	88	16	-	-	SYM
ejpam-4618	88	17	260	260	NUM
ejpam-4618	88	18	258	258	NUM
ejpam-4618	88	19	(	(	PUNCT
ejpam-4618	88	20	(	(	PUNCT
ejpam-4618	88	21	πi	πi	ADP
ejpam-4618	88	22	◦	◦	VERB
ejpam-4618	88	23	ô)(u	ô)(u	ADJ
ejpam-4618	88	24	)	)	PUNCT
ejpam-4618	88	25	≥	≥	NOUN
ejpam-4618	88	26	(	(	PUNCT
ejpam-4618	88	27	πi	πi	ADP
ejpam-4618	88	28	◦	◦	VERB
ejpam-4618	88	29	ô)(v))(∀u	ô)(v))(∀u	ADJ
ejpam-4618	88	30	,	,	PUNCT
ejpam-4618	88	31	v	v	ADP
ejpam-4618	88	32	∈	∈	PROPN
ejpam-4618	88	33	x	x	NOUN
ejpam-4618	88	34	,	,	PUNCT
ejpam-4618	88	35	i	i	NOUN
ejpam-4618	88	36	=	=	NOUN
ejpam-4618	88	37	1	1	NUM
ejpam-4618	88	38	,	,	PUNCT
ejpam-4618	88	39	2	2	NUM
ejpam-4618	88	40	,	,	PUNCT
ejpam-4618	88	41	...	...	PUNCT
ejpam-4618	88	42	,	,	PUNCT
ejpam-4618	88	43	m	m	NOUN
ejpam-4618	88	44	)	)	PUNCT
ejpam-4618	88	45	proof	proof	NOUN
ejpam-4618	88	46	.	.	PUNCT
ejpam-4618	89	1	if	if	SCONJ
ejpam-4618	89	2	u	u	PROPN
ejpam-4618	89	3	≤	≤	X
ejpam-4618	89	4	v	v	NOUN
ejpam-4618	89	5	,	,	PUNCT
ejpam-4618	89	6	then	then	ADV
ejpam-4618	89	7	v	v	X
ejpam-4618	89	8	∗	∗	NOUN
ejpam-4618	89	9	u	u	NOUN
ejpam-4618	89	10	=	=	NOUN
ejpam-4618	89	11	0	0	NUM
ejpam-4618	90	1	and	and	CCONJ
ejpam-4618	90	2	(	(	PUNCT
ejpam-4618	90	3	ku	ku	PROPN
ejpam-4618	90	4	3	3	NUM
ejpam-4618	90	5	)	)	PUNCT
ejpam-4618	90	6	0	0	NUM
ejpam-4618	90	7	∗	∗	NOUN
ejpam-4618	90	8	u	u	NOUN
ejpam-4618	90	9	=	=	X
ejpam-4618	90	10	u.	u.	PROPN
ejpam-4618	90	11	since	since	SCONJ
ejpam-4618	90	12	ô	ô	NUM
ejpam-4618	90	13	is	be	AUX
ejpam-4618	90	14	an	an	DET
ejpam-4618	90	15	m	m	ADJ
ejpam-4618	90	16	-	-	ADJ
ejpam-4618	90	17	polar	polar	ADJ
ejpam-4618	90	18	fuzzy	fuzzy	ADJ
ejpam-4618	90	19	ku	ku	NOUN
ejpam-4618	90	20	-	-	PUNCT
ejpam-4618	90	21	ideal	ideal	NOUN
ejpam-4618	90	22	of	of	ADP
ejpam-4618	90	23	x	x	SYM
ejpam-4618	90	24	,	,	PUNCT
ejpam-4618	90	25	we	we	PRON
ejpam-4618	90	26	get	get	VERB
ejpam-4618	90	27	ô(0	ô(0	NOUN
ejpam-4618	90	28	∗	∗	NOUN
ejpam-4618	90	29	u	u	NOUN
ejpam-4618	90	30	)	)	PUNCT
ejpam-4618	90	31	=	=	SYM
ejpam-4618	90	32	ô(u	ô(u	NOUN
ejpam-4618	90	33	)	)	PUNCT
ejpam-4618	90	34	≥	≥	NOUN
ejpam-4618	90	35	min{ô(0	min{ô(0	NOUN
ejpam-4618	90	36	∗	∗	NOUN
ejpam-4618	90	37	(	(	PUNCT
ejpam-4618	90	38	u	u	NOUN
ejpam-4618	90	39	∗	∗	NOUN
ejpam-4618	90	40	v	v	NOUN
ejpam-4618	90	41	)	)	PUNCT
ejpam-4618	90	42	)	)	PUNCT
ejpam-4618	90	43	,	,	PUNCT
ejpam-4618	90	44	o(v	o(v	NOUN
ejpam-4618	90	45	)	)	PUNCT
ejpam-4618	90	46	}	}	PUNCT
ejpam-4618	91	1	=	=	PUNCT
ejpam-4618	91	2	min{ô(0	min{ô(0	NOUN
ejpam-4618	91	3	∗	∗	NOUN
ejpam-4618	91	4	0	0	NUM
ejpam-4618	91	5	)	)	PUNCT
ejpam-4618	91	6	,	,	PUNCT
ejpam-4618	91	7	o(v	o(v	NOUN
ejpam-4618	91	8	)	)	PUNCT
ejpam-4618	91	9	}	}	PUNCT
ejpam-4618	91	10	=	=	SYM
ejpam-4618	91	11	min{ô(0	min{ô(0	NOUN
ejpam-4618	91	12	)	)	PUNCT
ejpam-4618	91	13	,	,	PUNCT
ejpam-4618	91	14	o(v	o(v	NOUN
ejpam-4618	91	15	)	)	PUNCT
ejpam-4618	91	16	}	}	PUNCT
ejpam-4618	91	17	=	=	PUNCT
ejpam-4618	91	18	ô(v	ô(v	NOUN
ejpam-4618	91	19	)	)	PUNCT
ejpam-4618	91	20	.	.	PUNCT
ejpam-4618	92	1	for	for	ADP
ejpam-4618	92	2	all	all	DET
ejpam-4618	92	3	u	u	NOUN
ejpam-4618	92	4	,	,	PUNCT
ejpam-4618	92	5	v	v	NOUN
ejpam-4618	92	6	∈	∈	NOUN
ejpam-4618	92	7	x.	x.	NOUN
ejpam-4618	92	8	proposition	proposition	NOUN
ejpam-4618	92	9	4	4	NUM
ejpam-4618	92	10	.	.	PUNCT
ejpam-4618	92	11	let	let	VERB
ejpam-4618	92	12	ô	ô	X
ejpam-4618	92	13	be	be	AUX
ejpam-4618	92	14	an	an	DET
ejpam-4618	92	15	m	m	ADJ
ejpam-4618	92	16	-	-	ADJ
ejpam-4618	92	17	polar	polar	ADJ
ejpam-4618	92	18	fuzzy	fuzzy	ADJ
ejpam-4618	92	19	ku	ku	NOUN
ejpam-4618	92	20	-	-	PUNCT
ejpam-4618	92	21	ideal	ideal	NOUN
ejpam-4618	92	22	of	of	ADP
ejpam-4618	92	23	x.	x.	NOUN
ejpam-4618	92	24	if	if	SCONJ
ejpam-4618	92	25	u	u	PROPN
ejpam-4618	92	26	∗	∗	VERB
ejpam-4618	92	27	v	v	NOUN
ejpam-4618	92	28	≤	≤	NUM
ejpam-4618	92	29	w	w	NOUN
ejpam-4618	92	30	,	,	PUNCT
ejpam-4618	92	31	holds	hold	VERB
ejpam-4618	92	32	in	in	ADP
ejpam-4618	92	33	x	x	PUNCT
ejpam-4618	92	34	then	then	ADV
ejpam-4618	92	35	,	,	PUNCT
ejpam-4618	92	36	(	(	PUNCT
ejpam-4618	92	37	ô(v	ô(v	ADV
ejpam-4618	92	38	)	)	PUNCT
ejpam-4618	92	39	≥	≥	NOUN
ejpam-4618	92	40	min{ô(u	min{ô(u	PROPN
ejpam-4618	92	41	)	)	PUNCT
ejpam-4618	92	42	,	,	PUNCT
ejpam-4618	92	43	ô(w)})(∀u	ô(w)})(∀u	PROPN
ejpam-4618	92	44	,	,	PUNCT
ejpam-4618	92	45	v	v	NOUN
ejpam-4618	92	46	,	,	PUNCT
ejpam-4618	92	47	w	w	PROPN
ejpam-4618	92	48	∈	∈	PROPN
ejpam-4618	92	49	x	x	X
ejpam-4618	92	50	)	)	PUNCT
ejpam-4618	92	51	(	(	PUNCT
ejpam-4618	92	52	6	6	NUM
ejpam-4618	92	53	)	)	PUNCT
ejpam-4618	92	54	that	that	PRON
ejpam-4618	92	55	is	be	AUX
ejpam-4618	92	56	,	,	PUNCT
ejpam-4618	92	57	(	(	PUNCT
ejpam-4618	92	58	(	(	PUNCT
ejpam-4618	92	59	πi	πi	ADP
ejpam-4618	92	60	◦	◦	NOUN
ejpam-4618	92	61	ô)(v	ô)(v	NOUN
ejpam-4618	92	62	)	)	PUNCT
ejpam-4618	92	63	≥	≥	NOUN
ejpam-4618	92	64	min{(πi	min{(πi	AUX
ejpam-4618	92	65	◦	◦	VERB
ejpam-4618	92	66	ô)(u	ô)(u	ADJ
ejpam-4618	92	67	)	)	PUNCT
ejpam-4618	92	68	,	,	PUNCT
ejpam-4618	92	69	(	(	PUNCT
ejpam-4618	92	70	πi	πi	ADP
ejpam-4618	92	71	◦	◦	NOUN
ejpam-4618	92	72	ô)(w)})(∀u	ô)(w)})(∀u	ADJ
ejpam-4618	92	73	,	,	PUNCT
ejpam-4618	92	74	v	v	NOUN
ejpam-4618	92	75	,	,	PUNCT
ejpam-4618	92	76	w	w	PROPN
ejpam-4618	92	77	∈	∈	PROPN
ejpam-4618	93	1	x	x	X
ejpam-4618	93	2	,	,	PUNCT
ejpam-4618	93	3	i	i	NOUN
ejpam-4618	93	4	=	=	NOUN
ejpam-4618	93	5	1	1	NUM
ejpam-4618	93	6	,	,	PUNCT
ejpam-4618	93	7	2	2	NUM
ejpam-4618	93	8	,	,	PUNCT
ejpam-4618	93	9	...	...	PUNCT
ejpam-4618	93	10	,	,	PUNCT
ejpam-4618	93	11	m	m	NOUN
ejpam-4618	93	12	)	)	PUNCT
ejpam-4618	93	13	proof	proof	NOUN
ejpam-4618	93	14	.	.	PUNCT
ejpam-4618	94	1	assume	assume	VERB
ejpam-4618	94	2	that	that	SCONJ
ejpam-4618	94	3	the	the	DET
ejpam-4618	94	4	inequality	inequality	NOUN
ejpam-4618	94	5	u	u	PROPN
ejpam-4618	94	6	∗	∗	NOUN
ejpam-4618	94	7	v	v	NOUN
ejpam-4618	94	8	≤	≤	NUM
ejpam-4618	94	9	w	w	NOUN
ejpam-4618	94	10	,	,	PUNCT
ejpam-4618	94	11	holds	hold	VERB
ejpam-4618	94	12	in	in	ADP
ejpam-4618	94	13	x.	x.	NOUN
ejpam-4618	94	14	then	then	ADV
ejpam-4618	94	15	w	w	PROPN
ejpam-4618	94	16	∗	∗	NOUN
ejpam-4618	94	17	(	(	PUNCT
ejpam-4618	94	18	u	u	NOUN
ejpam-4618	94	19	∗	∗	NOUN
ejpam-4618	94	20	v	v	NOUN
ejpam-4618	94	21	)	)	PUNCT
ejpam-4618	94	22	=	=	SYM
ejpam-4618	94	23	0	0	PUNCT
ejpam-4618	95	1	and	and	CCONJ
ejpam-4618	95	2	(	(	PUNCT
ejpam-4618	95	3	4	4	X
ejpam-4618	95	4	)	)	PUNCT
ejpam-4618	95	5	ô(u∗v	ô(u∗v	PROPN
ejpam-4618	95	6	)	)	PUNCT
ejpam-4618	95	7	≥	≥	NOUN
ejpam-4618	95	8	min{ô(u∗(w∗v	min{ô(u∗(w∗v	NOUN
ejpam-4618	95	9	)	)	PUNCT
ejpam-4618	95	10	)	)	PUNCT
ejpam-4618	95	11	,	,	PUNCT
ejpam-4618	95	12	ô(w	ô(w	NOUN
ejpam-4618	95	13	)	)	PUNCT
ejpam-4618	95	14	}	}	PUNCT
ejpam-4618	95	15	=	=	SYM
ejpam-4618	95	16	min{ô(w∗(u∗v	min{ô(w∗(u∗v	PROPN
ejpam-4618	95	17	)	)	PUNCT
ejpam-4618	95	18	)	)	PUNCT
ejpam-4618	95	19	,	,	PUNCT
ejpam-4618	95	20	ô(w	ô(w	NOUN
ejpam-4618	95	21	)	)	PUNCT
ejpam-4618	95	22	}	}	PUNCT
ejpam-4618	95	23	=	=	SYM
ejpam-4618	95	24	min{ô(0	min{ô(0	NOUN
ejpam-4618	95	25	)	)	PUNCT
ejpam-4618	95	26	,	,	PUNCT
ejpam-4618	95	27	ô(w	ô(w	NOUN
ejpam-4618	95	28	)	)	PUNCT
ejpam-4618	95	29	}	}	PUNCT
ejpam-4618	96	1	=	=	SYM
ejpam-4618	96	2	ô(w	ô(w	NUM
ejpam-4618	96	3	)	)	PUNCT
ejpam-4618	96	4	(	(	PUNCT
ejpam-4618	96	5	7	7	X
ejpam-4618	96	6	)	)	PUNCT
ejpam-4618	96	7	now	now	ADV
ejpam-4618	96	8	,	,	PUNCT
ejpam-4618	96	9	ô(0	ô(0	NOUN
ejpam-4618	96	10	∗	∗	NOUN
ejpam-4618	96	11	v	v	NOUN
ejpam-4618	96	12	)	)	PUNCT
ejpam-4618	96	13	=	=	PUNCT
ejpam-4618	97	1	ô(v	ô(v	X
ejpam-4618	97	2	)	)	PUNCT
ejpam-4618	97	3	=	=	SYM
ejpam-4618	97	4	min{ô(0	min{ô(0	NOUN
ejpam-4618	97	5	∗	∗	NOUN
ejpam-4618	97	6	(	(	PUNCT
ejpam-4618	97	7	u	u	NOUN
ejpam-4618	97	8	∗	∗	NOUN
ejpam-4618	97	9	v	v	NOUN
ejpam-4618	97	10	)	)	PUNCT
ejpam-4618	97	11	)	)	PUNCT
ejpam-4618	97	12	,	,	PUNCT
ejpam-4618	97	13	ô(u	ô(u	NOUN
ejpam-4618	97	14	)	)	PUNCT
ejpam-4618	97	15	}	}	PUNCT
ejpam-4618	97	16	=	=	PUNCT
ejpam-4618	97	17	min{ô(u	min{ô(u	PROPN
ejpam-4618	97	18	∗	∗	NOUN
ejpam-4618	97	19	v	v	NOUN
ejpam-4618	97	20	)	)	PUNCT
ejpam-4618	97	21	,	,	PUNCT
ejpam-4618	97	22	ô(u	ô(u	NOUN
ejpam-4618	97	23	)	)	PUNCT
ejpam-4618	97	24	}	}	PUNCT
ejpam-4618	97	25	≥	≥	PROPN
ejpam-4618	97	26	min{ô(w	min{ô(w	PROPN
ejpam-4618	97	27	)	)	PUNCT
ejpam-4618	97	28	,	,	PUNCT
ejpam-4618	97	29	ô(u	ô(u	NOUN
ejpam-4618	97	30	)	)	PUNCT
ejpam-4618	97	31	}	}	PUNCT
ejpam-4618	97	32	(	(	PUNCT
ejpam-4618	97	33	by	by	ADP
ejpam-4618	97	34	using	use	VERB
ejpam-4618	97	35	(	(	PUNCT
ejpam-4618	97	36	7	7	NUM
ejpam-4618	97	37	)	)	PUNCT
ejpam-4618	97	38	)	)	PUNCT
ejpam-4618	97	39	,	,	PUNCT
ejpam-4618	97	40	i.e.	i.e.	X
ejpam-4618	97	41	ô(v	ô(v	X
ejpam-4618	97	42	)	)	PUNCT
ejpam-4618	97	43	≥	≥	NOUN
ejpam-4618	97	44	min{ô(u	min{ô(u	PROPN
ejpam-4618	97	45	)	)	PUNCT
ejpam-4618	97	46	,	,	PUNCT
ejpam-4618	97	47	ô(w	ô(w	NOUN
ejpam-4618	97	48	)	)	PUNCT
ejpam-4618	97	49	}	}	PUNCT
ejpam-4618	97	50	.	.	PUNCT
ejpam-4618	98	1	this	this	PRON
ejpam-4618	98	2	completes	complete	VERB
ejpam-4618	98	3	the	the	DET
ejpam-4618	98	4	proof	proof	NOUN
ejpam-4618	98	5	.	.	PUNCT
ejpam-4618	99	1	theorem	theorem	ADJ
ejpam-4618	99	2	4	4	NUM
ejpam-4618	99	3	.	.	PUNCT
ejpam-4618	100	1	if	if	SCONJ
ejpam-4618	100	2	ô	ô	NUM
ejpam-4618	100	3	is	be	AUX
ejpam-4618	100	4	an	an	DET
ejpam-4618	100	5	m	m	ADJ
ejpam-4618	100	6	-	-	ADJ
ejpam-4618	100	7	polar	polar	ADJ
ejpam-4618	100	8	fuzzy	fuzzy	ADJ
ejpam-4618	100	9	ku	ku	NOUN
ejpam-4618	100	10	-	-	PUNCT
ejpam-4618	100	11	subalgebra	subalgebra	NOUN
ejpam-4618	100	12	of	of	ADP
ejpam-4618	100	13	x	x	PUNCT
ejpam-4618	100	14	satisfies	satisfie	NOUN
ejpam-4618	100	15	the	the	DET
ejpam-4618	100	16	condition	condition	NOUN
ejpam-4618	100	17	in	in	ADP
ejpam-4618	100	18	proposition	proposition	NOUN
ejpam-4618	100	19	4	4	NUM
ejpam-4618	100	20	,	,	PUNCT
ejpam-4618	100	21	then	then	ADV
ejpam-4618	100	22	ô	ô	NUM
ejpam-4618	100	23	is	be	AUX
ejpam-4618	100	24	an	an	DET
ejpam-4618	100	25	m	m	ADJ
ejpam-4618	100	26	-	-	ADJ
ejpam-4618	100	27	polar	polar	ADJ
ejpam-4618	100	28	fuzzy	fuzzy	ADJ
ejpam-4618	100	29	ku	ku	NOUN
ejpam-4618	100	30	-	-	PUNCT
ejpam-4618	100	31	ideal	ideal	NOUN
ejpam-4618	100	32	of	of	ADP
ejpam-4618	100	33	x.	x.	NOUN
ejpam-4618	100	34	proof	proof	NOUN
ejpam-4618	100	35	.	.	PUNCT
ejpam-4618	101	1	let	let	VERB
ejpam-4618	101	2	ô	ô	X
ejpam-4618	101	3	be	be	AUX
ejpam-4618	101	4	an	an	DET
ejpam-4618	101	5	m	m	ADJ
ejpam-4618	101	6	-	-	ADJ
ejpam-4618	101	7	polar	polar	ADJ
ejpam-4618	101	8	fuzzy	fuzzy	ADJ
ejpam-4618	101	9	ku	ku	NOUN
ejpam-4618	101	10	-	-	PUNCT
ejpam-4618	101	11	subalgebra	subalgebra	NOUN
ejpam-4618	101	12	of	of	ADP
ejpam-4618	101	13	x	x	PUNCT
ejpam-4618	101	14	satisfies	satisfie	NOUN
ejpam-4618	101	15	the	the	DET
ejpam-4618	101	16	condition	condition	NOUN
ejpam-4618	101	17	in	in	ADP
ejpam-4618	101	18	proposition	proposition	NOUN
ejpam-4618	101	19	4	4	NUM
ejpam-4618	101	20	and	and	CCONJ
ejpam-4618	101	21	by	by	ADP
ejpam-4618	101	22	lemma	lemma	PROPN
ejpam-4618	101	23	2	2	NUM
ejpam-4618	101	24	.	.	X
ejpam-4618	102	1	we	we	PRON
ejpam-4618	102	2	have	have	VERB
ejpam-4618	102	3	ô(0	ô(0	NUM
ejpam-4618	102	4	)	)	PUNCT
ejpam-4618	102	5	≥	≥	NOUN
ejpam-4618	102	6	ô(u	ô(u	NOUN
ejpam-4618	102	7	)	)	PUNCT
ejpam-4618	102	8	for	for	ADP
ejpam-4618	102	9	all	all	PRON
ejpam-4618	102	10	u	u	PROPN
ejpam-4618	102	11	∈	∈	NOUN
ejpam-4618	102	12	x.	x.	NOUN
ejpam-4618	102	13	by	by	ADP
ejpam-4618	102	14	theorem	theorem	NOUN
ejpam-4618	102	15	1(3	1(3	NUM
ejpam-4618	102	16	)	)	PUNCT
ejpam-4618	102	17	,	,	PUNCT
ejpam-4618	102	18	we	we	PRON
ejpam-4618	102	19	have	have	VERB
ejpam-4618	102	20	(	(	PUNCT
ejpam-4618	102	21	u	u	NOUN
ejpam-4618	102	22	∗	∗	X
ejpam-4618	102	23	(	(	PUNCT
ejpam-4618	102	24	v	v	NOUN
ejpam-4618	102	25	∗	∗	NOUN
ejpam-4618	102	26	w	w	NOUN
ejpam-4618	102	27	)	)	PUNCT
ejpam-4618	102	28	)	)	PUNCT
ejpam-4618	102	29	∗	∗	NOUN
ejpam-4618	102	30	(	(	PUNCT
ejpam-4618	102	31	u	u	NOUN
ejpam-4618	102	32	∗	∗	PROPN
ejpam-4618	102	33	w	w	NOUN
ejpam-4618	102	34	)	)	PUNCT
ejpam-4618	102	35	≤	≤	NUM
ejpam-4618	102	36	v	v	NOUN
ejpam-4618	102	37	,	,	PUNCT
ejpam-4618	102	38	for	for	ADP
ejpam-4618	102	39	all	all	DET
ejpam-4618	102	40	u	u	NOUN
ejpam-4618	102	41	,	,	PUNCT
ejpam-4618	102	42	v	v	NOUN
ejpam-4618	102	43	,	,	PUNCT
ejpam-4618	102	44	w	w	PROPN
ejpam-4618	102	45	∈	∈	PROPN
ejpam-4618	102	46	x.	x.	NOUN
ejpam-4618	103	1	it	it	PRON
ejpam-4618	103	2	follows	follow	VERB
ejpam-4618	103	3	from	from	ADP
ejpam-4618	103	4	proposition	proposition	NOUN
ejpam-4618	103	5	4	4	NUM
ejpam-4618	103	6	,	,	PUNCT
ejpam-4618	103	7	that	that	SCONJ
ejpam-4618	103	8	ô(u	ô(u	ADP
ejpam-4618	103	9	∗w	∗w	PROPN
ejpam-4618	103	10	)	)	PUNCT
ejpam-4618	103	11	≥	≥	NOUN
ejpam-4618	103	12	min{ô(u	min{ô(u	PROPN
ejpam-4618	103	13	∗	∗	NOUN
ejpam-4618	103	14	(	(	PUNCT
ejpam-4618	103	15	v	v	NOUN
ejpam-4618	103	16	∗w	∗w	ADJ
ejpam-4618	103	17	)	)	PUNCT
ejpam-4618	103	18	)	)	PUNCT
ejpam-4618	103	19	,	,	PUNCT
ejpam-4618	103	20	ô(v	ô(v	ADV
ejpam-4618	103	21	)	)	PUNCT
ejpam-4618	103	22	}	}	PUNCT
ejpam-4618	103	23	for	for	ADP
ejpam-4618	103	24	all	all	DET
ejpam-4618	103	25	u	u	NOUN
ejpam-4618	103	26	,	,	PUNCT
ejpam-4618	103	27	v	v	NOUN
ejpam-4618	103	28	,	,	PUNCT
ejpam-4618	103	29	w	w	PROPN
ejpam-4618	103	30	∈	∈	PROPN
ejpam-4618	103	31	x.	x.	NOUN
ejpam-4618	103	32	therefore	therefore	ADV
ejpam-4618	103	33	,	,	PUNCT
ejpam-4618	103	34	ô	ô	NUM
ejpam-4618	103	35	is	be	AUX
ejpam-4618	103	36	an	an	DET
ejpam-4618	103	37	m	m	ADJ
ejpam-4618	103	38	-	-	ADJ
ejpam-4618	103	39	polar	polar	ADJ
ejpam-4618	103	40	fuzzy	fuzzy	ADJ
ejpam-4618	103	41	ku	ku	NOUN
ejpam-4618	103	42	-	-	PUNCT
ejpam-4618	103	43	ideal	ideal	NOUN
ejpam-4618	103	44	of	of	ADP
ejpam-4618	103	45	x.	x.	NOUN
ejpam-4618	103	46	proposition	proposition	NOUN
ejpam-4618	103	47	5	5	NUM
ejpam-4618	103	48	.	.	PUNCT
ejpam-4618	104	1	every	every	DET
ejpam-4618	104	2	m	m	ADJ
ejpam-4618	104	3	-	-	ADJ
ejpam-4618	104	4	polar	polar	ADJ
ejpam-4618	104	5	fuzzy	fuzzy	ADJ
ejpam-4618	104	6	ku	ku	NOUN
ejpam-4618	104	7	-	-	PUNCT
ejpam-4618	104	8	ideal	ideal	NOUN
ejpam-4618	104	9	of	of	ADP
ejpam-4618	104	10	x	x	SYM
ejpam-4618	104	11	is	be	AUX
ejpam-4618	104	12	an	an	DET
ejpam-4618	104	13	m	m	ADJ
ejpam-4618	104	14	-	-	ADJ
ejpam-4618	104	15	polar	polar	ADJ
ejpam-4618	104	16	fuzzy	fuzzy	ADJ
ejpam-4618	104	17	ideal	ideal	NOUN
ejpam-4618	104	18	.	.	PUNCT
ejpam-4618	105	1	proof	proof	NOUN
ejpam-4618	105	2	.	.	PUNCT
ejpam-4618	106	1	straightforward	straightforward	ADJ
ejpam-4618	106	2	.	.	PUNCT
ejpam-4618	107	1	proposition	proposition	NOUN
ejpam-4618	107	2	6	6	NUM
ejpam-4618	107	3	.	.	PUNCT
ejpam-4618	108	1	if	if	SCONJ
ejpam-4618	108	2	ô	ô	NUM
ejpam-4618	108	3	is	be	AUX
ejpam-4618	108	4	an	an	DET
ejpam-4618	108	5	m	m	ADJ
ejpam-4618	108	6	-	-	ADJ
ejpam-4618	108	7	polar	polar	ADJ
ejpam-4618	108	8	fuzzy	fuzzy	ADJ
ejpam-4618	108	9	ku	ku	NOUN
ejpam-4618	108	10	-	-	PUNCT
ejpam-4618	108	11	ideal	ideal	NOUN
ejpam-4618	108	12	of	of	ADP
ejpam-4618	108	13	x	x	PRON
ejpam-4618	108	14	,	,	PUNCT
ejpam-4618	108	15	then	then	ADV
ejpam-4618	108	16	(	(	PUNCT
ejpam-4618	108	17	ô(u	ô(u	PROPN
ejpam-4618	108	18	∗	∗	X
ejpam-4618	108	19	(	(	PUNCT
ejpam-4618	108	20	u	u	NOUN
ejpam-4618	108	21	∗	∗	NOUN
ejpam-4618	108	22	v	v	NOUN
ejpam-4618	108	23	)	)	PUNCT
ejpam-4618	108	24	)	)	PUNCT
ejpam-4618	108	25	≥	≥	X
ejpam-4618	108	26	ô(v))(∀u	ô(v))(∀u	PROPN
ejpam-4618	108	27	,	,	PUNCT
ejpam-4618	108	28	v	v	NOUN
ejpam-4618	108	29	∈	∈	PROPN
ejpam-4618	108	30	x	x	X
ejpam-4618	108	31	)	)	PUNCT
ejpam-4618	108	32	(	(	PUNCT
ejpam-4618	108	33	8)	8)	NUM
ejpam-4618	108	34	that	that	PRON
ejpam-4618	108	35	is	be	AUX
ejpam-4618	108	36	,	,	PUNCT
ejpam-4618	108	37	(	(	PUNCT
ejpam-4618	108	38	(	(	PUNCT
ejpam-4618	108	39	πi	πi	ADP
ejpam-4618	108	40	◦	◦	VERB
ejpam-4618	108	41	ô)(u	ô)(u	ADJ
ejpam-4618	108	42	∗	∗	NOUN
ejpam-4618	108	43	(	(	PUNCT
ejpam-4618	108	44	u	u	NOUN
ejpam-4618	108	45	∗	∗	NOUN
ejpam-4618	108	46	v	v	NOUN
ejpam-4618	108	47	)	)	PUNCT
ejpam-4618	108	48	)	)	PUNCT
ejpam-4618	108	49	≥	≥	NOUN
ejpam-4618	108	50	(	(	PUNCT
ejpam-4618	108	51	πi	πi	ADP
ejpam-4618	108	52	◦	◦	VERB
ejpam-4618	108	53	ô)(v))(∀u	ô)(v))(∀u	ADJ
ejpam-4618	108	54	,	,	PUNCT
ejpam-4618	108	55	v	v	ADP
ejpam-4618	108	56	∈	∈	PROPN
ejpam-4618	108	57	x	x	NOUN
ejpam-4618	108	58	,	,	PUNCT
ejpam-4618	108	59	i	i	NOUN
ejpam-4618	108	60	=	=	NOUN
ejpam-4618	108	61	1	1	NUM
ejpam-4618	108	62	,	,	PUNCT
ejpam-4618	108	63	2	2	NUM
ejpam-4618	108	64	,	,	PUNCT
ejpam-4618	108	65	...	...	PUNCT
ejpam-4618	108	66	,	,	PUNCT
ejpam-4618	108	67	m	m	NOUN
ejpam-4618	108	68	)	)	PUNCT
ejpam-4618	108	69	references	reference	VERB
ejpam-4618	108	70	259	259	NUM
ejpam-4618	108	71	proof	proof	NOUN
ejpam-4618	108	72	.	.	PUNCT
ejpam-4618	109	1	let	let	VERB
ejpam-4618	109	2	ô	ô	X
ejpam-4618	109	3	be	be	AUX
ejpam-4618	109	4	an	an	DET
ejpam-4618	109	5	m	m	ADJ
ejpam-4618	109	6	-	-	ADJ
ejpam-4618	109	7	polar	polar	ADJ
ejpam-4618	109	8	fuzzy	fuzzy	ADJ
ejpam-4618	109	9	ku	ku	NOUN
ejpam-4618	109	10	-	-	PUNCT
ejpam-4618	109	11	ideal	ideal	NOUN
ejpam-4618	109	12	of	of	ADP
ejpam-4618	109	13	a	a	DET
ejpam-4618	109	14	ku	ku	NOUN
ejpam-4618	109	15	-	-	PUNCT
ejpam-4618	109	16	algebra	algebra	NOUN
ejpam-4618	109	17	x	x	PUNCT
ejpam-4618	109	18	and	and	CCONJ
ejpam-4618	109	19	let	let	VERB
ejpam-4618	109	20	u	u	NOUN
ejpam-4618	109	21	,	,	PUNCT
ejpam-4618	109	22	v	v	NOUN
ejpam-4618	109	23	,	,	PUNCT
ejpam-4618	109	24	w	w	PROPN
ejpam-4618	109	25	∈	∈	PROPN
ejpam-4618	109	26	x.	x.	NOUN
ejpam-4618	109	27	taking	take	VERB
ejpam-4618	109	28	w	w	NOUN
ejpam-4618	109	29	=	=	PUNCT
ejpam-4618	109	30	u	u	PROPN
ejpam-4618	109	31	∗	∗	NOUN
ejpam-4618	109	32	v	v	NOUN
ejpam-4618	109	33	in	in	ADP
ejpam-4618	109	34	(	(	PUNCT
ejpam-4618	109	35	4	4	NUM
ejpam-4618	109	36	)	)	PUNCT
ejpam-4618	109	37	and	and	CCONJ
ejpam-4618	109	38	using	use	VERB
ejpam-4618	109	39	(	(	PUNCT
ejpam-4618	109	40	ku	ku	PROPN
ejpam-4618	109	41	2	2	NUM
ejpam-4618	109	42	)	)	PUNCT
ejpam-4618	109	43	,	,	PUNCT
ejpam-4618	109	44	we	we	PRON
ejpam-4618	109	45	get	get	VERB
ejpam-4618	109	46	ô(u	ô(u	ADP
ejpam-4618	109	47	∗	∗	X
ejpam-4618	109	48	(	(	PUNCT
ejpam-4618	109	49	u	u	NOUN
ejpam-4618	109	50	∗	∗	PROPN
ejpam-4618	109	51	v	v	NOUN
ejpam-4618	109	52	)	)	PUNCT
ejpam-4618	109	53	≥	≥	NOUN
ejpam-4618	109	54	min{ô(u	min{ô(u	PROPN
ejpam-4618	109	55	∗	∗	NOUN
ejpam-4618	109	56	(	(	PUNCT
ejpam-4618	109	57	v	v	NOUN
ejpam-4618	109	58	∗	∗	NOUN
ejpam-4618	109	59	(	(	PUNCT
ejpam-4618	109	60	u	u	NOUN
ejpam-4618	109	61	∗	∗	NOUN
ejpam-4618	109	62	v	v	NOUN
ejpam-4618	109	63	)	)	PUNCT
ejpam-4618	109	64	)	)	PUNCT
ejpam-4618	109	65	)	)	PUNCT
ejpam-4618	109	66	,	,	PUNCT
ejpam-4618	109	67	ô(v	ô(v	ADV
ejpam-4618	109	68	)	)	PUNCT
ejpam-4618	109	69	}	}	PUNCT
ejpam-4618	109	70	=	=	SYM
ejpam-4618	110	1	min{ô(u	min{ô(u	PROPN
ejpam-4618	110	2	∗	∗	NOUN
ejpam-4618	110	3	(	(	PUNCT
ejpam-4618	110	4	u	u	NOUN
ejpam-4618	110	5	∗	∗	X
ejpam-4618	110	6	(	(	PUNCT
ejpam-4618	110	7	v	v	NOUN
ejpam-4618	110	8	∗	∗	NOUN
ejpam-4618	110	9	v	v	NOUN
ejpam-4618	110	10	)	)	PUNCT
ejpam-4618	110	11	)	)	PUNCT
ejpam-4618	110	12	)	)	PUNCT
ejpam-4618	110	13	,	,	PUNCT
ejpam-4618	110	14	ô(v	ô(v	ADV
ejpam-4618	110	15	)	)	PUNCT
ejpam-4618	110	16	}	}	PUNCT
ejpam-4618	110	17	=	=	SYM
ejpam-4618	111	1	min{ô(u	min{ô(u	PROPN
ejpam-4618	111	2	∗	∗	NOUN
ejpam-4618	111	3	(	(	PUNCT
ejpam-4618	111	4	u	u	NOUN
ejpam-4618	111	5	∗	∗	NOUN
ejpam-4618	111	6	0	0	NUM
ejpam-4618	111	7	)	)	PUNCT
ejpam-4618	111	8	)	)	PUNCT
ejpam-4618	111	9	,	,	PUNCT
ejpam-4618	111	10	ô(v	ô(v	ADV
ejpam-4618	111	11	)	)	PUNCT
ejpam-4618	111	12	}	}	PUNCT
ejpam-4618	111	13	=	=	SYM
ejpam-4618	111	14	min{ô(0	min{ô(0	NOUN
ejpam-4618	111	15	)	)	PUNCT
ejpam-4618	111	16	)	)	PUNCT
ejpam-4618	111	17	,	,	PUNCT
ejpam-4618	111	18	ô(v	ô(v	ADV
ejpam-4618	111	19	)	)	PUNCT
ejpam-4618	111	20	}	}	PUNCT
ejpam-4618	111	21	=	=	SYM
ejpam-4618	111	22	ô(v	ô(v	ADV
ejpam-4618	111	23	)	)	PUNCT
ejpam-4618	111	24	theorem	theorem	NOUN
ejpam-4618	111	25	5	5	NUM
ejpam-4618	111	26	.	.	PUNCT
ejpam-4618	112	1	if	if	SCONJ
ejpam-4618	112	2	ô	ô	NUM
ejpam-4618	112	3	is	be	AUX
ejpam-4618	112	4	an	an	DET
ejpam-4618	112	5	m	m	ADJ
ejpam-4618	112	6	-	-	ADJ
ejpam-4618	112	7	polar	polar	ADJ
ejpam-4618	112	8	fuzzy	fuzzy	ADJ
ejpam-4618	112	9	ku	ku	NOUN
ejpam-4618	112	10	-	-	PUNCT
ejpam-4618	112	11	ideal	ideal	NOUN
ejpam-4618	112	12	of	of	ADP
ejpam-4618	112	13	x	x	PRON
ejpam-4618	112	14	,	,	PUNCT
ejpam-4618	112	15	then	then	ADV
ejpam-4618	112	16	the	the	DET
ejpam-4618	112	17	set	set	NOUN
ejpam-4618	112	18	b	b	NOUN
ejpam-4618	112	19	=	=	SYM
ejpam-4618	112	20	{	{	PUNCT
ejpam-4618	112	21	u	u	NOUN
ejpam-4618	112	22	∈	∈	PROPN
ejpam-4618	112	23	x	x	X
ejpam-4618	112	24	:	:	PUNCT
ejpam-4618	112	25	ô(u	ô(u	NOUN
ejpam-4618	112	26	)	)	PUNCT
ejpam-4618	112	27	=	=	SYM
ejpam-4618	112	28	ô(0	ô(0	NUM
ejpam-4618	112	29	)	)	PUNCT
ejpam-4618	112	30	}	}	PUNCT
ejpam-4618	112	31	is	be	AUX
ejpam-4618	112	32	an	an	DET
ejpam-4618	112	33	m	m	ADJ
ejpam-4618	112	34	-	-	ADJ
ejpam-4618	112	35	polar	polar	ADJ
ejpam-4618	112	36	ku	ku	NOUN
ejpam-4618	112	37	-	-	PUNCT
ejpam-4618	112	38	ideal	ideal	NOUN
ejpam-4618	112	39	.	.	PUNCT
ejpam-4618	113	1	proof	proof	NOUN
ejpam-4618	113	2	.	.	PUNCT
ejpam-4618	114	1	since	since	SCONJ
ejpam-4618	114	2	0	0	NUM
ejpam-4618	114	3	∈	∈	PROPN
ejpam-4618	114	4	x	x	NOUN
ejpam-4618	114	5	,	,	PUNCT
ejpam-4618	114	6	then	then	ADV
ejpam-4618	114	7	ô(0	ô(0	NUM
ejpam-4618	114	8	)	)	PUNCT
ejpam-4618	114	9	=	=	SYM
ejpam-4618	114	10	ô(0	ô(0	NOUN
ejpam-4618	114	11	)	)	PUNCT
ejpam-4618	114	12	implies	imply	VERB
ejpam-4618	114	13	0	0	NUM
ejpam-4618	114	14	∈	∈	PROPN
ejpam-4618	114	15	b	b	NOUN
ejpam-4618	114	16	,	,	PUNCT
ejpam-4618	114	17	so	so	PROPN
ejpam-4618	114	18	b	b	PROPN
ejpam-4618	114	19	̸=	̸=	PROPN
ejpam-4618	114	20	ϕ.	ϕ.	NOUN
ejpam-4618	114	21	let	let	VERB
ejpam-4618	114	22	u	u	PRON
ejpam-4618	114	23	∗	∗	X
ejpam-4618	114	24	(	(	PUNCT
ejpam-4618	114	25	v	v	NOUN
ejpam-4618	114	26	∗	∗	NOUN
ejpam-4618	114	27	w	w	NOUN
ejpam-4618	114	28	)	)	PUNCT
ejpam-4618	114	29	∈	∈	PROPN
ejpam-4618	114	30	b	b	PROPN
ejpam-4618	114	31	and	and	CCONJ
ejpam-4618	114	32	v	v	ADP
ejpam-4618	114	33	∈	∈	PROPN
ejpam-4618	114	34	b	b	PROPN
ejpam-4618	114	35	implies	imply	VERB
ejpam-4618	114	36	ô(u	ô(u	PROPN
ejpam-4618	114	37	∗	∗	X
ejpam-4618	114	38	(	(	PUNCT
ejpam-4618	114	39	v	v	NOUN
ejpam-4618	114	40	∗	∗	NOUN
ejpam-4618	114	41	w	w	NOUN
ejpam-4618	114	42	)	)	PUNCT
ejpam-4618	114	43	)	)	PUNCT
ejpam-4618	115	1	=	=	SYM
ejpam-4618	115	2	ô(0	ô(0	NOUN
ejpam-4618	115	3	)	)	PUNCT
ejpam-4618	115	4	and	and	CCONJ
ejpam-4618	115	5	ô(v	ô(v	X
ejpam-4618	115	6	)	)	PUNCT
ejpam-4618	115	7	=	=	SYM
ejpam-4618	115	8	ô(0	ô(0	NUM
ejpam-4618	115	9	)	)	PUNCT
ejpam-4618	115	10	.	.	PUNCT
ejpam-4618	116	1	since	since	SCONJ
ejpam-4618	116	2	ô	ô	NUM
ejpam-4618	116	3	is	be	AUX
ejpam-4618	116	4	an	an	DET
ejpam-4618	116	5	m	m	ADJ
ejpam-4618	116	6	-	-	ADJ
ejpam-4618	116	7	polar	polar	ADJ
ejpam-4618	116	8	fuzzy	fuzzy	ADJ
ejpam-4618	116	9	ku	ku	NOUN
ejpam-4618	116	10	-	-	PUNCT
ejpam-4618	116	11	ideal	ideal	NOUN
ejpam-4618	116	12	of	of	ADP
ejpam-4618	116	13	x	x	PRON
ejpam-4618	116	14	,	,	PUNCT
ejpam-4618	116	15	then	then	ADV
ejpam-4618	116	16	(	(	PUNCT
ejpam-4618	116	17	ô(u	ô(u	ADP
ejpam-4618	116	18	∗	∗	NOUN
ejpam-4618	116	19	w	w	NOUN
ejpam-4618	116	20	)	)	PUNCT
ejpam-4618	116	21	)	)	PUNCT
ejpam-4618	116	22	≥	≥	NOUN
ejpam-4618	116	23	min{ô(u	min{ô(u	PROPN
ejpam-4618	116	24	∗	∗	NOUN
ejpam-4618	116	25	(	(	PUNCT
ejpam-4618	116	26	v	v	NOUN
ejpam-4618	116	27	∗	∗	NOUN
ejpam-4618	116	28	w	w	NOUN
ejpam-4618	116	29	)	)	PUNCT
ejpam-4618	116	30	)	)	PUNCT
ejpam-4618	116	31	,	,	PUNCT
ejpam-4618	116	32	ô(v	ô(v	ADV
ejpam-4618	116	33	)	)	PUNCT
ejpam-4618	116	34	}	}	PUNCT
ejpam-4618	116	35	=	=	PUNCT
ejpam-4618	116	36	ô(0	ô(0	NUM
ejpam-4618	116	37	)	)	PUNCT
ejpam-4618	116	38	.	.	PUNCT
ejpam-4618	117	1	but	but	CCONJ
ejpam-4618	117	2	ô(0	ô(0	NUM
ejpam-4618	117	3	)	)	PUNCT
ejpam-4618	117	4	≥	≥	NOUN
ejpam-4618	117	5	ô(u∗w	ô(u∗w	NOUN
ejpam-4618	117	6	)	)	PUNCT
ejpam-4618	117	7	.	.	PUNCT
ejpam-4618	118	1	then	then	ADV
ejpam-4618	118	2	ô(0	ô(0	NUM
ejpam-4618	118	3	)	)	PUNCT
ejpam-4618	119	1	=	=	SYM
ejpam-4618	119	2	ô(u∗w	ô(u∗w	NOUN
ejpam-4618	119	3	)	)	PUNCT
ejpam-4618	119	4	,	,	PUNCT
ejpam-4618	119	5	it	it	PRON
ejpam-4618	119	6	follows	follow	VERB
ejpam-4618	119	7	that	that	SCONJ
ejpam-4618	119	8	u∗w	u∗w	PROPN
ejpam-4618	119	9	∈	∈	PROPN
ejpam-4618	119	10	b	b	NOUN
ejpam-4618	119	11	,	,	PUNCT
ejpam-4618	119	12	for	for	ADP
ejpam-4618	119	13	all	all	DET
ejpam-4618	119	14	u	u	NOUN
ejpam-4618	119	15	,	,	PUNCT
ejpam-4618	119	16	v	v	NOUN
ejpam-4618	119	17	,	,	PUNCT
ejpam-4618	119	18	w	w	PROPN
ejpam-4618	119	19	∈	∈	PROPN
ejpam-4618	119	20	x.	x.	NOUN
ejpam-4618	119	21	hence	hence	ADV
ejpam-4618	119	22	,	,	PUNCT
ejpam-4618	119	23	the	the	DET
ejpam-4618	119	24	set	set	PROPN
ejpam-4618	119	25	b	b	PROPN
ejpam-4618	119	26	is	be	AUX
ejpam-4618	119	27	an	an	DET
ejpam-4618	119	28	m	m	ADJ
ejpam-4618	119	29	-	-	ADJ
ejpam-4618	119	30	polar	polar	ADJ
ejpam-4618	119	31	ku	ku	NOUN
ejpam-4618	119	32	-	-	PUNCT
ejpam-4618	119	33	ideal	ideal	NOUN
ejpam-4618	119	34	.	.	PUNCT
ejpam-4618	120	1	4	4	X
ejpam-4618	120	2	.	.	X
ejpam-4618	120	3	conclusion	conclusion	NOUN
ejpam-4618	120	4	an	an	DET
ejpam-4618	120	5	m	m	ADJ
ejpam-4618	120	6	-	-	ADJ
ejpam-4618	120	7	polar	polar	ADJ
ejpam-4618	120	8	fuzzy	fuzzy	ADJ
ejpam-4618	120	9	model	model	NOUN
ejpam-4618	120	10	is	be	AUX
ejpam-4618	120	11	a	a	DET
ejpam-4618	120	12	generalized	generalized	ADJ
ejpam-4618	120	13	form	form	NOUN
ejpam-4618	120	14	of	of	ADP
ejpam-4618	120	15	a	a	DET
ejpam-4618	120	16	bipolar	bipolar	ADJ
ejpam-4618	120	17	fuzzy	fuzzy	ADJ
ejpam-4618	120	18	model	model	NOUN
ejpam-4618	120	19	.	.	PUNCT
ejpam-4618	121	1	the	the	DET
ejpam-4618	121	2	m	m	ADJ
ejpam-4618	121	3	-	-	ADJ
ejpam-4618	121	4	polar	polar	ADJ
ejpam-4618	121	5	fuzzy	fuzzy	ADJ
ejpam-4618	121	6	models	model	NOUN
ejpam-4618	121	7	provide	provide	VERB
ejpam-4618	121	8	more	more	ADJ
ejpam-4618	121	9	precision	precision	NOUN
ejpam-4618	121	10	,	,	PUNCT
ejpam-4618	121	11	flexibility	flexibility	NOUN
ejpam-4618	121	12	and	and	CCONJ
ejpam-4618	121	13	compatibility	compatibility	NOUN
ejpam-4618	121	14	to	to	ADP
ejpam-4618	121	15	the	the	DET
ejpam-4618	121	16	system	system	NOUN
ejpam-4618	121	17	when	when	SCONJ
ejpam-4618	121	18	more	more	ADJ
ejpam-4618	121	19	than	than	ADP
ejpam-4618	121	20	one	one	NUM
ejpam-4618	121	21	agreement	agreement	NOUN
ejpam-4618	121	22	is	be	AUX
ejpam-4618	121	23	to	to	PART
ejpam-4618	121	24	be	be	AUX
ejpam-4618	121	25	dealt	deal	VERB
ejpam-4618	121	26	with	with	ADP
ejpam-4618	121	27	.	.	PUNCT
ejpam-4618	122	1	this	this	DET
ejpam-4618	122	2	article	article	NOUN
ejpam-4618	122	3	discussed	discuss	VERB
ejpam-4618	122	4	the	the	DET
ejpam-4618	122	5	ku	ku	NOUN
ejpam-4618	122	6	-	-	PUNCT
ejpam-4618	122	7	ideal	ideal	NOUN
ejpam-4618	122	8	of	of	ADP
ejpam-4618	122	9	kualgebras	kualgebra	NOUN
ejpam-4618	122	10	based	base	VERB
ejpam-4618	122	11	on	on	ADP
ejpam-4618	122	12	m	m	ADJ
ejpam-4618	122	13	-	-	ADJ
ejpam-4618	122	14	polar	polar	ADJ
ejpam-4618	122	15	fuzzy	fuzzy	ADJ
ejpam-4618	122	16	sets	set	NOUN
ejpam-4618	122	17	.	.	PUNCT
ejpam-4618	123	1	the	the	DET
ejpam-4618	123	2	notions	notion	NOUN
ejpam-4618	123	3	of	of	ADP
ejpam-4618	123	4	m	m	NOUN
ejpam-4618	123	5	-	-	ADJ
ejpam-4618	123	6	polar	polar	ADJ
ejpam-4618	123	7	fuzzy	fuzzy	ADJ
ejpam-4618	123	8	ku	ku	NOUN
ejpam-4618	123	9	-	-	PUNCT
ejpam-4618	123	10	subalgebras	subalgebras	PROPN
ejpam-4618	123	11	and	and	CCONJ
ejpam-4618	123	12	an	an	DET
ejpam-4618	123	13	m	m	ADJ
ejpam-4618	123	14	-	-	ADJ
ejpam-4618	123	15	polar	polar	ADJ
ejpam-4618	123	16	fuzzy	fuzzy	ADJ
ejpam-4618	123	17	ku	ku	NOUN
ejpam-4618	123	18	-	-	PUNCT
ejpam-4618	123	19	ideals	ideal	NOUN
ejpam-4618	123	20	were	be	AUX
ejpam-4618	123	21	introduced	introduce	VERB
ejpam-4618	123	22	,	,	PUNCT
ejpam-4618	123	23	and	and	CCONJ
ejpam-4618	123	24	several	several	ADJ
ejpam-4618	123	25	properties	property	NOUN
ejpam-4618	123	26	were	be	AUX
ejpam-4618	123	27	investigated	investigate	VERB
ejpam-4618	123	28	.	.	PUNCT
ejpam-4618	124	1	5	5	X
ejpam-4618	124	2	.	.	X
ejpam-4618	124	3	compliance	compliance	NOUN
ejpam-4618	124	4	with	with	ADP
ejpam-4618	124	5	ethical	ethical	ADJ
ejpam-4618	124	6	standards	standard	NOUN
ejpam-4618	124	7	conflict	conflict	NOUN
ejpam-4618	124	8	of	of	ADP
ejpam-4618	124	9	interest	interest	NOUN
ejpam-4618	124	10	:	:	PUNCT
ejpam-4618	124	11	the	the	DET
ejpam-4618	124	12	author	author	NOUN
ejpam-4618	124	13	declares	declare	VERB
ejpam-4618	124	14	that	that	SCONJ
ejpam-4618	124	15	there	there	PRON
ejpam-4618	124	16	is	be	VERB
ejpam-4618	124	17	no	no	DET
ejpam-4618	124	18	conflict	conflict	NOUN
ejpam-4618	124	19	of	of	ADP
ejpam-4618	124	20	interest	interest	NOUN
ejpam-4618	124	21	regarding	regard	VERB
ejpam-4618	124	22	the	the	DET
ejpam-4618	124	23	publication	publication	NOUN
ejpam-4618	124	24	of	of	ADP
ejpam-4618	124	25	this	this	DET
ejpam-4618	124	26	paper	paper	NOUN
ejpam-4618	124	27	.	.	PUNCT
ejpam-4618	125	1	references	reference	NOUN
ejpam-4618	125	2	[	[	X
ejpam-4618	125	3	1	1	NUM
ejpam-4618	125	4	]	]	X
ejpam-4618	125	5	akram	akram	PROPN
ejpam-4618	125	6	and	and	CCONJ
ejpam-4618	125	7	farooq	farooq	PROPN
ejpam-4618	125	8	,	,	PUNCT
ejpam-4618	125	9	m	m	ADJ
ejpam-4618	125	10	-	-	ADJ
ejpam-4618	125	11	polar	polar	ADJ
ejpam-4618	125	12	fuzzy	fuzzy	ADJ
ejpam-4618	125	13	lie	lie	NOUN
ejpam-4618	125	14	ideals	ideal	NOUN
ejpam-4618	125	15	of	of	ADP
ejpam-4618	125	16	lie	lie	NOUN
ejpam-4618	125	17	algebras	algebra	NOUN
ejpam-4618	125	18	.	.	PUNCT
ejpam-4618	126	1	quasigroups	quasigroups	PROPN
ejpam-4618	126	2	relat	relat	PROPN
ejpam-4618	126	3	.	.	PUNCT
ejpam-4618	127	1	syst	syst	PROPN
ejpam-4618	127	2	.	.	PUNCT
ejpam-4618	128	1	24(2	24(2	NUM
ejpam-4618	128	2	)	)	PUNCT
ejpam-4618	128	3	,	,	PUNCT
ejpam-4618	128	4	141	141	NUM
ejpam-4618	128	5	-	-	SYM
ejpam-4618	128	6	150	150	NUM
ejpam-4618	128	7	,	,	PUNCT
ejpam-4618	128	8	2016	2016	NUM
ejpam-4618	128	9	.	.	PUNCT
ejpam-4618	129	1	[	[	X
ejpam-4618	129	2	2	2	NUM
ejpam-4618	129	3	]	]	X
ejpam-4618	129	4	akram	akram	PROPN
ejpam-4618	129	5	,	,	PUNCT
ejpam-4618	129	6	farooq	farooq	PROPN
ejpam-4618	129	7	,	,	PUNCT
ejpam-4618	129	8	and	and	CCONJ
ejpam-4618	129	9	shum	shum	ADJ
ejpam-4618	129	10	,	,	PUNCT
ejpam-4618	129	11	on	on	ADP
ejpam-4618	129	12	m	m	ADJ
ejpam-4618	129	13	-	-	ADJ
ejpam-4618	129	14	polar	polar	ADJ
ejpam-4618	129	15	fuzzy	fuzzy	ADJ
ejpam-4618	129	16	lie	lie	NOUN
ejpam-4618	129	17	subalgebras	subalgebras	PROPN
ejpam-4618	129	18	,	,	PUNCT
ejpam-4618	129	19	ital	ital	PROPN
ejpam-4618	129	20	.	.	PUNCT
ejpam-4618	130	1	j.	j.	PROPN
ejpam-4618	130	2	pure	pure	PROPN
ejpam-4618	130	3	appl	appl	PROPN
ejpam-4618	130	4	.	.	PUNCT
ejpam-4618	131	1	math	math	NOUN
ejpam-4618	131	2	,	,	PUNCT
ejpam-4618	131	3	36	36	NUM
ejpam-4618	131	4	,	,	PUNCT
ejpam-4618	131	5	445	445	NUM
ejpam-4618	131	6	-	-	SYM
ejpam-4618	131	7	454	454	NUM
ejpam-4618	131	8	,	,	PUNCT
ejpam-4618	131	9	2016	2016	NUM
ejpam-4618	131	10	.	.	PUNCT
ejpam-4618	132	1	[	[	X
ejpam-4618	132	2	3	3	NUM
ejpam-4618	132	3	]	]	X
ejpam-4618	132	4	akram	akram	NOUN
ejpam-4618	132	5	and	and	CCONJ
ejpam-4618	132	6	sarwar	sarwar	PROPN
ejpam-4618	132	7	.	.	PUNCT
ejpam-4618	133	1	,	,	PUNCT
ejpam-4618	133	2	new	new	ADJ
ejpam-4618	133	3	applications	application	NOUN
ejpam-4618	133	4	of	of	ADP
ejpam-4618	133	5	m	m	ADJ
ejpam-4618	133	6	-	-	ADJ
ejpam-4618	133	7	polar	polar	ADJ
ejpam-4618	133	8	fuzzy	fuzzy	ADJ
ejpam-4618	133	9	competition	competition	NOUN
ejpam-4618	133	10	graphs	graph	NOUN
ejpam-4618	133	11	.	.	PUNCT
ejpam-4618	134	1	,	,	PUNCT
ejpam-4618	134	2	new	new	ADJ
ejpam-4618	134	3	math	math	NOUN
ejpam-4618	134	4	.	.	PUNCT
ejpam-4618	135	1	nat	nat	PROPN
ejpam-4618	135	2	.	.	PUNCT
ejpam-4618	136	1	comput	comput	PROPN
ejpam-4618	136	2	.	.	PUNCT
ejpam-4618	136	3	,	,	PUNCT
ejpam-4618	136	4	14(2	14(2	NUM
ejpam-4618	136	5	)	)	PUNCT
ejpam-4618	136	6	,	,	PUNCT
ejpam-4618	136	7	249	249	NUM
ejpam-4618	136	8	-	-	SYM
ejpam-4618	136	9	276	276	NUM
ejpam-4618	136	10	,	,	PUNCT
ejpam-4618	136	11	2018	2018	NUM
ejpam-4618	136	12	.	.	PUNCT
ejpam-4618	137	1	references	reference	NOUN
ejpam-4618	137	2	260	260	NUM
ejpam-4618	138	1	[	[	X
ejpam-4618	138	2	4	4	NUM
ejpam-4618	138	3	]	]	X
ejpam-4618	138	4	al	al	PROPN
ejpam-4618	138	5	-	-	PUNCT
ejpam-4618	138	6	masarwah	masarwah	PROPN
ejpam-4618	138	7	and	and	CCONJ
ejpam-4618	138	8	ahmad	ahmad	PROPN
ejpam-4618	138	9	,	,	PUNCT
ejpam-4618	138	10	m	m	NOUN
ejpam-4618	138	11	-	-	ADJ
ejpam-4618	138	12	polar	polar	ADJ
ejpam-4618	138	13	fuzzy	fuzzy	ADJ
ejpam-4618	138	14	ideals	ideal	NOUN
ejpam-4618	138	15	of	of	ADP
ejpam-4618	138	16	bck	bck	PROPN
ejpam-4618	138	17	/	/	SYM
ejpam-4618	138	18	bci	bci	NOUN
ejpam-4618	138	19	-	-	PUNCT
ejpam-4618	138	20	algebras	algebra	NOUN
ejpam-4618	138	21	.	.	PROPN
ejpam-4618	138	22	,	,	PUNCT
ejpam-4618	138	23	journal	journal	NOUN
ejpam-4618	138	24	of	of	ADP
ejpam-4618	138	25	king	king	PROPN
ejpam-4618	138	26	saud	saud	PROPN
ejpam-4618	138	27	university	university	PROPN
ejpam-4618	138	28	science	science	NOUN
ejpam-4618	138	29	,	,	PUNCT
ejpam-4618	138	30	31	31	NUM
ejpam-4618	138	31	,	,	PUNCT
ejpam-4618	138	32	1220	1220	NUM
ejpam-4618	138	33	-	-	SYM
ejpam-4618	138	34	1226	1226	NUM
ejpam-4618	138	35	.	.	PUNCT
ejpam-4618	139	1	2019	2019	NUM
ejpam-4618	140	1	[	[	SYM
ejpam-4618	140	2	5	5	NUM
ejpam-4618	140	3	]	]	X
ejpam-4618	140	4	chen	chen	PROPN
ejpam-4618	140	5	,	,	PUNCT
ejpam-4618	140	6	li	li	PROPN
ejpam-4618	140	7	,	,	PUNCT
ejpam-4618	140	8	ma	ma	PROPN
ejpam-4618	140	9	and	and	CCONJ
ejpam-4618	140	10	wang	wang	PROPN
ejpam-4618	140	11	,	,	PUNCT
ejpam-4618	140	12	m	m	NOUN
ejpam-4618	140	13	-	-	ADJ
ejpam-4618	140	14	polar	polar	ADJ
ejpam-4618	140	15	fuzzy	fuzzy	ADJ
ejpam-4618	140	16	sets	set	NOUN
ejpam-4618	140	17	:	:	PUNCT
ejpam-4618	140	18	an	an	DET
ejpam-4618	140	19	extension	extension	NOUN
ejpam-4618	140	20	of	of	ADP
ejpam-4618	140	21	bipolar	bipolar	ADJ
ejpam-4618	140	22	fuzzy	fuzzy	ADJ
ejpam-4618	140	23	sets	set	NOUN
ejpam-4618	140	24	.	.	PUNCT
ejpam-4618	140	25	,	,	PUNCT
ejpam-4618	141	1	sci	sci	PROPN
ejpam-4618	141	2	world	world	PROPN
ejpam-4618	141	3	j	j	PROPN
ejpam-4618	141	4	.	.	PUNCT
ejpam-4618	141	5	,	,	PUNCT
ejpam-4618	141	6	416	416	NUM
ejpam-4618	141	7	-	-	SYM
ejpam-4618	141	8	530	530	NUM
ejpam-4618	141	9	2014	2014	NUM
ejpam-4618	141	10	.	.	PUNCT
ejpam-4618	142	1	[	[	X
ejpam-4618	142	2	6	6	NUM
ejpam-4618	142	3	]	]	PUNCT
ejpam-4618	142	4	gulistan	gulistan	PROPN
ejpam-4618	142	5	,	,	PUNCT
ejpam-4618	142	6	shahzad	shahzad	PROPN
ejpam-4618	142	7	and	and	CCONJ
ejpam-4618	142	8	ahmed	ahmed	PROPN
ejpam-4618	142	9	(	(	PUNCT
ejpam-4618	142	10	2014).1	2014).1	NUM
ejpam-4618	142	11	-	-	SYM
ejpam-4618	142	12	11	11	NUM
ejpam-4618	142	13	.	.	PUNCT
ejpam-4618	142	14	,	,	PUNCT
ejpam-4618	142	15	on	on	ADP
ejpam-4618	142	16	(	(	PUNCT
ejpam-4618	142	17	α	α	NOUN
ejpam-4618	142	18	,	,	PUNCT
ejpam-4618	142	19	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-4618	142	20	ku	ku	NOUN
ejpam-4618	142	21	-	-	PUNCT
ejpam-4618	142	22	ideals	ideal	NOUN
ejpam-4618	142	23	of	of	ADP
ejpam-4618	142	24	kualgebras	kualgebra	NOUN
ejpam-4618	142	25	,	,	PUNCT
ejpam-4618	142	26	,	,	PUNCT
ejpam-4618	142	27	afr	afr	PROPN
ejpam-4618	142	28	.	.	PUNCT
ejpam-4618	142	29	mat	mat	PROPN
ejpam-4618	142	30	.	.	PROPN
ejpam-4618	142	31	,	,	PUNCT
ejpam-4618	142	32	,	,	PUNCT
ejpam-4618	142	33	1	1	NUM
ejpam-4618	142	34	-	-	SYM
ejpam-4618	142	35	11	11	NUM
ejpam-4618	142	36	2014	2014	NUM
ejpam-4618	142	37	.	.	PUNCT
ejpam-4618	143	1	[	[	X
ejpam-4618	143	2	7	7	NUM
ejpam-4618	143	3	]	]	X
ejpam-4618	143	4	mostafa	mostafa	PROPN
ejpam-4618	143	5	,	,	PUNCT
ejpam-4618	143	6	abd	abd	PROPN
ejpam-4618	143	7	-	-	PUNCT
ejpam-4618	143	8	elnaby	elnaby	PROPN
ejpam-4618	143	9	and	and	CCONJ
ejpam-4618	143	10	yousef	yousef	PROPN
ejpam-4618	143	11	,	,	PUNCT
ejpam-4618	143	12	fuzzy	fuzzy	ADJ
ejpam-4618	143	13	ideals	ideal	NOUN
ejpam-4618	143	14	of	of	ADP
ejpam-4618	143	15	ku	ku	PROPN
ejpam-4618	143	16	-	-	PUNCT
ejpam-4618	143	17	algebras	algebras	PROPN
ejpam-4618	143	18	.	.	PROPN
ejpam-4618	143	19	,	,	PUNCT
ejpam-4618	143	20	int	int	PROPN
ejpam-4618	143	21	.	.	PUNCT
ejpam-4618	143	22	math	math	PROPN
ejpam-4618	143	23	.	.	PUNCT
ejpam-4618	144	1	forum	forum	PROPN
ejpam-4618	144	2	,	,	PUNCT
ejpam-4618	144	3	6(63	6(63	NUM
ejpam-4618	144	4	)	)	PUNCT
ejpam-4618	144	5	3139	3139	NUM
ejpam-4618	144	6	-	-	SYM
ejpam-4618	144	7	3149	3149	NUM
ejpam-4618	144	8	,	,	PUNCT
ejpam-4618	144	9	2011	2011	NUM
ejpam-4618	144	10	.	.	PUNCT
ejpam-4618	145	1	[	[	X
ejpam-4618	145	2	8	8	NUM
ejpam-4618	145	3	]	]	X
ejpam-4618	145	4	prabpayak	prabpayak	NOUN
ejpam-4618	145	5	and	and	CCONJ
ejpam-4618	145	6	leerawat	leerawat	NOUN
ejpam-4618	145	7	.	.	PUNCT
ejpam-4618	145	8	,	,	PUNCT
ejpam-4618	145	9	on	on	ADP
ejpam-4618	145	10	ideals	ideal	NOUN
ejpam-4618	145	11	and	and	CCONJ
ejpam-4618	145	12	congruence	congruence	NOUN
ejpam-4618	145	13	in	in	ADP
ejpam-4618	145	14	ku	ku	PROPN
ejpam-4618	145	15	-	-	PUNCT
ejpam-4618	145	16	algebras	algebras	PROPN
ejpam-4618	145	17	,	,	PUNCT
ejpam-4618	145	18	,	,	PUNCT
ejpam-4618	145	19	scientia	scientia	PROPN
ejpam-4618	145	20	magna	magna	PROPN
ejpam-4618	145	21	,	,	PUNCT
ejpam-4618	145	22	international	international	ADJ
ejpam-4618	145	23	book	book	NOUN
ejpam-4618	145	24	series	series	NOUN
ejpam-4618	145	25	,	,	PUNCT
ejpam-4618	145	26	,	,	PUNCT
ejpam-4618	145	27	vol.5	vol.5	ADV
ejpam-4618	145	28	,	,	PUNCT
ejpam-4618	145	29	no.1	no.1	NUM
ejpam-4618	145	30	,	,	PUNCT
ejpam-4618	145	31	,	,	PUNCT
ejpam-4618	145	32	54	54	NUM
ejpam-4618	145	33	-	-	SYM
ejpam-4618	145	34	57	57	NUM
ejpam-4618	145	35	,	,	PUNCT
ejpam-4618	145	36	2009	2009	NUM
ejpam-4618	145	37	.	.	PUNCT
ejpam-4618	146	1	[	[	X
ejpam-4618	146	2	9	9	NUM
ejpam-4618	146	3	]	]	X
ejpam-4618	146	4	prabpayak	prabpayak	NOUN
ejpam-4618	146	5	and	and	CCONJ
ejpam-4618	146	6	leerawat	leerawat	VERB
ejpam-4618	146	7	.	.	PUNCT
ejpam-4618	146	8	,	,	PUNCT
ejpam-4618	146	9	on	on	ADP
ejpam-4618	146	10	isomorphisms	isomorphism	NOUN
ejpam-4618	146	11	of	of	ADP
ejpam-4618	146	12	ku	ku	PROPN
ejpam-4618	146	13	-	-	PUNCT
ejpam-4618	146	14	algebras	algebras	PROPN
ejpam-4618	146	15	,	,	PUNCT
ejpam-4618	146	16	,	,	PUNCT
ejpam-4618	146	17	scientia	scientia	PROPN
ejpam-4618	146	18	magna	magna	PROPN
ejpam-4618	146	19	,	,	PUNCT
ejpam-4618	146	20	international	international	ADJ
ejpam-4618	146	21	book	book	NOUN
ejpam-4618	146	22	series	series	NOUN
ejpam-4618	146	23	,	,	PUNCT
ejpam-4618	146	24	5	5	NUM
ejpam-4618	146	25	,	,	PUNCT
ejpam-4618	146	26	no	no	DET
ejpam-4618	146	27	.3	.3	NUM
ejpam-4618	146	28	.	.	PUNCT
ejpam-4618	146	29	,	,	PUNCT
ejpam-4618	146	30	25	25	NUM
ejpam-4618	146	31	-	-	SYM
ejpam-4618	146	32	31	31	NUM
ejpam-4618	146	33	.	.	NUM
ejpam-4618	146	34	,	,	PUNCT
ejpam-4618	146	35	2009	2009	NUM
ejpam-4618	146	36	.	.	PUNCT
ejpam-4618	147	1	[	[	X
ejpam-4618	147	2	10	10	NUM
ejpam-4618	147	3	]	]	X
ejpam-4618	147	4	yaqoob	yaqoob	NOUN
ejpam-4618	147	5	,	,	PUNCT
ejpam-4618	147	6	mostafa	mostafa	PROPN
ejpam-4618	147	7	and	and	CCONJ
ejpam-4618	147	8	ansari	ansari	PROPN
ejpam-4618	147	9	,	,	PUNCT
ejpam-4618	147	10	on	on	ADP
ejpam-4618	147	11	cubic	cubic	ADJ
ejpam-4618	147	12	ku	ku	PROPN
ejpam-4618	147	13	-	-	PUNCT
ejpam-4618	147	14	ideals	ideal	NOUN
ejpam-4618	147	15	of	of	ADP
ejpam-4618	147	16	ku	ku	PROPN
ejpam-4618	147	17	-	-	PUNCT
ejpam-4618	147	18	algebras	algebras	PROPN
ejpam-4618	147	19	,	,	PUNCT
ejpam-4618	147	20	,	,	PUNCT
ejpam-4618	147	21	isrn	isrn	PROPN
ejpam-4618	147	22	algebra	algebra	PROPN
ejpam-4618	147	23	,	,	PUNCT
ejpam-4618	147	24	article	article	NOUN
ejpam-4618	147	25	id935905	id935905	VERB
ejpam-4618	147	26	,	,	PUNCT
ejpam-4618	147	27	10	10	NUM
ejpam-4618	147	28	pages	page	NOUN
ejpam-4618	147	29	.	.	PUNCT
ejpam-4618	147	30	,	,	PUNCT
ejpam-4618	147	31	2013	2013	NUM
ejpam-4618	148	1	[	[	X
ejpam-4618	148	2	11	11	NUM
ejpam-4618	148	3	]	]	X
ejpam-4618	148	4	zadeh	zadeh	NOUN
ejpam-4618	148	5	,	,	PUNCT
ejpam-4618	148	6	fuzzy	fuzzy	ADJ
ejpam-4618	148	7	sets	set	NOUN
ejpam-4618	148	8	,	,	PUNCT
ejpam-4618	148	9	inf	inf	NOUN
ejpam-4618	148	10	control	control	NOUN
ejpam-4618	148	11	,	,	PUNCT
ejpam-4618	148	12	8	8	NUM
ejpam-4618	148	13	,	,	PUNCT
ejpam-4618	148	14	338	338	NUM
ejpam-4618	148	15	-	-	SYM
ejpam-4618	148	16	353	353	NUM
ejpam-4618	148	17	1965	1965	NUM
ejpam-4618	148	18	.	.	PUNCT