id	sid	tid	token	lemma	pos
ejpam-5255	1	1	european	european	PROPN
ejpam-5255	1	2	journal	journal	PROPN
ejpam-5255	1	3	of	of	ADP
ejpam-5255	1	4	pure	pure	ADJ
ejpam-5255	1	5	and	and	CCONJ
ejpam-5255	1	6	applied	apply	VERB
ejpam-5255	1	7	mathematics	mathematic	NOUN
ejpam-5255	1	8	vol	vol	NOUN
ejpam-5255	1	9	.	.	PROPN
ejpam-5255	2	1	17	17	NUM
ejpam-5255	2	2	,	,	PUNCT
ejpam-5255	2	3	no	no	INTJ
ejpam-5255	2	4	.	.	NOUN
ejpam-5255	2	5	3	3	NUM
ejpam-5255	2	6	,	,	PUNCT
ejpam-5255	2	7	2024	2024	NUM
ejpam-5255	2	8	,	,	PUNCT
ejpam-5255	2	9	1831	1831	NUM
ejpam-5255	2	10	-	-	SYM
ejpam-5255	2	11	1841	1841	NUM
ejpam-5255	2	12	issn	issn	VERB
ejpam-5255	2	13	1307	1307	NUM
ejpam-5255	2	14	-	-	SYM
ejpam-5255	2	15	5543	5543	NUM
ejpam-5255	2	16	–	–	PUNCT
ejpam-5255	2	17	ejpam.com	ejpam.com	X
ejpam-5255	2	18	published	publish	VERB
ejpam-5255	2	19	by	by	ADP
ejpam-5255	2	20	new	new	PROPN
ejpam-5255	2	21	york	york	PROPN
ejpam-5255	2	22	business	business	PROPN
ejpam-5255	2	23	global	global	ADJ
ejpam-5255	2	24	bipolar	bipolar	ADJ
ejpam-5255	2	25	fuzzy	fuzzy	ADJ
ejpam-5255	2	26	commutative	commutative	ADJ
ejpam-5255	2	27	ideals	ideal	NOUN
ejpam-5255	2	28	in	in	ADP
ejpam-5255	2	29	bck	bck	PROPN
ejpam-5255	2	30	-	-	PUNCT
ejpam-5255	2	31	algebras	algebras	PROPN
ejpam-5255	2	32	areej	areej	PROPN
ejpam-5255	2	33	almuhaimeed1,∗	almuhaimeed1,∗	PROPN
ejpam-5255	2	34	,	,	PUNCT
ejpam-5255	2	35	halimah	halimah	NOUN
ejpam-5255	2	36	alshehri2	alshehri2	NOUN
ejpam-5255	2	37	1	1	NUM
ejpam-5255	2	38	department	department	NOUN
ejpam-5255	2	39	of	of	ADP
ejpam-5255	2	40	mathematics	mathematic	NOUN
ejpam-5255	2	41	,	,	PUNCT
ejpam-5255	2	42	college	college	NOUN
ejpam-5255	2	43	of	of	ADP
ejpam-5255	2	44	science	science	PROPN
ejpam-5255	2	45	,	,	PUNCT
ejpam-5255	2	46	taibah	taibah	PROPN
ejpam-5255	2	47	university	university	PROPN
ejpam-5255	2	48	,	,	PUNCT
ejpam-5255	2	49	madinah	madinah	PROPN
ejpam-5255	2	50	,	,	PUNCT
ejpam-5255	2	51	saudi	saudi	PROPN
ejpam-5255	2	52	arabia	arabia	PROPN
ejpam-5255	2	53	2	2	NUM
ejpam-5255	2	54	department	department	NOUN
ejpam-5255	2	55	of	of	ADP
ejpam-5255	2	56	computer	computer	NOUN
ejpam-5255	2	57	science	science	NOUN
ejpam-5255	2	58	and	and	CCONJ
ejpam-5255	2	59	engineering	engineering	NOUN
ejpam-5255	2	60	,	,	PUNCT
ejpam-5255	2	61	faculty	faculty	NOUN
ejpam-5255	2	62	applied	apply	VERB
ejpam-5255	2	63	studies	study	NOUN
ejpam-5255	2	64	and	and	CCONJ
ejpam-5255	2	65	community	community	NOUN
ejpam-5255	2	66	service	service	NOUN
ejpam-5255	2	67	,	,	PUNCT
ejpam-5255	2	68	king	king	PROPN
ejpam-5255	2	69	saud	saud	PROPN
ejpam-5255	2	70	university	university	PROPN
ejpam-5255	2	71	,	,	PUNCT
ejpam-5255	2	72	riyadh	riyadh	PROPN
ejpam-5255	2	73	,	,	PUNCT
ejpam-5255	2	74	saudi	saudi	PROPN
ejpam-5255	2	75	arabia	arabia	PROPN
ejpam-5255	2	76	abstract	abstract	NOUN
ejpam-5255	2	77	.	.	PUNCT
ejpam-5255	3	1	this	this	DET
ejpam-5255	3	2	paper	paper	NOUN
ejpam-5255	3	3	presents	present	VERB
ejpam-5255	3	4	the	the	DET
ejpam-5255	3	5	notions	notion	NOUN
ejpam-5255	3	6	of	of	ADP
ejpam-5255	3	7	bipolar	bipolar	ADJ
ejpam-5255	3	8	fuzzy	fuzzy	ADJ
ejpam-5255	3	9	commutative	commutative	ADJ
ejpam-5255	3	10	ideals	ideal	NOUN
ejpam-5255	3	11	of	of	ADP
ejpam-5255	3	12	a	a	DET
ejpam-5255	3	13	bck	bck	NOUN
ejpam-5255	3	14	-	-	PUNCT
ejpam-5255	3	15	algebra	algebra	NOUN
ejpam-5255	3	16	.	.	PUNCT
ejpam-5255	4	1	we	we	PRON
ejpam-5255	4	2	also	also	ADV
ejpam-5255	4	3	study	study	VERB
ejpam-5255	4	4	various	various	ADJ
ejpam-5255	4	5	properties	property	NOUN
ejpam-5255	4	6	regarding	regard	VERB
ejpam-5255	4	7	this	this	DET
ejpam-5255	4	8	concept	concept	NOUN
ejpam-5255	4	9	.	.	PUNCT
ejpam-5255	5	1	we	we	PRON
ejpam-5255	5	2	prove	prove	VERB
ejpam-5255	5	3	various	various	ADJ
ejpam-5255	5	4	characterisations	characterisation	NOUN
ejpam-5255	5	5	for	for	ADP
ejpam-5255	5	6	bipolar	bipolar	ADJ
ejpam-5255	5	7	fuzzy	fuzzy	ADJ
ejpam-5255	5	8	commutative	commutative	ADJ
ejpam-5255	5	9	ideals	ideal	NOUN
ejpam-5255	5	10	.	.	PUNCT
ejpam-5255	6	1	moreover	moreover	ADV
ejpam-5255	6	2	,	,	PUNCT
ejpam-5255	6	3	a	a	DET
ejpam-5255	6	4	relationship	relationship	NOUN
ejpam-5255	6	5	between	between	ADP
ejpam-5255	6	6	bipolar	bipolar	ADJ
ejpam-5255	6	7	fuzzy	fuzzy	ADJ
ejpam-5255	6	8	commutative	commutative	ADJ
ejpam-5255	6	9	ideals	ideal	NOUN
ejpam-5255	6	10	and	and	CCONJ
ejpam-5255	6	11	bipolar	bipolar	ADJ
ejpam-5255	6	12	fuzzy	fuzzy	ADJ
ejpam-5255	6	13	positive	positive	ADJ
ejpam-5255	6	14	implicative	implicative	ADJ
ejpam-5255	6	15	ideals	ideal	NOUN
ejpam-5255	6	16	is	be	AUX
ejpam-5255	6	17	presented	present	VERB
ejpam-5255	6	18	.	.	PUNCT
ejpam-5255	7	1	in	in	ADP
ejpam-5255	7	2	addition	addition	NOUN
ejpam-5255	7	3	,	,	PUNCT
ejpam-5255	7	4	we	we	PRON
ejpam-5255	7	5	present	present	VERB
ejpam-5255	7	6	the	the	DET
ejpam-5255	7	7	notation	notation	NOUN
ejpam-5255	7	8	of	of	ADP
ejpam-5255	7	9	bipolar	bipolar	ADJ
ejpam-5255	7	10	fuzzy	fuzzy	ADJ
ejpam-5255	7	11	characteristic	characteristic	ADJ
ejpam-5255	7	12	commutative	commutative	ADJ
ejpam-5255	7	13	ideal	ideal	NOUN
ejpam-5255	7	14	.	.	PUNCT
ejpam-5255	8	1	then	then	ADV
ejpam-5255	8	2	a	a	DET
ejpam-5255	8	3	relationship	relationship	NOUN
ejpam-5255	8	4	between	between	ADP
ejpam-5255	8	5	a	a	DET
ejpam-5255	8	6	bipolar	bipolar	ADJ
ejpam-5255	8	7	fuzzy	fuzzy	ADJ
ejpam-5255	8	8	characteristic	characteristic	ADJ
ejpam-5255	8	9	commutative	commutative	ADJ
ejpam-5255	8	10	ideal	ideal	NOUN
ejpam-5255	8	11	and	and	CCONJ
ejpam-5255	8	12	its	its	PRON
ejpam-5255	8	13	level	level	NOUN
ejpam-5255	8	14	-	-	PUNCT
ejpam-5255	8	15	cut	cut	VERB
ejpam-5255	8	16	commutative	commutative	ADJ
ejpam-5255	8	17	ideals	ideal	NOUN
ejpam-5255	8	18	is	be	AUX
ejpam-5255	8	19	provided	provide	VERB
ejpam-5255	8	20	.	.	PUNCT
ejpam-5255	9	1	2020	2020	NUM
ejpam-5255	9	2	mathematics	mathematic	NOUN
ejpam-5255	9	3	subject	subject	NOUN
ejpam-5255	9	4	classifications	classification	NOUN
ejpam-5255	9	5	:	:	PUNCT
ejpam-5255	9	6	4d05	4d05	NUM
ejpam-5255	9	7	,	,	PUNCT
ejpam-5255	9	8	03e72	03e72	NUM
ejpam-5255	9	9	,	,	PUNCT
ejpam-5255	9	10	08a72	08a72	NOUN
ejpam-5255	9	11	key	key	ADJ
ejpam-5255	9	12	words	word	NOUN
ejpam-5255	9	13	and	and	CCONJ
ejpam-5255	9	14	phrases	phrase	NOUN
ejpam-5255	9	15	:	:	PUNCT
ejpam-5255	9	16	bipolar	bipolar	ADJ
ejpam-5255	9	17	fuzzy	fuzzy	ADJ
ejpam-5255	9	18	ideal	ideal	NOUN
ejpam-5255	9	19	,	,	PUNCT
ejpam-5255	9	20	commutative	commutative	ADJ
ejpam-5255	9	21	ideal	ideal	NOUN
ejpam-5255	9	22	,	,	PUNCT
ejpam-5255	9	23	positive	positive	ADJ
ejpam-5255	9	24	implicative	implicative	ADJ
ejpam-5255	9	25	ideal	ideal	NOUN
ejpam-5255	9	26	,	,	PUNCT
ejpam-5255	9	27	characteristic	characteristic	ADJ
ejpam-5255	9	28	ideal	ideal	NOUN
ejpam-5255	9	29	1	1	NUM
ejpam-5255	9	30	.	.	PUNCT
ejpam-5255	10	1	introduction	introduction	NOUN
ejpam-5255	10	2	zadeh	zadeh	PROPN
ejpam-5255	10	3	introduced	introduce	VERB
ejpam-5255	10	4	the	the	DET
ejpam-5255	10	5	notation	notation	NOUN
ejpam-5255	10	6	of	of	ADP
ejpam-5255	10	7	fuzzy	fuzzy	ADJ
ejpam-5255	10	8	sets	set	NOUN
ejpam-5255	10	9	[	[	X
ejpam-5255	10	10	16	16	NUM
ejpam-5255	10	11	]	]	PUNCT
ejpam-5255	10	12	.	.	PUNCT
ejpam-5255	11	1	in	in	ADP
ejpam-5255	11	2	[	[	X
ejpam-5255	11	3	6	6	NUM
ejpam-5255	11	4	]	]	PUNCT
ejpam-5255	11	5	,	,	PUNCT
ejpam-5255	11	6	bck	bck	PROPN
ejpam-5255	11	7	-	-	PUNCT
ejpam-5255	11	8	algebra	algebra	NOUN
ejpam-5255	11	9	was	be	AUX
ejpam-5255	11	10	provided	provide	VERB
ejpam-5255	11	11	as	as	ADP
ejpam-5255	11	12	a	a	DET
ejpam-5255	11	13	logical	logical	ADJ
ejpam-5255	11	14	class	class	NOUN
ejpam-5255	11	15	of	of	ADP
ejpam-5255	11	16	algebras	algebras	PROPN
ejpam-5255	11	17	.	.	PUNCT
ejpam-5255	12	1	this	this	PRON
ejpam-5255	12	2	encouraged	encourage	VERB
ejpam-5255	12	3	many	many	ADJ
ejpam-5255	12	4	researchers	researcher	NOUN
ejpam-5255	12	5	to	to	PART
ejpam-5255	12	6	apply	apply	VERB
ejpam-5255	12	7	fuzzy	fuzzy	ADJ
ejpam-5255	12	8	sets	set	NOUN
ejpam-5255	12	9	to	to	PART
ejpam-5255	12	10	bck	bck	VERB
ejpam-5255	12	11	-	-	PUNCT
ejpam-5255	12	12	algebras	algebras	X
ejpam-5255	12	13	,	,	PUNCT
ejpam-5255	12	14	see	see	VERB
ejpam-5255	12	15	[	[	X
ejpam-5255	12	16	8	8	NUM
ejpam-5255	12	17	]	]	PUNCT
ejpam-5255	12	18	,	,	PUNCT
ejpam-5255	13	1	[	[	X
ejpam-5255	13	2	12	12	NUM
ejpam-5255	13	3	]	]	PUNCT
ejpam-5255	13	4	,	,	PUNCT
ejpam-5255	13	5	[	[	X
ejpam-5255	13	6	13	13	NUM
ejpam-5255	13	7	]	]	PUNCT
ejpam-5255	13	8	,	,	PUNCT
ejpam-5255	13	9	[	[	X
ejpam-5255	13	10	14	14	NUM
ejpam-5255	13	11	]	]	PUNCT
ejpam-5255	13	12	,	,	PUNCT
ejpam-5255	14	1	[	[	X
ejpam-5255	14	2	15	15	NUM
ejpam-5255	14	3	]	]	PUNCT
ejpam-5255	14	4	and	and	CCONJ
ejpam-5255	14	5	[	[	X
ejpam-5255	14	6	10	10	NUM
ejpam-5255	14	7	]	]	PUNCT
ejpam-5255	14	8	then	then	ADV
ejpam-5255	14	9	many	many	ADJ
ejpam-5255	14	10	concepts	concept	NOUN
ejpam-5255	14	11	which	which	PRON
ejpam-5255	14	12	related	relate	VERB
ejpam-5255	14	13	to	to	ADP
ejpam-5255	14	14	fuzzy	fuzzy	ADJ
ejpam-5255	14	15	sets	set	NOUN
ejpam-5255	14	16	have	have	AUX
ejpam-5255	14	17	been	be	AUX
ejpam-5255	14	18	widely	widely	ADV
ejpam-5255	14	19	investigated	investigate	VERB
ejpam-5255	14	20	.	.	PUNCT
ejpam-5255	15	1	one	one	NUM
ejpam-5255	15	2	of	of	ADP
ejpam-5255	15	3	the	the	DET
ejpam-5255	15	4	most	most	ADV
ejpam-5255	15	5	interesting	interesting	ADJ
ejpam-5255	15	6	concepts	concept	NOUN
ejpam-5255	15	7	is	be	AUX
ejpam-5255	15	8	a	a	DET
ejpam-5255	15	9	bipolar	bipolar	ADJ
ejpam-5255	15	10	fuzzy	fuzzy	ADJ
ejpam-5255	15	11	set	set	NOUN
ejpam-5255	15	12	.	.	PUNCT
ejpam-5255	16	1	the	the	DET
ejpam-5255	16	2	importance	importance	NOUN
ejpam-5255	16	3	of	of	ADP
ejpam-5255	16	4	it	it	PRON
ejpam-5255	16	5	lies	lie	VERB
ejpam-5255	16	6	behind	behind	ADP
ejpam-5255	16	7	its	its	PRON
ejpam-5255	16	8	properties	property	NOUN
ejpam-5255	16	9	and	and	CCONJ
ejpam-5255	16	10	applications	application	NOUN
ejpam-5255	16	11	,	,	PUNCT
ejpam-5255	16	12	for	for	ADP
ejpam-5255	16	13	example	example	NOUN
ejpam-5255	16	14	see	see	VERB
ejpam-5255	16	15	[	[	X
ejpam-5255	16	16	3	3	NUM
ejpam-5255	16	17	]	]	PUNCT
ejpam-5255	16	18	.	.	PUNCT
ejpam-5255	17	1	many	many	ADJ
ejpam-5255	17	2	papers	paper	NOUN
ejpam-5255	17	3	provide	provide	VERB
ejpam-5255	17	4	the	the	DET
ejpam-5255	17	5	study	study	NOUN
ejpam-5255	17	6	of	of	ADP
ejpam-5255	17	7	bipolar	bipolar	ADJ
ejpam-5255	17	8	fuzzy	fuzzy	ADJ
ejpam-5255	17	9	set	set	NOUN
ejpam-5255	17	10	theory	theory	NOUN
ejpam-5255	17	11	in	in	ADP
ejpam-5255	17	12	several	several	ADJ
ejpam-5255	17	13	algebraic	algebraic	ADJ
ejpam-5255	17	14	structures	structure	NOUN
ejpam-5255	17	15	see	see	VERB
ejpam-5255	17	16	[	[	X
ejpam-5255	17	17	5	5	NUM
ejpam-5255	17	18	]	]	PUNCT
ejpam-5255	17	19	,	,	PUNCT
ejpam-5255	17	20	[	[	X
ejpam-5255	17	21	2	2	NUM
ejpam-5255	17	22	]	]	PUNCT
ejpam-5255	17	23	,	,	PUNCT
ejpam-5255	17	24	[	[	X
ejpam-5255	17	25	11	11	NUM
ejpam-5255	17	26	]	]	PUNCT
ejpam-5255	17	27	,	,	PUNCT
ejpam-5255	17	28	[	[	X
ejpam-5255	17	29	1	1	NUM
ejpam-5255	17	30	]	]	PUNCT
ejpam-5255	17	31	,	,	PUNCT
ejpam-5255	17	32	[	[	X
ejpam-5255	17	33	9	9	NUM
ejpam-5255	17	34	]	]	PUNCT
ejpam-5255	17	35	and	and	CCONJ
ejpam-5255	17	36	[	[	X
ejpam-5255	17	37	4	4	NUM
ejpam-5255	17	38	]	]	PUNCT
ejpam-5255	17	39	.	.	PUNCT
ejpam-5255	18	1	this	this	DET
ejpam-5255	18	2	paper	paper	NOUN
ejpam-5255	18	3	provides	provide	VERB
ejpam-5255	18	4	bipolar	bipolar	ADJ
ejpam-5255	18	5	fuzzy	fuzzy	ADJ
ejpam-5255	18	6	commutative	commutative	ADJ
ejpam-5255	18	7	ideals	ideal	NOUN
ejpam-5255	18	8	of	of	ADP
ejpam-5255	18	9	a	a	DET
ejpam-5255	18	10	bck	bck	NOUN
ejpam-5255	18	11	-	-	PUNCT
ejpam-5255	18	12	algebra	algebra	NOUN
ejpam-5255	18	13	.	.	PUNCT
ejpam-5255	19	1	we	we	PRON
ejpam-5255	19	2	also	also	ADV
ejpam-5255	19	3	study	study	VERB
ejpam-5255	19	4	various	various	ADJ
ejpam-5255	19	5	properties	property	NOUN
ejpam-5255	19	6	regarding	regard	VERB
ejpam-5255	19	7	this	this	DET
ejpam-5255	19	8	concept	concept	NOUN
ejpam-5255	19	9	.	.	PUNCT
ejpam-5255	20	1	we	we	PRON
ejpam-5255	20	2	prove	prove	VERB
ejpam-5255	20	3	various	various	ADJ
ejpam-5255	20	4	characterisations	characterisation	NOUN
ejpam-5255	20	5	for	for	ADP
ejpam-5255	20	6	bipolar	bipolar	ADJ
ejpam-5255	20	7	fuzzy	fuzzy	ADJ
ejpam-5255	20	8	commutative	commutative	ADJ
ejpam-5255	20	9	ideals	ideal	NOUN
ejpam-5255	20	10	.	.	PUNCT
ejpam-5255	21	1	moreover	moreover	ADV
ejpam-5255	21	2	,	,	PUNCT
ejpam-5255	21	3	we	we	PRON
ejpam-5255	21	4	provide	provide	VERB
ejpam-5255	21	5	a	a	DET
ejpam-5255	21	6	relationship	relationship	NOUN
ejpam-5255	21	7	between	between	ADP
ejpam-5255	21	8	bipolar	bipolar	ADJ
ejpam-5255	21	9	fuzzy	fuzzy	ADJ
ejpam-5255	21	10	commutative	commutative	ADJ
ejpam-5255	21	11	ideals	ideal	NOUN
ejpam-5255	21	12	and	and	CCONJ
ejpam-5255	21	13	bipolar	bipolar	ADJ
ejpam-5255	21	14	fuzzy	fuzzy	ADJ
ejpam-5255	21	15	positive	positive	ADJ
ejpam-5255	21	16	implicative	implicative	ADJ
ejpam-5255	21	17	ideals	ideal	NOUN
ejpam-5255	21	18	.	.	PUNCT
ejpam-5255	22	1	in	in	ADP
ejpam-5255	22	2	addition	addition	NOUN
ejpam-5255	22	3	,	,	PUNCT
ejpam-5255	22	4	we	we	PRON
ejpam-5255	22	5	present	present	VERB
ejpam-5255	22	6	the	the	DET
ejpam-5255	22	7	notation	notation	NOUN
ejpam-5255	22	8	of	of	ADP
ejpam-5255	22	9	bipolar	bipolar	ADJ
ejpam-5255	22	10	fuzzy	fuzzy	ADJ
ejpam-5255	22	11	characteristic	characteristic	ADJ
ejpam-5255	22	12	commutative	commutative	ADJ
ejpam-5255	22	13	ideal	ideal	NOUN
ejpam-5255	22	14	and	and	CCONJ
ejpam-5255	22	15	give	give	VERB
ejpam-5255	22	16	a	a	DET
ejpam-5255	22	17	relationship	relationship	NOUN
ejpam-5255	22	18	∗corresponding	∗corresponde	VERB
ejpam-5255	22	19	author	author	NOUN
ejpam-5255	22	20	.	.	PUNCT
ejpam-5255	23	1	doi	doi	NOUN
ejpam-5255	23	2	:	:	PUNCT
ejpam-5255	23	3	https://doi.org/10.29020/nybg.ejpam.v17i3.5255	https://doi.org/10.29020/nybg.ejpam.v17i3.5255	ADP
ejpam-5255	23	4	email	email	NOUN
ejpam-5255	23	5	addresses	address	NOUN
ejpam-5255	23	6	:	:	PUNCT
ejpam-5255	23	7	aamuhaimeed@taibahu.edu.sa	aamuhaimeed@taibahu.edu.sa	NOUN
ejpam-5255	23	8	(	(	PUNCT
ejpam-5255	23	9	a.	a.	NOUN
ejpam-5255	23	10	almuhaimeed	almuhaimeed	PROPN
ejpam-5255	23	11	)	)	PUNCT
ejpam-5255	23	12	,	,	PUNCT
ejpam-5255	23	13	haalshehri@ksu.edu.sa	haalshehri@ksu.edu.sa	PROPN
ejpam-5255	23	14	(	(	PUNCT
ejpam-5255	23	15	h.	h.	PROPN
ejpam-5255	23	16	alshehri	alshehri	PROPN
ejpam-5255	23	17	)	)	PUNCT
ejpam-5255	23	18	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5255	23	19	1831	1831	NUM
ejpam-5255	24	1	©	©	ADP
ejpam-5255	24	2	2024	2024	NUM
ejpam-5255	24	3	ejpam	ejpam	NOUN
ejpam-5255	24	4	all	all	DET
ejpam-5255	24	5	rights	right	NOUN
ejpam-5255	24	6	reserved	reserve	VERB
ejpam-5255	24	7	.	.	PUNCT
ejpam-5255	25	1	a.	a.	PROPN
ejpam-5255	25	2	almuhaimeed	almuhaimeed	PROPN
ejpam-5255	25	3	,	,	PUNCT
ejpam-5255	25	4	h.	h.	PROPN
ejpam-5255	25	5	alshehri	alshehri	PROPN
ejpam-5255	25	6	/	/	SYM
ejpam-5255	25	7	eur	eur	PROPN
ejpam-5255	25	8	.	.	PUNCT
ejpam-5255	26	1	j.	j.	PROPN
ejpam-5255	26	2	pure	pure	PROPN
ejpam-5255	26	3	appl	appl	PROPN
ejpam-5255	26	4	.	.	PROPN
ejpam-5255	26	5	math	math	PROPN
ejpam-5255	26	6	,	,	PUNCT
ejpam-5255	26	7	17	17	NUM
ejpam-5255	26	8	(	(	PUNCT
ejpam-5255	26	9	3	3	NUM
ejpam-5255	26	10	)	)	PUNCT
ejpam-5255	26	11	(	(	PUNCT
ejpam-5255	26	12	2024	2024	NUM
ejpam-5255	26	13	)	)	PUNCT
ejpam-5255	26	14	,	,	PUNCT
ejpam-5255	26	15	1831	1831	NUM
ejpam-5255	26	16	-	-	SYM
ejpam-5255	26	17	1841	1841	NUM
ejpam-5255	26	18	1832	1832	NUM
ejpam-5255	26	19	between	between	ADP
ejpam-5255	26	20	a	a	DET
ejpam-5255	26	21	bipolar	bipolar	ADJ
ejpam-5255	26	22	fuzzy	fuzzy	ADJ
ejpam-5255	26	23	characteristic	characteristic	ADJ
ejpam-5255	26	24	commutative	commutative	ADJ
ejpam-5255	26	25	ideal	ideal	NOUN
ejpam-5255	26	26	and	and	CCONJ
ejpam-5255	26	27	its	its	PRON
ejpam-5255	26	28	level	level	NOUN
ejpam-5255	26	29	-	-	PUNCT
ejpam-5255	26	30	cut	cut	VERB
ejpam-5255	26	31	commutative	commutative	ADJ
ejpam-5255	26	32	ideals	ideal	NOUN
ejpam-5255	26	33	.	.	PUNCT
ejpam-5255	27	1	our	our	PRON
ejpam-5255	27	2	motivation	motivation	NOUN
ejpam-5255	27	3	is	be	AUX
ejpam-5255	27	4	to	to	PART
ejpam-5255	27	5	investigate	investigate	VERB
ejpam-5255	27	6	bipolar	bipolar	ADJ
ejpam-5255	27	7	fuzzy	fuzzy	ADJ
ejpam-5255	27	8	commutative	commutative	ADJ
ejpam-5255	27	9	ideals	ideal	NOUN
ejpam-5255	27	10	in	in	ADP
ejpam-5255	27	11	bck	bck	NOUN
ejpam-5255	27	12	-	-	PUNCT
ejpam-5255	27	13	algebras	algebras	PROPN
ejpam-5255	27	14	and	and	CCONJ
ejpam-5255	27	15	their	their	PRON
ejpam-5255	27	16	properties	property	NOUN
ejpam-5255	27	17	.	.	PUNCT
ejpam-5255	28	1	this	this	PRON
ejpam-5255	28	2	can	can	AUX
ejpam-5255	28	3	lead	lead	VERB
ejpam-5255	28	4	to	to	PART
ejpam-5255	28	5	study	study	VERB
ejpam-5255	28	6	this	this	DET
ejpam-5255	28	7	concept	concept	NOUN
ejpam-5255	28	8	in	in	ADP
ejpam-5255	28	9	soft	soft	ADJ
ejpam-5255	28	10	case	case	NOUN
ejpam-5255	28	11	and	and	CCONJ
ejpam-5255	28	12	apply	apply	VERB
ejpam-5255	28	13	the	the	DET
ejpam-5255	28	14	results	result	NOUN
ejpam-5255	28	15	in	in	ADP
ejpam-5255	28	16	decision	decision	NOUN
ejpam-5255	28	17	making	make	VERB
ejpam-5255	28	18	algorithms	algorithm	NOUN
ejpam-5255	28	19	.	.	PUNCT
ejpam-5255	29	1	2	2	X
ejpam-5255	29	2	.	.	X
ejpam-5255	29	3	preliminaries	preliminary	NOUN
ejpam-5255	29	4	in	in	ADP
ejpam-5255	29	5	[	[	X
ejpam-5255	29	6	7	7	NUM
ejpam-5255	29	7	]	]	PUNCT
ejpam-5255	29	8	,	,	PUNCT
ejpam-5255	29	9	a	a	DET
ejpam-5255	29	10	bck	bck	NOUN
ejpam-5255	29	11	-	-	PUNCT
ejpam-5255	29	12	algebra	algebra	NOUN
ejpam-5255	29	13	is	be	AUX
ejpam-5255	29	14	defined	define	VERB
ejpam-5255	29	15	as	as	SCONJ
ejpam-5255	29	16	follows	follow	VERB
ejpam-5255	29	17	:	:	PUNCT
ejpam-5255	29	18	consider	consider	VERB
ejpam-5255	29	19	the	the	DET
ejpam-5255	29	20	algebra	algebra	NOUN
ejpam-5255	29	21	(	(	PUNCT
ejpam-5255	29	22	x	x	X
ejpam-5255	29	23	,	,	PUNCT
ejpam-5255	29	24	∗	∗	NOUN
ejpam-5255	29	25	,	,	PUNCT
ejpam-5255	29	26	0	0	NUM
ejpam-5255	29	27	)	)	PUNCT
ejpam-5255	29	28	.	.	PUNCT
ejpam-5255	30	1	suppose	suppose	VERB
ejpam-5255	30	2	that	that	SCONJ
ejpam-5255	30	3	,	,	PUNCT
ejpam-5255	30	4	for	for	ADP
ejpam-5255	30	5	all	all	DET
ejpam-5255	30	6	a	a	DET
ejpam-5255	30	7	,	,	PUNCT
ejpam-5255	30	8	g	g	NOUN
ejpam-5255	30	9	,	,	PUNCT
ejpam-5255	30	10	w	w	PROPN
ejpam-5255	30	11	∈	∈	PROPN
ejpam-5255	30	12	x	x	NOUN
ejpam-5255	30	13	,	,	PUNCT
ejpam-5255	30	14	we	we	PRON
ejpam-5255	30	15	have	have	VERB
ejpam-5255	30	16	(	(	PUNCT
ejpam-5255	30	17	bck1	bck1	PROPN
ejpam-5255	30	18	)	)	PUNCT
ejpam-5255	30	19	(	(	PUNCT
ejpam-5255	30	20	(	(	PUNCT
ejpam-5255	30	21	a	a	DET
ejpam-5255	30	22	∗	∗	NOUN
ejpam-5255	30	23	g	g	NOUN
ejpam-5255	30	24	)	)	PUNCT
ejpam-5255	30	25	∗	∗	NOUN
ejpam-5255	30	26	(	(	PUNCT
ejpam-5255	30	27	a	a	DET
ejpam-5255	30	28	∗	∗	NOUN
ejpam-5255	30	29	w	w	NOUN
ejpam-5255	30	30	)	)	PUNCT
ejpam-5255	30	31	∗	∗	NOUN
ejpam-5255	30	32	(	(	PUNCT
ejpam-5255	30	33	w	w	NOUN
ejpam-5255	30	34	∗	∗	NOUN
ejpam-5255	30	35	g	g	NOUN
ejpam-5255	30	36	)	)	PUNCT
ejpam-5255	30	37	=	=	SYM
ejpam-5255	31	1	0	0	NUM
ejpam-5255	31	2	)	)	PUNCT
ejpam-5255	31	3	,	,	PUNCT
ejpam-5255	31	4	(	(	PUNCT
ejpam-5255	31	5	bck2	bck2	PROPN
ejpam-5255	31	6	)	)	PUNCT
ejpam-5255	31	7	(	(	PUNCT
ejpam-5255	31	8	a	a	DET
ejpam-5255	31	9	∗	∗	NOUN
ejpam-5255	31	10	(	(	PUNCT
ejpam-5255	31	11	a	a	DET
ejpam-5255	31	12	∗	∗	NOUN
ejpam-5255	31	13	g	g	NOUN
ejpam-5255	31	14	)	)	PUNCT
ejpam-5255	31	15	)	)	PUNCT
ejpam-5255	31	16	∗	∗	NOUN
ejpam-5255	31	17	g	g	NOUN
ejpam-5255	31	18	=	=	SYM
ejpam-5255	31	19	0	0	NUM
ejpam-5255	31	20	,	,	PUNCT
ejpam-5255	31	21	(	(	PUNCT
ejpam-5255	31	22	bck3	bck3	PROPN
ejpam-5255	31	23	)	)	PUNCT
ejpam-5255	31	24	a	a	DET
ejpam-5255	31	25	∗	∗	NOUN
ejpam-5255	31	26	a	a	DET
ejpam-5255	31	27	=	=	NOUN
ejpam-5255	31	28	0	0	NUM
ejpam-5255	31	29	,	,	PUNCT
ejpam-5255	31	30	(	(	PUNCT
ejpam-5255	31	31	bck4	bck4	PROPN
ejpam-5255	31	32	)	)	PUNCT
ejpam-5255	31	33	0	0	NUM
ejpam-5255	32	1	∗	∗	NOUN
ejpam-5255	32	2	a	a	DET
ejpam-5255	32	3	=	=	NOUN
ejpam-5255	32	4	0	0	NUM
ejpam-5255	32	5	,	,	PUNCT
ejpam-5255	32	6	(	(	PUNCT
ejpam-5255	32	7	bck5	bck5	PROPN
ejpam-5255	32	8	)	)	PUNCT
ejpam-5255	32	9	if	if	SCONJ
ejpam-5255	32	10	a	a	DET
ejpam-5255	32	11	∗	∗	NOUN
ejpam-5255	32	12	g	g	NOUN
ejpam-5255	32	13	=	=	SYM
ejpam-5255	32	14	0	0	NUM
ejpam-5255	32	15	and	and	CCONJ
ejpam-5255	32	16	g	g	PROPN
ejpam-5255	32	17	∗	∗	NOUN
ejpam-5255	32	18	a	a	DET
ejpam-5255	32	19	=	=	SYM
ejpam-5255	32	20	0	0	NUM
ejpam-5255	32	21	,	,	PUNCT
ejpam-5255	32	22	then	then	ADV
ejpam-5255	32	23	we	we	PRON
ejpam-5255	32	24	have	have	VERB
ejpam-5255	32	25	f	f	NOUN
ejpam-5255	32	26	=	=	NOUN
ejpam-5255	32	27	a.	a.	NOUN
ejpam-5255	33	1	then	then	ADV
ejpam-5255	33	2	x	x	PRON
ejpam-5255	33	3	is	be	AUX
ejpam-5255	33	4	a	a	DET
ejpam-5255	33	5	bck	bck	NOUN
ejpam-5255	33	6	-	-	PUNCT
ejpam-5255	33	7	algebra	algebra	NOUN
ejpam-5255	33	8	.	.	PUNCT
ejpam-5255	34	1	the	the	DET
ejpam-5255	34	2	subsequent	subsequent	ADJ
ejpam-5255	34	3	properties	property	NOUN
ejpam-5255	34	4	are	be	AUX
ejpam-5255	34	5	applicable	applicable	ADJ
ejpam-5255	34	6	to	to	ADP
ejpam-5255	34	7	the	the	DET
ejpam-5255	34	8	operation	operation	NOUN
ejpam-5255	34	9	∗	∗	NOUN
ejpam-5255	34	10	,	,	PUNCT
ejpam-5255	34	11	for	for	ADP
ejpam-5255	34	12	all	all	DET
ejpam-5255	34	13	a	a	DET
ejpam-5255	34	14	,	,	PUNCT
ejpam-5255	34	15	g	g	NOUN
ejpam-5255	34	16	,	,	PUNCT
ejpam-5255	34	17	w	w	PROPN
ejpam-5255	34	18	∈	∈	PROPN
ejpam-5255	34	19	x	x	X
ejpam-5255	34	20	:	:	PUNCT
ejpam-5255	34	21	(	(	PUNCT
ejpam-5255	34	22	a	a	DET
ejpam-5255	34	23	∗	∗	NOUN
ejpam-5255	34	24	g	g	NOUN
ejpam-5255	34	25	)	)	PUNCT
ejpam-5255	34	26	∗	∗	NOUN
ejpam-5255	34	27	w	w	NOUN
ejpam-5255	34	28	=	=	PUNCT
ejpam-5255	34	29	(	(	PUNCT
ejpam-5255	34	30	a	a	DET
ejpam-5255	34	31	∗	∗	NOUN
ejpam-5255	34	32	w	w	NOUN
ejpam-5255	34	33	)	)	PUNCT
ejpam-5255	34	34	∗	∗	NOUN
ejpam-5255	34	35	g	g	NOUN
ejpam-5255	34	36	(	(	PUNCT
ejpam-5255	34	37	1	1	NUM
ejpam-5255	34	38	)	)	PUNCT
ejpam-5255	34	39	(	(	PUNCT
ejpam-5255	34	40	a	a	DET
ejpam-5255	34	41	∗	∗	NOUN
ejpam-5255	34	42	w	w	NOUN
ejpam-5255	34	43	)	)	PUNCT
ejpam-5255	34	44	∗	∗	NOUN
ejpam-5255	34	45	(	(	PUNCT
ejpam-5255	34	46	a	a	DET
ejpam-5255	34	47	∗	∗	NOUN
ejpam-5255	34	48	(	(	PUNCT
ejpam-5255	34	49	a	a	DET
ejpam-5255	34	50	∗	∗	NOUN
ejpam-5255	34	51	w	w	NOUN
ejpam-5255	34	52	)	)	PUNCT
ejpam-5255	34	53	)	)	PUNCT
ejpam-5255	35	1	=	=	PRON
ejpam-5255	35	2	(	(	PUNCT
ejpam-5255	35	3	a	a	DET
ejpam-5255	35	4	∗	∗	NOUN
ejpam-5255	35	5	w	w	NOUN
ejpam-5255	35	6	)	)	PUNCT
ejpam-5255	35	7	∗	∗	NOUN
ejpam-5255	35	8	w	w	NOUN
ejpam-5255	35	9	(	(	PUNCT
ejpam-5255	35	10	2	2	X
ejpam-5255	35	11	)	)	PUNCT
ejpam-5255	35	12	we	we	PRON
ejpam-5255	35	13	will	will	AUX
ejpam-5255	35	14	use	use	VERB
ejpam-5255	35	15	these	these	DET
ejpam-5255	35	16	properties	property	NOUN
ejpam-5255	35	17	in	in	ADP
ejpam-5255	35	18	later	later	ADJ
ejpam-5255	35	19	sections	section	NOUN
ejpam-5255	35	20	.	.	PUNCT
ejpam-5255	36	1	a	a	DET
ejpam-5255	36	2	relation	relation	NOUN
ejpam-5255	36	3	≤	≤	NUM
ejpam-5255	36	4	is	be	AUX
ejpam-5255	36	5	defined	define	VERB
ejpam-5255	36	6	on	on	ADP
ejpam-5255	36	7	x	x	PUNCT
ejpam-5255	36	8	as	as	SCONJ
ejpam-5255	36	9	follows	follow	VERB
ejpam-5255	36	10	:	:	PUNCT
ejpam-5255	36	11	a	a	DET
ejpam-5255	36	12	≤	≤	NUM
ejpam-5255	36	13	g	g	ADP
ejpam-5255	36	14	⇐	⇐	ADJ
ejpam-5255	36	15	⇒	⇒	NOUN
ejpam-5255	36	16	a	a	DET
ejpam-5255	36	17	∗	∗	NOUN
ejpam-5255	36	18	g	g	NOUN
ejpam-5255	36	19	=	=	SYM
ejpam-5255	36	20	0	0	NUM
ejpam-5255	36	21	definition	definition	NOUN
ejpam-5255	36	22	1	1	NUM
ejpam-5255	36	23	.	.	PUNCT
ejpam-5255	37	1	[	[	X
ejpam-5255	37	2	7	7	X
ejpam-5255	37	3	]	]	X
ejpam-5255	37	4	an	an	DET
ejpam-5255	37	5	ideal	ideal	NOUN
ejpam-5255	37	6	i	i	PRON
ejpam-5255	37	7	of	of	ADP
ejpam-5255	37	8	x	x	PUNCT
ejpam-5255	37	9	is	be	AUX
ejpam-5255	37	10	a	a	DET
ejpam-5255	37	11	non	non	ADJ
ejpam-5255	37	12	-	-	ADJ
ejpam-5255	37	13	empty	empty	ADJ
ejpam-5255	37	14	subset	subset	NOUN
ejpam-5255	37	15	satisfying	satisfy	VERB
ejpam-5255	37	16	two	two	NUM
ejpam-5255	37	17	conditions	condition	NOUN
ejpam-5255	37	18	:	:	PUNCT
ejpam-5255	37	19	(	(	PUNCT
ejpam-5255	37	20	1	1	X
ejpam-5255	37	21	)	)	PUNCT
ejpam-5255	37	22	0	0	NUM
ejpam-5255	38	1	∈	∈	PROPN
ejpam-5255	38	2	i.	i.	NOUN
ejpam-5255	38	3	(	(	PUNCT
ejpam-5255	38	4	2	2	NUM
ejpam-5255	38	5	)	)	PUNCT
ejpam-5255	38	6	if	if	SCONJ
ejpam-5255	38	7	a	a	DET
ejpam-5255	38	8	∗	∗	NOUN
ejpam-5255	38	9	g	g	X
ejpam-5255	38	10	∈	∈	PROPN
ejpam-5255	39	1	i	i	PRON
ejpam-5255	39	2	and	and	CCONJ
ejpam-5255	39	3	g	g	PROPN
ejpam-5255	39	4	∈	∈	PROPN
ejpam-5255	40	1	i	i	PRON
ejpam-5255	40	2	,	,	PUNCT
ejpam-5255	40	3	then	then	ADV
ejpam-5255	40	4	a	a	DET
ejpam-5255	40	5	∈	∈	PROPN
ejpam-5255	40	6	i.	i.	NOUN
ejpam-5255	40	7	definition	definition	NOUN
ejpam-5255	40	8	2	2	NUM
ejpam-5255	40	9	.	.	PUNCT
ejpam-5255	41	1	[	[	X
ejpam-5255	41	2	10	10	NUM
ejpam-5255	41	3	]	]	X
ejpam-5255	41	4	a	a	DET
ejpam-5255	41	5	bipolar	bipolar	ADJ
ejpam-5255	41	6	fuzzy	fuzzy	NOUN
ejpam-5255	41	7	set	set	VERB
ejpam-5255	41	8	ϱ	ϱ	ADP
ejpam-5255	41	9	:	:	PUNCT
ejpam-5255	41	10	=	=	SYM
ejpam-5255	41	11	(	(	PUNCT
ejpam-5255	41	12	x	x	NOUN
ejpam-5255	41	13	,	,	PUNCT
ejpam-5255	41	14	ϱ+	ϱ+	X
ejpam-5255	41	15	,	,	PUNCT
ejpam-5255	41	16	ϱ−	ϱ−	ADJ
ejpam-5255	41	17	)	)	PUNCT
ejpam-5255	41	18	that	that	PRON
ejpam-5255	41	19	satisfies	satisfy	VERB
ejpam-5255	41	20	the	the	DET
ejpam-5255	41	21	following	follow	VERB
ejpam-5255	41	22	conditions	condition	NOUN
ejpam-5255	41	23	:	:	PUNCT
ejpam-5255	41	24	(	(	PUNCT
ejpam-5255	41	25	1	1	X
ejpam-5255	41	26	)	)	PUNCT
ejpam-5255	41	27	ϱ−(0	ϱ−(0	PROPN
ejpam-5255	41	28	)	)	PUNCT
ejpam-5255	41	29	≤	≤	NUM
ejpam-5255	41	30	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	41	31	)	)	PUNCT
ejpam-5255	41	32	,	,	PUNCT
ejpam-5255	41	33	ϱ+(0	ϱ+(0	NOUN
ejpam-5255	41	34	)	)	PUNCT
ejpam-5255	41	35	≥	≥	NOUN
ejpam-5255	41	36	ϱ+(a	ϱ+(a	NOUN
ejpam-5255	41	37	)	)	PUNCT
ejpam-5255	41	38	for	for	ADP
ejpam-5255	41	39	all	all	DET
ejpam-5255	41	40	a	a	DET
ejpam-5255	41	41	∈	∈	NOUN
ejpam-5255	41	42	x.	x.	NOUN
ejpam-5255	41	43	(	(	PUNCT
ejpam-5255	41	44	2	2	NUM
ejpam-5255	41	45	)	)	PUNCT
ejpam-5255	41	46	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	41	47	)	)	PUNCT
ejpam-5255	41	48	≤	≤	NUM
ejpam-5255	42	1	max	max	PROPN
ejpam-5255	42	2	{	{	PUNCT
ejpam-5255	42	3	ϱ−(a∗g	ϱ−(a∗g	PROPN
ejpam-5255	42	4	)	)	PUNCT
ejpam-5255	42	5	,	,	PUNCT
ejpam-5255	42	6	ϱ−(g	ϱ−(g	PROPN
ejpam-5255	42	7	)	)	PUNCT
ejpam-5255	42	8	}	}	PUNCT
ejpam-5255	42	9	and	and	CCONJ
ejpam-5255	42	10	ϱ+(a	ϱ+(a	NUM
ejpam-5255	42	11	)	)	PUNCT
ejpam-5255	42	12	≥	≥	NOUN
ejpam-5255	42	13	min	min	NOUN
ejpam-5255	42	14	{	{	PUNCT
ejpam-5255	42	15	ϱ+(a∗g	ϱ+(a∗g	PROPN
ejpam-5255	42	16	)	)	PUNCT
ejpam-5255	42	17	,	,	PUNCT
ejpam-5255	42	18	ϱ+(g	ϱ+(g	PROPN
ejpam-5255	42	19	)	)	PUNCT
ejpam-5255	42	20	}	}	PUNCT
ejpam-5255	42	21	for	for	ADP
ejpam-5255	42	22	all	all	DET
ejpam-5255	42	23	a	a	PRON
ejpam-5255	42	24	,	,	PUNCT
ejpam-5255	42	25	g	g	PROPN
ejpam-5255	42	26	∈	∈	PROPN
ejpam-5255	42	27	x.	x.	NOUN
ejpam-5255	42	28	is	be	AUX
ejpam-5255	42	29	called	call	VERB
ejpam-5255	42	30	a	a	DET
ejpam-5255	42	31	bipolar	bipolar	ADJ
ejpam-5255	42	32	fuzzy	fuzzy	ADJ
ejpam-5255	42	33	ideal	ideal	NOUN
ejpam-5255	42	34	.	.	PUNCT
ejpam-5255	43	1	definition	definition	NOUN
ejpam-5255	43	2	3	3	NUM
ejpam-5255	43	3	.	.	PUNCT
ejpam-5255	44	1	[	[	X
ejpam-5255	44	2	10	10	NUM
ejpam-5255	44	3	]	]	X
ejpam-5255	44	4	a	a	DET
ejpam-5255	44	5	bipolar	bipolar	ADJ
ejpam-5255	44	6	fuzzy	fuzzy	NOUN
ejpam-5255	44	7	subset	subset	VERB
ejpam-5255	44	8	ϱ	ϱ	ADP
ejpam-5255	44	9	:	:	PUNCT
ejpam-5255	44	10	=	=	SYM
ejpam-5255	44	11	(	(	PUNCT
ejpam-5255	44	12	x	x	NOUN
ejpam-5255	44	13	,	,	PUNCT
ejpam-5255	44	14	ϱ+	ϱ+	X
ejpam-5255	44	15	,	,	PUNCT
ejpam-5255	44	16	ϱ−	ϱ−	VERB
ejpam-5255	44	17	)	)	PUNCT
ejpam-5255	44	18	is	be	AUX
ejpam-5255	44	19	called	call	VERB
ejpam-5255	44	20	a	a	DET
ejpam-5255	44	21	bipolar	bipolar	ADJ
ejpam-5255	44	22	fuzzy	fuzzy	ADJ
ejpam-5255	44	23	subalgebra	subalgebra	NOUN
ejpam-5255	44	24	if	if	SCONJ
ejpam-5255	44	25	it	it	PRON
ejpam-5255	44	26	meets	meet	VERB
ejpam-5255	44	27	two	two	NUM
ejpam-5255	44	28	criteria	criterion	NOUN
ejpam-5255	44	29	for	for	ADP
ejpam-5255	44	30	all	all	DET
ejpam-5255	44	31	a	a	PRON
ejpam-5255	44	32	,	,	PUNCT
ejpam-5255	44	33	g	g	PROPN
ejpam-5255	44	34	∈	∈	PROPN
ejpam-5255	44	35	x	x	X
ejpam-5255	44	36	:	:	PUNCT
ejpam-5255	44	37	(	(	PUNCT
ejpam-5255	44	38	1	1	X
ejpam-5255	44	39	)	)	PUNCT
ejpam-5255	44	40	ϱ+(a	ϱ+(a	NOUN
ejpam-5255	44	41	∗	∗	NOUN
ejpam-5255	44	42	g	g	NOUN
ejpam-5255	44	43	)	)	PUNCT
ejpam-5255	44	44	≥	≥	NOUN
ejpam-5255	44	45	min{ϱ+(a	min{ϱ+(a	NOUN
ejpam-5255	44	46	)	)	PUNCT
ejpam-5255	44	47	,	,	PUNCT
ejpam-5255	44	48	ϱ+(g	ϱ+(g	PROPN
ejpam-5255	44	49	)	)	PUNCT
ejpam-5255	44	50	}	}	PUNCT
ejpam-5255	44	51	.	.	PUNCT
ejpam-5255	45	1	(	(	PUNCT
ejpam-5255	45	2	2	2	X
ejpam-5255	45	3	)	)	PUNCT
ejpam-5255	45	4	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	45	5	∗	∗	NOUN
ejpam-5255	45	6	g	g	NOUN
ejpam-5255	45	7	)	)	PUNCT
ejpam-5255	45	8	≤	≤	NUM
ejpam-5255	45	9	max{ϱ−(a	max{ϱ−(a	NOUN
ejpam-5255	45	10	)	)	PUNCT
ejpam-5255	45	11	,	,	PUNCT
ejpam-5255	45	12	ϱ−(g	ϱ−(g	PROPN
ejpam-5255	45	13	)	)	PUNCT
ejpam-5255	45	14	}	}	PUNCT
ejpam-5255	45	15	.	.	PUNCT
ejpam-5255	46	1	this	this	DET
ejpam-5255	46	2	paper	paper	NOUN
ejpam-5255	46	3	interests	interest	NOUN
ejpam-5255	46	4	in	in	ADP
ejpam-5255	46	5	some	some	DET
ejpam-5255	46	6	types	type	NOUN
ejpam-5255	46	7	of	of	ADP
ejpam-5255	46	8	bipolar	bipolar	ADJ
ejpam-5255	46	9	fuzzy	fuzzy	ADJ
ejpam-5255	46	10	ideals	ideal	NOUN
ejpam-5255	46	11	of	of	ADP
ejpam-5255	46	12	a	a	DET
ejpam-5255	46	13	bck	bck	NOUN
ejpam-5255	46	14	-	-	PUNCT
ejpam-5255	46	15	algebra	algebra	NOUN
ejpam-5255	46	16	.	.	PUNCT
ejpam-5255	47	1	a.	a.	NOUN
ejpam-5255	47	2	almuhaimeed	almuhaimeed	PROPN
ejpam-5255	47	3	,	,	PUNCT
ejpam-5255	47	4	h.	h.	PROPN
ejpam-5255	47	5	alshehri	alshehri	PROPN
ejpam-5255	47	6	/	/	SYM
ejpam-5255	47	7	eur	eur	PROPN
ejpam-5255	47	8	.	.	PUNCT
ejpam-5255	48	1	j.	j.	PROPN
ejpam-5255	48	2	pure	pure	PROPN
ejpam-5255	48	3	appl	appl	PROPN
ejpam-5255	48	4	.	.	PROPN
ejpam-5255	48	5	math	math	PROPN
ejpam-5255	48	6	,	,	PUNCT
ejpam-5255	48	7	17	17	NUM
ejpam-5255	48	8	(	(	PUNCT
ejpam-5255	48	9	3	3	NUM
ejpam-5255	48	10	)	)	PUNCT
ejpam-5255	48	11	(	(	PUNCT
ejpam-5255	48	12	2024	2024	NUM
ejpam-5255	48	13	)	)	PUNCT
ejpam-5255	48	14	,	,	PUNCT
ejpam-5255	48	15	1831	1831	NUM
ejpam-5255	48	16	-	-	SYM
ejpam-5255	48	17	1841	1841	NUM
ejpam-5255	48	18	1833	1833	NUM
ejpam-5255	48	19	3	3	X
ejpam-5255	48	20	.	.	X
ejpam-5255	48	21	bipolar	bipolar	ADJ
ejpam-5255	48	22	fuzzy	fuzzy	ADJ
ejpam-5255	48	23	commutative	commutative	ADJ
ejpam-5255	48	24	ideal	ideal	NOUN
ejpam-5255	48	25	let	let	VERB
ejpam-5255	48	26	ϱ	ϱ	ADP
ejpam-5255	48	27	:	:	PUNCT
ejpam-5255	48	28	=	=	SYM
ejpam-5255	48	29	(	(	PUNCT
ejpam-5255	48	30	x	x	NOUN
ejpam-5255	48	31	,	,	PUNCT
ejpam-5255	48	32	ϱ+	ϱ+	X
ejpam-5255	48	33	,	,	PUNCT
ejpam-5255	48	34	ϱ−	ϱ−	NUM
ejpam-5255	48	35	)	)	PUNCT
ejpam-5255	48	36	be	be	VERB
ejpam-5255	48	37	a	a	DET
ejpam-5255	48	38	bipolar	bipolar	ADJ
ejpam-5255	48	39	fuzzy	fuzzy	ADJ
ejpam-5255	48	40	set	set	NOUN
ejpam-5255	48	41	.	.	PUNCT
ejpam-5255	49	1	a	a	DET
ejpam-5255	49	2	bipolar	bipolar	ADJ
ejpam-5255	49	3	fuzzy	fuzzy	ADJ
ejpam-5255	49	4	set	set	VERB
ejpam-5255	49	5	in	in	ADP
ejpam-5255	49	6	x	x	PROPN
ejpam-5255	49	7	is	be	AUX
ejpam-5255	49	8	said	say	VERB
ejpam-5255	49	9	to	to	PART
ejpam-5255	49	10	be	be	AUX
ejpam-5255	49	11	a	a	DET
ejpam-5255	49	12	bipolar	bipolar	ADJ
ejpam-5255	49	13	fuzzy	fuzzy	ADJ
ejpam-5255	49	14	commutative	commutative	ADJ
ejpam-5255	49	15	ideal	ideal	NOUN
ejpam-5255	49	16	of	of	ADP
ejpam-5255	49	17	x	x	PRON
ejpam-5255	49	18	if	if	SCONJ
ejpam-5255	49	19	(	(	PUNCT
ejpam-5255	49	20	1	1	NUM
ejpam-5255	49	21	)	)	PUNCT
ejpam-5255	49	22	ϱ+(0	ϱ+(0	NOUN
ejpam-5255	49	23	)	)	PUNCT
ejpam-5255	49	24	≥	≥	NOUN
ejpam-5255	49	25	ϱ+(a	ϱ+(a	NOUN
ejpam-5255	49	26	)	)	PUNCT
ejpam-5255	49	27	and	and	CCONJ
ejpam-5255	49	28	ϱ−(0	ϱ−(0	VERB
ejpam-5255	49	29	)	)	PUNCT
ejpam-5255	49	30	≤	≤	NUM
ejpam-5255	49	31	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	49	32	)	)	PUNCT
ejpam-5255	49	33	.	.	PUNCT
ejpam-5255	50	1	(	(	PUNCT
ejpam-5255	50	2	2	2	X
ejpam-5255	50	3	)	)	PUNCT
ejpam-5255	50	4	ϱ+(a	ϱ+(a	NOUN
ejpam-5255	50	5	∗	∗	NOUN
ejpam-5255	50	6	(	(	PUNCT
ejpam-5255	50	7	g	g	NOUN
ejpam-5255	50	8	∗	∗	NOUN
ejpam-5255	50	9	(	(	PUNCT
ejpam-5255	50	10	g	g	PROPN
ejpam-5255	50	11	∗	∗	X
ejpam-5255	50	12	a	a	NOUN
ejpam-5255	50	13	)	)	PUNCT
ejpam-5255	50	14	)	)	PUNCT
ejpam-5255	50	15	)	)	PUNCT
ejpam-5255	51	1	≥	≥	NOUN
ejpam-5255	51	2	min{ϱ+((a	min{ϱ+((a	NOUN
ejpam-5255	51	3	∗	∗	NOUN
ejpam-5255	51	4	g	g	NOUN
ejpam-5255	51	5	)	)	PUNCT
ejpam-5255	51	6	∗	∗	PROPN
ejpam-5255	51	7	w	w	PROPN
ejpam-5255	51	8	)	)	PUNCT
ejpam-5255	51	9	,	,	PUNCT
ejpam-5255	51	10	ϱ+(w	ϱ+(w	PROPN
ejpam-5255	51	11	)	)	PUNCT
ejpam-5255	51	12	}	}	PUNCT
ejpam-5255	51	13	.	.	PUNCT
ejpam-5255	52	1	(	(	PUNCT
ejpam-5255	52	2	3	3	X
ejpam-5255	52	3	)	)	PUNCT
ejpam-5255	52	4	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	52	5	∗	∗	NOUN
ejpam-5255	52	6	(	(	PUNCT
ejpam-5255	52	7	g	g	NOUN
ejpam-5255	52	8	∗	∗	NOUN
ejpam-5255	52	9	(	(	PUNCT
ejpam-5255	52	10	g	g	PROPN
ejpam-5255	52	11	∗	∗	X
ejpam-5255	52	12	a	a	NOUN
ejpam-5255	52	13	)	)	PUNCT
ejpam-5255	52	14	)	)	PUNCT
ejpam-5255	52	15	)	)	PUNCT
ejpam-5255	53	1	≤	≤	NUM
ejpam-5255	53	2	max{ϱ−((a	max{ϱ−((a	NOUN
ejpam-5255	53	3	∗	∗	VERB
ejpam-5255	53	4	g	g	NOUN
ejpam-5255	53	5	)	)	PUNCT
ejpam-5255	53	6	∗	∗	PROPN
ejpam-5255	53	7	w	w	PROPN
ejpam-5255	53	8	)	)	PUNCT
ejpam-5255	53	9	,	,	PUNCT
ejpam-5255	53	10	ϱ−(w	ϱ−(w	PROPN
ejpam-5255	53	11	)	)	PUNCT
ejpam-5255	53	12	}	}	PUNCT
ejpam-5255	53	13	.	.	PUNCT
ejpam-5255	54	1	for	for	ADP
ejpam-5255	54	2	all	all	DET
ejpam-5255	54	3	a	a	PRON
ejpam-5255	54	4	,	,	PUNCT
ejpam-5255	54	5	g	g	NOUN
ejpam-5255	54	6	,	,	PUNCT
ejpam-5255	54	7	w	w	PROPN
ejpam-5255	54	8	∈	∈	PROPN
ejpam-5255	54	9	x	x	X
ejpam-5255	54	10	:	:	PUNCT
ejpam-5255	54	11	example	example	NOUN
ejpam-5255	54	12	1	1	X
ejpam-5255	54	13	.	.	PUNCT
ejpam-5255	55	1	let	let	VERB
ejpam-5255	55	2	(	(	PUNCT
ejpam-5255	55	3	x	x	X
ejpam-5255	55	4	;	;	PUNCT
ejpam-5255	55	5	∗	∗	NOUN
ejpam-5255	55	6	,	,	PUNCT
ejpam-5255	55	7	0	0	NUM
ejpam-5255	55	8	)	)	PUNCT
ejpam-5255	55	9	be	be	AUX
ejpam-5255	55	10	a	a	DET
ejpam-5255	55	11	bck	bck	NOUN
ejpam-5255	55	12	-	-	PUNCT
ejpam-5255	55	13	algebra	algebra	NOUN
ejpam-5255	55	14	given	give	VERB
ejpam-5255	55	15	in	in	ADP
ejpam-5255	55	16	table	table	NOUN
ejpam-5255	55	17	1	1	NUM
ejpam-5255	55	18	as	as	SCONJ
ejpam-5255	55	19	follows	follow	VERB
ejpam-5255	55	20	:	:	PUNCT
ejpam-5255	55	21	*	*	PUNCT
ejpam-5255	55	22	0	0	NUM
ejpam-5255	56	1	r	r	NOUN
ejpam-5255	56	2	s	s	PROPN
ejpam-5255	56	3	t	t	NOUN
ejpam-5255	56	4	0	0	NUM
ejpam-5255	56	5	0	0	NUM
ejpam-5255	56	6	0	0	NUM
ejpam-5255	56	7	0	0	NUM
ejpam-5255	56	8	0	0	NUM
ejpam-5255	57	1	r	r	NOUN
ejpam-5255	57	2	r	r	NOUN
ejpam-5255	57	3	0	0	NUM
ejpam-5255	57	4	0	0	NUM
ejpam-5255	57	5	r	r	NOUN
ejpam-5255	57	6	s	s	NOUN
ejpam-5255	57	7	s	s	NOUN
ejpam-5255	57	8	r	r	NOUN
ejpam-5255	57	9	0	0	NUM
ejpam-5255	57	10	s	s	NOUN
ejpam-5255	57	11	t	t	NOUN
ejpam-5255	57	12	t	t	PROPN
ejpam-5255	57	13	t	t	PROPN
ejpam-5255	57	14	t	t	PROPN
ejpam-5255	57	15	0	0	NUM
ejpam-5255	57	16	table	table	NOUN
ejpam-5255	57	17	1	1	NUM
ejpam-5255	57	18	:	:	PUNCT
ejpam-5255	57	19	:	:	PUNCT
ejpam-5255	57	20	tabular	tabular	PROPN
ejpam-5255	57	21	representation	representation	NOUN
ejpam-5255	57	22	of	of	ADP
ejpam-5255	57	23	a	a	DET
ejpam-5255	57	24	bck	bck	NOUN
ejpam-5255	57	25	-	-	PUNCT
ejpam-5255	57	26	algebra	algebra	NOUN
ejpam-5255	57	27	x	x	PUNCT
ejpam-5255	57	28	in	in	ADP
ejpam-5255	57	29	example	example	NOUN
ejpam-5255	57	30	1	1	NUM
ejpam-5255	57	31	consider	consider	VERB
ejpam-5255	57	32	the	the	DET
ejpam-5255	57	33	bipolar	bipolar	ADJ
ejpam-5255	57	34	fuzzy	fuzzy	ADJ
ejpam-5255	57	35	set	set	VERB
ejpam-5255	57	36	ϱ	ϱ	ADP
ejpam-5255	57	37	:	:	PUNCT
ejpam-5255	57	38	=	=	SYM
ejpam-5255	57	39	(	(	PUNCT
ejpam-5255	57	40	x	x	NOUN
ejpam-5255	57	41	,	,	PUNCT
ejpam-5255	57	42	ϱ+	ϱ+	X
ejpam-5255	57	43	,	,	PUNCT
ejpam-5255	57	44	ϱ−	ϱ−	NUM
ejpam-5255	57	45	)	)	PUNCT
ejpam-5255	57	46	represented	represent	VERB
ejpam-5255	57	47	by	by	ADP
ejpam-5255	57	48	:	:	PUNCT
ejpam-5255	57	49	*	*	SYM
ejpam-5255	57	50	0	0	NUM
ejpam-5255	57	51	r	r	NOUN
ejpam-5255	57	52	s	s	PROPN
ejpam-5255	57	53	t	t	NOUN
ejpam-5255	57	54	ϱ−	ϱ−	PROPN
ejpam-5255	57	55	-0.5	-0.5	PROPN
ejpam-5255	57	56	-0.5	-0.5	PROPN
ejpam-5255	57	57	-0.5	-0.5	PROPN
ejpam-5255	57	58	-0.4	-0.4	NOUN
ejpam-5255	57	59	ϱ+	ϱ+	PUNCT
ejpam-5255	57	60	0.8	0.8	NUM
ejpam-5255	57	61	0.8	0.8	NUM
ejpam-5255	57	62	0.8	0.8	NUM
ejpam-5255	57	63	0.6	0.6	NUM
ejpam-5255	57	64	table	table	NOUN
ejpam-5255	57	65	2	2	NUM
ejpam-5255	57	66	:	:	PUNCT
ejpam-5255	57	67	:	:	PUNCT
ejpam-5255	57	68	tabular	tabular	PROPN
ejpam-5255	57	69	representation	representation	NOUN
ejpam-5255	57	70	of	of	ADP
ejpam-5255	57	71	a	a	DET
ejpam-5255	57	72	bipolar	bipolar	ADJ
ejpam-5255	57	73	fuzzy	fuzzy	ADJ
ejpam-5255	57	74	set	set	VERB
ejpam-5255	57	75	in	in	ADP
ejpam-5255	57	76	example	example	NOUN
ejpam-5255	57	77	1	1	NUM
ejpam-5255	57	78	direct	direct	ADJ
ejpam-5255	57	79	calculations	calculation	NOUN
ejpam-5255	57	80	implies	imply	VERB
ejpam-5255	57	81	that	that	SCONJ
ejpam-5255	57	82	ϱ	ϱ	ADP
ejpam-5255	57	83	:	:	PUNCT
ejpam-5255	57	84	=	=	SYM
ejpam-5255	57	85	(	(	PUNCT
ejpam-5255	57	86	x	x	NOUN
ejpam-5255	57	87	,	,	PUNCT
ejpam-5255	57	88	ϱ+	ϱ+	X
ejpam-5255	57	89	,	,	PUNCT
ejpam-5255	57	90	ϱ−	ϱ−	VERB
ejpam-5255	57	91	)	)	PUNCT
ejpam-5255	57	92	is	be	AUX
ejpam-5255	57	93	a	a	DET
ejpam-5255	57	94	bipolar	bipolar	ADJ
ejpam-5255	57	95	fuzzy	fuzzy	ADJ
ejpam-5255	57	96	commutative	commutative	ADJ
ejpam-5255	57	97	ideal	ideal	NOUN
ejpam-5255	57	98	of	of	ADP
ejpam-5255	57	99	x.	x.	NOUN
ejpam-5255	57	100	the	the	DET
ejpam-5255	57	101	following	follow	VERB
ejpam-5255	57	102	theorem	theorem	NOUN
ejpam-5255	57	103	provides	provide	VERB
ejpam-5255	57	104	equivalent	equivalent	ADJ
ejpam-5255	57	105	statements	statement	NOUN
ejpam-5255	57	106	regarding	regard	VERB
ejpam-5255	57	107	a	a	DET
ejpam-5255	57	108	bipolar	bipolar	ADJ
ejpam-5255	57	109	fuzzy	fuzzy	ADJ
ejpam-5255	57	110	commutative	commutative	ADJ
ejpam-5255	57	111	ideal	ideal	NOUN
ejpam-5255	57	112	.	.	PUNCT
ejpam-5255	58	1	theorem	theorem	NOUN
ejpam-5255	58	2	1	1	NUM
ejpam-5255	58	3	.	.	PUNCT
ejpam-5255	59	1	let	let	VERB
ejpam-5255	59	2	x	x	PRON
ejpam-5255	59	3	be	be	AUX
ejpam-5255	59	4	a	a	DET
ejpam-5255	59	5	bck	bck	NOUN
ejpam-5255	59	6	-	-	PUNCT
ejpam-5255	59	7	algebra	algebra	NOUN
ejpam-5255	59	8	and	and	CCONJ
ejpam-5255	59	9	ϱ	ϱ	ADP
ejpam-5255	59	10	:	:	PUNCT
ejpam-5255	59	11	=	=	SYM
ejpam-5255	59	12	(	(	PUNCT
ejpam-5255	59	13	x	x	NOUN
ejpam-5255	59	14	,	,	PUNCT
ejpam-5255	59	15	ϱ+	ϱ+	X
ejpam-5255	59	16	,	,	PUNCT
ejpam-5255	59	17	ϱ−	ϱ−	ADJ
ejpam-5255	59	18	)	)	PUNCT
ejpam-5255	59	19	a	a	DET
ejpam-5255	59	20	bipolar	bipolar	ADJ
ejpam-5255	59	21	fuzzy	fuzzy	ADJ
ejpam-5255	59	22	ideal	ideal	NOUN
ejpam-5255	59	23	of	of	ADP
ejpam-5255	59	24	x.	x.	NOUN
ejpam-5255	59	25	then	then	ADV
ejpam-5255	59	26	ϱ	ϱ	PROPN
ejpam-5255	59	27	is	be	AUX
ejpam-5255	59	28	a	a	DET
ejpam-5255	59	29	bipolar	bipolar	ADJ
ejpam-5255	59	30	fuzzy	fuzzy	ADJ
ejpam-5255	59	31	commutative	commutative	ADJ
ejpam-5255	59	32	ideal	ideal	NOUN
ejpam-5255	60	1	if	if	SCONJ
ejpam-5255	60	2	and	and	CCONJ
ejpam-5255	60	3	only	only	ADV
ejpam-5255	60	4	if	if	SCONJ
ejpam-5255	60	5	ϱ+(a	ϱ+(a	PROPN
ejpam-5255	60	6	∗	∗	NOUN
ejpam-5255	60	7	(	(	PUNCT
ejpam-5255	60	8	g	g	NOUN
ejpam-5255	60	9	∗	∗	NOUN
ejpam-5255	60	10	(	(	PUNCT
ejpam-5255	60	11	g	g	PROPN
ejpam-5255	60	12	∗	∗	X
ejpam-5255	60	13	a	a	NOUN
ejpam-5255	60	14	)	)	PUNCT
ejpam-5255	60	15	)	)	PUNCT
ejpam-5255	60	16	)	)	PUNCT
ejpam-5255	60	17	≥	≥	NOUN
ejpam-5255	60	18	ϱ+(a	ϱ+(a	NOUN
ejpam-5255	60	19	∗	∗	NOUN
ejpam-5255	60	20	g	g	NOUN
ejpam-5255	60	21	)	)	PUNCT
ejpam-5255	60	22	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	60	23	∗	∗	NOUN
ejpam-5255	60	24	(	(	PUNCT
ejpam-5255	60	25	g	g	NOUN
ejpam-5255	60	26	∗	∗	NOUN
ejpam-5255	60	27	(	(	PUNCT
ejpam-5255	60	28	g	g	PROPN
ejpam-5255	60	29	∗	∗	X
ejpam-5255	60	30	a	a	NOUN
ejpam-5255	60	31	)	)	PUNCT
ejpam-5255	60	32	)	)	PUNCT
ejpam-5255	60	33	)	)	PUNCT
ejpam-5255	60	34	≤	≤	NUM
ejpam-5255	60	35	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	60	36	∗	∗	NOUN
ejpam-5255	60	37	g	g	NOUN
ejpam-5255	60	38	)	)	PUNCT
ejpam-5255	60	39	(	(	PUNCT
ejpam-5255	60	40	3	3	X
ejpam-5255	60	41	)	)	PUNCT
ejpam-5255	60	42	proof	proof	NOUN
ejpam-5255	60	43	.	.	PUNCT
ejpam-5255	61	1	suppose	suppose	VERB
ejpam-5255	61	2	that	that	SCONJ
ejpam-5255	61	3	ϱ	ϱ	PROPN
ejpam-5255	61	4	is	be	AUX
ejpam-5255	61	5	a	a	DET
ejpam-5255	61	6	bipolar	bipolar	ADJ
ejpam-5255	61	7	fuzzy	fuzzy	ADJ
ejpam-5255	61	8	commutative	commutative	ADJ
ejpam-5255	61	9	ideal	ideal	NOUN
ejpam-5255	61	10	.	.	PUNCT
ejpam-5255	62	1	thus	thus	ADV
ejpam-5255	62	2	,	,	PUNCT
ejpam-5255	62	3	by	by	ADP
ejpam-5255	62	4	definition	definition	NOUN
ejpam-5255	62	5	,	,	PUNCT
ejpam-5255	62	6	we	we	PRON
ejpam-5255	62	7	have	have	VERB
ejpam-5255	62	8	ϱ+(a	ϱ+(a	NOUN
ejpam-5255	62	9	∗	∗	NOUN
ejpam-5255	62	10	(	(	PUNCT
ejpam-5255	62	11	g	g	NOUN
ejpam-5255	62	12	∗	∗	NOUN
ejpam-5255	62	13	(	(	PUNCT
ejpam-5255	62	14	g	g	PROPN
ejpam-5255	62	15	∗	∗	X
ejpam-5255	62	16	a	a	NOUN
ejpam-5255	62	17	)	)	PUNCT
ejpam-5255	62	18	)	)	PUNCT
ejpam-5255	62	19	)	)	PUNCT
ejpam-5255	63	1	≥	≥	PROPN
ejpam-5255	63	2	min	min	NOUN
ejpam-5255	63	3	{	{	PUNCT
ejpam-5255	63	4	ϱ+((a	ϱ+((a	ADP
ejpam-5255	63	5	∗	∗	NOUN
ejpam-5255	63	6	g	g	NOUN
ejpam-5255	63	7	)	)	PUNCT
ejpam-5255	63	8	∗	∗	PROPN
ejpam-5255	63	9	w	w	PROPN
ejpam-5255	63	10	)	)	PUNCT
ejpam-5255	63	11	,	,	PUNCT
ejpam-5255	63	12	ϱ+(w	ϱ+(w	PROPN
ejpam-5255	63	13	)	)	PUNCT
ejpam-5255	63	14	}	}	PUNCT
ejpam-5255	63	15	,	,	PUNCT
ejpam-5255	63	16	a.	a.	NOUN
ejpam-5255	63	17	almuhaimeed	almuhaimeed	PROPN
ejpam-5255	63	18	,	,	PUNCT
ejpam-5255	63	19	h.	h.	PROPN
ejpam-5255	63	20	alshehri	alshehri	PROPN
ejpam-5255	63	21	/	/	SYM
ejpam-5255	63	22	eur	eur	PROPN
ejpam-5255	63	23	.	.	PUNCT
ejpam-5255	64	1	j.	j.	PROPN
ejpam-5255	64	2	pure	pure	PROPN
ejpam-5255	64	3	appl	appl	PROPN
ejpam-5255	64	4	.	.	PROPN
ejpam-5255	64	5	math	math	PROPN
ejpam-5255	64	6	,	,	PUNCT
ejpam-5255	64	7	17	17	NUM
ejpam-5255	64	8	(	(	PUNCT
ejpam-5255	64	9	3	3	NUM
ejpam-5255	64	10	)	)	PUNCT
ejpam-5255	64	11	(	(	PUNCT
ejpam-5255	64	12	2024	2024	NUM
ejpam-5255	64	13	)	)	PUNCT
ejpam-5255	64	14	,	,	PUNCT
ejpam-5255	64	15	1831	1831	NUM
ejpam-5255	64	16	-	-	SYM
ejpam-5255	64	17	1841	1841	NUM
ejpam-5255	64	18	1834	1834	NUM
ejpam-5255	64	19	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	64	20	∗	∗	NOUN
ejpam-5255	64	21	(	(	PUNCT
ejpam-5255	64	22	g	g	NOUN
ejpam-5255	64	23	∗	∗	NOUN
ejpam-5255	64	24	(	(	PUNCT
ejpam-5255	64	25	g	g	PROPN
ejpam-5255	64	26	∗	∗	X
ejpam-5255	64	27	a	a	NOUN
ejpam-5255	64	28	)	)	PUNCT
ejpam-5255	64	29	)	)	PUNCT
ejpam-5255	64	30	)	)	PUNCT
ejpam-5255	65	1	≤	≤	NUM
ejpam-5255	65	2	max	max	PROPN
ejpam-5255	65	3	{	{	PUNCT
ejpam-5255	65	4	ϱ−((a	ϱ−((a	PROPN
ejpam-5255	65	5	∗	∗	NOUN
ejpam-5255	65	6	g	g	NOUN
ejpam-5255	65	7	)	)	PUNCT
ejpam-5255	65	8	∗	∗	PROPN
ejpam-5255	65	9	w	w	PROPN
ejpam-5255	65	10	)	)	PUNCT
ejpam-5255	65	11	,	,	PUNCT
ejpam-5255	65	12	ϱ−(w	ϱ−(w	PROPN
ejpam-5255	65	13	)	)	PUNCT
ejpam-5255	65	14	}	}	PUNCT
ejpam-5255	65	15	.	.	PUNCT
ejpam-5255	66	1	set	set	VERB
ejpam-5255	66	2	w	w	PROPN
ejpam-5255	66	3	=	=	NOUN
ejpam-5255	66	4	0	0	NUM
ejpam-5255	66	5	,	,	PUNCT
ejpam-5255	66	6	use	use	VERB
ejpam-5255	66	7	the	the	DET
ejpam-5255	66	8	fact	fact	NOUN
ejpam-5255	66	9	that	that	SCONJ
ejpam-5255	66	10	a	a	DET
ejpam-5255	66	11	∗	∗	NOUN
ejpam-5255	66	12	0	0	NUM
ejpam-5255	66	13	=	=	SYM
ejpam-5255	66	14	0	0	NUM
ejpam-5255	66	15	for	for	ADP
ejpam-5255	66	16	all	all	DET
ejpam-5255	66	17	a	a	DET
ejpam-5255	66	18	∈	∈	NOUN
ejpam-5255	66	19	x	x	PUNCT
ejpam-5255	66	20	and	and	CCONJ
ejpam-5255	66	21	condition	condition	NOUN
ejpam-5255	66	22	1	1	NUM
ejpam-5255	66	23	from	from	ADP
ejpam-5255	66	24	definition	definition	NOUN
ejpam-5255	66	25	,	,	PUNCT
ejpam-5255	66	26	the	the	DET
ejpam-5255	66	27	condition	condition	NOUN
ejpam-5255	66	28	in	in	ADP
ejpam-5255	66	29	(	(	PUNCT
ejpam-5255	66	30	1	1	X
ejpam-5255	66	31	)	)	PUNCT
ejpam-5255	66	32	holds	hold	VERB
ejpam-5255	66	33	.	.	PUNCT
ejpam-5255	67	1	conversely	conversely	ADV
ejpam-5255	67	2	,	,	PUNCT
ejpam-5255	67	3	assume	assume	VERB
ejpam-5255	67	4	that	that	SCONJ
ejpam-5255	67	5	ϱ	ϱ	NOUN
ejpam-5255	67	6	satisfies	satisfie	NOUN
ejpam-5255	67	7	the	the	DET
ejpam-5255	67	8	conditions	condition	NOUN
ejpam-5255	67	9	in	in	ADP
ejpam-5255	67	10	(	(	PUNCT
ejpam-5255	67	11	1	1	NUM
ejpam-5255	67	12	)	)	PUNCT
ejpam-5255	67	13	.	.	PUNCT
ejpam-5255	68	1	that	that	SCONJ
ejpam-5255	68	2	ϱ	ϱ	PROPN
ejpam-5255	68	3	is	be	AUX
ejpam-5255	68	4	a	a	DET
ejpam-5255	68	5	bipolar	bipolar	ADJ
ejpam-5255	68	6	fuzzy	fuzzy	ADJ
ejpam-5255	68	7	ideal	ideal	NOUN
ejpam-5255	68	8	implies	imply	VERB
ejpam-5255	68	9	that	that	SCONJ
ejpam-5255	68	10	ϱ+(a	ϱ+(a	PROPN
ejpam-5255	68	11	∗	∗	NOUN
ejpam-5255	68	12	g	g	NOUN
ejpam-5255	68	13	)	)	PUNCT
ejpam-5255	68	14	≥	≥	PROPN
ejpam-5255	68	15	min	min	NOUN
ejpam-5255	68	16	{	{	PUNCT
ejpam-5255	68	17	ϱ+((a	ϱ+((a	ADP
ejpam-5255	68	18	∗	∗	NOUN
ejpam-5255	68	19	g	g	NOUN
ejpam-5255	68	20	)	)	PUNCT
ejpam-5255	68	21	∗	∗	PROPN
ejpam-5255	68	22	w	w	PROPN
ejpam-5255	68	23	)	)	PUNCT
ejpam-5255	68	24	,	,	PUNCT
ejpam-5255	68	25	ϱ+(w	ϱ+(w	PROPN
ejpam-5255	68	26	)	)	PUNCT
ejpam-5255	68	27	}	}	PUNCT
ejpam-5255	68	28	,	,	PUNCT
ejpam-5255	68	29	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	68	30	∗	∗	NOUN
ejpam-5255	68	31	g	g	NOUN
ejpam-5255	68	32	)	)	PUNCT
ejpam-5255	68	33	≤	≤	NUM
ejpam-5255	68	34	max	max	PROPN
ejpam-5255	68	35	{	{	PUNCT
ejpam-5255	68	36	ϱ−((a	ϱ−((a	PROPN
ejpam-5255	68	37	∗	∗	NOUN
ejpam-5255	68	38	g	g	NOUN
ejpam-5255	68	39	)	)	PUNCT
ejpam-5255	68	40	∗	∗	PROPN
ejpam-5255	68	41	w	w	PROPN
ejpam-5255	68	42	)	)	PUNCT
ejpam-5255	68	43	,	,	PUNCT
ejpam-5255	68	44	ϱ−(w	ϱ−(w	PROPN
ejpam-5255	68	45	)	)	PUNCT
ejpam-5255	68	46	}	}	PUNCT
ejpam-5255	68	47	.	.	PUNCT
ejpam-5255	69	1	applying	apply	VERB
ejpam-5255	69	2	(	(	PUNCT
ejpam-5255	69	3	1	1	NUM
ejpam-5255	69	4	)	)	PUNCT
ejpam-5255	69	5	,	,	PUNCT
ejpam-5255	69	6	ϱ+(a	ϱ+(a	PROPN
ejpam-5255	69	7	∗	∗	NOUN
ejpam-5255	69	8	(	(	PUNCT
ejpam-5255	69	9	g	g	NOUN
ejpam-5255	69	10	∗	∗	NOUN
ejpam-5255	69	11	(	(	PUNCT
ejpam-5255	69	12	g	g	PROPN
ejpam-5255	69	13	∗	∗	X
ejpam-5255	69	14	a	a	NOUN
ejpam-5255	69	15	)	)	PUNCT
ejpam-5255	69	16	)	)	PUNCT
ejpam-5255	69	17	)	)	PUNCT
ejpam-5255	70	1	≥	≥	PROPN
ejpam-5255	70	2	min	min	NOUN
ejpam-5255	70	3	{	{	PUNCT
ejpam-5255	70	4	ϱ+((a	ϱ+((a	ADP
ejpam-5255	70	5	∗	∗	NOUN
ejpam-5255	70	6	g	g	NOUN
ejpam-5255	70	7	)	)	PUNCT
ejpam-5255	70	8	∗	∗	PROPN
ejpam-5255	70	9	w	w	PROPN
ejpam-5255	70	10	)	)	PUNCT
ejpam-5255	70	11	,	,	PUNCT
ejpam-5255	70	12	ϱ+(w	ϱ+(w	PROPN
ejpam-5255	70	13	)	)	PUNCT
ejpam-5255	70	14	}	}	PUNCT
ejpam-5255	70	15	,	,	PUNCT
ejpam-5255	70	16	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	70	17	∗	∗	NOUN
ejpam-5255	70	18	(	(	PUNCT
ejpam-5255	70	19	g	g	NOUN
ejpam-5255	70	20	∗	∗	NOUN
ejpam-5255	70	21	(	(	PUNCT
ejpam-5255	70	22	g	g	PROPN
ejpam-5255	70	23	∗	∗	X
ejpam-5255	70	24	a	a	NOUN
ejpam-5255	70	25	)	)	PUNCT
ejpam-5255	70	26	)	)	PUNCT
ejpam-5255	70	27	)	)	PUNCT
ejpam-5255	71	1	≤	≤	NUM
ejpam-5255	71	2	max	max	PROPN
ejpam-5255	71	3	{	{	PUNCT
ejpam-5255	71	4	ϱ−((a	ϱ−((a	PROPN
ejpam-5255	71	5	∗	∗	NOUN
ejpam-5255	71	6	g	g	NOUN
ejpam-5255	71	7	)	)	PUNCT
ejpam-5255	71	8	∗	∗	PROPN
ejpam-5255	71	9	w	w	PROPN
ejpam-5255	71	10	)	)	PUNCT
ejpam-5255	71	11	,	,	PUNCT
ejpam-5255	71	12	ϱ−(w	ϱ−(w	PROPN
ejpam-5255	71	13	)	)	PUNCT
ejpam-5255	71	14	}	}	PUNCT
ejpam-5255	71	15	,	,	PUNCT
ejpam-5255	71	16	and	and	CCONJ
ejpam-5255	71	17	so	so	ADV
ejpam-5255	71	18	ϱ	ϱ	NOUN
ejpam-5255	71	19	is	be	AUX
ejpam-5255	71	20	a	a	DET
ejpam-5255	71	21	bipolar	bipolar	ADJ
ejpam-5255	71	22	fuzzy	fuzzy	ADJ
ejpam-5255	71	23	commutative	commutative	ADJ
ejpam-5255	71	24	ideal	ideal	NOUN
ejpam-5255	71	25	.	.	PUNCT
ejpam-5255	72	1	theorem	theorem	NOUN
ejpam-5255	72	2	2	2	NUM
ejpam-5255	72	3	.	.	X
ejpam-5255	73	1	every	every	DET
ejpam-5255	73	2	bipolar	bipolar	ADJ
ejpam-5255	73	3	fuzzy	fuzzy	ADJ
ejpam-5255	73	4	commutative	commutative	ADJ
ejpam-5255	73	5	ideal	ideal	NOUN
ejpam-5255	73	6	of	of	ADP
ejpam-5255	73	7	a	a	DET
ejpam-5255	73	8	bck	bck	NOUN
ejpam-5255	73	9	-	-	PUNCT
ejpam-5255	73	10	algebra	algebra	NOUN
ejpam-5255	73	11	x	x	PUNCT
ejpam-5255	73	12	is	be	AUX
ejpam-5255	73	13	a	a	DET
ejpam-5255	73	14	bipolar	bipolar	ADJ
ejpam-5255	73	15	fuzzy	fuzzy	ADJ
ejpam-5255	73	16	ideal	ideal	NOUN
ejpam-5255	73	17	of	of	ADP
ejpam-5255	73	18	x.	x.	NOUN
ejpam-5255	73	19	proof	proof	NOUN
ejpam-5255	73	20	.	.	PUNCT
ejpam-5255	74	1	let	let	VERB
ejpam-5255	74	2	ϱ	ϱ	PART
ejpam-5255	74	3	be	be	AUX
ejpam-5255	74	4	a	a	DET
ejpam-5255	74	5	bipolar	bipolar	ADJ
ejpam-5255	74	6	fuzzy	fuzzy	ADJ
ejpam-5255	74	7	commutative	commutative	ADJ
ejpam-5255	74	8	ideal	ideal	NOUN
ejpam-5255	74	9	of	of	ADP
ejpam-5255	74	10	x	x	X
ejpam-5255	74	11	and	and	CCONJ
ejpam-5255	74	12	a	a	PRON
ejpam-5255	74	13	,	,	PUNCT
ejpam-5255	74	14	w	w	PROPN
ejpam-5255	74	15	∈	∈	PROPN
ejpam-5255	74	16	x	x	NOUN
ejpam-5255	74	17	,	,	PUNCT
ejpam-5255	74	18	then	then	ADV
ejpam-5255	74	19	we	we	PRON
ejpam-5255	74	20	have	have	VERB
ejpam-5255	74	21	min{ϱ+(a	min{ϱ+(a	PROPN
ejpam-5255	74	22	∗	∗	NOUN
ejpam-5255	74	23	w	w	NOUN
ejpam-5255	74	24	)	)	PUNCT
ejpam-5255	74	25	,	,	PUNCT
ejpam-5255	74	26	ϱ+(w	ϱ+(w	PROPN
ejpam-5255	74	27	)	)	PUNCT
ejpam-5255	74	28	}	}	PUNCT
ejpam-5255	75	1	=	=	SYM
ejpam-5255	75	2	min{ϱ+((a	min{ϱ+((a	NOUN
ejpam-5255	75	3	∗	∗	NOUN
ejpam-5255	75	4	0	0	NUM
ejpam-5255	75	5	)	)	PUNCT
ejpam-5255	75	6	∗	∗	PROPN
ejpam-5255	75	7	w	w	PROPN
ejpam-5255	75	8	)	)	PUNCT
ejpam-5255	75	9	,	,	PUNCT
ejpam-5255	75	10	ϱ+(w	ϱ+(w	PROPN
ejpam-5255	75	11	)	)	PUNCT
ejpam-5255	75	12	}	}	PUNCT
ejpam-5255	75	13	≤	≤	NOUN
ejpam-5255	75	14	ϱ+(a	ϱ+(a	NOUN
ejpam-5255	75	15	∗	∗	NOUN
ejpam-5255	75	16	(	(	PUNCT
ejpam-5255	75	17	0	0	NUM
ejpam-5255	75	18	∗	∗	NOUN
ejpam-5255	75	19	(	(	PUNCT
ejpam-5255	75	20	0	0	NUM
ejpam-5255	75	21	∗	∗	NOUN
ejpam-5255	75	22	a	a	NOUN
ejpam-5255	75	23	)	)	PUNCT
ejpam-5255	75	24	)	)	PUNCT
ejpam-5255	75	25	)	)	PUNCT
ejpam-5255	76	1	=	=	SYM
ejpam-5255	76	2	ϱ+(a	ϱ+(a	NOUN
ejpam-5255	76	3	)	)	PUNCT
ejpam-5255	76	4	and	and	CCONJ
ejpam-5255	76	5	max{ϱ−(a	max{ϱ−(a	PROPN
ejpam-5255	76	6	∗	∗	PROPN
ejpam-5255	76	7	w	w	PROPN
ejpam-5255	76	8	)	)	PUNCT
ejpam-5255	76	9	,	,	PUNCT
ejpam-5255	76	10	ϱ−(w	ϱ−(w	PROPN
ejpam-5255	76	11	)	)	PUNCT
ejpam-5255	76	12	}	}	PUNCT
ejpam-5255	76	13	=	=	SYM
ejpam-5255	76	14	max{ϱ−((a	max{ϱ−((a	PROPN
ejpam-5255	76	15	∗	∗	NOUN
ejpam-5255	76	16	0	0	NUM
ejpam-5255	76	17	)	)	PUNCT
ejpam-5255	76	18	∗	∗	NOUN
ejpam-5255	76	19	w	w	PROPN
ejpam-5255	76	20	)	)	PUNCT
ejpam-5255	76	21	,	,	PUNCT
ejpam-5255	76	22	ϱ−(w	ϱ−(w	PROPN
ejpam-5255	76	23	)	)	PUNCT
ejpam-5255	76	24	}	}	PUNCT
ejpam-5255	76	25	≥	≥	X
ejpam-5255	76	26	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	76	27	∗	∗	NOUN
ejpam-5255	76	28	(	(	PUNCT
ejpam-5255	76	29	0	0	NUM
ejpam-5255	76	30	∗	∗	NOUN
ejpam-5255	76	31	(	(	PUNCT
ejpam-5255	76	32	0	0	NUM
ejpam-5255	76	33	∗	∗	PROPN
ejpam-5255	76	34	w	w	NOUN
ejpam-5255	76	35	)	)	PUNCT
ejpam-5255	76	36	)	)	PUNCT
ejpam-5255	76	37	)	)	PUNCT
ejpam-5255	77	1	=	=	SYM
ejpam-5255	77	2	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	77	3	)	)	PUNCT
ejpam-5255	77	4	therefore	therefore	ADV
ejpam-5255	77	5	,	,	PUNCT
ejpam-5255	77	6	ϱ	ϱ	PROPN
ejpam-5255	77	7	is	be	AUX
ejpam-5255	77	8	a	a	DET
ejpam-5255	77	9	bipolar	bipolar	ADJ
ejpam-5255	77	10	fuzzy	fuzzy	ADJ
ejpam-5255	77	11	ideal	ideal	NOUN
ejpam-5255	77	12	.	.	PUNCT
ejpam-5255	78	1	the	the	DET
ejpam-5255	78	2	following	follow	VERB
ejpam-5255	78	3	corollary	corollary	ADJ
ejpam-5255	78	4	results	result	NOUN
ejpam-5255	78	5	from	from	ADP
ejpam-5255	78	6	the	the	DET
ejpam-5255	78	7	aforementioned	aforementioned	ADJ
ejpam-5255	78	8	theorem	theorem	NOUN
ejpam-5255	78	9	:	:	PUNCT
ejpam-5255	78	10	corollary	corollary	ADJ
ejpam-5255	78	11	1	1	NUM
ejpam-5255	78	12	.	.	PUNCT
ejpam-5255	79	1	every	every	DET
ejpam-5255	79	2	bipolar	bipolar	ADJ
ejpam-5255	79	3	fuzzy	fuzzy	ADJ
ejpam-5255	79	4	commutative	commutative	ADJ
ejpam-5255	79	5	ideal	ideal	NOUN
ejpam-5255	79	6	of	of	ADP
ejpam-5255	79	7	a	a	DET
ejpam-5255	79	8	bck	bck	NOUN
ejpam-5255	79	9	-	-	PUNCT
ejpam-5255	79	10	algebra	algebra	NOUN
ejpam-5255	79	11	x	x	PUNCT
ejpam-5255	79	12	is	be	AUX
ejpam-5255	79	13	a	a	DET
ejpam-5255	79	14	bipolar	bipolar	ADJ
ejpam-5255	79	15	fuzzy	fuzzy	ADJ
ejpam-5255	79	16	subalgebra	subalgebra	NOUN
ejpam-5255	79	17	of	of	ADP
ejpam-5255	79	18	x.	x.	NOUN
ejpam-5255	79	19	proof	proof	NOUN
ejpam-5255	79	20	.	.	PUNCT
ejpam-5255	80	1	clear	clear	ADJ
ejpam-5255	80	2	.	.	PUNCT
ejpam-5255	81	1	remark	remark	PROPN
ejpam-5255	81	2	1	1	NUM
ejpam-5255	81	3	.	.	PUNCT
ejpam-5255	82	1	a	a	DET
ejpam-5255	82	2	bipolar	bipolar	ADJ
ejpam-5255	82	3	fuzzy	fuzzy	ADJ
ejpam-5255	82	4	ideal	ideal	NOUN
ejpam-5255	82	5	of	of	ADP
ejpam-5255	82	6	a	a	DET
ejpam-5255	82	7	bck	bck	NOUN
ejpam-5255	82	8	-	-	PUNCT
ejpam-5255	82	9	algebra	algebra	NOUN
ejpam-5255	82	10	need	need	AUX
ejpam-5255	82	11	not	not	PART
ejpam-5255	82	12	be	be	AUX
ejpam-5255	82	13	a	a	DET
ejpam-5255	82	14	bipolar	bipolar	ADJ
ejpam-5255	82	15	fuzzy	fuzzy	ADJ
ejpam-5255	82	16	commutative	commutative	ADJ
ejpam-5255	82	17	ideal	ideal	NOUN
ejpam-5255	82	18	.	.	PUNCT
ejpam-5255	83	1	a	a	DET
ejpam-5255	83	2	counter	counter	ADJ
ejpam-5255	83	3	example	example	NOUN
ejpam-5255	83	4	is	be	AUX
ejpam-5255	83	5	given	give	VERB
ejpam-5255	83	6	as	as	SCONJ
ejpam-5255	83	7	follows	follow	VERB
ejpam-5255	83	8	:	:	PUNCT
ejpam-5255	83	9	consider	consider	VERB
ejpam-5255	83	10	the	the	DET
ejpam-5255	83	11	bck	bck	NOUN
ejpam-5255	83	12	-	-	PUNCT
ejpam-5255	83	13	algebra	algebra	NOUN
ejpam-5255	83	14	x	x	PUNCT
ejpam-5255	83	15	given	give	VERB
ejpam-5255	83	16	in	in	ADP
ejpam-5255	83	17	table	table	NOUN
ejpam-5255	83	18	3	3	NUM
ejpam-5255	83	19	:	:	PUNCT
ejpam-5255	83	20	a.	a.	NOUN
ejpam-5255	83	21	almuhaimeed	almuhaimeed	PROPN
ejpam-5255	83	22	,	,	PUNCT
ejpam-5255	83	23	h.	h.	PROPN
ejpam-5255	83	24	alshehri	alshehri	PROPN
ejpam-5255	83	25	/	/	SYM
ejpam-5255	83	26	eur	eur	PROPN
ejpam-5255	83	27	.	.	PUNCT
ejpam-5255	84	1	j.	j.	PROPN
ejpam-5255	84	2	pure	pure	PROPN
ejpam-5255	84	3	appl	appl	PROPN
ejpam-5255	84	4	.	.	PROPN
ejpam-5255	84	5	math	math	PROPN
ejpam-5255	84	6	,	,	PUNCT
ejpam-5255	84	7	17	17	NUM
ejpam-5255	84	8	(	(	PUNCT
ejpam-5255	84	9	3	3	NUM
ejpam-5255	84	10	)	)	PUNCT
ejpam-5255	84	11	(	(	PUNCT
ejpam-5255	84	12	2024	2024	NUM
ejpam-5255	84	13	)	)	PUNCT
ejpam-5255	84	14	,	,	PUNCT
ejpam-5255	84	15	1831	1831	NUM
ejpam-5255	84	16	-	-	SYM
ejpam-5255	84	17	1841	1841	NUM
ejpam-5255	84	18	1835	1835	NUM
ejpam-5255	84	19	*	*	SYM
ejpam-5255	84	20	0	0	NUM
ejpam-5255	84	21	1	1	NUM
ejpam-5255	84	22	2	2	NUM
ejpam-5255	84	23	3	3	NUM
ejpam-5255	84	24	4	4	NUM
ejpam-5255	84	25	0	0	NUM
ejpam-5255	84	26	0	0	NUM
ejpam-5255	84	27	0	0	NUM
ejpam-5255	84	28	0	0	NUM
ejpam-5255	84	29	0	0	NUM
ejpam-5255	84	30	0	0	NUM
ejpam-5255	84	31	1	1	NUM
ejpam-5255	84	32	1	1	NUM
ejpam-5255	84	33	0	0	NUM
ejpam-5255	84	34	1	1	NUM
ejpam-5255	84	35	0	0	NUM
ejpam-5255	84	36	0	0	NUM
ejpam-5255	84	37	2	2	NUM
ejpam-5255	84	38	2	2	NUM
ejpam-5255	84	39	2	2	NUM
ejpam-5255	84	40	0	0	NUM
ejpam-5255	84	41	0	0	NUM
ejpam-5255	84	42	0	0	NUM
ejpam-5255	84	43	3	3	NUM
ejpam-5255	84	44	3	3	NUM
ejpam-5255	84	45	3	3	NUM
ejpam-5255	84	46	3	3	NUM
ejpam-5255	84	47	0	0	NUM
ejpam-5255	84	48	0	0	NUM
ejpam-5255	84	49	4	4	NUM
ejpam-5255	84	50	4	4	NUM
ejpam-5255	84	51	4	4	NUM
ejpam-5255	84	52	4	4	NUM
ejpam-5255	84	53	3	3	NUM
ejpam-5255	84	54	0	0	NUM
ejpam-5255	84	55	table	table	NOUN
ejpam-5255	84	56	3	3	NUM
ejpam-5255	84	57	:	:	PUNCT
ejpam-5255	84	58	:	:	PUNCT
ejpam-5255	84	59	tabular	tabular	PROPN
ejpam-5255	84	60	representation	representation	NOUN
ejpam-5255	84	61	of	of	ADP
ejpam-5255	84	62	a	a	DET
ejpam-5255	84	63	bck	bck	NOUN
ejpam-5255	84	64	-	-	PUNCT
ejpam-5255	84	65	algebra	algebra	NOUN
ejpam-5255	84	66	x	x	PUNCT
ejpam-5255	84	67	described	describe	VERB
ejpam-5255	84	68	above	above	ADP
ejpam-5255	84	69	*	*	PUNCT
ejpam-5255	84	70	0	0	NUM
ejpam-5255	84	71	1	1	NUM
ejpam-5255	84	72	2	2	NUM
ejpam-5255	84	73	3	3	NUM
ejpam-5255	84	74	4	4	NUM
ejpam-5255	84	75	ϱ−	ϱ−	NOUN
ejpam-5255	84	76	-0.6	-0.6	X
ejpam-5255	84	77	-0.5	-0.5	X
ejpam-5255	84	78	-0.5	-0.5	PROPN
ejpam-5255	84	79	-0.4	-0.4	PROPN
ejpam-5255	84	80	-0.4	-0.4	NOUN
ejpam-5255	84	81	ϱ+	ϱ+	PUNCT
ejpam-5255	84	82	0.8	0.8	NUM
ejpam-5255	84	83	0.7	0.7	NUM
ejpam-5255	84	84	0.7	0.7	NUM
ejpam-5255	84	85	0.7	0.7	NUM
ejpam-5255	84	86	0.6	0.6	NUM
ejpam-5255	84	87	table	table	NOUN
ejpam-5255	84	88	4	4	NUM
ejpam-5255	84	89	:	:	PUNCT
ejpam-5255	84	90	:	:	PUNCT
ejpam-5255	84	91	tabular	tabular	PROPN
ejpam-5255	84	92	representation	representation	NOUN
ejpam-5255	84	93	of	of	ADP
ejpam-5255	84	94	a	a	DET
ejpam-5255	84	95	bipolar	bipolar	ADJ
ejpam-5255	84	96	fuzzy	fuzzy	NOUN
ejpam-5255	84	97	set	set	VERB
ejpam-5255	84	98	ϱ	ϱ	ADP
ejpam-5255	84	99	described	describe	VERB
ejpam-5255	84	100	above	above	ADV
ejpam-5255	84	101	let	let	VERB
ejpam-5255	84	102	ϱ	ϱ	VERB
ejpam-5255	84	103	:	:	PUNCT
ejpam-5255	84	104	=	=	SYM
ejpam-5255	84	105	(	(	PUNCT
ejpam-5255	84	106	x	x	NOUN
ejpam-5255	84	107	,	,	PUNCT
ejpam-5255	84	108	ϱ+	ϱ+	X
ejpam-5255	84	109	,	,	PUNCT
ejpam-5255	84	110	ϱ−	ϱ−	NUM
ejpam-5255	84	111	)	)	PUNCT
ejpam-5255	84	112	be	be	VERB
ejpam-5255	84	113	a	a	DET
ejpam-5255	84	114	bipolar	bipolar	ADJ
ejpam-5255	84	115	fuzzy	fuzzy	ADJ
ejpam-5255	84	116	set	set	VERB
ejpam-5255	84	117	in	in	ADP
ejpam-5255	84	118	x	x	PUNCT
ejpam-5255	84	119	given	give	VERB
ejpam-5255	84	120	by	by	ADP
ejpam-5255	84	121	table	table	NOUN
ejpam-5255	84	122	4	4	NUM
ejpam-5255	84	123	:	:	PUNCT
ejpam-5255	84	124	thus	thus	ADV
ejpam-5255	84	125	ϱ	ϱ	NOUN
ejpam-5255	84	126	is	be	AUX
ejpam-5255	84	127	a	a	DET
ejpam-5255	84	128	bipolar	bipolar	ADJ
ejpam-5255	84	129	fuzzy	fuzzy	ADJ
ejpam-5255	84	130	ideal	ideal	NOUN
ejpam-5255	84	131	.	.	PUNCT
ejpam-5255	85	1	however	however	ADV
ejpam-5255	85	2	,	,	PUNCT
ejpam-5255	85	3	ϱ	ϱ	PROPN
ejpam-5255	85	4	is	be	AUX
ejpam-5255	85	5	not	not	PART
ejpam-5255	85	6	a	a	DET
ejpam-5255	85	7	bipolar	bipolar	ADJ
ejpam-5255	85	8	fuzzy	fuzzy	ADJ
ejpam-5255	85	9	commutative	commutative	ADJ
ejpam-5255	85	10	ideal	ideal	NOUN
ejpam-5255	85	11	as	as	ADP
ejpam-5255	85	12	ϱ+(2	ϱ+(2	NOUN
ejpam-5255	85	13	∗	∗	NOUN
ejpam-5255	85	14	(	(	PUNCT
ejpam-5255	85	15	3	3	NUM
ejpam-5255	85	16	∗	∗	NOUN
ejpam-5255	85	17	(	(	PUNCT
ejpam-5255	85	18	3	3	NUM
ejpam-5255	85	19	∗	∗	NOUN
ejpam-5255	85	20	2	2	NUM
ejpam-5255	85	21	)	)	PUNCT
ejpam-5255	85	22	)	)	PUNCT
ejpam-5255	85	23	)	)	PUNCT
ejpam-5255	86	1	=	=	PUNCT
ejpam-5255	86	2	0.7	0.7	NUM
ejpam-5255	86	3	<	<	X
ejpam-5255	86	4	0.8	0.8	NUM
ejpam-5255	86	5	=	=	SYM
ejpam-5255	86	6	min{ϱ+((2	min{ϱ+((2	NOUN
ejpam-5255	86	7	∗	∗	NOUN
ejpam-5255	86	8	3	3	NUM
ejpam-5255	86	9	)	)	PUNCT
ejpam-5255	86	10	∗	∗	NOUN
ejpam-5255	86	11	0	0	NUM
ejpam-5255	86	12	)	)	PUNCT
ejpam-5255	86	13	,	,	PUNCT
ejpam-5255	86	14	ϱ+(0	ϱ+(0	NOUN
ejpam-5255	86	15	)	)	PUNCT
ejpam-5255	86	16	}	}	PUNCT
ejpam-5255	86	17	.	.	PUNCT
ejpam-5255	87	1	proposition	proposition	NOUN
ejpam-5255	87	2	1	1	NUM
ejpam-5255	87	3	.	.	PUNCT
ejpam-5255	88	1	every	every	DET
ejpam-5255	88	2	bipolar	bipolar	ADJ
ejpam-5255	88	3	fuzzy	fuzzy	ADJ
ejpam-5255	88	4	commutative	commutative	ADJ
ejpam-5255	88	5	ideal	ideal	NOUN
ejpam-5255	88	6	of	of	ADP
ejpam-5255	88	7	a	a	DET
ejpam-5255	88	8	bck	bck	NOUN
ejpam-5255	88	9	-	-	PUNCT
ejpam-5255	88	10	algebra	algebra	NOUN
ejpam-5255	88	11	is	be	AUX
ejpam-5255	88	12	order	order	NOUN
ejpam-5255	88	13	preserving	preserve	VERB
ejpam-5255	88	14	.	.	PUNCT
ejpam-5255	89	1	proof	proof	NOUN
ejpam-5255	89	2	.	.	PUNCT
ejpam-5255	90	1	assume	assume	VERB
ejpam-5255	90	2	that	that	SCONJ
ejpam-5255	90	3	ϱ	ϱ	ADP
ejpam-5255	90	4	:	:	PUNCT
ejpam-5255	90	5	=	=	SYM
ejpam-5255	90	6	(	(	PUNCT
ejpam-5255	90	7	x	x	NOUN
ejpam-5255	90	8	,	,	PUNCT
ejpam-5255	90	9	ϱ+	ϱ+	X
ejpam-5255	90	10	,	,	PUNCT
ejpam-5255	90	11	ϱ−	ϱ−	VERB
ejpam-5255	90	12	)	)	PUNCT
ejpam-5255	90	13	is	be	AUX
ejpam-5255	90	14	a	a	DET
ejpam-5255	90	15	bipolar	bipolar	ADJ
ejpam-5255	90	16	fuzzy	fuzzy	ADJ
ejpam-5255	90	17	commutative	commutative	ADJ
ejpam-5255	90	18	ideal	ideal	NOUN
ejpam-5255	90	19	and	and	CCONJ
ejpam-5255	90	20	let	let	VERB
ejpam-5255	90	21	a	a	DET
ejpam-5255	90	22	,	,	PUNCT
ejpam-5255	90	23	g	g	NOUN
ejpam-5255	90	24	,	,	PUNCT
ejpam-5255	90	25	w	w	PROPN
ejpam-5255	90	26	∈	∈	PROPN
ejpam-5255	90	27	x	x	PUNCT
ejpam-5255	90	28	in	in	ADP
ejpam-5255	90	29	which	which	PRON
ejpam-5255	90	30	a	a	DET
ejpam-5255	90	31	≤	≤	ADJ
ejpam-5255	90	32	w.	w.	NOUN
ejpam-5255	90	33	then	then	ADV
ejpam-5255	90	34	a	a	DET
ejpam-5255	90	35	∗	∗	NOUN
ejpam-5255	90	36	w	w	NOUN
ejpam-5255	90	37	=	=	NOUN
ejpam-5255	90	38	0	0	PROPN
ejpam-5255	90	39	.	.	PUNCT
ejpam-5255	91	1	that	that	SCONJ
ejpam-5255	91	2	ϱ+(a	ϱ+(a	PROPN
ejpam-5255	91	3	∗	∗	NOUN
ejpam-5255	91	4	(	(	PUNCT
ejpam-5255	91	5	g	g	NOUN
ejpam-5255	91	6	∗	∗	NOUN
ejpam-5255	91	7	(	(	PUNCT
ejpam-5255	91	8	g	g	PROPN
ejpam-5255	91	9	∗	∗	X
ejpam-5255	91	10	a	a	NOUN
ejpam-5255	91	11	)	)	PUNCT
ejpam-5255	91	12	)	)	PUNCT
ejpam-5255	91	13	)	)	PUNCT
ejpam-5255	92	1	≥	≥	PROPN
ejpam-5255	92	2	min	min	NOUN
ejpam-5255	92	3	{	{	PUNCT
ejpam-5255	92	4	ϱ+((a	ϱ+((a	ADP
ejpam-5255	92	5	∗	∗	NOUN
ejpam-5255	92	6	g	g	NOUN
ejpam-5255	92	7	)	)	PUNCT
ejpam-5255	92	8	∗	∗	PROPN
ejpam-5255	92	9	w	w	PROPN
ejpam-5255	92	10	)	)	PUNCT
ejpam-5255	92	11	,	,	PUNCT
ejpam-5255	92	12	ϱ+(w	ϱ+(w	PROPN
ejpam-5255	92	13	)	)	PUNCT
ejpam-5255	92	14	}	}	PUNCT
ejpam-5255	92	15	,	,	PUNCT
ejpam-5255	92	16	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	92	17	∗	∗	NOUN
ejpam-5255	92	18	(	(	PUNCT
ejpam-5255	92	19	g	g	NOUN
ejpam-5255	92	20	∗	∗	NOUN
ejpam-5255	92	21	(	(	PUNCT
ejpam-5255	92	22	g	g	PROPN
ejpam-5255	92	23	∗	∗	X
ejpam-5255	92	24	a	a	NOUN
ejpam-5255	92	25	)	)	PUNCT
ejpam-5255	92	26	)	)	PUNCT
ejpam-5255	92	27	)	)	PUNCT
ejpam-5255	93	1	≤	≤	NUM
ejpam-5255	93	2	max{ϱ−((a	max{ϱ−((a	NOUN
ejpam-5255	93	3	∗	∗	VERB
ejpam-5255	93	4	g	g	NOUN
ejpam-5255	93	5	)	)	PUNCT
ejpam-5255	93	6	∗	∗	PROPN
ejpam-5255	93	7	w	w	PROPN
ejpam-5255	93	8	)	)	PUNCT
ejpam-5255	93	9	,	,	PUNCT
ejpam-5255	93	10	ϱ−(w	ϱ−(w	PROPN
ejpam-5255	93	11	)	)	PUNCT
ejpam-5255	93	12	}	}	PUNCT
ejpam-5255	93	13	,	,	PUNCT
ejpam-5255	93	14	and	and	CCONJ
ejpam-5255	93	15	setting	set	VERB
ejpam-5255	93	16	g	g	NOUN
ejpam-5255	93	17	=	=	SYM
ejpam-5255	93	18	0	0	NUM
ejpam-5255	93	19	implies	imply	VERB
ejpam-5255	93	20	that	that	SCONJ
ejpam-5255	93	21	ϱ+(a	ϱ+(a	NOUN
ejpam-5255	93	22	)	)	PUNCT
ejpam-5255	94	1	=	=	SYM
ejpam-5255	94	2	ϱ+(a	ϱ+(a	NOUN
ejpam-5255	94	3	∗	∗	NOUN
ejpam-5255	94	4	(	(	PUNCT
ejpam-5255	94	5	0	0	NUM
ejpam-5255	94	6	∗	∗	NOUN
ejpam-5255	94	7	(	(	PUNCT
ejpam-5255	94	8	0	0	NUM
ejpam-5255	94	9	∗	∗	NOUN
ejpam-5255	94	10	a	a	NOUN
ejpam-5255	94	11	)	)	PUNCT
ejpam-5255	94	12	)	)	PUNCT
ejpam-5255	94	13	)	)	PUNCT
ejpam-5255	94	14	≥	≥	NOUN
ejpam-5255	94	15	min{ϱ+((a	min{ϱ+((a	NOUN
ejpam-5255	94	16	∗	∗	NOUN
ejpam-5255	94	17	0	0	NUM
ejpam-5255	94	18	)	)	PUNCT
ejpam-5255	94	19	∗	∗	PROPN
ejpam-5255	94	20	w	w	PROPN
ejpam-5255	94	21	)	)	PUNCT
ejpam-5255	94	22	,	,	PUNCT
ejpam-5255	94	23	ϱ+(w	ϱ+(w	PROPN
ejpam-5255	94	24	)	)	PUNCT
ejpam-5255	94	25	}	}	PUNCT
ejpam-5255	94	26	=	=	SYM
ejpam-5255	94	27	min{ϱ+(a	min{ϱ+(a	PROPN
ejpam-5255	94	28	∗	∗	NOUN
ejpam-5255	94	29	w	w	NOUN
ejpam-5255	94	30	)	)	PUNCT
ejpam-5255	94	31	,	,	PUNCT
ejpam-5255	94	32	ϱ+(w	ϱ+(w	PROPN
ejpam-5255	94	33	)	)	PUNCT
ejpam-5255	94	34	}	}	PUNCT
ejpam-5255	94	35	=	=	SYM
ejpam-5255	94	36	min{ϱ+(0	min{ϱ+(0	NOUN
ejpam-5255	94	37	)	)	PUNCT
ejpam-5255	94	38	,	,	PUNCT
ejpam-5255	94	39	ϱ+(w	ϱ+(w	PROPN
ejpam-5255	94	40	)	)	PUNCT
ejpam-5255	94	41	}	}	PUNCT
ejpam-5255	94	42	=	=	SYM
ejpam-5255	94	43	ϱ+(w	ϱ+(w	NUM
ejpam-5255	94	44	)	)	PUNCT
ejpam-5255	94	45	,	,	PUNCT
ejpam-5255	94	46	and	and	CCONJ
ejpam-5255	94	47	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	94	48	)	)	PUNCT
ejpam-5255	94	49	=	=	SYM
ejpam-5255	94	50	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	94	51	∗	∗	NOUN
ejpam-5255	94	52	(	(	PUNCT
ejpam-5255	94	53	0	0	NUM
ejpam-5255	94	54	∗	∗	NOUN
ejpam-5255	94	55	(	(	PUNCT
ejpam-5255	94	56	0	0	NUM
ejpam-5255	94	57	∗	∗	NOUN
ejpam-5255	94	58	a	a	NOUN
ejpam-5255	94	59	)	)	PUNCT
ejpam-5255	94	60	)	)	PUNCT
ejpam-5255	94	61	)	)	PUNCT
ejpam-5255	94	62	≤	≤	NUM
ejpam-5255	95	1	max	max	PROPN
ejpam-5255	95	2	{	{	PUNCT
ejpam-5255	95	3	ϱ−((a	ϱ−((a	PROPN
ejpam-5255	95	4	∗	∗	NOUN
ejpam-5255	95	5	0	0	NUM
ejpam-5255	95	6	)	)	PUNCT
ejpam-5255	95	7	∗	∗	NOUN
ejpam-5255	95	8	w	w	PROPN
ejpam-5255	95	9	)	)	PUNCT
ejpam-5255	95	10	,	,	PUNCT
ejpam-5255	95	11	ϱ−(w	ϱ−(w	PROPN
ejpam-5255	95	12	)	)	PUNCT
ejpam-5255	95	13	}	}	PUNCT
ejpam-5255	95	14	=	=	SYM
ejpam-5255	95	15	max	max	X
ejpam-5255	95	16	{	{	PUNCT
ejpam-5255	95	17	ϱ−(a	ϱ−(a	PROPN
ejpam-5255	95	18	∗	∗	NOUN
ejpam-5255	95	19	w	w	NOUN
ejpam-5255	95	20	)	)	PUNCT
ejpam-5255	95	21	,	,	PUNCT
ejpam-5255	95	22	ϱ−(w	ϱ−(w	PROPN
ejpam-5255	95	23	)	)	PUNCT
ejpam-5255	95	24	}	}	PUNCT
ejpam-5255	96	1	=	=	SYM
ejpam-5255	96	2	max	max	PROPN
ejpam-5255	96	3	{	{	PUNCT
ejpam-5255	96	4	ϱ−(0	ϱ−(0	PROPN
ejpam-5255	96	5	)	)	PUNCT
ejpam-5255	96	6	,	,	PUNCT
ejpam-5255	96	7	ϱ−(w	ϱ−(w	PROPN
ejpam-5255	96	8	)	)	PUNCT
ejpam-5255	96	9	}	}	PUNCT
ejpam-5255	96	10	a.	a.	NOUN
ejpam-5255	96	11	almuhaimeed	almuhaimeed	PROPN
ejpam-5255	96	12	,	,	PUNCT
ejpam-5255	96	13	h.	h.	PROPN
ejpam-5255	96	14	alshehri	alshehri	PROPN
ejpam-5255	96	15	/	/	SYM
ejpam-5255	96	16	eur	eur	PROPN
ejpam-5255	96	17	.	.	PUNCT
ejpam-5255	97	1	j.	j.	PROPN
ejpam-5255	97	2	pure	pure	PROPN
ejpam-5255	97	3	appl	appl	PROPN
ejpam-5255	97	4	.	.	PROPN
ejpam-5255	97	5	math	math	PROPN
ejpam-5255	97	6	,	,	PUNCT
ejpam-5255	97	7	17	17	NUM
ejpam-5255	97	8	(	(	PUNCT
ejpam-5255	97	9	3	3	NUM
ejpam-5255	97	10	)	)	PUNCT
ejpam-5255	97	11	(	(	PUNCT
ejpam-5255	97	12	2024	2024	NUM
ejpam-5255	97	13	)	)	PUNCT
ejpam-5255	97	14	,	,	PUNCT
ejpam-5255	97	15	1831	1831	NUM
ejpam-5255	97	16	-	-	SYM
ejpam-5255	97	17	1841	1841	NUM
ejpam-5255	97	18	1836	1836	NUM
ejpam-5255	97	19	=	=	SYM
ejpam-5255	97	20	ϱ−(w	ϱ−(w	PROPN
ejpam-5255	97	21	)	)	PUNCT
ejpam-5255	97	22	therefore	therefore	ADV
ejpam-5255	97	23	,	,	PUNCT
ejpam-5255	97	24	ϱ	ϱ	ADP
ejpam-5255	97	25	:	:	PUNCT
ejpam-5255	97	26	=	=	SYM
ejpam-5255	97	27	(	(	PUNCT
ejpam-5255	97	28	x	x	NOUN
ejpam-5255	97	29	,	,	PUNCT
ejpam-5255	97	30	ϱ+	ϱ+	X
ejpam-5255	97	31	,	,	PUNCT
ejpam-5255	97	32	ϱ−	ϱ−	VERB
ejpam-5255	97	33	)	)	PUNCT
ejpam-5255	97	34	is	be	AUX
ejpam-5255	97	35	order	order	NOUN
ejpam-5255	97	36	preserving	preserve	VERB
ejpam-5255	97	37	as	as	SCONJ
ejpam-5255	97	38	required	require	VERB
ejpam-5255	97	39	.	.	PUNCT
ejpam-5255	98	1	lemma	lemma	PROPN
ejpam-5255	98	2	1	1	NUM
ejpam-5255	98	3	.	.	PUNCT
ejpam-5255	98	4	suppose	suppose	VERB
ejpam-5255	98	5	that	that	SCONJ
ejpam-5255	98	6	ϱ	ϱ	ADP
ejpam-5255	98	7	:	:	PUNCT
ejpam-5255	98	8	=	=	SYM
ejpam-5255	98	9	(	(	PUNCT
ejpam-5255	98	10	x	x	NOUN
ejpam-5255	98	11	,	,	PUNCT
ejpam-5255	98	12	ϱ+	ϱ+	X
ejpam-5255	98	13	,	,	PUNCT
ejpam-5255	98	14	ϱ−	ϱ−	VERB
ejpam-5255	98	15	)	)	PUNCT
ejpam-5255	98	16	is	be	AUX
ejpam-5255	98	17	a	a	DET
ejpam-5255	98	18	bipolar	bipolar	ADJ
ejpam-5255	98	19	fuzzy	fuzzy	ADJ
ejpam-5255	98	20	ideal	ideal	NOUN
ejpam-5255	98	21	.	.	PUNCT
ejpam-5255	99	1	if	if	SCONJ
ejpam-5255	99	2	a	a	DET
ejpam-5255	99	3	∗	∗	NOUN
ejpam-5255	99	4	g	g	NOUN
ejpam-5255	99	5	≤	≤	NOUN
ejpam-5255	99	6	w	w	NOUN
ejpam-5255	99	7	holds	hold	VERB
ejpam-5255	99	8	for	for	ADP
ejpam-5255	99	9	some	some	PRON
ejpam-5255	99	10	a	a	PRON
ejpam-5255	99	11	,	,	PUNCT
ejpam-5255	99	12	g	g	NOUN
ejpam-5255	99	13	,	,	PUNCT
ejpam-5255	99	14	w	w	PROPN
ejpam-5255	99	15	∈	∈	PROPN
ejpam-5255	99	16	x	x	X
ejpam-5255	99	17	,	,	PUNCT
ejpam-5255	99	18	then	then	ADV
ejpam-5255	99	19	ϱ+(a	ϱ+(a	PROPN
ejpam-5255	99	20	)	)	PUNCT
ejpam-5255	99	21	≥	≥	NOUN
ejpam-5255	99	22	{	{	PUNCT
ejpam-5255	99	23	ϱ+(g	ϱ+(g	PROPN
ejpam-5255	99	24	)	)	PUNCT
ejpam-5255	99	25	,	,	PUNCT
ejpam-5255	99	26	ϱ+(w	ϱ+(w	PROPN
ejpam-5255	99	27	)	)	PUNCT
ejpam-5255	99	28	}	}	PUNCT
ejpam-5255	99	29	,	,	PUNCT
ejpam-5255	99	30	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	99	31	)	)	PUNCT
ejpam-5255	99	32	≤	≤	NUM
ejpam-5255	100	1	max	max	PROPN
ejpam-5255	100	2	{	{	PUNCT
ejpam-5255	100	3	ϱ−(g	ϱ−(g	PROPN
ejpam-5255	100	4	)	)	PUNCT
ejpam-5255	100	5	,	,	PUNCT
ejpam-5255	100	6	ϱ−(w	ϱ−(w	PROPN
ejpam-5255	100	7	)	)	PUNCT
ejpam-5255	100	8	}	}	PUNCT
ejpam-5255	100	9	.	.	PUNCT
ejpam-5255	101	1	recall	recall	VERB
ejpam-5255	101	2	that	that	SCONJ
ejpam-5255	101	3	a	a	DET
ejpam-5255	101	4	bck	bck	NOUN
ejpam-5255	101	5	-	-	PUNCT
ejpam-5255	101	6	algebra	algebra	NOUN
ejpam-5255	101	7	x	x	VERB
ejpam-5255	101	8	is	be	AUX
ejpam-5255	101	9	commutative	commutative	ADJ
ejpam-5255	101	10	if	if	SCONJ
ejpam-5255	101	11	it	it	PRON
ejpam-5255	101	12	satisfies	satisfy	VERB
ejpam-5255	101	13	the	the	DET
ejpam-5255	101	14	condition	condition	NOUN
ejpam-5255	101	15	:	:	PUNCT
ejpam-5255	101	16	f	f	PROPN
ejpam-5255	101	17	∗	∗	NOUN
ejpam-5255	101	18	(	(	PUNCT
ejpam-5255	101	19	f	f	PROPN
ejpam-5255	101	20	∗	∗	X
ejpam-5255	101	21	g	g	NOUN
ejpam-5255	101	22	)	)	PUNCT
ejpam-5255	102	1	=	=	SYM
ejpam-5255	102	2	g	g	PROPN
ejpam-5255	102	3	∗	∗	NOUN
ejpam-5255	102	4	(	(	PUNCT
ejpam-5255	102	5	g	g	PROPN
ejpam-5255	102	6	∗	∗	X
ejpam-5255	102	7	f	f	NOUN
ejpam-5255	102	8	)	)	PUNCT
ejpam-5255	102	9	.	.	PUNCT
ejpam-5255	103	1	theorem	theorem	NOUN
ejpam-5255	103	2	3	3	X
ejpam-5255	103	3	.	.	PUNCT
ejpam-5255	104	1	let	let	VERB
ejpam-5255	104	2	x	x	PRON
ejpam-5255	104	3	be	be	AUX
ejpam-5255	104	4	a	a	DET
ejpam-5255	104	5	commutative	commutative	ADJ
ejpam-5255	104	6	bck	bck	NOUN
ejpam-5255	104	7	-	-	PUNCT
ejpam-5255	104	8	algebra	algebra	NOUN
ejpam-5255	104	9	.	.	PUNCT
ejpam-5255	105	1	then	then	ADV
ejpam-5255	105	2	every	every	DET
ejpam-5255	105	3	bipolar	bipolar	ADJ
ejpam-5255	105	4	fuzzy	fuzzy	ADJ
ejpam-5255	105	5	ideal	ideal	NOUN
ejpam-5255	105	6	of	of	ADP
ejpam-5255	105	7	x	x	PUNCT
ejpam-5255	105	8	is	be	AUX
ejpam-5255	105	9	a	a	DET
ejpam-5255	105	10	bipolar	bipolar	ADJ
ejpam-5255	105	11	fuzzy	fuzzy	ADJ
ejpam-5255	105	12	commutative	commutative	ADJ
ejpam-5255	105	13	ideal	ideal	NOUN
ejpam-5255	105	14	.	.	PUNCT
ejpam-5255	106	1	proof	proof	NOUN
ejpam-5255	106	2	.	.	PUNCT
ejpam-5255	107	1	let	let	VERB
ejpam-5255	107	2	ϱ	ϱ	VERB
ejpam-5255	107	3	:	:	PUNCT
ejpam-5255	107	4	=	=	SYM
ejpam-5255	107	5	(	(	PUNCT
ejpam-5255	107	6	x	x	NOUN
ejpam-5255	107	7	,	,	PUNCT
ejpam-5255	107	8	ϱ+	ϱ+	X
ejpam-5255	107	9	,	,	PUNCT
ejpam-5255	107	10	ϱ−	ϱ−	NUM
ejpam-5255	107	11	)	)	PUNCT
ejpam-5255	107	12	be	be	VERB
ejpam-5255	107	13	a	a	DET
ejpam-5255	107	14	bipolar	bipolar	ADJ
ejpam-5255	107	15	fuzzy	fuzzy	ADJ
ejpam-5255	107	16	ideal	ideal	NOUN
ejpam-5255	107	17	and	and	CCONJ
ejpam-5255	107	18	a	a	DET
ejpam-5255	107	19	,	,	PUNCT
ejpam-5255	107	20	g	g	NOUN
ejpam-5255	107	21	,	,	PUNCT
ejpam-5255	107	22	w	w	PROPN
ejpam-5255	107	23	∈	∈	PROPN
ejpam-5255	107	24	x.	x.	NOUN
ejpam-5255	107	25	then	then	ADV
ejpam-5255	107	26	(	(	PUNCT
ejpam-5255	107	27	(	(	PUNCT
ejpam-5255	107	28	a	a	DET
ejpam-5255	107	29	∗	∗	NOUN
ejpam-5255	107	30	(	(	PUNCT
ejpam-5255	107	31	g	g	NOUN
ejpam-5255	107	32	∗	∗	NOUN
ejpam-5255	107	33	(	(	PUNCT
ejpam-5255	107	34	g	g	PROPN
ejpam-5255	107	35	∗	∗	X
ejpam-5255	107	36	a	a	NOUN
ejpam-5255	107	37	)	)	PUNCT
ejpam-5255	107	38	)	)	PUNCT
ejpam-5255	107	39	)	)	PUNCT
ejpam-5255	108	1	∗	∗	NOUN
ejpam-5255	108	2	(	(	PUNCT
ejpam-5255	108	3	(	(	PUNCT
ejpam-5255	108	4	a	a	DET
ejpam-5255	108	5	∗	∗	NOUN
ejpam-5255	108	6	g	g	NOUN
ejpam-5255	108	7	)	)	PUNCT
ejpam-5255	108	8	∗	∗	PROPN
ejpam-5255	108	9	w	w	NOUN
ejpam-5255	108	10	)	)	PUNCT
ejpam-5255	108	11	)	)	PUNCT
ejpam-5255	108	12	∗	∗	NOUN
ejpam-5255	108	13	w	w	NOUN
ejpam-5255	108	14	=	=	SYM
ejpam-5255	108	15	(	(	PUNCT
ejpam-5255	108	16	(	(	PUNCT
ejpam-5255	108	17	a	a	DET
ejpam-5255	108	18	∗	∗	NOUN
ejpam-5255	108	19	(	(	PUNCT
ejpam-5255	108	20	g	g	NOUN
ejpam-5255	108	21	∗	∗	NOUN
ejpam-5255	108	22	(	(	PUNCT
ejpam-5255	108	23	g	g	PROPN
ejpam-5255	108	24	∗	∗	X
ejpam-5255	108	25	a	a	NOUN
ejpam-5255	108	26	)	)	PUNCT
ejpam-5255	108	27	)	)	PUNCT
ejpam-5255	108	28	)	)	PUNCT
ejpam-5255	108	29	∗	∗	PROPN
ejpam-5255	108	30	w	w	NOUN
ejpam-5255	108	31	)	)	PUNCT
ejpam-5255	108	32	∗	∗	NOUN
ejpam-5255	108	33	(	(	PUNCT
ejpam-5255	108	34	(	(	PUNCT
ejpam-5255	108	35	a	a	DET
ejpam-5255	108	36	∗	∗	NOUN
ejpam-5255	108	37	g	g	NOUN
ejpam-5255	108	38	)	)	PUNCT
ejpam-5255	108	39	∗	∗	PROPN
ejpam-5255	108	40	w	w	NOUN
ejpam-5255	108	41	)	)	PUNCT
ejpam-5255	108	42	≤	≤	NOUN
ejpam-5255	108	43	(	(	PUNCT
ejpam-5255	108	44	a	a	DET
ejpam-5255	108	45	∗	∗	NOUN
ejpam-5255	108	46	(	(	PUNCT
ejpam-5255	108	47	g	g	NOUN
ejpam-5255	108	48	∗	∗	NOUN
ejpam-5255	108	49	(	(	PUNCT
ejpam-5255	108	50	g	g	PROPN
ejpam-5255	108	51	∗	∗	X
ejpam-5255	108	52	a	a	NOUN
ejpam-5255	108	53	)	)	PUNCT
ejpam-5255	108	54	)	)	PUNCT
ejpam-5255	108	55	)	)	PUNCT
ejpam-5255	108	56	∗	∗	NOUN
ejpam-5255	108	57	(	(	PUNCT
ejpam-5255	108	58	a	a	DET
ejpam-5255	108	59	∗	∗	NOUN
ejpam-5255	108	60	g	g	NOUN
ejpam-5255	108	61	)	)	PUNCT
ejpam-5255	108	62	=	=	SYM
ejpam-5255	108	63	(	(	PUNCT
ejpam-5255	108	64	a	a	DET
ejpam-5255	108	65	∗	∗	NOUN
ejpam-5255	108	66	(	(	PUNCT
ejpam-5255	108	67	a	a	DET
ejpam-5255	108	68	∗	∗	NOUN
ejpam-5255	108	69	g	g	NOUN
ejpam-5255	108	70	)	)	PUNCT
ejpam-5255	108	71	)	)	PUNCT
ejpam-5255	108	72	∗	∗	NOUN
ejpam-5255	108	73	(	(	PUNCT
ejpam-5255	108	74	g	g	NOUN
ejpam-5255	108	75	∗	∗	NOUN
ejpam-5255	108	76	(	(	PUNCT
ejpam-5255	108	77	g	g	PROPN
ejpam-5255	108	78	∗	∗	X
ejpam-5255	108	79	a	a	NOUN
ejpam-5255	108	80	)	)	PUNCT
ejpam-5255	108	81	)	)	PUNCT
ejpam-5255	109	1	=	=	SYM
ejpam-5255	109	2	0	0	NUM
ejpam-5255	109	3	,	,	PUNCT
ejpam-5255	109	4	and	and	CCONJ
ejpam-5255	109	5	so	so	ADV
ejpam-5255	109	6	(	(	PUNCT
ejpam-5255	109	7	a	a	DET
ejpam-5255	109	8	∗	∗	NOUN
ejpam-5255	109	9	(	(	PUNCT
ejpam-5255	109	10	g	g	NOUN
ejpam-5255	109	11	∗	∗	NOUN
ejpam-5255	109	12	(	(	PUNCT
ejpam-5255	109	13	g	g	PROPN
ejpam-5255	109	14	∗	∗	X
ejpam-5255	109	15	a	a	NOUN
ejpam-5255	109	16	)	)	PUNCT
ejpam-5255	109	17	)	)	PUNCT
ejpam-5255	109	18	)	)	PUNCT
ejpam-5255	109	19	∗	∗	NOUN
ejpam-5255	109	20	(	(	PUNCT
ejpam-5255	109	21	(	(	PUNCT
ejpam-5255	109	22	a	a	DET
ejpam-5255	109	23	∗	∗	NOUN
ejpam-5255	109	24	g	g	NOUN
ejpam-5255	109	25	)	)	PUNCT
ejpam-5255	109	26	∗	∗	PROPN
ejpam-5255	109	27	w	w	NOUN
ejpam-5255	109	28	)	)	PUNCT
ejpam-5255	109	29	≤	≤	NOUN
ejpam-5255	109	30	w.	w.	NOUN
ejpam-5255	109	31	applying	apply	VERB
ejpam-5255	109	32	lemma	lemma	PROPN
ejpam-5255	109	33	1	1	NUM
ejpam-5255	109	34	,	,	PUNCT
ejpam-5255	109	35	ϱ+(a	ϱ+(a	NOUN
ejpam-5255	109	36	∗	∗	NOUN
ejpam-5255	109	37	(	(	PUNCT
ejpam-5255	109	38	g	g	NOUN
ejpam-5255	109	39	∗	∗	NOUN
ejpam-5255	109	40	(	(	PUNCT
ejpam-5255	109	41	g	g	PROPN
ejpam-5255	109	42	∗	∗	X
ejpam-5255	109	43	a	a	NOUN
ejpam-5255	109	44	)	)	PUNCT
ejpam-5255	109	45	)	)	PUNCT
ejpam-5255	109	46	)	)	PUNCT
ejpam-5255	110	1	≥	≥	NOUN
ejpam-5255	110	2	min{ϱ+((a	min{ϱ+((a	NOUN
ejpam-5255	110	3	∗	∗	NOUN
ejpam-5255	110	4	g	g	NOUN
ejpam-5255	110	5	)	)	PUNCT
ejpam-5255	110	6	∗	∗	PROPN
ejpam-5255	110	7	w	w	PROPN
ejpam-5255	110	8	)	)	PUNCT
ejpam-5255	110	9	,	,	PUNCT
ejpam-5255	110	10	ϱ+(w	ϱ+(w	PROPN
ejpam-5255	110	11	)	)	PUNCT
ejpam-5255	110	12	}	}	PUNCT
ejpam-5255	110	13	,	,	PUNCT
ejpam-5255	110	14	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	110	15	∗	∗	NOUN
ejpam-5255	110	16	(	(	PUNCT
ejpam-5255	110	17	g	g	NOUN
ejpam-5255	110	18	∗	∗	NOUN
ejpam-5255	110	19	(	(	PUNCT
ejpam-5255	110	20	g	g	PROPN
ejpam-5255	110	21	∗	∗	X
ejpam-5255	110	22	a	a	NOUN
ejpam-5255	110	23	)	)	PUNCT
ejpam-5255	110	24	)	)	PUNCT
ejpam-5255	110	25	)	)	PUNCT
ejpam-5255	111	1	≤	≤	NUM
ejpam-5255	111	2	max	max	PROPN
ejpam-5255	111	3	{	{	PUNCT
ejpam-5255	111	4	ϱ−((a	ϱ−((a	PROPN
ejpam-5255	111	5	∗	∗	NOUN
ejpam-5255	111	6	g	g	NOUN
ejpam-5255	111	7	)	)	PUNCT
ejpam-5255	111	8	∗	∗	PROPN
ejpam-5255	111	9	w	w	PROPN
ejpam-5255	111	10	)	)	PUNCT
ejpam-5255	111	11	,	,	PUNCT
ejpam-5255	111	12	ϱ−(w	ϱ−(w	PROPN
ejpam-5255	111	13	)	)	PUNCT
ejpam-5255	111	14	}	}	PUNCT
ejpam-5255	111	15	.	.	PUNCT
ejpam-5255	112	1	therefore	therefore	ADV
ejpam-5255	112	2	,	,	PUNCT
ejpam-5255	112	3	ϱ	ϱ	PROPN
ejpam-5255	112	4	is	be	AUX
ejpam-5255	112	5	a	a	DET
ejpam-5255	112	6	bipolar	bipolar	ADJ
ejpam-5255	112	7	fuzzy	fuzzy	ADJ
ejpam-5255	112	8	commutative	commutative	ADJ
ejpam-5255	112	9	ideal	ideal	NOUN
ejpam-5255	112	10	.	.	PUNCT
ejpam-5255	113	1	suppose	suppose	VERB
ejpam-5255	113	2	that	that	SCONJ
ejpam-5255	113	3	ϱ	ϱ	ADP
ejpam-5255	113	4	:	:	PUNCT
ejpam-5255	113	5	=	=	SYM
ejpam-5255	113	6	(	(	PUNCT
ejpam-5255	113	7	x	x	NOUN
ejpam-5255	113	8	,	,	PUNCT
ejpam-5255	113	9	ϱ+	ϱ+	X
ejpam-5255	113	10	,	,	PUNCT
ejpam-5255	113	11	ϱ−	ϱ−	VERB
ejpam-5255	113	12	)	)	PUNCT
ejpam-5255	113	13	is	be	AUX
ejpam-5255	113	14	a	a	DET
ejpam-5255	113	15	bipolar	bipolar	ADJ
ejpam-5255	113	16	fuzzy	fuzzy	ADJ
ejpam-5255	113	17	set	set	NOUN
ejpam-5255	113	18	.	.	PUNCT
ejpam-5255	114	1	let	let	VERB
ejpam-5255	114	2	(	(	PUNCT
ejpam-5255	114	3	α	α	X
ejpam-5255	114	4	,	,	PUNCT
ejpam-5255	114	5	β	β	NOUN
ejpam-5255	114	6	)	)	PUNCT
ejpam-5255	114	7	∈	∈	PROPN
ejpam-5255	115	1	[	[	X
ejpam-5255	115	2	−1	−1	NOUN
ejpam-5255	115	3	,	,	PUNCT
ejpam-5255	115	4	0)×(0	0)×(0	NUM
ejpam-5255	115	5	,	,	PUNCT
ejpam-5255	115	6	1	1	NUM
ejpam-5255	115	7	]	]	PUNCT
ejpam-5255	115	8	.	.	PUNCT
ejpam-5255	116	1	recall	recall	VERB
ejpam-5255	116	2	that	that	DET
ejpam-5255	116	3	n(ϱ;α	n(ϱ;α	NOUN
ejpam-5255	116	4	)	)	PUNCT
ejpam-5255	117	1	=	=	PRON
ejpam-5255	117	2	{	{	PUNCT
ejpam-5255	117	3	a	a	DET
ejpam-5255	117	4	∈	∈	NOUN
ejpam-5255	117	5	x	x	X
ejpam-5255	117	6	:	:	PUNCT
ejpam-5255	117	7	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	117	8	)	)	PUNCT
ejpam-5255	117	9	≤	≤	NOUN
ejpam-5255	117	10	α	α	X
ejpam-5255	117	11	}	}	PUNCT
ejpam-5255	117	12	is	be	AUX
ejpam-5255	117	13	said	say	VERB
ejpam-5255	117	14	to	to	PART
ejpam-5255	117	15	be	be	AUX
ejpam-5255	117	16	the	the	DET
ejpam-5255	117	17	negative	negative	ADJ
ejpam-5255	117	18	α	α	NOUN
ejpam-5255	117	19	-	-	NOUN
ejpam-5255	117	20	cut	cut	NOUN
ejpam-5255	117	21	of	of	ADP
ejpam-5255	117	22	ϱ	ϱ	NOUN
ejpam-5255	117	23	,	,	PUNCT
ejpam-5255	117	24	and	and	CCONJ
ejpam-5255	117	25	p	p	X
ejpam-5255	117	26	(	(	PUNCT
ejpam-5255	117	27	ϱ;β	ϱ;β	PROPN
ejpam-5255	117	28	)	)	PUNCT
ejpam-5255	117	29	=	=	PRON
ejpam-5255	118	1	{	{	PUNCT
ejpam-5255	118	2	r	r	NOUN
ejpam-5255	118	3	∈	∈	PROPN
ejpam-5255	118	4	x	x	X
ejpam-5255	118	5	:	:	PUNCT
ejpam-5255	118	6	ϱ+(r	ϱ+(r	NUM
ejpam-5255	118	7	)	)	PUNCT
ejpam-5255	118	8	≥	≥	X
ejpam-5255	118	9	β	β	X
ejpam-5255	118	10	}	}	PUNCT
ejpam-5255	118	11	is	be	AUX
ejpam-5255	118	12	called	call	VERB
ejpam-5255	118	13	the	the	DET
ejpam-5255	118	14	positive	positive	ADJ
ejpam-5255	118	15	β	β	NOUN
ejpam-5255	118	16	-	-	NOUN
ejpam-5255	118	17	cut	cut	NOUN
ejpam-5255	118	18	of	of	ADP
ejpam-5255	118	19	ϱ.	ϱ.	NOUN
ejpam-5255	118	20	theorem	theorem	PROPN
ejpam-5255	118	21	4	4	X
ejpam-5255	118	22	.	.	PUNCT
ejpam-5255	119	1	let	let	VERB
ejpam-5255	119	2	ϱ	ϱ	VERB
ejpam-5255	119	3	:	:	PUNCT
ejpam-5255	119	4	=	=	SYM
ejpam-5255	119	5	(	(	PUNCT
ejpam-5255	119	6	x	x	NOUN
ejpam-5255	119	7	,	,	PUNCT
ejpam-5255	119	8	ϱ+	ϱ+	X
ejpam-5255	119	9	,	,	PUNCT
ejpam-5255	119	10	ϱ−	ϱ−	NUM
ejpam-5255	119	11	)	)	PUNCT
ejpam-5255	119	12	be	be	VERB
ejpam-5255	119	13	a	a	DET
ejpam-5255	119	14	bipolar	bipolar	ADJ
ejpam-5255	119	15	fuzzy	fuzzy	ADJ
ejpam-5255	119	16	set	set	NOUN
ejpam-5255	119	17	.	.	PUNCT
ejpam-5255	120	1	then	then	ADV
ejpam-5255	120	2	ϱ	ϱ	PROPN
ejpam-5255	120	3	is	be	AUX
ejpam-5255	120	4	a	a	DET
ejpam-5255	120	5	bipolar	bipolar	ADJ
ejpam-5255	120	6	fuzzy	fuzzy	ADJ
ejpam-5255	120	7	commutative	commutative	ADJ
ejpam-5255	120	8	ideal	ideal	NOUN
ejpam-5255	120	9	of	of	ADP
ejpam-5255	120	10	x	x	SYM
ejpam-5255	120	11	if	if	SCONJ
ejpam-5255	120	12	and	and	CCONJ
ejpam-5255	120	13	only	only	ADV
ejpam-5255	120	14	if	if	SCONJ
ejpam-5255	120	15	both	both	CCONJ
ejpam-5255	120	16	the	the	DET
ejpam-5255	120	17	non	non	ADJ
ejpam-5255	120	18	-	-	ADJ
ejpam-5255	120	19	empty	empty	ADJ
ejpam-5255	120	20	negative	negative	ADJ
ejpam-5255	120	21	α	α	NOUN
ejpam-5255	120	22	-	-	ADJ
ejpam-5255	120	23	cut	cut	NOUN
ejpam-5255	120	24	and	and	CCONJ
ejpam-5255	120	25	the	the	DET
ejpam-5255	120	26	nonempty	nonempty	ADJ
ejpam-5255	120	27	positive	positive	ADJ
ejpam-5255	120	28	β	β	NOUN
ejpam-5255	120	29	-	-	NOUN
ejpam-5255	120	30	cut	cut	NOUN
ejpam-5255	120	31	of	of	ADP
ejpam-5255	120	32	ϱ	ϱ	NOUN
ejpam-5255	120	33	are	be	AUX
ejpam-5255	120	34	commutative	commutative	ADJ
ejpam-5255	120	35	ideals	ideal	NOUN
ejpam-5255	120	36	of	of	ADP
ejpam-5255	120	37	x	x	PUNCT
ejpam-5255	120	38	for	for	ADP
ejpam-5255	120	39	all	all	DET
ejpam-5255	120	40	(	(	PUNCT
ejpam-5255	120	41	α	α	NOUN
ejpam-5255	120	42	,	,	PUNCT
ejpam-5255	120	43	β	β	NOUN
ejpam-5255	120	44	)	)	PUNCT
ejpam-5255	120	45	∈	∈	PROPN
ejpam-5255	121	1	[	[	X
ejpam-5255	121	2	−1	−1	NOUN
ejpam-5255	121	3	,	,	PUNCT
ejpam-5255	121	4	0)×	0)×	NUM
ejpam-5255	121	5	(	(	PUNCT
ejpam-5255	121	6	0	0	NUM
ejpam-5255	121	7	,	,	PUNCT
ejpam-5255	121	8	1	1	NUM
ejpam-5255	121	9	]	]	PUNCT
ejpam-5255	121	10	.	.	PUNCT
ejpam-5255	122	1	a.	a.	PROPN
ejpam-5255	122	2	almuhaimeed	almuhaimeed	PROPN
ejpam-5255	122	3	,	,	PUNCT
ejpam-5255	122	4	h.	h.	PROPN
ejpam-5255	122	5	alshehri	alshehri	PROPN
ejpam-5255	122	6	/	/	SYM
ejpam-5255	122	7	eur	eur	PROPN
ejpam-5255	122	8	.	.	PUNCT
ejpam-5255	123	1	j.	j.	PROPN
ejpam-5255	123	2	pure	pure	PROPN
ejpam-5255	123	3	appl	appl	PROPN
ejpam-5255	123	4	.	.	PROPN
ejpam-5255	123	5	math	math	PROPN
ejpam-5255	123	6	,	,	PUNCT
ejpam-5255	123	7	17	17	NUM
ejpam-5255	123	8	(	(	PUNCT
ejpam-5255	123	9	3	3	NUM
ejpam-5255	123	10	)	)	PUNCT
ejpam-5255	123	11	(	(	PUNCT
ejpam-5255	123	12	2024	2024	NUM
ejpam-5255	123	13	)	)	PUNCT
ejpam-5255	123	14	,	,	PUNCT
ejpam-5255	123	15	1831	1831	NUM
ejpam-5255	123	16	-	-	SYM
ejpam-5255	123	17	1841	1841	NUM
ejpam-5255	123	18	1837	1837	NUM
ejpam-5255	123	19	proof	proof	NOUN
ejpam-5255	123	20	.	.	PUNCT
ejpam-5255	124	1	let	let	VERB
ejpam-5255	124	2	ϱ	ϱ	PART
ejpam-5255	124	3	be	be	AUX
ejpam-5255	124	4	a	a	DET
ejpam-5255	124	5	bipolar	bipolar	ADJ
ejpam-5255	124	6	fuzzy	fuzzy	ADJ
ejpam-5255	124	7	commutative	commutative	ADJ
ejpam-5255	124	8	ideal	ideal	NOUN
ejpam-5255	124	9	of	of	ADP
ejpam-5255	124	10	x.	x.	NOUN
ejpam-5255	124	11	for	for	ADP
ejpam-5255	124	12	any	any	DET
ejpam-5255	124	13	fixed	fixed	ADJ
ejpam-5255	124	14	α	α	NOUN
ejpam-5255	124	15	∈	∈	PROPN
ejpam-5255	125	1	[	[	X
ejpam-5255	125	2	−1	−1	NOUN
ejpam-5255	125	3	,	,	PUNCT
ejpam-5255	125	4	0	0	NUM
ejpam-5255	125	5	)	)	PUNCT
ejpam-5255	125	6	and	and	CCONJ
ejpam-5255	125	7	β	β	X
ejpam-5255	125	8	∈	∈	PROPN
ejpam-5255	125	9	(	(	PUNCT
ejpam-5255	125	10	0	0	NUM
ejpam-5255	125	11	,	,	PUNCT
ejpam-5255	125	12	1	1	NUM
ejpam-5255	125	13	]	]	PUNCT
ejpam-5255	125	14	,	,	PUNCT
ejpam-5255	125	15	if	if	SCONJ
ejpam-5255	125	16	ϱ−(0	ϱ−(0	VERB
ejpam-5255	125	17	)	)	PUNCT
ejpam-5255	125	18	≤	≤	NOUN
ejpam-5255	125	19	α	α	NOUN
ejpam-5255	125	20	and	and	CCONJ
ejpam-5255	125	21	ϱ+(0	ϱ+(0	ADJ
ejpam-5255	125	22	)	)	PUNCT
ejpam-5255	125	23	≥	≥	NOUN
ejpam-5255	125	24	β	β	X
ejpam-5255	125	25	,	,	PUNCT
ejpam-5255	125	26	it	it	PRON
ejpam-5255	125	27	implies	imply	VERB
ejpam-5255	125	28	that	that	SCONJ
ejpam-5255	125	29	0	0	NUM
ejpam-5255	125	30	∈	∈	PROPN
ejpam-5255	125	31	ϱ−(α	ϱ−(α	NOUN
ejpam-5255	125	32	)	)	PUNCT
ejpam-5255	125	33	and	and	CCONJ
ejpam-5255	125	34	0	0	NUM
ejpam-5255	125	35	∈	∈	NOUN
ejpam-5255	125	36	ϱ+(β	ϱ+(β	PROPN
ejpam-5255	125	37	)	)	PUNCT
ejpam-5255	125	38	.	.	PUNCT
ejpam-5255	126	1	thus	thus	ADV
ejpam-5255	126	2	the	the	DET
ejpam-5255	126	3	first	first	ADJ
ejpam-5255	126	4	condition	condition	NOUN
ejpam-5255	126	5	holds	hold	VERB
ejpam-5255	126	6	.	.	PUNCT
ejpam-5255	127	1	let	let	VERB
ejpam-5255	127	2	(	(	PUNCT
ejpam-5255	127	3	(	(	PUNCT
ejpam-5255	127	4	a	a	DET
ejpam-5255	127	5	∗	∗	NOUN
ejpam-5255	127	6	g	g	NOUN
ejpam-5255	127	7	)	)	PUNCT
ejpam-5255	127	8	∗	∗	PROPN
ejpam-5255	127	9	w	w	NOUN
ejpam-5255	127	10	)	)	PUNCT
ejpam-5255	127	11	∈	∈	PROPN
ejpam-5255	127	12	ϱ−(α	ϱ−(α	NOUN
ejpam-5255	127	13	)	)	PUNCT
ejpam-5255	127	14	∩	∩	NOUN
ejpam-5255	127	15	ϱ+(β	ϱ+(β	NOUN
ejpam-5255	127	16	)	)	PUNCT
ejpam-5255	127	17	and	and	CCONJ
ejpam-5255	127	18	t	t	PROPN
ejpam-5255	127	19	∈	∈	PROPN
ejpam-5255	127	20	ϱ−(α	ϱ−(α	NOUN
ejpam-5255	127	21	)	)	PUNCT
ejpam-5255	127	22	∩	∩	NOUN
ejpam-5255	127	23	ϱ+(β	ϱ+(β	NOUN
ejpam-5255	127	24	)	)	PUNCT
ejpam-5255	127	25	.	.	PUNCT
ejpam-5255	128	1	it	it	PRON
ejpam-5255	128	2	follows	follow	VERB
ejpam-5255	128	3	that	that	PRON
ejpam-5255	128	4	ϱ−((a	ϱ−((a	ADP
ejpam-5255	128	5	∗	∗	NOUN
ejpam-5255	128	6	g	g	NOUN
ejpam-5255	128	7	)	)	PUNCT
ejpam-5255	128	8	∗	∗	PROPN
ejpam-5255	128	9	w	w	NOUN
ejpam-5255	128	10	)	)	PUNCT
ejpam-5255	128	11	≤	≤	NOUN
ejpam-5255	128	12	α	α	X
ejpam-5255	128	13	,	,	PUNCT
ejpam-5255	128	14	ϱ+((a	ϱ+((a	ADP
ejpam-5255	128	15	∗	∗	NOUN
ejpam-5255	128	16	g	g	NOUN
ejpam-5255	128	17	)	)	PUNCT
ejpam-5255	128	18	∗	∗	PROPN
ejpam-5255	128	19	w	w	PROPN
ejpam-5255	128	20	)	)	PUNCT
ejpam-5255	128	21	≥	≥	PROPN
ejpam-5255	128	22	β	β	X
ejpam-5255	128	23	and	and	CCONJ
ejpam-5255	128	24	ϱ−(w	ϱ−(w	PROPN
ejpam-5255	128	25	)	)	PUNCT
ejpam-5255	128	26	≤	≤	NOUN
ejpam-5255	128	27	α	α	X
ejpam-5255	128	28	,	,	PUNCT
ejpam-5255	128	29	ϱ+(w	ϱ+(w	PROPN
ejpam-5255	128	30	)	)	PUNCT
ejpam-5255	128	31	≥	≥	NOUN
ejpam-5255	128	32	β	β	NOUN
ejpam-5255	128	33	.	.	PUNCT
ejpam-5255	129	1	by	by	ADP
ejpam-5255	129	2	definition	definition	NOUN
ejpam-5255	129	3	,	,	PUNCT
ejpam-5255	129	4	we	we	PRON
ejpam-5255	129	5	obtain	obtain	VERB
ejpam-5255	129	6	ϱ+(a	ϱ+(a	ADJ
ejpam-5255	129	7	∗	∗	NOUN
ejpam-5255	129	8	(	(	PUNCT
ejpam-5255	129	9	g	g	NOUN
ejpam-5255	129	10	∗	∗	NOUN
ejpam-5255	129	11	(	(	PUNCT
ejpam-5255	129	12	g	g	PROPN
ejpam-5255	129	13	∗	∗	X
ejpam-5255	129	14	a	a	NOUN
ejpam-5255	129	15	)	)	PUNCT
ejpam-5255	129	16	)	)	PUNCT
ejpam-5255	129	17	)	)	PUNCT
ejpam-5255	129	18	≥	≥	PROPN
ejpam-5255	129	19	min	min	NOUN
ejpam-5255	129	20	{	{	PUNCT
ejpam-5255	129	21	ϱ+((a	ϱ+((a	ADP
ejpam-5255	129	22	∗	∗	NOUN
ejpam-5255	129	23	g	g	NOUN
ejpam-5255	129	24	)	)	PUNCT
ejpam-5255	129	25	∗	∗	PROPN
ejpam-5255	129	26	w	w	PROPN
ejpam-5255	129	27	)	)	PUNCT
ejpam-5255	129	28	,	,	PUNCT
ejpam-5255	129	29	ϱ+(w	ϱ+(w	PROPN
ejpam-5255	129	30	)	)	PUNCT
ejpam-5255	129	31	}	}	PUNCT
ejpam-5255	129	32	≥	≥	NUM
ejpam-5255	129	33	β	β	X
ejpam-5255	129	34	,	,	PUNCT
ejpam-5255	129	35	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	129	36	∗	∗	NOUN
ejpam-5255	129	37	(	(	PUNCT
ejpam-5255	129	38	g	g	NOUN
ejpam-5255	129	39	∗	∗	NOUN
ejpam-5255	129	40	(	(	PUNCT
ejpam-5255	129	41	g	g	PROPN
ejpam-5255	129	42	∗	∗	X
ejpam-5255	129	43	a	a	NOUN
ejpam-5255	129	44	)	)	PUNCT
ejpam-5255	129	45	)	)	PUNCT
ejpam-5255	129	46	)	)	PUNCT
ejpam-5255	130	1	≤	≤	NUM
ejpam-5255	131	1	max	max	PROPN
ejpam-5255	131	2	{	{	PUNCT
ejpam-5255	131	3	ϱ−((a	ϱ−((a	PROPN
ejpam-5255	131	4	∗	∗	NOUN
ejpam-5255	131	5	g	g	NOUN
ejpam-5255	131	6	)	)	PUNCT
ejpam-5255	131	7	∗	∗	PROPN
ejpam-5255	131	8	w	w	PROPN
ejpam-5255	131	9	)	)	PUNCT
ejpam-5255	131	10	,	,	PUNCT
ejpam-5255	131	11	ϱ−(w	ϱ−(w	PROPN
ejpam-5255	131	12	)	)	PUNCT
ejpam-5255	131	13	}	}	PUNCT
ejpam-5255	131	14	≤	≤	NUM
ejpam-5255	131	15	α	α	X
ejpam-5255	131	16	.	.	PUNCT
ejpam-5255	132	1	therefore	therefore	ADV
ejpam-5255	132	2	,	,	PUNCT
ejpam-5255	132	3	ϱ	ϱ	PROPN
ejpam-5255	132	4	is	be	AUX
ejpam-5255	132	5	a	a	DET
ejpam-5255	132	6	commutative	commutative	ADJ
ejpam-5255	132	7	ideal	ideal	NOUN
ejpam-5255	132	8	of	of	ADP
ejpam-5255	132	9	x.	x.	NOUN
ejpam-5255	132	10	now	now	ADV
ejpam-5255	132	11	,	,	PUNCT
ejpam-5255	132	12	assume	assume	VERB
ejpam-5255	132	13	that	that	SCONJ
ejpam-5255	132	14	ϱ	ϱ	ADP
ejpam-5255	132	15	:	:	PUNCT
ejpam-5255	132	16	=	=	SYM
ejpam-5255	132	17	(	(	PUNCT
ejpam-5255	132	18	x	x	NOUN
ejpam-5255	132	19	,	,	PUNCT
ejpam-5255	132	20	ϱ+	ϱ+	X
ejpam-5255	132	21	,	,	PUNCT
ejpam-5255	132	22	ϱ−	ϱ−	VERB
ejpam-5255	132	23	)	)	PUNCT
ejpam-5255	132	24	is	be	AUX
ejpam-5255	132	25	a	a	DET
ejpam-5255	132	26	commutative	commutative	ADJ
ejpam-5255	132	27	ideal	ideal	NOUN
ejpam-5255	132	28	of	of	ADP
ejpam-5255	132	29	x	x	PUNCT
ejpam-5255	132	30	for	for	ADP
ejpam-5255	132	31	all	all	DET
ejpam-5255	132	32	(	(	PUNCT
ejpam-5255	132	33	α	α	NOUN
ejpam-5255	132	34	,	,	PUNCT
ejpam-5255	132	35	β	β	NOUN
ejpam-5255	132	36	)	)	PUNCT
ejpam-5255	132	37	∈	∈	PROPN
ejpam-5255	133	1	[	[	X
ejpam-5255	133	2	−1	−1	NOUN
ejpam-5255	133	3	,	,	PUNCT
ejpam-5255	133	4	0)(0	0)(0	NUM
ejpam-5255	133	5	,	,	PUNCT
ejpam-5255	133	6	1	1	NUM
ejpam-5255	133	7	]	]	PUNCT
ejpam-5255	133	8	,	,	PUNCT
ejpam-5255	133	9	ϱ−(0	ϱ−(0	PROPN
ejpam-5255	133	10	)	)	PUNCT
ejpam-5255	133	11	≤	≤	NOUN
ejpam-5255	133	12	α	α	NOUN
ejpam-5255	133	13	and	and	CCONJ
ejpam-5255	133	14	ϱ+(0	ϱ+(0	ADJ
ejpam-5255	133	15	)	)	PUNCT
ejpam-5255	133	16	≥	≥	NOUN
ejpam-5255	133	17	β	β	AUX
ejpam-5255	133	18	.	.	PUNCT
ejpam-5255	133	19	suppose	suppose	VERB
ejpam-5255	133	20	that	that	SCONJ
ejpam-5255	133	21	ϱ−((a∗g)∗w	ϱ−((a∗g)∗w	NUM
ejpam-5255	133	22	)	)	PUNCT
ejpam-5255	134	1	=	=	SYM
ejpam-5255	134	2	α1	α1	PROPN
ejpam-5255	134	3	and	and	CCONJ
ejpam-5255	134	4	ϱ−(w	ϱ−(w	PROPN
ejpam-5255	134	5	)	)	PUNCT
ejpam-5255	134	6	=	=	VERB
ejpam-5255	135	1	α2	α2	ADJ
ejpam-5255	135	2	for	for	ADP
ejpam-5255	135	3	some	some	PRON
ejpam-5255	135	4	a	a	PRON
ejpam-5255	135	5	,	,	PUNCT
ejpam-5255	135	6	g	g	NOUN
ejpam-5255	135	7	,	,	PUNCT
ejpam-5255	135	8	w	w	PROPN
ejpam-5255	135	9	∈	∈	PROPN
ejpam-5255	135	10	x.	x.	NOUN
ejpam-5255	136	1	this	this	PRON
ejpam-5255	136	2	implies	imply	VERB
ejpam-5255	136	3	that	that	SCONJ
ejpam-5255	136	4	(	(	PUNCT
ejpam-5255	136	5	(	(	PUNCT
ejpam-5255	136	6	a	a	DET
ejpam-5255	136	7	∗	∗	NOUN
ejpam-5255	136	8	g	g	NOUN
ejpam-5255	136	9	)	)	PUNCT
ejpam-5255	136	10	∗	∗	PROPN
ejpam-5255	136	11	w	w	NOUN
ejpam-5255	136	12	)	)	PUNCT
ejpam-5255	136	13	∈	∈	PROPN
ejpam-5255	136	14	ϱ−(α1	ϱ−(α1	NUM
ejpam-5255	136	15	)	)	PUNCT
ejpam-5255	136	16	and	and	CCONJ
ejpam-5255	136	17	w	w	PROPN
ejpam-5255	136	18	∈	∈	PROPN
ejpam-5255	136	19	ϱ−(α2	ϱ−(α2	ADP
ejpam-5255	136	20	)	)	PUNCT
ejpam-5255	136	21	.	.	PUNCT
ejpam-5255	137	1	without	without	ADP
ejpam-5255	137	2	loss	loss	NOUN
ejpam-5255	137	3	of	of	ADP
ejpam-5255	137	4	generality	generality	NOUN
ejpam-5255	137	5	,	,	PUNCT
ejpam-5255	137	6	let	let	VERB
ejpam-5255	137	7	α1	α1	PROPN
ejpam-5255	137	8	≤	≤	ADV
ejpam-5255	137	9	α2	α2	PROPN
ejpam-5255	137	10	.	.	PUNCT
ejpam-5255	138	1	let	let	VERB
ejpam-5255	138	2	ϱ+((a	ϱ+((a	ADP
ejpam-5255	138	3	∗	∗	NOUN
ejpam-5255	138	4	g	g	NOUN
ejpam-5255	138	5	)	)	PUNCT
ejpam-5255	138	6	∗	∗	PROPN
ejpam-5255	138	7	w	w	NOUN
ejpam-5255	138	8	)	)	PUNCT
ejpam-5255	138	9	=	=	SYM
ejpam-5255	138	10	β1	β1	PROPN
ejpam-5255	138	11	,	,	PUNCT
ejpam-5255	138	12	ϱ+(w	ϱ+(w	NUM
ejpam-5255	138	13	)	)	PUNCT
ejpam-5255	139	1	=	=	VERB
ejpam-5255	139	2	β2	β2	NOUN
ejpam-5255	139	3	and	and	CCONJ
ejpam-5255	139	4	β1	β1	PROPN
ejpam-5255	139	5	≥	≥	NUM
ejpam-5255	139	6	β2	β2	PROPN
ejpam-5255	139	7	.	.	PUNCT
ejpam-5255	140	1	then	then	ADV
ejpam-5255	140	2	ϱ−(α2	ϱ−(α2	ADP
ejpam-5255	140	3	)	)	PUNCT
ejpam-5255	140	4	≤	≤	NOUN
ejpam-5255	140	5	ϱ−(α1	ϱ−(α1	NUM
ejpam-5255	140	6	)	)	PUNCT
ejpam-5255	140	7	which	which	PRON
ejpam-5255	140	8	means	mean	VERB
ejpam-5255	140	9	that	that	SCONJ
ejpam-5255	140	10	w	w	PROPN
ejpam-5255	140	11	∈	∈	PROPN
ejpam-5255	140	12	ϱ−(α1	ϱ−(α1	NUM
ejpam-5255	140	13	)	)	PUNCT
ejpam-5255	140	14	.	.	PUNCT
ejpam-5255	141	1	now	now	ADV
ejpam-5255	141	2	,	,	PUNCT
ejpam-5255	141	3	ϱ+(β1	ϱ+(β1	PROPN
ejpam-5255	141	4	)	)	PUNCT
ejpam-5255	141	5	≥	≥	NOUN
ejpam-5255	141	6	ϱ+(β2	ϱ+(β2	NOUN
ejpam-5255	141	7	)	)	PUNCT
ejpam-5255	141	8	and	and	CCONJ
ejpam-5255	141	9	hence	hence	ADV
ejpam-5255	141	10	w	w	PROPN
ejpam-5255	141	11	∈	∈	PROPN
ejpam-5255	141	12	ϱ+(β1	ϱ+(β1	PROPN
ejpam-5255	141	13	)	)	PUNCT
ejpam-5255	141	14	.	.	PUNCT
ejpam-5255	142	1	that	that	SCONJ
ejpam-5255	142	2	ϱ	ϱ	ADP
ejpam-5255	142	3	−(α1	−(α1	NOUN
ejpam-5255	142	4	)	)	PUNCT
ejpam-5255	142	5	and	and	CCONJ
ejpam-5255	142	6	ϱ+(β1	ϱ+(β1	NOUN
ejpam-5255	142	7	)	)	PUNCT
ejpam-5255	142	8	are	be	AUX
ejpam-5255	142	9	commutative	commutative	ADJ
ejpam-5255	142	10	ideals	ideal	NOUN
ejpam-5255	142	11	of	of	ADP
ejpam-5255	142	12	x	x	NOUN
ejpam-5255	142	13	,	,	PUNCT
ejpam-5255	142	14	implies	imply	VERB
ejpam-5255	142	15	that	that	SCONJ
ejpam-5255	142	16	a	a	DET
ejpam-5255	142	17	∗	∗	NOUN
ejpam-5255	142	18	(	(	PUNCT
ejpam-5255	142	19	g	g	NOUN
ejpam-5255	142	20	∗	∗	NOUN
ejpam-5255	142	21	(	(	PUNCT
ejpam-5255	142	22	g	g	PROPN
ejpam-5255	142	23	∗	∗	X
ejpam-5255	142	24	a	a	PRON
ejpam-5255	142	25	)	)	PUNCT
ejpam-5255	142	26	)	)	PUNCT
ejpam-5255	142	27	∈	∈	PROPN
ejpam-5255	142	28	(	(	PUNCT
ejpam-5255	142	29	ϱ−(α1	ϱ−(α1	NOUN
ejpam-5255	142	30	)	)	PUNCT
ejpam-5255	142	31	∩	∩	ADJ
ejpam-5255	142	32	ϱ+(β1	ϱ+(β1	NOUN
ejpam-5255	142	33	)	)	PUNCT
ejpam-5255	142	34	)	)	PUNCT
ejpam-5255	142	35	.	.	PUNCT
ejpam-5255	143	1	hence	hence	ADV
ejpam-5255	143	2	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	143	3	∗	∗	NOUN
ejpam-5255	143	4	(	(	PUNCT
ejpam-5255	143	5	g	g	NOUN
ejpam-5255	143	6	∗	∗	NOUN
ejpam-5255	143	7	(	(	PUNCT
ejpam-5255	143	8	g	g	PROPN
ejpam-5255	143	9	∗	∗	X
ejpam-5255	143	10	a	a	NOUN
ejpam-5255	143	11	)	)	PUNCT
ejpam-5255	143	12	)	)	PUNCT
ejpam-5255	143	13	)	)	PUNCT
ejpam-5255	144	1	≤	≤	NUM
ejpam-5255	144	2	α1	α1	PROPN
ejpam-5255	144	3	=	=	SYM
ejpam-5255	144	4	min	min	NOUN
ejpam-5255	144	5	{	{	PUNCT
ejpam-5255	144	6	ϱ−((a	ϱ−((a	ADP
ejpam-5255	144	7	∗	∗	NOUN
ejpam-5255	144	8	g	g	NOUN
ejpam-5255	144	9	)	)	PUNCT
ejpam-5255	144	10	∗	∗	PROPN
ejpam-5255	144	11	w	w	PROPN
ejpam-5255	144	12	)	)	PUNCT
ejpam-5255	144	13	,	,	PUNCT
ejpam-5255	144	14	ϱ−(h	ϱ−(h	PROPN
ejpam-5255	144	15	)	)	PUNCT
ejpam-5255	144	16	}	}	PUNCT
ejpam-5255	144	17	,	,	PUNCT
ejpam-5255	144	18	ϱ+(a	ϱ+(a	PROPN
ejpam-5255	144	19	∗	∗	NOUN
ejpam-5255	144	20	(	(	PUNCT
ejpam-5255	144	21	g	g	NOUN
ejpam-5255	144	22	∗	∗	NOUN
ejpam-5255	144	23	(	(	PUNCT
ejpam-5255	144	24	g	g	PROPN
ejpam-5255	144	25	∗	∗	X
ejpam-5255	144	26	a	a	NOUN
ejpam-5255	144	27	)	)	PUNCT
ejpam-5255	144	28	)	)	PUNCT
ejpam-5255	144	29	)	)	PUNCT
ejpam-5255	144	30	≥	≥	X
ejpam-5255	144	31	β1	β1	NOUN
ejpam-5255	144	32	=	=	SYM
ejpam-5255	144	33	max	max	PROPN
ejpam-5255	144	34	{	{	PUNCT
ejpam-5255	144	35	ϱ+((a	ϱ+((a	ADP
ejpam-5255	144	36	∗	∗	NOUN
ejpam-5255	144	37	g	g	NOUN
ejpam-5255	144	38	)	)	PUNCT
ejpam-5255	144	39	∗	∗	PROPN
ejpam-5255	144	40	w	w	PROPN
ejpam-5255	144	41	)	)	PUNCT
ejpam-5255	144	42	,	,	PUNCT
ejpam-5255	144	43	ϱ+(w	ϱ+(w	PROPN
ejpam-5255	144	44	)	)	PUNCT
ejpam-5255	144	45	}	}	PUNCT
ejpam-5255	144	46	.	.	PUNCT
ejpam-5255	145	1	thus	thus	ADV
ejpam-5255	145	2	ϱ−(0	ϱ−(0	VERB
ejpam-5255	145	3	)	)	PUNCT
ejpam-5255	145	4	≤	≤	NUM
ejpam-5255	145	5	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	145	6	)	)	PUNCT
ejpam-5255	145	7	and	and	CCONJ
ejpam-5255	145	8	ϱ+(0	ϱ+(0	ADJ
ejpam-5255	145	9	)	)	PUNCT
ejpam-5255	145	10	≥	≥	NOUN
ejpam-5255	145	11	ϱ+(a	ϱ+(a	NOUN
ejpam-5255	145	12	)	)	PUNCT
ejpam-5255	145	13	.	.	PUNCT
ejpam-5255	146	1	hence	hence	ADV
ejpam-5255	146	2	,	,	PUNCT
ejpam-5255	146	3	ϱ	ϱ	PROPN
ejpam-5255	146	4	is	be	AUX
ejpam-5255	146	5	a	a	DET
ejpam-5255	146	6	bipolar	bipolar	ADJ
ejpam-5255	146	7	fuzzy	fuzzy	ADJ
ejpam-5255	146	8	commutative	commutative	ADJ
ejpam-5255	146	9	ideal	ideal	NOUN
ejpam-5255	146	10	as	as	SCONJ
ejpam-5255	146	11	required	require	VERB
ejpam-5255	146	12	.	.	PUNCT
ejpam-5255	147	1	4	4	X
ejpam-5255	147	2	.	.	X
ejpam-5255	147	3	characteristic	characteristic	ADJ
ejpam-5255	147	4	commutative	commutative	ADJ
ejpam-5255	147	5	ideals	ideal	NOUN
ejpam-5255	147	6	recall	recall	VERB
ejpam-5255	147	7	that	that	SCONJ
ejpam-5255	147	8	a	a	DET
ejpam-5255	147	9	commutative	commutative	ADJ
ejpam-5255	147	10	ideal	ideal	NOUN
ejpam-5255	147	11	a	a	PRON
ejpam-5255	147	12	of	of	ADP
ejpam-5255	147	13	a	a	DET
ejpam-5255	147	14	bck	bck	NOUN
ejpam-5255	147	15	-	-	PUNCT
ejpam-5255	147	16	algebra	algebra	NOUN
ejpam-5255	147	17	x	x	PUNCT
ejpam-5255	147	18	that	that	PRON
ejpam-5255	147	19	satisfies	satisfy	VERB
ejpam-5255	147	20	:	:	PUNCT
ejpam-5255	147	21	if	if	SCONJ
ejpam-5255	147	22	for	for	ADP
ejpam-5255	147	23	every	every	DET
ejpam-5255	147	24	φ	φ	PROPN
ejpam-5255	147	25	∈	∈	PROPN
ejpam-5255	147	26	aut(x	aut(x	PROPN
ejpam-5255	147	27	)	)	PUNCT
ejpam-5255	147	28	,	,	PUNCT
ejpam-5255	147	29	the	the	DET
ejpam-5255	147	30	set	set	NOUN
ejpam-5255	147	31	of	of	ADP
ejpam-5255	147	32	all	all	DET
ejpam-5255	147	33	automorphisms	automorphism	NOUN
ejpam-5255	147	34	of	of	ADP
ejpam-5255	147	35	x	x	PRON
ejpam-5255	147	36	,	,	PUNCT
ejpam-5255	147	37	we	we	PRON
ejpam-5255	147	38	have	have	AUX
ejpam-5255	147	39	φ(a	φ(a	ADJ
ejpam-5255	147	40	)	)	PUNCT
ejpam-5255	147	41	=	=	SYM
ejpam-5255	148	1	a	a	PRON
ejpam-5255	148	2	,	,	PUNCT
ejpam-5255	148	3	is	be	AUX
ejpam-5255	148	4	said	say	VERB
ejpam-5255	148	5	to	to	PART
ejpam-5255	148	6	be	be	AUX
ejpam-5255	148	7	a	a	DET
ejpam-5255	148	8	characteristic	characteristic	ADJ
ejpam-5255	148	9	commutative	commutative	ADJ
ejpam-5255	148	10	ideal	ideal	NOUN
ejpam-5255	148	11	of	of	ADP
ejpam-5255	148	12	x	x	PUNCT
ejpam-5255	148	13	a	a	DET
ejpam-5255	148	14	bipolar	bipolar	ADJ
ejpam-5255	148	15	fuzzy	fuzzy	ADJ
ejpam-5255	148	16	commutative	commutative	ADJ
ejpam-5255	148	17	ideal	ideal	NOUN
ejpam-5255	148	18	a	a	PRON
ejpam-5255	148	19	of	of	ADP
ejpam-5255	148	20	a	a	DET
ejpam-5255	148	21	bck	bck	NOUN
ejpam-5255	148	22	-	-	PUNCT
ejpam-5255	148	23	algebra	algebra	NOUN
ejpam-5255	148	24	x	x	PUNCT
ejpam-5255	148	25	is	be	AUX
ejpam-5255	148	26	said	say	VERB
ejpam-5255	148	27	to	to	PART
ejpam-5255	148	28	be	be	AUX
ejpam-5255	148	29	a	a	DET
ejpam-5255	148	30	bipolar	bipolar	ADJ
ejpam-5255	148	31	fuzzy	fuzzy	ADJ
ejpam-5255	148	32	characteristic	characteristic	ADJ
ejpam-5255	148	33	commutative	commutative	ADJ
ejpam-5255	148	34	ideal	ideal	NOUN
ejpam-5255	148	35	of	of	ADP
ejpam-5255	148	36	x	x	PRON
ejpam-5255	148	37	if	if	SCONJ
ejpam-5255	148	38	for	for	ADP
ejpam-5255	148	39	every	every	DET
ejpam-5255	148	40	φ	φ	PROPN
ejpam-5255	148	41	∈	∈	PROPN
ejpam-5255	148	42	aut(x	aut(x	PROPN
ejpam-5255	148	43	)	)	PUNCT
ejpam-5255	148	44	we	we	PRON
ejpam-5255	148	45	have	have	VERB
ejpam-5255	148	46	ϱ+(φ(a	ϱ+(φ(a	NOUN
ejpam-5255	148	47	)	)	PUNCT
ejpam-5255	148	48	)	)	PUNCT
ejpam-5255	149	1	=	=	SYM
ejpam-5255	149	2	ϱ+(a	ϱ+(a	NOUN
ejpam-5255	149	3	)	)	PUNCT
ejpam-5255	149	4	and	and	CCONJ
ejpam-5255	149	5	ϱ−(φ(a	ϱ−(φ(a	NOUN
ejpam-5255	149	6	)	)	PUNCT
ejpam-5255	149	7	)	)	PUNCT
ejpam-5255	150	1	=	=	SYM
ejpam-5255	150	2	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	150	3	)	)	PUNCT
ejpam-5255	150	4	.	.	PUNCT
ejpam-5255	151	1	theorem	theorem	NOUN
ejpam-5255	151	2	5	5	NUM
ejpam-5255	151	3	.	.	PUNCT
ejpam-5255	152	1	let	let	VERB
ejpam-5255	152	2	x	x	PRON
ejpam-5255	152	3	be	be	AUX
ejpam-5255	152	4	a	a	DET
ejpam-5255	152	5	bck	bck	NOUN
ejpam-5255	152	6	-	-	PUNCT
ejpam-5255	152	7	algebra	algebra	NOUN
ejpam-5255	152	8	and	and	CCONJ
ejpam-5255	152	9	let	let	VERB
ejpam-5255	152	10	ϱ	ϱ	VERB
ejpam-5255	152	11	:	:	PUNCT
ejpam-5255	152	12	=	=	SYM
ejpam-5255	152	13	(	(	PUNCT
ejpam-5255	152	14	x	x	NOUN
ejpam-5255	152	15	,	,	PUNCT
ejpam-5255	152	16	ϱ+	ϱ+	X
ejpam-5255	152	17	,	,	PUNCT
ejpam-5255	152	18	ϱ−	ϱ−	NUM
ejpam-5255	152	19	)	)	PUNCT
ejpam-5255	152	20	be	be	VERB
ejpam-5255	152	21	a	a	DET
ejpam-5255	152	22	bipolar	bipolar	ADJ
ejpam-5255	152	23	fuzzy	fuzzy	ADJ
ejpam-5255	152	24	characteristic	characteristic	ADJ
ejpam-5255	152	25	commutative	commutative	ADJ
ejpam-5255	152	26	ideal	ideal	NOUN
ejpam-5255	152	27	of	of	ADP
ejpam-5255	152	28	x.	x.	NOUN
ejpam-5255	152	29	then	then	ADV
ejpam-5255	152	30	any	any	DET
ejpam-5255	152	31	negative	negative	ADJ
ejpam-5255	152	32	α	α	NOUN
ejpam-5255	152	33	-	-	ADJ
ejpam-5255	152	34	cut	cut	VERB
ejpam-5255	152	35	and	and	CCONJ
ejpam-5255	152	36	positive	positive	ADJ
ejpam-5255	152	37	β	β	NOUN
ejpam-5255	152	38	-	-	ADJ
ejpam-5255	152	39	cut	cut	VERB
ejpam-5255	152	40	commutative	commutative	ADJ
ejpam-5255	152	41	ideals	ideal	NOUN
ejpam-5255	152	42	of	of	ADP
ejpam-5255	152	43	ϱ	ϱ	NOUN
ejpam-5255	152	44	are	be	AUX
ejpam-5255	152	45	characteristic	characteristic	ADJ
ejpam-5255	152	46	commutative	commutative	ADJ
ejpam-5255	152	47	ideals	ideal	NOUN
ejpam-5255	152	48	of	of	ADP
ejpam-5255	152	49	x.	x.	PROPN
ejpam-5255	152	50	a.	a.	PROPN
ejpam-5255	152	51	almuhaimeed	almuhaimeed	PROPN
ejpam-5255	152	52	,	,	PUNCT
ejpam-5255	152	53	h.	h.	PROPN
ejpam-5255	152	54	alshehri	alshehri	PROPN
ejpam-5255	152	55	/	/	SYM
ejpam-5255	152	56	eur	eur	PROPN
ejpam-5255	152	57	.	.	PUNCT
ejpam-5255	153	1	j.	j.	PROPN
ejpam-5255	153	2	pure	pure	PROPN
ejpam-5255	153	3	appl	appl	PROPN
ejpam-5255	153	4	.	.	PROPN
ejpam-5255	153	5	math	math	PROPN
ejpam-5255	153	6	,	,	PUNCT
ejpam-5255	153	7	17	17	NUM
ejpam-5255	153	8	(	(	PUNCT
ejpam-5255	153	9	3	3	NUM
ejpam-5255	153	10	)	)	PUNCT
ejpam-5255	153	11	(	(	PUNCT
ejpam-5255	153	12	2024	2024	NUM
ejpam-5255	153	13	)	)	PUNCT
ejpam-5255	153	14	,	,	PUNCT
ejpam-5255	153	15	1831	1831	NUM
ejpam-5255	153	16	-	-	SYM
ejpam-5255	153	17	1841	1841	NUM
ejpam-5255	153	18	1838	1838	NUM
ejpam-5255	153	19	proof	proof	NOUN
ejpam-5255	153	20	.	.	PUNCT
ejpam-5255	154	1	consider	consider	VERB
ejpam-5255	154	2	a	a	DET
ejpam-5255	154	3	negative	negative	ADJ
ejpam-5255	154	4	α	α	NOUN
ejpam-5255	154	5	-	-	PUNCT
ejpam-5255	154	6	cut	cut	VERB
ejpam-5255	154	7	,	,	PUNCT
ejpam-5255	154	8	n(ϱ;α	n(ϱ;α	NOUN
ejpam-5255	154	9	)	)	PUNCT
ejpam-5255	154	10	=	=	NOUN
ejpam-5255	154	11	ϱα	ϱα	PROPN
ejpam-5255	154	12	and	and	CCONJ
ejpam-5255	154	13	a	a	DET
ejpam-5255	154	14	positive	positive	ADJ
ejpam-5255	154	15	β	β	NOUN
ejpam-5255	154	16	-	-	NOUN
ejpam-5255	154	17	cut	cut	ADJ
ejpam-5255	154	18	,	,	PUNCT
ejpam-5255	154	19	p	p	X
ejpam-5255	154	20	(	(	PUNCT
ejpam-5255	154	21	ϱ;β	ϱ;β	PROPN
ejpam-5255	154	22	)	)	PUNCT
ejpam-5255	155	1	=	=	PRON
ejpam-5255	155	2	ϱβ	ϱβ	NOUN
ejpam-5255	155	3	of	of	ADP
ejpam-5255	155	4	ϱ.	ϱ.	NOUN
ejpam-5255	155	5	let	let	VERB
ejpam-5255	155	6	α	α	PRON
ejpam-5255	155	7	,	,	PUNCT
ejpam-5255	155	8	β	β	PROPN
ejpam-5255	155	9	∈	∈	PROPN
ejpam-5255	155	10	im(f	im(f	NOUN
ejpam-5255	155	11	)	)	PUNCT
ejpam-5255	155	12	,	,	PUNCT
ejpam-5255	155	13	the	the	DET
ejpam-5255	155	14	image	image	NOUN
ejpam-5255	155	15	of	of	ADP
ejpam-5255	155	16	ϱ	ϱ	PROPN
ejpam-5255	155	17	,	,	PUNCT
ejpam-5255	155	18	φ	φ	PROPN
ejpam-5255	155	19	∈	∈	PROPN
ejpam-5255	155	20	aut(x	aut(x	PROPN
ejpam-5255	155	21	)	)	PUNCT
ejpam-5255	155	22	,	,	PUNCT
ejpam-5255	155	23	r	r	NOUN
ejpam-5255	155	24	∈	∈	PRON
ejpam-5255	155	25	ϱβ	ϱβ	VERB
ejpam-5255	155	26	and	and	CCONJ
ejpam-5255	155	27	s	s	PROPN
ejpam-5255	155	28	∈	∈	PROPN
ejpam-5255	155	29	ϱα	ϱα	PROPN
ejpam-5255	155	30	.	.	PUNCT
ejpam-5255	156	1	that	that	SCONJ
ejpam-5255	156	2	ϱ	ϱ	PROPN
ejpam-5255	156	3	is	be	AUX
ejpam-5255	156	4	a	a	DET
ejpam-5255	156	5	bipolar	bipolar	ADJ
ejpam-5255	156	6	fuzzy	fuzzy	ADJ
ejpam-5255	156	7	characteristic	characteristic	ADJ
ejpam-5255	156	8	commutative	commutative	ADJ
ejpam-5255	156	9	ideal	ideal	NOUN
ejpam-5255	156	10	of	of	ADP
ejpam-5255	156	11	x	x	PRON
ejpam-5255	156	12	,	,	PUNCT
ejpam-5255	156	13	implies	imply	VERB
ejpam-5255	156	14	that	that	SCONJ
ejpam-5255	156	15	ϱ+(φ(r	ϱ+(φ(r	NOUN
ejpam-5255	156	16	)	)	PUNCT
ejpam-5255	156	17	)	)	PUNCT
ejpam-5255	157	1	=	=	SYM
ejpam-5255	157	2	ϱ+(r	ϱ+(r	PROPN
ejpam-5255	157	3	)	)	PUNCT
ejpam-5255	157	4	≥	≥	NOUN
ejpam-5255	157	5	β	β	X
ejpam-5255	157	6	,	,	PUNCT
ejpam-5255	157	7	ϱ−(φ(s	ϱ−(φ(s	PROPN
ejpam-5255	157	8	)	)	PUNCT
ejpam-5255	157	9	)	)	PUNCT
ejpam-5255	158	1	=	=	SYM
ejpam-5255	158	2	ϱ−(s	ϱ−(s	PROPN
ejpam-5255	158	3	)	)	PUNCT
ejpam-5255	158	4	≤	≤	NOUN
ejpam-5255	158	5	α	α	X
ejpam-5255	158	6	.	.	PUNCT
ejpam-5255	159	1	then	then	ADV
ejpam-5255	159	2	φ(r	φ(r	ADJ
ejpam-5255	159	3	)	)	PUNCT
ejpam-5255	159	4	∈	∈	PROPN
ejpam-5255	159	5	ϱβ	ϱβ	NOUN
ejpam-5255	159	6	and	and	CCONJ
ejpam-5255	159	7	φ(s	φ(s	NOUN
ejpam-5255	159	8	)	)	PUNCT
ejpam-5255	159	9	∈	∈	PROPN
ejpam-5255	159	10	ϱα	ϱα	PROPN
ejpam-5255	159	11	which	which	PRON
ejpam-5255	159	12	means	mean	VERB
ejpam-5255	159	13	that	that	SCONJ
ejpam-5255	159	14	φ(ϱβ	φ(ϱβ	NOUN
ejpam-5255	159	15	)	)	PUNCT
ejpam-5255	159	16	⊆	⊆	NUM
ejpam-5255	159	17	ϱβ	ϱβ	NOUN
ejpam-5255	159	18	and	and	CCONJ
ejpam-5255	159	19	φ(ϱα	φ(ϱα	NUM
ejpam-5255	159	20	)	)	PUNCT
ejpam-5255	159	21	⊆	⊆	NUM
ejpam-5255	159	22	ϱα	ϱα	NOUN
ejpam-5255	159	23	.	.	PUNCT
ejpam-5255	160	1	on	on	ADP
ejpam-5255	160	2	the	the	DET
ejpam-5255	160	3	other	other	ADJ
ejpam-5255	160	4	hand	hand	NOUN
ejpam-5255	160	5	,	,	PUNCT
ejpam-5255	160	6	let	let	VERB
ejpam-5255	160	7	r	r	NOUN
ejpam-5255	160	8	∈	∈	PRON
ejpam-5255	160	9	ϱβ	ϱβ	VERB
ejpam-5255	160	10	,	,	PUNCT
ejpam-5255	160	11	s	s	PART
ejpam-5255	160	12	∈	∈	PROPN
ejpam-5255	160	13	ϱα	ϱα	PROPN
ejpam-5255	160	14	and	and	CCONJ
ejpam-5255	160	15	a	a	DET
ejpam-5255	160	16	,	,	PUNCT
ejpam-5255	160	17	b	b	X
ejpam-5255	160	18	∈	∈	PROPN
ejpam-5255	160	19	x	x	PUNCT
ejpam-5255	160	20	in	in	ADP
ejpam-5255	160	21	which	which	PRON
ejpam-5255	160	22	φ(a	φ(a	ADJ
ejpam-5255	160	23	)	)	PUNCT
ejpam-5255	160	24	=	=	SYM
ejpam-5255	160	25	r	r	NOUN
ejpam-5255	160	26	and	and	CCONJ
ejpam-5255	160	27	φ(b	φ(b	PROPN
ejpam-5255	160	28	)	)	PUNCT
ejpam-5255	160	29	=	=	VERB
ejpam-5255	161	1	s.	s.	PROPN
ejpam-5255	161	2	it	it	PRON
ejpam-5255	161	3	follows	follow	VERB
ejpam-5255	161	4	that	that	SCONJ
ejpam-5255	161	5	ϱ+(a	ϱ+(a	NOUN
ejpam-5255	161	6	)	)	PUNCT
ejpam-5255	161	7	=	=	SYM
ejpam-5255	161	8	ϱ+(φ(a	ϱ+(φ(a	NOUN
ejpam-5255	161	9	)	)	PUNCT
ejpam-5255	161	10	)	)	PUNCT
ejpam-5255	162	1	=	=	SYM
ejpam-5255	162	2	ϱ+(r	ϱ+(r	PROPN
ejpam-5255	162	3	)	)	PUNCT
ejpam-5255	162	4	≥	≥	NOUN
ejpam-5255	162	5	β	β	X
ejpam-5255	162	6	,	,	PUNCT
ejpam-5255	162	7	ϱ−(b	ϱ−(b	PROPN
ejpam-5255	162	8	)	)	PUNCT
ejpam-5255	162	9	=	=	SYM
ejpam-5255	162	10	ϱ−(φ(b	ϱ−(φ(b	PROPN
ejpam-5255	162	11	)	)	PUNCT
ejpam-5255	162	12	)	)	PUNCT
ejpam-5255	163	1	=	=	SYM
ejpam-5255	163	2	ϱ−(s	ϱ−(s	PROPN
ejpam-5255	163	3	)	)	PUNCT
ejpam-5255	163	4	≤	≤	NOUN
ejpam-5255	163	5	α	α	NUM
ejpam-5255	163	6	,	,	PUNCT
ejpam-5255	163	7	and	and	CCONJ
ejpam-5255	163	8	thus	thus	ADV
ejpam-5255	163	9	a	a	DET
ejpam-5255	163	10	∈	∈	NOUN
ejpam-5255	163	11	ϱβ	ϱβ	NOUN
ejpam-5255	163	12	and	and	CCONJ
ejpam-5255	163	13	b	b	PROPN
ejpam-5255	163	14	∈	∈	PROPN
ejpam-5255	163	15	ϱα	ϱα	PROPN
ejpam-5255	163	16	.	.	PUNCT
ejpam-5255	164	1	now	now	ADV
ejpam-5255	164	2	,	,	PUNCT
ejpam-5255	164	3	r	r	NOUN
ejpam-5255	164	4	=	=	SYM
ejpam-5255	164	5	φ(a	φ(a	ADJ
ejpam-5255	164	6	)	)	PUNCT
ejpam-5255	164	7	∈	∈	PROPN
ejpam-5255	164	8	φ(ϱβ	φ(ϱβ	PROPN
ejpam-5255	164	9	)	)	PUNCT
ejpam-5255	164	10	,	,	PUNCT
ejpam-5255	164	11	s	s	PART
ejpam-5255	164	12	=	=	ADJ
ejpam-5255	164	13	φ(b	φ(b	PROPN
ejpam-5255	164	14	)	)	PUNCT
ejpam-5255	164	15	∈	∈	PROPN
ejpam-5255	164	16	φ(ϱα	φ(ϱα	NOUN
ejpam-5255	164	17	)	)	PUNCT
ejpam-5255	164	18	,	,	PUNCT
ejpam-5255	164	19	and	and	CCONJ
ejpam-5255	164	20	hence	hence	ADV
ejpam-5255	164	21	ϱβ	ϱβ	VERB
ejpam-5255	164	22	⊆	⊆	NUM
ejpam-5255	164	23	φ(ϱβ	φ(ϱβ	NOUN
ejpam-5255	164	24	)	)	PUNCT
ejpam-5255	164	25	and	and	CCONJ
ejpam-5255	164	26	ϱα	ϱα	PROPN
ejpam-5255	164	27	⊆	⊆	NUM
ejpam-5255	164	28	φ(ϱα	φ(ϱα	NOUN
ejpam-5255	164	29	)	)	PUNCT
ejpam-5255	164	30	.	.	PUNCT
ejpam-5255	165	1	therefore	therefore	ADV
ejpam-5255	165	2	,	,	PUNCT
ejpam-5255	165	3	ϱβ	ϱβ	ADJ
ejpam-5255	165	4	and	and	CCONJ
ejpam-5255	165	5	ϱα	ϱα	PROPN
ejpam-5255	165	6	are	be	AUX
ejpam-5255	165	7	bipolar	bipolar	ADJ
ejpam-5255	165	8	characteristic	characteristic	ADJ
ejpam-5255	165	9	commutative	commutative	ADJ
ejpam-5255	165	10	ideals	ideal	NOUN
ejpam-5255	165	11	of	of	ADP
ejpam-5255	165	12	x	x	PUNCT
ejpam-5255	165	13	as	as	SCONJ
ejpam-5255	165	14	required	require	VERB
ejpam-5255	165	15	.	.	PUNCT
ejpam-5255	166	1	theorem	theorem	ADJ
ejpam-5255	166	2	6	6	NUM
ejpam-5255	166	3	.	.	PUNCT
ejpam-5255	167	1	let	let	VERB
ejpam-5255	167	2	x	x	PRON
ejpam-5255	167	3	be	be	AUX
ejpam-5255	167	4	a	a	DET
ejpam-5255	167	5	bck	bck	NOUN
ejpam-5255	167	6	-	-	PUNCT
ejpam-5255	167	7	algebra	algebra	NOUN
ejpam-5255	167	8	and	and	CCONJ
ejpam-5255	167	9	let	let	VERB
ejpam-5255	167	10	ϱ	ϱ	VERB
ejpam-5255	167	11	:	:	PUNCT
ejpam-5255	167	12	=	=	SYM
ejpam-5255	167	13	(	(	PUNCT
ejpam-5255	167	14	x	x	NOUN
ejpam-5255	167	15	,	,	PUNCT
ejpam-5255	167	16	ϱ+	ϱ+	X
ejpam-5255	167	17	,	,	PUNCT
ejpam-5255	167	18	ϱ−	ϱ−	NUM
ejpam-5255	167	19	)	)	PUNCT
ejpam-5255	167	20	be	be	VERB
ejpam-5255	167	21	a	a	DET
ejpam-5255	167	22	bipolar	bipolar	ADJ
ejpam-5255	167	23	fuzzy	fuzzy	ADJ
ejpam-5255	167	24	commutative	commutative	ADJ
ejpam-5255	167	25	ideal	ideal	NOUN
ejpam-5255	167	26	of	of	ADP
ejpam-5255	167	27	x.	x.	NOUN
ejpam-5255	167	28	if	if	SCONJ
ejpam-5255	167	29	each	each	DET
ejpam-5255	167	30	negative	negative	ADJ
ejpam-5255	167	31	α	α	NOUN
ejpam-5255	167	32	-	-	PUNCT
ejpam-5255	167	33	cut	cut	VERB
ejpam-5255	167	34	and	and	CCONJ
ejpam-5255	167	35	each	each	DET
ejpam-5255	167	36	positive	positive	ADJ
ejpam-5255	167	37	β	β	NOUN
ejpam-5255	167	38	-	-	ADJ
ejpam-5255	167	39	cut	cut	VERB
ejpam-5255	167	40	commutative	commutative	ADJ
ejpam-5255	167	41	ideal	ideal	NOUN
ejpam-5255	167	42	of	of	ADP
ejpam-5255	167	43	ϱ	ϱ	PROPN
ejpam-5255	167	44	is	be	AUX
ejpam-5255	167	45	a	a	DET
ejpam-5255	167	46	characteristic	characteristic	ADJ
ejpam-5255	167	47	commutative	commutative	ADJ
ejpam-5255	167	48	ideal	ideal	NOUN
ejpam-5255	167	49	of	of	ADP
ejpam-5255	167	50	x	x	PRON
ejpam-5255	167	51	,	,	PUNCT
ejpam-5255	167	52	then	then	ADV
ejpam-5255	167	53	ϱ	ϱ	PROPN
ejpam-5255	167	54	is	be	AUX
ejpam-5255	167	55	a	a	DET
ejpam-5255	167	56	bipolar	bipolar	ADJ
ejpam-5255	167	57	fuzzy	fuzzy	ADJ
ejpam-5255	167	58	characteristic	characteristic	ADJ
ejpam-5255	167	59	commutative	commutative	ADJ
ejpam-5255	167	60	ideal	ideal	NOUN
ejpam-5255	167	61	of	of	ADP
ejpam-5255	167	62	x.	x.	NOUN
ejpam-5255	167	63	proof	proof	NOUN
ejpam-5255	167	64	.	.	PUNCT
ejpam-5255	168	1	let	let	VERB
ejpam-5255	168	2	r	r	NOUN
ejpam-5255	168	3	∈	∈	PROPN
ejpam-5255	168	4	x	x	NOUN
ejpam-5255	168	5	,	,	PUNCT
ejpam-5255	168	6	φ	φ	PROPN
ejpam-5255	168	7	∈	∈	PROPN
ejpam-5255	168	8	aut(x	aut(x	PROPN
ejpam-5255	168	9	)	)	PUNCT
ejpam-5255	168	10	and	and	CCONJ
ejpam-5255	168	11	f(r	f(r	NOUN
ejpam-5255	168	12	)	)	PUNCT
ejpam-5255	169	1	=	=	SYM
ejpam-5255	169	2	β	β	X
ejpam-5255	169	3	.	.	PUNCT
ejpam-5255	170	1	it	it	PRON
ejpam-5255	170	2	follows	follow	VERB
ejpam-5255	170	3	that	that	SCONJ
ejpam-5255	170	4	r	r	NOUN
ejpam-5255	170	5	∈	∈	PRON
ejpam-5255	170	6	ϱβ	ϱβ	NOUN
ejpam-5255	170	7	and	and	CCONJ
ejpam-5255	170	8	r	r	NOUN
ejpam-5255	170	9	∈	∈	PROPN
ejpam-5255	170	10	ϱγ	ϱγ	ADV
ejpam-5255	170	11	for	for	ADP
ejpam-5255	170	12	all	all	DET
ejpam-5255	170	13	γ	γ	X
ejpam-5255	170	14	>	>	X
ejpam-5255	170	15	β	β	PROPN
ejpam-5255	170	16	.	.	PUNCT
ejpam-5255	171	1	that	that	DET
ejpam-5255	171	2	φ(ϱβ	φ(ϱβ	NOUN
ejpam-5255	171	3	)	)	PUNCT
ejpam-5255	171	4	=	=	SYM
ejpam-5255	172	1	ϱβ	ϱβ	ADJ
ejpam-5255	172	2	,	,	PUNCT
ejpam-5255	172	3	implies	imply	VERB
ejpam-5255	172	4	that	that	SCONJ
ejpam-5255	172	5	φ(r	φ(r	ADJ
ejpam-5255	172	6	)	)	PUNCT
ejpam-5255	172	7	∈	∈	PROPN
ejpam-5255	172	8	ϱβ	ϱβ	NOUN
ejpam-5255	172	9	.	.	PUNCT
ejpam-5255	172	10	thus	thus	ADV
ejpam-5255	172	11	ϱ+(φ(r	ϱ+(φ(r	NOUN
ejpam-5255	172	12	)	)	PUNCT
ejpam-5255	172	13	)	)	PUNCT
ejpam-5255	172	14	≥	≥	PROPN
ejpam-5255	173	1	β	β	X
ejpam-5255	173	2	.	.	PUNCT
ejpam-5255	174	1	let	let	VERB
ejpam-5255	174	2	γ	γ	X
ejpam-5255	174	3	=	=	VERB
ejpam-5255	174	4	ϱ+(φ(r	ϱ+(φ(r	PROPN
ejpam-5255	174	5	)	)	PUNCT
ejpam-5255	174	6	)	)	PUNCT
ejpam-5255	174	7	.	.	PUNCT
ejpam-5255	175	1	then	then	ADV
ejpam-5255	175	2	φ(r	φ(r	ADJ
ejpam-5255	175	3	)	)	PUNCT
ejpam-5255	175	4	∈	∈	PROPN
ejpam-5255	175	5	ϱγ	ϱγ	NOUN
ejpam-5255	175	6	=	=	SYM
ejpam-5255	175	7	φ(ϱγ	φ(ϱγ	PROPN
ejpam-5255	175	8	)	)	PUNCT
ejpam-5255	175	9	by	by	ADP
ejpam-5255	175	10	hypothesis	hypothesis	NOUN
ejpam-5255	175	11	.	.	PUNCT
ejpam-5255	176	1	that	that	SCONJ
ejpam-5255	176	2	φ	φ	PROPN
ejpam-5255	176	3	∈	∈	PROPN
ejpam-5255	176	4	aut(x	aut(x	PROPN
ejpam-5255	176	5	)	)	PUNCT
ejpam-5255	176	6	,	,	PUNCT
ejpam-5255	176	7	implies	imply	VERB
ejpam-5255	176	8	that	that	SCONJ
ejpam-5255	176	9	φ	φ	PROPN
ejpam-5255	176	10	is	be	AUX
ejpam-5255	176	11	injective	injective	ADJ
ejpam-5255	176	12	and	and	CCONJ
ejpam-5255	176	13	so	so	ADV
ejpam-5255	176	14	r	r	NOUN
ejpam-5255	176	15	∈	∈	PROPN
ejpam-5255	176	16	ϱγ	ϱγ	ADV
ejpam-5255	176	17	.	.	PUNCT
ejpam-5255	177	1	this	this	PRON
ejpam-5255	177	2	means	mean	VERB
ejpam-5255	177	3	that	that	SCONJ
ejpam-5255	177	4	γ	γ	PROPN
ejpam-5255	177	5	can	can	AUX
ejpam-5255	177	6	not	not	PART
ejpam-5255	177	7	be	be	AUX
ejpam-5255	177	8	greater	great	ADJ
ejpam-5255	177	9	than	than	ADP
ejpam-5255	177	10	β	β	NOUN
ejpam-5255	177	11	and	and	CCONJ
ejpam-5255	177	12	hence	hence	ADV
ejpam-5255	177	13	ϱ+(φ(r	ϱ+(φ(r	NOUN
ejpam-5255	177	14	)	)	PUNCT
ejpam-5255	177	15	)	)	PUNCT
ejpam-5255	178	1	=	=	PUNCT
ejpam-5255	178	2	β	β	X
ejpam-5255	178	3	=	=	SYM
ejpam-5255	178	4	ϱ+(r	ϱ+(r	PROPN
ejpam-5255	178	5	)	)	PUNCT
ejpam-5255	178	6	.	.	PUNCT
ejpam-5255	179	1	similarly	similarly	ADV
ejpam-5255	179	2	,	,	PUNCT
ejpam-5255	179	3	we	we	PRON
ejpam-5255	179	4	can	can	AUX
ejpam-5255	179	5	show	show	VERB
ejpam-5255	179	6	that	that	SCONJ
ejpam-5255	179	7	ϱ−(φ(r	ϱ−(φ(r	PROPN
ejpam-5255	179	8	)	)	PUNCT
ejpam-5255	179	9	)	)	PUNCT
ejpam-5255	180	1	=	=	SYM
ejpam-5255	180	2	α	α	NOUN
ejpam-5255	180	3	=	=	SYM
ejpam-5255	180	4	ϱ−(r	ϱ−(r	PROPN
ejpam-5255	180	5	)	)	PUNCT
ejpam-5255	180	6	for	for	ADP
ejpam-5255	180	7	every	every	DET
ejpam-5255	180	8	r	r	NOUN
ejpam-5255	180	9	∈	∈	PROPN
ejpam-5255	180	10	x	x	NOUN
ejpam-5255	180	11	,	,	PUNCT
ejpam-5255	180	12	φ	φ	PROPN
ejpam-5255	180	13	∈	∈	PROPN
ejpam-5255	180	14	aut(x	aut(x	PROPN
ejpam-5255	180	15	)	)	PUNCT
ejpam-5255	180	16	and	and	CCONJ
ejpam-5255	180	17	f(r	f(r	NOUN
ejpam-5255	180	18	)	)	PUNCT
ejpam-5255	181	1	=	=	SYM
ejpam-5255	181	2	α	α	X
ejpam-5255	181	3	.	.	PUNCT
ejpam-5255	182	1	therefore	therefore	ADV
ejpam-5255	182	2	,	,	PUNCT
ejpam-5255	182	3	ϱ	ϱ	PROPN
ejpam-5255	182	4	is	be	AUX
ejpam-5255	182	5	a	a	DET
ejpam-5255	182	6	bipolar	bipolar	ADJ
ejpam-5255	182	7	fuzzy	fuzzy	ADJ
ejpam-5255	182	8	characteristic	characteristic	ADJ
ejpam-5255	182	9	commutative	commutative	ADJ
ejpam-5255	182	10	ideal	ideal	NOUN
ejpam-5255	182	11	of	of	ADP
ejpam-5255	182	12	x.	x.	PROPN
ejpam-5255	182	13	5	5	NUM
ejpam-5255	182	14	.	.	X
ejpam-5255	182	15	bipolar	bipolar	ADJ
ejpam-5255	182	16	fuzzy	fuzzy	ADJ
ejpam-5255	182	17	commutative	commutative	ADJ
ejpam-5255	182	18	positive	positive	ADJ
ejpam-5255	182	19	ideal	ideal	NOUN
ejpam-5255	182	20	recall	recall	NOUN
ejpam-5255	182	21	that	that	SCONJ
ejpam-5255	182	22	a	a	DET
ejpam-5255	182	23	bipolar	bipolar	ADJ
ejpam-5255	182	24	fuzzy	fuzzy	ADJ
ejpam-5255	182	25	ideal	ideal	NOUN
ejpam-5255	182	26	of	of	ADP
ejpam-5255	182	27	a	a	DET
ejpam-5255	182	28	bck	bck	NOUN
ejpam-5255	182	29	-	-	PUNCT
ejpam-5255	182	30	algebra	algebra	NOUN
ejpam-5255	182	31	x	x	PUNCT
ejpam-5255	182	32	is	be	AUX
ejpam-5255	182	33	said	say	VERB
ejpam-5255	182	34	to	to	PART
ejpam-5255	182	35	be	be	AUX
ejpam-5255	182	36	a	a	DET
ejpam-5255	182	37	bipolar	bipolar	ADJ
ejpam-5255	182	38	fuzzy	fuzzy	ADJ
ejpam-5255	182	39	positive	positive	ADJ
ejpam-5255	182	40	implicative	implicative	ADJ
ejpam-5255	182	41	ideal	ideal	NOUN
ejpam-5255	182	42	of	of	ADP
ejpam-5255	182	43	x	x	PRON
ejpam-5255	182	44	if	if	SCONJ
ejpam-5255	182	45	ϱ+(a	ϱ+(a	PROPN
ejpam-5255	182	46	∗	∗	NOUN
ejpam-5255	182	47	w	w	NOUN
ejpam-5255	182	48	)	)	PUNCT
ejpam-5255	182	49	≥	≥	PROPN
ejpam-5255	182	50	min	min	NOUN
ejpam-5255	182	51	{	{	PUNCT
ejpam-5255	182	52	ϱ+((a	ϱ+((a	ADP
ejpam-5255	182	53	∗	∗	NOUN
ejpam-5255	182	54	g	g	NOUN
ejpam-5255	182	55	)	)	PUNCT
ejpam-5255	182	56	∗	∗	PROPN
ejpam-5255	182	57	w	w	PROPN
ejpam-5255	182	58	)	)	PUNCT
ejpam-5255	182	59	,	,	PUNCT
ejpam-5255	182	60	ϱ+(g	ϱ+(g	ADP
ejpam-5255	182	61	∗	∗	NOUN
ejpam-5255	182	62	w	w	NOUN
ejpam-5255	182	63	)	)	PUNCT
ejpam-5255	182	64	}	}	PUNCT
ejpam-5255	182	65	,	,	PUNCT
ejpam-5255	182	66	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	182	67	∗	∗	NOUN
ejpam-5255	182	68	w	w	NOUN
ejpam-5255	182	69	)	)	PUNCT
ejpam-5255	182	70	≤	≤	NUM
ejpam-5255	182	71	max	max	PROPN
ejpam-5255	182	72	{	{	PUNCT
ejpam-5255	182	73	ϱ−((a	ϱ−((a	PROPN
ejpam-5255	182	74	∗	∗	NOUN
ejpam-5255	182	75	g	g	NOUN
ejpam-5255	182	76	)	)	PUNCT
ejpam-5255	182	77	∗	∗	PROPN
ejpam-5255	182	78	w	w	NOUN
ejpam-5255	182	79	)	)	PUNCT
ejpam-5255	182	80	)	)	PUNCT
ejpam-5255	182	81	,	,	PUNCT
ejpam-5255	182	82	ϱ−(g	ϱ−(g	VERB
ejpam-5255	182	83	∗	∗	NOUN
ejpam-5255	182	84	w	w	NOUN
ejpam-5255	182	85	)	)	PUNCT
ejpam-5255	182	86	}	}	PUNCT
ejpam-5255	182	87	.	.	PUNCT
ejpam-5255	183	1	if	if	SCONJ
ejpam-5255	183	2	x	x	PRON
ejpam-5255	183	3	is	be	AUX
ejpam-5255	183	4	positive	positive	ADJ
ejpam-5255	183	5	implicative	implicative	ADJ
ejpam-5255	183	6	,	,	PUNCT
ejpam-5255	183	7	then	then	ADV
ejpam-5255	183	8	we	we	PRON
ejpam-5255	183	9	have	have	VERB
ejpam-5255	183	10	(	(	PUNCT
ejpam-5255	183	11	a	a	DET
ejpam-5255	183	12	∗	∗	NOUN
ejpam-5255	183	13	w	w	NOUN
ejpam-5255	183	14	)	)	PUNCT
ejpam-5255	183	15	∗	∗	NOUN
ejpam-5255	183	16	(	(	PUNCT
ejpam-5255	183	17	g	g	PROPN
ejpam-5255	183	18	∗	∗	X
ejpam-5255	183	19	w	w	NOUN
ejpam-5255	183	20	)	)	PUNCT
ejpam-5255	183	21	=	=	SYM
ejpam-5255	183	22	(	(	PUNCT
ejpam-5255	183	23	(	(	PUNCT
ejpam-5255	183	24	a	a	DET
ejpam-5255	183	25	∗	∗	NOUN
ejpam-5255	183	26	g	g	NOUN
ejpam-5255	183	27	)	)	PUNCT
ejpam-5255	183	28	∗	∗	PROPN
ejpam-5255	183	29	w	w	PROPN
ejpam-5255	183	30	)	)	PUNCT
ejpam-5255	183	31	(	(	PUNCT
ejpam-5255	183	32	4	4	X
ejpam-5255	183	33	)	)	PUNCT
ejpam-5255	183	34	a.	a.	NOUN
ejpam-5255	183	35	almuhaimeed	almuhaimeed	NOUN
ejpam-5255	183	36	,	,	PUNCT
ejpam-5255	183	37	h.	h.	PROPN
ejpam-5255	183	38	alshehri	alshehri	PROPN
ejpam-5255	183	39	/	/	SYM
ejpam-5255	183	40	eur	eur	PROPN
ejpam-5255	183	41	.	.	PUNCT
ejpam-5255	184	1	j.	j.	PROPN
ejpam-5255	184	2	pure	pure	PROPN
ejpam-5255	184	3	appl	appl	PROPN
ejpam-5255	184	4	.	.	PROPN
ejpam-5255	184	5	math	math	PROPN
ejpam-5255	184	6	,	,	PUNCT
ejpam-5255	184	7	17	17	NUM
ejpam-5255	184	8	(	(	PUNCT
ejpam-5255	184	9	3	3	NUM
ejpam-5255	184	10	)	)	PUNCT
ejpam-5255	184	11	(	(	PUNCT
ejpam-5255	184	12	2024	2024	NUM
ejpam-5255	184	13	)	)	PUNCT
ejpam-5255	184	14	,	,	PUNCT
ejpam-5255	184	15	1831	1831	NUM
ejpam-5255	184	16	-	-	SYM
ejpam-5255	184	17	1841	1841	NUM
ejpam-5255	184	18	1839	1839	NUM
ejpam-5255	184	19	theorem	theorem	VERB
ejpam-5255	184	20	7	7	NUM
ejpam-5255	184	21	.	.	PUNCT
ejpam-5255	185	1	let	let	VERB
ejpam-5255	185	2	x	x	PRON
ejpam-5255	185	3	be	be	AUX
ejpam-5255	185	4	positive	positive	ADJ
ejpam-5255	185	5	implicative	implicative	NOUN
ejpam-5255	185	6	.	.	PUNCT
ejpam-5255	186	1	a	a	DET
ejpam-5255	186	2	bipolar	bipolar	ADJ
ejpam-5255	186	3	fuzzy	fuzzy	ADJ
ejpam-5255	186	4	commutative	commutative	ADJ
ejpam-5255	186	5	ideal	ideal	NOUN
ejpam-5255	186	6	of	of	ADP
ejpam-5255	186	7	x	x	PUNCT
ejpam-5255	186	8	is	be	AUX
ejpam-5255	186	9	a	a	DET
ejpam-5255	186	10	bipolar	bipolar	ADJ
ejpam-5255	186	11	fuzzy	fuzzy	ADJ
ejpam-5255	186	12	positive	positive	ADJ
ejpam-5255	186	13	implicative	implicative	ADJ
ejpam-5255	186	14	ideal	ideal	NOUN
ejpam-5255	186	15	of	of	ADP
ejpam-5255	186	16	x	x	PRON
ejpam-5255	186	17	if	if	SCONJ
ejpam-5255	186	18	and	and	CCONJ
ejpam-5255	186	19	only	only	ADV
ejpam-5255	186	20	if	if	SCONJ
ejpam-5255	186	21	:	:	PUNCT
ejpam-5255	186	22	ϱ+(a	ϱ+(a	NOUN
ejpam-5255	186	23	∗	∗	NOUN
ejpam-5255	186	24	g	g	NOUN
ejpam-5255	186	25	)	)	PUNCT
ejpam-5255	186	26	≥	≥	NOUN
ejpam-5255	186	27	ϱ+((a	ϱ+((a	ADP
ejpam-5255	186	28	∗	∗	NOUN
ejpam-5255	186	29	g	g	NOUN
ejpam-5255	186	30	)	)	PUNCT
ejpam-5255	186	31	∗	∗	PROPN
ejpam-5255	186	32	w	w	NOUN
ejpam-5255	186	33	)	)	PUNCT
ejpam-5255	186	34	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	186	35	∗	∗	NOUN
ejpam-5255	186	36	g	g	NOUN
ejpam-5255	186	37	)	)	PUNCT
ejpam-5255	186	38	≤	≤	NOUN
ejpam-5255	186	39	ϱ−((a	ϱ−((a	ADP
ejpam-5255	186	40	∗	∗	NOUN
ejpam-5255	186	41	g	g	NOUN
ejpam-5255	186	42	)	)	PUNCT
ejpam-5255	186	43	∗	∗	PROPN
ejpam-5255	186	44	w	w	PROPN
ejpam-5255	186	45	)	)	PUNCT
ejpam-5255	186	46	(	(	PUNCT
ejpam-5255	186	47	5	5	X
ejpam-5255	186	48	)	)	PUNCT
ejpam-5255	186	49	proof	proof	NOUN
ejpam-5255	186	50	.	.	PUNCT
ejpam-5255	187	1	let	let	VERB
ejpam-5255	187	2	ϱ	ϱ	VERB
ejpam-5255	187	3	:	:	PUNCT
ejpam-5255	187	4	=	=	SYM
ejpam-5255	187	5	(	(	PUNCT
ejpam-5255	187	6	x	x	NOUN
ejpam-5255	187	7	,	,	PUNCT
ejpam-5255	187	8	ϱ+	ϱ+	X
ejpam-5255	187	9	,	,	PUNCT
ejpam-5255	187	10	ϱ−	ϱ−	NUM
ejpam-5255	187	11	)	)	PUNCT
ejpam-5255	187	12	be	be	VERB
ejpam-5255	187	13	a	a	DET
ejpam-5255	187	14	bipolar	bipolar	ADJ
ejpam-5255	187	15	fuzzy	fuzzy	ADJ
ejpam-5255	187	16	commutative	commutative	ADJ
ejpam-5255	187	17	ideal	ideal	NOUN
ejpam-5255	187	18	of	of	ADP
ejpam-5255	187	19	x.	x.	PROPN
ejpam-5255	187	20	assume	assume	VERB
ejpam-5255	187	21	that	that	SCONJ
ejpam-5255	187	22	ϱ	ϱ	NOUN
ejpam-5255	187	23	is	be	AUX
ejpam-5255	187	24	a	a	DET
ejpam-5255	187	25	bipolar	bipolar	ADJ
ejpam-5255	187	26	fuzzy	fuzzy	ADJ
ejpam-5255	187	27	positive	positive	ADJ
ejpam-5255	187	28	implicative	implicative	ADJ
ejpam-5255	187	29	ideal	ideal	NOUN
ejpam-5255	187	30	.	.	PUNCT
ejpam-5255	188	1	then	then	ADV
ejpam-5255	188	2	ϱ+(m	ϱ+(m	VERB
ejpam-5255	188	3	∗	∗	NOUN
ejpam-5255	188	4	p	p	NOUN
ejpam-5255	188	5	)	)	PUNCT
ejpam-5255	188	6	≥	≥	NOUN
ejpam-5255	188	7	min	min	NOUN
ejpam-5255	188	8	{	{	PUNCT
ejpam-5255	188	9	ϱ+((a	ϱ+((a	ADP
ejpam-5255	188	10	∗	∗	NOUN
ejpam-5255	188	11	g	g	NOUN
ejpam-5255	188	12	)	)	PUNCT
ejpam-5255	188	13	∗	∗	PROPN
ejpam-5255	188	14	w	w	PROPN
ejpam-5255	188	15	)	)	PUNCT
ejpam-5255	188	16	,	,	PUNCT
ejpam-5255	188	17	ϱ+(n	ϱ+(n	PROPN
ejpam-5255	188	18	∗	∗	NOUN
ejpam-5255	188	19	p	p	NOUN
ejpam-5255	188	20	)	)	PUNCT
ejpam-5255	188	21	}	}	PUNCT
ejpam-5255	188	22	,	,	PUNCT
ejpam-5255	188	23	ϱ−(m	ϱ−(m	PROPN
ejpam-5255	188	24	∗	∗	VERB
ejpam-5255	188	25	p	p	NOUN
ejpam-5255	188	26	)	)	PUNCT
ejpam-5255	188	27	≤	≤	NUM
ejpam-5255	188	28	max	max	PROPN
ejpam-5255	188	29	{	{	PUNCT
ejpam-5255	188	30	ϱ−((a	ϱ−((a	PROPN
ejpam-5255	188	31	∗	∗	NOUN
ejpam-5255	188	32	g	g	NOUN
ejpam-5255	188	33	)	)	PUNCT
ejpam-5255	188	34	∗	∗	PROPN
ejpam-5255	188	35	w	w	NOUN
ejpam-5255	188	36	)	)	PUNCT
ejpam-5255	188	37	,	,	PUNCT
ejpam-5255	188	38	ϱ−(n	ϱ−(n	NOUN
ejpam-5255	188	39	∗	∗	VERB
ejpam-5255	188	40	p	p	NOUN
ejpam-5255	188	41	)	)	PUNCT
ejpam-5255	188	42	}	}	PUNCT
ejpam-5255	188	43	.	.	PUNCT
ejpam-5255	189	1	put	put	VERB
ejpam-5255	189	2	p	p	NOUN
ejpam-5255	189	3	=	=	PUNCT
ejpam-5255	189	4	n	n	CCONJ
ejpam-5255	189	5	,	,	PUNCT
ejpam-5255	189	6	we	we	PRON
ejpam-5255	189	7	obtain	obtain	VERB
ejpam-5255	189	8	ϱ+(m	ϱ+(m	NOUN
ejpam-5255	189	9	∗	∗	NOUN
ejpam-5255	189	10	n	n	CCONJ
ejpam-5255	189	11	)	)	PUNCT
ejpam-5255	189	12	≥	≥	PROPN
ejpam-5255	189	13	min	min	NOUN
ejpam-5255	189	14	{	{	PUNCT
ejpam-5255	189	15	ϱ+((a	ϱ+((a	ADP
ejpam-5255	189	16	∗	∗	NOUN
ejpam-5255	189	17	g	g	NOUN
ejpam-5255	189	18	)	)	PUNCT
ejpam-5255	189	19	∗	∗	PROPN
ejpam-5255	189	20	w	w	PROPN
ejpam-5255	189	21	)	)	PUNCT
ejpam-5255	189	22	,	,	PUNCT
ejpam-5255	189	23	ϱ+(0	ϱ+(0	NOUN
ejpam-5255	189	24	)	)	PUNCT
ejpam-5255	189	25	}	}	PUNCT
ejpam-5255	189	26	,	,	PUNCT
ejpam-5255	189	27	ϱ−(m	ϱ−(m	PROPN
ejpam-5255	189	28	∗	∗	NOUN
ejpam-5255	189	29	n	n	CCONJ
ejpam-5255	189	30	)	)	PUNCT
ejpam-5255	189	31	≤	≤	NUM
ejpam-5255	189	32	max	max	PROPN
ejpam-5255	189	33	{	{	PUNCT
ejpam-5255	189	34	ϱ−((a	ϱ−((a	PROPN
ejpam-5255	189	35	∗	∗	NOUN
ejpam-5255	189	36	g	g	NOUN
ejpam-5255	189	37	)	)	PUNCT
ejpam-5255	189	38	∗	∗	PROPN
ejpam-5255	189	39	w	w	PROPN
ejpam-5255	189	40	)	)	PUNCT
ejpam-5255	189	41	,	,	PUNCT
ejpam-5255	189	42	ϱ−(0	ϱ−(0	PROPN
ejpam-5255	189	43	)	)	PUNCT
ejpam-5255	189	44	}	}	PUNCT
ejpam-5255	189	45	.	.	PUNCT
ejpam-5255	190	1	thus	thus	ADV
ejpam-5255	190	2	we	we	PRON
ejpam-5255	190	3	have	have	VERB
ejpam-5255	190	4	ϱ+(m	ϱ+(m	NOUN
ejpam-5255	190	5	∗	∗	NOUN
ejpam-5255	190	6	n	n	CCONJ
ejpam-5255	190	7	)	)	PUNCT
ejpam-5255	190	8	≥	≥	NOUN
ejpam-5255	190	9	ϱ+((a	ϱ+((a	ADP
ejpam-5255	190	10	∗	∗	NOUN
ejpam-5255	190	11	g	g	NOUN
ejpam-5255	190	12	)	)	PUNCT
ejpam-5255	190	13	∗	∗	PROPN
ejpam-5255	190	14	w	w	NOUN
ejpam-5255	190	15	)	)	PUNCT
ejpam-5255	190	16	ϱ−(m	ϱ−(m	PROPN
ejpam-5255	190	17	∗	∗	NOUN
ejpam-5255	190	18	n	n	CCONJ
ejpam-5255	190	19	)	)	PUNCT
ejpam-5255	190	20	≤	≤	NOUN
ejpam-5255	190	21	ϱ−((a	ϱ−((a	ADP
ejpam-5255	190	22	∗	∗	NOUN
ejpam-5255	190	23	g	g	NOUN
ejpam-5255	190	24	)	)	PUNCT
ejpam-5255	190	25	∗	∗	PROPN
ejpam-5255	190	26	w	w	NOUN
ejpam-5255	190	27	)	)	PUNCT
ejpam-5255	190	28	conversely	conversely	ADV
ejpam-5255	190	29	,	,	PUNCT
ejpam-5255	190	30	suppose	suppose	VERB
ejpam-5255	190	31	that	that	SCONJ
ejpam-5255	190	32	(	(	PUNCT
ejpam-5255	190	33	5	5	X
ejpam-5255	190	34	)	)	PUNCT
ejpam-5255	190	35	holds	hold	VERB
ejpam-5255	190	36	.	.	PUNCT
ejpam-5255	191	1	we	we	PRON
ejpam-5255	191	2	have	have	VERB
ejpam-5255	191	3	ϱ+(a	ϱ+(a	PROPN
ejpam-5255	191	4	∗	∗	NOUN
ejpam-5255	191	5	w	w	NOUN
ejpam-5255	191	6	)	)	PUNCT
ejpam-5255	191	7	≥	≥	NOUN
ejpam-5255	191	8	ϱ+((a	ϱ+((a	ADP
ejpam-5255	191	9	∗	∗	NOUN
ejpam-5255	191	10	g	g	NOUN
ejpam-5255	191	11	)	)	PUNCT
ejpam-5255	191	12	∗	∗	PROPN
ejpam-5255	191	13	w	w	PROPN
ejpam-5255	191	14	)	)	PUNCT
ejpam-5255	191	15	by(5	by(5	NOUN
ejpam-5255	191	16	)	)	PUNCT
ejpam-5255	191	17	=	=	PUNCT
ejpam-5255	191	18	ϱ+((a	ϱ+((a	ADP
ejpam-5255	191	19	∗	∗	PROPN
ejpam-5255	191	20	w	w	NOUN
ejpam-5255	191	21	)	)	PUNCT
ejpam-5255	191	22	∗	∗	NOUN
ejpam-5255	191	23	(	(	PUNCT
ejpam-5255	191	24	a	a	DET
ejpam-5255	191	25	∗	∗	NOUN
ejpam-5255	191	26	(	(	PUNCT
ejpam-5255	191	27	a	a	DET
ejpam-5255	191	28	∗	∗	NOUN
ejpam-5255	191	29	w	w	NOUN
ejpam-5255	191	30	)	)	PUNCT
ejpam-5255	191	31	)	)	PUNCT
ejpam-5255	191	32	)	)	PUNCT
ejpam-5255	191	33	by(2	by(2	NOUN
ejpam-5255	191	34	)	)	PUNCT
ejpam-5255	191	35	≥	≥	NOUN
ejpam-5255	191	36	ϱ+((a	ϱ+((a	ADP
ejpam-5255	191	37	∗	∗	PROPN
ejpam-5255	191	38	w	w	PROPN
ejpam-5255	191	39	)	)	PUNCT
ejpam-5255	191	40	∗	∗	NOUN
ejpam-5255	191	41	(	(	PUNCT
ejpam-5255	191	42	a	a	DET
ejpam-5255	191	43	∗	∗	NOUN
ejpam-5255	191	44	(	(	PUNCT
ejpam-5255	191	45	a	a	DET
ejpam-5255	191	46	∗	∗	NOUN
ejpam-5255	191	47	(	(	PUNCT
ejpam-5255	191	48	a	a	DET
ejpam-5255	191	49	∗	∗	NOUN
ejpam-5255	191	50	w	w	NOUN
ejpam-5255	191	51	)	)	PUNCT
ejpam-5255	191	52	)	)	PUNCT
ejpam-5255	191	53	)	)	PUNCT
ejpam-5255	191	54	)	)	PUNCT
ejpam-5255	192	1	x	x	X
ejpam-5255	192	2	is	be	AUX
ejpam-5255	192	3	positive	positive	ADJ
ejpam-5255	192	4	implicative	implicative	ADJ
ejpam-5255	192	5	≥	≥	NOUN
ejpam-5255	192	6	min	min	PROPN
ejpam-5255	192	7	{	{	PUNCT
ejpam-5255	192	8	ϱ+(((a	ϱ+(((a	PROPN
ejpam-5255	192	9	∗	∗	PROPN
ejpam-5255	192	10	w	w	NOUN
ejpam-5255	192	11	)	)	PUNCT
ejpam-5255	192	12	∗	∗	NOUN
ejpam-5255	192	13	a	a	PRON
ejpam-5255	192	14	)	)	PUNCT
ejpam-5255	192	15	∗	∗	NOUN
ejpam-5255	192	16	g	g	NOUN
ejpam-5255	192	17	)	)	PUNCT
ejpam-5255	192	18	,	,	PUNCT
ejpam-5255	192	19	ϱ+(g	ϱ+(g	PROPN
ejpam-5255	192	20	)	)	PUNCT
ejpam-5255	192	21	}	}	PUNCT
ejpam-5255	192	22	ϱ	ϱ	PROPN
ejpam-5255	192	23	is	be	AUX
ejpam-5255	192	24	a	a	DET
ejpam-5255	192	25	bipolar	bipolar	ADJ
ejpam-5255	192	26	fuzzy	fuzzy	ADJ
ejpam-5255	192	27	commutative	commutative	ADJ
ejpam-5255	192	28	ideal	ideal	NOUN
ejpam-5255	192	29	=	=	SYM
ejpam-5255	192	30	min	min	PROPN
ejpam-5255	192	31	{	{	PUNCT
ejpam-5255	192	32	ϱ+(((a	ϱ+(((a	PROPN
ejpam-5255	192	33	∗	∗	NOUN
ejpam-5255	192	34	w	w	NOUN
ejpam-5255	192	35	)	)	PUNCT
ejpam-5255	192	36	∗	∗	NOUN
ejpam-5255	192	37	g	g	NOUN
ejpam-5255	192	38	)	)	PUNCT
ejpam-5255	192	39	∗	∗	NOUN
ejpam-5255	192	40	a	a	NOUN
ejpam-5255	192	41	)	)	PUNCT
ejpam-5255	192	42	,	,	PUNCT
ejpam-5255	192	43	ϱ+(g	ϱ+(g	NOUN
ejpam-5255	192	44	)	)	PUNCT
ejpam-5255	192	45	}	}	PUNCT
ejpam-5255	192	46	by(1	by(1	ADJ
ejpam-5255	192	47	)	)	PUNCT
ejpam-5255	192	48	=	=	SYM
ejpam-5255	192	49	min	min	NOUN
ejpam-5255	192	50	{	{	PUNCT
ejpam-5255	192	51	ϱ+((a	ϱ+((a	ADP
ejpam-5255	192	52	∗	∗	NOUN
ejpam-5255	192	53	g	g	NOUN
ejpam-5255	192	54	)	)	PUNCT
ejpam-5255	192	55	∗	∗	PROPN
ejpam-5255	192	56	w	w	PROPN
ejpam-5255	192	57	)	)	PUNCT
ejpam-5255	192	58	∗	∗	NOUN
ejpam-5255	192	59	a	a	NOUN
ejpam-5255	192	60	)	)	PUNCT
ejpam-5255	192	61	,	,	PUNCT
ejpam-5255	192	62	ϱ+(g	ϱ+(g	NOUN
ejpam-5255	192	63	)	)	PUNCT
ejpam-5255	192	64	}	}	PUNCT
ejpam-5255	192	65	by(1	by(1	NOUN
ejpam-5255	192	66	)	)	PUNCT
ejpam-5255	192	67	≥	≥	NOUN
ejpam-5255	192	68	min	min	NOUN
ejpam-5255	192	69	{	{	PUNCT
ejpam-5255	192	70	ϱ+((a	ϱ+((a	ADP
ejpam-5255	192	71	∗	∗	NOUN
ejpam-5255	192	72	g	g	NOUN
ejpam-5255	192	73	)	)	PUNCT
ejpam-5255	192	74	∗	∗	PROPN
ejpam-5255	192	75	w	w	PROPN
ejpam-5255	192	76	)	)	PUNCT
ejpam-5255	192	77	,	,	PUNCT
ejpam-5255	192	78	ϱ+(g	ϱ+(g	PROPN
ejpam-5255	192	79	)	)	PUNCT
ejpam-5255	192	80	}	}	PUNCT
ejpam-5255	192	81	by	by	ADP
ejpam-5255	192	82	proposition	proposition	NOUN
ejpam-5255	192	83	1	1	NUM
ejpam-5255	192	84	=	=	SYM
ejpam-5255	192	85	min	min	NOUN
ejpam-5255	192	86	{	{	PUNCT
ejpam-5255	192	87	ϱ+((a	ϱ+((a	ADP
ejpam-5255	192	88	∗	∗	NOUN
ejpam-5255	192	89	(	(	PUNCT
ejpam-5255	192	90	g	g	PROPN
ejpam-5255	192	91	∗	∗	PROPN
ejpam-5255	192	92	w	w	NOUN
ejpam-5255	192	93	)	)	PUNCT
ejpam-5255	192	94	)	)	PUNCT
ejpam-5255	192	95	∗	∗	PROPN
ejpam-5255	192	96	w	w	PROPN
ejpam-5255	192	97	)	)	PUNCT
ejpam-5255	192	98	,	,	PUNCT
ejpam-5255	192	99	ϱ+(g	ϱ+(g	ADP
ejpam-5255	192	100	∗	∗	PROPN
ejpam-5255	192	101	w	w	NOUN
ejpam-5255	192	102	)	)	PUNCT
ejpam-5255	192	103	}	}	PUNCT
ejpam-5255	192	104	(	(	PUNCT
ejpam-5255	192	105	set	set	VERB
ejpam-5255	192	106	g	g	NOUN
ejpam-5255	192	107	=	=	SYM
ejpam-5255	192	108	g	g	PROPN
ejpam-5255	192	109	∗	∗	NOUN
ejpam-5255	192	110	w	w	NOUN
ejpam-5255	192	111	)	)	PUNCT
ejpam-5255	192	112	=	=	SYM
ejpam-5255	192	113	min	min	NOUN
ejpam-5255	192	114	{	{	PUNCT
ejpam-5255	192	115	ϱ+((a	ϱ+((a	ADP
ejpam-5255	192	116	∗	∗	NOUN
ejpam-5255	192	117	w	w	NOUN
ejpam-5255	192	118	)	)	PUNCT
ejpam-5255	192	119	∗	∗	NOUN
ejpam-5255	192	120	(	(	PUNCT
ejpam-5255	192	121	g	g	PROPN
ejpam-5255	192	122	∗	∗	PROPN
ejpam-5255	192	123	w	w	NOUN
ejpam-5255	192	124	)	)	PUNCT
ejpam-5255	192	125	)	)	PUNCT
ejpam-5255	192	126	,	,	PUNCT
ejpam-5255	192	127	ϱ+(g	ϱ+(g	ADP
ejpam-5255	192	128	∗	∗	PROPN
ejpam-5255	192	129	w	w	NOUN
ejpam-5255	192	130	)	)	PUNCT
ejpam-5255	192	131	}	}	PUNCT
ejpam-5255	192	132	by(1	by(1	ADJ
ejpam-5255	192	133	)	)	PUNCT
ejpam-5255	192	134	=	=	SYM
ejpam-5255	192	135	min	min	NOUN
ejpam-5255	192	136	{	{	PUNCT
ejpam-5255	192	137	ϱ+((a	ϱ+((a	ADP
ejpam-5255	192	138	∗	∗	NOUN
ejpam-5255	192	139	g	g	NOUN
ejpam-5255	192	140	)	)	PUNCT
ejpam-5255	192	141	∗	∗	PROPN
ejpam-5255	192	142	w	w	PROPN
ejpam-5255	192	143	)	)	PUNCT
ejpam-5255	192	144	,	,	PUNCT
ejpam-5255	192	145	ϱ+(g	ϱ+(g	ADP
ejpam-5255	192	146	∗	∗	PROPN
ejpam-5255	192	147	w	w	NOUN
ejpam-5255	192	148	)	)	PUNCT
ejpam-5255	192	149	}	}	PUNCT
ejpam-5255	192	150	by(4	by(4	NOUN
ejpam-5255	192	151	)	)	PUNCT
ejpam-5255	192	152	on	on	ADP
ejpam-5255	192	153	the	the	DET
ejpam-5255	192	154	other	other	ADJ
ejpam-5255	192	155	hand	hand	NOUN
ejpam-5255	192	156	,	,	PUNCT
ejpam-5255	192	157	ϱ−(a	ϱ−(a	NOUN
ejpam-5255	192	158	∗	∗	NOUN
ejpam-5255	192	159	w	w	NOUN
ejpam-5255	192	160	)	)	PUNCT
ejpam-5255	192	161	≤	≤	NOUN
ejpam-5255	192	162	ϱ−((a	ϱ−((a	ADP
ejpam-5255	192	163	∗	∗	NOUN
ejpam-5255	192	164	w	w	NOUN
ejpam-5255	192	165	)	)	PUNCT
ejpam-5255	192	166	∗	∗	PROPN
ejpam-5255	192	167	w	w	PROPN
ejpam-5255	192	168	)	)	PUNCT
ejpam-5255	192	169	by(5	by(5	NOUN
ejpam-5255	192	170	)	)	PUNCT
ejpam-5255	192	171	=	=	SYM
ejpam-5255	192	172	ϱ−((a	ϱ−((a	ADP
ejpam-5255	192	173	∗	∗	NOUN
ejpam-5255	192	174	w	w	NOUN
ejpam-5255	192	175	)	)	PUNCT
ejpam-5255	192	176	∗	∗	NOUN
ejpam-5255	192	177	(	(	PUNCT
ejpam-5255	192	178	a	a	DET
ejpam-5255	192	179	∗	∗	NOUN
ejpam-5255	192	180	(	(	PUNCT
ejpam-5255	192	181	a	a	DET
ejpam-5255	192	182	∗	∗	NOUN
ejpam-5255	192	183	w	w	NOUN
ejpam-5255	192	184	)	)	PUNCT
ejpam-5255	192	185	)	)	PUNCT
ejpam-5255	192	186	)	)	PUNCT
ejpam-5255	192	187	by(2	by(2	NOUN
ejpam-5255	192	188	)	)	PUNCT
ejpam-5255	192	189	≤	≤	NOUN
ejpam-5255	193	1	ϱ−((a	ϱ−((a	ADP
ejpam-5255	193	2	∗	∗	NOUN
ejpam-5255	193	3	w	w	NOUN
ejpam-5255	193	4	)	)	PUNCT
ejpam-5255	193	5	∗	∗	NOUN
ejpam-5255	193	6	(	(	PUNCT
ejpam-5255	193	7	a	a	DET
ejpam-5255	193	8	∗	∗	NOUN
ejpam-5255	193	9	(	(	PUNCT
ejpam-5255	193	10	a	a	DET
ejpam-5255	193	11	∗	∗	NOUN
ejpam-5255	193	12	(	(	PUNCT
ejpam-5255	193	13	a	a	DET
ejpam-5255	193	14	∗	∗	NOUN
ejpam-5255	193	15	w	w	NOUN
ejpam-5255	193	16	)	)	PUNCT
ejpam-5255	193	17	)	)	PUNCT
ejpam-5255	193	18	)	)	PUNCT
ejpam-5255	193	19	)	)	PUNCT
ejpam-5255	194	1	x	x	X
ejpam-5255	194	2	is	be	AUX
ejpam-5255	194	3	positive	positive	ADJ
ejpam-5255	194	4	implicative	implicative	ADJ
ejpam-5255	194	5	≤	≤	NUM
ejpam-5255	194	6	max	max	PROPN
ejpam-5255	194	7	{	{	PUNCT
ejpam-5255	194	8	ϱ−((a	ϱ−((a	PROPN
ejpam-5255	194	9	∗	∗	PROPN
ejpam-5255	194	10	w	w	NOUN
ejpam-5255	194	11	)	)	PUNCT
ejpam-5255	194	12	∗	∗	NOUN
ejpam-5255	194	13	a	a	PRON
ejpam-5255	194	14	)	)	PUNCT
ejpam-5255	194	15	∗	∗	NOUN
ejpam-5255	194	16	g	g	NOUN
ejpam-5255	194	17	)	)	PUNCT
ejpam-5255	194	18	,	,	PUNCT
ejpam-5255	194	19	ϱ−(g	ϱ−(g	PROPN
ejpam-5255	194	20	)	)	PUNCT
ejpam-5255	194	21	}	}	PUNCT
ejpam-5255	194	22	ϱ	ϱ	PROPN
ejpam-5255	194	23	is	be	AUX
ejpam-5255	194	24	a	a	DET
ejpam-5255	194	25	bipolar	bipolar	ADJ
ejpam-5255	194	26	fuzzy	fuzzy	ADJ
ejpam-5255	194	27	commutative	commutative	ADJ
ejpam-5255	194	28	ideal	ideal	ADJ
ejpam-5255	194	29	references	reference	NOUN
ejpam-5255	194	30	1840	1840	NUM
ejpam-5255	194	31	=	=	SYM
ejpam-5255	194	32	max	max	PROPN
ejpam-5255	194	33	{	{	PUNCT
ejpam-5255	194	34	ϱ−(((a	ϱ−(((a	PROPN
ejpam-5255	194	35	∗	∗	X
ejpam-5255	194	36	w	w	NOUN
ejpam-5255	194	37	)	)	PUNCT
ejpam-5255	194	38	∗	∗	NOUN
ejpam-5255	194	39	g	g	NOUN
ejpam-5255	194	40	)	)	PUNCT
ejpam-5255	194	41	∗	∗	NOUN
ejpam-5255	194	42	a	a	PRON
ejpam-5255	194	43	)	)	PUNCT
ejpam-5255	194	44	,	,	PUNCT
ejpam-5255	194	45	ϱ−(g	ϱ−(g	PROPN
ejpam-5255	194	46	)	)	PUNCT
ejpam-5255	194	47	}	}	PUNCT
ejpam-5255	194	48	by(1	by(1	VERB
ejpam-5255	194	49	)	)	PUNCT
ejpam-5255	194	50	=	=	SYM
ejpam-5255	194	51	max	max	PROPN
ejpam-5255	194	52	{	{	PUNCT
ejpam-5255	194	53	ϱ−(((a	ϱ−(((a	PROPN
ejpam-5255	194	54	∗	∗	VERB
ejpam-5255	194	55	g	g	NOUN
ejpam-5255	194	56	)	)	PUNCT
ejpam-5255	194	57	∗	∗	PROPN
ejpam-5255	194	58	w	w	PROPN
ejpam-5255	194	59	)	)	PUNCT
ejpam-5255	194	60	∗	∗	NOUN
ejpam-5255	194	61	a	a	NOUN
ejpam-5255	194	62	)	)	PUNCT
ejpam-5255	194	63	,	,	PUNCT
ejpam-5255	194	64	ϱ−(g	ϱ−(g	PROPN
ejpam-5255	194	65	)	)	PUNCT
ejpam-5255	194	66	}	}	PUNCT
ejpam-5255	194	67	by(1	by(1	NOUN
ejpam-5255	194	68	)	)	PUNCT
ejpam-5255	194	69	≤	≤	NUM
ejpam-5255	194	70	max	max	PROPN
ejpam-5255	194	71	{	{	PUNCT
ejpam-5255	194	72	ϱ−((a	ϱ−((a	PROPN
ejpam-5255	194	73	∗	∗	NOUN
ejpam-5255	194	74	g	g	NOUN
ejpam-5255	194	75	)	)	PUNCT
ejpam-5255	194	76	∗	∗	PROPN
ejpam-5255	194	77	w	w	PROPN
ejpam-5255	194	78	)	)	PUNCT
ejpam-5255	194	79	,	,	PUNCT
ejpam-5255	194	80	ϱ−(g	ϱ−(g	PROPN
ejpam-5255	194	81	)	)	PUNCT
ejpam-5255	194	82	}	}	PUNCT
ejpam-5255	194	83	by	by	ADP
ejpam-5255	194	84	proposition	proposition	NOUN
ejpam-5255	194	85	1	1	NUM
ejpam-5255	194	86	=	=	SYM
ejpam-5255	194	87	max	max	X
ejpam-5255	194	88	{	{	PUNCT
ejpam-5255	194	89	ϱ−(((a	ϱ−(((a	PROPN
ejpam-5255	194	90	∗	∗	NOUN
ejpam-5255	194	91	(	(	PUNCT
ejpam-5255	194	92	g	g	PROPN
ejpam-5255	194	93	∗	∗	PROPN
ejpam-5255	194	94	w	w	NOUN
ejpam-5255	194	95	)	)	PUNCT
ejpam-5255	194	96	)	)	PUNCT
ejpam-5255	194	97	∗	∗	PROPN
ejpam-5255	194	98	w	w	PROPN
ejpam-5255	194	99	)	)	PUNCT
ejpam-5255	194	100	,	,	PUNCT
ejpam-5255	194	101	ϱ−(g	ϱ−(g	VERB
ejpam-5255	194	102	∗	∗	NOUN
ejpam-5255	194	103	w	w	NOUN
ejpam-5255	194	104	)	)	PUNCT
ejpam-5255	194	105	}	}	PUNCT
ejpam-5255	194	106	(	(	PUNCT
ejpam-5255	194	107	set	set	VERB
ejpam-5255	194	108	g	g	NOUN
ejpam-5255	194	109	=	=	SYM
ejpam-5255	194	110	g	g	PROPN
ejpam-5255	194	111	∗	∗	NOUN
ejpam-5255	194	112	w	w	NOUN
ejpam-5255	194	113	)	)	PUNCT
ejpam-5255	194	114	=	=	SYM
ejpam-5255	194	115	max	max	X
ejpam-5255	194	116	{	{	PUNCT
ejpam-5255	194	117	ϱ−((a	ϱ−((a	PROPN
ejpam-5255	194	118	∗	∗	PROPN
ejpam-5255	194	119	w	w	NOUN
ejpam-5255	194	120	)	)	PUNCT
ejpam-5255	194	121	∗	∗	NOUN
ejpam-5255	194	122	(	(	PUNCT
ejpam-5255	194	123	g	g	PROPN
ejpam-5255	194	124	∗	∗	PROPN
ejpam-5255	194	125	w	w	NOUN
ejpam-5255	194	126	)	)	PUNCT
ejpam-5255	194	127	)	)	PUNCT
ejpam-5255	194	128	,	,	PUNCT
ejpam-5255	194	129	ϱ−(g	ϱ−(g	VERB
ejpam-5255	194	130	∗	∗	NOUN
ejpam-5255	194	131	w	w	NOUN
ejpam-5255	194	132	)	)	PUNCT
ejpam-5255	194	133	}	}	PUNCT
ejpam-5255	194	134	by(1	by(1	ADJ
ejpam-5255	194	135	)	)	PUNCT
ejpam-5255	194	136	=	=	SYM
ejpam-5255	194	137	max	max	X
ejpam-5255	194	138	{	{	PUNCT
ejpam-5255	194	139	ϱ−((a	ϱ−((a	PROPN
ejpam-5255	194	140	∗	∗	NOUN
ejpam-5255	194	141	g	g	NOUN
ejpam-5255	194	142	)	)	PUNCT
ejpam-5255	194	143	∗	∗	PROPN
ejpam-5255	194	144	w	w	PROPN
ejpam-5255	194	145	)	)	PUNCT
ejpam-5255	194	146	,	,	PUNCT
ejpam-5255	194	147	ϱ−(g	ϱ−(g	VERB
ejpam-5255	194	148	∗	∗	NOUN
ejpam-5255	194	149	w	w	NOUN
ejpam-5255	194	150	)	)	PUNCT
ejpam-5255	194	151	}	}	PUNCT
ejpam-5255	194	152	by(4	by(4	NOUN
ejpam-5255	194	153	)	)	PUNCT
ejpam-5255	194	154	this	this	PRON
ejpam-5255	194	155	completes	complete	VERB
ejpam-5255	194	156	the	the	DET
ejpam-5255	194	157	proof	proof	NOUN
ejpam-5255	194	158	.	.	PUNCT
ejpam-5255	195	1	references	reference	NOUN
ejpam-5255	195	2	[	[	X
ejpam-5255	195	3	1	1	NUM
ejpam-5255	195	4	]	]	X
ejpam-5255	195	5	d	d	PROPN
ejpam-5255	195	6	al	al	PROPN
ejpam-5255	195	7	-	-	PUNCT
ejpam-5255	195	8	kadi	kadi	PROPN
ejpam-5255	195	9	and	and	CCONJ
ejpam-5255	195	10	g	g	NOUN
ejpam-5255	195	11	muhiuddin	muhiuddin	NOUN
ejpam-5255	195	12	.	.	PUNCT
ejpam-5255	196	1	bipolar	bipolar	ADJ
ejpam-5255	196	2	fuzzy	fuzzy	ADJ
ejpam-5255	196	3	bci	bci	ADJ
ejpam-5255	196	4	-	-	ADJ
ejpam-5255	196	5	implicative	implicative	ADJ
ejpam-5255	196	6	ideals	ideal	NOUN
ejpam-5255	196	7	of	of	ADP
ejpam-5255	196	8	bci	bci	NOUN
ejpam-5255	196	9	-	-	PUNCT
ejpam-5255	196	10	algebras	algebras	PROPN
ejpam-5255	196	11	.	.	PUNCT
ejpam-5255	197	1	ann	ann	AUX
ejpam-5255	197	2	.	.	PUNCT
ejpam-5255	197	3	commun	commun	PROPN
ejpam-5255	197	4	.	.	PUNCT
ejpam-5255	198	1	math	math	PROPN
ejpam-5255	198	2	,	,	PUNCT
ejpam-5255	198	3	3(1):88–96	3(1):88–96	NUM
ejpam-5255	198	4	,	,	PUNCT
ejpam-5255	198	5	2020	2020	NUM
ejpam-5255	198	6	.	.	PUNCT
ejpam-5255	199	1	[	[	X
ejpam-5255	199	2	2	2	X
ejpam-5255	199	3	]	]	X
ejpam-5255	199	4	anas	anas	PROPN
ejpam-5255	199	5	al	al	PROPN
ejpam-5255	199	6	-	-	PROPN
ejpam-5255	199	7	masarwah	masarwah	PROPN
ejpam-5255	199	8	and	and	CCONJ
ejpam-5255	199	9	abd	abd	PROPN
ejpam-5255	199	10	ghafur	ghafur	NOUN
ejpam-5255	199	11	ahmad	ahmad	PROPN
ejpam-5255	199	12	.	.	PUNCT
ejpam-5255	200	1	doubt	doubt	VERB
ejpam-5255	200	2	bipolar	bipolar	ADJ
ejpam-5255	200	3	fuzzy	fuzzy	ADJ
ejpam-5255	200	4	subalgebras	subalgebra	NOUN
ejpam-5255	200	5	and	and	CCONJ
ejpam-5255	200	6	ideals	ideal	NOUN
ejpam-5255	200	7	in	in	ADP
ejpam-5255	200	8	bck	bck	PROPN
ejpam-5255	200	9	/	/	SYM
ejpam-5255	200	10	bci	bci	NOUN
ejpam-5255	200	11	-	-	PUNCT
ejpam-5255	200	12	algebras	algebras	X
ejpam-5255	200	13	.	.	PUNCT
ejpam-5255	201	1	j.	j.	PROPN
ejpam-5255	201	2	math	math	PROPN
ejpam-5255	201	3	.	.	PUNCT
ejpam-5255	202	1	anal	anal	PROPN
ejpam-5255	202	2	,	,	PUNCT
ejpam-5255	202	3	9(3):9–27	9(3):9–27	PROPN
ejpam-5255	202	4	,	,	PUNCT
ejpam-5255	202	5	2018	2018	NUM
ejpam-5255	202	6	.	.	PUNCT
ejpam-5255	203	1	[	[	X
ejpam-5255	203	2	3	3	X
ejpam-5255	203	3	]	]	X
ejpam-5255	203	4	ghous	ghous	ADJ
ejpam-5255	203	5	ali	ali	PROPN
ejpam-5255	203	6	,	,	PUNCT
ejpam-5255	203	7	g	g	PROPN
ejpam-5255	203	8	muhiuddin	muhiuddin	PROPN
ejpam-5255	203	9	,	,	PUNCT
ejpam-5255	203	10	arooj	arooj	PROPN
ejpam-5255	203	11	adeel	adeel	PROPN
ejpam-5255	203	12	,	,	PUNCT
ejpam-5255	203	13	and	and	CCONJ
ejpam-5255	203	14	muhammad	muhammad	PROPN
ejpam-5255	203	15	zain	zain	PROPN
ejpam-5255	203	16	ul	ul	PROPN
ejpam-5255	203	17	abidin	abidin	PROPN
ejpam-5255	203	18	.	.	PUNCT
ejpam-5255	204	1	ranking	rank	VERB
ejpam-5255	204	2	effectiveness	effectiveness	NOUN
ejpam-5255	204	3	of	of	ADP
ejpam-5255	204	4	covid-19	covid-19	PROPN
ejpam-5255	204	5	tests	test	NOUN
ejpam-5255	204	6	using	use	VERB
ejpam-5255	204	7	fuzzy	fuzzy	ADJ
ejpam-5255	204	8	bipolar	bipolar	ADJ
ejpam-5255	204	9	soft	soft	ADJ
ejpam-5255	204	10	expert	expert	NOUN
ejpam-5255	204	11	sets	set	NOUN
ejpam-5255	204	12	.	.	PUNCT
ejpam-5255	205	1	mathematical	mathematical	ADJ
ejpam-5255	205	2	problems	problem	NOUN
ejpam-5255	205	3	in	in	ADP
ejpam-5255	205	4	engineering	engineering	NOUN
ejpam-5255	205	5	,	,	PUNCT
ejpam-5255	205	6	2021	2021	NUM
ejpam-5255	205	7	,	,	PUNCT
ejpam-5255	205	8	2021	2021	NUM
ejpam-5255	205	9	.	.	PUNCT
ejpam-5255	206	1	[	[	X
ejpam-5255	206	2	4	4	NUM
ejpam-5255	206	3	]	]	X
ejpam-5255	206	4	h	h	NOUN
ejpam-5255	206	5	alshehri	alshehri	ADJ
ejpam-5255	206	6	and	and	CCONJ
ejpam-5255	206	7	a	a	DET
ejpam-5255	206	8	almuhaimeed	almuhaimeed	NOUN
ejpam-5255	206	9	.	.	PUNCT
ejpam-5255	207	1	on	on	ADP
ejpam-5255	207	2	bipolar	bipolar	ADJ
ejpam-5255	207	3	fuzzy	fuzzy	ADJ
ejpam-5255	207	4	d	d	NOUN
ejpam-5255	207	5	-	-	PUNCT
ejpam-5255	207	6	subalgebras	subalgebras	PROPN
ejpam-5255	207	7	and	and	CCONJ
ejpam-5255	207	8	d	d	NOUN
ejpam-5255	207	9	-	-	NOUN
ejpam-5255	207	10	ideals	ideal	NOUN
ejpam-5255	207	11	.	.	PUNCT
ejpam-5255	208	1	jp	jp	PROPN
ejpam-5255	208	2	journal	journal	PROPN
ejpam-5255	208	3	of	of	ADP
ejpam-5255	208	4	algebra	algebra	PROPN
ejpam-5255	208	5	,	,	PUNCT
ejpam-5255	208	6	number	number	NOUN
ejpam-5255	208	7	theory	theory	NOUN
ejpam-5255	208	8	and	and	CCONJ
ejpam-5255	208	9	applications	application	NOUN
ejpam-5255	208	10	,	,	PUNCT
ejpam-5255	208	11	45(1):73–83	45(1):73–83	NUM
ejpam-5255	208	12	,	,	PUNCT
ejpam-5255	208	13	2020	2020	NUM
ejpam-5255	208	14	.	.	PUNCT
ejpam-5255	209	1	[	[	X
ejpam-5255	209	2	5	5	X
ejpam-5255	209	3	]	]	PUNCT
ejpam-5255	209	4	isabelle	isabelle	PROPN
ejpam-5255	209	5	bloch	bloch	PROPN
ejpam-5255	209	6	.	.	PUNCT
ejpam-5255	210	1	lattices	lattice	NOUN
ejpam-5255	210	2	of	of	ADP
ejpam-5255	210	3	fuzzy	fuzzy	ADJ
ejpam-5255	210	4	sets	set	NOUN
ejpam-5255	210	5	and	and	CCONJ
ejpam-5255	210	6	bipolar	bipolar	ADJ
ejpam-5255	210	7	fuzzy	fuzzy	ADJ
ejpam-5255	210	8	sets	set	NOUN
ejpam-5255	210	9	,	,	PUNCT
ejpam-5255	210	10	and	and	CCONJ
ejpam-5255	210	11	mathematical	mathematical	ADJ
ejpam-5255	210	12	morphology	morphology	NOUN
ejpam-5255	210	13	.	.	PUNCT
ejpam-5255	211	1	information	information	NOUN
ejpam-5255	211	2	sciences	sciences	PROPN
ejpam-5255	211	3	,	,	PUNCT
ejpam-5255	211	4	181(10):2002–2015	181(10):2002–2015	NUM
ejpam-5255	211	5	,	,	PUNCT
ejpam-5255	211	6	2011	2011	NUM
ejpam-5255	211	7	.	.	PUNCT
ejpam-5255	212	1	[	[	X
ejpam-5255	212	2	6	6	NUM
ejpam-5255	212	3	]	]	X
ejpam-5255	212	4	y	y	PROPN
ejpam-5255	212	5	imai	imai	PROPN
ejpam-5255	212	6	and	and	CCONJ
ejpam-5255	212	7	k	k	PROPN
ejpam-5255	212	8	iseki	iseki	PROPN
ejpam-5255	212	9	.	.	PUNCT
ejpam-5255	213	1	on	on	ADP
ejpam-5255	213	2	axioms	axiom	NOUN
ejpam-5255	213	3	of	of	ADP
ejpam-5255	213	4	proportional	proportional	ADJ
ejpam-5255	213	5	calculi	calculi	PROPN
ejpam-5255	213	6	xiv	xiv	PROPN
ejpam-5255	213	7	proc	proc	PROPN
ejpam-5255	213	8	.	.	PUNCT
ejpam-5255	214	1	japan	japan	PROPN
ejpam-5255	214	2	acad	acad	PROPN
ejpam-5255	214	3	,	,	PUNCT
ejpam-5255	214	4	42:19–22	42:19–22	NUM
ejpam-5255	214	5	,	,	PUNCT
ejpam-5255	214	6	1966	1966	NUM
ejpam-5255	214	7	.	.	PUNCT
ejpam-5255	215	1	[	[	X
ejpam-5255	215	2	7	7	X
ejpam-5255	215	3	]	]	PUNCT
ejpam-5255	215	4	k.	k.	PROPN
ejpam-5255	215	5	is6ki	is6ki	PROPN
ejpam-5255	215	6	and	and	CCONJ
ejpam-5255	215	7	s.	s.	PROPN
ejpam-5255	215	8	tanaka	tanaka	PROPN
ejpam-5255	215	9	.	.	PUNCT
ejpam-5255	216	1	an	an	DET
ejpam-5255	216	2	introduction	introduction	NOUN
ejpam-5255	216	3	to	to	ADP
ejpam-5255	216	4	the	the	DET
ejpam-5255	216	5	theory	theory	NOUN
ejpam-5255	216	6	of	of	ADP
ejpam-5255	216	7	bck	bck	NOUN
ejpam-5255	216	8	-	-	PUNCT
ejpam-5255	216	9	algebra	algebra	NOUN
ejpam-5255	216	10	.	.	PUNCT
ejpam-5255	217	1	math	math	NOUN
ejpam-5255	217	2	,	,	PUNCT
ejpam-5255	217	3	23:1–26	23:1–26	NUM
ejpam-5255	217	4	,	,	PUNCT
ejpam-5255	217	5	1978	1978	NUM
ejpam-5255	217	6	.	.	PUNCT
ejpam-5255	218	1	[	[	X
ejpam-5255	218	2	8	8	NUM
ejpam-5255	218	3	]	]	X
ejpam-5255	218	4	yb	yb	PROPN
ejpam-5255	218	5	jun	jun	PROPN
ejpam-5255	218	6	,	,	PUNCT
ejpam-5255	218	7	hs	hs	PROPN
ejpam-5255	218	8	kim	kim	PROPN
ejpam-5255	218	9	,	,	PUNCT
ejpam-5255	218	10	and	and	CCONJ
ejpam-5255	218	11	ds	ds	PROPN
ejpam-5255	218	12	yoo	yoo	PROPN
ejpam-5255	218	13	.	.	PUNCT
ejpam-5255	219	1	fuzzy	fuzzy	ADJ
ejpam-5255	219	2	bck	bck	PROPN
ejpam-5255	219	3	-	-	PUNCT
ejpam-5255	219	4	algebras	algebras	PROPN
ejpam-5255	219	5	.	.	PUNCT
ejpam-5255	220	1	scientiae	scientiae	PROPN
ejpam-5255	220	2	mathematicae	mathematicae	PROPN
ejpam-5255	220	3	japonicae	japonicae	PROPN
ejpam-5255	220	4	,	,	PUNCT
ejpam-5255	220	5	36:935–942	36:935–942	NUM
ejpam-5255	220	6	,	,	PUNCT
ejpam-5255	220	7	1991	1991	NUM
ejpam-5255	220	8	.	.	PUNCT
ejpam-5255	221	1	[	[	X
ejpam-5255	221	2	9	9	NUM
ejpam-5255	221	3	]	]	X
ejpam-5255	221	4	young	young	ADJ
ejpam-5255	221	5	bae	bae	PROPN
ejpam-5255	221	6	jun	jun	PROPN
ejpam-5255	221	7	,	,	PUNCT
ejpam-5255	221	8	tahsin	tahsin	PROPN
ejpam-5255	221	9	oner	oner	NOUN
ejpam-5255	221	10	,	,	PUNCT
ejpam-5255	221	11	duygu	duygu	PROPN
ejpam-5255	221	12	selin	selin	PROPN
ejpam-5255	221	13	turan	turan	PROPN
ejpam-5255	221	14	,	,	PUNCT
ejpam-5255	221	15	and	and	CCONJ
ejpam-5255	221	16	burak	burak	PROPN
ejpam-5255	221	17	ordin	ordin	PROPN
ejpam-5255	221	18	.	.	PUNCT
ejpam-5255	221	19	bipolar	bipolar	ADJ
ejpam-5255	221	20	-	-	PUNCT
ejpam-5255	221	21	valued	value	VERB
ejpam-5255	221	22	fuzzy	fuzzy	ADJ
ejpam-5255	221	23	filters	filter	NOUN
ejpam-5255	221	24	of	of	ADP
ejpam-5255	221	25	sheffer	sheffer	PROPN
ejpam-5255	221	26	stroke	stroke	PROPN
ejpam-5255	221	27	bl	bl	PROPN
ejpam-5255	221	28	-	-	PUNCT
ejpam-5255	221	29	algebras	algebras	PROPN
ejpam-5255	221	30	.	.	PUNCT
ejpam-5255	222	1	new	new	ADJ
ejpam-5255	222	2	mathematics	mathematic	NOUN
ejpam-5255	222	3	and	and	CCONJ
ejpam-5255	222	4	natural	natural	ADJ
ejpam-5255	222	5	computation	computation	NOUN
ejpam-5255	222	6	,	,	PUNCT
ejpam-5255	222	7	pages	page	NOUN
ejpam-5255	222	8	1–17	1–17	PROPN
ejpam-5255	222	9	,	,	PUNCT
ejpam-5255	222	10	2023	2023	NUM
ejpam-5255	222	11	.	.	PUNCT
ejpam-5255	223	1	[	[	X
ejpam-5255	223	2	10	10	NUM
ejpam-5255	223	3	]	]	X
ejpam-5255	223	4	kyoung	kyoung	PROPN
ejpam-5255	223	5	ja	ja	PROPN
ejpam-5255	223	6	lee	lee	PROPN
ejpam-5255	223	7	.	.	PUNCT
ejpam-5255	224	1	bipolar	bipolar	ADJ
ejpam-5255	224	2	fuzzy	fuzzy	ADJ
ejpam-5255	224	3	subalgebras	subalgebra	NOUN
ejpam-5255	224	4	and	and	CCONJ
ejpam-5255	224	5	bipolar	bipolar	ADJ
ejpam-5255	224	6	fuzzy	fuzzy	ADJ
ejpam-5255	224	7	ideals	ideal	NOUN
ejpam-5255	224	8	of	of	ADP
ejpam-5255	224	9	bck	bck	PROPN
ejpam-5255	224	10	/	/	SYM
ejpam-5255	224	11	bcialgebras	bcialgebra	NOUN
ejpam-5255	224	12	.	.	PUNCT
ejpam-5255	225	1	bull	bull	NOUN
ejpam-5255	225	2	.	.	PUNCT
ejpam-5255	226	1	malays	malays	PROPN
ejpam-5255	226	2	.	.	PUNCT
ejpam-5255	227	1	math	math	NOUN
ejpam-5255	227	2	.	.	PUNCT
ejpam-5255	228	1	sci	sci	PROPN
ejpam-5255	228	2	.	.	PROPN
ejpam-5255	228	3	soc	soc	PROPN
ejpam-5255	228	4	,	,	PUNCT
ejpam-5255	228	5	32(3):361–373	32(3):361–373	PROPN
ejpam-5255	228	6	,	,	PUNCT
ejpam-5255	228	7	2009	2009	NUM
ejpam-5255	228	8	.	.	PUNCT
ejpam-5255	229	1	references	reference	NOUN
ejpam-5255	229	2	1841	1841	NUM
ejpam-5255	230	1	[	[	X
ejpam-5255	230	2	11	11	NUM
ejpam-5255	230	3	]	]	X
ejpam-5255	230	4	kyoung	kyoung	PROPN
ejpam-5255	230	5	-	-	PUNCT
ejpam-5255	230	6	ja	ja	PROPN
ejpam-5255	230	7	lee	lee	PROPN
ejpam-5255	230	8	and	and	CCONJ
ejpam-5255	230	9	young	young	ADJ
ejpam-5255	230	10	-	-	PUNCT
ejpam-5255	230	11	bae	bae	PROPN
ejpam-5255	230	12	jun	jun	PROPN
ejpam-5255	230	13	.	.	PUNCT
ejpam-5255	230	14	bipolar	bipolar	ADJ
ejpam-5255	230	15	fuzzy	fuzzy	ADJ
ejpam-5255	230	16	a	a	NOUN
ejpam-5255	230	17	-	-	PUNCT
ejpam-5255	230	18	ideals	ideal	NOUN
ejpam-5255	230	19	of	of	ADP
ejpam-5255	230	20	bci	bci	NOUN
ejpam-5255	230	21	-	-	PUNCT
ejpam-5255	230	22	algebras	algebra	NOUN
ejpam-5255	230	23	.	.	PUNCT
ejpam-5255	231	1	communications	communication	NOUN
ejpam-5255	231	2	of	of	ADP
ejpam-5255	231	3	the	the	DET
ejpam-5255	231	4	korean	korean	ADJ
ejpam-5255	231	5	mathematical	mathematical	ADJ
ejpam-5255	231	6	society	society	NOUN
ejpam-5255	231	7	,	,	PUNCT
ejpam-5255	231	8	26(4):531–542	26(4):531–542	PROPN
ejpam-5255	231	9	,	,	PUNCT
ejpam-5255	231	10	2011	2011	NUM
ejpam-5255	231	11	.	.	PUNCT
ejpam-5255	232	1	[	[	X
ejpam-5255	232	2	12	12	NUM
ejpam-5255	232	3	]	]	X
ejpam-5255	232	4	jie	jie	PROPN
ejpam-5255	232	5	meng	meng	PROPN
ejpam-5255	232	6	and	and	CCONJ
ejpam-5255	232	7	xiu	xiu	PROPN
ejpam-5255	232	8	-	-	PUNCT
ejpam-5255	232	9	e	e	PROPN
ejpam-5255	232	10	guo	guo	PROPN
ejpam-5255	232	11	.	.	PUNCT
ejpam-5255	233	1	on	on	ADP
ejpam-5255	233	2	fuzzy	fuzzy	ADJ
ejpam-5255	233	3	ideals	ideal	NOUN
ejpam-5255	233	4	in	in	ADP
ejpam-5255	233	5	bck	bck	PROPN
ejpam-5255	233	6	/	/	SYM
ejpam-5255	233	7	bci	bci	NOUN
ejpam-5255	233	8	-	-	PUNCT
ejpam-5255	233	9	algebras	algebras	X
ejpam-5255	233	10	.	.	PUNCT
ejpam-5255	234	1	fuzzy	fuzzy	ADJ
ejpam-5255	234	2	sets	set	NOUN
ejpam-5255	234	3	and	and	CCONJ
ejpam-5255	234	4	systems	system	NOUN
ejpam-5255	234	5	,	,	PUNCT
ejpam-5255	234	6	149(3):509–525	149(3):509–525	NUM
ejpam-5255	234	7	,	,	PUNCT
ejpam-5255	234	8	2005	2005	NUM
ejpam-5255	234	9	.	.	PUNCT
ejpam-5255	235	1	[	[	X
ejpam-5255	235	2	13	13	NUM
ejpam-5255	235	3	]	]	X
ejpam-5255	235	4	g	g	PROPN
ejpam-5255	235	5	muhiuddin	muhiuddin	PROPN
ejpam-5255	235	6	,	,	PUNCT
ejpam-5255	235	7	m	m	VERB
ejpam-5255	235	8	mohseni	mohseni	NOUN
ejpam-5255	235	9	takallo	takallo	PROPN
ejpam-5255	235	10	,	,	PUNCT
ejpam-5255	235	11	ra	ra	PROPN
ejpam-5255	235	12	borzooei	borzooei	PROPN
ejpam-5255	235	13	,	,	PUNCT
ejpam-5255	235	14	and	and	CCONJ
ejpam-5255	235	15	yb	yb	PROPN
ejpam-5255	235	16	jun	jun	PROPN
ejpam-5255	235	17	.	.	PROPN
ejpam-5255	236	1	m	m	PROPN
ejpam-5255	236	2	-	-	ADJ
ejpam-5255	236	3	polar	polar	ADJ
ejpam-5255	236	4	fuzzy	fuzzy	ADJ
ejpam-5255	236	5	q	q	NOUN
ejpam-5255	236	6	-	-	PUNCT
ejpam-5255	236	7	ideals	ideal	NOUN
ejpam-5255	236	8	in	in	ADP
ejpam-5255	236	9	bci	bci	NOUN
ejpam-5255	236	10	-	-	PUNCT
ejpam-5255	236	11	algebras	algebras	PROPN
ejpam-5255	236	12	.	.	PUNCT
ejpam-5255	237	1	journal	journal	PROPN
ejpam-5255	237	2	of	of	ADP
ejpam-5255	237	3	king	king	PROPN
ejpam-5255	237	4	saud	saud	PROPN
ejpam-5255	237	5	university	university	PROPN
ejpam-5255	237	6	-	-	PUNCT
ejpam-5255	237	7	science	science	NOUN
ejpam-5255	237	8	,	,	PUNCT
ejpam-5255	237	9	32(6):2803–2809	32(6):2803–2809	PROPN
ejpam-5255	237	10	,	,	PUNCT
ejpam-5255	237	11	2020	2020	NUM
ejpam-5255	237	12	.	.	PUNCT
ejpam-5255	238	1	[	[	X
ejpam-5255	238	2	14	14	NUM
ejpam-5255	238	3	]	]	PUNCT
ejpam-5255	238	4	tahsin	tahsin	PROPN
ejpam-5255	238	5	oner	oner	NOUN
ejpam-5255	238	6	,	,	PUNCT
ejpam-5255	238	7	t	t	PROPN
ejpam-5255	238	8	kalkan	kalkan	PROPN
ejpam-5255	238	9	,	,	PUNCT
ejpam-5255	238	10	and	and	CCONJ
ejpam-5255	238	11	arsham	arsham	PROPN
ejpam-5255	238	12	borumand	borumand	PROPN
ejpam-5255	238	13	saeid	saeid	PROPN
ejpam-5255	238	14	.	.	PUNCT
ejpam-5255	239	1	(	(	PUNCT
ejpam-5255	239	2	anti	anti	ADJ
ejpam-5255	239	3	)	)	PUNCT
ejpam-5255	239	4	fuzzy	fuzzy	ADJ
ejpam-5255	239	5	ideals	ideal	NOUN
ejpam-5255	239	6	of	of	ADP
ejpam-5255	239	7	sheffer	sheffer	PROPN
ejpam-5255	239	8	stroke	stroke	NOUN
ejpam-5255	239	9	bck	bck	PROPN
ejpam-5255	239	10	-	-	PUNCT
ejpam-5255	239	11	algebras	algebras	PROPN
ejpam-5255	239	12	.	.	PUNCT
ejpam-5255	240	1	journal	journal	PROPN
ejpam-5255	240	2	of	of	ADP
ejpam-5255	240	3	algebraic	algebraic	PROPN
ejpam-5255	240	4	systems	system	NOUN
ejpam-5255	240	5	,	,	PUNCT
ejpam-5255	240	6	11(1):105–135	11(1):105–135	PROPN
ejpam-5255	240	7	,	,	PUNCT
ejpam-5255	240	8	2023	2023	NUM
ejpam-5255	240	9	.	.	PUNCT
ejpam-5255	241	1	[	[	X
ejpam-5255	241	2	15	15	NUM
ejpam-5255	241	3	]	]	X
ejpam-5255	241	4	tahsin	tahsin	NOUN
ejpam-5255	241	5	oner	oner	NOUN
ejpam-5255	241	6	,	,	PUNCT
ejpam-5255	241	7	tugce	tugce	NOUN
ejpam-5255	241	8	kalkan	kalkan	PROPN
ejpam-5255	241	9	,	,	PUNCT
ejpam-5255	241	10	and	and	CCONJ
ejpam-5255	241	11	arsham	arsham	PROPN
ejpam-5255	241	12	borumand	borumand	PROPN
ejpam-5255	241	13	saeid	saeid	PROPN
ejpam-5255	241	14	.	.	PUNCT
ejpam-5255	242	1	class	class	NOUN
ejpam-5255	242	2	of	of	ADP
ejpam-5255	242	3	sheffer	sheffer	PROPN
ejpam-5255	242	4	stroke	stroke	PROPN
ejpam-5255	242	5	bckalgebras	bckalgebras	PROPN
ejpam-5255	242	6	.	.	PUNCT
ejpam-5255	243	1	analele	analele	PROPN
ejpam-5255	243	2	ştiinţifice	ştiinţifice	PROPN
ejpam-5255	243	3	ale	ale	NOUN
ejpam-5255	243	4	universităţii	universităţii	PROPN
ejpam-5255	243	5	”	"	PUNCT
ejpam-5255	243	6	ovidius	ovidius	ADJ
ejpam-5255	243	7	”	"	PUNCT
ejpam-5255	243	8	constanţa	constanţa	NOUN
ejpam-5255	243	9	.	.	PUNCT
ejpam-5255	244	1	seria	seria	PROPN
ejpam-5255	244	2	matematică	matematică	PROPN
ejpam-5255	244	3	,	,	PUNCT
ejpam-5255	244	4	30(1):247–269	30(1):247–269	NOUN
ejpam-5255	244	5	,	,	PUNCT
ejpam-5255	244	6	2022	2022	NUM
ejpam-5255	244	7	.	.	PUNCT
ejpam-5255	245	1	[	[	X
ejpam-5255	245	2	16	16	NUM
ejpam-5255	245	3	]	]	PUNCT
ejpam-5255	245	4	lotfi	lotfi	X
ejpam-5255	245	5	a	a	DET
ejpam-5255	245	6	zadeh	zadeh	PROPN
ejpam-5255	245	7	.	.	PUNCT
ejpam-5255	245	8	fuzzy	fuzzy	ADJ
ejpam-5255	245	9	sets	set	NOUN
ejpam-5255	245	10	.	.	PUNCT
ejpam-5255	246	1	information	information	NOUN
ejpam-5255	246	2	and	and	CCONJ
ejpam-5255	246	3	control	control	NOUN
ejpam-5255	246	4	,	,	PUNCT
ejpam-5255	246	5	8(3):338–353	8(3):338–353	NUM
ejpam-5255	246	6	,	,	PUNCT
ejpam-5255	246	7	1965	1965	NUM
ejpam-5255	246	8	.	.	PUNCT