id	sid	tid	token	lemma	pos
cana-3289	1	1	communications	communication	NOUN
cana-3289	1	2	on	on	ADP
cana-3289	1	3	applied	apply	VERB
cana-3289	1	4	nonlinear	nonlinear	ADJ
cana-3289	1	5	analysis	analysis	NOUN
cana-3289	1	6	issn	issn	NOUN
cana-3289	1	7	:	:	PUNCT
cana-3289	1	8	1074	1074	NUM
cana-3289	1	9	-	-	PUNCT
cana-3289	1	10	133x	133x	NUM
cana-3289	1	11	vol	vol	NOUN
cana-3289	1	12	32	32	NUM
cana-3289	1	13	no	no	NOUN
cana-3289	1	14	.	.	PUNCT
cana-3289	2	1	6s	6s	NUM
cana-3289	2	2	(	(	PUNCT
cana-3289	2	3	2025	2025	NUM
cana-3289	2	4	)	)	PUNCT
cana-3289	2	5	241	241	NUM
cana-3289	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3289	2	7	a	a	DET
cana-3289	2	8	study	study	NOUN
cana-3289	2	9	of	of	ADP
cana-3289	2	10	bipolar	bipolar	ADJ
cana-3289	2	11	fuzzy	fuzzy	ADJ
cana-3289	2	12	prime	prime	ADJ
cana-3289	2	13	ideals	ideal	NOUN
cana-3289	2	14	of	of	ADP
cana-3289	2	15	a	a	DET
cana-3289	2	16	lattice	lattice	NOUN
cana-3289	2	17	1venkata	1venkata	NUM
cana-3289	2	18	kalyani	kalyani	PROPN
cana-3289	2	19	u	u	NOUN
cana-3289	2	20	,	,	PUNCT
cana-3289	2	21	2b.v.s.n	2b.v.s.n	PROPN
cana-3289	2	22	.	.	PUNCT
cana-3289	3	1	hari	hari	PROPN
cana-3289	3	2	prasad	prasad	PROPN
cana-3289	3	3	,	,	PUNCT
cana-3289	3	4	3eswarlal	3eswarlal	NUM
cana-3289	3	5	.	.	PUNCT
cana-3289	4	1	t	t	PROPN
cana-3289	4	2	,	,	PUNCT
cana-3289	4	3	4aiyared	4aiyared	NUM
cana-3289	4	4	iampan	iampan	NOUN
cana-3289	4	5	1assistant	1assistant	NUM
cana-3289	4	6	professor	professor	NOUN
cana-3289	4	7	,	,	PUNCT
cana-3289	4	8	department	department	NOUN
cana-3289	4	9	of	of	ADP
cana-3289	4	10	mathematics	mathematics	PROPN
cana-3289	4	11	and	and	CCONJ
cana-3289	4	12	statistics	statistic	NOUN
cana-3289	4	13	,	,	PUNCT
cana-3289	4	14	vignan	vignan	NOUN
cana-3289	4	15	’s	’s	PART
cana-3289	4	16	foundation	foundation	PROPN
cana-3289	4	17	for	for	ADP
cana-3289	4	18	science	science	NOUN
cana-3289	4	19	,	,	PUNCT
cana-3289	4	20	technology	technology	NOUN
cana-3289	4	21	and	and	CCONJ
cana-3289	4	22	research	research	NOUN
cana-3289	4	23	,	,	PUNCT
cana-3289	4	24	vadlamudi	vadlamudi	NOUN
cana-3289	4	25	,	,	PUNCT
cana-3289	4	26	guntur-522213	guntur-522213	NOUN
cana-3289	4	27	,	,	PUNCT
cana-3289	4	28	india	india	PROPN
cana-3289	4	29	.	.	PUNCT
cana-3289	4	30	email	email	NOUN
cana-3289	4	31	:	:	PUNCT
cana-3289	4	32	u.v.kalyani@gmail.com	u.v.kalyani@gmail.com	X
cana-3289	4	33	2professor	2professor	NUM
cana-3289	4	34	,	,	PUNCT
cana-3289	4	35	department	department	NOUN
cana-3289	4	36	of	of	ADP
cana-3289	4	37	mathematics	mathematic	NOUN
cana-3289	4	38	,	,	PUNCT
cana-3289	4	39	vasireddy	vasireddy	NOUN
cana-3289	4	40	venkatadri	venkatadri	PROPN
cana-3289	4	41	institute	institute	PROPN
cana-3289	4	42	of	of	ADP
cana-3289	4	43	technology	technology	PROPN
cana-3289	4	44	,	,	PUNCT
cana-3289	4	45	nambur	nambur	PROPN
cana-3289	4	46	-522508	-522508	PROPN
cana-3289	4	47	,	,	PUNCT
cana-3289	4	48	india	india	PROPN
cana-3289	4	49	.	.	PUNCT
cana-3289	4	50	email	email	NOUN
cana-3289	4	51	:	:	PUNCT
cana-3289	5	1	bvsnhariprasad@gmail.com	bvsnhariprasad@gmail.com	NOUN
cana-3289	5	2	3associate	3associate	NUM
cana-3289	5	3	professor	professor	NOUN
cana-3289	5	4	,	,	PUNCT
cana-3289	5	5	department	department	NOUN
cana-3289	5	6	of	of	ADP
cana-3289	5	7	mathematics	mathematic	NOUN
cana-3289	5	8	,	,	PUNCT
cana-3289	5	9	koneru	koneru	PROPN
cana-3289	5	10	lakshmaiah	lakshmaiah	PROPN
cana-3289	5	11	education	education	PROPN
cana-3289	5	12	foundation	foundation	PROPN
cana-3289	5	13	,	,	PUNCT
cana-3289	5	14	vaddeswaram	vaddeswaram	PROPN
cana-3289	5	15	,	,	PUNCT
cana-3289	5	16	guntur	guntur	PROPN
cana-3289	5	17	,	,	PUNCT
cana-3289	5	18	ap	ap	PROPN
cana-3289	5	19	,	,	PUNCT
cana-3289	5	20	india	india	PROPN
cana-3289	5	21	.	.	PUNCT
cana-3289	5	22	email	email	NOUN
cana-3289	5	23	:	:	PUNCT
cana-3289	5	24	eswarlal@kluniversity.in	eswarlal@kluniversity.in	PROPN
cana-3289	5	25	4department	4department	NUM
cana-3289	5	26	of	of	ADP
cana-3289	5	27	mathematics	mathematic	NOUN
cana-3289	5	28	,	,	PUNCT
cana-3289	5	29	school	school	NOUN
cana-3289	5	30	of	of	ADP
cana-3289	5	31	science	science	NOUN
cana-3289	5	32	,	,	PUNCT
cana-3289	5	33	university	university	NOUN
cana-3289	5	34	of	of	ADP
cana-3289	5	35	phayao	phayao	NOUN
cana-3289	5	36	,	,	PUNCT
cana-3289	5	37	mae	mae	PROPN
cana-3289	5	38	ka	ka	PROPN
cana-3289	5	39	,	,	PUNCT
cana-3289	5	40	mueang	mueang	PROPN
cana-3289	5	41	,	,	PUNCT
cana-3289	5	42	phayao	phayao	NOUN
cana-3289	5	43	56000	56000	NUM
cana-3289	5	44	,	,	PUNCT
cana-3289	5	45	thailand	thailand	PROPN
cana-3289	5	46	.	.	PUNCT
cana-3289	6	1	email	email	NOUN
cana-3289	6	2	:	:	PUNCT
cana-3289	6	3	aiyared.ia@up.ac.th	aiyared.ia@up.ac.th	NOUN
cana-3289	6	4	article	article	NOUN
cana-3289	6	5	history	history	NOUN
cana-3289	6	6	:	:	PUNCT
cana-3289	6	7	received	receive	VERB
cana-3289	6	8	:	:	PUNCT
cana-3289	6	9	15	15	NUM
cana-3289	6	10	-	-	SYM
cana-3289	6	11	10	10	NUM
cana-3289	6	12	-	-	PUNCT
cana-3289	6	13	2024	2024	NUM
cana-3289	6	14	revised	revise	VERB
cana-3289	6	15	:	:	PUNCT
cana-3289	6	16	02	02	NUM
cana-3289	6	17	-	-	SYM
cana-3289	6	18	12	12	NUM
cana-3289	6	19	-	-	PUNCT
cana-3289	6	20	2024	2024	NUM
cana-3289	6	21	accepted	accept	VERB
cana-3289	6	22	:	:	PUNCT
cana-3289	6	23	10	10	NUM
cana-3289	6	24	-	-	SYM
cana-3289	6	25	12	12	NUM
cana-3289	6	26	-	-	PUNCT
cana-3289	6	27	2024	2024	NUM
cana-3289	6	28	abstract	abstract	NOUN
cana-3289	6	29	:	:	PUNCT
cana-3289	6	30	this	this	DET
cana-3289	6	31	study	study	NOUN
cana-3289	6	32	explores	explore	VERB
cana-3289	6	33	the	the	DET
cana-3289	6	34	investigation	investigation	NOUN
cana-3289	6	35	of	of	ADP
cana-3289	6	36	bipolar	bipolar	ADJ
cana-3289	6	37	fuzzy	fuzzy	ADJ
cana-3289	6	38	prime	prime	ADJ
cana-3289	6	39	ideals	ideal	NOUN
cana-3289	6	40	(	(	PUNCT
cana-3289	6	41	bfpi	bfpi	ADJ
cana-3289	6	42	)	)	PUNCT
cana-3289	6	43	in	in	ADP
cana-3289	6	44	lattices	lattice	NOUN
cana-3289	6	45	.	.	PUNCT
cana-3289	7	1	we	we	PRON
cana-3289	7	2	provide	provide	VERB
cana-3289	7	3	a	a	DET
cana-3289	7	4	detailed	detailed	ADJ
cana-3289	7	5	exploration	exploration	NOUN
cana-3289	7	6	of	of	ADP
cana-3289	7	7	their	their	PRON
cana-3289	7	8	properties	property	NOUN
cana-3289	7	9	,	,	PUNCT
cana-3289	7	10	characterizations	characterization	NOUN
cana-3289	7	11	,	,	PUNCT
cana-3289	7	12	and	and	CCONJ
cana-3289	7	13	associated	associated	ADJ
cana-3289	7	14	homomorphisms	homomorphism	NOUN
cana-3289	7	15	.	.	PUNCT
cana-3289	8	1	introduction	introduction	NOUN
cana-3289	8	2	:	:	PUNCT
cana-3289	8	3	fuzzy	fuzzy	ADJ
cana-3289	8	4	set	set	NOUN
cana-3289	8	5	theory	theory	NOUN
cana-3289	8	6	,	,	PUNCT
cana-3289	8	7	introduced	introduce	VERB
cana-3289	8	8	by	by	ADP
cana-3289	8	9	zadeh	zadeh	PROPN
cana-3289	8	10	l.a	l.a	PROPN
cana-3289	8	11	.	.	PROPN
cana-3289	8	12	,	,	PUNCT
cana-3289	8	13	is	be	AUX
cana-3289	8	14	grounded	ground	VERB
cana-3289	8	15	in	in	ADP
cana-3289	8	16	the	the	DET
cana-3289	8	17	concept	concept	NOUN
cana-3289	8	18	of	of	ADP
cana-3289	8	19	membership	membership	NOUN
cana-3289	8	20	functions	function	NOUN
cana-3289	8	21	where	where	SCONJ
cana-3289	8	22	each	each	DET
cana-3289	8	23	element	element	NOUN
cana-3289	8	24	in	in	ADP
cana-3289	8	25	a	a	DET
cana-3289	8	26	set	set	NOUN
cana-3289	8	27	is	be	AUX
cana-3289	8	28	assigned	assign	VERB
cana-3289	8	29	a	a	DET
cana-3289	8	30	membership	membership	NOUN
cana-3289	8	31	degree	degree	NOUN
cana-3289	8	32	ranging	range	VERB
cana-3289	8	33	between	between	ADP
cana-3289	8	34	0	0	NUM
cana-3289	8	35	and	and	CCONJ
cana-3289	8	36	1	1	NUM
cana-3289	8	37	.	.	PUNCT
cana-3289	9	1	although	although	SCONJ
cana-3289	9	2	this	this	DET
cana-3289	9	3	model	model	NOUN
cana-3289	9	4	effectively	effectively	ADV
cana-3289	9	5	combines	combine	VERB
cana-3289	9	6	supporting	support	VERB
cana-3289	9	7	and	and	CCONJ
cana-3289	9	8	opposing	oppose	VERB
cana-3289	9	9	evidence	evidence	NOUN
cana-3289	9	10	for	for	ADP
cana-3289	9	11	element	element	NOUN
cana-3289	9	12	membership	membership	NOUN
cana-3289	9	13	,	,	PUNCT
cana-3289	9	14	it	it	PRON
cana-3289	9	15	lacks	lack	VERB
cana-3289	9	16	explicit	explicit	ADJ
cana-3289	9	17	representation	representation	NOUN
cana-3289	9	18	of	of	ADP
cana-3289	9	19	the	the	DET
cana-3289	9	20	uncertainty	uncertainty	NOUN
cana-3289	9	21	or	or	CCONJ
cana-3289	9	22	dual	dual	ADJ
cana-3289	9	23	nature	nature	NOUN
cana-3289	9	24	of	of	ADP
cana-3289	9	25	these	these	DET
cana-3289	9	26	evidence	evidence	NOUN
cana-3289	9	27	.	.	PUNCT
cana-3289	10	1	to	to	PART
cana-3289	10	2	address	address	VERB
cana-3289	10	3	this	this	DET
cana-3289	10	4	limitation	limitation	NOUN
cana-3289	10	5	,	,	PUNCT
cana-3289	10	6	gau	gau	NOUN
cana-3289	10	7	and	and	CCONJ
cana-3289	10	8	buehrer	buehrer	NOUN
cana-3289	10	9	introduced	introduce	VERB
cana-3289	10	10	the	the	DET
cana-3289	10	11	concept	concept	NOUN
cana-3289	10	12	of	of	ADP
cana-3289	10	13	vague	vague	ADJ
cana-3289	10	14	sets	set	NOUN
cana-3289	10	15	,	,	PUNCT
cana-3289	10	16	characterized	characterize	VERB
cana-3289	10	17	by	by	ADP
cana-3289	10	18	two	two	NUM
cana-3289	10	19	functions	function	NOUN
cana-3289	10	20	:	:	PUNCT
cana-3289	10	21	one	one	NUM
cana-3289	10	22	for	for	ADP
cana-3289	10	23	membership	membership	NOUN
cana-3289	10	24	and	and	CCONJ
cana-3289	10	25	another	another	PRON
cana-3289	10	26	for	for	ADP
cana-3289	10	27	non	non	ADJ
cana-3289	10	28	-	-	NOUN
cana-3289	10	29	membership	membership	NOUN
cana-3289	10	30	,	,	PUNCT
cana-3289	10	31	where	where	SCONJ
cana-3289	10	32	their	their	PRON
cana-3289	10	33	sum	sum	NOUN
cana-3289	10	34	does	do	AUX
cana-3289	10	35	not	not	PART
cana-3289	10	36	exceed	exceed	VERB
cana-3289	10	37	one	one	NUM
cana-3289	10	38	.	.	PUNCT
cana-3289	11	1	further	further	ADJ
cana-3289	11	2	contributions	contribution	NOUN
cana-3289	11	3	to	to	ADP
cana-3289	11	4	fuzzy	fuzzy	ADJ
cana-3289	11	5	set	set	NOUN
cana-3289	11	6	theory	theory	NOUN
cana-3289	11	7	came	come	VERB
cana-3289	11	8	from	from	ADP
cana-3289	11	9	atanassov	atanassov	NOUN
cana-3289	11	10	's	's	PART
cana-3289	11	11	intuitionistic	intuitionistic	ADJ
cana-3289	11	12	fuzzy	fuzzy	ADJ
cana-3289	11	13	sets	set	NOUN
cana-3289	11	14	and	and	CCONJ
cana-3289	11	15	bustince	bustince	NOUN
cana-3289	11	16	and	and	CCONJ
cana-3289	11	17	burillo	burillo	PROPN
cana-3289	11	18	's	's	PART
cana-3289	11	19	work	work	NOUN
cana-3289	11	20	showing	show	VERB
cana-3289	11	21	their	their	PRON
cana-3289	11	22	mathematical	mathematical	ADJ
cana-3289	11	23	equivalence	equivalence	NOUN
cana-3289	11	24	to	to	ADP
cana-3289	11	25	vague	vague	ADJ
cana-3289	11	26	sets	set	NOUN
cana-3289	11	27	.	.	PUNCT
cana-3289	12	1	the	the	DET
cana-3289	12	2	dual	dual	ADJ
cana-3289	12	3	-	-	PUNCT
cana-3289	12	4	function	function	NOUN
cana-3289	12	5	approach	approach	NOUN
cana-3289	12	6	of	of	ADP
cana-3289	12	7	vague	vague	ADJ
cana-3289	12	8	sets	set	NOUN
cana-3289	12	9	has	have	AUX
cana-3289	12	10	been	be	AUX
cana-3289	12	11	applied	apply	VERB
cana-3289	12	12	extensively	extensively	ADV
cana-3289	12	13	in	in	ADP
cana-3289	12	14	decisionmaking	decisionmake	VERB
cana-3289	12	15	,	,	PUNCT
cana-3289	12	16	control	control	NOUN
cana-3289	12	17	systems	system	NOUN
cana-3289	12	18	,	,	PUNCT
cana-3289	12	19	and	and	CCONJ
cana-3289	12	20	fault	fault	VERB
cana-3289	12	21	diagnosis	diagnosis	NOUN
cana-3289	12	22	.	.	PUNCT
cana-3289	13	1	lattice	lattice	PROPN
cana-3289	13	2	theory	theory	NOUN
cana-3289	13	3	has	have	AUX
cana-3289	13	4	also	also	ADV
cana-3289	13	5	benefited	benefit	VERB
cana-3289	13	6	from	from	ADP
cana-3289	13	7	these	these	DET
cana-3289	13	8	advancements	advancement	NOUN
cana-3289	13	9	,	,	PUNCT
cana-3289	13	10	with	with	ADP
cana-3289	13	11	ajmal	ajmal	PROPN
cana-3289	13	12	and	and	CCONJ
cana-3289	13	13	thomas	thomas	PROPN
cana-3289	13	14	pioneering	pioneer	VERB
cana-3289	13	15	fuzzy	fuzzy	ADJ
cana-3289	13	16	sublattice	sublattice	NOUN
cana-3289	13	17	theory	theory	NOUN
cana-3289	13	18	,	,	PUNCT
cana-3289	13	19	and	and	CCONJ
cana-3289	13	20	later	later	ADV
cana-3289	13	21	works	work	NOUN
cana-3289	13	22	exploring	explore	VERB
cana-3289	13	23	intuitionistic	intuitionistic	ADJ
cana-3289	13	24	fuzzy	fuzzy	ADJ
cana-3289	13	25	lattices	lattice	NOUN
cana-3289	13	26	and	and	CCONJ
cana-3289	13	27	vague	vague	ADJ
cana-3289	13	28	lattices	lattice	NOUN
cana-3289	13	29	.	.	PUNCT
cana-3289	14	1	bipolar	bipolar	ADJ
cana-3289	14	2	fuzzy	fuzzy	ADJ
cana-3289	14	3	sets	set	NOUN
cana-3289	14	4	(	(	PUNCT
cana-3289	14	5	bfs	bfs	NOUN
cana-3289	14	6	)	)	PUNCT
cana-3289	14	7	,	,	PUNCT
cana-3289	14	8	introduced	introduce	VERB
cana-3289	14	9	by	by	ADP
cana-3289	14	10	lee	lee	PROPN
cana-3289	14	11	k.m	k.m	PROPN
cana-3289	14	12	.	.	PROPN
cana-3289	14	13	,	,	PUNCT
cana-3289	14	14	extended	extend	VERB
cana-3289	14	15	fuzzy	fuzzy	ADJ
cana-3289	14	16	sets	set	NOUN
cana-3289	14	17	by	by	ADP
cana-3289	14	18	incorporating	incorporate	VERB
cana-3289	14	19	dual	dual	ADJ
cana-3289	14	20	notions	notion	NOUN
cana-3289	14	21	of	of	ADP
cana-3289	14	22	positive	positive	ADJ
cana-3289	14	23	and	and	CCONJ
cana-3289	14	24	negative	negative	ADJ
cana-3289	14	25	membership	membership	NOUN
cana-3289	14	26	values	value	NOUN
cana-3289	14	27	within	within	ADP
cana-3289	14	28	a	a	DET
cana-3289	14	29	range	range	NOUN
cana-3289	14	30	of	of	ADP
cana-3289	14	31	[	[	X
cana-3289	14	32	-1	-1	X
cana-3289	14	33	,	,	PUNCT
cana-3289	14	34	1	1	NUM
cana-3289	14	35	]	]	PUNCT
cana-3289	14	36	.	.	PUNCT
cana-3289	15	1	this	this	DET
cana-3289	15	2	extension	extension	NOUN
cana-3289	15	3	enables	enable	VERB
cana-3289	15	4	interpretations	interpretation	NOUN
cana-3289	15	5	of	of	ADP
cana-3289	15	6	bipolar	bipolar	ADJ
cana-3289	15	7	information	information	NOUN
cana-3289	15	8	,	,	PUNCT
cana-3289	15	9	making	make	VERB
cana-3289	15	10	bfs	bfs	VERB
cana-3289	15	11	a	a	DET
cana-3289	15	12	valuable	valuable	ADJ
cana-3289	15	13	tool	tool	NOUN
cana-3289	15	14	in	in	ADP
cana-3289	15	15	decision	decision	NOUN
cana-3289	15	16	-	-	PUNCT
cana-3289	15	17	making	make	VERB
cana-3289	15	18	and	and	CCONJ
cana-3289	15	19	information	information	NOUN
cana-3289	15	20	processing	processing	NOUN
cana-3289	15	21	.	.	PUNCT
cana-3289	16	1	objectives	objective	NOUN
cana-3289	16	2	:	:	PUNCT
cana-3289	16	3	introduction	introduction	NOUN
cana-3289	16	4	of	of	ADP
cana-3289	16	5	bipolar	bipolar	ADJ
cana-3289	16	6	fuzzy	fuzzy	ADJ
cana-3289	16	7	prime	prime	ADJ
cana-3289	16	8	ideals	ideal	NOUN
cana-3289	16	9	of	of	ADP
cana-3289	16	10	a	a	DET
cana-3289	16	11	lattice	lattice	NOUN
cana-3289	16	12	,	,	PUNCT
cana-3289	16	13	study	study	NOUN
cana-3289	16	14	of	of	ADP
cana-3289	16	15	their	their	PRON
cana-3289	16	16	characterizations	characterization	NOUN
cana-3289	16	17	and	and	CCONJ
cana-3289	16	18	associated	associated	ADJ
cana-3289	16	19	homomorphisms	homomorphism	NOUN
cana-3289	16	20	.	.	PUNCT
cana-3289	17	1	keywords	keyword	NOUN
cana-3289	17	2	:	:	PUNCT
cana-3289	17	3	bipolar	bipolar	ADJ
cana-3289	17	4	fuzzy	fuzzy	ADJ
cana-3289	17	5	ideal	ideal	NOUN
cana-3289	17	6	,	,	PUNCT
cana-3289	17	7	bipolar	bipolar	ADJ
cana-3289	17	8	fuzzy	fuzzy	ADJ
cana-3289	17	9	prime	prime	ADJ
cana-3289	17	10	ideal	ideal	NOUN
cana-3289	17	11	,	,	PUNCT
cana-3289	17	12	bipolar	bipolar	ADJ
cana-3289	17	13	fuzzy	fuzzy	ADJ
cana-3289	17	14	homomorphism	homomorphism	NOUN
cana-3289	17	15	,	,	PUNCT
cana-3289	17	16	bipolar	bipolar	ADJ
cana-3289	17	17	fuzzy	fuzzy	ADJ
cana-3289	17	18	magnified	magnify	VERB
cana-3289	17	19	translation	translation	NOUN
cana-3289	17	20	.	.	PUNCT
cana-3289	18	1	1	1	X
cana-3289	18	2	.	.	X
cana-3289	18	3	introduction	introduction	NOUN
cana-3289	18	4	fuzzy	fuzzy	ADJ
cana-3289	18	5	set	set	NOUN
cana-3289	18	6	theory	theory	NOUN
cana-3289	18	7	,	,	PUNCT
cana-3289	18	8	introduced	introduce	VERB
cana-3289	18	9	by	by	ADP
cana-3289	18	10	zadeh	zadeh	PROPN
cana-3289	18	11	l.a	l.a	PROPN
cana-3289	18	12	.	.	PUNCT
cana-3289	19	1	[	[	X
cana-3289	19	2	1	1	NUM
cana-3289	19	3	]	]	PUNCT
cana-3289	19	4	,	,	PUNCT
cana-3289	19	5	is	be	AUX
cana-3289	19	6	grounded	ground	VERB
cana-3289	19	7	in	in	ADP
cana-3289	19	8	the	the	DET
cana-3289	19	9	concept	concept	NOUN
cana-3289	19	10	of	of	ADP
cana-3289	19	11	membership	membership	NOUN
cana-3289	19	12	functions	function	NOUN
cana-3289	19	13	where	where	SCONJ
cana-3289	19	14	each	each	DET
cana-3289	19	15	element	element	NOUN
cana-3289	19	16	in	in	ADP
cana-3289	19	17	a	a	DET
cana-3289	19	18	set	set	NOUN
cana-3289	19	19	is	be	AUX
cana-3289	19	20	assigned	assign	VERB
cana-3289	19	21	a	a	DET
cana-3289	19	22	membership	membership	NOUN
cana-3289	19	23	degree	degree	NOUN
cana-3289	19	24	ranging	range	VERB
cana-3289	19	25	between	between	ADP
cana-3289	19	26	0	0	NUM
cana-3289	19	27	and	and	CCONJ
cana-3289	19	28	1	1	NUM
cana-3289	19	29	.	.	PUNCT
cana-3289	20	1	although	although	SCONJ
cana-3289	20	2	this	this	DET
cana-3289	20	3	model	model	NOUN
cana-3289	20	4	effectively	effectively	ADV
cana-3289	20	5	combines	combine	VERB
cana-3289	20	6	supporting	support	VERB
cana-3289	20	7	and	and	CCONJ
cana-3289	20	8	opposing	oppose	VERB
cana-3289	20	9	evidence	evidence	NOUN
cana-3289	20	10	for	for	ADP
cana-3289	20	11	element	element	NOUN
cana-3289	20	12	membership	membership	NOUN
cana-3289	20	13	,	,	PUNCT
cana-3289	20	14	it	it	PRON
cana-3289	20	15	lacks	lack	VERB
cana-3289	20	16	mailto:bvsnhariprasad@gmail.com	mailto:bvsnhariprasad@gmail.com	NOUN
cana-3289	20	17	mailto:eswarlal@kluniversity.in	mailto:eswarlal@kluniversity.in	NOUN
cana-3289	20	18	communications	communication	NOUN
cana-3289	20	19	on	on	ADP
cana-3289	20	20	applied	apply	VERB
cana-3289	20	21	nonlinear	nonlinear	ADJ
cana-3289	20	22	analysis	analysis	NOUN
cana-3289	20	23	issn	issn	NOUN
cana-3289	20	24	:	:	PUNCT
cana-3289	20	25	1074	1074	NUM
cana-3289	20	26	-	-	PUNCT
cana-3289	20	27	133x	133x	NUM
cana-3289	20	28	vol	vol	NOUN
cana-3289	20	29	32	32	NUM
cana-3289	20	30	no	no	NOUN
cana-3289	20	31	.	.	PUNCT
cana-3289	21	1	6s	6s	NUM
cana-3289	21	2	(	(	PUNCT
cana-3289	21	3	2025	2025	NUM
cana-3289	21	4	)	)	PUNCT
cana-3289	21	5	242	242	NUM
cana-3289	21	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3289	21	7	explicit	explicit	ADJ
cana-3289	21	8	representation	representation	NOUN
cana-3289	21	9	of	of	ADP
cana-3289	21	10	the	the	DET
cana-3289	21	11	uncertainty	uncertainty	NOUN
cana-3289	21	12	or	or	CCONJ
cana-3289	21	13	dual	dual	ADJ
cana-3289	21	14	nature	nature	NOUN
cana-3289	21	15	of	of	ADP
cana-3289	21	16	these	these	DET
cana-3289	21	17	evidences	evidence	NOUN
cana-3289	21	18	.	.	PUNCT
cana-3289	22	1	to	to	PART
cana-3289	22	2	address	address	VERB
cana-3289	22	3	this	this	DET
cana-3289	22	4	limitation	limitation	NOUN
cana-3289	22	5	,	,	PUNCT
cana-3289	22	6	gau	gau	NOUN
cana-3289	22	7	and	and	CCONJ
cana-3289	22	8	buehrer	buehrer	NOUN
cana-3289	23	1	[	[	X
cana-3289	23	2	2	2	NUM
cana-3289	23	3	]	]	PUNCT
cana-3289	23	4	introduced	introduce	VERB
cana-3289	23	5	the	the	DET
cana-3289	23	6	concept	concept	NOUN
cana-3289	23	7	of	of	ADP
cana-3289	23	8	vague	vague	ADJ
cana-3289	23	9	sets	set	NOUN
cana-3289	23	10	,	,	PUNCT
cana-3289	23	11	characterized	characterize	VERB
cana-3289	23	12	by	by	ADP
cana-3289	23	13	two	two	NUM
cana-3289	23	14	functions	function	NOUN
cana-3289	23	15	:	:	PUNCT
cana-3289	23	16	one	one	NUM
cana-3289	23	17	for	for	ADP
cana-3289	23	18	membership	membership	NOUN
cana-3289	23	19	and	and	CCONJ
cana-3289	23	20	another	another	PRON
cana-3289	23	21	for	for	ADP
cana-3289	23	22	non	non	ADJ
cana-3289	23	23	-	-	NOUN
cana-3289	23	24	membership	membership	NOUN
cana-3289	23	25	,	,	PUNCT
cana-3289	23	26	where	where	SCONJ
cana-3289	23	27	their	their	PRON
cana-3289	23	28	sum	sum	NOUN
cana-3289	23	29	does	do	AUX
cana-3289	23	30	not	not	PART
cana-3289	23	31	exceed	exceed	VERB
cana-3289	23	32	one	one	NUM
cana-3289	23	33	.	.	PUNCT
cana-3289	24	1	further	further	ADJ
cana-3289	24	2	contributions	contribution	NOUN
cana-3289	24	3	to	to	ADP
cana-3289	24	4	fuzzy	fuzzy	ADJ
cana-3289	24	5	set	set	NOUN
cana-3289	24	6	theory	theory	NOUN
cana-3289	24	7	came	come	VERB
cana-3289	24	8	from	from	ADP
cana-3289	24	9	atanassov	atanassov	NOUN
cana-3289	24	10	's	's	PART
cana-3289	24	11	intuitionistic	intuitionistic	ADJ
cana-3289	24	12	fuzzy	fuzzy	ADJ
cana-3289	24	13	sets	set	NOUN
cana-3289	24	14	[	[	X
cana-3289	24	15	5]and	5]and	NUM
cana-3289	24	16	bustince	bustince	NOUN
cana-3289	24	17	and	and	CCONJ
cana-3289	24	18	burillo	burillo	NOUN
cana-3289	24	19	's	's	PART
cana-3289	25	1	[	[	X
cana-3289	25	2	3]work	3]work	NUM
cana-3289	25	3	showing	show	VERB
cana-3289	25	4	their	their	PRON
cana-3289	25	5	mathematical	mathematical	ADJ
cana-3289	25	6	equivalence	equivalence	NOUN
cana-3289	25	7	to	to	ADP
cana-3289	25	8	vague	vague	ADJ
cana-3289	25	9	sets	set	NOUN
cana-3289	25	10	.	.	PUNCT
cana-3289	26	1	the	the	DET
cana-3289	26	2	dualfunction	dualfunction	NOUN
cana-3289	26	3	approach	approach	NOUN
cana-3289	26	4	of	of	ADP
cana-3289	26	5	vague	vague	ADJ
cana-3289	26	6	sets	set	NOUN
cana-3289	26	7	has	have	AUX
cana-3289	26	8	been	be	AUX
cana-3289	26	9	applied	apply	VERB
cana-3289	26	10	extensively	extensively	ADV
cana-3289	26	11	in	in	ADP
cana-3289	26	12	decision	decision	NOUN
cana-3289	26	13	-	-	PUNCT
cana-3289	26	14	making	making	NOUN
cana-3289	26	15	,	,	PUNCT
cana-3289	26	16	control	control	NOUN
cana-3289	26	17	systems	system	NOUN
cana-3289	26	18	,	,	PUNCT
cana-3289	26	19	and	and	CCONJ
cana-3289	26	20	fault	fault	VERB
cana-3289	26	21	diagnosis	diagnosis	NOUN
cana-3289	26	22	.	.	PUNCT
cana-3289	27	1	lattice	lattice	PROPN
cana-3289	27	2	theory	theory	NOUN
cana-3289	27	3	has	have	AUX
cana-3289	27	4	also	also	ADV
cana-3289	27	5	benefited	benefit	VERB
cana-3289	27	6	from	from	ADP
cana-3289	27	7	these	these	DET
cana-3289	27	8	advancements	advancement	NOUN
cana-3289	27	9	,	,	PUNCT
cana-3289	27	10	with	with	ADP
cana-3289	27	11	ajmal	ajmal	PROPN
cana-3289	27	12	and	and	CCONJ
cana-3289	27	13	thomas	thomas	PROPN
cana-3289	27	14	pioneering	pioneer	VERB
cana-3289	27	15	fuzzy	fuzzy	ADJ
cana-3289	27	16	sublattice	sublattice	NOUN
cana-3289	27	17	theory	theory	NOUN
cana-3289	27	18	,	,	PUNCT
cana-3289	27	19	and	and	CCONJ
cana-3289	27	20	later	later	ADV
cana-3289	27	21	works	work	NOUN
cana-3289	27	22	exploring	explore	VERB
cana-3289	27	23	intuitionistic	intuitionistic	ADJ
cana-3289	27	24	fuzzy	fuzzy	ADJ
cana-3289	27	25	lattices	lattice	NOUN
cana-3289	27	26	and	and	CCONJ
cana-3289	27	27	vague	vague	ADJ
cana-3289	27	28	lattices	lattice	NOUN
cana-3289	27	29	.	.	PUNCT
cana-3289	28	1	bipolar	bipolar	ADJ
cana-3289	28	2	fuzzy	fuzzy	ADJ
cana-3289	28	3	sets	set	NOUN
cana-3289	28	4	(	(	PUNCT
cana-3289	28	5	bfs	bfs	NOUN
cana-3289	28	6	)	)	PUNCT
cana-3289	28	7	,	,	PUNCT
cana-3289	28	8	introduced	introduce	VERB
cana-3289	28	9	by	by	ADP
cana-3289	28	10	lee	lee	PROPN
cana-3289	28	11	k.m.[6	k.m.[6	PROPN
cana-3289	28	12	]	]	PUNCT
cana-3289	28	13	,	,	PUNCT
cana-3289	28	14	extended	extend	VERB
cana-3289	28	15	fuzzy	fuzzy	ADJ
cana-3289	28	16	sets	set	NOUN
cana-3289	28	17	by	by	ADP
cana-3289	28	18	incorporating	incorporate	VERB
cana-3289	28	19	dual	dual	ADJ
cana-3289	28	20	notions	notion	NOUN
cana-3289	28	21	of	of	ADP
cana-3289	28	22	positive	positive	ADJ
cana-3289	28	23	and	and	CCONJ
cana-3289	28	24	negative	negative	ADJ
cana-3289	28	25	membership	membership	NOUN
cana-3289	28	26	values	value	NOUN
cana-3289	28	27	within	within	ADP
cana-3289	28	28	a	a	DET
cana-3289	28	29	range	range	NOUN
cana-3289	28	30	of	of	ADP
cana-3289	28	31	[	[	X
cana-3289	28	32	-1	-1	X
cana-3289	28	33	,	,	PUNCT
cana-3289	28	34	1	1	NUM
cana-3289	28	35	]	]	PUNCT
cana-3289	28	36	.	.	PUNCT
cana-3289	29	1	this	this	DET
cana-3289	29	2	extension	extension	NOUN
cana-3289	29	3	enables	enable	VERB
cana-3289	29	4	interpretations	interpretation	NOUN
cana-3289	29	5	of	of	ADP
cana-3289	29	6	bipolar	bipolar	ADJ
cana-3289	29	7	information	information	NOUN
cana-3289	29	8	,	,	PUNCT
cana-3289	29	9	making	make	VERB
cana-3289	29	10	bfs	bfs	VERB
cana-3289	29	11	a	a	DET
cana-3289	29	12	valuable	valuable	ADJ
cana-3289	29	13	tool	tool	NOUN
cana-3289	29	14	in	in	ADP
cana-3289	29	15	decision	decision	NOUN
cana-3289	29	16	-	-	PUNCT
cana-3289	29	17	making	make	VERB
cana-3289	29	18	and	and	CCONJ
cana-3289	29	19	information	information	NOUN
cana-3289	29	20	processing	processing	NOUN
cana-3289	29	21	.	.	PUNCT
cana-3289	30	1	in	in	ADP
cana-3289	30	2	particular	particular	ADJ
cana-3289	30	3	,	,	PUNCT
cana-3289	30	4	ajmal	ajmal	ADJ
cana-3289	30	5	.	.	PUNCT
cana-3289	31	1	n	n	PRON
cana-3289	31	2	and	and	CCONJ
cana-3289	31	3	thomas.k.v	thomas.k.v	ADJ
cana-3289	32	1	[	[	X
cana-3289	32	2	7	7	X
cana-3289	32	3	]	]	PUNCT
cana-3289	32	4	both	both	PRON
cana-3289	32	5	explored	explore	VERB
cana-3289	32	6	theory	theory	NOUN
cana-3289	32	7	of	of	ADP
cana-3289	32	8	(	(	PUNCT
cana-3289	32	9	fl)fuzzy	fl)fuzzy	NOUN
cana-3289	32	10	sublattice	sublattice	VERB
cana-3289	32	11	and	and	CCONJ
cana-3289	32	12	introduced	introduce	VERB
cana-3289	32	13	the	the	DET
cana-3289	32	14	idea	idea	NOUN
cana-3289	32	15	of	of	ADP
cana-3289	32	16	fuzzy	fuzzy	ADJ
cana-3289	32	17	sets	set	NOUN
cana-3289	32	18	to	to	PART
cana-3289	32	19	lattice	lattice	VERB
cana-3289	32	20	theory	theory	NOUN
cana-3289	32	21	.	.	PUNCT
cana-3289	33	1	after	after	ADP
cana-3289	33	2	then	then	ADV
cana-3289	33	3	,	,	PUNCT
cana-3289	33	4	in	in	ADP
cana-3289	33	5	2011	2011	NUM
cana-3289	33	6	,	,	PUNCT
cana-3289	33	7	thomas.k.v	thomas.k.v	ADJ
cana-3289	33	8	and	and	CCONJ
cana-3289	33	9	nair	nair	PROPN
cana-3289	33	10	l.s.[4	l.s.[4	PROPN
cana-3289	33	11	]	]	PUNCT
cana-3289	33	12	presented	present	VERB
cana-3289	33	13	idea	idea	NOUN
cana-3289	33	14	of	of	ADP
cana-3289	33	15	intuitionistic	intuitionistic	ADJ
cana-3289	33	16	fuzzy	fuzzy	ADJ
cana-3289	33	17	lattices	lattice	NOUN
cana-3289	33	18	(	(	PUNCT
cana-3289	33	19	ifls	ifls	NOUN
cana-3289	33	20	)	)	PUNCT
cana-3289	33	21	.	.	PUNCT
cana-3289	34	1	in	in	ADP
cana-3289	34	2	2017	2017	NUM
cana-3289	34	3	,	,	PUNCT
cana-3289	34	4	milles	mille	NOUN
cana-3289	34	5	s	s	PART
cana-3289	34	6	[	[	X
cana-3289	34	7	13	13	NUM
cana-3289	34	8	]	]	PUNCT
cana-3289	34	9	investigated	investigate	VERB
cana-3289	34	10	the	the	DET
cana-3289	34	11	characterization	characterization	NOUN
cana-3289	34	12	of	of	ADP
cana-3289	34	13	ifis	ifis	NOUN
cana-3289	34	14	and	and	CCONJ
cana-3289	34	15	iffs	iff	NOUN
cana-3289	34	16	based	base	VERB
cana-3289	34	17	on	on	ADP
cana-3289	34	18	lattice	lattice	NOUN
cana-3289	34	19	operations	operation	NOUN
cana-3289	34	20	.	.	PUNCT
cana-3289	35	1	rao.r.p	rao.r.p	NOUN
cana-3289	36	1	[	[	X
cana-3289	36	2	15	15	NUM
cana-3289	36	3	]	]	PUNCT
cana-3289	36	4	later	later	ADV
cana-3289	36	5	researched	research	VERB
cana-3289	36	6	rough	rough	ADJ
cana-3289	36	7	vague	vague	ADJ
cana-3289	36	8	lattices	lattice	NOUN
cana-3289	36	9	in	in	ADP
cana-3289	36	10	2019	2019	NUM
cana-3289	36	11	.	.	PUNCT
cana-3289	37	1	nageswara	nageswara	PROPN
cana-3289	37	2	rao.b	rao.b	PROPN
cana-3289	37	3	.	.	PROPN
cana-3289	37	4	,	,	PUNCT
cana-3289	37	5	ramakrishna	ramakrishna	PROPN
cana-3289	37	6	n	n	PROPN
cana-3289	37	7	and	and	CCONJ
cana-3289	37	8	eswarlal.t	eswarlal.t	PUNCT
cana-3289	38	1	[	[	X
cana-3289	38	2	14	14	NUM
cana-3289	38	3	]	]	PUNCT
cana-3289	38	4	introduced	introduce	VERB
cana-3289	38	5	vague	vague	ADJ
cana-3289	38	6	lattices(vl	lattices(vl	PROPN
cana-3289	38	7	)	)	PUNCT
cana-3289	38	8	in	in	ADP
cana-3289	38	9	2020	2020	NUM
cana-3289	38	10	.	.	PUNCT
cana-3289	39	1	the	the	DET
cana-3289	39	2	principal	principal	ADJ
cana-3289	39	3	ifi	ifi	PROPN
cana-3289	39	4	and	and	CCONJ
cana-3289	39	5	iff	iff	VERB
cana-3289	39	6	on	on	ADP
cana-3289	39	7	a	a	DET
cana-3289	39	8	lattice	lattice	NOUN
cana-3289	39	9	were	be	AUX
cana-3289	39	10	the	the	DET
cana-3289	39	11	subject	subject	NOUN
cana-3289	39	12	of	of	ADP
cana-3289	39	13	boudaoud.s	boudaoud.s	PROPN
cana-3289	39	14	,	,	PUNCT
cana-3289	39	15	zedam.l	zedam.l	PROPN
cana-3289	39	16	and	and	CCONJ
cana-3289	39	17	milles	mille	NOUN
cana-3289	39	18	s	s	PART
cana-3289	39	19	[	[	X
cana-3289	39	20	10	10	NUM
cana-3289	39	21	]	]	PUNCT
cana-3289	39	22	study	study	NOUN
cana-3289	39	23	in	in	ADP
cana-3289	39	24	2020	2020	NUM
cana-3289	39	25	.	.	PUNCT
cana-3289	40	1	milles.s.[11	milles.s.[11	NOUN
cana-3289	40	2	]	]	PUNCT
cana-3289	40	3	as	as	ADV
cana-3289	40	4	well	well	ADV
cana-3289	40	5	as	as	ADP
cana-3289	40	6	studied	study	VERB
cana-3289	40	7	the	the	DET
cana-3289	40	8	lattice	lattice	NOUN
cana-3289	40	9	of	of	ADP
cana-3289	40	10	(	(	PUNCT
cana-3289	40	11	ift)intuitionistic	ift)intuitionistic	ADJ
cana-3289	40	12	fuzzy	fuzzy	ADJ
cana-3289	40	13	topologies	topology	NOUN
cana-3289	40	14	produced	produce	VERB
cana-3289	40	15	by	by	ADP
cana-3289	40	16	intuitionistic	intuitionistic	ADJ
cana-3289	40	17	fuzzy	fuzzy	ADJ
cana-3289	40	18	relations	relation	NOUN
cana-3289	40	19	in	in	ADP
cana-3289	40	20	2020	2020	NUM
cana-3289	40	21	.	.	PUNCT
cana-3289	41	1	on	on	ADP
cana-3289	41	2	residual	residual	ADJ
cana-3289	41	3	lattices	lattice	NOUN
cana-3289	41	4	,	,	PUNCT
cana-3289	41	5	zhang	zhang	PROPN
cana-3289	41	6	h	h	PROPN
cana-3289	41	7	and	and	CCONJ
cana-3289	41	8	qingguo.li	qingguo.li	X
cana-3289	42	1	[	[	X
cana-3289	42	2	12	12	NUM
cana-3289	42	3	]	]	PUNCT
cana-3289	42	4	researched	research	VERB
cana-3289	42	5	the	the	DET
cana-3289	42	6	(	(	PUNCT
cana-3289	42	7	iff)intuitionistic	iff)intuitionistic	ADJ
cana-3289	42	8	fuzzy	fuzzy	ADJ
cana-3289	42	9	filter	filter	NOUN
cana-3289	42	10	theory	theory	NOUN
cana-3289	42	11	.	.	PUNCT
cana-3289	43	1	bipolar	bipolar	ADJ
cana-3289	43	2	vague	vague	ADJ
cana-3289	43	3	cosets	coset	NOUN
cana-3289	44	1	[	[	X
cana-3289	44	2	9	9	NUM
cana-3289	44	3	]	]	PUNCT
cana-3289	44	4	,	,	PUNCT
cana-3289	44	5	homomorphism	homomorphism	NOUN
cana-3289	44	6	on	on	ADP
cana-3289	44	7	bipolar	bipolar	ADJ
cana-3289	44	8	vague	vague	ADJ
cana-3289	44	9	normal	normal	ADJ
cana-3289	44	10	groups	group	NOUN
cana-3289	44	11	[	[	X
cana-3289	44	12	8	8	NUM
cana-3289	44	13	]	]	PUNCT
cana-3289	44	14	were	be	AUX
cana-3289	44	15	studied	study	VERB
cana-3289	44	16	by	by	ADP
cana-3289	44	17	venkata	venkata	PROPN
cana-3289	44	18	kalyani	kalyani	PROPN
cana-3289	44	19	u	u	PROPN
cana-3289	44	20	in	in	ADP
cana-3289	44	21	2020	2020	NUM
cana-3289	44	22	.	.	PUNCT
cana-3289	45	1	later	later	ADV
cana-3289	45	2	,	,	PUNCT
cana-3289	45	3	bipolar	bipolar	ADJ
cana-3289	45	4	fuzzy	fuzzy	ADJ
cana-3289	45	5	sublattices	sublattice	NOUN
cana-3289	45	6	were	be	AUX
cana-3289	45	7	introduced	introduce	VERB
cana-3289	45	8	in	in	ADP
cana-3289	45	9	2023	2023	NUM
cana-3289	45	10	by	by	ADP
cana-3289	45	11	venkata	venkata	PROPN
cana-3289	45	12	kalyani	kalyani	PROPN
cana-3289	45	13	u	u	PROPN
cana-3289	45	14	and	and	CCONJ
cana-3289	45	15	studied	study	VERB
cana-3289	45	16	bipolar	bipolar	ADJ
cana-3289	45	17	fuzzy	fuzzy	ADJ
cana-3289	45	18	ideals	ideal	NOUN
cana-3289	45	19	of	of	ADP
cana-3289	45	20	a	a	DET
cana-3289	45	21	lattice	lattice	NOUN
cana-3289	46	1	[	[	X
cana-3289	46	2	16	16	NUM
cana-3289	46	3	]	]	PUNCT
cana-3289	46	4	.	.	PUNCT
cana-3289	47	1	venkata	venkata	PROPN
cana-3289	47	2	kalyani	kalyani	PROPN
cana-3289	47	3	u	u	PROPN
cana-3289	47	4	studied	study	VERB
cana-3289	47	5	the	the	DET
cana-3289	47	6	bipolar	bipolar	ADJ
cana-3289	47	7	magnified	magnify	VERB
cana-3289	47	8	fuzzy	fuzzy	ADJ
cana-3289	47	9	translation	translation	NOUN
cana-3289	47	10	of	of	ADP
cana-3289	47	11	a	a	DET
cana-3289	47	12	lattice	lattice	NOUN
cana-3289	47	13	[	[	X
cana-3289	47	14	17	17	NUM
cana-3289	47	15	]	]	PUNCT
cana-3289	47	16	in	in	ADP
cana-3289	47	17	2024	2024	NUM
cana-3289	47	18	.	.	PUNCT
cana-3289	48	1	now	now	ADV
cana-3289	48	2	,	,	PUNCT
cana-3289	48	3	in	in	ADP
cana-3289	48	4	this	this	DET
cana-3289	48	5	paper	paper	NOUN
cana-3289	48	6	we	we	PRON
cana-3289	48	7	introduce	introduce	VERB
cana-3289	48	8	the	the	DET
cana-3289	48	9	theory	theory	NOUN
cana-3289	48	10	of	of	ADP
cana-3289	48	11	bipolar	bipolar	ADJ
cana-3289	48	12	fuzzy	fuzzy	ADJ
cana-3289	48	13	prime	prime	NOUN
cana-3289	48	14	ideals(bfpi	ideals(bfpi	PROPN
cana-3289	48	15	)	)	PUNCT
cana-3289	48	16	and	and	CCONJ
cana-3289	48	17	their	their	PRON
cana-3289	48	18	characterization	characterization	NOUN
cana-3289	48	19	using	use	VERB
cana-3289	48	20	level	level	NOUN
cana-3289	48	21	sets	set	NOUN
cana-3289	48	22	and	and	CCONJ
cana-3289	48	23	homomorphism	homomorphism	NOUN
cana-3289	48	24	of	of	ADP
cana-3289	48	25	bfpi	bfpi	NOUN
cana-3289	48	26	of	of	ADP
cana-3289	48	27	a	a	DET
cana-3289	48	28	lattice	lattice	NOUN
cana-3289	48	29	.	.	PUNCT
cana-3289	49	1	2	2	X
cana-3289	49	2	.	.	NUM
cana-3289	49	3	preliminaries	preliminary	NOUN
cana-3289	49	4	definition	definition	NOUN
cana-3289	49	5	2.1[7	2.1[7	NOUN
cana-3289	49	6	]	]	X
cana-3289	49	7	:	:	PUNCT
cana-3289	49	8	“	"	PUNCT
cana-3289	49	9	a	a	DET
cana-3289	49	10	poset	poset	NOUN
cana-3289	49	11	(	(	PUNCT
cana-3289	49	12	𝔏	𝔏	PROPN
cana-3289	49	13	,	,	PUNCT
cana-3289	49	14	≤	≤	NUM
cana-3289	49	15	)	)	PUNCT
cana-3289	49	16	is	be	AUX
cana-3289	49	17	called	call	VERB
cana-3289	49	18	a	a	DET
cana-3289	49	19	lattice	lattice	NOUN
cana-3289	49	20	if	if	SCONJ
cana-3289	49	21	sup	sup	NOUN
cana-3289	49	22	{	{	PUNCT
cana-3289	49	23	𝑝	𝑝	NOUN
cana-3289	49	24	,	,	PUNCT
cana-3289	49	25	𝑞	𝑞	X
cana-3289	49	26	}	}	PUNCT
cana-3289	49	27	(	(	PUNCT
cana-3289	49	28	also	also	ADV
cana-3289	49	29	denoted	denote	VERB
cana-3289	49	30	by	by	ADP
cana-3289	49	31	𝑝	𝑝	PROPN
cana-3289	49	32	∨	∨	NUM
cana-3289	49	33	𝑞	𝑞	PROPN
cana-3289	49	34	)	)	PUNCT
cana-3289	49	35	and	and	CCONJ
cana-3289	49	36	inf{𝑝	inf{𝑝	PROPN
cana-3289	49	37	,	,	PUNCT
cana-3289	49	38	𝑞	𝑞	AUX
cana-3289	49	39	}	}	PUNCT
cana-3289	49	40	(	(	PUNCT
cana-3289	49	41	also	also	ADV
cana-3289	49	42	denoted	denote	VERB
cana-3289	49	43	by	by	ADP
cana-3289	49	44	𝑝	𝑝	PROPN
cana-3289	49	45	∧	∧	PROPN
cana-3289	49	46	𝑞	𝑞	NOUN
cana-3289	49	47	)	)	PUNCT
cana-3289	49	48	exists	exist	VERB
cana-3289	49	49	for	for	ADP
cana-3289	49	50	every	every	DET
cana-3289	49	51	pair	pair	NOUN
cana-3289	49	52	of	of	ADP
cana-3289	49	53	elements	element	NOUN
cana-3289	49	54	𝑝	𝑝	PROPN
cana-3289	49	55	,	,	PUNCT
cana-3289	49	56	𝑞	𝑞	NOUN
cana-3289	49	57	in	in	ADP
cana-3289	49	58	𝔏.	𝔏.	PROPN
cana-3289	49	59	”	"	PUNCT
cana-3289	49	60	definition	definition	NOUN
cana-3289	49	61	2.2[1	2.2[1	NUM
cana-3289	49	62	]	]	X
cana-3289	49	63	:	:	PUNCT
cana-3289	49	64	“	"	PUNCT
cana-3289	49	65	let	let	VERB
cana-3289	49	66	ϝ	ϝ	NOUN
cana-3289	49	67	be	be	AUX
cana-3289	49	68	any	any	DET
cana-3289	49	69	non	non	ADJ
cana-3289	49	70	-	-	ADJ
cana-3289	49	71	empty	empty	ADJ
cana-3289	49	72	set	set	NOUN
cana-3289	49	73	.	.	PUNCT
cana-3289	50	1	a	a	DET
cana-3289	50	2	mapping	mapping	NOUN
cana-3289	50	3	𝜓	𝜓	NOUN
cana-3289	50	4	:	:	PUNCT
cana-3289	50	5	ϝ	ϝ	NOUN
cana-3289	50	6	→	→	SYM
cana-3289	50	7	[	[	X
cana-3289	50	8	0,1	0,1	NUM
cana-3289	50	9	]	]	PUNCT
cana-3289	50	10	is	be	AUX
cana-3289	50	11	called	call	VERB
cana-3289	50	12	a	a	DET
cana-3289	50	13	fuzzy	fuzzy	ADJ
cana-3289	50	14	subset	subset	NOUN
cana-3289	50	15	of	of	ADP
cana-3289	50	16	ϝ.	ϝ.	NOUN
cana-3289	50	17	”	"	PUNCT
cana-3289	50	18	definition	definition	NOUN
cana-3289	50	19	2.3[1	2.3[1	NUM
cana-3289	50	20	]	]	PUNCT
cana-3289	50	21	:	:	PUNCT
cana-3289	50	22	“	"	PUNCT
cana-3289	50	23	let	let	VERB
cana-3289	50	24	𝜓	𝜓	NOUN
cana-3289	50	25	:	:	PUNCT
cana-3289	50	26	ϝ	ϝ	NOUN
cana-3289	50	27	→	→	SYM
cana-3289	50	28	[	[	X
cana-3289	50	29	0,1	0,1	NUM
cana-3289	50	30	]	]	PUNCT
cana-3289	50	31	be	be	VERB
cana-3289	50	32	any	any	DET
cana-3289	50	33	fs	fs	PROPN
cana-3289	50	34	.	.	PUNCT
cana-3289	51	1	then	then	ADV
cana-3289	51	2	the	the	DET
cana-3289	51	3	set	set	NOUN
cana-3289	51	4	{	{	PUNCT
cana-3289	51	5	𝜓(𝑝)/𝑝	𝜓(𝑝)/𝑝	PROPN
cana-3289	51	6	∈	∈	PROPN
cana-3289	51	7	ϝ	ϝ	NOUN
cana-3289	51	8	}	}	PUNCT
cana-3289	51	9	is	be	AUX
cana-3289	51	10	called	call	VERB
cana-3289	51	11	the	the	DET
cana-3289	51	12	image	image	NOUN
cana-3289	51	13	of	of	ADP
cana-3289	51	14	𝜓	𝜓	NOUN
cana-3289	51	15	and	and	CCONJ
cana-3289	51	16	is	be	AUX
cana-3289	51	17	denoted	denote	VERB
cana-3289	51	18	by	by	ADP
cana-3289	51	19	im(𝜓	im(𝜓	NOUN
cana-3289	51	20	)	)	PUNCT
cana-3289	51	21	.	.	PUNCT
cana-3289	52	1	for	for	ADP
cana-3289	52	2	𝑡	𝑡	PROPN
cana-3289	52	3	∈	∈	PROPN
cana-3289	52	4	[	[	X
cana-3289	52	5	0,1	0,1	NUM
cana-3289	52	6	]	]	PUNCT
cana-3289	52	7	,	,	PUNCT
cana-3289	52	8	𝜓𝑡	𝜓𝑡	NOUN
cana-3289	52	9	=	=	VERB
cana-3289	52	10	{	{	PUNCT
cana-3289	52	11	𝑝	𝑝	PROPN
cana-3289	52	12	∈	∈	PROPN
cana-3289	52	13	ϝ/𝜓(𝑝	ϝ/𝜓(𝑝	NOUN
cana-3289	52	14	)	)	PUNCT
cana-3289	52	15	≥	≥	NOUN
cana-3289	52	16	𝑡	𝑡	PROPN
cana-3289	52	17	}	}	PUNCT
cana-3289	52	18	is	be	AUX
cana-3289	52	19	called	call	VERB
cana-3289	52	20	a	a	DET
cana-3289	52	21	level	level	NOUN
cana-3289	52	22	subset	subset	NOUN
cana-3289	52	23	of	of	ADP
cana-3289	52	24	𝜓.	𝜓.	PROPN
cana-3289	52	25	”	"	PUNCT
cana-3289	52	26	definition	definition	NOUN
cana-3289	52	27	2.4[2	2.4[2	NUM
cana-3289	52	28	]	]	X
cana-3289	52	29	:	:	PUNCT
cana-3289	52	30	“	"	PUNCT
cana-3289	52	31	a	a	DET
cana-3289	52	32	vague	vague	ADJ
cana-3289	52	33	set	set	NOUN
cana-3289	52	34	𝜅	𝜅	NOUN
cana-3289	52	35	in	in	ADP
cana-3289	52	36	the	the	DET
cana-3289	52	37	universe	universe	NOUN
cana-3289	52	38	of	of	ADP
cana-3289	52	39	discourse	discourse	NOUN
cana-3289	52	40	ϝ	ϝ	NOUN
cana-3289	52	41	is	be	AUX
cana-3289	52	42	characterized	characterize	VERB
cana-3289	52	43	by	by	ADP
cana-3289	52	44	two	two	NUM
cana-3289	52	45	𝑀ship	𝑀ship	NOUN
cana-3289	52	46	functions	function	NOUN
cana-3289	52	47	given	give	VERB
cana-3289	52	48	by	by	ADP
cana-3289	52	49	(	(	PUNCT
cana-3289	52	50	i)a	i)a	ADJ
cana-3289	52	51	truth	truth	NOUN
cana-3289	52	52	𝑀ship	𝑀ship	NOUN
cana-3289	52	53	function	function	NOUN
cana-3289	52	54	𝑡𝜅	𝑡𝜅	ADP
cana-3289	52	55	:	:	PUNCT
cana-3289	52	56	ϝ	ϝ	NOUN
cana-3289	52	57	→	→	SYM
cana-3289	52	58	[	[	X
cana-3289	52	59	0,1	0,1	NUM
cana-3289	52	60	]	]	PUNCT
cana-3289	52	61	and	and	CCONJ
cana-3289	52	62	(	(	PUNCT
cana-3289	52	63	ii	ii	NOUN
cana-3289	52	64	)	)	PUNCT
cana-3289	52	65	a	a	DET
cana-3289	52	66	false	false	ADJ
cana-3289	52	67	𝑀ship	𝑀ship	NOUN
cana-3289	52	68	function	function	NOUN
cana-3289	52	69	𝑓𝜅	𝑓𝜅	NOUN
cana-3289	52	70	:	:	PUNCT
cana-3289	52	71	ϝ	ϝ	NOUN
cana-3289	52	72	→	→	SYM
cana-3289	52	73	[	[	X
cana-3289	52	74	0,1	0,1	NUM
cana-3289	52	75	]	]	PUNCT
cana-3289	52	76	,	,	PUNCT
cana-3289	52	77	where	where	SCONJ
cana-3289	52	78	𝑡𝜅(𝑝	𝑡𝜅(𝑝	NUM
cana-3289	52	79	)	)	PUNCT
cana-3289	52	80	is	be	AUX
cana-3289	52	81	a	a	DET
cana-3289	52	82	lower	low	ADJ
cana-3289	52	83	bound	bind	VERB
cana-3289	52	84	of	of	ADP
cana-3289	52	85	the	the	DET
cana-3289	52	86	grade	grade	NOUN
cana-3289	52	87	of	of	ADP
cana-3289	52	88	𝑀ship	𝑀ship	PROPN
cana-3289	52	89	of	of	ADP
cana-3289	52	90	𝑝	𝑝	PROPN
cana-3289	52	91	derived	derive	VERB
cana-3289	52	92	from	from	ADP
cana-3289	52	93	the	the	DET
cana-3289	52	94	evidence	evidence	NOUN
cana-3289	52	95	for	for	ADP
cana-3289	52	96	𝑝	𝑝	NOUN
cana-3289	52	97	and	and	CCONJ
cana-3289	52	98	𝑓𝜅(𝑝	𝑓𝜅(𝑝	NUM
cana-3289	52	99	)	)	PUNCT
cana-3289	52	100	is	be	AUX
cana-3289	52	101	a	a	DET
cana-3289	52	102	lower	lower	ADV
cana-3289	52	103	bound	bind	VERB
cana-3289	52	104	on	on	ADP
cana-3289	52	105	the	the	DET
cana-3289	52	106	negation	negation	NOUN
cana-3289	52	107	of	of	ADP
cana-3289	52	108	𝑝	𝑝	NOUN
cana-3289	52	109	derived	derive	VERB
cana-3289	52	110	from	from	ADP
cana-3289	52	111	the	the	DET
cana-3289	52	112	evidence	evidence	NOUN
cana-3289	52	113	against	against	ADP
cana-3289	52	114	𝑝	𝑝	PROPN
cana-3289	52	115	,	,	PUNCT
cana-3289	52	116	with	with	ADP
cana-3289	52	117	𝑡𝜅(𝑝	𝑡𝜅(𝑝	NUM
cana-3289	52	118	)	)	PUNCT
cana-3289	53	1	+	+	CCONJ
cana-3289	53	2	𝑓𝜅(𝑝	𝑓𝜅(𝑝	X
cana-3289	53	3	)	)	PUNCT
cana-3289	53	4	≤	≤	NOUN
cana-3289	53	5	1	1	NUM
cana-3289	53	6	"	"	PUNCT
cana-3289	53	7	.	.	PUNCT
cana-3289	54	1	communications	communication	NOUN
cana-3289	54	2	on	on	ADP
cana-3289	54	3	applied	apply	VERB
cana-3289	54	4	nonlinear	nonlinear	ADJ
cana-3289	54	5	analysis	analysis	NOUN
cana-3289	54	6	issn	issn	NOUN
cana-3289	54	7	:	:	PUNCT
cana-3289	54	8	1074	1074	NUM
cana-3289	54	9	-	-	PUNCT
cana-3289	54	10	133x	133x	NUM
cana-3289	54	11	vol	vol	NOUN
cana-3289	54	12	32	32	NUM
cana-3289	54	13	no	no	NOUN
cana-3289	54	14	.	.	PUNCT
cana-3289	55	1	6s	6s	NUM
cana-3289	55	2	(	(	PUNCT
cana-3289	55	3	2025	2025	NUM
cana-3289	55	4	)	)	PUNCT
cana-3289	55	5	243	243	NUM
cana-3289	55	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3289	55	7	we	we	PRON
cana-3289	55	8	give	give	VERB
cana-3289	55	9	below	below	ADP
cana-3289	55	10	a	a	DET
cana-3289	55	11	formation	formation	NOUN
cana-3289	55	12	of	of	ADP
cana-3289	55	13	the	the	DET
cana-3289	55	14	definition	definition	NOUN
cana-3289	55	15	of	of	ADP
cana-3289	55	16	vague	vague	ADJ
cana-3289	55	17	set	set	NOUN
cana-3289	55	18	in	in	ADP
cana-3289	55	19	the	the	DET
cana-3289	55	20	following	following	ADJ
cana-3289	55	21	way	way	NOUN
cana-3289	55	22	,	,	PUNCT
cana-3289	55	23	which	which	PRON
cana-3289	55	24	makes	make	VERB
cana-3289	55	25	atanassov	atanassov	ADJ
cana-3289	55	26	,	,	PUNCT
cana-3289	55	27	k.t.s	k.t.s	NOUN
cana-3289	55	28	intuitionistic	intuitionistic	ADJ
cana-3289	55	29	fuzzy	fuzzy	ADJ
cana-3289	55	30	sets	set	NOUN
cana-3289	55	31	and	and	CCONJ
cana-3289	55	32	gau	gau	NOUN
cana-3289	55	33	,	,	PUNCT
cana-3289	55	34	w.l	w.l	PROPN
cana-3289	55	35	.	.	PROPN
cana-3289	55	36	and	and	CCONJ
cana-3289	55	37	buehrer	buehrer	NOUN
cana-3289	55	38	,	,	PUNCT
cana-3289	55	39	d.j.[2	d.j.[2	PROPN
cana-3289	55	40	]	]	PUNCT
cana-3289	55	41	vague	vague	ADJ
cana-3289	55	42	sets	set	NOUN
cana-3289	55	43	in	in	ADP
cana-3289	55	44	a	a	DET
cana-3289	55	45	mathematically	mathematically	ADV
cana-3289	55	46	equivalent	equivalent	ADJ
cana-3289	55	47	form	form	NOUN
cana-3289	55	48	.	.	PUNCT
cana-3289	56	1	definition	definition	NOUN
cana-3289	56	2	2.5[2	2.5[2	NUM
cana-3289	56	3	]	]	PUNCT
cana-3289	56	4	:	:	PUNCT
cana-3289	56	5	“	"	PUNCT
cana-3289	56	6	let	let	VERB
cana-3289	56	7	𝜅	𝜅	PART
cana-3289	56	8	be	be	AUX
cana-3289	56	9	a	a	DET
cana-3289	56	10	vague	vague	ADJ
cana-3289	56	11	set	set	NOUN
cana-3289	56	12	of	of	ADP
cana-3289	56	13	a	a	DET
cana-3289	56	14	universe	universe	ADJ
cana-3289	56	15	ϝ	ϝ	NOUN
cana-3289	56	16	with	with	ADP
cana-3289	56	17	true	true	ADJ
cana-3289	56	18	𝑀ship	𝑀ship	NOUN
cana-3289	56	19	function	function	NOUN
cana-3289	56	20	𝑡𝜅	𝑡𝜅	ADP
cana-3289	56	21	and	and	CCONJ
cana-3289	56	22	false	false	ADJ
cana-3289	56	23	𝑀ship	𝑀ship	NOUN
cana-3289	56	24	function	function	NOUN
cana-3289	56	25	𝑓𝜅.	𝑓𝜅.	PROPN
cana-3289	56	26	for	for	ADP
cana-3289	56	27	𝛼	𝛼	PROPN
cana-3289	56	28	,	,	PUNCT
cana-3289	56	29	υ	υ	PROPN
cana-3289	56	30	∈	∈	PROPN
cana-3289	57	1	[	[	X
cana-3289	57	2	0,1	0,1	NUM
cana-3289	57	3	]	]	PUNCT
cana-3289	57	4	with	with	ADP
cana-3289	57	5	𝛼	𝛼	PROPN
cana-3289	57	6	≤	≤	NUM
cana-3289	57	7	υ	υ	NOUN
cana-3289	57	8	,	,	PUNCT
cana-3289	57	9	the	the	DET
cana-3289	57	10	(	(	PUNCT
cana-3289	57	11	𝛼	𝛼	PROPN
cana-3289	57	12	,	,	PUNCT
cana-3289	57	13	υ	υ	NOUN
cana-3289	57	14	)	)	PUNCT
cana-3289	57	15	cut	cut	ADJ
cana-3289	57	16	or	or	CCONJ
cana-3289	57	17	vague	vague	ADJ
cana-3289	57	18	cut	cut	NOUN
cana-3289	57	19	of	of	ADP
cana-3289	57	20	a	a	DET
cana-3289	57	21	vague	vague	ADJ
cana-3289	57	22	set	set	NOUN
cana-3289	57	23	𝜅	𝜅	PUNCT
cana-3289	57	24	is	be	AUX
cana-3289	57	25	the	the	DET
cana-3289	57	26	crisp	crisp	ADJ
cana-3289	57	27	subset	subset	NOUN
cana-3289	57	28	of	of	ADP
cana-3289	57	29	ϝ	ϝ	NOUN
cana-3289	57	30	is	be	AUX
cana-3289	57	31	given	give	VERB
cana-3289	57	32	by	by	ADP
cana-3289	57	33	𝜅(𝛼,υ	𝜅(𝛼,υ	NOUN
cana-3289	57	34	)	)	PUNCT
cana-3289	57	35	=	=	PRON
cana-3289	57	36	{	{	PUNCT
cana-3289	57	37	𝑝	𝑝	PROPN
cana-3289	57	38	∈	∈	PROPN
cana-3289	57	39	ϝ/𝑉𝜅(𝑝	ϝ/𝑉𝜅(𝑝	NOUN
cana-3289	57	40	)	)	PUNCT
cana-3289	57	41	≥	≥	PRON
cana-3289	58	1	[	[	X
cana-3289	58	2	𝛼	𝛼	X
cana-3289	58	3	,	,	PUNCT
cana-3289	58	4	υ	υ	NOUN
cana-3289	58	5	]	]	X
cana-3289	58	6	}	}	PUNCT
cana-3289	58	7	i.e.	i.e.	ADV
cana-3289	58	8	,	,	PUNCT
cana-3289	58	9	𝜅(𝛼,υ	𝜅(𝛼,υ	NUM
cana-3289	58	10	)	)	PUNCT
cana-3289	58	11	=	=	PRON
cana-3289	58	12	{	{	PUNCT
cana-3289	58	13	𝑝	𝑝	PROPN
cana-3289	58	14	∈	∈	PROPN
cana-3289	58	15	ϝ/𝑡𝜅(𝑝	ϝ/𝑡𝜅(𝑝	PROPN
cana-3289	58	16	)	)	PUNCT
cana-3289	58	17	≥	≥	NOUN
cana-3289	58	18	𝛼	𝛼	NOUN
cana-3289	58	19	and	and	CCONJ
cana-3289	58	20	1	1	NUM
cana-3289	58	21	−	−	NOUN
cana-3289	58	22	𝑓𝜅(𝑝	𝑓𝜅(𝑝	NUM
cana-3289	58	23	)	)	PUNCT
cana-3289	58	24	≥	≥	NOUN
cana-3289	58	25	υ	υ	NOUN
cana-3289	58	26	}	}	PUNCT
cana-3289	58	27	.	.	PUNCT
cana-3289	58	28	”	"	PUNCT
cana-3289	59	1	definition	definition	NOUN
cana-3289	59	2	2.6[2	2.6[2	NUM
cana-3289	59	3	]	]	PUNCT
cana-3289	59	4	:	:	PUNCT
cana-3289	59	5	“	"	PUNCT
cana-3289	59	6	the	the	DET
cana-3289	59	7	𝛼-cut	𝛼-cut	NOUN
cana-3289	59	8	,	,	PUNCT
cana-3289	59	9	𝜅𝛼	𝜅𝛼	ADP
cana-3289	59	10	of	of	ADP
cana-3289	59	11	the	the	DET
cana-3289	59	12	vague	vague	ADJ
cana-3289	59	13	set	set	NOUN
cana-3289	59	14	𝜅	𝜅	PUNCT
cana-3289	59	15	is	be	AUX
cana-3289	59	16	the	the	DET
cana-3289	59	17	(	(	PUNCT
cana-3289	59	18	𝛼	𝛼	PROPN
cana-3289	59	19	,	,	PUNCT
cana-3289	59	20	𝛼	𝛼	NOUN
cana-3289	59	21	)	)	PUNCT
cana-3289	59	22	-cut	-cut	VERB
cana-3289	59	23	of	of	ADP
cana-3289	59	24	𝜅	𝜅	PROPN
cana-3289	59	25	and	and	CCONJ
cana-3289	59	26	hence	hence	ADV
cana-3289	59	27	given	give	VERB
cana-3289	59	28	by	by	ADP
cana-3289	59	29	𝜅𝛼	𝜅𝛼	ADP
cana-3289	59	30	=	=	VERB
cana-3289	59	31	{	{	PUNCT
cana-3289	59	32	𝑝	𝑝	PROPN
cana-3289	59	33	∈	∈	PROPN
cana-3289	59	34	ϝ/𝑡𝜅(𝑝	ϝ/𝑡𝜅(𝑝	PROPN
cana-3289	59	35	)	)	PUNCT
cana-3289	59	36	≥	≥	X
cana-3289	59	37	𝛼	𝛼	NOUN
cana-3289	59	38	}	}	PUNCT
cana-3289	59	39	.	.	PUNCT
cana-3289	59	40	”	"	PUNCT
cana-3289	60	1	definition	definition	NOUN
cana-3289	60	2	2.7[6	2.7[6	NUM
cana-3289	60	3	]	]	X
cana-3289	60	4	:	:	PUNCT
cana-3289	60	5	“	"	PUNCT
cana-3289	60	6	suppose	suppose	VERB
cana-3289	60	7	ϝ	ϝ	PRON
cana-3289	60	8	be	be	AUX
cana-3289	60	9	a	a	DET
cana-3289	60	10	universal	universal	ADJ
cana-3289	60	11	set	set	NOUN
cana-3289	60	12	.	.	PUNCT
cana-3289	61	1	a	a	DET
cana-3289	61	2	(	(	PUNCT
cana-3289	61	3	𝐵𝐹𝑆	𝐵𝐹𝑆	NOUN
cana-3289	61	4	)	)	PUNCT
cana-3289	61	5	bipolar	bipolar	ADJ
cana-3289	61	6	fuzzy	fuzzy	ADJ
cana-3289	61	7	set	set	NOUN
cana-3289	61	8	𝔹	𝔹	NOUN
cana-3289	61	9	in	in	ADP
cana-3289	61	10	ϝ	ϝ	NOUN
cana-3289	61	11	is	be	AUX
cana-3289	61	12	an	an	DET
cana-3289	61	13	object	object	NOUN
cana-3289	61	14	having	have	VERB
cana-3289	61	15	the	the	DET
cana-3289	61	16	form	form	NOUN
cana-3289	61	17	𝔹	𝔹	NOUN
cana-3289	61	18	=	=	SYM
cana-3289	61	19	{	{	PUNCT
cana-3289	61	20	<	<	X
cana-3289	61	21	ℏ	ℏ	PROPN
cana-3289	61	22	,	,	PUNCT
cana-3289	61	23	𝔹𝑃(ℏ	𝔹𝑃(ℏ	NOUN
cana-3289	61	24	)	)	PUNCT
cana-3289	61	25	,	,	PUNCT
cana-3289	61	26	𝔹𝑁(ℏ	𝔹𝑁(ℏ	NOUN
cana-3289	61	27	)	)	PUNCT
cana-3289	61	28	>	>	PUNCT
cana-3289	61	29	/ℏ	/ℏ	PUNCT
cana-3289	62	1	∈	∈	PROPN
cana-3289	62	2	ϝ	ϝ	NOUN
cana-3289	62	3	}	}	PUNCT
cana-3289	62	4	where	where	SCONJ
cana-3289	62	5	𝔹𝑃	𝔹𝑃	NOUN
cana-3289	62	6	:	:	PUNCT
cana-3289	62	7	ϝ	ϝ	NOUN
cana-3289	62	8	→	→	SYM
cana-3289	62	9	[	[	X
cana-3289	62	10	0,1	0,1	NUM
cana-3289	62	11	]	]	PUNCT
cana-3289	62	12	and	and	CCONJ
cana-3289	62	13	𝔹𝑁	𝔹𝑁	PROPN
cana-3289	62	14	:	:	PUNCT
cana-3289	62	15	ϝ	ϝ	NOUN
cana-3289	62	16	→	→	SYM
cana-3289	62	17	[	[	X
cana-3289	62	18	−1,0	−1,0	X
cana-3289	62	19	]	]	X
cana-3289	62	20	are	be	AUX
cana-3289	62	21	a	a	DET
cana-3289	62	22	positive	positive	ADJ
cana-3289	62	23	and	and	CCONJ
cana-3289	62	24	negative	negative	ADJ
cana-3289	62	25	𝑀ship	𝑀ship	NOUN
cana-3289	62	26	functions	function	NOUN
cana-3289	62	27	,	,	PUNCT
cana-3289	62	28	respectively	respectively	ADV
cana-3289	62	29	.	.	PUNCT
cana-3289	62	30	”	"	PUNCT
cana-3289	63	1	definition	definition	NOUN
cana-3289	63	2	2.8[6	2.8[6	NUM
cana-3289	63	3	]	]	PUNCT
cana-3289	63	4	:	:	PUNCT
cana-3289	63	5	“	"	PUNCT
cana-3289	63	6	let	let	VERB
cana-3289	63	7	ϝ	ϝ	NOUN
cana-3289	63	8	be	be	AUX
cana-3289	63	9	a	a	DET
cana-3289	63	10	nonempty	nonempty	ADJ
cana-3289	63	11	set	set	NOUN
cana-3289	63	12	,	,	PUNCT
cana-3289	63	13	and	and	CCONJ
cana-3289	63	14	let	let	VERB
cana-3289	63	15	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	63	16	,	,	PUNCT
cana-3289	63	17	𝔹𝜔	𝔹𝜔	PROPN
cana-3289	63	18	∈	∈	PROPN
cana-3289	63	19	𝐵𝑃𝐹𝑆(ϝ	𝐵𝑃𝐹𝑆(ϝ	NOUN
cana-3289	63	20	)	)	PUNCT
cana-3289	63	21	.	.	PUNCT
cana-3289	64	1	(	(	PUNCT
cana-3289	64	2	i	i	NOUN
cana-3289	64	3	)	)	PUNCT
cana-3289	65	1	𝔹𝜗	𝔹𝜗	NOUN
cana-3289	65	2	is	be	AUX
cana-3289	65	3	a	a	DET
cana-3289	65	4	subset	subset	NOUN
cana-3289	65	5	of	of	ADP
cana-3289	65	6	𝔹𝜔	𝔹𝜔	PROPN
cana-3289	65	7	,	,	PUNCT
cana-3289	65	8	denoted	denote	VERB
cana-3289	65	9	by	by	ADP
cana-3289	65	10	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	65	11	⊆	⊆	NUM
cana-3289	65	12	𝔹𝜔	𝔹𝜔	PROPN
cana-3289	65	13	,	,	PUNCT
cana-3289	65	14	if	if	SCONJ
cana-3289	65	15	for	for	ADP
cana-3289	65	16	each	each	DET
cana-3289	65	17	ℏ	ℏ	PROPN
cana-3289	65	18	∈	∈	PROPN
cana-3289	65	19	ϝ	ϝ	NOUN
cana-3289	65	20	,	,	PUNCT
cana-3289	66	1	𝔹𝜗	𝔹𝜗	NOUN
cana-3289	66	2	𝑃(ℏ	𝑃(ℏ	X
cana-3289	66	3	)	)	PUNCT
cana-3289	66	4	≤	≤	PUNCT
cana-3289	67	1	𝔹𝜔	𝔹𝜔	VERB
cana-3289	67	2	𝑃	𝑃	PROPN
cana-3289	67	3	(	(	PUNCT
cana-3289	67	4	ℏ	ℏ	NOUN
cana-3289	67	5	)	)	PUNCT
cana-3289	67	6	and	and	CCONJ
cana-3289	68	1	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	68	2	𝑁(ℏ	𝑁(ℏ	NOUN
cana-3289	68	3	)	)	PUNCT
cana-3289	68	4	≥	≥	NOUN
cana-3289	68	5	𝔹𝜔	𝔹𝜔	PROPN
cana-3289	68	6	𝑁	𝑁	PROPN
cana-3289	68	7	(	(	PUNCT
cana-3289	68	8	ℏ	ℏ	PROPN
cana-3289	68	9	)	)	PUNCT
cana-3289	68	10	.	.	PUNCT
cana-3289	68	11	(	(	PUNCT
cana-3289	68	12	ii	ii	X
cana-3289	68	13	)	)	PUNCT
cana-3289	68	14	the	the	DET
cana-3289	68	15	complement	complement	NOUN
cana-3289	68	16	of	of	ADP
cana-3289	68	17	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	68	18	,	,	PUNCT
cana-3289	68	19	denoted	denote	VERB
cana-3289	68	20	by	by	ADP
cana-3289	68	21	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	68	22	𝑐	𝑐	NOUN
cana-3289	68	23	=	=	PUNCT
cana-3289	68	24	(	(	PUNCT
cana-3289	68	25	(	(	PUNCT
cana-3289	68	26	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	68	27	𝑐	𝑐	NOUN
cana-3289	68	28	)	)	PUNCT
cana-3289	68	29	𝑁	𝑁	PROPN
cana-3289	68	30	,	,	PUNCT
cana-3289	68	31	(	(	PUNCT
cana-3289	68	32	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	68	33	𝑐	𝑐	NOUN
cana-3289	68	34	)	)	PUNCT
cana-3289	68	35	𝑃	𝑃	NOUN
cana-3289	68	36	)	)	PUNCT
cana-3289	68	37	,	,	PUNCT
cana-3289	68	38	is	be	AUX
cana-3289	68	39	a	a	DET
cana-3289	68	40	𝐵𝐹𝑆	𝐵𝐹𝑆	NOUN
cana-3289	68	41	in	in	ADP
cana-3289	68	42	ϝ	ϝ	NOUN
cana-3289	68	43	defined	define	VERB
cana-3289	68	44	as	as	ADP
cana-3289	68	45	:	:	PUNCT
cana-3289	68	46	for	for	ADP
cana-3289	68	47	each	each	DET
cana-3289	68	48	ℏ	ℏ	PROPN
cana-3289	68	49	∈	∈	PROPN
cana-3289	68	50	ϝ	ϝ	NOUN
cana-3289	68	51	,	,	PUNCT
cana-3289	68	52	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	68	53	𝑐	𝑐	PROPN
cana-3289	68	54	(	(	PUNCT
cana-3289	68	55	ℏ	ℏ	NOUN
cana-3289	68	56	)	)	PUNCT
cana-3289	68	57	=	=	SYM
cana-3289	68	58	(	(	PUNCT
cana-3289	68	59	−1	−1	NOUN
cana-3289	68	60	−	−	NOUN
cana-3289	69	1	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	69	2	𝑁(ℏ),1	𝑁(ℏ),1	NOUN
cana-3289	69	3	−	−	NOUN
cana-3289	69	4	𝔹𝜗	𝔹𝜗	NOUN
cana-3289	69	5	𝑃(ℏ	𝑃(ℏ	NOUN
cana-3289	69	6	)	)	PUNCT
cana-3289	69	7	)	)	PUNCT
cana-3289	69	8	,	,	PUNCT
cana-3289	69	9	i.e.	i.e.	X
cana-3289	69	10	,	,	PUNCT
cana-3289	69	11	(	(	PUNCT
cana-3289	69	12	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	69	13	𝑐	𝑐	NOUN
cana-3289	69	14	)	)	PUNCT
cana-3289	69	15	𝑃(ℏ	𝑃(ℏ	NUM
cana-3289	69	16	)	)	PUNCT
cana-3289	69	17	=	=	SYM
cana-3289	69	18	1	1	NUM
cana-3289	69	19	−	−	NOUN
cana-3289	70	1	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	70	2	𝑁(ℏ	𝑁(ℏ	NOUN
cana-3289	70	3	)	)	PUNCT
cana-3289	70	4	,	,	PUNCT
cana-3289	70	5	(	(	PUNCT
cana-3289	70	6	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	70	7	𝑐	𝑐	NOUN
cana-3289	70	8	)	)	PUNCT
cana-3289	70	9	𝑁(ℏ	𝑁(ℏ	X
cana-3289	70	10	)	)	PUNCT
cana-3289	70	11	=	=	SYM
cana-3289	70	12	−1	−1	NOUN
cana-3289	70	13	−	−	PROPN
cana-3289	71	1	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	71	2	𝑁(ℏ	𝑁(ℏ	NOUN
cana-3289	71	3	)	)	PUNCT
cana-3289	71	4	.	.	PUNCT
cana-3289	72	1	(	(	PUNCT
cana-3289	72	2	iii	iii	X
cana-3289	72	3	)	)	PUNCT
cana-3289	72	4	the	the	DET
cana-3289	72	5	intersection	intersection	NOUN
cana-3289	72	6	of	of	ADP
cana-3289	72	7	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	72	8	and	and	CCONJ
cana-3289	72	9	𝔹𝜔	𝔹𝜔	PROPN
cana-3289	72	10	,	,	PUNCT
cana-3289	72	11	denoted	denote	VERB
cana-3289	72	12	by	by	ADP
cana-3289	72	13	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	72	14	∩	∩	NOUN
cana-3289	72	15	𝔹𝜔	𝔹𝜔	NOUN
cana-3289	72	16	,	,	PUNCT
cana-3289	72	17	is	be	AUX
cana-3289	72	18	a	a	DET
cana-3289	72	19	𝐵𝐹𝑆	𝐵𝐹𝑆	NOUN
cana-3289	72	20	in	in	ADP
cana-3289	72	21	ϝ	ϝ	NOUN
cana-3289	72	22	defined	define	VERB
cana-3289	72	23	as	as	ADP
cana-3289	72	24	:	:	PUNCT
cana-3289	72	25	for	for	ADP
cana-3289	72	26	each	each	DET
cana-3289	72	27	ℏ	ℏ	PROPN
cana-3289	72	28	∈	∈	PROPN
cana-3289	72	29	ϝ	ϝ	NOUN
cana-3289	72	30	,	,	PUNCT
cana-3289	72	31	(	(	PUNCT
cana-3289	72	32	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	72	33	∩	∩	ADJ
cana-3289	72	34	𝔹𝜔)(ℏ	𝔹𝜔)(ℏ	NOUN
cana-3289	72	35	)	)	PUNCT
cana-3289	72	36	=	=	NOUN
cana-3289	73	1	(	(	PUNCT
cana-3289	73	2	𝔹𝜗	𝔹𝜗	ADP
cana-3289	73	3	𝑁(ℏ	𝑁(ℏ	NOUN
cana-3289	73	4	)	)	PUNCT
cana-3289	73	5	∨	∨	NOUN
cana-3289	74	1	𝔹𝜔	𝔹𝜔	PROPN
cana-3289	74	2	𝑁	𝑁	PROPN
cana-3289	74	3	(	(	PUNCT
cana-3289	74	4	ℏ	ℏ	NOUN
cana-3289	74	5	)	)	PUNCT
cana-3289	74	6	,	,	PUNCT
cana-3289	74	7	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	74	8	𝑃(ℏ	𝑃(ℏ	X
cana-3289	74	9	)	)	PUNCT
cana-3289	74	10	∧	∧	NOUN
cana-3289	74	11	𝔹𝜔	𝔹𝜔	PROPN
cana-3289	74	12	𝑃	𝑃	PROPN
cana-3289	74	13	(	(	PUNCT
cana-3289	74	14	ℏ	ℏ	NOUN
cana-3289	74	15	)	)	PUNCT
cana-3289	74	16	)	)	PUNCT
cana-3289	74	17	.	.	PUNCT
cana-3289	75	1	(	(	PUNCT
cana-3289	75	2	iv	iv	X
cana-3289	75	3	)	)	PUNCT
cana-3289	75	4	the	the	DET
cana-3289	75	5	union	union	NOUN
cana-3289	75	6	of	of	ADP
cana-3289	75	7	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	75	8	and	and	CCONJ
cana-3289	75	9	𝔹𝜔	𝔹𝜔	PROPN
cana-3289	75	10	,	,	PUNCT
cana-3289	75	11	denoted	denote	VERB
cana-3289	75	12	by	by	ADP
cana-3289	75	13	𝔹𝜗	𝔹𝜗	NOUN
cana-3289	75	14	∪	∪	ADJ
cana-3289	75	15	𝔹𝜔	𝔹𝜔	NOUN
cana-3289	75	16	,	,	PUNCT
cana-3289	75	17	is	be	AUX
cana-3289	75	18	a	a	DET
cana-3289	75	19	𝐵𝐹𝑆	𝐵𝐹𝑆	NOUN
cana-3289	75	20	in	in	ADP
cana-3289	75	21	ϝ	ϝ	NOUN
cana-3289	75	22	defined	define	VERB
cana-3289	75	23	as	as	ADP
cana-3289	75	24	:	:	PUNCT
cana-3289	75	25	for	for	ADP
cana-3289	75	26	each	each	DET
cana-3289	75	27	ℏ	ℏ	PROPN
cana-3289	75	28	∈	∈	PROPN
cana-3289	75	29	ϝ	ϝ	NOUN
cana-3289	75	30	,	,	PUNCT
cana-3289	76	1	(	(	PUNCT
cana-3289	76	2	𝔹𝜗	𝔹𝜗	ADP
cana-3289	76	3	∪	∪	ADJ
cana-3289	76	4	𝔹𝜔)(ℏ	𝔹𝜔)(ℏ	NOUN
cana-3289	76	5	)	)	PUNCT
cana-3289	76	6	=	=	NOUN
cana-3289	76	7	(	(	PUNCT
cana-3289	76	8	𝔹𝜗	𝔹𝜗	ADP
cana-3289	76	9	𝑁(ℏ	𝑁(ℏ	NOUN
cana-3289	76	10	)	)	PUNCT
cana-3289	76	11	∧	∧	NOUN
cana-3289	76	12	𝔹𝜔	𝔹𝜔	PROPN
cana-3289	76	13	𝑁	𝑁	PROPN
cana-3289	76	14	(	(	PUNCT
cana-3289	76	15	ℏ	ℏ	NOUN
cana-3289	76	16	)	)	PUNCT
cana-3289	76	17	,	,	PUNCT
cana-3289	76	18	𝔹𝜗	𝔹𝜗	PROPN
cana-3289	76	19	𝑃(ℏ	𝑃(ℏ	X
cana-3289	76	20	)	)	PUNCT
cana-3289	76	21	∨	∨	NUM
cana-3289	76	22	𝔹𝜔	𝔹𝜔	PROPN
cana-3289	76	23	𝑃	𝑃	PROPN
cana-3289	76	24	(	(	PUNCT
cana-3289	76	25	ℏ	ℏ	NOUN
cana-3289	76	26	)	)	PUNCT
cana-3289	76	27	)	)	PUNCT
cana-3289	76	28	.	.	PUNCT
cana-3289	76	29	"	"	PUNCT
cana-3289	77	1	definition:2.9[17	definition:2.9[17	PROPN
cana-3289	77	2	]	]	X
cana-3289	77	3	:	:	PUNCT
cana-3289	77	4	“	"	PUNCT
cana-3289	77	5	let	let	VERB
cana-3289	77	6	𝔅	𝔅	NOUN
cana-3289	77	7	=	=	NOUN
cana-3289	77	8	<	<	X
cana-3289	77	9	𝔅𝑃	𝔅𝑃	PROPN
cana-3289	77	10	,	,	PUNCT
cana-3289	77	11	𝔅𝑁	𝔅𝑁	PROPN
cana-3289	77	12	>	>	PUNCT
cana-3289	77	13	be	be	AUX
cana-3289	77	14	a	a	DET
cana-3289	77	15	bfs	bfs	NOUN
cana-3289	77	16	in	in	ADP
cana-3289	77	17	ϝ	ϝ	NOUN
cana-3289	77	18	and	and	CCONJ
cana-3289	77	19	(	(	PUNCT
cana-3289	77	20	𝛼	𝛼	NOUN
cana-3289	77	21	,	,	PUNCT
cana-3289	77	22	𝜔	𝜔	NOUN
cana-3289	77	23	)	)	PUNCT
cana-3289	77	24	∈	∈	NOUN
cana-3289	78	1	[	[	X
cana-3289	78	2	0,1	0,1	NUM
cana-3289	78	3	]	]	PUNCT
cana-3289	78	4	,	,	PUNCT
cana-3289	78	5	(	(	PUNCT
cana-3289	78	6	𝜃	𝜃	NOUN
cana-3289	78	7	,	,	PUNCT
cana-3289	78	8	𝜗	𝜗	NOUN
cana-3289	78	9	)	)	PUNCT
cana-3289	78	10	∈	∈	PROPN
cana-3289	79	1	[	[	X
cana-3289	79	2	∇,0	∇,0	X
cana-3289	79	3	]	]	X
cana-3289	79	4	×	×	NOUN
cana-3289	80	1	[	[	X
cana-3289	80	2	0	0	NUM
cana-3289	80	3	,	,	PUNCT
cana-3289	80	4	△	△	X
cana-3289	80	5	]	]	PUNCT
cana-3289	80	6	.	.	PUNCT
cana-3289	81	1	by	by	ADP
cana-3289	81	2	a	a	DET
cana-3289	81	3	bfmt	bfmt	NOUN
cana-3289	81	4	of	of	ADP
cana-3289	81	5	𝐵	𝐵	NOUN
cana-3289	81	6	=	=	PROPN
cana-3289	81	7	<	<	X
cana-3289	81	8	𝔅𝑁	𝔅𝑁	PROPN
cana-3289	81	9	,	,	PUNCT
cana-3289	81	10	𝔅𝑃	𝔅𝑃	PROPN
cana-3289	81	11	>	>	X
cana-3289	81	12	,	,	PUNCT
cana-3289	81	13	we	we	PRON
cana-3289	81	14	mean	mean	VERB
cana-3289	81	15	a	a	DET
cana-3289	81	16	bfs	bfs	NOUN
cana-3289	81	17	𝑀	𝑀	NOUN
cana-3289	81	18	=	=	PUNCT
cana-3289	81	19	{	{	PUNCT
cana-3289	81	20	<	<	X
cana-3289	81	21	𝑟	𝑟	NOUN
cana-3289	81	22	,	,	PUNCT
cana-3289	81	23	𝔅(𝜔,𝜗	𝔅(𝜔,𝜗	NOUN
cana-3289	81	24	)	)	PUNCT
cana-3289	81	25	𝑃	𝑃	PROPN
cana-3289	81	26	(	(	PUNCT
cana-3289	81	27	𝑟	𝑟	NOUN
cana-3289	81	28	)	)	PUNCT
cana-3289	81	29	,	,	PUNCT
cana-3289	81	30	𝔅(𝛼,𝜃	𝔅(𝛼,𝜃	NUM
cana-3289	81	31	)	)	PUNCT
cana-3289	81	32	𝑁	𝑁	PROPN
cana-3289	81	33	(	(	PUNCT
cana-3289	81	34	𝑟	𝑟	NOUN
cana-3289	81	35	)	)	PUNCT
cana-3289	81	36	>	>	PUNCT
cana-3289	81	37	:	:	PUNCT
cana-3289	81	38	𝑟	𝑟	X
cana-3289	81	39	∈	∈	NOUN
cana-3289	81	40	ϝ	ϝ	NOUN
cana-3289	81	41	}	}	PUNCT
cana-3289	81	42	or	or	CCONJ
cana-3289	81	43	simply	simply	ADV
cana-3289	81	44	as	as	ADP
cana-3289	81	45	𝑀	𝑀	PROPN
cana-3289	81	46	=	=	PUNCT
cana-3289	81	47	{	{	PUNCT
cana-3289	81	48	<	<	X
cana-3289	81	49	𝑟	𝑟	NOUN
cana-3289	81	50	,	,	PUNCT
cana-3289	81	51	𝔅𝑀	𝔅𝑀	NOUN
cana-3289	81	52	𝑃	𝑃	PROPN
cana-3289	81	53	(	(	PUNCT
cana-3289	81	54	𝑟	𝑟	NOUN
cana-3289	81	55	)	)	PUNCT
cana-3289	81	56	,	,	PUNCT
cana-3289	81	57	𝔅𝑀	𝔅𝑀	PROPN
cana-3289	81	58	𝑁	𝑁	PROPN
cana-3289	81	59	(	(	PUNCT
cana-3289	81	60	𝑟	𝑟	NOUN
cana-3289	81	61	)	)	PUNCT
cana-3289	81	62	>	>	PUNCT
cana-3289	81	63	:	:	PUNCT
cana-3289	81	64	𝑟	𝑟	X
cana-3289	81	65	∈	∈	NOUN
cana-3289	81	66	ϝ	ϝ	NOUN
cana-3289	81	67	}	}	PUNCT
cana-3289	81	68	,	,	PUNCT
cana-3289	81	69	where	where	SCONJ
cana-3289	81	70	𝔅𝑀	𝔅𝑀	NOUN
cana-3289	81	71	𝑃	𝑃	NOUN
cana-3289	81	72	(	(	PUNCT
cana-3289	81	73	𝑟	𝑟	NOUN
cana-3289	81	74	)	)	PUNCT
cana-3289	81	75	=	=	SYM
cana-3289	81	76	𝔅(𝜔,𝜗	𝔅(𝜔,𝜗	NOUN
cana-3289	81	77	)	)	PUNCT
cana-3289	81	78	𝑃	𝑃	NOUN
cana-3289	81	79	:	:	PUNCT
cana-3289	81	80	ϝ	ϝ	NOUN
cana-3289	81	81	→	→	SYM
cana-3289	81	82	[	[	X
cana-3289	81	83	0,1	0,1	NUM
cana-3289	81	84	]	]	PUNCT
cana-3289	81	85	and	and	CCONJ
cana-3289	81	86	𝔅𝑀	𝔅𝑀	NOUN
cana-3289	81	87	𝑁	𝑁	PROPN
cana-3289	81	88	=	=	PUNCT
cana-3289	81	89	𝔅(𝛼,𝜃	𝔅(𝛼,𝜃	ADJ
cana-3289	81	90	)	)	PUNCT
cana-3289	81	91	𝑁	𝑁	NOUN
cana-3289	81	92	:	:	PUNCT
cana-3289	81	93	ϝ	ϝ	NOUN
cana-3289	81	94	→	→	SYM
cana-3289	81	95	[	[	X
cana-3289	81	96	−1,0	−1,0	X
cana-3289	81	97	]	]	PUNCT
cana-3289	81	98	and	and	CCONJ
cana-3289	81	99	defined	define	VERB
cana-3289	81	100	by	by	ADP
cana-3289	81	101	𝔅𝑀	𝔅𝑀	PROPN
cana-3289	81	102	𝑃	𝑃	PROPN
cana-3289	81	103	(	(	PUNCT
cana-3289	81	104	𝑟	𝑟	NOUN
cana-3289	81	105	)	)	PUNCT
cana-3289	81	106	=	=	SYM
cana-3289	81	107	𝔅(𝜔,𝜗	𝔅(𝜔,𝜗	NOUN
cana-3289	81	108	)	)	PUNCT
cana-3289	81	109	𝑃	𝑃	PROPN
cana-3289	81	110	(	(	PUNCT
cana-3289	81	111	𝑟	𝑟	NOUN
cana-3289	81	112	)	)	PUNCT
cana-3289	81	113	=	=	PUNCT
cana-3289	81	114	𝜔𝔅𝑃(𝑟	𝜔𝔅𝑃(𝑟	ADJ
cana-3289	81	115	)	)	PUNCT
cana-3289	81	116	+	+	NUM
cana-3289	81	117	𝜗	𝜗	NOUN
cana-3289	81	118	for	for	ADP
cana-3289	81	119	all	all	PRON
cana-3289	81	120	𝑟	𝑟	PRON
cana-3289	81	121	∈	∈	NOUN
cana-3289	81	122	ϝ	ϝ	NOUN
cana-3289	81	123	and	and	CCONJ
cana-3289	81	124	𝔅𝑀	𝔅𝑀	PROPN
cana-3289	81	125	𝑁	𝑁	PROPN
cana-3289	81	126	(	(	PUNCT
cana-3289	81	127	𝑟	𝑟	NOUN
cana-3289	81	128	)	)	PUNCT
cana-3289	81	129	=	=	SYM
cana-3289	81	130	𝔅(𝛼,𝜃	𝔅(𝛼,𝜃	X
cana-3289	81	131	)	)	PUNCT
cana-3289	81	132	𝑁	𝑁	PROPN
cana-3289	81	133	(	(	PUNCT
cana-3289	81	134	𝑟	𝑟	NOUN
cana-3289	81	135	)	)	PUNCT
cana-3289	81	136	=	=	SYM
cana-3289	81	137	𝛼𝔅𝑁(𝑟	𝛼𝔅𝑁(𝑟	NUM
cana-3289	81	138	)	)	PUNCT
cana-3289	81	139	+	+	CCONJ
cana-3289	81	140	𝜃.	𝜃.	NOUN
cana-3289	81	141	”	"	PUNCT
cana-3289	81	142	definition:2.10[16	definition:2.10[16	PROPN
cana-3289	81	143	]	]	X
cana-3289	81	144	:	:	PUNCT
cana-3289	81	145	“	"	PUNCT
cana-3289	81	146	suppose	suppose	VERB
cana-3289	81	147	𝔅	𝔅	PROPN
cana-3289	81	148	=	=	SYM
cana-3289	81	149	(	(	PUNCT
cana-3289	81	150	𝔅𝑃	𝔅𝑃	PROPN
cana-3289	81	151	,	,	PUNCT
cana-3289	81	152	𝔅𝑁	𝔅𝑁	PROPN
cana-3289	81	153	)	)	PUNCT
cana-3289	81	154	is	be	AUX
cana-3289	81	155	a	a	DET
cana-3289	81	156	bfs	bfs	NOUN
cana-3289	81	157	in	in	ADP
cana-3289	81	158	𝔏	𝔏	PROPN
cana-3289	81	159	where	where	SCONJ
cana-3289	81	160	𝔅𝑃	𝔅𝑃	PROPN
cana-3289	81	161	:	:	PUNCT
cana-3289	81	162	ϝ	ϝ	NOUN
cana-3289	81	163	→	→	SYM
cana-3289	82	1	[	[	X
cana-3289	82	2	0,1	0,1	NUM
cana-3289	82	3	]	]	PUNCT
cana-3289	82	4	and	and	CCONJ
cana-3289	82	5	𝔅𝑁	𝔅𝑁	PROPN
cana-3289	82	6	:	:	PUNCT
cana-3289	82	7	ϝ	ϝ	NOUN
cana-3289	82	8	→	→	SYM
cana-3289	82	9	[	[	X
cana-3289	82	10	−1,0]then	−1,0]then	ADP
cana-3289	82	11	𝔅	𝔅	PROPN
cana-3289	82	12	is	be	AUX
cana-3289	82	13	known	know	VERB
cana-3289	82	14	to	to	PART
cana-3289	82	15	be	be	AUX
cana-3289	82	16	a	a	DET
cana-3289	82	17	bfl(bipolar	bfl(bipolar	ADJ
cana-3289	82	18	fuzzy	fuzzy	ADJ
cana-3289	82	19	sublattice	sublattice	NOUN
cana-3289	82	20	)	)	PUNCT
cana-3289	82	21	of	of	ADP
cana-3289	82	22	𝔏	𝔏	PROPN
cana-3289	82	23	when	when	SCONJ
cana-3289	82	24	the	the	DET
cana-3289	82	25	following	follow	VERB
cana-3289	82	26	conditions	condition	NOUN
cana-3289	82	27	are	be	AUX
cana-3289	82	28	fulfilled	fulfil	VERB
cana-3289	82	29	for	for	ADP
cana-3289	82	30	all	all	DET
cana-3289	82	31	ℏ	ℏ	PROPN
cana-3289	82	32	,	,	PUNCT
cana-3289	83	1	𝑚	𝑚	PROPN
cana-3289	83	2	∈	∈	PROPN
cana-3289	83	3	𝔏	𝔏	PROPN
cana-3289	83	4	,	,	PUNCT
cana-3289	83	5	(	(	PUNCT
cana-3289	83	6	𝑖)𝔅𝑃(ℏ⋁𝑚	𝑖)𝔅𝑃(ℏ⋁𝑚	PROPN
cana-3289	83	7	)	)	PUNCT
cana-3289	83	8	≥	≥	NOUN
cana-3289	83	9	min{𝔅𝑃(ℏ	min{𝔅𝑃(ℏ	PROPN
cana-3289	83	10	)	)	PUNCT
cana-3289	83	11	,	,	PUNCT
cana-3289	83	12	𝔅𝑃(𝑚	𝔅𝑃(𝑚	NOUN
cana-3289	83	13	)	)	PUNCT
cana-3289	83	14	}	}	PUNCT
cana-3289	83	15	,	,	PUNCT
cana-3289	83	16	(	(	PUNCT
cana-3289	83	17	ii	ii	NOUN
cana-3289	83	18	)	)	PUNCT
cana-3289	83	19	𝔅𝑃(ℏ	𝔅𝑃(ℏ	PROPN
cana-3289	83	20	∧	∧	PROPN
cana-3289	83	21	𝑚	𝑚	NOUN
cana-3289	83	22	)	)	PUNCT
cana-3289	83	23	≥	≥	NOUN
cana-3289	83	24	min{𝔅𝑃(ℏ	min{𝔅𝑃(ℏ	PROPN
cana-3289	83	25	)	)	PUNCT
cana-3289	83	26	,	,	PUNCT
cana-3289	83	27	𝔅𝑃(𝑚	𝔅𝑃(𝑚	NOUN
cana-3289	83	28	)	)	PUNCT
cana-3289	83	29	}	}	PUNCT
cana-3289	83	30	.	.	PUNCT
cana-3289	84	1	(	(	PUNCT
cana-3289	84	2	iii	iii	X
cana-3289	84	3	)	)	PUNCT
cana-3289	84	4	𝔅𝑁(ℏ⋁𝑚	𝔅𝑁(ℏ⋁𝑚	NOUN
cana-3289	84	5	)	)	PUNCT
cana-3289	84	6	≤	≤	NOUN
cana-3289	85	1	max{𝔅𝑁(ℏ	max{𝔅𝑁(ℏ	NOUN
cana-3289	85	2	)	)	PUNCT
cana-3289	85	3	,	,	PUNCT
cana-3289	85	4	𝔅𝑁(𝑚	𝔅𝑁(𝑚	PROPN
cana-3289	85	5	)	)	PUNCT
cana-3289	85	6	}	}	PUNCT
cana-3289	85	7	,	,	PUNCT
cana-3289	85	8	(	(	PUNCT
cana-3289	85	9	iv	iv	X
cana-3289	85	10	)	)	PUNCT
cana-3289	85	11	𝔅𝑁(ℏ	𝔅𝑁(ℏ	PROPN
cana-3289	85	12	∧	∧	PROPN
cana-3289	85	13	𝑚	𝑚	PROPN
cana-3289	85	14	)	)	PUNCT
cana-3289	85	15	≤	≤	NOUN
cana-3289	86	1	max{𝔅𝑁(ℏ	max{𝔅𝑁(ℏ	NOUN
cana-3289	86	2	)	)	PUNCT
cana-3289	86	3	,	,	PUNCT
cana-3289	86	4	𝔅𝑁(𝑚	𝔅𝑁(𝑚	PROPN
cana-3289	86	5	)	)	PUNCT
cana-3289	86	6	}	}	PUNCT
cana-3289	86	7	.	.	PUNCT
cana-3289	86	8	”	"	PUNCT
cana-3289	87	1	definition:2.11[16	definition:2.11[16	PROPN
cana-3289	87	2	]	]	X
cana-3289	87	3	:	:	PUNCT
cana-3289	87	4	“	"	PUNCT
cana-3289	87	5	suppose	suppose	VERB
cana-3289	87	6	𝔅	𝔅	PROPN
cana-3289	87	7	=	=	SYM
cana-3289	87	8	(	(	PUNCT
cana-3289	87	9	𝔅𝑃	𝔅𝑃	PROPN
cana-3289	87	10	,	,	PUNCT
cana-3289	87	11	𝔅𝑁	𝔅𝑁	PROPN
cana-3289	87	12	)	)	PUNCT
cana-3289	87	13	is	be	AUX
cana-3289	87	14	a	a	DET
cana-3289	87	15	bfs	bfs	NOUN
cana-3289	87	16	in	in	ADP
cana-3289	87	17	𝔏	𝔏	PROPN
cana-3289	87	18	where	where	SCONJ
cana-3289	87	19	𝔅𝑃	𝔅𝑃	PROPN
cana-3289	87	20	:	:	PUNCT
cana-3289	87	21	ϝ	ϝ	NOUN
cana-3289	87	22	→	→	SYM
cana-3289	88	1	[	[	X
cana-3289	88	2	0,1	0,1	NUM
cana-3289	88	3	]	]	PUNCT
cana-3289	88	4	and	and	CCONJ
cana-3289	88	5	𝔅𝑁	𝔅𝑁	PROPN
cana-3289	88	6	:	:	PUNCT
cana-3289	88	7	ϝ	ϝ	NOUN
cana-3289	88	8	→	→	SYM
cana-3289	88	9	[	[	X
cana-3289	88	10	−1,0]then	−1,0]then	ADP
cana-3289	88	11	𝔅	𝔅	PROPN
cana-3289	88	12	is	be	AUX
cana-3289	88	13	known	know	VERB
cana-3289	88	14	to	to	PART
cana-3289	88	15	be	be	AUX
cana-3289	88	16	a	a	DET
cana-3289	88	17	bfil(bipolar	bfil(bipolar	ADJ
cana-3289	88	18	fuzzy	fuzzy	ADJ
cana-3289	88	19	ideal	ideal	NOUN
cana-3289	88	20	)	)	PUNCT
cana-3289	88	21	of	of	ADP
cana-3289	88	22	𝔏	𝔏	PROPN
cana-3289	88	23	when	when	SCONJ
cana-3289	88	24	the	the	DET
cana-3289	88	25	following	follow	VERB
cana-3289	88	26	conditions	condition	NOUN
cana-3289	88	27	are	be	AUX
cana-3289	88	28	fulfilled	fulfil	VERB
cana-3289	88	29	for	for	ADP
cana-3289	88	30	all	all	DET
cana-3289	88	31	ℏ	ℏ	PROPN
cana-3289	88	32	,	,	PUNCT
cana-3289	88	33	𝑚	𝑚	PROPN
cana-3289	88	34	∈	∈	PROPN
cana-3289	88	35	𝔏	𝔏	PROPN
cana-3289	88	36	,	,	PUNCT
cana-3289	88	37	communications	communication	NOUN
cana-3289	88	38	on	on	ADP
cana-3289	88	39	applied	apply	VERB
cana-3289	88	40	nonlinear	nonlinear	ADJ
cana-3289	88	41	analysis	analysis	NOUN
cana-3289	88	42	issn	issn	NOUN
cana-3289	88	43	:	:	PUNCT
cana-3289	88	44	1074	1074	NUM
cana-3289	88	45	-	-	PUNCT
cana-3289	88	46	133x	133x	NUM
cana-3289	88	47	vol	vol	NOUN
cana-3289	88	48	32	32	NUM
cana-3289	88	49	no	no	NOUN
cana-3289	88	50	.	.	PUNCT
cana-3289	89	1	6s	6s	NUM
cana-3289	89	2	(	(	PUNCT
cana-3289	89	3	2025	2025	NUM
cana-3289	89	4	)	)	PUNCT
cana-3289	89	5	244	244	NUM
cana-3289	90	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3289	90	2	(	(	PUNCT
cana-3289	90	3	𝑖)𝔅𝑃(ℏ⋁𝑚	𝑖)𝔅𝑃(ℏ⋁𝑚	PROPN
cana-3289	90	4	)	)	PUNCT
cana-3289	90	5	≥	≥	NOUN
cana-3289	90	6	min{𝔅𝑃(ℏ	min{𝔅𝑃(ℏ	PROPN
cana-3289	90	7	)	)	PUNCT
cana-3289	90	8	,	,	PUNCT
cana-3289	90	9	𝔅𝑃(𝑚	𝔅𝑃(𝑚	NOUN
cana-3289	90	10	)	)	PUNCT
cana-3289	90	11	}	}	PUNCT
cana-3289	90	12	,	,	PUNCT
cana-3289	90	13	(	(	PUNCT
cana-3289	90	14	ii	ii	NOUN
cana-3289	90	15	)	)	PUNCT
cana-3289	90	16	𝔅𝑃(ℏ	𝔅𝑃(ℏ	PROPN
cana-3289	90	17	∧	∧	PROPN
cana-3289	90	18	𝑚	𝑚	NOUN
cana-3289	90	19	)	)	PUNCT
cana-3289	90	20	≥	≥	NOUN
cana-3289	90	21	max{𝔅𝑃(ℏ	max{𝔅𝑃(ℏ	PROPN
cana-3289	90	22	)	)	PUNCT
cana-3289	90	23	,	,	PUNCT
cana-3289	90	24	𝔅𝑃(𝑚	𝔅𝑃(𝑚	NOUN
cana-3289	90	25	)	)	PUNCT
cana-3289	90	26	}	}	PUNCT
cana-3289	90	27	.	.	PUNCT
cana-3289	91	1	(	(	PUNCT
cana-3289	91	2	iii	iii	X
cana-3289	91	3	)	)	PUNCT
cana-3289	91	4	𝔅𝑁(ℏ⋁𝑚	𝔅𝑁(ℏ⋁𝑚	NOUN
cana-3289	91	5	)	)	PUNCT
cana-3289	91	6	≤	≤	NOUN
cana-3289	92	1	max{𝔅𝑁(ℏ	max{𝔅𝑁(ℏ	NOUN
cana-3289	92	2	)	)	PUNCT
cana-3289	92	3	,	,	PUNCT
cana-3289	92	4	𝔅𝑁(𝑚	𝔅𝑁(𝑚	PROPN
cana-3289	92	5	)	)	PUNCT
cana-3289	92	6	}	}	PUNCT
cana-3289	92	7	,	,	PUNCT
cana-3289	92	8	(	(	PUNCT
cana-3289	92	9	iv	iv	X
cana-3289	92	10	)	)	PUNCT
cana-3289	92	11	𝔅𝑁(ℏ	𝔅𝑁(ℏ	PROPN
cana-3289	92	12	∧	∧	PROPN
cana-3289	92	13	𝑚	𝑚	PROPN
cana-3289	92	14	)	)	PUNCT
cana-3289	92	15	≤	≤	NOUN
cana-3289	92	16	min{𝔅𝑁(ℏ	min{𝔅𝑁(ℏ	PROPN
cana-3289	92	17	)	)	PUNCT
cana-3289	92	18	,	,	PUNCT
cana-3289	92	19	𝔅𝑁(𝑚	𝔅𝑁(𝑚	PROPN
cana-3289	92	20	)	)	PUNCT
cana-3289	92	21	}	}	PUNCT
cana-3289	92	22	.	.	PUNCT
cana-3289	92	23	”	"	PUNCT
cana-3289	93	1	3	3	X
cana-3289	93	2	.	.	X
cana-3289	93	3	bipolar	bipolar	ADJ
cana-3289	93	4	fuzzy	fuzzy	ADJ
cana-3289	93	5	prime	prime	ADJ
cana-3289	93	6	ideals	ideal	NOUN
cana-3289	93	7	of	of	ADP
cana-3289	93	8	a	a	DET
cana-3289	93	9	lattice	lattice	NOUN
cana-3289	93	10	in	in	ADP
cana-3289	93	11	this	this	DET
cana-3289	93	12	section	section	NOUN
cana-3289	93	13	,	,	PUNCT
cana-3289	93	14	we	we	PRON
cana-3289	93	15	explore	explore	VERB
cana-3289	93	16	and	and	CCONJ
cana-3289	93	17	study	study	VERB
cana-3289	93	18	bipolar	bipolar	ADJ
cana-3289	93	19	fuzzy	fuzzy	ADJ
cana-3289	93	20	prime	prime	ADJ
cana-3289	93	21	ideals	ideal	NOUN
cana-3289	93	22	(	(	PUNCT
cana-3289	93	23	bfpi	bfpi	ADJ
cana-3289	93	24	)	)	PUNCT
cana-3289	93	25	of	of	ADP
cana-3289	93	26	𝔏	𝔏	PROPN
cana-3289	93	27	,	,	PUNCT
cana-3289	93	28	their	their	PRON
cana-3289	93	29	characterizations	characterization	NOUN
cana-3289	93	30	by	by	ADP
cana-3289	93	31	using	use	VERB
cana-3289	93	32	level	level	NOUN
cana-3289	93	33	subsets	subset	NOUN
cana-3289	93	34	,	,	PUNCT
cana-3289	93	35	homomorphism	homomorphism	NOUN
cana-3289	93	36	and	and	CCONJ
cana-3289	93	37	anti	anti	NOUN
cana-3289	93	38	-	-	NOUN
cana-3289	93	39	homomorphism	homomorphism	NOUN
cana-3289	93	40	of	of	ADP
cana-3289	93	41	bfpis	bfpis	NOUN
cana-3289	93	42	.	.	PUNCT
cana-3289	94	1	now	now	ADV
cana-3289	94	2	,	,	PUNCT
cana-3289	94	3	we	we	PRON
cana-3289	94	4	introduce	introduce	VERB
cana-3289	94	5	the	the	DET
cana-3289	94	6	following	following	NOUN
cana-3289	94	7	.	.	PUNCT
cana-3289	95	1	definition	definition	NOUN
cana-3289	95	2	3.1	3.1	NUM
cana-3289	95	3	:	:	PUNCT
cana-3289	95	4	suppose	suppose	VERB
cana-3289	95	5	𝔅	𝔅	PROPN
cana-3289	95	6	=	=	SYM
cana-3289	95	7	(	(	PUNCT
cana-3289	95	8	𝔅𝑃	𝔅𝑃	PROPN
cana-3289	95	9	,	,	PUNCT
cana-3289	95	10	𝔅𝑁	𝔅𝑁	PROPN
cana-3289	95	11	)	)	PUNCT
cana-3289	95	12	is	be	AUX
cana-3289	95	13	a	a	DET
cana-3289	95	14	bfi	bfi	PROPN
cana-3289	95	15	in	in	ADP
cana-3289	95	16	𝔏.	𝔏.	PROPN
cana-3289	95	17	then	then	ADV
cana-3289	95	18	𝔅	𝔅	PROPN
cana-3289	95	19	is	be	AUX
cana-3289	95	20	known	know	VERB
cana-3289	95	21	to	to	PART
cana-3289	95	22	be	be	AUX
cana-3289	95	23	a	a	DET
cana-3289	95	24	bfpi	bfpi	NOUN
cana-3289	95	25	of	of	ADP
cana-3289	95	26	𝔏	𝔏	NOUN
cana-3289	95	27	when	when	SCONJ
cana-3289	95	28	the	the	DET
cana-3289	95	29	following	follow	VERB
cana-3289	95	30	conditions	condition	NOUN
cana-3289	95	31	are	be	AUX
cana-3289	95	32	fulfilled	fulfil	VERB
cana-3289	95	33	for	for	ADP
cana-3289	95	34	all	all	DET
cana-3289	95	35	ℏ	ℏ	PROPN
cana-3289	95	36	,	,	PUNCT
cana-3289	95	37	𝑚	𝑚	PROPN
cana-3289	95	38	∈	∈	PROPN
cana-3289	95	39	𝔏	𝔏	PROPN
cana-3289	95	40	,	,	PUNCT
cana-3289	95	41	(	(	PUNCT
cana-3289	95	42	i	i	NOUN
cana-3289	95	43	)	)	PUNCT
cana-3289	95	44	𝔅𝑃(ℏ	𝔅𝑃(ℏ	PROPN
cana-3289	95	45	∧	∧	PROPN
cana-3289	95	46	𝑚	𝑚	NOUN
cana-3289	95	47	)	)	PUNCT
cana-3289	95	48	≤	≤	NUM
cana-3289	95	49	max{𝔅𝑃(ℏ	max{𝔅𝑃(ℏ	NOUN
cana-3289	95	50	)	)	PUNCT
cana-3289	95	51	,	,	PUNCT
cana-3289	95	52	𝔅𝑃(𝑚	𝔅𝑃(𝑚	NOUN
cana-3289	95	53	)	)	PUNCT
cana-3289	95	54	}	}	PUNCT
cana-3289	95	55	,	,	PUNCT
cana-3289	95	56	(	(	PUNCT
cana-3289	95	57	ii	ii	NOUN
cana-3289	95	58	)	)	PUNCT
cana-3289	95	59	𝔅𝑁(ℏ	𝔅𝑁(ℏ	PROPN
cana-3289	95	60	∧	∧	PROPN
cana-3289	95	61	𝑚	𝑚	PROPN
cana-3289	95	62	)	)	PUNCT
cana-3289	95	63	≥	≥	NOUN
cana-3289	95	64	min{𝔅𝑁(ℏ	min{𝔅𝑁(ℏ	NOUN
cana-3289	95	65	)	)	PUNCT
cana-3289	95	66	,	,	PUNCT
cana-3289	95	67	𝔅𝑁(𝑚	𝔅𝑁(𝑚	PROPN
cana-3289	95	68	)	)	PUNCT
cana-3289	95	69	}	}	PUNCT
cana-3289	95	70	.	.	PUNCT
cana-3289	96	1	example	example	NOUN
cana-3289	96	2	3.2	3.2	NUM
cana-3289	96	3	:	:	PUNCT
cana-3289	96	4	consider	consider	VERB
cana-3289	96	5	the	the	DET
cana-3289	96	6	lattice	lattice	NOUN
cana-3289	96	7	𝔏	𝔏	PROPN
cana-3289	96	8	of	of	ADP
cana-3289	96	9	"	"	PUNCT
cana-3289	96	10	divisors	divisor	NOUN
cana-3289	96	11	of	of	ADP
cana-3289	96	12	6	6	NUM
cana-3289	96	13	"	"	PUNCT
cana-3289	96	14	.	.	PUNCT
cana-3289	97	1	we	we	PRON
cana-3289	97	2	get	get	VERB
cana-3289	97	3	𝔏	𝔏	NOUN
cana-3289	97	4	=	=	PUNCT
cana-3289	97	5	{	{	PUNCT
cana-3289	97	6	1,2,3,6	1,2,3,6	NUM
cana-3289	97	7	}	}	PUNCT
cana-3289	97	8	.	.	PUNCT
cana-3289	98	1	suppose	suppose	VERB
cana-3289	99	1	𝔅	𝔅	NOUN
cana-3289	99	2	=	=	SYM
cana-3289	99	3	{	{	PUNCT
cana-3289	99	4	<	<	X
cana-3289	99	5	1,0.7	1,0.7	NOUN
cana-3289	99	6	,	,	PUNCT
cana-3289	99	7	−0.3	−0.3	PROPN
cana-3289	99	8	>	>	X
cana-3289	99	9	,	,	PUNCT
cana-3289	99	10	<	<	X
cana-3289	99	11	2,0.4	2,0.4	NUM
cana-3289	99	12	,	,	PUNCT
cana-3289	99	13	−0.3	−0.3	PROPN
cana-3289	99	14	>	>	X
cana-3289	99	15	,	,	PUNCT
cana-3289	99	16	<	<	X
cana-3289	99	17	3,0.7	3,0.7	NUM
cana-3289	99	18	,	,	PUNCT
cana-3289	99	19	−0.1	−0.1	PROPN
cana-3289	99	20	>	>	X
cana-3289	99	21	,	,	PUNCT
cana-3289	99	22	<	<	X
cana-3289	99	23	6,0.4	6,0.4	NUM
cana-3289	99	24	,	,	PUNCT
cana-3289	99	25	−0.1	−0.1	PROPN
cana-3289	99	26	>	>	PUNCT
cana-3289	99	27	}	}	PUNCT
cana-3289	99	28	.	.	PUNCT
cana-3289	100	1	we	we	PRON
cana-3289	100	2	can	can	AUX
cana-3289	100	3	easily	easily	ADV
cana-3289	100	4	prove	prove	VERB
cana-3289	100	5	that	that	SCONJ
cana-3289	100	6	𝔅	𝔅	NOUN
cana-3289	100	7	is	be	AUX
cana-3289	100	8	a	a	DET
cana-3289	100	9	bfpi	bfpi	NOUN
cana-3289	100	10	of	of	ADP
cana-3289	100	11	𝔏.	𝔏.	PROPN
cana-3289	100	12	theorem	theorem	VERB
cana-3289	100	13	3.3	3.3	NUM
cana-3289	100	14	:	:	PUNCT
cana-3289	100	15	suppose	suppose	VERB
cana-3289	100	16	𝔅	𝔅	PROPN
cana-3289	100	17	=	=	SYM
cana-3289	100	18	(	(	PUNCT
cana-3289	100	19	𝔅𝑃	𝔅𝑃	PROPN
cana-3289	100	20	,	,	PUNCT
cana-3289	100	21	𝔅𝑁	𝔅𝑁	PROPN
cana-3289	100	22	)	)	PUNCT
cana-3289	100	23	be	be	AUX
cana-3289	100	24	a	a	DET
cana-3289	100	25	bfs(𝔏	bfs(𝔏	NOUN
cana-3289	100	26	)	)	PUNCT
cana-3289	100	27	.	.	PUNCT
cana-3289	101	1	then	then	ADV
cana-3289	101	2	it	it	PRON
cana-3289	101	3	holds	hold	VERB
cana-3289	101	4	that	that	SCONJ
cana-3289	101	5	𝔅	𝔅	NOUN
cana-3289	101	6	is	be	AUX
cana-3289	101	7	a	a	DET
cana-3289	101	8	bfpi	bfpi	NOUN
cana-3289	101	9	on	on	ADP
cana-3289	101	10	𝔏	𝔏	PROPN
cana-3289	101	11	iff	iff	VERB
cana-3289	101	12	the	the	DET
cana-3289	101	13	following	follow	VERB
cana-3289	101	14	four	four	NUM
cana-3289	101	15	statements	statement	NOUN
cana-3289	101	16	hold	hold	VERB
cana-3289	101	17	:	:	PUNCT
cana-3289	101	18	(	(	PUNCT
cana-3289	101	19	i	i	NOUN
cana-3289	101	20	)	)	PUNCT
cana-3289	101	21	𝔅𝑃(ℏ	𝔅𝑃(ℏ	PROPN
cana-3289	101	22	∨	∨	NUM
cana-3289	101	23	𝑚	𝑚	NOUN
cana-3289	101	24	)	)	PUNCT
cana-3289	101	25	=	=	SYM
cana-3289	101	26	min{𝔅𝑃(ℏ	min{𝔅𝑃(ℏ	PROPN
cana-3289	101	27	)	)	PUNCT
cana-3289	101	28	,	,	PUNCT
cana-3289	101	29	𝔅𝑃(𝑚	𝔅𝑃(𝑚	NOUN
cana-3289	101	30	)	)	PUNCT
cana-3289	101	31	}	}	PUNCT
cana-3289	101	32	,	,	PUNCT
cana-3289	101	33	(	(	PUNCT
cana-3289	101	34	ii	ii	NOUN
cana-3289	101	35	)	)	PUNCT
cana-3289	101	36	𝔅𝑃(ℏ	𝔅𝑃(ℏ	NOUN
cana-3289	101	37	∧	∧	PROPN
cana-3289	101	38	𝑚	𝑚	NOUN
cana-3289	101	39	)	)	PUNCT
cana-3289	101	40	=	=	SYM
cana-3289	101	41	max{𝔅𝑃(ℏ	max{𝔅𝑃(ℏ	PROPN
cana-3289	101	42	)	)	PUNCT
cana-3289	101	43	,	,	PUNCT
cana-3289	101	44	𝔅𝑃(𝑚	𝔅𝑃(𝑚	NOUN
cana-3289	101	45	)	)	PUNCT
cana-3289	101	46	}	}	PUNCT
cana-3289	101	47	,	,	PUNCT
cana-3289	101	48	(	(	PUNCT
cana-3289	101	49	iii	iii	X
cana-3289	101	50	)	)	PUNCT
cana-3289	101	51	𝔅𝑁(ℏ	𝔅𝑁(ℏ	NOUN
cana-3289	101	52	∨	∨	NUM
cana-3289	101	53	𝑚	𝑚	PROPN
cana-3289	101	54	)	)	PUNCT
cana-3289	101	55	=	=	SYM
cana-3289	101	56	max{𝔅𝑁(ℏ	max{𝔅𝑁(ℏ	NOUN
cana-3289	101	57	)	)	PUNCT
cana-3289	101	58	,	,	PUNCT
cana-3289	101	59	𝔅𝑁(𝑚	𝔅𝑁(𝑚	PROPN
cana-3289	101	60	)	)	PUNCT
cana-3289	101	61	}	}	PUNCT
cana-3289	101	62	,	,	PUNCT
cana-3289	101	63	(	(	PUNCT
cana-3289	101	64	iv	iv	X
cana-3289	101	65	)	)	PUNCT
cana-3289	101	66	𝔅𝑁(ℏ	𝔅𝑁(ℏ	NOUN
cana-3289	101	67	∧	∧	PROPN
cana-3289	101	68	𝑚	𝑚	PROPN
cana-3289	101	69	)	)	PUNCT
cana-3289	101	70	=	=	SYM
cana-3289	101	71	min{𝔅𝑁(ℏ	min{𝔅𝑁(ℏ	PROPN
cana-3289	101	72	)	)	PUNCT
cana-3289	101	73	,	,	PUNCT
cana-3289	101	74	𝔅𝑁(𝑚	𝔅𝑁(𝑚	PROPN
cana-3289	101	75	)	)	PUNCT
cana-3289	101	76	}	}	PUNCT
cana-3289	101	77	for	for	ADP
cana-3289	101	78	any	any	DET
cana-3289	101	79	ℏ	ℏ	NOUN
cana-3289	101	80	,	,	PUNCT
cana-3289	101	81	𝑚	𝑚	PROPN
cana-3289	101	82	∈	∈	PROPN
cana-3289	101	83	𝔏.	𝔏.	PROPN
cana-3289	101	84	proof	proof	NOUN
cana-3289	101	85	:	:	PUNCT
cana-3289	101	86	proof	proof	NOUN
cana-3289	101	87	is	be	AUX
cana-3289	101	88	clear	clear	ADJ
cana-3289	101	89	.	.	PUNCT
cana-3289	102	1	theorem	theorem	VERB
cana-3289	102	2	3.4	3.4	NUM
cana-3289	102	3	:	:	PUNCT
cana-3289	102	4	if	if	SCONJ
cana-3289	102	5	𝜅	𝜅	PRON
cana-3289	102	6	and	and	CCONJ
cana-3289	102	7	𝔅	𝔅	NOUN
cana-3289	102	8	are	be	AUX
cana-3289	102	9	two	two	NUM
cana-3289	102	10	bfpis	bfpis	NOUN
cana-3289	102	11	of	of	ADP
cana-3289	102	12	a	a	DET
cana-3289	102	13	lattice	lattice	NOUN
cana-3289	102	14	𝔏	𝔏	PROPN
cana-3289	102	15	,	,	PUNCT
cana-3289	102	16	then	then	ADV
cana-3289	102	17	𝜅	𝜅	PRON
cana-3289	102	18	∩	∩	ADJ
cana-3289	102	19	𝔅	𝔅	NOUN
cana-3289	102	20	is	be	AUX
cana-3289	102	21	a	a	DET
cana-3289	102	22	bfpi	bfpi	NOUN
cana-3289	102	23	of	of	ADP
cana-3289	102	24	𝔏.	𝔏.	PROPN
cana-3289	102	25	proof	proof	NOUN
cana-3289	102	26	:	:	PUNCT
cana-3289	102	27	suppose	suppose	VERB
cana-3289	102	28	𝜅	𝜅	X
cana-3289	102	29	=	=	X
cana-3289	102	30	(	(	PUNCT
cana-3289	102	31	𝜅𝑃	𝜅𝑃	NUM
cana-3289	102	32	,	,	PUNCT
cana-3289	102	33	𝜅𝑁	𝜅𝑁	ADJ
cana-3289	102	34	)	)	PUNCT
cana-3289	102	35	and	and	CCONJ
cana-3289	102	36	𝔅	𝔅	NOUN
cana-3289	102	37	=	=	SYM
cana-3289	102	38	(	(	PUNCT
cana-3289	102	39	𝔅𝑃	𝔅𝑃	PROPN
cana-3289	102	40	,	,	PUNCT
cana-3289	102	41	𝔅𝑁	𝔅𝑁	PROPN
cana-3289	102	42	)	)	PUNCT
cana-3289	102	43	be	be	VERB
cana-3289	102	44	two	two	NUM
cana-3289	102	45	bfpis	bfpis	NOUN
cana-3289	102	46	of	of	ADP
cana-3289	102	47	𝔏.	𝔏.	PROPN
cana-3289	102	48	now	now	ADV
cana-3289	102	49	,	,	PUNCT
cana-3289	102	50	(	(	PUNCT
cana-3289	102	51	𝜅	𝜅	PRON
cana-3289	102	52	∩	∩	ADJ
cana-3289	102	53	𝔅)𝑃(ℏ	𝔅)𝑃(ℏ	NOUN
cana-3289	102	54	∧	∧	PROPN
cana-3289	102	55	𝑚	𝑚	NOUN
cana-3289	102	56	)	)	PUNCT
cana-3289	102	57	=	=	SYM
cana-3289	102	58	min{𝜅𝑃(ℏ	min{𝜅𝑃(ℏ	X
cana-3289	102	59	∧	∧	PROPN
cana-3289	102	60	𝑚	𝑚	NOUN
cana-3289	102	61	)	)	PUNCT
cana-3289	102	62	,	,	PUNCT
cana-3289	102	63	𝔅𝑃(ℏ	𝔅𝑃(ℏ	PROPN
cana-3289	102	64	∧	∧	PROPN
cana-3289	102	65	𝑚	𝑚	NOUN
cana-3289	102	66	)	)	PUNCT
cana-3289	102	67	}	}	PUNCT
cana-3289	102	68	≤	≤	NUM
cana-3289	102	69	min{max{𝜅𝑃(ℏ	min{max{𝜅𝑃(ℏ	NOUN
cana-3289	102	70	)	)	PUNCT
cana-3289	102	71	,	,	PUNCT
cana-3289	102	72	𝜅𝑃(𝑚	𝜅𝑃(𝑚	NUM
cana-3289	102	73	)	)	PUNCT
cana-3289	102	74	}	}	PUNCT
cana-3289	102	75	,	,	PUNCT
cana-3289	102	76	max{𝔅𝑃(ℏ	max{𝔅𝑃(ℏ	PROPN
cana-3289	102	77	)	)	PUNCT
cana-3289	102	78	,	,	PUNCT
cana-3289	102	79	𝔅𝑃(𝑚	𝔅𝑃(𝑚	NOUN
cana-3289	102	80	)	)	PUNCT
cana-3289	102	81	}	}	PUNCT
cana-3289	102	82	=	=	SYM
cana-3289	102	83	max{min{𝜅𝑃(ℏ	max{min{𝜅𝑃(ℏ	NOUN
cana-3289	102	84	)	)	PUNCT
cana-3289	102	85	,	,	PUNCT
cana-3289	102	86	𝔅𝑃(ℏ	𝔅𝑃(ℏ	PROPN
cana-3289	102	87	)	)	PUNCT
cana-3289	102	88	}	}	PUNCT
cana-3289	102	89	,	,	PUNCT
cana-3289	102	90	min{𝜅𝑃(𝑚	min{𝜅𝑃(𝑚	NOUN
cana-3289	102	91	)	)	PUNCT
cana-3289	102	92	,	,	PUNCT
cana-3289	102	93	𝔅𝑃(𝑚	𝔅𝑃(𝑚	NOUN
cana-3289	102	94	)	)	PUNCT
cana-3289	102	95	}	}	PUNCT
cana-3289	103	1	=	=	SYM
cana-3289	103	2	max{(𝜅	max{(𝜅	NOUN
cana-3289	103	3	∩	∩	X
cana-3289	103	4	𝔅)𝑃(ℏ	𝔅)𝑃(ℏ	NOUN
cana-3289	103	5	)	)	PUNCT
cana-3289	103	6	,	,	PUNCT
cana-3289	103	7	(	(	PUNCT
cana-3289	103	8	𝜅	𝜅	PROPN
cana-3289	103	9	∩	∩	ADJ
cana-3289	103	10	𝔅)𝑃(𝑚	𝔅)𝑃(𝑚	NOUN
cana-3289	103	11	)	)	PUNCT
cana-3289	103	12	}	}	PUNCT
cana-3289	103	13	.	.	PUNCT
cana-3289	104	1	thus	thus	ADV
cana-3289	104	2	(	(	PUNCT
cana-3289	104	3	𝜅	𝜅	X
cana-3289	104	4	∩	∩	ADJ
cana-3289	104	5	𝔅)𝑃(ℏ	𝔅)𝑃(ℏ	NOUN
cana-3289	104	6	∧	∧	PROPN
cana-3289	104	7	𝑚	𝑚	NOUN
cana-3289	104	8	)	)	PUNCT
cana-3289	104	9	≤	≤	NOUN
cana-3289	104	10	max{𝜅	max{𝜅	NOUN
cana-3289	104	11	∩	∩	NOUN
cana-3289	104	12	𝔅)𝑃(ℏ	𝔅)𝑃(ℏ	NOUN
cana-3289	104	13	)	)	PUNCT
cana-3289	104	14	,	,	PUNCT
cana-3289	104	15	𝜅	𝜅	PRON
cana-3289	104	16	∩	∩	ADJ
cana-3289	104	17	𝔅)𝑃(𝑚	𝔅)𝑃(𝑚	NOUN
cana-3289	104	18	)	)	PUNCT
cana-3289	104	19	}	}	PUNCT
cana-3289	104	20	for	for	ADP
cana-3289	104	21	all	all	DET
cana-3289	104	22	ℏ	ℏ	PROPN
cana-3289	104	23	,	,	PUNCT
cana-3289	104	24	𝑚	𝑚	PROPN
cana-3289	104	25	∈	∈	PROPN
cana-3289	104	26	𝔏.	𝔏.	PROPN
cana-3289	104	27	now	now	ADV
cana-3289	104	28	,	,	PUNCT
cana-3289	104	29	(	(	PUNCT
cana-3289	104	30	𝜅	𝜅	X
cana-3289	104	31	∩	∩	NOUN
cana-3289	104	32	𝔅)𝑁(ℏ	𝔅)𝑁(ℏ	ADP
cana-3289	104	33	∧	∧	PROPN
cana-3289	104	34	𝑚	𝑚	PROPN
cana-3289	104	35	)	)	PUNCT
cana-3289	104	36	=	=	NOUN
cana-3289	104	37	max{𝜅𝑁(ℏ	max{𝜅𝑁(ℏ	X
cana-3289	104	38	∧	∧	PROPN
cana-3289	104	39	𝑚	𝑚	PROPN
cana-3289	104	40	)	)	PUNCT
cana-3289	104	41	,	,	PUNCT
cana-3289	104	42	𝔅𝑁(ℏ	𝔅𝑁(ℏ	PROPN
cana-3289	104	43	∧	∧	PROPN
cana-3289	104	44	𝑚	𝑚	PROPN
cana-3289	104	45	)	)	PUNCT
cana-3289	104	46	}	}	PUNCT
cana-3289	104	47	≥	≥	X
cana-3289	104	48	max{min{𝜅𝑁(ℏ	max{min{𝜅𝑁(ℏ	NOUN
cana-3289	104	49	)	)	PUNCT
cana-3289	104	50	,	,	PUNCT
cana-3289	104	51	𝜅𝑁(𝑚	𝜅𝑁(𝑚	PROPN
cana-3289	104	52	)	)	PUNCT
cana-3289	104	53	}	}	PUNCT
cana-3289	104	54	,	,	PUNCT
cana-3289	104	55	min{𝔅𝑁(ℏ	min{𝔅𝑁(ℏ	PROPN
cana-3289	104	56	)	)	PUNCT
cana-3289	104	57	,	,	PUNCT
cana-3289	104	58	𝔅𝑁(𝑚	𝔅𝑁(𝑚	PROPN
cana-3289	104	59	)	)	PUNCT
cana-3289	104	60	}	}	PUNCT
cana-3289	104	61	=	=	SYM
cana-3289	104	62	min{max{𝜅𝑁(ℏ	min{max{𝜅𝑁(ℏ	NUM
cana-3289	104	63	)	)	PUNCT
cana-3289	104	64	,	,	PUNCT
cana-3289	104	65	𝔅𝑁(ℏ	𝔅𝑁(ℏ	PROPN
cana-3289	104	66	)	)	PUNCT
cana-3289	104	67	}	}	PUNCT
cana-3289	104	68	,	,	PUNCT
cana-3289	104	69	max{𝜅𝑁(𝑚	max{𝜅𝑁(𝑚	PROPN
cana-3289	104	70	)	)	PUNCT
cana-3289	104	71	,	,	PUNCT
cana-3289	104	72	𝔅𝑁(𝑚	𝔅𝑁(𝑚	PROPN
cana-3289	104	73	)	)	PUNCT
cana-3289	104	74	}	}	PUNCT
cana-3289	104	75	=	=	SYM
cana-3289	104	76	min{(𝜅	min{(𝜅	NOUN
cana-3289	104	77	∩	∩	PROPN
cana-3289	104	78	𝔅)𝑁(ℏ	𝔅)𝑁(ℏ	PROPN
cana-3289	104	79	)	)	PUNCT
cana-3289	104	80	,	,	PUNCT
cana-3289	104	81	(	(	PUNCT
cana-3289	104	82	𝜅	𝜅	X
cana-3289	104	83	∩	∩	ADJ
cana-3289	104	84	𝔅)𝑁(𝑚	𝔅)𝑁(𝑚	NOUN
cana-3289	104	85	)	)	PUNCT
cana-3289	104	86	}	}	PUNCT
cana-3289	104	87	.	.	PUNCT
cana-3289	105	1	communications	communication	NOUN
cana-3289	105	2	on	on	ADP
cana-3289	105	3	applied	apply	VERB
cana-3289	105	4	nonlinear	nonlinear	ADJ
cana-3289	105	5	analysis	analysis	NOUN
cana-3289	105	6	issn	issn	NOUN
cana-3289	105	7	:	:	PUNCT
cana-3289	105	8	1074	1074	NUM
cana-3289	105	9	-	-	PUNCT
cana-3289	105	10	133x	133x	NUM
cana-3289	105	11	vol	vol	NOUN
cana-3289	105	12	32	32	NUM
cana-3289	105	13	no	no	NOUN
cana-3289	105	14	.	.	PUNCT
cana-3289	106	1	6s	6s	NUM
cana-3289	106	2	(	(	PUNCT
cana-3289	106	3	2025	2025	NUM
cana-3289	106	4	)	)	PUNCT
cana-3289	106	5	245	245	NUM
cana-3289	106	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3289	106	7	thus	thus	ADV
cana-3289	106	8	(	(	PUNCT
cana-3289	106	9	𝜅	𝜅	X
cana-3289	106	10	∩	∩	NOUN
cana-3289	106	11	𝔅)𝑁(ℏ	𝔅)𝑁(ℏ	ADP
cana-3289	106	12	∧	∧	PROPN
cana-3289	106	13	𝑚	𝑚	PROPN
cana-3289	106	14	)	)	PUNCT
cana-3289	106	15	≥	≥	NOUN
cana-3289	106	16	min{𝜅	min{𝜅	PROPN
cana-3289	106	17	∩	∩	PROPN
cana-3289	106	18	𝔅)𝑁(ℏ	𝔅)𝑁(ℏ	PROPN
cana-3289	106	19	)	)	PUNCT
cana-3289	106	20	,	,	PUNCT
cana-3289	106	21	𝜅	𝜅	PRON
cana-3289	106	22	∩	∩	ADJ
cana-3289	106	23	𝔅)𝑁(𝑚	𝔅)𝑁(𝑚	NOUN
cana-3289	106	24	)	)	PUNCT
cana-3289	106	25	}	}	PUNCT
cana-3289	106	26	for	for	ADP
cana-3289	106	27	all	all	DET
cana-3289	106	28	ℏ	ℏ	PROPN
cana-3289	106	29	,	,	PUNCT
cana-3289	106	30	𝑚	𝑚	PROPN
cana-3289	106	31	∈	∈	NOUN
cana-3289	106	32	𝔏	𝔏	NOUN
cana-3289	106	33	hence	hence	ADV
cana-3289	106	34	,	,	PUNCT
cana-3289	106	35	𝜅	𝜅	DET
cana-3289	106	36	∩	∩	ADJ
cana-3289	106	37	𝔅	𝔅	NOUN
cana-3289	106	38	is	be	AUX
cana-3289	106	39	a	a	DET
cana-3289	106	40	bfpi	bfpi	NOUN
cana-3289	106	41	of	of	ADP
cana-3289	106	42	𝔏.	𝔏.	PROPN
cana-3289	106	43	remark	remark	NOUN
cana-3289	106	44	3.5	3.5	NUM
cana-3289	106	45	:	:	PUNCT
cana-3289	106	46	union	union	NOUN
cana-3289	106	47	of	of	ADP
cana-3289	106	48	two	two	NUM
cana-3289	106	49	bfpis	bfpis	NOUN
cana-3289	106	50	of	of	ADP
cana-3289	106	51	a	a	DET
cana-3289	106	52	lattice	lattice	NOUN
cana-3289	106	53	𝔏	𝔏	NOUN
cana-3289	106	54	need	need	AUX
cana-3289	106	55	not	not	PART
cana-3289	106	56	be	be	AUX
cana-3289	106	57	a	a	DET
cana-3289	106	58	bfpi	bfpi	NOUN
cana-3289	106	59	.	.	PUNCT
cana-3289	107	1	since	since	SCONJ
cana-3289	107	2	from	from	ADP
cana-3289	107	3	[	[	PUNCT
cana-3289	107	4	]	]	X
cana-3289	107	5	union	union	NOUN
cana-3289	107	6	of	of	ADP
cana-3289	107	7	two	two	NUM
cana-3289	107	8	bfis	bfis	NOUN
cana-3289	107	9	neednot	neednot	NOUN
cana-3289	107	10	be	be	AUX
cana-3289	107	11	a	a	DET
cana-3289	107	12	bfi	bfi	PROPN
cana-3289	107	13	,	,	PUNCT
cana-3289	107	14	hence	hence	ADV
cana-3289	107	15	,	,	PUNCT
cana-3289	107	16	𝜅	𝜅	PRON
cana-3289	107	17	∪	∪	ADJ
cana-3289	107	18	𝔅	𝔅	NOUN
cana-3289	107	19	need	need	AUX
cana-3289	107	20	not	not	PART
cana-3289	107	21	be	be	AUX
cana-3289	107	22	a	a	DET
cana-3289	107	23	bfpi	bfpi	NOUN
cana-3289	107	24	of	of	ADP
cana-3289	107	25	𝔏.	𝔏.	PROPN
cana-3289	107	26	theorem	theorem	VERB
cana-3289	107	27	3.6	3.6	NUM
cana-3289	107	28	:	:	PUNCT
cana-3289	107	29	the	the	DET
cana-3289	107	30	arbitrary	arbitrary	ADJ
cana-3289	107	31	intersection	intersection	NOUN
cana-3289	107	32	of	of	ADP
cana-3289	107	33	bfpis	bfpis	NOUN
cana-3289	107	34	of	of	ADP
cana-3289	107	35	a	a	DET
cana-3289	107	36	complete	complete	ADJ
cana-3289	107	37	lattice	lattice	NOUN
cana-3289	107	38	satisfying	satisfy	VERB
cana-3289	107	39	infinite	infinite	ADJ
cana-3289	107	40	meet	meet	NOUN
cana-3289	107	41	distributive	distributive	ADJ
cana-3289	107	42	law	law	NOUN
cana-3289	107	43	𝔏	𝔏	PROPN
cana-3289	107	44	is	be	AUX
cana-3289	107	45	also	also	ADV
cana-3289	107	46	a	a	DET
cana-3289	107	47	bfpi	bfpi	NOUN
cana-3289	107	48	of	of	ADP
cana-3289	107	49	𝔏.	𝔏.	PROPN
cana-3289	107	50	proof	proof	NOUN
cana-3289	107	51	:	:	PUNCT
cana-3289	107	52	proof	proof	NOUN
cana-3289	107	53	is	be	AUX
cana-3289	107	54	clear	clear	ADJ
cana-3289	107	55	.	.	PUNCT
cana-3289	108	1	theorem	theorem	VERB
cana-3289	108	2	3.7	3.7	NUM
cana-3289	108	3	:	:	PUNCT
cana-3289	108	4	let	let	VERB
cana-3289	108	5	b	b	X
cana-3289	108	6	=	=	NOUN
cana-3289	108	7	<	<	X
cana-3289	108	8	bp	bp	PROPN
cana-3289	108	9	,	,	PUNCT
cana-3289	108	10	bn	bn	X
cana-3289	108	11	>	>	X
cana-3289	108	12	∈	∈	PROPN
cana-3289	108	13	bfs(𝔏	bfs(𝔏	NUM
cana-3289	108	14	)	)	PUNCT
cana-3289	108	15	.	.	PUNCT
cana-3289	109	1	then	then	ADV
cana-3289	109	2	b	b	PROPN
cana-3289	109	3	is	be	AUX
cana-3289	109	4	bfpi	bfpi	ADJ
cana-3289	109	5	of	of	ADP
cana-3289	109	6	𝔏	𝔏	PROPN
cana-3289	109	7	if	if	SCONJ
cana-3289	109	8	and	and	CCONJ
cana-3289	109	9	only	only	ADV
cana-3289	109	10	if	if	SCONJ
cana-3289	109	11	the	the	DET
cana-3289	109	12	nonempty	nonempty	ADJ
cana-3289	109	13	level	level	NOUN
cana-3289	109	14	subset	subset	NOUN
cana-3289	109	15	b	b	NOUN
cana-3289	109	16	(	(	PUNCT
cana-3289	109	17	α	α	PROPN
cana-3289	109	18	,	,	PUNCT
cana-3289	109	19	ω	ω	NOUN
cana-3289	109	20	)	)	PUNCT
cana-3289	109	21	is	be	AUX
cana-3289	109	22	a	a	DET
cana-3289	109	23	prime	prime	ADJ
cana-3289	109	24	ideal	ideal	NOUN
cana-3289	109	25	of	of	ADP
cana-3289	109	26	𝔏	𝔏	PROPN
cana-3289	109	27	for	for	ADP
cana-3289	109	28	each	each	DET
cana-3289	109	29	α	α	NOUN
cana-3289	109	30	∈	∈	PROPN
cana-3289	110	1	[	[	X
cana-3289	110	2	0	0	NUM
cana-3289	110	3	,	,	PUNCT
cana-3289	110	4	1	1	NUM
cana-3289	110	5	]	]	PUNCT
cana-3289	110	6	and	and	CCONJ
cana-3289	110	7	ω	ω	NUM
cana-3289	110	8	∈	∈	PROPN
cana-3289	111	1	[	[	X
cana-3289	111	2	−1	−1	NOUN
cana-3289	111	3	,	,	PUNCT
cana-3289	111	4	0	0	NUM
cana-3289	111	5	]	]	PUNCT
cana-3289	111	6	.	.	PUNCT
cana-3289	112	1	proof	proof	NOUN
cana-3289	112	2	:	:	PUNCT
cana-3289	112	3	suppose	suppose	VERB
cana-3289	112	4	that	that	SCONJ
cana-3289	112	5	b	b	X
cana-3289	112	6	=	=	NOUN
cana-3289	112	7	<	<	X
cana-3289	112	8	bp	bp	PROPN
cana-3289	112	9	,	,	PUNCT
cana-3289	112	10	bn	bn	CCONJ
cana-3289	112	11	>	>	X
cana-3289	112	12	is	be	AUX
cana-3289	112	13	bfpi(𝔏	bfpi(𝔏	PROPN
cana-3289	112	14	)	)	PUNCT
cana-3289	112	15	.	.	PUNCT
cana-3289	113	1	let	let	VERB
cana-3289	113	2	ℏ	ℏ	VERB
cana-3289	113	3	,	,	PUNCT
cana-3289	113	4	m	m	PROPN
cana-3289	113	5	∈	∈	PROPN
cana-3289	113	6	b	b	PROPN
cana-3289	113	7	(	(	PUNCT
cana-3289	113	8	α	α	PROPN
cana-3289	113	9	,	,	PUNCT
cana-3289	113	10	ω	ω	NOUN
cana-3289	113	11	)	)	PUNCT
cana-3289	113	12	.	.	PUNCT
cana-3289	114	1	⇒	⇒	PROPN
cana-3289	114	2	bp	bp	PROPN
cana-3289	114	3	(	(	PUNCT
cana-3289	114	4	ℏ	ℏ	PROPN
cana-3289	114	5	)	)	PUNCT
cana-3289	114	6	≥	≥	NOUN
cana-3289	114	7	α	α	NOUN
cana-3289	114	8	,	,	PUNCT
cana-3289	114	9	bp	bp	PROPN
cana-3289	114	10	(	(	PUNCT
cana-3289	114	11	m	m	PROPN
cana-3289	114	12	)	)	PUNCT
cana-3289	114	13	≥	≥	NOUN
cana-3289	114	14	α	α	NOUN
cana-3289	114	15	and	and	CCONJ
cana-3289	114	16	bn	bn	INTJ
cana-3289	114	17	(	(	PUNCT
cana-3289	114	18	ℏ	ℏ	NOUN
cana-3289	114	19	)	)	PUNCT
cana-3289	114	20	≤	≤	NOUN
cana-3289	114	21	ω	ω	PROPN
cana-3289	114	22	,	,	PUNCT
cana-3289	114	23	bn	bn	PROPN
cana-3289	114	24	(	(	PUNCT
cana-3289	114	25	m	m	NOUN
cana-3289	114	26	)	)	PUNCT
cana-3289	114	27	≤	≤	NUM
cana-3289	114	28	ω	ω	X
cana-3289	114	29	.	.	PUNCT
cana-3289	115	1	to	to	PART
cana-3289	115	2	show	show	VERB
cana-3289	115	3	that	that	PRON
cana-3289	115	4	b	b	X
cana-3289	115	5	(	(	PUNCT
cana-3289	115	6	α	α	PROPN
cana-3289	115	7	,	,	PUNCT
cana-3289	115	8	ω	ω	NOUN
cana-3289	115	9	)	)	PUNCT
cana-3289	115	10	is	be	AUX
cana-3289	115	11	a	a	DET
cana-3289	115	12	prime	prime	ADJ
cana-3289	115	13	ideal	ideal	NOUN
cana-3289	115	14	of	of	ADP
cana-3289	115	15	𝔏	𝔏	PROPN
cana-3289	115	16	,	,	PUNCT
cana-3289	115	17	we	we	PRON
cana-3289	115	18	show	show	VERB
cana-3289	115	19	that	that	SCONJ
cana-3289	115	20	for	for	ADP
cana-3289	115	21	each	each	DET
cana-3289	115	22	ℏ	ℏ	NOUN
cana-3289	115	23	,	,	PUNCT
cana-3289	115	24	m	m	NOUN
cana-3289	115	25	∈	∈	NOUN
cana-3289	115	26	𝔏	𝔏	NOUN
cana-3289	115	27	and	and	CCONJ
cana-3289	115	28	ℏ	ℏ	PRON
cana-3289	115	29	∧	∧	PROPN
cana-3289	115	30	m	m	PROPN
cana-3289	115	31	∈	∈	PROPN
cana-3289	115	32	b	b	PROPN
cana-3289	115	33	(	(	PUNCT
cana-3289	115	34	α	α	PROPN
cana-3289	115	35	,	,	PUNCT
cana-3289	115	36	ω	ω	NOUN
cana-3289	115	37	)	)	PUNCT
cana-3289	115	38	then	then	ADV
cana-3289	115	39	either	either	CCONJ
cana-3289	115	40	ℏ	ℏ	PROPN
cana-3289	115	41	∈	∈	PROPN
cana-3289	115	42	b	b	PROPN
cana-3289	115	43	(	(	PUNCT
cana-3289	115	44	α	α	PROPN
cana-3289	115	45	,	,	PUNCT
cana-3289	115	46	ω	ω	NOUN
cana-3289	115	47	)	)	PUNCT
cana-3289	115	48	or	or	CCONJ
cana-3289	115	49	s	s	PROPN
cana-3289	115	50	∈	∈	PROPN
cana-3289	115	51	b	b	PROPN
cana-3289	115	52	(	(	PUNCT
cana-3289	115	53	α	α	PROPN
cana-3289	115	54	,	,	PUNCT
cana-3289	115	55	ω	ω	NOUN
cana-3289	115	56	)	)	PUNCT
cana-3289	115	57	.	.	PUNCT
cana-3289	116	1	suppose	suppose	VERB
cana-3289	116	2	ℏ	ℏ	X
cana-3289	116	3	,	,	PUNCT
cana-3289	116	4	m	m	NOUN
cana-3289	116	5	∈	∈	NOUN
cana-3289	116	6	𝔏	𝔏	NOUN
cana-3289	116	7	and	and	CCONJ
cana-3289	116	8	ℏ	ℏ	PRON
cana-3289	116	9	∧	∧	PROPN
cana-3289	116	10	s	s	PART
cana-3289	116	11	∈	∈	PROPN
cana-3289	116	12	b	b	PROPN
cana-3289	116	13	(	(	PUNCT
cana-3289	116	14	α	α	PROPN
cana-3289	116	15	,	,	PUNCT
cana-3289	116	16	ω	ω	NOUN
cana-3289	116	17	)	)	PUNCT
cana-3289	116	18	.	.	PUNCT
cana-3289	117	1	then	then	ADV
cana-3289	117	2	bp	bp	PROPN
cana-3289	117	3	(	(	PUNCT
cana-3289	117	4	ℏ	ℏ	PROPN
cana-3289	117	5	∧	∧	PROPN
cana-3289	117	6	m	m	PROPN
cana-3289	117	7	)	)	PUNCT
cana-3289	117	8	≥	≥	PROPN
cana-3289	117	9	α	α	NOUN
cana-3289	117	10	,	,	PUNCT
cana-3289	117	11	bn	bn	INTJ
cana-3289	117	12	(	(	PUNCT
cana-3289	117	13	ℏ	ℏ	PROPN
cana-3289	117	14	∧	∧	PROPN
cana-3289	117	15	m	m	NOUN
cana-3289	117	16	)	)	PUNCT
cana-3289	117	17	≤	≤	NOUN
cana-3289	117	18	ω	ω	NUM
cana-3289	117	19	⇒	⇒	PROPN
cana-3289	117	20	max	max	PROPN
cana-3289	117	21	{	{	PUNCT
cana-3289	117	22	bp	bp	PROPN
cana-3289	117	23	(	(	PUNCT
cana-3289	117	24	ℏ	ℏ	PROPN
cana-3289	117	25	)	)	PUNCT
cana-3289	117	26	,	,	PUNCT
cana-3289	117	27	bp	bp	PROPN
cana-3289	117	28	(	(	PUNCT
cana-3289	117	29	m	m	NOUN
cana-3289	117	30	)	)	PUNCT
cana-3289	117	31	}	}	PUNCT
cana-3289	117	32	≥	≥	PROPN
cana-3289	117	33	α	α	NOUN
cana-3289	117	34	,	,	PUNCT
cana-3289	117	35	min	min	PROPN
cana-3289	117	36	{	{	PUNCT
cana-3289	117	37	bp	bp	PROPN
cana-3289	117	38	(	(	PUNCT
cana-3289	117	39	ℏ	ℏ	PROPN
cana-3289	117	40	)	)	PUNCT
cana-3289	117	41	,	,	PUNCT
cana-3289	117	42	bp	bp	PROPN
cana-3289	117	43	(	(	PUNCT
cana-3289	117	44	m	m	NOUN
cana-3289	117	45	)	)	PUNCT
cana-3289	117	46	}	}	PUNCT
cana-3289	117	47	≤	≤	NUM
cana-3289	117	48	ω	ω	PROPN
cana-3289	117	49	(	(	PUNCT
cana-3289	117	50	since	since	SCONJ
cana-3289	117	51	b	b	PROPN
cana-3289	117	52	is	be	AUX
cana-3289	117	53	a	a	DET
cana-3289	117	54	bfpi	bfpi	NOUN
cana-3289	117	55	of	of	ADP
cana-3289	117	56	𝔏	𝔏	PROPN
cana-3289	117	57	)	)	PUNCT
cana-3289	117	58	⇒	⇒	PROPN
cana-3289	117	59	bp	bp	PROPN
cana-3289	117	60	(	(	PUNCT
cana-3289	117	61	ℏ	ℏ	PROPN
cana-3289	117	62	)	)	PUNCT
cana-3289	117	63	≥	≥	NOUN
cana-3289	117	64	α	α	NOUN
cana-3289	117	65	or	or	CCONJ
cana-3289	117	66	bp	bp	PROPN
cana-3289	117	67	(	(	PUNCT
cana-3289	117	68	m	m	PROPN
cana-3289	117	69	)	)	PUNCT
cana-3289	117	70	≥	≥	PROPN
cana-3289	117	71	α	α	NOUN
cana-3289	117	72	,	,	PUNCT
cana-3289	117	73	bn	bn	INTJ
cana-3289	117	74	(	(	PUNCT
cana-3289	117	75	ℏ	ℏ	NOUN
cana-3289	117	76	)	)	PUNCT
cana-3289	117	77	≤	≤	NOUN
cana-3289	117	78	ω	ω	NUM
cana-3289	117	79	or	or	CCONJ
cana-3289	117	80	bn	bn	ADJ
cana-3289	117	81	(	(	PUNCT
cana-3289	117	82	m	m	NOUN
cana-3289	117	83	)	)	PUNCT
cana-3289	117	84	≤	≤	NOUN
cana-3289	117	85	ω	ω	NUM
cana-3289	117	86	⇒	⇒	PROPN
cana-3289	117	87	bp	bp	PROPN
cana-3289	117	88	(	(	PUNCT
cana-3289	117	89	ℏ	ℏ	PROPN
cana-3289	117	90	)	)	PUNCT
cana-3289	117	91	≥	≥	NOUN
cana-3289	117	92	α	α	NOUN
cana-3289	117	93	and	and	CCONJ
cana-3289	117	94	bn	bn	INTJ
cana-3289	117	95	(	(	PUNCT
cana-3289	117	96	ℏ	ℏ	NOUN
cana-3289	117	97	)	)	PUNCT
cana-3289	117	98	≤	≤	NOUN
cana-3289	117	99	ω	ω	NUM
cana-3289	117	100	or	or	CCONJ
cana-3289	117	101	bp	bp	PROPN
cana-3289	117	102	(	(	PUNCT
cana-3289	117	103	m	m	PROPN
cana-3289	117	104	)	)	PUNCT
cana-3289	117	105	≥	≥	NOUN
cana-3289	117	106	α	α	NOUN
cana-3289	117	107	and	and	CCONJ
cana-3289	117	108	bn	bn	PROPN
cana-3289	117	109	(	(	PUNCT
cana-3289	117	110	m	m	NOUN
cana-3289	117	111	)	)	PUNCT
cana-3289	117	112	≤	≤	NUM
cana-3289	117	113	ω	ω	NUM
cana-3289	117	114	thus	thus	ADV
cana-3289	117	115	ℏ	ℏ	PROPN
cana-3289	117	116	∈	∈	PROPN
cana-3289	117	117	b	b	PROPN
cana-3289	117	118	(	(	PUNCT
cana-3289	117	119	α	α	PROPN
cana-3289	117	120	,	,	PUNCT
cana-3289	117	121	ω	ω	NOUN
cana-3289	117	122	)	)	PUNCT
cana-3289	117	123	or	or	CCONJ
cana-3289	117	124	s	s	PROPN
cana-3289	117	125	∈	∈	PROPN
cana-3289	117	126	b	b	PROPN
cana-3289	117	127	(	(	PUNCT
cana-3289	117	128	α	α	PROPN
cana-3289	117	129	,	,	PUNCT
cana-3289	117	130	ω	ω	NOUN
cana-3289	117	131	)	)	PUNCT
cana-3289	117	132	hence	hence	ADV
cana-3289	117	133	,	,	PUNCT
cana-3289	117	134	b	b	PROPN
cana-3289	117	135	(	(	PUNCT
cana-3289	117	136	α	α	PROPN
cana-3289	117	137	,	,	PUNCT
cana-3289	117	138	ω	ω	NOUN
cana-3289	117	139	)	)	PUNCT
cana-3289	117	140	is	be	AUX
cana-3289	117	141	a	a	DET
cana-3289	117	142	prime	prime	ADJ
cana-3289	117	143	ideal	ideal	NOUN
cana-3289	117	144	of	of	ADP
cana-3289	117	145	𝔏.	𝔏.	PROPN
cana-3289	117	146	in	in	ADP
cana-3289	117	147	converse	converse	NOUN
cana-3289	117	148	assume	assume	VERB
cana-3289	117	149	that	that	SCONJ
cana-3289	117	150	b	b	X
cana-3289	117	151	(	(	PUNCT
cana-3289	117	152	α	α	PROPN
cana-3289	117	153	,	,	PUNCT
cana-3289	117	154	ω	ω	NOUN
cana-3289	117	155	)	)	PUNCT
cana-3289	117	156	is	be	AUX
cana-3289	117	157	a	a	DET
cana-3289	117	158	prime	prime	ADJ
cana-3289	117	159	ideal	ideal	NOUN
cana-3289	117	160	of	of	ADP
cana-3289	117	161	𝔏.	𝔏.	PROPN
cana-3289	117	162	i.e	i.e	X
cana-3289	117	163	to	to	ADP
cana-3289	117	164	any	any	DET
cana-3289	117	165	ℏ	ℏ	NOUN
cana-3289	117	166	,	,	PUNCT
cana-3289	117	167	m	m	NOUN
cana-3289	117	168	∈	∈	NOUN
cana-3289	117	169	𝔏	𝔏	NOUN
cana-3289	117	170	and	and	CCONJ
cana-3289	117	171	ℏ	ℏ	PRON
cana-3289	117	172	∧	∧	PROPN
cana-3289	117	173	s	s	PART
cana-3289	117	174	∈	∈	PROPN
cana-3289	117	175	b	b	PROPN
cana-3289	117	176	(	(	PUNCT
cana-3289	117	177	α	α	PROPN
cana-3289	117	178	,	,	PUNCT
cana-3289	117	179	ω	ω	NOUN
cana-3289	117	180	)	)	PUNCT
cana-3289	118	1	then	then	ADV
cana-3289	118	2	either	either	CCONJ
cana-3289	118	3	ℏ	ℏ	PROPN
cana-3289	118	4	∈	∈	PROPN
cana-3289	118	5	b	b	PROPN
cana-3289	118	6	(	(	PUNCT
cana-3289	118	7	α	α	PROPN
cana-3289	118	8	,	,	PUNCT
cana-3289	118	9	ω	ω	NOUN
cana-3289	118	10	)	)	PUNCT
cana-3289	118	11	or	or	CCONJ
cana-3289	118	12	s	s	PROPN
cana-3289	118	13	∈	∈	PROPN
cana-3289	118	14	b	b	PROPN
cana-3289	118	15	(	(	PUNCT
cana-3289	118	16	α	α	PROPN
cana-3289	118	17	,	,	PUNCT
cana-3289	118	18	ω	ω	NOUN
cana-3289	118	19	)	)	PUNCT
cana-3289	118	20	.	.	PUNCT
cana-3289	119	1	we	we	PRON
cana-3289	119	2	have	have	VERB
cana-3289	119	3	to	to	PART
cana-3289	119	4	show	show	VERB
cana-3289	119	5	that	that	SCONJ
cana-3289	119	6	b	b	NOUN
cana-3289	119	7	is	be	AUX
cana-3289	119	8	a	a	DET
cana-3289	119	9	bfpi	bfpi	NOUN
cana-3289	119	10	of	of	ADP
cana-3289	119	11	𝔏.	𝔏.	PROPN
cana-3289	119	12	assume	assume	VERB
cana-3289	119	13	that	that	SCONJ
cana-3289	119	14	b	b	NOUN
cana-3289	119	15	is	be	AUX
cana-3289	119	16	not	not	PART
cana-3289	119	17	a	a	DET
cana-3289	119	18	bfpi	bfpi	NOUN
cana-3289	119	19	of	of	ADP
cana-3289	119	20	𝔏.	𝔏.	PROPN
cana-3289	119	21	thus	thus	ADV
cana-3289	119	22	bp	bp	PROPN
cana-3289	119	23	(	(	PUNCT
cana-3289	119	24	ℏ	ℏ	PROPN
cana-3289	119	25	∧	∧	PROPN
cana-3289	119	26	m	m	PROPN
cana-3289	119	27	)	)	PUNCT
cana-3289	119	28	>	>	X
cana-3289	119	29	max	max	PROPN
cana-3289	119	30	{	{	PUNCT
cana-3289	119	31	bp	bp	PROPN
cana-3289	119	32	(	(	PUNCT
cana-3289	119	33	ℏ	ℏ	PROPN
cana-3289	119	34	)	)	PUNCT
cana-3289	119	35	,	,	PUNCT
cana-3289	119	36	bp	bp	PROPN
cana-3289	119	37	(	(	PUNCT
cana-3289	119	38	m	m	NOUN
cana-3289	119	39	)	)	PUNCT
cana-3289	119	40	}	}	PUNCT
cana-3289	119	41	and	and	CCONJ
cana-3289	119	42	communications	communication	NOUN
cana-3289	119	43	on	on	ADP
cana-3289	119	44	applied	apply	VERB
cana-3289	119	45	nonlinear	nonlinear	ADJ
cana-3289	119	46	analysis	analysis	NOUN
cana-3289	119	47	issn	issn	NOUN
cana-3289	119	48	:	:	PUNCT
cana-3289	119	49	1074	1074	NUM
cana-3289	119	50	-	-	PUNCT
cana-3289	119	51	133x	133x	NUM
cana-3289	119	52	vol	vol	NOUN
cana-3289	119	53	32	32	NUM
cana-3289	119	54	no	no	NOUN
cana-3289	119	55	.	.	PUNCT
cana-3289	120	1	6s	6s	NUM
cana-3289	120	2	(	(	PUNCT
cana-3289	120	3	2025	2025	NUM
cana-3289	120	4	)	)	PUNCT
cana-3289	120	5	246	246	NUM
cana-3289	120	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3289	120	7	bn	bn	NOUN
cana-3289	120	8	(	(	PUNCT
cana-3289	120	9	ℏ	ℏ	PROPN
cana-3289	120	10	∧	∧	PROPN
cana-3289	120	11	m	m	NOUN
cana-3289	120	12	)	)	PUNCT
cana-3289	120	13	<	<	X
cana-3289	120	14	min	min	X
cana-3289	120	15	{	{	PUNCT
cana-3289	120	16	bn	bn	INTJ
cana-3289	120	17	(	(	PUNCT
cana-3289	120	18	ℏ	ℏ	PROPN
cana-3289	120	19	)	)	PUNCT
cana-3289	120	20	,	,	PUNCT
cana-3289	120	21	bn	bn	INTJ
cana-3289	120	22	(	(	PUNCT
cana-3289	120	23	m	m	NOUN
cana-3289	120	24	)	)	PUNCT
cana-3289	120	25	}	}	PUNCT
cana-3289	120	26	⇒	⇒	PROPN
cana-3289	120	27	bp	bp	PROPN
cana-3289	120	28	(	(	PUNCT
cana-3289	120	29	ℏ	ℏ	PROPN
cana-3289	120	30	∧	∧	PROPN
cana-3289	120	31	m	m	PROPN
cana-3289	120	32	)	)	PUNCT
cana-3289	120	33	>	>	X
cana-3289	120	34	bp	bp	PROPN
cana-3289	120	35	(	(	PUNCT
cana-3289	120	36	ℏ	ℏ	PROPN
cana-3289	120	37	)	)	PUNCT
cana-3289	120	38	and	and	CCONJ
cana-3289	120	39	bp	bp	PROPN
cana-3289	120	40	(	(	PUNCT
cana-3289	120	41	ℏ	ℏ	PROPN
cana-3289	120	42	∧	∧	PROPN
cana-3289	120	43	m	m	PROPN
cana-3289	120	44	)	)	PUNCT
cana-3289	120	45	>	>	X
cana-3289	120	46	bp	bp	PROPN
cana-3289	120	47	(	(	PUNCT
cana-3289	120	48	m	m	PROPN
cana-3289	120	49	)	)	PUNCT
cana-3289	120	50	,	,	PUNCT
cana-3289	120	51	bn	bn	INTJ
cana-3289	120	52	(	(	PUNCT
cana-3289	120	53	ℏ	ℏ	PROPN
cana-3289	120	54	∧	∧	PROPN
cana-3289	120	55	m	m	NOUN
cana-3289	120	56	)	)	PUNCT
cana-3289	120	57	<	<	X
cana-3289	120	58	bn	bn	INTJ
cana-3289	120	59	(	(	PUNCT
cana-3289	120	60	ℏ	ℏ	PROPN
cana-3289	120	61	)	)	PUNCT
cana-3289	120	62	and	and	CCONJ
cana-3289	120	63	bn	bn	INTJ
cana-3289	120	64	(	(	PUNCT
cana-3289	120	65	ℏ	ℏ	PROPN
cana-3289	120	66	∧	∧	PROPN
cana-3289	120	67	m	m	NOUN
cana-3289	120	68	)	)	PUNCT
cana-3289	120	69	<	<	X
cana-3289	120	70	bn	bn	INTJ
cana-3289	120	71	(	(	PUNCT
cana-3289	120	72	m	m	PROPN
cana-3289	120	73	)	)	PUNCT
cana-3289	120	74	.	.	PUNCT
cana-3289	120	75	suppose	suppose	VERB
cana-3289	120	76	that	that	SCONJ
cana-3289	120	77	bp	bp	PROPN
cana-3289	120	78	(	(	PUNCT
cana-3289	120	79	ℏ	ℏ	PROPN
cana-3289	120	80	∧	∧	PROPN
cana-3289	120	81	m	m	NOUN
cana-3289	120	82	)	)	PUNCT
cana-3289	120	83	=	=	SYM
cana-3289	120	84	α	α	PROPN
cana-3289	120	85	and	and	CCONJ
cana-3289	120	86	bn	bn	INTJ
cana-3289	120	87	(	(	PUNCT
cana-3289	120	88	ℏ	ℏ	PROPN
cana-3289	120	89	∧	∧	PROPN
cana-3289	120	90	m	m	NOUN
cana-3289	120	91	)	)	PUNCT
cana-3289	120	92	=	=	SYM
cana-3289	120	93	ω	ω	PROPN
cana-3289	120	94	⇒	⇒	PROPN
cana-3289	120	95	bp	bp	PROPN
cana-3289	120	96	(	(	PUNCT
cana-3289	120	97	ℏ	ℏ	PROPN
cana-3289	120	98	)	)	PUNCT
cana-3289	120	99	<	<	X
cana-3289	120	100	α	α	PROPN
cana-3289	120	101	and	and	CCONJ
cana-3289	120	102	bn	bn	INTJ
cana-3289	120	103	(	(	PUNCT
cana-3289	120	104	ℏ	ℏ	PROPN
cana-3289	120	105	)	)	PUNCT
cana-3289	120	106	>	>	X
cana-3289	120	107	ω	ω	PROPN
cana-3289	120	108	,	,	PUNCT
cana-3289	120	109	bp	bp	PROPN
cana-3289	120	110	(	(	PUNCT
cana-3289	120	111	m	m	PROPN
cana-3289	120	112	)	)	PUNCT
cana-3289	120	113	<	<	X
cana-3289	120	114	α	α	PROPN
cana-3289	120	115	and	and	CCONJ
cana-3289	120	116	bn	bn	PROPN
cana-3289	120	117	(	(	PUNCT
cana-3289	120	118	m	m	PROPN
cana-3289	120	119	)	)	PUNCT
cana-3289	120	120	>	>	X
cana-3289	120	121	ω	ω	X
cana-3289	120	122	.	.	PUNCT
cana-3289	120	123	hence	hence	ADV
cana-3289	120	124	ℏ	ℏ	PROPN
cana-3289	120	125	,	,	PUNCT
cana-3289	120	126	m	m	VERB
cana-3289	120	127	∈/	∈/	PROPN
cana-3289	120	128	b	b	PROPN
cana-3289	120	129	(	(	PUNCT
cana-3289	120	130	α	α	PROPN
cana-3289	120	131	,	,	PUNCT
cana-3289	120	132	ω	ω	NOUN
cana-3289	120	133	)	)	PUNCT
cana-3289	120	134	.	.	PUNCT
cana-3289	121	1	this	this	PRON
cana-3289	121	2	is	be	AUX
cana-3289	121	3	a	a	DET
cana-3289	121	4	contradiction	contradiction	NOUN
cana-3289	121	5	to	to	ADP
cana-3289	121	6	the	the	DET
cana-3289	121	7	fact	fact	NOUN
cana-3289	121	8	that	that	SCONJ
cana-3289	121	9	b	b	X
cana-3289	121	10	(	(	PUNCT
cana-3289	121	11	α	α	PROPN
cana-3289	121	12	,	,	PUNCT
cana-3289	121	13	ω	ω	NOUN
cana-3289	121	14	)	)	PUNCT
cana-3289	121	15	is	be	AUX
cana-3289	121	16	a	a	DET
cana-3289	121	17	prime	prime	ADJ
cana-3289	121	18	ideal	ideal	NOUN
cana-3289	121	19	of	of	ADP
cana-3289	121	20	𝔏	𝔏	PROPN
cana-3289	121	21	for	for	ADP
cana-3289	121	22	any	any	DET
cana-3289	121	23	α	α	NOUN
cana-3289	121	24	∈	∈	PROPN
cana-3289	122	1	[	[	X
cana-3289	122	2	0	0	NUM
cana-3289	122	3	,	,	PUNCT
cana-3289	122	4	1	1	NUM
cana-3289	122	5	]	]	PUNCT
cana-3289	122	6	and	and	CCONJ
cana-3289	122	7	ω	ω	NUM
cana-3289	122	8	∈	∈	PROPN
cana-3289	123	1	[	[	X
cana-3289	123	2	−1	−1	NOUN
cana-3289	123	3	,	,	PUNCT
cana-3289	123	4	0	0	NUM
cana-3289	123	5	]	]	PUNCT
cana-3289	123	6	.	.	PUNCT
cana-3289	124	1	hence	hence	ADV
cana-3289	124	2	b	b	PROPN
cana-3289	124	3	is	be	AUX
cana-3289	124	4	a	a	DET
cana-3289	124	5	bfpi	bfpi	NOUN
cana-3289	124	6	of	of	ADP
cana-3289	124	7	𝔏.	𝔏.	PROPN
cana-3289	124	8	theorem	theorem	VERB
cana-3289	124	9	3.8	3.8	NUM
cana-3289	124	10	:	:	PUNCT
cana-3289	124	11	let	let	VERB
cana-3289	124	12	𝔏	𝔏	PRON
cana-3289	124	13	be	be	AUX
cana-3289	124	14	a	a	DET
cana-3289	124	15	lattice	lattice	NOUN
cana-3289	124	16	and	and	CCONJ
cana-3289	124	17	b	b	NOUN
cana-3289	124	18	∈	∈	PROPN
cana-3289	124	19	bfs	bfs	NOUN
cana-3289	124	20	(	(	PUNCT
cana-3289	124	21	𝔏	𝔏	PROPN
cana-3289	124	22	)	)	PUNCT
cana-3289	124	23	.	.	PUNCT
cana-3289	125	1	if	if	SCONJ
cana-3289	125	2	b	b	NOUN
cana-3289	125	3	,	,	PUNCT
cana-3289	125	4	bfpi	bfpi	ADJ
cana-3289	125	5	of	of	ADP
cana-3289	125	6	𝔏	𝔏	PROPN
cana-3289	125	7	then	then	ADV
cana-3289	125	8	we	we	PRON
cana-3289	125	9	have	have	VERB
cana-3289	125	10	supp(b	supp(b	PROPN
cana-3289	125	11	)	)	PUNCT
cana-3289	125	12	forms	form	VERB
cana-3289	125	13	a	a	DET
cana-3289	125	14	crisp	crisp	ADJ
cana-3289	125	15	prime	prime	ADJ
cana-3289	125	16	ideal	ideal	NOUN
cana-3289	125	17	of	of	ADP
cana-3289	125	18	𝔏.	𝔏.	PROPN
cana-3289	125	19	proof	proof	NOUN
cana-3289	125	20	:	:	PUNCT
cana-3289	125	21	suppose	suppose	VERB
cana-3289	125	22	b	b	X
cana-3289	125	23	=	=	PRON
cana-3289	125	24	{	{	PUNCT
cana-3289	125	25	<	<	X
cana-3289	125	26	ℏ	ℏ	PROPN
cana-3289	125	27	,	,	PUNCT
cana-3289	125	28	bp	bp	PROPN
cana-3289	125	29	(	(	PUNCT
cana-3289	125	30	ℏ	ℏ	PROPN
cana-3289	125	31	)	)	PUNCT
cana-3289	125	32	,	,	PUNCT
cana-3289	125	33	bn	bn	INTJ
cana-3289	125	34	(	(	PUNCT
cana-3289	125	35	ℏ	ℏ	PROPN
cana-3289	125	36	)	)	PUNCT
cana-3289	125	37	>	>	X
cana-3289	125	38	/ℏ	/ℏ	PUNCT
cana-3289	125	39	∈	∈	PROPN
cana-3289	125	40	𝔏	𝔏	PROPN
cana-3289	125	41	}	}	PUNCT
cana-3289	125	42	∈	∈	PROPN
cana-3289	125	43	bfs	bfs	NOUN
cana-3289	125	44	(	(	PUNCT
cana-3289	125	45	𝔏	𝔏	PROPN
cana-3289	125	46	)	)	PUNCT
cana-3289	125	47	.	.	PUNCT
cana-3289	126	1	given	give	VERB
cana-3289	126	2	b	b	PROPN
cana-3289	126	3	is	be	AUX
cana-3289	126	4	a	a	DET
cana-3289	126	5	bfpi	bfpi	NOUN
cana-3289	126	6	of	of	ADP
cana-3289	126	7	𝔏.	𝔏.	PROPN
cana-3289	126	8	from	from	ADP
cana-3289	126	9	[	[	X
cana-3289	126	10	]	]	X
cana-3289	126	11	we	we	PRON
cana-3289	126	12	have	have	AUX
cana-3289	126	13	supp(b	supp(b	PROPN
cana-3289	126	14	)	)	PUNCT
cana-3289	126	15	is	be	AUX
cana-3289	126	16	a	a	DET
cana-3289	126	17	crisp	crisp	ADJ
cana-3289	126	18	ideal	ideal	NOUN
cana-3289	126	19	in	in	ADP
cana-3289	126	20	𝔏.	𝔏.	PROPN
cana-3289	126	21	now	now	ADV
cana-3289	126	22	we	we	PRON
cana-3289	126	23	prove	prove	VERB
cana-3289	126	24	that	that	SCONJ
cana-3289	126	25	supp(b	supp(b	NOUN
cana-3289	126	26	)	)	PUNCT
cana-3289	126	27	is	be	AUX
cana-3289	126	28	prime	prime	ADJ
cana-3289	126	29	.	.	PUNCT
cana-3289	127	1	suppose	suppose	VERB
cana-3289	127	2	ℏ	ℏ	X
cana-3289	127	3	,	,	PUNCT
cana-3289	127	4	m	m	NOUN
cana-3289	127	5	∈	∈	NOUN
cana-3289	127	6	𝔏	𝔏	NOUN
cana-3289	127	7	so	so	SCONJ
cana-3289	127	8	that	that	SCONJ
cana-3289	127	9	ℏ	ℏ	PROPN
cana-3289	127	10	∧	∧	PROPN
cana-3289	127	11	s	s	PART
cana-3289	127	12	∈	∈	PROPN
cana-3289	127	13	supp(b	supp(b	PROPN
cana-3289	127	14	)	)	PUNCT
cana-3289	127	15	.	.	PUNCT
cana-3289	128	1	thus	thus	ADV
cana-3289	128	2	bp	bp	PROPN
cana-3289	128	3	(	(	PUNCT
cana-3289	128	4	ℏ	ℏ	PROPN
cana-3289	128	5	∧	∧	PROPN
cana-3289	128	6	m	m	NOUN
cana-3289	128	7	)	)	PUNCT
cana-3289	128	8	̸=	̸=	NOUN
cana-3289	128	9	0	0	NUM
cana-3289	128	10	or	or	CCONJ
cana-3289	128	11	bn	bn	ADJ
cana-3289	128	12	(	(	PUNCT
cana-3289	128	13	ℏ	ℏ	PROPN
cana-3289	128	14	∧	∧	PROPN
cana-3289	128	15	m	m	NOUN
cana-3289	128	16	)	)	PUNCT
cana-3289	128	17	̸=	̸=	PROPN
cana-3289	128	18	0	0	NUM
cana-3289	128	19	.	.	PUNCT
cana-3289	129	1	but	but	CCONJ
cana-3289	129	2	bp	bp	PROPN
cana-3289	129	3	(	(	PUNCT
cana-3289	129	4	ℏ	ℏ	PROPN
cana-3289	129	5	∧	∧	PROPN
cana-3289	129	6	m	m	NOUN
cana-3289	129	7	)	)	PUNCT
cana-3289	129	8	=	=	SYM
cana-3289	129	9	max	max	PROPN
cana-3289	129	10	{	{	PUNCT
cana-3289	129	11	bp	bp	PROPN
cana-3289	129	12	(	(	PUNCT
cana-3289	129	13	ℏ	ℏ	PROPN
cana-3289	129	14	)	)	PUNCT
cana-3289	129	15	,	,	PUNCT
cana-3289	129	16	bp	bp	PROPN
cana-3289	129	17	(	(	PUNCT
cana-3289	129	18	m	m	PROPN
cana-3289	129	19	)	)	PUNCT
cana-3289	129	20	}	}	PUNCT
cana-3289	129	21	(	(	PUNCT
cana-3289	129	22	since	since	SCONJ
cana-3289	129	23	b	b	NOUN
cana-3289	129	24	is	be	AUX
cana-3289	129	25	bfpi	bfpi	ADJ
cana-3289	129	26	of	of	ADP
cana-3289	129	27	𝔏	𝔏	PROPN
cana-3289	129	28	)	)	PUNCT
cana-3289	129	29	.	.	PUNCT
cana-3289	130	1	⇒	⇒	NOUN
cana-3289	130	2	either	either	CCONJ
cana-3289	130	3	bp	bp	PROPN
cana-3289	130	4	(	(	PUNCT
cana-3289	130	5	ℏ	ℏ	NOUN
cana-3289	130	6	)	)	PUNCT
cana-3289	130	7	≠0	≠0	NOUN
cana-3289	130	8	,	,	PUNCT
cana-3289	130	9	or	or	CCONJ
cana-3289	130	10	bp	bp	PROPN
cana-3289	130	11	(	(	PUNCT
cana-3289	130	12	m	m	NOUN
cana-3289	130	13	)	)	PUNCT
cana-3289	130	14	≠	≠	PROPN
cana-3289	130	15	0	0	NUM
cana-3289	131	1	likewise	likewise	ADV
cana-3289	131	2	,	,	PUNCT
cana-3289	131	3	we	we	PRON
cana-3289	131	4	have	have	VERB
cana-3289	131	5	either	either	CCONJ
cana-3289	131	6	bn	bn	INTJ
cana-3289	131	7	(	(	PUNCT
cana-3289	131	8	ℏ	ℏ	NOUN
cana-3289	131	9	)	)	PUNCT
cana-3289	131	10	≠	≠	PROPN
cana-3289	131	11	0	0	NUM
cana-3289	131	12	,	,	PUNCT
cana-3289	131	13	or	or	CCONJ
cana-3289	131	14	bn	bn	INTJ
cana-3289	131	15	(	(	PUNCT
cana-3289	131	16	m	m	NOUN
cana-3289	131	17	)	)	PUNCT
cana-3289	131	18	≠	≠	PROPN
cana-3289	131	19	0	0	NUM
cana-3289	131	20	⇒	⇒	NOUN
cana-3289	131	21	either	either	CCONJ
cana-3289	131	22	ℏ	ℏ	PROPN
cana-3289	131	23	∈	∈	PROPN
cana-3289	131	24	supp(b	supp(b	PROPN
cana-3289	131	25	)	)	PUNCT
cana-3289	131	26	or	or	CCONJ
cana-3289	131	27	m	m	PROPN
cana-3289	131	28	∈	∈	PROPN
cana-3289	131	29	supp(b	supp(b	PROPN
cana-3289	131	30	)	)	PUNCT
cana-3289	131	31	.	.	PUNCT
cana-3289	132	1	hence	hence	ADV
cana-3289	132	2	supp(b	supp(b	PROPN
cana-3289	132	3	)	)	PUNCT
cana-3289	132	4	is	be	AUX
cana-3289	132	5	a	a	DET
cana-3289	132	6	prime	prime	ADJ
cana-3289	132	7	ideal	ideal	NOUN
cana-3289	132	8	of	of	ADP
cana-3289	132	9	𝔏.	𝔏.	PROPN
cana-3289	132	10	remark	remark	NOUN
cana-3289	132	11	3.9	3.9	NUM
cana-3289	132	12	:	:	PUNCT
cana-3289	132	13	converse	converse	NOUN
cana-3289	132	14	part	part	NOUN
cana-3289	132	15	of	of	ADP
cana-3289	132	16	the	the	DET
cana-3289	132	17	above	above	ADJ
cana-3289	132	18	theorem	theorem	NOUN
cana-3289	132	19	does	do	AUX
cana-3289	132	20	not	not	PART
cana-3289	132	21	hold	hold	VERB
cana-3289	132	22	in	in	ADP
cana-3289	132	23	general	general	ADJ
cana-3289	132	24	.	.	PUNCT
cana-3289	133	1	suppose	suppose	VERB
cana-3289	133	2	b	b	X
cana-3289	133	3	=	=	PRON
cana-3289	133	4	{	{	PUNCT
cana-3289	133	5	<	<	X
cana-3289	133	6	1	1	NUM
cana-3289	133	7	,	,	PUNCT
cana-3289	133	8	0.5	0.5	NUM
cana-3289	133	9	,	,	PUNCT
cana-3289	133	10	−0.1	−0.1	PROPN
cana-3289	133	11	>	>	X
cana-3289	133	12	,	,	PUNCT
cana-3289	133	13	<	<	X
cana-3289	133	14	2	2	NUM
cana-3289	133	15	,	,	PUNCT
cana-3289	133	16	0.7	0.7	NUM
cana-3289	133	17	,	,	PUNCT
cana-3289	133	18	−0.2	−0.2	PROPN
cana-3289	133	19	>	>	PUNCT
cana-3289	133	20	,	,	PUNCT
cana-3289	133	21	<	<	X
cana-3289	133	22	3	3	NUM
cana-3289	133	23	,	,	PUNCT
cana-3289	133	24	0.8	0.8	NUM
cana-3289	133	25	,	,	PUNCT
cana-3289	133	26	−0.05	−0.05	NOUN
cana-3289	133	27	>	>	X
cana-3289	133	28	,	,	PUNCT
cana-3289	133	29	<	<	X
cana-3289	133	30	6	6	NUM
cana-3289	133	31	,	,	PUNCT
cana-3289	133	32	0.4	0.4	NUM
cana-3289	133	33	,	,	PUNCT
cana-3289	133	34	−0.01	−0.01	INTJ
cana-3289	133	35	>	>	X
cana-3289	133	36	}	}	PUNCT
cana-3289	133	37	be	be	AUX
cana-3289	133	38	a	a	DET
cana-3289	133	39	bf	bf	NOUN
cana-3289	133	40	subset	subset	NOUN
cana-3289	133	41	in	in	ADP
cana-3289	133	42	𝔏	𝔏	PROPN
cana-3289	133	43	=	=	PUNCT
cana-3289	133	44	{	{	PUNCT
cana-3289	133	45	1	1	NUM
cana-3289	133	46	,	,	PUNCT
cana-3289	133	47	2	2	NUM
cana-3289	133	48	,	,	PUNCT
cana-3289	133	49	3	3	NUM
cana-3289	133	50	,	,	PUNCT
cana-3289	133	51	6	6	NUM
cana-3289	133	52	}	}	PUNCT
cana-3289	133	53	with	with	ADP
cana-3289	133	54	divisors	divisor	NOUN
cana-3289	133	55	of	of	ADP
cana-3289	133	56	6	6	NUM
cana-3289	133	57	.	.	PUNCT
cana-3289	134	1	supp(b	supp(b	NUM
cana-3289	134	2	)	)	PUNCT
cana-3289	135	1	=	=	PRON
cana-3289	135	2	{	{	PUNCT
cana-3289	135	3	1	1	NUM
cana-3289	135	4	,	,	PUNCT
cana-3289	135	5	2	2	NUM
cana-3289	135	6	,	,	PUNCT
cana-3289	135	7	3	3	NUM
cana-3289	135	8	,	,	PUNCT
cana-3289	135	9	6	6	NUM
cana-3289	135	10	}	}	PUNCT
cana-3289	135	11	is	be	AUX
cana-3289	135	12	a	a	DET
cana-3289	135	13	prime	prime	ADJ
cana-3289	135	14	ideal	ideal	NOUN
cana-3289	135	15	of	of	ADP
cana-3289	135	16	𝔏.	𝔏.	PROPN
cana-3289	135	17	but	but	CCONJ
cana-3289	135	18	bp	bp	PROPN
cana-3289	135	19	(	(	PUNCT
cana-3289	135	20	2	2	NUM
cana-3289	135	21	∧	∧	PROPN
cana-3289	135	22	3	3	NUM
cana-3289	135	23	)	)	PUNCT
cana-3289	135	24	=	=	SYM
cana-3289	135	25	bp	bp	PROPN
cana-3289	135	26	(	(	PUNCT
cana-3289	135	27	1	1	NUM
cana-3289	135	28	)	)	PUNCT
cana-3289	135	29	=	=	SYM
cana-3289	135	30	0.5	0.5	NUM
cana-3289	135	31	≥	≥	NOUN
cana-3289	135	32	max{bp	max{bp	NOUN
cana-3289	135	33	(	(	PUNCT
cana-3289	135	34	2	2	NUM
cana-3289	135	35	)	)	PUNCT
cana-3289	135	36	,	,	PUNCT
cana-3289	135	37	bp	bp	PROPN
cana-3289	135	38	(	(	PUNCT
cana-3289	135	39	8)	8)	NUM
cana-3289	135	40	}	}	PUNCT
cana-3289	135	41	=	=	PUNCT
cana-3289	135	42	0.8	0.8	NUM
cana-3289	135	43	⇒	⇒	NOUN
cana-3289	135	44	b	b	NOUN
cana-3289	135	45	is	be	AUX
cana-3289	135	46	not	not	PART
cana-3289	135	47	a	a	DET
cana-3289	135	48	bfpi	bfpi	NOUN
cana-3289	135	49	of	of	ADP
cana-3289	135	50	𝔏.	𝔏.	PROPN
cana-3289	135	51	theorem	theorem	NOUN
cana-3289	135	52	3.10	3.10	NUM
cana-3289	135	53	:	:	PUNCT
cana-3289	135	54	let	let	VERB
cana-3289	135	55	b	b	X
cana-3289	135	56	be	be	AUX
cana-3289	135	57	a	a	DET
cana-3289	135	58	bfs	bfs	NOUN
cana-3289	135	59	of	of	ADP
cana-3289	135	60	lattice	lattice	PROPN
cana-3289	135	61	𝔏.	𝔏.	PROPN
cana-3289	135	62	then	then	ADV
cana-3289	135	63	,	,	PUNCT
cana-3289	135	64	b	b	PROPN
cana-3289	135	65	is	be	AUX
cana-3289	135	66	a	a	DET
cana-3289	135	67	bfpi	bfpi	NOUN
cana-3289	135	68	of	of	ADP
cana-3289	135	69	𝔏	𝔏	PROPN
cana-3289	135	70	if	if	SCONJ
cana-3289	135	71	and	and	CCONJ
cana-3289	135	72	only	only	ADV
cana-3289	135	73	if	if	SCONJ
cana-3289	135	74	f	f	PROPN
cana-3289	135	75	the	the	DET
cana-3289	135	76	bfmt	bfmt	NOUN
cana-3289	135	77	m	m	VERB
cana-3289	135	78	of	of	ADP
cana-3289	136	1	b	b	PROPN
cana-3289	136	2	form	form	NOUN
cana-3289	136	3	bf	bf	NOUN
cana-3289	136	4	prime	prime	ADJ
cana-3289	136	5	ideal	ideal	NOUN
cana-3289	136	6	of	of	ADP
cana-3289	136	7	𝔏.	𝔏.	PROPN
cana-3289	136	8	proof	proof	NOUN
cana-3289	136	9	:	:	PUNCT
cana-3289	136	10	assume	assume	VERB
cana-3289	136	11	that	that	SCONJ
cana-3289	136	12	𝔅	𝔅	NOUN
cana-3289	136	13	is	be	AUX
cana-3289	136	14	a	a	DET
cana-3289	136	15	bf	bf	NOUN
cana-3289	136	16	primeideal	primeideal	NOUN
cana-3289	136	17	of	of	ADP
cana-3289	136	18	𝔏	𝔏	PROPN
cana-3289	136	19	and	and	CCONJ
cana-3289	136	20	𝑀	𝑀	PROPN
cana-3289	136	21	be	be	VERB
cana-3289	136	22	a	a	DET
cana-3289	136	23	bfmt	bfmt	NOUN
cana-3289	136	24	of	of	ADP
cana-3289	136	25	𝔅.	𝔅.	PROPN
cana-3289	136	26	now	now	ADV
cana-3289	136	27	we	we	PRON
cana-3289	136	28	have	have	VERB
cana-3289	136	29	to	to	PART
cana-3289	136	30	show	show	VERB
cana-3289	136	31	that	that	SCONJ
cana-3289	136	32	𝑀	𝑀	PROPN
cana-3289	136	33	is	be	AUX
cana-3289	136	34	a	a	DET
cana-3289	136	35	bfpi	bfpi	NOUN
cana-3289	136	36	of	of	ADP
cana-3289	136	37	𝔏.	𝔏.	PROPN
cana-3289	136	38	let	let	VERB
cana-3289	136	39	ℏ	ℏ	VERB
cana-3289	136	40	,	,	PUNCT
cana-3289	136	41	𝑚	𝑚	PROPN
cana-3289	136	42	∈	∈	PROPN
cana-3289	136	43	𝔏.	𝔏.	PROPN
cana-3289	136	44	communications	communication	NOUN
cana-3289	136	45	on	on	ADP
cana-3289	136	46	applied	apply	VERB
cana-3289	136	47	nonlinear	nonlinear	ADJ
cana-3289	136	48	analysis	analysis	NOUN
cana-3289	136	49	issn	issn	NOUN
cana-3289	136	50	:	:	PUNCT
cana-3289	136	51	1074	1074	NUM
cana-3289	136	52	-	-	PUNCT
cana-3289	136	53	133x	133x	NUM
cana-3289	136	54	vol	vol	NOUN
cana-3289	136	55	32	32	NUM
cana-3289	136	56	no	no	NOUN
cana-3289	136	57	.	.	PUNCT
cana-3289	137	1	6s	6s	NUM
cana-3289	137	2	(	(	PUNCT
cana-3289	137	3	2025	2025	NUM
cana-3289	137	4	)	)	PUNCT
cana-3289	137	5	247	247	NUM
cana-3289	137	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3289	137	7	now	now	ADV
cana-3289	137	8	,	,	PUNCT
cana-3289	137	9	consider	consider	VERB
cana-3289	137	10	𝔅(𝜔,𝜗	𝔅(𝜔,𝜗	NOUN
cana-3289	137	11	)	)	PUNCT
cana-3289	137	12	𝑃	𝑃	PROPN
cana-3289	137	13	(	(	PUNCT
cana-3289	137	14	ℏ	ℏ	NOUN
cana-3289	137	15	∧	∧	PROPN
cana-3289	137	16	𝑚	𝑚	NOUN
cana-3289	137	17	)	)	PUNCT
cana-3289	137	18	=	=	SYM
cana-3289	137	19	𝜔𝔅𝑃(ℏ	𝜔𝔅𝑃(ℏ	PUNCT
cana-3289	137	20	∧	∧	PROPN
cana-3289	137	21	𝑚	𝑚	NOUN
cana-3289	137	22	)	)	PUNCT
cana-3289	138	1	+	+	CCONJ
cana-3289	138	2	𝜗.	𝜗.	PROPN
cana-3289	138	3	≤	≤	PROPN
cana-3289	138	4	𝜔	𝜔	PRON
cana-3289	138	5	max{𝔅𝑃(ℏ	max{𝔅𝑃(ℏ	PROPN
cana-3289	138	6	)	)	PUNCT
cana-3289	138	7	,	,	PUNCT
cana-3289	138	8	𝔅𝑃(𝑚	𝔅𝑃(𝑚	NOUN
cana-3289	138	9	)	)	PUNCT
cana-3289	138	10	}	}	PUNCT
cana-3289	139	1	+	+	NUM
cana-3289	139	2	𝜗	𝜗	X
cana-3289	139	3	。	。	PUNCT
cana-3289	139	4	=	=	SYM
cana-3289	139	5	max{𝜔𝔅𝑃(ℏ	max{𝜔𝔅𝑃(ℏ	PROPN
cana-3289	139	6	)	)	PUNCT
cana-3289	139	7	+	+	NUM
cana-3289	139	8	𝜗	𝜗	NOUN
cana-3289	139	9	,	,	PUNCT
cana-3289	139	10	𝜔𝔅𝑃(𝑚	𝜔𝔅𝑃(𝑚	NOUN
cana-3289	139	11	)	)	PUNCT
cana-3289	139	12	+	+	NUM
cana-3289	139	13	𝜗	𝜗	NOUN
cana-3289	139	14	}	}	PUNCT
cana-3289	139	15	=	=	SYM
cana-3289	139	16	max{𝔅(𝜔,𝜗	max{𝔅(𝜔,𝜗	PROPN
cana-3289	139	17	)	)	PUNCT
cana-3289	139	18	𝑃	𝑃	PROPN
cana-3289	139	19	(	(	PUNCT
cana-3289	139	20	ℏ	ℏ	NOUN
cana-3289	139	21	)	)	PUNCT
cana-3289	139	22	,	,	PUNCT
cana-3289	139	23	𝔅(𝜔,𝜗	𝔅(𝜔,𝜗	NOUN
cana-3289	139	24	)	)	PUNCT
cana-3289	139	25	𝑃	𝑃	PROPN
cana-3289	139	26	(	(	PUNCT
cana-3289	139	27	𝑚	𝑚	NOUN
cana-3289	139	28	)	)	PUNCT
cana-3289	139	29	}	}	PUNCT
cana-3289	139	30	thus	thus	ADV
cana-3289	139	31	𝔅(𝜔,𝜗	𝔅(𝜔,𝜗	NOUN
cana-3289	139	32	)	)	PUNCT
cana-3289	139	33	𝑃	𝑃	PROPN
cana-3289	139	34	(	(	PUNCT
cana-3289	139	35	ℏ	ℏ	NOUN
cana-3289	139	36	∧	∧	PROPN
cana-3289	139	37	𝑚	𝑚	NOUN
cana-3289	139	38	)	)	PUNCT
cana-3289	139	39	≤	≤	NOUN
cana-3289	139	40	max{𝔅(𝜔,𝜗	max{𝔅(𝜔,𝜗	NOUN
cana-3289	139	41	)	)	PUNCT
cana-3289	139	42	𝑃	𝑃	PROPN
cana-3289	139	43	(	(	PUNCT
cana-3289	139	44	ℏ	ℏ	NOUN
cana-3289	139	45	)	)	PUNCT
cana-3289	139	46	,	,	PUNCT
cana-3289	139	47	𝔅(𝜔,𝜗	𝔅(𝜔,𝜗	NOUN
cana-3289	139	48	)	)	PUNCT
cana-3289	139	49	𝑃	𝑃	PROPN
cana-3289	139	50	(	(	PUNCT
cana-3289	139	51	𝑚	𝑚	NOUN
cana-3289	139	52	)	)	PUNCT
cana-3289	139	53	}	}	PUNCT
cana-3289	139	54	now	now	ADV
cana-3289	139	55	𝔅(𝛼,𝜃	𝔅(𝛼,𝜃	X
cana-3289	139	56	)	)	PUNCT
cana-3289	139	57	𝑁	𝑁	PROPN
cana-3289	139	58	(	(	PUNCT
cana-3289	139	59	ℏ	ℏ	PROPN
cana-3289	139	60	∧	∧	PROPN
cana-3289	139	61	𝑚	𝑚	NOUN
cana-3289	139	62	)	)	PUNCT
cana-3289	139	63	=	=	SYM
cana-3289	139	64	𝛼𝔅𝑁(ℏ	𝛼𝔅𝑁(ℏ	NUM
cana-3289	139	65	∧	∧	PROPN
cana-3289	139	66	𝑚	𝑚	NOUN
cana-3289	139	67	)	)	PUNCT
cana-3289	139	68	+	+	CCONJ
cana-3289	139	69	𝜃.	𝜃.	ADP
cana-3289	139	70	≥	≥	NOUN
cana-3289	139	71	𝛼min{𝔅𝑁(ℏ	𝛼min{𝔅𝑁(ℏ	NUM
cana-3289	139	72	)	)	PUNCT
cana-3289	139	73	,	,	PUNCT
cana-3289	139	74	𝔅𝑁(𝑚	𝔅𝑁(𝑚	PROPN
cana-3289	139	75	)	)	PUNCT
cana-3289	139	76	}	}	PUNCT
cana-3289	139	77	+	+	CCONJ
cana-3289	139	78	𝜃.	𝜃.	NOUN
cana-3289	139	79	=	=	SYM
cana-3289	139	80	min{𝛼𝔅𝑁(ℏ	min{𝛼𝔅𝑁(ℏ	NOUN
cana-3289	139	81	)	)	PUNCT
cana-3289	139	82	+	+	NUM
cana-3289	139	83	𝜃	𝜃	NOUN
cana-3289	139	84	,	,	PUNCT
cana-3289	139	85	𝛼𝔅𝑁(𝑚	𝛼𝔅𝑁(𝑚	NOUN
cana-3289	139	86	)	)	PUNCT
cana-3289	139	87	+	+	CCONJ
cana-3289	139	88	𝜃	𝜃	X
cana-3289	139	89	}	}	PUNCT
cana-3289	139	90	=	=	SYM
cana-3289	139	91	min{𝔅(𝛼,𝜃	min{𝔅(𝛼,𝜃	PROPN
cana-3289	139	92	)	)	PUNCT
cana-3289	139	93	𝑁	𝑁	PROPN
cana-3289	139	94	(	(	PUNCT
cana-3289	139	95	ℏ	ℏ	PROPN
cana-3289	139	96	)	)	PUNCT
cana-3289	139	97	,	,	PUNCT
cana-3289	139	98	𝔅(𝛼,𝜃	𝔅(𝛼,𝜃	NUM
cana-3289	139	99	)	)	PUNCT
cana-3289	140	1	𝑁	𝑁	PROPN
cana-3289	140	2	(	(	PUNCT
cana-3289	140	3	𝑚	𝑚	NOUN
cana-3289	140	4	)	)	PUNCT
cana-3289	140	5	}	}	PUNCT
cana-3289	140	6	thus	thus	ADV
cana-3289	140	7	𝔅(𝛼,𝜃	𝔅(𝛼,𝜃	X
cana-3289	140	8	)	)	PUNCT
cana-3289	140	9	𝑁	𝑁	PROPN
cana-3289	140	10	(	(	PUNCT
cana-3289	140	11	ℏ	ℏ	PROPN
cana-3289	140	12	∧	∧	PROPN
cana-3289	140	13	𝑚	𝑚	NOUN
cana-3289	140	14	)	)	PUNCT
cana-3289	140	15	≥	≥	NOUN
cana-3289	140	16	min{𝔅(𝛼,𝜃	min{𝔅(𝛼,𝜃	NOUN
cana-3289	140	17	)	)	PUNCT
cana-3289	141	1	𝑁	𝑁	PROPN
cana-3289	141	2	(	(	PUNCT
cana-3289	141	3	ℏ	ℏ	PROPN
cana-3289	141	4	)	)	PUNCT
cana-3289	141	5	,	,	PUNCT
cana-3289	141	6	𝔅(𝛼,𝜃	𝔅(𝛼,𝜃	NUM
cana-3289	141	7	)	)	PUNCT
cana-3289	141	8	𝑁	𝑁	PROPN
cana-3289	141	9	(	(	PUNCT
cana-3289	141	10	𝑚	𝑚	NOUN
cana-3289	141	11	)	)	PUNCT
cana-3289	141	12	}	}	PUNCT
cana-3289	141	13	hence	hence	ADV
cana-3289	141	14	the	the	DET
cana-3289	141	15	bfmt	bfmt	NOUN
cana-3289	141	16	𝑀	𝑀	PROPN
cana-3289	141	17	of	of	ADP
cana-3289	141	18	𝔅	𝔅	PROPN
cana-3289	141	19	is	be	AUX
cana-3289	141	20	again	again	ADV
cana-3289	141	21	a	a	DET
cana-3289	141	22	bfpi	bfpi	NOUN
cana-3289	141	23	of	of	ADP
cana-3289	141	24	𝔏.	𝔏.	PROPN
cana-3289	141	25	in	in	ADP
cana-3289	141	26	converse	converse	NOUN
cana-3289	141	27	assume	assume	VERB
cana-3289	141	28	that	that	SCONJ
cana-3289	141	29	𝑀	𝑀	PROPN
cana-3289	141	30	a	a	DET
cana-3289	141	31	bfpi	bfpi	NOUN
cana-3289	141	32	of	of	ADP
cana-3289	141	33	𝔏.	𝔏.	PROPN
cana-3289	141	34	now	now	ADV
cana-3289	141	35	,	,	PUNCT
cana-3289	141	36	consider	consider	VERB
cana-3289	141	37	𝔅𝑃(ℏ	𝔅𝑃(ℏ	NOUN
cana-3289	141	38	∧	∧	PROPN
cana-3289	141	39	𝑚	𝑚	NOUN
cana-3289	141	40	)	)	PUNCT
cana-3289	142	1	=	=	SYM
cana-3289	142	2	1	1	NUM
cana-3289	142	3	𝜔	𝜔	PRON
cana-3289	142	4	(	(	PUNCT
cana-3289	142	5	𝔅(𝜔,𝜗	𝔅(𝜔,𝜗	NOUN
cana-3289	142	6	)	)	PUNCT
cana-3289	142	7	𝑃	𝑃	PROPN
cana-3289	142	8	(	(	PUNCT
cana-3289	142	9	ℏ	ℏ	NOUN
cana-3289	142	10	∧	∧	PROPN
cana-3289	142	11	𝑚	𝑚	NOUN
cana-3289	142	12	)	)	PUNCT
cana-3289	142	13	−	−	ADP
cana-3289	142	14	𝜗	𝜗	NOUN
cana-3289	142	15	)	)	PUNCT
cana-3289	142	16	≤	≤	NOUN
cana-3289	142	17	1	1	NUM
cana-3289	142	18	𝜔	𝜔	DET
cana-3289	142	19	(	(	PUNCT
cana-3289	142	20	max{𝔅(𝜔,𝜗	max{𝔅(𝜔,𝜗	NOUN
cana-3289	142	21	)	)	PUNCT
cana-3289	142	22	𝑃	𝑃	PROPN
cana-3289	142	23	(	(	PUNCT
cana-3289	142	24	ℏ	ℏ	NOUN
cana-3289	142	25	)	)	PUNCT
cana-3289	142	26	,	,	PUNCT
cana-3289	142	27	𝔅(𝜔,𝜗	𝔅(𝜔,𝜗	NOUN
cana-3289	142	28	)	)	PUNCT
cana-3289	142	29	𝑃	𝑃	PROPN
cana-3289	142	30	(	(	PUNCT
cana-3289	142	31	𝑚	𝑚	NOUN
cana-3289	142	32	)	)	PUNCT
cana-3289	142	33	}	}	PUNCT
cana-3289	142	34	−	−	ADP
cana-3289	142	35	𝜗	𝜗	NOUN
cana-3289	142	36	)	)	PUNCT
cana-3289	142	37	=	=	SYM
cana-3289	142	38	max	max	NOUN
cana-3289	142	39	{	{	PUNCT
cana-3289	142	40	1	1	NUM
cana-3289	142	41	𝜔	𝜔	PRON
cana-3289	142	42	(	(	PUNCT
cana-3289	142	43	𝔅(𝜔,𝜗	𝔅(𝜔,𝜗	NOUN
cana-3289	142	44	)	)	PUNCT
cana-3289	142	45	𝑃	𝑃	PROPN
cana-3289	142	46	(	(	PUNCT
cana-3289	142	47	ℏ	ℏ	NOUN
cana-3289	142	48	)	)	PUNCT
cana-3289	142	49	−	−	ADP
cana-3289	142	50	𝜗	𝜗	NOUN
cana-3289	142	51	)	)	PUNCT
cana-3289	142	52	,	,	PUNCT
cana-3289	142	53	1	1	NUM
cana-3289	142	54	𝜔	𝜔	PRON
cana-3289	142	55	(	(	PUNCT
cana-3289	142	56	𝔅(𝜔,𝜗	𝔅(𝜔,𝜗	NOUN
cana-3289	142	57	)	)	PUNCT
cana-3289	142	58	𝑃	𝑃	NOUN
cana-3289	142	59	(	(	PUNCT
cana-3289	142	60	𝑚	𝑚	NOUN
cana-3289	142	61	)	)	PUNCT
cana-3289	142	62	−	−	ADP
cana-3289	142	63	𝜗	𝜗	NOUN
cana-3289	142	64	)	)	PUNCT
cana-3289	142	65	}	}	PUNCT
cana-3289	142	66	=	=	SYM
cana-3289	142	67	max{𝔅𝑃(ℏ	max{𝔅𝑃(ℏ	NOUN
cana-3289	142	68	)	)	PUNCT
cana-3289	142	69	,	,	PUNCT
cana-3289	142	70	𝔅𝑃(𝑚	𝔅𝑃(𝑚	NOUN
cana-3289	142	71	)	)	PUNCT
cana-3289	142	72	}	}	PUNCT
cana-3289	142	73	.	.	PUNCT
cana-3289	143	1	thus	thus	ADV
cana-3289	143	2	𝔅𝑃(ℏ	𝔅𝑃(ℏ	ADJ
cana-3289	143	3	∧	∧	PROPN
cana-3289	143	4	𝑚	𝑚	NOUN
cana-3289	143	5	)	)	PUNCT
cana-3289	143	6	≤	≤	NUM
cana-3289	143	7	max{𝔅𝑃(ℏ	max{𝔅𝑃(ℏ	NOUN
cana-3289	143	8	)	)	PUNCT
cana-3289	143	9	,	,	PUNCT
cana-3289	143	10	𝔅𝑃(𝑚	𝔅𝑃(𝑚	NOUN
cana-3289	143	11	)	)	PUNCT
cana-3289	143	12	}	}	PUNCT
cana-3289	143	13	.	.	PUNCT
cana-3289	144	1	likewise	likewise	ADV
cana-3289	144	2	,	,	PUNCT
cana-3289	144	3	we	we	PRON
cana-3289	144	4	can	can	AUX
cana-3289	144	5	prove	prove	VERB
cana-3289	144	6	𝔅𝑁(ℏ	𝔅𝑁(ℏ	PROPN
cana-3289	144	7	∧	∧	PROPN
cana-3289	144	8	𝑚	𝑚	PROPN
cana-3289	144	9	)	)	PUNCT
cana-3289	144	10	≥	≥	NOUN
cana-3289	144	11	min{𝔅𝑁(ℏ	min{𝔅𝑁(ℏ	NOUN
cana-3289	144	12	)	)	PUNCT
cana-3289	144	13	,	,	PUNCT
cana-3289	144	14	𝔅𝑁(𝑚	𝔅𝑁(𝑚	PROPN
cana-3289	144	15	)	)	PUNCT
cana-3289	144	16	}	}	PUNCT
cana-3289	144	17	.	.	PUNCT
cana-3289	145	1	thus	thus	ADV
cana-3289	145	2	𝑀	𝑀	PROPN
cana-3289	145	3	is	be	AUX
cana-3289	145	4	a	a	DET
cana-3289	145	5	bf	bf	NOUN
cana-3289	145	6	prime	prime	ADJ
cana-3289	145	7	ideal	ideal	NOUN
cana-3289	145	8	of	of	ADP
cana-3289	145	9	𝐿.	𝐿.	NOUN
cana-3289	145	10	hence	hence	ADV
cana-3289	145	11	the	the	DET
cana-3289	145	12	proof	proof	NOUN
cana-3289	145	13	.	.	PUNCT
cana-3289	146	1	theorem	theorem	VERB
cana-3289	146	2	3.11	3.11	NUM
cana-3289	146	3	:	:	PUNCT
cana-3289	146	4	let	let	VERB
cana-3289	146	5	𝜛	𝜛	X
cana-3289	146	6	:	:	PUNCT
cana-3289	146	7	𝔏	𝔏	PROPN
cana-3289	146	8	→	→	PUNCT
cana-3289	146	9	𝔏1	𝔏1	PROPN
cana-3289	146	10	be	be	AUX
cana-3289	146	11	a	a	DET
cana-3289	146	12	lattice	lattice	NOUN
cana-3289	146	13	epimorphism	epimorphism	NOUN
cana-3289	146	14	.	.	PUNCT
cana-3289	147	1	if	if	SCONJ
cana-3289	147	2	𝔅	𝔅	PROPN
cana-3289	147	3	is	be	AUX
cana-3289	147	4	a	a	DET
cana-3289	147	5	bfpi	bfpi	NOUN
cana-3289	147	6	of	of	ADP
cana-3289	147	7	𝔏	𝔏	PROPN
cana-3289	147	8	,	,	PUNCT
cana-3289	147	9	then	then	ADV
cana-3289	147	10	𝜛(𝔅	𝜛(𝔅	NOUN
cana-3289	147	11	)	)	PUNCT
cana-3289	147	12	is	be	AUX
cana-3289	147	13	a	a	DET
cana-3289	147	14	bfpi	bfpi	NOUN
cana-3289	147	15	of	of	ADP
cana-3289	147	16	𝔏1	𝔏1	PROPN
cana-3289	147	17	.	.	PUNCT
cana-3289	148	1	proof	proof	NOUN
cana-3289	148	2	:	:	PUNCT
cana-3289	148	3	assume	assume	VERB
cana-3289	148	4	𝔅	𝔅	PROPN
cana-3289	148	5	=	=	SYM
cana-3289	148	6	(	(	PUNCT
cana-3289	148	7	𝔅𝑃	𝔅𝑃	PROPN
cana-3289	148	8	,	,	PUNCT
cana-3289	148	9	𝔅𝑁	𝔅𝑁	PROPN
cana-3289	148	10	)	)	PUNCT
cana-3289	148	11	a	a	DET
cana-3289	148	12	bfpi	bfpi	NOUN
cana-3289	148	13	of	of	ADP
cana-3289	148	14	𝔏.	𝔏.	PROPN
cana-3289	148	15	then	then	ADV
cana-3289	148	16	by	by	ADP
cana-3289	148	17	known	known	ADJ
cana-3289	148	18	th	th	X
cana-3289	148	19	we	we	PRON
cana-3289	148	20	know	know	VERB
cana-3289	148	21	that	that	SCONJ
cana-3289	148	22	𝜛(𝔅	𝜛(𝔅	PROPN
cana-3289	148	23	)	)	PUNCT
cana-3289	148	24	is	be	AUX
cana-3289	148	25	a	a	DET
cana-3289	148	26	bfi	bfi	PROPN
cana-3289	148	27	in	in	ADP
cana-3289	148	28	𝔏1	𝔏1	PROPN
cana-3289	148	29	.	.	PUNCT
cana-3289	149	1	now	now	ADV
cana-3289	149	2	we	we	PRON
cana-3289	149	3	show	show	VERB
cana-3289	149	4	that	that	SCONJ
cana-3289	149	5	𝜛(𝔅	𝜛(𝔅	NOUN
cana-3289	149	6	)	)	PUNCT
cana-3289	149	7	is	be	AUX
cana-3289	149	8	prime	prime	ADJ
cana-3289	149	9	.	.	PUNCT
cana-3289	150	1	let	let	VERB
cana-3289	150	2	𝑠	𝑠	INTJ
cana-3289	150	3	,	,	PUNCT
cana-3289	150	4	𝑧	𝑧	DET
cana-3289	150	5	∈	∈	PROPN
cana-3289	150	6	𝔏1	𝔏1	NOUN
cana-3289	150	7	.	.	PUNCT
cana-3289	151	1	then	then	ADV
cana-3289	151	2	communications	communication	NOUN
cana-3289	151	3	on	on	ADP
cana-3289	151	4	applied	apply	VERB
cana-3289	151	5	nonlinear	nonlinear	ADJ
cana-3289	151	6	analysis	analysis	NOUN
cana-3289	151	7	issn	issn	NOUN
cana-3289	151	8	:	:	PUNCT
cana-3289	151	9	1074	1074	NUM
cana-3289	151	10	-	-	PUNCT
cana-3289	151	11	133x	133x	NUM
cana-3289	151	12	vol	vol	NOUN
cana-3289	151	13	32	32	NUM
cana-3289	151	14	no	no	NOUN
cana-3289	151	15	.	.	PUNCT
cana-3289	152	1	6s	6s	NUM
cana-3289	152	2	(	(	PUNCT
cana-3289	152	3	2025	2025	NUM
cana-3289	152	4	)	)	PUNCT
cana-3289	152	5	248	248	NUM
cana-3289	152	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3289	152	7	𝜛(𝔅𝑃)(𝑠	𝜛(𝔅𝑃)(𝑠	PRON
cana-3289	152	8	∧	∧	PROPN
cana-3289	152	9	𝑧	𝑧	NOUN
cana-3289	152	10	)	)	PUNCT
cana-3289	152	11	=	=	SYM
cana-3289	152	12	sup{𝔅𝑃(ℏ	sup{𝔅𝑃(ℏ	PROPN
cana-3289	152	13	)	)	PUNCT
cana-3289	152	14	∣	∣	ADJ
cana-3289	152	15	ℏ	ℏ	NOUN
cana-3289	152	16	∈	∈	NOUN
cana-3289	152	17	𝜛−1(𝑠	𝜛−1(𝑠	PROPN
cana-3289	152	18	∧	∧	PROPN
cana-3289	152	19	𝑧	𝑧	NOUN
cana-3289	152	20	)	)	PUNCT
cana-3289	152	21	}	}	PUNCT
cana-3289	152	22	=	=	SYM
cana-3289	152	23	sup{𝔅𝑃(𝑢	sup{𝔅𝑃(𝑢	PROPN
cana-3289	152	24	∧	∧	PROPN
cana-3289	152	25	𝜉	𝜉	NOUN
cana-3289	152	26	)	)	PUNCT
cana-3289	152	27	∣	∣	VERB
cana-3289	152	28	𝑢	𝑢	PROPN
cana-3289	152	29	∈	∈	PROPN
cana-3289	152	30	𝜛−1(𝑚	𝜛−1(𝑚	NOUN
cana-3289	152	31	)	)	PUNCT
cana-3289	152	32	,	,	PUNCT
cana-3289	152	33	𝜉	𝜉	PROPN
cana-3289	152	34	∈	∈	PROPN
cana-3289	152	35	𝜛−1(𝑧	𝜛−1(𝑧	PROPN
cana-3289	152	36	)	)	PUNCT
cana-3289	152	37	}	}	PUNCT
cana-3289	152	38	=	=	SYM
cana-3289	152	39	sup{max{𝔅𝑃(𝑢	sup{max{𝔅𝑃(𝑢	PROPN
cana-3289	152	40	)	)	PUNCT
cana-3289	152	41	,	,	PUNCT
cana-3289	152	42	𝔅𝑃(𝜉	𝔅𝑃(𝜉	X
cana-3289	152	43	)	)	PUNCT
cana-3289	152	44	}	}	PUNCT
cana-3289	152	45	∣	∣	VERB
cana-3289	152	46	𝑢	𝑢	PRON
cana-3289	152	47	∈	∈	PROPN
cana-3289	152	48	𝜛−1(𝑚	𝜛−1(𝑚	NOUN
cana-3289	152	49	)	)	PUNCT
cana-3289	152	50	,	,	PUNCT
cana-3289	152	51	𝜉	𝜉	PROPN
cana-3289	152	52	∈	∈	PROPN
cana-3289	152	53	𝜛−1(𝑧	𝜛−1(𝑧	PROPN
cana-3289	152	54	)	)	PUNCT
cana-3289	152	55	}	}	PUNCT
cana-3289	152	56	=	=	SYM
cana-3289	152	57	max	max	X
cana-3289	152	58	{	{	PUNCT
cana-3289	152	59	sup{𝔅𝑃(𝑢	sup{𝔅𝑃(𝑢	PROPN
cana-3289	152	60	)	)	PUNCT
cana-3289	152	61	∣	∣	VERB
cana-3289	152	62	𝑢	𝑢	PROPN
cana-3289	152	63	∈	∈	PROPN
cana-3289	152	64	𝜛−1(𝑚	𝜛−1(𝑚	NOUN
cana-3289	152	65	)	)	PUNCT
cana-3289	152	66	}	}	PUNCT
cana-3289	152	67	,	,	PUNCT
cana-3289	152	68	sup{𝔅𝑃(𝜉	sup{𝔅𝑃(𝜉	PROPN
cana-3289	152	69	)	)	PUNCT
cana-3289	152	70	∣	∣	ADJ
cana-3289	152	71	𝜉	𝜉	X
cana-3289	152	72	∈	∈	PROPN
cana-3289	152	73	𝜛−1(𝑧	𝜛−1(𝑧	PROPN
cana-3289	152	74	)	)	PUNCT
cana-3289	152	75	}	}	PUNCT
cana-3289	152	76	}	}	PUNCT
cana-3289	152	77	=	=	SYM
cana-3289	152	78	max{𝜛(𝔅𝑃)(𝑚	max{𝜛(𝔅𝑃)(𝑚	PROPN
cana-3289	152	79	)	)	PUNCT
cana-3289	152	80	,	,	PUNCT
cana-3289	152	81	𝜛(𝔅𝑃)(𝑧	𝜛(𝔅𝑃)(𝑧	NOUN
cana-3289	152	82	)	)	PUNCT
cana-3289	152	83	}	}	PUNCT
cana-3289	152	84	.	.	PUNCT
cana-3289	153	1	𝜛(𝔅𝑁)(𝑠	𝜛(𝔅𝑁)(𝑠	X
cana-3289	153	2	∧	∧	NOUN
cana-3289	153	3	𝑧	𝑧	NOUN
cana-3289	153	4	)	)	PUNCT
cana-3289	153	5	=	=	SYM
cana-3289	153	6	inf{𝔅𝑁(ℏ	inf{𝔅𝑁(ℏ	NOUN
cana-3289	153	7	)	)	PUNCT
cana-3289	153	8	∣	∣	ADJ
cana-3289	153	9	ℏ	ℏ	NOUN
cana-3289	153	10	∈	∈	NOUN
cana-3289	153	11	𝜛−1(𝑠	𝜛−1(𝑠	PROPN
cana-3289	153	12	∧	∧	PROPN
cana-3289	153	13	𝑧	𝑧	NOUN
cana-3289	153	14	)	)	PUNCT
cana-3289	153	15	}	}	PUNCT
cana-3289	153	16	=	=	PUNCT
cana-3289	153	17	inf{𝔅𝑁(𝑢	inf{𝔅𝑁(𝑢	PROPN
cana-3289	153	18	∧	∧	PROPN
cana-3289	153	19	𝜉	𝜉	NOUN
cana-3289	153	20	)	)	PUNCT
cana-3289	153	21	∣	∣	VERB
cana-3289	153	22	𝑢	𝑢	PROPN
cana-3289	153	23	∈	∈	PROPN
cana-3289	153	24	𝜛−1(𝑚	𝜛−1(𝑚	NOUN
cana-3289	153	25	)	)	PUNCT
cana-3289	153	26	,	,	PUNCT
cana-3289	153	27	𝜉	𝜉	PROPN
cana-3289	153	28	∈	∈	PROPN
cana-3289	153	29	𝜛−1(𝑧	𝜛−1(𝑧	PROPN
cana-3289	153	30	)	)	PUNCT
cana-3289	153	31	}	}	PUNCT
cana-3289	153	32	=	=	SYM
cana-3289	153	33	inf{min{𝔅𝑁(𝑢	inf{min{𝔅𝑁(𝑢	PROPN
cana-3289	153	34	)	)	PUNCT
cana-3289	153	35	,	,	PUNCT
cana-3289	153	36	𝔅𝑁(𝜉	𝔅𝑁(𝜉	PROPN
cana-3289	153	37	)	)	PUNCT
cana-3289	153	38	}	}	PUNCT
cana-3289	153	39	∣	∣	VERB
cana-3289	153	40	𝑢	𝑢	PRON
cana-3289	153	41	∈	∈	PROPN
cana-3289	153	42	𝜛−1(𝑚	𝜛−1(𝑚	NOUN
cana-3289	153	43	)	)	PUNCT
cana-3289	153	44	,	,	PUNCT
cana-3289	153	45	𝜉	𝜉	PROPN
cana-3289	153	46	∈	∈	PROPN
cana-3289	153	47	𝜛−1(𝑧	𝜛−1(𝑧	PROPN
cana-3289	153	48	)	)	PUNCT
cana-3289	153	49	}	}	PUNCT
cana-3289	153	50	=	=	SYM
cana-3289	153	51	min	min	NOUN
cana-3289	153	52	{	{	PUNCT
cana-3289	153	53	inf{𝔅𝑁(𝑢	inf{𝔅𝑁(𝑢	NOUN
cana-3289	153	54	)	)	PUNCT
cana-3289	153	55	∣	∣	ADJ
cana-3289	153	56	𝑢	𝑢	PROPN
cana-3289	153	57	∈	∈	PROPN
cana-3289	153	58	𝜛−1(𝑚	𝜛−1(𝑚	NOUN
cana-3289	153	59	)	)	PUNCT
cana-3289	153	60	}	}	PUNCT
cana-3289	153	61	,	,	PUNCT
cana-3289	153	62	inf{𝔅𝑁(𝜉	inf{𝔅𝑁(𝜉	NOUN
cana-3289	153	63	)	)	PUNCT
cana-3289	153	64	∣	∣	ADJ
cana-3289	153	65	𝜉	𝜉	X
cana-3289	153	66	∈	∈	PROPN
cana-3289	153	67	𝜛−1(𝑧	𝜛−1(𝑧	PROPN
cana-3289	153	68	)	)	PUNCT
cana-3289	153	69	}	}	PUNCT
cana-3289	153	70	}	}	PUNCT
cana-3289	154	1	=	=	SYM
cana-3289	154	2	min{𝜛(𝔅𝑁)(𝑚	min{𝜛(𝔅𝑁)(𝑚	PROPN
cana-3289	154	3	)	)	PUNCT
cana-3289	154	4	,	,	PUNCT
cana-3289	154	5	𝜛(𝔅𝑁)(𝑧	𝜛(𝔅𝑁)(𝑧	NOUN
cana-3289	154	6	)	)	PUNCT
cana-3289	154	7	}	}	PUNCT
cana-3289	154	8	,	,	PUNCT
cana-3289	154	9	hence	hence	ADV
cana-3289	154	10	,	,	PUNCT
cana-3289	154	11	𝜛(𝔅	𝜛(𝔅	PROPN
cana-3289	154	12	)	)	PUNCT
cana-3289	154	13	is	be	AUX
cana-3289	154	14	a	a	DET
cana-3289	154	15	bfpi	bfpi	NOUN
cana-3289	154	16	of	of	ADP
cana-3289	154	17	𝔏1	𝔏1	PROPN
cana-3289	154	18	.	.	PUNCT
cana-3289	155	1	theorem	theorem	VERB
cana-3289	155	2	3.12	3.12	NUM
cana-3289	155	3	:	:	PUNCT
cana-3289	155	4	let	let	VERB
cana-3289	155	5	𝜛	𝜛	X
cana-3289	155	6	:	:	PUNCT
cana-3289	155	7	𝔏	𝔏	PROPN
cana-3289	155	8	→	→	PUNCT
cana-3289	155	9	𝔏1	𝔏1	PROPN
cana-3289	155	10	be	be	AUX
cana-3289	155	11	a	a	DET
cana-3289	155	12	lattice	lattice	ADJ
cana-3289	155	13	homomorphism	homomorphism	NOUN
cana-3289	155	14	.	.	PUNCT
cana-3289	156	1	if	if	SCONJ
cana-3289	156	2	𝐶	𝐶	PROPN
cana-3289	156	3	is	be	AUX
cana-3289	156	4	a	a	DET
cana-3289	156	5	bfpi	bfpi	NOUN
cana-3289	156	6	of	of	ADP
cana-3289	156	7	𝔏1	𝔏1	PROPN
cana-3289	156	8	,	,	PUNCT
cana-3289	156	9	then	then	ADV
cana-3289	156	10	𝜛−1(𝐶	𝜛−1(𝐶	VERB
cana-3289	156	11	)	)	PUNCT
cana-3289	156	12	is	be	AUX
cana-3289	156	13	a	a	DET
cana-3289	156	14	bfpi	bfpi	NOUN
cana-3289	156	15	of	of	ADP
cana-3289	156	16	𝔏.	𝔏.	PROPN
cana-3289	156	17	proof	proof	NOUN
cana-3289	156	18	:	:	PUNCT
cana-3289	156	19	suppose	suppose	VERB
cana-3289	156	20	𝐶	𝐶	PROPN
cana-3289	156	21	=	=	SYM
cana-3289	156	22	(	(	PUNCT
cana-3289	156	23	𝐶𝑃	𝐶𝑃	PROPN
cana-3289	156	24	,	,	PUNCT
cana-3289	156	25	𝐶𝑁	𝐶𝑁	PROPN
cana-3289	156	26	)	)	PUNCT
cana-3289	156	27	be	be	VERB
cana-3289	156	28	a	a	DET
cana-3289	156	29	bfpi	bfpi	NOUN
cana-3289	156	30	of	of	ADP
cana-3289	156	31	𝔏1	𝔏1	PROPN
cana-3289	156	32	.	.	PUNCT
cana-3289	157	1	then	then	ADV
cana-3289	157	2	by	by	ADP
cana-3289	157	3	known	know	VERB
cana-3289	157	4	theorem	theorem	VERB
cana-3289	157	5	[	[	X
cana-3289	157	6	]	]	X
cana-3289	157	7	,	,	PUNCT
cana-3289	157	8	we	we	PRON
cana-3289	157	9	know	know	VERB
cana-3289	157	10	that	that	SCONJ
cana-3289	157	11	𝜛−1(𝐶	𝜛−1(𝐶	VERB
cana-3289	157	12	)	)	PUNCT
cana-3289	157	13	is	be	AUX
cana-3289	157	14	a	a	DET
cana-3289	157	15	bfi	bfi	PROPN
cana-3289	157	16	in	in	ADP
cana-3289	157	17	𝔏.	𝔏.	PROPN
cana-3289	157	18	now	now	ADV
cana-3289	157	19	we	we	PRON
cana-3289	157	20	show	show	VERB
cana-3289	157	21	that	that	SCONJ
cana-3289	157	22	𝜛−1(𝐶	𝜛−1(𝐶	VERB
cana-3289	157	23	)	)	PUNCT
cana-3289	157	24	is	be	AUX
cana-3289	157	25	prime	prime	ADJ
cana-3289	157	26	.	.	PUNCT
cana-3289	158	1	let	let	VERB
cana-3289	158	2	ℏ	ℏ	VERB
cana-3289	158	3	,	,	PUNCT
cana-3289	159	1	𝑚	𝑚	PROPN
cana-3289	159	2	∈	∈	PROPN
cana-3289	159	3	𝔏.	𝔏.	PROPN
cana-3289	159	4	then	then	ADV
cana-3289	159	5	𝜛−1(𝐶𝑃)(ℏ	𝜛−1(𝐶𝑃)(ℏ	X
cana-3289	159	6	∧	∧	PROPN
cana-3289	159	7	𝑚	𝑚	NOUN
cana-3289	159	8	)	)	PUNCT
cana-3289	159	9	=	=	PUNCT
cana-3289	159	10	𝐶𝑃(𝜛(ℏ	𝐶𝑃(𝜛(ℏ	NOUN
cana-3289	159	11	∧	∧	PROPN
cana-3289	159	12	𝑚	𝑚	NOUN
cana-3289	159	13	)	)	PUNCT
cana-3289	159	14	)	)	PUNCT
cana-3289	159	15	=	=	SYM
cana-3289	159	16	𝐶𝑃{(𝜛(ℏ	𝐶𝑃{(𝜛(ℏ	X
cana-3289	159	17	)	)	PUNCT
cana-3289	159	18	∧	∧	NOUN
cana-3289	159	19	𝜛(𝑚	𝜛(𝑚	NOUN
cana-3289	159	20	)	)	PUNCT
cana-3289	159	21	}	}	PUNCT
cana-3289	159	22	=	=	SYM
cana-3289	159	23	max{𝐶𝑃(𝜛(ℏ	max{𝐶𝑃(𝜛(ℏ	PROPN
cana-3289	159	24	)	)	PUNCT
cana-3289	159	25	)	)	PUNCT
cana-3289	159	26	,	,	PUNCT
cana-3289	159	27	𝐶𝑃(𝜛(𝑚	𝐶𝑃(𝜛(𝑚	NOUN
cana-3289	159	28	)	)	PUNCT
cana-3289	159	29	)	)	PUNCT
cana-3289	159	30	}	}	PUNCT
cana-3289	160	1	=	=	SYM
cana-3289	160	2	max{𝜛−1(𝐶𝑃)(ℏ	max{𝜛−1(𝐶𝑃)(ℏ	PROPN
cana-3289	160	3	)	)	PUNCT
cana-3289	160	4	,	,	PUNCT
cana-3289	160	5	𝜛−1(𝐶𝑃)(𝑚	𝜛−1(𝐶𝑃)(𝑚	PROPN
cana-3289	160	6	)	)	PUNCT
cana-3289	160	7	}	}	PUNCT
cana-3289	161	1	𝜛−1(𝐶𝑁)(ℏ	𝜛−1(𝐶𝑁)(ℏ	X
cana-3289	161	2	∧	∧	NOUN
cana-3289	161	3	𝑚	𝑚	NOUN
cana-3289	161	4	)	)	PUNCT
cana-3289	161	5	=	=	SYM
cana-3289	161	6	𝐶𝑁(𝜛(ℏ	𝐶𝑁(𝜛(ℏ	NOUN
cana-3289	161	7	∧	∧	PROPN
cana-3289	161	8	𝑚	𝑚	NOUN
cana-3289	161	9	)	)	PUNCT
cana-3289	161	10	)	)	PUNCT
cana-3289	161	11	=	=	SYM
cana-3289	161	12	𝐶𝑁{(𝜛(ℏ	𝐶𝑁{(𝜛(ℏ	NOUN
cana-3289	161	13	)	)	PUNCT
cana-3289	161	14	∧	∧	NOUN
cana-3289	161	15	𝜛(𝑚	𝜛(𝑚	NOUN
cana-3289	161	16	)	)	PUNCT
cana-3289	161	17	}	}	PUNCT
cana-3289	161	18	=	=	SYM
cana-3289	161	19	min{𝐶𝑁(𝜛(ℏ	min{𝐶𝑁(𝜛(ℏ	PROPN
cana-3289	161	20	)	)	PUNCT
cana-3289	161	21	)	)	PUNCT
cana-3289	161	22	,	,	PUNCT
cana-3289	161	23	𝐶𝑁(𝜛(𝑚	𝐶𝑁(𝜛(𝑚	NOUN
cana-3289	161	24	)	)	PUNCT
cana-3289	161	25	)	)	PUNCT
cana-3289	161	26	}	}	PUNCT
cana-3289	161	27	=	=	SYM
cana-3289	161	28	min{𝜛−1(𝐶𝑁)(ℏ	min{𝜛−1(𝐶𝑁)(ℏ	PROPN
cana-3289	161	29	)	)	PUNCT
cana-3289	161	30	,	,	PUNCT
cana-3289	161	31	𝜛−1(𝐶𝑁)(𝑚	𝜛−1(𝐶𝑁)(𝑚	PROPN
cana-3289	161	32	)	)	PUNCT
cana-3289	161	33	}	}	PUNCT
cana-3289	161	34	hence	hence	ADV
cana-3289	161	35	,	,	PUNCT
cana-3289	161	36	𝜛−1(𝐶	𝜛−1(𝐶	X
cana-3289	161	37	)	)	PUNCT
cana-3289	161	38	is	be	AUX
cana-3289	161	39	a	a	DET
cana-3289	161	40	bfpi	bfpi	NOUN
cana-3289	161	41	of	of	ADP
cana-3289	161	42	𝔏.	𝔏.	PROPN
cana-3289	161	43	4.conclusion	4.conclusion	PROPN
cana-3289	161	44	this	this	DET
cana-3289	161	45	study	study	NOUN
cana-3289	161	46	explores	explore	VERB
cana-3289	161	47	the	the	DET
cana-3289	161	48	investigation	investigation	NOUN
cana-3289	161	49	of	of	ADP
cana-3289	161	50	bipolar	bipolar	ADJ
cana-3289	161	51	fuzzy	fuzzy	ADJ
cana-3289	161	52	prime	prime	ADJ
cana-3289	161	53	ideals	ideal	NOUN
cana-3289	161	54	(	(	PUNCT
cana-3289	161	55	bfpi	bfpi	ADJ
cana-3289	161	56	)	)	PUNCT
cana-3289	161	57	in	in	ADP
cana-3289	161	58	lattices	lattice	NOUN
cana-3289	161	59	.	.	PUNCT
cana-3289	162	1	we	we	PRON
cana-3289	162	2	provide	provide	VERB
cana-3289	162	3	a	a	DET
cana-3289	162	4	detailed	detailed	ADJ
cana-3289	162	5	exploration	exploration	NOUN
cana-3289	162	6	of	of	ADP
cana-3289	162	7	their	their	PRON
cana-3289	162	8	properties	property	NOUN
cana-3289	162	9	,	,	PUNCT
cana-3289	162	10	characterizations	characterization	NOUN
cana-3289	162	11	,	,	PUNCT
cana-3289	162	12	and	and	CCONJ
cana-3289	162	13	associated	associated	ADJ
cana-3289	162	14	homomorphisms	homomorphism	NOUN
cana-3289	162	15	.	.	PUNCT
cana-3289	163	1	references	reference	NOUN
cana-3289	163	2	:	:	PUNCT
cana-3289	164	1	[	[	X
cana-3289	164	2	1	1	NUM
cana-3289	164	3	]	]	X
cana-3289	164	4	gau	gau	NOUN
cana-3289	164	5	,	,	PUNCT
cana-3289	164	6	w.l	w.l	PROPN
cana-3289	164	7	.	.	PROPN
cana-3289	164	8	,	,	PUNCT
cana-3289	164	9	buehrer	buehrer	PROPN
cana-3289	164	10	,	,	PUNCT
cana-3289	164	11	d.j	d.j	PROPN
cana-3289	164	12	.	.	PROPN
cana-3289	164	13	:	:	PUNCT
cana-3289	164	14	vague	vague	ADJ
cana-3289	164	15	sets	set	NOUN
cana-3289	164	16	.	.	PUNCT
cana-3289	165	1	ieee	ieee	NOUN
cana-3289	165	2	transactions	transaction	NOUN
cana-3289	165	3	on	on	ADP
cana-3289	165	4	systems	system	NOUN
cana-3289	165	5	,	,	PUNCT
cana-3289	165	6	man	man	NOUN
cana-3289	165	7	,	,	PUNCT
cana-3289	165	8	and	and	CCONJ
cana-3289	165	9	cybernetics	cybernetic	NOUN
cana-3289	165	10	23	23	NUM
cana-3289	165	11	(	(	PUNCT
cana-3289	165	12	1993	1993	NUM
cana-3289	165	13	)	)	PUNCT
cana-3289	166	1	610–614	610–614	NUM
cana-3289	166	2	.	.	PUNCT
cana-3289	167	1	[	[	X
cana-3289	167	2	2	2	NUM
cana-3289	167	3	]	]	X
cana-3289	167	4	l.a	l.a	PROPN
cana-3289	167	5	.	.	PROPN
cana-3289	167	6	zadeh	zadeh	PROPN
cana-3289	167	7	,	,	PUNCT
cana-3289	167	8	fuzzy	fuzzy	ADJ
cana-3289	167	9	sets	set	NOUN
cana-3289	167	10	,	,	PUNCT
cana-3289	167	11	inform	inform	NOUN
cana-3289	167	12	.	.	PUNCT
cana-3289	168	1	control	control	NOUN
cana-3289	168	2	.	.	PUNCT
cana-3289	169	1	8	8	NUM
cana-3289	169	2	(	(	PUNCT
cana-3289	169	3	1965	1965	NUM
cana-3289	169	4	)	)	PUNCT
cana-3289	169	5	,	,	PUNCT
cana-3289	169	6	338	338	NUM
cana-3289	169	7	-	-	SYM
cana-3289	169	8	353	353	NUM
cana-3289	169	9	.	.	PUNCT
cana-3289	170	1	https://doi.org/10.1016/s0019-9958(65	https://doi.org/10.1016/s0019-9958(65	PROPN
cana-3289	170	2	)	)	PUNCT
cana-3289	170	3	90241	90241	NUM
cana-3289	170	4	-	-	SYM
cana-3289	170	5	x.	x.	NOUN
cana-3289	171	1	[	[	X
cana-3289	171	2	3	3	NUM
cana-3289	171	3	]	]	ADJ
cana-3289	171	4	bustince	bustince	NOUN
cana-3289	171	5	,	,	PUNCT
cana-3289	171	6	h.	h.	PROPN
cana-3289	171	7	,	,	PUNCT
cana-3289	171	8	burillo	burillo	PROPN
cana-3289	171	9	,	,	PUNCT
cana-3289	171	10	p.	p.	NOUN
cana-3289	171	11	:	:	PUNCT
cana-3289	172	1	vague	vague	ADJ
cana-3289	172	2	sets	set	NOUN
cana-3289	172	3	are	be	AUX
cana-3289	172	4	intuitionistic	intuitionistic	ADJ
cana-3289	172	5	fuzzy	fuzzy	ADJ
cana-3289	172	6	sets	set	NOUN
cana-3289	172	7	.	.	PUNCT
cana-3289	173	1	fuzzy	fuzzy	ADJ
cana-3289	173	2	sets	set	NOUN
cana-3289	173	3	and	and	CCONJ
cana-3289	173	4	systems	system	NOUN
cana-3289	173	5	79	79	NUM
cana-3289	173	6	(	(	PUNCT
cana-3289	173	7	1996	1996	NUM
cana-3289	173	8	)	)	PUNCT
cana-3289	174	1	403–405	403–405	NUM
cana-3289	175	1	[	[	X
cana-3289	175	2	4	4	NUM
cana-3289	175	3	]	]	X
cana-3289	175	4	k.v	k.v	PROPN
cana-3289	175	5	.	.	PROPN
cana-3289	175	6	thomas	thomas	PROPN
cana-3289	175	7	,	,	PUNCT
cana-3289	175	8	l.s	l.s	PROPN
cana-3289	175	9	.	.	PROPN
cana-3289	175	10	nair	nair	PROPN
cana-3289	175	11	,	,	PUNCT
cana-3289	175	12	intuitionistic	intuitionistic	ADJ
cana-3289	175	13	fuzzy	fuzzy	ADJ
cana-3289	175	14	sublattices	sublattice	NOUN
cana-3289	175	15	and	and	CCONJ
cana-3289	175	16	ideals	ideal	NOUN
cana-3289	175	17	,	,	PUNCT
cana-3289	175	18	fuzzy	fuzzy	ADJ
cana-3289	175	19	inform	inform	NOUN
cana-3289	175	20	.	.	PUNCT
cana-3289	176	1	eng	eng	PROPN
cana-3289	176	2	.	.	PROPN
cana-3289	176	3	3	3	NUM
cana-3289	176	4	(	(	PUNCT
cana-3289	176	5	2011	2011	NUM
cana-3289	176	6	)	)	PUNCT
cana-3289	176	7	,	,	PUNCT
cana-3289	176	8	321	321	NUM
cana-3289	176	9	-	-	SYM
cana-3289	176	10	331	331	NUM
cana-3289	176	11	.	.	PUNCT
cana-3289	177	1	https	https	NOUN
cana-3289	177	2	:	:	PUNCT
cana-3289	177	3	//doi.org/10.1007	//doi.org/10.1007	PROPN
cana-3289	177	4	/	/	SYM
cana-3289	177	5	s12543	s12543	PROPN
cana-3289	177	6	-	-	PUNCT
cana-3289	177	7	011	011	NUM
cana-3289	177	8	-	-	PUNCT
cana-3289	177	9	0086	0086	NUM
cana-3289	177	10	-	-	PUNCT
cana-3289	177	11	5	5	NUM
cana-3289	177	12	.	.	PUNCT
cana-3289	178	1	[	[	X
cana-3289	178	2	5	5	X
cana-3289	178	3	]	]	X
cana-3289	178	4	k.t	k.t	PROPN
cana-3289	178	5	.	.	PROPN
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cana-3289	178	9	fuzzy	fuzzy	ADJ
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cana-3289	178	11	,	,	PUNCT
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cana-3289	178	13	sets	set	NOUN
cana-3289	178	14	syst	syst	NOUN
cana-3289	178	15	.	.	PUNCT
cana-3289	179	1	20	20	NUM
cana-3289	179	2	(	(	PUNCT
cana-3289	179	3	1986	1986	NUM
cana-3289	179	4	)	)	PUNCT
cana-3289	179	5	,	,	PUNCT
cana-3289	179	6	87	87	NUM
cana-3289	179	7	-	-	SYM
cana-3289	179	8	96	96	NUM
cana-3289	179	9	.	.	PUNCT
cana-3289	180	1	https://doi.org/10.1016/	https://doi.org/10.1016/	PROPN
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cana-3289	180	3	-	-	PUNCT
cana-3289	180	4	3	3	NUM
cana-3289	180	5	.	.	PUNCT
cana-3289	181	1	https://doi.org/10.1016/s0019-9958(65	https://doi.org/10.1016/s0019-9958(65	PROPN
cana-3289	181	2	)	)	PUNCT
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cana-3289	181	6	-	-	PUNCT
cana-3289	181	7	011	011	NUM
cana-3289	181	8	-	-	PUNCT
cana-3289	181	9	0086	0086	NUM
cana-3289	181	10	-	-	PUNCT
cana-3289	181	11	5	5	NUM
cana-3289	181	12	https://doi.org/10.1016/	https://doi.org/10.1016/	PROPN
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cana-3289	181	14	on	on	ADP
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cana-3289	181	17	analysis	analysis	NOUN
cana-3289	181	18	issn	issn	NOUN
cana-3289	181	19	:	:	PUNCT
cana-3289	181	20	1074	1074	NUM
cana-3289	181	21	-	-	PUNCT
cana-3289	181	22	133x	133x	NUM
cana-3289	181	23	vol	vol	NOUN
cana-3289	181	24	32	32	NUM
cana-3289	181	25	no	no	NOUN
cana-3289	181	26	.	.	PUNCT
cana-3289	182	1	6s	6s	NUM
cana-3289	182	2	(	(	PUNCT
cana-3289	182	3	2025	2025	NUM
cana-3289	182	4	)	)	PUNCT
cana-3289	182	5	249	249	NUM
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cana-3289	183	1	[	[	X
cana-3289	183	2	6	6	NUM
cana-3289	183	3	]	]	X
cana-3289	183	4	k.m	k.m	PROPN
cana-3289	183	5	.	.	PROPN
cana-3289	183	6	lee	lee	PROPN
cana-3289	183	7	,	,	PUNCT
cana-3289	183	8	bipolar	bipolar	ADV
cana-3289	183	9	-	-	PUNCT
cana-3289	183	10	valued	value	VERB
cana-3289	183	11	fuzzy	fuzzy	ADJ
cana-3289	183	12	sets	set	NOUN
cana-3289	183	13	and	and	CCONJ
cana-3289	183	14	their	their	PRON
cana-3289	183	15	operations	operation	NOUN
cana-3289	183	16	,	,	PUNCT
cana-3289	183	17	in	in	ADP
cana-3289	183	18	:	:	PUNCT
cana-3289	183	19	proc	proc	NOUN
cana-3289	183	20	.	.	PUNCT
cana-3289	184	1	int	int	NOUN
cana-3289	184	2	.	.	PUNCT
cana-3289	184	3	conf	conf	PROPN
cana-3289	184	4	.	.	PUNCT
cana-3289	185	1	on	on	ADP
cana-3289	185	2	intelligent	intelligent	ADJ
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cana-3289	185	4	,	,	PUNCT
cana-3289	185	5	bangkok	bangkok	PROPN
cana-3289	185	6	,	,	PUNCT
cana-3289	185	7	thailand	thailand	PROPN
cana-3289	185	8	,	,	PUNCT
cana-3289	185	9	(	(	PUNCT
cana-3289	185	10	2000	2000	NUM
cana-3289	185	11	)	)	PUNCT
cana-3289	185	12	,	,	PUNCT
cana-3289	185	13	307	307	NUM
cana-3289	185	14	-	-	SYM
cana-3289	185	15	312	312	NUM
cana-3289	185	16	.	.	PUNCT
cana-3289	186	1	[	[	X
cana-3289	186	2	7	7	X
cana-3289	186	3	]	]	X
cana-3289	186	4	n.	n.	NOUN
cana-3289	186	5	ajmal	ajmal	PROPN
cana-3289	186	6	,	,	PUNCT
cana-3289	186	7	k.v	k.v	PROPN
cana-3289	186	8	.	.	PROPN
cana-3289	186	9	thomas	thomas	PROPN
cana-3289	186	10	,	,	PUNCT
cana-3289	186	11	fuzzy	fuzzy	ADJ
cana-3289	186	12	lattices	lattice	NOUN
cana-3289	186	13	,	,	PUNCT
cana-3289	186	14	inform	inform	NOUN
cana-3289	186	15	.	.	PUNCT
cana-3289	187	1	sci	sci	PROPN
cana-3289	187	2	.	.	PROPN
cana-3289	187	3	79	79	NUM
cana-3289	187	4	(	(	PUNCT
cana-3289	187	5	1994	1994	NUM
cana-3289	187	6	)	)	PUNCT
cana-3289	187	7	,	,	PUNCT
cana-3289	187	8	271	271	NUM
cana-3289	187	9	-	-	SYM
cana-3289	187	10	291	291	NUM
cana-3289	187	11	.	.	PUNCT
cana-3289	188	1	https://doi.org/10.1016/	https://doi.org/10.1016/	PROPN
cana-3289	188	2	0020	0020	NUM
cana-3289	188	3	-	-	PUNCT
cana-3289	188	4	0255(94	0255(94	NUM
cana-3289	188	5	)	)	PUNCT
cana-3289	188	6	90124	90124	NUM
cana-3289	188	7	-	-	SYM
cana-3289	188	8	4	4	NUM
cana-3289	188	9	.	.	PUNCT
cana-3289	189	1	[	[	X
cana-3289	189	2	8	8	NUM
cana-3289	189	3	]	]	X
cana-3289	189	4	u.	u.	PROPN
cana-3289	189	5	venkata	venkata	PROPN
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cana-3289	189	7	,	,	PUNCT
cana-3289	189	8	t.	t.	PROPN
cana-3289	189	9	eswarlal	eswarlal	PROPN
cana-3289	189	10	,	,	PUNCT
cana-3289	189	11	homomorphism	homomorphism	NOUN
cana-3289	189	12	on	on	ADP
cana-3289	189	13	bipolar	bipolar	ADJ
cana-3289	189	14	vague	vague	ADJ
cana-3289	189	15	normal	normal	ADJ
cana-3289	189	16	groups	group	NOUN
cana-3289	189	17	,	,	PUNCT
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cana-3289	189	19	.	.	PUNCT
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cana-3289	189	21	.	.	PUNCT
cana-3289	189	22	,	,	PUNCT
cana-3289	190	1	sci	sci	PROPN
cana-3289	190	2	.	.	PUNCT
cana-3289	191	1	j.	j.	PROPN
cana-3289	191	2	9	9	NUM
cana-3289	191	3	(	(	PUNCT
cana-3289	191	4	2020	2020	NUM
cana-3289	191	5	)	)	PUNCT
cana-3289	191	6	,	,	PUNCT
cana-3289	191	7	3315	3315	NUM
cana-3289	191	8	-	-	SYM
cana-3289	191	9	3324	3324	NUM
cana-3289	191	10	.	.	PUNCT
cana-3289	192	1	https://doi.org/10.37418/amsj.9.6.11	https://doi.org/10.37418/amsj.9.6.11	VERB
cana-3289	192	2	.	.	PUNCT
cana-3289	193	1	[	[	X
cana-3289	193	2	9	9	NUM
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cana-3289	193	4	u.	u.	PROPN
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cana-3289	193	7	,	,	PUNCT
cana-3289	193	8	t.	t.	PROPN
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cana-3289	193	10	,	,	PUNCT
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cana-3289	193	14	,	,	PUNCT
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cana-3289	193	16	.	.	PUNCT
cana-3289	193	17	math	math	PROPN
cana-3289	193	18	.	.	PUNCT
cana-3289	193	19	,	,	PUNCT
cana-3289	193	20	sci	sci	PROPN
cana-3289	193	21	.	.	PUNCT
cana-3289	194	1	j.	j.	PROPN
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cana-3289	194	3	(	(	PUNCT
cana-3289	194	4	2020	2020	NUM
cana-3289	194	5	)	)	PUNCT
cana-3289	194	6	,	,	PUNCT
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cana-3289	194	8	-	-	SYM
cana-3289	194	9	6787	6787	NUM
cana-3289	194	10	.	.	PUNCT
cana-3289	195	1	https://doi	https://doi	PROPN
cana-3289	195	2	.	.	PUNCT
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cana-3289	195	6	.	.	PUNCT
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cana-3289	196	22	,	,	PUNCT
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cana-3289	196	26	.	.	PUNCT
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cana-3289	198	2	(	(	PUNCT
cana-3289	198	3	2020	2020	NUM
cana-3289	198	4	)	)	PUNCT
cana-3289	198	5	,	,	PUNCT
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cana-3289	198	7	-	-	SYM
cana-3289	198	8	88	88	NUM
cana-3289	198	9	.	.	PUNCT
cana-3289	199	1	https://doi.org/10.7151/dmgaa.1325	https://doi.org/10.7151/dmgaa.1325	PROPN
cana-3289	199	2	.	.	PUNCT
cana-3289	200	1	[	[	X
cana-3289	200	2	11	11	NUM
cana-3289	200	3	]	]	PUNCT
cana-3289	200	4	s.	s.	PROPN
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cana-3289	200	18	,	,	PUNCT
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cana-3289	200	20	.	.	PUNCT
cana-3289	200	21	appl	appl	PROPN
cana-3289	200	22	.	.	PROPN
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cana-3289	200	24	.	.	PUNCT
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cana-3289	201	2	(	(	PUNCT
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cana-3289	201	4	)	)	PUNCT
cana-3289	201	5	,	,	PUNCT
cana-3289	201	6	942	942	NUM
cana-3289	201	7	-	-	SYM
cana-3289	201	8	956	956	NUM
cana-3289	201	9	.	.	PUNCT
cana-3289	202	1	https://digitalcommons.pvamu.edu/aam/vol15/iss2/13	https://digitalcommons.pvamu.edu/aam/vol15/iss2/13	ADJ
cana-3289	202	2	.	.	PUNCT
cana-3289	203	1	[	[	X
cana-3289	203	2	12	12	NUM
cana-3289	203	3	]	]	PUNCT
cana-3289	203	4	h.	h.	PROPN
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cana-3289	203	6	,	,	PUNCT
cana-3289	203	7	q.	q.	PROPN
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cana-3289	203	9	,	,	PUNCT
cana-3289	203	10	intuitionistic	intuitionistic	ADJ
cana-3289	203	11	fuzzy	fuzzy	ADJ
cana-3289	203	12	filter	filter	NOUN
cana-3289	203	13	theory	theory	NOUN
cana-3289	203	14	on	on	ADP
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cana-3289	203	16	lattices	lattice	NOUN
cana-3289	203	17	,	,	PUNCT
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cana-3289	203	20	.	.	PUNCT
cana-3289	204	1	23	23	NUM
cana-3289	204	2	(	(	PUNCT
cana-3289	204	3	2018	2018	NUM
cana-3289	204	4	)	)	PUNCT
cana-3289	204	5	,	,	PUNCT
cana-3289	204	6	6777	6777	NUM
cana-3289	204	7	-	-	SYM
cana-3289	204	8	6783	6783	NUM
cana-3289	204	9	.	.	PUNCT
cana-3289	205	1	https://doi.org/10.1007/s00500-018-3647-2	https://doi.org/10.1007/s00500-018-3647-2	NUM
cana-3289	205	2	.	.	PUNCT
cana-3289	206	1	[	[	X
cana-3289	206	2	13	13	NUM
cana-3289	206	3	]	]	PUNCT
cana-3289	206	4	s.	s.	PROPN
cana-3289	206	5	milles	milles	PROPN
cana-3289	206	6	,	,	PUNCT
cana-3289	206	7	l.	l.	PROPN
cana-3289	206	8	zedam	zedam	PROPN
cana-3289	206	9	,	,	PUNCT
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cana-3289	206	12	,	,	PUNCT
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cana-3289	206	18	and	and	CCONJ
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cana-3289	206	20	based	base	VERB
cana-3289	206	21	on	on	ADP
cana-3289	206	22	lattice	lattice	NOUN
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cana-3289	206	30	.	.	PUNCT
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cana-3289	207	2	(	(	PUNCT
cana-3289	207	3	2017	2017	NUM
cana-3289	207	4	)	)	PUNCT
cana-3289	207	5	,	,	PUNCT
cana-3289	207	6	143	143	NUM
cana-3289	207	7	-	-	SYM
cana-3289	207	8	159	159	NUM
cana-3289	207	9	.	.	PUNCT
cana-3289	208	1	https://doi.org/10.5899/2017/jfsva-00399	https://doi.org/10.5899/2017/jfsva-00399	NOUN
cana-3289	208	2	.	.	PUNCT
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cana-3289	209	2	14	14	NUM
cana-3289	209	3	]	]	X
cana-3289	209	4	b.	b.	PROPN
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cana-3289	209	6	,	,	PUNCT
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cana-3289	209	9	,	,	PUNCT
cana-3289	209	10	t.	t.	PROPN
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cana-3289	209	12	,	,	PUNCT
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cana-3289	209	15	,	,	PUNCT
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cana-3289	209	17	rosenthaliana	rosenthaliana	PROPN
cana-3289	209	18	,	,	PUNCT
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cana-3289	209	20	(	(	PUNCT
cana-3289	209	21	2020	2020	NUM
cana-3289	209	22	)	)	PUNCT
cana-3289	209	23	,	,	PUNCT
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cana-3289	209	25	-	-	SYM
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cana-3289	209	27	.	.	PUNCT
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cana-3289	210	2	15	15	NUM
cana-3289	210	3	]	]	X
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cana-3289	210	7	,	,	PUNCT
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cana-3289	210	10	,	,	PUNCT
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cana-3289	210	14	,	,	PUNCT
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cana-3289	210	16	vague	vague	ADJ
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cana-3289	211	4	(	(	PUNCT
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cana-3289	211	6	)	)	PUNCT
cana-3289	211	7	,	,	PUNCT
cana-3289	211	8	115	115	NUM
cana-3289	211	9	-	-	SYM
cana-3289	211	10	124	124	NUM
cana-3289	211	11	.	.	PUNCT
cana-3289	212	1	[	[	X
cana-3289	212	2	16	16	NUM
cana-3289	212	3	]	]	X
cana-3289	212	4	uv	uv	NOUN
cana-3289	212	5	,	,	PUNCT
cana-3289	212	6	kalyani	kalyani	PROPN
cana-3289	212	7	&	&	CCONJ
cana-3289	212	8	tamma	tamma	PROPN
cana-3289	212	9	,	,	PUNCT
cana-3289	212	10	eswarlal	eswarlal	PROPN
cana-3289	212	11	&	&	CCONJ
cana-3289	212	12	jacob	jacob	PROPN
cana-3289	212	13	,	,	PUNCT
cana-3289	212	14	kavikumar	kavikumar	PROPN
cana-3289	212	15	&	&	CCONJ
cana-3289	212	16	iampan	iampan	PROPN
cana-3289	212	17	,	,	PUNCT
cana-3289	212	18	aiyared	aiyare	VERB
cana-3289	212	19	.	.	PUNCT
cana-3289	213	1	(	(	PUNCT
cana-3289	213	2	2022	2022	NUM
cana-3289	213	3	)	)	PUNCT
cana-3289	213	4	.	.	PUNCT
cana-3289	214	1	bipolar	bipolar	ADJ
cana-3289	214	2	fuzzy	fuzzy	ADJ
cana-3289	214	3	sublattices	sublattice	NOUN
cana-3289	214	4	and	and	CCONJ
cana-3289	214	5	ideals	ideal	NOUN
cana-3289	214	6	.	.	PUNCT
cana-3289	215	1	international	international	ADJ
cana-3289	215	2	journal	journal	NOUN
cana-3289	215	3	of	of	ADP
cana-3289	215	4	analysis	analysis	NOUN
cana-3289	215	5	and	and	CCONJ
cana-3289	215	6	applications	application	NOUN
cana-3289	215	7	.	.	PUNCT
cana-3289	216	1	20	20	NUM
cana-3289	216	2	.	.	X
cana-3289	217	1	45	45	NUM
cana-3289	217	2	.	.	X
cana-3289	218	1	10.28924/2291	10.28924/2291	NUM
cana-3289	218	2	-	-	PUNCT
cana-3289	218	3	8639	8639	NUM
cana-3289	218	4	-	-	PUNCT
cana-3289	218	5	20	20	NUM
cana-3289	218	6	-	-	PUNCT
cana-3289	218	7	2022	2022	NUM
cana-3289	218	8	-	-	SYM
cana-3289	218	9	45	45	NUM
cana-3289	218	10	.	.	PUNCT
cana-3289	219	1	[	[	X
cana-3289	219	2	17	17	NUM
cana-3289	219	3	]	]	X
cana-3289	219	4	uv	uv	NOUN
cana-3289	219	5	,	,	PUNCT
cana-3289	219	6	kalyani	kalyani	PROPN
cana-3289	219	7	&	&	CCONJ
cana-3289	219	8	iampan	iampan	PROPN
cana-3289	219	9	,	,	PUNCT
cana-3289	219	10	aiyared	aiyared	PROPN
cana-3289	219	11	&	&	CCONJ
cana-3289	219	12	tamma	tamma	PROPN
cana-3289	219	13	,	,	PUNCT
cana-3289	219	14	eswarlal	eswarlal	PROPN
cana-3289	219	15	.	.	PUNCT
cana-3289	220	1	(	(	PUNCT
cana-3289	220	2	2024	2024	NUM
cana-3289	220	3	)	)	PUNCT
cana-3289	220	4	.	.	PUNCT
cana-3289	221	1	bipolar	bipolar	ADJ
cana-3289	221	2	fuzzy	fuzzy	ADJ
cana-3289	221	3	magnified	magnify	VERB
cana-3289	221	4	translation	translation	NOUN
cana-3289	221	5	of	of	ADP
cana-3289	221	6	a	a	DET
cana-3289	221	7	lattice	lattice	NOUN
cana-3289	221	8	.	.	PUNCT
cana-3289	222	1	international	international	ADJ
cana-3289	222	2	journal	journal	NOUN
cana-3289	222	3	of	of	ADP
cana-3289	222	4	analysis	analysis	NOUN
cana-3289	222	5	and	and	CCONJ
cana-3289	222	6	applications	application	NOUN
cana-3289	222	7	.	.	PUNCT
cana-3289	223	1	22	22	NUM
cana-3289	223	2	.	.	X
cana-3289	224	1	195	195	NUM
cana-3289	224	2	.	.	NOUN
cana-3289	225	1	10.28924/2291	10.28924/2291	NUM
cana-3289	225	2	-	-	PUNCT
cana-3289	225	3	8639	8639	NUM
cana-3289	225	4	-	-	PUNCT
cana-3289	225	5	22	22	NUM
cana-3289	225	6	-	-	PUNCT
cana-3289	225	7	2024	2024	NUM
cana-3289	225	8	-	-	SYM
cana-3289	225	9	195	195	NUM
cana-3289	225	10	.	.	PUNCT
cana-3289	226	1	https://doi.org/10.1016/	https://doi.org/10.1016/	PROPN
cana-3289	226	2	https://doi.org/10.37418/amsj.9.6.11	https://doi.org/10.37418/amsj.9.6.11	ADV
cana-3289	226	3	https://doi/	https://doi/	ADP
cana-3289	226	4	https://doi.org/10.7151/dmgaa.1325	https://doi.org/10.7151/dmgaa.1325	PROPN
cana-3289	226	5	https://digitalcommons.pvamu.edu/aam/vol15/iss2/13	https://digitalcommons.pvamu.edu/aam/vol15/iss2/13	PROPN
cana-3289	226	6	https://doi.org/10.1007/s00500-018-3647-2	https://doi.org/10.1007/s00500-018-3647-2	NUM
cana-3289	226	7	https://doi.org/10.5899/2017/jfsva-00399	https://doi.org/10.5899/2017/jfsva-00399	NOUN