id	sid	tid	token	lemma	pos
cana-2746	1	1	communications	communication	NOUN
cana-2746	1	2	on	on	ADP
cana-2746	1	3	applied	apply	VERB
cana-2746	1	4	nonlinear	nonlinear	ADJ
cana-2746	1	5	analysis	analysis	NOUN
cana-2746	1	6	issn	issn	NOUN
cana-2746	1	7	:	:	PUNCT
cana-2746	1	8	1074	1074	NUM
cana-2746	1	9	-	-	PUNCT
cana-2746	1	10	133x	133x	NUM
cana-2746	1	11	vol	vol	NOUN
cana-2746	1	12	32	32	NUM
cana-2746	1	13	no	no	NOUN
cana-2746	1	14	.	.	PUNCT
cana-2746	2	1	4s	4s	NUM
cana-2746	2	2	(	(	PUNCT
cana-2746	2	3	2025	2025	NUM
cana-2746	2	4	)	)	PUNCT
cana-2746	2	5	150	150	NUM
cana-2746	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2746	2	7	characterization	characterization	NOUN
cana-2746	2	8	of	of	ADP
cana-2746	2	9	pythagorean	pythagorean	PROPN
cana-2746	2	10	fuzzy	fuzzy	ADJ
cana-2746	2	11	bi	bi	ADJ
cana-2746	2	12	-	-	ADJ
cana-2746	2	13	interior	interior	ADJ
cana-2746	2	14	ideal	ideal	NOUN
cana-2746	2	15	and	and	CCONJ
cana-2746	2	16	bi	bi	NOUN
cana-2746	2	17	-	-	ADJ
cana-2746	2	18	quasiideal	quasiideal	ADJ
cana-2746	2	19	in	in	ADP
cana-2746	2	20	𝚪-semirings	𝚪-semirings	PROPN
cana-2746	2	21	t.	t.	NOUN
cana-2746	2	22	anitha1	anitha1	NOUN
cana-2746	2	23	,	,	PUNCT
cana-2746	2	24	y.	y.	NOUN
cana-2746	2	25	lavanya2	lavanya2	NOUN
cana-2746	3	1	1department	1department	NUM
cana-2746	3	2	of	of	ADP
cana-2746	3	3	mathematics	mathematics	PROPN
cana-2746	3	4	,	,	PUNCT
cana-2746	3	5	annamalai	annamalai	PROPN
cana-2746	3	6	university	university	PROPN
cana-2746	3	7	,	,	PUNCT
cana-2746	3	8	annamalainagar	annamalainagar	NOUN
cana-2746	3	9	,	,	PUNCT
cana-2746	3	10	608002	608002	NUM
cana-2746	3	11	.	.	PUNCT
cana-2746	4	1	anitha81t@gmail.com	anitha81t@gmail.com	PROPN
cana-2746	4	2	2department	2department	NUM
cana-2746	4	3	of	of	ADP
cana-2746	4	4	mathematics	mathematics	PROPN
cana-2746	4	5	,	,	PUNCT
cana-2746	4	6	annamalai	annamalai	PROPN
cana-2746	4	7	university	university	PROPN
cana-2746	4	8	,	,	PUNCT
cana-2746	4	9	annamalainagar	annamalainagar	NOUN
cana-2746	4	10	,	,	PUNCT
cana-2746	4	11	608002	608002	NUM
cana-2746	4	12	.	.	PUNCT
cana-2746	5	1	lavanyaannamalaiuniversity@gmail.com	lavanyaannamalaiuniversity@gmail.com	X
cana-2746	5	2	article	article	NOUN
cana-2746	5	3	history	history	NOUN
cana-2746	5	4	:	:	PUNCT
cana-2746	5	5	received	receive	VERB
cana-2746	5	6	:	:	PUNCT
cana-2746	5	7	19	19	NUM
cana-2746	5	8	-	-	PUNCT
cana-2746	5	9	09	09	NUM
cana-2746	5	10	-	-	PUNCT
cana-2746	5	11	2024	2024	NUM
cana-2746	5	12	revised	revise	VERB
cana-2746	5	13	:	:	PUNCT
cana-2746	5	14	24	24	NUM
cana-2746	5	15	-	-	SYM
cana-2746	5	16	11	11	NUM
cana-2746	5	17	-	-	PUNCT
cana-2746	5	18	2024	2024	NUM
cana-2746	5	19	accepted	accept	VERB
cana-2746	5	20	:	:	PUNCT
cana-2746	5	21	01	01	NUM
cana-2746	5	22	-	-	SYM
cana-2746	5	23	12	12	NUM
cana-2746	5	24	-	-	PUNCT
cana-2746	5	25	2024	2024	NUM
cana-2746	5	26	abstract	abstract	NOUN
cana-2746	5	27	:	:	PUNCT
cana-2746	5	28	in	in	ADP
cana-2746	5	29	this	this	DET
cana-2746	5	30	paper	paper	NOUN
cana-2746	5	31	,	,	PUNCT
cana-2746	5	32	we	we	PRON
cana-2746	5	33	introduce	introduce	VERB
cana-2746	5	34	the	the	DET
cana-2746	5	35	pythagorean	pythagorean	ADJ
cana-2746	5	36	fuzzy	fuzzy	ADJ
cana-2746	5	37	bi	bi	ADJ
cana-2746	5	38	-	-	ADJ
cana-2746	5	39	interior	interior	ADJ
cana-2746	5	40	-	-	PUNCT
cana-2746	5	41	ideals	ideal	NOUN
cana-2746	5	42	and	and	CCONJ
cana-2746	5	43	pythagorean	pythagorean	VERB
cana-2746	6	1	fuzzy	fuzzy	ADJ
cana-2746	6	2	bi	bi	ADJ
cana-2746	6	3	-	-	ADJ
cana-2746	6	4	quasi	quasi	NOUN
cana-2746	6	5	-	-	NOUN
cana-2746	7	1	ideals	ideal	NOUN
cana-2746	7	2	in	in	ADP
cana-2746	7	3	γ	γ	X
cana-2746	7	4	semiring	semiring	NOUN
cana-2746	7	5	.	.	PUNCT
cana-2746	8	1	more	more	ADV
cana-2746	8	2	over	over	ADP
cana-2746	8	3	we	we	PRON
cana-2746	8	4	prove	prove	VERB
cana-2746	8	5	the	the	DET
cana-2746	8	6	every	every	DET
cana-2746	8	7	pythagorean	pythagorean	PROPN
cana-2746	8	8	fuzzy	fuzzy	NOUN
cana-2746	8	9	left	leave	VERB
cana-2746	8	10	and	and	CCONJ
cana-2746	8	11	right	right	ADJ
cana-2746	8	12	ideals	ideal	NOUN
cana-2746	8	13	are	be	AUX
cana-2746	8	14	pythagorean	pythagorean	PROPN
cana-2746	8	15	fuzzy	fuzzy	ADJ
cana-2746	8	16	bi	bi	ADJ
cana-2746	8	17	-	-	ADJ
cana-2746	8	18	interior	interior	ADJ
cana-2746	8	19	-ideal	-ideal	NOUN
cana-2746	8	20	in	in	ADP
cana-2746	8	21	γ	γ	X
cana-2746	8	22	semiring	semiring	NOUN
cana-2746	8	23	.	.	PUNCT
cana-2746	9	1	also	also	ADV
cana-2746	9	2	we	we	PRON
cana-2746	9	3	study	study	VERB
cana-2746	9	4	the	the	DET
cana-2746	9	5	notion	notion	NOUN
cana-2746	9	6	of	of	ADP
cana-2746	9	7	pythagorean	pythagorean	PROPN
cana-2746	9	8	fuzzy	fuzzy	ADJ
cana-2746	9	9	bi	bi	ADJ
cana-2746	9	10	-	-	ADJ
cana-2746	9	11	quasi	quasi	ADJ
cana-2746	9	12	-	-	NOUN
cana-2746	9	13	ideal	ideal	ADJ
cana-2746	9	14	in	in	ADP
cana-2746	9	15	γ	γ	X
cana-2746	9	16	semiring	semire	VERB
cana-2746	9	17	and	and	CCONJ
cana-2746	9	18	characterize	characterize	VERB
cana-2746	9	19	pythagorean	pythagorean	PROPN
cana-2746	9	20	fuzzy	fuzzy	ADJ
cana-2746	9	21	bi	bi	ADJ
cana-2746	9	22	-	-	ADJ
cana-2746	9	23	quasi	quasi	ADJ
cana-2746	9	24	-	-	NOUN
cana-2746	9	25	ideal	ideal	ADJ
cana-2746	9	26	in	in	ADP
cana-2746	9	27	γ	γ	X
cana-2746	9	28	semiring	semiring	NOUN
cana-2746	9	29	.	.	PUNCT
cana-2746	10	1	keywords	keyword	NOUN
cana-2746	10	2	:	:	PUNCT
cana-2746	10	3	fuzzy	fuzzy	ADJ
cana-2746	10	4	set	set	NOUN
cana-2746	10	5	,	,	PUNCT
cana-2746	10	6	bi	bi	NOUN
cana-2746	10	7	-	-	ADJ
cana-2746	10	8	ideal	ideal	ADJ
cana-2746	10	9	,	,	PUNCT
cana-2746	10	10	semiring	semire	VERB
cana-2746	10	11	1	1	NUM
cana-2746	10	12	.	.	PUNCT
cana-2746	10	13	introduction	introduction	NOUN
cana-2746	10	14	as	as	ADP
cana-2746	10	15	a	a	DET
cana-2746	10	16	generalization	generalization	NOUN
cana-2746	10	17	of	of	ADP
cana-2746	10	18	ring	ring	NOUN
cana-2746	10	19	,	,	PUNCT
cana-2746	10	20	the	the	DET
cana-2746	10	21	notion	notion	NOUN
cana-2746	10	22	of	of	ADP
cana-2746	10	23	a	a	DET
cana-2746	10	24	γ	γ	PROPN
cana-2746	10	25	ring	ring	NOUN
cana-2746	10	26	was	be	AUX
cana-2746	10	27	introduced	introduce	VERB
cana-2746	10	28	by	by	ADP
cana-2746	10	29	nobusawa	nobusawa	PROPN
cana-2746	11	1	[	[	X
cana-2746	11	2	21	21	NUM
cana-2746	11	3	]	]	PUNCT
cana-2746	11	4	in	in	ADP
cana-2746	11	5	1964	1964	NUM
cana-2746	11	6	and	and	CCONJ
cana-2746	11	7	iseki	iseki	PROPN
cana-2746	11	8	[	[	X
cana-2746	11	9	7	7	NUM
cana-2746	11	10	,	,	PUNCT
cana-2746	11	11	8	8	NUM
cana-2746	11	12	,	,	PUNCT
cana-2746	11	13	9	9	NUM
cana-2746	11	14	]	]	PUNCT
cana-2746	11	15	studied	study	VERB
cana-2746	11	16	the	the	DET
cana-2746	11	17	ideal	ideal	ADJ
cana-2746	11	18	theory	theory	NOUN
cana-2746	11	19	in	in	ADP
cana-2746	11	20	semiring	semiring	NOUN
cana-2746	11	21	.	.	PUNCT
cana-2746	12	1	in	in	ADP
cana-2746	12	2	1995	1995	NUM
cana-2746	12	3	,	,	PUNCT
cana-2746	12	4	murali	murali	PROPN
cana-2746	12	5	krishna	krishna	PROPN
cana-2746	12	6	rao	rao	PROPN
cana-2746	13	1	[	[	X
cana-2746	13	2	23	23	NUM
cana-2746	13	3	,	,	PUNCT
cana-2746	13	4	24	24	NUM
cana-2746	13	5	]	]	PUNCT
cana-2746	13	6	introduced	introduce	VERB
cana-2746	13	7	the	the	DET
cana-2746	13	8	notion	notion	NOUN
cana-2746	13	9	of	of	ADP
cana-2746	13	10	a	a	DET
cana-2746	13	11	γ	γ	X
cana-2746	13	12	semiring	semiring	NOUN
cana-2746	13	13	as	as	ADP
cana-2746	13	14	a	a	DET
cana-2746	13	15	generalization	generalization	NOUN
cana-2746	13	16	of	of	ADP
cana-2746	13	17	γ	γ	PROPN
cana-2746	13	18	-	-	PUNCT
cana-2746	13	19	ring	ring	NOUN
cana-2746	13	20	,	,	PUNCT
cana-2746	13	21	ring	ring	NOUN
cana-2746	13	22	,	,	PUNCT
cana-2746	13	23	ternary	ternary	ADJ
cana-2746	13	24	semiring	semiring	NOUN
cana-2746	13	25	and	and	CCONJ
cana-2746	13	26	semiring	semiring	NOUN
cana-2746	13	27	.	.	PUNCT
cana-2746	14	1	ahsan	ahsan	PROPN
cana-2746	14	2	et.al	et.al	PROPN
cana-2746	15	1	[	[	X
cana-2746	15	2	1	1	NUM
cana-2746	15	3	]	]	PUNCT
cana-2746	15	4	introduced	introduce	VERB
cana-2746	15	5	the	the	DET
cana-2746	15	6	concept	concept	NOUN
cana-2746	15	7	of	of	ADP
cana-2746	15	8	fuzzy	fuzzy	ADJ
cana-2746	15	9	semirings	semiring	NOUN
cana-2746	15	10	.	.	PUNCT
cana-2746	16	1	the	the	DET
cana-2746	16	2	concept	concept	NOUN
cana-2746	16	3	of	of	ADP
cana-2746	16	4	soft	soft	ADJ
cana-2746	16	5	set	set	NOUN
cana-2746	16	6	was	be	AUX
cana-2746	16	7	established	establish	VERB
cana-2746	16	8	by	by	ADP
cana-2746	16	9	molodtsov	molodtsov	NOUN
cana-2746	16	10	[	[	X
cana-2746	16	11	20	20	NUM
cana-2746	16	12	]	]	PUNCT
cana-2746	16	13	,	,	PUNCT
cana-2746	16	14	which	which	PRON
cana-2746	16	15	deals	deal	VERB
cana-2746	16	16	with	with	ADP
cana-2746	16	17	parametrized	parametrized	ADJ
cana-2746	16	18	values	value	NOUN
cana-2746	16	19	of	of	ADP
cana-2746	16	20	the	the	DET
cana-2746	16	21	alternative	alternative	NOUN
cana-2746	16	22	.	.	PUNCT
cana-2746	17	1	maji	maji	PROPN
cana-2746	17	2	et	et	PROPN
cana-2746	17	3	al	al	PROPN
cana-2746	17	4	.	.	PUNCT
cana-2746	18	1	[	[	X
cana-2746	18	2	17	17	NUM
cana-2746	18	3	,	,	PUNCT
cana-2746	18	4	18	18	NUM
cana-2746	18	5	,	,	PUNCT
cana-2746	18	6	19	19	NUM
cana-2746	18	7	]	]	PUNCT
cana-2746	18	8	investigated	investigate	VERB
cana-2746	18	9	the	the	DET
cana-2746	18	10	soft	soft	ADJ
cana-2746	18	11	set	set	ADJ
cana-2746	18	12	views	view	NOUN
cana-2746	18	13	on	on	ADP
cana-2746	18	14	decision	decision	NOUN
cana-2746	18	15	-	-	PUNCT
cana-2746	18	16	making	make	VERB
cana-2746	18	17	issues	issue	NOUN
cana-2746	18	18	and	and	CCONJ
cana-2746	18	19	defined	define	VERB
cana-2746	18	20	some	some	DET
cana-2746	18	21	important	important	ADJ
cana-2746	18	22	concepts	concept	NOUN
cana-2746	18	23	for	for	ADP
cana-2746	18	24	soft	soft	ADJ
cana-2746	18	25	set	set	NOUN
cana-2746	18	26	with	with	ADP
cana-2746	18	27	their	their	PRON
cana-2746	18	28	properties	property	NOUN
cana-2746	18	29	.	.	PUNCT
cana-2746	19	1	maji	maji	PROPN
cana-2746	19	2	et	et	PROPN
cana-2746	19	3	al.[18	al.[18	PROPN
cana-2746	19	4	]	]	PUNCT
cana-2746	19	5	offered	offer	VERB
cana-2746	19	6	the	the	DET
cana-2746	19	7	notion	notion	NOUN
cana-2746	19	8	of	of	ADP
cana-2746	19	9	the	the	DET
cana-2746	19	10	fuzzy	fuzzy	ADJ
cana-2746	19	11	soft	soft	ADJ
cana-2746	19	12	set	set	NOUN
cana-2746	19	13	by	by	ADP
cana-2746	19	14	merging	merge	VERB
cana-2746	19	15	two	two	NUM
cana-2746	19	16	existing	exist	VERB
cana-2746	19	17	notions	notion	NOUN
cana-2746	19	18	fuzzy	fuzzy	ADJ
cana-2746	19	19	sets	set	NOUN
cana-2746	19	20	and	and	CCONJ
cana-2746	19	21	soft	soft	ADJ
cana-2746	19	22	sets	set	NOUN
cana-2746	19	23	.	.	PUNCT
cana-2746	20	1	peng	peng	PROPN
cana-2746	20	2	et	et	PROPN
cana-2746	20	3	al	al	PROPN
cana-2746	20	4	.	.	PUNCT
cana-2746	21	1	[	[	X
cana-2746	21	2	22	22	NUM
cana-2746	21	3	]	]	PUNCT
cana-2746	21	4	protracted	protract	VERB
cana-2746	21	5	the	the	DET
cana-2746	21	6	idea	idea	NOUN
cana-2746	21	7	of	of	ADP
cana-2746	21	8	intuitionistic	intuitionistic	ADJ
cana-2746	21	9	fuzzy	fuzzy	ADJ
cana-2746	21	10	soft	soft	ADJ
cana-2746	21	11	set	set	NOUN
cana-2746	21	12	to	to	PART
cana-2746	21	13	pythagorean	pythagorean	VERB
cana-2746	21	14	fuzzy	fuzzy	ADJ
cana-2746	21	15	soft	soft	ADJ
cana-2746	21	16	set	set	VERB
cana-2746	21	17	by	by	ADP
cana-2746	21	18	upgrading	upgrade	VERB
cana-2746	21	19	the	the	DET
cana-2746	21	20	conditions	condition	NOUN
cana-2746	21	21	.	.	PUNCT
cana-2746	22	1	the	the	DET
cana-2746	22	2	fuzzy	fuzzy	ADJ
cana-2746	22	3	set	set	NOUN
cana-2746	22	4	was	be	AUX
cana-2746	22	5	studied	study	VERB
cana-2746	22	6	by	by	ADP
cana-2746	22	7	zadeh’s[36	zadeh’s[36	PROPN
cana-2746	22	8	]	]	PUNCT
cana-2746	22	9	in	in	ADP
cana-2746	22	10	his	his	PRON
cana-2746	22	11	seminal	seminal	ADJ
cana-2746	22	12	paper	paper	NOUN
cana-2746	22	13	.	.	PUNCT
cana-2746	23	1	pythagorean	pythagorean	PROPN
cana-2746	23	2	fuzzy	fuzzy	ADJ
cana-2746	23	3	sets	set	NOUN
cana-2746	23	4	[	[	X
cana-2746	23	5	34][35	34][35	NUM
cana-2746	23	6	]	]	PUNCT
cana-2746	23	7	characterized	characterize	VERB
cana-2746	23	8	by	by	ADP
cana-2746	23	9	the	the	DET
cana-2746	23	10	condition	condition	NOUN
cana-2746	23	11	that	that	SCONJ
cana-2746	23	12	the	the	DET
cana-2746	23	13	sum	sum	NOUN
cana-2746	23	14	of	of	ADP
cana-2746	23	15	the	the	DET
cana-2746	23	16	squares	square	NOUN
cana-2746	23	17	of	of	ADP
cana-2746	23	18	membership	membership	NOUN
cana-2746	23	19	and	and	CCONJ
cana-2746	23	20	non	non	ADJ
cana-2746	23	21	-	-	ADJ
cana-2746	23	22	membership	membership	ADJ
cana-2746	23	23	degrees	degree	NOUN
cana-2746	23	24	is	be	AUX
cana-2746	23	25	less	less	ADJ
cana-2746	23	26	than	than	ADP
cana-2746	23	27	or	or	CCONJ
cana-2746	23	28	equal	equal	ADJ
cana-2746	23	29	to	to	ADP
cana-2746	23	30	one	one	NUM
cana-2746	23	31	,	,	PUNCT
cana-2746	23	32	have	have	AUX
cana-2746	23	33	been	be	AUX
cana-2746	23	34	extensively	extensively	ADV
cana-2746	23	35	investigated	investigate	VERB
cana-2746	23	36	.	.	PUNCT
cana-2746	24	1	numerous	numerous	ADJ
cana-2746	24	2	authors	author	NOUN
cana-2746	24	3	have	have	AUX
cana-2746	24	4	explored	explore	VERB
cana-2746	24	5	the	the	DET
cana-2746	24	6	algebraic	algebraic	ADJ
cana-2746	24	7	properties	property	NOUN
cana-2746	24	8	of	of	ADP
cana-2746	24	9	pythagorean	pythagorean	ADJ
cana-2746	24	10	fuzzy	fuzzy	ADJ
cana-2746	24	11	ideals	ideal	NOUN
cana-2746	24	12	.	.	PUNCT
cana-2746	25	1	this	this	DET
cana-2746	25	2	paper	paper	NOUN
cana-2746	25	3	is	be	AUX
cana-2746	25	4	structured	structure	VERB
cana-2746	25	5	into	into	ADP
cana-2746	25	6	three	three	NUM
cana-2746	25	7	sections	section	NOUN
cana-2746	25	8	.	.	PUNCT
cana-2746	26	1	the	the	DET
cana-2746	26	2	initial	initial	ADJ
cana-2746	26	3	two	two	NUM
cana-2746	26	4	sections	section	NOUN
cana-2746	26	5	provide	provide	VERB
cana-2746	26	6	an	an	DET
cana-2746	26	7	introduction	introduction	NOUN
cana-2746	26	8	and	and	CCONJ
cana-2746	26	9	lay	lie	VERB
cana-2746	26	10	down	down	ADP
cana-2746	26	11	the	the	DET
cana-2746	26	12	preliminary	preliminary	ADJ
cana-2746	26	13	concepts	concept	NOUN
cana-2746	26	14	.	.	PUNCT
cana-2746	27	1	the	the	DET
cana-2746	27	2	third	third	ADJ
cana-2746	27	3	section	section	NOUN
cana-2746	27	4	deals	deal	NOUN
cana-2746	27	5	with	with	ADP
cana-2746	27	6	pythagorean	pythagorean	PROPN
cana-2746	27	7	fuzzy	fuzzy	ADJ
cana-2746	27	8	bi	bi	ADJ
cana-2746	27	9	-	-	ADJ
cana-2746	27	10	interior	interior	ADJ
cana-2746	27	11	ideal	ideal	NOUN
cana-2746	27	12	and	and	CCONJ
cana-2746	27	13	its	its	PRON
cana-2746	27	14	properties	property	NOUN
cana-2746	27	15	in	in	ADP
cana-2746	27	16	γ	γ	X
cana-2746	27	17	semiring	semiring	NOUN
cana-2746	27	18	.	.	PUNCT
cana-2746	28	1	also	also	ADV
cana-2746	28	2	characterize	characterize	VERB
cana-2746	28	3	pythagorean	pythagorean	PROPN
cana-2746	28	4	fuzzy	fuzzy	ADJ
cana-2746	28	5	soft	soft	ADJ
cana-2746	28	6	bi	bi	ADJ
cana-2746	28	7	-	-	ADJ
cana-2746	28	8	interior	interior	ADJ
cana-2746	28	9	ideal	ideal	NOUN
cana-2746	28	10	.	.	PUNCT
cana-2746	29	1	fourth	fourth	ADJ
cana-2746	29	2	section	section	NOUN
cana-2746	29	3	deals	deal	VERB
cana-2746	29	4	with	with	ADP
cana-2746	29	5	the	the	DET
cana-2746	29	6	pythagorean	pythagorean	PROPN
cana-2746	29	7	fuzzy	fuzzy	ADJ
cana-2746	29	8	bi	bi	ADJ
cana-2746	29	9	-	-	ADJ
cana-2746	29	10	quasi	quasi	ADJ
cana-2746	29	11	-	-	ADJ
cana-2746	29	12	ideal	ideal	ADJ
cana-2746	29	13	and	and	CCONJ
cana-2746	29	14	prove	prove	VERB
cana-2746	29	15	some	some	DET
cana-2746	29	16	important	important	ADJ
cana-2746	29	17	properties	property	NOUN
cana-2746	29	18	.	.	PUNCT
cana-2746	30	1	2	2	X
cana-2746	30	2	.	.	X
cana-2746	30	3	preliminaries	preliminary	NOUN
cana-2746	30	4	this	this	DET
cana-2746	30	5	section	section	NOUN
cana-2746	30	6	deals	deal	VERB
cana-2746	30	7	with	with	ADP
cana-2746	30	8	the	the	DET
cana-2746	30	9	basic	basic	ADJ
cana-2746	30	10	definitions	definition	NOUN
cana-2746	30	11	.	.	PUNCT
cana-2746	31	1	a	a	DET
cana-2746	31	2	semiring	semiring	NOUN
cana-2746	31	3	is	be	AUX
cana-2746	31	4	a	a	DET
cana-2746	31	5	set	set	ADJ
cana-2746	31	6	𝑆	𝑆	PROPN
cana-2746	31	7	with	with	ADP
cana-2746	31	8	two	two	NUM
cana-2746	31	9	binary	binary	ADJ
cana-2746	31	10	operations	operation	NOUN
cana-2746	31	11	+	+	CCONJ
cana-2746	31	12	and	and	CCONJ
cana-2746	31	13	.	.	PUNCT
cana-2746	32	1	on	on	ADP
cana-2746	32	2	𝑆	𝑆	PROPN
cana-2746	32	3	called	call	VERB
cana-2746	32	4	addition	addition	NOUN
cana-2746	32	5	and	and	CCONJ
cana-2746	32	6	multiplications	multiplication	NOUN
cana-2746	32	7	such	such	ADJ
cana-2746	32	8	that	that	PRON
cana-2746	32	9	,	,	PUNCT
cana-2746	32	10	(	(	PUNCT
cana-2746	32	11	i	i	NOUN
cana-2746	32	12	)	)	PUNCT
cana-2746	32	13	(	(	PUNCT
cana-2746	32	14	𝑆	𝑆	PROPN
cana-2746	32	15	,	,	PUNCT
cana-2746	32	16	+	+	PUNCT
cana-2746	32	17	)	)	PUNCT
cana-2746	32	18	is	be	AUX
cana-2746	32	19	a	a	DET
cana-2746	32	20	semigroup	semigroup	NOUN
cana-2746	32	21	,	,	PUNCT
cana-2746	32	22	(	(	PUNCT
cana-2746	32	23	ii)(𝑆	ii)(𝑆	VERB
cana-2746	32	24	,	,	PUNCT
cana-2746	32	25	.	.	PUNCT
cana-2746	32	26	)	)	PUNCT
cana-2746	33	1	is	be	AUX
cana-2746	33	2	a	a	DET
cana-2746	33	3	semigroup	semigroup	NOUN
cana-2746	33	4	and	and	CCONJ
cana-2746	33	5	(	(	PUNCT
cana-2746	33	6	iii	iii	X
cana-2746	33	7	)	)	PUNCT
cana-2746	33	8	𝑎(𝑏	𝑎(𝑏	PROPN
cana-2746	33	9	+	+	PROPN
cana-2746	33	10	𝑐	𝑐	X
cana-2746	33	11	)	)	PUNCT
cana-2746	33	12	=	=	PUNCT
cana-2746	34	1	𝑎𝑏	𝑎𝑏	PROPN
cana-2746	34	2	+	+	CCONJ
cana-2746	34	3	𝑎𝑐	𝑎𝑐	X
cana-2746	34	4	and	and	CCONJ
cana-2746	34	5	(	(	PUNCT
cana-2746	34	6	𝑎	𝑎	PROPN
cana-2746	34	7	+	+	X
cana-2746	34	8	𝑏)𝑐	𝑏)𝑐	NOUN
cana-2746	34	9	=	=	SYM
cana-2746	34	10	𝑎𝑐	𝑎𝑐	INTJ
cana-2746	34	11	+	+	CCONJ
cana-2746	34	12	𝑏𝑐	𝑏𝑐	PROPN
cana-2746	34	13	for	for	ADP
cana-2746	34	14	all	all	PRON
cana-2746	34	15	𝑎	𝑎	PROPN
cana-2746	34	16	,	,	PUNCT
cana-2746	34	17	𝑏	𝑏	NOUN
cana-2746	34	18	,	,	PUNCT
cana-2746	34	19	𝑐	𝑐	PROPN
cana-2746	34	20	∈	∈	NOUN
cana-2746	34	21	𝑆.	𝑆.	PROPN
cana-2746	34	22	mailto:anitha81t@gmail.com	mailto:anitha81t@gmail.com	X
cana-2746	34	23	mailto:lavanyaannamalaiuniversity@gmail.com	mailto:lavanyaannamalaiuniversity@gmail.com	X
cana-2746	34	24	communications	communication	NOUN
cana-2746	34	25	on	on	ADP
cana-2746	34	26	applied	apply	VERB
cana-2746	34	27	nonlinear	nonlinear	ADJ
cana-2746	34	28	analysis	analysis	NOUN
cana-2746	34	29	issn	issn	NOUN
cana-2746	34	30	:	:	PUNCT
cana-2746	34	31	1074	1074	NUM
cana-2746	34	32	-	-	PUNCT
cana-2746	34	33	133x	133x	NUM
cana-2746	34	34	vol	vol	NOUN
cana-2746	34	35	32	32	NUM
cana-2746	34	36	no	no	NOUN
cana-2746	34	37	.	.	PUNCT
cana-2746	35	1	4s	4s	NUM
cana-2746	35	2	(	(	PUNCT
cana-2746	35	3	2025	2025	NUM
cana-2746	35	4	)	)	PUNCT
cana-2746	35	5	151	151	NUM
cana-2746	35	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2746	35	7	a	a	DET
cana-2746	35	8	nonempty	nonempty	ADV
cana-2746	35	9	subset	subset	VERB
cana-2746	35	10	𝐴	𝐴	PROPN
cana-2746	35	11	of	of	ADP
cana-2746	35	12	a	a	DET
cana-2746	35	13	semiring	semire	VERB
cana-2746	35	14	𝑆	𝑆	PROPN
cana-2746	35	15	is	be	AUX
cana-2746	35	16	called	call	VERB
cana-2746	35	17	a	a	DET
cana-2746	35	18	left	left	ADJ
cana-2746	35	19	(	(	PUNCT
cana-2746	35	20	right	right	ADJ
cana-2746	35	21	)	)	PUNCT
cana-2746	35	22	ideal	ideal	NOUN
cana-2746	35	23	of	of	ADP
cana-2746	35	24	𝑆	𝑆	PROPN
cana-2746	35	25	if	if	SCONJ
cana-2746	35	26	𝐴	𝐴	PROPN
cana-2746	35	27	is	be	AUX
cana-2746	35	28	closed	close	VERB
cana-2746	35	29	under	under	ADP
cana-2746	35	30	addition	addition	NOUN
cana-2746	35	31	and	and	CCONJ
cana-2746	35	32	𝑆𝐴	𝑆𝐴	PROPN
cana-2746	35	33	⊆	⊆	NUM
cana-2746	35	34	𝐴(𝐴𝑆	𝐴(𝐴𝑆	PROPN
cana-2746	35	35	⊆	⊆	NUM
cana-2746	35	36	𝐴	𝐴	PROPN
cana-2746	35	37	)	)	PUNCT
cana-2746	35	38	.	.	PUNCT
cana-2746	36	1	𝐴	𝐴	PROPN
cana-2746	36	2	is	be	AUX
cana-2746	36	3	an	an	DET
cana-2746	36	4	ideal	ideal	NOUN
cana-2746	36	5	of	of	ADP
cana-2746	36	6	𝑆	𝑆	PROPN
cana-2746	36	7	if	if	SCONJ
cana-2746	36	8	it	it	PRON
cana-2746	36	9	is	be	AUX
cana-2746	36	10	both	both	CCONJ
cana-2746	36	11	a	a	DET
cana-2746	36	12	left	left	NOUN
cana-2746	36	13	and	and	CCONJ
cana-2746	36	14	a	a	DET
cana-2746	36	15	right	right	ADJ
cana-2746	36	16	ideal	ideal	NOUN
cana-2746	36	17	of	of	ADP
cana-2746	36	18	the	the	DET
cana-2746	36	19	semiring	semire	VERB
cana-2746	36	20	𝑆.	𝑆.	NOUN
cana-2746	36	21	definition	definition	NOUN
cana-2746	36	22	2.1	2.1	NUM
cana-2746	36	23	[	[	X
cana-2746	36	24	3	3	NUM
cana-2746	36	25	]	]	X
cana-2746	36	26	if	if	SCONJ
cana-2746	36	27	(	(	PUNCT
cana-2746	36	28	𝑆	𝑆	PROPN
cana-2746	36	29	,	,	PUNCT
cana-2746	36	30	+	+	NOUN
cana-2746	36	31	)	)	PUNCT
cana-2746	36	32	and	and	CCONJ
cana-2746	36	33	(	(	PUNCT
cana-2746	36	34	𝛤	𝛤	PROPN
cana-2746	36	35	,	,	PUNCT
cana-2746	36	36	+	+	ADJ
cana-2746	36	37	)	)	PUNCT
cana-2746	36	38	be	be	VERB
cana-2746	36	39	two	two	NUM
cana-2746	36	40	commutative	commutative	ADJ
cana-2746	36	41	semigroups	semigroup	NOUN
cana-2746	36	42	then	then	ADV
cana-2746	36	43	s	s	VERB
cana-2746	36	44	is	be	AUX
cana-2746	36	45	called	call	VERB
cana-2746	36	46	a	a	DET
cana-2746	36	47	𝛤	𝛤	PROPN
cana-2746	36	48	semiring	semire	VERB
cana-2746	36	49	if	if	SCONJ
cana-2746	36	50	there	there	PRON
cana-2746	36	51	exists	exist	VERB
cana-2746	36	52	a	a	DET
cana-2746	36	53	structure	structure	NOUN
cana-2746	36	54	𝑆	𝑆	PROPN
cana-2746	36	55	×	×	NOUN
cana-2746	36	56	𝛤	𝛤	PROPN
cana-2746	36	57	×	×	NOUN
cana-2746	36	58	𝑆	𝑆	PROPN
cana-2746	36	59	denoted	denote	VERB
cana-2746	36	60	by	by	ADP
cana-2746	36	61	𝛼𝛾𝛽	𝛼𝛾𝛽	NOUN
cana-2746	36	62	for	for	ADP
cana-2746	36	63	all	all	DET
cana-2746	36	64	𝛼	𝛼	PROPN
cana-2746	36	65	,	,	PUNCT
cana-2746	36	66	𝛽	𝛽	PROPN
cana-2746	36	67	∈	∈	PROPN
cana-2746	36	68	𝑆	𝑆	PROPN
cana-2746	36	69	and	and	CCONJ
cana-2746	36	70	𝛾	𝛾	ADP
cana-2746	36	71	∈	∈	NOUN
cana-2746	36	72	𝛤	𝛤	NOUN
cana-2746	36	73	satisfying	satisfy	VERB
cana-2746	36	74	the	the	DET
cana-2746	36	75	following	follow	VERB
cana-2746	36	76	properties	property	NOUN
cana-2746	36	77	,	,	PUNCT
cana-2746	36	78	1	1	NUM
cana-2746	36	79	.	.	NUM
cana-2746	36	80	𝛼𝛾(𝛽	𝛼𝛾(𝛽	NUM
cana-2746	36	81	+	+	CCONJ
cana-2746	36	82	𝜈	𝜈	X
cana-2746	36	83	)	)	PUNCT
cana-2746	36	84	=	=	SYM
cana-2746	37	1	𝛼𝛾𝛽	𝛼𝛾𝛽	PROPN
cana-2746	37	2	+	+	CCONJ
cana-2746	37	3	𝛼𝛾𝜈	𝛼𝛾𝜈	ADJ
cana-2746	37	4	,	,	PUNCT
cana-2746	37	5	2	2	NUM
cana-2746	37	6	.	.	PUNCT
cana-2746	37	7	(	(	PUNCT
cana-2746	37	8	𝛽	𝛽	NOUN
cana-2746	37	9	+	+	CCONJ
cana-2746	38	1	𝜈)𝛾𝛼	𝜈)𝛾𝛼	PROPN
cana-2746	38	2	=	=	PUNCT
cana-2746	38	3	𝛽𝛾𝛼	𝛽𝛾𝛼	PROPN
cana-2746	38	4	+	+	CCONJ
cana-2746	38	5	𝜈𝛾𝛼	𝜈𝛾𝛼	PROPN
cana-2746	38	6	,	,	PUNCT
cana-2746	38	7	3	3	NUM
cana-2746	38	8	.	.	X
cana-2746	38	9	𝛼(𝛾	𝛼(𝛾	NUM
cana-2746	38	10	+	+	CCONJ
cana-2746	38	11	𝛾1)𝜈	𝛾1)𝜈	NOUN
cana-2746	38	12	=	=	SYM
cana-2746	38	13	𝛼𝛾𝜈	𝛼𝛾𝜈	PROPN
cana-2746	39	1	+	+	CCONJ
cana-2746	39	2	𝛼𝛾1𝜈	𝛼𝛾1𝜈	NOUN
cana-2746	39	3	,	,	PUNCT
cana-2746	39	4	4	4	NUM
cana-2746	39	5	.	.	NUM
cana-2746	39	6	𝛼𝛾(𝛽𝛾1𝜈	𝛼𝛾(𝛽𝛾1𝜈	NOUN
cana-2746	39	7	)	)	PUNCT
cana-2746	40	1	=	=	PUNCT
cana-2746	40	2	(	(	PUNCT
cana-2746	40	3	𝛼𝛾𝛽)𝛾1𝜈	𝛼𝛾𝛽)𝛾1𝜈	NUM
cana-2746	40	4	for	for	ADP
cana-2746	40	5	all	all	DET
cana-2746	40	6	𝛼	𝛼	PROPN
cana-2746	40	7	,	,	PUNCT
cana-2746	40	8	𝛽	𝛽	PROPN
cana-2746	40	9	,	,	PUNCT
cana-2746	40	10	𝜈	𝜈	PROPN
cana-2746	40	11	∈	∈	PROPN
cana-2746	40	12	𝑆	𝑆	PROPN
cana-2746	40	13	and	and	CCONJ
cana-2746	40	14	𝛾	𝛾	PROPN
cana-2746	40	15	,	,	PUNCT
cana-2746	40	16	𝛾1	𝛾1	PROPN
cana-2746	40	17	∈	∈	PROPN
cana-2746	40	18	γ	γ	PROPN
cana-2746	40	19	.	.	PROPN
cana-2746	40	20	definition	definition	NOUN
cana-2746	40	21	2.2	2.2	NUM
cana-2746	40	22	[	[	X
cana-2746	40	23	3	3	NUM
cana-2746	40	24	]	]	PUNCT
cana-2746	40	25	define	define	VERB
cana-2746	40	26	addition	addition	NOUN
cana-2746	40	27	in	in	ADP
cana-2746	40	28	the	the	DET
cana-2746	40	29	following	follow	VERB
cana-2746	40	30	way	way	NOUN
cana-2746	40	31	𝐴	𝐴	PROPN
cana-2746	40	32	,	,	PUNCT
cana-2746	40	33	𝐵	𝐵	PROPN
cana-2746	40	34	∈	∈	PROPN
cana-2746	40	35	𝑆,𝛾	𝑆,𝛾	PUNCT
cana-2746	41	1	∈	∈	PROPN
cana-2746	41	2	𝛤	𝛤	PROPN
cana-2746	41	3	,	,	PUNCT
cana-2746	41	4	let	let	VERB
cana-2746	41	5	𝐴𝛾𝐵	𝐴𝛾𝐵	NOUN
cana-2746	41	6	denote	denote	VERB
cana-2746	41	7	the	the	DET
cana-2746	41	8	ideal	ideal	NOUN
cana-2746	41	9	generated	generate	VERB
cana-2746	41	10	by	by	ADP
cana-2746	41	11	{	{	PUNCT
cana-2746	41	12	𝛼𝛾𝛽/𝛼	𝛼𝛾𝛽/𝛼	PROPN
cana-2746	41	13	,	,	PUNCT
cana-2746	41	14	𝛽	𝛽	PROPN
cana-2746	41	15	∈	∈	PROPN
cana-2746	41	16	𝑆	𝑆	PROPN
cana-2746	41	17	}	}	PUNCT
cana-2746	41	18	.	.	PUNCT
cana-2746	42	1	then	then	ADV
cana-2746	42	2	𝑆	𝑆	PROPN
cana-2746	42	3	is	be	AUX
cana-2746	42	4	a	a	DET
cana-2746	42	5	𝛤	𝛤	PROPN
cana-2746	42	6	semiring	semire	VERB
cana-2746	42	7	.	.	PUNCT
cana-2746	43	1	definition	definition	NOUN
cana-2746	43	2	2.3	2.3	NUM
cana-2746	43	3	[	[	X
cana-2746	43	4	3	3	X
cana-2746	43	5	]	]	PUNCT
cana-2746	43	6	a	a	DET
cana-2746	43	7	𝛤	𝛤	PROPN
cana-2746	43	8	semiring	semire	VERB
cana-2746	43	9	s	s	NOUN
cana-2746	43	10	is	be	AUX
cana-2746	43	11	said	say	VERB
cana-2746	43	12	to	to	PART
cana-2746	43	13	be	be	AUX
cana-2746	43	14	commutative	commutative	ADJ
cana-2746	43	15	if	if	SCONJ
cana-2746	43	16	𝛼𝛾𝛽	𝛼𝛾𝛽	PROPN
cana-2746	43	17	=	=	SYM
cana-2746	43	18	𝛽𝛾𝛼	𝛽𝛾𝛼	PROPN
cana-2746	43	19	,	,	PUNCT
cana-2746	43	20	for	for	ADP
cana-2746	43	21	all	all	DET
cana-2746	43	22	𝛼	𝛼	PROPN
cana-2746	43	23	,	,	PUNCT
cana-2746	43	24	𝛽	𝛽	PROPN
cana-2746	43	25	∈	∈	PROPN
cana-2746	43	26	𝑆	𝑆	PROPN
cana-2746	43	27	and	and	CCONJ
cana-2746	43	28	𝛾	𝛾	ADP
cana-2746	43	29	∈	∈	PROPN
cana-2746	43	30	𝛤.	𝛤.	PROPN
cana-2746	43	31	definition	definition	NOUN
cana-2746	43	32	2.4	2.4	NUM
cana-2746	44	1	[	[	X
cana-2746	44	2	3	3	X
cana-2746	44	3	]	]	PUNCT
cana-2746	44	4	a	a	DET
cana-2746	44	5	𝛤-semiring	𝛤-semiring	PROPN
cana-2746	44	6	s	s	PART
cana-2746	44	7	is	be	AUX
cana-2746	44	8	said	say	VERB
cana-2746	44	9	to	to	PART
cana-2746	44	10	have	have	VERB
cana-2746	44	11	a	a	DET
cana-2746	44	12	zero	zero	NUM
cana-2746	44	13	element	element	NOUN
cana-2746	44	14	if	if	SCONJ
cana-2746	44	15	0𝛽𝛼	0𝛽𝛼	ADJ
cana-2746	44	16	=	=	SYM
cana-2746	45	1	0	0	PUNCT
cana-2746	45	2	=	=	SYM
cana-2746	45	3	𝛼𝛽0	𝛼𝛽0	PROPN
cana-2746	45	4	and	and	CCONJ
cana-2746	45	5	𝛼	𝛼	PRON
cana-2746	45	6	+	+	NOUN
cana-2746	45	7	0	0	NUM
cana-2746	45	8	=	=	SYM
cana-2746	45	9	𝛼	𝛼	NOUN
cana-2746	45	10	=	=	SYM
cana-2746	45	11	0	0	PUNCT
cana-2746	45	12	+	+	CCONJ
cana-2746	45	13	𝛼	𝛼	X
cana-2746	45	14	,	,	PUNCT
cana-2746	45	15	for	for	ADP
cana-2746	45	16	all	all	DET
cana-2746	45	17	𝛼	𝛼	PRON
cana-2746	45	18	∈	∈	PROPN
cana-2746	45	19	𝑆	𝑆	PROPN
cana-2746	45	20	and	and	CCONJ
cana-2746	45	21	𝛾	𝛾	ADP
cana-2746	45	22	∈	∈	PROPN
cana-2746	45	23	𝛤.	𝛤.	PROPN
cana-2746	45	24	definition	definition	NOUN
cana-2746	45	25	2.5	2.5	NUM
cana-2746	45	26	[	[	X
cana-2746	45	27	3	3	NUM
cana-2746	45	28	]	]	X
cana-2746	45	29	s	s	VERB
cana-2746	45	30	is	be	AUX
cana-2746	45	31	said	say	VERB
cana-2746	45	32	to	to	PART
cana-2746	45	33	have	have	VERB
cana-2746	45	34	a	a	DET
cana-2746	45	35	identity	identity	NOUN
cana-2746	45	36	element	element	NOUN
cana-2746	45	37	if	if	SCONJ
cana-2746	45	38	there	there	PRON
cana-2746	45	39	exists	exist	VERB
cana-2746	45	40	𝛾	𝛾	PROPN
cana-2746	45	41	∈	∈	NOUN
cana-2746	45	42	𝛤	𝛤	PROPN
cana-2746	45	43	such	such	ADJ
cana-2746	45	44	that	that	DET
cana-2746	45	45	1𝛾𝛼	1𝛾𝛼	NOUN
cana-2746	45	46	=	=	SYM
cana-2746	45	47	𝛼	𝛼	X
cana-2746	45	48	=	=	SYM
cana-2746	45	49	𝛼𝛾1	𝛼𝛾1	NOUN
cana-2746	45	50	,	,	PUNCT
cana-2746	45	51	for	for	ADP
cana-2746	45	52	all	all	DET
cana-2746	45	53	𝛼	𝛼	DET
cana-2746	45	54	∈	∈	NOUN
cana-2746	45	55	𝑆.	𝑆.	NOUN
cana-2746	45	56	definition	definition	NOUN
cana-2746	45	57	2.6	2.6	NUM
cana-2746	45	58	[	[	X
cana-2746	45	59	3	3	NUM
cana-2746	45	60	]	]	X
cana-2746	45	61	s	s	VERB
cana-2746	45	62	is	be	AUX
cana-2746	45	63	said	say	VERB
cana-2746	45	64	to	to	PART
cana-2746	45	65	have	have	VERB
cana-2746	45	66	a	a	DET
cana-2746	45	67	strong	strong	ADJ
cana-2746	45	68	identity	identity	NOUN
cana-2746	45	69	element	element	NOUN
cana-2746	45	70	if	if	SCONJ
cana-2746	45	71	for	for	ADP
cana-2746	45	72	all	all	DET
cana-2746	45	73	𝛼	𝛼	PRON
cana-2746	45	74	∈	∈	PROPN
cana-2746	45	75	𝑆	𝑆	PROPN
cana-2746	45	76	,	,	PUNCT
cana-2746	45	77	1𝛾𝛼	1𝛾𝛼	NOUN
cana-2746	45	78	=	=	SYM
cana-2746	45	79	𝛼	𝛼	NOUN
cana-2746	45	80	=	=	PUNCT
cana-2746	45	81	𝛼𝛾1	𝛼𝛾1	PROPN
cana-2746	45	82	for	for	ADP
cana-2746	45	83	all	all	PRON
cana-2746	45	84	𝛾	𝛾	ADP
cana-2746	45	85	∈	∈	ADJ
cana-2746	45	86	𝛤.	𝛤.	PROPN
cana-2746	45	87	definition	definition	NOUN
cana-2746	45	88	2.7	2.7	NUM
cana-2746	45	89	[	[	X
cana-2746	45	90	3	3	X
cana-2746	45	91	]	]	PUNCT
cana-2746	45	92	a	a	DET
cana-2746	45	93	nonempty	nonempty	NOUN
cana-2746	45	94	subset	subset	VERB
cana-2746	45	95	r	r	NOUN
cana-2746	45	96	of	of	ADP
cana-2746	45	97	a	a	DET
cana-2746	45	98	𝛤	𝛤	PROPN
cana-2746	45	99	semiring	semire	VERB
cana-2746	45	100	s	s	NOUN
cana-2746	45	101	is	be	AUX
cana-2746	45	102	said	say	VERB
cana-2746	45	103	to	to	PART
cana-2746	45	104	be	be	AUX
cana-2746	45	105	a	a	DET
cana-2746	45	106	sub𝛤	sub𝛤	ADJ
cana-2746	45	107	semiring	semiring	NOUN
cana-2746	45	108	of	of	ADP
cana-2746	45	109	s	s	PRON
cana-2746	45	110	if	if	SCONJ
cana-2746	45	111	(	(	PUNCT
cana-2746	45	112	𝑅	𝑅	NOUN
cana-2746	45	113	,	,	PUNCT
cana-2746	45	114	+	+	NOUN
cana-2746	45	115	)	)	PUNCT
cana-2746	45	116	is	be	AUX
cana-2746	45	117	a	a	DET
cana-2746	45	118	sub	sub	NOUN
cana-2746	45	119	semigroup	semigroup	NOUN
cana-2746	45	120	of	of	ADP
cana-2746	45	121	(	(	PUNCT
cana-2746	45	122	𝑆	𝑆	PROPN
cana-2746	45	123	,	,	PUNCT
cana-2746	45	124	+	+	NOUN
cana-2746	45	125	)	)	PUNCT
cana-2746	45	126	and	and	CCONJ
cana-2746	45	127	𝛼𝛾𝛽	𝛼𝛾𝛽	PROPN
cana-2746	45	128	∈	∈	PROPN
cana-2746	45	129	𝑆	𝑆	PROPN
cana-2746	45	130	for	for	ADP
cana-2746	45	131	all	all	DET
cana-2746	45	132	𝛼	𝛼	PROPN
cana-2746	45	133	,	,	PUNCT
cana-2746	45	134	𝛽	𝛽	PROPN
cana-2746	45	135	∈	∈	PROPN
cana-2746	45	136	𝑆	𝑆	PROPN
cana-2746	45	137	and	and	CCONJ
cana-2746	45	138	𝛾	𝛾	ADP
cana-2746	45	139	∈	∈	PROPN
cana-2746	45	140	𝛤.	𝛤.	PROPN
cana-2746	45	141	definition	definition	NOUN
cana-2746	45	142	2.8	2.8	NUM
cana-2746	45	143	[	[	X
cana-2746	45	144	3	3	X
cana-2746	45	145	]	]	PUNCT
cana-2746	45	146	a	a	DET
cana-2746	45	147	nonempty	nonempty	NOUN
cana-2746	45	148	subset	subset	VERB
cana-2746	45	149	r	r	NOUN
cana-2746	45	150	of	of	ADP
cana-2746	45	151	a	a	DET
cana-2746	45	152	𝛤	𝛤	PROPN
cana-2746	45	153	semiring	semire	VERB
cana-2746	45	154	s	s	VERB
cana-2746	45	155	is	be	AUX
cana-2746	45	156	called	call	VERB
cana-2746	45	157	an	an	DET
cana-2746	45	158	ideal	ideal	NOUN
cana-2746	45	159	if	if	SCONJ
cana-2746	45	160	𝛼	𝛼	NOUN
cana-2746	45	161	,	,	PUNCT
cana-2746	45	162	𝛽	𝛽	PROPN
cana-2746	45	163	∈	∈	PROPN
cana-2746	45	164	𝑅	𝑅	PROPN
cana-2746	45	165	implies	imply	VERB
cana-2746	45	166	𝛼	𝛼	PRON
cana-2746	45	167	+	+	X
cana-2746	45	168	𝛽	𝛽	NOUN
cana-2746	45	169	∈	∈	PROPN
cana-2746	45	170	𝑅	𝑅	PROPN
cana-2746	45	171	and	and	CCONJ
cana-2746	45	172	𝑎	𝑎	PROPN
cana-2746	45	173	∈	∈	PROPN
cana-2746	45	174	𝑅	𝑅	PROPN
cana-2746	45	175	,	,	PUNCT
cana-2746	45	176	𝛼	𝛼	PROPN
cana-2746	45	177	∈	∈	PROPN
cana-2746	45	178	𝑆	𝑆	PROPN
cana-2746	45	179	and	and	CCONJ
cana-2746	45	180	𝛾	𝛾	ADP
cana-2746	45	181	∈	∈	NOUN
cana-2746	46	1	𝛤	𝛤	PROPN
cana-2746	46	2	implies	imply	VERB
cana-2746	46	3	𝛼𝛾𝑎	𝛼𝛾𝑎	PROPN
cana-2746	46	4	∈	∈	PROPN
cana-2746	46	5	𝑅	𝑅	PROPN
cana-2746	46	6	and	and	CCONJ
cana-2746	46	7	𝑎𝛼𝛾	𝑎𝛼𝛾	NOUN
cana-2746	46	8	∈	∈	NOUN
cana-2746	46	9	𝑅.	𝑅.	NOUN
cana-2746	46	10	definition	definition	NOUN
cana-2746	46	11	2.9	2.9	NUM
cana-2746	46	12	[	[	X
cana-2746	46	13	11	11	NUM
cana-2746	46	14	]	]	PUNCT
cana-2746	46	15	let	let	VERB
cana-2746	46	16	𝑈	𝑈	PROPN
cana-2746	46	17	be	be	AUX
cana-2746	46	18	the	the	DET
cana-2746	46	19	universe	universe	NOUN
cana-2746	46	20	and	and	CCONJ
cana-2746	46	21	𝐸	𝐸	PROPN
cana-2746	46	22	be	be	VERB
cana-2746	46	23	the	the	DET
cana-2746	46	24	set	set	NOUN
cana-2746	46	25	of	of	ADP
cana-2746	46	26	parameters	parameter	NOUN
cana-2746	46	27	.	.	PUNCT
cana-2746	47	1	let	let	VERB
cana-2746	47	2	𝑃(𝑈	𝑃(𝑈	NOUN
cana-2746	47	3	)	)	PUNCT
cana-2746	47	4	denote	denote	VERB
cana-2746	47	5	the	the	DET
cana-2746	47	6	power	power	NOUN
cana-2746	47	7	set	set	NOUN
cana-2746	47	8	of	of	ADP
cana-2746	47	9	𝑈	𝑈	PROPN
cana-2746	47	10	and	and	CCONJ
cana-2746	47	11	𝐴	𝐴	PROPN
cana-2746	48	1	⊂	⊂	PROPN
cana-2746	48	2	𝐸.	𝐸.	VERB
cana-2746	48	3	a	a	DET
cana-2746	48	4	pair	pair	NOUN
cana-2746	48	5	(	(	PUNCT
cana-2746	48	6	𝐹	𝐹	PROPN
cana-2746	48	7	,	,	PUNCT
cana-2746	48	8	𝐴	𝐴	PROPN
cana-2746	48	9	)	)	PUNCT
cana-2746	48	10	is	be	AUX
cana-2746	48	11	called	call	VERB
cana-2746	48	12	a	a	DET
cana-2746	48	13	soft	soft	ADJ
cana-2746	48	14	set	set	NOUN
cana-2746	48	15	over	over	ADP
cana-2746	48	16	𝑈	𝑈	PROPN
cana-2746	48	17	,	,	PUNCT
cana-2746	48	18	where	where	SCONJ
cana-2746	48	19	𝐹	𝐹	PROPN
cana-2746	48	20	is	be	AUX
cana-2746	48	21	a	a	DET
cana-2746	48	22	mapping	mapping	NOUN
cana-2746	48	23	given	give	VERB
cana-2746	48	24	by	by	ADP
cana-2746	48	25	𝐹	𝐹	PROPN
cana-2746	48	26	:	:	PUNCT
cana-2746	48	27	𝐴	𝐴	PROPN
cana-2746	48	28	→	→	SYM
cana-2746	48	29	𝑃(𝑈	𝑃(𝑈	NOUN
cana-2746	48	30	)	)	PUNCT
cana-2746	48	31	.	.	PUNCT
cana-2746	49	1	definition	definition	NOUN
cana-2746	49	2	2.10	2.10	NUM
cana-2746	49	3	[	[	X
cana-2746	49	4	23	23	NUM
cana-2746	49	5	]	]	PUNCT
cana-2746	49	6	a	a	DET
cana-2746	49	7	nonempty	nonempty	ADV
cana-2746	49	8	set	set	VERB
cana-2746	49	9	𝐴	𝐴	PROPN
cana-2746	49	10	of	of	ADP
cana-2746	49	11	𝑆	𝑆	PROPN
cana-2746	49	12	is	be	AUX
cana-2746	49	13	called	call	VERB
cana-2746	49	14	a	a	DET
cana-2746	49	15	𝛤	𝛤	PROPN
cana-2746	49	16	subsemiring	subsemire	VERB
cana-2746	49	17	of	of	ADP
cana-2746	49	18	𝑆	𝑆	PROPN
cana-2746	49	19	if	if	SCONJ
cana-2746	49	20	(	(	PUNCT
cana-2746	49	21	𝐴	𝐴	PROPN
cana-2746	49	22	,	,	PUNCT
cana-2746	49	23	+	+	PUNCT
cana-2746	49	24	)	)	PUNCT
cana-2746	49	25	is	be	AUX
cana-2746	49	26	a	a	DET
cana-2746	49	27	subsemigroup	subsemigroup	NOUN
cana-2746	49	28	of	of	ADP
cana-2746	49	29	(	(	PUNCT
cana-2746	49	30	𝐴	𝐴	PROPN
cana-2746	49	31	,	,	PUNCT
cana-2746	49	32	+	+	ADJ
cana-2746	49	33	)	)	PUNCT
cana-2746	49	34	and	and	CCONJ
cana-2746	49	35	𝐴𝛤𝐴	𝐴𝛤𝐴	PROPN
cana-2746	49	36	⊆	⊆	NUM
cana-2746	49	37	𝐴.	𝐴.	NOUN
cana-2746	49	38	definition	definition	NOUN
cana-2746	49	39	2.11	2.11	NUM
cana-2746	49	40	[	[	SYM
cana-2746	49	41	30	30	NUM
cana-2746	49	42	]	]	X
cana-2746	49	43	a	a	PRON
cana-2746	49	44	is	be	AUX
cana-2746	49	45	called	call	VERB
cana-2746	49	46	a	a	DET
cana-2746	49	47	quasi	quasi	NOUN
cana-2746	49	48	-	-	NOUN
cana-2746	49	49	ideal	ideal	NOUN
cana-2746	49	50	of	of	ADP
cana-2746	49	51	𝑆	𝑆	PROPN
cana-2746	49	52	if	if	SCONJ
cana-2746	49	53	𝐴	𝐴	PROPN
cana-2746	49	54	is	be	AUX
cana-2746	49	55	a	a	DET
cana-2746	49	56	𝛤	𝛤	PROPN
cana-2746	49	57	subsemiring	subsemire	VERB
cana-2746	49	58	of	of	ADP
cana-2746	49	59	𝑆	𝑆	PROPN
cana-2746	49	60	and	and	CCONJ
cana-2746	49	61	𝐴𝛤𝑆	𝐴𝛤𝑆	PROPN
cana-2746	49	62	∩	∩	NOUN
cana-2746	49	63	𝑆𝛤𝐴	𝑆𝛤𝐴	PROPN
cana-2746	49	64	⊆	⊆	NUM
cana-2746	49	65	𝐴.	𝐴.	PROPN
cana-2746	49	66	definition	definition	NOUN
cana-2746	49	67	2.12	2.12	NUM
cana-2746	49	68	[	[	X
cana-2746	49	69	30	30	NUM
cana-2746	49	70	]	]	X
cana-2746	49	71	a	a	PRON
cana-2746	49	72	is	be	AUX
cana-2746	49	73	called	call	VERB
cana-2746	49	74	a	a	DET
cana-2746	49	75	bi	bi	NOUN
cana-2746	49	76	-	-	NOUN
cana-2746	49	77	ideal	ideal	NOUN
cana-2746	49	78	of	of	ADP
cana-2746	49	79	𝑆	𝑆	PROPN
cana-2746	49	80	if	if	SCONJ
cana-2746	49	81	𝐴	𝐴	PROPN
cana-2746	49	82	is	be	AUX
cana-2746	49	83	a	a	DET
cana-2746	49	84	𝛤	𝛤	PROPN
cana-2746	49	85	subsemiring	subsemire	VERB
cana-2746	49	86	of	of	ADP
cana-2746	49	87	𝑆	𝑆	PROPN
cana-2746	49	88	and	and	CCONJ
cana-2746	49	89	𝐴𝛤𝑆𝛤𝐴	𝐴𝛤𝑆𝛤𝐴	PROPN
cana-2746	49	90	⊆	⊆	NUM
cana-2746	49	91	𝐴.	𝐴.	PROPN
cana-2746	49	92	definition	definition	NOUN
cana-2746	49	93	2.13	2.13	NUM
cana-2746	50	1	[	[	SYM
cana-2746	50	2	30	30	NUM
cana-2746	50	3	]	]	X
cana-2746	50	4	a	a	PRON
cana-2746	50	5	is	be	AUX
cana-2746	50	6	called	call	VERB
cana-2746	50	7	an	an	DET
cana-2746	50	8	interior	interior	NOUN
cana-2746	50	9	-	-	PUNCT
cana-2746	50	10	ideal	ideal	NOUN
cana-2746	50	11	of	of	ADP
cana-2746	50	12	𝑆	𝑆	PROPN
cana-2746	50	13	if	if	SCONJ
cana-2746	50	14	𝐵	𝐵	PROPN
cana-2746	50	15	is	be	AUX
cana-2746	50	16	a	a	DET
cana-2746	50	17	𝛤	𝛤	PROPN
cana-2746	50	18	subsemiring	subsemire	VERB
cana-2746	50	19	of	of	ADP
cana-2746	50	20	𝑆	𝑆	PROPN
cana-2746	50	21	and	and	CCONJ
cana-2746	50	22	𝑆𝛤𝐴𝛤𝑆	𝑆𝛤𝐴𝛤𝑆	PROPN
cana-2746	50	23	⊆	⊆	NUM
cana-2746	50	24	𝐴.	𝐴.	PROPN
cana-2746	50	25	definition	definition	NOUN
cana-2746	50	26	2.14	2.14	NUM
cana-2746	51	1	[	[	X
cana-2746	51	2	30	30	NUM
cana-2746	51	3	]	]	X
cana-2746	51	4	a	a	PRON
cana-2746	51	5	is	be	AUX
cana-2746	51	6	called	call	VERB
cana-2746	51	7	a	a	DET
cana-2746	51	8	left(right	left(right	PROPN
cana-2746	51	9	)	)	PUNCT
cana-2746	51	10	ideal	ideal	NOUN
cana-2746	51	11	of	of	ADP
cana-2746	51	12	𝑆	𝑆	PROPN
cana-2746	51	13	if	if	SCONJ
cana-2746	51	14	𝐴	𝐴	PROPN
cana-2746	51	15	is	be	AUX
cana-2746	51	16	a	a	DET
cana-2746	51	17	𝛤	𝛤	PROPN
cana-2746	51	18	subsemiring	subsemire	VERB
cana-2746	51	19	of	of	ADP
cana-2746	51	20	𝑆	𝑆	PROPN
cana-2746	51	21	and	and	CCONJ
cana-2746	51	22	𝑆𝛤𝐴	𝑆𝛤𝐴	PROPN
cana-2746	51	23	⊆	⊆	NUM
cana-2746	51	24	𝐴(𝐴𝛤𝑆	𝐴(𝐴𝛤𝑆	PROPN
cana-2746	51	25	⊆	⊆	NUM
cana-2746	51	26	𝐴	𝐴	PROPN
cana-2746	51	27	)	)	PUNCT
cana-2746	51	28	.	.	PUNCT
cana-2746	52	1	definition	definition	NOUN
cana-2746	52	2	2.15	2.15	NUM
cana-2746	52	3	[	[	X
cana-2746	52	4	30	30	NUM
cana-2746	52	5	]	]	X
cana-2746	52	6	a	a	PRON
cana-2746	52	7	is	be	AUX
cana-2746	52	8	called	call	VERB
cana-2746	52	9	an	an	DET
cana-2746	52	10	ideal	ideal	NOUN
cana-2746	52	11	of	of	ADP
cana-2746	52	12	𝑆	𝑆	PROPN
cana-2746	52	13	if	if	SCONJ
cana-2746	52	14	𝐴	𝐴	PROPN
cana-2746	52	15	is	be	AUX
cana-2746	52	16	a	a	DET
cana-2746	52	17	𝛤	𝛤	PROPN
cana-2746	52	18	subsemiring	subsemire	VERB
cana-2746	52	19	of	of	ADP
cana-2746	52	20	𝑆	𝑆	PROPN
cana-2746	52	21	and	and	CCONJ
cana-2746	52	22	𝑆𝛤𝐴	𝑆𝛤𝐴	PROPN
cana-2746	52	23	⊆	⊆	PROPN
cana-2746	52	24	𝐴	𝐴	PROPN
cana-2746	52	25	,	,	PUNCT
cana-2746	52	26	𝐴𝛤𝑆	𝐴𝛤𝑆	PROPN
cana-2746	52	27	⊆	⊆	NUM
cana-2746	52	28	𝐴.	𝐴.	PROPN
cana-2746	52	29	communications	communication	NOUN
cana-2746	52	30	on	on	ADP
cana-2746	52	31	applied	apply	VERB
cana-2746	52	32	nonlinear	nonlinear	ADJ
cana-2746	52	33	analysis	analysis	NOUN
cana-2746	52	34	issn	issn	NOUN
cana-2746	52	35	:	:	PUNCT
cana-2746	52	36	1074	1074	NUM
cana-2746	52	37	-	-	PUNCT
cana-2746	52	38	133x	133x	NUM
cana-2746	52	39	vol	vol	NOUN
cana-2746	52	40	32	32	NUM
cana-2746	52	41	no	no	NOUN
cana-2746	52	42	.	.	PUNCT
cana-2746	53	1	4s	4s	NUM
cana-2746	53	2	(	(	PUNCT
cana-2746	53	3	2025	2025	NUM
cana-2746	53	4	)	)	PUNCT
cana-2746	53	5	152	152	NUM
cana-2746	53	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2746	53	7	definition	definition	NOUN
cana-2746	53	8	2.16	2.16	NUM
cana-2746	54	1	[	[	X
cana-2746	54	2	30	30	NUM
cana-2746	54	3	]	]	X
cana-2746	54	4	a	a	PRON
cana-2746	54	5	is	be	AUX
cana-2746	54	6	called	call	VERB
cana-2746	54	7	a	a	DET
cana-2746	54	8	left(right	left(right	PROPN
cana-2746	54	9	)	)	PUNCT
cana-2746	54	10	bi	bi	ADJ
cana-2746	54	11	-	-	ADJ
cana-2746	54	12	quasi	quasi	NOUN
cana-2746	54	13	-	-	NOUN
cana-2746	54	14	ideal	ideal	NOUN
cana-2746	54	15	of	of	ADP
cana-2746	54	16	𝑆	𝑆	PROPN
cana-2746	54	17	if	if	SCONJ
cana-2746	54	18	𝐴	𝐴	PROPN
cana-2746	54	19	is	be	AUX
cana-2746	54	20	a	a	DET
cana-2746	54	21	𝛤-subsemiring	𝛤-subsemiring	PROPN
cana-2746	54	22	of	of	ADP
cana-2746	54	23	𝑆	𝑆	PROPN
cana-2746	54	24	and	and	CCONJ
cana-2746	54	25	𝐴𝛤𝑆	𝐴𝛤𝑆	PROPN
cana-2746	54	26	∩	∩	NOUN
cana-2746	54	27	𝑆𝛤𝑆𝛤𝐴	𝑆𝛤𝑆𝛤𝐴	PROPN
cana-2746	54	28	⊆	⊆	NUM
cana-2746	54	29	𝐴(𝐴𝛤𝑆	𝐴(𝐴𝛤𝑆	NOUN
cana-2746	54	30	∩	∩	NOUN
cana-2746	54	31	𝑆𝛤𝐴𝛤𝑆	𝑆𝛤𝐴𝛤𝑆	PROPN
cana-2746	54	32	⊆	⊆	NUM
cana-2746	54	33	𝐴	𝐴	PROPN
cana-2746	54	34	)	)	PUNCT
cana-2746	54	35	.	.	PUNCT
cana-2746	55	1	definition	definition	NOUN
cana-2746	55	2	2.17	2.17	NUM
cana-2746	56	1	[	[	X
cana-2746	56	2	9	9	NUM
cana-2746	56	3	]	]	PUNCT
cana-2746	56	4	let	let	VERB
cana-2746	56	5	𝑈	𝑈	PROPN
cana-2746	56	6	be	be	AUX
cana-2746	56	7	the	the	DET
cana-2746	56	8	initial	initial	ADJ
cana-2746	56	9	universe	universe	NOUN
cana-2746	56	10	.	.	PUNCT
cana-2746	57	1	𝐸	𝐸	PROPN
cana-2746	57	2	be	be	VERB
cana-2746	57	3	the	the	DET
cana-2746	57	4	set	set	NOUN
cana-2746	57	5	of	of	ADP
cana-2746	57	6	parameters	parameter	NOUN
cana-2746	57	7	and	and	CCONJ
cana-2746	57	8	𝐹𝑆(𝑈	𝐹𝑆(𝑈	NOUN
cana-2746	57	9	)	)	PUNCT
cana-2746	57	10	denote	denote	VERB
cana-2746	57	11	the	the	DET
cana-2746	57	12	fuzzy	fuzzy	ADJ
cana-2746	57	13	power	power	NOUN
cana-2746	57	14	set	set	NOUN
cana-2746	57	15	of	of	ADP
cana-2746	57	16	𝑈	𝑈	PROPN
cana-2746	57	17	and	and	CCONJ
cana-2746	57	18	𝐴	𝐴	PROPN
cana-2746	58	1	⊂	⊂	PROPN
cana-2746	58	2	𝐸.	𝐸.	VERB
cana-2746	58	3	a	a	DET
cana-2746	58	4	pair	pair	NOUN
cana-2746	58	5	(	(	PUNCT
cana-2746	58	6	𝐹	𝐹	PROPN
cana-2746	58	7	,	,	PUNCT
cana-2746	58	8	𝐴	𝐴	PROPN
cana-2746	58	9	)	)	PUNCT
cana-2746	58	10	is	be	AUX
cana-2746	58	11	called	call	VERB
cana-2746	58	12	a	a	DET
cana-2746	58	13	fuzzy	fuzzy	ADJ
cana-2746	58	14	soft	soft	ADJ
cana-2746	58	15	set	set	NOUN
cana-2746	58	16	over	over	ADP
cana-2746	58	17	𝑈	𝑈	PROPN
cana-2746	58	18	,	,	PUNCT
cana-2746	58	19	where	where	SCONJ
cana-2746	58	20	𝐹	𝐹	PROPN
cana-2746	58	21	is	be	AUX
cana-2746	58	22	a	a	DET
cana-2746	58	23	mapping	mapping	NOUN
cana-2746	58	24	given	give	VERB
cana-2746	58	25	by	by	ADP
cana-2746	58	26	𝐹	𝐹	PROPN
cana-2746	58	27	:	:	PUNCT
cana-2746	58	28	𝐴	𝐴	PROPN
cana-2746	58	29	→	→	SYM
cana-2746	58	30	𝐹𝑆(𝑈	𝐹𝑆(𝑈	PROPN
cana-2746	58	31	)	)	PUNCT
cana-2746	58	32	.	.	PUNCT
cana-2746	59	1	a	a	DET
cana-2746	59	2	fuzzy	fuzzy	ADJ
cana-2746	59	3	soft	soft	ADJ
cana-2746	59	4	set	set	NOUN
cana-2746	59	5	is	be	AUX
cana-2746	59	6	a	a	DET
cana-2746	59	7	parameterized	parameterized	ADJ
cana-2746	59	8	family	family	NOUN
cana-2746	59	9	of	of	ADP
cana-2746	59	10	fuzzy	fuzzy	ADJ
cana-2746	59	11	subsets	subset	NOUN
cana-2746	59	12	of	of	ADP
cana-2746	59	13	𝑈.	𝑈.	ADJ
cana-2746	59	14	definition	definition	NOUN
cana-2746	59	15	2.18	2.18	NUM
cana-2746	59	16	[	[	X
cana-2746	59	17	16	16	NUM
cana-2746	59	18	]	]	PUNCT
cana-2746	59	19	let	let	VERB
cana-2746	59	20	𝑋	𝑋	NOUN
cana-2746	59	21	be	be	AUX
cana-2746	59	22	a	a	DET
cana-2746	59	23	non	non	X
cana-2746	59	24	empty	empty	ADJ
cana-2746	59	25	set	set	NOUN
cana-2746	59	26	.	.	PUNCT
cana-2746	60	1	a	a	DET
cana-2746	60	2	pythagorean	pythagorean	PROPN
cana-2746	60	3	fuzzy	fuzzy	ADJ
cana-2746	60	4	set	set	VERB
cana-2746	60	5	𝔄	𝔄	NOUN
cana-2746	60	6	in	in	ADP
cana-2746	60	7	𝑋	𝑋	PROPN
cana-2746	60	8	is	be	AUX
cana-2746	60	9	given	give	VERB
cana-2746	60	10	by	by	ADP
cana-2746	60	11	𝔄	𝔄	PROPN
cana-2746	60	12	=	=	SYM
cana-2746	60	13	{	{	PUNCT
cana-2746	60	14	𝛼	𝛼	NOUN
cana-2746	60	15	,	,	PUNCT
cana-2746	60	16	𝔄𝑥(𝛼	𝔄𝑥(𝛼	NOUN
cana-2746	60	17	)	)	PUNCT
cana-2746	60	18	,	,	PUNCT
cana-2746	60	19	𝔄𝑦(𝛼)/𝛼	𝔄𝑦(𝛼)/𝛼	PROPN
cana-2746	60	20	∈	∈	PROPN
cana-2746	60	21	𝑋	𝑋	PROPN
cana-2746	60	22	}	}	PUNCT
cana-2746	60	23	where	where	SCONJ
cana-2746	60	24	𝔄𝑥	𝔄𝑥	PROPN
cana-2746	60	25	:	:	PUNCT
cana-2746	60	26	𝑋	𝑋	NOUN
cana-2746	60	27	→	→	SYM
cana-2746	60	28	[	[	X
cana-2746	60	29	0,1	0,1	NUM
cana-2746	60	30	]	]	PUNCT
cana-2746	60	31	and	and	CCONJ
cana-2746	60	32	𝔄𝑦	𝔄𝑦	PROPN
cana-2746	60	33	:	:	PUNCT
cana-2746	60	34	𝑋	𝑋	NOUN
cana-2746	60	35	→	→	SYM
cana-2746	61	1	[	[	X
cana-2746	61	2	0,1	0,1	NUM
cana-2746	61	3	]	]	PUNCT
cana-2746	61	4	represent	represent	VERB
cana-2746	61	5	the	the	DET
cana-2746	61	6	degree	degree	NOUN
cana-2746	61	7	of	of	ADP
cana-2746	61	8	membership	membership	NOUN
cana-2746	61	9	and	and	CCONJ
cana-2746	61	10	degree	degree	NOUN
cana-2746	61	11	of	of	ADP
cana-2746	61	12	non	non	ADJ
cana-2746	61	13	membership	membership	NOUN
cana-2746	61	14	of	of	ADP
cana-2746	61	15	𝔄	𝔄	PROPN
cana-2746	61	16	respectively	respectively	ADV
cana-2746	61	17	.	.	PUNCT
cana-2746	62	1	also	also	ADV
cana-2746	62	2	,	,	PUNCT
cana-2746	62	3	𝔄𝑥	𝔄𝑥	PROPN
cana-2746	62	4	and	and	CCONJ
cana-2746	62	5	𝔄𝑦	𝔄𝑦	PROPN
cana-2746	62	6	satisfies	satisfy	VERB
cana-2746	62	7	the	the	DET
cana-2746	62	8	condition	condition	NOUN
cana-2746	62	9	(	(	PUNCT
cana-2746	62	10	𝔄𝑥)2	𝔄𝑥)2	PROPN
cana-2746	63	1	+	+	CCONJ
cana-2746	63	2	(	(	PUNCT
cana-2746	63	3	𝔄𝑦)2	𝔄𝑦)2	PROPN
cana-2746	63	4	≤	≤	ADJ
cana-2746	63	5	1	1	NUM
cana-2746	63	6	for	for	ADP
cana-2746	63	7	all	all	DET
cana-2746	63	8	𝛼	𝛼	PRON
cana-2746	63	9	∈	∈	NOUN
cana-2746	63	10	𝑋.	𝑋.	PROPN
cana-2746	63	11	definition	definition	NOUN
cana-2746	63	12	2.19	2.19	NUM
cana-2746	63	13	[	[	X
cana-2746	63	14	13	13	NUM
cana-2746	63	15	]	]	PUNCT
cana-2746	63	16	let	let	VERB
cana-2746	63	17	𝑈	𝑈	PROPN
cana-2746	63	18	be	be	AUX
cana-2746	63	19	the	the	DET
cana-2746	63	20	initial	initial	ADJ
cana-2746	63	21	universe	universe	NOUN
cana-2746	63	22	.	.	PUNCT
cana-2746	64	1	𝐸	𝐸	PROPN
cana-2746	64	2	be	be	VERB
cana-2746	64	3	the	the	DET
cana-2746	64	4	set	set	NOUN
cana-2746	64	5	of	of	ADP
cana-2746	64	6	parameters	parameter	NOUN
cana-2746	64	7	and	and	CCONJ
cana-2746	64	8	𝑃𝐹𝑆(𝑈	𝑃𝐹𝑆(𝑈	NOUN
cana-2746	64	9	)	)	PUNCT
cana-2746	64	10	denote	denote	VERB
cana-2746	64	11	the	the	DET
cana-2746	64	12	pythagorean	pythagorean	PROPN
cana-2746	64	13	fuzzy	fuzzy	ADJ
cana-2746	64	14	power	power	NOUN
cana-2746	64	15	set	set	NOUN
cana-2746	64	16	of	of	ADP
cana-2746	64	17	𝑈	𝑈	PROPN
cana-2746	64	18	and	and	CCONJ
cana-2746	64	19	𝐴	𝐴	PROPN
cana-2746	64	20	⊂	⊂	PROPN
cana-2746	64	21	𝐸.	𝐸.	VERB
cana-2746	64	22	a	a	DET
cana-2746	64	23	pair	pair	NOUN
cana-2746	64	24	(	(	PUNCT
cana-2746	64	25	𝐹	𝐹	PROPN
cana-2746	64	26	,	,	PUNCT
cana-2746	64	27	𝐴	𝐴	PROPN
cana-2746	64	28	)	)	PUNCT
cana-2746	64	29	is	be	AUX
cana-2746	64	30	called	call	VERB
cana-2746	64	31	a	a	DET
cana-2746	64	32	pythagorean	pythagorean	ADJ
cana-2746	64	33	fuzzy	fuzzy	ADJ
cana-2746	64	34	soft	soft	ADJ
cana-2746	64	35	set	set	NOUN
cana-2746	64	36	over	over	ADP
cana-2746	64	37	𝑈	𝑈	PROPN
cana-2746	64	38	,	,	PUNCT
cana-2746	64	39	where	where	SCONJ
cana-2746	64	40	𝐹	𝐹	PROPN
cana-2746	64	41	is	be	AUX
cana-2746	64	42	a	a	DET
cana-2746	64	43	mapping	mapping	NOUN
cana-2746	64	44	given	give	VERB
cana-2746	64	45	by	by	ADP
cana-2746	64	46	𝐹	𝐹	PROPN
cana-2746	64	47	:	:	PUNCT
cana-2746	64	48	𝐴	𝐴	PROPN
cana-2746	64	49	→	→	SYM
cana-2746	64	50	𝑃𝐹𝑆(𝑈	𝑃𝐹𝑆(𝑈	NOUN
cana-2746	64	51	)	)	PUNCT
cana-2746	64	52	.	.	PUNCT
cana-2746	65	1	a	a	DET
cana-2746	65	2	pythagorean	pythagorean	PROPN
cana-2746	65	3	fuzzy	fuzzy	ADJ
cana-2746	65	4	soft	soft	ADJ
cana-2746	65	5	set	set	NOUN
cana-2746	65	6	is	be	AUX
cana-2746	65	7	a	a	DET
cana-2746	65	8	parameterized	parameterized	ADJ
cana-2746	65	9	family	family	NOUN
cana-2746	65	10	of	of	ADP
cana-2746	65	11	fuzzy	fuzzy	ADJ
cana-2746	65	12	subsets	subset	NOUN
cana-2746	65	13	of	of	ADP
cana-2746	65	14	𝑈.	𝑈.	ADJ
cana-2746	65	15	definition	definition	NOUN
cana-2746	65	16	2.20	2.20	NUM
cana-2746	65	17	[	[	X
cana-2746	65	18	13	13	NUM
cana-2746	65	19	]	]	PUNCT
cana-2746	65	20	let	let	VERB
cana-2746	65	21	(	(	PUNCT
cana-2746	65	22	𝐹	𝐹	PROPN
cana-2746	65	23	,	,	PUNCT
cana-2746	65	24	𝐴	𝐴	PROPN
cana-2746	65	25	)	)	PUNCT
cana-2746	65	26	and	and	CCONJ
cana-2746	65	27	(	(	PUNCT
cana-2746	65	28	𝐺	𝐺	PROPN
cana-2746	65	29	,	,	PUNCT
cana-2746	65	30	𝐵	𝐵	PROPN
cana-2746	65	31	)	)	PUNCT
cana-2746	65	32	be	be	VERB
cana-2746	65	33	two	two	NUM
cana-2746	65	34	pythagorean	pythagorean	ADJ
cana-2746	65	35	fuzzy	fuzzy	ADJ
cana-2746	65	36	soft	soft	ADJ
cana-2746	65	37	sets	set	NOUN
cana-2746	65	38	over	over	ADP
cana-2746	65	39	𝑈.	𝑈.	PROPN
cana-2746	65	40	then	then	ADV
cana-2746	65	41	the	the	DET
cana-2746	65	42	union	union	NOUN
cana-2746	65	43	of	of	ADP
cana-2746	65	44	(	(	PUNCT
cana-2746	65	45	𝐹	𝐹	PROPN
cana-2746	65	46	,	,	PUNCT
cana-2746	65	47	𝐴	𝐴	PROPN
cana-2746	65	48	)	)	PUNCT
cana-2746	65	49	is	be	AUX
cana-2746	65	50	called	call	VERB
cana-2746	65	51	a	a	DET
cana-2746	65	52	pythagorean	pythagorean	ADJ
cana-2746	65	53	fuzzy	fuzzy	ADJ
cana-2746	65	54	soft	soft	ADJ
cana-2746	65	55	subset	subset	NOUN
cana-2746	65	56	of	of	ADP
cana-2746	65	57	(	(	PUNCT
cana-2746	65	58	𝐺	𝐺	PROPN
cana-2746	65	59	,	,	PUNCT
cana-2746	65	60	𝐵	𝐵	PROPN
cana-2746	65	61	)	)	PUNCT
cana-2746	65	62	if	if	SCONJ
cana-2746	65	63	1	1	NUM
cana-2746	65	64	.	.	X
cana-2746	66	1	𝐴	𝐴	PROPN
cana-2746	66	2	⊂	⊂	PROPN
cana-2746	66	3	𝐵	𝐵	PROPN
cana-2746	66	4	2	2	NUM
cana-2746	66	5	.	.	PUNCT
cana-2746	66	6	𝐹(𝛼	𝐹(𝛼	NUM
cana-2746	66	7	)	)	PUNCT
cana-2746	66	8	is	be	AUX
cana-2746	66	9	a	a	DET
cana-2746	66	10	pythagorean	pythagorean	ADJ
cana-2746	66	11	fuzzy	fuzzy	ADJ
cana-2746	66	12	subset	subset	NOUN
cana-2746	66	13	of	of	ADP
cana-2746	66	14	𝐺(𝛼	𝐺(𝛼	NOUN
cana-2746	66	15	)	)	PUNCT
cana-2746	66	16	,	,	PUNCT
cana-2746	66	17	for	for	ADP
cana-2746	66	18	all	all	DET
cana-2746	66	19	𝛼	𝛼	PRON
cana-2746	66	20	∈	∈	NOUN
cana-2746	66	21	𝐴.	𝐴.	NOUN
cana-2746	66	22	definition	definition	NOUN
cana-2746	66	23	2.21	2.21	NUM
cana-2746	66	24	[	[	X
cana-2746	66	25	13	13	NUM
cana-2746	66	26	]	]	PUNCT
cana-2746	66	27	let	let	VERB
cana-2746	66	28	(	(	PUNCT
cana-2746	66	29	𝐹	𝐹	PROPN
cana-2746	66	30	,	,	PUNCT
cana-2746	66	31	𝐴	𝐴	PROPN
cana-2746	66	32	)	)	PUNCT
cana-2746	66	33	and	and	CCONJ
cana-2746	66	34	(	(	PUNCT
cana-2746	66	35	𝐺	𝐺	PROPN
cana-2746	66	36	,	,	PUNCT
cana-2746	66	37	𝐵	𝐵	PROPN
cana-2746	66	38	)	)	PUNCT
cana-2746	66	39	be	be	VERB
cana-2746	66	40	two	two	NUM
cana-2746	66	41	pythagorean	pythagorean	ADJ
cana-2746	66	42	fuzzy	fuzzy	ADJ
cana-2746	66	43	soft	soft	ADJ
cana-2746	66	44	sets	set	NOUN
cana-2746	66	45	over	over	ADP
cana-2746	66	46	𝑈.	𝑈.	PROPN
cana-2746	66	47	(	(	PUNCT
cana-2746	66	48	𝐹	𝐹	PROPN
cana-2746	66	49	,	,	PUNCT
cana-2746	66	50	𝐴)𝐴𝑁𝐷(𝐺	𝐴)𝐴𝑁𝐷(𝐺	PROPN
cana-2746	66	51	,	,	PUNCT
cana-2746	66	52	𝐵	𝐵	NOUN
cana-2746	66	53	)	)	PUNCT
cana-2746	66	54	denoted	denote	VERB
cana-2746	66	55	by	by	ADP
cana-2746	66	56	(	(	PUNCT
cana-2746	66	57	𝐹	𝐹	PROPN
cana-2746	66	58	,	,	PUNCT
cana-2746	66	59	𝐴	𝐴	PROPN
cana-2746	66	60	)	)	PUNCT
cana-2746	66	61	∧	∧	PROPN
cana-2746	66	62	(	(	PUNCT
cana-2746	66	63	𝐺	𝐺	PROPN
cana-2746	66	64	,	,	PUNCT
cana-2746	66	65	𝐵	𝐵	PROPN
cana-2746	66	66	)	)	PUNCT
cana-2746	66	67	,	,	PUNCT
cana-2746	66	68	is	be	AUX
cana-2746	66	69	defined	define	VERB
cana-2746	66	70	by	by	ADP
cana-2746	66	71	(	(	PUNCT
cana-2746	66	72	𝐹	𝐹	PROPN
cana-2746	66	73	,	,	PUNCT
cana-2746	66	74	𝐴	𝐴	PROPN
cana-2746	66	75	)	)	PUNCT
cana-2746	66	76	∧	∧	PROPN
cana-2746	66	77	(	(	PUNCT
cana-2746	66	78	𝐺	𝐺	PROPN
cana-2746	66	79	,	,	PUNCT
cana-2746	66	80	𝐵	𝐵	PROPN
cana-2746	66	81	)	)	PUNCT
cana-2746	66	82	=	=	SYM
cana-2746	66	83	(	(	PUNCT
cana-2746	66	84	𝐻	𝐻	PROPN
cana-2746	66	85	,	,	PUNCT
cana-2746	66	86	𝐴	𝐴	PROPN
cana-2746	66	87	×	×	PROPN
cana-2746	66	88	𝐵	𝐵	PROPN
cana-2746	66	89	)	)	PUNCT
cana-2746	66	90	,	,	PUNCT
cana-2746	66	91	where	where	SCONJ
cana-2746	66	92	𝐻(𝛼	𝐻(𝛼	X
cana-2746	66	93	,	,	PUNCT
cana-2746	66	94	𝛽	𝛽	NOUN
cana-2746	66	95	)	)	PUNCT
cana-2746	66	96	=	=	SYM
cana-2746	66	97	𝐹(𝛼	𝐹(𝛼	NUM
cana-2746	66	98	)	)	PUNCT
cana-2746	66	99	∩	∩	NOUN
cana-2746	66	100	𝐺(𝛽	𝐺(𝛽	NUM
cana-2746	66	101	)	)	PUNCT
cana-2746	66	102	,	,	PUNCT
cana-2746	66	103	for	for	ADP
cana-2746	66	104	all	all	DET
cana-2746	66	105	(	(	PUNCT
cana-2746	66	106	𝛼	𝛼	PROPN
cana-2746	66	107	,	,	PUNCT
cana-2746	66	108	𝛽	𝛽	NOUN
cana-2746	66	109	)	)	PUNCT
cana-2746	66	110	∈	∈	PROPN
cana-2746	66	111	𝐴	𝐴	PROPN
cana-2746	66	112	×	×	NOUN
cana-2746	66	113	𝐵.	𝐵.	PROPN
cana-2746	66	114	definition	definition	NOUN
cana-2746	66	115	2.22	2.22	NUM
cana-2746	66	116	[	[	SYM
cana-2746	66	117	13	13	NUM
cana-2746	66	118	]	]	PUNCT
cana-2746	66	119	let	let	VERB
cana-2746	66	120	(	(	PUNCT
cana-2746	66	121	𝐹	𝐹	PROPN
cana-2746	66	122	,	,	PUNCT
cana-2746	66	123	𝐴	𝐴	PROPN
cana-2746	66	124	)	)	PUNCT
cana-2746	66	125	and	and	CCONJ
cana-2746	66	126	(	(	PUNCT
cana-2746	66	127	𝐺	𝐺	PROPN
cana-2746	66	128	,	,	PUNCT
cana-2746	66	129	𝐵	𝐵	PROPN
cana-2746	66	130	)	)	PUNCT
cana-2746	66	131	be	be	VERB
cana-2746	66	132	two	two	NUM
cana-2746	66	133	pythagorean	pythagorean	ADJ
cana-2746	66	134	fuzzy	fuzzy	ADJ
cana-2746	66	135	soft	soft	ADJ
cana-2746	66	136	sets	set	NOUN
cana-2746	66	137	over	over	ADP
cana-2746	66	138	𝑈.	𝑈.	PROPN
cana-2746	66	139	(	(	PUNCT
cana-2746	66	140	𝐹	𝐹	PROPN
cana-2746	66	141	,	,	PUNCT
cana-2746	66	142	𝐴)𝑂𝑅(𝐺	𝐴)𝑂𝑅(𝐺	PROPN
cana-2746	66	143	,	,	PUNCT
cana-2746	66	144	𝐵	𝐵	NOUN
cana-2746	66	145	)	)	PUNCT
cana-2746	66	146	denoted	denote	VERB
cana-2746	66	147	by	by	ADP
cana-2746	66	148	(	(	PUNCT
cana-2746	66	149	𝐹	𝐹	PROPN
cana-2746	66	150	,	,	PUNCT
cana-2746	66	151	𝐴	𝐴	PROPN
cana-2746	66	152	)	)	PUNCT
cana-2746	66	153	∨	∨	PROPN
cana-2746	66	154	(	(	PUNCT
cana-2746	66	155	𝐺	𝐺	NOUN
cana-2746	66	156	,	,	PUNCT
cana-2746	66	157	𝐵	𝐵	PROPN
cana-2746	66	158	)	)	PUNCT
cana-2746	66	159	,	,	PUNCT
cana-2746	66	160	is	be	AUX
cana-2746	66	161	defined	define	VERB
cana-2746	66	162	by	by	ADP
cana-2746	66	163	(	(	PUNCT
cana-2746	66	164	𝐹	𝐹	PROPN
cana-2746	66	165	,	,	PUNCT
cana-2746	66	166	𝐴	𝐴	PROPN
cana-2746	66	167	)	)	PUNCT
cana-2746	66	168	∨	∨	PROPN
cana-2746	66	169	(	(	PUNCT
cana-2746	66	170	𝐺	𝐺	NOUN
cana-2746	66	171	,	,	PUNCT
cana-2746	66	172	𝐵	𝐵	NOUN
cana-2746	66	173	)	)	PUNCT
cana-2746	67	1	=	=	SYM
cana-2746	67	2	(	(	PUNCT
cana-2746	67	3	𝐻	𝐻	PROPN
cana-2746	67	4	,	,	PUNCT
cana-2746	67	5	𝐴	𝐴	PROPN
cana-2746	67	6	×	×	PROPN
cana-2746	67	7	𝐵	𝐵	PROPN
cana-2746	67	8	)	)	PUNCT
cana-2746	67	9	,	,	PUNCT
cana-2746	67	10	where	where	SCONJ
cana-2746	67	11	𝐻(𝛼	𝐻(𝛼	X
cana-2746	67	12	,	,	PUNCT
cana-2746	67	13	𝛽	𝛽	NOUN
cana-2746	67	14	)	)	PUNCT
cana-2746	67	15	=	=	SYM
cana-2746	68	1	𝐹(𝛼	𝐹(𝛼	NUM
cana-2746	68	2	)	)	PUNCT
cana-2746	68	3	∪	∪	NOUN
cana-2746	68	4	𝐺(𝛽	𝐺(𝛽	NOUN
cana-2746	68	5	)	)	PUNCT
cana-2746	68	6	,	,	PUNCT
cana-2746	68	7	for	for	ADP
cana-2746	68	8	all	all	DET
cana-2746	68	9	(	(	PUNCT
cana-2746	68	10	𝛼	𝛼	PROPN
cana-2746	68	11	,	,	PUNCT
cana-2746	68	12	𝛽	𝛽	NOUN
cana-2746	68	13	)	)	PUNCT
cana-2746	68	14	∈	∈	PROPN
cana-2746	68	15	𝐴	𝐴	PROPN
cana-2746	68	16	×	×	NOUN
cana-2746	68	17	𝐵.	𝐵.	PROPN
cana-2746	68	18	definition	definition	NOUN
cana-2746	68	19	2.23	2.23	NUM
cana-2746	68	20	[	[	X
cana-2746	68	21	13	13	NUM
cana-2746	68	22	]	]	PUNCT
cana-2746	68	23	the	the	DET
cana-2746	68	24	intersection	intersection	NOUN
cana-2746	68	25	of	of	ADP
cana-2746	68	26	two	two	NUM
cana-2746	68	27	pythagorean	pythagorean	ADJ
cana-2746	68	28	fuzzy	fuzzy	ADJ
cana-2746	68	29	soft	soft	ADJ
cana-2746	68	30	sets	set	NOUN
cana-2746	68	31	(	(	PUNCT
cana-2746	68	32	𝐹	𝐹	PROPN
cana-2746	68	33	,	,	PUNCT
cana-2746	68	34	𝐴	𝐴	PROPN
cana-2746	68	35	)	)	PUNCT
cana-2746	68	36	and	and	CCONJ
cana-2746	68	37	(	(	PUNCT
cana-2746	68	38	𝐺	𝐺	PROPN
cana-2746	68	39	,	,	PUNCT
cana-2746	68	40	𝐵	𝐵	PROPN
cana-2746	68	41	)	)	PUNCT
cana-2746	68	42	over	over	ADP
cana-2746	68	43	a	a	DET
cana-2746	68	44	universe	universe	NOUN
cana-2746	68	45	𝑈	𝑈	NOUN
cana-2746	68	46	is	be	AUX
cana-2746	68	47	a	a	DET
cana-2746	68	48	pythagorean	pythagorean	ADJ
cana-2746	68	49	fuzzy	fuzzy	ADJ
cana-2746	68	50	soft	soft	ADJ
cana-2746	68	51	set	set	NOUN
cana-2746	68	52	denoted	denote	VERB
cana-2746	68	53	by	by	ADP
cana-2746	68	54	(	(	PUNCT
cana-2746	68	55	𝐻	𝐻	PROPN
cana-2746	68	56	,	,	PUNCT
cana-2746	68	57	𝐶	𝐶	PROPN
cana-2746	68	58	)	)	PUNCT
cana-2746	68	59	,	,	PUNCT
cana-2746	68	60	where	where	SCONJ
cana-2746	68	61	𝐶	𝐶	PROPN
cana-2746	68	62	=	=	SYM
cana-2746	68	63	𝐴	𝐴	PROPN
cana-2746	68	64	∩	∩	NOUN
cana-2746	68	65	𝐵	𝐵	PROPN
cana-2746	68	66	and	and	CCONJ
cana-2746	68	67	𝐻(𝛼	𝐻(𝛼	NOUN
cana-2746	68	68	)	)	PUNCT
cana-2746	68	69	=	=	SYM
cana-2746	68	70	{	{	PUNCT
cana-2746	68	71	𝐹(𝛼	𝐹(𝛼	NUM
cana-2746	68	72	)	)	PUNCT
cana-2746	68	73	if𝛼	if𝛼	NOUN
cana-2746	68	74	∈	∈	PROPN
cana-2746	68	75	𝐴	𝐴	PROPN
cana-2746	68	76	−	−	PROPN
cana-2746	68	77	𝐵	𝐵	PROPN
cana-2746	68	78	𝐺(𝛼	𝐺(𝛼	NOUN
cana-2746	68	79	)	)	PUNCT
cana-2746	68	80	if𝛼	if𝛼	NOUN
cana-2746	68	81	∈	∈	PROPN
cana-2746	68	82	𝐵	𝐵	PROPN
cana-2746	68	83	−	−	PROPN
cana-2746	68	84	𝐴	𝐴	PROPN
cana-2746	68	85	𝑚𝑖𝑛{𝐹(𝛼	𝑚𝑖𝑛{𝐹(𝛼	PROPN
cana-2746	68	86	)	)	PUNCT
cana-2746	68	87	,	,	PUNCT
cana-2746	68	88	𝐺(𝛼	𝐺(𝛼	NOUN
cana-2746	68	89	)	)	PUNCT
cana-2746	68	90	}	}	PUNCT
cana-2746	68	91	if𝛼	if𝛼	NOUN
cana-2746	68	92	∈	∈	PROPN
cana-2746	68	93	𝐴	𝐴	PROPN
cana-2746	68	94	∩	∩	ADJ
cana-2746	68	95	𝐵	𝐵	PROPN
cana-2746	68	96	for	for	ADP
cana-2746	68	97	all	all	DET
cana-2746	68	98	𝛼	𝛼	PRON
cana-2746	68	99	∈	∈	NOUN
cana-2746	68	100	𝐶.	𝐶.	NOUN
cana-2746	68	101	it	it	PRON
cana-2746	68	102	is	be	AUX
cana-2746	68	103	denoted	denote	VERB
cana-2746	68	104	by	by	ADP
cana-2746	68	105	(	(	PUNCT
cana-2746	68	106	𝐻	𝐻	PROPN
cana-2746	68	107	,	,	PUNCT
cana-2746	68	108	𝐶	𝐶	PROPN
cana-2746	68	109	)	)	PUNCT
cana-2746	68	110	=	=	PUNCT
cana-2746	68	111	(	(	PUNCT
cana-2746	68	112	𝐹	𝐹	PROPN
cana-2746	68	113	,	,	PUNCT
cana-2746	68	114	𝐴	𝐴	PROPN
cana-2746	68	115	)	)	PUNCT
cana-2746	68	116	∩	∩	NOUN
cana-2746	68	117	(	(	PUNCT
cana-2746	68	118	𝐹	𝐹	PROPN
cana-2746	68	119	,	,	PUNCT
cana-2746	68	120	𝐵	𝐵	PROPN
cana-2746	68	121	)	)	PUNCT
cana-2746	68	122	.	.	PUNCT
cana-2746	69	1	definition	definition	NOUN
cana-2746	69	2	2.24	2.24	NUM
cana-2746	70	1	[	[	X
cana-2746	70	2	13	13	NUM
cana-2746	70	3	]	]	PUNCT
cana-2746	70	4	the	the	DET
cana-2746	70	5	union	union	NOUN
cana-2746	70	6	of	of	ADP
cana-2746	70	7	two	two	NUM
cana-2746	70	8	pythagorean	pythagorean	ADJ
cana-2746	70	9	fuzzy	fuzzy	ADJ
cana-2746	70	10	soft	soft	ADJ
cana-2746	70	11	sets	set	NOUN
cana-2746	70	12	(	(	PUNCT
cana-2746	70	13	𝐹	𝐹	PROPN
cana-2746	70	14	,	,	PUNCT
cana-2746	70	15	𝐴	𝐴	PROPN
cana-2746	70	16	)	)	PUNCT
cana-2746	70	17	and	and	CCONJ
cana-2746	70	18	(	(	PUNCT
cana-2746	70	19	𝐺	𝐺	PROPN
cana-2746	70	20	,	,	PUNCT
cana-2746	70	21	𝐵	𝐵	PROPN
cana-2746	70	22	)	)	PUNCT
cana-2746	70	23	over	over	ADP
cana-2746	70	24	a	a	DET
cana-2746	70	25	universe	universe	NOUN
cana-2746	70	26	𝑈	𝑈	NOUN
cana-2746	70	27	is	be	AUX
cana-2746	70	28	a	a	DET
cana-2746	70	29	pythagorean	pythagorean	ADJ
cana-2746	70	30	fuzzy	fuzzy	ADJ
cana-2746	70	31	soft	soft	ADJ
cana-2746	70	32	set	set	NOUN
cana-2746	70	33	denoted	denote	VERB
cana-2746	70	34	by	by	ADP
cana-2746	70	35	(	(	PUNCT
cana-2746	70	36	𝐻	𝐻	PROPN
cana-2746	70	37	,	,	PUNCT
cana-2746	70	38	𝐶	𝐶	PROPN
cana-2746	70	39	)	)	PUNCT
cana-2746	70	40	,	,	PUNCT
cana-2746	70	41	where	where	SCONJ
cana-2746	70	42	𝐶	𝐶	PROPN
cana-2746	70	43	=	=	PROPN
cana-2746	70	44	𝐴	𝐴	PROPN
cana-2746	70	45	∪	∪	VERB
cana-2746	70	46	𝐵	𝐵	PROPN
cana-2746	70	47	and	and	CCONJ
cana-2746	70	48	𝐻(𝛼	𝐻(𝛼	NOUN
cana-2746	70	49	)	)	PUNCT
cana-2746	70	50	=	=	SYM
cana-2746	70	51	{	{	PUNCT
cana-2746	70	52	𝐹(𝛼	𝐹(𝛼	NUM
cana-2746	70	53	)	)	PUNCT
cana-2746	70	54	if𝛼	if𝛼	NOUN
cana-2746	70	55	∈	∈	PROPN
cana-2746	70	56	𝐴	𝐴	PROPN
cana-2746	70	57	−	−	PROPN
cana-2746	70	58	𝐵	𝐵	PROPN
cana-2746	70	59	𝐺(𝛼	𝐺(𝛼	NOUN
cana-2746	70	60	)	)	PUNCT
cana-2746	70	61	if𝛼	if𝛼	NOUN
cana-2746	70	62	∈	∈	PROPN
cana-2746	70	63	𝐵	𝐵	PROPN
cana-2746	70	64	−	−	PROPN
cana-2746	70	65	𝐴	𝐴	PROPN
cana-2746	70	66	𝑚𝑖𝑛{𝐹(𝛼	𝑚𝑖𝑛{𝐹(𝛼	PROPN
cana-2746	70	67	)	)	PUNCT
cana-2746	70	68	,	,	PUNCT
cana-2746	70	69	𝐺(𝛼	𝐺(𝛼	NOUN
cana-2746	70	70	)	)	PUNCT
cana-2746	70	71	}	}	PUNCT
cana-2746	70	72	if𝛼	if𝛼	NOUN
cana-2746	70	73	∈	∈	PROPN
cana-2746	70	74	𝐴	𝐴	PROPN
cana-2746	70	75	∪	∪	VERB
cana-2746	70	76	𝐵	𝐵	NOUN
cana-2746	70	77	for	for	ADP
cana-2746	70	78	all	all	DET
cana-2746	70	79	𝛼	𝛼	PRON
cana-2746	70	80	∈	∈	NOUN
cana-2746	70	81	𝐶.	𝐶.	NOUN
cana-2746	70	82	it	it	PRON
cana-2746	70	83	is	be	AUX
cana-2746	70	84	denoted	denote	VERB
cana-2746	70	85	by	by	ADP
cana-2746	70	86	(	(	PUNCT
cana-2746	70	87	𝐻	𝐻	PROPN
cana-2746	70	88	,	,	PUNCT
cana-2746	70	89	𝐶	𝐶	PROPN
cana-2746	70	90	)	)	PUNCT
cana-2746	70	91	=	=	PUNCT
cana-2746	70	92	(	(	PUNCT
cana-2746	70	93	𝐹	𝐹	PROPN
cana-2746	70	94	,	,	PUNCT
cana-2746	70	95	𝐴	𝐴	PROPN
cana-2746	70	96	)	)	PUNCT
cana-2746	70	97	∪	∪	NOUN
cana-2746	70	98	(	(	PUNCT
cana-2746	70	99	𝐹	𝐹	PROPN
cana-2746	70	100	,	,	PUNCT
cana-2746	70	101	𝐵	𝐵	PROPN
cana-2746	70	102	)	)	PUNCT
cana-2746	70	103	.	.	PUNCT
cana-2746	71	1	communications	communication	NOUN
cana-2746	71	2	on	on	ADP
cana-2746	71	3	applied	apply	VERB
cana-2746	71	4	nonlinear	nonlinear	ADJ
cana-2746	71	5	analysis	analysis	NOUN
cana-2746	71	6	issn	issn	NOUN
cana-2746	71	7	:	:	PUNCT
cana-2746	71	8	1074	1074	NUM
cana-2746	71	9	-	-	PUNCT
cana-2746	71	10	133x	133x	NUM
cana-2746	71	11	vol	vol	NOUN
cana-2746	71	12	32	32	NUM
cana-2746	71	13	no	no	NOUN
cana-2746	71	14	.	.	PUNCT
cana-2746	72	1	4s	4s	NUM
cana-2746	72	2	(	(	PUNCT
cana-2746	72	3	2025	2025	NUM
cana-2746	72	4	)	)	PUNCT
cana-2746	72	5	153	153	NUM
cana-2746	72	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2746	72	7	definition	definition	NOUN
cana-2746	72	8	2.25	2.25	NUM
cana-2746	72	9	[	[	X
cana-2746	72	10	13	13	NUM
cana-2746	72	11	]	]	PUNCT
cana-2746	72	12	let	let	VERB
cana-2746	72	13	(	(	PUNCT
cana-2746	72	14	𝐹	𝐹	PROPN
cana-2746	72	15	,	,	PUNCT
cana-2746	72	16	𝐴	𝐴	PROPN
cana-2746	72	17	)	)	PUNCT
cana-2746	72	18	and	and	CCONJ
cana-2746	72	19	(	(	PUNCT
cana-2746	72	20	𝐺	𝐺	PROPN
cana-2746	72	21	,	,	PUNCT
cana-2746	72	22	𝐵	𝐵	PROPN
cana-2746	72	23	)	)	PUNCT
cana-2746	72	24	be	be	VERB
cana-2746	72	25	two	two	NUM
cana-2746	72	26	pythagorean	pythagorean	ADJ
cana-2746	72	27	fuzzy	fuzzy	ADJ
cana-2746	72	28	soft	soft	ADJ
cana-2746	72	29	sets	set	NOUN
cana-2746	72	30	over	over	ADP
cana-2746	72	31	𝑈	𝑈	PROPN
cana-2746	72	32	such	such	ADJ
cana-2746	72	33	that	that	SCONJ
cana-2746	72	34	𝐴	𝐴	PROPN
cana-2746	72	35	∪	∪	AUX
cana-2746	72	36	𝐵	𝐵	PROPN
cana-2746	72	37	≠	≠	PROPN
cana-2746	72	38	∅.	∅.	ADP
cana-2746	72	39	the	the	DET
cana-2746	72	40	bi	bi	NOUN
cana-2746	72	41	-	-	NOUN
cana-2746	72	42	union	union	NOUN
cana-2746	72	43	of	of	ADP
cana-2746	72	44	(	(	PUNCT
cana-2746	72	45	𝐹	𝐹	PROPN
cana-2746	72	46	,	,	PUNCT
cana-2746	72	47	𝐴	𝐴	PROPN
cana-2746	72	48	)	)	PUNCT
cana-2746	72	49	and	and	CCONJ
cana-2746	72	50	(	(	PUNCT
cana-2746	72	51	𝐺	𝐺	PROPN
cana-2746	72	52	,	,	PUNCT
cana-2746	72	53	𝐵	𝐵	PROPN
cana-2746	72	54	)	)	PUNCT
cana-2746	72	55	is	be	AUX
cana-2746	72	56	defined	define	VERB
cana-2746	72	57	to	to	PART
cana-2746	72	58	be	be	AUX
cana-2746	72	59	the	the	DET
cana-2746	72	60	pythagorean	pythagorean	PROPN
cana-2746	72	61	fuzzy	fuzzy	ADJ
cana-2746	72	62	soft	soft	ADJ
cana-2746	72	63	set	set	NOUN
cana-2746	72	64	(	(	PUNCT
cana-2746	72	65	𝐻	𝐻	PROPN
cana-2746	72	66	,	,	PUNCT
cana-2746	72	67	𝐶	𝐶	PROPN
cana-2746	72	68	)	)	PUNCT
cana-2746	72	69	,	,	PUNCT
cana-2746	72	70	where	where	SCONJ
cana-2746	72	71	𝐶	𝐶	PROPN
cana-2746	72	72	=	=	PROPN
cana-2746	72	73	𝐴	𝐴	PROPN
cana-2746	72	74	∪	∪	VERB
cana-2746	72	75	𝐵	𝐵	PROPN
cana-2746	72	76	and	and	CCONJ
cana-2746	72	77	𝐻(𝛼	𝐻(𝛼	NOUN
cana-2746	72	78	)	)	PUNCT
cana-2746	72	79	=	=	SYM
cana-2746	72	80	𝐹(𝛼	𝐹(𝛼	NUM
cana-2746	72	81	)	)	PUNCT
cana-2746	72	82	∪	∪	ADP
cana-2746	72	83	𝐺(𝛼	𝐺(𝛼	NOUN
cana-2746	72	84	)	)	PUNCT
cana-2746	72	85	for	for	ADP
cana-2746	72	86	all	all	DET
cana-2746	72	87	𝛼	𝛼	PRON
cana-2746	72	88	∈	∈	NOUN
cana-2746	72	89	𝐶.	𝐶.	NOUN
cana-2746	72	90	it	it	PRON
cana-2746	72	91	is	be	AUX
cana-2746	72	92	denoted	denote	VERB
cana-2746	72	93	by	by	ADP
cana-2746	72	94	(	(	PUNCT
cana-2746	72	95	𝐻	𝐻	PROPN
cana-2746	72	96	,	,	PUNCT
cana-2746	72	97	𝐶	𝐶	PROPN
cana-2746	72	98	)	)	PUNCT
cana-2746	72	99	=	=	PUNCT
cana-2746	72	100	(	(	PUNCT
cana-2746	72	101	𝐹	𝐹	PROPN
cana-2746	72	102	,	,	PUNCT
cana-2746	72	103	𝐴	𝐴	PROPN
cana-2746	72	104	)	)	PUNCT
cana-2746	72	105	⊔	⊔	PROPN
cana-2746	73	1	(	(	PUNCT
cana-2746	73	2	𝐺	𝐺	PROPN
cana-2746	73	3	,	,	PUNCT
cana-2746	73	4	𝐵	𝐵	PROPN
cana-2746	73	5	)	)	PUNCT
cana-2746	73	6	.	.	PUNCT
cana-2746	74	1	definition	definition	NOUN
cana-2746	74	2	2.26	2.26	NUM
cana-2746	74	3	[	[	X
cana-2746	74	4	13	13	NUM
cana-2746	74	5	]	]	PUNCT
cana-2746	74	6	let	let	VERB
cana-2746	74	7	(	(	PUNCT
cana-2746	74	8	𝐹	𝐹	PROPN
cana-2746	74	9	,	,	PUNCT
cana-2746	74	10	𝐴	𝐴	PROPN
cana-2746	74	11	)	)	PUNCT
cana-2746	74	12	and	and	CCONJ
cana-2746	74	13	(	(	PUNCT
cana-2746	74	14	𝐺	𝐺	PROPN
cana-2746	74	15	,	,	PUNCT
cana-2746	74	16	𝐵	𝐵	PROPN
cana-2746	74	17	)	)	PUNCT
cana-2746	74	18	be	be	VERB
cana-2746	74	19	two	two	NUM
cana-2746	74	20	pythagorean	pythagorean	ADJ
cana-2746	74	21	fuzzy	fuzzy	ADJ
cana-2746	74	22	soft	soft	ADJ
cana-2746	74	23	sets	set	NOUN
cana-2746	74	24	over	over	ADP
cana-2746	74	25	𝑈	𝑈	PROPN
cana-2746	75	1	such	such	ADJ
cana-2746	75	2	that	that	SCONJ
cana-2746	75	3	𝐴	𝐴	PROPN
cana-2746	75	4	∩	∩	ADJ
cana-2746	75	5	𝐵	𝐵	PROPN
cana-2746	75	6	≠	≠	PROPN
cana-2746	75	7	∅.	∅.	ADP
cana-2746	75	8	the	the	DET
cana-2746	75	9	bi	bi	NOUN
cana-2746	75	10	-	-	NOUN
cana-2746	75	11	union	union	NOUN
cana-2746	75	12	of	of	ADP
cana-2746	75	13	(	(	PUNCT
cana-2746	75	14	𝐹	𝐹	PROPN
cana-2746	75	15	,	,	PUNCT
cana-2746	75	16	𝐴	𝐴	PROPN
cana-2746	75	17	)	)	PUNCT
cana-2746	75	18	and	and	CCONJ
cana-2746	75	19	(	(	PUNCT
cana-2746	75	20	𝐺	𝐺	PROPN
cana-2746	75	21	,	,	PUNCT
cana-2746	75	22	𝐵	𝐵	PROPN
cana-2746	75	23	)	)	PUNCT
cana-2746	75	24	is	be	AUX
cana-2746	75	25	defined	define	VERB
cana-2746	75	26	to	to	PART
cana-2746	75	27	be	be	AUX
cana-2746	75	28	the	the	DET
cana-2746	75	29	pythagorean	pythagorean	PROPN
cana-2746	75	30	fuzzy	fuzzy	ADJ
cana-2746	75	31	soft	soft	ADJ
cana-2746	75	32	set	set	NOUN
cana-2746	75	33	(	(	PUNCT
cana-2746	75	34	𝐻	𝐻	PROPN
cana-2746	75	35	,	,	PUNCT
cana-2746	75	36	𝐶	𝐶	PROPN
cana-2746	75	37	)	)	PUNCT
cana-2746	75	38	,	,	PUNCT
cana-2746	75	39	where	where	SCONJ
cana-2746	75	40	𝐶	𝐶	PROPN
cana-2746	75	41	=	=	SYM
cana-2746	75	42	𝐴	𝐴	PROPN
cana-2746	75	43	∩	∩	NOUN
cana-2746	75	44	𝐵	𝐵	PROPN
cana-2746	75	45	and	and	CCONJ
cana-2746	75	46	𝐻(𝛼	𝐻(𝛼	NOUN
cana-2746	75	47	)	)	PUNCT
cana-2746	75	48	=	=	SYM
cana-2746	75	49	𝐹(𝛼	𝐹(𝛼	NUM
cana-2746	75	50	)	)	PUNCT
cana-2746	75	51	∩	∩	NOUN
cana-2746	75	52	𝐺(𝛼	𝐺(𝛼	NOUN
cana-2746	75	53	)	)	PUNCT
cana-2746	75	54	for	for	ADP
cana-2746	75	55	all	all	DET
cana-2746	75	56	𝛼	𝛼	PRON
cana-2746	75	57	∈	∈	NOUN
cana-2746	75	58	𝐶.	𝐶.	NOUN
cana-2746	75	59	it	it	PRON
cana-2746	75	60	is	be	AUX
cana-2746	75	61	denoted	denote	VERB
cana-2746	75	62	by	by	ADP
cana-2746	75	63	(	(	PUNCT
cana-2746	75	64	𝐻	𝐻	PROPN
cana-2746	75	65	,	,	PUNCT
cana-2746	75	66	𝐶	𝐶	PROPN
cana-2746	75	67	)	)	PUNCT
cana-2746	76	1	=	=	PUNCT
cana-2746	76	2	(	(	PUNCT
cana-2746	76	3	𝐹	𝐹	PROPN
cana-2746	76	4	,	,	PUNCT
cana-2746	76	5	𝐴	𝐴	PROPN
cana-2746	76	6	)	)	PUNCT
cana-2746	76	7	⊓	⊓	PROPN
cana-2746	76	8	(	(	PUNCT
cana-2746	76	9	𝐺	𝐺	PROPN
cana-2746	76	10	,	,	PUNCT
cana-2746	76	11	𝐵	𝐵	PROPN
cana-2746	76	12	)	)	PUNCT
cana-2746	76	13	.	.	PUNCT
cana-2746	77	1	definition	definition	NOUN
cana-2746	77	2	2.27	2.27	NUM
cana-2746	78	1	[	[	X
cana-2746	78	2	13	13	NUM
cana-2746	78	3	]	]	PUNCT
cana-2746	78	4	let	let	VERB
cana-2746	78	5	(	(	PUNCT
cana-2746	78	6	𝐹	𝐹	PROPN
cana-2746	78	7	,	,	PUNCT
cana-2746	78	8	𝐴	𝐴	PROPN
cana-2746	78	9	)	)	PUNCT
cana-2746	78	10	and	and	CCONJ
cana-2746	78	11	(	(	PUNCT
cana-2746	78	12	𝐺	𝐺	PROPN
cana-2746	78	13	,	,	PUNCT
cana-2746	78	14	𝐵	𝐵	PROPN
cana-2746	78	15	)	)	PUNCT
cana-2746	78	16	two	two	NUM
cana-2746	78	17	pythagorean	pythagorean	PROPN
cana-2746	78	18	fuzzy	fuzzy	ADJ
cana-2746	78	19	soft	soft	ADJ
cana-2746	78	20	sets	set	NOUN
cana-2746	78	21	over	over	ADP
cana-2746	78	22	a	a	DET
cana-2746	78	23	universe	universe	ADJ
cana-2746	78	24	𝑈.	𝑈.	NOUN
cana-2746	78	25	the	the	DET
cana-2746	78	26	product	product	NOUN
cana-2746	78	27	of	of	ADP
cana-2746	78	28	(	(	PUNCT
cana-2746	78	29	𝐹	𝐹	PROPN
cana-2746	78	30	,	,	PUNCT
cana-2746	78	31	𝐴	𝐴	PROPN
cana-2746	78	32	)	)	PUNCT
cana-2746	78	33	and	and	CCONJ
cana-2746	78	34	(	(	PUNCT
cana-2746	78	35	𝐺	𝐺	PROPN
cana-2746	78	36	,	,	PUNCT
cana-2746	78	37	𝐵	𝐵	PROPN
cana-2746	78	38	)	)	PUNCT
cana-2746	78	39	is	be	AUX
cana-2746	78	40	defined	define	VERB
cana-2746	78	41	to	to	PART
cana-2746	78	42	be	be	AUX
cana-2746	78	43	the	the	DET
cana-2746	78	44	pythagorean	pythagorean	PROPN
cana-2746	78	45	fuzzy	fuzzy	ADJ
cana-2746	78	46	soft	soft	ADJ
cana-2746	78	47	set	set	NOUN
cana-2746	78	48	denoted	denote	VERB
cana-2746	78	49	by	by	ADP
cana-2746	78	50	(	(	PUNCT
cana-2746	78	51	𝐹	𝐹	PROPN
cana-2746	78	52	∘	∘	PROPN
cana-2746	78	53	𝐺	𝐺	PROPN
cana-2746	78	54	,	,	PUNCT
cana-2746	78	55	𝐶	𝐶	PROPN
cana-2746	78	56	)	)	PUNCT
cana-2746	78	57	,	,	PUNCT
cana-2746	78	58	where	where	SCONJ
cana-2746	78	59	𝐶	𝐶	PROPN
cana-2746	78	60	=	=	PROPN
cana-2746	78	61	𝐴	𝐴	PROPN
cana-2746	78	62	∪	∪	VERB
cana-2746	78	63	𝐵	𝐵	PROPN
cana-2746	78	64	and	and	CCONJ
cana-2746	78	65	𝐴𝑥(𝐹∘𝐺)(𝛼)(𝑖	𝐴𝑥(𝐹∘𝐺)(𝛼)(𝑖	NOUN
cana-2746	78	66	)	)	PUNCT
cana-2746	78	67	=	=	PRON
cana-2746	78	68	{	{	PUNCT
cana-2746	78	69	𝐴𝑥(𝐹)(𝛼)(𝑖	𝐴𝑥(𝐹)(𝛼)(𝑖	NOUN
cana-2746	78	70	)	)	PUNCT
cana-2746	78	71	if𝛼	if𝛼	NOUN
cana-2746	78	72	∈	∈	PROPN
cana-2746	78	73	𝐴	𝐴	PROPN
cana-2746	78	74	−	−	PROPN
cana-2746	78	75	𝐵	𝐵	PROPN
cana-2746	78	76	𝐴𝑥(𝐺)(𝛼)(𝑖	𝐴𝑥(𝐺)(𝛼)(𝑖	PROPN
cana-2746	78	77	)	)	PUNCT
cana-2746	78	78	if𝛼	if𝛼	NOUN
cana-2746	78	79	∈	∈	PROPN
cana-2746	78	80	𝐵	𝐵	PROPN
cana-2746	78	81	−	−	PROPN
cana-2746	78	82	𝐴	𝐴	PROPN
cana-2746	78	83	sup	sup	PROPN
cana-2746	78	84	𝑖=𝑎𝑏	𝑖=𝑎𝑏	PROPN
cana-2746	78	85	𝑚𝑖𝑛{𝐴𝑥(𝐹)(𝛼)(𝑖	𝑚𝑖𝑛{𝐴𝑥(𝐹)(𝛼)(𝑖	NUM
cana-2746	78	86	)	)	PUNCT
cana-2746	78	87	,	,	PUNCT
cana-2746	78	88	𝐴𝑥(𝐺)(𝛼)(𝑖	𝐴𝑥(𝐺)(𝛼)(𝑖	PROPN
cana-2746	78	89	)	)	PUNCT
cana-2746	78	90	}	}	PUNCT
cana-2746	78	91	if𝛼	if𝛼	NOUN
cana-2746	78	92	∈	∈	PROPN
cana-2746	78	93	𝐴	𝐴	PROPN
cana-2746	78	94	∩	∩	ADJ
cana-2746	78	95	𝐵	𝐵	NOUN
cana-2746	78	96	and	and	CCONJ
cana-2746	78	97	𝐴𝑦(𝐹∘𝐺)(𝛼)(𝑖	𝐴𝑦(𝐹∘𝐺)(𝛼)(𝑖	NOUN
cana-2746	78	98	)	)	PUNCT
cana-2746	78	99	=	=	PRON
cana-2746	78	100	{	{	PUNCT
cana-2746	78	101	𝐴𝑦(𝐹)(𝛼)(𝑖	𝐴𝑦(𝐹)(𝛼)(𝑖	NOUN
cana-2746	78	102	)	)	PUNCT
cana-2746	78	103	if𝛼	if𝛼	NOUN
cana-2746	78	104	∈	∈	PROPN
cana-2746	78	105	𝐴	𝐴	PROPN
cana-2746	78	106	−	−	PROPN
cana-2746	78	107	𝐵	𝐵	PROPN
cana-2746	78	108	𝐴𝑦(𝐺)(𝛼)(𝑖	𝐴𝑦(𝐺)(𝛼)(𝑖	NOUN
cana-2746	78	109	)	)	PUNCT
cana-2746	78	110	if𝛼	if𝛼	NOUN
cana-2746	78	111	∈	∈	PROPN
cana-2746	78	112	𝐵	𝐵	PROPN
cana-2746	78	113	−	−	PROPN
cana-2746	78	114	𝐴	𝐴	PROPN
cana-2746	78	115	inf	inf	PROPN
cana-2746	78	116	𝑖=𝑎𝑏	𝑖=𝑎𝑏	PROPN
cana-2746	78	117	𝑚𝑎𝑥{𝐴𝑦(𝐹)(𝛼)(𝑖	𝑚𝑎𝑥{𝐴𝑦(𝐹)(𝛼)(𝑖	PROPN
cana-2746	78	118	)	)	PUNCT
cana-2746	78	119	,	,	PUNCT
cana-2746	78	120	𝐴𝑦(𝐺)(𝛼)(𝑖	𝐴𝑦(𝐺)(𝛼)(𝑖	NOUN
cana-2746	78	121	)	)	PUNCT
cana-2746	78	122	}	}	PUNCT
cana-2746	78	123	if𝛼	if𝛼	NOUN
cana-2746	78	124	∈	∈	PROPN
cana-2746	78	125	𝐴	𝐴	PROPN
cana-2746	78	126	∩	∩	ADJ
cana-2746	78	127	𝐵	𝐵	PROPN
cana-2746	78	128	for	for	ADP
cana-2746	78	129	all	all	DET
cana-2746	78	130	𝛼	𝛼	PROPN
cana-2746	78	131	∈	∈	PROPN
cana-2746	78	132	𝐶	𝐶	PROPN
cana-2746	78	133	and	and	CCONJ
cana-2746	78	134	𝑖	𝑖	ADP
cana-2746	78	135	∈	∈	PROPN
cana-2746	78	136	𝑈	𝑈	PROPN
cana-2746	78	137	.	.	PUNCT
cana-2746	79	1	it	it	PRON
cana-2746	79	2	is	be	AUX
cana-2746	79	3	denoted	denote	VERB
cana-2746	79	4	by	by	ADP
cana-2746	79	5	(	(	PUNCT
cana-2746	79	6	𝐹	𝐹	PROPN
cana-2746	79	7	∘	∘	PROPN
cana-2746	79	8	𝐺	𝐺	PROPN
cana-2746	79	9	,	,	PUNCT
cana-2746	79	10	𝐶	𝐶	PROPN
cana-2746	79	11	)	)	PUNCT
cana-2746	79	12	=	=	PUNCT
cana-2746	79	13	(	(	PUNCT
cana-2746	79	14	𝐹	𝐹	PROPN
cana-2746	79	15	,	,	PUNCT
cana-2746	79	16	𝐴	𝐴	PROPN
cana-2746	79	17	)	)	PUNCT
cana-2746	79	18	∘	∘	PROPN
cana-2746	79	19	(	(	PUNCT
cana-2746	79	20	𝐺	𝐺	NOUN
cana-2746	79	21	,	,	PUNCT
cana-2746	79	22	𝐵	𝐵	PROPN
cana-2746	79	23	)	)	PUNCT
cana-2746	79	24	.	.	PUNCT
cana-2746	80	1	definition	definition	NOUN
cana-2746	80	2	2.28	2.28	NUM
cana-2746	80	3	[	[	X
cana-2746	80	4	28	28	NUM
cana-2746	80	5	]	]	X
cana-2746	80	6	a	a	DET
cana-2746	80	7	fuzzy	fuzzy	NOUN
cana-2746	80	8	subset	subset	VERB
cana-2746	80	9	a	a	PRON
cana-2746	80	10	of	of	ADP
cana-2746	80	11	s	s	PRON
cana-2746	80	12	is	be	AUX
cana-2746	80	13	called	call	VERB
cana-2746	80	14	a	a	DET
cana-2746	80	15	fuzzy	fuzzy	ADJ
cana-2746	80	16	bi	bi	ADJ
cana-2746	80	17	-	-	ADJ
cana-2746	80	18	interior	interior	ADJ
cana-2746	80	19	-	-	PUNCT
cana-2746	80	20	ideal	ideal	NOUN
cana-2746	80	21	if	if	SCONJ
cana-2746	80	22	𝑆𝐴𝑆	𝑆𝐴𝑆	PROPN
cana-2746	80	23	∩	∩	NOUN
cana-2746	80	24	𝐴𝑆𝐴	𝐴𝑆𝐴	PROPN
cana-2746	80	25	⊆	⊆	NUM
cana-2746	80	26	𝐴.	𝐴.	PROPN
cana-2746	80	27	3	3	NUM
cana-2746	80	28	.	.	PUNCT
cana-2746	81	1	pythagorean	pythagorean	PROPN
cana-2746	81	2	fuzzy	fuzzy	ADJ
cana-2746	81	3	bi	bi	ADJ
cana-2746	81	4	-	-	ADJ
cana-2746	81	5	interior	interior	ADJ
cana-2746	81	6	-	-	PUNCT
cana-2746	81	7	ideals	ideal	NOUN
cana-2746	81	8	in	in	ADP
cana-2746	81	9	𝚪-semiring	𝚪-semire	VERB
cana-2746	81	10	this	this	DET
cana-2746	81	11	section	section	NOUN
cana-2746	81	12	deals	deal	VERB
cana-2746	81	13	with	with	ADP
cana-2746	81	14	the	the	DET
cana-2746	81	15	pythagorean	pythagorean	ADJ
cana-2746	81	16	fuzzy	fuzzy	ADJ
cana-2746	81	17	bi	bi	ADJ
cana-2746	81	18	-	-	ADJ
cana-2746	81	19	interior	interior	ADJ
cana-2746	81	20	-	-	PUNCT
cana-2746	81	21	ideals	ideal	NOUN
cana-2746	81	22	in	in	ADP
cana-2746	81	23	γ	γ	NOUN
cana-2746	81	24	-	-	ADJ
cana-2746	81	25	semiring	semiring	ADJ
cana-2746	81	26	𝑆.	𝑆.	NOUN
cana-2746	81	27	definition	definition	NOUN
cana-2746	81	28	3.1	3.1	NUM
cana-2746	81	29	a	a	DET
cana-2746	81	30	pfs	pfs	ADJ
cana-2746	81	31	𝐴	𝐴	PROPN
cana-2746	81	32	=	=	SYM
cana-2746	81	33	(	(	PUNCT
cana-2746	81	34	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	81	35	,	,	PUNCT
cana-2746	81	36	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	81	37	)	)	PUNCT
cana-2746	81	38	of	of	ADP
cana-2746	81	39	𝑆	𝑆	PROPN
cana-2746	81	40	is	be	AUX
cana-2746	81	41	said	say	VERB
cana-2746	81	42	to	to	PART
cana-2746	81	43	be	be	AUX
cana-2746	81	44	a	a	DET
cana-2746	81	45	𝑃𝐹𝐵𝐼𝐼	𝑃𝐹𝐵𝐼𝐼	NOUN
cana-2746	81	46	of	of	ADP
cana-2746	81	47	𝑆	𝑆	PROPN
cana-2746	81	48	if	if	SCONJ
cana-2746	81	49	the	the	DET
cana-2746	81	50	following	follow	VERB
cana-2746	81	51	conditions	condition	NOUN
cana-2746	81	52	are	be	AUX
cana-2746	81	53	holds	hold	NOUN
cana-2746	81	54	:	:	PUNCT
cana-2746	82	1	1	1	X
cana-2746	82	2	.	.	X
cana-2746	83	1	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	83	2	+	+	CCONJ
cana-2746	83	3	𝑦	𝑦	NOUN
cana-2746	83	4	)	)	PUNCT
cana-2746	83	5	≥	≥	PROPN
cana-2746	83	6	𝑚𝑖𝑛{𝐴𝜇(𝑥	𝑚𝑖𝑛{𝐴𝜇(𝑥	PROPN
cana-2746	83	7	)	)	PUNCT
cana-2746	83	8	,	,	PUNCT
cana-2746	83	9	𝐴𝜇(𝑦	𝐴𝜇(𝑦	NOUN
cana-2746	83	10	)	)	PUNCT
cana-2746	83	11	}	}	PUNCT
cana-2746	84	1	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	85	1	+	+	NUM
cana-2746	85	2	𝑦	𝑦	X
cana-2746	85	3	)	)	PUNCT
cana-2746	85	4	≤	≤	NUM
cana-2746	85	5	𝑚𝑎𝑥{𝐴𝜈(𝑥	𝑚𝑎𝑥{𝐴𝜈(𝑥	PROPN
cana-2746	85	6	)	)	PUNCT
cana-2746	85	7	,	,	PUNCT
cana-2746	85	8	𝐴𝜈(𝑦	𝐴𝜈(𝑦	PROPN
cana-2746	85	9	)	)	PUNCT
cana-2746	85	10	}	}	PUNCT
cana-2746	85	11	2	2	NUM
cana-2746	85	12	.	.	X
cana-2746	86	1	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	86	2	∘	∘	PROPN
cana-2746	86	3	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	86	4	∘	∘	NOUN
cana-2746	86	5	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	86	6	∩	∩	NOUN
cana-2746	86	7	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	86	8	∘	∘	ADV
cana-2746	86	9	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	86	10	∘	∘	PROPN
cana-2746	86	11	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	86	12	⊇	⊇	PROPN
cana-2746	86	13	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	86	14	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	86	15	∘	∘	NOUN
cana-2746	86	16	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	86	17	∘	∘	ADJ
cana-2746	86	18	𝜒𝑆	𝜒𝑆	NOUN
cana-2746	86	19	∩	∩	NOUN
cana-2746	86	20	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	86	21	∘	∘	NOUN
cana-2746	86	22	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	86	23	∘	∘	NOUN
cana-2746	86	24	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	86	25	⊆	⊆	PROPN
cana-2746	86	26	𝐴𝜈.	𝐴𝜈.	PROPN
cana-2746	86	27	theorem	theorem	VERB
cana-2746	86	28	3.2	3.2	NUM
cana-2746	86	29	every	every	DET
cana-2746	86	30	pf	pf	NOUN
cana-2746	86	31	left	leave	VERB
cana-2746	86	32	ideal	ideal	NOUN
cana-2746	86	33	of	of	ADP
cana-2746	86	34	𝑆	𝑆	PROPN
cana-2746	86	35	is	be	AUX
cana-2746	86	36	a	a	DET
cana-2746	86	37	pfbii	pfbii	NOUN
cana-2746	86	38	of	of	ADP
cana-2746	86	39	𝑆.	𝑆.	ADJ
cana-2746	86	40	proof	proof	NOUN
cana-2746	86	41	.	.	PUNCT
cana-2746	87	1	let	let	VERB
cana-2746	87	2	𝐴	𝐴	PROPN
cana-2746	87	3	be	be	AUX
cana-2746	87	4	a	a	DET
cana-2746	87	5	𝑃𝐹	𝑃𝐹	NOUN
cana-2746	87	6	left	leave	VERB
cana-2746	87	7	ideal	ideal	NOUN
cana-2746	87	8	of	of	ADP
cana-2746	87	9	𝑆	𝑆	PROPN
cana-2746	87	10	and	and	CCONJ
cana-2746	87	11	𝑥	𝑥	DET
cana-2746	87	12	∈	∈	PROPN
cana-2746	87	13	𝑆	𝑆	PROPN
cana-2746	87	14	,	,	PUNCT
cana-2746	87	15	𝛼	𝛼	PROPN
cana-2746	87	16	,	,	PUNCT
cana-2746	87	17	𝛽	𝛽	PROPN
cana-2746	87	18	∈	∈	PROPN
cana-2746	87	19	γ	γ	X
cana-2746	87	20	.	.	PROPN
cana-2746	87	21	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	87	22	∘	∘	PROPN
cana-2746	87	23	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	87	24	)	)	PUNCT
cana-2746	88	1	=	=	SYM
cana-2746	88	2	sup	sup	NOUN
cana-2746	88	3	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PUNCT
cana-2746	88	4	{	{	PUNCT
cana-2746	88	5	𝑚𝑖𝑛{𝜒𝑆(𝑎	𝑚𝑖𝑛{𝜒𝑆(𝑎	NOUN
cana-2746	88	6	)	)	PUNCT
cana-2746	88	7	,	,	PUNCT
cana-2746	88	8	𝐴𝜇(𝑏	𝐴𝜇(𝑏	NOUN
cana-2746	88	9	)	)	PUNCT
cana-2746	88	10	}	}	PUNCT
cana-2746	88	11	}	}	PUNCT
cana-2746	88	12	=	=	SYM
cana-2746	88	13	sup	sup	NOUN
cana-2746	88	14	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	88	15	{	{	PUNCT
cana-2746	88	16	𝑚𝑖𝑛{1	𝑚𝑖𝑛{1	ADJ
cana-2746	88	17	,	,	PUNCT
cana-2746	88	18	𝐴𝜇(𝑏	𝐴𝜇(𝑏	NOUN
cana-2746	88	19	)	)	PUNCT
cana-2746	88	20	}	}	PUNCT
cana-2746	88	21	}	}	PUNCT
cana-2746	88	22	=	=	SYM
cana-2746	88	23	sup	sup	NOUN
cana-2746	88	24	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	88	25	{	{	PUNCT
cana-2746	88	26	𝐴𝜇(𝑏	𝐴𝜇(𝑏	NOUN
cana-2746	88	27	)	)	PUNCT
cana-2746	88	28	}	}	PUNCT
cana-2746	88	29	communications	communication	NOUN
cana-2746	88	30	on	on	ADP
cana-2746	88	31	applied	apply	VERB
cana-2746	88	32	nonlinear	nonlinear	ADJ
cana-2746	88	33	analysis	analysis	NOUN
cana-2746	88	34	issn	issn	NOUN
cana-2746	88	35	:	:	PUNCT
cana-2746	88	36	1074	1074	NUM
cana-2746	88	37	-	-	PUNCT
cana-2746	88	38	133x	133x	NUM
cana-2746	88	39	vol	vol	NOUN
cana-2746	88	40	32	32	NUM
cana-2746	88	41	no	no	NOUN
cana-2746	88	42	.	.	PUNCT
cana-2746	89	1	4s	4s	NUM
cana-2746	89	2	(	(	PUNCT
cana-2746	89	3	2025	2025	NUM
cana-2746	89	4	)	)	PUNCT
cana-2746	89	5	154	154	NUM
cana-2746	89	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2746	89	7	≥	≥	NOUN
cana-2746	89	8	sup	sup	NOUN
cana-2746	89	9	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PUNCT
cana-2746	89	10	{	{	PUNCT
cana-2746	89	11	𝐴𝜇(𝑎𝑏	𝐴𝜇(𝑎𝑏	ADJ
cana-2746	89	12	)	)	PUNCT
cana-2746	89	13	}	}	PUNCT
cana-2746	89	14	=	=	SYM
cana-2746	89	15	sup	sup	NOUN
cana-2746	89	16	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	89	17	{	{	PUNCT
cana-2746	89	18	𝐴𝜇(𝑥	𝐴𝜇(𝑥	NOUN
cana-2746	89	19	)	)	PUNCT
cana-2746	89	20	}	}	PUNCT
cana-2746	89	21	=	=	SYM
cana-2746	89	22	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	89	23	)	)	PUNCT
cana-2746	89	24	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	89	25	∘	∘	PROPN
cana-2746	89	26	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	89	27	∘	∘	PROPN
cana-2746	89	28	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	89	29	)	)	PUNCT
cana-2746	90	1	=	=	NOUN
cana-2746	90	2	sup	sup	NOUN
cana-2746	90	3	𝑥=𝑢𝛼𝑣𝛽𝑠	𝑥=𝑢𝛼𝑣𝛽𝑠	NOUN
cana-2746	90	4	{	{	PUNCT
cana-2746	90	5	𝑚𝑖𝑛{𝐴𝜇(𝑢	𝑚𝑖𝑛{𝐴𝜇(𝑢	NOUN
cana-2746	90	6	)	)	PUNCT
cana-2746	90	7	,	,	PUNCT
cana-2746	90	8	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	90	9	∘	∘	PROPN
cana-2746	90	10	𝐴𝜇(𝑣𝛽𝑠	𝐴𝜇(𝑣𝛽𝑠	NOUN
cana-2746	90	11	)	)	PUNCT
cana-2746	90	12	}	}	PUNCT
cana-2746	90	13	}	}	PUNCT
cana-2746	90	14	≥	≥	PROPN
cana-2746	90	15	sup	sup	NUM
cana-2746	90	16	𝑥=𝑢𝛼𝑣𝛽𝑠	𝑥=𝑢𝛼𝑣𝛽𝑠	NOUN
cana-2746	90	17	{	{	PUNCT
cana-2746	90	18	𝑚𝑖𝑛{𝐴𝜇(𝑢	𝑚𝑖𝑛{𝐴𝜇(𝑢	NOUN
cana-2746	90	19	)	)	PUNCT
cana-2746	90	20	,	,	PUNCT
cana-2746	90	21	𝐴𝜇(𝑣𝛽𝑠	𝐴𝜇(𝑣𝛽𝑠	NOUN
cana-2746	90	22	)	)	PUNCT
cana-2746	90	23	}	}	PUNCT
cana-2746	90	24	}	}	PUNCT
cana-2746	90	25	=	=	SYM
cana-2746	90	26	𝐴𝜇(𝑥	𝐴𝜇(𝑥	NOUN
cana-2746	90	27	)	)	PUNCT
cana-2746	90	28	now	now	ADV
cana-2746	90	29	𝜒𝑆	𝜒𝑆	VERB
cana-2746	90	30	∘	∘	PROPN
cana-2746	90	31	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	90	32	∘	∘	NOUN
cana-2746	90	33	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	90	34	∩	∩	NOUN
cana-2746	90	35	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	90	36	∘	∘	ADV
cana-2746	90	37	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	90	38	∘	∘	PROPN
cana-2746	90	39	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	90	40	)	)	PUNCT
cana-2746	90	41	=	=	PUNCT
cana-2746	90	42	𝑚𝑖𝑛{𝜒𝑆	𝑚𝑖𝑛{𝜒𝑆	VERB
cana-2746	90	43	∘	∘	NUM
cana-2746	90	44	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	90	45	∘	∘	ADJ
cana-2746	90	46	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	90	47	,	,	PUNCT
cana-2746	90	48	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	90	49	∘	∘	PROPN
cana-2746	90	50	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	90	51	∘	∘	PROPN
cana-2746	90	52	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	90	53	}	}	PUNCT
cana-2746	90	54	≥	≥	NOUN
cana-2746	90	55	𝑚𝑖𝑛{𝜒𝑆	𝑚𝑖𝑛{𝜒𝑆	NOUN
cana-2746	90	56	∘	∘	NUM
cana-2746	90	57	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	90	58	∘	∘	ADJ
cana-2746	90	59	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	90	60	,	,	PUNCT
cana-2746	90	61	𝐴𝜇(𝑥	𝐴𝜇(𝑥	NOUN
cana-2746	90	62	)	)	PUNCT
cana-2746	90	63	}	}	PUNCT
cana-2746	90	64	=	=	SYM
cana-2746	90	65	𝐴𝜇(𝑥	𝐴𝜇(𝑥	NOUN
cana-2746	90	66	)	)	PUNCT
cana-2746	90	67	hence	hence	ADV
cana-2746	90	68	𝜒𝑆	𝜒𝑆	VERB
cana-2746	91	1	∘	∘	PROPN
cana-2746	91	2	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	91	3	∘	∘	NOUN
cana-2746	91	4	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	91	5	∩	∩	NOUN
cana-2746	91	6	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	91	7	∘	∘	ADV
cana-2746	91	8	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	91	9	∘	∘	PROPN
cana-2746	91	10	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	91	11	⊇	⊇	PROPN
cana-2746	91	12	𝐴𝜇.	𝐴𝜇.	INTJ
cana-2746	92	1	next	next	ADV
cana-2746	92	2	we	we	PRON
cana-2746	92	3	have	have	VERB
cana-2746	92	4	to	to	PART
cana-2746	92	5	prove	prove	VERB
cana-2746	92	6	for	for	ADP
cana-2746	92	7	non	non	NOUN
cana-2746	92	8	membership	membership	NOUN
cana-2746	92	9	function	function	NOUN
cana-2746	92	10	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	92	11	∘	∘	PROPN
cana-2746	92	12	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	92	13	)	)	PUNCT
cana-2746	92	14	=	=	SYM
cana-2746	92	15	inf	inf	NOUN
cana-2746	92	16	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	X
cana-2746	92	17	{	{	PUNCT
cana-2746	92	18	𝑚𝑎𝑥{𝜒𝑆(𝑎	𝑚𝑎𝑥{𝜒𝑆(𝑎	NOUN
cana-2746	92	19	)	)	PUNCT
cana-2746	92	20	,	,	PUNCT
cana-2746	92	21	𝐴𝜈(𝑏	𝐴𝜈(𝑏	NOUN
cana-2746	92	22	)	)	PUNCT
cana-2746	92	23	}	}	PUNCT
cana-2746	92	24	}	}	PUNCT
cana-2746	92	25	=	=	SYM
cana-2746	92	26	inf	inf	NOUN
cana-2746	93	1	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	93	2	{	{	PUNCT
cana-2746	93	3	𝑚𝑎𝑥{0	𝑚𝑎𝑥{0	NOUN
cana-2746	93	4	,	,	PUNCT
cana-2746	93	5	𝐴𝜈(𝑏	𝐴𝜈(𝑏	NOUN
cana-2746	93	6	)	)	PUNCT
cana-2746	93	7	}	}	PUNCT
cana-2746	93	8	}	}	PUNCT
cana-2746	93	9	=	=	SYM
cana-2746	93	10	inf	inf	NOUN
cana-2746	93	11	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	93	12	{	{	PUNCT
cana-2746	93	13	𝐴𝜈(𝑏	𝐴𝜈(𝑏	NOUN
cana-2746	93	14	)	)	PUNCT
cana-2746	93	15	}	}	PUNCT
cana-2746	93	16	≤	≤	NUM
cana-2746	93	17	inf	inf	NOUN
cana-2746	93	18	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	93	19	{	{	PUNCT
cana-2746	93	20	𝐴𝜈(𝑎𝑏	𝐴𝜈(𝑎𝑏	NOUN
cana-2746	93	21	)	)	PUNCT
cana-2746	93	22	}	}	PUNCT
cana-2746	93	23	=	=	SYM
cana-2746	93	24	inf	inf	NOUN
cana-2746	93	25	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	X
cana-2746	93	26	{	{	PUNCT
cana-2746	93	27	𝐴𝜈(𝑥	𝐴𝜈(𝑥	NOUN
cana-2746	93	28	)	)	PUNCT
cana-2746	93	29	}	}	PUNCT
cana-2746	93	30	=	=	SYM
cana-2746	93	31	𝐴𝜈(𝑥	𝐴𝜈(𝑥	X
cana-2746	93	32	)	)	PUNCT
cana-2746	94	1	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	94	2	∘	∘	NOUN
cana-2746	94	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	94	4	∘	∘	PROPN
cana-2746	94	5	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	94	6	)	)	PUNCT
cana-2746	94	7	=	=	SYM
cana-2746	94	8	inf	inf	ADJ
cana-2746	94	9	𝑥=𝑢𝛼𝑣𝛽𝑠	𝑥=𝑢𝛼𝑣𝛽𝑠	NOUN
cana-2746	94	10	{	{	PUNCT
cana-2746	94	11	𝑚𝑎𝑥{𝐴𝜈(𝑢	𝑚𝑎𝑥{𝐴𝜈(𝑢	NOUN
cana-2746	94	12	)	)	PUNCT
cana-2746	94	13	,	,	PUNCT
cana-2746	94	14	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	94	15	∘	∘	PROPN
cana-2746	94	16	𝐴𝜈(𝑣𝛽𝑠	𝐴𝜈(𝑣𝛽𝑠	PROPN
cana-2746	94	17	)	)	PUNCT
cana-2746	94	18	}	}	PUNCT
cana-2746	94	19	}	}	PUNCT
cana-2746	94	20	≤	≤	NUM
cana-2746	94	21	inf	inf	ADJ
cana-2746	94	22	𝑥=𝑢𝛼𝑣𝛽𝑠	𝑥=𝑢𝛼𝑣𝛽𝑠	NOUN
cana-2746	94	23	{	{	PUNCT
cana-2746	94	24	𝑚𝑎𝑥{𝐴𝜈(𝑢	𝑚𝑎𝑥{𝐴𝜈(𝑢	NOUN
cana-2746	94	25	)	)	PUNCT
cana-2746	94	26	,	,	PUNCT
cana-2746	94	27	𝐴𝜈(𝑣𝛽𝑠	𝐴𝜈(𝑣𝛽𝑠	PROPN
cana-2746	94	28	)	)	PUNCT
cana-2746	94	29	}	}	PUNCT
cana-2746	94	30	}	}	PUNCT
cana-2746	94	31	=	=	SYM
cana-2746	94	32	𝐴𝜈(𝑥	𝐴𝜈(𝑥	X
cana-2746	94	33	)	)	PUNCT
cana-2746	94	34	now	now	ADV
cana-2746	94	35	𝜒𝑆	𝜒𝑆	VERB
cana-2746	94	36	∘	∘	NOUN
cana-2746	94	37	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	94	38	∘	∘	ADJ
cana-2746	94	39	𝜒𝑆	𝜒𝑆	NOUN
cana-2746	94	40	∩	∩	NOUN
cana-2746	94	41	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	94	42	∘	∘	NOUN
cana-2746	94	43	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	94	44	∘	∘	PROPN
cana-2746	94	45	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	94	46	)	)	PUNCT
cana-2746	94	47	=	=	PUNCT
cana-2746	94	48	𝑚𝑎𝑥{𝜒𝑆	𝑚𝑎𝑥{𝜒𝑆	NOUN
cana-2746	94	49	∘	∘	VERB
cana-2746	95	1	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	95	2	∘	∘	ADJ
cana-2746	95	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	95	4	,	,	PUNCT
cana-2746	95	5	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	95	6	∘	∘	PROPN
cana-2746	95	7	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	95	8	∘	∘	NUM
cana-2746	95	9	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	95	10	}	}	PUNCT
cana-2746	95	11	≤	≤	ADJ
cana-2746	95	12	𝑚𝑎𝑥{𝜒𝑆	𝑚𝑎𝑥{𝜒𝑆	NOUN
cana-2746	95	13	∘	∘	NOUN
cana-2746	95	14	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	95	15	∘	∘	ADJ
cana-2746	95	16	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	95	17	,	,	PUNCT
cana-2746	95	18	𝐴𝜈(𝑥	𝐴𝜈(𝑥	NOUN
cana-2746	95	19	)	)	PUNCT
cana-2746	95	20	}	}	PUNCT
cana-2746	95	21	=	=	SYM
cana-2746	95	22	𝐴𝜈(𝑥	𝐴𝜈(𝑥	NOUN
cana-2746	95	23	)	)	PUNCT
cana-2746	95	24	hence	hence	ADV
cana-2746	95	25	𝜒𝑆	𝜒𝑆	VERB
cana-2746	95	26	∘	∘	NOUN
cana-2746	95	27	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	95	28	∘	∘	ADJ
cana-2746	95	29	𝜒𝑆	𝜒𝑆	NOUN
cana-2746	95	30	∩	∩	NOUN
cana-2746	95	31	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	95	32	∘	∘	NOUN
cana-2746	95	33	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	95	34	∘	∘	NOUN
cana-2746	95	35	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	95	36	⊆	⊆	NUM
cana-2746	95	37	𝐴𝜈.	𝐴𝜈.	PROPN
cana-2746	95	38	communications	communication	NOUN
cana-2746	95	39	on	on	ADP
cana-2746	95	40	applied	apply	VERB
cana-2746	95	41	nonlinear	nonlinear	ADJ
cana-2746	95	42	analysis	analysis	NOUN
cana-2746	95	43	issn	issn	NOUN
cana-2746	95	44	:	:	PUNCT
cana-2746	95	45	1074	1074	NUM
cana-2746	95	46	-	-	PUNCT
cana-2746	95	47	133x	133x	NUM
cana-2746	95	48	vol	vol	NOUN
cana-2746	95	49	32	32	NUM
cana-2746	95	50	no	no	NOUN
cana-2746	95	51	.	.	PUNCT
cana-2746	96	1	4s	4s	NUM
cana-2746	96	2	(	(	PUNCT
cana-2746	96	3	2025	2025	NUM
cana-2746	96	4	)	)	PUNCT
cana-2746	96	5	155	155	NUM
cana-2746	96	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2746	96	7	theorem	theorem	VERB
cana-2746	96	8	3.3	3.3	NUM
cana-2746	96	9	every	every	DET
cana-2746	96	10	pf	pf	PROPN
cana-2746	96	11	right	right	ADJ
cana-2746	96	12	ideal	ideal	NOUN
cana-2746	96	13	of	of	ADP
cana-2746	96	14	s	s	PROPN
cana-2746	96	15	is	be	AUX
cana-2746	96	16	a	a	DET
cana-2746	96	17	pfbii	pfbii	NOUN
cana-2746	96	18	of	of	ADP
cana-2746	96	19	s.	s.	PROPN
cana-2746	96	20	proof	proof	PROPN
cana-2746	96	21	.	.	PUNCT
cana-2746	97	1	let	let	VERB
cana-2746	97	2	a	a	DET
cana-2746	97	3	be	be	AUX
cana-2746	97	4	a	a	DET
cana-2746	97	5	pf	pf	NOUN
cana-2746	97	6	right	right	ADJ
cana-2746	97	7	ideal	ideal	NOUN
cana-2746	97	8	of	of	ADP
cana-2746	97	9	𝑆	𝑆	PROPN
cana-2746	97	10	and	and	CCONJ
cana-2746	97	11	𝑥	𝑥	PRON
cana-2746	97	12	∈	∈	PROPN
cana-2746	97	13	𝑆,𝛼	𝑆,𝛼	NOUN
cana-2746	97	14	,	,	PUNCT
cana-2746	97	15	𝛽	𝛽	PROPN
cana-2746	97	16	∈	∈	PROPN
cana-2746	97	17	γ	γ	X
cana-2746	97	18	.	.	PROPN
cana-2746	97	19	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	97	20	)	)	PUNCT
cana-2746	97	21	∘	∘	PROPN
cana-2746	97	22	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	97	23	=	=	SYM
cana-2746	97	24	sup	sup	NOUN
cana-2746	97	25	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	97	26	{	{	PUNCT
cana-2746	97	27	𝑚𝑖𝑛{𝐴𝜇(𝑎	𝑚𝑖𝑛{𝐴𝜇(𝑎	PROPN
cana-2746	97	28	)	)	PUNCT
cana-2746	97	29	,	,	PUNCT
cana-2746	97	30	𝜒𝑆(𝑏	𝜒𝑆(𝑏	NUM
cana-2746	97	31	)	)	PUNCT
cana-2746	97	32	}	}	PUNCT
cana-2746	97	33	}	}	PUNCT
cana-2746	97	34	=	=	SYM
cana-2746	97	35	sup	sup	NOUN
cana-2746	97	36	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	97	37	{	{	PUNCT
cana-2746	97	38	𝑚𝑖𝑛{𝐴𝜇(𝑎),1	𝑚𝑖𝑛{𝐴𝜇(𝑎),1	NOUN
cana-2746	97	39	}	}	PUNCT
cana-2746	97	40	}	}	PUNCT
cana-2746	97	41	=	=	SYM
cana-2746	97	42	sup	sup	NOUN
cana-2746	97	43	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	97	44	{	{	PUNCT
cana-2746	97	45	𝐴𝜇(𝑎	𝐴𝜇(𝑎	PROPN
cana-2746	97	46	)	)	PUNCT
cana-2746	97	47	}	}	PUNCT
cana-2746	97	48	≤	≤	NUM
cana-2746	97	49	sup	sup	NOUN
cana-2746	97	50	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PUNCT
cana-2746	97	51	{	{	PUNCT
cana-2746	97	52	𝐴𝜇(𝑎𝑏	𝐴𝜇(𝑎𝑏	ADJ
cana-2746	97	53	)	)	PUNCT
cana-2746	97	54	}	}	PUNCT
cana-2746	97	55	=	=	SYM
cana-2746	97	56	sup	sup	NOUN
cana-2746	97	57	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	97	58	{	{	PUNCT
cana-2746	97	59	𝐴𝜇(𝑥	𝐴𝜇(𝑥	NOUN
cana-2746	97	60	)	)	PUNCT
cana-2746	97	61	}	}	PUNCT
cana-2746	97	62	=	=	SYM
cana-2746	97	63	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	97	64	)	)	PUNCT
cana-2746	97	65	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	97	66	∘	∘	PROPN
cana-2746	97	67	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	97	68	∘	∘	PROPN
cana-2746	97	69	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	97	70	)	)	PUNCT
cana-2746	97	71	=	=	NOUN
cana-2746	97	72	sup	sup	NOUN
cana-2746	97	73	𝑥=𝑢𝛼𝑣𝛽𝑠	𝑥=𝑢𝛼𝑣𝛽𝑠	NOUN
cana-2746	97	74	{	{	PUNCT
cana-2746	97	75	𝑚𝑖𝑛{𝐴𝜇	𝑚𝑖𝑛{𝐴𝜇	PROPN
cana-2746	97	76	∘	∘	X
cana-2746	97	77	𝜒𝑆(𝑢𝛼𝑣	𝜒𝑆(𝑢𝛼𝑣	PROPN
cana-2746	97	78	)	)	PUNCT
cana-2746	97	79	,	,	PUNCT
cana-2746	97	80	𝐴𝜇(𝑠	𝐴𝜇(𝑠	NOUN
cana-2746	97	81	)	)	PUNCT
cana-2746	97	82	}	}	PUNCT
cana-2746	97	83	}	}	PUNCT
cana-2746	97	84	≤	≤	NUM
cana-2746	97	85	sup	sup	NOUN
cana-2746	97	86	𝑥=𝑢𝛼𝑣𝛽𝑠	𝑥=𝑢𝛼𝑣𝛽𝑠	NOUN
cana-2746	97	87	{	{	PUNCT
cana-2746	97	88	𝑚𝑖𝑛{𝐴𝜇(𝑢𝛼𝑣	𝑚𝑖𝑛{𝐴𝜇(𝑢𝛼𝑣	PROPN
cana-2746	97	89	)	)	PUNCT
cana-2746	97	90	,	,	PUNCT
cana-2746	97	91	𝐴𝜇(𝑠	𝐴𝜇(𝑠	NOUN
cana-2746	97	92	)	)	PUNCT
cana-2746	97	93	}	}	PUNCT
cana-2746	97	94	}	}	PUNCT
cana-2746	97	95	=	=	SYM
cana-2746	97	96	𝐴𝜇(𝑥	𝐴𝜇(𝑥	NOUN
cana-2746	97	97	)	)	PUNCT
cana-2746	97	98	now	now	ADV
cana-2746	97	99	𝜒𝑆	𝜒𝑆	VERB
cana-2746	97	100	∘	∘	PROPN
cana-2746	98	1	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	98	2	∘	∘	NOUN
cana-2746	98	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	98	4	∩	∩	NOUN
cana-2746	98	5	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	98	6	∘	∘	ADV
cana-2746	98	7	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	98	8	∘	∘	PROPN
cana-2746	98	9	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	98	10	)	)	PUNCT
cana-2746	98	11	=	=	PUNCT
cana-2746	98	12	𝑚𝑖𝑛{𝜒𝑆	𝑚𝑖𝑛{𝜒𝑆	VERB
cana-2746	98	13	∘	∘	NUM
cana-2746	98	14	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	98	15	∘	∘	ADJ
cana-2746	98	16	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	98	17	,	,	PUNCT
cana-2746	98	18	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	98	19	∘	∘	PROPN
cana-2746	98	20	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	98	21	∘	∘	PROPN
cana-2746	98	22	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	98	23	}	}	PUNCT
cana-2746	98	24	≥	≥	NOUN
cana-2746	98	25	𝑚𝑖𝑛{𝜒𝑆	𝑚𝑖𝑛{𝜒𝑆	NOUN
cana-2746	98	26	∘	∘	NUM
cana-2746	98	27	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	98	28	∘	∘	ADJ
cana-2746	98	29	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	98	30	,	,	PUNCT
cana-2746	98	31	𝐴𝜇(𝑥	𝐴𝜇(𝑥	NOUN
cana-2746	98	32	)	)	PUNCT
cana-2746	98	33	}	}	PUNCT
cana-2746	98	34	=	=	SYM
cana-2746	98	35	𝐴𝜇(𝑥	𝐴𝜇(𝑥	NOUN
cana-2746	98	36	)	)	PUNCT
cana-2746	98	37	hence	hence	ADV
cana-2746	98	38	𝜒𝑆	𝜒𝑆	VERB
cana-2746	98	39	∘	∘	PROPN
cana-2746	98	40	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	98	41	∘	∘	NOUN
cana-2746	98	42	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	98	43	∩	∩	NOUN
cana-2746	98	44	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	98	45	∘	∘	ADV
cana-2746	98	46	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	98	47	∘	∘	PROPN
cana-2746	98	48	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	98	49	⊇	⊇	PROPN
cana-2746	99	1	𝐴𝜇.	𝐴𝜇.	INTJ
cana-2746	99	2	next	next	ADV
cana-2746	99	3	we	we	PRON
cana-2746	99	4	have	have	VERB
cana-2746	99	5	to	to	PART
cana-2746	99	6	prove	prove	VERB
cana-2746	99	7	for	for	ADP
cana-2746	99	8	non	non	ADJ
cana-2746	99	9	membership	membership	PROPN
cana-2746	99	10	function	function	PROPN
cana-2746	99	11	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	99	12	)	)	PUNCT
cana-2746	100	1	∘	∘	PROPN
cana-2746	100	2	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	100	3	=	=	SYM
cana-2746	100	4	inf	inf	PROPN
cana-2746	100	5	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	100	6	{	{	PUNCT
cana-2746	100	7	𝑚𝑎𝑥{𝐴𝜈(𝑎	𝑚𝑎𝑥{𝐴𝜈(𝑎	PROPN
cana-2746	100	8	)	)	PUNCT
cana-2746	100	9	,	,	PUNCT
cana-2746	100	10	𝜒𝑆(𝑏	𝜒𝑆(𝑏	NUM
cana-2746	100	11	)	)	PUNCT
cana-2746	100	12	}	}	PUNCT
cana-2746	100	13	}	}	PUNCT
cana-2746	100	14	=	=	SYM
cana-2746	100	15	inf	inf	NOUN
cana-2746	100	16	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	100	17	{	{	PUNCT
cana-2746	100	18	𝑚𝑎𝑥{𝐴𝜈(𝑎),0	𝑚𝑎𝑥{𝐴𝜈(𝑎),0	ADV
cana-2746	100	19	}	}	PUNCT
cana-2746	100	20	}	}	PUNCT
cana-2746	100	21	=	=	SYM
cana-2746	100	22	inf	inf	NOUN
cana-2746	100	23	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	100	24	{	{	PUNCT
cana-2746	100	25	𝐴𝜈(𝑎	𝐴𝜈(𝑎	PROPN
cana-2746	100	26	)	)	PUNCT
cana-2746	100	27	}	}	PUNCT
cana-2746	100	28	≤	≤	NUM
cana-2746	100	29	inf	inf	NOUN
cana-2746	100	30	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	100	31	{	{	PUNCT
cana-2746	100	32	𝐴𝜈(𝑎𝑏	𝐴𝜈(𝑎𝑏	NOUN
cana-2746	100	33	)	)	PUNCT
cana-2746	100	34	}	}	PUNCT
cana-2746	100	35	=	=	SYM
cana-2746	100	36	inf	inf	NOUN
cana-2746	100	37	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	X
cana-2746	100	38	{	{	PUNCT
cana-2746	100	39	𝐴𝜈(𝑥	𝐴𝜈(𝑥	NOUN
cana-2746	100	40	)	)	PUNCT
cana-2746	100	41	}	}	PUNCT
cana-2746	100	42	=	=	SYM
cana-2746	100	43	𝐴𝜈(𝑥	𝐴𝜈(𝑥	X
cana-2746	100	44	)	)	PUNCT
cana-2746	101	1	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	101	2	∘	∘	NOUN
cana-2746	101	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	101	4	∘	∘	PROPN
cana-2746	101	5	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	101	6	)	)	PUNCT
cana-2746	101	7	=	=	SYM
cana-2746	101	8	inf	inf	ADJ
cana-2746	101	9	𝑥=𝑢𝛼𝑣𝛽𝑠	𝑥=𝑢𝛼𝑣𝛽𝑠	NOUN
cana-2746	101	10	{	{	PUNCT
cana-2746	101	11	𝑚𝑎𝑥{𝐴𝜈	𝑚𝑎𝑥{𝐴𝜈	NOUN
cana-2746	101	12	∘	∘	X
cana-2746	101	13	𝜒𝑆(𝑢𝛼𝑣	𝜒𝑆(𝑢𝛼𝑣	PROPN
cana-2746	101	14	)	)	PUNCT
cana-2746	101	15	,	,	PUNCT
cana-2746	101	16	𝐴𝜈(𝑠	𝐴𝜈(𝑠	PROPN
cana-2746	101	17	)	)	PUNCT
cana-2746	101	18	}	}	PUNCT
cana-2746	101	19	}	}	PUNCT
cana-2746	101	20	≤	≤	NUM
cana-2746	101	21	inf	inf	ADJ
cana-2746	101	22	𝑥=𝑢𝛼𝑣𝛽𝑠	𝑥=𝑢𝛼𝑣𝛽𝑠	NOUN
cana-2746	101	23	{	{	PUNCT
cana-2746	101	24	𝑚𝑎𝑥{𝐴𝜈(𝑢𝛼𝑣	𝑚𝑎𝑥{𝐴𝜈(𝑢𝛼𝑣	NOUN
cana-2746	101	25	)	)	PUNCT
cana-2746	101	26	,	,	PUNCT
cana-2746	101	27	𝐴𝜈(𝑠	𝐴𝜈(𝑠	PROPN
cana-2746	101	28	)	)	PUNCT
cana-2746	101	29	}	}	PUNCT
cana-2746	101	30	}	}	PUNCT
cana-2746	101	31	communications	communication	NOUN
cana-2746	101	32	on	on	ADP
cana-2746	101	33	applied	apply	VERB
cana-2746	101	34	nonlinear	nonlinear	ADJ
cana-2746	101	35	analysis	analysis	NOUN
cana-2746	101	36	issn	issn	NOUN
cana-2746	101	37	:	:	PUNCT
cana-2746	101	38	1074	1074	NUM
cana-2746	101	39	-	-	PUNCT
cana-2746	101	40	133x	133x	NUM
cana-2746	101	41	vol	vol	NOUN
cana-2746	101	42	32	32	NUM
cana-2746	101	43	no	no	NOUN
cana-2746	101	44	.	.	PUNCT
cana-2746	102	1	4s	4s	NUM
cana-2746	102	2	(	(	PUNCT
cana-2746	102	3	2025	2025	NUM
cana-2746	102	4	)	)	PUNCT
cana-2746	102	5	156	156	NUM
cana-2746	102	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2746	102	7	=	=	SYM
cana-2746	102	8	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	102	9	)	)	PUNCT
cana-2746	102	10	now	now	ADV
cana-2746	102	11	𝜒𝑆	𝜒𝑆	VERB
cana-2746	102	12	∘	∘	NOUN
cana-2746	102	13	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	102	14	∘	∘	ADJ
cana-2746	102	15	𝜒𝑆	𝜒𝑆	NOUN
cana-2746	102	16	∩	∩	NOUN
cana-2746	102	17	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	102	18	∘	∘	NOUN
cana-2746	102	19	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	102	20	∘	∘	PROPN
cana-2746	102	21	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	102	22	)	)	PUNCT
cana-2746	102	23	=	=	PUNCT
cana-2746	102	24	𝑚𝑎𝑥{𝜒𝑆	𝑚𝑎𝑥{𝜒𝑆	NOUN
cana-2746	102	25	∘	∘	VERB
cana-2746	103	1	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	103	2	∘	∘	ADJ
cana-2746	103	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	103	4	,	,	PUNCT
cana-2746	103	5	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	103	6	∘	∘	PROPN
cana-2746	103	7	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	103	8	∘	∘	NUM
cana-2746	103	9	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	103	10	}	}	PUNCT
cana-2746	103	11	≤	≤	ADJ
cana-2746	103	12	𝑚𝑎𝑥{𝜒𝑆	𝑚𝑎𝑥{𝜒𝑆	NOUN
cana-2746	103	13	∘	∘	NOUN
cana-2746	103	14	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	103	15	∘	∘	ADJ
cana-2746	103	16	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	103	17	,	,	PUNCT
cana-2746	103	18	𝐴𝜈(𝑥	𝐴𝜈(𝑥	NOUN
cana-2746	103	19	)	)	PUNCT
cana-2746	103	20	}	}	PUNCT
cana-2746	103	21	=	=	SYM
cana-2746	103	22	𝐴𝜈(𝑥	𝐴𝜈(𝑥	NOUN
cana-2746	103	23	)	)	PUNCT
cana-2746	103	24	hence	hence	ADV
cana-2746	103	25	𝜒𝑆	𝜒𝑆	VERB
cana-2746	103	26	∘	∘	NOUN
cana-2746	103	27	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	103	28	∘	∘	ADJ
cana-2746	103	29	𝜒𝑆	𝜒𝑆	NOUN
cana-2746	103	30	∩	∩	NOUN
cana-2746	103	31	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	103	32	∘	∘	NOUN
cana-2746	103	33	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	103	34	∘	∘	NOUN
cana-2746	103	35	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	103	36	⊆	⊆	NUM
cana-2746	103	37	𝐴𝜈.	𝐴𝜈.	PROPN
cana-2746	103	38	corollary	corollary	NOUN
cana-2746	103	39	3.4	3.4	NUM
cana-2746	103	40	every	every	DET
cana-2746	103	41	pfi	pfi	NOUN
cana-2746	103	42	is	be	AUX
cana-2746	103	43	pfbii	pfbii	NOUN
cana-2746	103	44	of	of	ADP
cana-2746	103	45	s.	s.	PROPN
cana-2746	103	46	proof	proof	PROPN
cana-2746	103	47	.	.	PUNCT
cana-2746	104	1	by	by	ADP
cana-2746	104	2	theorem	theorem	ADJ
cana-2746	104	3	3.2	3.2	NUM
cana-2746	104	4	and	and	CCONJ
cana-2746	104	5	3.4	3.4	NUM
cana-2746	104	6	proof	proof	NOUN
cana-2746	104	7	is	be	AUX
cana-2746	104	8	obvious	obvious	ADJ
cana-2746	104	9	.	.	PUNCT
cana-2746	105	1	theorem	theorem	VERB
cana-2746	105	2	3.5	3.5	NUM
cana-2746	105	3	let	let	VERB
cana-2746	105	4	𝐵	𝐵	PRON
cana-2746	105	5	be	be	AUX
cana-2746	105	6	a	a	DET
cana-2746	105	7	nonempty	nonempty	ADJ
cana-2746	105	8	subset	subset	NOUN
cana-2746	105	9	of	of	ADP
cana-2746	105	10	𝑆.	𝑆.	PROPN
cana-2746	105	11	then	then	ADV
cana-2746	105	12	𝐵	𝐵	NOUN
cana-2746	105	13	is	be	AUX
cana-2746	105	14	a	a	DET
cana-2746	105	15	bi	bi	ADJ
cana-2746	105	16	-	-	ADJ
cana-2746	105	17	interior	interior	ADJ
cana-2746	105	18	-	-	PUNCT
cana-2746	105	19	ideal	ideal	NOUN
cana-2746	105	20	of	of	ADP
cana-2746	105	21	𝑆	𝑆	PROPN
cana-2746	105	22	⟺	⟺	PROPN
cana-2746	105	23	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	105	24	is	be	AUX
cana-2746	105	25	an	an	DET
cana-2746	105	26	pfbii	pfbii	NOUN
cana-2746	105	27	of	of	ADP
cana-2746	105	28	s.	s.	PROPN
cana-2746	105	29	proof	proof	PROPN
cana-2746	105	30	.	.	PUNCT
cana-2746	106	1	assume	assume	VERB
cana-2746	106	2	that	that	SCONJ
cana-2746	106	3	𝐵	𝐵	NOUN
cana-2746	106	4	is	be	AUX
cana-2746	106	5	a	a	DET
cana-2746	106	6	bi	bi	ADJ
cana-2746	106	7	-	-	ADJ
cana-2746	106	8	interior	interior	ADJ
cana-2746	106	9	-	-	PUNCT
cana-2746	106	10	ideal	ideal	NOUN
cana-2746	106	11	of	of	ADP
cana-2746	106	12	𝑆.	𝑆.	PROPN
cana-2746	106	13	then	then	ADV
cana-2746	106	14	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	106	15	is	be	AUX
cana-2746	106	16	an	an	DET
cana-2746	106	17	𝑃𝐹	𝑃𝐹	NOUN
cana-2746	106	18	sub	sub	NOUN
cana-2746	106	19	-	-	ADJ
cana-2746	106	20	γ	γ	ADJ
cana-2746	106	21	semiring	semiring	NOUN
cana-2746	106	22	of	of	ADP
cana-2746	106	23	𝑆.	𝑆.	PROPN
cana-2746	106	24	by	by	ADP
cana-2746	106	25	hypothesis	hypothesis	NOUN
cana-2746	106	26	we	we	PRON
cana-2746	106	27	’ve	’ve	VERB
cana-2746	106	28	𝑆γ𝐵γ𝑆	𝑆γ𝐵γ𝑆	PROPN
cana-2746	106	29	∩	∩	NOUN
cana-2746	106	30	𝐵γ𝑆γ𝐵	𝐵γ𝑆γ𝐵	VERB
cana-2746	106	31	⊆	⊆	NUM
cana-2746	106	32	𝐵.	𝐵.	PROPN
cana-2746	106	33	then	then	ADV
cana-2746	106	34	,	,	PUNCT
cana-2746	107	1	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	107	2	∘	∘	ADJ
cana-2746	107	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	107	4	∘	∘	PROPN
cana-2746	107	5	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	107	6	∩	∩	ADJ
cana-2746	107	7	𝜒𝐵	𝜒𝐵	X
cana-2746	107	8	∘	∘	ADJ
cana-2746	107	9	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	107	10	∘	∘	PROPN
cana-2746	107	11	𝜒𝐵	𝜒𝐵	X
cana-2746	107	12	=	=	SYM
cana-2746	107	13	𝜒𝑆γ𝐵γ𝑆	𝜒𝑆γ𝐵γ𝑆	NOUN
cana-2746	107	14	∩	∩	NOUN
cana-2746	107	15	𝜒𝐵γ𝑆γ𝐵	𝜒𝐵γ𝑆γ𝐵	NUM
cana-2746	107	16	=	=	PUNCT
cana-2746	107	17	𝜒𝑆γ𝐵γ𝑆∩𝐵γ𝑆γ𝐵	𝜒𝑆γ𝐵γ𝑆∩𝐵γ𝑆γ𝐵	ADJ
cana-2746	107	18	⊆	⊆	NUM
cana-2746	107	19	𝜒𝐵	𝜒𝐵	NOUN
cana-2746	107	20	hence	hence	ADV
cana-2746	107	21	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	107	22	is	be	AUX
cana-2746	107	23	a	a	DET
cana-2746	107	24	𝑃𝐹𝐵𝐼𝐼	𝑃𝐹𝐵𝐼𝐼	NOUN
cana-2746	107	25	of	of	ADP
cana-2746	107	26	𝑆.	𝑆.	NOUN
cana-2746	107	27	conversely	conversely	ADV
cana-2746	107	28	,	,	PUNCT
cana-2746	107	29	let	let	VERB
cana-2746	107	30	us	we	PRON
cana-2746	107	31	assume	assume	VERB
cana-2746	107	32	that	that	SCONJ
cana-2746	107	33	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	107	34	is	be	AUX
cana-2746	107	35	a	a	DET
cana-2746	107	36	𝑃𝐹𝐵𝐼𝐼	𝑃𝐹𝐵𝐼𝐼	NOUN
cana-2746	107	37	of	of	ADP
cana-2746	107	38	𝑆.	𝑆.	PROPN
cana-2746	107	39	then	then	ADV
cana-2746	107	40	𝐵	𝐵	NOUN
cana-2746	107	41	is	be	AUX
cana-2746	107	42	a	a	DET
cana-2746	107	43	sub	sub	NOUN
cana-2746	107	44	-	-	ADJ
cana-2746	107	45	γ	γ	ADJ
cana-2746	107	46	semiring	semiring	NOUN
cana-2746	107	47	of	of	ADP
cana-2746	107	48	𝑆.	𝑆.	PROPN
cana-2746	107	49	we	we	PRON
cana-2746	107	50	have	have	VERB
cana-2746	107	51	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	107	52	∘	∘	PROPN
cana-2746	107	53	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	107	54	∘	∘	PROPN
cana-2746	107	55	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	107	56	∩	∩	ADJ
cana-2746	107	57	𝜒𝐵	𝜒𝐵	NOUN
cana-2746	107	58	∘	∘	ADJ
cana-2746	107	59	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	107	60	∘	∘	PROPN
cana-2746	107	61	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	108	1	⊆	⊆	NUM
cana-2746	108	2	𝜒𝐵	𝜒𝐵	NOUN
cana-2746	108	3	𝜒𝑆γ𝐵γ𝑆	𝜒𝑆γ𝐵γ𝑆	ADJ
cana-2746	108	4	∩	∩	NOUN
cana-2746	108	5	𝜒𝐵γ𝑆γ𝐵	𝜒𝐵γ𝑆γ𝐵	NUM
cana-2746	108	6	⊆	⊆	NUM
cana-2746	108	7	𝜒𝐵	𝜒𝐵	NOUN
cana-2746	108	8	𝜒𝑆𝐵𝑆∩𝐵𝑆𝐵	𝜒𝑆𝐵𝑆∩𝐵𝑆𝐵	ADJ
cana-2746	108	9	⊆	⊆	NUM
cana-2746	108	10	𝜒𝐵	𝜒𝐵	NOUN
cana-2746	108	11	𝑆𝐵𝑆	𝑆𝐵𝑆	PROPN
cana-2746	108	12	∩	∩	ADJ
cana-2746	108	13	𝐵𝑆𝐵	𝐵𝑆𝐵	PROPN
cana-2746	108	14	⊆	⊆	NUM
cana-2746	108	15	𝐵	𝐵	NOUN
cana-2746	108	16	hence	hence	ADV
cana-2746	108	17	𝐵	𝐵	PROPN
cana-2746	108	18	is	be	AUX
cana-2746	108	19	a	a	DET
cana-2746	108	20	bi	bi	ADJ
cana-2746	108	21	-	-	ADJ
cana-2746	108	22	interior	interior	ADJ
cana-2746	108	23	-	-	PUNCT
cana-2746	108	24	ideal	ideal	NOUN
cana-2746	108	25	of	of	ADP
cana-2746	108	26	𝑆.	𝑆.	PROPN
cana-2746	108	27	theorem	theorem	ADJ
cana-2746	108	28	3.6	3.6	NUM
cana-2746	108	29	let	let	VERB
cana-2746	108	30	b	b	NOUN
cana-2746	108	31	be	be	AUX
cana-2746	108	32	a	a	DET
cana-2746	108	33	nonempty	nonempty	ADJ
cana-2746	108	34	subset	subset	NOUN
cana-2746	108	35	of	of	ADP
cana-2746	108	36	𝑆.	𝑆.	PROPN
cana-2746	108	37	then	then	ADV
cana-2746	108	38	𝐵	𝐵	NOUN
cana-2746	108	39	is	be	AUX
cana-2746	108	40	a	a	DET
cana-2746	108	41	pfbii	pfbii	NOUN
cana-2746	108	42	of	of	ADP
cana-2746	108	43	𝑆	𝑆	PROPN
cana-2746	108	44	⟺	⟺	PROPN
cana-2746	108	45	the	the	DET
cana-2746	108	46	nonempty	nonempty	ADJ
cana-2746	108	47	level	level	NOUN
cana-2746	108	48	subset	subset	NOUN
cana-2746	108	49	of	of	ADP
cana-2746	108	50	𝐵	𝐵	NOUN
cana-2746	108	51	is	be	AUX
cana-2746	108	52	a	a	DET
cana-2746	108	53	bi	bi	ADJ
cana-2746	108	54	-	-	ADJ
cana-2746	108	55	interior	interior	ADJ
cana-2746	108	56	-	-	PUNCT
cana-2746	108	57	ideal	ideal	NOUN
cana-2746	108	58	of	of	ADP
cana-2746	108	59	𝑆	𝑆	PROPN
cana-2746	108	60	for	for	ADP
cana-2746	108	61	every	every	DET
cana-2746	108	62	𝑡	𝑡	PROPN
cana-2746	108	63	∈	∈	PROPN
cana-2746	109	1	[	[	X
cana-2746	109	2	0,1	0,1	NUM
cana-2746	109	3	]	]	PUNCT
cana-2746	109	4	.	.	PUNCT
cana-2746	110	1	proof	proof	NOUN
cana-2746	110	2	.	.	PUNCT
cana-2746	111	1	assume	assume	VERB
cana-2746	111	2	that	that	SCONJ
cana-2746	111	3	𝐵	𝐵	NOUN
cana-2746	111	4	is	be	AUX
cana-2746	111	5	a	a	DET
cana-2746	111	6	𝑃𝐹𝐵𝐼𝐼	𝑃𝐹𝐵𝐼𝐼	NOUN
cana-2746	111	7	of	of	ADP
cana-2746	111	8	𝑆.	𝑆.	PROPN
cana-2746	111	9	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	111	10	≠	≠	PROPN
cana-2746	111	11	𝜙,𝑡	𝜙,𝑡	VERB
cana-2746	111	12	∈	∈	PROPN
cana-2746	111	13	[	[	X
cana-2746	111	14	0,1	0,1	NUM
cana-2746	111	15	]	]	PUNCT
cana-2746	111	16	and	and	CCONJ
cana-2746	111	17	𝑎	𝑎	NOUN
cana-2746	111	18	,	,	PUNCT
cana-2746	111	19	𝑏	𝑏	PROPN
cana-2746	111	20	∈	∈	PROPN
cana-2746	111	21	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	111	22	then	then	ADV
cana-2746	111	23	,	,	PUNCT
cana-2746	111	24	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	111	25	(	(	PUNCT
cana-2746	111	26	𝑎	𝑎	NOUN
cana-2746	111	27	)	)	PUNCT
cana-2746	111	28	≥	≥	NUM
cana-2746	111	29	𝑡	𝑡	PROPN
cana-2746	111	30	,	,	PUNCT
cana-2746	111	31	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	111	32	(	(	PUNCT
cana-2746	111	33	𝑏	𝑏	NOUN
cana-2746	111	34	)	)	PUNCT
cana-2746	111	35	≥	≥	NOUN
cana-2746	111	36	𝑡	𝑡	PROPN
cana-2746	111	37	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	111	38	(	(	PUNCT
cana-2746	111	39	𝑎	𝑎	PROPN
cana-2746	111	40	+	+	NOUN
cana-2746	111	41	𝑏	𝑏	NOUN
cana-2746	111	42	)	)	PUNCT
cana-2746	111	43	≥	≥	NOUN
cana-2746	111	44	𝑚𝑖𝑛{𝐵𝜇𝑡	𝑚𝑖𝑛{𝐵𝜇𝑡	PROPN
cana-2746	111	45	(	(	PUNCT
cana-2746	111	46	𝑎	𝑎	NOUN
cana-2746	111	47	)	)	PUNCT
cana-2746	111	48	,	,	PUNCT
cana-2746	111	49	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	111	50	(	(	PUNCT
cana-2746	111	51	𝑏	𝑏	NOUN
cana-2746	111	52	)	)	PUNCT
cana-2746	111	53	}	}	PUNCT
cana-2746	111	54	≥	≥	X
cana-2746	111	55	𝑡	𝑡	X
cana-2746	111	56	𝑎	𝑎	X
cana-2746	111	57	+	+	X
cana-2746	111	58	𝑏	𝑏	PROPN
cana-2746	111	59	∈	∈	PROPN
cana-2746	111	60	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	111	61	.	.	PUNCT
cana-2746	112	1	let	let	VERB
cana-2746	112	2	𝑥	𝑥	PRON
cana-2746	112	3	∈	∈	PROPN
cana-2746	112	4	𝑆γ𝐵𝜇𝑡	𝑆γ𝐵𝜇𝑡	PROPN
cana-2746	112	5	γ𝑆	γ𝑆	PROPN
cana-2746	112	6	∩	∩	NOUN
cana-2746	112	7	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	112	8	γ𝑆γ𝐵𝜇𝑡	γ𝑆γ𝐵𝜇𝑡	PROPN
cana-2746	112	9	.	.	PUNCT
cana-2746	113	1	then	then	ADV
cana-2746	113	2	𝑥	𝑥	VERB
cana-2746	113	3	=	=	SYM
cana-2746	113	4	𝑏𝛼𝑎𝛽𝑢	𝑏𝛼𝑎𝛽𝑢	NOUN
cana-2746	113	5	=	=	SYM
cana-2746	113	6	𝑐𝛾𝑑𝛿𝑒	𝑐𝛾𝑑𝛿𝑒	PROPN
cana-2746	113	7	,	,	PUNCT
cana-2746	113	8	𝑏	𝑏	NOUN
cana-2746	113	9	,	,	PUNCT
cana-2746	113	10	𝑢	𝑢	PROPN
cana-2746	113	11	,	,	PUNCT
cana-2746	113	12	𝑑	𝑑	PROPN
cana-2746	113	13	∈	∈	PROPN
cana-2746	113	14	𝑆	𝑆	PROPN
cana-2746	113	15	and	and	CCONJ
cana-2746	113	16	𝑎	𝑎	PROPN
cana-2746	113	17	,	,	PUNCT
cana-2746	113	18	𝑐	𝑐	PROPN
cana-2746	113	19	,	,	PUNCT
cana-2746	113	20	𝑒	𝑒	PROPN
cana-2746	113	21	∈	∈	PROPN
cana-2746	113	22	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	113	23	,	,	PUNCT
cana-2746	113	24	𝛼	𝛼	PROPN
cana-2746	113	25	,	,	PUNCT
cana-2746	113	26	𝛽	𝛽	NOUN
cana-2746	113	27	,	,	PUNCT
cana-2746	113	28	𝛾	𝛾	PROPN
cana-2746	113	29	,	,	PUNCT
cana-2746	113	30	𝛿	𝛿	PRON
cana-2746	113	31	∈	∈	PROPN
cana-2746	113	32	γ	γ	X
cana-2746	113	33	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	113	34	∘	∘	NOUN
cana-2746	113	35	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	113	36	∘	∘	PROPN
cana-2746	113	37	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	113	38	≥	≥	NUM
cana-2746	113	39	𝑡	𝑡	PROPN
cana-2746	113	40	communications	communication	NOUN
cana-2746	113	41	on	on	ADP
cana-2746	113	42	applied	apply	VERB
cana-2746	113	43	nonlinear	nonlinear	ADJ
cana-2746	113	44	analysis	analysis	NOUN
cana-2746	113	45	issn	issn	NOUN
cana-2746	113	46	:	:	PUNCT
cana-2746	113	47	1074	1074	NUM
cana-2746	113	48	-	-	PUNCT
cana-2746	113	49	133x	133x	NUM
cana-2746	113	50	vol	vol	NOUN
cana-2746	113	51	32	32	NUM
cana-2746	113	52	no	no	NOUN
cana-2746	113	53	.	.	PUNCT
cana-2746	114	1	4s	4s	NUM
cana-2746	114	2	(	(	PUNCT
cana-2746	114	3	2025	2025	NUM
cana-2746	114	4	)	)	PUNCT
cana-2746	114	5	157	157	NUM
cana-2746	114	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2746	114	7	and	and	CCONJ
cana-2746	114	8	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	114	9	∘	∘	PROPN
cana-2746	115	1	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	115	2	∘	∘	PROPN
cana-2746	115	3	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	115	4	≥	≥	NOUN
cana-2746	115	5	𝑡.	𝑡.	NOUN
cana-2746	115	6	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	115	7	(	(	PUNCT
cana-2746	115	8	𝑥	𝑥	NOUN
cana-2746	115	9	)	)	PUNCT
cana-2746	115	10	≥	≥	NOUN
cana-2746	115	11	𝑡	𝑡	VERB
cana-2746	115	12	hence	hence	ADV
cana-2746	115	13	𝑥	𝑥	PRON
cana-2746	115	14	∈	∈	NOUN
cana-2746	115	15	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	115	16	conversely	conversely	ADV
cana-2746	115	17	,	,	PUNCT
cana-2746	115	18	suppose	suppose	VERB
cana-2746	115	19	that	that	SCONJ
cana-2746	115	20	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	115	21	is	be	AUX
cana-2746	115	22	a	a	DET
cana-2746	115	23	bi	bi	ADJ
cana-2746	115	24	-	-	ADJ
cana-2746	115	25	interior	interior	ADJ
cana-2746	115	26	-	-	PUNCT
cana-2746	115	27	ideal	ideal	NOUN
cana-2746	115	28	of	of	ADP
cana-2746	115	29	𝑆	𝑆	PROPN
cana-2746	115	30	for	for	ADP
cana-2746	115	31	all	all	DET
cana-2746	115	32	𝑡	𝑡	ADP
cana-2746	115	33	∈	∈	PROPN
cana-2746	115	34	[	[	X
cana-2746	115	35	0,1	0,1	NUM
cana-2746	115	36	]	]	PUNCT
cana-2746	115	37	let	let	VERB
cana-2746	115	38	𝑎	𝑎	NOUN
cana-2746	115	39	,	,	PUNCT
cana-2746	115	40	𝑏	𝑏	PROPN
cana-2746	115	41	∈	∈	PROPN
cana-2746	115	42	𝑆	𝑆	PROPN
cana-2746	115	43	,	,	PUNCT
cana-2746	115	44	𝛼	𝛼	PROPN
cana-2746	115	45	∈	∈	PROPN
cana-2746	115	46	γ	γ	X
cana-2746	115	47	,	,	PUNCT
cana-2746	115	48	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	115	49	(	(	PUNCT
cana-2746	115	50	𝑎	𝑎	NOUN
cana-2746	115	51	)	)	PUNCT
cana-2746	115	52	=	=	SYM
cana-2746	115	53	𝑡1	𝑡1	NOUN
cana-2746	115	54	,	,	PUNCT
cana-2746	115	55	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	115	56	(	(	PUNCT
cana-2746	115	57	𝑏	𝑏	NOUN
cana-2746	115	58	)	)	PUNCT
cana-2746	115	59	=	=	SYM
cana-2746	115	60	𝑡2	𝑡2	ADJ
cana-2746	115	61	and	and	CCONJ
cana-2746	115	62	𝑡1	𝑡1	NOUN
cana-2746	115	63	≥	≥	NOUN
cana-2746	115	64	𝑡2	𝑡2	PROPN
cana-2746	115	65	.	.	PUNCT
cana-2746	116	1	then	then	ADV
cana-2746	116	2	𝑎	𝑎	X
cana-2746	116	3	,	,	PUNCT
cana-2746	116	4	𝑏	𝑏	PROPN
cana-2746	116	5	∈	∈	PROPN
cana-2746	116	6	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	116	7	then	then	ADV
cana-2746	116	8	,	,	PUNCT
cana-2746	116	9	𝑎	𝑎	PROPN
cana-2746	116	10	+	+	NOUN
cana-2746	116	11	𝑏	𝑏	DET
cana-2746	116	12	∈	∈	PROPN
cana-2746	116	13	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	116	14	.	.	PUNCT
cana-2746	117	1	therefore	therefore	ADV
cana-2746	117	2	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	117	3	=	=	PROPN
cana-2746	117	4	≥	≥	NOUN
cana-2746	117	5	𝑡2	𝑡2	NOUN
cana-2746	117	6	=	=	SYM
cana-2746	117	7	𝑚𝑖𝑛{𝐵𝜇𝑡	𝑚𝑖𝑛{𝐵𝜇𝑡	PROPN
cana-2746	117	8	(	(	PUNCT
cana-2746	117	9	𝑎	𝑎	NOUN
cana-2746	117	10	)	)	PUNCT
cana-2746	117	11	,	,	PUNCT
cana-2746	117	12	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	117	13	(	(	PUNCT
cana-2746	117	14	𝑏	𝑏	NOUN
cana-2746	117	15	)	)	PUNCT
cana-2746	117	16	}	}	PUNCT
cana-2746	117	17	.	.	PUNCT
cana-2746	118	1	hence	hence	ADV
cana-2746	118	2	we	we	PRON
cana-2746	118	3	have	have	VERB
cana-2746	118	4	,	,	PUNCT
cana-2746	118	5	𝑆γ𝐵𝜇𝑡	𝑆γ𝐵𝜇𝑡	PROPN
cana-2746	118	6	γ𝑆	γ𝑆	PROPN
cana-2746	118	7	∩	∩	NOUN
cana-2746	118	8	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	118	9	γ𝑆γ𝐵𝜇𝑡	γ𝑆γ𝐵𝜇𝑡	PROPN
cana-2746	118	10	⊆	⊆	NUM
cana-2746	118	11	𝐵𝜇𝑡	𝐵𝜇𝑡	PROPN
cana-2746	118	12	.	.	PUNCT
cana-2746	119	1	similarly	similarly	ADV
cana-2746	119	2	we	we	PRON
cana-2746	119	3	prove	prove	VERB
cana-2746	119	4	the	the	DET
cana-2746	119	5	non	non	ADJ
cana-2746	119	6	membership	membership	NOUN
cana-2746	119	7	function	function	NOUN
cana-2746	119	8	.	.	PUNCT
cana-2746	120	1	theorem	theorem	VERB
cana-2746	120	2	3.7	3.7	NUM
cana-2746	120	3	if	if	SCONJ
cana-2746	120	4	𝐴	𝐴	PROPN
cana-2746	120	5	and	and	CCONJ
cana-2746	120	6	b	b	NOUN
cana-2746	120	7	are	be	AUX
cana-2746	120	8	pfbii	pfbii	NOUN
cana-2746	120	9	of	of	ADP
cana-2746	120	10	𝑆	𝑆	PROPN
cana-2746	120	11	then	then	ADV
cana-2746	120	12	𝐴	𝐴	PROPN
cana-2746	120	13	∩	∩	ADJ
cana-2746	120	14	𝐵	𝐵	NOUN
cana-2746	120	15	is	be	AUX
cana-2746	120	16	pfbii	pfbii	NOUN
cana-2746	120	17	of	of	ADP
cana-2746	120	18	𝑆.	𝑆.	ADJ
cana-2746	120	19	proof	proof	NOUN
cana-2746	120	20	.	.	PUNCT
cana-2746	121	1	let	let	VERB
cana-2746	121	2	𝐴	𝐴	PROPN
cana-2746	121	3	and	and	CCONJ
cana-2746	121	4	𝐵	𝐵	PROPN
cana-2746	121	5	are	be	AUX
cana-2746	121	6	𝑃𝐹𝐵𝐼𝐼	𝑃𝐹𝐵𝐼𝐼	NOUN
cana-2746	121	7	of	of	ADP
cana-2746	121	8	𝑆	𝑆	PROPN
cana-2746	121	9	and	and	CCONJ
cana-2746	121	10	𝑥	𝑥	PROPN
cana-2746	121	11	,	,	PUNCT
cana-2746	121	12	𝑦	𝑦	NOUN
cana-2746	121	13	∈	∈	PROPN
cana-2746	121	14	𝑆	𝑆	PROPN
cana-2746	121	15	and	and	CCONJ
cana-2746	121	16	𝛼	𝛼	NOUN
cana-2746	121	17	,	,	PUNCT
cana-2746	121	18	𝛽	𝛽	PROPN
cana-2746	121	19	∈	∈	PROPN
cana-2746	121	20	γ	γ	X
cana-2746	121	21	.	.	PUNCT
cana-2746	122	1	(	(	PUNCT
cana-2746	122	2	𝐴𝜇	𝐴𝜇	NOUN
cana-2746	122	3	∩	∩	ADJ
cana-2746	122	4	𝐵𝜇)(𝑥	𝐵𝜇)(𝑥	PROPN
cana-2746	122	5	+	+	CCONJ
cana-2746	122	6	𝑦	𝑦	NOUN
cana-2746	122	7	)	)	PUNCT
cana-2746	122	8	=	=	SYM
cana-2746	123	1	𝑚𝑖𝑛{𝐴𝜇(𝑥	𝑚𝑖𝑛{𝐴𝜇(𝑥	NOUN
cana-2746	123	2	+	+	CCONJ
cana-2746	123	3	𝑦	𝑦	NOUN
cana-2746	123	4	)	)	PUNCT
cana-2746	123	5	,	,	PUNCT
cana-2746	123	6	𝐵𝜇(𝑥	𝐵𝜇(𝑥	PROPN
cana-2746	123	7	+	+	PROPN
cana-2746	123	8	𝑦	𝑦	NOUN
cana-2746	123	9	)	)	PUNCT
cana-2746	123	10	}	}	PUNCT
cana-2746	123	11	≥	≥	PROPN
cana-2746	123	12	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-2746	123	13	{	{	PUNCT
cana-2746	123	14	𝑚𝑖𝑛{𝐴𝜇(𝑥	𝑚𝑖𝑛{𝐴𝜇(𝑥	NOUN
cana-2746	123	15	)	)	PUNCT
cana-2746	123	16	,	,	PUNCT
cana-2746	123	17	𝐴𝜇(𝑦	𝐴𝜇(𝑦	NOUN
cana-2746	123	18	)	)	PUNCT
cana-2746	123	19	}	}	PUNCT
cana-2746	123	20	,	,	PUNCT
cana-2746	123	21	𝑚𝑖𝑛{𝐵𝜇(𝑥	𝑚𝑖𝑛{𝐵𝜇(𝑥	NOUN
cana-2746	123	22	)	)	PUNCT
cana-2746	123	23	,	,	PUNCT
cana-2746	123	24	𝐵𝜇(𝑦	𝐵𝜇(𝑦	NOUN
cana-2746	123	25	)	)	PUNCT
cana-2746	123	26	}	}	PUNCT
cana-2746	123	27	}	}	PUNCT
cana-2746	123	28	=	=	SYM
cana-2746	123	29	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-2746	123	30	{	{	PUNCT
cana-2746	123	31	𝑚𝑖𝑛{𝐴𝜇(𝑥	𝑚𝑖𝑛{𝐴𝜇(𝑥	NOUN
cana-2746	123	32	)	)	PUNCT
cana-2746	123	33	,	,	PUNCT
cana-2746	123	34	𝐵𝜇(𝑥	𝐵𝜇(𝑥	NOUN
cana-2746	123	35	)	)	PUNCT
cana-2746	123	36	}	}	PUNCT
cana-2746	123	37	,	,	PUNCT
cana-2746	123	38	𝑚𝑖𝑛{𝐴𝜇(𝑦	𝑚𝑖𝑛{𝐴𝜇(𝑦	PROPN
cana-2746	123	39	)	)	PUNCT
cana-2746	123	40	,	,	PUNCT
cana-2746	123	41	𝐵𝜇(𝑦	𝐵𝜇(𝑦	NOUN
cana-2746	123	42	)	)	PUNCT
cana-2746	123	43	}	}	PUNCT
cana-2746	123	44	}	}	PUNCT
cana-2746	123	45	=	=	SYM
cana-2746	123	46	𝑚𝑖𝑛{(𝐴𝜇	𝑚𝑖𝑛{(𝐴𝜇	PROPN
cana-2746	123	47	∩	∩	ADJ
cana-2746	123	48	𝐵𝜇)(𝑥	𝐵𝜇)(𝑥	NOUN
cana-2746	123	49	)	)	PUNCT
cana-2746	123	50	,	,	PUNCT
cana-2746	123	51	(	(	PUNCT
cana-2746	123	52	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	123	53	∩	∩	ADJ
cana-2746	123	54	𝐵𝜇)(𝑦	𝐵𝜇)(𝑦	NOUN
cana-2746	123	55	)	)	PUNCT
cana-2746	123	56	}	}	PUNCT
cana-2746	123	57	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	123	58	∘	∘	NOUN
cana-2746	123	59	(	(	PUNCT
cana-2746	123	60	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	123	61	∩	∩	ADJ
cana-2746	123	62	𝐵𝜇)(𝑥	𝐵𝜇)(𝑥	NOUN
cana-2746	123	63	)	)	PUNCT
cana-2746	123	64	=	=	SYM
cana-2746	123	65	sup	sup	NOUN
cana-2746	123	66	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PUNCT
cana-2746	123	67	{	{	PUNCT
cana-2746	123	68	𝑚𝑖𝑛{𝜒𝑆(𝑎	𝑚𝑖𝑛{𝜒𝑆(𝑎	NOUN
cana-2746	123	69	)	)	PUNCT
cana-2746	123	70	,	,	PUNCT
cana-2746	123	71	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	123	72	∩	∩	ADJ
cana-2746	123	73	𝐵𝜇(𝑏	𝐵𝜇(𝑏	NOUN
cana-2746	123	74	)	)	PUNCT
cana-2746	123	75	}	}	PUNCT
cana-2746	123	76	}	}	PUNCT
cana-2746	123	77	=	=	SYM
cana-2746	123	78	sup	sup	NOUN
cana-2746	123	79	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	123	80	{	{	PUNCT
cana-2746	123	81	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-2746	123	82	{	{	PUNCT
cana-2746	123	83	𝜒𝑆(𝑎	𝜒𝑆(𝑎	NOUN
cana-2746	123	84	)	)	PUNCT
cana-2746	123	85	,	,	PUNCT
cana-2746	123	86	𝑚𝑖𝑛{𝐴𝜇(𝑏	𝑚𝑖𝑛{𝐴𝜇(𝑏	PROPN
cana-2746	123	87	)	)	PUNCT
cana-2746	123	88	,	,	PUNCT
cana-2746	123	89	𝐵𝜇(𝑏	𝐵𝜇(𝑏	NOUN
cana-2746	123	90	)	)	PUNCT
cana-2746	123	91	}	}	PUNCT
cana-2746	123	92	}	}	PUNCT
cana-2746	123	93	}	}	PUNCT
cana-2746	123	94	=	=	SYM
cana-2746	123	95	sup	sup	NOUN
cana-2746	123	96	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	123	97	{	{	PUNCT
cana-2746	123	98	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-2746	123	99	{	{	PUNCT
cana-2746	123	100	𝑚𝑖𝑛{𝜒𝑆(𝑎	𝑚𝑖𝑛{𝜒𝑆(𝑎	NOUN
cana-2746	123	101	)	)	PUNCT
cana-2746	123	102	,	,	PUNCT
cana-2746	123	103	𝐴𝜇(𝑏	𝐴𝜇(𝑏	NOUN
cana-2746	123	104	)	)	PUNCT
cana-2746	123	105	}	}	PUNCT
cana-2746	123	106	,	,	PUNCT
cana-2746	123	107	𝑚𝑖𝑛{𝜒𝑆(𝑎	𝑚𝑖𝑛{𝜒𝑆(𝑎	NOUN
cana-2746	123	108	)	)	PUNCT
cana-2746	123	109	,	,	PUNCT
cana-2746	123	110	𝐵𝜇(𝑏	𝐵𝜇(𝑏	NOUN
cana-2746	123	111	)	)	PUNCT
cana-2746	123	112	}	}	PUNCT
cana-2746	123	113	}	}	PUNCT
cana-2746	123	114	}	}	PUNCT
cana-2746	123	115	=	=	SYM
cana-2746	123	116	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-2746	123	117	{	{	PUNCT
cana-2746	123	118	sup	sup	NOUN
cana-2746	123	119	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	123	120	{	{	PUNCT
cana-2746	123	121	𝑚𝑖𝑛{𝜒𝑆(𝑎	𝑚𝑖𝑛{𝜒𝑆(𝑎	NOUN
cana-2746	123	122	)	)	PUNCT
cana-2746	123	123	,	,	PUNCT
cana-2746	123	124	𝐴𝜇(𝑏	𝐴𝜇(𝑏	NOUN
cana-2746	123	125	)	)	PUNCT
cana-2746	123	126	}	}	PUNCT
cana-2746	123	127	}	}	PUNCT
cana-2746	123	128	,	,	PUNCT
cana-2746	123	129	sup	sup	NOUN
cana-2746	123	130	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PRON
cana-2746	123	131	{	{	PUNCT
cana-2746	123	132	𝑚𝑖𝑛{𝜒𝑆(𝑎	𝑚𝑖𝑛{𝜒𝑆(𝑎	NOUN
cana-2746	123	133	)	)	PUNCT
cana-2746	123	134	,	,	PUNCT
cana-2746	123	135	𝐵𝜇(𝑏	𝐵𝜇(𝑏	NOUN
cana-2746	123	136	)	)	PUNCT
cana-2746	123	137	}	}	PUNCT
cana-2746	123	138	}	}	PUNCT
cana-2746	123	139	}	}	PUNCT
cana-2746	123	140	=	=	PUNCT
cana-2746	123	141	𝑚𝑖𝑛{𝜒𝑆	𝑚𝑖𝑛{𝜒𝑆	VERB
cana-2746	123	142	∘	∘	PROPN
cana-2746	123	143	𝐴𝜇(𝑥	𝐴𝜇(𝑥	NOUN
cana-2746	123	144	)	)	PUNCT
cana-2746	123	145	,	,	PUNCT
cana-2746	123	146	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	123	147	∘	∘	PROPN
cana-2746	123	148	𝐵𝜇(𝑥	𝐵𝜇(𝑥	NOUN
cana-2746	123	149	)	)	PUNCT
cana-2746	123	150	}	}	PUNCT
cana-2746	123	151	=	=	SYM
cana-2746	123	152	(	(	PUNCT
cana-2746	123	153	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	123	154	∘	∘	PROPN
cana-2746	123	155	𝐴𝜇	𝐴𝜇	NOUN
cana-2746	123	156	)	)	PUNCT
cana-2746	123	157	∩	∩	NOUN
cana-2746	123	158	(	(	PUNCT
cana-2746	123	159	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	123	160	∘	∘	NOUN
cana-2746	123	161	𝐵𝜇)(𝑥	𝐵𝜇)(𝑥	NOUN
cana-2746	123	162	)	)	PUNCT
cana-2746	123	163	(	(	PUNCT
cana-2746	123	164	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	123	165	∩	∩	ADJ
cana-2746	123	166	𝐵𝜇	𝐵𝜇	NOUN
cana-2746	123	167	)	)	PUNCT
cana-2746	123	168	∘	∘	PROPN
cana-2746	123	169	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	123	170	∘	∘	PROPN
cana-2746	123	171	(	(	PUNCT
cana-2746	123	172	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	123	173	∩	∩	ADJ
cana-2746	123	174	𝐵𝜇)(𝑥	𝐵𝜇)(𝑥	NOUN
cana-2746	123	175	)	)	PUNCT
cana-2746	123	176	=	=	SYM
cana-2746	123	177	sup	sup	NOUN
cana-2746	123	178	𝑥=𝑎𝛼𝑏𝛽𝑐	𝑥=𝑎𝛼𝑏𝛽𝑐	ADP
cana-2746	123	179	{	{	PUNCT
cana-2746	123	180	𝑚𝑖𝑛{𝐴𝜇	𝑚𝑖𝑛{𝐴𝜇	PROPN
cana-2746	123	181	∩	∩	ADJ
cana-2746	123	182	𝐵𝜇(𝑎	𝐵𝜇(𝑎	NOUN
cana-2746	123	183	)	)	PUNCT
cana-2746	123	184	,	,	PUNCT
cana-2746	123	185	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	123	186	∘	∘	NOUN
cana-2746	123	187	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	123	188	∩	∩	ADJ
cana-2746	123	189	𝐵𝜇(𝑏𝛽𝑐	𝐵𝜇(𝑏𝛽𝑐	NOUN
cana-2746	123	190	)	)	PUNCT
cana-2746	123	191	}	}	PUNCT
cana-2746	123	192	}	}	PUNCT
cana-2746	123	193	=	=	SYM
cana-2746	123	194	sup	sup	NOUN
cana-2746	123	195	𝑥=𝑎𝛼𝑏𝛽𝑐	𝑥=𝑎𝛼𝑏𝛽𝑐	ADP
cana-2746	123	196	{	{	PUNCT
cana-2746	123	197	𝑚𝑖𝑛{(𝐴𝜇	𝑚𝑖𝑛{(𝐴𝜇	PROPN
cana-2746	123	198	∩	∩	ADJ
cana-2746	123	199	𝐵𝜇)(𝑎	𝐵𝜇)(𝑎	NOUN
cana-2746	123	200	)	)	PUNCT
cana-2746	123	201	,	,	PUNCT
cana-2746	123	202	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	123	203	∘	∘	NOUN
cana-2746	123	204	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	123	205	∩	∩	NOUN
cana-2746	123	206	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	123	207	∘	∘	PROPN
cana-2746	123	208	𝐵𝜇(𝑏𝛽𝑐	𝐵𝜇(𝑏𝛽𝑐	NOUN
cana-2746	123	209	)	)	PUNCT
cana-2746	123	210	}	}	PUNCT
cana-2746	123	211	}	}	PUNCT
cana-2746	123	212	=	=	SYM
cana-2746	123	213	sup	sup	NOUN
cana-2746	123	214	𝑥=𝑎𝛼𝑏𝛽𝑐	𝑥=𝑎𝛼𝑏𝛽𝑐	ADP
cana-2746	123	215	{	{	PUNCT
cana-2746	123	216	𝑚𝑖𝑛	𝑚𝑖𝑛	PROPN
cana-2746	123	217	{	{	PUNCT
cana-2746	123	218	𝑚𝑖𝑛{𝐴𝜇(𝑎	𝑚𝑖𝑛{𝐴𝜇(𝑎	PROPN
cana-2746	123	219	)	)	PUNCT
cana-2746	123	220	,	,	PUNCT
cana-2746	123	221	𝐵𝜇(𝑎	𝐵𝜇(𝑎	NOUN
cana-2746	123	222	)	)	PUNCT
cana-2746	123	223	}	}	PUNCT
cana-2746	123	224	,	,	PUNCT
cana-2746	123	225	𝑚𝑖𝑛{(𝜒𝑆	𝑚𝑖𝑛{(𝜒𝑆	PUNCT
cana-2746	123	226	∘	∘	PROPN
cana-2746	123	227	𝐴𝜇)(𝑏𝛽𝑐	𝐴𝜇)(𝑏𝛽𝑐	NOUN
cana-2746	123	228	)	)	PUNCT
cana-2746	123	229	,	,	PUNCT
cana-2746	123	230	(	(	PUNCT
cana-2746	123	231	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	123	232	∘	∘	PROPN
cana-2746	123	233	𝐵𝜇)(𝑏𝛽𝑐	𝐵𝜇)(𝑏𝛽𝑐	NOUN
cana-2746	123	234	)	)	PUNCT
cana-2746	123	235	}	}	PUNCT
cana-2746	123	236	}	}	PUNCT
cana-2746	123	237	}	}	PUNCT
cana-2746	123	238	=	=	PUNCT
cana-2746	123	239	𝑚𝑖𝑛{𝐴𝜇	𝑚𝑖𝑛{𝐴𝜇	PROPN
cana-2746	123	240	∘	∘	X
cana-2746	123	241	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	123	242	∘	∘	PROPN
cana-2746	123	243	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	123	244	)	)	PUNCT
cana-2746	123	245	,	,	PUNCT
cana-2746	123	246	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	123	247	∘	∘	ADJ
cana-2746	123	248	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	123	249	∘	∘	PROPN
cana-2746	123	250	𝐵𝜇(𝑥	𝐵𝜇(𝑥	NOUN
cana-2746	123	251	)	)	PUNCT
cana-2746	123	252	}	}	PUNCT
cana-2746	123	253	therefore	therefore	ADV
cana-2746	123	254	(	(	PUNCT
cana-2746	123	255	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	123	256	∩	∩	ADJ
cana-2746	123	257	𝐵𝜇	𝐵𝜇	NOUN
cana-2746	123	258	)	)	PUNCT
cana-2746	123	259	∘	∘	PROPN
cana-2746	123	260	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	123	261	∘	∘	PROPN
cana-2746	123	262	(	(	PUNCT
cana-2746	123	263	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	123	264	∩	∩	ADJ
cana-2746	123	265	𝐵𝜇	𝐵𝜇	NOUN
cana-2746	123	266	)	)	PUNCT
cana-2746	123	267	=	=	PUNCT
cana-2746	124	1	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	124	2	∘	∘	X
cana-2746	124	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	124	4	∘	∘	NOUN
cana-2746	124	5	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	124	6	∩	∩	PUNCT
cana-2746	124	7	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	124	8	∘	∘	ADJ
cana-2746	124	9	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	124	10	∘	∘	NOUN
cana-2746	124	11	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	124	12	communications	communication	NOUN
cana-2746	124	13	on	on	ADP
cana-2746	124	14	applied	apply	VERB
cana-2746	124	15	nonlinear	nonlinear	ADJ
cana-2746	124	16	analysis	analysis	NOUN
cana-2746	124	17	issn	issn	NOUN
cana-2746	124	18	:	:	PUNCT
cana-2746	124	19	1074	1074	NUM
cana-2746	124	20	-	-	PUNCT
cana-2746	124	21	133x	133x	NUM
cana-2746	124	22	vol	vol	NOUN
cana-2746	124	23	32	32	NUM
cana-2746	124	24	no	no	NOUN
cana-2746	124	25	.	.	PUNCT
cana-2746	125	1	4s	4s	NUM
cana-2746	125	2	(	(	PUNCT
cana-2746	125	3	2025	2025	NUM
cana-2746	125	4	)	)	PUNCT
cana-2746	125	5	158	158	NUM
cana-2746	125	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2746	125	7	similarly	similarly	ADV
cana-2746	125	8	we	we	PRON
cana-2746	125	9	prove	prove	VERB
cana-2746	125	10	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	125	11	∘	∘	NOUN
cana-2746	125	12	𝐴𝜇	𝐴𝜇	NOUN
cana-2746	125	13	∩	∩	PUNCT
cana-2746	126	1	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	126	2	∘	∘	NOUN
cana-2746	126	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	126	4	=	=	PUNCT
cana-2746	126	5	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	126	6	∘	∘	X
cana-2746	126	7	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	126	8	∘	∘	NOUN
cana-2746	126	9	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	126	10	∩	∩	PUNCT
cana-2746	126	11	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	126	12	∘	∘	ADJ
cana-2746	126	13	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	126	14	∘	∘	PROPN
cana-2746	126	15	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	126	16	hence	hence	ADV
cana-2746	126	17	𝜒𝑆	𝜒𝑆	VERB
cana-2746	126	18	∘	∘	NOUN
cana-2746	126	19	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	126	20	∩	∩	PUNCT
cana-2746	126	21	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	126	22	∘	∘	ADJ
cana-2746	126	23	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	126	24	∩	∩	NOUN
cana-2746	126	25	𝐴𝜇	𝐴𝜇	NOUN
cana-2746	126	26	∩	∩	NOUN
cana-2746	126	27	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	126	28	∘	∘	ADJ
cana-2746	126	29	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	126	30	∘	∘	NOUN
cana-2746	126	31	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	126	32	∩	∩	NOUN
cana-2746	126	33	𝐵𝜇	𝐵𝜇	NOUN
cana-2746	126	34	=	=	SYM
cana-2746	126	35	(	(	PUNCT
cana-2746	126	36	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	126	37	∘	∘	PROPN
cana-2746	126	38	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	126	39	∘	∘	ADJ
cana-2746	126	40	𝜒𝑆	𝜒𝑆	NOUN
cana-2746	126	41	)	)	PUNCT
cana-2746	126	42	∩	∩	NOUN
cana-2746	126	43	(	(	PUNCT
cana-2746	126	44	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	126	45	∘	∘	PROPN
cana-2746	126	46	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	126	47	∘	∘	PROPN
cana-2746	126	48	𝐴𝜇	𝐴𝜇	NOUN
cana-2746	126	49	)	)	PUNCT
cana-2746	126	50	∩	∩	NOUN
cana-2746	126	51	(	(	PUNCT
cana-2746	126	52	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	126	53	∘	∘	PROPN
cana-2746	126	54	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	126	55	∘	∘	ADJ
cana-2746	126	56	𝜒𝑆	𝜒𝑆	NOUN
cana-2746	126	57	)	)	PUNCT
cana-2746	126	58	∩	∩	NOUN
cana-2746	126	59	(	(	PUNCT
cana-2746	126	60	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	126	61	∘	∘	ADJ
cana-2746	126	62	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	126	63	∘	∘	PROPN
cana-2746	126	64	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	126	65	)	)	PUNCT
cana-2746	126	66	⊇	⊇	NOUN
cana-2746	126	67	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	126	68	∩	∩	PUNCT
cana-2746	126	69	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	126	70	similarly	similarly	ADV
cana-2746	126	71	we	we	PRON
cana-2746	126	72	can	can	AUX
cana-2746	126	73	prove	prove	VERB
cana-2746	126	74	for	for	ADP
cana-2746	126	75	non	non	NOUN
cana-2746	126	76	membership	membership	NOUN
cana-2746	126	77	(	(	PUNCT
cana-2746	126	78	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	126	79	∩	∩	NOUN
cana-2746	126	80	𝐵𝜈)(𝑥	𝐵𝜈)(𝑥	PROPN
cana-2746	126	81	+	+	PROPN
cana-2746	126	82	𝑦	𝑦	NOUN
cana-2746	126	83	)	)	PUNCT
cana-2746	126	84	=	=	SYM
cana-2746	127	1	𝑚𝑎𝑥{𝐴𝜈(𝑥	𝑚𝑎𝑥{𝐴𝜈(𝑥	PROPN
cana-2746	127	2	+	+	NUM
cana-2746	127	3	𝑦	𝑦	NOUN
cana-2746	127	4	)	)	PUNCT
cana-2746	127	5	,	,	PUNCT
cana-2746	127	6	𝐵𝜈(𝑥	𝐵𝜈(𝑥	PROPN
cana-2746	127	7	+	+	PUNCT
cana-2746	127	8	𝑦	𝑦	X
cana-2746	127	9	)	)	PUNCT
cana-2746	127	10	}	}	PUNCT
cana-2746	127	11	≤	≤	NUM
cana-2746	127	12	𝑚𝑎𝑥{𝑚𝑎𝑥{𝐴𝜈(𝑥	𝑚𝑎𝑥{𝑚𝑎𝑥{𝐴𝜈(𝑥	NOUN
cana-2746	127	13	)	)	PUNCT
cana-2746	127	14	,	,	PUNCT
cana-2746	127	15	𝐴𝜈(𝑦	𝐴𝜈(𝑦	NOUN
cana-2746	127	16	)	)	PUNCT
cana-2746	127	17	}	}	PUNCT
cana-2746	127	18	,	,	PUNCT
cana-2746	127	19	𝑚𝑎𝑥{𝐵𝜈(𝑥	𝑚𝑎𝑥{𝐵𝜈(𝑥	PROPN
cana-2746	127	20	)	)	PUNCT
cana-2746	127	21	,	,	PUNCT
cana-2746	127	22	𝐵𝜈(𝑦	𝐵𝜈(𝑦	PROPN
cana-2746	127	23	)	)	PUNCT
cana-2746	127	24	}	}	PUNCT
cana-2746	127	25	}	}	PUNCT
cana-2746	127	26	=	=	SYM
cana-2746	127	27	𝑚𝑎𝑥{𝑚𝑎𝑥{𝐴𝜈(𝑥	𝑚𝑎𝑥{𝑚𝑎𝑥{𝐴𝜈(𝑥	PROPN
cana-2746	127	28	)	)	PUNCT
cana-2746	127	29	,	,	PUNCT
cana-2746	127	30	𝐵𝜈(𝑥	𝐵𝜈(𝑥	PROPN
cana-2746	127	31	)	)	PUNCT
cana-2746	127	32	}	}	PUNCT
cana-2746	127	33	,	,	PUNCT
cana-2746	127	34	𝑚𝑎𝑥{𝐴𝜈(𝑦	𝑚𝑎𝑥{𝐴𝜈(𝑦	PROPN
cana-2746	127	35	)	)	PUNCT
cana-2746	127	36	,	,	PUNCT
cana-2746	127	37	𝐵𝜈(𝑦	𝐵𝜈(𝑦	PROPN
cana-2746	127	38	)	)	PUNCT
cana-2746	127	39	}	}	PUNCT
cana-2746	127	40	}	}	PUNCT
cana-2746	127	41	=	=	SYM
cana-2746	127	42	𝑚𝑎𝑥{(𝐴𝜈	𝑚𝑎𝑥{(𝐴𝜈	PROPN
cana-2746	127	43	∩	∩	X
cana-2746	127	44	𝐵𝜈)(𝑥	𝐵𝜈)(𝑥	PROPN
cana-2746	127	45	)	)	PUNCT
cana-2746	127	46	,	,	PUNCT
cana-2746	127	47	(	(	PUNCT
cana-2746	127	48	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	127	49	∩	∩	NOUN
cana-2746	127	50	𝐵𝜈)(𝑦	𝐵𝜈)(𝑦	NOUN
cana-2746	127	51	)	)	PUNCT
cana-2746	127	52	}	}	PUNCT
cana-2746	127	53	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	127	54	∘	∘	NOUN
cana-2746	127	55	(	(	PUNCT
cana-2746	127	56	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	127	57	∩	∩	NOUN
cana-2746	127	58	𝐵𝜈)(𝑥	𝐵𝜈)(𝑥	NOUN
cana-2746	127	59	)	)	PUNCT
cana-2746	127	60	=	=	PROPN
cana-2746	127	61	inf	inf	NOUN
cana-2746	127	62	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	X
cana-2746	127	63	{	{	PUNCT
cana-2746	127	64	𝑚𝑎𝑥{𝜒𝑆(𝑎	𝑚𝑎𝑥{𝜒𝑆(𝑎	NUM
cana-2746	127	65	)	)	PUNCT
cana-2746	127	66	,	,	PUNCT
cana-2746	127	67	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	127	68	∩	∩	ADJ
cana-2746	127	69	𝐵𝜇(𝑏	𝐵𝜇(𝑏	NOUN
cana-2746	127	70	)	)	PUNCT
cana-2746	127	71	}	}	PUNCT
cana-2746	127	72	}	}	PUNCT
cana-2746	127	73	=	=	SYM
cana-2746	127	74	inf	inf	NOUN
cana-2746	127	75	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	X
cana-2746	127	76	{	{	PUNCT
cana-2746	127	77	𝑚𝑎𝑥	𝑚𝑎𝑥	X
cana-2746	127	78	{	{	PUNCT
cana-2746	127	79	𝜒𝑆(𝑎	𝜒𝑆(𝑎	NOUN
cana-2746	127	80	)	)	PUNCT
cana-2746	127	81	,	,	PUNCT
cana-2746	127	82	𝑚𝑎𝑥{𝐴𝜈(𝑏	𝑚𝑎𝑥{𝐴𝜈(𝑏	PROPN
cana-2746	127	83	)	)	PUNCT
cana-2746	127	84	,	,	PUNCT
cana-2746	127	85	𝐵𝜇(𝑏	𝐵𝜇(𝑏	NOUN
cana-2746	127	86	)	)	PUNCT
cana-2746	127	87	}	}	PUNCT
cana-2746	127	88	}	}	PUNCT
cana-2746	127	89	}	}	PUNCT
cana-2746	127	90	=	=	SYM
cana-2746	127	91	inf	inf	NOUN
cana-2746	127	92	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	127	93	{	{	PUNCT
cana-2746	127	94	𝑚𝑎𝑥{𝑚𝑎𝑥{𝜒𝑆(𝑎	𝑚𝑎𝑥{𝑚𝑎𝑥{𝜒𝑆(𝑎	NOUN
cana-2746	127	95	)	)	PUNCT
cana-2746	127	96	,	,	PUNCT
cana-2746	127	97	𝐴𝜈(𝑏	𝐴𝜈(𝑏	NOUN
cana-2746	127	98	)	)	PUNCT
cana-2746	127	99	}	}	PUNCT
cana-2746	127	100	,	,	PUNCT
cana-2746	127	101	𝑚𝑎𝑥{𝜒𝑆(𝑎	𝑚𝑎𝑥{𝜒𝑆(𝑎	NUM
cana-2746	127	102	)	)	PUNCT
cana-2746	127	103	,	,	PUNCT
cana-2746	127	104	𝐵𝜈(𝑏	𝐵𝜈(𝑏	NOUN
cana-2746	127	105	)	)	PUNCT
cana-2746	127	106	}	}	PUNCT
cana-2746	127	107	}	}	PUNCT
cana-2746	127	108	}	}	PUNCT
cana-2746	127	109	=	=	PUNCT
cana-2746	127	110	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-2746	127	111	{	{	PUNCT
cana-2746	127	112	inf	inf	NOUN
cana-2746	127	113	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	127	114	{	{	PUNCT
cana-2746	127	115	𝑚𝑎𝑥{𝜒𝑆(𝑎	𝑚𝑎𝑥{𝜒𝑆(𝑎	NOUN
cana-2746	127	116	)	)	PUNCT
cana-2746	127	117	,	,	PUNCT
cana-2746	127	118	𝐴𝜈(𝑏	𝐴𝜈(𝑏	NOUN
cana-2746	127	119	)	)	PUNCT
cana-2746	127	120	}	}	PUNCT
cana-2746	127	121	}	}	PUNCT
cana-2746	127	122	,	,	PUNCT
cana-2746	127	123	inf	inf	PROPN
cana-2746	127	124	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	X
cana-2746	127	125	{	{	PUNCT
cana-2746	127	126	𝑚𝑎𝑥{𝜒𝑆(𝑎	𝑚𝑎𝑥{𝜒𝑆(𝑎	NOUN
cana-2746	127	127	)	)	PUNCT
cana-2746	127	128	,	,	PUNCT
cana-2746	127	129	𝐵𝜈(𝑏	𝐵𝜈(𝑏	NOUN
cana-2746	127	130	)	)	PUNCT
cana-2746	127	131	}	}	PUNCT
cana-2746	127	132	}	}	PUNCT
cana-2746	127	133	}	}	PUNCT
cana-2746	127	134	=	=	SYM
cana-2746	127	135	𝑚𝑎𝑥{𝜒𝑆	𝑚𝑎𝑥{𝜒𝑆	ADJ
cana-2746	127	136	∘	∘	PROPN
cana-2746	127	137	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	127	138	)	)	PUNCT
cana-2746	127	139	,	,	PUNCT
cana-2746	127	140	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	127	141	∘	∘	PROPN
cana-2746	127	142	𝐵𝜈(𝑥	𝐵𝜈(𝑥	PROPN
cana-2746	127	143	)	)	PUNCT
cana-2746	127	144	}	}	PUNCT
cana-2746	127	145	=	=	SYM
cana-2746	127	146	(	(	PUNCT
cana-2746	127	147	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	127	148	∘	∘	NUM
cana-2746	127	149	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	127	150	)	)	PUNCT
cana-2746	127	151	∩	∩	NOUN
cana-2746	127	152	(	(	PUNCT
cana-2746	127	153	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	127	154	∘	∘	NOUN
cana-2746	127	155	𝐵𝜈)(𝑥	𝐵𝜈)(𝑥	ADV
cana-2746	127	156	)	)	PUNCT
cana-2746	127	157	(	(	PUNCT
cana-2746	127	158	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	127	159	∩	∩	NOUN
cana-2746	127	160	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	127	161	)	)	PUNCT
cana-2746	127	162	∘	∘	PROPN
cana-2746	127	163	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	127	164	∘	∘	X
cana-2746	127	165	(	(	PUNCT
cana-2746	127	166	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	127	167	∩	∩	NOUN
cana-2746	127	168	𝐵𝜈)(𝑥	𝐵𝜈)(𝑥	NOUN
cana-2746	127	169	)	)	PUNCT
cana-2746	127	170	=	=	SYM
cana-2746	127	171	inf	inf	PROPN
cana-2746	127	172	𝑥=𝑎𝛼𝑏𝛽𝑐	𝑥=𝑎𝛼𝑏𝛽𝑐	ADP
cana-2746	127	173	{	{	PUNCT
cana-2746	127	174	𝑚𝑎𝑥{𝐴𝜈	𝑚𝑎𝑥{𝐴𝜈	NOUN
cana-2746	127	175	∩	∩	NOUN
cana-2746	127	176	𝐵𝜈(𝑎	𝐵𝜈(𝑎	NOUN
cana-2746	127	177	)	)	PUNCT
cana-2746	127	178	,	,	PUNCT
cana-2746	127	179	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	127	180	∘	∘	NOUN
cana-2746	127	181	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	127	182	∩	∩	ADJ
cana-2746	127	183	𝐵𝜈(𝑏𝛽𝑐	𝐵𝜈(𝑏𝛽𝑐	NOUN
cana-2746	127	184	)	)	PUNCT
cana-2746	127	185	}	}	PUNCT
cana-2746	127	186	}	}	PUNCT
cana-2746	127	187	=	=	SYM
cana-2746	127	188	inf	inf	PROPN
cana-2746	127	189	𝑥=𝑎𝛼𝑏𝛽𝑐	𝑥=𝑎𝛼𝑏𝛽𝑐	ADP
cana-2746	127	190	{	{	PUNCT
cana-2746	127	191	𝑚𝑎𝑥{(𝐴𝜈	𝑚𝑎𝑥{(𝐴𝜈	PROPN
cana-2746	127	192	∩	∩	ADJ
cana-2746	127	193	𝐵𝜈)(𝑎	𝐵𝜈)(𝑎	NOUN
cana-2746	127	194	)	)	PUNCT
cana-2746	127	195	,	,	PUNCT
cana-2746	127	196	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	127	197	∘	∘	ADJ
cana-2746	127	198	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	127	199	∩	∩	NOUN
cana-2746	127	200	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	127	201	∘	∘	PROPN
cana-2746	127	202	𝐵𝜈(𝑏𝛽𝑐	𝐵𝜈(𝑏𝛽𝑐	PROPN
cana-2746	127	203	)	)	PUNCT
cana-2746	127	204	}	}	PUNCT
cana-2746	127	205	}	}	PUNCT
cana-2746	127	206	=	=	SYM
cana-2746	127	207	inf	inf	NOUN
cana-2746	127	208	𝑥=𝑎𝛼𝑏𝛽𝑐	𝑥=𝑎𝛼𝑏𝛽𝑐	ADP
cana-2746	127	209	{	{	PUNCT
cana-2746	127	210	𝑚𝑎𝑥{𝑚𝑎𝑥{𝐴𝜈(𝑎	𝑚𝑎𝑥{𝑚𝑎𝑥{𝐴𝜈(𝑎	NOUN
cana-2746	127	211	)	)	PUNCT
cana-2746	127	212	,	,	PUNCT
cana-2746	127	213	𝐵𝜈(𝑎	𝐵𝜈(𝑎	PROPN
cana-2746	127	214	)	)	PUNCT
cana-2746	127	215	}	}	PUNCT
cana-2746	127	216	,	,	PUNCT
cana-2746	127	217	𝑚𝑖𝑛{(𝜒𝑆	𝑚𝑖𝑛{(𝜒𝑆	PUNCT
cana-2746	127	218	∘	∘	PROPN
cana-2746	127	219	𝐴𝜈)(𝑏𝛽𝑐	𝐴𝜈)(𝑏𝛽𝑐	NOUN
cana-2746	127	220	)	)	PUNCT
cana-2746	127	221	,	,	PUNCT
cana-2746	127	222	(	(	PUNCT
cana-2746	127	223	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	127	224	∘	∘	PROPN
cana-2746	127	225	𝐵𝜈)(𝑏𝛽𝑐	𝐵𝜈)(𝑏𝛽𝑐	NOUN
cana-2746	127	226	)	)	PUNCT
cana-2746	127	227	}	}	PUNCT
cana-2746	127	228	}	}	PUNCT
cana-2746	127	229	}	}	PUNCT
cana-2746	127	230	=	=	PUNCT
cana-2746	128	1	𝑚𝑎𝑥{𝐴𝜈	𝑚𝑎𝑥{𝐴𝜈	PRON
cana-2746	128	2	∘	∘	X
cana-2746	128	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	128	4	∘	∘	PROPN
cana-2746	128	5	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	128	6	)	)	PUNCT
cana-2746	128	7	,	,	PUNCT
cana-2746	128	8	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	128	9	∘	∘	PROPN
cana-2746	128	10	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	128	11	∘	∘	PROPN
cana-2746	128	12	𝐵𝜈(𝑥	𝐵𝜈(𝑥	PROPN
cana-2746	128	13	)	)	PUNCT
cana-2746	128	14	}	}	PUNCT
cana-2746	128	15	therefore	therefore	ADV
cana-2746	128	16	(	(	PUNCT
cana-2746	128	17	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	128	18	∩	∩	NOUN
cana-2746	128	19	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	128	20	)	)	PUNCT
cana-2746	128	21	∘	∘	PROPN
cana-2746	128	22	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	128	23	∘	∘	X
cana-2746	128	24	(	(	PUNCT
cana-2746	128	25	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	128	26	∩	∩	NOUN
cana-2746	128	27	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	128	28	)	)	PUNCT
cana-2746	128	29	=	=	PUNCT
cana-2746	129	1	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	129	2	∘	∘	NOUN
cana-2746	129	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	129	4	∘	∘	NOUN
cana-2746	129	5	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	129	6	∩	∩	NOUN
cana-2746	129	7	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	129	8	∘	∘	PROPN
cana-2746	129	9	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	129	10	∘	∘	NOUN
cana-2746	130	1	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	130	2	similarly	similarly	ADV
cana-2746	130	3	we	we	PRON
cana-2746	130	4	prove	prove	VERB
cana-2746	130	5	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	130	6	∘	∘	NOUN
cana-2746	130	7	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	130	8	∩	∩	NOUN
cana-2746	130	9	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	130	10	∘	∘	ADJ
cana-2746	130	11	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	130	12	=	=	PUNCT
cana-2746	130	13	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	130	14	∘	∘	NOUN
cana-2746	131	1	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	131	2	∘	∘	NOUN
cana-2746	131	3	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	131	4	∩	∩	NOUN
cana-2746	131	5	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	131	6	∘	∘	PROPN
cana-2746	131	7	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	131	8	∘	∘	NOUN
cana-2746	131	9	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	131	10	hence	hence	ADV
cana-2746	131	11	𝜒𝑆	𝜒𝑆	VERB
cana-2746	131	12	∘	∘	NOUN
cana-2746	131	13	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	131	14	∩	∩	ADJ
cana-2746	131	15	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	131	16	∘	∘	ADJ
cana-2746	131	17	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	131	18	∩	∩	ADJ
cana-2746	131	19	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	131	20	∩	∩	NOUN
cana-2746	131	21	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	131	22	∘	∘	PROPN
cana-2746	131	23	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	131	24	∘	∘	NOUN
cana-2746	132	1	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	132	2	∩	∩	ADJ
cana-2746	132	3	𝐵𝜇	𝐵𝜇	NOUN
cana-2746	132	4	=	=	SYM
cana-2746	132	5	(	(	PUNCT
cana-2746	132	6	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	132	7	∘	∘	NOUN
cana-2746	132	8	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	132	9	∘	∘	ADJ
cana-2746	132	10	𝜒𝑆	𝜒𝑆	NOUN
cana-2746	132	11	)	)	PUNCT
cana-2746	132	12	∩	∩	NOUN
cana-2746	132	13	(	(	PUNCT
cana-2746	132	14	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	132	15	∘	∘	PROPN
cana-2746	132	16	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	132	17	∘	∘	NUM
cana-2746	132	18	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	132	19	)	)	PUNCT
cana-2746	132	20	∩	∩	NOUN
cana-2746	132	21	(	(	PUNCT
cana-2746	132	22	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	132	23	∘	∘	PROPN
cana-2746	132	24	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	132	25	∘	∘	ADJ
cana-2746	132	26	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	132	27	)	)	PUNCT
cana-2746	132	28	∩	∩	NOUN
cana-2746	132	29	(	(	PUNCT
cana-2746	132	30	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	132	31	∘	∘	PROPN
cana-2746	132	32	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	132	33	∘	∘	PROPN
cana-2746	132	34	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	132	35	)	)	PUNCT
cana-2746	132	36	⊆	⊆	NUM
cana-2746	132	37	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	132	38	∩	∩	NOUN
cana-2746	132	39	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	132	40	communications	communication	NOUN
cana-2746	132	41	on	on	ADP
cana-2746	132	42	applied	apply	VERB
cana-2746	132	43	nonlinear	nonlinear	ADJ
cana-2746	132	44	analysis	analysis	NOUN
cana-2746	132	45	issn	issn	NOUN
cana-2746	132	46	:	:	PUNCT
cana-2746	132	47	1074	1074	NUM
cana-2746	132	48	-	-	PUNCT
cana-2746	132	49	133x	133x	NUM
cana-2746	132	50	vol	vol	NOUN
cana-2746	132	51	32	32	NUM
cana-2746	132	52	no	no	NOUN
cana-2746	132	53	.	.	PUNCT
cana-2746	133	1	4s	4s	NUM
cana-2746	133	2	(	(	PUNCT
cana-2746	133	3	2025	2025	NUM
cana-2746	133	4	)	)	PUNCT
cana-2746	133	5	159	159	NUM
cana-2746	133	6	https://internationalpubls.com	https://internationalpubls.com	SYM
cana-2746	133	7	4	4	NUM
cana-2746	133	8	pythagorean	pythagorean	NOUN
cana-2746	133	9	fuzzy	fuzzy	ADJ
cana-2746	133	10	bi	bi	ADJ
cana-2746	133	11	-	-	ADJ
cana-2746	133	12	quasi	quasi	NOUN
cana-2746	133	13	-	-	NOUN
cana-2746	133	14	ideals	ideal	NOUN
cana-2746	133	15	in	in	ADP
cana-2746	133	16	𝚪-semiring	𝚪-semire	VERB
cana-2746	133	17	this	this	DET
cana-2746	133	18	section	section	NOUN
cana-2746	133	19	deals	deal	VERB
cana-2746	133	20	with	with	ADP
cana-2746	133	21	the	the	DET
cana-2746	133	22	pythagorean	pythagorean	ADJ
cana-2746	133	23	fuzzy	fuzzy	ADJ
cana-2746	133	24	bi	bi	ADJ
cana-2746	133	25	-	-	ADJ
cana-2746	133	26	interior	interior	ADJ
cana-2746	133	27	-	-	PUNCT
cana-2746	133	28	ideals	ideal	NOUN
cana-2746	133	29	in	in	ADP
cana-2746	133	30	γ	γ	NOUN
cana-2746	133	31	-	-	ADJ
cana-2746	133	32	semiring	semiring	ADJ
cana-2746	133	33	𝑆.	𝑆.	NOUN
cana-2746	133	34	definition	definition	NOUN
cana-2746	133	35	4.1	4.1	NUM
cana-2746	133	36	a	a	DET
cana-2746	133	37	pfs	pfs	ADJ
cana-2746	133	38	𝐴	𝐴	PROPN
cana-2746	133	39	=	=	SYM
cana-2746	133	40	(	(	PUNCT
cana-2746	133	41	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	133	42	,	,	PUNCT
cana-2746	133	43	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	133	44	)	)	PUNCT
cana-2746	133	45	of	of	ADP
cana-2746	133	46	𝑆	𝑆	PROPN
cana-2746	133	47	is	be	AUX
cana-2746	133	48	said	say	VERB
cana-2746	133	49	to	to	PART
cana-2746	133	50	be	be	AUX
cana-2746	133	51	a	a	DET
cana-2746	133	52	pflbqi	pflbqi	NOUN
cana-2746	133	53	of	of	ADP
cana-2746	133	54	𝑆	𝑆	PROPN
cana-2746	133	55	if	if	SCONJ
cana-2746	133	56	the	the	DET
cana-2746	133	57	following	follow	VERB
cana-2746	133	58	conditions	condition	NOUN
cana-2746	133	59	are	be	AUX
cana-2746	133	60	holds	hold	NOUN
cana-2746	133	61	:	:	PUNCT
cana-2746	134	1	1	1	X
cana-2746	134	2	.	.	X
cana-2746	135	1	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	135	2	+	+	CCONJ
cana-2746	135	3	𝑦	𝑦	NOUN
cana-2746	135	4	)	)	PUNCT
cana-2746	135	5	≥	≥	PROPN
cana-2746	135	6	𝑚𝑖𝑛{𝐴𝜇(𝑥	𝑚𝑖𝑛{𝐴𝜇(𝑥	PROPN
cana-2746	135	7	)	)	PUNCT
cana-2746	135	8	,	,	PUNCT
cana-2746	135	9	𝐴𝜇(𝑦	𝐴𝜇(𝑦	NOUN
cana-2746	135	10	)	)	PUNCT
cana-2746	135	11	}	}	PUNCT
cana-2746	136	1	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	137	1	+	+	NUM
cana-2746	137	2	𝑦	𝑦	X
cana-2746	137	3	)	)	PUNCT
cana-2746	137	4	≤	≤	NUM
cana-2746	137	5	𝑚𝑎𝑥{𝐴𝜈(𝑥	𝑚𝑎𝑥{𝐴𝜈(𝑥	PROPN
cana-2746	137	6	)	)	PUNCT
cana-2746	137	7	,	,	PUNCT
cana-2746	137	8	𝐴𝜈(𝑦	𝐴𝜈(𝑦	PROPN
cana-2746	137	9	)	)	PUNCT
cana-2746	137	10	}	}	PUNCT
cana-2746	137	11	2	2	NUM
cana-2746	137	12	.	.	X
cana-2746	138	1	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	138	2	∘	∘	NOUN
cana-2746	138	3	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	138	4	∩	∩	ADJ
cana-2746	138	5	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	138	6	∘	∘	ADV
cana-2746	138	7	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	138	8	∘	∘	PROPN
cana-2746	138	9	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	138	10	⊇	⊇	PROPN
cana-2746	138	11	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	138	12	𝜒𝑆	𝜒𝑆	VERB
cana-2746	138	13	∘	∘	NOUN
cana-2746	138	14	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	138	15	∩	∩	NOUN
cana-2746	138	16	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	138	17	∘	∘	NOUN
cana-2746	138	18	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	138	19	∘	∘	NOUN
cana-2746	138	20	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	138	21	⊆	⊆	NUM
cana-2746	138	22	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	138	23	definition	definition	NOUN
cana-2746	138	24	4.2	4.2	NUM
cana-2746	138	25	a	a	DET
cana-2746	138	26	pfs	pfs	ADJ
cana-2746	138	27	𝐴	𝐴	PROPN
cana-2746	138	28	=	=	SYM
cana-2746	138	29	(	(	PUNCT
cana-2746	138	30	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	138	31	,	,	PUNCT
cana-2746	138	32	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	138	33	)	)	PUNCT
cana-2746	138	34	of	of	ADP
cana-2746	138	35	𝑆	𝑆	PROPN
cana-2746	138	36	is	be	AUX
cana-2746	138	37	said	say	VERB
cana-2746	138	38	to	to	PART
cana-2746	138	39	be	be	AUX
cana-2746	138	40	a	a	DET
cana-2746	138	41	pfrbqi	pfrbqi	NOUN
cana-2746	138	42	of	of	ADP
cana-2746	138	43	𝑆	𝑆	PROPN
cana-2746	138	44	if	if	SCONJ
cana-2746	138	45	the	the	DET
cana-2746	138	46	following	follow	VERB
cana-2746	138	47	conditions	condition	NOUN
cana-2746	138	48	are	be	AUX
cana-2746	138	49	holds	hold	NOUN
cana-2746	138	50	:	:	PUNCT
cana-2746	139	1	1	1	X
cana-2746	139	2	.	.	X
cana-2746	140	1	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	140	2	+	+	CCONJ
cana-2746	140	3	𝑦	𝑦	NOUN
cana-2746	140	4	)	)	PUNCT
cana-2746	140	5	≥	≥	PROPN
cana-2746	140	6	𝑚𝑖𝑛{𝐴𝜇(𝑥	𝑚𝑖𝑛{𝐴𝜇(𝑥	PROPN
cana-2746	140	7	)	)	PUNCT
cana-2746	140	8	,	,	PUNCT
cana-2746	140	9	𝐴𝜇(𝑦	𝐴𝜇(𝑦	NOUN
cana-2746	140	10	)	)	PUNCT
cana-2746	140	11	}	}	PUNCT
cana-2746	141	1	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	142	1	+	+	NUM
cana-2746	142	2	𝑦	𝑦	X
cana-2746	142	3	)	)	PUNCT
cana-2746	142	4	≤	≤	NUM
cana-2746	142	5	𝑚𝑎𝑥{𝐴𝜈(𝑥	𝑚𝑎𝑥{𝐴𝜈(𝑥	PROPN
cana-2746	142	6	)	)	PUNCT
cana-2746	142	7	,	,	PUNCT
cana-2746	142	8	𝐴𝜈(𝑦	𝐴𝜈(𝑦	PROPN
cana-2746	142	9	)	)	PUNCT
cana-2746	142	10	}	}	PUNCT
cana-2746	142	11	2	2	X
cana-2746	142	12	.	.	PUNCT
cana-2746	143	1	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	143	2	∘	∘	X
cana-2746	143	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	143	4	∩	∩	NOUN
cana-2746	143	5	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	143	6	∘	∘	ADV
cana-2746	143	7	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	143	8	∘	∘	PROPN
cana-2746	143	9	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	143	10	⊇	⊇	NOUN
cana-2746	143	11	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	143	12	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	143	13	∘	∘	ADJ
cana-2746	143	14	𝜒𝑆	𝜒𝑆	NOUN
cana-2746	143	15	∩	∩	NOUN
cana-2746	143	16	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	143	17	∘	∘	NOUN
cana-2746	143	18	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	143	19	∘	∘	NOUN
cana-2746	143	20	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	143	21	⊆	⊆	NUM
cana-2746	143	22	𝐴𝜈.	𝐴𝜈.	PROPN
cana-2746	143	23	definition	definition	NOUN
cana-2746	143	24	4.3	4.3	NUM
cana-2746	143	25	a	a	DET
cana-2746	143	26	pfs	pfs	PROPN
cana-2746	143	27	a	a	DET
cana-2746	143	28	=	=	X
cana-2746	143	29	(	(	PUNCT
cana-2746	143	30	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	143	31	,	,	PUNCT
cana-2746	143	32	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	143	33	)	)	PUNCT
cana-2746	143	34	of	of	ADP
cana-2746	143	35	𝑆	𝑆	PROPN
cana-2746	143	36	is	be	AUX
cana-2746	143	37	said	say	VERB
cana-2746	143	38	to	to	PART
cana-2746	143	39	be	be	AUX
cana-2746	143	40	a	a	DET
cana-2746	143	41	pfbqi	pfbqi	NOUN
cana-2746	143	42	of	of	ADP
cana-2746	143	43	𝑆	𝑆	PROPN
cana-2746	143	44	if	if	SCONJ
cana-2746	143	45	it	it	PRON
cana-2746	143	46	is	be	AUX
cana-2746	143	47	both	both	PRON
cana-2746	143	48	pythagorean	pythagorean	ADJ
cana-2746	143	49	fuzzy	fuzzy	ADJ
cana-2746	143	50	left	leave	VERB
cana-2746	143	51	bi	bi	ADJ
cana-2746	143	52	-	-	ADJ
cana-2746	143	53	quasi	quasi	ADJ
cana-2746	143	54	-	-	ADJ
cana-2746	143	55	ideal	ideal	ADJ
cana-2746	143	56	and	and	CCONJ
cana-2746	143	57	right	right	ADJ
cana-2746	143	58	bi	bi	ADJ
cana-2746	143	59	-	-	ADJ
cana-2746	143	60	quasi	quasi	ADJ
cana-2746	143	61	-	-	NOUN
cana-2746	143	62	ideal	ideal	NOUN
cana-2746	143	63	of	of	ADP
cana-2746	143	64	𝑆.	𝑆.	PROPN
cana-2746	143	65	theorem	theorem	VERB
cana-2746	143	66	4.4	4.4	NUM
cana-2746	143	67	every	every	DET
cana-2746	143	68	pf	pf	NOUN
cana-2746	143	69	left	leave	VERB
cana-2746	143	70	ideal	ideal	NOUN
cana-2746	143	71	of	of	ADP
cana-2746	143	72	𝑆	𝑆	PROPN
cana-2746	143	73	is	be	AUX
cana-2746	143	74	a	a	DET
cana-2746	143	75	pflbqi	pflbqi	NOUN
cana-2746	143	76	of	of	ADP
cana-2746	143	77	𝑆.	𝑆.	PROPN
cana-2746	143	78	proof	proof	NOUN
cana-2746	143	79	.	.	PUNCT
cana-2746	144	1	let	let	VERB
cana-2746	144	2	𝐴	𝐴	PROPN
cana-2746	144	3	be	be	AUX
cana-2746	144	4	a	a	DET
cana-2746	144	5	𝑃𝐹	𝑃𝐹	NOUN
cana-2746	144	6	left	leave	VERB
cana-2746	144	7	ideal	ideal	NOUN
cana-2746	144	8	of	of	ADP
cana-2746	144	9	𝑆	𝑆	PROPN
cana-2746	144	10	and	and	CCONJ
cana-2746	144	11	𝑥	𝑥	PRON
cana-2746	144	12	∈	∈	PROPN
cana-2746	144	13	𝑆,𝛼	𝑆,𝛼	NOUN
cana-2746	144	14	,	,	PUNCT
cana-2746	145	1	𝛽	𝛽	PROPN
cana-2746	145	2	∈	∈	PROPN
cana-2746	145	3	γ	γ	X
cana-2746	145	4	.	.	PROPN
cana-2746	145	5	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	145	6	∘	∘	PROPN
cana-2746	145	7	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	145	8	)	)	PUNCT
cana-2746	145	9	=	=	SYM
cana-2746	145	10	sup	sup	NOUN
cana-2746	145	11	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PUNCT
cana-2746	145	12	{	{	PUNCT
cana-2746	145	13	𝑚𝑖𝑛{𝜒𝑆(𝑎	𝑚𝑖𝑛{𝜒𝑆(𝑎	NOUN
cana-2746	145	14	)	)	PUNCT
cana-2746	145	15	,	,	PUNCT
cana-2746	145	16	𝐴𝜇(𝑏	𝐴𝜇(𝑏	NOUN
cana-2746	145	17	)	)	PUNCT
cana-2746	145	18	}	}	PUNCT
cana-2746	145	19	}	}	PUNCT
cana-2746	145	20	=	=	SYM
cana-2746	145	21	sup	sup	NOUN
cana-2746	145	22	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	145	23	{	{	PUNCT
cana-2746	145	24	𝐴𝜇(𝑏	𝐴𝜇(𝑏	NOUN
cana-2746	145	25	)	)	PUNCT
cana-2746	145	26	}	}	PUNCT
cana-2746	145	27	≥	≥	PROPN
cana-2746	145	28	sup	sup	NOUN
cana-2746	145	29	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	145	30	{	{	PUNCT
cana-2746	145	31	𝐴𝜇(𝑎𝛼𝑏	𝐴𝜇(𝑎𝛼𝑏	PROPN
cana-2746	145	32	)	)	PUNCT
cana-2746	145	33	}	}	PUNCT
cana-2746	145	34	=	=	SYM
cana-2746	145	35	sup	sup	NOUN
cana-2746	145	36	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	145	37	{	{	PUNCT
cana-2746	145	38	𝐴𝜇(𝑥	𝐴𝜇(𝑥	NOUN
cana-2746	145	39	)	)	PUNCT
cana-2746	145	40	}	}	PUNCT
cana-2746	145	41	=	=	SYM
cana-2746	145	42	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	145	43	)	)	PUNCT
cana-2746	145	44	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	145	45	∘	∘	PROPN
cana-2746	145	46	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	145	47	∘	∘	PROPN
cana-2746	145	48	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	145	49	)	)	PUNCT
cana-2746	145	50	=	=	SYM
cana-2746	146	1	sup	sup	NOUN
cana-2746	146	2	𝑥=𝑢𝛼𝑣𝛽𝑠	𝑥=𝑢𝛼𝑣𝛽𝑠	NOUN
cana-2746	146	3	{	{	PUNCT
cana-2746	146	4	𝑚𝑖𝑛{𝐴𝜇(𝑢𝛼𝑣	𝑚𝑖𝑛{𝐴𝜇(𝑢𝛼𝑣	PROPN
cana-2746	146	5	)	)	PUNCT
cana-2746	146	6	∘	∘	PROPN
cana-2746	146	7	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	146	8	,	,	PUNCT
cana-2746	146	9	𝐴𝜇(𝑠	𝐴𝜇(𝑠	NOUN
cana-2746	146	10	)	)	PUNCT
cana-2746	146	11	}	}	PUNCT
cana-2746	146	12	}	}	PUNCT
cana-2746	146	13	≥	≥	PROPN
cana-2746	146	14	sup	sup	NUM
cana-2746	146	15	𝑥=𝑢𝛼𝑣𝛽𝑠	𝑥=𝑢𝛼𝑣𝛽𝑠	NOUN
cana-2746	146	16	{	{	PUNCT
cana-2746	146	17	𝑚𝑖𝑛{𝐴𝜇(𝑢𝛼𝑣	𝑚𝑖𝑛{𝐴𝜇(𝑢𝛼𝑣	PROPN
cana-2746	146	18	)	)	PUNCT
cana-2746	146	19	,	,	PUNCT
cana-2746	146	20	𝐴𝜇(𝑠	𝐴𝜇(𝑠	NOUN
cana-2746	146	21	)	)	PUNCT
cana-2746	146	22	}	}	PUNCT
cana-2746	146	23	}	}	PUNCT
cana-2746	146	24	=	=	SYM
cana-2746	146	25	𝐴𝜇(𝑥	𝐴𝜇(𝑥	NOUN
cana-2746	146	26	)	)	PUNCT
cana-2746	146	27	hence	hence	ADV
cana-2746	146	28	𝜒𝑆	𝜒𝑆	VERB
cana-2746	146	29	∘	∘	NOUN
cana-2746	146	30	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	146	31	∩	∩	ADJ
cana-2746	146	32	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	146	33	∘	∘	ADV
cana-2746	146	34	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	146	35	∘	∘	PROPN
cana-2746	146	36	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	146	37	⊇	⊇	PROPN
cana-2746	146	38	𝐴𝜇.	𝐴𝜇.	INTJ
cana-2746	147	1	next	next	ADV
cana-2746	147	2	we	we	PRON
cana-2746	147	3	have	have	VERB
cana-2746	147	4	to	to	PART
cana-2746	147	5	prove	prove	VERB
cana-2746	147	6	for	for	SCONJ
cana-2746	147	7	non	non	ADJ
cana-2746	147	8	membership	membership	NOUN
cana-2746	147	9	function	function	NOUN
cana-2746	147	10	communications	communication	NOUN
cana-2746	147	11	on	on	ADP
cana-2746	147	12	applied	apply	VERB
cana-2746	147	13	nonlinear	nonlinear	ADJ
cana-2746	147	14	analysis	analysis	NOUN
cana-2746	147	15	issn	issn	NOUN
cana-2746	147	16	:	:	PUNCT
cana-2746	147	17	1074	1074	NUM
cana-2746	147	18	-	-	PUNCT
cana-2746	147	19	133x	133x	NUM
cana-2746	147	20	vol	vol	NOUN
cana-2746	147	21	32	32	NUM
cana-2746	147	22	no	no	NOUN
cana-2746	147	23	.	.	PUNCT
cana-2746	148	1	4s	4s	NUM
cana-2746	148	2	(	(	PUNCT
cana-2746	148	3	2025	2025	NUM
cana-2746	148	4	)	)	PUNCT
cana-2746	148	5	160	160	NUM
cana-2746	148	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2746	148	7	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	148	8	∘	∘	PROPN
cana-2746	148	9	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	148	10	)	)	PUNCT
cana-2746	148	11	=	=	SYM
cana-2746	148	12	inf	inf	NOUN
cana-2746	148	13	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	X
cana-2746	148	14	{	{	PUNCT
cana-2746	148	15	𝑚𝑎𝑥{𝜒𝑆(𝑎	𝑚𝑎𝑥{𝜒𝑆(𝑎	NOUN
cana-2746	148	16	)	)	PUNCT
cana-2746	148	17	,	,	PUNCT
cana-2746	148	18	𝐴𝜈(𝑏	𝐴𝜈(𝑏	NOUN
cana-2746	148	19	)	)	PUNCT
cana-2746	148	20	}	}	PUNCT
cana-2746	148	21	}	}	PUNCT
cana-2746	148	22	=	=	SYM
cana-2746	148	23	inf	inf	NOUN
cana-2746	148	24	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	148	25	{	{	PUNCT
cana-2746	148	26	𝐴𝜈(𝑏	𝐴𝜈(𝑏	NOUN
cana-2746	148	27	)	)	PUNCT
cana-2746	148	28	}	}	PUNCT
cana-2746	148	29	≤	≤	NUM
cana-2746	148	30	inf	inf	NOUN
cana-2746	148	31	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	148	32	{	{	PUNCT
cana-2746	148	33	𝐴𝜈(𝑎𝛼𝑏	𝐴𝜈(𝑎𝛼𝑏	PROPN
cana-2746	148	34	)	)	PUNCT
cana-2746	148	35	}	}	PUNCT
cana-2746	148	36	=	=	SYM
cana-2746	148	37	inf	inf	NOUN
cana-2746	148	38	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	X
cana-2746	148	39	{	{	PUNCT
cana-2746	148	40	𝐴𝜈(𝑥	𝐴𝜈(𝑥	NOUN
cana-2746	148	41	)	)	PUNCT
cana-2746	148	42	}	}	PUNCT
cana-2746	148	43	=	=	SYM
cana-2746	148	44	𝐴𝜈(𝑥	𝐴𝜈(𝑥	X
cana-2746	148	45	)	)	PUNCT
cana-2746	149	1	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	149	2	∘	∘	NOUN
cana-2746	149	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	149	4	∘	∘	PROPN
cana-2746	149	5	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	149	6	)	)	PUNCT
cana-2746	149	7	=	=	SYM
cana-2746	149	8	inf	inf	ADJ
cana-2746	149	9	𝑥=𝑢𝛼𝑣𝛽𝑠	𝑥=𝑢𝛼𝑣𝛽𝑠	NOUN
cana-2746	149	10	{	{	PUNCT
cana-2746	149	11	𝑚𝑎𝑥{𝐴𝜈(𝑢	𝑚𝑎𝑥{𝐴𝜈(𝑢	NOUN
cana-2746	149	12	)	)	PUNCT
cana-2746	149	13	,	,	PUNCT
cana-2746	149	14	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	149	15	∘	∘	PROPN
cana-2746	149	16	𝐴𝜈(𝑣𝛽𝑠	𝐴𝜈(𝑣𝛽𝑠	PROPN
cana-2746	149	17	)	)	PUNCT
cana-2746	149	18	}	}	PUNCT
cana-2746	149	19	}	}	PUNCT
cana-2746	149	20	≤	≤	NUM
cana-2746	149	21	inf	inf	ADJ
cana-2746	149	22	𝑥=𝑢𝛼𝑣𝛽𝑠	𝑥=𝑢𝛼𝑣𝛽𝑠	NOUN
cana-2746	149	23	{	{	PUNCT
cana-2746	149	24	𝑚𝑎𝑥{𝐴𝜈(𝑢	𝑚𝑎𝑥{𝐴𝜈(𝑢	NOUN
cana-2746	149	25	)	)	PUNCT
cana-2746	149	26	,	,	PUNCT
cana-2746	149	27	𝐴𝜈(𝑣𝛽𝑠	𝐴𝜈(𝑣𝛽𝑠	PROPN
cana-2746	149	28	)	)	PUNCT
cana-2746	149	29	}	}	PUNCT
cana-2746	149	30	}	}	PUNCT
cana-2746	149	31	=	=	SYM
cana-2746	149	32	𝐴𝜈(𝑥	𝐴𝜈(𝑥	NOUN
cana-2746	149	33	)	)	PUNCT
cana-2746	149	34	hence	hence	ADV
cana-2746	149	35	𝜒𝑆	𝜒𝑆	VERB
cana-2746	149	36	∘	∘	NOUN
cana-2746	149	37	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	149	38	∩	∩	NOUN
cana-2746	149	39	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	149	40	∘	∘	NOUN
cana-2746	149	41	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	149	42	∘	∘	NOUN
cana-2746	149	43	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	149	44	⊆	⊆	PROPN
cana-2746	149	45	𝐴𝜈.	𝐴𝜈.	PROPN
cana-2746	149	46	theorem	theorem	VERB
cana-2746	149	47	4.5	4.5	NUM
cana-2746	149	48	every	every	DET
cana-2746	149	49	pf	pf	PROPN
cana-2746	149	50	right	right	ADJ
cana-2746	149	51	ideal	ideal	NOUN
cana-2746	149	52	of	of	ADP
cana-2746	149	53	𝑆	𝑆	PROPN
cana-2746	149	54	is	be	AUX
cana-2746	149	55	a	a	DET
cana-2746	149	56	pfrbqi	pfrbqi	NOUN
cana-2746	149	57	of	of	ADP
cana-2746	149	58	𝑆.	𝑆.	PROPN
cana-2746	149	59	proof	proof	NOUN
cana-2746	149	60	.	.	PUNCT
cana-2746	150	1	let	let	VERB
cana-2746	150	2	𝐴	𝐴	PROPN
cana-2746	150	3	be	be	AUX
cana-2746	150	4	a	a	DET
cana-2746	150	5	𝑃𝐹	𝑃𝐹	NOUN
cana-2746	150	6	right	right	ADJ
cana-2746	150	7	ideal	ideal	NOUN
cana-2746	150	8	of	of	ADP
cana-2746	150	9	𝑆	𝑆	PROPN
cana-2746	150	10	and	and	CCONJ
cana-2746	150	11	𝑥	𝑥	PRON
cana-2746	150	12	∈	∈	PROPN
cana-2746	150	13	𝑆,𝛼	𝑆,𝛼	NOUN
cana-2746	150	14	,	,	PUNCT
cana-2746	150	15	𝛽	𝛽	PROPN
cana-2746	150	16	∈	∈	PROPN
cana-2746	150	17	γ	γ	X
cana-2746	150	18	.	.	PUNCT
cana-2746	151	1	𝐴𝜇	𝐴𝜇	NOUN
cana-2746	151	2	∘	∘	NOUN
cana-2746	151	3	𝜒𝑆(𝑥	𝜒𝑆(𝑥	NUM
cana-2746	151	4	)	)	PUNCT
cana-2746	151	5	=	=	SYM
cana-2746	151	6	sup	sup	NOUN
cana-2746	151	7	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	151	8	{	{	PUNCT
cana-2746	151	9	𝑚𝑖𝑛{𝐴𝜇(𝑎	𝑚𝑖𝑛{𝐴𝜇(𝑎	PROPN
cana-2746	151	10	)	)	PUNCT
cana-2746	151	11	,	,	PUNCT
cana-2746	151	12	𝜒𝑆(𝑏	𝜒𝑆(𝑏	NUM
cana-2746	151	13	)	)	PUNCT
cana-2746	151	14	}	}	PUNCT
cana-2746	151	15	}	}	PUNCT
cana-2746	151	16	=	=	SYM
cana-2746	151	17	sup	sup	NOUN
cana-2746	151	18	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	151	19	{	{	PUNCT
cana-2746	151	20	𝐴𝜇(𝑎	𝐴𝜇(𝑎	PROPN
cana-2746	151	21	)	)	PUNCT
cana-2746	151	22	}	}	PUNCT
cana-2746	151	23	≥	≥	NOUN
cana-2746	151	24	sup	sup	NOUN
cana-2746	151	25	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	151	26	{	{	PUNCT
cana-2746	151	27	𝐴𝜇(𝑎𝛼𝑏	𝐴𝜇(𝑎𝛼𝑏	PROPN
cana-2746	151	28	)	)	PUNCT
cana-2746	151	29	}	}	PUNCT
cana-2746	151	30	=	=	SYM
cana-2746	151	31	sup	sup	NOUN
cana-2746	151	32	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	151	33	{	{	PUNCT
cana-2746	151	34	𝐴𝜇(𝑥	𝐴𝜇(𝑥	NOUN
cana-2746	151	35	)	)	PUNCT
cana-2746	151	36	}	}	PUNCT
cana-2746	151	37	=	=	SYM
cana-2746	151	38	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	151	39	)	)	PUNCT
cana-2746	151	40	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	151	41	∘	∘	PROPN
cana-2746	151	42	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	151	43	∘	∘	PROPN
cana-2746	151	44	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	151	45	)	)	PUNCT
cana-2746	151	46	=	=	NOUN
cana-2746	151	47	sup	sup	NOUN
cana-2746	151	48	𝑥=𝑢𝛼𝑣𝛽𝑠	𝑥=𝑢𝛼𝑣𝛽𝑠	NOUN
cana-2746	151	49	{	{	PUNCT
cana-2746	151	50	𝑚𝑖𝑛{𝐴𝜇	𝑚𝑖𝑛{𝐴𝜇	PROPN
cana-2746	151	51	∘	∘	X
cana-2746	151	52	𝜒𝑆(𝑢𝛼𝑣	𝜒𝑆(𝑢𝛼𝑣	PROPN
cana-2746	151	53	)	)	PUNCT
cana-2746	152	1	,	,	PUNCT
cana-2746	152	2	𝐴𝜇(𝑠	𝐴𝜇(𝑠	NOUN
cana-2746	152	3	)	)	PUNCT
cana-2746	152	4	}	}	PUNCT
cana-2746	152	5	}	}	PUNCT
cana-2746	152	6	≥	≥	PROPN
cana-2746	152	7	sup	sup	NUM
cana-2746	152	8	𝑥=𝑢𝛼𝑣𝛽𝑠	𝑥=𝑢𝛼𝑣𝛽𝑠	NOUN
cana-2746	152	9	{	{	PUNCT
cana-2746	152	10	𝑚𝑖𝑛{𝐴𝜇(𝑢𝛼𝑣	𝑚𝑖𝑛{𝐴𝜇(𝑢𝛼𝑣	PROPN
cana-2746	152	11	)	)	PUNCT
cana-2746	152	12	,	,	PUNCT
cana-2746	152	13	𝐴𝜇(𝑠	𝐴𝜇(𝑠	NOUN
cana-2746	152	14	)	)	PUNCT
cana-2746	152	15	}	}	PUNCT
cana-2746	152	16	}	}	PUNCT
cana-2746	152	17	=	=	SYM
cana-2746	152	18	𝐴𝜇(𝑥	𝐴𝜇(𝑥	NOUN
cana-2746	152	19	)	)	PUNCT
cana-2746	152	20	hence	hence	ADV
cana-2746	152	21	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	152	22	∘	∘	NOUN
cana-2746	152	23	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	152	24	∩	∩	NOUN
cana-2746	152	25	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	152	26	∘	∘	ADV
cana-2746	152	27	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	152	28	∘	∘	PROPN
cana-2746	152	29	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	152	30	⊇	⊇	PROPN
cana-2746	153	1	𝐴𝜇.	𝐴𝜇.	INTJ
cana-2746	153	2	next	next	ADV
cana-2746	153	3	we	we	PRON
cana-2746	153	4	have	have	VERB
cana-2746	153	5	to	to	PART
cana-2746	153	6	prove	prove	VERB
cana-2746	153	7	for	for	SCONJ
cana-2746	153	8	non	non	ADJ
cana-2746	153	9	membership	membership	NOUN
cana-2746	153	10	function	function	NOUN
cana-2746	153	11	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	153	12	∘	∘	ADJ
cana-2746	153	13	𝜒𝑆(𝑥	𝜒𝑆(𝑥	NUM
cana-2746	153	14	)	)	PUNCT
cana-2746	153	15	=	=	SYM
cana-2746	153	16	inf	inf	NOUN
cana-2746	153	17	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	153	18	{	{	PUNCT
cana-2746	153	19	𝑚𝑎𝑥{𝐴𝜇(𝑎	𝑚𝑎𝑥{𝐴𝜇(𝑎	NOUN
cana-2746	153	20	)	)	PUNCT
cana-2746	153	21	,	,	PUNCT
cana-2746	153	22	𝜒𝑆(𝑏	𝜒𝑆(𝑏	NUM
cana-2746	153	23	)	)	PUNCT
cana-2746	153	24	}	}	PUNCT
cana-2746	153	25	}	}	PUNCT
cana-2746	153	26	=	=	SYM
cana-2746	153	27	inf	inf	NOUN
cana-2746	153	28	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	153	29	{	{	PUNCT
cana-2746	153	30	𝐴𝜈(𝑎	𝐴𝜈(𝑎	PROPN
cana-2746	153	31	)	)	PUNCT
cana-2746	153	32	}	}	PUNCT
cana-2746	153	33	≤	≤	NUM
cana-2746	153	34	inf	inf	NOUN
cana-2746	153	35	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	153	36	{	{	PUNCT
cana-2746	153	37	𝐴𝜈(𝑎𝛼𝑏	𝐴𝜈(𝑎𝛼𝑏	PROPN
cana-2746	153	38	)	)	PUNCT
cana-2746	153	39	}	}	PUNCT
cana-2746	153	40	=	=	SYM
cana-2746	153	41	inf	inf	NOUN
cana-2746	153	42	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	X
cana-2746	153	43	{	{	PUNCT
cana-2746	153	44	𝐴𝜈(𝑥	𝐴𝜈(𝑥	NOUN
cana-2746	153	45	)	)	PUNCT
cana-2746	153	46	}	}	PUNCT
cana-2746	153	47	communications	communication	NOUN
cana-2746	153	48	on	on	ADP
cana-2746	153	49	applied	apply	VERB
cana-2746	153	50	nonlinear	nonlinear	ADJ
cana-2746	153	51	analysis	analysis	NOUN
cana-2746	153	52	issn	issn	NOUN
cana-2746	153	53	:	:	PUNCT
cana-2746	153	54	1074	1074	NUM
cana-2746	153	55	-	-	PUNCT
cana-2746	153	56	133x	133x	NUM
cana-2746	153	57	vol	vol	NOUN
cana-2746	153	58	32	32	NUM
cana-2746	153	59	no	no	NOUN
cana-2746	153	60	.	.	PUNCT
cana-2746	154	1	4s	4s	NUM
cana-2746	154	2	(	(	PUNCT
cana-2746	154	3	2025	2025	NUM
cana-2746	154	4	)	)	PUNCT
cana-2746	154	5	161	161	NUM
cana-2746	154	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2746	154	7	=	=	SYM
cana-2746	154	8	𝐴𝜈(𝑥	𝐴𝜈(𝑥	X
cana-2746	154	9	)	)	PUNCT
cana-2746	155	1	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	155	2	∘	∘	NOUN
cana-2746	155	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	155	4	∘	∘	PROPN
cana-2746	155	5	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	155	6	)	)	PUNCT
cana-2746	155	7	=	=	SYM
cana-2746	155	8	inf	inf	ADJ
cana-2746	155	9	𝑥=𝑢𝛼𝑣𝛽𝑠	𝑥=𝑢𝛼𝑣𝛽𝑠	NOUN
cana-2746	155	10	{	{	PUNCT
cana-2746	155	11	𝑚𝑎𝑥{𝐴𝜈	𝑚𝑎𝑥{𝐴𝜈	NOUN
cana-2746	155	12	∘	∘	X
cana-2746	155	13	𝜒𝑆(𝑢𝛼𝑣	𝜒𝑆(𝑢𝛼𝑣	PROPN
cana-2746	155	14	)	)	PUNCT
cana-2746	155	15	,	,	PUNCT
cana-2746	155	16	𝐴𝜈(𝑠	𝐴𝜈(𝑠	PROPN
cana-2746	155	17	)	)	PUNCT
cana-2746	155	18	}	}	PUNCT
cana-2746	155	19	}	}	PUNCT
cana-2746	155	20	≤	≤	NUM
cana-2746	155	21	inf	inf	ADJ
cana-2746	155	22	𝑥=𝑢𝛼𝑣𝛽𝑠	𝑥=𝑢𝛼𝑣𝛽𝑠	NOUN
cana-2746	155	23	{	{	PUNCT
cana-2746	155	24	𝑚𝑎𝑥{𝐴𝜈(𝑢𝛼𝑣	𝑚𝑎𝑥{𝐴𝜈(𝑢𝛼𝑣	NOUN
cana-2746	155	25	)	)	PUNCT
cana-2746	155	26	,	,	PUNCT
cana-2746	155	27	𝐴𝜈(𝑠	𝐴𝜈(𝑠	PROPN
cana-2746	155	28	)	)	PUNCT
cana-2746	155	29	}	}	PUNCT
cana-2746	155	30	}	}	PUNCT
cana-2746	155	31	=	=	SYM
cana-2746	155	32	𝐴𝜈(𝑥	𝐴𝜈(𝑥	NOUN
cana-2746	155	33	)	)	PUNCT
cana-2746	155	34	.	.	PUNCT
cana-2746	156	1	hence	hence	ADV
cana-2746	156	2	𝐴𝜈	𝐴𝜈	ADJ
cana-2746	156	3	∘	∘	ADJ
cana-2746	156	4	𝜒𝑆	𝜒𝑆	NOUN
cana-2746	156	5	∩	∩	NOUN
cana-2746	156	6	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	156	7	∘	∘	NOUN
cana-2746	156	8	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	156	9	∘	∘	NOUN
cana-2746	156	10	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	156	11	⊆	⊆	PROPN
cana-2746	156	12	𝐴𝜈.	𝐴𝜈.	PROPN
cana-2746	156	13	theorem	theorem	VERB
cana-2746	156	14	4.6	4.6	NUM
cana-2746	156	15	every	every	DET
cana-2746	156	16	pf	pf	NOUN
cana-2746	156	17	left	leave	VERB
cana-2746	156	18	ideal	ideal	NOUN
cana-2746	156	19	of	of	ADP
cana-2746	156	20	𝑆	𝑆	PROPN
cana-2746	156	21	is	be	AUX
cana-2746	156	22	a	a	DET
cana-2746	156	23	pfrbqi	pfrbqi	NOUN
cana-2746	156	24	of	of	ADP
cana-2746	156	25	𝑆.	𝑆.	PROPN
cana-2746	156	26	proof	proof	NOUN
cana-2746	156	27	.	.	PUNCT
cana-2746	157	1	let	let	VERB
cana-2746	157	2	𝐴	𝐴	PROPN
cana-2746	157	3	be	be	AUX
cana-2746	157	4	a	a	DET
cana-2746	157	5	𝑃𝐹	𝑃𝐹	NOUN
cana-2746	157	6	left	leave	VERB
cana-2746	157	7	ideal	ideal	NOUN
cana-2746	157	8	of	of	ADP
cana-2746	157	9	𝑆	𝑆	PROPN
cana-2746	157	10	and	and	CCONJ
cana-2746	157	11	𝑥	𝑥	PRON
cana-2746	157	12	∈	∈	PROPN
cana-2746	157	13	𝑆,𝛼	𝑆,𝛼	NOUN
cana-2746	157	14	,	,	PUNCT
cana-2746	158	1	𝛽	𝛽	PROPN
cana-2746	158	2	∈	∈	PROPN
cana-2746	158	3	γ	γ	X
cana-2746	158	4	.	.	PROPN
cana-2746	158	5	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	158	6	∘	∘	PROPN
cana-2746	158	7	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	158	8	)	)	PUNCT
cana-2746	158	9	=	=	SYM
cana-2746	158	10	sup	sup	NOUN
cana-2746	158	11	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PUNCT
cana-2746	158	12	{	{	PUNCT
cana-2746	158	13	𝑚𝑖𝑛{𝜒𝑆(𝑎	𝑚𝑖𝑛{𝜒𝑆(𝑎	NOUN
cana-2746	158	14	)	)	PUNCT
cana-2746	158	15	,	,	PUNCT
cana-2746	158	16	𝐴𝜇(𝑏	𝐴𝜇(𝑏	NOUN
cana-2746	158	17	)	)	PUNCT
cana-2746	158	18	}	}	PUNCT
cana-2746	158	19	}	}	PUNCT
cana-2746	158	20	=	=	SYM
cana-2746	158	21	sup	sup	NOUN
cana-2746	158	22	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	158	23	{	{	PUNCT
cana-2746	158	24	𝐴𝜇(𝑏	𝐴𝜇(𝑏	NOUN
cana-2746	158	25	)	)	PUNCT
cana-2746	158	26	}	}	PUNCT
cana-2746	158	27	≥	≥	PROPN
cana-2746	158	28	sup	sup	NOUN
cana-2746	158	29	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	158	30	{	{	PUNCT
cana-2746	158	31	𝐴𝜇(𝑎𝛼𝑏	𝐴𝜇(𝑎𝛼𝑏	PROPN
cana-2746	158	32	)	)	PUNCT
cana-2746	158	33	}	}	PUNCT
cana-2746	158	34	=	=	SYM
cana-2746	158	35	sup	sup	NOUN
cana-2746	158	36	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	158	37	{	{	PUNCT
cana-2746	158	38	𝐴𝜇(𝑥	𝐴𝜇(𝑥	NOUN
cana-2746	158	39	)	)	PUNCT
cana-2746	158	40	}	}	PUNCT
cana-2746	158	41	=	=	SYM
cana-2746	158	42	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	158	43	)	)	PUNCT
cana-2746	158	44	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	158	45	∘	∘	PROPN
cana-2746	158	46	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	158	47	∘	∘	PROPN
cana-2746	158	48	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	158	49	)	)	PUNCT
cana-2746	158	50	=	=	NOUN
cana-2746	158	51	sup	sup	NOUN
cana-2746	158	52	𝑥=𝑢𝛼𝑣𝛽𝑠	𝑥=𝑢𝛼𝑣𝛽𝑠	NOUN
cana-2746	158	53	{	{	PUNCT
cana-2746	158	54	𝑚𝑖𝑛{𝐴𝜇(𝑠	𝑚𝑖𝑛{𝐴𝜇(𝑠	PROPN
cana-2746	158	55	)	)	PUNCT
cana-2746	158	56	,	,	PUNCT
cana-2746	158	57	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	158	58	∘	∘	PROPN
cana-2746	158	59	𝐴𝜇(𝑣𝛽𝑠	𝐴𝜇(𝑣𝛽𝑠	NOUN
cana-2746	158	60	)	)	PUNCT
cana-2746	158	61	}	}	PUNCT
cana-2746	158	62	}	}	PUNCT
cana-2746	158	63	≥	≥	NUM
cana-2746	158	64	sup	sup	NUM
cana-2746	158	65	𝑥=𝑢𝛼𝑣𝛽𝑠	𝑥=𝑢𝛼𝑣𝛽𝑠	NOUN
cana-2746	158	66	{	{	PUNCT
cana-2746	158	67	𝑚𝑖𝑛{𝐴𝜇(𝑠	𝑚𝑖𝑛{𝐴𝜇(𝑠	PROPN
cana-2746	158	68	)	)	PUNCT
cana-2746	158	69	,	,	PUNCT
cana-2746	158	70	𝐴𝜇(𝑣𝛽𝑠	𝐴𝜇(𝑣𝛽𝑠	NOUN
cana-2746	158	71	)	)	PUNCT
cana-2746	158	72	}	}	PUNCT
cana-2746	158	73	}	}	PUNCT
cana-2746	158	74	=	=	SYM
cana-2746	158	75	𝐴𝜇(𝑥	𝐴𝜇(𝑥	NOUN
cana-2746	158	76	)	)	PUNCT
cana-2746	158	77	now	now	ADV
cana-2746	158	78	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	158	79	∘	∘	NOUN
cana-2746	158	80	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	158	81	∩	∩	NOUN
cana-2746	158	82	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	158	83	∘	∘	ADV
cana-2746	158	84	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	158	85	∘	∘	PROPN
cana-2746	158	86	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	158	87	)	)	PUNCT
cana-2746	158	88	=	=	SYM
cana-2746	159	1	𝑚𝑖𝑛{𝐴𝜇	𝑚𝑖𝑛{𝐴𝜇	PROPN
cana-2746	159	2	∘	∘	ADJ
cana-2746	159	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	159	4	,	,	PUNCT
cana-2746	159	5	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	159	6	∘	∘	PROPN
cana-2746	159	7	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	159	8	∘	∘	PROPN
cana-2746	159	9	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	159	10	)	)	PUNCT
cana-2746	159	11	}	}	PUNCT
cana-2746	159	12	≥	≥	NOUN
cana-2746	159	13	𝑚𝑖𝑛{𝐴𝜇	𝑚𝑖𝑛{𝐴𝜇	PROPN
cana-2746	159	14	∘	∘	X
cana-2746	159	15	𝜒𝑆(𝑥	𝜒𝑆(𝑥	NOUN
cana-2746	159	16	)	)	PUNCT
cana-2746	159	17	,	,	PUNCT
cana-2746	159	18	𝐴𝜇(𝑥	𝐴𝜇(𝑥	NOUN
cana-2746	159	19	)	)	PUNCT
cana-2746	159	20	}	}	PUNCT
cana-2746	159	21	=	=	SYM
cana-2746	159	22	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	159	23	)	)	PUNCT
cana-2746	159	24	.	.	PUNCT
cana-2746	160	1	hence	hence	ADV
cana-2746	160	2	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	160	3	is	be	AUX
cana-2746	160	4	a	a	DET
cana-2746	160	5	𝑃𝐹𝑅𝐵𝑄𝐼	𝑃𝐹𝑅𝐵𝑄𝐼	PROPN
cana-2746	160	6	of	of	ADP
cana-2746	160	7	𝑆.	𝑆.	NOUN
cana-2746	160	8	similarly	similarly	ADV
cana-2746	160	9	we	we	PRON
cana-2746	160	10	can	can	AUX
cana-2746	160	11	prove	prove	VERB
cana-2746	160	12	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	160	13	is	be	AUX
cana-2746	160	14	a	a	DET
cana-2746	160	15	𝑃𝐹𝑅𝐵𝑄𝐼	𝑃𝐹𝑅𝐵𝑄𝐼	PROPN
cana-2746	160	16	of	of	ADP
cana-2746	160	17	𝑆.	𝑆.	PROPN
cana-2746	160	18	theorem	theorem	VERB
cana-2746	160	19	4.7	4.7	NUM
cana-2746	160	20	every	every	DET
cana-2746	160	21	pf	pf	PROPN
cana-2746	160	22	right	right	ADJ
cana-2746	160	23	ideal	ideal	NOUN
cana-2746	160	24	of	of	ADP
cana-2746	160	25	𝑆	𝑆	PROPN
cana-2746	160	26	is	be	AUX
cana-2746	160	27	a	a	DET
cana-2746	160	28	pflbqi	pflbqi	NOUN
cana-2746	160	29	of	of	ADP
cana-2746	160	30	𝑆.	𝑆.	PROPN
cana-2746	160	31	proof	proof	NOUN
cana-2746	160	32	.	.	PUNCT
cana-2746	161	1	proof	proof	NOUN
cana-2746	161	2	is	be	AUX
cana-2746	161	3	straight	straight	ADV
cana-2746	161	4	forward	forward	ADV
cana-2746	161	5	.	.	PUNCT
cana-2746	162	1	corollary	corollary	ADJ
cana-2746	162	2	4.8	4.8	NUM
cana-2746	163	1	every	every	DET
cana-2746	163	2	pf	pf	PROPN
cana-2746	163	3	left(right	left(right	PROPN
cana-2746	163	4	)	)	PUNCT
cana-2746	163	5	ideal	ideal	NOUN
cana-2746	163	6	of	of	ADP
cana-2746	163	7	𝑆	𝑆	PROPN
cana-2746	163	8	is	be	AUX
cana-2746	163	9	a	a	DET
cana-2746	163	10	pfrbqi	pfrbqi	NOUN
cana-2746	163	11	of	of	ADP
cana-2746	163	12	𝑆.	𝑆.	PROPN
cana-2746	163	13	theorem	theorem	NOUN
cana-2746	163	14	4.9	4.9	NUM
cana-2746	163	15	let	let	VERB
cana-2746	163	16	𝐵	𝐵	PRON
cana-2746	163	17	be	be	AUX
cana-2746	163	18	a	a	DET
cana-2746	163	19	nonempty	nonempty	ADJ
cana-2746	163	20	subset	subset	NOUN
cana-2746	163	21	of	of	ADP
cana-2746	163	22	𝑆.	𝑆.	PROPN
cana-2746	163	23	then	then	ADV
cana-2746	163	24	𝐵	𝐵	NOUN
cana-2746	163	25	is	be	AUX
cana-2746	164	1	a	a	DET
cana-2746	164	2	right	right	ADJ
cana-2746	164	3	bi	bi	ADJ
cana-2746	164	4	-	-	ADJ
cana-2746	164	5	quasi	quasi	ADJ
cana-2746	164	6	-	-	NOUN
cana-2746	164	7	ideal	ideal	NOUN
cana-2746	164	8	of	of	ADP
cana-2746	164	9	𝑆	𝑆	PROPN
cana-2746	164	10	⟺	⟺	PROPN
cana-2746	164	11	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	164	12	is	be	AUX
cana-2746	164	13	an	an	DET
cana-2746	164	14	pfrbqi	pfrbqi	NOUN
cana-2746	164	15	of	of	ADP
cana-2746	164	16	𝑆.	𝑆.	PROPN
cana-2746	164	17	proof	proof	NOUN
cana-2746	164	18	.	.	PUNCT
cana-2746	165	1	assume	assume	VERB
cana-2746	165	2	that	that	SCONJ
cana-2746	165	3	𝐵	𝐵	NOUN
cana-2746	165	4	is	be	AUX
cana-2746	165	5	a	a	DET
cana-2746	165	6	right	right	ADJ
cana-2746	165	7	bi	bi	ADJ
cana-2746	165	8	-	-	ADJ
cana-2746	165	9	quasi	quasi	ADJ
cana-2746	165	10	-	-	NOUN
cana-2746	165	11	ideal	ideal	NOUN
cana-2746	165	12	of	of	ADP
cana-2746	165	13	𝑆.	𝑆.	PROPN
cana-2746	165	14	then	then	ADV
cana-2746	165	15	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	165	16	is	be	AUX
cana-2746	165	17	an	an	DET
cana-2746	165	18	𝑃𝐹	𝑃𝐹	NOUN
cana-2746	165	19	subsemiring	subsemiring	NOUN
cana-2746	165	20	of	of	ADP
cana-2746	165	21	𝑆.	𝑆.	NOUN
cana-2746	165	22	communications	communication	NOUN
cana-2746	165	23	on	on	ADP
cana-2746	165	24	applied	apply	VERB
cana-2746	165	25	nonlinear	nonlinear	ADJ
cana-2746	165	26	analysis	analysis	NOUN
cana-2746	165	27	issn	issn	NOUN
cana-2746	165	28	:	:	PUNCT
cana-2746	165	29	1074	1074	NUM
cana-2746	165	30	-	-	PUNCT
cana-2746	165	31	133x	133x	NUM
cana-2746	165	32	vol	vol	NOUN
cana-2746	165	33	32	32	NUM
cana-2746	165	34	no	no	NOUN
cana-2746	165	35	.	.	PUNCT
cana-2746	166	1	4s	4s	NUM
cana-2746	166	2	(	(	PUNCT
cana-2746	166	3	2025	2025	NUM
cana-2746	166	4	)	)	PUNCT
cana-2746	166	5	162	162	NUM
cana-2746	166	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2746	166	7	by	by	ADP
cana-2746	166	8	hypothesis	hypothesis	NOUN
cana-2746	166	9	we	we	PRON
cana-2746	166	10	’ve	’ve	VERB
cana-2746	166	11	𝑆γ𝐵	𝑆γ𝐵	PROPN
cana-2746	166	12	∩	∩	NOUN
cana-2746	166	13	𝐵γ𝑆γ𝐵	𝐵γ𝑆γ𝐵	ADJ
cana-2746	166	14	⊆	⊆	NUM
cana-2746	166	15	𝐵.	𝐵.	PROPN
cana-2746	166	16	then	then	ADV
cana-2746	166	17	,	,	PUNCT
cana-2746	166	18	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	166	19	∘	∘	PROPN
cana-2746	166	20	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	166	21	∩	∩	ADJ
cana-2746	166	22	𝜒𝐵	𝜒𝐵	X
cana-2746	166	23	∘	∘	ADJ
cana-2746	166	24	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	166	25	∘	∘	PROPN
cana-2746	166	26	𝜒𝐵	𝜒𝐵	X
cana-2746	166	27	=	=	SYM
cana-2746	166	28	𝜒𝑆γ𝐵	𝜒𝑆γ𝐵	PROPN
cana-2746	166	29	∩	∩	NOUN
cana-2746	166	30	𝜒𝐵γ𝑆γ𝐵	𝜒𝐵γ𝑆γ𝐵	NUM
cana-2746	166	31	=	=	SYM
cana-2746	166	32	𝜒𝑆γ𝐵∩𝐵γ𝑆γ𝐵	𝜒𝑆γ𝐵∩𝐵γ𝑆γ𝐵	NOUN
cana-2746	166	33	⊆	⊆	NUM
cana-2746	166	34	𝜒𝐵	𝜒𝐵	NOUN
cana-2746	166	35	hence	hence	ADV
cana-2746	166	36	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	166	37	is	be	AUX
cana-2746	166	38	an	an	DET
cana-2746	166	39	𝑃𝐹𝑅𝐵𝑄𝐼	𝑃𝐹𝑅𝐵𝑄𝐼	PROPN
cana-2746	166	40	of	of	ADP
cana-2746	166	41	𝑆.	𝑆.	NOUN
cana-2746	166	42	conversely	conversely	ADV
cana-2746	166	43	,	,	PUNCT
cana-2746	166	44	let	let	VERB
cana-2746	166	45	us	we	PRON
cana-2746	166	46	assume	assume	VERB
cana-2746	166	47	that	that	SCONJ
cana-2746	166	48	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	166	49	is	be	AUX
cana-2746	166	50	a	a	DET
cana-2746	166	51	𝑃𝐹𝑅𝐵𝑄𝐼	𝑃𝐹𝑅𝐵𝑄𝐼	PROPN
cana-2746	166	52	of	of	ADP
cana-2746	166	53	𝑆.	𝑆.	PROPN
cana-2746	166	54	then	then	ADV
cana-2746	166	55	𝐵	𝐵	NOUN
cana-2746	166	56	is	be	AUX
cana-2746	166	57	a	a	DET
cana-2746	166	58	subsemiring	subsemiring	NOUN
cana-2746	166	59	of	of	ADP
cana-2746	166	60	𝑆.	𝑆.	PROPN
cana-2746	166	61	we	we	PRON
cana-2746	166	62	have	have	VERB
cana-2746	166	63	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	166	64	∘	∘	PROPN
cana-2746	166	65	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	166	66	∩	∩	ADJ
cana-2746	167	1	𝜒𝐵	𝜒𝐵	X
cana-2746	167	2	∘	∘	ADJ
cana-2746	167	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	167	4	∘	∘	PROPN
cana-2746	167	5	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	167	6	⊆	⊆	NUM
cana-2746	167	7	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	167	8	𝜒𝑆γ𝐵	𝜒𝑆γ𝐵	NOUN
cana-2746	167	9	∩	∩	NOUN
cana-2746	167	10	𝜒𝐵γ𝑆γ𝐵	𝜒𝐵γ𝑆γ𝐵	NUM
cana-2746	167	11	⊆	⊆	NUM
cana-2746	167	12	𝜒𝐵	𝜒𝐵	NOUN
cana-2746	167	13	hence	hence	ADV
cana-2746	167	14	𝐵	𝐵	NOUN
cana-2746	167	15	is	be	AUX
cana-2746	167	16	a	a	DET
cana-2746	167	17	bi	bi	ADJ
cana-2746	167	18	-	-	ADJ
cana-2746	167	19	quasi	quasi	NOUN
cana-2746	167	20	-	-	NOUN
cana-2746	167	21	ideal	ideal	NOUN
cana-2746	167	22	of	of	ADP
cana-2746	167	23	𝑆.	𝑆.	PROPN
cana-2746	167	24	theorem	theorem	NOUN
cana-2746	167	25	4.10	4.10	NUM
cana-2746	167	26	let	let	VERB
cana-2746	167	27	𝐵	𝐵	PRON
cana-2746	167	28	be	be	AUX
cana-2746	167	29	a	a	DET
cana-2746	167	30	nonempty	nonempty	ADJ
cana-2746	167	31	subset	subset	NOUN
cana-2746	167	32	of	of	ADP
cana-2746	167	33	𝑆.	𝑆.	PROPN
cana-2746	167	34	then	then	ADV
cana-2746	167	35	𝐵	𝐵	NOUN
cana-2746	167	36	is	be	AUX
cana-2746	167	37	a	a	DET
cana-2746	167	38	left	left	ADJ
cana-2746	167	39	bi	bi	ADJ
cana-2746	167	40	-	-	ADJ
cana-2746	167	41	quasi	quasi	NOUN
cana-2746	167	42	-	-	NOUN
cana-2746	167	43	ideal	ideal	NOUN
cana-2746	167	44	of	of	ADP
cana-2746	167	45	𝑆	𝑆	PROPN
cana-2746	167	46	⟺	⟺	PROPN
cana-2746	167	47	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	167	48	is	be	AUX
cana-2746	167	49	an	an	DET
cana-2746	167	50	pflbqi	pflbqi	NOUN
cana-2746	167	51	of	of	ADP
cana-2746	167	52	𝑆.	𝑆.	PROPN
cana-2746	167	53	proof	proof	NOUN
cana-2746	167	54	.	.	PUNCT
cana-2746	168	1	assume	assume	VERB
cana-2746	168	2	that	that	SCONJ
cana-2746	168	3	𝐵	𝐵	NOUN
cana-2746	168	4	is	be	AUX
cana-2746	168	5	a	a	DET
cana-2746	168	6	left	left	ADJ
cana-2746	168	7	bi	bi	ADJ
cana-2746	168	8	-	-	ADJ
cana-2746	168	9	quasi	quasi	NOUN
cana-2746	168	10	-	-	NOUN
cana-2746	168	11	ideal	ideal	NOUN
cana-2746	168	12	of	of	ADP
cana-2746	168	13	𝑆.	𝑆.	PROPN
cana-2746	168	14	then	then	ADV
cana-2746	168	15	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	168	16	is	be	AUX
cana-2746	168	17	an	an	DET
cana-2746	168	18	𝑃𝐹	𝑃𝐹	NOUN
cana-2746	168	19	sub	sub	NOUN
cana-2746	168	20	γ	γ	X
cana-2746	168	21	semi	semi	ADP
cana-2746	168	22	ring	ring	NOUN
cana-2746	168	23	of	of	ADP
cana-2746	168	24	𝑆.	𝑆.	NOUN
cana-2746	168	25	by	by	ADP
cana-2746	168	26	hypothesis	hypothesis	NOUN
cana-2746	168	27	we	we	PRON
cana-2746	168	28	’ve	’ve	VERB
cana-2746	168	29	𝐵γ𝑆	𝐵γ𝑆	PROPN
cana-2746	168	30	∩	∩	NOUN
cana-2746	168	31	𝐵γ𝑆γ𝐵	𝐵γ𝑆γ𝐵	VERB
cana-2746	168	32	⊆	⊆	NUM
cana-2746	168	33	𝐵.	𝐵.	PROPN
cana-2746	168	34	then	then	ADV
cana-2746	168	35	,	,	PUNCT
cana-2746	168	36	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	168	37	∘	∘	ADJ
cana-2746	168	38	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	168	39	∩	∩	ADJ
cana-2746	168	40	𝜒𝐵	𝜒𝐵	NOUN
cana-2746	168	41	∘	∘	ADJ
cana-2746	168	42	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	168	43	∘	∘	PROPN
cana-2746	168	44	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	169	1	=	=	SYM
cana-2746	169	2	𝜒𝐵γ𝑆	𝜒𝐵γ𝑆	PROPN
cana-2746	169	3	∩	∩	NOUN
cana-2746	169	4	𝜒𝐵γ𝑆γ𝐵	𝜒𝐵γ𝑆γ𝐵	NUM
cana-2746	169	5	=	=	PRON
cana-2746	169	6	𝜒𝐵γ𝑆∩𝐵γ𝑆γ𝐵	𝜒𝐵γ𝑆∩𝐵γ𝑆γ𝐵	VERB
cana-2746	169	7	⊆	⊆	NUM
cana-2746	169	8	𝜒𝐵	𝜒𝐵	NOUN
cana-2746	169	9	hence	hence	ADV
cana-2746	169	10	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	169	11	is	be	AUX
cana-2746	169	12	an	an	DET
cana-2746	169	13	𝑃𝐹𝑅𝐵𝑄𝐼	𝑃𝐹𝑅𝐵𝑄𝐼	PROPN
cana-2746	169	14	of	of	ADP
cana-2746	169	15	𝑆.	𝑆.	NOUN
cana-2746	169	16	conversely	conversely	ADV
cana-2746	169	17	,	,	PUNCT
cana-2746	169	18	let	let	VERB
cana-2746	169	19	us	we	PRON
cana-2746	169	20	assume	assume	VERB
cana-2746	169	21	that	that	SCONJ
cana-2746	169	22	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	169	23	is	be	AUX
cana-2746	169	24	a	a	DET
cana-2746	169	25	𝑃𝐹𝑅𝐵𝑄𝐼	𝑃𝐹𝑅𝐵𝑄𝐼	PROPN
cana-2746	169	26	of	of	ADP
cana-2746	169	27	𝑆.	𝑆.	PROPN
cana-2746	169	28	then	then	ADV
cana-2746	169	29	𝐵	𝐵	NOUN
cana-2746	169	30	is	be	AUX
cana-2746	169	31	a	a	DET
cana-2746	169	32	sub	sub	NOUN
cana-2746	169	33	γ	γ	X
cana-2746	169	34	semi	semi	ADP
cana-2746	169	35	ring	ring	NOUN
cana-2746	169	36	of	of	ADP
cana-2746	169	37	𝑆.	𝑆.	PROPN
cana-2746	169	38	we	we	PRON
cana-2746	169	39	have	have	VERB
cana-2746	169	40	𝜒𝐵	𝜒𝐵	VERB
cana-2746	169	41	∘	∘	ADJ
cana-2746	169	42	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	169	43	∩	∩	ADJ
cana-2746	169	44	𝜒𝐵	𝜒𝐵	NOUN
cana-2746	169	45	∘	∘	ADJ
cana-2746	169	46	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	169	47	∘	∘	PROPN
cana-2746	169	48	𝜒𝐵	𝜒𝐵	PROPN
cana-2746	169	49	⊆	⊆	NUM
cana-2746	169	50	𝜒𝐵	𝜒𝐵	NOUN
cana-2746	169	51	𝜒𝐵γ𝑆	𝜒𝐵γ𝑆	PROPN
cana-2746	169	52	∩	∩	NOUN
cana-2746	169	53	𝜒𝐵γ𝑆γ𝐵	𝜒𝐵γ𝑆γ𝐵	NUM
cana-2746	169	54	⊆	⊆	NUM
cana-2746	169	55	𝜒𝐵	𝜒𝐵	NOUN
cana-2746	169	56	hence	hence	ADV
cana-2746	169	57	𝐵	𝐵	NOUN
cana-2746	169	58	is	be	AUX
cana-2746	170	1	a	a	DET
cana-2746	170	2	bi	bi	ADJ
cana-2746	170	3	-	-	ADJ
cana-2746	170	4	quasi	quasi	NOUN
cana-2746	170	5	-	-	NOUN
cana-2746	170	6	ideal	ideal	NOUN
cana-2746	170	7	of	of	ADP
cana-2746	170	8	𝑆.	𝑆.	PROPN
cana-2746	170	9	theorem	theorem	NOUN
cana-2746	170	10	4.11	4.11	NUM
cana-2746	170	11	if	if	SCONJ
cana-2746	170	12	𝐴	𝐴	PROPN
cana-2746	170	13	and	and	CCONJ
cana-2746	170	14	𝐵	𝐵	PROPN
cana-2746	170	15	are	be	AUX
cana-2746	170	16	pflbqi	pflbqi	NOUN
cana-2746	170	17	of	of	ADP
cana-2746	170	18	𝑆	𝑆	PROPN
cana-2746	170	19	then	then	ADV
cana-2746	170	20	𝐴	𝐴	PROPN
cana-2746	170	21	∩	∩	ADJ
cana-2746	170	22	𝐵	𝐵	NOUN
cana-2746	170	23	is	be	AUX
cana-2746	170	24	pflbqi	pflbqi	NOUN
cana-2746	170	25	of	of	ADP
cana-2746	170	26	𝑆.	𝑆.	PROPN
cana-2746	170	27	proof	proof	NOUN
cana-2746	170	28	.	.	PUNCT
cana-2746	171	1	let	let	VERB
cana-2746	171	2	𝐴	𝐴	PROPN
cana-2746	171	3	and	and	CCONJ
cana-2746	171	4	𝐵	𝐵	PROPN
cana-2746	171	5	are	be	AUX
cana-2746	171	6	𝑃𝐹𝐿𝐵𝑄𝐼	𝑃𝐹𝐿𝐵𝑄𝐼	NOUN
cana-2746	171	7	of	of	ADP
cana-2746	171	8	𝑆	𝑆	PROPN
cana-2746	171	9	and	and	CCONJ
cana-2746	171	10	𝑥	𝑥	PROPN
cana-2746	171	11	,	,	PUNCT
cana-2746	171	12	𝑦	𝑦	NOUN
cana-2746	171	13	∈	∈	PROPN
cana-2746	171	14	𝑆	𝑆	PROPN
cana-2746	171	15	and	and	CCONJ
cana-2746	171	16	𝛼	𝛼	NOUN
cana-2746	171	17	,	,	PUNCT
cana-2746	171	18	𝛽	𝛽	PROPN
cana-2746	171	19	∈	∈	PROPN
cana-2746	171	20	γ	γ	X
cana-2746	171	21	.	.	PUNCT
cana-2746	172	1	(	(	PUNCT
cana-2746	172	2	𝐴𝜇	𝐴𝜇	NOUN
cana-2746	172	3	∩	∩	ADJ
cana-2746	172	4	𝐵𝜇)(𝑥	𝐵𝜇)(𝑥	PROPN
cana-2746	172	5	+	+	CCONJ
cana-2746	172	6	𝑦	𝑦	NOUN
cana-2746	172	7	)	)	PUNCT
cana-2746	172	8	=	=	SYM
cana-2746	173	1	𝑚𝑖𝑛{𝐴𝜇(𝑥	𝑚𝑖𝑛{𝐴𝜇(𝑥	NOUN
cana-2746	173	2	+	+	CCONJ
cana-2746	173	3	𝑦	𝑦	NOUN
cana-2746	173	4	)	)	PUNCT
cana-2746	173	5	,	,	PUNCT
cana-2746	173	6	𝐵𝜇(𝑥	𝐵𝜇(𝑥	PROPN
cana-2746	173	7	+	+	PROPN
cana-2746	173	8	𝑦	𝑦	NOUN
cana-2746	173	9	)	)	PUNCT
cana-2746	173	10	}	}	PUNCT
cana-2746	173	11	≥	≥	PROPN
cana-2746	173	12	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-2746	173	13	{	{	PUNCT
cana-2746	173	14	𝑚𝑖𝑛{𝐴𝜇(𝑥	𝑚𝑖𝑛{𝐴𝜇(𝑥	NOUN
cana-2746	173	15	)	)	PUNCT
cana-2746	173	16	,	,	PUNCT
cana-2746	173	17	𝐴𝜇(𝑦	𝐴𝜇(𝑦	NOUN
cana-2746	173	18	)	)	PUNCT
cana-2746	173	19	}	}	PUNCT
cana-2746	173	20	,	,	PUNCT
cana-2746	173	21	𝑚𝑖𝑛{𝐵𝜇(𝑥	𝑚𝑖𝑛{𝐵𝜇(𝑥	NOUN
cana-2746	173	22	)	)	PUNCT
cana-2746	173	23	,	,	PUNCT
cana-2746	173	24	𝐵𝜇(𝑦	𝐵𝜇(𝑦	NOUN
cana-2746	173	25	)	)	PUNCT
cana-2746	173	26	}	}	PUNCT
cana-2746	173	27	}	}	PUNCT
cana-2746	173	28	=	=	SYM
cana-2746	173	29	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-2746	173	30	{	{	PUNCT
cana-2746	173	31	𝑚𝑖𝑛{𝐴𝜇(𝑥	𝑚𝑖𝑛{𝐴𝜇(𝑥	NOUN
cana-2746	173	32	)	)	PUNCT
cana-2746	173	33	,	,	PUNCT
cana-2746	173	34	𝐵𝜇(𝑥	𝐵𝜇(𝑥	NOUN
cana-2746	173	35	)	)	PUNCT
cana-2746	173	36	}	}	PUNCT
cana-2746	173	37	,	,	PUNCT
cana-2746	173	38	𝑚𝑖𝑛{𝐴𝜇(𝑦	𝑚𝑖𝑛{𝐴𝜇(𝑦	PROPN
cana-2746	173	39	)	)	PUNCT
cana-2746	173	40	,	,	PUNCT
cana-2746	173	41	𝐵𝜇(𝑦	𝐵𝜇(𝑦	NOUN
cana-2746	173	42	)	)	PUNCT
cana-2746	173	43	}	}	PUNCT
cana-2746	173	44	}	}	PUNCT
cana-2746	173	45	=	=	SYM
cana-2746	173	46	𝑚𝑖𝑛{(𝐴𝜇	𝑚𝑖𝑛{(𝐴𝜇	PROPN
cana-2746	173	47	∩	∩	ADJ
cana-2746	173	48	𝐵𝜇)(𝑥	𝐵𝜇)(𝑥	NOUN
cana-2746	173	49	)	)	PUNCT
cana-2746	173	50	,	,	PUNCT
cana-2746	173	51	(	(	PUNCT
cana-2746	173	52	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	173	53	∩	∩	ADJ
cana-2746	173	54	𝐵𝜇)(𝑦	𝐵𝜇)(𝑦	NOUN
cana-2746	173	55	)	)	PUNCT
cana-2746	173	56	}	}	PUNCT
cana-2746	173	57	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	173	58	∘	∘	NOUN
cana-2746	173	59	(	(	PUNCT
cana-2746	173	60	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	173	61	∩	∩	ADJ
cana-2746	173	62	𝐵𝜇)(𝑥	𝐵𝜇)(𝑥	NOUN
cana-2746	173	63	)	)	PUNCT
cana-2746	173	64	=	=	SYM
cana-2746	173	65	sup	sup	NOUN
cana-2746	173	66	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PUNCT
cana-2746	173	67	{	{	PUNCT
cana-2746	173	68	𝑚𝑖𝑛{𝜒𝑆(𝑎	𝑚𝑖𝑛{𝜒𝑆(𝑎	NOUN
cana-2746	173	69	)	)	PUNCT
cana-2746	173	70	,	,	PUNCT
cana-2746	173	71	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	173	72	∩	∩	ADJ
cana-2746	173	73	𝐵𝜇(𝑏	𝐵𝜇(𝑏	NOUN
cana-2746	173	74	)	)	PUNCT
cana-2746	173	75	}	}	PUNCT
cana-2746	173	76	}	}	PUNCT
cana-2746	173	77	=	=	SYM
cana-2746	173	78	sup	sup	NOUN
cana-2746	173	79	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	173	80	{	{	PUNCT
cana-2746	173	81	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-2746	173	82	{	{	PUNCT
cana-2746	173	83	𝜒𝑆(𝑎	𝜒𝑆(𝑎	NOUN
cana-2746	173	84	)	)	PUNCT
cana-2746	173	85	,	,	PUNCT
cana-2746	173	86	𝑚𝑖𝑛{𝐴𝜇(𝑏	𝑚𝑖𝑛{𝐴𝜇(𝑏	PROPN
cana-2746	173	87	)	)	PUNCT
cana-2746	173	88	,	,	PUNCT
cana-2746	173	89	𝐵𝜇(𝑏	𝐵𝜇(𝑏	NOUN
cana-2746	173	90	)	)	PUNCT
cana-2746	173	91	}	}	PUNCT
cana-2746	173	92	}	}	PUNCT
cana-2746	173	93	}	}	PUNCT
cana-2746	173	94	communications	communication	NOUN
cana-2746	173	95	on	on	ADP
cana-2746	173	96	applied	apply	VERB
cana-2746	173	97	nonlinear	nonlinear	ADJ
cana-2746	173	98	analysis	analysis	NOUN
cana-2746	173	99	issn	issn	NOUN
cana-2746	173	100	:	:	PUNCT
cana-2746	173	101	1074	1074	NUM
cana-2746	173	102	-	-	PUNCT
cana-2746	173	103	133x	133x	NUM
cana-2746	173	104	vol	vol	NOUN
cana-2746	173	105	32	32	NUM
cana-2746	173	106	no	no	NOUN
cana-2746	173	107	.	.	PUNCT
cana-2746	174	1	4s	4s	NUM
cana-2746	174	2	(	(	PUNCT
cana-2746	174	3	2025	2025	NUM
cana-2746	174	4	)	)	PUNCT
cana-2746	174	5	163	163	NUM
cana-2746	174	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2746	174	7	=	=	SYM
cana-2746	174	8	sup	sup	NOUN
cana-2746	174	9	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	174	10	{	{	PUNCT
cana-2746	174	11	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-2746	174	12	{	{	PUNCT
cana-2746	174	13	𝑚𝑖𝑛{𝜒𝑆(𝑎	𝑚𝑖𝑛{𝜒𝑆(𝑎	NOUN
cana-2746	174	14	)	)	PUNCT
cana-2746	174	15	,	,	PUNCT
cana-2746	174	16	𝐴𝜇(𝑏	𝐴𝜇(𝑏	NOUN
cana-2746	174	17	)	)	PUNCT
cana-2746	174	18	}	}	PUNCT
cana-2746	174	19	,	,	PUNCT
cana-2746	174	20	𝑚𝑖𝑛{𝜒𝑆(𝑎	𝑚𝑖𝑛{𝜒𝑆(𝑎	NOUN
cana-2746	174	21	)	)	PUNCT
cana-2746	174	22	,	,	PUNCT
cana-2746	174	23	𝐵𝜇(𝑏	𝐵𝜇(𝑏	NOUN
cana-2746	174	24	)	)	PUNCT
cana-2746	174	25	}	}	PUNCT
cana-2746	174	26	}	}	PUNCT
cana-2746	174	27	}	}	PUNCT
cana-2746	174	28	=	=	SYM
cana-2746	174	29	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-2746	174	30	{	{	PUNCT
cana-2746	174	31	sup	sup	NOUN
cana-2746	174	32	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	174	33	{	{	PUNCT
cana-2746	174	34	𝑚𝑖𝑛{𝜒𝑆(𝑎	𝑚𝑖𝑛{𝜒𝑆(𝑎	NOUN
cana-2746	174	35	)	)	PUNCT
cana-2746	174	36	,	,	PUNCT
cana-2746	174	37	𝐴𝜇(𝑏	𝐴𝜇(𝑏	NOUN
cana-2746	174	38	)	)	PUNCT
cana-2746	174	39	}	}	PUNCT
cana-2746	174	40	}	}	PUNCT
cana-2746	174	41	,	,	PUNCT
cana-2746	174	42	sup	sup	NOUN
cana-2746	174	43	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PRON
cana-2746	174	44	{	{	PUNCT
cana-2746	174	45	𝑚𝑖𝑛{𝜒𝑆(𝑎	𝑚𝑖𝑛{𝜒𝑆(𝑎	NOUN
cana-2746	174	46	)	)	PUNCT
cana-2746	174	47	,	,	PUNCT
cana-2746	174	48	𝐵𝜇(𝑏	𝐵𝜇(𝑏	NOUN
cana-2746	174	49	)	)	PUNCT
cana-2746	174	50	}	}	PUNCT
cana-2746	174	51	}	}	PUNCT
cana-2746	174	52	}	}	PUNCT
cana-2746	174	53	=	=	PUNCT
cana-2746	174	54	𝑚𝑖𝑛{𝜒𝑆	𝑚𝑖𝑛{𝜒𝑆	VERB
cana-2746	174	55	∘	∘	PROPN
cana-2746	174	56	𝐴𝜇(𝑥	𝐴𝜇(𝑥	NOUN
cana-2746	174	57	)	)	PUNCT
cana-2746	174	58	,	,	PUNCT
cana-2746	174	59	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	174	60	∘	∘	PROPN
cana-2746	174	61	𝐵𝜇(𝑥	𝐵𝜇(𝑥	NOUN
cana-2746	174	62	)	)	PUNCT
cana-2746	174	63	}	}	PUNCT
cana-2746	174	64	=	=	SYM
cana-2746	174	65	(	(	PUNCT
cana-2746	174	66	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	174	67	∘	∘	PROPN
cana-2746	174	68	𝐴𝜇	𝐴𝜇	NOUN
cana-2746	174	69	)	)	PUNCT
cana-2746	174	70	∩	∩	NOUN
cana-2746	174	71	(	(	PUNCT
cana-2746	174	72	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	174	73	∘	∘	NOUN
cana-2746	174	74	𝐵𝜇)(𝑥	𝐵𝜇)(𝑥	NOUN
cana-2746	174	75	)	)	PUNCT
cana-2746	174	76	(	(	PUNCT
cana-2746	174	77	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	174	78	∩	∩	ADJ
cana-2746	174	79	𝐵𝜇	𝐵𝜇	NOUN
cana-2746	174	80	)	)	PUNCT
cana-2746	174	81	∘	∘	PROPN
cana-2746	174	82	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	174	83	∘	∘	PROPN
cana-2746	174	84	(	(	PUNCT
cana-2746	174	85	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	174	86	∩	∩	ADJ
cana-2746	174	87	𝐵𝜇)(𝑥	𝐵𝜇)(𝑥	NOUN
cana-2746	174	88	)	)	PUNCT
cana-2746	174	89	=	=	SYM
cana-2746	174	90	sup	sup	NOUN
cana-2746	174	91	𝑥=𝑎𝛼𝑏𝛽𝑐	𝑥=𝑎𝛼𝑏𝛽𝑐	ADP
cana-2746	174	92	{	{	PUNCT
cana-2746	174	93	𝑚𝑖𝑛{𝐴𝜇	𝑚𝑖𝑛{𝐴𝜇	PROPN
cana-2746	174	94	∩	∩	ADJ
cana-2746	174	95	𝐵𝜇(𝑎	𝐵𝜇(𝑎	NOUN
cana-2746	174	96	)	)	PUNCT
cana-2746	174	97	,	,	PUNCT
cana-2746	174	98	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	174	99	∘	∘	NOUN
cana-2746	174	100	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	174	101	∩	∩	ADJ
cana-2746	174	102	𝐵𝜇(𝑏𝛽𝑐	𝐵𝜇(𝑏𝛽𝑐	NOUN
cana-2746	174	103	)	)	PUNCT
cana-2746	174	104	}	}	PUNCT
cana-2746	174	105	}	}	PUNCT
cana-2746	174	106	=	=	SYM
cana-2746	174	107	sup	sup	NOUN
cana-2746	174	108	𝑥=𝑎𝛼𝑏𝛽𝑐	𝑥=𝑎𝛼𝑏𝛽𝑐	ADP
cana-2746	174	109	{	{	PUNCT
cana-2746	174	110	𝑚𝑖𝑛{(𝐴𝜇	𝑚𝑖𝑛{(𝐴𝜇	PROPN
cana-2746	174	111	∩	∩	ADJ
cana-2746	174	112	𝐵𝜇)(𝑎	𝐵𝜇)(𝑎	NOUN
cana-2746	174	113	)	)	PUNCT
cana-2746	174	114	,	,	PUNCT
cana-2746	174	115	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	174	116	∘	∘	NOUN
cana-2746	174	117	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	174	118	∩	∩	NOUN
cana-2746	174	119	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	174	120	∘	∘	PROPN
cana-2746	174	121	𝐵𝜇(𝑏𝛽𝑐	𝐵𝜇(𝑏𝛽𝑐	NOUN
cana-2746	174	122	)	)	PUNCT
cana-2746	174	123	}	}	PUNCT
cana-2746	174	124	}	}	PUNCT
cana-2746	174	125	=	=	SYM
cana-2746	174	126	sup	sup	NOUN
cana-2746	174	127	𝑥=𝑎𝛼𝑏𝛽𝑐	𝑥=𝑎𝛼𝑏𝛽𝑐	ADP
cana-2746	174	128	{	{	PUNCT
cana-2746	174	129	𝑚𝑖𝑛	𝑚𝑖𝑛	PROPN
cana-2746	174	130	{	{	PUNCT
cana-2746	174	131	𝑚𝑖𝑛{𝐴𝜇(𝑎	𝑚𝑖𝑛{𝐴𝜇(𝑎	PROPN
cana-2746	174	132	)	)	PUNCT
cana-2746	174	133	,	,	PUNCT
cana-2746	174	134	𝐵𝜇(𝑎	𝐵𝜇(𝑎	NOUN
cana-2746	174	135	)	)	PUNCT
cana-2746	174	136	}	}	PUNCT
cana-2746	174	137	,	,	PUNCT
cana-2746	174	138	𝑚𝑖𝑛{(𝜒𝑆	𝑚𝑖𝑛{(𝜒𝑆	PUNCT
cana-2746	174	139	∘	∘	PROPN
cana-2746	174	140	𝐴𝜇)(𝑏𝛽𝑐	𝐴𝜇)(𝑏𝛽𝑐	NOUN
cana-2746	174	141	)	)	PUNCT
cana-2746	174	142	,	,	PUNCT
cana-2746	174	143	(	(	PUNCT
cana-2746	174	144	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	174	145	∘	∘	PROPN
cana-2746	174	146	𝐵𝜇)(𝑏𝛽𝑐	𝐵𝜇)(𝑏𝛽𝑐	NOUN
cana-2746	174	147	)	)	PUNCT
cana-2746	174	148	}	}	PUNCT
cana-2746	174	149	}	}	PUNCT
cana-2746	174	150	}	}	PUNCT
cana-2746	174	151	=	=	PUNCT
cana-2746	174	152	𝑚𝑖𝑛{𝐴𝜇	𝑚𝑖𝑛{𝐴𝜇	PROPN
cana-2746	174	153	∘	∘	X
cana-2746	174	154	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	174	155	∘	∘	PROPN
cana-2746	174	156	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	174	157	)	)	PUNCT
cana-2746	174	158	,	,	PUNCT
cana-2746	174	159	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	174	160	∘	∘	ADJ
cana-2746	174	161	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	174	162	∘	∘	PROPN
cana-2746	174	163	𝐵𝜇(𝑥	𝐵𝜇(𝑥	NOUN
cana-2746	174	164	)	)	PUNCT
cana-2746	174	165	}	}	PUNCT
cana-2746	174	166	therefore	therefore	ADV
cana-2746	174	167	(	(	PUNCT
cana-2746	174	168	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	174	169	∩	∩	ADJ
cana-2746	174	170	𝐵𝜇	𝐵𝜇	NOUN
cana-2746	174	171	)	)	PUNCT
cana-2746	174	172	∘	∘	PROPN
cana-2746	174	173	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	174	174	∘	∘	PROPN
cana-2746	174	175	(	(	PUNCT
cana-2746	174	176	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	174	177	∩	∩	ADJ
cana-2746	174	178	𝐵𝜇	𝐵𝜇	NOUN
cana-2746	174	179	)	)	PUNCT
cana-2746	174	180	=	=	PUNCT
cana-2746	175	1	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	175	2	∘	∘	X
cana-2746	175	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	175	4	∘	∘	NOUN
cana-2746	175	5	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	175	6	∩	∩	PUNCT
cana-2746	175	7	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	175	8	∘	∘	ADJ
cana-2746	175	9	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	175	10	∘	∘	PROPN
cana-2746	175	11	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	175	12	hence	hence	ADV
cana-2746	175	13	𝜒𝑆	𝜒𝑆	VERB
cana-2746	175	14	∘	∘	NOUN
cana-2746	175	15	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	175	16	∩	∩	PUNCT
cana-2746	175	17	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	175	18	∘	∘	ADJ
cana-2746	175	19	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	175	20	∩	∩	NOUN
cana-2746	175	21	𝐴𝜇	𝐴𝜇	NOUN
cana-2746	175	22	∩	∩	NOUN
cana-2746	175	23	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	175	24	∘	∘	ADJ
cana-2746	175	25	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	175	26	∘	∘	NOUN
cana-2746	175	27	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	175	28	∩	∩	NOUN
cana-2746	175	29	𝐵𝜇	𝐵𝜇	NOUN
cana-2746	175	30	=	=	SYM
cana-2746	175	31	(	(	PUNCT
cana-2746	175	32	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	175	33	∘	∘	PROPN
cana-2746	175	34	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	175	35	∘	∘	ADJ
cana-2746	175	36	𝜒𝑆	𝜒𝑆	NOUN
cana-2746	175	37	)	)	PUNCT
cana-2746	175	38	∩	∩	NOUN
cana-2746	175	39	(	(	PUNCT
cana-2746	175	40	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	175	41	∘	∘	PROPN
cana-2746	175	42	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	175	43	∘	∘	PROPN
cana-2746	175	44	𝐴𝜇	𝐴𝜇	NOUN
cana-2746	175	45	)	)	PUNCT
cana-2746	175	46	∩	∩	NOUN
cana-2746	175	47	(	(	PUNCT
cana-2746	175	48	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	175	49	∘	∘	PROPN
cana-2746	175	50	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	175	51	∘	∘	ADJ
cana-2746	175	52	𝜒𝑆	𝜒𝑆	NOUN
cana-2746	175	53	)	)	PUNCT
cana-2746	175	54	∩	∩	NOUN
cana-2746	175	55	(	(	PUNCT
cana-2746	175	56	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	175	57	∘	∘	ADJ
cana-2746	175	58	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	175	59	∘	∘	PROPN
cana-2746	175	60	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	175	61	)	)	PUNCT
cana-2746	175	62	⊇	⊇	NOUN
cana-2746	175	63	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	175	64	∩	∩	PUNCT
cana-2746	175	65	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	175	66	similarly	similarly	ADV
cana-2746	175	67	we	we	PRON
cana-2746	175	68	can	can	AUX
cana-2746	175	69	prove	prove	VERB
cana-2746	175	70	for	for	ADP
cana-2746	175	71	non	non	NOUN
cana-2746	175	72	membership	membership	NOUN
cana-2746	175	73	(	(	PUNCT
cana-2746	175	74	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	175	75	∩	∩	NOUN
cana-2746	175	76	𝐵𝜈)(𝑥	𝐵𝜈)(𝑥	PROPN
cana-2746	175	77	+	+	PROPN
cana-2746	175	78	𝑦	𝑦	NOUN
cana-2746	175	79	)	)	PUNCT
cana-2746	175	80	=	=	SYM
cana-2746	176	1	𝑚𝑎𝑥{𝐴𝜈(𝑥	𝑚𝑎𝑥{𝐴𝜈(𝑥	PROPN
cana-2746	176	2	+	+	NUM
cana-2746	176	3	𝑦	𝑦	NOUN
cana-2746	176	4	)	)	PUNCT
cana-2746	176	5	,	,	PUNCT
cana-2746	176	6	𝐵𝜈(𝑥	𝐵𝜈(𝑥	PROPN
cana-2746	176	7	+	+	PUNCT
cana-2746	176	8	𝑦	𝑦	X
cana-2746	176	9	)	)	PUNCT
cana-2746	176	10	}	}	PUNCT
cana-2746	176	11	≤	≤	NUM
cana-2746	176	12	𝑚𝑎𝑥{𝑚𝑎𝑥{𝐴𝜈(𝑥	𝑚𝑎𝑥{𝑚𝑎𝑥{𝐴𝜈(𝑥	NOUN
cana-2746	176	13	)	)	PUNCT
cana-2746	176	14	,	,	PUNCT
cana-2746	176	15	𝐴𝜈(𝑦	𝐴𝜈(𝑦	NOUN
cana-2746	176	16	)	)	PUNCT
cana-2746	176	17	}	}	PUNCT
cana-2746	176	18	,	,	PUNCT
cana-2746	176	19	𝑚𝑎𝑥{𝐵𝜈(𝑥	𝑚𝑎𝑥{𝐵𝜈(𝑥	PROPN
cana-2746	176	20	)	)	PUNCT
cana-2746	176	21	,	,	PUNCT
cana-2746	176	22	𝐵𝜈(𝑦	𝐵𝜈(𝑦	PROPN
cana-2746	176	23	)	)	PUNCT
cana-2746	176	24	}	}	PUNCT
cana-2746	176	25	}	}	PUNCT
cana-2746	176	26	=	=	SYM
cana-2746	176	27	𝑚𝑎𝑥{𝑚𝑎𝑥{𝐴𝜈(𝑥	𝑚𝑎𝑥{𝑚𝑎𝑥{𝐴𝜈(𝑥	PROPN
cana-2746	176	28	)	)	PUNCT
cana-2746	176	29	,	,	PUNCT
cana-2746	176	30	𝐵𝜈(𝑥	𝐵𝜈(𝑥	PROPN
cana-2746	176	31	)	)	PUNCT
cana-2746	176	32	}	}	PUNCT
cana-2746	176	33	,	,	PUNCT
cana-2746	176	34	𝑚𝑎𝑥{𝐴𝜈(𝑦	𝑚𝑎𝑥{𝐴𝜈(𝑦	PROPN
cana-2746	176	35	)	)	PUNCT
cana-2746	176	36	,	,	PUNCT
cana-2746	176	37	𝐵𝜈(𝑦	𝐵𝜈(𝑦	PROPN
cana-2746	176	38	)	)	PUNCT
cana-2746	176	39	}	}	PUNCT
cana-2746	176	40	}	}	PUNCT
cana-2746	176	41	=	=	SYM
cana-2746	176	42	𝑚𝑎𝑥{(𝐴𝜈	𝑚𝑎𝑥{(𝐴𝜈	PROPN
cana-2746	176	43	∩	∩	X
cana-2746	176	44	𝐵𝜈)(𝑥	𝐵𝜈)(𝑥	PROPN
cana-2746	176	45	)	)	PUNCT
cana-2746	176	46	,	,	PUNCT
cana-2746	176	47	(	(	PUNCT
cana-2746	176	48	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	176	49	∩	∩	NOUN
cana-2746	176	50	𝐵𝜈)(𝑦	𝐵𝜈)(𝑦	NOUN
cana-2746	176	51	)	)	PUNCT
cana-2746	176	52	}	}	PUNCT
cana-2746	176	53	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	176	54	∘	∘	NOUN
cana-2746	176	55	(	(	PUNCT
cana-2746	176	56	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	176	57	∩	∩	NOUN
cana-2746	176	58	𝐵𝜈)(𝑥	𝐵𝜈)(𝑥	NOUN
cana-2746	176	59	)	)	PUNCT
cana-2746	176	60	=	=	PROPN
cana-2746	176	61	inf	inf	NOUN
cana-2746	176	62	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	X
cana-2746	176	63	{	{	PUNCT
cana-2746	176	64	𝑚𝑎𝑥{𝜒𝑆(𝑎	𝑚𝑎𝑥{𝜒𝑆(𝑎	NUM
cana-2746	176	65	)	)	PUNCT
cana-2746	176	66	,	,	PUNCT
cana-2746	176	67	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	176	68	∩	∩	ADJ
cana-2746	176	69	𝐵𝜇(𝑏	𝐵𝜇(𝑏	NOUN
cana-2746	176	70	)	)	PUNCT
cana-2746	176	71	}	}	PUNCT
cana-2746	176	72	}	}	PUNCT
cana-2746	176	73	=	=	SYM
cana-2746	176	74	inf	inf	NOUN
cana-2746	176	75	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	X
cana-2746	176	76	{	{	PUNCT
cana-2746	176	77	𝑚𝑎𝑥	𝑚𝑎𝑥	X
cana-2746	176	78	{	{	PUNCT
cana-2746	176	79	𝜒𝑆(𝑎	𝜒𝑆(𝑎	NOUN
cana-2746	176	80	)	)	PUNCT
cana-2746	176	81	,	,	PUNCT
cana-2746	176	82	𝑚𝑎𝑥{𝐴𝜈(𝑏	𝑚𝑎𝑥{𝐴𝜈(𝑏	PROPN
cana-2746	176	83	)	)	PUNCT
cana-2746	176	84	,	,	PUNCT
cana-2746	176	85	𝐵𝜇(𝑏	𝐵𝜇(𝑏	NOUN
cana-2746	176	86	)	)	PUNCT
cana-2746	176	87	}	}	PUNCT
cana-2746	176	88	}	}	PUNCT
cana-2746	176	89	}	}	PUNCT
cana-2746	176	90	=	=	SYM
cana-2746	176	91	inf	inf	NOUN
cana-2746	176	92	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	176	93	{	{	PUNCT
cana-2746	176	94	𝑚𝑎𝑥{𝑚𝑎𝑥{𝜒𝑆(𝑎	𝑚𝑎𝑥{𝑚𝑎𝑥{𝜒𝑆(𝑎	NOUN
cana-2746	176	95	)	)	PUNCT
cana-2746	176	96	,	,	PUNCT
cana-2746	176	97	𝐴𝜈(𝑏	𝐴𝜈(𝑏	NOUN
cana-2746	176	98	)	)	PUNCT
cana-2746	176	99	}	}	PUNCT
cana-2746	176	100	,	,	PUNCT
cana-2746	176	101	𝑚𝑎𝑥{𝜒𝑆(𝑎	𝑚𝑎𝑥{𝜒𝑆(𝑎	NUM
cana-2746	176	102	)	)	PUNCT
cana-2746	176	103	,	,	PUNCT
cana-2746	176	104	𝐵𝜈(𝑏	𝐵𝜈(𝑏	NOUN
cana-2746	176	105	)	)	PUNCT
cana-2746	176	106	}	}	PUNCT
cana-2746	176	107	}	}	PUNCT
cana-2746	176	108	}	}	PUNCT
cana-2746	176	109	=	=	PUNCT
cana-2746	176	110	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-2746	176	111	{	{	PUNCT
cana-2746	176	112	inf	inf	NOUN
cana-2746	176	113	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	176	114	{	{	PUNCT
cana-2746	176	115	𝑚𝑎𝑥{𝜒𝑆(𝑎	𝑚𝑎𝑥{𝜒𝑆(𝑎	NOUN
cana-2746	176	116	)	)	PUNCT
cana-2746	176	117	,	,	PUNCT
cana-2746	176	118	𝐴𝜈(𝑏	𝐴𝜈(𝑏	NOUN
cana-2746	176	119	)	)	PUNCT
cana-2746	176	120	}	}	PUNCT
cana-2746	176	121	}	}	PUNCT
cana-2746	176	122	,	,	PUNCT
cana-2746	176	123	inf	inf	PROPN
cana-2746	176	124	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	X
cana-2746	176	125	{	{	PUNCT
cana-2746	176	126	𝑚𝑎𝑥{𝜒𝑆(𝑎	𝑚𝑎𝑥{𝜒𝑆(𝑎	NOUN
cana-2746	176	127	)	)	PUNCT
cana-2746	176	128	,	,	PUNCT
cana-2746	176	129	𝐵𝜈(𝑏	𝐵𝜈(𝑏	NOUN
cana-2746	176	130	)	)	PUNCT
cana-2746	176	131	}	}	PUNCT
cana-2746	176	132	}	}	PUNCT
cana-2746	176	133	}	}	PUNCT
cana-2746	176	134	=	=	SYM
cana-2746	176	135	𝑚𝑎𝑥{𝜒𝑆	𝑚𝑎𝑥{𝜒𝑆	ADJ
cana-2746	176	136	∘	∘	PROPN
cana-2746	176	137	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	176	138	)	)	PUNCT
cana-2746	176	139	,	,	PUNCT
cana-2746	176	140	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	176	141	∘	∘	PROPN
cana-2746	176	142	𝐵𝜈(𝑥	𝐵𝜈(𝑥	PROPN
cana-2746	176	143	)	)	PUNCT
cana-2746	176	144	}	}	PUNCT
cana-2746	176	145	=	=	SYM
cana-2746	176	146	(	(	PUNCT
cana-2746	176	147	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	176	148	∘	∘	NUM
cana-2746	176	149	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	176	150	)	)	PUNCT
cana-2746	176	151	∩	∩	NOUN
cana-2746	176	152	(	(	PUNCT
cana-2746	176	153	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	176	154	∘	∘	PROPN
cana-2746	176	155	𝐵𝜈)(𝑥	𝐵𝜈)(𝑥	NOUN
cana-2746	176	156	)	)	PUNCT
cana-2746	176	157	communications	communication	NOUN
cana-2746	176	158	on	on	ADP
cana-2746	176	159	applied	apply	VERB
cana-2746	176	160	nonlinear	nonlinear	ADJ
cana-2746	176	161	analysis	analysis	NOUN
cana-2746	176	162	issn	issn	NOUN
cana-2746	176	163	:	:	PUNCT
cana-2746	176	164	1074	1074	NUM
cana-2746	176	165	-	-	PUNCT
cana-2746	176	166	133x	133x	NUM
cana-2746	176	167	vol	vol	NOUN
cana-2746	176	168	32	32	NUM
cana-2746	176	169	no	no	NOUN
cana-2746	176	170	.	.	PUNCT
cana-2746	177	1	4s	4s	NUM
cana-2746	177	2	(	(	PUNCT
cana-2746	177	3	2025	2025	NUM
cana-2746	177	4	)	)	PUNCT
cana-2746	177	5	164	164	NUM
cana-2746	177	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2746	177	7	(	(	PUNCT
cana-2746	177	8	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	177	9	∩	∩	NOUN
cana-2746	177	10	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	177	11	)	)	PUNCT
cana-2746	177	12	∘	∘	PROPN
cana-2746	177	13	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	177	14	∘	∘	X
cana-2746	177	15	(	(	PUNCT
cana-2746	177	16	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	177	17	∩	∩	NOUN
cana-2746	177	18	𝐵𝜈)(𝑥	𝐵𝜈)(𝑥	NOUN
cana-2746	177	19	)	)	PUNCT
cana-2746	177	20	=	=	SYM
cana-2746	177	21	inf	inf	PROPN
cana-2746	177	22	𝑥=𝑎𝛼𝑏𝛽𝑐	𝑥=𝑎𝛼𝑏𝛽𝑐	ADP
cana-2746	177	23	{	{	PUNCT
cana-2746	177	24	𝑚𝑎𝑥{𝐴𝜈	𝑚𝑎𝑥{𝐴𝜈	NOUN
cana-2746	177	25	∩	∩	NOUN
cana-2746	177	26	𝐵𝜈(𝑎	𝐵𝜈(𝑎	NOUN
cana-2746	177	27	)	)	PUNCT
cana-2746	177	28	,	,	PUNCT
cana-2746	177	29	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	177	30	∘	∘	NOUN
cana-2746	177	31	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	177	32	∩	∩	ADJ
cana-2746	177	33	𝐵𝜈(𝑏𝛽𝑐	𝐵𝜈(𝑏𝛽𝑐	NOUN
cana-2746	177	34	)	)	PUNCT
cana-2746	177	35	}	}	PUNCT
cana-2746	177	36	}	}	PUNCT
cana-2746	177	37	=	=	SYM
cana-2746	177	38	inf	inf	PROPN
cana-2746	177	39	𝑥=𝑎𝛼𝑏𝛽𝑐	𝑥=𝑎𝛼𝑏𝛽𝑐	ADP
cana-2746	177	40	{	{	PUNCT
cana-2746	177	41	𝑚𝑎𝑥{(𝐴𝜈	𝑚𝑎𝑥{(𝐴𝜈	PROPN
cana-2746	177	42	∩	∩	ADJ
cana-2746	177	43	𝐵𝜈)(𝑎	𝐵𝜈)(𝑎	NOUN
cana-2746	177	44	)	)	PUNCT
cana-2746	177	45	,	,	PUNCT
cana-2746	177	46	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	177	47	∘	∘	ADJ
cana-2746	177	48	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	177	49	∩	∩	NOUN
cana-2746	177	50	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	177	51	∘	∘	PROPN
cana-2746	177	52	𝐵𝜈(𝑏𝛽𝑐	𝐵𝜈(𝑏𝛽𝑐	PROPN
cana-2746	177	53	)	)	PUNCT
cana-2746	177	54	}	}	PUNCT
cana-2746	177	55	}	}	PUNCT
cana-2746	177	56	=	=	SYM
cana-2746	177	57	inf	inf	NOUN
cana-2746	177	58	𝑥=𝑎𝛼𝑏𝛽𝑐	𝑥=𝑎𝛼𝑏𝛽𝑐	ADP
cana-2746	177	59	{	{	PUNCT
cana-2746	177	60	𝑚𝑎𝑥{𝑚𝑎𝑥{𝐴𝜈(𝑎	𝑚𝑎𝑥{𝑚𝑎𝑥{𝐴𝜈(𝑎	NOUN
cana-2746	177	61	)	)	PUNCT
cana-2746	177	62	,	,	PUNCT
cana-2746	177	63	𝐵𝜈(𝑎	𝐵𝜈(𝑎	PROPN
cana-2746	177	64	)	)	PUNCT
cana-2746	177	65	}	}	PUNCT
cana-2746	177	66	,	,	PUNCT
cana-2746	177	67	𝑚𝑎𝑥{(𝜒𝑆	𝑚𝑎𝑥{(𝜒𝑆	ADP
cana-2746	177	68	∘	∘	PROPN
cana-2746	177	69	𝐴𝜈)(𝑏𝛽𝑐	𝐴𝜈)(𝑏𝛽𝑐	NOUN
cana-2746	177	70	)	)	PUNCT
cana-2746	177	71	,	,	PUNCT
cana-2746	177	72	(	(	PUNCT
cana-2746	177	73	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	177	74	∘	∘	PROPN
cana-2746	177	75	𝐵𝜈)(𝑏𝛽𝑐	𝐵𝜈)(𝑏𝛽𝑐	NOUN
cana-2746	177	76	)	)	PUNCT
cana-2746	177	77	}	}	PUNCT
cana-2746	177	78	}	}	PUNCT
cana-2746	177	79	}	}	PUNCT
cana-2746	177	80	=	=	PUNCT
cana-2746	178	1	𝑚𝑎𝑥{𝐴𝜈	𝑚𝑎𝑥{𝐴𝜈	PRON
cana-2746	178	2	∘	∘	X
cana-2746	178	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	178	4	∘	∘	PROPN
cana-2746	178	5	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	178	6	)	)	PUNCT
cana-2746	178	7	,	,	PUNCT
cana-2746	178	8	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	178	9	∘	∘	PROPN
cana-2746	178	10	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	178	11	∘	∘	PROPN
cana-2746	178	12	𝐵𝜈(𝑥	𝐵𝜈(𝑥	PROPN
cana-2746	178	13	)	)	PUNCT
cana-2746	178	14	}	}	PUNCT
cana-2746	178	15	therefore	therefore	ADV
cana-2746	178	16	(	(	PUNCT
cana-2746	178	17	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	178	18	∩	∩	NOUN
cana-2746	178	19	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	178	20	)	)	PUNCT
cana-2746	178	21	∘	∘	PROPN
cana-2746	178	22	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	178	23	∘	∘	X
cana-2746	178	24	(	(	PUNCT
cana-2746	178	25	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	178	26	∩	∩	NOUN
cana-2746	178	27	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	178	28	)	)	PUNCT
cana-2746	178	29	=	=	PUNCT
cana-2746	179	1	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	179	2	∘	∘	NOUN
cana-2746	179	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	179	4	∘	∘	NOUN
cana-2746	179	5	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	179	6	∩	∩	NOUN
cana-2746	179	7	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	179	8	∘	∘	PROPN
cana-2746	179	9	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	179	10	∘	∘	NOUN
cana-2746	179	11	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	179	12	hence	hence	ADV
cana-2746	179	13	𝜒𝑆	𝜒𝑆	VERB
cana-2746	179	14	∘	∘	NOUN
cana-2746	179	15	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	179	16	∩	∩	ADJ
cana-2746	179	17	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	179	18	∘	∘	ADJ
cana-2746	179	19	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	179	20	∩	∩	ADJ
cana-2746	179	21	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	179	22	∩	∩	NOUN
cana-2746	179	23	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	179	24	∘	∘	PROPN
cana-2746	179	25	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	179	26	∘	∘	NOUN
cana-2746	180	1	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	180	2	∩	∩	ADJ
cana-2746	180	3	𝐵𝜇	𝐵𝜇	NOUN
cana-2746	180	4	=	=	SYM
cana-2746	180	5	(	(	PUNCT
cana-2746	180	6	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	180	7	∘	∘	NOUN
cana-2746	180	8	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	180	9	∘	∘	ADJ
cana-2746	180	10	𝜒𝑆	𝜒𝑆	NOUN
cana-2746	180	11	)	)	PUNCT
cana-2746	180	12	∩	∩	NOUN
cana-2746	180	13	(	(	PUNCT
cana-2746	180	14	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	180	15	∘	∘	PROPN
cana-2746	180	16	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	180	17	∘	∘	NUM
cana-2746	180	18	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	180	19	)	)	PUNCT
cana-2746	180	20	∩	∩	NOUN
cana-2746	180	21	(	(	PUNCT
cana-2746	180	22	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	180	23	∘	∘	PROPN
cana-2746	180	24	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	180	25	∘	∘	ADJ
cana-2746	180	26	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	180	27	)	)	PUNCT
cana-2746	180	28	∩	∩	NOUN
cana-2746	180	29	(	(	PUNCT
cana-2746	180	30	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	180	31	∘	∘	PROPN
cana-2746	180	32	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	180	33	∘	∘	PROPN
cana-2746	180	34	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	180	35	)	)	PUNCT
cana-2746	180	36	⊆	⊆	NUM
cana-2746	180	37	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	180	38	∩	∩	ADJ
cana-2746	180	39	𝐵𝜈.	𝐵𝜈.	PROPN
cana-2746	180	40	theorem	theorem	VERB
cana-2746	180	41	4.12	4.12	NUM
cana-2746	180	42	if	if	SCONJ
cana-2746	180	43	𝐴	𝐴	PROPN
cana-2746	180	44	and	and	CCONJ
cana-2746	180	45	𝐵	𝐵	PROPN
cana-2746	180	46	are	be	AUX
cana-2746	180	47	pfrbqi	pfrbqi	ADJ
cana-2746	180	48	of	of	ADP
cana-2746	180	49	𝑆	𝑆	PROPN
cana-2746	180	50	then	then	ADV
cana-2746	180	51	𝐴	𝐴	PROPN
cana-2746	180	52	∩	∩	ADJ
cana-2746	180	53	𝐵	𝐵	NOUN
cana-2746	180	54	is	be	AUX
cana-2746	180	55	pfrbqi	pfrbqi	ADJ
cana-2746	180	56	of	of	ADP
cana-2746	180	57	𝑆.	𝑆.	PROPN
cana-2746	180	58	proof	proof	NOUN
cana-2746	180	59	.	.	PUNCT
cana-2746	181	1	let	let	VERB
cana-2746	181	2	𝐴	𝐴	PROPN
cana-2746	181	3	and	and	CCONJ
cana-2746	181	4	𝐵	𝐵	NOUN
cana-2746	181	5	are	be	AUX
cana-2746	181	6	𝑃𝐹𝑅𝐵𝑄𝐼	𝑃𝐹𝑅𝐵𝑄𝐼	PROPN
cana-2746	181	7	of	of	ADP
cana-2746	181	8	𝑆	𝑆	PROPN
cana-2746	181	9	and	and	CCONJ
cana-2746	181	10	𝑥	𝑥	PROPN
cana-2746	181	11	,	,	PUNCT
cana-2746	181	12	𝑦	𝑦	NOUN
cana-2746	181	13	∈	∈	PROPN
cana-2746	181	14	𝑆	𝑆	PROPN
cana-2746	181	15	and	and	CCONJ
cana-2746	181	16	𝛼	𝛼	NOUN
cana-2746	181	17	,	,	PUNCT
cana-2746	181	18	𝛽	𝛽	PROPN
cana-2746	181	19	∈	∈	PROPN
cana-2746	181	20	γ	γ	X
cana-2746	181	21	.	.	PUNCT
cana-2746	182	1	(	(	PUNCT
cana-2746	182	2	𝐴𝜇	𝐴𝜇	NOUN
cana-2746	182	3	∩	∩	ADJ
cana-2746	182	4	𝐵𝜇)(𝑥	𝐵𝜇)(𝑥	PROPN
cana-2746	182	5	+	+	CCONJ
cana-2746	182	6	𝑦	𝑦	NOUN
cana-2746	182	7	)	)	PUNCT
cana-2746	182	8	=	=	SYM
cana-2746	183	1	𝑚𝑖𝑛{𝐴𝜇(𝑥	𝑚𝑖𝑛{𝐴𝜇(𝑥	NOUN
cana-2746	183	2	+	+	CCONJ
cana-2746	183	3	𝑦	𝑦	NOUN
cana-2746	183	4	)	)	PUNCT
cana-2746	183	5	,	,	PUNCT
cana-2746	183	6	𝐵𝜇(𝑥	𝐵𝜇(𝑥	PROPN
cana-2746	183	7	+	+	PROPN
cana-2746	183	8	𝑦	𝑦	NOUN
cana-2746	183	9	)	)	PUNCT
cana-2746	183	10	}	}	PUNCT
cana-2746	183	11	≥	≥	PROPN
cana-2746	183	12	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-2746	183	13	{	{	PUNCT
cana-2746	183	14	𝑚𝑖𝑛{𝐴𝜇(𝑥	𝑚𝑖𝑛{𝐴𝜇(𝑥	NOUN
cana-2746	183	15	)	)	PUNCT
cana-2746	183	16	,	,	PUNCT
cana-2746	183	17	𝐴𝜇(𝑦	𝐴𝜇(𝑦	NOUN
cana-2746	183	18	)	)	PUNCT
cana-2746	183	19	}	}	PUNCT
cana-2746	183	20	,	,	PUNCT
cana-2746	183	21	𝑚𝑖𝑛{𝐵𝜇(𝑥	𝑚𝑖𝑛{𝐵𝜇(𝑥	NOUN
cana-2746	183	22	)	)	PUNCT
cana-2746	183	23	,	,	PUNCT
cana-2746	183	24	𝐵𝜇(𝑦	𝐵𝜇(𝑦	NOUN
cana-2746	183	25	)	)	PUNCT
cana-2746	183	26	}	}	PUNCT
cana-2746	183	27	}	}	PUNCT
cana-2746	183	28	=	=	SYM
cana-2746	183	29	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-2746	183	30	{	{	PUNCT
cana-2746	183	31	𝑚𝑖𝑛{𝐴𝜇(𝑥	𝑚𝑖𝑛{𝐴𝜇(𝑥	NOUN
cana-2746	183	32	)	)	PUNCT
cana-2746	183	33	,	,	PUNCT
cana-2746	183	34	𝐵𝜇(𝑥	𝐵𝜇(𝑥	NOUN
cana-2746	183	35	)	)	PUNCT
cana-2746	183	36	}	}	PUNCT
cana-2746	183	37	,	,	PUNCT
cana-2746	183	38	𝑚𝑖𝑛{𝐴𝜇(𝑦	𝑚𝑖𝑛{𝐴𝜇(𝑦	PROPN
cana-2746	183	39	)	)	PUNCT
cana-2746	183	40	,	,	PUNCT
cana-2746	183	41	𝐵𝜇(𝑦	𝐵𝜇(𝑦	NOUN
cana-2746	183	42	)	)	PUNCT
cana-2746	183	43	}	}	PUNCT
cana-2746	183	44	}	}	PUNCT
cana-2746	183	45	=	=	SYM
cana-2746	183	46	𝑚𝑖𝑛{(𝐴𝜇	𝑚𝑖𝑛{(𝐴𝜇	PROPN
cana-2746	183	47	∩	∩	ADJ
cana-2746	183	48	𝐵𝜇)(𝑥	𝐵𝜇)(𝑥	NOUN
cana-2746	183	49	)	)	PUNCT
cana-2746	183	50	,	,	PUNCT
cana-2746	183	51	(	(	PUNCT
cana-2746	183	52	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	183	53	∩	∩	ADJ
cana-2746	183	54	𝐵𝜇)(𝑦	𝐵𝜇)(𝑦	NOUN
cana-2746	183	55	)	)	PUNCT
cana-2746	183	56	}	}	PUNCT
cana-2746	183	57	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	183	58	∘	∘	NOUN
cana-2746	183	59	(	(	PUNCT
cana-2746	183	60	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	183	61	∩	∩	ADJ
cana-2746	183	62	𝐵𝜇)(𝑥	𝐵𝜇)(𝑥	NOUN
cana-2746	183	63	)	)	PUNCT
cana-2746	183	64	=	=	SYM
cana-2746	183	65	sup	sup	NOUN
cana-2746	183	66	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PUNCT
cana-2746	183	67	{	{	PUNCT
cana-2746	183	68	𝑚𝑖𝑛{𝜒𝑆(𝑎	𝑚𝑖𝑛{𝜒𝑆(𝑎	NOUN
cana-2746	183	69	)	)	PUNCT
cana-2746	183	70	,	,	PUNCT
cana-2746	183	71	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	183	72	∩	∩	ADJ
cana-2746	183	73	𝐵𝜇(𝑏	𝐵𝜇(𝑏	NOUN
cana-2746	183	74	)	)	PUNCT
cana-2746	183	75	}	}	PUNCT
cana-2746	183	76	}	}	PUNCT
cana-2746	183	77	=	=	SYM
cana-2746	183	78	sup	sup	NOUN
cana-2746	183	79	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	183	80	{	{	PUNCT
cana-2746	183	81	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-2746	183	82	{	{	PUNCT
cana-2746	183	83	𝜒𝑆(𝑎	𝜒𝑆(𝑎	NOUN
cana-2746	183	84	)	)	PUNCT
cana-2746	183	85	,	,	PUNCT
cana-2746	183	86	𝑚𝑖𝑛{𝐴𝜇(𝑏	𝑚𝑖𝑛{𝐴𝜇(𝑏	PROPN
cana-2746	183	87	)	)	PUNCT
cana-2746	183	88	,	,	PUNCT
cana-2746	183	89	𝐵𝜇(𝑏	𝐵𝜇(𝑏	NOUN
cana-2746	183	90	)	)	PUNCT
cana-2746	183	91	}	}	PUNCT
cana-2746	183	92	}	}	PUNCT
cana-2746	183	93	}	}	PUNCT
cana-2746	183	94	=	=	SYM
cana-2746	183	95	sup	sup	NOUN
cana-2746	183	96	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	NOUN
cana-2746	183	97	{	{	PUNCT
cana-2746	183	98	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-2746	183	99	{	{	PUNCT
cana-2746	183	100	𝑚𝑖𝑛{𝜒𝑆(𝑎	𝑚𝑖𝑛{𝜒𝑆(𝑎	NOUN
cana-2746	183	101	)	)	PUNCT
cana-2746	183	102	,	,	PUNCT
cana-2746	183	103	𝐴𝜇(𝑏	𝐴𝜇(𝑏	NOUN
cana-2746	183	104	)	)	PUNCT
cana-2746	183	105	}	}	PUNCT
cana-2746	183	106	,	,	PUNCT
cana-2746	183	107	𝑚𝑖𝑛{𝜒𝑆(𝑎	𝑚𝑖𝑛{𝜒𝑆(𝑎	NOUN
cana-2746	183	108	)	)	PUNCT
cana-2746	183	109	,	,	PUNCT
cana-2746	183	110	𝐵𝜇(𝑏	𝐵𝜇(𝑏	NOUN
cana-2746	183	111	)	)	PUNCT
cana-2746	183	112	}	}	PUNCT
cana-2746	183	113	}	}	PUNCT
cana-2746	183	114	}	}	PUNCT
cana-2746	183	115	=	=	SYM
cana-2746	183	116	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
cana-2746	183	117	{	{	PUNCT
cana-2746	183	118	sup	sup	NOUN
cana-2746	183	119	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	183	120	{	{	PUNCT
cana-2746	183	121	𝑚𝑖𝑛{𝜒𝑆(𝑎	𝑚𝑖𝑛{𝜒𝑆(𝑎	NOUN
cana-2746	183	122	)	)	PUNCT
cana-2746	183	123	,	,	PUNCT
cana-2746	183	124	𝐴𝜇(𝑏	𝐴𝜇(𝑏	NOUN
cana-2746	183	125	)	)	PUNCT
cana-2746	183	126	}	}	PUNCT
cana-2746	183	127	}	}	PUNCT
cana-2746	183	128	,	,	PUNCT
cana-2746	183	129	sup	sup	NOUN
cana-2746	183	130	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PRON
cana-2746	183	131	{	{	PUNCT
cana-2746	183	132	𝑚𝑖𝑛{𝜒𝑆(𝑎	𝑚𝑖𝑛{𝜒𝑆(𝑎	NOUN
cana-2746	183	133	)	)	PUNCT
cana-2746	183	134	,	,	PUNCT
cana-2746	183	135	𝐵𝜇(𝑏	𝐵𝜇(𝑏	NOUN
cana-2746	183	136	)	)	PUNCT
cana-2746	183	137	}	}	PUNCT
cana-2746	183	138	}	}	PUNCT
cana-2746	183	139	}	}	PUNCT
cana-2746	183	140	=	=	PUNCT
cana-2746	183	141	𝑚𝑖𝑛{𝜒𝑆	𝑚𝑖𝑛{𝜒𝑆	VERB
cana-2746	183	142	∘	∘	PROPN
cana-2746	183	143	𝐴𝜇(𝑥	𝐴𝜇(𝑥	NOUN
cana-2746	183	144	)	)	PUNCT
cana-2746	183	145	,	,	PUNCT
cana-2746	183	146	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	183	147	∘	∘	PROPN
cana-2746	183	148	𝐵𝜇(𝑥	𝐵𝜇(𝑥	NOUN
cana-2746	183	149	)	)	PUNCT
cana-2746	183	150	}	}	PUNCT
cana-2746	183	151	=	=	SYM
cana-2746	183	152	(	(	PUNCT
cana-2746	183	153	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	183	154	∘	∘	PROPN
cana-2746	183	155	𝐴𝜇	𝐴𝜇	NOUN
cana-2746	183	156	)	)	PUNCT
cana-2746	183	157	∩	∩	NOUN
cana-2746	183	158	(	(	PUNCT
cana-2746	183	159	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	183	160	∘	∘	NOUN
cana-2746	183	161	𝐵𝜇)(𝑥	𝐵𝜇)(𝑥	NOUN
cana-2746	183	162	)	)	PUNCT
cana-2746	183	163	.	.	PUNCT
cana-2746	184	1	(	(	PUNCT
cana-2746	184	2	𝐴𝜇	𝐴𝜇	NOUN
cana-2746	184	3	∩	∩	ADJ
cana-2746	184	4	𝐵𝜇	𝐵𝜇	NOUN
cana-2746	184	5	)	)	PUNCT
cana-2746	184	6	∘	∘	PROPN
cana-2746	184	7	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	184	8	∘	∘	PROPN
cana-2746	184	9	(	(	PUNCT
cana-2746	184	10	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	184	11	∩	∩	ADJ
cana-2746	184	12	𝐵𝜇)(𝑥	𝐵𝜇)(𝑥	NOUN
cana-2746	184	13	)	)	PUNCT
cana-2746	184	14	=	=	SYM
cana-2746	184	15	sup	sup	NOUN
cana-2746	184	16	𝑥=𝑎𝛼𝑏𝛽𝑐	𝑥=𝑎𝛼𝑏𝛽𝑐	ADP
cana-2746	184	17	{	{	PUNCT
cana-2746	184	18	𝑚𝑖𝑛{𝐴𝜇	𝑚𝑖𝑛{𝐴𝜇	PROPN
cana-2746	184	19	∩	∩	ADJ
cana-2746	184	20	𝐵𝜇(𝑎	𝐵𝜇(𝑎	NOUN
cana-2746	184	21	)	)	PUNCT
cana-2746	184	22	,	,	PUNCT
cana-2746	184	23	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	184	24	∘	∘	NOUN
cana-2746	184	25	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	184	26	∩	∩	ADJ
cana-2746	184	27	𝐵𝜇(𝑏𝛽𝑐	𝐵𝜇(𝑏𝛽𝑐	NOUN
cana-2746	184	28	)	)	PUNCT
cana-2746	184	29	}	}	PUNCT
cana-2746	184	30	}	}	PUNCT
cana-2746	184	31	=	=	SYM
cana-2746	184	32	sup	sup	NOUN
cana-2746	184	33	𝑥=𝑎𝛼𝑏𝛽𝑐	𝑥=𝑎𝛼𝑏𝛽𝑐	ADP
cana-2746	184	34	{	{	PUNCT
cana-2746	184	35	𝑚𝑖𝑛{(𝐴𝜇	𝑚𝑖𝑛{(𝐴𝜇	PROPN
cana-2746	184	36	∩	∩	ADJ
cana-2746	184	37	𝐵𝜇)(𝑎	𝐵𝜇)(𝑎	NOUN
cana-2746	184	38	)	)	PUNCT
cana-2746	184	39	,	,	PUNCT
cana-2746	184	40	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	184	41	∘	∘	NOUN
cana-2746	184	42	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	184	43	∩	∩	NOUN
cana-2746	184	44	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	184	45	∘	∘	PROPN
cana-2746	184	46	𝐵𝜇(𝑏𝛽𝑐	𝐵𝜇(𝑏𝛽𝑐	NOUN
cana-2746	184	47	)	)	PUNCT
cana-2746	184	48	}	}	PUNCT
cana-2746	184	49	}	}	PUNCT
cana-2746	184	50	=	=	SYM
cana-2746	184	51	sup	sup	NOUN
cana-2746	184	52	𝑥=𝑎𝛼𝑏𝛽𝑐	𝑥=𝑎𝛼𝑏𝛽𝑐	ADP
cana-2746	184	53	{	{	PUNCT
cana-2746	184	54	𝑚𝑖𝑛	𝑚𝑖𝑛	PROPN
cana-2746	184	55	{	{	PUNCT
cana-2746	184	56	𝑚𝑖𝑛{𝐴𝜇(𝑎	𝑚𝑖𝑛{𝐴𝜇(𝑎	PROPN
cana-2746	184	57	)	)	PUNCT
cana-2746	184	58	,	,	PUNCT
cana-2746	184	59	𝐵𝜇(𝑎	𝐵𝜇(𝑎	NOUN
cana-2746	184	60	)	)	PUNCT
cana-2746	184	61	}	}	PUNCT
cana-2746	184	62	,	,	PUNCT
cana-2746	184	63	𝑚𝑖𝑛{(𝜒𝑆	𝑚𝑖𝑛{(𝜒𝑆	PUNCT
cana-2746	184	64	∘	∘	PROPN
cana-2746	184	65	𝐴𝜇)(𝑏𝛽𝑐	𝐴𝜇)(𝑏𝛽𝑐	NOUN
cana-2746	184	66	)	)	PUNCT
cana-2746	184	67	,	,	PUNCT
cana-2746	184	68	(	(	PUNCT
cana-2746	184	69	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	184	70	∘	∘	PROPN
cana-2746	184	71	𝐵𝜇)(𝑏𝛽𝑐	𝐵𝜇)(𝑏𝛽𝑐	NOUN
cana-2746	184	72	)	)	PUNCT
cana-2746	184	73	}	}	PUNCT
cana-2746	184	74	}	}	PUNCT
cana-2746	184	75	}	}	PUNCT
cana-2746	184	76	communications	communication	NOUN
cana-2746	184	77	on	on	ADP
cana-2746	184	78	applied	apply	VERB
cana-2746	184	79	nonlinear	nonlinear	ADJ
cana-2746	184	80	analysis	analysis	NOUN
cana-2746	184	81	issn	issn	NOUN
cana-2746	184	82	:	:	PUNCT
cana-2746	184	83	1074	1074	NUM
cana-2746	184	84	-	-	PUNCT
cana-2746	184	85	133x	133x	NUM
cana-2746	184	86	vol	vol	NOUN
cana-2746	184	87	32	32	NUM
cana-2746	184	88	no	no	NOUN
cana-2746	184	89	.	.	PUNCT
cana-2746	185	1	4s	4s	NUM
cana-2746	185	2	(	(	PUNCT
cana-2746	185	3	2025	2025	NUM
cana-2746	185	4	)	)	PUNCT
cana-2746	186	1	165	165	NUM
cana-2746	186	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-2746	186	3	=	=	PUNCT
cana-2746	186	4	𝑚𝑖𝑛{𝐴𝜇	𝑚𝑖𝑛{𝐴𝜇	PROPN
cana-2746	186	5	∘	∘	X
cana-2746	186	6	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	186	7	∘	∘	PROPN
cana-2746	186	8	𝐴𝜇(𝑥	𝐴𝜇(𝑥	PROPN
cana-2746	186	9	)	)	PUNCT
cana-2746	186	10	,	,	PUNCT
cana-2746	186	11	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	186	12	∘	∘	ADJ
cana-2746	186	13	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	186	14	∘	∘	PROPN
cana-2746	186	15	𝐵𝜇(𝑥	𝐵𝜇(𝑥	NOUN
cana-2746	186	16	)	)	PUNCT
cana-2746	186	17	}	}	PUNCT
cana-2746	186	18	therefore	therefore	ADV
cana-2746	186	19	(	(	PUNCT
cana-2746	186	20	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	186	21	∩	∩	ADJ
cana-2746	186	22	𝐵𝜇	𝐵𝜇	NOUN
cana-2746	186	23	)	)	PUNCT
cana-2746	186	24	∘	∘	PROPN
cana-2746	186	25	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	186	26	∘	∘	PROPN
cana-2746	186	27	(	(	PUNCT
cana-2746	186	28	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	186	29	∩	∩	ADJ
cana-2746	186	30	𝐵𝜇	𝐵𝜇	NOUN
cana-2746	186	31	)	)	PUNCT
cana-2746	186	32	=	=	PUNCT
cana-2746	187	1	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	187	2	∘	∘	X
cana-2746	187	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	187	4	∘	∘	NOUN
cana-2746	187	5	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	187	6	∩	∩	PUNCT
cana-2746	187	7	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	187	8	∘	∘	ADJ
cana-2746	187	9	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	187	10	∘	∘	PROPN
cana-2746	187	11	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	187	12	hence	hence	ADV
cana-2746	187	13	𝜒𝑆	𝜒𝑆	VERB
cana-2746	187	14	∘	∘	NOUN
cana-2746	187	15	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	187	16	∩	∩	PUNCT
cana-2746	187	17	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	187	18	∘	∘	ADJ
cana-2746	187	19	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	187	20	∩	∩	NOUN
cana-2746	187	21	𝐴𝜇	𝐴𝜇	NOUN
cana-2746	187	22	∩	∩	NOUN
cana-2746	187	23	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	187	24	∘	∘	ADJ
cana-2746	187	25	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	187	26	∘	∘	NOUN
cana-2746	187	27	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	187	28	∩	∩	NOUN
cana-2746	187	29	𝐵𝜇	𝐵𝜇	NOUN
cana-2746	187	30	=	=	SYM
cana-2746	187	31	(	(	PUNCT
cana-2746	187	32	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	187	33	∘	∘	PROPN
cana-2746	187	34	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	187	35	∘	∘	ADJ
cana-2746	187	36	𝜒𝑆	𝜒𝑆	NOUN
cana-2746	187	37	)	)	PUNCT
cana-2746	187	38	∩	∩	NOUN
cana-2746	187	39	(	(	PUNCT
cana-2746	187	40	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	187	41	∘	∘	PROPN
cana-2746	187	42	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	187	43	∘	∘	PROPN
cana-2746	187	44	𝐴𝜇	𝐴𝜇	NOUN
cana-2746	187	45	)	)	PUNCT
cana-2746	187	46	∩	∩	NOUN
cana-2746	187	47	(	(	PUNCT
cana-2746	187	48	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	187	49	∘	∘	PROPN
cana-2746	187	50	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	187	51	∘	∘	ADJ
cana-2746	187	52	𝜒𝑆	𝜒𝑆	NOUN
cana-2746	187	53	)	)	PUNCT
cana-2746	187	54	∩	∩	NOUN
cana-2746	187	55	(	(	PUNCT
cana-2746	187	56	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	187	57	∘	∘	ADJ
cana-2746	187	58	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	187	59	∘	∘	PROPN
cana-2746	187	60	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	187	61	)	)	PUNCT
cana-2746	187	62	⊇	⊇	NOUN
cana-2746	187	63	𝐴𝜇	𝐴𝜇	PROPN
cana-2746	187	64	∩	∩	PUNCT
cana-2746	187	65	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	187	66	similarly	similarly	ADV
cana-2746	187	67	we	we	PRON
cana-2746	187	68	can	can	AUX
cana-2746	187	69	prove	prove	VERB
cana-2746	187	70	for	for	ADP
cana-2746	187	71	non	non	NOUN
cana-2746	187	72	membership	membership	NOUN
cana-2746	187	73	(	(	PUNCT
cana-2746	187	74	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	187	75	∩	∩	NOUN
cana-2746	187	76	𝐵𝜈)(𝑥	𝐵𝜈)(𝑥	PROPN
cana-2746	187	77	+	+	PROPN
cana-2746	187	78	𝑦	𝑦	NOUN
cana-2746	187	79	)	)	PUNCT
cana-2746	187	80	=	=	SYM
cana-2746	188	1	𝑚𝑎𝑥{𝐴𝜈(𝑥	𝑚𝑎𝑥{𝐴𝜈(𝑥	PROPN
cana-2746	188	2	+	+	NUM
cana-2746	188	3	𝑦	𝑦	NOUN
cana-2746	188	4	)	)	PUNCT
cana-2746	188	5	,	,	PUNCT
cana-2746	188	6	𝐵𝜈(𝑥	𝐵𝜈(𝑥	PROPN
cana-2746	188	7	+	+	PUNCT
cana-2746	188	8	𝑦	𝑦	X
cana-2746	188	9	)	)	PUNCT
cana-2746	188	10	}	}	PUNCT
cana-2746	188	11	≤	≤	NUM
cana-2746	188	12	𝑚𝑎𝑥{𝑚𝑎𝑥{𝐴𝜈(𝑥	𝑚𝑎𝑥{𝑚𝑎𝑥{𝐴𝜈(𝑥	NOUN
cana-2746	188	13	)	)	PUNCT
cana-2746	188	14	,	,	PUNCT
cana-2746	188	15	𝐴𝜈(𝑦	𝐴𝜈(𝑦	NOUN
cana-2746	188	16	)	)	PUNCT
cana-2746	188	17	}	}	PUNCT
cana-2746	188	18	,	,	PUNCT
cana-2746	188	19	𝑚𝑎𝑥{𝐵𝜈(𝑥	𝑚𝑎𝑥{𝐵𝜈(𝑥	PROPN
cana-2746	188	20	)	)	PUNCT
cana-2746	188	21	,	,	PUNCT
cana-2746	188	22	𝐵𝜈(𝑦	𝐵𝜈(𝑦	PROPN
cana-2746	188	23	)	)	PUNCT
cana-2746	188	24	}	}	PUNCT
cana-2746	188	25	}	}	PUNCT
cana-2746	188	26	=	=	SYM
cana-2746	188	27	𝑚𝑎𝑥{𝑚𝑎𝑥{𝐴𝜈(𝑥	𝑚𝑎𝑥{𝑚𝑎𝑥{𝐴𝜈(𝑥	PROPN
cana-2746	188	28	)	)	PUNCT
cana-2746	188	29	,	,	PUNCT
cana-2746	188	30	𝐵𝜈(𝑥	𝐵𝜈(𝑥	PROPN
cana-2746	188	31	)	)	PUNCT
cana-2746	188	32	}	}	PUNCT
cana-2746	188	33	,	,	PUNCT
cana-2746	188	34	𝑚𝑎𝑥{𝐴𝜈(𝑦	𝑚𝑎𝑥{𝐴𝜈(𝑦	PROPN
cana-2746	188	35	)	)	PUNCT
cana-2746	188	36	,	,	PUNCT
cana-2746	188	37	𝐵𝜈(𝑦	𝐵𝜈(𝑦	PROPN
cana-2746	188	38	)	)	PUNCT
cana-2746	188	39	}	}	PUNCT
cana-2746	188	40	}	}	PUNCT
cana-2746	188	41	=	=	SYM
cana-2746	188	42	𝑚𝑎𝑥{(𝐴𝜈	𝑚𝑎𝑥{(𝐴𝜈	PROPN
cana-2746	188	43	∩	∩	X
cana-2746	188	44	𝐵𝜈)(𝑥	𝐵𝜈)(𝑥	PROPN
cana-2746	188	45	)	)	PUNCT
cana-2746	188	46	,	,	PUNCT
cana-2746	188	47	(	(	PUNCT
cana-2746	188	48	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	188	49	∩	∩	NOUN
cana-2746	188	50	𝐵𝜈)(𝑦	𝐵𝜈)(𝑦	NOUN
cana-2746	188	51	)	)	PUNCT
cana-2746	188	52	}	}	PUNCT
cana-2746	188	53	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	188	54	∘	∘	NOUN
cana-2746	188	55	(	(	PUNCT
cana-2746	188	56	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	188	57	∩	∩	NOUN
cana-2746	188	58	𝐵𝜈)(𝑥	𝐵𝜈)(𝑥	NOUN
cana-2746	188	59	)	)	PUNCT
cana-2746	188	60	=	=	PROPN
cana-2746	188	61	inf	inf	NOUN
cana-2746	188	62	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	X
cana-2746	188	63	{	{	PUNCT
cana-2746	188	64	𝑚𝑎𝑥{𝜒𝑆(𝑎	𝑚𝑎𝑥{𝜒𝑆(𝑎	NUM
cana-2746	188	65	)	)	PUNCT
cana-2746	188	66	,	,	PUNCT
cana-2746	188	67	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	188	68	∩	∩	ADJ
cana-2746	188	69	𝐵𝜇(𝑏	𝐵𝜇(𝑏	NOUN
cana-2746	188	70	)	)	PUNCT
cana-2746	188	71	}	}	PUNCT
cana-2746	188	72	}	}	PUNCT
cana-2746	188	73	=	=	SYM
cana-2746	188	74	inf	inf	NOUN
cana-2746	188	75	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	X
cana-2746	188	76	{	{	PUNCT
cana-2746	188	77	𝑚𝑎𝑥	𝑚𝑎𝑥	X
cana-2746	188	78	{	{	PUNCT
cana-2746	188	79	𝜒𝑆(𝑎	𝜒𝑆(𝑎	NOUN
cana-2746	188	80	)	)	PUNCT
cana-2746	188	81	,	,	PUNCT
cana-2746	188	82	𝑚𝑎𝑥{𝐴𝜈(𝑏	𝑚𝑎𝑥{𝐴𝜈(𝑏	PROPN
cana-2746	188	83	)	)	PUNCT
cana-2746	188	84	,	,	PUNCT
cana-2746	188	85	𝐵𝜇(𝑏	𝐵𝜇(𝑏	NOUN
cana-2746	188	86	)	)	PUNCT
cana-2746	188	87	}	}	PUNCT
cana-2746	188	88	}	}	PUNCT
cana-2746	188	89	}	}	PUNCT
cana-2746	188	90	=	=	SYM
cana-2746	188	91	inf	inf	NOUN
cana-2746	188	92	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	188	93	{	{	PUNCT
cana-2746	188	94	𝑚𝑎𝑥{𝑚𝑎𝑥{𝜒𝑆(𝑎	𝑚𝑎𝑥{𝑚𝑎𝑥{𝜒𝑆(𝑎	NOUN
cana-2746	188	95	)	)	PUNCT
cana-2746	188	96	,	,	PUNCT
cana-2746	188	97	𝐴𝜈(𝑏	𝐴𝜈(𝑏	NOUN
cana-2746	188	98	)	)	PUNCT
cana-2746	188	99	}	}	PUNCT
cana-2746	188	100	,	,	PUNCT
cana-2746	188	101	𝑚𝑎𝑥{𝜒𝑆(𝑎	𝑚𝑎𝑥{𝜒𝑆(𝑎	NUM
cana-2746	188	102	)	)	PUNCT
cana-2746	188	103	,	,	PUNCT
cana-2746	188	104	𝐵𝜈(𝑏	𝐵𝜈(𝑏	NOUN
cana-2746	188	105	)	)	PUNCT
cana-2746	188	106	}	}	PUNCT
cana-2746	188	107	}	}	PUNCT
cana-2746	188	108	}	}	PUNCT
cana-2746	188	109	=	=	PUNCT
cana-2746	188	110	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-2746	188	111	{	{	PUNCT
cana-2746	188	112	inf	inf	NOUN
cana-2746	188	113	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	PROPN
cana-2746	188	114	{	{	PUNCT
cana-2746	188	115	𝑚𝑎𝑥{𝜒𝑆(𝑎	𝑚𝑎𝑥{𝜒𝑆(𝑎	NOUN
cana-2746	188	116	)	)	PUNCT
cana-2746	188	117	,	,	PUNCT
cana-2746	188	118	𝐴𝜈(𝑏	𝐴𝜈(𝑏	NOUN
cana-2746	188	119	)	)	PUNCT
cana-2746	188	120	}	}	PUNCT
cana-2746	188	121	}	}	PUNCT
cana-2746	188	122	,	,	PUNCT
cana-2746	188	123	inf	inf	PROPN
cana-2746	188	124	𝑥=𝑎𝛼𝑏	𝑥=𝑎𝛼𝑏	X
cana-2746	188	125	{	{	PUNCT
cana-2746	188	126	𝑚𝑎𝑥{𝜒𝑆(𝑎	𝑚𝑎𝑥{𝜒𝑆(𝑎	NOUN
cana-2746	188	127	)	)	PUNCT
cana-2746	188	128	,	,	PUNCT
cana-2746	188	129	𝐵𝜈(𝑏	𝐵𝜈(𝑏	NOUN
cana-2746	188	130	)	)	PUNCT
cana-2746	188	131	}	}	PUNCT
cana-2746	188	132	}	}	PUNCT
cana-2746	188	133	}	}	PUNCT
cana-2746	188	134	=	=	SYM
cana-2746	188	135	𝑚𝑎𝑥{𝜒𝑆	𝑚𝑎𝑥{𝜒𝑆	ADJ
cana-2746	188	136	∘	∘	PROPN
cana-2746	188	137	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	188	138	)	)	PUNCT
cana-2746	188	139	,	,	PUNCT
cana-2746	188	140	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	188	141	∘	∘	PROPN
cana-2746	188	142	𝐵𝜈(𝑥	𝐵𝜈(𝑥	PROPN
cana-2746	188	143	)	)	PUNCT
cana-2746	188	144	}	}	PUNCT
cana-2746	188	145	=	=	SYM
cana-2746	188	146	(	(	PUNCT
cana-2746	188	147	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	188	148	∘	∘	NUM
cana-2746	188	149	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	188	150	)	)	PUNCT
cana-2746	188	151	∩	∩	NOUN
cana-2746	188	152	(	(	PUNCT
cana-2746	188	153	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	188	154	∘	∘	NOUN
cana-2746	188	155	𝐵𝜈)(𝑥	𝐵𝜈)(𝑥	NOUN
cana-2746	188	156	)	)	PUNCT
cana-2746	188	157	.	.	PUNCT
cana-2746	189	1	(	(	PUNCT
cana-2746	189	2	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	189	3	∩	∩	NOUN
cana-2746	189	4	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	189	5	)	)	PUNCT
cana-2746	189	6	∘	∘	PROPN
cana-2746	189	7	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	189	8	∘	∘	X
cana-2746	189	9	(	(	PUNCT
cana-2746	189	10	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	189	11	∩	∩	NOUN
cana-2746	189	12	𝐵𝜈)(𝑥	𝐵𝜈)(𝑥	NOUN
cana-2746	189	13	)	)	PUNCT
cana-2746	189	14	=	=	SYM
cana-2746	189	15	inf	inf	PROPN
cana-2746	189	16	𝑥=𝑎𝛼𝑏𝛽𝑐	𝑥=𝑎𝛼𝑏𝛽𝑐	ADP
cana-2746	189	17	{	{	PUNCT
cana-2746	189	18	𝑚𝑎𝑥{𝐴𝜈	𝑚𝑎𝑥{𝐴𝜈	NOUN
cana-2746	189	19	∩	∩	NOUN
cana-2746	189	20	𝐵𝜈(𝑎	𝐵𝜈(𝑎	NOUN
cana-2746	189	21	)	)	PUNCT
cana-2746	189	22	,	,	PUNCT
cana-2746	189	23	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	189	24	∘	∘	NOUN
cana-2746	189	25	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	189	26	∩	∩	ADJ
cana-2746	189	27	𝐵𝜈(𝑏𝛽𝑐	𝐵𝜈(𝑏𝛽𝑐	NOUN
cana-2746	189	28	)	)	PUNCT
cana-2746	189	29	}	}	PUNCT
cana-2746	189	30	}	}	PUNCT
cana-2746	189	31	=	=	SYM
cana-2746	189	32	inf	inf	PROPN
cana-2746	189	33	𝑥=𝑎𝛼𝑏𝛽𝑐	𝑥=𝑎𝛼𝑏𝛽𝑐	ADP
cana-2746	189	34	{	{	PUNCT
cana-2746	189	35	𝑚𝑎𝑥{(𝐴𝜈	𝑚𝑎𝑥{(𝐴𝜈	PROPN
cana-2746	189	36	∩	∩	ADJ
cana-2746	189	37	𝐵𝜈)(𝑎	𝐵𝜈)(𝑎	NOUN
cana-2746	189	38	)	)	PUNCT
cana-2746	189	39	,	,	PUNCT
cana-2746	189	40	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	189	41	∘	∘	ADJ
cana-2746	189	42	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	189	43	∩	∩	NOUN
cana-2746	189	44	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	189	45	∘	∘	PROPN
cana-2746	189	46	𝐵𝜈(𝑏𝛽𝑐	𝐵𝜈(𝑏𝛽𝑐	PROPN
cana-2746	189	47	)	)	PUNCT
cana-2746	189	48	}	}	PUNCT
cana-2746	189	49	}	}	PUNCT
cana-2746	189	50	=	=	SYM
cana-2746	189	51	inf	inf	NOUN
cana-2746	189	52	𝑥=𝑎𝛼𝑏𝛽𝑐	𝑥=𝑎𝛼𝑏𝛽𝑐	ADP
cana-2746	189	53	{	{	PUNCT
cana-2746	189	54	𝑚𝑎𝑥{𝑚𝑎𝑥{𝐴𝜈(𝑎	𝑚𝑎𝑥{𝑚𝑎𝑥{𝐴𝜈(𝑎	NOUN
cana-2746	189	55	)	)	PUNCT
cana-2746	189	56	,	,	PUNCT
cana-2746	189	57	𝐵𝜈(𝑎	𝐵𝜈(𝑎	PROPN
cana-2746	189	58	)	)	PUNCT
cana-2746	189	59	}	}	PUNCT
cana-2746	189	60	,	,	PUNCT
cana-2746	189	61	𝑚𝑎𝑥{(𝜒𝑆	𝑚𝑎𝑥{(𝜒𝑆	ADP
cana-2746	189	62	∘	∘	PROPN
cana-2746	189	63	𝐴𝜈)(𝑏𝛽𝑐	𝐴𝜈)(𝑏𝛽𝑐	NOUN
cana-2746	189	64	)	)	PUNCT
cana-2746	189	65	,	,	PUNCT
cana-2746	189	66	(	(	PUNCT
cana-2746	189	67	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	189	68	∘	∘	PROPN
cana-2746	189	69	𝐵𝜈)(𝑏𝛽𝑐	𝐵𝜈)(𝑏𝛽𝑐	NOUN
cana-2746	189	70	)	)	PUNCT
cana-2746	189	71	}	}	PUNCT
cana-2746	189	72	}	}	PUNCT
cana-2746	189	73	}	}	PUNCT
cana-2746	189	74	=	=	PUNCT
cana-2746	189	75	𝑚𝑎𝑥{𝐴𝜈	𝑚𝑎𝑥{𝐴𝜈	DET
cana-2746	189	76	∘	∘	X
cana-2746	189	77	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	189	78	∘	∘	PROPN
cana-2746	189	79	𝐴𝜈(𝑥	𝐴𝜈(𝑥	PROPN
cana-2746	189	80	)	)	PUNCT
cana-2746	189	81	,	,	PUNCT
cana-2746	189	82	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	189	83	∘	∘	PROPN
cana-2746	189	84	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	189	85	∘	∘	PROPN
cana-2746	189	86	𝐵𝜈(𝑥	𝐵𝜈(𝑥	PROPN
cana-2746	189	87	)	)	PUNCT
cana-2746	189	88	}	}	PUNCT
cana-2746	189	89	therefore	therefore	ADV
cana-2746	189	90	(	(	PUNCT
cana-2746	189	91	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	189	92	∩	∩	NOUN
cana-2746	189	93	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	189	94	)	)	PUNCT
cana-2746	189	95	∘	∘	PROPN
cana-2746	189	96	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	189	97	∘	∘	X
cana-2746	189	98	(	(	PUNCT
cana-2746	189	99	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	189	100	∩	∩	NOUN
cana-2746	189	101	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	189	102	)	)	PUNCT
cana-2746	189	103	=	=	PUNCT
cana-2746	190	1	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	190	2	∘	∘	NOUN
cana-2746	190	3	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	190	4	∘	∘	NOUN
cana-2746	190	5	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	190	6	∩	∩	NOUN
cana-2746	190	7	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	190	8	∘	∘	PROPN
cana-2746	190	9	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	190	10	∘	∘	NOUN
cana-2746	190	11	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	190	12	hence	hence	ADV
cana-2746	190	13	𝜒𝑆	𝜒𝑆	VERB
cana-2746	190	14	∘	∘	NOUN
cana-2746	190	15	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	190	16	∩	∩	ADJ
cana-2746	190	17	𝐵𝜇	𝐵𝜇	PROPN
cana-2746	190	18	∘	∘	ADJ
cana-2746	190	19	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	190	20	∩	∩	ADJ
cana-2746	190	21	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	190	22	∩	∩	NOUN
cana-2746	190	23	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	190	24	∘	∘	PROPN
cana-2746	190	25	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	190	26	∘	∘	NOUN
cana-2746	191	1	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	191	2	∩	∩	ADJ
cana-2746	191	3	𝐵𝜇	𝐵𝜇	NOUN
cana-2746	191	4	=	=	SYM
cana-2746	191	5	(	(	PUNCT
cana-2746	191	6	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	191	7	∘	∘	NOUN
cana-2746	191	8	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	191	9	∘	∘	ADJ
cana-2746	191	10	𝜒𝑆	𝜒𝑆	NOUN
cana-2746	191	11	)	)	PUNCT
cana-2746	191	12	∩	∩	NOUN
cana-2746	191	13	(	(	PUNCT
cana-2746	191	14	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	191	15	∘	∘	PROPN
cana-2746	191	16	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	191	17	∘	∘	NUM
cana-2746	191	18	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	191	19	)	)	PUNCT
cana-2746	191	20	∩	∩	NOUN
cana-2746	191	21	(	(	PUNCT
cana-2746	191	22	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	191	23	∘	∘	PROPN
cana-2746	191	24	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	191	25	∘	∘	ADJ
cana-2746	191	26	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	191	27	)	)	PUNCT
cana-2746	191	28	∩	∩	NOUN
cana-2746	191	29	(	(	PUNCT
cana-2746	191	30	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	191	31	∘	∘	PROPN
cana-2746	191	32	𝜒𝑆	𝜒𝑆	PROPN
cana-2746	191	33	∘	∘	PROPN
cana-2746	191	34	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	191	35	)	)	PUNCT
cana-2746	191	36	⊆	⊆	NUM
cana-2746	191	37	𝐴𝜈	𝐴𝜈	PROPN
cana-2746	191	38	∩	∩	NOUN
cana-2746	191	39	𝐵𝜈	𝐵𝜈	PROPN
cana-2746	191	40	communications	communication	NOUN
cana-2746	191	41	on	on	ADP
cana-2746	191	42	applied	apply	VERB
cana-2746	191	43	nonlinear	nonlinear	ADJ
cana-2746	191	44	analysis	analysis	NOUN
cana-2746	191	45	issn	issn	NOUN
cana-2746	191	46	:	:	PUNCT
cana-2746	191	47	1074	1074	NUM
cana-2746	191	48	-	-	PUNCT
cana-2746	191	49	133x	133x	NUM
cana-2746	191	50	vol	vol	NOUN
cana-2746	191	51	32	32	NUM
cana-2746	191	52	no	no	NOUN
cana-2746	191	53	.	.	PUNCT
cana-2746	192	1	4s	4s	NUM
cana-2746	192	2	(	(	PUNCT
cana-2746	192	3	2025	2025	NUM
cana-2746	192	4	)	)	PUNCT
cana-2746	192	5	166	166	NUM
cana-2746	192	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2746	192	7	5	5	NUM
cana-2746	192	8	conclusion	conclusion	NOUN
cana-2746	192	9	this	this	DET
cana-2746	192	10	paper	paper	NOUN
cana-2746	192	11	deals	deal	NOUN
cana-2746	192	12	with	with	ADP
cana-2746	192	13	the	the	DET
cana-2746	192	14	concept	concept	NOUN
cana-2746	192	15	of	of	ADP
cana-2746	192	16	pythagorean	pythagorean	PROPN
cana-2746	192	17	fuzzy	fuzzy	ADJ
cana-2746	192	18	bi	bi	ADJ
cana-2746	192	19	-	-	ADJ
cana-2746	192	20	interior	interior	ADJ
cana-2746	192	21	-	-	PUNCT
cana-2746	192	22	ideal	ideal	NOUN
cana-2746	192	23	,	,	PUNCT
cana-2746	192	24	pythagorean	pythagorean	PROPN
cana-2746	192	25	fuzzy	fuzzy	ADJ
cana-2746	192	26	soft	soft	ADJ
cana-2746	192	27	biinterior	biinterior	NOUN
cana-2746	192	28	-	-	PUNCT
cana-2746	192	29	ideal	ideal	NOUN
cana-2746	192	30	and	and	CCONJ
cana-2746	192	31	pythagorean	pythagorean	VERB
cana-2746	193	1	fuzzy	fuzzy	ADJ
cana-2746	193	2	bi	bi	ADJ
cana-2746	193	3	-	-	ADJ
cana-2746	193	4	quasi	quasi	NOUN
cana-2746	193	5	-	-	NOUN
cana-2746	194	1	ideals	ideal	NOUN
cana-2746	194	2	in	in	ADP
cana-2746	194	3	γ	γ	NOUN
cana-2746	194	4	-	-	NOUN
cana-2746	194	5	semirings	semiring	NOUN
cana-2746	194	6	.	.	PUNCT
cana-2746	195	1	some	some	DET
cana-2746	195	2	properties	property	NOUN
cana-2746	195	3	of	of	ADP
cana-2746	195	4	these	these	DET
cana-2746	195	5	ideals	ideal	NOUN
cana-2746	195	6	are	be	AUX
cana-2746	195	7	studied	study	VERB
cana-2746	195	8	.	.	PUNCT
cana-2746	196	1	additionally	additionally	ADV
cana-2746	196	2	,	,	PUNCT
cana-2746	196	3	we	we	PRON
cana-2746	196	4	aim	aim	VERB
cana-2746	196	5	to	to	PART
cana-2746	196	6	extend	extend	VERB
cana-2746	196	7	this	this	DET
cana-2746	196	8	structure	structure	NOUN
cana-2746	196	9	to	to	ADP
cana-2746	196	10	various	various	ADJ
cana-2746	196	11	other	other	ADJ
cana-2746	196	12	algebraic	algebraic	ADJ
cana-2746	196	13	structures	structure	NOUN
cana-2746	196	14	and	and	CCONJ
cana-2746	196	15	look	look	VERB
cana-2746	196	16	into	into	ADP
cana-2746	196	17	its	its	PRON
cana-2746	196	18	applications	application	NOUN
cana-2746	196	19	in	in	ADP
cana-2746	196	20	real	real	ADJ
cana-2746	196	21	-	-	PUNCT
cana-2746	196	22	life	life	NOUN
cana-2746	196	23	scenarios	scenario	NOUN
cana-2746	196	24	in	in	ADP
cana-2746	196	25	future	future	ADJ
cana-2746	196	26	research	research	NOUN
cana-2746	196	27	endeavors	endeavor	NOUN
cana-2746	196	28	.	.	PUNCT
cana-2746	197	1	refrences	refrence	VERB
cana-2746	198	1	[	[	X
cana-2746	198	2	1	1	X
cana-2746	198	3	]	]	PUNCT
cana-2746	198	4	j.	j.	PROPN
cana-2746	198	5	ahsan	ahsan	PROPN
cana-2746	198	6	,	,	PUNCT
cana-2746	198	7	k.	k.	PROPN
cana-2746	198	8	saifullah	saifullah	PROPN
cana-2746	198	9	,	,	PUNCT
cana-2746	198	10	and	and	CCONJ
cana-2746	198	11	m.f	m.f	PROPN
cana-2746	198	12	.	.	PROPN
cana-2746	198	13	khan	khan	PROPN
cana-2746	198	14	,	,	PUNCT
cana-2746	198	15	fuzzy	fuzzy	ADJ
cana-2746	198	16	semirings	semiring	NOUN
cana-2746	198	17	,	,	PUNCT
cana-2746	198	18	fuzzy	fuzzy	ADJ
cana-2746	198	19	sets	set	NOUN
cana-2746	198	20	and	and	CCONJ
cana-2746	198	21	systems	system	NOUN
cana-2746	198	22	,	,	PUNCT
cana-2746	198	23	(	(	PUNCT
cana-2746	198	24	1993	1993	NUM
cana-2746	198	25	)	)	PUNCT
cana-2746	198	26	,	,	PUNCT
cana-2746	198	27	302	302	NUM
cana-2746	198	28	-	-	SYM
cana-2746	198	29	309	309	NUM
cana-2746	198	30	.	.	PUNCT
cana-2746	199	1	[	[	X
cana-2746	199	2	2	2	X
cana-2746	199	3	]	]	PUNCT
cana-2746	199	4	k.	k.	PROPN
cana-2746	199	5	t.	t.	PROPN
cana-2746	199	6	atanassov	atanassov	PROPN
cana-2746	199	7	,	,	PUNCT
cana-2746	199	8	intuitionistic	intuitionistic	ADJ
cana-2746	199	9	fuzzy	fuzzy	ADJ
cana-2746	199	10	sets	set	NOUN
cana-2746	199	11	.	.	PUNCT
cana-2746	200	1	fuzzy	fuzzy	ADJ
cana-2746	200	2	sets	set	NOUN
cana-2746	200	3	and	and	CCONJ
cana-2746	200	4	systems	system	NOUN
cana-2746	200	5	,	,	PUNCT
cana-2746	200	6	20	20	NUM
cana-2746	200	7	,	,	PUNCT
cana-2746	200	8	(	(	PUNCT
cana-2746	200	9	1986	1986	NUM
cana-2746	200	10	)	)	PUNCT
cana-2746	200	11	,	,	PUNCT
cana-2746	200	12	87	87	NUM
cana-2746	200	13	-	-	SYM
cana-2746	200	14	96	96	NUM
cana-2746	200	15	.	.	PUNCT
cana-2746	201	1	[	[	X
cana-2746	201	2	3	3	X
cana-2746	201	3	]	]	X
cana-2746	201	4	y.	y.	NOUN
cana-2746	201	5	bhargavi	bhargavi	PROPN
cana-2746	201	6	and	and	CCONJ
cana-2746	201	7	t.	t.	PROPN
cana-2746	201	8	eswarlal	eswarlal	PROPN
cana-2746	201	9	,	,	PUNCT
cana-2746	201	10	fuzzy	fuzzy	ADJ
cana-2746	201	11	γ	γ	NOUN
cana-2746	201	12	-	-	PUNCT
cana-2746	201	13	semirings	semiring	NOUN
cana-2746	201	14	,	,	PUNCT
cana-2746	201	15	international	international	ADJ
cana-2746	201	16	journal	journal	NOUN
cana-2746	201	17	of	of	ADP
cana-2746	201	18	pure	pure	ADJ
cana-2746	201	19	and	and	CCONJ
cana-2746	201	20	applied	applied	ADJ
cana-2746	201	21	mathematics	mathematic	NOUN
cana-2746	201	22	,	,	PUNCT
cana-2746	201	23	98	98	NUM
cana-2746	201	24	,	,	PUNCT
cana-2746	201	25	(	(	PUNCT
cana-2746	201	26	2015	2015	NUM
cana-2746	201	27	)	)	PUNCT
cana-2746	201	28	,	,	PUNCT
cana-2746	201	29	339	339	NUM
cana-2746	201	30	-	-	SYM
cana-2746	201	31	349	349	NUM
cana-2746	201	32	.	.	PUNCT
cana-2746	202	1	[	[	X
cana-2746	202	2	4	4	X
cana-2746	202	3	]	]	PUNCT
cana-2746	202	4	t.	t.	PROPN
cana-2746	202	5	k.	k.	PROPN
cana-2746	202	6	dutta	dutta	PROPN
cana-2746	202	7	,	,	PUNCT
cana-2746	202	8	s.	s.	PROPN
cana-2746	202	9	k.	k.	PROPN
cana-2746	202	10	sardar	sardar	PROPN
cana-2746	202	11	,	,	PUNCT
cana-2746	202	12	and	and	CCONJ
cana-2746	202	13	s.	s.	PROPN
cana-2746	202	14	goswami	goswami	PROPN
cana-2746	202	15	,	,	PUNCT
cana-2746	202	16	(	(	PUNCT
cana-2746	202	17	2011	2011	NUM
cana-2746	202	18	)	)	PUNCT
cana-2746	202	19	,	,	PUNCT
cana-2746	202	20	operations	operation	NOUN
cana-2746	202	21	on	on	ADP
cana-2746	202	22	fuzzy	fuzzy	ADJ
cana-2746	202	23	ideals	ideal	NOUN
cana-2746	202	24	of	of	ADP
cana-2746	202	25	𝛾	𝛾	PROPN
cana-2746	202	26	semirings	semiring	NOUN
cana-2746	202	27	,	,	PUNCT
cana-2746	202	28	proceedings	proceeding	NOUN
cana-2746	202	29	of	of	ADP
cana-2746	202	30	national	national	ADJ
cana-2746	202	31	seminar	seminar	NOUN
cana-2746	202	32	on	on	ADP
cana-2746	202	33	algebra	algebra	NOUN
cana-2746	202	34	,	,	PUNCT
cana-2746	202	35	analysis	analysis	NOUN
cana-2746	202	36	and	and	CCONJ
cana-2746	202	37	discrete	discrete	ADJ
cana-2746	202	38	mathematics	mathematic	NOUN
cana-2746	202	39	arxiv:1101.4791v1	arxiv:1101.4791v1	NOUN
cana-2746	202	40	[	[	X
cana-2746	202	41	math.gm	math.gm	X
cana-2746	202	42	]	]	PUNCT
cana-2746	202	43	.	.	PUNCT
cana-2746	203	1	[	[	X
cana-2746	203	2	5	5	X
cana-2746	203	3	]	]	X
cana-2746	203	4	h.	h.	PROPN
cana-2746	203	5	garg	garg	PROPN
cana-2746	203	6	,	,	PUNCT
cana-2746	203	7	a	a	DET
cana-2746	203	8	novel	novel	ADJ
cana-2746	203	9	accuracy	accuracy	NOUN
cana-2746	203	10	function	function	NOUN
cana-2746	203	11	under	under	ADP
cana-2746	203	12	interval	interval	NOUN
cana-2746	203	13	-	-	PUNCT
cana-2746	203	14	valued	value	VERB
cana-2746	203	15	pythagorean	pythagorean	PROPN
cana-2746	203	16	fuzzy	fuzzy	ADJ
cana-2746	203	17	environment	environment	NOUN
cana-2746	203	18	for	for	ADP
cana-2746	203	19	solving	solve	VERB
cana-2746	203	20	multicriteria	multicriteria	PROPN
cana-2746	203	21	decision	decision	NOUN
cana-2746	203	22	making	making	NOUN
cana-2746	203	23	problem	problem	NOUN
cana-2746	203	24	.	.	PUNCT
cana-2746	204	1	j	j	PROPN
cana-2746	204	2	intell	intell	PROPN
cana-2746	204	3	fuzzy	fuzzy	ADJ
cana-2746	204	4	syst	syst	PROPN
cana-2746	204	5	.	.	PUNCT
cana-2746	205	1	31(1	31(1	NUM
cana-2746	205	2	)	)	PUNCT
cana-2746	205	3	,	,	PUNCT
cana-2746	205	4	529	529	NUM
cana-2746	205	5	-	-	SYM
cana-2746	205	6	540	540	NUM
cana-2746	205	7	,	,	PUNCT
cana-2746	205	8	(	(	PUNCT
cana-2746	205	9	2016	2016	NUM
cana-2746	205	10	)	)	PUNCT
cana-2746	205	11	.	.	PUNCT
cana-2746	206	1	[	[	X
cana-2746	206	2	6	6	NUM
cana-2746	206	3	]	]	X
cana-2746	206	4	h.	h.	PROPN
cana-2746	206	5	garg	garg	PROPN
cana-2746	206	6	,	,	PUNCT
cana-2746	206	7	a	a	DET
cana-2746	206	8	new	new	ADJ
cana-2746	206	9	generalized	generalized	ADJ
cana-2746	206	10	pythagorean	pythagorean	NOUN
cana-2746	206	11	fuzzy	fuzzy	ADJ
cana-2746	206	12	information	information	NOUN
cana-2746	206	13	aggregation	aggregation	NOUN
cana-2746	206	14	using	use	VERB
cana-2746	206	15	einstein	einstein	ADJ
cana-2746	206	16	operations	operation	NOUN
cana-2746	206	17	and	and	CCONJ
cana-2746	206	18	its	its	PRON
cana-2746	206	19	application	application	NOUN
cana-2746	206	20	to	to	ADP
cana-2746	206	21	decision	decision	NOUN
cana-2746	206	22	making	making	NOUN
cana-2746	206	23	.	.	PUNCT
cana-2746	207	1	int	int	PROPN
cana-2746	207	2	j	j	PROPN
cana-2746	207	3	intell	intell	PROPN
cana-2746	207	4	syst	syst	PROPN
cana-2746	207	5	.	.	PUNCT
cana-2746	208	1	31(7	31(7	PROPN
cana-2746	208	2	)	)	PUNCT
cana-2746	208	3	,	,	PUNCT
cana-2746	208	4	886	886	NUM
cana-2746	208	5	-	-	SYM
cana-2746	208	6	920	920	NUM
cana-2746	208	7	(	(	PUNCT
cana-2746	208	8	2016	2016	NUM
cana-2746	208	9	)	)	PUNCT
cana-2746	208	10	.	.	PUNCT
cana-2746	209	1	[	[	X
cana-2746	209	2	7	7	X
cana-2746	209	3	]	]	PUNCT
cana-2746	209	4	k.	k.	PROPN
cana-2746	209	5	iseki	iseki	PROPN
cana-2746	209	6	,	,	PUNCT
cana-2746	209	7	quasi	quasi	NOUN
cana-2746	209	8	-	-	NOUN
cana-2746	209	9	ideals	ideal	NOUN
cana-2746	209	10	in	in	ADP
cana-2746	209	11	semirings	semiring	NOUN
cana-2746	209	12	without	without	ADP
cana-2746	209	13	zero	zero	NUM
cana-2746	209	14	,	,	PUNCT
cana-2746	209	15	proc	proc	NOUN
cana-2746	209	16	.	.	PUNCT
cana-2746	210	1	japan	japan	PROPN
cana-2746	210	2	acad	acad	PROPN
cana-2746	210	3	.	.	PROPN
cana-2746	210	4	,	,	PUNCT
cana-2746	210	5	34	34	NUM
cana-2746	210	6	(	(	PUNCT
cana-2746	210	7	1958	1958	NUM
cana-2746	210	8	)	)	PUNCT
cana-2746	210	9	,	,	PUNCT
cana-2746	210	10	79	79	NUM
cana-2746	210	11	-	-	SYM
cana-2746	210	12	84	84	NUM
cana-2746	210	13	.	.	PUNCT
cana-2746	211	1	[	[	X
cana-2746	211	2	8	8	NUM
cana-2746	211	3	]	]	PUNCT
cana-2746	211	4	k.	k.	PROPN
cana-2746	211	5	iseki	iseki	PROPN
cana-2746	211	6	,	,	PUNCT
cana-2746	211	7	ideal	ideal	ADJ
cana-2746	211	8	theory	theory	NOUN
cana-2746	211	9	of	of	ADP
cana-2746	211	10	semiring	semiring	NOUN
cana-2746	211	11	,	,	PUNCT
cana-2746	211	12	proc	proc	PROPN
cana-2746	211	13	.	.	PUNCT
cana-2746	212	1	japan	japan	PROPN
cana-2746	212	2	acad	acad	PROPN
cana-2746	212	3	.	.	PROPN
cana-2746	212	4	,	,	PUNCT
cana-2746	212	5	32	32	NUM
cana-2746	212	6	(	(	PUNCT
cana-2746	212	7	1956	1956	NUM
cana-2746	212	8	)	)	PUNCT
cana-2746	212	9	,	,	PUNCT
cana-2746	212	10	554	554	NUM
cana-2746	212	11	-	-	SYM
cana-2746	212	12	559	559	NUM
cana-2746	212	13	.	.	PUNCT
cana-2746	213	1	[	[	X
cana-2746	213	2	9	9	NUM
cana-2746	213	3	]	]	PUNCT
cana-2746	213	4	k.	k.	PROPN
cana-2746	213	5	iseki	iseki	PROPN
cana-2746	213	6	,	,	PUNCT
cana-2746	213	7	ideal	ideal	NOUN
cana-2746	213	8	in	in	ADP
cana-2746	213	9	semirings	semirings	PROPN
cana-2746	213	10	proc	proc	PROPN
cana-2746	213	11	.	.	PUNCT
cana-2746	214	1	japan	japan	PROPN
cana-2746	214	2	acad	acad	PROPN
cana-2746	214	3	.	.	PROPN
cana-2746	214	4	,	,	PUNCT
cana-2746	214	5	34	34	NUM
cana-2746	214	6	(	(	PUNCT
cana-2746	214	7	1958	1958	NUM
cana-2746	214	8	)	)	PUNCT
cana-2746	214	9	,	,	PUNCT
cana-2746	214	10	29	29	NUM
cana-2746	214	11	-	-	SYM
cana-2746	214	12	31	31	NUM
cana-2746	214	13	.	.	PUNCT
cana-2746	215	1	[	[	X
cana-2746	215	2	10	10	NUM
cana-2746	215	3	]	]	PUNCT
cana-2746	215	4	k.	k.	PROPN
cana-2746	215	5	izuka	izuka	PROPN
cana-2746	215	6	,	,	PUNCT
cana-2746	215	7	on	on	ADP
cana-2746	215	8	the	the	DET
cana-2746	215	9	jacobson	jacobson	PROPN
cana-2746	215	10	radical	radical	PROPN
cana-2746	215	11	of	of	ADP
cana-2746	215	12	a	a	DET
cana-2746	215	13	semiring	semiring	NOUN
cana-2746	215	14	,	,	PUNCT
cana-2746	215	15	tohoku	tohoku	PROPN
cana-2746	215	16	,	,	PUNCT
cana-2746	215	17	math	math	NOUN
cana-2746	215	18	.	.	PUNCT
cana-2746	216	1	j.	j.	PROPN
cana-2746	216	2	,	,	PUNCT
cana-2746	216	3	11	11	NUM
cana-2746	216	4	(	(	PUNCT
cana-2746	216	5	2	2	NUM
cana-2746	216	6	)	)	PUNCT
cana-2746	216	7	(	(	PUNCT
cana-2746	216	8	1959	1959	NUM
cana-2746	216	9	)	)	PUNCT
cana-2746	216	10	,	,	PUNCT
cana-2746	216	11	409	409	NUM
cana-2746	216	12	-	-	SYM
cana-2746	216	13	421	421	NUM
cana-2746	216	14	.	.	PUNCT
cana-2746	217	1	[	[	X
cana-2746	217	2	11	11	NUM
cana-2746	217	3	]	]	PUNCT
cana-2746	217	4	r.	r.	PROPN
cana-2746	217	5	d.	d.	PROPN
cana-2746	217	6	jagatap	jagatap	PROPN
cana-2746	217	7	,	,	PUNCT
cana-2746	217	8	y.	y.	PROPN
cana-2746	217	9	s.	s.	PROPN
cana-2746	217	10	pawar	pawar	PROPN
cana-2746	217	11	,	,	PUNCT
cana-2746	217	12	quasi	quasi	NOUN
cana-2746	217	13	-	-	NOUN
cana-2746	217	14	ideals	ideal	NOUN
cana-2746	217	15	and	and	CCONJ
cana-2746	217	16	minimal	minimal	ADJ
cana-2746	217	17	quasi	quasi	NOUN
cana-2746	217	18	-	-	NOUN
cana-2746	217	19	ideals	ideal	NOUN
cana-2746	217	20	in	in	ADP
cana-2746	217	21	g	g	PROPN
cana-2746	217	22	-semirings	-semiring	NOUN
cana-2746	217	23	,	,	PUNCT
cana-2746	217	24	novi	novi	PROPN
cana-2746	217	25	sad	sad	PROPN
cana-2746	217	26	j.	j.	PROPN
cana-2746	217	27	math	math	PROPN
cana-2746	217	28	.	.	PUNCT
cana-2746	217	29	,	,	PUNCT
cana-2746	217	30	39	39	NUM
cana-2746	217	31	(	(	PUNCT
cana-2746	217	32	2	2	NUM
cana-2746	217	33	)	)	PUNCT
cana-2746	217	34	,	,	PUNCT
cana-2746	217	35	(	(	PUNCT
cana-2746	217	36	2009	2009	NUM
cana-2746	217	37	)	)	PUNCT
cana-2746	217	38	,	,	PUNCT
cana-2746	217	39	79	79	NUM
cana-2746	217	40	-	-	SYM
cana-2746	217	41	87	87	NUM
cana-2746	217	42	.	.	PUNCT
cana-2746	218	1	[	[	X
cana-2746	218	2	12	12	NUM
cana-2746	218	3	]	]	X
cana-2746	218	4	r.d	r.d	PROPN
cana-2746	218	5	.	.	PROPN
cana-2746	218	6	jagatap	jagatap	PROPN
cana-2746	218	7	and	and	CCONJ
cana-2746	218	8	y.	y.	PROPN
cana-2746	218	9	s.	s.	PROPN
cana-2746	218	10	pawar	pawar	PROPN
cana-2746	218	11	,	,	PUNCT
cana-2746	218	12	bi	bi	NOUN
cana-2746	218	13	-	-	NOUN
cana-2746	218	14	ideals	ideal	NOUN
cana-2746	218	15	in	in	ADP
cana-2746	218	16	g	g	PROPN
cana-2746	218	17	-semirings	-semiring	NOUN
cana-2746	218	18	,	,	PUNCT
cana-2746	218	19	bull	bull	NOUN
cana-2746	218	20	.	.	PUNCT
cana-2746	219	1	inter	inter	PROPN
cana-2746	219	2	.	.	PUNCT
cana-2746	220	1	math	math	NOUN
cana-2746	220	2	.	.	PUNCT
cana-2746	221	1	virtual	virtual	ADJ
cana-2746	221	2	inst	inst	PROPN
cana-2746	221	3	.	.	PROPN
cana-2746	221	4	,	,	PUNCT
cana-2746	221	5	6	6	NUM
cana-2746	221	6	(	(	PUNCT
cana-2746	221	7	2	2	NUM
cana-2746	221	8	)	)	PUNCT
cana-2746	221	9	,	,	PUNCT
cana-2746	221	10	(	(	PUNCT
cana-2746	221	11	2016	2016	NUM
cana-2746	221	12	)	)	PUNCT
cana-2746	221	13	,	,	PUNCT
cana-2746	221	14	169	169	NUM
cana-2746	221	15	-	-	SYM
cana-2746	221	16	179	179	NUM
cana-2746	221	17	.	.	PUNCT
cana-2746	222	1	[	[	X
cana-2746	222	2	13	13	NUM
cana-2746	222	3	]	]	X
cana-2746	222	4	n.	n.	PROPN
cana-2746	222	5	kuroki	kuroki	PROPN
cana-2746	222	6	,	,	PUNCT
cana-2746	222	7	on	on	ADP
cana-2746	222	8	fuzzy	fuzzy	ADJ
cana-2746	222	9	semigroups	semigroup	NOUN
cana-2746	222	10	,	,	PUNCT
cana-2746	222	11	information	information	NOUN
cana-2746	222	12	sciences	science	NOUN
cana-2746	222	13	,	,	PUNCT
cana-2746	222	14	53	53	NUM
cana-2746	222	15	(	(	PUNCT
cana-2746	222	16	3	3	NUM
cana-2746	222	17	)	)	PUNCT
cana-2746	222	18	(	(	PUNCT
cana-2746	222	19	1991	1991	NUM
cana-2746	222	20	)	)	PUNCT
cana-2746	222	21	,	,	PUNCT
cana-2746	222	22	203	203	NUM
cana-2746	222	23	-	-	SYM
cana-2746	222	24	236	236	NUM
cana-2746	222	25	.	.	PUNCT
cana-2746	223	1	[	[	X
cana-2746	223	2	14	14	NUM
cana-2746	223	3	]	]	X
cana-2746	223	4	s.	s.	PROPN
cana-2746	223	5	lajos	lajos	PROPN
cana-2746	223	6	on	on	ADP
cana-2746	223	7	the	the	DET
cana-2746	223	8	bi	bi	NOUN
cana-2746	223	9	-	-	NOUN
cana-2746	223	10	ideals	ideal	NOUN
cana-2746	223	11	in	in	ADP
cana-2746	223	12	semigroups	semigroup	NOUN
cana-2746	223	13	,	,	PUNCT
cana-2746	223	14	proc	proc	NOUN
cana-2746	223	15	.	.	PUNCT
cana-2746	224	1	japan	japan	PROPN
cana-2746	224	2	acad	acad	PROPN
cana-2746	224	3	.	.	PROPN
cana-2746	224	4	,	,	PUNCT
cana-2746	224	5	45	45	NUM
cana-2746	224	6	,	,	PUNCT
cana-2746	224	7	(	(	PUNCT
cana-2746	224	8	1969	1969	NUM
cana-2746	224	9	)	)	PUNCT
cana-2746	224	10	,	,	PUNCT
cana-2746	224	11	710	710	NUM
cana-2746	224	12	-	-	SYM
cana-2746	224	13	712	712	NUM
cana-2746	224	14	.	.	PUNCT
cana-2746	225	1	[	[	X
cana-2746	225	2	15	15	NUM
cana-2746	225	3	]	]	X
cana-2746	225	4	z.	z.	PROPN
cana-2746	225	5	kong	kong	PROPN
cana-2746	225	6	,	,	PUNCT
cana-2746	225	7	l.	l.	PROPN
cana-2746	225	8	gao	gao	PROPN
cana-2746	225	9	,	,	PUNCT
cana-2746	225	10	and	and	CCONJ
cana-2746	225	11	l.	l.	PROPN
cana-2746	225	12	wang	wang	PROPN
cana-2746	225	13	,	,	PUNCT
cana-2746	225	14	comment	comment	NOUN
cana-2746	225	15	on	on	ADP
cana-2746	225	16	a	a	DET
cana-2746	225	17	fuzzy	fuzzy	ADJ
cana-2746	225	18	soft	soft	ADJ
cana-2746	225	19	set	set	ADJ
cana-2746	225	20	theoretic	theoretic	ADJ
cana-2746	225	21	approach	approach	NOUN
cana-2746	225	22	to	to	ADP
cana-2746	225	23	decision	decision	NOUN
cana-2746	225	24	making	making	NOUN
cana-2746	225	25	problems	problem	NOUN
cana-2746	225	26	,	,	PUNCT
cana-2746	225	27	journal	journal	NOUN
cana-2746	225	28	of	of	ADP
cana-2746	225	29	computational	computational	ADJ
cana-2746	225	30	and	and	CCONJ
cana-2746	225	31	applied	applied	ADJ
cana-2746	225	32	mathematics	mathematic	NOUN
cana-2746	225	33	,	,	PUNCT
cana-2746	225	34	223	223	NUM
cana-2746	225	35	,	,	PUNCT
cana-2746	225	36	2	2	NUM
cana-2746	225	37	,	,	PUNCT
cana-2746	225	38	540	540	NUM
cana-2746	225	39	-	-	SYM
cana-2746	225	40	542	542	NUM
cana-2746	225	41	,	,	PUNCT
cana-2746	225	42	2009	2009	NUM
cana-2746	225	43	.	.	PUNCT
cana-2746	226	1	[	[	X
cana-2746	226	2	16	16	NUM
cana-2746	226	3	]	]	PUNCT
cana-2746	226	4	p.	p.	PROPN
cana-2746	226	5	k.	k.	PROPN
cana-2746	227	1	maji	maji	PROPN
cana-2746	227	2	,	,	PUNCT
cana-2746	227	3	r.	r.	PROPN
cana-2746	227	4	k.	k.	PROPN
cana-2746	227	5	biswas	biswas	PROPN
cana-2746	227	6	,	,	PUNCT
cana-2746	227	7	and	and	CCONJ
cana-2746	227	8	a.	a.	PROPN
cana-2746	227	9	roy	roy	PROPN
cana-2746	227	10	,	,	PUNCT
cana-2746	227	11	soft	soft	ADJ
cana-2746	227	12	set	set	NOUN
cana-2746	227	13	theory	theory	NOUN
cana-2746	227	14	,	,	PUNCT
cana-2746	227	15	computer	computer	NOUN
cana-2746	227	16	and	and	CCONJ
cana-2746	227	17	mathematics	mathematic	NOUN
cana-2746	227	18	with	with	ADP
cana-2746	227	19	applications	application	NOUN
cana-2746	227	20	,	,	PUNCT
cana-2746	227	21	45	45	NUM
cana-2746	227	22	,	,	PUNCT
cana-2746	227	23	4	4	NUM
cana-2746	227	24	-	-	SYM
cana-2746	227	25	5	5	NUM
cana-2746	227	26	,	,	PUNCT
cana-2746	227	27	(	(	PUNCT
cana-2746	227	28	2003	2003	NUM
cana-2746	227	29	)	)	PUNCT
cana-2746	227	30	,	,	PUNCT
cana-2746	227	31	555	555	NUM
cana-2746	227	32	-	-	SYM
cana-2746	227	33	562	562	NUM
cana-2746	227	34	.	.	PUNCT
cana-2746	228	1	[	[	X
cana-2746	228	2	17	17	NUM
cana-2746	228	3	]	]	PUNCT
cana-2746	228	4	p.	p.	PROPN
cana-2746	228	5	k.	k.	PROPN
cana-2746	229	1	maji	maji	PROPN
cana-2746	229	2	,	,	PUNCT
cana-2746	229	3	r.	r.	PROPN
cana-2746	229	4	biswas	biswas	PROPN
cana-2746	229	5	,	,	PUNCT
cana-2746	229	6	and	and	CCONJ
cana-2746	229	7	a.	a.	PROPN
cana-2746	229	8	r.	r.	PROPN
cana-2746	229	9	roy	roy	PROPN
cana-2746	229	10	,	,	PUNCT
cana-2746	229	11	fuzzy	fuzzy	ADJ
cana-2746	229	12	soft	soft	ADJ
cana-2746	229	13	sets	set	NOUN
cana-2746	229	14	,	,	PUNCT
cana-2746	229	15	journalof	journalof	ADJ
cana-2746	229	16	fuzzy	fuzzy	ADJ
cana-2746	229	17	mathematics	mathematic	NOUN
cana-2746	229	18	,	,	PUNCT
cana-2746	229	19	9	9	NUM
cana-2746	229	20	,	,	PUNCT
cana-2746	229	21	(	(	PUNCT
cana-2746	229	22	2001	2001	NUM
cana-2746	229	23	)	)	PUNCT
cana-2746	229	24	,	,	PUNCT
cana-2746	229	25	589	589	NUM
cana-2746	229	26	-	-	SYM
cana-2746	229	27	602	602	NUM
cana-2746	229	28	.	.	PUNCT
cana-2746	230	1	[	[	X
cana-2746	230	2	18	18	NUM
cana-2746	230	3	]	]	PUNCT
cana-2746	230	4	p.	p.	PROPN
cana-2746	230	5	k.	k.	PROPN
cana-2746	231	1	maji	maji	PROPN
cana-2746	231	2	,	,	PUNCT
cana-2746	231	3	r.	r.	PROPN
cana-2746	231	4	biswas	biswas	PROPN
cana-2746	231	5	,	,	PUNCT
cana-2746	231	6	and	and	CCONJ
cana-2746	231	7	a.	a.	PROPN
cana-2746	231	8	r.	r.	PROPN
cana-2746	231	9	roy	roy	PROPN
cana-2746	231	10	,	,	PUNCT
cana-2746	231	11	intuitionistic	intuitionistic	ADJ
cana-2746	231	12	fuzzy	fuzzy	ADJ
cana-2746	231	13	soft	soft	ADJ
cana-2746	231	14	sets	set	NOUN
cana-2746	231	15	,	,	PUNCT
cana-2746	231	16	journal	journal	NOUN
cana-2746	231	17	of	of	ADP
cana-2746	231	18	fuzzy	fuzzy	ADJ
cana-2746	231	19	mathematics	mathematic	NOUN
cana-2746	231	20	,	,	PUNCT
cana-2746	231	21	9	9	NUM
cana-2746	231	22	,	,	PUNCT
cana-2746	231	23	3,(2001	3,(2001	NOUN
cana-2746	231	24	)	)	PUNCT
cana-2746	231	25	,	,	PUNCT
cana-2746	231	26	677692	677692	NUM
cana-2746	231	27	.	.	PUNCT
cana-2746	232	1	[	[	X
cana-2746	232	2	19	19	NUM
cana-2746	232	3	]	]	X
cana-2746	232	4	d.	d.	PROPN
cana-2746	232	5	molodtsov	molodtsov	PROPN
cana-2746	232	6	,	,	PUNCT
cana-2746	232	7	soft	soft	ADJ
cana-2746	232	8	set	set	NOUN
cana-2746	232	9	theory	theory	NOUN
cana-2746	232	10	-	-	PUNCT
cana-2746	232	11	first	first	ADJ
cana-2746	232	12	results	result	NOUN
cana-2746	232	13	,	,	PUNCT
cana-2746	232	14	computers	computer	NOUN
cana-2746	232	15	and	and	CCONJ
cana-2746	232	16	mathematics	mathematic	NOUN
cana-2746	232	17	with	with	ADP
cana-2746	232	18	applications	application	NOUN
cana-2746	232	19	,	,	PUNCT
cana-2746	232	20	37	37	NUM
cana-2746	232	21	,	,	PUNCT
cana-2746	232	22	4	4	NUM
cana-2746	232	23	-	-	SYM
cana-2746	232	24	5	5	NUM
cana-2746	232	25	,	,	PUNCT
cana-2746	232	26	(	(	PUNCT
cana-2746	232	27	1991	1991	NUM
cana-2746	232	28	)	)	PUNCT
cana-2746	232	29	,	,	PUNCT
cana-2746	232	30	19	19	NUM
cana-2746	232	31	-	-	SYM
cana-2746	232	32	31	31	NUM
cana-2746	232	33	.	.	PUNCT
cana-2746	233	1	[	[	X
cana-2746	233	2	20	20	NUM
cana-2746	233	3	]	]	PUNCT
cana-2746	233	4	nobusawa	nobusawa	PROPN
cana-2746	233	5	,	,	PUNCT
cana-2746	233	6	n.	n.	NOUN
cana-2746	233	7	on	on	ADP
cana-2746	233	8	a	a	DET
cana-2746	233	9	generalization	generalization	NOUN
cana-2746	233	10	of	of	ADP
cana-2746	233	11	the	the	DET
cana-2746	233	12	ring	ring	NOUN
cana-2746	233	13	theory	theory	NOUN
cana-2746	233	14	,	,	PUNCT
cana-2746	233	15	osaka	osaka	PROPN
cana-2746	233	16	j.	j.	PROPN
cana-2746	233	17	math	math	PROPN
cana-2746	233	18	,	,	PUNCT
cana-2746	233	19	1	1	NUM
cana-2746	233	20	,	,	PUNCT
cana-2746	233	21	(	(	PUNCT
cana-2746	233	22	1964	1964	NUM
cana-2746	233	23	)	)	PUNCT
cana-2746	233	24	,	,	PUNCT
cana-2746	233	25	81	81	NUM
cana-2746	233	26	-	-	SYM
cana-2746	233	27	89	89	NUM
cana-2746	233	28	.	.	PUNCT
cana-2746	234	1	[	[	X
cana-2746	234	2	21	21	NUM
cana-2746	234	3	]	]	PUNCT
cana-2746	234	4	x.	x.	PROPN
cana-2746	234	5	peng	peng	PROPN
cana-2746	234	6	,	,	PUNCT
cana-2746	234	7	y.	y.	PROPN
cana-2746	234	8	yang	yang	PROPN
cana-2746	234	9	,	,	PUNCT
cana-2746	234	10	and	and	CCONJ
cana-2746	234	11	j.	j.	PROPN
cana-2746	234	12	song	song	PROPN
cana-2746	234	13	,	,	PUNCT
cana-2746	234	14	pythagorean	pythagorean	PROPN
cana-2746	234	15	fuzzy	fuzzy	ADJ
cana-2746	234	16	soft	soft	ADJ
cana-2746	234	17	set	set	NOUN
cana-2746	234	18	and	and	CCONJ
cana-2746	234	19	its	its	PRON
cana-2746	234	20	application	application	NOUN
cana-2746	234	21	,	,	PUNCT
cana-2746	234	22	computer	computer	NOUN
cana-2746	234	23	engineering	engineering	NOUN
cana-2746	234	24	,	,	PUNCT
cana-2746	234	25	41	41	NUM
cana-2746	234	26	,	,	PUNCT
cana-2746	234	27	7	7	NUM
cana-2746	234	28	,	,	PUNCT
cana-2746	234	29	(	(	PUNCT
cana-2746	234	30	2015	2015	NUM
cana-2746	234	31	)	)	PUNCT
cana-2746	234	32	,	,	PUNCT
cana-2746	234	33	224	224	NUM
cana-2746	234	34	-	-	SYM
cana-2746	234	35	229	229	NUM
cana-2746	234	36	.	.	PUNCT
cana-2746	235	1	[	[	X
cana-2746	235	2	22	22	NUM
cana-2746	235	3	]	]	SYM
cana-2746	235	4	rao	rao	PROPN
cana-2746	235	5	,	,	PUNCT
cana-2746	235	6	m.	m.	NOUN
cana-2746	235	7	m.	m.	PROPN
cana-2746	235	8	k.	k.	PROPN
cana-2746	235	9	γ	γ	PROPN
cana-2746	235	10	-	-	PUNCT
cana-2746	235	11	semirings	semiring	NOUN
cana-2746	235	12	-	-	PUNCT
cana-2746	235	13	i	i	PROPN
cana-2746	235	14	,	,	PUNCT
cana-2746	235	15	south	south	PROPN
cana-2746	235	16	east	east	PROPN
cana-2746	235	17	asian	asian	PROPN
cana-2746	235	18	bull	bull	PROPN
cana-2746	235	19	.	.	PUNCT
cana-2746	236	1	of	of	ADP
cana-2746	236	2	math	math	NOUN
cana-2746	236	3	.	.	PUNCT
cana-2746	237	1	19	19	NUM
cana-2746	237	2	,	,	PUNCT
cana-2746	237	3	(	(	PUNCT
cana-2746	237	4	1995	1995	NUM
cana-2746	237	5	)	)	PUNCT
cana-2746	237	6	,	,	PUNCT
cana-2746	237	7	49	49	NUM
cana-2746	237	8	-	-	SYM
cana-2746	237	9	54	54	NUM
cana-2746	237	10	.	.	PUNCT
cana-2746	238	1	[	[	X
cana-2746	238	2	23	23	NUM
cana-2746	238	3	]	]	PUNCT
cana-2746	238	4	m.	m.	NOUN
cana-2746	238	5	murali	murali	PROPN
cana-2746	238	6	krishna	krishna	PROPN
cana-2746	238	7	rao	rao	PROPN
cana-2746	238	8	,	,	PUNCT
cana-2746	238	9	g	g	PROPN
cana-2746	238	10	-semirings	-semirings	PROPN
cana-2746	238	11	-	-	PUNCT
cana-2746	238	12	ii	ii	NOUN
cana-2746	238	13	,	,	PUNCT
cana-2746	238	14	southeast	southeast	ADJ
cana-2746	238	15	asian	asian	ADJ
cana-2746	238	16	bulletin	bulletin	NOUN
cana-2746	238	17	of	of	ADP
cana-2746	238	18	mathematics	mathematic	NOUN
cana-2746	238	19	,	,	PUNCT
cana-2746	238	20	21	21	NUM
cana-2746	238	21	(	(	PUNCT
cana-2746	238	22	3	3	NUM
cana-2746	238	23	)	)	PUNCT
cana-2746	238	24	(	(	PUNCT
cana-2746	238	25	1997	1997	NUM
cana-2746	238	26	)	)	PUNCT
cana-2746	238	27	,	,	PUNCT
cana-2746	238	28	281	281	NUM
cana-2746	238	29	-	-	SYM
cana-2746	238	30	287	287	NUM
cana-2746	238	31	.	.	PUNCT
cana-2746	239	1	[	[	X
cana-2746	239	2	24	24	NUM
cana-2746	239	3	]	]	PUNCT
cana-2746	239	4	m.	m.	NOUN
cana-2746	239	5	muralikrishnarao	muralikrishnarao	PROPN
cana-2746	239	6	,	,	PUNCT
cana-2746	239	7	the	the	DET
cana-2746	239	8	jacobson	jacobson	PROPN
cana-2746	239	9	radical	radical	PROPN
cana-2746	239	10	of	of	ADP
cana-2746	239	11	g	g	PROPN
cana-2746	239	12	-semiring	-semiring	NOUN
cana-2746	239	13	,	,	PUNCT
cana-2746	239	14	south	south	PROPN
cana-2746	239	15	east	east	ADJ
cana-2746	239	16	asian	asian	ADJ
cana-2746	239	17	bulletin	bulletin	NOUN
cana-2746	239	18	of	of	ADP
cana-2746	239	19	mathematics	mathematic	NOUN
cana-2746	239	20	,	,	PUNCT
cana-2746	239	21	23	23	NUM
cana-2746	239	22	(	(	PUNCT
cana-2746	239	23	1999	1999	NUM
cana-2746	239	24	)	)	PUNCT
cana-2746	239	25	,	,	PUNCT
cana-2746	239	26	127	127	NUM
cana-2746	239	27	-	-	SYM
cana-2746	239	28	134	134	NUM
cana-2746	239	29	.	.	PUNCT
cana-2746	240	1	[	[	X
cana-2746	240	2	25	25	NUM
cana-2746	240	3	]	]	PUNCT
cana-2746	240	4	m.	m.	NOUN
cana-2746	240	5	murali	murali	PROPN
cana-2746	240	6	krishna	krishna	PROPN
cana-2746	240	7	rao	rao	PROPN
cana-2746	240	8	,	,	PUNCT
cana-2746	240	9	g	g	NOUN
cana-2746	240	10	-semiring	-semiring	NOUN
cana-2746	240	11	with	with	ADP
cana-2746	240	12	identity	identity	NOUN
cana-2746	240	13	,	,	PUNCT
cana-2746	240	14	discussiones	discussione	NOUN
cana-2746	240	15	mathematicae	mathematicae	VERB
cana-2746	240	16	general	general	ADJ
cana-2746	240	17	algebra	algebra	PROPN
cana-2746	240	18	and	and	CCONJ
cana-2746	240	19	applications	application	NOUN
cana-2746	240	20	.	.	PUNCT
cana-2746	241	1	,	,	PUNCT
cana-2746	241	2	37	37	NUM
cana-2746	241	3	(	(	PUNCT
cana-2746	241	4	2017	2017	NUM
cana-2746	241	5	)	)	PUNCT
cana-2746	241	6	189207	189207	NUM
cana-2746	241	7	.	.	PUNCT
cana-2746	242	1	[	[	X
cana-2746	242	2	26	26	NUM
cana-2746	242	3	]	]	PUNCT
cana-2746	242	4	m.	m.	NOUN
cana-2746	242	5	murali	murali	PROPN
cana-2746	242	6	krishna	krishna	PROPN
cana-2746	242	7	rao	rao	PROPN
cana-2746	242	8	,	,	PUNCT
cana-2746	242	9	ideals	ideal	NOUN
cana-2746	242	10	in	in	ADP
cana-2746	242	11	ordered	order	VERB
cana-2746	242	12	g	g	PROPN
cana-2746	242	13	-semirings	-semiring	NOUN
cana-2746	242	14	,	,	PUNCT
cana-2746	242	15	discussiones	discussione	NOUN
cana-2746	242	16	mathematicae	mathematicae	VERB
cana-2746	242	17	general	general	ADJ
cana-2746	242	18	algebra	algebra	PROPN
cana-2746	242	19	and	and	CCONJ
cana-2746	242	20	applications	application	NOUN
cana-2746	242	21	,	,	PUNCT
cana-2746	242	22	38	38	NUM
cana-2746	242	23	(	(	PUNCT
cana-2746	242	24	2018	2018	NUM
cana-2746	242	25	)	)	PUNCT
cana-2746	242	26	,	,	PUNCT
cana-2746	242	27	47	47	NUM
cana-2746	242	28	-	-	SYM
cana-2746	242	29	68	68	NUM
cana-2746	242	30	.	.	PUNCT
cana-2746	243	1	[	[	X
cana-2746	243	2	27	27	NUM
cana-2746	243	3	]	]	X
cana-2746	243	4	m.	m.	NOUN
cana-2746	243	5	murali	murali	PROPN
cana-2746	243	6	krishna	krishna	PROPN
cana-2746	243	7	rao	rao	PROPN
cana-2746	243	8	,	,	PUNCT
cana-2746	243	9	bi	bi	ADJ
cana-2746	243	10	-	-	ADJ
cana-2746	243	11	interior	interior	ADJ
cana-2746	243	12	ideals	ideal	NOUN
cana-2746	243	13	in	in	ADP
cana-2746	243	14	semigroups	semigroup	NOUN
cana-2746	243	15	,	,	PUNCT
cana-2746	243	16	discus	discus	NOUN
cana-2746	243	17	-	-	PUNCT
cana-2746	243	18	siones	sione	NOUN
cana-2746	243	19	mathematicae	mathematicae	VERB
cana-2746	243	20	general	general	ADJ
cana-2746	243	21	algebra	algebra	PROPN
cana-2746	243	22	and	and	CCONJ
cana-2746	243	23	applications	application	NOUN
cana-2746	243	24	,	,	PUNCT
cana-2746	243	25	38	38	NUM
cana-2746	243	26	(	(	PUNCT
cana-2746	243	27	2018	2018	NUM
cana-2746	243	28	)	)	PUNCT
cana-2746	243	29	,	,	PUNCT
cana-2746	243	30	69	69	NUM
cana-2746	243	31	-	-	SYM
cana-2746	243	32	78	78	NUM
cana-2746	243	33	.	.	PUNCT
cana-2746	244	1	[	[	X
cana-2746	244	2	28	28	NUM
cana-2746	244	3	]	]	X
cana-2746	244	4	m.	m.	NOUN
cana-2746	244	5	murali	murali	PROPN
cana-2746	244	6	krishna	krishna	PROPN
cana-2746	244	7	rao	rao	PROPN
cana-2746	244	8	,	,	PUNCT
cana-2746	244	9	t	t	NOUN
cana-2746	244	10	-	-	PUNCT
cana-2746	244	11	fuzzy	fuzzy	ADJ
cana-2746	244	12	ideals	ideal	NOUN
cana-2746	244	13	in	in	ADP
cana-2746	244	14	ordered	order	VERB
cana-2746	244	15	g	g	PROPN
cana-2746	244	16	-semirings	-semiring	NOUN
cana-2746	244	17	,	,	PUNCT
cana-2746	244	18	annals	annal	NOUN
cana-2746	244	19	of	of	ADP
cana-2746	244	20	fuzzy	fuzzy	ADJ
cana-2746	244	21	mathematics	mathematic	NOUN
cana-2746	244	22	and	and	CCONJ
cana-2746	244	23	informatics	informatic	NOUN
cana-2746	244	24	,	,	PUNCT
cana-2746	244	25	13(2	13(2	PROPN
cana-2746	244	26	)	)	PUNCT
cana-2746	244	27	,	,	PUNCT
cana-2746	244	28	(	(	PUNCT
cana-2746	244	29	2017	2017	NUM
cana-2746	244	30	)	)	PUNCT
cana-2746	244	31	,	,	PUNCT
cana-2746	244	32	253	253	NUM
cana-2746	244	33	-	-	SYM
cana-2746	244	34	276	276	NUM
cana-2746	244	35	.	.	PUNCT
cana-2746	244	36	communications	communication	NOUN
cana-2746	244	37	on	on	ADP
cana-2746	244	38	applied	apply	VERB
cana-2746	244	39	nonlinear	nonlinear	ADJ
cana-2746	244	40	analysis	analysis	NOUN
cana-2746	244	41	issn	issn	NOUN
cana-2746	244	42	:	:	PUNCT
cana-2746	244	43	1074	1074	NUM
cana-2746	244	44	-	-	PUNCT
cana-2746	244	45	133x	133x	NUM
cana-2746	244	46	vol	vol	NOUN
cana-2746	244	47	32	32	NUM
cana-2746	244	48	no	no	NOUN
cana-2746	244	49	.	.	PUNCT
cana-2746	245	1	4s	4s	NUM
cana-2746	245	2	(	(	PUNCT
cana-2746	245	3	2025	2025	NUM
cana-2746	245	4	)	)	PUNCT
cana-2746	245	5	167	167	NUM
cana-2746	245	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2746	246	1	[	[	X
cana-2746	246	2	29	29	NUM
cana-2746	246	3	]	]	PUNCT
cana-2746	246	4	m.	m.	NOUN
cana-2746	246	5	murali	murali	PROPN
cana-2746	246	6	krishna	krishna	PROPN
cana-2746	246	7	rao	rao	PROPN
cana-2746	246	8	,	,	PUNCT
cana-2746	246	9	left	leave	VERB
cana-2746	246	10	bi	bi	ADJ
cana-2746	246	11	-	-	ADJ
cana-2746	246	12	quasi	quasi	ADJ
cana-2746	246	13	ideals	ideal	NOUN
cana-2746	246	14	of	of	ADP
cana-2746	246	15	semirings	semiring	NOUN
cana-2746	246	16	,	,	PUNCT
cana-2746	246	17	bull	bull	NOUN
cana-2746	246	18	.	.	PUNCT
cana-2746	247	1	int	int	NOUN
cana-2746	247	2	.	.	PUNCT
cana-2746	248	1	math	math	NOUN
cana-2746	248	2	.	.	PUNCT
cana-2746	249	1	virtual	virtual	ADJ
cana-2746	249	2	inst	inst	NOUN
cana-2746	249	3	8	8	NUM
cana-2746	249	4	(	(	PUNCT
cana-2746	249	5	2018	2018	NUM
cana-2746	249	6	)	)	PUNCT
cana-2746	249	7	,	,	PUNCT
cana-2746	249	8	45	45	NUM
cana-2746	249	9	-	-	SYM
cana-2746	249	10	53	53	NUM
cana-2746	249	11	.	.	PUNCT
cana-2746	250	1	[	[	X
cana-2746	250	2	30	30	NUM
cana-2746	250	3	]	]	X
cana-2746	250	4	m.	m.	NOUN
cana-2746	250	5	murali	murali	PROPN
cana-2746	250	6	krishna	krishna	PROPN
cana-2746	250	7	rao	rao	PROPN
cana-2746	250	8	,	,	PUNCT
cana-2746	250	9	bi	bi	ADJ
cana-2746	250	10	-	-	ADJ
cana-2746	250	11	quasi	quasi	NOUN
cana-2746	250	12	-	-	NOUN
cana-2746	250	13	ideals	ideal	NOUN
cana-2746	250	14	and	and	CCONJ
cana-2746	250	15	fuzzy	fuzzy	ADJ
cana-2746	250	16	bi	bi	NOUN
cana-2746	250	17	-	-	NOUN
cana-2746	250	18	quasiideals	quasiideal	NOUN
cana-2746	250	19	of	of	ADP
cana-2746	250	20	γ	γ	NOUN
cana-2746	250	21	-	-	PUNCT
cana-2746	250	22	semigroups	semigroup	NOUN
cana-2746	250	23	,	,	PUNCT
cana-2746	250	24	bull	bull	NOUN
cana-2746	250	25	.	.	PUNCT
cana-2746	251	1	int	int	NOUN
cana-2746	251	2	.	.	PUNCT
cana-2746	252	1	math	math	NOUN
cana-2746	252	2	.	.	PUNCT
cana-2746	253	1	virtual	virtual	ADJ
cana-2746	253	2	inst	inst	PROPN
cana-2746	253	3	.	.	PUNCT
cana-2746	254	1	,7	,7	PUNCT
cana-2746	254	2	(	(	PUNCT
cana-2746	254	3	2	2	NUM
cana-2746	254	4	)	)	PUNCT
cana-2746	254	5	,	,	PUNCT
cana-2746	254	6	(	(	PUNCT
cana-2746	254	7	2017	2017	NUM
cana-2746	254	8	)	)	PUNCT
cana-2746	254	9	,	,	PUNCT
cana-2746	254	10	231	231	NUM
cana-2746	254	11	-	-	SYM
cana-2746	254	12	242	242	NUM
cana-2746	254	13	.	.	PUNCT
cana-2746	255	1	[	[	X
cana-2746	255	2	31	31	NUM
cana-2746	255	3	]	]	PUNCT
cana-2746	255	4	m.	m.	NOUN
cana-2746	255	5	murali	murali	PROPN
cana-2746	255	6	krishna	krishna	PROPN
cana-2746	255	7	rao	rao	PROPN
cana-2746	255	8	,	,	PUNCT
cana-2746	255	9	b.	b.	PROPN
cana-2746	255	10	venkateswarlu	venkateswarlu	PROPN
cana-2746	255	11	and	and	CCONJ
cana-2746	255	12	n.	n.	PROPN
cana-2746	255	13	rafi	rafi	PROPN
cana-2746	255	14	,	,	PUNCT
cana-2746	255	15	left	leave	VERB
cana-2746	255	16	bi	bi	ADJ
cana-2746	255	17	-	-	ADJ
cana-2746	255	18	quasi	quasi	NOUN
cana-2746	255	19	-	-	NOUN
cana-2746	255	20	ideals	ideal	NOUN
cana-2746	255	21	of	of	ADP
cana-2746	255	22	g	g	PROPN
cana-2746	255	23	-semirings	-semiring	NOUN
cana-2746	255	24	,	,	PUNCT
cana-2746	255	25	asia	asia	PROPN
cana-2746	255	26	pacific	pacific	PROPN
cana-2746	255	27	journal	journal	PROPN
cana-2746	255	28	of	of	ADP
cana-2746	255	29	mathematics	mathematics	PROPN
cana-2746	255	30	,	,	PUNCT
cana-2746	255	31	vol	vol	NOUN
cana-2746	255	32	.	.	PROPN
cana-2746	255	33	4	4	NUM
cana-2746	255	34	,	,	PUNCT
cana-2746	255	35	no	no	INTJ
cana-2746	255	36	.	.	NOUN
cana-2746	255	37	2	2	NUM
cana-2746	255	38	(	(	PUNCT
cana-2746	255	39	2017	2017	NUM
cana-2746	255	40	)	)	PUNCT
cana-2746	255	41	,	,	PUNCT
cana-2746	255	42	144	144	NUM
cana-2746	255	43	-	-	SYM
cana-2746	255	44	153	153	NUM
cana-2746	255	45	.	.	PUNCT
cana-2746	256	1	[	[	X
cana-2746	256	2	32	32	NUM
cana-2746	256	3	]	]	PUNCT
cana-2746	256	4	rosenfeld	rosenfeld	PROPN
cana-2746	256	5	,	,	PUNCT
cana-2746	256	6	a.	a.	NOUN
cana-2746	256	7	fuzzy	fuzzy	ADJ
cana-2746	256	8	groups	group	NOUN
cana-2746	256	9	,	,	PUNCT
cana-2746	256	10	j.	j.	PROPN
cana-2746	256	11	math	math	PROPN
cana-2746	256	12	analysis	analysis	NOUN
cana-2746	256	13	applications	application	NOUN
cana-2746	256	14	35	35	NUM
cana-2746	256	15	,	,	PUNCT
cana-2746	256	16	(	(	PUNCT
cana-2746	256	17	1971	1971	NUM
cana-2746	256	18	)	)	PUNCT
cana-2746	256	19	,	,	PUNCT
cana-2746	256	20	512	512	NUM
cana-2746	256	21	-	-	SYM
cana-2746	256	22	519	519	NUM
cana-2746	256	23	.	.	PUNCT
cana-2746	257	1	[	[	X
cana-2746	257	2	33	33	NUM
cana-2746	257	3	]	]	X
cana-2746	257	4	yager	yager	NOUN
cana-2746	257	5	,	,	PUNCT
cana-2746	257	6	r.r	r.r	PROPN
cana-2746	257	7	,	,	PUNCT
cana-2746	257	8	pythagorean	pythagorean	ADJ
cana-2746	257	9	fuzzy	fuzzy	ADJ
cana-2746	257	10	subsets	subset	NOUN
cana-2746	257	11	,	,	PUNCT
cana-2746	257	12	in	in	ADP
cana-2746	257	13	:	:	PUNCT
cana-2746	257	14	proceedings	proceeding	NOUN
cana-2746	257	15	of	of	ADP
cana-2746	257	16	joint	joint	ADJ
cana-2746	257	17	ifsa	ifsa	PROPN
cana-2746	257	18	world	world	PROPN
cana-2746	257	19	congress	congress	PROPN
cana-2746	257	20	and	and	CCONJ
cana-2746	257	21	nafips	nafip	NOUN
cana-2746	257	22	annual	annual	ADJ
cana-2746	257	23	meeting	meeting	NOUN
cana-2746	257	24	,	,	PUNCT
cana-2746	257	25	edmonton	edmonton	PROPN
cana-2746	257	26	.	.	PUNCT
cana-2746	258	1	canada	canada	PROPN
cana-2746	258	2	,	,	PUNCT
cana-2746	258	3	(	(	PUNCT
cana-2746	258	4	2013	2013	NUM
cana-2746	258	5	)	)	PUNCT
cana-2746	258	6	,	,	PUNCT
cana-2746	258	7	57	57	NUM
cana-2746	258	8	-	-	SYM
cana-2746	258	9	61	61	NUM
cana-2746	258	10	.	.	PUNCT
cana-2746	259	1	[	[	X
cana-2746	259	2	34	34	NUM
cana-2746	259	3	]	]	SYM
cana-2746	259	4	yager	yager	NOUN
cana-2746	259	5	,	,	PUNCT
cana-2746	259	6	r.r	r.r	PROPN
cana-2746	259	7	.	.	PROPN
cana-2746	259	8	:	:	PUNCT
cana-2746	259	9	pythagorean	pythagorean	PROPN
cana-2746	259	10	membership	membership	NOUN
cana-2746	259	11	grades	grade	NOUN
cana-2746	259	12	in	in	ADP
cana-2746	259	13	multicriteria	multicriteria	PROPN
cana-2746	259	14	decision	decision	NOUN
cana-2746	259	15	making	making	NOUN
cana-2746	259	16	.	.	PUNCT
cana-2746	260	1	ieee	ieee	PROPN
cana-2746	260	2	trans	trans	PROPN
cana-2746	260	3	.	.	PUNCT
cana-2746	260	4	fuzzy	fuzzy	ADJ
cana-2746	260	5	syst	syst	PROPN
cana-2746	260	6	.	.	PUNCT
cana-2746	261	1	22(4	22(4	NUM
cana-2746	261	2	)	)	PUNCT
cana-2746	261	3	,	,	PUNCT
cana-2746	261	4	(	(	PUNCT
cana-2746	261	5	2014	2014	NUM
cana-2746	261	6	)	)	PUNCT
cana-2746	261	7	,	,	PUNCT
cana-2746	261	8	958	958	NUM
cana-2746	261	9	-	-	SYM
cana-2746	261	10	965	965	NUM
cana-2746	261	11	.	.	PUNCT
cana-2746	262	1	[	[	X
cana-2746	262	2	35	35	NUM
cana-2746	262	3	]	]	X
cana-2746	262	4	l.	l.	PROPN
cana-2746	262	5	a	a	DET
cana-2746	262	6	zadeh	zadeh	PROPN
cana-2746	262	7	,	,	PUNCT
cana-2746	262	8	fuzzy	fuzzy	ADJ
cana-2746	262	9	sets	set	NOUN
cana-2746	262	10	.	.	PUNCT
cana-2746	263	1	inform	inform	NOUN
cana-2746	263	2	and	and	CCONJ
cana-2746	263	3	control	control	NOUN
cana-2746	263	4	.	.	PUNCT
cana-2746	264	1	8	8	NUM
cana-2746	264	2	(	(	PUNCT
cana-2746	264	3	1965	1965	NUM
cana-2746	264	4	)	)	PUNCT
cana-2746	264	5	338	338	NUM
cana-2746	264	6	-	-	SYM
cana-2746	264	7	353	353	NUM
cana-2746	264	8	.	.	PUNCT