id	sid	tid	token	lemma	pos
cana-2405	1	1	communications	communication	NOUN
cana-2405	1	2	on	on	ADP
cana-2405	1	3	applied	apply	VERB
cana-2405	1	4	nonlinear	nonlinear	ADJ
cana-2405	1	5	analysis	analysis	NOUN
cana-2405	1	6	issn	issn	NOUN
cana-2405	1	7	:	:	PUNCT
cana-2405	1	8	1074	1074	NUM
cana-2405	1	9	-	-	PUNCT
cana-2405	1	10	133x	133x	NUM
cana-2405	1	11	vol	vol	NOUN
cana-2405	1	12	32	32	NUM
cana-2405	1	13	no	no	NOUN
cana-2405	1	14	.	.	PUNCT
cana-2405	2	1	2s	2s	NUM
cana-2405	2	2	(	(	PUNCT
cana-2405	2	3	2025	2025	NUM
cana-2405	2	4	)	)	PUNCT
cana-2405	2	5	312	312	NUM
cana-2405	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2405	2	7	pythagorean	pythagorean	ADJ
cana-2405	2	8	fuzzy	fuzzy	ADJ
cana-2405	2	9	ideals	ideal	NOUN
cana-2405	2	10	in	in	ADP
cana-2405	2	11	semiring	semire	VERB
cana-2405	2	12	1	1	NUM
cana-2405	2	13	j.jayaraj	j.jayaraj	NOUN
cana-2405	2	14	,	,	PUNCT
cana-2405	2	15	2	2	NUM
cana-2405	2	16	r.raghu	r.raghu	NOUN
cana-2405	2	17	,	,	PUNCT
cana-2405	2	18	3r.ezhilarasi	3r.ezhilarasi	NUM
cana-2405	2	19	1	1	NUM
cana-2405	2	20	assistant	assistant	NOUN
cana-2405	2	21	professor	professor	NOUN
cana-2405	2	22	,	,	PUNCT
cana-2405	2	23	department	department	NOUN
cana-2405	2	24	of	of	ADP
cana-2405	2	25	mathematics	mathematics	PROPN
cana-2405	2	26	,	,	PUNCT
cana-2405	2	27	dr	dr	PROPN
cana-2405	2	28	.	.	PROPN
cana-2405	2	29	puratchithalaivar	puratchithalaivar	PROPN
cana-2405	2	30	m.g.r	m.g.r	PROPN
cana-2405	2	31	government	government	NOUN
cana-2405	2	32	arts	arts	PROPN
cana-2405	2	33	and	and	CCONJ
cana-2405	2	34	science	science	PROPN
cana-2405	2	35	college	college	PROPN
cana-2405	2	36	,	,	PUNCT
cana-2405	2	37	kudavasal612601	kudavasal612601	PROPN
cana-2405	2	38	,	,	PUNCT
cana-2405	2	39	india	india	PROPN
cana-2405	2	40	.	.	PUNCT
cana-2405	3	1	e.mail	e.mail	ADV
cana-2405	3	2	:	:	PUNCT
cana-2405	3	3	joe.jayaraj@gmail.com	joe.jayaraj@gmail.com	PROPN
cana-2405	3	4	2	2	NUM
cana-2405	3	5	department	department	NOUN
cana-2405	3	6	of	of	ADP
cana-2405	3	7	mathematics	mathematic	NOUN
cana-2405	3	8	,	,	PUNCT
cana-2405	3	9	thiru	thiru	PROPN
cana-2405	3	10	.	.	PUNCT
cana-2405	4	1	a.	a.	PROPN
cana-2405	4	2	govindasamy	govindasamy	PROPN
cana-2405	4	3	government	government	NOUN
cana-2405	4	4	arts	arts	PROPN
cana-2405	4	5	college	college	NOUN
cana-2405	4	6	,	,	PUNCT
cana-2405	4	7	tindivanam-604307	tindivanam-604307	ADJ
cana-2405	4	8	,	,	PUNCT
cana-2405	4	9	india	india	PROPN
cana-2405	4	10	.	.	PUNCT
cana-2405	5	1	e.mail	e.mail	ADV
cana-2405	5	2	:	:	PUNCT
cana-2405	5	3	c.r.raghu89@gmail.com	c.r.raghu89@gmail.com	X
cana-2405	5	4	1,2department	1,2department	NUM
cana-2405	5	5	of	of	ADP
cana-2405	5	6	mathematics	mathematic	NOUN
cana-2405	5	7	,	,	PUNCT
cana-2405	5	8	annamalai	annamalai	PROPN
cana-2405	5	9	university	university	PROPN
cana-2405	5	10	,	,	PUNCT
cana-2405	5	11	annamalainagar	annamalainagar	NOUN
cana-2405	5	12	,	,	PUNCT
cana-2405	5	13	608002	608002	NUM
cana-2405	5	14	,	,	PUNCT
cana-2405	5	15	3asscociate	3asscociate	PROPN
cana-2405	5	16	professor	professor	NOUN
cana-2405	5	17	,	,	PUNCT
cana-2405	5	18	department	department	NOUN
cana-2405	5	19	of	of	ADP
cana-2405	5	20	mathematics	mathematic	NOUN
cana-2405	5	21	,	,	PUNCT
cana-2405	5	22	arignar	arignar	ADJ
cana-2405	5	23	anna	anna	PROPN
cana-2405	5	24	government	government	PROPN
cana-2405	5	25	arts	arts	PROPN
cana-2405	5	26	college	college	PROPN
cana-2405	5	27	,	,	PUNCT
cana-2405	5	28	villupuram-605	villupuram-605	NOUN
cana-2405	5	29	602	602	NUM
cana-2405	5	30	,	,	PUNCT
cana-2405	5	31	india	india	PROPN
cana-2405	5	32	.	.	PUNCT
cana-2405	6	1	e.mail	e.mail	ADV
cana-2405	6	2	:	:	PUNCT
cana-2405	7	1	rearasi@gmail.com	rearasi@gmail.com	X
cana-2405	7	2	article	article	NOUN
cana-2405	7	3	history	history	NOUN
cana-2405	7	4	:	:	PUNCT
cana-2405	7	5	received	receive	VERB
cana-2405	7	6	:	:	PUNCT
cana-2405	7	7	15	15	NUM
cana-2405	7	8	-	-	SYM
cana-2405	7	9	09	09	NUM
cana-2405	7	10	-	-	PUNCT
cana-2405	7	11	2024	2024	NUM
cana-2405	7	12	revised	revise	VERB
cana-2405	7	13	:	:	PUNCT
cana-2405	7	14	23	23	NUM
cana-2405	7	15	-	-	SYM
cana-2405	7	16	10	10	NUM
cana-2405	7	17	-	-	PUNCT
cana-2405	7	18	2024	2024	NUM
cana-2405	7	19	accepted	accept	VERB
cana-2405	7	20	:	:	PUNCT
cana-2405	7	21	03	03	NUM
cana-2405	7	22	-	-	SYM
cana-2405	7	23	11	11	NUM
cana-2405	7	24	-	-	PUNCT
cana-2405	7	25	2024	2024	NUM
cana-2405	7	26	abstract	abstract	NOUN
cana-2405	7	27	:	:	PUNCT
cana-2405	7	28	in	in	ADP
cana-2405	7	29	this	this	DET
cana-2405	7	30	paper	paper	NOUN
cana-2405	7	31	,	,	PUNCT
cana-2405	7	32	we	we	PRON
cana-2405	7	33	introduce	introduce	VERB
cana-2405	7	34	the	the	DET
cana-2405	7	35	notion	notion	NOUN
cana-2405	7	36	of	of	ADP
cana-2405	7	37	pythagorean	pythagorean	PROPN
cana-2405	7	38	fuzzy	fuzzy	ADJ
cana-2405	7	39	ideals	ideal	NOUN
cana-2405	7	40	in	in	ADP
cana-2405	7	41	semiring	semire	VERB
cana-2405	7	42	some	some	DET
cana-2405	7	43	interesting	interesting	ADJ
cana-2405	7	44	properties	property	NOUN
cana-2405	7	45	,	,	PUNCT
cana-2405	7	46	results	result	NOUN
cana-2405	7	47	are	be	AUX
cana-2405	7	48	discussed	discuss	VERB
cana-2405	7	49	in	in	ADP
cana-2405	7	50	this	this	DET
cana-2405	7	51	paper	paper	NOUN
cana-2405	7	52	.	.	PUNCT
cana-2405	8	1	keywords	keyword	NOUN
cana-2405	8	2	:	:	PUNCT
cana-2405	8	3	pythagorean	pythagorean	NOUN
cana-2405	8	4	,	,	PUNCT
cana-2405	8	5	fuzzy	fuzzy	ADJ
cana-2405	8	6	set	set	NOUN
cana-2405	8	7	,	,	PUNCT
cana-2405	8	8	semiring	semire	VERB
cana-2405	8	9	1	1	NUM
cana-2405	8	10	introduction	introduction	NOUN
cana-2405	8	11	nobusawa[5	nobusawa[5	PROPN
cana-2405	8	12	]	]	PUNCT
cana-2405	8	13	studied	study	VERB
cana-2405	8	14	the	the	DET
cana-2405	8	15	concept	concept	NOUN
cana-2405	8	16	of	of	ADP
cana-2405	8	17	gamma	gamma	NOUN
cana-2405	8	18	semiring	semire	VERB
cana-2405	8	19	as	as	ADP
cana-2405	8	20	a	a	DET
cana-2405	8	21	generalization	generalization	NOUN
cana-2405	8	22	of	of	ADP
cana-2405	8	23	ring	ring	NOUN
cana-2405	8	24	after	after	SCONJ
cana-2405	8	25	that	that	PRON
cana-2405	8	26	sen	sen	PROPN
cana-2405	8	27	introduced	introduce	VERB
cana-2405	8	28	the	the	DET
cana-2405	8	29	gamma	gamma	NOUN
cana-2405	8	30	semigroups	semigroup	NOUN
cana-2405	8	31	as	as	ADP
cana-2405	8	32	a	a	DET
cana-2405	8	33	generalization	generalization	NOUN
cana-2405	8	34	of	of	ADP
cana-2405	8	35	gamma	gamma	NOUN
cana-2405	8	36	groups	group	NOUN
cana-2405	8	37	.	.	PUNCT
cana-2405	9	1	murali	murali	PROPN
cana-2405	9	2	krishna	krishna	PROPN
cana-2405	9	3	rao[6	rao[6	NOUN
cana-2405	9	4	]	]	X
cana-2405	9	5	in	in	ADP
cana-2405	9	6	1995	1995	NUM
cana-2405	9	7	introduced	introduce	VERB
cana-2405	9	8	the	the	DET
cana-2405	9	9	notion	notion	NOUN
cana-2405	9	10	of	of	ADP
cana-2405	9	11	gamma	gamma	NOUN
cana-2405	9	12	semiring	semire	VERB
cana-2405	9	13	as	as	ADP
cana-2405	9	14	a	a	DET
cana-2405	9	15	generalization	generalization	NOUN
cana-2405	9	16	of	of	ADP
cana-2405	9	17	gamma	gamma	PROPN
cana-2405	9	18	ring	ring	NOUN
cana-2405	9	19	,	,	PUNCT
cana-2405	9	20	ring	ring	NOUN
cana-2405	9	21	,	,	PUNCT
cana-2405	9	22	ternary	ternary	ADJ
cana-2405	9	23	semiring	semiring	NOUN
cana-2405	9	24	and	and	CCONJ
cana-2405	9	25	semiring.the	semiring.the	DET
cana-2405	9	26	important	important	ADJ
cana-2405	9	27	reason	reason	NOUN
cana-2405	9	28	for	for	ADP
cana-2405	9	29	development	development	NOUN
cana-2405	9	30	of	of	ADP
cana-2405	9	31	gamma	gamma	NOUN
cana-2405	9	32	semiring	semiring	NOUN
cana-2405	9	33	is	be	AUX
cana-2405	9	34	a	a	DET
cana-2405	9	35	generalization	generalization	NOUN
cana-2405	9	36	of	of	ADP
cana-2405	9	37	results	result	NOUN
cana-2405	9	38	of	of	ADP
cana-2405	9	39	rings	ring	NOUN
cana-2405	9	40	,	,	PUNCT
cana-2405	9	41	gamma	gamma	NOUN
cana-2405	9	42	rings	ring	NOUN
cana-2405	9	43	,	,	PUNCT
cana-2405	9	44	semirings	semiring	NOUN
cana-2405	9	45	,	,	PUNCT
cana-2405	9	46	semigroup	semigroup	ADJ
cana-2405	9	47	and	and	CCONJ
cana-2405	9	48	ternary	ternary	ADJ
cana-2405	9	49	semirings	semiring	NOUN
cana-2405	9	50	.	.	PUNCT
cana-2405	10	1	zadeh[12	zadeh[12	PROPN
cana-2405	10	2	]	]	X
cana-2405	10	3	studied	study	VERB
cana-2405	10	4	the	the	DET
cana-2405	10	5	notion	notion	NOUN
cana-2405	10	6	of	of	ADP
cana-2405	10	7	fuzzy	fuzzy	ADJ
cana-2405	10	8	set	set	NOUN
cana-2405	10	9	theory	theory	NOUN
cana-2405	10	10	.	.	PUNCT
cana-2405	11	1	atanassov	atanassov	PROPN
cana-2405	12	1	[	[	X
cana-2405	12	2	2	2	NUM
cana-2405	12	3	]	]	PUNCT
cana-2405	12	4	introduced	introduce	VERB
cana-2405	12	5	intuitionistic	intuitionistic	ADJ
cana-2405	12	6	fuzzy	fuzzy	ADJ
cana-2405	12	7	sets	set	NOUN
cana-2405	12	8	as	as	ADP
cana-2405	12	9	a	a	DET
cana-2405	12	10	generalization	generalization	NOUN
cana-2405	12	11	of	of	ADP
cana-2405	12	12	fuzzy	fuzzy	ADJ
cana-2405	12	13	sets	set	NOUN
cana-2405	12	14	.	.	PUNCT
cana-2405	13	1	in	in	ADP
cana-2405	13	2	intuitionistic	intuitionistic	ADJ
cana-2405	13	3	,	,	PUNCT
cana-2405	13	4	the	the	DET
cana-2405	13	5	sum	sum	NOUN
cana-2405	13	6	of	of	ADP
cana-2405	13	7	membership	membership	NOUN
cana-2405	13	8	degree	degree	NOUN
cana-2405	13	9	and	and	CCONJ
cana-2405	13	10	non	non	ADJ
cana-2405	13	11	-	-	ADJ
cana-2405	13	12	membership	membership	ADJ
cana-2405	13	13	degree	degree	NOUN
cana-2405	13	14	should	should	AUX
cana-2405	13	15	not	not	PART
cana-2405	13	16	exceed	exceed	VERB
cana-2405	13	17	one	one	NUM
cana-2405	13	18	.	.	PUNCT
cana-2405	14	1	yager	yager	NOUN
cana-2405	15	1	[	[	X
cana-2405	15	2	10	10	NUM
cana-2405	15	3	]	]	PUNCT
cana-2405	15	4	initially	initially	ADV
cana-2405	15	5	introduced	introduce	VERB
cana-2405	15	6	the	the	DET
cana-2405	15	7	concept	concept	NOUN
cana-2405	15	8	of	of	ADP
cana-2405	15	9	pythagorean	pythagorean	PROPN
cana-2405	15	10	fuzzy	fuzzy	ADJ
cana-2405	15	11	sets	set	NOUN
cana-2405	15	12	.	.	PUNCT
cana-2405	16	1	in	in	ADP
cana-2405	16	2	a	a	DET
cana-2405	16	3	pythagorean	pythagorean	ADJ
cana-2405	16	4	fuzzy	fuzzy	ADJ
cana-2405	16	5	sets	set	NOUN
cana-2405	16	6	,	,	PUNCT
cana-2405	16	7	the	the	DET
cana-2405	16	8	sum	sum	NOUN
cana-2405	16	9	of	of	ADP
cana-2405	16	10	the	the	DET
cana-2405	16	11	squared	square	VERB
cana-2405	16	12	membership	membership	NOUN
cana-2405	16	13	and	and	CCONJ
cana-2405	16	14	non	non	ADJ
cana-2405	16	15	-	-	ADJ
cana-2405	16	16	membership	membership	ADJ
cana-2405	16	17	degrees	degree	NOUN
cana-2405	16	18	satisfies	satisfy	VERB
cana-2405	16	19	the	the	DET
cana-2405	16	20	condition	condition	NOUN
cana-2405	16	21	.	.	PUNCT
cana-2405	17	1	more	more	ADV
cana-2405	17	2	recently	recently	ADV
cana-2405	17	3	,	,	PUNCT
cana-2405	17	4	yager	yager	NOUN
cana-2405	18	1	[	[	X
cana-2405	18	2	10	10	NUM
cana-2405	18	3	,	,	PUNCT
cana-2405	18	4	11	11	NUM
cana-2405	18	5	]	]	PUNCT
cana-2405	18	6	proposed	propose	VERB
cana-2405	18	7	pythagorean	pythagorean	PROPN
cana-2405	18	8	fuzzy	fuzzy	ADJ
cana-2405	18	9	sets	set	NOUN
cana-2405	18	10	as	as	ADP
cana-2405	18	11	a	a	DET
cana-2405	18	12	powerful	powerful	ADJ
cana-2405	18	13	tool	tool	NOUN
cana-2405	18	14	for	for	ADP
cana-2405	18	15	effectively	effectively	ADV
cana-2405	18	16	managing	manage	VERB
cana-2405	18	17	uncertainty	uncertainty	NOUN
cana-2405	18	18	or	or	CCONJ
cana-2405	18	19	imprecise	imprecise	ADJ
cana-2405	18	20	information	information	NOUN
cana-2405	18	21	in	in	ADP
cana-2405	18	22	real	real	ADJ
cana-2405	18	23	-	-	PUNCT
cana-2405	18	24	world	world	NOUN
cana-2405	18	25	scenarios	scenario	NOUN
cana-2405	18	26	.	.	PUNCT
cana-2405	19	1	these	these	DET
cana-2405	19	2	sets	set	VERB
cana-2405	19	3	enforce	enforce	VERB
cana-2405	19	4	a	a	DET
cana-2405	19	5	constraint	constraint	NOUN
cana-2405	19	6	where	where	SCONJ
cana-2405	19	7	the	the	DET
cana-2405	19	8	sum	sum	NOUN
cana-2405	19	9	of	of	ADP
cana-2405	19	10	squares	square	NOUN
cana-2405	19	11	of	of	ADP
cana-2405	19	12	membership	membership	NOUN
cana-2405	19	13	and	and	CCONJ
cana-2405	19	14	non	non	ADJ
cana-2405	19	15	-	-	ADJ
cana-2405	19	16	membership	membership	ADJ
cana-2405	19	17	degrees	degree	NOUN
cana-2405	19	18	is	be	AUX
cana-2405	19	19	less	less	ADJ
cana-2405	19	20	than	than	ADP
cana-2405	19	21	or	or	CCONJ
cana-2405	19	22	equal	equal	ADJ
cana-2405	19	23	to	to	ADP
cana-2405	19	24	1	1	NUM
cana-2405	19	25	.	.	PUNCT
cana-2405	20	1	pythagorean	pythagorean	PROPN
cana-2405	20	2	fuzzy	fuzzy	ADJ
cana-2405	20	3	sets	set	NOUN
cana-2405	20	4	have	have	AUX
cana-2405	20	5	showcased	showcase	VERB
cana-2405	20	6	remarkable	remarkable	ADJ
cana-2405	20	7	efficacy	efficacy	NOUN
cana-2405	20	8	in	in	ADP
cana-2405	20	9	navigating	navigate	VERB
cana-2405	20	10	uncertainties	uncertainty	NOUN
cana-2405	20	11	,	,	PUNCT
cana-2405	20	12	prompting	prompt	VERB
cana-2405	20	13	a	a	DET
cana-2405	20	14	surge	surge	NOUN
cana-2405	20	15	of	of	ADP
cana-2405	20	16	scholarly	scholarly	ADJ
cana-2405	20	17	exploration	exploration	NOUN
cana-2405	20	18	across	across	ADP
cana-2405	20	19	diverse	diverse	ADJ
cana-2405	20	20	research	research	NOUN
cana-2405	20	21	avenues	avenue	NOUN
cana-2405	20	22	,	,	PUNCT
cana-2405	20	23	resulting	result	VERB
cana-2405	20	24	in	in	ADP
cana-2405	20	25	significant	significant	ADJ
cana-2405	20	26	progress	progress	NOUN
cana-2405	20	27	.	.	PUNCT
cana-2405	21	1	the	the	DET
cana-2405	21	2	conceptualization	conceptualization	NOUN
cana-2405	21	3	of	of	ADP
cana-2405	21	4	pythagorean	pythagorean	PROPN
cana-2405	21	5	fuzzy	fuzzy	ADJ
cana-2405	21	6	sets	set	NOUN
cana-2405	21	7	facilitates	facilitate	VERB
cana-2405	21	8	a	a	DET
cana-2405	21	9	more	more	ADV
cana-2405	21	10	comprehensive	comprehensive	ADJ
cana-2405	21	11	and	and	CCONJ
cana-2405	21	12	accurate	accurate	ADJ
cana-2405	21	13	portrayal	portrayal	NOUN
cana-2405	21	14	of	of	ADP
cana-2405	21	15	uncertain	uncertain	ADJ
cana-2405	21	16	information	information	NOUN
cana-2405	21	17	when	when	SCONJ
cana-2405	21	18	juxtaposed	juxtapose	VERB
cana-2405	21	19	with	with	ADP
cana-2405	21	20	intuitionistic	intuitionistic	ADJ
cana-2405	21	21	fuzzy	fuzzy	ADJ
cana-2405	21	22	sets	set	NOUN
cana-2405	21	23	.	.	PUNCT
cana-2405	22	1	across	across	ADP
cana-2405	22	2	various	various	ADJ
cana-2405	22	3	disciplines	discipline	NOUN
cana-2405	22	4	,	,	PUNCT
cana-2405	22	5	academics	academic	NOUN
cana-2405	22	6	have	have	AUX
cana-2405	22	7	meticulously	meticulously	ADV
cana-2405	22	8	examined	examine	VERB
cana-2405	22	9	the	the	DET
cana-2405	22	10	algebraic	algebraic	ADJ
cana-2405	22	11	attributes	attribute	NOUN
cana-2405	22	12	of	of	ADP
cana-2405	22	13	pythagorean	pythagorean	PROPN
cana-2405	22	14	fuzzy	fuzzy	ADJ
cana-2405	22	15	sets	set	NOUN
cana-2405	22	16	,	,	PUNCT
cana-2405	22	17	shedding	shed	VERB
cana-2405	22	18	light	light	NOUN
cana-2405	22	19	on	on	ADP
cana-2405	22	20	their	their	PRON
cana-2405	22	21	practical	practical	ADJ
cana-2405	22	22	applications	application	NOUN
cana-2405	22	23	and	and	CCONJ
cana-2405	22	24	foundational	foundational	ADJ
cana-2405	22	25	theoretical	theoretical	ADJ
cana-2405	22	26	constructs	construct	NOUN
cana-2405	22	27	.	.	PUNCT
cana-2405	23	1	many	many	ADJ
cana-2405	23	2	authors	author	NOUN
cana-2405	23	3	studied	study	VERB
cana-2405	23	4	the	the	DET
cana-2405	23	5	algebraic	algebraic	ADJ
cana-2405	23	6	structures	structure	NOUN
cana-2405	23	7	of	of	ADP
cana-2405	23	8	pythagorean	pythagorean	PROPN
cana-2405	23	9	fuzzy	fuzzy	ADJ
cana-2405	23	10	sets	set	VERB
cana-2405	23	11	this	this	DET
cana-2405	23	12	paper	paper	NOUN
cana-2405	23	13	is	be	AUX
cana-2405	23	14	structured	structure	VERB
cana-2405	23	15	into	into	ADP
cana-2405	23	16	three	three	NUM
cana-2405	23	17	sections	section	NOUN
cana-2405	23	18	.	.	PUNCT
cana-2405	24	1	the	the	DET
cana-2405	24	2	first	first	ADJ
cana-2405	24	3	and	and	CCONJ
cana-2405	24	4	second	second	ADJ
cana-2405	24	5	sections	section	NOUN
cana-2405	24	6	serve	serve	VERB
cana-2405	24	7	as	as	ADP
cana-2405	24	8	the	the	DET
cana-2405	24	9	introduction	introduction	NOUN
cana-2405	24	10	and	and	CCONJ
cana-2405	24	11	cover	cover	VERB
cana-2405	24	12	basic	basic	ADJ
cana-2405	24	13	results	result	NOUN
cana-2405	24	14	pertinent	pertinent	ADJ
cana-2405	24	15	to	to	ADP
cana-2405	24	16	the	the	DET
cana-2405	24	17	paper	paper	NOUN
cana-2405	24	18	’s	’s	PART
cana-2405	24	19	topic	topic	NOUN
cana-2405	24	20	.	.	PUNCT
cana-2405	25	1	in	in	ADP
cana-2405	25	2	the	the	DET
cana-2405	25	3	third	third	ADJ
cana-2405	25	4	section	section	NOUN
cana-2405	25	5	,	,	PUNCT
cana-2405	25	6	we	we	PRON
cana-2405	25	7	introduce	introduce	VERB
cana-2405	25	8	pythagorean	pythagorean	ADJ
cana-2405	25	9	fuzzy	fuzzy	ADJ
cana-2405	25	10	communications	communication	NOUN
cana-2405	25	11	on	on	ADP
cana-2405	25	12	applied	apply	VERB
cana-2405	25	13	nonlinear	nonlinear	ADJ
cana-2405	25	14	analysis	analysis	NOUN
cana-2405	25	15	issn	issn	NOUN
cana-2405	25	16	:	:	PUNCT
cana-2405	25	17	1074	1074	NUM
cana-2405	25	18	-	-	PUNCT
cana-2405	25	19	133x	133x	NUM
cana-2405	25	20	vol	vol	NOUN
cana-2405	25	21	32	32	NUM
cana-2405	25	22	no	no	NOUN
cana-2405	25	23	.	.	PUNCT
cana-2405	26	1	2s	2s	NUM
cana-2405	26	2	(	(	PUNCT
cana-2405	26	3	2025	2025	NUM
cana-2405	26	4	)	)	PUNCT
cana-2405	26	5	313	313	NUM
cana-2405	26	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2405	26	7	ideals	ideal	NOUN
cana-2405	26	8	in	in	ADP
cana-2405	26	9	semirings	semiring	NOUN
cana-2405	26	10	and	and	CCONJ
cana-2405	26	11	some	some	DET
cana-2405	26	12	interesting	interesting	ADJ
cana-2405	26	13	properties	property	NOUN
cana-2405	26	14	this	this	DET
cana-2405	26	15	ideals	ideal	NOUN
cana-2405	26	16	are	be	AUX
cana-2405	26	17	discussed	discuss	VERB
cana-2405	26	18	.	.	PUNCT
cana-2405	27	1	2	2	NUM
cana-2405	27	2	preliminaries	preliminary	NOUN
cana-2405	27	3	in	in	ADP
cana-2405	27	4	this	this	DET
cana-2405	27	5	section	section	NOUN
cana-2405	27	6	we	we	PRON
cana-2405	27	7	present	present	VERB
cana-2405	27	8	the	the	DET
cana-2405	27	9	basic	basic	ADJ
cana-2405	27	10	concepts	concept	NOUN
cana-2405	27	11	related	relate	VERB
cana-2405	27	12	to	to	ADP
cana-2405	27	13	this	this	DET
cana-2405	27	14	paper	paper	NOUN
cana-2405	27	15	.	.	PUNCT
cana-2405	28	1	definition	definition	NOUN
cana-2405	28	2	2.1	2.1	NUM
cana-2405	28	3	a	a	DET
cana-2405	28	4	nonempty	nonempty	ADV
cana-2405	28	5	set	set	VERB
cana-2405	28	6	𝑆	𝑆	PROPN
cana-2405	28	7	is	be	AUX
cana-2405	28	8	said	say	VERB
cana-2405	28	9	to	to	PART
cana-2405	28	10	be	be	AUX
cana-2405	28	11	a	a	DET
cana-2405	28	12	semi	semi	NOUN
cana-2405	28	13	-	-	NOUN
cana-2405	28	14	ring	ring	NOUN
cana-2405	28	15	with	with	ADP
cana-2405	28	16	respect	respect	NOUN
cana-2405	28	17	to	to	ADP
cana-2405	28	18	two	two	NUM
cana-2405	28	19	binary	binary	ADJ
cana-2405	28	20	compositions	composition	NOUN
cana-2405	28	21	,	,	PUNCT
cana-2405	28	22	addition	addition	NOUN
cana-2405	28	23	and	and	CCONJ
cana-2405	28	24	multiplication	multiplication	NOUN
cana-2405	28	25	defined	define	VERB
cana-2405	28	26	on	on	ADP
cana-2405	28	27	it	it	PRON
cana-2405	28	28	,	,	PUNCT
cana-2405	28	29	if	if	SCONJ
cana-2405	28	30	the	the	DET
cana-2405	28	31	following	follow	VERB
cana-2405	28	32	conditions	condition	NOUN
cana-2405	28	33	are	be	AUX
cana-2405	28	34	satisfied	satisfied	ADJ
cana-2405	28	35	:	:	PUNCT
cana-2405	28	36	1	1	X
cana-2405	28	37	.	.	X
cana-2405	28	38	(	(	PUNCT
cana-2405	28	39	𝑆	𝑆	PROPN
cana-2405	28	40	,	,	PUNCT
cana-2405	28	41	+	+	PUNCT
cana-2405	28	42	)	)	PUNCT
cana-2405	28	43	is	be	AUX
cana-2405	28	44	a	a	DET
cana-2405	28	45	commutative	commutative	ADJ
cana-2405	28	46	semigroup	semigroup	NOUN
cana-2405	28	47	with	with	ADP
cana-2405	28	48	zero	zero	NUM
cana-2405	28	49	.	.	PUNCT
cana-2405	29	1	2	2	NUM
cana-2405	29	2	.	.	X
cana-2405	29	3	(	(	PUNCT
cana-2405	29	4	𝑆	𝑆	PROPN
cana-2405	29	5	,	,	PUNCT
cana-2405	29	6	.	.	PUNCT
cana-2405	29	7	)	)	PUNCT
cana-2405	29	8	is	be	AUX
cana-2405	29	9	a	a	DET
cana-2405	29	10	semigroup	semigroup	NOUN
cana-2405	29	11	.	.	PUNCT
cana-2405	30	1	3	3	X
cana-2405	30	2	.	.	X
cana-2405	30	3	for	for	ADP
cana-2405	30	4	any	any	DET
cana-2405	30	5	three	three	NUM
cana-2405	30	6	elements	element	NOUN
cana-2405	30	7	𝑎	𝑎	NOUN
cana-2405	30	8	,	,	PUNCT
cana-2405	30	9	𝑏	𝑏	NOUN
cana-2405	30	10	,	,	PUNCT
cana-2405	30	11	𝑐	𝑐	PROPN
cana-2405	30	12	∈	∈	PROPN
cana-2405	30	13	𝑆	𝑆	PROPN
cana-2405	30	14	,	,	PUNCT
cana-2405	30	15	the	the	DET
cana-2405	30	16	left	left	ADJ
cana-2405	30	17	distributive	distributive	ADJ
cana-2405	30	18	law	law	NOUN
cana-2405	30	19	𝑎.	𝑎.	NOUN
cana-2405	30	20	(	(	PUNCT
cana-2405	30	21	𝑏	𝑏	PROPN
cana-2405	30	22	+	+	NOUN
cana-2405	30	23	𝑐	𝑐	X
cana-2405	30	24	)	)	PUNCT
cana-2405	30	25	=	=	PUNCT
cana-2405	31	1	𝑎.	𝑎.	NOUN
cana-2405	31	2	𝑏	𝑏	PROPN
cana-2405	32	1	+	+	CCONJ
cana-2405	32	2	𝑎.	𝑎.	PROPN
cana-2405	32	3	𝑐	𝑐	PROPN
cana-2405	32	4	and	and	CCONJ
cana-2405	32	5	the	the	DET
cana-2405	32	6	right	right	ADJ
cana-2405	32	7	distributive	distributive	ADJ
cana-2405	32	8	law	law	NOUN
cana-2405	32	9	(	(	PUNCT
cana-2405	32	10	𝑏	𝑏	PROPN
cana-2405	32	11	+	+	NOUN
cana-2405	32	12	𝑐	𝑐	NOUN
cana-2405	32	13	)	)	PUNCT
cana-2405	32	14	.	.	PUNCT
cana-2405	33	1	𝑎	𝑎	X
cana-2405	33	2	=	=	NOUN
cana-2405	33	3	𝑏.	𝑏.	NOUN
cana-2405	33	4	𝑎	𝑎	PROPN
cana-2405	33	5	+	+	X
cana-2405	33	6	𝑐.	𝑐.	NOUN
cana-2405	33	7	𝑎.	𝑎.	NOUN
cana-2405	33	8	4	4	X
cana-2405	33	9	.	.	PUNCT
cana-2405	34	1	𝑠.	𝑠.	NOUN
cana-2405	34	2	0	0	PUNCT
cana-2405	35	1	=	=	SYM
cana-2405	35	2	0	0	NUM
cana-2405	35	3	.	.	PUNCT
cana-2405	36	1	𝑠	𝑠	X
cana-2405	36	2	,	,	PUNCT
cana-2405	36	3	for	for	ADP
cana-2405	36	4	all	all	PRON
cana-2405	36	5	𝑠	𝑠	PRON
cana-2405	36	6	∈	∈	NOUN
cana-2405	36	7	𝑆.	𝑆.	NOUN
cana-2405	36	8	definition	definition	NOUN
cana-2405	36	9	2.2	2.2	NUM
cana-2405	36	10	a	a	DET
cana-2405	36	11	nonempty	nonempty	NOUN
cana-2405	36	12	subset	subset	VERB
cana-2405	36	13	ℐ	ℐ	PRON
cana-2405	36	14	of	of	ADP
cana-2405	36	15	a	a	DET
cana-2405	36	16	semi	semi	ADJ
cana-2405	36	17	-	-	ADJ
cana-2405	36	18	ring	ring	ADJ
cana-2405	36	19	𝑆	𝑆	PROPN
cana-2405	36	20	is	be	AUX
cana-2405	36	21	called	call	VERB
cana-2405	36	22	an	an	DET
cana-2405	36	23	ideal	ideal	NOUN
cana-2405	36	24	if	if	SCONJ
cana-2405	36	25	1	1	NUM
cana-2405	36	26	.	.	X
cana-2405	37	1	𝑎	𝑎	X
cana-2405	37	2	,	,	PUNCT
cana-2405	37	3	𝑏	𝑏	PROPN
cana-2405	37	4	∈	∈	PROPN
cana-2405	37	5	ℐ	ℐ	PRON
cana-2405	37	6	implies	imply	VERB
cana-2405	37	7	𝑎	𝑎	PROPN
cana-2405	37	8	+	+	NOUN
cana-2405	37	9	𝑏	𝑏	NOUN
cana-2405	37	10	∈	∈	NOUN
cana-2405	37	11	ℐ	ℐ	PROPN
cana-2405	37	12	2	2	NUM
cana-2405	37	13	.	.	PUNCT
cana-2405	38	1	𝑎	𝑎	DET
cana-2405	38	2	∈	∈	NOUN
cana-2405	38	3	ℐ	ℐ	PROPN
cana-2405	38	4	,	,	PUNCT
cana-2405	38	5	𝑠	𝑠	PROPN
cana-2405	38	6	∈	∈	PROPN
cana-2405	38	7	𝑆	𝑆	PROPN
cana-2405	38	8	implies	imply	VERB
cana-2405	38	9	𝑠.	𝑠.	NOUN
cana-2405	38	10	𝑎	𝑎	PRON
cana-2405	38	11	∈	∈	NOUN
cana-2405	38	12	ℐ	ℐ	PRON
cana-2405	38	13	and	and	CCONJ
cana-2405	38	14	𝑎.	𝑎.	VERB
cana-2405	38	15	𝑠	𝑠	PROPN
cana-2405	38	16	∈	∈	PROPN
cana-2405	39	1	ℐ	ℐ	PRON
cana-2405	39	2	definition	definition	NOUN
cana-2405	39	3	2.3	2.3	NUM
cana-2405	39	4	let	let	VERB
cana-2405	39	5	𝜇	𝜇	ADP
cana-2405	39	6	be	be	AUX
cana-2405	39	7	a	a	DET
cana-2405	39	8	nonempty	nonempty	ADJ
cana-2405	39	9	fuzzy	fuzzy	ADJ
cana-2405	39	10	subset	subset	NOUN
cana-2405	39	11	of	of	ADP
cana-2405	39	12	a	a	DET
cana-2405	39	13	semi	semi	ADJ
cana-2405	39	14	-	-	ADJ
cana-2405	39	15	ring	ring	ADJ
cana-2405	39	16	𝑆.	𝑆.	NOUN
cana-2405	39	17	then	then	ADV
cana-2405	39	18	𝜇	𝜇	SCONJ
cana-2405	39	19	is	be	AUX
cana-2405	39	20	called	call	VERB
cana-2405	39	21	a	a	DET
cana-2405	39	22	fuzzy	fuzzy	ADJ
cana-2405	39	23	left(right	left(right	PROPN
cana-2405	39	24	)	)	PUNCT
cana-2405	39	25	ideal	ideal	NOUN
cana-2405	39	26	of	of	ADP
cana-2405	39	27	𝑆	𝑆	PROPN
cana-2405	39	28	if	if	SCONJ
cana-2405	39	29	for	for	ADP
cana-2405	39	30	all	all	PRON
cana-2405	39	31	𝑖	𝑖	ADP
cana-2405	39	32	,	,	PUNCT
cana-2405	39	33	𝑗	𝑗	PROPN
cana-2405	39	34	∈	∈	NOUN
cana-2405	39	35	𝑆.	𝑆.	PROPN
cana-2405	39	36	1	1	NUM
cana-2405	39	37	.	.	PUNCT
cana-2405	40	1	𝜇(𝑖	𝜇(𝑖	PROPN
cana-2405	40	2	+	+	CCONJ
cana-2405	40	3	𝑗	𝑗	NOUN
cana-2405	40	4	)	)	PUNCT
cana-2405	40	5	≥	≥	NOUN
cana-2405	40	6	min{𝜇(𝑖	min{𝜇(𝑖	PROPN
cana-2405	40	7	)	)	PUNCT
cana-2405	40	8	,	,	PUNCT
cana-2405	40	9	𝜇(𝑗	𝜇(𝑗	PROPN
cana-2405	40	10	)	)	PUNCT
cana-2405	40	11	}	}	PUNCT
cana-2405	40	12	2	2	NUM
cana-2405	40	13	.	.	X
cana-2405	41	1	𝜇(𝑖𝑗	𝜇(𝑖𝑗	ADV
cana-2405	41	2	)	)	PUNCT
cana-2405	42	1	≥	≥	NOUN
cana-2405	42	2	𝜇(𝑗)(𝑟𝑒𝑠𝑝.	𝜇(𝑗)(𝑟𝑒𝑠𝑝.	NOUN
cana-2405	42	3	𝑟𝑖𝑔ℎ𝑡	𝑟𝑖𝑔ℎ𝑡	NOUN
cana-2405	42	4	)	)	PUNCT
cana-2405	42	5	𝜇(𝑖𝑗	𝜇(𝑖𝑗	NUM
cana-2405	42	6	)	)	PUNCT
cana-2405	42	7	≥	≥	NOUN
cana-2405	42	8	𝜇(𝑖	𝜇(𝑖	NUM
cana-2405	42	9	)	)	PUNCT
cana-2405	42	10	)	)	PUNCT
cana-2405	43	1	a	a	DET
cana-2405	43	2	fuzzy	fuzzy	ADJ
cana-2405	43	3	ideal	ideal	NOUN
cana-2405	43	4	of	of	ADP
cana-2405	43	5	a	a	DET
cana-2405	43	6	semi	semi	ADJ
cana-2405	43	7	-	-	ADJ
cana-2405	43	8	ring	ring	ADJ
cana-2405	43	9	𝑆	𝑆	PROPN
cana-2405	43	10	is	be	AUX
cana-2405	43	11	a	a	DET
cana-2405	43	12	nonempty	nonempty	ADJ
cana-2405	43	13	fuzzy	fuzzy	ADJ
cana-2405	43	14	subset	subset	NOUN
cana-2405	43	15	of	of	ADP
cana-2405	43	16	𝑆	𝑆	PROPN
cana-2405	43	17	which	which	PRON
cana-2405	43	18	is	be	AUX
cana-2405	43	19	both	both	CCONJ
cana-2405	43	20	a	a	DET
cana-2405	43	21	fuzzy	fuzzy	ADJ
cana-2405	43	22	left	leave	VERB
cana-2405	43	23	ideal	ideal	NOUN
cana-2405	43	24	and	and	CCONJ
cana-2405	43	25	a	a	DET
cana-2405	43	26	fuzzy	fuzzy	ADJ
cana-2405	43	27	right	right	ADJ
cana-2405	43	28	ideal	ideal	NOUN
cana-2405	43	29	of	of	ADP
cana-2405	43	30	𝑆.	𝑆.	PROPN
cana-2405	43	31	3	3	NUM
cana-2405	43	32	pythagorean	pythagorean	NOUN
cana-2405	43	33	fuzzy	fuzzy	ADJ
cana-2405	43	34	ideals	ideal	NOUN
cana-2405	43	35	in	in	ADP
cana-2405	43	36	semiring	semire	VERB
cana-2405	43	37	in	in	ADP
cana-2405	43	38	this	this	DET
cana-2405	43	39	section	section	NOUN
cana-2405	43	40	𝑆	𝑆	PROPN
cana-2405	43	41	denotes	denote	VERB
cana-2405	43	42	semiring(s	semiring(s	PROPN
cana-2405	43	43	)	)	PUNCT
cana-2405	43	44	.	.	PUNCT
cana-2405	44	1	definition	definition	NOUN
cana-2405	44	2	3.1	3.1	NUM
cana-2405	44	3	let	let	VERB
cana-2405	44	4	𝑃	𝑃	VERB
cana-2405	44	5	=	=	SYM
cana-2405	44	6	(	(	PUNCT
cana-2405	44	7	𝜇𝑃	𝜇𝑃	PROPN
cana-2405	44	8	,	,	PUNCT
cana-2405	44	9	𝜗𝑃	𝜗𝑃	NOUN
cana-2405	44	10	)	)	PUNCT
cana-2405	44	11	be	be	VERB
cana-2405	44	12	a	a	DET
cana-2405	44	13	pythagorean	pythagorean	ADJ
cana-2405	44	14	fuzzy	fuzzy	ADJ
cana-2405	44	15	subset	subset	NOUN
cana-2405	44	16	of	of	ADP
cana-2405	44	17	a	a	DET
cana-2405	44	18	semiring	semire	VERB
cana-2405	44	19	𝑆	𝑆	PROPN
cana-2405	44	20	and	and	CCONJ
cana-2405	44	21	∀𝑥	∀𝑥	NOUN
cana-2405	44	22	,	,	PUNCT
cana-2405	44	23	𝑦	𝑦	NOUN
cana-2405	44	24	∈	∈	NOUN
cana-2405	44	25	𝑆.	𝑆.	PROPN
cana-2405	44	26	(	(	PUNCT
cana-2405	44	27	𝑖	𝑖	NOUN
cana-2405	44	28	)	)	PUNCT
cana-2405	45	1	𝜇𝑃(𝑥	𝜇𝑃(𝑥	PROPN
cana-2405	45	2	+	+	NUM
cana-2405	45	3	𝑦	𝑦	NOUN
cana-2405	45	4	)	)	PUNCT
cana-2405	45	5	≥	≥	NOUN
cana-2405	45	6	min{𝜇𝑃(𝑥	min{𝜇𝑃(𝑥	X
cana-2405	45	7	)	)	PUNCT
cana-2405	45	8	,	,	PUNCT
cana-2405	45	9	𝜇𝑃(𝑦	𝜇𝑃(𝑦	PROPN
cana-2405	45	10	)	)	PUNCT
cana-2405	45	11	}	}	PUNCT
cana-2405	45	12	;	;	PUNCT
cana-2405	45	13	𝜗𝑃(𝑥	𝜗𝑃(𝑥	PROPN
cana-2405	45	14	+	+	SYM
cana-2405	45	15	𝑦	𝑦	X
cana-2405	45	16	)	)	PUNCT
cana-2405	45	17	≤	≤	NOUN
cana-2405	45	18	max{𝜗𝑃(𝑥	max{𝜗𝑃(𝑥	NUM
cana-2405	45	19	)	)	PUNCT
cana-2405	45	20	,	,	PUNCT
cana-2405	45	21	𝜗𝑃(𝑦	𝜗𝑃(𝑦	NUM
cana-2405	45	22	)	)	PUNCT
cana-2405	45	23	}	}	PUNCT
cana-2405	45	24	(	(	PUNCT
cana-2405	45	25	𝑖𝑖	𝑖𝑖	NOUN
cana-2405	45	26	)	)	PUNCT
cana-2405	45	27	𝜇𝑃(𝑥𝑦	𝜇𝑃(𝑥𝑦	PROPN
cana-2405	45	28	)	)	PUNCT
cana-2405	45	29	≥	≥	NOUN
cana-2405	45	30	min{𝜇𝑃(𝑥	min{𝜇𝑃(𝑥	NOUN
cana-2405	45	31	)	)	PUNCT
cana-2405	45	32	,	,	PUNCT
cana-2405	45	33	𝜇𝑃(𝑦	𝜇𝑃(𝑦	PROPN
cana-2405	45	34	)	)	PUNCT
cana-2405	45	35	}	}	PUNCT
cana-2405	45	36	;	;	PUNCT
cana-2405	45	37	𝜗𝑃(𝑥𝑦	𝜗𝑃(𝑥𝑦	X
cana-2405	45	38	)	)	PUNCT
cana-2405	45	39	≤	≤	NOUN
cana-2405	45	40	max{𝜗𝑃(𝑥	max{𝜗𝑃(𝑥	NUM
cana-2405	45	41	)	)	PUNCT
cana-2405	45	42	,	,	PUNCT
cana-2405	45	43	𝜗𝑃(𝑦	𝜗𝑃(𝑦	NUM
cana-2405	45	44	)	)	PUNCT
cana-2405	45	45	}	}	PUNCT
cana-2405	45	46	then	then	ADV
cana-2405	45	47	𝑃	𝑃	VERB
cana-2405	45	48	=	=	SYM
cana-2405	45	49	(	(	PUNCT
cana-2405	45	50	𝜇𝑃	𝜇𝑃	NOUN
cana-2405	45	51	,	,	PUNCT
cana-2405	45	52	𝜗𝑃	𝜗𝑃	NOUN
cana-2405	45	53	)	)	PUNCT
cana-2405	45	54	is	be	AUX
cana-2405	45	55	called	call	VERB
cana-2405	45	56	a	a	DET
cana-2405	45	57	pythagorean	pythagorean	ADJ
cana-2405	45	58	fuzzy	fuzzy	ADJ
cana-2405	45	59	subsemiring	subsemiring	NOUN
cana-2405	45	60	of	of	ADP
cana-2405	45	61	𝑅.	𝑅.	NOUN
cana-2405	45	62	definition	definition	NOUN
cana-2405	45	63	3.2	3.2	NUM
cana-2405	45	64	let	let	VERB
cana-2405	45	65	𝑃	𝑃	VERB
cana-2405	45	66	=	=	SYM
cana-2405	45	67	(	(	PUNCT
cana-2405	45	68	𝜇𝑃	𝜇𝑃	PROPN
cana-2405	45	69	,	,	PUNCT
cana-2405	45	70	𝜗𝑃	𝜗𝑃	NOUN
cana-2405	45	71	)	)	PUNCT
cana-2405	45	72	of	of	ADP
cana-2405	45	73	𝑆	𝑆	PROPN
cana-2405	45	74	is	be	AUX
cana-2405	45	75	called	call	VERB
cana-2405	45	76	a	a	DET
cana-2405	45	77	pythagorean	pythagorean	ADJ
cana-2405	45	78	fuzzy	fuzzy	ADJ
cana-2405	45	79	left	leave	VERB
cana-2405	45	80	ideal	ideal	NOUN
cana-2405	45	81	of	of	ADP
cana-2405	45	82	𝑆	𝑆	PROPN
cana-2405	45	83	,	,	PUNCT
cana-2405	45	84	if	if	SCONJ
cana-2405	45	85	𝑃	𝑃	NOUN
cana-2405	45	86	satisfies	satisfy	VERB
cana-2405	45	87	the	the	DET
cana-2405	45	88	following	follow	VERB
cana-2405	45	89	conditions	condition	NOUN
cana-2405	45	90	(	(	PUNCT
cana-2405	45	91	𝑖	𝑖	X
cana-2405	45	92	)	)	PUNCT
cana-2405	45	93	𝜇𝑃(𝑥	𝜇𝑃(𝑥	PROPN
cana-2405	45	94	+	+	NUM
cana-2405	45	95	𝑦	𝑦	NOUN
cana-2405	45	96	)	)	PUNCT
cana-2405	45	97	≥	≥	NOUN
cana-2405	45	98	min{𝜇𝑃(𝑥	min{𝜇𝑃(𝑥	X
cana-2405	45	99	)	)	PUNCT
cana-2405	45	100	,	,	PUNCT
cana-2405	45	101	𝜇𝑃(𝑦	𝜇𝑃(𝑦	PROPN
cana-2405	45	102	)	)	PUNCT
cana-2405	45	103	}	}	PUNCT
cana-2405	45	104	;	;	PUNCT
cana-2405	45	105	𝜗𝑃(𝑥	𝜗𝑃(𝑥	PROPN
cana-2405	45	106	+	+	SYM
cana-2405	45	107	𝑦	𝑦	X
cana-2405	45	108	)	)	PUNCT
cana-2405	45	109	≤	≤	NOUN
cana-2405	45	110	max{𝜗𝑃(𝑥	max{𝜗𝑃(𝑥	NUM
cana-2405	45	111	)	)	PUNCT
cana-2405	45	112	,	,	PUNCT
cana-2405	45	113	𝜗𝑃(𝑦	𝜗𝑃(𝑦	NUM
cana-2405	45	114	)	)	PUNCT
cana-2405	45	115	}	}	PUNCT
cana-2405	45	116	(	(	PUNCT
cana-2405	45	117	𝑖𝑖	𝑖𝑖	NOUN
cana-2405	45	118	)	)	PUNCT
cana-2405	45	119	𝜇𝑃(𝑥𝑦	𝜇𝑃(𝑥𝑦	PROPN
cana-2405	45	120	)	)	PUNCT
cana-2405	45	121	≥	≥	NOUN
cana-2405	45	122	𝜇𝑃(𝑦	𝜇𝑃(𝑦	PROPN
cana-2405	45	123	)	)	PUNCT
cana-2405	45	124	;	;	PUNCT
cana-2405	45	125	𝜗𝑃(𝑥𝑦	𝜗𝑃(𝑥𝑦	X
cana-2405	45	126	)	)	PUNCT
cana-2405	45	127	≤	≤	NUM
cana-2405	45	128	𝜗𝑃(𝑦	𝜗𝑃(𝑦	PROPN
cana-2405	45	129	)	)	PUNCT
cana-2405	45	130	definition	definition	NOUN
cana-2405	45	131	3.3	3.3	NUM
cana-2405	45	132	let	let	VERB
cana-2405	45	133	𝑃	𝑃	VERB
cana-2405	45	134	=	=	SYM
cana-2405	45	135	(	(	PUNCT
cana-2405	45	136	𝜇𝑃	𝜇𝑃	PROPN
cana-2405	45	137	,	,	PUNCT
cana-2405	45	138	𝜗𝑃	𝜗𝑃	NOUN
cana-2405	45	139	)	)	PUNCT
cana-2405	45	140	of	of	ADP
cana-2405	45	141	𝑆	𝑆	PROPN
cana-2405	45	142	is	be	AUX
cana-2405	45	143	called	call	VERB
cana-2405	45	144	a	a	DET
cana-2405	45	145	pythagorean	pythagorean	ADJ
cana-2405	45	146	fuzzy	fuzzy	ADJ
cana-2405	45	147	right	right	ADJ
cana-2405	45	148	ideal	ideal	NOUN
cana-2405	45	149	of	of	ADP
cana-2405	45	150	𝑆	𝑆	PROPN
cana-2405	45	151	,	,	PUNCT
cana-2405	45	152	if	if	SCONJ
cana-2405	45	153	𝑃	𝑃	NOUN
cana-2405	45	154	satisfies	satisfy	VERB
cana-2405	45	155	the	the	DET
cana-2405	45	156	following	follow	VERB
cana-2405	45	157	conditions	condition	NOUN
cana-2405	45	158	(	(	PUNCT
cana-2405	45	159	𝑖	𝑖	X
cana-2405	45	160	)	)	PUNCT
cana-2405	45	161	𝜇𝑃(𝑥	𝜇𝑃(𝑥	PROPN
cana-2405	45	162	+	+	NUM
cana-2405	45	163	𝑦	𝑦	NOUN
cana-2405	45	164	)	)	PUNCT
cana-2405	45	165	≥	≥	NOUN
cana-2405	45	166	min{𝜇𝑃(𝑥	min{𝜇𝑃(𝑥	X
cana-2405	45	167	)	)	PUNCT
cana-2405	45	168	,	,	PUNCT
cana-2405	45	169	𝜇𝑃(𝑦	𝜇𝑃(𝑦	PROPN
cana-2405	45	170	)	)	PUNCT
cana-2405	45	171	}	}	PUNCT
cana-2405	45	172	;	;	PUNCT
cana-2405	45	173	𝜗𝑃(𝑥	𝜗𝑃(𝑥	PROPN
cana-2405	45	174	+	+	SYM
cana-2405	45	175	𝑦	𝑦	X
cana-2405	45	176	)	)	PUNCT
cana-2405	45	177	≤	≤	NOUN
cana-2405	45	178	max{𝜗𝑃(𝑥	max{𝜗𝑃(𝑥	NUM
cana-2405	45	179	)	)	PUNCT
cana-2405	45	180	,	,	PUNCT
cana-2405	45	181	𝜗𝑃(𝑦	𝜗𝑃(𝑦	NUM
cana-2405	45	182	)	)	PUNCT
cana-2405	45	183	}	}	PUNCT
cana-2405	45	184	communications	communication	NOUN
cana-2405	45	185	on	on	ADP
cana-2405	45	186	applied	apply	VERB
cana-2405	45	187	nonlinear	nonlinear	ADJ
cana-2405	45	188	analysis	analysis	NOUN
cana-2405	45	189	issn	issn	NOUN
cana-2405	45	190	:	:	PUNCT
cana-2405	45	191	1074	1074	NUM
cana-2405	45	192	-	-	PUNCT
cana-2405	45	193	133x	133x	NUM
cana-2405	45	194	vol	vol	NOUN
cana-2405	45	195	32	32	NUM
cana-2405	45	196	no	no	NOUN
cana-2405	45	197	.	.	PUNCT
cana-2405	46	1	2s	2s	NUM
cana-2405	46	2	(	(	PUNCT
cana-2405	46	3	2025	2025	NUM
cana-2405	46	4	)	)	PUNCT
cana-2405	46	5	314	314	NUM
cana-2405	46	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2405	46	7	(	(	PUNCT
cana-2405	46	8	𝑖𝑖	𝑖𝑖	NOUN
cana-2405	46	9	)	)	PUNCT
cana-2405	46	10	𝜇𝑃(𝑥𝑦	𝜇𝑃(𝑥𝑦	PROPN
cana-2405	46	11	)	)	PUNCT
cana-2405	46	12	≥	≥	NOUN
cana-2405	46	13	𝜇𝑃(𝑥	𝜇𝑃(𝑥	PROPN
cana-2405	46	14	)	)	PUNCT
cana-2405	46	15	;	;	PUNCT
cana-2405	46	16	𝜗𝑃(𝑥𝑦	𝜗𝑃(𝑥𝑦	X
cana-2405	46	17	)	)	PUNCT
cana-2405	46	18	≤	≤	NOUN
cana-2405	46	19	𝜗𝑃(𝑥	𝜗𝑃(𝑥	PROPN
cana-2405	46	20	)	)	PUNCT
cana-2405	46	21	.	.	PUNCT
cana-2405	47	1	theorem	theorem	VERB
cana-2405	47	2	3.4	3.4	NUM
cana-2405	47	3	intersection	intersection	NOUN
cana-2405	47	4	of	of	ADP
cana-2405	47	5	a	a	DET
cana-2405	47	6	non	non	X
cana-2405	47	7	empty	empty	ADJ
cana-2405	47	8	collection	collection	NOUN
cana-2405	47	9	of	of	ADP
cana-2405	47	10	pythagorean	pythagorean	PROPN
cana-2405	47	11	fuzzy	fuzzy	ADJ
cana-2405	48	1	right	right	NOUN
cana-2405	48	2	(	(	PUNCT
cana-2405	48	3	resp	resp	NOUN
cana-2405	48	4	.	.	PUNCT
cana-2405	49	1	left	left	ADJ
cana-2405	49	2	)	)	PUNCT
cana-2405	49	3	ideals	ideal	NOUN
cana-2405	49	4	is	be	AUX
cana-2405	49	5	also	also	ADV
cana-2405	49	6	a	a	DET
cana-2405	49	7	pythagorean	pythagorean	ADJ
cana-2405	50	1	fuzzy	fuzzy	ADJ
cana-2405	50	2	right	right	NOUN
cana-2405	50	3	(	(	PUNCT
cana-2405	50	4	resp	resp	NOUN
cana-2405	50	5	.	.	PUNCT
cana-2405	51	1	left	left	ADJ
cana-2405	51	2	)	)	PUNCT
cana-2405	51	3	ideal	ideal	NOUN
cana-2405	51	4	of	of	ADP
cana-2405	51	5	𝑆.	𝑆.	PROPN
cana-2405	51	6	proof	proof	NOUN
cana-2405	51	7	.	.	PUNCT
cana-2405	52	1	let	let	VERB
cana-2405	52	2	{	{	PUNCT
cana-2405	52	3	𝑃𝑖	𝑃𝑖	VERB
cana-2405	52	4	=	=	SYM
cana-2405	52	5	(	(	PUNCT
cana-2405	52	6	𝜇𝑖	𝜇𝑖	ADP
cana-2405	52	7	,	,	PUNCT
cana-2405	52	8	𝜗𝑖)|𝑖	𝜗𝑖)|𝑖	NOUN
cana-2405	52	9	∈	∈	PROPN
cana-2405	52	10	𝐼	𝐼	PROPN
cana-2405	52	11	}	}	PUNCT
cana-2405	52	12	be	be	AUX
cana-2405	52	13	a	a	DET
cana-2405	52	14	non	non	X
cana-2405	52	15	empty	empty	ADJ
cana-2405	52	16	family	family	NOUN
cana-2405	52	17	of	of	ADP
cana-2405	52	18	pythagorean	pythagorean	PROPN
cana-2405	52	19	fuzzy	fuzzy	ADJ
cana-2405	52	20	right	right	ADJ
cana-2405	52	21	ideals	ideal	NOUN
cana-2405	52	22	of	of	ADP
cana-2405	52	23	𝑆	𝑆	PROPN
cana-2405	52	24	and	and	CCONJ
cana-2405	52	25	𝑥	𝑥	PROPN
cana-2405	52	26	,	,	PUNCT
cana-2405	52	27	𝑦	𝑦	PRON
cana-2405	52	28	∈	∈	NOUN
cana-2405	52	29	𝑆.	𝑆.	PROPN
cana-2405	52	30	then	then	ADV
cana-2405	52	31	⋂	⋂	PROPN
cana-2405	52	32	𝑖∈𝐼	𝑖∈𝐼	X
cana-2405	52	33	𝜇𝑖(𝑥	𝜇𝑖(𝑥	X
cana-2405	53	1	+	+	SYM
cana-2405	53	2	𝑦	𝑦	X
cana-2405	53	3	)	)	PUNCT
cana-2405	53	4	=	=	VERB
cana-2405	53	5	inf	inf	NOUN
cana-2405	53	6	𝑖∈𝐼	𝑖∈𝐼	X
cana-2405	53	7	{	{	PUNCT
cana-2405	53	8	𝜇𝑖(𝑥	𝜇𝑖(𝑥	PUNCT
cana-2405	53	9	+	+	CCONJ
cana-2405	53	10	𝑦	𝑦	X
cana-2405	53	11	)	)	PUNCT
cana-2405	53	12	}	}	PUNCT
cana-2405	53	13	≥	≥	NOUN
cana-2405	53	14	inf	inf	NOUN
cana-2405	53	15	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	53	16	{	{	PUNCT
cana-2405	53	17	min{𝜇𝑖(𝑥	min{𝜇𝑖(𝑥	PROPN
cana-2405	53	18	)	)	PUNCT
cana-2405	53	19	,	,	PUNCT
cana-2405	53	20	𝜇𝑖(𝑦	𝜇𝑖(𝑦	NOUN
cana-2405	53	21	)	)	PUNCT
cana-2405	53	22	}	}	PUNCT
cana-2405	53	23	}	}	PUNCT
cana-2405	53	24	=	=	PUNCT
cana-2405	53	25	min{inf	min{inf	VERB
cana-2405	53	26	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	53	27	𝜇𝑖(𝑥	𝜇𝑖(𝑥	NUM
cana-2405	53	28	)	)	PUNCT
cana-2405	53	29	,	,	PUNCT
cana-2405	53	30	inf	inf	NOUN
cana-2405	53	31	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	53	32	𝜇𝑖(𝑦	𝜇𝑖(𝑦	NOUN
cana-2405	53	33	)	)	PUNCT
cana-2405	53	34	}	}	PUNCT
cana-2405	54	1	=	=	NOUN
cana-2405	54	2	min{⋂𝑖∈𝐼	min{⋂𝑖∈𝐼	ADJ
cana-2405	54	3	𝜇𝑖(𝑥	𝜇𝑖(𝑥	NUM
cana-2405	54	4	)	)	PUNCT
cana-2405	54	5	,	,	PUNCT
cana-2405	54	6	⋂𝑖∈𝐼	⋂𝑖∈𝐼	VERB
cana-2405	54	7	𝜇𝑖(𝑦	𝜇𝑖(𝑦	NOUN
cana-2405	54	8	)	)	PUNCT
cana-2405	54	9	}	}	PUNCT
cana-2405	54	10	.	.	PUNCT
cana-2405	55	1	also	also	ADV
cana-2405	55	2	⋂	⋂	PROPN
cana-2405	55	3	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	55	4	𝜗𝑖(𝑥	𝜗𝑖(𝑥	X
cana-2405	55	5	+	+	NUM
cana-2405	55	6	𝑦	𝑦	X
cana-2405	55	7	)	)	PUNCT
cana-2405	55	8	=	=	SYM
cana-2405	55	9	sup	sup	NOUN
cana-2405	55	10	𝑖∈𝐼	𝑖∈𝐼	X
cana-2405	55	11	{	{	PUNCT
cana-2405	55	12	𝜗𝑖(𝑥	𝜗𝑖(𝑥	X
cana-2405	55	13	+	+	NUM
cana-2405	55	14	𝑦	𝑦	X
cana-2405	55	15	)	)	PUNCT
cana-2405	55	16	}	}	PUNCT
cana-2405	55	17	≤	≤	NUM
cana-2405	55	18	sup	sup	NOUN
cana-2405	55	19	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	55	20	{	{	PUNCT
cana-2405	55	21	max{𝜗𝑖(𝑥	max{𝜗𝑖(𝑥	PROPN
cana-2405	55	22	)	)	PUNCT
cana-2405	55	23	,	,	PUNCT
cana-2405	55	24	𝜗𝑖(𝑦	𝜗𝑖(𝑦	NOUN
cana-2405	55	25	)	)	PUNCT
cana-2405	55	26	}	}	PUNCT
cana-2405	55	27	}	}	PUNCT
cana-2405	55	28	=	=	PUNCT
cana-2405	55	29	max{sup	max{sup	X
cana-2405	55	30	𝑖∈𝐼	𝑖∈𝐼	X
cana-2405	55	31	𝜗𝑖(𝑥	𝜗𝑖(𝑥	NUM
cana-2405	55	32	)	)	PUNCT
cana-2405	55	33	,	,	PUNCT
cana-2405	55	34	sup	sup	NOUN
cana-2405	55	35	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	55	36	𝜗𝑖(𝑦	𝜗𝑖(𝑦	PRON
cana-2405	55	37	)	)	PUNCT
cana-2405	55	38	}	}	PUNCT
cana-2405	56	1	=	=	NOUN
cana-2405	56	2	max{⋂𝑖∈𝐼	max{⋂𝑖∈𝐼	ADJ
cana-2405	56	3	𝜗𝑖(𝑥	𝜗𝑖(𝑥	NUM
cana-2405	56	4	)	)	PUNCT
cana-2405	56	5	,	,	PUNCT
cana-2405	56	6	⋂𝑖∈𝐼	⋂𝑖∈𝐼	VERB
cana-2405	56	7	𝜗𝑖(𝑦	𝜗𝑖(𝑦	NOUN
cana-2405	56	8	)	)	PUNCT
cana-2405	56	9	}	}	PUNCT
cana-2405	56	10	.	.	PUNCT
cana-2405	57	1	moreover	moreover	ADV
cana-2405	57	2	⋂	⋂	PROPN
cana-2405	57	3	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	57	4	𝜇𝑖(𝑥𝑦	𝜇𝑖(𝑥𝑦	PROPN
cana-2405	57	5	)	)	PUNCT
cana-2405	57	6	=	=	SYM
cana-2405	57	7	inf	inf	NOUN
cana-2405	57	8	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	57	9	{	{	PUNCT
cana-2405	57	10	𝜇𝑖(𝑥𝑦	𝜇𝑖(𝑥𝑦	PROPN
cana-2405	57	11	)	)	PUNCT
cana-2405	57	12	}	}	PUNCT
cana-2405	57	13	≥	≥	X
cana-2405	57	14	inf	inf	NOUN
cana-2405	57	15	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	57	16	{	{	PUNCT
cana-2405	57	17	𝜇𝑖(𝑥	𝜇𝑖(𝑥	NUM
cana-2405	57	18	)	)	PUNCT
cana-2405	57	19	}	}	PUNCT
cana-2405	58	1	=	=	SYM
cana-2405	58	2	⋂𝑖∈𝐼	⋂𝑖∈𝐼	NOUN
cana-2405	58	3	𝜇𝑖(𝑥	𝜇𝑖(𝑥	NUM
cana-2405	58	4	)	)	PUNCT
cana-2405	58	5	.	.	PUNCT
cana-2405	59	1	finally	finally	ADV
cana-2405	59	2	⋂	⋂	PROPN
cana-2405	59	3	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	59	4	𝜗𝑖(𝑥𝑦	𝜗𝑖(𝑥𝑦	PROPN
cana-2405	59	5	)	)	PUNCT
cana-2405	60	1	=	=	SYM
cana-2405	60	2	sup	sup	NOUN
cana-2405	60	3	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	60	4	{	{	PUNCT
cana-2405	60	5	𝜗𝑖(𝑥𝑦	𝜗𝑖(𝑥𝑦	NOUN
cana-2405	60	6	)	)	PUNCT
cana-2405	60	7	}	}	PUNCT
cana-2405	60	8	≤	≤	NUM
cana-2405	60	9	sup	sup	NOUN
cana-2405	60	10	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	60	11	{	{	PUNCT
cana-2405	60	12	𝜗𝑖(𝑥	𝜗𝑖(𝑥	NUM
cana-2405	60	13	)	)	PUNCT
cana-2405	60	14	}	}	PUNCT
cana-2405	60	15	=	=	SYM
cana-2405	60	16	⋂𝑖∈𝐼	⋂𝑖∈𝐼	NOUN
cana-2405	60	17	𝜗𝑖(𝑥	𝜗𝑖(𝑥	NUM
cana-2405	60	18	)	)	PUNCT
cana-2405	60	19	.	.	PUNCT
cana-2405	61	1	hence	hence	ADV
cana-2405	61	2	⋂𝑖∈𝐼	⋂𝑖∈𝐼	VERB
cana-2405	61	3	𝑃𝑖	𝑃𝑖	SCONJ
cana-2405	61	4	is	be	AUX
cana-2405	61	5	a	a	DET
cana-2405	61	6	pythagorean	pythagorean	ADJ
cana-2405	61	7	fuzzy	fuzzy	ADJ
cana-2405	61	8	right	right	ADJ
cana-2405	61	9	ideals	ideal	NOUN
cana-2405	61	10	of	of	ADP
cana-2405	61	11	𝑆.	𝑆.	PROPN
cana-2405	61	12	similarly	similarly	ADV
cana-2405	61	13	,	,	PUNCT
cana-2405	61	14	we	we	PRON
cana-2405	61	15	can	can	AUX
cana-2405	61	16	prove	prove	VERB
cana-2405	61	17	the	the	DET
cana-2405	61	18	result	result	NOUN
cana-2405	61	19	for	for	ADP
cana-2405	61	20	pythagorean	pythagorean	PROPN
cana-2405	61	21	fuzzy	fuzzy	PROPN
cana-2405	61	22	left	leave	VERB
cana-2405	61	23	ideal	ideal	NOUN
cana-2405	61	24	also	also	ADV
cana-2405	61	25	.	.	PUNCT
cana-2405	62	1	theorem	theorem	VERB
cana-2405	62	2	3.5	3.5	NUM
cana-2405	62	3	union	union	NOUN
cana-2405	62	4	of	of	ADP
cana-2405	62	5	a	a	DET
cana-2405	62	6	non	non	X
cana-2405	62	7	empty	empty	ADJ
cana-2405	62	8	collection	collection	NOUN
cana-2405	62	9	of	of	ADP
cana-2405	62	10	pythagorean	pythagorean	PROPN
cana-2405	62	11	fuzzy	fuzzy	ADJ
cana-2405	63	1	right	right	NOUN
cana-2405	63	2	(	(	PUNCT
cana-2405	63	3	resp	resp	NOUN
cana-2405	63	4	.	.	PUNCT
cana-2405	64	1	left	left	ADJ
cana-2405	64	2	)	)	PUNCT
cana-2405	64	3	ideals	ideal	NOUN
cana-2405	64	4	is	be	AUX
cana-2405	64	5	also	also	ADV
cana-2405	64	6	a	a	DET
cana-2405	64	7	communications	communication	NOUN
cana-2405	64	8	on	on	ADP
cana-2405	64	9	applied	apply	VERB
cana-2405	64	10	nonlinear	nonlinear	ADJ
cana-2405	64	11	analysis	analysis	NOUN
cana-2405	64	12	issn	issn	NOUN
cana-2405	64	13	:	:	PUNCT
cana-2405	64	14	1074	1074	NUM
cana-2405	64	15	-	-	PUNCT
cana-2405	64	16	133x	133x	NUM
cana-2405	64	17	vol	vol	NOUN
cana-2405	64	18	32	32	NUM
cana-2405	64	19	no	no	NOUN
cana-2405	64	20	.	.	PUNCT
cana-2405	65	1	2s	2s	NUM
cana-2405	65	2	(	(	PUNCT
cana-2405	65	3	2025	2025	NUM
cana-2405	65	4	)	)	PUNCT
cana-2405	65	5	315	315	NUM
cana-2405	65	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2405	65	7	pythagorean	pythagorean	PROPN
cana-2405	65	8	fuzzy	fuzzy	ADJ
cana-2405	66	1	right	right	ADV
cana-2405	66	2	(	(	PUNCT
cana-2405	66	3	resp	resp	NOUN
cana-2405	66	4	.	.	PUNCT
cana-2405	67	1	left	left	ADJ
cana-2405	67	2	)	)	PUNCT
cana-2405	67	3	ideal	ideal	NOUN
cana-2405	67	4	of	of	ADP
cana-2405	67	5	𝑆.	𝑆.	PROPN
cana-2405	67	6	proof	proof	NOUN
cana-2405	67	7	.	.	PUNCT
cana-2405	68	1	let	let	VERB
cana-2405	68	2	{	{	PUNCT
cana-2405	68	3	𝑃𝑖	𝑃𝑖	VERB
cana-2405	68	4	=	=	SYM
cana-2405	68	5	(	(	PUNCT
cana-2405	68	6	𝜇𝑖	𝜇𝑖	ADP
cana-2405	68	7	,	,	PUNCT
cana-2405	68	8	𝜗𝑖)|𝑖	𝜗𝑖)|𝑖	NOUN
cana-2405	68	9	∈	∈	PROPN
cana-2405	68	10	𝐼	𝐼	PROPN
cana-2405	68	11	}	}	PUNCT
cana-2405	68	12	be	be	AUX
cana-2405	68	13	a	a	DET
cana-2405	68	14	non	non	X
cana-2405	68	15	empty	empty	ADJ
cana-2405	68	16	family	family	NOUN
cana-2405	68	17	of	of	ADP
cana-2405	68	18	pythagorean	pythagorean	PROPN
cana-2405	68	19	fuzzy	fuzzy	ADJ
cana-2405	68	20	right	right	ADJ
cana-2405	68	21	ideals	ideal	NOUN
cana-2405	68	22	of	of	ADP
cana-2405	68	23	𝑆	𝑆	PROPN
cana-2405	68	24	and	and	CCONJ
cana-2405	68	25	𝑥	𝑥	PROPN
cana-2405	68	26	,	,	PUNCT
cana-2405	68	27	𝑦	𝑦	PRON
cana-2405	68	28	∈	∈	NOUN
cana-2405	68	29	𝑆.	𝑆.	NOUN
cana-2405	68	30	then	then	ADV
cana-2405	68	31	⋃	⋃	NOUN
cana-2405	68	32	𝑖∈𝐼	𝑖∈𝐼	NOUN
cana-2405	68	33	𝜇𝑖(𝑥	𝜇𝑖(𝑥	NUM
cana-2405	68	34	+	+	SYM
cana-2405	68	35	𝑦	𝑦	X
cana-2405	68	36	)	)	PUNCT
cana-2405	68	37	=	=	SYM
cana-2405	68	38	sup	sup	NOUN
cana-2405	68	39	𝑖∈𝐼	𝑖∈𝐼	X
cana-2405	68	40	{	{	PUNCT
cana-2405	68	41	𝜇𝑖(𝑥	𝜇𝑖(𝑥	PUNCT
cana-2405	68	42	+	+	CCONJ
cana-2405	68	43	𝑦	𝑦	X
cana-2405	68	44	)	)	PUNCT
cana-2405	68	45	}	}	PUNCT
cana-2405	68	46	≤	≤	NUM
cana-2405	68	47	sup	sup	NOUN
cana-2405	68	48	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	68	49	{	{	PUNCT
cana-2405	68	50	max{𝜇𝑖(𝑥	max{𝜇𝑖(𝑥	PROPN
cana-2405	68	51	)	)	PUNCT
cana-2405	68	52	,	,	PUNCT
cana-2405	68	53	𝜇𝑖(𝑦	𝜇𝑖(𝑦	NOUN
cana-2405	68	54	)	)	PUNCT
cana-2405	68	55	}	}	PUNCT
cana-2405	68	56	}	}	PUNCT
cana-2405	68	57	=	=	PUNCT
cana-2405	68	58	max{sup	max{sup	X
cana-2405	68	59	𝑖∈𝐼	𝑖∈𝐼	X
cana-2405	68	60	𝜇𝑖(𝑥	𝜇𝑖(𝑥	NUM
cana-2405	68	61	)	)	PUNCT
cana-2405	68	62	,	,	PUNCT
cana-2405	68	63	sup	sup	NOUN
cana-2405	68	64	𝑖∈𝐼	𝑖∈𝐼	NUM
cana-2405	68	65	𝜇𝑖(𝑦	𝜇𝑖(𝑦	NOUN
cana-2405	68	66	)	)	PUNCT
cana-2405	68	67	}	}	PUNCT
cana-2405	68	68	=	=	SYM
cana-2405	68	69	max{⋃𝑖∈𝐼	max{⋃𝑖∈𝐼	PROPN
cana-2405	68	70	𝜇𝑖(𝑥	𝜇𝑖(𝑥	NUM
cana-2405	68	71	)	)	PUNCT
cana-2405	68	72	,	,	PUNCT
cana-2405	68	73	⋃𝑖∈𝐼	⋃𝑖∈𝐼	X
cana-2405	68	74	𝜇𝑖(𝑦	𝜇𝑖(𝑦	NUM
cana-2405	68	75	)	)	PUNCT
cana-2405	68	76	}	}	PUNCT
cana-2405	68	77	.	.	PUNCT
cana-2405	69	1	also	also	ADV
cana-2405	69	2	⋃	⋃	VERB
cana-2405	69	3	𝑖∈𝐼	𝑖∈𝐼	NOUN
cana-2405	69	4	𝜗𝑖(𝑥	𝜗𝑖(𝑥	X
cana-2405	69	5	+	+	NUM
cana-2405	69	6	𝑦	𝑦	X
cana-2405	69	7	)	)	PUNCT
cana-2405	69	8	=	=	VERB
cana-2405	69	9	inf	inf	NOUN
cana-2405	69	10	𝑖∈𝐼	𝑖∈𝐼	X
cana-2405	69	11	{	{	PUNCT
cana-2405	69	12	𝜗𝑖(𝑥	𝜗𝑖(𝑥	X
cana-2405	69	13	+	+	NUM
cana-2405	69	14	𝑦	𝑦	X
cana-2405	69	15	)	)	PUNCT
cana-2405	69	16	}	}	PUNCT
cana-2405	69	17	≥	≥	NOUN
cana-2405	69	18	inf	inf	NOUN
cana-2405	69	19	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	69	20	{	{	PUNCT
cana-2405	69	21	min{𝜗𝑖(𝑥	min{𝜗𝑖(𝑥	PROPN
cana-2405	69	22	)	)	PUNCT
cana-2405	69	23	,	,	PUNCT
cana-2405	69	24	𝜗𝑖(𝑦	𝜗𝑖(𝑦	NOUN
cana-2405	69	25	)	)	PUNCT
cana-2405	69	26	}	}	PUNCT
cana-2405	69	27	}	}	PUNCT
cana-2405	69	28	=	=	PUNCT
cana-2405	69	29	min{inf	min{inf	VERB
cana-2405	69	30	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	69	31	𝜗𝑖(𝑥	𝜗𝑖(𝑥	NUM
cana-2405	69	32	)	)	PUNCT
cana-2405	69	33	,	,	PUNCT
cana-2405	69	34	inf	inf	NOUN
cana-2405	69	35	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	69	36	𝜗𝑖(𝑦	𝜗𝑖(𝑦	NOUN
cana-2405	69	37	)	)	PUNCT
cana-2405	69	38	}	}	PUNCT
cana-2405	69	39	=	=	SYM
cana-2405	69	40	min{⋃𝑖∈𝐼	min{⋃𝑖∈𝐼	X
cana-2405	69	41	𝜗𝑖(𝑥	𝜗𝑖(𝑥	NUM
cana-2405	69	42	)	)	PUNCT
cana-2405	69	43	,	,	PUNCT
cana-2405	69	44	⋃𝑖∈𝐼	⋃𝑖∈𝐼	NOUN
cana-2405	69	45	𝜗𝑖(𝑦	𝜗𝑖(𝑦	NUM
cana-2405	69	46	)	)	PUNCT
cana-2405	69	47	}	}	PUNCT
cana-2405	69	48	.	.	PUNCT
cana-2405	70	1	moreover	moreover	ADV
cana-2405	70	2	⋃	⋃	PROPN
cana-2405	70	3	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	70	4	𝜇𝑖(𝑥𝑦	𝜇𝑖(𝑥𝑦	NOUN
cana-2405	70	5	)	)	PUNCT
cana-2405	70	6	=	=	SYM
cana-2405	70	7	sup	sup	NOUN
cana-2405	70	8	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	70	9	{	{	PUNCT
cana-2405	70	10	𝜇𝑖(𝑥𝑦	𝜇𝑖(𝑥𝑦	PROPN
cana-2405	70	11	)	)	PUNCT
cana-2405	70	12	}	}	PUNCT
cana-2405	70	13	≤	≤	NUM
cana-2405	70	14	sup	sup	NOUN
cana-2405	70	15	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	70	16	{	{	PUNCT
cana-2405	70	17	𝜇𝑖(𝑥	𝜇𝑖(𝑥	NUM
cana-2405	70	18	)	)	PUNCT
cana-2405	70	19	}	}	PUNCT
cana-2405	70	20	=	=	NOUN
cana-2405	70	21	⋃𝑖∈𝐼	⋃𝑖∈𝐼	NOUN
cana-2405	70	22	𝜇𝑖(𝑥	𝜇𝑖(𝑥	NUM
cana-2405	70	23	)	)	PUNCT
cana-2405	70	24	.	.	PUNCT
cana-2405	71	1	finally	finally	ADV
cana-2405	71	2	⋃	⋃	ADP
cana-2405	71	3	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	71	4	𝜗𝑖(𝑥𝑦	𝜗𝑖(𝑥𝑦	ADJ
cana-2405	71	5	)	)	PUNCT
cana-2405	71	6	=	=	SYM
cana-2405	71	7	inf	inf	NOUN
cana-2405	71	8	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	71	9	{	{	PUNCT
cana-2405	71	10	𝜗𝑖(𝑥𝑦	𝜗𝑖(𝑥𝑦	NOUN
cana-2405	71	11	)	)	PUNCT
cana-2405	71	12	}	}	PUNCT
cana-2405	71	13	≥	≥	X
cana-2405	71	14	inf	inf	NOUN
cana-2405	71	15	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	71	16	{	{	PUNCT
cana-2405	71	17	𝜗𝑖(𝑥	𝜗𝑖(𝑥	X
cana-2405	71	18	)	)	PUNCT
cana-2405	71	19	}	}	PUNCT
cana-2405	71	20	=	=	NOUN
cana-2405	71	21	⋃𝑖∈𝐼	⋃𝑖∈𝐼	NOUN
cana-2405	71	22	𝜗𝑖(𝑥	𝜗𝑖(𝑥	NUM
cana-2405	71	23	)	)	PUNCT
cana-2405	71	24	.	.	PUNCT
cana-2405	72	1	hence	hence	ADV
cana-2405	72	2	⋃𝑖∈𝐼	⋃𝑖∈𝐼	PUNCT
cana-2405	72	3	𝑃𝑖	𝑃𝑖	SCONJ
cana-2405	72	4	is	be	AUX
cana-2405	72	5	a	a	DET
cana-2405	72	6	pythagorean	pythagorean	ADJ
cana-2405	72	7	fuzzy	fuzzy	ADJ
cana-2405	72	8	right	right	ADJ
cana-2405	72	9	ideals	ideal	NOUN
cana-2405	72	10	of	of	ADP
cana-2405	72	11	𝑆.	𝑆.	PROPN
cana-2405	72	12	similarly	similarly	ADV
cana-2405	72	13	,	,	PUNCT
cana-2405	72	14	we	we	PRON
cana-2405	72	15	can	can	AUX
cana-2405	72	16	prove	prove	VERB
cana-2405	72	17	the	the	DET
cana-2405	72	18	result	result	NOUN
cana-2405	72	19	for	for	ADP
cana-2405	72	20	pythagorean	pythagorean	PROPN
cana-2405	72	21	fuzzy	fuzzy	PROPN
cana-2405	72	22	left	leave	VERB
cana-2405	72	23	ideal	ideal	NOUN
cana-2405	72	24	also	also	ADV
cana-2405	72	25	.	.	PUNCT
cana-2405	73	1	definition	definition	NOUN
cana-2405	73	2	3.6	3.6	NUM
cana-2405	73	3	let	let	VERB
cana-2405	73	4	𝑃1	𝑃1	NOUN
cana-2405	73	5	=	=	SYM
cana-2405	73	6	(	(	PUNCT
cana-2405	73	7	𝜇1	𝜇1	ADJ
cana-2405	73	8	,	,	PUNCT
cana-2405	73	9	𝜗1	𝜗1	NOUN
cana-2405	73	10	)	)	PUNCT
cana-2405	73	11	and	and	CCONJ
cana-2405	73	12	𝑃2	𝑃2	NOUN
cana-2405	73	13	=	=	SYM
cana-2405	73	14	(	(	PUNCT
cana-2405	73	15	𝜇2	𝜇2	PROPN
cana-2405	73	16	,	,	PUNCT
cana-2405	73	17	𝜗2	𝜗2	PROPN
cana-2405	73	18	)	)	PUNCT
cana-2405	73	19	pythagorean	pythagorean	ADJ
cana-2405	73	20	fuzzy	fuzzy	ADJ
cana-2405	73	21	subsets	subset	NOUN
cana-2405	73	22	of	of	ADP
cana-2405	73	23	𝑆.	𝑆.	PROPN
cana-2405	73	24	the	the	DET
cana-2405	73	25	cartesian	cartesian	ADJ
cana-2405	73	26	product	product	NOUN
cana-2405	73	27	of	of	ADP
cana-2405	73	28	𝑃1	𝑃1	NOUN
cana-2405	73	29	and	and	CCONJ
cana-2405	73	30	𝑃2	𝑃2	NOUN
cana-2405	73	31	is	be	AUX
cana-2405	73	32	defined	define	VERB
cana-2405	73	33	by	by	ADP
cana-2405	73	34	(	(	PUNCT
cana-2405	73	35	i	i	NOUN
cana-2405	73	36	)	)	PUNCT
cana-2405	73	37	𝜇1	𝜇1	PROPN
cana-2405	73	38	×	×	PROPN
cana-2405	73	39	𝜇2(𝑥	𝜇2(𝑥	PROPN
cana-2405	73	40	,	,	PUNCT
cana-2405	73	41	𝑦	𝑦	NOUN
cana-2405	73	42	)	)	PUNCT
cana-2405	73	43	=	=	SYM
cana-2405	73	44	min{𝜇1(𝑥	min{𝜇1(𝑥	PROPN
cana-2405	73	45	)	)	PUNCT
cana-2405	73	46	,	,	PUNCT
cana-2405	73	47	𝜇2(𝑥	𝜇2(𝑥	NOUN
cana-2405	73	48	)	)	PUNCT
cana-2405	73	49	}	}	PUNCT
cana-2405	73	50	(	(	PUNCT
cana-2405	73	51	ii	ii	NOUN
cana-2405	73	52	)	)	PUNCT
cana-2405	73	53	𝜗1	𝜗1	X
cana-2405	73	54	×	×	PROPN
cana-2405	73	55	𝜗2(𝑥	𝜗2(𝑥	PROPN
cana-2405	73	56	,	,	PUNCT
cana-2405	73	57	𝑦	𝑦	NOUN
cana-2405	73	58	)	)	PUNCT
cana-2405	73	59	=	=	SYM
cana-2405	73	60	max{𝜗1(𝑥	max{𝜗1(𝑥	NOUN
cana-2405	73	61	)	)	PUNCT
cana-2405	73	62	,	,	PUNCT
cana-2405	73	63	𝜗2(𝑥	𝜗2(𝑥	NOUN
cana-2405	73	64	)	)	PUNCT
cana-2405	73	65	}	}	PUNCT
cana-2405	73	66	,	,	PUNCT
cana-2405	73	67	for	for	ADP
cana-2405	73	68	all	all	PRON
cana-2405	73	69	𝑥	𝑥	PROPN
cana-2405	73	70	,	,	PUNCT
cana-2405	73	71	𝑦	𝑦	NOUN
cana-2405	73	72	∈	∈	NOUN
cana-2405	73	73	𝑆.	𝑆.	NOUN
cana-2405	73	74	communications	communication	NOUN
cana-2405	73	75	on	on	ADP
cana-2405	73	76	applied	apply	VERB
cana-2405	73	77	nonlinear	nonlinear	ADJ
cana-2405	73	78	analysis	analysis	NOUN
cana-2405	73	79	issn	issn	NOUN
cana-2405	73	80	:	:	PUNCT
cana-2405	73	81	1074	1074	NUM
cana-2405	73	82	-	-	PUNCT
cana-2405	73	83	133x	133x	NUM
cana-2405	73	84	vol	vol	NOUN
cana-2405	73	85	32	32	NUM
cana-2405	73	86	no	no	NOUN
cana-2405	73	87	.	.	PUNCT
cana-2405	74	1	2s	2s	NUM
cana-2405	74	2	(	(	PUNCT
cana-2405	74	3	2025	2025	NUM
cana-2405	74	4	)	)	PUNCT
cana-2405	74	5	316	316	NUM
cana-2405	74	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2405	74	7	theorem	theorem	VERB
cana-2405	74	8	3.7	3.7	NUM
cana-2405	74	9	let	let	VERB
cana-2405	74	10	𝑃1	𝑃1	NOUN
cana-2405	74	11	and	and	CCONJ
cana-2405	74	12	𝑃2	𝑃2	NOUN
cana-2405	74	13	be	be	AUX
cana-2405	74	14	a	a	DET
cana-2405	74	15	pythagorean	pythagorean	ADJ
cana-2405	74	16	fuzzy	fuzzy	ADJ
cana-2405	74	17	left	leave	VERB
cana-2405	74	18	ideals	ideal	NOUN
cana-2405	74	19	of	of	ADP
cana-2405	74	20	semiring	semire	VERB
cana-2405	74	21	𝑆.	𝑆.	PROPN
cana-2405	74	22	then	then	ADV
cana-2405	74	23	𝑃1	𝑃1	NOUN
cana-2405	74	24	×	×	PROPN
cana-2405	74	25	𝑃2	𝑃2	PROPN
cana-2405	74	26	is	be	AUX
cana-2405	74	27	a	a	DET
cana-2405	74	28	pythagorean	pythagorean	ADJ
cana-2405	74	29	fuzzy	fuzzy	ADJ
cana-2405	74	30	left	leave	VERB
cana-2405	74	31	ideal	ideal	NOUN
cana-2405	74	32	of	of	ADP
cana-2405	74	33	𝑆	𝑆	PROPN
cana-2405	74	34	×	×	NOUN
cana-2405	74	35	𝑆.	𝑆.	NOUN
cana-2405	74	36	proof	proof	NOUN
cana-2405	74	37	.	.	PUNCT
cana-2405	75	1	let	let	VERB
cana-2405	75	2	(	(	PUNCT
cana-2405	75	3	𝑥1	𝑥1	NOUN
cana-2405	75	4	,	,	PUNCT
cana-2405	75	5	𝑥2	𝑥2	NOUN
cana-2405	75	6	)	)	PUNCT
cana-2405	75	7	,	,	PUNCT
cana-2405	75	8	(	(	PUNCT
cana-2405	75	9	𝑦1	𝑦1	NOUN
cana-2405	75	10	,	,	PUNCT
cana-2405	76	1	𝑦2	𝑦2	NOUN
cana-2405	76	2	)	)	PUNCT
cana-2405	76	3	∈	∈	PROPN
cana-2405	76	4	𝑆	𝑆	PROPN
cana-2405	76	5	×	×	NOUN
cana-2405	76	6	𝑆.	𝑆.	PROPN
cana-2405	76	7	then	then	ADV
cana-2405	76	8	(	(	PUNCT
cana-2405	76	9	𝜇1	𝜇1	PROPN
cana-2405	76	10	×	×	PROPN
cana-2405	76	11	𝜇2)((𝑥1	𝜇2)((𝑥1	PUNCT
cana-2405	76	12	,	,	PUNCT
cana-2405	76	13	𝑥2	𝑥2	NOUN
cana-2405	76	14	)	)	PUNCT
cana-2405	76	15	+	+	CCONJ
cana-2405	76	16	(	(	PUNCT
cana-2405	76	17	𝑦1	𝑦1	PROPN
cana-2405	76	18	,	,	PUNCT
cana-2405	76	19	𝑦2	𝑦2	NOUN
cana-2405	76	20	)	)	PUNCT
cana-2405	76	21	)	)	PUNCT
cana-2405	77	1	=	=	PUNCT
cana-2405	77	2	(	(	PUNCT
cana-2405	77	3	𝜇1	𝜇1	ADJ
cana-2405	77	4	×	×	PROPN
cana-2405	77	5	𝜇2)(𝑥1	𝜇2)(𝑥1	NOUN
cana-2405	77	6	+	+	CCONJ
cana-2405	77	7	𝑦1	𝑦1	NOUN
cana-2405	77	8	,	,	PUNCT
cana-2405	77	9	𝑥2	𝑥2	NOUN
cana-2405	77	10	+	+	CCONJ
cana-2405	77	11	𝑦2	𝑦2	NOUN
cana-2405	77	12	)	)	PUNCT
cana-2405	77	13	=	=	SYM
cana-2405	78	1	min{𝜇1(𝑥1	min{𝜇1(𝑥1	PUNCT
cana-2405	79	1	+	+	NUM
cana-2405	79	2	𝑦1	𝑦1	PROPN
cana-2405	79	3	)	)	PUNCT
cana-2405	79	4	,	,	PUNCT
cana-2405	79	5	𝜇2(𝑥2	𝜇2(𝑥2	PROPN
cana-2405	79	6	+	+	NUM
cana-2405	79	7	𝑦2	𝑦2	NOUN
cana-2405	79	8	)	)	PUNCT
cana-2405	79	9	}	}	PUNCT
cana-2405	79	10	≥	≥	PROPN
cana-2405	79	11	min{min{𝜇1(𝑥1	min{min{𝜇1(𝑥1	ADJ
cana-2405	79	12	)	)	PUNCT
cana-2405	79	13	,	,	PUNCT
cana-2405	79	14	𝜇1(𝑦1	𝜇1(𝑦1	PROPN
cana-2405	79	15	)	)	PUNCT
cana-2405	79	16	}	}	PUNCT
cana-2405	79	17	,	,	PUNCT
cana-2405	79	18	min{𝜇2(𝑥2	min{𝜇2(𝑥2	NUM
cana-2405	79	19	)	)	PUNCT
cana-2405	79	20	,	,	PUNCT
cana-2405	79	21	𝜇2(𝑦2	𝜇2(𝑦2	NOUN
cana-2405	79	22	)	)	PUNCT
cana-2405	79	23	}	}	PUNCT
cana-2405	79	24	}	}	PUNCT
cana-2405	79	25	=	=	SYM
cana-2405	79	26	min{min{𝜇1(𝑥1	min{min{𝜇1(𝑥1	ADJ
cana-2405	79	27	)	)	PUNCT
cana-2405	79	28	,	,	PUNCT
cana-2405	79	29	𝜇2(𝑥2	𝜇2(𝑥2	PROPN
cana-2405	79	30	)	)	PUNCT
cana-2405	79	31	}	}	PUNCT
cana-2405	79	32	,	,	PUNCT
cana-2405	79	33	min{𝜇1(𝑦1	min{𝜇1(𝑦1	PROPN
cana-2405	79	34	)	)	PUNCT
cana-2405	79	35	,	,	PUNCT
cana-2405	79	36	𝜇2(𝑦2	𝜇2(𝑦2	NOUN
cana-2405	79	37	)	)	PUNCT
cana-2405	79	38	}	}	PUNCT
cana-2405	79	39	}	}	PUNCT
cana-2405	79	40	=	=	PUNCT
cana-2405	79	41	min{(𝜇1	min{(𝜇1	NOUN
cana-2405	79	42	×	×	PROPN
cana-2405	79	43	𝜇2)(𝑥1	𝜇2)(𝑥1	ADJ
cana-2405	79	44	,	,	PUNCT
cana-2405	79	45	𝑥2	𝑥2	NOUN
cana-2405	79	46	)	)	PUNCT
cana-2405	79	47	,	,	PUNCT
cana-2405	79	48	(	(	PUNCT
cana-2405	79	49	𝜇1	𝜇1	PROPN
cana-2405	79	50	×	×	PROPN
cana-2405	79	51	𝜇2)(𝑦1	𝜇2)(𝑦1	PROPN
cana-2405	79	52	,	,	PUNCT
cana-2405	79	53	𝑦2	𝑦2	PROPN
cana-2405	79	54	)	)	PUNCT
cana-2405	79	55	}	}	PUNCT
cana-2405	79	56	.	.	PUNCT
cana-2405	80	1	(	(	PUNCT
cana-2405	80	2	𝜇1	𝜇1	PROPN
cana-2405	80	3	×	×	PROPN
cana-2405	80	4	𝜇2)((𝑥1	𝜇2)((𝑥1	PROPN
cana-2405	80	5	,	,	PUNCT
cana-2405	80	6	𝑥2)(𝑦1	𝑥2)(𝑦1	PROPN
cana-2405	80	7	,	,	PUNCT
cana-2405	80	8	𝑦2	𝑦2	PROPN
cana-2405	80	9	)	)	PUNCT
cana-2405	80	10	)	)	PUNCT
cana-2405	81	1	=	=	PUNCT
cana-2405	81	2	(	(	PUNCT
cana-2405	81	3	𝜇1	𝜇1	PROPN
cana-2405	81	4	×	×	PROPN
cana-2405	81	5	𝜇2)(𝑥1𝑦1	𝜇2)(𝑥1𝑦1	PROPN
cana-2405	81	6	,	,	PUNCT
cana-2405	81	7	𝑥2𝑦2	𝑥2𝑦2	X
cana-2405	81	8	)	)	PUNCT
cana-2405	81	9	=	=	SYM
cana-2405	81	10	min{𝜇1(𝑥1𝑦1	min{𝜇1(𝑥1𝑦1	NOUN
cana-2405	81	11	)	)	PUNCT
cana-2405	81	12	,	,	PUNCT
cana-2405	81	13	𝜇2(𝑥2𝑦2	𝜇2(𝑥2𝑦2	PROPN
cana-2405	81	14	)	)	PUNCT
cana-2405	81	15	}	}	PUNCT
cana-2405	81	16	≥	≥	NOUN
cana-2405	81	17	min{𝜇1(𝑦1	min{𝜇1(𝑦1	PROPN
cana-2405	81	18	)	)	PUNCT
cana-2405	81	19	,	,	PUNCT
cana-2405	81	20	𝜇2(𝑦2	𝜇2(𝑦2	NOUN
cana-2405	81	21	)	)	PUNCT
cana-2405	81	22	}	}	PUNCT
cana-2405	81	23	=	=	SYM
cana-2405	81	24	(	(	PUNCT
cana-2405	81	25	𝜇1	𝜇1	PROPN
cana-2405	81	26	×	×	PROPN
cana-2405	81	27	𝜇2)(𝑦1	𝜇2)(𝑦1	PROPN
cana-2405	81	28	,	,	PUNCT
cana-2405	81	29	𝑦2	𝑦2	PROPN
cana-2405	81	30	)	)	PUNCT
cana-2405	81	31	.	.	PUNCT
cana-2405	82	1	(	(	PUNCT
cana-2405	82	2	𝜗1	𝜗1	X
cana-2405	82	3	×	×	NOUN
cana-2405	82	4	𝜗2)((𝑥1	𝜗2)((𝑥1	NUM
cana-2405	82	5	,	,	PUNCT
cana-2405	82	6	𝑥2	𝑥2	NOUN
cana-2405	82	7	)	)	PUNCT
cana-2405	83	1	+	+	CCONJ
cana-2405	83	2	(	(	PUNCT
cana-2405	83	3	𝑦1	𝑦1	PROPN
cana-2405	83	4	,	,	PUNCT
cana-2405	83	5	𝑦2	𝑦2	NOUN
cana-2405	83	6	)	)	PUNCT
cana-2405	83	7	)	)	PUNCT
cana-2405	84	1	=	=	PRON
cana-2405	84	2	(	(	PUNCT
cana-2405	84	3	𝜗1	𝜗1	X
cana-2405	84	4	×	×	X
cana-2405	84	5	𝜗2)(𝑥1	𝜗2)(𝑥1	X
cana-2405	84	6	+	+	CCONJ
cana-2405	84	7	𝑦1	𝑦1	NOUN
cana-2405	84	8	,	,	PUNCT
cana-2405	84	9	𝑥2	𝑥2	NOUN
cana-2405	84	10	+	+	CCONJ
cana-2405	84	11	𝑦2	𝑦2	NOUN
cana-2405	84	12	)	)	PUNCT
cana-2405	84	13	=	=	SYM
cana-2405	85	1	max{𝜗1(𝑥1	max{𝜗1(𝑥1	PROPN
cana-2405	85	2	+	+	NUM
cana-2405	85	3	𝑦1	𝑦1	PROPN
cana-2405	85	4	)	)	PUNCT
cana-2405	85	5	,	,	PUNCT
cana-2405	85	6	𝜗2(𝑥2	𝜗2(𝑥2	NOUN
cana-2405	85	7	+	+	CCONJ
cana-2405	85	8	𝑦2	𝑦2	NOUN
cana-2405	85	9	)	)	PUNCT
cana-2405	85	10	}	}	PUNCT
cana-2405	85	11	≤	≤	NUM
cana-2405	85	12	max{max{𝜗1(𝑥1	max{max{𝜗1(𝑥1	ADV
cana-2405	85	13	)	)	PUNCT
cana-2405	85	14	,	,	PUNCT
cana-2405	85	15	𝜗1(𝑦1	𝜗1(𝑦1	PROPN
cana-2405	85	16	)	)	PUNCT
cana-2405	85	17	}	}	PUNCT
cana-2405	85	18	,	,	PUNCT
cana-2405	85	19	max{𝜗2(𝑥2	max{𝜗2(𝑥2	PROPN
cana-2405	85	20	)	)	PUNCT
cana-2405	85	21	,	,	PUNCT
cana-2405	85	22	𝜗2(𝑦2	𝜗2(𝑦2	NOUN
cana-2405	85	23	)	)	PUNCT
cana-2405	85	24	}	}	PUNCT
cana-2405	85	25	}	}	PUNCT
cana-2405	85	26	=	=	SYM
cana-2405	85	27	max{max{𝜗1(𝑥1	max{max{𝜗1(𝑥1	ADJ
cana-2405	85	28	)	)	PUNCT
cana-2405	85	29	,	,	PUNCT
cana-2405	85	30	𝜗2(𝑥2	𝜗2(𝑥2	NOUN
cana-2405	85	31	)	)	PUNCT
cana-2405	85	32	}	}	PUNCT
cana-2405	85	33	,	,	PUNCT
cana-2405	85	34	max{𝜗1(𝑦1	max{𝜗1(𝑦1	PROPN
cana-2405	85	35	)	)	PUNCT
cana-2405	85	36	,	,	PUNCT
cana-2405	85	37	𝜗2(𝑦2	𝜗2(𝑦2	NOUN
cana-2405	85	38	)	)	PUNCT
cana-2405	85	39	}	}	PUNCT
cana-2405	85	40	}	}	PUNCT
cana-2405	85	41	=	=	SYM
cana-2405	85	42	max{(𝜗1	max{(𝜗1	X
cana-2405	85	43	×	×	NOUN
cana-2405	85	44	𝜗2)(𝑥1	𝜗2)(𝑥1	X
cana-2405	85	45	,	,	PUNCT
cana-2405	85	46	𝑥2	𝑥2	NOUN
cana-2405	85	47	)	)	PUNCT
cana-2405	85	48	,	,	PUNCT
cana-2405	85	49	(	(	PUNCT
cana-2405	85	50	𝜗1	𝜗1	X
cana-2405	85	51	×	×	PROPN
cana-2405	85	52	𝜗2)(𝑦1	𝜗2)(𝑦1	PROPN
cana-2405	85	53	,	,	PUNCT
cana-2405	85	54	𝑦2	𝑦2	PROPN
cana-2405	85	55	)	)	PUNCT
cana-2405	85	56	}	}	PUNCT
cana-2405	85	57	.	.	PUNCT
cana-2405	86	1	(	(	PUNCT
cana-2405	86	2	𝜗1	𝜗1	X
cana-2405	86	3	×	×	NOUN
cana-2405	86	4	𝜗2)((𝑥1	𝜗2)((𝑥1	NUM
cana-2405	86	5	,	,	PUNCT
cana-2405	86	6	𝑥2)(𝑦1	𝑥2)(𝑦1	PROPN
cana-2405	86	7	,	,	PUNCT
cana-2405	86	8	𝑦2	𝑦2	PROPN
cana-2405	86	9	)	)	PUNCT
cana-2405	86	10	)	)	PUNCT
cana-2405	87	1	=	=	PRON
cana-2405	87	2	(	(	PUNCT
cana-2405	87	3	𝜗1	𝜗1	X
cana-2405	87	4	×	×	PROPN
cana-2405	87	5	𝜗2)(𝑥1𝑦1	𝜗2)(𝑥1𝑦1	NOUN
cana-2405	87	6	,	,	PUNCT
cana-2405	87	7	𝑥2𝑦2	𝑥2𝑦2	NOUN
cana-2405	87	8	)	)	PUNCT
cana-2405	87	9	=	=	SYM
cana-2405	87	10	max{𝜗1(𝑥1𝑦1	max{𝜗1(𝑥1𝑦1	NOUN
cana-2405	87	11	)	)	PUNCT
cana-2405	87	12	,	,	PUNCT
cana-2405	87	13	𝜗2(𝑥2𝑦2	𝜗2(𝑥2𝑦2	PROPN
cana-2405	87	14	)	)	PUNCT
cana-2405	87	15	}	}	PUNCT
cana-2405	87	16	≤	≤	NUM
cana-2405	87	17	max{𝜗1(𝑦1	max{𝜗1(𝑦1	PROPN
cana-2405	87	18	)	)	PUNCT
cana-2405	87	19	,	,	PUNCT
cana-2405	87	20	𝜗2(𝑦2	𝜗2(𝑦2	NOUN
cana-2405	87	21	)	)	PUNCT
cana-2405	87	22	}	}	PUNCT
cana-2405	87	23	=	=	SYM
cana-2405	87	24	(	(	PUNCT
cana-2405	87	25	𝜗1	𝜗1	X
cana-2405	87	26	×	×	PROPN
cana-2405	87	27	𝜗2)(𝑦1	𝜗2)(𝑦1	PROPN
cana-2405	87	28	,	,	PUNCT
cana-2405	87	29	𝑦2	𝑦2	PROPN
cana-2405	87	30	)	)	PUNCT
cana-2405	87	31	.	.	PUNCT
cana-2405	88	1	therefore	therefore	ADV
cana-2405	88	2	𝑃1	𝑃1	NOUN
cana-2405	88	3	×	×	PROPN
cana-2405	88	4	𝑃2	𝑃2	PROPN
cana-2405	88	5	is	be	AUX
cana-2405	88	6	a	a	DET
cana-2405	88	7	pythagorean	pythagorean	ADJ
cana-2405	88	8	fuzzy	fuzzy	ADJ
cana-2405	88	9	left	leave	VERB
cana-2405	88	10	ideal	ideal	NOUN
cana-2405	88	11	of	of	ADP
cana-2405	88	12	𝑆	𝑆	PROPN
cana-2405	88	13	×	×	NOUN
cana-2405	88	14	𝑆.	𝑆.	PROPN
cana-2405	88	15	theorem	theorem	NOUN
cana-2405	88	16	3.8	3.8	NUM
cana-2405	88	17	let	let	VERB
cana-2405	88	18	𝑃	𝑃	PRON
cana-2405	88	19	be	be	AUX
cana-2405	88	20	a	a	DET
cana-2405	88	21	pythagorean	pythagorean	ADJ
cana-2405	88	22	fuzzy	fuzzy	ADJ
cana-2405	88	23	subset	subset	NOUN
cana-2405	88	24	of	of	ADP
cana-2405	88	25	semiring	semiring	NOUN
cana-2405	88	26	.	.	PUNCT
cana-2405	89	1	then	then	ADV
cana-2405	89	2	𝑃	𝑃	PROPN
cana-2405	89	3	is	be	AUX
cana-2405	89	4	a	a	DET
cana-2405	89	5	pythagorean	pythagorean	ADJ
cana-2405	89	6	fuzzy	fuzzy	ADJ
cana-2405	89	7	left	leave	VERB
cana-2405	89	8	ideal	ideal	NOUN
cana-2405	89	9	of	of	ADP
cana-2405	89	10	𝑆	𝑆	PROPN
cana-2405	89	11	if	if	SCONJ
cana-2405	89	12	and	and	CCONJ
cana-2405	89	13	only	only	ADV
cana-2405	89	14	if	if	SCONJ
cana-2405	89	15	𝑃	𝑃	NOUN
cana-2405	89	16	×	×	NOUN
cana-2405	89	17	𝑃	𝑃	NOUN
cana-2405	89	18	is	be	AUX
cana-2405	89	19	a	a	DET
cana-2405	89	20	pythagorean	pythagorean	ADJ
cana-2405	89	21	fuzzy	fuzzy	ADJ
cana-2405	89	22	left	leave	VERB
cana-2405	89	23	ideal	ideal	NOUN
cana-2405	89	24	of	of	ADP
cana-2405	89	25	𝑆	𝑆	PROPN
cana-2405	89	26	×	×	NOUN
cana-2405	89	27	𝑆.	𝑆.	NOUN
cana-2405	89	28	proof	proof	NOUN
cana-2405	89	29	.	.	PUNCT
cana-2405	90	1	consider	consider	VERB
cana-2405	90	2	𝑃	𝑃	NOUN
cana-2405	90	3	is	be	AUX
cana-2405	90	4	a	a	DET
cana-2405	90	5	pythagorean	pythagorean	ADJ
cana-2405	90	6	fuzzy	fuzzy	ADJ
cana-2405	90	7	left	leave	VERB
cana-2405	90	8	ideal	ideal	NOUN
cana-2405	90	9	of	of	ADP
cana-2405	90	10	𝑆.	𝑆.	PROPN
cana-2405	90	11	then	then	ADV
cana-2405	90	12	by	by	ADP
cana-2405	90	13	previous	previous	ADJ
cana-2405	90	14	theorem	theorem	ADJ
cana-2405	90	15	𝑃	𝑃	NOUN
cana-2405	90	16	×	×	NOUN
cana-2405	90	17	𝑆.	𝑆.	NOUN
cana-2405	90	18	conversely	conversely	ADV
cana-2405	90	19	𝑃	𝑃	VERB
cana-2405	90	20	×	×	NOUN
cana-2405	90	21	𝑃	𝑃	NOUN
cana-2405	90	22	is	be	AUX
cana-2405	90	23	a	a	DET
cana-2405	90	24	pythagorean	pythagorean	ADJ
cana-2405	90	25	fuzzy	fuzzy	ADJ
cana-2405	90	26	left	leave	VERB
cana-2405	90	27	ideal	ideal	NOUN
cana-2405	90	28	of	of	ADP
cana-2405	90	29	𝑆	𝑆	PROPN
cana-2405	90	30	×	×	PROPN
cana-2405	90	31	𝑆	𝑆	PROPN
cana-2405	90	32	,	,	PUNCT
cana-2405	90	33	for	for	ADP
cana-2405	90	34	all	all	DET
cana-2405	90	35	𝑥1	𝑥1	NOUN
cana-2405	90	36	,	,	PUNCT
cana-2405	90	37	𝑥2	𝑥2	NOUN
cana-2405	90	38	,	,	PUNCT
cana-2405	90	39	𝑦1	𝑦1	PROPN
cana-2405	90	40	,	,	PUNCT
cana-2405	90	41	𝑦2	𝑦2	PROPN
cana-2405	90	42	∈	∈	PROPN
cana-2405	90	43	𝑆.	𝑆.	PROPN
cana-2405	90	44	then	then	ADV
cana-2405	90	45	min{𝜇(𝑥1	min{𝜇(𝑥1	NOUN
cana-2405	90	46	+	+	CCONJ
cana-2405	90	47	𝑦1	𝑦1	NOUN
cana-2405	90	48	)	)	PUNCT
cana-2405	90	49	,	,	PUNCT
cana-2405	90	50	𝜇(𝑥2	𝜇(𝑥2	NOUN
cana-2405	90	51	+	+	CCONJ
cana-2405	90	52	𝑦2	𝑦2	NOUN
cana-2405	90	53	)	)	PUNCT
cana-2405	90	54	}	}	PUNCT
cana-2405	91	1	=	=	PUNCT
cana-2405	91	2	𝜇	𝜇	ADP
cana-2405	91	3	×	×	NOUN
cana-2405	91	4	𝜇(𝑥1	𝜇(𝑥1	NOUN
cana-2405	91	5	+	+	CCONJ
cana-2405	91	6	𝑦1	𝑦1	NOUN
cana-2405	91	7	,	,	PUNCT
cana-2405	91	8	𝑥2	𝑥2	NOUN
cana-2405	91	9	+	+	CCONJ
cana-2405	91	10	𝑦2	𝑦2	NOUN
cana-2405	91	11	)	)	PUNCT
cana-2405	91	12	=	=	PRON
cana-2405	91	13	(	(	PUNCT
cana-2405	91	14	𝜇	𝜇	ADP
cana-2405	91	15	×	×	PROPN
cana-2405	91	16	𝜇){(𝑥1	𝜇){(𝑥1	ADJ
cana-2405	91	17	,	,	PUNCT
cana-2405	91	18	𝑥2	𝑥2	NOUN
cana-2405	91	19	)	)	PUNCT
cana-2405	91	20	+	+	CCONJ
cana-2405	91	21	(	(	PUNCT
cana-2405	91	22	𝑦1	𝑦1	PROPN
cana-2405	91	23	,	,	PUNCT
cana-2405	91	24	𝑦2	𝑦2	NOUN
cana-2405	91	25	)	)	PUNCT
cana-2405	91	26	}	}	PUNCT
cana-2405	91	27	≥	≥	VERB
cana-2405	91	28	min{(𝜇	min{(𝜇	ADJ
cana-2405	91	29	×	×	NOUN
cana-2405	91	30	𝜇)(𝑥1	𝜇)(𝑥1	NOUN
cana-2405	91	31	,	,	PUNCT
cana-2405	91	32	𝑥2	𝑥2	NOUN
cana-2405	91	33	)	)	PUNCT
cana-2405	91	34	,	,	PUNCT
cana-2405	91	35	(	(	PUNCT
cana-2405	91	36	𝜇	𝜇	SCONJ
cana-2405	91	37	×	×	PROPN
cana-2405	91	38	𝜇)(𝑦1	𝜇)(𝑦1	PROPN
cana-2405	91	39	,	,	PUNCT
cana-2405	91	40	𝑦2	𝑦2	PROPN
cana-2405	91	41	)	)	PUNCT
cana-2405	91	42	}	}	PUNCT
cana-2405	91	43	communications	communication	NOUN
cana-2405	91	44	on	on	ADP
cana-2405	91	45	applied	apply	VERB
cana-2405	91	46	nonlinear	nonlinear	ADJ
cana-2405	91	47	analysis	analysis	NOUN
cana-2405	91	48	issn	issn	NOUN
cana-2405	91	49	:	:	PUNCT
cana-2405	91	50	1074	1074	NUM
cana-2405	91	51	-	-	PUNCT
cana-2405	91	52	133x	133x	NUM
cana-2405	91	53	vol	vol	NOUN
cana-2405	91	54	32	32	NUM
cana-2405	91	55	no	no	NOUN
cana-2405	91	56	.	.	PUNCT
cana-2405	92	1	2s	2s	NUM
cana-2405	92	2	(	(	PUNCT
cana-2405	92	3	2025	2025	NUM
cana-2405	92	4	)	)	PUNCT
cana-2405	92	5	317	317	NUM
cana-2405	92	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2405	92	7	=	=	SYM
cana-2405	92	8	min{min{𝜇(𝑥1	min{min{𝜇(𝑥1	X
cana-2405	92	9	)	)	PUNCT
cana-2405	92	10	,	,	PUNCT
cana-2405	92	11	𝜇(𝑥2	𝜇(𝑥2	NOUN
cana-2405	92	12	)	)	PUNCT
cana-2405	92	13	}	}	PUNCT
cana-2405	92	14	,	,	PUNCT
cana-2405	92	15	min{𝜇(𝑦1	min{𝜇(𝑦1	PROPN
cana-2405	92	16	)	)	PUNCT
cana-2405	92	17	,	,	PUNCT
cana-2405	92	18	𝜇(𝑦2	𝜇(𝑦2	NOUN
cana-2405	92	19	)	)	PUNCT
cana-2405	92	20	}	}	PUNCT
cana-2405	92	21	}	}	PUNCT
cana-2405	92	22	next	next	ADV
cana-2405	92	23	,	,	PUNCT
cana-2405	92	24	we	we	PRON
cana-2405	92	25	have	have	VERB
cana-2405	92	26	min{𝜇(𝑥1𝑦1	min{𝜇(𝑥1𝑦1	PROPN
cana-2405	92	27	)	)	PUNCT
cana-2405	92	28	,	,	PUNCT
cana-2405	92	29	𝜇(𝑥2𝑦2	𝜇(𝑥2𝑦2	PROPN
cana-2405	92	30	)	)	PUNCT
cana-2405	92	31	}	}	PUNCT
cana-2405	93	1	=	=	SYM
cana-2405	93	2	(	(	PUNCT
cana-2405	93	3	𝜇	𝜇	ADP
cana-2405	93	4	×	×	PROPN
cana-2405	93	5	𝜇)(𝑥1𝑦1	𝜇)(𝑥1𝑦1	NOUN
cana-2405	93	6	,	,	PUNCT
cana-2405	93	7	𝑥2𝑦2	𝑥2𝑦2	X
cana-2405	93	8	)	)	PUNCT
cana-2405	93	9	=	=	SYM
cana-2405	93	10	(	(	PUNCT
cana-2405	93	11	𝜇	𝜇	ADP
cana-2405	93	12	×	×	PROPN
cana-2405	93	13	𝜇){(𝑥1	𝜇){(𝑥1	PROPN
cana-2405	93	14	,	,	PUNCT
cana-2405	93	15	𝑥2)(𝑦1	𝑥2)(𝑦1	PROPN
cana-2405	93	16	,	,	PUNCT
cana-2405	93	17	𝑦2	𝑦2	PROPN
cana-2405	93	18	)	)	PUNCT
cana-2405	93	19	}	}	PUNCT
cana-2405	93	20	=	=	SYM
cana-2405	93	21	(	(	PUNCT
cana-2405	93	22	𝜇	𝜇	ADP
cana-2405	93	23	×	×	PROPN
cana-2405	93	24	𝜇){𝑦1	𝜇){𝑦1	ADJ
cana-2405	93	25	,	,	PUNCT
cana-2405	93	26	𝑦2	𝑦2	NOUN
cana-2405	93	27	}	}	PUNCT
cana-2405	93	28	=	=	SYM
cana-2405	93	29	min{𝜇(𝑦1	min{𝜇(𝑦1	PROPN
cana-2405	93	30	)	)	PUNCT
cana-2405	93	31	,	,	PUNCT
cana-2405	93	32	𝜇(𝑦2	𝜇(𝑦2	NOUN
cana-2405	93	33	)	)	PUNCT
cana-2405	93	34	}	}	PUNCT
cana-2405	93	35	also	also	ADV
cana-2405	93	36	max{𝜗(𝑥1	max{𝜗(𝑥1	VERB
cana-2405	93	37	+	+	CCONJ
cana-2405	93	38	𝑦1	𝑦1	NOUN
cana-2405	93	39	)	)	PUNCT
cana-2405	93	40	,	,	PUNCT
cana-2405	93	41	𝜗(𝑥2	𝜗(𝑥2	NOUN
cana-2405	93	42	+	+	NUM
cana-2405	93	43	𝑦2	𝑦2	NOUN
cana-2405	93	44	)	)	PUNCT
cana-2405	93	45	}	}	PUNCT
cana-2405	93	46	=	=	SYM
cana-2405	93	47	𝜗	𝜗	X
cana-2405	93	48	×	×	NOUN
cana-2405	93	49	𝜗(𝑥1	𝜗(𝑥1	ADJ
cana-2405	93	50	+	+	CCONJ
cana-2405	93	51	𝑦1	𝑦1	NOUN
cana-2405	93	52	,	,	PUNCT
cana-2405	93	53	𝑥2	𝑥2	NOUN
cana-2405	93	54	+	+	CCONJ
cana-2405	93	55	𝑦2	𝑦2	NOUN
cana-2405	93	56	)	)	PUNCT
cana-2405	93	57	=	=	PUNCT
cana-2405	93	58	(	(	PUNCT
cana-2405	93	59	𝜗	𝜗	X
cana-2405	93	60	×	×	NOUN
cana-2405	93	61	𝜗){(𝑥1	𝜗){(𝑥1	ADJ
cana-2405	93	62	,	,	PUNCT
cana-2405	93	63	𝑥2	𝑥2	NOUN
cana-2405	93	64	)	)	PUNCT
cana-2405	93	65	+	+	CCONJ
cana-2405	93	66	(	(	PUNCT
cana-2405	93	67	𝑦1	𝑦1	PROPN
cana-2405	93	68	,	,	PUNCT
cana-2405	93	69	𝑦2	𝑦2	NOUN
cana-2405	93	70	)	)	PUNCT
cana-2405	93	71	}	}	PUNCT
cana-2405	93	72	≤	≤	NUM
cana-2405	93	73	max{(𝜗	max{(𝜗	NOUN
cana-2405	93	74	×	×	NOUN
cana-2405	93	75	𝜗)(𝑥1	𝜗)(𝑥1	ADJ
cana-2405	93	76	,	,	PUNCT
cana-2405	93	77	𝑥2	𝑥2	NOUN
cana-2405	93	78	)	)	PUNCT
cana-2405	93	79	,	,	PUNCT
cana-2405	93	80	(	(	PUNCT
cana-2405	93	81	𝜗	𝜗	PROPN
cana-2405	93	82	×	×	PROPN
cana-2405	93	83	𝜗)(𝑦1	𝜗)(𝑦1	PROPN
cana-2405	93	84	,	,	PUNCT
cana-2405	93	85	𝑦2	𝑦2	PROPN
cana-2405	93	86	)	)	PUNCT
cana-2405	93	87	}	}	PUNCT
cana-2405	93	88	=	=	SYM
cana-2405	93	89	max{max{𝜗(𝑥1	max{max{𝜗(𝑥1	ADJ
cana-2405	93	90	)	)	PUNCT
cana-2405	93	91	,	,	PUNCT
cana-2405	93	92	𝜗(𝑥2	𝜗(𝑥2	NOUN
cana-2405	93	93	)	)	PUNCT
cana-2405	93	94	}	}	PUNCT
cana-2405	93	95	,	,	PUNCT
cana-2405	93	96	max{𝜗(𝑦1	max{𝜗(𝑦1	PROPN
cana-2405	93	97	)	)	PUNCT
cana-2405	93	98	,	,	PUNCT
cana-2405	93	99	𝜗(𝑦2	𝜗(𝑦2	NOUN
cana-2405	93	100	)	)	PUNCT
cana-2405	93	101	}	}	PUNCT
cana-2405	93	102	}	}	PUNCT
cana-2405	93	103	and	and	CCONJ
cana-2405	93	104	max{𝜗(𝑥1𝑦1	max{𝜗(𝑥1𝑦1	PROPN
cana-2405	93	105	)	)	PUNCT
cana-2405	93	106	,	,	PUNCT
cana-2405	93	107	𝜗(𝑥2𝑦2	𝜗(𝑥2𝑦2	PROPN
cana-2405	93	108	)	)	PUNCT
cana-2405	93	109	}	}	PUNCT
cana-2405	93	110	=	=	SYM
cana-2405	93	111	(	(	PUNCT
cana-2405	93	112	𝜗	𝜗	PROPN
cana-2405	93	113	×	×	PROPN
cana-2405	93	114	𝜗)(𝑥1𝑦1	𝜗)(𝑥1𝑦1	NOUN
cana-2405	93	115	,	,	PUNCT
cana-2405	93	116	𝑥2𝑦2	𝑥2𝑦2	X
cana-2405	93	117	)	)	PUNCT
cana-2405	93	118	=	=	SYM
cana-2405	93	119	(	(	PUNCT
cana-2405	93	120	𝜗	𝜗	X
cana-2405	93	121	×	×	NOUN
cana-2405	93	122	𝜗){(𝑥1	𝜗){(𝑥1	PROPN
cana-2405	93	123	,	,	PUNCT
cana-2405	93	124	𝑥2)(𝑦1	𝑥2)(𝑦1	PROPN
cana-2405	93	125	,	,	PUNCT
cana-2405	93	126	𝑦2	𝑦2	PROPN
cana-2405	93	127	)	)	PUNCT
cana-2405	93	128	}	}	PUNCT
cana-2405	93	129	=	=	SYM
cana-2405	93	130	(	(	PUNCT
cana-2405	93	131	𝜗	𝜗	NUM
cana-2405	93	132	×	×	NOUN
cana-2405	93	133	𝜗){𝑦1	𝜗){𝑦1	ADJ
cana-2405	93	134	,	,	PUNCT
cana-2405	93	135	𝑦2	𝑦2	NOUN
cana-2405	93	136	}	}	PUNCT
cana-2405	93	137	=	=	SYM
cana-2405	93	138	max{𝜗(𝑦1	max{𝜗(𝑦1	PROPN
cana-2405	93	139	)	)	PUNCT
cana-2405	93	140	,	,	PUNCT
cana-2405	93	141	𝜗(𝑦2	𝜗(𝑦2	NOUN
cana-2405	93	142	)	)	PUNCT
cana-2405	93	143	}	}	PUNCT
cana-2405	93	144	hence	hence	ADV
cana-2405	93	145	𝑃	𝑃	PROPN
cana-2405	93	146	is	be	AUX
cana-2405	93	147	a	a	DET
cana-2405	93	148	pythagorean	pythagorean	ADJ
cana-2405	93	149	fuzzy	fuzzy	ADJ
cana-2405	93	150	left	leave	VERB
cana-2405	93	151	ideal	ideal	NOUN
cana-2405	93	152	of	of	ADP
cana-2405	93	153	𝑆.	𝑆.	PROPN
cana-2405	93	154	theorem	theorem	NOUN
cana-2405	93	155	3.9	3.9	NUM
cana-2405	93	156	if	if	SCONJ
cana-2405	93	157	𝑃1	𝑃1	NOUN
cana-2405	93	158	,	,	PUNCT
cana-2405	93	159	𝑃2	𝑃2	PROPN
cana-2405	93	160	be	be	AUX
cana-2405	93	161	any	any	DET
cana-2405	93	162	two	two	NUM
cana-2405	93	163	pythagorean	pythagorean	ADJ
cana-2405	93	164	fuzzy	fuzzy	ADJ
cana-2405	93	165	ideals	ideal	NOUN
cana-2405	93	166	of	of	ADP
cana-2405	93	167	semiring	semire	VERB
cana-2405	93	168	𝑆	𝑆	PROPN
cana-2405	93	169	,	,	PUNCT
cana-2405	93	170	then	then	ADV
cana-2405	93	171	𝑃1	𝑃1	NOUN
cana-2405	93	172	+	+	CCONJ
cana-2405	93	173	𝑃2	𝑃2	NOUN
cana-2405	93	174	is	be	AUX
cana-2405	93	175	also	also	ADV
cana-2405	93	176	so	so	ADV
cana-2405	93	177	.	.	PUNCT
cana-2405	94	1	proof	proof	NOUN
cana-2405	94	2	.	.	PUNCT
cana-2405	95	1	consider	consider	VERB
cana-2405	95	2	𝑃1	𝑃1	NOUN
cana-2405	95	3	,	,	PUNCT
cana-2405	95	4	𝑃2	𝑃2	PROPN
cana-2405	95	5	are	be	AUX
cana-2405	95	6	any	any	DET
cana-2405	95	7	two	two	NUM
cana-2405	95	8	pythagorean	pythagorean	ADJ
cana-2405	95	9	fuzzy	fuzzy	ADJ
cana-2405	95	10	ideals	ideal	NOUN
cana-2405	95	11	of	of	ADP
cana-2405	95	12	semiring	semire	VERB
cana-2405	95	13	𝑆	𝑆	PROPN
cana-2405	95	14	and	and	CCONJ
cana-2405	95	15	𝑥	𝑥	PROPN
cana-2405	95	16	,	,	PUNCT
cana-2405	95	17	𝑦	𝑦	PRON
cana-2405	95	18	∈	∈	NOUN
cana-2405	95	19	𝑆.	𝑆.	PROPN
cana-2405	95	20	then	then	ADV
cana-2405	95	21	(	(	PUNCT
cana-2405	95	22	𝜇1	𝜇1	NOUN
cana-2405	95	23	+	+	PROPN
cana-2405	96	1	𝜇2)(𝑥	𝜇2)(𝑥	PROPN
cana-2405	96	2	+	+	NUM
cana-2405	96	3	𝑦	𝑦	X
cana-2405	96	4	)	)	PUNCT
cana-2405	96	5	=	=	SYM
cana-2405	96	6	sup	sup	NOUN
cana-2405	96	7	𝑥+𝑦≤𝑐+𝑑	𝑥+𝑦≤𝑐+𝑑	NOUN
cana-2405	96	8	{	{	PUNCT
cana-2405	96	9	min{𝜇1(𝑐	min{𝜇1(𝑐	PROPN
cana-2405	96	10	)	)	PUNCT
cana-2405	96	11	,	,	PUNCT
cana-2405	96	12	𝜇2(𝑑	𝜇2(𝑑	PROPN
cana-2405	96	13	)	)	PUNCT
cana-2405	96	14	}	}	PUNCT
cana-2405	96	15	}	}	PUNCT
cana-2405	96	16	≥	≥	PROPN
cana-2405	96	17	sup	sup	PROPN
cana-2405	96	18	𝑥+𝑦≤(𝑎1+𝑏1)+(𝑎2+𝑏2)=(𝑎1+𝑎2)+(𝑏1+𝑏2	𝑥+𝑦≤(𝑎1+𝑏1)+(𝑎2+𝑏2)=(𝑎1+𝑎2)+(𝑏1+𝑏2	PROPN
cana-2405	96	19	)	)	PUNCT
cana-2405	96	20	{	{	PUNCT
cana-2405	96	21	min{𝜇1(𝑎1	min{𝜇1(𝑎1	PROPN
cana-2405	96	22	+	+	NUM
cana-2405	96	23	𝑎2	𝑎2	PROPN
cana-2405	96	24	)	)	PUNCT
cana-2405	96	25	,	,	PUNCT
cana-2405	96	26	𝜇2(𝑏1	𝜇2(𝑏1	NOUN
cana-2405	96	27	+	+	CCONJ
cana-2405	96	28	𝑏2	𝑏2	NOUN
cana-2405	96	29	)	)	PUNCT
cana-2405	96	30	}	}	PUNCT
cana-2405	96	31	}	}	PUNCT
cana-2405	96	32	≥	≥	NOUN
cana-2405	96	33	sup{min{𝜇1(𝑎1	sup{min{𝜇1(𝑎1	PROPN
cana-2405	96	34	)	)	PUNCT
cana-2405	96	35	,	,	PUNCT
cana-2405	96	36	𝜇2(𝑎2	𝜇2(𝑎2	NOUN
cana-2405	96	37	)	)	PUNCT
cana-2405	96	38	}	}	PUNCT
cana-2405	96	39	,	,	PUNCT
cana-2405	96	40	min{𝜇2(𝑏1	min{𝜇2(𝑏1	PROPN
cana-2405	96	41	)	)	PUNCT
cana-2405	96	42	,	,	PUNCT
cana-2405	96	43	𝜇2(𝑏2	𝜇2(𝑏2	NOUN
cana-2405	96	44	)	)	PUNCT
cana-2405	96	45	}	}	PUNCT
cana-2405	96	46	}	}	PUNCT
cana-2405	96	47	=	=	SYM
cana-2405	96	48	min	min	NOUN
cana-2405	96	49	{	{	PUNCT
cana-2405	96	50	sup	sup	NOUN
cana-2405	96	51	𝑥≤𝑎1+𝑏1	𝑥≤𝑎1+𝑏1	NOUN
cana-2405	96	52	{	{	PUNCT
cana-2405	96	53	min{𝜇1(𝑎1	min{𝜇1(𝑎1	PROPN
cana-2405	96	54	)	)	PUNCT
cana-2405	96	55	,	,	PUNCT
cana-2405	96	56	𝜇2(𝑏1	𝜇2(𝑏1	NOUN
cana-2405	96	57	)	)	PUNCT
cana-2405	96	58	}	}	PUNCT
cana-2405	96	59	}	}	PUNCT
cana-2405	96	60	,	,	PUNCT
cana-2405	96	61	sup	sup	NOUN
cana-2405	96	62	𝑦≤𝑎2+𝑏2	𝑦≤𝑎2+𝑏2	PRON
cana-2405	96	63	{	{	PUNCT
cana-2405	96	64	min{𝜇1(𝑎2	min{𝜇1(𝑎2	NUM
cana-2405	96	65	)	)	PUNCT
cana-2405	96	66	,	,	PUNCT
cana-2405	96	67	𝜇2(𝑏2	𝜇2(𝑏2	NOUN
cana-2405	96	68	)	)	PUNCT
cana-2405	96	69	}	}	PUNCT
cana-2405	96	70	}	}	PUNCT
cana-2405	96	71	}	}	PUNCT
cana-2405	96	72	=	=	SYM
cana-2405	96	73	min{(𝜇1	min{(𝜇1	NOUN
cana-2405	96	74	+	+	CCONJ
cana-2405	96	75	𝜇2)(𝑥	𝜇2)(𝑥	NOUN
cana-2405	96	76	)	)	PUNCT
cana-2405	96	77	,	,	PUNCT
cana-2405	96	78	(	(	PUNCT
cana-2405	96	79	𝜇1	𝜇1	PROPN
cana-2405	96	80	+	+	CCONJ
cana-2405	96	81	𝜇2)(𝑦	𝜇2)(𝑦	PROPN
cana-2405	96	82	)	)	PUNCT
cana-2405	96	83	}	}	PUNCT
cana-2405	97	1	also	also	ADV
cana-2405	97	2	(	(	PUNCT
cana-2405	97	3	𝜗1	𝜗1	X
cana-2405	97	4	+	+	X
cana-2405	97	5	𝜗2)(𝑥	𝜗2)(𝑥	NOUN
cana-2405	97	6	+	+	NUM
cana-2405	97	7	𝑦	𝑦	X
cana-2405	97	8	)	)	PUNCT
cana-2405	97	9	=	=	SYM
cana-2405	97	10	inf	inf	NOUN
cana-2405	97	11	𝑥+𝑦≤𝑐+𝑑	𝑥+𝑦≤𝑐+𝑑	NOUN
cana-2405	97	12	{	{	PUNCT
cana-2405	97	13	max{𝜗1(𝑐	max{𝜗1(𝑐	PROPN
cana-2405	97	14	)	)	PUNCT
cana-2405	97	15	,	,	PUNCT
cana-2405	97	16	𝜗2(𝑑	𝜗2(𝑑	X
cana-2405	97	17	)	)	PUNCT
cana-2405	97	18	}	}	PUNCT
cana-2405	97	19	}	}	PUNCT
cana-2405	97	20	≤	≤	NUM
cana-2405	97	21	inf	inf	PROPN
cana-2405	97	22	𝑥+𝑦≤(𝑎1+𝑏1)+(𝑎2+𝑏2)=(𝑎1+𝑎2)+(𝑏1+𝑏2	𝑥+𝑦≤(𝑎1+𝑏1)+(𝑎2+𝑏2)=(𝑎1+𝑎2)+(𝑏1+𝑏2	NOUN
cana-2405	97	23	)	)	PUNCT
cana-2405	97	24	{	{	PUNCT
cana-2405	97	25	max{𝜗1(𝑎1	max{𝜗1(𝑎1	PROPN
cana-2405	97	26	+	+	CCONJ
cana-2405	97	27	𝑎2	𝑎2	PROPN
cana-2405	97	28	)	)	PUNCT
cana-2405	97	29	,	,	PUNCT
cana-2405	97	30	𝜗2(𝑏1	𝜗2(𝑏1	X
cana-2405	97	31	+	+	X
cana-2405	97	32	𝑏2	𝑏2	NOUN
cana-2405	97	33	)	)	PUNCT
cana-2405	97	34	}	}	PUNCT
cana-2405	97	35	}	}	PUNCT
cana-2405	97	36	≤	≤	ADJ
cana-2405	97	37	inf{max{𝜗1(𝑎1	inf{max{𝜗1(𝑎1	NOUN
cana-2405	97	38	)	)	PUNCT
cana-2405	97	39	,	,	PUNCT
cana-2405	97	40	𝜗2(𝑎2	𝜗2(𝑎2	NOUN
cana-2405	97	41	)	)	PUNCT
cana-2405	97	42	}	}	PUNCT
cana-2405	97	43	,	,	PUNCT
cana-2405	97	44	max{𝜗2(𝑏1	max{𝜗2(𝑏1	PROPN
cana-2405	97	45	)	)	PUNCT
cana-2405	97	46	,	,	PUNCT
cana-2405	97	47	𝜗2(𝑏2	𝜗2(𝑏2	NOUN
cana-2405	97	48	)	)	PUNCT
cana-2405	97	49	}	}	PUNCT
cana-2405	97	50	}	}	PUNCT
cana-2405	97	51	communications	communication	NOUN
cana-2405	97	52	on	on	ADP
cana-2405	97	53	applied	apply	VERB
cana-2405	97	54	nonlinear	nonlinear	ADJ
cana-2405	97	55	analysis	analysis	NOUN
cana-2405	97	56	issn	issn	NOUN
cana-2405	97	57	:	:	PUNCT
cana-2405	97	58	1074	1074	NUM
cana-2405	97	59	-	-	PUNCT
cana-2405	97	60	133x	133x	NUM
cana-2405	97	61	vol	vol	NOUN
cana-2405	97	62	32	32	NUM
cana-2405	97	63	no	no	NOUN
cana-2405	97	64	.	.	PUNCT
cana-2405	98	1	2s	2s	NUM
cana-2405	98	2	(	(	PUNCT
cana-2405	98	3	2025	2025	NUM
cana-2405	98	4	)	)	PUNCT
cana-2405	98	5	318	318	NUM
cana-2405	98	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2405	98	7	=	=	SYM
cana-2405	98	8	max	max	PROPN
cana-2405	98	9	{	{	PUNCT
cana-2405	98	10	inf	inf	PROPN
cana-2405	98	11	𝑥≤𝑎1+𝑏1	𝑥≤𝑎1+𝑏1	PROPN
cana-2405	98	12	{	{	PUNCT
cana-2405	98	13	max{𝜗1(𝑎1	max{𝜗1(𝑎1	PROPN
cana-2405	98	14	)	)	PUNCT
cana-2405	98	15	,	,	PUNCT
cana-2405	98	16	𝜗2(𝑏1	𝜗2(𝑏1	NOUN
cana-2405	98	17	)	)	PUNCT
cana-2405	98	18	}	}	PUNCT
cana-2405	98	19	}	}	PUNCT
cana-2405	98	20	,	,	PUNCT
cana-2405	98	21	inf	inf	PROPN
cana-2405	98	22	𝑦≤𝑎2+𝑏2	𝑦≤𝑎2+𝑏2	PUNCT
cana-2405	98	23	{	{	PUNCT
cana-2405	98	24	max{𝜗1(𝑎2	max{𝜗1(𝑎2	NUM
cana-2405	98	25	)	)	PUNCT
cana-2405	98	26	,	,	PUNCT
cana-2405	98	27	𝜗2(𝑏2	𝜗2(𝑏2	NOUN
cana-2405	98	28	)	)	PUNCT
cana-2405	98	29	}	}	PUNCT
cana-2405	98	30	}	}	PUNCT
cana-2405	98	31	}	}	PUNCT
cana-2405	98	32	=	=	SYM
cana-2405	98	33	max{(𝜗1	max{(𝜗1	NOUN
cana-2405	98	34	+	+	X
cana-2405	98	35	𝜗2)(𝑥	𝜗2)(𝑥	PROPN
cana-2405	98	36	)	)	PUNCT
cana-2405	98	37	,	,	PUNCT
cana-2405	98	38	(	(	PUNCT
cana-2405	98	39	𝜗1	𝜗1	X
cana-2405	98	40	+	+	X
cana-2405	98	41	𝜗2)(𝑦	𝜗2)(𝑦	NOUN
cana-2405	98	42	)	)	PUNCT
cana-2405	98	43	}	}	PUNCT
cana-2405	98	44	now	now	ADV
cana-2405	98	45	let	let	VERB
cana-2405	98	46	as	as	SCONJ
cana-2405	98	47	consider	consider	VERB
cana-2405	98	48	𝑃1	𝑃1	NOUN
cana-2405	98	49	,	,	PUNCT
cana-2405	98	50	𝑃2	𝑃2	PROPN
cana-2405	98	51	are	be	AUX
cana-2405	98	52	pythagorean	pythagorean	PROPN
cana-2405	98	53	fuzzy	fuzzy	ADJ
cana-2405	98	54	right	right	ADJ
cana-2405	98	55	ideals	ideal	NOUN
cana-2405	98	56	and	and	CCONJ
cana-2405	98	57	we	we	PRON
cana-2405	98	58	have	have	VERB
cana-2405	98	59	(	(	PUNCT
cana-2405	98	60	𝜇1	𝜇1	NOUN
cana-2405	98	61	+	+	CCONJ
cana-2405	98	62	𝜇2)(𝑥𝑦	𝜇2)(𝑥𝑦	PROPN
cana-2405	98	63	)	)	PUNCT
cana-2405	99	1	=	=	SYM
cana-2405	99	2	sup	sup	NOUN
cana-2405	99	3	𝑥+𝑦≤𝑐+𝑑	𝑥+𝑦≤𝑐+𝑑	NOUN
cana-2405	99	4	{	{	PUNCT
cana-2405	99	5	min{𝜇1(𝑐	min{𝜇1(𝑐	PROPN
cana-2405	99	6	)	)	PUNCT
cana-2405	99	7	,	,	PUNCT
cana-2405	99	8	𝜇2(𝑑	𝜇2(𝑑	PROPN
cana-2405	99	9	)	)	PUNCT
cana-2405	99	10	}	}	PUNCT
cana-2405	99	11	}	}	PUNCT
cana-2405	99	12	≥	≥	PROPN
cana-2405	99	13	sup	sup	NOUN
cana-2405	99	14	𝑥𝑦≤(𝑥1+𝑥2)𝑦	𝑥𝑦≤(𝑥1+𝑥2)𝑦	NOUN
cana-2405	99	15	{	{	PUNCT
cana-2405	99	16	min{𝜇1(𝑥1𝑦	min{𝜇1(𝑥1𝑦	PROPN
cana-2405	99	17	)	)	PUNCT
cana-2405	99	18	,	,	PUNCT
cana-2405	99	19	𝜇2(𝑥2𝑦	𝜇2(𝑥2𝑦	PROPN
cana-2405	99	20	)	)	PUNCT
cana-2405	99	21	}	}	PUNCT
cana-2405	99	22	}	}	PUNCT
cana-2405	99	23	≥	≥	PROPN
cana-2405	99	24	sup	sup	NOUN
cana-2405	99	25	𝑥≤(𝑥1+𝑥2	𝑥≤(𝑥1+𝑥2	PROPN
cana-2405	99	26	)	)	PUNCT
cana-2405	99	27	{	{	PUNCT
cana-2405	99	28	min{𝜇1(𝑥1	min{𝜇1(𝑥1	PROPN
cana-2405	99	29	)	)	PUNCT
cana-2405	99	30	,	,	PUNCT
cana-2405	99	31	𝜇2(𝑥2	𝜇2(𝑥2	PROPN
cana-2405	99	32	)	)	PUNCT
cana-2405	99	33	}	}	PUNCT
cana-2405	99	34	}	}	PUNCT
cana-2405	99	35	=	=	SYM
cana-2405	99	36	(	(	PUNCT
cana-2405	99	37	𝜇1	𝜇1	NOUN
cana-2405	99	38	+	+	CCONJ
cana-2405	99	39	𝜇2)(𝑥	𝜇2)(𝑥	NOUN
cana-2405	99	40	)	)	PUNCT
cana-2405	99	41	.	.	PUNCT
cana-2405	100	1	and	and	CCONJ
cana-2405	100	2	(	(	PUNCT
cana-2405	100	3	𝜗1	𝜗1	X
cana-2405	100	4	+	+	X
cana-2405	100	5	𝜗2)(𝑥𝑦	𝜗2)(𝑥𝑦	PROPN
cana-2405	100	6	)	)	PUNCT
cana-2405	100	7	=	=	SYM
cana-2405	100	8	inf	inf	NOUN
cana-2405	100	9	𝑥+𝑦≤𝑐+𝑑	𝑥+𝑦≤𝑐+𝑑	NOUN
cana-2405	100	10	{	{	PUNCT
cana-2405	100	11	max{𝜗1(𝑐	max{𝜗1(𝑐	PROPN
cana-2405	100	12	)	)	PUNCT
cana-2405	100	13	,	,	PUNCT
cana-2405	100	14	𝜗2(𝑑	𝜗2(𝑑	X
cana-2405	100	15	)	)	PUNCT
cana-2405	100	16	}	}	PUNCT
cana-2405	100	17	}	}	PUNCT
cana-2405	100	18	≤	≤	NUM
cana-2405	100	19	inf	inf	PROPN
cana-2405	100	20	𝑥𝑦≤(𝑥1+𝑥2)𝑦	𝑥𝑦≤(𝑥1+𝑥2)𝑦	PROPN
cana-2405	100	21	{	{	PUNCT
cana-2405	100	22	max{𝜗1(𝑥1𝑦	max{𝜗1(𝑥1𝑦	PROPN
cana-2405	100	23	)	)	PUNCT
cana-2405	100	24	,	,	PUNCT
cana-2405	100	25	𝜗2(𝑥2𝑦	𝜗2(𝑥2𝑦	X
cana-2405	100	26	)	)	PUNCT
cana-2405	100	27	}	}	PUNCT
cana-2405	100	28	}	}	PUNCT
cana-2405	100	29	≤	≤	NUM
cana-2405	100	30	inf	inf	PROPN
cana-2405	100	31	𝑥≤(𝑥1+𝑥2	𝑥≤(𝑥1+𝑥2	PROPN
cana-2405	100	32	)	)	PUNCT
cana-2405	100	33	{	{	PUNCT
cana-2405	100	34	max{𝜗1(𝑥1	max{𝜗1(𝑥1	PROPN
cana-2405	100	35	)	)	PUNCT
cana-2405	100	36	,	,	PUNCT
cana-2405	100	37	𝜗2(𝑥2	𝜗2(𝑥2	NOUN
cana-2405	100	38	)	)	PUNCT
cana-2405	100	39	}	}	PUNCT
cana-2405	100	40	}	}	PUNCT
cana-2405	100	41	=	=	SYM
cana-2405	100	42	(	(	PUNCT
cana-2405	100	43	𝜗1	𝜗1	X
cana-2405	100	44	+	+	X
cana-2405	100	45	𝜗2)(𝑥	𝜗2)(𝑥	PROPN
cana-2405	100	46	)	)	PUNCT
cana-2405	100	47	.	.	PUNCT
cana-2405	101	1	similarly	similarly	ADV
cana-2405	101	2	assuming	assume	VERB
cana-2405	101	3	𝑃1	𝑃1	NOUN
cana-2405	101	4	,	,	PUNCT
cana-2405	101	5	𝑃2	𝑃2	PROPN
cana-2405	101	6	are	be	AUX
cana-2405	101	7	pythagorean	pythagorean	PROPN
cana-2405	101	8	fuzzy	fuzzy	ADJ
cana-2405	101	9	left	leave	VERB
cana-2405	101	10	ideal	ideal	ADJ
cana-2405	101	11	,	,	PUNCT
cana-2405	101	12	we	we	PRON
cana-2405	101	13	can	can	AUX
cana-2405	101	14	show	show	VERB
cana-2405	101	15	that	that	SCONJ
cana-2405	101	16	(	(	PUNCT
cana-2405	101	17	𝑃1	𝑃1	NOUN
cana-2405	101	18	+	+	SYM
cana-2405	101	19	𝑃2)(𝑥𝑦	𝑃2)(𝑥𝑦	NOUN
cana-2405	101	20	)	)	PUNCT
cana-2405	101	21	≥	≥	NUM
cana-2405	101	22	(	(	PUNCT
cana-2405	101	23	𝑃1	𝑃1	NOUN
cana-2405	101	24	+	+	CCONJ
cana-2405	101	25	𝑃2)(𝑦	𝑃2)(𝑦	NOUN
cana-2405	101	26	)	)	PUNCT
cana-2405	101	27	also	also	ADV
cana-2405	101	28	(	(	PUNCT
cana-2405	101	29	𝜇1	𝜇1	PROPN
cana-2405	101	30	+	+	CCONJ
cana-2405	101	31	𝜇2)(𝑥	𝜇2)(𝑥	NOUN
cana-2405	101	32	)	)	PUNCT
cana-2405	101	33	=	=	SYM
cana-2405	102	1	sup	sup	NOUN
cana-2405	102	2	𝑥≤𝑥1+𝑥2	𝑥≤𝑥1+𝑥2	PUNCT
cana-2405	102	3	{	{	PUNCT
cana-2405	102	4	min{𝜇1(𝑥1	min{𝜇1(𝑥1	PROPN
cana-2405	102	5	)	)	PUNCT
cana-2405	102	6	,	,	PUNCT
cana-2405	102	7	𝜇2(𝑥2	𝜇2(𝑥2	PROPN
cana-2405	102	8	)	)	PUNCT
cana-2405	102	9	}	}	PUNCT
cana-2405	102	10	}	}	PUNCT
cana-2405	102	11	≥	≥	NUM
cana-2405	102	12	sup	sup	NUM
cana-2405	102	13	𝑥≤𝑦≤𝑦1+𝑦2	𝑥≤𝑦≤𝑦1+𝑦2	NOUN
cana-2405	102	14	{	{	PUNCT
cana-2405	102	15	min{𝜇1(𝑦1	min{𝜇1(𝑦1	PROPN
cana-2405	102	16	)	)	PUNCT
cana-2405	102	17	,	,	PUNCT
cana-2405	102	18	𝜇2(𝑦2	𝜇2(𝑦2	NOUN
cana-2405	102	19	)	)	PUNCT
cana-2405	102	20	}	}	PUNCT
cana-2405	102	21	}	}	PUNCT
cana-2405	102	22	=	=	SYM
cana-2405	102	23	sup	sup	NOUN
cana-2405	102	24	𝑦≤𝑦1+𝑦2	𝑦≤𝑦1+𝑦2	PUNCT
cana-2405	102	25	{	{	PUNCT
cana-2405	102	26	min{𝜇1(𝑦1	min{𝜇1(𝑦1	PROPN
cana-2405	102	27	)	)	PUNCT
cana-2405	102	28	,	,	PUNCT
cana-2405	102	29	𝜇2(𝑦2	𝜇2(𝑦2	NOUN
cana-2405	102	30	)	)	PUNCT
cana-2405	102	31	}	}	PUNCT
cana-2405	102	32	}	}	PUNCT
cana-2405	102	33	=	=	SYM
cana-2405	102	34	(	(	PUNCT
cana-2405	102	35	𝜇1	𝜇1	NOUN
cana-2405	102	36	+	+	CCONJ
cana-2405	102	37	𝜇2)(𝑦	𝜇2)(𝑦	PROPN
cana-2405	102	38	)	)	PUNCT
cana-2405	102	39	and	and	CCONJ
cana-2405	102	40	(	(	PUNCT
cana-2405	102	41	𝜗1	𝜗1	X
cana-2405	102	42	+	+	X
cana-2405	102	43	𝜗2)(𝑥	𝜗2)(𝑥	ADJ
cana-2405	102	44	)	)	PUNCT
cana-2405	102	45	=	=	PROPN
cana-2405	102	46	inf	inf	PROPN
cana-2405	102	47	𝑥≤𝑥1+𝑥2	𝑥≤𝑥1+𝑥2	PUNCT
cana-2405	102	48	{	{	PUNCT
cana-2405	102	49	max{𝜗1(𝑥1	max{𝜗1(𝑥1	PROPN
cana-2405	102	50	)	)	PUNCT
cana-2405	102	51	,	,	PUNCT
cana-2405	102	52	𝜗2(𝑥2	𝜗2(𝑥2	NOUN
cana-2405	102	53	)	)	PUNCT
cana-2405	102	54	}	}	PUNCT
cana-2405	102	55	}	}	PUNCT
cana-2405	102	56	≤	≤	NUM
cana-2405	102	57	inf	inf	PROPN
cana-2405	102	58	𝑥≤𝑦≤𝑦1+𝑦2	𝑥≤𝑦≤𝑦1+𝑦2	PROPN
cana-2405	102	59	{	{	PUNCT
cana-2405	102	60	max{𝜗1(𝑦1	max{𝜗1(𝑦1	PROPN
cana-2405	102	61	)	)	PUNCT
cana-2405	102	62	,	,	PUNCT
cana-2405	102	63	𝜗2(𝑦2	𝜗2(𝑦2	NOUN
cana-2405	102	64	)	)	PUNCT
cana-2405	102	65	}	}	PUNCT
cana-2405	102	66	}	}	PUNCT
cana-2405	102	67	=	=	SYM
cana-2405	102	68	inf	inf	PROPN
cana-2405	102	69	𝑦≤𝑦1+𝑦2	𝑦≤𝑦1+𝑦2	NUM
cana-2405	102	70	{	{	PUNCT
cana-2405	102	71	max{𝜗1(𝑦1	max{𝜗1(𝑦1	PROPN
cana-2405	102	72	)	)	PUNCT
cana-2405	102	73	,	,	PUNCT
cana-2405	102	74	𝜗2(𝑦2	𝜗2(𝑦2	NOUN
cana-2405	102	75	)	)	PUNCT
cana-2405	102	76	}	}	PUNCT
cana-2405	102	77	}	}	PUNCT
cana-2405	102	78	=	=	SYM
cana-2405	102	79	(	(	PUNCT
cana-2405	102	80	𝜗1	𝜗1	X
cana-2405	102	81	+	+	X
cana-2405	102	82	𝜗2)(𝑦	𝜗2)(𝑦	NOUN
cana-2405	102	83	)	)	PUNCT
cana-2405	102	84	hence	hence	ADV
cana-2405	102	85	𝑃1	𝑃1	NOUN
cana-2405	102	86	+	+	CCONJ
cana-2405	102	87	𝑃2	𝑃2	NOUN
cana-2405	102	88	is	be	AUX
cana-2405	102	89	a	a	DET
cana-2405	102	90	pythagorean	pythagorean	ADJ
cana-2405	102	91	fuzzy	fuzzy	ADJ
cana-2405	102	92	ideal	ideal	NOUN
cana-2405	102	93	of	of	ADP
cana-2405	102	94	𝑆.	𝑆.	PROPN
cana-2405	102	95	theorem	theorem	VERB
cana-2405	102	96	3.10	3.10	NUM
cana-2405	102	97	if	if	SCONJ
cana-2405	102	98	𝑃1	𝑃1	NOUN
cana-2405	102	99	,	,	PUNCT
cana-2405	102	100	𝑃2	𝑃2	PROPN
cana-2405	102	101	be	be	AUX
cana-2405	102	102	any	any	DET
cana-2405	102	103	two	two	NUM
cana-2405	102	104	pythagorean	pythagorean	ADJ
cana-2405	102	105	fuzzy	fuzzy	ADJ
cana-2405	102	106	ideals	ideal	NOUN
cana-2405	102	107	of	of	ADP
cana-2405	102	108	semiring	semire	VERB
cana-2405	102	109	𝑆	𝑆	PROPN
cana-2405	102	110	,	,	PUNCT
cana-2405	102	111	then	then	ADV
cana-2405	102	112	𝑃1	𝑃1	NOUN
cana-2405	102	113	∘	∘	PROPN
cana-2405	102	114	𝑃2	𝑃2	PROPN
cana-2405	102	115	is	be	AUX
cana-2405	102	116	also	also	ADV
cana-2405	102	117	so	so	ADV
cana-2405	102	118	.	.	PUNCT
cana-2405	103	1	proof	proof	NOUN
cana-2405	103	2	.	.	PUNCT
cana-2405	104	1	let	let	VERB
cana-2405	104	2	𝑃1	𝑃1	NOUN
cana-2405	104	3	,	,	PUNCT
cana-2405	104	4	𝑃2	𝑃2	PROPN
cana-2405	104	5	are	be	AUX
cana-2405	104	6	any	any	DET
cana-2405	104	7	two	two	NUM
cana-2405	104	8	pythagorean	pythagorean	ADJ
cana-2405	104	9	fuzzy	fuzzy	ADJ
cana-2405	104	10	ideals	ideal	NOUN
cana-2405	104	11	of	of	ADP
cana-2405	104	12	semiring	semire	VERB
cana-2405	104	13	𝑆	𝑆	PROPN
cana-2405	104	14	and	and	CCONJ
cana-2405	104	15	𝑥	𝑥	PROPN
cana-2405	104	16	,	,	PUNCT
cana-2405	104	17	𝑦	𝑦	NOUN
cana-2405	104	18	∈	∈	NOUN
cana-2405	104	19	𝑆.	𝑆.	NOUN
cana-2405	104	20	communications	communication	NOUN
cana-2405	104	21	on	on	ADP
cana-2405	104	22	applied	apply	VERB
cana-2405	104	23	nonlinear	nonlinear	ADJ
cana-2405	104	24	analysis	analysis	NOUN
cana-2405	104	25	issn	issn	NOUN
cana-2405	104	26	:	:	PUNCT
cana-2405	104	27	1074	1074	NUM
cana-2405	104	28	-	-	PUNCT
cana-2405	104	29	133x	133x	NUM
cana-2405	104	30	vol	vol	NOUN
cana-2405	104	31	32	32	NUM
cana-2405	104	32	no	no	NOUN
cana-2405	104	33	.	.	PUNCT
cana-2405	105	1	2s	2s	NUM
cana-2405	105	2	(	(	PUNCT
cana-2405	105	3	2025	2025	NUM
cana-2405	105	4	)	)	PUNCT
cana-2405	105	5	319	319	NUM
cana-2405	105	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2405	105	7	then	then	ADV
cana-2405	105	8	(	(	PUNCT
cana-2405	105	9	𝜇1	𝜇1	PROPN
cana-2405	105	10	∘	∘	NOUN
cana-2405	105	11	𝜇2)(𝑥	𝜇2)(𝑥	PROPN
cana-2405	105	12	+	+	NUM
cana-2405	105	13	𝑦	𝑦	NOUN
cana-2405	105	14	)	)	PUNCT
cana-2405	105	15	=	=	SYM
cana-2405	105	16	sup	sup	NOUN
cana-2405	105	17	𝑥+𝑦≤𝑐+𝑑	𝑥+𝑦≤𝑐+𝑑	NOUN
cana-2405	105	18	{	{	PUNCT
cana-2405	105	19	min{𝜇1(𝑐	min{𝜇1(𝑐	PROPN
cana-2405	105	20	)	)	PUNCT
cana-2405	105	21	,	,	PUNCT
cana-2405	105	22	𝜇2(𝑑	𝜇2(𝑑	PROPN
cana-2405	105	23	)	)	PUNCT
cana-2405	105	24	}	}	PUNCT
cana-2405	105	25	}	}	PUNCT
cana-2405	105	26	≥	≥	PROPN
cana-2405	105	27	sup	sup	NOUN
cana-2405	105	28	𝑥+𝑦≤(𝑐1𝑑1)+(𝑐2𝑑2)≤(𝑐1+𝑐2)(𝑑1+𝑑2	𝑥+𝑦≤(𝑐1𝑑1)+(𝑐2𝑑2)≤(𝑐1+𝑐2)(𝑑1+𝑑2	NOUN
cana-2405	105	29	)	)	PUNCT
cana-2405	105	30	{	{	PUNCT
cana-2405	105	31	min{𝜇1(𝑐1	min{𝜇1(𝑐1	NUM
cana-2405	105	32	+	+	NUM
cana-2405	105	33	𝑐2	𝑐2	NOUN
cana-2405	105	34	)	)	PUNCT
cana-2405	105	35	,	,	PUNCT
cana-2405	105	36	𝜇2(𝑑1	𝜇2(𝑑1	ADV
cana-2405	105	37	+	+	CCONJ
cana-2405	105	38	𝑑2	𝑑2	NOUN
cana-2405	105	39	)	)	PUNCT
cana-2405	105	40	}	}	PUNCT
cana-2405	105	41	}	}	PUNCT
cana-2405	105	42	≥	≥	NOUN
cana-2405	105	43	sup{min{𝜇1(𝑐1	sup{min{𝜇1(𝑐1	VERB
cana-2405	105	44	)	)	PUNCT
cana-2405	105	45	,	,	PUNCT
cana-2405	105	46	𝜇1(𝑐2	𝜇1(𝑐2	ADV
cana-2405	105	47	)	)	PUNCT
cana-2405	105	48	}	}	PUNCT
cana-2405	105	49	,	,	PUNCT
cana-2405	105	50	min{𝜇2(𝑑1	min{𝜇2(𝑑1	NOUN
cana-2405	105	51	)	)	PUNCT
cana-2405	105	52	,	,	PUNCT
cana-2405	105	53	𝜇2(𝑑2	𝜇2(𝑑2	NOUN
cana-2405	105	54	)	)	PUNCT
cana-2405	105	55	}	}	PUNCT
cana-2405	105	56	}	}	PUNCT
cana-2405	105	57	=	=	SYM
cana-2405	105	58	min	min	NOUN
cana-2405	105	59	{	{	PUNCT
cana-2405	105	60	sup	sup	NOUN
cana-2405	105	61	𝑥≤𝑐1𝑑1	𝑥≤𝑐1𝑑1	ADV
cana-2405	105	62	{	{	PUNCT
cana-2405	105	63	min{𝜇1(𝑐1	min{𝜇1(𝑐1	NUM
cana-2405	105	64	)	)	PUNCT
cana-2405	105	65	,	,	PUNCT
cana-2405	105	66	𝜇2(𝑑1	𝜇2(𝑑1	NOUN
cana-2405	105	67	)	)	PUNCT
cana-2405	105	68	}	}	PUNCT
cana-2405	105	69	}	}	PUNCT
cana-2405	105	70	,	,	PUNCT
cana-2405	105	71	sup	sup	INTJ
cana-2405	105	72	𝑦≤𝑐2𝑑2	𝑦≤𝑐2𝑑2	ADV
cana-2405	105	73	{	{	PUNCT
cana-2405	105	74	min{𝜇1(𝑐2	min{𝜇1(𝑐2	PROPN
cana-2405	105	75	)	)	PUNCT
cana-2405	105	76	,	,	PUNCT
cana-2405	105	77	𝜇2(𝑑2	𝜇2(𝑑2	NOUN
cana-2405	105	78	)	)	PUNCT
cana-2405	105	79	}	}	PUNCT
cana-2405	105	80	}	}	PUNCT
cana-2405	105	81	}	}	PUNCT
cana-2405	105	82	=	=	SYM
cana-2405	105	83	min{(𝜇1	min{(𝜇1	NOUN
cana-2405	105	84	∘	∘	NOUN
cana-2405	105	85	𝜇2)(𝑥	𝜇2)(𝑥	NOUN
cana-2405	105	86	)	)	PUNCT
cana-2405	105	87	,	,	PUNCT
cana-2405	105	88	(	(	PUNCT
cana-2405	105	89	𝜇1	𝜇1	PROPN
cana-2405	105	90	∘	∘	PROPN
cana-2405	105	91	𝜇2)(𝑦	𝜇2)(𝑦	PROPN
cana-2405	105	92	)	)	PUNCT
cana-2405	105	93	}	}	PUNCT
cana-2405	105	94	also	also	ADV
cana-2405	105	95	(	(	PUNCT
cana-2405	105	96	𝜗1	𝜗1	X
cana-2405	105	97	∘	∘	X
cana-2405	105	98	𝜗2)(𝑥	𝜗2)(𝑥	PROPN
cana-2405	105	99	+	+	NUM
cana-2405	105	100	𝑦	𝑦	X
cana-2405	105	101	)	)	PUNCT
cana-2405	105	102	=	=	SYM
cana-2405	105	103	inf	inf	NOUN
cana-2405	105	104	𝑥+𝑦≤𝑐+𝑑	𝑥+𝑦≤𝑐+𝑑	NOUN
cana-2405	105	105	{	{	PUNCT
cana-2405	105	106	max{𝜗1(𝑐	max{𝜗1(𝑐	PROPN
cana-2405	105	107	)	)	PUNCT
cana-2405	105	108	,	,	PUNCT
cana-2405	105	109	𝜗2(𝑑	𝜗2(𝑑	X
cana-2405	105	110	)	)	PUNCT
cana-2405	105	111	}	}	PUNCT
cana-2405	105	112	}	}	PUNCT
cana-2405	105	113	≤	≤	NUM
cana-2405	105	114	inf	inf	ADJ
cana-2405	105	115	𝑥+𝑦≤(𝑐1𝑑1)+(𝑐2𝑑2)≤(𝑐1+𝑐2)(𝑑1+𝑑2	𝑥+𝑦≤(𝑐1𝑑1)+(𝑐2𝑑2)≤(𝑐1+𝑐2)(𝑑1+𝑑2	NOUN
cana-2405	105	116	)	)	PUNCT
cana-2405	105	117	{	{	PUNCT
cana-2405	105	118	max{𝜗1(𝑐1	max{𝜗1(𝑐1	NOUN
cana-2405	105	119	+	+	NUM
cana-2405	105	120	𝑐2	𝑐2	NOUN
cana-2405	105	121	)	)	PUNCT
cana-2405	105	122	,	,	PUNCT
cana-2405	105	123	𝜗2(𝑑1	𝜗2(𝑑1	ADV
cana-2405	105	124	+	+	CCONJ
cana-2405	105	125	𝑑2	𝑑2	VERB
cana-2405	105	126	)	)	PUNCT
cana-2405	105	127	}	}	PUNCT
cana-2405	105	128	}	}	PUNCT
cana-2405	105	129	≤	≤	PROPN
cana-2405	105	130	inf{max{𝜗1(𝑐1	inf{max{𝜗1(𝑐1	PROPN
cana-2405	105	131	)	)	PUNCT
cana-2405	105	132	,	,	PUNCT
cana-2405	105	133	𝜗1(𝑐2	𝜗1(𝑐2	NOUN
cana-2405	105	134	)	)	PUNCT
cana-2405	105	135	}	}	PUNCT
cana-2405	105	136	,	,	PUNCT
cana-2405	105	137	max{𝜗2(𝑑1	max{𝜗2(𝑑1	PROPN
cana-2405	105	138	)	)	PUNCT
cana-2405	105	139	,	,	PUNCT
cana-2405	105	140	𝜗2(𝑑2	𝜗2(𝑑2	PROPN
cana-2405	105	141	)	)	PUNCT
cana-2405	105	142	}	}	PUNCT
cana-2405	105	143	}	}	PUNCT
cana-2405	105	144	=	=	SYM
cana-2405	105	145	max	max	X
cana-2405	105	146	{	{	PUNCT
cana-2405	105	147	inf	inf	PROPN
cana-2405	105	148	𝑥≤𝑐1𝑑1	𝑥≤𝑐1𝑑1	X
cana-2405	105	149	{	{	PUNCT
cana-2405	105	150	max{𝜗1(𝑐1	max{𝜗1(𝑐1	NOUN
cana-2405	105	151	)	)	PUNCT
cana-2405	105	152	,	,	PUNCT
cana-2405	105	153	𝜗2(𝑑1	𝜗2(𝑑1	NOUN
cana-2405	105	154	)	)	PUNCT
cana-2405	105	155	}	}	PUNCT
cana-2405	105	156	}	}	PUNCT
cana-2405	105	157	,	,	PUNCT
cana-2405	105	158	inf	inf	PROPN
cana-2405	105	159	𝑦≤𝑐2𝑑2	𝑦≤𝑐2𝑑2	PROPN
cana-2405	105	160	{	{	PUNCT
cana-2405	105	161	max{𝜗1(𝑐2	max{𝜗1(𝑐2	PROPN
cana-2405	105	162	)	)	PUNCT
cana-2405	105	163	,	,	PUNCT
cana-2405	105	164	𝜗2(𝑑2	𝜗2(𝑑2	PROPN
cana-2405	105	165	)	)	PUNCT
cana-2405	105	166	}	}	PUNCT
cana-2405	105	167	}	}	PUNCT
cana-2405	105	168	}	}	PUNCT
cana-2405	105	169	=	=	SYM
cana-2405	105	170	max{(𝜗1	max{(𝜗1	NOUN
cana-2405	105	171	∘	∘	NOUN
cana-2405	105	172	𝜗2)(𝑥	𝜗2)(𝑥	PROPN
cana-2405	105	173	)	)	PUNCT
cana-2405	105	174	,	,	PUNCT
cana-2405	105	175	(	(	PUNCT
cana-2405	105	176	𝜗1]𝑐𝑖𝑟𝑐𝜗2)(𝑦	𝜗1]𝑐𝑖𝑟𝑐𝜗2)(𝑦	NOUN
cana-2405	105	177	)	)	PUNCT
cana-2405	105	178	}	}	PUNCT
cana-2405	105	179	now	now	ADV
cana-2405	105	180	let	let	VERB
cana-2405	105	181	as	as	SCONJ
cana-2405	105	182	consider	consider	VERB
cana-2405	105	183	𝑃1	𝑃1	NOUN
cana-2405	105	184	,	,	PUNCT
cana-2405	105	185	𝑃2	𝑃2	PROPN
cana-2405	105	186	are	be	AUX
cana-2405	105	187	pythagorean	pythagorean	PROPN
cana-2405	105	188	fuzzy	fuzzy	ADJ
cana-2405	105	189	right	right	ADJ
cana-2405	105	190	ideals	ideal	NOUN
cana-2405	105	191	and	and	CCONJ
cana-2405	105	192	we	we	PRON
cana-2405	105	193	have	have	VERB
cana-2405	105	194	(	(	PUNCT
cana-2405	105	195	𝜇1	𝜇1	ADV
cana-2405	105	196	∘	∘	NOUN
cana-2405	105	197	𝜇2)(𝑥𝑦	𝜇2)(𝑥𝑦	X
cana-2405	105	198	)	)	PUNCT
cana-2405	106	1	=	=	SYM
cana-2405	106	2	sup	sup	NOUN
cana-2405	106	3	𝑥𝑦≤𝑐𝑑	𝑥𝑦≤𝑐𝑑	PRON
cana-2405	106	4	{	{	PUNCT
cana-2405	106	5	min{𝜇1(𝑐	min{𝜇1(𝑐	PROPN
cana-2405	106	6	)	)	PUNCT
cana-2405	106	7	,	,	PUNCT
cana-2405	106	8	𝜇2(𝑑	𝜇2(𝑑	PROPN
cana-2405	106	9	)	)	PUNCT
cana-2405	106	10	}	}	PUNCT
cana-2405	106	11	}	}	PUNCT
cana-2405	106	12	≥	≥	NUM
cana-2405	106	13	sup	sup	PROPN
cana-2405	106	14	𝑥𝑦≤(𝑥1𝑥2)𝑦	𝑥𝑦≤(𝑥1𝑥2)𝑦	PROPN
cana-2405	106	15	{	{	PUNCT
cana-2405	106	16	min{𝜇1(𝑥1𝑦	min{𝜇1(𝑥1𝑦	PROPN
cana-2405	106	17	)	)	PUNCT
cana-2405	106	18	,	,	PUNCT
cana-2405	106	19	𝜇2(𝑥2𝑦	𝜇2(𝑥2𝑦	PROPN
cana-2405	106	20	)	)	PUNCT
cana-2405	106	21	}	}	PUNCT
cana-2405	106	22	}	}	PUNCT
cana-2405	106	23	≥	≥	PROPN
cana-2405	106	24	sup	sup	NOUN
cana-2405	106	25	𝑥≤(𝑥1𝑥2	𝑥≤(𝑥1𝑥2	NOUN
cana-2405	106	26	)	)	PUNCT
cana-2405	106	27	{	{	PUNCT
cana-2405	106	28	min{𝜇1(𝑥1	min{𝜇1(𝑥1	PROPN
cana-2405	106	29	)	)	PUNCT
cana-2405	106	30	,	,	PUNCT
cana-2405	106	31	𝜇2(𝑥2	𝜇2(𝑥2	PROPN
cana-2405	106	32	)	)	PUNCT
cana-2405	106	33	}	}	PUNCT
cana-2405	106	34	}	}	PUNCT
cana-2405	106	35	=	=	SYM
cana-2405	106	36	(	(	PUNCT
cana-2405	106	37	𝜇1	𝜇1	PROPN
cana-2405	106	38	∘	∘	NOUN
cana-2405	106	39	𝜇2)(𝑥	𝜇2)(𝑥	NOUN
cana-2405	106	40	)	)	PUNCT
cana-2405	106	41	.	.	PUNCT
cana-2405	107	1	and	and	CCONJ
cana-2405	107	2	(	(	PUNCT
cana-2405	107	3	𝜗1	𝜗1	X
cana-2405	107	4	∘	∘	PROPN
cana-2405	107	5	𝜗2)(𝑥𝑦	𝜗2)(𝑥𝑦	PROPN
cana-2405	107	6	)	)	PUNCT
cana-2405	107	7	=	=	SYM
cana-2405	107	8	inf	inf	PROPN
cana-2405	107	9	𝑥𝑦≤𝑐𝑑	𝑥𝑦≤𝑐𝑑	X
cana-2405	107	10	{	{	PUNCT
cana-2405	107	11	max{𝜗1(𝑐	max{𝜗1(𝑐	PROPN
cana-2405	107	12	)	)	PUNCT
cana-2405	107	13	,	,	PUNCT
cana-2405	107	14	𝜗2(𝑑	𝜗2(𝑑	X
cana-2405	107	15	)	)	PUNCT
cana-2405	107	16	}	}	PUNCT
cana-2405	107	17	}	}	PUNCT
cana-2405	107	18	≤	≤	NUM
cana-2405	107	19	inf	inf	ADJ
cana-2405	107	20	𝑥𝑦≤(𝑥1𝑥2)𝑦	𝑥𝑦≤(𝑥1𝑥2)𝑦	NOUN
cana-2405	107	21	{	{	PUNCT
cana-2405	107	22	max{𝜗1(𝑥1𝑦	max{𝜗1(𝑥1𝑦	PROPN
cana-2405	107	23	)	)	PUNCT
cana-2405	107	24	,	,	PUNCT
cana-2405	107	25	𝜗2(𝑥2𝑦	𝜗2(𝑥2𝑦	X
cana-2405	107	26	)	)	PUNCT
cana-2405	107	27	}	}	PUNCT
cana-2405	107	28	}	}	PUNCT
cana-2405	107	29	≤	≤	NUM
cana-2405	107	30	inf	inf	PROPN
cana-2405	107	31	𝑥≤(𝑥1𝑥2	𝑥≤(𝑥1𝑥2	NOUN
cana-2405	107	32	)	)	PUNCT
cana-2405	107	33	{	{	PUNCT
cana-2405	107	34	max{𝜗1(𝑥1	max{𝜗1(𝑥1	PROPN
cana-2405	107	35	)	)	PUNCT
cana-2405	107	36	,	,	PUNCT
cana-2405	107	37	𝜗2(𝑥2	𝜗2(𝑥2	NOUN
cana-2405	107	38	)	)	PUNCT
cana-2405	107	39	}	}	PUNCT
cana-2405	107	40	}	}	PUNCT
cana-2405	107	41	=	=	SYM
cana-2405	107	42	(	(	PUNCT
cana-2405	107	43	𝜗1	𝜗1	X
cana-2405	107	44	∘	∘	NOUN
cana-2405	107	45	𝜗2)(𝑥	𝜗2)(𝑥	PROPN
cana-2405	107	46	)	)	PUNCT
cana-2405	107	47	.	.	PUNCT
cana-2405	108	1	similarly	similarly	ADV
cana-2405	108	2	assuming	assume	VERB
cana-2405	108	3	𝑃1	𝑃1	NOUN
cana-2405	108	4	,	,	PUNCT
cana-2405	108	5	𝑃2	𝑃2	PROPN
cana-2405	108	6	are	be	AUX
cana-2405	108	7	pythagorean	pythagorean	PROPN
cana-2405	108	8	fuzzy	fuzzy	ADJ
cana-2405	108	9	left	leave	VERB
cana-2405	108	10	ideal	ideal	ADJ
cana-2405	108	11	,	,	PUNCT
cana-2405	108	12	we	we	PRON
cana-2405	108	13	can	can	AUX
cana-2405	108	14	show	show	VERB
cana-2405	108	15	that	that	SCONJ
cana-2405	108	16	(	(	PUNCT
cana-2405	108	17	𝑃1	𝑃1	NOUN
cana-2405	108	18	∘	∘	NOUN
cana-2405	108	19	𝑃2)(𝑥𝑦	𝑃2)(𝑥𝑦	NOUN
cana-2405	108	20	)	)	PUNCT
cana-2405	108	21	≥	≥	NUM
cana-2405	108	22	(	(	PUNCT
cana-2405	108	23	𝑃1	𝑃1	NOUN
cana-2405	108	24	∘	∘	PROPN
cana-2405	108	25	𝑃2)(𝑦	𝑃2)(𝑦	NOUN
cana-2405	108	26	)	)	PUNCT
cana-2405	108	27	also	also	ADV
cana-2405	108	28	(	(	PUNCT
cana-2405	108	29	𝜇1	𝜇1	PROPN
cana-2405	108	30	∘	∘	NOUN
cana-2405	108	31	𝜇2)(𝑥	𝜇2)(𝑥	NOUN
cana-2405	108	32	)	)	PUNCT
cana-2405	108	33	=	=	SYM
cana-2405	108	34	sup	sup	NOUN
cana-2405	108	35	𝑥≤𝑥1𝑥2	𝑥≤𝑥1𝑥2	NOUN
cana-2405	108	36	{	{	PUNCT
cana-2405	108	37	min{𝜇1(𝑥1	min{𝜇1(𝑥1	PROPN
cana-2405	108	38	)	)	PUNCT
cana-2405	108	39	,	,	PUNCT
cana-2405	108	40	𝜇2(𝑥2	𝜇2(𝑥2	PROPN
cana-2405	108	41	)	)	PUNCT
cana-2405	108	42	}	}	PUNCT
cana-2405	108	43	}	}	PUNCT
cana-2405	108	44	≥	≥	PROPN
cana-2405	108	45	sup	sup	NOUN
cana-2405	108	46	𝑥≤𝑦≤𝑦1𝑦2	𝑥≤𝑦≤𝑦1𝑦2	PROPN
cana-2405	108	47	{	{	PUNCT
cana-2405	108	48	min{𝜇1(𝑦1	min{𝜇1(𝑦1	PROPN
cana-2405	108	49	)	)	PUNCT
cana-2405	108	50	,	,	PUNCT
cana-2405	108	51	𝜇2(𝑦2	𝜇2(𝑦2	NOUN
cana-2405	108	52	)	)	PUNCT
cana-2405	108	53	}	}	PUNCT
cana-2405	108	54	}	}	PUNCT
cana-2405	108	55	communications	communication	NOUN
cana-2405	108	56	on	on	ADP
cana-2405	108	57	applied	apply	VERB
cana-2405	108	58	nonlinear	nonlinear	ADJ
cana-2405	108	59	analysis	analysis	NOUN
cana-2405	108	60	issn	issn	NOUN
cana-2405	108	61	:	:	PUNCT
cana-2405	108	62	1074	1074	NUM
cana-2405	108	63	-	-	PUNCT
cana-2405	108	64	133x	133x	NUM
cana-2405	108	65	vol	vol	NOUN
cana-2405	108	66	32	32	NUM
cana-2405	108	67	no	no	NOUN
cana-2405	108	68	.	.	PUNCT
cana-2405	109	1	2s	2s	NUM
cana-2405	109	2	(	(	PUNCT
cana-2405	109	3	2025	2025	NUM
cana-2405	109	4	)	)	PUNCT
cana-2405	109	5	320	320	NUM
cana-2405	109	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2405	109	7	=	=	PUNCT
cana-2405	109	8	sup	sup	NUM
cana-2405	109	9	𝑦≤𝑦1𝑦2	𝑦≤𝑦1𝑦2	NOUN
cana-2405	109	10	{	{	PUNCT
cana-2405	109	11	min{𝜇1(𝑦1	min{𝜇1(𝑦1	PROPN
cana-2405	109	12	)	)	PUNCT
cana-2405	109	13	,	,	PUNCT
cana-2405	109	14	𝜇2(𝑦2	𝜇2(𝑦2	NOUN
cana-2405	109	15	)	)	PUNCT
cana-2405	109	16	}	}	PUNCT
cana-2405	109	17	}	}	PUNCT
cana-2405	109	18	=	=	SYM
cana-2405	109	19	(	(	PUNCT
cana-2405	109	20	𝜇1	𝜇1	PROPN
cana-2405	109	21	∘	∘	PROPN
cana-2405	109	22	𝜇2)(𝑦	𝜇2)(𝑦	PROPN
cana-2405	109	23	)	)	PUNCT
cana-2405	109	24	and	and	CCONJ
cana-2405	109	25	(	(	PUNCT
cana-2405	109	26	𝜗1	𝜗1	X
cana-2405	109	27	∘	∘	NOUN
cana-2405	109	28	𝜗2)(𝑥	𝜗2)(𝑥	PROPN
cana-2405	109	29	)	)	PUNCT
cana-2405	109	30	=	=	SYM
cana-2405	109	31	inf	inf	PROPN
cana-2405	109	32	𝑥≤𝑥1𝑥2	𝑥≤𝑥1𝑥2	NOUN
cana-2405	109	33	{	{	PUNCT
cana-2405	109	34	max{𝜗1(𝑥1	max{𝜗1(𝑥1	PROPN
cana-2405	109	35	)	)	PUNCT
cana-2405	109	36	,	,	PUNCT
cana-2405	109	37	𝜗2(𝑥2	𝜗2(𝑥2	NOUN
cana-2405	109	38	)	)	PUNCT
cana-2405	109	39	}	}	PUNCT
cana-2405	109	40	}	}	PUNCT
cana-2405	109	41	≤	≤	NUM
cana-2405	109	42	inf	inf	NOUN
cana-2405	109	43	𝑥≤𝑦≤𝑦1𝑦2	𝑥≤𝑦≤𝑦1𝑦2	PROPN
cana-2405	109	44	{	{	PUNCT
cana-2405	109	45	max{𝜗1(𝑦1	max{𝜗1(𝑦1	PROPN
cana-2405	109	46	)	)	PUNCT
cana-2405	109	47	,	,	PUNCT
cana-2405	109	48	𝜗2(𝑦2	𝜗2(𝑦2	NOUN
cana-2405	109	49	)	)	PUNCT
cana-2405	109	50	}	}	PUNCT
cana-2405	109	51	}	}	PUNCT
cana-2405	109	52	=	=	SYM
cana-2405	109	53	inf	inf	PROPN
cana-2405	109	54	𝑦≤𝑦1𝑦2	𝑦≤𝑦1𝑦2	X
cana-2405	109	55	{	{	PUNCT
cana-2405	109	56	max{𝜗1(𝑦1	max{𝜗1(𝑦1	PROPN
cana-2405	109	57	)	)	PUNCT
cana-2405	109	58	,	,	PUNCT
cana-2405	109	59	𝜗2(𝑦2	𝜗2(𝑦2	NOUN
cana-2405	109	60	)	)	PUNCT
cana-2405	109	61	}	}	PUNCT
cana-2405	109	62	}	}	PUNCT
cana-2405	109	63	=	=	SYM
cana-2405	109	64	(	(	PUNCT
cana-2405	109	65	𝜗1	𝜗1	X
cana-2405	109	66	∘	∘	X
cana-2405	109	67	𝜗2)(𝑦	𝜗2)(𝑦	NOUN
cana-2405	109	68	)	)	PUNCT
cana-2405	109	69	hence	hence	ADV
cana-2405	109	70	𝑃1	𝑃1	NOUN
cana-2405	109	71	∘	∘	PROPN
cana-2405	109	72	𝑃2	𝑃2	PROPN
cana-2405	109	73	is	be	AUX
cana-2405	109	74	a	a	DET
cana-2405	109	75	pythagorean	pythagorean	ADJ
cana-2405	109	76	fuzzy	fuzzy	ADJ
cana-2405	109	77	ideal	ideal	NOUN
cana-2405	109	78	of	of	ADP
cana-2405	109	79	𝑆.	𝑆.	PROPN
cana-2405	109	80	definition	definition	NOUN
cana-2405	109	81	3.11	3.11	NUM
cana-2405	109	82	a	a	DET
cana-2405	109	83	pythagorean	pythagorean	ADJ
cana-2405	109	84	fuzzy	fuzzy	ADJ
cana-2405	109	85	subset	subset	NOUN
cana-2405	109	86	𝑃	𝑃	NOUN
cana-2405	109	87	=	=	SYM
cana-2405	109	88	(	(	PUNCT
cana-2405	109	89	𝜇	𝜇	X
cana-2405	109	90	,	,	PUNCT
cana-2405	109	91	𝜗	𝜗	NOUN
cana-2405	109	92	)	)	PUNCT
cana-2405	109	93	is	be	AUX
cana-2405	109	94	called	call	VERB
cana-2405	109	95	a	a	DET
cana-2405	109	96	pythagorean	pythagorean	ADJ
cana-2405	109	97	fuzzy	fuzzy	ADJ
cana-2405	109	98	bi	bi	NOUN
cana-2405	109	99	-	-	NOUN
cana-2405	109	100	ideal	ideal	NOUN
cana-2405	109	101	of	of	ADP
cana-2405	109	102	𝑆	𝑆	PROPN
cana-2405	109	103	,	,	PUNCT
cana-2405	109	104	for	for	ADP
cana-2405	109	105	all	all	PRON
cana-2405	109	106	𝑥	𝑥	PROPN
cana-2405	109	107	,	,	PUNCT
cana-2405	109	108	𝑦	𝑦	NOUN
cana-2405	109	109	,	,	PUNCT
cana-2405	109	110	𝑧	𝑧	PRON
cana-2405	109	111	∈	∈	PROPN
cana-2405	109	112	𝑆.	𝑆.	PROPN
cana-2405	109	113	(	(	PUNCT
cana-2405	109	114	i	i	NOUN
cana-2405	109	115	)	)	PUNCT
cana-2405	110	1	𝜇(𝑥	𝜇(𝑥	PROPN
cana-2405	110	2	+	+	PROPN
cana-2405	110	3	𝑦	𝑦	X
cana-2405	110	4	)	)	PUNCT
cana-2405	110	5	≥	≥	NOUN
cana-2405	110	6	min{𝜇(𝑥	min{𝜇(𝑥	PROPN
cana-2405	110	7	)	)	PUNCT
cana-2405	110	8	,	,	PUNCT
cana-2405	110	9	𝜇(𝑦)};𝜗(𝑥	𝜇(𝑦)};𝜗(𝑥	NOUN
cana-2405	110	10	+	+	CCONJ
cana-2405	110	11	𝑦	𝑦	NOUN
cana-2405	110	12	)	)	PUNCT
cana-2405	110	13	≤	≤	NOUN
cana-2405	110	14	max{𝜗(𝑥	max{𝜗(𝑥	PROPN
cana-2405	110	15	)	)	PUNCT
cana-2405	110	16	,	,	PUNCT
cana-2405	110	17	𝜗(𝑦	𝜗(𝑦	PROPN
cana-2405	110	18	)	)	PUNCT
cana-2405	110	19	}	}	PUNCT
cana-2405	110	20	(	(	PUNCT
cana-2405	110	21	ii	ii	NOUN
cana-2405	110	22	)	)	PUNCT
cana-2405	110	23	𝜇(𝑥𝑦	𝜇(𝑥𝑦	PROPN
cana-2405	110	24	)	)	PUNCT
cana-2405	110	25	≥	≥	NOUN
cana-2405	110	26	min{𝜇(𝑥	min{𝜇(𝑥	PROPN
cana-2405	110	27	)	)	PUNCT
cana-2405	110	28	,	,	PUNCT
cana-2405	110	29	𝜇(𝑦)};𝜗(𝑥𝑦	𝜇(𝑦)};𝜗(𝑥𝑦	NUM
cana-2405	110	30	)	)	PUNCT
cana-2405	110	31	≤	≤	NOUN
cana-2405	110	32	max{𝜗(𝑥	max{𝜗(𝑥	PROPN
cana-2405	110	33	)	)	PUNCT
cana-2405	110	34	,	,	PUNCT
cana-2405	110	35	𝜗(𝑦	𝜗(𝑦	PROPN
cana-2405	110	36	)	)	PUNCT
cana-2405	110	37	}	}	PUNCT
cana-2405	110	38	(	(	PUNCT
cana-2405	110	39	iii	iii	X
cana-2405	110	40	)	)	PUNCT
cana-2405	110	41	𝜇(𝑥𝑦𝑧	𝜇(𝑥𝑦𝑧	PROPN
cana-2405	110	42	)	)	PUNCT
cana-2405	110	43	≥	≥	NOUN
cana-2405	110	44	min{𝜇(𝑥	min{𝜇(𝑥	PROPN
cana-2405	110	45	)	)	PUNCT
cana-2405	110	46	,	,	PUNCT
cana-2405	110	47	𝜇(𝑧)};𝜗(𝑥𝑦𝑧	𝜇(𝑧)};𝜗(𝑥𝑦𝑧	PROPN
cana-2405	110	48	)	)	PUNCT
cana-2405	110	49	≤	≤	NOUN
cana-2405	111	1	max{𝜗(𝑥	max{𝜗(𝑥	PROPN
cana-2405	111	2	)	)	PUNCT
cana-2405	111	3	,	,	PUNCT
cana-2405	111	4	𝜗(𝑧	𝜗(𝑧	NOUN
cana-2405	111	5	)	)	PUNCT
cana-2405	111	6	}	}	PUNCT
cana-2405	111	7	theorem	theorem	VERB
cana-2405	111	8	3.12	3.12	NUM
cana-2405	111	9	intersection	intersection	NOUN
cana-2405	111	10	of	of	ADP
cana-2405	111	11	a	a	DET
cana-2405	111	12	non	non	X
cana-2405	111	13	empty	empty	ADJ
cana-2405	111	14	collection	collection	NOUN
cana-2405	111	15	of	of	ADP
cana-2405	111	16	pythagorean	pythagorean	PROPN
cana-2405	111	17	fuzzy	fuzzy	ADJ
cana-2405	111	18	bi	bi	NOUN
cana-2405	111	19	-	-	NOUN
cana-2405	111	20	ideals	ideal	NOUN
cana-2405	111	21	is	be	AUX
cana-2405	111	22	also	also	ADV
cana-2405	111	23	pythagorean	pythagorean	ADJ
cana-2405	111	24	fuzzy	fuzzy	ADJ
cana-2405	111	25	bi	bi	NOUN
cana-2405	111	26	-	-	NOUN
cana-2405	111	27	ideal	ideal	NOUN
cana-2405	111	28	of	of	ADP
cana-2405	111	29	𝑆.	𝑆.	ADJ
cana-2405	111	30	proof	proof	NOUN
cana-2405	111	31	.	.	PUNCT
cana-2405	112	1	let	let	VERB
cana-2405	112	2	{	{	PUNCT
cana-2405	112	3	𝑃𝑖	𝑃𝑖	VERB
cana-2405	112	4	=	=	SYM
cana-2405	112	5	(	(	PUNCT
cana-2405	112	6	𝜇𝑖	𝜇𝑖	ADP
cana-2405	112	7	,	,	PUNCT
cana-2405	112	8	𝜗𝑖)|𝑖	𝜗𝑖)|𝑖	NOUN
cana-2405	112	9	∈	∈	PROPN
cana-2405	112	10	𝐼	𝐼	PROPN
cana-2405	112	11	}	}	PUNCT
cana-2405	112	12	be	be	AUX
cana-2405	112	13	a	a	DET
cana-2405	112	14	family	family	NOUN
cana-2405	112	15	of	of	ADP
cana-2405	112	16	pythagorean	pythagorean	PROPN
cana-2405	112	17	fuzzy	fuzzy	ADJ
cana-2405	112	18	bi	bi	NOUN
cana-2405	112	19	-	-	NOUN
cana-2405	112	20	ideals	ideal	NOUN
cana-2405	112	21	of	of	ADP
cana-2405	112	22	𝑆	𝑆	PROPN
cana-2405	112	23	and	and	CCONJ
cana-2405	112	24	𝑥	𝑥	PROPN
cana-2405	112	25	,	,	PUNCT
cana-2405	112	26	𝑦	𝑦	PRON
cana-2405	112	27	∈	∈	NOUN
cana-2405	112	28	𝑆.	𝑆.	PROPN
cana-2405	112	29	then	then	ADV
cana-2405	112	30	⋂	⋂	PROPN
cana-2405	112	31	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	112	32	𝜇𝑖(𝑥𝑦𝑧	𝜇𝑖(𝑥𝑦𝑧	PROPN
cana-2405	112	33	)	)	PUNCT
cana-2405	112	34	=	=	SYM
cana-2405	112	35	inf	inf	NOUN
cana-2405	112	36	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	112	37	{	{	PUNCT
cana-2405	112	38	𝜇𝑖(𝑥𝑦𝑧	𝜇𝑖(𝑥𝑦𝑧	PROPN
cana-2405	112	39	)	)	PUNCT
cana-2405	112	40	}	}	PUNCT
cana-2405	112	41	≥	≥	NOUN
cana-2405	112	42	inf	inf	NOUN
cana-2405	112	43	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	112	44	{	{	PUNCT
cana-2405	112	45	min{𝜇𝑖(𝑥	min{𝜇𝑖(𝑥	PROPN
cana-2405	112	46	)	)	PUNCT
cana-2405	112	47	,	,	PUNCT
cana-2405	112	48	𝜇𝑖(𝑧	𝜇𝑖(𝑧	NUM
cana-2405	112	49	)	)	PUNCT
cana-2405	112	50	}	}	PUNCT
cana-2405	112	51	}	}	PUNCT
cana-2405	112	52	=	=	PUNCT
cana-2405	112	53	min{inf	min{inf	VERB
cana-2405	112	54	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	112	55	𝜇𝑖(𝑥	𝜇𝑖(𝑥	NUM
cana-2405	112	56	)	)	PUNCT
cana-2405	112	57	,	,	PUNCT
cana-2405	112	58	inf	inf	NOUN
cana-2405	112	59	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	112	60	𝜇𝑖(𝑧	𝜇𝑖(𝑧	NUM
cana-2405	112	61	)	)	PUNCT
cana-2405	112	62	}	}	PUNCT
cana-2405	112	63	=	=	NOUN
cana-2405	112	64	min{⋂𝑖∈𝐼	min{⋂𝑖∈𝐼	ADJ
cana-2405	112	65	𝜇𝑖(𝑥	𝜇𝑖(𝑥	NUM
cana-2405	112	66	)	)	PUNCT
cana-2405	112	67	,	,	PUNCT
cana-2405	112	68	⋂𝑖∈𝐼	⋂𝑖∈𝐼	VERB
cana-2405	112	69	𝜇𝑖(𝑧	𝜇𝑖(𝑧	NUM
cana-2405	112	70	)	)	PUNCT
cana-2405	112	71	}	}	PUNCT
cana-2405	112	72	.	.	PUNCT
cana-2405	113	1	finally	finally	ADV
cana-2405	113	2	⋂	⋂	PROPN
cana-2405	113	3	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	113	4	𝜗𝑖(𝑥𝑦𝑧	𝜗𝑖(𝑥𝑦𝑧	NOUN
cana-2405	113	5	)	)	PUNCT
cana-2405	113	6	=	=	SYM
cana-2405	113	7	sup	sup	NOUN
cana-2405	113	8	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	113	9	{	{	PUNCT
cana-2405	113	10	𝜗𝑖(𝑥𝑦𝑧	𝜗𝑖(𝑥𝑦𝑧	NOUN
cana-2405	113	11	)	)	PUNCT
cana-2405	113	12	}	}	PUNCT
cana-2405	113	13	≤	≤	NUM
cana-2405	113	14	sup	sup	NOUN
cana-2405	113	15	𝑖∈𝐼	𝑖∈𝐼	PROPN
cana-2405	113	16	{	{	PUNCT
cana-2405	113	17	max{𝜗𝑖(𝑥	max{𝜗𝑖(𝑥	PROPN
cana-2405	113	18	)	)	PUNCT
cana-2405	113	19	,	,	PUNCT
cana-2405	113	20	𝜗𝑖(𝑧	𝜗𝑖(𝑧	NUM
cana-2405	113	21	)	)	PUNCT
cana-2405	113	22	}	}	PUNCT
cana-2405	113	23	}	}	PUNCT
cana-2405	113	24	=	=	PUNCT
cana-2405	113	25	max{sup	max{sup	X
cana-2405	113	26	𝑖∈𝐼	𝑖∈𝐼	X
cana-2405	113	27	𝜗𝑖(𝑥	𝜗𝑖(𝑥	NUM
cana-2405	113	28	)	)	PUNCT
cana-2405	113	29	,	,	PUNCT
cana-2405	113	30	sup	sup	NOUN
cana-2405	113	31	𝑖∈𝐼	𝑖∈𝐼	NUM
cana-2405	113	32	𝜗𝑖(𝑧	𝜗𝑖(𝑧	NUM
cana-2405	113	33	)	)	PUNCT
cana-2405	113	34	}	}	PUNCT
cana-2405	113	35	=	=	SYM
cana-2405	113	36	max{⋂𝑖∈𝐼	max{⋂𝑖∈𝐼	ADJ
cana-2405	113	37	𝜗𝑖(𝑥	𝜗𝑖(𝑥	NUM
cana-2405	113	38	)	)	PUNCT
cana-2405	113	39	,	,	PUNCT
cana-2405	113	40	⋂𝑖∈𝐼	⋂𝑖∈𝐼	NOUN
cana-2405	113	41	𝜗𝑖(𝑧	𝜗𝑖(𝑧	NUM
cana-2405	113	42	)	)	PUNCT
cana-2405	113	43	}	}	PUNCT
cana-2405	113	44	.	.	PUNCT
cana-2405	114	1	hence	hence	ADV
cana-2405	114	2	𝑃𝑖	𝑃𝑖	SCONJ
cana-2405	114	3	is	be	AUX
cana-2405	114	4	a	a	DET
cana-2405	114	5	pythagorean	pythagorean	ADJ
cana-2405	114	6	fuzzy	fuzzy	ADJ
cana-2405	114	7	bi	bi	NOUN
cana-2405	114	8	-	-	NOUN
cana-2405	114	9	ideal	ideal	NOUN
cana-2405	114	10	of	of	ADP
cana-2405	114	11	𝑆.	𝑆.	NOUN
cana-2405	114	12	communications	communication	NOUN
cana-2405	114	13	on	on	ADP
cana-2405	114	14	applied	apply	VERB
cana-2405	114	15	nonlinear	nonlinear	ADJ
cana-2405	114	16	analysis	analysis	NOUN
cana-2405	114	17	issn	issn	NOUN
cana-2405	114	18	:	:	PUNCT
cana-2405	114	19	1074	1074	NUM
cana-2405	114	20	-	-	PUNCT
cana-2405	114	21	133x	133x	NUM
cana-2405	114	22	vol	vol	NOUN
cana-2405	114	23	32	32	NUM
cana-2405	114	24	no	no	NOUN
cana-2405	114	25	.	.	PUNCT
cana-2405	115	1	2s	2s	NUM
cana-2405	115	2	(	(	PUNCT
cana-2405	115	3	2025	2025	NUM
cana-2405	115	4	)	)	PUNCT
cana-2405	115	5	321	321	NUM
cana-2405	115	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2405	115	7	theorem	theorem	VERB
cana-2405	115	8	3.13	3.13	NUM
cana-2405	115	9	let	let	VERB
cana-2405	115	10	𝑃1	𝑃1	NOUN
cana-2405	115	11	and	and	CCONJ
cana-2405	115	12	𝑃2	𝑃2	NOUN
cana-2405	115	13	be	be	AUX
cana-2405	115	14	two	two	NUM
cana-2405	115	15	pythagorean	pythagorean	ADJ
cana-2405	115	16	fuzzy	fuzzy	ADJ
cana-2405	115	17	bi	bi	NOUN
cana-2405	115	18	-	-	NOUN
cana-2405	115	19	ideal	ideal	NOUN
cana-2405	115	20	of	of	ADP
cana-2405	115	21	𝑆.	𝑆.	PROPN
cana-2405	115	22	then	then	ADV
cana-2405	115	23	𝑃	𝑃	NOUN
cana-2405	115	24	is	be	AUX
cana-2405	115	25	a	a	DET
cana-2405	115	26	pythagorean	pythagorean	ADJ
cana-2405	115	27	fuzzy	fuzzy	ADJ
cana-2405	115	28	bi	bi	NOUN
cana-2405	115	29	-	-	NOUN
cana-2405	115	30	ideal	ideal	NOUN
cana-2405	115	31	of	of	ADP
cana-2405	115	32	𝑆.	𝑆.	ADJ
cana-2405	115	33	proof	proof	NOUN
cana-2405	115	34	.	.	PUNCT
cana-2405	116	1	let	let	VERB
cana-2405	116	2	𝑥	𝑥	PRON
cana-2405	116	3	,	,	PUNCT
cana-2405	116	4	𝑦	𝑦	NOUN
cana-2405	116	5	,	,	PUNCT
cana-2405	116	6	𝑧	𝑧	DET
cana-2405	116	7	∈	∈	PROPN
cana-2405	116	8	𝑆	𝑆	PROPN
cana-2405	116	9	(	(	PUNCT
cana-2405	116	10	𝜇1	𝜇1	PROPN
cana-2405	116	11	⋅	⋅	PROPN
cana-2405	116	12	𝜇2)(𝑥	𝜇2)(𝑥	PROPN
cana-2405	116	13	+	+	NUM
cana-2405	116	14	𝑦	𝑦	X
cana-2405	116	15	)	)	PUNCT
cana-2405	116	16	=	=	SYM
cana-2405	116	17	min{𝜇1(𝑥	min{𝜇1(𝑥	PROPN
cana-2405	116	18	+	+	CCONJ
cana-2405	116	19	𝑦	𝑦	NOUN
cana-2405	116	20	)	)	PUNCT
cana-2405	116	21	,	,	PUNCT
cana-2405	116	22	𝜇2(𝑥	𝜇2(𝑥	ADP
cana-2405	116	23	+	+	NUM
cana-2405	116	24	𝑦	𝑦	X
cana-2405	116	25	)	)	PUNCT
cana-2405	116	26	}	}	PUNCT
cana-2405	116	27	≥	≥	NOUN
cana-2405	116	28	min{min{𝜇1(𝑥	min{min{𝜇1(𝑥	NOUN
cana-2405	116	29	)	)	PUNCT
cana-2405	116	30	,	,	PUNCT
cana-2405	116	31	𝜇1(𝑦	𝜇1(𝑦	PROPN
cana-2405	116	32	)	)	PUNCT
cana-2405	116	33	}	}	PUNCT
cana-2405	116	34	,	,	PUNCT
cana-2405	116	35	min{𝜇2(𝑥	min{𝜇2(𝑥	PROPN
cana-2405	116	36	)	)	PUNCT
cana-2405	116	37	,	,	PUNCT
cana-2405	116	38	𝜇2(𝑦	𝜇2(𝑦	PROPN
cana-2405	116	39	)	)	PUNCT
cana-2405	116	40	}	}	PUNCT
cana-2405	116	41	}	}	PUNCT
cana-2405	116	42	=	=	SYM
cana-2405	116	43	min{min{𝜇1(𝑥	min{min{𝜇1(𝑥	NOUN
cana-2405	116	44	)	)	PUNCT
cana-2405	116	45	,	,	PUNCT
cana-2405	116	46	𝜇2(𝑥	𝜇2(𝑥	NOUN
cana-2405	116	47	)	)	PUNCT
cana-2405	116	48	}	}	PUNCT
cana-2405	116	49	,	,	PUNCT
cana-2405	116	50	min{𝜇1(𝑦	min{𝜇1(𝑦	PROPN
cana-2405	116	51	)	)	PUNCT
cana-2405	116	52	,	,	PUNCT
cana-2405	116	53	𝜇2(𝑦	𝜇2(𝑦	PROPN
cana-2405	116	54	)	)	PUNCT
cana-2405	116	55	}	}	PUNCT
cana-2405	116	56	}	}	PUNCT
cana-2405	116	57	=	=	SYM
cana-2405	116	58	min{(𝜇1	min{(𝜇1	PROPN
cana-2405	116	59	⋅	⋅	PROPN
cana-2405	116	60	𝜇2)(𝑥	𝜇2)(𝑥	PROPN
cana-2405	116	61	)	)	PUNCT
cana-2405	116	62	,	,	PUNCT
cana-2405	116	63	(	(	PUNCT
cana-2405	116	64	𝜇1	𝜇1	PROPN
cana-2405	116	65	⋅	⋅	PROPN
cana-2405	116	66	𝜇2)(𝑦	𝜇2)(𝑦	PROPN
cana-2405	116	67	)	)	PUNCT
cana-2405	116	68	}	}	PUNCT
cana-2405	116	69	and	and	CCONJ
cana-2405	116	70	(	(	PUNCT
cana-2405	116	71	𝜗1	𝜗1	X
cana-2405	116	72	⋅	⋅	PROPN
cana-2405	116	73	𝜗2)(𝑥	𝜗2)(𝑥	PROPN
cana-2405	116	74	+	+	NUM
cana-2405	116	75	𝑦	𝑦	X
cana-2405	116	76	)	)	PUNCT
cana-2405	116	77	=	=	SYM
cana-2405	116	78	max{𝜗1(𝑥	max{𝜗1(𝑥	NOUN
cana-2405	116	79	+	+	CCONJ
cana-2405	116	80	𝑦	𝑦	X
cana-2405	116	81	)	)	PUNCT
cana-2405	116	82	,	,	PUNCT
cana-2405	116	83	𝜗2(𝑥	𝜗2(𝑥	X
cana-2405	116	84	+	+	NUM
cana-2405	116	85	𝑦	𝑦	NOUN
cana-2405	116	86	)	)	PUNCT
cana-2405	116	87	}	}	PUNCT
cana-2405	116	88	≤	≤	NUM
cana-2405	116	89	max{max{𝜗1(𝑥	max{max{𝜗1(𝑥	PROPN
cana-2405	116	90	)	)	PUNCT
cana-2405	116	91	,	,	PUNCT
cana-2405	116	92	𝜗1(𝑦	𝜗1(𝑦	PROPN
cana-2405	116	93	)	)	PUNCT
cana-2405	116	94	}	}	PUNCT
cana-2405	116	95	,	,	PUNCT
cana-2405	116	96	max{𝜗2(𝑥	max{𝜗2(𝑥	PROPN
cana-2405	116	97	)	)	PUNCT
cana-2405	116	98	,	,	PUNCT
cana-2405	116	99	𝜗2(𝑦	𝜗2(𝑦	NUM
cana-2405	116	100	)	)	PUNCT
cana-2405	116	101	}	}	PUNCT
cana-2405	116	102	}	}	PUNCT
cana-2405	116	103	=	=	SYM
cana-2405	116	104	max{max{𝜗1(𝑥	max{max{𝜗1(𝑥	PROPN
cana-2405	116	105	)	)	PUNCT
cana-2405	116	106	,	,	PUNCT
cana-2405	116	107	𝜗2(𝑥	𝜗2(𝑥	NOUN
cana-2405	116	108	)	)	PUNCT
cana-2405	116	109	}	}	PUNCT
cana-2405	116	110	,	,	PUNCT
cana-2405	116	111	max{𝜗1(𝑦	max{𝜗1(𝑦	PROPN
cana-2405	116	112	)	)	PUNCT
cana-2405	116	113	,	,	PUNCT
cana-2405	116	114	𝜗2(𝑦	𝜗2(𝑦	NUM
cana-2405	116	115	)	)	PUNCT
cana-2405	116	116	}	}	PUNCT
cana-2405	116	117	}	}	PUNCT
cana-2405	116	118	=	=	SYM
cana-2405	116	119	max{(𝜗1	max{(𝜗1	X
cana-2405	116	120	⋅	⋅	PROPN
cana-2405	116	121	𝜗2)(𝑥	𝜗2)(𝑥	PROPN
cana-2405	116	122	)	)	PUNCT
cana-2405	116	123	,	,	PUNCT
cana-2405	116	124	(	(	PUNCT
cana-2405	116	125	𝜗1	𝜗1	X
cana-2405	116	126	⋅	⋅	PROPN
cana-2405	116	127	𝜗2)(𝑦	𝜗2)(𝑦	PROPN
cana-2405	116	128	)	)	PUNCT
cana-2405	116	129	}	}	PUNCT
cana-2405	117	1	next	next	ADV
cana-2405	117	2	(	(	PUNCT
cana-2405	117	3	𝜇1	𝜇1	PROPN
cana-2405	117	4	⋅	⋅	PROPN
cana-2405	117	5	𝜇2)(𝑥𝑦	𝜇2)(𝑥𝑦	PROPN
cana-2405	117	6	)	)	PUNCT
cana-2405	117	7	=	=	SYM
cana-2405	117	8	min{𝜇1(𝑥𝑦	min{𝜇1(𝑥𝑦	NOUN
cana-2405	117	9	)	)	PUNCT
cana-2405	117	10	,	,	PUNCT
cana-2405	117	11	𝜇2(𝑥𝑦	𝜇2(𝑥𝑦	PROPN
cana-2405	117	12	)	)	PUNCT
cana-2405	117	13	}	}	PUNCT
cana-2405	117	14	≥	≥	NUM
cana-2405	117	15	min{min{𝜇1(𝑥	min{min{𝜇1(𝑥	NOUN
cana-2405	117	16	)	)	PUNCT
cana-2405	117	17	,	,	PUNCT
cana-2405	117	18	𝜇1(𝑦	𝜇1(𝑦	PROPN
cana-2405	117	19	)	)	PUNCT
cana-2405	117	20	}	}	PUNCT
cana-2405	117	21	,	,	PUNCT
cana-2405	117	22	min{𝜇2(𝑥	min{𝜇2(𝑥	PROPN
cana-2405	117	23	)	)	PUNCT
cana-2405	117	24	,	,	PUNCT
cana-2405	117	25	𝜇2(𝑦	𝜇2(𝑦	PROPN
cana-2405	117	26	)	)	PUNCT
cana-2405	117	27	}	}	PUNCT
cana-2405	117	28	}	}	PUNCT
cana-2405	117	29	=	=	SYM
cana-2405	117	30	min{min{𝜇1(𝑥	min{min{𝜇1(𝑥	NOUN
cana-2405	117	31	)	)	PUNCT
cana-2405	117	32	,	,	PUNCT
cana-2405	117	33	𝜇2(𝑥	𝜇2(𝑥	NOUN
cana-2405	117	34	)	)	PUNCT
cana-2405	117	35	}	}	PUNCT
cana-2405	117	36	,	,	PUNCT
cana-2405	117	37	min{𝜇1(𝑦	min{𝜇1(𝑦	PROPN
cana-2405	117	38	)	)	PUNCT
cana-2405	117	39	,	,	PUNCT
cana-2405	117	40	𝜇2(𝑦	𝜇2(𝑦	PROPN
cana-2405	117	41	)	)	PUNCT
cana-2405	117	42	}	}	PUNCT
cana-2405	117	43	}	}	PUNCT
cana-2405	117	44	=	=	SYM
cana-2405	117	45	min{(𝜇1	min{(𝜇1	NOUN
cana-2405	117	46	⋅	⋅	PROPN
cana-2405	117	47	𝜇2(𝑥	𝜇2(𝑥	NOUN
cana-2405	117	48	)	)	PUNCT
cana-2405	117	49	)	)	PUNCT
cana-2405	117	50	,	,	PUNCT
cana-2405	117	51	(	(	PUNCT
cana-2405	117	52	𝜇1	𝜇1	PROPN
cana-2405	117	53	⋅	⋅	PROPN
cana-2405	117	54	𝜇2(𝑦	𝜇2(𝑦	NOUN
cana-2405	117	55	)	)	PUNCT
cana-2405	117	56	)	)	PUNCT
cana-2405	117	57	}	}	PUNCT
cana-2405	117	58	and	and	CCONJ
cana-2405	117	59	(	(	PUNCT
cana-2405	117	60	𝜗1	𝜗1	X
cana-2405	117	61	⋅	⋅	PROPN
cana-2405	117	62	𝜗2)(𝑥𝑦	𝜗2)(𝑥𝑦	PROPN
cana-2405	117	63	)	)	PUNCT
cana-2405	117	64	=	=	SYM
cana-2405	117	65	max{𝜗1(𝑥𝑦	max{𝜗1(𝑥𝑦	PROPN
cana-2405	117	66	)	)	PUNCT
cana-2405	117	67	,	,	PUNCT
cana-2405	117	68	𝜗2(𝑥𝑦	𝜗2(𝑥𝑦	NUM
cana-2405	117	69	)	)	PUNCT
cana-2405	117	70	}	}	PUNCT
cana-2405	117	71	≤	≤	NUM
cana-2405	117	72	max{max{𝜗1(𝑥	max{max{𝜗1(𝑥	PROPN
cana-2405	117	73	)	)	PUNCT
cana-2405	117	74	,	,	PUNCT
cana-2405	117	75	𝜗1(𝑦	𝜗1(𝑦	PROPN
cana-2405	117	76	)	)	PUNCT
cana-2405	117	77	}	}	PUNCT
cana-2405	117	78	,	,	PUNCT
cana-2405	117	79	max{𝜗2(𝑥	max{𝜗2(𝑥	PROPN
cana-2405	117	80	)	)	PUNCT
cana-2405	117	81	,	,	PUNCT
cana-2405	117	82	𝜗2(𝑦	𝜗2(𝑦	NUM
cana-2405	117	83	)	)	PUNCT
cana-2405	117	84	}	}	PUNCT
cana-2405	117	85	}	}	PUNCT
cana-2405	117	86	=	=	SYM
cana-2405	117	87	max{max{𝜗1(𝑥	max{max{𝜗1(𝑥	PROPN
cana-2405	117	88	)	)	PUNCT
cana-2405	117	89	,	,	PUNCT
cana-2405	117	90	𝜗2(𝑥	𝜗2(𝑥	NOUN
cana-2405	117	91	)	)	PUNCT
cana-2405	117	92	}	}	PUNCT
cana-2405	117	93	,	,	PUNCT
cana-2405	117	94	max{𝜗1(𝑦	max{𝜗1(𝑦	PROPN
cana-2405	117	95	)	)	PUNCT
cana-2405	117	96	,	,	PUNCT
cana-2405	117	97	𝜗2(𝑦	𝜗2(𝑦	NUM
cana-2405	117	98	)	)	PUNCT
cana-2405	117	99	}	}	PUNCT
cana-2405	117	100	}	}	PUNCT
cana-2405	117	101	=	=	SYM
cana-2405	117	102	max{(𝜗1	max{(𝜗1	X
cana-2405	117	103	⋅	⋅	PROPN
cana-2405	117	104	𝜗2(𝑥	𝜗2(𝑥	NOUN
cana-2405	117	105	)	)	PUNCT
cana-2405	117	106	)	)	PUNCT
cana-2405	117	107	,	,	PUNCT
cana-2405	117	108	(	(	PUNCT
cana-2405	117	109	𝜗1	𝜗1	X
cana-2405	117	110	⋅	⋅	PROPN
cana-2405	117	111	𝜗2(𝑦	𝜗2(𝑦	NUM
cana-2405	117	112	)	)	PUNCT
cana-2405	117	113	)	)	PUNCT
cana-2405	117	114	}	}	PUNCT
cana-2405	117	115	also	also	ADV
cana-2405	117	116	(	(	PUNCT
cana-2405	117	117	𝜇1	𝜇1	PROPN
cana-2405	117	118	⋅	⋅	PROPN
cana-2405	117	119	𝜇2)(𝑥𝑦𝑧	𝜇2)(𝑥𝑦𝑧	PROPN
cana-2405	117	120	)	)	PUNCT
cana-2405	117	121	=	=	SYM
cana-2405	118	1	min{𝜇1(𝑥𝑦𝑧	min{𝜇1(𝑥𝑦𝑧	VERB
cana-2405	118	2	)	)	PUNCT
cana-2405	118	3	,	,	PUNCT
cana-2405	118	4	𝜇2(𝑥𝑦𝑧	𝜇2(𝑥𝑦𝑧	NOUN
cana-2405	118	5	)	)	PUNCT
cana-2405	118	6	}	}	PUNCT
cana-2405	118	7	≥	≥	NOUN
cana-2405	118	8	min{min{𝜇1(𝑥	min{min{𝜇1(𝑥	NOUN
cana-2405	118	9	)	)	PUNCT
cana-2405	118	10	,	,	PUNCT
cana-2405	118	11	𝜇1(𝑧	𝜇1(𝑧	NUM
cana-2405	118	12	)	)	PUNCT
cana-2405	118	13	}	}	PUNCT
cana-2405	118	14	,	,	PUNCT
cana-2405	118	15	min{𝜇2(𝑥	min{𝜇2(𝑥	PROPN
cana-2405	118	16	)	)	PUNCT
cana-2405	118	17	,	,	PUNCT
cana-2405	118	18	𝜇2(𝑧	𝜇2(𝑧	NOUN
cana-2405	118	19	)	)	PUNCT
cana-2405	118	20	}	}	PUNCT
cana-2405	118	21	}	}	PUNCT
cana-2405	118	22	=	=	SYM
cana-2405	118	23	min{min{𝜇1(𝑥	min{min{𝜇1(𝑥	NOUN
cana-2405	118	24	)	)	PUNCT
cana-2405	118	25	,	,	PUNCT
cana-2405	118	26	𝜇2(𝑥	𝜇2(𝑥	NOUN
cana-2405	118	27	)	)	PUNCT
cana-2405	118	28	}	}	PUNCT
cana-2405	118	29	,	,	PUNCT
cana-2405	118	30	min{𝜇1(𝑧	min{𝜇1(𝑧	PROPN
cana-2405	118	31	)	)	PUNCT
cana-2405	118	32	,	,	PUNCT
cana-2405	118	33	𝜇2(𝑧	𝜇2(𝑧	NOUN
cana-2405	118	34	)	)	PUNCT
cana-2405	118	35	}	}	PUNCT
cana-2405	118	36	}	}	PUNCT
cana-2405	118	37	=	=	SYM
cana-2405	118	38	min{(𝜇1	min{(𝜇1	NOUN
cana-2405	118	39	⋅	⋅	PROPN
cana-2405	118	40	𝜇2(𝑥	𝜇2(𝑥	NOUN
cana-2405	118	41	)	)	PUNCT
cana-2405	118	42	)	)	PUNCT
cana-2405	118	43	,	,	PUNCT
cana-2405	118	44	(	(	PUNCT
cana-2405	118	45	𝜇1	𝜇1	PROPN
cana-2405	118	46	⋅	⋅	PROPN
cana-2405	118	47	𝜇2(𝑥	𝜇2(𝑥	NOUN
cana-2405	118	48	)	)	PUNCT
cana-2405	118	49	)	)	PUNCT
cana-2405	118	50	}	}	PUNCT
cana-2405	118	51	and	and	CCONJ
cana-2405	118	52	(	(	PUNCT
cana-2405	118	53	𝜗1	𝜗1	X
cana-2405	118	54	⋅	⋅	X
cana-2405	118	55	𝜗2)(𝑥𝑦𝑧	𝜗2)(𝑥𝑦𝑧	NOUN
cana-2405	118	56	)	)	PUNCT
cana-2405	118	57	=	=	SYM
cana-2405	118	58	max{𝜗1(𝑥𝑦𝑧	max{𝜗1(𝑥𝑦𝑧	NOUN
cana-2405	118	59	)	)	PUNCT
cana-2405	118	60	,	,	PUNCT
cana-2405	118	61	𝜗2(𝑥𝑦𝑧	𝜗2(𝑥𝑦𝑧	VERB
cana-2405	118	62	)	)	PUNCT
cana-2405	118	63	}	}	PUNCT
cana-2405	118	64	≤	≤	NUM
cana-2405	118	65	max{max{𝜗1(𝑥	max{max{𝜗1(𝑥	PROPN
cana-2405	118	66	)	)	PUNCT
cana-2405	118	67	,	,	PUNCT
cana-2405	118	68	𝜗1(𝑧	𝜗1(𝑧	PROPN
cana-2405	118	69	)	)	PUNCT
cana-2405	118	70	}	}	PUNCT
cana-2405	118	71	,	,	PUNCT
cana-2405	118	72	max{𝜗2(𝑥	max{𝜗2(𝑥	PROPN
cana-2405	118	73	)	)	PUNCT
cana-2405	118	74	,	,	PUNCT
cana-2405	118	75	𝜗2(𝑧	𝜗2(𝑧	PROPN
cana-2405	118	76	)	)	PUNCT
cana-2405	118	77	}	}	PUNCT
cana-2405	118	78	}	}	PUNCT
cana-2405	118	79	=	=	SYM
cana-2405	118	80	max{max{𝜗1(𝑥	max{max{𝜗1(𝑥	PROPN
cana-2405	118	81	)	)	PUNCT
cana-2405	118	82	,	,	PUNCT
cana-2405	118	83	𝜗2(𝑥	𝜗2(𝑥	NOUN
cana-2405	118	84	)	)	PUNCT
cana-2405	118	85	}	}	PUNCT
cana-2405	118	86	,	,	PUNCT
cana-2405	118	87	max{𝜗1(𝑧	max{𝜗1(𝑧	PROPN
cana-2405	118	88	)	)	PUNCT
cana-2405	118	89	,	,	PUNCT
cana-2405	118	90	𝜗2(𝑧	𝜗2(𝑧	PROPN
cana-2405	118	91	)	)	PUNCT
cana-2405	118	92	}	}	PUNCT
cana-2405	118	93	}	}	PUNCT
cana-2405	118	94	=	=	SYM
cana-2405	118	95	max{(𝜗1	max{(𝜗1	X
cana-2405	118	96	⋅	⋅	PROPN
cana-2405	118	97	𝜗2(𝑥	𝜗2(𝑥	NOUN
cana-2405	118	98	)	)	PUNCT
cana-2405	118	99	)	)	PUNCT
cana-2405	118	100	,	,	PUNCT
cana-2405	118	101	(	(	PUNCT
cana-2405	118	102	𝜗1	𝜗1	X
cana-2405	118	103	⋅	⋅	PROPN
cana-2405	118	104	𝜗2(𝑧	𝜗2(𝑧	NOUN
cana-2405	118	105	)	)	PUNCT
cana-2405	118	106	)	)	PUNCT
cana-2405	118	107	}	}	PUNCT
cana-2405	118	108	hence	hence	ADV
cana-2405	118	109	𝑃1	𝑃1	NOUN
cana-2405	118	110	and	and	CCONJ
cana-2405	118	111	𝑃2	𝑃2	NOUN
cana-2405	118	112	is	be	AUX
cana-2405	118	113	a	a	DET
cana-2405	118	114	pythagorean	pythagorean	ADJ
cana-2405	118	115	fuzzy	fuzzy	ADJ
cana-2405	118	116	bi	bi	NOUN
cana-2405	118	117	-	-	NOUN
cana-2405	118	118	ideal	ideal	NOUN
cana-2405	118	119	of	of	ADP
cana-2405	118	120	𝑆.	𝑆.	PROPN
cana-2405	118	121	definition	definition	NOUN
cana-2405	118	122	3.14	3.14	NUM
cana-2405	118	123	the	the	DET
cana-2405	118	124	product	product	NOUN
cana-2405	118	125	of	of	ADP
cana-2405	118	126	𝑃1	𝑃1	NOUN
cana-2405	118	127	and	and	CCONJ
cana-2405	118	128	𝑃2	𝑃2	NOUN
cana-2405	118	129	is	be	AUX
cana-2405	118	130	a	a	DET
cana-2405	118	131	pythagorean	pythagorean	ADJ
cana-2405	118	132	fuzzy	fuzzy	NOUN
cana-2405	118	133	subset	subset	VERB
cana-2405	118	134	𝑃1	𝑃1	NOUN
cana-2405	118	135	∘	∘	PROPN
cana-2405	118	136	𝑃2	𝑃2	PROPN
cana-2405	118	137	:	:	PUNCT
cana-2405	118	138	𝑆	𝑆	PROPN
cana-2405	118	139	→	→	SYM
cana-2405	118	140	[	[	X
cana-2405	118	141	0,1	0,1	NUM
cana-2405	118	142	]	]	PUNCT
cana-2405	118	143	by	by	ADP
cana-2405	118	144	communications	communication	NOUN
cana-2405	118	145	on	on	ADP
cana-2405	118	146	applied	apply	VERB
cana-2405	118	147	nonlinear	nonlinear	ADJ
cana-2405	118	148	analysis	analysis	NOUN
cana-2405	118	149	issn	issn	NOUN
cana-2405	118	150	:	:	PUNCT
cana-2405	118	151	1074	1074	NUM
cana-2405	118	152	-	-	PUNCT
cana-2405	118	153	133x	133x	NUM
cana-2405	118	154	vol	vol	NOUN
cana-2405	118	155	32	32	NUM
cana-2405	118	156	no	no	NOUN
cana-2405	118	157	.	.	PUNCT
cana-2405	119	1	2s	2s	NUM
cana-2405	119	2	(	(	PUNCT
cana-2405	119	3	2025	2025	NUM
cana-2405	119	4	)	)	PUNCT
cana-2405	119	5	322	322	NUM
cana-2405	119	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2405	119	7	(	(	PUNCT
cana-2405	119	8	𝜇1	𝜇1	PROPN
cana-2405	119	9	∘	∘	NOUN
cana-2405	119	10	𝜇2)(𝑎	𝜇2)(𝑎	PROPN
cana-2405	119	11	)	)	PUNCT
cana-2405	119	12	=	=	SYM
cana-2405	119	13	sup	sup	NOUN
cana-2405	119	14	𝑎=𝑏𝑐	𝑎=𝑏𝑐	NOUN
cana-2405	119	15	{	{	PUNCT
cana-2405	119	16	min{𝜇1(𝑏	min{𝜇1(𝑏	PROPN
cana-2405	119	17	)	)	PUNCT
cana-2405	119	18	,	,	PUNCT
cana-2405	119	19	𝜇2(𝑐	𝜇2(𝑐	PROPN
cana-2405	119	20	)	)	PUNCT
cana-2405	119	21	}	}	PUNCT
cana-2405	119	22	}	}	PUNCT
cana-2405	119	23	(	(	PUNCT
cana-2405	119	24	𝜗1	𝜗1	X
cana-2405	119	25	∘	∘	X
cana-2405	119	26	𝜗2)(𝑎	𝜗2)(𝑎	PROPN
cana-2405	119	27	)	)	PUNCT
cana-2405	119	28	=	=	SYM
cana-2405	119	29	inf	inf	NOUN
cana-2405	119	30	𝑎=𝑏𝑐	𝑎=𝑏𝑐	NOUN
cana-2405	119	31	{	{	PUNCT
cana-2405	119	32	max{𝜗1(𝑏	max{𝜗1(𝑏	PROPN
cana-2405	119	33	)	)	PUNCT
cana-2405	119	34	,	,	PUNCT
cana-2405	119	35	𝜗2(𝑐	𝜗2(𝑐	NUM
cana-2405	119	36	)	)	PUNCT
cana-2405	119	37	}	}	PUNCT
cana-2405	119	38	}	}	PUNCT
cana-2405	119	39	theorem	theorem	VERB
cana-2405	119	40	3.15	3.15	NUM
cana-2405	119	41	if	if	SCONJ
cana-2405	119	42	𝑃1	𝑃1	NOUN
cana-2405	119	43	,	,	PUNCT
cana-2405	119	44	𝑃2	𝑃2	PROPN
cana-2405	119	45	be	be	AUX
cana-2405	119	46	any	any	DET
cana-2405	119	47	two	two	NUM
cana-2405	119	48	pythagorean	pythagorean	ADJ
cana-2405	119	49	fuzzy	fuzzy	ADJ
cana-2405	119	50	bi	bi	NOUN
cana-2405	119	51	-	-	NOUN
cana-2405	119	52	ideals	ideal	NOUN
cana-2405	119	53	of	of	ADP
cana-2405	119	54	semiring	semire	VERB
cana-2405	119	55	𝑆	𝑆	PROPN
cana-2405	119	56	,	,	PUNCT
cana-2405	119	57	then	then	ADV
cana-2405	119	58	𝑃1	𝑃1	NOUN
cana-2405	119	59	∘	∘	PROPN
cana-2405	119	60	𝑃2	𝑃2	PROPN
cana-2405	119	61	is	be	AUX
cana-2405	119	62	a	a	DET
cana-2405	119	63	pythagorean	pythagorean	ADJ
cana-2405	119	64	fuzzy	fuzzy	ADJ
cana-2405	119	65	bi	bi	NOUN
cana-2405	119	66	-	-	NOUN
cana-2405	119	67	ideal	ideal	NOUN
cana-2405	119	68	of	of	ADP
cana-2405	119	69	𝑆.	𝑆.	ADJ
cana-2405	119	70	proof	proof	NOUN
cana-2405	119	71	.	.	PUNCT
cana-2405	120	1	let	let	VERB
cana-2405	120	2	𝑃1	𝑃1	NOUN
cana-2405	120	3	,	,	PUNCT
cana-2405	120	4	𝑃2	𝑃2	PROPN
cana-2405	120	5	are	be	AUX
cana-2405	120	6	any	any	DET
cana-2405	120	7	two	two	NUM
cana-2405	120	8	pythagorean	pythagorean	ADJ
cana-2405	120	9	fuzzy	fuzzy	ADJ
cana-2405	120	10	ideals	ideal	NOUN
cana-2405	120	11	of	of	ADP
cana-2405	120	12	semiring	semire	VERB
cana-2405	120	13	𝑆	𝑆	PROPN
cana-2405	120	14	and	and	CCONJ
cana-2405	120	15	𝑥	𝑥	PROPN
cana-2405	120	16	,	,	PUNCT
cana-2405	120	17	𝑦	𝑦	PRON
cana-2405	120	18	∈	∈	NOUN
cana-2405	120	19	𝑆.	𝑆.	PROPN
cana-2405	120	20	then	then	ADV
cana-2405	120	21	(	(	PUNCT
cana-2405	120	22	𝜇1	𝜇1	PROPN
cana-2405	120	23	∘	∘	NOUN
cana-2405	120	24	𝜇2)(𝑥	𝜇2)(𝑥	PROPN
cana-2405	120	25	+	+	NUM
cana-2405	120	26	𝑦	𝑦	NOUN
cana-2405	120	27	)	)	PUNCT
cana-2405	120	28	=	=	SYM
cana-2405	120	29	sup	sup	NOUN
cana-2405	120	30	𝑥+𝑦≤𝑐+𝑑	𝑥+𝑦≤𝑐+𝑑	NOUN
cana-2405	120	31	{	{	PUNCT
cana-2405	120	32	min{𝜇1(𝑐	min{𝜇1(𝑐	PROPN
cana-2405	120	33	)	)	PUNCT
cana-2405	120	34	,	,	PUNCT
cana-2405	120	35	𝜇2(𝑑	𝜇2(𝑑	PROPN
cana-2405	120	36	)	)	PUNCT
cana-2405	120	37	}	}	PUNCT
cana-2405	120	38	}	}	PUNCT
cana-2405	120	39	≥	≥	PROPN
cana-2405	120	40	sup	sup	NOUN
cana-2405	120	41	𝑥+𝑦≤(𝑐1𝑑1)+(𝑐2𝑑2)≤(𝑐1+𝑐2)(𝑑1+𝑑2	𝑥+𝑦≤(𝑐1𝑑1)+(𝑐2𝑑2)≤(𝑐1+𝑐2)(𝑑1+𝑑2	NOUN
cana-2405	120	42	)	)	PUNCT
cana-2405	120	43	{	{	PUNCT
cana-2405	120	44	min{𝜇1(𝑐1	min{𝜇1(𝑐1	NUM
cana-2405	120	45	+	+	NUM
cana-2405	120	46	𝑐2	𝑐2	NOUN
cana-2405	120	47	)	)	PUNCT
cana-2405	120	48	,	,	PUNCT
cana-2405	120	49	𝜇2(𝑑1	𝜇2(𝑑1	ADV
cana-2405	120	50	+	+	CCONJ
cana-2405	120	51	𝑑2	𝑑2	NOUN
cana-2405	120	52	)	)	PUNCT
cana-2405	120	53	}	}	PUNCT
cana-2405	120	54	}	}	PUNCT
cana-2405	120	55	≥	≥	NOUN
cana-2405	120	56	sup{min{𝜇1(𝑐1	sup{min{𝜇1(𝑐1	VERB
cana-2405	120	57	)	)	PUNCT
cana-2405	120	58	,	,	PUNCT
cana-2405	120	59	𝜇1(𝑐1	𝜇1(𝑐1	ADV
cana-2405	120	60	)	)	PUNCT
cana-2405	120	61	}	}	PUNCT
cana-2405	120	62	,	,	PUNCT
cana-2405	120	63	min{𝜇2(𝑑1	min{𝜇2(𝑑1	NOUN
cana-2405	120	64	)	)	PUNCT
cana-2405	120	65	,	,	PUNCT
cana-2405	120	66	𝜇2(𝑑2	𝜇2(𝑑2	NOUN
cana-2405	120	67	)	)	PUNCT
cana-2405	120	68	}	}	PUNCT
cana-2405	120	69	}	}	PUNCT
cana-2405	120	70	=	=	SYM
cana-2405	120	71	min	min	NOUN
cana-2405	120	72	{	{	PUNCT
cana-2405	120	73	sup	sup	NOUN
cana-2405	120	74	𝑥≤𝑐1𝑑1	𝑥≤𝑐1𝑑1	ADV
cana-2405	120	75	{	{	PUNCT
cana-2405	120	76	min{𝜇1(𝑐1	min{𝜇1(𝑐1	NUM
cana-2405	120	77	)	)	PUNCT
cana-2405	120	78	,	,	PUNCT
cana-2405	120	79	𝜇2(𝑑1	𝜇2(𝑑1	NOUN
cana-2405	120	80	)	)	PUNCT
cana-2405	120	81	}	}	PUNCT
cana-2405	120	82	}	}	PUNCT
cana-2405	120	83	,	,	PUNCT
cana-2405	120	84	sup	sup	INTJ
cana-2405	120	85	𝑦≤𝑐2𝑑2	𝑦≤𝑐2𝑑2	ADV
cana-2405	120	86	{	{	PUNCT
cana-2405	120	87	min{𝜇1(𝑐2	min{𝜇1(𝑐2	PROPN
cana-2405	120	88	)	)	PUNCT
cana-2405	120	89	,	,	PUNCT
cana-2405	120	90	𝜇2(𝑑2	𝜇2(𝑑2	NOUN
cana-2405	120	91	)	)	PUNCT
cana-2405	120	92	}	}	PUNCT
cana-2405	120	93	}	}	PUNCT
cana-2405	120	94	}	}	PUNCT
cana-2405	120	95	=	=	SYM
cana-2405	120	96	min{(𝜇1	min{(𝜇1	NOUN
cana-2405	120	97	∘	∘	NOUN
cana-2405	120	98	𝜇2)(𝑥	𝜇2)(𝑥	NOUN
cana-2405	120	99	)	)	PUNCT
cana-2405	120	100	,	,	PUNCT
cana-2405	120	101	(	(	PUNCT
cana-2405	120	102	𝜇1	𝜇1	PROPN
cana-2405	120	103	∘	∘	PROPN
cana-2405	120	104	𝜇2)(𝑦	𝜇2)(𝑦	PROPN
cana-2405	120	105	)	)	PUNCT
cana-2405	120	106	}	}	PUNCT
cana-2405	120	107	also	also	ADV
cana-2405	120	108	(	(	PUNCT
cana-2405	120	109	𝜗1	𝜗1	X
cana-2405	120	110	∘	∘	X
cana-2405	120	111	𝜗2)(𝑥	𝜗2)(𝑥	PROPN
cana-2405	120	112	+	+	NUM
cana-2405	120	113	𝑦	𝑦	X
cana-2405	120	114	)	)	PUNCT
cana-2405	120	115	=	=	SYM
cana-2405	120	116	inf	inf	NOUN
cana-2405	120	117	𝑥+𝑦≤𝑐+𝑑	𝑥+𝑦≤𝑐+𝑑	NOUN
cana-2405	120	118	{	{	PUNCT
cana-2405	120	119	max{𝜗1(𝑐	max{𝜗1(𝑐	PROPN
cana-2405	120	120	)	)	PUNCT
cana-2405	120	121	,	,	PUNCT
cana-2405	120	122	𝜗2(𝑑	𝜗2(𝑑	X
cana-2405	120	123	)	)	PUNCT
cana-2405	120	124	}	}	PUNCT
cana-2405	120	125	}	}	PUNCT
cana-2405	120	126	≤	≤	NUM
cana-2405	120	127	inf	inf	ADJ
cana-2405	120	128	𝑥+𝑦≤(𝑐1𝑑1)+(𝑐2𝑑2)≤(𝑐1+𝑐2)(𝑑1+𝑑2	𝑥+𝑦≤(𝑐1𝑑1)+(𝑐2𝑑2)≤(𝑐1+𝑐2)(𝑑1+𝑑2	NOUN
cana-2405	120	129	)	)	PUNCT
cana-2405	120	130	{	{	PUNCT
cana-2405	120	131	max{𝜗1(𝑐1	max{𝜗1(𝑐1	NOUN
cana-2405	120	132	+	+	NUM
cana-2405	120	133	𝑐2	𝑐2	NOUN
cana-2405	120	134	)	)	PUNCT
cana-2405	120	135	,	,	PUNCT
cana-2405	120	136	𝜗2(𝑑1	𝜗2(𝑑1	ADV
cana-2405	120	137	+	+	CCONJ
cana-2405	120	138	𝑑2	𝑑2	VERB
cana-2405	120	139	)	)	PUNCT
cana-2405	120	140	}	}	PUNCT
cana-2405	120	141	}	}	PUNCT
cana-2405	120	142	≤	≤	PROPN
cana-2405	120	143	inf{max{𝜗1(𝑐1	inf{max{𝜗1(𝑐1	PROPN
cana-2405	120	144	)	)	PUNCT
cana-2405	120	145	,	,	PUNCT
cana-2405	120	146	𝜗1(𝑐2	𝜗1(𝑐2	NOUN
cana-2405	120	147	)	)	PUNCT
cana-2405	120	148	}	}	PUNCT
cana-2405	120	149	,	,	PUNCT
cana-2405	120	150	max{𝜗2(𝑑1	max{𝜗2(𝑑1	PROPN
cana-2405	120	151	)	)	PUNCT
cana-2405	120	152	,	,	PUNCT
cana-2405	120	153	𝜗2(𝑑2	𝜗2(𝑑2	PROPN
cana-2405	120	154	)	)	PUNCT
cana-2405	120	155	}	}	PUNCT
cana-2405	120	156	}	}	PUNCT
cana-2405	120	157	=	=	SYM
cana-2405	120	158	max	max	X
cana-2405	120	159	{	{	PUNCT
cana-2405	120	160	inf	inf	PROPN
cana-2405	120	161	𝑥≤𝑐1𝑑1	𝑥≤𝑐1𝑑1	X
cana-2405	120	162	{	{	PUNCT
cana-2405	120	163	max{𝜗1(𝑐1	max{𝜗1(𝑐1	NOUN
cana-2405	120	164	)	)	PUNCT
cana-2405	120	165	,	,	PUNCT
cana-2405	120	166	𝜗2(𝑑1	𝜗2(𝑑1	NOUN
cana-2405	120	167	)	)	PUNCT
cana-2405	120	168	}	}	PUNCT
cana-2405	120	169	}	}	PUNCT
cana-2405	120	170	,	,	PUNCT
cana-2405	120	171	inf	inf	PROPN
cana-2405	120	172	𝑦≤𝑐2𝑑2	𝑦≤𝑐2𝑑2	PROPN
cana-2405	120	173	{	{	PUNCT
cana-2405	120	174	max{𝜗1(𝑐2	max{𝜗1(𝑐2	PROPN
cana-2405	120	175	)	)	PUNCT
cana-2405	120	176	,	,	PUNCT
cana-2405	120	177	𝜗2(𝑑2	𝜗2(𝑑2	PROPN
cana-2405	120	178	)	)	PUNCT
cana-2405	120	179	}	}	PUNCT
cana-2405	120	180	}	}	PUNCT
cana-2405	120	181	}	}	PUNCT
cana-2405	120	182	=	=	SYM
cana-2405	120	183	max{(𝜗1	max{(𝜗1	NOUN
cana-2405	120	184	∘	∘	NOUN
cana-2405	120	185	𝜗2)(𝑥	𝜗2)(𝑥	PROPN
cana-2405	120	186	)	)	PUNCT
cana-2405	120	187	,	,	PUNCT
cana-2405	120	188	(	(	PUNCT
cana-2405	120	189	𝜗1]𝑐𝑖𝑟𝑐𝜗2)(𝑦	𝜗1]𝑐𝑖𝑟𝑐𝜗2)(𝑦	NOUN
cana-2405	120	190	)	)	PUNCT
cana-2405	120	191	}	}	PUNCT
cana-2405	120	192	now	now	ADV
cana-2405	120	193	let	let	VERB
cana-2405	120	194	as	as	SCONJ
cana-2405	120	195	consider	consider	VERB
cana-2405	120	196	𝑃1	𝑃1	NOUN
cana-2405	120	197	,	,	PUNCT
cana-2405	120	198	𝑃2	𝑃2	PROPN
cana-2405	120	199	are	be	AUX
cana-2405	120	200	pythagorean	pythagorean	PROPN
cana-2405	120	201	fuzzy	fuzzy	ADJ
cana-2405	120	202	right	right	ADJ
cana-2405	120	203	ideals	ideal	NOUN
cana-2405	120	204	and	and	CCONJ
cana-2405	120	205	we	we	PRON
cana-2405	120	206	have	have	VERB
cana-2405	120	207	(	(	PUNCT
cana-2405	120	208	𝜇1	𝜇1	ADV
cana-2405	120	209	∘	∘	NOUN
cana-2405	120	210	𝜇2)(𝑥𝑦	𝜇2)(𝑥𝑦	X
cana-2405	120	211	)	)	PUNCT
cana-2405	121	1	=	=	SYM
cana-2405	121	2	sup	sup	NOUN
cana-2405	121	3	𝑥𝑦≤𝑐𝑑	𝑥𝑦≤𝑐𝑑	PRON
cana-2405	121	4	{	{	PUNCT
cana-2405	121	5	min{𝜇1(𝑐	min{𝜇1(𝑐	PROPN
cana-2405	121	6	)	)	PUNCT
cana-2405	121	7	,	,	PUNCT
cana-2405	121	8	𝜇2(𝑑	𝜇2(𝑑	PROPN
cana-2405	121	9	)	)	PUNCT
cana-2405	121	10	}	}	PUNCT
cana-2405	121	11	}	}	PUNCT
cana-2405	121	12	≥	≥	NUM
cana-2405	121	13	sup	sup	PROPN
cana-2405	121	14	𝑥𝑦≤(𝑥1𝑥2)𝑦	𝑥𝑦≤(𝑥1𝑥2)𝑦	PROPN
cana-2405	121	15	{	{	PUNCT
cana-2405	121	16	min{𝜇1(𝑥1𝑦	min{𝜇1(𝑥1𝑦	PROPN
cana-2405	121	17	)	)	PUNCT
cana-2405	121	18	,	,	PUNCT
cana-2405	121	19	𝜇2(𝑥2𝑦	𝜇2(𝑥2𝑦	PROPN
cana-2405	121	20	)	)	PUNCT
cana-2405	121	21	}	}	PUNCT
cana-2405	121	22	}	}	PUNCT
cana-2405	121	23	≥	≥	PROPN
cana-2405	121	24	sup	sup	NOUN
cana-2405	121	25	𝑥≤(𝑥1𝑥2	𝑥≤(𝑥1𝑥2	NOUN
cana-2405	121	26	)	)	PUNCT
cana-2405	121	27	{	{	PUNCT
cana-2405	121	28	min{𝜇1(𝑥1	min{𝜇1(𝑥1	PROPN
cana-2405	121	29	)	)	PUNCT
cana-2405	121	30	,	,	PUNCT
cana-2405	121	31	𝜇2(𝑥2	𝜇2(𝑥2	PROPN
cana-2405	121	32	)	)	PUNCT
cana-2405	121	33	}	}	PUNCT
cana-2405	121	34	}	}	PUNCT
cana-2405	121	35	=	=	SYM
cana-2405	121	36	(	(	PUNCT
cana-2405	121	37	𝜇1	𝜇1	PROPN
cana-2405	121	38	∘	∘	NOUN
cana-2405	121	39	𝜇2)(𝑥	𝜇2)(𝑥	NOUN
cana-2405	121	40	)	)	PUNCT
cana-2405	121	41	.	.	PUNCT
cana-2405	122	1	and	and	CCONJ
cana-2405	122	2	(	(	PUNCT
cana-2405	122	3	𝜗1	𝜗1	X
cana-2405	122	4	∘	∘	PROPN
cana-2405	122	5	𝜗2)(𝑥𝑦	𝜗2)(𝑥𝑦	PROPN
cana-2405	122	6	)	)	PUNCT
cana-2405	122	7	=	=	SYM
cana-2405	122	8	inf	inf	PROPN
cana-2405	122	9	𝑥𝑦≤𝑐𝑑	𝑥𝑦≤𝑐𝑑	X
cana-2405	122	10	{	{	PUNCT
cana-2405	122	11	max{𝜗1(𝑐	max{𝜗1(𝑐	PROPN
cana-2405	122	12	)	)	PUNCT
cana-2405	122	13	,	,	PUNCT
cana-2405	122	14	𝜗2(𝑑	𝜗2(𝑑	X
cana-2405	122	15	)	)	PUNCT
cana-2405	122	16	}	}	PUNCT
cana-2405	122	17	}	}	PUNCT
cana-2405	122	18	≤	≤	NUM
cana-2405	122	19	inf	inf	ADJ
cana-2405	122	20	𝑥𝑦≤(𝑥1𝑥2)𝑦	𝑥𝑦≤(𝑥1𝑥2)𝑦	NOUN
cana-2405	122	21	{	{	PUNCT
cana-2405	122	22	max{𝜗1(𝑥1𝑦	max{𝜗1(𝑥1𝑦	PROPN
cana-2405	122	23	)	)	PUNCT
cana-2405	122	24	,	,	PUNCT
cana-2405	122	25	𝜗2(𝑥2𝑦	𝜗2(𝑥2𝑦	X
cana-2405	122	26	)	)	PUNCT
cana-2405	122	27	}	}	PUNCT
cana-2405	122	28	}	}	PUNCT
cana-2405	122	29	≤	≤	NUM
cana-2405	122	30	inf	inf	PROPN
cana-2405	122	31	𝑥≤(𝑥1𝑥2	𝑥≤(𝑥1𝑥2	NOUN
cana-2405	122	32	)	)	PUNCT
cana-2405	122	33	{	{	PUNCT
cana-2405	122	34	max{𝜗1(𝑥1	max{𝜗1(𝑥1	PROPN
cana-2405	122	35	)	)	PUNCT
cana-2405	122	36	,	,	PUNCT
cana-2405	122	37	𝜗2(𝑥2	𝜗2(𝑥2	NOUN
cana-2405	122	38	)	)	PUNCT
cana-2405	122	39	}	}	PUNCT
cana-2405	122	40	}	}	PUNCT
cana-2405	122	41	communications	communication	NOUN
cana-2405	122	42	on	on	ADP
cana-2405	122	43	applied	apply	VERB
cana-2405	122	44	nonlinear	nonlinear	ADJ
cana-2405	122	45	analysis	analysis	NOUN
cana-2405	122	46	issn	issn	NOUN
cana-2405	122	47	:	:	PUNCT
cana-2405	122	48	1074	1074	NUM
cana-2405	122	49	-	-	PUNCT
cana-2405	122	50	133x	133x	NUM
cana-2405	122	51	vol	vol	NOUN
cana-2405	122	52	32	32	NUM
cana-2405	122	53	no	no	NOUN
cana-2405	122	54	.	.	PUNCT
cana-2405	123	1	2s	2s	NUM
cana-2405	123	2	(	(	PUNCT
cana-2405	123	3	2025	2025	NUM
cana-2405	123	4	)	)	PUNCT
cana-2405	123	5	323	323	NUM
cana-2405	123	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2405	123	7	=	=	SYM
cana-2405	123	8	(	(	PUNCT
cana-2405	123	9	𝜗1	𝜗1	X
cana-2405	123	10	∘	∘	NOUN
cana-2405	123	11	𝜗2)(𝑥	𝜗2)(𝑥	PROPN
cana-2405	123	12	)	)	PUNCT
cana-2405	123	13	.	.	PUNCT
cana-2405	124	1	similarly	similarly	ADV
cana-2405	124	2	assuming	assume	VERB
cana-2405	124	3	𝑃1	𝑃1	NOUN
cana-2405	124	4	,	,	PUNCT
cana-2405	124	5	𝑃2	𝑃2	PROPN
cana-2405	124	6	are	be	AUX
cana-2405	124	7	pythagorean	pythagorean	PROPN
cana-2405	124	8	fuzzy	fuzzy	ADJ
cana-2405	124	9	left	leave	VERB
cana-2405	124	10	ideal	ideal	ADJ
cana-2405	124	11	,	,	PUNCT
cana-2405	124	12	we	we	PRON
cana-2405	124	13	can	can	AUX
cana-2405	124	14	show	show	VERB
cana-2405	124	15	that	that	SCONJ
cana-2405	124	16	(	(	PUNCT
cana-2405	124	17	𝑃1	𝑃1	NOUN
cana-2405	124	18	∘	∘	NOUN
cana-2405	124	19	𝑃2)(𝑥𝑦	𝑃2)(𝑥𝑦	NOUN
cana-2405	124	20	)	)	PUNCT
cana-2405	124	21	≥	≥	NUM
cana-2405	124	22	(	(	PUNCT
cana-2405	124	23	𝑃1	𝑃1	NOUN
cana-2405	124	24	∘	∘	PROPN
cana-2405	124	25	𝑃2)(𝑦	𝑃2)(𝑦	NOUN
cana-2405	124	26	)	)	PUNCT
cana-2405	124	27	also	also	ADV
cana-2405	124	28	(	(	PUNCT
cana-2405	124	29	𝜇1	𝜇1	PROPN
cana-2405	124	30	∘	∘	NOUN
cana-2405	124	31	𝜇2)(𝑥	𝜇2)(𝑥	NOUN
cana-2405	124	32	)	)	PUNCT
cana-2405	124	33	=	=	SYM
cana-2405	124	34	sup	sup	NOUN
cana-2405	124	35	𝑥≤𝑥1𝑥2	𝑥≤𝑥1𝑥2	NOUN
cana-2405	124	36	{	{	PUNCT
cana-2405	124	37	min{𝜇1(𝑥1	min{𝜇1(𝑥1	PROPN
cana-2405	124	38	)	)	PUNCT
cana-2405	124	39	,	,	PUNCT
cana-2405	124	40	𝜇2(𝑥2	𝜇2(𝑥2	PROPN
cana-2405	124	41	)	)	PUNCT
cana-2405	124	42	}	}	PUNCT
cana-2405	124	43	}	}	PUNCT
cana-2405	124	44	≥	≥	PROPN
cana-2405	124	45	sup	sup	NOUN
cana-2405	124	46	𝑥≤𝑦≤𝑦1𝑦2	𝑥≤𝑦≤𝑦1𝑦2	PROPN
cana-2405	124	47	{	{	PUNCT
cana-2405	124	48	min{𝜇1(𝑦1	min{𝜇1(𝑦1	PROPN
cana-2405	124	49	)	)	PUNCT
cana-2405	124	50	,	,	PUNCT
cana-2405	124	51	𝜇2(𝑦2	𝜇2(𝑦2	NOUN
cana-2405	124	52	)	)	PUNCT
cana-2405	124	53	}	}	PUNCT
cana-2405	124	54	}	}	PUNCT
cana-2405	124	55	=	=	PUNCT
cana-2405	124	56	sup	sup	NUM
cana-2405	124	57	𝑦≤𝑦1𝑦2	𝑦≤𝑦1𝑦2	NOUN
cana-2405	124	58	{	{	PUNCT
cana-2405	124	59	min{𝜇1(𝑦1	min{𝜇1(𝑦1	PROPN
cana-2405	124	60	)	)	PUNCT
cana-2405	124	61	,	,	PUNCT
cana-2405	124	62	𝜇2(𝑦2	𝜇2(𝑦2	NOUN
cana-2405	124	63	)	)	PUNCT
cana-2405	124	64	}	}	PUNCT
cana-2405	124	65	}	}	PUNCT
cana-2405	124	66	=	=	SYM
cana-2405	124	67	(	(	PUNCT
cana-2405	124	68	𝜇1	𝜇1	PROPN
cana-2405	124	69	∘	∘	PROPN
cana-2405	124	70	𝜇2)(𝑦	𝜇2)(𝑦	PROPN
cana-2405	124	71	)	)	PUNCT
cana-2405	124	72	and	and	CCONJ
cana-2405	124	73	(	(	PUNCT
cana-2405	124	74	𝜗1	𝜗1	X
cana-2405	124	75	∘	∘	NOUN
cana-2405	124	76	𝜗2)(𝑥	𝜗2)(𝑥	PROPN
cana-2405	124	77	)	)	PUNCT
cana-2405	124	78	=	=	SYM
cana-2405	124	79	inf	inf	PROPN
cana-2405	124	80	𝑥≤𝑥1𝑥2	𝑥≤𝑥1𝑥2	NOUN
cana-2405	124	81	{	{	PUNCT
cana-2405	124	82	max{𝜗1(𝑥1	max{𝜗1(𝑥1	PROPN
cana-2405	124	83	)	)	PUNCT
cana-2405	124	84	,	,	PUNCT
cana-2405	124	85	𝜗2(𝑥2	𝜗2(𝑥2	NOUN
cana-2405	124	86	)	)	PUNCT
cana-2405	124	87	}	}	PUNCT
cana-2405	124	88	}	}	PUNCT
cana-2405	124	89	≤	≤	NUM
cana-2405	124	90	inf	inf	NOUN
cana-2405	124	91	𝑥≤𝑦≤𝑦1𝑦2	𝑥≤𝑦≤𝑦1𝑦2	PROPN
cana-2405	124	92	{	{	PUNCT
cana-2405	124	93	max{𝜗1(𝑦1	max{𝜗1(𝑦1	PROPN
cana-2405	124	94	)	)	PUNCT
cana-2405	124	95	,	,	PUNCT
cana-2405	124	96	𝜗2(𝑦2	𝜗2(𝑦2	NOUN
cana-2405	124	97	)	)	PUNCT
cana-2405	124	98	}	}	PUNCT
cana-2405	124	99	}	}	PUNCT
cana-2405	124	100	=	=	SYM
cana-2405	124	101	inf	inf	PROPN
cana-2405	124	102	𝑦≤𝑦1𝑦2	𝑦≤𝑦1𝑦2	X
cana-2405	124	103	{	{	PUNCT
cana-2405	124	104	max{𝜗1(𝑦1	max{𝜗1(𝑦1	PROPN
cana-2405	124	105	)	)	PUNCT
cana-2405	124	106	,	,	PUNCT
cana-2405	124	107	𝜗2(𝑦2	𝜗2(𝑦2	NOUN
cana-2405	124	108	)	)	PUNCT
cana-2405	124	109	}	}	PUNCT
cana-2405	124	110	}	}	PUNCT
cana-2405	124	111	=	=	SYM
cana-2405	124	112	(	(	PUNCT
cana-2405	124	113	𝜗1	𝜗1	X
cana-2405	124	114	∘	∘	X
cana-2405	124	115	𝜗2)(𝑦	𝜗2)(𝑦	NOUN
cana-2405	124	116	)	)	PUNCT
cana-2405	124	117	hence	hence	ADV
cana-2405	124	118	𝑃1	𝑃1	NOUN
cana-2405	124	119	∘	∘	PROPN
cana-2405	124	120	𝑃2	𝑃2	PROPN
cana-2405	124	121	is	be	AUX
cana-2405	124	122	a	a	DET
cana-2405	124	123	pythagorean	pythagorean	ADJ
cana-2405	124	124	fuzzy	fuzzy	ADJ
cana-2405	124	125	ideal	ideal	NOUN
cana-2405	124	126	of	of	ADP
cana-2405	124	127	𝑆.	𝑆.	ADJ
cana-2405	124	128	references	reference	NOUN
cana-2405	124	129	[	[	X
cana-2405	124	130	1	1	NUM
cana-2405	124	131	]	]	X
cana-2405	124	132	ahsan	ahsan	PROPN
cana-2405	124	133	,	,	PUNCT
cana-2405	124	134	j.	j.	PROPN
cana-2405	124	135	saifullah	saifullah	PROPN
cana-2405	124	136	,	,	PUNCT
cana-2405	124	137	k.	k.	PROPN
cana-2405	124	138	and	and	CCONJ
cana-2405	124	139	khan	khan	PROPN
cana-2405	124	140	,	,	PUNCT
cana-2405	124	141	m.f.(1993	m.f.(1993	PROPN
cana-2405	124	142	)	)	PUNCT
cana-2405	124	143	,	,	PUNCT
cana-2405	124	144	fuzzy	fuzzy	ADJ
cana-2405	124	145	semirings	semiring	NOUN
cana-2405	124	146	,	,	PUNCT
cana-2405	124	147	fuzzy	fuzzy	ADJ
cana-2405	124	148	sets	set	NOUN
cana-2405	124	149	and	and	CCONJ
cana-2405	124	150	systems	system	NOUN
cana-2405	124	151	,	,	PUNCT
cana-2405	124	152	302	302	NUM
cana-2405	124	153	-	-	SYM
cana-2405	124	154	309	309	NUM
cana-2405	124	155	.	.	PUNCT
cana-2405	125	1	[	[	X
cana-2405	125	2	2	2	NUM
cana-2405	125	3	]	]	PUNCT
cana-2405	125	4	atanassov	atanassov	NOUN
cana-2405	125	5	,	,	PUNCT
cana-2405	125	6	k.	k.	PROPN
cana-2405	125	7	t.	t.	PROPN
cana-2405	125	8	(	(	PUNCT
cana-2405	125	9	1986	1986	NUM
cana-2405	125	10	)	)	PUNCT
cana-2405	125	11	intuitionistic	intuitionistic	ADJ
cana-2405	125	12	fuzzy	fuzzy	ADJ
cana-2405	125	13	sets	set	NOUN
cana-2405	125	14	.	.	PUNCT
cana-2405	126	1	fuzzy	fuzzy	ADJ
cana-2405	126	2	sets	set	NOUN
cana-2405	126	3	and	and	CCONJ
cana-2405	126	4	systems,20	systems,20	NOUN
cana-2405	126	5	,	,	PUNCT
cana-2405	126	6	87	87	NUM
cana-2405	126	7	-	-	SYM
cana-2405	126	8	96	96	NUM
cana-2405	126	9	.	.	PUNCT
cana-2405	127	1	[	[	X
cana-2405	127	2	3	3	NUM
cana-2405	127	3	]	]	PUNCT
cana-2405	127	4	bhargavi	bhargavi	NOUN
cana-2405	127	5	y	y	PROPN
cana-2405	127	6	and	and	CCONJ
cana-2405	127	7	eswarlal	eswarlal	PROPN
cana-2405	127	8	t	t	PROPN
cana-2405	127	9	(	(	PUNCT
cana-2405	127	10	2015	2015	NUM
cana-2405	127	11	)	)	PUNCT
cana-2405	127	12	,	,	PUNCT
cana-2405	127	13	fuzzy	fuzzy	ADJ
cana-2405	127	14	γ	γ	PROPN
cana-2405	127	15	-semirings	-semiring	NOUN
cana-2405	127	16	,	,	PUNCT
cana-2405	127	17	international	international	ADJ
cana-2405	127	18	journal	journal	NOUN
cana-2405	127	19	of	of	ADP
cana-2405	127	20	pure	pure	ADJ
cana-2405	127	21	and	and	CCONJ
cana-2405	127	22	applied	apply	VERB
cana-2405	127	23	mathematics,98,339	mathematics,98,339	PROPN
cana-2405	127	24	-	-	PUNCT
cana-2405	127	25	349	349	NUM
cana-2405	127	26	.	.	PUNCT
cana-2405	128	1	[	[	X
cana-2405	128	2	4	4	NUM
cana-2405	128	3	]	]	X
cana-2405	128	4	dutta	dutta	NOUN
cana-2405	128	5	,	,	PUNCT
cana-2405	128	6	t.	t.	PROPN
cana-2405	128	7	k.	k.	PROPN
cana-2405	128	8	,	,	PUNCT
cana-2405	128	9	sardar	sardar	PROPN
cana-2405	128	10	,	,	PUNCT
cana-2405	128	11	s.	s.	PROPN
cana-2405	128	12	k.	k.	PROPN
cana-2405	128	13	and	and	CCONJ
cana-2405	128	14	goswami	goswami	PROPN
cana-2405	128	15	,	,	PUNCT
cana-2405	128	16	s.	s.	PROPN
cana-2405	128	17	(	(	PUNCT
cana-2405	128	18	2011),operations	2011),operation	NOUN
cana-2405	128	19	on	on	ADP
cana-2405	128	20	fuzzy	fuzzy	ADJ
cana-2405	128	21	ideals	ideal	NOUN
cana-2405	128	22	of	of	ADP
cana-2405	128	23	γ	γ	PROPN
cana-2405	128	24	semirings	semiring	NOUN
cana-2405	128	25	,	,	PUNCT
cana-2405	128	26	proceedings	proceeding	NOUN
cana-2405	128	27	of	of	ADP
cana-2405	128	28	national	national	ADJ
cana-2405	128	29	seminar	seminar	NOUN
cana-2405	128	30	on	on	ADP
cana-2405	128	31	algebra	algebra	NOUN
cana-2405	128	32	,	,	PUNCT
cana-2405	128	33	analysis	analysis	NOUN
cana-2405	128	34	and	and	CCONJ
cana-2405	128	35	discrete	discrete	ADJ
cana-2405	128	36	mathematics	mathematic	NOUN
cana-2405	128	37	arxiv:1101.4791v1	arxiv:1101.4791v1	NOUN
cana-2405	128	38	[	[	X
cana-2405	128	39	math.gm	math.gm	X
cana-2405	128	40	]	]	PUNCT
cana-2405	128	41	.	.	PUNCT
cana-2405	129	1	[	[	X
cana-2405	129	2	5	5	NUM
cana-2405	129	3	]	]	PUNCT
cana-2405	129	4	nobusawa	nobusawa	PROPN
cana-2405	129	5	,	,	PUNCT
cana-2405	129	6	n.(1964	n.(1964	PROPN
cana-2405	129	7	)	)	PUNCT
cana-2405	129	8	,	,	PUNCT
cana-2405	129	9	on	on	ADP
cana-2405	129	10	a	a	DET
cana-2405	129	11	generalization	generalization	NOUN
cana-2405	129	12	of	of	ADP
cana-2405	129	13	the	the	DET
cana-2405	129	14	ring	ring	NOUN
cana-2405	129	15	theory	theory	NOUN
cana-2405	129	16	,	,	PUNCT
cana-2405	129	17	osaka	osaka	PROPN
cana-2405	129	18	j.	j.	PROPN
cana-2405	129	19	math	math	PROPN
cana-2405	129	20	,	,	PUNCT
cana-2405	129	21	1,81	1,81	PROPN
cana-2405	129	22	-	-	SYM
cana-2405	129	23	89	89	NUM
cana-2405	129	24	.	.	PUNCT
cana-2405	130	1	[	[	X
cana-2405	130	2	6	6	NUM
cana-2405	130	3	]	]	SYM
cana-2405	130	4	rao	rao	PROPN
cana-2405	130	5	,	,	PUNCT
cana-2405	130	6	m.	m.	NOUN
cana-2405	130	7	m.	m.	NOUN
cana-2405	130	8	k.(1995),γ	k.(1995),γ	PROPN
cana-2405	130	9	-	-	PUNCT
cana-2405	130	10	semirings	semiring	NOUN
cana-2405	130	11	-	-	PUNCT
cana-2405	130	12	i	i	PROPN
cana-2405	130	13	,	,	PUNCT
cana-2405	130	14	south	south	PROPN
cana-2405	130	15	east	east	PROPN
cana-2405	130	16	asian	asian	PROPN
cana-2405	130	17	bull	bull	PROPN
cana-2405	130	18	.	.	PUNCT
cana-2405	131	1	of	of	ADP
cana-2405	131	2	math	math	NOUN
cana-2405	131	3	.	.	PUNCT
cana-2405	132	1	19	19	NUM
cana-2405	132	2	,	,	PUNCT
cana-2405	132	3	49	49	NUM
cana-2405	132	4	-	-	SYM
cana-2405	132	5	54	54	NUM
cana-2405	132	6	.	.	PUNCT
cana-2405	133	1	[	[	X
cana-2405	133	2	7	7	NUM
cana-2405	133	3	]	]	X
cana-2405	133	4	rosenfeld	rosenfeld	PROPN
cana-2405	133	5	,	,	PUNCT
cana-2405	133	6	a.(1971	a.(1971	PROPN
cana-2405	133	7	)	)	PUNCT
cana-2405	133	8	,	,	PUNCT
cana-2405	133	9	fuzzy	fuzzy	ADJ
cana-2405	133	10	groups	group	NOUN
cana-2405	133	11	,	,	PUNCT
cana-2405	133	12	j.	j.	PROPN
cana-2405	133	13	math	math	PROPN
cana-2405	133	14	analysis	analysis	NOUN
cana-2405	133	15	applications	application	NOUN
cana-2405	133	16	35	35	NUM
cana-2405	133	17	,	,	PUNCT
cana-2405	133	18	512	512	NUM
cana-2405	133	19	-	-	SYM
cana-2405	133	20	519	519	NUM
cana-2405	133	21	.	.	PUNCT
cana-2405	134	1	[	[	X
cana-2405	134	2	8	8	NUM
cana-2405	134	3	]	]	X
cana-2405	134	4	sen	sen	PROPN
cana-2405	134	5	,	,	PUNCT
cana-2405	134	6	m.k	m.k	PROPN
cana-2405	134	7	.	.	PROPN
cana-2405	135	1	and	and	CCONJ
cana-2405	135	2	saha	saha	PROPN
cana-2405	135	3	n.k.(1986	n.k.(1986	PROPN
cana-2405	135	4	)	)	PUNCT
cana-2405	135	5	,	,	PUNCT
cana-2405	135	6	on	on	ADP
cana-2405	135	7	γ	γ	PROPN
cana-2405	135	8	semigroup	semigroup	PROPN
cana-2405	135	9	,	,	PUNCT
cana-2405	135	10	bull	bull	NOUN
cana-2405	135	11	.	.	PUNCT
cana-2405	136	1	calcutta	calcutta	PROPN
cana-2405	136	2	math	math	NOUN
cana-2405	136	3	,	,	PUNCT
cana-2405	136	4	180	180	NUM
cana-2405	136	5	-	-	SYM
cana-2405	136	6	186	186	NUM
cana-2405	136	7	.	.	PUNCT
cana-2405	137	1	[	[	X
cana-2405	137	2	9	9	NUM
cana-2405	137	3	]	]	PUNCT
cana-2405	137	4	sharma	sharma	NOUN
cana-2405	137	5	,	,	PUNCT
cana-2405	137	6	tilak	tilak	PROPN
cana-2405	137	7	raj	raj	PROPN
cana-2405	137	8	and	and	CCONJ
cana-2405	137	9	kumar	kumar	PROPN
cana-2405	137	10	rajesh(2023	rajesh(2023	PROPN
cana-2405	137	11	)	)	PUNCT
cana-2405	137	12	,	,	PUNCT
cana-2405	137	13	intuitionistic	intuitionistic	ADJ
cana-2405	137	14	fuzzy	fuzzy	ADJ
cana-2405	137	15	ideals	ideal	NOUN
cana-2405	137	16	of	of	ADP
cana-2405	137	17	γ	γ	PROPN
cana-2405	137	18	semirings	semiring	NOUN
cana-2405	137	19	,	,	PUNCT
cana-2405	137	20	communicated	communicate	VERB
cana-2405	137	21	to	to	ADP
cana-2405	137	22	springer	springer	NOUN
cana-2405	137	23	proceedings	proceeding	NOUN
cana-2405	137	24	.	.	PUNCT
cana-2405	138	1	[	[	X
cana-2405	138	2	10	10	NUM
cana-2405	138	3	]	]	X
cana-2405	138	4	yager	yager	NOUN
cana-2405	138	5	,	,	PUNCT
cana-2405	138	6	r.r	r.r	PROPN
cana-2405	138	7	,	,	PUNCT
cana-2405	138	8	pythagorean	pythagorean	ADJ
cana-2405	138	9	fuzzy	fuzzy	ADJ
cana-2405	138	10	subsets	subset	NOUN
cana-2405	138	11	,	,	PUNCT
cana-2405	138	12	in	in	ADP
cana-2405	138	13	:	:	PUNCT
cana-2405	138	14	proceedings	proceeding	NOUN
cana-2405	138	15	of	of	ADP
cana-2405	138	16	joint	joint	ADJ
cana-2405	138	17	ifsa	ifsa	PROPN
cana-2405	138	18	world	world	PROPN
cana-2405	138	19	congress	congress	PROPN
cana-2405	138	20	and	and	CCONJ
cana-2405	138	21	nafips	nafip	NOUN
cana-2405	138	22	annual	annual	ADJ
cana-2405	138	23	meeting	meeting	NOUN
cana-2405	138	24	,	,	PUNCT
cana-2405	138	25	edmonton	edmonton	PROPN
cana-2405	138	26	.	.	PUNCT
cana-2405	139	1	canada	canada	PROPN
cana-2405	139	2	,	,	PUNCT
cana-2405	139	3	57	57	NUM
cana-2405	139	4	-	-	PUNCT
cana-2405	139	5	61(2013	61(2013	NUM
cana-2405	139	6	)	)	PUNCT
cana-2405	139	7	.	.	PUNCT
cana-2405	140	1	[	[	X
cana-2405	140	2	11	11	NUM
cana-2405	140	3	]	]	SYM
cana-2405	140	4	yager	yager	NOUN
cana-2405	140	5	,	,	PUNCT
cana-2405	140	6	r.r	r.r	PROPN
cana-2405	140	7	.	.	PROPN
cana-2405	140	8	:	:	PUNCT
cana-2405	140	9	pythagorean	pythagorean	PROPN
cana-2405	140	10	membership	membership	NOUN
cana-2405	140	11	grades	grade	NOUN
cana-2405	140	12	in	in	ADP
cana-2405	140	13	multicriteria	multicriteria	PROPN
cana-2405	140	14	decision	decision	NOUN
cana-2405	140	15	making	making	NOUN
cana-2405	140	16	.	.	PUNCT
cana-2405	141	1	ieee	ieee	PROPN
cana-2405	141	2	trans	trans	PROPN
cana-2405	141	3	.	.	PUNCT
cana-2405	141	4	fuzzy	fuzzy	ADJ
cana-2405	141	5	syst	syst	PROPN
cana-2405	141	6	.	.	PUNCT
cana-2405	142	1	22(4	22(4	NUM
cana-2405	142	2	)	)	PUNCT
cana-2405	142	3	,	,	PUNCT
cana-2405	142	4	958	958	NUM
cana-2405	142	5	-	-	NOUN
cana-2405	142	6	965(2014	965(2014	NUM
cana-2405	142	7	)	)	PUNCT
cana-2405	142	8	.	.	PUNCT
cana-2405	143	1	[	[	X
cana-2405	143	2	12	12	NUM
cana-2405	143	3	]	]	X
cana-2405	143	4	l.	l.	PROPN
cana-2405	143	5	a	a	DET
cana-2405	143	6	zadeh	zadeh	PROPN
cana-2405	143	7	,	,	PUNCT
cana-2405	143	8	fuzzy	fuzzy	ADJ
cana-2405	143	9	sets	set	NOUN
cana-2405	143	10	.	.	PUNCT
cana-2405	144	1	inform	inform	NOUN
cana-2405	144	2	and	and	CCONJ
cana-2405	144	3	control	control	NOUN
cana-2405	144	4	.	.	PUNCT
cana-2405	145	1	8	8	NUM
cana-2405	145	2	(	(	PUNCT
cana-2405	145	3	1965	1965	NUM
cana-2405	145	4	)	)	PUNCT
cana-2405	145	5	338	338	NUM
cana-2405	145	6	-	-	SYM
cana-2405	145	7	353	353	NUM
cana-2405	145	8	.	.	PUNCT