id	sid	tid	token	lemma	pos
cana-3969	1	1	communications	communication	NOUN
cana-3969	1	2	on	on	ADP
cana-3969	1	3	applied	apply	VERB
cana-3969	1	4	nonlinear	nonlinear	ADJ
cana-3969	1	5	analysis	analysis	NOUN
cana-3969	1	6	issn	issn	NOUN
cana-3969	1	7	:	:	PUNCT
cana-3969	1	8	1074	1074	NUM
cana-3969	1	9	-	-	PUNCT
cana-3969	1	10	133x	133x	NUM
cana-3969	1	11	vol	vol	NOUN
cana-3969	1	12	32	32	NUM
cana-3969	1	13	no	no	NOUN
cana-3969	1	14	.	.	PUNCT
cana-3969	2	1	9s	9s	NUM
cana-3969	2	2	(	(	PUNCT
cana-3969	2	3	2025	2025	NUM
cana-3969	2	4	)	)	PUNCT
cana-3969	3	1	635	635	NUM
cana-3969	3	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-3969	3	3	impact	impact	NOUN
cana-3969	3	4	fuzzy	fuzzy	ADJ
cana-3969	3	5	ideal	ideal	ADJ
cana-3969	3	6	extension	extension	NOUN
cana-3969	3	7	in	in	ADP
cana-3969	3	8	terms	term	NOUN
cana-3969	3	9	of	of	ADP
cana-3969	3	10	gamma	gamma	PROPN
cana-3969	3	11	semigroup	semigroup	PROPN
cana-3969	3	12	1vasantha.gadipally	1vasantha.gadipally	ADV
cana-3969	3	13	,	,	PUNCT
cana-3969	3	14	2dr	2dr	ADJ
cana-3969	3	15	.	.	PUNCT
cana-3969	4	1	sri	sri	PROPN
cana-3969	4	2	lakshmi	lakshmi	PROPN
cana-3969	4	3	t	t	PROPN
cana-3969	4	4	1gitam	1gitam	NUM
cana-3969	4	5	,	,	PUNCT
cana-3969	4	6	vishakapatnam	vishakapatnam	PROPN
cana-3969	4	7	,	,	PUNCT
cana-3969	4	8	ap	ap	PROPN
cana-3969	4	9	,	,	PUNCT
cana-3969	4	10	india	india	PROPN
cana-3969	4	11	,	,	PUNCT
cana-3969	4	12	vgadipal@gitam.in	vgadipal@gitam.in	NOUN
cana-3969	4	13	2gitam	2gitam	NUM
cana-3969	4	14	,	,	PUNCT
cana-3969	4	15	vishakapatnam	vishakapatnam	PROPN
cana-3969	4	16	,	,	PUNCT
cana-3969	4	17	ap	ap	PROPN
cana-3969	4	18	,	,	PUNCT
cana-3969	4	19	india	india	PROPN
cana-3969	4	20	,	,	PUNCT
cana-3969	4	21	stalasil@gitam.in	stalasil@gitam.in	PROPN
cana-3969	4	22	article	article	NOUN
cana-3969	4	23	history	history	NOUN
cana-3969	4	24	:	:	PUNCT
cana-3969	4	25	received	receive	VERB
cana-3969	4	26	:	:	PUNCT
cana-3969	4	27	12	12	NUM
cana-3969	4	28	-	-	SYM
cana-3969	4	29	11	11	NUM
cana-3969	4	30	-	-	PUNCT
cana-3969	4	31	2024	2024	NUM
cana-3969	4	32	revised	revise	VERB
cana-3969	4	33	:	:	PUNCT
cana-3969	4	34	24	24	NUM
cana-3969	4	35	-	-	SYM
cana-3969	4	36	12	12	NUM
cana-3969	4	37	-	-	PUNCT
cana-3969	4	38	2024	2024	NUM
cana-3969	4	39	accepted	accept	VERB
cana-3969	4	40	:	:	PUNCT
cana-3969	4	41	16	16	NUM
cana-3969	4	42	-	-	SYM
cana-3969	4	43	01	01	NUM
cana-3969	4	44	-	-	PUNCT
cana-3969	4	45	2025	2025	NUM
cana-3969	4	46	abstract	abstract	NOUN
cana-3969	4	47	:	:	PUNCT
cana-3969	4	48	our	our	PRON
cana-3969	4	49	exploration	exploration	NOUN
cana-3969	4	50	into	into	ADP
cana-3969	4	51	the	the	DET
cana-3969	4	52	properties	property	NOUN
cana-3969	4	53	of	of	ADP
cana-3969	4	54	some	some	DET
cana-3969	4	55	fuzzy	fuzzy	ADJ
cana-3969	4	56	semigroups	semigroup	NOUN
cana-3969	4	57	and	and	CCONJ
cana-3969	4	58	ideals	ideal	NOUN
cana-3969	4	59	is	be	AUX
cana-3969	4	60	a	a	DET
cana-3969	4	61	deeply	deeply	ADV
cana-3969	4	62	collaborative	collaborative	ADJ
cana-3969	4	63	effort	effort	NOUN
cana-3969	4	64	involving	involve	VERB
cana-3969	4	65	the	the	DET
cana-3969	4	66	contributions	contribution	NOUN
cana-3969	4	67	of	of	ADP
cana-3969	4	68	many	many	ADJ
cana-3969	4	69	researchers	researcher	NOUN
cana-3969	4	70	in	in	ADP
cana-3969	4	71	the	the	DET
cana-3969	4	72	field	field	NOUN
cana-3969	4	73	.	.	PUNCT
cana-3969	5	1	this	this	DET
cana-3969	5	2	collective	collective	ADJ
cana-3969	5	3	endeavour	endeavour	NOUN
cana-3969	5	4	,	,	PUNCT
cana-3969	5	5	which	which	PRON
cana-3969	5	6	builds	build	VERB
cana-3969	5	7	on	on	ADP
cana-3969	5	8	the	the	DET
cana-3969	5	9	work	work	NOUN
cana-3969	5	10	of	of	ADP
cana-3969	5	11	our	our	PRON
cana-3969	5	12	peers	peer	NOUN
cana-3969	5	13	,	,	PUNCT
cana-3969	5	14	aims	aim	VERB
cana-3969	5	15	to	to	PART
cana-3969	5	16	deepen	deepen	VERB
cana-3969	5	17	our	our	PRON
cana-3969	5	18	understanding	understanding	NOUN
cana-3969	5	19	of	of	ADP
cana-3969	5	20	fuzzy	fuzzy	ADJ
cana-3969	5	21	ideal	ideal	ADJ
cana-3969	5	22	semigroup	semigroup	NOUN
cana-3969	5	23	and	and	CCONJ
cana-3969	5	24	fuzzy	fuzzy	ADJ
cana-3969	5	25	extension	extension	NOUN
cana-3969	5	26	in	in	ADP
cana-3969	5	27	terms	term	NOUN
cana-3969	5	28	of	of	ADP
cana-3969	5	29	gamma	gamma	NOUN
cana-3969	5	30	semigroups	semigroup	NOUN
cana-3969	5	31	.	.	PUNCT
cana-3969	6	1	using	use	VERB
cana-3969	6	2	a	a	DET
cana-3969	6	3	star	star	NOUN
cana-3969	6	4	zeta	zeta	NOUN
cana-3969	6	5	,	,	PUNCT
cana-3969	6	6	we	we	PRON
cana-3969	6	7	define	define	VERB
cana-3969	6	8	,	,	PUNCT
cana-3969	6	9	characterise	characterise	NOUN
cana-3969	6	10	and	and	CCONJ
cana-3969	6	11	describe	describe	VERB
cana-3969	6	12	the	the	DET
cana-3969	6	13	different	different	ADJ
cana-3969	6	14	classes	class	NOUN
cana-3969	6	15	of	of	ADP
cana-3969	6	16	fuzzy	fuzzy	ADJ
cana-3969	6	17	ideal	ideal	ADJ
cana-3969	6	18	extension	extension	NOUN
cana-3969	6	19	in	in	ADP
cana-3969	6	20	the	the	DET
cana-3969	6	21	gamma	gamma	PROPN
cana-3969	6	22	semigroup	semigroup	PROPN
cana-3969	6	23	,	,	PUNCT
cana-3969	6	24	resulting	result	VERB
cana-3969	6	25	from	from	ADP
cana-3969	6	26	our	our	PRON
cana-3969	6	27	shared	share	VERB
cana-3969	6	28	research	research	NOUN
cana-3969	6	29	efforts	effort	NOUN
cana-3969	6	30	.	.	PUNCT
cana-3969	7	1	investigating	investigate	VERB
cana-3969	7	2	star	star	NOUN
cana-3969	7	3	zeta	zeta	NOUN
cana-3969	7	4	and	and	CCONJ
cana-3969	7	5	fuzzy	fuzzy	ADJ
cana-3969	7	6	ideal	ideal	ADJ
cana-3969	7	7	properties	property	NOUN
cana-3969	7	8	and	and	CCONJ
cana-3969	7	9	their	their	PRON
cana-3969	7	10	results	result	NOUN
cana-3969	7	11	is	be	AUX
cana-3969	7	12	a	a	DET
cana-3969	7	13	testament	testament	NOUN
cana-3969	7	14	to	to	ADP
cana-3969	7	15	the	the	DET
cana-3969	7	16	power	power	NOUN
cana-3969	7	17	of	of	ADP
cana-3969	7	18	shared	shared	ADJ
cana-3969	7	19	knowledge	knowledge	NOUN
cana-3969	7	20	in	in	ADP
cana-3969	7	21	our	our	PRON
cana-3969	7	22	academic	academic	ADJ
cana-3969	7	23	community	community	NOUN
cana-3969	7	24	.	.	PUNCT
cana-3969	8	1	the	the	DET
cana-3969	8	2	description	description	NOUN
cana-3969	8	3	of	of	ADP
cana-3969	8	4	many	many	ADJ
cana-3969	8	5	properties	property	NOUN
cana-3969	8	6	of	of	ADP
cana-3969	8	7	fuzzy	fuzzy	ADJ
cana-3969	8	8	prime	prime	ADJ
cana-3969	8	9	ideal	ideal	NOUN
cana-3969	8	10	and	and	CCONJ
cana-3969	8	11	fuzzy	fuzzy	ADJ
cana-3969	8	12	semiprime	semiprime	NOUN
cana-3969	8	13	in	in	ADP
cana-3969	8	14	the	the	DET
cana-3969	8	15	gamma	gamma	NOUN
cana-3969	8	16	function	function	NOUN
cana-3969	8	17	further	far	ADV
cana-3969	8	18	underscores	underscore	VERB
cana-3969	8	19	the	the	DET
cana-3969	8	20	collaborative	collaborative	ADJ
cana-3969	8	21	nature	nature	NOUN
cana-3969	8	22	of	of	ADP
cana-3969	8	23	scholarly	scholarly	ADJ
cana-3969	8	24	research	research	NOUN
cana-3969	8	25	,	,	PUNCT
cana-3969	8	26	making	make	VERB
cana-3969	8	27	each	each	DET
cana-3969	8	28	member	member	NOUN
cana-3969	8	29	of	of	ADP
cana-3969	8	30	our	our	PRON
cana-3969	8	31	community	community	NOUN
cana-3969	8	32	feel	feel	VERB
cana-3969	8	33	included	include	VERB
cana-3969	8	34	and	and	CCONJ
cana-3969	8	35	valued	value	VERB
cana-3969	8	36	.	.	PUNCT
cana-3969	9	1	keywords	keyword	NOUN
cana-3969	9	2	:	:	PUNCT
cana-3969	9	3	fuzzy	fuzzy	ADJ
cana-3969	9	4	subsemigroups	subsemigroup	NOUN
cana-3969	9	5	,	,	PUNCT
cana-3969	9	6	fuzzy	fuzzy	ADJ
cana-3969	9	7	ideal	ideal	ADJ
cana-3969	9	8	,	,	PUNCT
cana-3969	9	9	fuzzy	fuzzy	ADJ
cana-3969	9	10	prime	prime	ADJ
cana-3969	9	11	ideal	ideal	ADJ
cana-3969	9	12	,	,	PUNCT
cana-3969	9	13	fuzzy	fuzzy	ADJ
cana-3969	9	14	ideal	ideal	ADJ
cana-3969	9	15	extension	extension	NOUN
cana-3969	9	16	semigroup	semigroup	NOUN
cana-3969	9	17	,	,	PUNCT
cana-3969	9	18	star	star	NOUN
cana-3969	9	19	zeta	zeta	NOUN
cana-3969	9	20	of	of	ADP
cana-3969	9	21	fuzzy	fuzzy	ADJ
cana-3969	9	22	semigroup	semigroup	NOUN
cana-3969	9	23	.	.	PUNCT
cana-3969	10	1	1	1	X
cana-3969	10	2	.	.	X
cana-3969	10	3	introduction	introduction	NOUN
cana-3969	10	4	the	the	DET
cana-3969	10	5	work	work	NOUN
cana-3969	10	6	aims	aim	VERB
cana-3969	10	7	to	to	PART
cana-3969	10	8	explain	explain	VERB
cana-3969	10	9	the	the	DET
cana-3969	10	10	terms	term	NOUN
cana-3969	10	11	used	use	VERB
cana-3969	10	12	by	by	ADP
cana-3969	10	13	the	the	DET
cana-3969	10	14	authors	author	NOUN
cana-3969	10	15	and	and	CCONJ
cana-3969	10	16	a	a	DET
cana-3969	10	17	quick	quick	ADJ
cana-3969	10	18	review	review	NOUN
cana-3969	10	19	of	of	ADP
cana-3969	10	20	semigroup	semigroup	PROPN
cana-3969	10	21	theory	theory	NOUN
cana-3969	10	22	.	.	PUNCT
cana-3969	11	1	this	this	DET
cana-3969	11	2	study	study	NOUN
cana-3969	11	3	examined	examine	VERB
cana-3969	11	4	the	the	DET
cana-3969	11	5	fundamental	fundamental	ADJ
cana-3969	11	6	properties	property	NOUN
cana-3969	11	7	and	and	CCONJ
cana-3969	11	8	variations	variation	NOUN
cana-3969	11	9	of	of	ADP
cana-3969	11	10	semigroups	semigroup	NOUN
cana-3969	11	11	.	.	PUNCT
cana-3969	12	1	early	early	ADJ
cana-3969	12	2	authors	author	NOUN
cana-3969	12	3	described	describe	VERB
cana-3969	12	4	an	an	DET
cana-3969	12	5	extensive	extensive	ADJ
cana-3969	12	6	range	range	NOUN
cana-3969	12	7	of	of	ADP
cana-3969	12	8	semigroup	semigroup	ADJ
cana-3969	12	9	kinds	kind	NOUN
cana-3969	12	10	and	and	CCONJ
cana-3969	12	11	verified	verify	VERB
cana-3969	12	12	various	various	ADJ
cana-3969	12	13	procedures	procedure	NOUN
cana-3969	12	14	.	.	PUNCT
cana-3969	13	1	we	we	PRON
cana-3969	13	2	studied	study	VERB
cana-3969	13	3	the	the	DET
cana-3969	13	4	roots	root	NOUN
cana-3969	13	5	of	of	ADP
cana-3969	13	6	semigroups	semigroup	NOUN
cana-3969	13	7	and	and	CCONJ
cana-3969	13	8	worked	work	VERB
cana-3969	13	9	on	on	ADP
cana-3969	13	10	perplexing	perplex	VERB
cana-3969	13	11	assertions	assertion	NOUN
cana-3969	13	12	in	in	ADP
cana-3969	13	13	regular	regular	ADJ
cana-3969	13	14	semigroups	semigroup	NOUN
cana-3969	13	15	.	.	PUNCT
cana-3969	14	1	we	we	PRON
cana-3969	14	2	found	find	VERB
cana-3969	14	3	properties	property	NOUN
cana-3969	14	4	related	relate	VERB
cana-3969	14	5	to	to	ADP
cana-3969	14	6	the	the	DET
cana-3969	14	7	semigroup	semigroup	PROPN
cana-3969	14	8	theory	theory	NOUN
cana-3969	14	9	written	write	VERB
cana-3969	14	10	by	by	ADP
cana-3969	14	11	four	four	NUM
cana-3969	14	12	critical	critical	ADJ
cana-3969	14	13	writers	writer	NOUN
cana-3969	14	14	.	.	PUNCT
cana-3969	15	1	anton	anton	NOUN
cana-3969	15	2	schushewitch	schushewitch	NOUN
cana-3969	15	3	is	be	AUX
cana-3969	15	4	the	the	DET
cana-3969	15	5	first	first	ADJ
cana-3969	15	6	semigroup	semigroup	ADJ
cana-3969	15	7	theorist	theorist	NOUN
cana-3969	15	8	in	in	ADP
cana-3969	15	9	history	history	NOUN
cana-3969	15	10	.	.	PUNCT
cana-3969	16	1	in	in	ADP
cana-3969	16	2	1941	1941	NUM
cana-3969	16	3	,	,	PUNCT
cana-3969	16	4	clifford	clifford	PROPN
cana-3969	16	5	proved	prove	VERB
cana-3969	16	6	that	that	SCONJ
cana-3969	16	7	'	'	PUNCT
cana-3969	16	8	if	if	SCONJ
cana-3969	16	9	s	s	VERB
cana-3969	16	10	is	be	AUX
cana-3969	16	11	a	a	DET
cana-3969	16	12	collection	collection	NOUN
cana-3969	16	13	of	of	ADP
cana-3969	16	14	groups	group	NOUN
cana-3969	16	15	,	,	PUNCT
cana-3969	16	16	then	then	ADV
cana-3969	16	17	it	it	PRON
cana-3969	16	18	is	be	AUX
cana-3969	16	19	a	a	DET
cana-3969	16	20	semilattice	semilattice	NOUN
cana-3969	16	21	of	of	ADP
cana-3969	16	22	completely	completely	ADV
cana-3969	16	23	simple	simple	ADJ
cana-3969	16	24	semigroups	semigroup	NOUN
cana-3969	17	1	[	[	X
cana-3969	17	2	1	1	NUM
cana-3969	17	3	-	-	SYM
cana-3969	17	4	3	3	NUM
cana-3969	17	5	]	]	PUNCT
cana-3969	17	6	.	.	PUNCT
cana-3969	17	7	'	'	PUNCT
cana-3969	18	1	furthermore	furthermore	ADV
cana-3969	18	2	,	,	PUNCT
cana-3969	18	3	he	he	PRON
cana-3969	18	4	proved	prove	VERB
cana-3969	18	5	that	that	SCONJ
cana-3969	18	6	'	'	PUNCT
cana-3969	18	7	a	a	DET
cana-3969	18	8	band	band	NOUN
cana-3969	18	9	is	be	AUX
cana-3969	18	10	a	a	DET
cana-3969	18	11	semilattice	semilattice	NOUN
cana-3969	18	12	of	of	ADP
cana-3969	18	13	rectangular	rectangular	ADJ
cana-3969	18	14	bands	band	NOUN
cana-3969	18	15	.	.	PUNCT
cana-3969	18	16	'	'	PUNCT
cana-3969	19	1	vagner	vagner	NOUN
cana-3969	19	2	did	do	AUX
cana-3969	19	3	,	,	PUNCT
cana-3969	19	4	however	however	ADV
cana-3969	19	5	,	,	PUNCT
cana-3969	19	6	provide	provide	VERB
cana-3969	19	7	the	the	DET
cana-3969	19	8	opposite	opposite	ADJ
cana-3969	19	9	semigroup	semigroup	NOUN
cana-3969	19	10	[	[	X
cana-3969	19	11	4	4	NUM
cana-3969	19	12	-	-	SYM
cana-3969	19	13	7	7	NUM
cana-3969	19	14	]	]	PUNCT
cana-3969	19	15	.	.	PUNCT
cana-3969	20	1	this	this	DET
cana-3969	20	2	paper	paper	NOUN
cana-3969	20	3	introduces	introduce	NOUN
cana-3969	20	4	and	and	CCONJ
cana-3969	20	5	defines	define	VERB
cana-3969	20	6	a	a	DET
cana-3969	20	7	new	new	ADJ
cana-3969	20	8	concept	concept	NOUN
cana-3969	20	9	of	of	ADP
cana-3969	20	10	ṱ	ṱ	NOUN
cana-3969	20	11	-	-	PUNCT
cana-3969	20	12	norms	norm	NOUN
cana-3969	20	13	and	and	CCONJ
cana-3969	20	14	ṧ	ṧ	NOUN
cana-3969	20	15	-	-	NOUN
cana-3969	20	16	norms	norm	NOUN
cana-3969	20	17	(	(	PUNCT
cana-3969	20	18	notations	notation	NOUN
cana-3969	20	19	used	use	VERB
cana-3969	20	20	to	to	PART
cana-3969	20	21	represent	represent	VERB
cana-3969	20	22	certain	certain	ADJ
cana-3969	20	23	operations	operation	NOUN
cana-3969	20	24	in	in	ADP
cana-3969	20	25	fuzzy	fuzzy	ADJ
cana-3969	20	26	set	set	NOUN
cana-3969	20	27	theory	theory	NOUN
cana-3969	20	28	)	)	PUNCT
cana-3969	20	29	and	and	CCONJ
cana-3969	20	30	their	their	PRON
cana-3969	20	31	properties	property	NOUN
cana-3969	20	32	,	,	PUNCT
cana-3969	20	33	which	which	PRON
cana-3969	20	34	we	we	PRON
cana-3969	20	35	denote	denote	VERB
cana-3969	20	36	throughout	throughout	ADP
cana-3969	20	37	the	the	DET
cana-3969	20	38	paper	paper	NOUN
cana-3969	20	39	,	,	PUNCT
cana-3969	20	40	bringing	bring	VERB
cana-3969	20	41	a	a	DET
cana-3969	20	42	novel	novel	ADJ
cana-3969	20	43	perspective	perspective	NOUN
cana-3969	20	44	to	to	ADP
cana-3969	20	45	the	the	DET
cana-3969	20	46	field	field	NOUN
cana-3969	20	47	.	.	PUNCT
cana-3969	21	1	zadeh	zadeh	PROPN
cana-3969	21	2	introduced	introduce	VERB
cana-3969	21	3	the	the	DET
cana-3969	21	4	essential	essential	ADJ
cana-3969	21	5	concept	concept	NOUN
cana-3969	21	6	of	of	ADP
cana-3969	21	7	a	a	DET
cana-3969	21	8	fuzzy	fuzzy	ADJ
cana-3969	21	9	set	set	NOUN
cana-3969	21	10	in	in	ADP
cana-3969	21	11	1965	1965	NUM
cana-3969	21	12	[	[	X
cana-3969	21	13	8,9	8,9	NUM
cana-3969	21	14	]	]	PUNCT
cana-3969	21	15	,	,	PUNCT
cana-3969	21	16	leading	lead	VERB
cana-3969	21	17	to	to	ADP
cana-3969	21	18	insightful	insightful	ADJ
cana-3969	21	19	discoveries	discovery	NOUN
cana-3969	21	20	and	and	CCONJ
cana-3969	21	21	practical	practical	ADJ
cana-3969	21	22	uses	use	NOUN
cana-3969	21	23	in	in	ADP
cana-3969	21	24	various	various	ADJ
cana-3969	21	25	scientific	scientific	ADJ
cana-3969	21	26	fields	field	NOUN
cana-3969	21	27	.	.	PUNCT
cana-3969	22	1	this	this	DET
cana-3969	22	2	seminal	seminal	ADJ
cana-3969	22	3	work	work	NOUN
cana-3969	22	4	paved	pave	VERB
cana-3969	22	5	the	the	DET
cana-3969	22	6	way	way	NOUN
cana-3969	22	7	for	for	ADP
cana-3969	22	8	numerous	numerous	ADJ
cana-3969	22	9	writers	writer	NOUN
cana-3969	22	10	who	who	PRON
cana-3969	22	11	wrote	write	VERB
cana-3969	22	12	after	after	ADP
cana-3969	22	13	that	that	DET
cana-3969	22	14	time	time	NOUN
cana-3969	22	15	,	,	PUNCT
cana-3969	22	16	confirming	confirm	VERB
cana-3969	22	17	the	the	DET
cana-3969	22	18	necessity	necessity	NOUN
cana-3969	22	19	and	and	CCONJ
cana-3969	22	20	value	value	NOUN
cana-3969	22	21	of	of	ADP
cana-3969	22	22	the	the	DET
cana-3969	22	23	concept	concept	NOUN
cana-3969	22	24	.	.	PUNCT
cana-3969	23	1	rosenfeld	rosenfeld	PROPN
cana-3969	23	2	extended	extend	VERB
cana-3969	23	3	several	several	ADJ
cana-3969	23	4	group	group	NOUN
cana-3969	23	5	results	result	NOUN
cana-3969	23	6	to	to	PART
cana-3969	23	7	include	include	VERB
cana-3969	23	8	ambiguous	ambiguous	ADJ
cana-3969	23	9	groupings	grouping	NOUN
cana-3969	23	10	.	.	PUNCT
cana-3969	24	1	additionally	additionally	ADV
cana-3969	24	2	,	,	PUNCT
cana-3969	24	3	he	he	PRON
cana-3969	24	4	suggested	suggest	VERB
cana-3969	24	5	noting	note	VERB
cana-3969	24	6	fuzzy	fuzzy	ADJ
cana-3969	24	7	groups	group	NOUN
cana-3969	24	8	[	[	X
cana-3969	24	9	10	10	NUM
cana-3969	24	10	]	]	PUNCT
cana-3969	24	11	.	.	PUNCT
cana-3969	25	1	wu	wu	PROPN
cana-3969	25	2	studied	study	VERB
cana-3969	25	3	and	and	CCONJ
cana-3969	25	4	introduced	introduce	VERB
cana-3969	25	5	traditional	traditional	ADJ
cana-3969	25	6	fuzzy	fuzzy	ADJ
cana-3969	25	7	subgroups	subgroup	NOUN
cana-3969	25	8	.	.	PUNCT
cana-3969	26	1	rosenfeld	rosenfeld	PROPN
cana-3969	26	2	also	also	ADV
cana-3969	26	3	demonstrated	demonstrate	VERB
cana-3969	26	4	that	that	SCONJ
cana-3969	26	5	the	the	DET
cana-3969	26	6	fuzzy	fuzzy	ADJ
cana-3969	26	7	subgroup	subgroup	NOUN
cana-3969	26	8	's	's	PART
cana-3969	26	9	homomorphic	homomorphic	ADJ
cana-3969	26	10	image	image	NOUN
cana-3969	26	11	equals	equal	VERB
cana-3969	26	12	one	one	NUM
cana-3969	26	13	.	.	PUNCT
cana-3969	27	1	therefore	therefore	ADV
cana-3969	27	2	,	,	PUNCT
cana-3969	27	3	anthony	anthony	PROPN
cana-3969	27	4	and	and	CCONJ
cana-3969	27	5	sherwood	sherwood	PROPN
cana-3969	27	6	should	should	AUX
cana-3969	27	7	have	have	AUX
cana-3969	27	8	utilised	utilise	VERB
cana-3969	27	9	these	these	DET
cana-3969	27	10	properties	property	NOUN
cana-3969	27	11	to	to	PART
cana-3969	27	12	study	study	VERB
cana-3969	27	13	fuzzy	fuzzy	ADJ
cana-3969	27	14	homomorphisms	homomorphism	NOUN
cana-3969	27	15	.	.	PUNCT
cana-3969	28	1	initially	initially	ADV
cana-3969	28	2	,	,	PUNCT
cana-3969	28	3	authors	authors	PROPN
cana-3969	28	4	imtiaz	imtiaz	PROPN
cana-3969	28	5	,	,	PUNCT
cana-3969	28	6	a.	a.	PROPN
cana-3969	28	7	,	,	PUNCT
cana-3969	28	8	alolaiyan	alolaiyan	PROPN
cana-3969	28	9	,	,	PUNCT
cana-3969	28	10	h.	h.	PROPN
cana-3969	28	11	,	,	PUNCT
cana-3969	28	12	and	and	CCONJ
cana-3969	28	13	shuaib	shuaib	NOUN
cana-3969	28	14	investigated	investigate	VERB
cana-3969	28	15	the	the	DET
cana-3969	28	16	idea	idea	NOUN
cana-3969	28	17	of	of	ADP
cana-3969	28	18	applications	application	NOUN
cana-3969	28	19	of	of	ADP
cana-3969	28	20	conjunctive	conjunctive	ADJ
cana-3969	28	21	complex	complex	ADJ
cana-3969	28	22	fuzzy	fuzzy	ADJ
cana-3969	28	23	or	or	CCONJ
cana-3969	28	24	vague	vague	ADJ
cana-3969	28	25	cosets	coset	NOUN
cana-3969	28	26	and	and	CCONJ
cana-3969	28	27	mailto:vgadipal@gitam.in	mailto:vgadipal@gitam.in	PROPN
cana-3969	28	28	communications	communication	NOUN
cana-3969	28	29	on	on	ADP
cana-3969	28	30	applied	apply	VERB
cana-3969	28	31	nonlinear	nonlinear	ADJ
cana-3969	28	32	analysis	analysis	NOUN
cana-3969	28	33	issn	issn	NOUN
cana-3969	28	34	:	:	PUNCT
cana-3969	28	35	1074	1074	NUM
cana-3969	28	36	-	-	PUNCT
cana-3969	28	37	133x	133x	NUM
cana-3969	28	38	vol	vol	NOUN
cana-3969	28	39	32	32	NUM
cana-3969	28	40	no	no	NOUN
cana-3969	28	41	.	.	PUNCT
cana-3969	29	1	9s	9s	NUM
cana-3969	29	2	(	(	PUNCT
cana-3969	29	3	2025	2025	NUM
cana-3969	29	4	)	)	PUNCT
cana-3969	29	5	636	636	NUM
cana-3969	29	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3969	29	7	their	their	PRON
cana-3969	29	8	interrelation	interrelation	NOUN
cana-3969	29	9	within	within	ADP
cana-3969	29	10	fuzzy	fuzzy	ADJ
cana-3969	29	11	normal	normal	ADJ
cana-3969	29	12	subgroups	subgroup	NOUN
cana-3969	29	13	,	,	PUNCT
cana-3969	29	14	proving	prove	VERB
cana-3969	29	15	how	how	SCONJ
cana-3969	29	16	impact	impact	NOUN
cana-3969	29	17	in	in	ADP
cana-3969	29	18	sylow	sylow	NOUN
cana-3969	29	19	theory[11	theory[11	NOUN
cana-3969	29	20	-	-	SYM
cana-3969	29	21	19	19	NUM
cana-3969	29	22	]	]	PUNCT
cana-3969	29	23	.	.	PUNCT
cana-3969	30	1	these	these	DET
cana-3969	30	2	historical	historical	ADJ
cana-3969	30	3	developments	development	NOUN
cana-3969	30	4	in	in	ADP
cana-3969	30	5	the	the	DET
cana-3969	30	6	field	field	NOUN
cana-3969	30	7	of	of	ADP
cana-3969	30	8	fuzzy	fuzzy	ADJ
cana-3969	30	9	set	set	NOUN
cana-3969	30	10	theory	theory	NOUN
cana-3969	30	11	and	and	CCONJ
cana-3969	30	12	algebraic	algebraic	ADJ
cana-3969	30	13	structures	structure	NOUN
cana-3969	30	14	have	have	AUX
cana-3969	30	15	laid	lay	VERB
cana-3969	30	16	the	the	DET
cana-3969	30	17	foundation	foundation	NOUN
cana-3969	30	18	for	for	ADP
cana-3969	30	19	our	our	PRON
cana-3969	30	20	current	current	ADJ
cana-3969	30	21	research	research	NOUN
cana-3969	30	22	.	.	PUNCT
cana-3969	31	1	for	for	ADP
cana-3969	31	2	example	example	NOUN
cana-3969	31	3	,	,	PUNCT
cana-3969	31	4	the	the	DET
cana-3969	31	5	introduction	introduction	NOUN
cana-3969	31	6	of	of	ADP
cana-3969	31	7	fuzzy	fuzzy	ADJ
cana-3969	31	8	(	(	PUNCT
cana-3969	31	9	subgroupoids	subgroupoid	NOUN
cana-3969	31	10	)	)	PUNCT
cana-3969	31	11	subgroups	subgroup	NOUN
cana-3969	31	12	and	and	CCONJ
cana-3969	31	13	fuzzy	fuzzy	ADJ
cana-3969	31	14	(	(	PUNCT
cana-3969	31	15	left	left	ADJ
cana-3969	31	16	,	,	PUNCT
cana-3969	31	17	fight	fight	NOUN
cana-3969	31	18	)	)	PUNCT
cana-3969	31	19	ideals	ideal	NOUN
cana-3969	31	20	in	in	ADP
cana-3969	31	21	the	the	DET
cana-3969	31	22	seminal	seminal	ADJ
cana-3969	31	23	publication	publication	NOUN
cana-3969	31	24	of	of	ADP
cana-3969	31	25	[	[	PUNCT
cana-3969	31	26	20	20	NUM
cana-3969	31	27	-	-	SYM
cana-3969	31	28	24	24	NUM
cana-3969	31	29	]	]	PUNCT
cana-3969	31	30	yiarayong	yiarayong	NOUN
cana-3969	31	31	marked	mark	VERB
cana-3969	31	32	the	the	DET
cana-3969	31	33	beginning	beginning	NOUN
cana-3969	31	34	of	of	ADP
cana-3969	31	35	the	the	DET
cana-3969	31	36	study	study	NOUN
cana-3969	31	37	of	of	ADP
cana-3969	31	38	fuzzy	fuzzy	ADJ
cana-3969	31	39	algebraic	algebraic	ADJ
cana-3969	31	40	structures	structure	NOUN
cana-3969	31	41	.	.	PUNCT
cana-3969	32	1	fuzzy	fuzzy	ADJ
cana-3969	32	2	set	set	NOUN
cana-3969	32	3	theory	theory	NOUN
cana-3969	32	4	has	have	AUX
cana-3969	32	5	now	now	ADV
cana-3969	32	6	been	be	AUX
cana-3969	32	7	extended	extend	VERB
cana-3969	32	8	to	to	ADP
cana-3969	32	9	semigroups	semigroup	NOUN
cana-3969	32	10	by	by	ADP
cana-3969	32	11	several	several	ADJ
cana-3969	32	12	writers	writer	NOUN
cana-3969	32	13	.	.	PUNCT
cana-3969	33	1	for	for	ADP
cana-3969	33	2	instance	instance	NOUN
cana-3969	33	3	,	,	PUNCT
cana-3969	33	4	mursaleen	mursaleen	PROPN
cana-3969	33	5	,	,	PUNCT
cana-3969	33	6	srivastava	srivastava	PROPN
cana-3969	33	7	,	,	PUNCT
cana-3969	33	8	and	and	CCONJ
cana-3969	33	9	sunil	sunil	NOUN
cana-3969	33	10	[	[	X
cana-3969	33	11	32	32	NUM
cana-3969	33	12	]	]	PUNCT
cana-3969	33	13	investigated	investigate	VERB
cana-3969	33	14	particular	particular	ADJ
cana-3969	33	15	novel	novel	ADJ
cana-3969	33	16	spaces	space	NOUN
cana-3969	33	17	of	of	ADP
cana-3969	33	18	statistically	statistically	ADV
cana-3969	33	19	convergent	convergent	ADJ
cana-3969	33	20	and	and	CCONJ
cana-3969	33	21	strongly	strongly	ADV
cana-3969	33	22	summable	summable	ADJ
cana-3969	33	23	sequences	sequence	NOUN
cana-3969	33	24	of	of	ADP
cana-3969	33	25	fuzzy	fuzzy	ADJ
cana-3969	33	26	numbers.[25	numbers.[25	PROPN
cana-3969	33	27	-	-	SYM
cana-3969	33	28	31	31	NUM
cana-3969	33	29	]	]	PUNCT
cana-3969	33	30	jun	jun	PROPN
cana-3969	33	31	,	,	PUNCT
cana-3969	33	32	song	song	NOUN
cana-3969	33	33	,	,	PUNCT
cana-3969	33	34	and	and	CCONJ
cana-3969	33	35	muhiuddin	muhiuddin	AUX
cana-3969	33	36	established	establish	VERB
cana-3969	33	37	the	the	DET
cana-3969	33	38	concept	concept	NOUN
cana-3969	33	39	of	of	ADP
cana-3969	33	40	hybrid	hybrid	ADJ
cana-3969	33	41	structure	structure	NOUN
cana-3969	33	42	in	in	ADP
cana-3969	33	43	a	a	DET
cana-3969	33	44	set	set	NOUN
cana-3969	33	45	of	of	ADP
cana-3969	33	46	parameters	parameter	NOUN
cana-3969	33	47	over	over	ADP
cana-3969	33	48	an	an	DET
cana-3969	33	49	initial	initial	ADJ
cana-3969	33	50	universe	universe	NOUN
cana-3969	33	51	set	set	VERB
cana-3969	33	52	as	as	ADP
cana-3969	33	53	a	a	DET
cana-3969	33	54	parallel	parallel	ADJ
cana-3969	33	55	circuit	circuit	NOUN
cana-3969	33	56	of	of	ADP
cana-3969	33	57	fuzzy	fuzzy	ADJ
cana-3969	33	58	sets	set	NOUN
cana-3969	33	59	and	and	CCONJ
cana-3969	33	60	soft	soft	ADJ
cana-3969	33	61	sets	set	NOUN
cana-3969	33	62	(	(	PUNCT
cana-3969	33	63	or	or	CCONJ
cana-3969	33	64	hesitant	hesitant	ADJ
cana-3969	33	65	fuzzy	fuzzy	ADJ
cana-3969	33	66	sets	set	NOUN
cana-3969	33	67	)	)	PUNCT
cana-3969	34	1	[	[	X
cana-3969	34	2	34	34	NUM
cana-3969	34	3	-	-	SYM
cana-3969	34	4	38	38	NUM
cana-3969	34	5	]	]	PUNCT
cana-3969	34	6	.	.	PUNCT
cana-3969	35	1	they	they	PRON
cana-3969	35	2	used	use	VERB
cana-3969	35	3	it	it	PRON
cana-3969	35	4	on	on	ADP
cana-3969	35	5	linear	linear	ADJ
cana-3969	35	6	spaces	space	NOUN
cana-3969	35	7	and	and	CCONJ
cana-3969	35	8	bck	bck	VERB
cana-3969	35	9	/	/	SYM
cana-3969	35	10	bci	bci	PROPN
cana-3969	35	11	algebras	algebra	NOUN
cana-3969	35	12	.	.	PUNCT
cana-3969	36	1	a	a	DET
cana-3969	36	2	russian	russian	ADJ
cana-3969	36	3	scientist	scientist	NOUN
cana-3969	36	4	(	(	PUNCT
cana-3969	36	5	1999	1999	NUM
cana-3969	36	6	)	)	PUNCT
cana-3969	36	7	presented	present	VERB
cana-3969	36	8	the	the	DET
cana-3969	36	9	soft	soft	ADJ
cana-3969	36	10	set	set	NOUN
cana-3969	36	11	theory	theory	NOUN
cana-3969	36	12	as	as	ADP
cana-3969	36	13	a	a	DET
cana-3969	36	14	novel	novel	ADJ
cana-3969	36	15	mathematical	mathematical	ADJ
cana-3969	36	16	technique	technique	NOUN
cana-3969	36	17	for	for	ADP
cana-3969	36	18	handling	handle	VERB
cana-3969	36	19	uncertainties.[33	uncertainties.[33	NOUN
cana-3969	36	20	]	]	X
cana-3969	36	21	torra	torra	PROPN
cana-3969	36	22	,	,	PUNCT
cana-3969	36	23	2010	2010	NUM
cana-3969	36	24	;	;	PUNCT
cana-3969	36	25	torra	torra	PROPN
cana-3969	36	26	&	&	CCONJ
cana-3969	36	27	narukawa	narukawa	PROPN
cana-3969	36	28	,	,	PUNCT
cana-3969	36	29	a	a	DET
cana-3969	36	30	generalisation	generalisation	NOUN
cana-3969	36	31	of	of	ADP
cana-3969	36	32	zadeh	zadeh	PROPN
cana-3969	36	33	’s	’s	PART
cana-3969	36	34	fuzzy	fuzzy	ADJ
cana-3969	36	35	or	or	CCONJ
cana-3969	36	36	vague	vague	ADJ
cana-3969	36	37	set	set	NOUN
cana-3969	36	38	.	.	PUNCT
cana-3969	37	1	the	the	DET
cana-3969	37	2	hesitant	hesitant	ADJ
cana-3969	37	3	fuzzy	fuzzy	ADJ
cana-3969	37	4	set	set	NOUN
cana-3969	37	5	is	be	AUX
cana-3969	37	6	handy	handy	ADJ
cana-3969	37	7	for	for	ADP
cana-3969	37	8	expressing	express	VERB
cana-3969	37	9	people	people	NOUN
cana-3969	37	10	's	's	PART
cana-3969	37	11	hesitancy	hesitancy	NOUN
cana-3969	37	12	in	in	ADP
cana-3969	37	13	daily	daily	ADJ
cana-3969	37	14	life	life	NOUN
cana-3969	37	15	,	,	PUNCT
cana-3969	37	16	and	and	CCONJ
cana-3969	37	17	it	it	PRON
cana-3969	37	18	is	be	AUX
cana-3969	37	19	a	a	DET
cana-3969	37	20	convenient	convenient	ADJ
cana-3969	37	21	tool	tool	NOUN
cana-3969	37	22	to	to	PART
cana-3969	37	23	deal	deal	VERB
cana-3969	37	24	with	with	ADP
cana-3969	37	25	uncertainty	uncertainty	NOUN
cana-3969	37	26	,	,	PUNCT
cana-3969	37	27	which	which	PRON
cana-3969	37	28	can	can	AUX
cana-3969	37	29	be	be	AUX
cana-3969	37	30	accurately	accurately	ADV
cana-3969	37	31	and	and	CCONJ
cana-3969	37	32	perfectly	perfectly	ADV
cana-3969	37	33	described	describe	VERB
cana-3969	37	34	in	in	ADP
cana-3969	37	35	terms	term	NOUN
cana-3969	37	36	of	of	ADP
cana-3969	37	37	decision	decision	NOUN
cana-3969	37	38	-	-	PUNCT
cana-3969	37	39	makers	maker	NOUN
cana-3969	37	40	'	'	PART
cana-3969	37	41	opinions	opinion	NOUN
cana-3969	37	42	.	.	PUNCT
cana-3969	38	1	we	we	PRON
cana-3969	38	2	introduce	introduce	VERB
cana-3969	38	3	and	and	CCONJ
cana-3969	38	4	define	define	VERB
cana-3969	38	5	a	a	DET
cana-3969	38	6	new	new	ADJ
cana-3969	38	7	concept	concept	NOUN
cana-3969	38	8	of	of	ADP
cana-3969	38	9	ṱ	ṱ	NOUN
cana-3969	38	10	-	-	PUNCT
cana-3969	38	11	norms	norm	NOUN
cana-3969	38	12	and	and	CCONJ
cana-3969	38	13	ṧ	ṧ	NOUN
cana-3969	38	14	-	-	NOUN
cana-3969	38	15	norms	norm	NOUN
cana-3969	38	16	(	(	PUNCT
cana-3969	38	17	notations	notation	NOUN
cana-3969	38	18	used	use	VERB
cana-3969	38	19	to	to	PART
cana-3969	38	20	represent	represent	VERB
cana-3969	38	21	certain	certain	ADJ
cana-3969	38	22	operations	operation	NOUN
cana-3969	38	23	in	in	ADP
cana-3969	38	24	fuzzy	fuzzy	ADJ
cana-3969	38	25	set	set	NOUN
cana-3969	38	26	theory	theory	NOUN
cana-3969	38	27	)	)	PUNCT
cana-3969	38	28	and	and	CCONJ
cana-3969	38	29	their	their	PRON
cana-3969	38	30	properties	property	NOUN
cana-3969	38	31	,	,	PUNCT
cana-3969	38	32	which	which	PRON
cana-3969	38	33	we	we	PRON
cana-3969	38	34	denote	denote	VERB
cana-3969	38	35	throughout	throughout	ADP
cana-3969	38	36	the	the	DET
cana-3969	38	37	paper	paper	NOUN
cana-3969	38	38	,	,	PUNCT
cana-3969	38	39	bringing	bring	VERB
cana-3969	38	40	a	a	DET
cana-3969	38	41	novel	novel	ADJ
cana-3969	38	42	perspective	perspective	NOUN
cana-3969	38	43	to	to	ADP
cana-3969	38	44	the	the	DET
cana-3969	38	45	field	field	NOUN
cana-3969	38	46	.	.	PUNCT
cana-3969	39	1	this	this	DET
cana-3969	39	2	paper	paper	NOUN
cana-3969	39	3	introduces	introduce	VERB
cana-3969	39	4	the	the	DET
cana-3969	39	5	star	star	NOUN
cana-3969	39	6	of	of	ADP
cana-3969	39	7	zeta	zeta	PROPN
cana-3969	39	8	,	,	PUNCT
cana-3969	39	9	a	a	DET
cana-3969	39	10	key	key	ADJ
cana-3969	39	11	concept	concept	NOUN
cana-3969	39	12	in	in	ADP
cana-3969	39	13	our	our	PRON
cana-3969	39	14	research	research	NOUN
cana-3969	39	15	,	,	PUNCT
cana-3969	39	16	and	and	CCONJ
cana-3969	39	17	its	its	PRON
cana-3969	39	18	application	application	NOUN
cana-3969	39	19	to	to	ADP
cana-3969	39	20	fuzzy	fuzzy	ADJ
cana-3969	39	21	ideal	ideal	ADJ
cana-3969	39	22	extension	extension	NOUN
cana-3969	39	23	in	in	ADP
cana-3969	39	24	gamma	gamma	NOUN
cana-3969	39	25	semigroups	semigroup	NOUN
cana-3969	39	26	.	.	PUNCT
cana-3969	40	1	this	this	DET
cana-3969	40	2	extension	extension	NOUN
cana-3969	40	3	is	be	AUX
cana-3969	40	4	significant	significant	ADJ
cana-3969	40	5	as	as	SCONJ
cana-3969	40	6	it	it	PRON
cana-3969	40	7	provides	provide	VERB
cana-3969	40	8	a	a	DET
cana-3969	40	9	new	new	ADJ
cana-3969	40	10	perspective	perspective	NOUN
cana-3969	40	11	on	on	ADP
cana-3969	40	12	the	the	DET
cana-3969	40	13	properties	property	NOUN
cana-3969	40	14	and	and	CCONJ
cana-3969	40	15	extensions	extension	NOUN
cana-3969	40	16	of	of	ADP
cana-3969	40	17	fuzzy	fuzzy	ADJ
cana-3969	40	18	semigroups	semigroup	NOUN
cana-3969	40	19	.	.	PUNCT
cana-3969	41	1	it	it	PRON
cana-3969	41	2	has	have	VERB
cana-3969	41	3	practical	practical	ADJ
cana-3969	41	4	applications	application	NOUN
cana-3969	41	5	in	in	ADP
cana-3969	41	6	various	various	ADJ
cana-3969	41	7	scientific	scientific	ADJ
cana-3969	41	8	fields	field	NOUN
cana-3969	41	9	,	,	PUNCT
cana-3969	41	10	such	such	ADJ
cana-3969	41	11	as	as	ADP
cana-3969	41	12	data	datum	NOUN
cana-3969	41	13	analysis	analysis	NOUN
cana-3969	41	14	,	,	PUNCT
cana-3969	41	15	pattern	pattern	NOUN
cana-3969	41	16	recognition	recognition	NOUN
cana-3969	41	17	,	,	PUNCT
cana-3969	41	18	and	and	CCONJ
cana-3969	41	19	decision	decision	NOUN
cana-3969	41	20	-	-	PUNCT
cana-3969	41	21	making	making	NOUN
cana-3969	41	22	under	under	ADP
cana-3969	41	23	uncertainty	uncertainty	NOUN
cana-3969	41	24	.	.	PUNCT
cana-3969	42	1	we	we	PRON
cana-3969	42	2	aim	aim	VERB
cana-3969	42	3	to	to	PART
cana-3969	42	4	inspire	inspire	VERB
cana-3969	42	5	and	and	CCONJ
cana-3969	42	6	motivate	motivate	VERB
cana-3969	42	7	further	further	ADJ
cana-3969	42	8	research	research	NOUN
cana-3969	42	9	and	and	CCONJ
cana-3969	42	10	innovation	innovation	NOUN
cana-3969	42	11	by	by	ADP
cana-3969	42	12	highlighting	highlight	VERB
cana-3969	42	13	these	these	DET
cana-3969	42	14	practical	practical	ADJ
cana-3969	42	15	applications	application	NOUN
cana-3969	42	16	.	.	PUNCT
cana-3969	43	1	we	we	PRON
cana-3969	43	2	present	present	VERB
cana-3969	43	3	the	the	DET
cana-3969	43	4	notions	notion	NOUN
cana-3969	43	5	of	of	ADP
cana-3969	43	6	fuzzy	fuzzy	ADJ
cana-3969	43	7	subsemigroups	subsemigroup	NOUN
cana-3969	43	8	and	and	CCONJ
cana-3969	43	9	discuss	discuss	VERB
cana-3969	43	10	their	their	PRON
cana-3969	43	11	potential	potential	ADJ
cana-3969	43	12	real	real	ADJ
cana-3969	43	13	-	-	PUNCT
cana-3969	43	14	world	world	NOUN
cana-3969	43	15	applications	application	NOUN
cana-3969	43	16	.	.	PUNCT
cana-3969	44	1	using	use	VERB
cana-3969	44	2	these	these	DET
cana-3969	44	3	notions	notion	NOUN
cana-3969	44	4	,	,	PUNCT
cana-3969	44	5	we	we	PRON
cana-3969	44	6	consider	consider	VERB
cana-3969	44	7	characterisations	characterisation	NOUN
cana-3969	44	8	of	of	ADP
cana-3969	44	9	sub	sub	NOUN
cana-3969	44	10	-	-	NOUN
cana-3969	44	11	semigroups	semigroup	NOUN
cana-3969	44	12	and	and	CCONJ
cana-3969	44	13	extensions	extension	NOUN
cana-3969	44	14	of	of	ADP
cana-3969	44	15	fuzzy	fuzzy	ADJ
cana-3969	44	16	semigroups	semigroup	NOUN
cana-3969	44	17	.	.	PUNCT
cana-3969	45	1	we	we	PRON
cana-3969	45	2	also	also	ADV
cana-3969	45	3	introduce	introduce	VERB
cana-3969	45	4	the	the	DET
cana-3969	45	5	concept	concept	NOUN
cana-3969	45	6	of	of	ADP
cana-3969	45	7	the	the	DET
cana-3969	45	8	dot(product	dot(product	NOUN
cana-3969	45	9	)	)	PUNCT
cana-3969	45	10	with	with	ADP
cana-3969	45	11	properties	property	NOUN
cana-3969	45	12	and	and	CCONJ
cana-3969	45	13	star	star	NOUN
cana-3969	45	14	zeta	zeta	PROPN
cana-3969	45	15	with	with	ADP
cana-3969	45	16	properties	property	NOUN
cana-3969	45	17	developed	develop	VERB
cana-3969	45	18	in	in	ADP
cana-3969	45	19	this	this	DET
cana-3969	45	20	paper	paper	NOUN
cana-3969	45	21	and	and	CCONJ
cana-3969	45	22	discuss	discuss	VERB
cana-3969	45	23	characterisations	characterisation	NOUN
cana-3969	45	24	of	of	ADP
cana-3969	45	25	the	the	DET
cana-3969	45	26	fuzzy	fuzzy	ADJ
cana-3969	45	27	ideal	ideal	NOUN
cana-3969	45	28	of	of	ADP
cana-3969	45	29	gamma	gamma	NOUN
cana-3969	45	30	semigroups	semigroup	NOUN
cana-3969	45	31	.	.	PUNCT
cana-3969	46	1	2	2	X
cana-3969	46	2	.	.	X
cana-3969	46	3	objectives	objective	NOUN
cana-3969	46	4	this	this	DET
cana-3969	46	5	study	study	NOUN
cana-3969	46	6	comprehensively	comprehensively	ADV
cana-3969	46	7	explores	explore	VERB
cana-3969	46	8	semigroups	semigroup	NOUN
cana-3969	46	9	'	'	PART
cana-3969	46	10	fundamental	fundamental	ADJ
cana-3969	46	11	properties	property	NOUN
cana-3969	46	12	and	and	CCONJ
cana-3969	46	13	variations	variation	NOUN
cana-3969	46	14	to	to	PART
cana-3969	46	15	review	review	VERB
cana-3969	46	16	semigroup	semigroup	PROPN
cana-3969	46	17	theory	theory	NOUN
cana-3969	46	18	and	and	CCONJ
cana-3969	46	19	its	its	PRON
cana-3969	46	20	historical	historical	ADJ
cana-3969	46	21	evolution	evolution	NOUN
cana-3969	46	22	.	.	PUNCT
cana-3969	47	1	by	by	ADP
cana-3969	47	2	examining	examine	VERB
cana-3969	47	3	the	the	DET
cana-3969	47	4	contributions	contribution	NOUN
cana-3969	47	5	of	of	ADP
cana-3969	47	6	significant	significant	ADJ
cana-3969	47	7	theorists	theorist	NOUN
cana-3969	47	8	,	,	PUNCT
cana-3969	47	9	this	this	DET
cana-3969	47	10	study	study	NOUN
cana-3969	47	11	aims	aim	VERB
cana-3969	47	12	to	to	PART
cana-3969	47	13	comprehend	comprehend	VERB
cana-3969	47	14	the	the	DET
cana-3969	47	15	evolution	evolution	NOUN
cana-3969	47	16	of	of	ADP
cana-3969	47	17	semigroups	semigroup	NOUN
cana-3969	47	18	and	and	CCONJ
cana-3969	47	19	their	their	PRON
cana-3969	47	20	structural	structural	ADJ
cana-3969	47	21	classifications	classification	NOUN
cana-3969	47	22	.	.	PUNCT
cana-3969	48	1	fuzzy	fuzzy	ADJ
cana-3969	48	2	algebraic	algebraic	ADJ
cana-3969	48	3	structures	structure	NOUN
cana-3969	48	4	are	be	AUX
cana-3969	48	5	also	also	ADV
cana-3969	48	6	analysed	analyse	VERB
cana-3969	48	7	in	in	ADP
cana-3969	48	8	the	the	DET
cana-3969	48	9	research	research	NOUN
cana-3969	48	10	,	,	PUNCT
cana-3969	48	11	including	include	VERB
cana-3969	48	12	fuzzy	fuzzy	ADJ
cana-3969	48	13	subgroups	subgroup	NOUN
cana-3969	48	14	,	,	PUNCT
cana-3969	48	15	fuzzy	fuzzy	ADJ
cana-3969	48	16	ideals	ideal	NOUN
cana-3969	48	17	and	and	CCONJ
cana-3969	48	18	their	their	PRON
cana-3969	48	19	extensions	extension	NOUN
cana-3969	48	20	,	,	PUNCT
cana-3969	48	21	and	and	CCONJ
cana-3969	48	22	the	the	DET
cana-3969	48	23	use	use	NOUN
cana-3969	48	24	of	of	ADP
cana-3969	48	25	fuzzy	fuzzy	ADJ
cana-3969	48	26	set	set	NOUN
cana-3969	48	27	theory	theory	NOUN
cana-3969	48	28	with	with	ADP
cana-3969	48	29	semigroups	semigroup	NOUN
cana-3969	48	30	.	.	PUNCT
cana-3969	49	1	this	this	DET
cana-3969	49	2	study	study	NOUN
cana-3969	49	3	provides	provide	VERB
cana-3969	49	4	fresh	fresh	ADJ
cana-3969	49	5	insights	insight	NOUN
cana-3969	49	6	into	into	ADP
cana-3969	49	7	semigroup	semigroup	PROPN
cana-3969	49	8	features	feature	NOUN
cana-3969	49	9	by	by	ADP
cana-3969	49	10	presenting	present	VERB
cana-3969	49	11	and	and	CCONJ
cana-3969	49	12	describing	describe	VERB
cana-3969	49	13	new	new	ADJ
cana-3969	49	14	concepts	concept	NOUN
cana-3969	49	15	,	,	PUNCT
cana-3969	49	16	including	include	VERB
cana-3969	49	17	fuzzy	fuzzy	ADJ
cana-3969	49	18	ideal	ideal	ADJ
cana-3969	49	19	extension	extension	NOUN
cana-3969	49	20	,	,	PUNCT
cana-3969	49	21	star	star	NOUN
cana-3969	49	22	zeta	zeta	PROPN
cana-3969	49	23	,	,	PUNCT
cana-3969	49	24	ṱ	ṱ	NOUN
cana-3969	49	25	-	-	PUNCT
cana-3969	49	26	norms	norm	NOUN
cana-3969	49	27	,	,	PUNCT
cana-3969	49	28	ṧ	ṧ	NOUN
cana-3969	49	29	-	-	NOUN
cana-3969	49	30	norms	norm	NOUN
cana-3969	49	31	,	,	PUNCT
cana-3969	49	32	and	and	CCONJ
cana-3969	49	33	the	the	DET
cana-3969	49	34	dot	dot	NOUN
cana-3969	49	35	product	product	NOUN
cana-3969	49	36	in	in	ADP
cana-3969	49	37	the	the	DET
cana-3969	49	38	context	context	NOUN
cana-3969	49	39	of	of	ADP
cana-3969	49	40	gamma	gamma	NOUN
cana-3969	49	41	semigroups	semigroup	NOUN
cana-3969	49	42	.	.	PUNCT
cana-3969	50	1	with	with	ADP
cana-3969	50	2	two	two	NUM
cana-3969	50	3	theorems	theorem	NOUN
cana-3969	50	4	and	and	CCONJ
cana-3969	50	5	eleven	eleven	NUM
cana-3969	50	6	lemmas	lemma	NOUN
cana-3969	50	7	that	that	PRON
cana-3969	50	8	expand	expand	VERB
cana-3969	50	9	our	our	PRON
cana-3969	50	10	theoretical	theoretical	ADJ
cana-3969	50	11	and	and	CCONJ
cana-3969	50	12	applied	apply	VERB
cana-3969	50	13	understanding	understanding	NOUN
cana-3969	50	14	of	of	ADP
cana-3969	50	15	semigroup	semigroup	PROPN
cana-3969	50	16	structures	structure	NOUN
cana-3969	50	17	,	,	PUNCT
cana-3969	50	18	the	the	DET
cana-3969	50	19	study	study	NOUN
cana-3969	50	20	provides	provide	VERB
cana-3969	50	21	strong	strong	ADJ
cana-3969	50	22	mathematical	mathematical	ADJ
cana-3969	50	23	evidence	evidence	NOUN
cana-3969	50	24	in	in	ADP
cana-3969	50	25	favour	favour	NOUN
cana-3969	50	26	of	of	ADP
cana-3969	50	27	these	these	DET
cana-3969	50	28	concepts	concept	NOUN
cana-3969	50	29	.	.	PUNCT
cana-3969	51	1	the	the	DET
cana-3969	51	2	paper	paper	NOUN
cana-3969	51	3	also	also	ADV
cana-3969	51	4	highlights	highlight	VERB
cana-3969	51	5	the	the	DET
cana-3969	51	6	valuable	valuable	ADJ
cana-3969	51	7	applications	application	NOUN
cana-3969	51	8	of	of	ADP
cana-3969	51	9	fuzzy	fuzzy	ADJ
cana-3969	51	10	semigroups	semigroup	NOUN
cana-3969	51	11	,	,	PUNCT
cana-3969	51	12	emphasising	emphasise	VERB
cana-3969	51	13	how	how	SCONJ
cana-3969	51	14	they	they	PRON
cana-3969	51	15	can	can	AUX
cana-3969	51	16	be	be	AUX
cana-3969	51	17	used	use	VERB
cana-3969	51	18	in	in	ADP
cana-3969	51	19	various	various	ADJ
cana-3969	51	20	scientific	scientific	ADJ
cana-3969	51	21	domains	domain	NOUN
cana-3969	51	22	.	.	PUNCT
cana-3969	52	1	this	this	DET
cana-3969	52	2	work	work	NOUN
cana-3969	52	3	intends	intend	VERB
cana-3969	52	4	to	to	PART
cana-3969	52	5	stimulate	stimulate	VERB
cana-3969	52	6	additional	additional	ADJ
cana-3969	52	7	research	research	NOUN
cana-3969	52	8	and	and	CCONJ
cana-3969	52	9	creativity	creativity	NOUN
cana-3969	52	10	by	by	ADP
cana-3969	52	11	characterising	characterise	VERB
cana-3969	52	12	sub	sub	NOUN
cana-3969	52	13	-	-	NOUN
cana-3969	52	14	semigroups	semigroup	NOUN
cana-3969	52	15	and	and	CCONJ
cana-3969	52	16	expanding	expand	VERB
cana-3969	52	17	fuzzy	fuzzy	ADJ
cana-3969	52	18	semigroup	semigroup	NOUN
cana-3969	52	19	theory	theory	NOUN
cana-3969	52	20	,	,	PUNCT
cana-3969	52	21	promoting	promote	VERB
cana-3969	52	22	a	a	DET
cana-3969	52	23	deeper	deep	ADJ
cana-3969	52	24	investigation	investigation	NOUN
cana-3969	52	25	of	of	ADP
cana-3969	52	26	algebraic	algebraic	ADJ
cana-3969	52	27	structures	structure	NOUN
cana-3969	52	28	and	and	CCONJ
cana-3969	52	29	their	their	PRON
cana-3969	52	30	computing	computing	NOUN
cana-3969	52	31	applications	application	NOUN
cana-3969	52	32	.	.	PUNCT
cana-3969	53	1	communications	communication	NOUN
cana-3969	53	2	on	on	ADP
cana-3969	53	3	applied	apply	VERB
cana-3969	53	4	nonlinear	nonlinear	ADJ
cana-3969	53	5	analysis	analysis	NOUN
cana-3969	53	6	issn	issn	NOUN
cana-3969	53	7	:	:	PUNCT
cana-3969	53	8	1074	1074	NUM
cana-3969	53	9	-	-	PUNCT
cana-3969	53	10	133x	133x	NUM
cana-3969	53	11	vol	vol	NOUN
cana-3969	53	12	32	32	NUM
cana-3969	53	13	no	no	NOUN
cana-3969	53	14	.	.	PUNCT
cana-3969	54	1	9s	9s	NUM
cana-3969	54	2	(	(	PUNCT
cana-3969	54	3	2025	2025	NUM
cana-3969	54	4	)	)	PUNCT
cana-3969	55	1	637	637	NUM
cana-3969	55	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-3969	55	3	3	3	X
cana-3969	55	4	.	.	PUNCT
cana-3969	55	5	preliminaries	preliminary	NOUN
cana-3969	55	6	definition	definition	NOUN
cana-3969	55	7	3.1	3.1	NUM
cana-3969	55	8	:	:	PUNCT
cana-3969	56	1	[	[	X
cana-3969	56	2	10	10	NUM
cana-3969	56	3	]	]	X
cana-3969	56	4	a	a	DET
cana-3969	56	5	non	non	ADJ
cana-3969	56	6	-	-	ADJ
cana-3969	56	7	empty	empty	ADJ
cana-3969	56	8	set	set	NOUN
cana-3969	56	9	ꟊ	ꟊ	PUNCT
cana-3969	56	10	is	be	AUX
cana-3969	56	11	called	call	VERB
cana-3969	56	12	a	a	DET
cana-3969	56	13	ternary	ternary	ADJ
cana-3969	56	14	semigroup	semigroup	NOUN
cana-3969	56	15	if	if	SCONJ
cana-3969	56	16	there	there	PRON
cana-3969	56	17	exists	exist	VERB
cana-3969	56	18	a	a	DET
cana-3969	56	19	mapping	mapping	NOUN
cana-3969	56	20	ꟊxꟊxꟊ	ꟊxꟊxꟊ	NOUN
cana-3969	56	21	→	→	PUNCT
cana-3969	56	22	ꟊ	ꟊ	X
cana-3969	56	23	is	be	AUX
cana-3969	56	24	defined	define	VERB
cana-3969	56	25	by	by	ADP
cana-3969	56	26	𝑎𝑏𝑐	𝑎𝑏𝑐	PROPN
cana-3969	56	27	∈	∈	PROPN
cana-3969	56	28	ꟊ	ꟊ	X
cana-3969	56	29	and	and	CCONJ
cana-3969	56	30	(	(	PUNCT
cana-3969	56	31	𝑎𝑏𝑐)𝑑𝑒	𝑎𝑏𝑐)𝑑𝑒	NOUN
cana-3969	56	32	=	=	SYM
cana-3969	56	33	𝑎(𝑏𝑐𝑑)𝑒	𝑎(𝑏𝑐𝑑)𝑒	NOUN
cana-3969	56	34	=	=	PUNCT
cana-3969	56	35	𝑎𝑏(𝑐𝑑𝑒	𝑎𝑏(𝑐𝑑𝑒	NOUN
cana-3969	56	36	)	)	PUNCT
cana-3969	56	37	for	for	ADP
cana-3969	56	38	all	all	DET
cana-3969	56	39	a	a	DET
cana-3969	56	40	,	,	PUNCT
cana-3969	56	41	b	b	NOUN
cana-3969	56	42	,	,	PUNCT
cana-3969	56	43	c	c	NOUN
cana-3969	56	44	,	,	PUNCT
cana-3969	56	45	d	d	NOUN
cana-3969	56	46	,	,	PUNCT
cana-3969	56	47	e	e	PROPN
cana-3969	56	48	∈	∈	PROPN
cana-3969	56	49	ꟊ.	ꟊ.	ADP
cana-3969	56	50	example	example	NOUN
cana-3969	56	51	:	:	PUNCT
cana-3969	56	52	let	let	VERB
cana-3969	56	53	ꟊ	ꟊ	PRON
cana-3969	56	54	=	=	PRON
cana-3969	56	55	{	{	PUNCT
cana-3969	56	56	x√5	x√5	PROPN
cana-3969	56	57	/	/	SYM
cana-3969	56	58	x∈	x∈	PROPN
cana-3969	57	1	z	z	PROPN
cana-3969	57	2	}	}	PUNCT
cana-3969	57	3	where	where	SCONJ
cana-3969	57	4	z	z	NOUN
cana-3969	57	5	is	be	AUX
cana-3969	57	6	the	the	DET
cana-3969	57	7	set	set	NOUN
cana-3969	57	8	of	of	ADP
cana-3969	57	9	negative	negative	ADJ
cana-3969	57	10	odd	odd	ADJ
cana-3969	57	11	integers	integer	NOUN
cana-3969	57	12	.	.	PUNCT
cana-3969	58	1	then	then	ADV
cana-3969	58	2	ꟊ	ꟊ	PRON
cana-3969	58	3	is	be	AUX
cana-3969	58	4	a	a	DET
cana-3969	58	5	ternary	ternary	ADJ
cana-3969	58	6	semigroup	semigroup	NOUN
cana-3969	58	7	under	under	ADP
cana-3969	58	8	usual	usual	ADJ
cana-3969	58	9	multiplication	multiplication	NOUN
cana-3969	58	10	.	.	PUNCT
cana-3969	59	1	definition	definition	NOUN
cana-3969	59	2	3.2	3.2	NUM
cana-3969	59	3	:	:	PUNCT
cana-3969	60	1	[	[	X
cana-3969	60	2	12	12	NUM
cana-3969	60	3	]	]	PUNCT
cana-3969	60	4	a	a	DET
cana-3969	60	5	non	non	ADJ
cana-3969	60	6	-	-	ADJ
cana-3969	60	7	empty	empty	ADJ
cana-3969	60	8	set	set	ADJ
cana-3969	60	9	l	l	NOUN
cana-3969	60	10	of	of	ADP
cana-3969	60	11	a	a	DET
cana-3969	60	12	ternary	ternary	ADJ
cana-3969	60	13	semigroup	semigroup	NOUN
cana-3969	60	14	ꟊ	ꟊ	X
cana-3969	60	15	is	be	AUX
cana-3969	60	16	called	call	VERB
cana-3969	60	17	(	(	PUNCT
cana-3969	60	18	i	i	NOUN
cana-3969	60	19	)	)	PUNCT
cana-3969	60	20	a	a	DET
cana-3969	60	21	left	left	ADJ
cana-3969	60	22	ideal	ideal	NOUN
cana-3969	60	23	of	of	ADP
cana-3969	60	24	ꟊ	ꟊ	PRON
cana-3969	60	25	if	if	SCONJ
cana-3969	60	26	ꟊꟊl	ꟊꟊl	PROPN
cana-3969	60	27	⊆	⊆	NUM
cana-3969	60	28	l	l	NOUN
cana-3969	60	29	(	(	PUNCT
cana-3969	60	30	ii	ii	NOUN
cana-3969	60	31	)	)	PUNCT
cana-3969	60	32	an	an	DET
cana-3969	60	33	interior	interior	ADJ
cana-3969	60	34	ideal	ideal	NOUN
cana-3969	60	35	of	of	ADP
cana-3969	60	36	ꟊ	ꟊ	PRON
cana-3969	60	37	if	if	SCONJ
cana-3969	60	38	ꟊlꟊ	ꟊlꟊ	ADJ
cana-3969	60	39	⊆	⊆	NUM
cana-3969	60	40	l	l	NOUN
cana-3969	60	41	(	(	PUNCT
cana-3969	60	42	iii	iii	NOUN
cana-3969	60	43	)	)	PUNCT
cana-3969	60	44	a	a	DET
cana-3969	60	45	proper	proper	ADJ
cana-3969	60	46	ideal	ideal	NOUN
cana-3969	60	47	of	of	ADP
cana-3969	60	48	ꟊ	ꟊ	PRON
cana-3969	60	49	lꟊꟊ	lꟊꟊ	NOUN
cana-3969	60	50	⊆	⊆	NUM
cana-3969	60	51	l	l	NOUN
cana-3969	60	52	(	(	PUNCT
cana-3969	60	53	iv	iv	X
cana-3969	60	54	)	)	PUNCT
cana-3969	60	55	an	an	DET
cana-3969	60	56	ideal	ideal	NOUN
cana-3969	60	57	of	of	ADP
cana-3969	60	58	ꟊ	ꟊ	PRON
cana-3969	60	59	if	if	SCONJ
cana-3969	60	60	l	l	NOUN
cana-3969	60	61	is	be	AUX
cana-3969	60	62	a	a	DET
cana-3969	60	63	left	left	ADJ
cana-3969	60	64	ideal	ideal	NOUN
cana-3969	60	65	,	,	PUNCT
cana-3969	60	66	a	a	DET
cana-3969	60	67	right	right	ADJ
cana-3969	60	68	ideal	ideal	NOUN
cana-3969	60	69	and	and	CCONJ
cana-3969	60	70	an	an	DET
cana-3969	60	71	interior	interior	ADJ
cana-3969	60	72	ideal	ideal	NOUN
cana-3969	60	73	of	of	ADP
cana-3969	60	74	ꟊ	ꟊ	PRON
cana-3969	60	75	an	an	DET
cana-3969	60	76	ideal	ideal	ADJ
cana-3969	60	77	l	l	NOUN
cana-3969	60	78	of	of	ADP
cana-3969	60	79	a	a	DET
cana-3969	60	80	ternary	ternary	ADJ
cana-3969	60	81	semigroup	semigroup	NOUN
cana-3969	60	82	ꟊ	ꟊ	PRON
cana-3969	60	83	is	be	AUX
cana-3969	60	84	a	a	DET
cana-3969	60	85	proper	proper	ADJ
cana-3969	60	86	ideal	ideal	NOUN
cana-3969	60	87	if	if	SCONJ
cana-3969	60	88	l.	l.	PROPN
cana-3969	60	89	≠	≠	PROPN
cana-3969	60	90	ꟊ.	ꟊ.	ADP
cana-3969	60	91	definition	definition	NOUN
cana-3969	60	92	3.3	3.3	NUM
cana-3969	60	93	:	:	PUNCT
cana-3969	60	94	let	let	VERB
cana-3969	60	95	ζ	ζ	NOUN
cana-3969	60	96	be	be	AUX
cana-3969	60	97	a	a	DET
cana-3969	60	98	fuzzy	fuzzy	ADJ
cana-3969	60	99	subset	subset	NOUN
cana-3969	60	100	of	of	ADP
cana-3969	60	101	ternary	ternary	ADJ
cana-3969	60	102	semigroup	semigroup	PROPN
cana-3969	60	103	ꟊ	ꟊ	NOUN
cana-3969	60	104	,	,	PUNCT
cana-3969	60	105	then	then	ADV
cana-3969	60	106	(	(	PUNCT
cana-3969	60	107	ꟊ,ζ	ꟊ,ζ	CCONJ
cana-3969	60	108	)	)	PUNCT
cana-3969	60	109	is	be	AUX
cana-3969	60	110	called	call	VERB
cana-3969	60	111	fuzzy	fuzzy	ADJ
cana-3969	60	112	ternary	ternary	ADJ
cana-3969	60	113	semigroup	semigroup	NOUN
cana-3969	60	114	if	if	SCONJ
cana-3969	60	115	ζ(pqr	ζ(pqr	NOUN
cana-3969	60	116	)	)	PUNCT
cana-3969	60	117	≥	≥	NOUN
cana-3969	60	118	min{ζ(p),ζ(q),ζ(r	min{ζ(p),ζ(q),ζ(r	NOUN
cana-3969	60	119	)	)	PUNCT
cana-3969	60	120	}	}	PUNCT
cana-3969	60	121	for	for	ADP
cana-3969	60	122	all	all	DET
cana-3969	60	123	p	p	NOUN
cana-3969	60	124	,	,	PUNCT
cana-3969	60	125	q	q	ADJ
cana-3969	60	126	,	,	PUNCT
cana-3969	60	127	r,∈	r,∈	NOUN
cana-3969	60	128	ꟊ.	ꟊ.	NOUN
cana-3969	60	129	example	example	NOUN
cana-3969	60	130	:	:	PUNCT
cana-3969	60	131	consider	consider	VERB
cana-3969	60	132	set	set	VERB
cana-3969	60	133	ꟊ	ꟊ	NOUN
cana-3969	60	134	=	=	PUNCT
cana-3969	60	135	{	{	PUNCT
cana-3969	60	136	p	p	X
cana-3969	60	137	,	,	PUNCT
cana-3969	60	138	q	q	ADJ
cana-3969	60	139	,	,	PUNCT
cana-3969	60	140	r	r	NOUN
cana-3969	60	141	}	}	PUNCT
cana-3969	60	142	with	with	ADP
cana-3969	60	143	the	the	DET
cana-3969	60	144	ternary	ternary	ADJ
cana-3969	60	145	operation	operation	NOUN
cana-3969	60	146	ʘ	ʘ	PROPN
cana-3969	60	147	and	and	CCONJ
cana-3969	60	148	assigned	assign	VERB
cana-3969	60	149	membership	membership	NOUN
cana-3969	60	150	values	value	NOUN
cana-3969	60	151	of	of	ADP
cana-3969	60	152	fuzzy	fuzzy	ADJ
cana-3969	60	153	set	set	VERB
cana-3969	60	154	ζ	ζ	NOUN
cana-3969	60	155	as	as	SCONJ
cana-3969	60	156	follows	follow	VERB
cana-3969	60	157	.	.	PUNCT
cana-3969	61	1	table	table	NOUN
cana-3969	61	2	of	of	ADP
cana-3969	61	3	ternary	ternary	ADJ
cana-3969	61	4	operation	operation	NOUN
cana-3969	61	5	:	:	PUNCT
cana-3969	62	1	ʘ	ʘ	PROPN
cana-3969	62	2	p	p	NOUN
cana-3969	62	3	q	q	PROPN
cana-3969	62	4	r	r	NOUN
cana-3969	62	5	p	p	X
cana-3969	62	6	p	p	X
cana-3969	62	7	q	q	NOUN
cana-3969	62	8	r	r	NOUN
cana-3969	62	9	q	q	X
cana-3969	62	10	q	q	NOUN
cana-3969	62	11	r	r	NOUN
cana-3969	62	12	p	p	NOUN
cana-3969	62	13	r	r	NOUN
cana-3969	62	14	r	r	NOUN
cana-3969	62	15	p	p	NOUN
cana-3969	62	16	q	q	NOUN
cana-3969	62	17	table-1(caylley	table-1(caylley	NOUN
cana-3969	62	18	’s	’s	PART
cana-3969	62	19	table	table	NOUN
cana-3969	62	20	of	of	ADP
cana-3969	62	21	p	p	X
cana-3969	62	22	,	,	PUNCT
cana-3969	62	23	q	q	NOUN
cana-3969	62	24	r	r	NOUN
cana-3969	62	25	)	)	PUNCT
cana-3969	62	26	and	and	CCONJ
cana-3969	62	27	ζ(p	ζ(p	NUM
cana-3969	62	28	)	)	PUNCT
cana-3969	63	1	=	=	SYM
cana-3969	63	2	0.8	0.8	NUM
cana-3969	63	3	;	;	PUNCT
cana-3969	63	4	ζ(q	ζ(q	PROPN
cana-3969	63	5	)	)	PUNCT
cana-3969	63	6	=	=	SYM
cana-3969	63	7	0.6	0.6	NUM
cana-3969	63	8	;	;	PUNCT
cana-3969	63	9	ζ(r	ζ(r	NOUN
cana-3969	63	10	)	)	PUNCT
cana-3969	63	11	=	=	PUNCT
cana-3969	63	12	0.4	0.4	NUM
cana-3969	63	13	the	the	DET
cana-3969	63	14	graph	graph	NOUN
cana-3969	63	15	shows	show	VERB
cana-3969	63	16	this	this	DET
cana-3969	63	17	ternary	ternary	ADJ
cana-3969	63	18	operation	operation	NOUN
cana-3969	63	19	and	and	CCONJ
cana-3969	63	20	membership	membership	NOUN
cana-3969	63	21	function	function	NOUN
cana-3969	63	22	as	as	ADP
cana-3969	63	23	visualisation	visualisation	NOUN
cana-3969	63	24	.	.	PUNCT
cana-3969	64	1	let	let	VERB
cana-3969	64	2	us	we	PRON
cana-3969	64	3	assume	assume	VERB
cana-3969	64	4	that	that	SCONJ
cana-3969	64	5	nodes	node	NOUN
cana-3969	64	6	represent	represent	VERB
cana-3969	64	7	elements	element	NOUN
cana-3969	64	8	p	p	X
cana-3969	64	9	,	,	PUNCT
cana-3969	64	10	q	q	ADJ
cana-3969	64	11	,	,	PUNCT
cana-3969	64	12	r	r	NOUN
cana-3969	64	13	,	,	PUNCT
cana-3969	64	14	and	and	CCONJ
cana-3969	64	15	edges	edge	NOUN
cana-3969	64	16	represent	represent	VERB
cana-3969	64	17	the	the	DET
cana-3969	64	18	ternary	ternary	ADJ
cana-3969	64	19	operation	operation	NOUN
cana-3969	64	20	;	;	PUNCT
cana-3969	64	21	the	the	DET
cana-3969	64	22	membership	membership	NOUN
cana-3969	64	23	values	value	NOUN
cana-3969	64	24	can	can	AUX
cana-3969	64	25	be	be	AUX
cana-3969	64	26	indicated	indicate	VERB
cana-3969	64	27	next	next	ADV
cana-3969	64	28	to	to	ADP
cana-3969	64	29	the	the	DET
cana-3969	64	30	nodes	node	NOUN
cana-3969	64	31	.	.	PUNCT
cana-3969	65	1	a	a	DET
cana-3969	65	2	textual	textual	ADJ
cana-3969	65	3	representation	representation	NOUN
cana-3969	65	4	of	of	ADP
cana-3969	65	5	the	the	DET
cana-3969	65	6	fuzzy	fuzzy	ADJ
cana-3969	65	7	ternary	ternary	ADJ
cana-3969	65	8	semigroup	semigroup	NOUN
cana-3969	65	9	is	be	AUX
cana-3969	65	10	below	below	ADV
cana-3969	65	11	.	.	PUNCT
cana-3969	66	1	nodes	nod	VERB
cana-3969	66	2	:	:	PUNCT
cana-3969	66	3	p	p	X
cana-3969	66	4	(	(	PUNCT
cana-3969	66	5	0.8	0.8	NUM
cana-3969	66	6	)	)	PUNCT
cana-3969	66	7	;	;	PUNCT
cana-3969	66	8	q	q	X
cana-3969	66	9	(	(	PUNCT
cana-3969	66	10	0.6	0.6	NUM
cana-3969	66	11	)	)	PUNCT
cana-3969	66	12	;	;	PUNCT
cana-3969	66	13	r	r	NOUN
cana-3969	66	14	(	(	PUNCT
cana-3969	66	15	0.4	0.4	NUM
cana-3969	66	16	)	)	PUNCT
cana-3969	66	17	edges	edge	NOUN
cana-3969	66	18	(	(	PUNCT
cana-3969	66	19	ternary	ternary	ADJ
cana-3969	66	20	operation	operation	NOUN
cana-3969	66	21	)	)	PUNCT
cana-3969	66	22	:	:	PUNCT
cana-3969	67	1	(	(	PUNCT
cana-3969	67	2	p	p	X
cana-3969	67	3	,	,	PUNCT
cana-3969	67	4	p	p	X
cana-3969	67	5	,	,	PUNCT
cana-3969	67	6	p	p	NOUN
cana-3969	67	7	)	)	PUNCT
cana-3969	67	8	it	it	PRON
cana-3969	67	9	tends	tend	VERB
cana-3969	67	10	to	to	ADP
cana-3969	67	11	p	p	X
cana-3969	67	12	;	;	PUNCT
cana-3969	67	13	(	(	PUNCT
cana-3969	67	14	p	p	X
cana-3969	67	15	,	,	PUNCT
cana-3969	67	16	q	q	ADJ
cana-3969	67	17	,	,	PUNCT
cana-3969	67	18	r	r	NOUN
cana-3969	67	19	)	)	PUNCT
cana-3969	67	20	it	it	PRON
cana-3969	67	21	tends	tend	VERB
cana-3969	67	22	to	to	PART
cana-3969	67	23	q	q	VERB
cana-3969	67	24	;	;	PUNCT
cana-3969	67	25	(	(	PUNCT
cana-3969	67	26	p	p	X
cana-3969	67	27	,	,	PUNCT
cana-3969	67	28	r	r	NOUN
cana-3969	67	29	,	,	PUNCT
cana-3969	67	30	q	q	NOUN
cana-3969	67	31	)	)	PUNCT
cana-3969	67	32	is	be	AUX
cana-3969	67	33	tends	tend	VERB
cana-3969	67	34	to	to	PART
cana-3969	67	35	r	r	VERB
cana-3969	67	36	;	;	PUNCT
cana-3969	67	37	(	(	PUNCT
cana-3969	67	38	q	q	X
cana-3969	67	39	,	,	PUNCT
cana-3969	67	40	q	q	ADJ
cana-3969	67	41	,	,	PUNCT
cana-3969	67	42	q	q	X
cana-3969	67	43	)	)	PUNCT
cana-3969	67	44	is	be	AUX
cana-3969	67	45	tends	tend	VERB
cana-3969	67	46	to	to	PART
cana-3969	67	47	r	r	VERB
cana-3969	67	48	;	;	PUNCT
cana-3969	67	49	(	(	PUNCT
cana-3969	67	50	r	r	NOUN
cana-3969	67	51	,	,	PUNCT
cana-3969	67	52	r	r	NOUN
cana-3969	67	53	,	,	PUNCT
cana-3969	67	54	r	r	NOUN
cana-3969	67	55	)	)	PUNCT
cana-3969	67	56	is	be	AUX
cana-3969	67	57	tends	tend	VERB
cana-3969	67	58	to	to	ADP
cana-3969	67	59	p	p	PROPN
cana-3969	67	60	(	(	PUNCT
cana-3969	67	61	q	q	NOUN
cana-3969	67	62	,	,	PUNCT
cana-3969	67	63	r	r	NOUN
cana-3969	67	64	,	,	PUNCT
cana-3969	67	65	p	p	NOUN
cana-3969	67	66	)	)	PUNCT
cana-3969	67	67	is	be	AUX
cana-3969	67	68	tends	tend	VERB
cana-3969	67	69	to	to	PART
cana-3969	67	70	q	q	PROPN
cana-3969	68	1	and	and	CCONJ
cana-3969	68	2	so	so	ADV
cana-3969	68	3	on	on	ADV
cana-3969	68	4	.	.	PUNCT
cana-3969	69	1	using	use	VERB
cana-3969	69	2	the	the	DET
cana-3969	69	3	above	above	ADJ
cana-3969	69	4	steps	step	NOUN
cana-3969	69	5	,	,	PUNCT
cana-3969	69	6	we	we	PRON
cana-3969	69	7	can	can	AUX
cana-3969	69	8	draw	draw	VERB
cana-3969	69	9	nodes	node	NOUN
cana-3969	69	10	and	and	CCONJ
cana-3969	69	11	edges	edge	NOUN
cana-3969	69	12	with	with	ADP
cana-3969	69	13	labelled	label	VERB
cana-3969	69	14	membership	membership	NOUN
cana-3969	69	15	values	value	NOUN
cana-3969	69	16	and	and	CCONJ
cana-3969	69	17	operation	operation	NOUN
cana-3969	69	18	results	result	NOUN
cana-3969	69	19	.	.	PUNCT
cana-3969	70	1	this	this	DET
cana-3969	70	2	visualisation	visualisation	NOUN
cana-3969	70	3	technique	technique	NOUN
cana-3969	70	4	is	be	AUX
cana-3969	70	5	a	a	DET
cana-3969	70	6	helpful	helpful	ADJ
cana-3969	70	7	tool	tool	NOUN
cana-3969	70	8	for	for	ADP
cana-3969	70	9	understanding	understand	VERB
cana-3969	70	10	the	the	DET
cana-3969	70	11	relationships	relationship	NOUN
cana-3969	70	12	and	and	CCONJ
cana-3969	70	13	operations	operation	NOUN
cana-3969	70	14	within	within	ADP
cana-3969	70	15	a	a	DET
cana-3969	70	16	fuzzy	fuzzy	ADJ
cana-3969	70	17	ternary	ternary	ADJ
cana-3969	70	18	semigroup	semigroup	NOUN
cana-3969	70	19	.	.	PUNCT
cana-3969	71	1	this	this	DET
cana-3969	71	2	section	section	NOUN
cana-3969	71	3	provides	provide	VERB
cana-3969	71	4	a	a	DET
cana-3969	71	5	step	step	NOUN
cana-3969	71	6	-	-	PUNCT
cana-3969	71	7	by	by	ADP
cana-3969	71	8	-	-	PUNCT
cana-3969	71	9	step	step	NOUN
cana-3969	71	10	guide	guide	NOUN
cana-3969	71	11	on	on	ADP
cana-3969	71	12	creating	create	VERB
cana-3969	71	13	such	such	ADJ
cana-3969	71	14	visualisations	visualisation	NOUN
cana-3969	71	15	and	and	CCONJ
cana-3969	71	16	a	a	DET
cana-3969	71	17	detailed	detailed	ADJ
cana-3969	71	18	example	example	NOUN
cana-3969	71	19	of	of	ADP
cana-3969	71	20	the	the	DET
cana-3969	71	21	algorithm	algorithm	NOUN
cana-3969	71	22	.	.	PUNCT
cana-3969	72	1	communications	communication	NOUN
cana-3969	72	2	on	on	ADP
cana-3969	72	3	applied	apply	VERB
cana-3969	72	4	nonlinear	nonlinear	ADJ
cana-3969	72	5	analysis	analysis	NOUN
cana-3969	72	6	issn	issn	NOUN
cana-3969	72	7	:	:	PUNCT
cana-3969	72	8	1074	1074	NUM
cana-3969	72	9	-	-	PUNCT
cana-3969	72	10	133x	133x	NUM
cana-3969	72	11	vol	vol	NOUN
cana-3969	72	12	32	32	NUM
cana-3969	72	13	no	no	NOUN
cana-3969	72	14	.	.	PUNCT
cana-3969	73	1	9s	9s	NUM
cana-3969	73	2	(	(	PUNCT
cana-3969	73	3	2025	2025	NUM
cana-3969	73	4	)	)	PUNCT
cana-3969	74	1	638	638	NUM
cana-3969	74	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-3969	74	3	figure	figure	NOUN
cana-3969	74	4	(	(	PUNCT
cana-3969	74	5	algorithm	algorithm	NOUN
cana-3969	74	6	)	)	PUNCT
cana-3969	74	7	definition	definition	NOUN
cana-3969	74	8	3.4	3.4	NUM
cana-3969	74	9	:	:	PUNCT
cana-3969	74	10	a	a	DET
cana-3969	74	11	non	non	ADJ
cana-3969	74	12	-	-	ADJ
cana-3969	74	13	empty	empty	ADJ
cana-3969	74	14	set	set	NOUN
cana-3969	74	15	ꟊ	ꟊ	PUNCT
cana-3969	74	16	is	be	AUX
cana-3969	74	17	called	call	VERB
cana-3969	74	18	ternary	ternary	ADJ
cana-3969	74	19	gamma	gamma	NOUN
cana-3969	74	20	semigroup	semigroup	NOUN
cana-3969	74	21	if	if	SCONJ
cana-3969	74	22	there	there	PRON
cana-3969	74	23	exists	exist	VERB
cana-3969	74	24	a	a	DET
cana-3969	74	25	map	map	NOUN
cana-3969	74	26	ꟊγꟊγꟊ.	ꟊγꟊγꟊ.	NOUN
cana-3969	74	27	→	→	PUNCT
cana-3969	74	28	ꟊ	ꟊ	PRON
cana-3969	74	29	is	be	AUX
cana-3969	74	30	ternary	ternary	ADJ
cana-3969	74	31	gamma	gamma	NOUN
cana-3969	74	32	semigroup	semigroup	PROPN
cana-3969	74	33	;	;	PUNCT
cana-3969	74	34	then	then	ADV
cana-3969	74	35	it	it	PRON
cana-3969	74	36	is	be	AUX
cana-3969	74	37	satisfies	satisfie	NOUN
cana-3969	74	38	(	(	PUNCT
cana-3969	74	39	i	i	NOUN
cana-3969	74	40	)	)	PUNCT
cana-3969	74	41	(	(	PUNCT
cana-3969	74	42	p	p	NOUN
cana-3969	74	43	αq	αq	ADP
cana-3969	74	44	β	β	NOUN
cana-3969	74	45	r	r	NOUN
cana-3969	74	46	)	)	PUNCT
cana-3969	74	47	γ	γ	PROPN
cana-3969	74	48	s	s	PROPN
cana-3969	74	49	δ	δ	NOUN
cana-3969	74	50	t	t	NOUN
cana-3969	74	51	=	=	SYM
cana-3969	74	52	p	p	X
cana-3969	74	53	α(q	α(q	PROPN
cana-3969	74	54	β	β	X
cana-3969	74	55	r	r	NOUN
cana-3969	74	56	γ	γ	PROPN
cana-3969	74	57	s	s	PART
cana-3969	74	58	)	)	PUNCT
cana-3969	74	59	δ	δ	PROPN
cana-3969	74	60	t	t	NOUN
cana-3969	75	1	=	=	PUNCT
cana-3969	75	2	p	p	NOUN
cana-3969	75	3	αq	αq	INTJ
cana-3969	75	4	β	β	NOUN
cana-3969	75	5	(	(	PUNCT
cana-3969	75	6	r	r	NOUN
cana-3969	75	7	γ	γ	PROPN
cana-3969	75	8	s	s	PROPN
cana-3969	75	9	δ	δ	PROPN
cana-3969	75	10	t	t	PROPN
cana-3969	75	11	)	)	PUNCT
cana-3969	75	12	(	(	PUNCT
cana-3969	75	13	ii	ii	NOUN
cana-3969	75	14	)	)	PUNCT
cana-3969	75	15	p	p	NOUN
cana-3969	75	16	α	α	PROPN
cana-3969	75	17	q	q	X
cana-3969	75	18	β	β	NOUN
cana-3969	75	19	r	r	NOUN
cana-3969	75	20	∈	∈	PROPN
cana-3969	75	21	ꟊ	ꟊ	X
cana-3969	75	22	for	for	ADP
cana-3969	75	23	all	all	DET
cana-3969	75	24	p	p	NOUN
cana-3969	75	25	,	,	PUNCT
cana-3969	75	26	q	q	NOUN
cana-3969	75	27	,	,	PUNCT
cana-3969	75	28	r	r	NOUN
cana-3969	75	29	,	,	PUNCT
cana-3969	75	30	s	s	PROPN
cana-3969	75	31	,	,	PUNCT
cana-3969	75	32	t	t	PROPN
cana-3969	75	33	∈	∈	PROPN
cana-3969	75	34	ꟊ	ꟊ	X
cana-3969	75	35	and	and	CCONJ
cana-3969	75	36	α	α	NOUN
cana-3969	75	37	,	,	PUNCT
cana-3969	75	38	β	β	X
cana-3969	75	39	,	,	PUNCT
cana-3969	75	40	γ	γ	PROPN
cana-3969	75	41	,	,	PUNCT
cana-3969	75	42	δ	δ	PROPN
cana-3969	75	43	∈	∈	PROPN
cana-3969	75	44	γ	γ	PROPN
cana-3969	75	45	.	.	PROPN
cana-3969	75	46	definition	definition	NOUN
cana-3969	75	47	3.5	3.5	NUM
cana-3969	75	48	:	:	PUNCT
cana-3969	75	49	let	let	VERB
cana-3969	75	50	ζ	ζ	NOUN
cana-3969	75	51	be	be	AUX
cana-3969	75	52	a	a	DET
cana-3969	75	53	fuzzy	fuzzy	ADJ
cana-3969	75	54	subset	subset	NOUN
cana-3969	75	55	of	of	ADP
cana-3969	75	56	the	the	DET
cana-3969	75	57	ternary	ternary	ADJ
cana-3969	75	58	gamma	gamma	NOUN
cana-3969	75	59	semigroup	semigroup	PROPN
cana-3969	75	60	ꟊ	ꟊ	PROPN
cana-3969	75	61	,	,	PUNCT
cana-3969	75	62	then	then	ADV
cana-3969	75	63	it	it	PRON
cana-3969	75	64	is	be	AUX
cana-3969	75	65	called	call	VERB
cana-3969	75	66	fuzzy	fuzzy	ADJ
cana-3969	75	67	ternary	ternary	ADJ
cana-3969	75	68	gamma	gamma	NOUN
cana-3969	75	69	semigroup	semigroup	PROPN
cana-3969	75	70	if	if	SCONJ
cana-3969	75	71	𝜁(𝑝𝛼𝑞𝛽𝑟	𝜁(𝑝𝛼𝑞𝛽𝑟	NOUN
cana-3969	75	72	)	)	PUNCT
cana-3969	75	73	≥	≥	NOUN
cana-3969	75	74	⋁(𝑝𝛼𝑞𝛽𝑟)𝑚𝑖𝑛	⋁(𝑝𝛼𝑞𝛽𝑟)𝑚𝑖𝑛	PROPN
cana-3969	75	75	{	{	PUNCT
cana-3969	75	76	𝜁(𝑝	𝜁(𝑝	PROPN
cana-3969	75	77	)	)	PUNCT
cana-3969	75	78	,	,	PUNCT
cana-3969	75	79	𝜁(𝑞	𝜁(𝑞	NOUN
cana-3969	75	80	)	)	PUNCT
cana-3969	75	81	,	,	PUNCT
cana-3969	75	82	𝜁(𝑟)}.if	𝜁(𝑟)}.if	PROPN
cana-3969	75	83	ζ	ζ	PROPN
cana-3969	75	84	is	be	AUX
cana-3969	75	85	the	the	DET
cana-3969	75	86	fuzzy	fuzzy	ADJ
cana-3969	75	87	interior	interior	ADJ
cana-3969	75	88	ideal	ideal	NOUN
cana-3969	75	89	of	of	ADP
cana-3969	75	90	ternary	ternary	ADJ
cana-3969	75	91	gamma	gamma	NOUN
cana-3969	75	92	semigroup	semigroup	PROPN
cana-3969	75	93	ꟊ	ꟊ	PROPN
cana-3969	75	94	,	,	PUNCT
cana-3969	75	95	then	then	ADV
cana-3969	75	96	ζ(pαqβr	ζ(pαqβr	NOUN
cana-3969	75	97	)	)	PUNCT
cana-3969	75	98	≥	≥	PROPN
cana-3969	75	99	ζ(q	ζ(q	PROPN
cana-3969	75	100	)	)	PUNCT
cana-3969	75	101	;	;	PUNCT
cana-3969	75	102	if	if	SCONJ
cana-3969	75	103	ζ	ζ	NOUN
cana-3969	75	104	is	be	AUX
cana-3969	75	105	the	the	DET
cana-3969	75	106	fuzzy	fuzzy	ADJ
cana-3969	75	107	left	leave	VERB
cana-3969	75	108	ideal	ideal	NOUN
cana-3969	75	109	of	of	ADP
cana-3969	75	110	ternary	ternary	ADJ
cana-3969	75	111	gamma	gamma	NOUN
cana-3969	75	112	semigroup	semigroup	PROPN
cana-3969	75	113	ꟊ	ꟊ	PROPN
cana-3969	75	114	,	,	PUNCT
cana-3969	75	115	then	then	ADV
cana-3969	75	116	ζ(pαqβr	ζ(pαqβr	NOUN
cana-3969	75	117	)	)	PUNCT
cana-3969	75	118	≥	≥	NOUN
cana-3969	75	119	ζ(r	ζ(r	NOUN
cana-3969	75	120	)	)	PUNCT
cana-3969	75	121	;	;	PUNCT
cana-3969	75	122	if	if	SCONJ
cana-3969	75	123	ζ	ζ	NOUN
cana-3969	75	124	is	be	AUX
cana-3969	75	125	the	the	DET
cana-3969	75	126	fuzzy	fuzzy	ADJ
cana-3969	75	127	right	right	ADJ
cana-3969	75	128	ideal	ideal	NOUN
cana-3969	75	129	of	of	ADP
cana-3969	75	130	ternary	ternary	ADJ
cana-3969	75	131	gamma	gamma	NOUN
cana-3969	75	132	semigroup	semigroup	PROPN
cana-3969	75	133	ꟊ	ꟊ	PROPN
cana-3969	75	134	,	,	PUNCT
cana-3969	75	135	then	then	ADV
cana-3969	75	136	ζ(pαqβr	ζ(pαqβr	NOUN
cana-3969	75	137	)	)	PUNCT
cana-3969	75	138	≥	≥	NOUN
cana-3969	75	139	ζ(p	ζ(p	PROPN
cana-3969	75	140	)	)	PUNCT
cana-3969	75	141	.	.	PUNCT
cana-3969	76	1	definition	definition	NOUN
cana-3969	76	2	3.6	3.6	NUM
cana-3969	76	3	:	:	PUNCT
cana-3969	76	4	a	a	DET
cana-3969	76	5	non	non	ADJ
cana-3969	76	6	-	-	ADJ
cana-3969	76	7	empty	empty	ADJ
cana-3969	76	8	set	set	NOUN
cana-3969	76	9	ꞗ	ꞗ	PROPN
cana-3969	76	10	of	of	ADP
cana-3969	76	11	ꟊ	ꟊ	PROPN
cana-3969	76	12	is	be	AUX
cana-3969	76	13	called	call	VERB
cana-3969	76	14	generalised	generalised	ADJ
cana-3969	76	15	bi	bi	NOUN
cana-3969	76	16	-	-	NOUN
cana-3969	76	17	ideal	ideal	ADJ
cana-3969	76	18	if	if	SCONJ
cana-3969	76	19	ꞗꟊꞗꟊꞗ	ꞗꟊꞗꟊꞗ	NOUN
cana-3969	76	20	⊆	⊆	NUM
cana-3969	76	21	ꞗ.the	ꞗ.the	PRON
cana-3969	76	22	generalised	generalise	VERB
cana-3969	76	23	bi	bi	NOUN
cana-3969	76	24	-	-	NOUN
cana-3969	76	25	ideal	ideal	NOUN
cana-3969	76	26	ꞗ	ꞗ	X
cana-3969	76	27	of	of	ADP
cana-3969	76	28	ꟊ	ꟊ	PROPN
cana-3969	76	29	is	be	AUX
cana-3969	76	30	called	call	VERB
cana-3969	76	31	bi	bi	NOUN
cana-3969	76	32	-	-	NOUN
cana-3969	76	33	ideal	ideal	NOUN
cana-3969	76	34	if	if	SCONJ
cana-3969	76	35	ꞗꞗꞗ	ꞗꞗꞗ	PROPN
cana-3969	76	36	⊆	⊆	NUM
cana-3969	76	37	ꞗ	ꞗ	NOUN
cana-3969	76	38	.	.	PUNCT
cana-3969	76	39	a	a	DET
cana-3969	76	40	mapping	mapping	NOUN
cana-3969	76	41	ζ:ӽ	ζ:ӽ	NOUN
cana-3969	76	42	→	→	PUNCT
cana-3969	77	1	[	[	X
cana-3969	77	2	0,1	0,1	NUM
cana-3969	77	3	]	]	PUNCT
cana-3969	77	4	is	be	AUX
cana-3969	77	5	called	call	VERB
cana-3969	77	6	a	a	DET
cana-3969	77	7	fuzzy	fuzzy	ADJ
cana-3969	77	8	set	set	NOUN
cana-3969	77	9	of	of	ADP
cana-3969	77	10	ӽ.	ӽ.	NOUN
cana-3969	77	11	the	the	DET
cana-3969	77	12	fuzzy	fuzzy	ADJ
cana-3969	77	13	set	set	VERB
cana-3969	77	14	ζ	ζ	NOUN
cana-3969	77	15	of	of	ADP
cana-3969	77	16	ꟊ	ꟊ	PROPN
cana-3969	77	17	is	be	AUX
cana-3969	77	18	called	call	VERB
cana-3969	77	19	generalised	generalise	VERB
cana-3969	77	20	fuzzy	fuzzy	ADJ
cana-3969	77	21	bi	bi	NOUN
cana-3969	77	22	-	-	NOUN
cana-3969	77	23	ideal	ideal	ADJ
cana-3969	77	24	if	if	SCONJ
cana-3969	77	25	ζ(pαqβr	ζ(pαqβr	NOUN
cana-3969	77	26	)	)	PUNCT
cana-3969	77	27	≥	≥	NOUN
cana-3969	77	28	min{ζ(p),ζ(q),ζ(r	min{ζ(p),ζ(q),ζ(r	NOUN
cana-3969	77	29	)	)	PUNCT
cana-3969	77	30	}	}	PUNCT
cana-3969	77	31	.	.	PUNCT
cana-3969	78	1	the	the	DET
cana-3969	78	2	generalised	generalise	VERB
cana-3969	78	3	fuzzy	fuzzy	ADJ
cana-3969	78	4	bi	bi	ADJ
cana-3969	78	5	-	-	ADJ
cana-3969	78	6	ideal	ideal	ADJ
cana-3969	78	7	ζ	ζ	NOUN
cana-3969	78	8	of	of	ADP
cana-3969	78	9	ꟊ	ꟊ	PROPN
cana-3969	78	10	is	be	AUX
cana-3969	78	11	called	call	VERB
cana-3969	78	12	a	a	DET
cana-3969	78	13	fuzzy	fuzzy	ADJ
cana-3969	78	14	bi	bi	NOUN
cana-3969	78	15	-	-	NOUN
cana-3969	78	16	ideal	ideal	ADJ
cana-3969	78	17	if	if	SCONJ
cana-3969	78	18	ζ(pαqβr	ζ(pαqβr	NOUN
cana-3969	78	19	)	)	PUNCT
cana-3969	78	20	≥	≥	NOUN
cana-3969	78	21	min{ζ(p),ζ(q),ζ(r	min{ζ(p),ζ(q),ζ(r	NOUN
cana-3969	78	22	)	)	PUNCT
cana-3969	78	23	}	}	PUNCT
cana-3969	78	24	.	.	PUNCT
cana-3969	79	1	the	the	DET
cana-3969	79	2	fuzzy	fuzzy	ADJ
cana-3969	79	3	set	set	VERB
cana-3969	79	4	ζ	ζ	NOUN
cana-3969	79	5	of	of	ADP
cana-3969	79	6	ꟊ	ꟊ	PROPN
cana-3969	79	7	is	be	AUX
cana-3969	79	8	called	call	VERB
cana-3969	79	9	generalised	generalised	ADJ
cana-3969	79	10	anti	anti	ADJ
cana-3969	79	11	-	-	ADJ
cana-3969	79	12	fuzzy	fuzzy	ADJ
cana-3969	79	13	bi	bi	NOUN
cana-3969	79	14	-	-	NOUN
cana-3969	79	15	ideal	ideal	ADJ
cana-3969	79	16	if	if	SCONJ
cana-3969	79	17	ζ(pαqβr	ζ(pαqβr	NOUN
cana-3969	79	18	)	)	PUNCT
cana-3969	79	19	≤	≤	NUM
cana-3969	79	20	max{ζ(p),ζ(q),ζ(r	max{ζ(p),ζ(q),ζ(r	NOUN
cana-3969	79	21	)	)	PUNCT
cana-3969	79	22	}	}	PUNCT
cana-3969	79	23	.	.	PUNCT
cana-3969	80	1	the	the	DET
cana-3969	80	2	generalised	generalise	VERB
cana-3969	80	3	fuzzy	fuzzy	ADJ
cana-3969	80	4	biideal	biideal	NOUN
cana-3969	80	5	ζ	ζ	NOUN
cana-3969	80	6	of	of	ADP
cana-3969	80	7	ꟊ	ꟊ	PROPN
cana-3969	80	8	is	be	AUX
cana-3969	80	9	called	call	VERB
cana-3969	80	10	an	an	DET
cana-3969	80	11	anti	anti	ADJ
cana-3969	80	12	-	-	ADJ
cana-3969	80	13	fuzzy	fuzzy	ADJ
cana-3969	80	14	bi	bi	NOUN
cana-3969	80	15	-	-	NOUN
cana-3969	80	16	ideal	ideal	ADJ
cana-3969	80	17	if	if	SCONJ
cana-3969	80	18	ζ(pαqβr	ζ(pαqβr	NOUN
cana-3969	80	19	)	)	PUNCT
cana-3969	80	20	≤	≤	NUM
cana-3969	80	21	max{ζ(p),ζ(q),ζ(r	max{ζ(p),ζ(q),ζ(r	NOUN
cana-3969	80	22	)	)	PUNCT
cana-3969	80	23	}	}	PUNCT
cana-3969	80	24	.	.	PUNCT
cana-3969	81	1	4	4	X
cana-3969	81	2	.	.	X
cana-3969	81	3	methods	method	NOUN
cana-3969	81	4	4.1.star	4.1.star	PROPN
cana-3969	81	5	fuzzing	fuzze	VERB
cana-3969	81	6	semigroups	semigroup	NOUN
cana-3969	81	7	:	:	PUNCT
cana-3969	81	8	let	let	VERB
cana-3969	81	9	ξ	ξ	X
cana-3969	81	10	∈	∈	VERB
cana-3969	81	11	𝐹𝑇	𝐹𝑇	PROPN
cana-3969	81	12	and	and	CCONJ
cana-3969	81	13	x	x	PART
cana-3969	81	14	∈	∈	PROPN
cana-3969	81	15	s	s	AUX
cana-3969	81	16	then	then	ADV
cana-3969	81	17	we	we	PRON
cana-3969	81	18	defined	define	VERB
cana-3969	81	19	ξ	ξ	X
cana-3969	81	20	*	*	PUNCT
cana-3969	81	21	=	=	SYM
cana-3969	81	22	{	{	PUNCT
cana-3969	81	23	x∈s	x∈s	NOUN
cana-3969	81	24	such	such	ADJ
cana-3969	81	25	that	that	DET
cana-3969	81	26	ξ(x	ξ(x	NOUN
cana-3969	81	27	)	)	PUNCT
cana-3969	81	28	=	=	SYM
cana-3969	81	29	ξ(0	ξ(0	NOUN
cana-3969	81	30	)	)	PUNCT
cana-3969	81	31	}	}	PUNCT
cana-3969	81	32	1	1	X
cana-3969	81	33	.	.	PUNCT
cana-3969	82	1	lemma	lemma	PROPN
cana-3969	82	2	:	:	PUNCT
cana-3969	82	3	let	let	VERB
cana-3969	82	4	ξ	ξ	X
cana-3969	82	5	∈	∈	PROPN
cana-3969	82	6	𝐹𝑇	𝐹𝑇	PROPN
cana-3969	82	7	(	(	PUNCT
cana-3969	82	8	s	s	NOUN
cana-3969	82	9	)	)	PUNCT
cana-3969	82	10	then	then	ADV
cana-3969	82	11	(	(	PUNCT
cana-3969	82	12	i	i	NOUN
cana-3969	82	13	)	)	PUNCT
cana-3969	82	14	(	(	PUNCT
cana-3969	82	15	ξi	ξi	X
cana-3969	82	16	)	)	PUNCT
cana-3969	82	17	*	*	NOUN
cana-3969	82	18	⊆ξ	⊆ξ	PROPN
cana-3969	82	19	*	*	VERB
cana-3969	82	20	for	for	ADP
cana-3969	82	21	all	all	DET
cana-3969	82	22	i	i	PRON
cana-3969	82	23	∈∞	∈∞	PROPN
cana-3969	82	24	(	(	PUNCT
cana-3969	82	25	ii	ii	NOUN
cana-3969	82	26	)	)	PUNCT
cana-3969	82	27	(	(	PUNCT
cana-3969	82	28	ξ(i	ξ(i	PROPN
cana-3969	82	29	)	)	PUNCT
cana-3969	82	30	)	)	PUNCT
cana-3969	82	31	*	*	PUNCT
cana-3969	83	1	⊆	⊆	NUM
cana-3969	83	2	ξ	ξ	X
cana-3969	83	3	*	*	PUNCT
cana-3969	83	4	let	let	VERB
cana-3969	83	5	x	x	X
cana-3969	83	6	∈	∈	PROPN
cana-3969	83	7	(	(	PUNCT
cana-3969	83	8	ξi	ξi	NOUN
cana-3969	83	9	)	)	PUNCT
cana-3969	83	10	*	*	PUNCT
cana-3969	84	1	then	then	ADV
cana-3969	84	2	ξi	ξi	INTJ
cana-3969	84	3	(	(	PUNCT
cana-3969	84	4	x	x	X
cana-3969	84	5	)	)	PUNCT
cana-3969	84	6	=	=	SYM
cana-3969	84	7	ξi	ξi	PROPN
cana-3969	84	8	(	(	PUNCT
cana-3969	84	9	0	0	NUM
cana-3969	84	10	)	)	PUNCT
cana-3969	84	11	=	=	SYM
cana-3969	84	12	ξ(0	ξ(0	X
cana-3969	84	13	)	)	PUNCT
cana-3969	84	14	∵ξi	∵ξi	PROPN
cana-3969	84	15	(	(	PUNCT
cana-3969	84	16	x	x	NOUN
cana-3969	84	17	)	)	PUNCT
cana-3969	84	18	≤	≤	PUNCT
cana-3969	84	19	ξ(1	ξ(1	PROPN
cana-3969	84	20	)	)	PUNCT
cana-3969	84	21	ξ(x	ξ(x	NOUN
cana-3969	84	22	)	)	PUNCT
cana-3969	84	23	=	=	SYM
cana-3969	84	24	ξ(0	ξ(0	X
cana-3969	84	25	)	)	PUNCT
cana-3969	84	26	then	then	ADV
cana-3969	84	27	x∈ξ	x∈ξ	PROPN
cana-3969	84	28	*	*	SYM
cana-3969	84	29	communications	communication	NOUN
cana-3969	84	30	on	on	ADP
cana-3969	84	31	applied	apply	VERB
cana-3969	84	32	nonlinear	nonlinear	ADJ
cana-3969	84	33	analysis	analysis	NOUN
cana-3969	84	34	issn	issn	NOUN
cana-3969	84	35	:	:	PUNCT
cana-3969	84	36	1074	1074	NUM
cana-3969	84	37	-	-	PUNCT
cana-3969	84	38	133x	133x	NUM
cana-3969	84	39	vol	vol	NOUN
cana-3969	84	40	32	32	NUM
cana-3969	84	41	no	no	NOUN
cana-3969	84	42	.	.	PUNCT
cana-3969	85	1	9s	9s	NUM
cana-3969	85	2	(	(	PUNCT
cana-3969	85	3	2025	2025	NUM
cana-3969	85	4	)	)	PUNCT
cana-3969	85	5	639	639	NUM
cana-3969	85	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3969	85	7	here	here	ADV
cana-3969	85	8	(	(	PUNCT
cana-3969	85	9	ξi	ξi	NOUN
cana-3969	85	10	)	)	PUNCT
cana-3969	85	11	*	*	PUNCT
cana-3969	86	1	⊆	⊆	NUM
cana-3969	86	2	ξ	ξ	X
cana-3969	86	3	*	*	VERB
cana-3969	86	4	similarly	similarly	ADV
cana-3969	86	5	we	we	PRON
cana-3969	86	6	can	can	AUX
cana-3969	86	7	prove	prove	VERB
cana-3969	86	8	(	(	PUNCT
cana-3969	86	9	ii	ii	NOUN
cana-3969	86	10	)	)	PUNCT
cana-3969	86	11	(	(	PUNCT
cana-3969	86	12	ξ(i	ξ(i	PROPN
cana-3969	86	13	)	)	PUNCT
cana-3969	86	14	)	)	PUNCT
cana-3969	86	15	*	*	PUNCT
cana-3969	87	1	⊆	⊆	NUM
cana-3969	87	2	ξ	ξ	X
cana-3969	87	3	*	*	PROPN
cana-3969	87	4	2	2	NUM
cana-3969	87	5	.	.	PUNCT
cana-3969	87	6	lemma	lemma	PROPN
cana-3969	87	7	:	:	PUNCT
cana-3969	87	8	let	let	VERB
cana-3969	87	9	ξ	ξ	X
cana-3969	87	10	∈	∈	PROPN
cana-3969	87	11	f(s	f(	NOUN
cana-3969	87	12	)	)	PUNCT
cana-3969	87	13	and	and	CCONJ
cana-3969	87	14	k∈n	k∈n	PROPN
cana-3969	87	15	then	then	ADV
cana-3969	87	16	𝜉𝐾+1(𝑥1	𝜉𝐾+1(𝑥1	PROPN
cana-3969	87	17	,	,	PUNCT
cana-3969	87	18	𝑥2	𝑥2	NOUN
cana-3969	87	19	,	,	PUNCT
cana-3969	87	20	…	…	PUNCT
cana-3969	87	21	,	,	PUNCT
cana-3969	87	22	𝑥𝑘+1	𝑥𝑘+1	NOUN
cana-3969	87	23	)	)	PUNCT
cana-3969	87	24	=	=	SYM
cana-3969	88	1	λ𝑖=1	λ𝑖=1	PROPN
cana-3969	88	2	𝐾+1𝜉(𝑥𝑖	𝐾+1𝜉(𝑥𝑖	ADJ
cana-3969	88	3	)	)	PUNCT
cana-3969	88	4	if	if	SCONJ
cana-3969	88	5	x1,x2,	x1,x2,	PROPN
cana-3969	88	6	…	…	SYM
cana-3969	88	7	,xk	,xk	PUNCT
cana-3969	88	8	∈	∈	PROPN
cana-3969	88	9	s	s	VERB
cana-3969	88	10	then	then	ADV
cana-3969	88	11	ξk	ξk	ADP
cana-3969	88	12	(	(	PUNCT
cana-3969	88	13	x1,x2,	x1,x2,	PROPN
cana-3969	88	14	…	…	SYM
cana-3969	88	15	,xk)≥𝛬(𝑖	,xk)≥𝛬(𝑖	NOUN
cana-3969	88	16	=	=	NOUN
cana-3969	88	17	1	1	X
cana-3969	88	18	)	)	PUNCT
cana-3969	88	19	𝐾	𝐾	PROPN
cana-3969	88	20	ξ(xi	ξ(xi	PROPN
cana-3969	88	21	)	)	PUNCT
cana-3969	88	22	clearly	clearly	ADV
cana-3969	88	23	,	,	PUNCT
cana-3969	88	24	the	the	DET
cana-3969	88	25	result	result	NOUN
cana-3969	88	26	is	be	AUX
cana-3969	88	27	valid	valid	ADJ
cana-3969	88	28	for	for	ADP
cana-3969	88	29	k	k	PROPN
cana-3969	88	30	=	=	SYM
cana-3969	88	31	1	1	NUM
cana-3969	88	32	assume	assume	VERB
cana-3969	88	33	that	that	SCONJ
cana-3969	88	34	it	it	PRON
cana-3969	88	35	is	be	AUX
cana-3969	88	36	valid	valid	ADJ
cana-3969	88	37	for	for	ADP
cana-3969	88	38	k	k	PROPN
cana-3969	88	39	≥	≥	PROPN
cana-3969	88	40	1	1	NUM
cana-3969	88	41	now	now	ADV
cana-3969	88	42	ξ(k+1	ξ(k+1	PRON
cana-3969	88	43	)	)	PUNCT
cana-3969	88	44	(	(	PUNCT
cana-3969	88	45	x1,x2,	x1,x2,	NOUN
cana-3969	88	46	…	…	SYM
cana-3969	88	47	,xk+1	,xk+1	X
cana-3969	88	48	)	)	PUNCT
cana-3969	88	49	=	=	PUNCT
cana-3969	89	1	ξk	ξk	PROPN
cana-3969	89	2	.	.	PROPN
cana-3969	89	3	ξ(x1,x2,	ξ(x1,x2,	NUM
cana-3969	89	4	…	…	PUNCT
cana-3969	89	5	,xk+1	,xk+1	PUNCT
cana-3969	89	6	)	)	PUNCT
cana-3969	89	7	⋏	⋏	ADV
cana-3969	89	8	ξ(x(k+1	ξ(x(k+1	NOUN
cana-3969	89	9	)	)	PUNCT
cana-3969	89	10	)	)	PUNCT
cana-3969	90	1	=	=	PUNCT
cana-3969	91	1	𝛬(𝑖	𝛬(𝑖	X
cana-3969	91	2	=	=	NOUN
cana-3969	91	3	1	1	X
cana-3969	91	4	)	)	PUNCT
cana-3969	91	5	𝐾	𝐾	PROPN
cana-3969	91	6	ξ(xi)⋏ξ(x(k+1	ξ(xi)⋏ξ(x(k+1	NOUN
cana-3969	91	7	)	)	PUNCT
cana-3969	91	8	)	)	PUNCT
cana-3969	92	1	=	=	PUNCT
cana-3969	93	1	𝛬(𝑖	𝛬(𝑖	X
cana-3969	93	2	=	=	NOUN
cana-3969	93	3	1	1	X
cana-3969	93	4	)	)	PUNCT
cana-3969	93	5	𝐾	𝐾	PROPN
cana-3969	93	6	ξ(xi	ξ(xi	PROPN
cana-3969	93	7	)	)	PUNCT
cana-3969	93	8	3	3	NUM
cana-3969	93	9	.	.	PUNCT
cana-3969	94	1	lemma	lemma	PROPN
cana-3969	94	2	:	:	PUNCT
cana-3969	94	3	let	let	VERB
cana-3969	94	4	ξ,µ∈f(s	ξ,µ∈f(s	PRON
cana-3969	94	5	)	)	PUNCT
cana-3969	94	6	prove	prove	VERB
cana-3969	94	7	that	that	SCONJ
cana-3969	94	8	𝜉∗⋂µ	𝜉∗⋂µ	ADJ
cana-3969	94	9	∗	∗	NOUN
cana-3969	94	10	⊆	⊆	NUM
cana-3969	94	11	(	(	PUNCT
cana-3969	94	12	𝜉	𝜉	NOUN
cana-3969	94	13	∩	∩	NOUN
cana-3969	94	14	µ)∗	µ)∗	PROPN
cana-3969	94	15	let	let	VERB
cana-3969	94	16	x∈𝜉∗⋂µ	x∈𝜉∗⋂µ	PROPN
cana-3969	94	17	*	*	PUNCT
cana-3969	94	18	then	then	ADV
cana-3969	94	19	ξ(x	ξ(x	NOUN
cana-3969	94	20	)	)	PUNCT
cana-3969	94	21	=	=	SYM
cana-3969	94	22	ξ(0	ξ(0	NOUN
cana-3969	94	23	)	)	PUNCT
cana-3969	94	24	and	and	CCONJ
cana-3969	94	25	µ(x	µ(x	NOUN
cana-3969	94	26	)	)	PUNCT
cana-3969	94	27	=	=	SYM
cana-3969	94	28	µ(0	µ(0	NOUN
cana-3969	94	29	)	)	PUNCT
cana-3969	94	30	now	now	ADV
cana-3969	94	31	(	(	PUNCT
cana-3969	94	32	ξ∩µ)(x	ξ∩µ)(x	NOUN
cana-3969	94	33	)	)	PUNCT
cana-3969	94	34	=	=	SYM
cana-3969	94	35	ξ(x)∩µ(x	ξ(x)∩µ(x	NOUN
cana-3969	94	36	)	)	PUNCT
cana-3969	94	37	=	=	SYM
cana-3969	94	38	ξ(0)∩µ(0	ξ(0)∩µ(0	X
cana-3969	94	39	)	)	PUNCT
cana-3969	94	40	=	=	SYM
cana-3969	94	41	(	(	PUNCT
cana-3969	94	42	ξ∩µ)(0	ξ∩µ)(0	NOUN
cana-3969	94	43	)	)	PUNCT
cana-3969	94	44	x∈(ξ∩µ	x∈(ξ∩µ	NOUN
cana-3969	94	45	)	)	PUNCT
cana-3969	94	46	*	*	PUNCT
cana-3969	94	47	hence	hence	ADV
cana-3969	94	48	𝜉∗	𝜉∗	PROPN
cana-3969	94	49	∩	∩	X
cana-3969	94	50	µ	µ	X
cana-3969	94	51	∗	∗	NOUN
cana-3969	94	52	⊆	⊆	NUM
cana-3969	94	53	(	(	PUNCT
cana-3969	94	54	𝜉	𝜉	X
cana-3969	94	55	∩	∩	NOUN
cana-3969	94	56	µ)∗	µ)∗	PROPN
cana-3969	94	57	definition	definition	NOUN
cana-3969	94	58	4.1	4.1	NUM
cana-3969	94	59	:	:	PUNCT
cana-3969	94	60	let	let	VERB
cana-3969	94	61	ξ	ξ	X
cana-3969	94	62	,	,	PUNCT
cana-3969	94	63	ζ∈fs	ζ∈fs	NOUN
cana-3969	94	64	define	define	NOUN
cana-3969	94	65	ζ⨁ξ	ζ⨁ξ	PROPN
cana-3969	94	66	as	as	SCONJ
cana-3969	94	67	follows	follow	VERB
cana-3969	94	68	(	(	PUNCT
cana-3969	94	69	𝜁⨁𝜉)(𝑥	𝜁⨁𝜉)(𝑥	NOUN
cana-3969	94	70	)	)	PUNCT
cana-3969	94	71	=	=	SYM
cana-3969	95	1	∨	∨	X
cana-3969	95	2	{	{	PUNCT
cana-3969	95	3	𝜁(𝑦	𝜁(𝑦	ADJ
cana-3969	95	4	)	)	PUNCT
cana-3969	95	5	∧	∧	PROPN
cana-3969	95	6	𝜉(𝑧)/𝑦	𝜉(𝑧)/𝑦	PROPN
cana-3969	95	7	∈	∈	PROPN
cana-3969	95	8	𝑆	𝑆	PROPN
cana-3969	95	9	,	,	PUNCT
cana-3969	95	10	𝑦	𝑦	NOUN
cana-3969	95	11	+	+	CCONJ
cana-3969	95	12	𝑧	𝑧	NOUN
cana-3969	95	13	=	=	SYM
cana-3969	95	14	𝑥	𝑥	NOUN
cana-3969	95	15	}	}	PUNCT
cana-3969	95	16	(	(	PUNCT
cana-3969	95	17	𝜁	𝜁	PROPN
cana-3969	95	18	⊙	⊙	X
cana-3969	95	19	𝜉)(𝑥	𝜉)(𝑥	NOUN
cana-3969	95	20	)	)	PUNCT
cana-3969	96	1	=	=	PRON
cana-3969	96	2	∨	∨	X
cana-3969	96	3	{	{	PUNCT
cana-3969	96	4	λ𝑖=1	λ𝑖=1	PROPN
cana-3969	96	5	𝑛	𝑛	PROPN
cana-3969	96	6	𝜁(𝑟𝑖	𝜁(𝑟𝑖	PROPN
cana-3969	96	7	)	)	PUNCT
cana-3969	96	8	⋏	⋏	PROPN
cana-3969	96	9	𝜉(𝑥𝑖)/𝑟𝑖	𝜉(𝑥𝑖)/𝑟𝑖	VERB
cana-3969	96	10	∈	∈	PROPN
cana-3969	96	11	𝑅1	𝑅1	NOUN
cana-3969	96	12	𝑥𝑖	𝑥𝑖	PROPN
cana-3969	96	13	∈	∈	PROPN
cana-3969	96	14	𝑆	𝑆	PROPN
cana-3969	96	15	,	,	PUNCT
cana-3969	96	16	ℎ	ℎ	VERB
cana-3969	96	17	≤	≤	NOUN
cana-3969	96	18	𝑖	𝑖	SYM
cana-3969	96	19	≤	≤	NUM
cana-3969	96	20	𝑛	𝑛	NOUN
cana-3969	96	21	,	,	PUNCT
cana-3969	96	22	𝑛	𝑛	PRON
cana-3969	96	23	∈	∈	PROPN
cana-3969	96	24	𝑁	𝑁	PROPN
cana-3969	96	25	,	,	PUNCT
cana-3969	96	26	∑	∑	ADV
cana-3969	96	27	𝑟𝑖𝑥𝑖	𝑟𝑖𝑥𝑖	NOUN
cana-3969	96	28	=	=	PUNCT
cana-3969	96	29	𝑥ℎ	𝑥ℎ	NOUN
cana-3969	96	30	𝑖=1	𝑖=1	PROPN
cana-3969	96	31	}	}	PUNCT
cana-3969	96	32	4	4	NUM
cana-3969	96	33	.	.	X
cana-3969	97	1	lemma	lemma	PROPN
cana-3969	97	2	:	:	PUNCT
cana-3969	97	3	1	1	X
cana-3969	97	4	.	.	PUNCT
cana-3969	97	5	(	(	PUNCT
cana-3969	97	6	𝑟𝜁)(𝑟𝑥	𝑟𝜁)(𝑟𝑥	NOUN
cana-3969	97	7	)	)	PUNCT
cana-3969	97	8	≥	≥	NOUN
cana-3969	97	9	𝜁(𝑥	𝜁(𝑥	NOUN
cana-3969	97	10	)	)	PUNCT
cana-3969	97	11	;	;	PUNCT
cana-3969	97	12	∀𝑥	∀𝑥	PROPN
cana-3969	97	13	∈	∈	PROPN
cana-3969	97	14	𝑆	𝑆	PROPN
cana-3969	97	15	2	2	NUM
cana-3969	97	16	.	.	PUNCT
cana-3969	97	17	𝜉(𝑟𝑥	𝜉(𝑟𝑥	NOUN
cana-3969	97	18	)	)	PUNCT
cana-3969	97	19	≥	≥	NOUN
cana-3969	97	20	𝜁(𝑥	𝜁(𝑥	NOUN
cana-3969	97	21	)	)	PUNCT
cana-3969	97	22	∀	∀	PUNCT
cana-3969	98	1	𝑥	𝑥	DET
cana-3969	98	2	∈	∈	NOUN
cana-3969	98	3	𝑆	𝑆	PROPN
cana-3969	98	4	=	=	PUNCT
cana-3969	98	5	>	>	X
cana-3969	98	6	𝑟	𝑟	X
cana-3969	98	7	𝜁	𝜁	PROPN
cana-3969	98	8	⊆	⊆	NUM
cana-3969	98	9	𝜉	𝜉	SYM
cana-3969	98	10	3	3	NUM
cana-3969	98	11	.	.	PUNCT
cana-3969	98	12	(	(	PUNCT
cana-3969	98	13	𝑟𝜁	𝑟𝜁	X
cana-3969	98	14	+	+	PUNCT
cana-3969	98	15	𝑠𝜉)(𝑟𝑥	𝑠𝜉)(𝑟𝑥	PROPN
cana-3969	98	16	+	+	NUM
cana-3969	98	17	𝑠𝑦	𝑠𝑦	NOUN
cana-3969	98	18	)	)	PUNCT
cana-3969	98	19	≥	≥	NOUN
cana-3969	98	20	𝜁(𝑥	𝜁(𝑥	NOUN
cana-3969	98	21	)	)	PUNCT
cana-3969	98	22	∧	∧	PROPN
cana-3969	98	23	𝜉(𝑦	𝜉(𝑦	NOUN
cana-3969	98	24	)	)	PUNCT
cana-3969	98	25	∀𝑥	∀𝑥	NOUN
cana-3969	98	26	,	,	PUNCT
cana-3969	98	27	𝑦	𝑦	NOUN
cana-3969	98	28	∈	∈	NOUN
cana-3969	98	29	𝑆	𝑆	PROPN
cana-3969	98	30	4	4	NUM
cana-3969	98	31	.	.	NOUN
cana-3969	98	32	𝜉(𝑟𝑥	𝜉(𝑟𝑥	NOUN
cana-3969	98	33	+	+	NUM
cana-3969	98	34	𝑠𝑦	𝑠𝑦	NOUN
cana-3969	98	35	)	)	PUNCT
cana-3969	98	36	≥	≥	NOUN
cana-3969	98	37	𝜁(𝑥	𝜁(𝑥	NOUN
cana-3969	98	38	)	)	PUNCT
cana-3969	98	39	∧	∧	NOUN
cana-3969	98	40	𝜎(𝑦	𝜎(𝑦	NOUN
cana-3969	98	41	)	)	PUNCT
cana-3969	98	42	∀𝑥	∀𝑥	PROPN
cana-3969	98	43	,	,	PUNCT
cana-3969	98	44	𝑦	𝑦	NOUN
cana-3969	98	45	∈	∈	NOUN
cana-3969	98	46	𝑆	𝑆	PROPN
cana-3969	98	47	=	=	SYM
cana-3969	98	48	>	>	X
cana-3969	98	49	𝑟𝜁	𝑟𝜁	PRON
cana-3969	98	50	+	+	CCONJ
cana-3969	99	1	𝑆𝜎	𝑆𝜎	PROPN
cana-3969	99	2	⊆	⊆	NUM
cana-3969	99	3	𝜉	𝜉	ADP
cana-3969	99	4	communications	communication	NOUN
cana-3969	99	5	on	on	ADP
cana-3969	99	6	applied	apply	VERB
cana-3969	99	7	nonlinear	nonlinear	ADJ
cana-3969	99	8	analysis	analysis	NOUN
cana-3969	99	9	issn	issn	NOUN
cana-3969	99	10	:	:	PUNCT
cana-3969	99	11	1074	1074	NUM
cana-3969	99	12	-	-	PUNCT
cana-3969	99	13	133x	133x	NUM
cana-3969	99	14	vol	vol	NOUN
cana-3969	99	15	32	32	NUM
cana-3969	99	16	no	no	NOUN
cana-3969	99	17	.	.	PUNCT
cana-3969	100	1	9s	9s	NUM
cana-3969	100	2	(	(	PUNCT
cana-3969	100	3	2025	2025	NUM
cana-3969	100	4	)	)	PUNCT
cana-3969	100	5	640	640	NUM
cana-3969	100	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3969	100	7	proof	proof	NOUN
cana-3969	100	8	:	:	PUNCT
cana-3969	100	9	(	(	PUNCT
cana-3969	100	10	1	1	NUM
cana-3969	100	11	)	)	PUNCT
cana-3969	100	12	.	.	PUNCT
cana-3969	101	1	(	(	PUNCT
cana-3969	101	2	𝑥𝜁)(𝑟𝑥	𝑥𝜁)(𝑟𝑥	NOUN
cana-3969	101	3	)	)	PUNCT
cana-3969	101	4	≥	≥	X
cana-3969	101	5	𝜁(𝑥	𝜁(𝑥	NOUN
cana-3969	101	6	)	)	PUNCT
cana-3969	101	7	∀𝑥	∀𝑥	PROPN
cana-3969	101	8	∈	∈	PROPN
cana-3969	101	9	𝑆	𝑆	PROPN
cana-3969	101	10	(	(	PUNCT
cana-3969	101	11	𝑥𝜁)(𝑟𝑥	𝑥𝜁)(𝑟𝑥	X
cana-3969	101	12	)	)	PUNCT
cana-3969	101	13	=	=	SYM
cana-3969	101	14	𝑉	𝑉	PROPN
cana-3969	101	15	{	{	PUNCT
cana-3969	101	16	𝜁(𝑦	𝜁(𝑦	PROPN
cana-3969	101	17	)	)	PUNCT
cana-3969	102	1	⁄	⁄	ADP
cana-3969	102	2	𝑦	𝑦	NUM
cana-3969	102	3	∈	∈	PROPN
cana-3969	102	4	𝑆	𝑆	PROPN
cana-3969	102	5	,	,	PUNCT
cana-3969	102	6	𝑟𝑦	𝑟𝑦	X
cana-3969	102	7	=	=	PUNCT
cana-3969	102	8	𝑟𝑥	𝑟𝑥	PROPN
cana-3969	102	9	}	}	PUNCT
cana-3969	102	10	≥	≥	NOUN
cana-3969	102	11	ζ(x	ζ(x	NOUN
cana-3969	102	12	)	)	PUNCT
cana-3969	102	13	∀x∈s	∀x∈s	PROPN
cana-3969	102	14	(	(	PUNCT
cana-3969	102	15	2	2	NUM
cana-3969	102	16	)	)	PUNCT
cana-3969	102	17	.	.	PUNCT
cana-3969	103	1	𝐼𝑓	𝐼𝑓	ADJ
cana-3969	103	2	𝜉(𝑟𝑥	𝜉(𝑟𝑥	NOUN
cana-3969	103	3	)	)	PUNCT
cana-3969	103	4	≥	≥	NOUN
cana-3969	103	5	𝜁(𝑥	𝜁(𝑥	NOUN
cana-3969	103	6	)	)	PUNCT
cana-3969	103	7	∀𝑥	∀𝑥	PROPN
cana-3969	103	8	∈	∈	PROPN
cana-3969	103	9	𝑆	𝑆	PROPN
cana-3969	103	10	then	then	ADV
cana-3969	103	11	(	(	PUNCT
cana-3969	103	12	𝑟𝜁)(𝑥	𝑟𝜁)(𝑥	PROPN
cana-3969	103	13	)	)	PUNCT
cana-3969	103	14	=	=	SYM
cana-3969	103	15	𝑉{𝜁(𝑦	𝑉{𝜁(𝑦	X
cana-3969	103	16	)	)	PUNCT
cana-3969	103	17	⁄	⁄	ADP
cana-3969	103	18	𝑦	𝑦	NUM
cana-3969	103	19	∈	∈	PROPN
cana-3969	103	20	𝑆	𝑆	PROPN
cana-3969	103	21	,	,	PUNCT
cana-3969	103	22	𝑟𝑦	𝑟𝑦	NOUN
cana-3969	103	23	=	=	SYM
cana-3969	103	24	𝑥	𝑥	PROPN
cana-3969	103	25	}	}	PUNCT
cana-3969	103	26	≤	≤	NUM
cana-3969	103	27	𝑉{𝜉(𝑟𝑦	𝑉{𝜉(𝑟𝑦	PROPN
cana-3969	103	28	)	)	PUNCT
cana-3969	104	1	⁄	⁄	ADP
cana-3969	104	2	𝑦	𝑦	NUM
cana-3969	104	3	∈	∈	PROPN
cana-3969	104	4	𝑆	𝑆	PROPN
cana-3969	104	5	,	,	PUNCT
cana-3969	104	6	𝑟𝑦	𝑟𝑦	NOUN
cana-3969	104	7	=	=	SYM
cana-3969	104	8	𝑥	𝑥	PROPN
cana-3969	104	9	}	}	PUNCT
cana-3969	104	10	≤	≤	NOUN
cana-3969	104	11	𝜉(𝑥	𝜉(𝑥	NOUN
cana-3969	104	12	)	)	PUNCT
cana-3969	104	13	∀	∀	PUNCT
cana-3969	105	1	𝑥	𝑥	PRON
cana-3969	105	2	∈	∈	PROPN
cana-3969	105	3	𝑦	𝑦	NOUN
cana-3969	105	4	𝑟𝜁	𝑟𝜁	PRON
cana-3969	105	5	⊆	⊆	NUM
cana-3969	105	6	𝜉	𝜉	ADP
cana-3969	105	7	𝐼𝑓	𝐼𝑓	NOUN
cana-3969	105	8	𝑟𝜁	𝑟𝜁	ADP
cana-3969	105	9	⊆	⊆	NUM
cana-3969	105	10	𝜉	𝜉	NOUN
cana-3969	105	11	then	then	ADV
cana-3969	105	12	𝜉(𝑟𝑥	𝜉(𝑟𝑥	NUM
cana-3969	105	13	)	)	PUNCT
cana-3969	105	14	≥	≥	NOUN
cana-3969	105	15	𝑟𝜁(𝑟𝑥	𝑟𝜁(𝑟𝑥	NOUN
cana-3969	105	16	)	)	PUNCT
cana-3969	105	17	≥	≥	NOUN
cana-3969	105	18	𝜁(𝑥	𝜁(𝑥	NOUN
cana-3969	105	19	)	)	PUNCT
cana-3969	105	20	∀𝑥	∀𝑥	PROPN
cana-3969	105	21	∈	∈	PROPN
cana-3969	105	22	𝑆	𝑆	PROPN
cana-3969	105	23	(	(	PUNCT
cana-3969	105	24	3	3	NUM
cana-3969	105	25	)	)	PUNCT
cana-3969	105	26	.	.	PUNCT
cana-3969	106	1	by	by	ADP
cana-3969	106	2	definition	definition	NOUN
cana-3969	106	3	of	of	ADP
cana-3969	106	4	rζ	rζ	NOUN
cana-3969	106	5	consider	consider	VERB
cana-3969	106	6	⨁	⨁	PROPN
cana-3969	106	7	(	(	PUNCT
cana-3969	106	8	𝑟𝜁	𝑟𝜁	X
cana-3969	106	9	+	+	PUNCT
cana-3969	106	10	𝑠𝜉)(𝑟𝑥	𝑠𝜉)(𝑟𝑥	PROPN
cana-3969	106	11	+	+	NUM
cana-3969	106	12	𝑠𝑦	𝑠𝑦	NOUN
cana-3969	106	13	)	)	PUNCT
cana-3969	106	14	≥	≥	NOUN
cana-3969	106	15	(	(	PUNCT
cana-3969	106	16	𝑟𝜁)(𝑟𝑥	𝑟𝜁)(𝑟𝑥	NOUN
cana-3969	106	17	)	)	PUNCT
cana-3969	106	18	∧	∧	PROPN
cana-3969	106	19	(	(	PUNCT
cana-3969	106	20	𝑠𝜉)(𝑠𝑦	𝑠𝜉)(𝑠𝑦	PROPN
cana-3969	106	21	)	)	PUNCT
cana-3969	106	22	≥	≥	NOUN
cana-3969	106	23	𝜁(𝑥	𝜁(𝑥	NOUN
cana-3969	106	24	)	)	PUNCT
cana-3969	106	25	∧	∧	PROPN
cana-3969	106	26	𝜉(𝑦	𝜉(𝑦	NOUN
cana-3969	106	27	)	)	PUNCT
cana-3969	106	28	∀	∀	PUNCT
cana-3969	107	1	𝑥	𝑥	NOUN
cana-3969	107	2	,	,	PUNCT
cana-3969	107	3	𝑦	𝑦	NOUN
cana-3969	107	4	∈	∈	PROPN
cana-3969	107	5	𝑆	𝑆	PROPN
cana-3969	107	6	(	(	PUNCT
cana-3969	107	7	4	4	NUM
cana-3969	107	8	)	)	PUNCT
cana-3969	107	9	.	.	PUNCT
cana-3969	108	1	suppose	suppose	VERB
cana-3969	108	2	that	that	SCONJ
cana-3969	108	3	𝜉(𝑟𝑥	𝜉(𝑟𝑥	NOUN
cana-3969	108	4	+	+	CCONJ
cana-3969	108	5	𝑠𝑦	𝑠𝑦	NOUN
cana-3969	108	6	)	)	PUNCT
cana-3969	108	7	≥	≥	NOUN
cana-3969	108	8	𝜁(𝑥	𝜁(𝑥	NOUN
cana-3969	108	9	)	)	PUNCT
cana-3969	108	10	∧	∧	NOUN
cana-3969	108	11	𝜎(𝑦	𝜎(𝑦	PROPN
cana-3969	108	12	)	)	PUNCT
cana-3969	108	13	∀	∀	PUNCT
cana-3969	109	1	𝑥	𝑥	NOUN
cana-3969	109	2	,	,	PUNCT
cana-3969	109	3	𝑦	𝑦	NOUN
cana-3969	109	4	∈	∈	PROPN
cana-3969	109	5	𝑆	𝑆	PROPN
cana-3969	109	6	then	then	ADV
cana-3969	109	7	(	(	PUNCT
cana-3969	109	8	𝑟𝜁	𝑟𝜁	X
cana-3969	109	9	+	+	NOUN
cana-3969	109	10	𝑠𝜎)(𝑧	𝑠𝜎)(𝑧	NOUN
cana-3969	109	11	)	)	PUNCT
cana-3969	109	12	=	=	SYM
cana-3969	109	13	𝑉{(𝑟𝜁)(𝑢	𝑉{(𝑟𝜁)(𝑢	NOUN
cana-3969	109	14	)	)	PUNCT
cana-3969	109	15	∧	∧	PROPN
cana-3969	109	16	(	(	PUNCT
cana-3969	109	17	𝑠𝜎)((𝑣	𝑠𝜎)((𝑣	NOUN
cana-3969	109	18	)	)	PUNCT
cana-3969	109	19	)	)	PUNCT
cana-3969	110	1	⁄	⁄	PROPN
cana-3969	110	2	(	(	PUNCT
cana-3969	110	3	𝑢	𝑢	X
cana-3969	110	4	,	,	PUNCT
cana-3969	110	5	𝑣	𝑣	NOUN
cana-3969	110	6	)	)	PUNCT
cana-3969	110	7	∈	∈	PROPN
cana-3969	110	8	𝑆	𝑆	PROPN
cana-3969	110	9	;	;	PUNCT
cana-3969	110	10	𝑢	𝑢	X
cana-3969	110	11	+	+	X
cana-3969	110	12	𝑣	𝑣	X
cana-3969	110	13	=	=	PUNCT
cana-3969	110	14	𝑧	𝑧	NOUN
cana-3969	110	15	}	}	PUNCT
cana-3969	110	16	=	=	SYM
cana-3969	110	17	𝑉{(𝑉{𝜁(𝑥)	𝑉{(𝑉{𝜁(𝑥)	VERB
cana-3969	110	18	│	│	ADJ
cana-3969	110	19	𝑥	𝑥	PRON
cana-3969	110	20	∈	∈	PROPN
cana-3969	110	21	𝑆	𝑆	PROPN
cana-3969	110	22	,	,	PUNCT
cana-3969	110	23	𝑟𝑥	𝑟𝑥	VERB
cana-3969	110	24	=	=	PUNCT
cana-3969	110	25	𝑢	𝑢	X
cana-3969	110	26	}	}	PUNCT
cana-3969	110	27	)	)	PUNCT
cana-3969	110	28	∧	∧	PROPN
cana-3969	110	29	(	(	PUNCT
cana-3969	110	30	𝑉{𝜎(𝑦	𝑉{𝜎(𝑦	ADJ
cana-3969	110	31	)	)	PUNCT
cana-3969	111	1	⁄	⁄	ADP
cana-3969	111	2	𝑦	𝑦	NUM
cana-3969	111	3	∈	∈	PROPN
cana-3969	111	4	𝑆	𝑆	PROPN
cana-3969	111	5	,	,	PUNCT
cana-3969	111	6	𝑠𝑦	𝑠𝑦	ADP
cana-3969	111	7	=	=	SYM
cana-3969	111	8	𝑣	𝑣	X
cana-3969	111	9	}	}	PUNCT
cana-3969	111	10	)	)	PUNCT
cana-3969	112	1	⁄	⁄	ADP
cana-3969	112	2	𝑢	𝑢	X
cana-3969	112	3	,	,	PUNCT
cana-3969	112	4	𝑣	𝑣	PRON
cana-3969	112	5	∈	∈	PROPN
cana-3969	112	6	𝑆	𝑆	PROPN
cana-3969	112	7	,	,	PUNCT
cana-3969	112	8	𝑢	𝑢	PROPN
cana-3969	112	9	+	+	X
cana-3969	112	10	𝑣	𝑣	X
cana-3969	112	11	=	=	PUNCT
cana-3969	112	12	𝑧	𝑧	NOUN
cana-3969	112	13	}	}	PUNCT
cana-3969	112	14	=	=	SYM
cana-3969	112	15	𝑉{𝜁(𝑥	𝑉{𝜁(𝑥	NOUN
cana-3969	112	16	)	)	PUNCT
cana-3969	112	17	∧	∧	PROPN
cana-3969	112	18	𝜎(𝑦)|𝑥	𝜎(𝑦)|𝑥	PROPN
cana-3969	112	19	,	,	PUNCT
cana-3969	112	20	𝑦	𝑦	NOUN
cana-3969	112	21	∈	∈	PROPN
cana-3969	112	22	𝑆	𝑆	PROPN
cana-3969	112	23	,	,	PUNCT
cana-3969	112	24	𝑟𝑥	𝑟𝑥	ADP
cana-3969	112	25	+	+	ADJ
cana-3969	112	26	𝑠𝑦	𝑠𝑦	X
cana-3969	112	27	=	=	SYM
cana-3969	112	28	𝑧	𝑧	NOUN
cana-3969	112	29	}	}	PUNCT
cana-3969	112	30	=	=	SYM
cana-3969	112	31	𝜉(𝑥	𝜉(𝑥	NOUN
cana-3969	112	32	)	)	PUNCT
cana-3969	112	33	∀	∀	PUNCT
cana-3969	113	1	𝑧	𝑧	DET
cana-3969	113	2	∈	∈	PROPN
cana-3969	113	3	𝑆	𝑆	PROPN
cana-3969	113	4	hence	hence	ADV
cana-3969	113	5	𝑟𝜁	𝑟𝜁	PRON
cana-3969	113	6	+	+	CCONJ
cana-3969	113	7	𝑠𝜎	𝑠𝜎	NOUN
cana-3969	113	8	⊆	⊆	NUM
cana-3969	113	9	𝜉	𝜉	NOUN
cana-3969	113	10	conversely	conversely	ADV
cana-3969	113	11	,	,	PUNCT
cana-3969	113	12	suppose	suppose	VERB
cana-3969	113	13	that	that	PRON
cana-3969	113	14	.	.	PUNCT
cana-3969	114	1	𝑟𝜁	𝑟𝜁	X
cana-3969	114	2	+	+	CCONJ
cana-3969	114	3	𝑠𝜎	𝑠𝜎	PROPN
cana-3969	115	1	⊆	⊆	NUM
cana-3969	115	2	𝜉	𝜉	NOUN
cana-3969	115	3	𝜉(𝑟𝑥	𝜉(𝑟𝑥	PROPN
cana-3969	115	4	+	+	NUM
cana-3969	115	5	𝑠𝑦	𝑠𝑦	NOUN
cana-3969	115	6	)	)	PUNCT
cana-3969	115	7	≥	≥	NOUN
cana-3969	115	8	(	(	PUNCT
cana-3969	115	9	𝑟𝜁	𝑟𝜁	X
cana-3969	115	10	+	+	CCONJ
cana-3969	115	11	𝑠𝜎)(𝑟𝑥	𝑠𝜎)(𝑟𝑥	PROPN
cana-3969	115	12	+	+	NUM
cana-3969	115	13	𝑠𝑦	𝑠𝑦	NOUN
cana-3969	115	14	)	)	PUNCT
cana-3969	115	15	≥	≥	NOUN
cana-3969	115	16	(	(	PUNCT
cana-3969	115	17	𝑟𝜁)(𝑟𝑥	𝑟𝜁)(𝑟𝑥	NOUN
cana-3969	115	18	)	)	PUNCT
cana-3969	115	19	∧	∧	PROPN
cana-3969	115	20	(	(	PUNCT
cana-3969	115	21	𝑠𝜎)(𝑠𝑦	𝑠𝜎)(𝑠𝑦	PROPN
cana-3969	115	22	)	)	PUNCT
cana-3969	115	23	≥	≥	NOUN
cana-3969	115	24	𝜁(𝑥	𝜁(𝑥	NOUN
cana-3969	115	25	)	)	PUNCT
cana-3969	115	26	∧	∧	PROPN
cana-3969	115	27	𝜎(𝑦)[∵	𝜎(𝑦)[∵	PROPN
cana-3969	115	28	(	(	PUNCT
cana-3969	115	29	1	1	NUM
cana-3969	115	30	)	)	PUNCT
cana-3969	115	31	]	]	PUNCT
cana-3969	116	1	communications	communication	NOUN
cana-3969	116	2	on	on	ADP
cana-3969	116	3	applied	apply	VERB
cana-3969	116	4	nonlinear	nonlinear	ADJ
cana-3969	116	5	analysis	analysis	NOUN
cana-3969	116	6	issn	issn	NOUN
cana-3969	116	7	:	:	PUNCT
cana-3969	116	8	1074	1074	NUM
cana-3969	116	9	-	-	PUNCT
cana-3969	116	10	133x	133x	NUM
cana-3969	116	11	vol	vol	NOUN
cana-3969	116	12	32	32	NUM
cana-3969	116	13	no	no	NOUN
cana-3969	116	14	.	.	PUNCT
cana-3969	117	1	9s	9s	NUM
cana-3969	117	2	(	(	PUNCT
cana-3969	117	3	2025	2025	NUM
cana-3969	117	4	)	)	PUNCT
cana-3969	117	5	641	641	NUM
cana-3969	117	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3969	117	7	∀	∀	X
cana-3969	118	1	𝑥	𝑥	NOUN
cana-3969	118	2	,	,	PUNCT
cana-3969	118	3	𝑦	𝑦	NOUN
cana-3969	118	4	∈	∈	NOUN
cana-3969	118	5	𝑆	𝑆	PROPN
cana-3969	118	6	5	5	NUM
cana-3969	118	7	.	.	PUNCT
cana-3969	119	1	lemma	lemma	PROPN
cana-3969	119	2	:	:	PUNCT
cana-3969	119	3	prove	prove	VERB
cana-3969	119	4	that	that	SCONJ
cana-3969	119	5	(	(	PUNCT
cana-3969	119	6	𝜁	𝜁	PROPN
cana-3969	119	7	⊙	⊙	X
cana-3969	119	8	𝜎)(𝑥	𝜎)(𝑥	ADV
cana-3969	119	9	+	+	CCONJ
cana-3969	119	10	𝑦	𝑦	X
cana-3969	119	11	)	)	PUNCT
cana-3969	119	12	≥	≥	NOUN
cana-3969	119	13	(	(	PUNCT
cana-3969	119	14	𝜁	𝜁	PROPN
cana-3969	119	15	⊙	⊙	NOUN
cana-3969	119	16	𝜎)(𝑥	𝜎)(𝑥	ADV
cana-3969	119	17	)	)	PUNCT
cana-3969	120	1	∧	∧	PROPN
cana-3969	120	2	(	(	PUNCT
cana-3969	120	3	𝜁	𝜁	PROPN
cana-3969	120	4	⊙	⊙	PROPN
cana-3969	120	5	𝜎)(𝑦)(𝜁	𝜎)(𝑦)(𝜁	PROPN
cana-3969	120	6	⊙	⊙	PROPN
cana-3969	120	7	𝜎)(𝑟𝑥	𝜎)(𝑟𝑥	NUM
cana-3969	120	8	)	)	PUNCT
cana-3969	121	1	=	=	PUNCT
cana-3969	121	2	𝑉{𝛬(𝑖	𝑉{𝛬(𝑖	NOUN
cana-3969	121	3	=	=	SYM
cana-3969	121	4	1	1	NUM
cana-3969	121	5	)	)	PUNCT
cana-3969	121	6	𝑛	𝑛	PRON
cana-3969	121	7	𝜁(𝑠𝑖	𝜁(𝑠𝑖	X
cana-3969	121	8	)	)	PUNCT
cana-3969	121	9	∧	∧	PROPN
cana-3969	121	10	𝜁(𝑧𝑖)𝑠𝑖	𝜁(𝑧𝑖)𝑠𝑖	NOUN
cana-3969	121	11	,	,	PUNCT
cana-3969	121	12	𝑧	𝑧	PROPN
cana-3969	121	13	∈	∈	PROPN
cana-3969	121	14	𝑆	𝑆	PROPN
cana-3969	121	15	;	;	PUNCT
cana-3969	121	16	1	1	NUM
cana-3969	121	17	≤	≤	NUM
cana-3969	121	18	𝑖	𝑖	SYM
cana-3969	121	19	≤	≤	NUM
cana-3969	121	20	𝑛	𝑛	NOUN
cana-3969	121	21	,	,	PUNCT
cana-3969	121	22	𝑛	𝑛	DET
cana-3969	121	23	∈	∈	PROPN
cana-3969	121	24	𝑁	𝑁	PROPN
cana-3969	121	25	,	,	PUNCT
cana-3969	121	26	∑	∑	ADV
cana-3969	121	27	𝑠𝑖	𝑠𝑖	NOUN
cana-3969	121	28	𝑧𝑖	𝑧𝑖	INTJ
cana-3969	121	29	𝑛	𝑛	PRON
cana-3969	121	30	𝑖=1	𝑖=1	PUNCT
cana-3969	121	31	=	=	SYM
cana-3969	121	32	𝑟𝑥	𝑟𝑥	PROPN
cana-3969	121	33	}	}	PUNCT
cana-3969	121	34	≥	≥	NOUN
cana-3969	121	35	ѵ{𝛬(𝑖	ѵ{𝛬(𝑖	NOUN
cana-3969	121	36	=	=	SYM
cana-3969	121	37	1	1	X
cana-3969	121	38	)	)	PUNCT
cana-3969	121	39	𝑛	𝑛	NOUN
cana-3969	121	40	(	(	PUNCT
cana-3969	121	41	𝜁(𝑟	𝜁(𝑟	PROPN
cana-3969	121	42	𝑟𝑖	𝑟𝑖	NOUN
cana-3969	121	43	)	)	PUNCT
cana-3969	121	44	∧	∧	NOUN
cana-3969	121	45	𝜎(𝑥𝑖	𝜎(𝑥𝑖	PROPN
cana-3969	121	46	)	)	PUNCT
cana-3969	121	47	)	)	PUNCT
cana-3969	121	48	𝑟𝑖	𝑟𝑖	X
cana-3969	121	49	,	,	PUNCT
cana-3969	121	50	𝑥𝑖	𝑥𝑖	PROPN
cana-3969	121	51	∈	∈	PROPN
cana-3969	121	52	𝑆	𝑆	PROPN
cana-3969	121	53	;	;	PUNCT
cana-3969	121	54	1	1	NUM
cana-3969	121	55	≤	≤	NUM
cana-3969	121	56	𝑖	𝑖	SYM
cana-3969	121	57	≤	≤	NUM
cana-3969	121	58	𝑛	𝑛	NOUN
cana-3969	121	59	,	,	PUNCT
cana-3969	121	60	𝑛	𝑛	DET
cana-3969	121	61	∈	∈	PROPN
cana-3969	121	62	𝑁	𝑁	PROPN
cana-3969	121	63	,	,	PUNCT
cana-3969	121	64	∑	∑	ADV
cana-3969	121	65	(	(	PUNCT
cana-3969	121	66	𝑟𝑟𝑖)𝑥𝑖	𝑟𝑟𝑖)𝑥𝑖	VERB
cana-3969	121	67	𝑛	𝑛	PRON
cana-3969	121	68	𝑖=1	𝑖=1	PUNCT
cana-3969	121	69	=	=	SYM
cana-3969	121	70	𝑟𝑥	𝑟𝑥	PROPN
cana-3969	121	71	}	}	PUNCT
cana-3969	121	72	≥	≥	NOUN
cana-3969	121	73	ѵ{𝛬(𝑖	ѵ{𝛬(𝑖	NOUN
cana-3969	121	74	=	=	SYM
cana-3969	121	75	1	1	X
cana-3969	121	76	)	)	PUNCT
cana-3969	121	77	𝑛	𝑛	PROPN
cana-3969	121	78	(	(	PUNCT
cana-3969	121	79	𝜁	𝜁	PROPN
cana-3969	121	80	(	(	PUNCT
cana-3969	121	81	𝑟𝑖	𝑟𝑖	NOUN
cana-3969	121	82	)	)	PUNCT
cana-3969	121	83	∧	∧	NOUN
cana-3969	121	84	𝜎(𝑥𝑖	𝜎(𝑥𝑖	PROPN
cana-3969	121	85	)	)	PUNCT
cana-3969	121	86	)	)	PUNCT
cana-3969	121	87	𝑟𝑖	𝑟𝑖	X
cana-3969	121	88	,	,	PUNCT
cana-3969	121	89	𝑥𝑖	𝑥𝑖	PROPN
cana-3969	121	90	∈	∈	PROPN
cana-3969	121	91	𝑆	𝑆	PROPN
cana-3969	121	92	;	;	PUNCT
cana-3969	121	93	1	1	NUM
cana-3969	121	94	≤	≤	NUM
cana-3969	121	95	𝑖	𝑖	SYM
cana-3969	121	96	≤	≤	NUM
cana-3969	121	97	𝑛	𝑛	NOUN
cana-3969	121	98	,	,	PUNCT
cana-3969	121	99	𝑛	𝑛	DET
cana-3969	121	100	∈	∈	PROPN
cana-3969	121	101	𝑁	𝑁	PROPN
cana-3969	121	102	,	,	PUNCT
cana-3969	121	103	∑	∑	ADV
cana-3969	121	104	(	(	PUNCT
cana-3969	121	105	𝑟𝑟𝑖)𝑥𝑖	𝑟𝑟𝑖)𝑥𝑖	VERB
cana-3969	121	106	𝑛	𝑛	PRON
cana-3969	121	107	𝑖=1	𝑖=1	PUNCT
cana-3969	121	108	=	=	SYM
cana-3969	121	109	𝑟𝑥	𝑟𝑥	PROPN
cana-3969	121	110	}	}	PUNCT
cana-3969	121	111	=	=	PUNCT
cana-3969	121	112	(	(	PUNCT
cana-3969	121	113	𝜁	𝜁	PROPN
cana-3969	121	114	⊙	⊙	NOUN
cana-3969	121	115	𝜎)(𝑥	𝜎)(𝑥	ADV
cana-3969	121	116	)	)	PUNCT
cana-3969	121	117	…	…	PUNCT
cana-3969	121	118	…	…	PUNCT
cana-3969	121	119	…	…	PUNCT
cana-3969	121	120	.	.	PUNCT
cana-3969	121	121	.	.	PUNCT
cana-3969	122	1	(	(	PUNCT
cana-3969	122	2	1	1	X
cana-3969	122	3	)	)	PUNCT
cana-3969	122	4	similarly	similarly	ADV
cana-3969	122	5	(	(	PUNCT
cana-3969	122	6	𝜁	𝜁	PROPN
cana-3969	122	7	⊙	⊙	PROPN
cana-3969	122	8	𝜎)(𝑟𝑦	𝜎)(𝑟𝑦	PROPN
cana-3969	122	9	)	)	PUNCT
cana-3969	122	10	≥	≥	PROPN
cana-3969	122	11	(	(	PUNCT
cana-3969	122	12	𝜁	𝜁	PROPN
cana-3969	122	13	⊙	⊙	PROPN
cana-3969	122	14	𝜎)(𝑦	𝜎)(𝑦	NOUN
cana-3969	122	15	)	)	PUNCT
cana-3969	122	16	…	…	PUNCT
cana-3969	122	17	…	…	PUNCT
cana-3969	122	18	…	…	PUNCT
cana-3969	122	19	…	…	PUNCT
cana-3969	122	20	(	(	PUNCT
cana-3969	122	21	2	2	NUM
cana-3969	122	22	)	)	PUNCT
cana-3969	122	23	by	by	ADP
cana-3969	122	24	definition	definition	NOUN
cana-3969	122	25	additive	additive	NOUN
cana-3969	122	26	of	of	ADP
cana-3969	122	27	(	(	PUNCT
cana-3969	122	28	1	1	NUM
cana-3969	122	29	)	)	PUNCT
cana-3969	122	30	&	&	CCONJ
cana-3969	122	31	(	(	PUNCT
cana-3969	122	32	2	2	NUM
cana-3969	122	33	)	)	PUNCT
cana-3969	122	34	(	(	PUNCT
cana-3969	122	35	𝜁	𝜁	PROPN
cana-3969	122	36	⊙	⊙	X
cana-3969	122	37	𝜎)(𝑟𝑥	𝜎)(𝑟𝑥	NUM
cana-3969	122	38	+	+	CCONJ
cana-3969	122	39	𝑟𝑦	𝑟𝑦	X
cana-3969	122	40	)	)	PUNCT
cana-3969	122	41	≥	≥	NOUN
cana-3969	122	42	(	(	PUNCT
cana-3969	122	43	𝜁	𝜁	PROPN
cana-3969	122	44	⊙	⊙	NOUN
cana-3969	122	45	𝜎)(𝑥	𝜎)(𝑥	ADV
cana-3969	122	46	)	)	PUNCT
cana-3969	122	47	∧	∧	PROPN
cana-3969	122	48	(	(	PUNCT
cana-3969	122	49	𝜁	𝜁	PROPN
cana-3969	122	50	⊙	⊙	PROPN
cana-3969	122	51	𝜎)(𝑦	𝜎)(𝑦	NOUN
cana-3969	122	52	)	)	PUNCT
cana-3969	122	53	put	put	VERB
cana-3969	122	54	𝑟	𝑟	NOUN
cana-3969	122	55	=	=	SYM
cana-3969	122	56	1	1	NUM
cana-3969	122	57	(	(	PUNCT
cana-3969	122	58	𝜁	𝜁	PROPN
cana-3969	122	59	⊙	⊙	X
cana-3969	122	60	𝜎)(𝑥	𝜎)(𝑥	ADV
cana-3969	123	1	+	+	CCONJ
cana-3969	123	2	𝑦	𝑦	X
cana-3969	123	3	)	)	PUNCT
cana-3969	123	4	≥	≥	NOUN
cana-3969	123	5	(	(	PUNCT
cana-3969	123	6	𝜁	𝜁	PROPN
cana-3969	123	7	⊙	⊙	NOUN
cana-3969	123	8	𝜎)(𝑥	𝜎)(𝑥	ADV
cana-3969	123	9	)	)	PUNCT
cana-3969	124	1	∧	∧	PROPN
cana-3969	124	2	(	(	PUNCT
cana-3969	124	3	𝜁	𝜁	PROPN
cana-3969	124	4	⊙	⊙	PROPN
cana-3969	124	5	𝜎)(𝑦	𝜎)(𝑦	NOUN
cana-3969	124	6	)	)	PUNCT
cana-3969	124	7	definition	definition	NOUN
cana-3969	124	8	4.2	4.2	NUM
cana-3969	124	9	:	:	PUNCT
cana-3969	125	1	[	[	X
cana-3969	125	2	13	13	NUM
cana-3969	125	3	]	]	PUNCT
cana-3969	125	4	a	a	DET
cana-3969	125	5	fuzzy	fuzzy	ADJ
cana-3969	125	6	ideal	ideal	ADJ
cana-3969	125	7	ζ	ζ	NOUN
cana-3969	125	8	of	of	ADP
cana-3969	125	9	a	a	DET
cana-3969	125	10	γsemigroup	γsemigroup	NOUN
cana-3969	125	11	ꟊ	ꟊ	NOUN
cana-3969	125	12	is	be	AUX
cana-3969	125	13	called	call	VERB
cana-3969	125	14	a	a	DET
cana-3969	125	15	fuzzy	fuzzy	ADJ
cana-3969	125	16	prime	prime	ADJ
cana-3969	125	17	ideal	ideal	NOUN
cana-3969	125	18	if	if	SCONJ
cana-3969	125	19	𝜁(𝑥𝛾𝑦)(𝛾∈𝛤	𝜁(𝑥𝛾𝑦)(𝛾∈𝛤	ADP
cana-3969	125	20	)	)	PUNCT
cana-3969	125	21	𝑖𝑛𝑓	𝑖𝑛𝑓	NOUN
cana-3969	126	1	=	=	PUNCT
cana-3969	126	2	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-3969	126	3	{	{	PUNCT
cana-3969	126	4	𝜁(𝑥	𝜁(𝑥	NOUN
cana-3969	126	5	)	)	PUNCT
cana-3969	126	6	,	,	PUNCT
cana-3969	126	7	𝜁(𝑦	𝜁(𝑦	PROPN
cana-3969	126	8	)	)	PUNCT
cana-3969	126	9	}	}	PUNCT
cana-3969	126	10	for	for	ADP
cana-3969	126	11	all	all	DET
cana-3969	126	12	x	x	NOUN
cana-3969	126	13	,	,	PUNCT
cana-3969	126	14	y	y	PROPN
cana-3969	126	15	∈	∈	PROPN
cana-3969	126	16	ꟊ	ꟊ	X
cana-3969	126	17	and	and	CCONJ
cana-3969	126	18	γ∈	γ∈	X
cana-3969	126	19	γ	γ	PROPN
cana-3969	126	20	.	.	PROPN
cana-3969	126	21	example	example	NOUN
cana-3969	126	22	:	:	PUNCT
cana-3969	126	23	let	let	VERB
cana-3969	126	24	ꟊ	ꟊ	PRON
cana-3969	126	25	be	be	AUX
cana-3969	126	26	the	the	DET
cana-3969	126	27	set	set	NOUN
cana-3969	126	28	of	of	ADP
cana-3969	126	29	all	all	DET
cana-3969	126	30	1x2	1x2	NUM
cana-3969	126	31	matrices	matrix	NOUN
cana-3969	126	32	,	,	PUNCT
cana-3969	126	33	γ	γ	X
cana-3969	126	34	be	be	VERB
cana-3969	126	35	the	the	DET
cana-3969	126	36	set	set	NOUN
cana-3969	126	37	of	of	ADP
cana-3969	126	38	all	all	DET
cana-3969	126	39	2x1	2x1	NUM
cana-3969	126	40	matrices	matrix	NOUN
cana-3969	126	41	,	,	PUNCT
cana-3969	126	42	then	then	ADV
cana-3969	126	43	ꟊ	ꟊ	PRON
cana-3969	126	44	is	be	AUX
cana-3969	126	45	γ	γ	X
cana-3969	126	46	-	-	PUNCT
cana-3969	126	47	semigroup	semigroup	NOUN
cana-3969	126	48	where	where	SCONJ
cana-3969	126	49	p	p	X
cana-3969	126	50	,	,	PUNCT
cana-3969	126	51	q∈ꟊ	q∈ꟊ	NOUN
cana-3969	126	52	;	;	PUNCT
cana-3969	126	53	α	α	X
cana-3969	126	54	,	,	PUNCT
cana-3969	126	55	β∈γ	β∈γ	NOUN
cana-3969	126	56	which	which	PRON
cana-3969	126	57	denotes	denote	VERB
cana-3969	126	58	the	the	DET
cana-3969	126	59	usual	usual	ADJ
cana-3969	126	60	matrix	matrix	NOUN
cana-3969	126	61	product	product	NOUN
cana-3969	126	62	.	.	PUNCT
cana-3969	127	1	let	let	VERB
cana-3969	127	2	ζ	ζ	PRON
cana-3969	127	3	:	:	PUNCT
cana-3969	127	4	ꟊ	ꟊ	X
cana-3969	127	5	→	→	SYM
cana-3969	127	6	[	[	X
cana-3969	127	7	0,1	0,1	NUM
cana-3969	127	8	]	]	PUNCT
cana-3969	127	9	be	be	AUX
cana-3969	127	10	defined	define	VERB
cana-3969	127	11	by	by	ADP
cana-3969	127	12	𝜁(𝑝	𝜁(𝑝	PROPN
cana-3969	127	13	)	)	PUNCT
cana-3969	128	1	=	=	PRON
cana-3969	128	2	{	{	PUNCT
cana-3969	128	3	0.1	0.1	NUM
cana-3969	128	4	𝑖𝑓	𝑖𝑓	NOUN
cana-3969	129	1	[	[	X
cana-3969	129	2	0	0	NUM
cana-3969	129	3	0	0	NUM
cana-3969	129	4	]	]	PUNCT
cana-3969	129	5	0.3	0.3	NUM
cana-3969	129	6	0.𝑤	0.𝑤	NOUN
cana-3969	129	7	let	let	VERB
cana-3969	129	8	𝑝	𝑝	NOUN
cana-3969	129	9	=	=	PUNCT
cana-3969	130	1	[	[	X
cana-3969	130	2	1	1	NUM
cana-3969	130	3	0	0	NUM
cana-3969	130	4	]	]	PUNCT
cana-3969	130	5	;	;	PUNCT
cana-3969	130	6	𝑞	𝑞	X
cana-3969	130	7	=	=	PUNCT
cana-3969	131	1	[	[	X
cana-3969	131	2	0	0	NUM
cana-3969	131	3	0	0	NUM
cana-3969	131	4	]	]	PUNCT
cana-3969	131	5	and	and	CCONJ
cana-3969	131	6	𝛾	𝛾	X
cana-3969	131	7	=	=	PUNCT
cana-3969	131	8	[	[	PUNCT
cana-3969	131	9	1	1	NUM
cana-3969	131	10	1	1	NUM
cana-3969	131	11	]	]	PUNCT
cana-3969	131	12	then	then	ADV
cana-3969	131	13	𝜁(𝑝𝛾𝑞	𝜁(𝑝𝛾𝑞	NOUN
cana-3969	131	14	)	)	PUNCT
cana-3969	131	15	=	=	SYM
cana-3969	131	16	𝜁	𝜁	PROPN
cana-3969	131	17	(	(	PUNCT
cana-3969	131	18	[	[	X
cana-3969	131	19	1	1	NUM
cana-3969	131	20	0	0	NUM
cana-3969	131	21	]	]	X
cana-3969	131	22	[	[	PUNCT
cana-3969	131	23	1	1	NUM
cana-3969	131	24	1	1	NUM
cana-3969	131	25	]	]	PUNCT
cana-3969	132	1	[	[	X
cana-3969	132	2	0	0	NUM
cana-3969	132	3	0	0	NUM
cana-3969	132	4	]	]	PUNCT
cana-3969	132	5	)	)	PUNCT
cana-3969	132	6	=	=	PRON
cana-3969	132	7	𝜁([1][0	𝜁([1][0	NOUN
cana-3969	132	8	0	0	NUM
cana-3969	132	9	]	]	PUNCT
cana-3969	132	10	)	)	PUNCT
cana-3969	132	11	=	=	SYM
cana-3969	132	12	𝜁([0	𝜁([0	NOUN
cana-3969	132	13	0	0	NUM
cana-3969	132	14	]	]	PUNCT
cana-3969	132	15	)	)	PUNCT
cana-3969	132	16	=	=	SYM
cana-3969	132	17	0.1	0.1	NUM
cana-3969	132	18	𝑀𝑎𝑥	𝑀𝑎𝑥	PROPN
cana-3969	132	19	{	{	PUNCT
cana-3969	132	20	𝜁(𝑝	𝜁(𝑝	PROPN
cana-3969	132	21	)	)	PUNCT
cana-3969	132	22	,	,	PUNCT
cana-3969	132	23	𝜁(𝑞	𝜁(𝑞	NOUN
cana-3969	132	24	)	)	PUNCT
cana-3969	132	25	}	}	PUNCT
cana-3969	132	26	=	=	PUNCT
cana-3969	132	27	𝑚𝑎𝑥	𝑚𝑎𝑥	X
cana-3969	132	28	{	{	PUNCT
cana-3969	132	29	0.3,0.1	0.3,0.1	NOUN
cana-3969	132	30	}	}	PUNCT
cana-3969	132	31	=	=	NOUN
cana-3969	132	32	0.3	0.3	NUM
cana-3969	132	33	𝜁(𝑥𝛾𝑦	𝜁(𝑥𝛾𝑦	NUM
cana-3969	132	34	)	)	PUNCT
cana-3969	132	35	≤	≤	NUM
cana-3969	132	36	𝑚𝑎𝑥	𝑚𝑎𝑥	NOUN
cana-3969	132	37	{	{	PUNCT
cana-3969	132	38	𝜁(𝑥	𝜁(𝑥	NOUN
cana-3969	132	39	)	)	PUNCT
cana-3969	132	40	,	,	PUNCT
cana-3969	132	41	𝜁(𝑦	𝜁(𝑦	PROPN
cana-3969	132	42	)	)	PUNCT
cana-3969	132	43	}	}	PUNCT
cana-3969	132	44	definition	definition	NOUN
cana-3969	132	45	4.3	4.3	NUM
cana-3969	132	46	:	:	PUNCT
cana-3969	132	47	a	a	DET
cana-3969	132	48	fuzzy	fuzzy	ADJ
cana-3969	132	49	ideal	ideal	ADJ
cana-3969	132	50	ζ	ζ	NOUN
cana-3969	132	51	of	of	ADP
cana-3969	132	52	a	a	DET
cana-3969	132	53	ternary	ternary	ADJ
cana-3969	132	54	γsemigroup	γsemigroup	NOUN
cana-3969	132	55	ꟊ	ꟊ	INTJ
cana-3969	132	56	is	be	AUX
cana-3969	132	57	called	call	VERB
cana-3969	132	58	a	a	DET
cana-3969	132	59	fuzzy	fuzzy	ADJ
cana-3969	132	60	prime	prime	ADJ
cana-3969	132	61	ideal	ideal	NOUN
cana-3969	132	62	if	if	SCONJ
cana-3969	132	63	inf	inf	NOUN
cana-3969	132	64	¦	¦	X
cana-3969	132	65	(γ∈γ	(γ∈γ	NUM
cana-3969	132	66	)	)	PUNCT
cana-3969	132	67	ζ(pαqβr	ζ(pαqβr	NOUN
cana-3969	132	68	)	)	PUNCT
cana-3969	132	69	=	=	SYM
cana-3969	132	70	max	max	PROPN
cana-3969	132	71	{	{	PUNCT
cana-3969	132	72	ζ(p	ζ(p	PROPN
cana-3969	132	73	)	)	PUNCT
cana-3969	132	74	,	,	PUNCT
cana-3969	132	75	ζ(q),ζ(r	ζ(q),ζ(r	NOUN
cana-3969	132	76	)	)	PUNCT
cana-3969	132	77	}	}	PUNCT
cana-3969	132	78	for	for	ADP
cana-3969	132	79	all	all	DET
cana-3969	132	80	p	p	NOUN
cana-3969	132	81	,	,	PUNCT
cana-3969	132	82	q	q	ADJ
cana-3969	132	83	,	,	PUNCT
cana-3969	132	84	r	r	NOUN
cana-3969	132	85	∈	∈	PROPN
cana-3969	132	86	ꟊ	ꟊ	X
cana-3969	132	87	and	and	CCONJ
cana-3969	132	88	γ∈	γ∈	X
cana-3969	132	89	γ	γ	PROPN
cana-3969	132	90	.	.	PROPN
cana-3969	132	91	example	example	NOUN
cana-3969	132	92	:	:	PUNCT
cana-3969	132	93	let	let	VERB
cana-3969	132	94	ꟊ	ꟊ	PRON
cana-3969	132	95	be	be	AUX
cana-3969	132	96	the	the	DET
cana-3969	132	97	set	set	NOUN
cana-3969	132	98	of	of	ADP
cana-3969	132	99	all	all	DET
cana-3969	132	100	1x2	1x2	NUM
cana-3969	132	101	matrices	matrix	NOUN
cana-3969	132	102	,	,	PUNCT
cana-3969	132	103	γ	γ	X
cana-3969	132	104	be	be	VERB
cana-3969	132	105	the	the	DET
cana-3969	132	106	set	set	NOUN
cana-3969	132	107	of	of	ADP
cana-3969	132	108	all	all	DET
cana-3969	132	109	2x1	2x1	NUM
cana-3969	132	110	matrices	matrix	NOUN
cana-3969	132	111	,	,	PUNCT
cana-3969	132	112	then	then	ADV
cana-3969	132	113	ꟊ	ꟊ	PRON
cana-3969	132	114	is	be	AUX
cana-3969	132	115	ternary	ternary	ADJ
cana-3969	132	116	γsemigroup	γsemigroup	NOUN
cana-3969	132	117	where	where	SCONJ
cana-3969	132	118	p	p	X
cana-3969	132	119	,	,	PUNCT
cana-3969	132	120	q∈ꟊ	q∈ꟊ	NOUN
cana-3969	132	121	;	;	PUNCT
cana-3969	132	122	α	α	X
cana-3969	132	123	,	,	PUNCT
cana-3969	132	124	β∈γ	β∈γ	NOUN
cana-3969	132	125	which	which	PRON
cana-3969	132	126	denotes	denote	VERB
cana-3969	132	127	the	the	DET
cana-3969	132	128	usual	usual	ADJ
cana-3969	132	129	matrix	matrix	NOUN
cana-3969	132	130	product	product	NOUN
cana-3969	132	131	.	.	PUNCT
cana-3969	133	1	let	let	VERB
cana-3969	133	2	ζ	ζ	PRON
cana-3969	133	3	:	:	PUNCT
cana-3969	133	4	ꟊ	ꟊ	X
cana-3969	133	5	→	→	SYM
cana-3969	133	6	[	[	X
cana-3969	133	7	0,1	0,1	NUM
cana-3969	133	8	]	]	PUNCT
cana-3969	133	9	be	be	AUX
cana-3969	133	10	defined	define	VERB
cana-3969	133	11	by	by	ADP
cana-3969	133	12	𝜁(𝑝	𝜁(𝑝	PROPN
cana-3969	133	13	)	)	PUNCT
cana-3969	134	1	=	=	PRON
cana-3969	134	2	{	{	PUNCT
cana-3969	134	3	0.5	0.5	NUM
cana-3969	134	4	𝑖𝑓	𝑖𝑓	NOUN
cana-3969	135	1	[	[	X
cana-3969	135	2	0	0	NUM
cana-3969	135	3	0	0	NUM
cana-3969	135	4	]	]	X
cana-3969	135	5	0.9	0.9	NUM
cana-3969	135	6	𝑖𝑓	𝑖𝑓	NOUN
cana-3969	136	1	[	[	X
cana-3969	136	2	1	1	NUM
cana-3969	136	3	2	2	NUM
cana-3969	136	4	]	]	PUNCT
cana-3969	136	5	0.7	0.7	NUM
cana-3969	136	6	0	0	NUM
cana-3969	136	7	.	.	PUNCT
cana-3969	137	1	𝑤	𝑤	ADP
cana-3969	137	2	communications	communication	NOUN
cana-3969	137	3	on	on	ADP
cana-3969	137	4	applied	apply	VERB
cana-3969	137	5	nonlinear	nonlinear	ADJ
cana-3969	137	6	analysis	analysis	NOUN
cana-3969	137	7	issn	issn	NOUN
cana-3969	137	8	:	:	PUNCT
cana-3969	137	9	1074	1074	NUM
cana-3969	137	10	-	-	PUNCT
cana-3969	137	11	133x	133x	NUM
cana-3969	137	12	vol	vol	NOUN
cana-3969	137	13	32	32	NUM
cana-3969	137	14	no	no	NOUN
cana-3969	137	15	.	.	PUNCT
cana-3969	138	1	9s	9s	NUM
cana-3969	138	2	(	(	PUNCT
cana-3969	138	3	2025	2025	NUM
cana-3969	138	4	)	)	PUNCT
cana-3969	138	5	642	642	NUM
cana-3969	138	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3969	138	7	let	let	VERB
cana-3969	138	8	𝑝	𝑝	NOUN
cana-3969	138	9	=	=	PUNCT
cana-3969	139	1	[	[	X
cana-3969	139	2	0	0	NUM
cana-3969	139	3	0	0	NUM
cana-3969	139	4	]	]	PUNCT
cana-3969	139	5	;	;	PUNCT
cana-3969	139	6	𝑞	𝑞	X
cana-3969	139	7	=	=	PUNCT
cana-3969	140	1	[	[	X
cana-3969	140	2	1	1	NUM
cana-3969	140	3	2	2	NUM
cana-3969	140	4	]	]	PUNCT
cana-3969	140	5	;	;	PUNCT
cana-3969	140	6	𝑟	𝑟	X
cana-3969	140	7	=	=	PUNCT
cana-3969	141	1	[	[	X
cana-3969	141	2	2	2	NUM
cana-3969	141	3	1	1	NUM
cana-3969	141	4	]	]	PUNCT
cana-3969	141	5	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-3969	141	6	𝛼	𝛼	NOUN
cana-3969	141	7	=	=	X
cana-3969	141	8	[	[	PUNCT
cana-3969	141	9	1	1	NUM
cana-3969	141	10	1	1	NUM
cana-3969	141	11	]	]	PUNCT
cana-3969	141	12	;	;	PUNCT
cana-3969	142	1	𝛽	𝛽	X
cana-3969	142	2	=	=	PUNCT
cana-3969	142	3	[	[	PUNCT
cana-3969	142	4	1	1	NUM
cana-3969	142	5	0	0	NUM
cana-3969	142	6	]	]	PUNCT
cana-3969	142	7	then	then	ADV
cana-3969	142	8	𝜁(𝑝𝛼𝑞𝛽𝑟	𝜁(𝑝𝛼𝑞𝛽𝑟	NOUN
cana-3969	142	9	)	)	PUNCT
cana-3969	142	10	=	=	SYM
cana-3969	142	11	𝜁	𝜁	PROPN
cana-3969	142	12	(	(	PUNCT
cana-3969	142	13	[	[	X
cana-3969	142	14	0	0	NUM
cana-3969	142	15	0	0	NUM
cana-3969	142	16	]	]	X
cana-3969	142	17	[	[	PUNCT
cana-3969	142	18	1	1	NUM
cana-3969	142	19	1	1	NUM
cana-3969	142	20	]	]	PUNCT
cana-3969	143	1	[	[	X
cana-3969	143	2	1	1	NUM
cana-3969	143	3	2	2	NUM
cana-3969	143	4	]	]	PUNCT
cana-3969	143	5	[	[	PUNCT
cana-3969	143	6	1	1	NUM
cana-3969	143	7	0	0	NUM
cana-3969	143	8	]	]	PUNCT
cana-3969	144	1	[	[	X
cana-3969	144	2	2	2	NUM
cana-3969	144	3	1	1	NUM
cana-3969	144	4	]	]	PUNCT
cana-3969	144	5	)	)	PUNCT
cana-3969	145	1	=	=	SYM
cana-3969	145	2	𝜁([0][1][2	𝜁([0][1][2	PROPN
cana-3969	145	3	1	1	NUM
cana-3969	145	4	]	]	PUNCT
cana-3969	145	5	)	)	PUNCT
cana-3969	145	6	=	=	VERB
cana-3969	145	7	𝜁([0	𝜁([0	NOUN
cana-3969	145	8	0	0	NUM
cana-3969	145	9	]	]	PUNCT
cana-3969	145	10	)	)	PUNCT
cana-3969	145	11	=	=	SYM
cana-3969	145	12	0.5	0.5	NUM
cana-3969	145	13	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-3969	145	14	{	{	PUNCT
cana-3969	145	15	𝜁(𝑝	𝜁(𝑝	PROPN
cana-3969	145	16	)	)	PUNCT
cana-3969	145	17	,	,	PUNCT
cana-3969	145	18	𝜁(𝑞	𝜁(𝑞	NOUN
cana-3969	145	19	)	)	PUNCT
cana-3969	145	20	,	,	PUNCT
cana-3969	145	21	𝜁(𝑟	𝜁(𝑟	NUM
cana-3969	145	22	)	)	PUNCT
cana-3969	145	23	}	}	PUNCT
cana-3969	145	24	=	=	PUNCT
cana-3969	145	25	𝑚𝑎𝑥	𝑚𝑎𝑥	X
cana-3969	145	26	{	{	PUNCT
cana-3969	145	27	0.5,0.9,0.7	0.5,0.9,0.7	NOUN
cana-3969	145	28	}	}	PUNCT
cana-3969	145	29	=	=	NOUN
cana-3969	145	30	0.9	0.9	NUM
cana-3969	145	31	𝜁(𝑝𝛼𝑞𝛽𝑟	𝜁(𝑝𝛼𝑞𝛽𝑟	NOUN
cana-3969	145	32	)	)	PUNCT
cana-3969	145	33	≤	≤	NUM
cana-3969	145	34	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-3969	145	35	{	{	PUNCT
cana-3969	145	36	𝜁(𝑝	𝜁(𝑝	PROPN
cana-3969	145	37	)	)	PUNCT
cana-3969	145	38	,	,	PUNCT
cana-3969	145	39	𝜁(𝑞	𝜁(𝑞	NOUN
cana-3969	145	40	)	)	PUNCT
cana-3969	145	41	,	,	PUNCT
cana-3969	145	42	𝜁(𝑟	𝜁(𝑟	NUM
cana-3969	145	43	)	)	PUNCT
cana-3969	145	44	}	}	PUNCT
cana-3969	145	45	definition	definition	NOUN
cana-3969	145	46	4.4:[18	4.4:[18	NUM
cana-3969	145	47	]	]	PUNCT
cana-3969	145	48	a	a	DET
cana-3969	145	49	fuzzy	fuzzy	ADJ
cana-3969	145	50	ideal	ideal	ADJ
cana-3969	145	51	ζ	ζ	NOUN
cana-3969	145	52	of	of	ADP
cana-3969	145	53	a	a	DET
cana-3969	145	54	γsemigroup	γsemigroup	NOUN
cana-3969	145	55	ꟊ	ꟊ	NOUN
cana-3969	145	56	is	be	AUX
cana-3969	145	57	called	call	VERB
cana-3969	145	58	a	a	DET
cana-3969	145	59	fuzzy	fuzzy	ADJ
cana-3969	145	60	semiprime	semiprime	NOUN
cana-3969	145	61	ideal	ideal	NOUN
cana-3969	145	62	if	if	SCONJ
cana-3969	145	63	ζ(x	ζ(x	NOUN
cana-3969	145	64	)	)	PUNCT
cana-3969	145	65	≥	≥	NOUN
cana-3969	145	66	𝑖𝑛𝑓	𝑖𝑛𝑓	NOUN
cana-3969	145	67	𝛾∈𝛤	𝛾∈𝛤	PROPN
cana-3969	145	68	ζ(xγx	ζ(xγx	NOUN
cana-3969	145	69	)	)	PUNCT
cana-3969	145	70	.	.	PUNCT
cana-3969	146	1	definition	definition	NOUN
cana-3969	146	2	4.5	4.5	NUM
cana-3969	146	3	:	:	PUNCT
cana-3969	146	4	a	a	DET
cana-3969	146	5	fuzzy	fuzzy	ADJ
cana-3969	146	6	ideal	ideal	ADJ
cana-3969	146	7	ζ	ζ	NOUN
cana-3969	146	8	of	of	ADP
cana-3969	146	9	a	a	DET
cana-3969	146	10	ternary	ternary	ADJ
cana-3969	146	11	γsemigroup	γsemigroup	NOUN
cana-3969	146	12	ꟊ	ꟊ	INTJ
cana-3969	146	13	is	be	AUX
cana-3969	146	14	called	call	VERB
cana-3969	146	15	a	a	DET
cana-3969	146	16	fuzzy	fuzzy	ADJ
cana-3969	146	17	semiprime	semiprime	NOUN
cana-3969	146	18	ideal	ideal	NOUN
cana-3969	146	19	if	if	SCONJ
cana-3969	146	20	𝜁(𝑥	𝜁(𝑥	NOUN
cana-3969	146	21	)	)	PUNCT
cana-3969	146	22	≥	≥	NOUN
cana-3969	147	1	𝑖𝑛𝑓	𝑖𝑛𝑓	NOUN
cana-3969	147	2	𝛼,𝛽∈𝛤	𝛼,𝛽∈𝛤	PROPN
cana-3969	147	3	𝜁(𝑥𝛼𝑥𝛽𝑥	𝜁(𝑥𝛼𝑥𝛽𝑥	NOUN
cana-3969	147	4	)	)	PUNCT
cana-3969	147	5	.	.	PUNCT
cana-3969	148	1	6.lemma	6.lemma	NUM
cana-3969	148	2	:	:	PUNCT
cana-3969	149	1	[	[	X
cana-3969	149	2	19	19	NUM
cana-3969	149	3	]	]	PUNCT
cana-3969	149	4	let	let	VERB
cana-3969	149	5	ꟊ	ꟊ	PRON
cana-3969	149	6	be	be	AUX
cana-3969	149	7	a	a	DET
cana-3969	149	8	γ	γ	NOUN
cana-3969	149	9	-	-	PUNCT
cana-3969	149	10	semigroup	semigroup	NOUN
cana-3969	149	11	and	and	CCONJ
cana-3969	149	12	∅≠i⊆	∅≠i⊆	NOUN
cana-3969	149	13	ꟊ.	ꟊ.	ADV
cana-3969	150	1	then	then	ADV
cana-3969	150	2	i	i	PRON
cana-3969	150	3	is	be	AUX
cana-3969	150	4	a	a	DET
cana-3969	150	5	prime	prime	ADJ
cana-3969	150	6	ideal	ideal	NOUN
cana-3969	150	7	(	(	PUNCT
cana-3969	150	8	semiprime	semiprime	NOUN
cana-3969	150	9	ideal	ideal	NOUN
cana-3969	150	10	)	)	PUNCT
cana-3969	150	11	of	of	ADP
cana-3969	150	12	ꟊ	ꟊ	PRON
cana-3969	150	13	iff	iff	PROPN
cana-3969	150	14	𝜁𝐼	𝜁𝐼	PROPN
cana-3969	150	15	is	be	AUX
cana-3969	150	16	a	a	DET
cana-3969	150	17	fuzzy	fuzzy	ADJ
cana-3969	150	18	prime	prime	ADJ
cana-3969	150	19	ideal	ideal	NOUN
cana-3969	150	20	(	(	PUNCT
cana-3969	150	21	respectively	respectively	ADV
cana-3969	150	22	fuzzy	fuzzy	ADJ
cana-3969	150	23	semiprime	semiprime	NOUN
cana-3969	150	24	ideal	ideal	NOUN
cana-3969	150	25	)	)	PUNCT
cana-3969	150	26	of	of	ADP
cana-3969	150	27	ꟊ	ꟊ	NOUN
cana-3969	150	28	,	,	PUNCT
cana-3969	150	29	where	where	SCONJ
cana-3969	150	30	𝜁𝐼	𝜁𝐼	NOUN
cana-3969	150	31	is	be	AUX
cana-3969	150	32	the	the	DET
cana-3969	150	33	characteristic	characteristic	ADJ
cana-3969	150	34	function	function	NOUN
cana-3969	150	35	of	of	ADP
cana-3969	150	36	i.	i.	PROPN
cana-3969	150	37	definition	definition	NOUN
cana-3969	150	38	4.6	4.6	NUM
cana-3969	150	39	:	:	PUNCT
cana-3969	151	1	[	[	X
cana-3969	151	2	35	35	NUM
cana-3969	151	3	]	]	PUNCT
cana-3969	151	4	let	let	VERB
cana-3969	151	5	ꟊ	ꟊ	PRON
cana-3969	151	6	be	be	AUX
cana-3969	151	7	γsemigroup	γsemigroup	NOUN
cana-3969	151	8	,	,	PUNCT
cana-3969	151	9	ζ	ζ	NOUN
cana-3969	151	10	be	be	VERB
cana-3969	151	11	a	a	DET
cana-3969	151	12	fuzzy	fuzzy	ADJ
cana-3969	151	13	subset	subset	NOUN
cana-3969	151	14	of	of	ADP
cana-3969	151	15	ꟊ	ꟊ	PROPN
cana-3969	151	16	and	and	CCONJ
cana-3969	151	17	x∈	x∈	PROPN
cana-3969	152	1	ꟊ	ꟊ	INTJ
cana-3969	152	2	then	then	ADV
cana-3969	152	3	the	the	DET
cana-3969	152	4	fuzzy	fuzzy	NOUN
cana-3969	152	5	subset	subset	VERB
cana-3969	152	6	<	<	X
cana-3969	152	7	x	x	NOUN
cana-3969	152	8	,	,	PUNCT
cana-3969	152	9	ζ	ζ	NOUN
cana-3969	152	10	>	>	X
cana-3969	152	11	:	:	PUNCT
cana-3969	152	12	ꟊ→	ꟊ→	PROPN
cana-3969	153	1	[	[	X
cana-3969	153	2	0,1	0,1	NUM
cana-3969	153	3	]	]	PUNCT
cana-3969	153	4	defined	define	VERB
cana-3969	153	5	by	by	ADP
cana-3969	153	6	<	<	X
cana-3969	153	7	𝑥	𝑥	PROPN
cana-3969	153	8	,	,	PUNCT
cana-3969	153	9	𝜁	𝜁	X
cana-3969	153	10	>	>	X
cana-3969	153	11	(	(	PUNCT
cana-3969	153	12	𝑦	𝑦	NOUN
cana-3969	153	13	)	)	PUNCT
cana-3969	153	14	=	=	SYM
cana-3969	154	1	𝑖𝑛𝑓	𝑖𝑛𝑓	PROPN
cana-3969	154	2	𝛾∈𝛤	𝛾∈𝛤	PROPN
cana-3969	154	3	𝜁(𝑥𝛾𝑦	𝜁(𝑥𝛾𝑦	NOUN
cana-3969	154	4	)	)	PUNCT
cana-3969	154	5	is	be	AUX
cana-3969	154	6	called	call	VERB
cana-3969	154	7	the	the	DET
cana-3969	154	8	extension	extension	NOUN
cana-3969	154	9	of	of	ADP
cana-3969	154	10	ζ	ζ	NOUN
cana-3969	154	11	by	by	ADP
cana-3969	154	12	x.	x.	NOUN
cana-3969	154	13	note	note	NOUN
cana-3969	154	14	:	:	PUNCT
cana-3969	154	15	for	for	ADP
cana-3969	154	16	a	a	DET
cana-3969	154	17	fuzzy	fuzzy	ADJ
cana-3969	154	18	subset	subset	VERB
cana-3969	154	19	ζ	ζ	NOUN
cana-3969	154	20	of	of	ADP
cana-3969	154	21	ℛ(ω	ℛ(ω	NUM
cana-3969	154	22	of	of	ADP
cana-3969	154	23	ℒ	ℒ	PROPN
cana-3969	154	24	)	)	PUNCT
cana-3969	154	25	a	a	DET
cana-3969	154	26	fuzzy	fuzzy	ADJ
cana-3969	154	27	subset	subset	VERB
cana-3969	154	28	ζ	ζ	NOUN
cana-3969	154	29	*	*	X
cana-3969	154	30	(	(	PUNCT
cana-3969	154	31	ωº	ωº	NOUN
cana-3969	154	32	)	)	PUNCT
cana-3969	154	33	of	of	ADP
cana-3969	154	34	ꟊ	ꟊ	PRON
cana-3969	154	35	by	by	ADP
cana-3969	154	36	ζ	ζ	X
cana-3969	154	37	*	*	PUNCT
cana-3969	154	38	(	(	PUNCT
cana-3969	154	39	𝑎	𝑎	NOUN
cana-3969	154	40	)	)	PUNCT
cana-3969	154	41	=	=	SYM
cana-3969	154	42	𝑖𝑛𝑓	𝑖𝑛𝑓	PROPN
cana-3969	154	43	¦	¦	PROPN
cana-3969	154	44	(𝛾	(𝛾	PROPN
cana-3969	154	45	∈	∈	PROPN
cana-3969	154	46	𝛤	𝛤	PROPN
cana-3969	154	47	)	)	PUNCT
cana-3969	154	48	𝜁	𝜁	PROPN
cana-3969	154	49	(	(	PUNCT
cana-3969	154	50	[	[	X
cana-3969	154	51	𝛾	𝛾	NOUN
cana-3969	154	52	,	,	PUNCT
cana-3969	154	53	𝑎	𝑎	NOUN
cana-3969	154	54	]	]	PUNCT
cana-3969	154	55	)	)	PUNCT
cana-3969	154	56	;	;	PUNCT
cana-3969	155	1	ωº	ωº	NUM
cana-3969	155	2	(	(	PUNCT
cana-3969	155	3	𝑎	𝑎	NOUN
cana-3969	155	4	)	)	PUNCT
cana-3969	155	5	=	=	SYM
cana-3969	155	6	𝑖𝑛𝑓	𝑖𝑛𝑓	PROPN
cana-3969	155	7	𝛾∈𝛤	𝛾∈𝛤	PROPN
cana-3969	155	8	ω	ω	NOUN
cana-3969	155	9	(	(	PUNCT
cana-3969	155	10	[	[	X
cana-3969	155	11	𝑎	𝑎	X
cana-3969	155	12	,	,	PUNCT
cana-3969	155	13	𝛾	𝛾	NOUN
cana-3969	155	14	]	]	PUNCT
cana-3969	155	15	)	)	PUNCT
cana-3969	155	16	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-3969	155	17	ղ	ղ	DET
cana-3969	155	18	∗′([𝛼,𝑎])=	∗′([𝛼,𝑎])=	ADJ
cana-3969	155	19	𝑖𝑛𝑓	𝑖𝑛𝑓	NOUN
cana-3969	155	20	𝑠∈ꟊ	𝑠∈ꟊ	NOUN
cana-3969	155	21	ղ	ղ	PROPN
cana-3969	155	22	(	(	PUNCT
cana-3969	155	23	𝑠𝛼𝑎	𝑠𝛼𝑎	PROPN
cana-3969	155	24	)	)	PUNCT
cana-3969	155	25	;	;	PUNCT
cana-3969	155	26	ղ𝑜′([𝑎,𝛼	ղ𝑜′([𝑎,𝛼	NUM
cana-3969	155	27	]	]	PUNCT
cana-3969	155	28	)	)	PUNCT
cana-3969	155	29	=	=	SYM
cana-3969	156	1	𝑖𝑛𝑓	𝑖𝑛𝑓	PROPN
cana-3969	156	2	𝑠∈ꟊ	𝑠∈ꟊ	NOUN
cana-3969	156	3	ղ	ղ	PROPN
cana-3969	156	4	(	(	PUNCT
cana-3969	156	5	𝑎𝛼𝑠	𝑎𝛼𝑠	NOUN
cana-3969	156	6	)	)	PUNCT
cana-3969	156	7	.	.	PUNCT
cana-3969	157	1	7	7	X
cana-3969	157	2	.	.	X
cana-3969	157	3	lemma	lemma	PROPN
cana-3969	157	4	:	:	PUNCT
cana-3969	157	5	let	let	VERB
cana-3969	157	6	ζ	ζ	NOUN
cana-3969	157	7	be	be	AUX
cana-3969	157	8	a	a	DET
cana-3969	157	9	non	non	ADJ
cana-3969	157	10	-	-	ADJ
cana-3969	157	11	empty	empty	ADJ
cana-3969	157	12	fuzzy	fuzzy	ADJ
cana-3969	157	13	subset	subset	NOUN
cana-3969	157	14	of	of	ADP
cana-3969	157	15	commutative	commutative	ADJ
cana-3969	157	16	γ	γ	PROPN
cana-3969	157	17	-	-	PUNCT
cana-3969	157	18	semigroup	semigroup	NOUN
cana-3969	157	19	ꟊ	ꟊ	NOUN
cana-3969	157	20	,	,	PUNCT
cana-3969	157	21	then	then	ADV
cana-3969	157	22	for	for	ADP
cana-3969	157	23	all	all	DET
cana-3969	157	24	x∈ꟊ	x∈ꟊ	PROPN
cana-3969	157	25	(	(	PUNCT
cana-3969	157	26	𝑖	𝑖	X
cana-3969	157	27	)	)	PUNCT
cana-3969	157	28	<	<	X
cana-3969	158	1	𝑥	𝑥	X
cana-3969	158	2	,	,	PUNCT
cana-3969	158	3	𝜁	𝜁	PROPN
cana-3969	158	4	>	>	X
cana-3969	158	5	∗′	∗′	PROPN
cana-3969	158	6	⊆	⊆	NUM
cana-3969	158	7	<	<	X
cana-3969	159	1	[	[	X
cana-3969	159	2	𝛼	𝛼	X
cana-3969	159	3	,	,	PUNCT
cana-3969	159	4	𝑥	𝑥	X
cana-3969	159	5	]	]	X
cana-3969	159	6	,	,	PUNCT
cana-3969	159	7	𝜁∗1	𝜁∗1	PROPN
cana-3969	159	8	>	>	X
cana-3969	159	9	𝑓𝑜𝑟	𝑓𝑜𝑟	ADJ
cana-3969	159	10	𝑎𝑙𝑙	𝑎𝑙𝑙	X
cana-3969	159	11	𝛼𝜖𝛤	𝛼𝜖𝛤	PROPN
cana-3969	159	12	(	(	PUNCT
cana-3969	159	13	𝑖𝑖	𝑖𝑖	NOUN
cana-3969	159	14	)	)	PUNCT
cana-3969	159	15	<	<	X
cana-3969	160	1	𝑥	𝑥	X
cana-3969	160	2	,	,	PUNCT
cana-3969	160	3	𝜁	𝜁	PROPN
cana-3969	160	4	>	>	PUNCT
cana-3969	160	5	∗′	∗′	PROPN
cana-3969	160	6	=	=	PUNCT
cana-3969	160	7	𝑖𝑛𝑓	𝑖𝑛𝑓	NOUN
cana-3969	161	1	𝛼𝜖𝛤	𝛼𝜖𝛤	PROPN
cana-3969	161	2	<	<	X
cana-3969	162	1	[	[	X
cana-3969	162	2	𝑥	𝑥	X
cana-3969	162	3	,	,	PUNCT
cana-3969	162	4	𝛼	𝛼	X
cana-3969	162	5	]	]	X
cana-3969	162	6	,	,	PUNCT
cana-3969	162	7	𝜁∗1	𝜁∗1	PROPN
cana-3969	162	8	>	>	X
cana-3969	162	9	𝑓𝑜𝑟	𝑓𝑜𝑟	ADJ
cana-3969	162	10	𝑎𝑙𝑙	𝑎𝑙𝑙	X
cana-3969	162	11	𝛼𝜖𝛤	𝛼𝜖𝛤	PROPN
cana-3969	162	12	(	(	PUNCT
cana-3969	162	13	i	i	NOUN
cana-3969	162	14	)	)	PUNCT
cana-3969	162	15	let	let	VERB
cana-3969	162	16	[	[	X
cana-3969	162	17	β	β	X
cana-3969	162	18	,	,	PUNCT
cana-3969	162	19	y	y	PROPN
cana-3969	162	20	]	]	X
cana-3969	162	21	∈	∈	PROPN
cana-3969	163	1	ꟊ	ꟊ	X
cana-3969	163	2	then	then	ADV
cana-3969	163	3	<	<	X
cana-3969	163	4	𝑥	𝑥	X
cana-3969	163	5	,	,	PUNCT
cana-3969	163	6	𝜁	𝜁	PROPN
cana-3969	163	7	>	>	X
cana-3969	163	8	∗′	∗′	PROPN
cana-3969	163	9	(	(	PUNCT
cana-3969	163	10	[	[	X
cana-3969	163	11	𝛽	𝛽	NOUN
cana-3969	163	12	,	,	PUNCT
cana-3969	163	13	𝑦	𝑦	NOUN
cana-3969	163	14	]	]	X
cana-3969	163	15	)	)	PUNCT
cana-3969	163	16	=	=	PUNCT
cana-3969	164	1	𝑖𝑛𝑓𝑠∈𝑆	𝑖𝑛𝑓𝑠∈𝑆	PUNCT
cana-3969	164	2	<	<	X
cana-3969	165	1	𝑥	𝑥	X
cana-3969	165	2	,	,	PUNCT
cana-3969	165	3	𝜁	𝜁	X
cana-3969	165	4	>	>	X
cana-3969	165	5	(	(	PUNCT
cana-3969	165	6	𝑠𝛽𝑦	𝑠𝛽𝑦	PROPN
cana-3969	165	7	)	)	PUNCT
cana-3969	165	8	=	=	SYM
cana-3969	165	9	𝑖𝑛𝑓𝑠∈𝑆	𝑖𝑛𝑓𝑠∈𝑆	PUNCT
cana-3969	165	10	𝑖𝑛𝑓𝛾∈𝛤𝜁(𝑥𝛾𝑠𝛽𝑦	𝑖𝑛𝑓𝛾∈𝛤𝜁(𝑥𝛾𝑠𝛽𝑦	ADJ
cana-3969	165	11	)	)	PUNCT
cana-3969	165	12	again	again	ADV
cana-3969	165	13	<	<	X
cana-3969	166	1	[	[	X
cana-3969	166	2	𝛼	𝛼	X
cana-3969	166	3	,	,	PUNCT
cana-3969	166	4	𝑥	𝑥	X
cana-3969	166	5	]	]	X
cana-3969	166	6	,	,	PUNCT
cana-3969	166	7	𝜁∗′	𝜁∗′	NUM
cana-3969	166	8	>	>	X
cana-3969	166	9	(	(	PUNCT
cana-3969	166	10	[	[	X
cana-3969	166	11	𝛽	𝛽	NOUN
cana-3969	166	12	,	,	PUNCT
cana-3969	166	13	𝑦	𝑦	NOUN
cana-3969	166	14	]	]	X
cana-3969	166	15	)	)	PUNCT
cana-3969	166	16	=	=	PUNCT
cana-3969	166	17	𝜁∗([𝛼	𝜁∗([𝛼	ADJ
cana-3969	166	18	,	,	PUNCT
cana-3969	166	19	𝑥][𝛽	𝑥][𝛽	NOUN
cana-3969	166	20	,	,	PUNCT
cana-3969	166	21	𝑦	𝑦	NOUN
cana-3969	166	22	]	]	X
cana-3969	166	23	)	)	PUNCT
cana-3969	166	24	=	=	SYM
cana-3969	166	25	𝜁∗′([𝛼	𝜁∗′([𝛼	PROPN
cana-3969	166	26	,	,	PUNCT
cana-3969	166	27	𝑥𝛽𝑦	𝑥𝛽𝑦	NOUN
cana-3969	166	28	]	]	PUNCT
cana-3969	166	29	)	)	PUNCT
cana-3969	166	30	=	=	SYM
cana-3969	166	31	𝑖𝑛𝑓𝑠∈𝑆𝜁(𝑠𝛼𝑥𝛽𝑦	𝑖𝑛𝑓𝑠∈𝑆𝜁(𝑠𝛼𝑥𝛽𝑦	NOUN
cana-3969	166	32	)	)	PUNCT
cana-3969	167	1	𝑆𝑖𝑛𝑐𝑒	𝑆𝑖𝑛𝑐𝑒	PROPN
cana-3969	167	2	𝑖𝑛𝑓𝛾∈𝛤𝑖𝑛𝑓𝑠∈𝑆𝜁(𝑥𝛾𝑠𝛽𝑦	𝑖𝑛𝑓𝛾∈𝛤𝑖𝑛𝑓𝑠∈𝑆𝜁(𝑥𝛾𝑠𝛽𝑦	PROPN
cana-3969	167	3	)	)	PUNCT
cana-3969	167	4	≤	≤	NUM
cana-3969	167	5	𝑖𝑛𝑓𝑠∈𝑆𝜁(𝑥𝛼𝑠𝛽𝑦	𝑖𝑛𝑓𝑠∈𝑆𝜁(𝑥𝛼𝑠𝛽𝑦	NOUN
cana-3969	167	6	)	)	PUNCT
cana-3969	167	7	communications	communication	NOUN
cana-3969	167	8	on	on	ADP
cana-3969	167	9	applied	apply	VERB
cana-3969	167	10	nonlinear	nonlinear	ADJ
cana-3969	167	11	analysis	analysis	NOUN
cana-3969	167	12	issn	issn	NOUN
cana-3969	167	13	:	:	PUNCT
cana-3969	167	14	1074	1074	NUM
cana-3969	167	15	-	-	PUNCT
cana-3969	167	16	133x	133x	NUM
cana-3969	167	17	vol	vol	NOUN
cana-3969	167	18	32	32	NUM
cana-3969	167	19	no	no	NOUN
cana-3969	167	20	.	.	PUNCT
cana-3969	168	1	9s	9s	NUM
cana-3969	168	2	(	(	PUNCT
cana-3969	168	3	2025	2025	NUM
cana-3969	168	4	)	)	PUNCT
cana-3969	168	5	643	643	NUM
cana-3969	169	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3969	169	2	<	<	X
cana-3969	169	3	𝑥	𝑥	X
cana-3969	169	4	,	,	PUNCT
cana-3969	169	5	𝜁	𝜁	PROPN
cana-3969	169	6	>	>	X
cana-3969	169	7	∗′	∗′	PROPN
cana-3969	169	8	(	(	PUNCT
cana-3969	169	9	[	[	X
cana-3969	169	10	𝛽	𝛽	NOUN
cana-3969	169	11	,	,	PUNCT
cana-3969	169	12	𝑦	𝑦	NOUN
cana-3969	169	13	]	]	X
cana-3969	169	14	)	)	PUNCT
cana-3969	169	15	≤	≤	NOUN
cana-3969	169	16	<	<	X
cana-3969	170	1	[	[	X
cana-3969	170	2	𝛼	𝛼	X
cana-3969	170	3	,	,	PUNCT
cana-3969	170	4	𝑥	𝑥	X
cana-3969	170	5	]	]	X
cana-3969	170	6	,	,	PUNCT
cana-3969	170	7	𝜁∗′	𝜁∗′	NUM
cana-3969	170	8	>	>	X
cana-3969	170	9	(	(	PUNCT
cana-3969	170	10	[	[	X
cana-3969	170	11	𝛽	𝛽	NOUN
cana-3969	170	12	,	,	PUNCT
cana-3969	170	13	𝑦	𝑦	NOUN
cana-3969	170	14	]	]	X
cana-3969	170	15	)	)	PUNCT
cana-3969	171	1	𝐻𝑒𝑛𝑐𝑒	𝐻𝑒𝑛𝑐𝑒	PROPN
cana-3969	171	2	,	,	PUNCT
cana-3969	171	3	<	<	X
cana-3969	171	4	𝑥	𝑥	X
cana-3969	171	5	,	,	PUNCT
cana-3969	171	6	𝜁	𝜁	PROPN
cana-3969	171	7	>	>	X
cana-3969	171	8	∗	∗	VERB
cana-3969	171	9	⊆	⊆	NUM
cana-3969	171	10	<	<	X
cana-3969	172	1	[	[	X
cana-3969	172	2	𝛼	𝛼	X
cana-3969	172	3	,	,	PUNCT
cana-3969	172	4	𝑥	𝑥	X
cana-3969	172	5	]	]	X
cana-3969	172	6	,	,	PUNCT
cana-3969	172	7	𝜁∗′	𝜁∗′	NUM
cana-3969	172	8	>	>	X
cana-3969	172	9	(	(	PUNCT
cana-3969	172	10	ii	ii	NOUN
cana-3969	172	11	)	)	PUNCT
cana-3969	172	12	let	let	VERB
cana-3969	172	13	[	[	X
cana-3969	172	14	β	β	X
cana-3969	172	15	,	,	PUNCT
cana-3969	172	16	y]∈r	y]∈r	PROPN
cana-3969	172	17	then	then	ADV
cana-3969	172	18	𝑖𝑛𝑓	𝑖𝑛𝑓	X
cana-3969	172	19	<	<	X
cana-3969	173	1	[	[	X
cana-3969	173	2	𝛼	𝛼	X
cana-3969	173	3	,	,	PUNCT
cana-3969	173	4	𝑥	𝑥	X
cana-3969	173	5	]	]	X
cana-3969	173	6	,	,	PUNCT
cana-3969	173	7	𝜁∗′	𝜁∗′	NUM
cana-3969	173	8	>	>	X
cana-3969	173	9	(	(	PUNCT
cana-3969	173	10	[	[	X
cana-3969	173	11	𝛽	𝛽	NOUN
cana-3969	173	12	,	,	PUNCT
cana-3969	173	13	𝑦	𝑦	NOUN
cana-3969	173	14	]	]	X
cana-3969	173	15	)	)	PUNCT
cana-3969	173	16	=	=	SYM
cana-3969	173	17	𝑖𝑛𝑓𝛼∈𝛤𝜁∗′([𝛼	𝑖𝑛𝑓𝛼∈𝛤𝜁∗′([𝛼	VERB
cana-3969	173	18	,	,	PUNCT
cana-3969	173	19	𝑥	𝑥	X
cana-3969	173	20	]	]	X
cana-3969	174	1	[	[	X
cana-3969	174	2	𝛽	𝛽	NOUN
cana-3969	174	3	,	,	PUNCT
cana-3969	174	4	𝑦	𝑦	NOUN
cana-3969	174	5	]	]	X
cana-3969	174	6	)	)	PUNCT
cana-3969	174	7	=	=	SYM
cana-3969	174	8	𝑖𝑛𝑓𝛼∈𝛤𝜁∗′([𝛼	𝑖𝑛𝑓𝛼∈𝛤𝜁∗′([𝛼	VERB
cana-3969	174	9	,	,	PUNCT
cana-3969	174	10	𝑥𝛽𝑦	𝑥𝛽𝑦	NOUN
cana-3969	174	11	]	]	PUNCT
cana-3969	174	12	)	)	PUNCT
cana-3969	174	13	=	=	SYM
cana-3969	174	14	𝑖𝑛𝑓𝛼∈𝛤	𝑖𝑛𝑓𝛼∈𝛤	PUNCT
cana-3969	174	15	𝑖𝑛𝑓𝑠∈𝑆	𝑖𝑛𝑓𝑠∈𝑆	NUM
cana-3969	174	16	𝜁([𝑠𝛼𝑥𝛽𝑦	𝜁([𝑠𝛼𝑥𝛽𝑦	NOUN
cana-3969	174	17	]	]	PUNCT
cana-3969	174	18	)	)	PUNCT
cana-3969	174	19	=	=	PUNCT
cana-3969	174	20	𝑖𝑛𝑓𝑠∈𝑆	𝑖𝑛𝑓𝑠∈𝑆	PUNCT
cana-3969	174	21	<	<	X
cana-3969	174	22	𝑥	𝑥	X
cana-3969	174	23	,	,	PUNCT
cana-3969	174	24	𝜁	𝜁	X
cana-3969	174	25	>	>	X
cana-3969	174	26	(	(	PUNCT
cana-3969	174	27	𝑠𝛽𝑦	𝑠𝛽𝑦	NOUN
cana-3969	174	28	)	)	PUNCT
cana-3969	174	29	=	=	NOUN
cana-3969	174	30	<	<	X
cana-3969	174	31	𝑥	𝑥	PROPN
cana-3969	174	32	,	,	PUNCT
cana-3969	174	33	𝜁	𝜁	PROPN
cana-3969	174	34	>	>	X
cana-3969	174	35	∗′	∗′	PROPN
cana-3969	174	36	(	(	PUNCT
cana-3969	174	37	[	[	X
cana-3969	174	38	𝛽	𝛽	NOUN
cana-3969	174	39	,	,	PUNCT
cana-3969	174	40	𝑦	𝑦	NOUN
cana-3969	174	41	]	]	X
cana-3969	174	42	)	)	PUNCT
cana-3969	175	1	𝑇ℎ𝑢𝑠	𝑇ℎ𝑢𝑠	PROPN
cana-3969	175	2	<	<	X
cana-3969	175	3	𝑥	𝑥	PROPN
cana-3969	175	4	,	,	PUNCT
cana-3969	175	5	𝜁	𝜁	PROPN
cana-3969	175	6	>	>	X
cana-3969	175	7	∗′=	∗′=	PROPN
cana-3969	175	8	𝑖𝑛𝑓𝛼∈𝛤	𝑖𝑛𝑓𝛼∈𝛤	PUNCT
cana-3969	175	9	<	<	X
cana-3969	175	10	[	[	X
cana-3969	175	11	𝛼	𝛼	X
cana-3969	175	12	,	,	PUNCT
cana-3969	175	13	𝑥	𝑥	X
cana-3969	175	14	]	]	X
cana-3969	175	15	,	,	PUNCT
cana-3969	175	16	𝜁∗′	𝜁∗′	ADP
cana-3969	175	17	>	>	X
cana-3969	175	18	5	5	X
cana-3969	175	19	.	.	PUNCT
cana-3969	175	20	results	result	NOUN
cana-3969	175	21	1	1	NUM
cana-3969	175	22	.	.	PUNCT
cana-3969	176	1	let	let	VERB
cana-3969	176	2	σ	σ	NOUN
cana-3969	176	3	be	be	AUX
cana-3969	176	4	a	a	DET
cana-3969	176	5	non	non	ADJ
cana-3969	176	6	-	-	ADJ
cana-3969	176	7	empty	empty	ADJ
cana-3969	176	8	fuzzy	fuzzy	ADJ
cana-3969	176	9	subset	subset	NOUN
cana-3969	176	10	of	of	ADP
cana-3969	176	11	the	the	DET
cana-3969	176	12	right	right	ADJ
cana-3969	176	13	operator	operator	NOUN
cana-3969	176	14	semi	semi	NOUN
cana-3969	176	15	-	-	NOUN
cana-3969	176	16	groups	group	NOUN
cana-3969	176	17	of	of	ADP
cana-3969	176	18	a	a	DET
cana-3969	176	19	γ	γ	NOUN
cana-3969	176	20	-	-	PUNCT
cana-3969	176	21	semigroup	semigroup	NOUN
cana-3969	176	22	ꟊ.	ꟊ.	NOUN
cana-3969	176	23	then	then	ADV
cana-3969	176	24	∀	∀	VERB
cana-3969	177	1	𝑥	𝑥	PRON
cana-3969	177	2	∈	∈	PROPN
cana-3969	177	3	ꟊ	ꟊ	X
cana-3969	177	4	,	,	PUNCT
cana-3969	177	5	<	<	X
cana-3969	177	6	[	[	X
cana-3969	177	7	𝛽	𝛽	NOUN
cana-3969	177	8	,	,	PUNCT
cana-3969	177	9	𝑥	𝑥	X
cana-3969	177	10	]	]	X
cana-3969	177	11	,	,	PUNCT
cana-3969	177	12	𝜎	𝜎	PROPN
cana-3969	177	13	>	>	X
cana-3969	177	14	∗	∗	X
cana-3969	177	15	≥	≥	NOUN
cana-3969	177	16	<	<	X
cana-3969	177	17	𝑥	𝑥	PROPN
cana-3969	177	18	,	,	PUNCT
cana-3969	177	19	𝜎∗	𝜎∗	PROPN
cana-3969	177	20	>	>	SYM
cana-3969	177	21	∀𝛽	∀𝛽	PROPN
cana-3969	177	22	∈	∈	PROPN
cana-3969	177	23	𝛤	𝛤	PROPN
cana-3969	177	24	let	let	VERB
cana-3969	177	25	p∈ꟊ	p∈ꟊ	NOUN
cana-3969	177	26	then	then	ADV
cana-3969	177	27	<	<	X
cana-3969	177	28	[	[	X
cana-3969	177	29	𝛽	𝛽	PROPN
cana-3969	177	30	,	,	PUNCT
cana-3969	177	31	𝑥	𝑥	X
cana-3969	177	32	]	]	X
cana-3969	177	33	,	,	PUNCT
cana-3969	177	34	𝜎	𝜎	PROPN
cana-3969	177	35	>	>	X
cana-3969	177	36	∗	∗	X
cana-3969	177	37	(	(	PUNCT
cana-3969	177	38	𝑝	𝑝	NOUN
cana-3969	177	39	)	)	PUNCT
cana-3969	177	40	=	=	PUNCT
cana-3969	178	1	𝑖𝑛𝑓𝛼∈𝛤	𝑖𝑛𝑓𝛼∈𝛤	PUNCT
cana-3969	179	1	<	<	X
cana-3969	180	1	[	[	X
cana-3969	180	2	𝛽	𝛽	NOUN
cana-3969	180	3	,	,	PUNCT
cana-3969	180	4	𝑥	𝑥	X
cana-3969	180	5	]	]	X
cana-3969	180	6	,	,	PUNCT
cana-3969	180	7	𝜎	𝜎	X
cana-3969	180	8	>	>	X
cana-3969	180	9	(	(	PUNCT
cana-3969	180	10	[	[	X
cana-3969	180	11	𝛾	𝛾	NOUN
cana-3969	180	12	,	,	PUNCT
cana-3969	180	13	𝑝	𝑝	NOUN
cana-3969	180	14	]	]	PUNCT
cana-3969	180	15	)	)	PUNCT
cana-3969	180	16	=	=	SYM
cana-3969	180	17	𝑖𝑛𝑓𝛼∈𝛤	𝑖𝑛𝑓𝛼∈𝛤	PUNCT
cana-3969	180	18	𝜎([𝛽	𝜎([𝛽	NUM
cana-3969	180	19	,	,	PUNCT
cana-3969	180	20	𝑥	𝑥	X
cana-3969	180	21	]	]	X
cana-3969	181	1	[	[	X
cana-3969	181	2	𝛾	𝛾	ADP
cana-3969	181	3	,	,	PUNCT
cana-3969	181	4	𝑝	𝑝	NOUN
cana-3969	181	5	]	]	PUNCT
cana-3969	181	6	)	)	PUNCT
cana-3969	181	7	=	=	PUNCT
cana-3969	181	8	𝑖𝑛𝑓𝛼∈𝜎	𝑖𝑛𝑓𝛼∈𝜎	ADJ
cana-3969	181	9	𝜎([𝛽	𝜎([𝛽	NUM
cana-3969	181	10	,	,	PUNCT
cana-3969	181	11	𝑥𝛾𝑝	𝑥𝛾𝑝	NOUN
cana-3969	181	12	]	]	PUNCT
cana-3969	181	13	)	)	PUNCT
cana-3969	182	1	𝐴𝑔𝑎𝑖𝑛	𝐴𝑔𝑎𝑖𝑛	PROPN
cana-3969	182	2	<	<	X
cana-3969	182	3	𝑥	𝑥	PROPN
cana-3969	182	4	,	,	PUNCT
cana-3969	182	5	𝜎∗	𝜎∗	PROPN
cana-3969	182	6	>	>	X
cana-3969	182	7	(	(	PUNCT
cana-3969	182	8	𝑝	𝑝	NOUN
cana-3969	182	9	)	)	PUNCT
cana-3969	182	10	=	=	SYM
cana-3969	182	11	𝑖𝑛𝑓𝛾∈𝛤𝜎∗(𝑥𝛾𝑝	𝑖𝑛𝑓𝛾∈𝛤𝜎∗(𝑥𝛾𝑝	NOUN
cana-3969	182	12	)	)	PUNCT
cana-3969	182	13	=	=	PUNCT
cana-3969	183	1	𝑖𝑛𝑓𝛾∈𝛤	𝑖𝑛𝑓𝛾∈𝛤	PROPN
cana-3969	183	2	𝑖𝑛𝑓𝛽∈𝛤	𝑖𝑛𝑓𝛽∈𝛤	NOUN
cana-3969	183	3	𝜎[(𝛽	𝜎[(𝛽	NOUN
cana-3969	183	4	,	,	PUNCT
cana-3969	183	5	𝑥𝛾𝑝	𝑥𝛾𝑝	NOUN
cana-3969	183	6	)	)	PUNCT
cana-3969	183	7	]	]	PUNCT
cana-3969	184	1	=	=	PUNCT
cana-3969	184	2	𝑖𝑛𝑓𝛽∈𝛤	𝑖𝑛𝑓𝛽∈𝛤	PUNCT
cana-3969	184	3	𝑖𝑛𝑓𝛾∈𝛤	𝑖𝑛𝑓𝛾∈𝛤	PROPN
cana-3969	184	4	𝜎[(𝛽	𝜎[(𝛽	NOUN
cana-3969	184	5	,	,	PUNCT
cana-3969	184	6	𝑥𝛾𝑝	𝑥𝛾𝑝	NOUN
cana-3969	184	7	)	)	PUNCT
cana-3969	184	8	]	]	PUNCT
cana-3969	185	1	∵	∵	PROPN
cana-3969	185	2	𝑖𝑛𝑓𝛾∈𝛤	𝑖𝑛𝑓𝛾∈𝛤	PUNCT
cana-3969	185	3	𝜎[(𝛽	𝜎[(𝛽	PROPN
cana-3969	185	4	,	,	PUNCT
cana-3969	185	5	𝑥𝛾𝑝	𝑥𝛾𝑝	NOUN
cana-3969	185	6	)	)	PUNCT
cana-3969	185	7	]	]	PUNCT
cana-3969	185	8	≥	≥	X
cana-3969	185	9	𝑖𝑛𝑓𝛽∈𝛤	𝑖𝑛𝑓𝛽∈𝛤	PUNCT
cana-3969	185	10	𝑖𝑛𝑓𝛾∈𝛤	𝑖𝑛𝑓𝛾∈𝛤	PROPN
cana-3969	185	11	𝜎[(𝛽	𝜎[(𝛽	NOUN
cana-3969	185	12	,	,	PUNCT
cana-3969	185	13	𝑥𝛾𝑝	𝑥𝛾𝑝	NOUN
cana-3969	185	14	)	)	PUNCT
cana-3969	185	15	]	]	PUNCT
cana-3969	186	1	we	we	PRON
cana-3969	186	2	have	have	VERB
cana-3969	186	3	<	<	X
cana-3969	186	4	[	[	X
cana-3969	186	5	𝛽	𝛽	NOUN
cana-3969	186	6	,	,	PUNCT
cana-3969	186	7	𝑥	𝑥	X
cana-3969	186	8	]	]	X
cana-3969	186	9	,	,	PUNCT
cana-3969	186	10	𝜎	𝜎	PROPN
cana-3969	186	11	>	>	X
cana-3969	186	12	∗	∗	X
cana-3969	186	13	(	(	PUNCT
cana-3969	186	14	𝑝	𝑝	NOUN
cana-3969	186	15	)	)	PUNCT
cana-3969	186	16	≥	≥	NOUN
cana-3969	186	17	<	<	X
cana-3969	186	18	𝑥	𝑥	PROPN
cana-3969	186	19	,	,	PUNCT
cana-3969	186	20	𝜎∗	𝜎∗	PROPN
cana-3969	186	21	>	>	X
cana-3969	186	22	(	(	PUNCT
cana-3969	186	23	𝑝	𝑝	NOUN
cana-3969	186	24	)	)	PUNCT
cana-3969	186	25	consequently	consequently	ADV
cana-3969	186	26	<	<	X
cana-3969	187	1	[	[	X
cana-3969	187	2	𝛽	𝛽	NOUN
cana-3969	187	3	,	,	PUNCT
cana-3969	187	4	𝑥	𝑥	X
cana-3969	187	5	]	]	X
cana-3969	187	6	,	,	PUNCT
cana-3969	187	7	𝜎	𝜎	PROPN
cana-3969	187	8	>	>	X
cana-3969	187	9	∗	∗	X
cana-3969	187	10	⊇	⊇	X
cana-3969	187	11	<	<	X
cana-3969	187	12	𝑥	𝑥	PROPN
cana-3969	187	13	,	,	PUNCT
cana-3969	187	14	𝜎∗	𝜎∗	PROPN
cana-3969	187	15	>	>	X
cana-3969	187	16	2	2	X
cana-3969	187	17	.	.	PUNCT
cana-3969	187	18	let	let	VERB
cana-3969	187	19	{	{	PUNCT
cana-3969	187	20	𝐴𝛼}𝛼∈𝐴	𝐴𝛼}𝛼∈𝐴	PROPN
cana-3969	187	21	be	be	AUX
cana-3969	187	22	a	a	DET
cana-3969	187	23	family	family	NOUN
cana-3969	187	24	of	of	ADP
cana-3969	187	25	ideals	ideal	NOUN
cana-3969	187	26	of	of	ADP
cana-3969	187	27	a	a	DET
cana-3969	187	28	γ	γ	NOUN
cana-3969	187	29	-	-	PUNCT
cana-3969	187	30	semigroup	semigroup	NOUN
cana-3969	187	31	ꟊ.	ꟊ.	NOUN
cana-3969	188	1	then	then	ADV
cana-3969	188	2	〖(∩𝛼∈∧	〖(∩𝛼∈∧	VERB
cana-3969	188	3	𝐴𝛼)∗′	𝐴𝛼)∗′	PROPN
cana-3969	188	4	=	=	SYM
cana-3969	188	5	∩𝛼∈∧	∩𝛼∈∧	NOUN
cana-3969	188	6	𝐴𝛼	𝐴𝛼	PROPN
cana-3969	188	7	∗	∗	NOUN
cana-3969	188	8	′	′	NOUN
cana-3969	188	9	let	let	VERB
cana-3969	189	1	[	[	X
cana-3969	189	2	𝛼	𝛼	X
cana-3969	189	3	,	,	PUNCT
cana-3969	189	4	𝑥](∩𝛼∈∧	𝑥](∩𝛼∈∧	PUNCT
cana-3969	189	5	𝐴𝛼)∗′	𝐴𝛼)∗′	PROPN
cana-3969	189	6	then	then	ADV
cana-3969	189	7	𝑠𝛼𝑥	𝑠𝛼𝑥	PROPN
cana-3969	189	8	∈	∈	PROPN
cana-3969	189	9	∩𝛼∈∧	∩𝛼∈∧	NOUN
cana-3969	189	10	𝐴𝛼	𝐴𝛼	PROPN
cana-3969	189	11	∀	∀	NOUN
cana-3969	189	12	𝑠	𝑠	ADP
cana-3969	189	13	∈	∈	PROPN
cana-3969	189	14	𝑆	𝑆	PROPN
cana-3969	189	15	here	here	ADV
cana-3969	189	16	𝑠	𝑠	PROPN
cana-3969	189	17	∈	∈	PROPN
cana-3969	189	18	𝑆	𝑆	PROPN
cana-3969	189	19	communications	communication	NOUN
cana-3969	189	20	on	on	ADP
cana-3969	189	21	applied	apply	VERB
cana-3969	189	22	nonlinear	nonlinear	ADJ
cana-3969	189	23	analysis	analysis	NOUN
cana-3969	189	24	issn	issn	NOUN
cana-3969	189	25	:	:	PUNCT
cana-3969	189	26	1074	1074	NUM
cana-3969	189	27	-	-	PUNCT
cana-3969	189	28	133x	133x	NUM
cana-3969	189	29	vol	vol	NOUN
cana-3969	189	30	32	32	NUM
cana-3969	189	31	no	no	NOUN
cana-3969	189	32	.	.	PUNCT
cana-3969	190	1	9s	9s	NUM
cana-3969	190	2	(	(	PUNCT
cana-3969	190	3	2025	2025	NUM
cana-3969	190	4	)	)	PUNCT
cana-3969	190	5	644	644	NUM
cana-3969	190	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3969	190	7	𝑠𝛼𝑥	𝑠𝛼𝑥	NOUN
cana-3969	190	8	∈	∈	PROPN
cana-3969	190	9	𝐴𝛼	𝐴𝛼	PROPN
cana-3969	190	10	∀	∀	NOUN
cana-3969	190	11	𝛼	𝛼	NOUN
cana-3969	190	12	∈∧	∈∧	NOUN
cana-3969	190	13	[	[	X
cana-3969	190	14	𝛼	𝛼	X
cana-3969	190	15	,	,	PUNCT
cana-3969	190	16	𝑥	𝑥	NOUN
cana-3969	190	17	]	]	X
cana-3969	190	18	∩𝛼∈∧	∩𝛼∈∧	NOUN
cana-3969	190	19	𝐴𝛼	𝐴𝛼	PROPN
cana-3969	190	20	∗′	∗′	PROPN
cana-3969	190	21	here	here	ADV
cana-3969	190	22	(	(	PUNCT
cana-3969	190	23	∩𝛼∈∧	∩𝛼∈∧	NOUN
cana-3969	190	24	𝐴𝛼)∗′	𝐴𝛼)∗′	PROPN
cana-3969	190	25	⊆	⊆	NUM
cana-3969	190	26	∩𝛼∈∧	∩𝛼∈∧	NOUN
cana-3969	190	27	𝐴𝛼	𝐴𝛼	PROPN
cana-3969	190	28	∗′	∗′	ADJ
cana-3969	190	29	−	−	PROPN
cana-3969	190	30	(	(	PUNCT
cana-3969	190	31	1	1	X
cana-3969	190	32	)	)	PUNCT
cana-3969	190	33	we	we	PRON
cana-3969	190	34	can	can	AUX
cana-3969	190	35	deduce	deduce	VERB
cana-3969	190	36	that	that	DET
cana-3969	190	37	∩𝛼∈∧	∩𝛼∈∧	NOUN
cana-3969	190	38	𝐴𝛼	𝐴𝛼	PROPN
cana-3969	190	39	∗′	∗′	ADJ
cana-3969	190	40	⊆	⊆	NUM
cana-3969	190	41	(	(	PUNCT
cana-3969	190	42	∩𝛼∈∧	∩𝛼∈∧	NOUN
cana-3969	191	1	𝐴𝛼)∗′	𝐴𝛼)∗′	PROPN
cana-3969	191	2	[	[	X
cana-3969	191	3	𝛼	𝛼	NOUN
cana-3969	191	4	,	,	PUNCT
cana-3969	191	5	𝑥	𝑥	X
cana-3969	191	6	]	]	X
cana-3969	191	7	∈∩𝛼∈∧	∈∩𝛼∈∧	NOUN
cana-3969	192	1	𝐴𝛼	𝐴𝛼	PROPN
cana-3969	192	2	∗′	∗′	PROPN
cana-3969	192	3	then	then	ADV
cana-3969	192	4	𝑠𝛼𝑥	𝑠𝛼𝑥	ADJ
cana-3969	192	5	∈	∈	NOUN
cana-3969	192	6	𝐴𝛼	𝐴𝛼	PROPN
cana-3969	192	7	∀	∀	NOUN
cana-3969	192	8	𝛼	𝛼	NOUN
cana-3969	192	9	∈∧	∈∧	NOUN
cana-3969	192	10	𝑠𝛼𝑥	𝑠𝛼𝑥	NOUN
cana-3969	192	11	∈	∈	PROPN
cana-3969	192	12	∩𝛼∈∧	∩𝛼∈∧	NOUN
cana-3969	192	13	𝐴𝛼	𝐴𝛼	PROPN
cana-3969	192	14	∀	∀	NOUN
cana-3969	192	15	𝑠	𝑠	ADP
cana-3969	192	16	∈	∈	PROPN
cana-3969	192	17	𝑆	𝑆	PROPN
cana-3969	193	1	[	[	X
cana-3969	193	2	𝛼	𝛼	X
cana-3969	193	3	,	,	PUNCT
cana-3969	193	4	𝑥	𝑥	X
cana-3969	193	5	]	]	X
cana-3969	193	6	∈	∈	PROPN
cana-3969	193	7	(	(	PUNCT
cana-3969	193	8	∩𝛼∈∧	∩𝛼∈∧	NOUN
cana-3969	193	9	𝐴𝛼)∗	𝐴𝛼)∗	PROPN
cana-3969	193	10	′	′	NUM
cana-3969	193	11	∩𝛼∈∧	∩𝛼∈∧	NOUN
cana-3969	193	12	𝐴𝛼	𝐴𝛼	PROPN
cana-3969	193	13	∗′	∗′	ADJ
cana-3969	193	14	⊆	⊆	NUM
cana-3969	193	15	(	(	PUNCT
cana-3969	193	16	∩𝛼∈∧	∩𝛼∈∧	NOUN
cana-3969	193	17	𝐴𝛼	𝐴𝛼	PROPN
cana-3969	193	18	)	)	PUNCT
cana-3969	193	19	∗′	∗′	PROPN
cana-3969	193	20	−	−	PROPN
cana-3969	193	21	(	(	PUNCT
cana-3969	193	22	2	2	NUM
cana-3969	193	23	)	)	PUNCT
cana-3969	193	24	from	from	ADP
cana-3969	193	25	(	(	PUNCT
cana-3969	193	26	1	1	NUM
cana-3969	193	27	)	)	PUNCT
cana-3969	193	28	&	&	CCONJ
cana-3969	193	29	(	(	PUNCT
cana-3969	193	30	2	2	X
cana-3969	193	31	)	)	PUNCT
cana-3969	193	32	we	we	PRON
cana-3969	193	33	get	get	VERB
cana-3969	193	34	∩𝛼∈∧	∩𝛼∈∧	NOUN
cana-3969	193	35	𝐴𝛼	𝐴𝛼	PROPN
cana-3969	193	36	∗′	∗′	PROPN
cana-3969	193	37	=	=	SYM
cana-3969	193	38	(	(	PUNCT
cana-3969	193	39	∩𝛼∈∧	∩𝛼∈∧	NOUN
cana-3969	193	40	𝐴𝛼)∗′	𝐴𝛼)∗′	PROPN
cana-3969	193	41	3	3	X
cana-3969	193	42	.	.	PUNCT
cana-3969	194	1	let	let	VERB
cana-3969	194	2	ꟊ	ꟊ	PRON
cana-3969	194	3	be	be	AUX
cana-3969	194	4	a	a	DET
cana-3969	194	5	γ	γ	NOUN
cana-3969	194	6	-	-	PUNCT
cana-3969	194	7	semigroups	semigroup	NOUN
cana-3969	194	8	,	,	PUNCT
cana-3969	194	9	r	r	VERB
cana-3969	194	10	its	its	PRON
cana-3969	194	11	right	right	ADJ
cana-3969	194	12	operator	operator	NOUN
cana-3969	194	13	semigroup	semigroup	NOUN
cana-3969	194	14	and	and	CCONJ
cana-3969	194	15	𝜉	𝜉	NOUN
cana-3969	194	16	=	=	VERB
cana-3969	194	17	𝑖𝑛	𝑖𝑛	NOUN
cana-3969	194	18	𝑓{𝜉𝑖	𝑓{𝜉𝑖	PROPN
cana-3969	194	19	:	:	PUNCT
cana-3969	194	20	𝑖	𝑖	SYM
cana-3969	194	21	∈	∈	PROPN
cana-3969	194	22	𝐼	𝐼	PROPN
cana-3969	194	23	}	}	PUNCT
cana-3969	194	24	a	a	DET
cana-3969	194	25	nonempty	nonempty	ADJ
cana-3969	194	26	family	family	NOUN
cana-3969	194	27	of	of	ADP
cana-3969	194	28	the	the	DET
cana-3969	194	29	fuzzy	fuzzy	ADJ
cana-3969	194	30	subset	subset	NOUN
cana-3969	194	31	of	of	ADP
cana-3969	194	32	ꟊ.	ꟊ.	NOUN
cana-3969	194	33	then	then	ADV
cana-3969	194	34	𝜉∗′	𝜉∗′	ADP
cana-3969	195	1	=	=	NOUN
cana-3969	195	2	𝑖𝑛	𝑖𝑛	X
cana-3969	195	3	𝑓	𝑓	PROPN
cana-3969	195	4	{	{	PUNCT
cana-3969	195	5	𝜉	𝜉	X
cana-3969	195	6	𝑝∗′	𝑝∗′	NOUN
cana-3969	195	7	:	:	PUNCT
cana-3969	195	8	𝑖	𝑖	SYM
cana-3969	195	9	∈	∈	PROPN
cana-3969	195	10	𝐼	𝐼	PROPN
cana-3969	195	11	}	}	PUNCT
cana-3969	195	12	fuzzy	fuzzy	ADJ
cana-3969	195	13	ideal	ideal	ADJ
cana-3969	195	14	extension	extension	NOUN
cana-3969	195	15	of	of	ADP
cana-3969	195	16	γ	γ	PROPN
cana-3969	195	17	-	-	PUNCT
cana-3969	195	18	semigroup	semigroup	NOUN
cana-3969	195	19	.	.	PUNCT
cana-3969	196	1	let	let	VERB
cana-3969	197	1	[	[	X
cana-3969	197	2	α	α	X
cana-3969	197	3	,	,	PUNCT
cana-3969	197	4	x]∈r	x]∈r	PROPN
cana-3969	197	5	then	then	ADV
cana-3969	197	6	𝜉∗′[𝛼	𝜉∗′[𝛼	VERB
cana-3969	197	7	,	,	PUNCT
cana-3969	197	8	𝑥	𝑥	X
cana-3969	197	9	]	]	X
cana-3969	197	10	=	=	PUNCT
cana-3969	197	11	𝑖𝑛	𝑖𝑛	X
cana-3969	197	12	𝑓{𝜉𝑖	𝑓{𝜉𝑖	PROPN
cana-3969	197	13	:	:	PUNCT
cana-3969	197	14	𝑖	𝑖	SYM
cana-3969	197	15	∈	∈	PROPN
cana-3969	197	16	𝐼}∗′	𝐼}∗′	PROPN
cana-3969	198	1	[	[	X
cana-3969	198	2	𝛼	𝛼	X
cana-3969	198	3	,	,	PUNCT
cana-3969	198	4	𝑥	𝑥	X
cana-3969	198	5	]	]	X
cana-3969	198	6	=	=	PUNCT
cana-3969	198	7	𝑖𝑛𝑓𝑠∈𝑆(𝑖𝑛	𝑖𝑛𝑓𝑠∈𝑆(𝑖𝑛	PROPN
cana-3969	198	8	𝑓{𝜉𝑖	𝑓{𝜉𝑖	PROPN
cana-3969	198	9	:	:	PUNCT
cana-3969	198	10	𝑖	𝑖	SYM
cana-3969	198	11	∈	∈	PROPN
cana-3969	198	12	𝐼}(𝑠𝛼𝑥	𝐼}(𝑠𝛼𝑥	NOUN
cana-3969	198	13	)	)	PUNCT
cana-3969	198	14	)	)	PUNCT
cana-3969	199	1	=	=	SYM
cana-3969	200	1	𝑖𝑛𝑓𝑠∈𝑆	𝑖𝑛𝑓𝑠∈𝑆	NUM
cana-3969	200	2	𝑖𝑛𝑓𝑖∈𝐼𝜉𝑖(𝑠𝛼𝑥	𝑖𝑛𝑓𝑖∈𝐼𝜉𝑖(𝑠𝛼𝑥	NOUN
cana-3969	200	3	)	)	PUNCT
cana-3969	201	1	now	now	ADV
cana-3969	201	2	𝑖𝑛𝑓	𝑖𝑛𝑓	X
cana-3969	201	3	{	{	PUNCT
cana-3969	201	4	𝜉𝑖	𝜉𝑖	ADP
cana-3969	201	5	∗′	∗′	PROPN
cana-3969	201	6	𝑖	𝑖	SYM
cana-3969	201	7	∈	∈	PROPN
cana-3969	201	8	𝐼	𝐼	PROPN
cana-3969	201	9	}	}	PUNCT
cana-3969	201	10	[	[	X
cana-3969	201	11	𝛼	𝛼	X
cana-3969	201	12	,	,	PUNCT
cana-3969	201	13	𝑥	𝑥	X
cana-3969	201	14	]	]	X
cana-3969	201	15	=	=	SYM
cana-3969	201	16	𝑖𝑛𝑓𝑖∈𝐼	𝑖𝑛𝑓𝑖∈𝐼	PROPN
cana-3969	201	17	(	(	PUNCT
cana-3969	201	18	𝜉𝑖	𝜉𝑖	ADP
cana-3969	201	19	∗′[𝛼	∗′[𝛼	PROPN
cana-3969	201	20	,	,	PUNCT
cana-3969	201	21	𝑥	𝑥	X
cana-3969	201	22	]	]	X
cana-3969	201	23	)	)	PUNCT
cana-3969	201	24	=	=	SYM
cana-3969	201	25	𝑖𝑛𝑓𝑖∈𝐼𝑖𝑛𝑓𝑠∈𝑆𝜉𝑖	𝑖𝑛𝑓𝑖∈𝐼𝑖𝑛𝑓𝑠∈𝑆𝜉𝑖	X
cana-3969	201	26	∗′(𝑠𝛼𝑥	∗′(𝑠𝛼𝑥	PROPN
cana-3969	201	27	)	)	PUNCT
cana-3969	201	28	=	=	PUNCT
cana-3969	202	1	𝑖𝑛𝑓𝑖∈𝐼𝑖𝑛𝑓𝑠∈𝑆𝜉𝑖(𝑠𝛼𝑥	𝑖𝑛𝑓𝑖∈𝐼𝑖𝑛𝑓𝑠∈𝑆𝜉𝑖(𝑠𝛼𝑥	PROPN
cana-3969	202	2	)	)	PUNCT
cana-3969	202	3	=	=	SYM
cana-3969	202	4	𝑖𝑛𝑓𝑠∈𝑆𝑖𝑛𝑓𝑖∈𝐼𝜉𝑖(𝑠𝛼𝑥	𝑖𝑛𝑓𝑠∈𝑆𝑖𝑛𝑓𝑖∈𝐼𝜉𝑖(𝑠𝛼𝑥	X
cana-3969	202	5	)	)	PUNCT
cana-3969	202	6	∴𝜉∗′	∴𝜉∗′	PROPN
cana-3969	202	7	is	be	AUX
cana-3969	202	8	the	the	DET
cana-3969	202	9	fuzzy	fuzzy	ADJ
cana-3969	202	10	ideal	ideal	ADJ
cana-3969	202	11	extension	extension	NOUN
cana-3969	202	12	of	of	ADP
cana-3969	202	13	γ	γ	PROPN
cana-3969	202	14	-	-	PUNCT
cana-3969	202	15	semigroup	semigroup	NOUN
cana-3969	202	16	.	.	PUNCT
cana-3969	203	1	4	4	X
cana-3969	203	2	.	.	X
cana-3969	203	3	suppose	suppose	VERB
cana-3969	203	4	ξ	ξ	X
cana-3969	203	5	:	:	PUNCT
cana-3969	203	6	s→[0,1	s→[0,1	NOUN
cana-3969	203	7	]	]	X
cana-3969	203	8	the	the	DET
cana-3969	203	9	fuzzy	fuzzy	ADJ
cana-3969	203	10	subset	subset	NOUN
cana-3969	203	11	of	of	ADP
cana-3969	203	12	semigroup	semigroup	PROPN
cana-3969	203	13	and	and	CCONJ
cana-3969	203	14	then	then	ADV
cana-3969	203	15	(	(	PUNCT
cana-3969	203	16	i	i	NOUN
cana-3969	203	17	)	)	PUNCT
cana-3969	203	18	(	(	PUNCT
cana-3969	203	19	𝜉𝑖	𝜉𝑖	X
cana-3969	203	20	)	)	PUNCT
cana-3969	203	21	∗	∗	NOUN
cana-3969	203	22	=	=	PUNCT
cana-3969	203	23	(	(	PUNCT
cana-3969	203	24	𝜉∗)𝑖	𝜉∗)𝑖	VERB
cana-3969	203	25	∀	∀	X
cana-3969	203	26	𝑖	𝑖	SYM
cana-3969	203	27	∈	∈	NOUN
cana-3969	203	28	𝑁	𝑁	PROPN
cana-3969	203	29	(	(	PUNCT
cana-3969	203	30	𝑖𝑖	𝑖𝑖	NOUN
cana-3969	203	31	)	)	PUNCT
cana-3969	203	32	(	(	PUNCT
cana-3969	203	33	𝜉(𝑖	𝜉(𝑖	PROPN
cana-3969	203	34	)	)	PUNCT
cana-3969	203	35	)	)	PUNCT
cana-3969	203	36	∗	∗	NOUN
cana-3969	203	37	=	=	PUNCT
cana-3969	203	38	(	(	PUNCT
cana-3969	203	39	𝜉∗)𝑖	𝜉∗)𝑖	VERB
cana-3969	203	40	proof	proof	NOUN
cana-3969	203	41	:	:	PUNCT
cana-3969	203	42	(	(	PUNCT
cana-3969	203	43	i	i	NOUN
cana-3969	203	44	)	)	PUNCT
cana-3969	203	45	if	if	SCONJ
cana-3969	203	46	𝑖	𝑖	PRON
cana-3969	203	47	=	=	SYM
cana-3969	203	48	1	1	NUM
cana-3969	203	49	communications	communication	NOUN
cana-3969	203	50	on	on	ADP
cana-3969	203	51	applied	apply	VERB
cana-3969	203	52	nonlinear	nonlinear	ADJ
cana-3969	203	53	analysis	analysis	NOUN
cana-3969	203	54	issn	issn	NOUN
cana-3969	203	55	:	:	PUNCT
cana-3969	203	56	1074	1074	NUM
cana-3969	203	57	-	-	PUNCT
cana-3969	203	58	133x	133x	NUM
cana-3969	203	59	vol	vol	NOUN
cana-3969	203	60	32	32	NUM
cana-3969	204	1	no	no	NOUN
cana-3969	204	2	.	.	PUNCT
cana-3969	205	1	9s	9s	NUM
cana-3969	205	2	(	(	PUNCT
cana-3969	205	3	2025	2025	NUM
cana-3969	205	4	)	)	PUNCT
cana-3969	205	5	645	645	NUM
cana-3969	205	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3969	205	7	then	then	ADV
cana-3969	205	8	(	(	PUNCT
cana-3969	205	9	𝜉1)∗	𝜉1)∗	NOUN
cana-3969	205	10	=	=	PUNCT
cana-3969	205	11	(	(	PUNCT
cana-3969	205	12	𝜉∗)1	𝜉∗)1	X
cana-3969	205	13	→	→	SYM
cana-3969	205	14	𝜉∗	𝜉∗	NOUN
cana-3969	205	15	=	=	NOUN
cana-3969	205	16	𝜉∗	𝜉∗	NOUN
cana-3969	205	17	∴	∴	NOUN
cana-3969	205	18	𝑖	𝑖	PUNCT
cana-3969	206	1	=	=	SYM
cana-3969	206	2	1	1	NUM
cana-3969	206	3	the	the	DET
cana-3969	206	4	results	result	NOUN
cana-3969	206	5	are	be	AUX
cana-3969	206	6	true	true	ADJ
cana-3969	206	7	we	we	PRON
cana-3969	206	8	assume	assume	VERB
cana-3969	206	9	that	that	SCONJ
cana-3969	206	10	it	it	PRON
cana-3969	206	11	is	be	AUX
cana-3969	206	12	true	true	ADJ
cana-3969	206	13	for	for	SCONJ
cana-3969	206	14	𝑖	𝑖	PUNCT
cana-3969	206	15	>	>	X
cana-3969	206	16	1	1	NUM
cana-3969	206	17	let	let	VERB
cana-3969	206	18	𝑥	𝑥	X
cana-3969	206	19	𝜖	𝜖	X
cana-3969	206	20	(	(	PUNCT
cana-3969	206	21	𝜉𝑖+1	𝜉𝑖+1	NOUN
cana-3969	206	22	)	)	PUNCT
cana-3969	206	23	∗	∗	NOUN
cana-3969	206	24	then	then	ADV
cana-3969	206	25	𝜉𝑖+1(𝑥	𝜉𝑖+1(𝑥	NUM
cana-3969	206	26	)	)	PUNCT
cana-3969	207	1	=	=	SYM
cana-3969	207	2	𝜉𝑖+1	𝜉𝑖+1	NOUN
cana-3969	207	3	(	(	PUNCT
cana-3969	207	4	0	0	NUM
cana-3969	207	5	)	)	PUNCT
cana-3969	207	6	=	=	SYM
cana-3969	207	7	𝜉(0	𝜉(0	NOUN
cana-3969	207	8	)	)	PUNCT
cana-3969	207	9	but	but	CCONJ
cana-3969	207	10	𝜉𝑖+1	𝜉𝑖+1	NOUN
cana-3969	207	11	(	(	PUNCT
cana-3969	207	12	𝑥	𝑥	NOUN
cana-3969	207	13	)	)	PUNCT
cana-3969	207	14	=	=	SYM
cana-3969	207	15	𝑉{𝜉𝑖	𝑉{𝜉𝑖	PROPN
cana-3969	207	16	(	(	PUNCT
cana-3969	207	17	𝑦)𝜉(𝑧	𝑦)𝜉(𝑧	PROPN
cana-3969	207	18	)	)	PUNCT
cana-3969	207	19	/𝑦	/𝑦	NOUN
cana-3969	207	20	,	,	PUNCT
cana-3969	207	21	𝑧	𝑧	PROPN
cana-3969	207	22	∈	∈	PROPN
cana-3969	207	23	𝑆	𝑆	PROPN
cana-3969	207	24	𝑥	𝑥	NOUN
cana-3969	207	25	=	=	SYM
cana-3969	207	26	𝑦𝑧	𝑦𝑧	PROPN
cana-3969	207	27	}	}	PUNCT
cana-3969	207	28	𝜉(0	𝜉(0	PROPN
cana-3969	207	29	)	)	PUNCT
cana-3969	207	30	thus	thus	ADV
cana-3969	207	31	𝜉𝑖	𝜉𝑖	X
cana-3969	207	32	(	(	PUNCT
cana-3969	207	33	𝑦	𝑦	NOUN
cana-3969	207	34	)	)	PUNCT
cana-3969	207	35	=	=	SYM
cana-3969	207	36	𝜉(0	𝜉(0	PROPN
cana-3969	207	37	)	)	PUNCT
cana-3969	207	38	=	=	SYM
cana-3969	207	39	𝜉(𝑧	𝜉(𝑧	PROPN
cana-3969	207	40	)	)	PUNCT
cana-3969	207	41	for	for	ADP
cana-3969	207	42	some	some	DET
cana-3969	207	43	𝑦	𝑦	NOUN
cana-3969	207	44	,	,	PUNCT
cana-3969	207	45	𝑧	𝑧	PRON
cana-3969	207	46	∈	∈	PROPN
cana-3969	207	47	𝑆	𝑆	PROPN
cana-3969	207	48	∵	∵	NOUN
cana-3969	207	49	𝜉𝑖	𝜉𝑖	PROPN
cana-3969	207	50	(	(	PUNCT
cana-3969	207	51	𝑦	𝑦	NOUN
cana-3969	207	52	)	)	PUNCT
cana-3969	207	53	≤	≤	NOUN
cana-3969	207	54	𝜉𝑖	𝜉𝑖	ADP
cana-3969	207	55	(	(	PUNCT
cana-3969	207	56	0	0	NUM
cana-3969	207	57	)	)	PUNCT
cana-3969	207	58	=	=	SYM
cana-3969	207	59	𝜉(0	𝜉(0	NOUN
cana-3969	207	60	)	)	PUNCT
cana-3969	207	61	𝜉𝑖	𝜉𝑖	PROPN
cana-3969	207	62	𝑖s	𝑖s	PUNCT
cana-3969	207	63	finite	finite	VERB
cana-3969	207	64	-	-	PUNCT
cana-3969	207	65	valued	value	VERB
cana-3969	207	66	∀	∀	NOUN
cana-3969	207	67	𝑖	𝑖	X
cana-3969	207	68	≥	≥	NOUN
cana-3969	207	69	1	1	NUM
cana-3969	207	70	thus	thus	ADV
cana-3969	207	71	𝑥	𝑥	ADP
cana-3969	207	72	=	=	SYM
cana-3969	207	73	𝑦𝑧	𝑦𝑧	PROPN
cana-3969	207	74	for	for	ADP
cana-3969	207	75	some	some	DET
cana-3969	207	76	𝑦	𝑦	NOUN
cana-3969	207	77	∈	∈	NOUN
cana-3969	207	78	(	(	PUNCT
cana-3969	207	79	𝜉𝑖	𝜉𝑖	NOUN
cana-3969	207	80	)	)	PUNCT
cana-3969	207	81	∗	∗	NOUN
cana-3969	207	82	=	=	PUNCT
cana-3969	207	83	(	(	PUNCT
cana-3969	207	84	𝜉∗	𝜉∗	NOUN
cana-3969	207	85	)	)	PUNCT
cana-3969	207	86	𝑖	𝑖	PROPN
cana-3969	207	87	and	and	CCONJ
cana-3969	207	88	𝑧	𝑧	PRON
cana-3969	207	89	∈	∈	NOUN
cana-3969	207	90	𝜉∗	𝜉∗	NOUN
cana-3969	207	91	hence	hence	ADV
cana-3969	207	92	𝑥	𝑥	X
cana-3969	207	93	∈	∈	NOUN
cana-3969	207	94	(	(	PUNCT
cana-3969	207	95	𝜉∗	𝜉∗	NOUN
cana-3969	207	96	)	)	PUNCT
cana-3969	207	97	𝑖	𝑖	SYM
cana-3969	208	1	𝜉∗	𝜉∗	NOUN
cana-3969	208	2	=	=	PUNCT
cana-3969	208	3	(	(	PUNCT
cana-3969	208	4	𝜉∗	𝜉∗	NOUN
cana-3969	208	5	)	)	PUNCT
cana-3969	208	6	𝑖+1	𝑖+1	X
cana-3969	208	7	thus	thus	ADV
cana-3969	208	8	(	(	PUNCT
cana-3969	208	9	𝜉(𝑖+1	𝜉(𝑖+1	PROPN
cana-3969	208	10	)	)	PUNCT
cana-3969	208	11	)	)	PUNCT
cana-3969	209	1	∗	∗	NOUN
cana-3969	209	2	⊆	⊆	NUM
cana-3969	209	3	(	(	PUNCT
cana-3969	209	4	𝜉∗	𝜉∗	NOUN
cana-3969	209	5	)	)	PUNCT
cana-3969	209	6	(	(	PUNCT
cana-3969	209	7	𝑖+1	𝑖+1	NUM
cana-3969	209	8	)	)	PUNCT
cana-3969	209	9	now	now	ADV
cana-3969	209	10	let	let	VERB
cana-3969	209	11	𝑥	𝑥	X
cana-3969	209	12	=	=	PUNCT
cana-3969	209	13	∑	∑	NOUN
cana-3969	209	14	𝑥𝑗1	𝑥𝑗1	NOUN
cana-3969	209	15	,	,	PUNCT
cana-3969	209	16	𝑥𝑗2	𝑥𝑗2	ADJ
cana-3969	209	17	…	…	PUNCT
cana-3969	209	18	𝑥𝑗1	𝑥𝑗1	NOUN
cana-3969	209	19	𝑟	𝑟	NOUN
cana-3969	209	20	(	(	PUNCT
cana-3969	209	21	𝑗=1	𝑗=1	NOUN
cana-3969	209	22	)	)	PUNCT
cana-3969	209	23	∈	∈	PROPN
cana-3969	209	24	(	(	PUNCT
cana-3969	209	25	𝜉∗	𝜉∗	NOUN
cana-3969	209	26	)	)	PUNCT
cana-3969	209	27	𝑖+1	𝑖+1	PUNCT
cana-3969	209	28	where	where	SCONJ
cana-3969	209	29	𝑥𝑗𝑘	𝑥𝑗𝑘	NOUN
cana-3969	209	30	∈	∈	PROPN
cana-3969	209	31	𝜉∗∀	𝜉∗∀	X
cana-3969	209	32	𝑘	𝑘	X
cana-3969	209	33	=	=	NOUN
cana-3969	209	34	1	1	NUM
cana-3969	209	35	,	,	PUNCT
cana-3969	209	36	2	2	NUM
cana-3969	209	37	,	,	PUNCT
cana-3969	209	38	…	…	PUNCT
cana-3969	209	39	𝑖	𝑖	X
cana-3969	210	1	+	+	NOUN
cana-3969	210	2	1	1	NUM
cana-3969	210	3	𝑗	𝑗	NOUN
cana-3969	210	4	=	=	SYM
cana-3969	210	5	1,2	1,2	NUM
cana-3969	210	6	,	,	PUNCT
cana-3969	210	7	…	…	PUNCT
cana-3969	210	8	𝑟	𝑟	X
cana-3969	210	9	then	then	ADV
cana-3969	210	10	𝜉𝑖+1	𝜉𝑖+1	PROPN
cana-3969	210	11	(	(	PUNCT
cana-3969	210	12	𝑥	𝑥	NOUN
cana-3969	210	13	)	)	PUNCT
cana-3969	210	14	≥	≥	NOUN
cana-3969	210	15	⋀	⋀	PUNCT
cana-3969	210	16	𝜉𝑖+1(𝑥𝑗1	𝜉𝑖+1(𝑥𝑗1	X
cana-3969	210	17	𝑥𝑗2	𝑥𝑗2	ADJ
cana-3969	210	18	…	…	PUNCT
cana-3969	210	19	𝑥𝑗+1)𝑟	𝑥𝑗+1)𝑟	NOUN
cana-3969	210	20	𝑗	𝑗	NOUN
cana-3969	210	21	=	=	SYM
cana-3969	210	22	1	1	NUM
cana-3969	210	23	≥	≥	NOUN
cana-3969	210	24	⋀	⋀	PROPN
cana-3969	210	25	(	(	PUNCT
cana-3969	210	26	⋀	⋀	PROPN
cana-3969	210	27	𝜉(𝑥𝑗𝑘)𝑖+1	𝜉(𝑥𝑗𝑘)𝑖+1	X
cana-3969	210	28	𝑘	𝑘	X
cana-3969	210	29	=	=	SYM
cana-3969	210	30	1	1	NUM
cana-3969	210	31	)	)	PUNCT
cana-3969	210	32	𝑟	𝑟	NOUN
cana-3969	210	33	𝑗	𝑗	X
cana-3969	210	34	=	=	SYM
cana-3969	210	35	1	1	NUM
cana-3969	210	36	=	=	SYM
cana-3969	210	37	𝜉(0	𝜉(0	NOUN
cana-3969	210	38	)	)	PUNCT
cana-3969	210	39	thus	thus	ADV
cana-3969	210	40	𝜉𝑖+1	𝜉𝑖+1	VERB
cana-3969	210	41	(	(	PUNCT
cana-3969	210	42	𝑥	𝑥	NOUN
cana-3969	210	43	)	)	PUNCT
cana-3969	210	44	=	=	SYM
cana-3969	210	45	𝜉(0	𝜉(0	PROPN
cana-3969	210	46	)	)	PUNCT
cana-3969	210	47	=	=	NOUN
cana-3969	210	48	𝜉𝑖+1(0	𝜉𝑖+1(0	NOUN
cana-3969	210	49	)	)	PUNCT
cana-3969	210	50	hence	hence	ADV
cana-3969	210	51	𝑥	𝑥	X
cana-3969	210	52	∈	∈	NOUN
cana-3969	210	53	(	(	PUNCT
cana-3969	210	54	𝜉𝑖+1	𝜉𝑖+1	NOUN
cana-3969	210	55	)	)	PUNCT
cana-3969	210	56	∗	∗	NOUN
cana-3969	210	57	thus	thus	ADV
cana-3969	210	58	(	(	PUNCT
cana-3969	210	59	𝜉∗)𝑖+1	𝜉∗)𝑖+1	PROPN
cana-3969	210	60	⊆	⊆	NUM
cana-3969	210	61	(	(	PUNCT
cana-3969	210	62	𝜉𝑖+1	𝜉𝑖+1	NOUN
cana-3969	210	63	)	)	PUNCT
cana-3969	210	64	∗	∗	NOUN
cana-3969	210	65	hence	hence	ADV
cana-3969	210	66	(	(	PUNCT
cana-3969	210	67	〖𝜉𝑖+1)〗∗	〖𝜉𝑖+1)〗∗	PROPN
cana-3969	210	68	=	=	SYM
cana-3969	210	69	(	(	PUNCT
cana-3969	210	70	𝜉∗)𝑖+1	𝜉∗)𝑖+1	PROPN
cana-3969	210	71	clearly	clearly	ADV
cana-3969	210	72	,	,	PUNCT
cana-3969	210	73	the	the	DET
cana-3969	210	74	result	result	NOUN
cana-3969	210	75	is	be	AUX
cana-3969	210	76	true	true	ADJ
cana-3969	210	77	for	for	ADP
cana-3969	210	78	𝑖	𝑖	SYM
cana-3969	210	79	=	=	SYM
cana-3969	210	80	1	1	NUM
cana-3969	210	81	assume	assume	VERB
cana-3969	210	82	that	that	SCONJ
cana-3969	210	83	it	it	PRON
cana-3969	210	84	is	be	AUX
cana-3969	210	85	true	true	ADJ
cana-3969	210	86	for	for	SCONJ
cana-3969	210	87	𝑖	𝑖	PRON
cana-3969	210	88	≥	≥	NOUN
cana-3969	210	89	1	1	NUM
cana-3969	210	90	let	let	VERB
cana-3969	210	91	𝑥	𝑥	PRON
cana-3969	210	92	∈	∈	PROPN
cana-3969	210	93	(	(	PUNCT
cana-3969	210	94	𝜉𝑖+1	𝜉𝑖+1	NOUN
cana-3969	210	95	)	)	PUNCT
cana-3969	210	96	∗	∗	NOUN
cana-3969	210	97	then	then	ADV
cana-3969	210	98	𝜉𝑖+1(𝑥	𝜉𝑖+1(𝑥	NUM
cana-3969	210	99	)	)	PUNCT
cana-3969	210	100	=	=	PUNCT
cana-3969	210	101	𝜉𝑖+1(0	𝜉𝑖+1(0	X
cana-3969	210	102	)	)	PUNCT
cana-3969	210	103	=	=	SYM
cana-3969	210	104	𝜉(0	𝜉(0	NOUN
cana-3969	210	105	)	)	PUNCT
cana-3969	210	106	communications	communication	NOUN
cana-3969	210	107	on	on	ADP
cana-3969	210	108	applied	apply	VERB
cana-3969	210	109	nonlinear	nonlinear	ADJ
cana-3969	210	110	analysis	analysis	NOUN
cana-3969	210	111	issn	issn	NOUN
cana-3969	210	112	:	:	PUNCT
cana-3969	210	113	1074	1074	NUM
cana-3969	210	114	-	-	PUNCT
cana-3969	210	115	133x	133x	NUM
cana-3969	210	116	vol	vol	NOUN
cana-3969	210	117	32	32	NUM
cana-3969	210	118	no	no	NOUN
cana-3969	210	119	.	.	PUNCT
cana-3969	211	1	9s	9s	NUM
cana-3969	211	2	(	(	PUNCT
cana-3969	211	3	2025	2025	NUM
cana-3969	211	4	)	)	PUNCT
cana-3969	211	5	646	646	NUM
cana-3969	211	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3969	211	7	but	but	CCONJ
cana-3969	211	8	𝜉𝑖+1(𝑥	𝜉𝑖+1(𝑥	NUM
cana-3969	211	9	)	)	PUNCT
cana-3969	211	10	=	=	SYM
cana-3969	211	11	𝑉{⋀	𝑉{⋀	NOUN
cana-3969	211	12	𝜉𝑖(𝑦𝑘	𝜉𝑖(𝑦𝑘	ADJ
cana-3969	211	13	)	)	PUNCT
cana-3969	211	14	∧	∧	PROPN
cana-3969	211	15	𝜉(𝑧𝑘))𝑛	𝜉(𝑧𝑘))𝑛	NOUN
cana-3969	211	16	𝑘=1	𝑘=1	PROPN
cana-3969	211	17	/	/	SYM
cana-3969	211	18	𝑦𝑘	𝑦𝑘	PROPN
cana-3969	211	19	,	,	PUNCT
cana-3969	211	20	𝑧𝑘	𝑧𝑘	PROPN
cana-3969	211	21	∈	∈	PROPN
cana-3969	211	22	𝑆	𝑆	PROPN
cana-3969	211	23	,	,	PUNCT
cana-3969	211	24	1	1	NUM
cana-3969	211	25	≤	≤	NOUN
cana-3969	211	26	𝑘	𝑘	DET
cana-3969	211	27	≤	≤	NUM
cana-3969	211	28	𝑛	𝑛	NOUN
cana-3969	211	29	,	,	PUNCT
cana-3969	211	30	𝑛	𝑛	PRON
cana-3969	211	31	∈	∈	PROPN
cana-3969	211	32	𝑁;∑	𝑁;∑	PROPN
cana-3969	211	33	𝑦𝑘𝑧𝑘	𝑦𝑘𝑧𝑘	NOUN
cana-3969	211	34	=	=	PUNCT
cana-3969	211	35	𝑥	𝑥	NOUN
cana-3969	211	36	}	}	PUNCT
cana-3969	211	37	=	=	SYM
cana-3969	211	38	𝑛	𝑛	PRON
cana-3969	211	39	𝑘=1	𝑘=1	NOUN
cana-3969	211	40	𝜉(0	𝜉(0	PROPN
cana-3969	211	41	)	)	PUNCT
cana-3969	211	42	∵	∵	NOUN
cana-3969	211	43	𝜉𝑖	𝜉𝑖	PROPN
cana-3969	211	44	is	be	AUX
cana-3969	211	45	finite	finite	ADJ
cana-3969	211	46	-	-	PUNCT
cana-3969	211	47	valued	value	VERB
cana-3969	211	48	∀	∀	NOUN
cana-3969	211	49	𝑖	𝑖	X
cana-3969	211	50	≥	≥	NOUN
cana-3969	211	51	1	1	NUM
cana-3969	211	52	this	this	PRON
cana-3969	211	53	implies	imply	VERB
cana-3969	211	54	that	that	SCONJ
cana-3969	211	55	𝜉𝑖(𝑦𝑘	𝜉𝑖(𝑦𝑘	PROPN
cana-3969	211	56	)	)	PUNCT
cana-3969	211	57	=	=	SYM
cana-3969	211	58	𝜉(0	𝜉(0	PROPN
cana-3969	211	59	)	)	PUNCT
cana-3969	211	60	=	=	SYM
cana-3969	211	61	𝜉(𝑧𝑘	𝜉(𝑧𝑘	NOUN
cana-3969	211	62	)	)	PUNCT
cana-3969	211	63	∀𝑘	∀𝑘	NOUN
cana-3969	211	64	,	,	PUNCT
cana-3969	211	65	1	1	NUM
cana-3969	211	66	≤	≤	NOUN
cana-3969	211	67	𝑘	𝑘	DET
cana-3969	211	68	≤	≤	NOUN
cana-3969	211	69	𝑛	𝑛	DET
cana-3969	211	70	thus	thus	ADV
cana-3969	211	71	𝑧𝑘	𝑧𝑘	ADP
cana-3969	211	72	∈	∈	PROPN
cana-3969	211	73	𝜉𝑘	𝜉𝑘	PROPN
cana-3969	211	74	∀	∀	X
cana-3969	211	75	𝑘	𝑘	NOUN
cana-3969	211	76	,	,	PUNCT
cana-3969	211	77	1	1	NUM
cana-3969	211	78	≤	≤	NUM
cana-3969	211	79	𝑘	𝑘	DET
cana-3969	211	80	≤	≤	NUM
cana-3969	211	81	𝑛	𝑛	PRON
cana-3969	211	82	also	also	ADV
cana-3969	211	83	𝜉(0	𝜉(0	PROPN
cana-3969	211	84	)	)	PUNCT
cana-3969	211	85	≥	≥	NOUN
cana-3969	211	86	𝜉(𝑖)(0	𝜉(𝑖)(0	NUM
cana-3969	211	87	)	)	PUNCT
cana-3969	211	88	≥	≥	NOUN
cana-3969	211	89	𝜉(𝑖)(𝑦𝑘	𝜉(𝑖)(𝑦𝑘	NUM
cana-3969	211	90	)	)	PUNCT
cana-3969	211	91	=	=	SYM
cana-3969	211	92	𝜉(0	𝜉(0	NOUN
cana-3969	211	93	)	)	PUNCT
cana-3969	211	94	so	so	ADV
cana-3969	211	95	𝜉(𝑖)(0	𝜉(𝑖)(0	PROPN
cana-3969	211	96	)	)	PUNCT
cana-3969	211	97	=	=	SYM
cana-3969	211	98	𝜉(𝑖)(𝑦𝑘	𝜉(𝑖)(𝑦𝑘	NOUN
cana-3969	211	99	)	)	PUNCT
cana-3969	211	100	i.e.	i.e.	X
cana-3969	211	101	𝑦𝑘	𝑦𝑘	X
cana-3969	211	102	∈	∈	PROPN
cana-3969	211	103	(	(	PUNCT
cana-3969	211	104	𝜉𝑖	𝜉𝑖	NOUN
cana-3969	211	105	)	)	PUNCT
cana-3969	211	106	∗	∗	NOUN
cana-3969	211	107	∀𝑘	∀𝑘	NOUN
cana-3969	211	108	,	,	PUNCT
cana-3969	211	109	𝑖	𝑖	ADP
cana-3969	211	110	≤	≤	NOUN
cana-3969	211	111	𝑘	𝑘	DET
cana-3969	211	112	≤	≤	NUM
cana-3969	211	113	𝑛	𝑛	PRON
cana-3969	211	114	hence	hence	ADV
cana-3969	211	115	𝑥	𝑥	X
cana-3969	211	116	=	=	PUNCT
cana-3969	211	117	∑	∑	PUNCT
cana-3969	211	118	𝑦𝑘𝑧𝑘	𝑦𝑘𝑧𝑘	ADJ
cana-3969	211	119	𝑛	𝑛	PRON
cana-3969	211	120	𝑘	𝑘	NOUN
cana-3969	211	121	=	=	SYM
cana-3969	211	122	1	1	NUM
cana-3969	211	123	∈	∈	NOUN
cana-3969	211	124	(	(	PUNCT
cana-3969	211	125	𝜉∗)(𝑖+1)𝑤ℎ𝑒𝑟𝑒	𝜉∗)(𝑖+1)𝑤ℎ𝑒𝑟𝑒	NOUN
cana-3969	211	126	𝑥𝑗𝑘	𝑥𝑗𝑘	X
cana-3969	211	127	∈	∈	NOUN
cana-3969	211	128	𝜉∗∀	𝜉∗∀	X
cana-3969	211	129	𝑘	𝑘	X
cana-3969	212	1	=	=	NOUN
cana-3969	212	2	1,2	1,2	NUM
cana-3969	212	3	,	,	PUNCT
cana-3969	212	4	…	…	PUNCT
cana-3969	212	5	𝜉(𝑖+1)(𝑥	𝜉(𝑖+1)(𝑥	NOUN
cana-3969	212	6	)	)	PUNCT
cana-3969	212	7	≥	≥	NOUN
cana-3969	213	1	⋀	⋀	PROPN
cana-3969	213	2	𝜉(𝑖+1)(𝑥𝑗1	𝜉(𝑖+1)(𝑥𝑗1	VERB
cana-3969	213	3	𝑥𝑗2	𝑥𝑗2	ADJ
cana-3969	213	4	…	…	PUNCT
cana-3969	213	5	𝑥𝑗+1)𝑟	𝑥𝑗+1)𝑟	NOUN
cana-3969	213	6	𝑗	𝑗	NOUN
cana-3969	213	7	=	=	SYM
cana-3969	213	8	1	1	NUM
cana-3969	213	9	≥	≥	NOUN
cana-3969	213	10	⋀	⋀	PROPN
cana-3969	213	11	⋀	⋀	PROPN
cana-3969	213	12	𝜉(𝑥𝑗𝑘)𝑖+1	𝜉(𝑥𝑗𝑘)𝑖+1	VERB
cana-3969	213	13	𝑘	𝑘	X
cana-3969	213	14	=	=	SYM
cana-3969	213	15	1	1	NUM
cana-3969	213	16	𝑟	𝑟	SYM
cana-3969	213	17	𝑗	𝑗	NOUN
cana-3969	213	18	=	=	SYM
cana-3969	213	19	1	1	NUM
cana-3969	213	20	=	=	SYM
cana-3969	213	21	𝜉(0	𝜉(0	NOUN
cana-3969	213	22	)	)	PUNCT
cana-3969	213	23	thus	thus	ADV
cana-3969	213	24	𝜉(𝑖+1)(𝑥	𝜉(𝑖+1)(𝑥	NOUN
cana-3969	213	25	)	)	PUNCT
cana-3969	213	26	=	=	SYM
cana-3969	213	27	𝜉(0	𝜉(0	NOUN
cana-3969	213	28	)	)	PUNCT
cana-3969	213	29	=	=	PUNCT
cana-3969	214	1	𝜉(𝑖+1)(0	𝜉(𝑖+1)(0	NOUN
cana-3969	214	2	)	)	PUNCT
cana-3969	214	3	hence	hence	ADV
cana-3969	214	4	𝑥	𝑥	X
cana-3969	214	5	∈	∈	PROPN
cana-3969	214	6	(	(	PUNCT
cana-3969	214	7	𝜉(𝑖+1	𝜉(𝑖+1	PROPN
cana-3969	214	8	)	)	PUNCT
cana-3969	214	9	)	)	PUNCT
cana-3969	214	10	∗	∗	NOUN
cana-3969	214	11	thus	thus	ADV
cana-3969	214	12	(	(	PUNCT
cana-3969	214	13	𝜉∗)𝑖+1	𝜉∗)𝑖+1	PROPN
cana-3969	214	14	⊆	⊆	NUM
cana-3969	214	15	(	(	PUNCT
cana-3969	214	16	𝜉(𝑖+1	𝜉(𝑖+1	PROPN
cana-3969	214	17	)	)	PUNCT
cana-3969	214	18	)	)	PUNCT
cana-3969	214	19	∗	∗	NOUN
cana-3969	214	20	hence	hence	ADV
cana-3969	214	21	(	(	PUNCT
cana-3969	214	22	𝜉(𝑖+1	𝜉(𝑖+1	PROPN
cana-3969	214	23	)	)	PUNCT
cana-3969	214	24	)	)	PUNCT
cana-3969	214	25	∗	∗	NOUN
cana-3969	214	26	=	=	SYM
cana-3969	214	27	(	(	PUNCT
cana-3969	214	28	𝜉∗)𝑖+1	𝜉∗)𝑖+1	PROPN
cana-3969	214	29	definition	definition	NOUN
cana-3969	214	30	6.1	6.1	NUM
cana-3969	214	31	:	:	PUNCT
cana-3969	214	32	let	let	VERB
cana-3969	214	33	𝜉	𝜉	PRON
cana-3969	214	34	∈	∈	PROPN
cana-3969	214	35	𝐹𝑆	𝐹𝑆	PROPN
cana-3969	214	36	define	define	VERB
cana-3969	214	37	𝜉𝑛	𝜉𝑛	PRON
cana-3969	214	38	and	and	CCONJ
cana-3969	214	39	𝜉𝑛	𝜉𝑛	NOUN
cana-3969	214	40	as	as	SCONJ
cana-3969	214	41	follows	follow	VERB
cana-3969	214	42	when	when	SCONJ
cana-3969	214	43	𝑥	𝑥	DET
cana-3969	214	44	∈	∈	PROPN
cana-3969	214	45	𝑁	𝑁	PROPN
cana-3969	214	46	,	,	PUNCT
cana-3969	214	47	𝑛	𝑛	PROPN
cana-3969	214	48	>	>	X
cana-3969	214	49	1	1	NUM
cana-3969	214	50	;	;	PUNCT
cana-3969	214	51	𝜉𝐼	𝜉𝐼	NOUN
cana-3969	214	52	=	=	SYM
cana-3969	214	53	𝜉	𝜉	X
cana-3969	214	54	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-3969	214	55	𝜉𝑛	𝜉𝑛	X
cana-3969	214	56	=	=	SYM
cana-3969	214	57	𝜉1	𝜉1	PROPN
cana-3969	214	58	°	°	PROPN
cana-3969	214	59	𝜉𝑛−1	𝜉𝑛−1	PROPN
cana-3969	214	60	𝜉(𝐼	𝜉(𝐼	X
cana-3969	214	61	)	)	PUNCT
cana-3969	215	1	=	=	SYM
cana-3969	215	2	𝜉	𝜉	X
cana-3969	215	3	𝜉(𝑛	𝜉(𝑛	NOUN
cana-3969	215	4	)	)	PUNCT
cana-3969	215	5	=	=	SYM
cana-3969	215	6	𝜉(1)𝜉(𝑛−1	𝜉(1)𝜉(𝑛−1	PROPN
cana-3969	215	7	)	)	PUNCT
cana-3969	215	8	definition	definition	NOUN
cana-3969	215	9	6.2	6.2	NUM
cana-3969	215	10	:	:	PUNCT
cana-3969	215	11	let	let	VERB
cana-3969	215	12	𝜉1	𝜉1	PROPN
cana-3969	215	13	,	,	PUNCT
cana-3969	215	14	𝜉2	𝜉2	PROPN
cana-3969	215	15	𝐹𝑆.	𝐹𝑆.	AUX
cana-3969	215	16	define	define	VERB
cana-3969	215	17	𝜉1𝜉2	𝜉1𝜉2	PUNCT
cana-3969	215	18	∈	∈	PROPN
cana-3969	215	19	𝐹𝑆	𝐹𝑆	PROPN
cana-3969	215	20	∀	∀	X
cana-3969	216	1	𝑥	𝑥	DET
cana-3969	216	2	∈	∈	PROPN
cana-3969	216	3	𝑆	𝑆	PROPN
cana-3969	216	4	(	(	PUNCT
cana-3969	216	5	𝜉1𝜉2)(𝑥	𝜉1𝜉2)(𝑥	PROPN
cana-3969	216	6	)	)	PUNCT
cana-3969	216	7	=	=	SYM
cana-3969	216	8	𝑉{⋀	𝑉{⋀	NOUN
cana-3969	216	9	(	(	PUNCT
cana-3969	216	10	𝜉(𝑦𝑖	𝜉(𝑦𝑖	NUM
cana-3969	216	11	)	)	PUNCT
cana-3969	216	12	∧	∧	PROPN
cana-3969	216	13	𝜉(𝑧𝑖))|	𝜉(𝑧𝑖))|	PROPN
cana-3969	216	14	𝑦𝑖𝑧𝑖	𝑦𝑖𝑧𝑖	NOUN
cana-3969	216	15	∈	∈	PROPN
cana-3969	216	16	𝑆𝑛	𝑆𝑛	PROPN
cana-3969	216	17	𝑖=1	𝑖=1	PROPN
cana-3969	216	18	and	and	CCONJ
cana-3969	216	19	1	1	NUM
cana-3969	216	20	≤	≤	NUM
cana-3969	216	21	𝑖	𝑖	SYM
cana-3969	216	22	≤	≤	NUM
cana-3969	216	23	𝑛	𝑛	NOUN
cana-3969	216	24	,	,	PUNCT
cana-3969	216	25	𝑛	𝑛	PRON
cana-3969	216	26	∈	∈	PROPN
cana-3969	216	27	𝑁	𝑁	PROPN
cana-3969	216	28	,	,	PUNCT
cana-3969	216	29	∑	∑	ADV
cana-3969	216	30	𝑦𝑖𝑧𝑖	𝑦𝑖𝑧𝑖	NOUN
cana-3969	216	31	=	=	NOUN
cana-3969	216	32	𝑥𝑛	𝑥𝑛	VERB
cana-3969	216	33	𝑖=1	𝑖=1	PUNCT
cana-3969	216	34	}	}	PUNCT
cana-3969	216	35	5	5	X
cana-3969	216	36	.	.	PUNCT
cana-3969	217	1	if	if	SCONJ
cana-3969	217	2	s	s	NOUN
cana-3969	217	3	is	be	AUX
cana-3969	217	4	commutative	commutative	ADJ
cana-3969	217	5	then	then	ADV
cana-3969	217	6	𝜉1𝜉2	𝜉1𝜉2	VERB
cana-3969	217	7	=	=	SYM
cana-3969	217	8	𝜉2𝜉1	𝜉2𝜉1	X
cana-3969	217	9	(	(	PUNCT
cana-3969	217	10	𝜉1𝜉2	𝜉1𝜉2	NOUN
cana-3969	217	11	)	)	PUNCT
cana-3969	217	12	(	(	PUNCT
cana-3969	217	13	𝑥	𝑥	NOUN
cana-3969	217	14	)	)	PUNCT
cana-3969	217	15	=	=	SYM
cana-3969	217	16	∨	∨	X
cana-3969	217	17	{	{	PUNCT
cana-3969	217	18	⋀	⋀	PROPN
cana-3969	217	19	(	(	PUNCT
cana-3969	217	20	𝜉1(𝑦𝑖	𝜉1(𝑦𝑖	PROPN
cana-3969	217	21	)	)	PUNCT
cana-3969	217	22	∧	∧	PROPN
cana-3969	217	23	𝜉2(𝑧𝑖)|𝑦𝑖	𝜉2(𝑧𝑖)|𝑦𝑖	NOUN
cana-3969	217	24	,	,	PUNCT
cana-3969	217	25	𝑧𝑖	𝑧𝑖	X
cana-3969	218	1	∈	∈	PUNCT
cana-3969	219	1	𝑆𝑛	𝑆𝑛	ADJ
cana-3969	219	2	𝑖	𝑖	NOUN
cana-3969	219	3	=	=	NOUN
cana-3969	219	4	1	1	NUM
cana-3969	219	5	1	1	NUM
cana-3969	219	6	≤	≤	NUM
cana-3969	219	7	𝑖	𝑖	SYM
cana-3969	219	8	≤	≤	NUM
cana-3969	219	9	𝑛	𝑛	NOUN
cana-3969	219	10	,	,	PUNCT
cana-3969	219	11	𝑛	𝑛	PRON
cana-3969	219	12	∈	∈	PROPN
cana-3969	219	13	𝑁	𝑁	PROPN
cana-3969	219	14	,	,	PUNCT
cana-3969	219	15	∑	∑	PUNCT
cana-3969	219	16	〖𝑦𝑖𝑧𝑖	〖𝑦𝑖𝑧𝑖	PROPN
cana-3969	219	17	=	=	PUNCT
cana-3969	219	18	𝑥}𝑛	𝑥}𝑛	NOUN
cana-3969	219	19	𝑖	𝑖	NOUN
cana-3969	219	20	=	=	SYM
cana-3969	219	21	1	1	NUM
cana-3969	219	22	=	=	SYM
cana-3969	219	23	∨	∨	X
cana-3969	219	24	{	{	PUNCT
cana-3969	219	25	⋀	⋀	PROPN
cana-3969	219	26	(	(	PUNCT
cana-3969	219	27	𝜉2(𝑧𝑖	𝜉2(𝑧𝑖	PROPN
cana-3969	219	28	)	)	PUNCT
cana-3969	219	29	∧	∧	PROPN
cana-3969	219	30	𝜉1(𝑦𝑖	𝜉1(𝑦𝑖	PROPN
cana-3969	219	31	)	)	PUNCT
cana-3969	219	32	|𝑛	|𝑛	X
cana-3969	220	1	𝑖	𝑖	NOUN
cana-3969	221	1	=	=	SYM
cana-3969	222	1	1	1	NUM
cana-3969	222	2	𝑦𝑖	𝑦𝑖	INTJ
cana-3969	222	3	,	,	PUNCT
cana-3969	222	4	𝑧𝑖	𝑧𝑖	NOUN
cana-3969	222	5	∈	∈	PROPN
cana-3969	222	6	𝑆	𝑆	PROPN
cana-3969	222	7	1	1	NUM
cana-3969	222	8	≤	≤	PROPN
cana-3969	222	9	𝑖	𝑖	SYM
cana-3969	222	10	≤	≤	NUM
cana-3969	222	11	𝑛	𝑛	NOUN
cana-3969	222	12	,	,	PUNCT
cana-3969	222	13	𝑛	𝑛	DET
cana-3969	222	14	∈	∈	PROPN
cana-3969	222	15	𝑁	𝑁	PROPN
cana-3969	222	16	,	,	PUNCT
cana-3969	222	17	∑	∑	ADP
cana-3969	222	18	〖𝑧𝑖𝑦𝑖	〖𝑧𝑖𝑦𝑖	PROPN
cana-3969	222	19	=	=	PUNCT
cana-3969	222	20	𝑥}𝑛	𝑥}𝑛	PROPN
cana-3969	222	21	𝑖	𝑖	NOUN
cana-3969	222	22	=	=	SYM
cana-3969	222	23	1	1	NUM
cana-3969	222	24	=	=	SYM
cana-3969	222	25	(	(	PUNCT
cana-3969	222	26	𝜉2𝜉1)(𝑥	𝜉2𝜉1)(𝑥	PROPN
cana-3969	222	27	)	)	PUNCT
cana-3969	222	28	𝜉1𝜉2	𝜉1𝜉2	AUX
cana-3969	222	29	=	=	SYM
cana-3969	222	30	𝜉2𝜉1	𝜉2𝜉1	PROPN
cana-3969	222	31	communications	communication	NOUN
cana-3969	222	32	on	on	ADP
cana-3969	222	33	applied	apply	VERB
cana-3969	222	34	nonlinear	nonlinear	ADJ
cana-3969	222	35	analysis	analysis	NOUN
cana-3969	222	36	issn	issn	NOUN
cana-3969	222	37	:	:	PUNCT
cana-3969	222	38	1074	1074	NUM
cana-3969	222	39	-	-	PUNCT
cana-3969	222	40	133x	133x	NUM
cana-3969	222	41	vol	vol	NOUN
cana-3969	222	42	32	32	NUM
cana-3969	222	43	no	no	NOUN
cana-3969	222	44	.	.	PUNCT
cana-3969	223	1	9s	9s	NUM
cana-3969	223	2	(	(	PUNCT
cana-3969	223	3	2025	2025	NUM
cana-3969	223	4	)	)	PUNCT
cana-3969	223	5	647	647	NUM
cana-3969	223	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3969	223	7	𝜉∗	𝜉∗	NOUN
cana-3969	223	8	=	=	PUNCT
cana-3969	223	9	{	{	PUNCT
cana-3969	223	10	𝑥	𝑥	NOUN
cana-3969	223	11	∈	∈	PROPN
cana-3969	223	12	𝑅|𝜉(𝑥	𝑅|𝜉(𝑥	NOUN
cana-3969	223	13	)	)	PUNCT
cana-3969	223	14	=	=	SYM
cana-3969	223	15	𝜉(0	𝜉(0	NOUN
cana-3969	223	16	)	)	PUNCT
cana-3969	223	17	}	}	PUNCT
cana-3969	223	18	6	6	NUM
cana-3969	223	19	.	.	PUNCT
cana-3969	224	1	let	let	VERB
cana-3969	224	2	𝜉	𝜉	PRON
cana-3969	224	3	∈	∈	VERB
cana-3969	224	4	𝐹𝑆	𝐹𝑆	PROPN
cana-3969	224	5	and	and	CCONJ
cana-3969	224	6	𝑘	𝑘	PRON
cana-3969	224	7	∈	∈	PROPN
cana-3969	224	8	𝑁	𝑁	PROPN
cana-3969	224	9	,	,	PUNCT
cana-3969	224	10	𝑖𝑓	𝑖𝑓	NUM
cana-3969	224	11	𝑥1	𝑥1	NOUN
cana-3969	224	12	,	,	PUNCT
cana-3969	224	13	𝑥2	𝑥2	NOUN
cana-3969	224	14	,	,	PUNCT
cana-3969	224	15	…	…	PUNCT
cana-3969	225	1	,	,	PUNCT
cana-3969	225	2	𝑥𝑘	𝑥𝑘	PROPN
cana-3969	225	3	∈	∈	PROPN
cana-3969	225	4	𝑆	𝑆	PROPN
cana-3969	225	5	then	then	ADV
cana-3969	225	6	1	1	X
cana-3969	225	7	.	.	PUNCT
cana-3969	226	1	𝜉∗	𝜉∗	PROPN
cana-3969	226	2	𝑘(𝑥1	𝑘(𝑥1	ADJ
cana-3969	226	3	,	,	PUNCT
cana-3969	226	4	𝑥2	𝑥2	NOUN
cana-3969	226	5	,	,	PUNCT
cana-3969	226	6	…	…	PUNCT
cana-3969	226	7	,	,	PUNCT
cana-3969	226	8	𝑥𝑘	𝑥𝑘	X
cana-3969	226	9	)	)	PUNCT
cana-3969	226	10	≥	≥	NOUN
cana-3969	227	1	⋀	⋀	PROPN
cana-3969	227	2	𝜉∗(𝑥𝑖)𝑘	𝜉∗(𝑥𝑖)𝑘	PART
cana-3969	227	3	𝑖	𝑖	NOUN
cana-3969	227	4	=	=	NOUN
cana-3969	227	5	1	1	NUM
cana-3969	227	6	2	2	NUM
cana-3969	227	7	.	.	PUNCT
cana-3969	228	1	𝜉∗	𝜉∗	PROPN
cana-3969	228	2	(	(	PUNCT
cana-3969	228	3	𝑘)(𝑥1	𝑘)(𝑥1	ADJ
cana-3969	228	4	,	,	PUNCT
cana-3969	228	5	𝑥2	𝑥2	NOUN
cana-3969	228	6	,	,	PUNCT
cana-3969	228	7	…	…	PUNCT
cana-3969	228	8	,	,	PUNCT
cana-3969	228	9	𝑥𝑘	𝑥𝑘	X
cana-3969	228	10	)	)	PUNCT
cana-3969	228	11	≥	≥	NOUN
cana-3969	229	1	⋀	⋀	PROPN
cana-3969	229	2	𝜉∗(𝑥𝑖)𝑘	𝜉∗(𝑥𝑖)𝑘	PART
cana-3969	229	3	𝑖	𝑖	NOUN
cana-3969	229	4	=	=	SYM
cana-3969	229	5	1	1	NUM
cana-3969	229	6	proof	proof	NOUN
cana-3969	229	7	:	:	PUNCT
cana-3969	229	8	1	1	X
cana-3969	229	9	.	.	PUNCT
cana-3969	229	10	clearly	clearly	ADV
cana-3969	229	11	,	,	PUNCT
cana-3969	229	12	the	the	DET
cana-3969	229	13	result	result	NOUN
cana-3969	229	14	is	be	AUX
cana-3969	229	15	true	true	ADJ
cana-3969	229	16	for	for	ADP
cana-3969	229	17	𝑘	𝑘	NOUN
cana-3969	229	18	=	=	SYM
cana-3969	229	19	1	1	NUM
cana-3969	229	20	assume	assume	VERB
cana-3969	229	21	that	that	SCONJ
cana-3969	229	22	it	it	PRON
cana-3969	229	23	is	be	AUX
cana-3969	229	24	true	true	ADJ
cana-3969	229	25	for	for	ADP
cana-3969	229	26	𝑘	𝑘	PROPN
cana-3969	229	27	>	>	SYM
cana-3969	229	28	1	1	NUM
cana-3969	229	29	now	now	ADV
cana-3969	229	30	𝜉∗	𝜉∗	PROPN
cana-3969	229	31	𝑘+1(𝑥1	𝑘+1(𝑥1	ADJ
cana-3969	229	32	,	,	PUNCT
cana-3969	229	33	𝑥2	𝑥2	NOUN
cana-3969	229	34	,	,	PUNCT
cana-3969	229	35	…	…	PUNCT
cana-3969	229	36	,	,	PUNCT
cana-3969	229	37	𝑥𝑘+1	𝑥𝑘+1	NOUN
cana-3969	229	38	)	)	PUNCT
cana-3969	229	39	=	=	PUNCT
cana-3969	230	1	(	(	PUNCT
cana-3969	230	2	𝜉∗	𝜉∗	NOUN
cana-3969	230	3	𝑘	𝑘	PRON
cana-3969	230	4	∘	∘	PROPN
cana-3969	230	5	𝜉∗)(𝑥1	𝜉∗)(𝑥1	NOUN
cana-3969	230	6	,	,	PUNCT
cana-3969	230	7	𝑥2	𝑥2	NOUN
cana-3969	230	8	,	,	PUNCT
cana-3969	230	9	…	…	PUNCT
cana-3969	230	10	,	,	PUNCT
cana-3969	230	11	𝑥𝑘+1	𝑥𝑘+1	X
cana-3969	230	12	)	)	PUNCT
cana-3969	230	13	≥	≥	NOUN
cana-3969	230	14	𝜉∗	𝜉∗	PROPN
cana-3969	230	15	𝑘(𝑥1	𝑘(𝑥1	ADJ
cana-3969	230	16	,	,	PUNCT
cana-3969	230	17	𝑥2	𝑥2	NOUN
cana-3969	230	18	,	,	PUNCT
cana-3969	230	19	…	…	PUNCT
cana-3969	230	20	,	,	PUNCT
cana-3969	230	21	𝑥𝑘	𝑥𝑘	NOUN
cana-3969	230	22	)	)	PUNCT
cana-3969	230	23	∧	∧	PROPN
cana-3969	230	24	𝜉∗(𝑥𝑘+1	𝜉∗(𝑥𝑘+1	PROPN
cana-3969	230	25	)	)	PUNCT
cana-3969	230	26	≥	≥	NOUN
cana-3969	231	1	𝜉∗	𝜉∗	PROPN
cana-3969	231	2	𝑘(0,0	𝑘(0,0	NOUN
cana-3969	231	3	,	,	PUNCT
cana-3969	231	4	…	…	PUNCT
cana-3969	231	5	,	,	PUNCT
cana-3969	231	6	0	0	X
cana-3969	231	7	)	)	PUNCT
cana-3969	231	8	∧	∧	PROPN
cana-3969	231	9	𝜉∗(𝑥𝑘+1	𝜉∗(𝑥𝑘+1	PROPN
cana-3969	231	10	)	)	PUNCT
cana-3969	231	11	𝜉∗	𝜉∗	PROPN
cana-3969	231	12	𝑘(𝑥1	𝑘(𝑥1	ADJ
cana-3969	231	13	,	,	PUNCT
cana-3969	231	14	𝑥2	𝑥2	NOUN
cana-3969	231	15	,	,	PUNCT
cana-3969	231	16	…	…	PUNCT
cana-3969	231	17	,	,	PUNCT
cana-3969	231	18	𝑥𝑘	𝑥𝑘	X
cana-3969	231	19	)	)	PUNCT
cana-3969	231	20	≥	≥	NOUN
cana-3969	232	1	⋀	⋀	PROPN
cana-3969	232	2	𝜉∗(𝑥𝑖)𝑘	𝜉∗(𝑥𝑖)𝑘	PART
cana-3969	232	3	𝑖	𝑖	NOUN
cana-3969	232	4	=	=	NOUN
cana-3969	232	5	1	1	NUM
cana-3969	232	6	2	2	NUM
cana-3969	232	7	.	.	PUNCT
cana-3969	233	1	this	this	DET
cana-3969	233	2	proof	proof	NOUN
cana-3969	233	3	follows	follow	VERB
cana-3969	233	4	the	the	DET
cana-3969	233	5	same	same	ADJ
cana-3969	233	6	as	as	ADP
cana-3969	233	7	lemma	lemma	PROPN
cana-3969	233	8	1	1	NUM
cana-3969	233	9	.	.	NOUN
cana-3969	233	10	6	6	NUM
cana-3969	233	11	.	.	X
cana-3969	233	12	discussion	discussion	NOUN
cana-3969	233	13	a	a	DET
cana-3969	233	14	fuzzy	fuzzy	ADJ
cana-3969	233	15	ternary	ternary	ADJ
cana-3969	233	16	gamma	gamma	NOUN
cana-3969	233	17	semigroup	semigroup	PROPN
cana-3969	233	18	is	be	AUX
cana-3969	233	19	a	a	DET
cana-3969	233	20	mathematical	mathematical	ADJ
cana-3969	233	21	structure	structure	NOUN
cana-3969	233	22	that	that	PRON
cana-3969	233	23	combines	combine	VERB
cana-3969	233	24	concepts	concept	NOUN
cana-3969	233	25	from	from	ADP
cana-3969	233	26	fuzzy	fuzzy	ADJ
cana-3969	233	27	set	set	NOUN
cana-3969	233	28	theory	theory	NOUN
cana-3969	233	29	,	,	PUNCT
cana-3969	233	30	ternary	ternary	ADJ
cana-3969	233	31	algebra	algebra	NOUN
cana-3969	233	32	,	,	PUNCT
cana-3969	233	33	and	and	CCONJ
cana-3969	233	34	gamma	gamma	PROPN
cana-3969	233	35	semigroup	semigroup	PROPN
cana-3969	233	36	.	.	PUNCT
cana-3969	234	1	we	we	PRON
cana-3969	234	2	discussed	discuss	VERB
cana-3969	234	3	that	that	PRON
cana-3969	234	4	fuzzy	fuzzy	ADJ
cana-3969	234	5	ternary	ternary	ADJ
cana-3969	234	6	gamma	gamma	NOUN
cana-3969	234	7	semigroups	semigroup	NOUN
cana-3969	234	8	can	can	AUX
cana-3969	234	9	be	be	AUX
cana-3969	234	10	used	use	VERB
cana-3969	234	11	to	to	PART
cana-3969	234	12	model	model	VERB
cana-3969	234	13	fuzzy	fuzzy	ADJ
cana-3969	234	14	inference	inference	NOUN
cana-3969	234	15	systems	system	NOUN
cana-3969	234	16	.	.	PUNCT
cana-3969	235	1	fuzzy	fuzzy	ADJ
cana-3969	235	2	ternary	ternary	ADJ
cana-3969	235	3	gamma	gamma	NOUN
cana-3969	235	4	semigroups	semigroup	NOUN
cana-3969	235	5	can	can	AUX
cana-3969	235	6	be	be	AUX
cana-3969	235	7	applied	apply	VERB
cana-3969	235	8	to	to	ADP
cana-3969	235	9	computer	computer	NOUN
cana-3969	235	10	science	science	NOUN
cana-3969	235	11	,	,	PUNCT
cana-3969	235	12	particularly	particularly	ADV
cana-3969	235	13	in	in	ADP
cana-3969	235	14	studying	study	VERB
cana-3969	235	15	fuzzy	fuzzy	ADJ
cana-3969	235	16	automata	automata	NOUN
cana-3969	235	17	and	and	CCONJ
cana-3969	235	18	fuzzy	fuzzy	ADJ
cana-3969	235	19	languages	language	NOUN
cana-3969	235	20	.	.	PUNCT
cana-3969	236	1	fuzzy	fuzzy	ADJ
cana-3969	236	2	ternary	ternary	ADJ
cana-3969	236	3	gamma	gamma	NOUN
cana-3969	236	4	semigroup	semigroup	PROPN
cana-3969	236	5	may	may	AUX
cana-3969	236	6	have	have	VERB
cana-3969	236	7	applications	application	NOUN
cana-3969	236	8	in	in	ADP
cana-3969	236	9	cryptography	cryptography	NOUN
cana-3969	236	10	,	,	PUNCT
cana-3969	236	11	especially	especially	ADV
cana-3969	236	12	in	in	ADP
cana-3969	236	13	developing	develop	VERB
cana-3969	236	14	fuzzy	fuzzy	ADJ
cana-3969	236	15	cryptographic	cryptographic	ADJ
cana-3969	236	16	protocols	protocol	NOUN
cana-3969	236	17	.	.	PUNCT
cana-3969	237	1	new	new	ADJ
cana-3969	237	2	problems	problem	NOUN
cana-3969	237	3	are	be	AUX
cana-3969	237	4	explored	explore	VERB
cana-3969	237	5	in	in	ADP
cana-3969	237	6	studying	study	VERB
cana-3969	237	7	fuzzy	fuzzy	ADJ
cana-3969	237	8	ternary	ternary	ADJ
cana-3969	237	9	gamma	gamma	NOUN
cana-3969	237	10	semigroups	semigroup	NOUN
cana-3969	237	11	,	,	PUNCT
cana-3969	237	12	a	a	DET
cana-3969	237	13	relatively	relatively	ADV
cana-3969	237	14	new	new	ADJ
cana-3969	237	15	research	research	NOUN
cana-3969	237	16	area	area	NOUN
cana-3969	237	17	.	.	PUNCT
cana-3969	238	1	❖	❖	AUX
cana-3969	238	2	clarify	clarify	VERB
cana-3969	238	3	key	key	ADJ
cana-3969	238	4	terms	term	NOUN
cana-3969	238	5	:	:	PUNCT
cana-3969	238	6	while	while	SCONJ
cana-3969	238	7	"	"	PUNCT
cana-3969	238	8	star	star	NOUN
cana-3969	238	9	zeta	zeta	NOUN
cana-3969	238	10	"	"	PUNCT
cana-3969	238	11	and	and	CCONJ
cana-3969	238	12	"	"	PUNCT
cana-3969	238	13	gamma	gamma	PROPN
cana-3969	238	14	semigroup	semigroup	PROPN
cana-3969	238	15	"	"	PUNCT
cana-3969	238	16	are	be	AUX
cana-3969	238	17	central	central	ADJ
cana-3969	238	18	concepts	concept	NOUN
cana-3969	238	19	,	,	PUNCT
cana-3969	238	20	a	a	DET
cana-3969	238	21	brief	brief	ADJ
cana-3969	238	22	clarification	clarification	NOUN
cana-3969	238	23	of	of	ADP
cana-3969	238	24	their	their	PRON
cana-3969	238	25	role	role	NOUN
cana-3969	238	26	in	in	ADP
cana-3969	238	27	fuzzy	fuzzy	ADJ
cana-3969	238	28	semigroups	semigroup	NOUN
cana-3969	238	29	could	could	AUX
cana-3969	238	30	help	help	VERB
cana-3969	238	31	readers	reader	NOUN
cana-3969	238	32	unfamiliar	unfamiliar	ADJ
cana-3969	238	33	with	with	ADP
cana-3969	238	34	the	the	DET
cana-3969	238	35	topic	topic	NOUN
cana-3969	238	36	.	.	PUNCT
cana-3969	239	1	❖	❖	AUX
cana-3969	239	2	strengthen	strengthen	VERB
cana-3969	239	3	logical	logical	ADJ
cana-3969	239	4	progression	progression	NOUN
cana-3969	239	5	:	:	PUNCT
cana-3969	239	6	ensure	ensure	VERB
cana-3969	239	7	the	the	DET
cana-3969	239	8	discussion	discussion	NOUN
cana-3969	239	9	flows	flow	VERB
cana-3969	239	10	from	from	ADP
cana-3969	239	11	definitions	definition	NOUN
cana-3969	239	12	to	to	ADP
cana-3969	239	13	properties	property	NOUN
cana-3969	239	14	,	,	PUNCT
cana-3969	239	15	followed	follow	VERB
cana-3969	239	16	by	by	ADP
cana-3969	239	17	implications	implication	NOUN
cana-3969	239	18	and	and	CCONJ
cana-3969	239	19	the	the	DET
cana-3969	239	20	collaborative	collaborative	ADJ
cana-3969	239	21	nature	nature	NOUN
cana-3969	239	22	of	of	ADP
cana-3969	239	23	the	the	DET
cana-3969	239	24	work	work	NOUN
cana-3969	239	25	.	.	PUNCT
cana-3969	240	1	references	reference	NOUN
cana-3969	240	2	[	[	X
cana-3969	240	3	1	1	NUM
cana-3969	240	4	]	]	X
cana-3969	240	5	razaq	razaq	ADJ
cana-3969	240	6	,	,	PUNCT
cana-3969	240	7	a.	a.	NOUN
cana-3969	240	8	,	,	PUNCT
cana-3969	240	9	&	&	CCONJ
cana-3969	240	10	alhamzi	alhamzi	PROPN
cana-3969	240	11	,	,	PUNCT
cana-3969	240	12	g.	g.	PROPN
cana-3969	240	13	(	(	PUNCT
cana-3969	240	14	2023	2023	NUM
cana-3969	240	15	)	)	PUNCT
cana-3969	240	16	.	.	PUNCT
cana-3969	241	1	on	on	ADP
cana-3969	241	2	pythagorean	pythagorean	PROPN
cana-3969	241	3	fuzzy	fuzzy	ADJ
cana-3969	241	4	ideals	ideal	NOUN
cana-3969	241	5	of	of	ADP
cana-3969	241	6	a	a	DET
cana-3969	241	7	classical	classical	ADJ
cana-3969	241	8	ring	ring	NOUN
cana-3969	241	9	.	.	PUNCT
cana-3969	242	1	aims	aim	VERB
cana-3969	242	2	math	math	NOUN
cana-3969	242	3	,	,	PUNCT
cana-3969	242	4	8(2	8(2	NUM
cana-3969	242	5	)	)	PUNCT
cana-3969	242	6	,	,	PUNCT
cana-3969	242	7	4280	4280	NUM
cana-3969	242	8	-	-	SYM
cana-3969	242	9	4303	4303	NUM
cana-3969	242	10	.	.	PUNCT
cana-3969	243	1	[	[	X
cana-3969	243	2	2	2	NUM
cana-3969	243	3	]	]	X
cana-3969	243	4	clifford	clifford	PROPN
cana-3969	243	5	a.h	a.h	PROPN
cana-3969	243	6	and	and	CCONJ
cana-3969	243	7	g.	g.	PROPN
cana-3969	243	8	b.	b.	PROPN
cana-3969	243	9	preston	preston	PROPN
cana-3969	243	10	,	,	PUNCT
cana-3969	243	11	the	the	DET
cana-3969	243	12	algebraic	algebraic	ADJ
cana-3969	243	13	theory	theory	NOUN
cana-3969	243	14	of	of	ADP
cana-3969	243	15	semigroups	semigroup	NOUN
cana-3969	243	16	,	,	PUNCT
cana-3969	243	17	vol	vol	NOUN
cana-3969	243	18	.	.	PUNCT
cana-3969	244	1	i	i	PRON
cana-3969	244	2	,	,	PUNCT
cana-3969	244	3	mathematical	mathematical	ADJ
cana-3969	244	4	surveys	survey	NOUN
cana-3969	244	5	no	no	NOUN
cana-3969	244	6	.	.	NOUN
cana-3969	244	7	7	7	NUM
cana-3969	244	8	,	,	PUNCT
cana-3969	244	9	amer	amer	PROPN
cana-3969	244	10	.	.	PROPN
cana-3969	244	11	math	math	PROPN
cana-3969	244	12	.	.	PUNCT
cana-3969	245	1	soc	soc	PROPN
cana-3969	245	2	.	.	PUNCT
cana-3969	245	3	,	,	PUNCT
cana-3969	245	4	providence	providence	NOUN
cana-3969	245	5	,	,	PUNCT
cana-3969	245	6	1961	1961	NUM
cana-3969	245	7	.	.	PUNCT
cana-3969	246	1	[	[	X
cana-3969	246	2	3	3	X
cana-3969	246	3	]	]	X
cana-3969	246	4	clifford	clifford	PROPN
cana-3969	246	5	a.h	a.h	PROPN
cana-3969	246	6	and	and	CCONJ
cana-3969	246	7	g.	g.	PROPN
cana-3969	246	8	b.	b.	PROPN
cana-3969	246	9	preston	preston	PROPN
cana-3969	246	10	,	,	PUNCT
cana-3969	246	11	the	the	DET
cana-3969	246	12	algebraic	algebraic	ADJ
cana-3969	246	13	theory	theory	NOUN
cana-3969	246	14	of	of	ADP
cana-3969	246	15	semigroups	semigroup	NOUN
cana-3969	246	16	,	,	PUNCT
cana-3969	246	17	vol	vol	NOUN
cana-3969	246	18	.	.	PUNCT
cana-3969	246	19	ii	ii	PROPN
cana-3969	246	20	,	,	PUNCT
cana-3969	246	21	mathematical	mathematical	ADJ
cana-3969	246	22	surveys	survey	NOUN
cana-3969	246	23	no	no	NOUN
cana-3969	246	24	.	.	NOUN
cana-3969	246	25	7	7	NUM
cana-3969	246	26	,	,	PUNCT
cana-3969	246	27	amer	amer	PROPN
cana-3969	246	28	.	.	PROPN
cana-3969	246	29	math	math	PROPN
cana-3969	246	30	soc	soc	PROPN
cana-3969	246	31	.	.	PUNCT
cana-3969	246	32	,	,	PUNCT
cana-3969	246	33	providence	providence	NOUN
cana-3969	246	34	,	,	PUNCT
cana-3969	246	35	1967	1967	NUM
cana-3969	246	36	.	.	PUNCT
cana-3969	247	1	[	[	X
cana-3969	247	2	4	4	X
cana-3969	247	3	]	]	X
cana-3969	247	4	j.m	j.m	ADJ
cana-3969	247	5	howie	howie	NOUN
cana-3969	247	6	,	,	PUNCT
cana-3969	247	7	fundamentals	fundamental	NOUN
cana-3969	247	8	of	of	ADP
cana-3969	247	9	semigroup	semigroup	PROPN
cana-3969	247	10	theory	theory	NOUN
cana-3969	247	11	,	,	PUNCT
cana-3969	247	12	clarendon	clarendon	PROPN
cana-3969	247	13	press	press	PROPN
cana-3969	247	14	,	,	PUNCT
cana-3969	247	15	oxford	oxford	NOUN
cana-3969	247	16	,	,	PUNCT
cana-3969	247	17	1995	1995	NUM
cana-3969	247	18	.	.	PUNCT
cana-3969	248	1	[	[	X
cana-3969	248	2	5	5	X
cana-3969	248	3	]	]	X
cana-3969	248	4	j.m	j.m	ADJ
cana-3969	248	5	howie	howie	NOUN
cana-3969	248	6	an	an	DET
cana-3969	248	7	introduction	introduction	NOUN
cana-3969	248	8	to	to	ADP
cana-3969	248	9	semigroup	semigroup	PROPN
cana-3969	248	10	theory	theory	NOUN
cana-3969	248	11	,	,	PUNCT
cana-3969	248	12	academic	academic	ADJ
cana-3969	248	13	press	press	NOUN
cana-3969	248	14	,	,	PUNCT
cana-3969	248	15	london	london	PROPN
cana-3969	248	16	,	,	PUNCT
cana-3969	248	17	1976	1976	NUM
cana-3969	248	18	.	.	PUNCT
cana-3969	249	1	ijirt	ijirt	NOUN
cana-3969	249	2	154793	154793	NUM
cana-3969	249	3	623	623	NUM
cana-3969	249	4	.	.	PUNCT
cana-3969	250	1	[	[	X
cana-3969	250	2	6	6	NUM
cana-3969	250	3	]	]	PUNCT
cana-3969	250	4	vasantha	vasantha	NOUN
cana-3969	250	5	g	g	PROPN
cana-3969	250	6	,	,	PUNCT
cana-3969	250	7	t	t	PROPN
cana-3969	250	8	srilakshmi	srilakshmi	NOUN
cana-3969	250	9	;	;	PUNCT
cana-3969	250	10	“	"	PUNCT
cana-3969	250	11	important	important	ADJ
cana-3969	250	12	role	role	NOUN
cana-3969	250	13	of	of	ADP
cana-3969	250	14	idempotent	idempotent	NOUN
cana-3969	250	15	and	and	CCONJ
cana-3969	250	16	regular	regular	ADJ
cana-3969	250	17	classes	class	NOUN
cana-3969	250	18	in	in	ADP
cana-3969	250	19	the	the	DET
cana-3969	250	20	distinguished	distinguished	ADJ
cana-3969	250	21	semigroup	semigroup	PROPN
cana-3969	250	22	theory	theory	NOUN
cana-3969	250	23	”	"	PUNCT
cana-3969	250	24	in	in	ADP
cana-3969	250	25	volume	volume	NOUN
cana-3969	250	26	15	15	NUM
cana-3969	250	27	–	–	PUNCT
cana-3969	250	28	issue	issue	NOUN
cana-3969	250	29	viii	viii	VERB
cana-3969	250	30	august	august	PROPN
cana-3969	250	31	2022	2022	NUM
cana-3969	250	32	–	–	PUNCT
cana-3969	250	33	jac	jac	NOUN
cana-3969	250	34	:	:	PUNCT
cana-3969	250	35	a	a	DET
cana-3969	250	36	journal	journal	NOUN
cana-3969	250	37	of	of	ADP
cana-3969	250	38	composition	composition	NOUN
cana-3969	250	39	theory(jct	theory(jct	NOUN
cana-3969	250	40	)	)	PUNCT
cana-3969	250	41	.	.	PUNCT
cana-3969	251	1	https://jctjournal.com/volume-15-issue-viii-august-2022;issn	https://jctjournal.com/volume-15-issue-viii-august-2022;issn	PROPN
cana-3969	251	2	no.:0731	no.:0731	PROPN
cana-3969	251	3	-	-	PUNCT
cana-3969	251	4	6755	6755	NUM
cana-3969	251	5	.	.	PUNCT
cana-3969	252	1	[	[	X
cana-3969	252	2	7	7	X
cana-3969	252	3	]	]	PUNCT
cana-3969	252	4	vasantha	vasantha	NOUN
cana-3969	252	5	g	g	PROPN
cana-3969	252	6	,	,	PUNCT
cana-3969	252	7	t	t	PROPN
cana-3969	252	8	sri	sri	PROPN
cana-3969	252	9	lakshmi	lakshmi	PROPN
cana-3969	252	10	;	;	PUNCT
cana-3969	252	11	background	background	NOUN
cana-3969	252	12	history	history	NOUN
cana-3969	252	13	of	of	ADP
cana-3969	252	14	semigroup	semigroup	PROPN
cana-3969	252	15	theory	theory	NOUN
cana-3969	252	16	in	in	ADP
cana-3969	252	17	algebra	algebra	NOUN
cana-3969	252	18	in	in	ADP
cana-3969	252	19	“	"	PUNCT
cana-3969	252	20	the	the	DET
cana-3969	252	21	international	international	ADJ
cana-3969	252	22	journal	journal	NOUN
cana-3969	252	23	of	of	ADP
cana-3969	252	24	innovative	innovative	ADJ
cana-3969	252	25	research	research	NOUN
cana-3969	252	26	in	in	ADP
cana-3969	252	27	technology	technology	NOUN
cana-3969	252	28	”	"	PUNCT
cana-3969	252	29	.	.	PUNCT
cana-3969	253	1	ijirt	ijirt	NOUN
cana-3969	253	2	|	|	ADV
cana-3969	253	3	volume	volume	NOUN
cana-3969	253	4	8	8	NUM
cana-3969	253	5	issue	issue	NOUN
cana-3969	253	6	12	12	NUM
cana-3969	253	7	|	|	ADV
cana-3969	253	8	issn	issn	VERB
cana-3969	253	9	:	:	PUNCT
cana-3969	253	10	23496002;htt://ijirt.org	23496002;htt://ijirt.org	NUM
cana-3969	253	11	/	/	SYM
cana-3969	253	12	master	master	NOUN
cana-3969	253	13	/	/	SYM
cana-3969	253	14	publishedpaper	publishedpaper	NOUN
cana-3969	253	15	/	/	SYM
cana-3969	253	16	ijirt154793_paper.pdf	ijirt154793_paper.pdf	NOUN
cana-3969	253	17	.	.	PUNCT
cana-3969	254	1	[	[	X
cana-3969	254	2	8	8	NUM
cana-3969	254	3	]	]	X
cana-3969	254	4	ponizovskii	ponizovskii	PROPN
cana-3969	254	5	,	,	PUNCT
cana-3969	254	6	j.s	j.s	PROPN
cana-3969	254	7	..	..	PUNCT
cana-3969	254	8	"	"	PUNCT
cana-3969	254	9	on	on	ADP
cana-3969	254	10	a	a	DET
cana-3969	254	11	type	type	NOUN
cana-3969	254	12	of	of	ADP
cana-3969	254	13	matrix	matrix	NOUN
cana-3969	254	14	semigroup	semigroup	NOUN
cana-3969	254	15	..	..	PUNCT
cana-3969	254	16	"	"	PUNCT
cana-3969	254	17	semigroup	semigroup	PROPN
cana-3969	254	18	forum	forum	PROPN
cana-3969	254	19	44.1	44.1	NUM
cana-3969	254	20	(	(	PUNCT
cana-3969	254	21	1992	1992	NUM
cana-3969	254	22	):	):	PUNCT
cana-3969	254	23	125	125	NUM
cana-3969	254	24	128	128	NUM
cana-3969	254	25	.	.	PUNCT
cana-3969	254	26	communications	communication	NOUN
cana-3969	254	27	on	on	ADP
cana-3969	254	28	applied	apply	VERB
cana-3969	254	29	nonlinear	nonlinear	ADJ
cana-3969	254	30	analysis	analysis	NOUN
cana-3969	254	31	issn	issn	NOUN
cana-3969	254	32	:	:	PUNCT
cana-3969	254	33	1074	1074	NUM
cana-3969	254	34	-	-	PUNCT
cana-3969	254	35	133x	133x	NUM
cana-3969	254	36	vol	vol	NOUN
cana-3969	254	37	32	32	NUM
cana-3969	254	38	no	no	NOUN
cana-3969	254	39	.	.	PUNCT
cana-3969	255	1	9s	9s	NUM
cana-3969	255	2	(	(	PUNCT
cana-3969	255	3	2025	2025	NUM
cana-3969	255	4	)	)	PUNCT
cana-3969	255	5	648	648	NUM
cana-3969	255	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3969	255	7	hollings	holling	NOUN
cana-3969	255	8	,	,	PUNCT
cana-3969	255	9	c.	c.	PROPN
cana-3969	255	10	(	(	PUNCT
cana-3969	255	11	2009	2009	NUM
cana-3969	255	12	)	)	PUNCT
cana-3969	255	13	.	.	PUNCT
cana-3969	256	1	the	the	DET
cana-3969	256	2	early	early	ADJ
cana-3969	256	3	development	development	NOUN
cana-3969	256	4	of	of	ADP
cana-3969	256	5	the	the	DET
cana-3969	256	6	algebraic	algebraic	ADJ
cana-3969	256	7	theory	theory	NOUN
cana-3969	256	8	of	of	ADP
cana-3969	256	9	semigroups	semigroup	NOUN
cana-3969	256	10	.	.	PUNCT
cana-3969	257	1	archive	archive	NOUN
cana-3969	257	2	for	for	ADP
cana-3969	257	3	history	history	NOUN
cana-3969	257	4	of	of	ADP
cana-3969	257	5	exact	exact	ADJ
cana-3969	257	6	sciences	science	NOUN
cana-3969	257	7	,	,	PUNCT
cana-3969	257	8	63	63	NUM
cana-3969	257	9	,	,	PUNCT
cana-3969	257	10	497	497	NUM
cana-3969	257	11	-	-	SYM
cana-3969	257	12	536	536	NUM
cana-3969	257	13	[	[	PUNCT
cana-3969	257	14	9	9	NUM
cana-3969	257	15	]	]	SYM
cana-3969	257	16	zadeh	zadeh	PROPN
cana-3969	257	17	.	.	PUNCT
cana-3969	258	1	l.	l.	PROPN
cana-3969	258	2	(	(	PUNCT
cana-3969	258	3	1965	1965	NUM
cana-3969	258	4	)	)	PUNCT
cana-3969	258	5	.	.	PUNCT
cana-3969	259	1	fuzzy	fuzzy	ADJ
cana-3969	259	2	sets	set	NOUN
cana-3969	259	3	,	,	PUNCT
cana-3969	259	4	information	information	NOUN
cana-3969	259	5	,	,	PUNCT
cana-3969	259	6	and	and	CCONJ
cana-3969	259	7	control	control	NOUN
cana-3969	259	8	,	,	PUNCT
cana-3969	259	9	338	338	NUM
cana-3969	259	10	-3353	-3353	NOUN
cana-3969	259	11	.	.	PUNCT
cana-3969	260	1	[	[	X
cana-3969	260	2	10	10	NUM
cana-3969	260	3	]	]	X
cana-3969	260	4	rosenfeld	rosenfeld	PROPN
cana-3969	260	5	,	,	PUNCT
cana-3969	260	6	a.	a.	NOUN
cana-3969	260	7	(	(	PUNCT
cana-3969	260	8	1971	1971	NUM
cana-3969	260	9	)	)	PUNCT
cana-3969	260	10	.	.	PUNCT
cana-3969	261	1	fuzzy	fuzzy	ADJ
cana-3969	261	2	groups	group	NOUN
cana-3969	261	3	.	.	PUNCT
cana-3969	262	1	journal	journal	PROPN
cana-3969	262	2	of	of	ADP
cana-3969	262	3	mathematical	mathematical	ADJ
cana-3969	262	4	analysis	analysis	NOUN
cana-3969	262	5	and	and	CCONJ
cana-3969	262	6	applications	application	NOUN
cana-3969	262	7	,	,	PUNCT
cana-3969	262	8	35	35	NUM
cana-3969	262	9	,	,	PUNCT
cana-3969	262	10	512–517	512–517	NUM
cana-3969	262	11	.	.	PUNCT
cana-3969	263	1	[	[	X
cana-3969	263	2	11	11	NUM
cana-3969	263	3	]	]	X
cana-3969	263	4	imtiaz	imtiaz	PROPN
cana-3969	263	5	,	,	PUNCT
cana-3969	263	6	a.	a.	PROPN
cana-3969	263	7	,	,	PUNCT
cana-3969	263	8	alolaiyan	alolaiyan	PROPN
cana-3969	263	9	,	,	PUNCT
cana-3969	263	10	h.	h.	PROPN
cana-3969	263	11	,	,	PUNCT
cana-3969	263	12	shuaib	shuaib	PROPN
cana-3969	263	13	,	,	PUNCT
cana-3969	263	14	u.	u.	PROPN
cana-3969	263	15	,	,	PUNCT
cana-3969	263	16	razaq	razaq	NOUN
cana-3969	263	17	,	,	PUNCT
cana-3969	263	18	a.	a.	NOUN
cana-3969	263	19	,	,	PUNCT
cana-3969	263	20	&	&	CCONJ
cana-3969	263	21	liu	liu	PROPN
cana-3969	263	22	,	,	PUNCT
cana-3969	263	23	j.	j.	PROPN
cana-3969	263	24	b.	b.	PROPN
cana-3969	263	25	(	(	PUNCT
cana-3969	263	26	2024	2024	NUM
cana-3969	263	27	)	)	PUNCT
cana-3969	263	28	.	.	PUNCT
cana-3969	264	1	applications	application	NOUN
cana-3969	264	2	of	of	ADP
cana-3969	264	3	conjunctive	conjunctive	ADJ
cana-3969	264	4	complex	complex	ADJ
cana-3969	264	5	fuzzy	fuzzy	ADJ
cana-3969	264	6	subgroups	subgroup	NOUN
cana-3969	264	7	to	to	PART
cana-3969	264	8	sylow	sylow	VERB
cana-3969	264	9	theory	theory	NOUN
cana-3969	264	10	.	.	PUNCT
cana-3969	265	1	aims	aim	VERB
cana-3969	265	2	mathematics	mathematic	NOUN
cana-3969	265	3	,	,	PUNCT
cana-3969	265	4	9(1	9(1	NUM
cana-3969	265	5	)	)	PUNCT
cana-3969	265	6	,	,	PUNCT
cana-3969	265	7	38	38	NUM
cana-3969	265	8	-	-	SYM
cana-3969	265	9	54	54	NUM
cana-3969	265	10	.	.	PUNCT
cana-3969	266	1	[	[	X
cana-3969	266	2	12	12	NUM
cana-3969	266	3	]	]	X
cana-3969	266	4	al	al	PROPN
cana-3969	266	5	-	-	PUNCT
cana-3969	266	6	masarwah	masarwah	PROPN
cana-3969	266	7	,	,	PUNCT
cana-3969	266	8	a.	a.	PROPN
cana-3969	266	9	,	,	PUNCT
cana-3969	266	10	&	&	CCONJ
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cana-3969	266	12	,	,	PUNCT
cana-3969	266	13	m.	m.	NOUN
cana-3969	266	14	(	(	PUNCT
cana-3969	266	15	2023	2023	NUM
cana-3969	266	16	)	)	PUNCT
cana-3969	266	17	.	.	PUNCT
cana-3969	267	1	operational	operational	ADJ
cana-3969	267	2	algebraic	algebraic	ADJ
cana-3969	267	3	properties	property	NOUN
cana-3969	267	4	and	and	CCONJ
cana-3969	267	5	subsemigroups	subsemigroup	NOUN
cana-3969	267	6	of	of	ADP
cana-3969	267	7	semigroups	semigroup	NOUN
cana-3969	267	8	given	give	VERB
cana-3969	267	9	k	k	NOUN
cana-3969	267	10	-	-	PUNCT
cana-3969	267	11	folded	fold	VERB
cana-3969	267	12	n	n	CCONJ
cana-3969	267	13	-	-	PUNCT
cana-3969	267	14	structures	structure	NOUN
cana-3969	267	15	.	.	PUNCT
cana-3969	268	1	aims	aim	VERB
cana-3969	268	2	math	math	NOUN
cana-3969	268	3	,	,	PUNCT
cana-3969	268	4	8(9	8(9	NUM
cana-3969	268	5	)	)	PUNCT
cana-3969	268	6	,	,	PUNCT
cana-3969	268	7	22081	22081	NUM
cana-3969	268	8	-	-	SYM
cana-3969	268	9	22096	22096	NUM
cana-3969	268	10	.	.	PUNCT
cana-3969	269	1	[	[	X
cana-3969	269	2	13	13	NUM
cana-3969	269	3	]	]	PUNCT
cana-3969	269	4	preethi	preethi	ADV
cana-3969	269	5	,	,	PUNCT
cana-3969	269	6	d.	d.	PROPN
cana-3969	269	7	,	,	PUNCT
cana-3969	269	8	vimala	vimala	PROPN
cana-3969	269	9	,	,	PUNCT
cana-3969	269	10	j.	j.	PROPN
cana-3969	269	11	,	,	PUNCT
cana-3969	269	12	&	&	CCONJ
cana-3969	269	13	rajareega	rajareega	PROPN
cana-3969	269	14	,	,	PUNCT
cana-3969	269	15	s.	s.	PROPN
cana-3969	269	16	(	(	PUNCT
cana-3969	269	17	2020	2020	NUM
cana-3969	269	18	)	)	PUNCT
cana-3969	269	19	.	.	PUNCT
cana-3969	270	1	a	a	DET
cana-3969	270	2	systematic	systematic	ADJ
cana-3969	270	3	study	study	NOUN
cana-3969	270	4	in	in	ADP
cana-3969	270	5	the	the	DET
cana-3969	270	6	applications	application	NOUN
cana-3969	270	7	of	of	ADP
cana-3969	270	8	fuzzy	fuzzy	ADJ
cana-3969	270	9	hyperlattice	hyperlattice	NOUN
cana-3969	270	10	.	.	PUNCT
cana-3969	271	1	aims	aim	VERB
cana-3969	271	2	mathematics	mathematic	NOUN
cana-3969	271	3	,	,	PUNCT
cana-3969	271	4	6(2	6(2	NUM
cana-3969	271	5	)	)	PUNCT
cana-3969	271	6	,	,	PUNCT
cana-3969	271	7	1695	1695	NUM
cana-3969	271	8	-	-	SYM
cana-3969	271	9	1705	1705	NUM
cana-3969	271	10	.	.	PUNCT
cana-3969	272	1	[	[	X
cana-3969	272	2	14	14	NUM
cana-3969	272	3	]	]	SYM
cana-3969	272	4	ali	ali	PROPN
cana-3969	272	5	,	,	PUNCT
cana-3969	272	6	a.	a.	PROPN
cana-3969	272	7	,	,	PUNCT
cana-3969	272	8	mateen	mateen	PROPN
cana-3969	272	9	,	,	PUNCT
cana-3969	272	10	m.	m.	PROPN
cana-3969	272	11	h.	h.	PROPN
cana-3969	272	12	,	,	PUNCT
cana-3969	272	13	xin	xin	PROPN
cana-3969	272	14	,	,	PUNCT
cana-3969	272	15	q.	q.	PROPN
cana-3969	272	16	,	,	PUNCT
cana-3969	272	17	alsuraiheed	alsuraiheed	NOUN
cana-3969	272	18	,	,	PUNCT
cana-3969	272	19	t.	t.	PROPN
cana-3969	272	20	,	,	PUNCT
cana-3969	272	21	&	&	CCONJ
cana-3969	272	22	alhamzi	alhamzi	PROPN
cana-3969	272	23	,	,	PUNCT
cana-3969	272	24	g.	g.	PROPN
cana-3969	272	25	(	(	PUNCT
cana-3969	272	26	2024	2024	NUM
cana-3969	272	27	)	)	PUNCT
cana-3969	272	28	.	.	PUNCT
cana-3969	273	1	$	$	SYM
cana-3969	273	2	(	(	PUNCT
cana-3969	273	3	\epsilon,\delta	\epsilon,\delta	NOUN
cana-3969	273	4	)	)	PUNCT
cana-3969	273	5	$	$	SYM
cana-3969	273	6	-complex	-complex	ADJ
cana-3969	273	7	anti	anti	ADJ
cana-3969	273	8	fuzzy	fuzzy	ADJ
cana-3969	273	9	subgroups	subgroup	NOUN
cana-3969	273	10	and	and	CCONJ
cana-3969	273	11	their	their	PRON
cana-3969	273	12	applications	application	NOUN
cana-3969	273	13	.	.	PUNCT
cana-3969	274	1	aims	aim	VERB
cana-3969	274	2	mathematics	mathematic	NOUN
cana-3969	274	3	,	,	PUNCT
cana-3969	274	4	9(5	9(5	NUM
cana-3969	274	5	)	)	PUNCT
cana-3969	274	6	,	,	PUNCT
cana-3969	274	7	11580	11580	NUM
cana-3969	274	8	-	-	SYM
cana-3969	274	9	11595	11595	NUM
cana-3969	274	10	.	.	PUNCT
cana-3969	275	1	[	[	X
cana-3969	275	2	15	15	NUM
cana-3969	275	3	]	]	X
cana-3969	275	4	ali	ali	PROPN
cana-3969	275	5	,	,	PUNCT
cana-3969	275	6	a.	a.	PROPN
cana-3969	275	7	,	,	PUNCT
cana-3969	275	8	ameer	ameer	PROPN
cana-3969	275	9	,	,	PUNCT
cana-3969	275	10	e.	e.	PROPN
cana-3969	275	11	,	,	PUNCT
cana-3969	275	12	aiadi	aiadi	PROPN
cana-3969	275	13	,	,	PUNCT
cana-3969	275	14	s.	s.	PROPN
cana-3969	275	15	s.	s.	PROPN
cana-3969	275	16	,	,	PUNCT
cana-3969	275	17	tariq	tariq	PROPN
cana-3969	275	18	,	,	PUNCT
cana-3969	275	19	m.	m.	NOUN
cana-3969	275	20	,	,	PUNCT
cana-3969	275	21	arshad	arshad	ADJ
cana-3969	275	22	,	,	PUNCT
cana-3969	275	23	m.	m.	NOUN
cana-3969	275	24	,	,	PUNCT
cana-3969	275	25	mlaiki	mlaiki	PROPN
cana-3969	275	26	,	,	PUNCT
cana-3969	275	27	n.	n.	NOUN
cana-3969	275	28	,	,	PUNCT
cana-3969	275	29	&	&	CCONJ
cana-3969	275	30	shatanawi	shatanawi	PROPN
cana-3969	275	31	,	,	PUNCT
cana-3969	275	32	w.	w.	NOUN
cana-3969	275	33	(	(	PUNCT
cana-3969	275	34	2022	2022	NUM
cana-3969	275	35	)	)	PUNCT
cana-3969	275	36	.	.	PUNCT
cana-3969	276	1	new	new	ADJ
cana-3969	276	2	extension	extension	NOUN
cana-3969	276	3	to	to	ADP
cana-3969	276	4	fuzzy	fuzzy	ADJ
cana-3969	276	5	dynamic	dynamic	ADJ
cana-3969	276	6	system	system	NOUN
cana-3969	276	7	and	and	CCONJ
cana-3969	276	8	fuzzy	fuzzy	ADJ
cana-3969	276	9	fixed	fix	VERB
cana-3969	276	10	point	point	NOUN
cana-3969	276	11	results	result	NOUN
cana-3969	276	12	with	with	ADP
cana-3969	276	13	an	an	DET
cana-3969	276	14	application	application	NOUN
cana-3969	276	15	.	.	PUNCT
cana-3969	277	1	aims	aim	VERB
cana-3969	277	2	math	math	NOUN
cana-3969	277	3	,	,	PUNCT
cana-3969	277	4	8	8	NUM
cana-3969	277	5	,	,	PUNCT
cana-3969	277	6	1208	1208	NUM
cana-3969	277	7	-	-	SYM
cana-3969	277	8	1229	1229	NUM
cana-3969	277	9	.	.	PUNCT
cana-3969	278	1	[	[	X
cana-3969	278	2	16	16	NUM
cana-3969	278	3	]	]	PUNCT
cana-3969	278	4	ullah	ullah	PROPN
cana-3969	278	5	,	,	PUNCT
cana-3969	278	6	a.	a.	PROPN
cana-3969	278	7	,	,	PUNCT
cana-3969	278	8	ibrahim	ibrahim	PROPN
cana-3969	278	9	,	,	PUNCT
cana-3969	278	10	m.	m.	NOUN
cana-3969	278	11	,	,	PUNCT
cana-3969	278	12	&	&	CCONJ
cana-3969	278	13	saeed	saeed	PROPN
cana-3969	278	14	,	,	PUNCT
cana-3969	278	15	t.	t.	PROPN
cana-3969	278	16	(	(	PUNCT
cana-3969	278	17	2022	2022	NUM
cana-3969	278	18	)	)	PUNCT
cana-3969	278	19	.	.	PUNCT
cana-3969	279	1	fuzzy	fuzzy	ADJ
cana-3969	279	2	cosets	coset	NOUN
cana-3969	279	3	in	in	ADP
cana-3969	279	4	ag	ag	PROPN
cana-3969	279	5	-	-	PUNCT
cana-3969	279	6	groups	group	NOUN
cana-3969	279	7	.	.	PUNCT
cana-3969	280	1	aims	aim	VERB
cana-3969	280	2	mathematics	mathematic	NOUN
cana-3969	280	3	,	,	PUNCT
cana-3969	280	4	7(3	7(3	NUM
cana-3969	280	5	)	)	PUNCT
cana-3969	280	6	,	,	PUNCT
cana-3969	280	7	3321	3321	NUM
cana-3969	280	8	-	-	SYM
cana-3969	280	9	3344	3344	NUM
cana-3969	280	10	.	.	PUNCT
cana-3969	281	1	[	[	X
cana-3969	281	2	17	17	NUM
cana-3969	281	3	]	]	X
cana-3969	281	4	sezer	sezer	NOUN
cana-3969	281	5	,	,	PUNCT
cana-3969	281	6	a.	a.	PROPN
cana-3969	281	7	s.	s.	PROPN
cana-3969	281	8	(	(	PUNCT
cana-3969	281	9	2014	2014	NUM
cana-3969	281	10	)	)	PUNCT
cana-3969	281	11	.	.	PUNCT
cana-3969	282	1	a	a	DET
cana-3969	282	2	new	new	ADJ
cana-3969	282	3	approach	approach	NOUN
cana-3969	282	4	to	to	ADP
cana-3969	282	5	la	la	ADJ
cana-3969	282	6	-	-	PUNCT
cana-3969	282	7	semigroup	semigroup	PROPN
cana-3969	282	8	theory	theory	NOUN
cana-3969	282	9	via	via	ADP
cana-3969	282	10	the	the	DET
cana-3969	282	11	soft	soft	ADJ
cana-3969	282	12	sets	set	NOUN
cana-3969	282	13	.	.	PUNCT
cana-3969	283	1	journal	journal	NOUN
cana-3969	283	2	of	of	ADP
cana-3969	283	3	intelligent	intelligent	ADJ
cana-3969	283	4	&	&	CCONJ
cana-3969	283	5	fuzzy	fuzzy	ADJ
cana-3969	283	6	systems	system	NOUN
cana-3969	283	7	,	,	PUNCT
cana-3969	283	8	26(5	26(5	NUM
cana-3969	283	9	)	)	PUNCT
cana-3969	283	10	,	,	PUNCT
cana-3969	283	11	2483	2483	NUM
cana-3969	283	12	-	-	SYM
cana-3969	283	13	2495	2495	NUM
cana-3969	283	14	.	.	PUNCT
cana-3969	284	1	[	[	X
cana-3969	284	2	18	18	NUM
cana-3969	284	3	]	]	X
cana-3969	284	4	feng	feng	PROPN
cana-3969	284	5	,	,	PUNCT
cana-3969	284	6	x.	x.	PROPN
cana-3969	284	7	,	,	PUNCT
cana-3969	284	8	tang	tang	PROPN
cana-3969	284	9	,	,	PUNCT
cana-3969	284	10	j.	j.	PROPN
cana-3969	284	11	,	,	PUNCT
cana-3969	284	12	davvaz	davvaz	PROPN
cana-3969	284	13	,	,	PUNCT
cana-3969	284	14	b.	b.	PROPN
cana-3969	284	15	,	,	PUNCT
cana-3969	284	16	&	&	CCONJ
cana-3969	284	17	luo	luo	PROPN
cana-3969	284	18	,	,	PUNCT
cana-3969	284	19	y.	y.	PROPN
cana-3969	284	20	(	(	PUNCT
cana-3969	284	21	2017	2017	NUM
cana-3969	284	22	)	)	PUNCT
cana-3969	284	23	.	.	PUNCT
cana-3969	285	1	a	a	DET
cana-3969	285	2	novel	novel	ADJ
cana-3969	285	3	study	study	NOUN
cana-3969	285	4	on	on	ADP
cana-3969	285	5	fuzzy	fuzzy	ADJ
cana-3969	285	6	ideals	ideal	NOUN
cana-3969	285	7	and	and	CCONJ
cana-3969	285	8	fuzzy	fuzzy	ADJ
cana-3969	285	9	filters	filter	NOUN
cana-3969	285	10	of	of	ADP
cana-3969	285	11	ordered*semigroups	ordered*semigroup	NOUN
cana-3969	285	12	.	.	PUNCT
cana-3969	286	1	journal	journal	PROPN
cana-3969	286	2	of	of	ADP
cana-3969	286	3	intelligent	intelligent	ADJ
cana-3969	286	4	&	&	CCONJ
cana-3969	286	5	fuzzy	fuzzy	ADJ
cana-3969	286	6	systems	system	NOUN
cana-3969	286	7	,	,	PUNCT
cana-3969	286	8	33(1	33(1	NUM
cana-3969	286	9	)	)	PUNCT
cana-3969	286	10	,	,	PUNCT
cana-3969	286	11	423	423	NUM
cana-3969	286	12	-	-	SYM
cana-3969	286	13	431	431	NUM
cana-3969	286	14	.	.	PUNCT
cana-3969	287	1	[	[	X
cana-3969	287	2	19	19	NUM
cana-3969	287	3	]	]	X
cana-3969	287	4	khan	khan	PROPN
cana-3969	287	5	,	,	PUNCT
cana-3969	287	6	f.	f.	PROPN
cana-3969	287	7	m.	m.	PROPN
cana-3969	287	8	,	,	PUNCT
cana-3969	287	9	bibi	bibi	NOUN
cana-3969	287	10	,	,	PUNCT
cana-3969	287	11	n.	n.	PROPN
cana-3969	287	12	,	,	PUNCT
cana-3969	287	13	xin	xin	PROPN
cana-3969	287	14	,	,	PUNCT
cana-3969	287	15	x.	x.	PROPN
cana-3969	287	16	l.	l.	PROPN
cana-3969	287	17	,	,	PUNCT
cana-3969	287	18	&	&	CCONJ
cana-3969	287	19	alam	alam	PROPN
cana-3969	287	20	,	,	PUNCT
cana-3969	287	21	a.	a.	NOUN
cana-3969	287	22	(	(	PUNCT
cana-3969	287	23	2022	2022	NUM
cana-3969	287	24	)	)	PUNCT
cana-3969	287	25	.	.	PUNCT
cana-3969	288	1	rough	rough	ADJ
cana-3969	288	2	fermatean	fermatean	ADJ
cana-3969	288	3	fuzzy	fuzzy	ADJ
cana-3969	288	4	ideals	ideal	NOUN
cana-3969	288	5	in	in	ADP
cana-3969	288	6	semigroups	semigroup	NOUN
cana-3969	288	7	.	.	PUNCT
cana-3969	289	1	journal	journal	NOUN
cana-3969	289	2	of	of	ADP
cana-3969	289	3	intelligent	intelligent	ADJ
cana-3969	289	4	&	&	CCONJ
cana-3969	289	5	fuzzy	fuzzy	ADJ
cana-3969	289	6	systems	system	NOUN
cana-3969	289	7	,	,	PUNCT
cana-3969	289	8	42(6	42(6	NOUN
cana-3969	289	9	)	)	PUNCT
cana-3969	289	10	,	,	PUNCT
cana-3969	289	11	5741	5741	NUM
cana-3969	289	12	-	-	SYM
cana-3969	289	13	5752	5752	NUM
cana-3969	289	14	.	.	PUNCT
cana-3969	290	1	[	[	X
cana-3969	290	2	20	20	NUM
cana-3969	290	3	]	]	SYM
cana-3969	290	4	yiarayong	yiarayong	NOUN
cana-3969	290	5	,	,	PUNCT
cana-3969	290	6	p.	p.	NOUN
cana-3969	290	7	(	(	PUNCT
cana-3969	290	8	2021	2021	NUM
cana-3969	290	9	)	)	PUNCT
cana-3969	290	10	.	.	PUNCT
cana-3969	291	1	on	on	ADP
cana-3969	291	2	2	2	NUM
cana-3969	291	3	-	-	PUNCT
cana-3969	291	4	absorbing	absorb	VERB
cana-3969	291	5	bipolar	bipolar	ADJ
cana-3969	291	6	fuzzy	fuzzy	ADJ
cana-3969	291	7	ideals	ideal	NOUN
cana-3969	291	8	over	over	ADP
cana-3969	291	9	la	la	NOUN
cana-3969	291	10	-	-	PUNCT
cana-3969	291	11	semigroups	semigroup	NOUN
cana-3969	291	12	.	.	PUNCT
cana-3969	292	1	journal	journal	NOUN
cana-3969	292	2	of	of	ADP
cana-3969	292	3	intelligent	intelligent	ADJ
cana-3969	292	4	&	&	CCONJ
cana-3969	292	5	fuzzy	fuzzy	ADJ
cana-3969	292	6	systems	system	NOUN
cana-3969	292	7	,	,	PUNCT
cana-3969	292	8	41(2	41(2	NUM
cana-3969	292	9	)	)	PUNCT
cana-3969	292	10	,	,	PUNCT
cana-3969	292	11	3173	3173	NUM
cana-3969	292	12	-	-	SYM
cana-3969	292	13	3181	3181	NUM
cana-3969	292	14	.	.	PUNCT
cana-3969	293	1	[	[	X
cana-3969	293	2	21	21	NUM
cana-3969	293	3	]	]	X
cana-3969	293	4	zhang	zhang	PROPN
cana-3969	293	5	,	,	PUNCT
cana-3969	293	6	x.	x.	PROPN
cana-3969	293	7	,	,	PUNCT
cana-3969	293	8	wu	wu	PROPN
cana-3969	293	9	,	,	PUNCT
cana-3969	293	10	x.	x.	PROPN
cana-3969	293	11	,	,	PUNCT
cana-3969	293	12	mao	mao	PROPN
cana-3969	293	13	,	,	PUNCT
cana-3969	293	14	x.	x.	PROPN
cana-3969	293	15	,	,	PUNCT
cana-3969	293	16	smarandache	smarandache	PROPN
cana-3969	293	17	,	,	PUNCT
cana-3969	293	18	f.	f.	PROPN
cana-3969	293	19	,	,	PUNCT
cana-3969	293	20	&	&	CCONJ
cana-3969	293	21	park	park	PROPN
cana-3969	293	22	,	,	PUNCT
cana-3969	293	23	c.	c.	PROPN
cana-3969	293	24	(	(	PUNCT
cana-3969	293	25	2019	2019	NUM
cana-3969	293	26	)	)	PUNCT
cana-3969	293	27	.	.	PUNCT
cana-3969	294	1	on	on	ADP
cana-3969	294	2	neutrosophic	neutrosophic	ADJ
cana-3969	294	3	extended	extend	VERB
cana-3969	294	4	triplet	triplet	NOUN
cana-3969	294	5	groups	group	NOUN
cana-3969	294	6	(	(	PUNCT
cana-3969	294	7	loops	loop	NOUN
cana-3969	294	8	)	)	PUNCT
cana-3969	294	9	and	and	CCONJ
cana-3969	294	10	abel	abel	PROPN
cana-3969	294	11	-	-	PUNCT
cana-3969	294	12	grassmann	grassmann	PROPN
cana-3969	294	13	’s	’s	PART
cana-3969	294	14	groupoids	groupoid	NOUN
cana-3969	294	15	(	(	PUNCT
cana-3969	294	16	ag	ag	NOUN
cana-3969	294	17	-	-	PUNCT
cana-3969	294	18	groupoids	groupoid	NOUN
cana-3969	294	19	)	)	PUNCT
cana-3969	294	20	.	.	PUNCT
cana-3969	295	1	journal	journal	PROPN
cana-3969	295	2	of	of	ADP
cana-3969	295	3	intelligent	intelligent	ADJ
cana-3969	295	4	&	&	CCONJ
cana-3969	295	5	fuzzy	fuzzy	ADJ
cana-3969	295	6	systems	system	NOUN
cana-3969	295	7	,	,	PUNCT
cana-3969	295	8	37(4	37(4	NOUN
cana-3969	295	9	)	)	PUNCT
cana-3969	295	10	,	,	PUNCT
cana-3969	295	11	5743	5743	NUM
cana-3969	295	12	-	-	SYM
cana-3969	295	13	5753	5753	NUM
cana-3969	295	14	.	.	PUNCT
cana-3969	296	1	[	[	X
cana-3969	296	2	22	22	NUM
cana-3969	296	3	]	]	X
cana-3969	296	4	budimirović	budimirović	PROPN
cana-3969	296	5	,	,	PUNCT
cana-3969	296	6	b.	b.	PROPN
cana-3969	296	7	,	,	PUNCT
cana-3969	296	8	budimirović	budimirović	PROPN
cana-3969	296	9	,	,	PUNCT
cana-3969	296	10	v.	v.	ADV
cana-3969	296	11	,	,	PUNCT
cana-3969	296	12	šešelja	šešelja	PROPN
cana-3969	296	13	,	,	PUNCT
cana-3969	296	14	b.	b.	PROPN
cana-3969	296	15	,	,	PUNCT
cana-3969	296	16	&	&	CCONJ
cana-3969	296	17	tepavčević	tepavčević	PROPN
cana-3969	296	18	,	,	PUNCT
cana-3969	296	19	a.	a.	NOUN
cana-3969	296	20	(	(	PUNCT
cana-3969	296	21	2014	2014	NUM
cana-3969	296	22	)	)	PUNCT
cana-3969	296	23	.	.	PUNCT
cana-3969	297	1	fuzzy	fuzzy	ADJ
cana-3969	297	2	identities	identity	NOUN
cana-3969	297	3	with	with	ADP
cana-3969	297	4	application	application	NOUN
cana-3969	297	5	to	to	ADP
cana-3969	297	6	fuzzy	fuzzy	ADJ
cana-3969	297	7	semigroups	semigroup	NOUN
cana-3969	297	8	.	.	PUNCT
cana-3969	298	1	information	information	NOUN
cana-3969	298	2	sciences	sciences	PROPN
cana-3969	298	3	,	,	PUNCT
cana-3969	298	4	266	266	NUM
cana-3969	298	5	,	,	PUNCT
cana-3969	298	6	148	148	NUM
cana-3969	298	7	-	-	SYM
cana-3969	298	8	159	159	NUM
cana-3969	298	9	.	.	PUNCT
cana-3969	299	1	[	[	X
cana-3969	299	2	23	23	NUM
cana-3969	299	3	]	]	SYM
cana-3969	299	4	yiarayong	yiarayong	NOUN
cana-3969	299	5	,	,	PUNCT
cana-3969	299	6	p.	p.	NOUN
cana-3969	299	7	(	(	PUNCT
cana-3969	299	8	2022	2022	NUM
cana-3969	299	9	)	)	PUNCT
cana-3969	299	10	.	.	PUNCT
cana-3969	300	1	on	on	ADP
cana-3969	300	2	bipolar	bipolar	ADV
cana-3969	300	3	-	-	PUNCT
cana-3969	300	4	valued	value	VERB
cana-3969	300	5	fuzzy	fuzzy	ADJ
cana-3969	300	6	quasi	quasi	ADJ
cana-3969	300	7	-	-	ADJ
cana-3969	300	8	semiprime	semiprime	ADJ
cana-3969	300	9	ideals	ideal	NOUN
cana-3969	300	10	of	of	ADP
cana-3969	300	11	la	la	NOUN
cana-3969	300	12	-	-	PUNCT
cana-3969	300	13	semigroups	semigroup	NOUN
cana-3969	300	14	.	.	PUNCT
cana-3969	301	1	afrika	afrika	PROPN
cana-3969	301	2	matematika	matematika	PROPN
cana-3969	301	3	,	,	PUNCT
cana-3969	301	4	33(3	33(3	NOUN
cana-3969	301	5	)	)	PUNCT
cana-3969	301	6	,	,	PUNCT
cana-3969	301	7	81	81	NUM
cana-3969	301	8	.	.	PUNCT
cana-3969	302	1	[	[	X
cana-3969	302	2	24	24	NUM
cana-3969	302	3	]	]	SYM
cana-3969	302	4	jun	jun	PROPN
cana-3969	302	5	,	,	PUNCT
cana-3969	302	6	y.	y.	PROPN
cana-3969	302	7	b.	b.	PROPN
cana-3969	302	8	,	,	PUNCT
cana-3969	302	9	song	song	NOUN
cana-3969	302	10	,	,	PUNCT
cana-3969	302	11	s.	s.	PROPN
cana-3969	302	12	z.	z.	PROPN
cana-3969	302	13	,	,	PUNCT
cana-3969	302	14	&	&	CCONJ
cana-3969	302	15	muhiuddin	muhiuddin	PROPN
cana-3969	302	16	,	,	PUNCT
cana-3969	302	17	g.	g.	PROPN
cana-3969	302	18	(	(	PUNCT
cana-3969	302	19	2016	2016	NUM
cana-3969	302	20	)	)	PUNCT
cana-3969	302	21	.	.	PUNCT
cana-3969	303	1	hesitant	hesitant	ADJ
cana-3969	303	2	fuzzy	fuzzy	ADJ
cana-3969	303	3	semigroups	semigroup	NOUN
cana-3969	303	4	with	with	ADP
cana-3969	303	5	a	a	DET
cana-3969	303	6	frontier	frontier	NOUN
cana-3969	303	7	.	.	PUNCT
cana-3969	304	1	journal	journal	PROPN
cana-3969	304	2	of	of	ADP
cana-3969	304	3	intelligent	intelligent	ADJ
cana-3969	304	4	&	&	CCONJ
cana-3969	304	5	fuzzy	fuzzy	ADJ
cana-3969	304	6	systems	system	NOUN
cana-3969	304	7	,	,	PUNCT
cana-3969	304	8	30(3	30(3	NUM
cana-3969	304	9	)	)	PUNCT
cana-3969	304	10	,	,	PUNCT
cana-3969	304	11	1613	1613	NUM
cana-3969	304	12	-	-	SYM
cana-3969	304	13	1618	1618	NUM
cana-3969	304	14	.	.	PUNCT
cana-3969	305	1	[	[	X
cana-3969	305	2	25	25	NUM
cana-3969	305	3	]	]	X
cana-3969	305	4	habib	habib	PROPN
cana-3969	305	5	,	,	PUNCT
cana-3969	305	6	s.	s.	PROPN
cana-3969	305	7	,	,	PUNCT
cana-3969	305	8	muhammad	muhammad	PROPN
cana-3969	305	9	khan	khan	PROPN
cana-3969	305	10	,	,	PUNCT
cana-3969	305	11	f.	f.	PROPN
cana-3969	305	12	,	,	PUNCT
cana-3969	305	13	&	&	CCONJ
cana-3969	305	14	yufeng	yufeng	PROPN
cana-3969	305	15	,	,	PUNCT
cana-3969	305	16	n.	n.	PROPN
cana-3969	305	17	(	(	PUNCT
cana-3969	305	18	2019	2019	NUM
cana-3969	305	19	)	)	PUNCT
cana-3969	305	20	.	.	PUNCT
cana-3969	306	1	a	a	DET
cana-3969	306	2	new	new	ADJ
cana-3969	306	3	concept	concept	NOUN
cana-3969	306	4	of	of	ADP
cana-3969	306	5	possibility	possibility	NOUN
cana-3969	306	6	fuzzy	fuzzy	ADJ
cana-3969	306	7	soft	soft	ADJ
cana-3969	306	8	ordered	order	VERB
cana-3969	306	9	semigroups	semigroup	NOUN
cana-3969	306	10	via	via	ADP
cana-3969	306	11	its	its	PRON
cana-3969	306	12	applications	application	NOUN
cana-3969	306	13	.	.	PUNCT
cana-3969	307	1	journal	journal	NOUN
cana-3969	307	2	of	of	ADP
cana-3969	307	3	intelligent	intelligent	ADJ
cana-3969	307	4	&	&	CCONJ
cana-3969	307	5	fuzzy	fuzzy	ADJ
cana-3969	307	6	systems	system	NOUN
cana-3969	307	7	,	,	PUNCT
cana-3969	307	8	36(4	36(4	NUM
cana-3969	307	9	)	)	PUNCT
cana-3969	307	10	,	,	PUNCT
cana-3969	307	11	3685	3685	NUM
cana-3969	307	12	-	-	SYM
cana-3969	307	13	3696	3696	NUM
cana-3969	307	14	.	.	PUNCT
cana-3969	308	1	[	[	X
cana-3969	308	2	26	26	NUM
cana-3969	308	3	]	]	PUNCT
cana-3969	308	4	muhiuddin	muhiuddin	PROPN
cana-3969	308	5	,	,	PUNCT
cana-3969	308	6	g.	g.	PROPN
cana-3969	308	7	,	,	PUNCT
cana-3969	308	8	mahboob	mahboob	PROPN
cana-3969	308	9	,	,	PUNCT
cana-3969	308	10	a.	a.	PROPN
cana-3969	308	11	,	,	PUNCT
cana-3969	308	12	khan	khan	PROPN
cana-3969	308	13	,	,	PUNCT
cana-3969	308	14	n.	n.	PROPN
cana-3969	308	15	m.	m.	NOUN
cana-3969	308	16	,	,	PUNCT
cana-3969	308	17	&	&	CCONJ
cana-3969	308	18	al	al	PROPN
cana-3969	308	19	-	-	PUNCT
cana-3969	308	20	kadi	kadi	PROPN
cana-3969	308	21	,	,	PUNCT
cana-3969	308	22	d.	d.	PROPN
cana-3969	308	23	(	(	PUNCT
cana-3969	308	24	2021	2021	NUM
cana-3969	308	25	)	)	PUNCT
cana-3969	308	26	.	.	PUNCT
cana-3969	309	1	new	new	ADJ
cana-3969	309	2	types	type	NOUN
cana-3969	309	3	of	of	ADP
cana-3969	309	4	fuzzy	fuzzy	ADJ
cana-3969	309	5	(	(	PUNCT
cana-3969	309	6	m	m	PROPN
cana-3969	309	7	,	,	PUNCT
cana-3969	309	8	n)-ideals	n)-ideal	NOUN
cana-3969	309	9	in	in	ADP
cana-3969	309	10	ordered	order	VERB
cana-3969	309	11	semigroups	semigroup	NOUN
cana-3969	309	12	.	.	PUNCT
cana-3969	310	1	journal	journal	NOUN
cana-3969	310	2	of	of	ADP
cana-3969	310	3	intelligent	intelligent	ADJ
cana-3969	310	4	&	&	CCONJ
cana-3969	310	5	fuzzy	fuzzy	ADJ
cana-3969	310	6	systems	system	NOUN
cana-3969	310	7	,	,	PUNCT
cana-3969	310	8	41(6	41(6	NOUN
cana-3969	310	9	)	)	PUNCT
cana-3969	310	10	,	,	PUNCT
cana-3969	310	11	6561	6561	NUM
cana-3969	310	12	-	-	SYM
cana-3969	310	13	6574	6574	NUM
cana-3969	310	14	.	.	PUNCT
cana-3969	311	1	[	[	X
cana-3969	311	2	27	27	NUM
cana-3969	311	3	]	]	X
cana-3969	311	4	khan	khan	PROPN
cana-3969	311	5	,	,	PUNCT
cana-3969	311	6	a.	a.	NOUN
cana-3969	311	7	,	,	PUNCT
cana-3969	311	8	sarmin	sarmin	NOUN
cana-3969	311	9	,	,	PUNCT
cana-3969	311	10	n.	n.	PROPN
cana-3969	311	11	h.	h.	PROPN
cana-3969	311	12	,	,	PUNCT
cana-3969	311	13	davvaz	davvaz	PROPN
cana-3969	311	14	,	,	PUNCT
cana-3969	311	15	b.	b.	PROPN
cana-3969	311	16	,	,	PUNCT
cana-3969	311	17	&	&	CCONJ
cana-3969	311	18	khan	khan	PROPN
cana-3969	311	19	,	,	PUNCT
cana-3969	311	20	f.	f.	PROPN
cana-3969	311	21	m.	m.	PROPN
cana-3969	311	22	(	(	PUNCT
cana-3969	311	23	2012	2012	NUM
cana-3969	311	24	)	)	PUNCT
cana-3969	311	25	.	.	PUNCT
cana-3969	312	1	new	new	ADJ
cana-3969	312	2	types	type	NOUN
cana-3969	312	3	of	of	ADP
cana-3969	312	4	fuzzy	fuzzy	ADJ
cana-3969	312	5	bi	bi	NOUN
cana-3969	312	6	-	-	NOUN
cana-3969	312	7	ideals	ideal	NOUN
cana-3969	312	8	in	in	ADP
cana-3969	312	9	ordered	order	VERB
cana-3969	312	10	semigroups	semigroup	NOUN
cana-3969	312	11	.	.	PUNCT
cana-3969	313	1	neural	neural	ADJ
cana-3969	313	2	computing	computing	NOUN
cana-3969	313	3	and	and	CCONJ
cana-3969	313	4	applications	application	NOUN
cana-3969	313	5	,	,	PUNCT
cana-3969	313	6	21(suppl	21(suppl	NUM
cana-3969	313	7	1	1	NUM
cana-3969	313	8	)	)	PUNCT
cana-3969	313	9	,	,	PUNCT
cana-3969	313	10	295	295	NUM
cana-3969	313	11	-	-	SYM
cana-3969	313	12	305	305	NUM
cana-3969	313	13	.	.	PUNCT
cana-3969	314	1	[	[	X
cana-3969	314	2	28	28	NUM
cana-3969	314	3	]	]	SYM
cana-3969	314	4	li	li	PROPN
cana-3969	314	5	,	,	PUNCT
cana-3969	314	6	c.	c.	PROPN
cana-3969	314	7	,	,	PUNCT
cana-3969	314	8	xu	xu	PROPN
cana-3969	314	9	,	,	PUNCT
cana-3969	314	10	b.	b.	PROPN
cana-3969	314	11	,	,	PUNCT
cana-3969	314	12	&	&	CCONJ
cana-3969	314	13	huang	huang	PROPN
cana-3969	314	14	,	,	PUNCT
cana-3969	314	15	h.	h.	PROPN
cana-3969	314	16	(	(	PUNCT
cana-3969	314	17	2020	2020	NUM
cana-3969	314	18	)	)	PUNCT
cana-3969	314	19	.	.	PUNCT
cana-3969	315	1	bipolar	bipolar	ADJ
cana-3969	315	2	fuzzy	fuzzy	ADJ
cana-3969	315	3	abundant	abundant	ADJ
cana-3969	315	4	semigroups	semigroup	NOUN
cana-3969	315	5	with	with	ADP
cana-3969	315	6	applications	application	NOUN
cana-3969	315	7	.	.	PUNCT
cana-3969	316	1	journal	journal	NOUN
cana-3969	316	2	of	of	ADP
cana-3969	316	3	intelligent	intelligent	ADJ
cana-3969	316	4	&	&	CCONJ
cana-3969	316	5	fuzzy	fuzzy	ADJ
cana-3969	316	6	systems	system	NOUN
cana-3969	316	7	,	,	PUNCT
cana-3969	316	8	39(1	39(1	NUM
cana-3969	316	9	)	)	PUNCT
cana-3969	316	10	,	,	PUNCT
cana-3969	316	11	167	167	NUM
cana-3969	316	12	-	-	SYM
cana-3969	316	13	176	176	NUM
cana-3969	316	14	.	.	PUNCT
cana-3969	317	1	[	[	X
cana-3969	317	2	29	29	NUM
cana-3969	317	3	]	]	X
cana-3969	317	4	rehman	rehman	NOUN
cana-3969	317	5	,	,	PUNCT
cana-3969	317	6	n.	n.	NOUN
cana-3969	317	7	,	,	PUNCT
cana-3969	317	8	&	&	CCONJ
cana-3969	317	9	shabir	shabir	PROPN
cana-3969	317	10	,	,	PUNCT
cana-3969	317	11	m.	m.	NOUN
cana-3969	317	12	(	(	PUNCT
cana-3969	317	13	2014	2014	NUM
cana-3969	317	14	)	)	PUNCT
cana-3969	317	15	.	.	PUNCT
cana-3969	318	1	some	some	DET
cana-3969	318	2	characterizations	characterization	NOUN
cana-3969	318	3	of	of	ADP
cana-3969	318	4	ternary	ternary	ADJ
cana-3969	318	5	semigroups	semigroup	NOUN
cana-3969	318	6	by	by	ADP
cana-3969	318	7	the	the	DET
cana-3969	318	8	properties	property	NOUN
cana-3969	318	9	of	of	ADP
cana-3969	318	10	their	their	PRON
cana-3969	318	11	$	$	SYM
cana-3969	318	12	\left	\left	PROPN
cana-3969	318	13	(	(	PUNCT
cana-3969	318	14	\in	\in	X
cana-3969	318	15	_	_	PUNCT
cana-3969	318	16	{	{	PUNCT
cana-3969	318	17	\gamma},\in	\gamma},\in	X
cana-3969	318	18	_	_	PUNCT
cana-3969	318	19	{	{	PUNCT
cana-3969	318	20	\gamma}\vee	\gamma}\vee	PROPN
cana-3969	318	21	q	q	NOUN
cana-3969	318	22	_	_	PRON
cana-3969	318	23	{	{	PUNCT
cana-3969	318	24	\delta}\right	\delta}\right	NOUN
cana-3969	318	25	)	)	PUNCT
cana-3969	318	26	$	$	SYM
cana-3969	318	27	-fuzzy	-fuzzy	NOUN
cana-3969	318	28	ideals	ideal	NOUN
cana-3969	318	29	.	.	PUNCT
cana-3969	319	1	journal	journal	NOUN
cana-3969	319	2	of	of	ADP
cana-3969	319	3	intelligent	intelligent	ADJ
cana-3969	319	4	&	&	CCONJ
cana-3969	319	5	fuzzy	fuzzy	ADJ
cana-3969	319	6	systems	system	NOUN
cana-3969	319	7	,	,	PUNCT
cana-3969	319	8	26(5	26(5	NUM
cana-3969	319	9	)	)	PUNCT
cana-3969	319	10	,	,	PUNCT
cana-3969	319	11	2107	2107	NUM
cana-3969	319	12	-	-	SYM
cana-3969	319	13	2117	2117	NUM
cana-3969	319	14	.	.	PUNCT
cana-3969	320	1	[	[	X
cana-3969	320	2	30	30	NUM
cana-3969	320	3	]	]	SYM
cana-3969	320	4	li	li	PROPN
cana-3969	320	5	,	,	PUNCT
cana-3969	320	6	c.	c.	PROPN
cana-3969	320	7	,	,	PUNCT
cana-3969	320	8	xu	xu	PROPN
cana-3969	320	9	,	,	PUNCT
cana-3969	320	10	b.	b.	PROPN
cana-3969	320	11	,	,	PUNCT
cana-3969	320	12	&	&	CCONJ
cana-3969	320	13	huang	huang	PROPN
cana-3969	320	14	,	,	PUNCT
cana-3969	320	15	h.	h.	PROPN
cana-3969	320	16	(	(	PUNCT
cana-3969	320	17	2021	2021	NUM
cana-3969	320	18	)	)	PUNCT
cana-3969	320	19	.	.	PUNCT
cana-3969	321	1	a	a	DET
cana-3969	321	2	new	new	ADJ
cana-3969	321	3	characterisation	characterisation	NOUN
cana-3969	321	4	of	of	ADP
cana-3969	321	5	fuzzy	fuzzy	ADJ
cana-3969	321	6	ideals	ideal	NOUN
cana-3969	321	7	of	of	ADP
cana-3969	321	8	semigroups	semigroup	NOUN
cana-3969	321	9	and	and	CCONJ
cana-3969	321	10	its	its	PRON
cana-3969	321	11	applications	application	NOUN
cana-3969	321	12	.	.	PUNCT
cana-3969	321	13	automatika	automatika	PROPN
cana-3969	321	14	,	,	PUNCT
cana-3969	321	15	62(3	62(3	NOUN
cana-3969	321	16	-	-	SYM
cana-3969	321	17	4	4	NUM
cana-3969	321	18	)	)	PUNCT
cana-3969	321	19	,	,	PUNCT
cana-3969	321	20	407	407	NUM
cana-3969	321	21	-	-	SYM
cana-3969	321	22	414	414	NUM
cana-3969	321	23	.	.	PUNCT
cana-3969	322	1	[	[	X
cana-3969	322	2	31	31	NUM
cana-3969	322	3	]	]	SYM
cana-3969	322	4	anis	anis	PROPN
cana-3969	322	5	,	,	PUNCT
cana-3969	322	6	saima	saima	PROPN
cana-3969	322	7	,	,	PUNCT
cana-3969	322	8	madad	madad	PROPN
cana-3969	322	9	khan	khan	PROPN
cana-3969	322	10	,	,	PUNCT
cana-3969	322	11	and	and	CCONJ
cana-3969	322	12	young	young	ADJ
cana-3969	322	13	bae	bae	PROPN
cana-3969	322	14	jun	jun	PROPN
cana-3969	322	15	.	.	PUNCT
cana-3969	323	1	"	"	PUNCT
cana-3969	323	2	hybrid	hybrid	ADJ
cana-3969	323	3	ideals	ideal	NOUN
cana-3969	323	4	in	in	ADP
cana-3969	323	5	semigroups	semigroup	NOUN
cana-3969	323	6	.	.	PUNCT
cana-3969	323	7	"	"	PUNCT
cana-3969	324	1	cogent	cogent	NOUN
cana-3969	324	2	mathematics	mathematic	NOUN
cana-3969	324	3	4.1	4.1	NUM
cana-3969	324	4	(	(	PUNCT
cana-3969	324	5	2017	2017	NUM
cana-3969	324	6	):	):	PUNCT
cana-3969	324	7	1352117	1352117	NUM
cana-3969	324	8	.	.	PUNCT
cana-3969	325	1	[	[	X
cana-3969	325	2	32	32	NUM
cana-3969	325	3	]	]	X
cana-3969	325	4	mursaleen	mursaleen	PROPN
cana-3969	325	5	,	,	PUNCT
cana-3969	325	6	m.	m.	NOUN
cana-3969	325	7	,	,	PUNCT
cana-3969	325	8	srivastava	srivastava	PROPN
cana-3969	325	9	,	,	PUNCT
cana-3969	325	10	h.	h.	PROPN
cana-3969	325	11	m.	m.	PROPN
cana-3969	325	12	,	,	PUNCT
cana-3969	325	13	&	&	CCONJ
cana-3969	325	14	sharma	sharma	PROPN
cana-3969	325	15	,	,	PUNCT
cana-3969	325	16	s.	s.	PROPN
cana-3969	325	17	k.	k.	PROPN
cana-3969	325	18	(	(	PUNCT
cana-3969	325	19	2016	2016	NUM
cana-3969	325	20	)	)	PUNCT
cana-3969	325	21	.	.	PUNCT
cana-3969	325	22	generalised	generalise	VERB
cana-3969	325	23	statistically	statistically	ADV
cana-3969	325	24	convergent	convergent	ADJ
cana-3969	325	25	sequences	sequence	NOUN
cana-3969	325	26	of	of	ADP
cana-3969	325	27	fuzzy	fuzzy	ADJ
cana-3969	325	28	numbers	number	NOUN
cana-3969	325	29	.	.	PUNCT
cana-3969	326	1	journal	journal	NOUN
cana-3969	326	2	of	of	ADP
cana-3969	326	3	intelligent	intelligent	ADJ
cana-3969	326	4	&	&	CCONJ
cana-3969	326	5	fuzzy	fuzzy	ADJ
cana-3969	326	6	systems	system	NOUN
cana-3969	326	7	,	,	PUNCT
cana-3969	326	8	30	30	NUM
cana-3969	326	9	,	,	PUNCT
cana-3969	326	10	1511–1518	1511–1518	NUM
cana-3969	326	11	.	.	PUNCT
cana-3969	327	1	[	[	X
cana-3969	327	2	33	33	NUM
cana-3969	327	3	]	]	X
cana-3969	327	4	torra	torra	PROPN
cana-3969	327	5	,	,	PUNCT
cana-3969	327	6	v.	v.	PROPN
cana-3969	327	7	(	(	PUNCT
cana-3969	327	8	2010	2010	NUM
cana-3969	327	9	)	)	PUNCT
cana-3969	327	10	.	.	PUNCT
cana-3969	328	1	hesitant	hesitant	ADJ
cana-3969	328	2	fuzzy	fuzzy	ADJ
cana-3969	328	3	sets	set	NOUN
cana-3969	328	4	.	.	PUNCT
cana-3969	329	1	international	international	ADJ
cana-3969	329	2	journal	journal	NOUN
cana-3969	329	3	of	of	ADP
cana-3969	329	4	computational	computational	ADJ
cana-3969	329	5	intelligence	intelligence	NOUN
cana-3969	329	6	systems	system	NOUN
cana-3969	329	7	,	,	PUNCT
cana-3969	329	8	25	25	NUM
cana-3969	329	9	,	,	PUNCT
cana-3969	329	10	529–539	529–539	NUM
cana-3969	329	11	[	[	X
cana-3969	329	12	34	34	NUM
cana-3969	329	13	]	]	X
cana-3969	329	14	torra	torra	PROPN
cana-3969	329	15	,	,	PUNCT
cana-3969	329	16	v.	v.	ADV
cana-3969	329	17	,	,	PUNCT
cana-3969	329	18	&	&	CCONJ
cana-3969	329	19	narukawa	narukawa	PROPN
cana-3969	329	20	,	,	PUNCT
cana-3969	329	21	y.	y.	PROPN
cana-3969	329	22	(	(	PUNCT
cana-3969	329	23	2009	2009	NUM
cana-3969	329	24	)	)	PUNCT
cana-3969	329	25	.	.	PUNCT
cana-3969	330	1	on	on	ADP
cana-3969	330	2	hesitant	hesitant	ADJ
cana-3969	330	3	fuzzy	fuzzy	ADJ
cana-3969	330	4	sets	set	NOUN
cana-3969	330	5	and	and	CCONJ
cana-3969	330	6	decisions	decision	NOUN
cana-3969	330	7	.	.	PUNCT
cana-3969	331	1	in	in	ADP
cana-3969	331	2	the	the	DET
cana-3969	331	3	18th	18th	ADJ
cana-3969	331	4	ieee	ieee	NOUN
cana-3969	331	5	international	international	ADJ
cana-3969	331	6	conference	conference	NOUN
cana-3969	331	7	on	on	ADP
cana-3969	331	8	fuzzy	fuzzy	ADJ
cana-3969	331	9	systems	system	NOUN
cana-3969	331	10	(	(	PUNCT
cana-3969	331	11	1378–1382	1378–1382	NUM
cana-3969	331	12	)	)	PUNCT
cana-3969	331	13	.	.	PUNCT
cana-3969	332	1	jeju	jeju	PROPN
cana-3969	332	2	island	island	PROPN
cana-3969	332	3	zadeh	zadeh	PROPN
cana-3969	332	4	,	,	PUNCT
cana-3969	332	5	l.	l.	PROPN
cana-3969	332	6	a.	a.	PROPN
cana-3969	332	7	(	(	PUNCT
cana-3969	332	8	1965	1965	NUM
cana-3969	332	9	)	)	PUNCT
cana-3969	332	10	.	.	PUNCT
cana-3969	333	1	fuzzy	fuzzy	ADJ
cana-3969	333	2	sets	set	NOUN
cana-3969	333	3	.	.	PUNCT
cana-3969	334	1	information	information	NOUN
cana-3969	334	2	and	and	CCONJ
cana-3969	334	3	control	control	NOUN
cana-3969	334	4	,	,	PUNCT
cana-3969	334	5	8	8	NUM
cana-3969	334	6	,	,	PUNCT
cana-3969	334	7	338	338	NUM
cana-3969	334	8	.	.	PUNCT
cana-3969	335	1	communications	communication	NOUN
cana-3969	335	2	on	on	ADP
cana-3969	335	3	applied	apply	VERB
cana-3969	335	4	nonlinear	nonlinear	ADJ
cana-3969	335	5	analysis	analysis	NOUN
cana-3969	335	6	issn	issn	NOUN
cana-3969	335	7	:	:	PUNCT
cana-3969	335	8	1074	1074	NUM
cana-3969	335	9	-	-	PUNCT
cana-3969	335	10	133x	133x	NUM
cana-3969	335	11	vol	vol	NOUN
cana-3969	335	12	32	32	NUM
cana-3969	335	13	no	no	NOUN
cana-3969	335	14	.	.	PUNCT
cana-3969	336	1	9s	9s	NUM
cana-3969	336	2	(	(	PUNCT
cana-3969	336	3	2025	2025	NUM
cana-3969	336	4	)	)	PUNCT
cana-3969	336	5	649	649	NUM
cana-3969	336	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3969	337	1	[	[	X
cana-3969	337	2	35	35	NUM
cana-3969	337	3	]	]	X
cana-3969	337	4	hedayati	hedayati	ADJ
cana-3969	337	5	,	,	PUNCT
cana-3969	337	6	2012	2012	NUM
cana-3969	337	7	intuitionistic	intuitionistic	ADJ
cana-3969	337	8	(	(	PUNCT
cana-3969	337	9	s	s	PROPN
cana-3969	337	10	,	,	PUNCT
cana-3969	337	11	t)-fuzzy	t)-fuzzy	PUNCT
cana-3969	337	12	(	(	PUNCT
cana-3969	337	13	1	1	NUM
cana-3969	337	14	,	,	PUNCT
cana-3969	337	15	2)-ideals	2)-ideals	NUM
cana-3969	337	16	of	of	ADP
cana-3969	337	17	semigroups	semigroup	NOUN
cana-3969	337	18	with	with	ADP
cana-3969	337	19	interval	interval	NOUN
cana-3969	337	20	-	-	PUNCT
cana-3969	337	21	valued	value	VERB
cana-3969	337	22	membership	membership	NOUN
cana-3969	337	23	functions	function	NOUN
cana-3969	337	24	hinternational	hinternational	ADJ
cana-3969	337	25	journal	journal	NOUN
cana-3969	337	26	of	of	ADP
cana-3969	337	27	fuzzy	fuzzy	ADJ
cana-3969	337	28	systems	system	NOUN
cana-3969	337	29	(	(	PUNCT
cana-3969	337	30	2012	2012	NUM
cana-3969	337	31	)	)	PUNCT
cana-3969	337	32	14(1	14(1	NUM
cana-3969	337	33	)	)	PUNCT
cana-3969	338	1	[	[	X
cana-3969	338	2	36	36	NUM
cana-3969	338	3	]	]	PUNCT
cana-3969	338	4	singer	singer	NOUN
cana-3969	338	5	,	,	PUNCT
cana-3969	338	6	2007linearly	2007linearly	PROPN
cana-3969	338	7	ordered	order	VERB
cana-3969	338	8	semigroups	semigroup	NOUN
cana-3969	338	9	for	for	ADP
cana-3969	338	10	fuzzy	fuzzy	ADJ
cana-3969	338	11	set	set	NOUN
cana-3969	338	12	theory	theory	NOUN
cana-3969	338	13	dannals	dannal	NOUN
cana-3969	338	14	of	of	ADP
cana-3969	338	15	mathematics	mathematic	NOUN
cana-3969	338	16	and	and	CCONJ
cana-3969	338	17	artificial	artificial	ADJ
cana-3969	338	18	intelligence	intelligence	NOUN
cana-3969	338	19	(	(	PUNCT
cana-3969	338	20	2007	2007	NUM
cana-3969	338	21	)	)	PUNCT
cana-3969	338	22	49(1	49(1	NOUN
cana-3969	338	23	-	-	PUNCT
cana-3969	338	24	4	4	NUM
cana-3969	338	25	)	)	PUNCT
cana-3969	338	26	.	.	PUNCT
cana-3969	339	1	[	[	X
cana-3969	339	2	37	37	NUM
cana-3969	339	3	]	]	X
cana-3969	339	4	murthy	murthy	PROPN
cana-3969	339	5	,	,	PUNCT
cana-3969	339	6	a.	a.	PROPN
cana-3969	339	7	,	,	PUNCT
cana-3969	339	8	deen	deen	PROPN
cana-3969	339	9	,	,	PUNCT
cana-3969	339	10	m.	m.	NOUN
cana-3969	339	11	,	,	PUNCT
cana-3969	339	12	&	&	CCONJ
cana-3969	339	13	fang	fang	PROPN
cana-3969	339	14	,	,	PUNCT
cana-3969	339	15	q.	q.	PROPN
cana-3969	339	16	(	(	PUNCT
cana-3969	339	17	2015	2015	NUM
cana-3969	339	18	)	)	PUNCT
cana-3969	339	19	.	.	PUNCT
cana-3969	340	1	led	lead	VERB
cana-3969	340	2	array	array	NOUN
cana-3969	340	3	design	design	NOUN
cana-3969	340	4	with	with	ADP
cana-3969	340	5	image	image	NOUN
cana-3969	340	6	-	-	PUNCT
cana-3969	340	7	based	base	VERB
cana-3969	340	8	planar	planar	ADJ
cana-3969	340	9	verification	verification	NOUN
cana-3969	340	10	.	.	PUNCT
cana-3969	341	1	international	international	ADJ
cana-3969	341	2	journal	journal	PROPN
cana-3969	341	3	of	of	ADP
cana-3969	341	4	electronics	electronic	NOUN
cana-3969	341	5	communication	communication	NOUN
cana-3969	341	6	and	and	CCONJ
cana-3969	341	7	computer	computer	NOUN
cana-3969	341	8	engineering	engineering	NOUN
cana-3969	341	9	,	,	PUNCT
cana-3969	341	10	6(5	6(5	NUM
cana-3969	341	11	)	)	PUNCT
cana-3969	341	12	,	,	PUNCT
cana-3969	341	13	565	565	NUM
cana-3969	341	14	-	-	SYM
cana-3969	341	15	571	571	NUM
cana-3969	341	16	.	.	PUNCT
cana-3969	342	1	[	[	X
cana-3969	342	2	38	38	NUM
cana-3969	342	3	]	]	X
cana-3969	342	4	kumar	kumar	PROPN
cana-3969	342	5	,	,	PUNCT
cana-3969	342	6	v.	v.	PROPN
cana-3969	342	7	,	,	PUNCT
cana-3969	342	8	gupta	gupta	PROPN
cana-3969	342	9	,	,	PUNCT
cana-3969	342	10	h.	h.	PROPN
cana-3969	342	11	,	,	PUNCT
cana-3969	342	12	&	&	CCONJ
cana-3969	342	13	sharma	sharma	PROPN
cana-3969	342	14	,	,	PUNCT
cana-3969	342	15	s.	s.	PROPN
cana-3969	342	16	(	(	PUNCT
cana-3969	342	17	2014	2014	NUM
cana-3969	342	18	)	)	PUNCT
cana-3969	342	19	.	.	PUNCT
cana-3969	343	1	neutralising	neutralise	VERB
cana-3969	343	2	the	the	DET
cana-3969	343	3	impact	impact	NOUN
cana-3969	343	4	of	of	ADP
cana-3969	343	5	account	account	NOUN
cana-3969	343	6	or	or	CCONJ
cana-3969	343	7	service	service	NOUN
cana-3969	343	8	traffic	traffic	NOUN
cana-3969	343	9	hijacking	hijacking	NOUN
cana-3969	343	10	in	in	ADP
cana-3969	343	11	cloud	cloud	NOUN
cana-3969	343	12	computing	computing	NOUN
cana-3969	343	13	.	.	PUNCT
cana-3969	344	1	international	international	ADJ
cana-3969	344	2	journal	journal	PROPN
cana-3969	344	3	of	of	ADP
cana-3969	344	4	electronics	electronic	NOUN
cana-3969	344	5	communication	communication	NOUN
cana-3969	344	6	and	and	CCONJ
cana-3969	344	7	computer	computer	NOUN
cana-3969	344	8	engineering	engineering	NOUN
cana-3969	344	9	,	,	PUNCT
cana-3969	344	10	5(1	5(1	NUM
cana-3969	344	11	)	)	PUNCT
cana-3969	344	12	,	,	PUNCT
cana-3969	344	13	206	206	NUM
cana-3969	344	14	-	-	SYM
cana-3969	344	15	209	209	NUM
cana-3969	344	16	.	.	PUNCT