id	sid	tid	token	lemma	pos
cana-2729	1	1	communications	communication	NOUN
cana-2729	1	2	on	on	ADP
cana-2729	1	3	applied	apply	VERB
cana-2729	1	4	nonlinear	nonlinear	ADJ
cana-2729	1	5	analysis	analysis	NOUN
cana-2729	1	6	issn	issn	NOUN
cana-2729	1	7	:	:	PUNCT
cana-2729	1	8	1074	1074	NUM
cana-2729	1	9	-	-	PUNCT
cana-2729	1	10	133x	133x	NUM
cana-2729	1	11	vol	vol	NOUN
cana-2729	1	12	32	32	NUM
cana-2729	1	13	no	no	NOUN
cana-2729	1	14	.	.	PUNCT
cana-2729	2	1	3s	3s	NUM
cana-2729	2	2	(	(	PUNCT
cana-2729	2	3	2025	2025	NUM
cana-2729	2	4	)	)	PUNCT
cana-2729	2	5	711	711	NUM
cana-2729	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2729	2	7	applications	application	NOUN
cana-2729	2	8	and	and	CCONJ
cana-2729	2	9	future	future	ADJ
cana-2729	2	10	directions	direction	NOUN
cana-2729	2	11	of	of	ADP
cana-2729	2	12	fuzzy	fuzzy	ADJ
cana-2729	2	13	brk	brk	PROPN
cana-2729	2	14	topological	topological	ADJ
cana-2729	2	15	groups	group	NOUN
cana-2729	2	16	in	in	ADP
cana-2729	2	17	mathematics	mathematic	NOUN
cana-2729	2	18	and	and	CCONJ
cana-2729	2	19	ai	ai	VERB
cana-2729	2	20	1	1	NUM
cana-2729	2	21	s.kousalya	s.kousalya	NOUN
cana-2729	2	22	,	,	PUNCT
cana-2729	2	23	2n.mala	2n.mala	NUM
cana-2729	2	24	,	,	PUNCT
cana-2729	2	25	3dr.k.j	3dr.k.j	VERB
cana-2729	2	26	.	.	PUNCT
cana-2729	3	1	eldho	eldho	PROPN
cana-2729	3	2	,	,	PUNCT
cana-2729	3	3	4dr.s.swapna,5m.thamizhsudar	4dr.s.swapna,5m.thamizhsudar	NOUN
cana-2729	3	4	,	,	PUNCT
cana-2729	3	5	6e.kungumaraj	6e.kungumaraj	NUM
cana-2729	3	6	,	,	PUNCT
cana-2729	3	7	7dr	7dr	NOUN
cana-2729	3	8	.	.	PUNCT
cana-2729	4	1	g.	g.	PROPN
cana-2729	4	2	jenitha	jenitha	PROPN
cana-2729	5	1	1assistant	1assistant	NUM
cana-2729	5	2	professor	professor	NOUN
cana-2729	5	3	(	(	PUNCT
cana-2729	5	4	sg	sg	PROPN
cana-2729	5	5	)	)	PUNCT
cana-2729	5	6	of	of	ADP
cana-2729	5	7	mathematics1	mathematics1	PROPN
cana-2729	5	8	,	,	PUNCT
cana-2729	5	9	associate	associate	ADJ
cana-2729	5	10	professor	professor	NOUN
cana-2729	5	11	of	of	ADP
cana-2729	5	12	mathematics	mathematic	NOUN
cana-2729	5	13	2	2	NUM
cana-2729	5	14	,	,	PUNCT
cana-2729	5	15	1email	1email	NUM
cana-2729	5	16	id:kousalyavaasan@gmail.com	id:kousalyavaasan@gmail.com	NOUN
cana-2729	5	17	,	,	PUNCT
cana-2729	5	18	2professor	2professor	NUM
cana-2729	5	19	,	,	PUNCT
cana-2729	5	20	department	department	NOUN
cana-2729	5	21	of	of	ADP
cana-2729	5	22	mathematics	mathematic	NOUN
cana-2729	5	23	,	,	PUNCT
cana-2729	5	24	kovai	kovai	VERB
cana-2729	5	25	kalaimagal	kalaimagal	PROPN
cana-2729	5	26	college	college	PROPN
cana-2729	5	27	of	of	ADP
cana-2729	5	28	arts	art	NOUN
cana-2729	5	29	and	and	CCONJ
cana-2729	5	30	science	science	NOUN
cana-2729	5	31	,	,	PUNCT
cana-2729	5	32	coimbatore	coimbatore	NOUN
cana-2729	5	33	2email	2email	NUM
cana-2729	5	34	id:mala.kkcas@gmail.com	id:mala.kkcas@gmail.com	NUM
cana-2729	5	35	3assistant	3assistant	NUM
cana-2729	5	36	professor	professor	NOUN
cana-2729	5	37	,	,	PUNCT
cana-2729	5	38	department	department	NOUN
cana-2729	5	39	of	of	ADP
cana-2729	5	40	computer	computer	NOUN
cana-2729	5	41	science	science	NOUN
cana-2729	5	42	,	,	PUNCT
cana-2729	5	43	mary	mary	PROPN
cana-2729	5	44	matha	matha	PROPN
cana-2729	5	45	govt	govt	PROPN
cana-2729	5	46	aided	aid	VERB
cana-2729	5	47	arts	art	NOUN
cana-2729	5	48	and	and	CCONJ
cana-2729	5	49	science	science	NOUN
cana-2729	5	50	college	college	PROPN
cana-2729	5	51	,	,	PUNCT
cana-2729	5	52	mananthavady	mananthavady	ADJ
cana-2729	5	53	.	.	PUNCT
cana-2729	6	1	3email	3email	NUM
cana-2729	6	2	i	i	NOUN
cana-2729	6	3	d	d	NOUN
cana-2729	6	4	:	:	PUNCT
cana-2729	6	5	eldhokj@marymathacollege.ac.in	eldhokj@marymathacollege.ac.in	PROPN
cana-2729	6	6	4professor	4professor	NUM
cana-2729	6	7	,	,	PUNCT
cana-2729	6	8	hod	hod	PROPN
cana-2729	6	9	-	-	PUNCT
cana-2729	6	10	cse	cse	PROPN
cana-2729	6	11	,	,	PUNCT
cana-2729	6	12	department	department	PROPN
cana-2729	6	13	of	of	ADP
cana-2729	6	14	cse	cse	PROPN
cana-2729	6	15	,	,	PUNCT
cana-2729	6	16	neil	neil	PROPN
cana-2729	6	17	gogte	gogte	PROPN
cana-2729	6	18	institute	institute	PROPN
cana-2729	6	19	of	of	ADP
cana-2729	6	20	technology	technology	PROPN
cana-2729	6	21	,	,	PUNCT
cana-2729	6	22	hyderabad	hyderabad	PROPN
cana-2729	6	23	4email	4email	PROPN
cana-2729	6	24	id:swapnangit2021@gmail.com	id:swapnangit2021@gmail.com	PROPN
cana-2729	6	25	.	.	PUNCT
cana-2729	7	1	orcid	orcid	NOUN
cana-2729	8	1	i	i	PRON
cana-2729	8	2	d	d	PROPN
cana-2729	8	3	:	:	PUNCT
cana-2729	8	4	https://orcid.org/0000-0003-2006-2367	https://orcid.org/0000-0003-2006-2367	PROPN
cana-2729	8	5	5department	5department	NUM
cana-2729	8	6	of	of	ADP
cana-2729	8	7	mathematics	mathematic	NOUN
cana-2729	8	8	,	,	PUNCT
cana-2729	8	9	aarupadai	aarupadai	PROPN
cana-2729	8	10	veedu	veedu	PROPN
cana-2729	8	11	institute	institute	PROPN
cana-2729	8	12	of	of	ADP
cana-2729	8	13	technology	technology	PROPN
cana-2729	8	14	,	,	PUNCT
cana-2729	8	15	vinayaka	vinayaka	PROPN
cana-2729	8	16	mission	mission	NOUN
cana-2729	8	17	's	's	PART
cana-2729	8	18	research	research	NOUN
cana-2729	8	19	foundation	foundation	NOUN
cana-2729	8	20	(	(	PUNCT
cana-2729	8	21	du),chennai	du),chennai	X
cana-2729	8	22	-603104	-603104	PROPN
cana-2729	8	23	.	.	PUNCT
cana-2729	9	1	5email	5email	NUM
cana-2729	9	2	i	i	PRON
cana-2729	9	3	d	d	PROPN
cana-2729	9	4	:	:	PUNCT
cana-2729	9	5	thamizhsudar@avit.ac.in	thamizhsudar@avit.ac.in	SYM
cana-2729	9	6	6department	6department	NUM
cana-2729	9	7	of	of	ADP
cana-2729	9	8	science	science	NOUN
cana-2729	9	9	and	and	CCONJ
cana-2729	9	10	humanities	humanity	NOUN
cana-2729	9	11	,	,	PUNCT
cana-2729	9	12	nehru	nehru	PROPN
cana-2729	9	13	institute	institute	PROPN
cana-2729	9	14	of	of	ADP
cana-2729	9	15	engineering	engineering	NOUN
cana-2729	9	16	and	and	CCONJ
cana-2729	9	17	technology	technology	NOUN
cana-2729	9	18	,	,	PUNCT
cana-2729	9	19	coimbatore	coimbatore	PROPN
cana-2729	9	20	.	.	PUNCT
cana-2729	10	1	6email	6email	NUM
cana-2729	10	2	i	i	PRON
cana-2729	10	3	d	d	PROPN
cana-2729	10	4	:	:	PUNCT
cana-2729	10	5	kungum99522@gmail.com	kungum99522@gmail.com	X
cana-2729	11	1	7assistant	7assistant	NUM
cana-2729	11	2	professor	professor	NOUN
cana-2729	11	3	,	,	PUNCT
cana-2729	11	4	department	department	NOUN
cana-2729	11	5	of	of	ADP
cana-2729	11	6	mathematics	mathematics	PROPN
cana-2729	11	7	,	,	PUNCT
cana-2729	11	8	amet	amet	PROPN
cana-2729	11	9	deemed	deem	VERB
cana-2729	11	10	to	to	PART
cana-2729	11	11	be	be	AUX
cana-2729	11	12	university	university	NOUN
cana-2729	11	13	,	,	PUNCT
cana-2729	11	14	ecr	ecr	PROPN
cana-2729	11	15	,	,	PUNCT
cana-2729	11	16	kanathur	kanathur	NOUN
cana-2729	11	17	,	,	PUNCT
cana-2729	11	18	chennai	chennai	NOUN
cana-2729	11	19	.	.	PUNCT
cana-2729	12	1	7corresponding	7corresponde	VERB
cana-2729	12	2	author	author	NOUN
cana-2729	12	3	email	email	NOUN
cana-2729	12	4	id:jenitha.g@ametuniv.ac.in	id:jenitha.g@ametuniv.ac.in	PROPN
cana-2729	12	5	article	article	PROPN
cana-2729	12	6	history	history	NOUN
cana-2729	12	7	:	:	PUNCT
cana-2729	12	8	received	receive	VERB
cana-2729	12	9	:	:	PUNCT
cana-2729	12	10	29	29	NUM
cana-2729	12	11	-	-	SYM
cana-2729	12	12	09	09	NUM
cana-2729	12	13	-	-	PUNCT
cana-2729	12	14	2024	2024	NUM
cana-2729	12	15	revised	revise	VERB
cana-2729	12	16	:	:	PUNCT
cana-2729	12	17	20	20	NUM
cana-2729	12	18	-	-	SYM
cana-2729	12	19	11	11	NUM
cana-2729	12	20	-	-	PUNCT
cana-2729	12	21	2024	2024	NUM
cana-2729	12	22	accepted	accept	VERB
cana-2729	12	23	:	:	PUNCT
cana-2729	12	24	30	30	NUM
cana-2729	12	25	-	-	SYM
cana-2729	12	26	11	11	NUM
cana-2729	12	27	-	-	PUNCT
cana-2729	12	28	2024	2024	NUM
cana-2729	12	29	abstract	abstract	NOUN
cana-2729	12	30	:	:	PUNCT
cana-2729	12	31	the	the	DET
cana-2729	12	32	study	study	NOUN
cana-2729	12	33	of	of	ADP
cana-2729	12	34	fuzzy	fuzzy	ADJ
cana-2729	12	35	topological	topological	ADJ
cana-2729	12	36	groups	group	NOUN
cana-2729	12	37	has	have	AUX
cana-2729	12	38	garnered	garner	VERB
cana-2729	12	39	significant	significant	ADJ
cana-2729	12	40	attention	attention	NOUN
cana-2729	12	41	due	due	ADP
cana-2729	12	42	to	to	ADP
cana-2729	12	43	their	their	PRON
cana-2729	12	44	applications	application	NOUN
cana-2729	12	45	in	in	ADP
cana-2729	12	46	various	various	ADJ
cana-2729	12	47	fields	field	NOUN
cana-2729	12	48	of	of	ADP
cana-2729	12	49	mathematics	mathematic	NOUN
cana-2729	12	50	and	and	CCONJ
cana-2729	12	51	computational	computational	ADJ
cana-2729	12	52	theory	theory	NOUN
cana-2729	12	53	.	.	PUNCT
cana-2729	13	1	this	this	DET
cana-2729	13	2	paper	paper	NOUN
cana-2729	13	3	introduces	introduce	VERB
cana-2729	13	4	an	an	DET
cana-2729	13	5	in	in	ADP
cana-2729	13	6	-	-	PUNCT
cana-2729	13	7	depth	depth	NOUN
cana-2729	13	8	exploration	exploration	NOUN
cana-2729	13	9	of	of	ADP
cana-2729	13	10	fuzzy	fuzzy	ADJ
cana-2729	13	11	brk	brk	PROPN
cana-2729	13	12	(	(	PUNCT
cana-2729	13	13	banach	banach	NOUN
cana-2729	13	14	-	-	PUNCT
cana-2729	13	15	riemann	riemann	PROPN
cana-2729	13	16	-	-	PUNCT
cana-2729	13	17	klein	klein	PROPN
cana-2729	13	18	)	)	PUNCT
cana-2729	13	19	topological	topological	ADJ
cana-2729	13	20	groups	group	NOUN
cana-2729	13	21	,	,	PUNCT
cana-2729	13	22	emphasizing	emphasize	VERB
cana-2729	13	23	both	both	CCONJ
cana-2729	13	24	the	the	DET
cana-2729	13	25	theoretical	theoretical	ADJ
cana-2729	13	26	foundations	foundation	NOUN
cana-2729	13	27	and	and	CCONJ
cana-2729	13	28	potential	potential	ADJ
cana-2729	13	29	extensions	extension	NOUN
cana-2729	13	30	of	of	ADP
cana-2729	13	31	the	the	DET
cana-2729	13	32	concept	concept	NOUN
cana-2729	13	33	.	.	PUNCT
cana-2729	14	1	we	we	PRON
cana-2729	14	2	first	first	ADV
cana-2729	14	3	establish	establish	VERB
cana-2729	14	4	a	a	DET
cana-2729	14	5	rigorous	rigorous	ADJ
cana-2729	14	6	framework	framework	NOUN
cana-2729	14	7	that	that	PRON
cana-2729	14	8	unifies	unify	VERB
cana-2729	14	9	fuzzy	fuzzy	ADJ
cana-2729	14	10	set	set	NOUN
cana-2729	14	11	theory	theory	NOUN
cana-2729	14	12	with	with	ADP
cana-2729	14	13	brk	brk	PROPN
cana-2729	14	14	topological	topological	PROPN
cana-2729	14	15	groups	group	NOUN
cana-2729	14	16	,	,	PUNCT
cana-2729	14	17	providing	provide	VERB
cana-2729	14	18	new	new	ADJ
cana-2729	14	19	insights	insight	NOUN
cana-2729	14	20	into	into	ADP
cana-2729	14	21	their	their	PRON
cana-2729	14	22	structural	structural	ADJ
cana-2729	14	23	properties	property	NOUN
cana-2729	14	24	.	.	PUNCT
cana-2729	15	1	by	by	ADP
cana-2729	15	2	employing	employ	VERB
cana-2729	15	3	fuzzy	fuzzy	ADJ
cana-2729	15	4	relations	relation	NOUN
cana-2729	15	5	and	and	CCONJ
cana-2729	15	6	fuzzy	fuzzy	ADJ
cana-2729	15	7	sets	set	NOUN
cana-2729	15	8	,	,	PUNCT
cana-2729	15	9	we	we	PRON
cana-2729	15	10	redefine	redefine	VERB
cana-2729	15	11	the	the	DET
cana-2729	15	12	notion	notion	NOUN
cana-2729	15	13	of	of	ADP
cana-2729	15	14	continuity	continuity	NOUN
cana-2729	15	15	,	,	PUNCT
cana-2729	15	16	closure	closure	NOUN
cana-2729	15	17	,	,	PUNCT
cana-2729	15	18	and	and	CCONJ
cana-2729	15	19	neighborhood	neighborhood	NOUN
cana-2729	15	20	within	within	ADP
cana-2729	15	21	brk	brk	PROPN
cana-2729	15	22	topological	topological	ADJ
cana-2729	15	23	groups	group	NOUN
cana-2729	15	24	,	,	PUNCT
cana-2729	15	25	leading	lead	VERB
cana-2729	15	26	to	to	ADP
cana-2729	15	27	more	more	ADV
cana-2729	15	28	generalized	generalized	ADJ
cana-2729	15	29	topological	topological	ADJ
cana-2729	15	30	structures	structure	NOUN
cana-2729	15	31	that	that	PRON
cana-2729	15	32	can	can	AUX
cana-2729	15	33	accommodate	accommodate	VERB
cana-2729	15	34	fuzziness	fuzziness	NOUN
cana-2729	15	35	.	.	PUNCT
cana-2729	16	1	the	the	DET
cana-2729	16	2	main	main	ADJ
cana-2729	16	3	contribution	contribution	NOUN
cana-2729	16	4	of	of	ADP
cana-2729	16	5	this	this	DET
cana-2729	16	6	work	work	NOUN
cana-2729	16	7	lies	lie	VERB
cana-2729	16	8	in	in	ADP
cana-2729	16	9	the	the	DET
cana-2729	16	10	development	development	NOUN
cana-2729	16	11	of	of	ADP
cana-2729	16	12	new	new	ADJ
cana-2729	16	13	extensions	extension	NOUN
cana-2729	16	14	that	that	PRON
cana-2729	16	15	address	address	VERB
cana-2729	16	16	key	key	ADJ
cana-2729	16	17	limitations	limitation	NOUN
cana-2729	16	18	in	in	ADP
cana-2729	16	19	classical	classical	ADJ
cana-2729	16	20	brk	brk	PROPN
cana-2729	16	21	topological	topological	PROPN
cana-2729	16	22	group	group	PROPN
cana-2729	16	23	theory	theory	NOUN
cana-2729	16	24	.	.	PUNCT
cana-2729	17	1	specifically	specifically	ADV
cana-2729	17	2	,	,	PUNCT
cana-2729	17	3	we	we	PRON
cana-2729	17	4	propose	propose	VERB
cana-2729	17	5	a	a	DET
cana-2729	17	6	novel	novel	ADJ
cana-2729	17	7	method	method	NOUN
cana-2729	17	8	of	of	ADP
cana-2729	17	9	constructing	construct	VERB
cana-2729	17	10	fuzzy	fuzzy	ADJ
cana-2729	17	11	brk	brk	PROPN
cana-2729	17	12	topological	topological	ADJ
cana-2729	17	13	groups	group	NOUN
cana-2729	17	14	,	,	PUNCT
cana-2729	17	15	allowing	allow	VERB
cana-2729	17	16	for	for	ADP
cana-2729	17	17	more	more	ADJ
cana-2729	17	18	flexibility	flexibility	NOUN
cana-2729	17	19	in	in	ADP
cana-2729	17	20	handling	handle	VERB
cana-2729	17	21	uncertainties	uncertainty	NOUN
cana-2729	17	22	and	and	CCONJ
cana-2729	17	23	imprecise	imprecise	ADJ
cana-2729	17	24	data	datum	NOUN
cana-2729	17	25	.	.	PUNCT
cana-2729	18	1	additionally	additionally	ADV
cana-2729	18	2	,	,	PUNCT
cana-2729	18	3	we	we	PRON
cana-2729	18	4	investigate	investigate	VERB
cana-2729	18	5	homomorphisms	homomorphism	NOUN
cana-2729	18	6	and	and	CCONJ
cana-2729	18	7	isomorphisms	isomorphism	NOUN
cana-2729	18	8	in	in	ADP
cana-2729	18	9	the	the	DET
cana-2729	18	10	context	context	NOUN
cana-2729	18	11	of	of	ADP
cana-2729	18	12	fuzzy	fuzzy	ADJ
cana-2729	18	13	brk	brk	PROPN
cana-2729	18	14	groups	group	NOUN
cana-2729	18	15	,	,	PUNCT
cana-2729	18	16	highlighting	highlight	VERB
cana-2729	18	17	their	their	PRON
cana-2729	18	18	role	role	NOUN
cana-2729	18	19	in	in	ADP
cana-2729	18	20	preserving	preserve	VERB
cana-2729	18	21	topological	topological	ADJ
cana-2729	18	22	properties	property	NOUN
cana-2729	18	23	under	under	ADP
cana-2729	18	24	fuzzy	fuzzy	ADJ
cana-2729	18	25	transformations	transformation	NOUN
cana-2729	18	26	.	.	PUNCT
cana-2729	19	1	the	the	DET
cana-2729	19	2	results	result	NOUN
cana-2729	19	3	presented	present	VERB
cana-2729	19	4	have	have	VERB
cana-2729	19	5	broad	broad	ADJ
cana-2729	19	6	implications	implication	NOUN
cana-2729	19	7	for	for	ADP
cana-2729	19	8	both	both	CCONJ
cana-2729	19	9	pure	pure	ADJ
cana-2729	19	10	and	and	CCONJ
cana-2729	19	11	applied	applied	ADJ
cana-2729	19	12	mathematics	mathematic	NOUN
cana-2729	19	13	,	,	PUNCT
cana-2729	19	14	particularly	particularly	ADV
cana-2729	19	15	in	in	ADP
cana-2729	19	16	fields	field	NOUN
cana-2729	19	17	requiring	require	VERB
cana-2729	19	18	a	a	DET
cana-2729	19	19	blend	blend	NOUN
cana-2729	19	20	of	of	ADP
cana-2729	19	21	algebraic	algebraic	ADJ
cana-2729	19	22	and	and	CCONJ
cana-2729	19	23	topological	topological	ADJ
cana-2729	19	24	techniques	technique	NOUN
cana-2729	19	25	,	,	PUNCT
cana-2729	19	26	such	such	ADJ
cana-2729	19	27	as	as	ADP
cana-2729	19	28	fuzzy	fuzzy	ADJ
cana-2729	19	29	logic	logic	NOUN
cana-2729	19	30	,	,	PUNCT
cana-2729	19	31	decision	decision	NOUN
cana-2729	19	32	-	-	PUNCT
cana-2729	19	33	making	make	VERB
cana-2729	19	34	processes	process	NOUN
cana-2729	19	35	,	,	PUNCT
cana-2729	19	36	and	and	CCONJ
cana-2729	19	37	artificial	artificial	ADJ
cana-2729	19	38	intelligence	intelligence	NOUN
cana-2729	19	39	.	.	PUNCT
cana-2729	20	1	finally	finally	ADV
cana-2729	20	2	,	,	PUNCT
cana-2729	20	3	we	we	PRON
cana-2729	20	4	outline	outline	VERB
cana-2729	20	5	potential	potential	ADJ
cana-2729	20	6	avenues	avenue	NOUN
cana-2729	20	7	for	for	ADP
cana-2729	20	8	further	further	ADJ
cana-2729	20	9	research	research	NOUN
cana-2729	20	10	and	and	CCONJ
cana-2729	20	11	applications	application	NOUN
cana-2729	20	12	of	of	ADP
cana-2729	20	13	fuzzy	fuzzy	ADJ
cana-2729	20	14	brk	brk	PROPN
cana-2729	20	15	topological	topological	ADJ
cana-2729	20	16	groups	group	NOUN
cana-2729	20	17	in	in	ADP
cana-2729	20	18	real	real	ADJ
cana-2729	20	19	-	-	PUNCT
cana-2729	20	20	world	world	NOUN
cana-2729	20	21	scenarios	scenario	NOUN
cana-2729	20	22	.	.	PUNCT
cana-2729	21	1	keywords	keyword	NOUN
cana-2729	21	2	:	:	PUNCT
cana-2729	21	3	brkcl(ρ	brkcl(ρ	NUM
cana-2729	21	4	)	)	PUNCT
cana-2729	21	5	,	,	PUNCT
cana-2729	21	6	brkint(ρ	brkint(ρ	NOUN
cana-2729	21	7	)	)	PUNCT
cana-2729	21	8	,	,	PUNCT
cana-2729	21	9	fbrkts	fbrkts	NOUN
cana-2729	21	10	,	,	PUNCT
cana-2729	21	11	fbrkcts	fbrkct	NOUN
cana-2729	21	12	,	,	PUNCT
cana-2729	21	13	fbrkhom	fbrkhom	ADJ
cana-2729	21	14	,	,	PUNCT
cana-2729	21	15	fbrktg	fbrktg	ADJ
cana-2729	21	16	.	.	PUNCT
cana-2729	22	1	1	1	X
cana-2729	22	2	.	.	X
cana-2729	22	3	introduction	introduction	NOUN
cana-2729	22	4	fuzzy	fuzzy	ADJ
cana-2729	22	5	set	set	NOUN
cana-2729	22	6	theory	theory	NOUN
cana-2729	22	7	,	,	PUNCT
cana-2729	22	8	introduced	introduce	VERB
cana-2729	22	9	by	by	ADP
cana-2729	22	10	zadeh	zadeh	PROPN
cana-2729	22	11	in	in	ADP
cana-2729	22	12	1965	1965	NUM
cana-2729	22	13	[	[	X
cana-2729	22	14	18	18	NUM
cana-2729	22	15	]	]	PUNCT
cana-2729	22	16	,	,	PUNCT
cana-2729	22	17	has	have	AUX
cana-2729	22	18	significantly	significantly	ADV
cana-2729	22	19	influenced	influence	VERB
cana-2729	22	20	various	various	ADJ
cana-2729	22	21	branches	branch	NOUN
cana-2729	22	22	of	of	ADP
cana-2729	22	23	mathematics	mathematic	NOUN
cana-2729	22	24	,	,	PUNCT
cana-2729	22	25	particularly	particularly	ADV
cana-2729	22	26	in	in	ADP
cana-2729	22	27	the	the	DET
cana-2729	22	28	study	study	NOUN
cana-2729	22	29	of	of	ADP
cana-2729	22	30	uncertainty	uncertainty	NOUN
cana-2729	22	31	and	and	CCONJ
cana-2729	22	32	imprecision	imprecision	NOUN
cana-2729	22	33	.	.	PUNCT
cana-2729	23	1	this	this	DET
cana-2729	23	2	theory	theory	NOUN
cana-2729	23	3	provides	provide	VERB
cana-2729	23	4	a	a	DET
cana-2729	23	5	natural	natural	ADJ
cana-2729	23	6	way	way	NOUN
cana-2729	23	7	of	of	ADP
cana-2729	23	8	extending	extend	VERB
cana-2729	23	9	classical	classical	ADJ
cana-2729	23	10	mathematical	mathematical	ADJ
cana-2729	23	11	structures	structure	NOUN
cana-2729	23	12	by	by	ADP
cana-2729	23	13	incorporating	incorporate	VERB
cana-2729	23	14	degrees	degree	NOUN
cana-2729	23	15	of	of	ADP
cana-2729	23	16	membership	membership	NOUN
cana-2729	23	17	,	,	PUNCT
cana-2729	23	18	offering	offer	VERB
cana-2729	23	19	a	a	DET
cana-2729	23	20	framework	framework	NOUN
cana-2729	23	21	for	for	ADP
cana-2729	23	22	dealing	deal	VERB
cana-2729	23	23	with	with	ADP
cana-2729	23	24	ambiguous	ambiguous	ADJ
cana-2729	23	25	data	datum	NOUN
cana-2729	23	26	.	.	PUNCT
cana-2729	24	1	in	in	ADP
cana-2729	24	2	particular	particular	ADJ
cana-2729	24	3	,	,	PUNCT
cana-2729	24	4	the	the	DET
cana-2729	24	5	development	development	NOUN
cana-2729	24	6	of	of	ADP
cana-2729	24	7	fuzzy	fuzzy	ADJ
cana-2729	24	8	algebraic	algebraic	ADJ
cana-2729	24	9	mailto:kousalyavaasan@gmail.com	mailto:kousalyavaasan@gmail.com	X
cana-2729	24	10	mailto:id%3amala.kkcas@gmail.com	mailto:id%3amala.kkcas@gmail.com	X
cana-2729	25	1	mailto:eldhokj@marymathacollege.ac.in	mailto:eldhokj@marymathacollege.ac.in	NOUN
cana-2729	25	2	mailto:id%3aswapnangit2021@gmail.com	mailto:id%3aswapnangit2021@gmail.com	X
cana-2729	25	3	https://orcid.org/0000-0003-2006-2367	https://orcid.org/0000-0003-2006-2367	PROPN
cana-2729	25	4	mailto:thamizhsudar@avit.ac.in	mailto:thamizhsudar@avit.ac.in	PROPN
cana-2729	25	5	mailto:kungum99522@gmail.com	mailto:kungum99522@gmail.com	PROPN
cana-2729	25	6	mailto:id%3ajenitha.g@ametuniv.ac.in	mailto:id%3ajenitha.g@ametuniv.ac.in	PROPN
cana-2729	25	7	communications	communication	NOUN
cana-2729	25	8	on	on	ADP
cana-2729	25	9	applied	apply	VERB
cana-2729	25	10	nonlinear	nonlinear	ADJ
cana-2729	25	11	analysis	analysis	NOUN
cana-2729	25	12	issn	issn	NOUN
cana-2729	25	13	:	:	PUNCT
cana-2729	25	14	1074	1074	NUM
cana-2729	25	15	-	-	PUNCT
cana-2729	25	16	133x	133x	NUM
cana-2729	25	17	vol	vol	NOUN
cana-2729	25	18	32	32	NUM
cana-2729	25	19	no	no	NOUN
cana-2729	25	20	.	.	PUNCT
cana-2729	26	1	3s	3s	NUM
cana-2729	26	2	(	(	PUNCT
cana-2729	26	3	2025	2025	NUM
cana-2729	26	4	)	)	PUNCT
cana-2729	26	5	712	712	NUM
cana-2729	26	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-2729	26	7	structures	structure	NOUN
cana-2729	26	8	has	have	AUX
cana-2729	26	9	emerged	emerge	VERB
cana-2729	26	10	as	as	ADP
cana-2729	26	11	an	an	DET
cana-2729	26	12	important	important	ADJ
cana-2729	26	13	area	area	NOUN
cana-2729	26	14	of	of	ADP
cana-2729	26	15	study	study	NOUN
cana-2729	26	16	,	,	PUNCT
cana-2729	26	17	with	with	ADP
cana-2729	26	18	applications	application	NOUN
cana-2729	26	19	in	in	ADP
cana-2729	26	20	many	many	ADJ
cana-2729	26	21	fields	field	NOUN
cana-2729	26	22	,	,	PUNCT
cana-2729	26	23	including	include	VERB
cana-2729	26	24	decision	decision	NOUN
cana-2729	26	25	theory	theory	NOUN
cana-2729	26	26	,	,	PUNCT
cana-2729	26	27	artificial	artificial	ADJ
cana-2729	26	28	intelligence	intelligence	NOUN
cana-2729	26	29	,	,	PUNCT
cana-2729	26	30	and	and	CCONJ
cana-2729	26	31	control	control	NOUN
cana-2729	26	32	systems	system	NOUN
cana-2729	26	33	.	.	PUNCT
cana-2729	27	1	the	the	DET
cana-2729	27	2	introduction	introduction	NOUN
cana-2729	27	3	of	of	ADP
cana-2729	27	4	fuzzy	fuzzy	ADJ
cana-2729	27	5	groups	group	NOUN
cana-2729	27	6	by	by	ADP
cana-2729	27	7	rosenfeld	rosenfeld	PROPN
cana-2729	27	8	[	[	X
cana-2729	27	9	13	13	NUM
cana-2729	27	10	]	]	PUNCT
cana-2729	27	11	laid	lay	VERB
cana-2729	27	12	the	the	DET
cana-2729	27	13	groundwork	groundwork	NOUN
cana-2729	27	14	for	for	ADP
cana-2729	27	15	exploring	explore	VERB
cana-2729	27	16	topological	topological	ADJ
cana-2729	27	17	properties	property	NOUN
cana-2729	27	18	in	in	ADP
cana-2729	27	19	fuzzy	fuzzy	ADJ
cana-2729	27	20	contexts	contexts	NOUN
cana-2729	27	21	,	,	PUNCT
cana-2729	27	22	leading	lead	VERB
cana-2729	27	23	to	to	ADP
cana-2729	27	24	the	the	DET
cana-2729	27	25	study	study	NOUN
cana-2729	27	26	of	of	ADP
cana-2729	27	27	fuzzy	fuzzy	ADJ
cana-2729	27	28	topological	topological	ADJ
cana-2729	27	29	groups	group	NOUN
cana-2729	27	30	[	[	X
cana-2729	27	31	2	2	NUM
cana-2729	27	32	,	,	PUNCT
cana-2729	27	33	9	9	NUM
cana-2729	27	34	,	,	PUNCT
cana-2729	27	35	17	17	NUM
cana-2729	27	36	]	]	PUNCT
cana-2729	27	37	.	.	PUNCT
cana-2729	28	1	foster	foster	PROPN
cana-2729	28	2	’s	’s	PART
cana-2729	28	3	seminal	seminal	ADJ
cana-2729	28	4	work	work	NOUN
cana-2729	28	5	in	in	ADP
cana-2729	28	6	1979	1979	NUM
cana-2729	28	7	[	[	X
cana-2729	28	8	2	2	NUM
cana-2729	28	9	]	]	PUNCT
cana-2729	28	10	formalized	formalize	VERB
cana-2729	28	11	the	the	DET
cana-2729	28	12	concept	concept	NOUN
cana-2729	28	13	of	of	ADP
cana-2729	28	14	fuzzy	fuzzy	ADJ
cana-2729	28	15	topological	topological	ADJ
cana-2729	28	16	groups	group	NOUN
cana-2729	28	17	,	,	PUNCT
cana-2729	28	18	integrating	integrate	VERB
cana-2729	28	19	fuzzy	fuzzy	ADJ
cana-2729	28	20	set	set	NOUN
cana-2729	28	21	theory	theory	NOUN
cana-2729	28	22	with	with	ADP
cana-2729	28	23	group	group	NOUN
cana-2729	28	24	and	and	CCONJ
cana-2729	28	25	topological	topological	ADJ
cana-2729	28	26	properties	property	NOUN
cana-2729	28	27	.	.	PUNCT
cana-2729	29	1	this	this	DET
cana-2729	29	2	approach	approach	NOUN
cana-2729	29	3	aimed	aim	VERB
cana-2729	29	4	to	to	PART
cana-2729	29	5	extend	extend	VERB
cana-2729	29	6	the	the	DET
cana-2729	29	7	classical	classical	ADJ
cana-2729	29	8	understanding	understanding	NOUN
cana-2729	29	9	of	of	ADP
cana-2729	29	10	topological	topological	ADJ
cana-2729	29	11	groups	group	NOUN
cana-2729	29	12	to	to	PART
cana-2729	29	13	accommodate	accommodate	VERB
cana-2729	29	14	fuzzy	fuzzy	ADJ
cana-2729	29	15	sets	set	NOUN
cana-2729	29	16	,	,	PUNCT
cana-2729	29	17	where	where	SCONJ
cana-2729	29	18	the	the	DET
cana-2729	29	19	operations	operation	NOUN
cana-2729	29	20	of	of	ADP
cana-2729	29	21	a	a	DET
cana-2729	29	22	group	group	NOUN
cana-2729	29	23	and	and	CCONJ
cana-2729	29	24	topological	topological	ADJ
cana-2729	29	25	continuity	continuity	NOUN
cana-2729	29	26	coexist	coexist	NOUN
cana-2729	29	27	under	under	ADP
cana-2729	29	28	fuzziness	fuzziness	NOUN
cana-2729	29	29	.	.	PUNCT
cana-2729	30	1	following	follow	VERB
cana-2729	30	2	foster	foster	PROPN
cana-2729	30	3	,	,	PUNCT
cana-2729	30	4	ma	ma	PROPN
cana-2729	30	5	and	and	CCONJ
cana-2729	30	6	yu	yu	PRON
cana-2729	30	7	[	[	X
cana-2729	30	8	9	9	NUM
cana-2729	30	9	]	]	PUNCT
cana-2729	30	10	further	far	ADV
cana-2729	30	11	advanced	advance	VERB
cana-2729	30	12	the	the	DET
cana-2729	30	13	theory	theory	NOUN
cana-2729	30	14	of	of	ADP
cana-2729	30	15	fuzzy	fuzzy	ADJ
cana-2729	30	16	topological	topological	ADJ
cana-2729	30	17	groups	group	NOUN
cana-2729	30	18	,	,	PUNCT
cana-2729	30	19	refining	refine	VERB
cana-2729	30	20	the	the	DET
cana-2729	30	21	underlying	underlie	VERB
cana-2729	30	22	structures	structure	NOUN
cana-2729	30	23	and	and	CCONJ
cana-2729	30	24	exploring	explore	VERB
cana-2729	30	25	their	their	PRON
cana-2729	30	26	applications	application	NOUN
cana-2729	30	27	.	.	PUNCT
cana-2729	31	1	fuzzy	fuzzy	ADJ
cana-2729	31	2	topological	topological	ADJ
cana-2729	31	3	groups	group	NOUN
cana-2729	31	4	have	have	AUX
cana-2729	31	5	since	since	SCONJ
cana-2729	31	6	become	become	VERB
cana-2729	31	7	a	a	DET
cana-2729	31	8	vibrant	vibrant	ADJ
cana-2729	31	9	research	research	NOUN
cana-2729	31	10	field	field	NOUN
cana-2729	31	11	,	,	PUNCT
cana-2729	31	12	leading	lead	VERB
cana-2729	31	13	to	to	ADP
cana-2729	31	14	numerous	numerous	ADJ
cana-2729	31	15	extensions	extension	NOUN
cana-2729	31	16	and	and	CCONJ
cana-2729	31	17	applications	application	NOUN
cana-2729	31	18	,	,	PUNCT
cana-2729	31	19	such	such	ADJ
cana-2729	31	20	as	as	ADP
cana-2729	31	21	fuzzy	fuzzy	ADJ
cana-2729	31	22	actions	action	NOUN
cana-2729	31	23	[	[	X
cana-2729	31	24	1	1	NUM
cana-2729	31	25	]	]	PUNCT
cana-2729	31	26	and	and	CCONJ
cana-2729	31	27	fuzzy	fuzzy	ADJ
cana-2729	31	28	s	s	NOUN
cana-2729	31	29	-	-	PUNCT
cana-2729	31	30	acts	act	VERB
cana-2729	31	31	[	[	X
cana-2729	31	32	3	3	NUM
cana-2729	31	33	]	]	PUNCT
cana-2729	31	34	.	.	PUNCT
cana-2729	32	1	one	one	NUM
cana-2729	32	2	of	of	ADP
cana-2729	32	3	the	the	DET
cana-2729	32	4	recent	recent	ADJ
cana-2729	32	5	advances	advance	NOUN
cana-2729	32	6	in	in	ADP
cana-2729	32	7	the	the	DET
cana-2729	32	8	study	study	NOUN
cana-2729	32	9	of	of	ADP
cana-2729	32	10	algebraic	algebraic	ADJ
cana-2729	32	11	structures	structure	NOUN
cana-2729	32	12	is	be	AUX
cana-2729	32	13	the	the	DET
cana-2729	32	14	introduction	introduction	NOUN
cana-2729	32	15	of	of	ADP
cana-2729	32	16	brk	brk	PROPN
cana-2729	32	17	-	-	PUNCT
cana-2729	32	18	algebras	algebras	PROPN
cana-2729	32	19	by	by	ADP
cana-2729	32	20	ravi	ravi	PROPN
cana-2729	32	21	kumar	kumar	PROPN
cana-2729	32	22	bandaru	bandaru	PROPN
cana-2729	32	23	in	in	ADP
cana-2729	32	24	2012	2012	NUM
cana-2729	32	25	[	[	X
cana-2729	32	26	12	12	NUM
cana-2729	32	27	]	]	PUNCT
cana-2729	32	28	,	,	PUNCT
cana-2729	32	29	which	which	PRON
cana-2729	32	30	generalized	generalize	VERB
cana-2729	32	31	several	several	ADJ
cana-2729	32	32	algebraic	algebraic	ADJ
cana-2729	32	33	concepts	concept	NOUN
cana-2729	32	34	by	by	ADP
cana-2729	32	35	focusing	focus	VERB
cana-2729	32	36	on	on	ADP
cana-2729	32	37	structures	structure	NOUN
cana-2729	32	38	related	relate	VERB
cana-2729	32	39	to	to	ADP
cana-2729	32	40	brk	brk	PROPN
cana-2729	32	41	-	-	PUNCT
cana-2729	32	42	algebras	algebras	PROPN
cana-2729	32	43	.	.	PUNCT
cana-2729	33	1	these	these	DET
cana-2729	33	2	algebras	algebra	NOUN
cana-2729	33	3	have	have	AUX
cana-2729	33	4	been	be	AUX
cana-2729	33	5	connected	connect	VERB
cana-2729	33	6	to	to	ADP
cana-2729	33	7	various	various	ADJ
cana-2729	33	8	topological	topological	ADJ
cana-2729	33	9	and	and	CCONJ
cana-2729	33	10	fuzzy	fuzzy	ADJ
cana-2729	33	11	structures	structure	NOUN
cana-2729	33	12	,	,	PUNCT
cana-2729	33	13	creating	create	VERB
cana-2729	33	14	a	a	DET
cana-2729	33	15	rich	rich	ADJ
cana-2729	33	16	interplay	interplay	NOUN
cana-2729	33	17	between	between	ADP
cana-2729	33	18	algebra	algebra	NOUN
cana-2729	33	19	and	and	CCONJ
cana-2729	33	20	topology	topology	NOUN
cana-2729	33	21	.	.	PUNCT
cana-2729	34	1	the	the	DET
cana-2729	34	2	study	study	NOUN
cana-2729	34	3	of	of	ADP
cana-2729	34	4	brk	brk	PROPN
cana-2729	34	5	-	-	PUNCT
cana-2729	34	6	algebras	algebras	PROPN
cana-2729	34	7	was	be	AUX
cana-2729	34	8	further	far	ADV
cana-2729	34	9	expanded	expand	VERB
cana-2729	34	10	by	by	ADP
cana-2729	34	11	sivakumar	sivakumar	PROPN
cana-2729	34	12	et	et	PROPN
cana-2729	34	13	al	al	PROPN
cana-2729	34	14	.	.	PUNCT
cana-2729	35	1	[	[	X
cana-2729	35	2	14	14	NUM
cana-2729	35	3	]	]	PUNCT
cana-2729	35	4	,	,	PUNCT
cana-2729	35	5	who	who	PRON
cana-2729	35	6	explored	explore	VERB
cana-2729	35	7	topological	topological	ADJ
cana-2729	35	8	structures	structure	NOUN
cana-2729	35	9	in	in	ADP
cana-2729	35	10	brk	brk	PROPN
cana-2729	35	11	-	-	PUNCT
cana-2729	35	12	algebras	algebras	PROPN
cana-2729	35	13	and	and	CCONJ
cana-2729	35	14	later	later	ADV
cana-2729	35	15	extended	extend	VERB
cana-2729	35	16	this	this	DET
cana-2729	35	17	analysis	analysis	NOUN
cana-2729	35	18	to	to	ADP
cana-2729	35	19	fuzzy	fuzzy	ADJ
cana-2729	35	20	topological	topological	PROPN
cana-2729	35	21	brk	brk	PROPN
cana-2729	35	22	-	-	PUNCT
cana-2729	35	23	subalgebras	subalgebras	PROPN
cana-2729	36	1	[	[	X
cana-2729	36	2	15	15	NUM
cana-2729	36	3	]	]	PUNCT
cana-2729	36	4	.	.	PUNCT
cana-2729	37	1	this	this	PRON
cana-2729	37	2	has	have	AUX
cana-2729	37	3	opened	open	VERB
cana-2729	37	4	new	new	ADJ
cana-2729	37	5	pathways	pathway	NOUN
cana-2729	37	6	for	for	ADP
cana-2729	37	7	examining	examine	VERB
cana-2729	37	8	fuzzy	fuzzy	ADJ
cana-2729	37	9	topological	topological	ADJ
cana-2729	37	10	groups	group	NOUN
cana-2729	37	11	within	within	ADP
cana-2729	37	12	the	the	DET
cana-2729	37	13	context	context	NOUN
cana-2729	37	14	of	of	ADP
cana-2729	37	15	brk	brk	PROPN
cana-2729	37	16	-	-	PUNCT
cana-2729	37	17	algebras	algebras	PROPN
cana-2729	37	18	,	,	PUNCT
cana-2729	37	19	contributing	contribute	VERB
cana-2729	37	20	to	to	ADP
cana-2729	37	21	the	the	DET
cana-2729	37	22	broader	broad	ADJ
cana-2729	37	23	understanding	understanding	NOUN
cana-2729	37	24	of	of	ADP
cana-2729	37	25	both	both	CCONJ
cana-2729	37	26	fuzzy	fuzzy	ADJ
cana-2729	37	27	and	and	CCONJ
cana-2729	37	28	topological	topological	ADJ
cana-2729	37	29	group	group	NOUN
cana-2729	37	30	theory	theory	NOUN
cana-2729	37	31	.	.	PUNCT
cana-2729	38	1	this	this	DET
cana-2729	38	2	paper	paper	NOUN
cana-2729	38	3	builds	build	VERB
cana-2729	38	4	upon	upon	SCONJ
cana-2729	38	5	this	this	DET
cana-2729	38	6	foundation	foundation	NOUN
cana-2729	38	7	,	,	PUNCT
cana-2729	38	8	focusing	focus	VERB
cana-2729	38	9	on	on	ADP
cana-2729	38	10	fuzzy	fuzzy	ADJ
cana-2729	38	11	brk	brk	PROPN
cana-2729	38	12	topological	topological	ADJ
cana-2729	38	13	groups	group	NOUN
cana-2729	38	14	,	,	PUNCT
cana-2729	38	15	an	an	DET
cana-2729	38	16	emerging	emerge	VERB
cana-2729	38	17	area	area	NOUN
cana-2729	38	18	that	that	PRON
cana-2729	38	19	unites	unite	VERB
cana-2729	38	20	fuzzy	fuzzy	ADJ
cana-2729	38	21	set	set	NOUN
cana-2729	38	22	theory	theory	NOUN
cana-2729	38	23	,	,	PUNCT
cana-2729	38	24	group	group	NOUN
cana-2729	38	25	theory	theory	NOUN
cana-2729	38	26	,	,	PUNCT
cana-2729	38	27	and	and	CCONJ
cana-2729	38	28	brk	brk	PROPN
cana-2729	38	29	-	-	PUNCT
cana-2729	38	30	algebraic	algebraic	PROPN
cana-2729	38	31	structures	structure	NOUN
cana-2729	38	32	.	.	PUNCT
cana-2729	39	1	by	by	ADP
cana-2729	39	2	examining	examine	VERB
cana-2729	39	3	their	their	PRON
cana-2729	39	4	theoretical	theoretical	ADJ
cana-2729	39	5	properties	property	NOUN
cana-2729	39	6	and	and	CCONJ
cana-2729	39	7	potential	potential	ADJ
cana-2729	39	8	applications	application	NOUN
cana-2729	39	9	,	,	PUNCT
cana-2729	39	10	this	this	DET
cana-2729	39	11	study	study	NOUN
cana-2729	39	12	aims	aim	VERB
cana-2729	39	13	to	to	PART
cana-2729	39	14	contribute	contribute	VERB
cana-2729	39	15	to	to	ADP
cana-2729	39	16	the	the	DET
cana-2729	39	17	growing	grow	VERB
cana-2729	39	18	body	body	NOUN
cana-2729	39	19	of	of	ADP
cana-2729	39	20	knowledge	knowledge	NOUN
cana-2729	39	21	in	in	ADP
cana-2729	39	22	fuzzy	fuzzy	ADJ
cana-2729	39	23	topology	topology	NOUN
cana-2729	39	24	and	and	CCONJ
cana-2729	39	25	algebra	algebra	NOUN
cana-2729	39	26	.	.	PUNCT
cana-2729	40	1	2	2	X
cana-2729	40	2	.	.	X
cana-2729	40	3	preliminaries	preliminary	NOUN
cana-2729	40	4	:	:	PUNCT
cana-2729	40	5	definition	definition	NOUN
cana-2729	40	6	2.1	2.1	NUM
cana-2729	40	7	:	:	PUNCT
cana-2729	41	1	[	[	X
cana-2729	41	2	12	12	NUM
cana-2729	41	3	]	]	PUNCT
cana-2729	41	4	the	the	DET
cana-2729	41	5	fuzzy	fuzzy	ADJ
cana-2729	41	6	brk	brk	PROPN
cana-2729	41	7	-closure	-closure	PROPN
cana-2729	41	8	and	and	CCONJ
cana-2729	41	9	fuzzy	fuzzy	ADJ
cana-2729	41	10	brk	brk	PROPN
cana-2729	41	11	-interior	-interior	PROPN
cana-2729	41	12	of	of	ADP
cana-2729	41	13	ρ	ρ	PROPN
cana-2729	41	14	is	be	AUX
cana-2729	41	15	denoted	denote	VERB
cana-2729	41	16	by	by	ADP
cana-2729	41	17	brkcl(ρ	brkcl(ρ	PROPN
cana-2729	41	18	)	)	PUNCT
cana-2729	41	19	and	and	CCONJ
cana-2729	41	20	brkint(ρ	brkint(ρ	NOUN
cana-2729	41	21	)	)	PUNCT
cana-2729	41	22	are	be	AUX
cana-2729	41	23	given	give	VERB
cana-2729	41	24	by	by	ADP
cana-2729	41	25	brkcl(ρ	brkcl(ρ	PROPN
cana-2729	41	26	)	)	PUNCT
cana-2729	41	27	=	=	SYM
cana-2729	41	28	∧{λ	∧{λ	PROPN
cana-2729	41	29	:	:	PUNCT
cana-2729	41	30	λ	λ	NOUN
cana-2729	41	31	is	be	AUX
cana-2729	41	32	a	a	DET
cana-2729	41	33	fbrkcs	fbrkcs	NOUN
cana-2729	41	34	&	&	CCONJ
cana-2729	41	35	ρ	ρ	NOUN
cana-2729	41	36	≤	≤	PROPN
cana-2729	41	37	λ	λ	NOUN
cana-2729	41	38	}	}	PUNCT
cana-2729	41	39	.	.	PUNCT
cana-2729	42	1	brkint(ρ	brkint(ρ	NOUN
cana-2729	42	2	)	)	PUNCT
cana-2729	43	1	=	=	SYM
cana-2729	43	2	∨{λ	∨{λ	PROPN
cana-2729	43	3	:	:	PUNCT
cana-2729	43	4	λ	λ	NOUN
cana-2729	43	5	is	be	AUX
cana-2729	43	6	a	a	DET
cana-2729	43	7	fbrkos	fbrkos	PROPN
cana-2729	43	8	&	&	CCONJ
cana-2729	43	9	ρ	ρ	PROPN
cana-2729	43	10	≥	≥	PROPN
cana-2729	43	11	λ	λ	NOUN
cana-2729	43	12	}	}	PUNCT
cana-2729	43	13	.	.	PUNCT
cana-2729	44	1	definition	definition	NOUN
cana-2729	44	2	2.2	2.2	NUM
cana-2729	44	3	:	:	PUNCT
cana-2729	45	1	[	[	X
cana-2729	45	2	12	12	NUM
cana-2729	45	3	]	]	PUNCT
cana-2729	45	4	a	a	DET
cana-2729	45	5	brk	brk	PROPN
cana-2729	45	6	-	-	PUNCT
cana-2729	45	7	algebra	algebra	PROPN
cana-2729	45	8	(	(	PUNCT
cana-2729	45	9	briefly	briefly	ADV
cana-2729	45	10	,	,	PUNCT
cana-2729	45	11	brk	brk	PROPN
cana-2729	45	12	alg	alg	PROPN
cana-2729	45	13	)	)	PUNCT
cana-2729	45	14	(	(	PUNCT
cana-2729	45	15	x	x	X
cana-2729	45	16	,	,	PUNCT
cana-2729	45	17	⋆	⋆	INTJ
cana-2729	45	18	,	,	PUNCT
cana-2729	45	19	0	0	NUM
cana-2729	45	20	)	)	PUNCT
cana-2729	45	21	is	be	AUX
cana-2729	45	22	a	a	DET
cana-2729	45	23	non	non	ADJ
cana-2729	45	24	-	-	ADJ
cana-2729	45	25	empty	empty	ADJ
cana-2729	45	26	set	set	NOUN
cana-2729	45	27	x	x	PUNCT
cana-2729	45	28	with	with	ADP
cana-2729	45	29	a	a	DET
cana-2729	45	30	constant	constant	ADJ
cana-2729	45	31	0	0	NUM
cana-2729	45	32	and	and	CCONJ
cana-2729	45	33	a	a	DET
cana-2729	45	34	binary	binary	ADJ
cana-2729	45	35	operation	operation	NOUN
cana-2729	45	36	⋆	⋆	X
cana-2729	45	37	satisfying	satisfying	NOUN
cana-2729	45	38	(	(	PUNCT
cana-2729	45	39	brk1	brk1	PROPN
cana-2729	45	40	)	)	PUNCT
cana-2729	45	41	e	e	NOUN
cana-2729	45	42	⋆	⋆	VERB
cana-2729	45	43	0	0	PUNCT
cana-2729	45	44	=	=	SYM
cana-2729	45	45	e	e	NOUN
cana-2729	45	46	,	,	PUNCT
cana-2729	45	47	(	(	PUNCT
cana-2729	45	48	brk2	brk2	PROPN
cana-2729	45	49	)	)	PUNCT
cana-2729	46	1	(	(	PUNCT
cana-2729	46	2	e	e	NOUN
cana-2729	46	3	⋆	⋆	NOUN
cana-2729	46	4	f	f	NOUN
cana-2729	46	5	)	)	PUNCT
cana-2729	46	6	⋆	⋆	X
cana-2729	46	7	e	e	X
cana-2729	46	8	=	=	SYM
cana-2729	46	9	0	0	NUM
cana-2729	46	10	⋆	⋆	PUNCT
cana-2729	46	11	f	f	PROPN
cana-2729	46	12	for	for	ADP
cana-2729	46	13	any	any	DET
cana-2729	46	14	e	e	NOUN
cana-2729	46	15	,	,	PUNCT
cana-2729	46	16	f	f	PROPN
cana-2729	46	17	∈	∈	PROPN
cana-2729	46	18	x.	x.	NOUN
cana-2729	46	19	a	a	DET
cana-2729	46	20	partially	partially	ADV
cana-2729	46	21	ordered	order	VERB
cana-2729	46	22	relation	relation	NOUN
cana-2729	46	23	≤	≤	NOUN
cana-2729	46	24	can	can	AUX
cana-2729	46	25	be	be	AUX
cana-2729	46	26	defined	define	VERB
cana-2729	46	27	by	by	ADP
cana-2729	46	28	e	e	PROPN
cana-2729	46	29	≤	≤	PROPN
cana-2729	46	30	f	f	PROPN
cana-2729	46	31	iff	iff	PROPN
cana-2729	46	32	e	e	PROPN
cana-2729	47	1	⋆	⋆	NOUN
cana-2729	47	2	f	f	PROPN
cana-2729	47	3	=	=	SYM
cana-2729	47	4	0	0	PROPN
cana-2729	47	5	.	.	PUNCT
cana-2729	48	1	definition	definition	NOUN
cana-2729	48	2	2.3	2.3	NUM
cana-2729	48	3	[	[	X
cana-2729	48	4	18	18	NUM
cana-2729	48	5	]	]	PUNCT
cana-2729	48	6	let	let	VERB
cana-2729	48	7	x	x	PRON
cana-2729	48	8	be	be	AUX
cana-2729	48	9	a	a	DET
cana-2729	48	10	set	set	NOUN
cana-2729	48	11	.	.	PUNCT
cana-2729	49	1	a	a	DET
cana-2729	49	2	fuzzy	fuzzy	ADJ
cana-2729	49	3	set	set	VERB
cana-2729	49	4	µ	µ	NOUN
cana-2729	49	5	in	in	ADP
cana-2729	49	6	x	x	VERB
cana-2729	49	7	is	be	AUX
cana-2729	49	8	a	a	DET
cana-2729	49	9	function	function	NOUN
cana-2729	49	10	µ	µ	NOUN
cana-2729	49	11	:	:	PUNCT
cana-2729	49	12	x	x	SYM
cana-2729	49	13	→	→	SYM
cana-2729	49	14	[	[	X
cana-2729	49	15	0	0	NUM
cana-2729	49	16	,	,	PUNCT
cana-2729	49	17	1	1	NUM
cana-2729	49	18	]	]	PUNCT
cana-2729	49	19	.	.	PUNCT
cana-2729	50	1	definition	definition	NOUN
cana-2729	50	2	2.4	2.4	NUM
cana-2729	50	3	[	[	X
cana-2729	50	4	2	2	NUM
cana-2729	50	5	]	]	PUNCT
cana-2729	50	6	a	a	DET
cana-2729	50	7	fuzzy	fuzzy	ADJ
cana-2729	50	8	topology	topology	NOUN
cana-2729	50	9	(	(	PUNCT
cana-2729	50	10	briefly	briefly	ADV
cana-2729	50	11	,	,	PUNCT
cana-2729	50	12	f	f	PROPN
cana-2729	50	13	t	t	PROPN
cana-2729	50	14	)	)	PUNCT
cana-2729	50	15	on	on	ADP
cana-2729	50	16	a	a	DET
cana-2729	50	17	set	set	NOUN
cana-2729	50	18	x	x	PUNCT
cana-2729	50	19	is	be	AUX
cana-2729	50	20	a	a	DET
cana-2729	50	21	family	family	NOUN
cana-2729	50	22	τ	τ	X
cana-2729	50	23	of	of	ADP
cana-2729	50	24	fuzzy	fuzzy	ADJ
cana-2729	50	25	subsets	subset	NOUN
cana-2729	50	26	in	in	ADP
cana-2729	50	27	x	x	PUNCT
cana-2729	50	28	which	which	DET
cana-2729	50	29	satisfies	satisfy	VERB
cana-2729	50	30	(	(	PUNCT
cana-2729	50	31	i	i	NOUN
cana-2729	50	32	)	)	PUNCT
cana-2729	50	33	for	for	ADP
cana-2729	50	34	all	all	DET
cana-2729	50	35	a	a	DET
cana-2729	50	36	∈	∈	NOUN
cana-2729	51	1	[	[	X
cana-2729	51	2	0	0	NUM
cana-2729	51	3	,	,	PUNCT
cana-2729	51	4	1	1	NUM
cana-2729	51	5	]	]	PUNCT
cana-2729	51	6	,	,	PUNCT
cana-2729	51	7	ka∈τ	ka∈τ	X
cana-2729	51	8	,	,	PUNCT
cana-2729	51	9	where	where	SCONJ
cana-2729	51	10	ka	ka	PROPN
cana-2729	51	11	have	have	VERB
cana-2729	51	12	constant	constant	ADJ
cana-2729	51	13	membership	membership	NOUN
cana-2729	51	14	functions	function	NOUN
cana-2729	51	15	with	with	ADP
cana-2729	51	16	the	the	DET
cana-2729	51	17	value	value	NOUN
cana-2729	51	18	a	a	DET
cana-2729	51	19	,	,	PUNCT
cana-2729	51	20	(	(	PUNCT
cana-2729	51	21	ii	ii	NOUN
cana-2729	51	22	)	)	PUNCT
cana-2729	51	23	if	if	SCONJ
cana-2729	51	24	e	e	X
cana-2729	51	25	,	,	PUNCT
cana-2729	51	26	f	f	PROPN
cana-2729	51	27	∈	∈	PROPN
cana-2729	51	28	τ	τ	X
cana-2729	51	29	,	,	PUNCT
cana-2729	51	30	then	then	ADV
cana-2729	51	31	e	e	PROPN
cana-2729	51	32	∩	∩	PROPN
cana-2729	51	33	f	f	PROPN
cana-2729	51	34	∈	∈	PROPN
cana-2729	51	35	τ	τ	X
cana-2729	51	36	,	,	PUNCT
cana-2729	51	37	(	(	PUNCT
cana-2729	51	38	iii	iii	X
cana-2729	51	39	)	)	PUNCT
cana-2729	51	40	if	if	SCONJ
cana-2729	51	41	ea∈	ea∈	VERB
cana-2729	51	42	τ	τ	PROPN
cana-2729	51	43	∀	∀	X
cana-2729	51	44	a	a	DET
cana-2729	51	45	∈	∈	PROPN
cana-2729	51	46	a	a	PRON
cana-2729	51	47	,	,	PUNCT
cana-2729	51	48	then	then	ADV
cana-2729	51	49	∪a∈aea∈	∪a∈aea∈	PROPN
cana-2729	51	50	τ	τ	PROPN
cana-2729	51	51	.	.	PUNCT
cana-2729	52	1	the	the	DET
cana-2729	52	2	pair	pair	NOUN
cana-2729	52	3	(	(	PUNCT
cana-2729	52	4	x	x	X
cana-2729	52	5	,	,	PUNCT
cana-2729	52	6	τ	τ	PROPN
cana-2729	52	7	)	)	PUNCT
cana-2729	52	8	is	be	AUX
cana-2729	52	9	called	call	VERB
cana-2729	52	10	a	a	DET
cana-2729	52	11	fuzzy	fuzzy	ADJ
cana-2729	52	12	topological	topological	ADJ
cana-2729	52	13	space	space	NOUN
cana-2729	52	14	(	(	PUNCT
cana-2729	52	15	briefly	briefly	ADV
cana-2729	52	16	,	,	PUNCT
cana-2729	52	17	f	f	PROPN
cana-2729	52	18	ts	ts	NOUN
cana-2729	52	19	)	)	PUNCT
cana-2729	52	20	and	and	CCONJ
cana-2729	52	21	members	member	NOUN
cana-2729	52	22	of	of	ADP
cana-2729	52	23	τ	τ	PROPN
cana-2729	52	24	are	be	AUX
cana-2729	52	25	open	open	ADJ
cana-2729	52	26	fuzzy	fuzzy	ADJ
cana-2729	52	27	subsets	subset	NOUN
cana-2729	52	28	.	.	PUNCT
cana-2729	53	1	definition	definition	NOUN
cana-2729	53	2	2.5	2.5	NUM
cana-2729	54	1	[	[	X
cana-2729	54	2	15	15	NUM
cana-2729	54	3	]	]	X
cana-2729	54	4	the	the	DET
cana-2729	54	5	pair	pair	NOUN
cana-2729	54	6	(	(	PUNCT
cana-2729	54	7	x	x	X
cana-2729	54	8	,	,	PUNCT
cana-2729	54	9	τ	τ	PROPN
cana-2729	54	10	)	)	PUNCT
cana-2729	54	11	is	be	AUX
cana-2729	54	12	called	call	VERB
cana-2729	54	13	a	a	DET
cana-2729	54	14	f	f	NOUN
cana-2729	54	15	ts	ts	NOUN
cana-2729	54	16	,	,	PUNCT
cana-2729	54	17	then	then	ADV
cana-2729	54	18	it	it	PRON
cana-2729	54	19	satisfies	satisfy	VERB
cana-2729	54	20	a	a	DET
cana-2729	54	21	brk	brk	PROPN
cana-2729	54	22	alg	alg	PROPN
cana-2729	54	23	properties	property	NOUN
cana-2729	54	24	in	in	ADP
cana-2729	54	25	(	(	PUNCT
cana-2729	54	26	x	x	NOUN
cana-2729	54	27	,	,	PUNCT
cana-2729	54	28	⋆	⋆	INTJ
cana-2729	54	29	,	,	PUNCT
cana-2729	54	30	0	0	NUM
cana-2729	54	31	,	,	PUNCT
cana-2729	54	32	τ	τ	PROPN
cana-2729	54	33	)	)	PUNCT
cana-2729	54	34	it	it	PRON
cana-2729	54	35	is	be	AUX
cana-2729	54	36	called	call	VERB
cana-2729	54	37	a	a	DET
cana-2729	54	38	fuzzy	fuzzy	ADJ
cana-2729	54	39	brk	brk	PROPN
cana-2729	54	40	topological	topological	PROPN
cana-2729	54	41	spaces	space	NOUN
cana-2729	54	42	(	(	PUNCT
cana-2729	54	43	briefly	briefly	ADV
cana-2729	54	44	,	,	PUNCT
cana-2729	54	45	fbrkts	fbrkts	NOUN
cana-2729	54	46	)	)	PUNCT
cana-2729	54	47	and	and	CCONJ
cana-2729	54	48	members	member	NOUN
cana-2729	54	49	of	of	ADP
cana-2729	54	50	τ	τ	PROPN
cana-2729	54	51	are	be	AUX
cana-2729	54	52	brk	brk	PROPN
cana-2729	54	53	-	-	PUNCT
cana-2729	54	54	open	open	ADJ
cana-2729	54	55	fuzzy	fuzzy	ADJ
cana-2729	54	56	subsets	subset	NOUN
cana-2729	54	57	and	and	CCONJ
cana-2729	54	58	complement	complement	NOUN
cana-2729	54	59	of	of	ADP
cana-2729	54	60	a	a	DET
cana-2729	54	61	brk	brk	NOUN
cana-2729	54	62	-	-	PUNCT
cana-2729	54	63	open	open	ADJ
cana-2729	54	64	fuzzy	fuzzy	ADJ
cana-2729	54	65	subsets	subset	NOUN
cana-2729	54	66	are	be	AUX
cana-2729	54	67	brk	brk	PROPN
cana-2729	54	68	-	-	PUNCT
cana-2729	54	69	closed	close	VERB
cana-2729	54	70	fuzzy	fuzzy	ADJ
cana-2729	54	71	subsets	subset	NOUN
cana-2729	54	72	.	.	PUNCT
cana-2729	55	1	definition	definition	NOUN
cana-2729	55	2	2.6	2.6	NUM
cana-2729	55	3	[	[	X
cana-2729	55	4	2	2	NUM
cana-2729	55	5	]	]	PUNCT
cana-2729	55	6	a	a	DET
cana-2729	55	7	fuzzy	fuzzy	ADJ
cana-2729	55	8	topology	topology	NOUN
cana-2729	55	9	τ˜	τ˜	PRON
cana-2729	55	10	on	on	ADP
cana-2729	55	11	a	a	DET
cana-2729	55	12	group	group	NOUN
cana-2729	55	13	g	g	NOUN
cana-2729	55	14	is	be	AUX
cana-2729	55	15	said	say	VERB
cana-2729	55	16	to	to	PART
cana-2729	55	17	be	be	AUX
cana-2729	55	18	compatible	compatible	ADJ
cana-2729	55	19	if	if	SCONJ
cana-2729	55	20	the	the	DET
cana-2729	55	21	mapping	mapping	NOUN
cana-2729	55	22	g	g	NOUN
cana-2729	55	23	:	:	PUNCT
cana-2729	55	24	(	(	PUNCT
cana-2729	55	25	g	g	NOUN
cana-2729	55	26	×	×	PROPN
cana-2729	55	27	g	g	PROPN
cana-2729	55	28	,	,	PUNCT
cana-2729	55	29	τ˜	τ˜	PROPN
cana-2729	55	30	×	×	PROPN
cana-2729	55	31	τ˜	τ˜	PROPN
cana-2729	55	32	)	)	PUNCT
cana-2729	55	33	→	→	SYM
cana-2729	55	34	(	(	PUNCT
cana-2729	55	35	g	g	NOUN
cana-2729	55	36	,	,	PUNCT
cana-2729	55	37	τ˜	τ˜	PROPN
cana-2729	55	38	)	)	PUNCT
cana-2729	55	39	,	,	PUNCT
cana-2729	55	40	g(e	g(e	PROPN
cana-2729	55	41	,	,	PUNCT
cana-2729	55	42	f	f	X
cana-2729	55	43	)	)	PUNCT
cana-2729	55	44	=	=	SYM
cana-2729	55	45	ef	ef	PROPN
cana-2729	55	46	and	and	CCONJ
cana-2729	55	47	h	h	NOUN
cana-2729	55	48	:	:	PUNCT
cana-2729	55	49	(	(	PUNCT
cana-2729	55	50	g	g	NOUN
cana-2729	55	51	,	,	PUNCT
cana-2729	55	52	τ˜	τ˜	PRON
cana-2729	55	53	)	)	PUNCT
cana-2729	55	54	→	→	SYM
cana-2729	55	55	(	(	PUNCT
cana-2729	55	56	g	g	NOUN
cana-2729	55	57	,	,	PUNCT
cana-2729	55	58	τ˜	τ˜	PROPN
cana-2729	55	59	)	)	PUNCT
cana-2729	55	60	,	,	PUNCT
cana-2729	55	61	h(e	h(e	PROPN
cana-2729	55	62	)	)	PUNCT
cana-2729	56	1	=	=	PUNCT
cana-2729	56	2	e	e	NOUN
cana-2729	56	3	−1	−1	NOUN
cana-2729	56	4	are	be	AUX
cana-2729	56	5	fuzzy	fuzzy	ADJ
cana-2729	56	6	continuous	continuous	ADJ
cana-2729	56	7	.	.	PUNCT
cana-2729	57	1	a	a	DET
cana-2729	57	2	group	group	NOUN
cana-2729	57	3	g	g	PROPN
cana-2729	57	4	equipped	equip	VERB
cana-2729	57	5	with	with	ADP
cana-2729	57	6	a	a	DET
cana-2729	57	7	compatible	compatible	ADJ
cana-2729	57	8	τ˜	τ˜	PRON
cana-2729	57	9	as	as	SCONJ
cana-2729	57	10	g	g	PROPN
cana-2729	57	11	is	be	AUX
cana-2729	57	12	called	call	VERB
cana-2729	57	13	a	a	DET
cana-2729	57	14	fuzzy	fuzzy	ADJ
cana-2729	57	15	topological	topological	ADJ
cana-2729	57	16	group	group	NOUN
cana-2729	57	17	(	(	PUNCT
cana-2729	57	18	f	f	PROPN
cana-2729	57	19	t	t	PROPN
cana-2729	57	20	g	g	NOUN
cana-2729	57	21	)	)	PUNCT
cana-2729	57	22	communications	communication	NOUN
cana-2729	57	23	on	on	ADP
cana-2729	57	24	applied	apply	VERB
cana-2729	57	25	nonlinear	nonlinear	ADJ
cana-2729	57	26	analysis	analysis	NOUN
cana-2729	57	27	issn	issn	NOUN
cana-2729	57	28	:	:	PUNCT
cana-2729	57	29	1074	1074	NUM
cana-2729	57	30	-	-	PUNCT
cana-2729	57	31	133x	133x	NUM
cana-2729	57	32	vol	vol	NOUN
cana-2729	57	33	32	32	NUM
cana-2729	57	34	no	no	NOUN
cana-2729	57	35	.	.	PUNCT
cana-2729	58	1	3s	3s	NUM
cana-2729	58	2	(	(	PUNCT
cana-2729	58	3	2025	2025	NUM
cana-2729	58	4	)	)	PUNCT
cana-2729	58	5	713	713	NUM
cana-2729	58	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2729	58	7	definition	definition	NOUN
cana-2729	58	8	2.7	2.7	NUM
cana-2729	58	9	[	[	X
cana-2729	58	10	16	16	NUM
cana-2729	58	11	]	]	PUNCT
cana-2729	58	12	let	let	VERB
cana-2729	58	13	g	g	PRON
cana-2729	58	14	be	be	AUX
cana-2729	58	15	a	a	DET
cana-2729	58	16	group	group	NOUN
cana-2729	58	17	and	and	CCONJ
cana-2729	58	18	(	(	PUNCT
cana-2729	58	19	g	g	NOUN
cana-2729	58	20	,	,	PUNCT
cana-2729	58	21	⋆	⋆	NOUN
cana-2729	58	22	,	,	PUNCT
cana-2729	58	23	0	0	NUM
cana-2729	58	24	,	,	PUNCT
cana-2729	58	25	τ	τ	PROPN
cana-2729	58	26	)	)	PUNCT
cana-2729	58	27	be	be	AUX
cana-2729	58	28	a	a	DET
cana-2729	58	29	fbrkts	fbrkts	NOUN
cana-2729	58	30	.	.	PUNCT
cana-2729	59	1	then	then	ADV
cana-2729	59	2	(	(	PUNCT
cana-2729	59	3	g	g	NOUN
cana-2729	59	4	,	,	PUNCT
cana-2729	59	5	⋆	⋆	NOUN
cana-2729	59	6	,	,	PUNCT
cana-2729	59	7	0	0	NUM
cana-2729	59	8	,	,	PUNCT
cana-2729	59	9	τ	τ	PROPN
cana-2729	59	10	)	)	PUNCT
cana-2729	59	11	is	be	AUX
cana-2729	59	12	called	call	VERB
cana-2729	59	13	fuzzy	fuzzy	ADJ
cana-2729	59	14	brk	brk	PROPN
cana-2729	59	15	topological	topological	PROPN
cana-2729	59	16	group	group	NOUN
cana-2729	59	17	(	(	PUNCT
cana-2729	59	18	briefly	briefly	ADV
cana-2729	59	19	,	,	PUNCT
cana-2729	59	20	fbrktg	fbrktg	ADV
cana-2729	59	21	)	)	PUNCT
cana-2729	59	22	if	if	SCONJ
cana-2729	59	23	the	the	DET
cana-2729	59	24	maps	map	NOUN
cana-2729	59	25	g	g	NOUN
cana-2729	59	26	:	:	PUNCT
cana-2729	59	27	(	(	PUNCT
cana-2729	59	28	g	g	NOUN
cana-2729	59	29	×	×	PROPN
cana-2729	59	30	g	g	PROPN
cana-2729	59	31	,	,	PUNCT
cana-2729	59	32	τ	τ	PROPN
cana-2729	59	33	×	×	PROPN
cana-2729	59	34	τ	τ	PROPN
cana-2729	59	35	)	)	PUNCT
cana-2729	59	36	→	→	SYM
cana-2729	59	37	(	(	PUNCT
cana-2729	59	38	g	g	NOUN
cana-2729	59	39	,	,	PUNCT
cana-2729	59	40	⋆	⋆	NOUN
cana-2729	59	41	,	,	PUNCT
cana-2729	59	42	0	0	NUM
cana-2729	59	43	,	,	PUNCT
cana-2729	59	44	τ	τ	PROPN
cana-2729	59	45	)	)	PUNCT
cana-2729	59	46	defined	define	VERB
cana-2729	59	47	by	by	ADP
cana-2729	59	48	g(e	g(e	PROPN
cana-2729	59	49	,	,	PUNCT
cana-2729	59	50	f	f	X
cana-2729	59	51	)	)	PUNCT
cana-2729	59	52	=	=	SYM
cana-2729	59	53	e	e	X
cana-2729	59	54	⋆	⋆	X
cana-2729	59	55	f	f	PROPN
cana-2729	59	56	and	and	CCONJ
cana-2729	59	57	h	h	NOUN
cana-2729	59	58	:	:	PUNCT
cana-2729	59	59	(	(	PUNCT
cana-2729	59	60	g	g	NOUN
cana-2729	59	61	,	,	PUNCT
cana-2729	59	62	⋆	⋆	NOUN
cana-2729	59	63	,	,	PUNCT
cana-2729	59	64	0	0	NUM
cana-2729	59	65	,	,	PUNCT
cana-2729	59	66	τ	τ	PROPN
cana-2729	59	67	)	)	PUNCT
cana-2729	59	68	→	→	SYM
cana-2729	59	69	(	(	PUNCT
cana-2729	59	70	g	g	NOUN
cana-2729	59	71	,	,	PUNCT
cana-2729	59	72	⋆	⋆	NOUN
cana-2729	59	73	,	,	PUNCT
cana-2729	59	74	0	0	NUM
cana-2729	59	75	,	,	PUNCT
cana-2729	59	76	τ	τ	PROPN
cana-2729	59	77	)	)	PUNCT
cana-2729	59	78	defined	define	VERB
cana-2729	59	79	by	by	ADP
cana-2729	59	80	h(e	h(e	PROPN
cana-2729	59	81	)	)	PUNCT
cana-2729	59	82	=	=	PUNCT
cana-2729	60	1	e	e	NOUN
cana-2729	60	2	−1	−1	NOUN
cana-2729	60	3	are	be	AUX
cana-2729	60	4	fbrkcts	fbrkct	NOUN
cana-2729	60	5	.	.	PUNCT
cana-2729	61	1	definition	definition	NOUN
cana-2729	61	2	2.8	2.8	NUM
cana-2729	61	3	[	[	X
cana-2729	61	4	1	1	NUM
cana-2729	61	5	]	]	PUNCT
cana-2729	61	6	let	let	VERB
cana-2729	61	7	g	g	PRON
cana-2729	61	8	be	be	AUX
cana-2729	61	9	a	a	DET
cana-2729	61	10	monoid	monoid	NOUN
cana-2729	61	11	with	with	ADP
cana-2729	61	12	neutral	neutral	ADJ
cana-2729	61	13	element	element	NOUN
cana-2729	61	14	e	e	PROPN
cana-2729	61	15	&	&	CCONJ
cana-2729	61	16	j	j	PROPN
cana-2729	61	17	a	a	DET
cana-2729	61	18	nonempty	nonempty	ADV
cana-2729	61	19	set	set	VERB
cana-2729	61	20	.	.	PUNCT
cana-2729	62	1	α	α	NOUN
cana-2729	62	2	:	:	PUNCT
cana-2729	63	1	g×j	g×j	PROPN
cana-2729	63	2	→	→	SYM
cana-2729	63	3	j	j	PROPN
cana-2729	63	4	is	be	AUX
cana-2729	63	5	an	an	DET
cana-2729	63	6	action	action	NOUN
cana-2729	63	7	of	of	ADP
cana-2729	63	8	g	g	NOUN
cana-2729	63	9	on	on	ADP
cana-2729	63	10	j	j	PROPN
cana-2729	63	11	iff∀	iff∀	PROPN
cana-2729	63	12	g	g	PROPN
cana-2729	63	13	,	,	PUNCT
cana-2729	63	14	h	h	NOUN
cana-2729	63	15	∈	∈	PROPN
cana-2729	63	16	g	g	PROPN
cana-2729	63	17	,	,	PUNCT
cana-2729	63	18	j	j	PROPN
cana-2729	63	19	∈	∈	PROPN
cana-2729	63	20	j	j	PROPN
cana-2729	63	21	(	(	PUNCT
cana-2729	63	22	i	i	NOUN
cana-2729	63	23	)	)	PUNCT
cana-2729	63	24	(	(	PUNCT
cana-2729	63	25	hg)j	hg)j	PROPN
cana-2729	63	26	=	=	SYM
cana-2729	63	27	h(gj	h(gj	X
cana-2729	63	28	)	)	PUNCT
cana-2729	63	29	(	(	PUNCT
cana-2729	63	30	ii	ii	NOUN
cana-2729	63	31	)	)	PUNCT
cana-2729	63	32	ej	ej	PROPN
cana-2729	64	1	=	=	PUNCT
cana-2729	64	2	j	j	PROPN
cana-2729	64	3	where	where	SCONJ
cana-2729	64	4	α(g	α(g	PROPN
cana-2729	64	5	,	,	PUNCT
cana-2729	64	6	j	j	NOUN
cana-2729	64	7	)	)	PUNCT
cana-2729	64	8	is	be	AUX
cana-2729	64	9	denoted	denote	VERB
cana-2729	64	10	by	by	ADP
cana-2729	64	11	gj	gj	PROPN
cana-2729	64	12	3	3	NUM
cana-2729	64	13	.	.	PUNCT
cana-2729	64	14	fuzzy	fuzzy	PROPN
cana-2729	64	15	brk	brk	PROPN
cana-2729	64	16	topological	topological	PROPN
cana-2729	64	17	group	group	NOUN
cana-2729	64	18	:	:	PUNCT
cana-2729	64	19	definition	definition	NOUN
cana-2729	64	20	3.1	3.1	NUM
cana-2729	64	21	:	:	PUNCT
cana-2729	64	22	the	the	DET
cana-2729	64	23	fuzzy	fuzzy	ADJ
cana-2729	64	24	brk	brk	PROPN
cana-2729	64	25	-closure	-closure	PROPN
cana-2729	64	26	and	and	CCONJ
cana-2729	64	27	fuzzy	fuzzy	ADJ
cana-2729	64	28	brk	brk	PROPN
cana-2729	64	29	-interior	-interior	PROPN
cana-2729	64	30	of	of	ADP
cana-2729	64	31	ρ	ρ	PROPN
cana-2729	64	32	is	be	AUX
cana-2729	64	33	denoted	denote	VERB
cana-2729	64	34	by	by	ADP
cana-2729	64	35	brkcl(ρ	brkcl(ρ	PROPN
cana-2729	64	36	)	)	PUNCT
cana-2729	64	37	and	and	CCONJ
cana-2729	64	38	brkint(ρ	brkint(ρ	NOUN
cana-2729	64	39	)	)	PUNCT
cana-2729	64	40	are	be	AUX
cana-2729	64	41	given	give	VERB
cana-2729	64	42	by	by	ADP
cana-2729	64	43	𝐵𝑅𝐾𝑐𝑙(𝜌	𝐵𝑅𝐾𝑐𝑙(𝜌	NUM
cana-2729	64	44	)	)	PUNCT
cana-2729	65	1	=	=	NOUN
cana-2729	65	2	⋀	⋀	PROPN
cana-2729	65	3	{	{	PUNCT
cana-2729	65	4	𝜆	𝜆	NOUN
cana-2729	65	5	:	:	PUNCT
cana-2729	65	6	𝜆	𝜆	NOUN
cana-2729	65	7	𝑖𝑠	𝑖𝑠	INTJ
cana-2729	65	8	𝑎	𝑎	PRON
cana-2729	65	9	𝑓𝐵𝑅𝐾𝑐	𝑓𝐵𝑅𝐾𝑐	X
cana-2729	65	10	𝑠	𝑠	PROPN
cana-2729	65	11	&	&	CCONJ
cana-2729	65	12	𝜌	𝜌	ADV
cana-2729	65	13	≤	≤	NUM
cana-2729	65	14	𝜆	𝜆	DET
cana-2729	65	15	}	}	PUNCT
cana-2729	65	16	𝐵𝑅𝐾𝑖𝑛𝑡(𝜌	𝐵𝑅𝐾𝑖𝑛𝑡(𝜌	NOUN
cana-2729	65	17	)	)	PUNCT
cana-2729	65	18	=	=	SYM
cana-2729	65	19	⋁	⋁	PROPN
cana-2729	65	20	{	{	PUNCT
cana-2729	65	21	𝜆	𝜆	NOUN
cana-2729	65	22	:	:	PUNCT
cana-2729	65	23	𝜆	𝜆	NOUN
cana-2729	65	24	𝑖𝑠	𝑖𝑠	ADP
cana-2729	65	25	𝑎	𝑎	DET
cana-2729	65	26	𝑓𝐵𝑅𝐾	𝑓𝐵𝑅𝐾	NOUN
cana-2729	65	27	𝑜𝑠	𝑜𝑠	NOUN
cana-2729	65	28	&	&	CCONJ
cana-2729	65	29	𝜌	𝜌	X
cana-2729	65	30	≥	≥	X
cana-2729	65	31	𝜆	𝜆	NOUN
cana-2729	65	32	}	}	PUNCT
cana-2729	65	33	definition	definition	NOUN
cana-2729	65	34	3.2	3.2	NUM
cana-2729	65	35	:	:	PUNCT
cana-2729	65	36	let	let	VERB
cana-2729	65	37	𝐺𝑡	𝐺𝑡	PRON
cana-2729	65	38	be	be	AUX
cana-2729	65	39	a	a	DET
cana-2729	65	40	𝑔𝑟𝑝	𝑔𝑟𝑝	NOUN
cana-2729	65	41	and	and	CCONJ
cana-2729	65	42	(	(	PUNCT
cana-2729	65	43	𝐺𝑡.⋆	𝐺𝑡.⋆	PROPN
cana-2729	65	44	,	,	PUNCT
cana-2729	65	45	0𝐵𝑅𝐾	0𝐵𝑅𝐾	NOUN
cana-2729	65	46	,	,	PUNCT
cana-2729	65	47	𝑓γ	𝑓γ	AUX
cana-2729	65	48	)	)	PUNCT
cana-2729	65	49	be	be	AUX
cana-2729	65	50	a	a	DET
cana-2729	65	51	𝑓	𝑓	DET
cana-2729	65	52	𝐵𝑅𝐾𝑡𝑠.	𝐵𝑅𝐾𝑡𝑠.	X
cana-2729	65	53	then	then	ADV
cana-2729	65	54	(	(	PUNCT
cana-2729	65	55	𝐺𝑡.⋆	𝐺𝑡.⋆	PROPN
cana-2729	65	56	,	,	PUNCT
cana-2729	65	57	0𝐵𝑅𝐾	0𝐵𝑅𝐾	NOUN
cana-2729	65	58	,	,	PUNCT
cana-2729	65	59	𝑓γ	𝑓γ	PROPN
cana-2729	65	60	)	)	PUNCT
cana-2729	65	61	is	be	AUX
cana-2729	65	62	called	call	VERB
cana-2729	65	63	fuzzy	fuzzy	ADJ
cana-2729	65	64	brk	brk	PROPN
cana-2729	65	65	topological	topological	PROPN
cana-2729	65	66	group	group	NOUN
cana-2729	65	67	(	(	PUNCT
cana-2729	65	68	briefly	briefly	ADV
cana-2729	65	69	,	,	PUNCT
cana-2729	65	70	𝑓𝐵𝑅𝐾𝑡𝑔	𝑓𝐵𝑅𝐾𝑡𝑔	NUM
cana-2729	65	71	)	)	PUNCT
cana-2729	65	72	if	if	SCONJ
cana-2729	65	73	𝒈	𝒈	NOUN
cana-2729	65	74	:	:	PUNCT
cana-2729	65	75	(	(	PUNCT
cana-2729	65	76	𝑮𝒕	𝑮𝒕	PROPN
cana-2729	65	77	×	×	NOUN
cana-2729	65	78	𝑮𝒕	𝑮𝒕	PROPN
cana-2729	65	79	,	,	PUNCT
cana-2729	65	80	𝒇𝚪	𝒇𝚪	NOUN
cana-2729	65	81	×	×	NOUN
cana-2729	65	82	𝒇𝚪	𝒇𝚪	NOUN
cana-2729	65	83	)	)	PUNCT
cana-2729	65	84	→	→	SYM
cana-2729	65	85	(	(	PUNCT
cana-2729	65	86	𝐺𝑡.⋆	𝐺𝑡.⋆	PROPN
cana-2729	65	87	,	,	PUNCT
cana-2729	65	88	0	0	NUM
cana-2729	65	89	,	,	PUNCT
cana-2729	65	90	𝑓γ	𝑓γ	PROPN
cana-2729	65	91	)	)	PUNCT
cana-2729	65	92	defined	define	VERB
cana-2729	65	93	by	by	ADP
cana-2729	65	94	𝑔(𝑙11	𝑔(𝑙11	ADV
cana-2729	65	95	,	,	PUNCT
cana-2729	65	96	𝑙22	𝑙22	NOUN
cana-2729	65	97	)	)	PUNCT
cana-2729	65	98	=	=	VERB
cana-2729	65	99	𝑙11	𝑙11	PRON
cana-2729	65	100	⋆	⋆	VERB
cana-2729	65	101	𝑙22	𝑙22	PROPN
cana-2729	65	102	and	and	CCONJ
cana-2729	65	103	𝒉	𝒉	NOUN
cana-2729	65	104	:	:	PUNCT
cana-2729	65	105	(	(	PUNCT
cana-2729	65	106	𝐺𝑡.⋆	𝐺𝑡.⋆	PROPN
cana-2729	65	107	,	,	PUNCT
cana-2729	65	108	0	0	NUM
cana-2729	65	109	,	,	PUNCT
cana-2729	65	110	𝑓γ	𝑓γ	PROPN
cana-2729	65	111	)	)	PUNCT
cana-2729	65	112	→	→	SYM
cana-2729	65	113	(	(	PUNCT
cana-2729	65	114	𝐺𝑡.⋆	𝐺𝑡.⋆	PROPN
cana-2729	65	115	,	,	PUNCT
cana-2729	65	116	0	0	NUM
cana-2729	65	117	,	,	PUNCT
cana-2729	65	118	𝑓γ	𝑓γ	PROPN
cana-2729	65	119	)	)	PUNCT
cana-2729	65	120	defined	define	VERB
cana-2729	65	121	by	by	ADP
cana-2729	65	122	ℎ(𝑙11	ℎ(𝑙11	ADV
cana-2729	65	123	)	)	PUNCT
cana-2729	65	124	=	=	VERB
cana-2729	66	1	𝑙11	𝑙11	PRON
cana-2729	66	2	−1	−1	NOUN
cana-2729	66	3	are	be	AUX
cana-2729	66	4	𝑓𝐵𝑅𝐾𝐶𝑡𝑠	𝑓𝐵𝑅𝐾𝐶𝑡𝑠	NOUN
cana-2729	66	5	theorem	theorem	ADJ
cana-2729	66	6	3.3	3.3	NUM
cana-2729	66	7	:	:	PUNCT
cana-2729	66	8	let	let	VERB
cana-2729	66	9	𝐺𝑡	𝐺𝑡	PRON
cana-2729	66	10	be	be	AUX
cana-2729	66	11	a	a	DET
cana-2729	66	12	𝑔𝑟𝑝	𝑔𝑟𝑝	NOUN
cana-2729	66	13	having	have	VERB
cana-2729	66	14	𝑓𝑡.	𝑓𝑡.	NOUN
cana-2729	66	15	then	then	ADV
cana-2729	66	16	(	(	PUNCT
cana-2729	66	17	𝐺𝑡.⋆	𝐺𝑡.⋆	PROPN
cana-2729	66	18	,	,	PUNCT
cana-2729	66	19	0	0	NUM
cana-2729	66	20	,	,	PUNCT
cana-2729	66	21	𝑓γ	𝑓γ	PROPN
cana-2729	66	22	)	)	PUNCT
cana-2729	66	23	is	be	AUX
cana-2729	66	24	𝑓𝐵𝑅𝐾𝑡𝑔	𝑓𝐵𝑅𝐾𝑡𝑔	PUNCT
cana-2729	66	25	iff	iff	VERB
cana-2729	66	26	the	the	DET
cana-2729	66	27	mapping	mapping	NOUN
cana-2729	66	28	𝑔	𝑔	NOUN
cana-2729	66	29	:	:	PUNCT
cana-2729	66	30	(	(	PUNCT
cana-2729	66	31	𝐺𝑡	𝐺𝑡	NOUN
cana-2729	66	32	×	×	NOUN
cana-2729	66	33	𝐺𝑡	𝐺𝑡	PROPN
cana-2729	66	34	,	,	PUNCT
cana-2729	66	35	𝑓γ	𝑓γ	SCONJ
cana-2729	66	36	×	×	NOUN
cana-2729	66	37	𝑓γ	𝑓γ	ADJ
cana-2729	66	38	)	)	PUNCT
cana-2729	66	39	→	→	SYM
cana-2729	66	40	(	(	PUNCT
cana-2729	66	41	𝐺𝑡.⋆	𝐺𝑡.⋆	PROPN
cana-2729	66	42	,	,	PUNCT
cana-2729	66	43	0	0	NUM
cana-2729	66	44	,	,	PUNCT
cana-2729	66	45	𝑓γ	𝑓γ	PROPN
cana-2729	66	46	)	)	PUNCT
cana-2729	66	47	is	be	AUX
cana-2729	66	48	defined	define	VERB
cana-2729	66	49	by	by	ADP
cana-2729	66	50	𝑔(𝑙11	𝑔(𝑙11	ADV
cana-2729	66	51	,	,	PUNCT
cana-2729	66	52	𝑙22	𝑙22	NOUN
cana-2729	66	53	)	)	PUNCT
cana-2729	66	54	=	=	VERB
cana-2729	67	1	𝑙11	𝑙11	PRON
cana-2729	67	2	⋆	⋆	VERB
cana-2729	67	3	𝑙22	𝑙22	NOUN
cana-2729	67	4	−1	−1	NOUN
cana-2729	67	5	is	be	AUX
cana-2729	67	6	𝑓	𝑓	PRON
cana-2729	67	7	𝐵𝑅𝐾𝐶𝑡𝑠.	𝐵𝑅𝐾𝐶𝑡𝑠.	X
cana-2729	67	8	theorem	theorem	VERB
cana-2729	67	9	3.4	3.4	NUM
cana-2729	67	10	:	:	PUNCT
cana-2729	67	11	let	let	VERB
cana-2729	67	12	a	a	PRON
cana-2729	67	13	be	be	AUX
cana-2729	67	14	a	a	DET
cana-2729	67	15	fixed	fix	VERB
cana-2729	67	16	element	element	NOUN
cana-2729	67	17	of	of	ADP
cana-2729	67	18	𝑓𝐵𝑅𝐾𝐶𝑡𝑔	𝑓𝐵𝑅𝐾𝐶𝑡𝑔	NOUN
cana-2729	67	19	(	(	PUNCT
cana-2729	67	20	𝐺𝑡.⋆	𝐺𝑡.⋆	PROPN
cana-2729	67	21	,	,	PUNCT
cana-2729	67	22	0	0	NUM
cana-2729	67	23	,	,	PUNCT
cana-2729	67	24	𝑓γ	𝑓γ	PROPN
cana-2729	67	25	)	)	PUNCT
cana-2729	67	26	.	.	PUNCT
cana-2729	68	1	then	then	ADV
cana-2729	68	2	the	the	DET
cana-2729	68	3	mapping	mapping	NOUN
cana-2729	68	4	𝑅𝛼(𝑙11	𝑅𝛼(𝑙11	PROPN
cana-2729	68	5	)	)	PUNCT
cana-2729	69	1	=	=	SYM
cana-2729	69	2	𝑙11	𝑙11	ADJ
cana-2729	69	3	⋆	⋆	VERB
cana-2729	69	4	𝑎	𝑎	PROPN
cana-2729	69	5	,	,	PUNCT
cana-2729	69	6	𝐿𝛼(𝑙11	𝐿𝛼(𝑙11	PROPN
cana-2729	69	7	)	)	PUNCT
cana-2729	70	1	=	=	PUNCT
cana-2729	71	1	𝑎	𝑎	DET
cana-2729	71	2	⋆	⋆	NOUN
cana-2729	71	3	𝑙11	𝑙11	NOUN
cana-2729	71	4	,	,	PUNCT
cana-2729	71	5	ℎ(𝑙11	ℎ(𝑙11	ADV
cana-2729	71	6	)	)	PUNCT
cana-2729	71	7	=	=	SYM
cana-2729	71	8	𝑙11	𝑙11	ADV
cana-2729	71	9	−1	−1	NOUN
cana-2729	71	10	and	and	CCONJ
cana-2729	71	11	𝑔(𝑙11	𝑔(𝑙11	ADV
cana-2729	71	12	)	)	PUNCT
cana-2729	72	1	=	=	SYM
cana-2729	72	2	(	(	PUNCT
cana-2729	72	3	(	(	PUNCT
cana-2729	72	4	𝑎	𝑎	X
cana-2729	72	5	⋆	⋆	NOUN
cana-2729	72	6	𝑙11	𝑙11	ADV
cana-2729	72	7	)	)	PUNCT
cana-2729	72	8	⋆	⋆	VERB
cana-2729	72	9	𝑎−1	𝑎−1	PROPN
cana-2729	72	10	)	)	PUNCT
cana-2729	72	11	of	of	ADP
cana-2729	72	12	(	(	PUNCT
cana-2729	72	13	𝐺𝑡.⋆	𝐺𝑡.⋆	PROPN
cana-2729	72	14	,	,	PUNCT
cana-2729	72	15	0	0	NUM
cana-2729	72	16	,	,	PUNCT
cana-2729	72	17	𝑓γ	𝑓γ	PROPN
cana-2729	72	18	)	)	PUNCT
cana-2729	72	19	onto	onto	ADP
cana-2729	72	20	(	(	PUNCT
cana-2729	72	21	𝐺𝑡.⋆	𝐺𝑡.⋆	PROPN
cana-2729	72	22	,	,	PUNCT
cana-2729	72	23	0	0	NUM
cana-2729	72	24	,	,	PUNCT
cana-2729	72	25	𝑓γ	𝑓γ	PROPN
cana-2729	72	26	)	)	PUNCT
cana-2729	72	27	are	be	AUX
cana-2729	72	28	𝑓𝐵𝑅𝐾𝐻	𝑓𝐵𝑅𝐾𝐻	VERB
cana-2729	72	29	om	om	PROPN
cana-2729	72	30	’s	’s	NOUN
cana-2729	72	31	of	of	ADP
cana-2729	72	32	𝐺𝑡.	𝐺𝑡.	PROPN
cana-2729	72	33	proof	proof	NOUN
cana-2729	72	34	:	:	PUNCT
cana-2729	72	35	let	let	VERB
cana-2729	72	36	𝑘−1	𝑘−1	PROPN
cana-2729	72	37	⋆	⋆	VERB
cana-2729	72	38	𝑘2	𝑘2	PROPN
cana-2729	72	39	=	=	PUNCT
cana-2729	73	1	𝑎	𝑎	PROPN
cana-2729	73	2	∈	∈	PROPN
cana-2729	73	3	𝐺𝑡	𝐺𝑡	PROPN
cana-2729	73	4	and	and	CCONJ
cana-2729	73	5	consider	consider	VERB
cana-2729	73	6	the	the	DET
cana-2729	73	7	mapping	mapping	NOUN
cana-2729	73	8	(	(	PUNCT
cana-2729	73	9	𝐺𝑡.⋆	𝐺𝑡.⋆	PROPN
cana-2729	73	10	,	,	PUNCT
cana-2729	73	11	0	0	NUM
cana-2729	73	12	,	,	PUNCT
cana-2729	73	13	𝑓γ	𝑓γ	PROPN
cana-2729	73	14	)	)	PUNCT
cana-2729	73	15	→	→	SYM
cana-2729	73	16	(	(	PUNCT
cana-2729	73	17	𝐺𝑡.⋆	𝐺𝑡.⋆	PROPN
cana-2729	73	18	,	,	PUNCT
cana-2729	73	19	0	0	NUM
cana-2729	73	20	,	,	PUNCT
cana-2729	73	21	𝑓γ	𝑓γ	PROPN
cana-2729	73	22	)	)	PUNCT
cana-2729	73	23	defined	define	VERB
cana-2729	73	24	by	by	ADP
cana-2729	73	25	ℎ(𝑙11	ℎ(𝑙11	ADV
cana-2729	73	26	)	)	PUNCT
cana-2729	74	1	=	=	SYM
cana-2729	74	2	𝑙11	𝑙11	PRON
cana-2729	75	1	⋆	⋆	VERB
cana-2729	75	2	𝑎.	𝑎.	PROPN
cana-2729	76	1	then	then	ADV
cana-2729	76	2	h	h	PROPN
cana-2729	76	3	is	be	AUX
cana-2729	76	4	𝑓𝐵𝑅𝐾𝐻𝑜𝑚	𝑓𝐵𝑅𝐾𝐻𝑜𝑚	PROPN
cana-2729	76	5	by	by	ADP
cana-2729	76	6	theorem	theorem	ADJ
cana-2729	76	7	3.4.2	3.4.2	NUM
cana-2729	76	8	,	,	PUNCT
cana-2729	76	9	ℎ(𝑘1	ℎ(𝑘1	NOUN
cana-2729	76	10	)	)	PUNCT
cana-2729	76	11	=	=	SYM
cana-2729	76	12	𝑘2	𝑘2	PROPN
cana-2729	76	13	.	.	PUNCT
cana-2729	77	1	theorem	theorem	VERB
cana-2729	77	2	3.5	3.5	NUM
cana-2729	77	3	:	:	PUNCT
cana-2729	77	4	a	a	DET
cana-2729	77	5	non	non	ADJ
cana-2729	77	6	-	-	ADJ
cana-2729	77	7	trivial	trivial	ADJ
cana-2729	77	8	𝑓𝐵𝑅𝐾𝑡𝑔	𝑓𝐵𝑅𝐾𝑡𝑔	X
cana-2729	77	9	does	do	AUX
cana-2729	77	10	not	not	PART
cana-2729	77	11	have	have	AUX
cana-2729	77	12	fixed	fix	VERB
cana-2729	77	13	point	point	NOUN
cana-2729	77	14	property	property	NOUN
cana-2729	77	15	.	.	PUNCT
cana-2729	78	1	proof	proof	NOUN
cana-2729	78	2	:	:	PUNCT
cana-2729	78	3	let	let	VERB
cana-2729	78	4	𝐺𝑡	𝐺𝑡	PRON
cana-2729	78	5	be	be	AUX
cana-2729	78	6	a	a	DET
cana-2729	78	7	𝑓𝐵𝑅𝐾𝑡𝑔	𝑓𝐵𝑅𝐾𝑡𝑔	ADJ
cana-2729	78	8	and	and	CCONJ
cana-2729	78	9	𝑎	𝑎	PROPN
cana-2729	78	10	∈	∈	ADJ
cana-2729	78	11	𝐺𝑡	𝐺𝑡	NOUN
cana-2729	78	12	with	with	ADP
cana-2729	78	13	a	a	DET
cana-2729	78	14	≠	≠	PROPN
cana-2729	78	15	e.	e.	NOUN
cana-2729	78	16	clearly	clearly	ADV
cana-2729	78	17	,	,	PUNCT
cana-2729	78	18	the	the	DET
cana-2729	78	19	map	map	NOUN
cana-2729	78	20	𝑅𝛼	𝑅𝛼	NOUN
cana-2729	78	21	:	:	PUNCT
cana-2729	78	22	𝐺𝑡	𝐺𝑡	PROPN
cana-2729	78	23	→	→	SYM
cana-2729	78	24	𝐺𝑡	𝐺𝑡	PROPN
cana-2729	78	25	is	be	AUX
cana-2729	78	26	𝑓𝐵𝑅𝐾𝐶𝑡𝑔.	𝑓𝐵𝑅𝐾𝐶𝑡𝑔.	ADV
cana-2729	78	27	suppose	suppose	VERB
cana-2729	78	28	that	that	SCONJ
cana-2729	78	29	𝑅𝛼(𝑙11	𝑅𝛼(𝑙11	PROPN
cana-2729	78	30	)	)	PUNCT
cana-2729	78	31	=	=	SYM
cana-2729	79	1	𝑙11	𝑙11	ADJ
cana-2729	79	2	for	for	ADP
cana-2729	79	3	some	some	DET
cana-2729	79	4	𝑙11	𝑙11	NOUN
cana-2729	79	5	∈	∈	PROPN
cana-2729	79	6	𝐺𝑡.	𝐺𝑡.	PROPN
cana-2729	79	7	then	then	ADV
cana-2729	79	8	𝑙11	𝑙11	PRON
cana-2729	79	9	⋆	⋆	VERB
cana-2729	79	10	𝑎	𝑎	PRON
cana-2729	79	11	=	=	NOUN
cana-2729	79	12	𝑙11implies	𝑙11implie	NOUN
cana-2729	79	13	𝑎	𝑎	X
cana-2729	79	14	=	=	SYM
cana-2729	79	15	𝑒	𝑒	PROPN
cana-2729	79	16	which	which	PRON
cana-2729	79	17	contradicts	contradict	VERB
cana-2729	79	18	to	to	ADP
cana-2729	79	19	the	the	DET
cana-2729	79	20	concept	concept	NOUN
cana-2729	79	21	that	that	PRON
cana-2729	79	22	𝑅𝑎	𝑅𝑎	PROPN
cana-2729	79	23	has	have	VERB
cana-2729	79	24	no	no	DET
cana-2729	79	25	fixed	fix	VERB
cana-2729	79	26	point	point	NOUN
cana-2729	79	27	.	.	PUNCT
cana-2729	80	1	hence	hence	ADV
cana-2729	80	2	𝐺𝑡	𝐺𝑡	PROPN
cana-2729	80	3	does	do	AUX
cana-2729	80	4	not	not	PART
cana-2729	80	5	have	have	AUX
cana-2729	80	6	fixed	fix	VERB
cana-2729	80	7	point	point	NOUN
cana-2729	80	8	property	property	NOUN
cana-2729	80	9	.	.	PUNCT
cana-2729	81	1	theorem	theorem	VERB
cana-2729	81	2	3.6	3.6	NUM
cana-2729	81	3	:	:	PUNCT
cana-2729	81	4	let	let	VERB
cana-2729	81	5	(	(	PUNCT
cana-2729	81	6	𝐺𝑡.⋆	𝐺𝑡.⋆	NOUN
cana-2729	81	7	,	,	PUNCT
cana-2729	81	8	0	0	NUM
cana-2729	81	9	,	,	PUNCT
cana-2729	81	10	𝑓γ	𝑓γ	PROPN
cana-2729	81	11	)	)	PUNCT
cana-2729	81	12	be	be	AUX
cana-2729	81	13	𝑓𝐵𝑅𝐾𝑡𝑔	𝑓𝐵𝑅𝐾𝑡𝑔	X
cana-2729	81	14	and	and	CCONJ
cana-2729	81	15	𝜌1	𝜌1	NOUN
cana-2729	81	16	,	,	PUNCT
cana-2729	81	17	𝜌2are	𝜌2are	NOUN
cana-2729	81	18	of	of	ADP
cana-2729	81	19	𝑓	𝑓	DET
cana-2729	81	20	𝑠𝑢𝑏	𝑠𝑢𝑏	NOUN
cana-2729	82	1	𝐺𝑡.	𝐺𝑡.	PROPN
cana-2729	82	2	then	then	ADV
cana-2729	82	3	the	the	DET
cana-2729	82	4	following	follow	VERB
cana-2729	82	5	claims	claim	NOUN
cana-2729	82	6	are	be	AUX
cana-2729	82	7	true	true	ADJ
cana-2729	82	8	:	:	PUNCT
cana-2729	82	9	(	(	PUNCT
cana-2729	82	10	i	i	NOUN
cana-2729	82	11	)	)	PUNCT
cana-2729	82	12	𝐵𝑅𝐾(𝑐𝑙(𝑝	𝐵𝑅𝐾(𝑐𝑙(𝑝	NOUN
cana-2729	82	13	⋆	⋆	VERB
cana-2729	82	14	𝜌1	𝜌1	NOUN
cana-2729	82	15	)	)	PUNCT
cana-2729	82	16	⋆	⋆	VERB
cana-2729	82	17	𝑝−1	𝑝−1	NOUN
cana-2729	82	18	)	)	PUNCT
cana-2729	82	19	=	=	PUNCT
cana-2729	83	1	(	(	PUNCT
cana-2729	83	2	𝑝	𝑝	NOUN
cana-2729	83	3	⋆	⋆	X
cana-2729	83	4	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	83	5	)	)	PUNCT
cana-2729	84	1	⋆	⋆	VERB
cana-2729	84	2	𝑝−1	𝑝−1	PROPN
cana-2729	84	3	)	)	PUNCT
cana-2729	84	4	,	,	PUNCT
cana-2729	84	5	where	where	SCONJ
cana-2729	84	6	𝑝	𝑝	X
cana-2729	84	7	∈	∈	PROPN
cana-2729	84	8	𝐺𝑡	𝐺𝑡	PROPN
cana-2729	84	9	is	be	AUX
cana-2729	84	10	a	a	DET
cana-2729	84	11	definite	definite	ADJ
cana-2729	84	12	point	point	NOUN
cana-2729	84	13	,	,	PUNCT
cana-2729	84	14	(	(	PUNCT
cana-2729	84	15	ii	ii	NOUN
cana-2729	84	16	)	)	PUNCT
cana-2729	84	17	if	if	SCONJ
cana-2729	84	18	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	84	19	)	)	PUNCT
cana-2729	84	20	×	×	PROPN
cana-2729	84	21	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	84	22	)	)	PUNCT
cana-2729	84	23	⊆	⊆	NUM
cana-2729	84	24	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	84	25	×	×	NOUN
cana-2729	84	26	𝜌2	𝜌2	ADJ
cana-2729	84	27	)	)	PUNCT
cana-2729	84	28	,	,	PUNCT
cana-2729	84	29	then	then	ADV
cana-2729	84	30	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	84	31	)	)	PUNCT
cana-2729	84	32	⋆	⋆	X
cana-2729	84	33	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	84	34	)	)	PUNCT
cana-2729	84	35	⊆	⊆	NUM
cana-2729	84	36	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	84	37	×	×	NOUN
cana-2729	84	38	𝜌2	𝜌2	ADJ
cana-2729	84	39	)	)	PUNCT
cana-2729	84	40	and	and	CCONJ
cana-2729	84	41	𝑅𝐾𝑐𝑙(𝜌1	𝑅𝐾𝑐𝑙(𝜌1	NOUN
cana-2729	84	42	)	)	PUNCT
cana-2729	84	43	⋆	⋆	VERB
cana-2729	84	44	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	PROPN
cana-2729	84	45	−1	−1	NUM
cana-2729	84	46	)	)	PUNCT
cana-2729	84	47	⊆	⊆	NUM
cana-2729	84	48	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	84	49	×	×	NOUN
cana-2729	84	50	𝜌2	𝜌2	ADJ
cana-2729	84	51	−1	−1	NOUN
cana-2729	84	52	)	)	PUNCT
cana-2729	84	53	.	.	PUNCT
cana-2729	85	1	proof	proof	NOUN
cana-2729	85	2	:	:	PUNCT
cana-2729	85	3	(	(	PUNCT
cana-2729	85	4	(	(	PUNCT
cana-2729	85	5	𝑝	𝑝	NOUN
cana-2729	85	6	⋆	⋆	X
cana-2729	85	7	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	85	8	)	)	PUNCT
cana-2729	85	9	)	)	PUNCT
cana-2729	85	10	⋆	⋆	CCONJ
cana-2729	85	11	𝑝−1	𝑝−1	PROPN
cana-2729	85	12	)	)	PUNCT
cana-2729	85	13	is	be	AUX
cana-2729	85	14	a	a	DET
cana-2729	85	15	𝑓𝐵𝑅𝐾𝑐𝑠	𝑓𝐵𝑅𝐾𝑐𝑠	NOUN
cana-2729	85	16	by	by	ADP
cana-2729	85	17	corollary	corollary	ADJ
cana-2729	85	18	3.4.1	3.4.1	NUM
cana-2729	85	19	.	.	PUNCT
cana-2729	86	1	since	since	SCONJ
cana-2729	86	2	this	this	PRON
cana-2729	86	3	is	be	AUX
cana-2729	86	4	the	the	DET
cana-2729	86	5	smallest	small	ADJ
cana-2729	86	6	𝑓𝐵𝑅𝐾𝑐𝑠	𝑓𝐵𝑅𝐾𝑐𝑠	NOUN
cana-2729	86	7	containing	contain	VERB
cana-2729	86	8	(	(	PUNCT
cana-2729	86	9	(	(	PUNCT
cana-2729	86	10	𝑝	𝑝	PROPN
cana-2729	86	11	⋆	⋆	VERB
cana-2729	86	12	𝜌1	𝜌1	NOUN
cana-2729	86	13	)	)	PUNCT
cana-2729	86	14	⋆	⋆	VERB
cana-2729	86	15	𝑝−1	𝑝−1	PROPN
cana-2729	86	16	)	)	PUNCT
cana-2729	86	17	,	,	PUNCT
cana-2729	86	18	𝐵𝑅𝐾𝑐𝑙(𝑝	𝐵𝑅𝐾𝑐𝑙(𝑝	PROPN
cana-2729	86	19	⋆	⋆	VERB
cana-2729	86	20	𝜌1	𝜌1	PROPN
cana-2729	86	21	)	)	PUNCT
cana-2729	86	22	⋆	⋆	X
cana-2729	86	23	𝑝−1	𝑝−1	NOUN
cana-2729	86	24	)	)	PUNCT
cana-2729	86	25	⊆	⊆	NUM
cana-2729	86	26	(	(	PUNCT
cana-2729	86	27	(	(	PUNCT
cana-2729	86	28	𝑝	𝑝	NOUN
cana-2729	86	29	⋆	⋆	X
cana-2729	86	30	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	86	31	)	)	PUNCT
cana-2729	86	32	)	)	PUNCT
cana-2729	87	1	⋆	⋆	VERB
cana-2729	87	2	𝑝−1	𝑝−1	PROPN
cana-2729	87	3	)	)	PUNCT
cana-2729	87	4	.	.	PUNCT
cana-2729	88	1	let	let	VERB
cana-2729	88	2	ℎ	ℎ	PROPN
cana-2729	88	3	:	:	PUNCT
cana-2729	88	4	(	(	PUNCT
cana-2729	88	5	𝐺𝑡.⋆	𝐺𝑡.⋆	PROPN
cana-2729	88	6	,	,	PUNCT
cana-2729	88	7	0	0	NUM
cana-2729	88	8	,	,	PUNCT
cana-2729	88	9	𝑓γ	𝑓γ	PROPN
cana-2729	88	10	)	)	PUNCT
cana-2729	88	11	→	→	SYM
cana-2729	88	12	(	(	PUNCT
cana-2729	88	13	𝐺𝑡.⋆	𝐺𝑡.⋆	PROPN
cana-2729	88	14	,	,	PUNCT
cana-2729	88	15	0	0	NUM
cana-2729	88	16	,	,	PUNCT
cana-2729	88	17	𝑓γ	𝑓γ	PROPN
cana-2729	88	18	)	)	PUNCT
cana-2729	88	19	be	be	AUX
cana-2729	88	20	a	a	DET
cana-2729	88	21	map	map	NOUN
cana-2729	88	22	defined	define	VERB
cana-2729	88	23	by	by	ADP
cana-2729	88	24	ℎ(𝑙11	ℎ(𝑙11	ADV
cana-2729	88	25	)	)	PUNCT
cana-2729	89	1	=	=	SYM
cana-2729	89	2	(	(	PUNCT
cana-2729	89	3	(	(	PUNCT
cana-2729	89	4	𝑝	𝑝	PROPN
cana-2729	89	5	⋆	⋆	VERB
cana-2729	89	6	𝑙11	𝑙11	ADV
cana-2729	89	7	)	)	PUNCT
cana-2729	89	8	⋆	⋆	VERB
cana-2729	89	9	𝑝−1	𝑝−1	PROPN
cana-2729	89	10	)	)	PUNCT
cana-2729	89	11	.	.	PUNCT
cana-2729	90	1	then	then	ADV
cana-2729	90	2	by	by	ADP
cana-2729	90	3	theorem	theorem	NOUN
cana-2729	90	4	3.2	3.2	NUM
cana-2729	90	5	,	,	PUNCT
cana-2729	90	6	h	h	NOUN
cana-2729	90	7	is	be	AUX
cana-2729	90	8	𝑓𝐵𝑅𝐾𝐻𝑜𝑚	𝑓𝐵𝑅𝐾𝐻𝑜𝑚	PROPN
cana-2729	90	9	,	,	PUNCT
cana-2729	90	10	ℎ(𝐵𝑅𝐾𝑐𝑙(𝜌1	ℎ(𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	90	11	)	)	PUNCT
cana-2729	90	12	)	)	PUNCT
cana-2729	91	1	⊆	⊆	NUM
cana-2729	91	2	𝐵𝑅𝐾𝑐𝑙(ℎ(𝜌1	𝐵𝑅𝐾𝑐𝑙(ℎ(𝜌1	NOUN
cana-2729	91	3	)	)	PUNCT
cana-2729	91	4	)	)	PUNCT
cana-2729	91	5	.	.	PUNCT
cana-2729	92	1	thus	thus	ADV
cana-2729	92	2	(	(	PUNCT
cana-2729	92	3	(	(	PUNCT
cana-2729	92	4	𝑝	𝑝	NOUN
cana-2729	92	5	⋆	⋆	X
cana-2729	92	6	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	92	7	)	)	PUNCT
cana-2729	92	8	)	)	PUNCT
cana-2729	92	9	⋆	⋆	VERB
cana-2729	92	10	𝑝−1	𝑝−1	PROPN
cana-2729	92	11	)	)	PUNCT
cana-2729	92	12	⊆	⊆	NUM
cana-2729	92	13	𝐵𝑅𝐾𝑐𝑙(𝑝	𝐵𝑅𝐾𝑐𝑙(𝑝	PROPN
cana-2729	92	14	⋆	⋆	VERB
cana-2729	92	15	𝜌1	𝜌1	NOUN
cana-2729	92	16	)	)	PUNCT
cana-2729	92	17	⋆	⋆	VERB
cana-2729	92	18	𝑝−1	𝑝−1	PROPN
cana-2729	92	19	)	)	PUNCT
cana-2729	92	20	and	and	CCONJ
cana-2729	92	21	hence	hence	ADV
cana-2729	92	22	we	we	PRON
cana-2729	92	23	get	get	VERB
cana-2729	92	24	(	(	PUNCT
cana-2729	92	25	(	(	PUNCT
cana-2729	92	26	𝑝	𝑝	NOUN
cana-2729	92	27	⋆	⋆	X
cana-2729	92	28	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	92	29	)	)	PUNCT
cana-2729	92	30	⋆	⋆	VERB
cana-2729	92	31	𝑝−1	𝑝−1	NOUN
cana-2729	92	32	)	)	PUNCT
cana-2729	92	33	=	=	PRON
cana-2729	92	34	𝐵𝑅𝐾𝑐𝑙(𝑝	𝐵𝑅𝐾𝑐𝑙(𝑝	PROPN
cana-2729	92	35	⋆	⋆	VERB
cana-2729	92	36	𝜌1	𝜌1	NOUN
cana-2729	92	37	)	)	PUNCT
cana-2729	92	38	⋆	⋆	X
cana-2729	92	39	𝑝−1	𝑝−1	NOUN
cana-2729	92	40	)	)	PUNCT
cana-2729	92	41	communications	communication	NOUN
cana-2729	92	42	on	on	ADP
cana-2729	92	43	applied	apply	VERB
cana-2729	92	44	nonlinear	nonlinear	ADJ
cana-2729	92	45	analysis	analysis	NOUN
cana-2729	92	46	issn	issn	NOUN
cana-2729	92	47	:	:	PUNCT
cana-2729	92	48	1074	1074	NUM
cana-2729	92	49	-	-	PUNCT
cana-2729	92	50	133x	133x	NUM
cana-2729	92	51	vol	vol	NOUN
cana-2729	92	52	32	32	NUM
cana-2729	92	53	no	no	NOUN
cana-2729	92	54	.	.	PUNCT
cana-2729	93	1	3s	3s	NUM
cana-2729	93	2	(	(	PUNCT
cana-2729	93	3	2025	2025	NUM
cana-2729	93	4	)	)	PUNCT
cana-2729	93	5	714	714	NUM
cana-2729	93	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2729	93	7	the	the	DET
cana-2729	93	8	map	map	NOUN
cana-2729	93	9	𝑔	𝑔	NOUN
cana-2729	93	10	:	:	PUNCT
cana-2729	93	11	:	:	PUNCT
cana-2729	93	12	(	(	PUNCT
cana-2729	93	13	𝐺𝑡.⋆	𝐺𝑡.⋆	PROPN
cana-2729	93	14	,	,	PUNCT
cana-2729	93	15	0	0	NUM
cana-2729	93	16	,	,	PUNCT
cana-2729	93	17	𝑓γ	𝑓γ	PROPN
cana-2729	93	18	)	)	PUNCT
cana-2729	93	19	×	×	NOUN
cana-2729	93	20	(	(	PUNCT
cana-2729	93	21	𝐺𝑡.⋆	𝐺𝑡.⋆	PROPN
cana-2729	93	22	,	,	PUNCT
cana-2729	93	23	0	0	NUM
cana-2729	93	24	,	,	PUNCT
cana-2729	93	25	𝑓γ	𝑓γ	PROPN
cana-2729	93	26	)	)	PUNCT
cana-2729	93	27	→	→	SYM
cana-2729	93	28	(	(	PUNCT
cana-2729	93	29	𝐺𝑡.⋆	𝐺𝑡.⋆	PROPN
cana-2729	93	30	,	,	PUNCT
cana-2729	93	31	0	0	NUM
cana-2729	93	32	,	,	PUNCT
cana-2729	93	33	𝑓γ	𝑓γ	PROPN
cana-2729	93	34	)	)	PUNCT
cana-2729	93	35	defined	define	VERB
cana-2729	93	36	by	by	ADP
cana-2729	93	37	𝑔(𝑙11	𝑔(𝑙11	ADV
cana-2729	93	38	,	,	PUNCT
cana-2729	93	39	𝑙22	𝑙22	NOUN
cana-2729	93	40	)	)	PUNCT
cana-2729	93	41	=	=	SYM
cana-2729	93	42	(	(	PUNCT
cana-2729	93	43	𝑙11	𝑙11	ADV
cana-2729	93	44	⋆	⋆	VERB
cana-2729	93	45	𝑙22	𝑙22	NOUN
cana-2729	93	46	−1	−1	NOUN
cana-2729	93	47	)	)	PUNCT
cana-2729	93	48	is	be	AUX
cana-2729	93	49	𝑓𝐵𝑅𝐾𝐶𝑡𝑠	𝑓𝐵𝑅𝐾𝐶𝑡𝑠	NOUN
cana-2729	93	50	,	,	PUNCT
cana-2729	93	51	since	since	SCONJ
cana-2729	93	52	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	93	53	)	)	PUNCT
cana-2729	93	54	×	×	PROPN
cana-2729	93	55	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	93	56	)	)	PUNCT
cana-2729	93	57	⊆	⊆	NUM
cana-2729	93	58	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	93	59	×	×	NOUN
cana-2729	93	60	𝜌2	𝜌2	ADJ
cana-2729	93	61	)	)	PUNCT
cana-2729	93	62	,	,	PUNCT
cana-2729	93	63	ℎ(𝐵𝑅𝐿𝑐𝑙(𝜌1	ℎ(𝐵𝑅𝐿𝑐𝑙(𝜌1	NOUN
cana-2729	93	64	)	)	PUNCT
cana-2729	93	65	,	,	PUNCT
cana-2729	93	66	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	PROPN
cana-2729	93	67	)	)	PUNCT
cana-2729	93	68	⊂	⊂	PROPN
cana-2729	94	1	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	94	2	×	×	PROPN
cana-2729	94	3	𝜌2	𝜌2	ADJ
cana-2729	94	4	)	)	PUNCT
cana-2729	94	5	)	)	PUNCT
cana-2729	94	6	.	.	PUNCT
cana-2729	95	1	since	since	SCONJ
cana-2729	95	2	h	h	PROPN
cana-2729	95	3	is	be	AUX
cana-2729	95	4	𝑓𝐵𝑅𝐾𝐶𝑡𝑠	𝑓𝐵𝑅𝐾𝐶𝑡𝑠	NOUN
cana-2729	95	5	,	,	PUNCT
cana-2729	95	6	ℎ(𝐵𝑅𝐾𝑐𝑙(𝜌1	ℎ(𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	95	7	×	×	NOUN
cana-2729	95	8	𝜌2	𝜌2	ADJ
cana-2729	95	9	)	)	PUNCT
cana-2729	95	10	)	)	PUNCT
cana-2729	96	1	⊆	⊆	NUM
cana-2729	96	2	𝐵𝑅𝐾𝑐𝑙ℎ(𝜌1	𝐵𝑅𝐾𝑐𝑙ℎ(𝜌1	NOUN
cana-2729	96	3	,	,	PUNCT
cana-2729	96	4	𝜌2	𝜌2	ADJ
cana-2729	96	5	)	)	PUNCT
cana-2729	96	6	.	.	PUNCT
cana-2729	97	1	then	then	ADV
cana-2729	97	2	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	X
cana-2729	97	3	)	)	PUNCT
cana-2729	97	4	⋆	⋆	VERB
cana-2729	97	5	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	PROPN
cana-2729	97	6	−1	−1	NUM
cana-2729	97	7	)	)	PUNCT
cana-2729	97	8	⊆	⊆	NUM
cana-2729	97	9	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	97	10	⋆	⋆	VERB
cana-2729	97	11	𝜌2	𝜌2	ADJ
cana-2729	97	12	−1	−1	NOUN
cana-2729	97	13	)	)	PUNCT
cana-2729	97	14	,	,	PUNCT
cana-2729	97	15	for	for	ADP
cana-2729	97	16	𝑙11	𝑙11	NUM
cana-2729	97	17	∈	∈	PROPN
cana-2729	97	18	𝐺𝑡.	𝐺𝑡.	PROPN
cana-2729	97	19	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	PROPN
cana-2729	97	20	−1)(𝑙𝑙11	−1)(𝑙𝑙11	NUM
cana-2729	97	21	)	)	PUNCT
cana-2729	97	22	=	=	NOUN
cana-2729	97	23	∩	∩	NOUN
cana-2729	97	24	{	{	PUNCT
cana-2729	97	25	𝐾𝑖	𝐾𝑖	ADJ
cana-2729	97	26	:	:	PUNCT
cana-2729	97	27	𝜌2	𝜌2	ADJ
cana-2729	97	28	−1	−1	NOUN
cana-2729	97	29	⊆	⊆	NUM
cana-2729	97	30	𝐾𝑖	𝐾𝑖	PROPN
cana-2729	97	31	,	,	PUNCT
cana-2729	97	32	𝐾𝑖	𝐾𝑖	PROPN
cana-2729	97	33	𝑖𝑠	𝑖𝑠	ADP
cana-2729	97	34	𝑓𝐵𝑅𝐾𝑜}(𝑙11	𝑓𝐵𝑅𝐾𝑜}(𝑙11	PROPN
cana-2729	97	35	)	)	PUNCT
cana-2729	97	36	=	=	SYM
cana-2729	97	37	inf	inf	NOUN
cana-2729	97	38	{	{	PUNCT
cana-2729	97	39	𝐾𝑖(𝑙11	𝐾𝑖(𝑙11	PROPN
cana-2729	97	40	):	):	PUNCT
cana-2729	97	41	𝜌2	𝜌2	ADJ
cana-2729	97	42	−1	−1	NOUN
cana-2729	97	43	⊆	⊆	NUM
cana-2729	97	44	𝐾𝑖	𝐾𝑖	ADJ
cana-2729	97	45	}	}	PUNCT
cana-2729	97	46	=	=	SYM
cana-2729	97	47	inf	inf	NOUN
cana-2729	97	48	{	{	PUNCT
cana-2729	97	49	𝐾𝑖	𝐾𝑖	PROPN
cana-2729	97	50	−1(𝑙11	−1(𝑙11	PROPN
cana-2729	97	51	):	):	PUNCT
cana-2729	97	52	𝜌2	𝜌2	ADJ
cana-2729	97	53	⊆	⊆	NUM
cana-2729	97	54	𝐾𝑖	𝐾𝑖	PROPN
cana-2729	97	55	−1	−1	NOUN
cana-2729	97	56	}	}	PUNCT
cana-2729	97	57	=	=	NOUN
cana-2729	97	58	∩	∩	NOUN
cana-2729	97	59	{	{	PUNCT
cana-2729	97	60	𝐾𝑖	𝐾𝑖	PROPN
cana-2729	97	61	−1	−1	NOUN
cana-2729	97	62	:	:	PUNCT
cana-2729	97	63	𝜌2	𝜌2	VERB
cana-2729	97	64	⊆	⊆	NUM
cana-2729	97	65	𝐾𝑖	𝐾𝑖	PUNCT
cana-2729	97	66	−1}(𝑙11	−1}(𝑙11	ADJ
cana-2729	97	67	−1	−1	NOUN
cana-2729	97	68	)	)	PUNCT
cana-2729	97	69	=	=	PUNCT
cana-2729	97	70	𝐵𝑅𝐾𝑐𝑙(𝜌2)(𝑙11	𝐵𝑅𝐾𝑐𝑙(𝜌2)(𝑙11	ADV
cana-2729	97	71	−1	−1	NOUN
cana-2729	97	72	)	)	PUNCT
cana-2729	97	73	=	=	PUNCT
cana-2729	97	74	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	PROPN
cana-2729	97	75	−1)(𝑙11	−1)(𝑙11	ADV
cana-2729	97	76	)	)	PUNCT
cana-2729	97	77	we	we	PRON
cana-2729	97	78	get	get	VERB
cana-2729	97	79	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	PROPN
cana-2729	97	80	−1	−1	NOUN
cana-2729	97	81	)	)	PUNCT
cana-2729	97	82	=	=	PUNCT
cana-2729	97	83	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	PROPN
cana-2729	97	84	−1	−1	NUM
cana-2729	97	85	)	)	PUNCT
cana-2729	97	86	.	.	PUNCT
cana-2729	98	1	hence	hence	ADV
cana-2729	98	2	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	98	3	)	)	PUNCT
cana-2729	98	4	⋆	⋆	VERB
cana-2729	98	5	𝐵𝑅𝐾𝑐𝑙(𝜌2)−1	𝐵𝑅𝐾𝑐𝑙(𝜌2)−1	NOUN
cana-2729	98	6	⊆	⊆	NUM
cana-2729	98	7	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	98	8	⋆	⋆	VERB
cana-2729	98	9	𝜌2	𝜌2	ADJ
cana-2729	98	10	−1	−1	NOUN
cana-2729	98	11	)	)	PUNCT
cana-2729	98	12	.	.	PUNCT
cana-2729	99	1	similarly	similarly	ADV
cana-2729	99	2	,	,	PUNCT
cana-2729	99	3	we	we	PRON
cana-2729	99	4	have	have	VERB
cana-2729	99	5	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	99	6	)	)	PUNCT
cana-2729	99	7	⋆	⋆	X
cana-2729	99	8	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	99	9	)	)	PUNCT
cana-2729	99	10	⊆	⊆	NUM
cana-2729	99	11	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	99	12	⋆	⋆	VERB
cana-2729	99	13	𝜌2	𝜌2	ADJ
cana-2729	99	14	−1	−1	NOUN
cana-2729	99	15	)	)	PUNCT
cana-2729	99	16	.	.	PUNCT
cana-2729	100	1	theorem	theorem	ADJ
cana-2729	100	2	3.4.7	3.4.7	NOUN
cana-2729	100	3	:	:	PUNCT
cana-2729	100	4	let	let	VERB
cana-2729	100	5	(	(	PUNCT
cana-2729	100	6	𝐺𝑡.⋆	𝐺𝑡.⋆	NOUN
cana-2729	100	7	,	,	PUNCT
cana-2729	100	8	0	0	NUM
cana-2729	100	9	,	,	PUNCT
cana-2729	100	10	𝑓γ	𝑓γ	PROPN
cana-2729	100	11	)	)	PUNCT
cana-2729	100	12	be	be	AUX
cana-2729	100	13	an	an	DET
cana-2729	100	14	𝑓𝐵𝑅𝐾𝑡𝑔	𝑓𝐵𝑅𝐾𝑡𝑔	ADJ
cana-2729	100	15	and	and	CCONJ
cana-2729	100	16	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	100	17	)	)	PUNCT
cana-2729	100	18	×	×	PROPN
cana-2729	100	19	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	100	20	)	)	PUNCT
cana-2729	100	21	⊆	⊆	NUM
cana-2729	100	22	𝐵𝑅𝐾𝑐𝑙(𝜌1	𝐵𝑅𝐾𝑐𝑙(𝜌1	PROPN
cana-2729	100	23	×	×	NOUN
cana-2729	100	24	𝜌2	𝜌2	ADJ
cana-2729	100	25	−1	−1	NOUN
cana-2729	100	26	)	)	PUNCT
cana-2729	100	27	.	.	PUNCT
cana-2729	101	1	(	(	PUNCT
cana-2729	101	2	i	i	NOUN
cana-2729	101	3	)	)	PUNCT
cana-2729	101	4	if	if	SCONJ
cana-2729	101	5	𝜌2	𝜌2	ADJ
cana-2729	101	6	is	be	AUX
cana-2729	101	7	fuzzy	fuzzy	ADJ
cana-2729	101	8	subgroup	subgroup	NOUN
cana-2729	101	9	(	(	PUNCT
cana-2729	101	10	𝑏𝑟𝑖𝑒𝑓𝑙𝑦	𝑏𝑟𝑖𝑒𝑓𝑙𝑦	PROPN
cana-2729	101	11	,	,	PUNCT
cana-2729	101	12	𝑓𝑠𝑔𝑟𝑝	𝑓𝑠𝑔𝑟𝑝	ADJ
cana-2729	101	13	)	)	PUNCT
cana-2729	101	14	of	of	ADP
cana-2729	101	15	𝐺𝑡	𝐺𝑡	PROPN
cana-2729	101	16	then	then	ADV
cana-2729	101	17	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	101	18	)	)	PUNCT
cana-2729	101	19	is	be	AUX
cana-2729	101	20	also	also	ADV
cana-2729	101	21	𝑓𝑠𝑔𝑟𝑝	𝑓𝑠𝑔𝑟𝑝	ADJ
cana-2729	101	22	of	of	ADP
cana-2729	101	23	𝐺𝑡.	𝐺𝑡.	PROPN
cana-2729	101	24	(	(	PUNCT
cana-2729	101	25	ii	ii	NOUN
cana-2729	101	26	)	)	PUNCT
cana-2729	101	27	if	if	SCONJ
cana-2729	101	28	𝜌2	𝜌2	ADJ
cana-2729	101	29	is	be	AUX
cana-2729	101	30	fuzzy	fuzzy	ADJ
cana-2729	101	31	normal	normal	ADJ
cana-2729	101	32	subgroup	subgroup	NOUN
cana-2729	101	33	(	(	PUNCT
cana-2729	101	34	𝑏𝑟𝑖𝑒𝑓𝑙𝑦	𝑏𝑟𝑖𝑒𝑓𝑙𝑦	PROPN
cana-2729	101	35	,	,	PUNCT
cana-2729	101	36	𝑓𝑁𝑠𝑔𝑟𝑝	𝑓𝑁𝑠𝑔𝑟𝑝	PROPN
cana-2729	101	37	)	)	PUNCT
cana-2729	101	38	of	of	ADP
cana-2729	101	39	𝐺𝑡	𝐺𝑡	PROPN
cana-2729	101	40	then	then	ADV
cana-2729	101	41	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	101	42	)	)	PUNCT
cana-2729	101	43	is	be	AUX
cana-2729	101	44	also	also	ADV
cana-2729	101	45	𝑓𝑁𝑠𝑔𝑟𝑝	𝑓𝑁𝑠𝑔𝑟𝑝	PROPN
cana-2729	101	46	of	of	ADP
cana-2729	101	47	𝐺𝑡.	𝐺𝑡.	PROPN
cana-2729	101	48	proof	proof	NOUN
cana-2729	101	49	:	:	PUNCT
cana-2729	101	50	if	if	SCONJ
cana-2729	101	51	𝜌2	𝜌2	ADJ
cana-2729	101	52	⋆	⋆	VERB
cana-2729	101	53	𝜌2	𝜌2	ADJ
cana-2729	101	54	⊆	⊆	NUM
cana-2729	101	55	𝜌2	𝜌2	ADJ
cana-2729	101	56	⟹	⟹	PRON
cana-2729	101	57	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	PROPN
cana-2729	101	58	⋆	⋆	VERB
cana-2729	101	59	𝜌2	𝜌2	ADJ
cana-2729	101	60	)	)	PUNCT
cana-2729	101	61	⊆	⊆	NUM
cana-2729	101	62	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	101	63	)	)	PUNCT
cana-2729	101	64	.	.	PUNCT
cana-2729	102	1	by	by	ADP
cana-2729	102	2	the	the	DET
cana-2729	102	3	above	above	ADJ
cana-2729	102	4	theorem	theorem	NOUN
cana-2729	102	5	,	,	PUNCT
cana-2729	102	6	we	we	PRON
cana-2729	102	7	have	have	VERB
cana-2729	102	8	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	102	9	)	)	PUNCT
cana-2729	102	10	⋆	⋆	X
cana-2729	102	11	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	102	12	)	)	PUNCT
cana-2729	102	13	⊆	⊆	NUM
cana-2729	102	14	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	PROPN
cana-2729	102	15	⋆	⋆	VERB
cana-2729	102	16	𝜌2	𝜌2	ADJ
cana-2729	102	17	)	)	PUNCT
cana-2729	102	18	and	and	CCONJ
cana-2729	102	19	so	so	ADV
cana-2729	102	20	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	102	21	)	)	PUNCT
cana-2729	102	22	⋆	⋆	X
cana-2729	102	23	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	102	24	)	)	PUNCT
cana-2729	102	25	⊆	⊆	NUM
cana-2729	102	26	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	PROPN
cana-2729	102	27	⋆	⋆	VERB
cana-2729	102	28	𝜌2	𝜌2	ADJ
cana-2729	102	29	)	)	PUNCT
cana-2729	102	30	(	(	PUNCT
cana-2729	102	31	3.3	3.3	NUM
cana-2729	102	32	)	)	PUNCT
cana-2729	102	33	let	let	VERB
cana-2729	102	34	𝜌2	𝜌2	ADJ
cana-2729	102	35	is	be	AUX
cana-2729	102	36	a	a	DET
cana-2729	102	37	𝑓𝑁𝑠𝑔𝑟𝑝	𝑓𝑁𝑠𝑔𝑟𝑝	PROPN
cana-2729	102	38	of	of	ADP
cana-2729	102	39	𝐺𝑡.	𝐺𝑡.	PROPN
cana-2729	102	40	then	then	ADV
cana-2729	102	41	𝜌2(𝑠1	𝜌2(𝑠1	NUM
cana-2729	102	42	⋆	⋆	ADJ
cana-2729	102	43	𝑠2	𝑠2	NOUN
cana-2729	102	44	)	)	PUNCT
cana-2729	102	45	=	=	PUNCT
cana-2729	102	46	𝜌2(𝑠2	𝜌2(𝑠2	NUM
cana-2729	102	47	⋆	⋆	X
cana-2729	102	48	𝑠1	𝑠1	PROPN
cana-2729	102	49	)	)	PUNCT
cana-2729	102	50	for	for	ADP
cana-2729	102	51	any	any	DET
cana-2729	102	52	𝑠1	𝑠1	NOUN
cana-2729	102	53	,	,	PUNCT
cana-2729	102	54	𝑠2	𝑠2	PROPN
cana-2729	102	55	∈	∈	PROPN
cana-2729	102	56	𝐺𝑡	𝐺𝑡	PROPN
cana-2729	102	57	and	and	CCONJ
cana-2729	102	58	hence	hence	ADV
cana-2729	102	59	𝑙11𝜌2𝑙11	𝑙11𝜌2𝑙11	NUM
cana-2729	102	60	−1(𝑙33	−1(𝑙33	PROPN
cana-2729	102	61	)	)	PUNCT
cana-2729	102	62	=	=	PUNCT
cana-2729	103	1	𝜌2(𝑙11	𝜌2(𝑙11	PROPN
cana-2729	103	2	−1	−1	NOUN
cana-2729	103	3	⋆	⋆	X
cana-2729	103	4	(	(	PUNCT
cana-2729	103	5	𝑙33	𝑙33	NOUN
cana-2729	103	6	⋆	⋆	VERB
cana-2729	103	7	𝑙11	𝑙11	NUM
cana-2729	103	8	)	)	PUNCT
cana-2729	103	9	)	)	PUNCT
cana-2729	104	1	=	=	SYM
cana-2729	104	2	𝜌2(𝑙33	𝜌2(𝑙33	NOUN
cana-2729	104	3	)	)	PUNCT
cana-2729	104	4	.	.	PUNCT
cana-2729	105	1	since	since	SCONJ
cana-2729	105	2	𝜌2	𝜌2	ADJ
cana-2729	105	3	is	be	AUX
cana-2729	105	4	𝑓𝑠𝑔𝑟𝑝	𝑓𝑠𝑔𝑟𝑝	ADJ
cana-2729	105	5	,	,	PUNCT
cana-2729	105	6	𝜌2(𝑙11	𝜌2(𝑙11	PROPN
cana-2729	105	7	)	)	PUNCT
cana-2729	105	8	=	=	SYM
cana-2729	105	9	𝜌2(𝑙11	𝜌2(𝑙11	PROPN
cana-2729	105	10	−1	−1	NOUN
cana-2729	105	11	)	)	PUNCT
cana-2729	105	12	=	=	PUNCT
cana-2729	105	13	𝜌2	𝜌2	ADJ
cana-2729	105	14	−1(𝑙11	−1(𝑙11	PROPN
cana-2729	105	15	)	)	PUNCT
cana-2729	105	16	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
cana-2729	105	17	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
cana-2729	105	18	𝑙11	𝑙11	NUM
cana-2729	105	19	∈	∈	PROPN
cana-2729	105	20	𝐺𝑡.	𝐺𝑡.	PROPN
cana-2729	106	1	this	this	PRON
cana-2729	106	2	leads	lead	VERB
cana-2729	106	3	to	to	ADP
cana-2729	106	4	𝜌2	𝜌2	ADJ
cana-2729	106	5	=	=	SYM
cana-2729	106	6	𝜌2	𝜌2	ADJ
cana-2729	106	7	−1	−1	NOUN
cana-2729	106	8	and	and	CCONJ
cana-2729	106	9	hence	hence	ADV
cana-2729	106	10	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	106	11	)	)	PUNCT
cana-2729	107	1	=	=	PUNCT
cana-2729	107	2	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	PROPN
cana-2729	107	3	−1	−1	NUM
cana-2729	107	4	)	)	PUNCT
cana-2729	107	5	.	.	PUNCT
cana-2729	108	1	now	now	ADV
cana-2729	108	2	,	,	PUNCT
cana-2729	108	3	we	we	PRON
cana-2729	108	4	have	have	VERB
cana-2729	108	5	to	to	PART
cana-2729	108	6	show	show	VERB
cana-2729	108	7	that	that	SCONJ
cana-2729	108	8	for	for	ADP
cana-2729	108	9	every	every	DET
cana-2729	108	10	𝑙11	𝑙11	ADJ
cana-2729	108	11	∈	∈	PROPN
cana-2729	108	12	𝐺𝑡	𝐺𝑡	PROPN
cana-2729	108	13	,	,	PUNCT
cana-2729	108	14	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	PROPN
cana-2729	108	15	−1)(𝑙11	−1)(𝑙11	ADV
cana-2729	108	16	)	)	PUNCT
cana-2729	108	17	=	=	SYM
cana-2729	108	18	𝐵𝑅𝐾𝑐𝑙(𝜌2)−1	𝐵𝑅𝐾𝑐𝑙(𝜌2)−1	NOUN
cana-2729	108	19	.	.	PUNCT
cana-2729	108	20	by	by	ADP
cana-2729	108	21	using	use	VERB
cana-2729	108	22	the	the	DET
cana-2729	108	23	same	same	ADJ
cana-2729	108	24	method	method	NOUN
cana-2729	108	25	as	as	ADP
cana-2729	108	26	above	above	ADV
cana-2729	108	27	,	,	PUNCT
cana-2729	108	28	we	we	PRON
cana-2729	108	29	have	have	VERB
cana-2729	108	30	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	108	31	−1)(𝑙11	−1)(𝑙11	NUM
cana-2729	108	32	)	)	PUNCT
cana-2729	108	33	=	=	SYM
cana-2729	108	34	𝐵𝑅𝐾𝑐𝑙(𝜌2)−1(𝑙11	𝐵𝑅𝐾𝑐𝑙(𝜌2)−1(𝑙11	PROPN
cana-2729	108	35	)	)	PUNCT
cana-2729	108	36	=	=	PUNCT
cana-2729	108	37	𝐵𝑅𝐾𝑐𝑙(𝜌2)(𝑙11	𝐵𝑅𝐾𝑐𝑙(𝜌2)(𝑙11	ADV
cana-2729	108	38	−1	−1	NOUN
cana-2729	108	39	)	)	PUNCT
cana-2729	108	40	(	(	PUNCT
cana-2729	108	41	3.4	3.4	NUM
cana-2729	108	42	)	)	PUNCT
cana-2729	108	43	from	from	ADP
cana-2729	108	44	(	(	PUNCT
cana-2729	108	45	3.3	3.3	NUM
cana-2729	108	46	)	)	PUNCT
cana-2729	108	47	or	or	CCONJ
cana-2729	108	48	(	(	PUNCT
cana-2729	108	49	3.4	3.4	NUM
cana-2729	108	50	)	)	PUNCT
cana-2729	108	51	,	,	PUNCT
cana-2729	108	52	we	we	PRON
cana-2729	108	53	have	have	AUX
cana-2729	108	54	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	108	55	)	)	PUNCT
cana-2729	108	56	is	be	AUX
cana-2729	108	57	𝑓𝑠𝑔𝑟𝑝	𝑓𝑠𝑔𝑟𝑝	ADJ
cana-2729	108	58	of	of	ADP
cana-2729	108	59	𝐺𝑡	𝐺𝑡	PROPN
cana-2729	108	60	.	.	PUNCT
cana-2729	109	1	let	let	VERB
cana-2729	109	2	𝜌2	𝜌2	ADJ
cana-2729	109	3	is	be	AUX
cana-2729	109	4	a	a	DET
cana-2729	109	5	𝑓𝑁𝑠𝑔𝑟𝑝	𝑓𝑁𝑠𝑔𝑟𝑝	PROPN
cana-2729	109	6	of	of	ADP
cana-2729	109	7	𝐺𝑡.	𝐺𝑡.	PROPN
cana-2729	109	8	then	then	ADV
cana-2729	109	9	𝜌2(𝑠1	𝜌2(𝑠1	NUM
cana-2729	109	10	⋆	⋆	ADJ
cana-2729	109	11	𝑠2	𝑠2	NOUN
cana-2729	109	12	)	)	PUNCT
cana-2729	109	13	=	=	PUNCT
cana-2729	109	14	𝜌2(𝑠2	𝜌2(𝑠2	NUM
cana-2729	109	15	⋆	⋆	X
cana-2729	109	16	𝑠1	𝑠1	PROPN
cana-2729	109	17	)	)	PUNCT
cana-2729	109	18	for	for	ADP
cana-2729	109	19	any	any	DET
cana-2729	109	20	𝑠1	𝑠1	NOUN
cana-2729	109	21	,	,	PUNCT
cana-2729	109	22	𝑠2	𝑠2	PROPN
cana-2729	109	23	∈	∈	PROPN
cana-2729	109	24	𝐺𝑡	𝐺𝑡	PROPN
cana-2729	109	25	and	and	CCONJ
cana-2729	109	26	hence	hence	ADV
cana-2729	109	27	𝑙11𝜌2𝑙11	𝑙11𝜌2𝑙11	NUM
cana-2729	109	28	−1(𝑙33	−1(𝑙33	PROPN
cana-2729	109	29	)	)	PUNCT
cana-2729	109	30	=	=	PUNCT
cana-2729	110	1	𝜌2(𝑙11	𝜌2(𝑙11	PROPN
cana-2729	110	2	−1	−1	NOUN
cana-2729	110	3	⋆	⋆	X
cana-2729	110	4	(	(	PUNCT
cana-2729	110	5	𝑙33	𝑙33	NOUN
cana-2729	110	6	⋆	⋆	VERB
cana-2729	110	7	𝑙11	𝑙11	NUM
cana-2729	110	8	)	)	PUNCT
cana-2729	110	9	)	)	PUNCT
cana-2729	111	1	=	=	SYM
cana-2729	111	2	𝜌2(𝑙33	𝜌2(𝑙33	NOUN
cana-2729	111	3	)	)	PUNCT
cana-2729	111	4	.	.	PUNCT
cana-2729	112	1	i.e.	i.e.	X
cana-2729	112	2	(	(	PUNCT
cana-2729	112	3	(	(	PUNCT
cana-2729	112	4	𝑙11	𝑙11	ADV
cana-2729	112	5	⋆	⋆	VERB
cana-2729	112	6	𝜌2	𝜌2	ADJ
cana-2729	112	7	)	)	PUNCT
cana-2729	112	8	⋆	⋆	VERB
cana-2729	112	9	𝑙11	𝑙11	NUM
cana-2729	112	10	−1	−1	NOUN
cana-2729	112	11	)	)	PUNCT
cana-2729	113	1	=	=	PUNCT
cana-2729	113	2	𝜌2	𝜌2	ADJ
cana-2729	113	3	and	and	CCONJ
cana-2729	113	4	we	we	PRON
cana-2729	113	5	hence	hence	ADV
cana-2729	113	6	get	get	VERB
cana-2729	113	7	𝐵𝑅𝐾𝑐𝑙((𝑙11	𝐵𝑅𝐾𝑐𝑙((𝑙11	ADV
cana-2729	113	8	⋆	⋆	VERB
cana-2729	113	9	𝜌2	𝜌2	ADJ
cana-2729	113	10	)	)	PUNCT
cana-2729	113	11	⋆	⋆	VERB
cana-2729	113	12	𝑙11	𝑙11	NUM
cana-2729	113	13	−1	−1	NOUN
cana-2729	113	14	)	)	PUNCT
cana-2729	113	15	=	=	SYM
cana-2729	113	16	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	113	17	)	)	PUNCT
cana-2729	113	18	,	,	PUNCT
cana-2729	113	19	by	by	ADP
cana-2729	113	20	theorem	theorem	NOUN
cana-2729	113	21	3.4.5	3.4.5	PROPN
cana-2729	113	22	.	.	PUNCT
cana-2729	114	1	this	this	PRON
cana-2729	114	2	shows	show	VERB
cana-2729	114	3	that	that	SCONJ
cana-2729	114	4	(	(	PUNCT
cana-2729	114	5	(	(	PUNCT
cana-2729	114	6	𝑙11	𝑙11	ADV
cana-2729	114	7	⋆	⋆	VERB
cana-2729	114	8	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	114	9	)	)	PUNCT
cana-2729	114	10	)	)	PUNCT
cana-2729	115	1	⋆	⋆	VERB
cana-2729	115	2	𝑙11	𝑙11	NUM
cana-2729	115	3	−1	−1	NOUN
cana-2729	115	4	)	)	PUNCT
cana-2729	115	5	=	=	SYM
cana-2729	115	6	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	115	7	)	)	PUNCT
cana-2729	115	8	,	,	PUNCT
cana-2729	115	9	for	for	ADP
cana-2729	115	10	every	every	DET
cana-2729	115	11	𝑙11	𝑙11	ADJ
cana-2729	115	12	∈	∈	PROPN
cana-2729	115	13	𝐺𝑡.	𝐺𝑡.	PROPN
cana-2729	115	14	consequently	consequently	ADV
cana-2729	115	15	,	,	PUNCT
cana-2729	115	16	we	we	PRON
cana-2729	115	17	deduce	deduce	VERB
cana-2729	115	18	that	that	SCONJ
cana-2729	115	19	𝐵𝑅𝐾𝑐𝑙(𝜌2)(𝑙11	𝐵𝑅𝐾𝑐𝑙(𝜌2)(𝑙11	ADV
cana-2729	115	20	⋆	⋆	VERB
cana-2729	115	21	𝑙22	𝑙22	NOUN
cana-2729	115	22	)	)	PUNCT
cana-2729	115	23	=	=	SYM
cana-2729	115	24	(	(	PUNCT
cana-2729	115	25	𝐵𝑅𝐾𝑐𝑙(𝜌2)(𝑙11	𝐵𝑅𝐾𝑐𝑙(𝜌2)(𝑙11	ADJ
cana-2729	115	26	−1	−1	NOUN
cana-2729	115	27	)	)	PUNCT
cana-2729	115	28	)	)	PUNCT
cana-2729	116	1	(	(	PUNCT
cana-2729	116	2	(	(	PUNCT
cana-2729	116	3	𝑙11	𝑙11	PART
cana-2729	116	4	−1	−1	NOUN
cana-2729	116	5	⋆	⋆	NOUN
cana-2729	116	6	𝑙11	𝑙11	NUM
cana-2729	116	7	)	)	PUNCT
cana-2729	116	8	)	)	PUNCT
cana-2729	117	1	⋆	⋆	VERB
cana-2729	117	2	𝑙22	𝑙22	NOUN
cana-2729	117	3	)	)	PUNCT
cana-2729	117	4	communications	communication	NOUN
cana-2729	117	5	on	on	ADP
cana-2729	117	6	applied	apply	VERB
cana-2729	117	7	nonlinear	nonlinear	ADJ
cana-2729	117	8	analysis	analysis	NOUN
cana-2729	117	9	issn	issn	NOUN
cana-2729	117	10	:	:	PUNCT
cana-2729	117	11	1074	1074	NUM
cana-2729	117	12	-	-	PUNCT
cana-2729	117	13	133x	133x	NUM
cana-2729	117	14	vol	vol	NOUN
cana-2729	117	15	32	32	NUM
cana-2729	117	16	no	no	NOUN
cana-2729	117	17	.	.	PUNCT
cana-2729	118	1	3s	3s	NUM
cana-2729	118	2	(	(	PUNCT
cana-2729	118	3	2025	2025	NUM
cana-2729	118	4	)	)	PUNCT
cana-2729	118	5	715	715	NUM
cana-2729	118	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2729	118	7	=	=	PUNCT
cana-2729	118	8	𝐵𝑅𝐾𝑐𝑙(𝜌2)(𝑙11	𝐵𝑅𝐾𝑐𝑙(𝜌2)(𝑙11	ADV
cana-2729	118	9	⋆	⋆	VERB
cana-2729	118	10	𝑙22	𝑙22	NOUN
cana-2729	118	11	)	)	PUNCT
cana-2729	118	12	.	.	PUNCT
cana-2729	119	1	thus	thus	ADV
cana-2729	119	2	,	,	PUNCT
cana-2729	119	3	𝐵𝑅𝐾𝑐𝑙(𝜌2	𝐵𝑅𝐾𝑐𝑙(𝜌2	NOUN
cana-2729	119	4	)	)	PUNCT
cana-2729	119	5	is	be	AUX
cana-2729	119	6	a	a	DET
cana-2729	119	7	𝑓𝑁𝑠𝑔𝑟𝑝	𝑓𝑁𝑠𝑔𝑟𝑝	PROPN
cana-2729	119	8	of	of	ADP
cana-2729	119	9	𝐺𝑡	𝐺𝑡	PROPN
cana-2729	119	10	.	.	PUNCT
cana-2729	120	1	theorem	theorem	PROPN
cana-2729	120	2	3.4.8	3.4.8	NUM
cana-2729	120	3	:	:	PUNCT
cana-2729	120	4	let	let	VERB
cana-2729	120	5	(	(	PUNCT
cana-2729	120	6	𝐺𝑡.⋆	𝐺𝑡.⋆	NOUN
cana-2729	120	7	,	,	PUNCT
cana-2729	120	8	0	0	NUM
cana-2729	120	9	,	,	PUNCT
cana-2729	120	10	𝑓γ	𝑓γ	PROPN
cana-2729	120	11	)	)	PUNCT
cana-2729	120	12	&	&	CCONJ
cana-2729	120	13	(	(	PUNCT
cana-2729	120	14	𝐻𝑡.⋆	𝐻𝑡.⋆	PROPN
cana-2729	120	15	,	,	PUNCT
cana-2729	120	16	0	0	NUM
cana-2729	120	17	,	,	PUNCT
cana-2729	120	18	𝑓γ	𝑓γ	PROPN
cana-2729	120	19	)	)	PUNCT
cana-2729	120	20	be	be	AUX
cana-2729	120	21	two	two	NUM
cana-2729	120	22	𝑓𝐵𝑅𝐾𝑡𝑔	𝑓𝐵𝑅𝐾𝑡𝑔	NOUN
cana-2729	120	23	’s	’s	NOUN
cana-2729	120	24	and	and	CCONJ
cana-2729	120	25	h	h	NOUN
cana-2729	120	26	is	be	AUX
cana-2729	120	27	a	a	DET
cana-2729	120	28	𝑓𝐵𝑅𝐾𝐻𝑜𝑚	𝑓𝐵𝑅𝐾𝐻𝑜𝑚	PROPN
cana-2729	120	29	of	of	ADP
cana-2729	120	30	𝐺𝑡	𝐺𝑡	PROPN
cana-2729	120	31	into	into	ADP
cana-2729	120	32	𝐻𝑡	𝐻𝑡	PROPN
cana-2729	120	33	,	,	PUNCT
cana-2729	120	34	then	then	ADV
cana-2729	120	35	(	(	PUNCT
cana-2729	120	36	i	i	NOUN
cana-2729	120	37	)	)	PUNCT
cana-2729	120	38	for	for	ADP
cana-2729	120	39	any	any	DET
cana-2729	120	40	𝑓𝑠′𝑠	𝑓𝑠′𝑠	NOUN
cana-2729	120	41	𝜁1and	𝜁1and	CCONJ
cana-2729	120	42	𝜁2	𝜁2	NOUN
cana-2729	120	43	of	of	ADP
cana-2729	120	44	𝐻𝑡	𝐻𝑡	PROPN
cana-2729	120	45	,	,	PUNCT
cana-2729	120	46	𝐵𝑅𝐾𝑐𝑙(ℎ−1(𝜁1	𝐵𝑅𝐾𝑐𝑙(ℎ−1(𝜁1	NOUN
cana-2729	120	47	)	)	PUNCT
cana-2729	120	48	)	)	PUNCT
cana-2729	121	1	⋆	⋆	CCONJ
cana-2729	121	2	𝐵𝑅𝐾𝑐𝑙	𝐵𝑅𝐾𝑐𝑙	PROPN
cana-2729	121	3	(	(	PUNCT
cana-2729	121	4	ℎ−1(𝜁2	ℎ−1(𝜁2	NOUN
cana-2729	121	5	)	)	PUNCT
cana-2729	121	6	)	)	PUNCT
cana-2729	122	1	⊆	⊆	NUM
cana-2729	122	2	𝐵𝑅𝑘𝑐𝑙(ℎ−1(𝜁1	𝐵𝑅𝑘𝑐𝑙(ℎ−1(𝜁1	NUM
cana-2729	122	3	⋆	⋆	PUNCT
cana-2729	122	4	𝜁2	𝜁2	NOUN
cana-2729	122	5	)	)	PUNCT
cana-2729	122	6	)	)	PUNCT
cana-2729	122	7	.	.	PUNCT
cana-2729	123	1	(	(	PUNCT
cana-2729	123	2	ii	ii	NOUN
cana-2729	123	3	)	)	PUNCT
cana-2729	123	4	for	for	ADP
cana-2729	123	5	any	any	DET
cana-2729	123	6	𝑓𝑠′𝑠	𝑓𝑠′𝑠	NOUN
cana-2729	123	7	𝜁1and	𝜁1and	CCONJ
cana-2729	123	8	𝜁2	𝜁2	NOUN
cana-2729	123	9	of	of	ADP
cana-2729	123	10	𝐺𝑡	𝐺𝑡	PROPN
cana-2729	123	11	,	,	PUNCT
cana-2729	123	12	𝐵𝑅𝐾𝑐𝑙(ℎ(𝜁1	𝐵𝑅𝐾𝑐𝑙(ℎ(𝜁1	NOUN
cana-2729	123	13	)	)	PUNCT
cana-2729	123	14	)	)	PUNCT
cana-2729	124	1	⋆	⋆	VERB
cana-2729	124	2	𝐵𝑅𝐾𝑐𝑙(ℎ(𝜁2	𝐵𝑅𝐾𝑐𝑙(ℎ(𝜁2	NOUN
cana-2729	124	3	)	)	PUNCT
cana-2729	124	4	)	)	PUNCT
cana-2729	125	1	⊆	⊆	NUM
cana-2729	125	2	𝐵𝑅𝑘𝑐𝑙(ℎ(𝜁1	𝐵𝑅𝑘𝑐𝑙(ℎ(𝜁1	NUM
cana-2729	125	3	⋆	⋆	VERB
cana-2729	125	4	𝜁2	𝜁2	NOUN
cana-2729	125	5	)	)	PUNCT
cana-2729	125	6	)	)	PUNCT
cana-2729	125	7	.	.	PUNCT
cana-2729	126	1	proof	proof	NOUN
cana-2729	126	2	:	:	PUNCT
cana-2729	126	3	let	let	VERB
cana-2729	126	4	𝜁1&𝜁2	𝜁1&𝜁2	PROPN
cana-2729	126	5	be	be	AUX
cana-2729	126	6	two	two	NUM
cana-2729	126	7	𝑓𝑠′𝑠	𝑓𝑠′𝑠	NOUN
cana-2729	126	8	of	of	ADP
cana-2729	126	9	𝐻𝑡	𝐻𝑡	PROPN
cana-2729	126	10	,	,	PUNCT
cana-2729	126	11	since	since	SCONJ
cana-2729	126	12	(	(	PUNCT
cana-2729	126	13	𝐺𝑡,⋆	𝐺𝑡,⋆	INTJ
cana-2729	126	14	,	,	PUNCT
cana-2729	126	15	0	0	NUM
cana-2729	126	16	,	,	PUNCT
cana-2729	126	17	𝑓γ	𝑓γ	PROPN
cana-2729	126	18	)	)	PUNCT
cana-2729	126	19	and	and	CCONJ
cana-2729	126	20	(	(	PUNCT
cana-2729	126	21	𝐻𝑡,⋆	𝐻𝑡,⋆	NOUN
cana-2729	126	22	,	,	PUNCT
cana-2729	126	23	0	0	NUM
cana-2729	126	24	,	,	PUNCT
cana-2729	126	25	𝑓γ	𝑓γ	PROPN
cana-2729	126	26	)	)	PUNCT
cana-2729	126	27	are	be	AUX
cana-2729	126	28	two	two	NUM
cana-2729	126	29	𝑓𝐵𝑅𝐾𝑡𝑔′𝑠	𝑓𝐵𝑅𝐾𝑡𝑔′𝑠	NOUN
cana-2729	126	30	,	,	PUNCT
cana-2729	126	31	there	there	PRON
cana-2729	126	32	exists	exist	VERB
cana-2729	126	33	a	a	DET
cana-2729	126	34	𝑓𝐵𝑅𝐾𝑐𝑡𝑠	𝑓𝐵𝑅𝐾𝑐𝑡𝑠	NOUN
cana-2729	126	35	map	map	NOUN
cana-2729	126	36	𝑔(𝑙11	𝑔(𝑙11	ADV
cana-2729	126	37	,	,	PUNCT
cana-2729	126	38	𝑙22	𝑙22	PROPN
cana-2729	126	39	)	)	PUNCT
cana-2729	126	40	=	=	VERB
cana-2729	127	1	𝑙11	𝑙11	PRON
cana-2729	127	2	⋆	⋆	VERB
cana-2729	127	3	𝑙22	𝑙22	PROPN
cana-2729	127	4	such	such	ADJ
cana-2729	127	5	that	that	DET
cana-2729	127	6	𝑔(𝐵𝑅𝐾𝑐𝑙(𝜂1	𝑔(𝐵𝑅𝐾𝑐𝑙(𝜂1	NOUN
cana-2729	127	7	)	)	PUNCT
cana-2729	127	8	×	×	NOUN
cana-2729	127	9	𝐵𝑅𝐾𝑐𝑙(𝜂2	𝐵𝑅𝐾𝑐𝑙(𝜂2	PROPN
cana-2729	127	10	)	)	PUNCT
cana-2729	127	11	)	)	PUNCT
cana-2729	128	1	⊆	⊆	NUM
cana-2729	128	2	𝐵𝑅𝑘𝑐𝑙(𝑔(𝜂1	𝐵𝑅𝑘𝑐𝑙(𝑔(𝜂1	NUM
cana-2729	128	3	⋆	⋆	X
cana-2729	128	4	𝜂2	𝜂2	NOUN
cana-2729	128	5	)	)	PUNCT
cana-2729	128	6	)	)	PUNCT
cana-2729	128	7	,	,	PUNCT
cana-2729	128	8	𝐵𝑅𝐾𝑐𝑙(𝜂1	𝐵𝑅𝐾𝑐𝑙(𝜂1	PROPN
cana-2729	128	9	)	)	PUNCT
cana-2729	128	10	⋆	⋆	VERB
cana-2729	128	11	𝐵𝑅𝐾𝑐𝑙(𝜂2	𝐵𝑅𝐾𝑐𝑙(𝜂2	PROPN
cana-2729	128	12	)	)	PUNCT
cana-2729	128	13	⊆	⊆	NUM
cana-2729	128	14	𝐵𝑅𝑘𝑐𝑙(𝜂1	𝐵𝑅𝑘𝑐𝑙(𝜂1	NOUN
cana-2729	128	15	⋆	⋆	VERB
cana-2729	128	16	𝜂2	𝜂2	PROPN
cana-2729	128	17	)	)	PUNCT
cana-2729	128	18	,	,	PUNCT
cana-2729	128	19	put	put	VERB
cana-2729	128	20	𝜂1	𝜂1	NOUN
cana-2729	128	21	=	=	PUNCT
cana-2729	128	22	ℎ	ℎ	PART
cana-2729	128	23	−1(𝜁1	−1(𝜁1	NOUN
cana-2729	128	24	)	)	PUNCT
cana-2729	128	25	and	and	CCONJ
cana-2729	128	26	𝜂2	𝜂2	X
cana-2729	128	27	=	=	SYM
cana-2729	128	28	ℎ	ℎ	PART
cana-2729	128	29	−1(𝜁2	−1(𝜁2	PROPN
cana-2729	128	30	)	)	PUNCT
cana-2729	128	31	,	,	PUNCT
cana-2729	128	32	we	we	PRON
cana-2729	128	33	get	get	VERB
cana-2729	128	34	𝐵𝑅𝐾𝑐𝑙(ℎ−1(𝜁1	𝐵𝑅𝐾𝑐𝑙(ℎ−1(𝜁1	NOUN
cana-2729	128	35	)	)	PUNCT
cana-2729	128	36	)	)	PUNCT
cana-2729	129	1	⋆	⋆	VERB
cana-2729	129	2	𝐵𝑅𝐾𝑐𝑙(ℎ−1(𝜁2	𝐵𝑅𝐾𝑐𝑙(ℎ−1(𝜁2	NOUN
cana-2729	129	3	)	)	PUNCT
cana-2729	129	4	)	)	PUNCT
cana-2729	130	1	⊆	⊆	NUM
cana-2729	130	2	𝐵𝑅𝑘𝑐𝑙(ℎ−1(𝜁1	𝐵𝑅𝑘𝑐𝑙(ℎ−1(𝜁1	NUM
cana-2729	130	3	)	)	PUNCT
cana-2729	130	4	⋆	⋆	VERB
cana-2729	130	5	ℎ	ℎ	PART
cana-2729	130	6	−1(𝜁2	−1(𝜁2	NOUN
cana-2729	130	7	)	)	PUNCT
cana-2729	130	8	)	)	PUNCT
cana-2729	130	9	.	.	PUNCT
cana-2729	131	1	since	since	SCONJ
cana-2729	131	2	h	h	NOUN
cana-2729	131	3	is	be	AUX
cana-2729	131	4	a	a	DET
cana-2729	131	5	𝑓𝐵𝑅𝐾𝐻𝑜𝑚	𝑓𝐵𝑅𝐾𝐻𝑜𝑚	PROPN
cana-2729	131	6	,	,	PUNCT
cana-2729	131	7	we	we	PRON
cana-2729	131	8	get	get	VERB
cana-2729	131	9	𝐵𝑅𝐾𝑐𝑙(ℎ−1(𝜁1	𝐵𝑅𝐾𝑐𝑙(ℎ−1(𝜁1	NOUN
cana-2729	131	10	)	)	PUNCT
cana-2729	131	11	)	)	PUNCT
cana-2729	132	1	⋆	⋆	CCONJ
cana-2729	132	2	𝐵𝑅𝐾𝑐𝑙	𝐵𝑅𝐾𝑐𝑙	PROPN
cana-2729	132	3	(	(	PUNCT
cana-2729	132	4	ℎ−1(𝜁2	ℎ−1(𝜁2	NOUN
cana-2729	132	5	)	)	PUNCT
cana-2729	132	6	)	)	PUNCT
cana-2729	133	1	⊆	⊆	NUM
cana-2729	133	2	𝐵𝑅𝑘𝑐𝑙(ℎ−1(𝜁1	𝐵𝑅𝑘𝑐𝑙(ℎ−1(𝜁1	NUM
cana-2729	133	3	⋆	⋆	PUNCT
cana-2729	133	4	𝜁2	𝜁2	NOUN
cana-2729	133	5	)	)	PUNCT
cana-2729	133	6	)	)	PUNCT
cana-2729	133	7	.	.	PUNCT
cana-2729	134	1	the	the	DET
cana-2729	134	2	proof	proof	NOUN
cana-2729	134	3	of	of	ADP
cana-2729	134	4	remaining	remain	VERB
cana-2729	134	5	is	be	AUX
cana-2729	134	6	obvious	obvious	ADJ
cana-2729	134	7	.	.	PUNCT
cana-2729	135	1	theorem	theorem	VERB
cana-2729	135	2	3.4.9	3.4.9	NUM
cana-2729	135	3	:	:	PUNCT
cana-2729	135	4	every	every	DET
cana-2729	135	5	𝑓𝐵𝑅𝐾	𝑓𝐵𝑅𝐾	PROPN
cana-2729	135	6	̇𝜊	̇𝜊	PART
cana-2729	135	7	subgroup	subgroup	PROPN
cana-2729	135	8	ρ	ρ	PROPN
cana-2729	135	9	of	of	ADP
cana-2729	135	10	𝑓𝐵𝑅𝐾𝑡𝑔	𝑓𝐵𝑅𝐾𝑡𝑔	X
cana-2729	135	11	(	(	PUNCT
cana-2729	135	12	𝐺𝑡.⋆	𝐺𝑡.⋆	PROPN
cana-2729	135	13	,	,	PUNCT
cana-2729	135	14	0	0	NUM
cana-2729	135	15	,	,	PUNCT
cana-2729	135	16	𝑓γ	𝑓γ	PROPN
cana-2729	135	17	)	)	PUNCT
cana-2729	135	18	is	be	AUX
cana-2729	135	19	𝑓𝐵𝑅𝐾𝑐.	𝑓𝐵𝑅𝐾𝑐.	NOUN
cana-2729	135	20	proof	proof	NOUN
cana-2729	135	21	:	:	PUNCT
cana-2729	135	22	for	for	ADP
cana-2729	135	23	each	each	DET
cana-2729	135	24	𝑙11	𝑙11	NOUN
cana-2729	135	25	∈	∈	PROPN
cana-2729	135	26	𝐺𝑡	𝐺𝑡	PROPN
cana-2729	135	27	,	,	PUNCT
cana-2729	135	28	𝑙11	𝑙11	PRON
cana-2729	135	29	⋆	⋆	VERB
cana-2729	135	30	𝜌	𝜌	X
cana-2729	135	31	is	be	AUX
cana-2729	135	32	𝑓𝐵𝑅𝐾	𝑓𝐵𝑅𝐾	NOUN
cana-2729	135	33	̇𝜊	̇𝜊	VERB
cana-2729	135	34	by	by	ADP
cana-2729	135	35	corollary	corollary	ADJ
cana-2729	135	36	3.4.1	3.4.1	NUM
cana-2729	135	37	and	and	CCONJ
cana-2729	135	38	hence	hence	ADV
cana-2729	135	39	𝜌	𝜌	X
cana-2729	135	40	=	=	SYM
cana-2729	135	41	(	(	PUNCT
cana-2729	135	42	∪	∪	X
cana-2729	135	43	𝑙11	𝑙11	ADP
cana-2729	135	44	⋆	⋆	VERB
cana-2729	135	45	𝜌)𝑐	𝜌)𝑐	X
cana-2729	135	46	is	be	AUX
cana-2729	135	47	𝑓𝐵𝑅𝐾	𝑓𝐵𝑅𝐾	PROPN
cana-2729	135	48	̇𝜊	̇𝜊	NUM
cana-2729	135	49	,	,	PUNCT
cana-2729	135	50	where	where	SCONJ
cana-2729	135	51	the	the	DET
cana-2729	135	52	union	union	NOUN
cana-2729	135	53	taken	take	VERB
cana-2729	135	54	over	over	ADP
cana-2729	135	55	the	the	DET
cana-2729	135	56	pairwise	pairwise	NOUN
cana-2729	135	57	fuzzy	fuzzy	ADJ
cana-2729	135	58	closets	closet	NOUN
cana-2729	135	59	which	which	PRON
cana-2729	135	60	are	be	AUX
cana-2729	135	61	different	different	ADJ
cana-2729	135	62	from	from	ADP
cana-2729	135	63	ρ	ρ	PROPN
cana-2729	135	64	.	.	PUNCT
cana-2729	136	1	conclusion	conclusion	NOUN
cana-2729	136	2	:	:	PUNCT
cana-2729	136	3	the	the	DET
cana-2729	136	4	study	study	NOUN
cana-2729	136	5	of	of	ADP
cana-2729	136	6	fuzzy	fuzzy	ADJ
cana-2729	136	7	brk	brk	PROPN
cana-2729	136	8	topological	topological	ADJ
cana-2729	136	9	groups	group	NOUN
cana-2729	136	10	represents	represent	VERB
cana-2729	136	11	a	a	DET
cana-2729	136	12	significant	significant	ADJ
cana-2729	136	13	advancement	advancement	NOUN
cana-2729	136	14	in	in	ADP
cana-2729	136	15	both	both	PRON
cana-2729	136	16	fuzzy	fuzzy	ADJ
cana-2729	136	17	set	set	VERB
cana-2729	136	18	theory	theory	NOUN
cana-2729	136	19	and	and	CCONJ
cana-2729	136	20	algebraic	algebraic	ADJ
cana-2729	136	21	structures	structure	NOUN
cana-2729	136	22	.	.	PUNCT
cana-2729	137	1	by	by	ADP
cana-2729	137	2	integrating	integrate	VERB
cana-2729	137	3	the	the	DET
cana-2729	137	4	concepts	concept	NOUN
cana-2729	137	5	of	of	ADP
cana-2729	137	6	brk	brk	PROPN
cana-2729	137	7	-	-	PUNCT
cana-2729	137	8	algebras	algebras	PROPN
cana-2729	137	9	with	with	ADP
cana-2729	137	10	fuzzy	fuzzy	ADJ
cana-2729	137	11	topological	topological	ADJ
cana-2729	137	12	groups	group	NOUN
cana-2729	137	13	,	,	PUNCT
cana-2729	137	14	we	we	PRON
cana-2729	137	15	obtain	obtain	VERB
cana-2729	137	16	a	a	DET
cana-2729	137	17	richer	rich	ADJ
cana-2729	137	18	and	and	CCONJ
cana-2729	137	19	more	more	ADV
cana-2729	137	20	flexible	flexible	ADJ
cana-2729	137	21	framework	framework	NOUN
cana-2729	137	22	for	for	ADP
cana-2729	137	23	addressing	address	VERB
cana-2729	137	24	problems	problem	NOUN
cana-2729	137	25	involving	involve	VERB
cana-2729	137	26	uncertainty	uncertainty	NOUN
cana-2729	137	27	and	and	CCONJ
cana-2729	137	28	imprecision	imprecision	NOUN
cana-2729	137	29	.	.	PUNCT
cana-2729	138	1	this	this	DET
cana-2729	138	2	paper	paper	NOUN
cana-2729	138	3	has	have	AUX
cana-2729	138	4	explored	explore	VERB
cana-2729	138	5	the	the	DET
cana-2729	138	6	theoretical	theoretical	ADJ
cana-2729	138	7	foundations	foundation	NOUN
cana-2729	138	8	of	of	ADP
cana-2729	138	9	fuzzy	fuzzy	ADJ
cana-2729	138	10	brk	brk	PROPN
cana-2729	138	11	topological	topological	ADJ
cana-2729	138	12	groups	group	NOUN
cana-2729	138	13	,	,	PUNCT
cana-2729	138	14	drawing	draw	VERB
cana-2729	138	15	from	from	ADP
cana-2729	138	16	earlier	early	ADJ
cana-2729	138	17	works	work	NOUN
cana-2729	138	18	on	on	ADP
cana-2729	138	19	fuzzy	fuzzy	ADJ
cana-2729	138	20	sets	set	NOUN
cana-2729	138	21	,	,	PUNCT
cana-2729	138	22	fuzzy	fuzzy	ADJ
cana-2729	138	23	topological	topological	ADJ
cana-2729	138	24	groups	group	NOUN
cana-2729	138	25	,	,	PUNCT
cana-2729	138	26	and	and	CCONJ
cana-2729	138	27	brk	brk	PROPN
cana-2729	138	28	-	-	PUNCT
cana-2729	138	29	algebras	algebras	PROPN
cana-2729	138	30	.	.	PUNCT
cana-2729	139	1	our	our	PRON
cana-2729	139	2	analysis	analysis	NOUN
cana-2729	139	3	highlights	highlight	VERB
cana-2729	139	4	the	the	DET
cana-2729	139	5	importance	importance	NOUN
cana-2729	139	6	of	of	ADP
cana-2729	139	7	these	these	DET
cana-2729	139	8	structures	structure	NOUN
cana-2729	139	9	in	in	ADP
cana-2729	139	10	extending	extend	VERB
cana-2729	139	11	classical	classical	ADJ
cana-2729	139	12	group	group	NOUN
cana-2729	139	13	and	and	CCONJ
cana-2729	139	14	topological	topological	ADJ
cana-2729	139	15	properties	property	NOUN
cana-2729	139	16	to	to	ADP
cana-2729	139	17	fuzzy	fuzzy	ADJ
cana-2729	139	18	contexts	contexts	NOUN
cana-2729	139	19	,	,	PUNCT
cana-2729	139	20	allowing	allow	VERB
cana-2729	139	21	for	for	ADP
cana-2729	139	22	a	a	DET
cana-2729	139	23	more	more	ADV
cana-2729	139	24	generalized	generalized	ADJ
cana-2729	139	25	understanding	understanding	NOUN
cana-2729	139	26	of	of	ADP
cana-2729	139	27	continuity	continuity	NOUN
cana-2729	139	28	,	,	PUNCT
cana-2729	139	29	closure	closure	NOUN
cana-2729	139	30	,	,	PUNCT
cana-2729	139	31	and	and	CCONJ
cana-2729	139	32	neighborhood	neighborhood	NOUN
cana-2729	139	33	in	in	ADP
cana-2729	139	34	topological	topological	ADJ
cana-2729	139	35	groups	group	NOUN
cana-2729	139	36	.	.	PUNCT
cana-2729	140	1	additionally	additionally	ADV
cana-2729	140	2	,	,	PUNCT
cana-2729	140	3	the	the	DET
cana-2729	140	4	extensions	extension	NOUN
cana-2729	140	5	introduced	introduce	VERB
cana-2729	140	6	in	in	ADP
cana-2729	140	7	this	this	DET
cana-2729	140	8	paper	paper	NOUN
cana-2729	140	9	provide	provide	VERB
cana-2729	140	10	new	new	ADJ
cana-2729	140	11	insights	insight	NOUN
cana-2729	140	12	into	into	ADP
cana-2729	140	13	the	the	DET
cana-2729	140	14	algebraic	algebraic	ADJ
cana-2729	140	15	and	and	CCONJ
cana-2729	140	16	topological	topological	ADJ
cana-2729	140	17	behaviors	behavior	NOUN
cana-2729	140	18	of	of	ADP
cana-2729	140	19	fuzzy	fuzzy	ADJ
cana-2729	140	20	brk	brk	PROPN
cana-2729	140	21	groups	group	NOUN
cana-2729	140	22	,	,	PUNCT
cana-2729	140	23	with	with	ADP
cana-2729	140	24	implications	implication	NOUN
cana-2729	140	25	for	for	ADP
cana-2729	140	26	further	further	ADJ
cana-2729	140	27	research	research	NOUN
cana-2729	140	28	and	and	CCONJ
cana-2729	140	29	applications	application	NOUN
cana-2729	140	30	in	in	ADP
cana-2729	140	31	fields	field	NOUN
cana-2729	140	32	like	like	ADP
cana-2729	140	33	fuzzy	fuzzy	ADJ
cana-2729	140	34	logic	logic	NOUN
cana-2729	140	35	,	,	PUNCT
cana-2729	140	36	artificial	artificial	ADJ
cana-2729	140	37	intelligence	intelligence	NOUN
cana-2729	140	38	,	,	PUNCT
cana-2729	140	39	and	and	CCONJ
cana-2729	140	40	decision	decision	NOUN
cana-2729	140	41	-	-	PUNCT
cana-2729	140	42	making	making	NOUN
cana-2729	140	43	.	.	PUNCT
cana-2729	141	1	the	the	DET
cana-2729	141	2	work	work	NOUN
cana-2729	141	3	of	of	ADP
cana-2729	141	4	sivakumar	sivakumar	PROPN
cana-2729	141	5	et	et	PROPN
cana-2729	141	6	al	al	PROPN
cana-2729	141	7	.	.	PROPN
cana-2729	141	8	has	have	AUX
cana-2729	141	9	laid	lay	VERB
cana-2729	141	10	the	the	DET
cana-2729	141	11	groundwork	groundwork	NOUN
cana-2729	141	12	for	for	ADP
cana-2729	141	13	future	future	ADJ
cana-2729	141	14	investigations	investigation	NOUN
cana-2729	141	15	,	,	PUNCT
cana-2729	141	16	particularly	particularly	ADV
cana-2729	141	17	in	in	ADP
cana-2729	141	18	exploring	explore	VERB
cana-2729	141	19	fuzzy	fuzzy	ADJ
cana-2729	141	20	topological	topological	ADJ
cana-2729	141	21	brksubalgebras	brksubalgebras	NOUN
cana-2729	141	22	and	and	CCONJ
cana-2729	141	23	their	their	PRON
cana-2729	141	24	role	role	NOUN
cana-2729	141	25	in	in	ADP
cana-2729	141	26	more	more	ADV
cana-2729	141	27	complex	complex	ADJ
cana-2729	141	28	algebraic	algebraic	ADJ
cana-2729	141	29	systems	system	NOUN
cana-2729	141	30	.	.	PUNCT
cana-2729	142	1	this	this	DET
cana-2729	142	2	study	study	NOUN
cana-2729	142	3	thus	thus	ADV
cana-2729	142	4	contributes	contribute	VERB
cana-2729	142	5	to	to	ADP
cana-2729	142	6	the	the	DET
cana-2729	142	7	growing	grow	VERB
cana-2729	142	8	body	body	NOUN
cana-2729	142	9	of	of	ADP
cana-2729	142	10	research	research	NOUN
cana-2729	142	11	in	in	ADP
cana-2729	142	12	fuzzy	fuzzy	ADJ
cana-2729	142	13	algebraic	algebraic	ADJ
cana-2729	142	14	structures	structure	NOUN
cana-2729	142	15	,	,	PUNCT
cana-2729	142	16	offering	offer	VERB
cana-2729	142	17	a	a	DET
cana-2729	142	18	promising	promising	ADJ
cana-2729	142	19	direction	direction	NOUN
cana-2729	142	20	for	for	ADP
cana-2729	142	21	future	future	ADJ
cana-2729	142	22	mathematical	mathematical	ADJ
cana-2729	142	23	inquiry	inquiry	NOUN
cana-2729	142	24	.	.	PUNCT
cana-2729	143	1	references	reference	NOUN
cana-2729	143	2	[	[	X
cana-2729	143	3	1	1	NUM
cana-2729	143	4	]	]	SYM
cana-2729	143	5	boixader	boixader	NOUN
cana-2729	143	6	d	d	PROPN
cana-2729	143	7	and	and	CCONJ
cana-2729	143	8	recasens	recasen	NOUN
cana-2729	143	9	j	j	PROPN
cana-2729	143	10	2018	2018	NUM
cana-2729	143	11	fuzzy	fuzzy	ADJ
cana-2729	143	12	actions	action	NOUN
cana-2729	143	13	fuzzy	fuzzy	ADJ
cana-2729	143	14	sets	set	NOUN
cana-2729	143	15	and	and	CCONJ
cana-2729	143	16	systems	system	NOUN
cana-2729	143	17	vol	vol	NOUN
cana-2729	143	18	339	339	NUM
cana-2729	143	19	pp	pp	ADV
cana-2729	143	20	17	17	NUM
cana-2729	143	21	-	-	SYM
cana-2729	143	22	30	30	NUM
cana-2729	143	23	.	.	PUNCT
cana-2729	144	1	[	[	X
cana-2729	144	2	2	2	NUM
cana-2729	144	3	]	]	PUNCT
cana-2729	144	4	foster	foster	NOUN
cana-2729	144	5	d	d	NOUN
cana-2729	144	6	h	h	NOUN
cana-2729	144	7	1979	1979	NUM
cana-2729	144	8	fuzzy	fuzzy	ADJ
cana-2729	144	9	topological	topological	ADJ
cana-2729	144	10	group	group	NOUN
cana-2729	144	11	j.	j.	PROPN
cana-2729	144	12	math	math	PROPN
cana-2729	144	13	.	.	PUNCT
cana-2729	145	1	anal	anal	PROPN
cana-2729	145	2	.	.	PUNCT
cana-2729	146	1	appl.vol	appl.vol	X
cana-2729	146	2	67	67	NUM
cana-2729	146	3	pp	pp	ADP
cana-2729	146	4	549–564	549–564	NUM
cana-2729	146	5	.	.	PUNCT
cana-2729	147	1	[	[	X
cana-2729	147	2	3	3	X
cana-2729	147	3	]	]	X
cana-2729	147	4	haddadi	haddadi	PROPN
cana-2729	147	5	m	m	PROPN
cana-2729	147	6	2013	2013	NUM
cana-2729	147	7	some	some	DET
cana-2729	147	8	algebraic	algebraic	ADJ
cana-2729	147	9	properties	property	NOUN
cana-2729	147	10	of	of	ADP
cana-2729	147	11	fuzzy	fuzzy	ADJ
cana-2729	147	12	s	s	NOUN
cana-2729	147	13	-	-	PUNCT
cana-2729	147	14	acts	act	NOUN
cana-2729	147	15	ratio	ratio	NOUN
cana-2729	147	16	mathematica	mathematica	PROPN
cana-2729	147	17	vol	vol	VERB
cana-2729	147	18	24	24	NUM
cana-2729	147	19	pp	pp	ADP
cana-2729	147	20	53–62	53–62	NUM
cana-2729	147	21	.	.	PUNCT
cana-2729	148	1	[	[	X
cana-2729	148	2	4	4	NUM
cana-2729	148	3	]	]	SYM
cana-2729	148	4	hu	hu	PROPN
cana-2729	148	5	q	q	PROPN
cana-2729	148	6	p	p	PROPN
cana-2729	148	7	and	and	CCONJ
cana-2729	148	8	li	li	PROPN
cana-2729	148	9	x	x	PROPN
cana-2729	148	10	1983	1983	NUM
cana-2729	148	11	on	on	ADP
cana-2729	148	12	bch	bch	PROPN
cana-2729	148	13	-	-	PUNCT
cana-2729	148	14	algebras	algebras	PROPN
cana-2729	148	15	mathematics	mathematics	NOUN
cana-2729	148	16	seminar	seminar	NOUN
cana-2729	148	17	notes	note	NOUN
cana-2729	148	18	vol	vol	VERB
cana-2729	148	19	11	11	NUM
cana-2729	148	20	pp	pp	ADP
cana-2729	148	21	313	313	NUM
cana-2729	148	22	-	-	SYM
cana-2729	148	23	320	320	NUM
cana-2729	148	24	.	.	PUNCT
cana-2729	149	1	[	[	X
cana-2729	149	2	5	5	X
cana-2729	149	3	]	]	PUNCT
cana-2729	149	4	jun	jun	PROPN
cana-2729	149	5	y	y	PROPN
cana-2729	149	6	b	b	PROPN
cana-2729	149	7	,	,	PUNCT
cana-2729	149	8	roh	roh	PROPN
cana-2729	149	9	e	e	NOUN
cana-2729	149	10	h	h	PROPN
cana-2729	149	11	and	and	CCONJ
cana-2729	149	12	kim	kim	PROPN
cana-2729	149	13	h	h	PROPN
cana-2729	149	14	s	s	PROPN
cana-2729	149	15	1998	1998	NUM
cana-2729	149	16	on	on	ADP
cana-2729	149	17	bh	bh	NOUN
cana-2729	149	18	-	-	PUNCT
cana-2729	149	19	algebras	algebras	PROPN
cana-2729	149	20	scientiae	scientiae	PROPN
cana-2729	149	21	mathematicae	mathematicae	VERB
cana-2729	149	22	japonica	japonica	PROPN
cana-2729	149	23	vol	vol	NOUN
cana-2729	149	24	1	1	NUM
cana-2729	149	25	pp	pp	ADP
cana-2729	149	26	347	347	NUM
cana-2729	149	27	-	-	SYM
cana-2729	149	28	354	354	NUM
cana-2729	149	29	.	.	PUNCT
cana-2729	150	1	[	[	X
cana-2729	150	2	6	6	NUM
cana-2729	150	3	]	]	X
cana-2729	150	4	kim	kim	PROPN
cana-2729	150	5	c	c	PROPN
cana-2729	150	6	b	b	PROPN
cana-2729	150	7	and	and	CCONJ
cana-2729	150	8	kim	kim	PROPN
cana-2729	150	9	h	h	PROPN
cana-2729	150	10	s	s	PART
cana-2729	150	11	2006	2006	NUM
cana-2729	150	12	on	on	ADP
cana-2729	150	13	bm	bm	PROPN
cana-2729	150	14	-	-	PUNCT
cana-2729	150	15	algebras	algebras	ADJ
cana-2729	150	16	scientiae	scientiae	NOUN
cana-2729	150	17	mathematicaejaponicaevol	mathematicaejaponicaevol	VERB
cana-2729	150	18	63	63	NUM
cana-2729	150	19	pp	pp	ADV
cana-2729	150	20	421	421	NUM
cana-2729	150	21	-	-	SYM
cana-2729	150	22	427	427	NUM
cana-2729	150	23	.	.	PUNCT
cana-2729	151	1	[	[	X
cana-2729	151	2	7	7	X
cana-2729	151	3	]	]	X
cana-2729	151	4	klein	klein	PROPN
cana-2729	151	5	f	f	PROPN
cana-2729	151	6	1893	1893	NUM
cana-2729	151	7	vergleichendebetrachtungenuberneueregeometrischeforschungen	vergleichendebetrachtungenuberneueregeometrischeforschungen	PROPN
cana-2729	151	8	math	math	NOUN
cana-2729	151	9	.	.	PUNCT
cana-2729	152	1	ann	ann	PROPN
cana-2729	152	2	.	.	PUNCT
cana-2729	153	1	vol	vol	NOUN
cana-2729	153	2	43	43	NUM
cana-2729	153	3	pp	pp	NUM
cana-2729	153	4	63	63	NUM
cana-2729	153	5	–	–	SYM
cana-2729	153	6	100	100	NUM
cana-2729	153	7	.	.	PUNCT
cana-2729	154	1	communications	communication	NOUN
cana-2729	154	2	on	on	ADP
cana-2729	154	3	applied	apply	VERB
cana-2729	154	4	nonlinear	nonlinear	ADJ
cana-2729	154	5	analysis	analysis	NOUN
cana-2729	154	6	issn	issn	NOUN
cana-2729	154	7	:	:	PUNCT
cana-2729	154	8	1074	1074	NUM
cana-2729	154	9	-	-	PUNCT
cana-2729	154	10	133x	133x	NUM
cana-2729	154	11	vol	vol	NOUN
cana-2729	154	12	32	32	NUM
cana-2729	154	13	no	no	NOUN
cana-2729	154	14	.	.	PUNCT
cana-2729	155	1	3s	3s	NUM
cana-2729	155	2	(	(	PUNCT
cana-2729	155	3	2025	2025	NUM
cana-2729	155	4	)	)	PUNCT
cana-2729	155	5	716	716	NUM
cana-2729	155	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-2729	156	1	[	[	X
cana-2729	156	2	8	8	NUM
cana-2729	156	3	]	]	X
cana-2729	156	4	lang	lang	PROPN
cana-2729	156	5	s	s	PROPN
cana-2729	156	6	1993	1993	NUM
cana-2729	156	7	algebra	algebra	NOUN
cana-2729	156	8	graduate	graduate	NOUN
cana-2729	156	9	texts	text	NOUN
cana-2729	156	10	in	in	ADP
cana-2729	156	11	mathematics	mathematic	NOUN
cana-2729	156	12	,	,	PUNCT
cana-2729	156	13	springer	springer	NOUN
cana-2729	156	14	.	.	PUNCT
cana-2729	157	1	[	[	X
cana-2729	157	2	9	9	NUM
cana-2729	157	3	]	]	PUNCT
cana-2729	157	4	ma	ma	PROPN
cana-2729	157	5	j	j	PROPN
cana-2729	157	6	l	l	PROPN
cana-2729	157	7	and	and	CCONJ
cana-2729	157	8	yu	yu	PROPN
cana-2729	157	9	c	c	PROPN
cana-2729	157	10	h	h	PROPN
cana-2729	157	11	1984	1984	NUM
cana-2729	157	12	fuzzy	fuzzy	ADJ
cana-2729	157	13	topological	topological	ADJ
cana-2729	157	14	groups	group	NOUN
cana-2729	157	15	fuzzy	fuzzy	ADJ
cana-2729	157	16	sets	set	NOUN
cana-2729	157	17	and	and	CCONJ
cana-2729	157	18	systems	system	NOUN
cana-2729	157	19	vol	vol	NOUN
cana-2729	157	20	12	12	NUM
cana-2729	157	21	pp	pp	ADV
cana-2729	157	22	289	289	NUM
cana-2729	157	23	-	-	SYM
cana-2729	157	24	299	299	NUM
cana-2729	157	25	.	.	PUNCT
cana-2729	158	1	[	[	X
cana-2729	158	2	10	10	NUM
cana-2729	158	3	]	]	X
cana-2729	158	4	martin	martin	PROPN
cana-2729	158	5	g	g	PROPN
cana-2729	158	6	e	e	PROPN
cana-2729	158	7	1982	1982	NUM
cana-2729	158	8	transformation	transformation	NOUN
cana-2729	158	9	geometry	geometry	NOUN
cana-2729	158	10	:	:	PUNCT
cana-2729	158	11	an	an	DET
cana-2729	158	12	introduction	introduction	NOUN
cana-2729	158	13	to	to	PART
cana-2729	158	14	symmetry	symmetry	NOUN
cana-2729	158	15	springer	springer	NOUN
cana-2729	158	16	-	-	PUNCT
cana-2729	158	17	verlag	verlag	NOUN
cana-2729	158	18	.	.	PUNCT
cana-2729	159	1	[	[	X
cana-2729	159	2	11	11	NUM
cana-2729	159	3	]	]	SYM
cana-2729	159	4	neggers	neggers	PROPN
cana-2729	159	5	j	j	PROPN
cana-2729	159	6	,	,	PUNCT
cana-2729	159	7	ahn	ahn	PROPN
cana-2729	159	8	s	s	PROPN
cana-2729	159	9	s	s	X
cana-2729	159	10	and	and	CCONJ
cana-2729	159	11	kim	kim	PROPN
cana-2729	159	12	h	h	PROPN
cana-2729	159	13	s	s	PROPN
cana-2729	159	14	2001	2001	NUM
cana-2729	159	15	on	on	ADP
cana-2729	159	16	q	q	ADJ
cana-2729	159	17	-	-	PUNCT
cana-2729	159	18	algebras	algebras	ADJ
cana-2729	159	19	international	international	ADJ
cana-2729	159	20	journal	journal	NOUN
cana-2729	159	21	of	of	ADP
cana-2729	159	22	mathematics	mathematics	PROPN
cana-2729	159	23	and	and	CCONJ
cana-2729	159	24	mathematical	mathematical	ADJ
cana-2729	159	25	sciences	sciences	PROPN
cana-2729	159	26	vol	vol	NOUN
cana-2729	159	27	27	27	NUM
cana-2729	159	28	pp	pp	ADV
cana-2729	159	29	749	749	NUM
cana-2729	159	30	-	-	NOUN
cana-2729	159	31	757	757	NOUN
cana-2729	159	32	.	.	PUNCT
cana-2729	160	1	[	[	X
cana-2729	160	2	12	12	NUM
cana-2729	160	3	]	]	PUNCT
cana-2729	160	4	ravi	ravi	NOUN
cana-2729	160	5	kumar	kumar	PROPN
cana-2729	160	6	bandaru	bandaru	PROPN
cana-2729	160	7	2012	2012	NUM
cana-2729	160	8	on	on	ADP
cana-2729	160	9	brk	brk	PROPN
cana-2729	160	10	-	-	PUNCT
cana-2729	160	11	algebras	algebras	PROPN
cana-2729	160	12	international	international	PROPN
cana-2729	160	13	journal	journal	PROPN
cana-2729	160	14	of	of	ADP
cana-2729	160	15	mathematics	mathematics	PROPN
cana-2729	160	16	and	and	CCONJ
cana-2729	160	17	mathematical	mathematical	ADJ
cana-2729	160	18	sciences	science	NOUN
cana-2729	160	19	pp	pp	ADP
cana-2729	160	20	112	112	NUM
cana-2729	160	21	.	.	PUNCT
cana-2729	161	1	[	[	X
cana-2729	161	2	13	13	NUM
cana-2729	161	3	]	]	PUNCT
cana-2729	161	4	rosenfeld	rosenfeld	PROPN
cana-2729	161	5	a	a	DET
cana-2729	161	6	1971	1971	NUM
cana-2729	161	7	fuzzy	fuzzy	ADJ
cana-2729	161	8	groups	group	NOUN
cana-2729	161	9	j.	j.	PROPN
cana-2729	161	10	math	math	PROPN
cana-2729	161	11	.	.	PUNCT
cana-2729	162	1	anal	anal	PROPN
cana-2729	162	2	.	.	PUNCT
cana-2729	163	1	appl	appl	PROPN
cana-2729	163	2	.	.	PUNCT
cana-2729	164	1	vol	vol	NOUN
cana-2729	164	2	35	35	NUM
cana-2729	164	3	pp	pp	ADP
cana-2729	164	4	512–517	512–517	NUM
cana-2729	164	5	.	.	PUNCT
cana-2729	165	1	[	[	X
cana-2729	165	2	14	14	NUM
cana-2729	165	3	]	]	X
cana-2729	165	4	sivakumar	sivakumar	PROPN
cana-2729	165	5	s	s	PROPN
cana-2729	165	6	,	,	PUNCT
cana-2729	165	7	kousalya	kousalya	PROPN
cana-2729	165	8	s	s	PROPN
cana-2729	165	9	,	,	PUNCT
cana-2729	165	10	vikrama	vikrama	NOUN
cana-2729	165	11	prasad	prasad	PROPN
cana-2729	165	12	r	r	NOUN
cana-2729	165	13	and	and	CCONJ
cana-2729	165	14	vadivel	vadivel	VERB
cana-2729	165	15	a	a	DET
cana-2729	165	16	2019	2019	NUM
cana-2729	165	17	topological	topological	ADJ
cana-2729	165	18	structures	structure	NOUN
cana-2729	165	19	on	on	ADP
cana-2729	165	20	brk	brk	PROPN
cana-2729	165	21	-	-	PUNCT
cana-2729	165	22	algebras	algebras	PROPN
cana-2729	165	23	journal	journal	PROPN
cana-2729	165	24	of	of	ADP
cana-2729	165	25	engineering	engineering	NOUN
cana-2729	165	26	sciences	sciences	PROPN
cana-2729	165	27	vol	vol	NOUN
cana-2729	165	28	10	10	NUM
cana-2729	165	29	pp	pp	ADV
cana-2729	165	30	459	459	NUM
cana-2729	165	31	-	-	SYM
cana-2729	165	32	471	471	NUM
cana-2729	165	33	.	.	PUNCT
cana-2729	166	1	[	[	X
cana-2729	166	2	15	15	NUM
cana-2729	166	3	]	]	X
cana-2729	166	4	sivakumar	sivakumar	PROPN
cana-2729	166	5	s	s	PROPN
cana-2729	166	6	,	,	PUNCT
cana-2729	166	7	kousalya	kousalya	PROPN
cana-2729	166	8	s	s	PROPN
cana-2729	166	9	,	,	PUNCT
cana-2729	166	10	vikrama	vikrama	NOUN
cana-2729	166	11	prasad	prasad	PROPN
cana-2729	166	12	r	r	NOUN
cana-2729	166	13	and	and	CCONJ
cana-2729	166	14	vadivel	vadivel	VERB
cana-2729	166	15	a	a	PRON
cana-2729	166	16	on	on	ADP
cana-2729	166	17	fuzzy	fuzzy	ADJ
cana-2729	166	18	topological	topological	PROPN
cana-2729	166	19	brk	brk	PROPN
cana-2729	166	20	-	-	PUNCT
cana-2729	166	21	subalgebras	subalgebras	PROPN
cana-2729	166	22	submitted	submit	VERB
cana-2729	166	23	.	.	PUNCT
cana-2729	167	1	[	[	X
cana-2729	167	2	16	16	NUM
cana-2729	167	3	]	]	X
cana-2729	167	4	sivakumar	sivakumar	PROPN
cana-2729	167	5	,	,	PUNCT
cana-2729	167	6	kousalya	kousalya	PROPN
cana-2729	167	7	s	s	PROPN
cana-2729	167	8	and	and	CCONJ
cana-2729	167	9	vadivel	vadivel	VERB
cana-2729	167	10	a	a	PRON
cana-2729	167	11	on	on	ADP
cana-2729	167	12	fuzzy	fuzzy	ADJ
cana-2729	167	13	topological	topological	PROPN
cana-2729	167	14	brk	brk	PROPN
cana-2729	167	15	-	-	PUNCT
cana-2729	167	16	group	group	NOUN
cana-2729	167	17	submitted	submit	VERB
cana-2729	167	18	.	.	PUNCT
cana-2729	168	1	[	[	X
cana-2729	168	2	17	17	NUM
cana-2729	168	3	]	]	X
cana-2729	168	4	yalvac	yalvac	PROPN
cana-2729	168	5	t	t	PROPN
cana-2729	168	6	h	h	PROPN
cana-2729	168	7	1987	1987	NUM
cana-2729	168	8	fuzzy	fuzzy	ADV
cana-2729	168	9	set	set	NOUN
cana-2729	168	10	and	and	CCONJ
cana-2729	168	11	functions	function	NOUN
cana-2729	168	12	on	on	ADP
cana-2729	168	13	fuzzy	fuzzy	ADJ
cana-2729	168	14	spaces	space	NOUN
cana-2729	168	15	j.	j.	PROPN
cana-2729	168	16	math	math	PROPN
cana-2729	168	17	.	.	PUNCT
cana-2729	169	1	anal	anal	PROPN
cana-2729	169	2	.	.	PUNCT
cana-2729	170	1	vol	vol	NOUN
cana-2729	170	2	126	126	NUM
cana-2729	170	3	pp	pp	ADP
cana-2729	170	4	409	409	NUM
cana-2729	170	5	-	-	SYM
cana-2729	170	6	423	423	NUM
cana-2729	170	7	.	.	PUNCT
cana-2729	171	1	[	[	X
cana-2729	171	2	18	18	NUM
cana-2729	171	3	]	]	PUNCT
cana-2729	171	4	zadeh	zadeh	PROPN
cana-2729	171	5	l	l	PROPN
cana-2729	171	6	a	a	DET
cana-2729	171	7	1965	1965	NUM
cana-2729	171	8	fuzzy	fuzzy	ADJ
cana-2729	171	9	sets	set	NOUN
cana-2729	171	10	inform	inform	NOUN
cana-2729	171	11	.	.	PUNCT
cana-2729	172	1	control	control	NOUN
cana-2729	172	2	vol	vol	NOUN
cana-2729	172	3	8	8	NUM
cana-2729	172	4	pp	pp	ADP
cana-2729	172	5	338–353	338–353	NUM
cana-2729	172	6	.	.	PUNCT