id	sid	tid	token	lemma	pos
cana-610	1	1	communications	communication	NOUN
cana-610	1	2	on	on	ADP
cana-610	1	3	applied	apply	VERB
cana-610	1	4	nonlinear	nonlinear	ADJ
cana-610	1	5	analysis	analysis	NOUN
cana-610	1	6	issn	issn	NOUN
cana-610	1	7	:	:	PUNCT
cana-610	1	8	1074	1074	NUM
cana-610	1	9	-	-	PUNCT
cana-610	1	10	133x	133x	NUM
cana-610	1	11	vol	vol	NOUN
cana-610	1	12	31	31	NUM
cana-610	1	13	no	no	NOUN
cana-610	1	14	.	.	NOUN
cana-610	1	15	2	2	NUM
cana-610	1	16	(	(	PUNCT
cana-610	1	17	2024	2024	NUM
cana-610	1	18	)	)	PUNCT
cana-610	1	19	409	409	NUM
cana-610	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-610	1	21	prime	prime	ADJ
cana-610	1	22	spectrum	spectrum	NOUN
cana-610	1	23	graph	graph	NOUN
cana-610	1	24	of	of	ADP
cana-610	1	25	c	c	NOUN
cana-610	1	26	-	-	PUNCT
cana-610	1	27	lattices	lattice	NOUN
cana-610	1	28	lakpa	lakpa	PROPN
cana-610	1	29	sherpa1	sherpa1	NOUN
cana-610	1	30	,	,	PUNCT
cana-610	1	31	vilas	vilas	PROPN
cana-610	1	32	kharat2	kharat2	PROPN
cana-610	1	33	,	,	PUNCT
cana-610	1	34	manish	manish	PROPN
cana-610	1	35	agalave3	agalave3	PROPN
cana-610	1	36	,	,	PUNCT
cana-610	1	37	ganesh	ganesh	PROPN
cana-610	1	38	gandal4	gandal4	NOUN
cana-610	1	39	,	,	PUNCT
cana-610	1	40	narayan	narayan	NOUN
cana-610	1	41	phadatare5	phadatare5	NOUN
cana-610	2	1	1,2department	1,2department	NUM
cana-610	2	2	of	of	ADP
cana-610	2	3	mathematics	mathematic	NOUN
cana-610	2	4	,	,	PUNCT
cana-610	2	5	savitribai	savitribai	VERB
cana-610	2	6	phule	phule	PROPN
cana-610	2	7	pune	pune	PROPN
cana-610	2	8	university	university	NOUN
cana-610	2	9	,	,	PUNCT
cana-610	2	10	pune-411	pune-411	NOUN
cana-610	2	11	007	007	NUM
cana-610	2	12	(	(	PUNCT
cana-610	2	13	india	india	PROPN
cana-610	2	14	)	)	PUNCT
cana-610	3	1	3department	3department	NUM
cana-610	3	2	of	of	ADP
cana-610	3	3	mathematics	mathematic	NOUN
cana-610	3	4	,	,	PUNCT
cana-610	3	5	fergusson	fergusson	NOUN
cana-610	3	6	college(autonomus	college(autonomus	PROPN
cana-610	3	7	)	)	PUNCT
cana-610	3	8	,	,	PUNCT
cana-610	3	9	pune-411	pune-411	NOUN
cana-610	3	10	004	004	NUM
cana-610	3	11	(	(	PUNCT
cana-610	3	12	india	india	PROPN
cana-610	3	13	)	)	PUNCT
cana-610	3	14	4smt	4smt	PROPN
cana-610	3	15	kashibai	kashibai	VERB
cana-610	3	16	navale	navale	PROPN
cana-610	3	17	college	college	PROPN
cana-610	3	18	of	of	ADP
cana-610	3	19	engineering	engineering	PROPN
cana-610	3	20	,	,	PUNCT
cana-610	3	21	pune	pune	PROPN
cana-610	3	22	(	(	PUNCT
cana-610	3	23	india	india	PROPN
cana-610	3	24	)	)	PUNCT
cana-610	4	1	5bharati	5bharati	PROPN
cana-610	4	2	vidyapeeth	vidyapeeth	PROPN
cana-610	4	3	deemed	deem	VERB
cana-610	4	4	to	to	PART
cana-610	4	5	be	be	AUX
cana-610	4	6	university	university	NOUN
cana-610	4	7	college	college	NOUN
cana-610	4	8	of	of	ADP
cana-610	4	9	engineering	engineering	NOUN
cana-610	4	10	,	,	PUNCT
cana-610	4	11	pune-411	pune-411	NOUN
cana-610	4	12	043	043	NUM
cana-610	4	13	(	(	PUNCT
cana-610	4	14	india	india	PROPN
cana-610	4	15	)	)	PUNCT
cana-610	4	16	csherpaap@gmail.com,laddoo1@yahoo.com,manishagalave@gmail.com,ganeshgandal2001@gmail.com	csherpaap@gmail.com,laddoo1@yahoo.com,manishagalave@gmail.com,ganeshgandal2001@gmail.com	X
cana-610	5	1	nmphadatare@bvucoep	nmphadatare@bvucoep	PROPN
cana-610	5	2	.	.	PUNCT
cana-610	5	3	article	article	PROPN
cana-610	5	4	history	history	NOUN
cana-610	5	5	:	:	PUNCT
cana-610	5	6	received	receive	VERB
cana-610	5	7	:	:	PUNCT
cana-610	5	8	18	18	NUM
cana-610	5	9	-	-	PUNCT
cana-610	5	10	02	02	NUM
cana-610	5	11	-	-	PUNCT
cana-610	5	12	2024	2024	NUM
cana-610	5	13	revised	revise	VERB
cana-610	5	14	:	:	PUNCT
cana-610	5	15	20	20	NUM
cana-610	5	16	-	-	PUNCT
cana-610	5	17	04	04	NUM
cana-610	5	18	-	-	PUNCT
cana-610	5	19	2024	2024	NUM
cana-610	5	20	accepted	accept	VERB
cana-610	5	21	:	:	PUNCT
cana-610	5	22	12	12	NUM
cana-610	5	23	-	-	PUNCT
cana-610	5	24	05	05	NUM
cana-610	5	25	-	-	PUNCT
cana-610	5	26	2024	2024	NUM
cana-610	5	27	abstract	abstract	NOUN
cana-610	5	28	:	:	PUNCT
cana-610	5	29	let	let	VERB
cana-610	5	30	£	£	PART
cana-610	5	31	be	be	AUX
cana-610	5	32	a	a	DET
cana-610	5	33	c	c	NOUN
cana-610	5	34	-	-	PUNCT
cana-610	5	35	lattice	lattice	NOUN
cana-610	5	36	.	.	PUNCT
cana-610	6	1	let	let	VERB
cana-610	6	2	σ(£	σ(£	PROPN
cana-610	6	3	)	)	PUNCT
cana-610	6	4	be	be	VERB
cana-610	6	5	the	the	DET
cana-610	6	6	set	set	NOUN
cana-610	6	7	of	of	ADP
cana-610	6	8	all	all	DET
cana-610	6	9	prime	prime	ADJ
cana-610	6	10	elements	element	NOUN
cana-610	6	11	of	of	ADP
cana-610	6	12	£	£	NUM
cana-610	6	13	and	and	CCONJ
cana-610	6	14	m(£	m(£	PROPN
cana-610	6	15	)	)	PUNCT
cana-610	6	16	be	be	AUX
cana-610	6	17	the	the	DET
cana-610	6	18	collection	collection	NOUN
cana-610	6	19	of	of	ADP
cana-610	6	20	all	all	DET
cana-610	6	21	maximal	maximal	ADJ
cana-610	6	22	elements	element	NOUN
cana-610	6	23	of	of	ADP
cana-610	6	24	£	£	SYM
cana-610	6	25	.	.	PUNCT
cana-610	7	1	for	for	ADP
cana-610	7	2	s	s	NOUN
cana-610	7	3	⊆m(£	⊆m(£	NOUN
cana-610	7	4	)	)	PUNCT
cana-610	7	5	,	,	PUNCT
cana-610	7	6	we	we	PRON
cana-610	7	7	introduce	introduce	VERB
cana-610	7	8	the	the	DET
cana-610	7	9	new	new	ADJ
cana-610	7	10	graph	graph	NOUN
cana-610	7	11	called	call	VERB
cana-610	7	12	s	s	NOUN
cana-610	7	13	-	-	PUNCT
cana-610	7	14	join	join	NOUN
cana-610	7	15	graph	graph	NOUN
cana-610	7	16	on	on	ADP
cana-610	7	17	σ(£	σ(£	PROPN
cana-610	7	18	)	)	PUNCT
cana-610	7	19	,	,	PUNCT
cana-610	7	20	denoted	denote	VERB
cana-610	7	21	by	by	ADP
cana-610	7	22	γs(σ(£	γs(σ(£	NOUN
cana-610	7	23	)	)	PUNCT
cana-610	7	24	)	)	PUNCT
cana-610	7	25	.	.	PUNCT
cana-610	8	1	we	we	PRON
cana-610	8	2	have	have	AUX
cana-610	8	3	studied	study	VERB
cana-610	8	4	properties	property	NOUN
cana-610	8	5	like	like	ADP
cana-610	8	6	connectivity	connectivity	NOUN
cana-610	8	7	,	,	PUNCT
cana-610	8	8	diameter	diameter	NOUN
cana-610	8	9	and	and	CCONJ
cana-610	8	10	domination	domination	NOUN
cana-610	8	11	number	number	NOUN
cana-610	8	12	of	of	ADP
cana-610	8	13	the	the	DET
cana-610	8	14	graph	graph	NOUN
cana-610	8	15	σ(£	σ(£	PROPN
cana-610	8	16	)	)	PUNCT
cana-610	8	17	.	.	PUNCT
cana-610	9	1	in	in	ADP
cana-610	9	2	this	this	DET
cana-610	9	3	paper	paper	NOUN
cana-610	9	4	,	,	PUNCT
cana-610	9	5	we	we	PRON
cana-610	9	6	established	establish	VERB
cana-610	9	7	that	that	SCONJ
cana-610	9	8	the	the	DET
cana-610	9	9	topological	topological	ADJ
cana-610	9	10	space	space	NOUN
cana-610	9	11	σ(£	σ(£	PROPN
cana-610	9	12	)	)	PUNCT
cana-610	9	13	is	be	AUX
cana-610	9	14	connected	connect	VERB
cana-610	9	15	if	if	SCONJ
cana-610	9	16	and	and	CCONJ
cana-610	9	17	only	only	ADV
cana-610	9	18	if	if	SCONJ
cana-610	9	19	the	the	DET
cana-610	9	20	graph	graph	NOUN
cana-610	9	21	γs(σ(£	γs(σ(£	NOUN
cana-610	9	22	)	)	PUNCT
cana-610	9	23	)	)	PUNCT
cana-610	9	24	is	be	AUX
cana-610	9	25	connected	connect	VERB
cana-610	9	26	.	.	PUNCT
cana-610	10	1	keywords	keyword	NOUN
cana-610	10	2	:	:	PUNCT
cana-610	10	3	prime	prime	ADJ
cana-610	10	4	element	element	NOUN
cana-610	10	5	;	;	PUNCT
cana-610	10	6	maximal	maximal	ADJ
cana-610	10	7	element	element	NOUN
cana-610	10	8	;	;	PUNCT
cana-610	10	9	s	s	X
cana-610	10	10	-	-	PUNCT
cana-610	10	11	join	join	NOUN
cana-610	10	12	graph	graph	NOUN
cana-610	10	13	.	.	PUNCT
cana-610	11	1	1	1	X
cana-610	11	2	.	.	X
cana-610	11	3	introduction	introduction	NOUN
cana-610	11	4	the	the	DET
cana-610	11	5	study	study	NOUN
cana-610	11	6	of	of	ADP
cana-610	11	7	commutative	commutative	ADJ
cana-610	11	8	rings	ring	NOUN
cana-610	11	9	indeed	indeed	ADV
cana-610	11	10	enriched	enrich	VERB
cana-610	11	11	by	by	ADP
cana-610	11	12	using	use	VERB
cana-610	11	13	graph	graph	NOUN
cana-610	11	14	theory	theory	NOUN
cana-610	11	15	techniques	technique	NOUN
cana-610	11	16	.	.	PUNCT
cana-610	12	1	in	in	ADP
cana-610	12	2	1988	1988	NUM
cana-610	12	3	,	,	PUNCT
cana-610	12	4	i.	i.	PROPN
cana-610	12	5	beck	beck	PROPN
cana-610	12	6	proved	prove	VERB
cana-610	12	7	that	that	SCONJ
cana-610	12	8	how	how	SCONJ
cana-610	12	9	graph	graph	NOUN
cana-610	12	10	theory	theory	NOUN
cana-610	12	11	can	can	AUX
cana-610	12	12	be	be	AUX
cana-610	12	13	applied	apply	VERB
cana-610	12	14	to	to	ADP
cana-610	12	15	the	the	DET
cana-610	12	16	study	study	NOUN
cana-610	12	17	of	of	ADP
cana-610	12	18	commutative	commutative	ADJ
cana-610	12	19	rings	ring	NOUN
cana-610	12	20	.	.	PUNCT
cana-610	13	1	according	accord	VERB
cana-610	13	2	to	to	ADP
cana-610	13	3	i.	i.	PROPN
cana-610	13	4	beck	beck	PROPN
cana-610	13	5	,	,	PUNCT
cana-610	13	6	the	the	DET
cana-610	13	7	zero	zero	NUM
cana-610	13	8	-	-	PUNCT
cana-610	13	9	divisor	divisor	NOUN
cana-610	13	10	graph	graph	NOUN
cana-610	13	11	of	of	ADP
cana-610	13	12	a	a	DET
cana-610	13	13	commutative	commutative	ADJ
cana-610	13	14	ring	ring	NOUN
cana-610	13	15	is	be	AUX
cana-610	13	16	a	a	DET
cana-610	13	17	graph	graph	NOUN
cana-610	13	18	where	where	SCONJ
cana-610	13	19	the	the	DET
cana-610	13	20	vertices	vertex	NOUN
cana-610	13	21	represent	represent	VERB
cana-610	13	22	the	the	DET
cana-610	13	23	elements	element	NOUN
cana-610	13	24	of	of	ADP
cana-610	13	25	the	the	DET
cana-610	13	26	ring	ring	NOUN
cana-610	13	27	,	,	PUNCT
cana-610	13	28	and	and	CCONJ
cana-610	13	29	two	two	NUM
cana-610	13	30	vertices	vertex	NOUN
cana-610	13	31	are	be	AUX
cana-610	13	32	connected	connect	VERB
cana-610	13	33	by	by	ADP
cana-610	13	34	an	an	DET
cana-610	13	35	edge	edge	NOUN
cana-610	13	36	if	if	SCONJ
cana-610	13	37	their	their	PRON
cana-610	13	38	product	product	NOUN
cana-610	13	39	is	be	AUX
cana-610	13	40	zero	zero	NUM
cana-610	13	41	(	(	PUNCT
cana-610	13	42	see	see	VERB
cana-610	13	43	[	[	X
cana-610	13	44	5	5	NUM
cana-610	13	45	]	]	NUM
cana-610	13	46	)	)	PUNCT
cana-610	13	47	.	.	PUNCT
cana-610	14	1	further	far	ADV
cana-610	14	2	,	,	PUNCT
cana-610	14	3	by	by	ADP
cana-610	14	4	employing	employ	VERB
cana-610	14	5	these	these	DET
cana-610	14	6	graph	graph	NOUN
cana-610	14	7	-	-	PUNCT
cana-610	14	8	theoretic	theoretic	NOUN
cana-610	14	9	approach	approach	NOUN
cana-610	14	10	,	,	PUNCT
cana-610	14	11	many	many	ADJ
cana-610	14	12	researchers	researcher	NOUN
cana-610	14	13	visually	visually	ADV
cana-610	14	14	represented	represent	VERB
cana-610	14	15	and	and	CCONJ
cana-610	14	16	analyzed	analyze	VERB
cana-610	14	17	various	various	ADJ
cana-610	14	18	aspects	aspect	NOUN
cana-610	14	19	of	of	ADP
cana-610	14	20	commutative	commutative	ADJ
cana-610	14	21	rings	ring	NOUN
cana-610	14	22	,	,	PUNCT
cana-610	14	23	providing	provide	VERB
cana-610	14	24	additional	additional	ADJ
cana-610	14	25	insights	insight	NOUN
cana-610	14	26	into	into	ADP
cana-610	14	27	their	their	PRON
cana-610	14	28	structure	structure	NOUN
cana-610	14	29	,	,	PUNCT
cana-610	14	30	properties	property	NOUN
cana-610	14	31	,	,	PUNCT
cana-610	14	32	and	and	CCONJ
cana-610	14	33	relationships	relationship	NOUN
cana-610	14	34	(	(	PUNCT
cana-610	14	35	see	see	VERB
cana-610	14	36	[	[	X
cana-610	14	37	1]-[4	1]-[4	NOUN
cana-610	14	38	]	]	X
cana-610	14	39	,	,	PUNCT
cana-610	15	1	[	[	X
cana-610	15	2	9]-[10	9]-[10	NOUN
cana-610	15	3	]	]	PUNCT
cana-610	15	4	,	,	PUNCT
cana-610	15	5	[	[	X
cana-610	15	6	12	12	NUM
cana-610	15	7	]	]	PUNCT
cana-610	15	8	)	)	PUNCT
cana-610	15	9	.	.	PUNCT
cana-610	16	1	the	the	DET
cana-610	16	2	ideals	ideal	NOUN
cana-610	16	3	of	of	ADP
cana-610	16	4	ring	ring	NOUN
cana-610	16	5	play	play	VERB
cana-610	16	6	a	a	DET
cana-610	16	7	fundamental	fundamental	ADJ
cana-610	16	8	role	role	NOUN
cana-610	16	9	in	in	ADP
cana-610	16	10	the	the	DET
cana-610	16	11	study	study	NOUN
cana-610	16	12	of	of	ADP
cana-610	16	13	ring	ring	NOUN
cana-610	16	14	structure	structure	NOUN
cana-610	16	15	.	.	PUNCT
cana-610	17	1	therefore	therefore	ADV
cana-610	17	2	m.	m.	NOUN
cana-610	17	3	behboodi	behboodi	PROPN
cana-610	17	4	et	et	PROPN
cana-610	17	5	.	.	PUNCT
cana-610	18	1	al	al	PROPN
cana-610	18	2	.	.	PROPN
cana-610	18	3	introduced	introduce	VERB
cana-610	18	4	and	and	CCONJ
cana-610	18	5	studied	study	VERB
cana-610	18	6	annihilating	annihilate	VERB
cana-610	18	7	-	-	PUNCT
cana-610	18	8	ideal	ideal	NOUN
cana-610	18	9	graph	graph	NOUN
cana-610	18	10	whose	whose	DET
cana-610	18	11	vertices	vertex	NOUN
cana-610	18	12	are	be	AUX
cana-610	18	13	annihilating	annihilate	VERB
cana-610	18	14	ideals	ideal	NOUN
cana-610	18	15	of	of	ADP
cana-610	18	16	a	a	DET
cana-610	18	17	commutative	commutative	ADJ
cana-610	18	18	ring	ring	NOUN
cana-610	18	19	with	with	ADP
cana-610	18	20	unity	unity	NOUN
cana-610	18	21	(	(	PUNCT
cana-610	18	22	see	see	VERB
cana-610	18	23	[	[	X
cana-610	18	24	6]-[7	6]-[7	NOUN
cana-610	18	25	]	]	X
cana-610	18	26	)	)	PUNCT
cana-610	18	27	.	.	PUNCT
cana-610	19	1	the	the	DET
cana-610	19	2	set	set	NOUN
cana-610	19	3	of	of	ADP
cana-610	19	4	ideals	ideal	NOUN
cana-610	19	5	of	of	ADP
cana-610	19	6	a	a	DET
cana-610	19	7	ring	ring	NOUN
cana-610	19	8	is	be	AUX
cana-610	19	9	naturally	naturally	ADV
cana-610	19	10	endowed	endow	VERB
cana-610	19	11	with	with	ADP
cana-610	19	12	a	a	DET
cana-610	19	13	lattice	lattice	NOUN
cana-610	19	14	structure	structure	NOUN
cana-610	19	15	.	.	PUNCT
cana-610	20	1	it	it	PRON
cana-610	20	2	is	be	AUX
cana-610	20	3	very	very	ADV
cana-610	20	4	much	much	ADV
cana-610	20	5	interesting	interesting	ADJ
cana-610	20	6	that	that	SCONJ
cana-610	20	7	the	the	DET
cana-610	20	8	set	set	NOUN
cana-610	20	9	of	of	ADP
cana-610	20	10	ideals	ideal	NOUN
cana-610	20	11	of	of	ADP
cana-610	20	12	a	a	DET
cana-610	20	13	ring	ring	NOUN
cana-610	20	14	,	,	PUNCT
cana-610	20	15	denoted	denote	VERB
cana-610	20	16	as	as	ADP
cana-610	20	17	id(r	id(r	NOUN
cana-610	20	18	)	)	PUNCT
cana-610	20	19	,	,	PUNCT
cana-610	20	20	forms	form	VERB
cana-610	20	21	a	a	DET
cana-610	20	22	multiplicative	multiplicative	ADJ
cana-610	20	23	lattice	lattice	NOUN
cana-610	20	24	.	.	PUNCT
cana-610	21	1	definition	definition	NOUN
cana-610	21	2	1.1	1.1	NUM
cana-610	21	3	.	.	PUNCT
cana-610	22	1	a	a	DET
cana-610	22	2	multiplicative	multiplicative	ADJ
cana-610	22	3	lattice	lattice	NOUN
cana-610	22	4	is	be	AUX
cana-610	22	5	denoted	denote	VERB
cana-610	22	6	as	as	ADP
cana-610	22	7	(	(	PUNCT
cana-610	22	8	£	£	PROPN
cana-610	22	9	,	,	PUNCT
cana-610	22	10	0,1,∗	0,1,∗	PROPN
cana-610	22	11	)	)	PUNCT
cana-610	22	12	,	,	PUNCT
cana-610	22	13	where	where	SCONJ
cana-610	22	14	£	£	NOUN
cana-610	22	15	is	be	AUX
cana-610	22	16	a	a	DET
cana-610	22	17	complete	complete	ADJ
cana-610	22	18	lattice	lattice	NOUN
cana-610	22	19	with	with	ADP
cana-610	22	20	least	least	ADJ
cana-610	22	21	element	element	ADJ
cana-610	22	22	0	0	NUM
cana-610	22	23	,	,	PUNCT
cana-610	22	24	greatest	great	ADJ
cana-610	22	25	element	element	NOUN
cana-610	22	26	1	1	NUM
cana-610	22	27	and	and	CCONJ
cana-610	22	28	∗	∗	NOUN
cana-610	22	29	is	be	AUX
cana-610	22	30	a	a	DET
cana-610	22	31	binary	binary	ADJ
cana-610	22	32	operation	operation	NOUN
cana-610	22	33	defined	define	VERB
cana-610	22	34	on	on	ADP
cana-610	22	35	£	£	PROPN
cana-610	22	36	that	that	PRON
cana-610	22	37	satisfies	satisfy	VERB
cana-610	22	38	the	the	DET
cana-610	22	39	following	follow	VERB
cana-610	22	40	properties	property	NOUN
cana-610	22	41	for	for	ADP
cana-610	22	42	all	all	DET
cana-610	22	43	a	a	DET
cana-610	22	44	,	,	PUNCT
cana-610	22	45	b	b	NOUN
cana-610	22	46	,	,	PUNCT
cana-610	22	47	c	c	PROPN
cana-610	22	48	∈	∈	PROPN
cana-610	23	1	£	£	NOUN
cana-610	23	2	:	:	PUNCT
cana-610	23	3	1	1	NUM
cana-610	23	4	.	.	PUNCT
cana-610	23	5	a	a	DET
cana-610	23	6	∗	∗	NOUN
cana-610	23	7	b	b	NOUN
cana-610	23	8	≤	≤	NOUN
cana-610	23	9	a	a	DET
cana-610	23	10	∧	∧	PROPN
cana-610	23	11	b.	b.	PROPN
cana-610	23	12	2	2	NUM
cana-610	23	13	.	.	PUNCT
cana-610	23	14	a	a	DET
cana-610	23	15	∗	∗	NOUN
cana-610	23	16	b	b	NOUN
cana-610	23	17	=	=	SYM
cana-610	23	18	b	b	PROPN
cana-610	23	19	∗	∗	X
cana-610	23	20	a.	a.	NOUN
cana-610	23	21	communications	communication	NOUN
cana-610	23	22	on	on	ADP
cana-610	23	23	applied	apply	VERB
cana-610	23	24	nonlinear	nonlinear	ADJ
cana-610	23	25	analysis	analysis	NOUN
cana-610	23	26	issn	issn	NOUN
cana-610	23	27	:	:	PUNCT
cana-610	23	28	1074	1074	NUM
cana-610	23	29	-	-	PUNCT
cana-610	23	30	133x	133x	NUM
cana-610	23	31	vol	vol	NOUN
cana-610	23	32	31	31	NUM
cana-610	23	33	no	no	NOUN
cana-610	23	34	.	.	NOUN
cana-610	23	35	2	2	NUM
cana-610	23	36	(	(	PUNCT
cana-610	23	37	2024	2024	NUM
cana-610	23	38	)	)	PUNCT
cana-610	23	39	410	410	NUM
cana-610	23	40	https://internationalpubls.com	https://internationalpubls.com	X
cana-610	23	41	3	3	X
cana-610	23	42	.	.	PUNCT
cana-610	24	1	(	(	PUNCT
cana-610	24	2	a	a	DET
cana-610	24	3	∗	∗	NOUN
cana-610	24	4	b	b	NOUN
cana-610	24	5	)	)	PUNCT
cana-610	24	6	∗	∗	NOUN
cana-610	24	7	c	c	NOUN
cana-610	24	8	=	=	PUNCT
cana-610	24	9	a	a	DET
cana-610	24	10	∗	∗	NOUN
cana-610	24	11	(	(	PUNCT
cana-610	24	12	b	b	NOUN
cana-610	24	13	∗	∗	NOUN
cana-610	24	14	c	c	NOUN
cana-610	24	15	)	)	PUNCT
cana-610	24	16	.	.	PUNCT
cana-610	25	1	4	4	X
cana-610	25	2	.	.	X
cana-610	25	3	a	a	DET
cana-610	25	4	∗	∗	NOUN
cana-610	25	5	(	(	PUNCT
cana-610	25	6	∨α∈ibα	∨α∈ibα	X
cana-610	25	7	)	)	PUNCT
cana-610	25	8	=	=	SYM
cana-610	26	1	∨α∈i(a	∨α∈i(a	PROPN
cana-610	26	2	∗	∗	NOUN
cana-610	26	3	bα	bα	PROPN
cana-610	26	4	)	)	PUNCT
cana-610	26	5	,	,	PUNCT
cana-610	26	6	where	where	SCONJ
cana-610	26	7	bα	bα	PROPN
cana-610	26	8	∈	∈	PROPN
cana-610	26	9	£	£	PROPN
cana-610	27	1	and	and	CCONJ
cana-610	27	2	i	i	PRON
cana-610	27	3	is	be	AUX
cana-610	27	4	an	an	DET
cana-610	27	5	indexing	indexing	NOUN
cana-610	27	6	set	set	NOUN
cana-610	27	7	.	.	PUNCT
cana-610	28	1	5	5	X
cana-610	28	2	.	.	X
cana-610	28	3	a	a	DET
cana-610	28	4	∗	∗	NOUN
cana-610	28	5	1	1	NUM
cana-610	28	6	=	=	SYM
cana-610	28	7	a.	a.	NOUN
cana-610	28	8	henceforth	henceforth	ADV
cana-610	28	9	,	,	PUNCT
cana-610	28	10	we	we	PRON
cana-610	28	11	write	write	VERB
cana-610	28	12	a	a	DET
cana-610	28	13	∗	∗	NOUN
cana-610	28	14	b	b	NOUN
cana-610	28	15	=	=	SYM
cana-610	28	16	ab	ab	PROPN
cana-610	28	17	for	for	ADP
cana-610	28	18	the	the	DET
cana-610	28	19	sake	sake	NOUN
cana-610	28	20	of	of	ADP
cana-610	28	21	convenience	convenience	NOUN
cana-610	28	22	only	only	ADV
cana-610	28	23	.	.	PUNCT
cana-610	29	1	a	a	DET
cana-610	29	2	member	member	NOUN
cana-610	29	3	a	a	PRON
cana-610	29	4	∈	∈	PROPN
cana-610	29	5	£	£	NOUN
cana-610	29	6	is	be	AUX
cana-610	29	7	called	call	VERB
cana-610	29	8	compact	compact	ADJ
cana-610	29	9	if	if	SCONJ
cana-610	29	10	a	a	DET
cana-610	29	11	≤	≤	NUM
cana-610	29	12	∨β∈i	∨β∈i	NOUN
cana-610	30	1	aβ	aβ	NOUN
cana-610	30	2	implies	imply	VERB
cana-610	30	3	a	a	DET
cana-610	30	4	≤	≤	PROPN
cana-610	30	5	⋁𝑖=0	⋁𝑖=0	PROPN
cana-610	30	6	𝑛	𝑛	DET
cana-610	30	7	𝑎β𝑖	𝑎β𝑖	NOUN
cana-610	30	8	.	.	PUNCT
cana-610	31	1	let	let	VERB
cana-610	31	2	£	£	SYM
cana-610	31	3	c	c	VERB
cana-610	31	4	the	the	DET
cana-610	31	5	set	set	NOUN
cana-610	31	6	of	of	ADP
cana-610	31	7	all	all	DET
cana-610	31	8	compact	compact	ADJ
cana-610	31	9	elements	element	NOUN
cana-610	31	10	of	of	ADP
cana-610	31	11	£	£	PROPN
cana-610	31	12	.	.	PUNCT
cana-610	32	1	a	a	DET
cana-610	32	2	multiplicative	multiplicative	ADJ
cana-610	32	3	lattice	lattice	NOUN
cana-610	32	4	£	£	PROPN
cana-610	32	5	is	be	AUX
cana-610	32	6	called	call	VERB
cana-610	32	7	compactly	compactly	ADV
cana-610	32	8	generated	generate	VERB
cana-610	32	9	if	if	SCONJ
cana-610	32	10	each	each	DET
cana-610	32	11	a	a	DET
cana-610	32	12	∈	∈	PROPN
cana-610	32	13	£	£	NOUN
cana-610	32	14	is	be	AUX
cana-610	32	15	of	of	ADP
cana-610	32	16	the	the	DET
cana-610	32	17	form	form	NOUN
cana-610	32	18	∨bi	∨bi	VERB
cana-610	32	19	for	for	ADP
cana-610	32	20	bi	bi	PROPN
cana-610	32	21	∈	∈	PROPN
cana-610	32	22	£	£	PROPN
cana-610	32	23	c.	c.	NOUN
cana-610	32	24	by	by	ADP
cana-610	32	25	c	c	NOUN
cana-610	32	26	-	-	PUNCT
cana-610	32	27	lattice	lattice	NOUN
cana-610	32	28	£	£	SYM
cana-610	32	29	,	,	PUNCT
cana-610	32	30	we	we	PRON
cana-610	32	31	mean	mean	VERB
cana-610	32	32	a	a	DET
cana-610	32	33	multiplicative	multiplicative	ADJ
cana-610	32	34	lattice	lattice	NOUN
cana-610	32	35	(	(	PUNCT
cana-610	32	36	£	£	NOUN
cana-610	32	37	,	,	PUNCT
cana-610	32	38	0,1,•	0,1,•	NOUN
cana-610	32	39	)	)	PUNCT
cana-610	32	40	which	which	PRON
cana-610	32	41	is	be	AUX
cana-610	32	42	generated	generate	VERB
cana-610	32	43	under	under	ADP
cana-610	32	44	join	join	NOUN
cana-610	32	45	by	by	ADP
cana-610	32	46	a	a	DET
cana-610	32	47	multiplicatively	multiplicatively	ADV
cana-610	32	48	closed	close	VERB
cana-610	32	49	set	set	VERB
cana-610	32	50	c	c	NOUN
cana-610	32	51	of	of	ADP
cana-610	32	52	compact	compact	ADJ
cana-610	32	53	elements	element	NOUN
cana-610	32	54	and	and	CCONJ
cana-610	32	55	the	the	DET
cana-610	32	56	greatest	great	ADJ
cana-610	32	57	element	element	NOUN
cana-610	32	58	1	1	NUM
cana-610	32	59	is	be	AUX
cana-610	32	60	compact	compact	ADJ
cana-610	32	61	as	as	ADV
cana-610	32	62	well	well	ADV
cana-610	32	63	as	as	ADP
cana-610	32	64	multiplicative	multiplicative	ADJ
cana-610	32	65	identity	identity	NOUN
cana-610	32	66	.	.	PUNCT
cana-610	33	1	an	an	DET
cana-610	33	2	element	element	NOUN
cana-610	33	3	p	p	PROPN
cana-610	33	4	∈	∈	PROPN
cana-610	33	5	£	£	NOUN
cana-610	33	6	is	be	AUX
cana-610	33	7	said	say	VERB
cana-610	33	8	to	to	PART
cana-610	33	9	be	be	AUX
cana-610	33	10	proper	proper	ADJ
cana-610	33	11	if	if	SCONJ
cana-610	33	12	p	p	NOUN
cana-610	33	13	<	<	NOUN
cana-610	33	14	1	1	NUM
cana-610	33	15	.	.	PUNCT
cana-610	34	1	in	in	ADP
cana-610	34	2	a	a	DET
cana-610	34	3	multiplicative	multiplicative	ADJ
cana-610	34	4	lattice	lattice	NOUN
cana-610	34	5	£	£	PROPN
cana-610	34	6	,	,	PUNCT
cana-610	34	7	a	a	DET
cana-610	34	8	proper	proper	ADJ
cana-610	34	9	element	element	NOUN
cana-610	34	10	m	m	VERB
cana-610	34	11	is	be	AUX
cana-610	34	12	maximal	maximal	ADJ
cana-610	34	13	,	,	PUNCT
cana-610	34	14	if	if	SCONJ
cana-610	34	15	m	m	NOUN
cana-610	34	16	is	be	AUX
cana-610	34	17	not	not	PART
cana-610	34	18	properly	properly	ADV
cana-610	34	19	contained	contain	VERB
cana-610	34	20	within	within	ADP
cana-610	34	21	any	any	DET
cana-610	34	22	other	other	ADJ
cana-610	34	23	element	element	NOUN
cana-610	34	24	of	of	ADP
cana-610	34	25	£	£	NOUN
cana-610	34	26	under	under	ADP
cana-610	34	27	the	the	DET
cana-610	34	28	partial	partial	ADJ
cana-610	34	29	order	order	NOUN
cana-610	34	30	relation	relation	NOUN
cana-610	34	31	≤.	≤.	NOUN
cana-610	34	32	we	we	PRON
cana-610	34	33	denote	denote	VERB
cana-610	34	34	m(£	m(£	PROPN
cana-610	34	35	)	)	PUNCT
cana-610	34	36	,	,	PUNCT
cana-610	34	37	the	the	DET
cana-610	34	38	collection	collection	NOUN
cana-610	34	39	of	of	ADP
cana-610	34	40	all	all	DET
cana-610	34	41	maximal	maximal	ADJ
cana-610	34	42	elements	element	NOUN
cana-610	34	43	of	of	ADP
cana-610	34	44	£	£	SYM
cana-610	34	45	.	.	PUNCT
cana-610	35	1	in	in	ADP
cana-610	35	2	a	a	DET
cana-610	35	3	multiplicative	multiplicative	ADJ
cana-610	35	4	lattice	lattice	NOUN
cana-610	35	5	£	£	SYM
cana-610	35	6	,	,	PUNCT
cana-610	35	7	if	if	SCONJ
cana-610	35	8	greatest	great	ADJ
cana-610	35	9	element	element	NOUN
cana-610	35	10	1	1	NUM
cana-610	35	11	is	be	AUX
cana-610	35	12	compact	compact	ADJ
cana-610	35	13	then	then	ADV
cana-610	35	14	each	each	DET
cana-610	35	15	a	a	DET
cana-610	35	16	<	<	NOUN
cana-610	35	17	1	1	NUM
cana-610	35	18	lies	lie	NOUN
cana-610	35	19	below	below	ADP
cana-610	35	20	some	some	DET
cana-610	35	21	m	m	NOUN
cana-610	35	22	∈	∈	PROPN
cana-610	35	23	m(£	m(£	PROPN
cana-610	35	24	)	)	PUNCT
cana-610	35	25	.	.	PUNCT
cana-610	36	1	if	if	SCONJ
cana-610	36	2	m(£	m(£	PROPN
cana-610	36	3	)	)	PUNCT
cana-610	36	4	=	=	PRON
cana-610	36	5	{	{	PUNCT
cana-610	36	6	m	m	NOUN
cana-610	36	7	}	}	PUNCT
cana-610	36	8	,	,	PUNCT
cana-610	36	9	then	then	ADV
cana-610	36	10	£	£	PROPN
cana-610	36	11	is	be	AUX
cana-610	36	12	called	call	VERB
cana-610	36	13	as	as	ADP
cana-610	36	14	local	local	ADJ
cana-610	36	15	.	.	PUNCT
cana-610	37	1	for	for	ADP
cana-610	37	2	a	a	DET
cana-610	37	3	,	,	PUNCT
cana-610	37	4	b	b	PROPN
cana-610	37	5	∈	∈	PROPN
cana-610	37	6	£	£	PROPN
cana-610	37	7	,	,	PUNCT
cana-610	37	8	a	a	DET
cana-610	37	9	proper	proper	ADJ
cana-610	37	10	element	element	NOUN
cana-610	37	11	p	p	PROPN
cana-610	37	12	∈	∈	PROPN
cana-610	37	13	m	m	VERB
cana-610	37	14	is	be	AUX
cana-610	37	15	said	say	VERB
cana-610	37	16	to	to	PART
cana-610	37	17	be	be	AUX
cana-610	37	18	prime	prime	ADJ
cana-610	37	19	,	,	PUNCT
cana-610	37	20	if	if	SCONJ
cana-610	37	21	ab	ab	PROPN
cana-610	37	22	≤	≤	PROPN
cana-610	37	23	p	p	X
cana-610	37	24	,	,	PUNCT
cana-610	37	25	then	then	ADV
cana-610	37	26	a	a	DET
cana-610	37	27	≤	≤	ADJ
cana-610	37	28	p	p	NOUN
cana-610	37	29	or	or	CCONJ
cana-610	37	30	b	b	NOUN
cana-610	37	31	≤	≤	NOUN
cana-610	37	32	p.	p.	NOUN
cana-610	37	33	let	let	VERB
cana-610	37	34	σ(£	σ(£	PROPN
cana-610	37	35	)	)	PUNCT
cana-610	37	36	the	the	DET
cana-610	37	37	collection	collection	NOUN
cana-610	37	38	of	of	ADP
cana-610	37	39	all	all	DET
cana-610	37	40	prime	prime	ADJ
cana-610	37	41	elements	element	NOUN
cana-610	37	42	of	of	ADP
cana-610	37	43	£	£	SYM
cana-610	37	44	.	.	PUNCT
cana-610	38	1	as	as	SCONJ
cana-610	38	2	each	each	DET
cana-610	38	3	maximal	maximal	ADJ
cana-610	38	4	element	element	NOUN
cana-610	38	5	is	be	AUX
cana-610	38	6	prime	prime	ADJ
cana-610	38	7	,	,	PUNCT
cana-610	38	8	we	we	PRON
cana-610	38	9	have	have	VERB
cana-610	38	10	m(£	m(£	PROPN
cana-610	38	11	)	)	PUNCT
cana-610	38	12	⊆	⊆	NUM
cana-610	38	13	σ(£	σ(£	PROPN
cana-610	38	14	)	)	PUNCT
cana-610	38	15	.	.	PUNCT
cana-610	39	1	a	a	DET
cana-610	39	2	multiplicative	multiplicative	ADJ
cana-610	39	3	lattice	lattice	NOUN
cana-610	39	4	£	£	PROPN
cana-610	39	5	is	be	AUX
cana-610	39	6	said	say	VERB
cana-610	39	7	to	to	PART
cana-610	39	8	be	be	AUX
cana-610	39	9	domain	domain	NOUN
cana-610	39	10	if	if	SCONJ
cana-610	39	11	0	0	NUM
cana-610	39	12	∈	∈	PROPN
cana-610	39	13	σ(£	σ(£	PROPN
cana-610	39	14	)	)	PUNCT
cana-610	39	15	.	.	PUNCT
cana-610	40	1	in	in	ADP
cana-610	40	2	this	this	DET
cana-610	40	3	paper	paper	NOUN
cana-610	40	4	,	,	PUNCT
cana-610	40	5	we	we	PRON
cana-610	40	6	used	use	VERB
cana-610	40	7	a	a	DET
cana-610	40	8	non	non	ADJ
cana-610	40	9	-	-	ADJ
cana-610	40	10	empty	empty	ADJ
cana-610	40	11	subset	subset	NOUN
cana-610	40	12	s	s	NOUN
cana-610	40	13	of	of	ADP
cana-610	40	14	m(£	m(£	PROPN
cana-610	40	15	)	)	PUNCT
cana-610	40	16	and	and	CCONJ
cana-610	40	17	defined	define	VERB
cana-610	40	18	a	a	DET
cana-610	40	19	new	new	ADJ
cana-610	40	20	simple	simple	ADJ
cana-610	40	21	,	,	PUNCT
cana-610	40	22	undirected	undirected	ADJ
cana-610	40	23	graph	graph	NOUN
cana-610	40	24	called	call	VERB
cana-610	40	25	the	the	DET
cana-610	40	26	s	s	NOUN
cana-610	40	27	-	-	PUNCT
cana-610	40	28	join	join	ADJ
cana-610	40	29	graph	graph	NOUN
cana-610	40	30	γs(σ(£	γs(σ(£	NOUN
cana-610	40	31	)	)	PUNCT
cana-610	40	32	)	)	PUNCT
cana-610	40	33	with	with	ADP
cana-610	40	34	the	the	DET
cana-610	40	35	vertex	vertex	NOUN
cana-610	40	36	set	set	VERB
cana-610	40	37	σ(£	σ(£	PROPN
cana-610	40	38	)	)	PUNCT
cana-610	40	39	,	,	PUNCT
cana-610	40	40	where	where	SCONJ
cana-610	40	41	σ(£	σ(£	NOUN
cana-610	40	42	)	)	PUNCT
cana-610	40	43	is	be	AUX
cana-610	40	44	the	the	DET
cana-610	40	45	collection	collection	NOUN
cana-610	40	46	of	of	ADP
cana-610	40	47	all	all	DET
cana-610	40	48	prime	prime	ADJ
cana-610	40	49	elements	element	NOUN
cana-610	40	50	of	of	ADP
cana-610	40	51	£	£	NUM
cana-610	40	52	and	and	CCONJ
cana-610	40	53	two	two	NUM
cana-610	40	54	distinct	distinct	ADJ
cana-610	40	55	vertices	vertex	NOUN
cana-610	40	56	a	a	PRON
cana-610	40	57	and	and	CCONJ
cana-610	40	58	b	b	NOUN
cana-610	40	59	are	be	AUX
cana-610	40	60	adjacent	adjacent	ADJ
cana-610	40	61	i.e.	i.e.	X
cana-610	40	62	,	,	PUNCT
cana-610	40	63	a	a	DET
cana-610	40	64	∼	∼	NOUN
cana-610	40	65	b	b	NOUN
cana-610	40	66	if	if	SCONJ
cana-610	40	67	and	and	CCONJ
cana-610	40	68	only	only	ADV
cana-610	40	69	if	if	SCONJ
cana-610	40	70	a	a	DET
cana-610	40	71	∨	∨	NUM
cana-610	40	72	b	b	NOUN
cana-610	40	73	≤	≤	NUM
cana-610	40	74	m	m	VERB
cana-610	40	75	for	for	ADP
cana-610	40	76	some	some	DET
cana-610	40	77	m	m	NOUN
cana-610	40	78	∈	∈	PROPN
cana-610	40	79	s.	s.	PROPN
cana-610	40	80	here	here	ADV
cana-610	40	81	,	,	PUNCT
cana-610	40	82	we	we	PRON
cana-610	40	83	study	study	VERB
cana-610	40	84	some	some	DET
cana-610	40	85	basic	basic	ADJ
cana-610	40	86	properties	property	NOUN
cana-610	40	87	like	like	ADP
cana-610	40	88	connectivity	connectivity	NOUN
cana-610	40	89	,	,	PUNCT
cana-610	40	90	girth	girth	NOUN
cana-610	40	91	and	and	CCONJ
cana-610	40	92	clique	clique	ADJ
cana-610	40	93	number	number	NOUN
cana-610	40	94	of	of	ADP
cana-610	40	95	the	the	DET
cana-610	40	96	graph	graph	NOUN
cana-610	40	97	γs(σ(£	γs(σ(£	NOUN
cana-610	40	98	)	)	PUNCT
cana-610	40	99	)	)	PUNCT
cana-610	40	100	.	.	PUNCT
cana-610	41	1	throughout	throughout	ADP
cana-610	41	2	this	this	DET
cana-610	41	3	paper	paper	NOUN
cana-610	41	4	,	,	PUNCT
cana-610	41	5	multiplicative	multiplicative	PROPN
cana-610	41	6	lattice	lattice	PROPN
cana-610	41	7	£	£	PROPN
cana-610	41	8	assumed	assume	VERB
cana-610	41	9	to	to	PART
cana-610	41	10	be	be	AUX
cana-610	41	11	c	c	NOUN
cana-610	41	12	-	-	PUNCT
cana-610	41	13	lattice	lattice	NOUN
cana-610	41	14	£	£	PROPN
cana-610	41	15	.	.	PUNCT
cana-610	41	16	by	by	ADP
cana-610	41	17	σ(£	σ(£	PROPN
cana-610	41	18	)	)	PUNCT
cana-610	41	19	and	and	CCONJ
cana-610	41	20	m(£	m(£	PROPN
cana-610	41	21	)	)	PUNCT
cana-610	41	22	,	,	PUNCT
cana-610	41	23	we	we	PRON
cana-610	41	24	mean	mean	VERB
cana-610	41	25	the	the	DET
cana-610	41	26	collection	collection	NOUN
cana-610	41	27	of	of	ADP
cana-610	41	28	all	all	DET
cana-610	41	29	prime	prime	ADJ
cana-610	41	30	elements	element	NOUN
cana-610	41	31	,	,	PUNCT
cana-610	41	32	and	and	CCONJ
cana-610	41	33	the	the	DET
cana-610	41	34	collection	collection	NOUN
cana-610	41	35	of	of	ADP
cana-610	41	36	all	all	DET
cana-610	41	37	maximal	maximal	ADJ
cana-610	41	38	elements	element	NOUN
cana-610	41	39	of	of	ADP
cana-610	41	40	£	£	SYM
cana-610	41	41	,	,	PUNCT
cana-610	41	42	respectively	respectively	ADV
cana-610	41	43	.	.	PUNCT
cana-610	42	1	2	2	X
cana-610	42	2	.	.	X
cana-610	42	3	graph	graph	NOUN
cana-610	42	4	theoretic	theoretic	ADJ
cana-610	42	5	definitions	definition	NOUN
cana-610	42	6	let	let	VERB
cana-610	42	7	the	the	DET
cana-610	42	8	undirected	undirected	ADJ
cana-610	42	9	graph	graph	NOUN
cana-610	42	10	g	g	PROPN
cana-610	42	11	=	=	PUNCT
cana-610	42	12	(	(	PUNCT
cana-610	42	13	v	v	NOUN
cana-610	42	14	,	,	PUNCT
cana-610	42	15	e	e	NOUN
cana-610	42	16	)	)	PUNCT
cana-610	42	17	,	,	PUNCT
cana-610	42	18	where	where	SCONJ
cana-610	42	19	v	v	NOUN
cana-610	42	20	=	=	SYM
cana-610	42	21	v	v	NOUN
cana-610	42	22	(	(	PUNCT
cana-610	42	23	g	g	NOUN
cana-610	42	24	)	)	PUNCT
cana-610	42	25	is	be	AUX
cana-610	42	26	the	the	DET
cana-610	42	27	set	set	NOUN
cana-610	42	28	of	of	ADP
cana-610	42	29	vertices	vertex	NOUN
cana-610	42	30	of	of	ADP
cana-610	42	31	g	g	NOUN
cana-610	42	32	and	and	CCONJ
cana-610	42	33	e	e	PROPN
cana-610	42	34	=	=	PROPN
cana-610	42	35	e(g	e(g	PROPN
cana-610	42	36	)	)	PUNCT
cana-610	42	37	is	be	AUX
cana-610	42	38	the	the	DET
cana-610	42	39	set	set	NOUN
cana-610	42	40	of	of	ADP
cana-610	42	41	edges	edge	NOUN
cana-610	42	42	of	of	ADP
cana-610	42	43	g.	g.	PROPN
cana-610	42	44	a	a	DET
cana-610	42	45	graph	graph	NOUN
cana-610	42	46	with	with	ADP
cana-610	42	47	empty	empty	ADJ
cana-610	42	48	vertex	vertex	NOUN
cana-610	42	49	set	set	NOUN
cana-610	42	50	is	be	AUX
cana-610	42	51	called	call	VERB
cana-610	42	52	an	an	DET
cana-610	42	53	empty	empty	ADJ
cana-610	42	54	graph	graph	NOUN
cana-610	42	55	.	.	PUNCT
cana-610	43	1	let	let	VERB
cana-610	43	2	b	b	X
cana-610	43	3	∈	∈	PROPN
cana-610	43	4	v	v	NOUN
cana-610	43	5	,	,	PUNCT
cana-610	43	6	the	the	DET
cana-610	43	7	number	number	NOUN
cana-610	43	8	of	of	ADP
cana-610	43	9	edges	edge	NOUN
cana-610	43	10	incident	incident	NOUN
cana-610	43	11	on	on	ADP
cana-610	43	12	b	b	PROPN
cana-610	43	13	is	be	AUX
cana-610	43	14	called	call	VERB
cana-610	43	15	degree	degree	NOUN
cana-610	43	16	of	of	ADP
cana-610	43	17	a	a	DET
cana-610	43	18	vertex	vertex	NOUN
cana-610	43	19	b	b	NOUN
cana-610	43	20	and	and	CCONJ
cana-610	43	21	it	it	PRON
cana-610	43	22	is	be	AUX
cana-610	43	23	denoted	denote	VERB
cana-610	43	24	by	by	ADP
cana-610	43	25	deg(b	deg(b	PROPN
cana-610	43	26	)	)	PUNCT
cana-610	43	27	.	.	PUNCT
cana-610	44	1	in	in	ADP
cana-610	44	2	a	a	DET
cana-610	44	3	graph	graph	NOUN
cana-610	44	4	g	g	NOUN
cana-610	44	5	,	,	PUNCT
cana-610	44	6	d(a	d(a	PROPN
cana-610	44	7	,	,	PUNCT
cana-610	44	8	c	c	NOUN
cana-610	44	9	)	)	PUNCT
cana-610	44	10	represents	represent	VERB
cana-610	44	11	the	the	DET
cana-610	44	12	length	length	NOUN
cana-610	44	13	of	of	ADP
cana-610	44	14	shortest	short	ADJ
cana-610	44	15	path	path	NOUN
cana-610	44	16	between	between	ADP
cana-610	44	17	a	a	PRON
cana-610	44	18	and	and	CCONJ
cana-610	44	19	c.	c.	NOUN
cana-610	44	20	note	note	VERB
cana-610	44	21	that	that	SCONJ
cana-610	44	22	,	,	PUNCT
cana-610	44	23	d(a	d(a	PROPN
cana-610	44	24	,	,	PUNCT
cana-610	44	25	c	c	NOUN
cana-610	44	26	)	)	PUNCT
cana-610	44	27	=	=	SYM
cana-610	45	1	∞	∞	PROPN
cana-610	45	2	,	,	PUNCT
cana-610	45	3	if	if	SCONJ
cana-610	45	4	there	there	PRON
cana-610	45	5	is	be	VERB
cana-610	45	6	no	no	DET
cana-610	45	7	path	path	NOUN
cana-610	45	8	between	between	ADP
cana-610	45	9	a	a	DET
cana-610	45	10	and	and	CCONJ
cana-610	45	11	c.	c.	NOUN
cana-610	45	12	the	the	DET
cana-610	45	13	diameter	diameter	NOUN
cana-610	45	14	of	of	ADP
cana-610	45	15	a	a	DET
cana-610	45	16	graph	graph	NOUN
cana-610	45	17	g	g	NOUN
cana-610	45	18	is	be	AUX
cana-610	45	19	defined	define	VERB
cana-610	45	20	as	as	ADP
cana-610	45	21	diam(g	diam(g	NOUN
cana-610	45	22	)	)	PUNCT
cana-610	46	1	=	=	SYM
cana-610	46	2	sup{d(a	sup{d(a	PROPN
cana-610	46	3	,	,	PUNCT
cana-610	46	4	c)|a	c)|a	VERB
cana-610	46	5	,	,	PUNCT
cana-610	46	6	c	c	PROPN
cana-610	46	7	∈	∈	PROPN
cana-610	46	8	v	v	NOUN
cana-610	46	9	(	(	PUNCT
cana-610	46	10	g	g	NOUN
cana-610	46	11	)	)	PUNCT
cana-610	46	12	}	}	PUNCT
cana-610	46	13	.	.	PUNCT
cana-610	47	1	the	the	DET
cana-610	47	2	length	length	NOUN
cana-610	47	3	of	of	ADP
cana-610	47	4	shortest	short	ADJ
cana-610	47	5	cycle	cycle	NOUN
cana-610	47	6	in	in	ADP
cana-610	47	7	g	g	PROPN
cana-610	47	8	is	be	AUX
cana-610	47	9	called	call	VERB
cana-610	47	10	the	the	DET
cana-610	47	11	girth	girth	NOUN
cana-610	47	12	of	of	ADP
cana-610	47	13	g	g	NOUN
cana-610	47	14	,	,	PUNCT
cana-610	47	15	denoted	denote	VERB
cana-610	47	16	by	by	ADP
cana-610	47	17	gr(g	gr(g	PROPN
cana-610	47	18	)	)	PUNCT
cana-610	47	19	.	.	PUNCT
cana-610	48	1	a	a	DET
cana-610	48	2	clique	clique	NOUN
cana-610	48	3	of	of	ADP
cana-610	48	4	graph	graph	NOUN
cana-610	48	5	is	be	AUX
cana-610	48	6	its	its	PRON
cana-610	48	7	maximal	maximal	ADJ
cana-610	48	8	complete	complete	ADJ
cana-610	48	9	subgraph	subgraph	NOUN
cana-610	48	10	.	.	PUNCT
cana-610	49	1	for	for	ADP
cana-610	49	2	a	a	DET
cana-610	49	3	graph	graph	NOUN
cana-610	49	4	g	g	NOUN
cana-610	49	5	,	,	PUNCT
cana-610	49	6	a	a	DET
cana-610	49	7	subset	subset	NOUN
cana-610	49	8	s	s	VERB
cana-610	49	9	⊆	⊆	NUM
cana-610	49	10	v	v	NOUN
cana-610	49	11	(	(	PUNCT
cana-610	49	12	g	g	NOUN
cana-610	49	13	)	)	PUNCT
cana-610	49	14	is	be	AUX
cana-610	49	15	supposed	suppose	VERB
cana-610	49	16	to	to	PART
cana-610	49	17	be	be	AUX
cana-610	49	18	independent	independent	ADJ
cana-610	49	19	,	,	PUNCT
cana-610	49	20	if	if	SCONJ
cana-610	49	21	no	no	DET
cana-610	49	22	two	two	NUM
cana-610	49	23	vertices	vertex	NOUN
cana-610	49	24	in	in	ADP
cana-610	49	25	s	s	NOUN
cana-610	49	26	are	be	AUX
cana-610	49	27	adjacent	adjacent	ADJ
cana-610	49	28	.	.	PUNCT
cana-610	50	1	the	the	DET
cana-610	50	2	independence	independence	NOUN
cana-610	50	3	number	number	NOUN
cana-610	50	4	α(g	α(g	NUM
cana-610	50	5	)	)	PUNCT
cana-610	50	6	is	be	AUX
cana-610	50	7	the	the	DET
cana-610	50	8	maximum	maximum	ADJ
cana-610	50	9	size	size	NOUN
cana-610	50	10	of	of	ADP
cana-610	50	11	an	an	DET
cana-610	50	12	independent	independent	ADJ
cana-610	50	13	set	set	NOUN
cana-610	50	14	in	in	ADP
cana-610	50	15	g.	g.	PROPN
cana-610	50	16	let	let	VERB
cana-610	50	17	∅	∅	NOUN
cana-610	50	18	≠	≠	PROPN
cana-610	50	19	s	s	PART
cana-610	50	20	⊆	⊆	NUM
cana-610	50	21	v.	v.	ADP
cana-610	50	22	if	if	SCONJ
cana-610	50	23	each	each	DET
cana-610	50	24	vertex	vertex	NOUN
cana-610	50	25	in	in	ADP
cana-610	50	26	v−s	v−	NOUN
cana-610	50	27	is	be	AUX
cana-610	50	28	adjacent	adjacent	ADJ
cana-610	50	29	to	to	ADP
cana-610	50	30	some	some	DET
cana-610	50	31	vertex	vertex	NOUN
cana-610	50	32	in	in	ADP
cana-610	50	33	s	s	PROPN
cana-610	50	34	,	,	PUNCT
cana-610	50	35	then	then	ADV
cana-610	50	36	s	s	VERB
cana-610	50	37	is	be	AUX
cana-610	50	38	called	call	VERB
cana-610	50	39	a	a	DET
cana-610	50	40	dominating	dominating	NOUN
cana-610	50	41	set	set	NOUN
cana-610	50	42	.	.	PUNCT
cana-610	51	1	number	number	NOUN
cana-610	51	2	of	of	ADP
cana-610	51	3	vertices	vertex	NOUN
cana-610	51	4	in	in	ADP
cana-610	51	5	smallest	small	ADJ
cana-610	51	6	dominating	dominating	NOUN
cana-610	51	7	set	set	NOUN
cana-610	51	8	is	be	AUX
cana-610	51	9	called	call	VERB
cana-610	51	10	domination	domination	NOUN
cana-610	51	11	number	number	NOUN
cana-610	51	12	and	and	CCONJ
cana-610	51	13	it	it	PRON
cana-610	51	14	is	be	AUX
cana-610	51	15	denoted	denote	VERB
cana-610	51	16	by	by	ADP
cana-610	51	17	γ(g	γ(g	PROPN
cana-610	51	18	)	)	PUNCT
cana-610	51	19	.	.	PUNCT
cana-610	52	1	for	for	ADP
cana-610	52	2	more	more	ADJ
cana-610	52	3	information	information	NOUN
cana-610	52	4	on	on	ADP
cana-610	52	5	graph	graph	NOUN
cana-610	52	6	theory	theory	NOUN
cana-610	52	7	,	,	PUNCT
cana-610	52	8	the	the	DET
cana-610	52	9	reader	reader	NOUN
cana-610	52	10	may	may	AUX
cana-610	52	11	refer	refer	VERB
cana-610	52	12	(	(	PUNCT
cana-610	52	13	[	[	X
cana-610	52	14	11	11	NUM
cana-610	52	15	]	]	PUNCT
cana-610	52	16	,	,	PUNCT
cana-610	52	17	[	[	X
cana-610	52	18	14	14	NUM
cana-610	52	19	]	]	SYM
cana-610	52	20	)	)	PUNCT
cana-610	52	21	.	.	PUNCT
cana-610	53	1	communications	communication	NOUN
cana-610	53	2	on	on	ADP
cana-610	53	3	applied	apply	VERB
cana-610	53	4	nonlinear	nonlinear	ADJ
cana-610	53	5	analysis	analysis	NOUN
cana-610	53	6	issn	issn	NOUN
cana-610	53	7	:	:	PUNCT
cana-610	53	8	1074	1074	NUM
cana-610	53	9	-	-	PUNCT
cana-610	53	10	133x	133x	NUM
cana-610	53	11	vol	vol	NOUN
cana-610	53	12	31	31	NUM
cana-610	53	13	no	no	NOUN
cana-610	53	14	.	.	NOUN
cana-610	53	15	2	2	NUM
cana-610	53	16	(	(	PUNCT
cana-610	53	17	2024	2024	NUM
cana-610	53	18	)	)	PUNCT
cana-610	53	19	411	411	NUM
cana-610	53	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-610	53	21	3	3	X
cana-610	53	22	.	.	X
cana-610	54	1	s	s	X
cana-610	54	2	-	-	PUNCT
cana-610	54	3	join	join	VERB
cana-610	54	4	graph	graph	NOUN
cana-610	54	5	γs((σ(£	γs((σ(£	PROPN
cana-610	54	6	)	)	PUNCT
cana-610	54	7	)	)	PUNCT
cana-610	55	1	we	we	PRON
cana-610	55	2	start	start	VERB
cana-610	55	3	this	this	DET
cana-610	55	4	section	section	NOUN
cana-610	55	5	with	with	ADP
cana-610	55	6	the	the	DET
cana-610	55	7	following	follow	VERB
cana-610	55	8	definition	definition	NOUN
cana-610	55	9	.	.	PUNCT
cana-610	56	1	definition	definition	NOUN
cana-610	56	2	3.1	3.1	NUM
cana-610	56	3	.	.	PUNCT
cana-610	57	1	let	let	VERB
cana-610	57	2	m(£	m(£	PROPN
cana-610	57	3	)	)	PUNCT
cana-610	57	4	be	be	AUX
cana-610	57	5	the	the	DET
cana-610	57	6	set	set	NOUN
cana-610	57	7	of	of	ADP
cana-610	57	8	all	all	DET
cana-610	57	9	maximal	maximal	ADJ
cana-610	57	10	elements	element	NOUN
cana-610	57	11	of	of	ADP
cana-610	57	12	c	c	NOUN
cana-610	57	13	-	-	PUNCT
cana-610	57	14	lattice	lattice	NOUN
cana-610	57	15	£	£	NOUN
cana-610	57	16	and	and	CCONJ
cana-610	58	1	∅	∅	NOUN
cana-610	58	2	≠	≠	PROPN
cana-610	58	3	s	s	PART
cana-610	58	4	⊆	⊆	NUM
cana-610	58	5	m(£	m(£	PROPN
cana-610	58	6	)	)	PUNCT
cana-610	58	7	.	.	PUNCT
cana-610	59	1	the	the	DET
cana-610	59	2	s	s	NOUN
cana-610	59	3	-	-	PUNCT
cana-610	59	4	join	join	ADJ
cana-610	59	5	graph	graph	NOUN
cana-610	59	6	γs(σ(£	γs(σ(£	NOUN
cana-610	59	7	)	)	PUNCT
cana-610	59	8	)	)	PUNCT
cana-610	59	9	is	be	AUX
cana-610	59	10	simple	simple	ADJ
cana-610	59	11	,	,	PUNCT
cana-610	59	12	undirected	undirected	ADJ
cana-610	59	13	graph	graph	NOUN
cana-610	59	14	with	with	ADP
cana-610	59	15	vertex	vertex	NOUN
cana-610	59	16	set	set	VERB
cana-610	59	17	σ(£	σ(£	PROPN
cana-610	59	18	)	)	PUNCT
cana-610	59	19	and	and	CCONJ
cana-610	59	20	two	two	NUM
cana-610	59	21	distinct	distinct	ADJ
cana-610	59	22	vertices	vertex	NOUN
cana-610	59	23	a	a	PRON
cana-610	59	24	and	and	CCONJ
cana-610	59	25	b	b	NOUN
cana-610	59	26	are	be	AUX
cana-610	59	27	adjacent	adjacent	ADJ
cana-610	59	28	if	if	SCONJ
cana-610	59	29	and	and	CCONJ
cana-610	59	30	only	only	ADV
cana-610	59	31	if	if	SCONJ
cana-610	59	32	a	a	DET
cana-610	59	33	∨	∨	NUM
cana-610	59	34	b	b	NOUN
cana-610	59	35	≤	≤	NUM
cana-610	59	36	m	m	VERB
cana-610	59	37	for	for	ADP
cana-610	59	38	some	some	DET
cana-610	59	39	m	m	NOUN
cana-610	59	40	∈	∈	PROPN
cana-610	59	41	s.	s.	PROPN
cana-610	59	42	example	example	NOUN
cana-610	59	43	3.2	3.2	NUM
cana-610	59	44	.	.	PUNCT
cana-610	60	1	the	the	DET
cana-610	60	2	lattice	lattice	NOUN
cana-610	60	3	in	in	ADP
cana-610	60	4	figure	figure	NOUN
cana-610	60	5	(	(	PUNCT
cana-610	60	6	1	1	NUM
cana-610	60	7	)	)	PUNCT
cana-610	60	8	is	be	AUX
cana-610	60	9	a	a	DET
cana-610	60	10	multiplicative	multiplicative	ADJ
cana-610	60	11	lattice	lattice	NOUN
cana-610	60	12	£	£	NOUN
cana-610	60	13	and	and	CCONJ
cana-610	60	14	figure	figure	NOUN
cana-610	60	15	(	(	PUNCT
cana-610	60	16	2	2	NUM
cana-610	60	17	)	)	PUNCT
cana-610	60	18	represents	represent	VERB
cana-610	60	19	the	the	DET
cana-610	60	20	s	s	NOUN
cana-610	60	21	-	-	PUNCT
cana-610	60	22	join	join	ADJ
cana-610	60	23	graph	graph	NOUN
cana-610	60	24	γs(σ(£	γs(σ(£	NOUN
cana-610	60	25	)	)	PUNCT
cana-610	60	26	)	)	PUNCT
cana-610	60	27	with	with	ADP
cana-610	60	28	the	the	DET
cana-610	60	29	vertex	vertex	NOUN
cana-610	60	30	set	set	VERB
cana-610	60	31	σ(l	σ(l	NOUN
cana-610	60	32	)	)	PUNCT
cana-610	60	33	=	=	PRON
cana-610	60	34	{	{	PUNCT
cana-610	60	35	a	a	X
cana-610	60	36	,	,	PUNCT
cana-610	60	37	c	c	NOUN
cana-610	60	38	,	,	PUNCT
cana-610	60	39	d	d	NOUN
cana-610	60	40	}	}	PUNCT
cana-610	60	41	and	and	CCONJ
cana-610	60	42	s	s	AUX
cana-610	60	43	=	=	X
cana-610	60	44	{	{	PUNCT
cana-610	60	45	c	c	NOUN
cana-610	60	46	,	,	PUNCT
cana-610	60	47	d	d	NOUN
cana-610	60	48	}	}	PUNCT
cana-610	60	49	.	.	PUNCT
cana-610	61	1	figure	figure	NOUN
cana-610	61	2	(	(	PUNCT
cana-610	61	3	1	1	NUM
cana-610	61	4	)	)	PUNCT
cana-610	61	5	multiplicative	multiplicative	ADJ
cana-610	61	6	lattice	lattice	NOUN
cana-610	61	7	£	£	PROPN
cana-610	61	8	a	a	DET
cana-610	61	9	d	d	X
cana-610	61	10	c	c	NOUN
cana-610	61	11	figure	figure	NOUN
cana-610	61	12	(	(	PUNCT
cana-610	61	13	2	2	NUM
cana-610	61	14	)	)	PUNCT
cana-610	61	15	γs(σ(£	γs(σ(£	NOUN
cana-610	61	16	)	)	PUNCT
cana-610	61	17	)	)	PUNCT
cana-610	61	18	definition	definition	NOUN
cana-610	61	19	3.3	3.3	NUM
cana-610	61	20	.	.	PUNCT
cana-610	62	1	a	a	DET
cana-610	62	2	non	non	ADJ
cana-610	62	3	-	-	ADJ
cana-610	62	4	empty	empty	ADJ
cana-610	62	5	subset	subset	NOUN
cana-610	62	6	s	s	NOUN
cana-610	62	7	of	of	ADP
cana-610	62	8	m(£	m(£	PROPN
cana-610	62	9	)	)	PUNCT
cana-610	62	10	is	be	AUX
cana-610	62	11	said	say	VERB
cana-610	62	12	to	to	PART
cana-610	62	13	be	be	AUX
cana-610	62	14	small	small	ADJ
cana-610	62	15	with	with	ADP
cana-610	62	16	respect	respect	NOUN
cana-610	62	17	to	to	ADP
cana-610	62	18	∧	∧	PROPN
cana-610	62	19	in	in	ADP
cana-610	62	20	short	short	ADJ
cana-610	62	21	∧min	∧min	ADJ
cana-610	62	22	set	set	NOUN
cana-610	62	23	,	,	PUNCT
cana-610	62	24	if	if	SCONJ
cana-610	62	25	for	for	ADP
cana-610	62	26	any	any	DET
cana-610	62	27	a	a	PRON
cana-610	62	28	,	,	PUNCT
cana-610	62	29	b	b	PROPN
cana-610	62	30	∈	∈	PROPN
cana-610	62	31	s	s	X
cana-610	62	32	,	,	PUNCT
cana-610	62	33	there	there	PRON
cana-610	62	34	is	be	VERB
cana-610	62	35	no	no	DET
cana-610	62	36	0	0	NUM
cana-610	62	37	≠	≠	PROPN
cana-610	62	38	p	p	PROPN
cana-610	62	39	∈	∈	PROPN
cana-610	62	40	σ(£	σ(£	PROPN
cana-610	62	41	)	)	PUNCT
cana-610	62	42	such	such	ADJ
cana-610	62	43	that	that	SCONJ
cana-610	62	44	p	p	PROPN
cana-610	62	45	≤	≤	NOUN
cana-610	62	46	a	a	DET
cana-610	62	47	∧	∧	PROPN
cana-610	62	48	b.	b.	PROPN
cana-610	62	49	proposition	proposition	NOUN
cana-610	62	50	3.4	3.4	NUM
cana-610	62	51	.	.	PUNCT
cana-610	63	1	let	let	VERB
cana-610	63	2	£	£	PART
cana-610	63	3	be	be	AUX
cana-610	63	4	a	a	DET
cana-610	63	5	c	c	NOUN
cana-610	63	6	-	-	PUNCT
cana-610	63	7	lattice	lattice	NOUN
cana-610	63	8	and	and	CCONJ
cana-610	63	9	s	s	AUX
cana-610	63	10	be	be	AUX
cana-610	63	11	a	a	DET
cana-610	63	12	∧-min	∧-min	NOUN
cana-610	63	13	set	set	VERB
cana-610	63	14	.	.	PUNCT
cana-610	64	1	if	if	SCONJ
cana-610	64	2	the	the	DET
cana-610	64	3	s	s	NOUN
cana-610	64	4	-	-	PUNCT
cana-610	64	5	join	join	ADJ
cana-610	64	6	graph	graph	NOUN
cana-610	64	7	γs(σ(£	γs(σ(£	NOUN
cana-610	64	8	)	)	PUNCT
cana-610	64	9	)	)	PUNCT
cana-610	64	10	is	be	AUX
cana-610	64	11	connected	connect	VERB
cana-610	64	12	,	,	PUNCT
cana-610	64	13	then	then	ADV
cana-610	64	14	£	£	PROPN
cana-610	64	15	is	be	AUX
cana-610	64	16	a	a	DET
cana-610	64	17	local	local	ADJ
cana-610	64	18	or	or	CCONJ
cana-610	64	19	a	a	DET
cana-610	64	20	domain	domain	NOUN
cana-610	64	21	.	.	PUNCT
cana-610	65	1	proof	proof	NOUN
cana-610	65	2	.	.	PUNCT
cana-610	66	1	suppose	suppose	VERB
cana-610	66	2	that	that	SCONJ
cana-610	66	3	zero	zero	NUM
cana-610	66	4	element	element	NOUN
cana-610	66	5	of	of	ADP
cana-610	66	6	£	£	NOUN
cana-610	66	7	is	be	AUX
cana-610	66	8	not	not	PART
cana-610	66	9	a	a	DET
cana-610	66	10	prime	prime	NOUN
cana-610	66	11	and	and	CCONJ
cana-610	66	12	m1,m2	m1,m2	PROPN
cana-610	66	13	∈	∈	PROPN
cana-610	66	14	m(£	m(£	PROPN
cana-610	66	15	)	)	PUNCT
cana-610	66	16	.	.	PUNCT
cana-610	67	1	it	it	PRON
cana-610	67	2	is	be	AUX
cana-610	67	3	given	give	VERB
cana-610	67	4	that	that	SCONJ
cana-610	67	5	the	the	DET
cana-610	67	6	sjoin	sjoin	NOUN
cana-610	67	7	graph	graph	NOUN
cana-610	67	8	γs(σ(£	γs(σ(£	NOUN
cana-610	67	9	)	)	PUNCT
cana-610	67	10	)	)	PUNCT
cana-610	67	11	is	be	AUX
cana-610	67	12	connected	connect	VERB
cana-610	67	13	,	,	PUNCT
cana-610	67	14	we	we	PRON
cana-610	67	15	have	have	VERB
cana-610	67	16	a	a	DET
cana-610	67	17	path	path	NOUN
cana-610	67	18	m1	m1	NOUN
cana-610	67	19	∼	∼	NOUN
cana-610	67	20	p1	p1	NOUN
cana-610	67	21	∼	∼	NOUN
cana-610	67	22	•••	•••	ADV
cana-610	67	23	∼	∼	NOUN
cana-610	67	24	pn	pn	NOUN
cana-610	67	25	∼	∼	NOUN
cana-610	67	26	m2	m2	PROPN
cana-610	67	27	between	between	ADP
cana-610	67	28	m1	m1	PROPN
cana-610	67	29	and	and	CCONJ
cana-610	67	30	m2	m2	PROPN
cana-610	67	31	.	.	PUNCT
cana-610	68	1	clearly	clearly	ADV
cana-610	68	2	,	,	PUNCT
cana-610	68	3	all	all	PRON
cana-610	68	4	pi	pi	NOUN
cana-610	68	5	’s	’s	ADV
cana-610	68	6	are	be	AUX
cana-610	68	7	non	non	ADJ
cana-610	68	8	-	-	ADJ
cana-610	68	9	zero	zero	NUM
cana-610	68	10	members	member	NOUN
cana-610	68	11	of	of	ADP
cana-610	68	12	σ(£	σ(£	PROPN
cana-610	68	13	)	)	PUNCT
cana-610	68	14	.	.	PUNCT
cana-610	69	1	since	since	SCONJ
cana-610	69	2	m1	m1	PROPN
cana-610	69	3	is	be	AUX
cana-610	69	4	adjacent	adjacent	ADJ
cana-610	69	5	to	to	ADP
cana-610	69	6	p1	p1	NOUN
cana-610	69	7	and	and	CCONJ
cana-610	69	8	p1	p1	NOUN
cana-610	69	9	is	be	AUX
cana-610	69	10	adjacent	adjacent	ADJ
cana-610	69	11	to	to	ADP
cana-610	69	12	p2	p2	PROPN
cana-610	69	13	,	,	PUNCT
cana-610	69	14	we	we	PRON
cana-610	69	15	have	have	AUX
cana-610	69	16	p1	p1	NOUN
cana-610	69	17	≤	≤	NOUN
cana-610	69	18	m1	m1	NOUN
cana-610	69	19	and	and	CCONJ
cana-610	69	20	there	there	PRON
cana-610	69	21	exists	exist	VERB
cana-610	69	22	m	m	VERB
cana-610	69	23	∈	∈	NOUN
cana-610	69	24	s	s	NOUN
cana-610	69	25	with	with	ADP
cana-610	69	26	p1∨p2	p1∨p2	ADP
cana-610	69	27	≤	≤	NUM
cana-610	69	28	m.	m.	NOUN
cana-610	69	29	this	this	PRON
cana-610	69	30	implies	imply	VERB
cana-610	69	31	that	that	SCONJ
cana-610	69	32	,	,	PUNCT
cana-610	69	33	p1	p1	NOUN
cana-610	69	34	≤	≤	PUNCT
cana-610	69	35	m1	m1	PROPN
cana-610	69	36	∧	∧	PROPN
cana-610	69	37	m.	m.	NOUN
cana-610	69	38	since	since	SCONJ
cana-610	69	39	s	s	PROPN
cana-610	69	40	is	be	AUX
cana-610	69	41	∧-min	∧-min	NOUN
cana-610	69	42	set	set	VERB
cana-610	69	43	and	and	CCONJ
cana-610	69	44	p1	p1	PROPN
cana-610	69	45	≠	≠	PROPN
cana-610	69	46	0	0	NUM
cana-610	69	47	implies	imply	VERB
cana-610	69	48	that	that	SCONJ
cana-610	69	49	m1	m1	PROPN
cana-610	69	50	=	=	PUNCT
cana-610	69	51	m.	m.	NOUN
cana-610	69	52	by	by	ADP
cana-610	69	53	the	the	DET
cana-610	69	54	similar	similar	ADJ
cana-610	69	55	arguments	argument	NOUN
cana-610	69	56	,	,	PUNCT
cana-610	69	57	we	we	PRON
cana-610	69	58	have	have	VERB
cana-610	69	59	m1	m1	PROPN
cana-610	69	60	=	=	SYM
cana-610	69	61	m2	m2	PROPN
cana-610	69	62	.	.	PUNCT
cana-610	70	1	hence	hence	ADV
cana-610	70	2	,	,	PUNCT
cana-610	70	3	£	£	PROPN
cana-610	70	4	is	be	AUX
cana-610	70	5	a	a	DET
cana-610	70	6	local	local	ADJ
cana-610	70	7	.	.	PUNCT
cana-610	71	1	proposition	proposition	NOUN
cana-610	71	2	3.5	3.5	NUM
cana-610	71	3	.	.	PUNCT
cana-610	72	1	if	if	SCONJ
cana-610	72	2	£	£	NOUN
cana-610	72	3	is	be	AUX
cana-610	72	4	a	a	DET
cana-610	72	5	c	c	NOUN
cana-610	72	6	-	-	PUNCT
cana-610	72	7	lattice	lattice	NOUN
cana-610	72	8	which	which	PRON
cana-610	72	9	is	be	AUX
cana-610	72	10	either	either	CCONJ
cana-610	72	11	a	a	DET
cana-610	72	12	local	local	ADJ
cana-610	72	13	or	or	CCONJ
cana-610	72	14	a	a	DET
cana-610	72	15	domain	domain	NOUN
cana-610	72	16	s	s	X
cana-610	72	17	is	be	AUX
cana-610	72	18	non	non	ADJ
cana-610	72	19	-	-	ADJ
cana-610	72	20	empty	empty	ADJ
cana-610	72	21	subset	subset	NOUN
cana-610	72	22	of	of	ADP
cana-610	72	23	m(£	m(£	PROPN
cana-610	72	24	)	)	PUNCT
cana-610	72	25	,	,	PUNCT
cana-610	72	26	then	then	ADV
cana-610	72	27	the	the	DET
cana-610	72	28	s	s	NOUN
cana-610	72	29	-	-	PUNCT
cana-610	72	30	join	join	NOUN
cana-610	72	31	graph	graph	NOUN
cana-610	72	32	γs(σ(£	γs(σ(£	NOUN
cana-610	72	33	)	)	PUNCT
cana-610	72	34	)	)	PUNCT
cana-610	73	1	is	be	AUX
cana-610	73	2	connected	connect	VERB
cana-610	73	3	.	.	PUNCT
cana-610	74	1	proof	proof	NOUN
cana-610	74	2	.	.	PUNCT
cana-610	75	1	suppose	suppose	VERB
cana-610	75	2	that	that	SCONJ
cana-610	75	3	£	£	NOUN
cana-610	75	4	is	be	AUX
cana-610	75	5	a	a	DET
cana-610	75	6	c	c	NOUN
cana-610	75	7	-	-	PUNCT
cana-610	75	8	lattice	lattice	NOUN
cana-610	75	9	which	which	PRON
cana-610	75	10	is	be	AUX
cana-610	75	11	a	a	DET
cana-610	75	12	domain	domain	NOUN
cana-610	75	13	,	,	PUNCT
cana-610	75	14	then	then	ADV
cana-610	75	15	0	0	NUM
cana-610	75	16	∈	∈	PROPN
cana-610	75	17	σ(£	σ(£	PROPN
cana-610	75	18	)	)	PUNCT
cana-610	75	19	.	.	PUNCT
cana-610	76	1	therefore	therefore	ADV
cana-610	76	2	for	for	ADP
cana-610	76	3	any	any	DET
cana-610	76	4	prime	prime	ADJ
cana-610	76	5	elements	element	NOUN
cana-610	76	6	p	p	NOUN
cana-610	76	7	and	and	CCONJ
cana-610	76	8	q	q	NOUN
cana-610	76	9	other	other	ADJ
cana-610	76	10	than	than	ADP
cana-610	76	11	0	0	NUM
cana-610	76	12	,	,	PUNCT
cana-610	76	13	we	we	PRON
cana-610	76	14	have	have	VERB
cana-610	76	15	,	,	PUNCT
cana-610	76	16	a	a	DET
cana-610	76	17	path	path	NOUN
cana-610	76	18	p	p	NOUN
cana-610	76	19	∼	∼	NOUN
cana-610	76	20	0	0	PUNCT
cana-610	76	21	∼	∼	NOUN
cana-610	76	22	q.	q.	NOUN
cana-610	76	23	hence	hence	ADV
cana-610	76	24	,	,	PUNCT
cana-610	76	25	γs(σ(£	γs(σ(£	NOUN
cana-610	76	26	)	)	PUNCT
cana-610	76	27	)	)	PUNCT
cana-610	76	28	is	be	AUX
cana-610	76	29	connected	connect	VERB
cana-610	76	30	.	.	PUNCT
cana-610	77	1	now	now	ADV
cana-610	77	2	,	,	PUNCT
cana-610	77	3	if	if	SCONJ
cana-610	77	4	£	£	PROPN
cana-610	77	5	is	be	AUX
cana-610	77	6	a	a	DET
cana-610	77	7	c	c	NOUN
cana-610	77	8	-	-	PUNCT
cana-610	77	9	lattice	lattice	NOUN
cana-610	77	10	which	which	PRON
cana-610	77	11	is	be	AUX
cana-610	77	12	a	a	DET
cana-610	77	13	local	local	ADJ
cana-610	77	14	with	with	ADP
cana-610	77	15	m(£	m(£	PROPN
cana-610	77	16	)	)	PUNCT
cana-610	77	17	=	=	PRON
cana-610	77	18	{	{	PUNCT
cana-610	77	19	m	m	NOUN
cana-610	77	20	}	}	PUNCT
cana-610	77	21	.	.	PUNCT
cana-610	78	1	then	then	ADV
cana-610	78	2	for	for	ADP
cana-610	78	3	any	any	DET
cana-610	78	4	p1,p2	p1,p2	PROPN
cana-610	78	5	∈	∈	PROPN
cana-610	78	6	σ(£	σ(£	PROPN
cana-610	78	7	)	)	PUNCT
cana-610	78	8	,	,	PUNCT
cana-610	78	9	we	we	PRON
cana-610	78	10	have	have	VERB
cana-610	78	11	p1	p1	NOUN
cana-610	78	12	∨	∨	NUM
cana-610	78	13	p2	p2	PROPN
cana-610	78	14	≤	≤	PROPN
cana-610	78	15	m.	m.	NOUN
cana-610	78	16	therefore	therefore	ADV
cana-610	78	17	,	,	PUNCT
cana-610	78	18	p1	p1	NOUN
cana-610	78	19	∼	∼	NOUN
cana-610	78	20	p2	p2	NOUN
cana-610	78	21	.	.	PUNCT
cana-610	79	1	consequently	consequently	ADV
cana-610	79	2	,	,	PUNCT
cana-610	79	3	γs(σ(£	γs(σ(£	NOUN
cana-610	79	4	)	)	PUNCT
cana-610	79	5	)	)	PUNCT
cana-610	79	6	is	be	AUX
cana-610	79	7	connected	connect	VERB
cana-610	79	8	.	.	PUNCT
cana-610	79	9	0	0	PUNCT
cana-610	80	1	a	a	DET
cana-610	80	2	b	b	NOUN
cana-610	80	3	c	c	NOUN
cana-610	80	4	d	d	SYM
cana-610	80	5	1	1	NUM
cana-610	80	6	communications	communication	NOUN
cana-610	80	7	on	on	ADP
cana-610	80	8	applied	apply	VERB
cana-610	80	9	nonlinear	nonlinear	ADJ
cana-610	80	10	analysis	analysis	NOUN
cana-610	80	11	issn	issn	NOUN
cana-610	80	12	:	:	PUNCT
cana-610	80	13	1074	1074	NUM
cana-610	80	14	-	-	PUNCT
cana-610	80	15	133x	133x	NUM
cana-610	80	16	vol	vol	NOUN
cana-610	80	17	31	31	NUM
cana-610	80	18	no	no	NOUN
cana-610	80	19	.	.	NOUN
cana-610	80	20	2	2	NUM
cana-610	80	21	(	(	PUNCT
cana-610	80	22	2024	2024	NUM
cana-610	80	23	)	)	PUNCT
cana-610	80	24	412	412	NUM
cana-610	80	25	https://internationalpubls.com	https://internationalpubls.com	X
cana-610	80	26	from	from	ADP
cana-610	80	27	the	the	DET
cana-610	80	28	proposition	proposition	NOUN
cana-610	80	29	3.5	3.5	NUM
cana-610	80	30	,	,	PUNCT
cana-610	80	31	we	we	PRON
cana-610	80	32	have	have	AUX
cana-610	80	33	following	follow	VERB
cana-610	80	34	corollary	corollary	ADJ
cana-610	80	35	.	.	PUNCT
cana-610	81	1	corollary	corollary	ADJ
cana-610	81	2	3.6	3.6	NUM
cana-610	81	3	.	.	PUNCT
cana-610	82	1	let	let	VERB
cana-610	82	2	£	£	PART
cana-610	82	3	be	be	AUX
cana-610	82	4	a	a	DET
cana-610	82	5	c	c	NOUN
cana-610	82	6	-	-	PUNCT
cana-610	82	7	lattice	lattice	NOUN
cana-610	82	8	.	.	PUNCT
cana-610	83	1	if	if	SCONJ
cana-610	83	2	£	£	NOUN
cana-610	83	3	is	be	AUX
cana-610	83	4	a	a	DET
cana-610	83	5	local	local	ADJ
cana-610	83	6	or	or	CCONJ
cana-610	83	7	a	a	DET
cana-610	83	8	domain	domain	NOUN
cana-610	83	9	,	,	PUNCT
cana-610	83	10	then	then	ADV
cana-610	83	11	diam(γs(σ(£	diam(γs(σ(£	ADJ
cana-610	83	12	)	)	PUNCT
cana-610	83	13	)	)	PUNCT
cana-610	83	14	)	)	PUNCT
cana-610	84	1	≤2	≤2	NOUN
cana-610	84	2	.	.	PUNCT
cana-610	85	1	in	in	ADP
cana-610	85	2	the	the	DET
cana-610	85	3	following	following	NOUN
cana-610	85	4	theorem	theorem	NOUN
cana-610	85	5	,	,	PUNCT
cana-610	85	6	we	we	PRON
cana-610	85	7	have	have	AUX
cana-610	85	8	obtained	obtain	VERB
cana-610	85	9	a	a	DET
cana-610	85	10	characterization	characterization	NOUN
cana-610	85	11	of	of	ADP
cana-610	85	12	a	a	DET
cana-610	85	13	local	local	ADJ
cana-610	85	14	c	c	NOUN
cana-610	85	15	-	-	PUNCT
cana-610	85	16	lattice	lattice	NOUN
cana-610	85	17	£	£	PROPN
cana-610	85	18	.	.	PUNCT
cana-610	86	1	theorem	theorem	VERB
cana-610	86	2	3.7	3.7	NUM
cana-610	86	3	.	.	PUNCT
cana-610	87	1	let	let	VERB
cana-610	87	2	£	£	PART
cana-610	87	3	be	be	AUX
cana-610	87	4	a	a	DET
cana-610	87	5	c	c	NOUN
cana-610	87	6	-	-	PUNCT
cana-610	87	7	lattice	lattice	NOUN
cana-610	87	8	and	and	CCONJ
cana-610	87	9	let	let	VERB
cana-610	87	10	γs(σ(£	γs(σ(£	NOUN
cana-610	87	11	)	)	PUNCT
cana-610	87	12	)	)	PUNCT
cana-610	88	1	be	be	AUX
cana-610	88	2	the	the	DET
cana-610	88	3	s	s	NOUN
cana-610	88	4	-	-	PUNCT
cana-610	88	5	join	join	ADJ
cana-610	88	6	graph	graph	NOUN
cana-610	88	7	of	of	ADP
cana-610	88	8	£	£	PROPN
cana-610	88	9	.	.	PUNCT
cana-610	89	1	then	then	ADV
cana-610	89	2	the	the	DET
cana-610	89	3	graph	graph	NOUN
cana-610	89	4	γs(σ(£	γs(σ(£	NOUN
cana-610	89	5	)	)	PUNCT
cana-610	89	6	)	)	PUNCT
cana-610	89	7	is	be	AUX
cana-610	89	8	complete	complete	ADJ
cana-610	89	9	if	if	SCONJ
cana-610	89	10	and	and	CCONJ
cana-610	89	11	only	only	ADV
cana-610	89	12	if	if	SCONJ
cana-610	89	13	£	£	NOUN
cana-610	89	14	is	be	AUX
cana-610	89	15	a	a	DET
cana-610	89	16	local	local	ADJ
cana-610	89	17	.	.	PUNCT
cana-610	90	1	proof	proof	NOUN
cana-610	90	2	.	.	PUNCT
cana-610	91	1	suppose	suppose	VERB
cana-610	91	2	that	that	SCONJ
cana-610	91	3	,	,	PUNCT
cana-610	91	4	γs(σ(£	γs(σ(£	NOUN
cana-610	91	5	)	)	PUNCT
cana-610	91	6	)	)	PUNCT
cana-610	91	7	is	be	AUX
cana-610	91	8	a	a	DET
cana-610	91	9	complete	complete	ADJ
cana-610	91	10	graph	graph	NOUN
cana-610	91	11	of	of	ADP
cana-610	91	12	a	a	DET
cana-610	91	13	c	c	NOUN
cana-610	91	14	-	-	PUNCT
cana-610	91	15	lattice	lattice	NOUN
cana-610	91	16	£	£	NOUN
cana-610	91	17	with	with	ADP
cana-610	91	18	m1,m2	m1,m2	PROPN
cana-610	91	19	∈	∈	PROPN
cana-610	91	20	m(£	m(£	PROPN
cana-610	91	21	)	)	PUNCT
cana-610	91	22	.	.	PUNCT
cana-610	92	1	since	since	SCONJ
cana-610	92	2	the	the	DET
cana-610	92	3	graph	graph	NOUN
cana-610	92	4	γs(σ(£	γs(σ(£	NOUN
cana-610	92	5	)	)	PUNCT
cana-610	92	6	)	)	PUNCT
cana-610	92	7	is	be	AUX
cana-610	92	8	complete	complete	ADJ
cana-610	92	9	and	and	CCONJ
cana-610	92	10	every	every	DET
cana-610	92	11	maximal	maximal	ADJ
cana-610	92	12	element	element	NOUN
cana-610	92	13	of	of	ADP
cana-610	92	14	£	£	NOUN
cana-610	92	15	is	be	AUX
cana-610	92	16	prime	prime	ADJ
cana-610	92	17	,	,	PUNCT
cana-610	92	18	we	we	PRON
cana-610	92	19	have	have	VERB
cana-610	92	20	m1	m1	NOUN
cana-610	92	21	∼	∼	NOUN
cana-610	92	22	m2	m2	PROPN
cana-610	92	23	,	,	PUNCT
cana-610	92	24	implies	imply	VERB
cana-610	92	25	that	that	SCONJ
cana-610	92	26	m1∨m2	m1∨m2	PROPN
cana-610	92	27	≤	≤	NUM
cana-610	92	28	m	m	VERB
cana-610	92	29	for	for	ADP
cana-610	92	30	some	some	DET
cana-610	92	31	m	m	NOUN
cana-610	92	32	∈	∈	PROPN
cana-610	92	33	m(£	m(£	PROPN
cana-610	92	34	)	)	PUNCT
cana-610	93	1	such	such	ADJ
cana-610	93	2	that	that	SCONJ
cana-610	93	3	m	m	VERB
cana-610	93	4	≠	≠	PROPN
cana-610	93	5	1	1	NUM
cana-610	93	6	,	,	PUNCT
cana-610	93	7	a	a	DET
cana-610	93	8	contradiction	contradiction	NOUN
cana-610	93	9	to	to	ADP
cana-610	93	10	the	the	DET
cana-610	93	11	fact	fact	NOUN
cana-610	93	12	that	that	SCONJ
cana-610	93	13	m1	m1	PROPN
cana-610	93	14	∨	∨	NUM
cana-610	93	15	m2	m2	PROPN
cana-610	93	16	=	=	SYM
cana-610	93	17	1	1	NUM
cana-610	93	18	so	so	SCONJ
cana-610	93	19	that	that	SCONJ
cana-610	93	20	m1	m1	PROPN
cana-610	93	21	=	=	SYM
cana-610	93	22	m2	m2	PROPN
cana-610	93	23	.	.	PUNCT
cana-610	93	24	hence	hence	ADV
cana-610	93	25	,	,	PUNCT
cana-610	93	26	£	£	PROPN
cana-610	93	27	is	be	AUX
cana-610	93	28	a	a	DET
cana-610	93	29	local	local	ADJ
cana-610	93	30	c	c	NOUN
cana-610	93	31	-	-	PUNCT
cana-610	93	32	lattice	lattice	NOUN
cana-610	93	33	.	.	PUNCT
cana-610	94	1	conversely	conversely	ADV
cana-610	94	2	,	,	PUNCT
cana-610	94	3	suppose	suppose	VERB
cana-610	94	4	that	that	SCONJ
cana-610	94	5	£	£	NOUN
cana-610	94	6	is	be	AUX
cana-610	94	7	a	a	DET
cana-610	94	8	local	local	ADJ
cana-610	94	9	c	c	NOUN
cana-610	94	10	-	-	PUNCT
cana-610	94	11	lattice	lattice	NOUN
cana-610	94	12	with	with	ADP
cana-610	94	13	the	the	DET
cana-610	94	14	maximal	maximal	ADJ
cana-610	94	15	element	element	NOUN
cana-610	94	16	m.	m.	NOUN
cana-610	94	17	then	then	ADV
cana-610	94	18	for	for	ADP
cana-610	94	19	any	any	DET
cana-610	94	20	p1,p2	p1,p2	PROPN
cana-610	94	21	∈	∈	PROPN
cana-610	94	22	σ(£	σ(£	PROPN
cana-610	94	23	)	)	PUNCT
cana-610	94	24	,	,	PUNCT
cana-610	94	25	we	we	PRON
cana-610	94	26	have	have	AUX
cana-610	94	27	p1	p1	NOUN
cana-610	94	28	∨	∨	NUM
cana-610	94	29	p2	p2	PROPN
cana-610	94	30	≤	≤	NOUN
cana-610	94	31	m	m	ADP
cana-610	94	32	and	and	CCONJ
cana-610	94	33	hence	hence	ADV
cana-610	94	34	p1	p1	NOUN
cana-610	94	35	and	and	CCONJ
cana-610	94	36	p2	p2	PROPN
cana-610	94	37	are	be	AUX
cana-610	94	38	adjacent	adjacent	ADJ
cana-610	94	39	.	.	PUNCT
cana-610	95	1	consequently	consequently	ADV
cana-610	95	2	,	,	PUNCT
cana-610	95	3	the	the	DET
cana-610	95	4	s	s	NOUN
cana-610	95	5	-	-	PUNCT
cana-610	95	6	join	join	ADJ
cana-610	95	7	graph	graph	NOUN
cana-610	95	8	γs(σ(£	γs(σ(£	NOUN
cana-610	95	9	)	)	PUNCT
cana-610	95	10	)	)	PUNCT
cana-610	95	11	is	be	AUX
cana-610	95	12	complete	complete	ADJ
cana-610	95	13	.	.	PUNCT
cana-610	96	1	proposition	proposition	NOUN
cana-610	96	2	3.8	3.8	NUM
cana-610	96	3	.	.	PUNCT
cana-610	97	1	if	if	SCONJ
cana-610	97	2	a	a	DET
cana-610	97	3	c	c	NOUN
cana-610	97	4	-	-	PUNCT
cana-610	97	5	lattice	lattice	NOUN
cana-610	97	6	£	£	PROPN
cana-610	97	7	is	be	AUX
cana-610	97	8	such	such	ADJ
cana-610	97	9	that	that	SCONJ
cana-610	97	10	m(£	m(£	PROPN
cana-610	97	11	)	)	PUNCT
cana-610	97	12	=	=	SYM
cana-610	97	13	σ(£	σ(£	PROPN
cana-610	97	14	)	)	PUNCT
cana-610	97	15	−	−	PROPN
cana-610	97	16	{	{	PUNCT
cana-610	97	17	0	0	NUM
cana-610	97	18	}	}	PUNCT
cana-610	97	19	,	,	PUNCT
cana-610	97	20	with	with	ADP
cana-610	97	21	unique	unique	ADJ
cana-610	97	22	atom	atom	NOUN
cana-610	97	23	,	,	PUNCT
cana-610	97	24	then	then	ADV
cana-610	97	25	γs(σ(£	γs(σ(£	NOUN
cana-610	97	26	)	)	PUNCT
cana-610	97	27	)	)	PUNCT
cana-610	97	28	is	be	AUX
cana-610	97	29	a	a	DET
cana-610	97	30	star	star	NOUN
cana-610	97	31	graph	graph	NOUN
cana-610	97	32	.	.	PUNCT
cana-610	98	1	proof	proof	NOUN
cana-610	98	2	.	.	PUNCT
cana-610	99	1	suppose	suppose	VERB
cana-610	99	2	that	that	SCONJ
cana-610	99	3	a	a	DET
cana-610	99	4	c	c	NOUN
cana-610	99	5	-	-	PUNCT
cana-610	99	6	lattice	lattice	NOUN
cana-610	99	7	£	£	PROPN
cana-610	99	8	is	be	AUX
cana-610	99	9	a	a	DET
cana-610	99	10	domain	domain	NOUN
cana-610	99	11	.	.	PUNCT
cana-610	100	1	so	so	ADV
cana-610	100	2	that	that	SCONJ
cana-610	100	3	,	,	PUNCT
cana-610	100	4	0	0	NUM
cana-610	100	5	∈	∈	PROPN
cana-610	100	6	σ(£	σ(£	PROPN
cana-610	100	7	)	)	PUNCT
cana-610	100	8	.	.	PUNCT
cana-610	101	1	also	also	ADV
cana-610	101	2	,	,	PUNCT
cana-610	101	3	any	any	DET
cana-610	101	4	two	two	NUM
cana-610	101	5	non	non	ADJ
cana-610	101	6	-	-	ADJ
cana-610	101	7	zero	zero	NUM
cana-610	101	8	elements	element	NOUN
cana-610	101	9	a	a	DET
cana-610	101	10	,	,	PUNCT
cana-610	101	11	b	b	X
cana-610	101	12	∈	∈	PROPN
cana-610	101	13	£	£	PROPN
cana-610	101	14	are	be	AUX
cana-610	101	15	co	co	ADJ
cana-610	101	16	-	-	ADJ
cana-610	101	17	maximal	maximal	ADJ
cana-610	101	18	.	.	PUNCT
cana-610	102	1	therefore	therefore	ADV
cana-610	102	2	,	,	PUNCT
cana-610	102	3	for	for	ADP
cana-610	102	4	any	any	DET
cana-610	102	5	0	0	NUM
cana-610	102	6	≠	≠	PROPN
cana-610	102	7	a	a	DET
cana-610	102	8	∈	∈	ADJ
cana-610	102	9	£	£	NOUN
cana-610	102	10	,	,	PUNCT
cana-610	102	11	we	we	PRON
cana-610	102	12	have	have	VERB
cana-610	102	13	a	a	DET
cana-610	102	14	path	path	NOUN
cana-610	102	15	0	0	PUNCT
cana-610	102	16	∼	∼	NOUN
cana-610	102	17	a.	a.	NOUN
cana-610	102	18	also	also	ADV
cana-610	102	19	,	,	PUNCT
cana-610	102	20	there	there	PRON
cana-610	102	21	does	do	AUX
cana-610	102	22	not	not	PART
cana-610	102	23	exist	exist	VERB
cana-610	102	24	edge	edge	NOUN
cana-610	102	25	between	between	ADP
cana-610	102	26	any	any	DET
cana-610	102	27	two	two	NUM
cana-610	102	28	non	non	ADJ
cana-610	102	29	-	-	ADJ
cana-610	102	30	zero	zero	NUM
cana-610	102	31	elements	element	NOUN
cana-610	102	32	of	of	ADP
cana-610	102	33	£	£	SYM
cana-610	102	34	.	.	PUNCT
cana-610	103	1	hence	hence	ADV
cana-610	103	2	,	,	PUNCT
cana-610	103	3	γs(σ(£	γs(σ(£	NOUN
cana-610	103	4	)	)	PUNCT
cana-610	103	5	)	)	PUNCT
cana-610	103	6	is	be	AUX
cana-610	103	7	a	a	DET
cana-610	103	8	star	star	NOUN
cana-610	103	9	graph	graph	NOUN
cana-610	103	10	.	.	PUNCT
cana-610	104	1	the	the	DET
cana-610	104	2	following	follow	VERB
cana-610	104	3	definition	definition	NOUN
cana-610	104	4	of	of	ADP
cana-610	104	5	dimension	dimension	NOUN
cana-610	104	6	of	of	ADP
cana-610	104	7	a	a	DET
cana-610	104	8	c	c	NOUN
cana-610	104	9	-	-	PUNCT
cana-610	104	10	lattice	lattice	NOUN
cana-610	104	11	£	£	PROPN
cana-610	104	12	is	be	AUX
cana-610	104	13	used	use	VERB
cana-610	104	14	in	in	ADP
cana-610	104	15	results	result	NOUN
cana-610	104	16	that	that	PRON
cana-610	104	17	follows	follow	VERB
cana-610	104	18	.	.	PUNCT
cana-610	105	1	definition	definition	NOUN
cana-610	105	2	3.9	3.9	NUM
cana-610	105	3	.	.	PUNCT
cana-610	106	1	let	let	VERB
cana-610	106	2	£	£	PART
cana-610	106	3	be	be	AUX
cana-610	106	4	a	a	DET
cana-610	106	5	c	c	NOUN
cana-610	106	6	-	-	PUNCT
cana-610	106	7	lattice	lattice	NOUN
cana-610	106	8	.	.	PUNCT
cana-610	107	1	the	the	DET
cana-610	107	2	dimension	dimension	NOUN
cana-610	107	3	of	of	ADP
cana-610	107	4	£	£	SYM
cana-610	107	5	,	,	PUNCT
cana-610	107	6	denoted	denote	VERB
cana-610	107	7	as	as	ADP
cana-610	107	8	dim(£	dim(£	NOUN
cana-610	107	9	)	)	PUNCT
cana-610	107	10	,	,	PUNCT
cana-610	107	11	is	be	AUX
cana-610	107	12	the	the	DET
cana-610	107	13	supremum	supremum	NOUN
cana-610	107	14	of	of	ADP
cana-610	107	15	the	the	DET
cana-610	107	16	lengths	length	NOUN
cana-610	107	17	of	of	ADP
cana-610	107	18	chains	chain	NOUN
cana-610	107	19	of	of	ADP
cana-610	107	20	members	member	NOUN
cana-610	107	21	of	of	ADP
cana-610	107	22	σ(£	σ(£	PROPN
cana-610	107	23	)	)	PUNCT
cana-610	107	24	.	.	PUNCT
cana-610	108	1	theorem	theorem	VERB
cana-610	108	2	3.10	3.10	NUM
cana-610	108	3	.	.	PUNCT
cana-610	109	1	let	let	VERB
cana-610	109	2	£	£	PART
cana-610	109	3	be	be	AUX
cana-610	109	4	a	a	DET
cana-610	109	5	c	c	NOUN
cana-610	109	6	-	-	PUNCT
cana-610	109	7	lattice	lattice	NOUN
cana-610	109	8	which	which	PRON
cana-610	109	9	is	be	AUX
cana-610	109	10	not	not	PART
cana-610	109	11	a	a	DET
cana-610	109	12	local	local	ADJ
cana-610	109	13	and	and	CCONJ
cana-610	109	14	let	let	VERB
cana-610	109	15	s	s	PRON
cana-610	109	16	be	be	AUX
cana-610	109	17	a	a	DET
cana-610	109	18	∧-min	∧-min	ADJ
cana-610	109	19	set	set	NOUN
cana-610	109	20	of	of	ADP
cana-610	109	21	£	£	PROPN
cana-610	109	22	.	.	PUNCT
cana-610	110	1	if	if	SCONJ
cana-610	110	2	a	a	DET
cana-610	110	3	sjoin	sjoin	NOUN
cana-610	110	4	graph	graph	NOUN
cana-610	110	5	γs(σ(£	γs(σ(£	NOUN
cana-610	110	6	)	)	PUNCT
cana-610	110	7	)	)	PUNCT
cana-610	110	8	is	be	AUX
cana-610	110	9	a	a	DET
cana-610	110	10	star	star	NOUN
cana-610	110	11	graph	graph	NOUN
cana-610	110	12	,	,	PUNCT
cana-610	110	13	then	then	ADV
cana-610	110	14	£	£	PROPN
cana-610	110	15	is	be	AUX
cana-610	110	16	a	a	DET
cana-610	110	17	domain	domain	NOUN
cana-610	110	18	with	with	ADP
cana-610	110	19	dim(£	dim(£	NOUN
cana-610	110	20	)	)	PUNCT
cana-610	110	21	≤	≤	NUM
cana-610	110	22	1	1	NUM
cana-610	110	23	.	.	PUNCT
cana-610	110	24	proof	proof	NOUN
cana-610	110	25	.	.	PUNCT
cana-610	111	1	suppose	suppose	VERB
cana-610	111	2	that	that	SCONJ
cana-610	111	3	γs(σ(£	γs(σ(£	NOUN
cana-610	111	4	)	)	PUNCT
cana-610	111	5	)	)	PUNCT
cana-610	111	6	is	be	AUX
cana-610	111	7	a	a	DET
cana-610	111	8	star	star	NOUN
cana-610	111	9	graph	graph	NOUN
cana-610	111	10	of	of	ADP
cana-610	111	11	£	£	SYM
cana-610	111	12	.	.	PUNCT
cana-610	112	1	hence	hence	ADV
cana-610	112	2	,	,	PUNCT
cana-610	112	3	there	there	PRON
cana-610	112	4	is	be	VERB
cana-610	112	5	a	a	DET
cana-610	112	6	vertex	vertex	NOUN
cana-610	112	7	p	p	NOUN
cana-610	112	8	in	in	ADP
cana-610	112	9	the	the	DET
cana-610	112	10	graph	graph	NOUN
cana-610	112	11	γs(σ(£	γs(σ(£	NOUN
cana-610	112	12	)	)	PUNCT
cana-610	112	13	)	)	PUNCT
cana-610	112	14	such	such	ADJ
cana-610	112	15	that	that	SCONJ
cana-610	112	16	p	p	ADJ
cana-610	112	17	∼	∼	NOUN
cana-610	112	18	q	q	NOUN
cana-610	112	19	,	,	PUNCT
cana-610	112	20	for	for	ADP
cana-610	112	21	all	all	DET
cana-610	112	22	q	q	PROPN
cana-610	112	23	∈	∈	PROPN
cana-610	112	24	σ(£	σ(£	PROPN
cana-610	112	25	)	)	PUNCT
cana-610	112	26	.	.	PUNCT
cana-610	113	1	therefore	therefore	ADV
cana-610	113	2	,	,	PUNCT
cana-610	113	3	for	for	ADP
cana-610	113	4	m1,m2	m1,m2	PROPN
cana-610	113	5	∈	∈	PROPN
cana-610	113	6	m(£	m(£	PROPN
cana-610	113	7	)	)	PUNCT
cana-610	113	8	,	,	PUNCT
cana-610	113	9	we	we	PRON
cana-610	113	10	have	have	VERB
cana-610	113	11	p	p	NOUN
cana-610	113	12	∼	∼	NOUN
cana-610	113	13	m1	m1	NOUN
cana-610	113	14	and	and	CCONJ
cana-610	113	15	p	p	NOUN
cana-610	113	16	∼	∼	NOUN
cana-610	113	17	m2	m2	PROPN
cana-610	113	18	.	.	PUNCT
cana-610	114	1	since	since	SCONJ
cana-610	114	2	m1,m2	m1,m2	PROPN
cana-610	114	3	∈	∈	PROPN
cana-610	114	4	m(£	m(£	PROPN
cana-610	114	5	)	)	PUNCT
cana-610	114	6	,	,	PUNCT
cana-610	114	7	p	p	PROPN
cana-610	114	8	∨	∨	NOUN
cana-610	114	9	m1	m1	NOUN
cana-610	114	10	=	=	SYM
cana-610	114	11	m1	m1	PROPN
cana-610	114	12	and	and	CCONJ
cana-610	114	13	p	p	NOUN
cana-610	114	14	∨	∨	NUM
cana-610	114	15	m2	m2	PROPN
cana-610	114	16	=	=	PROPN
cana-610	114	17	m2	m2	PROPN
cana-610	114	18	.	.	PUNCT
cana-610	115	1	this	this	PRON
cana-610	115	2	implies	imply	VERB
cana-610	115	3	that	that	SCONJ
cana-610	115	4	p	p	PROPN
cana-610	115	5	≤	≤	VERB
cana-610	115	6	m1	m1	NOUN
cana-610	115	7	∧	∧	PROPN
cana-610	115	8	m2	m2	PROPN
cana-610	115	9	.	.	PUNCT
cana-610	116	1	but	but	CCONJ
cana-610	116	2	it	it	PRON
cana-610	116	3	is	be	AUX
cana-610	116	4	given	give	VERB
cana-610	116	5	that	that	SCONJ
cana-610	116	6	s	s	NOUN
cana-610	116	7	is	be	AUX
cana-610	116	8	a	a	DET
cana-610	116	9	∧-min	∧-min	ADJ
cana-610	116	10	set	set	NOUN
cana-610	116	11	,	,	PUNCT
cana-610	116	12	therefore	therefore	ADV
cana-610	116	13	p	p	X
cana-610	116	14	=	=	NOUN
cana-610	116	15	0	0	NUM
cana-610	116	16	.	.	PUNCT
cana-610	117	1	hence	hence	ADV
cana-610	117	2	,	,	PUNCT
cana-610	117	3	0	0	NUM
cana-610	117	4	is	be	AUX
cana-610	117	5	a	a	DET
cana-610	117	6	prime	prime	ADJ
cana-610	117	7	element	element	NOUN
cana-610	117	8	of	of	ADP
cana-610	117	9	£	£	SYM
cana-610	117	10	,	,	PUNCT
cana-610	117	11	consequently	consequently	ADV
cana-610	117	12	,	,	PUNCT
cana-610	117	13	£	£	PROPN
cana-610	117	14	is	be	AUX
cana-610	117	15	a	a	DET
cana-610	117	16	domain	domain	NOUN
cana-610	117	17	.	.	PUNCT
cana-610	118	1	now	now	ADV
cana-610	118	2	,	,	PUNCT
cana-610	118	3	suppose	suppose	VERB
cana-610	118	4	that	that	SCONJ
cana-610	118	5	dim(£	dim(£	VERB
cana-610	118	6	)	)	PUNCT
cana-610	118	7	≥	≥	NOUN
cana-610	118	8	2	2	NUM
cana-610	118	9	,	,	PUNCT
cana-610	118	10	then	then	ADV
cana-610	118	11	there	there	PRON
cana-610	118	12	exists	exist	VERB
cana-610	118	13	a	a	DET
cana-610	118	14	chain	chain	NOUN
cana-610	118	15	of	of	ADP
cana-610	118	16	prime	prime	ADJ
cana-610	118	17	elements	element	NOUN
cana-610	118	18	having	have	VERB
cana-610	118	19	length	length	NOUN
cana-610	118	20	at	at	ADV
cana-610	118	21	least	least	ADV
cana-610	118	22	two	two	NUM
cana-610	118	23	,	,	PUNCT
cana-610	118	24	say	say	INTJ
cana-610	118	25	,	,	PUNCT
cana-610	118	26	p1	p1	PROPN
cana-610	118	27	≤	≤	NUM
cana-610	118	28	p2	p2	PROPN
cana-610	118	29	≤	≤	NUM
cana-610	118	30	p3	p3	NOUN
cana-610	118	31	.	.	PUNCT
cana-610	119	1	it	it	PRON
cana-610	119	2	is	be	AUX
cana-610	119	3	clear	clear	ADJ
cana-610	119	4	that	that	SCONJ
cana-610	119	5	p1	p1	NOUN
cana-610	119	6	=	=	NOUN
cana-610	119	7	0	0	X
cana-610	119	8	.	.	PUNCT
cana-610	120	1	since	since	SCONJ
cana-610	120	2	γs(σ(£	γs(σ(£	NOUN
cana-610	120	3	)	)	PUNCT
cana-610	120	4	)	)	PUNCT
cana-610	120	5	is	be	AUX
cana-610	120	6	a	a	DET
cana-610	120	7	star	star	NOUN
cana-610	120	8	graph	graph	NOUN
cana-610	120	9	,	,	PUNCT
cana-610	120	10	we	we	PRON
cana-610	120	11	have	have	VERB
cana-610	120	12	p1	p1	NOUN
cana-610	120	13	∼	∼	NOUN
cana-610	120	14	p2	p2	NOUN
cana-610	120	15	,	,	PUNCT
cana-610	120	16	by	by	ADP
cana-610	120	17	definition	definition	NOUN
cana-610	120	18	there	there	PRON
cana-610	120	19	exists	exist	VERB
cana-610	120	20	m	m	VERB
cana-610	120	21	∈	∈	PROPN
cana-610	120	22	m(£	m(£	PROPN
cana-610	120	23	)	)	PUNCT
cana-610	120	24	such	such	ADJ
cana-610	120	25	that	that	SCONJ
cana-610	120	26	p1	p1	PROPN
cana-610	120	27	∨	∨	NUM
cana-610	120	28	p2	p2	PROPN
cana-610	120	29	≤	≤	NUM
cana-610	120	30	m	m	PROPN
cana-610	120	31	,	,	PUNCT
cana-610	120	32	i.e.	i.e.	X
cana-610	120	33	,	,	PUNCT
cana-610	120	34	p1	p1	NOUN
cana-610	120	35	∨	∨	NOUN
cana-610	120	36	p2	p2	X
cana-610	120	37	=	=	SYM
cana-610	120	38	p2	p2	PROPN
cana-610	120	39	≤	≤	NUM
cana-610	120	40	m.	m.	NOUN
cana-610	120	41	if	if	SCONJ
cana-610	120	42	p2≠	p2≠	PROPN
cana-610	120	43	m	m	PROPN
cana-610	120	44	,	,	PUNCT
cana-610	120	45	then	then	ADV
cana-610	120	46	we	we	PRON
cana-610	120	47	have	have	VERB
cana-610	120	48	a	a	DET
cana-610	120	49	cycle	cycle	NOUN
cana-610	120	50	p1	p1	NOUN
cana-610	120	51	∼	∼	NOUN
cana-610	120	52	p2	p2	NOUN
cana-610	120	53	∼	∼	NOUN
cana-610	120	54	m	m	VERB
cana-610	120	55	∼	∼	NOUN
cana-610	120	56	•••	•••	ADV
cana-610	120	57	∼	∼	NOUN
cana-610	120	58	p1	p1	NOUN
cana-610	120	59	,	,	PUNCT
cana-610	120	60	a	a	DET
cana-610	120	61	contradiction	contradiction	NOUN
cana-610	120	62	to	to	ADP
cana-610	120	63	the	the	DET
cana-610	120	64	fact	fact	NOUN
cana-610	120	65	that	that	SCONJ
cana-610	120	66	the	the	DET
cana-610	120	67	graph	graph	NOUN
cana-610	120	68	γs(σ(£	γs(σ(£	NOUN
cana-610	120	69	)	)	PUNCT
cana-610	120	70	)	)	PUNCT
cana-610	120	71	is	be	AUX
cana-610	120	72	a	a	DET
cana-610	120	73	star	star	NOUN
cana-610	120	74	graph	graph	NOUN
cana-610	120	75	.	.	PUNCT
cana-610	121	1	if	if	SCONJ
cana-610	121	2	p2	p2	PROPN
cana-610	121	3	=	=	SYM
cana-610	121	4	m	m	PROPN
cana-610	121	5	,	,	PUNCT
cana-610	121	6	then	then	ADV
cana-610	121	7	we	we	PRON
cana-610	121	8	have	have	AUX
cana-610	121	9	p1	p1	NOUN
cana-610	121	10	≤	≤	ADJ
cana-610	121	11	p2	p2	NOUN
cana-610	121	12	=	=	SYM
cana-610	121	13	m	m	VERB
cana-610	121	14	≤	≤	NOUN
cana-610	121	15	p3	p3	NOUN
cana-610	121	16	which	which	PRON
cana-610	121	17	not	not	PART
cana-610	121	18	possible	possible	ADJ
cana-610	121	19	because	because	SCONJ
cana-610	121	20	m	m	NOUN
cana-610	121	21	is	be	AUX
cana-610	121	22	maximal	maximal	ADJ
cana-610	121	23	.	.	PUNCT
cana-610	122	1	consequently	consequently	ADV
cana-610	122	2	,	,	PUNCT
cana-610	122	3	dim(£	dim(£	PROPN
cana-610	122	4	)	)	PUNCT
cana-610	122	5	≤	≤	NUM
cana-610	122	6	1	1	NUM
cana-610	122	7	.	.	PUNCT
cana-610	123	1	proposition	proposition	NOUN
cana-610	123	2	3.11	3.11	NUM
cana-610	123	3	.	.	PUNCT
cana-610	124	1	let	let	VERB
cana-610	124	2	£	£	PART
cana-610	124	3	be	be	AUX
cana-610	124	4	a	a	DET
cana-610	124	5	c	c	NOUN
cana-610	124	6	-	-	PUNCT
cana-610	124	7	lattice	lattice	NOUN
cana-610	124	8	and	and	CCONJ
cana-610	124	9	γs(σ(£	γs(σ(£	NOUN
cana-610	124	10	)	)	PUNCT
cana-610	124	11	)	)	PUNCT
cana-610	124	12	be	be	AUX
cana-610	124	13	a	a	DET
cana-610	124	14	s	s	NOUN
cana-610	124	15	-	-	PUNCT
cana-610	124	16	join	join	NOUN
cana-610	124	17	graph	graph	NOUN
cana-610	124	18	of	of	ADP
cana-610	124	19	£	£	PROPN
cana-610	124	20	.	.	PUNCT
cana-610	125	1	if	if	SCONJ
cana-610	125	2	γs(σ(£	γs(σ(£	NOUN
cana-610	125	3	)	)	PUNCT
cana-610	125	4	)	)	PUNCT
cana-610	125	5	is	be	AUX
cana-610	125	6	a	a	DET
cana-610	125	7	star	star	NOUN
cana-610	125	8	graph	graph	NOUN
cana-610	125	9	,	,	PUNCT
cana-610	125	10	then	then	ADV
cana-610	125	11	s	s	PART
cana-610	125	12	=	=	SYM
cana-610	125	13	m(£	m(£	PROPN
cana-610	125	14	)	)	PUNCT
cana-610	125	15	.	.	PUNCT
cana-610	126	1	proof	proof	NOUN
cana-610	126	2	.	.	PUNCT
cana-610	127	1	suppose	suppose	VERB
cana-610	127	2	that	that	SCONJ
cana-610	127	3	the	the	DET
cana-610	127	4	s	s	NOUN
cana-610	127	5	-	-	PUNCT
cana-610	127	6	join	join	ADJ
cana-610	127	7	graph	graph	NOUN
cana-610	127	8	γs(σ(£	γs(σ(£	NOUN
cana-610	127	9	)	)	PUNCT
cana-610	127	10	)	)	PUNCT
cana-610	127	11	of	of	ADP
cana-610	127	12	£	£	PROPN
cana-610	127	13	is	be	AUX
cana-610	127	14	a	a	DET
cana-610	127	15	star	star	NOUN
cana-610	127	16	graph	graph	NOUN
cana-610	127	17	.	.	PUNCT
cana-610	128	1	then	then	ADV
cana-610	128	2	,	,	PUNCT
cana-610	128	3	we	we	PRON
cana-610	128	4	have	have	VERB
cana-610	128	5	a	a	DET
cana-610	128	6	fixed	fix	VERB
cana-610	128	7	vertex	vertex	NOUN
cana-610	128	8	,	,	PUNCT
cana-610	128	9	say	say	VERB
cana-610	128	10	a	a	PRON
cana-610	128	11	,	,	PUNCT
cana-610	128	12	in	in	ADP
cana-610	128	13	the	the	DET
cana-610	128	14	graph	graph	NOUN
cana-610	128	15	γs(σ(£	γs(σ(£	NOUN
cana-610	128	16	)	)	PUNCT
cana-610	128	17	)	)	PUNCT
cana-610	128	18	such	such	ADJ
cana-610	128	19	that	that	SCONJ
cana-610	128	20	a	a	DET
cana-610	128	21	∼	∼	NOUN
cana-610	128	22	b	b	NOUN
cana-610	128	23	for	for	ADP
cana-610	128	24	each	each	DET
cana-610	128	25	vertex	vertex	NOUN
cana-610	128	26	b	b	NOUN
cana-610	128	27	in	in	ADP
cana-610	128	28	γs(σ(£	γs(σ(£	NOUN
cana-610	128	29	)	)	PUNCT
cana-610	128	30	)	)	PUNCT
cana-610	128	31	.	.	PUNCT
cana-610	129	1	therefore	therefore	ADV
cana-610	129	2	,	,	PUNCT
cana-610	129	3	communications	communication	NOUN
cana-610	129	4	on	on	ADP
cana-610	129	5	applied	apply	VERB
cana-610	129	6	nonlinear	nonlinear	ADJ
cana-610	129	7	analysis	analysis	NOUN
cana-610	129	8	issn	issn	NOUN
cana-610	129	9	:	:	PUNCT
cana-610	129	10	1074	1074	NUM
cana-610	129	11	-	-	PUNCT
cana-610	129	12	133x	133x	NUM
cana-610	129	13	vol	vol	NOUN
cana-610	129	14	31	31	NUM
cana-610	129	15	no	no	NOUN
cana-610	129	16	.	.	NOUN
cana-610	129	17	2	2	NUM
cana-610	129	18	(	(	PUNCT
cana-610	129	19	2024	2024	NUM
cana-610	129	20	)	)	PUNCT
cana-610	129	21	413	413	NUM
cana-610	129	22	https://internationalpubls.com	https://internationalpubls.com	X
cana-610	129	23	for	for	ADP
cana-610	129	24	any	any	DET
cana-610	129	25	m	m	NOUN
cana-610	129	26	∈	∈	PROPN
cana-610	129	27	m(£	m(£	PROPN
cana-610	129	28	)	)	PUNCT
cana-610	129	29	,	,	PUNCT
cana-610	129	30	we	we	PRON
cana-610	129	31	have	have	VERB
cana-610	129	32	a	a	DET
cana-610	129	33	∼	∼	NOUN
cana-610	129	34	m.	m.	NOUN
cana-610	129	35	this	this	PRON
cana-610	129	36	implies	imply	VERB
cana-610	129	37	that	that	SCONJ
cana-610	129	38	,	,	PUNCT
cana-610	129	39	a	a	DET
cana-610	129	40	∨	∨	NUM
cana-610	129	41	m	m	PROPN
cana-610	129	42	≤	≤	NOUN
cana-610	129	43	m1	m1	NOUN
cana-610	129	44	,	,	PUNCT
cana-610	129	45	for	for	ADP
cana-610	129	46	some	some	DET
cana-610	129	47	m1	m1	PROPN
cana-610	129	48	∈	∈	PROPN
cana-610	129	49	s.	s.	PROPN
cana-610	129	50	as	as	ADP
cana-610	129	51	,	,	PUNCT
cana-610	129	52	m	m	VERB
cana-610	129	53	∈	∈	PROPN
cana-610	129	54	m(£	m(£	PROPN
cana-610	129	55	)	)	PUNCT
cana-610	129	56	,	,	PUNCT
cana-610	129	57	implies	imply	VERB
cana-610	129	58	that	that	SCONJ
cana-610	129	59	m	m	PROPN
cana-610	129	60	=	=	SYM
cana-610	129	61	m1	m1	NOUN
cana-610	129	62	.	.	PUNCT
cana-610	130	1	hence	hence	ADV
cana-610	130	2	,	,	PUNCT
cana-610	130	3	s	s	PART
cana-610	130	4	=	=	SYM
cana-610	130	5	m(£	m(£	PROPN
cana-610	130	6	)	)	PUNCT
cana-610	130	7	.	.	PUNCT
cana-610	131	1	we	we	PRON
cana-610	131	2	need	need	VERB
cana-610	131	3	the	the	DET
cana-610	131	4	following	follow	VERB
cana-610	131	5	proposition	proposition	NOUN
cana-610	131	6	to	to	PART
cana-610	131	7	establish	establish	VERB
cana-610	131	8	the	the	DET
cana-610	131	9	some	some	DET
cana-610	131	10	important	important	ADJ
cana-610	131	11	theorem	theorem	NOUN
cana-610	131	12	ahead	ahead	ADV
cana-610	131	13	.	.	PUNCT
cana-610	132	1	proposition	proposition	NOUN
cana-610	132	2	3.12	3.12	NUM
cana-610	132	3	.	.	PUNCT
cana-610	133	1	[	[	X
cana-610	133	2	13	13	NUM
cana-610	133	3	]	]	PUNCT
cana-610	133	4	let	let	VERB
cana-610	133	5	£	£	PRON
cana-610	133	6	be	be	AUX
cana-610	133	7	a	a	DET
cana-610	133	8	multiplicative	multiplicative	ADJ
cana-610	133	9	lattice	lattice	NOUN
cana-610	133	10	with	with	ADP
cana-610	133	11	greatest	great	ADJ
cana-610	133	12	element	element	NOUN
cana-610	133	13	1	1	NUM
cana-610	133	14	is	be	AUX
cana-610	133	15	compact	compact	ADJ
cana-610	133	16	.	.	PUNCT
cana-610	134	1	then	then	ADV
cana-610	134	2	for	for	ADP
cana-610	134	3	any	any	DET
cana-610	134	4	a	a	DET
cana-610	134	5	∈	∈	NOUN
cana-610	134	6	£	£	NOUN
cana-610	134	7	such	such	ADJ
cana-610	134	8	that	that	SCONJ
cana-610	134	9	a	a	DET
cana-610	134	10	≠	≠	PROPN
cana-610	134	11	1	1	NUM
cana-610	134	12	,	,	PUNCT
cana-610	134	13	there	there	PRON
cana-610	134	14	exists	exist	VERB
cana-610	134	15	m	m	VERB
cana-610	134	16	∈	∈	PROPN
cana-610	134	17	m(£	m(£	PROPN
cana-610	134	18	)	)	PUNCT
cana-610	134	19	such	such	ADJ
cana-610	134	20	that	that	SCONJ
cana-610	134	21	a	a	DET
cana-610	134	22	≤	≤	ADJ
cana-610	134	23	m.	m.	NOUN
cana-610	134	24	theorem	theorem	VERB
cana-610	134	25	3.13	3.13	NUM
cana-610	134	26	.	.	PUNCT
cana-610	135	1	let	let	VERB
cana-610	135	2	£	£	PART
cana-610	135	3	be	be	AUX
cana-610	135	4	a	a	DET
cana-610	135	5	c	c	NOUN
cana-610	135	6	-	-	PUNCT
cana-610	135	7	lattice	lattice	NOUN
cana-610	135	8	and	and	CCONJ
cana-610	135	9	γs(σ(£	γs(σ(£	NOUN
cana-610	135	10	)	)	PUNCT
cana-610	135	11	)	)	PUNCT
cana-610	135	12	be	be	AUX
cana-610	135	13	a	a	DET
cana-610	135	14	s	s	NOUN
cana-610	135	15	-	-	PUNCT
cana-610	135	16	join	join	NOUN
cana-610	135	17	graph	graph	NOUN
cana-610	135	18	of	of	ADP
cana-610	135	19	£	£	PROPN
cana-610	135	20	.	.	PUNCT
cana-610	136	1	if	if	SCONJ
cana-610	136	2	dim(£	dim(£	NOUN
cana-610	136	3	)	)	PUNCT
cana-610	136	4	≥	≥	NOUN
cana-610	136	5	2	2	NUM
cana-610	136	6	,	,	PUNCT
cana-610	136	7	then	then	ADV
cana-610	136	8	gr(γs(σ(£	gr(γs(σ(£	PROPN
cana-610	136	9	)	)	PUNCT
cana-610	136	10	)	)	PUNCT
cana-610	136	11	)	)	PUNCT
cana-610	137	1	=	=	SYM
cana-610	137	2	3	3	X
cana-610	137	3	.	.	X
cana-610	137	4	proof	proof	NOUN
cana-610	137	5	.	.	PUNCT
cana-610	138	1	suppose	suppose	VERB
cana-610	138	2	that	that	SCONJ
cana-610	138	3	a0	a0	PROPN
cana-610	138	4	≤	≤	PROPN
cana-610	138	5	a1	a1	NOUN
cana-610	138	6	be	be	AUX
cana-610	138	7	a	a	DET
cana-610	138	8	chain	chain	NOUN
cana-610	138	9	of	of	ADP
cana-610	138	10	members	member	NOUN
cana-610	138	11	of	of	ADP
cana-610	138	12	σ(£	σ(£	PROPN
cana-610	138	13	)	)	PUNCT
cana-610	138	14	.	.	PUNCT
cana-610	139	1	since	since	SCONJ
cana-610	139	2	£	£	PROPN
cana-610	139	3	is	be	AUX
cana-610	139	4	a	a	DET
cana-610	139	5	c	c	NOUN
cana-610	139	6	-	-	PUNCT
cana-610	139	7	lattice	lattice	NOUN
cana-610	139	8	,	,	PUNCT
cana-610	139	9	we	we	PRON
cana-610	139	10	have	have	VERB
cana-610	139	11	greatest	great	ADJ
cana-610	139	12	element	element	NOUN
cana-610	139	13	1	1	NUM
cana-610	139	14	is	be	AUX
cana-610	139	15	compact	compact	ADJ
cana-610	139	16	.	.	PUNCT
cana-610	140	1	therefore	therefore	ADV
cana-610	140	2	,	,	PUNCT
cana-610	140	3	by	by	ADP
cana-610	140	4	proposition	proposition	NOUN
cana-610	140	5	3.12	3.12	NUM
cana-610	140	6	,	,	PUNCT
cana-610	140	7	there	there	PRON
cana-610	140	8	exists	exist	VERB
cana-610	140	9	a	a	DET
cana-610	140	10	maximal	maximal	ADJ
cana-610	140	11	element	element	NOUN
cana-610	140	12	m	m	VERB
cana-610	140	13	such	such	ADJ
cana-610	140	14	that	that	DET
cana-610	140	15	a1	a1	NOUN
cana-610	140	16	≤	≤	ADJ
cana-610	140	17	m.	m.	NOUN
cana-610	140	18	now	now	ADV
cana-610	140	19	,	,	PUNCT
cana-610	140	20	we	we	PRON
cana-610	140	21	have	have	VERB
cana-610	140	22	a0	a0	NOUN
cana-610	140	23	∼	∼	NOUN
cana-610	140	24	a1	a1	NOUN
cana-610	140	25	∼	∼	NOUN
cana-610	140	26	m	m	NOUN
cana-610	140	27	∼	∼	NOUN
cana-610	140	28	a0	a0	NOUN
cana-610	140	29	,	,	PUNCT
cana-610	140	30	since	since	SCONJ
cana-610	140	31	a0	a0	PROPN
cana-610	140	32	∨	∨	NUM
cana-610	140	33	a1	a1	PROPN
cana-610	140	34	≤	≤	PUNCT
cana-610	140	35	m	m	PROPN
cana-610	140	36	and	and	CCONJ
cana-610	140	37	a0	a0	PROPN
cana-610	140	38	∨	∨	NUM
cana-610	140	39	m	m	PROPN
cana-610	140	40	=	=	NOUN
cana-610	140	41	m.	m.	NOUN
cana-610	140	42	theorem	theorem	VERB
cana-610	140	43	3.14	3.14	NUM
cana-610	140	44	.	.	PUNCT
cana-610	141	1	let	let	VERB
cana-610	141	2	£	£	PART
cana-610	141	3	be	be	AUX
cana-610	141	4	a	a	DET
cana-610	141	5	c	c	NOUN
cana-610	141	6	-	-	PUNCT
cana-610	141	7	lattice	lattice	NOUN
cana-610	141	8	with	with	ADP
cana-610	141	9	dim(£	dim(£	NOUN
cana-610	141	10	)	)	PUNCT
cana-610	141	11	=	=	SYM
cana-610	142	1	1	1	X
cana-610	142	2	.	.	PUNCT
cana-610	143	1	if	if	SCONJ
cana-610	143	2	a	a	DET
cana-610	143	3	s	s	NOUN
cana-610	143	4	-	-	PUNCT
cana-610	143	5	join	join	NOUN
cana-610	143	6	graph	graph	NOUN
cana-610	143	7	γs(σ(£	γs(σ(£	NOUN
cana-610	143	8	)	)	PUNCT
cana-610	143	9	)	)	PUNCT
cana-610	143	10	of	of	ADP
cana-610	143	11	£	£	PROPN
cana-610	143	12	contains	contain	VERB
cana-610	143	13	a	a	DET
cana-610	143	14	cycle	cycle	NOUN
cana-610	143	15	,	,	PUNCT
cana-610	143	16	then	then	ADV
cana-610	143	17	£	£	PROPN
cana-610	143	18	is	be	AUX
cana-610	143	19	not	not	PART
cana-610	143	20	a	a	DET
cana-610	143	21	domain	domain	NOUN
cana-610	143	22	.	.	PUNCT
cana-610	144	1	proof	proof	NOUN
cana-610	144	2	.	.	PUNCT
cana-610	145	1	suppose	suppose	VERB
cana-610	145	2	on	on	ADP
cana-610	145	3	the	the	DET
cana-610	145	4	contrary	contrary	NOUN
cana-610	145	5	that	that	SCONJ
cana-610	145	6	0	0	NUM
cana-610	145	7	∈	∈	PROPN
cana-610	145	8	σ(£	σ(£	PROPN
cana-610	145	9	)	)	PUNCT
cana-610	145	10	.	.	PUNCT
cana-610	146	1	since	since	SCONJ
cana-610	146	2	dim(£	dim(£	NOUN
cana-610	146	3	)	)	PUNCT
cana-610	146	4	=	=	SYM
cana-610	146	5	1	1	NUM
cana-610	146	6	and	and	CCONJ
cana-610	146	7	graph	graph	NOUN
cana-610	146	8	γs(σ(£	γs(σ(£	NOUN
cana-610	146	9	)	)	PUNCT
cana-610	146	10	)	)	PUNCT
cana-610	146	11	of	of	ADP
cana-610	146	12	£	£	PROPN
cana-610	146	13	contains	contain	VERB
cana-610	146	14	a	a	DET
cana-610	146	15	cycle	cycle	NOUN
cana-610	146	16	,	,	PUNCT
cana-610	146	17	by	by	ADP
cana-610	146	18	theorem	theorem	NOUN
cana-610	146	19	3.13	3.13	NUM
cana-610	146	20	we	we	PRON
cana-610	146	21	have	have	VERB
cana-610	146	22	a	a	DET
cana-610	146	23	cycle	cycle	NOUN
cana-610	146	24	a0	a0	NOUN
cana-610	146	25	∼	∼	NOUN
cana-610	146	26	a1	a1	NOUN
cana-610	146	27	∼	∼	NOUN
cana-610	146	28	a2	a2	NOUN
cana-610	146	29	∼	∼	NOUN
cana-610	146	30	•••	•••	ADV
cana-610	146	31	∼	∼	NOUN
cana-610	146	32	a0	a0	NOUN
cana-610	146	33	in	in	ADP
cana-610	146	34	γs(σ(£	γs(σ(£	NOUN
cana-610	146	35	)	)	PUNCT
cana-610	146	36	)	)	PUNCT
cana-610	146	37	.	.	PUNCT
cana-610	147	1	as	as	ADP
cana-610	147	2	a0	a0	NOUN
cana-610	147	3	∼	∼	NOUN
cana-610	147	4	a1	a1	NOUN
cana-610	147	5	,	,	PUNCT
cana-610	147	6	there	there	PRON
cana-610	147	7	exist	exist	VERB
cana-610	147	8	m	m	PROPN
cana-610	147	9	∈	∈	NOUN
cana-610	147	10	m(£	m(£	PROPN
cana-610	147	11	)	)	PUNCT
cana-610	147	12	,	,	PUNCT
cana-610	147	13	such	such	ADJ
cana-610	147	14	that	that	SCONJ
cana-610	147	15	a0	a0	PROPN
cana-610	147	16	∨	∨	NUM
cana-610	147	17	a1	a1	PROPN
cana-610	147	18	≤	≤	ADJ
cana-610	147	19	m.	m.	NOUN
cana-610	147	20	here	here	ADV
cana-610	147	21	either	either	CCONJ
cana-610	147	22	a0	a0	NOUN
cana-610	147	23	or	or	CCONJ
cana-610	147	24	a1	a1	NOUN
cana-610	147	25	are	be	AUX
cana-610	147	26	non	non	ADJ
cana-610	147	27	-	-	ADJ
cana-610	147	28	zero	zero	NUM
cana-610	147	29	member	member	NOUN
cana-610	147	30	of	of	ADP
cana-610	147	31	σ(£	σ(£	PROPN
cana-610	147	32	)	)	PUNCT
cana-610	147	33	.	.	PUNCT
cana-610	148	1	suppose	suppose	VERB
cana-610	148	2	that	that	SCONJ
cana-610	148	3	a0	a0	PROPN
cana-610	148	4	̸=	̸=	PROPN
cana-610	148	5	0	0	NUM
cana-610	148	6	.	.	PUNCT
cana-610	149	1	since	since	SCONJ
cana-610	149	2	0	0	NUM
cana-610	149	3	∈	∈	PROPN
cana-610	149	4	σ(£	σ(£	PROPN
cana-610	149	5	)	)	PUNCT
cana-610	149	6	and	and	CCONJ
cana-610	149	7	dim(£	dim(£	NOUN
cana-610	149	8	)	)	PUNCT
cana-610	149	9	=	=	SYM
cana-610	149	10	1	1	NUM
cana-610	149	11	,	,	PUNCT
cana-610	149	12	we	we	PRON
cana-610	149	13	have	have	VERB
cana-610	149	14	a1	a1	NOUN
cana-610	149	15	is	be	AUX
cana-610	149	16	maximal	maximal	ADJ
cana-610	149	17	element	element	NOUN
cana-610	149	18	.	.	PUNCT
cana-610	150	1	in	in	ADP
cana-610	150	2	fact	fact	NOUN
cana-610	150	3	,	,	PUNCT
cana-610	150	4	a1	a1	NOUN
cana-610	150	5	=	=	NOUN
cana-610	150	6	m.	m.	NOUN
cana-610	150	7	similarly	similarly	ADV
cana-610	150	8	,	,	PUNCT
cana-610	150	9	we	we	PRON
cana-610	150	10	can	can	AUX
cana-610	150	11	prove	prove	VERB
cana-610	150	12	that	that	SCONJ
cana-610	150	13	a2	a2	PROPN
cana-610	150	14	is	be	AUX
cana-610	150	15	also	also	ADV
cana-610	150	16	maximal	maximal	ADJ
cana-610	150	17	element	element	NOUN
cana-610	150	18	.	.	PUNCT
cana-610	151	1	hence	hence	ADV
cana-610	151	2	a1	a1	NOUN
cana-610	151	3	and	and	CCONJ
cana-610	151	4	a2	a2	PROPN
cana-610	151	5	are	be	AUX
cana-610	151	6	comaximal	comaximal	ADJ
cana-610	151	7	elements	element	NOUN
cana-610	151	8	of	of	ADP
cana-610	151	9	£	£	SYM
cana-610	151	10	,	,	PUNCT
cana-610	151	11	which	which	PRON
cana-610	151	12	is	be	AUX
cana-610	151	13	contradiction	contradiction	NOUN
cana-610	151	14	to	to	ADP
cana-610	151	15	the	the	DET
cana-610	151	16	fact	fact	NOUN
cana-610	151	17	that	that	SCONJ
cana-610	151	18	a1	a1	NOUN
cana-610	151	19	∼	∼	NOUN
cana-610	151	20	a2	a2	NOUN
cana-610	151	21	.	.	PUNCT
cana-610	152	1	consequently	consequently	ADV
cana-610	152	2	,	,	PUNCT
cana-610	152	3	0	0	NUM
cana-610	152	4	∈	∈	PROPN
cana-610	152	5	σ(£	σ(£	PROPN
cana-610	152	6	)	)	PUNCT
cana-610	152	7	.	.	PUNCT
cana-610	153	1	note	note	NOUN
cana-610	153	2	,	,	PUNCT
cana-610	153	3	σmin(£	σmin(£	NOUN
cana-610	153	4	)	)	PUNCT
cana-610	153	5	denotes	denote	VERB
cana-610	153	6	the	the	DET
cana-610	153	7	set	set	NOUN
cana-610	153	8	of	of	ADP
cana-610	153	9	all	all	DET
cana-610	153	10	minimal	minimal	ADJ
cana-610	153	11	prime	prime	ADJ
cana-610	153	12	elements	element	NOUN
cana-610	153	13	of	of	ADP
cana-610	153	14	£	£	SYM
cana-610	153	15	.	.	PUNCT
cana-610	154	1	theorem	theorem	VERB
cana-610	154	2	3.15	3.15	NUM
cana-610	154	3	.	.	PUNCT
cana-610	155	1	let	let	VERB
cana-610	155	2	£	£	PART
cana-610	155	3	be	be	AUX
cana-610	155	4	a	a	DET
cana-610	155	5	c	c	NOUN
cana-610	155	6	-	-	PUNCT
cana-610	155	7	lattice	lattice	NOUN
cana-610	155	8	with	with	ADP
cana-610	155	9	|σmin(£)|	|σmin(£)|	SYM
cana-610	155	10	≤	≤	NUM
cana-610	155	11	∞.	∞.	PROPN
cana-610	155	12	then	then	ADV
cana-610	155	13	γ[γs(σ(£	γ[γs(σ(£	NOUN
cana-610	155	14	)	)	PUNCT
cana-610	155	15	)	)	PUNCT
cana-610	155	16	]	]	PUNCT
cana-610	156	1	≤	≤	NUM
cana-610	156	2	|σmin(£)|	|σmin(£)|	NOUN
cana-610	156	3	.	.	PUNCT
cana-610	157	1	proof	proof	NOUN
cana-610	157	2	.	.	PUNCT
cana-610	158	1	suppose	suppose	VERB
cana-610	158	2	on	on	ADP
cana-610	158	3	the	the	DET
cana-610	158	4	contrary	contrary	NOUN
cana-610	158	5	that	that	PRON
cana-610	158	6	κ	κ	PROPN
cana-610	158	7	=	=	PRON
cana-610	158	8	{	{	PUNCT
cana-610	158	9	p1,p2,•••,pn	p1,p2,•••,pn	NOUN
cana-610	158	10	}	}	PUNCT
cana-610	158	11	be	be	AUX
cana-610	158	12	a	a	DET
cana-610	158	13	dominating	dominating	NOUN
cana-610	158	14	set	set	NOUN
cana-610	158	15	in	in	ADP
cana-610	158	16	the	the	DET
cana-610	158	17	s	s	NOUN
cana-610	158	18	-	-	PUNCT
cana-610	158	19	join	join	NOUN
cana-610	158	20	graph	graph	NOUN
cana-610	158	21	γs(σ(£	γs(σ(£	NOUN
cana-610	158	22	)	)	PUNCT
cana-610	158	23	)	)	PUNCT
cana-610	158	24	of	of	ADP
cana-610	158	25	£	£	SYM
cana-610	158	26	.	.	PUNCT
cana-610	159	1	by	by	ADP
cana-610	159	2	definition	definition	NOUN
cana-610	159	3	,	,	PUNCT
cana-610	159	4	for	for	ADP
cana-610	159	5	any	any	DET
cana-610	159	6	q	q	PROPN
cana-610	159	7	∈	∈	PROPN
cana-610	159	8	σ(£	σ(£	PROPN
cana-610	159	9	)	)	PUNCT
cana-610	159	10	−	−	PROPN
cana-610	159	11	κ	κ	NOUN
cana-610	159	12	there	there	PRON
cana-610	159	13	is	be	VERB
cana-610	159	14	pi	pi	PROPN
cana-610	159	15	∈	∈	PROPN
cana-610	159	16	κ	κ	ADP
cana-610	159	17	such	such	DET
cana-610	159	18	that	that	PRON
cana-610	159	19	q	q	NOUN
cana-610	159	20	is	be	AUX
cana-610	159	21	adjacent	adjacent	ADJ
cana-610	159	22	to	to	PART
cana-610	159	23	pi	pi	VERB
cana-610	159	24	for	for	ADP
cana-610	159	25	some	some	DET
cana-610	159	26	1	1	NUM
cana-610	159	27	≤	≤	NUM
cana-610	159	28	i	i	PRON
cana-610	159	29	≤	≤	ADJ
cana-610	159	30	n.	n.	NOUN
cana-610	159	31	let	let	VERB
cana-610	159	32	q1,q2,•••	q1,q2,•••	PROPN
cana-610	159	33	,	,	PUNCT
cana-610	159	34	qr	qr	PROPN
cana-610	159	35	be	be	AUX
cana-610	159	36	the	the	DET
cana-610	159	37	distinct	distinct	ADJ
cana-610	159	38	members	member	NOUN
cana-610	159	39	of	of	ADP
cana-610	159	40	σmin(£	σmin(£	PROPN
cana-610	159	41	)	)	PUNCT
cana-610	159	42	.	.	PUNCT
cana-610	160	1	for	for	ADP
cana-610	160	2	each	each	DET
cana-610	160	3	pi	pi	NOUN
cana-610	160	4	∈	∈	PROPN
cana-610	160	5	κ	κ	NOUN
cana-610	160	6	there	there	PRON
cana-610	160	7	is	be	VERB
cana-610	160	8	at	at	ADV
cana-610	160	9	least	least	ADV
cana-610	160	10	one	one	NUM
cana-610	160	11	qj	qj	NOUN
cana-610	160	12	(	(	PUNCT
cana-610	160	13	1	1	NUM
cana-610	160	14	≤	≤	NUM
cana-610	160	15	j	j	NOUN
cana-610	160	16	≤	≤	ADJ
cana-610	160	17	r	r	NOUN
cana-610	160	18	)	)	PUNCT
cana-610	160	19	with	with	ADP
cana-610	160	20	qj	qj	PROPN
cana-610	160	21	≤	≤	NUM
cana-610	160	22	pi	pi	NOUN
cana-610	160	23	.	.	PUNCT
cana-610	161	1	this	this	PRON
cana-610	161	2	implies	imply	VERB
cana-610	161	3	that	that	SCONJ
cana-610	161	4	r	r	NOUN
cana-610	161	5	≤	≤	NUM
cana-610	161	6	n.	n.	NOUN
cana-610	161	7	now	now	ADV
cana-610	161	8	we	we	PRON
cana-610	161	9	prove	prove	VERB
cana-610	161	10	that	that	SCONJ
cana-610	161	11	τ	τ	PROPN
cana-610	161	12	=	=	PRON
cana-610	161	13	{	{	PUNCT
cana-610	161	14	q1,q2,•••	q1,q2,•••	PROPN
cana-610	161	15	,	,	PUNCT
cana-610	161	16	qr	qr	PROPN
cana-610	161	17	}	}	PUNCT
cana-610	161	18	is	be	AUX
cana-610	161	19	also	also	ADV
cana-610	161	20	a	a	DET
cana-610	161	21	dominating	dominating	NOUN
cana-610	161	22	set	set	NOUN
cana-610	161	23	.	.	PUNCT
cana-610	162	1	suppose	suppose	VERB
cana-610	162	2	that	that	SCONJ
cana-610	162	3	p	p	PROPN
cana-610	162	4	∈	∈	PROPN
cana-610	162	5	σ(£	σ(£	PROPN
cana-610	162	6	)	)	PUNCT
cana-610	162	7	such	such	ADJ
cana-610	162	8	that	that	SCONJ
cana-610	162	9	p	p	PROPN
cana-610	162	10	∉	∉	PROPN
cana-610	162	11	τ	τ	PROPN
cana-610	162	12	.	.	PUNCT
cana-610	163	1	if	if	SCONJ
cana-610	163	2	p	p	PROPN
cana-610	163	3	∉	∉	PROPN
cana-610	163	4	κ	κ	X
cana-610	163	5	.	.	PUNCT
cana-610	164	1	since	since	SCONJ
cana-610	164	2	κ	κ	PROPN
cana-610	164	3	is	be	AUX
cana-610	164	4	a	a	DET
cana-610	164	5	dominating	dominating	NOUN
cana-610	164	6	set	set	NOUN
cana-610	164	7	,	,	PUNCT
cana-610	164	8	we	we	PRON
cana-610	164	9	have	have	VERB
cana-610	164	10	pi	pi	PROPN
cana-610	164	11	∈	∈	PROPN
cana-610	164	12	κ	κ	NOUN
cana-610	164	13	such	such	ADJ
cana-610	164	14	that	that	SCONJ
cana-610	164	15	p	p	ADJ
cana-610	164	16	∼	∼	NOUN
cana-610	164	17	p1	p1	NOUN
cana-610	164	18	.	.	PUNCT
cana-610	165	1	by	by	ADP
cana-610	165	2	definition	definition	NOUN
cana-610	165	3	there	there	PRON
cana-610	165	4	exists	exist	VERB
cana-610	165	5	m′	m′	NOUN
cana-610	165	6	∈	∈	PROPN
cana-610	165	7	m(£	m(£	PROPN
cana-610	165	8	)	)	PUNCT
cana-610	165	9	such	such	ADJ
cana-610	165	10	that	that	SCONJ
cana-610	165	11	p	p	PROPN
cana-610	165	12	∨	∨	NUM
cana-610	165	13	pi	pi	PROPN
cana-610	165	14	≤	≤	PROPN
cana-610	165	15	m′.	m′.	NOUN
cana-610	165	16	according	accord	VERB
cana-610	165	17	to	to	ADP
cana-610	165	18	the	the	DET
cana-610	165	19	construction	construction	NOUN
cana-610	165	20	of	of	ADP
cana-610	165	21	the	the	DET
cana-610	165	22	set	set	NOUN
cana-610	165	23	τ	τ	X
cana-610	165	24	=	=	PRON
cana-610	165	25	{	{	PUNCT
cana-610	165	26	q1,q2,•••	q1,q2,•••	PROPN
cana-610	165	27	,	,	PUNCT
cana-610	165	28	qr	qr	PROPN
cana-610	165	29	}	}	PUNCT
cana-610	165	30	,	,	PUNCT
cana-610	165	31	there	there	PRON
cana-610	165	32	are	be	VERB
cana-610	165	33	some	some	DET
cana-610	165	34	qj	qj	PROPN
cana-610	165	35	∈	∈	PROPN
cana-610	165	36	τ	τ	X
cana-610	165	37	with	with	ADP
cana-610	165	38	qj	qj	PROPN
cana-610	165	39	≤	≤	NUM
cana-610	165	40	pi	pi	NOUN
cana-610	165	41	,	,	PUNCT
cana-610	165	42	therefore	therefore	ADV
cana-610	165	43	p	p	X
cana-610	165	44	∨	∨	PROPN
cana-610	165	45	qj	qj	PROPN
cana-610	165	46	≤	≤	NUM
cana-610	165	47	m′	m′	NOUN
cana-610	165	48	and	and	CCONJ
cana-610	165	49	hence	hence	ADV
cana-610	165	50	p	p	X
cana-610	165	51	∼	∼	NOUN
cana-610	165	52	qj	qj	NOUN
cana-610	165	53	for	for	ADP
cana-610	165	54	some	some	DET
cana-610	165	55	qj	qj	PROPN
cana-610	165	56	∈	∈	PROPN
cana-610	165	57	τ	τ	PROPN
cana-610	165	58	.	.	PUNCT
cana-610	166	1	now	now	ADV
cana-610	166	2	,	,	PUNCT
cana-610	166	3	suppose	suppose	VERB
cana-610	166	4	p	p	PROPN
cana-610	166	5	∈	∈	PROPN
cana-610	166	6	κ	κ	X
cana-610	166	7	.	.	PUNCT
cana-610	167	1	by	by	ADP
cana-610	167	2	the	the	DET
cana-610	167	3	definition	definition	NOUN
cana-610	167	4	of	of	ADP
cana-610	167	5	τ	τ	PROPN
cana-610	167	6	,	,	PUNCT
cana-610	167	7	there	there	PRON
cana-610	167	8	exists	exist	VERB
cana-610	167	9	qi	qi	PROPN
cana-610	167	10	∈	∈	PROPN
cana-610	167	11	τ	τ	X
cana-610	167	12	such	such	ADJ
cana-610	167	13	that	that	SCONJ
cana-610	167	14	qi	qi	PROPN
cana-610	167	15	≤	≤	PROPN
cana-610	167	16	p.	p.	NOUN
cana-610	167	17	since	since	SCONJ
cana-610	167	18	the	the	DET
cana-610	167	19	greatest	great	ADJ
cana-610	167	20	element	element	NOUN
cana-610	167	21	1	1	NUM
cana-610	167	22	is	be	AUX
cana-610	167	23	compact	compact	ADJ
cana-610	167	24	,	,	PUNCT
cana-610	167	25	by	by	ADP
cana-610	167	26	proposition	proposition	NOUN
cana-610	167	27	3.12	3.12	NUM
cana-610	167	28	we	we	PRON
cana-610	167	29	have	have	VERB
cana-610	167	30	m	m	NOUN
cana-610	167	31	∈	∈	PROPN
cana-610	167	32	m(£	m(£	PROPN
cana-610	167	33	)	)	PUNCT
cana-610	167	34	such	such	ADJ
cana-610	167	35	that	that	SCONJ
cana-610	167	36	p	p	PROPN
cana-610	167	37	≤	≤	NUM
cana-610	167	38	m	m	NOUN
cana-610	167	39	,	,	PUNCT
cana-610	167	40	therefore	therefore	ADV
cana-610	167	41	qi	qi	PROPN
cana-610	167	42	∨	∨	PROPN
cana-610	167	43	p	p	PROPN
cana-610	167	44	≤	≤	ADJ
cana-610	167	45	m.	m.	NOUN
cana-610	167	46	this	this	PRON
cana-610	167	47	implies	imply	VERB
cana-610	167	48	that	that	SCONJ
cana-610	167	49	qi	qi	PROPN
cana-610	167	50	∼	∼	NOUN
cana-610	167	51	p	p	NOUN
cana-610	167	52	,	,	PUNCT
cana-610	167	53	consequently	consequently	ADV
cana-610	167	54	,	,	PUNCT
cana-610	167	55	τ	τ	PROPN
cana-610	167	56	is	be	AUX
cana-610	167	57	a	a	DET
cana-610	167	58	dominating	dominating	NOUN
cana-610	167	59	set	set	NOUN
cana-610	167	60	.	.	PUNCT
cana-610	168	1	in	in	ADP
cana-610	168	2	[	[	X
cana-610	168	3	8	8	NUM
cana-610	168	4	]	]	PUNCT
cana-610	168	5	,	,	PUNCT
cana-610	168	6	f.	f.	PROPN
cana-610	168	7	callialp	callialp	PROPN
cana-610	168	8	et	et	PROPN
cana-610	168	9	.	.	PUNCT
cana-610	169	1	al	al	PROPN
cana-610	169	2	.	.	PROPN
cana-610	169	3	established	establish	VERB
cana-610	169	4	some	some	DET
cana-610	169	5	results	result	NOUN
cana-610	169	6	on	on	ADP
cana-610	169	7	the	the	DET
cana-610	169	8	zariski	zariski	NOUN
cana-610	169	9	topology	topology	NOUN
cana-610	169	10	over	over	ADP
cana-610	169	11	σ(£	σ(£	PROPN
cana-610	169	12	)	)	PUNCT
cana-610	169	13	.	.	PUNCT
cana-610	170	1	for	for	ADP
cana-610	170	2	a	a	DET
cana-610	170	3	∈	∈	ADJ
cana-610	170	4	£	£	NOUN
cana-610	170	5	,	,	PUNCT
cana-610	170	6	define	define	VERB
cana-610	170	7	ϑ(a	ϑ(a	VERB
cana-610	170	8	)	)	PUNCT
cana-610	170	9	=	=	PRON
cana-610	170	10	{	{	PUNCT
cana-610	170	11	p	p	NOUN
cana-610	170	12	∈	∈	PROPN
cana-610	170	13	σ(£)|a	σ(£)|a	VERB
cana-610	170	14	≤	≤	NOUN
cana-610	170	15	p	p	X
cana-610	170	16	}	}	PUNCT
cana-610	170	17	.	.	PUNCT
cana-610	171	1	f.	f.	PROPN
cana-610	171	2	callialp	callialp	PROPN
cana-610	171	3	et	et	PROPN
cana-610	171	4	.	.	PUNCT
cana-610	172	1	al	al	PROPN
cana-610	172	2	.	.	PROPN
cana-610	172	3	introduced	introduce	VERB
cana-610	172	4	a	a	DET
cana-610	172	5	topology	topology	NOUN
cana-610	172	6	on	on	ADP
cana-610	172	7	σ(£	σ(£	PROPN
cana-610	172	8	)	)	PUNCT
cana-610	172	9	with	with	ADP
cana-610	172	10	the	the	DET
cana-610	172	11	collection	collection	NOUN
cana-610	172	12	of	of	ADP
cana-610	172	13	all	all	DET
cana-610	172	14	closed	closed	ADJ
cana-610	172	15	set	set	NOUN
cana-610	172	16	{	{	PUNCT
cana-610	172	17	ϑ(a)|a	ϑ(a)|a	NOUN
cana-610	172	18	∈	∈	PROPN
cana-610	173	1	£	£	AUX
cana-610	173	2	}	}	PUNCT
cana-610	173	3	using	use	VERB
cana-610	173	4	the	the	DET
cana-610	173	5	following	follow	VERB
cana-610	173	6	proposition	proposition	NOUN
cana-610	173	7	3.16	3.16	NUM
cana-610	173	8	(	(	PUNCT
cana-610	173	9	see	see	VERB
cana-610	173	10	[	[	X
cana-610	173	11	8	8	NUM
cana-610	173	12	]	]	NUM
cana-610	173	13	)	)	PUNCT
cana-610	173	14	.	.	PUNCT
cana-610	174	1	proposition	proposition	NOUN
cana-610	174	2	3.16	3.16	NUM
cana-610	174	3	.	.	PUNCT
cana-610	175	1	[	[	X
cana-610	175	2	8	8	NUM
cana-610	175	3	]	]	PUNCT
cana-610	175	4	let	let	VERB
cana-610	175	5	£	£	PRON
cana-610	175	6	be	be	AUX
cana-610	175	7	a	a	DET
cana-610	175	8	c	c	NOUN
cana-610	175	9	-	-	PUNCT
cana-610	175	10	lattice	lattice	NOUN
cana-610	175	11	and	and	CCONJ
cana-610	175	12	for	for	ADP
cana-610	175	13	a	a	DET
cana-610	175	14	∈	∈	ADJ
cana-610	175	15	£	£	NOUN
cana-610	175	16	,	,	PUNCT
cana-610	175	17	let	let	VERB
cana-610	175	18	ϑ(a	ϑ(a	VERB
cana-610	175	19	)	)	PUNCT
cana-610	175	20	=	=	PRON
cana-610	175	21	{	{	PUNCT
cana-610	175	22	p	p	NOUN
cana-610	175	23	∈	∈	PROPN
cana-610	175	24	σ(£)|a	σ(£)|a	VERB
cana-610	175	25	≤	≤	NOUN
cana-610	175	26	p	p	X
cana-610	175	27	}	}	PUNCT
cana-610	175	28	.	.	PUNCT
cana-610	176	1	then	then	ADV
cana-610	176	2	the	the	DET
cana-610	176	3	following	following	ADJ
cana-610	176	4	axioms	axiom	NOUN
cana-610	176	5	hold	hold	VERB
cana-610	176	6	:	:	PUNCT
cana-610	176	7	communications	communication	NOUN
cana-610	176	8	on	on	ADP
cana-610	176	9	applied	apply	VERB
cana-610	176	10	nonlinear	nonlinear	ADJ
cana-610	176	11	analysis	analysis	NOUN
cana-610	176	12	issn	issn	NOUN
cana-610	176	13	:	:	PUNCT
cana-610	176	14	1074	1074	NUM
cana-610	176	15	-	-	PUNCT
cana-610	176	16	133x	133x	NUM
cana-610	176	17	vol	vol	NOUN
cana-610	176	18	31	31	NUM
cana-610	176	19	no	no	NOUN
cana-610	176	20	.	.	NOUN
cana-610	176	21	2	2	NUM
cana-610	176	22	(	(	PUNCT
cana-610	176	23	2024	2024	NUM
cana-610	176	24	)	)	PUNCT
cana-610	176	25	414	414	NUM
cana-610	176	26	https://internationalpubls.com	https://internationalpubls.com	X
cana-610	176	27	1	1	X
cana-610	176	28	.	.	PUNCT
cana-610	177	1	ϑ(0	ϑ(0	NOUN
cana-610	177	2	)	)	PUNCT
cana-610	177	3	=	=	SYM
cana-610	177	4	σ(£	σ(£	PROPN
cana-610	177	5	)	)	PUNCT
cana-610	177	6	and	and	CCONJ
cana-610	177	7	ϑ(1	ϑ(1	PROPN
cana-610	177	8	)	)	PUNCT
cana-610	177	9	=	=	NOUN
cana-610	177	10	∅.	∅.	PRON
cana-610	177	11	2	2	NUM
cana-610	177	12	.	.	PUNCT
cana-610	177	13	∩α∈	∩α∈	NOUN
cana-610	177	14	△	△	NOUN
cana-610	177	15	ϑ(aα	ϑ(aα	ADJ
cana-610	177	16	)	)	PUNCT
cana-610	177	17	=	=	SYM
cana-610	177	18	ϑ(∨α∈	ϑ(∨α∈	PROPN
cana-610	177	19	△	△	X
cana-610	177	20	aα	aα	NOUN
cana-610	177	21	)	)	PUNCT
cana-610	177	22	for	for	ADP
cana-610	177	23	any	any	DET
cana-610	177	24	index	index	NOUN
cana-610	177	25	set	set	VERB
cana-610	177	26	△	△	PROPN
cana-610	177	27	.	.	PROPN
cana-610	178	1	3	3	NUM
cana-610	178	2	.	.	PUNCT
cana-610	178	3	ϑ(p	ϑ(p	PROPN
cana-610	178	4	)	)	PUNCT
cana-610	178	5	∪	∪	ADP
cana-610	178	6	ϑ(q	ϑ(q	PROPN
cana-610	178	7	)	)	PUNCT
cana-610	179	1	=	=	SYM
cana-610	179	2	ϑ(p	ϑ(p	PROPN
cana-610	179	3	∧	∧	PROPN
cana-610	179	4	q	q	NOUN
cana-610	179	5	)	)	PUNCT
cana-610	179	6	=	=	SYM
cana-610	179	7	ϑ(pq	ϑ(pq	X
cana-610	179	8	)	)	PUNCT
cana-610	179	9	for	for	ADP
cana-610	179	10	p	p	NOUN
cana-610	179	11	,	,	PUNCT
cana-610	179	12	q	q	NOUN
cana-610	179	13	∈	∈	PROPN
cana-610	179	14	£	£	SYM
cana-610	179	15	.	.	PUNCT
cana-610	180	1	in	in	ADP
cana-610	180	2	the	the	DET
cana-610	180	3	next	next	ADJ
cana-610	180	4	theorem	theorem	NOUN
cana-610	180	5	3.17	3.17	NUM
cana-610	180	6	,	,	PUNCT
cana-610	180	7	we	we	PRON
cana-610	180	8	studied	study	VERB
cana-610	180	9	the	the	DET
cana-610	180	10	connected	connected	ADJ
cana-610	180	11	topological	topological	ADJ
cana-610	180	12	space	space	NOUN
cana-610	180	13	σ(£	σ(£	PROPN
cana-610	180	14	)	)	PUNCT
cana-610	180	15	.	.	PUNCT
cana-610	181	1	theorem	theorem	VERB
cana-610	181	2	3.17	3.17	NUM
cana-610	181	3	.	.	PUNCT
cana-610	182	1	let	let	VERB
cana-610	182	2	£	£	PART
cana-610	182	3	be	be	AUX
cana-610	182	4	a	a	DET
cana-610	182	5	c	c	NOUN
cana-610	182	6	-	-	PUNCT
cana-610	182	7	lattice	lattice	NOUN
cana-610	182	8	and	and	CCONJ
cana-610	182	9	|σmin(£)|	|σmin(£)|	PRON
cana-610	182	10	≤	≤	ADJ
cana-610	182	11	∞.	∞.	PROPN
cana-610	182	12	then	then	ADV
cana-610	182	13	σ(£	σ(£	PROPN
cana-610	182	14	)	)	PUNCT
cana-610	182	15	is	be	AUX
cana-610	182	16	connected	connect	VERB
cana-610	182	17	if	if	SCONJ
cana-610	182	18	and	and	CCONJ
cana-610	182	19	only	only	ADV
cana-610	182	20	if	if	SCONJ
cana-610	182	21	the	the	DET
cana-610	182	22	s	s	NOUN
cana-610	182	23	-	-	PUNCT
cana-610	182	24	join	join	ADJ
cana-610	182	25	graph	graph	NOUN
cana-610	182	26	γs(σ(£	γs(σ(£	NOUN
cana-610	182	27	)	)	PUNCT
cana-610	182	28	)	)	PUNCT
cana-610	182	29	of	of	ADP
cana-610	182	30	£	£	PROPN
cana-610	182	31	is	be	AUX
cana-610	182	32	connected	connect	VERB
cana-610	182	33	and	and	CCONJ
cana-610	182	34	diam(γs(σ(£	diam(γs(σ(£	ADJ
cana-610	182	35	)	)	PUNCT
cana-610	182	36	)	)	PUNCT
cana-610	182	37	)	)	PUNCT
cana-610	183	1	≤	≤	NUM
cana-610	183	2	2|σmin(£)|	2|σmin(£)|	NUM
cana-610	183	3	.	.	PUNCT
cana-610	184	1	proof	proof	NOUN
cana-610	184	2	.	.	PUNCT
cana-610	185	1	suppose	suppose	VERB
cana-610	185	2	that	that	SCONJ
cana-610	185	3	σ(£	σ(£	PROPN
cana-610	185	4	)	)	PUNCT
cana-610	185	5	is	be	AUX
cana-610	185	6	not	not	PART
cana-610	185	7	connected	connect	VERB
cana-610	185	8	.	.	PUNCT
cana-610	186	1	if	if	SCONJ
cana-610	186	2	γs(σ(£	γs(σ(£	NOUN
cana-610	186	3	)	)	PUNCT
cana-610	186	4	)	)	PUNCT
cana-610	186	5	is	be	AUX
cana-610	186	6	disconnected	disconnect	VERB
cana-610	186	7	,	,	PUNCT
cana-610	186	8	then	then	ADV
cana-610	186	9	nothing	nothing	PRON
cana-610	186	10	to	to	PART
cana-610	186	11	prove	prove	VERB
cana-610	186	12	.	.	PUNCT
cana-610	187	1	suppose	suppose	VERB
cana-610	187	2	that	that	SCONJ
cana-610	187	3	γs(σ(£	γs(σ(£	NOUN
cana-610	187	4	)	)	PUNCT
cana-610	187	5	)	)	PUNCT
cana-610	187	6	is	be	AUX
cana-610	187	7	connected	connect	VERB
cana-610	187	8	,	,	PUNCT
cana-610	187	9	then	then	ADV
cana-610	187	10	for	for	ADP
cana-610	187	11	some	some	PRON
cana-610	187	12	a	a	PRON
cana-610	187	13	,	,	PUNCT
cana-610	187	14	b	b	PROPN
cana-610	187	15	∈	∈	PROPN
cana-610	187	16	£	£	SYM
cana-610	187	17	,	,	PUNCT
cana-610	187	18	σ(£	σ(£	PROPN
cana-610	187	19	)	)	PUNCT
cana-610	187	20	=	=	SYM
cana-610	187	21	ϑ(a	ϑ(a	VERB
cana-610	187	22	)	)	PUNCT
cana-610	187	23	∪	∪	ADP
cana-610	187	24	ϑ(b	ϑ(b	PROPN
cana-610	187	25	)	)	PUNCT
cana-610	187	26	and	and	CCONJ
cana-610	187	27	ϑ(a	ϑ(a	VERB
cana-610	187	28	)	)	PUNCT
cana-610	187	29	∩	∩	NOUN
cana-610	187	30	ϑ(b	ϑ(b	ADJ
cana-610	187	31	)	)	PUNCT
cana-610	187	32	=	=	PUNCT
cana-610	187	33	∅.	∅.	AUX
cana-610	187	34	let	let	VERB
cana-610	187	35	p	p	PRON
cana-610	187	36	,	,	PUNCT
cana-610	187	37	q	q	PROPN
cana-610	187	38	∈	∈	PROPN
cana-610	187	39	σ(£	σ(£	PROPN
cana-610	187	40	)	)	PUNCT
cana-610	187	41	such	such	ADJ
cana-610	187	42	that	that	SCONJ
cana-610	187	43	p	p	PROPN
cana-610	187	44	∈	∈	PROPN
cana-610	187	45	ϑ(a	ϑ(a	VERB
cana-610	187	46	)	)	PUNCT
cana-610	187	47	and	and	CCONJ
cana-610	187	48	q	q	NOUN
cana-610	187	49	∈	∈	PROPN
cana-610	187	50	ϑ(b	ϑ(b	PROPN
cana-610	187	51	)	)	PUNCT
cana-610	187	52	.	.	PUNCT
cana-610	188	1	since	since	SCONJ
cana-610	188	2	γs(σ(£	γs(σ(£	NOUN
cana-610	188	3	)	)	PUNCT
cana-610	188	4	)	)	PUNCT
cana-610	188	5	is	be	AUX
cana-610	188	6	the	the	DET
cana-610	188	7	connected	connected	ADJ
cana-610	188	8	graph	graph	NOUN
cana-610	188	9	,	,	PUNCT
cana-610	188	10	we	we	PRON
cana-610	188	11	have	have	VERB
cana-610	188	12	a	a	DET
cana-610	188	13	path	path	NOUN
cana-610	188	14	p	p	NOUN
cana-610	188	15	∼	∼	NOUN
cana-610	188	16	p1	p1	NOUN
cana-610	188	17	∼	∼	NOUN
cana-610	188	18	p2	p2	NOUN
cana-610	188	19	∼	∼	NOUN
cana-610	188	20	p3	p3	NOUN
cana-610	188	21	∼	∼	NOUN
cana-610	188	22	•••	•••	ADV
cana-610	188	23	∼	∼	NOUN
cana-610	188	24	pn	pn	NOUN
cana-610	188	25	∼	∼	NOUN
cana-610	188	26	q	q	NOUN
cana-610	188	27	between	between	ADP
cana-610	188	28	p	p	PROPN
cana-610	188	29	and	and	CCONJ
cana-610	188	30	q.	q.	NOUN
cana-610	188	31	as	as	ADP
cana-610	188	32	p	p	PRON
cana-610	188	33	∼	∼	NOUN
cana-610	188	34	p1	p1	NOUN
cana-610	188	35	,	,	PUNCT
cana-610	188	36	by	by	ADP
cana-610	188	37	definition	definition	NOUN
cana-610	188	38	there	there	PRON
cana-610	188	39	exists	exist	VERB
cana-610	188	40	m1	m1	PROPN
cana-610	188	41	∈	∈	PROPN
cana-610	188	42	m(£	m(£	PROPN
cana-610	188	43	)	)	PUNCT
cana-610	188	44	such	such	ADJ
cana-610	188	45	that	that	SCONJ
cana-610	188	46	p	p	PROPN
cana-610	188	47	∨	∨	NUM
cana-610	188	48	p1	p1	NOUN
cana-610	188	49	≤	≤	NOUN
cana-610	188	50	m1	m1	NOUN
cana-610	188	51	.	.	PUNCT
cana-610	189	1	this	this	PRON
cana-610	189	2	implies	imply	VERB
cana-610	189	3	that	that	SCONJ
cana-610	189	4	p1	p1	PROPN
cana-610	189	5	∈	∈	PROPN
cana-610	189	6	ϑ(a	ϑ(a	PROPN
cana-610	189	7	)	)	PUNCT
cana-610	189	8	since	since	SCONJ
cana-610	189	9	p	p	X
cana-610	189	10	,	,	PUNCT
cana-610	189	11	m1	m1	PROPN
cana-610	189	12	∈	∈	PROPN
cana-610	189	13	ϑ(a	ϑ(a	VERB
cana-610	189	14	)	)	PUNCT
cana-610	189	15	.	.	PUNCT
cana-610	190	1	using	use	VERB
cana-610	190	2	similar	similar	ADJ
cana-610	190	3	arguments	argument	NOUN
cana-610	190	4	,	,	PUNCT
cana-610	190	5	we	we	PRON
cana-610	190	6	can	can	AUX
cana-610	190	7	prove	prove	VERB
cana-610	190	8	that	that	SCONJ
cana-610	190	9	pn	pn	PROPN
cana-610	190	10	∈	∈	PROPN
cana-610	190	11	ϑ(a	ϑ(a	PROPN
cana-610	190	12	)	)	PUNCT
cana-610	190	13	.	.	PUNCT
cana-610	191	1	also	also	ADV
cana-610	191	2	,	,	PUNCT
cana-610	191	3	note	note	VERB
cana-610	191	4	that	that	SCONJ
cana-610	191	5	pn	pn	PROPN
cana-610	191	6	∼	∼	NOUN
cana-610	191	7	q.	q.	PROPN
cana-610	191	8	therefore	therefore	ADV
cana-610	191	9	by	by	ADP
cana-610	191	10	definition	definition	NOUN
cana-610	191	11	,	,	PUNCT
cana-610	191	12	there	there	PRON
cana-610	191	13	exists	exist	VERB
cana-610	191	14	mn	mn	PROPN
cana-610	191	15	∈	∈	PROPN
cana-610	191	16	m(£	m(£	PROPN
cana-610	191	17	)	)	PUNCT
cana-610	191	18	such	such	ADJ
cana-610	191	19	that	that	SCONJ
cana-610	191	20	pn	pn	PROPN
cana-610	191	21	∨	∨	NUM
cana-610	191	22	q	q	PROPN
cana-610	191	23	≤	≤	PROPN
cana-610	191	24	mn	mn	PROPN
cana-610	191	25	.	.	PUNCT
cana-610	192	1	this	this	PRON
cana-610	192	2	implies	imply	VERB
cana-610	192	3	that	that	SCONJ
cana-610	192	4	mn	mn	PROPN
cana-610	192	5	∈	∈	PROPN
cana-610	192	6	ϑ(a	ϑ(a	VERB
cana-610	192	7	)	)	PUNCT
cana-610	192	8	because	because	SCONJ
cana-610	192	9	pn	pn	PROPN
cana-610	192	10	≤	≤	PROPN
cana-610	192	11	mn	mn	PROPN
cana-610	192	12	.	.	PUNCT
cana-610	193	1	also	also	ADV
cana-610	193	2	,	,	PUNCT
cana-610	193	3	note	note	VERB
cana-610	193	4	that	that	SCONJ
cana-610	193	5	q	q	PROPN
cana-610	193	6	≤	≤	NUM
cana-610	193	7	mn	mn	PROPN
cana-610	193	8	,	,	PUNCT
cana-610	193	9	therefore	therefore	ADV
cana-610	193	10	we	we	PRON
cana-610	193	11	have	have	VERB
cana-610	193	12	mn	mn	PROPN
cana-610	193	13	∈	∈	PROPN
cana-610	193	14	ϑ(b	ϑ(b	PROPN
cana-610	193	15	)	)	PUNCT
cana-610	193	16	,	,	PUNCT
cana-610	193	17	a	a	DET
cana-610	193	18	contradiction	contradiction	NOUN
cana-610	193	19	to	to	ADP
cana-610	193	20	the	the	DET
cana-610	193	21	fact	fact	NOUN
cana-610	193	22	that	that	SCONJ
cana-610	193	23	ϑ(a	ϑ(a	VERB
cana-610	193	24	)	)	PUNCT
cana-610	193	25	∩	∩	NOUN
cana-610	193	26	ϑ(b	ϑ(b	ADJ
cana-610	193	27	)	)	PUNCT
cana-610	193	28	=	=	NOUN
cana-610	193	29	∅.	∅.	ADP
cana-610	193	30	consequently	consequently	ADV
cana-610	193	31	,	,	PUNCT
cana-610	193	32	σ(£	σ(£	PROPN
cana-610	193	33	)	)	PUNCT
cana-610	193	34	is	be	AUX
cana-610	193	35	connected	connect	VERB
cana-610	193	36	.	.	PUNCT
cana-610	194	1	conversely	conversely	ADV
cana-610	194	2	,	,	PUNCT
cana-610	194	3	suppose	suppose	VERB
cana-610	194	4	that	that	SCONJ
cana-610	194	5	the	the	DET
cana-610	194	6	space	space	NOUN
cana-610	194	7	σ(£	σ(£	PROPN
cana-610	194	8	)	)	PUNCT
cana-610	194	9	is	be	AUX
cana-610	194	10	connected	connect	VERB
cana-610	194	11	and	and	CCONJ
cana-610	194	12	σmin(£	σmin(£	NOUN
cana-610	194	13	)	)	PUNCT
cana-610	194	14	=	=	PRON
cana-610	194	15	{	{	PUNCT
cana-610	194	16	a1,a2,•••	a1,a2,•••	PROPN
cana-610	194	17	,	,	PUNCT
cana-610	194	18	ar	ar	PROPN
cana-610	194	19	}	}	PUNCT
cana-610	194	20	.	.	PUNCT
cana-610	195	1	case	case	NOUN
cana-610	195	2	i	i	NOUN
cana-610	195	3	)	)	PUNCT
cana-610	195	4	suppose	suppose	VERB
cana-610	195	5	r	r	NOUN
cana-610	195	6	=	=	SYM
cana-610	195	7	2	2	NUM
cana-610	195	8	.	.	PUNCT
cana-610	196	1	for	for	ADP
cana-610	196	2	p	p	NOUN
cana-610	196	3	,	,	PUNCT
cana-610	196	4	q	q	PROPN
cana-610	196	5	∈	∈	PROPN
cana-610	196	6	σ(£	σ(£	PROPN
cana-610	196	7	)	)	PUNCT
cana-610	196	8	,	,	PUNCT
cana-610	196	9	there	there	PRON
cana-610	196	10	exist	exist	VERB
cana-610	196	11	a1,a2	a1,a2	PROPN
cana-610	196	12	∈	∈	PROPN
cana-610	196	13	σmin(£	σmin(£	PROPN
cana-610	196	14	)	)	PUNCT
cana-610	196	15	such	such	ADJ
cana-610	196	16	that	that	DET
cana-610	196	17	a1	a1	NOUN
cana-610	196	18	≤	≤	PUNCT
cana-610	196	19	p	p	NOUN
cana-610	196	20	and	and	CCONJ
cana-610	196	21	a2	a2	PROPN
cana-610	196	22	≤	≤	PROPN
cana-610	196	23	q.	q.	PROPN
cana-610	196	24	since	since	SCONJ
cana-610	196	25	σ(£	σ(£	PROPN
cana-610	196	26	)	)	PUNCT
cana-610	196	27	is	be	AUX
cana-610	196	28	connected	connect	VERB
cana-610	196	29	,	,	PUNCT
cana-610	196	30	we	we	PRON
cana-610	196	31	have	have	VERB
cana-610	196	32	a3	a3	NOUN
cana-610	196	33	∈	∈	PROPN
cana-610	196	34	ϑ(a1)∩ϑ(a2	ϑ(a1)∩ϑ(a2	NOUN
cana-610	196	35	)	)	PUNCT
cana-610	196	36	.	.	PUNCT
cana-610	197	1	also	also	ADV
cana-610	197	2	,	,	PUNCT
cana-610	197	3	since	since	SCONJ
cana-610	197	4	the	the	DET
cana-610	197	5	greatest	great	ADJ
cana-610	197	6	element	element	NOUN
cana-610	197	7	1	1	NUM
cana-610	197	8	is	be	AUX
cana-610	197	9	compact	compact	ADJ
cana-610	197	10	,	,	PUNCT
cana-610	197	11	by	by	ADP
cana-610	197	12	proposition	proposition	NOUN
cana-610	197	13	3.12	3.12	NUM
cana-610	197	14	we	we	PRON
cana-610	197	15	have	have	VERB
cana-610	197	16	a	a	DET
cana-610	197	17	path	path	NOUN
cana-610	197	18	p	p	NOUN
cana-610	197	19	∼	∼	NOUN
cana-610	197	20	a1	a1	NOUN
cana-610	197	21	∼	∼	NOUN
cana-610	197	22	a3	a3	NOUN
cana-610	197	23	∼	∼	NOUN
cana-610	197	24	a2	a2	NOUN
cana-610	197	25	∼	∼	NOUN
cana-610	197	26	q	q	NOUN
cana-610	197	27	of	of	ADP
cana-610	197	28	length	length	NOUN
cana-610	197	29	4	4	NUM
cana-610	197	30	=	=	SYM
cana-610	197	31	2|σmin(£)|	2|σmin(£)|	NUM
cana-610	197	32	between	between	ADP
cana-610	197	33	p	p	PROPN
cana-610	197	34	and	and	CCONJ
cana-610	197	35	p.	p.	NOUN
cana-610	197	36	case	case	NOUN
cana-610	197	37	ii	ii	X
cana-610	197	38	)	)	PUNCT
cana-610	197	39	suppose	suppose	VERB
cana-610	197	40	that	that	SCONJ
cana-610	197	41	r	r	NOUN
cana-610	197	42	>	>	X
cana-610	197	43	2	2	NUM
cana-610	197	44	.	.	PUNCT
cana-610	198	1	let	let	VERB
cana-610	198	2	p	p	PRON
cana-610	198	3	,	,	PUNCT
cana-610	198	4	q	q	PROPN
cana-610	198	5	∈	∈	PROPN
cana-610	198	6	σ(£	σ(£	PROPN
cana-610	198	7	)	)	PUNCT
cana-610	198	8	such	such	ADJ
cana-610	198	9	that	that	PRON
cana-610	198	10	for	for	ADP
cana-610	198	11	1	1	NUM
cana-610	198	12	≤	≤	NUM
cana-610	198	13	i	i	PROPN
cana-610	198	14	≤	≤	PROPN
cana-610	198	15	l1	l1	PROPN
cana-610	198	16	,	,	PUNCT
cana-610	198	17	p	p	PROPN
cana-610	198	18	∈	∈	PROPN
cana-610	198	19	ϑ(ai	ϑ(ai	NOUN
cana-610	198	20	)	)	PUNCT
cana-610	198	21	,	,	PUNCT
cana-610	198	22	for	for	ADP
cana-610	198	23	l1	l1	PROPN
cana-610	198	24	≤	≤	PROPN
cana-610	198	25	i	i	PRON
cana-610	198	26	≤	≤	NUM
cana-610	198	27	l2	l2	NOUN
cana-610	198	28	,	,	PUNCT
cana-610	198	29	q	q	PROPN
cana-610	198	30	∈	∈	PROPN
cana-610	198	31	ϑ(ai	ϑ(ai	NOUN
cana-610	198	32	)	)	PUNCT
cana-610	198	33	and	and	CCONJ
cana-610	198	34	for	for	ADP
cana-610	198	35	l2	l2	NOUN
cana-610	198	36	≤	≤	NUM
cana-610	198	37	i	i	PRON
cana-610	198	38	≤	≤	ADJ
cana-610	198	39	r	r	NOUN
cana-610	198	40	,	,	PUNCT
cana-610	198	41	p	p	X
cana-610	198	42	,	,	PUNCT
cana-610	198	43	q	q	PROPN
cana-610	198	44	∉	∉	PROPN
cana-610	198	45	ϑ(ai	ϑ(ai	PROPN
cana-610	198	46	)	)	PUNCT
cana-610	198	47	.	.	PUNCT
cana-610	199	1	but	but	CCONJ
cana-610	199	2	σ(£	σ(£	PROPN
cana-610	199	3	)	)	PUNCT
cana-610	199	4	is	be	AUX
cana-610	199	5	connected	connect	VERB
cana-610	199	6	,	,	PUNCT
cana-610	199	7	therefore	therefore	ADV
cana-610	199	8	there	there	PRON
cana-610	199	9	exist	exist	VERB
cana-610	199	10	1	1	NUM
cana-610	199	11	≤	≤	PROPN
cana-610	199	12	i1	i1	PROPN
cana-610	199	13	≤	≤	PROPN
cana-610	200	1	•••	•••	ADV
cana-610	200	2	≤	≤	NUM
cana-610	200	3	ik	ik	PROPN
cana-610	200	4	≤	≤	PROPN
cana-610	200	5	r	r	NOUN
cana-610	200	6	with	with	ADP
cana-610	200	7	ϑ(𝑎𝑖𝑙	ϑ(𝑎𝑖𝑙	PROPN
cana-610	200	8	)	)	PUNCT
cana-610	200	9	∩	∩	NOUN
cana-610	200	10	ϑ(𝑎𝑖𝑙+1	ϑ(𝑎𝑖𝑙+1	NOUN
cana-610	200	11	)	)	PUNCT
cana-610	200	12	≠	≠	PROPN
cana-610	200	13	∅	∅	NOUN
cana-610	200	14	for	for	ADP
cana-610	200	15	1	1	NUM
cana-610	200	16	≤	≤	NUM
cana-610	200	17	l	l	NOUN
cana-610	200	18	≤	≤	X
cana-610	200	19	k	k	PROPN
cana-610	200	20	and	and	CCONJ
cana-610	200	21	ϑ(∨𝑖=1	ϑ(∨𝑖=1	PROPN
cana-610	200	22	𝑙1	𝑙1	PROPN
cana-610	200	23	𝑎𝑖	𝑎𝑖	PART
cana-610	200	24	)	)	PUNCT
cana-610	200	25	∩	∩	ADJ
cana-610	200	26	ϑ(𝑎𝑖1	ϑ(𝑎𝑖1	X
cana-610	200	27	)	)	PUNCT
cana-610	200	28	≠	≠	PROPN
cana-610	200	29	∅	∅	NOUN
cana-610	200	30	,	,	PUNCT
cana-610	200	31	ϑ(∨𝑖=𝑙1	ϑ(∨𝑖=𝑙1	PROPN
cana-610	200	32	+	+	PROPN
cana-610	200	33	1	1	NUM
cana-610	200	34	𝑙2	𝑙2	PROPN
cana-610	200	35	𝑎𝑖	𝑎𝑖	NOUN
cana-610	200	36	)	)	PUNCT
cana-610	200	37	∩	∩	NOUN
cana-610	200	38	ϑ(𝑎𝑖𝑘	ϑ(𝑎𝑖𝑘	NOUN
cana-610	200	39	)	)	PUNCT
cana-610	200	40	≠	≠	PROPN
cana-610	200	41	∅.	∅.	ADP
cana-610	200	42	this	this	PRON
cana-610	200	43	implies	imply	VERB
cana-610	200	44	that	that	SCONJ
cana-610	200	45	there	there	PRON
cana-610	200	46	is	be	VERB
cana-610	200	47	a	a	DET
cana-610	200	48	path	path	NOUN
cana-610	200	49	between	between	ADP
cana-610	200	50	p	p	NOUN
cana-610	200	51	and	and	CCONJ
cana-610	200	52	q	q	NOUN
cana-610	200	53	with	with	ADP
cana-610	200	54	length	length	NOUN
cana-610	200	55	at	at	ADP
cana-610	200	56	most	most	ADJ
cana-610	200	57	2|σmin(£)|	2|σmin(£)|	NUM
cana-610	200	58	.	.	PUNCT
cana-610	201	1	consequently	consequently	ADV
cana-610	201	2	,	,	PUNCT
cana-610	201	3	the	the	DET
cana-610	201	4	graph	graph	NOUN
cana-610	201	5	γs(σ(£	γs(σ(£	NOUN
cana-610	201	6	)	)	PUNCT
cana-610	201	7	)	)	PUNCT
cana-610	201	8	of	of	ADP
cana-610	201	9	£	£	SYM
cana-610	201	10	connected	connect	VERB
cana-610	201	11	and	and	CCONJ
cana-610	201	12	diam(γs(σ(£	diam(γs(σ(£	ADJ
cana-610	201	13	)	)	PUNCT
cana-610	201	14	)	)	PUNCT
cana-610	201	15	)	)	PUNCT
cana-610	202	1	≤	≤	NUM
cana-610	202	2	2|σmin(£)|	2|σmin(£)|	NUM
cana-610	202	3	.	.	PUNCT
cana-610	203	1	references	reference	NOUN
cana-610	203	2	[	[	X
cana-610	203	3	1	1	X
cana-610	203	4	]	]	PUNCT
cana-610	203	5	s.	s.	PROPN
cana-610	203	6	akbari	akbari	PROPN
cana-610	203	7	and	and	CCONJ
cana-610	203	8	a.	a.	NOUN
cana-610	203	9	mohammadian	mohammadian	NOUN
cana-610	203	10	,	,	PUNCT
cana-610	203	11	on	on	ADP
cana-610	203	12	the	the	DET
cana-610	203	13	zero	zero	NUM
cana-610	203	14	-	-	PUNCT
cana-610	203	15	divisor	divisor	NOUN
cana-610	203	16	graph	graph	NOUN
cana-610	203	17	of	of	ADP
cana-610	203	18	a	a	DET
cana-610	203	19	commutative	commutative	ADJ
cana-610	203	20	ring	ring	NOUN
cana-610	203	21	,	,	PUNCT
cana-610	203	22	j.	j.	PROPN
cana-610	203	23	algebra	algebra	PROPN
cana-610	203	24	,	,	PUNCT
cana-610	203	25	274(2)(2004),847	274(2)(2004),847	PROPN
cana-610	203	26	-	-	PUNCT
cana-610	203	27	855	855	NUM
cana-610	203	28	.	.	PUNCT
cana-610	204	1	[	[	X
cana-610	204	2	2	2	X
cana-610	204	3	]	]	PUNCT
cana-610	204	4	d.	d.	PROPN
cana-610	204	5	f.	f.	PROPN
cana-610	204	6	anderson	anderson	PROPN
cana-610	204	7	,	,	PUNCT
cana-610	204	8	r.	r.	PROPN
cana-610	204	9	levy	levy	PROPN
cana-610	204	10	and	and	CCONJ
cana-610	204	11	j.	j.	PROPN
cana-610	204	12	shapiro	shapiro	PROPN
cana-610	204	13	,	,	PUNCT
cana-610	204	14	zero	zero	NUM
cana-610	204	15	-	-	PUNCT
cana-610	204	16	divisor	divisor	NOUN
cana-610	204	17	graphs	graph	NOUN
cana-610	204	18	,	,	PUNCT
cana-610	204	19	von	von	PROPN
cana-610	204	20	neumann	neumann	PROPN
cana-610	204	21	regular	regular	PROPN
cana-610	204	22	rings	ring	NOUN
cana-610	204	23	,	,	PUNCT
cana-610	204	24	and	and	CCONJ
cana-610	204	25	boolean	boolean	ADJ
cana-610	204	26	algebras	algebra	NOUN
cana-610	204	27	,	,	PUNCT
cana-610	204	28	j.pure	j.pure	NOUN
cana-610	204	29	appl	appl	NOUN
cana-610	204	30	.	.	PUNCT
cana-610	205	1	algebra	algebra	PROPN
cana-610	205	2	,	,	PUNCT
cana-610	205	3	180(2003	180(2003	NUM
cana-610	205	4	)	)	PUNCT
cana-610	205	5	,	,	PUNCT
cana-610	205	6	221	221	NUM
cana-610	205	7	-	-	SYM
cana-610	205	8	241	241	NUM
cana-610	205	9	.	.	PUNCT
cana-610	206	1	[	[	X
cana-610	206	2	3	3	X
cana-610	206	3	]	]	X
cana-610	206	4	f.f	f.f	PROPN
cana-610	206	5	.	.	PROPN
cana-610	206	6	anderson	anderson	PROPN
cana-610	206	7	and	and	CCONJ
cana-610	206	8	m.	m.	PROPN
cana-610	206	9	naseer	naseer	PROPN
cana-610	206	10	,	,	PUNCT
cana-610	206	11	beck	beck	PROPN
cana-610	206	12	’s	’s	PART
cana-610	206	13	coloring	coloring	NOUN
cana-610	206	14	of	of	ADP
cana-610	206	15	a	a	DET
cana-610	206	16	commutative	commutative	ADJ
cana-610	206	17	ring	ring	NOUN
cana-610	206	18	,	,	PUNCT
cana-610	206	19	j.	j.	PROPN
cana-610	206	20	algebra	algebra	PROPN
cana-610	206	21	,	,	PUNCT
cana-610	206	22	159(1993	159(1993	NUM
cana-610	206	23	)	)	PUNCT
cana-610	206	24	,	,	PUNCT
cana-610	206	25	500	500	NUM
cana-610	206	26	-	-	SYM
cana-610	206	27	514	514	NUM
cana-610	206	28	.	.	PUNCT
cana-610	207	1	[	[	X
cana-610	207	2	4	4	X
cana-610	207	3	]	]	X
cana-610	207	4	d.	d.	PROPN
cana-610	207	5	f.	f.	PROPN
cana-610	207	6	anderson	anderson	PROPN
cana-610	207	7	and	and	CCONJ
cana-610	207	8	p.	p.	PROPN
cana-610	207	9	s.	s.	PROPN
cana-610	207	10	livingston	livingston	PROPN
cana-610	207	11	,	,	PUNCT
cana-610	207	12	the	the	DET
cana-610	207	13	zero	zero	NUM
cana-610	207	14	-	-	PUNCT
cana-610	207	15	divisor	divisor	NOUN
cana-610	207	16	graph	graph	NOUN
cana-610	207	17	of	of	ADP
cana-610	207	18	a	a	DET
cana-610	207	19	commutative	commutative	ADJ
cana-610	207	20	ring	ring	NOUN
cana-610	207	21	,	,	PUNCT
cana-610	207	22	j.	j.	PROPN
cana-610	207	23	algebra	algebra	PROPN
cana-610	207	24	,	,	PUNCT
cana-610	207	25	217(2)(1999	217(2)(1999	NUM
cana-610	207	26	)	)	PUNCT
cana-610	207	27	,	,	PUNCT
cana-610	207	28	434	434	NUM
cana-610	207	29	-	-	SYM
cana-610	207	30	447	447	NUM
cana-610	207	31	.	.	PUNCT
cana-610	208	1	[	[	X
cana-610	208	2	5	5	NUM
cana-610	208	3	]	]	PUNCT
cana-610	208	4	i.	i.	PROPN
cana-610	208	5	beck	beck	PROPN
cana-610	208	6	,	,	PUNCT
cana-610	208	7	coloring	coloring	NOUN
cana-610	208	8	of	of	ADP
cana-610	208	9	commutative	commutative	ADJ
cana-610	208	10	rings	ring	NOUN
cana-610	208	11	,	,	PUNCT
cana-610	208	12	j.	j.	PROPN
cana-610	208	13	algebra	algebra	PROPN
cana-610	208	14	,	,	PUNCT
cana-610	208	15	116(1988	116(1988	NUM
cana-610	208	16	)	)	PUNCT
cana-610	208	17	,	,	PUNCT
cana-610	208	18	208	208	NUM
cana-610	208	19	-	-	SYM
cana-610	208	20	226	226	NUM
cana-610	208	21	.	.	PUNCT
cana-610	209	1	[	[	X
cana-610	209	2	6	6	NUM
cana-610	209	3	]	]	PUNCT
cana-610	209	4	m.	m.	NOUN
cana-610	209	5	behboodi	behboodi	NOUN
cana-610	209	6	and	and	CCONJ
cana-610	209	7	z.	z.	PROPN
cana-610	209	8	rakeei	rakeei	PROPN
cana-610	209	9	z	z	PROPN
cana-610	209	10	,	,	PUNCT
cana-610	209	11	the	the	DET
cana-610	209	12	annihilating	annihilate	VERB
cana-610	209	13	-	-	PUNCT
cana-610	209	14	ideal	ideal	ADJ
cana-610	209	15	graph	graph	NOUN
cana-610	209	16	of	of	ADP
cana-610	209	17	commutative	commutative	ADJ
cana-610	209	18	rings	ring	NOUN
cana-610	209	19	-	-	PUNCT
cana-610	209	20	i	i	PROPN
cana-610	209	21	,	,	PUNCT
cana-610	209	22	j.	j.	PROPN
cana-610	209	23	algebra	algebra	PROPN
cana-610	209	24	appl	appl	PROPN
cana-610	209	25	.	.	PROPN
cana-610	209	26	,	,	PUNCT
cana-610	209	27	10(4	10(4	NUM
cana-610	209	28	)	)	PUNCT
cana-610	209	29	(	(	PUNCT
cana-610	209	30	2011	2011	NUM
cana-610	209	31	)	)	PUNCT
cana-610	209	32	,	,	PUNCT
cana-610	209	33	727	727	NUM
cana-610	209	34	-	-	SYM
cana-610	209	35	739	739	NUM
cana-610	209	36	.	.	PUNCT
cana-610	210	1	[	[	X
cana-610	210	2	7	7	X
cana-610	210	3	]	]	PUNCT
cana-610	210	4	m.	m.	NOUN
cana-610	210	5	behboodi	behboodi	NOUN
cana-610	210	6	and	and	CCONJ
cana-610	210	7	z.	z.	PROPN
cana-610	210	8	rakeei	rakeei	PROPN
cana-610	210	9	z	z	PROPN
cana-610	210	10	,	,	PUNCT
cana-610	210	11	the	the	DET
cana-610	210	12	annihilating	annihilate	VERB
cana-610	210	13	-	-	PUNCT
cana-610	210	14	ideal	ideal	ADJ
cana-610	210	15	graph	graph	NOUN
cana-610	210	16	of	of	ADP
cana-610	210	17	commutative	commutative	ADJ
cana-610	210	18	rings	ring	NOUN
cana-610	210	19	-	-	PUNCT
cana-610	210	20	ii	ii	PROPN
cana-610	210	21	,	,	PUNCT
cana-610	210	22	j.	j.	PROPN
cana-610	210	23	algebra	algebra	PROPN
cana-610	210	24	appl	appl	PROPN
cana-610	210	25	.	.	PROPN
cana-610	210	26	,	,	PUNCT
cana-610	210	27	10(4	10(4	NUM
cana-610	210	28	)	)	PUNCT
cana-610	210	29	(	(	PUNCT
cana-610	210	30	2011	2011	NUM
cana-610	210	31	)	)	PUNCT
cana-610	210	32	,	,	PUNCT
cana-610	210	33	741	741	NUM
cana-610	210	34	-	-	SYM
cana-610	210	35	753	753	NUM
cana-610	210	36	.	.	PUNCT
cana-610	211	1	[	[	X
cana-610	211	2	8	8	NUM
cana-610	211	3	]	]	X
cana-610	211	4	f.	f.	PROPN
cana-610	211	5	callialp	callialp	PROPN
cana-610	211	6	,	,	PUNCT
cana-610	211	7	g.ulucak	g.ulucak	ADJ
cana-610	211	8	and	and	CCONJ
cana-610	211	9	u.tekir	u.tekir	NOUN
cana-610	211	10	,	,	PUNCT
cana-610	211	11	on	on	ADP
cana-610	211	12	the	the	DET
cana-610	211	13	zariski	zariski	NOUN
cana-610	211	14	topology	topology	NOUN
cana-610	211	15	over	over	ADP
cana-610	211	16	an	an	DET
cana-610	211	17	l	l	NOUN
cana-610	211	18	-	-	NOUN
cana-610	211	19	module	module	NOUN
cana-610	211	20	m	m	PROPN
cana-610	211	21	,	,	PUNCT
cana-610	211	22	turk	turk	PROPN
cana-610	211	23	.	.	PUNCT
cana-610	212	1	j.	j.	PROPN
cana-610	212	2	math	math	PROPN
cana-610	212	3	.	.	PUNCT
cana-610	212	4	,	,	PUNCT
cana-610	212	5	communications	communication	NOUN
cana-610	212	6	on	on	ADP
cana-610	212	7	applied	apply	VERB
cana-610	212	8	nonlinear	nonlinear	ADJ
cana-610	212	9	analysis	analysis	NOUN
cana-610	212	10	issn	issn	NOUN
cana-610	212	11	:	:	PUNCT
cana-610	212	12	1074	1074	NUM
cana-610	212	13	-	-	PUNCT
cana-610	212	14	133x	133x	NUM
cana-610	212	15	vol	vol	NOUN
cana-610	212	16	31	31	NUM
cana-610	212	17	no	no	NOUN
cana-610	212	18	.	.	NOUN
cana-610	212	19	2	2	NUM
cana-610	212	20	(	(	PUNCT
cana-610	212	21	2024	2024	NUM
cana-610	212	22	)	)	PUNCT
cana-610	213	1	415	415	NUM
cana-610	213	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-610	213	3	41(2017	41(2017	NUM
cana-610	213	4	)	)	PUNCT
cana-610	213	5	,	,	PUNCT
cana-610	213	6	326	326	NUM
cana-610	213	7	-	-	SYM
cana-610	213	8	336	336	NUM
cana-610	213	9	.	.	PUNCT
cana-610	214	1	[	[	X
cana-610	214	2	9	9	NUM
cana-610	214	3	]	]	X
cana-610	214	4	g.	g.	PROPN
cana-610	214	5	gandal	gandal	PROPN
cana-610	214	6	,	,	PUNCT
cana-610	214	7	r.	r.	PROPN
cana-610	214	8	mary	mary	PROPN
cana-610	214	9	jothi	jothi	PROPN
cana-610	214	10	and	and	CCONJ
cana-610	214	11	n.	n.	PROPN
cana-610	214	12	phadatare	phadatare	NOUN
cana-610	214	13	,	,	PUNCT
cana-610	214	14	residual	residual	ADJ
cana-610	214	15	division	division	NOUN
cana-610	214	16	graph	graph	NOUN
cana-610	214	17	of	of	ADP
cana-610	214	18	lattice	lattice	NOUN
cana-610	214	19	modules	module	NOUN
cana-610	214	20	,	,	PUNCT
cana-610	214	21	j.	j.	PROPN
cana-610	214	22	maths	maths	PROPN
cana-610	214	23	.	.	PROPN
cana-610	214	24	,	,	PUNCT
cana-610	214	25	2022(2022	2022(2022	NUM
cana-610	214	26	)	)	PUNCT
cana-610	214	27	,	,	PUNCT
cana-610	214	28	article	article	NOUN
cana-610	214	29	i	i	PROPN
cana-610	214	30	d	d	PROPN
cana-610	214	31	2892841	2892841	NUM
cana-610	214	32	,	,	PUNCT
cana-610	214	33	6	6	NUM
cana-610	214	34	pages	page	NOUN
cana-610	214	35	.	.	PUNCT
cana-610	215	1	https://doi.org/10.1155/2022/2892841	https://doi.org/10.1155/2022/2892841	PROPN
cana-610	215	2	.	.	PUNCT
cana-610	216	1	[	[	X
cana-610	216	2	10	10	NUM
cana-610	216	3	]	]	X
cana-610	216	4	g.	g.	PROPN
cana-610	216	5	gandal	gandal	PROPN
cana-610	216	6	,	,	PUNCT
cana-610	216	7	r.	r.	PROPN
cana-610	216	8	mary	mary	PROPN
cana-610	216	9	jothi	jothi	PROPN
cana-610	216	10	and	and	CCONJ
cana-610	216	11	n.	n.	PROPN
cana-610	216	12	phadatare	phadatare	NOUN
cana-610	216	13	,	,	PUNCT
cana-610	216	14	on	on	ADP
cana-610	216	15	very	very	ADV
cana-610	216	16	strongly	strongly	ADV
cana-610	216	17	perfect	perfect	ADJ
cana-610	216	18	cartesian	cartesian	ADJ
cana-610	216	19	product	product	NOUN
cana-610	216	20	graphs	graph	NOUN
cana-610	216	21	,	,	PUNCT
cana-610	216	22	aims	aim	VERB
cana-610	216	23	maths	math	NOUN
cana-610	216	24	,	,	PUNCT
cana-610	216	25	7(2)(2021	7(2)(2021	NUM
cana-610	216	26	)	)	PUNCT
cana-610	216	27	,	,	PUNCT
cana-610	216	28	2634–2645	2634–2645	NUM
cana-610	216	29	.	.	PUNCT
cana-610	217	1	doi	doi	NOUN
cana-610	217	2	:	:	PUNCT
cana-610	217	3	10.3934	10.3934	NUM
cana-610	217	4	/	/	SYM
cana-610	217	5	math.2022148	math.2022148	NOUN
cana-610	217	6	.	.	PUNCT
cana-610	218	1	[	[	X
cana-610	218	2	11	11	NUM
cana-610	218	3	]	]	X
cana-610	218	4	f.	f.	PROPN
cana-610	218	5	harary	harary	PROPN
cana-610	218	6	,	,	PUNCT
cana-610	218	7	graph	graph	NOUN
cana-610	218	8	theory	theory	NOUN
cana-610	218	9	,	,	PUNCT
cana-610	218	10	narosa	narosa	PROPN
cana-610	218	11	,	,	PUNCT
cana-610	218	12	new	new	ADJ
cana-610	218	13	delhi	delhi	PROPN
cana-610	218	14	(	(	PUNCT
cana-610	218	15	1988	1988	NUM
cana-610	218	16	)	)	PUNCT
cana-610	218	17	.	.	PUNCT
cana-610	219	1	[	[	X
cana-610	219	2	12	12	NUM
cana-610	219	3	]	]	X
cana-610	219	4	n.	n.	NOUN
cana-610	219	5	phadatare	phadatare	NOUN
cana-610	219	6	,	,	PUNCT
cana-610	219	7	v.	v.	CCONJ
cana-610	219	8	kharat	kharat	PROPN
cana-610	219	9	and	and	CCONJ
cana-610	219	10	s.	s.	PROPN
cana-610	219	11	ballal	ballal	PROPN
cana-610	219	12	,	,	PUNCT
cana-610	219	13	semi	semi	ADJ
cana-610	219	14	-	-	ADJ
cana-610	219	15	complement	complement	ADJ
cana-610	219	16	graph	graph	NOUN
cana-610	219	17	of	of	ADP
cana-610	219	18	lattice	lattice	NOUN
cana-610	219	19	modules	module	NOUN
cana-610	219	20	,	,	PUNCT
cana-610	219	21	soft	soft	ADJ
cana-610	219	22	computing	computing	NOUN
cana-610	219	23	,	,	PUNCT
cana-610	219	24	23	23	NUM
cana-610	219	25	(	(	PUNCT
cana-610	219	26	2019	2019	NUM
cana-610	219	27	)	)	PUNCT
cana-610	219	28	,	,	PUNCT
cana-610	219	29	3973	3973	NUM
cana-610	219	30	-	-	SYM
cana-610	219	31	3978	3978	NUM
cana-610	219	32	.	.	PUNCT
cana-610	220	1	[	[	X
cana-610	220	2	13	13	NUM
cana-610	220	3	]	]	X
cana-610	220	4	n.	n.	PROPN
cana-610	220	5	k.	k.	PROPN
cana-610	220	6	thakare	thakare	PROPN
cana-610	220	7	,	,	PUNCT
cana-610	220	8	c.	c.	PROPN
cana-610	220	9	s.	s.	PROPN
cana-610	220	10	manjarekar	manjarekar	PROPN
cana-610	220	11	and	and	CCONJ
cana-610	220	12	s.	s.	PROPN
cana-610	220	13	maeda	maeda	PROPN
cana-610	220	14	,	,	PUNCT
cana-610	220	15	abstract	abstract	ADJ
cana-610	220	16	spectral	spectral	ADJ
cana-610	220	17	theory	theory	NOUN
cana-610	220	18	.	.	PUNCT
cana-610	221	1	ii	ii	PROPN
cana-610	221	2	:	:	PUNCT
cana-610	221	3	minimal	minimal	ADJ
cana-610	221	4	characters	character	NOUN
cana-610	221	5	and	and	CCONJ
cana-610	221	6	minimal	minimal	ADJ
cana-610	221	7	spectrums	spectrum	NOUN
cana-610	221	8	of	of	ADP
cana-610	221	9	multiplicative	multiplicative	ADJ
cana-610	221	10	lattices	lattice	NOUN
cana-610	221	11	,	,	PUNCT
cana-610	221	12	acta	acta	PROPN
cana-610	221	13	sci	sci	PROPN
cana-610	221	14	.	.	PROPN
cana-610	221	15	math	math	PROPN
cana-610	221	16	.	.	PUNCT
cana-610	221	17	,	,	PUNCT
cana-610	221	18	52(1988	52(1988	NUM
cana-610	221	19	)	)	PUNCT
cana-610	221	20	,	,	PUNCT
cana-610	221	21	53	53	NUM
cana-610	221	22	-	-	SYM
cana-610	221	23	67	67	NUM
cana-610	221	24	.	.	PUNCT
cana-610	222	1	[	[	X
cana-610	222	2	14	14	NUM
cana-610	222	3	]	]	SYM
cana-610	222	4	d.b	d.b	PROPN
cana-610	222	5	.	.	PROPN
cana-610	222	6	west	west	PROPN
cana-610	222	7	,	,	PUNCT
cana-610	222	8	introduction	introduction	NOUN
cana-610	222	9	to	to	AUX
cana-610	222	10	graph	graph	NOUN
cana-610	222	11	theory	theory	NOUN
cana-610	222	12	,	,	PUNCT
cana-610	222	13	second	second	ADJ
cana-610	222	14	ed	ed	NOUN
cana-610	222	15	.	.	PROPN
cana-610	222	16	,	,	PUNCT
cana-610	222	17	prentice	prentice	NOUN
cana-610	222	18	-	-	PUNCT
cana-610	222	19	hall	hall	NOUN
cana-610	222	20	of	of	ADP
cana-610	222	21	india	india	PROPN
cana-610	222	22	,	,	PUNCT
cana-610	222	23	new	new	PROPN
cana-610	222	24	delhi	delhi	PROPN
cana-610	222	25	,	,	PUNCT
cana-610	222	26	2002	2002	NUM
cana-610	222	27	.	.	PUNCT