id	sid	tid	token	lemma	pos
cana-408	1	1	communications	communication	NOUN
cana-408	1	2	on	on	ADP
cana-408	1	3	applied	apply	VERB
cana-408	1	4	nonlinear	nonlinear	ADJ
cana-408	1	5	analysis	analysis	NOUN
cana-408	1	6	issn	issn	NOUN
cana-408	1	7	:	:	PUNCT
cana-408	1	8	1074	1074	NUM
cana-408	1	9	-	-	PUNCT
cana-408	1	10	133x	133x	NUM
cana-408	1	11	vol	vol	NOUN
cana-408	1	12	31	31	NUM
cana-408	1	13	no	no	NOUN
cana-408	1	14	.	.	NOUN
cana-408	1	15	1	1	NUM
cana-408	1	16	(	(	PUNCT
cana-408	1	17	2024	2024	NUM
cana-408	1	18	)	)	PUNCT
cana-408	1	19	231	231	NUM
cana-408	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-408	1	21	product	product	NOUN
cana-408	1	22	of	of	ADP
cana-408	1	23	semi	semi	ADJ
cana-408	1	24	–	–	PUNCT
cana-408	1	25	lattices	lattice	NOUN
cana-408	1	26	of	of	ADP
cana-408	1	27	certain	certain	ADJ
cana-408	1	28	graphs	graph	NOUN
cana-408	1	29	raghupatruni	raghupatruni	PROPN
cana-408	1	30	sunil	sunil	PROPN
cana-408	1	31	kumar1	kumar1	PROPN
cana-408	1	32	,	,	PUNCT
cana-408	1	33	v.	v.	PROPN
cana-408	1	34	b.	b.	PROPN
cana-408	2	1	v.	v.	ADP
cana-408	2	2	n.	n.	PROPN
cana-408	2	3	prasad2	prasad2	PROPN
cana-408	2	4	,	,	PUNCT
cana-408	2	5	chudamani	chudamani	PROPN
cana-408	2	6	.	.	PUNCT
cana-408	3	1	r3	r3	PROPN
cana-408	3	2	,	,	PUNCT
cana-408	3	3	mudda	mudda	NOUN
cana-408	3	4	ramesh4	ramesh4	PROPN
cana-408	3	5	1research	1research	NUM
cana-408	3	6	scholar	scholar	NOUN
cana-408	3	7	,	,	PUNCT
cana-408	3	8	1,2department	1,2department	NUM
cana-408	3	9	of	of	ADP
cana-408	3	10	engineering	engineering	NOUN
cana-408	3	11	mathematics	mathematic	NOUN
cana-408	3	12	,	,	PUNCT
cana-408	3	13	koneru	koneru	PROPN
cana-408	3	14	lakshmaiah	lakshmaiah	PROPN
cana-408	3	15	education	education	PROPN
cana-408	3	16	foundation	foundation	PROPN
cana-408	3	17	,	,	PUNCT
cana-408	3	18	green	green	ADJ
cana-408	3	19	fields	field	NOUN
cana-408	3	20	,	,	PUNCT
cana-408	3	21	vaddeswaram	vaddeswaram	PROPN
cana-408	3	22	,	,	PUNCT
cana-408	3	23	a.p	a.p	PROPN
cana-408	3	24	.	.	PROPN
cana-408	3	25	,	,	PUNCT
cana-408	3	26	india-522302	india-522302	ADJ
cana-408	3	27	.	.	PUNCT
cana-408	4	1	3department	3department	NUM
cana-408	4	2	of	of	ADP
cana-408	4	3	mathematics	mathematic	NOUN
cana-408	4	4	,	,	PUNCT
cana-408	4	5	prasad	prasad	PROPN
cana-408	4	6	v	v	NUM
cana-408	4	7	potluri	potluri	PROPN
cana-408	4	8	siddhartha	siddhartha	PROPN
cana-408	4	9	institute	institute	PROPN
cana-408	4	10	of	of	ADP
cana-408	4	11	technology	technology	PROPN
cana-408	4	12	,	,	PUNCT
cana-408	4	13	vijayawada	vijayawada	PROPN
cana-408	4	14	,	,	PUNCT
cana-408	4	15	a.p	a.p	PROPN
cana-408	4	16	.	.	PROPN
cana-408	4	17	,	,	PUNCT
cana-408	4	18	india-520007	india-520007	ADJ
cana-408	4	19	.	.	PUNCT
cana-408	5	1	4department	4department	NUM
cana-408	5	2	of	of	ADP
cana-408	5	3	mathematics	mathematic	NOUN
cana-408	5	4	,	,	PUNCT
cana-408	5	5	s.	s.	PROPN
cana-408	5	6	s	s	PROPN
cana-408	5	7	&	&	CCONJ
cana-408	5	8	n	n	PROPN
cana-408	5	9	college	college	PROPN
cana-408	5	10	,	,	PUNCT
cana-408	5	11	narasaraopet	narasaraopet	PROPN
cana-408	5	12	,	,	PUNCT
cana-408	5	13	guntur	guntur	PROPN
cana-408	5	14	,	,	PUNCT
cana-408	5	15	a.p	a.p	PROPN
cana-408	5	16	.	.	PROPN
cana-408	5	17	,	,	PUNCT
cana-408	5	18	india-522601	india-522601	PROPN
cana-408	5	19	.	.	PUNCT
cana-408	5	20	article	article	NOUN
cana-408	5	21	history	history	NOUN
cana-408	5	22	:	:	PUNCT
cana-408	5	23	received	receive	VERB
cana-408	5	24	:	:	PUNCT
cana-408	5	25	22	22	NUM
cana-408	5	26	-	-	SYM
cana-408	5	27	10	10	NUM
cana-408	5	28	-	-	PUNCT
cana-408	5	29	2023	2023	NUM
cana-408	5	30	revised	revise	VERB
cana-408	5	31	:	:	PUNCT
cana-408	5	32	10	10	NUM
cana-408	5	33	-	-	SYM
cana-408	5	34	12	12	NUM
cana-408	5	35	-	-	PUNCT
cana-408	5	36	2023	2023	NUM
cana-408	5	37	accepted	accept	VERB
cana-408	5	38	:	:	PUNCT
cana-408	5	39	20	20	NUM
cana-408	5	40	-	-	SYM
cana-408	5	41	12	12	NUM
cana-408	5	42	-	-	PUNCT
cana-408	5	43	2023	2023	NUM
cana-408	5	44	abstract	abstract	NOUN
cana-408	5	45	:	:	PUNCT
cana-408	5	46	introduction	introduction	NOUN
cana-408	5	47	:	:	PUNCT
cana-408	5	48	in	in	ADP
cana-408	5	49	this	this	DET
cana-408	5	50	article	article	NOUN
cana-408	5	51	,	,	PUNCT
cana-408	5	52	author	author	NOUN
cana-408	5	53	tries	try	VERB
cana-408	5	54	to	to	PART
cana-408	5	55	construct	construct	VERB
cana-408	5	56	a	a	DET
cana-408	5	57	relation	relation	NOUN
cana-408	5	58	between	between	ADP
cana-408	5	59	graphs	graph	NOUN
cana-408	5	60	of	of	ADP
cana-408	5	61	product	product	NOUN
cana-408	5	62	of	of	ADP
cana-408	5	63	meetsemilattices	meetsemilattice	NOUN
cana-408	5	64	l	l	NOUN
cana-408	5	65	=	=	SYM
cana-408	5	66	l_1	l_1	PROPN
cana-408	5	67	x	x	SYM
cana-408	5	68	l_2	l_2	PROPN
cana-408	5	69	,	,	PUNCT
cana-408	5	70	where	where	SCONJ
cana-408	5	71	l_1	l_1	PROPN
cana-408	5	72	and	and	CCONJ
cana-408	5	73	l_2	l_2	NOUN
cana-408	5	74	are	be	AUX
cana-408	5	75	two	two	NUM
cana-408	5	76	semilattices	semilattice	NOUN
cana-408	5	77	and	and	CCONJ
cana-408	5	78	obtain	obtain	VERB
cana-408	5	79	some	some	DET
cana-408	5	80	properties	property	NOUN
cana-408	5	81	of	of	ADP
cana-408	5	82	such	such	ADJ
cana-408	5	83	graphs	graph	NOUN
cana-408	5	84	.	.	PUNCT
cana-408	6	1	author	author	NOUN
cana-408	6	2	investigated	investigate	VERB
cana-408	6	3	that	that	SCONJ
cana-408	6	4	for	for	ADP
cana-408	6	5	meet	meet	NOUN
cana-408	6	6	-	-	PUNCT
cana-408	6	7	semilattices	semilattice	NOUN
cana-408	6	8	l_1	l_1	PROPN
cana-408	6	9	and	and	CCONJ
cana-408	6	10	l_2	l_2	PROPN
cana-408	6	11	has	have	VERB
cana-408	6	12	a	a	DET
cana-408	6	13	cycle	cycle	NOUN
cana-408	6	14	of	of	ADP
cana-408	6	15	length	length	NOUN
cana-408	6	16	n-1	n-1	NOUN
cana-408	6	17	and	and	CCONJ
cana-408	6	18	n.	n.	NOUN
cana-408	6	19	objectives	objective	NOUN
cana-408	6	20	:	:	PUNCT
cana-408	6	21	author	author	NOUN
cana-408	6	22	reveals	reveal	VERB
cana-408	6	23	that	that	SCONJ
cana-408	6	24	if	if	SCONJ
cana-408	6	25	l_1	l_1	PROPN
cana-408	6	26	and	and	CCONJ
cana-408	6	27	l_2	l_2	VERB
cana-408	6	28	be	be	AUX
cana-408	6	29	two	two	NUM
cana-408	6	30	meet	meet	NOUN
cana-408	6	31	-	-	PUNCT
cana-408	6	32	semilattices	semilattice	NOUN
cana-408	6	33	with	with	ADP
cana-408	6	34	0	0	NUM
cana-408	6	35	and	and	CCONJ
cana-408	6	36	l	l	NOUN
cana-408	6	37	=	=	SYM
cana-408	6	38	l_1	l_1	PROPN
cana-408	6	39	x	x	SYM
cana-408	6	40	l_2	l_2	PROPN
cana-408	6	41	,	,	PUNCT
cana-408	6	42	then	then	ADV
cana-408	6	43	it	it	PRON
cana-408	6	44	is	be	AUX
cana-408	6	45	a	a	DET
cana-408	6	46	star	star	NOUN
cana-408	6	47	graph	graph	NOUN
cana-408	6	48	.	.	PUNCT
cana-408	7	1	in	in	ADP
cana-408	7	2	this	this	DET
cana-408	7	3	paper	paper	NOUN
cana-408	7	4	,	,	PUNCT
cana-408	7	5	we	we	PRON
cana-408	7	6	have	have	AUX
cana-408	7	7	covered	cover	VERB
cana-408	7	8	some	some	DET
cana-408	7	9	definitions	definition	NOUN
cana-408	7	10	,	,	PUNCT
cana-408	7	11	examples	example	NOUN
cana-408	7	12	and	and	CCONJ
cana-408	7	13	theorems	theorem	NOUN
cana-408	7	14	on	on	ADP
cana-408	7	15	zero	zero	NUM
cana-408	7	16	devisor	devisor	NOUN
cana-408	7	17	graph	graph	NOUN
cana-408	7	18	edge	edge	NOUN
cana-408	7	19	of	of	ADP
cana-408	7	20	a	a	DET
cana-408	7	21	4	4	NUM
cana-408	7	22	cycles	cycle	NOUN
cana-408	7	23	or	or	CCONJ
cana-408	7	24	a	a	DET
cana-408	7	25	5	5	NUM
cana-408	7	26	cycles	cycle	NOUN
cana-408	7	27	.	.	PUNCT
cana-408	8	1	γ(l	γ(l	NOUN
cana-408	8	2	)	)	PUNCT
cana-408	8	3	is	be	AUX
cana-408	8	4	a	a	DET
cana-408	8	5	star	star	NOUN
cana-408	8	6	graph	graph	NOUN
cana-408	8	7	.	.	PUNCT
cana-408	9	1	methods	method	NOUN
cana-408	9	2	:	:	PUNCT
cana-408	9	3	theorem	theorem	VERB
cana-408	9	4	1.1	1.1	NUM
cana-408	10	1	[	[	X
cana-408	10	2	9	9	NUM
cana-408	10	3	]	]	PUNCT
cana-408	10	4	the	the	DET
cana-408	10	5	zero	zero	NUM
cana-408	10	6	-	-	PUNCT
cana-408	10	7	divisor	divisor	NOUN
cana-408	10	8	graph	graph	NOUN
cana-408	10	9	of	of	ADP
cana-408	10	10	a	a	DET
cana-408	10	11	finite	finite	ADJ
cana-408	10	12	meet	meet	NOUN
cana-408	10	13	-	-	PUNCT
cana-408	10	14	semilattice	semilattice	NOUN
cana-408	10	15	with	with	ADP
cana-408	10	16	only	only	ADV
cana-408	10	17	one	one	NUM
cana-408	10	18	atom	atom	NOUN
cana-408	10	19	is	be	AUX
cana-408	10	20	the	the	DET
cana-408	10	21	empty	empty	ADJ
cana-408	10	22	graph	graph	NOUN
cana-408	10	23	.	.	PUNCT
cana-408	11	1	the	the	DET
cana-408	11	2	zero	zero	NUM
cana-408	11	3	-	-	PUNCT
cana-408	11	4	divisor	divisor	NOUN
cana-408	11	5	graph	graph	NOUN
cana-408	11	6	of	of	ADP
cana-408	11	7	the	the	DET
cana-408	11	8	meet	meet	NOUN
cana-408	11	9	semilattice	semilattice	NOUN
cana-408	11	10	is	be	AUX
cana-408	11	11	the	the	DET
cana-408	11	12	empty	empty	ADJ
cana-408	11	13	graph	graph	NOUN
cana-408	11	14	.	.	PUNCT
cana-408	12	1	however	however	ADV
cana-408	12	2	,	,	PUNCT
cana-408	12	3	this	this	PRON
cana-408	12	4	does	do	AUX
cana-408	12	5	not	not	PART
cana-408	12	6	hold	hold	VERB
cana-408	12	7	for	for	ADP
cana-408	12	8	infinite	infinite	ADJ
cana-408	12	9	meet	meet	NOUN
cana-408	12	10	-	-	PUNCT
cana-408	12	11	semilattices	semilattice	NOUN
cana-408	12	12	with	with	ADP
cana-408	12	13	one	one	NUM
cana-408	12	14	atom	atom	NOUN
cana-408	12	15	.	.	PUNCT
cana-408	13	1	for	for	ADP
cana-408	13	2	consider	consider	VERB
cana-408	13	3	,	,	PUNCT
cana-408	13	4	the	the	DET
cana-408	13	5	infinite	infinite	ADJ
cana-408	13	6	meetsemilattice	meetsemilattice	NOUN
cana-408	13	7	,	,	PUNCT
cana-408	13	8	where	where	SCONJ
cana-408	13	9	the	the	DET
cana-408	13	10	descending	descend	VERB
cana-408	13	11	dots	dot	NOUN
cana-408	13	12	represent	represent	VERB
cana-408	13	13	infinite	infinite	ADJ
cana-408	13	14	descending	descend	VERB
cana-408	13	15	chain	chain	NOUN
cana-408	13	16	.	.	PUNCT
cana-408	14	1	it	it	PRON
cana-408	14	2	has	have	VERB
cana-408	14	3	only	only	ADV
cana-408	14	4	one	one	NUM
cana-408	14	5	atom	atom	NOUN
cana-408	14	6	c	c	NOUN
cana-408	14	7	but	but	CCONJ
cana-408	14	8	its	its	PRON
cana-408	14	9	graph	graph	NOUN
cana-408	14	10	γ(l	γ(l	NOUN
cana-408	14	11	)	)	PUNCT
cana-408	14	12	is	be	AUX
cana-408	14	13	an	an	DET
cana-408	14	14	infinite	infinite	ADJ
cana-408	14	15	star	star	NOUN
cana-408	14	16	graph	graph	NOUN
cana-408	14	17	.	.	PUNCT
cana-408	15	1	theorem	theorem	VERB
cana-408	15	2	1.2	1.2	NUM
cana-408	15	3	[	[	X
cana-408	15	4	12	12	NUM
cana-408	15	5	]	]	PUNCT
cana-408	15	6	every	every	DET
cana-408	15	7	disconnected	disconnected	ADJ
cana-408	15	8	graph	graph	NOUN
cana-408	15	9	can	can	AUX
cana-408	15	10	not	not	PART
cana-408	15	11	be	be	AUX
cana-408	15	12	a	a	DET
cana-408	15	13	graph	graph	NOUN
cana-408	15	14	of	of	ADP
cana-408	15	15	any	any	DET
cana-408	15	16	meetsemilattice	meetsemilattice	NOUN
cana-408	15	17	l	l	NOUN
cana-408	15	18	with	with	ADP
cana-408	15	19	0	0	NUM
cana-408	15	20	.	.	PUNCT
cana-408	15	21	remark	remark	VERB
cana-408	15	22	1.2	1.2	NUM
cana-408	15	23	a	a	DET
cana-408	15	24	graphs	graph	NOUN
cana-408	15	25	of	of	ADP
cana-408	15	26	product	product	NOUN
cana-408	15	27	of	of	ADP
cana-408	15	28	meet	meet	NOUN
cana-408	15	29	-	-	PUNCT
cana-408	15	30	semilattices	semilattice	NOUN
cana-408	15	31	and	and	CCONJ
cana-408	15	32	obtain	obtain	VERB
cana-408	15	33	some	some	DET
cana-408	15	34	properties	property	NOUN
cana-408	15	35	of	of	ADP
cana-408	15	36	such	such	ADJ
cana-408	15	37	graphs	graph	NOUN
cana-408	15	38	.	.	PUNCT
cana-408	16	1	in	in	ADP
cana-408	16	2	this	this	DET
cana-408	16	3	section	section	NOUN
cana-408	16	4	,	,	PUNCT
cana-408	16	5	we	we	PRON
cana-408	16	6	consider	consider	VERB
cana-408	16	7	two	two	NUM
cana-408	16	8	integral	integral	ADJ
cana-408	16	9	meet	meet	NOUN
cana-408	16	10	-	-	PUNCT
cana-408	16	11	semilattices	semilattice	NOUN
cana-408	16	12	l_1	l_1	PROPN
cana-408	16	13	and	and	CCONJ
cana-408	16	14	l_2	l_2	VERB
cana-408	16	15	with	with	ADP
cana-408	16	16	l	l	NOUN
cana-408	16	17	∼=l_1	∼=l_1	X
cana-408	16	18	x	x	PUNCT
cana-408	16	19	l_2	l_2	VERB
cana-408	16	20	and	and	CCONJ
cana-408	16	21	show	show	VERB
cana-408	16	22	that	that	SCONJ
cana-408	16	23	if	if	SCONJ
cana-408	16	24	∣l_1∣	∣l_1∣	PUNCT
cana-408	16	25	=	=	SYM
cana-408	16	26	m+1	m+1	NUM
cana-408	16	27	,	,	PUNCT
cana-408	16	28	∣l_2∣	∣l_2∣	PUNCT
cana-408	16	29	=	=	SYM
cana-408	16	30	n+1	n+1	PROPN
cana-408	16	31	,	,	PUNCT
cana-408	16	32	then	then	ADV
cana-408	16	33	γ(l	γ(l	PROPN
cana-408	16	34	)	)	PUNCT
cana-408	16	35	is	be	AUX
cana-408	16	36	the	the	DET
cana-408	16	37	complete	complete	ADJ
cana-408	16	38	bipartite	bipartite	PROPN
cana-408	16	39	graph	graph	NOUN
cana-408	16	40	k	k	PROPN
cana-408	16	41	(	(	PUNCT
cana-408	16	42	m	m	PROPN
cana-408	16	43	,	,	PUNCT
cana-408	16	44	n	n	CCONJ
cana-408	16	45	)	)	PUNCT
cana-408	16	46	.	.	PUNCT
cana-408	17	1	results	result	NOUN
cana-408	17	2	:	:	PUNCT
cana-408	17	3	if	if	SCONJ
cana-408	17	4	l	l	NOUN
cana-408	17	5	does	do	AUX
cana-408	17	6	not	not	PART
cana-408	17	7	contain	contain	VERB
cana-408	17	8	any	any	DET
cana-408	17	9	atom	atom	NOUN
cana-408	17	10	,	,	PUNCT
cana-408	17	11	then	then	ADV
cana-408	17	12	any	any	DET
cana-408	17	13	edge	edge	NOUN
cana-408	17	14	in	in	ADP
cana-408	17	15	γ(l	γ(l	NOUN
cana-408	17	16	)	)	PUNCT
cana-408	17	17	is	be	AUX
cana-408	17	18	contained	contain	VERB
cana-408	17	19	in	in	ADP
cana-408	17	20	a	a	DET
cana-408	17	21	cycle	cycle	NOUN
cana-408	17	22	of	of	ADP
cana-408	17	23	length	length	NOUN
cana-408	17	24	≤	≤	NUM
cana-408	17	25	6	6	NUM
cana-408	17	26	,	,	PUNCT
cana-408	17	27	and	and	CCONJ
cana-408	17	28	therefore	therefore	ADV
cana-408	17	29	γ(l	γ(l	PROPN
cana-408	17	30	)	)	PUNCT
cana-408	17	31	is	be	AUX
cana-408	17	32	a	a	DET
cana-408	17	33	union	union	NOUN
cana-408	17	34	of	of	ADP
cana-408	17	35	4	4	NUM
cana-408	17	36	cycles	cycle	NOUN
cana-408	17	37	and	and	CCONJ
cana-408	17	38	5	5	NUM
cana-408	17	39	cycles	cycle	NOUN
cana-408	17	40	.	.	PUNCT
cana-408	18	1	let	let	VERB
cana-408	18	2	l	l	NOUN
cana-408	18	3	be	be	AUX
cana-408	18	4	a	a	DET
cana-408	18	5	meet	meet	NOUN
cana-408	18	6	-	-	PUNCT
cana-408	18	7	semilattice	semilattice	NOUN
cana-408	18	8	with	with	ADP
cana-408	18	9	0	0	NUM
cana-408	18	10	.	.	PUNCT
cana-408	19	1	if	if	SCONJ
cana-408	19	2	γ(l	γ(l	NOUN
cana-408	19	3	)	)	PUNCT
cana-408	19	4	contains	contain	VERB
cana-408	19	5	a	a	DET
cana-408	19	6	cycle	cycle	NOUN
cana-408	19	7	,	,	PUNCT
cana-408	19	8	then	then	ADV
cana-408	19	9	the	the	DET
cana-408	19	10	core	core	NOUN
cana-408	19	11	k	k	PROPN
cana-408	19	12	of	of	ADP
cana-408	19	13	γ(l	γ(l	PROPN
cana-408	19	14	)	)	PUNCT
cana-408	19	15	is	be	AUX
cana-408	19	16	a	a	DET
cana-408	19	17	union	union	NOUN
cana-408	19	18	of	of	ADP
cana-408	19	19	4cycles	4cycles	NUM
cana-408	19	20	and	and	CCONJ
cana-408	19	21	5	5	NUM
cana-408	19	22	–	–	PUNCT
cana-408	19	23	cycles	cycle	NOUN
cana-408	19	24	and	and	CCONJ
cana-408	19	25	any	any	DET
cana-408	19	26	vertex	vertex	NOUN
cana-408	19	27	in	in	ADP
cana-408	19	28	γ(l	γ(l	NOUN
cana-408	19	29	)	)	PUNCT
cana-408	19	30	is	be	AUX
cana-408	19	31	either	either	CCONJ
cana-408	19	32	a	a	DET
cana-408	19	33	vertex	vertex	NOUN
cana-408	19	34	of	of	ADP
cana-408	19	35	the	the	DET
cana-408	19	36	core	core	NOUN
cana-408	19	37	k	k	PROPN
cana-408	19	38	of	of	ADP
cana-408	19	39	γ(l	γ(l	PROPN
cana-408	19	40	)	)	PUNCT
cana-408	19	41	or	or	CCONJ
cana-408	19	42	is	be	AUX
cana-408	19	43	a	a	DET
cana-408	19	44	pendant	pendant	NOUN
cana-408	19	45	of	of	ADP
cana-408	19	46	γ(l	γ(l	NOUN
cana-408	19	47	)	)	PUNCT
cana-408	19	48	.	.	PUNCT
cana-408	20	1	let	let	VERB
cana-408	20	2	l_1	l_1	PROPN
cana-408	20	3	and	and	CCONJ
cana-408	20	4	l_2	l_2	AUX
cana-408	20	5	be	be	AUX
cana-408	20	6	two	two	NUM
cana-408	20	7	meet	meet	NOUN
cana-408	20	8	-	-	PUNCT
cana-408	20	9	semilattices	semilattice	NOUN
cana-408	20	10	with	with	ADP
cana-408	20	11	0	0	NUM
cana-408	20	12	and	and	CCONJ
cana-408	20	13	l	l	NOUN
cana-408	20	14	=	=	SYM
cana-408	20	15	l_1	l_1	PROPN
cana-408	20	16	x	x	SYM
cana-408	20	17	l_2	l_2	PROPN
cana-408	20	18	.	.	PUNCT
cana-408	21	1	then	then	ADV
cana-408	21	2	exactly	exactly	ADV
cana-408	21	3	one	one	NUM
cana-408	21	4	of	of	ADP
cana-408	21	5	the	the	DET
cana-408	21	6	following	follow	VERB
cana-408	21	7	holds	hold	VERB
cana-408	21	8	:	:	PUNCT
cana-408	21	9	1	1	X
cana-408	21	10	.	.	PUNCT
cana-408	21	11	γ(l	γ(l	NOUN
cana-408	21	12	)	)	PUNCT
cana-408	21	13	has	have	VERB
cana-408	21	14	a	a	DET
cana-408	21	15	cycle	cycle	NOUN
cana-408	21	16	of	of	ADP
cana-408	21	17	length	length	NOUN
cana-408	21	18	n-1	n-1	NOUN
cana-408	21	19	or	or	CCONJ
cana-408	21	20	n	n	PRON
cana-408	21	21	(	(	PUNCT
cana-408	21	22	that	that	PRON
cana-408	21	23	is	be	AUX
cana-408	21	24	gr	gr	DET
cana-408	21	25	γ(l)≤	γ(l)≤	NOUN
cana-408	21	26	n	n	CCONJ
cana-408	21	27	)	)	PUNCT
cana-408	21	28	.	.	PUNCT
cana-408	22	1	2	2	X
cana-408	22	2	.	.	X
cana-408	22	3	γ(l	γ(l	NOUN
cana-408	22	4	)	)	PUNCT
cana-408	22	5	is	be	AUX
cana-408	22	6	a	a	DET
cana-408	22	7	star	star	NOUN
cana-408	22	8	graph	graph	NOUN
cana-408	22	9	.	.	PUNCT
cana-408	23	1	conclusions	conclusion	NOUN
cana-408	23	2	:	:	PUNCT
cana-408	23	3	in	in	ADP
cana-408	23	4	this	this	DET
cana-408	23	5	article	article	NOUN
cana-408	23	6	we	we	PRON
cana-408	23	7	have	have	AUX
cana-408	23	8	studied	study	VERB
cana-408	23	9	the	the	DET
cana-408	23	10	concept	concept	NOUN
cana-408	23	11	of	of	ADP
cana-408	23	12	the	the	DET
cana-408	23	13	zero	zero	NUM
cana-408	23	14	-	-	PUNCT
cana-408	23	15	divisor	divisor	NOUN
cana-408	23	16	graph	graph	NOUN
cana-408	23	17	derived	derive	VERB
cana-408	23	18	from	from	ADP
cana-408	23	19	meet	meet	NOUN
cana-408	23	20	-	-	PUNCT
cana-408	23	21	semilattice	semilattice	NOUN
cana-408	23	22	l	l	NOUN
cana-408	23	23	with	with	ADP
cana-408	23	24	0	0	NUM
cana-408	23	25	on	on	ADP
cana-408	23	26	the	the	DET
cana-408	23	27	lines	line	NOUN
cana-408	23	28	of	of	ADP
cana-408	23	29	anderson	anderson	PROPN
cana-408	23	30	and	and	CCONJ
cana-408	23	31	livingston	livingston	PROPN
cana-408	24	1	[	[	X
cana-408	24	2	6	6	NUM
cana-408	24	3	]	]	PUNCT
cana-408	24	4	.	.	PUNCT
cana-408	25	1	also	also	ADV
cana-408	25	2	,	,	PUNCT
cana-408	25	3	we	we	PRON
cana-408	25	4	generalized	generalize	VERB
cana-408	25	5	certain	certain	ADJ
cana-408	25	6	results	result	NOUN
cana-408	25	7	from	from	ADP
cana-408	25	8	demeyer	demeyer	NOUN
cana-408	25	9	,	,	PUNCT
cana-408	25	10	mckenzie	mckenzie	NOUN
cana-408	25	11	and	and	CCONJ
cana-408	25	12	schneider	schneider	NOUN
cana-408	26	1	[	[	X
cana-408	26	2	18	18	NUM
cana-408	26	3	]	]	PUNCT
cana-408	26	4	to	to	PART
cana-408	26	5	meet	meet	VERB
cana-408	26	6	-	-	PUNCT
cana-408	26	7	semilattice	semilattice	NOUN
cana-408	26	8	l	l	NOUN
cana-408	26	9	with	with	ADP
cana-408	26	10	0	0	NUM
cana-408	26	11	.	.	PUNCT
cana-408	27	1	keywords	keyword	NOUN
cana-408	27	2	:	:	PUNCT
cana-408	27	3	lattices	lattice	NOUN
cana-408	27	4	,	,	PUNCT
cana-408	27	5	meet	meet	VERB
cana-408	27	6	semi	semi	ADJ
cana-408	27	7	lattice	lattice	PROPN
cana-408	27	8	,	,	PUNCT
cana-408	27	9	star	star	NOUN
cana-408	27	10	graph	graph	NOUN
cana-408	27	11	.	.	PUNCT
cana-408	28	1	1	1	X
cana-408	28	2	.	.	X
cana-408	28	3	introduction	introduction	NOUN
cana-408	28	4	graph	graph	NOUN
cana-408	28	5	theory	theory	NOUN
cana-408	28	6	is	be	AUX
cana-408	28	7	an	an	DET
cana-408	28	8	interesting	interesting	ADJ
cana-408	28	9	discipline	discipline	NOUN
cana-408	28	10	of	of	ADP
cana-408	28	11	human	human	ADJ
cana-408	28	12	enquiry	enquiry	NOUN
cana-408	28	13	.	.	PUNCT
cana-408	29	1	it	it	PRON
cana-408	29	2	has	have	VERB
cana-408	29	3	multiple	multiple	ADJ
cana-408	29	4	hundred	hundred	NUM
cana-408	29	5	significant	significant	ADJ
cana-408	29	6	subareas	subarea	NOUN
cana-408	29	7	.	.	PUNCT
cana-408	30	1	many	many	ADJ
cana-408	30	2	researchers	researcher	NOUN
cana-408	30	3	worked	work	VERB
cana-408	30	4	on	on	ADP
cana-408	30	5	two	two	NUM
cana-408	30	6	primary	primary	ADJ
cana-408	30	7	areas	area	NOUN
cana-408	30	8	of	of	ADP
cana-408	30	9	arithmetic	arithmetic	ADJ
cana-408	30	10	which	which	PRON
cana-408	30	11	are	be	AUX
cana-408	30	12	lattice	lattice	ADJ
cana-408	30	13	hypothesis	hypothesis	NOUN
cana-408	30	14	and	and	CCONJ
cana-408	30	15	theory	theory	NOUN
cana-408	30	16	hypothesis	hypothesis	NOUN
cana-408	30	17	.	.	PUNCT
cana-408	31	1	graph	graph	NOUN
cana-408	31	2	theory	theory	NOUN
cana-408	31	3	is	be	AUX
cana-408	31	4	a	a	DET
cana-408	31	5	thriving	thrive	VERB
cana-408	31	6	discipline	discipline	NOUN
cana-408	31	7	containing	contain	VERB
cana-408	31	8	a	a	DET
cana-408	31	9	group	group	NOUN
cana-408	31	10	of	of	ADP
cana-408	31	11	lovely	lovely	ADJ
cana-408	31	12	and	and	CCONJ
cana-408	31	13	strong	strong	ADJ
cana-408	31	14	hypotheses	hypothesis	NOUN
cana-408	31	15	of	of	ADP
cana-408	31	16	wide	wide	ADJ
cana-408	31	17	appropriateness	appropriateness	NOUN
cana-408	31	18	.	.	PUNCT
cana-408	32	1	its	its	PRON
cana-408	32	2	touchy	touchy	ADJ
cana-408	32	3	development	development	NOUN
cana-408	32	4	is	be	AUX
cana-408	32	5	fundamentally	fundamentally	ADV
cana-408	32	6	because	because	SCONJ
cana-408	32	7	of	of	ADP
cana-408	32	8	its	its	PRON
cana-408	32	9	job	job	NOUN
cana-408	32	10	as	as	ADP
cana-408	32	11	a	a	DET
cana-408	32	12	fundamental	fundamental	ADJ
cana-408	32	13	design	design	NOUN
cana-408	32	14	communications	communication	NOUN
cana-408	32	15	on	on	ADP
cana-408	32	16	applied	apply	VERB
cana-408	32	17	nonlinear	nonlinear	ADJ
cana-408	32	18	analysis	analysis	NOUN
cana-408	32	19	issn	issn	NOUN
cana-408	32	20	:	:	PUNCT
cana-408	32	21	1074	1074	NUM
cana-408	32	22	-	-	PUNCT
cana-408	32	23	133x	133x	NUM
cana-408	32	24	vol	vol	NOUN
cana-408	32	25	31	31	NUM
cana-408	32	26	no	no	NOUN
cana-408	32	27	.	.	NOUN
cana-408	32	28	1	1	NUM
cana-408	32	29	(	(	PUNCT
cana-408	32	30	2024	2024	NUM
cana-408	32	31	)	)	PUNCT
cana-408	32	32	232	232	NUM
cana-408	33	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-408	33	2	supporting	support	VERB
cana-408	33	3	current	current	ADJ
cana-408	33	4	applied	apply	VERB
cana-408	33	5	math	math	NOUN
cana-408	33	6	,	,	PUNCT
cana-408	33	7	software	software	NOUN
cana-408	33	8	engineering	engineering	NOUN
cana-408	33	9	,	,	PUNCT
cana-408	33	10	combinatorial	combinatorial	ADJ
cana-408	33	11	enhancement	enhancement	NOUN
cana-408	33	12	and	and	CCONJ
cana-408	33	13	activity	activity	NOUN
cana-408	33	14	research	research	NOUN
cana-408	33	15	specifically	specifically	ADV
cana-408	33	16	yet	yet	ADV
cana-408	33	17	in	in	ADP
cana-408	33	18	addition	addition	NOUN
cana-408	33	19	to	to	ADP
cana-408	33	20	its	its	PRON
cana-408	33	21	rising	rise	VERB
cana-408	33	22	applications	application	NOUN
cana-408	33	23	in	in	ADP
cana-408	33	24	the	the	DET
cana-408	33	25	more	more	ADV
cana-408	33	26	applied	applied	ADJ
cana-408	33	27	sciences	science	NOUN
cana-408	33	28	.	.	PUNCT
cana-408	34	1	likewise	likewise	ADV
cana-408	34	2	,	,	PUNCT
cana-408	34	3	graph	graph	NOUN
cana-408	34	4	theory	theory	NOUN
cana-408	34	5	hypothesis	hypothesis	NOUN
cana-408	34	6	is	be	AUX
cana-408	34	7	an	an	DET
cana-408	34	8	awesome	awesome	ADJ
cana-408	34	9	jungle	jungle	NOUN
cana-408	34	10	gym	gym	NOUN
cana-408	34	11	for	for	ADP
cana-408	34	12	the	the	DET
cana-408	34	13	investigation	investigation	NOUN
cana-408	34	14	of	of	ADP
cana-408	34	15	confirmation	confirmation	NOUN
cana-408	34	16	procedure	procedure	NOUN
cana-408	34	17	in	in	ADP
cana-408	34	18	discrete	discrete	ADJ
cana-408	34	19	mathematics	mathematic	NOUN
cana-408	34	20	and	and	CCONJ
cana-408	34	21	its	its	PRON
cana-408	34	22	outcomes	outcome	NOUN
cana-408	34	23	have	have	VERB
cana-408	34	24	applications	application	NOUN
cana-408	34	25	in	in	ADP
cana-408	34	26	numerous	numerous	ADJ
cana-408	34	27	space	space	NOUN
cana-408	34	28	of	of	ADP
cana-408	34	29	the	the	DET
cana-408	34	30	registering	registering	NOUN
cana-408	34	31	,	,	PUNCT
cana-408	34	32	social	social	ADJ
cana-408	34	33	and	and	CCONJ
cana-408	34	34	innate	innate	ADJ
cana-408	34	35	sciences	science	NOUN
cana-408	34	36	.	.	PUNCT
cana-408	35	1	numerous	numerous	ADJ
cana-408	35	2	issues	issue	NOUN
cana-408	35	3	of	of	ADP
cana-408	35	4	functional	functional	ADJ
cana-408	35	5	interest	interest	NOUN
cana-408	35	6	can	can	AUX
cana-408	35	7	be	be	AUX
cana-408	35	8	addressed	address	VERB
cana-408	35	9	by	by	ADP
cana-408	35	10	theories	theory	NOUN
cana-408	35	11	.	.	PUNCT
cana-408	36	1	the	the	DET
cana-408	36	2	paper	paper	NOUN
cana-408	36	3	composed	compose	VERB
cana-408	36	4	by	by	ADP
cana-408	36	5	leonhard	leonhard	PROPN
cana-408	36	6	euler	euler	PROPN
cana-408	36	7	on	on	ADP
cana-408	36	8	the	the	DET
cana-408	36	9	seven	seven	NUM
cana-408	36	10	bridge	bridge	NOUN
cana-408	36	11	of	of	ADP
cana-408	36	12	konigsberg	konigsberg	PROPN
cana-408	36	13	and	and	CCONJ
cana-408	36	14	distributed	distribute	VERB
cana-408	36	15	in	in	ADP
cana-408	36	16	1736	1736	NUM
cana-408	36	17	is	be	AUX
cana-408	36	18	viewed	view	VERB
cana-408	36	19	as	as	ADP
cana-408	36	20	the	the	DET
cana-408	36	21	principal	principal	ADJ
cana-408	36	22	paper	paper	NOUN
cana-408	36	23	throughout	throughout	ADP
cana-408	36	24	the	the	DET
cana-408	36	25	entire	entire	ADJ
cana-408	36	26	existence	existence	NOUN
cana-408	36	27	of	of	ADP
cana-408	36	28	diagram	diagram	NOUN
cana-408	36	29	hypothesis	hypothesis	NOUN
cana-408	36	30	.	.	PUNCT
cana-408	37	1	this	this	DET
cana-408	37	2	paper	paper	NOUN
cana-408	37	3	,	,	PUNCT
cana-408	37	4	as	as	ADV
cana-408	37	5	well	well	ADV
cana-408	37	6	as	as	ADP
cana-408	37	7	the	the	DET
cana-408	37	8	one	one	NOUN
cana-408	37	9	composed	compose	VERB
cana-408	37	10	by	by	ADP
cana-408	37	11	vandermonde	vandermonde	NOUN
cana-408	37	12	on	on	ADP
cana-408	37	13	the	the	DET
cana-408	37	14	knight	knight	NOUN
cana-408	37	15	issues	issue	NOUN
cana-408	37	16	continued	continue	VERB
cana-408	37	17	with	with	ADP
cana-408	37	18	the	the	DET
cana-408	37	19	examination	examination	NOUN
cana-408	37	20	situs	situs	PROPN
cana-408	37	21	started	start	VERB
cana-408	37	22	by	by	ADP
cana-408	37	23	leibnitz	leibnitz	PROPN
cana-408	37	24	.	.	PUNCT
cana-408	38	1	eulers	euler	NOUN
cana-408	38	2	equation	equation	NOUN
cana-408	38	3	relating	relate	VERB
cana-408	38	4	the	the	DET
cana-408	38	5	quantity	quantity	NOUN
cana-408	38	6	of	of	ADP
cana-408	38	7	edges	edge	NOUN
cana-408	38	8	,	,	PUNCT
cana-408	38	9	vertices	vertex	NOUN
cana-408	38	10	and	and	CCONJ
cana-408	38	11	appearances	appearance	NOUN
cana-408	38	12	of	of	ADP
cana-408	38	13	a	a	DET
cana-408	38	14	raised	raise	VERB
cana-408	38	15	polyhedron	polyhedron	NOUN
cana-408	38	16	was	be	AUX
cana-408	38	17	examined	examine	VERB
cana-408	38	18	and	and	CCONJ
cana-408	38	19	summed	sum	VERB
cana-408	38	20	up	up	ADP
cana-408	38	21	by	by	ADP
cana-408	38	22	cauchy	cauchy	NOUN
cana-408	38	23	and	and	CCONJ
cana-408	38	24	l'huillies	l'huillie	NOUN
cana-408	38	25	.	.	PUNCT
cana-408	39	1	the	the	DET
cana-408	39	2	principal	principal	ADJ
cana-408	39	3	course	course	NOUN
cana-408	39	4	reading	read	VERB
cana-408	39	5	on	on	ADP
cana-408	39	6	diagram	diagram	NOUN
cana-408	39	7	hypothesis	hypothesis	NOUN
cana-408	39	8	was	be	AUX
cana-408	39	9	composed	compose	VERB
cana-408	39	10	by	by	ADP
cana-408	39	11	denes	dene	NOUN
cana-408	39	12	konig	konig	PROPN
cana-408	39	13	which	which	PRON
cana-408	39	14	is	be	AUX
cana-408	39	15	distributed	distribute	VERB
cana-408	39	16	in	in	ADP
cana-408	39	17	1936	1936	NUM
cana-408	39	18	.	.	PUNCT
cana-408	40	1	garrett	garrett	PROPN
cana-408	40	2	birkhoff	birkhoff	PROPN
cana-408	40	3	's	's	PART
cana-408	40	4	work	work	NOUN
cana-408	40	5	during	during	ADP
cana-408	40	6	the	the	DET
cana-408	40	7	thirties	thirty	NOUN
cana-408	40	8	of	of	ADP
cana-408	40	9	the	the	DET
cana-408	40	10	20th	20th	ADJ
cana-408	40	11	century	century	NOUN
cana-408	40	12	began	begin	VERB
cana-408	40	13	the	the	DET
cana-408	40	14	overall	overall	ADJ
cana-408	40	15	improvement	improvement	NOUN
cana-408	40	16	of	of	ADP
cana-408	40	17	lattice	lattice	ADJ
cana-408	40	18	hypothesis	hypothesis	NOUN
cana-408	40	19	.	.	PUNCT
cana-408	41	1	in	in	ADP
cana-408	41	2	a	a	DET
cana-408	41	3	progression	progression	NOUN
cana-408	41	4	of	of	ADP
cana-408	41	5	papers	paper	NOUN
cana-408	41	6	,	,	PUNCT
cana-408	41	7	he	he	PRON
cana-408	41	8	exhibited	exhibit	VERB
cana-408	41	9	the	the	DET
cana-408	41	10	significance	significance	NOUN
cana-408	41	11	of	of	ADP
cana-408	41	12	lattice	lattice	ADJ
cana-408	41	13	hypothesis	hypothesis	NOUN
cana-408	41	14	.	.	PUNCT
cana-408	42	1	lattice	lattice	PROPN
cana-408	42	2	hypothesis	hypothesis	NOUN
cana-408	42	3	assumes	assume	VERB
cana-408	42	4	a	a	DET
cana-408	42	5	significant	significant	ADJ
cana-408	42	6	part	part	NOUN
cana-408	42	7	in	in	ADP
cana-408	42	8	numerous	numerous	ADJ
cana-408	42	9	areas	area	NOUN
cana-408	42	10	of	of	ADP
cana-408	42	11	math	math	NOUN
cana-408	42	12	for	for	ADP
cana-408	42	13	instance	instance	NOUN
cana-408	42	14	boolean	boolean	ADJ
cana-408	42	15	variable	variable	NOUN
cana-408	42	16	-	-	PUNCT
cana-408	42	17	based	base	VERB
cana-408	42	18	math	math	NOUN
cana-408	42	19	,	,	PUNCT
cana-408	42	20	rationale	rationale	NOUN
cana-408	42	21	and	and	CCONJ
cana-408	42	22	different	different	ADJ
cana-408	42	23	regions	region	NOUN
cana-408	42	24	like	like	ADP
cana-408	42	25	exchanging	exchange	VERB
cana-408	42	26	hypothesis	hypothesis	NOUN
cana-408	42	27	,	,	PUNCT
cana-408	42	28	software	software	NOUN
cana-408	42	29	engineering	engineering	NOUN
cana-408	42	30	,	,	PUNCT
cana-408	42	31	quantum	quantum	NOUN
cana-408	42	32	mechanics	mechanic	NOUN
cana-408	42	33	.	.	PUNCT
cana-408	43	1	an	an	DET
cana-408	43	2	alternate	alternate	ADJ
cana-408	43	3	part	part	NOUN
cana-408	43	4	of	of	ADP
cana-408	43	5	lattice	lattice	ADJ
cana-408	43	6	hypothesis	hypothesis	NOUN
cana-408	43	7	concerns	concern	VERB
cana-408	43	8	the	the	DET
cana-408	43	9	underpinning	underpinning	NOUN
cana-408	43	10	of	of	ADP
cana-408	43	11	set	set	ADJ
cana-408	43	12	hypothesis	hypothesis	NOUN
cana-408	43	13	(	(	PUNCT
cana-408	43	14	counting	count	VERB
cana-408	43	15	geotherm	geotherm	NOUN
cana-408	43	16	and	and	CCONJ
cana-408	43	17	genuine	genuine	ADJ
cana-408	43	18	investigation	investigation	NOUN
cana-408	43	19	)	)	PUNCT
cana-408	43	20	.	.	PUNCT
cana-408	44	1	lattice	lattice	PROPN
cana-408	44	2	have	have	VERB
cana-408	44	3	a	a	DET
cana-408	44	4	few	few	ADJ
cana-408	44	5	associations	association	NOUN
cana-408	44	6	with	with	ADP
cana-408	44	7	the	the	DET
cana-408	44	8	group	group	NOUN
cana-408	44	9	of	of	ADP
cana-408	44	10	gathering	gather	VERB
cana-408	44	11	like	like	ADP
cana-408	44	12	designs	design	NOUN
cana-408	44	13	since	since	SCONJ
cana-408	44	14	meet	meet	VERB
cana-408	44	15	and	and	CCONJ
cana-408	44	16	join	join	VERB
cana-408	44	17	both	both	PRON
cana-408	44	18	are	be	AUX
cana-408	44	19	commutative	commutative	ADJ
cana-408	44	20	and	and	CCONJ
cana-408	44	21	affiliated	affiliate	VERB
cana-408	44	22	a	a	DET
cana-408	44	23	cross	cross	NOUN
cana-408	44	24	section	section	NOUN
cana-408	44	25	can	can	AUX
cana-408	44	26	be	be	AUX
cana-408	44	27	seen	see	VERB
cana-408	44	28	as	as	ADP
cana-408	44	29	comprising	comprising	NOUN
cana-408	44	30	of	of	ADP
cana-408	44	31	two	two	NUM
cana-408	44	32	commutative	commutative	ADJ
cana-408	44	33	semigroups	semigroup	NOUN
cana-408	44	34	having	have	VERB
cana-408	44	35	a	a	DET
cana-408	44	36	similar	similar	ADJ
cana-408	44	37	space	space	NOUN
cana-408	44	38	.	.	PUNCT
cana-408	45	1	for	for	ADP
cana-408	45	2	a	a	DET
cana-408	45	3	limited	limited	ADJ
cana-408	45	4	cross	cross	NOUN
cana-408	45	5	section	section	NOUN
cana-408	45	6	,	,	PUNCT
cana-408	45	7	these	these	DET
cana-408	45	8	semigroups	semigroup	NOUN
cana-408	45	9	are	be	AUX
cana-408	45	10	truth	truth	NOUN
cana-408	45	11	be	be	AUX
cana-408	45	12	told	tell	VERB
cana-408	45	13	commutative	commutative	ADJ
cana-408	45	14	monoids	monoid	NOUN
cana-408	45	15	.	.	PUNCT
cana-408	46	1	the	the	DET
cana-408	46	2	logarithmic	logarithmic	ADJ
cana-408	46	3	translation	translation	NOUN
cana-408	46	4	of	of	ADP
cana-408	46	5	lattice	lattice	PROPN
cana-408	46	6	plays	play	NOUN
cana-408	46	7	and	and	CCONJ
cana-408	46	8	fundamental	fundamental	ADJ
cana-408	46	9	job	job	NOUN
cana-408	46	10	in	in	ADP
cana-408	46	11	widespread	widespread	ADJ
cana-408	46	12	polynomial	polynomial	ADJ
cana-408	46	13	math	math	NOUN
cana-408	46	14	.	.	PUNCT
cana-408	47	1	the	the	DET
cana-408	47	2	magnificence	magnificence	NOUN
cana-408	47	3	of	of	ADP
cana-408	47	4	cross	cross	NOUN
cana-408	47	5	section	section	NOUN
cana-408	47	6	hypothesis	hypothesis	NOUN
cana-408	47	7	gets	get	VERB
cana-408	47	8	to	to	ADP
cana-408	47	9	some	some	DET
cana-408	47	10	extent	extent	NOUN
cana-408	47	11	from	from	ADP
cana-408	47	12	the	the	DET
cana-408	47	13	outrageous	outrageous	ADJ
cana-408	47	14	straightforwardness	straightforwardness	NOUN
cana-408	47	15	of	of	ADP
cana-408	47	16	its	its	PRON
cana-408	47	17	fundamental	fundamental	ADJ
cana-408	47	18	ideas	idea	NOUN
cana-408	47	19	for	for	ADP
cana-408	47	20	instance	instance	NOUN
cana-408	47	21	poset	poset	NOUN
cana-408	47	22	,	,	PUNCT
cana-408	47	23	least	least	ADV
cana-408	47	24	upper	upper	ADJ
cana-408	47	25	bound	bind	VERB
cana-408	47	26	,	,	PUNCT
cana-408	47	27	most	most	ADV
cana-408	47	28	noteworthy	noteworthy	ADJ
cana-408	47	29	lower	lower	ADV
cana-408	47	30	bound	bind	VERB
cana-408	47	31	and	and	CCONJ
cana-408	47	32	so	so	ADV
cana-408	47	33	forth	forth	ADV
cana-408	47	34	.	.	PUNCT
cana-408	48	1	the	the	DET
cana-408	48	2	investigation	investigation	NOUN
cana-408	48	3	of	of	ADP
cana-408	48	4	logarithmic	logarithmic	ADJ
cana-408	48	5	chart	chart	NOUN
cana-408	48	6	hypothesis	hypothesis	NOUN
cana-408	48	7	is	be	AUX
cana-408	48	8	an	an	DET
cana-408	48	9	intriguing	intriguing	ADJ
cana-408	48	10	subject	subject	NOUN
cana-408	48	11	for	for	ADP
cana-408	48	12	mathematicians	mathematician	NOUN
cana-408	48	13	and	and	CCONJ
cana-408	48	14	returns	return	NOUN
cana-408	48	15	basically	basically	ADV
cana-408	48	16	to	to	ADP
cana-408	48	17	1973	1973	NUM
cana-408	48	18	,	,	PUNCT
cana-408	48	19	when	when	SCONJ
cana-408	48	20	n.	n.	PROPN
cana-408	48	21	biggs	biggs	PROPN
cana-408	48	22	see	see	VERB
cana-408	48	23	[	[	X
cana-408	48	24	9	9	NUM
cana-408	48	25	]	]	PUNCT
cana-408	48	26	distributed	distribute	VERB
cana-408	48	27	his	his	PRON
cana-408	48	28	book	book	NOUN
cana-408	48	29	on	on	ADP
cana-408	48	30	algebraic	algebraic	ADJ
cana-408	48	31	diagram	diagram	NOUN
cana-408	48	32	hypothesis	hypothesis	NOUN
cana-408	48	33	.	.	PUNCT
cana-408	49	1	as	as	SCONJ
cana-408	49	2	he	he	PRON
cana-408	49	3	wrote	write	VERB
cana-408	49	4	in	in	ADP
cana-408	49	5	the	the	DET
cana-408	49	6	prelude	prelude	NOUN
cana-408	49	7	of	of	ADP
cana-408	49	8	his	his	PRON
cana-408	49	9	book	book	NOUN
cana-408	49	10	,	,	PUNCT
cana-408	49	11	his	his	PRON
cana-408	49	12	point	point	NOUN
cana-408	49	13	was	be	AUX
cana-408	49	14	"	"	PUNCT
cana-408	49	15	to	to	PART
cana-408	49	16	make	make	VERB
cana-408	49	17	an	an	DET
cana-408	49	18	interpretation	interpretation	NOUN
cana-408	49	19	of	of	ADP
cana-408	49	20	properties	property	NOUN
cana-408	49	21	of	of	ADP
cana-408	49	22	charts	chart	NOUN
cana-408	49	23	into	into	ADP
cana-408	49	24	mathematical	mathematical	ADJ
cana-408	49	25	properties	property	NOUN
cana-408	49	26	and	and	CCONJ
cana-408	49	27	afterward	afterward	ADV
cana-408	49	28	utilizing	utilize	VERB
cana-408	49	29	the	the	DET
cana-408	49	30	outcomes	outcome	NOUN
cana-408	49	31	and	and	CCONJ
cana-408	49	32	strategies	strategy	NOUN
cana-408	49	33	for	for	ADP
cana-408	49	34	variable	variable	ADJ
cana-408	49	35	based	base	VERB
cana-408	49	36	math	math	NOUN
cana-408	49	37	,	,	PUNCT
cana-408	49	38	to	to	PART
cana-408	49	39	derive	derive	VERB
cana-408	49	40	hypotheses	hypothesis	NOUN
cana-408	49	41	about	about	ADP
cana-408	49	42	diagrams	diagram	NOUN
cana-408	49	43	"	"	PUNCT
cana-408	49	44	.	.	PUNCT
cana-408	50	1	despite	despite	SCONJ
cana-408	50	2	the	the	DET
cana-408	50	3	fact	fact	NOUN
cana-408	50	4	that	that	SCONJ
cana-408	50	5	biggs	biggs	PROPN
cana-408	50	6	talked	talk	VERB
cana-408	50	7	about	about	ADP
cana-408	50	8	mathematical	mathematical	ADJ
cana-408	50	9	techniques	technique	NOUN
cana-408	50	10	and	and	CCONJ
cana-408	50	11	variable	variable	NOUN
cana-408	50	12	based	base	VERB
cana-408	50	13	math	math	NOUN
cana-408	50	14	as	as	ADP
cana-408	50	15	a	a	DET
cana-408	50	16	rule	rule	NOUN
cana-408	50	17	,	,	PUNCT
cana-408	50	18	the	the	DET
cana-408	50	19	sort	sort	NOUN
cana-408	50	20	of	of	ADP
cana-408	50	21	variable	variable	NOUN
cana-408	50	22	based	base	VERB
cana-408	50	23	math	math	NOUN
cana-408	50	24	he	he	PRON
cana-408	50	25	truly	truly	ADV
cana-408	50	26	utilized	utilize	VERB
cana-408	50	27	was	be	AUX
cana-408	50	28	straight	straight	ADJ
cana-408	50	29	variable	variable	NOUN
cana-408	50	30	based	base	VERB
cana-408	50	31	math	math	NOUN
cana-408	50	32	and	and	CCONJ
cana-408	50	33	a	a	DET
cana-408	50	34	few	few	ADJ
cana-408	50	35	properties	property	NOUN
cana-408	50	36	of	of	ADP
cana-408	50	37	polynomials	polynomial	NOUN
cana-408	50	38	.	.	PUNCT
cana-408	51	1	in	in	ADP
cana-408	51	2	1993	1993	NUM
cana-408	51	3	anderson	anderson	PROPN
cana-408	51	4	and	and	CCONJ
cana-408	51	5	naseer	naseer	PROPN
cana-408	51	6	[	[	X
cana-408	51	7	2	2	X
cana-408	51	8	]	]	PUNCT
cana-408	51	9	addressed	address	VERB
cana-408	51	10	this	this	DET
cana-408	51	11	issue	issue	NOUN
cana-408	51	12	adversely	adversely	ADV
cana-408	51	13	and	and	CCONJ
cana-408	51	14	gave	give	VERB
cana-408	51	15	a	a	DET
cana-408	51	16	counter	counter	ADJ
cana-408	51	17	model	model	NOUN
cana-408	51	18	.	.	PUNCT
cana-408	52	1	they	they	PRON
cana-408	52	2	further	far	ADV
cana-408	52	3	concentrated	concentrate	VERB
cana-408	52	4	on	on	ADP
cana-408	52	5	the	the	DET
cana-408	52	6	zero	zero	NUM
cana-408	52	7	divisor	divisor	NOUN
cana-408	52	8	graphs	graph	NOUN
cana-408	52	9	of	of	ADP
cana-408	52	10	commutative	commutative	ADJ
cana-408	52	11	rings	ring	NOUN
cana-408	52	12	by	by	ADP
cana-408	52	13	altering	alter	VERB
cana-408	52	14	the	the	DET
cana-408	52	15	beck	beck	NOUN
cana-408	52	16	's	's	PART
cana-408	52	17	[	[	X
cana-408	52	18	7	7	NUM
cana-408	52	19	]	]	PUNCT
cana-408	52	20	definition	definition	NOUN
cana-408	52	21	.	.	PUNCT
cana-408	53	1	they	they	PRON
cana-408	53	2	considered	consider	VERB
cana-408	53	3	just	just	ADV
cana-408	53	4	nonzero	nonzero	ADJ
cana-408	53	5	zero	zero	NUM
cana-408	53	6	-	-	PUNCT
cana-408	53	7	divisors	divisor	NOUN
cana-408	53	8	as	as	ADP
cana-408	53	9	vertices	vertex	NOUN
cana-408	53	10	of	of	ADP
cana-408	53	11	graphs	graph	NOUN
cana-408	53	12	.	.	PUNCT
cana-408	54	1	in	in	ADP
cana-408	54	2	1999	1999	NUM
cana-408	54	3	anderson	anderson	PROPN
cana-408	54	4	and	and	CCONJ
cana-408	54	5	livingston	livingston	PROPN
cana-408	54	6	[	[	X
cana-408	54	7	4	4	X
cana-408	54	8	]	]	PUNCT
cana-408	54	9	changed	change	VERB
cana-408	54	10	the	the	DET
cana-408	54	11	meaning	meaning	NOUN
cana-408	54	12	of	of	ADP
cana-408	54	13	the	the	DET
cana-408	54	14	zero	zero	NUM
cana-408	54	15	-	-	PUNCT
cana-408	54	16	divisor	divisor	NOUN
cana-408	54	17	graph	graph	NOUN
cana-408	54	18	,	,	PUNCT
cana-408	54	19	characterizing	characterize	VERB
cana-408	54	20	the	the	DET
cana-408	54	21	vertices	vertex	NOUN
cana-408	54	22	of	of	ADP
cana-408	54	23	the	the	DET
cana-408	54	24	graph	graph	NOUN
cana-408	54	25	to	to	PART
cana-408	54	26	be	be	AUX
cana-408	54	27	the	the	DET
cana-408	54	28	nonzero	nonzero	ADJ
cana-408	54	29	zero	zero	NUM
cana-408	54	30	-	-	PUNCT
cana-408	54	31	divisors	divisor	NOUN
cana-408	54	32	of	of	ADP
cana-408	54	33	the	the	DET
cana-408	54	34	commutative	commutative	ADJ
cana-408	54	35	ring	ring	NOUN
cana-408	54	36	.	.	PUNCT
cana-408	55	1	afterward	afterward	ADV
cana-408	55	2	,	,	PUNCT
cana-408	55	3	demeyer	demeyer	NOUN
cana-408	55	4	,	,	PUNCT
cana-408	55	5	mckenzie	mckenzie	NOUN
cana-408	55	6	and	and	CCONJ
cana-408	55	7	schneider	schneider	NOUN
cana-408	55	8	in	in	ADP
cana-408	55	9	[	[	X
cana-408	55	10	10	10	NUM
cana-408	55	11	]	]	PUNCT
cana-408	55	12	concentrated	concentrate	VERB
cana-408	55	13	on	on	ADP
cana-408	55	14	graphs	graph	NOUN
cana-408	55	15	on	on	ADP
cana-408	55	16	commutative	commutative	ADJ
cana-408	55	17	semigroup	semigroup	NOUN
cana-408	55	18	with	with	ADP
cana-408	55	19	0	0	NUM
cana-408	55	20	.	.	PUNCT
cana-408	56	1	this	this	DET
cana-408	56	2	review	review	NOUN
cana-408	56	3	for	for	ADP
cana-408	56	4	semigroups	semigroup	NOUN
cana-408	56	5	was	be	AUX
cana-408	56	6	gone	go	VERB
cana-408	56	7	on	on	ADP
cana-408	56	8	by	by	ADP
cana-408	56	9	demeyer	demeyer	NOUN
cana-408	56	10	and	and	CCONJ
cana-408	56	11	demeyer	demeyer	NOUN
cana-408	56	12	in	in	ADP
cana-408	56	13	[	[	X
cana-408	56	14	9	9	NUM
cana-408	56	15	]	]	PUNCT
cana-408	56	16	.	.	PUNCT
cana-408	57	1	therefore	therefore	ADV
cana-408	57	2	,	,	PUNCT
cana-408	57	3	research	research	NOUN
cana-408	57	4	has	have	AUX
cana-408	57	5	moved	move	VERB
cana-408	57	6	in	in	ADP
cana-408	57	7	a	a	DET
cana-408	57	8	few	few	ADJ
cana-408	57	9	headings	heading	NOUN
cana-408	57	10	for	for	ADP
cana-408	57	11	instance	instance	NOUN
cana-408	57	12	,	,	PUNCT
cana-408	57	13	anderson	anderson	PROPN
cana-408	57	14	,	,	PUNCT
cana-408	57	15	r.	r.	PROPN
cana-408	57	16	levy	levy	PROPN
cana-408	57	17	and	and	CCONJ
cana-408	57	18	j.	j.	PROPN
cana-408	57	19	shapiro	shapiro	PROPN
cana-408	57	20	in	in	ADP
cana-408	57	21	[	[	X
cana-408	57	22	5	5	NUM
cana-408	57	23	]	]	PUNCT
cana-408	57	24	broadened	broaden	VERB
cana-408	57	25	the	the	DET
cana-408	57	26	outcomes	outcome	NOUN
cana-408	57	27	by	by	ADP
cana-408	57	28	taking	take	VERB
cana-408	57	29	a	a	DET
cana-408	57	30	gander	gander	NOUN
cana-408	57	31	at	at	ADP
cana-408	57	32	the	the	DET
cana-408	57	33	coterie	coterie	NOUN
cana-408	57	34	number	number	NOUN
cana-408	57	35	what	what	PRON
cana-408	57	36	's	be	AUX
cana-408	57	37	more	more	ADJ
cana-408	57	38	,	,	PUNCT
cana-408	57	39	planarity	planarity	NOUN
cana-408	57	40	of	of	ADP
cana-408	57	41	zero	zero	NUM
cana-408	57	42	-	-	PUNCT
cana-408	57	43	divisor	divisor	NOUN
cana-408	57	44	communications	communication	NOUN
cana-408	57	45	on	on	ADP
cana-408	57	46	applied	apply	VERB
cana-408	57	47	nonlinear	nonlinear	ADJ
cana-408	57	48	analysis	analysis	NOUN
cana-408	57	49	issn	issn	NOUN
cana-408	57	50	:	:	PUNCT
cana-408	57	51	1074	1074	NUM
cana-408	57	52	-	-	PUNCT
cana-408	57	53	133x	133x	NUM
cana-408	57	54	vol	vol	NOUN
cana-408	57	55	31	31	NUM
cana-408	57	56	no	no	NOUN
cana-408	57	57	.	.	NOUN
cana-408	57	58	1	1	NUM
cana-408	57	59	(	(	PUNCT
cana-408	57	60	2024	2024	NUM
cana-408	57	61	)	)	PUNCT
cana-408	57	62	233	233	NUM
cana-408	57	63	https://internationalpubls.com	https://internationalpubls.com	X
cana-408	57	64	charts	chart	NOUN
cana-408	57	65	,	,	PUNCT
cana-408	57	66	while	while	SCONJ
cana-408	57	67	r.	r.	PROPN
cana-408	57	68	akhtar	akhtar	PROPN
cana-408	57	69	and	and	CCONJ
cana-408	57	70	l.	l.	PROPN
cana-408	57	71	lee	lee	PROPN
cana-408	57	72	in	in	ADP
cana-408	57	73	[	[	X
cana-408	57	74	1	1	NUM
cana-408	57	75	]	]	PUNCT
cana-408	57	76	explored	explore	VERB
cana-408	57	77	the	the	DET
cana-408	57	78	properties	property	NOUN
cana-408	57	79	vital	vital	ADJ
cana-408	57	80	for	for	ADP
cana-408	57	81	a	a	DET
cana-408	57	82	zero	zero	NUM
cana-408	57	83	-	-	PUNCT
cana-408	57	84	divisor	divisor	NOUN
cana-408	57	85	diagram	diagram	NOUN
cana-408	57	86	to	to	PART
cana-408	57	87	be	be	AUX
cana-408	57	88	either	either	CCONJ
cana-408	57	89	a	a	DET
cana-408	57	90	planar	planar	ADJ
cana-408	57	91	or	or	CCONJ
cana-408	57	92	a	a	DET
cana-408	57	93	total	total	ADJ
cana-408	57	94	r	r	NOUN
cana-408	57	95	partite	partite	NOUN
cana-408	57	96	.	.	PUNCT
cana-408	58	1	redmond	redmond	NOUN
cana-408	58	2	in	in	ADP
cana-408	58	3	[	[	X
cana-408	58	4	14	14	NUM
cana-408	58	5	]	]	PUNCT
cana-408	58	6	,	,	PUNCT
cana-408	58	7	took	take	VERB
cana-408	58	8	a	a	DET
cana-408	58	9	gander	gander	NOUN
cana-408	58	10	at	at	ADP
cana-408	58	11	a	a	DET
cana-408	58	12	portion	portion	NOUN
cana-408	58	13	of	of	ADP
cana-408	58	14	the	the	DET
cana-408	58	15	progressions	progression	NOUN
cana-408	58	16	inferred	infer	VERB
cana-408	58	17	by	by	ADP
cana-408	58	18	the	the	DET
cana-408	58	19	zero	zero	NUM
cana-408	58	20	-	-	PUNCT
cana-408	58	21	divisor	divisor	NOUN
cana-408	58	22	diagram	diagram	NOUN
cana-408	58	23	on	on	ADP
cana-408	58	24	a	a	DET
cana-408	58	25	noncommutative	noncommutative	ADJ
cana-408	58	26	ring	ring	NOUN
cana-408	58	27	.	.	PUNCT
cana-408	59	1	the	the	DET
cana-408	59	2	zero	zero	NUM
cana-408	59	3	-	-	PUNCT
cana-408	59	4	divisor	divisor	NOUN
cana-408	59	5	chart	chart	NOUN
cana-408	59	6	of	of	ADP
cana-408	59	7	a	a	DET
cana-408	59	8	commutative	commutative	ADJ
cana-408	59	9	ring	ring	NOUN
cana-408	59	10	has	have	AUX
cana-408	59	11	additionally	additionally	ADV
cana-408	59	12	been	be	AUX
cana-408	59	13	concentrated	concentrate	VERB
cana-408	59	14	in	in	ADP
cana-408	59	15	(	(	PUNCT
cana-408	59	16	[	[	X
cana-408	59	17	3	3	NUM
cana-408	59	18	]	]	PUNCT
cana-408	59	19	,	,	PUNCT
cana-408	60	1	[	[	X
cana-408	60	2	12	12	NUM
cana-408	60	3	]	]	PUNCT
cana-408	60	4	,	,	PUNCT
cana-408	60	5	[	[	X
cana-408	60	6	13	13	NUM
cana-408	60	7	]	]	PUNCT
cana-408	60	8	,	,	PUNCT
cana-408	61	1	[	[	X
cana-408	61	2	16	16	NUM
cana-408	61	3	]	]	PUNCT
cana-408	61	4	)	)	PUNCT
cana-408	61	5	and	and	CCONJ
cana-408	61	6	the	the	DET
cana-408	61	7	zero	zero	NUM
cana-408	61	8	-	-	PUNCT
cana-408	61	9	divisor	divisor	NOUN
cana-408	61	10	diagram	diagram	NOUN
cana-408	61	11	idea	idea	NOUN
cana-408	61	12	has	have	AUX
cana-408	61	13	been	be	AUX
cana-408	61	14	reached	reach	VERB
cana-408	61	15	out	out	ADP
cana-408	61	16	to	to	ADP
cana-408	61	17	noncommutative	noncommutative	ADJ
cana-408	61	18	ring	ring	NOUN
cana-408	61	19	in	in	ADP
cana-408	61	20	[	[	X
cana-408	61	21	15	15	NUM
cana-408	61	22	]	]	PUNCT
cana-408	61	23	.	.	PUNCT
cana-408	62	1	mulay	mulay	NOUN
cana-408	62	2	in	in	ADP
cana-408	62	3	[	[	X
cana-408	62	4	13	13	NUM
cana-408	62	5	]	]	PUNCT
cana-408	62	6	utilizing	utilize	VERB
cana-408	62	7	anderson	anderson	PROPN
cana-408	62	8	and	and	CCONJ
cana-408	62	9	livingston	livingston	PROPN
cana-408	62	10	's	's	PART
cana-408	62	11	in	in	ADP
cana-408	62	12	[	[	X
cana-408	62	13	4	4	X
cana-408	62	14	]	]	SYM
cana-408	62	15	meaning	meaning	NOUN
cana-408	62	16	of	of	ADP
cana-408	62	17	the	the	DET
cana-408	62	18	zero	zero	NUM
cana-408	62	19	-	-	PUNCT
cana-408	62	20	divisor	divisor	NOUN
cana-408	62	21	graph	graph	NOUN
cana-408	62	22	examined	examine	VERB
cana-408	62	23	the	the	DET
cana-408	62	24	cycle	cycle	NOUN
cana-408	62	25	construction	construction	NOUN
cana-408	62	26	of	of	ADP
cana-408	62	27	γ(r	γ(r	PROPN
cana-408	62	28	)	)	PUNCT
cana-408	62	29	.	.	PUNCT
cana-408	63	1	m.	m.	PROPN
cana-408	63	2	axtell	axtell	PROPN
cana-408	63	3	et	et	PROPN
cana-408	63	4	al	al	PROPN
cana-408	63	5	.	.	PUNCT
cana-408	64	1	in	in	ADP
cana-408	64	2	[	[	X
cana-408	64	3	3	3	NUM
cana-408	64	4	]	]	PUNCT
cana-408	64	5	look	look	NOUN
cana-408	64	6	at	at	ADP
cana-408	64	7	the	the	DET
cana-408	64	8	conservation	conservation	NOUN
cana-408	64	9	of	of	ADP
cana-408	64	10	diagram	diagram	NOUN
cana-408	64	11	hypothetical	hypothetical	ADJ
cana-408	64	12	properties	property	NOUN
cana-408	64	13	of	of	ADP
cana-408	64	14	the	the	DET
cana-408	64	15	zero	zero	NUM
cana-408	64	16	-	-	PUNCT
cana-408	64	17	divisor	divisor	NOUN
cana-408	64	18	graph	graph	NOUN
cana-408	64	19	under	under	ADP
cana-408	64	20	expansion	expansion	NOUN
cana-408	64	21	to	to	ADP
cana-408	64	22	polynomial	polynomial	ADJ
cana-408	64	23	and	and	CCONJ
cana-408	64	24	power	power	NOUN
cana-408	64	25	series	series	PROPN
cana-408	64	26	rings	ring	NOUN
cana-408	64	27	.	.	PUNCT
cana-408	65	1	2	2	X
cana-408	65	2	.	.	X
cana-408	65	3	objectives	objective	NOUN
cana-408	65	4	definition	definition	NOUN
cana-408	65	5	1.1	1.1	NUM
cana-408	65	6	.	.	PUNCT
cana-408	66	1	[	[	X
cana-408	66	2	8	8	X
cana-408	66	3	]	]	X
cana-408	66	4	a	a	DET
cana-408	66	5	meet	meet	NOUN
cana-408	66	6	-	-	PUNCT
cana-408	66	7	semilattice	semilattice	NOUN
cana-408	66	8	(	(	PUNCT
cana-408	66	9	or	or	CCONJ
cana-408	66	10	lower	low	ADJ
cana-408	66	11	semilattice	semilattice	NOUN
cana-408	66	12	)	)	PUNCT
cana-408	66	13	is	be	AUX
cana-408	66	14	a	a	DET
cana-408	66	15	partially	partially	ADV
cana-408	66	16	ordered	order	VERB
cana-408	66	17	set	set	NOUN
cana-408	66	18	which	which	PRON
cana-408	66	19	has	have	VERB
cana-408	66	20	a	a	DET
cana-408	66	21	meet	meet	NOUN
cana-408	66	22	(	(	PUNCT
cana-408	66	23	or	or	CCONJ
cana-408	66	24	greatest	great	ADJ
cana-408	66	25	lower	lower	ADV
cana-408	66	26	bound	bind	VERB
cana-408	66	27	)	)	PUNCT
cana-408	66	28	for	for	ADP
cana-408	66	29	any	any	DET
cana-408	66	30	nonempty	nonempty	ADJ
cana-408	66	31	finite	finite	NOUN
cana-408	66	32	subset	subset	NOUN
cana-408	66	33	.	.	PUNCT
cana-408	67	1	remark	remark	PROPN
cana-408	67	2	1.1	1.1	NUM
cana-408	67	3	.	.	PUNCT
cana-408	68	1	every	every	DET
cana-408	68	2	join	join	NOUN
cana-408	68	3	-	-	PUNCT
cana-408	68	4	semilattice	semilattice	NOUN
cana-408	68	5	is	be	AUX
cana-408	68	6	a	a	DET
cana-408	68	7	meet	meet	NOUN
cana-408	68	8	-	-	PUNCT
cana-408	68	9	semilattice	semilattice	NOUN
cana-408	68	10	in	in	ADP
cana-408	68	11	the	the	DET
cana-408	68	12	inverse	inverse	NOUN
cana-408	68	13	order	order	NOUN
cana-408	68	14	and	and	CCONJ
cana-408	68	15	vice	vice	ADV
cana-408	68	16	versa	versa	ADV
cana-408	68	17	.	.	PUNCT
cana-408	69	1	definition	definition	NOUN
cana-408	69	2	1.2	1.2	NUM
cana-408	69	3	.	.	PUNCT
cana-408	70	1	[	[	X
cana-408	70	2	6	6	NUM
cana-408	70	3	]	]	PUNCT
cana-408	70	4	an	an	DET
cana-408	70	5	element	element	NOUN
cana-408	70	6	a	a	DET
cana-408	70	7	∈	∈	PROPN
cana-408	70	8	l	l	NOUN
cana-408	70	9	is	be	AUX
cana-408	70	10	called	call	VERB
cana-408	70	11	a	a	DET
cana-408	70	12	zero	zero	NUM
cana-408	70	13	-	-	PUNCT
cana-408	70	14	divisor	divisor	NOUN
cana-408	70	15	if	if	SCONJ
cana-408	70	16	there	there	PRON
cana-408	70	17	exists	exist	VERB
cana-408	70	18	a	a	DET
cana-408	70	19	nonzero	nonzero	NOUN
cana-408	70	20	element	element	NOUN
cana-408	70	21	bi∈	bi∈	PROPN
cana-408	70	22	l	l	NOUN
cana-408	70	23	such	such	ADJ
cana-408	70	24	that	that	DET
cana-408	70	25	a∧b	a∧b	NOUN
cana-408	70	26	=	=	SYM
cana-408	70	27	i0	i0	PROPN
cana-408	70	28	.	.	PUNCT
cana-408	71	1	we	we	PRON
cana-408	71	2	denote	denote	VERB
cana-408	71	3	by	by	ADP
cana-408	71	4	z(l	z(l	PROPN
cana-408	71	5	)	)	PUNCT
cana-408	71	6	the	the	DET
cana-408	71	7	set	set	NOUN
cana-408	71	8	of	of	ADP
cana-408	71	9	all	all	DET
cana-408	71	10	zerodivisors	zerodivisor	NOUN
cana-408	71	11	of	of	ADP
cana-408	71	12	l.	l.	PROPN
cana-408	71	13	definition	definition	PROPN
cana-408	71	14	i1.3	i1.3	PROPN
cana-408	71	15	.	.	PUNCT
cana-408	72	1	[	[	X
cana-408	72	2	10	10	NUM
cana-408	72	3	]	]	X
cana-408	72	4	ia	ia	NOUN
cana-408	72	5	graph	graph	NOUN
cana-408	72	6	γ(l	γ(l	PROPN
cana-408	72	7	)	)	PUNCT
cana-408	72	8	to	to	ADP
cana-408	72	9	l	l	NOUN
cana-408	72	10	with	with	ADP
cana-408	72	11	vertex	vertex	NOUN
cana-408	72	12	set	set	VERB
cana-408	72	13	z∗(l	z∗(l	NOUN
cana-408	72	14	)	)	PUNCT
cana-408	72	15	=	=	SYM
cana-408	72	16	z(l	z(l	PROPN
cana-408	72	17	)	)	PUNCT
cana-408	72	18	−	−	NOUN
cana-408	72	19	{	{	PUNCT
cana-408	72	20	0	0	NUM
cana-408	72	21	}	}	PUNCT
cana-408	72	22	,	,	PUNCT
cana-408	72	23	the	the	DET
cana-408	72	24	set	set	NOUN
cana-408	72	25	of	of	ADP
cana-408	72	26	all	all	DET
cana-408	72	27	non	non	ADJ
cana-408	72	28	-	-	ADJ
cana-408	72	29	zero	zero	NUM
cana-408	72	30	zero	zero	NUM
cana-408	72	31	-	-	PUNCT
cana-408	72	32	divisors	divisor	NOUN
cana-408	72	33	of	of	ADP
cana-408	72	34	l.	l.	PROPN
cana-408	72	35	two	two	NUM
cana-408	72	36	distinct	distinct	ADJ
cana-408	72	37	ix	ix	ADV
cana-408	72	38	,	,	PUNCT
cana-408	72	39	y	y	PROPN
cana-408	72	40	∈	∈	PROPN
cana-408	72	41	z∗(l	z∗(l	PROPN
cana-408	72	42	)	)	PUNCT
cana-408	72	43	are	be	AUX
cana-408	72	44	adjacent	adjacent	ADJ
cana-408	72	45	if	if	SCONJ
cana-408	72	46	and	and	CCONJ
cana-408	72	47	only	only	ADV
cana-408	72	48	if	if	SCONJ
cana-408	72	49	ix	ix	PROPN
cana-408	72	50	∧	∧	PROPN
cana-408	72	51	y	y	PROPN
cana-408	72	52	=	=	SYM
cana-408	72	53	i0	i0	PROPN
cana-408	72	54	and	and	CCONJ
cana-408	72	55	call	call	VERB
cana-408	72	56	this	this	DET
cana-408	72	57	graph	graph	NOUN
cana-408	72	58	as	as	ADP
cana-408	72	59	the	the	DET
cana-408	72	60	zero	zero	NUM
cana-408	72	61	-	-	PUNCT
cana-408	72	62	divisor	divisor	NOUN
cana-408	72	63	graph	graph	NOUN
cana-408	72	64	of	of	ADP
cana-408	72	65	l.	l.	PROPN
cana-408	72	66	remark	remark	PROPN
cana-408	72	67	i1.2	i1.2	PROPN
cana-408	72	68	γ(l	γ(l	PROPN
cana-408	72	69	)	)	PUNCT
cana-408	72	70	is	be	AUX
cana-408	72	71	connected	connect	VERB
cana-408	72	72	with	with	ADP
cana-408	72	73	diam	diam	PROPN
cana-408	72	74	γ(l	γ(l	PROPN
cana-408	72	75	)	)	PUNCT
cana-408	72	76	≤	≤	NOUN
cana-408	72	77	i3	i3	NOUN
cana-408	72	78	and	and	CCONJ
cana-408	72	79	if	if	SCONJ
cana-408	72	80	γ(l	γ(l	PROPN
cana-408	72	81	)	)	PUNCT
cana-408	72	82	contains	contain	VERB
cana-408	72	83	a	a	DET
cana-408	72	84	cycle	cycle	NOUN
cana-408	72	85	,	,	PUNCT
cana-408	72	86	then	then	ADV
cana-408	72	87	grγ(l	grγ(l	NOUN
cana-408	72	88	)	)	PUNCT
cana-408	72	89	≤	≤	NOUN
cana-408	72	90	i4	i4	PROPN
cana-408	72	91	.	.	PUNCT
cana-408	73	1	we	we	PRON
cana-408	73	2	show	show	VERB
cana-408	73	3	that	that	SCONJ
cana-408	73	4	if	if	SCONJ
cana-408	73	5	γ(l	γ(l	NOUN
cana-408	73	6	)	)	PUNCT
cana-408	73	7	contains	contain	VERB
cana-408	73	8	a	a	DET
cana-408	73	9	cycle	cycle	NOUN
cana-408	73	10	,	,	PUNCT
cana-408	73	11	then	then	ADV
cana-408	73	12	the	the	DET
cana-408	73	13	core	core	NOUN
cana-408	73	14	k	k	PROPN
cana-408	73	15	of	of	ADP
cana-408	73	16	γ(l	γ(l	PROPN
cana-408	73	17	)	)	PUNCT
cana-408	73	18	is	be	AUX
cana-408	73	19	a	a	DET
cana-408	73	20	union	union	NOUN
cana-408	73	21	of	of	ADP
cana-408	73	22	i3	i3	NOUN
cana-408	73	23	cycles	cycle	NOUN
cana-408	73	24	and	and	CCONJ
cana-408	73	25	i4	i4	PROPN
cana-408	73	26	cycles	cycle	NOUN
cana-408	73	27	.	.	PUNCT
cana-408	74	1	moreover	moreover	ADV
cana-408	74	2	,	,	PUNCT
cana-408	74	3	any	any	DET
cana-408	74	4	vertex	vertex	NOUN
cana-408	74	5	in	in	ADP
cana-408	74	6	γ(l	γ(l	NOUN
cana-408	74	7	)	)	PUNCT
cana-408	74	8	is	be	AUX
cana-408	74	9	either	either	CCONJ
cana-408	74	10	a	a	DET
cana-408	74	11	vertex	vertex	NOUN
cana-408	74	12	of	of	ADP
cana-408	74	13	the	the	DET
cana-408	74	14	core	core	NOUN
cana-408	74	15	k	k	PROPN
cana-408	74	16	of	of	ADP
cana-408	74	17	γ(l	γ(l	PROPN
cana-408	74	18	)	)	PUNCT
cana-408	74	19	or	or	CCONJ
cana-408	74	20	else	else	ADV
cana-408	74	21	is	be	AUX
cana-408	74	22	a	a	DET
cana-408	74	23	pendant	pendant	ADJ
cana-408	74	24	vertex	vertex	NOUN
cana-408	74	25	of	of	ADP
cana-408	74	26	γ(l	γ(l	NOUN
cana-408	74	27	)	)	PUNCT
cana-408	74	28	.	.	PUNCT
cana-408	75	1	it	it	PRON
cana-408	75	2	is	be	AUX
cana-408	75	3	also	also	ADV
cana-408	75	4	shown	show	VERB
cana-408	75	5	that	that	SCONJ
cana-408	75	6	if	if	SCONJ
cana-408	75	7	l	l	NOUN
cana-408	75	8	does	do	AUX
cana-408	75	9	not	not	PART
cana-408	75	10	contain	contain	VERB
cana-408	75	11	any	any	DET
cana-408	75	12	atom	atom	NOUN
cana-408	75	13	,	,	PUNCT
cana-408	75	14	then	then	ADV
cana-408	75	15	every	every	DET
cana-408	75	16	pair	pair	NOUN
cana-408	75	17	of	of	ADP
cana-408	75	18	vertices	vertex	NOUN
cana-408	75	19	in	in	ADP
cana-408	75	20	γ(l	γ(l	NOUN
cana-408	75	21	)	)	PUNCT
cana-408	75	22	is	be	AUX
cana-408	75	23	contained	contain	VERB
cana-408	75	24	in	in	ADP
cana-408	75	25	a	a	DET
cana-408	75	26	cycle	cycle	NOUN
cana-408	75	27	of	of	ADP
cana-408	75	28	length	length	NOUN
cana-408	75	29	≤	≤	NOUN
cana-408	75	30	i6	i6	NOUN
cana-408	75	31	.	.	PUNCT
cana-408	76	1	i	i	PRON
cana-408	76	2	definition	definition	NOUN
cana-408	76	3	i1.4	i1.4	NOUN
cana-408	76	4	.	.	PUNCT
cana-408	77	1	[	[	X
cana-408	77	2	1	1	X
cana-408	77	3	]	]	X
cana-408	77	4	let	let	VERB
cana-408	77	5	(	(	PUNCT
cana-408	77	6	l	l	NOUN
cana-408	77	7	,	,	PUNCT
cana-408	77	8	≤	≤	NUM
cana-408	77	9	)	)	PUNCT
cana-408	77	10	be	be	AUX
cana-408	77	11	a	a	DET
cana-408	77	12	meet	meet	ADJ
cana-408	77	13	-	-	PUNCT
cana-408	77	14	semilattice	semilattice	NOUN
cana-408	77	15	.	.	PUNCT
cana-408	78	1	for	for	ADP
cana-408	78	2	any	any	DET
cana-408	78	3	a	a	PRON
cana-408	78	4	,	,	PUNCT
cana-408	78	5	b	b	X
cana-408	78	6	∈	∈	PROPN
cana-408	78	7	l	l	NOUN
cana-408	78	8	either	either	CCONJ
cana-408	78	9	a	a	DET
cana-408	78	10	≤	≤	NUM
cana-408	78	11	b	b	NOUN
cana-408	78	12	or	or	CCONJ
cana-408	78	13	b	b	NOUN
cana-408	78	14	≤	≤	NOUN
cana-408	78	15	a	a	DET
cana-408	78	16	holds	hold	NOUN
cana-408	78	17	then	then	ADV
cana-408	78	18	(	(	PUNCT
cana-408	78	19	l	l	NOUN
cana-408	78	20	,	,	PUNCT
cana-408	78	21	≤	≤	NUM
cana-408	78	22	)	)	PUNCT
cana-408	78	23	is	be	AUX
cana-408	78	24	called	call	VERB
cana-408	78	25	a	a	DET
cana-408	78	26	chain	chain	NOUN
cana-408	78	27	.	.	PUNCT
cana-408	79	1	definition	definition	NOUN
cana-408	79	2	i1.5	i1.5	NOUN
cana-408	79	3	.	.	PUNCT
cana-408	80	1	[	[	X
cana-408	80	2	2	2	X
cana-408	80	3	]	]	PUNCT
cana-408	80	4	in	in	ADP
cana-408	80	5	a	a	DET
cana-408	80	6	lattice	lattice	ADJ
cana-408	80	7	l	l	NOUN
cana-408	80	8	with	with	ADP
cana-408	80	9	0	0	NUM
cana-408	80	10	,	,	PUNCT
cana-408	80	11	a	a	DET
cana-408	80	12	nonzero	nonzero	NOUN
cana-408	80	13	element	element	NOUN
cana-408	80	14	a	a	DET
cana-408	80	15	∈	∈	PROPN
cana-408	80	16	l	l	NOUN
cana-408	80	17	is	be	AUX
cana-408	80	18	called	call	VERB
cana-408	80	19	an	an	DET
cana-408	80	20	atom	atom	NOUN
cana-408	80	21	if	if	SCONJ
cana-408	80	22	there	there	PRON
cana-408	80	23	is	be	VERB
cana-408	80	24	no	no	DET
cana-408	80	25	x	x	SYM
cana-408	80	26	∈	∈	NOUN
cana-408	80	27	l	l	NOUN
cana-408	80	28	such	such	ADJ
cana-408	80	29	that	that	SCONJ
cana-408	80	30	0	0	NUM
cana-408	80	31	<	<	X
cana-408	80	32	x	x	X
cana-408	80	33	<	<	X
cana-408	80	34	a.	a.	NOUN
cana-408	80	35	definition	definition	NOUN
cana-408	80	36	i1.6	i1.6	NOUN
cana-408	80	37	.	.	PUNCT
cana-408	81	1	[	[	X
cana-408	81	2	4	4	X
cana-408	81	3	]	]	PUNCT
cana-408	81	4	let	let	VERB
cana-408	81	5	g	g	PRON
cana-408	81	6	be	be	AUX
cana-408	81	7	a	a	DET
cana-408	81	8	graph	graph	NOUN
cana-408	81	9	.	.	PUNCT
cana-408	82	1	for	for	ADP
cana-408	82	2	distinct	distinct	ADJ
cana-408	82	3	vertices	vertex	NOUN
cana-408	82	4	x	x	PUNCT
cana-408	82	5	and	and	CCONJ
cana-408	82	6	y	y	PROPN
cana-408	82	7	of	of	ADP
cana-408	82	8	g	g	PROPN
cana-408	82	9	,	,	PUNCT
cana-408	82	10	let	let	VERB
cana-408	82	11	d	d	X
cana-408	82	12	(	(	PUNCT
cana-408	82	13	x	x	NOUN
cana-408	82	14	,	,	PUNCT
cana-408	82	15	y	y	NOUN
cana-408	82	16	)	)	PUNCT
cana-408	82	17	be	be	VERB
cana-408	82	18	the	the	DET
cana-408	82	19	length	length	NOUN
cana-408	82	20	of	of	ADP
cana-408	82	21	the	the	DET
cana-408	82	22	shortest	short	ADJ
cana-408	82	23	path	path	NOUN
cana-408	82	24	from	from	ADP
cana-408	82	25	x	x	PUNCT
cana-408	82	26	to	to	ADP
cana-408	82	27	y	y	PRON
cana-408	82	28	;	;	PUNCT
cana-408	82	29	(	(	PUNCT
cana-408	82	30	d	d	X
cana-408	82	31	(	(	PUNCT
cana-408	82	32	x	x	NOUN
cana-408	82	33	,	,	PUNCT
cana-408	82	34	y	y	NOUN
cana-408	82	35	)	)	PUNCT
cana-408	83	1	=	=	SYM
cana-408	83	2	∞	∞	NOUN
cana-408	83	3	if	if	SCONJ
cana-408	83	4	there	there	PRON
cana-408	83	5	is	be	VERB
cana-408	83	6	no	no	DET
cana-408	83	7	such	such	ADJ
cana-408	83	8	path	path	NOUN
cana-408	83	9	)	)	PUNCT
cana-408	83	10	.	.	PUNCT
cana-408	84	1	the	the	DET
cana-408	84	2	diameter	diameter	NOUN
cana-408	84	3	of	of	ADP
cana-408	84	4	g	g	PROPN
cana-408	84	5	is	be	AUX
cana-408	84	6	diam	diam	PROPN
cana-408	84	7	g	g	PROPN
cana-408	84	8	=	=	NOUN
cana-408	84	9	sup	sup	NOUN
cana-408	84	10	{	{	PUNCT
cana-408	84	11	d	d	X
cana-408	84	12	(	(	PUNCT
cana-408	84	13	x	x	NOUN
cana-408	84	14	,	,	PUNCT
cana-408	84	15	y	y	NOUN
cana-408	84	16	)	)	PUNCT
cana-408	85	1	|	|	ADV
cana-408	85	2	x	x	PUNCT
cana-408	85	3	and	and	CCONJ
cana-408	85	4	y	y	PROPN
cana-408	85	5	are	be	AUX
cana-408	85	6	distinct	distinct	ADJ
cana-408	85	7	vertices	vertex	NOUN
cana-408	85	8	of	of	ADP
cana-408	85	9	g	g	NOUN
cana-408	85	10	}	}	PUNCT
cana-408	85	11	.	.	PUNCT
cana-408	86	1	definition	definition	NOUN
cana-408	86	2	i1.7	i1.7	PROPN
cana-408	86	3	.	.	PUNCT
cana-408	87	1	[	[	X
cana-408	87	2	6	6	NUM
cana-408	87	3	]	]	PUNCT
cana-408	87	4	the	the	DET
cana-408	87	5	graph	graph	NOUN
cana-408	87	6	of	of	ADP
cana-408	87	7	g	g	NOUN
cana-408	87	8	,	,	PUNCT
cana-408	87	9	denoted	denote	VERB
cana-408	87	10	by	by	ADP
cana-408	87	11	gr(g	gr(g	PROPN
cana-408	87	12	)	)	PUNCT
cana-408	87	13	,	,	PUNCT
cana-408	87	14	is	be	AUX
cana-408	87	15	defined	define	VERB
cana-408	87	16	as	as	ADP
cana-408	87	17	the	the	DET
cana-408	87	18	length	length	NOUN
cana-408	87	19	of	of	ADP
cana-408	87	20	the	the	DET
cana-408	87	21	shortest	short	ADJ
cana-408	87	22	cycle	cycle	NOUN
cana-408	87	23	in	in	ADP
cana-408	87	24	g.	g.	PROPN
cana-408	87	25	(	(	PUNCT
cana-408	87	26	gr(g	gr(g	PROPN
cana-408	87	27	)	)	PUNCT
cana-408	88	1	=	=	SYM
cana-408	88	2	∞	∞	NOUN
cana-408	88	3	if	if	SCONJ
cana-408	88	4	g	g	PROPN
cana-408	88	5	contains	contain	VERB
cana-408	88	6	no	no	DET
cana-408	88	7	cycles	cycle	NOUN
cana-408	88	8	)	)	PUNCT
cana-408	88	9	.	.	PUNCT
cana-408	89	1	definition	definition	NOUN
cana-408	89	2	i1.8	i1.8	PROPN
cana-408	89	3	.	.	PUNCT
cana-408	90	1	[	[	X
cana-408	90	2	11	11	NUM
cana-408	90	3	]	]	X
cana-408	90	4	a	a	DET
cana-408	90	5	graph	graph	NOUN
cana-408	90	6	g	g	NOUN
cana-408	90	7	is	be	AUX
cana-408	90	8	called	call	VERB
cana-408	90	9	a	a	DET
cana-408	90	10	star	star	NOUN
cana-408	90	11	graph	graph	NOUN
cana-408	90	12	if	if	SCONJ
cana-408	90	13	it	it	PRON
cana-408	90	14	has	have	VERB
cana-408	90	15	a	a	DET
cana-408	90	16	vertex	vertex	NOUN
cana-408	90	17	adjacent	adjacent	ADJ
cana-408	90	18	to	to	ADP
cana-408	90	19	every	every	DET
cana-408	90	20	other	other	ADJ
cana-408	90	21	vertex	vertex	NOUN
cana-408	90	22	and	and	CCONJ
cana-408	90	23	these	these	PRON
cana-408	90	24	are	be	AUX
cana-408	90	25	the	the	DET
cana-408	90	26	only	only	ADJ
cana-408	90	27	adjacency	adjacency	PROPN
cana-408	90	28	relations	relation	NOUN
cana-408	90	29	.	.	PUNCT
cana-408	91	1	https://en.wikipedia.org/wiki/meet_(mathematics	https://en.wikipedia.org/wiki/meet_(mathematic	NOUN
cana-408	91	2	)	)	PUNCT
cana-408	92	1	https://en.wikipedia.org/wiki/greatest_lower_bound	https://en.wikipedia.org/wiki/greatest_lower_bound	ADP
cana-408	92	2	https://en.wikipedia.org/wiki/inverse_order	https://en.wikipedia.org/wiki/inverse_order	NOUN
cana-408	92	3	communications	communication	NOUN
cana-408	92	4	on	on	ADP
cana-408	92	5	applied	apply	VERB
cana-408	92	6	nonlinear	nonlinear	ADJ
cana-408	92	7	analysis	analysis	NOUN
cana-408	92	8	issn	issn	NOUN
cana-408	92	9	:	:	PUNCT
cana-408	92	10	1074	1074	NUM
cana-408	92	11	-	-	PUNCT
cana-408	92	12	133x	133x	NUM
cana-408	92	13	vol	vol	NOUN
cana-408	92	14	31	31	NUM
cana-408	92	15	no	no	NOUN
cana-408	92	16	.	.	NOUN
cana-408	92	17	1	1	NUM
cana-408	92	18	(	(	PUNCT
cana-408	92	19	2024	2024	NUM
cana-408	92	20	)	)	PUNCT
cana-408	92	21	234	234	NUM
cana-408	92	22	https://internationalpubls.com	https://internationalpubls.com	X
cana-408	92	23	3	3	X
cana-408	92	24	.	.	PUNCT
cana-408	92	25	methods	method	NOUN
cana-408	92	26	theorem	theorem	VERB
cana-408	92	27	i1.1	i1.1	NOUN
cana-408	92	28	[	[	X
cana-408	92	29	9	9	NUM
cana-408	92	30	]	]	X
cana-408	92	31	ithe	ithe	ADJ
cana-408	92	32	izero	izero	NOUN
cana-408	92	33	-	-	PUNCT
cana-408	92	34	divisor	divisor	NOUN
cana-408	92	35	igraph	igraph	NOUN
cana-408	92	36	iof	iof	X
cana-408	92	37	ia	ia	PROPN
cana-408	92	38	ifinite	ifinite	PROPN
cana-408	92	39	imeet	imeet	NOUN
cana-408	92	40	-	-	PUNCT
cana-408	92	41	semilattice	semilattice	NOUN
cana-408	92	42	iwith	iwith	ADP
cana-408	92	43	ionly	ionly	ADV
cana-408	92	44	ione	ione	PROPN
cana-408	92	45	iatom	iatom	PROPN
cana-408	92	46	iis	iis	PROPN
cana-408	92	47	ithe	ithe	PROPN
cana-408	92	48	iempty	iempty	PROPN
cana-408	92	49	igraph	igraph	NOUN
cana-408	92	50	.	.	PUNCT
cana-408	93	1	ithe	ithe	PROPN
cana-408	93	2	izero	izero	NOUN
cana-408	93	3	-	-	PUNCT
cana-408	93	4	divisor	divisor	NOUN
cana-408	93	5	igraph	igraph	NOUN
cana-408	93	6	iof	iof	VERB
cana-408	93	7	ithe	ithe	PROPN
cana-408	93	8	imeet	imeet	NOUN
cana-408	93	9	semilattice	semilattice	NOUN
cana-408	93	10	iin	iin	NOUN
cana-408	93	11	ifigure	ifigure	NOUN
cana-408	93	12	i1.1	i1.1	PRON
cana-408	93	13	iis	iis	PROPN
cana-408	93	14	ithe	ithe	PROPN
cana-408	93	15	iempty	iempty	PROPN
cana-408	93	16	igraph	igraph	NOUN
cana-408	93	17	.	.	PUNCT
cana-408	94	1	however	however	ADV
cana-408	94	2	,	,	PUNCT
cana-408	94	3	ithis	ithis	PROPN
cana-408	94	4	idoes	idoe	NOUN
cana-408	94	5	inot	inot	VERB
cana-408	94	6	ihold	ihold	ADJ
cana-408	94	7	ifor	ifor	PROPN
cana-408	94	8	iinfinite	iinfinite	PROPN
cana-408	94	9	imeet	imeet	NOUN
cana-408	94	10	-	-	PUNCT
cana-408	94	11	semilattices	semilattice	NOUN
cana-408	94	12	iwith	iwith	ADP
cana-408	94	13	ione	ione	PROPN
cana-408	94	14	iatom	iatom	PROPN
cana-408	94	15	.	.	PUNCT
cana-408	95	1	ifor	ifor	PROPN
cana-408	95	2	iconsider	iconsider	PROPN
cana-408	95	3	,	,	PUNCT
cana-408	95	4	ithe	ithe	PRON
cana-408	95	5	iinfinite	iinfinite	PROPN
cana-408	95	6	imeet	imeet	NOUN
cana-408	95	7	-	-	PUNCT
cana-408	95	8	semilattice	semilattice	NOUN
cana-408	95	9	igiven	igiven	ADJ
cana-408	95	10	iin	iin	NOUN
cana-408	95	11	ifigure	ifigure	NOUN
cana-408	95	12	i1.2	i1.2	PROPN
cana-408	95	13	,	,	PUNCT
cana-408	95	14	iwhere	iwhere	ADV
cana-408	95	15	ithe	ithe	NOUN
cana-408	95	16	idescending	idescende	VERB
cana-408	95	17	idots	idot	NOUN
cana-408	95	18	irepresent	irepresent	ADJ
cana-408	95	19	iinfinite	iinfinite	PROPN
cana-408	95	20	idescending	idescende	VERB
cana-408	95	21	ichain	ichain	NOUN
cana-408	95	22	.	.	PUNCT
cana-408	96	1	iit	iit	PROPN
cana-408	96	2	ihas	iha	VERB
cana-408	96	3	ionly	ionly	ADV
cana-408	96	4	ione	ione	PROPN
cana-408	96	5	iatom	iatom	PROPN
cana-408	96	6	ic	ic	PROPN
cana-408	96	7	,	,	PUNCT
cana-408	96	8	ibut	ibut	PROPN
cana-408	96	9	iits	iit	VERB
cana-408	96	10	igraph	igraph	ADJ
cana-408	96	11	iγ(𝐿	iγ(𝐿	ADV
cana-408	96	12	)	)	PUNCT
cana-408	96	13	iis	iis	PROPN
cana-408	96	14	ian	ian	PROPN
cana-408	96	15	iinfinite	iinfinite	PROPN
cana-408	96	16	istar	istar	PROPN
cana-408	96	17	igraph	igraph	PROPN
cana-408	96	18	.	.	PUNCT
cana-408	97	1	i	i	PRON
cana-408	97	2	i	i	PRON
cana-408	98	1	i	i	PRON
cana-408	98	2	i	i	PRON
cana-408	99	1	i	i	PRON
cana-408	99	2	i	i	PRON
cana-408	100	1	i	i	PRON
cana-408	100	2	i	i	PRON
cana-408	101	1	i	i	PRON
cana-408	101	2	i	i	PRON
cana-408	102	1	i	i	PRON
cana-408	102	2	i	i	PRON
cana-408	103	1	i	i	PRON
cana-408	103	2	i	i	PRON
cana-408	104	1	i	i	PRON
cana-408	104	2	i	i	PRON
cana-408	105	1	i	i	PRON
cana-408	105	2	i	i	PRON
cana-408	106	1	i	i	PRON
cana-408	106	2	i	i	PRON
cana-408	107	1	i	i	PRON
cana-408	107	2	i	i	PRON
cana-408	108	1	i	i	PRON
cana-408	108	2	i	i	PRON
cana-408	109	1	i	i	PRON
cana-408	109	2	i	i	PRON
cana-408	110	1	i	i	PRON
cana-408	110	2	i	i	PRON
cana-408	111	1	i	i	PRON
cana-408	111	2	i	i	PRON
cana-408	112	1	i	i	PRON
cana-408	112	2	i	i	PRON
cana-408	113	1	i	i	PRON
cana-408	113	2	i	i	PRON
cana-408	114	1	i	i	PRON
cana-408	114	2	i	i	PRON
cana-408	115	1	i	i	PRON
cana-408	115	2	i	i	PRON
cana-408	116	1	i	i	PRON
cana-408	116	2	figure	figure	VERB
cana-408	116	3	i1.1	i1.1	DET
cana-408	116	4	ifigure	ifigure	NOUN
cana-408	116	5	i1.2	i1.2	NOUN
cana-408	116	6	theorem	theorem	VERB
cana-408	116	7	1.2	1.2	NUM
cana-408	116	8	[	[	X
cana-408	116	9	12	12	NUM
cana-408	116	10	]	]	PUNCT
cana-408	116	11	every	every	DET
cana-408	116	12	disconnected	disconnected	ADJ
cana-408	116	13	graph	graph	NOUN
cana-408	116	14	can	can	AUX
cana-408	116	15	not	not	PART
cana-408	116	16	be	be	AUX
cana-408	116	17	a	a	DET
cana-408	116	18	graph	graph	NOUN
cana-408	116	19	of	of	ADP
cana-408	116	20	any	any	DET
cana-408	116	21	meetsemilattice	meetsemilattice	NOUN
cana-408	116	22	l	l	NOUN
cana-408	116	23	with	with	ADP
cana-408	116	24	0	0	NUM
cana-408	116	25	.	.	PUNCT
cana-408	116	26	remark	remark	VERB
cana-408	116	27	1.2	1.2	NUM
cana-408	116	28	a	a	DET
cana-408	116	29	graphs	graph	NOUN
cana-408	116	30	of	of	ADP
cana-408	116	31	product	product	NOUN
cana-408	116	32	of	of	ADP
cana-408	116	33	imeet	imeet	NOUN
cana-408	116	34	-	-	PUNCT
cana-408	116	35	semilattices	semilattice	NOUN
cana-408	116	36	and	and	CCONJ
cana-408	116	37	obtain	obtain	VERB
cana-408	116	38	some	some	DET
cana-408	116	39	properties	property	NOUN
cana-408	116	40	of	of	ADP
cana-408	116	41	such	such	ADJ
cana-408	116	42	graphs	graph	NOUN
cana-408	116	43	.	.	PUNCT
cana-408	117	1	in	in	ADP
cana-408	117	2	this	this	DET
cana-408	117	3	section	section	NOUN
cana-408	117	4	,	,	PUNCT
cana-408	117	5	we	we	PRON
cana-408	117	6	consider	consider	VERB
cana-408	117	7	two	two	NUM
cana-408	117	8	integral	integral	ADJ
cana-408	117	9	meet	meet	NOUN
cana-408	117	10	-	-	PUNCT
cana-408	117	11	semilattices	semilattice	NOUN
cana-408	117	12	l_1	l_1	PROPN
cana-408	117	13	and	and	CCONJ
cana-408	117	14	l_2	l_2	VERB
cana-408	117	15	with	with	ADP
cana-408	117	16	l	l	NOUN
cana-408	117	17	∼=l_1	∼=l_1	X
cana-408	117	18	x	x	PUNCT
cana-408	117	19	l_2	l_2	VERB
cana-408	117	20	and	and	CCONJ
cana-408	117	21	show	show	VERB
cana-408	117	22	that	that	SCONJ
cana-408	117	23	if	if	SCONJ
cana-408	117	24	∣l_1∣=m+1	∣l_1∣=m+1	ADJ
cana-408	117	25	,	,	PUNCT
cana-408	117	26	∣l_2∣=	∣l_2∣=	NOUN
cana-408	117	27	n+1	n+1	PROPN
cana-408	117	28	,	,	PUNCT
cana-408	117	29	then	then	ADV
cana-408	117	30	γ(l	γ(l	PROPN
cana-408	117	31	)	)	PUNCT
cana-408	117	32	is	be	AUX
cana-408	117	33	the	the	DET
cana-408	117	34	complete	complete	ADJ
cana-408	117	35	bipartite	bipartite	PROPN
cana-408	117	36	graph	graph	NOUN
cana-408	117	37	k	k	PROPN
cana-408	117	38	(	(	PUNCT
cana-408	117	39	m	m	PROPN
cana-408	117	40	,	,	PUNCT
cana-408	117	41	n	n	CCONJ
cana-408	117	42	)	)	PUNCT
cana-408	117	43	.	.	PUNCT
cana-408	118	1	also	also	ADV
cana-408	118	2	,	,	PUNCT
cana-408	118	3	it	it	PRON
cana-408	118	4	is	be	AUX
cana-408	118	5	shown	show	VERB
cana-408	118	6	that	that	SCONJ
cana-408	118	7	gr	gr	INTJ
cana-408	118	8	(	(	PUNCT
cana-408	118	9	γ(l	γ(l	NOUN
cana-408	118	10	)	)	PUNCT
cana-408	118	11	)	)	PUNCT
cana-408	119	1	=	=	SYM
cana-408	119	2	∞	∞	NOUN
cana-408	119	3	if	if	SCONJ
cana-408	119	4	and	and	CCONJ
cana-408	119	5	only	only	ADV
cana-408	119	6	if	if	SCONJ
cana-408	119	7	either	either	CCONJ
cana-408	119	8	(	(	PUNCT
cana-408	119	9	i	i	NOUN
cana-408	119	10	)	)	PUNCT
cana-408	119	11	|	|	ADV
cana-408	119	12	γ(l	γ(l	NOUN
cana-408	119	13	)	)	PUNCT
cana-408	119	14	|	|	ADV
cana-408	119	15	≤	≤	NUM
cana-408	119	16	2	2	NUM
cana-408	119	17	or	or	CCONJ
cana-408	119	18	(	(	PUNCT
cana-408	119	19	ii	ii	NOUN
cana-408	119	20	)	)	PUNCT
cana-408	119	21	|	|	ADV
cana-408	119	22	γ(l	γ(l	NOUN
cana-408	119	23	)	)	PUNCT
cana-408	120	1	|	|	ADV
cana-408	120	2	=	=	SYM
cana-408	120	3	3	3	NUM
cana-408	120	4	and	and	CCONJ
cana-408	120	5	γ(l	γ(l	NOUN
cana-408	120	6	)	)	PUNCT
cana-408	120	7	is	be	AUX
cana-408	120	8	not	not	PART
cana-408	120	9	complete	complete	ADJ
cana-408	120	10	or	or	CCONJ
cana-408	120	11	(	(	PUNCT
cana-408	120	12	iii	iii	NOUN
cana-408	120	13	)	)	PUNCT
cana-408	120	14	l	l	NOUN
cana-408	120	15	∼	∼	NOUN
cana-408	120	16	=	=	NOUN
cana-408	120	17	〖	〖	NOUN
cana-408	120	18	c〗_2	c〗_2	X
cana-408	120	19	x	x	X
cana-408	120	20	l_1	l_1	PROPN
cana-408	120	21	,	,	PUNCT
cana-408	120	22	where	where	SCONJ
cana-408	120	23	l_1	l_1	PROPN
cana-408	120	24	an	an	DET
cana-408	120	25	integral	integral	ADJ
cana-408	120	26	meet	meet	NOUN
cana-408	120	27	-	-	PUNCT
cana-408	120	28	semilattice	semilattice	NOUN
cana-408	120	29	and	and	CCONJ
cana-408	120	30	c_2	c_2	NOUN
cana-408	120	31	is	be	AUX
cana-408	120	32	a	a	DET
cana-408	120	33	two	two	NUM
cana-408	120	34	-	-	PUNCT
cana-408	120	35	element	element	NOUN
cana-408	120	36	chain	chain	NOUN
cana-408	120	37	.	.	PUNCT
cana-408	121	1	in	in	ADP
cana-408	121	2	this	this	DET
cana-408	121	3	case	case	NOUN
cana-408	121	4	,	,	PUNCT
cana-408	121	5	γ(l	γ(l	PROPN
cana-408	121	6	)	)	PUNCT
cana-408	121	7	is	be	AUX
cana-408	121	8	a	a	DET
cana-408	121	9	star	star	NOUN
cana-408	121	10	igraph	igraph	NOUN
cana-408	121	11	.	.	PUNCT
cana-408	122	1	further	further	ADJ
cana-408	122	2	γ(l	γ(l	NOUN
cana-408	122	3	)	)	PUNCT
cana-408	122	4	has	have	VERB
cana-408	122	5	a	a	DET
cana-408	122	6	cycle	cycle	NOUN
cana-408	122	7	of	of	ADP
cana-408	122	8	length	length	NOUN
cana-408	122	9	3	3	NUM
cana-408	122	10	or	or	CCONJ
cana-408	122	11	4	4	NUM
cana-408	122	12	(	(	PUNCT
cana-408	122	13	i.e.	i.e.	X
cana-408	122	14	gr	gr	DET
cana-408	122	15	γ(l	γ(l	NOUN
cana-408	122	16	)	)	PUNCT
cana-408	122	17	≤	≤	NUM
cana-408	122	18	4	4	NUM
cana-408	122	19	)	)	PUNCT
cana-408	122	20	,	,	PUNCT
cana-408	122	21	and	and	CCONJ
cana-408	122	22	γ(l	γ(l	NOUN
cana-408	122	23	)	)	PUNCT
cana-408	122	24	is	be	AUX
cana-408	122	25	a	a	DET
cana-408	122	26	star	star	NOUN
cana-408	122	27	graph	graph	NOUN
cana-408	122	28	.	.	PUNCT
cana-408	123	1	results	result	NOUN
cana-408	123	2	theorem	theorem	VERB
cana-408	123	3	2.1	2.1	NUM
cana-408	123	4	.	.	PUNCT
cana-408	124	1	if	if	SCONJ
cana-408	124	2	l	l	NOUN
cana-408	124	3	does	do	AUX
cana-408	124	4	not	not	PART
cana-408	124	5	contain	contain	VERB
cana-408	124	6	any	any	DET
cana-408	124	7	atom	atom	NOUN
cana-408	124	8	,	,	PUNCT
cana-408	124	9	then	then	ADV
cana-408	124	10	any	any	DET
cana-408	124	11	edge	edge	NOUN
cana-408	124	12	in	in	ADP
cana-408	124	13	γ(l	γ(l	NOUN
cana-408	124	14	)	)	PUNCT
cana-408	124	15	is	be	AUX
cana-408	124	16	contained	contain	VERB
cana-408	124	17	in	in	ADP
cana-408	124	18	a	a	DET
cana-408	124	19	cycle	cycle	NOUN
cana-408	124	20	of	of	ADP
cana-408	124	21	length	length	NOUN
cana-408	124	22	≤	≤	NUM
cana-408	124	23	6	6	NUM
cana-408	124	24	,	,	PUNCT
cana-408	124	25	and	and	CCONJ
cana-408	124	26	therefore	therefore	ADV
cana-408	124	27	γ(l	γ(l	PROPN
cana-408	124	28	)	)	PUNCT
cana-408	124	29	is	be	AUX
cana-408	124	30	a	a	DET
cana-408	124	31	union	union	NOUN
cana-408	124	32	of	of	ADP
cana-408	124	33	4	4	NUM
cana-408	124	34	cycles	cycle	NOUN
cana-408	124	35	and	and	CCONJ
cana-408	124	36	5	5	NUM
cana-408	124	37	cycles	cycle	NOUN
cana-408	124	38	.	.	PUNCT
cana-408	125	1	corollary	corollary	ADJ
cana-408	125	2	2.1	2.1	NUM
cana-408	125	3	.	.	PUNCT
cana-408	126	1	for	for	ADP
cana-408	126	2	any	any	DET
cana-408	126	3	meet	meet	ADJ
cana-408	126	4	-	-	PUNCT
cana-408	126	5	semilattice	semilattice	NOUN
cana-408	126	6	l	l	NOUN
cana-408	126	7	,	,	PUNCT
cana-408	126	8	let	let	VERB
cana-408	126	9	k	k	X
cana-408	126	10	,	,	PUNCT
cana-408	126	11	the	the	DET
cana-408	126	12	core	core	NOUN
cana-408	126	13	(	(	PUNCT
cana-408	126	14	a	a	DET
cana-408	126	15	notion	notion	NOUN
cana-408	126	16	that	that	SCONJ
cana-408	126	17	describes	describe	VERB
cana-408	126	18	behavior	behavior	NOUN
cana-408	126	19	of	of	ADP
cana-408	126	20	a	a	DET
cana-408	126	21	graph	graph	NOUN
cana-408	126	22	)	)	PUNCT
cana-408	126	23	of	of	ADP
cana-408	126	24	γ(l	γ(l	PROPN
cana-408	126	25	)	)	PUNCT
cana-408	126	26	,	,	PUNCT
cana-408	126	27	be	be	AUX
cana-408	126	28	the	the	DET
cana-408	126	29	union	union	NOUN
cana-408	126	30	of	of	ADP
cana-408	126	31	cycles	cycle	NOUN
cana-408	126	32	in	in	ADP
cana-408	126	33	γ(l	γ(l	PROPN
cana-408	126	34	)	)	PUNCT
cana-408	126	35	.	.	PUNCT
cana-408	127	1	theorem	theorem	VERB
cana-408	127	2	2.2	2.2	NUM
cana-408	127	3	.	.	PUNCT
cana-408	128	1	let	let	VERB
cana-408	128	2	l	l	NOUN
cana-408	128	3	be	be	AUX
cana-408	128	4	a	a	DET
cana-408	128	5	meet	meet	NOUN
cana-408	128	6	-	-	PUNCT
cana-408	128	7	semilattice	semilattice	NOUN
cana-408	128	8	with	with	ADP
cana-408	128	9	0	0	NUM
cana-408	128	10	.	.	PUNCT
cana-408	129	1	if	if	SCONJ
cana-408	129	2	γ(l	γ(l	NOUN
cana-408	129	3	)	)	PUNCT
cana-408	129	4	contains	contain	VERB
cana-408	129	5	a	a	DET
cana-408	129	6	cycle	cycle	NOUN
cana-408	129	7	,	,	PUNCT
cana-408	129	8	then	then	ADV
cana-408	129	9	the	the	DET
cana-408	129	10	core	core	NOUN
cana-408	129	11	k	k	PROPN
cana-408	129	12	of	of	ADP
cana-408	129	13	γ(l	γ(l	PROPN
cana-408	129	14	)	)	PUNCT
cana-408	129	15	is	be	AUX
cana-408	129	16	a	a	DET
cana-408	129	17	union	union	NOUN
cana-408	129	18	of	of	ADP
cana-408	129	19	4cycles	4cycles	NUM
cana-408	129	20	and	and	CCONJ
cana-408	129	21	5	5	NUM
cana-408	129	22	–	–	PUNCT
cana-408	129	23	cycles	cycle	NOUN
cana-408	129	24	and	and	CCONJ
cana-408	129	25	any	any	DET
cana-408	129	26	vertex	vertex	NOUN
cana-408	129	27	in	in	ADP
cana-408	129	28	γ(l	γ(l	NOUN
cana-408	129	29	)	)	PUNCT
cana-408	129	30	is	be	AUX
cana-408	129	31	either	either	CCONJ
cana-408	129	32	a	a	DET
cana-408	129	33	vertex	vertex	NOUN
cana-408	129	34	of	of	ADP
cana-408	129	35	the	the	DET
cana-408	129	36	core	core	NOUN
cana-408	129	37	k	k	PROPN
cana-408	129	38	of	of	ADP
cana-408	129	39	γ(l	γ(l	PROPN
cana-408	129	40	)	)	PUNCT
cana-408	129	41	or	or	CCONJ
cana-408	129	42	is	be	AUX
cana-408	129	43	a	a	DET
cana-408	129	44	pendant	pendant	NOUN
cana-408	129	45	of	of	ADP
cana-408	129	46	γ(l	γ(l	NOUN
cana-408	129	47	)	)	PUNCT
cana-408	129	48	.	.	PUNCT
cana-408	130	1	remark	remark	VERB
cana-408	130	2	2.1	2.1	NUM
cana-408	130	3	.	.	PUNCT
cana-408	131	1	let	let	VERB
cana-408	131	2	l_1	l_1	PROPN
cana-408	131	3	and	and	CCONJ
cana-408	131	4	l_2	l_2	AUX
cana-408	131	5	be	be	AUX
cana-408	131	6	two	two	NUM
cana-408	131	7	meet	meet	NOUN
cana-408	131	8	-	-	PUNCT
cana-408	131	9	semilattices	semilattice	NOUN
cana-408	131	10	with	with	ADP
cana-408	131	11	0	0	NUM
cana-408	131	12	and	and	CCONJ
cana-408	131	13	l	l	NOUN
cana-408	132	1	=	=	SYM
cana-408	132	2	l_1	l_1	PROPN
cana-408	132	3	x	x	SYM
cana-408	132	4	l_2	l_2	PROPN
cana-408	132	5	,	,	PUNCT
cana-408	132	6	then	then	ADV
cana-408	132	7	γ(l	γ(l	PROPN
cana-408	132	8	)	)	PUNCT
cana-408	132	9	is	be	AUX
cana-408	132	10	star	star	NOUN
cana-408	132	11	graph	graph	NOUN
cana-408	132	12	if	if	SCONJ
cana-408	132	13	and	and	CCONJ
cana-408	132	14	only	only	ADV
cana-408	132	15	if	if	SCONJ
cana-408	132	16	one	one	NUM
cana-408	132	17	of	of	ADP
cana-408	132	18	the	the	DET
cana-408	132	19	l_1	l_1	PROPN
cana-408	132	20	or	or	CCONJ
cana-408	132	21	l_2	l_2	NOUN
cana-408	132	22	is	be	AUX
cana-408	132	23	c_2	c_2	ADJ
cana-408	132	24	and	and	CCONJ
cana-408	132	25	the	the	DET
cana-408	132	26	other	other	ADJ
cana-408	132	27	is	be	AUX
cana-408	132	28	an	an	DET
cana-408	132	29	integral	integral	ADJ
cana-408	132	30	meet	meet	NOUN
cana-408	132	31	-	-	PUNCT
cana-408	132	32	semilattice	semilattice	NOUN
cana-408	132	33	.	.	PUNCT
cana-408	133	1	communications	communication	NOUN
cana-408	133	2	on	on	ADP
cana-408	133	3	applied	apply	VERB
cana-408	133	4	nonlinear	nonlinear	ADJ
cana-408	133	5	analysis	analysis	NOUN
cana-408	133	6	issn	issn	NOUN
cana-408	133	7	:	:	PUNCT
cana-408	133	8	1074	1074	NUM
cana-408	133	9	-	-	PUNCT
cana-408	133	10	133x	133x	NUM
cana-408	133	11	vol	vol	NOUN
cana-408	133	12	31	31	NUM
cana-408	133	13	no	no	NOUN
cana-408	133	14	.	.	NOUN
cana-408	133	15	1	1	NUM
cana-408	133	16	(	(	PUNCT
cana-408	133	17	2024	2024	NUM
cana-408	133	18	)	)	PUNCT
cana-408	133	19	235	235	NUM
cana-408	133	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-408	133	21	theorem	theorem	VERB
cana-408	133	22	2.3	2.3	NUM
cana-408	133	23	.	.	PUNCT
cana-408	134	1	let	let	VERB
cana-408	134	2	l_1	l_1	PROPN
cana-408	134	3	and	and	CCONJ
cana-408	134	4	l_2	l_2	AUX
cana-408	134	5	be	be	AUX
cana-408	134	6	two	two	NUM
cana-408	134	7	meet	meet	NOUN
cana-408	134	8	-	-	PUNCT
cana-408	134	9	semilattices	semilattice	NOUN
cana-408	134	10	with	with	ADP
cana-408	134	11	0	0	NUM
cana-408	134	12	and	and	CCONJ
cana-408	134	13	l	l	NOUN
cana-408	134	14	=	=	SYM
cana-408	134	15	l_1	l_1	PROPN
cana-408	134	16	x	x	SYM
cana-408	134	17	l_2	l_2	PROPN
cana-408	134	18	.	.	PUNCT
cana-408	135	1	then	then	ADV
cana-408	135	2	exactly	exactly	ADV
cana-408	135	3	one	one	NUM
cana-408	135	4	of	of	ADP
cana-408	135	5	the	the	DET
cana-408	135	6	following	follow	VERB
cana-408	135	7	holds	hold	VERB
cana-408	135	8	:	:	PUNCT
cana-408	135	9	1	1	X
cana-408	135	10	.	.	PUNCT
cana-408	135	11	γ(l	γ(l	NOUN
cana-408	135	12	)	)	PUNCT
cana-408	135	13	has	have	VERB
cana-408	135	14	a	a	DET
cana-408	135	15	cycle	cycle	NOUN
cana-408	135	16	of	of	ADP
cana-408	135	17	length	length	NOUN
cana-408	135	18	n-1	n-1	NOUN
cana-408	135	19	or	or	CCONJ
cana-408	135	20	n	n	PRON
cana-408	135	21	(	(	PUNCT
cana-408	135	22	that	that	PRON
cana-408	135	23	is	be	AUX
cana-408	135	24	gr	gr	DET
cana-408	135	25	γ(l)≤	γ(l)≤	NOUN
cana-408	135	26	n	n	CCONJ
cana-408	135	27	)	)	PUNCT
cana-408	135	28	,	,	PUNCT
cana-408	135	29	2	2	X
cana-408	135	30	.	.	PUNCT
cana-408	135	31	γ(l	γ(l	NOUN
cana-408	135	32	)	)	PUNCT
cana-408	135	33	is	be	AUX
cana-408	135	34	a	a	DET
cana-408	135	35	star	star	NOUN
cana-408	135	36	graph	graph	NOUN
cana-408	135	37	.	.	PUNCT
cana-408	136	1	4	4	X
cana-408	136	2	.	.	X
cana-408	136	3	discussion	discussion	NOUN
cana-408	136	4	theorem	theorem	VERB
cana-408	136	5	2.1	2.1	NUM
cana-408	136	6	.	.	PUNCT
cana-408	137	1	if	if	SCONJ
cana-408	137	2	l	l	NOUN
cana-408	137	3	does	do	AUX
cana-408	137	4	not	not	PART
cana-408	137	5	contain	contain	VERB
cana-408	137	6	any	any	DET
cana-408	137	7	atom	atom	NOUN
cana-408	137	8	,	,	PUNCT
cana-408	137	9	then	then	ADV
cana-408	137	10	any	any	DET
cana-408	137	11	edge	edge	NOUN
cana-408	137	12	in	in	ADP
cana-408	137	13	γ(l	γ(l	NOUN
cana-408	137	14	)	)	PUNCT
cana-408	137	15	is	be	AUX
cana-408	137	16	contained	contain	VERB
cana-408	137	17	in	in	ADP
cana-408	137	18	a	a	DET
cana-408	137	19	cycle	cycle	NOUN
cana-408	137	20	of	of	ADP
cana-408	137	21	length	length	NOUN
cana-408	137	22	≤	≤	NUM
cana-408	137	23	6	6	NUM
cana-408	137	24	,	,	PUNCT
cana-408	137	25	and	and	CCONJ
cana-408	137	26	therefore	therefore	ADV
cana-408	137	27	γ(l	γ(l	PROPN
cana-408	137	28	)	)	PUNCT
cana-408	137	29	is	be	AUX
cana-408	137	30	a	a	DET
cana-408	137	31	union	union	NOUN
cana-408	137	32	of	of	ADP
cana-408	137	33	4	4	NUM
cana-408	137	34	cycles	cycle	NOUN
cana-408	137	35	and	and	CCONJ
cana-408	137	36	5	5	NUM
cana-408	137	37	cycles	cycle	NOUN
cana-408	137	38	.	.	PUNCT
cana-408	138	1	proof	proof	NOUN
cana-408	138	2	.	.	PUNCT
cana-408	139	1	let	let	VERB
cana-408	139	2	a−	a−	PROPN
cana-408	139	3	x	x	PART
cana-408	139	4	be	be	AUX
cana-408	139	5	an	an	DET
cana-408	139	6	edge	edge	NOUN
cana-408	139	7	in	in	ADP
cana-408	139	8	γ(l	γ(l	NOUN
cana-408	139	9	)	)	PUNCT
cana-408	139	10	.	.	PUNCT
cana-408	140	1	since	since	SCONJ
cana-408	140	2	γ(l	γ(l	PROPN
cana-408	140	3	)	)	PUNCT
cana-408	140	4	is	be	AUX
cana-408	140	5	connected	connect	VERB
cana-408	140	6	and	and	CCONJ
cana-408	140	7	|	|	ADV
cana-408	140	8	γ(l)|	γ(l)|	VERB
cana-408	140	9	≥	≥	NUM
cana-408	140	10	4	4	NUM
cana-408	140	11	,	,	PUNCT
cana-408	140	12	there	there	PRON
cana-408	140	13	exists	exist	VERB
cana-408	140	14	a	a	DET
cana-408	140	15	vertex	vertex	NOUN
cana-408	140	16	b	b	NOUN
cana-408	140	17	in	in	ADP
cana-408	140	18	γ(l	γ(l	NOUN
cana-408	140	19	)	)	PUNCT
cana-408	140	20	with	with	ADP
cana-408	140	21	a	a	DET
cana-408	140	22	−	−	NOUN
cana-408	140	23	x	x	SYM
cana-408	140	24	−	−	PROPN
cana-408	140	25	b	b	NOUN
cana-408	140	26	or	or	CCONJ
cana-408	140	27	x	x	SYM
cana-408	140	28	−	−	NOUN
cana-408	140	29	a	a	PRON
cana-408	140	30	–	–	PUNCT
cana-408	140	31	b	b	NOUN
cana-408	140	32	is	be	AUX
cana-408	140	33	a	a	DET
cana-408	140	34	path	path	NOUN
cana-408	140	35	in	in	ADP
cana-408	140	36	γ(l	γ(l	NOUN
cana-408	140	37	)	)	PUNCT
cana-408	140	38	.	.	PUNCT
cana-408	141	1	in	in	ADP
cana-408	141	2	the	the	DET
cana-408	141	3	first	first	ADJ
cana-408	141	4	case	case	NOUN
cana-408	141	5	,	,	PUNCT
cana-408	141	6	if	if	SCONJ
cana-408	141	7	b	b	PROPN
cana-408	141	8	∧	∧	PROPN
cana-408	141	9	a	a	DET
cana-408	141	10	=	=	X
cana-408	141	11	0	0	PUNCT
cana-408	141	12	then	then	ADV
cana-408	141	13	a	a	DET
cana-408	141	14	−	−	NOUN
cana-408	141	15	x	x	SYM
cana-408	141	16	−	−	PROPN
cana-408	141	17	b	b	X
cana-408	141	18	−	−	PROPN
cana-408	142	1	a	a	PRON
cana-408	142	2	is	be	AUX
cana-408	142	3	a	a	DET
cana-408	142	4	4	4	NUM
cana-408	142	5	cycle	cycle	NOUN
cana-408	142	6	.	.	PUNCT
cana-408	143	1	if	if	SCONJ
cana-408	143	2	b	b	PROPN
cana-408	143	3	∧	∧	PROPN
cana-408	143	4	a	a	DET
cana-408	143	5	≠	≠	PROPN
cana-408	143	6	0	0	NUM
cana-408	143	7	,	,	PUNCT
cana-408	143	8	since	since	SCONJ
cana-408	143	9	x	x	PRON
cana-408	143	10	is	be	AUX
cana-408	143	11	not	not	PART
cana-408	143	12	an	an	DET
cana-408	143	13	atom	atom	NOUN
cana-408	143	14	then	then	ADV
cana-408	143	15	there	there	PRON
cana-408	143	16	exists	exist	VERB
cana-408	143	17	a	a	DET
cana-408	143	18	nonzero	nonzero	NOUN
cana-408	143	19	c	c	NOUN
cana-408	143	20	<	<	X
cana-408	143	21	x.	x.	NOUN
cana-408	143	22	then	then	ADV
cana-408	143	23	a	a	DET
cana-408	143	24	∧	∧	PROPN
cana-408	143	25	c	c	NOUN
cana-408	143	26	=	=	SYM
cana-408	143	27	0	0	NUM
cana-408	143	28	,	,	PUNCT
cana-408	143	29	b	b	PROPN
cana-408	144	1	∧	∧	NOUN
cana-408	144	2	c	c	NOUN
cana-408	144	3	=	=	SYM
cana-408	144	4	0	0	NUM
cana-408	144	5	.	.	PUNCT
cana-408	145	1	hence	hence	ADV
cana-408	145	2	,	,	PUNCT
cana-408	145	3	a	a	DET
cana-408	145	4	−	−	NOUN
cana-408	145	5	x	x	SYM
cana-408	146	1	−	−	PROPN
cana-408	146	2	b	b	X
cana-408	146	3	−	−	PROPN
cana-408	147	1	c	c	NOUN
cana-408	147	2	–	–	PUNCT
cana-408	147	3	d	d	NOUN
cana-408	147	4	–	–	PUNCT
cana-408	147	5	a	a	PRON
cana-408	147	6	is	be	AUX
cana-408	147	7	a	a	DET
cana-408	147	8	cycle	cycle	NOUN
cana-408	147	9	of	of	ADP
cana-408	147	10	length	length	NOUN
cana-408	147	11	4	4	NUM
cana-408	147	12	.	.	PUNCT
cana-408	148	1	thus	thus	ADV
cana-408	148	2	,	,	PUNCT
cana-408	148	3	x	x	PRON
cana-408	148	4	is	be	AUX
cana-408	148	5	contained	contain	VERB
cana-408	148	6	in	in	ADP
cana-408	148	7	a	a	DET
cana-408	148	8	cycle	cycle	NOUN
cana-408	148	9	of	of	ADP
cana-408	148	10	length	length	NOUN
cana-408	148	11	≤	≤	NUM
cana-408	148	12	4	4	NUM
cana-408	148	13	,	,	PUNCT
cana-408	148	14	so	so	SCONJ
cana-408	148	15	a	a	DET
cana-408	148	16	−	−	NOUN
cana-408	148	17	x	x	PUNCT
cana-408	148	18	is	be	AUX
cana-408	148	19	an	an	DET
cana-408	148	20	edge	edge	NOUN
cana-408	148	21	of	of	ADP
cana-408	148	22	either	either	CCONJ
cana-408	148	23	a	a	DET
cana-408	148	24	4	4	NUM
cana-408	148	25	cycles	cycle	NOUN
cana-408	148	26	or	or	CCONJ
cana-408	148	27	a	a	DET
cana-408	148	28	5	5	NUM
cana-408	148	29	cycles	cycle	NOUN
cana-408	148	30	.	.	PUNCT
cana-408	149	1	in	in	ADP
cana-408	149	2	the	the	DET
cana-408	149	3	second	second	ADJ
cana-408	149	4	case	case	NOUN
cana-408	149	5	,	,	PUNCT
cana-408	149	6	if	if	SCONJ
cana-408	149	7	x	x	PUNCT
cana-408	149	8	∧	∧	NOUN
cana-408	149	9	b	b	NOUN
cana-408	149	10	=	=	SYM
cana-408	149	11	0	0	PROPN
cana-408	149	12	then	then	ADV
cana-408	149	13	a	a	DET
cana-408	149	14	−	−	NOUN
cana-408	149	15	x	x	SYM
cana-408	150	1	−	−	PROPN
cana-408	150	2	b	b	X
cana-408	150	3	−	−	PROPN
cana-408	151	1	c	c	NOUN
cana-408	151	2	–	–	PUNCT
cana-408	151	3	d	d	NOUN
cana-408	151	4	–	–	PUNCT
cana-408	151	5	a	a	PRON
cana-408	151	6	is	be	AUX
cana-408	151	7	a	a	DET
cana-408	151	8	4	4	NUM
cana-408	151	9	cycle	cycle	NOUN
cana-408	151	10	.	.	PUNCT
cana-408	152	1	if	if	SCONJ
cana-408	152	2	x	x	SYM
cana-408	152	3	∧	∧	PROPN
cana-408	152	4	b	b	PROPN
cana-408	152	5	≠	≠	PROPN
cana-408	152	6	0	0	NUM
cana-408	152	7	,	,	PUNCT
cana-408	152	8	since	since	SCONJ
cana-408	152	9	a	a	PRON
cana-408	152	10	is	be	AUX
cana-408	152	11	not	not	PART
cana-408	152	12	an	an	DET
cana-408	152	13	atom	atom	NOUN
cana-408	152	14	then	then	ADV
cana-408	152	15	there	there	PRON
cana-408	152	16	exists	exist	VERB
cana-408	152	17	a	a	DET
cana-408	152	18	nonzero	nonzero	NOUN
cana-408	152	19	d	d	X
cana-408	152	20	<	<	X
cana-408	152	21	a.	a.	NOUN
cana-408	153	1	then	then	ADV
cana-408	153	2	d	d	X
cana-408	153	3	∧	∧	NOUN
cana-408	153	4	x	x	X
cana-408	153	5	=	=	SYM
cana-408	153	6	0	0	NUM
cana-408	153	7	,	,	PUNCT
cana-408	153	8	d	d	PROPN
cana-408	153	9	∧	∧	PROPN
cana-408	153	10	b	b	PROPN
cana-408	153	11	=	=	SYM
cana-408	153	12	0	0	PROPN
cana-408	153	13	.	.	PUNCT
cana-408	154	1	hence	hence	ADV
cana-408	154	2	d	d	X
cana-408	154	3	−	−	PROPN
cana-408	155	1	x	x	PUNCT
cana-408	155	2	−	−	NOUN
cana-408	155	3	a	a	DET
cana-408	155	4	−	−	PROPN
cana-408	155	5	b	b	PROPN
cana-408	155	6	–	–	PUNCT
cana-408	155	7	c	c	X
cana-408	155	8	–	–	PUNCT
cana-408	155	9	d	d	NOUN
cana-408	155	10	is	be	AUX
cana-408	155	11	a	a	DET
cana-408	155	12	cycle	cycle	NOUN
cana-408	155	13	of	of	ADP
cana-408	155	14	length	length	NOUN
cana-408	155	15	6	6	NUM
cana-408	155	16	.	.	PUNCT
cana-408	156	1	thus	thus	ADV
cana-408	156	2	,	,	PUNCT
cana-408	156	3	a	a	PRON
cana-408	156	4	is	be	AUX
cana-408	156	5	contained	contain	VERB
cana-408	156	6	in	in	ADP
cana-408	156	7	a	a	DET
cana-408	156	8	cycle	cycle	NOUN
cana-408	156	9	of	of	ADP
cana-408	156	10	length	length	NOUN
cana-408	156	11	≤	≤	NUM
cana-408	156	12	6	6	NUM
cana-408	156	13	,	,	PUNCT
cana-408	156	14	so	so	CCONJ
cana-408	156	15	a−x	a−x	NOUN
cana-408	156	16	is	be	AUX
cana-408	156	17	an	an	DET
cana-408	156	18	edge	edge	NOUN
cana-408	156	19	of	of	ADP
cana-408	156	20	a	a	DET
cana-408	156	21	4	4	NUM
cana-408	156	22	cycle	cycle	NOUN
cana-408	156	23	.	.	PUNCT
cana-408	157	1	hence	hence	ADV
cana-408	157	2	a−x	a−x	NOUN
cana-408	157	3	is	be	AUX
cana-408	157	4	an	an	DET
cana-408	157	5	edge	edge	NOUN
cana-408	157	6	of	of	ADP
cana-408	157	7	a	a	DET
cana-408	157	8	4	4	NUM
cana-408	157	9	cycles	cycle	NOUN
cana-408	157	10	or	or	CCONJ
cana-408	157	11	a	a	DET
cana-408	157	12	5	5	NUM
cana-408	157	13	cycles	cycle	NOUN
cana-408	157	14	.	.	PUNCT
cana-408	158	1	corollary	corollary	ADJ
cana-408	158	2	2.1	2.1	NUM
cana-408	158	3	.	.	PUNCT
cana-408	159	1	for	for	ADP
cana-408	159	2	any	any	DET
cana-408	159	3	meet	meet	ADJ
cana-408	159	4	-	-	PUNCT
cana-408	159	5	semilattice	semilattice	NOUN
cana-408	159	6	l	l	NOUN
cana-408	159	7	,	,	PUNCT
cana-408	159	8	let	let	VERB
cana-408	159	9	k	k	X
cana-408	159	10	,	,	PUNCT
cana-408	159	11	the	the	DET
cana-408	159	12	core	core	NOUN
cana-408	159	13	(	(	PUNCT
cana-408	159	14	a	a	DET
cana-408	159	15	notion	notion	NOUN
cana-408	159	16	that	that	SCONJ
cana-408	159	17	describes	describe	VERB
cana-408	159	18	behavior	behavior	NOUN
cana-408	159	19	of	of	ADP
cana-408	159	20	a	a	DET
cana-408	159	21	graph	graph	NOUN
cana-408	159	22	)	)	PUNCT
cana-408	159	23	of	of	ADP
cana-408	159	24	γ(l	γ(l	PROPN
cana-408	159	25	)	)	PUNCT
cana-408	159	26	,	,	PUNCT
cana-408	159	27	be	be	AUX
cana-408	159	28	the	the	DET
cana-408	159	29	union	union	NOUN
cana-408	159	30	of	of	ADP
cana-408	159	31	cycles	cycle	NOUN
cana-408	159	32	in	in	ADP
cana-408	159	33	γ(l	γ(l	PROPN
cana-408	159	34	)	)	PUNCT
cana-408	159	35	.	.	PUNCT
cana-408	160	1	theorem	theorem	VERB
cana-408	160	2	2.2	2.2	NUM
cana-408	160	3	.	.	PUNCT
cana-408	161	1	let	let	VERB
cana-408	161	2	l	l	NOUN
cana-408	161	3	be	be	AUX
cana-408	161	4	a	a	DET
cana-408	161	5	meet	meet	NOUN
cana-408	161	6	-	-	PUNCT
cana-408	161	7	semilattice	semilattice	NOUN
cana-408	161	8	with	with	ADP
cana-408	161	9	0	0	NUM
cana-408	161	10	.	.	PUNCT
cana-408	162	1	if	if	SCONJ
cana-408	162	2	γ(l	γ(l	NOUN
cana-408	162	3	)	)	PUNCT
cana-408	162	4	contains	contain	VERB
cana-408	162	5	a	a	DET
cana-408	162	6	cycle	cycle	NOUN
cana-408	162	7	,	,	PUNCT
cana-408	162	8	then	then	ADV
cana-408	162	9	the	the	DET
cana-408	162	10	core	core	NOUN
cana-408	162	11	k	k	PROPN
cana-408	162	12	of	of	ADP
cana-408	162	13	γ(l	γ(l	PROPN
cana-408	162	14	)	)	PUNCT
cana-408	162	15	is	be	AUX
cana-408	162	16	a	a	DET
cana-408	162	17	union	union	NOUN
cana-408	162	18	of	of	ADP
cana-408	162	19	4cycles	4cycles	NUM
cana-408	162	20	and	and	CCONJ
cana-408	162	21	5	5	NUM
cana-408	162	22	–	–	PUNCT
cana-408	162	23	cycles	cycle	NOUN
cana-408	162	24	and	and	CCONJ
cana-408	162	25	any	any	DET
cana-408	162	26	vertex	vertex	NOUN
cana-408	162	27	in	in	ADP
cana-408	162	28	γ(l	γ(l	NOUN
cana-408	162	29	)	)	PUNCT
cana-408	162	30	is	be	AUX
cana-408	162	31	either	either	CCONJ
cana-408	162	32	a	a	DET
cana-408	162	33	vertex	vertex	NOUN
cana-408	162	34	of	of	ADP
cana-408	162	35	the	the	DET
cana-408	162	36	core	core	NOUN
cana-408	162	37	k	k	PROPN
cana-408	162	38	of	of	ADP
cana-408	162	39	γ(l	γ(l	PROPN
cana-408	162	40	)	)	PUNCT
cana-408	162	41	or	or	CCONJ
cana-408	162	42	is	be	AUX
cana-408	162	43	a	a	DET
cana-408	162	44	pendant	pendant	NOUN
cana-408	162	45	of	of	ADP
cana-408	162	46	γ(l	γ(l	NOUN
cana-408	162	47	)	)	PUNCT
cana-408	162	48	.	.	PUNCT
cana-408	163	1	proof	proof	NOUN
cana-408	163	2	.	.	PUNCT
cana-408	164	1	let	let	VERB
cana-408	164	2	a1	a1	NOUN
cana-408	164	3	∈	∈	PROPN
cana-408	164	4	k	k	PROPN
cana-408	164	5	and	and	CCONJ
cana-408	164	6	suppose	suppose	VERB
cana-408	164	7	that	that	SCONJ
cana-408	164	8	a1	a1	NOUN
cana-408	164	9	does	do	AUX
cana-408	164	10	not	not	PART
cana-408	164	11	belong	belong	VERB
cana-408	164	12	to	to	ADP
cana-408	164	13	any	any	DET
cana-408	164	14	4	4	NUM
cana-408	164	15	–	–	PUNCT
cana-408	164	16	cycles	cycle	NOUN
cana-408	164	17	or	or	CCONJ
cana-408	164	18	a	a	DET
cana-408	164	19	5	5	NUM
cana-408	164	20	cycle	cycle	NOUN
cana-408	164	21	in	in	ADP
cana-408	164	22	γ(l	γ(l	NOUN
cana-408	164	23	)	)	PUNCT
cana-408	164	24	.	.	PUNCT
cana-408	165	1	then	then	ADV
cana-408	165	2	a_1	a_1	PROPN
cana-408	165	3	is	be	AUX
cana-408	165	4	in	in	ADP
cana-408	165	5	some	some	DET
cana-408	165	6	in	in	ADP
cana-408	165	7	–	–	PUNCT
cana-408	165	8	cycle	cycle	NOUN
cana-408	165	9	a_1	a_1	NOUN
cana-408	165	10	,	,	PUNCT
cana-408	165	11	a_2	a_2	PROPN
cana-408	165	12	,	,	PUNCT
cana-408	165	13	a_3	a_3	PROPN
cana-408	165	14	⋯a_n	⋯a_n	PROPN
cana-408	165	15	,	,	PUNCT
cana-408	165	16	a_1	a_1	NOUN
cana-408	165	17	with	with	ADP
cana-408	165	18	in	in	ADP
cana-408	165	19	≥	≥	NOUN
cana-408	165	20	6	6	NUM
cana-408	165	21	.	.	PUNCT
cana-408	165	22	by	by	ADP
cana-408	165	23	theorem	theorem	NOUN
cana-408	165	24	2.1	2.1	NUM
cana-408	165	25	,	,	PUNCT
cana-408	165	26	a_1	a_1	NOUN
cana-408	165	27	is	be	AUX
cana-408	165	28	an	an	DET
cana-408	165	29	atom	atom	NOUN
cana-408	165	30	in	in	ADP
cana-408	165	31	l.	l.	PROPN
cana-408	165	32	then	then	ADV
cana-408	165	33	a_1	a_1	NOUN
cana-408	165	34	≤	≤	NUM
cana-408	165	35	a_5	a_5	NOUN
cana-408	165	36	implies	imply	VERB
cana-408	165	37	that	that	SCONJ
cana-408	165	38	a_1∧a_4	a_1∧a_4	PROPN
cana-408	165	39	=	=	PUNCT
cana-408	165	40	0	0	NUM
cana-408	165	41	,	,	PUNCT
cana-408	165	42	which	which	PRON
cana-408	165	43	is	be	AUX
cana-408	165	44	a	a	DET
cana-408	165	45	contradiction	contradiction	NOUN
cana-408	165	46	.	.	PUNCT
cana-408	166	1	hence	hence	ADV
cana-408	166	2	γ(l	γ(l	PROPN
cana-408	166	3	)	)	PUNCT
cana-408	166	4	is	be	AUX
cana-408	166	5	a	a	DET
cana-408	166	6	union	union	NOUN
cana-408	166	7	of	of	ADP
cana-408	166	8	4cycles	4cycles	NUM
cana-408	166	9	and	and	CCONJ
cana-408	166	10	5	5	NUM
cana-408	166	11	–	–	PUNCT
cana-408	166	12	cycles	cycle	NOUN
cana-408	166	13	.	.	PUNCT
cana-408	167	1	now	now	ADV
cana-408	167	2	suppose	suppose	VERB
cana-408	167	3	that	that	SCONJ
cana-408	167	4	a	a	PRON
cana-408	167	5	is	be	AUX
cana-408	167	6	any	any	DET
cana-408	167	7	vertex	vertex	NOUN
cana-408	167	8	in	in	ADP
cana-408	167	9	γ(l	γ(l	NOUN
cana-408	167	10	)	)	PUNCT
cana-408	167	11	.	.	PUNCT
cana-408	168	1	if	if	SCONJ
cana-408	168	2	a	a	DET
cana-408	168	3	∉k	∉k	PROPN
cana-408	168	4	and	and	CCONJ
cana-408	168	5	a	a	PRON
cana-408	168	6	is	be	AUX
cana-408	168	7	not	not	PART
cana-408	168	8	a	a	DET
cana-408	168	9	pendant	pendant	ADJ
cana-408	168	10	vertex	vertex	NOUN
cana-408	168	11	then	then	ADV
cana-408	168	12	the	the	DET
cana-408	168	13	following	follow	VERB
cana-408	168	14	possibility	possibility	NOUN
cana-408	168	15	holds	hold	VERB
cana-408	168	16	.	.	PUNCT
cana-408	169	1	(	(	PUNCT
cana-408	169	2	i	i	NOUN
cana-408	169	3	)	)	PUNCT
cana-408	169	4	a	a	PRON
cana-408	169	5	is	be	AUX
cana-408	169	6	contained	contain	VERB
cana-408	169	7	in	in	ADP
cana-408	169	8	a	a	DET
cana-408	169	9	path	path	NOUN
cana-408	169	10	of	of	ADP
cana-408	169	11	the	the	DET
cana-408	169	12	form	form	NOUN
cana-408	169	13	x	x	PUNCT
cana-408	169	14	−	−	PUNCT
cana-408	169	15	y	y	INTJ
cana-408	169	16	−	−	PROPN
cana-408	169	17	a	a	DET
cana-408	169	18	–	–	PUNCT
cana-408	169	19	b	b	NOUN
cana-408	169	20	–	–	PUNCT
cana-408	169	21	c	c	NOUN
cana-408	169	22	with	with	ADP
cana-408	169	23	c	c	PROPN
cana-408	169	24	∈	∈	PROPN
cana-408	169	25	k	k	PROPN
cana-408	169	26	or	or	CCONJ
cana-408	169	27	(	(	PUNCT
cana-408	169	28	ii	ii	NOUN
cana-408	169	29	)	)	PUNCT
cana-408	169	30	a	a	PRON
cana-408	169	31	is	be	AUX
cana-408	169	32	contained	contain	VERB
cana-408	169	33	in	in	ADP
cana-408	169	34	a	a	DET
cana-408	169	35	path	path	NOUN
cana-408	169	36	of	of	ADP
cana-408	169	37	the	the	DET
cana-408	169	38	form	form	NOUN
cana-408	169	39	x	x	PUNCT
cana-408	169	40	−	−	NOUN
cana-408	169	41	a	a	DET
cana-408	169	42	–	–	PUNCT
cana-408	169	43	b	b	NOUN
cana-408	169	44	–	–	PUNCT
cana-408	169	45	c	c	NOUN
cana-408	169	46	with	with	ADP
cana-408	169	47	c	c	PROPN
cana-408	169	48	∈	∈	PROPN
cana-408	169	49	k.	k.	PROPN
cana-408	169	50	since	since	SCONJ
cana-408	169	51	c	c	PROPN
cana-408	169	52	∈	∈	PROPN
cana-408	169	53	k	k	PROPN
cana-408	169	54	,	,	PUNCT
cana-408	169	55	b	b	PROPN
cana-408	169	56	is	be	AUX
cana-408	169	57	contained	contain	VERB
cana-408	169	58	in	in	ADP
cana-408	169	59	a	a	DET
cana-408	169	60	4	4	NUM
cana-408	169	61	cycles	cycle	NOUN
cana-408	169	62	or	or	CCONJ
cana-408	169	63	a	a	DET
cana-408	169	64	5	5	NUM
cana-408	169	65	cycles	cycle	NOUN
cana-408	169	66	,	,	PUNCT
cana-408	169	67	say	say	VERB
cana-408	169	68	b	b	X
cana-408	170	1	−	−	NOUN
cana-408	171	1	c	c	NOUN
cana-408	172	1	−	−	NOUN
cana-408	173	1	d	d	NOUN
cana-408	173	2	−	−	PROPN
cana-408	173	3	e	e	NOUN
cana-408	173	4	−	−	PROPN
cana-408	173	5	b	b	PROPN
cana-408	173	6	or	or	CCONJ
cana-408	173	7	b	b	NOUN
cana-408	173	8	−	−	PROPN
cana-408	173	9	c	c	NOUN
cana-408	174	1	−	−	NOUN
cana-408	174	2	d	d	NOUN
cana-408	174	3	−	−	PROPN
cana-408	175	1	e	e	NOUN
cana-408	175	2	–	–	PUNCT
cana-408	175	3	a	a	PRON
cana-408	175	4	–	–	PUNCT
cana-408	175	5	b.	b.	NOUN
cana-408	175	6	in	in	ADP
cana-408	175	7	(	(	PUNCT
cana-408	175	8	i	i	PROPN
cana-408	175	9	)	)	PUNCT
cana-408	175	10	,	,	PUNCT
cana-408	175	11	we	we	PRON
cana-408	175	12	get	get	VERB
cana-408	175	13	d	d	X
cana-408	175	14	(	(	PUNCT
cana-408	175	15	x	x	NOUN
cana-408	175	16	,	,	PUNCT
cana-408	175	17	c	c	NOUN
cana-408	175	18	)	)	PUNCT
cana-408	175	19	=	=	SYM
cana-408	176	1	4	4	NUM
cana-408	176	2	,	,	PUNCT
cana-408	176	3	contradicts	contradict	VERB
cana-408	176	4	|	|	ADV
cana-408	176	5	γ(l)|	γ(l)|	VERB
cana-408	176	6	≤	≤	ADJ
cana-408	176	7	4	4	NUM
cana-408	176	8	.	.	PUNCT
cana-408	177	1	hence	hence	ADV
cana-408	177	2	(	(	PUNCT
cana-408	177	3	i	i	NOUN
cana-408	177	4	)	)	PUNCT
cana-408	177	5	can	can	AUX
cana-408	177	6	not	not	PART
cana-408	177	7	hold	hold	VERB
cana-408	177	8	.	.	PUNCT
cana-408	178	1	in	in	ADP
cana-408	178	2	(	(	PUNCT
cana-408	178	3	ii	ii	NOUN
cana-408	178	4	)	)	PUNCT
cana-408	178	5	,	,	PUNCT
cana-408	178	6	we	we	PRON
cana-408	178	7	get	get	VERB
cana-408	178	8	x	x	PUNCT
cana-408	178	9	−	−	NOUN
cana-408	178	10	a	a	DET
cana-408	178	11	−	−	PROPN
cana-408	178	12	b	b	NOUN
cana-408	178	13	−	−	PROPN
cana-408	179	1	c	c	NOUN
cana-408	180	1	−	−	NOUN
cana-408	181	1	d	d	NOUN
cana-408	181	2	−	−	PROPN
cana-408	181	3	b	b	PROPN
cana-408	181	4	or	or	CCONJ
cana-408	181	5	x	x	SYM
cana-408	181	6	−	−	NOUN
cana-408	181	7	a	a	DET
cana-408	181	8	−	−	PROPN
cana-408	181	9	b	b	NOUN
cana-408	181	10	−	−	PROPN
cana-408	182	1	c	c	NOUN
cana-408	183	1	−	−	NOUN
cana-408	184	1	d	d	NOUN
cana-408	184	2	−	−	PROPN
cana-408	184	3	e	e	PROPN
cana-408	184	4	−	−	PROPN
cana-408	184	5	b.	b.	PROPN
cana-408	184	6	hence	hence	ADV
cana-408	184	7	by	by	ADP
cana-408	184	8	theorem	theorem	ADJ
cana-408	184	9	2.1	2.1	NUM
cana-408	184	10	.	.	PUNCT
cana-408	184	11	,	,	PUNCT
cana-408	184	12	a	a	PRON
cana-408	184	13	must	must	AUX
cana-408	184	14	be	be	AUX
cana-408	184	15	an	an	DET
cana-408	184	16	atom	atom	NOUN
cana-408	184	17	.	.	PUNCT
cana-408	185	1	therefore	therefore	ADV
cana-408	185	2	,	,	PUNCT
cana-408	185	3	a	a	DET
cana-408	185	4	∧	∧	PROPN
cana-408	185	5	c	c	NOUN
cana-408	185	6	=	=	NOUN
cana-408	185	7	a.	a.	NOUN
cana-408	185	8	communications	communication	NOUN
cana-408	185	9	on	on	ADP
cana-408	185	10	applied	apply	VERB
cana-408	185	11	nonlinear	nonlinear	ADJ
cana-408	185	12	analysis	analysis	NOUN
cana-408	185	13	issn	issn	NOUN
cana-408	185	14	:	:	PUNCT
cana-408	185	15	1074	1074	NUM
cana-408	185	16	-	-	PUNCT
cana-408	185	17	133x	133x	NUM
cana-408	185	18	vol	vol	NOUN
cana-408	185	19	31	31	NUM
cana-408	185	20	no	no	NOUN
cana-408	185	21	.	.	NOUN
cana-408	185	22	1	1	NUM
cana-408	185	23	(	(	PUNCT
cana-408	185	24	2024	2024	NUM
cana-408	185	25	)	)	PUNCT
cana-408	185	26	236	236	NUM
cana-408	185	27	https://internationalpubls.com	https://internationalpubls.com	X
cana-408	185	28	this	this	PRON
cana-408	185	29	gives	give	VERB
cana-408	185	30	a	a	DET
cana-408	185	31	∧	∧	PROPN
cana-408	185	32	d	d	NOUN
cana-408	185	33	=	=	SYM
cana-408	185	34	0	0	PROPN
cana-408	185	35	,	,	PUNCT
cana-408	185	36	a	a	DET
cana-408	185	37	contradiction	contradiction	NOUN
cana-408	185	38	as	as	ADP
cana-408	185	39	a	a	DET
cana-408	185	40	∉k	∉k	NOUN
cana-408	185	41	.	.	PUNCT
cana-408	186	1	thus	thus	ADV
cana-408	186	2	(	(	PUNCT
cana-408	186	3	ii	ii	NOUN
cana-408	186	4	)	)	PUNCT
cana-408	186	5	can	can	AUX
cana-408	186	6	not	not	PART
cana-408	186	7	hold	hold	VERB
cana-408	186	8	.	.	PUNCT
cana-408	187	1	hence	hence	ADV
cana-408	187	2	either	either	CCONJ
cana-408	187	3	a	a	DET
cana-408	187	4	∈	∈	PROPN
cana-408	187	5	k	k	NOUN
cana-408	187	6	or	or	CCONJ
cana-408	187	7	a	a	PRON
cana-408	187	8	is	be	AUX
cana-408	187	9	a	a	DET
cana-408	187	10	pendant	pendant	ADJ
cana-408	187	11	vertex	vertex	NOUN
cana-408	187	12	.	.	PUNCT
cana-408	188	1	remark	remark	NOUN
cana-408	188	2	2.1	2.1	NUM
cana-408	188	3	.	.	PUNCT
cana-408	189	1	let	let	VERB
cana-408	189	2	l_1	l_1	PROPN
cana-408	189	3	and	and	CCONJ
cana-408	189	4	l_2	l_2	AUX
cana-408	189	5	be	be	AUX
cana-408	189	6	two	two	NUM
cana-408	189	7	meet	meet	NOUN
cana-408	189	8	-	-	PUNCT
cana-408	189	9	semilattices	semilattice	NOUN
cana-408	189	10	with	with	ADP
cana-408	189	11	0	0	NUM
cana-408	189	12	and	and	CCONJ
cana-408	189	13	l	l	NOUN
cana-408	190	1	=	=	SYM
cana-408	190	2	l_1	l_1	PROPN
cana-408	190	3	x	x	SYM
cana-408	190	4	l_2	l_2	PROPN
cana-408	190	5	,	,	PUNCT
cana-408	190	6	then	then	ADV
cana-408	190	7	γ(l	γ(l	PROPN
cana-408	190	8	)	)	PUNCT
cana-408	190	9	is	be	AUX
cana-408	190	10	star	star	NOUN
cana-408	190	11	graph	graph	NOUN
cana-408	190	12	if	if	SCONJ
cana-408	190	13	and	and	CCONJ
cana-408	190	14	only	only	ADV
cana-408	190	15	if	if	SCONJ
cana-408	190	16	one	one	NUM
cana-408	190	17	of	of	ADP
cana-408	190	18	the	the	DET
cana-408	190	19	l_1	l_1	PROPN
cana-408	190	20	or	or	CCONJ
cana-408	190	21	l_2	l_2	NOUN
cana-408	190	22	is	be	AUX
cana-408	190	23	c_2	c_2	ADJ
cana-408	190	24	and	and	CCONJ
cana-408	190	25	the	the	DET
cana-408	190	26	other	other	ADJ
cana-408	190	27	is	be	AUX
cana-408	190	28	an	an	DET
cana-408	190	29	integral	integral	ADJ
cana-408	190	30	meet	meet	NOUN
cana-408	190	31	-	-	PUNCT
cana-408	190	32	semilattice	semilattice	NOUN
cana-408	190	33	.	.	PUNCT
cana-408	191	1	remark	remark	PROPN
cana-408	191	2	i2.2	i2.2	PROPN
cana-408	191	3	.	.	PUNCT
cana-408	192	1	ifor	ifor	PROPN
cana-408	192	2	iany	iany	PROPN
cana-408	192	3	istar	istar	PROPN
cana-408	192	4	igraph	igraph	PROPN
cana-408	192	5	iwith	iwith	PROPN
cana-408	192	6	in	in	ADP
cana-408	192	7	ielements	ielement	NOUN
cana-408	192	8	ithere	ithere	VERB
cana-408	192	9	icorresponds	icorrespond	NOUN
cana-408	192	10	ia	ia	PROPN
cana-408	192	11	imeet	imeet	NOUN
cana-408	192	12	-	-	PUNCT
cana-408	192	13	semilattice	semilattice	NOUN
cana-408	192	14	ias	ias	NOUN
cana-408	192	15	iin	iin	NOUN
cana-408	192	16	ifigure	ifigure	NOUN
cana-408	192	17	i2.1	i2.1	PRON
cana-408	192	18	figure	figure	NOUN
cana-408	192	19	i2.1	i2.1	VERB
cana-408	192	20	theorem	theorem	ADJ
cana-408	192	21	2.3	2.3	NUM
cana-408	192	22	.	.	PUNCT
cana-408	193	1	let	let	VERB
cana-408	193	2	l_1	l_1	PROPN
cana-408	193	3	and	and	CCONJ
cana-408	193	4	l_2	l_2	AUX
cana-408	193	5	be	be	AUX
cana-408	193	6	two	two	NUM
cana-408	193	7	meet	meet	NOUN
cana-408	193	8	-	-	PUNCT
cana-408	193	9	semilattices	semilattice	NOUN
cana-408	193	10	with	with	ADP
cana-408	193	11	0	0	NUM
cana-408	193	12	and	and	CCONJ
cana-408	193	13	l	l	NOUN
cana-408	193	14	=	=	SYM
cana-408	193	15	l_1	l_1	PROPN
cana-408	193	16	x	x	SYM
cana-408	193	17	l_2	l_2	PROPN
cana-408	193	18	.	.	PUNCT
cana-408	194	1	then	then	ADV
cana-408	194	2	exactly	exactly	ADV
cana-408	194	3	one	one	NUM
cana-408	194	4	of	of	ADP
cana-408	194	5	the	the	DET
cana-408	194	6	following	follow	VERB
cana-408	194	7	holds	hold	VERB
cana-408	194	8	:	:	PUNCT
cana-408	194	9	1	1	X
cana-408	194	10	.	.	PUNCT
cana-408	194	11	γ(l	γ(l	NOUN
cana-408	194	12	)	)	PUNCT
cana-408	194	13	has	have	VERB
cana-408	194	14	a	a	DET
cana-408	194	15	cycle	cycle	NOUN
cana-408	194	16	of	of	ADP
cana-408	194	17	length	length	NOUN
cana-408	194	18	n-1	n-1	NOUN
cana-408	194	19	or	or	CCONJ
cana-408	194	20	n	n	PRON
cana-408	194	21	(	(	PUNCT
cana-408	194	22	that	that	PRON
cana-408	194	23	is	be	AUX
cana-408	194	24	gr	gr	DET
cana-408	194	25	γ(l)≤	γ(l)≤	NOUN
cana-408	194	26	n	n	CCONJ
cana-408	194	27	)	)	PUNCT
cana-408	194	28	,	,	PUNCT
cana-408	194	29	2	2	X
cana-408	194	30	.	.	PUNCT
cana-408	194	31	γ(l	γ(l	NOUN
cana-408	194	32	)	)	PUNCT
cana-408	194	33	is	be	AUX
cana-408	194	34	a	a	DET
cana-408	194	35	star	star	NOUN
cana-408	194	36	graph	graph	NOUN
cana-408	194	37	.	.	PUNCT
cana-408	195	1	proof	proof	NOUN
cana-408	195	2	.	.	PUNCT
cana-408	196	1	suppose	suppose	VERB
cana-408	196	2	l	l	NOUN
cana-408	196	3	=	=	SYM
cana-408	196	4	l_1	l_1	PROPN
cana-408	196	5	x	x	SYM
cana-408	196	6	l_2	l_2	PROPN
cana-408	196	7	,	,	PUNCT
cana-408	196	8	where	where	SCONJ
cana-408	196	9	at	at	ADV
cana-408	196	10	least	least	ADV
cana-408	196	11	one	one	NUM
cana-408	196	12	of	of	ADP
cana-408	196	13	l_1	l_1	PROPN
cana-408	196	14	and	and	CCONJ
cana-408	196	15	l_2	l_2	PROPN
cana-408	196	16	is	be	AUX
cana-408	196	17	not	not	PART
cana-408	196	18	an	an	DET
cana-408	196	19	integral	integral	ADJ
cana-408	196	20	meet	meet	NOUN
cana-408	196	21	-	-	PUNCT
cana-408	196	22	semilattice	semilattice	NOUN
cana-408	196	23	,	,	PUNCT
cana-408	196	24	say	say	VERB
cana-408	196	25	l_1	l_1	PROPN
cana-408	196	26	is	be	AUX
cana-408	196	27	not	not	PART
cana-408	196	28	an	an	DET
cana-408	196	29	integral	integral	ADJ
cana-408	196	30	(	(	PUNCT
cana-408	196	31	join	join	NOUN
cana-408	196	32	)	)	PUNCT
cana-408	196	33	meet	meet	VERB
cana-408	196	34	-semilattice	-semilattice	NOUN
cana-408	196	35	.	.	PUNCT
cana-408	197	1	then	then	ADV
cana-408	197	2	there	there	PRON
cana-408	197	3	exists	exist	VERB
cana-408	197	4	nonzero	nonzero	PROPN
cana-408	197	5	a	a	PRON
cana-408	197	6	,	,	PUNCT
cana-408	197	7	b	b	PROPN
cana-408	197	8	∈	∈	PROPN
cana-408	197	9	l_1	l_1	PROPN
cana-408	197	10	,	,	PUNCT
cana-408	197	11	with	with	ADP
cana-408	197	12	a	a	DET
cana-408	197	13	∧	∧	PROPN
cana-408	197	14	b	b	NOUN
cana-408	197	15	=	=	SYM
cana-408	197	16	0	0	PUNCT
cana-408	197	17	and	and	CCONJ
cana-408	197	18	choose	choose	VERB
cana-408	197	19	nonzero	nonzero	PROPN
cana-408	197	20	c	c	PROPN
cana-408	197	21	∈	∈	PROPN
cana-408	197	22	l_2	l_2	PROPN
cana-408	197	23	.	.	PUNCT
cana-408	198	1	then	then	ADV
cana-408	198	2	(	(	PUNCT
cana-408	198	3	a	a	PRON
cana-408	198	4	,	,	PUNCT
cana-408	198	5	0	0	NUM
cana-408	198	6	)	)	PUNCT
cana-408	198	7	−	−	PROPN
cana-408	198	8	(	(	PUNCT
cana-408	198	9	b	b	NOUN
cana-408	198	10	,	,	PUNCT
cana-408	198	11	0	0	NUM
cana-408	198	12	)	)	PUNCT
cana-408	198	13	−	−	PROPN
cana-408	198	14	(	(	PUNCT
cana-408	198	15	0	0	NUM
cana-408	198	16	,	,	PUNCT
cana-408	198	17	c	c	NOUN
cana-408	198	18	)	)	PUNCT
cana-408	198	19	–	–	PUNCT
cana-408	198	20	…	…	PUNCT
cana-408	198	21	.	.	PUNCT
cana-408	198	22	form	form	VERB
cana-408	198	23	a	a	DET
cana-408	198	24	cycle	cycle	NOUN
cana-408	198	25	of	of	ADP
cana-408	198	26	length	length	NOUN
cana-408	198	27	n-1	n-1	NOUN
cana-408	198	28	in	in	ADP
cana-408	198	29	γ(l	γ(l	PROPN
cana-408	198	30	)	)	PUNCT
cana-408	198	31	.	.	PUNCT
cana-408	199	1	let	let	VERB
cana-408	199	2	l	l	NOUN
cana-408	199	3	=	=	SYM
cana-408	199	4	l_1	l_1	PROPN
cana-408	199	5	x	x	SYM
cana-408	199	6	l_2	l_2	PROPN
cana-408	199	7	,	,	PUNCT
cana-408	199	8	where	where	SCONJ
cana-408	199	9	l_1	l_1	PROPN
cana-408	199	10	and	and	CCONJ
cana-408	199	11	l_2	l_2	PROPN
cana-408	199	12	,	,	PUNCT
cana-408	199	13	both	both	PRON
cana-408	199	14	are	be	AUX
cana-408	199	15	integral	integral	ADJ
cana-408	199	16	meet	meet	NOUN
cana-408	199	17	semi	semi	NOUN
cana-408	199	18	-	-	NOUN
cana-408	199	19	lattices	lattice	NOUN
cana-408	199	20	with	with	ADP
cana-408	199	21	|l_1|	|l_1|	PROPN
cana-408	199	22	>	>	SYM
cana-408	199	23	n-2	n-2	PROPN
cana-408	199	24	,	,	PUNCT
cana-408	199	25	|l_2|	|l_2|	ADP
cana-408	199	26	>	>	X
cana-408	199	27	n-2	n-2	PROPN
cana-408	199	28	.	.	PUNCT
cana-408	200	1	let	let	VERB
cana-408	200	2	a	a	PRON
cana-408	200	3	,	,	PUNCT
cana-408	200	4	b	b	PROPN
cana-408	200	5	∈	∈	PROPN
cana-408	200	6	l_1	l_1	PROPN
cana-408	200	7	,	,	PUNCT
cana-408	200	8	and	and	CCONJ
cana-408	200	9	c	c	X
cana-408	200	10	,	,	PUNCT
cana-408	200	11	d	d	PROPN
cana-408	200	12	∈	∈	PROPN
cana-408	200	13	l_2	l_2	NOUN
cana-408	200	14	be	be	AUX
cana-408	200	15	ia	ia	X
cana-408	200	16	non	non	ADJ
cana-408	200	17	zero	zero	NUM
cana-408	200	18	elements	element	NOUN
cana-408	200	19	.	.	PUNCT
cana-408	201	1	then	then	ADV
cana-408	201	2	(	(	PUNCT
cana-408	201	3	a	a	PRON
cana-408	201	4	,	,	PUNCT
cana-408	201	5	0	0	NUM
cana-408	201	6	)	)	PUNCT
cana-408	201	7	−	−	PROPN
cana-408	201	8	(	(	PUNCT
cana-408	201	9	0	0	NUM
cana-408	201	10	,	,	PUNCT
cana-408	201	11	c	c	NOUN
cana-408	201	12	)	)	PUNCT
cana-408	202	1	−	−	PROPN
cana-408	202	2	(	(	PUNCT
cana-408	202	3	b	b	NOUN
cana-408	202	4	,	,	PUNCT
cana-408	202	5	0	0	NUM
cana-408	202	6	)	)	PUNCT
cana-408	202	7	−	−	PROPN
cana-408	202	8	(	(	PUNCT
cana-408	202	9	0	0	NUM
cana-408	202	10	,	,	PUNCT
cana-408	202	11	d	d	NOUN
cana-408	202	12	)	)	PUNCT
cana-408	202	13	−	−	NOUN
cana-408	202	14	…	…	PUNCT
cana-408	202	15	form	form	VERB
cana-408	202	16	a	a	DET
cana-408	202	17	cycle	cycle	NOUN
cana-408	202	18	of	of	ADP
cana-408	202	19	length	length	NOUN
cana-408	202	20	in	in	ADP
cana-408	202	21	γ(l	γ(l	NOUN
cana-408	202	22	)	)	PUNCT
cana-408	202	23	.	.	PUNCT
cana-408	203	1	let	let	VERB
cana-408	203	2	l	l	NOUN
cana-408	203	3	∼=	∼=	PROPN
cana-408	203	4	l_1	l_1	PROPN
cana-408	203	5	x	x	SYM
cana-408	203	6	l_2	l_2	PROPN
cana-408	203	7	,	,	PUNCT
cana-408	203	8	where	where	SCONJ
cana-408	203	9	either	either	CCONJ
cana-408	203	10	|l_1|	|l_1|	PROPN
cana-408	203	11	=	=	SYM
cana-408	203	12	n2	n2	NOUN
cana-408	203	13	or	or	CCONJ
cana-408	203	14	|l_2|	|l_2|	PROPN
cana-408	203	15	=	=	PUNCT
cana-408	203	16	n-2	n-2	PROPN
cana-408	203	17	.	.	PUNCT
cana-408	204	1	let	let	VERB
cana-408	204	2	|l_1|	|l_1|	PROPN
cana-408	204	3	=	=	SYM
cana-408	204	4	n-2	n-2	X
cana-408	204	5	and	and	CCONJ
cana-408	204	6	l_2	l_2	NOUN
cana-408	204	7	is	be	AUX
cana-408	204	8	an	an	DET
cana-408	204	9	integral	integral	ADJ
cana-408	204	10	meet	meet	NOUN
cana-408	204	11	-	-	PUNCT
cana-408	204	12	semilattice	semilattice	NOUN
cana-408	204	13	then	then	ADV
cana-408	204	14	by	by	ADP
cana-408	204	15	theorem	theorem	ADJ
cana-408	204	16	2.2	2.2	NUM
cana-408	204	17	,	,	PUNCT
cana-408	204	18	γ(l	γ(l	PROPN
cana-408	204	19	)	)	PUNCT
cana-408	204	20	is	be	AUX
cana-408	204	21	a	a	DET
cana-408	204	22	star	star	NOUN
cana-408	204	23	graph	graph	NOUN
cana-408	204	24	.	.	PUNCT
cana-408	205	1	references	reference	NOUN
cana-408	205	2	[	[	X
cana-408	205	3	1	1	NUM
cana-408	205	4	]	]	PUNCT
cana-408	205	5	r.	r.	PROPN
cana-408	205	6	iakhtar	iakhtar	PROPN
cana-408	205	7	iand	iand	PROPN
cana-408	205	8	il	il	PROPN
cana-408	205	9	.	.	PROPN
cana-408	205	10	ilee	ilee	PROPN
cana-408	205	11	,	,	PUNCT
cana-408	205	12	iconnectivity	iconnectivity	NOUN
cana-408	205	13	iof	iof	VERB
cana-408	205	14	ithe	ithe	DET
cana-408	205	15	izero	izero	NOUN
cana-408	205	16	-	-	PUNCT
cana-408	205	17	divisor	divisor	NOUN
cana-408	205	18	igraph	igraph	NOUN
cana-408	205	19	ifor	ifor	PROPN
cana-408	205	20	ifinite	ifinite	NOUN
cana-408	205	21	irings	iring	NOUN
cana-408	205	22	,	,	PUNCT
cana-408	205	23	idiscrete	idiscrete	NOUN
cana-408	205	24	maths	math	NOUN
cana-408	205	25	,	,	PUNCT
cana-408	205	26	296	296	NUM
cana-408	205	27	(	(	PUNCT
cana-408	205	28	2005	2005	NUM
cana-408	205	29	)	)	PUNCT
cana-408	205	30	,	,	PUNCT
cana-408	205	31	73	73	NUM
cana-408	205	32	–	–	SYM
cana-408	205	33	86	86	NUM
cana-408	205	34	.	.	PUNCT
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cana-408	206	2	on	on	ADP
cana-408	206	3	applied	apply	VERB
cana-408	206	4	nonlinear	nonlinear	ADJ
cana-408	206	5	analysis	analysis	NOUN
cana-408	206	6	issn	issn	NOUN
cana-408	206	7	:	:	PUNCT
cana-408	206	8	1074	1074	NUM
cana-408	206	9	-	-	PUNCT
cana-408	206	10	133x	133x	NUM
cana-408	206	11	vol	vol	NOUN
cana-408	206	12	31	31	NUM
cana-408	206	13	no	no	NOUN
cana-408	206	14	.	.	NOUN
cana-408	206	15	1	1	NUM
cana-408	206	16	(	(	PUNCT
cana-408	206	17	2024	2024	NUM
cana-408	206	18	)	)	PUNCT
cana-408	206	19	237	237	NUM
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cana-408	207	1	[	[	X
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cana-408	207	3	]	]	PUNCT
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cana-408	207	5	i	i	PROPN
cana-408	207	6	d.	d.	PROPN
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cana-408	207	8	iand	iand	PROPN
cana-408	208	1	i	i	PRON
cana-408	208	2	m.	m.	NOUN
cana-408	208	3	inaseer	inaseer	PROPN
cana-408	208	4	,	,	PUNCT
cana-408	208	5	ibeck	ibeck	AUX
cana-408	208	6	’s	’s	PART
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cana-408	208	8	iof	iof	PROPN
cana-408	208	9	ia	ia	PROPN
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cana-408	208	11	iring	iring	NOUN
cana-408	208	12	,	,	PUNCT
cana-408	208	13	ijournal	ijournal	ADJ
cana-408	208	14	iof	iof	ADJ
cana-408	208	15	ialgebra	ialgebra	NOUN
cana-408	208	16	,	,	PUNCT
cana-408	208	17	159	159	NUM
cana-408	208	18	(	(	PUNCT
cana-408	208	19	1993	1993	NUM
cana-408	208	20	)	)	PUNCT
cana-408	208	21	,	,	PUNCT
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cana-408	208	23	i514	i514	PROPN
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cana-408	209	1	[	[	X
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cana-408	209	3	]	]	PUNCT
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cana-408	209	12	.	.	PUNCT
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cana-408	210	2	i	i	PROPN
cana-408	210	3	,	,	PUNCT
cana-408	210	4	izero	izero	NOUN
cana-408	210	5	-	-	PUNCT
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cana-408	210	7	igraphs	igraph	NOUN
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cana-408	210	10	-	-	PUNCT
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cana-408	210	20	.	.	PUNCT
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cana-408	211	3	(	(	PUNCT
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cana-408	211	5	)	)	PUNCT
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cana-408	211	8	ii2050	ii2050	PROPN
cana-408	211	9	.	.	PUNCT
cana-408	212	1	[	[	X
cana-408	212	2	4	4	X
cana-408	212	3	]	]	X
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cana-408	213	2	.	.	PUNCT
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cana-408	214	2	,	,	PUNCT
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cana-408	214	10	ia	ia	PROPN
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cana-408	214	13	,	,	PUNCT
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cana-408	214	17	,	,	PUNCT
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cana-408	214	21	)	)	PUNCT
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cana-408	216	3	]	]	PUNCT
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cana-408	217	1	ianderson	ianderson	PROPN
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cana-408	217	12	-	-	PUNCT
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cana-408	218	6	)	)	PUNCT
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cana-408	219	3	]	]	PUNCT
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cana-408	220	5	)	)	PUNCT
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cana-408	221	1	[	[	X
cana-408	221	2	7	7	X
cana-408	221	3	]	]	X
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cana-408	221	5	ibeck	ibeck	PROPN
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cana-408	221	10	irings	iring	NOUN
cana-408	221	11	,	,	PUNCT
cana-408	221	12	ij	ij	NOUN
cana-408	221	13	.	.	PUNCT
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cana-408	221	15	,	,	PUNCT
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cana-408	221	17	(	(	PUNCT
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cana-408	221	19	)	)	PUNCT
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cana-408	222	1	[	[	X
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cana-408	222	3	]	]	X
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cana-408	222	11	edges	edge	NOUN
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cana-408	222	14	and	and	CCONJ
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cana-408	222	16	graphs	graph	NOUN
cana-408	222	17	,	,	PUNCT
cana-408	222	18	ihal	ihal	NOUN
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cana-408	222	22	i	i	PRON
cana-408	222	23	(	(	PUNCT
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cana-408	222	25	)	)	PUNCT
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cana-408	223	12	-	-	PUNCT
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cana-408	223	14	igraphs	igraph	NOUN
cana-408	223	15	iof	iof	NOUN
cana-408	223	16	isemigroups	isemigroup	NOUN
cana-408	223	17	,	,	PUNCT
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cana-408	223	19	iof	iof	ADJ
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cana-408	223	21	,	,	PUNCT
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cana-408	223	25	)	)	PUNCT
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cana-408	231	20	.	.	PUNCT
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