id	sid	tid	token	lemma	pos
cana-1295	1	1	communications	communication	NOUN
cana-1295	1	2	on	on	ADP
cana-1295	1	3	applied	apply	VERB
cana-1295	1	4	nonlinear	nonlinear	ADJ
cana-1295	1	5	analysis	analysis	NOUN
cana-1295	1	6	issn	issn	NOUN
cana-1295	1	7	:	:	PUNCT
cana-1295	1	8	1074	1074	NUM
cana-1295	1	9	-	-	PUNCT
cana-1295	1	10	133x	133x	NUM
cana-1295	1	11	vol	vol	NOUN
cana-1295	1	12	31	31	NUM
cana-1295	1	13	no	no	NOUN
cana-1295	1	14	.	.	PUNCT
cana-1295	2	1	7s	7	NOUN
cana-1295	2	2	(	(	PUNCT
cana-1295	2	3	2024	2024	NUM
cana-1295	2	4	)	)	PUNCT
cana-1295	2	5	203	203	NUM
cana-1295	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1295	2	7	concepts	concept	NOUN
cana-1295	2	8	of	of	ADP
cana-1295	2	9	bi	bi	ADJ
cana-1295	2	10	-	-	ADJ
cana-1295	2	11	ternary	ternary	ADJ
cana-1295	2	12	semi	semi	ADJ
cana-1295	2	13	groups	group	NOUN
cana-1295	2	14	1n	1n	NUM
cana-1295	2	15	.	.	PUNCT
cana-1295	3	1	sandhya	sandhya	PROPN
cana-1295	3	2	rani	rani	PROPN
cana-1295	3	3	,	,	PUNCT
cana-1295	3	4	*	*	PROPN
cana-1295	3	5	2	2	NUM
cana-1295	3	6	g.	g.	PROPN
cana-1295	3	7	srinivasa	srinivasa	PROPN
cana-1295	3	8	rao	rao	PROPN
cana-1295	3	9	,	,	PUNCT
cana-1295	3	10	*	*	PROPN
cana-1295	3	11	3ch	3ch	NOUN
cana-1295	3	12	.	.	PUNCT
cana-1295	4	1	ramprasad	ramprasad	ADJ
cana-1295	4	2	1research	1research	NUM
cana-1295	4	3	scholar	scholar	NOUN
cana-1295	4	4	,	,	PUNCT
cana-1295	4	5	department	department	NOUN
cana-1295	4	6	of	of	ADP
cana-1295	4	7	mathematics	mathematics	PROPN
cana-1295	4	8	&	&	CCONJ
cana-1295	4	9	statistics	statistic	NOUN
cana-1295	4	10	,	,	PUNCT
cana-1295	4	11	school	school	NOUN
cana-1295	4	12	of	of	ADP
cana-1295	4	13	applied	apply	VERB
cana-1295	4	14	sciences	sciences	PROPN
cana-1295	4	15	&	&	CCONJ
cana-1295	4	16	humanities	humanity	NOUN
cana-1295	4	17	,	,	PUNCT
cana-1295	4	18	vfstr	vfstr	NOUN
cana-1295	4	19	deemed	deem	VERB
cana-1295	4	20	to	to	PART
cana-1295	4	21	be	be	AUX
cana-1295	4	22	university	university	NOUN
cana-1295	4	23	,	,	PUNCT
cana-1295	4	24	vadlamudi	vadlamudi	NOUN
cana-1295	4	25	,	,	PUNCT
cana-1295	4	26	guntur	guntur	PROPN
cana-1295	4	27	,	,	PUNCT
cana-1295	4	28	a.p	a.p	PROPN
cana-1295	4	29	.	.	PROPN
cana-1295	4	30	,	,	PUNCT
cana-1295	4	31	india	india	PROPN
cana-1295	4	32	.	.	PUNCT
cana-1295	4	33	email	email	NOUN
cana-1295	4	34	:	:	PUNCT
cana-1295	4	35	n.sandhayarani@rguktrkv.ac.in	n.sandhayarani@rguktrkv.ac.in	PROPN
cana-1295	4	36	1mathematics	1mathematics	NUM
cana-1295	4	37	mentor	mentor	NOUN
cana-1295	4	38	,	,	PUNCT
cana-1295	4	39	rgukt	rgukt	NOUN
cana-1295	4	40	,	,	PUNCT
cana-1295	4	41	r.k.valley	r.k.valley	NOUN
cana-1295	4	42	,	,	PUNCT
cana-1295	4	43	idupulapaya	idupulapaya	PROPN
cana-1295	4	44	,	,	PUNCT
cana-1295	4	45	a.p	a.p	PROPN
cana-1295	4	46	.	.	PROPN
cana-1295	4	47	,	,	PUNCT
cana-1295	4	48	india	india	PROPN
cana-1295	4	49	.	.	PUNCT
cana-1295	5	1	2associate	2associate	NUM
cana-1295	5	2	professor	professor	NOUN
cana-1295	5	3	,	,	PUNCT
cana-1295	5	4	department	department	NOUN
cana-1295	5	5	of	of	ADP
cana-1295	5	6	mathematics	mathematics	PROPN
cana-1295	5	7	&	&	CCONJ
cana-1295	5	8	statistics	statistic	NOUN
cana-1295	5	9	,	,	PUNCT
cana-1295	5	10	school	school	NOUN
cana-1295	5	11	of	of	ADP
cana-1295	5	12	applied	apply	VERB
cana-1295	5	13	sciences	sciences	PROPN
cana-1295	5	14	&	&	CCONJ
cana-1295	5	15	humanities	humanity	NOUN
cana-1295	5	16	,	,	PUNCT
cana-1295	5	17	vfstr	vfstr	NOUN
cana-1295	5	18	deemed	deem	VERB
cana-1295	5	19	to	to	PART
cana-1295	5	20	be	be	AUX
cana-1295	5	21	university	university	NOUN
cana-1295	5	22	,	,	PUNCT
cana-1295	5	23	vadlamudi	vadlamudi	NOUN
cana-1295	5	24	,	,	PUNCT
cana-1295	5	25	guntur	guntur	PROPN
cana-1295	5	26	,	,	PUNCT
cana-1295	5	27	a.p	a.p	PROPN
cana-1295	5	28	.	.	PROPN
cana-1295	5	29	,	,	PUNCT
cana-1295	5	30	india.email:gsrinulakshmi77@gmail.com	india.email:gsrinulakshmi77@gmail.com	X
cana-1295	6	1	3associate	3associate	NUM
cana-1295	6	2	professor	professor	NOUN
cana-1295	6	3	,	,	PUNCT
cana-1295	6	4	department	department	NOUN
cana-1295	6	5	of	of	ADP
cana-1295	6	6	mathematics	mathematics	PROPN
cana-1295	6	7	,	,	PUNCT
cana-1295	6	8	vvit	vvit	PROPN
cana-1295	6	9	,	,	PUNCT
cana-1295	6	10	guntur	guntur	PROPN
cana-1295	6	11	,	,	PUNCT
cana-1295	6	12	a.p	a.p	PROPN
cana-1295	6	13	.	.	PROPN
cana-1295	6	14	,	,	PUNCT
cana-1295	6	15	india.email	india.email	PROPN
cana-1295	6	16	:	:	PUNCT
cana-1295	6	17	ramprasadchegu1984@gmail.com	ramprasadchegu1984@gmail.com	NOUN
cana-1295	6	18	article	article	NOUN
cana-1295	6	19	history	history	NOUN
cana-1295	6	20	:	:	PUNCT
cana-1295	6	21	received	receive	VERB
cana-1295	6	22	:	:	PUNCT
cana-1295	6	23	01	01	NUM
cana-1295	6	24	-	-	PUNCT
cana-1295	6	25	06	06	NUM
cana-1295	6	26	-	-	PUNCT
cana-1295	6	27	2024	2024	NUM
cana-1295	6	28	revised	revise	VERB
cana-1295	6	29	:	:	PUNCT
cana-1295	6	30	03	03	NUM
cana-1295	6	31	-	-	PUNCT
cana-1295	6	32	07	07	NUM
cana-1295	6	33	-	-	PUNCT
cana-1295	6	34	2024	2024	NUM
cana-1295	6	35	accepted	accept	VERB
cana-1295	6	36	:	:	PUNCT
cana-1295	6	37	29	29	NUM
cana-1295	6	38	-	-	SYM
cana-1295	6	39	07	07	NUM
cana-1295	6	40	-	-	PUNCT
cana-1295	6	41	2024	2024	NUM
cana-1295	6	42	abstract	abstract	NOUN
cana-1295	6	43	:	:	PUNCT
cana-1295	6	44	in	in	ADP
cana-1295	6	45	this	this	DET
cana-1295	6	46	paper	paper	NOUN
cana-1295	6	47	we	we	PRON
cana-1295	6	48	made	make	VERB
cana-1295	6	49	an	an	DET
cana-1295	6	50	attempt	attempt	NOUN
cana-1295	6	51	to	to	PART
cana-1295	6	52	study	study	VERB
cana-1295	6	53	the	the	DET
cana-1295	6	54	algebraic	algebraic	ADJ
cana-1295	6	55	structure	structure	NOUN
cana-1295	6	56	of	of	ADP
cana-1295	6	57	bi	bi	PROPN
cana-1295	6	58	-groups	-groups	PROPN
cana-1295	6	59	,	,	PUNCT
cana-1295	6	60	biternary	biternary	ADJ
cana-1295	6	61	semi	semi	ADJ
cana-1295	6	62	group	group	NOUN
cana-1295	6	63	,	,	PUNCT
cana-1295	6	64	bi	bi	ADJ
cana-1295	6	65	-	-	ADJ
cana-1295	6	66	ternary	ternary	ADJ
cana-1295	6	67	sub	sub	NOUN
cana-1295	6	68	semi	semi	NOUN
cana-1295	6	69	group	group	NOUN
cana-1295	6	70	,	,	PUNCT
cana-1295	6	71	ideals	ideal	NOUN
cana-1295	6	72	in	in	ADP
cana-1295	6	73	bi	bi	ADJ
cana-1295	6	74	ternary	ternary	NOUN
cana-1295	6	75	semi	semi	NOUN
cana-1295	6	76	group	group	NOUN
cana-1295	6	77	and	and	CCONJ
cana-1295	6	78	discussed	discuss	VERB
cana-1295	6	79	some	some	PRON
cana-1295	6	80	of	of	ADP
cana-1295	6	81	its	its	PRON
cana-1295	6	82	properties	property	NOUN
cana-1295	6	83	with	with	ADP
cana-1295	6	84	counter	counter	ADJ
cana-1295	6	85	examples	example	NOUN
cana-1295	6	86	.	.	PUNCT
cana-1295	7	1	keywords	keyword	NOUN
cana-1295	7	2	:	:	PUNCT
cana-1295	7	3	group	group	NOUN
cana-1295	7	4	,	,	PUNCT
cana-1295	7	5	bi	bi	NOUN
cana-1295	7	6	-	-	NOUN
cana-1295	7	7	groups	group	NOUN
cana-1295	7	8	,	,	PUNCT
cana-1295	7	9	bi	bi	ADJ
cana-1295	7	10	-	-	ADJ
cana-1295	7	11	ternary	ternary	ADJ
cana-1295	7	12	semi	semi	ADJ
cana-1295	7	13	groups	group	NOUN
cana-1295	7	14	.	.	PUNCT
cana-1295	8	1	introduction	introduction	NOUN
cana-1295	8	2	bi	bi	NOUN
cana-1295	8	3	-	-	NOUN
cana-1295	8	4	groups	group	NOUN
cana-1295	8	5	are	be	AUX
cana-1295	8	6	a	a	DET
cana-1295	8	7	particularly	particularly	ADV
cana-1295	8	8	useful	useful	ADJ
cana-1295	8	9	tool	tool	NOUN
cana-1295	8	10	since	since	SCONJ
cana-1295	8	11	they	they	PRON
cana-1295	8	12	provide	provide	VERB
cana-1295	8	13	solutions	solution	NOUN
cana-1295	8	14	to	to	ADP
cana-1295	8	15	a	a	DET
cana-1295	8	16	significant	significant	ADJ
cana-1295	8	17	difficulty	difficulty	NOUN
cana-1295	8	18	that	that	PRON
cana-1295	8	19	all	all	DET
cana-1295	8	20	groups	group	NOUN
cana-1295	8	21	encounter	encounter	VERB
cana-1295	8	22	,	,	PUNCT
cana-1295	8	23	namely	namely	ADV
cana-1295	8	24	that	that	SCONJ
cana-1295	8	25	the	the	DET
cana-1295	8	26	union	union	NOUN
cana-1295	8	27	of	of	ADP
cana-1295	8	28	two	two	NUM
cana-1295	8	29	subgroups	subgroup	NOUN
cana-1295	8	30	does	do	AUX
cana-1295	8	31	not	not	PART
cana-1295	8	32	create	create	VERB
cana-1295	8	33	any	any	DET
cana-1295	8	34	algebraic	algebraic	ADJ
cana-1295	8	35	framework	framework	NOUN
cana-1295	8	36	,	,	PUNCT
cana-1295	8	37	but	but	CCONJ
cana-1295	8	38	they	they	PRON
cana-1295	8	39	do	do	AUX
cana-1295	8	40	make	make	VERB
cana-1295	8	41	a	a	DET
cana-1295	8	42	good	good	ADJ
cana-1295	8	43	bi	bi	ADJ
cana-1295	8	44	-	-	ADJ
cana-1295	8	45	algebraic	algebraic	ADJ
cana-1295	8	46	framework	framework	NOUN
cana-1295	8	47	.	.	PUNCT
cana-1295	9	1	the	the	DET
cana-1295	9	2	bi	bi	PROPN
cana-1295	9	3	-	-	PROPN
cana-1295	9	4	group	group	NOUN
cana-1295	9	5	research	research	NOUN
cana-1295	9	6	was	be	AUX
cana-1295	9	7	conducted	conduct	VERB
cana-1295	9	8	from	from	ADP
cana-1295	9	9	1994	1994	NUM
cana-1295	9	10	to	to	ADP
cana-1295	9	11	1996	1996	NUM
cana-1295	9	12	.	.	PUNCT
cana-1295	10	1	maggu	maggu	NOUN
cana-1295	10	2	were	be	AUX
cana-1295	10	3	the	the	DET
cana-1295	10	4	initial	initial	ADJ
cana-1295	10	5	one	one	NOUN
cana-1295	10	6	to	to	PART
cana-1295	10	7	use	use	VERB
cana-1295	10	8	a	a	DET
cana-1295	10	9	syntax	syntax	NOUN
cana-1295	10	10	for	for	ADP
cana-1295	10	11	bi	bi	NOUN
cana-1295	10	12	-	-	NOUN
cana-1295	10	13	groups	group	NOUN
cana-1295	10	14	.	.	PUNCT
cana-1295	11	1	vasantha	vasantha	PROPN
cana-1295	11	2	kandaswamy	kandaswamy	PROPN
cana-1295	11	3	and	and	CCONJ
cana-1295	11	4	meiyappan	meiyappan	NOUN
cana-1295	11	5	expanded	expand	VERB
cana-1295	11	6	this	this	DET
cana-1295	11	7	theory	theory	NOUN
cana-1295	11	8	in	in	ADP
cana-1295	11	9	1997	1997	NUM
cana-1295	11	10	.	.	PUNCT
cana-1295	12	1	several	several	ADJ
cana-1295	12	2	individuals	individual	NOUN
cana-1295	12	3	propose	propose	VERB
cana-1295	12	4	adjustments	adjustment	NOUN
cana-1295	12	5	to	to	ADP
cana-1295	12	6	some	some	PRON
cana-1295	12	7	of	of	ADP
cana-1295	12	8	maggu	maggu	NOUN
cana-1295	12	9	's	's	PART
cana-1295	12	10	already	already	ADV
cana-1295	12	11	proven	prove	VERB
cana-1295	12	12	outcomes	outcome	NOUN
cana-1295	12	13	.	.	PUNCT
cana-1295	13	1	these	these	DET
cana-1295	13	2	conclusions	conclusion	NOUN
cana-1295	13	3	included	include	VERB
cana-1295	13	4	supplementary	supplementary	ADJ
cana-1295	13	5	bi	bi	ADJ
cana-1295	13	6	-	-	ADJ
cana-1295	13	7	group	group	NOUN
cana-1295	13	8	characterisation	characterisation	NOUN
cana-1295	13	9	results	result	NOUN
cana-1295	13	10	.	.	PUNCT
cana-1295	14	1	however	however	ADV
cana-1295	14	2	,	,	PUNCT
cana-1295	14	3	vasantha	vasantha	PROPN
cana-1295	14	4	kandaswamy	kandaswamy	PROPN
cana-1295	14	5	has	have	AUX
cana-1295	14	6	lately	lately	ADV
cana-1295	14	7	researched	research	VERB
cana-1295	14	8	the	the	DET
cana-1295	14	9	idea	idea	NOUN
cana-1295	14	10	of	of	ADP
cana-1295	14	11	bi	bi	ADJ
cana-1295	14	12	-	-	ADJ
cana-1295	14	13	algebraic	algebraic	ADJ
cana-1295	14	14	framework.agboola	framework.agboola	PROPN
cana-1295	14	15	and	and	CCONJ
cana-1295	14	16	akinola	akinola	PROPN
cana-1295	14	17	investigated	investigate	VERB
cana-1295	14	18	bi	bi	NOUN
cana-1295	14	19	-	-	NOUN
cana-1295	14	20	cosets	coset	NOUN
cana-1295	14	21	in	in	ADP
cana-1295	14	22	a	a	DET
cana-1295	14	23	bivector	bivector	NOUN
cana-1295	14	24	space	space	NOUN
cana-1295	14	25	..	..	PUNCT
cana-1295	15	1	i	i	PRON
cana-1295	15	2	expanded	expand	VERB
cana-1295	15	3	upon	upon	SCONJ
cana-1295	15	4	it	it	PRON
cana-1295	15	5	the	the	DET
cana-1295	15	6	ternary	ternary	ADJ
cana-1295	15	7	operation	operation	NOUN
cana-1295	15	8	is	be	AUX
cana-1295	15	9	one	one	NUM
cana-1295	15	10	of	of	ADP
cana-1295	15	11	the	the	DET
cana-1295	15	12	operations	operation	NOUN
cana-1295	15	13	in	in	ADP
cana-1295	15	14	algebraic	algebraic	ADJ
cana-1295	15	15	frame	frame	NOUN
cana-1295	15	16	work	work	NOUN
cana-1295	15	17	that	that	PRON
cana-1295	15	18	i	i	PRON
cana-1295	15	19	used	use	VERB
cana-1295	15	20	,	,	PUNCT
cana-1295	15	21	and	and	CCONJ
cana-1295	15	22	then	then	ADV
cana-1295	15	23	used	use	VERB
cana-1295	15	24	the	the	DET
cana-1295	15	25	same	same	ADJ
cana-1295	15	26	idea	idea	NOUN
cana-1295	15	27	in	in	ADP
cana-1295	15	28	every	every	DET
cana-1295	15	29	way	way	NOUN
cana-1295	15	30	that	that	PRON
cana-1295	15	31	i	i	PRON
cana-1295	15	32	could	could	AUX
cana-1295	15	33	.	.	PUNCT
cana-1295	16	1	g.	g.	PROPN
cana-1295	16	2	srinivasa	srinivasa	PROPN
cana-1295	16	3	rao	rao	PROPN
cana-1295	16	4	et.al[1	et.al[1	PROPN
cana-1295	16	5	-	-	PUNCT
cana-1295	16	6	2	2	NUM
cana-1295	16	7	,	,	PUNCT
cana-1295	16	8	13	13	NUM
cana-1295	16	9	-	-	SYM
cana-1295	16	10	18	18	NUM
cana-1295	16	11	]	]	PUNCT
cana-1295	16	12	discussed	discuss	VERB
cana-1295	16	13	about	about	ADP
cana-1295	16	14	ternary	ternary	ADJ
cana-1295	16	15	semirings	semiring	NOUN
cana-1295	16	16	,	,	PUNCT
cana-1295	16	17	ordered	order	VERB
cana-1295	16	18	ternary	ternary	ADJ
cana-1295	16	19	semirings	semiring	NOUN
cana-1295	16	20	and	and	CCONJ
cana-1295	16	21	gamma	gamma	NOUN
cana-1295	16	22	semirings	semiring	NOUN
cana-1295	16	23	and	and	CCONJ
cana-1295	16	24	their	their	PRON
cana-1295	16	25	properties	property	NOUN
cana-1295	16	26	.	.	PUNCT
cana-1295	17	1	1	1	X
cana-1295	17	2	.	.	X
cana-1295	17	3	preliminaries	preliminary	NOUN
cana-1295	17	4	:	:	PUNCT
cana-1295	17	5	definition	definition	NOUN
cana-1295	17	6	1.1	1.1	NUM
cana-1295	17	7	:	:	PUNCT
cana-1295	17	8	a	a	DET
cana-1295	17	9	non	non	ADJ
cana-1295	17	10	-	-	ADJ
cana-1295	17	11	empty	empty	ADJ
cana-1295	17	12	set𝐺and′	set𝐺and′	PROPN
cana-1295	17	13	∗	∗	NOUN
cana-1295	17	14	′is	′is	PROPN
cana-1295	17	15	a	a	DET
cana-1295	17	16	binary	binary	ADJ
cana-1295	17	17	operation	operation	NOUN
cana-1295	17	18	on	on	ADP
cana-1295	17	19	𝐺	𝐺	PROPN
cana-1295	17	20	if	if	SCONJ
cana-1295	17	21	it	it	PRON
cana-1295	17	22	satisfies	satisfy	VERB
cana-1295	17	23	the	the	DET
cana-1295	17	24	following	follow	VERB
cana-1295	17	25	conditions	condition	NOUN
cana-1295	17	26	,	,	PUNCT
cana-1295	17	27	then	then	ADV
cana-1295	17	28	algebraic	algebraic	PROPN
cana-1295	17	29	structure(𝐺,∗	structure(𝐺,∗	PROPN
cana-1295	17	30	)	)	PUNCT
cana-1295	17	31	is	be	AUX
cana-1295	17	32	called	call	VERB
cana-1295	17	33	a	a	DET
cana-1295	17	34	group	group	NOUN
cana-1295	17	35	.	.	PUNCT
cana-1295	18	1	𝑖	𝑖	X
cana-1295	18	2	)	)	PUNCT
cana-1295	18	3	𝑎	𝑎	NOUN
cana-1295	18	4	,	,	PUNCT
cana-1295	18	5	𝑏	𝑏	PROPN
cana-1295	18	6	∈	∈	PROPN
cana-1295	18	7	𝐺	𝐺	NOUN
cana-1295	18	8	⟹	⟹	NUM
cana-1295	18	9	𝑎	𝑎	NOUN
cana-1295	18	10	∗	∗	NOUN
cana-1295	18	11	𝑏	𝑏	PRON
cana-1295	18	12	∈	∈	PROPN
cana-1295	18	13	𝐺	𝐺	PROPN
cana-1295	18	14	𝑖𝑖	𝑖𝑖	NOUN
cana-1295	18	15	)	)	PUNCT
cana-1295	18	16	𝑎	𝑎	PRON
cana-1295	18	17	∗	∗	NOUN
cana-1295	18	18	(	(	PUNCT
cana-1295	18	19	𝑏	𝑏	PROPN
cana-1295	18	20	∗	∗	NOUN
cana-1295	18	21	𝑐	𝑐	NOUN
cana-1295	18	22	)	)	PUNCT
cana-1295	18	23	=	=	NOUN
cana-1295	18	24	(	(	PUNCT
cana-1295	18	25	𝑎	𝑎	NOUN
cana-1295	18	26	∗	∗	NOUN
cana-1295	18	27	𝑏	𝑏	NOUN
cana-1295	18	28	)	)	PUNCT
cana-1295	18	29	∗	∗	NOUN
cana-1295	18	30	𝑐∀𝑎	𝑐∀𝑎	NUM
cana-1295	18	31	,	,	PUNCT
cana-1295	18	32	𝑏	𝑏	NOUN
cana-1295	18	33	,	,	PUNCT
cana-1295	18	34	𝑐	𝑐	PROPN
cana-1295	18	35	∈	∈	PROPN
cana-1295	18	36	𝐺	𝐺	PROPN
cana-1295	18	37	𝑖𝑖𝑖	𝑖𝑖𝑖	NOUN
cana-1295	18	38	)	)	PUNCT
cana-1295	18	39	if	if	SCONJ
cana-1295	18	40	𝑒	𝑒	PROPN
cana-1295	18	41	∈	∈	PROPN
cana-1295	18	42	𝐺such	𝐺such	NOUN
cana-1295	18	43	that	that	SCONJ
cana-1295	18	44	𝑎	𝑎	DET
cana-1295	18	45	∗	∗	NOUN
cana-1295	18	46	𝑒	𝑒	PART
cana-1295	18	47	=	=	SYM
cana-1295	18	48	𝑒	𝑒	X
cana-1295	18	49	∗	∗	NOUN
cana-1295	18	50	𝑎	𝑎	X
cana-1295	18	51	=	=	SYM
cana-1295	18	52	𝑎∀𝑎	𝑎∀𝑎	NUM
cana-1295	18	53	∈	∈	PROPN
cana-1295	18	54	𝐺	𝐺	NOUN
cana-1295	18	55	𝑖𝑣	𝑖𝑣	NOUN
cana-1295	18	56	)	)	PUNCT
cana-1295	18	57	meant	mean	VERB
cana-1295	18	58	for	for	ADP
cana-1295	18	59	each	each	DET
cana-1295	18	60	𝑎	𝑎	PROPN
cana-1295	18	61	∈	∈	PROPN
cana-1295	18	62	𝐺	𝐺	NOUN
cana-1295	18	63	present	present	NOUN
cana-1295	18	64	exist	exist	VERB
cana-1295	18	65	part	part	NOUN
cana-1295	18	66	𝑏	𝑏	PROPN
cana-1295	18	67	∈	∈	PROPN
cana-1295	18	68	𝐺	𝐺	NOUN
cana-1295	18	69	such	such	ADJ
cana-1295	18	70	with	with	ADP
cana-1295	18	71	the	the	DET
cana-1295	18	72	aim	aim	NOUN
cana-1295	18	73	of	of	ADP
cana-1295	18	74	𝑎	𝑎	NOUN
cana-1295	18	75	∗	∗	NOUN
cana-1295	18	76	𝑏	𝑏	NOUN
cana-1295	18	77	=	=	SYM
cana-1295	18	78	𝑏	𝑏	PROPN
cana-1295	18	79	∗	∗	NOUN
cana-1295	18	80	𝑎	𝑎	X
cana-1295	18	81	=	=	X
cana-1295	18	82	𝑒	𝑒	ADJ
cana-1295	18	83	where𝑏	where𝑏	NOUN
cana-1295	18	84	=	=	PUNCT
cana-1295	18	85	𝑎−1	𝑎−1	PROPN
cana-1295	18	86	mailto:ramprasadchegu1984@gmail.com	mailto:ramprasadchegu1984@gmail.com	PROPN
cana-1295	18	87	communications	communication	NOUN
cana-1295	18	88	on	on	ADP
cana-1295	18	89	applied	apply	VERB
cana-1295	18	90	nonlinear	nonlinear	ADJ
cana-1295	18	91	analysis	analysis	NOUN
cana-1295	18	92	issn	issn	NOUN
cana-1295	18	93	:	:	PUNCT
cana-1295	18	94	1074	1074	NUM
cana-1295	18	95	-	-	PUNCT
cana-1295	18	96	133x	133x	NUM
cana-1295	18	97	vol	vol	NOUN
cana-1295	18	98	31	31	NUM
cana-1295	18	99	no	no	NOUN
cana-1295	18	100	.	.	PUNCT
cana-1295	19	1	7s	7	NOUN
cana-1295	19	2	(	(	PUNCT
cana-1295	19	3	2024	2024	NUM
cana-1295	19	4	)	)	PUNCT
cana-1295	19	5	204	204	NUM
cana-1295	19	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1295	19	7	definition	definition	NOUN
cana-1295	19	8	1.2	1.2	NUM
cana-1295	19	9	:	:	PUNCT
cana-1295	19	10	a	a	DET
cana-1295	19	11	non	non	ADJ
cana-1295	19	12	-	-	ADJ
cana-1295	19	13	empty	empty	ADJ
cana-1295	19	14	group	group	NOUN
cana-1295	19	15	𝐺and	𝐺and	PROPN
cana-1295	19	16	′	′	NOUN
cana-1295	19	17	∗	∗	NOUN
cana-1295	19	18	′	′	NUM
cana-1295	19	19	is	be	AUX
cana-1295	19	20	a	a	DET
cana-1295	19	21	dual	dual	ADJ
cana-1295	19	22	procedure	procedure	NOUN
cana-1295	19	23	on	on	ADP
cana-1295	19	24	𝐺	𝐺	PROPN
cana-1295	19	25	if	if	SCONJ
cana-1295	19	26	it	it	PRON
cana-1295	19	27	satisfies	satisfy	VERB
cana-1295	19	28	the	the	DET
cana-1295	19	29	following	follow	VERB
cana-1295	19	30	conditions	condition	NOUN
cana-1295	19	31	,	,	PUNCT
cana-1295	19	32	then	then	ADV
cana-1295	19	33	algebraic	algebraic	PROPN
cana-1295	19	34	structure(𝐺,∗	structure(𝐺,∗	PROPN
cana-1295	19	35	)	)	PUNCT
cana-1295	19	36	be	be	AUX
cana-1295	19	37	call	call	VERB
cana-1295	19	38	a	a	DET
cana-1295	19	39	semi	semi	ADJ
cana-1295	19	40	group	group	NOUN
cana-1295	19	41	.	.	PUNCT
cana-1295	20	1	𝑖)𝑎	𝑖)𝑎	PROPN
cana-1295	20	2	,	,	PUNCT
cana-1295	20	3	𝑏	𝑏	PROPN
cana-1295	20	4	∈	∈	PROPN
cana-1295	20	5	𝐺	𝐺	NOUN
cana-1295	20	6	⟹	⟹	NUM
cana-1295	20	7	𝑎	𝑎	NOUN
cana-1295	20	8	∗	∗	NOUN
cana-1295	20	9	𝑏	𝑏	PRON
cana-1295	20	10	∈	∈	PROPN
cana-1295	20	11	𝐺	𝐺	PROPN
cana-1295	20	12	𝑖𝑖)𝑎	𝑖𝑖)𝑎	PROPN
cana-1295	20	13	∗	∗	NOUN
cana-1295	20	14	(	(	PUNCT
cana-1295	20	15	𝑏	𝑏	PROPN
cana-1295	20	16	∗	∗	NOUN
cana-1295	20	17	𝑐	𝑐	NOUN
cana-1295	20	18	)	)	PUNCT
cana-1295	20	19	=	=	NOUN
cana-1295	20	20	(	(	PUNCT
cana-1295	20	21	𝑎	𝑎	NOUN
cana-1295	20	22	∗	∗	NOUN
cana-1295	20	23	𝑏	𝑏	NOUN
cana-1295	20	24	)	)	PUNCT
cana-1295	20	25	∗	∗	NOUN
cana-1295	20	26	𝑐∀𝑎	𝑐∀𝑎	NUM
cana-1295	20	27	,	,	PUNCT
cana-1295	20	28	𝑏	𝑏	NOUN
cana-1295	20	29	,	,	PUNCT
cana-1295	20	30	𝑐	𝑐	PROPN
cana-1295	20	31	∈	∈	PROPN
cana-1295	20	32	𝐺	𝐺	PROPN
cana-1295	20	33	definition	definition	NOUN
cana-1295	20	34	1.3	1.3	NUM
cana-1295	20	35	:	:	PUNCT
cana-1295	20	36	a	a	DET
cana-1295	20	37	non	non	ADJ
cana-1295	20	38	-	-	ADJ
cana-1295	20	39	empty	empty	ADJ
cana-1295	20	40	group	group	NOUN
cana-1295	20	41	be	be	AUX
cana-1295	20	42	a	a	DET
cana-1295	20	43	t.	t.	NOUN
cana-1295	20	44	then	then	ADV
cana-1295	20	45	𝑇	𝑇	PROPN
cana-1295	20	46	be	be	AUX
cana-1295	20	47	understood	understand	VERB
cana-1295	20	48	to	to	PART
cana-1295	20	49	exist	exist	VERB
cana-1295	20	50	a	a	DET
cana-1295	20	51	ternary	ternary	ADJ
cana-1295	20	52	semi	semi	ADJ
cana-1295	20	53	group	group	NOUN
cana-1295	20	54	from	from	ADP
cana-1295	20	55	𝑇	𝑇	PROPN
cana-1295	20	56	×	×	NOUN
cana-1295	20	57	𝑇	𝑇	PROPN
cana-1295	20	58	×	×	NOUN
cana-1295	20	59	𝑇	𝑇	PROPN
cana-1295	20	60	→	→	SYM
cana-1295	20	61	𝑇	𝑇	PROPN
cana-1295	20	62	which	which	PRON
cana-1295	20	63	map	map	VERB
cana-1295	20	64	[	[	X
cana-1295	20	65	𝑥1𝑥2𝑥3	𝑥1𝑥2𝑥3	NOUN
cana-1295	20	66	]	]	X
cana-1295	20	67	→	→	SYM
cana-1295	20	68	𝑥1𝑥2𝑥3	𝑥1𝑥2𝑥3	NOUN
cana-1295	20	69	satisfy	satisfy	VERB
cana-1295	20	70	the	the	DET
cana-1295	20	71	condition:[(𝑥1𝑥2𝑥3)𝑥4𝑥5	condition:[(𝑥1𝑥2𝑥3)𝑥4𝑥5	NOUN
cana-1295	20	72	]	]	X
cana-1295	20	73	=	=	PUNCT
cana-1295	21	1	[	[	X
cana-1295	21	2	𝑥1(𝑥2𝑥3𝑥4)𝑥5	𝑥1(𝑥2𝑥3𝑥4)𝑥5	X
cana-1295	21	3	]	]	X
cana-1295	21	4	=	=	PUNCT
cana-1295	22	1	[	[	X
cana-1295	22	2	𝑥1𝑥2(𝑥3𝑥4𝑥5	𝑥1𝑥2(𝑥3𝑥4𝑥5	PROPN
cana-1295	22	3	)	)	PUNCT
cana-1295	22	4	]	]	PUNCT
cana-1295	22	5	for	for	ADP
cana-1295	22	6	all	all	PRON
cana-1295	22	7	𝑥𝑖	𝑥𝑖	ADP
cana-1295	22	8	∈	∈	PROPN
cana-1295	22	9	𝑇	𝑇	PROPN
cana-1295	22	10	,	,	PUNCT
cana-1295	22	11	𝑖	𝑖	X
cana-1295	22	12	=	=	SYM
cana-1295	22	13	1𝑡𝑜5	1𝑡𝑜5	NUM
cana-1295	22	14	.	.	PUNCT
cana-1295	22	15	definition	definition	NOUN
cana-1295	22	16	1.4	1.4	NUM
cana-1295	22	17	:	:	PUNCT
cana-1295	22	18	a	a	DET
cana-1295	22	19	non	non	ADJ
cana-1295	22	20	-	-	ADJ
cana-1295	22	21	empty	empty	ADJ
cana-1295	22	22	set	set	NOUN
cana-1295	22	23	(	(	PUNCT
cana-1295	22	24	𝐺	𝐺	NOUN
cana-1295	22	25	,	,	PUNCT
cana-1295	22	26	+	+	PROPN
cana-1295	22	27	,	,	PUNCT
cana-1295	22	28	∙	∙	PROPN
cana-1295	22	29	)	)	PUNCT
cana-1295	22	30	with	with	ADP
cana-1295	22	31	dual	dual	ADJ
cana-1295	22	32	action	action	NOUN
cana-1295	22	33	two	two	NUM
cana-1295	22	34	are	be	AUX
cana-1295	22	35	′	′	NUM
cana-1295	23	1	+	+	CCONJ
cana-1295	23	2	′	′	NUM
cana-1295	23	3	&	&	CCONJ
cana-1295	23	4	′	′	NUM
cana-1295	24	1	∙	∙	PROPN
cana-1295	24	2	′	′	NUM
cana-1295	24	3	be	be	AUX
cana-1295	24	4	call	call	VERB
cana-1295	24	5	a	a	DET
cana-1295	24	6	bi	bi	NOUN
cana-1295	24	7	-	-	NOUN
cana-1295	24	8	group	group	NOUN
cana-1295	24	9	if	if	SCONJ
cana-1295	24	10	here	here	ADV
cana-1295	24	11	be	be	VERB
cana-1295	24	12	real	real	ADJ
cana-1295	24	13	two	two	NUM
cana-1295	24	14	suitable	suitable	ADJ
cana-1295	24	15	sub	sub	NOUN
cana-1295	24	16	sets	set	NOUN
cana-1295	24	17	𝐺1	𝐺1	NOUN
cana-1295	24	18	&	&	CCONJ
cana-1295	24	19	𝐺2	𝐺2	NOUN
cana-1295	24	20	of	of	ADP
cana-1295	24	21	𝐺	𝐺	PROPN
cana-1295	24	22	follow	follow	NOUN
cana-1295	24	23	𝑖	𝑖	SYM
cana-1295	24	24	)	)	PUNCT
cana-1295	24	25	𝐺	𝐺	NOUN
cana-1295	24	26	=	=	PUNCT
cana-1295	24	27	𝐺1	𝐺1	NOUN
cana-1295	24	28	∪	∪	ADJ
cana-1295	24	29	𝐺2	𝐺2	NOUN
cana-1295	24	30	𝑖𝑖	𝑖𝑖	NOUN
cana-1295	24	31	)	)	PUNCT
cana-1295	24	32	(	(	PUNCT
cana-1295	24	33	𝐺1+)is	𝐺1+)is	VERB
cana-1295	24	34	a	a	DET
cana-1295	24	35	group	group	NOUN
cana-1295	24	36	.	.	PUNCT
cana-1295	25	1	𝑖𝑖𝑖)(𝐺1,⋅	𝑖𝑖𝑖)(𝐺1,⋅	NOUN
cana-1295	25	2	)	)	PUNCT
cana-1295	25	3	is	be	AUX
cana-1295	25	4	group	group	NOUN
cana-1295	25	5	.	.	PUNCT
cana-1295	25	6	example	example	NOUN
cana-1295	25	7	1.5	1.5	NUM
cana-1295	25	8	:	:	PUNCT
cana-1295	25	9	let	let	VERB
cana-1295	25	10	𝐺	𝐺	PROPN
cana-1295	25	11	=	=	PUNCT
cana-1295	25	12	{	{	PUNCT
cana-1295	25	13	set	set	NOUN
cana-1295	25	14	of	of	ADP
cana-1295	25	15	integers	integer	NOUN
cana-1295	25	16	}	}	PUNCT
cana-1295	25	17	∪	∪	VERB
cana-1295	25	18	{	{	PUNCT
cana-1295	25	19	𝑖	𝑖	NOUN
cana-1295	25	20	,	,	PUNCT
cana-1295	25	21	−𝑖	−𝑖	ADJ
cana-1295	25	22	}	}	PUNCT
cana-1295	25	23	be	be	AUX
cana-1295	25	24	a	a	DET
cana-1295	25	25	bi	bi	ADJ
cana-1295	25	26	group	group	NOUN
cana-1295	25	27	.	.	PUNCT
cana-1295	26	1	definition	definition	NOUN
cana-1295	26	2	1.6	1.6	NUM
cana-1295	26	3	:	:	PUNCT
cana-1295	26	4	a	a	DET
cana-1295	26	5	non	non	ADJ
cana-1295	26	6	-	-	ADJ
cana-1295	26	7	empty	empty	ADJ
cana-1295	26	8	set	set	NOUN
cana-1295	26	9	(	(	PUNCT
cana-1295	26	10	𝐺	𝐺	NOUN
cana-1295	26	11	,	,	PUNCT
cana-1295	26	12	+	+	PROPN
cana-1295	26	13	,	,	PUNCT
cana-1295	26	14	∙	∙	PROPN
cana-1295	26	15	)	)	PUNCT
cana-1295	26	16	with	with	ADP
cana-1295	26	17	dual	dual	ADJ
cana-1295	26	18	action	action	NOUN
cana-1295	26	19	two	two	NUM
cana-1295	26	20	are	be	AUX
cana-1295	26	21	′	′	NUM
cana-1295	27	1	+	+	CCONJ
cana-1295	27	2	′	′	NUM
cana-1295	27	3	&	&	CCONJ
cana-1295	27	4	′	′	NUM
cana-1295	28	1	∙	∙	PROPN
cana-1295	28	2	′	′	NUM
cana-1295	28	3	is	be	AUX
cana-1295	28	4	call	call	VERB
cana-1295	28	5	a	a	DET
cana-1295	28	6	bi	bi	NOUN
cana-1295	28	7	semi	semi	NOUN
cana-1295	28	8	group	group	NOUN
cana-1295	28	9	if	if	SCONJ
cana-1295	28	10	here	here	ADV
cana-1295	28	11	be	be	VERB
cana-1295	28	12	real	real	ADJ
cana-1295	28	13	two	two	NUM
cana-1295	28	14	suitable	suitable	ADJ
cana-1295	28	15	sub	sub	NOUN
cana-1295	28	16	sets	set	NOUN
cana-1295	28	17	𝐺1	𝐺1	NOUN
cana-1295	28	18	&	&	CCONJ
cana-1295	28	19	𝐺2	𝐺2	NOUN
cana-1295	28	20	of	of	ADP
cana-1295	28	21	𝐺	𝐺	PROPN
cana-1295	28	22	as	as	SCONJ
cana-1295	28	23	follow	follow	VERB
cana-1295	28	24	𝑖)𝐺	𝑖)𝐺	PUNCT
cana-1295	29	1	=	=	PUNCT
cana-1295	29	2	𝐺1	𝐺1	NOUN
cana-1295	29	3	∪	∪	ADP
cana-1295	29	4	𝐺2	𝐺2	ADJ
cana-1295	29	5	𝑖𝑖)(𝐺1	𝑖𝑖)(𝐺1	NOUN
cana-1295	29	6	,	,	PUNCT
cana-1295	29	7	+	+	NOUN
cana-1295	29	8	)	)	PUNCT
cana-1295	29	9	be	be	AUX
cana-1295	29	10	semi	semi	ADV
cana-1295	29	11	group	group	NOUN
cana-1295	29	12	.	.	PUNCT
cana-1295	30	1	𝑖𝑖𝑖)(𝐺1,⋅	𝑖𝑖𝑖)(𝐺1,⋅	NOUN
cana-1295	30	2	)	)	PUNCT
cana-1295	30	3	be	be	VERB
cana-1295	30	4	semi	semi	ADV
cana-1295	30	5	group	group	NOUN
cana-1295	30	6	.	.	PUNCT
cana-1295	31	1	example	example	NOUN
cana-1295	31	2	1.7	1.7	NUM
cana-1295	31	3	:	:	PUNCT
cana-1295	31	4	let	let	VERB
cana-1295	31	5	𝐺	𝐺	PROPN
cana-1295	31	6	=	=	SYM
cana-1295	31	7	(	(	PUNCT
cana-1295	31	8	𝑍+	𝑍+	X
cana-1295	31	9	,	,	PUNCT
cana-1295	31	10	+	+	NOUN
cana-1295	31	11	)	)	PUNCT
cana-1295	31	12	∪	∪	X
cana-1295	31	13	(	(	PUNCT
cana-1295	31	14	{	{	PUNCT
cana-1295	31	15	1	1	NUM
cana-1295	31	16	,	,	PUNCT
cana-1295	31	17	−1	−1	NOUN
cana-1295	31	18	,	,	PUNCT
cana-1295	31	19	𝑖	𝑖	X
cana-1295	31	20	,	,	PUNCT
cana-1295	31	21	−𝑖},∙	−𝑖},∙	NOUN
cana-1295	31	22	)	)	PUNCT
cana-1295	31	23	is	be	AUX
cana-1295	31	24	a	a	DET
cana-1295	31	25	bi	bi	NOUN
cana-1295	31	26	semi	semi	NOUN
cana-1295	31	27	group	group	NOUN
cana-1295	31	28	.	.	PUNCT
cana-1295	32	1	2	2	X
cana-1295	32	2	.	.	X
cana-1295	32	3	main	main	ADJ
cana-1295	32	4	results	result	NOUN
cana-1295	32	5	:	:	PUNCT
cana-1295	32	6	definition	definition	NOUN
cana-1295	32	7	2.1	2.1	NUM
cana-1295	32	8	:	:	PUNCT
cana-1295	32	9	a	a	DET
cana-1295	32	10	non	non	ADJ
cana-1295	32	11	-	-	ADJ
cana-1295	32	12	empty	empty	ADJ
cana-1295	32	13	set	set	VERB
cana-1295	32	14	𝑇	𝑇	PROPN
cana-1295	32	15	=	=	NOUN
cana-1295	32	16	𝑇1	𝑇1	PROPN
cana-1295	32	17	∪	∪	ADJ
cana-1295	32	18	𝑇2	𝑇2	PROPN
cana-1295	32	19	be	be	AUX
cana-1295	32	20	a	a	DET
cana-1295	32	21	bi	bi	ADJ
cana-1295	32	22	ternary	ternary	NOUN
cana-1295	32	23	semi	semi	NOUN
cana-1295	32	24	group	group	NOUN
cana-1295	32	25	.	.	PUNCT
cana-1295	33	1	then	then	ADV
cana-1295	33	2	both	both	PRON
cana-1295	33	3	𝑇1	𝑇1	NOUN
cana-1295	33	4	and	and	CCONJ
cana-1295	33	5	𝑇2	𝑇2	NOUN
cana-1295	33	6	are	be	AUX
cana-1295	33	7	satisfy	satisfy	VERB
cana-1295	33	8	the	the	DET
cana-1295	33	9	conditions	condition	NOUN
cana-1295	33	10	of	of	ADP
cana-1295	33	11	ternary	ternary	ADJ
cana-1295	33	12	semi	semi	ADJ
cana-1295	33	13	group	group	NOUN
cana-1295	33	14	.	.	PUNCT
cana-1295	34	1	i.e.	i.e.	X
cana-1295	34	2	[	[	X
cana-1295	34	3	(	(	PUNCT
cana-1295	34	4	𝑥1𝑥2𝑥3)𝑥4𝑥5	𝑥1𝑥2𝑥3)𝑥4𝑥5	X
cana-1295	34	5	]	]	X
cana-1295	34	6	=	=	PUNCT
cana-1295	35	1	[	[	X
cana-1295	35	2	𝑥1(𝑥2𝑥3𝑥4)𝑥5	𝑥1(𝑥2𝑥3𝑥4)𝑥5	X
cana-1295	35	3	]	]	X
cana-1295	35	4	=	=	PUNCT
cana-1295	36	1	[	[	X
cana-1295	36	2	𝑥1𝑥2(𝑥3𝑥4𝑥5	𝑥1𝑥2(𝑥3𝑥4𝑥5	PROPN
cana-1295	36	3	)	)	PUNCT
cana-1295	36	4	]	]	PUNCT
cana-1295	37	1	∀𝑥𝑖	∀𝑥𝑖	VERB
cana-1295	37	2	∈	∈	NOUN
cana-1295	37	3	𝑇1	𝑇1	NOUN
cana-1295	37	4	similarly	similarly	ADV
cana-1295	37	5	[	[	X
cana-1295	37	6	(	(	PUNCT
cana-1295	37	7	𝑎𝑏𝑐)𝑑𝑒	𝑎𝑏𝑐)𝑑𝑒	NOUN
cana-1295	37	8	]	]	PUNCT
cana-1295	37	9	=	=	PUNCT
cana-1295	38	1	[	[	X
cana-1295	38	2	𝑎(𝑏𝑐𝑑)𝑒	𝑎(𝑏𝑐𝑑)𝑒	X
cana-1295	38	3	]	]	X
cana-1295	38	4	=	=	PUNCT
cana-1295	39	1	[	[	X
cana-1295	39	2	𝑎𝑏(𝑐𝑑𝑒)]∀𝑎	𝑎𝑏(𝑐𝑑𝑒)]∀𝑎	ADJ
cana-1295	39	3	,	,	PUNCT
cana-1295	39	4	𝑏	𝑏	NOUN
cana-1295	39	5	,	,	PUNCT
cana-1295	39	6	𝑐	𝑐	PROPN
cana-1295	39	7	,	,	PUNCT
cana-1295	39	8	𝑑	𝑑	NOUN
cana-1295	39	9	,	,	PUNCT
cana-1295	39	10	𝑒	𝑒	PROPN
cana-1295	39	11	∈	∈	PROPN
cana-1295	39	12	𝑇2	𝑇2	NOUN
cana-1295	39	13	set	set	NOUN
cana-1295	39	14	of	of	ADP
cana-1295	39	15	non	non	ADJ
cana-1295	39	16	-	-	ADJ
cana-1295	39	17	empty	empty	ADJ
cana-1295	39	18	is	be	AUX
cana-1295	39	19	𝑇	𝑇	PROPN
cana-1295	39	20	with	with	ADP
cana-1295	39	21	ternary	ternary	ADJ
cana-1295	39	22	growth	growth	NOUN
cana-1295	39	23	be	be	AUX
cana-1295	39	24	a	a	DET
cana-1295	39	25	mark	mark	NOUN
cana-1295	39	26	of	of	ADP
cana-1295	39	27	[	[	PUNCT
cana-1295	39	28	]	]	X
cana-1295	39	29	is	be	AUX
cana-1295	39	30	said	say	VERB
cana-1295	39	31	to	to	PART
cana-1295	39	32	be	be	AUX
cana-1295	39	33	a	a	DET
cana-1295	39	34	bi	bi	ADJ
cana-1295	39	35	-	-	ADJ
cana-1295	39	36	ternary	ternary	ADJ
cana-1295	39	37	semi	semi	ADJ
cana-1295	39	38	group	group	NOUN
cana-1295	39	39	if	if	SCONJ
cana-1295	39	40	𝑇	𝑇	PROPN
cana-1295	39	41	=	=	SYM
cana-1295	39	42	𝑇1	𝑇1	NOUN
cana-1295	39	43	∪	∪	VERB
cana-1295	39	44	𝑇2	𝑇2	PROPN
cana-1295	39	45	someplace	someplace	ADJ
cana-1295	39	46	𝑇1	𝑇1	PROPN
cana-1295	39	47	&	&	CCONJ
cana-1295	39	48	𝑇2	𝑇2	PROPN
cana-1295	39	49	be	be	AUX
cana-1295	39	50	proper	proper	ADJ
cana-1295	39	51	subsets	subset	NOUN
cana-1295	39	52	of	of	ADP
cana-1295	39	53	𝑇	𝑇	PROPN
cana-1295	39	54	such	such	ADJ
cana-1295	39	55	that	that	SCONJ
cana-1295	39	56	𝑖)𝑇1	𝑖)𝑇1	PRON
cana-1295	39	57	is	be	AUX
cana-1295	39	58	ternary	ternary	ADJ
cana-1295	39	59	semi	semi	ADJ
cana-1295	39	60	group	group	NOUN
cana-1295	39	61	.	.	PUNCT
cana-1295	40	1	𝑖𝑖)𝑇2	𝑖𝑖)𝑇2	NOUN
cana-1295	40	2	is	be	AUX
cana-1295	40	3	ternary	ternary	ADJ
cana-1295	40	4	semi	semi	ADJ
cana-1295	40	5	group	group	NOUN
cana-1295	40	6	.	.	PUNCT
cana-1295	40	7	example	example	NOUN
cana-1295	40	8	2.2	2.2	NUM
cana-1295	40	9	:	:	PUNCT
cana-1295	40	10	let	let	VERB
cana-1295	40	11	𝑇	𝑇	PROPN
cana-1295	40	12	=	=	SYM
cana-1295	40	13	{	{	PUNCT
cana-1295	40	14	𝑖	𝑖	PROPN
cana-1295	40	15	,	,	PUNCT
cana-1295	40	16	−𝑖	−𝑖	ADJ
cana-1295	40	17	}	}	PUNCT
cana-1295	40	18	∪	∪	X
cana-1295	40	19	{	{	PUNCT
cana-1295	40	20	[	[	PUNCT
cana-1295	40	21	0	0	NUM
cana-1295	40	22	0	0	NUM
cana-1295	40	23	0	0	NUM
cana-1295	40	24	0	0	NUM
cana-1295	40	25	]	]	PUNCT
cana-1295	40	26	,	,	PUNCT
cana-1295	40	27	[	[	PUNCT
cana-1295	40	28	1	1	NUM
cana-1295	40	29	0	0	NUM
cana-1295	40	30	0	0	NUM
cana-1295	40	31	0	0	NUM
cana-1295	40	32	]	]	PUNCT
cana-1295	40	33	,	,	PUNCT
cana-1295	40	34	[	[	PUNCT
cana-1295	40	35	1	1	NUM
cana-1295	40	36	0	0	NUM
cana-1295	40	37	0	0	NUM
cana-1295	40	38	1	1	NUM
cana-1295	40	39	]	]	PUNCT
cana-1295	40	40	,	,	PUNCT
cana-1295	40	41	[	[	PUNCT
cana-1295	40	42	0	0	NUM
cana-1295	40	43	1	1	NUM
cana-1295	40	44	0	0	NUM
cana-1295	40	45	0	0	NUM
cana-1295	40	46	]	]	PUNCT
cana-1295	40	47	,	,	PUNCT
cana-1295	40	48	[	[	PUNCT
cana-1295	40	49	0	0	NUM
cana-1295	40	50	0	0	NUM
cana-1295	40	51	1	1	NUM
cana-1295	40	52	0	0	NUM
cana-1295	40	53	]	]	PUNCT
cana-1295	40	54	,	,	PUNCT
cana-1295	40	55	[	[	PUNCT
cana-1295	40	56	0	0	NUM
cana-1295	40	57	0	0	NUM
cana-1295	40	58	0	0	NUM
cana-1295	40	59	1	1	NUM
cana-1295	40	60	]	]	PUNCT
cana-1295	40	61	}	}	PUNCT
cana-1295	40	62	then	then	ADV
cana-1295	40	63	𝑇	𝑇	PROPN
cana-1295	40	64	is	be	AUX
cana-1295	40	65	bi	bi	ADJ
cana-1295	40	66	ternary	ternary	NOUN
cana-1295	40	67	semi	semi	ADJ
cana-1295	40	68	group	group	NOUN
cana-1295	40	69	under	under	ADP
cana-1295	40	70	complex	complex	ADJ
cana-1295	40	71	multiplication	multiplication	NOUN
cana-1295	40	72	and	and	CCONJ
cana-1295	40	73	matrix	matrix	NOUN
cana-1295	40	74	multiplication	multiplication	NOUN
cana-1295	40	75	.	.	PUNCT
cana-1295	40	76	example	example	NOUN
cana-1295	40	77	2.3	2.3	NUM
cana-1295	40	78	:	:	PUNCT
cana-1295	40	79	let	let	VERB
cana-1295	40	80	𝑇	𝑇	PROPN
cana-1295	40	81	=	=	PUNCT
cana-1295	40	82	∁	∁	PROPN
cana-1295	40	83	∪	∪	ADJ
cana-1295	40	84	𝑍	𝑍	NOUN
cana-1295	40	85	is	be	AUX
cana-1295	40	86	of	of	ADP
cana-1295	40	87	bi	bi	ADJ
cana-1295	40	88	ternary	ternary	ADJ
cana-1295	40	89	semi	semi	PROPN
cana-1295	40	90	group	group	NOUN
cana-1295	40	91	.	.	PUNCT
cana-1295	41	1	example	example	NOUN
cana-1295	41	2	2.4	2.4	NUM
cana-1295	41	3	:	:	PUNCT
cana-1295	41	4	let	let	VERB
cana-1295	41	5	𝑇	𝑇	PROPN
cana-1295	41	6	=	=	PUNCT
cana-1295	41	7	𝐶0	𝐶0	PROPN
cana-1295	41	8	−	−	PROPN
cana-1295	41	9	∪	∪	ADJ
cana-1295	41	10	𝑍	𝑍	NOUN
cana-1295	41	11	is	be	AUX
cana-1295	41	12	of	of	ADP
cana-1295	41	13	bi	bi	ADJ
cana-1295	41	14	ternary	ternary	ADJ
cana-1295	41	15	semi	semi	NOUN
cana-1295	41	16	group	group	NOUN
cana-1295	41	17	.	.	PUNCT
cana-1295	42	1	communications	communication	NOUN
cana-1295	42	2	on	on	ADP
cana-1295	42	3	applied	apply	VERB
cana-1295	42	4	nonlinear	nonlinear	ADJ
cana-1295	42	5	analysis	analysis	NOUN
cana-1295	42	6	issn	issn	NOUN
cana-1295	42	7	:	:	PUNCT
cana-1295	42	8	1074	1074	NUM
cana-1295	42	9	-	-	PUNCT
cana-1295	42	10	133x	133x	NUM
cana-1295	42	11	vol	vol	NOUN
cana-1295	42	12	31	31	NUM
cana-1295	42	13	no	no	NOUN
cana-1295	42	14	.	.	PUNCT
cana-1295	43	1	7s	7	NOUN
cana-1295	43	2	(	(	PUNCT
cana-1295	43	3	2024	2024	NUM
cana-1295	43	4	)	)	PUNCT
cana-1295	43	5	205	205	NUM
cana-1295	43	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1295	43	7	example	example	NOUN
cana-1295	43	8	2.5	2.5	NUM
cana-1295	43	9	:	:	PUNCT
cana-1295	43	10	let	let	VERB
cana-1295	43	11	𝑇	𝑇	PROPN
cana-1295	43	12	=	=	PUNCT
cana-1295	43	13	𝑇1	𝑇1	NOUN
cana-1295	43	14	∪	∪	VERB
cana-1295	43	15	𝑇2	𝑇2	NOUN
cana-1295	43	16	where	where	SCONJ
cana-1295	43	17	if	if	SCONJ
cana-1295	43	18	𝑇1	𝑇1	NOUN
cana-1295	43	19	=	=	SYM
cana-1295	43	20	(	(	PUNCT
cana-1295	43	21	𝐶,∙	𝐶,∙	ADV
cana-1295	43	22	)	)	PUNCT
cana-1295	43	23	and	and	CCONJ
cana-1295	43	24	𝑇2	𝑇2	NOUN
cana-1295	43	25	=	=	SYM
cana-1295	43	26	(	(	PUNCT
cana-1295	43	27	3𝒁	3𝒁	NOUN
cana-1295	43	28	,	,	PUNCT
cana-1295	43	29	[	[	X
cana-1295	43	30	⬚	⬚	NOUN
cana-1295	43	31	]	]	PUNCT
cana-1295	43	32	)	)	PUNCT
cana-1295	43	33	are	be	AUX
cana-1295	43	34	twoternary	twoternary	NOUN
cana-1295	43	35	semi	semi	ADJ
cana-1295	43	36	group	group	NOUN
cana-1295	43	37	so	so	SCONJ
cana-1295	43	38	that	that	SCONJ
cana-1295	43	39	𝑇	𝑇	PROPN
cana-1295	43	40	is	be	AUX
cana-1295	43	41	bi	bi	ADJ
cana-1295	43	42	ternary	ternary	NOUN
cana-1295	43	43	semi	semi	NOUN
cana-1295	43	44	group	group	NOUN
cana-1295	43	45	.	.	PUNCT
cana-1295	44	1	definition	definition	NOUN
cana-1295	44	2	2.6	2.6	NUM
cana-1295	44	3	:	:	PUNCT
cana-1295	44	4	a	a	DET
cana-1295	44	5	nonempty	nonempty	ADJ
cana-1295	44	6	sub	sub	NOUN
cana-1295	44	7	set	set	VERB
cana-1295	44	8	𝑆	𝑆	PROPN
cana-1295	44	9	=	=	PUNCT
cana-1295	44	10	𝑆1	𝑆1	NOUN
cana-1295	44	11	∪	∪	ADJ
cana-1295	44	12	𝑆2	𝑆2	NOUN
cana-1295	44	13	of	of	ADP
cana-1295	44	14	a	a	DET
cana-1295	44	15	bi	bi	ADJ
cana-1295	44	16	ternary	ternary	NOUN
cana-1295	44	17	semi	semi	PROPN
cana-1295	44	18	group	group	NOUN
cana-1295	44	19	𝑇	𝑇	PROPN
cana-1295	44	20	be	be	AUX
cana-1295	44	21	understood	understand	VERB
cana-1295	44	22	to	to	PART
cana-1295	44	23	be	be	AUX
cana-1295	44	24	a	a	DET
cana-1295	44	25	bi	bi	ADJ
cana-1295	44	26	ternary	ternary	ADJ
cana-1295	44	27	sub	sub	NOUN
cana-1295	44	28	semi	semi	NOUN
cana-1295	44	29	group	group	NOUN
cana-1295	44	30	,	,	PUNCT
cana-1295	44	31	when	when	SCONJ
cana-1295	44	32	if	if	SCONJ
cana-1295	44	33	both	both	PRON
cana-1295	44	34	𝑆1&𝑆2	𝑆1&𝑆2	ADV
cana-1295	44	35	satisfy	satisfy	VERB
cana-1295	44	36	the	the	DET
cana-1295	44	37	condition	condition	NOUN
cana-1295	44	38	:	:	PUNCT
cana-1295	44	39	𝑎1𝑏1𝑐1∀𝑎1	𝑎1𝑏1𝑐1∀𝑎1	NOUN
cana-1295	44	40	,	,	PUNCT
cana-1295	44	41	𝑏1	𝑏1	NOUN
cana-1295	44	42	,	,	PUNCT
cana-1295	44	43	𝑐1	𝑐1	NOUN
cana-1295	44	44	∈	∈	NOUN
cana-1295	44	45	𝑆1	𝑆1	NOUN
cana-1295	44	46	𝑎2𝑏2𝑐2∀𝑎2	𝑎2𝑏2𝑐2∀𝑎2	PROPN
cana-1295	44	47	,	,	PUNCT
cana-1295	44	48	𝑏2	𝑏2	PROPN
cana-1295	44	49	,	,	PUNCT
cana-1295	44	50	𝑐2	𝑐2	NOUN
cana-1295	44	51	∈	∈	PROPN
cana-1295	44	52	𝑆2	𝑆2	NOUN
cana-1295	44	53	note	note	VERB
cana-1295	44	54	2.7	2.7	NUM
cana-1295	44	55	:	:	PUNCT
cana-1295	44	56	a	a	DET
cana-1295	44	57	non	non	ADJ
cana-1295	44	58	-	-	ADJ
cana-1295	44	59	blank	blank	ADJ
cana-1295	44	60	sub	sub	NOUN
cana-1295	44	61	group	group	NOUN
cana-1295	44	62	𝑆	𝑆	PROPN
cana-1295	44	63	=	=	PUNCT
cana-1295	44	64	𝑆1	𝑆1	NOUN
cana-1295	44	65	∪	∪	ADJ
cana-1295	44	66	𝑆2	𝑆2	NOUN
cana-1295	44	67	of	of	ADP
cana-1295	44	68	a	a	DET
cana-1295	44	69	bi	bi	ADJ
cana-1295	44	70	-	-	PUNCT
cana-1295	44	71	digit	digit	NOUN
cana-1295	44	72	group	group	NOUN
cana-1295	44	73	𝑇	𝑇	PROPN
cana-1295	44	74	be	be	AUX
cana-1295	44	75	tell	tell	ADJ
cana-1295	44	76	to	to	PART
cana-1295	44	77	be	be	AUX
cana-1295	44	78	a	a	DET
cana-1295	44	79	bi	bi	ADJ
cana-1295	44	80	ternary	ternary	ADJ
cana-1295	44	81	sub	sub	NOUN
cana-1295	44	82	semi	semi	NOUN
cana-1295	44	83	group	group	NOUN
cana-1295	44	84	if	if	SCONJ
cana-1295	44	85	and	and	CCONJ
cana-1295	45	1	only	only	ADV
cana-1295	45	2	if𝑆1𝑆1𝑆1	if𝑆1𝑆1𝑆1	ADJ
cana-1295	45	3	⊆	⊆	NUM
cana-1295	45	4	𝑆1	𝑆1	NOUN
cana-1295	45	5	𝑆2𝑆2𝑆2	𝑆2𝑆2𝑆2	PROPN
cana-1295	45	6	⊆	⊆	NUM
cana-1295	45	7	𝑆2	𝑆2	ADJ
cana-1295	45	8	example	example	NOUN
cana-1295	45	9	2.8	2.8	NUM
cana-1295	45	10	:	:	PUNCT
cana-1295	45	11	let	let	VERB
cana-1295	45	12	𝑇	𝑇	PROPN
cana-1295	45	13	=	=	PUNCT
cana-1295	45	14	𝑇1	𝑇1	NOUN
cana-1295	45	15	∪	∪	VERB
cana-1295	45	16	𝑇2	𝑇2	NOUN
cana-1295	45	17	where	where	SCONJ
cana-1295	45	18	𝑇1	𝑇1	NOUN
cana-1295	45	19	=	=	SYM
cana-1295	45	20	(	(	PUNCT
cana-1295	45	21	𝐶,∙	𝐶,∙	ADV
cana-1295	45	22	)	)	PUNCT
cana-1295	45	23	and	and	CCONJ
cana-1295	45	24	𝑇2	𝑇2	NOUN
cana-1295	45	25	=	=	SYM
cana-1295	45	26	(	(	PUNCT
cana-1295	45	27	𝒁	𝒁	PROPN
cana-1295	45	28	,	,	PUNCT
cana-1295	45	29	[	[	X
cana-1295	45	30	⬚	⬚	NOUN
cana-1295	45	31	]	]	PUNCT
cana-1295	45	32	)	)	PUNCT
cana-1295	45	33	are	be	AUX
cana-1295	45	34	two	two	NUM
cana-1295	45	35	ternary	ternary	ADJ
cana-1295	45	36	semi	semi	ADJ
cana-1295	45	37	groups	group	NOUN
cana-1295	45	38	.	.	PUNCT
cana-1295	46	1	let	let	VERB
cana-1295	46	2	𝑆	𝑆	PROPN
cana-1295	46	3	=	=	PUNCT
cana-1295	46	4	𝑆1	𝑆1	NOUN
cana-1295	46	5	∪	∪	ADJ
cana-1295	46	6	𝑆2	𝑆2	NOUN
cana-1295	46	7	where	where	SCONJ
cana-1295	46	8	𝑆1	𝑆1	NOUN
cana-1295	46	9	=	=	SYM
cana-1295	46	10	(	(	PUNCT
cana-1295	46	11	𝑪0	𝑪0	NOUN
cana-1295	46	12	−	−	PROPN
cana-1295	46	13	,	,	PUNCT
cana-1295	46	14	[	[	X
cana-1295	46	15	⬚	⬚	NOUN
cana-1295	46	16	]	]	PUNCT
cana-1295	46	17	)	)	PUNCT
cana-1295	46	18	and	and	CCONJ
cana-1295	46	19	𝑆2	𝑆2	PROPN
cana-1295	46	20	=	=	PUNCT
cana-1295	46	21	(	(	PUNCT
cana-1295	46	22	3𝒁	3𝒁	NOUN
cana-1295	46	23	,	,	PUNCT
cana-1295	46	24	[	[	X
cana-1295	46	25	⬚	⬚	NOUN
cana-1295	46	26	]	]	PUNCT
cana-1295	46	27	)	)	PUNCT
cana-1295	46	28	are	be	AUX
cana-1295	46	29	two	two	NUM
cana-1295	46	30	ternary	ternary	ADJ
cana-1295	46	31	sub	sub	NOUN
cana-1295	46	32	semi	semi	NOUN
cana-1295	46	33	-	-	NOUN
cana-1295	46	34	groups	group	NOUN
cana-1295	46	35	of	of	ADP
cana-1295	46	36	𝑇.so,𝑆	𝑇.so,𝑆	PROPN
cana-1295	46	37	⊂	⊂	PROPN
cana-1295	46	38	𝑇	𝑇	PROPN
cana-1295	46	39	is	be	AUX
cana-1295	46	40	ternary	ternary	ADJ
cana-1295	46	41	sub	sub	NOUN
cana-1295	46	42	semi	semi	NOUN
cana-1295	46	43	group	group	NOUN
cana-1295	46	44	of	of	ADP
cana-1295	46	45	𝑇.	𝑇.	PROPN
cana-1295	46	46	theorem	theorem	VERB
cana-1295	46	47	2.9	2.9	NUM
cana-1295	46	48	:	:	PUNCT
cana-1295	46	49	the	the	DET
cana-1295	46	50	non	non	ADJ
cana-1295	46	51	-	-	ADJ
cana-1295	46	52	empty	empty	ADJ
cana-1295	46	53	intersection	intersection	NOUN
cana-1295	46	54	of	of	ADP
cana-1295	46	55	two	two	NUM
cana-1295	46	56	bi	bi	ADJ
cana-1295	46	57	-	-	ADJ
cana-1295	46	58	ternary	ternary	ADJ
cana-1295	46	59	sub	sub	NOUN
cana-1295	46	60	semi	semi	ADJ
cana-1295	46	61	groups	group	NOUN
cana-1295	46	62	of	of	ADP
cana-1295	46	63	a	a	DET
cana-1295	46	64	biternary	biternary	ADJ
cana-1295	46	65	semi	semi	ADJ
cana-1295	46	66	group	group	PROPN
cana-1295	46	67	𝑇	𝑇	PROPN
cana-1295	46	68	is	be	AUX
cana-1295	46	69	a	a	DET
cana-1295	46	70	bi	bi	ADJ
cana-1295	46	71	-	-	ADJ
cana-1295	46	72	ternary	ternary	ADJ
cana-1295	46	73	sub	sub	NOUN
cana-1295	46	74	semi	semi	NOUN
cana-1295	46	75	-	-	NOUN
cana-1295	46	76	group	group	NOUN
cana-1295	46	77	of	of	ADP
cana-1295	46	78	𝑇.	𝑇.	PROPN
cana-1295	46	79	proof	proof	NOUN
cana-1295	46	80	.	.	PUNCT
cana-1295	47	1	:	:	PUNCT
cana-1295	47	2	let	let	VERB
cana-1295	47	3	𝑆1	𝑆1	NOUN
cana-1295	47	4	,	,	PUNCT
cana-1295	47	5	𝑆2	𝑆2	PROPN
cana-1295	47	6	be	be	AUX
cana-1295	47	7	two	two	NUM
cana-1295	47	8	bi	bi	ADJ
cana-1295	47	9	-	-	ADJ
cana-1295	47	10	ternary	ternary	ADJ
cana-1295	47	11	sub	sub	NOUN
cana-1295	47	12	semi	semi	ADJ
cana-1295	47	13	groups	group	NOUN
cana-1295	47	14	of	of	ADP
cana-1295	47	15	𝑇.	𝑇.	PROPN
cana-1295	47	16	let	let	VERB
cana-1295	47	17	𝑎	𝑎	NOUN
cana-1295	47	18	,	,	PUNCT
cana-1295	47	19	𝑏	𝑏	NOUN
cana-1295	47	20	,	,	PUNCT
cana-1295	47	21	𝑐	𝑐	PROPN
cana-1295	47	22	∈	∈	PROPN
cana-1295	47	23	𝑆1	𝑆1	NOUN
cana-1295	47	24	∩	∩	PROPN
cana-1295	47	25	𝑆2	𝑆2	PROPN
cana-1295	47	26	𝑎	𝑎	PROPN
cana-1295	47	27	,	,	PUNCT
cana-1295	47	28	𝑏	𝑏	NOUN
cana-1295	47	29	,	,	PUNCT
cana-1295	47	30	𝑐	𝑐	PROPN
cana-1295	47	31	∈	∈	PROPN
cana-1295	47	32	𝑆1	𝑆1	NOUN
cana-1295	47	33	∩	∩	ADJ
cana-1295	47	34	𝑆2	𝑆2	VERB
cana-1295	47	35	then	then	ADV
cana-1295	47	36	𝑎	𝑎	PROPN
cana-1295	47	37	,	,	PUNCT
cana-1295	47	38	𝑏	𝑏	NOUN
cana-1295	47	39	,	,	PUNCT
cana-1295	47	40	𝑐	𝑐	PROPN
cana-1295	47	41	∈	∈	PROPN
cana-1295	47	42	𝑆1	𝑆1	NOUN
cana-1295	47	43	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
cana-1295	47	44	𝑎	𝑎	PROPN
cana-1295	47	45	,	,	PUNCT
cana-1295	47	46	𝑏	𝑏	NOUN
cana-1295	47	47	,	,	PUNCT
cana-1295	47	48	𝑐	𝑐	PROPN
cana-1295	47	49	∈	∈	PROPN
cana-1295	47	50	𝑆2	𝑆2	PROPN
cana-1295	47	51	𝑎	𝑎	PROPN
cana-1295	47	52	,	,	PUNCT
cana-1295	47	53	𝑏	𝑏	NOUN
cana-1295	47	54	,	,	PUNCT
cana-1295	47	55	𝑐	𝑐	PROPN
cana-1295	47	56	∈	∈	PROPN
cana-1295	47	57	𝑆1	𝑆1	NOUN
cana-1295	47	58	,	,	PUNCT
cana-1295	47	59	𝑆1	𝑆1	NOUN
cana-1295	47	60	is	be	AUX
cana-1295	47	61	a	a	DET
cana-1295	47	62	bi	bi	ADJ
cana-1295	47	63	-	-	ADJ
cana-1295	47	64	ternary	ternary	ADJ
cana-1295	47	65	sub	sub	NOUN
cana-1295	47	66	semi	semi	NOUN
cana-1295	47	67	group	group	NOUN
cana-1295	47	68	of	of	ADP
cana-1295	47	69	𝑇	𝑇	PROPN
cana-1295	47	70	,	,	PUNCT
cana-1295	47	71	𝑎𝑏𝑐	𝑎𝑏𝑐	ADJ
cana-1295	47	72	∈	∈	PROPN
cana-1295	47	73	𝑆1	𝑆1	NOUN
cana-1295	47	74	𝑎	𝑎	PROPN
cana-1295	47	75	,	,	PUNCT
cana-1295	47	76	𝑏	𝑏	NOUN
cana-1295	47	77	,	,	PUNCT
cana-1295	47	78	𝑐	𝑐	PROPN
cana-1295	47	79	∈	∈	PROPN
cana-1295	47	80	𝑆2	𝑆2	NOUN
cana-1295	47	81	,	,	PUNCT
cana-1295	47	82	𝑆2	𝑆2	PROPN
cana-1295	47	83	is	be	AUX
cana-1295	47	84	a	a	DET
cana-1295	47	85	bi	bi	ADJ
cana-1295	47	86	-	-	ADJ
cana-1295	47	87	ternary	ternary	ADJ
cana-1295	47	88	sub	sub	NOUN
cana-1295	47	89	semi	semi	NOUN
cana-1295	47	90	group	group	NOUN
cana-1295	47	91	of	of	ADP
cana-1295	47	92	𝑇	𝑇	PROPN
cana-1295	47	93	,	,	PUNCT
cana-1295	47	94	𝑎𝑏𝑐	𝑎𝑏𝑐	PROPN
cana-1295	47	95	∈	∈	PROPN
cana-1295	47	96	𝑆2	𝑆2	NOUN
cana-1295	47	97	so	so	ADV
cana-1295	47	98	,	,	PUNCT
cana-1295	47	99	that𝑎𝑏𝑐	that𝑎𝑏𝑐	PROPN
cana-1295	47	100	∈	∈	PROPN
cana-1295	47	101	𝑆1	𝑆1	PROPN
cana-1295	47	102	,	,	PUNCT
cana-1295	47	103	𝑎𝑏𝑐	𝑎𝑏𝑐	PROPN
cana-1295	47	104	∈	∈	PROPN
cana-1295	47	105	𝑆2	𝑆2	NOUN
cana-1295	47	106	then	then	ADV
cana-1295	47	107	𝑎𝑏𝑐	𝑎𝑏𝑐	PROPN
cana-1295	47	108	∈	∈	PROPN
cana-1295	47	109	𝑆1	𝑆1	NOUN
cana-1295	47	110	∩	∩	ADJ
cana-1295	47	111	𝑆2	𝑆2	NOUN
cana-1295	47	112	therefore	therefore	ADV
cana-1295	47	113	𝑆1	𝑆1	NOUN
cana-1295	47	114	∩	∩	ADJ
cana-1295	47	115	𝑆2	𝑆2	PROPN
cana-1295	47	116	is	be	AUX
cana-1295	47	117	a	a	DET
cana-1295	47	118	bi	bi	ADJ
cana-1295	47	119	-	-	ADJ
cana-1295	47	120	ternary	ternary	ADJ
cana-1295	47	121	sub	sub	NOUN
cana-1295	47	122	semi	semi	NOUN
cana-1295	47	123	group	group	NOUN
cana-1295	47	124	of	of	ADP
cana-1295	47	125	𝑇.	𝑇.	PROPN
cana-1295	47	126	theorem	theorem	VERB
cana-1295	47	127	2.10	2.10	NUM
cana-1295	47	128	:	:	PUNCT
cana-1295	47	129	the	the	DET
cana-1295	47	130	meeting	meeting	NOUN
cana-1295	47	131	point	point	NOUN
cana-1295	47	132	of	of	ADP
cana-1295	47	133	some	some	DET
cana-1295	47	134	relations	relation	NOUN
cana-1295	47	135	of	of	ADP
cana-1295	47	136	bi	bi	ADJ
cana-1295	47	137	-	-	ADJ
cana-1295	47	138	ternary	ternary	ADJ
cana-1295	47	139	sub	sub	NOUN
cana-1295	47	140	semi	semi	ADJ
cana-1295	47	141	groups	group	NOUN
cana-1295	47	142	of	of	ADP
cana-1295	47	143	𝑇	𝑇	PROPN
cana-1295	47	144	is	be	AUX
cana-1295	47	145	the	the	DET
cana-1295	47	146	bi	bi	ADJ
cana-1295	47	147	ternary	ternary	ADJ
cana-1295	47	148	sub	sub	NOUN
cana-1295	47	149	semi	semi	ADJ
cana-1295	47	150	group	group	NOUN
cana-1295	47	151	of	of	ADP
cana-1295	47	152	𝑇.	𝑇.	PROPN
cana-1295	47	153	proof	proof	NOUN
cana-1295	47	154	:	:	PUNCT
cana-1295	47	155	let	let	VERB
cana-1295	47	156	{	{	PUNCT
cana-1295	47	157	𝑆𝛼}𝛼∈∆	𝑆𝛼}𝛼∈∆	VERB
cana-1295	47	158	be	be	AUX
cana-1295	47	159	a	a	DET
cana-1295	47	160	family	family	NOUN
cana-1295	47	161	of	of	ADP
cana-1295	47	162	bi	bi	ADJ
cana-1295	47	163	ternary	ternary	ADJ
cana-1295	47	164	sub	sub	NOUN
cana-1295	47	165	semi	semi	ADJ
cana-1295	47	166	groups	group	NOUN
cana-1295	47	167	of	of	ADP
cana-1295	47	168	t	t	PROPN
cana-1295	47	169	,	,	PUNCT
cana-1295	47	170	and	and	CCONJ
cana-1295	47	171	𝑆	𝑆	PROPN
cana-1295	47	172	=	=	SYM
cana-1295	47	173	⋂	⋂	PROPN
cana-1295	47	174	𝑆𝛼𝛼∈∆	𝑆𝛼𝛼∈∆	VERB
cana-1295	47	175	.	.	PUNCT
cana-1295	48	1	let	let	VERB
cana-1295	48	2	𝑎	𝑎	NOUN
cana-1295	48	3	,	,	PUNCT
cana-1295	48	4	𝑏	𝑏	NOUN
cana-1295	48	5	,	,	PUNCT
cana-1295	48	6	𝑐	𝑐	PROPN
cana-1295	48	7	∈	∈	PROPN
cana-1295	48	8	𝑆	𝑆	PROPN
cana-1295	48	9	⇒	⇒	VERB
cana-1295	48	10	𝑎	𝑎	PROPN
cana-1295	48	11	,	,	PUNCT
cana-1295	48	12	𝑏	𝑏	NOUN
cana-1295	48	13	,	,	PUNCT
cana-1295	48	14	𝑐	𝑐	PROPN
cana-1295	48	15	∈	∈	PROPN
cana-1295	48	16	⋂	⋂	PROPN
cana-1295	48	17	𝑆𝛼𝛼∈∆	𝑆𝛼𝛼∈∆	ADJ
cana-1295	48	18	⇒	⇒	NOUN
cana-1295	48	19	𝑎	𝑎	PROPN
cana-1295	48	20	,	,	PUNCT
cana-1295	48	21	𝑏	𝑏	NOUN
cana-1295	48	22	,	,	PUNCT
cana-1295	48	23	𝑐	𝑐	PROPN
cana-1295	48	24	∈	∈	NOUN
cana-1295	48	25	𝑆𝛼	𝑆𝛼	PROPN
cana-1295	48	26	∀	∀	NOUN
cana-1295	48	27	𝛼	𝛼	NOUN
cana-1295	48	28	∈	∈	NOUN
cana-1295	48	29	∆	∆	PROPN
cana-1295	48	30	𝑎	𝑎	X
cana-1295	48	31	,	,	PUNCT
cana-1295	48	32	𝑏	𝑏	NOUN
cana-1295	48	33	,	,	PUNCT
cana-1295	48	34	𝑐	𝑐	PROPN
cana-1295	48	35	∈	∈	PROPN
cana-1295	48	36	𝑆𝛼	𝑆𝛼	PROPN
cana-1295	48	37	,	,	PUNCT
cana-1295	48	38	𝑆𝛼	𝑆𝛼	PROPN
cana-1295	48	39	is	be	AUX
cana-1295	48	40	a	a	DET
cana-1295	48	41	bi	bi	ADJ
cana-1295	48	42	ternary	ternary	ADJ
cana-1295	48	43	sub	sub	NOUN
cana-1295	48	44	semi	semi	PROPN
cana-1295	48	45	group	group	PROPN
cana-1295	48	46	t	t	PROPN
cana-1295	48	47	then	then	ADV
cana-1295	48	48	𝑎𝑏𝑐	𝑎𝑏𝑐	PROPN
cana-1295	48	49	∈	∈	PROPN
cana-1295	48	50	𝑆𝛼	𝑆𝛼	PROPN
cana-1295	48	51	now	now	ADV
cana-1295	48	52	,	,	PUNCT
cana-1295	48	53	𝑎𝑏𝑐	𝑎𝑏𝑐	ADJ
cana-1295	48	54	∈	∈	PROPN
cana-1295	48	55	𝑆𝛼	𝑆𝛼	PROPN
cana-1295	48	56	,	,	PUNCT
cana-1295	48	57	∀𝛼	∀𝛼	PROPN
cana-1295	48	58	∈	∈	PROPN
cana-1295	48	59	∆	∆	PROPN
cana-1295	48	60	⇒	⇒	VERB
cana-1295	48	61	𝑎𝑏𝑐	𝑎𝑏𝑐	PROPN
cana-1295	48	62	∈	∈	PROPN
cana-1295	48	63	⋂	⋂	PROPN
cana-1295	48	64	𝑆𝛼𝛼∈∆	𝑆𝛼𝛼∈∆	VERB
cana-1295	48	65	then	then	ADV
cana-1295	48	66	𝑎𝑏𝑐	𝑎𝑏𝑐	PROPN
cana-1295	48	67	∈	∈	PROPN
cana-1295	48	68	𝑆.	𝑆.	PROPN
cana-1295	48	69	therefore	therefore	ADV
cana-1295	48	70	s	s	VERB
cana-1295	48	71	is	be	AUX
cana-1295	48	72	a	a	DET
cana-1295	48	73	bi	bi	ADJ
cana-1295	48	74	ternary	ternary	ADJ
cana-1295	48	75	sub	sub	NOUN
cana-1295	48	76	semi	semi	ADJ
cana-1295	48	77	group	group	NOUN
cana-1295	48	78	of	of	ADP
cana-1295	48	79	𝑇.	𝑇.	PROPN
cana-1295	48	80	theorem	theorem	VERB
cana-1295	48	81	2.11	2.11	NUM
cana-1295	48	82	:	:	PUNCT
cana-1295	48	83	let	let	VERB
cana-1295	48	84	𝑇	𝑇	PROPN
cana-1295	48	85	be	be	AUX
cana-1295	48	86	a	a	DET
cana-1295	48	87	bi	bi	ADJ
cana-1295	48	88	ternary	ternary	NOUN
cana-1295	48	89	semi	semi	ADJ
cana-1295	48	90	grouping	grouping	NOUN
cana-1295	48	91	&	&	CCONJ
cana-1295	48	92	𝐴	𝐴	PROPN
cana-1295	48	93	be	be	VERB
cana-1295	48	94	a	a	DET
cana-1295	48	95	non	non	ADJ
cana-1295	48	96	-	-	ADJ
cana-1295	48	97	empty	empty	ADJ
cana-1295	48	98	sub	sub	NOUN
cana-1295	48	99	set	set	NOUN
cana-1295	48	100	of	of	ADP
cana-1295	48	101	𝑇.	𝑇.	PROPN
cana-1295	48	102	then	then	ADV
cana-1295	48	103	<	<	X
cana-1295	48	104	𝐴	𝐴	PROPN
cana-1295	48	105	>	>	PUNCT
cana-1295	48	106	=	=	PUNCT
cana-1295	48	107	the	the	DET
cana-1295	48	108	intersection	intersection	NOUN
cana-1295	48	109	of	of	ADP
cana-1295	48	110	all	all	DET
cana-1295	48	111	bi	bi	ADJ
cana-1295	48	112	ternary	ternary	ADJ
cana-1295	48	113	sub	sub	NOUN
cana-1295	48	114	semi	semi	ADJ
cana-1295	48	115	groups	group	NOUN
cana-1295	48	116	of	of	ADP
cana-1295	48	117	𝑇containing𝐴.	𝑇containing𝐴.	VERB
cana-1295	48	118	proof	proof	NOUN
cana-1295	48	119	:	:	PUNCT
cana-1295	48	120	letting	let	VERB
cana-1295	48	121			AUX
cana-1295	48	122	be	be	AUX
cana-1295	48	123	a	a	DET
cana-1295	48	124	group	group	NOUN
cana-1295	48	125	comprising	comprise	VERB
cana-1295	48	126	all	all	DET
cana-1295	48	127	bi	bi	ADJ
cana-1295	48	128	-	-	ADJ
cana-1295	48	129	ternary	ternary	ADJ
cana-1295	48	130	primary	primary	ADJ
cana-1295	48	131	semi	semi	NOUN
cana-1295	48	132	-	-	NOUN
cana-1295	48	133	groups	group	NOUN
cana-1295	48	134	of	of	ADP
cana-1295	48	135	t	t	NOUN
cana-1295	48	136	containing	contain	VERB
cana-1295	48	137	a.	a.	NOUN
cana-1295	48	138	bi	bi	PROPN
cana-1295	48	139	ternary	ternary	NOUN
cana-1295	48	140	be	be	AUX
cana-1295	48	141	t	t	PROPN
cana-1295	48	142	,	,	PUNCT
cana-1295	48	143	sub	sub	X
cana-1295	48	144	semi	semi	ADJ
cana-1295	48	145	group	group	NOUN
cana-1295	48	146	of	of	ADP
cana-1295	48	147	`	`	PUNCT
cana-1295	48	148	𝑇containing𝐴.	𝑇containing𝐴.	VERB
cana-1295	48	149	𝑇	𝑇	PROPN
cana-1295	48	150	∈	∈	PROPN
cana-1295	48	151	∆	∆	X
cana-1295	48	152	,	,	PUNCT
cana-1295	48	153	accordingly	accordingly	ADV
cana-1295	48	154	∆≠	∆≠	PROPN
cana-1295	48	155	∅	∅	NOUN
cana-1295	48	156	agree	agree	VERB
cana-1295	48	157	𝑆∗	𝑆∗	X
cana-1295	48	158	=	=	PUNCT
cana-1295	48	159	⋂	⋂	PROPN
cana-1295	48	160	𝑆𝑠∈∆	𝑆𝑠∈∆	PROPN
cana-1295	48	161	giving	give	VERB
cana-1295	48	162	that	that	SCONJ
cana-1295	48	163	𝐴	𝐴	PROPN
cana-1295	48	164	⊆	⊆	NUM
cana-1295	48	165	𝑆	𝑆	PROPN
cana-1295	48	166	∀	∀	NOUN
cana-1295	48	167	𝑆	𝑆	PROPN
cana-1295	48	168	∈	∈	PROPN
cana-1295	48	169	∆	∆	PROPN
cana-1295	48	170	and	and	CCONJ
cana-1295	48	171	𝐴	𝐴	PROPN
cana-1295	48	172	⊆	⊆	NUM
cana-1295	48	173	𝑆∗	𝑆∗	PROPN
cana-1295	48	174	communications	communication	NOUN
cana-1295	48	175	on	on	ADP
cana-1295	48	176	applied	apply	VERB
cana-1295	48	177	nonlinear	nonlinear	ADJ
cana-1295	48	178	analysis	analysis	NOUN
cana-1295	48	179	issn	issn	NOUN
cana-1295	48	180	:	:	PUNCT
cana-1295	48	181	1074	1074	NUM
cana-1295	48	182	-	-	PUNCT
cana-1295	48	183	133x	133x	NUM
cana-1295	48	184	vol	vol	NOUN
cana-1295	48	185	31	31	NUM
cana-1295	48	186	no	no	NOUN
cana-1295	48	187	.	.	PUNCT
cana-1295	49	1	7s	7	NOUN
cana-1295	49	2	(	(	PUNCT
cana-1295	49	3	2024	2024	NUM
cana-1295	49	4	)	)	PUNCT
cana-1295	49	5	206	206	NUM
cana-1295	49	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1295	49	7	since	since	SCONJ
cana-1295	49	8	𝑆∗	𝑆∗	PROPN
cana-1295	49	9	is	be	AUX
cana-1295	49	10	the	the	DET
cana-1295	49	11	intersection	intersection	NOUN
cana-1295	49	12	of	of	ADP
cana-1295	49	13	bi	bi	ADJ
cana-1295	49	14	ternary	ternary	ADJ
cana-1295	49	15	sub	sub	NOUN
cana-1295	49	16	semi	semi	ADJ
cana-1295	49	17	groups	group	NOUN
cana-1295	49	18	of	of	ADP
cana-1295	49	19	𝑇.	𝑇.	PROPN
cana-1295	49	20	so,𝑆∗	so,𝑆∗	PROPN
cana-1295	49	21	is	be	AUX
cana-1295	49	22	a	a	DET
cana-1295	49	23	bi	bi	ADJ
cana-1295	49	24	ternary	ternary	ADJ
cana-1295	49	25	sub	sub	NOUN
cana-1295	49	26	semi	semi	NOUN
cana-1295	49	27	group	group	NOUN
cana-1295	49	28	of𝑇.	of𝑇.	PROPN
cana-1295	49	29	since	since	SCONJ
cana-1295	49	30	𝑆∗	𝑆∗	PROPN
cana-1295	49	31	⊆	⊆	NUM
cana-1295	49	32	𝑆	𝑆	PROPN
cana-1295	49	33	for	for	ADP
cana-1295	49	34	all	all	DET
cana-1295	49	35	𝑆	𝑆	PROPN
cana-1295	49	36	∈	∈	PROPN
cana-1295	49	37	∆	∆	PROPN
cana-1295	49	38	,	,	PUNCT
cana-1295	49	39	𝑆∗	𝑆∗	PROPN
cana-1295	49	40	is	be	AUX
cana-1295	49	41	the	the	DET
cana-1295	49	42	smallest	small	ADJ
cana-1295	49	43	bi	bi	ADJ
cana-1295	49	44	ternary	ternary	ADJ
cana-1295	49	45	sub	sub	NOUN
cana-1295	49	46	semi	semi	ADJ
cana-1295	49	47	group	group	NOUN
cana-1295	49	48	of	of	ADP
cana-1295	49	49	𝑇	𝑇	PROPN
cana-1295	49	50	containing	contain	VERB
cana-1295	49	51	𝐴.	𝐴.	PROPN
cana-1295	49	52	hence	hence	ADV
cana-1295	49	53	𝑆∗	𝑆∗	X
cana-1295	50	1	=	=	NOUN
cana-1295	50	2	<	<	X
cana-1295	50	3	𝐴	𝐴	PROPN
cana-1295	50	4	>	>	X
cana-1295	50	5	definition	definition	NOUN
cana-1295	50	6	2.12	2.12	NUM
cana-1295	50	7	:	:	PUNCT
cana-1295	50	8	a	a	DET
cana-1295	50	9	bi	bi	ADJ
cana-1295	50	10	-	-	ADJ
cana-1295	50	11	ternary	ternary	ADJ
cana-1295	50	12	semi	semi	ADJ
cana-1295	50	13	group	group	NOUN
cana-1295	50	14	𝑇	𝑇	PROPN
cana-1295	50	15	be	be	AUX
cana-1295	50	16	thought	think	VERB
cana-1295	50	17	concurrent	concurrent	ADJ
cana-1295	50	18	if	if	SCONJ
cana-1295	50	19	for	for	ADP
cana-1295	50	20	every	every	DET
cana-1295	50	21	𝑎	𝑎	NOUN
cana-1295	50	22	,	,	PUNCT
cana-1295	50	23	𝑏	𝑏	NOUN
cana-1295	50	24	,	,	PUNCT
cana-1295	50	25	𝑐	𝑐	PROPN
cana-1295	50	26	∈	∈	PROPN
cana-1295	50	27	𝑇	𝑇	PROPN
cana-1295	50	28	,	,	PUNCT
cana-1295	50	29	then	then	ADV
cana-1295	50	30	wehave	wehave	VERB
cana-1295	50	31	𝑎𝑏𝑐	𝑎𝑏𝑐	PROPN
cana-1295	50	32	=	=	SYM
cana-1295	50	33	𝑏𝑐𝑎	𝑏𝑐𝑎	PROPN
cana-1295	50	34	=	=	NOUN
cana-1295	50	35	𝑐𝑎𝑏	𝑐𝑎𝑏	NOUN
cana-1295	50	36	=	=	NOUN
cana-1295	50	37	𝑎𝑐𝑏	𝑎𝑐𝑏	PROPN
cana-1295	50	38	=	=	NOUN
cana-1295	50	39	𝑏𝑎𝑐	𝑏𝑎𝑐	NOUN
cana-1295	50	40	=	=	PUNCT
cana-1295	50	41	𝑐𝑏𝑎	𝑐𝑏𝑎	NOUN
cana-1295	50	42	3.ideals	3.ideals	NUM
cana-1295	50	43	in	in	ADP
cana-1295	50	44	biternary	biternary	ADJ
cana-1295	50	45	semi	semi	ADJ
cana-1295	50	46	groups	group	NOUN
cana-1295	50	47	:	:	PUNCT
cana-1295	50	48	left	leave	VERB
cana-1295	50	49	ideals	ideal	NOUN
cana-1295	50	50	in	in	ADP
cana-1295	50	51	bi	bi	ADJ
cana-1295	50	52	-	-	ADJ
cana-1295	50	53	ternary	ternary	ADJ
cana-1295	50	54	semi	semi	ADJ
cana-1295	50	55	groups	group	NOUN
cana-1295	50	56	definition	definition	NOUN
cana-1295	50	57	3.1	3.1	NUM
cana-1295	50	58	:	:	PUNCT
cana-1295	50	59	a	a	DET
cana-1295	50	60	nonempty	nonempty	ADJ
cana-1295	50	61	sub	sub	NOUN
cana-1295	50	62	set	set	NOUN
cana-1295	50	63	𝐴	𝐴	PROPN
cana-1295	50	64	=	=	PUNCT
cana-1295	50	65	𝐴1	𝐴1	PROPN
cana-1295	50	66	∪	∪	VERB
cana-1295	50	67	𝐴2	𝐴2	PROPN
cana-1295	50	68	of	of	ADP
cana-1295	50	69	a	a	DET
cana-1295	50	70	bi	bi	ADJ
cana-1295	50	71	-	-	ADJ
cana-1295	50	72	ternary	ternary	ADJ
cana-1295	50	73	semi	semi	ADJ
cana-1295	50	74	group	group	NOUN
cana-1295	50	75	𝑇	𝑇	PROPN
cana-1295	50	76	be	be	AUX
cana-1295	50	77	thought	think	VERB
cana-1295	50	78	concurrent	concurrent	ADJ
cana-1295	50	79	left	leave	VERB
cana-1295	50	80	ideal	ideal	NOUN
cana-1295	50	81	of	of	ADP
cana-1295	50	82	𝑇	𝑇	PROPN
cana-1295	50	83	if	if	SCONJ
cana-1295	50	84	both	both	DET
cana-1295	50	85	𝐴1	𝐴1	PROPN
cana-1295	50	86	and	and	CCONJ
cana-1295	50	87	𝐴2	𝐴2	PROPN
cana-1295	50	88	are	be	AUX
cana-1295	50	89	left	leave	VERB
cana-1295	50	90	ideals	ideal	NOUN
cana-1295	50	91	of	of	ADP
cana-1295	50	92	𝑇1	𝑇1	NOUN
cana-1295	50	93	and	and	CCONJ
cana-1295	50	94	𝑇2	𝑇2	NOUN
cana-1295	50	95	such	such	ADJ
cana-1295	50	96	that	that	SCONJ
cana-1295	50	97	if	if	SCONJ
cana-1295	50	98	𝑏	𝑏	PROPN
cana-1295	50	99	,	,	PUNCT
cana-1295	50	100	𝑐	𝑐	PROPN
cana-1295	50	101	∈	∈	PROPN
cana-1295	50	102	𝑇1	𝑇1	NOUN
cana-1295	50	103	,	,	PUNCT
cana-1295	50	104	𝑎	𝑎	PROPN
cana-1295	50	105	∈	∈	ADJ
cana-1295	50	106	𝐴1	𝐴1	PROPN
cana-1295	50	107	implies	imply	VERB
cana-1295	50	108	𝑏𝑐𝑎	𝑏𝑐𝑎	PROPN
cana-1295	50	109	∈	∈	PROPN
cana-1295	50	110	𝐴1	𝐴1	PROPN
cana-1295	50	111	similarly	similarly	ADV
cana-1295	50	112	,	,	PUNCT
cana-1295	50	113	𝑏	𝑏	PROPN
cana-1295	50	114	,	,	PUNCT
cana-1295	50	115	𝑐	𝑐	PROPN
cana-1295	50	116	∈	∈	PROPN
cana-1295	50	117	𝑇2	𝑇2	NOUN
cana-1295	50	118	,	,	PUNCT
cana-1295	50	119	𝑎	𝑎	PROPN
cana-1295	50	120	∈	∈	PROPN
cana-1295	50	121	𝐴2implies	𝐴2implie	VERB
cana-1295	50	122	𝑏𝑐𝑎	𝑏𝑐𝑎	PROPN
cana-1295	50	123	∈	∈	PROPN
cana-1295	50	124	𝐴2	𝐴2	PROPN
cana-1295	50	125	.	.	PUNCT
cana-1295	51	1	note	note	VERB
cana-1295	51	2	3.2	3.2	NUM
cana-1295	51	3	:	:	PUNCT
cana-1295	51	4	a	a	DET
cana-1295	51	5	unfilled	unfilled	ADJ
cana-1295	51	6	auxiliary	auxiliary	NOUN
cana-1295	51	7	set	set	NOUN
cana-1295	51	8	called	call	VERB
cana-1295	51	9	a	a	PRON
cana-1295	51	10	of	of	ADP
cana-1295	51	11	a	a	DET
cana-1295	51	12	biternary	biternary	ADJ
cana-1295	51	13	semigroup	semigroup	PROPN
cana-1295	51	14	t	t	PROPN
cana-1295	51	15	is	be	AUX
cana-1295	51	16	considered	consider	VERB
cana-1295	51	17	that	that	SCONJ
cana-1295	51	18	it	it	PRON
cana-1295	51	19	's	be	AUX
cana-1295	51	20	left	leave	VERB
cana-1295	51	21	idealistic	idealistic	ADJ
cana-1295	51	22	if	if	SCONJ
cana-1295	51	23	just	just	ADV
cana-1295	51	24	if	if	SCONJ
cana-1295	51	25	𝑇𝑇𝐴	𝑇𝑇𝐴	PROPN
cana-1295	51	26	⊆	⊆	NUM
cana-1295	51	27	𝐴.	𝐴.	PROPN
cana-1295	51	28	example	example	NOUN
cana-1295	52	1	3.3:𝑇	3.3:𝑇	NUM
cana-1295	52	2	=	=	SYM
cana-1295	52	3	𝑇1	𝑇1	PROPN
cana-1295	52	4	∪	∪	VERB
cana-1295	52	5	𝑇2	𝑇2	PROPN
cana-1295	52	6	be	be	AUX
cana-1295	52	7	a	a	DET
cana-1295	52	8	bi	bi	ADJ
cana-1295	52	9	ternary	ternary	NOUN
cana-1295	52	10	semi	semi	NOUN
cana-1295	52	11	group	group	NOUN
cana-1295	52	12	where	where	SCONJ
cana-1295	52	13	𝑇1	𝑇1	NOUN
cana-1295	52	14	=	=	SYM
cana-1295	52	15	{	{	PUNCT
cana-1295	52	16	[	[	PUNCT
cana-1295	52	17	𝑎	𝑎	NOUN
cana-1295	52	18	0	0	PUNCT
cana-1295	52	19	𝑏	𝑏	PROPN
cana-1295	52	20	𝑐	𝑐	PROPN
cana-1295	52	21	]	]	PUNCT
cana-1295	52	22	,	,	PUNCT
cana-1295	52	23	𝑎	𝑎	X
cana-1295	52	24	,	,	PUNCT
cana-1295	52	25	𝑏	𝑏	NOUN
cana-1295	52	26	,	,	PUNCT
cana-1295	52	27	𝑐	𝑐	PROPN
cana-1295	52	28	∈	∈	PROPN
cana-1295	52	29	𝑍	𝑍	PROPN
cana-1295	52	30	}	}	PUNCT
cana-1295	52	31	,	,	PUNCT
cana-1295	52	32	𝑇2	𝑇2	NOUN
cana-1295	52	33	=	=	SYM
cana-1295	52	34	{	{	PUNCT
cana-1295	52	35	[	[	PUNCT
cana-1295	52	36	𝑥	𝑥	NOUN
cana-1295	52	37	𝑦	𝑦	NOUN
cana-1295	52	38	0	0	PUNCT
cana-1295	52	39	𝑧	𝑧	NOUN
cana-1295	52	40	]	]	PUNCT
cana-1295	52	41	,	,	PUNCT
cana-1295	52	42	𝑥	𝑥	X
cana-1295	52	43	,	,	PUNCT
cana-1295	52	44	𝑦	𝑦	X
cana-1295	52	45	,	,	PUNCT
cana-1295	52	46	𝑧	𝑧	DET
cana-1295	52	47	∈	∈	NOUN
cana-1295	52	48	𝑍}are	𝑍}are	VERB
cana-1295	52	49	ternary	ternary	ADJ
cana-1295	52	50	semi	semi	ADJ
cana-1295	52	51	groups	group	NOUN
cana-1295	52	52	of	of	ADP
cana-1295	52	53	𝑇.	𝑇.	PROPN
cana-1295	52	54	𝐼	𝐼	PROPN
cana-1295	52	55	=	=	PUNCT
cana-1295	52	56	𝐼1	𝐼1	NOUN
cana-1295	52	57	∪	∪	X
cana-1295	52	58	𝐼2	𝐼2	NOUN
cana-1295	52	59	where	where	SCONJ
cana-1295	52	60	𝐼1	𝐼1	NOUN
cana-1295	52	61	=	=	SYM
cana-1295	52	62	{	{	PUNCT
cana-1295	52	63	[	[	PUNCT
cana-1295	52	64	𝑎	𝑎	NOUN
cana-1295	52	65	0	0	NUM
cana-1295	52	66	𝑏	𝑏	NOUN
cana-1295	52	67	0	0	NUM
cana-1295	52	68	]	]	PUNCT
cana-1295	52	69	,	,	PUNCT
cana-1295	52	70	𝑎	𝑎	X
cana-1295	52	71	,	,	PUNCT
cana-1295	52	72	𝑏	𝑏	PROPN
cana-1295	52	73	∈	∈	PROPN
cana-1295	52	74	𝑍	𝑍	PROPN
cana-1295	52	75	}	}	PUNCT
cana-1295	52	76	,	,	PUNCT
cana-1295	52	77	𝐼2	𝐼2	NOUN
cana-1295	52	78	=	=	SYM
cana-1295	52	79	{	{	PUNCT
cana-1295	52	80	[	[	PUNCT
cana-1295	52	81	𝑎	𝑎	PROPN
cana-1295	52	82	0	0	NUM
cana-1295	52	83	0	0	NUM
cana-1295	52	84	0	0	NUM
cana-1295	52	85	]	]	PUNCT
cana-1295	52	86	,	,	PUNCT
cana-1295	52	87	𝑎	𝑎	PROPN
cana-1295	52	88	∈	∈	PROPN
cana-1295	52	89	𝑍	𝑍	NOUN
cana-1295	52	90	}	}	PUNCT
cana-1295	52	91	are	be	AUX
cana-1295	52	92	left	leave	VERB
cana-1295	52	93	ideals	ideal	NOUN
cana-1295	52	94	of	of	ADP
cana-1295	52	95	𝑇1	𝑇1	NOUN
cana-1295	52	96	∩	∩	NOUN
cana-1295	52	97	𝑇2respectively	𝑇2respectively	ADV
cana-1295	52	98	.	.	PUNCT
cana-1295	53	1	thus	thus	ADV
cana-1295	53	2	𝐼	𝐼	PROPN
cana-1295	53	3	is	be	AUX
cana-1295	53	4	the	the	DET
cana-1295	53	5	left	leave	VERB
cana-1295	53	6	idealistic	idealistic	ADJ
cana-1295	53	7	of	of	ADP
cana-1295	53	8	bi	bi	ADJ
cana-1295	53	9	ternary	ternary	PROPN
cana-1295	53	10	semi	semi	PROPN
cana-1295	53	11	group	group	PROPN
cana-1295	53	12	𝑇.	𝑇.	PROPN
cana-1295	53	13	theorem	theorem	VERB
cana-1295	53	14	3.4	3.4	NUM
cana-1295	53	15	:	:	PUNCT
cana-1295	53	16	the	the	DET
cana-1295	53	17	intersection	intersection	NOUN
cana-1295	53	18	of	of	ADP
cana-1295	53	19	any	any	DET
cana-1295	53	20	two	two	NUM
cana-1295	53	21	left	left	ADJ
cana-1295	53	22	ideals	ideal	NOUN
cana-1295	53	23	be	be	VERB
cana-1295	53	24	a	a	DET
cana-1295	53	25	bi	bi	ADJ
cana-1295	53	26	-	-	ADJ
cana-1295	53	27	ternary	ternary	ADJ
cana-1295	53	28	semi	semi	ADJ
cana-1295	53	29	group	group	NOUN
cana-1295	53	30	𝑇	𝑇	PROPN
cana-1295	53	31	be	be	AUX
cana-1295	53	32	left	leave	VERB
cana-1295	53	33	t	t	PROPN
cana-1295	53	34	is	be	AUX
cana-1295	53	35	idealistic	idealistic	ADJ
cana-1295	53	36	.	.	PUNCT
cana-1295	54	1	proof	proof	NOUN
cana-1295	54	2	.	.	PUNCT
cana-1295	55	1	:	:	PUNCT
cana-1295	55	2	let	let	VERB
cana-1295	55	3	𝐴	𝐴	PROPN
cana-1295	55	4	,	,	PUNCT
cana-1295	55	5	𝐵	𝐵	PROPN
cana-1295	55	6	left	leave	VERB
cana-1295	55	7	idealistic	idealistic	ADJ
cana-1295	55	8	of	of	ADP
cana-1295	55	9	two	two	NUM
cana-1295	55	10	be𝑇.	be𝑇.	NOUN
cana-1295	55	11	let	let	VERB
cana-1295	55	12	𝑎	𝑎	PRON
cana-1295	55	13	∈	∈	PROPN
cana-1295	55	14	𝐴	𝐴	NOUN
cana-1295	55	15	∩	∩	ADJ
cana-1295	55	16	𝐵	𝐵	PROPN
cana-1295	55	17	and𝑏	and𝑏	NOUN
cana-1295	55	18	,	,	PUNCT
cana-1295	55	19	𝑐	𝑐	PROPN
cana-1295	55	20	∈	∈	PROPN
cana-1295	55	21	𝑇	𝑇	PROPN
cana-1295	55	22	.	.	PUNCT
cana-1295	56	1	if	if	SCONJ
cana-1295	56	2	𝑎	𝑎	PRON
cana-1295	56	3	∈	∈	PROPN
cana-1295	56	4	𝐴	𝐴	NOUN
cana-1295	56	5	∩	∩	ADJ
cana-1295	56	6	𝐵	𝐵	NOUN
cana-1295	56	7	then	then	ADV
cana-1295	56	8	𝑎	𝑎	PROPN
cana-1295	56	9	∈	∈	PROPN
cana-1295	56	10	𝐴	𝐴	NOUN
cana-1295	56	11	and	and	CCONJ
cana-1295	56	12	𝑎	𝑎	PROPN
cana-1295	56	13	∈	∈	PROPN
cana-1295	56	14	𝐵	𝐵	NOUN
cana-1295	56	15	𝑎	𝑎	PROPN
cana-1295	56	16	∈	∈	PROPN
cana-1295	56	17	𝐴	𝐴	NOUN
cana-1295	56	18	;	;	PUNCT
cana-1295	56	19	𝑏	𝑏	X
cana-1295	56	20	,	,	PUNCT
cana-1295	56	21	𝑐	𝑐	PROPN
cana-1295	56	22	∈	∈	PROPN
cana-1295	56	23	𝑇	𝑇	PROPN
cana-1295	56	24	,	,	PUNCT
cana-1295	56	25	𝐴	𝐴	PROPN
cana-1295	56	26	be	be	AUX
cana-1295	56	27	left	leave	VERB
cana-1295	56	28	idealistic	idealistic	ADJ
cana-1295	56	29	𝑇	𝑇	PROPN
cana-1295	56	30	then	then	ADV
cana-1295	56	31	𝑏𝑐𝑎	𝑏𝑐𝑎	NOUN
cana-1295	56	32	∈	∈	PROPN
cana-1295	56	33	𝐴.	𝐴.	NOUN
cana-1295	56	34	𝑎	𝑎	X
cana-1295	56	35	∈	∈	PROPN
cana-1295	56	36	𝐵	𝐵	NOUN
cana-1295	56	37	;	;	PUNCT
cana-1295	56	38	𝑏	𝑏	PROPN
cana-1295	56	39	,	,	PUNCT
cana-1295	56	40	𝑐	𝑐	PROPN
cana-1295	56	41	∈	∈	PROPN
cana-1295	56	42	𝑇	𝑇	PROPN
cana-1295	56	43	,	,	PUNCT
cana-1295	56	44	𝐵	𝐵	NOUN
cana-1295	56	45	be	be	AUX
cana-1295	56	46	left	leave	VERB
cana-1295	56	47	idealistic	idealistic	ADJ
cana-1295	56	48	of	of	ADP
cana-1295	56	49	𝑇	𝑇	PROPN
cana-1295	56	50	then	then	ADV
cana-1295	56	51	𝑏𝑐𝑎	𝑏𝑐𝑎	NOUN
cana-1295	56	52	∈	∈	PROPN
cana-1295	56	53	𝐵	𝐵	NOUN
cana-1295	56	54	therefore	therefore	ADV
cana-1295	56	55	,	,	PUNCT
cana-1295	56	56	𝑏𝑐𝑎	𝑏𝑐𝑎	PROPN
cana-1295	56	57	∈	∈	PROPN
cana-1295	56	58	𝐴	𝐴	PROPN
cana-1295	56	59	,	,	PUNCT
cana-1295	56	60	𝑏𝑐𝑎	𝑏𝑐𝑎	NOUN
cana-1295	56	61	∈	∈	PROPN
cana-1295	56	62	𝐵	𝐵	NOUN
cana-1295	56	63	⟹	⟹	NUM
cana-1295	56	64	𝑏𝑐𝑎	𝑏𝑐𝑎	NOUN
cana-1295	56	65	∈	∈	PROPN
cana-1295	56	66	𝐴	𝐴	PROPN
cana-1295	56	67	∩	∩	NOUN
cana-1295	56	68	𝐵.	𝐵.	PROPN
cana-1295	56	69	therefore	therefore	ADV
cana-1295	56	70	𝐴	𝐴	PROPN
cana-1295	56	71	∩	∩	ADJ
cana-1295	56	72	𝐵	𝐵	NOUN
cana-1295	56	73	is	be	AUX
cana-1295	56	74	a	a	DET
cana-1295	56	75	left	left	ADJ
cana-1295	56	76	ideals	ideal	NOUN
cana-1295	56	77	of	of	ADP
cana-1295	56	78	𝑇.	𝑇.	PROPN
cana-1295	56	79	theorem	theorem	ADJ
cana-1295	56	80	3.5	3.5	NUM
cana-1295	56	81	:	:	PUNCT
cana-1295	56	82	t	t	PROPN
cana-1295	56	83	's	's	PART
cana-1295	56	84	left	leave	VERB
cana-1295	56	85	ideally	ideally	ADV
cana-1295	56	86	suited	suit	VERB
cana-1295	56	87	represents	represent	VERB
cana-1295	56	88	the	the	DET
cana-1295	56	89	non	non	ADJ
cana-1295	56	90	-	-	ADJ
cana-1295	56	91	empty	empty	ADJ
cana-1295	56	92	intersection	intersection	NOUN
cana-1295	56	93	of	of	ADP
cana-1295	56	94	any	any	DET
cana-1295	56	95	set	set	NOUN
cana-1295	56	96	of	of	ADP
cana-1295	56	97	left	left	ADJ
cana-1295	56	98	ideals	ideal	NOUN
cana-1295	56	99	biternary	biternary	ADJ
cana-1295	56	100	semi	semi	ADJ
cana-1295	56	101	groups	group	NOUN
cana-1295	56	102	.	.	PUNCT
cana-1295	57	1	proof	proof	NOUN
cana-1295	57	2	:	:	PUNCT
cana-1295	57	3	let	let	VERB
cana-1295	57	4	𝐴𝛼	𝐴𝛼	VERB
cana-1295	57	5	,	,	PUNCT
cana-1295	57	6	𝛼	𝛼	PROPN
cana-1295	57	7	∈	∈	NOUN
cana-1295	57	8	∆	∆	PUNCT
cana-1295	57	9	be	be	VERB
cana-1295	57	10	group	group	NOUN
cana-1295	57	11	to	to	PART
cana-1295	57	12	be	be	AUX
cana-1295	57	13	left	leave	VERB
cana-1295	57	14	idealistic	idealistic	ADJ
cana-1295	57	15	of	of	ADP
cana-1295	57	16	𝑇	𝑇	PROPN
cana-1295	57	17	&	&	CCONJ
cana-1295	57	18	let	let	VERB
cana-1295	57	19	𝐴	𝐴	PROPN
cana-1295	57	20	=	=	SYM
cana-1295	57	21	⋂	⋂	PROPN
cana-1295	57	22	𝐴𝛼𝛼∈∆	𝐴𝛼𝛼∈∆	VERB
cana-1295	57	23	let𝑎	let𝑎	ADJ
cana-1295	57	24	∈	∈	PROPN
cana-1295	57	25	𝐴	𝐴	PROPN
cana-1295	57	26	;	;	PUNCT
cana-1295	57	27	𝑏	𝑏	PROPN
cana-1295	57	28	,	,	PUNCT
cana-1295	57	29	𝑐	𝑐	PROPN
cana-1295	57	30	∈	∈	PROPN
cana-1295	57	31	𝑇.	𝑇.	PROPN
cana-1295	57	32	now	now	ADV
cana-1295	57	33	𝑎	𝑎	PROPN
cana-1295	57	34	∈	∈	PROPN
cana-1295	57	35	𝐴	𝐴	PROPN
cana-1295	57	36	,	,	PUNCT
cana-1295	57	37	𝑎	𝑎	PROPN
cana-1295	57	38	∈	∈	PROPN
cana-1295	57	39	⋂	⋂	PROPN
cana-1295	57	40	𝐴𝛼𝛼∈∆	𝐴𝛼𝛼∈∆	VERB
cana-1295	57	41	⟹	⟹	NOUN
cana-1295	57	42	𝑎	𝑎	PRON
cana-1295	57	43	∈	∈	PROPN
cana-1295	57	44	𝐴𝛼	𝐴𝛼	NOUN
cana-1295	57	45	for	for	ADP
cana-1295	57	46	every𝛼	every𝛼	ADJ
cana-1295	57	47	∈	∈	NOUN
cana-1295	58	1	∆.	∆.	NOUN
cana-1295	58	2	𝑎	𝑎	PROPN
cana-1295	58	3	∈	∈	PROPN
cana-1295	58	4	𝐴𝛼	𝐴𝛼	PROPN
cana-1295	58	5	;	;	PUNCT
cana-1295	58	6	𝑏	𝑏	PROPN
cana-1295	58	7	,	,	PUNCT
cana-1295	58	8	𝑐	𝑐	PROPN
cana-1295	58	9	∈	∈	PROPN
cana-1295	58	10	𝑇	𝑇	PROPN
cana-1295	58	11	,	,	PUNCT
cana-1295	58	12	𝐴𝛼	𝐴𝛼	PROPN
cana-1295	58	13	be	be	AUX
cana-1295	58	14	left	leave	VERB
cana-1295	58	15	idealistic	idealistic	ADJ
cana-1295	58	16	of	of	ADP
cana-1295	58	17	left	left	NOUN
cana-1295	58	18	of	of	ADP
cana-1295	58	19	𝑇	𝑇	PROPN
cana-1295	58	20	⟹	⟹	PUNCT
cana-1295	58	21	𝑏𝑐𝑎	𝑏𝑐𝑎	NOUN
cana-1295	58	22	∈	∈	NOUN
cana-1295	58	23	𝐴𝛼	𝐴𝛼	ADP
cana-1295	58	24	𝑏𝑐𝑎	𝑏𝑐𝑎	NOUN
cana-1295	58	25	∈	∈	NOUN
cana-1295	58	26	𝐴𝛼	𝐴𝛼	NOUN
cana-1295	58	27	for	for	ADP
cana-1295	58	28	all	all	DET
cana-1295	58	29	𝛼	𝛼	PROPN
cana-1295	58	30	∈	∈	NOUN
cana-1295	58	31	∆⟹	∆⟹	VERB
cana-1295	58	32	𝑏𝑐𝑎	𝑏𝑐𝑎	NOUN
cana-1295	58	33	∈	∈	PROPN
cana-1295	58	34	⋂	⋂	PROPN
cana-1295	58	35	𝐴𝛼𝛼∈∆	𝐴𝛼𝛼∈∆	VERB
cana-1295	58	36	⟹	⟹	NUM
cana-1295	58	37	𝑏𝑐𝑎	𝑏𝑐𝑎	NOUN
cana-1295	58	38	∈	∈	PROPN
cana-1295	58	39	𝐴.	𝐴.	NOUN
cana-1295	58	40	communications	communication	NOUN
cana-1295	58	41	on	on	ADP
cana-1295	58	42	applied	apply	VERB
cana-1295	58	43	nonlinear	nonlinear	ADJ
cana-1295	58	44	analysis	analysis	NOUN
cana-1295	58	45	issn	issn	NOUN
cana-1295	58	46	:	:	PUNCT
cana-1295	58	47	1074	1074	NUM
cana-1295	58	48	-	-	PUNCT
cana-1295	58	49	133x	133x	NUM
cana-1295	58	50	vol	vol	NOUN
cana-1295	58	51	31	31	NUM
cana-1295	58	52	no	no	NOUN
cana-1295	58	53	.	.	PUNCT
cana-1295	59	1	7s	7	NOUN
cana-1295	59	2	(	(	PUNCT
cana-1295	59	3	2024	2024	NUM
cana-1295	59	4	)	)	PUNCT
cana-1295	59	5	207	207	NUM
cana-1295	59	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1295	59	7	therefore	therefore	ADV
cana-1295	59	8	𝐴	𝐴	PROPN
cana-1295	59	9	be	be	AUX
cana-1295	59	10	left	leave	VERB
cana-1295	59	11	idealistic	idealistic	ADJ
cana-1295	59	12	of	of	ADP
cana-1295	59	13	bi	bi	ADJ
cana-1295	59	14	ternary	ternary	PROPN
cana-1295	59	15	semi	semi	PROPN
cana-1295	59	16	group	group	PROPN
cana-1295	59	17	𝑇.	𝑇.	PROPN
cana-1295	59	18	theorem	theorem	VERB
cana-1295	59	19	3.6	3.6	NUM
cana-1295	59	20	:	:	PUNCT
cana-1295	59	21	the	the	DET
cana-1295	59	22	unification	unification	NOUN
cana-1295	59	23	of	of	ADP
cana-1295	59	24	other	other	ADJ
cana-1295	59	25	two	two	NUM
cana-1295	59	26	left	left	ADJ
cana-1295	59	27	ideals	ideal	NOUN
cana-1295	59	28	of	of	ADP
cana-1295	59	29	bi	bi	ADJ
cana-1295	59	30	-	-	ADJ
cana-1295	59	31	ternary	ternary	ADJ
cana-1295	59	32	semi	semi	ADJ
cana-1295	59	33	group	group	NOUN
cana-1295	59	34	𝑇	𝑇	PROPN
cana-1295	59	35	is	be	AUX
cana-1295	59	36	a	a	DET
cana-1295	59	37	left	left	ADJ
cana-1295	59	38	ideal	ideal	NOUN
cana-1295	59	39	of	of	ADP
cana-1295	59	40	𝑇.	𝑇.	PROPN
cana-1295	59	41	proof	proof	NOUN
cana-1295	59	42	:	:	PUNCT
cana-1295	59	43	let	let	VERB
cana-1295	59	44	𝐼1	𝐼1	NOUN
cana-1295	59	45	,	,	PUNCT
cana-1295	59	46	𝐼2	𝐼2	ADP
cana-1295	59	47	idealistic	idealistic	ADJ
cana-1295	59	48	left	leave	VERB
cana-1295	59	49	two	two	NUM
cana-1295	59	50	of	of	ADP
cana-1295	59	51	bi	bi	ADJ
cana-1295	59	52	ternary	ternary	NOUN
cana-1295	59	53	semi	semi	NOUN
cana-1295	60	1	group𝑇.	group𝑇.	PROPN
cana-1295	60	2	let	let	VERB
cana-1295	60	3	𝑎	𝑎	PRON
cana-1295	60	4	∈	∈	NOUN
cana-1295	60	5	𝐼1	𝐼1	NOUN
cana-1295	60	6	∪	∪	ADJ
cana-1295	60	7	𝐼2	𝐼2	NOUN
cana-1295	60	8	⟹	⟹	NUM
cana-1295	60	9	𝑎	𝑎	PROPN
cana-1295	60	10	∈	∈	NOUN
cana-1295	60	11	𝐼1	𝐼1	NOUN
cana-1295	60	12	or	or	CCONJ
cana-1295	60	13	𝑎	𝑎	PRON
cana-1295	60	14	∈	∈	NOUN
cana-1295	60	15	𝐼2	𝐼2	NOUN
cana-1295	60	16	or	or	CCONJ
cana-1295	60	17	both	both	PRON
cana-1295	60	18	and	and	CCONJ
cana-1295	60	19	𝛼	𝛼	X
cana-1295	60	20	,	,	PUNCT
cana-1295	60	21	𝛽	𝛽	PROPN
cana-1295	60	22	∈	∈	PROPN
cana-1295	60	23	𝑇	𝑇	PROPN
cana-1295	60	24	𝛼	𝛼	NOUN
cana-1295	60	25	,	,	PUNCT
cana-1295	60	26	𝛽	𝛽	PROPN
cana-1295	60	27	∈	∈	PROPN
cana-1295	60	28	𝑇	𝑇	PROPN
cana-1295	60	29	,	,	PUNCT
cana-1295	60	30	𝑎	𝑎	PROPN
cana-1295	60	31	∈	∈	NOUN
cana-1295	60	32	𝐼1	𝐼1	NOUN
cana-1295	60	33	⟹	⟹	PROPN
cana-1295	60	34	𝛼𝛽𝑎	𝛼𝛽𝑎	PROPN
cana-1295	60	35	∈	∈	PROPN
cana-1295	60	36	𝐼1	𝐼1	NOUN
cana-1295	60	37	𝛼	𝛼	NOUN
cana-1295	60	38	,	,	PUNCT
cana-1295	60	39	𝛽	𝛽	PROPN
cana-1295	60	40	∈	∈	PROPN
cana-1295	60	41	𝑇	𝑇	PROPN
cana-1295	60	42	,	,	PUNCT
cana-1295	60	43	𝑎	𝑎	PROPN
cana-1295	60	44	∈	∈	NOUN
cana-1295	60	45	𝐼2	𝐼2	NOUN
cana-1295	60	46	⟹	⟹	PROPN
cana-1295	60	47	𝛼𝛽𝑎	𝛼𝛽𝑎	PROPN
cana-1295	60	48	∈	∈	PROPN
cana-1295	60	49	𝐼2	𝐼2	NOUN
cana-1295	60	50	𝛼𝛽𝑎	𝛼𝛽𝑎	PROPN
cana-1295	60	51	∈	∈	PROPN
cana-1295	60	52	𝐼1	𝐼1	PROPN
cana-1295	60	53	,	,	PUNCT
cana-1295	60	54	𝛼𝛽𝑎	𝛼𝛽𝑎	PROPN
cana-1295	60	55	∈	∈	PROPN
cana-1295	60	56	𝐼2	𝐼2	NOUN
cana-1295	60	57	⟹	⟹	PROPN
cana-1295	60	58	𝛼𝛽𝑎	𝛼𝛽𝑎	PROPN
cana-1295	60	59	∈	∈	PROPN
cana-1295	60	60	𝐼1	𝐼1	NOUN
cana-1295	60	61	∪	∪	X
cana-1295	60	62	𝐼2	𝐼2	ADP
cana-1295	60	63	𝛼𝛽𝑎	𝛼𝛽𝑎	PROPN
cana-1295	60	64	∈	∈	PROPN
cana-1295	60	65	𝐼1	𝐼1	NOUN
cana-1295	60	66	∪	∪	X
cana-1295	60	67	𝐼2	𝐼2	NOUN
cana-1295	60	68	then𝐼1	then𝐼1	NOUN
cana-1295	60	69	∪	∪	ADP
cana-1295	60	70	𝐼2	𝐼2	NOUN
cana-1295	60	71	is	be	AUX
cana-1295	60	72	left	leave	VERB
cana-1295	60	73	ideal	ideal	NOUN
cana-1295	60	74	of	of	ADP
cana-1295	60	75	𝑇.	𝑇.	PROPN
cana-1295	60	76	theorem	theorem	VERB
cana-1295	60	77	3.7	3.7	NUM
cana-1295	60	78	:	:	PUNCT
cana-1295	60	79	any	any	DET
cana-1295	60	80	family	family	NOUN
cana-1295	60	81	combined	combine	VERB
cana-1295	60	82	of	of	ADP
cana-1295	60	83	left	left	ADJ
cana-1295	60	84	idealistic	idealistic	ADJ
cana-1295	60	85	of	of	ADP
cana-1295	60	86	bi	bi	ADJ
cana-1295	60	87	-	-	ADJ
cana-1295	60	88	ternary	ternary	ADJ
cana-1295	60	89	semi	semi	ADJ
cana-1295	60	90	group	group	NOUN
cana-1295	60	91	𝑇	𝑇	PROPN
cana-1295	60	92	be	be	AUX
cana-1295	60	93	left	leave	VERB
cana-1295	60	94	ideal	ideal	NOUN
cana-1295	60	95	of	of	ADP
cana-1295	60	96	𝑇.	𝑇.	PROPN
cana-1295	60	97	proof	proof	NOUN
cana-1295	60	98	.	.	PUNCT
cana-1295	61	1	:	:	PUNCT
cana-1295	61	2	agree	agree	VERB
cana-1295	61	3	to	to	ADP
cana-1295	61	4	𝐴𝛼	𝐴𝛼	PROPN
cana-1295	61	5	,	,	PUNCT
cana-1295	61	6	𝛼	𝛼	PROPN
cana-1295	61	7	∈	∈	NOUN
cana-1295	61	8	∆	∆	PUNCT
cana-1295	61	9	be	be	VERB
cana-1295	61	10	a	a	DET
cana-1295	61	11	related	relate	VERB
cana-1295	61	12	of	of	ADP
cana-1295	61	13	left	left	ADJ
cana-1295	61	14	ideals	ideal	NOUN
cana-1295	61	15	of	of	ADP
cana-1295	61	16	𝑇	𝑇	NOUN
cana-1295	61	17	and	and	CCONJ
cana-1295	61	18	let	let	VERB
cana-1295	61	19	𝐴	𝐴	PROPN
cana-1295	61	20	=	=	SYM
cana-1295	61	21	⋃	⋃	NOUN
cana-1295	61	22	𝐴𝛼𝛼∈∆	𝐴𝛼𝛼∈∆	NOUN
cana-1295	61	23	clearly	clearly	ADV
cana-1295	61	24	𝐴	𝐴	PROPN
cana-1295	61	25	is	be	AUX
cana-1295	61	26	a	a	DET
cana-1295	61	27	non	non	X
cana-1295	61	28	empty	empty	ADJ
cana-1295	61	29	sub	sub	NOUN
cana-1295	61	30	set	set	VERB
cana-1295	61	31	of𝑇.	of𝑇.	PRON
cana-1295	61	32	let𝑎	let𝑎	PROPN
cana-1295	61	33	∈	∈	PROPN
cana-1295	61	34	𝐴	𝐴	PROPN
cana-1295	61	35	;	;	PUNCT
cana-1295	61	36	𝑏	𝑏	PROPN
cana-1295	61	37	,	,	PUNCT
cana-1295	61	38	𝑐	𝑐	PROPN
cana-1295	61	39	∈	∈	PROPN
cana-1295	61	40	𝑇.	𝑇.	PROPN
cana-1295	61	41	now	now	ADV
cana-1295	61	42	𝑎	𝑎	PROPN
cana-1295	61	43	∈	∈	PROPN
cana-1295	61	44	𝐴	𝐴	PROPN
cana-1295	61	45	,	,	PUNCT
cana-1295	61	46	𝑎	𝑎	PROPN
cana-1295	61	47	∈	∈	PROPN
cana-1295	61	48	⋃	⋃	NOUN
cana-1295	61	49	𝐴𝛼𝛼∈∆	𝐴𝛼𝛼∈∆	NOUN
cana-1295	61	50	then	then	ADV
cana-1295	61	51	𝑎	𝑎	PROPN
cana-1295	61	52	∈	∈	NOUN
cana-1295	61	53	𝐴𝛼	𝐴𝛼	NOUN
cana-1295	61	54	for	for	ADP
cana-1295	61	55	some	some	DET
cana-1295	61	56	𝛼	𝛼	SYM
cana-1295	61	57	∈	∈	NOUN
cana-1295	61	58	∆	∆	PROPN
cana-1295	62	1	𝑎	𝑎	PRON
cana-1295	62	2	∈	∈	PROPN
cana-1295	62	3	𝐴𝛼	𝐴𝛼	PROPN
cana-1295	62	4	;	;	PUNCT
cana-1295	62	5	𝑏	𝑏	PROPN
cana-1295	62	6	,	,	PUNCT
cana-1295	62	7	𝑐	𝑐	PROPN
cana-1295	62	8	∈	∈	PROPN
cana-1295	62	9	𝑇	𝑇	PROPN
cana-1295	62	10	,	,	PUNCT
cana-1295	62	11	𝐴𝛼	𝐴𝛼	PROPN
cana-1295	62	12	be	be	AUX
cana-1295	62	13	left	leave	VERB
cana-1295	62	14	idealistic	idealistic	ADJ
cana-1295	62	15	of	of	ADP
cana-1295	62	16	𝑇	𝑇	PROPN
cana-1295	62	17	⟹	⟹	PUNCT
cana-1295	62	18	𝑏𝑐𝑎	𝑏𝑐𝑎	NOUN
cana-1295	62	19	∈	∈	NOUN
cana-1295	62	20	𝐴𝛼	𝐴𝛼	ADP
cana-1295	62	21	𝑏𝑐𝑎	𝑏𝑐𝑎	NOUN
cana-1295	62	22	∈	∈	NOUN
cana-1295	62	23	𝐴𝛼	𝐴𝛼	NOUN
cana-1295	62	24	for	for	ADP
cana-1295	62	25	all	all	DET
cana-1295	62	26	𝛼	𝛼	PROPN
cana-1295	62	27	∈	∈	NOUN
cana-1295	62	28	∆⟹	∆⟹	VERB
cana-1295	62	29	𝑏𝑐𝑎	𝑏𝑐𝑎	NOUN
cana-1295	62	30	∈	∈	PROPN
cana-1295	62	31	⋃	⋃	NOUN
cana-1295	62	32	𝐴𝛼𝛼∈∆	𝐴𝛼𝛼∈∆	ADJ
cana-1295	62	33	⟹	⟹	NUM
cana-1295	62	34	𝑏𝑐𝑎	𝑏𝑐𝑎	NOUN
cana-1295	62	35	∈	∈	PROPN
cana-1295	62	36	𝐴.	𝐴.	PROPN
cana-1295	62	37	therefore	therefore	ADV
cana-1295	62	38	𝐴	𝐴	PROPN
cana-1295	62	39	be	be	AUX
cana-1295	62	40	left	leave	VERB
cana-1295	62	41	idealistic	idealistic	ADJ
cana-1295	62	42	of	of	ADP
cana-1295	62	43	bi	bi	ADJ
cana-1295	62	44	ternary	ternary	PROPN
cana-1295	62	45	semi	semi	PROPN
cana-1295	62	46	group	group	PROPN
cana-1295	62	47	𝑇.	𝑇.	PROPN
cana-1295	62	48	lateral	lateral	ADJ
cana-1295	62	49	ideals	ideal	NOUN
cana-1295	62	50	in	in	ADP
cana-1295	62	51	bi	bi	ADJ
cana-1295	62	52	-	-	ADJ
cana-1295	62	53	ternary	ternary	ADJ
cana-1295	62	54	semi	semi	ADJ
cana-1295	62	55	groups	group	NOUN
cana-1295	62	56	:	:	PUNCT
cana-1295	62	57	definition	definition	NOUN
cana-1295	62	58	3.8	3.8	NUM
cana-1295	62	59	:	:	PUNCT
cana-1295	62	60	a	a	DET
cana-1295	62	61	non	non	NOUN
cana-1295	62	62	−	−	PROPN
cana-1295	62	63	blank	blank	ADJ
cana-1295	62	64	auxiliary	auxiliary	ADJ
cana-1295	62	65	set	set	VERB
cana-1295	62	66	𝐴	𝐴	PROPN
cana-1295	62	67	=	=	PUNCT
cana-1295	62	68	𝐴1	𝐴1	PROPN
cana-1295	62	69	∪	∪	VERB
cana-1295	62	70	𝐴2	𝐴2	PROPN
cana-1295	62	71	of	of	ADP
cana-1295	62	72	a	a	DET
cana-1295	62	73	bi	bi	ADJ
cana-1295	62	74	-	-	ADJ
cana-1295	62	75	ternary	ternary	ADJ
cana-1295	62	76	semi	semi	ADJ
cana-1295	62	77	group𝑇	group𝑇	PROPN
cana-1295	62	78	be	be	AUX
cana-1295	62	79	lateral	lateral	ADJ
cana-1295	62	80	ideal	ideal	NOUN
cana-1295	62	81	of	of	ADP
cana-1295	62	82	𝑇	𝑇	PROPN
cana-1295	62	83	if	if	SCONJ
cana-1295	62	84	both	both	DET
cana-1295	62	85	𝐴1	𝐴1	PROPN
cana-1295	62	86	and	and	CCONJ
cana-1295	62	87	𝐴2	𝐴2	PROPN
cana-1295	62	88	are	be	AUX
cana-1295	62	89	lateral	lateral	ADJ
cana-1295	62	90	ideals	ideal	NOUN
cana-1295	62	91	of	of	ADP
cana-1295	62	92	𝑇1	𝑇1	NOUN
cana-1295	62	93	and	and	CCONJ
cana-1295	62	94	𝑇2	𝑇2	NOUN
cana-1295	62	95	such	such	ADJ
cana-1295	62	96	that	that	SCONJ
cana-1295	62	97	if	if	SCONJ
cana-1295	62	98	𝑏	𝑏	PROPN
cana-1295	62	99	,	,	PUNCT
cana-1295	62	100	𝑐	𝑐	PROPN
cana-1295	62	101	∈	∈	PROPN
cana-1295	62	102	𝑇1	𝑇1	NOUN
cana-1295	62	103	,	,	PUNCT
cana-1295	62	104	𝑎	𝑎	PROPN
cana-1295	62	105	∈	∈	ADJ
cana-1295	62	106	𝐴1	𝐴1	PROPN
cana-1295	62	107	implies	imply	VERB
cana-1295	62	108	𝑏𝑎𝑐	𝑏𝑎𝑐	VERB
cana-1295	62	109	∈	∈	PROPN
cana-1295	62	110	𝐴1	𝐴1	PROPN
cana-1295	62	111	similarly	similarly	ADV
cana-1295	62	112	,	,	PUNCT
cana-1295	62	113	𝑏	𝑏	PROPN
cana-1295	62	114	,	,	PUNCT
cana-1295	62	115	𝑐	𝑐	PROPN
cana-1295	62	116	∈	∈	PROPN
cana-1295	62	117	𝑇2	𝑇2	NOUN
cana-1295	62	118	,	,	PUNCT
cana-1295	62	119	𝑎	𝑎	PROPN
cana-1295	62	120	∈	∈	NOUN
cana-1295	62	121	𝐴2	𝐴2	NOUN
cana-1295	62	122	implies	imply	VERB
cana-1295	62	123	𝑏𝑎𝑐	𝑏𝑎𝑐	VERB
cana-1295	62	124	∈	∈	PROPN
cana-1295	62	125	𝐴2	𝐴2	PROPN
cana-1295	62	126	.	.	PUNCT
cana-1295	63	1	note	note	VERB
cana-1295	63	2	3.9	3.9	NUM
cana-1295	63	3	:	:	PUNCT
cana-1295	63	4	a	a	DET
cana-1295	63	5	non	non	ADJ
cana-1295	63	6	-	-	ADJ
cana-1295	63	7	empty	empty	ADJ
cana-1295	63	8	auxiliary	auxiliary	ADJ
cana-1295	63	9	set	set	VERB
cana-1295	63	10	𝐴	𝐴	PROPN
cana-1295	63	11	of	of	ADP
cana-1295	63	12	a	a	DET
cana-1295	63	13	bi	bi	ADJ
cana-1295	63	14	-	-	ADJ
cana-1295	63	15	ternary	ternary	ADJ
cana-1295	63	16	semi	semi	ADJ
cana-1295	63	17	group𝑇	group𝑇	PROPN
cana-1295	63	18	be	be	AUX
cana-1295	63	19	to	to	ADP
cana-1295	63	20	lateral	lateral	ADJ
cana-1295	63	21	ideal	ideal	NOUN
cana-1295	64	1	if	if	SCONJ
cana-1295	64	2	and	and	CCONJ
cana-1295	64	3	only	only	ADV
cana-1295	64	4	if	if	SCONJ
cana-1295	64	5	𝑇𝐴𝑇	𝑇𝐴𝑇	PROPN
cana-1295	64	6	⊆	⊆	NUM
cana-1295	64	7	𝐴.	𝐴.	PROPN
cana-1295	64	8	example	example	NOUN
cana-1295	64	9	3.10	3.10	NUM
cana-1295	64	10	:	:	PUNCT
cana-1295	64	11	𝑇	𝑇	PROPN
cana-1295	64	12	=	=	SYM
cana-1295	64	13	𝑇1	𝑇1	PROPN
cana-1295	64	14	∪	∪	ADJ
cana-1295	64	15	𝑇2	𝑇2	PROPN
cana-1295	64	16	be	be	AUX
cana-1295	64	17	a	a	DET
cana-1295	64	18	bi	bi	ADJ
cana-1295	64	19	-	-	ADJ
cana-1295	64	20	ternary	ternary	ADJ
cana-1295	64	21	semi	semi	ADJ
cana-1295	64	22	group	group	NOUN
cana-1295	64	23	where𝑇1	where𝑇1	NOUN
cana-1295	64	24	=	=	SYM
cana-1295	64	25	{	{	PUNCT
cana-1295	64	26	[	[	PUNCT
cana-1295	64	27	𝑎	𝑎	NOUN
cana-1295	64	28	0	0	PUNCT
cana-1295	64	29	𝑏	𝑏	PROPN
cana-1295	64	30	𝑐	𝑐	PROPN
cana-1295	64	31	]	]	PUNCT
cana-1295	64	32	,	,	PUNCT
cana-1295	64	33	𝑎	𝑎	X
cana-1295	64	34	,	,	PUNCT
cana-1295	64	35	𝑏	𝑏	NOUN
cana-1295	64	36	,	,	PUNCT
cana-1295	64	37	𝑐	𝑐	PROPN
cana-1295	64	38	∈	∈	PROPN
cana-1295	64	39	𝑍	𝑍	PROPN
cana-1295	64	40	}	}	PUNCT
cana-1295	64	41	,	,	PUNCT
cana-1295	64	42	𝑇2	𝑇2	NOUN
cana-1295	64	43	=	=	SYM
cana-1295	64	44	{	{	PUNCT
cana-1295	64	45	[	[	PUNCT
cana-1295	64	46	𝑥	𝑥	NOUN
cana-1295	64	47	𝑦	𝑦	NOUN
cana-1295	64	48	0	0	PUNCT
cana-1295	64	49	𝑧	𝑧	VERB
cana-1295	64	50	]	]	PUNCT
cana-1295	64	51	,	,	PUNCT
cana-1295	64	52	𝑥	𝑥	X
cana-1295	64	53	,	,	PUNCT
cana-1295	64	54	𝑦	𝑦	NOUN
cana-1295	64	55	,	,	PUNCT
cana-1295	64	56	𝑧	𝑧	DET
cana-1295	64	57	∈	∈	PROPN
cana-1295	64	58	𝑍	𝑍	NOUN
cana-1295	64	59	}	}	PUNCT
cana-1295	64	60	are	be	AUX
cana-1295	64	61	ternary	ternary	ADJ
cana-1295	64	62	semi	semi	ADJ
cana-1295	64	63	groups	group	NOUN
cana-1295	64	64	of	of	ADP
cana-1295	64	65	t.	t.	NOUN
cana-1295	64	66	𝐼	𝐼	PROPN
cana-1295	64	67	=	=	PUNCT
cana-1295	64	68	𝐼1	𝐼1	PROPN
cana-1295	64	69	∪	∪	X
cana-1295	64	70	𝐼2	𝐼2	NOUN
cana-1295	64	71	where	where	SCONJ
cana-1295	64	72	𝐼1	𝐼1	NOUN
cana-1295	64	73	=	=	SYM
cana-1295	64	74	{	{	PUNCT
cana-1295	64	75	[	[	PUNCT
cana-1295	64	76	0	0	NUM
cana-1295	64	77	0	0	NUM
cana-1295	64	78	𝑏	𝑏	NOUN
cana-1295	64	79	0	0	NUM
cana-1295	64	80	]	]	PUNCT
cana-1295	64	81	,	,	PUNCT
cana-1295	64	82	𝑏	𝑏	DET
cana-1295	64	83	∈	∈	PROPN
cana-1295	64	84	𝑍	𝑍	PROPN
cana-1295	64	85	}	}	PUNCT
cana-1295	64	86	,	,	PUNCT
cana-1295	64	87	𝐼2	𝐼2	NOUN
cana-1295	64	88	=	=	SYM
cana-1295	64	89	{	{	PUNCT
cana-1295	64	90	[	[	PUNCT
cana-1295	64	91	0	0	NUM
cana-1295	65	1	𝑎	𝑎	X
cana-1295	65	2	0	0	NUM
cana-1295	65	3	0	0	NUM
cana-1295	65	4	]	]	PUNCT
cana-1295	65	5	,	,	PUNCT
cana-1295	65	6	𝑎	𝑎	PROPN
cana-1295	65	7	∈	∈	PROPN
cana-1295	65	8	𝑍	𝑍	NOUN
cana-1295	65	9	}	}	PUNCT
cana-1295	65	10	are	be	AUX
cana-1295	65	11	lateral	lateral	ADJ
cana-1295	65	12	ideals	ideal	NOUN
cana-1295	65	13	of	of	ADP
cana-1295	65	14	𝑇1&𝑇2respectively	𝑇1&𝑇2respectively	ADV
cana-1295	65	15	.	.	PUNCT
cana-1295	66	1	thus	thus	ADV
cana-1295	66	2	i	i	PRON
cana-1295	66	3	is	be	AUX
cana-1295	66	4	the	the	DET
cana-1295	66	5	lateral	lateral	ADJ
cana-1295	66	6	ideal	ideal	NOUN
cana-1295	66	7	of	of	ADP
cana-1295	66	8	bi	bi	ADJ
cana-1295	66	9	ternary	ternary	PROPN
cana-1295	66	10	semi	semi	PROPN
cana-1295	66	11	group	group	PROPN
cana-1295	66	12	t.	t.	PROPN
cana-1295	66	13	theorem	theorem	PROPN
cana-1295	66	14	3.11	3.11	NUM
cana-1295	66	15	:	:	PUNCT
cana-1295	66	16	the	the	DET
cana-1295	66	17	connection	connection	NOUN
cana-1295	66	18	by	by	ADP
cana-1295	66	19	two	two	NUM
cana-1295	66	20	lateral	lateral	ADJ
cana-1295	66	21	ideals	ideal	NOUN
cana-1295	66	22	of	of	ADP
cana-1295	66	23	a	a	DET
cana-1295	66	24	bi	bi	ADJ
cana-1295	66	25	-	-	ADJ
cana-1295	66	26	ternary	ternary	ADJ
cana-1295	66	27	semi	semi	ADJ
cana-1295	66	28	group	group	NOUN
cana-1295	66	29	𝑇	𝑇	PROPN
cana-1295	66	30	be	be	VERB
cana-1295	66	31	a	a	DET
cana-1295	66	32	lateral	lateral	ADJ
cana-1295	66	33	idealistic	idealistic	ADJ
cana-1295	66	34	of	of	ADP
cana-1295	66	35	𝑇.	𝑇.	PROPN
cana-1295	66	36	proof	proof	NOUN
cana-1295	66	37	:	:	PUNCT
cana-1295	66	38	agree	agree	VERB
cana-1295	66	39	𝐴	𝐴	PROPN
cana-1295	66	40	,	,	PUNCT
cana-1295	66	41	𝐵	𝐵	PROPN
cana-1295	66	42	be	be	VERB
cana-1295	66	43	two	two	NUM
cana-1295	66	44	lateral	lateral	ADJ
cana-1295	66	45	perfects	perfect	NOUN
cana-1295	66	46	of	of	ADP
cana-1295	66	47	𝑇.	𝑇.	PROPN
cana-1295	66	48	be	be	VERB
cana-1295	66	49	in	in	ADP
cana-1295	66	50	agreement	agreement	NOUN
cana-1295	66	51	𝑎	𝑎	PRON
cana-1295	66	52	∈	∈	PROPN
cana-1295	66	53	𝐴	𝐴	NOUN
cana-1295	66	54	∩	∩	ADJ
cana-1295	66	55	𝐵	𝐵	NOUN
cana-1295	66	56	and	and	CCONJ
cana-1295	66	57	𝑏	𝑏	NOUN
cana-1295	66	58	,	,	PUNCT
cana-1295	66	59	𝑐	𝑐	PROPN
cana-1295	66	60	∈	∈	PROPN
cana-1295	66	61	𝑇	𝑇	PROPN
cana-1295	66	62	.	.	PUNCT
cana-1295	67	1	if	if	SCONJ
cana-1295	67	2	𝑎	𝑎	PRON
cana-1295	67	3	∈	∈	PROPN
cana-1295	67	4	𝐴	𝐴	NOUN
cana-1295	67	5	∩	∩	ADJ
cana-1295	67	6	𝐵	𝐵	NOUN
cana-1295	67	7	then	then	ADV
cana-1295	67	8	𝑎	𝑎	PROPN
cana-1295	67	9	∈	∈	PROPN
cana-1295	67	10	𝐴	𝐴	NOUN
cana-1295	67	11	and	and	CCONJ
cana-1295	67	12	𝑎	𝑎	PROPN
cana-1295	67	13	∈	∈	NOUN
cana-1295	67	14	𝐵	𝐵	NOUN
cana-1295	67	15	communications	communication	NOUN
cana-1295	67	16	on	on	ADP
cana-1295	67	17	applied	apply	VERB
cana-1295	67	18	nonlinear	nonlinear	ADJ
cana-1295	67	19	analysis	analysis	NOUN
cana-1295	67	20	issn	issn	NOUN
cana-1295	67	21	:	:	PUNCT
cana-1295	67	22	1074	1074	NUM
cana-1295	67	23	-	-	PUNCT
cana-1295	67	24	133x	133x	NUM
cana-1295	67	25	vol	vol	NOUN
cana-1295	67	26	31	31	NUM
cana-1295	67	27	no	no	NOUN
cana-1295	67	28	.	.	PUNCT
cana-1295	68	1	7s	7	NOUN
cana-1295	68	2	(	(	PUNCT
cana-1295	68	3	2024	2024	NUM
cana-1295	68	4	)	)	PUNCT
cana-1295	68	5	208	208	NUM
cana-1295	68	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1295	68	7	𝑎	𝑎	PROPN
cana-1295	68	8	∈	∈	PROPN
cana-1295	68	9	𝐴	𝐴	PROPN
cana-1295	68	10	;	;	PUNCT
cana-1295	68	11	𝑏	𝑏	PROPN
cana-1295	68	12	,	,	PUNCT
cana-1295	68	13	𝑐	𝑐	PROPN
cana-1295	68	14	∈	∈	PROPN
cana-1295	68	15	𝑇	𝑇	PROPN
cana-1295	68	16	,	,	PUNCT
cana-1295	68	17	𝐴	𝐴	PROPN
cana-1295	68	18	is	be	AUX
cana-1295	68	19	a	a	DET
cana-1295	68	20	lateral	lateral	ADJ
cana-1295	68	21	perfect	perfect	NOUN
cana-1295	68	22	of	of	ADP
cana-1295	68	23	𝑇	𝑇	PROPN
cana-1295	68	24	then𝑏𝑎𝑐	then𝑏𝑎𝑐	NOUN
cana-1295	68	25	∈	∈	PROPN
cana-1295	68	26	𝐴.	𝐴.	PROPN
cana-1295	68	27	𝑎	𝑎	PRON
cana-1295	68	28	∈	∈	PROPN
cana-1295	68	29	𝐵	𝐵	NOUN
cana-1295	68	30	;	;	PUNCT
cana-1295	68	31	𝑏	𝑏	PROPN
cana-1295	68	32	,	,	PUNCT
cana-1295	68	33	𝑐	𝑐	PROPN
cana-1295	68	34	∈	∈	PROPN
cana-1295	68	35	𝑇	𝑇	PROPN
cana-1295	68	36	,	,	PUNCT
cana-1295	68	37	𝐵	𝐵	NOUN
cana-1295	68	38	is	be	AUX
cana-1295	68	39	a	a	DET
cana-1295	68	40	lateral	lateral	ADJ
cana-1295	68	41	perfect	perfect	NOUN
cana-1295	68	42	of	of	ADP
cana-1295	68	43	𝑇	𝑇	PROPN
cana-1295	68	44	then	then	ADV
cana-1295	68	45	𝑏𝑎𝑐	𝑏𝑎𝑐	VERB
cana-1295	68	46	∈	∈	PROPN
cana-1295	68	47	𝐵	𝐵	NOUN
cana-1295	68	48	hence	hence	ADV
cana-1295	68	49	,	,	PUNCT
cana-1295	68	50	𝑏𝑎𝑐	𝑏𝑎𝑐	VERB
cana-1295	68	51	∈	∈	PROPN
cana-1295	68	52	𝐴	𝐴	PROPN
cana-1295	68	53	,	,	PUNCT
cana-1295	68	54	𝑏𝑎𝑐	𝑏𝑎𝑐	VERB
cana-1295	68	55	∈	∈	PROPN
cana-1295	68	56	𝐵	𝐵	PROPN
cana-1295	68	57	⟹	⟹	NUM
cana-1295	68	58	𝑏𝑎𝑐	𝑏𝑎𝑐	VERB
cana-1295	68	59	∈	∈	PROPN
cana-1295	68	60	𝐴	𝐴	PROPN
cana-1295	68	61	∩	∩	NOUN
cana-1295	68	62	𝐵.	𝐵.	PROPN
cana-1295	68	63	therefore	therefore	ADV
cana-1295	68	64	𝐴	𝐴	PROPN
cana-1295	68	65	∩	∩	ADJ
cana-1295	68	66	𝐵	𝐵	NOUN
cana-1295	68	67	is	be	AUX
cana-1295	68	68	a	a	DET
cana-1295	68	69	on	on	ADP
cana-1295	68	70	the	the	DET
cana-1295	68	71	side	side	NOUN
cana-1295	68	72	perfect	perfect	NOUN
cana-1295	68	73	of	of	ADP
cana-1295	68	74	𝑇.	𝑇.	PROPN
cana-1295	68	75	theorem	theorem	VERB
cana-1295	68	76	3.12	3.12	NUM
cana-1295	68	77	:	:	PUNCT
cana-1295	68	78	the	the	DET
cana-1295	68	79	non	non	ADJ
cana-1295	68	80	-	-	ADJ
cana-1295	68	81	empty	empty	ADJ
cana-1295	68	82	joint	joint	NOUN
cana-1295	68	83	of	of	ADP
cana-1295	68	84	any	any	DET
cana-1295	68	85	family	family	NOUN
cana-1295	68	86	of	of	ADP
cana-1295	68	87	lateral	lateral	ADJ
cana-1295	68	88	ideals	ideal	NOUN
cana-1295	68	89	bi	bi	ADJ
cana-1295	68	90	ternary	ternary	PROPN
cana-1295	68	91	semi	semi	PROPN
cana-1295	68	92	group	group	NOUN
cana-1295	68	93	𝑇	𝑇	PROPN
cana-1295	68	94	is	be	AUX
cana-1295	68	95	a	a	DET
cana-1295	68	96	lateral	lateral	ADJ
cana-1295	68	97	perfect	perfect	NOUN
cana-1295	68	98	of	of	ADP
cana-1295	68	99	𝑇.	𝑇.	PROPN
cana-1295	68	100	proof	proof	NOUN
cana-1295	68	101	:	:	PUNCT
cana-1295	68	102	agree	agree	VERB
cana-1295	68	103	𝐴𝛼	𝐴𝛼	PROPN
cana-1295	68	104	,	,	PUNCT
cana-1295	68	105	𝛼	𝛼	PROPN
cana-1295	68	106	∈	∈	NOUN
cana-1295	68	107	∆	∆	PUNCT
cana-1295	68	108	be	be	VERB
cana-1295	68	109	a	a	DET
cana-1295	68	110	combined	combine	VERB
cana-1295	68	111	of	of	ADP
cana-1295	68	112	on	on	ADP
cana-1295	68	113	the	the	DET
cana-1295	68	114	side	side	NOUN
cana-1295	68	115	perfect	perfect	NOUN
cana-1295	68	116	of	of	ADP
cana-1295	68	117	𝑇	𝑇	NOUN
cana-1295	68	118	and	and	CCONJ
cana-1295	68	119	let	let	VERB
cana-1295	68	120	𝐴	𝐴	PROPN
cana-1295	68	121	=	=	SYM
cana-1295	68	122	⋂	⋂	PROPN
cana-1295	68	123	𝐴𝛼𝛼∈∆	𝐴𝛼𝛼∈∆	NOUN
cana-1295	68	124	let	let	VERB
cana-1295	68	125	𝑎	𝑎	PROPN
cana-1295	68	126	∈	∈	PROPN
cana-1295	68	127	𝐴	𝐴	PROPN
cana-1295	68	128	;	;	PUNCT
cana-1295	68	129	𝑏	𝑏	PROPN
cana-1295	68	130	,	,	PUNCT
cana-1295	68	131	𝑐	𝑐	PROPN
cana-1295	68	132	∈	∈	PROPN
cana-1295	68	133	𝑇.	𝑇.	PROPN
cana-1295	68	134	now	now	ADV
cana-1295	68	135	𝑎	𝑎	PROPN
cana-1295	68	136	∈	∈	PROPN
cana-1295	68	137	𝐴	𝐴	PROPN
cana-1295	68	138	,	,	PUNCT
cana-1295	68	139	𝑎	𝑎	PROPN
cana-1295	68	140	∈	∈	PROPN
cana-1295	68	141	⋂	⋂	PROPN
cana-1295	68	142	𝐴𝛼𝛼∈∆	𝐴𝛼𝛼∈∆	VERB
cana-1295	68	143	⟹	⟹	NOUN
cana-1295	68	144	𝑎	𝑎	PRON
cana-1295	68	145	∈	∈	PROPN
cana-1295	68	146	𝐴𝛼	𝐴𝛼	NOUN
cana-1295	68	147	for	for	ADP
cana-1295	68	148	every𝛼	every𝛼	ADJ
cana-1295	68	149	∈	∈	NOUN
cana-1295	69	1	∆.	∆.	NOUN
cana-1295	69	2	𝑎	𝑎	PROPN
cana-1295	69	3	∈	∈	PROPN
cana-1295	69	4	𝐴𝛼	𝐴𝛼	PROPN
cana-1295	69	5	;	;	PUNCT
cana-1295	69	6	𝑏	𝑏	PROPN
cana-1295	69	7	,	,	PUNCT
cana-1295	69	8	𝑐	𝑐	PROPN
cana-1295	69	9	∈	∈	PROPN
cana-1295	69	10	𝑇	𝑇	PROPN
cana-1295	69	11	,	,	PUNCT
cana-1295	69	12	𝐴𝛼	𝐴𝛼	PROPN
cana-1295	69	13	were	be	AUX
cana-1295	69	14	perfect	perfect	ADJ
cana-1295	69	15	on	on	ADP
cana-1295	69	16	side	side	NOUN
cana-1295	69	17	perfect	perfect	ADJ
cana-1295	69	18	&	&	CCONJ
cana-1295	69	19	𝑇	𝑇	PROPN
cana-1295	69	20	⟹	⟹	PUNCT
cana-1295	69	21	𝑏𝑎𝑐	𝑏𝑎𝑐	VERB
cana-1295	69	22	∈	∈	PROPN
cana-1295	70	1	𝐴𝛼	𝐴𝛼	ADP
cana-1295	70	2	𝑏𝑐𝑎	𝑏𝑐𝑎	NOUN
cana-1295	70	3	∈	∈	NOUN
cana-1295	70	4	𝐴𝛼	𝐴𝛼	NOUN
cana-1295	70	5	for	for	ADP
cana-1295	70	6	all	all	DET
cana-1295	70	7	𝛼	𝛼	PROPN
cana-1295	70	8	∈	∈	NOUN
cana-1295	70	9	∆⟹	∆⟹	NOUN
cana-1295	70	10	𝑏𝑎𝑐	𝑏𝑎𝑐	PROPN
cana-1295	70	11	∈	∈	PROPN
cana-1295	70	12	⋂	⋂	PROPN
cana-1295	70	13	𝐴𝛼𝛼∈∆	𝐴𝛼𝛼∈∆	VERB
cana-1295	70	14	⟹	⟹	NUM
cana-1295	70	15	𝑏𝑎𝑐	𝑏𝑎𝑐	VERB
cana-1295	70	16	∈	∈	PROPN
cana-1295	71	1	𝐴.	𝐴.	PROPN
cana-1295	72	1	thus	thus	ADV
cana-1295	72	2	𝐴	𝐴	PROPN
cana-1295	72	3	be	be	VERB
cana-1295	72	4	a	a	DET
cana-1295	72	5	lateral	lateral	ADJ
cana-1295	72	6	idealistic	idealistic	ADJ
cana-1295	72	7	of	of	ADP
cana-1295	72	8	bi	bi	ADJ
cana-1295	72	9	ternary	ternary	PROPN
cana-1295	72	10	semi	semi	PROPN
cana-1295	72	11	group	group	PROPN
cana-1295	72	12	𝑇.	𝑇.	PROPN
cana-1295	72	13	theorem	theorem	VERB
cana-1295	72	14	3.13	3.13	NUM
cana-1295	72	15	:	:	PUNCT
cana-1295	72	16	the	the	DET
cana-1295	72	17	amalgamation	amalgamation	NOUN
cana-1295	72	18	by	by	ADP
cana-1295	72	19	two	two	NUM
cana-1295	72	20	lateral	lateral	ADJ
cana-1295	72	21	ideals	ideal	NOUN
cana-1295	72	22	of	of	ADP
cana-1295	72	23	bi	bi	ADJ
cana-1295	72	24	-	-	ADJ
cana-1295	72	25	ternary	ternary	ADJ
cana-1295	72	26	semi	semi	ADJ
cana-1295	72	27	grouping	group	VERB
cana-1295	72	28	𝑇	𝑇	PROPN
cana-1295	72	29	is	be	AUX
cana-1295	72	30	a	a	DET
cana-1295	72	31	on	on	ADP
cana-1295	72	32	the	the	DET
cana-1295	72	33	side	side	NOUN
cana-1295	72	34	perfect	perfect	NOUN
cana-1295	72	35	of	of	ADP
cana-1295	72	36	𝑇.	𝑇.	PROPN
cana-1295	72	37	proof	proof	NOUN
cana-1295	72	38	:	:	PUNCT
cana-1295	72	39	agree	agree	VERB
cana-1295	72	40	to	to	PART
cana-1295	72	41	𝐼1	𝐼1	VERB
cana-1295	72	42	,	,	PUNCT
cana-1295	72	43	𝐼2	𝐼2	VERB
cana-1295	72	44	any	any	DET
cana-1295	72	45	two	two	NUM
cana-1295	72	46	on	on	ADP
cana-1295	72	47	the	the	DET
cana-1295	72	48	side	side	NOUN
cana-1295	72	49	perfect	perfect	NOUN
cana-1295	72	50	of	of	ADP
cana-1295	72	51	bi	bi	ADJ
cana-1295	72	52	-	-	ADJ
cana-1295	72	53	ternary	ternary	ADJ
cana-1295	72	54	semi	semi	ADJ
cana-1295	72	55	-	-	ADJ
cana-1295	72	56	grouping	grouping	ADJ
cana-1295	72	57	𝑇.	𝑇.	PROPN
cana-1295	72	58	let	let	VERB
cana-1295	72	59	𝑎	𝑎	PROPN
cana-1295	72	60	∈	∈	NOUN
cana-1295	72	61	𝐼1	𝐼1	NOUN
cana-1295	72	62	∪	∪	ADJ
cana-1295	72	63	𝐼2	𝐼2	NOUN
cana-1295	72	64	⟹	⟹	NUM
cana-1295	72	65	𝑎	𝑎	PROPN
cana-1295	72	66	∈	∈	NOUN
cana-1295	72	67	𝐼1	𝐼1	NOUN
cana-1295	72	68	or	or	CCONJ
cana-1295	72	69	𝑎	𝑎	PRON
cana-1295	72	70	∈	∈	NOUN
cana-1295	72	71	𝐼2	𝐼2	NOUN
cana-1295	72	72	or	or	CCONJ
cana-1295	72	73	both	both	PRON
cana-1295	72	74	and	and	CCONJ
cana-1295	72	75	𝛼	𝛼	X
cana-1295	72	76	,	,	PUNCT
cana-1295	72	77	𝛽	𝛽	PROPN
cana-1295	72	78	∈	∈	PROPN
cana-1295	72	79	𝑇	𝑇	PROPN
cana-1295	72	80	𝛼	𝛼	NOUN
cana-1295	72	81	,	,	PUNCT
cana-1295	72	82	𝛽	𝛽	PROPN
cana-1295	72	83	∈	∈	PROPN
cana-1295	72	84	𝑇	𝑇	PROPN
cana-1295	72	85	,	,	PUNCT
cana-1295	72	86	𝑎	𝑎	PROPN
cana-1295	72	87	∈	∈	NOUN
cana-1295	72	88	𝐼1	𝐼1	NOUN
cana-1295	72	89	⟹	⟹	PUNCT
cana-1295	72	90	𝛼𝑎𝛽	𝛼𝑎𝛽	PROPN
cana-1295	72	91	∈	∈	PROPN
cana-1295	72	92	𝐼1	𝐼1	NOUN
cana-1295	72	93	𝛼	𝛼	NOUN
cana-1295	72	94	,	,	PUNCT
cana-1295	72	95	𝛽	𝛽	PROPN
cana-1295	72	96	∈	∈	PROPN
cana-1295	72	97	𝑇	𝑇	PROPN
cana-1295	72	98	,	,	PUNCT
cana-1295	72	99	𝑎	𝑎	PROPN
cana-1295	72	100	∈	∈	NOUN
cana-1295	72	101	𝐼2	𝐼2	NOUN
cana-1295	72	102	⟹	⟹	PUNCT
cana-1295	72	103	𝛼𝑎𝛽	𝛼𝑎𝛽	PROPN
cana-1295	72	104	∈	∈	PROPN
cana-1295	72	105	𝐼2	𝐼2	PROPN
cana-1295	72	106	𝛼𝑎𝛽	𝛼𝑎𝛽	PROPN
cana-1295	72	107	∈	∈	PROPN
cana-1295	72	108	𝐼1	𝐼1	NOUN
cana-1295	72	109	,	,	PUNCT
cana-1295	72	110	𝛼𝑎𝛽	𝛼𝑎𝛽	PROPN
cana-1295	72	111	∈	∈	PROPN
cana-1295	72	112	𝐼2	𝐼2	PROPN
cana-1295	72	113	⟹	⟹	PUNCT
cana-1295	72	114	𝛼𝑎𝛽	𝛼𝑎𝛽	PROPN
cana-1295	72	115	∈	∈	PROPN
cana-1295	72	116	𝐼1	𝐼1	NOUN
cana-1295	72	117	∪	∪	X
cana-1295	72	118	𝐼2	𝐼2	PROPN
cana-1295	72	119	𝛼𝑎𝛽	𝛼𝑎𝛽	NOUN
cana-1295	72	120	∈	∈	PROPN
cana-1295	72	121	𝐼1	𝐼1	NOUN
cana-1295	72	122	∪	∪	X
cana-1295	72	123	𝐼2	𝐼2	NOUN
cana-1295	72	124	then𝐼1	then𝐼1	NOUN
cana-1295	72	125	∪	∪	ADP
cana-1295	72	126	𝐼2	𝐼2	NOUN
cana-1295	72	127	is	be	AUX
cana-1295	72	128	lateral	lateral	ADJ
cana-1295	72	129	ideal	ideal	NOUN
cana-1295	72	130	of	of	ADP
cana-1295	72	131	𝑇.	𝑇.	PROPN
cana-1295	72	132	theorem	theorem	VERB
cana-1295	72	133	3.14	3.14	NUM
cana-1295	72	134	:	:	PUNCT
cana-1295	72	135	the	the	DET
cana-1295	72	136	combined	combine	VERB
cana-1295	72	137	be	be	VERB
cana-1295	72	138	all	all	ADV
cana-1295	72	139	related	relate	VERB
cana-1295	72	140	of	of	ADP
cana-1295	72	141	lateral	lateral	ADJ
cana-1295	72	142	ideals	ideal	NOUN
cana-1295	72	143	be	be	VERB
cana-1295	72	144	bi	bi	ADJ
cana-1295	72	145	-	-	ADJ
cana-1295	72	146	ternary	ternary	ADJ
cana-1295	72	147	semi	semi	ADJ
cana-1295	72	148	grouping	group	VERB
cana-1295	72	149	𝑇	𝑇	PROPN
cana-1295	72	150	is	be	AUX
cana-1295	72	151	a	a	DET
cana-1295	72	152	lateral	lateral	ADJ
cana-1295	72	153	ideal	ideal	NOUN
cana-1295	72	154	of	of	ADP
cana-1295	72	155	𝑇.	𝑇.	PROPN
cana-1295	72	156	proof	proof	NOUN
cana-1295	72	157	:	:	PUNCT
cana-1295	72	158	agree	agree	VERB
cana-1295	72	159	to	to	ADP
cana-1295	72	160	𝐴𝛼	𝐴𝛼	PROPN
cana-1295	72	161	,	,	PUNCT
cana-1295	72	162	𝛼	𝛼	PROPN
cana-1295	72	163	∈	∈	NOUN
cana-1295	72	164	∆	∆	PROPN
cana-1295	72	165	become	become	VERB
cana-1295	72	166	a	a	DET
cana-1295	72	167	group	group	NOUN
cana-1295	72	168	of	of	ADP
cana-1295	72	169	transverse	transverse	NOUN
cana-1295	72	170	ideals	ideal	NOUN
cana-1295	72	171	.	.	PUNCT
cana-1295	73	1	of	of	ADP
cana-1295	73	2	𝑇	𝑇	PROPN
cana-1295	73	3	and	and	CCONJ
cana-1295	73	4	let	let	VERB
cana-1295	73	5	𝐴	𝐴	PROPN
cana-1295	73	6	=	=	SYM
cana-1295	73	7	⋃	⋃	NOUN
cana-1295	73	8	𝐴𝛼𝛼∈∆	𝐴𝛼𝛼∈∆	NOUN
cana-1295	73	9	clearly	clearly	ADV
cana-1295	73	10	𝐴	𝐴	PROPN
cana-1295	73	11	is	be	AUX
cana-1295	73	12	a	a	DET
cana-1295	73	13	non	non	ADJ
cana-1295	73	14	-	-	ADJ
cana-1295	73	15	blank	blank	ADJ
cana-1295	73	16	sub	sub	NOUN
cana-1295	73	17	set	set	NOUN
cana-1295	73	18	of	of	ADP
cana-1295	73	19	𝑇.	𝑇.	PROPN
cana-1295	73	20	let	let	VERB
cana-1295	73	21	𝑎	𝑎	PROPN
cana-1295	73	22	∈	∈	PROPN
cana-1295	73	23	𝐴	𝐴	PROPN
cana-1295	73	24	;	;	PUNCT
cana-1295	73	25	𝑏	𝑏	PROPN
cana-1295	73	26	,	,	PUNCT
cana-1295	73	27	𝑐	𝑐	PROPN
cana-1295	73	28	∈	∈	PROPN
cana-1295	73	29	𝑇.	𝑇.	PROPN
cana-1295	73	30	now	now	ADV
cana-1295	73	31	𝑎	𝑎	PROPN
cana-1295	73	32	∈	∈	PROPN
cana-1295	73	33	𝐴	𝐴	PROPN
cana-1295	73	34	,	,	PUNCT
cana-1295	73	35	𝑎	𝑎	PROPN
cana-1295	73	36	∈	∈	PROPN
cana-1295	73	37	⋃	⋃	NOUN
cana-1295	73	38	𝐴𝛼𝛼∈∆	𝐴𝛼𝛼∈∆	NOUN
cana-1295	73	39	then	then	ADV
cana-1295	73	40	𝑎	𝑎	PROPN
cana-1295	73	41	∈	∈	NOUN
cana-1295	73	42	𝐴𝛼	𝐴𝛼	NOUN
cana-1295	73	43	for	for	ADP
cana-1295	73	44	some	some	DET
cana-1295	73	45	𝛼	𝛼	SYM
cana-1295	73	46	∈	∈	NOUN
cana-1295	73	47	∆	∆	PROPN
cana-1295	74	1	𝑎	𝑎	PRON
cana-1295	74	2	∈	∈	PROPN
cana-1295	74	3	𝐴𝛼	𝐴𝛼	PROPN
cana-1295	74	4	;	;	PUNCT
cana-1295	74	5	𝑏	𝑏	PROPN
cana-1295	74	6	,	,	PUNCT
cana-1295	74	7	𝑐	𝑐	PROPN
cana-1295	74	8	∈	∈	PROPN
cana-1295	74	9	𝑇	𝑇	PROPN
cana-1295	74	10	,	,	PUNCT
cana-1295	74	11	𝐴𝛼	𝐴𝛼	PROPN
cana-1295	74	12	is	be	AUX
cana-1295	74	13	a	a	DET
cana-1295	74	14	transverse	transverse	NOUN
cana-1295	74	15	ideals	ideal	NOUN
cana-1295	74	16	of	of	ADP
cana-1295	74	17	𝑇	𝑇	PROPN
cana-1295	74	18	⟹	⟹	PUNCT
cana-1295	74	19	𝑏𝑎𝑐	𝑏𝑎𝑐	VERB
cana-1295	74	20	∈	∈	PROPN
cana-1295	75	1	𝐴𝛼	𝐴𝛼	ADP
cana-1295	75	2	𝑏𝑐𝑎	𝑏𝑐𝑎	NOUN
cana-1295	75	3	∈	∈	NOUN
cana-1295	75	4	𝐴𝛼	𝐴𝛼	NOUN
cana-1295	75	5	for	for	ADP
cana-1295	75	6	all	all	DET
cana-1295	75	7	𝛼	𝛼	PROPN
cana-1295	75	8	∈	∈	PROPN
cana-1295	75	9	∆⟹	∆⟹	NOUN
cana-1295	75	10	𝑏𝑎𝑐	𝑏𝑎𝑐	PROPN
cana-1295	75	11	∈	∈	PROPN
cana-1295	75	12	⋃	⋃	NOUN
cana-1295	75	13	𝐴𝛼𝛼∈∆	𝐴𝛼𝛼∈∆	NOUN
cana-1295	75	14	⟹	⟹	PUNCT
cana-1295	75	15	𝑏𝑎𝑐	𝑏𝑎𝑐	VERB
cana-1295	75	16	∈	∈	PROPN
cana-1295	76	1	𝐴.	𝐴.	PROPN
cana-1295	77	1	thus	thus	ADV
cana-1295	77	2	𝐴	𝐴	PROPN
cana-1295	77	3	is	be	AUX
cana-1295	77	4	a	a	DET
cana-1295	77	5	lateral	lateral	ADJ
cana-1295	77	6	ideals	ideal	NOUN
cana-1295	77	7	of	of	ADP
cana-1295	77	8	bi	bi	ADJ
cana-1295	77	9	ternary	ternary	PROPN
cana-1295	77	10	semi	semi	PROPN
cana-1295	77	11	group	group	PROPN
cana-1295	77	12	𝑇.	𝑇.	PROPN
cana-1295	77	13	right	right	ADJ
cana-1295	77	14	ideals	ideal	NOUN
cana-1295	77	15	in	in	ADP
cana-1295	77	16	bi	bi	ADJ
cana-1295	77	17	-	-	ADJ
cana-1295	77	18	ternary	ternary	ADJ
cana-1295	77	19	semi	semi	ADJ
cana-1295	77	20	groups	group	NOUN
cana-1295	77	21	:	:	PUNCT
cana-1295	77	22	definition	definition	NOUN
cana-1295	77	23	3.15	3.15	NUM
cana-1295	77	24	:	:	PUNCT
cana-1295	77	25	a	a	DET
cana-1295	77	26	non	non	ADJ
cana-1295	77	27	-	-	ADJ
cana-1295	77	28	empty	empty	ADJ
cana-1295	77	29	sub	sub	NOUN
cana-1295	77	30	set	set	VERB
cana-1295	77	31	𝐴	𝐴	PROPN
cana-1295	77	32	=	=	PUNCT
cana-1295	77	33	𝐴1	𝐴1	PROPN
cana-1295	77	34	∪	∪	VERB
cana-1295	77	35	𝐴2	𝐴2	PROPN
cana-1295	77	36	of	of	ADP
cana-1295	77	37	become	become	VERB
cana-1295	77	38	a	a	DET
cana-1295	77	39	family	family	NOUN
cana-1295	77	40	of	of	ADP
cana-1295	77	41	lateral	lateral	ADJ
cana-1295	77	42	ideas	idea	NOUN
cana-1295	77	43	.	.	PUNCT
cana-1295	78	1	a	a	DET
cana-1295	78	2	bi	bi	ADJ
cana-1295	78	3	-	-	ADJ
cana-1295	78	4	ternary	ternary	ADJ
cana-1295	78	5	semi	semi	ADJ
cana-1295	78	6	-	-	NOUN
cana-1295	78	7	group	group	ADJ
cana-1295	78	8	t	t	PROPN
cana-1295	78	9	is	be	AUX
cana-1295	78	10	considered	consider	VERB
cana-1295	78	11	the	the	DET
cana-1295	78	12	proper	proper	ADJ
cana-1295	78	13	optimum	optimum	NOUN
cana-1295	78	14	of	of	ADP
cana-1295	78	15	t	t	PROPN
cana-1295	78	16	if	if	SCONJ
cana-1295	78	17	equally	equally	ADV
cana-1295	78	18	𝐴1	𝐴1	PROPN
cana-1295	78	19	and	and	CCONJ
cana-1295	78	20	𝐴2	𝐴2	PROPN
cana-1295	78	21	are	be	AUX
cana-1295	78	22	right	right	ADJ
cana-1295	78	23	ideals	ideal	NOUN
cana-1295	78	24	of	of	ADP
cana-1295	78	25	𝑇1	𝑇1	NOUN
cana-1295	78	26	and	and	CCONJ
cana-1295	78	27	𝑇2	𝑇2	NOUN
cana-1295	78	28	such	such	ADJ
cana-1295	78	29	that	that	SCONJ
cana-1295	78	30	if	if	SCONJ
cana-1295	78	31	𝑏	𝑏	PROPN
cana-1295	78	32	,	,	PUNCT
cana-1295	78	33	𝑐	𝑐	PROPN
cana-1295	78	34	∈	∈	PROPN
cana-1295	78	35	𝑇1	𝑇1	NOUN
cana-1295	78	36	,	,	PUNCT
cana-1295	78	37	𝑎	𝑎	PROPN
cana-1295	78	38	∈	∈	ADJ
cana-1295	78	39	𝐴1	𝐴1	PROPN
cana-1295	78	40	implies	imply	VERB
cana-1295	78	41	𝑎𝑏𝑐	𝑎𝑏𝑐	PROPN
cana-1295	78	42	∈	∈	PROPN
cana-1295	78	43	𝐴1	𝐴1	PROPN
cana-1295	78	44	communications	communication	NOUN
cana-1295	78	45	on	on	ADP
cana-1295	78	46	applied	apply	VERB
cana-1295	78	47	nonlinear	nonlinear	ADJ
cana-1295	78	48	analysis	analysis	NOUN
cana-1295	78	49	issn	issn	NOUN
cana-1295	78	50	:	:	PUNCT
cana-1295	78	51	1074	1074	NUM
cana-1295	78	52	-	-	PUNCT
cana-1295	78	53	133x	133x	NUM
cana-1295	78	54	vol	vol	NOUN
cana-1295	78	55	31	31	NUM
cana-1295	78	56	no	no	NOUN
cana-1295	78	57	.	.	PUNCT
cana-1295	79	1	7s	7	NOUN
cana-1295	79	2	(	(	PUNCT
cana-1295	79	3	2024	2024	NUM
cana-1295	79	4	)	)	PUNCT
cana-1295	79	5	209	209	NUM
cana-1295	80	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1295	80	2	note	note	VERB
cana-1295	80	3	3.16	3.16	NUM
cana-1295	80	4	:	:	PUNCT
cana-1295	80	5	a	a	DET
cana-1295	80	6	non	non	ADJ
cana-1295	80	7	-	-	ADJ
cana-1295	80	8	empty	empty	ADJ
cana-1295	80	9	auxiliary	auxiliary	ADJ
cana-1295	80	10	set	set	NOUN
cana-1295	80	11	.	.	PUNCT
cana-1295	81	1	a	a	PRON
cana-1295	81	2	of	of	ADP
cana-1295	81	3	a	a	DET
cana-1295	81	4	bi	bi	ADJ
cana-1295	81	5	-	-	ADJ
cana-1295	81	6	ternary	ternary	ADJ
cana-1295	81	7	semi	semi	ADJ
cana-1295	81	8	-	-	NOUN
cana-1295	81	9	group	group	ADJ
cana-1295	81	10	t	t	PROPN
cana-1295	81	11	is	be	AUX
cana-1295	81	12	considered	consider	VERB
cana-1295	81	13	to	to	PART
cana-1295	81	14	be	be	AUX
cana-1295	81	15	correct	correct	ADJ
cana-1295	81	16	if	if	SCONJ
cana-1295	81	17	just	just	ADV
cana-1295	81	18	if	if	SCONJ
cana-1295	81	19	𝐴𝑇𝑇	𝐴𝑇𝑇	VERB
cana-1295	81	20	⊆	⊆	NUM
cana-1295	81	21	𝐴.	𝐴.	NOUN
cana-1295	81	22	example	example	NOUN
cana-1295	82	1	3.17:𝑇	3.17:𝑇	NUM
cana-1295	82	2	=	=	NOUN
cana-1295	82	3	𝑇1	𝑇1	PROPN
cana-1295	82	4	∪	∪	ADJ
cana-1295	82	5	𝑇2	𝑇2	PROPN
cana-1295	82	6	be	be	AUX
cana-1295	82	7	a	a	DET
cana-1295	82	8	bi	bi	ADJ
cana-1295	82	9	ternary	ternary	NOUN
cana-1295	82	10	semi	semi	NOUN
cana-1295	82	11	group	group	NOUN
cana-1295	82	12	where	where	SCONJ
cana-1295	82	13	𝑇1	𝑇1	NOUN
cana-1295	82	14	=	=	SYM
cana-1295	82	15	{	{	PUNCT
cana-1295	82	16	[	[	PUNCT
cana-1295	82	17	𝑎	𝑎	NOUN
cana-1295	82	18	0	0	PUNCT
cana-1295	82	19	𝑏	𝑏	PROPN
cana-1295	82	20	𝑐	𝑐	PROPN
cana-1295	82	21	]	]	PUNCT
cana-1295	82	22	,	,	PUNCT
cana-1295	82	23	𝑎	𝑎	X
cana-1295	82	24	,	,	PUNCT
cana-1295	82	25	𝑏	𝑏	NOUN
cana-1295	82	26	,	,	PUNCT
cana-1295	82	27	𝑐	𝑐	PROPN
cana-1295	82	28	∈	∈	PROPN
cana-1295	82	29	𝑍	𝑍	PROPN
cana-1295	82	30	}	}	PUNCT
cana-1295	82	31	,	,	PUNCT
cana-1295	82	32	𝑇2	𝑇2	NOUN
cana-1295	82	33	=	=	SYM
cana-1295	82	34	{	{	PUNCT
cana-1295	82	35	[	[	PUNCT
cana-1295	82	36	𝑥	𝑥	NOUN
cana-1295	82	37	𝑦	𝑦	NOUN
cana-1295	82	38	0	0	PUNCT
cana-1295	82	39	𝑧	𝑧	VERB
cana-1295	82	40	]	]	PUNCT
cana-1295	82	41	,	,	PUNCT
cana-1295	82	42	𝑥	𝑥	X
cana-1295	82	43	,	,	PUNCT
cana-1295	82	44	𝑦	𝑦	NOUN
cana-1295	82	45	,	,	PUNCT
cana-1295	82	46	𝑧	𝑧	DET
cana-1295	82	47	∈	∈	PROPN
cana-1295	82	48	𝑍	𝑍	NOUN
cana-1295	82	49	}	}	PUNCT
cana-1295	82	50	are	be	AUX
cana-1295	82	51	ternary	ternary	ADJ
cana-1295	82	52	semi	semi	ADJ
cana-1295	82	53	groups	group	NOUN
cana-1295	82	54	of	of	ADP
cana-1295	82	55	𝑇.	𝑇.	PROPN
cana-1295	82	56	𝐼	𝐼	PROPN
cana-1295	82	57	=	=	PUNCT
cana-1295	82	58	𝐼1	𝐼1	NOUN
cana-1295	82	59	∪	∪	X
cana-1295	82	60	𝐼2	𝐼2	NOUN
cana-1295	82	61	where	where	SCONJ
cana-1295	82	62	𝐼1	𝐼1	NOUN
cana-1295	82	63	=	=	SYM
cana-1295	82	64	{	{	PUNCT
cana-1295	82	65	[	[	PUNCT
cana-1295	82	66	𝑎	𝑎	PROPN
cana-1295	82	67	𝑏	𝑏	NOUN
cana-1295	82	68	0	0	NUM
cana-1295	82	69	0	0	NUM
cana-1295	82	70	]	]	PUNCT
cana-1295	82	71	,	,	PUNCT
cana-1295	82	72	𝑎	𝑎	X
cana-1295	82	73	,	,	PUNCT
cana-1295	82	74	𝑏	𝑏	PROPN
cana-1295	82	75	∈	∈	PROPN
cana-1295	82	76	𝑍	𝑍	PROPN
cana-1295	82	77	}	}	PUNCT
cana-1295	82	78	,	,	PUNCT
cana-1295	82	79	𝐼2	𝐼2	NOUN
cana-1295	82	80	=	=	SYM
cana-1295	82	81	{	{	PUNCT
cana-1295	82	82	[	[	PUNCT
cana-1295	82	83	0	0	NUM
cana-1295	83	1	𝑎	𝑎	X
cana-1295	83	2	0	0	NUM
cana-1295	83	3	0	0	NUM
cana-1295	83	4	]	]	PUNCT
cana-1295	83	5	,	,	PUNCT
cana-1295	83	6	𝑎	𝑎	PROPN
cana-1295	83	7	∈	∈	PROPN
cana-1295	83	8	𝑍	𝑍	NOUN
cana-1295	83	9	}	}	PUNCT
cana-1295	83	10	are	be	AUX
cana-1295	83	11	right	right	ADJ
cana-1295	83	12	ideals	ideal	NOUN
cana-1295	83	13	of	of	ADP
cana-1295	83	14	𝑇1	𝑇1	NOUN
cana-1295	83	15	&	&	CCONJ
cana-1295	83	16	𝑇2respectively	𝑇2respectively	ADV
cana-1295	83	17	.	.	PUNCT
cana-1295	84	1	thus	thus	ADV
cana-1295	84	2	𝐼is	𝐼is	VERB
cana-1295	84	3	the	the	DET
cana-1295	84	4	right	right	ADJ
cana-1295	84	5	ideal	ideal	NOUN
cana-1295	84	6	of	of	ADP
cana-1295	84	7	bi	bi	ADJ
cana-1295	84	8	ternary	ternary	PROPN
cana-1295	84	9	semi	semi	PROPN
cana-1295	84	10	group	group	PROPN
cana-1295	84	11	𝑇.	𝑇.	PROPN
cana-1295	84	12	theorem	theorem	VERB
cana-1295	84	13	3.18	3.18	NUM
cana-1295	84	14	:	:	PUNCT
cana-1295	84	15	t	t	PROPN
cana-1295	84	16	's	's	PART
cana-1295	84	17	right	right	ADJ
cana-1295	84	18	ideal	ideal	NOUN
cana-1295	84	19	is	be	AUX
cana-1295	84	20	the	the	DET
cana-1295	84	21	non	non	ADJ
cana-1295	84	22	-	-	ADJ
cana-1295	84	23	empty	empty	ADJ
cana-1295	84	24	confluence	confluence	NOUN
cana-1295	84	25	of	of	ADP
cana-1295	84	26	multiple	multiple	ADJ
cana-1295	84	27	right	right	ADJ
cana-1295	84	28	conceptions	conception	NOUN
cana-1295	84	29	of	of	ADP
cana-1295	84	30	a	a	DET
cana-1295	84	31	bi	bi	ADJ
cana-1295	84	32	-	-	ADJ
cana-1295	84	33	ternary	ternary	ADJ
cana-1295	84	34	semi	semi	ADJ
cana-1295	84	35	group	group	NOUN
cana-1295	84	36	.	.	PUNCT
cana-1295	85	1	proof	proof	NOUN
cana-1295	85	2	:	:	PUNCT
cana-1295	85	3	agree	agree	VERB
cana-1295	85	4	to	to	PART
cana-1295	85	5	𝐼1	𝐼1	VERB
cana-1295	85	6	,	,	PUNCT
cana-1295	85	7	𝐼2	𝐼2	NOUN
cana-1295	85	8	be	be	AUX
cana-1295	85	9	two	two	NUM
cana-1295	85	10	ideals	ideal	NOUN
cana-1295	85	11	of	of	ADP
cana-1295	85	12	bi	bi	ADJ
cana-1295	85	13	-	-	ADJ
cana-1295	85	14	ternary	ternary	ADJ
cana-1295	85	15	semi	semi	ADJ
cana-1295	85	16	group	group	PROPN
cana-1295	85	17	𝑇.	𝑇.	PROPN
cana-1295	85	18	let	let	VERB
cana-1295	85	19	𝑎	𝑎	NOUN
cana-1295	85	20	,	,	PUNCT
cana-1295	85	21	𝑏	𝑏	PROPN
cana-1295	85	22	∈	∈	PROPN
cana-1295	85	23	𝐼1	𝐼1	NOUN
cana-1295	85	24	∩	∩	ADJ
cana-1295	85	25	𝐼2&𝛼	𝐼2&𝛼	NOUN
cana-1295	85	26	,	,	PUNCT
cana-1295	85	27	𝛽	𝛽	PROPN
cana-1295	85	28	∈	∈	PROPN
cana-1295	85	29	𝑇	𝑇	PROPN
cana-1295	85	30	also	also	ADV
cana-1295	85	31	𝛼	𝛼	VERB
cana-1295	85	32	,	,	PUNCT
cana-1295	85	33	𝛽	𝛽	PROPN
cana-1295	85	34	∈	∈	PROPN
cana-1295	85	35	𝑇	𝑇	PROPN
cana-1295	85	36	,	,	PUNCT
cana-1295	85	37	𝑎	𝑎	PROPN
cana-1295	85	38	∈	∈	NOUN
cana-1295	85	39	𝐼1	𝐼1	NOUN
cana-1295	85	40	⟹	⟹	PROPN
cana-1295	85	41	𝛼𝛽𝑎	𝛼𝛽𝑎	PROPN
cana-1295	85	42	∈	∈	PROPN
cana-1295	85	43	𝐼1	𝐼1	NOUN
cana-1295	85	44	𝛼	𝛼	NOUN
cana-1295	85	45	,	,	PUNCT
cana-1295	85	46	𝛽	𝛽	PROPN
cana-1295	85	47	∈	∈	PROPN
cana-1295	85	48	𝑇	𝑇	PROPN
cana-1295	85	49	,	,	PUNCT
cana-1295	85	50	𝑎	𝑎	PROPN
cana-1295	85	51	∈	∈	NOUN
cana-1295	85	52	𝐼2	𝐼2	NOUN
cana-1295	85	53	⟹	⟹	PROPN
cana-1295	85	54	𝛼𝛽𝑎	𝛼𝛽𝑎	PROPN
cana-1295	85	55	∈	∈	PROPN
cana-1295	85	56	𝐼2	𝐼2	NOUN
cana-1295	85	57	𝛼𝛽𝑎	𝛼𝛽𝑎	PROPN
cana-1295	85	58	∈	∈	PROPN
cana-1295	85	59	𝐼1	𝐼1	PROPN
cana-1295	85	60	,	,	PUNCT
cana-1295	85	61	𝛼𝛽𝑎	𝛼𝛽𝑎	PROPN
cana-1295	85	62	∈	∈	PROPN
cana-1295	85	63	𝐼2	𝐼2	NOUN
cana-1295	85	64	⟹	⟹	PROPN
cana-1295	85	65	𝛼𝛽𝑎	𝛼𝛽𝑎	PROPN
cana-1295	85	66	∈	∈	PROPN
cana-1295	85	67	𝐼1	𝐼1	NOUN
cana-1295	85	68	∩	∩	NOUN
cana-1295	85	69	𝐼2	𝐼2	NOUN
cana-1295	85	70	hence	hence	ADV
cana-1295	85	71	𝐼1	𝐼1	NOUN
cana-1295	85	72	∩	∩	ADJ
cana-1295	85	73	𝐼2	𝐼2	NOUN
cana-1295	85	74	is	be	AUX
cana-1295	85	75	the	the	DET
cana-1295	85	76	right	right	ADJ
cana-1295	85	77	ideal	ideal	NOUN
cana-1295	85	78	of	of	ADP
cana-1295	85	79	bi	bi	ADJ
cana-1295	85	80	ternary	ternary	NOUN
cana-1295	85	81	semi	semi	PROPN
cana-1295	85	82	group𝑇.	group𝑇.	PROPN
cana-1295	85	83	theorem	theorem	VERB
cana-1295	85	84	3.19	3.19	NUM
cana-1295	85	85	:	:	PUNCT
cana-1295	85	86	the	the	DET
cana-1295	85	87	non	non	ADJ
cana-1295	85	88	-	-	ADJ
cana-1295	85	89	empty	empty	ADJ
cana-1295	85	90	intersections	intersection	NOUN
cana-1295	85	91	of	of	ADP
cana-1295	85	92	any	any	DET
cana-1295	85	93	group	group	NOUN
cana-1295	85	94	of	of	ADP
cana-1295	85	95	right	right	ADJ
cana-1295	85	96	conceptions	conception	NOUN
cana-1295	85	97	be	be	VERB
cana-1295	85	98	bi	bi	ADJ
cana-1295	85	99	-	-	ADJ
cana-1295	85	100	ternary	ternary	ADJ
cana-1295	85	101	semi	semi	ADJ
cana-1295	85	102	grouping	group	VERB
cana-1295	85	103	t	t	PROPN
cana-1295	85	104	is	be	AUX
cana-1295	85	105	a	a	DET
cana-1295	85	106	right	right	ADJ
cana-1295	85	107	ideal	ideal	NOUN
cana-1295	85	108	of	of	ADP
cana-1295	85	109	t	t	PROPN
cana-1295	85	110	..	..	PUNCT
cana-1295	85	111	theorem	theorem	VERB
cana-1295	85	112	3.20	3.20	NUM
cana-1295	85	113	:	:	PUNCT
cana-1295	85	114	the	the	DET
cana-1295	85	115	amalgamation	amalgamation	NOUN
cana-1295	85	116	of	of	ADP
cana-1295	85	117	neither	neither	PRON
cana-1295	85	118	of	of	ADP
cana-1295	85	119	the	the	DET
cana-1295	85	120	right	right	ADJ
cana-1295	85	121	ideals	ideal	NOUN
cana-1295	85	122	be	be	VERB
cana-1295	85	123	bi	bi	ADJ
cana-1295	85	124	-	-	NOUN
cana-1295	85	125	digit	digit	NOUN
cana-1295	85	126	semi	semi	ADJ
cana-1295	85	127	grouping	group	VERB
cana-1295	85	128	t	t	PROPN
cana-1295	85	129	were	be	AUX
cana-1295	85	130	right	right	ADJ
cana-1295	85	131	picture	picture	NOUN
cana-1295	85	132	of	of	ADP
cana-1295	85	133	t.	t.	NOUN
cana-1295	85	134	proof	proof	NOUN
cana-1295	85	135	:	:	PUNCT
cana-1295	85	136	agree	agree	VERB
cana-1295	85	137	𝐼1	𝐼1	NOUN
cana-1295	85	138	,	,	PUNCT
cana-1295	85	139	𝐼2	𝐼2	NOUN
cana-1295	85	140	be	be	AUX
cana-1295	85	141	multiple	multiple	ADJ
cana-1295	85	142	(	(	PUNCT
cana-1295	85	143	two	two	NUM
cana-1295	85	144	)	)	PUNCT
cana-1295	85	145	right	right	ADJ
cana-1295	85	146	ideals	ideal	NOUN
cana-1295	85	147	of	of	ADP
cana-1295	85	148	bi	bi	ADJ
cana-1295	85	149	ternary	ternary	NOUN
cana-1295	85	150	semi	semi	NOUN
cana-1295	86	1	group𝑇.	group𝑇.	PROPN
cana-1295	86	2	let	let	VERB
cana-1295	86	3	𝑎	𝑎	PRON
cana-1295	86	4	∈	∈	NOUN
cana-1295	86	5	𝐼1	𝐼1	NOUN
cana-1295	86	6	∪	∪	ADJ
cana-1295	86	7	𝐼2	𝐼2	NOUN
cana-1295	86	8	⟹	⟹	NUM
cana-1295	86	9	𝑎	𝑎	PROPN
cana-1295	86	10	∈	∈	NOUN
cana-1295	86	11	𝐼1	𝐼1	NOUN
cana-1295	86	12	or	or	CCONJ
cana-1295	86	13	𝑎	𝑎	PRON
cana-1295	86	14	∈	∈	NOUN
cana-1295	86	15	𝐼2	𝐼2	NOUN
cana-1295	86	16	or	or	CCONJ
cana-1295	86	17	both	both	PRON
cana-1295	86	18	and	and	CCONJ
cana-1295	86	19	𝛼	𝛼	X
cana-1295	86	20	,	,	PUNCT
cana-1295	86	21	𝛽	𝛽	PROPN
cana-1295	86	22	∈	∈	PROPN
cana-1295	86	23	𝑇	𝑇	PROPN
cana-1295	86	24	also𝛼	also𝛼	NOUN
cana-1295	86	25	,	,	PUNCT
cana-1295	86	26	𝛽	𝛽	PROPN
cana-1295	86	27	∈	∈	PROPN
cana-1295	86	28	𝑇	𝑇	PROPN
cana-1295	86	29	,	,	PUNCT
cana-1295	86	30	𝑎	𝑎	PROPN
cana-1295	86	31	∈	∈	NOUN
cana-1295	86	32	𝐼1	𝐼1	NOUN
cana-1295	86	33	⟹	⟹	PROPN
cana-1295	86	34	𝛼𝛽𝑎	𝛼𝛽𝑎	PROPN
cana-1295	86	35	∈	∈	PROPN
cana-1295	86	36	𝐼1	𝐼1	NOUN
cana-1295	86	37	𝛼	𝛼	NOUN
cana-1295	86	38	,	,	PUNCT
cana-1295	86	39	𝛽	𝛽	PROPN
cana-1295	86	40	∈	∈	PROPN
cana-1295	86	41	𝑇	𝑇	PROPN
cana-1295	86	42	,	,	PUNCT
cana-1295	86	43	𝑎	𝑎	PROPN
cana-1295	86	44	∈	∈	NOUN
cana-1295	86	45	𝐼2	𝐼2	NOUN
cana-1295	86	46	⟹	⟹	PROPN
cana-1295	86	47	𝛼𝛽𝑎	𝛼𝛽𝑎	PROPN
cana-1295	86	48	∈	∈	PROPN
cana-1295	86	49	𝐼2	𝐼2	NOUN
cana-1295	86	50	𝛼𝛽𝑎	𝛼𝛽𝑎	PROPN
cana-1295	86	51	∈	∈	PROPN
cana-1295	86	52	𝐼1	𝐼1	PROPN
cana-1295	86	53	,	,	PUNCT
cana-1295	86	54	𝛼𝛽𝑎	𝛼𝛽𝑎	PROPN
cana-1295	86	55	∈	∈	PROPN
cana-1295	86	56	𝐼2	𝐼2	NOUN
cana-1295	86	57	⟹	⟹	PROPN
cana-1295	86	58	𝛼𝛽𝑎	𝛼𝛽𝑎	PROPN
cana-1295	86	59	∈	∈	PROPN
cana-1295	86	60	𝐼1	𝐼1	NOUN
cana-1295	86	61	∪	∪	X
cana-1295	86	62	𝐼2	𝐼2	NOUN
cana-1295	86	63	𝛼	𝛼	NOUN
cana-1295	86	64	,	,	PUNCT
cana-1295	86	65	𝛽	𝛽	PROPN
cana-1295	86	66	∈	∈	PROPN
cana-1295	86	67	𝑇	𝑇	PROPN
cana-1295	86	68	,	,	PUNCT
cana-1295	86	69	𝑎	𝑎	PROPN
cana-1295	86	70	∈	∈	NOUN
cana-1295	86	71	𝐼1	𝐼1	NOUN
cana-1295	86	72	∪	∪	ADJ
cana-1295	86	73	𝐼2	𝐼2	NOUN
cana-1295	86	74	,	,	PUNCT
cana-1295	86	75	𝛼𝛽𝑎	𝛼𝛽𝑎	PROPN
cana-1295	86	76	∈	∈	PROPN
cana-1295	86	77	𝐼1	𝐼1	NOUN
cana-1295	86	78	∪	∪	X
cana-1295	86	79	𝐼2	𝐼2	NOUN
cana-1295	86	80	then𝐼1	then𝐼1	NOUN
cana-1295	86	81	∪	∪	ADP
cana-1295	86	82	𝐼2	𝐼2	NOUN
cana-1295	86	83	is	be	AUX
cana-1295	86	84	rightideal	rightideal	NOUN
cana-1295	86	85	of𝑇.	of𝑇.	NOUN
cana-1295	86	86	definition	definition	NOUN
cana-1295	86	87	3.21	3.21	NUM
cana-1295	86	88	:	:	PUNCT
cana-1295	86	89	a	a	DET
cana-1295	86	90	nonblank	nonblank	NOUN
cana-1295	86	91	set	set	VERB
cana-1295	86	92	𝐴	𝐴	PROPN
cana-1295	86	93	be	be	VERB
cana-1295	86	94	bi	bi	ADJ
cana-1295	86	95	ternary	ternary	NOUN
cana-1295	86	96	semi	semi	ADV
cana-1295	86	97	group𝑇	group𝑇	PROPN
cana-1295	86	98	is	be	AUX
cana-1295	86	99	said	say	VERB
cana-1295	86	100	to	to	PART
cana-1295	86	101	be	be	AUX
cana-1295	86	102	ternary	ternary	ADJ
cana-1295	86	103	ideal	ideal	NOUN
cana-1295	86	104	or	or	CCONJ
cana-1295	86	105	just	just	ADV
cana-1295	86	106	an	an	DET
cana-1295	86	107	perfect	perfect	NOUN
cana-1295	86	108	of	of	ADP
cana-1295	86	109	𝑇	𝑇	PROPN
cana-1295	86	110	if	if	SCONJ
cana-1295	86	111	𝑏	𝑏	PROPN
cana-1295	86	112	,	,	PUNCT
cana-1295	86	113	𝑐	𝑐	PROPN
cana-1295	86	114	∈	∈	PROPN
cana-1295	86	115	𝑇	𝑇	PROPN
cana-1295	86	116	,	,	PUNCT
cana-1295	86	117	𝑎	𝑎	PROPN
cana-1295	86	118	∈	∈	PROPN
cana-1295	86	119	𝐴then	𝐴then	PROPN
cana-1295	86	120	,	,	PUNCT
cana-1295	86	121	𝑏𝑐𝑎	𝑏𝑐𝑎	PROPN
cana-1295	86	122	∈	∈	PROPN
cana-1295	86	123	𝐴	𝐴	PROPN
cana-1295	86	124	,	,	PUNCT
cana-1295	86	125	𝑏𝑎𝑐	𝑏𝑎𝑐	VERB
cana-1295	86	126	∈	∈	PROPN
cana-1295	86	127	𝐴	𝐴	PROPN
cana-1295	86	128	,	,	PUNCT
cana-1295	86	129	𝑎𝑏𝑐	𝑎𝑏𝑐	PROPN
cana-1295	86	130	∈	∈	PROPN
cana-1295	86	131	𝐴.	𝐴.	PROPN
cana-1295	86	132	definition	definition	NOUN
cana-1295	86	133	3.22	3.22	NUM
cana-1295	86	134	:	:	PUNCT
cana-1295	86	135	an	an	DET
cana-1295	86	136	ideal	ideal	ADJ
cana-1295	86	137	𝐴	𝐴	NOUN
cana-1295	86	138	=	=	PUNCT
cana-1295	86	139	𝐴1	𝐴1	PROPN
cana-1295	86	140	∪	∪	VERB
cana-1295	86	141	𝐴2	𝐴2	PROPN
cana-1295	86	142	be	be	VERB
cana-1295	86	143	a	a	DET
cana-1295	86	144	bi	bi	ADJ
cana-1295	86	145	-	-	ADJ
cana-1295	86	146	ternary	ternary	ADJ
cana-1295	86	147	semi	semi	ADJ
cana-1295	86	148	group	group	NOUN
cana-1295	86	149	𝑇	𝑇	PROPN
cana-1295	86	150	be	be	AUX
cana-1295	86	151	maximal	maximal	ADV
cana-1295	86	152	perfect	perfect	ADJ
cana-1295	86	153	.	.	PUNCT
cana-1295	87	1	both	both	DET
cana-1295	87	2	𝐴1&𝐴2	𝐴1&𝐴2	NOUN
cana-1295	87	3	are	be	AUX
cana-1295	87	4	maximal	maximal	ADJ
cana-1295	87	5	ideals	ideal	NOUN
cana-1295	87	6	of	of	ADP
cana-1295	87	7	𝑇1&𝑇2	𝑇1&𝑇2	PUNCT
cana-1295	87	8	provided	provide	VERB
cana-1295	87	9	that	that	SCONJ
cana-1295	87	10	𝐴1&𝐴2	𝐴1&𝐴2	NOUN
cana-1295	87	11	any	any	DET
cana-1295	87	12	good	good	ADJ
cana-1295	87	13	ideals	ideal	NOUN
cana-1295	87	14	of	of	ADP
cana-1295	87	15	𝑇1&𝑇2	𝑇1&𝑇2	PRON
cana-1295	87	16	.	.	PUNCT
cana-1295	88	1	hence	hence	ADV
cana-1295	88	2	𝐴	𝐴	PROPN
cana-1295	88	3	is	be	AUX
cana-1295	88	4	maximal	maximal	ADJ
cana-1295	88	5	ideal	ideal	NOUN
cana-1295	88	6	of	of	ADP
cana-1295	88	7	bi	bi	ADJ
cana-1295	88	8	ternary	ternary	NOUN
cana-1295	88	9	semi	semi	NOUN
cana-1295	88	10	group𝑇.	group𝑇.	PROPN
cana-1295	88	11	communications	communication	NOUN
cana-1295	88	12	on	on	ADP
cana-1295	88	13	applied	apply	VERB
cana-1295	88	14	nonlinear	nonlinear	ADJ
cana-1295	88	15	analysis	analysis	NOUN
cana-1295	88	16	issn	issn	NOUN
cana-1295	88	17	:	:	PUNCT
cana-1295	88	18	1074	1074	NUM
cana-1295	88	19	-	-	PUNCT
cana-1295	88	20	133x	133x	NUM
cana-1295	88	21	vol	vol	NOUN
cana-1295	88	22	31	31	NUM
cana-1295	88	23	no	no	NOUN
cana-1295	88	24	.	.	PUNCT
cana-1295	89	1	7s	7	NOUN
cana-1295	89	2	(	(	PUNCT
cana-1295	89	3	2024	2024	NUM
cana-1295	89	4	)	)	PUNCT
cana-1295	89	5	210	210	NUM
cana-1295	89	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1295	89	7	definition3.23	definition3.23	PROPN
cana-1295	89	8	:	:	PUNCT
cana-1295	89	9	an	an	DET
cana-1295	89	10	ideal	ideal	ADJ
cana-1295	89	11	𝐴	𝐴	PROPN
cana-1295	89	12	be	be	VERB
cana-1295	89	13	bi	bi	ADJ
cana-1295	89	14	-	-	ADJ
cana-1295	89	15	ternary	ternary	ADJ
cana-1295	89	16	semi	semi	ADJ
cana-1295	89	17	group	group	PROPN
cana-1295	89	18	𝑇	𝑇	PROPN
cana-1295	89	19	,	,	PUNCT
cana-1295	89	20	the	the	DET
cana-1295	89	21	principal	principal	ADJ
cana-1295	89	22	bi	bi	ADJ
cana-1295	89	23	ternary	ternary	PROPN
cana-1295	89	24	ideal	ideal	NOUN
cana-1295	89	25	generated	generate	VERB
cana-1295	89	26	by	by	ADP
cana-1295	89	27	a	a	PRON
cana-1295	89	28	if	if	SCONJ
cana-1295	89	29	𝐴1&𝐴2	𝐴1&𝐴2	NOUN
cana-1295	89	30	are	be	AUX
cana-1295	89	31	ideals	ideal	NOUN
cana-1295	89	32	of	of	ADP
cana-1295	89	33	𝑇1&𝑇2	𝑇1&𝑇2	NOUN
cana-1295	89	34	generated	generate	VERB
cana-1295	89	35	by	by	ADP
cana-1295	89	36	{	{	PUNCT
cana-1295	89	37	𝑎	𝑎	NOUN
cana-1295	89	38	}	}	PUNCT
cana-1295	89	39	for	for	ADP
cana-1295	89	40	some	some	DET
cana-1295	89	41	𝑎	𝑎	PROPN
cana-1295	89	42	∈	∈	PROPN
cana-1295	89	43	𝑇.	𝑇.	PROPN
cana-1295	89	44	is	be	AUX
cana-1295	89	45	denoted	denote	VERB
cana-1295	89	46	by	by	ADP
cana-1295	89	47	〈	〈	NOUN
cana-1295	89	48	𝑎	𝑎	NOUN
cana-1295	89	49	〉	〉	NOUN
cana-1295	89	50	.	.	PUNCT
cana-1295	90	1	definition	definition	NOUN
cana-1295	90	2	3.24	3.24	NUM
cana-1295	90	3	:	:	PUNCT
cana-1295	90	4	an	an	DET
cana-1295	90	5	ideal	ideal	ADJ
cana-1295	90	6	𝐴	𝐴	PROPN
cana-1295	90	7	of	of	ADP
cana-1295	90	8	a	a	DET
cana-1295	90	9	bi	bi	ADJ
cana-1295	90	10	ternary	ternary	NOUN
cana-1295	90	11	semi	semi	ADV
cana-1295	90	12	group𝑇	group𝑇	PROPN
cana-1295	90	13	is	be	AUX
cana-1295	90	14	said	say	VERB
cana-1295	90	15	to	to	PART
cana-1295	90	16	be	be	AUX
cana-1295	90	17	globally	globally	ADV
cana-1295	90	18	idempotent.if	idempotent.if	X
cana-1295	90	19	𝑨𝟑	𝑨𝟑	NOUN
cana-1295	90	20	=	=	SYM
cana-1295	90	21	𝑨.	𝑨.	ADJ
cana-1295	90	22	definition	definition	NOUN
cana-1295	90	23	3.25	3.25	NUM
cana-1295	90	24	:	:	PUNCT
cana-1295	90	25	a	a	DET
cana-1295	90	26	bi	bi	ADJ
cana-1295	90	27	ternary	ternary	NOUN
cana-1295	90	28	semi	semi	ADV
cana-1295	90	29	group𝑇	group𝑇	PROPN
cana-1295	90	30	is	be	AUX
cana-1295	90	31	said	say	VERB
cana-1295	90	32	to	to	PART
cana-1295	90	33	be	be	AUX
cana-1295	90	34	globally	globally	ADV
cana-1295	90	35	idempotent.𝑻𝟑	idempotent.𝑻𝟑	X
cana-1295	90	36	=	=	SYM
cana-1295	90	37	𝑻.	𝑻.	PROPN
cana-1295	90	38	theorem	theorem	VERB
cana-1295	90	39	3.26	3.26	NUM
cana-1295	90	40	:	:	PUNCT
cana-1295	90	41	if	if	SCONJ
cana-1295	90	42	𝑇	𝑇	PROPN
cana-1295	90	43	is	be	AUX
cana-1295	90	44	a	a	DET
cana-1295	90	45	bi	bi	ADJ
cana-1295	90	46	ternary	ternary	NOUN
cana-1295	90	47	semi	semi	ADJ
cana-1295	90	48	group	group	NOUN
cana-1295	90	49	with	with	ADP
cana-1295	90	50	unity	unity	NOUN
cana-1295	90	51	1	1	NUM
cana-1295	90	52	then	then	ADV
cana-1295	90	53	the	the	DET
cana-1295	90	54	union	union	NOUN
cana-1295	90	55	of	of	ADP
cana-1295	90	56	all	all	DET
cana-1295	90	57	proper	proper	ADJ
cana-1295	90	58	ideals	ideal	NOUN
cana-1295	90	59	of	of	ADP
cana-1295	90	60	𝑇	𝑇	PROPN
cana-1295	90	61	is	be	AUX
cana-1295	90	62	the	the	DET
cana-1295	90	63	unique	unique	ADJ
cana-1295	90	64	maximal	maximal	ADJ
cana-1295	90	65	ideal	ideal	NOUN
cana-1295	90	66	of	of	ADP
cana-1295	90	67	𝑇.	𝑇.	PROPN
cana-1295	90	68	proof	proof	NOUN
cana-1295	90	69	:	:	PUNCT
cana-1295	90	70	let	let	VERB
cana-1295	90	71	𝑀	𝑀	PROPN
cana-1295	90	72	be	be	AUX
cana-1295	90	73	the	the	DET
cana-1295	90	74	union	union	NOUN
cana-1295	90	75	of	of	ADP
cana-1295	90	76	all	all	DET
cana-1295	90	77	proper	proper	ADJ
cana-1295	90	78	ideals	ideal	NOUN
cana-1295	90	79	of	of	ADP
cana-1295	90	80	𝑇.	𝑇.	PROPN
cana-1295	90	81	since	since	SCONJ
cana-1295	90	82	1	1	NUM
cana-1295	90	83	is	be	AUX
cana-1295	90	84	not	not	PART
cana-1295	90	85	an	an	DET
cana-1295	90	86	element	element	NOUN
cana-1295	90	87	of	of	ADP
cana-1295	90	88	any	any	DET
cana-1295	90	89	proper	proper	ADJ
cana-1295	90	90	ideal	ideal	NOUN
cana-1295	90	91	of	of	ADP
cana-1295	90	92	𝑇.	𝑇.	PROPN
cana-1295	90	93	1	1	NUM
cana-1295	90	94	∉	∉	PROPN
cana-1295	90	95	𝑀.	𝑀.	PROPN
cana-1295	90	96	therefore	therefore	ADV
cana-1295	90	97	𝑀	𝑀	PROPN
cana-1295	90	98	is	be	AUX
cana-1295	90	99	aproper	aproper	NOUN
cana-1295	90	100	sub	sub	NOUN
cana-1295	90	101	set	set	NOUN
cana-1295	90	102	of	of	ADP
cana-1295	90	103	𝑇.	𝑇.	PROPN
cana-1295	90	104	by	by	ADP
cana-1295	90	105	theorem	theorem	ADJ
cana-1295	90	106	union	union	NOUN
cana-1295	90	107	of	of	ADP
cana-1295	90	108	all	all	DET
cana-1295	90	109	ideals	ideal	NOUN
cana-1295	90	110	of	of	ADP
cana-1295	90	111	𝑇	𝑇	PROPN
cana-1295	90	112	is	be	AUX
cana-1295	90	113	an	an	DET
cana-1295	90	114	ideal	ideal	NOUN
cana-1295	90	115	of	of	ADP
cana-1295	90	116	𝑇.𝑀	𝑇.𝑀	PROPN
cana-1295	90	117	is	be	AUX
cana-1295	90	118	an	an	DET
cana-1295	90	119	ideal	ideal	NOUN
cana-1295	90	120	of	of	ADP
cana-1295	90	121	𝑇.	𝑇.	PROPN
cana-1295	90	122	thus	thus	ADV
cana-1295	90	123	𝑀	𝑀	PROPN
cana-1295	90	124	is	be	AUX
cana-1295	90	125	a	a	DET
cana-1295	90	126	proper	proper	ADJ
cana-1295	90	127	ideal	ideal	NOUN
cana-1295	90	128	of	of	ADP
cana-1295	90	129	𝑇.	𝑇.	PROPN
cana-1295	90	130	since	since	SCONJ
cana-1295	90	131	𝑀	𝑀	PROPN
cana-1295	90	132	contains	contain	VERB
cana-1295	90	133	all	all	DET
cana-1295	90	134	proper	proper	ADJ
cana-1295	90	135	ideals	ideal	NOUN
cana-1295	90	136	of	of	ADP
cana-1295	90	137	,	,	PUNCT
cana-1295	90	138	𝑀	𝑀	PROPN
cana-1295	90	139	is	be	AUX
cana-1295	90	140	a	a	DET
cana-1295	90	141	maximal	maximal	ADJ
cana-1295	90	142	ideal	ideal	NOUN
cana-1295	90	143	of	of	ADP
cana-1295	90	144	𝑇.	𝑇.	PROPN
cana-1295	90	145	if	if	SCONJ
cana-1295	90	146	𝑊	𝑊	PROPN
cana-1295	90	147	is	be	AUX
cana-1295	90	148	any	any	DET
cana-1295	90	149	maximal	maximal	ADJ
cana-1295	90	150	ideal	ideal	NOUN
cana-1295	90	151	of	of	ADP
cana-1295	90	152	,	,	PUNCT
cana-1295	90	153	then	then	ADV
cana-1295	91	1	𝑊	𝑊	PROPN
cana-1295	91	2	⊆	⊆	NUM
cana-1295	91	3	𝑀	𝑀	PROPN
cana-1295	91	4	⊆	⊆	NUM
cana-1295	91	5	𝑇	𝑇	PROPN
cana-1295	91	6	and	and	CCONJ
cana-1295	91	7	hence	hence	ADV
cana-1295	91	8	𝑊	𝑊	PROPN
cana-1295	91	9	=	=	NOUN
cana-1295	91	10	𝑀.	𝑀.	PROPN
cana-1295	91	11	therefore	therefore	ADV
cana-1295	91	12	𝑀is	𝑀is	PROPN
cana-1295	91	13	the	the	DET
cana-1295	91	14	unique	unique	ADJ
cana-1295	91	15	maximal	maximal	ADJ
cana-1295	91	16	ideal	ideal	NOUN
cana-1295	91	17	of	of	ADP
cana-1295	91	18	𝑇.	𝑇.	PROPN
cana-1295	91	19	definition	definition	NOUN
cana-1295	91	20	3.27	3.27	NUM
cana-1295	91	21	:	:	PUNCT
cana-1295	91	22	let	let	VERB
cana-1295	91	23	𝑇	𝑇	PROPN
cana-1295	91	24	be	be	AUX
cana-1295	91	25	a	a	DET
cana-1295	91	26	bi	bi	ADJ
cana-1295	91	27	ternary	ternary	NOUN
cana-1295	91	28	semi	semi	NOUN
cana-1295	91	29	group	group	NOUN
cana-1295	91	30	and	and	CCONJ
cana-1295	91	31	𝐴	𝐴	PROPN
cana-1295	91	32	be	be	VERB
cana-1295	91	33	non	non	X
cana-1295	91	34	empty	empty	ADJ
cana-1295	91	35	subset	subset	NOUN
cana-1295	91	36	of	of	ADP
cana-1295	91	37	𝑇.	𝑇.	PROPN
cana-1295	91	38	the	the	DET
cana-1295	91	39	smallest	small	ADJ
cana-1295	91	40	left	leave	VERB
cana-1295	91	41	ideal	ideal	NOUN
cana-1295	91	42	of	of	ADP
cana-1295	91	43	𝑇containing	𝑇containing	PROPN
cana-1295	91	44	𝐴	𝐴	PROPN
cana-1295	91	45	is	be	AUX
cana-1295	91	46	call	call	NOUN
cana-1295	91	47	left	leave	VERB
cana-1295	91	48	perfect	perfect	ADJ
cana-1295	91	49	of	of	ADP
cana-1295	91	50	𝑇generated	𝑇generate	VERB
cana-1295	91	51	by	by	ADP
cana-1295	91	52	𝑨.	𝑨.	PROPN
cana-1295	91	53	theorem	theorem	VERB
cana-1295	91	54	3.28	3.28	NUM
cana-1295	91	55	:	:	PUNCT
cana-1295	91	56	the	the	DET
cana-1295	91	57	intersection	intersection	NOUN
cana-1295	91	58	of	of	ADP
cana-1295	91	59	all	all	DET
cana-1295	91	60	left	leave	VERB
cana-1295	91	61	ideals	ideal	NOUN
cana-1295	91	62	of	of	ADP
cana-1295	91	63	t	t	PROPN
cana-1295	91	64	that	that	PRON
cana-1295	91	65	include	include	VERB
cana-1295	91	66	a	a	PRON
cana-1295	91	67	is	be	AUX
cana-1295	91	68	the	the	DET
cana-1295	91	69	left	left	ADJ
cana-1295	91	70	perfect	perfect	NOUN
cana-1295	91	71	of	of	ADP
cana-1295	91	72	a	a	DET
cana-1295	91	73	bi	bi	NOUN
cana-1295	91	74	modal	modal	NOUN
cana-1295	91	75	semigroup	semigroup	PROPN
cana-1295	91	76	t	t	PROPN
cana-1295	91	77	created	create	VERB
cana-1295	91	78	by	by	ADP
cana-1295	91	79	a	a	DET
cana-1295	91	80	non	non	ADJ
cana-1295	91	81	-	-	ADJ
cana-1295	91	82	empty	empty	ADJ
cana-1295	91	83	auxiliary	auxiliary	NOUN
cana-1295	91	84	set	set	VERB
cana-1295	91	85	a.	a.	NOUN
cana-1295	91	86	proof	proof	NOUN
cana-1295	91	87	:	:	PUNCT
cana-1295	91	88	agree	agree	VERB
cana-1295	91	89	of	of	ADP
cana-1295	91	90	∆	∆	PROPN
cana-1295	91	91	a	a	DET
cana-1295	91	92	put	put	NOUN
cana-1295	91	93	the	the	DET
cana-1295	91	94	left	left	ADJ
cana-1295	91	95	ideals	ideal	NOUN
cana-1295	91	96	of	of	ADP
cana-1295	91	97	𝑇	𝑇	PROPN
cana-1295	91	98	contaiing	contaie	VERB
cana-1295	91	99	𝐴.	𝐴.	PROPN
cana-1295	91	100	given	give	VERB
cana-1295	91	101	that	that	SCONJ
cana-1295	91	102	t	t	PROPN
cana-1295	91	103	is	be	AUX
cana-1295	91	104	a	a	DET
cana-1295	91	105	left	left	ADJ
cana-1295	91	106	perfect	perfect	ADJ
cana-1295	91	107	and	and	CCONJ
cana-1295	91	108	includes	include	VERB
cana-1295	91	109	a	a	DET
cana-1295	91	110	,	,	PUNCT
cana-1295	91	111	𝑇	𝑇	PROPN
cana-1295	91	112	∈	∈	PROPN
cana-1295	92	1	∆.	∆.	NOUN
cana-1295	92	2	so	so	ADV
cana-1295	92	3	∆≠	∆≠	PROPN
cana-1295	92	4	∅.	∅.	PROPN
cana-1295	92	5	agree	agree	VERB
cana-1295	92	6	𝑆∗	𝑆∗	X
cana-1295	92	7	=	=	PUNCT
cana-1295	92	8	⋂	⋂	PROPN
cana-1295	92	9	𝑆𝑠∈∆	𝑆𝑠∈∆	PROPN
cana-1295	92	10	.	.	PUNCT
cana-1295	93	1	seeing	see	VERB
cana-1295	93	2	as	as	SCONJ
cana-1295	93	3	𝐴	𝐴	PROPN
cana-1295	93	4	is	be	AUX
cana-1295	93	5	a	a	DET
cana-1295	93	6	resulting	result	VERB
cana-1295	93	7	put	put	NOUN
cana-1295	93	8	of	of	ADP
cana-1295	93	9	𝑆	𝑆	PROPN
cana-1295	93	10	for	for	ADP
cana-1295	93	11	all	all	DET
cana-1295	93	12	𝑆	𝑆	PROPN
cana-1295	93	13	∈	∈	PROPN
cana-1295	93	14	∆	∆	NOUN
cana-1295	93	15	,	,	PUNCT
cana-1295	93	16	𝐴	𝐴	PROPN
cana-1295	93	17	⊆	⊆	NUM
cana-1295	93	18	𝑆∗.	𝑆∗.	PROPN
cana-1295	93	19	𝑆∗	𝑆∗	PROPN
cana-1295	93	20	is	be	AUX
cana-1295	93	21	the	the	DET
cana-1295	93	22	left	left	ADJ
cana-1295	93	23	ideal	ideal	NOUN
cana-1295	93	24	of	of	ADP
cana-1295	93	25	𝑆	𝑆	PROPN
cana-1295	93	26	.	.	PUNCT
cana-1295	94	1	let	let	VERB
cana-1295	94	2	𝑃	𝑃	PRON
cana-1295	94	3	be	be	AUX
cana-1295	94	4	the	the	DET
cana-1295	94	5	left	left	ADJ
cana-1295	94	6	ideal	ideal	NOUN
cana-1295	94	7	of	of	ADP
cana-1295	94	8	𝑇	𝑇	PROPN
cana-1295	94	9	containing𝐴	containing𝐴	PROPN
cana-1295	94	10	.𝑃	.𝑃	X
cana-1295	94	11	be	be	AUX
cana-1295	94	12	the	the	DET
cana-1295	94	13	left	leave	VERB
cana-1295	94	14	perfect	perfect	NOUN
cana-1295	94	15	of	of	ADP
cana-1295	94	16	𝑇	𝑇	PROPN
cana-1295	94	17	.	.	PUNCT
cana-1295	95	1	clearly	clearly	ADV
cana-1295	95	2	𝐴	𝐴	PROPN
cana-1295	95	3	⊆	⊆	NUM
cana-1295	95	4	𝑃.therefore𝑃	𝑃.therefore𝑃	DET
cana-1295	95	5	∈	∈	NOUN
cana-1295	95	6	∆⟹	∆⟹	NOUN
cana-1295	95	7	𝑆∗	𝑆∗	X
cana-1295	95	8	⊆	⊆	NUM
cana-1295	95	9	𝑃	𝑃	NOUN
cana-1295	95	10	&	&	CCONJ
cana-1295	95	11	hence	hence	ADV
cana-1295	95	12	𝑆∗	𝑆∗	PROPN
cana-1295	95	13	be	be	AUX
cana-1295	95	14	the	the	DET
cana-1295	95	15	left	left	ADJ
cana-1295	95	16	perfect	perfect	NOUN
cana-1295	95	17	of	of	ADP
cana-1295	95	18	𝑇containing	𝑇containing	PROPN
cana-1295	95	19	𝐴.	𝐴.	PROPN
cana-1295	95	20	definition	definition	NOUN
cana-1295	95	21	3.34	3.34	NUM
cana-1295	95	22	:	:	PUNCT
cana-1295	95	23	preservative	preservative	ADJ
cana-1295	95	24	auxiliary	auxiliary	ADJ
cana-1295	95	25	semi	semi	ADJ
cana-1295	95	26	group	group	NOUN
cana-1295	95	27	𝑄	𝑄	PROPN
cana-1295	95	28	of	of	ADP
cana-1295	95	29	a	a	DET
cana-1295	95	30	bi	bi	ADJ
cana-1295	95	31	-	-	ADJ
cana-1295	95	32	ternary	ternary	ADJ
cana-1295	95	33	semiring	semire	VERB
cana-1295	95	34	𝑇	𝑇	PROPN
cana-1295	95	35	is	be	AUX
cana-1295	95	36	called	call	VERB
cana-1295	95	37	quasi	quasi	ADJ
cana-1295	95	38	ideal	ideal	NOUN
cana-1295	95	39	of	of	ADP
cana-1295	95	40	bi	bi	ADJ
cana-1295	95	41	ternary	ternary	NOUN
cana-1295	95	42	semiring	semiring	NOUN
cana-1295	95	43	t	t	NOUN
cana-1295	95	44	then	then	ADV
cana-1295	95	45	𝑄	𝑄	PROPN
cana-1295	95	46	=	=	PUNCT
cana-1295	95	47	𝑄1	𝑄1	PROPN
cana-1295	95	48	∪	∪	ADP
cana-1295	95	49	𝑄2	𝑄2	PROPN
cana-1295	95	50	mutually	mutually	ADV
cana-1295	95	51	𝑄1	𝑄1	PROPN
cana-1295	95	52	&	&	CCONJ
cana-1295	95	53	𝑄2	𝑄2	PROPN
cana-1295	95	54	are	be	AUX
cana-1295	95	55	quasi	quasi	NOUN
cana-1295	95	56	standards	standard	NOUN
cana-1295	95	57	of	of	ADP
cana-1295	95	58	𝑇1	𝑇1	PROPN
cana-1295	95	59	&	&	CCONJ
cana-1295	95	60	𝑇2	𝑇2	PROPN
cana-1295	95	61	if	if	SCONJ
cana-1295	95	62	𝑄1𝑇1	𝑄1𝑇1	PROPN
cana-1295	95	63	𝑇1	𝑇1	NOUN
cana-1295	95	64	∩	∩	NOUN
cana-1295	95	65	(	(	PUNCT
cana-1295	95	66	𝑇1𝑄1𝑇	𝑇1𝑄1𝑇	SYM
cana-1295	95	67	1	1	NUM
cana-1295	95	68	+	+	CCONJ
cana-1295	95	69	𝑇1	𝑇1	PROPN
cana-1295	95	70	𝑇1𝑄1𝑇1	𝑇1𝑄1𝑇1	PROPN
cana-1295	95	71	𝑇1	𝑇1	NOUN
cana-1295	95	72	)	)	PUNCT
cana-1295	95	73	∩	∩	NOUN
cana-1295	95	74	𝑇1	𝑇1	NOUN
cana-1295	95	75	𝑇1𝑄1	𝑇1𝑄1	PROPN
cana-1295	95	76	⊆	⊆	NUM
cana-1295	95	77	𝑄1	𝑄1	NOUN
cana-1295	95	78	𝑄2𝑇2	𝑄2𝑇2	PROPN
cana-1295	95	79	𝑇2	𝑇2	PROPN
cana-1295	95	80	∩	∩	PROPN
cana-1295	95	81	(	(	PUNCT
cana-1295	95	82	𝑇	𝑇	PROPN
cana-1295	95	83	2𝑄2𝑇	2𝑄2𝑇	PROPN
cana-1295	95	84	2	2	NUM
cana-1295	95	85	+	+	CCONJ
cana-1295	95	86	𝑇2	𝑇2	PROPN
cana-1295	95	87	𝑇2𝑄2𝑇2	𝑇2𝑄2𝑇2	PROPN
cana-1295	95	88	𝑇2	𝑇2	PROPN
cana-1295	95	89	)	)	PUNCT
cana-1295	96	1	∩	∩	PROPN
cana-1295	96	2	𝑇2	𝑇2	PROPN
cana-1295	96	3	𝑇2𝑄2	𝑇2𝑄2	PROPN
cana-1295	96	4	⊆	⊆	NUM
cana-1295	96	5	𝑄2	𝑄2	PROPN
cana-1295	96	6	.	.	PUNCT
cana-1295	97	1	again	again	ADV
cana-1295	97	2	𝑄	𝑄	PRON
cana-1295	97	3	is	be	AUX
cana-1295	97	4	a	a	DET
cana-1295	97	5	quasi	quasi	NOUN
cana-1295	97	6	-	-	NOUN
cana-1295	97	7	ideal	ideal	NOUN
cana-1295	97	8	of	of	ADP
cana-1295	97	9	𝑇.	𝑇.	PROPN
cana-1295	97	10	definition	definition	NOUN
cana-1295	97	11	3.35	3.35	NUM
cana-1295	97	12	:	:	PUNCT
cana-1295	97	13	a	a	DET
cana-1295	97	14	apt	apt	ADJ
cana-1295	97	15	perfect	perfect	NOUN
cana-1295	97	16	of	of	ADP
cana-1295	97	17	𝑃	𝑃	NOUN
cana-1295	97	18	=	=	NOUN
cana-1295	97	19	𝑃1	𝑃1	NOUN
cana-1295	97	20	∪	∪	VERB
cana-1295	97	21	𝑃2	𝑃2	NOUN
cana-1295	97	22	of	of	ADP
cana-1295	97	23	a	a	DET
cana-1295	97	24	commutative	commutative	ADJ
cana-1295	97	25	bi	bi	ADJ
cana-1295	97	26	ternary	ternary	NOUN
cana-1295	97	27	semi	semi	ADV
cana-1295	97	28	group𝑇	group𝑇	PROPN
cana-1295	97	29	is	be	AUX
cana-1295	97	30	prime	prime	ADJ
cana-1295	97	31	ideal	ideal	NOUN
cana-1295	97	32	if	if	SCONJ
cana-1295	97	33	,	,	PUNCT
cana-1295	97	34	together	together	ADV
cana-1295	97	35	𝑃1	𝑃1	NOUN
cana-1295	97	36	&	&	CCONJ
cana-1295	97	37	𝑃2	𝑃2	PROPN
cana-1295	97	38	were	be	AUX
cana-1295	97	39	main	main	ADJ
cana-1295	97	40	ideals	ideal	NOUN
cana-1295	97	41	of	of	ADP
cana-1295	97	42	𝑇1&𝑇2	𝑇1&𝑇2	NOUN
cana-1295	97	43	correspondingly	correspondingly	ADV
cana-1295	97	44	if	if	SCONJ
cana-1295	97	45	𝑋1	𝑋1	PROPN
cana-1295	97	46	,	,	PUNCT
cana-1295	97	47	𝑌1	𝑌1	PROPN
cana-1295	97	48	,	,	PUNCT
cana-1295	97	49	𝑍1	𝑍1	PROPN
cana-1295	97	50	are	be	AUX
cana-1295	97	51	ideals	ideal	NOUN
cana-1295	97	52	of	of	ADP
cana-1295	97	53	𝑇1and𝑋2	𝑇1and𝑋2	PROPN
cana-1295	97	54	,	,	PUNCT
cana-1295	97	55	𝑌2	𝑌2	PROPN
cana-1295	97	56	,	,	PUNCT
cana-1295	97	57	𝑍2	𝑍2	PROPN
cana-1295	97	58	are	be	AUX
cana-1295	97	59	ideals	ideal	NOUN
cana-1295	97	60	of𝑇2	of𝑇2	NOUN
cana-1295	98	1	such	such	ADJ
cana-1295	98	2	that	that	SCONJ
cana-1295	98	3	𝑋1𝑌1𝑍1	𝑋1𝑌1𝑍1	PROPN
cana-1295	98	4	⊆	⊆	NUM
cana-1295	98	5	𝑃1	𝑃1	NOUN
cana-1295	98	6	⟹	⟹	PUNCT
cana-1295	98	7	𝑋1	𝑋1	PROPN
cana-1295	98	8	⊆	⊆	NUM
cana-1295	98	9	𝑃1𝑜𝑟𝑌1	𝑃1𝑜𝑟𝑌1	PROPN
cana-1295	98	10	⊆	⊆	NUM
cana-1295	98	11	𝑃1𝑜𝑟	𝑃1𝑜𝑟	NOUN
cana-1295	98	12	𝑍1	𝑍1	NOUN
cana-1295	98	13	⊆	⊆	NUM
cana-1295	98	14	𝑃1	𝑃1	NOUN
cana-1295	98	15	𝑋2𝑌2𝑍2	𝑋2𝑌2𝑍2	ADP
cana-1295	98	16	⊆	⊆	NUM
cana-1295	98	17	𝑃2	𝑃2	NOUN
cana-1295	98	18	⟹	⟹	NUM
cana-1295	98	19	𝑋2	𝑋2	VERB
cana-1295	98	20	⊆	⊆	NUM
cana-1295	98	21	𝑃2𝑜𝑟𝑌2	𝑃2𝑜𝑟𝑌2	NUM
cana-1295	98	22	⊆	⊆	NUM
cana-1295	98	23	𝑃2𝑜𝑟	𝑃2𝑜𝑟	NOUN
cana-1295	98	24	𝑍2	𝑍2	VERB
cana-1295	98	25	⊆	⊆	NUM
cana-1295	98	26	𝑃2	𝑃2	DET
cana-1295	98	27	definition	definition	NOUN
cana-1295	98	28	3.36	3.36	NUM
cana-1295	98	29	:	:	PUNCT
cana-1295	98	30	a	a	DET
cana-1295	98	31	put	put	NOUN
cana-1295	98	32	perfect	perfect	ADJ
cana-1295	98	33	by	by	ADP
cana-1295	98	34	𝑃	𝑃	NOUN
cana-1295	98	35	=	=	NOUN
cana-1295	98	36	𝑃1	𝑃1	NOUN
cana-1295	98	37	∪	∪	VERB
cana-1295	98	38	𝑃2	𝑃2	NOUN
cana-1295	98	39	of	of	ADP
cana-1295	98	40	a	a	DET
cana-1295	98	41	commutative	commutative	ADJ
cana-1295	98	42	bi	bi	ADJ
cana-1295	98	43	ternary	ternary	PROPN
cana-1295	98	44	semi	semi	PROPN
cana-1295	98	45	group	group	NOUN
cana-1295	98	46	𝑇	𝑇	PROPN
cana-1295	98	47	is	be	AUX
cana-1295	98	48	completely	completely	ADV
cana-1295	98	49	prime	prime	ADJ
cana-1295	98	50	ideal	ideal	NOUN
cana-1295	98	51	if	if	SCONJ
cana-1295	98	52	,	,	PUNCT
cana-1295	98	53	together	together	ADV
cana-1295	98	54	𝑃1	𝑃1	NOUN
cana-1295	98	55	&	&	CCONJ
cana-1295	98	56	𝑃2	𝑃2	PROPN
cana-1295	98	57	are	be	AUX
cana-1295	98	58	completely	completely	ADV
cana-1295	98	59	prime	prime	ADJ
cana-1295	98	60	ideals	ideal	NOUN
cana-1295	98	61	of	of	ADP
cana-1295	98	62	𝑇1&𝑇2	𝑇1&𝑇2	NOUN
cana-1295	98	63	separately	separately	ADV
cana-1295	98	64	if	if	SCONJ
cana-1295	98	65	𝑥1	𝑥1	PROPN
cana-1295	98	66	,	,	PUNCT
cana-1295	98	67	𝑦1	𝑦1	PROPN
cana-1295	98	68	,	,	PUNCT
cana-1295	98	69	𝑧1	𝑧1	NOUN
cana-1295	98	70	are	be	AUX
cana-1295	98	71	of	of	ADP
cana-1295	98	72	𝑇1and𝑥2	𝑇1and𝑥2	NOUN
cana-1295	98	73	,	,	PUNCT
cana-1295	98	74	𝑦2	𝑦2	PROPN
cana-1295	98	75	,	,	PUNCT
cana-1295	98	76	𝑧2	𝑧2	PROPN
cana-1295	98	77	are	be	AUX
cana-1295	98	78	of𝑇2	of𝑇2	NOUN
cana-1295	98	79	such	such	ADJ
cana-1295	98	80	that	that	SCONJ
cana-1295	98	81	𝑥1𝑦1𝑧1	𝑥1𝑦1𝑧1	ADJ
cana-1295	98	82	∈	∈	PROPN
cana-1295	98	83	𝑃1	𝑃1	NOUN
cana-1295	98	84	⟹	⟹	NUM
cana-1295	98	85	𝑥1	𝑥1	PROPN
cana-1295	98	86	∈	∈	PROPN
cana-1295	98	87	𝑃1𝑜𝑟𝑦1	𝑃1𝑜𝑟𝑦1	NOUN
cana-1295	98	88	∈	∈	PROPN
cana-1295	98	89	𝑃1𝑜𝑟	𝑃1𝑜𝑟	PUNCT
cana-1295	98	90	𝑧1	𝑧1	VERB
cana-1295	98	91	∈	∈	PROPN
cana-1295	98	92	𝑃1	𝑃1	NOUN
cana-1295	98	93	𝑥2𝑦2𝑧2	𝑥2𝑦2𝑧2	PROPN
cana-1295	98	94	∈	∈	PROPN
cana-1295	98	95	𝑃2	𝑃2	PROPN
cana-1295	98	96	⟹	⟹	NUM
cana-1295	98	97	𝑥2	𝑥2	PROPN
cana-1295	98	98	∈	∈	PROPN
cana-1295	98	99	𝑃2𝑜𝑟𝑦2	𝑃2𝑜𝑟𝑦2	VERB
cana-1295	98	100	∈	∈	PROPN
cana-1295	98	101	𝑃2𝑜𝑟	𝑃2𝑜𝑟	PROPN
cana-1295	98	102	𝑧2	𝑧2	NOUN
cana-1295	98	103	∈	∈	PROPN
cana-1295	98	104	𝑃2	𝑃2	NOUN
cana-1295	98	105	communications	communication	NOUN
cana-1295	98	106	on	on	ADP
cana-1295	98	107	applied	apply	VERB
cana-1295	98	108	nonlinear	nonlinear	ADJ
cana-1295	98	109	analysis	analysis	NOUN
cana-1295	98	110	issn	issn	NOUN
cana-1295	98	111	:	:	PUNCT
cana-1295	98	112	1074	1074	NUM
cana-1295	98	113	-	-	PUNCT
cana-1295	98	114	133x	133x	NUM
cana-1295	98	115	vol	vol	NOUN
cana-1295	98	116	31	31	NUM
cana-1295	98	117	no	no	NOUN
cana-1295	98	118	.	.	PUNCT
cana-1295	99	1	7s	7	NOUN
cana-1295	99	2	(	(	PUNCT
cana-1295	99	3	2024	2024	NUM
cana-1295	99	4	)	)	PUNCT
cana-1295	99	5	211	211	NUM
cana-1295	99	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1295	99	7	theorem	theorem	VERB
cana-1295	99	8	3.37	3.37	NUM
cana-1295	99	9	:	:	PUNCT
cana-1295	99	10	let	let	VERB
cana-1295	99	11	𝑇	𝑇	PROPN
cana-1295	99	12	be	be	AUX
cana-1295	99	13	a	a	DET
cana-1295	99	14	bi	bi	ADJ
cana-1295	99	15	ternary	ternary	NOUN
cana-1295	99	16	semi	semi	ADJ
cana-1295	99	17	group	group	NOUN
cana-1295	99	18	with	with	ADP
cana-1295	99	19	identity	identity	NOUN
cana-1295	99	20	.	.	PUNCT
cana-1295	100	1	then	then	ADV
cana-1295	100	2	t	t	PROPN
cana-1295	100	3	's	's	PART
cana-1295	100	4	maximum	maximum	ADJ
cana-1295	100	5	notions	notion	NOUN
cana-1295	100	6	are	be	AUX
cana-1295	100	7	all	all	PRON
cana-1295	100	8	fundamental	fundamental	ADJ
cana-1295	100	9	concepts	concept	NOUN
cana-1295	100	10	.	.	PUNCT
cana-1295	101	1	proof	proof	NOUN
cana-1295	101	2	:	:	PUNCT
cana-1295	101	3	agree	agree	VERB
cana-1295	101	4	𝑃	𝑃	NOUN
cana-1295	101	5	be	be	VERB
cana-1295	101	6	a	a	DET
cana-1295	101	7	ultimate	ultimate	ADJ
cana-1295	101	8	perfect	perfect	NOUN
cana-1295	101	9	of	of	ADP
cana-1295	101	10	𝑇.	𝑇.	PROPN
cana-1295	101	11	let	let	VERB
cana-1295	101	12	𝐴	𝐴	PROPN
cana-1295	101	13	,	,	PUNCT
cana-1295	101	14	𝐵	𝐵	PROPN
cana-1295	101	15	,	,	PUNCT
cana-1295	101	16	𝐶	𝐶	PROPN
cana-1295	101	17	be	be	VERB
cana-1295	101	18	ideals	ideal	NOUN
cana-1295	101	19	of	of	ADP
cana-1295	101	20	𝑇	𝑇	PROPN
cana-1295	101	21	such	such	ADJ
cana-1295	101	22	that𝐴𝐵𝐶	that𝐴𝐵𝐶	PROPN
cana-1295	101	23	⊆	⊆	NUM
cana-1295	101	24	𝑃.suppose	𝑃.suppose	NUM
cana-1295	101	25	that	that	DET
cana-1295	101	26	𝐴	𝐴	PROPN
cana-1295	101	27	,	,	PUNCT
cana-1295	101	28	𝐵	𝐵	NOUN
cana-1295	101	29	not	not	PART
cana-1295	101	30	contained	contain	VERB
cana-1295	101	31	in	in	ADP
cana-1295	101	32	𝑃	𝑃	NOUN
cana-1295	101	33	.	.	PUNCT
cana-1295	102	1	then	then	ADV
cana-1295	102	2	𝐴	𝐴	PROPN
cana-1295	102	3	∪	∪	VERB
cana-1295	102	4	𝑃	𝑃	NOUN
cana-1295	102	5	=	=	SYM
cana-1295	102	6	𝑇	𝑇	PROPN
cana-1295	102	7	and	and	CCONJ
cana-1295	102	8	𝐵	𝐵	NOUN
cana-1295	102	9	∪	∪	VERB
cana-1295	102	10	𝑃	𝑃	NOUN
cana-1295	102	11	=	=	SYM
cana-1295	102	12	𝑇	𝑇	PROPN
cana-1295	102	13	.	.	PUNCT
cana-1295	103	1	as	as	ADP
cana-1295	103	2	𝑒	𝑒	PROPN
cana-1295	103	3	∈	∈	PROPN
cana-1295	103	4	𝑇	𝑇	PROPN
cana-1295	103	5	so	so	SCONJ
cana-1295	103	6	𝑒	𝑒	PROPN
cana-1295	103	7	∈	∈	PROPN
cana-1295	103	8	𝐴	𝐴	PROPN
cana-1295	103	9	∪	∪	NOUN
cana-1295	103	10	𝑃	𝑃	NOUN
cana-1295	103	11	and	and	CCONJ
cana-1295	103	12	𝑒	𝑒	PROPN
cana-1295	103	13	∈	∈	NOUN
cana-1295	103	14	𝐵	𝐵	NOUN
cana-1295	103	15	∪	∪	NOUN
cana-1295	103	16	𝑃	𝑃	NOUN
cana-1295	103	17	implies	imply	VERB
cana-1295	103	18	𝑒	𝑒	PROPN
cana-1295	103	19	∈	∈	PROPN
cana-1295	103	20	𝐴	𝐴	PROPN
cana-1295	103	21	or	or	CCONJ
cana-1295	103	22	𝑒	𝑒	PROPN
cana-1295	103	23	∈	∈	NOUN
cana-1295	103	24	𝑃	𝑃	NOUN
cana-1295	103	25	and	and	CCONJ
cana-1295	103	26	𝑒	𝑒	PROPN
cana-1295	103	27	∈	∈	PROPN
cana-1295	103	28	𝐵	𝐵	NOUN
cana-1295	103	29	or	or	CCONJ
cana-1295	103	30	𝑒	𝑒	PROPN
cana-1295	103	31	∈	∈	PROPN
cana-1295	103	32	𝑃.	𝑃.	PROPN
cana-1295	103	33	since	since	SCONJ
cana-1295	103	34	𝑒	𝑒	PROPN
cana-1295	103	35	∉	∉	X
cana-1295	103	36	𝑃	𝑃	VERB
cana-1295	103	37	so	so	ADV
cana-1295	103	38	𝑒	𝑒	PROPN
cana-1295	103	39	∈	∈	PROPN
cana-1295	103	40	𝐴	𝐴	PROPN
cana-1295	103	41	and	and	CCONJ
cana-1295	103	42	𝑒	𝑒	ADP
cana-1295	103	43	∈	∈	PROPN
cana-1295	103	44	𝐵	𝐵	NOUN
cana-1295	103	45	⇒	⇒	NOUN
cana-1295	103	46	𝐴	𝐴	PROPN
cana-1295	103	47	=	=	SYM
cana-1295	103	48	𝑇	𝑇	PROPN
cana-1295	103	49	and	and	CCONJ
cana-1295	103	50	𝐵	𝐵	NOUN
cana-1295	103	51	=	=	PROPN
cana-1295	103	52	𝑇.	𝑇.	PROPN
cana-1295	103	53	now	now	ADV
cana-1295	103	54	since	since	SCONJ
cana-1295	103	55	𝑒	𝑒	PROPN
cana-1295	103	56	∈	∈	PROPN
cana-1295	103	57	𝑇	𝑇	PROPN
cana-1295	103	58	,	,	PUNCT
cana-1295	103	59	so	so	ADV
cana-1295	103	60	𝑇𝑇𝐶	𝑇𝑇𝐶	PROPN
cana-1295	103	61	=	=	SYM
cana-1295	103	62	𝐶	𝐶	PROPN
cana-1295	103	63	and	and	CCONJ
cana-1295	103	64	𝐶	𝐶	PROPN
cana-1295	103	65	=	=	SYM
cana-1295	103	66	𝑇𝑇𝐶	𝑇𝑇𝐶	PROPN
cana-1295	103	67	=	=	SYM
cana-1295	103	68	𝐴𝐵𝐶	𝐴𝐵𝐶	PROPN
cana-1295	103	69	⊆	⊆	NUM
cana-1295	103	70	𝑃	𝑃	PROPN
cana-1295	103	71	suggests	suggest	VERB
cana-1295	103	72	𝐶	𝐶	PROPN
cana-1295	103	73	⊆	⊆	NUM
cana-1295	103	74	𝑃.	𝑃.	PROPN
cana-1295	103	75	henceforth	henceforth	ADV
cana-1295	103	76	𝑃	𝑃	NOUN
cana-1295	103	77	is	be	AUX
cana-1295	103	78	a	a	DET
cana-1295	103	79	main	main	ADJ
cana-1295	103	80	idealistic	idealistic	ADJ
cana-1295	103	81	of	of	ADP
cana-1295	103	82	𝑇.	𝑇.	PROPN
cana-1295	103	83	4.homomarphism	4.homomarphism	PROPN
cana-1295	103	84	on	on	ADP
cana-1295	103	85	bi	bi	ADJ
cana-1295	103	86	ternary	ternary	NOUN
cana-1295	103	87	semi	semi	PROPN
cana-1295	103	88	group	group	NOUN
cana-1295	103	89	:	:	PUNCT
cana-1295	103	90	definition	definition	NOUN
cana-1295	103	91	4.1	4.1	NUM
cana-1295	103	92	:	:	PUNCT
cana-1295	103	93	let	let	VERB
cana-1295	103	94	(	(	PUNCT
cana-1295	103	95	𝑆,∙,∗	𝑆,∙,∗	NOUN
cana-1295	103	96	)	)	PUNCT
cana-1295	103	97	𝑤ℎ𝑒𝑟𝑒	𝑤ℎ𝑒𝑟𝑒	NOUN
cana-1295	103	98	𝑆	𝑆	PROPN
cana-1295	103	99	=	=	SYM
cana-1295	103	100	(	(	PUNCT
cana-1295	103	101	𝑆1,∙	𝑆1,∙	PROPN
cana-1295	103	102	)	)	PUNCT
cana-1295	103	103	∪	∪	NOUN
cana-1295	103	104	(	(	PUNCT
cana-1295	103	105	𝑆2,∗	𝑆2,∗	ADJ
cana-1295	103	106	)	)	PUNCT
cana-1295	103	107	and	and	CCONJ
cana-1295	103	108	(	(	PUNCT
cana-1295	103	109	𝑇,∘,⊗	𝑇,∘,⊗	INTJ
cana-1295	103	110	)	)	PUNCT
cana-1295	103	111	=	=	SYM
cana-1295	104	1	(	(	PUNCT
cana-1295	104	2	𝑇1,∘	𝑇1,∘	PROPN
cana-1295	104	3	)	)	PUNCT
cana-1295	104	4	∪	∪	NOUN
cana-1295	104	5	(	(	PUNCT
cana-1295	104	6	𝑇2,⊗	𝑇2,⊗	NUM
cana-1295	104	7	)	)	PUNCT
cana-1295	104	8	be	be	AUX
cana-1295	104	9	two	two	NUM
cana-1295	104	10	bi	bi	ADJ
cana-1295	104	11	ternary	ternary	ADJ
cana-1295	104	12	semi	semi	ADJ
cana-1295	104	13	groups	group	NOUN
cana-1295	104	14	.	.	PUNCT
cana-1295	105	1	a	a	DET
cana-1295	105	2	plan	plan	NOUN
cana-1295	105	3	∅	∅	NOUN
cana-1295	105	4	=	=	PUNCT
cana-1295	105	5	∅1	∅1	VERB
cana-1295	105	6	∪	∪	ADP
cana-1295	105	7	∅2	∅2	NOUN
cana-1295	105	8	on	on	ADP
cana-1295	105	9	or	or	CCONJ
cana-1295	105	10	after	after	ADP
cana-1295	105	11	𝑆	𝑆	PROPN
cana-1295	105	12	to	to	ADP
cana-1295	105	13	𝑇	𝑇	PROPN
cana-1295	105	14	is	be	AUX
cana-1295	105	15	bi	bi	ADJ
cana-1295	105	16	ternary	ternary	NOUN
cana-1295	105	17	semi	semi	ADJ
cana-1295	105	18	group	group	NOUN
cana-1295	105	19	homomorphism	homomorphism	NOUN
cana-1295	105	20	.	.	PUNCT
cana-1295	106	1	if	if	SCONJ
cana-1295	106	2	∅1preserves	∅1preserve	NOUN
cana-1295	106	3	homomorphism	homomorphism	NOUN
cana-1295	106	4	from	from	ADP
cana-1295	106	5	𝑆1to𝑇1	𝑆1to𝑇1	PROPN
cana-1295	106	6	,	,	PUNCT
cana-1295	106	7	∅2	∅2	VERB
cana-1295	106	8	preserves	preserve	VERB
cana-1295	106	9	homomorphism	homomorphism	NOUN
cana-1295	106	10	from	from	ADP
cana-1295	106	11	𝑆2to	𝑆2to	PROPN
cana-1295	106	12	𝑇2	𝑇2	NOUN
cana-1295	106	13	such	such	ADJ
cana-1295	106	14	that	that	DET
cana-1295	106	15	1	1	NUM
cana-1295	106	16	)	)	PUNCT
cana-1295	106	17	∅(𝑎	∅(𝑎	NOUN
cana-1295	107	1	∙	∙	PROPN
cana-1295	107	2	𝑏	𝑏	X
cana-1295	107	3	∙	∙	PROPN
cana-1295	107	4	𝑐	𝑐	NOUN
cana-1295	107	5	)	)	PUNCT
cana-1295	107	6	=	=	SYM
cana-1295	107	7	∅(𝑎	∅(𝑎	NOUN
cana-1295	107	8	)	)	PUNCT
cana-1295	107	9	∘	∘	NOUN
cana-1295	107	10	∅	∅	NOUN
cana-1295	107	11	(	(	PUNCT
cana-1295	107	12	𝑏	𝑏	NOUN
cana-1295	107	13	)	)	PUNCT
cana-1295	107	14	∘	∘	PROPN
cana-1295	107	15	∅	∅	NOUN
cana-1295	107	16	(	(	PUNCT
cana-1295	107	17	𝑐	𝑐	NOUN
cana-1295	107	18	)	)	PUNCT
cana-1295	107	19	2	2	NUM
cana-1295	107	20	)	)	PUNCT
cana-1295	107	21	∅(𝑎	∅(𝑎	PROPN
cana-1295	107	22	∗	∗	NOUN
cana-1295	107	23	𝑏	𝑏	PROPN
cana-1295	107	24	∗	∗	NOUN
cana-1295	107	25	𝑐	𝑐	NOUN
cana-1295	107	26	)	)	PUNCT
cana-1295	107	27	=	=	NOUN
cana-1295	107	28	∅	∅	NOUN
cana-1295	107	29	(	(	PUNCT
cana-1295	107	30	𝑎	𝑎	PROPN
cana-1295	107	31	)	)	PUNCT
cana-1295	107	32	⨂	⨂	NOUN
cana-1295	107	33	∅	∅	NOUN
cana-1295	107	34	(	(	PUNCT
cana-1295	107	35	𝑏	𝑏	NOUN
cana-1295	107	36	)	)	PUNCT
cana-1295	107	37	⨂	⨂	NOUN
cana-1295	107	38	∅	∅	NOUN
cana-1295	107	39	(	(	PUNCT
cana-1295	107	40	𝑐	𝑐	NOUN
cana-1295	107	41	)	)	PUNCT
cana-1295	107	42	for	for	ADP
cana-1295	107	43	all	all	DET
cana-1295	107	44	𝑎	𝑎	PROPN
cana-1295	107	45	,	,	PUNCT
cana-1295	107	46	𝑏	𝑏	NOUN
cana-1295	107	47	,	,	PUNCT
cana-1295	107	48	𝑐	𝑐	PROPN
cana-1295	107	49	∈	∈	PROPN
cana-1295	107	50	𝑆	𝑆	PROPN
cana-1295	107	51	&	&	CCONJ
cana-1295	107	52	∅(𝑎	∅(𝑎	PROPN
cana-1295	107	53	)	)	PUNCT
cana-1295	107	54	,	,	PUNCT
cana-1295	107	55	∅(𝑏	∅(𝑏	NOUN
cana-1295	107	56	)	)	PUNCT
cana-1295	107	57	,	,	PUNCT
cana-1295	107	58	∅(𝑐	∅(𝑐	NOUN
cana-1295	107	59	)	)	PUNCT
cana-1295	107	60	∈	∈	PROPN
cana-1295	107	61	𝑇.	𝑇.	PROPN
cana-1295	107	62	definition	definition	NOUN
cana-1295	107	63	4.2	4.2	NUM
cana-1295	107	64	:	:	PUNCT
cana-1295	107	65	a	a	DET
cana-1295	107	66	bi	bi	ADJ
cana-1295	107	67	ternary	ternary	NOUN
cana-1295	107	68	semi	semi	NOUN
cana-1295	107	69	group	group	NOUN
cana-1295	107	70	homomorphism	homomorphism	NOUN
cana-1295	107	71	∅	∅	NOUN
cana-1295	107	72	:	:	PUNCT
cana-1295	107	73	𝑆	𝑆	PROPN
cana-1295	107	74	→	→	SYM
cana-1295	107	75	𝑇	𝑇	PROPN
cana-1295	107	76	is	be	AUX
cana-1295	107	77	onto	onto	ADP
cana-1295	107	78	homomorphism	homomorphism	NOUN
cana-1295	107	79	is	be	AUX
cana-1295	107	80	called	call	VERB
cana-1295	107	81	epimorphism	epimorphism	NOUN
cana-1295	107	82	∅	∅	NOUN
cana-1295	107	83	:	:	PUNCT
cana-1295	107	84	𝑆	𝑆	PROPN
cana-1295	107	85	→	→	SYM
cana-1295	107	86	𝑇.	𝑇.	PROPN
cana-1295	107	87	definition	definition	NOUN
cana-1295	107	88	4.3	4.3	NUM
cana-1295	107	89	:	:	PUNCT
cana-1295	107	90	a	a	DET
cana-1295	107	91	bi	bi	ADJ
cana-1295	107	92	ternary	ternary	NOUN
cana-1295	107	93	semi	semi	NOUN
cana-1295	107	94	group	group	NOUN
cana-1295	107	95	homomorphism	homomorphism	NOUN
cana-1295	107	96	∅	∅	NOUN
cana-1295	107	97	:	:	PUNCT
cana-1295	107	98	𝑆	𝑆	PROPN
cana-1295	107	99	→	→	SYM
cana-1295	107	100	𝑇	𝑇	PROPN
cana-1295	107	101	is	be	AUX
cana-1295	107	102	one	one	NUM
cana-1295	107	103	-	-	PUNCT
cana-1295	107	104	to	to	ADP
cana-1295	107	105	-	-	PUNCT
cana-1295	107	106	one	one	NUM
cana-1295	107	107	homomorphism	homomorphism	NOUN
cana-1295	107	108	is	be	AUX
cana-1295	107	109	called	call	VERB
cana-1295	107	110	monomorphism	monomorphism	NOUN
cana-1295	107	111	∅	∅	NOUN
cana-1295	107	112	:	:	PUNCT
cana-1295	107	113	𝑆	𝑆	PROPN
cana-1295	107	114	→	→	SYM
cana-1295	107	115	𝑇.	𝑇.	PROPN
cana-1295	107	116	definition	definition	NOUN
cana-1295	107	117	4.4	4.4	NUM
cana-1295	107	118	:	:	PUNCT
cana-1295	107	119	a	a	DET
cana-1295	107	120	bi	bi	ADJ
cana-1295	107	121	ternary	ternary	NOUN
cana-1295	107	122	semi	semi	NOUN
cana-1295	107	123	group	group	NOUN
cana-1295	107	124	homomorphism	homomorphism	NOUN
cana-1295	107	125	∅	∅	NOUN
cana-1295	107	126	:	:	PUNCT
cana-1295	108	1	𝑆	𝑆	PROPN
cana-1295	108	2	→	→	SYM
cana-1295	108	3	𝑇is	𝑇is	VERB
cana-1295	108	4	both	both	DET
cana-1295	108	5	one	one	NUM
cana-1295	108	6	-	-	PUNCT
cana-1295	108	7	to	to	ADP
cana-1295	108	8	-	-	PUNCT
cana-1295	108	9	one	one	NUM
cana-1295	108	10	and	and	CCONJ
cana-1295	108	11	onto	onto	ADP
cana-1295	108	12	(	(	PUNCT
cana-1295	108	13	i.e.	i.e.	X
cana-1295	108	14	bijective	bijective	ADJ
cana-1295	108	15	)	)	PUNCT
cana-1295	108	16	homomorphism	homomorphism	NOUN
cana-1295	108	17	is	be	AUX
cana-1295	108	18	called	call	VERB
cana-1295	108	19	isomorphism∅	isomorphism∅	NOUN
cana-1295	108	20	:	:	PUNCT
cana-1295	108	21	𝑆	𝑆	PROPN
cana-1295	108	22	→	→	SYM
cana-1295	108	23	𝑇.	𝑇.	PROPN
cana-1295	108	24	it	it	PRON
cana-1295	108	25	says	say	VERB
cana-1295	108	26	that	that	SCONJ
cana-1295	108	27	𝑆	𝑆	PROPN
cana-1295	108	28	is	be	AUX
cana-1295	108	29	isomorphic	isomorphic	ADJ
cana-1295	108	30	to	to	ADP
cana-1295	108	31	𝑇.	𝑇.	PROPN
cana-1295	108	32	𝑆	𝑆	PROPN
cana-1295	108	33	≅	≅	PROPN
cana-1295	108	34	𝑇.	𝑇.	PROPN
cana-1295	108	35	definition	definition	NOUN
cana-1295	108	36	4.5	4.5	NUM
cana-1295	108	37	:	:	PUNCT
cana-1295	108	38	a	a	DET
cana-1295	108	39	bi	bi	ADJ
cana-1295	108	40	ternary	ternary	NOUN
cana-1295	108	41	isomorphism	isomorphism	NOUN
cana-1295	108	42	∅	∅	NOUN
cana-1295	108	43	is	be	AUX
cana-1295	108	44	defined	define	VERB
cana-1295	108	45	from	from	ADP
cana-1295	108	46	same	same	ADJ
cana-1295	108	47	bi	bi	ADJ
cana-1295	108	48	ternary	ternary	NOUN
cana-1295	108	49	semiring	semire	VERB
cana-1295	108	50	onto	onto	ADP
cana-1295	108	51	itself	itself	PRON
cana-1295	108	52	is	be	AUX
cana-1295	108	53	called	call	VERB
cana-1295	108	54	an	an	DET
cana-1295	108	55	automorphism	automorphism	NOUN
cana-1295	108	56	.	.	PUNCT
cana-1295	109	1	definition	definition	NOUN
cana-1295	109	2	4.6	4.6	NUM
cana-1295	109	3	:	:	PUNCT
cana-1295	109	4	a	a	DET
cana-1295	109	5	mapping	mapping	NOUN
cana-1295	109	6	∅	∅	NOUN
cana-1295	109	7	:	:	PUNCT
cana-1295	109	8	𝑆	𝑆	PROPN
cana-1295	109	9	→	→	SYM
cana-1295	109	10	𝑇	𝑇	PROPN
cana-1295	109	11	is	be	AUX
cana-1295	109	12	bi	bi	ADJ
cana-1295	109	13	ternary	ternary	NOUN
cana-1295	109	14	semi	semi	NOUN
cana-1295	109	15	group	group	NOUN
cana-1295	109	16	homomorphism	homomorphism	NOUN
cana-1295	109	17	the	the	DET
cana-1295	109	18	kernel	kernel	NOUN
cana-1295	109	19	of	of	ADP
cana-1295	109	20	the	the	DET
cana-1295	109	21	homomorphism	homomorphism	NOUN
cana-1295	109	22	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	PROPN
cana-1295	109	23	is	be	AUX
cana-1295	109	24	defined	define	VERB
cana-1295	109	25	as	as	ADP
cana-1295	109	26	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	NOUN
cana-1295	109	27	=	=	SYM
cana-1295	109	28	{	{	PUNCT
cana-1295	109	29	𝑒	𝑒	PROPN
cana-1295	109	30	∈	∈	PROPN
cana-1295	109	31	𝑆	𝑆	PROPN
cana-1295	109	32	:	:	PUNCT
cana-1295	109	33	∅(𝑒	∅(𝑒	ADJ
cana-1295	109	34	)	)	PUNCT
cana-1295	109	35	=	=	SYM
cana-1295	109	36	𝑒′	𝑒′	NOUN
cana-1295	109	37	}	}	PUNCT
cana-1295	109	38	where	where	SCONJ
cana-1295	109	39	𝑒′	𝑒′	NOUN
cana-1295	109	40	∈	∈	PROPN
cana-1295	109	41	𝑇	𝑇	PROPN
cana-1295	109	42	is	be	AUX
cana-1295	109	43	the	the	DET
cana-1295	109	44	identity	identity	NOUN
cana-1295	109	45	of	of	ADP
cana-1295	109	46	𝑇.	𝑇.	PROPN
cana-1295	109	47	theorem	theorem	VERB
cana-1295	109	48	4.7	4.7	NUM
cana-1295	109	49	:	:	PUNCT
cana-1295	109	50	let	let	VERB
cana-1295	109	51	𝑆	𝑆	PROPN
cana-1295	109	52	=	=	PUNCT
cana-1295	109	53	𝑆1	𝑆1	NOUN
cana-1295	109	54	∪	∪	ADJ
cana-1295	109	55	𝑆2	𝑆2	NOUN
cana-1295	109	56	and𝑇	and𝑇	ADJ
cana-1295	109	57	=	=	VERB
cana-1295	109	58	𝑇1	𝑇1	NOUN
cana-1295	109	59	∪	∪	ADJ
cana-1295	109	60	𝑇2	𝑇2	NOUN
cana-1295	109	61	be	be	AUX
cana-1295	109	62	two	two	NUM
cana-1295	109	63	bi	bi	ADJ
cana-1295	109	64	ternary	ternary	ADJ
cana-1295	109	65	semi	semi	ADJ
cana-1295	109	66	groups	group	NOUN
cana-1295	109	67	with	with	ADP
cana-1295	109	68	∅	∅	NOUN
cana-1295	109	69	is	be	AUX
cana-1295	109	70	a	a	DET
cana-1295	109	71	bi	bi	ADJ
cana-1295	109	72	ternary	ternary	NOUN
cana-1295	109	73	semi	semi	ADJ
cana-1295	109	74	group	group	NOUN
cana-1295	109	75	homomorphism	homomorphism	NOUN
cana-1295	109	76	from	from	ADP
cana-1295	109	77	𝑆	𝑆	PROPN
cana-1295	109	78	→	→	SYM
cana-1295	109	79	𝑇	𝑇	PROPN
cana-1295	109	80	then	then	ADV
cana-1295	109	81	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	PROPN
cana-1295	110	1	=	=	SYM
cana-1295	110	2	𝑘𝑒𝑟∅1	𝑘𝑒𝑟∅1	PROPN
cana-1295	110	3	∪	∪	ADP
cana-1295	110	4	𝑘𝑒𝑟∅2	𝑘𝑒𝑟∅2	PROPN
cana-1295	110	5	is	be	AUX
cana-1295	110	6	ideal	ideal	ADJ
cana-1295	110	7	of	of	ADP
cana-1295	110	8	the	the	DET
cana-1295	110	9	set	set	ADJ
cana-1295	110	10	𝑆.	𝑆.	NOUN
cana-1295	110	11	proof	proof	NOUN
cana-1295	110	12	:	:	PUNCT
cana-1295	110	13	given	give	VERB
cana-1295	110	14	that	that	SCONJ
cana-1295	110	15	𝑆	𝑆	PROPN
cana-1295	110	16	is	be	AUX
cana-1295	110	17	a	a	DET
cana-1295	110	18	bi	bi	ADJ
cana-1295	110	19	ternary	ternary	NOUN
cana-1295	110	20	semi	semi	NOUN
cana-1295	110	21	group	group	NOUN
cana-1295	110	22	.	.	PUNCT
cana-1295	111	1	let	let	VERB
cana-1295	111	2	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	NOUN
cana-1295	111	3	=	=	PRON
cana-1295	111	4	{	{	PUNCT
cana-1295	111	5	𝑎	𝑎	NOUN
cana-1295	111	6	∈	∈	NOUN
cana-1295	111	7	∅	∅	NOUN
cana-1295	111	8	:	:	PUNCT
cana-1295	111	9	∅(𝑎	∅(𝑎	ADJ
cana-1295	111	10	)	)	PUNCT
cana-1295	111	11	=	=	SYM
cana-1295	111	12	0′	0′	X
cana-1295	111	13	}	}	PUNCT
cana-1295	111	14	we	we	PRON
cana-1295	111	15	prove	prove	VERB
cana-1295	111	16	that	that	SCONJ
cana-1295	111	17	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	PROPN
cana-1295	111	18	is	be	AUX
cana-1295	111	19	an	an	DET
cana-1295	111	20	ideal	ideal	NOUN
cana-1295	111	21	of	of	ADP
cana-1295	111	22	𝑆.	𝑆.	PROPN
cana-1295	111	23	we	we	PRON
cana-1295	111	24	first	first	ADV
cana-1295	111	25	prove	prove	VERB
cana-1295	111	26	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	PROPN
cana-1295	111	27	is	be	AUX
cana-1295	111	28	a	a	DET
cana-1295	111	29	non	non	X
cana-1295	111	30	empty	empty	ADJ
cana-1295	111	31	set	set	NOUN
cana-1295	111	32	.	.	PUNCT
cana-1295	112	1	let	let	VERB
cana-1295	112	2	us	we	PRON
cana-1295	112	3	assume	assume	VERB
cana-1295	112	4	0	0	NUM
cana-1295	112	5	is	be	AUX
cana-1295	112	6	the	the	DET
cana-1295	112	7	0(zero	0(zero	NOUN
cana-1295	112	8	)	)	PUNCT
cana-1295	112	9	section	section	NOUN
cana-1295	112	10	of	of	ADP
cana-1295	112	11	𝑆	𝑆	PROPN
cana-1295	112	12	,	,	PUNCT
cana-1295	112	13	also	also	ADV
cana-1295	112	14	0′	0′	NUM
cana-1295	112	15	is	be	AUX
cana-1295	112	16	the	the	DET
cana-1295	112	17	0(zero	0(zero	NOUN
cana-1295	112	18	)	)	PUNCT
cana-1295	112	19	section	section	NOUN
cana-1295	112	20	of	of	ADP
cana-1295	112	21	𝑇.	𝑇.	PROPN
cana-1295	112	22	let	let	VERB
cana-1295	112	23	0	0	NUM
cana-1295	112	24	∈	∈	PROPN
cana-1295	112	25	𝑆	𝑆	PROPN
cana-1295	112	26	⇒	⇒	VERB
cana-1295	112	27	0	0	NUM
cana-1295	113	1	∈	∈	PROPN
cana-1295	113	2	𝑆1	𝑆1	NOUN
cana-1295	113	3	∪	∪	X
cana-1295	113	4	𝑆2	𝑆2	X
cana-1295	113	5	⟹	⟹	PUNCT
cana-1295	113	6	0	0	NUM
cana-1295	113	7	∈	∈	PROPN
cana-1295	113	8	𝑆1	𝑆1	PROPN
cana-1295	113	9	&	&	CCONJ
cana-1295	113	10	0	0	NUM
cana-1295	113	11	∈	∈	PROPN
cana-1295	113	12	𝑆2	𝑆2	PROPN
cana-1295	113	13	⇒	⇒	PROPN
cana-1295	113	14	∅1(0	∅1(0	PROPN
cana-1295	113	15	)	)	PUNCT
cana-1295	113	16	=	=	PUNCT
cana-1295	113	17	0′&∅2(0	0′&∅2(0	X
cana-1295	113	18	)	)	PUNCT
cana-1295	113	19	=	=	PUNCT
cana-1295	113	20	0′	0′	PUNCT
cana-1295	114	1	thus	thus	ADV
cana-1295	114	2	0	0	X
cana-1295	114	3	∈	∈	PROPN
cana-1295	114	4	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	PROPN
cana-1295	114	5	.hence	.hence	PROPN
cana-1295	114	6	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	PROPN
cana-1295	114	7	is	be	AUX
cana-1295	114	8	non	non	X
cana-1295	114	9	empty	empty	ADJ
cana-1295	114	10	.	.	PUNCT
cana-1295	115	1	communications	communication	NOUN
cana-1295	115	2	on	on	ADP
cana-1295	115	3	applied	apply	VERB
cana-1295	115	4	nonlinear	nonlinear	ADJ
cana-1295	115	5	analysis	analysis	NOUN
cana-1295	115	6	issn	issn	NOUN
cana-1295	115	7	:	:	PUNCT
cana-1295	115	8	1074	1074	NUM
cana-1295	115	9	-	-	PUNCT
cana-1295	115	10	133x	133x	NUM
cana-1295	115	11	vol	vol	NOUN
cana-1295	115	12	31	31	NUM
cana-1295	115	13	no	no	NOUN
cana-1295	115	14	.	.	PUNCT
cana-1295	116	1	7s	7	NOUN
cana-1295	116	2	(	(	PUNCT
cana-1295	116	3	2024	2024	NUM
cana-1295	116	4	)	)	PUNCT
cana-1295	116	5	212	212	NUM
cana-1295	117	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1295	117	2	let	let	VERB
cana-1295	117	3	𝑥	𝑥	PRON
cana-1295	117	4	,	,	PUNCT
cana-1295	117	5	𝑦	𝑦	PRON
cana-1295	117	6	∈	∈	NOUN
cana-1295	117	7	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	NOUN
cana-1295	117	8	by	by	ADP
cana-1295	117	9	the	the	DET
cana-1295	117	10	definition	definition	NOUN
cana-1295	117	11	of	of	ADP
cana-1295	117	12	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	PROPN
cana-1295	117	13	if	if	SCONJ
cana-1295	117	14	𝑥	𝑥	PROPN
cana-1295	117	15	∈	∈	PROPN
cana-1295	117	16	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	NOUN
cana-1295	117	17	⇒	⇒	PROPN
cana-1295	117	18	∅(𝑥	∅(𝑥	PROPN
cana-1295	117	19	)	)	PUNCT
cana-1295	117	20	=	=	PUNCT
cana-1295	118	1	0′&∅(𝑦	0′&∅(𝑦	X
cana-1295	118	2	)	)	PUNCT
cana-1295	118	3	=	=	SYM
cana-1295	118	4	0′	0′	PROPN
cana-1295	119	1	let	let	VERB
cana-1295	119	2	𝑎	𝑎	NOUN
cana-1295	119	3	,	,	PUNCT
cana-1295	119	4	𝑏	𝑏	PROPN
cana-1295	119	5	∈	∈	PROPN
cana-1295	119	6	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	PROPN
cana-1295	119	7	&	&	CCONJ
cana-1295	119	8	𝑠	𝑠	PROPN
cana-1295	119	9	∈	∈	PROPN
cana-1295	119	10	𝑆	𝑆	PROPN
cana-1295	119	11	since	since	SCONJ
cana-1295	119	12	𝑎	𝑎	PROPN
cana-1295	119	13	,	,	PUNCT
cana-1295	119	14	𝑏	𝑏	PROPN
cana-1295	119	15	∈	∈	PROPN
cana-1295	119	16	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	NOUN
cana-1295	119	17	⇒	⇒	NOUN
cana-1295	119	18	∅(a	∅(a	PROPN
cana-1295	119	19	)	)	PUNCT
cana-1295	119	20	=	=	SYM
cana-1295	119	21	0′	0′	PROPN
cana-1295	119	22	&	&	CCONJ
cana-1295	119	23	∅(b	∅(b	PROPN
cana-1295	119	24	)	)	PUNCT
cana-1295	119	25	=	=	PUNCT
cana-1295	119	26	0′	0′	NUM
cana-1295	120	1	consider,∅(𝑎𝑏𝑠	consider,∅(𝑎𝑏𝑠	NUM
cana-1295	120	2	)	)	PUNCT
cana-1295	120	3	=	=	SYM
cana-1295	120	4	∅(𝑎)∅(𝑏)∅(𝑠	∅(𝑎)∅(𝑏)∅(𝑠	NOUN
cana-1295	120	5	)	)	PUNCT
cana-1295	120	6	=	=	NOUN
cana-1295	120	7	0′.	0′.	NOUN
cana-1295	120	8	0′.	0′.	X
cana-1295	120	9	∅(𝑠	∅(𝑠	PROPN
cana-1295	120	10	)	)	PUNCT
cana-1295	120	11	=	=	SYM
cana-1295	120	12	∅(𝑠	∅(𝑠	NOUN
cana-1295	120	13	)	)	PUNCT
cana-1295	120	14	=	=	SYM
cana-1295	120	15	0′	0′	PROPN
cana-1295	121	1	∴	∴	PROPN
cana-1295	121	2	𝑎𝑏𝑠	𝑎𝑏𝑠	PROPN
cana-1295	121	3	∈	∈	PROPN
cana-1295	121	4	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	NOUN
cana-1295	121	5	similarly	similarly	ADV
cana-1295	121	6	𝑠𝑎𝑏	𝑠𝑎𝑏	PROPN
cana-1295	121	7	∈	∈	PROPN
cana-1295	121	8	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	PROPN
cana-1295	121	9	,	,	PUNCT
cana-1295	121	10	𝑎𝑠𝑏	𝑎𝑠𝑏	PROPN
cana-1295	121	11	∈	∈	PROPN
cana-1295	121	12	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	NOUN
cana-1295	121	13	hence	hence	ADV
cana-1295	121	14	∀𝑎	∀𝑎	PROPN
cana-1295	121	15	,	,	PUNCT
cana-1295	121	16	𝑏	𝑏	PROPN
cana-1295	121	17	∈	∈	PROPN
cana-1295	121	18	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	NOUN
cana-1295	121	19	&	&	CCONJ
cana-1295	121	20	𝑠	𝑠	PROPN
cana-1295	121	21	∈	∈	PROPN
cana-1295	121	22	𝑆	𝑆	PROPN
cana-1295	121	23	𝑡ℎ𝑒𝑛	𝑡ℎ𝑒𝑛	VERB
cana-1295	121	24	𝑎𝑏𝑠	𝑎𝑏𝑠	PROPN
cana-1295	121	25	∈	∈	PROPN
cana-1295	121	26	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	PROPN
cana-1295	121	27	,	,	PUNCT
cana-1295	121	28	𝑠𝑎𝑏	𝑠𝑎𝑏	PROPN
cana-1295	121	29	∈	∈	PROPN
cana-1295	121	30	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	PROPN
cana-1295	121	31	,	,	PUNCT
cana-1295	121	32	𝑎𝑠𝑏	𝑎𝑠𝑏	PROPN
cana-1295	121	33	∈	∈	PROPN
cana-1295	121	34	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	NOUN
cana-1295	122	1	so	so	ADV
cana-1295	122	2	𝑘𝑒𝑟∅	𝑘𝑒𝑟∅	PROPN
cana-1295	122	3	is	be	AUX
cana-1295	122	4	an	an	DET
cana-1295	122	5	ideal	ideal	NOUN
cana-1295	122	6	of	of	ADP
cana-1295	122	7	s.	s.	PROPN
cana-1295	122	8	theorem	theorem	VERB
cana-1295	122	9	4.8	4.8	NUM
cana-1295	122	10	:	:	PUNCT
cana-1295	122	11	the	the	DET
cana-1295	122	12	homomorphic	homomorphic	ADJ
cana-1295	122	13	image	image	NOUN
cana-1295	122	14	of	of	ADP
cana-1295	122	15	an	an	DET
cana-1295	122	16	ideal	ideal	NOUN
cana-1295	122	17	is	be	AUX
cana-1295	122	18	an	an	DET
cana-1295	122	19	ideal	ideal	NOUN
cana-1295	122	20	.	.	PUNCT
cana-1295	123	1	if	if	SCONJ
cana-1295	123	2	∅	∅	NOUN
cana-1295	123	3	:	:	PUNCT
cana-1295	123	4	𝑆	𝑆	PROPN
cana-1295	123	5	→	→	SYM
cana-1295	123	6	𝑇	𝑇	PROPN
cana-1295	123	7	is	be	AUX
cana-1295	123	8	an	an	DET
cana-1295	123	9	bi	bi	ADJ
cana-1295	123	10	ternary	ternary	NOUN
cana-1295	123	11	semi	semi	PROPN
cana-1295	123	12	group	group	NOUN
cana-1295	123	13	onto	onto	ADP
cana-1295	123	14	homomorphism	homomorphism	PROPN
cana-1295	123	15	and	and	CCONJ
cana-1295	123	16	𝐴	𝐴	PROPN
cana-1295	123	17	is	be	AUX
cana-1295	123	18	an	an	DET
cana-1295	123	19	ideal	ideal	NOUN
cana-1295	123	20	of	of	ADP
cana-1295	123	21	𝑆	𝑆	PROPN
cana-1295	123	22	then	then	ADV
cana-1295	123	23	∅(𝐴	∅(𝐴	PUNCT
cana-1295	123	24	)	)	PUNCT
cana-1295	123	25	is	be	AUX
cana-1295	123	26	an	an	DET
cana-1295	123	27	ideal	ideal	NOUN
cana-1295	123	28	of	of	ADP
cana-1295	123	29	𝑇.	𝑇.	PROPN
cana-1295	123	30	proof	proof	NOUN
cana-1295	123	31	:	:	PUNCT
cana-1295	123	32	perfect	perfect	ADJ
cana-1295	123	33	0	0	NUM
cana-1295	123	34	&	&	CCONJ
cana-1295	123	35	0′	0′	NUM
cana-1295	124	1	be	be	AUX
cana-1295	124	2	the	the	DET
cana-1295	124	3	additive	additive	ADJ
cana-1295	124	4	identity	identity	NOUN
cana-1295	124	5	of	of	ADP
cana-1295	124	6	𝑆	𝑆	PROPN
cana-1295	124	7	&	&	CCONJ
cana-1295	124	8	𝑇	𝑇	PROPN
cana-1295	124	9	respectively	respectively	ADV
cana-1295	124	10	.	.	PUNCT
cana-1295	125	1	∅(𝐴	∅(𝐴	X
cana-1295	125	2	)	)	PUNCT
cana-1295	125	3	=	=	PRON
cana-1295	125	4	{	{	PUNCT
cana-1295	125	5	𝜇′	𝜇′	CCONJ
cana-1295	125	6	∈	∈	PROPN
cana-1295	125	7	𝑇	𝑇	PROPN
cana-1295	125	8	:	:	PUNCT
cana-1295	125	9	∃𝜇	∃𝜇	PROPN
cana-1295	125	10	∈	∈	PROPN
cana-1295	125	11	𝑆	𝑆	PROPN
cana-1295	125	12	𝑠.	𝑠.	NOUN
cana-1295	125	13	𝑡	𝑡	VERB
cana-1295	125	14	∅(𝜇	∅(𝜇	PROPN
cana-1295	125	15	)	)	PUNCT
cana-1295	125	16	=	=	SYM
cana-1295	125	17	𝜇′	𝜇′	CCONJ
cana-1295	125	18	}	}	PUNCT
cana-1295	125	19	0	0	NUM
cana-1295	125	20	∈	∈	PROPN
cana-1295	125	21	𝐴	𝐴	PROPN
cana-1295	125	22	𝑠.	𝑠.	NOUN
cana-1295	125	23	𝑡	𝑡	PROPN
cana-1295	125	24	∅(0	∅(0	PROPN
cana-1295	125	25	)	)	PUNCT
cana-1295	125	26	∈	∈	PROPN
cana-1295	125	27	∅(𝐴	∅(𝐴	NOUN
cana-1295	125	28	)	)	PUNCT
cana-1295	125	29	⇒	⇒	NOUN
cana-1295	125	30	0′	0′	PUNCT
cana-1295	126	1	∈	∈	PROPN
cana-1295	126	2	∅(𝐴	∅(𝐴	NOUN
cana-1295	126	3	)	)	PUNCT
cana-1295	126	4	where	where	SCONJ
cana-1295	126	5	∅(0	∅(0	NOUN
cana-1295	126	6	)	)	PUNCT
cana-1295	126	7	=	=	PUNCT
cana-1295	126	8	0′	0′	NUM
cana-1295	126	9	∈	∈	PROPN
cana-1295	126	10	𝑇	𝑇	PROPN
cana-1295	126	11	∴	∴	PROPN
cana-1295	126	12	∅(𝐴	∅(𝐴	NOUN
cana-1295	126	13	)	)	PUNCT
cana-1295	126	14	is	be	AUX
cana-1295	126	15	a	a	DET
cana-1295	126	16	non	non	ADJ
cana-1295	126	17	-	-	ADJ
cana-1295	126	18	empty	empty	ADJ
cana-1295	126	19	.	.	PUNCT
cana-1295	127	1	∴	∴	NOUN
cana-1295	127	2	∅(𝐴	∅(𝐴	PROPN
cana-1295	127	3	)	)	PUNCT
cana-1295	128	1	⊆	⊆	NUM
cana-1295	128	2	𝑇	𝑇	PROPN
cana-1295	128	3	let	let	VERB
cana-1295	128	4	𝑎′	𝑎′	NUM
cana-1295	128	5	,	,	PUNCT
cana-1295	128	6	𝑏′	𝑏′	NUM
cana-1295	128	7	∈	∈	PROPN
cana-1295	128	8	∅(𝐴)∃𝑎	∅(𝐴)∃𝑎	PROPN
cana-1295	128	9	,	,	PUNCT
cana-1295	128	10	𝑏	𝑏	PROPN
cana-1295	128	11	∈	∈	PROPN
cana-1295	128	12	𝐴	𝐴	PROPN
cana-1295	128	13	such	such	ADJ
cana-1295	128	14	that∅(𝑎	that∅(𝑎	NOUN
cana-1295	128	15	)	)	PUNCT
cana-1295	128	16	=	=	SYM
cana-1295	128	17	𝑎′	𝑎′	PRON
cana-1295	128	18	,	,	PUNCT
cana-1295	128	19	∅(𝑏	∅(𝑏	NOUN
cana-1295	128	20	)	)	PUNCT
cana-1295	128	21	=	=	SYM
cana-1295	128	22	𝑏′	𝑏′	PROPN
cana-1295	128	23	let	let	VERB
cana-1295	128	24	𝑎′	𝑎′	NUM
cana-1295	128	25	,	,	PUNCT
cana-1295	128	26	𝑏′	𝑏′	PROPN
cana-1295	128	27	∈	∈	PROPN
cana-1295	128	28	∅(𝐴	∅(𝐴	NOUN
cana-1295	128	29	)	)	PUNCT
cana-1295	128	30	&	&	CCONJ
cana-1295	128	31	𝑡	𝑡	PROPN
cana-1295	128	32	∈	∈	PROPN
cana-1295	128	33	𝑇.	𝑇.	PROPN
cana-1295	128	34	we	we	PRON
cana-1295	128	35	prove	prove	VERB
cana-1295	128	36	𝑎′𝑏′𝑡	𝑎′𝑏′𝑡	PROPN
cana-1295	128	37	∈	∈	PROPN
cana-1295	128	38	∅(𝐴	∅(𝐴	ADP
cana-1295	128	39	)	)	PUNCT
cana-1295	128	40	since	since	SCONJ
cana-1295	128	41	∅	∅	NOUN
cana-1295	128	42	is	be	AUX
cana-1295	128	43	onto	onto	ADP
cana-1295	128	44	∃𝑠	∃𝑠	PROPN
cana-1295	128	45	∈	∈	PROPN
cana-1295	128	46	𝑆	𝑆	PROPN
cana-1295	128	47	𝑠.	𝑠.	NOUN
cana-1295	128	48	𝑡	𝑡	PROPN
cana-1295	128	49	∅(𝑠	∅(𝑠	PROPN
cana-1295	128	50	)	)	PUNCT
cana-1295	128	51	=	=	SYM
cana-1295	128	52	𝑡	𝑡	NOUN
cana-1295	128	53	consider	consider	VERB
cana-1295	128	54	,	,	PUNCT
cana-1295	128	55	𝑎′𝑏′𝑡	𝑎′𝑏′𝑡	NOUN
cana-1295	128	56	=	=	SYM
cana-1295	128	57	∅(𝑎)∅(𝑏)∅(𝑠	∅(𝑎)∅(𝑏)∅(𝑠	NOUN
cana-1295	128	58	)	)	PUNCT
cana-1295	128	59	=	=	SYM
cana-1295	128	60	∅(𝑎𝑏𝑠	∅(𝑎𝑏𝑠	NOUN
cana-1295	128	61	)	)	PUNCT
cana-1295	128	62	∈	∈	PROPN
cana-1295	128	63	∅(𝐴	∅(𝐴	NOUN
cana-1295	128	64	)	)	PUNCT
cana-1295	128	65	since	since	SCONJ
cana-1295	128	66	𝑎	𝑎	X
cana-1295	128	67	,	,	PUNCT
cana-1295	128	68	𝑏	𝑏	PROPN
cana-1295	128	69	∈	∈	PROPN
cana-1295	128	70	𝐴	𝐴	PROPN
cana-1295	128	71	&	&	CCONJ
cana-1295	128	72	𝑠	𝑠	PROPN
cana-1295	128	73	∈	∈	PROPN
cana-1295	128	74	𝑆	𝑆	PROPN
cana-1295	128	75	⇒	⇒	NOUN
cana-1295	128	76	𝑎𝑏𝑠	𝑎𝑏𝑠	PROPN
cana-1295	128	77	∈	∈	PROPN
cana-1295	128	78	𝐴	𝐴	PROPN
cana-1295	128	79	has	have	VERB
cana-1295	128	80	a	a	DET
cana-1295	128	81	perfect	perfect	NOUN
cana-1295	128	82	of	of	ADP
cana-1295	128	83	s.	s.	PROPN
cana-1295	128	84	∴	∴	PROPN
cana-1295	128	85	𝑎′𝑏′𝑡	𝑎′𝑏′𝑡	PROPN
cana-1295	128	86	∈	∈	PROPN
cana-1295	128	87	∅(𝐴	∅(𝐴	NOUN
cana-1295	128	88	)	)	PUNCT
cana-1295	128	89	similarly	similarly	ADV
cana-1295	128	90	𝑡𝑎′𝑏′	𝑡𝑎′𝑏′	PROPN
cana-1295	128	91	∈	∈	PROPN
cana-1295	128	92	∅(𝐴)&𝑎′𝑡𝑏′	∅(𝐴)&𝑎′𝑡𝑏′	NOUN
cana-1295	128	93	∈	∈	PROPN
cana-1295	128	94	∅(𝐴	∅(𝐴	NOUN
cana-1295	128	95	)	)	PUNCT
cana-1295	128	96	hence	hence	ADV
cana-1295	128	97	∅(𝐴	∅(𝐴	PUNCT
cana-1295	128	98	)	)	PUNCT
cana-1295	128	99	is	be	AUX
cana-1295	128	100	the	the	DET
cana-1295	128	101	left	left	NOUN
cana-1295	128	102	,	,	PUNCT
cana-1295	128	103	on	on	ADP
cana-1295	128	104	the	the	DET
cana-1295	128	105	side	side	NOUN
cana-1295	128	106	,	,	PUNCT
cana-1295	128	107	right	right	ADV
cana-1295	128	108	perfect	perfect	ADJ
cana-1295	128	109	of	of	ADP
cana-1295	128	110	𝑇.	𝑇.	PROPN
cana-1295	128	111	hereafter	hereafter	NOUN
cana-1295	128	112	∅(𝐴	∅(𝐴	PRON
cana-1295	128	113	)	)	PUNCT
cana-1295	128	114	be	be	AUX
cana-1295	128	115	idealistic	idealistic	ADJ
cana-1295	128	116	of	of	ADP
cana-1295	128	117	𝑇.	𝑇.	PROPN
cana-1295	128	118	definition	definition	NOUN
cana-1295	128	119	5.1	5.1	NUM
cana-1295	128	120	:	:	PUNCT
cana-1295	128	121	bi	bi	ADJ
cana-1295	128	122	ternary	ternary	ADJ
cana-1295	128	123	semi	semi	ADJ
cana-1295	128	124	field	field	NOUN
cana-1295	128	125	:	:	PUNCT
cana-1295	128	126	let	let	VERB
cana-1295	128	127	𝑇	𝑇	PROPN
cana-1295	128	128	is	be	AUX
cana-1295	128	129	a	a	DET
cana-1295	128	130	bi	bi	ADJ
cana-1295	128	131	ternary	ternary	NOUN
cana-1295	128	132	semi	semi	ADJ
cana-1295	128	133	ring	ring	NOUN
cana-1295	128	134	,	,	PUNCT
cana-1295	128	135	𝑇	𝑇	PROPN
cana-1295	128	136	said	say	VERB
cana-1295	128	137	of	of	ADP
cana-1295	128	138	t	t	PROPN
cana-1295	128	139	is	be	AUX
cana-1295	128	140	a	a	DET
cana-1295	128	141	bi	bi	ADJ
cana-1295	128	142	-	-	NOUN
cana-1295	128	143	digit	digit	NOUN
cana-1295	128	144	semi	semi	ADJ
cana-1295	128	145	field	field	NOUN
cana-1295	128	146	condition	condition	NOUN
cana-1295	128	147	𝑇1	𝑇1	NOUN
cana-1295	128	148	is	be	AUX
cana-1295	128	149	a	a	DET
cana-1295	128	150	ternary	ternary	ADJ
cana-1295	128	151	semi	semi	ADJ
cana-1295	128	152	field	field	NOUN
cana-1295	128	153	&	&	CCONJ
cana-1295	128	154	𝑇2	𝑇2	PROPN
cana-1295	128	155	is	be	AUX
cana-1295	128	156	a	a	DET
cana-1295	128	157	ternary	ternary	ADJ
cana-1295	128	158	semi	semi	ADJ
cana-1295	128	159	field	field	NOUN
cana-1295	128	160	everyplace	everyplace	NOUN
cana-1295	128	161	𝑇	𝑇	PROPN
cana-1295	128	162	=	=	SYM
cana-1295	128	163	𝑇1	𝑇1	NOUN
cana-1295	128	164	∪	∪	VERB
cana-1295	128	165	𝑇2.i.e	𝑇2.i.e	ADV
cana-1295	128	166	.	.	PUNCT
cana-1295	129	1	equally	equally	ADV
cana-1295	129	2	𝑇1	𝑇1	PROPN
cana-1295	129	3	&	&	CCONJ
cana-1295	129	4	𝑇2	𝑇2	PROPN
cana-1295	129	5	stand	stand	NOUN
cana-1295	129	6	shifting	shift	VERB
cana-1295	129	7	ternary	ternary	ADJ
cana-1295	129	8	semi	semi	ADJ
cana-1295	129	9	rings	ring	NOUN
cana-1295	129	10	thru	thru	ADP
cana-1295	129	11	non	non	ADJ
cana-1295	129	12	zero(0	zero(0	NOUN
cana-1295	129	13	)	)	PUNCT
cana-1295	129	14	element	element	NOUN
cana-1295	129	15	has	have	VERB
cana-1295	129	16	multiplicative	multiplicative	ADJ
cana-1295	129	17	inverse	inverse	NOUN
cana-1295	129	18	.	.	PUNCT
cana-1295	130	1	communications	communication	NOUN
cana-1295	130	2	on	on	ADP
cana-1295	130	3	applied	apply	VERB
cana-1295	130	4	nonlinear	nonlinear	ADJ
cana-1295	130	5	analysis	analysis	NOUN
cana-1295	130	6	issn	issn	NOUN
cana-1295	130	7	:	:	PUNCT
cana-1295	130	8	1074	1074	NUM
cana-1295	130	9	-	-	PUNCT
cana-1295	130	10	133x	133x	NUM
cana-1295	130	11	vol	vol	NOUN
cana-1295	130	12	31	31	NUM
cana-1295	130	13	no	no	NOUN
cana-1295	130	14	.	.	PUNCT
cana-1295	131	1	7s	7	NOUN
cana-1295	131	2	(	(	PUNCT
cana-1295	131	3	2024	2024	NUM
cana-1295	131	4	)	)	PUNCT
cana-1295	131	5	213	213	NUM
cana-1295	131	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1295	131	7	references	reference	NOUN
cana-1295	131	8	:	:	PUNCT
cana-1295	131	9	[	[	X
cana-1295	131	10	1	1	NUM
cana-1295	131	11	]	]	X
cana-1295	131	12	rao	rao	PROPN
cana-1295	131	13	,	,	PUNCT
cana-1295	131	14	d.	d.	PROPN
cana-1295	131	15	m.	m.	PROPN
cana-1295	131	16	,	,	PUNCT
cana-1295	131	17	&	&	CCONJ
cana-1295	131	18	srinivasa	srinivasa	PROPN
cana-1295	131	19	rao	rao	PROPN
cana-1295	131	20	,	,	PUNCT
cana-1295	131	21	g.	g.	PROPN
cana-1295	131	22	(	(	PUNCT
cana-1295	131	23	2014	2014	NUM
cana-1295	131	24	)	)	PUNCT
cana-1295	131	25	.	.	PUNCT
cana-1295	132	1	concepts	concept	NOUN
cana-1295	132	2	on	on	ADP
cana-1295	132	3	ternary	ternary	ADJ
cana-1295	132	4	semirings	semiring	NOUN
cana-1295	132	5	.	.	PUNCT
cana-1295	133	1	international	international	ADJ
cana-1295	133	2	journal	journal	PROPN
cana-1295	133	3	of	of	ADP
cana-1295	133	4	modern	modern	ADJ
cana-1295	133	5	sciences	sciences	PROPN
cana-1295	133	6	and	and	CCONJ
cana-1295	133	7	engineering	engineering	NOUN
cana-1295	133	8	technology	technology	NOUN
cana-1295	133	9	,	,	PUNCT
cana-1295	133	10	1(7	1(7	NUM
cana-1295	133	11	)	)	PUNCT
cana-1295	133	12	,	,	PUNCT
cana-1295	133	13	105	105	NUM
cana-1295	133	14	-	-	SYM
cana-1295	133	15	110	110	NUM
cana-1295	133	16	.	.	PUNCT
cana-1295	134	1	[	[	X
cana-1295	134	2	2	2	NUM
cana-1295	134	3	]	]	X
cana-1295	134	4	rao	rao	PROPN
cana-1295	134	5	,	,	PUNCT
cana-1295	134	6	d.	d.	PROPN
cana-1295	134	7	m.	m.	PROPN
cana-1295	134	8	,	,	PUNCT
cana-1295	134	9	&	&	CCONJ
cana-1295	134	10	srinivasa	srinivasa	PROPN
cana-1295	134	11	rao	rao	PROPN
cana-1295	134	12	,	,	PUNCT
cana-1295	134	13	g.	g.	PROPN
cana-1295	134	14	(	(	PUNCT
cana-1295	134	15	2015	2015	NUM
cana-1295	134	16	)	)	PUNCT
cana-1295	134	17	.	.	PUNCT
cana-1295	135	1	structure	structure	NOUN
cana-1295	135	2	of	of	ADP
cana-1295	135	3	certain	certain	ADJ
cana-1295	135	4	ideals	ideal	NOUN
cana-1295	135	5	in	in	ADP
cana-1295	135	6	ternary	ternary	ADJ
cana-1295	135	7	semirings	semiring	NOUN
cana-1295	135	8	.	.	PUNCT
cana-1295	136	1	international	international	ADJ
cana-1295	136	2	journal	journal	NOUN
cana-1295	136	3	of	of	ADP
cana-1295	136	4	innovative	innovative	ADJ
cana-1295	136	5	science	science	NOUN
cana-1295	136	6	and	and	CCONJ
cana-1295	136	7	modern	modern	ADJ
cana-1295	136	8	engineering	engineering	NOUN
cana-1295	136	9	(	(	PUNCT
cana-1295	136	10	ijisme	ijisme	ADJ
cana-1295	136	11	)	)	PUNCT
cana-1295	136	12	,	,	PUNCT
cana-1295	136	13	3	3	NUM
cana-1295	136	14	,	,	PUNCT
cana-1295	136	15	49	49	NUM
cana-1295	136	16	-	-	SYM
cana-1295	136	17	56	56	NUM
cana-1295	136	18	.	.	PUNCT
cana-1295	137	1	[	[	X
cana-1295	137	2	3	3	NUM
cana-1295	137	3	]	]	X
cana-1295	137	4	sarala	sarala	NOUN
cana-1295	137	5	,	,	PUNCT
cana-1295	137	6	y.	y.	PROPN
cana-1295	137	7	,	,	PUNCT
cana-1295	137	8	anjaneyulu	anjaneyulu	VERB
cana-1295	137	9	,	,	PUNCT
cana-1295	137	10	a.	a.	NOUN
cana-1295	137	11	,	,	PUNCT
cana-1295	137	12	&	&	CCONJ
cana-1295	137	13	rao	rao	PROPN
cana-1295	137	14	,	,	PUNCT
cana-1295	137	15	d.	d.	PROPN
cana-1295	137	16	m.	m.	PROPN
cana-1295	137	17	(	(	PUNCT
cana-1295	137	18	2013	2013	NUM
cana-1295	137	19	)	)	PUNCT
cana-1295	137	20	.	.	PUNCT
cana-1295	138	1	ternary	ternary	ADJ
cana-1295	138	2	semi	semi	ADJ
cana-1295	138	3	groups	group	NOUN
cana-1295	138	4	.	.	PUNCT
cana-1295	139	1	international	international	ADJ
cana-1295	139	2	journal	journal	PROPN
cana-1295	139	3	of	of	ADP
cana-1295	139	4	mathematics	mathematics	PROPN
cana-1295	139	5	sciences	science	NOUN
cana-1295	139	6	,	,	PUNCT
cana-1295	139	7	technology	technology	NOUN
cana-1295	139	8	and	and	CCONJ
cana-1295	139	9	humanities	humanity	NOUN
cana-1295	139	10	,	,	PUNCT
cana-1295	139	11	76	76	NUM
cana-1295	139	12	,	,	PUNCT
cana-1295	139	13	848	848	NUM
cana-1295	139	14	-	-	SYM
cana-1295	139	15	859	859	NUM
cana-1295	139	16	.	.	PUNCT
cana-1295	140	1	[	[	X
cana-1295	140	2	4	4	NUM
cana-1295	140	3	]	]	X
cana-1295	140	4	dixit	dixit	PROPN
cana-1295	140	5	,	,	PUNCT
cana-1295	140	6	v.	v.	PROPN
cana-1295	140	7	n.	n.	PROPN
cana-1295	140	8	,	,	PUNCT
cana-1295	140	9	&	&	CCONJ
cana-1295	140	10	dewan	dewan	PROPN
cana-1295	140	11	,	,	PUNCT
cana-1295	140	12	s.	s.	PROPN
cana-1295	140	13	(	(	PUNCT
cana-1295	140	14	1995	1995	NUM
cana-1295	140	15	)	)	PUNCT
cana-1295	140	16	.	.	PUNCT
cana-1295	141	1	a	a	DET
cana-1295	141	2	note	note	NOUN
cana-1295	141	3	on	on	ADP
cana-1295	141	4	quasi	quasi	NOUN
cana-1295	141	5	and	and	CCONJ
cana-1295	141	6	bi	bi	NOUN
cana-1295	141	7	-	-	NOUN
cana-1295	141	8	ideals	ideal	NOUN
cana-1295	141	9	in	in	ADP
cana-1295	141	10	ternary	ternary	ADJ
cana-1295	141	11	semi	semi	ADJ
cana-1295	141	12	groups	group	NOUN
cana-1295	141	13	.	.	PUNCT
cana-1295	142	1	international	international	ADJ
cana-1295	142	2	journal	journal	PROPN
cana-1295	142	3	of	of	ADP
cana-1295	142	4	mathematics	mathematics	PROPN
cana-1295	142	5	and	and	CCONJ
cana-1295	142	6	mathematical	mathematical	ADJ
cana-1295	142	7	sciences	science	NOUN
cana-1295	142	8	,	,	PUNCT
cana-1295	142	9	18	18	NUM
cana-1295	142	10	,	,	PUNCT
cana-1295	142	11	501	501	NUM
cana-1295	142	12	-	-	SYM
cana-1295	142	13	508.5	508.5	NUM
cana-1295	142	14	.	.	PUNCT
cana-1295	143	1	kar	kar	PROPN
cana-1295	143	2	,	,	PUNCT
cana-1295	143	3	s.	s.	PROPN
cana-1295	143	4	,	,	PUNCT
cana-1295	143	5	&	&	CCONJ
cana-1295	143	6	maity	maity	PROPN
cana-1295	143	7	,	,	PUNCT
cana-1295	143	8	b.	b.	PROPN
cana-1295	143	9	k.	k.	PROPN
cana-1295	143	10	(	(	PUNCT
cana-1295	143	11	2011	2011	NUM
cana-1295	143	12	)	)	PUNCT
cana-1295	143	13	.	.	PUNCT
cana-1295	144	1	some	some	DET
cana-1295	144	2	ideals	ideal	NOUN
cana-1295	144	3	of	of	ADP
cana-1295	144	4	ternary	ternary	ADJ
cana-1295	144	5	semi	semi	ADJ
cana-1295	144	6	groups	group	NOUN
cana-1295	144	7	.	.	PUNCT
cana-1295	145	1	annals	annal	NOUN
cana-1295	145	2	of	of	ADP
cana-1295	145	3	the	the	DET
cana-1295	145	4	alexandru	alexandru	PROPN
cana-1295	145	5	ioan	ioan	PROPN
cana-1295	145	6	cuza	cuza	PROPN
cana-1295	145	7	university	university	NOUN
cana-1295	145	8	-	-	PUNCT
cana-1295	145	9	mathematics	mathematic	NOUN
cana-1295	145	10	,	,	PUNCT
cana-1295	145	11	57(2	57(2	NUM
cana-1295	145	12	)	)	PUNCT
cana-1295	145	13	,	,	PUNCT
cana-1295	145	14	247	247	NUM
cana-1295	145	15	-	-	SYM
cana-1295	145	16	258	258	NUM
cana-1295	145	17	.	.	PUNCT
cana-1295	146	1	[	[	X
cana-1295	146	2	5	5	NUM
cana-1295	146	3	]	]	X
cana-1295	146	4	chinram	chinram	PROPN
cana-1295	146	5	,	,	PUNCT
cana-1295	146	6	r.	r.	PROPN
cana-1295	146	7	,	,	PUNCT
cana-1295	146	8	baupradist	baupradist	NOUN
cana-1295	146	9	,	,	PUNCT
cana-1295	146	10	s.	s.	PROPN
cana-1295	146	11	,	,	PUNCT
cana-1295	146	12	&	&	CCONJ
cana-1295	146	13	saelee	saelee	PROPN
cana-1295	146	14	,	,	PUNCT
cana-1295	146	15	s.	s.	PROPN
cana-1295	146	16	(	(	PUNCT
cana-1295	146	17	2012	2012	NUM
cana-1295	146	18	)	)	PUNCT
cana-1295	146	19	.	.	PUNCT
cana-1295	147	1	minimal	minimal	ADJ
cana-1295	147	2	and	and	CCONJ
cana-1295	147	3	maximal	maximal	ADJ
cana-1295	147	4	bi	bi	NOUN
cana-1295	147	5	-	-	NOUN
cana-1295	147	6	ideals	ideal	NOUN
cana-1295	147	7	in	in	ADP
cana-1295	147	8	ordered	order	VERB
cana-1295	147	9	ternary	ternary	ADJ
cana-1295	147	10	semi	semi	ADJ
cana-1295	147	11	groups	group	NOUN
cana-1295	147	12	.	.	PUNCT
cana-1295	148	1	int	int	NOUN
cana-1295	148	2	.	.	PUNCT
cana-1295	149	1	j.	j.	PROPN
cana-1295	149	2	phys	phys	PROPN
cana-1295	149	3	.	.	PUNCT
cana-1295	150	1	sci	sci	PROPN
cana-1295	150	2	,	,	PUNCT
cana-1295	150	3	7	7	NUM
cana-1295	150	4	,	,	PUNCT
cana-1295	150	5	2674	2674	NUM
cana-1295	150	6	-	-	SYM
cana-1295	150	7	2681	2681	NUM
cana-1295	150	8	.	.	PUNCT
cana-1295	151	1	[	[	X
cana-1295	151	2	6	6	NUM
cana-1295	151	3	]	]	SYM
cana-1295	151	4	daddi	daddi	NOUN
cana-1295	151	5	,	,	PUNCT
cana-1295	151	6	v.	v.	PROPN
cana-1295	151	7	r.	r.	PROPN
cana-1295	151	8	,	,	PUNCT
cana-1295	151	9	&	&	CCONJ
cana-1295	151	10	pawar	pawar	PROPN
cana-1295	151	11	,	,	PUNCT
cana-1295	151	12	y.	y.	PROPN
cana-1295	151	13	s.	s.	PROPN
cana-1295	151	14	(	(	PUNCT
cana-1295	151	15	2012	2012	NUM
cana-1295	151	16	)	)	PUNCT
cana-1295	151	17	.	.	PUNCT
cana-1295	152	1	on	on	ADP
cana-1295	152	2	ordered	order	VERB
cana-1295	152	3	ternary	ternary	ADJ
cana-1295	152	4	semi	semi	ADJ
cana-1295	152	5	groups	group	NOUN
cana-1295	152	6	.	.	PUNCT
cana-1295	153	1	kyungpook	kyungpook	PROPN
cana-1295	153	2	mathematical	mathematical	PROPN
cana-1295	153	3	journal	journal	PROPN
cana-1295	153	4	,	,	PUNCT
cana-1295	153	5	52(4	52(4	NUM
cana-1295	153	6	)	)	PUNCT
cana-1295	153	7	.	.	PUNCT
cana-1295	154	1	[	[	X
cana-1295	154	2	7	7	NUM
cana-1295	154	3	]	]	X
cana-1295	154	4	sarala	sarala	NOUN
cana-1295	154	5	,	,	PUNCT
cana-1295	154	6	y.	y.	PROPN
cana-1295	154	7	,	,	PUNCT
cana-1295	154	8	anjaneyulu	anjaneyulu	VERB
cana-1295	154	9	,	,	PUNCT
cana-1295	154	10	a.	a.	NOUN
cana-1295	154	11	,	,	PUNCT
cana-1295	154	12	&	&	CCONJ
cana-1295	154	13	rao	rao	PROPN
cana-1295	154	14	,	,	PUNCT
cana-1295	154	15	m.	m.	PROPN
cana-1295	154	16	d.	d.	PROPN
cana-1295	154	17	(	(	PUNCT
cana-1295	154	18	2019	2019	NUM
cana-1295	154	19	)	)	PUNCT
cana-1295	154	20	,	,	PUNCT
cana-1295	154	21	prime	prime	ADJ
cana-1295	154	22	radicals	radical	NOUN
cana-1295	154	23	in	in	ADP
cana-1295	154	24	ternary	ternary	ADJ
cana-1295	154	25	semi	semi	ADJ
cana-1295	154	26	groups	group	NOUN
cana-1295	154	27	.	.	PUNCT
cana-1295	155	1	international	international	ADJ
cana-1295	155	2	journal	journal	NOUN
cana-1295	155	3	of	of	ADP
cana-1295	155	4	innovative	innovative	ADJ
cana-1295	155	5	technology	technology	NOUN
cana-1295	155	6	and	and	CCONJ
cana-1295	155	7	exploring	explore	VERB
cana-1295	155	8	engineering	engineering	NOUN
cana-1295	155	9	,	,	PUNCT
cana-1295	155	10	8(6	8(6	NUM
cana-1295	155	11	special	special	ADJ
cana-1295	155	12	issue	issue	NOUN
cana-1295	155	13	4	4	NUM
cana-1295	155	14	)	)	PUNCT
cana-1295	155	15	,	,	PUNCT
cana-1295	155	16	1403	1403	NUM
cana-1295	155	17	-	-	SYM
cana-1295	155	18	1404	1404	NUM
cana-1295	155	19	.	.	PUNCT
cana-1295	156	1	[	[	X
cana-1295	156	2	8	8	NUM
cana-1295	156	3	]	]	X
cana-1295	156	4	iampan	iampan	NOUN
cana-1295	156	5	,	,	PUNCT
cana-1295	156	6	a.	a.	NOUN
cana-1295	156	7	(	(	PUNCT
cana-1295	156	8	2007	2007	NUM
cana-1295	156	9	)	)	PUNCT
cana-1295	156	10	.	.	PUNCT
cana-1295	157	1	lateral	lateral	ADJ
cana-1295	157	2	ideals	ideal	NOUN
cana-1295	157	3	of	of	ADP
cana-1295	157	4	ternary	ternary	ADJ
cana-1295	157	5	semi	semi	ADJ
cana-1295	157	6	groups	group	NOUN
cana-1295	157	7	.	.	PUNCT
cana-1295	158	1	український	український	VERB
cana-1295	158	2	математичний	математичний	PROPN
cana-1295	158	3	вісник	вісник	NOUN
cana-1295	158	4	.	.	PUNCT
cana-1295	159	1	[	[	X
cana-1295	159	2	9	9	NUM
cana-1295	159	3	]	]	X
cana-1295	159	4	choosuwan	choosuwan	NOUN
cana-1295	159	5	,	,	PUNCT
cana-1295	159	6	p.	p.	NOUN
cana-1295	159	7	,	,	PUNCT
cana-1295	159	8	&	&	CCONJ
cana-1295	159	9	chinram	chinram	PROPN
cana-1295	159	10	,	,	PUNCT
cana-1295	159	11	r.	r.	PROPN
cana-1295	159	12	(	(	PUNCT
cana-1295	159	13	2012	2012	NUM
cana-1295	159	14	)	)	PUNCT
cana-1295	159	15	.	.	PUNCT
cana-1295	160	1	a	a	DET
cana-1295	160	2	study	study	NOUN
cana-1295	160	3	on	on	ADP
cana-1295	160	4	quasi	quasi	NOUN
cana-1295	160	5	-	-	NOUN
cana-1295	160	6	ideals	ideal	NOUN
cana-1295	160	7	in	in	ADP
cana-1295	160	8	ternary	ternary	ADJ
cana-1295	160	9	semi	semi	ADJ
cana-1295	160	10	groups	group	NOUN
cana-1295	160	11	.	.	PUNCT
cana-1295	161	1	int	int	NOUN
cana-1295	161	2	.	.	PUNCT
cana-1295	162	1	j.	j.	PROPN
cana-1295	162	2	pure	pure	PROPN
cana-1295	162	3	appl	appl	PROPN
cana-1295	162	4	.	.	PUNCT
cana-1295	162	5	math	math	PROPN
cana-1295	162	6	,	,	PUNCT
cana-1295	162	7	77(5	77(5	NOUN
cana-1295	162	8	)	)	PUNCT
cana-1295	162	9	,	,	PUNCT
cana-1295	162	10	639	639	NUM
cana-1295	162	11	-	-	SYM
cana-1295	162	12	647	647	NUM
cana-1295	163	1	[	[	SYM
cana-1295	163	2	10	10	NUM
cana-1295	163	3	]	]	PUNCT
cana-1295	163	4	vasantha	vasantha	NOUN
cana-1295	163	5	kandasamy	kandasamy	PROPN
cana-1295	163	6	w.b	w.b	PROPN
cana-1295	163	7	.	.	PROPN
cana-1295	163	8	,	,	PUNCT
cana-1295	163	9	bialgebraic	bialgebraic	NOUN
cana-1295	163	10	structures	structure	NOUN
cana-1295	163	11	and	and	CCONJ
cana-1295	163	12	smarandache	smarandache	NOUN
cana-1295	163	13	bialgebraic	bialgebraic	PROPN
cana-1295	163	14	structures	structure	NOUN
cana-1295	163	15	,	,	PUNCT
cana-1295	163	16	american	american	ADJ
cana-1295	163	17	research	research	PROPN
cana-1295	163	18	press	press	PROPN
cana-1295	163	19	,	,	PUNCT
cana-1295	163	20	rehoboth	rehoboth	NOUN
cana-1295	163	21	,	,	PUNCT
cana-1295	163	22	2003	2003	NUM
cana-1295	163	23	.	.	PUNCT
cana-1295	164	1	[	[	X
cana-1295	164	2	11	11	NUM
cana-1295	164	3	]	]	PUNCT
cana-1295	164	4	vasantha	vasantha	NOUN
cana-1295	164	5	kandasamy	kandasamy	PROPN
cana-1295	164	6	w.b	w.b	PROPN
cana-1295	164	7	.	.	PROPN
cana-1295	164	8	,	,	PUNCT
cana-1295	164	9	bivector	bivector	NOUN
cana-1295	164	10	spaces	space	NOUN
cana-1295	164	11	,	,	PUNCT
cana-1295	164	12	u.	u.	PROPN
cana-1295	164	13	sci	sci	PROPN
cana-1295	164	14	.	.	PUNCT
cana-1295	164	15	phy	phy	PROPN
cana-1295	164	16	.	.	PUNCT
cana-1295	165	1	sci	sci	PROPN
cana-1295	165	2	.	.	PROPN
cana-1295	165	3	,	,	PUNCT
cana-1295	165	4	(	(	PUNCT
cana-1295	165	5	11	11	NUM
cana-1295	165	6	)	)	PUNCT
cana-1295	165	7	(	(	PUNCT
cana-1295	165	8	1999	1999	NUM
cana-1295	165	9	)	)	PUNCT
cana-1295	165	10	,	,	PUNCT
cana-1295	165	11	186	186	NUM
cana-1295	165	12	-	-	SYM
cana-1295	165	13	190	190	NUM
cana-1295	165	14	.	.	PUNCT
cana-1295	166	1	[	[	X
cana-1295	166	2	12	12	NUM
cana-1295	166	3	]	]	X
cana-1295	166	4	g.	g.	PROPN
cana-1295	166	5	srinivasa	srinivasa	PROPN
cana-1295	166	6	rao	rao	PROPN
cana-1295	166	7	,	,	PUNCT
cana-1295	166	8	d.	d.	PROPN
cana-1295	166	9	madhusudhanarao	madhusudhanarao	PROPN
cana-1295	166	10	and	and	CCONJ
cana-1295	166	11	p.	p.	PROPN
cana-1295	166	12	siva	siva	PROPN
cana-1295	166	13	prasad	prasad	PROPN
cana-1295	166	14	,	,	PUNCT
cana-1295	166	15	simple	simple	ADJ
cana-1295	166	16	ternary	ternary	ADJ
cana-1295	166	17	semi	semi	NOUN
cana-1295	166	18	-	-	NOUN
cana-1295	166	19	rings	ring	NOUN
cana-1295	166	20	,	,	PUNCT
cana-1295	166	21	the	the	DET
cana-1295	166	22	global	global	ADJ
cana-1295	166	23	journal	journal	NOUN
cana-1295	166	24	of	of	ADP
cana-1295	166	25	mathematics	mathematics	PROPN
cana-1295	166	26	&	&	CCONJ
cana-1295	166	27	mathematical	mathematical	PROPN
cana-1295	166	28	sciences	sciences	PROPN
cana-1295	166	29	,	,	PUNCT
cana-1295	166	30	9(2	9(2	NUM
cana-1295	166	31	)	)	PUNCT
cana-1295	166	32	(	(	PUNCT
cana-1295	166	33	2016	2016	NUM
cana-1295	166	34	)	)	PUNCT
cana-1295	166	35	,	,	PUNCT
cana-1295	166	36	185	185	NUM
cana-1295	166	37	-	-	SYM
cana-1295	166	38	196	196	NUM
cana-1295	166	39	.	.	PUNCT
cana-1295	167	1	[	[	X
cana-1295	167	2	13	13	NUM
cana-1295	167	3	]	]	X
cana-1295	167	4	d.	d.	PROPN
cana-1295	167	5	madhusudhana	madhusudhana	PROPN
cana-1295	167	6	rao	rao	PROPN
cana-1295	167	7	,	,	PUNCT
cana-1295	167	8	g.	g.	PROPN
cana-1295	167	9	srinivasa	srinivasa	PROPN
cana-1295	167	10	rao	rao	PROPN
cana-1295	167	11	,	,	PUNCT
cana-1295	167	12	special	special	ADJ
cana-1295	167	13	elements	element	NOUN
cana-1295	167	14	in	in	ADP
cana-1295	167	15	ternary	ternary	ADJ
cana-1295	167	16	semi	semi	ADJ
cana-1295	167	17	rings	ring	NOUN
cana-1295	167	18	,	,	PUNCT
cana-1295	167	19	international	international	ADJ
cana-1295	167	20	journal	journal	NOUN
cana-1295	167	21	of	of	ADP
cana-1295	167	22	engineering	engineering	NOUN
cana-1295	167	23	research	research	NOUN
cana-1295	167	24	and	and	CCONJ
cana-1295	167	25	applications	application	NOUN
cana-1295	167	26	,	,	PUNCT
cana-1295	167	27	4(11	4(11	NUM
cana-1295	167	28	)	)	PUNCT
cana-1295	167	29	(	(	PUNCT
cana-1295	167	30	2014	2014	NUM
cana-1295	167	31	)	)	PUNCT
cana-1295	167	32	,	,	PUNCT
cana-1295	167	33	123	123	NUM
cana-1295	167	34	-	-	SYM
cana-1295	167	35	130	130	NUM
cana-1295	167	36	.	.	PUNCT
cana-1295	168	1	[	[	X
cana-1295	168	2	14	14	NUM
cana-1295	168	3	]	]	X
cana-1295	168	4	g.	g.	PROPN
cana-1295	168	5	srinivasa	srinivasa	PROPN
cana-1295	168	6	rao	rao	PROPN
cana-1295	168	7	,	,	PUNCT
cana-1295	168	8	d.	d.	PROPN
cana-1295	168	9	madhusudhana	madhusudhana	PROPN
cana-1295	168	10	rao	rao	PROPN
cana-1295	168	11	,	,	PUNCT
cana-1295	168	12	a	a	DET
cana-1295	168	13	study	study	NOUN
cana-1295	168	14	on	on	ADP
cana-1295	168	15	ternary	ternary	ADJ
cana-1295	168	16	semi	semi	ADJ
cana-1295	168	17	rings	ring	NOUN
cana-1295	168	18	,	,	PUNCT
cana-1295	168	19	int	int	NOUN
cana-1295	168	20	.	.	PUNCT
cana-1295	169	1	j.	j.	PROPN
cana-1295	169	2	of	of	ADP
cana-1295	169	3	math	math	PROPN
cana-1295	169	4	.	.	PUNCT
cana-1295	170	1	archive	archive	NOUN
cana-1295	170	2	,	,	PUNCT
cana-1295	170	3	5(12	5(12	NUM
cana-1295	170	4	)	)	PUNCT
cana-1295	170	5	(	(	PUNCT
cana-1295	170	6	2014	2014	NUM
cana-1295	170	7	)	)	PUNCT
cana-1295	170	8	,	,	PUNCT
cana-1295	170	9	24	24	NUM
cana-1295	170	10	-	-	SYM
cana-1295	170	11	30	30	NUM
cana-1295	170	12	.	.	PUNCT
cana-1295	171	1	[	[	X
cana-1295	171	2	15	15	NUM
cana-1295	171	3	]	]	X
cana-1295	171	4	g.	g.	PROPN
cana-1295	171	5	srinivasa	srinivasa	PROPN
cana-1295	171	6	rao	rao	PROPN
cana-1295	171	7	,	,	PUNCT
cana-1295	171	8	d.	d.	PROPN
cana-1295	171	9	madhusudhana	madhusudhana	PROPN
cana-1295	171	10	rao	rao	PROPN
cana-1295	171	11	,	,	PUNCT
cana-1295	171	12	characteristics	characteristic	NOUN
cana-1295	171	13	of	of	ADP
cana-1295	171	14	ternary	ternary	ADJ
cana-1295	171	15	semi	semi	ADJ
cana-1295	171	16	rings	ring	NOUN
cana-1295	171	17	,	,	PUNCT
cana-1295	171	18	int.j	int.j	PROPN
cana-1295	171	19	.	.	PROPN
cana-1295	171	20	of	of	ADP
cana-1295	171	21	engg	engg	PROPN
cana-1295	171	22	.	.	PUNCT
cana-1295	172	1	res	re	NOUN
cana-1295	172	2	.	.	PUNCT
cana-1295	172	3	and	and	CCONJ
cana-1295	172	4	mgt	mgt	PROPN
cana-1295	172	5	.	.	PUNCT
cana-1295	172	6	,	,	PUNCT
cana-1295	172	7	2(1	2(1	NUM
cana-1295	172	8	)	)	PUNCT
cana-1295	172	9	(	(	PUNCT
cana-1295	172	10	2015	2015	NUM
cana-1295	172	11	)	)	PUNCT
cana-1295	172	12	,	,	PUNCT
cana-1295	172	13	3	3	NUM
cana-1295	172	14	-	-	SYM
cana-1295	172	15	6	6	NUM
cana-1295	172	16	.	.	PUNCT
cana-1295	173	1	[	[	X
cana-1295	173	2	16	16	NUM
cana-1295	173	3	]	]	PUNCT
cana-1295	173	4	g.	g.	PROPN
cana-1295	173	5	srinivasa	srinivasa	PROPN
cana-1295	173	6	rao	rao	PROPN
cana-1295	173	7	,	,	PUNCT
cana-1295	173	8	a.	a.	PROPN
cana-1295	173	9	nagamalleswara	nagamalleswara	PROPN
cana-1295	173	10	rao	rao	PROPN
cana-1295	173	11	,	,	PUNCT
cana-1295	173	12	p.l.n	p.l.n	PROPN
cana-1295	173	13	.	.	PROPN
cana-1295	173	14	varma	varma	PROPN
cana-1295	173	15	,	,	PUNCT
cana-1295	173	16	d.madhusudhana	d.madhusudhana	PROPN
cana-1295	173	17	rao	rao	PROPN
cana-1295	173	18	,	,	PUNCT
cana-1295	173	19	ch	ch	NOUN
cana-1295	173	20	.	.	PROPN
cana-1295	173	21	ramprasad	ramprasad	ADJ
cana-1295	173	22	,	,	PUNCT
cana-1295	173	23	prime	prime	ADJ
cana-1295	173	24	biinterior	biinterior	PROPN
cana-1295	173	25	ideals	ideal	NOUN
cana-1295	173	26	in	in	ADP
cana-1295	173	27	tgsr	tgsr	ADJ
cana-1295	173	28	,	,	PUNCT
cana-1295	173	29	malaya	malaya	PROPN
cana-1295	173	30	journal	journal	PROPN
cana-1295	173	31	of	of	ADP
cana-1295	173	32	mathematika	mathematika	NOUN
cana-1295	173	33	,	,	PUNCT
cana-1295	173	34	vol.9	vol.9	PROPN
cana-1295	173	35	,	,	PUNCT
cana-1295	173	36	no.1	no.1	NUM
cana-1295	173	37	,	,	PUNCT
cana-1295	173	38	pp:542	pp:542	ADV
cana-1295	173	39	-	-	PUNCT
cana-1295	173	40	546	546	NUM
cana-1295	173	41	,	,	PUNCT
cana-1295	173	42	2021	2021	NUM
cana-1295	173	43	.	.	PUNCT
cana-1295	174	1	[	[	X
cana-1295	174	2	17	17	NUM
cana-1295	174	3	]	]	X
cana-1295	174	4	g.	g.	PROPN
cana-1295	174	5	srinivasa	srinivasa	PROPN
cana-1295	174	6	rao	rao	PROPN
cana-1295	174	7	,	,	PUNCT
cana-1295	174	8	a.	a.	PROPN
cana-1295	174	9	nagamalleswara	nagamalleswara	PROPN
cana-1295	174	10	rao	rao	PROPN
cana-1295	174	11	,	,	PUNCT
cana-1295	174	12	p.l.n	p.l.n	PROPN
cana-1295	174	13	.	.	PROPN
cana-1295	174	14	varma	varma	PROPN
cana-1295	174	15	,	,	PUNCT
cana-1295	174	16	d.	d.	PROPN
cana-1295	174	17	madhusudhana	madhusudhana	PROPN
cana-1295	174	18	rao	rao	PROPN
cana-1295	174	19	,	,	PUNCT
cana-1295	174	20	ch	ch	NOUN
cana-1295	174	21	.	.	PROPN
cana-1295	174	22	ramprasad	ramprasad	ADJ
cana-1295	174	23	,	,	PUNCT
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cana-1295	174	25	-	-	ADJ
cana-1295	174	26	interior	interior	ADJ
cana-1295	174	27	ideals	ideal	NOUN
cana-1295	174	28	in	in	ADP
cana-1295	174	29	tgsr	tgsr	ADJ
cana-1295	174	30	,	,	PUNCT
cana-1295	174	31	advances	advance	NOUN
cana-1295	174	32	in	in	ADP
cana-1295	174	33	mathematics	mathematics	NOUN
cana-1295	174	34	scientific	scientific	ADJ
cana-1295	174	35	journal	journal	NOUN
cana-1295	174	36	,	,	PUNCT
cana-1295	174	37	10	10	NUM
cana-1295	174	38	(	(	PUNCT
cana-1295	174	39	2021	2021	NUM
cana-1295	174	40	)	)	PUNCT
cana-1295	174	41	,	,	PUNCT
cana-1295	174	42	no.3	no.3	VERB
cana-1295	174	43	,	,	PUNCT
cana-1295	174	44	pp	pp	CCONJ
cana-1295	174	45	:	:	PUNCT
cana-1295	174	46	1183	1183	NUM
cana-1295	174	47	-	-	SYM
cana-1295	174	48	1195	1195	NUM
cana-1295	174	49	.	.	PUNCT