id	sid	tid	token	lemma	pos
cana-1327	1	1	title	title	NOUN
cana-1327	1	2	goes	go	VERB
cana-1327	1	3	here	here	ADV
cana-1327	1	4	communications	communication	NOUN
cana-1327	1	5	on	on	ADP
cana-1327	1	6	applied	apply	VERB
cana-1327	1	7	nonlinear	nonlinear	ADJ
cana-1327	1	8	analysis	analysis	NOUN
cana-1327	1	9	issn	issn	NOUN
cana-1327	1	10	:	:	PUNCT
cana-1327	1	11	1074	1074	NUM
cana-1327	1	12	-	-	PUNCT
cana-1327	1	13	133x	133x	NUM
cana-1327	1	14	vol	vol	NOUN
cana-1327	1	15	31	31	NUM
cana-1327	1	16	no	no	NOUN
cana-1327	1	17	.	.	PUNCT
cana-1327	2	1	7s	7	NOUN
cana-1327	2	2	(	(	PUNCT
cana-1327	2	3	2024	2024	NUM
cana-1327	2	4	)	)	PUNCT
cana-1327	2	5	478	478	NUM
cana-1327	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1327	2	7	pseudo	pseudo	NOUN
cana-1327	2	8	symmetric	symmetric	ADJ
cana-1327	2	9	ideals	ideal	NOUN
cana-1327	2	10	in	in	ADP
cana-1327	2	11	near	near	ADJ
cana-1327	2	12	subtraction	subtraction	NOUN
cana-1327	2	13	semigroups	semigroup	NOUN
cana-1327	2	14	jayarami	jayarami	PROPN
cana-1327	2	15	reddy.m1,2	reddy.m1,2	ADJ
cana-1327	2	16	a),siva	a),siva	PROPN
cana-1327	2	17	prasad.p3	prasad.p3	PROPN
cana-1327	2	18	,	,	PUNCT
cana-1327	2	19	b)and	b)and	ADV
cana-1327	2	20	madhusudana	madhusudana	VERB
cana-1327	2	21	rao.d4c	rao.d4c	PROPN
cana-1327	2	22	)	)	PUNCT
cana-1327	2	23	1research	1research	NUM
cana-1327	2	24	scholar	scholar	NOUN
cana-1327	2	25	,	,	PUNCT
cana-1327	2	26	vfstr	vfstr	NOUN
cana-1327	2	27	deemed	deem	VERB
cana-1327	2	28	to	to	PART
cana-1327	2	29	be	be	AUX
cana-1327	2	30	university	university	NOUN
cana-1327	2	31	,	,	PUNCT
cana-1327	2	32	guntur	guntur	PROPN
cana-1327	2	33	,	,	PUNCT
cana-1327	2	34	andhra	andhra	PROPN
cana-1327	2	35	pradesh	pradesh	PROPN
cana-1327	2	36	,	,	PUNCT
cana-1327	2	37	india	india	PROPN
cana-1327	2	38	2associate	2associate	PROPN
cana-1327	2	39	professor	professor	NOUN
cana-1327	2	40	,	,	PUNCT
cana-1327	2	41	ucet	ucet	PROPN
cana-1327	2	42	,	,	PUNCT
cana-1327	2	43	guntur	guntur	PROPN
cana-1327	2	44	,	,	PUNCT
cana-1327	2	45	ap	ap	PROPN
cana-1327	2	46	,	,	PUNCT
cana-1327	2	47	india	india	PROPN
cana-1327	2	48	3associate	3associate	NUM
cana-1327	2	49	professor	professor	NOUN
cana-1327	2	50	,	,	PUNCT
cana-1327	2	51	dept	dept	NOUN
cana-1327	2	52	of	of	ADP
cana-1327	2	53	cse	cse	PROPN
cana-1327	2	54	,	,	PUNCT
cana-1327	2	55	soci	soci	PROPN
cana-1327	2	56	,	,	PUNCT
cana-1327	2	57	vfstr	vfstr	NOUN
cana-1327	2	58	,	,	PUNCT
cana-1327	2	59	deemed	deem	VERB
cana-1327	2	60	to	to	PART
cana-1327	2	61	be	be	AUX
cana-1327	2	62	university	university	NOUN
cana-1327	2	63	,	,	PUNCT
cana-1327	2	64	guntur	guntur	PROPN
cana-1327	2	65	,	,	PUNCT
cana-1327	2	66	ap	ap	PROPN
cana-1327	2	67	,	,	PUNCT
cana-1327	2	68	india	india	PROPN
cana-1327	2	69	4professor	4professor	PROPN
cana-1327	2	70	of	of	ADP
cana-1327	2	71	mathematics	mathematic	NOUN
cana-1327	2	72	,	,	PUNCT
cana-1327	2	73	government	government	NOUN
cana-1327	2	74	college	college	NOUN
cana-1327	2	75	for	for	ADP
cana-1327	2	76	women	woman	NOUN
cana-1327	2	77	,	,	PUNCT
cana-1327	2	78	sambasiva	sambasiva	PROPN
cana-1327	2	79	pet	pet	PROPN
cana-1327	2	80	,	,	PUNCT
cana-1327	2	81	guntur	guntur	PROPN
cana-1327	2	82	,	,	PUNCT
cana-1327	2	83	ap	ap	PROPN
cana-1327	2	84	,	,	PUNCT
cana-1327	2	85	india	india	PROPN
cana-1327	2	86	a	a	PRON
cana-1327	2	87	)	)	PUNCT
cana-1327	2	88	jayaramireddy1979@gmail.com	jayaramireddy1979@gmail.com	PROPN
cana-1327	2	89	,	,	PUNCT
cana-1327	2	90	b	b	X
cana-1327	2	91	)	)	PUNCT
cana-1327	2	92	pusapatisivaprasad@gmail.com	pusapatisivaprasad@gmail.com	NOUN
cana-1327	2	93	,	,	PUNCT
cana-1327	2	94	c	c	NOUN
cana-1327	3	1	)	)	PUNCT
cana-1327	3	2	dmrmaths@gmail.com	dmrmaths@gmail.com	NOUN
cana-1327	3	3	article	article	NOUN
cana-1327	3	4	history	history	NOUN
cana-1327	3	5	:	:	PUNCT
cana-1327	3	6	received	receive	VERB
cana-1327	3	7	:	:	PUNCT
cana-1327	3	8	01	01	NUM
cana-1327	3	9	-	-	PUNCT
cana-1327	3	10	06	06	NUM
cana-1327	3	11	-	-	PUNCT
cana-1327	3	12	2024	2024	NUM
cana-1327	3	13	revised	revise	VERB
cana-1327	3	14	:	:	PUNCT
cana-1327	3	15	03	03	NUM
cana-1327	3	16	-	-	PUNCT
cana-1327	3	17	07	07	NUM
cana-1327	3	18	-	-	PUNCT
cana-1327	3	19	2024	2024	NUM
cana-1327	3	20	accepted	accept	VERB
cana-1327	3	21	:	:	PUNCT
cana-1327	3	22	29	29	NUM
cana-1327	3	23	-	-	SYM
cana-1327	3	24	07	07	NUM
cana-1327	3	25	-	-	PUNCT
cana-1327	3	26	2024	2024	NUM
cana-1327	3	27	abstract	abstract	NOUN
cana-1327	3	28	:	:	PUNCT
cana-1327	3	29	the	the	DET
cana-1327	3	30	current	current	ADJ
cana-1327	3	31	ongoing	ongoing	ADJ
cana-1327	3	32	work	work	NOUN
cana-1327	3	33	,	,	PUNCT
cana-1327	3	34	author	author	NOUN
cana-1327	3	35	introduce	introduce	VERB
cana-1327	3	36	the	the	DET
cana-1327	3	37	theory	theory	NOUN
cana-1327	3	38	of	of	ADP
cana-1327	3	39	pseudo	pseudo	NOUN
cana-1327	3	40	symmetric	symmetric	ADJ
cana-1327	3	41	,	,	PUNCT
cana-1327	3	42	psideal	psideal	NOUN
cana-1327	3	43	,	,	PUNCT
cana-1327	3	44	semiprime	semiprime	NOUN
cana-1327	3	45	ideal	ideal	NOUN
cana-1327	3	46	,	,	PUNCT
cana-1327	3	47	completely	completely	ADV
cana-1327	3	48	semi	semi	VERB
cana-1327	3	49	prime	prime	ADJ
cana-1327	3	50	ideal	ideal	NOUN
cana-1327	3	51	,	,	PUNCT
cana-1327	3	52	left	leave	VERB
cana-1327	3	53	pseudo	pseudo	NOUN
cana-1327	3	54	commutative	commutative	ADJ
cana-1327	3	55	,	,	PUNCT
cana-1327	3	56	right	right	ADJ
cana-1327	3	57	pseudo	pseudo	NOUN
cana-1327	3	58	commutative	commutative	ADJ
cana-1327	3	59	,	,	PUNCT
cana-1327	3	60	duo	duo	NOUN
cana-1327	3	61	nssgand	nssgand	NOUN
cana-1327	3	62	psnear	psnear	PROPN
cana-1327	3	63	subtraction	subtraction	NOUN
cana-1327	3	64	sg	sg	PROPN
cana-1327	3	65	.	.	PUNCT
cana-1327	4	1	keywords	keyword	NOUN
cana-1327	4	2	:	:	PUNCT
cana-1327	4	3	pseudo	pseudo	NOUN
cana-1327	4	4	symmetric	symmetric	NOUN
cana-1327	4	5	,	,	PUNCT
cana-1327	4	6	completely	completely	ADV
cana-1327	4	7	semiprme	semiprme	VERB
cana-1327	4	8	,	,	PUNCT
cana-1327	4	9	duo	duo	NOUN
cana-1327	4	10	semigroup	semigroup	NOUN
cana-1327	4	11	,	,	PUNCT
cana-1327	4	12	near	near	ADP
cana-1327	4	13	subtractionsemigroup	subtractionsemigroup	PROPN
cana-1327	4	14	.	.	PUNCT
cana-1327	5	1	1	1	X
cana-1327	5	2	.	.	X
cana-1327	5	3	introduction	introduction	NOUN
cana-1327	5	4	in	in	ADP
cana-1327	5	5	abstract	abstract	ADJ
cana-1327	5	6	algebra	algebra	NOUN
cana-1327	5	7	the	the	DET
cana-1327	5	8	theory	theory	NOUN
cana-1327	5	9	of	of	ADP
cana-1327	5	10	near	near	ADJ
cana-1327	5	11	subtraction	subtraction	NOUN
cana-1327	5	12	semi	semi	ADJ
cana-1327	5	13	groups	group	NOUN
cana-1327	5	14	is	be	AUX
cana-1327	5	15	a	a	DET
cana-1327	5	16	rapid	rapid	ADJ
cana-1327	5	17	departing	depart	VERB
cana-1327	5	18	branch	branch	NOUN
cana-1327	5	19	.	.	PUNCT
cana-1327	6	1	the	the	DET
cana-1327	6	2	subtraction	subtraction	NOUN
cana-1327	6	3	algebra	algebra	NOUN
cana-1327	6	4	was	be	AUX
cana-1327	6	5	introduced	introduce	VERB
cana-1327	6	6	by	by	ADP
cana-1327	6	7	abbott	abbott	PROPN
cana-1327	6	8	in1967	in1967	PROPN
cana-1327	6	9	byusingthe	byusingthe	PROPN
cana-1327	6	10	concept	concept	NOUN
cana-1327	6	11	of	of	ADP
cana-1327	6	12	subtraction	subtraction	NOUN
cana-1327	6	13	algebra	algebra	PROPN
cana-1327	6	14	,	,	PUNCT
cana-1327	6	15	schein	schein	PROPN
cana-1327	6	16	developed	develop	VERB
cana-1327	6	17	subtraction	subtraction	NOUN
cana-1327	6	18	sg	sg	NOUN
cana-1327	6	19	in	in	ADP
cana-1327	6	20	the	the	DET
cana-1327	6	21	year	year	NOUN
cana-1327	6	22	1992.in	1992.in	NOUN
cana-1327	6	23	the	the	DET
cana-1327	6	24	year	year	NOUN
cana-1327	6	25	of	of	ADP
cana-1327	6	26	2007	2007	NUM
cana-1327	6	27	dheena.p	dheena.p	NOUN
cana-1327	6	28	was	be	AUX
cana-1327	6	29	first	first	ADV
cana-1327	6	30	introduced	introduce	VERB
cana-1327	6	31	the	the	DET
cana-1327	6	32	concept	concept	NOUN
cana-1327	6	33	of	of	ADP
cana-1327	6	34	near	near	ADJ
cana-1327	6	35	subtraction	subtraction	NOUN
cana-1327	6	36	sg	sg	PROPN
cana-1327	6	37	.	.	PUNCT
cana-1327	7	1	basically	basically	ADV
cana-1327	7	2	,	,	PUNCT
cana-1327	7	3	the	the	DET
cana-1327	7	4	nssg	nssg	NOUN
cana-1327	7	5	is	be	AUX
cana-1327	7	6	a	a	DET
cana-1327	7	7	fundamental	fundamental	ADJ
cana-1327	7	8	notions	notion	NOUN
cana-1327	7	9	of	of	ADP
cana-1327	7	10	subtraction	subtraction	NOUN
cana-1327	7	11	algebra	algebra	NOUN
cana-1327	7	12	.	.	PUNCT
cana-1327	8	1	jun	jun	PROPN
cana-1327	8	2	et	et	PROPN
cana-1327	8	3	al	al	PROPN
cana-1327	8	4	.	.	PROPN
cana-1327	8	5	,	,	PUNCT
cana-1327	8	6	investigated	investigate	VERB
cana-1327	8	7	about	about	ADP
cana-1327	8	8	the	the	DET
cana-1327	8	9	theory	theory	NOUN
cana-1327	8	10	of	of	ADP
cana-1327	8	11	ideals	ideal	NOUN
cana-1327	8	12	in	in	ADP
cana-1327	8	13	near	near	ADJ
cana-1327	8	14	subtraction	subtraction	NOUN
cana-1327	8	15	algebra	algebra	NOUN
cana-1327	8	16	after	after	ADP
cana-1327	8	17	that	that	SCONJ
cana-1327	8	18	he	he	PRON
cana-1327	8	19	characterized	characterize	VERB
cana-1327	8	20	some	some	DET
cana-1327	8	21	elementary	elementary	ADJ
cana-1327	8	22	axioms	axiom	NOUN
cana-1327	8	23	.	.	PUNCT
cana-1327	9	1	later	later	ADV
cana-1327	9	2	on	on	ADP
cana-1327	9	3	some	some	DET
cana-1327	9	4	research	research	NOUN
cana-1327	9	5	scholars	scholar	NOUN
cana-1327	9	6	developed	develop	VERB
cana-1327	9	7	the	the	DET
cana-1327	9	8	basic	basic	ADJ
cana-1327	9	9	properties	property	NOUN
cana-1327	9	10	of	of	ADP
cana-1327	9	11	near	near	ADJ
cana-1327	9	12	subtraction	subtraction	NOUN
cana-1327	9	13	sg	sg	PROPN
cana-1327	9	14	.	.	PUNCT
cana-1327	10	1	later	later	ADV
cana-1327	10	2	they	they	PRON
cana-1327	10	3	establish	establish	VERB
cana-1327	10	4	a	a	DET
cana-1327	10	5	structure	structure	NOUN
cana-1327	10	6	in	in	ADP
cana-1327	10	7	nssg	nssg	ADJ
cana-1327	10	8	and	and	CCONJ
cana-1327	10	9	also	also	ADV
cana-1327	10	10	examine	examine	VERB
cana-1327	10	11	different	different	ADJ
cana-1327	10	12	kinds	kind	NOUN
cana-1327	10	13	of	of	ADP
cana-1327	10	14	near	near	ADJ
cana-1327	10	15	subtraction	subtraction	NOUN
cana-1327	10	16	algebra	algebra	NOUN
cana-1327	10	17	and	and	CCONJ
cana-1327	10	18	getting	get	VERB
cana-1327	10	19	the	the	DET
cana-1327	10	20	equivalent	equivalent	ADJ
cana-1327	10	21	conditions	condition	NOUN
cana-1327	10	22	for	for	ADP
cana-1327	10	23	regularity	regularity	NOUN
cana-1327	10	24	.	.	PUNCT
cana-1327	11	1	the	the	DET
cana-1327	11	2	fundamental	fundamental	ADJ
cana-1327	11	3	concept	concept	NOUN
cana-1327	11	4	of	of	ADP
cana-1327	11	5	this	this	DET
cana-1327	11	6	work	work	NOUN
cana-1327	11	7	is	be	AUX
cana-1327	11	8	to	to	PART
cana-1327	11	9	learn	learn	VERB
cana-1327	11	10	about	about	ADP
cana-1327	11	11	ps	ps	NOUN
cana-1327	11	12	ideals	ideal	NOUN
cana-1327	11	13	in	in	ADP
cana-1327	11	14	nssg	nssg	ADJ
cana-1327	11	15	and	and	CCONJ
cana-1327	11	16	psnssg	psnssg	VERB
cana-1327	11	17	and	and	CCONJ
cana-1327	11	18	some	some	PRON
cana-1327	11	19	of	of	ADP
cana-1327	11	20	the	the	DET
cana-1327	11	21	results	result	NOUN
cana-1327	11	22	were	be	AUX
cana-1327	11	23	proved	prove	VERB
cana-1327	11	24	regarding	regard	VERB
cana-1327	11	25	nssg	nssg	VERB
cana-1327	11	26	.	.	PUNCT
cana-1327	12	1	2	2	X
cana-1327	12	2	.	.	X
cana-1327	12	3	preliminaries	preliminary	NOUN
cana-1327	12	4	forfundamental	forfundamental	ADJ
cana-1327	12	5	conceptsmake	conceptsmake	NOUN
cana-1327	12	6	reference	reference	NOUN
cana-1327	12	7	to	to	ADP
cana-1327	12	8	(	(	PUNCT
cana-1327	12	9	1),(2),(3	1),(2),(3	X
cana-1327	12	10	)	)	PUNCT
cana-1327	12	11	and(4	and(4	ADJ
cana-1327	12	12	)	)	PUNCT
cana-1327	13	1	3	3	X
cana-1327	13	2	.	.	X
cana-1327	13	3	maincontent	maincontent	ADJ
cana-1327	13	4	definition	definition	NOUN
cana-1327	13	5	3.1	3.1	NUM
cana-1327	13	6	:	:	PUNCT
cana-1327	13	7	assume	assume	VERB
cana-1327	13	8	that	that	SCONJ
cana-1327	13	9	a	a	PRON
cana-1327	13	10	be	be	AUX
cana-1327	13	11	an	an	DET
cana-1327	13	12	ideal	ideal	NOUN
cana-1327	13	13	of	of	ADP
cana-1327	13	14	a	a	DET
cana-1327	13	15	nssg	nssg	VERB
cana-1327	13	16	x	x	PUNCT
cana-1327	13	17	which	which	PRON
cana-1327	13	18	is	be	AUX
cana-1327	13	19	said	say	VERB
cana-1327	13	20	to	to	PART
cana-1327	13	21	be	be	AUX
cana-1327	13	22	a	a	DET
cana-1327	13	23	ps	ps	NOUN
cana-1327	13	24	if	if	SCONJ
cana-1327	13	25	x	x	PROPN
cana-1327	13	26	,	,	PUNCT
cana-1327	13	27	yx	yx	PROPN
cana-1327	13	28	,	,	PUNCT
cana-1327	13	29	xya	xya	PUNCT
cana-1327	14	1	=	=	PROPN
cana-1327	14	2	>	>	X
cana-1327	14	3	xsya	xsya	PROPN
cana-1327	14	4	for	for	ADP
cana-1327	14	5	each	each	DET
cana-1327	14	6	sx	sx	NOUN
cana-1327	14	7	.	.	PUNCT
cana-1327	14	8	example3.2	example3.2	NOUN
cana-1327	14	9	:	:	PUNCT
cana-1327	14	10	assume	assume	VERB
cana-1327	14	11	that	that	SCONJ
cana-1327	14	12	x=	x=	PUNCT
cana-1327	15	1	{	{	PUNCT
cana-1327	15	2	0,1,2,3,4,5	0,1,2,3,4,5	NOUN
cana-1327	15	3	}	}	PUNCT
cana-1327	15	4	for	for	ADP
cana-1327	15	5	which	which	PRON
cana-1327	15	6	‘	'	PUNCT
cana-1327	15	7	-	-	PUNCT
cana-1327	15	8	,	,	PUNCT
cana-1327	15	9	’	'	PUNCT
cana-1327	15	10	and	and	CCONJ
cana-1327	15	11	‘	'	PUNCT
cana-1327	15	12	.	.	PUNCT
cana-1327	15	13	’	'	PUNCT
cana-1327	15	14	are	be	AUX
cana-1327	15	15	defined	define	VERB
cana-1327	15	16	by	by	ADP
cana-1327	15	17	0	0	NUM
cana-1327	15	18	1	1	NUM
cana-1327	15	19	2	2	NUM
cana-1327	15	20	3	3	NUM
cana-1327	15	21	4	4	NUM
cana-1327	15	22	5	5	NUM
cana-1327	15	23	0	0	NUM
cana-1327	15	24	0	0	NUM
cana-1327	15	25	0	0	NUM
cana-1327	15	26	0	0	NUM
cana-1327	15	27	0	0	NUM
cana-1327	15	28	0	0	NUM
cana-1327	15	29	0	0	NUM
cana-1327	15	30	1	1	NUM
cana-1327	15	31	1	1	NUM
cana-1327	15	32	0	0	NUM
cana-1327	15	33	3	3	NUM
cana-1327	15	34	4	4	NUM
cana-1327	15	35	3	3	NUM
cana-1327	15	36	1	1	NUM
cana-1327	15	37	communications	communication	NOUN
cana-1327	15	38	on	on	ADP
cana-1327	15	39	applied	apply	VERB
cana-1327	15	40	nonlinear	nonlinear	ADJ
cana-1327	15	41	analysis	analysis	NOUN
cana-1327	15	42	issn	issn	NOUN
cana-1327	15	43	:	:	PUNCT
cana-1327	15	44	1074	1074	NUM
cana-1327	15	45	-	-	PUNCT
cana-1327	15	46	133x	133x	NUM
cana-1327	15	47	vol	vol	NOUN
cana-1327	15	48	31	31	NUM
cana-1327	15	49	no	no	NOUN
cana-1327	15	50	.	.	PUNCT
cana-1327	16	1	7s	7	NOUN
cana-1327	16	2	(	(	PUNCT
cana-1327	16	3	2024	2024	NUM
cana-1327	16	4	)	)	PUNCT
cana-1327	16	5	479	479	NUM
cana-1327	17	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1327	17	2	2	2	NUM
cana-1327	17	3	2	2	NUM
cana-1327	17	4	5	5	NUM
cana-1327	17	5	0	0	NUM
cana-1327	17	6	2	2	NUM
cana-1327	17	7	5	5	NUM
cana-1327	17	8	4	4	NUM
cana-1327	17	9	3	3	NUM
cana-1327	17	10	3	3	NUM
cana-1327	17	11	0	0	NUM
cana-1327	17	12	3	3	NUM
cana-1327	17	13	0	0	NUM
cana-1327	17	14	3	3	NUM
cana-1327	17	15	3	3	NUM
cana-1327	17	16	4	4	NUM
cana-1327	17	17	4	4	NUM
cana-1327	17	18	0	0	NUM
cana-1327	17	19	0	0	NUM
cana-1327	17	20	4	4	NUM
cana-1327	17	21	0	0	NUM
cana-1327	17	22	4	4	NUM
cana-1327	17	23	5	5	NUM
cana-1327	17	24	5	5	NUM
cana-1327	17	25	5	5	NUM
cana-1327	17	26	0	0	NUM
cana-1327	17	27	5	5	NUM
cana-1327	17	28	5	5	NUM
cana-1327	17	29	0	0	NUM
cana-1327	17	30	.	.	PUNCT
cana-1327	17	31	0	0	NUM
cana-1327	17	32	1	1	NUM
cana-1327	17	33	2	2	NUM
cana-1327	17	34	3	3	NUM
cana-1327	17	35	4	4	NUM
cana-1327	17	36	5	5	NUM
cana-1327	17	37	0	0	NUM
cana-1327	17	38	0	0	NUM
cana-1327	17	39	0	0	NUM
cana-1327	17	40	0	0	NUM
cana-1327	17	41	0	0	NUM
cana-1327	17	42	0	0	NUM
cana-1327	17	43	0	0	NUM
cana-1327	17	44	1	1	NUM
cana-1327	17	45	0	0	NUM
cana-1327	17	46	1	1	NUM
cana-1327	17	47	4	4	NUM
cana-1327	17	48	3	3	NUM
cana-1327	17	49	4	4	NUM
cana-1327	17	50	0	0	NUM
cana-1327	17	51	2	2	NUM
cana-1327	17	52	0	0	NUM
cana-1327	17	53	4	4	NUM
cana-1327	17	54	2	2	NUM
cana-1327	17	55	0	0	NUM
cana-1327	17	56	4	4	NUM
cana-1327	17	57	5	5	NUM
cana-1327	17	58	3	3	NUM
cana-1327	17	59	0	0	NUM
cana-1327	17	60	3	3	NUM
cana-1327	17	61	0	0	NUM
cana-1327	17	62	3	3	NUM
cana-1327	17	63	0	0	NUM
cana-1327	17	64	0	0	NUM
cana-1327	17	65	4	4	NUM
cana-1327	17	66	0	0	NUM
cana-1327	17	67	4	4	NUM
cana-1327	17	68	4	4	NUM
cana-1327	17	69	0	0	NUM
cana-1327	17	70	4	4	NUM
cana-1327	17	71	0	0	NUM
cana-1327	17	72	5	5	NUM
cana-1327	17	73	0	0	NUM
cana-1327	17	74	0	0	NUM
cana-1327	17	75	5	5	NUM
cana-1327	17	76	0	0	NUM
cana-1327	17	77	0	0	NUM
cana-1327	17	78	5	5	NUM
cana-1327	17	79	then	then	ADV
cana-1327	17	80	(	(	PUNCT
cana-1327	17	81	a,-	a,-	NOUN
cana-1327	17	82	,	,	PUNCT
cana-1327	17	83	.	.	PUNCT
cana-1327	17	84	)	)	PUNCT
cana-1327	18	1	is	be	AUX
cana-1327	18	2	a	a	DET
cana-1327	18	3	nssg	nssg	ADJ
cana-1327	18	4	.	.	PUNCT
cana-1327	19	1	let	let	VERB
cana-1327	19	2	i	i	PRON
cana-1327	19	3	=	=	PUNCT
cana-1327	19	4	{	{	PUNCT
cana-1327	19	5	0,1,3,4	0,1,3,4	NUM
cana-1327	19	6	}	}	PUNCT
cana-1327	19	7	is	be	AUX
cana-1327	19	8	a	a	DET
cana-1327	19	9	ps	ps	NOUN
cana-1327	19	10	ideal	ideal	NOUN
cana-1327	19	11	of	of	ADP
cana-1327	19	12	x.	x.	PROPN
cana-1327	19	13	example3.3	example3.3	PROPN
cana-1327	19	14	:	:	PUNCT
cana-1327	19	15	letx={p	letx={p	PROPN
cana-1327	19	16	,	,	PUNCT
cana-1327	19	17	q	q	X
cana-1327	19	18	,	,	PUNCT
cana-1327	19	19	r	r	NOUN
cana-1327	19	20	}	}	PUNCT
cana-1327	19	21	for	for	ADP
cana-1327	19	22	which	which	PRON
cana-1327	19	23	‘	'	PUNCT
cana-1327	19	24	-	-	PUNCT
cana-1327	19	25	‘	'	PUNCT
cana-1327	19	26	and	and	CCONJ
cana-1327	19	27	‘	'	PUNCT
cana-1327	19	28	.	.	PUNCT
cana-1327	19	29	’	'	PUNCT
cana-1327	19	30	are	be	AUX
cana-1327	19	31	defined	define	VERB
cana-1327	19	32	as	as	SCONJ
cana-1327	19	33	follows	follow	VERB
cana-1327	19	34	:	:	PUNCT
cana-1327	19	35	p	p	X
cana-1327	19	36	q	q	X
cana-1327	19	37	r	r	NOUN
cana-1327	19	38	p	p	NOUN
cana-1327	19	39	p	p	PROPN
cana-1327	19	40	p	p	X
cana-1327	19	41	p	p	X
cana-1327	19	42	q	q	X
cana-1327	19	43	q	q	X
cana-1327	19	44	p	p	X
cana-1327	19	45	q	q	NOUN
cana-1327	19	46	r	r	NOUN
cana-1327	19	47	r	r	NOUN
cana-1327	19	48	r	r	NOUN
cana-1327	19	49	p	p	NOUN
cana-1327	19	50	.	.	PUNCT
cana-1327	20	1	p	p	X
cana-1327	20	2	q	q	NOUN
cana-1327	20	3	r	r	NOUN
cana-1327	20	4	p	p	NOUN
cana-1327	21	1	p	p	PROPN
cana-1327	21	2	p	p	X
cana-1327	21	3	p	p	NOUN
cana-1327	21	4	q	q	X
cana-1327	21	5	p	p	X
cana-1327	21	6	p	p	X
cana-1327	21	7	p	p	NOUN
cana-1327	21	8	r	r	NOUN
cana-1327	21	9	p	p	NOUN
cana-1327	21	10	q	q	NOUN
cana-1327	21	11	r	r	NOUN
cana-1327	21	12	then	then	ADV
cana-1327	21	13	(	(	PUNCT
cana-1327	21	14	x	x	X
cana-1327	21	15	,	,	PUNCT
cana-1327	21	16	-	-	PUNCT
cana-1327	21	17	,	,	PUNCT
cana-1327	21	18	.	.	PUNCT
cana-1327	21	19	)	)	PUNCT
cana-1327	21	20	is	be	AUX
cana-1327	21	21	a	a	DET
cana-1327	21	22	near	near	ADJ
cana-1327	21	23	subtraction	subtraction	NOUN
cana-1327	21	24	normal	normal	ADJ
cana-1327	21	25	sg	sg	NOUN
cana-1327	21	26	.	.	PUNCT
cana-1327	22	1	the	the	DET
cana-1327	22	2	ideals	ideal	NOUN
cana-1327	22	3	of	of	ADP
cana-1327	22	4	x	x	SYM
cana-1327	22	5	are	be	AUX
cana-1327	22	6	{	{	PUNCT
cana-1327	22	7	p},{p	p},{p	NOUN
cana-1327	22	8	,	,	PUNCT
cana-1327	22	9	q	q	NOUN
cana-1327	22	10	}	}	PUNCT
cana-1327	22	11	{	{	PUNCT
cana-1327	22	12	p	p	X
cana-1327	22	13	,	,	PUNCT
cana-1327	22	14	q	q	ADJ
cana-1327	22	15	,	,	PUNCT
cana-1327	22	16	r	r	NOUN
cana-1327	22	17	}	}	PUNCT
cana-1327	22	18	which	which	PRON
cana-1327	22	19	are	be	AUX
cana-1327	22	20	ps	ps	PROPN
cana-1327	22	21	.	.	PUNCT
cana-1327	22	22	theorem	theorem	VERB
cana-1327	22	23	3.4	3.4	NUM
cana-1327	22	24	:	:	PUNCT
cana-1327	22	25	the	the	DET
cana-1327	22	26	family	family	NOUN
cana-1327	22	27	of	of	ADP
cana-1327	22	28	non	non	ADJ
cana-1327	22	29	empty	empty	ADJ
cana-1327	22	30	intersection	intersection	NOUN
cana-1327	22	31	of	of	ADP
cana-1327	22	32	ps	ps	NOUN
cana-1327	22	33	ideals	ideal	NOUN
cana-1327	22	34	of	of	ADP
cana-1327	22	35	a	a	DET
cana-1327	22	36	nssg	nssg	ADJ
cana-1327	22	37	x	x	NOUN
cana-1327	22	38	a	a	DET
cana-1327	22	39	ps	ps	NOUN
cana-1327	22	40	is	be	AUX
cana-1327	22	41	an	an	DET
cana-1327	22	42	ideals	ideal	NOUN
cana-1327	22	43	of	of	ADP
cana-1327	22	44	x.	x.	NOUN
cana-1327	22	45	proof	proof	NOUN
cana-1327	22	46	:	:	PUNCT
cana-1327	22	47	assume	assume	VERB
cana-1327	22	48	that	that	SCONJ
cana-1327	22	49	{	{	PUNCT
cana-1327	22	50	𝐴𝛼}∝𝜖∆a	𝐴𝛼}∝𝜖∆a	PROPN
cana-1327	22	51	family	family	NOUN
cana-1327	22	52	of	of	ADP
cana-1327	22	53	psi	psi	NOUN
cana-1327	22	54	of	of	ADP
cana-1327	22	55	x	x	PUNCT
cana-1327	22	56	and	and	CCONJ
cana-1327	22	57	leta=⋂	leta=⋂	PROPN
cana-1327	22	58	𝐴𝛼𝛼𝜖∆	𝐴𝛼𝛼𝜖∆	PROPN
cana-1327	22	59	.	.	PUNCT
cana-1327	23	1	by	by	ADP
cana-1327	23	2	known	know	VERB
cana-1327	23	3	theorem	theorem	NOUN
cana-1327	23	4	,	,	PUNCT
cana-1327	23	5	a	a	PRON
cana-1327	23	6	is	be	AUX
cana-1327	23	7	an	an	DET
cana-1327	23	8	ideal	ideal	NOUN
cana-1327	23	9	of	of	ADP
cana-1327	23	10	x.	x.	PROPN
cana-1327	23	11	letx	letx	PROPN
cana-1327	23	12	,	,	PUNCT
cana-1327	23	13	y	y	PROPN
cana-1327	23	14	,	,	PUNCT
cana-1327	23	15	sx	sx	PROPN
cana-1327	23	16	,	,	PUNCT
cana-1327	23	17	xy∈	xy∈	PROPN
cana-1327	23	18	a	a	X
cana-1327	23	19	,	,	PUNCT
cana-1327	23	20	γy∈	γy∈	PROPN
cana-1327	23	21	a	a	DET
cana-1327	23	22	⇒xy∈⋂	⇒xy∈⋂	NOUN
cana-1327	23	23	𝐴𝛼𝛼𝜖∆	𝐴𝛼𝛼𝜖∆	NOUN
cana-1327	23	24	⇒	⇒	NOUN
cana-1327	23	25	xy	xy	PROPN
cana-1327	23	26	a	a	NOUN
cana-1327	23	27	for	for	ADP
cana-1327	23	28	each	each	DET
cana-1327	23	29	α∈	α∈	PROPN
cana-1327	23	30	δ	δ	PROPN
cana-1327	23	31	.	.	PUNCT
cana-1327	24	1	xy	xy	PROPN
cana-1327	24	2	a	a	ADV
cana-1327	24	3	,	,	PUNCT
cana-1327	24	4	a	a	ADV
cana-1327	24	5	is	be	AUX
cana-1327	24	6	a	a	DET
cana-1327	24	7	psi	psi	NOUN
cana-1327	24	8	of	of	ADP
cana-1327	24	9	x	x	PRON
cana-1327	24	10	,	,	PUNCT
cana-1327	24	11	sx⇒xsy∈𝐴𝛼	sx⇒xsy∈𝐴𝛼	ADP
cana-1327	24	12	⇒	⇒	NOUN
cana-1327	24	13	xsy	xsy	VERB
cana-1327	24	14	∈𝐴𝛼for	∈𝐴𝛼for	ADP
cana-1327	24	15	all	all	DET
cana-1327	24	16	α	α	PRON
cana-1327	24	17	∈δ	∈δ	PROPN
cana-1327	24	18	⇒xsy	⇒xsy	PROPN
cana-1327	24	19	∈⋂	∈⋂	NUM
cana-1327	24	20	𝐴𝛼𝛼𝜖∆	𝐴𝛼𝛼𝜖∆	PROPN
cana-1327	24	21	⇒xsy	⇒xsy	PROPN
cana-1327	24	22	∈	∈	PROPN
cana-1327	24	23	a.	a.	NOUN
cana-1327	24	24	communications	communication	NOUN
cana-1327	24	25	on	on	ADP
cana-1327	24	26	applied	apply	VERB
cana-1327	24	27	nonlinear	nonlinear	ADJ
cana-1327	24	28	analysis	analysis	NOUN
cana-1327	24	29	issn	issn	NOUN
cana-1327	24	30	:	:	PUNCT
cana-1327	24	31	1074	1074	NUM
cana-1327	24	32	-	-	PUNCT
cana-1327	24	33	133x	133x	NUM
cana-1327	24	34	vol	vol	NOUN
cana-1327	24	35	31	31	NUM
cana-1327	24	36	no	no	NOUN
cana-1327	24	37	.	.	PUNCT
cana-1327	25	1	7s	7	NOUN
cana-1327	25	2	(	(	PUNCT
cana-1327	25	3	2024	2024	NUM
cana-1327	25	4	)	)	PUNCT
cana-1327	25	5	480	480	NUM
cana-1327	25	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1327	25	7	∴a	∴a	PROPN
cana-1327	25	8	is	be	AUX
cana-1327	25	9	a	a	DET
cana-1327	25	10	psi	psi	NOUN
cana-1327	25	11	of	of	ADP
cana-1327	25	12	x.	x.	NOUN
cana-1327	25	13	theorem	theorem	NOUN
cana-1327	25	14	3.5	3.5	NUM
cana-1327	25	15	:	:	PUNCT
cana-1327	25	16	suppose	suppose	VERB
cana-1327	25	17	that	that	SCONJ
cana-1327	25	18	x	x	PRON
cana-1327	25	19	be	be	AUX
cana-1327	25	20	a	a	DET
cana-1327	25	21	nssg	nssg	ADJ
cana-1327	25	22	and	and	CCONJ
cana-1327	25	23	a	a	DET
cana-1327	25	24	be	be	NOUN
cana-1327	25	25	ideal	ideal	ADJ
cana-1327	25	26	in	in	ADP
cana-1327	25	27	x.	x.	NOUN
cana-1327	25	28	so	so	SCONJ
cana-1327	25	29	that	that	SCONJ
cana-1327	25	30	a	a	PRON
cana-1327	25	31	is	be	AUX
cana-1327	25	32	a	a	DET
cana-1327	25	33	ps	ps	NOUN
cana-1327	25	34	iff	iff	NOUN
cana-1327	25	35	for	for	ADP
cana-1327	25	36	all	all	DET
cana-1327	25	37	a	a	DET
cana-1327	25	38	∈	∈	NOUN
cana-1327	25	39	x.the	x.the	DET
cana-1327	25	40	seta	seta	PROPN
cana-1327	25	41	r	r	X
cana-1327	25	42	(	(	PUNCT
cana-1327	25	43	a	a	X
cana-1327	25	44	)	)	PUNCT
cana-1327	26	1	=	=	NOUN
cana-1327	26	2	{	{	PUNCT
cana-1327	26	3	x	x	SYM
cana-1327	26	4	∈	∈	PROPN
cana-1327	26	5	x.	x.	NOUN
cana-1327	26	6	:	:	PUNCT
cana-1327	26	7	ax	ax	NOUN
cana-1327	26	8	a	a	PRON
cana-1327	26	9	}	}	PUNCT
cana-1327	26	10	is	be	AUX
cana-1327	26	11	an	an	DET
cana-1327	26	12	ideal	ideal	NOUN
cana-1327	26	13	of	of	ADP
cana-1327	26	14	x.	x.	NOUN
cana-1327	26	15	proof	proof	NOUN
cana-1327	26	16	:	:	PUNCT
cana-1327	26	17	assume	assume	VERB
cana-1327	26	18	that	that	SCONJ
cana-1327	26	19	a	a	PRON
cana-1327	26	20	will	will	AUX
cana-1327	26	21	be	be	AUX
cana-1327	26	22	a	a	DET
cana-1327	26	23	psi	psi	NOUN
cana-1327	26	24	in	in	ADP
cana-1327	26	25	x	x	X
cana-1327	26	26	and	and	CCONJ
cana-1327	26	27	a∈	a∈	PROPN
cana-1327	26	28	x	x	PRON
cana-1327	26	29	let	let	VERB
cana-1327	26	30	x	x	X
cana-1327	26	31	,	,	PUNCT
cana-1327	26	32	y∈a	y∈a	ADJ
cana-1327	26	33	r	r	X
cana-1327	26	34	(	(	PUNCT
cana-1327	26	35	a	a	PRON
cana-1327	26	36	)	)	PUNCT
cana-1327	26	37	⇒	⇒	NOUN
cana-1327	26	38	ax	ax	NOUN
cana-1327	26	39	,	,	PUNCT
cana-1327	26	40	ay∈a	ay∈a	VERB
cana-1327	26	41	.	.	PUNCT
cana-1327	27	1	a(x	a(x	PROPN
cana-1327	27	2	–	–	PUNCT
cana-1327	27	3	y	y	NOUN
cana-1327	27	4	)	)	PUNCT
cana-1327	27	5	=	=	NOUN
cana-1327	27	6	ax	ax	NOUN
cana-1327	27	7	–	–	PUNCT
cana-1327	27	8	ay	ay	X
cana-1327	27	9	∈a	∈a	ADJ
cana-1327	27	10	,	,	PUNCT
cana-1327	27	11	since	since	SCONJ
cana-1327	27	12	ais	ais	PROPN
cana-1327	27	13	an	an	DET
cana-1327	27	14	ideal	ideal	NOUN
cana-1327	27	15	of	of	ADP
cana-1327	27	16	x	x	X
cana-1327	27	17	and	and	CCONJ
cana-1327	27	18	ax	ax	NOUN
cana-1327	27	19	,	,	PUNCT
cana-1327	27	20	ay∈a	ay∈a	VERB
cana-1327	27	21	.	.	PUNCT
cana-1327	28	1	in	in	ADP
cana-1327	28	2	this	this	DET
cana-1327	28	3	manner	manner	NOUN
cana-1327	28	4	x	x	X
cana-1327	28	5	–	–	PUNCT
cana-1327	28	6	y	y	PROPN
cana-1327	28	7	∈	∈	PROPN
cana-1327	28	8	a	a	DET
cana-1327	28	9	r	r	NOUN
cana-1327	28	10	(	(	PUNCT
cana-1327	28	11	a	a	NOUN
cana-1327	28	12	)	)	PUNCT
cana-1327	28	13	and	and	CCONJ
cana-1327	28	14	hence	hence	ADV
cana-1327	28	15	a	a	DET
cana-1327	28	16	r	r	NOUN
cana-1327	28	17	(	(	PUNCT
cana-1327	28	18	a	a	PRON
cana-1327	28	19	)	)	PUNCT
cana-1327	28	20	is	be	AUX
cana-1327	28	21	a	a	DET
cana-1327	28	22	sub	sub	NOUN
cana-1327	28	23	algebra	algebra	NOUN
cana-1327	28	24	of	of	ADP
cana-1327	28	25	x.	x.	NOUN
cana-1327	28	26	presently	presently	ADV
cana-1327	28	27	x∈	x∈	VERB
cana-1327	28	28	a	a	DET
cana-1327	28	29	r	r	NOUN
cana-1327	28	30	(	(	PUNCT
cana-1327	28	31	a	a	PRON
cana-1327	28	32	)	)	PUNCT
cana-1327	28	33	implies	imply	VERB
cana-1327	28	34	that	that	SCONJ
cana-1327	28	35	ax∈	ax∈	NOUN
cana-1327	28	36	a.	a.	NOUN
cana-1327	28	37	presently	presently	ADV
cana-1327	28	38	ax∈	ax∈	VERB
cana-1327	28	39	a	a	PRON
cana-1327	28	40	,	,	PUNCT
cana-1327	28	41	a	a	PRON
cana-1327	28	42	is	be	AUX
cana-1327	28	43	ps	ps	NOUN
cana-1327	28	44	implies	imply	VERB
cana-1327	28	45	asx∈	asx∈	NOUN
cana-1327	28	46	a	a	DET
cana-1327	28	47	implies	imply	VERB
cana-1327	28	48	that	that	SCONJ
cana-1327	28	49	sx∈a	sx∈a	ADJ
cana-1327	28	50	r	r	NOUN
cana-1327	28	51	(	(	PUNCT
cana-1327	28	52	a	a	PRON
cana-1327	28	53	)	)	PUNCT
cana-1327	28	54	presently	presently	ADV
cana-1327	28	55	ax∈	ax∈	VERB
cana-1327	28	56	a	a	PRON
cana-1327	28	57	,	,	PUNCT
cana-1327	28	58	s∈s	s∈s	NOUN
cana-1327	28	59	,	,	PUNCT
cana-1327	28	60	a	a	PRON
cana-1327	28	61	is	be	AUX
cana-1327	28	62	an	an	DET
cana-1327	28	63	ideal	ideal	NOUN
cana-1327	28	64	implies	imply	VERB
cana-1327	28	65	that	that	SCONJ
cana-1327	28	66	axs∈	axs∈	NUM
cana-1327	28	67	a	a	DET
cana-1327	28	68	implies	implie	NOUN
cana-1327	28	69	that	that	SCONJ
cana-1327	28	70	xs∈a	xs∈a	PROPN
cana-1327	28	71	r	r	PROPN
cana-1327	28	72	(	(	PUNCT
cana-1327	28	73	a	a	NOUN
cana-1327	28	74	)	)	PUNCT
cana-1327	28	75	consequently	consequently	ADV
cana-1327	28	76	a	a	DET
cana-1327	28	77	r	r	NOUN
cana-1327	28	78	(	(	PUNCT
cana-1327	28	79	a	a	PRON
cana-1327	28	80	)	)	PUNCT
cana-1327	28	81	is	be	AUX
cana-1327	28	82	an	an	DET
cana-1327	28	83	ideal	ideal	NOUN
cana-1327	28	84	in	in	ADP
cana-1327	28	85	x	x	PUNCT
cana-1327	28	86	for	for	ADP
cana-1327	28	87	each	each	DET
cana-1327	28	88	a∈	a∈	PROPN
cana-1327	28	89	x.	x.	NOUN
cana-1327	28	90	on	on	ADP
cana-1327	28	91	opposite	opposite	ADJ
cana-1327	28	92	way	way	NOUN
cana-1327	28	93	guess	guess	VERB
cana-1327	28	94	that	that	SCONJ
cana-1327	28	95	a	a	DET
cana-1327	28	96	r	r	NOUN
cana-1327	28	97	(	(	PUNCT
cana-1327	28	98	a	a	PRON
cana-1327	28	99	)	)	PUNCT
cana-1327	28	100	is	be	AUX
cana-1327	28	101	an	an	DET
cana-1327	28	102	ideal	ideal	NOUN
cana-1327	28	103	in	in	ADP
cana-1327	28	104	x	x	PUNCT
cana-1327	28	105	for	for	ADP
cana-1327	28	106	every	every	DET
cana-1327	28	107	a∈	a∈	PROPN
cana-1327	28	108	x	x	PUNCT
cana-1327	28	109	.	.	PUNCT
cana-1327	29	1	letx	letx	PROPN
cana-1327	29	2	,	,	PUNCT
cana-1327	29	3	y∈xand	y∈xand	PROPN
cana-1327	29	4	xy∈a	xy∈a	PROPN
cana-1327	29	5	.	.	PUNCT
cana-1327	30	1	xy∈a	xy∈a	PROPN
cana-1327	30	2	implies	imply	VERB
cana-1327	30	3	that	that	SCONJ
cana-1327	30	4	y∈a	y∈a	ADJ
cana-1327	30	5	r	r	NOUN
cana-1327	30	6	(	(	PUNCT
cana-1327	30	7	a	a	NOUN
cana-1327	30	8	)	)	PUNCT
cana-1327	30	9	y	y	PROPN
cana-1327	30	10	∈a	∈a	ADJ
cana-1327	30	11	r	r	NOUN
cana-1327	30	12	(	(	PUNCT
cana-1327	30	13	a	a	NOUN
cana-1327	30	14	)	)	PUNCT
cana-1327	30	15	,	,	PUNCT
cana-1327	30	16	a	a	DET
cana-1327	30	17	r	r	NOUN
cana-1327	30	18	(	(	PUNCT
cana-1327	30	19	a	a	PRON
cana-1327	30	20	)	)	PUNCT
cana-1327	30	21	is	be	AUX
cana-1327	30	22	an	an	DET
cana-1327	30	23	ideal	ideal	NOUN
cana-1327	30	24	implies	imply	VERB
cana-1327	30	25	sy	sy	PROPN
cana-1327	30	26	∈a	∈a	ADJ
cana-1327	30	27	r	r	NOUN
cana-1327	30	28	(	(	PUNCT
cana-1327	30	29	a	a	NOUN
cana-1327	30	30	)	)	PUNCT
cana-1327	30	31	for	for	ADP
cana-1327	30	32	every	every	DET
cana-1327	30	33	s	s	NOUN
cana-1327	30	34	∈x	∈x	NOUN
cana-1327	30	35	implies	imply	VERB
cana-1327	30	36	that	that	SCONJ
cana-1327	30	37	xsy∈a	xsy∈a	PROPN
cana-1327	30	38	for	for	ADP
cana-1327	30	39	each	each	DET
cana-1327	30	40	s∈x	s∈x	NOUN
cana-1327	30	41	∴	∴	PROPN
cana-1327	30	42	a	a	PRON
cana-1327	30	43	is	be	AUX
cana-1327	30	44	a	a	DET
cana-1327	30	45	psi	psi	NOUN
cana-1327	30	46	of	of	ADP
cana-1327	30	47	x.	x.	NOUN
cana-1327	30	48	theorem	theorem	VERB
cana-1327	30	49	3.6	3.6	NUM
cana-1327	30	50	:	:	PUNCT
cana-1327	30	51	let	let	VERB
cana-1327	30	52	x	x	PRON
cana-1327	30	53	be	be	AUX
cana-1327	30	54	a	a	DET
cana-1327	30	55	nssg	nssg	ADJ
cana-1327	30	56	and	and	CCONJ
cana-1327	30	57	a	a	PRON
cana-1327	30	58	is	be	AUX
cana-1327	30	59	an	an	DET
cana-1327	30	60	ideal	ideal	NOUN
cana-1327	30	61	in	in	ADP
cana-1327	30	62	x.	x.	NOUN
cana-1327	30	63	then	then	ADV
cana-1327	30	64	a	a	PRON
cana-1327	30	65	is	be	AUX
cana-1327	30	66	ps	ps	PROPN
cana-1327	30	67	iff	iff	PROPN
cana-1327	30	68	for	for	ADP
cana-1327	30	69	all	all	DET
cana-1327	30	70	a∈x	a∈x	NOUN
cana-1327	30	71	the	the	DET
cana-1327	30	72	set	set	NOUN
cana-1327	30	73	al(a	al(a	NUM
cana-1327	30	74	)	)	PUNCT
cana-1327	30	75	=	=	PRON
cana-1327	30	76	{	{	PUNCT
cana-1327	30	77	x∈x	x∈x	VERB
cana-1327	31	1	:	:	PUNCT
cana-1327	31	2	xa	xa	PROPN
cana-1327	31	3	∈	∈	PROPN
cana-1327	31	4	a	a	PRON
cana-1327	31	5	}	}	PUNCT
cana-1327	31	6	is	be	AUX
cana-1327	31	7	an	an	DET
cana-1327	31	8	ideal	ideal	NOUN
cana-1327	31	9	of	of	ADP
cana-1327	31	10	x.	x.	PROPN
cana-1327	31	11	xy∈a	xy∈a	PROPN
cana-1327	31	12	implies	imply	VERB
cana-1327	31	13	x∈	x∈	PROPN
cana-1327	31	14	al(y	al(y	NOUN
cana-1327	31	15	)	)	PUNCT
cana-1327	31	16	,	,	PUNCT
cana-1327	31	17	al(y	al(y	X
cana-1327	31	18	)	)	PUNCT
cana-1327	31	19	is	be	AUX
cana-1327	31	20	an	an	DET
cana-1327	31	21	ideal	ideal	NOUN
cana-1327	31	22	implies	imply	VERB
cana-1327	31	23	that	that	SCONJ
cana-1327	31	24	xs∈	xs∈	PROPN
cana-1327	31	25	al(y	al(y	NOUN
cana-1327	31	26	)	)	PUNCT
cana-1327	31	27	for	for	ADP
cana-1327	31	28	each	each	DET
cana-1327	31	29	s∈	s∈	NOUN
cana-1327	31	30	x	x	PRON
cana-1327	31	31	,	,	PUNCT
cana-1327	31	32	xsy	xsy	X
cana-1327	31	33	∈a	∈a	VERB
cana-1327	31	34	for	for	ADP
cana-1327	31	35	each	each	PRON
cana-1327	32	1	s	s	PART
cana-1327	32	2	∈x	∈x	NOUN
cana-1327	32	3	.	.	PUNCT
cana-1327	32	4	thusa	thusa	NOUN
cana-1327	32	5	will	will	AUX
cana-1327	32	6	be	be	AUX
cana-1327	32	7	a	a	DET
cana-1327	32	8	psi	psi	NOUN
cana-1327	32	9	of	of	ADP
cana-1327	32	10	x.	x.	NOUN
cana-1327	32	11	corollary3.7	corollary3.7	NOUN
cana-1327	32	12	:	:	PUNCT
cana-1327	32	13	suppose	suppose	VERB
cana-1327	32	14	x	x	PRON
cana-1327	32	15	be	be	AUX
cana-1327	32	16	a	a	DET
cana-1327	32	17	nssg	nssg	ADJ
cana-1327	32	18	and	and	CCONJ
cana-1327	32	19	a	a	PRON
cana-1327	32	20	is	be	AUX
cana-1327	32	21	an	an	DET
cana-1327	32	22	ideal	ideal	NOUN
cana-1327	32	23	in	in	ADP
cana-1327	32	24	x.	x.	NOUN
cana-1327	32	25	then	then	ADV
cana-1327	32	26	a	a	PRON
cana-1327	32	27	is	be	AUX
cana-1327	32	28	ps	ps	NOUN
cana-1327	32	29	iff	iff	PROPN
cana-1327	32	30	for	for	ADP
cana-1327	32	31	each	each	DET
cana-1327	32	32	a∈	a∈	PROPN
cana-1327	32	33	x	x	PUNCT
cana-1327	32	34	the	the	DET
cana-1327	32	35	set	set	NOUN
cana-1327	32	36	a	a	DET
cana-1327	32	37	(	(	PUNCT
cana-1327	32	38	a	a	NOUN
cana-1327	32	39	)	)	PUNCT
cana-1327	32	40	=	=	NOUN
cana-1327	32	41	{	{	PUNCT
cana-1327	32	42	x∈	x∈	PROPN
cana-1327	32	43	x	x	X
cana-1327	32	44	:	:	PUNCT
cana-1327	32	45	xa	xa	PROPN
cana-1327	32	46	,	,	PUNCT
cana-1327	32	47	ax	ax	NOUN
cana-1327	32	48	∈	∈	PROPN
cana-1327	32	49	a	a	PRON
cana-1327	32	50	}	}	PUNCT
cana-1327	32	51	is	be	AUX
cana-1327	32	52	an	an	DET
cana-1327	32	53	ideal	ideal	NOUN
cana-1327	32	54	in	in	ADP
cana-1327	32	55	x.	x.	NOUN
cana-1327	32	56	communications	communication	NOUN
cana-1327	32	57	on	on	ADP
cana-1327	32	58	applied	apply	VERB
cana-1327	32	59	nonlinear	nonlinear	ADJ
cana-1327	32	60	analysis	analysis	NOUN
cana-1327	32	61	issn	issn	NOUN
cana-1327	32	62	:	:	PUNCT
cana-1327	32	63	1074	1074	NUM
cana-1327	32	64	-	-	PUNCT
cana-1327	32	65	133x	133x	NUM
cana-1327	32	66	vol	vol	NOUN
cana-1327	32	67	31	31	NUM
cana-1327	32	68	no	no	NOUN
cana-1327	32	69	.	.	PUNCT
cana-1327	33	1	7s	7	NOUN
cana-1327	33	2	(	(	PUNCT
cana-1327	33	3	2024	2024	NUM
cana-1327	33	4	)	)	PUNCT
cana-1327	33	5	481	481	NUM
cana-1327	33	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1327	33	7	theorem3.8	theorem3.8	NUM
cana-1327	33	8	:	:	PUNCT
cana-1327	33	9	let	let	VERB
cana-1327	33	10	a	a	PRON
cana-1327	33	11	be	be	AUX
cana-1327	33	12	any	any	DET
cana-1327	33	13	psi	psi	NOUN
cana-1327	33	14	in	in	ADP
cana-1327	33	15	a	a	DET
cana-1327	33	16	nssg	nssg	ADJ
cana-1327	33	17	x	x	NOUN
cana-1327	33	18	and	and	CCONJ
cana-1327	33	19	x1	x1	PROPN
cana-1327	33	20	,	,	PUNCT
cana-1327	33	21	x2	x2	PROPN
cana-1327	33	22	…	…	PUNCT
cana-1327	33	23	.,xn	.,xn	PROPN
cana-1327	33	24	x.	x.	NOUN
cana-1327	33	25	then	then	ADV
cana-1327	34	1	1	1	X
cana-1327	34	2	)	)	PUNCT
cana-1327	34	3	x1x2∈	x1x2∈	PUNCT
cana-1327	35	1	a	a	DET
cana-1327	35	2	implies	imply	VERB
cana-1327	35	3	x2x1∈	x2x1∈	PROPN
cana-1327	35	4	a.	a.	NOUN
cana-1327	35	5	2	2	NUM
cana-1327	35	6	)	)	PUNCT
cana-1327	35	7	x1x2	x1x2	X
cana-1327	35	8	…	…	PUNCT
cana-1327	35	9	.xn	.xn	X
cana-1327	36	1	a	a	DET
cana-1327	36	2	if	if	NOUN
cana-1327	36	3	and	and	CCONJ
cana-1327	36	4	only	only	ADV
cana-1327	36	5	if	if	SCONJ
cana-1327	36	6	<	<	X
cana-1327	36	7	x1><x2>	x1><x2>	NUM
cana-1327	36	8	…	…	SYM
cana-1327	36	9	.<xn>	.<xn>	NOUN
cana-1327	36	10	a.	a.	NOUN
cana-1327	36	11	proof	proof	NOUN
cana-1327	36	12	:	:	PUNCT
cana-1327	36	13	1	1	X
cana-1327	36	14	)	)	PUNCT
cana-1327	36	15	suppose	suppose	VERB
cana-1327	36	16	x1x2∈	x1x2∈	PROPN
cana-1327	36	17	a.	a.	PROPN
cana-1327	36	18	then	then	ADV
cana-1327	36	19	(	(	PUNCT
cana-1327	36	20	x2x1	x2x1	X
cana-1327	36	21	)	)	PUNCT
cana-1327	36	22	2	2	NUM
cana-1327	36	23	=	=	SYM
cana-1327	36	24	x2	x2	PROPN
cana-1327	36	25	(	(	PUNCT
cana-1327	36	26	x1x2)x1∈	x1x2)x1∈	PROPN
cana-1327	36	27	a	a	X
cana-1327	36	28	and	and	CCONJ
cana-1327	36	29	hence	hence	ADV
cana-1327	36	30	x2x1∈	x2x1∈	PUNCT
cana-1327	36	31	a.	a.	NOUN
cana-1327	36	32	2	2	X
cana-1327	36	33	)	)	PUNCT
cana-1327	36	34	let	let	VERB
cana-1327	36	35	x1x2	x1x2	VERB
cana-1327	36	36	…	…	PUNCT
cana-1327	36	37	.xna	.xna	NUM
cana-1327	36	38	.	.	PUNCT
cana-1327	37	1	since	since	SCONJ
cana-1327	37	2	a	a	PRON
cana-1327	37	3	is	be	AUX
cana-1327	37	4	psi	psi	NOUN
cana-1327	37	5	of	of	ADP
cana-1327	37	6	x.	x.	NOUN
cana-1327	37	7	then	then	ADV
cana-1327	37	8	by	by	ADP
cana-1327	37	9	corollary	corollary	ADJ
cana-1327	37	10	3.6	3.6	NUM
cana-1327	37	11	,	,	PUNCT
cana-1327	37	12	x1∈al(x2x3	x1∈al(x2x3	PROPN
cana-1327	37	13	…	…	SYM
cana-1327	37	14	..	..	PUNCT
cana-1327	37	15	xn	xn	X
cana-1327	37	16	)	)	PUNCT
cana-1327	37	17	⇒<x1>⊆	⇒<x1>⊆	NOUN
cana-1327	37	18	al(x2x3	al(x2x3	PROPN
cana-1327	37	19	…	…	SYM
cana-1327	37	20	..	..	PUNCT
cana-1327	37	21	xn	xn	X
cana-1327	37	22	)	)	PUNCT
cana-1327	37	23	⇒<x1	⇒<x1	PROPN
cana-1327	37	24	>	>	PROPN
cana-1327	37	25	x2x3	x2x3	PROPN
cana-1327	37	26	…	…	PUNCT
cana-1327	37	27	..	..	PUNCT
cana-1327	37	28	xn⊆a	xn⊆a	PROPN
cana-1327	37	29	.	.	PUNCT
cana-1327	38	1	by	by	ADP
cana-1327	38	2	property	property	NOUN
cana-1327	38	3	(	(	PUNCT
cana-1327	38	4	1	1	NUM
cana-1327	38	5	)	)	PUNCT
cana-1327	38	6	,	,	PUNCT
cana-1327	38	7	we	we	PRON
cana-1327	38	8	have	have	VERB
cana-1327	38	9	x2x3	x2x3	NOUN
cana-1327	38	10	…	…	PUNCT
cana-1327	38	11	..	..	PUNCT
cana-1327	38	12	xn	xn	PROPN
cana-1327	39	1	<	<	X
cana-1327	39	2	x1>⊆	x1>⊆	PROPN
cana-1327	39	3	a.	a.	NOUN
cana-1327	39	4	now	now	ADV
cana-1327	39	5	x2∈	x2∈	PROPN
cana-1327	39	6	al(x3x4	al(x3x4	PROPN
cana-1327	39	7	…	…	PUNCT
cana-1327	39	8	..	..	PUNCT
cana-1327	39	9	xn	xn	PROPN
cana-1327	39	10	<	<	X
cana-1327	39	11	x1>)⇒<x2>⊆al(x3x4	x1>)⇒<x2>⊆al(x3x4	X
cana-1327	39	12	…	…	PUNCT
cana-1327	39	13	..	..	PUNCT
cana-1327	39	14	xn	xn	PROPN
cana-1327	39	15	<	<	X
cana-1327	39	16	x1	x1	PROPN
cana-1327	39	17	>	>	X
cana-1327	39	18	)	)	PUNCT
cana-1327	39	19	.	.	PUNCT
cana-1327	40	1	therefore	therefore	ADV
cana-1327	40	2	<	<	X
cana-1327	40	3	x2	x2	X
cana-1327	40	4	>	>	X
cana-1327	40	5	x3x4	x3x4	PROPN
cana-1327	40	6	…	…	SYM
cana-1327	40	7	..	..	PUNCT
cana-1327	40	8	xn	xn	PROPN
cana-1327	40	9	<	<	X
cana-1327	40	10	x1>⊆	x1>⊆	PROPN
cana-1327	40	11	a	a	DET
cana-1327	40	12	⇒x3x4	⇒x3x4	NUM
cana-1327	40	13	…	…	PUNCT
cana-1327	40	14	..	..	PUNCT
cana-1327	40	15	xn	xn	PUNCT
cana-1327	40	16	<	<	X
cana-1327	40	17	x1><x2>⊆	x1><x2>⊆	PROPN
cana-1327	40	18	a.	a.	NOUN
cana-1327	40	19	continuing	continue	VERB
cana-1327	40	20	this	this	DET
cana-1327	40	21	process	process	NOUN
cana-1327	40	22	we	we	PRON
cana-1327	40	23	have	have	VERB
cana-1327	40	24	x1x2	x1x2	PART
cana-1327	40	25	…	…	PUNCT
cana-1327	40	26	.xn	.xn	PROPN
cana-1327	40	27	a⇒<x1><x2>	a⇒<x1><x2>	PROPN
cana-1327	40	28	…	…	PUNCT
cana-1327	40	29	..	..	PUNCT
cana-1327	40	30	<xn>⊆	<xn>⊆	X
cana-1327	40	31	a.	a.	NOUN
cana-1327	40	32	conversely	conversely	ADV
cana-1327	40	33	suppose	suppose	VERB
cana-1327	40	34	that	that	SCONJ
cana-1327	40	35	<	<	X
cana-1327	40	36	x1><x2>	x1><x2>	PROPN
cana-1327	40	37	…	…	SYM
cana-1327	40	38	.<xn>	.<xn>	PROPN
cana-1327	40	39	a	a	PRON
cana-1327	40	40	then	then	ADV
cana-1327	40	41	x1x2	x1x2	X
cana-1327	40	42	…	…	PUNCT
cana-1327	40	43	.xn<x1><x2>	.xn<x1><x2>	PUNCT
cana-1327	40	44	…	…	SYM
cana-1327	40	45	.<xn>	.<xn>	PROPN
cana-1327	40	46	a.	a.	NOUN
cana-1327	40	47	thereforex1x2	thereforex1x2	PROPN
cana-1327	40	48	…	…	SYM
cana-1327	40	49	.xna	.xna	X
cana-1327	40	50	corollary3.9	corollary3.9	NOUN
cana-1327	40	51	:	:	PUNCT
cana-1327	40	52	if	if	SCONJ
cana-1327	40	53	a	a	PRON
cana-1327	40	54	is	be	AUX
cana-1327	40	55	a	a	DET
cana-1327	40	56	psiin	psiin	NOUN
cana-1327	40	57	a	a	DET
cana-1327	40	58	nssgx	nssgx	NOUN
cana-1327	40	59	,	,	PUNCT
cana-1327	40	60	then	then	ADV
cana-1327	40	61	for	for	ADP
cana-1327	40	62	eachnnana⇒<a	eachnnana⇒<a	PROPN
cana-1327	40	63	>	>	X
cana-1327	40	64	na	na	NOUN
cana-1327	40	65	.	.	PUNCT
cana-1327	41	1	proof	proof	NOUN
cana-1327	41	2	:	:	PUNCT
cana-1327	41	3	by	by	ADP
cana-1327	41	4	the	the	DET
cana-1327	41	5	abovecorollary	abovecorollary	ADJ
cana-1327	41	6	3.8	3.8	NUM
cana-1327	41	7	substitute	substitute	NOUN
cana-1327	41	8	a1=	a1=	PROPN
cana-1327	41	9	a2=	a2=	PROPN
cana-1327	41	10	a3	a3	NOUN
cana-1327	41	11	…	…	PUNCT
cana-1327	41	12	.=	.=	PROPN
cana-1327	41	13	an=	an=	VERB
cana-1327	41	14	a	a	DET
cana-1327	41	15	corollary3.10	corollary3.10	NOUN
cana-1327	41	16	:	:	PUNCT
cana-1327	41	17	let	let	VERB
cana-1327	41	18	a	a	PRON
cana-1327	41	19	be	be	AUX
cana-1327	41	20	a	a	DET
cana-1327	41	21	psiin	psiin	NOUN
cana-1327	41	22	a	a	DET
cana-1327	41	23	nssgx	nssgx	NOUN
cana-1327	41	24	.	.	PUNCT
cana-1327	42	1	if	if	SCONJ
cana-1327	42	2	ana	ana	NOUN
cana-1327	42	3	,	,	PUNCT
cana-1327	42	4	nnthen	nnthen	ADV
cana-1327	42	5	<	<	X
cana-1327	42	6	as	as	ADP
cana-1327	42	7	>	>	X
cana-1327	42	8	n	n	CCONJ
cana-1327	42	9	,	,	PUNCT
cana-1327	42	10	<	<	X
cana-1327	42	11	sa	sa	AUX
cana-1327	42	12	>	>	X
cana-1327	42	13	n	n	PROPN
cana-1327	42	14	x.	x.	PROPN
cana-1327	42	15	theorem	theorem	VERB
cana-1327	42	16	3.11	3.11	NUM
cana-1327	42	17	:	:	PUNCT
cana-1327	42	18	each	each	DET
cana-1327	42	19	cspi	cspi	NOUN
cana-1327	42	20	a	a	PRON
cana-1327	42	21	in	in	ADP
cana-1327	42	22	anssgx	anssgx	NOUN
cana-1327	42	23	is	be	AUX
cana-1327	42	24	a	a	DET
cana-1327	42	25	psiof	psiof	NOUN
cana-1327	42	26	x.	x.	NOUN
cana-1327	42	27	proof	proof	NOUN
cana-1327	42	28	:	:	PUNCT
cana-1327	42	29	assume	assume	VERB
cana-1327	42	30	that	that	SCONJ
cana-1327	42	31	a	a	DET
cana-1327	42	32	be	be	AUX
cana-1327	42	33	a	a	DET
cana-1327	42	34	cspi	cspi	NOUN
cana-1327	42	35	ofthe	ofthe	NOUN
cana-1327	42	36	nssg	nssg	ADJ
cana-1327	43	1	x.	x.	NOUN
cana-1327	43	2	let	let	VERB
cana-1327	43	3	x	x	X
cana-1327	43	4	,	,	PUNCT
cana-1327	43	5	y	y	PROPN
cana-1327	43	6	x	x	SYM
cana-1327	43	7	and	and	CCONJ
cana-1327	43	8	xya⇒(yx)2=(yx)(yx)=y(xy)x	xya⇒(yx)2=(yx)(yx)=y(xy)x	PROPN
cana-1327	43	9	a.	a.	NOUN
cana-1327	43	10	(	(	PUNCT
cana-1327	43	11	yx)2a	yx)2a	PROPN
cana-1327	43	12	,	,	PUNCT
cana-1327	43	13	a	a	PRON
cana-1327	43	14	is	be	AUX
cana-1327	43	15	cspi	cspi	NOUN
cana-1327	43	16	implies	imply	VERB
cana-1327	43	17	yxa	yxa	NOUN
cana-1327	43	18	.	.	PUNCT
cana-1327	44	1	if	if	SCONJ
cana-1327	44	2	s	s	PROPN
cana-1327	44	3	x	x	SYM
cana-1327	44	4	,	,	PUNCT
cana-1327	44	5	(	(	PUNCT
cana-1327	44	6	xsy)2=	xsy)2=	PROPN
cana-1327	44	7	(	(	PUNCT
cana-1327	44	8	xsy)(xsy	xsy)(xsy	PROPN
cana-1327	44	9	)	)	PUNCT
cana-1327	44	10	=	=	PUNCT
cana-1327	44	11	xs(yx)sya	xs(yx)sya	PROPN
cana-1327	44	12	.	.	PUNCT
cana-1327	45	1	(	(	PUNCT
cana-1327	45	2	xsy)2a	xsy)2a	PROPN
cana-1327	45	3	,	,	PUNCT
cana-1327	45	4	a	a	PRON
cana-1327	45	5	is	be	AUX
cana-1327	45	6	csp	csp	PROPN
cana-1327	45	7	implies	imply	VERB
cana-1327	45	8	that	that	SCONJ
cana-1327	45	9	xsya	xsya	PROPN
cana-1327	45	10	.	.	PUNCT
cana-1327	46	1	thus	thus	ADV
cana-1327	46	2	a	a	PRON
cana-1327	46	3	will	will	AUX
cana-1327	46	4	be	be	AUX
cana-1327	46	5	a	a	DET
cana-1327	46	6	psi	psi	NOUN
cana-1327	46	7	of	of	ADP
cana-1327	46	8	x.	x.	NOUN
cana-1327	46	9	note3.12	note3.12	PROPN
cana-1327	46	10	:	:	PUNCT
cana-1327	46	11	the	the	DET
cana-1327	46	12	reverse	reverse	ADJ
cana-1327	46	13	proof	proof	NOUN
cana-1327	46	14	of	of	ADP
cana-1327	46	15	the	the	DET
cana-1327	46	16	theorem	theorem	NOUN
cana-1327	46	17	3.11which	3.11which	PRON
cana-1327	46	18	is	be	AUX
cana-1327	46	19	absurdity	absurdity	NOUN
cana-1327	46	20	i.e.	i.e.	ADV
cana-1327	46	21	,	,	PUNCT
cana-1327	46	22	a	a	DET
cana-1327	46	23	psiofanssg	psiofanssg	NOUN
cana-1327	46	24	which	which	PRON
cana-1327	46	25	is	be	AUX
cana-1327	46	26	not	not	PART
cana-1327	46	27	to	to	PART
cana-1327	46	28	becsp	becsp	VERB
cana-1327	46	29	.	.	PUNCT
cana-1327	47	1	example	example	NOUN
cana-1327	47	2	3.13	3.13	NUM
cana-1327	47	3	:	:	PUNCT
cana-1327	47	4	:	:	PUNCT
cana-1327	47	5	:	:	PUNCT
cana-1327	47	6	let	let	VERB
cana-1327	47	7	x	x	PUNCT
cana-1327	47	8	=	=	PUNCT
cana-1327	47	9	{	{	PUNCT
cana-1327	47	10	p	p	X
cana-1327	47	11	,	,	PUNCT
cana-1327	47	12	q	q	ADJ
cana-1327	47	13	,	,	PUNCT
cana-1327	47	14	r	r	NOUN
cana-1327	47	15	}	}	PUNCT
cana-1327	47	16	.	.	PUNCT
cana-1327	48	1	define	define	VERB
cana-1327	48	2	a	a	DET
cana-1327	48	3	binary	binary	ADJ
cana-1327	48	4	operations	operation	NOUN
cana-1327	48	5	‘	'	PUNCT
cana-1327	48	6	-	-	PUNCT
cana-1327	48	7	‘	'	PUNCT
cana-1327	48	8	and	and	CCONJ
cana-1327	48	9	‘	'	PUNCT
cana-1327	48	10	.	.	PUNCT
cana-1327	48	11	’	'	PUNCT
cana-1327	49	1	on	on	ADP
cana-1327	49	2	xas	xas	PROPN
cana-1327	49	3	follows	follow	VERB
cana-1327	49	4	p	p	NOUN
cana-1327	49	5	q	q	NOUN
cana-1327	49	6	r	r	NOUN
cana-1327	49	7	r	r	NOUN
cana-1327	49	8	p	p	NOUN
cana-1327	49	9	p	p	X
cana-1327	49	10	p	p	X
cana-1327	49	11	q	q	X
cana-1327	49	12	q	q	X
cana-1327	49	13	p	p	X
cana-1327	49	14	q	q	NOUN
cana-1327	49	15	r	r	NOUN
cana-1327	49	16	r	r	NOUN
cana-1327	49	17	r	r	NOUN
cana-1327	49	18	p	p	NOUN
cana-1327	49	19	.	.	PUNCT
cana-1327	50	1	p	p	X
cana-1327	50	2	q	q	NOUN
cana-1327	50	3	r	r	NOUN
cana-1327	50	4	p	p	NOUN
cana-1327	50	5	p	p	X
cana-1327	50	6	p	p	X
cana-1327	50	7	p	p	NOUN
cana-1327	50	8	communications	communication	NOUN
cana-1327	50	9	on	on	ADP
cana-1327	50	10	applied	apply	VERB
cana-1327	50	11	nonlinear	nonlinear	ADJ
cana-1327	50	12	analysis	analysis	NOUN
cana-1327	50	13	issn	issn	NOUN
cana-1327	50	14	:	:	PUNCT
cana-1327	50	15	1074	1074	NUM
cana-1327	50	16	-	-	PUNCT
cana-1327	50	17	133x	133x	NUM
cana-1327	50	18	vol	vol	NOUN
cana-1327	50	19	31	31	NUM
cana-1327	50	20	no	no	NOUN
cana-1327	50	21	.	.	PUNCT
cana-1327	51	1	7s	7	NOUN
cana-1327	51	2	(	(	PUNCT
cana-1327	51	3	2024	2024	NUM
cana-1327	51	4	)	)	PUNCT
cana-1327	51	5	482	482	NUM
cana-1327	52	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1327	52	2	q	q	PUNCT
cana-1327	53	1	p	p	X
cana-1327	53	2	p	p	X
cana-1327	53	3	p	p	NOUN
cana-1327	53	4	r	r	NOUN
cana-1327	53	5	p	p	NOUN
cana-1327	53	6	q	q	NOUN
cana-1327	53	7	r	r	NOUN
cana-1327	53	8	clearly	clearly	ADV
cana-1327	53	9	(	(	PUNCT
cana-1327	53	10	x	x	NOUN
cana-1327	53	11	,	,	PUNCT
cana-1327	53	12	-	-	PUNCT
cana-1327	53	13	,	,	PUNCT
cana-1327	53	14	.	.	PUNCT
cana-1327	53	15	)	)	PUNCT
cana-1327	53	16	is	be	AUX
cana-1327	53	17	a	a	DET
cana-1327	53	18	nssg	nssg	ADJ
cana-1327	53	19	and{p	and{p	NOUN
cana-1327	53	20	}	}	PUNCT
cana-1327	53	21	,	,	PUNCT
cana-1327	53	22	{	{	PUNCT
cana-1327	53	23	p	p	X
cana-1327	53	24	,	,	PUNCT
cana-1327	53	25	q	q	ADJ
cana-1327	53	26	}	}	PUNCT
cana-1327	53	27	,	,	PUNCT
cana-1327	53	28	x	x	PRON
cana-1327	53	29	are	be	AUX
cana-1327	53	30	the	the	DET
cana-1327	53	31	ideal	ideal	NOUN
cana-1327	53	32	of	of	ADP
cana-1327	53	33	x.	x.	NOUN
cana-1327	53	34	here	here	ADV
cana-1327	53	35	pp{p}ppp	pp{p}ppp	PROPN
cana-1327	53	36	,	,	PUNCT
cana-1327	53	37	pqp	pqp	NOUN
cana-1327	53	38	,	,	PUNCT
cana-1327	53	39	prp{p	prp{p	PROPN
cana-1327	53	40	}	}	PUNCT
cana-1327	53	41	pq{p}ppq	pq{p}ppq	NOUN
cana-1327	53	42	,	,	PUNCT
cana-1327	53	43	pqq	pqq	NOUN
cana-1327	53	44	,	,	PUNCT
cana-1327	53	45	prq{p	prq{p	NOUN
cana-1327	53	46	}	}	PUNCT
cana-1327	53	47	pr{p}ppr	pr{p}ppr	NOUN
cana-1327	53	48	,	,	PUNCT
cana-1327	53	49	pqr	pqr	PROPN
cana-1327	53	50	,	,	PUNCT
cana-1327	53	51	prr{p	prr{p	PROPN
cana-1327	53	52	}	}	PUNCT
cana-1327	53	53	qp{p}qpp	qp{p}qpp	PROPN
cana-1327	53	54	,	,	PUNCT
cana-1327	53	55	qqp	qqp	PROPN
cana-1327	53	56	,	,	PUNCT
cana-1327	53	57	qrp{p	qrp{p	NOUN
cana-1327	53	58	}	}	PUNCT
cana-1327	53	59	qr{p}qpr	qr{p}qpr	PROPN
cana-1327	53	60	,	,	PUNCT
cana-1327	53	61	qqr	qqr	PROPN
cana-1327	53	62	,	,	PUNCT
cana-1327	53	63	qrr{p	qrr{p	PROPN
cana-1327	53	64	}	}	PUNCT
cana-1327	53	65	rp{p}rpp	rp{p}rpp	PROPN
cana-1327	53	66	,	,	PUNCT
cana-1327	53	67	rqp	rqp	PROPN
cana-1327	53	68	,	,	PUNCT
cana-1327	53	69	rrp{p	rrp{p	NOUN
cana-1327	53	70	}	}	PUNCT
cana-1327	53	71	.	.	PUNCT
cana-1327	54	1	therefore{p}is	therefore{p}is	PRON
cana-1327	54	2	a	a	DET
cana-1327	54	3	psi	psi	NOUN
cana-1327	54	4	in	in	ADP
cana-1327	54	5	x.	x.	NOUN
cana-1327	54	6	here	here	ADV
cana-1327	54	7	p2	p2	PROPN
cana-1327	54	8	=	=	PROPN
cana-1327	54	9	p{p}but	p{p}but	NOUN
cana-1327	54	10	b{p	b{p	NOUN
cana-1327	54	11	}	}	PUNCT
cana-1327	54	12	.	.	PUNCT
cana-1327	55	1	therefore{p}is	therefore{p}is	PRON
cana-1327	55	2	not	not	PART
cana-1327	55	3	a	a	DET
cana-1327	55	4	cspi	cspi	NOUN
cana-1327	55	5	.	.	PUNCT
cana-1327	56	1	corollary	corollary	ADJ
cana-1327	56	2	3.14	3.14	NUM
cana-1327	56	3	:	:	PUNCT
cana-1327	56	4	each	each	DET
cana-1327	56	5	cpi	cpi	NOUN
cana-1327	56	6	of	of	ADP
cana-1327	56	7	a	a	DET
cana-1327	56	8	nssg	nssg	ADJ
cana-1327	56	9	x	x	PUNCT
cana-1327	56	10	is	be	AUX
cana-1327	56	11	a	a	DET
cana-1327	56	12	psi	psi	NOUN
cana-1327	56	13	of	of	ADP
cana-1327	56	14	a	a	DET
cana-1327	56	15	nssgx	nssgx	NOUN
cana-1327	56	16	.	.	PUNCT
cana-1327	57	1	proof	proof	NOUN
cana-1327	57	2	:	:	PUNCT
cana-1327	57	3	let	let	VERB
cana-1327	57	4	a	a	PRON
cana-1327	57	5	is	be	AUX
cana-1327	57	6	a	a	DET
cana-1327	57	7	cpi	cpi	NOUN
cana-1327	57	8	of	of	ADP
cana-1327	57	9	a	a	DET
cana-1327	57	10	nssgx	nssgx	NOUN
cana-1327	57	11	.	.	PUNCT
cana-1327	58	1	by	by	ADP
cana-1327	58	2	known	know	VERB
cana-1327	58	3	theorem	theorem	NOUN
cana-1327	58	4	,	,	PUNCT
cana-1327	58	5	a	a	PRON
cana-1327	58	6	is	be	AUX
cana-1327	58	7	a	a	DET
cana-1327	58	8	cspiof	cspiof	NOUN
cana-1327	58	9	x.	x.	NOUN
cana-1327	58	10	by	by	ADP
cana-1327	58	11	known	know	VERB
cana-1327	58	12	theorem	theorem	NOUN
cana-1327	58	13	,	,	PUNCT
cana-1327	59	1	aispsiof	aispsiof	NOUN
cana-1327	59	2	x.	x.	NOUN
cana-1327	59	3	theorem3.15	theorem3.15	PROPN
cana-1327	59	4	:	:	PUNCT
cana-1327	59	5	if	if	SCONJ
cana-1327	59	6	a	a	PRON
cana-1327	59	7	is	be	AUX
cana-1327	59	8	apsiof	apsiof	NOUN
cana-1327	59	9	a	a	DET
cana-1327	59	10	nssgx	nssgx	NOUN
cana-1327	59	11	then	then	ADV
cana-1327	59	12	a2	a2	PROPN
cana-1327	59	13	=	=	SYM
cana-1327	59	14	{	{	PUNCT
cana-1327	59	15	x	x	PROPN
cana-1327	59	16	/	/	SYM
cana-1327	59	17	xn∈	xn∈	NUM
cana-1327	59	18	a	a	PRON
cana-1327	59	19	for	for	ADP
cana-1327	59	20	every	every	DET
cana-1327	59	21	n∈	n∈	NOUN
cana-1327	59	22	n	n	CCONJ
cana-1327	59	23	}	}	PUNCT
cana-1327	59	24	is	be	AUX
cana-1327	59	25	a	a	DET
cana-1327	59	26	cspi	cspi	NOUN
cana-1327	59	27	ofx	ofx	NOUN
cana-1327	59	28	.	.	PUNCT
cana-1327	60	1	proof	proof	NOUN
cana-1327	60	2	:	:	PUNCT
cana-1327	60	3	clearly	clearly	ADV
cana-1327	60	4	a	a	PROPN
cana-1327	60	5	a2	a2	PROPN
cana-1327	60	6	and	and	CCONJ
cana-1327	60	7	subsequently	subsequently	ADV
cana-1327	60	8	a2	a2	PROPN
cana-1327	60	9	is	be	AUX
cana-1327	60	10	a	a	DET
cana-1327	60	11	non	non	ADJ
cana-1327	60	12	-	-	ADJ
cana-1327	60	13	empty	empty	ADJ
cana-1327	60	14	subset	subset	ADJ
cana-1327	60	15	ofx	ofx	NOUN
cana-1327	60	16	.	.	PUNCT
cana-1327	61	1	assume	assume	VERB
cana-1327	61	2	thatx	thatx	PROPN
cana-1327	61	3	,	,	PUNCT
cana-1327	61	4	y	y	PROPN
cana-1327	61	5	∈a2	∈a2	PROPN
cana-1327	61	6	and	and	CCONJ
cana-1327	61	7	s∈	s∈	NOUN
cana-1327	61	8	x.	x.	NOUN
cana-1327	61	9	presently	presently	ADV
cana-1327	61	10	x	x	SYM
cana-1327	61	11	∈	∈	PROPN
cana-1327	61	12	a2x	a2x	PROPN
cana-1327	61	13	n	n	CCONJ
cana-1327	61	14	,	,	PUNCT
cana-1327	61	15	ym∈	ym∈	PROPN
cana-1327	61	16	a	a	PRON
cana-1327	61	17	for	for	ADP
cana-1327	61	18	some	some	DET
cana-1327	61	19	m	m	PROPN
cana-1327	61	20	,	,	PUNCT
cana-1327	61	21	n∈	n∈	NOUN
cana-1327	61	22	n	n	CCONJ
cana-1327	61	23	,	,	PUNCT
cana-1327	61	24	xn	xn	PROPN
cana-1327	61	25	,	,	PUNCT
cana-1327	61	26	ym∈	ym∈	PROPN
cana-1327	61	27	a	a	DET
cana-1327	61	28	⇒x	⇒x	NOUN
cana-1327	61	29	,	,	PUNCT
cana-1327	61	30	y	y	PROPN
cana-1327	61	31	∈	∈	PROPN
cana-1327	61	32	a	a	PRON
cana-1327	61	33	,	,	PUNCT
cana-1327	61	34	awill	awill	NOUN
cana-1327	61	35	be	be	AUX
cana-1327	61	36	a	a	DET
cana-1327	61	37	psi	psi	NOUN
cana-1327	61	38	of	of	ADP
cana-1327	61	39	ximplies	ximplie	NOUN
cana-1327	61	40	(	(	PUNCT
cana-1327	61	41	x	x	SYM
cana-1327	61	42	y	y	PROPN
cana-1327	61	43	)	)	PUNCT
cana-1327	61	44	∈	∈	PROPN
cana-1327	62	1	a⇒	a⇒	X
cana-1327	63	1	(	(	PUNCT
cana-1327	63	2	x	x	NOUN
cana-1327	63	3	y	y	PROPN
cana-1327	63	4	)	)	PUNCT
cana-1327	63	5	1∈	1∈	PROPN
cana-1327	63	6	aand	aand	PROPN
cana-1327	64	1	consequently	consequently	ADV
cana-1327	64	2	(	(	PUNCT
cana-1327	64	3	x	x	SYM
cana-1327	64	4	y	y	PROPN
cana-1327	64	5	)	)	PUNCT
cana-1327	64	6	∈	∈	PROPN
cana-1327	64	7	a2	a2	PROPN
cana-1327	64	8	.	.	PUNCT
cana-1327	65	1	hence	hence	ADV
cana-1327	65	2	a2	a2	PROPN
cana-1327	65	3	is	be	AUX
cana-1327	65	4	a	a	DET
cana-1327	65	5	sub	sub	NOUN
cana-1327	65	6	algebra	algebra	NOUN
cana-1327	65	7	of	of	ADP
cana-1327	65	8	x.	x.	NOUN
cana-1327	65	9	presently	presently	ADV
cana-1327	65	10	x	x	SYM
cana-1327	65	11	∈a2implies	∈a2implie	NOUN
cana-1327	65	12	xn∈a	xn∈a	PROPN
cana-1327	65	13	for	for	ADP
cana-1327	65	14	each	each	DET
cana-1327	65	15	n∈	n∈	NOUN
cana-1327	65	16	n.	n.	PROPN
cana-1327	65	17	xn∈a	xn∈a	PROPN
cana-1327	65	18	,	,	PUNCT
cana-1327	65	19	s∈	s∈	PROPN
cana-1327	65	20	x	x	PRON
cana-1327	65	21	,	,	PUNCT
cana-1327	65	22	awill	awill	ADV
cana-1327	65	23	be	be	AUX
cana-1327	65	24	a	a	DET
cana-1327	65	25	psi	psi	NOUN
cana-1327	65	26	of	of	ADP
cana-1327	65	27	x	x	NOUN
cana-1327	65	28	⇒(xs)n∈a	⇒(xs)n∈a	NOUN
cana-1327	65	29	,	,	PUNCT
cana-1327	65	30	(	(	PUNCT
cana-1327	65	31	sx)n∈a	sx)n∈a	NUM
cana-1327	65	32	implies	imply	VERB
cana-1327	65	33	that	that	SCONJ
cana-1327	65	34	xs	xs	PROPN
cana-1327	65	35	,	,	PUNCT
cana-1327	65	36	sx∈	sx∈	PROPN
cana-1327	65	37	a2	a2	PROPN
cana-1327	65	38	.	.	PUNCT
cana-1327	66	1	accordingly	accordingly	ADV
cana-1327	66	2	a2is	a2is	PUNCT
cana-1327	66	3	an	an	DET
cana-1327	66	4	ideal	ideal	NOUN
cana-1327	66	5	of	of	ADP
cana-1327	66	6	x.	x.	NOUN
cana-1327	66	7	assume	assume	VERB
cana-1327	66	8	that	that	SCONJ
cana-1327	66	9	x	x	SYM
cana-1327	66	10	∈	∈	PROPN
cana-1327	66	11	xand	xand	PROPN
cana-1327	66	12	x2∈a2	x2∈a2	PROPN
cana-1327	66	13	.	.	PUNCT
cana-1327	66	14	presently	presently	ADV
cana-1327	66	15	x2∈a2	x2∈a2	PROPN
cana-1327	66	16	implies	imply	VERB
cana-1327	66	17	that(x2)n∈	that(x2)n∈	NUM
cana-1327	66	18	afor	afor	ADP
cana-1327	66	19	each	each	DET
cana-1327	66	20	n∈	n∈	NOUN
cana-1327	66	21	n	n	ADP
cana-1327	66	22	x2n∈a	x2n∈a	PROPN
cana-1327	66	23	implies	imply	VERB
cana-1327	66	24	that	that	SCONJ
cana-1327	66	25	x	x	PROPN
cana-1327	66	26	∈a2	∈a2	PROPN
cana-1327	66	27	.	.	PUNCT
cana-1327	67	1	so	so	ADV
cana-1327	67	2	a2	a2	PROPN
cana-1327	67	3	is	be	AUX
cana-1327	67	4	a	a	DET
cana-1327	67	5	cspi	cspi	NOUN
cana-1327	67	6	of	of	ADP
cana-1327	67	7	x.	x.	NOUN
cana-1327	67	8	let	let	VERB
cana-1327	67	9	q	q	NOUN
cana-1327	67	10	to	to	PART
cana-1327	67	11	be	be	AUX
cana-1327	67	12	any	any	DET
cana-1327	67	13	cspi	cspi	NOUN
cana-1327	67	14	containing	contain	VERB
cana-1327	67	15	a.	a.	NOUN
cana-1327	67	16	let	let	VERB
cana-1327	67	17	x	x	PROPN
cana-1327	67	18	∈a2	∈a2	PROPN
cana-1327	67	19	.	.	PUNCT
cana-1327	68	1	then	then	ADV
cana-1327	68	2	,	,	PUNCT
cana-1327	68	3	at	at	ADP
cana-1327	68	4	that	that	DET
cana-1327	68	5	point	point	NOUN
cana-1327	68	6	,	,	PUNCT
cana-1327	68	7	xn∈a	xn∈a	PROPN
cana-1327	68	8	for	for	ADP
cana-1327	68	9	each	each	DET
cana-1327	68	10	n∈	n∈	NOUN
cana-1327	68	11	n.	n.	NOUN
cana-1327	68	12	by	by	ADP
cana-1327	68	13	corrolary	corrolary	ADJ
cana-1327	68	14	3.7	3.7	NUM
cana-1327	68	15	,	,	PUNCT
cana-1327	68	16	xn∈a	xn∈a	PROPN
cana-1327	68	17	implies	imply	VERB
cana-1327	68	18	that	that	SCONJ
cana-1327	68	19	<	<	X
cana-1327	68	20	x	x	X
cana-1327	68	21	>	>	X
cana-1327	68	22	naq	naq	NUM
cana-1327	68	23	.	.	PUNCT
cana-1327	69	1	since	since	SCONJ
cana-1327	69	2	q	q	PROPN
cana-1327	69	3	is	be	AUX
cana-1327	69	4	csp	csp	PROPN
cana-1327	69	5	<	<	X
cana-1327	69	6	x	x	X
cana-1327	69	7	>	>	X
cana-1327	69	8	nq	nq	PROPN
cana-1327	69	9	implies	imply	VERB
cana-1327	69	10	that	that	SCONJ
cana-1327	69	11	x	x	NOUN
cana-1327	69	12	q	q	X
cana-1327	69	13	.	.	PUNCT
cana-1327	70	1	in	in	ADP
cana-1327	70	2	this	this	DET
cana-1327	70	3	way	way	NOUN
cana-1327	70	4	a2	a2	PROPN
cana-1327	70	5	is	be	AUX
cana-1327	70	6	minimal	minimal	ADJ
cana-1327	70	7	cspi	cspi	NOUN
cana-1327	70	8	of	of	ADP
cana-1327	70	9	x	x	PUNCT
cana-1327	70	10	containing	contain	VERB
cana-1327	70	11	a.	a.	NOUN
cana-1327	70	12	theorem	theorem	NOUN
cana-1327	70	13	3	3	NUM
cana-1327	70	14	..	..	SYM
cana-1327	70	15	16	16	NUM
cana-1327	70	16	:	:	PUNCT
cana-1327	70	17	if	if	SCONJ
cana-1327	70	18	a	a	PRON
cana-1327	70	19	is	be	AUX
cana-1327	70	20	a	a	DET
cana-1327	70	21	psiof	psiof	NOUN
cana-1327	70	22	a	a	DET
cana-1327	70	23	nssg	nssg	ADJ
cana-1327	70	24	x	x	PUNCT
cana-1327	70	25	then	then	ADV
cana-1327	70	26	a2	a2	PROPN
cana-1327	70	27	=	=	SYM
cana-1327	70	28	a4	a4	PROPN
cana-1327	70	29	.	.	PUNCT
cana-1327	71	1	proof	proof	NOUN
cana-1327	71	2	:	:	PUNCT
cana-1327	71	3	by	by	ADP
cana-1327	71	4	known	know	VERB
cana-1327	71	5	theorem	theorem	NOUN
cana-1327	71	6	,	,	PUNCT
cana-1327	71	7	a4a2	a4a2	PROPN
cana-1327	71	8	.	.	PUNCT
cana-1327	72	1	supposexa2	supposexa2	PROPN
cana-1327	72	2	then	then	ADV
cana-1327	72	3	xna	xna	VERB
cana-1327	72	4	for	for	ADP
cana-1327	72	5	some	some	DET
cana-1327	72	6	nn	nn	NOUN
cana-1327	72	7	.	.	PUNCT
cana-1327	73	1	since	since	SCONJ
cana-1327	73	2	a	a	PRON
cana-1327	73	3	is	be	AUX
cana-1327	73	4	ps	ps	NOUN
cana-1327	73	5	,	,	PUNCT
cana-1327	73	6	xna<x	xna<x	PROPN
cana-1327	73	7	>	>	X
cana-1327	73	8	nax	nax	PROPN
cana-1327	73	9	a4	a4	PROPN
cana-1327	73	10	.	.	PUNCT
cana-1327	74	1	communications	communication	NOUN
cana-1327	74	2	on	on	ADP
cana-1327	74	3	applied	apply	VERB
cana-1327	74	4	nonlinear	nonlinear	ADJ
cana-1327	74	5	analysis	analysis	NOUN
cana-1327	74	6	issn	issn	NOUN
cana-1327	74	7	:	:	PUNCT
cana-1327	74	8	1074	1074	NUM
cana-1327	74	9	-	-	PUNCT
cana-1327	74	10	133x	133x	NUM
cana-1327	74	11	vol	vol	NOUN
cana-1327	74	12	31	31	NUM
cana-1327	74	13	no	no	NOUN
cana-1327	74	14	.	.	PUNCT
cana-1327	75	1	7s	7	NOUN
cana-1327	75	2	(	(	PUNCT
cana-1327	75	3	2024	2024	NUM
cana-1327	75	4	)	)	PUNCT
cana-1327	75	5	483	483	NUM
cana-1327	75	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1327	75	7	∴	∴	PROPN
cana-1327	75	8	a2a4	a2a4	PROPN
cana-1327	75	9	and	and	CCONJ
cana-1327	75	10	hence	hence	ADV
cana-1327	75	11	a2	a2	PROPN
cana-1327	75	12	=	=	SYM
cana-1327	75	13	a4	a4	PROPN
cana-1327	75	14	.	.	PUNCT
cana-1327	75	15	theorem	theorem	VERB
cana-1327	75	16	3.17	3.17	NUM
cana-1327	75	17	:	:	PUNCT
cana-1327	75	18	if	if	SCONJ
cana-1327	75	19	a	a	PRON
cana-1327	75	20	is	be	AUX
cana-1327	75	21	a	a	DET
cana-1327	75	22	psiof	psiof	NOUN
cana-1327	75	23	a	a	DET
cana-1327	75	24	nssg	nssg	ADJ
cana-1327	75	25	x	x	NOUN
cana-1327	75	26	then	then	ADV
cana-1327	75	27	a4	a4	INTJ
cana-1327	75	28	=	=	SYM
cana-1327	75	29	{	{	PUNCT
cana-1327	75	30	x	x	X
cana-1327	75	31	/	/	SYM
cana-1327	75	32	<	<	X
cana-1327	75	33	x	x	X
cana-1327	75	34	>	>	X
cana-1327	75	35	na	na	NOUN
cana-1327	75	36	forsomenn	forsomenn	PUNCT
cana-1327	75	37	}	}	PUNCT
cana-1327	75	38	is	be	AUX
cana-1327	75	39	the	the	DET
cana-1327	75	40	minimalsemiprimeideal	minimalsemiprimeideal	NOUN
cana-1327	75	41	of	of	ADP
cana-1327	75	42	x	x	PUNCT
cana-1327	75	43	containing	contain	VERB
cana-1327	75	44	a	a	PRON
cana-1327	75	45	.	.	PUNCT
cana-1327	76	1	theorem	theorem	ADJ
cana-1327	76	2	3.18	3.18	NUM
cana-1327	76	3	:	:	PUNCT
cana-1327	76	4	let	let	VERB
cana-1327	76	5	a	a	DET
cana-1327	76	6	bean	bean	NOUN
cana-1327	76	7	ideal	ideal	NOUN
cana-1327	76	8	of	of	ADP
cana-1327	76	9	a	a	DET
cana-1327	76	10	nssg	nssg	ADJ
cana-1327	76	11	x.	x.	NOUN
cana-1327	76	12	then	then	ADV
cana-1327	76	13	a	a	PRON
cana-1327	76	14	is	be	AUX
cana-1327	76	15	completely	completely	ADV
cana-1327	76	16	primeiff	primeiff	ADJ
cana-1327	76	17	a	a	PRON
cana-1327	76	18	is	be	AUX
cana-1327	76	19	primeand	primeand	NOUN
cana-1327	76	20	ps	ps	PROPN
cana-1327	76	21	.	.	PROPN
cana-1327	77	1	proof	proof	NOUN
cana-1327	77	2	:	:	PUNCT
cana-1327	77	3	assume	assume	VERB
cana-1327	77	4	a	a	PRON
cana-1327	77	5	will	will	AUX
cana-1327	77	6	be	be	AUX
cana-1327	77	7	a	a	DET
cana-1327	77	8	cpi	cpi	NOUN
cana-1327	77	9	.	.	PUNCT
cana-1327	78	1	by	by	ADP
cana-1327	78	2	known	known	ADJ
cana-1327	78	3	hypothesis	hypothesis	NOUN
cana-1327	78	4	,	,	PUNCT
cana-1327	78	5	a	a	PRON
cana-1327	78	6	is	be	AUX
cana-1327	78	7	prime	prime	ADJ
cana-1327	78	8	.	.	PUNCT
cana-1327	79	1	by	by	ADP
cana-1327	79	2	corollary	corollary	ADJ
cana-1327	79	3	3.14	3.14	NUM
cana-1327	79	4	,	,	PUNCT
cana-1327	79	5	a	a	PRON
cana-1327	79	6	will	will	AUX
cana-1327	79	7	be	be	AUX
cana-1327	79	8	a	a	DET
cana-1327	79	9	psi	psi	NOUN
cana-1327	79	10	of	of	ADP
cana-1327	79	11	x.	x.	NOUN
cana-1327	79	12	on	on	ADP
cana-1327	79	13	the	the	DET
cana-1327	79	14	other	other	ADJ
cana-1327	79	15	hand	hand	NOUN
cana-1327	79	16	assume	assume	VERB
cana-1327	79	17	that	that	SCONJ
cana-1327	79	18	a	a	PRON
cana-1327	79	19	is	be	AUX
cana-1327	79	20	prime	prime	ADJ
cana-1327	79	21	and	and	CCONJ
cana-1327	79	22	ps	ps	PROPN
cana-1327	79	23	.	.	PROPN
cana-1327	80	1	let	let	VERB
cana-1327	80	2	x	x	X
cana-1327	80	3	,	,	PUNCT
cana-1327	80	4	y∈	y∈	NOUN
cana-1327	80	5	x	x	PUNCT
cana-1327	80	6	and	and	CCONJ
cana-1327	80	7	xy∈	xy∈	PROPN
cana-1327	80	8	a	a	DET
cana-1327	80	9	.	.	PUNCT
cana-1327	81	1	xy∈	xy∈	X
cana-1327	82	1	a	a	PRON
cana-1327	82	2	,	,	PUNCT
cana-1327	82	3	a	a	PRON
cana-1327	82	4	will	will	AUX
cana-1327	82	5	be	be	AUX
cana-1327	82	6	a	a	DET
cana-1327	82	7	psi	psi	NOUN
cana-1327	82	8	of	of	ADP
cana-1327	82	9	x	x	PART
cana-1327	82	10	⇒xsy	⇒xsy	ADP
cana-1327	82	11	∈	∈	PROPN
cana-1327	82	12	a	a	PRON
cana-1327	82	13	for	for	ADP
cana-1327	82	14	all	all	PRON
cana-1327	82	15	s	s	PART
cana-1327	82	16	∈	∈	NOUN
cana-1327	82	17	x	x	SYM
cana-1327	82	18	<x>⊆	<x>⊆	NOUN
cana-1327	83	1	a	a	DET
cana-1327	83	2	or	or	CCONJ
cana-1327	83	3	<	<	X
cana-1327	83	4	y>⊆	y>⊆	NOUN
cana-1327	83	5	a	a	DET
cana-1327	83	6			NOUN
cana-1327	83	7	x	x	X
cana-1327	83	8	∈	∈	PROPN
cana-1327	83	9	x	x	X
cana-1327	83	10	or	or	CCONJ
cana-1327	83	11	y	y	PROPN
cana-1327	83	12	∈	∈	PROPN
cana-1327	83	13	a	a	PRON
cana-1327	83	14	.	.	PUNCT
cana-1327	84	1	consequently	consequently	ADV
cana-1327	84	2	a	a	PRON
cana-1327	84	3	is	be	AUX
cana-1327	84	4	completely	completely	ADV
cana-1327	84	5	prime	prime	ADJ
cana-1327	84	6	.	.	PUNCT
cana-1327	85	1	theorem.3.19	theorem.3.19	PROPN
cana-1327	85	2	:	:	PUNCT
cana-1327	85	3	let	let	VERB
cana-1327	85	4	a	a	PRON
cana-1327	85	5	be	be	AUX
cana-1327	85	6	an	an	DET
cana-1327	85	7	ideal	ideal	NOUN
cana-1327	85	8	of	of	ADP
cana-1327	85	9	a	a	DET
cana-1327	85	10	nssg	nssg	ADJ
cana-1327	85	11	x.	x.	NOUN
cana-1327	85	12	then	then	ADV
cana-1327	85	13	a	a	PRON
cana-1327	85	14	is	be	AUX
cana-1327	85	15	cspiffais	cspiffais	NOUN
cana-1327	85	16	semiprime	semiprime	NOUN
cana-1327	85	17	and	and	CCONJ
cana-1327	85	18	ps	ps	PROPN
cana-1327	85	19	.	.	PROPN
cana-1327	85	20	proof	proof	NOUN
cana-1327	85	21	:	:	PUNCT
cana-1327	85	22	assume	assume	VERB
cana-1327	85	23	that	that	SCONJ
cana-1327	85	24	a	a	PRON
cana-1327	85	25	is	be	AUX
cana-1327	85	26	completely	completely	ADV
cana-1327	85	27	semiprime	semiprime	NOUN
cana-1327	85	28	.	.	PUNCT
cana-1327	86	1	by	by	ADP
cana-1327	86	2	known	know	VERB
cana-1327	86	3	hypothesis	hypothesis	NOUN
cana-1327	86	4	,	,	PUNCT
cana-1327	86	5	a	a	PRON
cana-1327	86	6	is	be	AUX
cana-1327	86	7	semiprime	semiprime	NOUN
cana-1327	86	8	and	and	CCONJ
cana-1327	86	9	furthermore	furthermore	ADV
cana-1327	86	10	by	by	ADP
cana-1327	86	11	known	known	ADJ
cana-1327	86	12	hypothesis	hypothesis	NOUN
cana-1327	86	13	3.11	3.11	NUM
cana-1327	86	14	,	,	PUNCT
cana-1327	86	15	a	a	PRON
cana-1327	86	16	is	be	AUX
cana-1327	86	17	ps	ps	NOUN
cana-1327	86	18	.	.	PROPN
cana-1327	87	1	on	on	ADP
cana-1327	87	2	the	the	DET
cana-1327	87	3	other	other	ADJ
cana-1327	87	4	hand	hand	NOUN
cana-1327	87	5	suppose	suppose	VERB
cana-1327	87	6	that	that	SCONJ
cana-1327	87	7	a	a	PRON
cana-1327	87	8	is	be	AUX
cana-1327	87	9	semiprime	semiprime	NOUN
cana-1327	87	10	and	and	CCONJ
cana-1327	87	11	ps	ps	PROPN
cana-1327	88	1	.	.	PUNCT
cana-1327	88	2	let	let	VERB
cana-1327	88	3	x	x	PUNCT
cana-1327	88	4	∈	∈	PROPN
cana-1327	88	5	x	x	X
cana-1327	88	6	and	and	CCONJ
cana-1327	88	7	x2∈a	x2∈a	X
cana-1327	88	8	.	.	PUNCT
cana-1327	89	1	x2∈a	x2∈a	PROPN
cana-1327	89	2	,	,	PUNCT
cana-1327	89	3	a	a	PRON
cana-1327	89	4	is	is	NOUN
cana-1327	89	5	ps⇒<x2>⊆a	ps⇒<x2>⊆a	NOUN
cana-1327	89	6	⇒<x>⊆a	⇒<x>⊆a	PROPN
cana-1327	89	7	⇒x	⇒x	VERB
cana-1327	89	8	∈a	∈a	PROPN
cana-1327	89	9	.	.	PUNCT
cana-1327	90	1	communications	communication	NOUN
cana-1327	90	2	on	on	ADP
cana-1327	90	3	applied	apply	VERB
cana-1327	90	4	nonlinear	nonlinear	ADJ
cana-1327	90	5	analysis	analysis	NOUN
cana-1327	90	6	issn	issn	NOUN
cana-1327	90	7	:	:	PUNCT
cana-1327	90	8	1074	1074	NUM
cana-1327	90	9	-	-	PUNCT
cana-1327	90	10	133x	133x	NUM
cana-1327	90	11	vol	vol	NOUN
cana-1327	90	12	31	31	NUM
cana-1327	90	13	no	no	NOUN
cana-1327	90	14	.	.	PUNCT
cana-1327	91	1	7s	7	NOUN
cana-1327	91	2	(	(	PUNCT
cana-1327	91	3	2024	2024	NUM
cana-1327	91	4	)	)	PUNCT
cana-1327	91	5	484	484	NUM
cana-1327	91	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1327	91	7	in	in	ADP
cana-1327	91	8	this	this	DET
cana-1327	91	9	way	way	NOUN
cana-1327	91	10	a	a	PRON
cana-1327	91	11	is	be	AUX
cana-1327	91	12	csp	csp	PROPN
cana-1327	91	13	.	.	PROPN
cana-1327	91	14	4	4	NUM
cana-1327	91	15	.	.	X
cana-1327	91	16	pseudo	pseudo	NOUN
cana-1327	91	17	symmetric	symmetric	NOUN
cana-1327	91	18	near	near	ADP
cana-1327	91	19	subtraction	subtraction	NOUN
cana-1327	91	20	semigroup	semigroup	NOUN
cana-1327	91	21	definition	definition	NOUN
cana-1327	91	22	4.1	4.1	NUM
cana-1327	91	23	:	:	PUNCT
cana-1327	91	24	a	a	DET
cana-1327	91	25	nssg	nssg	ADJ
cana-1327	91	26	x	x	VERB
cana-1327	91	27	is	be	AUX
cana-1327	91	28	said	say	VERB
cana-1327	91	29	to	to	PART
cana-1327	91	30	be	be	AUX
cana-1327	91	31	psnssg	psnssg	VERB
cana-1327	91	32	provided	provide	VERB
cana-1327	91	33	every	every	DET
cana-1327	91	34	ideal	ideal	NOUN
cana-1327	91	35	in	in	ADP
cana-1327	91	36	x	x	PUNCT
cana-1327	91	37	is	be	AUX
cana-1327	91	38	a	a	DET
cana-1327	91	39	psi	psi	NOUN
cana-1327	91	40	.	.	PUNCT
cana-1327	91	41	example	example	NOUN
cana-1327	91	42	4.2	4.2	NUM
cana-1327	91	43	:	:	PUNCT
cana-1327	91	44	in	in	ADP
cana-1327	91	45	example	example	NOUN
cana-1327	91	46	3.3	3.3	NUM
cana-1327	91	47	,	,	PUNCT
cana-1327	91	48	the	the	DET
cana-1327	91	49	nssg	nssg	ADJ
cana-1327	91	50	x	x	VERB
cana-1327	91	51	is	be	AUX
cana-1327	91	52	a	a	DET
cana-1327	91	53	psnssg	psnssg	VERB
cana-1327	91	54	.	.	PUNCT
cana-1327	92	1	theorem	theorem	VERB
cana-1327	92	2	4.3	4.3	NUM
cana-1327	92	3	:	:	PUNCT
cana-1327	92	4	each	each	DET
cana-1327	92	5	left	leave	VERB
cana-1327	92	6	duo	duo	NOUN
cana-1327	92	7	nssg	nssg	VERB
cana-1327	92	8	x	x	PUNCT
cana-1327	92	9	is	be	AUX
cana-1327	92	10	a	a	DET
cana-1327	92	11	psssg	psssg	ADJ
cana-1327	92	12	.	.	PUNCT
cana-1327	93	1	theorem	theorem	VERB
cana-1327	93	2	4.4	4.4	NUM
cana-1327	93	3	:	:	PUNCT
cana-1327	93	4	each	each	DET
cana-1327	93	5	right	right	PROPN
cana-1327	93	6	duo	duo	NOUN
cana-1327	93	7	nssg	nssg	VERB
cana-1327	93	8	x	x	PUNCT
cana-1327	93	9	is	be	AUX
cana-1327	93	10	a	a	DET
cana-1327	93	11	psnssg	psnssg	VERB
cana-1327	93	12	.	.	PUNCT
cana-1327	94	1	corollary	corollary	ADJ
cana-1327	94	2	4.5	4.5	NUM
cana-1327	94	3	:	:	PUNCT
cana-1327	94	4	each	each	DET
cana-1327	94	5	duo	duo	NOUN
cana-1327	94	6	nssg	nssg	VERB
cana-1327	94	7	is	be	AUX
cana-1327	94	8	a	a	DET
cana-1327	94	9	psnssg	psnssg	VERB
cana-1327	94	10	.	.	PUNCT
cana-1327	95	1	proof	proof	NOUN
cana-1327	95	2	:	:	PUNCT
cana-1327	95	3	by	by	SCONJ
cana-1327	95	4	theorem	theorem	NOUN
cana-1327	95	5	4.3	4.3	NUM
cana-1327	95	6	,	,	PUNCT
cana-1327	95	7	and	and	CCONJ
cana-1327	95	8	4.4	4.4	NUM
cana-1327	95	9	,	,	PUNCT
cana-1327	95	10	we	we	PRON
cana-1327	95	11	conclude	conclude	VERB
cana-1327	95	12	that	that	SCONJ
cana-1327	95	13	every	every	DET
cana-1327	95	14	duonssgx	duonssgx	NOUN
cana-1327	95	15	is	be	AUX
cana-1327	95	16	a	a	DET
cana-1327	95	17	psnssg	psnssg	VERB
cana-1327	95	18	.	.	PUNCT
cana-1327	96	1	communications	communication	NOUN
cana-1327	96	2	on	on	ADP
cana-1327	96	3	applied	apply	VERB
cana-1327	96	4	nonlinear	nonlinear	ADJ
cana-1327	96	5	analysis	analysis	NOUN
cana-1327	96	6	issn	issn	NOUN
cana-1327	96	7	:	:	PUNCT
cana-1327	96	8	1074	1074	NUM
cana-1327	96	9	-	-	PUNCT
cana-1327	96	10	133x	133x	NUM
cana-1327	96	11	vol	vol	NOUN
cana-1327	96	12	31	31	NUM
cana-1327	96	13	no	no	NOUN
cana-1327	96	14	.	.	PUNCT
cana-1327	97	1	7s	7	NOUN
cana-1327	97	2	(	(	PUNCT
cana-1327	97	3	2024	2024	NUM
cana-1327	97	4	)	)	PUNCT
cana-1327	97	5	485	485	NUM
cana-1327	97	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1327	97	7	theorem	theorem	VERB
cana-1327	97	8	4.6	4.6	NUM
cana-1327	97	9	:	:	PUNCT
cana-1327	97	10	each	each	DET
cana-1327	97	11	left	leave	VERB
cana-1327	97	12	pseudo	pseudo	NOUN
cana-1327	97	13	commutative	commutative	ADJ
cana-1327	97	14	nssg	nssg	VERB
cana-1327	97	15	is	be	AUX
cana-1327	97	16	a	a	DET
cana-1327	97	17	psnssg	psnssg	VERB
cana-1327	97	18	.	.	PUNCT
cana-1327	98	1	proof	proof	NOUN
cana-1327	98	2	:	:	PUNCT
cana-1327	98	3	let	let	VERB
cana-1327	98	4	x	x	PRON
cana-1327	98	5	be	be	AUX
cana-1327	98	6	a	a	DET
cana-1327	98	7	left	left	ADJ
cana-1327	98	8	pseudo	pseudo	NOUN
cana-1327	98	9	commutativenssg	commutativenssg	VERB
cana-1327	98	10	and	and	CCONJ
cana-1327	98	11	a	a	DET
cana-1327	98	12	be	be	AUX
cana-1327	98	13	any	any	DET
cana-1327	98	14	ideal	ideal	NOUN
cana-1327	98	15	of	of	ADP
cana-1327	98	16	x.	x.	NOUN
cana-1327	98	17	let	let	VERB
cana-1327	98	18	x	x	X
cana-1327	98	19	,	,	PUNCT
cana-1327	98	20	y∈	y∈	NOUN
cana-1327	98	21	x	x	NOUN
cana-1327	98	22	,	,	PUNCT
cana-1327	98	23	xy∈a	xy∈a	VERB
cana-1327	98	24	and	and	CCONJ
cana-1327	98	25	s	s	PROPN
cana-1327	98	26	∈	∈	PROPN
cana-1327	98	27	x.	x.	NOUN
cana-1327	98	28	now	now	ADV
cana-1327	98	29	xsy	xsy	VERB
cana-1327	99	1	=	=	PUNCT
cana-1327	99	2	sxy=	sxy=	NUM
cana-1327	99	3	s(xy	s(xy	NOUN
cana-1327	99	4	)	)	PUNCT
cana-1327	99	5	∈a	∈a	NOUN
cana-1327	99	6	.	.	PUNCT
cana-1327	100	1	therefore	therefore	ADV
cana-1327	100	2	xsy∈a	xsy∈a	PROPN
cana-1327	100	3	for	for	ADP
cana-1327	100	4	all	all	DET
cana-1327	100	5	s	s	PART
cana-1327	100	6	∈	∈	PROPN
cana-1327	100	7	x.	x.	NOUN
cana-1327	100	8	thereforeais	thereforeais	PROPN
cana-1327	100	9	apsi	apsi	PROPN
cana-1327	100	10	.	.	PUNCT
cana-1327	101	1	therefore	therefore	ADV
cana-1327	101	2	x	x	X
cana-1327	101	3	is	be	AUX
cana-1327	101	4	a	a	DET
cana-1327	101	5	psnssg	psnssg	VERB
cana-1327	101	6	.	.	PUNCT
cana-1327	102	1	theorem	theorem	VERB
cana-1327	102	2	4.7	4.7	NUM
cana-1327	102	3	:	:	PUNCT
cana-1327	102	4	each	each	DET
cana-1327	102	5	right	right	ADJ
cana-1327	102	6	pseudo	pseudo	NOUN
cana-1327	102	7	commutative	commutative	ADJ
cana-1327	102	8	nssg	nssg	VERB
cana-1327	102	9	is	be	AUX
cana-1327	102	10	psnssg	psnssg	VERB
cana-1327	102	11	.	.	PUNCT
cana-1327	103	1	proof	proof	NOUN
cana-1327	103	2	:	:	PUNCT
cana-1327	103	3	let	let	VERB
cana-1327	103	4	x	x	PRON
cana-1327	103	5	be	be	AUX
cana-1327	103	6	a	a	DET
cana-1327	103	7	right	right	ADJ
cana-1327	103	8	pseudo	pseudo	NOUN
cana-1327	103	9	commutativenssg	commutativenssg	VERB
cana-1327	103	10	and	and	CCONJ
cana-1327	103	11	a	a	DET
cana-1327	103	12	be	be	AUX
cana-1327	103	13	any	any	DET
cana-1327	103	14	ideal	ideal	NOUN
cana-1327	103	15	of	of	ADP
cana-1327	103	16	x.	x.	PROPN
cana-1327	103	17	letx	letx	PROPN
cana-1327	103	18	,	,	PUNCT
cana-1327	103	19	y	y	PROPN
cana-1327	103	20	∈	∈	PROPN
cana-1327	103	21	x	x	AUX
cana-1327	103	22	,	,	PUNCT
cana-1327	103	23	xy	xy	PROPN
cana-1327	103	24	∈	∈	PROPN
cana-1327	103	25	a	a	DET
cana-1327	103	26	and	and	CCONJ
cana-1327	103	27	s	s	PROPN
cana-1327	103	28	∈	∈	NOUN
cana-1327	103	29	x.	x.	NOUN
cana-1327	103	30	now	now	ADV
cana-1327	103	31	xsy	xsy	VERB
cana-1327	104	1	=	=	SYM
cana-1327	104	2	xys	xys	NOUN
cana-1327	104	3	=	=	SYM
cana-1327	104	4	(	(	PUNCT
cana-1327	104	5	xy)s	xy)s	PROPN
cana-1327	104	6	∈	∈	PROPN
cana-1327	104	7	a.	a.	NOUN
cana-1327	104	8	thereforexsy	thereforexsy	NOUN
cana-1327	104	9	∈	∈	PROPN
cana-1327	104	10	a	a	PRON
cana-1327	104	11	for	for	ADP
cana-1327	104	12	all	all	PRON
cana-1327	104	13	s	s	PART
cana-1327	104	14	∈	∈	NOUN
cana-1327	104	15	x.	x.	NOUN
cana-1327	104	16	∴a	∴a	PROPN
cana-1327	104	17	is	be	AUX
cana-1327	104	18	a	a	DET
cana-1327	104	19	psi	psi	NOUN
cana-1327	104	20	.	.	PUNCT
cana-1327	105	1	hence	hence	ADV
cana-1327	105	2	x	x	X
cana-1327	105	3	is	be	AUX
cana-1327	105	4	a	a	DET
cana-1327	105	5	psnssg	psnssg	VERB
cana-1327	105	6	.	.	PUNCT
cana-1327	106	1	corollary	corollary	NOUN
cana-1327	106	2	4.8	4.8	NUM
cana-1327	106	3	:	:	PUNCT
cana-1327	106	4	each	each	DET
cana-1327	106	5	quasi	quasi	PROPN
cana-1327	106	6	commutative	commutative	ADJ
cana-1327	106	7	nssg	nssg	VERB
cana-1327	106	8	is	be	AUX
cana-1327	106	9	a	a	DET
cana-1327	106	10	psnssg	psnssg	VERB
cana-1327	106	11	.	.	PUNCT
cana-1327	107	1	proof	proof	NOUN
cana-1327	107	2	:	:	PUNCT
cana-1327	107	3	let	let	VERB
cana-1327	107	4	x	x	PRON
cana-1327	107	5	be	be	AUX
cana-1327	107	6	a	a	DET
cana-1327	107	7	quasi	quasi	NOUN
cana-1327	107	8	commutative	commutative	ADJ
cana-1327	107	9	sg	sg	PROPN
cana-1327	107	10	.	.	PUNCT
cana-1327	108	1	by	by	ADP
cana-1327	108	2	known	know	VERB
cana-1327	108	3	theorem	theorem	NOUN
cana-1327	108	4	,	,	PUNCT
cana-1327	108	5	x	x	PUNCT
cana-1327	108	6	is	be	AUX
cana-1327	108	7	a	a	DET
cana-1327	108	8	normal	normal	ADJ
cana-1327	108	9	nssg	nssg	NOUN
cana-1327	108	10	.	.	PUNCT
cana-1327	109	1	by	by	ADP
cana-1327	109	2	known	know	VERB
cana-1327	109	3	theorem	theorem	NOUN
cana-1327	109	4	,	,	PUNCT
cana-1327	109	5	x	x	PRON
cana-1327	109	6	is	be	AUX
cana-1327	109	7	a	a	DET
cana-1327	109	8	duo	duo	NOUN
cana-1327	109	9	nssg	nssg	VERB
cana-1327	109	10	.	.	PUNCT
cana-1327	110	1	by	by	ADP
cana-1327	110	2	corollary	corollary	ADJ
cana-1327	110	3	4.5	4.5	NUM
cana-1327	110	4	,	,	PUNCT
cana-1327	110	5	x	x	X
cana-1327	110	6	is	be	AUX
cana-1327	110	7	a	a	DET
cana-1327	110	8	ps	ps	NOUN
cana-1327	110	9	sg	sg	PROPN
cana-1327	110	10	.	.	PUNCT
cana-1327	111	1	corollary	corollary	NOUN
cana-1327	111	2	4.9	4.9	NUM
cana-1327	111	3	:	:	PUNCT
cana-1327	111	4	each	each	DET
cana-1327	111	5	generalized	generalize	VERB
cana-1327	111	6	commutativenssg	commutativenssg	VERB
cana-1327	111	7	is	be	AUX
cana-1327	111	8	a	a	DET
cana-1327	111	9	psnssg	psnssg	VERB
cana-1327	111	10	.	.	PUNCT
cana-1327	112	1	proof	proof	NOUN
cana-1327	112	2	:	:	PUNCT
cana-1327	112	3	suppose	suppose	VERB
cana-1327	112	4	that	that	SCONJ
cana-1327	112	5	s	s	AUX
cana-1327	112	6	be	be	AUX
cana-1327	112	7	a	a	DET
cana-1327	112	8	commutative	commutative	ADJ
cana-1327	112	9	nssg	nssg	NOUN
cana-1327	112	10	.	.	PUNCT
cana-1327	113	1	by	by	ADP
cana-1327	113	2	known	know	VERB
cana-1327	113	3	theorem	theorem	NOUN
cana-1327	113	4	,	,	PUNCT
cana-1327	113	5	x	x	PRON
cana-1327	113	6	is	be	AUX
cana-1327	113	7	a	a	DET
cana-1327	113	8	left	left	ADJ
cana-1327	113	9	duo	duo	NOUN
cana-1327	113	10	nssg	nssg	VERB
cana-1327	113	11	.	.	PUNCT
cana-1327	114	1	by	by	ADP
cana-1327	114	2	known	know	VERB
cana-1327	114	3	theorem	theorem	VERB
cana-1327	114	4	4.3	4.3	NUM
cana-1327	114	5	,	,	PUNCT
cana-1327	114	6	x	x	X
cana-1327	114	7	is	be	AUX
cana-1327	114	8	a	a	DET
cana-1327	114	9	psnssg	psnssg	VERB
cana-1327	114	10	.	.	PUNCT
cana-1327	115	1	corollary	corollary	ADJ
cana-1327	115	2	4.10	4.10	NUM
cana-1327	115	3	:	:	PUNCT
cana-1327	115	4	eachnssg	eachnssg	VERB
cana-1327	115	5	is	be	AUX
cana-1327	115	6	a	a	DET
cana-1327	115	7	psnssg	psnssg	VERB
cana-1327	115	8	.	.	PUNCT
cana-1327	116	1	proof	proof	NOUN
cana-1327	116	2	:	:	PUNCT
cana-1327	116	3	suppose	suppose	VERB
cana-1327	116	4	that	that	SCONJ
cana-1327	116	5	x	x	PRON
cana-1327	116	6	be	be	AUX
cana-1327	116	7	a	a	DET
cana-1327	116	8	normal	normal	ADJ
cana-1327	116	9	sg	sg	NOUN
cana-1327	116	10	.	.	PUNCT
cana-1327	117	1	by	by	ADP
cana-1327	117	2	known	know	VERB
cana-1327	117	3	theorem	theorem	NOUN
cana-1327	117	4	,	,	PUNCT
cana-1327	117	5	x	x	PRON
cana-1327	117	6	is	be	AUX
cana-1327	117	7	a	a	DET
cana-1327	117	8	duo	duo	NOUN
cana-1327	117	9	sg	sg	NOUN
cana-1327	117	10	.	.	PUNCT
cana-1327	118	1	by	by	ADP
cana-1327	118	2	corollary	corollary	ADJ
cana-1327	118	3	4.5	4.5	NUM
cana-1327	118	4	,	,	PUNCT
cana-1327	118	5	x	x	X
cana-1327	118	6	is	be	AUX
cana-1327	118	7	a	a	DET
cana-1327	118	8	psnssg	psnssg	VERB
cana-1327	118	9	.	.	PUNCT
cana-1327	119	1	theorem	theorem	NOUN
cana-1327	119	2	4.11	4.11	NUM
cana-1327	119	3	:	:	PUNCT
cana-1327	119	4	each	each	DET
cana-1327	119	5	idempotent	idempotent	NOUN
cana-1327	119	6	nssg	nssg	VERB
cana-1327	119	7	is	be	AUX
cana-1327	119	8	a	a	DET
cana-1327	119	9	psnssg	psnssg	ADJ
cana-1327	119	10	proof	proof	NOUN
cana-1327	119	11	:	:	PUNCT
cana-1327	119	12	suppose	suppose	VERB
cana-1327	119	13	that	that	SCONJ
cana-1327	119	14	x	x	PRON
cana-1327	119	15	be	be	AUX
cana-1327	119	16	an	an	DET
cana-1327	119	17	idempotent	idempotent	NOUN
cana-1327	119	18	nssg	nssg	VERB
cana-1327	119	19	and	and	CCONJ
cana-1327	119	20	a	a	DET
cana-1327	119	21	be	be	AUX
cana-1327	119	22	any	any	DET
cana-1327	119	23	ideal	ideal	NOUN
cana-1327	119	24	of	of	ADP
cana-1327	119	25	x.	x.	NOUN
cana-1327	119	26	let	let	VERB
cana-1327	119	27	x	x	PRON
cana-1327	119	28	,	,	PUNCT
cana-1327	119	29	y	y	PROPN
cana-1327	119	30	∈	∈	PROPN
cana-1327	119	31	x	x	X
cana-1327	119	32	,	,	PUNCT
cana-1327	119	33	xy	xy	PROPN
cana-1327	119	34	∈	∈	PROPN
cana-1327	119	35	a.	a.	NOUN
cana-1327	119	36	now	now	ADV
cana-1327	119	37	xy	xy	PROPN
cana-1327	119	38	∈	∈	PROPN
cana-1327	120	1	a⇒(yx)=	a⇒(yx)=	PROPN
cana-1327	120	2	(	(	PUNCT
cana-1327	120	3	yx)2	yx)2	NOUN
cana-1327	120	4	=	=	SYM
cana-1327	120	5	(	(	PUNCT
cana-1327	120	6	yx)(yx	yx)(yx	PROPN
cana-1327	120	7	)	)	PUNCT
cana-1327	120	8	=	=	SYM
cana-1327	121	1	y(xy)x∈	y(xy)x∈	NOUN
cana-1327	121	2	a.	a.	NOUN
cana-1327	121	3	thereforeyx∈	thereforeyx∈	NOUN
cana-1327	121	4	a.	a.	NOUN
cana-1327	121	5	if	if	SCONJ
cana-1327	121	6	s	s	VERB
cana-1327	121	7	∈	∈	PROPN
cana-1327	121	8	x	x	X
cana-1327	121	9	,	,	PUNCT
cana-1327	121	10	then	then	ADV
cana-1327	121	11	xsy	xsy	X
cana-1327	121	12	=	=	SYM
cana-1327	121	13	(	(	PUNCT
cana-1327	121	14	xsy)2	xsy)2	PROPN
cana-1327	121	15	=	=	SYM
cana-1327	121	16	(	(	PUNCT
cana-1327	121	17	xsy	xsy	PROPN
cana-1327	121	18	)	)	PUNCT
cana-1327	121	19	(	(	PUNCT
cana-1327	121	20	xsy	xsy	PROPN
cana-1327	121	21	)	)	PUNCT
cana-1327	121	22	=	=	SYM
cana-1327	121	23	xs(yx)sy	xs(yx)sy	NOUN
cana-1327	121	24	∈	∈	NOUN
cana-1327	121	25	a.	a.	NOUN
cana-1327	121	26	therefore	therefore	ADV
cana-1327	121	27	xsy	xsy	VERB
cana-1327	121	28	∈	∈	PROPN
cana-1327	121	29	a	a	PRON
cana-1327	121	30	for	for	ADP
cana-1327	121	31	every	every	DET
cana-1327	121	32	s	s	X
cana-1327	121	33	∈	∈	NOUN
cana-1327	121	34	x.	x.	NOUN
cana-1327	121	35	hence	hence	ADV
cana-1327	121	36	a	a	PRON
cana-1327	121	37	is	be	AUX
cana-1327	121	38	a	a	DET
cana-1327	121	39	psi	psi	NOUN
cana-1327	121	40	.	.	PUNCT
cana-1327	122	1	therefore	therefore	ADV
cana-1327	122	2	x	x	X
cana-1327	122	3	is	be	AUX
cana-1327	122	4	a	a	DET
cana-1327	122	5	pssg	pssg	NOUN
cana-1327	122	6	.	.	PUNCT
cana-1327	123	1	theorem	theorem	VERB
cana-1327	123	2	4.12	4.12	NUM
cana-1327	123	3	:	:	PUNCT
cana-1327	123	4	if	if	SCONJ
cana-1327	123	5	x	x	PRON
cana-1327	123	6	is	be	AUX
cana-1327	123	7	a	a	DET
cana-1327	123	8	nssg	nssg	VERB
cana-1327	123	9	in	in	ADP
cana-1327	123	10	which	which	PRON
cana-1327	123	11	every	every	DET
cana-1327	123	12	element	element	NOUN
cana-1327	123	13	is	be	AUX
cana-1327	123	14	a	a	DET
cana-1327	123	15	midunit	midunit	NOUN
cana-1327	123	16	then	then	ADV
cana-1327	123	17	x	x	PUNCT
cana-1327	123	18	is	be	AUX
cana-1327	123	19	a	a	DET
cana-1327	123	20	ps	ps	NOUN
cana-1327	123	21	nssg	nssg	VERB
cana-1327	123	22	.	.	PUNCT
cana-1327	124	1	proof	proof	NOUN
cana-1327	124	2	:	:	PUNCT
cana-1327	124	3	suppose	suppose	VERB
cana-1327	124	4	that	that	SCONJ
cana-1327	124	5	x	x	PRON
cana-1327	124	6	be	be	AUX
cana-1327	124	7	a	a	DET
cana-1327	124	8	nssg	nssg	VERB
cana-1327	124	9	in	in	ADP
cana-1327	124	10	which	which	PRON
cana-1327	124	11	every	every	DET
cana-1327	124	12	element	element	NOUN
cana-1327	124	13	is	be	AUX
cana-1327	124	14	a	a	DET
cana-1327	124	15	midunit	midunit	NOUN
cana-1327	124	16	and	and	CCONJ
cana-1327	124	17	a	a	DET
cana-1327	124	18	be	be	AUX
cana-1327	124	19	any	any	DET
cana-1327	124	20	ideal	ideal	ADJ
cana-1327	124	21	ofx	ofx	NOUN
cana-1327	124	22	.	.	PUNCT
cana-1327	125	1	if	if	SCONJ
cana-1327	125	2	s	s	X
cana-1327	125	3	∈	∈	PROPN
cana-1327	125	4	x	x	NOUN
cana-1327	125	5	,	,	PUNCT
cana-1327	125	6	then	then	ADV
cana-1327	125	7	s	s	VERB
cana-1327	125	8	is	be	AUX
cana-1327	125	9	a	a	DET
cana-1327	125	10	midunit	midunit	NOUN
cana-1327	125	11	and	and	CCONJ
cana-1327	125	12	hence	hence	ADV
cana-1327	125	13	xsy=	xsy=	PROPN
cana-1327	125	14	xy∈a	xy∈a	VERB
cana-1327	125	15	.	.	PUNCT
cana-1327	126	1	hence	hence	ADV
cana-1327	126	2	ais	ais	PROPN
cana-1327	126	3	apsi	apsi	NOUN
cana-1327	126	4	.	.	PUNCT
cana-1327	127	1	∴	∴	NOUN
cana-1327	127	2	x	x	PROPN
cana-1327	127	3	isa	isa	NOUN
cana-1327	127	4	pssg	pssg	NOUN
cana-1327	127	5	.	.	PUNCT
cana-1327	128	1	remarks	remark	VERB
cana-1327	128	2	:	:	PUNCT
cana-1327	128	3	sg	sg	PROPN
cana-1327	128	4	-	-	PUNCT
cana-1327	128	5	semigroup	semigroup	NOUN
cana-1327	128	6	,	,	PUNCT
cana-1327	128	7	ps	ps	NOUN
cana-1327	128	8	-	-	ADJ
cana-1327	128	9	pseudo	pseudo	NOUN
cana-1327	128	10	symmetric	symmetric	ADJ
cana-1327	128	11	,	,	PUNCT
cana-1327	128	12	psi	psi	NOUN
cana-1327	128	13	-	-	PUNCT
cana-1327	128	14	pseudo	pseudo	NOUN
cana-1327	128	15	symmetric	symmetric	ADJ
cana-1327	128	16	ideal	ideal	ADJ
cana-1327	128	17	nssg	nssg	VERB
cana-1327	128	18	-	-	PUNCT
cana-1327	128	19	near	near	ADP
cana-1327	128	20	subtraction	subtraction	NOUN
cana-1327	128	21	semigroup	semigroup	NOUN
cana-1327	128	22	andpsnssg	andpsnssg	VERB
cana-1327	128	23	-pseudo	-pseudo	NOUN
cana-1327	128	24	symmetric	symmetric	ADJ
cana-1327	128	25	near	near	ADP
cana-1327	128	26	subtraction	subtraction	NOUN
cana-1327	128	27	semigroup	semigroup	NOUN
cana-1327	128	28	,	,	PUNCT
cana-1327	128	29	cspcompletely	cspcompletely	ADJ
cana-1327	128	30	semiprime	semiprime	NOUN
cana-1327	128	31	,	,	PUNCT
cana-1327	128	32	cpi	cpi	NOUN
cana-1327	128	33	-	-	PUNCT
cana-1327	128	34	completely	completely	ADV
cana-1327	128	35	prime	prime	ADJ
cana-1327	128	36	ideal	ideal	NOUN
cana-1327	128	37	and	and	CCONJ
cana-1327	128	38	cspicompletely	cspicompletely	ADV
cana-1327	128	39	semiprime	semiprime	PROPN
cana-1327	128	40	ideal	ideal	NOUN
cana-1327	128	41	.	.	PUNCT
cana-1327	129	1	communications	communication	NOUN
cana-1327	129	2	on	on	ADP
cana-1327	129	3	applied	apply	VERB
cana-1327	129	4	nonlinear	nonlinear	ADJ
cana-1327	129	5	analysis	analysis	NOUN
cana-1327	129	6	issn	issn	NOUN
cana-1327	129	7	:	:	PUNCT
cana-1327	129	8	1074	1074	NUM
cana-1327	129	9	-	-	PUNCT
cana-1327	129	10	133x	133x	NUM
cana-1327	129	11	vol	vol	NOUN
cana-1327	129	12	31	31	NUM
cana-1327	129	13	no	no	NOUN
cana-1327	129	14	.	.	PUNCT
cana-1327	130	1	7s	7	NOUN
cana-1327	130	2	(	(	PUNCT
cana-1327	130	3	2024	2024	NUM
cana-1327	130	4	)	)	PUNCT
cana-1327	130	5	486	486	NUM
cana-1327	130	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-1327	130	7	references	reference	NOUN
cana-1327	130	8	[	[	X
cana-1327	130	9	1	1	NUM
cana-1327	130	10	]	]	PUNCT
cana-1327	130	11	m.jayarami	m.jayarami	ADJ
cana-1327	130	12	reddy	reddy	NOUN
cana-1327	130	13	,	,	PUNCT
cana-1327	130	14	p.siva	p.siva	NOUN
cana-1327	130	15	prasad	prasad	PROPN
cana-1327	130	16	,	,	PUNCT
cana-1327	130	17	d.madhusudana	d.madhusudana	PROPN
cana-1327	130	18	rao	rao	PROPN
cana-1327	130	19	.	.	PUNCT
cana-1327	130	20	,pseudo	,pseudo	PUNCT
cana-1327	130	21	integral	integral	ADJ
cana-1327	130	22	nssg	nssg	ADJ
cana-1327	130	23	,	,	PUNCT
cana-1327	130	24	natural	natural	ADJ
cana-1327	130	25	volatiles	volatile	NOUN
cana-1327	130	26	&	&	CCONJ
cana-1327	130	27	essential	essential	ADJ
cana-1327	130	28	oils2021;volume-8(issue	oils2021;volume-8(issue	PROPN
cana-1327	130	29	no-5):pp:5422	no-5):pp:5422	NOUN
cana-1327	130	30	-	-	PUNCT
cana-1327	130	31	542	542	NUM
cana-1327	130	32	.	.	PUNCT
cana-1327	131	1	[	[	X
cana-1327	131	2	2	2	NUM
cana-1327	131	3	]	]	PUNCT
cana-1327	131	4	m.	m.	NOUN
cana-1327	131	5	jayarami	jayarami	PROPN
cana-1327	131	6	reddy	reddy	PROPN
cana-1327	131	7	,	,	PUNCT
cana-1327	131	8	p.siva	p.siva	NOUN
cana-1327	131	9	prasad	prasad	PROPN
cana-1327	131	10	,	,	PUNCT
cana-1327	131	11	d.madhusudana	d.madhusudana	PROPN
cana-1327	131	12	rao	rao	PROPN
cana-1327	131	13	,	,	PUNCT
cana-1327	131	14	nssg	nssg	VERB
cana-1327	131	15	in	in	ADP
cana-1327	131	16	which	which	PRON
cana-1327	131	17	prime	prime	ADJ
cana-1327	131	18	ideals	ideal	NOUN
cana-1327	131	19	are	be	AUX
cana-1327	131	20	maximal	maximal	ADJ
cana-1327	131	21	-	-	PUNCT
cana-1327	131	22	ijournal	ijournal	NOUN
cana-1327	131	23	of	of	ADP
cana-1327	131	24	positive	positive	ADJ
cana-1327	131	25	school	school	NOUN
cana-1327	131	26	psychology	psychology	NOUN
cana-1327	131	27	,	,	PUNCT
cana-1327	131	28	2022	2022	NUM
cana-1327	131	29	,	,	PUNCT
cana-1327	131	30	vol	vol	NOUN
cana-1327	131	31	.	.	PROPN
cana-1327	132	1	6	6	NUM
cana-1327	132	2	,	,	PUNCT
cana-1327	132	3	no	no	INTJ
cana-1327	132	4	.	.	NOUN
cana-1327	132	5	5	5	NUM
cana-1327	132	6	,	,	PUNCT
cana-1327	132	7	2014	2014	NUM
cana-1327	132	8	–	–	PUNCT
cana-1327	132	9	2019	2019	NUM
cana-1327	132	10	.	.	PUNCT
cana-1327	133	1	[	[	X
cana-1327	133	2	3	3	X
cana-1327	133	3	]	]	PUNCT
cana-1327	133	4	d.madhusudana	d.madhusudana	PROPN
cana-1327	133	5	rao	rao	PROPN
cana-1327	133	6	,	,	PUNCT
cana-1327	133	7	a.	a.	PROPN
cana-1327	133	8	anjaneyulu	anjaneyulu	VERB
cana-1327	133	9	and	and	CCONJ
cana-1327	133	10	a.	a.	NOUN
cana-1327	133	11	gangadhara	gangadhara	PROPN
cana-1327	133	12	rao	rao	PROPN
cana-1327	133	13	,	,	PUNCT
cana-1327	133	14	prime	prime	ADJ
cana-1327	133	15	γ	γ	X
cana-1327	133	16	-	-	PUNCT
cana-1327	133	17	ideals	ideal	NOUN
cana-1327	133	18	inγ	inγ	NOUN
cana-1327	133	19	-	-	PUNCT
cana-1327	133	20	sgs	sgs	PROPN
cana-1327	133	21	international	international	ADJ
cana-1327	133	22	ejournal	ejournal	NOUN
cana-1327	133	23	of	of	ADP
cana-1327	133	24	mathematics	mathematic	NOUN
cana-1327	133	25	and	and	CCONJ
cana-1327	133	26	engineering	engineering	NOUN
cana-1327	133	27	138	138	NUM
cana-1327	133	28	(	(	PUNCT
cana-1327	133	29	2011	2011	NUM
cana-1327	133	30	)	)	PUNCT
cana-1327	133	31	1250	1250	NUM
cana-1327	133	32	-	-	SYM
cana-1327	133	33	1259	1259	NUM
cana-1327	133	34	.	.	PUNCT
cana-1327	134	1	[	[	X
cana-1327	134	2	4	4	X
cana-1327	134	3	]	]	PUNCT
cana-1327	134	4	d.madhusudana	d.madhusudana	PROPN
cana-1327	134	5	rao	rao	PROPN
cana-1327	134	6	,	,	PUNCT
cana-1327	134	7	a.	a.	PROPN
cana-1327	134	8	anjaneyulu	anjaneyulu	VERB
cana-1327	134	9	and	and	CCONJ
cana-1327	134	10	a.	a.	NOUN
cana-1327	134	11	gangadhara	gangadhara	PROPN
cana-1327	134	12	rao.prime	rao.prime	X
cana-1327	134	13	γ	γ	X
cana-1327	134	14	-	-	NOUN
cana-1327	134	15	ideals	ideal	NOUN
cana-1327	134	16	in	in	ADP
cana-1327	134	17	duo	duo	ADJ
cana-1327	134	18	γ	γ	PROPN
cana-1327	134	19	-	-	PUNCT
cana-1327	134	20	sgs	sgs	PROPN
cana-1327	134	21	-	-	PUNCT
cana-1327	134	22	international	international	ADJ
cana-1327	134	23	ejournal	ejournal	NOUN
cana-1327	134	24	of	of	ADP
cana-1327	134	25	mathematics	mathematic	NOUN
cana-1327	134	26	and	and	CCONJ
cana-1327	134	27	engineering	engineering	NOUN
cana-1327	134	28	174	174	NUM
cana-1327	134	29	(	(	PUNCT
cana-1327	134	30	2012	2012	NUM
cana-1327	134	31	)	)	PUNCT
cana-1327	134	32	1642	1642	NUM
cana-1327	134	33	-	-	SYM
cana-1327	134	34	1653	1653	NUM
cana-1327	134	35	.	.	PUNCT
cana-1327	135	1	[	[	X
cana-1327	135	2	5	5	X
cana-1327	135	3	]	]	PUNCT
cana-1327	135	4	d.madhusudana	d.madhusudana	PROPN
cana-1327	135	5	rao	rao	PROPN
cana-1327	135	6	,	,	PUNCT
cana-1327	135	7	a.	a.	PROPN
cana-1327	135	8	anjaneyulu	anjaneyulu	VERB
cana-1327	135	9	and	and	CCONJ
cana-1327	135	10	a.	a.	NOUN
cana-1327	135	11	gangadhara	gangadhara	PROPN
cana-1327	135	12	rao	rao	PROPN
cana-1327	135	13	,	,	PUNCT
cana-1327	135	14	primary	primary	ADJ
cana-1327	135	15	decomposition	decomposition	NOUN
cana-1327	135	16	in	in	ADP
cana-1327	135	17	aγ	aγ	ADP
cana-1327	135	18	-	-	PUNCT
cana-1327	135	19	sg	sg	ADP
cana-1327	135	20	-	-	PUNCT
cana-1327	135	21	international	international	ADJ
cana-1327	135	22	journal	journal	NOUN
cana-1327	135	23	of	of	ADP
cana-1327	135	24	mathematical	mathematical	ADJ
cana-1327	135	25	science	science	NOUN
cana-1327	135	26	,	,	PUNCT
cana-1327	135	27	technology	technology	NOUN
cana-1327	135	28	and	and	CCONJ
cana-1327	135	29	humanities	humanity	NOUN
cana-1327	135	30	46	46	NUM
cana-1327	135	31	(	(	PUNCT
cana-1327	135	32	2012	2012	NUM
cana-1327	135	33	)	)	PUNCT
cana-1327	135	34	466	466	NUM
cana-1327	135	35	-	-	PUNCT
cana-1327	135	36	479	479	NUM
cana-1327	135	37	.	.	PUNCT
cana-1327	136	1	[	[	X
cana-1327	136	2	6	6	X
cana-1327	136	3	]	]	PUNCT
cana-1327	136	4	p.siva	p.siva	PRON
cana-1327	136	5	prasad	prasad	PROPN
cana-1327	136	6	,	,	PUNCT
cana-1327	136	7	c.sreemannarayana	c.sreemannarayana	PROPN
cana-1327	136	8	,	,	PUNCT
cana-1327	136	9	d.madhusudana	d.madhusudana	PROPN
cana-1327	136	10	rao	rao	PROPN
cana-1327	136	11	,	,	PUNCT
cana-1327	136	12	t.nageswara	t.nageswara	X
cana-1327	136	13	rao	rao	NOUN
cana-1327	136	14	.	.	PUNCT
cana-1327	136	15	,	,	PUNCT
cana-1327	136	16	on	on	ADP
cana-1327	136	17	leternary	leternary	PROPN
cana-1327	136	18	sgs	sgs	PROPN
cana-1327	136	19	-	-	PUNCT
cana-1327	136	20	i)-international	i)-international	PROPN
cana-1327	136	21	journal	journal	NOUN
cana-1327	136	22	of	of	ADP
cana-1327	136	23	recent	recent	ADJ
cana-1327	136	24	technology	technology	NOUN
cana-1327	136	25	and	and	CCONJ
cana-1327	136	26	engineering	engineering	NOUN
cana-1327	136	27	(	(	PUNCT
cana-1327	136	28	ijrte),issn:2277	ijrte),issn:2277	NOUN
cana-1327	136	29	-	-	PUNCT
cana-1327	136	30	3878	3878	NUM
cana-1327	136	31	,	,	PUNCT
cana-1327	136	32	volume-7	volume-7	ADJ
cana-1327	136	33	issue	issue	NOUN
cana-1327	136	34	-	-	PUNCT
cana-1327	136	35	icetesm	icetesm	NOUN
cana-1327	136	36	-	-	PUNCT
cana-1327	136	37	mar-2019	mar-2019	NOUN
cana-1327	136	38	,	,	PUNCT
cana-1327	136	39	indexed	index	VERB
cana-1327	136	40	by	by	ADP
cana-1327	136	41	scopus	scopus	PROPN
cana-1327	136	42	,	,	PUNCT
cana-1327	136	43	impactfactor:5.93(year	impactfactor:5.93(year	PROPN
cana-1327	136	44	2018	2018	NUM
cana-1327	136	45	)	)	PUNCT
cana-1327	137	1	[	[	X
cana-1327	137	2	7	7	X
cana-1327	137	3	]	]	PUNCT
cana-1327	137	4	d.madhusudana	d.madhusudana	PROPN
cana-1327	137	5	rao	rao	PROPN
cana-1327	137	6	,	,	PUNCT
cana-1327	137	7	a.	a.	PROPN
cana-1327	137	8	anjaneyulu	anjaneyulu	VERB
cana-1327	137	9	and	and	CCONJ
cana-1327	137	10	a.	a.	NOUN
cana-1327	137	11	gangadhara	gangadhara	PROPN
cana-1327	137	12	rao	rao	PROPN
cana-1327	137	13	,	,	PUNCT
cana-1327	137	14	psγ	psγ	NOUN
cana-1327	137	15	-	-	NOUN
cana-1327	137	16	ideals	ideal	NOUN
cana-1327	137	17	in	in	ADP
cana-1327	137	18	γ	γ	PROPN
cana-1327	137	19	-	-	PUNCT
cana-1327	137	20	sgs	sgs	PROPN
cana-1327	137	21	international	international	ADJ
cana-1327	137	22	ejournal	ejournal	NOUN
cana-1327	137	23	of	of	ADP
cana-1327	137	24	mathematics	mathematic	NOUN
cana-1327	137	25	and	and	CCONJ
cana-1327	137	26	engineering	engineering	NOUN
cana-1327	137	27	166	166	NUM
cana-1327	137	28	(	(	PUNCT
cana-1327	137	29	2011	2011	NUM
cana-1327	137	30	)	)	PUNCT
cana-1327	137	31	1074	1074	NUM
cana-1327	137	32	-	-	SYM
cana-1327	137	33	1081	1081	NUM
cana-1327	137	34	.	.	PUNCT
cana-1327	138	1	[	[	X
cana-1327	138	2	8	8	X
cana-1327	138	3	]	]	PUNCT
cana-1327	138	4	d.madhusudana	d.madhusudana	PROPN
cana-1327	138	5	rao	rao	PROPN
cana-1327	138	6	,	,	PUNCT
cana-1327	138	7	a.	a.	PROPN
cana-1327	138	8	anjaneyulu	anjaneyulu	VERB
cana-1327	138	9	and	and	CCONJ
cana-1327	138	10	a.	a.	NOUN
cana-1327	138	11	gangadhara	gangadhara	PROPN
cana-1327	138	12	rao	rao	PROPN
cana-1327	138	13	,	,	PUNCT
cana-1327	138	14	semipseudosymmetric	semipseudosymmetric	ADJ
cana-1327	138	15	γ	γ	NOUN
cana-1327	138	16	-	-	NOUN
cana-1327	138	17	ideals	ideal	NOUN
cana-1327	138	18	in	in	ADP
cana-1327	138	19	γ	γ	PROPN
cana-1327	138	20	-	-	PUNCT
cana-1327	138	21	sgs	sgs	PROPN
cana-1327	138	22	–	–	PUNCT
cana-1327	138	23	international	international	ADJ
cana-1327	138	24	journal	journal	NOUN
cana-1327	138	25	of	of	ADP
cana-1327	138	26	mathematical	mathematical	ADJ
cana-1327	138	27	sciences	science	NOUN
cana-1327	138	28	,	,	PUNCT
cana-1327	138	29	technology	technology	NOUN
cana-1327	138	30	and	and	CCONJ
cana-1327	138	31	humanities	humanity	NOUN
cana-1327	138	32	18	18	NUM
cana-1327	138	33	(	(	PUNCT
cana-1327	138	34	2011	2011	NUM
cana-1327	138	35	)	)	PUNCT
cana-1327	138	36	183	183	NUM
cana-1327	138	37	-	-	SYM
cana-1327	138	38	192	192	NUM
cana-1327	138	39	.	.	PUNCT
cana-1327	139	1	[	[	X
cana-1327	139	2	9	9	X
cana-1327	139	3	]	]	PUNCT
cana-1327	139	4	d.madhusudana	d.madhusudana	PROPN
cana-1327	139	5	rao	rao	PROPN
cana-1327	139	6	,	,	PUNCT
cana-1327	139	7	a.	a.	PROPN
cana-1327	139	8	anjaneyulu	anjaneyulu	VERB
cana-1327	139	9	and	and	CCONJ
cana-1327	139	10	a.	a.	NOUN
cana-1327	139	11	gangadhara	gangadhara	PROPN
cana-1327	139	12	rao	rao	PROPN
cana-1327	139	13	,	,	PUNCT
cana-1327	139	14	n(a)-γ	n(a)-γ	NOUN
cana-1327	139	15	-	-	PUNCT
cana-1327	139	16	sgindian	sgindian	ADJ
cana-1327	139	17	journal	journal	NOUN
cana-1327	139	18	of	of	ADP
cana-1327	139	19	mathematics	mathematics	PROPN
cana-1327	139	20	and	and	CCONJ
cana-1327	139	21	mathematical	mathematical	ADJ
cana-1327	139	22	sciences	sciences	PROPN
cana-1327	139	23	vol	vol	NOUN
cana-1327	139	24	.	.	PROPN
cana-1327	140	1	7	7	NUM
cana-1327	140	2	,	,	PUNCT
cana-1327	140	3	no	no	INTJ
cana-1327	140	4	.	.	NOUN
cana-1327	141	1	2,(december	2,(december	NUM
cana-1327	141	2	2011	2011	NUM
cana-1327	141	3	)	)	PUNCT
cana-1327	141	4	:	:	PUNCT
cana-1327	142	1	75	75	NUM
cana-1327	142	2	-	-	SYM
cana-1327	142	3	83	83	NUM
cana-1327	142	4	.	.	PUNCT
cana-1327	143	1	[	[	X
cana-1327	143	2	10	10	NUM
cana-1327	143	3	]	]	PUNCT
cana-1327	143	4	d.madhusudana	d.madhusudana	PROPN
cana-1327	143	5	rao	rao	PROPN
cana-1327	143	6	,	,	PUNCT
cana-1327	143	7	a.	a.	PROPN
cana-1327	143	8	anjaneyulu	anjaneyulu	VERB
cana-1327	143	9	and	and	CCONJ
cana-1327	143	10	a.	a.	NOUN
cana-1327	143	11	gangadhara	gangadhara	PROPN
cana-1327	143	12	rao	rao	PROPN
cana-1327	143	13	,	,	PUNCT
cana-1327	143	14	pseudo	pseudo	NOUN
cana-1327	143	15	integral	integral	ADJ
cana-1327	143	16	γ	γ	PROPN
cana-1327	143	17	-	-	PUNCT
cana-1327	143	18	sg	sg	PROPN
cana-1327	143	19	–	–	PUNCT
cana-1327	143	20	international	international	ADJ
cana-1327	143	21	journal	journal	NOUN
cana-1327	143	22	of	of	ADP
cana-1327	143	23	mathematical	mathematical	ADJ
cana-1327	143	24	sciences	science	NOUN
cana-1327	143	25	,	,	PUNCT
cana-1327	143	26	technology	technology	NOUN
cana-1327	143	27	and	and	CCONJ
cana-1327	143	28	humanities	humanity	NOUN
cana-1327	143	29	12	12	NUM
cana-1327	143	30	(	(	PUNCT
cana-1327	143	31	2011	2011	NUM
cana-1327	143	32	)	)	PUNCT
cana-1327	143	33	118	118	NUM
cana-1327	143	34	-	-	SYM
cana-1327	143	35	124	124	NUM
cana-1327	143	36	.	.	PUNCT
cana-1327	144	1	[	[	X
cana-1327	144	2	11	11	NUM
cana-1327	144	3	]	]	SYM
cana-1327	144	4	11	11	NUM
cana-1327	144	5	.	.	PUNCT
cana-1327	145	1	d.madhusudana	d.madhusudana	PROPN
cana-1327	145	2	rao	rao	PROPN
cana-1327	145	3	,	,	PUNCT
cana-1327	145	4	a.	a.	PROPN
cana-1327	145	5	anjaneyulu	anjaneyulu	VERB
cana-1327	145	6	and	and	CCONJ
cana-1327	145	7	a.	a.	NOUN
cana-1327	145	8	gangadhara	gangadhara	PROPN
cana-1327	145	9	rao	rao	PROPN
cana-1327	145	10	,	,	PUNCT
cana-1327	145	11	primary	primary	ADJ
cana-1327	145	12	and	and	CCONJ
cana-1327	145	13	semiprimaryγ	semiprimaryγ	NOUN
cana-1327	145	14	-	-	PUNCT
cana-1327	145	15	ideals	ideal	NOUN
cana-1327	145	16	in	in	ADP
cana-1327	145	17	γ	γ	PROPN
cana-1327	145	18	-	-	PUNCT
cana-1327	145	19	sgs	sgs	PROPN
cana-1327	145	20	international	international	ADJ
cana-1327	145	21	journal	journal	PROPN
cana-1327	145	22	of	of	ADP
cana-1327	145	23	mathematical	mathematical	ADJ
cana-1327	145	24	sciences	science	NOUN
cana-1327	145	25	,	,	PUNCT
cana-1327	145	26	technology	technology	NOUN
cana-1327	145	27	and	and	CCONJ
cana-1327	145	28	humanities	humanity	NOUN
cana-1327	145	29	29	29	NUM
cana-1327	145	30	(	(	PUNCT
cana-1327	145	31	2012	2012	NUM
cana-1327	145	32	)	)	PUNCT
cana-1327	145	33	282	282	NUM
cana-1327	145	34	-	-	SYM
cana-1327	145	35	293	293	NUM
cana-1327	145	36	.	.	PUNCT