id	sid	tid	token	lemma	pos
cana-3487	1	1	communications	communication	NOUN
cana-3487	1	2	on	on	ADP
cana-3487	1	3	applied	apply	VERB
cana-3487	1	4	nonlinear	nonlinear	ADJ
cana-3487	1	5	analysis	analysis	NOUN
cana-3487	1	6	issn	issn	NOUN
cana-3487	1	7	:	:	PUNCT
cana-3487	1	8	1074	1074	NUM
cana-3487	1	9	-	-	PUNCT
cana-3487	1	10	133x	133x	NUM
cana-3487	1	11	vol	vol	NOUN
cana-3487	1	12	32	32	NUM
cana-3487	1	13	no	no	NOUN
cana-3487	1	14	.	.	PUNCT
cana-3487	2	1	7s	7	NOUN
cana-3487	2	2	(	(	PUNCT
cana-3487	2	3	2025	2025	NUM
cana-3487	2	4	)	)	PUNCT
cana-3487	2	5	799	799	NUM
cana-3487	3	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-3487	3	2	normal	normal	ADJ
cana-3487	3	3	be	be	VERB
cana-3487	3	4	-	-	PUNCT
cana-3487	3	5	algebras	algebras	PROPN
cana-3487	3	6	m.	m.	PROPN
cana-3487	3	7	bala	bala	PROPN
cana-3487	3	8	prabhakar1	prabhakar1	PROPN
cana-3487	3	9	*	*	PROPN
cana-3487	3	10	,	,	PUNCT
cana-3487	3	11	m.	m.	PROPN
cana-3487	3	12	sambasiva	sambasiva	PROPN
cana-3487	3	13	rao2	rao2	PROPN
cana-3487	3	14	,	,	PUNCT
cana-3487	3	15	s.	s.	PROPN
cana-3487	3	16	kalesha	kalesha	PROPN
cana-3487	3	17	vali3	vali3	PROPN
cana-3487	3	18	and	and	CCONJ
cana-3487	3	19	g.	g.	PROPN
cana-3487	3	20	v.	v.	ADP
cana-3487	3	21	ramana4	ramana4	PROPN
cana-3487	3	22	.	.	PUNCT
cana-3487	4	1	1associate	1associate	NUM
cana-3487	4	2	professor	professor	NOUN
cana-3487	4	3	,	,	PUNCT
cana-3487	4	4	department	department	NOUN
cana-3487	4	5	of	of	ADP
cana-3487	4	6	mathematics	mathematics	PROPN
cana-3487	4	7	,	,	PUNCT
cana-3487	4	8	aditya	aditya	PROPN
cana-3487	4	9	university	university	PROPN
cana-3487	4	10	,	,	PUNCT
cana-3487	4	11	surampalem	surampalem	PROPN
cana-3487	4	12	,	,	PUNCT
cana-3487	4	13	kakinada	kakinada	PROPN
cana-3487	4	14	,	,	PUNCT
cana-3487	4	15	andhra	andhra	PROPN
cana-3487	4	16	pradesh	pradesh	PROPN
cana-3487	4	17	,	,	PUNCT
cana-3487	4	18	india533437	india533437	PROPN
cana-3487	4	19	.	.	PUNCT
cana-3487	4	20	mail	mail	NOUN
cana-3487	4	21	:	:	PUNCT
cana-3487	4	22	prabhakar_mb@yahoo.co.in	prabhakar_mb@yahoo.co.in	PROPN
cana-3487	4	23	2professor	2professor	NUM
cana-3487	4	24	,	,	PUNCT
cana-3487	4	25	department	department	NOUN
cana-3487	4	26	of	of	ADP
cana-3487	4	27	mathematics	mathematic	NOUN
cana-3487	4	28	,	,	PUNCT
cana-3487	4	29	mvgr	mvgr	PROPN
cana-3487	4	30	college	college	PROPN
cana-3487	4	31	of	of	ADP
cana-3487	4	32	engineering(a	engineering(a	PROPN
cana-3487	4	33	)	)	PUNCT
cana-3487	4	34	,	,	PUNCT
cana-3487	4	35	chintalavalasa	chintalavalasa	PROPN
cana-3487	4	36	,	,	PUNCT
cana-3487	4	37	vizianagaram	vizianagaram	PROPN
cana-3487	4	38	,	,	PUNCT
cana-3487	4	39	andhra	andhra	PROPN
cana-3487	4	40	pradesh	pradesh	PROPN
cana-3487	4	41	,	,	PUNCT
cana-3487	4	42	india-535005	india-535005	NOUN
cana-3487	4	43	.	.	PUNCT
cana-3487	5	1	mail	mail	NOUN
cana-3487	5	2	:	:	PUNCT
cana-3487	5	3	mssraomaths35@rediffmail.com	mssraomaths35@rediffmail.com	X
cana-3487	5	4	3professor	3professor	NUM
cana-3487	5	5	,	,	PUNCT
cana-3487	5	6	department	department	NOUN
cana-3487	5	7	of	of	ADP
cana-3487	5	8	engineering	engineering	NOUN
cana-3487	5	9	mathematics	mathematic	NOUN
cana-3487	5	10	,	,	PUNCT
cana-3487	5	11	andhra	andhra	PROPN
cana-3487	5	12	university	university	PROPN
cana-3487	5	13	,	,	PUNCT
cana-3487	5	14	visakhapatnam	visakhapatnam	PROPN
cana-3487	5	15	,	,	PUNCT
cana-3487	5	16	andhra	andhra	PROPN
cana-3487	5	17	pradesh	pradesh	PROPN
cana-3487	5	18	,	,	PUNCT
cana-3487	5	19	india-530003	india-530003	ADJ
cana-3487	5	20	.	.	PUNCT
cana-3487	6	1	mail	mail	NOUN
cana-3487	6	2	:	:	PUNCT
cana-3487	7	1	valijntuv@gmail.com	valijntuv@gmail.com	X
cana-3487	8	1	4associate	4associate	NUM
cana-3487	8	2	professor	professor	NOUN
cana-3487	8	3	,	,	PUNCT
cana-3487	8	4	department	department	NOUN
cana-3487	8	5	of	of	ADP
cana-3487	8	6	mathematics	mathematics	PROPN
cana-3487	8	7	,	,	PUNCT
cana-3487	8	8	aditya	aditya	PROPN
cana-3487	8	9	university	university	PROPN
cana-3487	8	10	,	,	PUNCT
cana-3487	8	11	surampalem	surampalem	PROPN
cana-3487	8	12	,	,	PUNCT
cana-3487	8	13	kakinada	kakinada	PROPN
cana-3487	8	14	,	,	PUNCT
cana-3487	8	15	andhra	andhra	PROPN
cana-3487	8	16	pradesh	pradesh	PROPN
cana-3487	8	17	.	.	PROPN
cana-3487	8	18	,	,	PUNCT
cana-3487	8	19	india533437	india533437	PROPN
cana-3487	8	20	.	.	PUNCT
cana-3487	8	21	mail	mail	NOUN
cana-3487	8	22	:	:	PUNCT
cana-3487	9	1	ramanaginjala9@gmail.com	ramanaginjala9@gmail.com	X
cana-3487	9	2	*	*	PUNCT
cana-3487	9	3	corresponding	correspond	VERB
cana-3487	9	4	&	&	CCONJ
cana-3487	9	5	main	main	ADJ
cana-3487	9	6	author	author	NOUN
cana-3487	9	7	.	.	PUNCT
cana-3487	10	1	article	article	NOUN
cana-3487	10	2	history	history	NOUN
cana-3487	10	3	:	:	PUNCT
cana-3487	10	4	received	receive	VERB
cana-3487	10	5	:	:	PUNCT
cana-3487	10	6	28	28	NUM
cana-3487	10	7	-	-	SYM
cana-3487	10	8	10	10	NUM
cana-3487	10	9	-	-	PUNCT
cana-3487	10	10	2024	2024	NUM
cana-3487	10	11	revised:12	revised:12	NOUN
cana-3487	10	12	-	-	PUNCT
cana-3487	10	13	11	11	NUM
cana-3487	10	14	-	-	PUNCT
cana-3487	10	15	2024	2024	NUM
cana-3487	10	16	accepted:19	accepted:19	VERB
cana-3487	10	17	-	-	PUNCT
cana-3487	10	18	12	12	NUM
cana-3487	10	19	-	-	PUNCT
cana-3487	10	20	2024	2024	NUM
cana-3487	10	21	abstract	abstract	NOUN
cana-3487	10	22	:	:	PUNCT
cana-3487	10	23	in	in	ADP
cana-3487	10	24	this	this	DET
cana-3487	10	25	present	present	ADJ
cana-3487	10	26	article	article	NOUN
cana-3487	10	27	we	we	PRON
cana-3487	10	28	propounded	propound	VERB
cana-3487	10	29	the	the	DET
cana-3487	10	30	idea	idea	NOUN
cana-3487	10	31	of	of	ADP
cana-3487	10	32	normal	normal	ADJ
cana-3487	10	33	be	be	AUX
cana-3487	10	34	-	-	PUNCT
cana-3487	10	35	algebras	algebra	NOUN
cana-3487	10	36	,	,	PUNCT
cana-3487	10	37	derived	derive	VERB
cana-3487	10	38	some	some	DET
cana-3487	10	39	significant	significant	ADJ
cana-3487	10	40	properties	property	NOUN
cana-3487	10	41	of	of	ADP
cana-3487	10	42	normal	normal	ADJ
cana-3487	10	43	be	be	AUX
cana-3487	10	44	-	-	PUNCT
cana-3487	10	45	algebras	algebra	VERB
cana-3487	10	46	and	and	CCONJ
cana-3487	10	47	obtained	obtain	VERB
cana-3487	10	48	a	a	DET
cana-3487	10	49	set	set	NOUN
cana-3487	10	50	of	of	ADP
cana-3487	10	51	equivalent	equivalent	ADJ
cana-3487	10	52	conditions	condition	NOUN
cana-3487	10	53	indicating	indicate	VERB
cana-3487	10	54	when	when	SCONJ
cana-3487	10	55	a	a	DET
cana-3487	10	56	normal	normal	ADJ
cana-3487	10	57	be	be	NOUN
cana-3487	10	58	-	-	PUNCT
cana-3487	10	59	algebra	algebra	NOUN
cana-3487	10	60	assumes	assume	VERB
cana-3487	10	61	the	the	DET
cana-3487	10	62	characteristics	characteristic	NOUN
cana-3487	10	63	of	of	ADP
cana-3487	10	64	an	an	DET
cana-3487	10	65	involutory	involutory	NOUN
cana-3487	10	66	be	be	NOUN
cana-3487	10	67	-	-	PUNCT
cana-3487	10	68	algebra	algebra	NOUN
cana-3487	10	69	.	.	PUNCT
cana-3487	11	1	some	some	DET
cana-3487	11	2	sufficient	sufficient	ADJ
cana-3487	11	3	conditions	condition	NOUN
cana-3487	11	4	for	for	ADP
cana-3487	11	5	a	a	DET
cana-3487	11	6	be	be	NOUN
cana-3487	11	7	-	-	PUNCT
cana-3487	11	8	algebra	algebra	NOUN
cana-3487	11	9	to	to	PART
cana-3487	11	10	become	become	VERB
cana-3487	11	11	a	a	DET
cana-3487	11	12	normal	normal	ADJ
cana-3487	11	13	be	be	NOUN
cana-3487	11	14	-	-	PUNCT
cana-3487	11	15	algebra	algebra	NOUN
cana-3487	11	16	are	be	AUX
cana-3487	11	17	derived	derive	VERB
cana-3487	11	18	.	.	PUNCT
cana-3487	12	1	also	also	ADV
cana-3487	12	2	,	,	PUNCT
cana-3487	12	3	congruence	congruence	PROPN
cana-3487	12	4	relation	relation	NOUN
cana-3487	12	5	is	be	AUX
cana-3487	12	6	introduced	introduce	VERB
cana-3487	12	7	on	on	ADP
cana-3487	12	8	a	a	DET
cana-3487	12	9	normal	normal	ADJ
cana-3487	12	10	be	be	NOUN
cana-3487	12	11	-	-	PUNCT
cana-3487	12	12	algebra	algebra	NOUN
cana-3487	12	13	.	.	PUNCT
cana-3487	13	1	keywords	keyword	NOUN
cana-3487	13	2	:	:	PUNCT
cana-3487	13	3	be	be	AUX
cana-3487	13	4	-	-	PUNCT
cana-3487	13	5	algebra	algebra	NOUN
cana-3487	13	6	;	;	PUNCT
cana-3487	13	7	congruence	congruence	NOUN
cana-3487	13	8	;	;	PUNCT
cana-3487	13	9	involutory	involutory	NOUN
cana-3487	13	10	be	be	NOUN
cana-3487	13	11	-	-	PUNCT
cana-3487	13	12	algebra	algebra	NOUN
cana-3487	13	13	;	;	PUNCT
cana-3487	13	14	normal	normal	ADJ
cana-3487	13	15	be	be	NOUN
cana-3487	13	16	-	-	PUNCT
cana-3487	13	17	algebra	algebra	NOUN
cana-3487	13	18	;	;	PUNCT
cana-3487	13	19	transitive	transitive	ADJ
cana-3487	13	20	be	be	NOUN
cana-3487	13	21	-	-	PUNCT
cana-3487	13	22	algebra	algebra	NOUN
cana-3487	13	23	.	.	PUNCT
cana-3487	14	1	2020	2020	NUM
cana-3487	14	2	mathematics	mathematic	NOUN
cana-3487	14	3	subject	subject	ADJ
cana-3487	14	4	classification	classification	NOUN
cana-3487	14	5	:	:	PUNCT
cana-3487	14	6	03g25	03g25	NOUN
cana-3487	14	7	.	.	PUNCT
cana-3487	15	1	1	1	X
cana-3487	15	2	.	.	X
cana-3487	15	3	introduction	introduction	NOUN
cana-3487	15	4	.	.	PUNCT
cana-3487	16	1	in	in	ADP
cana-3487	16	2	[	[	X
cana-3487	16	3	11	11	NUM
cana-3487	16	4	]	]	PUNCT
cana-3487	16	5	,	,	PUNCT
cana-3487	16	6	h.	h.	PROPN
cana-3487	16	7	s.	s.	PROPN
cana-3487	16	8	kim	kim	PROPN
cana-3487	16	9	and	and	CCONJ
cana-3487	16	10	y.	y.	PROPN
cana-3487	16	11	h.	h.	PROPN
cana-3487	16	12	kim	kim	PROPN
cana-3487	16	13	were	be	AUX
cana-3487	16	14	introduced	introduce	VERB
cana-3487	16	15	the	the	DET
cana-3487	16	16	theory	theory	NOUN
cana-3487	16	17	of	of	ADP
cana-3487	16	18	be	be	NOUN
cana-3487	16	19	-	-	PUNCT
cana-3487	16	20	algebras	algebra	NOUN
cana-3487	16	21	.	.	PUNCT
cana-3487	17	1	be	be	AUX
cana-3487	17	2	-	-	PUNCT
cana-3487	17	3	algebras	algebras	PROPN
cana-3487	17	4	was	be	AUX
cana-3487	17	5	made	make	VERB
cana-3487	17	6	familiar	familiar	ADJ
cana-3487	17	7	to	to	PART
cana-3487	17	8	extend	extend	VERB
cana-3487	17	9	the	the	DET
cana-3487	17	10	class	class	NOUN
cana-3487	17	11	of	of	ADP
cana-3487	17	12	bck	bck	NOUN
cana-3487	17	13	-	-	PUNCT
cana-3487	17	14	algebras	algebras	PROPN
cana-3487	17	15	of	of	ADP
cana-3487	17	16	k.	k.	PROPN
cana-3487	17	17	iseki	iseki	PROPN
cana-3487	17	18	and	and	CCONJ
cana-3487	17	19	s.	s.	PROPN
cana-3487	17	20	tanaka	tanaka	PROPN
cana-3487	18	1	[	[	X
cana-3487	18	2	10	10	NUM
cana-3487	18	3	]	]	PUNCT
cana-3487	18	4	.	.	PUNCT
cana-3487	19	1	in	in	ADP
cana-3487	19	2	[	[	X
cana-3487	19	3	1	1	NUM
cana-3487	19	4	]	]	PUNCT
cana-3487	19	5	,	,	PUNCT
cana-3487	19	6	s.s	s.s	PROPN
cana-3487	19	7	.	.	PROPN
cana-3487	19	8	ahn	ahn	PROPN
cana-3487	19	9	and	and	CCONJ
cana-3487	19	10	y.	y.	PROPN
cana-3487	19	11	h.	h.	PROPN
cana-3487	19	12	kim	kim	PROPN
cana-3487	19	13	studied	study	VERB
cana-3487	19	14	some	some	DET
cana-3487	19	15	properties	property	NOUN
cana-3487	19	16	of	of	ADP
cana-3487	19	17	filters	filter	NOUN
cana-3487	19	18	of	of	ADP
cana-3487	19	19	be	be	NOUN
cana-3487	19	20	-	-	PUNCT
cana-3487	19	21	algebras	algebra	NOUN
cana-3487	19	22	and	and	CCONJ
cana-3487	19	23	by	by	ADP
cana-3487	19	24	b.	b.	PROPN
cana-3487	19	25	l.	l.	PROPN
cana-3487	19	26	meng	meng	PROPN
cana-3487	19	27	in	in	ADP
cana-3487	19	28	[	[	X
cana-3487	19	29	12	12	NUM
cana-3487	19	30	]	]	PUNCT
cana-3487	19	31	.	.	PUNCT
cana-3487	20	1	some	some	DET
cana-3487	20	2	relationships	relationship	NOUN
cana-3487	20	3	between	between	ADP
cana-3487	20	4	congruence	congruence	NOUN
cana-3487	20	5	relations	relation	NOUN
cana-3487	20	6	and	and	CCONJ
cana-3487	20	7	normal	normal	ADJ
cana-3487	20	8	filters	filter	NOUN
cana-3487	20	9	of	of	ADP
cana-3487	20	10	a	a	DET
cana-3487	20	11	be	be	NOUN
cana-3487	20	12	-	-	PUNCT
cana-3487	20	13	algebras	algebras	PROPN
cana-3487	20	14	was	be	AUX
cana-3487	20	15	discussed	discuss	VERB
cana-3487	20	16	by	by	ADP
cana-3487	20	17	a.	a.	NOUN
cana-3487	20	18	walendziak	walendziak	PROPN
cana-3487	20	19	in	in	ADP
cana-3487	20	20	[	[	X
cana-3487	20	21	14	14	NUM
cana-3487	20	22	]	]	PUNCT
cana-3487	20	23	.	.	PUNCT
cana-3487	21	1	in	in	ADP
cana-3487	21	2	[	[	X
cana-3487	21	3	13	13	NUM
cana-3487	21	4	]	]	PUNCT
cana-3487	21	5	,	,	PUNCT
cana-3487	21	6	p.	p.	PROPN
cana-3487	21	7	sun	sun	PROPN
cana-3487	21	8	investigated	investigate	VERB
cana-3487	21	9	homomorphism	homomorphism	NOUN
cana-3487	21	10	theorems	theorem	NOUN
cana-3487	21	11	via	via	ADP
cana-3487	21	12	dual	dual	ADJ
cana-3487	21	13	ideals	ideal	NOUN
cana-3487	21	14	of	of	ADP
cana-3487	21	15	bck	bck	NOUN
cana-3487	21	16	-	-	PUNCT
cana-3487	21	17	algebras	algebras	PROPN
cana-3487	21	18	.	.	PUNCT
cana-3487	22	1	in	in	ADP
cana-3487	22	2	[	[	X
cana-3487	22	3	9	9	NUM
cana-3487	22	4	]	]	PUNCT
cana-3487	22	5	,	,	PUNCT
cana-3487	22	6	z.	z.	PROPN
cana-3487	22	7	ciloglu	ciloglu	PROPN
cana-3487	22	8	and	and	CCONJ
cana-3487	22	9	y.	y.	PROPN
cana-3487	22	10	ceven	ceven	PROPN
cana-3487	22	11	introduced	introduce	VERB
cana-3487	22	12	the	the	DET
cana-3487	22	13	notion	notion	NOUN
cana-3487	22	14	of	of	ADP
cana-3487	22	15	commutative	commutative	ADJ
cana-3487	22	16	and	and	CCONJ
cana-3487	22	17	bounded	bound	VERB
cana-3487	22	18	be	be	NOUN
cana-3487	22	19	-	-	PUNCT
cana-3487	22	20	algebras	algebra	NOUN
cana-3487	22	21	.	.	PUNCT
cana-3487	23	1	in	in	ADP
cana-3487	23	2	[	[	X
cana-3487	23	3	8	8	NUM
cana-3487	23	4	]	]	PUNCT
cana-3487	23	5	,	,	PUNCT
cana-3487	23	6	r.	r.	PROPN
cana-3487	23	7	borzooei	borzooei	PROPN
cana-3487	23	8	et	et	PROPN
cana-3487	23	9	al	al	PROPN
cana-3487	23	10	.	.	PROPN
cana-3487	23	11	introduced	introduce	VERB
cana-3487	23	12	the	the	DET
cana-3487	23	13	notion	notion	NOUN
cana-3487	23	14	of	of	ADP
cana-3487	23	15	involutory	involutory	NOUN
cana-3487	23	16	be	be	AUX
cana-3487	23	17	-	-	PUNCT
cana-3487	23	18	algebras	algebra	NOUN
cana-3487	23	19	.	.	PUNCT
cana-3487	24	1	in	in	ADP
cana-3487	24	2	[	[	X
cana-3487	24	3	2	2	NUM
cana-3487	24	4	]	]	PUNCT
cana-3487	24	5	,	,	PUNCT
cana-3487	24	6	m.	m.	NOUN
cana-3487	24	7	bala	bala	PROPN
cana-3487	24	8	prabhakar	prabhakar	PROPN
cana-3487	24	9	,	,	PUNCT
cana-3487	24	10	s.	s.	PROPN
cana-3487	24	11	k.	k.	PROPN
cana-3487	24	12	vali	vali	PROPN
cana-3487	24	13	and	and	CCONJ
cana-3487	24	14	m.	m.	PROPN
cana-3487	24	15	sambasiva	sambasiva	PROPN
cana-3487	24	16	rao	rao	PROPN
cana-3487	24	17	were	be	AUX
cana-3487	24	18	introduced	introduce	VERB
cana-3487	24	19	the	the	DET
cana-3487	24	20	idea	idea	NOUN
cana-3487	24	21	of	of	ADP
cana-3487	24	22	closed	closed	ADJ
cana-3487	24	23	and	and	CCONJ
cana-3487	24	24	dense	dense	ADJ
cana-3487	24	25	elements	element	NOUN
cana-3487	24	26	of	of	ADP
cana-3487	24	27	be	be	NOUN
cana-3487	24	28	-	-	PUNCT
cana-3487	24	29	algebras	algebra	NOUN
cana-3487	24	30	.	.	PUNCT
cana-3487	25	1	also	also	ADV
cana-3487	25	2	,	,	PUNCT
cana-3487	25	3	these	these	DET
cana-3487	25	4	authors	author	NOUN
cana-3487	25	5	were	be	AUX
cana-3487	25	6	introduced	introduce	VERB
cana-3487	25	7	the	the	DET
cana-3487	25	8	concepts	concept	NOUN
cana-3487	25	9	of	of	ADP
cana-3487	25	10	ideals	ideal	NOUN
cana-3487	25	11	of	of	ADP
cana-3487	25	12	transitive	transitive	ADJ
cana-3487	25	13	be	be	AUX
cana-3487	25	14	-	-	PUNCT
cana-3487	25	15	algebras	algebras	ADJ
cana-3487	25	16	in	in	ADP
cana-3487	25	17	[	[	X
cana-3487	25	18	3	3	NUM
cana-3487	25	19	]	]	PUNCT
cana-3487	25	20	,	,	PUNCT
cana-3487	25	21	semi	semi	ADJ
cana-3487	25	22	maximal	maximal	ADJ
cana-3487	25	23	ideals	ideal	NOUN
cana-3487	25	24	of	of	ADP
cana-3487	25	25	be	be	NOUN
cana-3487	25	26	-	-	PUNCT
cana-3487	25	27	algebras	algebras	ADJ
cana-3487	25	28	in	in	ADP
cana-3487	25	29	[	[	X
cana-3487	25	30	4	4	NUM
cana-3487	25	31	]	]	PUNCT
cana-3487	25	32	,	,	PUNCT
cana-3487	25	33	maximal	maximal	ADJ
cana-3487	25	34	ideals	ideal	NOUN
cana-3487	25	35	of	of	ADP
cana-3487	25	36	transitive	transitive	ADJ
cana-3487	25	37	be	be	AUX
cana-3487	25	38	-	-	PUNCT
cana-3487	25	39	algebras	algebras	ADJ
cana-3487	25	40	in	in	ADP
cana-3487	25	41	[	[	X
cana-3487	25	42	5	5	NUM
cana-3487	25	43	]	]	PUNCT
cana-3487	25	44	,	,	PUNCT
cana-3487	25	45	prime	prime	ADJ
cana-3487	25	46	ideals	ideal	NOUN
cana-3487	25	47	of	of	ADP
cana-3487	25	48	transitive	transitive	ADJ
cana-3487	25	49	be	be	AUX
cana-3487	25	50	-	-	PUNCT
cana-3487	25	51	algebras	algebras	ADJ
cana-3487	25	52	in	in	ADP
cana-3487	25	53	[	[	X
cana-3487	25	54	6	6	NUM
cana-3487	25	55	]	]	PUNCT
cana-3487	25	56	and	and	CCONJ
cana-3487	25	57	generalized	generalize	VERB
cana-3487	25	58	lower	low	ADJ
cana-3487	25	59	sets	set	NOUN
cana-3487	25	60	of	of	ADP
cana-3487	25	61	transitive	transitive	ADJ
cana-3487	25	62	be	be	AUX
cana-3487	25	63	-	-	PUNCT
cana-3487	25	64	algebras	algebras	ADJ
cana-3487	25	65	in	in	ADP
cana-3487	25	66	[	[	X
cana-3487	25	67	7	7	NUM
cana-3487	25	68	]	]	PUNCT
cana-3487	25	69	.	.	PUNCT
cana-3487	26	1	in	in	ADP
cana-3487	26	2	this	this	DET
cana-3487	26	3	work	work	NOUN
cana-3487	26	4	,	,	PUNCT
cana-3487	26	5	the	the	DET
cana-3487	26	6	concept	concept	NOUN
cana-3487	26	7	of	of	ADP
cana-3487	26	8	normal	normal	ADJ
cana-3487	26	9	be	be	AUX
cana-3487	26	10	-	-	PUNCT
cana-3487	26	11	algebras	algebras	PROPN
cana-3487	26	12	is	be	AUX
cana-3487	26	13	introduced	introduce	VERB
cana-3487	26	14	,	,	PUNCT
cana-3487	26	15	derived	derive	VERB
cana-3487	26	16	some	some	DET
cana-3487	26	17	properties	property	NOUN
cana-3487	26	18	and	and	CCONJ
cana-3487	26	19	equivalent	equivalent	ADJ
cana-3487	26	20	conditions	condition	NOUN
cana-3487	26	21	.	.	PUNCT
cana-3487	27	1	a	a	DET
cana-3487	27	2	congruence	congruence	NOUN
cana-3487	27	3	is	be	AUX
cana-3487	27	4	introduced	introduce	VERB
cana-3487	27	5	on	on	ADP
cana-3487	27	6	a	a	DET
cana-3487	27	7	normal	normal	ADJ
cana-3487	27	8	be	be	NOUN
cana-3487	27	9	-	-	PUNCT
cana-3487	27	10	algebra	algebra	NOUN
cana-3487	27	11	.	.	PUNCT
cana-3487	28	1	2	2	X
cana-3487	28	2	.	.	X
cana-3487	28	3	preliminary	preliminary	ADJ
cana-3487	28	4	results	result	NOUN
cana-3487	28	5	.	.	PUNCT
cana-3487	29	1	this	this	DET
cana-3487	29	2	section	section	NOUN
cana-3487	29	3	outlines	outline	VERB
cana-3487	29	4	a	a	DET
cana-3487	29	5	combination	combination	NOUN
cana-3487	29	6	of	of	ADP
cana-3487	29	7	definitions	definition	NOUN
cana-3487	29	8	and	and	CCONJ
cana-3487	29	9	results	result	NOUN
cana-3487	29	10	,	,	PUNCT
cana-3487	29	11	formerly	formerly	ADV
cana-3487	29	12	sourced	source	VERB
cana-3487	29	13	from	from	ADP
cana-3487	29	14	existing	exist	VERB
cana-3487	29	15	papers	paper	NOUN
cana-3487	29	16	for	for	ADP
cana-3487	29	17	the	the	DET
cana-3487	29	18	readers	reader	NOUN
cana-3487	29	19	convenience	convenience	VERB
cana-3487	29	20	.	.	PUNCT
cana-3487	30	1	communications	communication	NOUN
cana-3487	30	2	on	on	ADP
cana-3487	30	3	applied	apply	VERB
cana-3487	30	4	nonlinear	nonlinear	ADJ
cana-3487	30	5	analysis	analysis	NOUN
cana-3487	30	6	issn	issn	NOUN
cana-3487	30	7	:	:	PUNCT
cana-3487	30	8	1074	1074	NUM
cana-3487	30	9	-	-	PUNCT
cana-3487	30	10	133x	133x	NUM
cana-3487	30	11	vol	vol	NOUN
cana-3487	30	12	32	32	NUM
cana-3487	30	13	no	no	NOUN
cana-3487	30	14	.	.	PUNCT
cana-3487	31	1	7s	7	NOUN
cana-3487	31	2	(	(	PUNCT
cana-3487	31	3	2025	2025	NUM
cana-3487	31	4	)	)	PUNCT
cana-3487	31	5	800	800	NUM
cana-3487	31	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3487	31	7	definition	definition	NOUN
cana-3487	31	8	1.1	1.1	NUM
cana-3487	31	9	.	.	PUNCT
cana-3487	32	1	[	[	X
cana-3487	32	2	11	11	NUM
cana-3487	32	3	]	]	PUNCT
cana-3487	32	4	an	an	DET
cana-3487	32	5	algebra	algebra	NOUN
cana-3487	32	6	(	(	PUNCT
cana-3487	32	7	𝑋,∗	𝑋,∗	PROPN
cana-3487	32	8	,	,	PUNCT
cana-3487	32	9	1)of	1)of	PROPN
cana-3487	32	10	type	type	NOUN
cana-3487	32	11	(	(	PUNCT
cana-3487	32	12	2	2	NUM
cana-3487	32	13	,	,	PUNCT
cana-3487	32	14	0	0	NUM
cana-3487	32	15	)	)	PUNCT
cana-3487	32	16	is	be	AUX
cana-3487	32	17	called	call	VERB
cana-3487	32	18	a	a	DET
cana-3487	32	19	be	be	NOUN
cana-3487	32	20	-	-	PUNCT
cana-3487	32	21	algebra	algebra	NOUN
cana-3487	32	22	,	,	PUNCT
cana-3487	32	23	if	if	SCONJ
cana-3487	32	24	it	it	PRON
cana-3487	32	25	satisfies	satisfy	VERB
cana-3487	32	26	the	the	DET
cana-3487	32	27	following	follow	VERB
cana-3487	32	28	properties	property	NOUN
cana-3487	32	29	:	:	PUNCT
cana-3487	32	30	(	(	PUNCT
cana-3487	32	31	1	1	X
cana-3487	32	32	)	)	PUNCT
cana-3487	32	33	𝑥	𝑥	NOUN
cana-3487	32	34	∗	∗	NOUN
cana-3487	32	35	𝑥	𝑥	NOUN
cana-3487	32	36	=	=	SYM
cana-3487	32	37	1	1	NUM
cana-3487	32	38	,	,	PUNCT
cana-3487	32	39	(	(	PUNCT
cana-3487	33	1	2	2	X
cana-3487	33	2	)	)	PUNCT
cana-3487	33	3	𝑥	𝑥	NOUN
cana-3487	33	4	∗	∗	NOUN
cana-3487	33	5	1	1	NUM
cana-3487	33	6	=	=	SYM
cana-3487	33	7	1	1	NUM
cana-3487	33	8	,	,	PUNCT
cana-3487	33	9	(	(	PUNCT
cana-3487	33	10	3	3	X
cana-3487	33	11	)	)	SYM
cana-3487	33	12	1	1	NUM
cana-3487	33	13	∗	∗	NOUN
cana-3487	33	14	𝑥	𝑥	NOUN
cana-3487	33	15	=	=	SYM
cana-3487	33	16	𝑥	𝑥	NOUN
cana-3487	33	17	,	,	PUNCT
cana-3487	33	18	(	(	PUNCT
cana-3487	33	19	4	4	X
cana-3487	33	20	)	)	PUNCT
cana-3487	33	21	𝑥	𝑥	NOUN
cana-3487	33	22	∗	∗	NOUN
cana-3487	33	23	(	(	PUNCT
cana-3487	33	24	𝑦	𝑦	NOUN
cana-3487	33	25	∗	∗	X
cana-3487	33	26	𝑧	𝑧	NOUN
cana-3487	33	27	)	)	PUNCT
cana-3487	33	28	=	=	SYM
cana-3487	33	29	𝑦	𝑦	NOUN
cana-3487	33	30	∗	∗	NOUN
cana-3487	33	31	(	(	PUNCT
cana-3487	33	32	𝑥	𝑥	PROPN
cana-3487	33	33	∗	∗	NOUN
cana-3487	33	34	𝑧	𝑧	NOUN
cana-3487	33	35	)	)	PUNCT
cana-3487	33	36	for	for	ADP
cana-3487	33	37	all	all	DET
cana-3487	33	38	𝑥	𝑥	PROPN
cana-3487	33	39	,	,	PUNCT
cana-3487	33	40	𝑦	𝑦	NOUN
cana-3487	33	41	,	,	PUNCT
cana-3487	33	42	𝑧	𝑧	DET
cana-3487	33	43	∈	∈	PROPN
cana-3487	33	44	𝑋.	𝑋.	PROPN
cana-3487	33	45	a	a	DET
cana-3487	33	46	be	be	NOUN
cana-3487	33	47	-	-	PUNCT
cana-3487	33	48	algebra	algebra	NOUN
cana-3487	33	49	𝑋	𝑋	NOUN
cana-3487	33	50	is	be	AUX
cana-3487	33	51	called	call	VERB
cana-3487	33	52	transitive	transitive	ADJ
cana-3487	33	53	if	if	SCONJ
cana-3487	33	54	𝑦	𝑦	PRON
cana-3487	33	55	∗	∗	VERB
cana-3487	33	56	𝑧	𝑧	PRON
cana-3487	33	57	≤	≤	NUM
cana-3487	33	58	(	(	PUNCT
cana-3487	33	59	𝑥	𝑥	NOUN
cana-3487	33	60	∗	∗	X
cana-3487	33	61	𝑦	𝑦	NOUN
cana-3487	33	62	)	)	PUNCT
cana-3487	33	63	∗	∗	NOUN
cana-3487	33	64	(	(	PUNCT
cana-3487	33	65	𝑥	𝑥	NOUN
cana-3487	33	66	∗	∗	NOUN
cana-3487	33	67	𝑧)for	𝑧)for	NUM
cana-3487	33	68	all	all	DET
cana-3487	33	69	𝑥	𝑥	PROPN
cana-3487	33	70	,	,	PUNCT
cana-3487	33	71	𝑦	𝑦	NOUN
cana-3487	33	72	,	,	PUNCT
cana-3487	33	73	𝑧	𝑧	DET
cana-3487	33	74	∈	∈	NOUN
cana-3487	33	75	𝑋.	𝑋.	PROPN
cana-3487	33	76	every	every	PRON
cana-3487	33	77	self	self	NOUN
cana-3487	33	78	-	-	PUNCT
cana-3487	33	79	distributive	distributive	ADJ
cana-3487	33	80	be	be	NOUN
cana-3487	33	81	-	-	PUNCT
cana-3487	33	82	algebra	algebra	NOUN
cana-3487	33	83	is	be	AUX
cana-3487	33	84	transitive	transitive	ADJ
cana-3487	33	85	.	.	PUNCT
cana-3487	34	1	we	we	PRON
cana-3487	34	2	introduce	introduce	VERB
cana-3487	34	3	a	a	DET
cana-3487	34	4	relation	relation	NOUN
cana-3487	34	5	≤	≤	NUM
cana-3487	34	6	on	on	ADP
cana-3487	34	7	a	a	DET
cana-3487	34	8	be	be	NOUN
cana-3487	34	9	-	-	PUNCT
cana-3487	34	10	algebra	algebra	NOUN
cana-3487	34	11	𝑋	𝑋	NOUN
cana-3487	34	12	by	by	ADP
cana-3487	34	13	𝑥	𝑥	DET
cana-3487	34	14	≤	≤	NUM
cana-3487	34	15	𝑦	𝑦	NOUN
cana-3487	34	16	if	if	SCONJ
cana-3487	35	1	and	and	CCONJ
cana-3487	35	2	only	only	ADV
cana-3487	35	3	if	if	SCONJ
cana-3487	35	4	𝑥	𝑥	PROPN
cana-3487	35	5	∗	∗	X
cana-3487	35	6	𝑦	𝑦	NOUN
cana-3487	35	7	=	=	SYM
cana-3487	35	8	1	1	NUM
cana-3487	35	9	for	for	ADP
cana-3487	35	10	all	all	PRON
cana-3487	35	11	for	for	ADP
cana-3487	35	12	all	all	DET
cana-3487	35	13	𝑥	𝑥	PROPN
cana-3487	35	14	,	,	PUNCT
cana-3487	35	15	𝑦	𝑦	PROPN
cana-3487	35	16	∈	∈	PROPN
cana-3487	35	17	𝑋.	𝑋.	PROPN
cana-3487	35	18	theorem	theorem	VERB
cana-3487	35	19	1.2	1.2	NUM
cana-3487	35	20	.	.	PUNCT
cana-3487	36	1	[	[	X
cana-3487	36	2	12	12	NUM
cana-3487	36	3	]	]	PUNCT
cana-3487	36	4	let	let	VERB
cana-3487	36	5	x	x	PRON
cana-3487	36	6	be	be	AUX
cana-3487	36	7	a	a	DET
cana-3487	36	8	transitive	transitive	ADJ
cana-3487	36	9	be	be	NOUN
cana-3487	36	10	-	-	PUNCT
cana-3487	36	11	algebra	algebra	NOUN
cana-3487	36	12	and	and	CCONJ
cana-3487	36	13	𝑥	𝑥	NOUN
cana-3487	36	14	,	,	PUNCT
cana-3487	36	15	𝑦	𝑦	NOUN
cana-3487	36	16	,	,	PUNCT
cana-3487	36	17	𝑧	𝑧	DET
cana-3487	36	18	∈	∈	PROPN
cana-3487	36	19	𝑋.	𝑋.	PROPN
cana-3487	36	20	then	then	ADV
cana-3487	36	21	(	(	PUNCT
cana-3487	36	22	1	1	X
cana-3487	36	23	)	)	SYM
cana-3487	36	24	1	1	NUM
cana-3487	36	25	≤	≤	NOUN
cana-3487	36	26	𝑥	𝑥	PROPN
cana-3487	36	27	implies	imply	VERB
cana-3487	36	28	𝑥	𝑥	X
cana-3487	36	29	=	=	SYM
cana-3487	36	30	1	1	NUM
cana-3487	36	31	,	,	PUNCT
cana-3487	36	32	(	(	PUNCT
cana-3487	36	33	2	2	X
cana-3487	36	34	)	)	PUNCT
cana-3487	36	35	𝑦	𝑦	NOUN
cana-3487	36	36	≤	≤	NOUN
cana-3487	36	37	𝑧	𝑧	PRON
cana-3487	36	38	implies	imply	VERB
cana-3487	36	39	𝑥	𝑥	PROPN
cana-3487	36	40	∗	∗	NOUN
cana-3487	36	41	𝑦	𝑦	NOUN
cana-3487	36	42	≤	≤	NUM
cana-3487	36	43	𝑥	𝑥	DET
cana-3487	36	44	∗	∗	NOUN
cana-3487	36	45	𝑧	𝑧	PROPN
cana-3487	36	46	and	and	CCONJ
cana-3487	36	47	𝑧	𝑧	PROPN
cana-3487	36	48	∗	∗	NOUN
cana-3487	36	49	𝑥	𝑥	X
cana-3487	36	50	≤	≤	NUM
cana-3487	36	51	𝑦	𝑦	NOUN
cana-3487	36	52	∗	∗	NOUN
cana-3487	36	53	𝑥.	𝑥.	ADJ
cana-3487	36	54	definition	definition	NOUN
cana-3487	36	55	1.3	1.3	NUM
cana-3487	36	56	.	.	PUNCT
cana-3487	37	1	[	[	X
cana-3487	37	2	11	11	NUM
cana-3487	37	3	]	]	X
cana-3487	37	4	a	a	DET
cana-3487	37	5	non	non	ADJ
cana-3487	37	6	-	-	ADJ
cana-3487	37	7	empty	empty	ADJ
cana-3487	37	8	subset	subset	NOUN
cana-3487	37	9	f	f	PROPN
cana-3487	37	10	of	of	ADP
cana-3487	37	11	a	a	DET
cana-3487	37	12	be	be	NOUN
cana-3487	37	13	-	-	PUNCT
cana-3487	37	14	algebra	algebra	NOUN
cana-3487	37	15	x	x	PUNCT
cana-3487	37	16	is	be	AUX
cana-3487	37	17	called	call	VERB
cana-3487	37	18	a	a	DET
cana-3487	37	19	filter	filter	NOUN
cana-3487	37	20	of	of	ADP
cana-3487	37	21	x	x	PRON
cana-3487	37	22	if	if	SCONJ
cana-3487	37	23	,	,	PUNCT
cana-3487	37	24	for	for	ADP
cana-3487	37	25	all	all	PRON
cana-3487	37	26	𝑥	𝑥	PROPN
cana-3487	37	27	,	,	PUNCT
cana-3487	37	28	𝑦	𝑦	NOUN
cana-3487	37	29	∈	∈	PROPN
cana-3487	37	30	𝑋	𝑋	PROPN
cana-3487	37	31	,	,	PUNCT
cana-3487	37	32	it	it	PRON
cana-3487	37	33	satisfies	satisfy	VERB
cana-3487	37	34	the	the	DET
cana-3487	37	35	following	follow	VERB
cana-3487	37	36	properties	property	NOUN
cana-3487	37	37	:	:	PUNCT
cana-3487	37	38	(	(	PUNCT
cana-3487	37	39	1)1	1)1	NUM
cana-3487	37	40	∈	∈	NOUN
cana-3487	37	41	𝐹	𝐹	PROPN
cana-3487	37	42	,	,	PUNCT
cana-3487	37	43	(	(	PUNCT
cana-3487	37	44	2	2	X
cana-3487	37	45	)	)	PUNCT
cana-3487	37	46	𝑥	𝑥	PRON
cana-3487	37	47	∈	∈	PROPN
cana-3487	37	48	𝐹and	𝐹and	PROPN
cana-3487	37	49	𝑥	𝑥	PROPN
cana-3487	37	50	∗	∗	NOUN
cana-3487	37	51	𝑦	𝑦	NOUN
cana-3487	37	52	∈	∈	NOUN
cana-3487	37	53	𝐹	𝐹	PRON
cana-3487	37	54	imply	imply	VERB
cana-3487	37	55	that	that	SCONJ
cana-3487	37	56	𝑦	𝑦	NOUN
cana-3487	37	57	∈	∈	PROPN
cana-3487	37	58	𝐹.	𝐹.	PROPN
cana-3487	37	59	definition	definition	NOUN
cana-3487	37	60	1.4	1.4	NUM
cana-3487	37	61	.	.	PUNCT
cana-3487	38	1	[	[	X
cana-3487	38	2	9	9	NUM
cana-3487	38	3	]	]	X
cana-3487	38	4	a	a	DET
cana-3487	38	5	be	be	NOUN
cana-3487	38	6	-	-	PUNCT
cana-3487	38	7	algebra	algebra	NOUN
cana-3487	38	8	𝑋	𝑋	NOUN
cana-3487	38	9	is	be	AUX
cana-3487	38	10	called	call	VERB
cana-3487	38	11	bounded	bounded	ADJ
cana-3487	38	12	be	be	NOUN
cana-3487	38	13	-	-	PUNCT
cana-3487	38	14	algebra	algebra	NOUN
cana-3487	38	15	,	,	PUNCT
cana-3487	38	16	if	if	SCONJ
cana-3487	38	17	there	there	PRON
cana-3487	38	18	exist	exist	VERB
cana-3487	38	19	an	an	DET
cana-3487	38	20	element	element	NOUN
cana-3487	38	21	0	0	PUNCT
cana-3487	38	22	satisfying	satisfy	VERB
cana-3487	38	23	0	0	NUM
cana-3487	38	24	≤	≤	NOUN
cana-3487	38	25	𝑥(𝑜𝑟	𝑥(𝑜𝑟	PROPN
cana-3487	38	26	0	0	NUM
cana-3487	38	27	∗	∗	NOUN
cana-3487	38	28	𝑥	𝑥	NOUN
cana-3487	38	29	=	=	SYM
cana-3487	38	30	1	1	NUM
cana-3487	38	31	)	)	PUNCT
cana-3487	38	32	for	for	ADP
cana-3487	38	33	all	all	DET
cana-3487	38	34	𝑥	𝑥	DET
cana-3487	38	35	∈	∈	PROPN
cana-3487	38	36	𝑋.	𝑋.	PROPN
cana-3487	38	37	define	define	VERB
cana-3487	38	38	an	an	DET
cana-3487	38	39	unary	unary	ADJ
cana-3487	38	40	operation	operation	NOUN
cana-3487	38	41	𝑁	𝑁	PROPN
cana-3487	38	42	on	on	ADP
cana-3487	38	43	𝑋	𝑋	NOUN
cana-3487	38	44	by	by	ADP
cana-3487	38	45	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	38	46	=	=	PUNCT
cana-3487	38	47	𝑥	𝑥	PROPN
cana-3487	38	48	∗	∗	NOUN
cana-3487	38	49	0	0	NUM
cana-3487	38	50	for	for	ADP
cana-3487	38	51	all	all	PRON
cana-3487	38	52	𝑥	𝑥	DET
cana-3487	38	53	∈	∈	PROPN
cana-3487	38	54	𝑋.	𝑋.	PROPN
cana-3487	38	55	clearly	clearly	ADV
cana-3487	38	56	,	,	PUNCT
cana-3487	38	57	0𝑁	0𝑁	NOUN
cana-3487	38	58	=	=	SYM
cana-3487	38	59	1and	1and	NUM
cana-3487	38	60	1𝑁	1𝑁	NOUN
cana-3487	38	61	=	=	SYM
cana-3487	38	62	0	0	X
cana-3487	38	63	.	.	PUNCT
cana-3487	38	64	theorem	theorem	VERB
cana-3487	38	65	1.5	1.5	NUM
cana-3487	38	66	.	.	PUNCT
cana-3487	39	1	[	[	X
cana-3487	39	2	9	9	NUM
cana-3487	39	3	]	]	PUNCT
cana-3487	39	4	let	let	VERB
cana-3487	39	5	𝑋	𝑋	NOUN
cana-3487	39	6	be	be	AUX
cana-3487	39	7	a	a	DET
cana-3487	39	8	transitive	transitive	ADJ
cana-3487	39	9	be	be	NOUN
cana-3487	39	10	-	-	PUNCT
cana-3487	39	11	algebra	algebra	NOUN
cana-3487	39	12	and	and	CCONJ
cana-3487	39	13	𝑥	𝑥	NOUN
cana-3487	39	14	,	,	PUNCT
cana-3487	39	15	𝑦	𝑦	NOUN
cana-3487	39	16	,	,	PUNCT
cana-3487	39	17	𝑧	𝑧	DET
cana-3487	39	18	∈	∈	PROPN
cana-3487	39	19	𝑋.	𝑋.	PROPN
cana-3487	39	20	then	then	ADV
cana-3487	39	21	(	(	PUNCT
cana-3487	39	22	1	1	X
cana-3487	39	23	)	)	PUNCT
cana-3487	39	24	0𝑁	0𝑁	NOUN
cana-3487	39	25	=	=	SYM
cana-3487	40	1	1and	1and	NUM
cana-3487	40	2	1𝑁	1𝑁	NOUN
cana-3487	40	3	=	=	SYM
cana-3487	40	4	0	0	NUM
cana-3487	40	5	,	,	PUNCT
cana-3487	40	6	(	(	PUNCT
cana-3487	40	7	2	2	X
cana-3487	40	8	)	)	PUNCT
cana-3487	40	9	𝑥	𝑥	PROPN
cana-3487	40	10	≤	≤	ADV
cana-3487	40	11	𝑥𝑁𝑁	𝑥𝑁𝑁	ADV
cana-3487	40	12	,	,	PUNCT
cana-3487	40	13	(	(	PUNCT
cana-3487	40	14	3	3	X
cana-3487	40	15	)	)	PUNCT
cana-3487	40	16	𝑥	𝑥	NOUN
cana-3487	40	17	∗	∗	NOUN
cana-3487	40	18	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	40	19	=	=	SYM
cana-3487	40	20	𝑦	𝑦	NOUN
cana-3487	40	21	∗	∗	NOUN
cana-3487	40	22	𝑥𝑁.	𝑥𝑁.	NOUN
cana-3487	40	23	definition	definition	NOUN
cana-3487	40	24	1.6	1.6	NUM
cana-3487	40	25	.	.	PUNCT
cana-3487	41	1	[	[	X
cana-3487	41	2	8	8	X
cana-3487	41	3	]	]	X
cana-3487	41	4	an	an	DET
cana-3487	41	5	element	element	NOUN
cana-3487	41	6	𝑥	𝑥	NOUN
cana-3487	41	7	of	of	ADP
cana-3487	41	8	a	a	DET
cana-3487	41	9	be	be	NOUN
cana-3487	41	10	-	-	PUNCT
cana-3487	41	11	algebra	algebra	NOUN
cana-3487	41	12	𝑋	𝑋	NOUN
cana-3487	41	13	is	be	AUX
cana-3487	41	14	called	call	VERB
cana-3487	41	15	involutory	involutory	ADJ
cana-3487	41	16	element	element	NOUN
cana-3487	41	17	if	if	SCONJ
cana-3487	41	18	𝑥𝑁𝑁	𝑥𝑁𝑁	ADV
cana-3487	41	19	=	=	PUNCT
cana-3487	41	20	𝑥.	𝑥.	ADJ
cana-3487	41	21	if	if	SCONJ
cana-3487	41	22	every	every	DET
cana-3487	41	23	element	element	NOUN
cana-3487	41	24	of	of	ADP
cana-3487	41	25	a	a	DET
cana-3487	41	26	be	be	NOUN
cana-3487	41	27	-	-	PUNCT
cana-3487	41	28	algebra	algebra	NOUN
cana-3487	41	29	𝑋	𝑋	NOUN
cana-3487	41	30	is	be	AUX
cana-3487	41	31	involutory	involutory	NOUN
cana-3487	41	32	,	,	PUNCT
cana-3487	41	33	then	then	ADV
cana-3487	41	34	𝑋	𝑋	PROPN
cana-3487	41	35	is	be	AUX
cana-3487	41	36	called	call	VERB
cana-3487	41	37	an	an	DET
cana-3487	41	38	involutory	involutory	NOUN
cana-3487	41	39	be	be	NOUN
cana-3487	41	40	-	-	PUNCT
cana-3487	41	41	algebra	algebra	NOUN
cana-3487	41	42	.	.	PUNCT
cana-3487	42	1	definition	definition	NOUN
cana-3487	42	2	1.7	1.7	NUM
cana-3487	42	3	.	.	PUNCT
cana-3487	43	1	[	[	X
cana-3487	43	2	2	2	X
cana-3487	43	3	]	]	PUNCT
cana-3487	43	4	an	an	DET
cana-3487	43	5	element	element	NOUN
cana-3487	43	6	𝑎	𝑎	NOUN
cana-3487	43	7	of	of	ADP
cana-3487	43	8	a	a	DET
cana-3487	43	9	be	be	NOUN
cana-3487	43	10	-	-	PUNCT
cana-3487	43	11	algebra	algebra	NOUN
cana-3487	43	12	𝑋	𝑋	NOUN
cana-3487	43	13	is	be	AUX
cana-3487	43	14	called	call	VERB
cana-3487	43	15	closed	closed	ADJ
cana-3487	43	16	element	element	NOUN
cana-3487	43	17	if	if	SCONJ
cana-3487	43	18	𝑎𝑁𝑁	𝑎𝑁𝑁	PROPN
cana-3487	43	19	=	=	PUNCT
cana-3487	43	20	𝑎.	𝑎.	NOUN
cana-3487	43	21	we	we	PRON
cana-3487	43	22	denote	denote	VERB
cana-3487	43	23	the	the	DET
cana-3487	43	24	set	set	NOUN
cana-3487	43	25	𝐶(𝑋	𝐶(𝑋	PROPN
cana-3487	43	26	)	)	PUNCT
cana-3487	43	27	=	=	PRON
cana-3487	43	28	{	{	PUNCT
cana-3487	43	29	𝑎	𝑎	NOUN
cana-3487	43	30	∈	∈	PROPN
cana-3487	43	31	𝑋/𝑎𝑁𝑁	𝑋/𝑎𝑁𝑁	NOUN
cana-3487	43	32	=	=	SYM
cana-3487	43	33	𝑎	𝑎	X
cana-3487	43	34	}	}	PUNCT
cana-3487	43	35	,	,	PUNCT
cana-3487	43	36	is	be	AUX
cana-3487	43	37	the	the	DET
cana-3487	43	38	set	set	NOUN
cana-3487	43	39	of	of	ADP
cana-3487	43	40	all	all	DET
cana-3487	43	41	closed	closed	ADJ
cana-3487	43	42	elements	element	NOUN
cana-3487	43	43	of	of	ADP
cana-3487	43	44	a	a	DET
cana-3487	43	45	be	be	NOUN
cana-3487	43	46	-	-	PUNCT
cana-3487	43	47	algebra	algebra	NOUN
cana-3487	43	48	𝑋.	𝑋.	PROPN
cana-3487	43	49	definition	definition	NOUN
cana-3487	43	50	1.8	1.8	NUM
cana-3487	43	51	.	.	PUNCT
cana-3487	44	1	[	[	X
cana-3487	44	2	7	7	X
cana-3487	44	3	]	]	PUNCT
cana-3487	44	4	let	let	VERB
cana-3487	44	5	𝑋	𝑋	NOUN
cana-3487	44	6	be	be	AUX
cana-3487	44	7	a	a	DET
cana-3487	44	8	bounded	bound	VERB
cana-3487	44	9	be	be	NOUN
cana-3487	44	10	-	-	PUNCT
cana-3487	44	11	algebra	algebra	NOUN
cana-3487	44	12	.	.	PUNCT
cana-3487	45	1	∅	∅	NOUN
cana-3487	45	2	≠	≠	PROPN
cana-3487	45	3	𝑆	𝑆	PROPN
cana-3487	45	4	⊆	⊆	NUM
cana-3487	45	5	𝑋	𝑋	PROPN
cana-3487	45	6	is	be	AUX
cana-3487	45	7	called	call	VERB
cana-3487	45	8	a	a	DET
cana-3487	45	9	bounded	bounded	ADJ
cana-3487	45	10	subalgebra	subalgebra	NOUN
cana-3487	45	11	if	if	SCONJ
cana-3487	45	12	𝑆	𝑆	PROPN
cana-3487	45	13	is	be	AUX
cana-3487	45	14	closed	close	VERB
cana-3487	45	15	under	under	ADP
cana-3487	45	16	the	the	DET
cana-3487	45	17	operations	operation	NOUN
cana-3487	45	18	∗	∗	NOUN
cana-3487	45	19	and	and	CCONJ
cana-3487	45	20	𝑁.	𝑁.	PROPN
cana-3487	45	21	in	in	ADP
cana-3487	45	22	particular	particular	ADJ
cana-3487	45	23	(	(	PUNCT
cana-3487	45	24	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	45	25	∗	∗	NOUN
cana-3487	45	26	𝑦𝑁)𝑁	𝑦𝑁)𝑁	PROPN
cana-3487	45	27	∈	∈	PROPN
cana-3487	45	28	𝑆	𝑆	PROPN
cana-3487	45	29	whenever	whenever	SCONJ
cana-3487	45	30	𝑥	𝑥	PROPN
cana-3487	45	31	,	,	PUNCT
cana-3487	45	32	𝑦	𝑦	NOUN
cana-3487	45	33	∈	∈	NOUN
cana-3487	45	34	𝑆.	𝑆.	PROPN
cana-3487	45	35	lemma	lemma	PROPN
cana-3487	45	36	1.9	1.9	NUM
cana-3487	45	37	.	.	PUNCT
cana-3487	46	1	[	[	X
cana-3487	46	2	3	3	X
cana-3487	46	3	]	]	PUNCT
cana-3487	46	4	let	let	VERB
cana-3487	46	5	𝑋	𝑋	NOUN
cana-3487	46	6	be	be	AUX
cana-3487	46	7	a	a	DET
cana-3487	46	8	transitive	transitive	ADJ
cana-3487	46	9	be	be	NOUN
cana-3487	46	10	-	-	PUNCT
cana-3487	46	11	algebra	algebra	NOUN
cana-3487	46	12	.	.	PUNCT
cana-3487	47	1	for	for	ADP
cana-3487	47	2	any	any	DET
cana-3487	47	3	𝑥	𝑥	PROPN
cana-3487	47	4	,	,	PUNCT
cana-3487	47	5	𝑦	𝑦	NOUN
cana-3487	47	6	,	,	PUNCT
cana-3487	47	7	𝑧	𝑧	DET
cana-3487	47	8	∈	∈	PROPN
cana-3487	47	9	𝑋	𝑋	PROPN
cana-3487	47	10	,	,	PUNCT
cana-3487	47	11	we	we	PRON
cana-3487	47	12	have	have	VERB
cana-3487	47	13	:	:	PUNCT
cana-3487	47	14	(	(	PUNCT
cana-3487	47	15	1	1	X
cana-3487	47	16	)	)	PUNCT
cana-3487	47	17	𝑥𝑁𝑁𝑁	𝑥𝑁𝑁𝑁	NOUN
cana-3487	47	18	≤	≤	VERB
cana-3487	48	1	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	48	2	,	,	PUNCT
cana-3487	48	3	(	(	PUNCT
cana-3487	48	4	2	2	X
cana-3487	48	5	)	)	PUNCT
cana-3487	49	1	𝑥	𝑥	NOUN
cana-3487	49	2	∗	∗	NOUN
cana-3487	49	3	𝑦	𝑦	NOUN
cana-3487	49	4	≤	≤	X
cana-3487	49	5	𝑦𝑁	𝑦𝑁	PROPN
cana-3487	49	6	∗	∗	NOUN
cana-3487	49	7	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	49	8	,	,	PUNCT
cana-3487	49	9	(	(	PUNCT
cana-3487	49	10	3	3	X
cana-3487	49	11	)	)	PUNCT
cana-3487	49	12	𝑥	𝑥	NOUN
cana-3487	49	13	∗	∗	NOUN
cana-3487	49	14	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	49	15	≤	≤	PROPN
cana-3487	49	16	𝑥𝑁𝑁	𝑥𝑁𝑁	PROPN
cana-3487	49	17	∗	∗	NOUN
cana-3487	49	18	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	49	19	,	,	PUNCT
cana-3487	49	20	(	(	PUNCT
cana-3487	49	21	4	4	NUM
cana-3487	49	22	)	)	PUNCT
cana-3487	49	23	(	(	PUNCT
cana-3487	49	24	𝑥	𝑥	NOUN
cana-3487	49	25	∗	∗	X
cana-3487	49	26	𝑦𝑁𝑁)𝑁𝑁	𝑦𝑁𝑁)𝑁𝑁	VERB
cana-3487	49	27	≤	≤	NUM
cana-3487	49	28	𝑥	𝑥	PRON
cana-3487	49	29	∗	∗	NOUN
cana-3487	49	30	𝑦𝑁𝑁	𝑦𝑁𝑁	NOUN
cana-3487	49	31	,	,	PUNCT
cana-3487	49	32	communications	communication	NOUN
cana-3487	49	33	on	on	ADP
cana-3487	49	34	applied	apply	VERB
cana-3487	49	35	nonlinear	nonlinear	ADJ
cana-3487	49	36	analysis	analysis	NOUN
cana-3487	49	37	issn	issn	NOUN
cana-3487	49	38	:	:	PUNCT
cana-3487	49	39	1074	1074	NUM
cana-3487	49	40	-	-	PUNCT
cana-3487	49	41	133x	133x	NUM
cana-3487	49	42	vol	vol	NOUN
cana-3487	49	43	32	32	NUM
cana-3487	49	44	no	no	NOUN
cana-3487	49	45	.	.	PUNCT
cana-3487	50	1	7s	7	NOUN
cana-3487	50	2	(	(	PUNCT
cana-3487	50	3	2025	2025	NUM
cana-3487	50	4	)	)	PUNCT
cana-3487	50	5	801	801	NUM
cana-3487	50	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3487	50	7	(	(	PUNCT
cana-3487	50	8	5	5	NUM
cana-3487	50	9	)	)	PUNCT
cana-3487	50	10	(	(	PUNCT
cana-3487	50	11	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	50	12	∗	∗	NOUN
cana-3487	50	13	𝑦𝑁)𝑁𝑁	𝑦𝑁)𝑁𝑁	PROPN
cana-3487	50	14	≤	≤	ADJ
cana-3487	50	15	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	50	16	∗	∗	NOUN
cana-3487	50	17	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	50	18	,	,	PUNCT
cana-3487	50	19	(	(	PUNCT
cana-3487	50	20	6	6	NUM
cana-3487	50	21	)	)	PUNCT
cana-3487	51	1	𝑥	𝑥	PROPN
cana-3487	51	2	≤	≤	PROPN
cana-3487	51	3	𝑦	𝑦	PRON
cana-3487	51	4	⇒	⇒	NOUN
cana-3487	51	5	𝑦𝑁	𝑦𝑁	PROPN
cana-3487	51	6	≤	≤	ADJ
cana-3487	51	7	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	51	8	,	,	PUNCT
cana-3487	51	9	(	(	PUNCT
cana-3487	51	10	7	7	X
cana-3487	51	11	)	)	PUNCT
cana-3487	51	12	𝑥	𝑥	PROPN
cana-3487	51	13	≤	≤	NUM
cana-3487	51	14	𝑦	𝑦	NUM
cana-3487	51	15	⇒	⇒	NOUN
cana-3487	51	16	𝑦	𝑦	PRON
cana-3487	51	17	∗	∗	NOUN
cana-3487	51	18	𝑧𝑁	𝑧𝑁	PROPN
cana-3487	51	19	≤	≤	ADV
cana-3487	51	20	𝑥	𝑥	PRON
cana-3487	51	21	∗	∗	NOUN
cana-3487	51	22	𝑧𝑁.	𝑧𝑁.	PROPN
cana-3487	51	23	3	3	NUM
cana-3487	51	24	.	.	PUNCT
cana-3487	51	25	main	main	ADJ
cana-3487	51	26	results	result	NOUN
cana-3487	51	27	(	(	PUNCT
cana-3487	51	28	normal	normal	ADJ
cana-3487	51	29	be	be	AUX
cana-3487	51	30	-	-	PUNCT
cana-3487	51	31	algebras	algebra	NOUN
cana-3487	51	32	)	)	PUNCT
cana-3487	51	33	.	.	PUNCT
cana-3487	52	1	in	in	ADP
cana-3487	52	2	this	this	DET
cana-3487	52	3	section	section	NOUN
cana-3487	52	4	,	,	PUNCT
cana-3487	52	5	the	the	DET
cana-3487	52	6	concept	concept	NOUN
cana-3487	52	7	of	of	ADP
cana-3487	52	8	normal	normal	ADJ
cana-3487	52	9	be	be	AUX
cana-3487	52	10	-	-	PUNCT
cana-3487	52	11	algebras	algebras	PROPN
cana-3487	52	12	is	be	AUX
cana-3487	52	13	introduced	introduce	VERB
cana-3487	52	14	.	.	PUNCT
cana-3487	53	1	some	some	DET
cana-3487	53	2	properties	property	NOUN
cana-3487	53	3	of	of	ADP
cana-3487	53	4	normal	normal	ADJ
cana-3487	53	5	bealgebras	bealgebra	NOUN
cana-3487	53	6	are	be	AUX
cana-3487	53	7	studied	study	VERB
cana-3487	53	8	.	.	PUNCT
cana-3487	54	1	some	some	DET
cana-3487	54	2	sufficient	sufficient	ADJ
cana-3487	54	3	conditions	condition	NOUN
cana-3487	54	4	for	for	ADP
cana-3487	54	5	a	a	DET
cana-3487	54	6	be	be	NOUN
cana-3487	54	7	-	-	PUNCT
cana-3487	54	8	algebra	algebra	NOUN
cana-3487	54	9	to	to	PART
cana-3487	54	10	become	become	VERB
cana-3487	54	11	a	a	DET
cana-3487	54	12	normal	normal	ADJ
cana-3487	54	13	be	be	NOUN
cana-3487	54	14	-	-	PUNCT
cana-3487	54	15	algebra	algebra	NOUN
cana-3487	54	16	are	be	AUX
cana-3487	54	17	derived	derive	VERB
cana-3487	54	18	.	.	PUNCT
cana-3487	55	1	definition	definition	NOUN
cana-3487	55	2	2.1	2.1	NUM
cana-3487	55	3	.	.	PUNCT
cana-3487	56	1	a	a	DET
cana-3487	56	2	bounded	bound	VERB
cana-3487	56	3	be	be	NOUN
cana-3487	56	4	-	-	PUNCT
cana-3487	56	5	algebra	algebra	NOUN
cana-3487	56	6	(	(	PUNCT
cana-3487	56	7	𝑋	𝑋	PROPN
cana-3487	56	8	,	,	PUNCT
cana-3487	56	9	∗	∗	NOUN
cana-3487	56	10	,	,	PUNCT
cana-3487	56	11	0	0	NUM
cana-3487	56	12	,	,	PUNCT
cana-3487	56	13	1	1	NUM
cana-3487	56	14	)	)	PUNCT
cana-3487	56	15	is	be	AUX
cana-3487	56	16	said	say	VERB
cana-3487	56	17	to	to	PART
cana-3487	56	18	be	be	AUX
cana-3487	56	19	a	a	DET
cana-3487	56	20	normal	normal	ADJ
cana-3487	56	21	be	be	NOUN
cana-3487	56	22	-	-	PUNCT
cana-3487	56	23	algebra	algebra	NOUN
cana-3487	56	24	,	,	PUNCT
cana-3487	56	25	if	if	SCONJ
cana-3487	56	26	it	it	PRON
cana-3487	56	27	satisfies	satisfy	VERB
cana-3487	56	28	the	the	DET
cana-3487	56	29	following	follow	VERB
cana-3487	56	30	properties	property	NOUN
cana-3487	56	31	for	for	ADP
cana-3487	56	32	all	all	PRON
cana-3487	56	33	𝑥	𝑥	PROPN
cana-3487	56	34	,	,	PUNCT
cana-3487	56	35	𝑦	𝑦	NOUN
cana-3487	56	36	∈	∈	PROPN
cana-3487	56	37	𝑋	𝑋	NOUN
cana-3487	56	38	:	:	PUNCT
cana-3487	56	39	(	(	PUNCT
cana-3487	56	40	n1	n1	NOUN
cana-3487	56	41	)	)	PUNCT
cana-3487	56	42	𝑥𝑁𝑁𝑁	𝑥𝑁𝑁𝑁	NOUN
cana-3487	56	43	=	=	SYM
cana-3487	57	1	𝑥𝑁	𝑥𝑁	ADJ
cana-3487	57	2	,	,	PUNCT
cana-3487	57	3	(	(	PUNCT
cana-3487	57	4	n2	n2	ADJ
cana-3487	57	5	)	)	PUNCT
cana-3487	57	6	(	(	PUNCT
cana-3487	57	7	𝑥	𝑥	NOUN
cana-3487	57	8	∗	∗	X
cana-3487	57	9	𝑦𝑁𝑁)𝑁𝑁	𝑦𝑁𝑁)𝑁𝑁	NOUN
cana-3487	57	10	=	=	SYM
cana-3487	57	11	𝑥	𝑥	NOUN
cana-3487	57	12	∗	∗	NOUN
cana-3487	57	13	𝑦𝑁𝑁.	𝑦𝑁𝑁.	PROPN
cana-3487	57	14	proposition	proposition	NOUN
cana-3487	57	15	2.2	2.2	NUM
cana-3487	57	16	.	.	PUNCT
cana-3487	58	1	let	let	VERB
cana-3487	58	2	(	(	PUNCT
cana-3487	58	3	𝑋	𝑋	NOUN
cana-3487	58	4	,	,	PUNCT
cana-3487	58	5	∗	∗	NOUN
cana-3487	58	6	,	,	PUNCT
cana-3487	58	7	0	0	NUM
cana-3487	58	8	,	,	PUNCT
cana-3487	58	9	1	1	NUM
cana-3487	58	10	)	)	PUNCT
cana-3487	58	11	be	be	AUX
cana-3487	58	12	a	a	DET
cana-3487	58	13	normal	normal	ADJ
cana-3487	58	14	be	be	NOUN
cana-3487	58	15	-	-	PUNCT
cana-3487	58	16	algebra	algebra	NOUN
cana-3487	58	17	.	.	PUNCT
cana-3487	59	1	for	for	ADP
cana-3487	59	2	any	any	DET
cana-3487	59	3	𝑥	𝑥	PROPN
cana-3487	59	4	,	,	PUNCT
cana-3487	59	5	𝑦	𝑦	NOUN
cana-3487	59	6	∈	∈	PROPN
cana-3487	59	7	𝑋	𝑋	PROPN
cana-3487	59	8	,	,	PUNCT
cana-3487	59	9	the	the	DET
cana-3487	59	10	following	follow	VERB
cana-3487	59	11	properties	property	NOUN
cana-3487	59	12	hold	hold	VERB
cana-3487	59	13	:	:	PUNCT
cana-3487	59	14	(	(	PUNCT
cana-3487	59	15	1	1	X
cana-3487	59	16	)	)	PUNCT
cana-3487	59	17	𝑥	𝑥	NOUN
cana-3487	59	18	∗	∗	NOUN
cana-3487	59	19	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	59	20	=	=	PRON
cana-3487	60	1	𝑥𝑁𝑁	𝑥𝑁𝑁	PROPN
cana-3487	60	2	∗	∗	NOUN
cana-3487	60	3	𝑦𝑁	𝑦𝑁	PROPN
cana-3487	60	4	,	,	PUNCT
cana-3487	60	5	(	(	PUNCT
cana-3487	60	6	2	2	X
cana-3487	60	7	)	)	PUNCT
cana-3487	60	8	𝑥	𝑥	NOUN
cana-3487	60	9	∗	∗	NOUN
cana-3487	60	10	𝑦𝑁𝑁	𝑦𝑁𝑁	NUM
cana-3487	60	11	=	=	PUNCT
cana-3487	60	12	𝑥𝑁𝑁	𝑥𝑁𝑁	ADJ
cana-3487	60	13	∗	∗	NOUN
cana-3487	60	14	𝑦𝑁𝑁	𝑦𝑁𝑁	PROPN
cana-3487	60	15	,	,	PUNCT
cana-3487	60	16	(	(	PUNCT
cana-3487	60	17	3	3	NUM
cana-3487	60	18	)	)	PUNCT
cana-3487	60	19	(	(	PUNCT
cana-3487	60	20	𝑥	𝑥	NOUN
cana-3487	60	21	∗	∗	NOUN
cana-3487	60	22	𝑦𝑁)𝑁𝑁	𝑦𝑁)𝑁𝑁	PROPN
cana-3487	60	23	=	=	PUNCT
cana-3487	60	24	𝑥	𝑥	PROPN
cana-3487	60	25	∗	∗	NOUN
cana-3487	60	26	𝑦𝑁.	𝑦𝑁.	NOUN
cana-3487	60	27	proof	proof	NOUN
cana-3487	60	28	.	.	PUNCT
cana-3487	61	1	(	(	PUNCT
cana-3487	61	2	1	1	NUM
cana-3487	61	3	)	)	PUNCT
cana-3487	61	4	.	.	PUNCT
cana-3487	62	1	let	let	VERB
cana-3487	62	2	𝑥	𝑥	PRON
cana-3487	62	3	,	,	PUNCT
cana-3487	62	4	𝑦	𝑦	PRON
cana-3487	62	5	∈	∈	PROPN
cana-3487	62	6	𝑋.	𝑋.	NOUN
cana-3487	62	7	then	then	ADV
cana-3487	62	8	𝑥𝑁𝑁	𝑥𝑁𝑁	PROPN
cana-3487	62	9	∗	∗	NOUN
cana-3487	62	10	𝑦𝑁	𝑦𝑁	PROPN
cana-3487	62	11	=	=	PUNCT
cana-3487	63	1	𝑥𝑁𝑁	𝑥𝑁𝑁	ADJ
cana-3487	63	2	∗	∗	NOUN
cana-3487	63	3	(	(	PUNCT
cana-3487	63	4	𝑦	𝑦	NOUN
cana-3487	63	5	∗	∗	NOUN
cana-3487	63	6	0	0	NUM
cana-3487	63	7	)	)	PUNCT
cana-3487	64	1	=	=	SYM
cana-3487	64	2	𝑦	𝑦	X
cana-3487	64	3	∗	∗	NOUN
cana-3487	64	4	(	(	PUNCT
cana-3487	64	5	𝑥𝑁𝑁	𝑥𝑁𝑁	ADV
cana-3487	64	6	∗	∗	NOUN
cana-3487	64	7	0	0	NUM
cana-3487	64	8	)	)	PUNCT
cana-3487	64	9	=	=	SYM
cana-3487	64	10	𝑦	𝑦	X
cana-3487	64	11	∗	∗	NOUN
cana-3487	64	12	𝑥𝑁𝑁𝑁	𝑥𝑁𝑁𝑁	NUM
cana-3487	64	13	=	=	SYM
cana-3487	64	14	𝑦	𝑦	NOUN
cana-3487	64	15	∗	∗	X
cana-3487	64	16	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	64	17	=	=	PUNCT
cana-3487	64	18	𝑦	𝑦	SYM
cana-3487	64	19	∗	∗	NOUN
cana-3487	64	20	(	(	PUNCT
cana-3487	64	21	𝑥	𝑥	NOUN
cana-3487	64	22	∗	∗	NOUN
cana-3487	64	23	0	0	NUM
cana-3487	64	24	)	)	PUNCT
cana-3487	64	25	=	=	SYM
cana-3487	64	26	𝑥	𝑥	PROPN
cana-3487	64	27	∗	∗	NOUN
cana-3487	64	28	(	(	PUNCT
cana-3487	64	29	𝑦	𝑦	NOUN
cana-3487	64	30	∗	∗	NOUN
cana-3487	64	31	0	0	NUM
cana-3487	64	32	)	)	PUNCT
cana-3487	64	33	=	=	SYM
cana-3487	64	34	𝑥	𝑥	PROPN
cana-3487	64	35	∗	∗	NOUN
cana-3487	64	36	𝑦𝑁.	𝑦𝑁.	NOUN
cana-3487	64	37	(	(	PUNCT
cana-3487	64	38	2	2	NUM
cana-3487	64	39	)	)	PUNCT
cana-3487	64	40	.	.	PUNCT
cana-3487	65	1	let	let	VERB
cana-3487	65	2	𝑥	𝑥	PRON
cana-3487	65	3	,	,	PUNCT
cana-3487	65	4	𝑦	𝑦	PRON
cana-3487	65	5	∈	∈	PROPN
cana-3487	65	6	𝑋.	𝑋.	NOUN
cana-3487	65	7	then	then	ADV
cana-3487	65	8	by	by	ADP
cana-3487	65	9	(	(	PUNCT
cana-3487	65	10	1	1	NUM
cana-3487	65	11	)	)	PUNCT
cana-3487	65	12	,	,	PUNCT
cana-3487	65	13	we	we	PRON
cana-3487	65	14	have	have	VERB
cana-3487	65	15	𝑥	𝑥	PROPN
cana-3487	65	16	∗	∗	NOUN
cana-3487	65	17	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	65	18	=	=	PRON
cana-3487	66	1	𝑥𝑁𝑁	𝑥𝑁𝑁	PROPN
cana-3487	66	2	∗	∗	VERB
cana-3487	66	3	𝑦𝑁.	𝑦𝑁.	PROPN
cana-3487	66	4	replace	replace	VERB
cana-3487	66	5	𝑦	𝑦	NOUN
cana-3487	66	6	by	by	ADP
cana-3487	66	7	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	66	8	,	,	PUNCT
cana-3487	66	9	we	we	PRON
cana-3487	66	10	get	get	VERB
cana-3487	66	11	𝑥	𝑥	DET
cana-3487	66	12	∗	∗	NOUN
cana-3487	66	13	𝑦𝑁𝑁	𝑦𝑁𝑁	NUM
cana-3487	66	14	=	=	PUNCT
cana-3487	66	15	𝑥	𝑥	PROPN
cana-3487	66	16	∗	∗	NOUN
cana-3487	66	17	(	(	PUNCT
cana-3487	66	18	𝑦𝑁)𝑁	𝑦𝑁)𝑁	PROPN
cana-3487	66	19	=	=	PUNCT
cana-3487	66	20	𝑥𝑁𝑁	𝑥𝑁𝑁	PROPN
cana-3487	66	21	∗	∗	X
cana-3487	66	22	𝑦𝑁𝑁.	𝑦𝑁𝑁.	PROPN
cana-3487	66	23	(	(	PUNCT
cana-3487	66	24	3	3	NUM
cana-3487	66	25	)	)	PUNCT
cana-3487	66	26	.	.	PUNCT
cana-3487	67	1	let	let	VERB
cana-3487	67	2	𝑥	𝑥	PRON
cana-3487	67	3	,	,	PUNCT
cana-3487	67	4	𝑦	𝑦	PRON
cana-3487	67	5	∈	∈	NOUN
cana-3487	67	6	𝑋.	𝑋.	PROPN
cana-3487	67	7	since	since	SCONJ
cana-3487	67	8	𝑋	𝑋	PROPN
cana-3487	67	9	is	be	AUX
cana-3487	67	10	normal	normal	ADJ
cana-3487	67	11	,	,	PUNCT
cana-3487	67	12	we	we	PRON
cana-3487	67	13	get	get	VERB
cana-3487	67	14	that	that	PRON
cana-3487	67	15	(	(	PUNCT
cana-3487	68	1	𝑥	𝑥	NOUN
cana-3487	68	2	∗	∗	NOUN
cana-3487	68	3	𝑦𝑁)𝑁𝑁	𝑦𝑁)𝑁𝑁	PROPN
cana-3487	68	4	=	=	PUNCT
cana-3487	68	5	(	(	PUNCT
cana-3487	68	6	𝑥	𝑥	NOUN
cana-3487	68	7	∗	∗	NOUN
cana-3487	68	8	𝑦𝑁𝑁𝑁)𝑁𝑁	𝑦𝑁𝑁𝑁)𝑁𝑁	NOUN
cana-3487	68	9	=	=	PUNCT
cana-3487	68	10	𝑥	𝑥	PROPN
cana-3487	68	11	∗	∗	NOUN
cana-3487	68	12	𝑦𝑁𝑁𝑁	𝑦𝑁𝑁𝑁	NUM
cana-3487	69	1	=	=	SYM
cana-3487	69	2	𝑥	𝑥	PROPN
cana-3487	69	3	∗	∗	NOUN
cana-3487	69	4	𝑦𝑁.	𝑦𝑁.	NOUN
cana-3487	69	5	theorem	theorem	NOUN
cana-3487	69	6	2.3	2.3	NUM
cana-3487	69	7	.	.	PUNCT
cana-3487	70	1	let	let	VERB
cana-3487	70	2	𝑋	𝑋	NOUN
cana-3487	70	3	be	be	AUX
cana-3487	70	4	a	a	DET
cana-3487	70	5	be	be	NOUN
cana-3487	70	6	-	-	PUNCT
cana-3487	70	7	algebra	algebra	NOUN
cana-3487	70	8	which	which	PRON
cana-3487	70	9	satisfies	satisfy	VERB
cana-3487	70	10	the	the	DET
cana-3487	70	11	following	follow	VERB
cana-3487	70	12	conditions	condition	NOUN
cana-3487	70	13	:	:	PUNCT
cana-3487	70	14	(	(	PUNCT
cana-3487	70	15	1	1	X
cana-3487	70	16	)	)	PUNCT
cana-3487	70	17	𝑥𝑁𝑁𝑁	𝑥𝑁𝑁𝑁	NOUN
cana-3487	70	18	=	=	SYM
cana-3487	71	1	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	71	2	,	,	PUNCT
cana-3487	71	3	(	(	PUNCT
cana-3487	71	4	2	2	NUM
cana-3487	71	5	)	)	PUNCT
cana-3487	71	6	(	(	PUNCT
cana-3487	71	7	𝑥	𝑥	NOUN
cana-3487	71	8	∗	∗	NOUN
cana-3487	71	9	𝑦)𝑁𝑁	𝑦)𝑁𝑁	NOUN
cana-3487	71	10	=	=	SYM
cana-3487	72	1	𝑥𝑁𝑁	𝑥𝑁𝑁	ADJ
cana-3487	72	2	∗	∗	NOUN
cana-3487	72	3	𝑦𝑁𝑁	𝑦𝑁𝑁	NOUN
cana-3487	72	4	for	for	ADP
cana-3487	72	5	all	all	PRON
cana-3487	72	6	𝑥	𝑥	PROPN
cana-3487	72	7	,	,	PUNCT
cana-3487	72	8	𝑦	𝑦	PRON
cana-3487	72	9	∈	∈	PROPN
cana-3487	72	10	𝑋.	𝑋.	NOUN
cana-3487	72	11	then	then	ADV
cana-3487	72	12	𝑋	𝑋	PROPN
cana-3487	72	13	is	be	AUX
cana-3487	72	14	a	a	DET
cana-3487	72	15	normal	normal	ADJ
cana-3487	72	16	be	be	NOUN
cana-3487	72	17	-	-	PUNCT
cana-3487	72	18	algebra	algebra	NOUN
cana-3487	72	19	.	.	PUNCT
cana-3487	73	1	proof	proof	NOUN
cana-3487	73	2	.	.	PUNCT
cana-3487	74	1	let	let	VERB
cana-3487	74	2	𝑥	𝑥	PRON
cana-3487	74	3	,	,	PUNCT
cana-3487	74	4	𝑦	𝑦	PRON
cana-3487	74	5	∈	∈	PROPN
cana-3487	74	6	𝑋.	𝑋.	PROPN
cana-3487	74	7	suppose	suppose	VERB
cana-3487	74	8	𝑋	𝑋	NOUN
cana-3487	74	9	is	be	AUX
cana-3487	74	10	satisfying	satisfy	VERB
cana-3487	74	11	the	the	DET
cana-3487	74	12	above	above	ADJ
cana-3487	74	13	two	two	NUM
cana-3487	74	14	conditions	condition	NOUN
cana-3487	74	15	.	.	PUNCT
cana-3487	75	1	then	then	ADV
cana-3487	75	2	(	(	PUNCT
cana-3487	75	3	𝑥	𝑥	NOUN
cana-3487	75	4	∗	∗	X
cana-3487	75	5	𝑦𝑁𝑁)𝑁𝑁	𝑦𝑁𝑁)𝑁𝑁	NOUN
cana-3487	75	6	=	=	SYM
cana-3487	75	7	(	(	PUNCT
cana-3487	75	8	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	75	9	∗	∗	VERB
cana-3487	75	10	𝑥𝑁)𝑁𝑁	𝑥𝑁)𝑁𝑁	NOUN
cana-3487	75	11	=	=	SYM
cana-3487	75	12	𝑦𝑁𝑁𝑁	𝑦𝑁𝑁𝑁	NOUN
cana-3487	75	13	∗	∗	NOUN
cana-3487	75	14	𝑥𝑁𝑁𝑁	𝑥𝑁𝑁𝑁	NOUN
cana-3487	76	1	=	=	SYM
cana-3487	76	2	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	76	3	∗	∗	VERB
cana-3487	76	4	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	76	5	=	=	PUNCT
cana-3487	76	6	𝑥	𝑥	PROPN
cana-3487	76	7	∗	∗	NOUN
cana-3487	76	8	𝑦𝑁𝑁.	𝑦𝑁𝑁.	PROPN
cana-3487	76	9	therefore	therefore	ADV
cana-3487	76	10	,	,	PUNCT
cana-3487	76	11	𝑋	𝑋	PROPN
cana-3487	76	12	is	be	AUX
cana-3487	76	13	a	a	DET
cana-3487	76	14	normal	normal	ADJ
cana-3487	76	15	be	be	NOUN
cana-3487	76	16	-	-	PUNCT
cana-3487	76	17	algebra	algebra	NOUN
cana-3487	76	18	.	.	PUNCT
cana-3487	77	1	communications	communication	NOUN
cana-3487	77	2	on	on	ADP
cana-3487	77	3	applied	apply	VERB
cana-3487	77	4	nonlinear	nonlinear	ADJ
cana-3487	77	5	analysis	analysis	NOUN
cana-3487	77	6	issn	issn	NOUN
cana-3487	77	7	:	:	PUNCT
cana-3487	77	8	1074	1074	NUM
cana-3487	77	9	-	-	PUNCT
cana-3487	77	10	133x	133x	NUM
cana-3487	77	11	vol	vol	NOUN
cana-3487	77	12	32	32	NUM
cana-3487	77	13	no	no	NOUN
cana-3487	77	14	.	.	PUNCT
cana-3487	78	1	7s	7	NOUN
cana-3487	78	2	(	(	PUNCT
cana-3487	78	3	2025	2025	NUM
cana-3487	78	4	)	)	PUNCT
cana-3487	78	5	802	802	NUM
cana-3487	78	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3487	78	7	proposition	proposition	NOUN
cana-3487	78	8	2.4	2.4	NUM
cana-3487	78	9	.	.	PUNCT
cana-3487	79	1	every	every	DET
cana-3487	79	2	involutory	involutory	NOUN
cana-3487	79	3	be	be	NOUN
cana-3487	79	4	-	-	PUNCT
cana-3487	79	5	algebra	algebra	NOUN
cana-3487	79	6	is	be	AUX
cana-3487	79	7	a	a	DET
cana-3487	79	8	normal	normal	ADJ
cana-3487	79	9	be	be	NOUN
cana-3487	79	10	-	-	PUNCT
cana-3487	79	11	algebra	algebra	NOUN
cana-3487	79	12	.	.	PUNCT
cana-3487	80	1	proof	proof	NOUN
cana-3487	80	2	.	.	PUNCT
cana-3487	81	1	assume	assume	VERB
cana-3487	81	2	that	that	SCONJ
cana-3487	81	3	𝑋	𝑋	PROPN
cana-3487	81	4	is	be	AUX
cana-3487	81	5	an	an	DET
cana-3487	81	6	involutory	involutory	NOUN
cana-3487	81	7	be	be	NOUN
cana-3487	81	8	-	-	PUNCT
cana-3487	81	9	algebra	algebra	NOUN
cana-3487	81	10	.	.	PUNCT
cana-3487	82	1	let	let	VERB
cana-3487	82	2	𝑥	𝑥	X
cana-3487	82	3	∈	∈	PROPN
cana-3487	82	4	𝑋.	𝑋.	PROPN
cana-3487	82	5	then	then	ADV
cana-3487	82	6	𝑥𝑁𝑁	𝑥𝑁𝑁	ADV
cana-3487	82	7	=	=	PUNCT
cana-3487	82	8	𝑥.	𝑥.	ADV
cana-3487	82	9	hence	hence	ADV
cana-3487	82	10	𝑥𝑁𝑁𝑁	𝑥𝑁𝑁𝑁	NOUN
cana-3487	82	11	=	=	SYM
cana-3487	82	12	(	(	PUNCT
cana-3487	82	13	𝑥𝑁𝑁)𝑁	𝑥𝑁𝑁)𝑁	NOUN
cana-3487	82	14	=	=	NOUN
cana-3487	82	15	𝑥𝑁.	𝑥𝑁.	NOUN
cana-3487	82	16	again	again	ADV
cana-3487	82	17	,	,	PUNCT
cana-3487	82	18	let	let	VERB
cana-3487	82	19	𝑥	𝑥	PRON
cana-3487	82	20	,	,	PUNCT
cana-3487	82	21	𝑦	𝑦	PRON
cana-3487	82	22	∈	∈	PROPN
cana-3487	82	23	𝑋.	𝑋.	PROPN
cana-3487	82	24	then	then	ADV
cana-3487	82	25	(	(	PUNCT
cana-3487	82	26	𝑥	𝑥	NOUN
cana-3487	82	27	∗	∗	X
cana-3487	82	28	𝑦𝑁𝑁)𝑁𝑁	𝑦𝑁𝑁)𝑁𝑁	NOUN
cana-3487	82	29	=	=	SYM
cana-3487	83	1	(	(	PUNCT
cana-3487	83	2	𝑥	𝑥	NOUN
cana-3487	83	3	∗	∗	NOUN
cana-3487	83	4	𝑦)𝑁𝑁	𝑦)𝑁𝑁	NOUN
cana-3487	84	1	=	=	SYM
cana-3487	84	2	𝑥	𝑥	PROPN
cana-3487	84	3	∗	∗	NOUN
cana-3487	84	4	𝑦	𝑦	NOUN
cana-3487	84	5	=	=	SYM
cana-3487	84	6	𝑥	𝑥	PROPN
cana-3487	84	7	∗	∗	NOUN
cana-3487	84	8	𝑦𝑁𝑁.	𝑦𝑁𝑁.	PROPN
cana-3487	84	9	therefore	therefore	ADV
cana-3487	84	10	,	,	PUNCT
cana-3487	84	11	𝑋	𝑋	PROPN
cana-3487	84	12	is	be	AUX
cana-3487	84	13	a	a	DET
cana-3487	84	14	normal	normal	ADJ
cana-3487	84	15	be	be	NOUN
cana-3487	84	16	-	-	PUNCT
cana-3487	84	17	algebra	algebra	NOUN
cana-3487	84	18	.	.	PUNCT
cana-3487	85	1	example	example	NOUN
cana-3487	85	2	2.5	2.5	NUM
cana-3487	85	3	.	.	PUNCT
cana-3487	86	1	let	let	VERB
cana-3487	86	2	𝑋	𝑋	PROPN
cana-3487	86	3	=	=	SYM
cana-3487	86	4	{	{	PUNCT
cana-3487	86	5	1	1	NUM
cana-3487	86	6	,	,	PUNCT
cana-3487	86	7	𝑎	𝑎	NOUN
cana-3487	86	8	,	,	PUNCT
cana-3487	86	9	𝑏	𝑏	NOUN
cana-3487	86	10	,	,	PUNCT
cana-3487	86	11	𝑐	𝑐	PROPN
cana-3487	86	12	,	,	PUNCT
cana-3487	86	13	𝑑	𝑑	NOUN
cana-3487	86	14	,	,	PUNCT
cana-3487	86	15	0	0	NUM
cana-3487	86	16	}	}	PUNCT
cana-3487	86	17	be	be	AUX
cana-3487	86	18	a	a	DET
cana-3487	86	19	set	set	NOUN
cana-3487	86	20	.	.	PUNCT
cana-3487	87	1	define	define	VERB
cana-3487	87	2	a	a	DET
cana-3487	87	3	binary	binary	ADJ
cana-3487	87	4	operation	operation	NOUN
cana-3487	87	5	∗	∗	NOUN
cana-3487	87	6	on	on	ADP
cana-3487	87	7	𝑋	𝑋	PROPN
cana-3487	87	8	as	as	SCONJ
cana-3487	87	9	follows	follow	VERB
cana-3487	87	10	:	:	PUNCT
cana-3487	87	11	*	*	PUNCT
cana-3487	87	12	1	1	NUM
cana-3487	87	13	a	a	DET
cana-3487	87	14	b	b	NOUN
cana-3487	87	15	c	c	NOUN
cana-3487	87	16	d	d	SYM
cana-3487	87	17	0	0	NUM
cana-3487	87	18	1	1	NUM
cana-3487	87	19	1	1	NUM
cana-3487	87	20	a	a	DET
cana-3487	87	21	b	b	NOUN
cana-3487	87	22	c	c	NOUN
cana-3487	87	23	d	d	NOUN
cana-3487	87	24	0	0	PUNCT
cana-3487	87	25	a	a	DET
cana-3487	87	26	1	1	NUM
cana-3487	87	27	1	1	NUM
cana-3487	87	28	a	a	DET
cana-3487	87	29	c	c	NOUN
cana-3487	87	30	c	c	NOUN
cana-3487	87	31	d	d	PROPN
cana-3487	87	32	b	b	PROPN
cana-3487	87	33	1	1	NUM
cana-3487	87	34	1	1	NUM
cana-3487	87	35	1	1	NUM
cana-3487	87	36	c	c	NOUN
cana-3487	87	37	c	c	NOUN
cana-3487	87	38	c	c	NOUN
cana-3487	87	39	c	c	PROPN
cana-3487	87	40	1	1	NUM
cana-3487	87	41	a	a	DET
cana-3487	87	42	b	b	PROPN
cana-3487	87	43	1	1	NUM
cana-3487	87	44	a	a	DET
cana-3487	87	45	b	b	PROPN
cana-3487	87	46	d	d	SYM
cana-3487	87	47	1	1	NUM
cana-3487	87	48	1	1	NUM
cana-3487	87	49	a	a	DET
cana-3487	87	50	1	1	NUM
cana-3487	87	51	1	1	NUM
cana-3487	87	52	a	a	DET
cana-3487	87	53	0	0	NUM
cana-3487	87	54	1	1	NUM
cana-3487	87	55	1	1	NUM
cana-3487	87	56	1	1	NUM
cana-3487	87	57	1	1	NUM
cana-3487	87	58	1	1	NUM
cana-3487	87	59	1	1	NUM
cana-3487	87	60	clearly	clearly	ADV
cana-3487	87	61	,	,	PUNCT
cana-3487	87	62	(	(	PUNCT
cana-3487	87	63	𝑋	𝑋	NOUN
cana-3487	87	64	,	,	PUNCT
cana-3487	87	65	∗	∗	NOUN
cana-3487	87	66	,	,	PUNCT
cana-3487	87	67	0	0	NUM
cana-3487	87	68	,	,	PUNCT
cana-3487	87	69	1	1	NUM
cana-3487	87	70	)	)	PUNCT
cana-3487	87	71	is	be	AUX
cana-3487	87	72	a	a	DET
cana-3487	87	73	bounded	bound	VERB
cana-3487	87	74	be	be	NOUN
cana-3487	87	75	-	-	PUNCT
cana-3487	87	76	algebra	algebra	NOUN
cana-3487	87	77	.	.	PUNCT
cana-3487	88	1	it	it	PRON
cana-3487	88	2	is	be	AUX
cana-3487	88	3	easy	easy	ADJ
cana-3487	88	4	to	to	PART
cana-3487	88	5	observe	observe	VERB
cana-3487	88	6	that	that	SCONJ
cana-3487	88	7	x	x	PRON
cana-3487	88	8	is	be	AUX
cana-3487	88	9	an	an	DET
cana-3487	88	10	involutory	involutory	NOUN
cana-3487	88	11	be	be	NOUN
cana-3487	88	12	-	-	PUNCT
cana-3487	88	13	algebra	algebra	NOUN
cana-3487	88	14	and	and	CCONJ
cana-3487	88	15	a	a	DET
cana-3487	88	16	normal	normal	ADJ
cana-3487	88	17	be	be	NOUN
cana-3487	88	18	-	-	PUNCT
cana-3487	88	19	algebra	algebra	NOUN
cana-3487	88	20	too	too	ADV
cana-3487	88	21	.	.	PUNCT
cana-3487	89	1	the	the	DET
cana-3487	89	2	converse	converse	NOUN
cana-3487	89	3	of	of	ADP
cana-3487	89	4	the	the	DET
cana-3487	89	5	above	above	ADJ
cana-3487	89	6	proposition	proposition	NOUN
cana-3487	89	7	is	be	AUX
cana-3487	89	8	not	not	PART
cana-3487	89	9	true	true	ADJ
cana-3487	89	10	.	.	PUNCT
cana-3487	90	1	i.e.	i.e.	X
cana-3487	90	2	a	a	DET
cana-3487	90	3	normal	normal	ADJ
cana-3487	90	4	be	be	NOUN
cana-3487	90	5	-	-	PUNCT
cana-3487	90	6	algebra	algebra	NOUN
cana-3487	90	7	need	need	AUX
cana-3487	90	8	not	not	PART
cana-3487	90	9	be	be	AUX
cana-3487	90	10	involutory	involutory	ADJ
cana-3487	90	11	.	.	PUNCT
cana-3487	91	1	for	for	SCONJ
cana-3487	91	2	this	this	PRON
cana-3487	91	3	consider	consider	VERB
cana-3487	91	4	the	the	DET
cana-3487	91	5	following	follow	VERB
cana-3487	91	6	example	example	NOUN
cana-3487	91	7	:	:	PUNCT
cana-3487	91	8	example	example	NOUN
cana-3487	91	9	2.6	2.6	NUM
cana-3487	91	10	.	.	PUNCT
cana-3487	92	1	let	let	VERB
cana-3487	92	2	𝑋	𝑋	PROPN
cana-3487	92	3	=	=	SYM
cana-3487	92	4	{	{	PUNCT
cana-3487	92	5	1	1	NUM
cana-3487	92	6	,	,	PUNCT
cana-3487	92	7	𝑎	𝑎	NOUN
cana-3487	92	8	,	,	PUNCT
cana-3487	92	9	𝑏	𝑏	NOUN
cana-3487	92	10	,	,	PUNCT
cana-3487	92	11	𝑐	𝑐	NOUN
cana-3487	92	12	,	,	PUNCT
cana-3487	92	13	0	0	NUM
cana-3487	92	14	}	}	PUNCT
cana-3487	92	15	be	be	AUX
cana-3487	92	16	a	a	DET
cana-3487	92	17	set	set	NOUN
cana-3487	92	18	.	.	PUNCT
cana-3487	93	1	define	define	VERB
cana-3487	93	2	a	a	DET
cana-3487	93	3	binary	binary	ADJ
cana-3487	93	4	operation	operation	NOUN
cana-3487	93	5	∗	∗	NOUN
cana-3487	93	6	on	on	ADP
cana-3487	93	7	𝑋	𝑋	PROPN
cana-3487	93	8	as	as	SCONJ
cana-3487	93	9	follows	follow	VERB
cana-3487	93	10	:	:	PUNCT
cana-3487	93	11	clearly	clearly	ADV
cana-3487	93	12	,	,	PUNCT
cana-3487	93	13	(	(	PUNCT
cana-3487	93	14	𝑋	𝑋	NOUN
cana-3487	93	15	,	,	PUNCT
cana-3487	93	16	∗	∗	NOUN
cana-3487	93	17	,	,	PUNCT
cana-3487	93	18	0	0	NUM
cana-3487	93	19	,	,	PUNCT
cana-3487	93	20	1	1	NUM
cana-3487	93	21	)	)	PUNCT
cana-3487	93	22	is	be	AUX
cana-3487	93	23	a	a	DET
cana-3487	93	24	normal	normal	ADJ
cana-3487	93	25	be	be	NOUN
cana-3487	93	26	-	-	PUNCT
cana-3487	93	27	algebra	algebra	NOUN
cana-3487	93	28	.	.	PUNCT
cana-3487	94	1	however	however	ADV
cana-3487	94	2	,	,	PUNCT
cana-3487	94	3	𝑋	𝑋	PROPN
cana-3487	94	4	is	be	AUX
cana-3487	94	5	not	not	PART
cana-3487	94	6	an	an	DET
cana-3487	94	7	involutory	involutory	NOUN
cana-3487	94	8	be	be	NOUN
cana-3487	94	9	-	-	PUNCT
cana-3487	94	10	algebra	algebra	NOUN
cana-3487	94	11	because	because	SCONJ
cana-3487	94	12	of	of	ADP
cana-3487	94	13	𝑎𝑁𝑁	𝑎𝑁𝑁	NOUN
cana-3487	94	14	=	=	PUNCT
cana-3487	95	1	0𝑁	0𝑁	NOUN
cana-3487	95	2	=	=	SYM
cana-3487	95	3	1	1	NUM
cana-3487	95	4	,	,	PUNCT
cana-3487	95	5	𝑏𝑁𝑁	𝑏𝑁𝑁	NOUN
cana-3487	95	6	=	=	SYM
cana-3487	95	7	0𝑁	0𝑁	NOUN
cana-3487	95	8	=	=	SYM
cana-3487	95	9	1	1	NUM
cana-3487	95	10	&	&	CCONJ
cana-3487	95	11	𝑐𝑁𝑁	𝑐𝑁𝑁	NOUN
cana-3487	95	12	=	=	NOUN
cana-3487	96	1	0𝑁	0𝑁	NOUN
cana-3487	96	2	=	=	SYM
cana-3487	96	3	1	1	X
cana-3487	96	4	.	.	PUNCT
cana-3487	96	5	in	in	ADP
cana-3487	96	6	the	the	DET
cana-3487	96	7	following	following	NOUN
cana-3487	96	8	theorem	theorem	NOUN
cana-3487	96	9	,	,	PUNCT
cana-3487	96	10	we	we	PRON
cana-3487	96	11	derive	derive	VERB
cana-3487	96	12	a	a	DET
cana-3487	96	13	set	set	NOUN
cana-3487	96	14	of	of	ADP
cana-3487	96	15	equivalent	equivalent	ADJ
cana-3487	96	16	conditions	condition	NOUN
cana-3487	96	17	for	for	ADP
cana-3487	96	18	a	a	DET
cana-3487	96	19	normal	normal	ADJ
cana-3487	96	20	be	be	NOUN
cana-3487	96	21	-	-	PUNCT
cana-3487	96	22	algebra	algebra	NOUN
cana-3487	96	23	to	to	PART
cana-3487	96	24	become	become	VERB
cana-3487	96	25	an	an	DET
cana-3487	96	26	involutory	involutory	NOUN
cana-3487	96	27	be	be	NOUN
cana-3487	96	28	-	-	PUNCT
cana-3487	96	29	algebra	algebra	NOUN
cana-3487	96	30	.	.	PUNCT
cana-3487	97	1	theorem	theorem	ADJ
cana-3487	97	2	2.7	2.7	NUM
cana-3487	97	3	.	.	PUNCT
cana-3487	98	1	let	let	VERB
cana-3487	98	2	(	(	PUNCT
cana-3487	98	3	𝑋	𝑋	NOUN
cana-3487	98	4	,	,	PUNCT
cana-3487	98	5	∗	∗	NOUN
cana-3487	98	6	,	,	PUNCT
cana-3487	98	7	0	0	NUM
cana-3487	98	8	,	,	PUNCT
cana-3487	98	9	1	1	NUM
cana-3487	98	10	)	)	PUNCT
cana-3487	98	11	is	be	AUX
cana-3487	98	12	a	a	DET
cana-3487	98	13	normal	normal	ADJ
cana-3487	98	14	be	be	NOUN
cana-3487	98	15	-	-	PUNCT
cana-3487	98	16	algebra	algebra	NOUN
cana-3487	98	17	.	.	PUNCT
cana-3487	99	1	then	then	ADV
cana-3487	99	2	the	the	DET
cana-3487	99	3	following	follow	VERB
cana-3487	99	4	conditions	condition	NOUN
cana-3487	99	5	are	be	AUX
cana-3487	99	6	equivalent	equivalent	ADJ
cana-3487	99	7	:	:	PUNCT
cana-3487	99	8	(	(	PUNCT
cana-3487	99	9	1	1	X
cana-3487	99	10	)	)	PUNCT
cana-3487	99	11	𝑋	𝑋	NOUN
cana-3487	99	12	is	be	AUX
cana-3487	99	13	involutory	involutory	ADJ
cana-3487	99	14	;	;	PUNCT
cana-3487	99	15	(	(	PUNCT
cana-3487	99	16	2	2	X
cana-3487	99	17	)	)	PUNCT
cana-3487	99	18	for	for	ADP
cana-3487	99	19	any	any	DET
cana-3487	99	20	𝑥	𝑥	PROPN
cana-3487	99	21	,	,	PUNCT
cana-3487	99	22	𝑦	𝑦	NOUN
cana-3487	99	23	∈	∈	PROPN
cana-3487	99	24	𝑋	𝑋	NOUN
cana-3487	99	25	,	,	PUNCT
cana-3487	99	26	𝑥𝑁	𝑥𝑁	ADJ
cana-3487	99	27	=	=	ADJ
cana-3487	99	28	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	99	29	implies	imply	VERB
cana-3487	99	30	𝑥	𝑥	X
cana-3487	99	31	=	=	SYM
cana-3487	99	32	𝑦	𝑦	NUM
cana-3487	99	33	;	;	PUNCT
cana-3487	99	34	(	(	PUNCT
cana-3487	99	35	3	3	X
cana-3487	99	36	)	)	PUNCT
cana-3487	99	37	for	for	ADP
cana-3487	99	38	all	all	PRON
cana-3487	99	39	𝑥	𝑥	DET
cana-3487	99	40	∈	∈	PROPN
cana-3487	99	41	𝑋	𝑋	NOUN
cana-3487	99	42	;	;	PUNCT
cana-3487	99	43	(	(	PUNCT
cana-3487	99	44	𝑥	𝑥	NOUN
cana-3487	99	45	∗	∗	NOUN
cana-3487	99	46	0	0	NUM
cana-3487	99	47	)	)	PUNCT
cana-3487	99	48	∗	∗	NOUN
cana-3487	99	49	0	0	NUM
cana-3487	100	1	=	=	SYM
cana-3487	100	2	(	(	PUNCT
cana-3487	100	3	0	0	NUM
cana-3487	100	4	∗	∗	NOUN
cana-3487	100	5	𝑥)𝑥.	𝑥)𝑥.	PUNCT
cana-3487	101	1	*	*	PUNCT
cana-3487	101	2	1	1	NUM
cana-3487	101	3	a	a	DET
cana-3487	101	4	b	b	NOUN
cana-3487	101	5	c	c	NOUN
cana-3487	101	6	0	0	NUM
cana-3487	101	7	1	1	NUM
cana-3487	101	8	1	1	NUM
cana-3487	101	9	a	a	DET
cana-3487	101	10	b	b	NOUN
cana-3487	101	11	c	c	NOUN
cana-3487	101	12	0	0	PUNCT
cana-3487	102	1	a	a	DET
cana-3487	102	2	1	1	NUM
cana-3487	102	3	1	1	NUM
cana-3487	102	4	b	b	SYM
cana-3487	102	5	b	b	PROPN
cana-3487	102	6	0	0	NUM
cana-3487	102	7	b	b	SYM
cana-3487	102	8	1	1	NUM
cana-3487	102	9	a	a	DET
cana-3487	102	10	1	1	NUM
cana-3487	102	11	a	a	DET
cana-3487	102	12	0	0	NUM
cana-3487	102	13	c	c	NOUN
cana-3487	102	14	1	1	NUM
cana-3487	102	15	1	1	NUM
cana-3487	102	16	1	1	NUM
cana-3487	102	17	1	1	NUM
cana-3487	102	18	0	0	NUM
cana-3487	102	19	0	0	NUM
cana-3487	102	20	1	1	NUM
cana-3487	102	21	1	1	NUM
cana-3487	102	22	1	1	NUM
cana-3487	102	23	1	1	NUM
cana-3487	102	24	1	1	NUM
cana-3487	102	25	communications	communication	NOUN
cana-3487	102	26	on	on	ADP
cana-3487	102	27	applied	apply	VERB
cana-3487	102	28	nonlinear	nonlinear	ADJ
cana-3487	102	29	analysis	analysis	NOUN
cana-3487	102	30	issn	issn	NOUN
cana-3487	102	31	:	:	PUNCT
cana-3487	102	32	1074	1074	NUM
cana-3487	102	33	-	-	PUNCT
cana-3487	102	34	133x	133x	NUM
cana-3487	102	35	vol	vol	NOUN
cana-3487	102	36	32	32	NUM
cana-3487	102	37	no	no	NOUN
cana-3487	102	38	.	.	PUNCT
cana-3487	103	1	7s	7	NOUN
cana-3487	103	2	(	(	PUNCT
cana-3487	103	3	2025	2025	NUM
cana-3487	103	4	)	)	PUNCT
cana-3487	103	5	803	803	NUM
cana-3487	103	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3487	103	7	proof	proof	NOUN
cana-3487	103	8	.	.	PUNCT
cana-3487	104	1	(	(	PUNCT
cana-3487	104	2	1	1	X
cana-3487	104	3	)	)	PUNCT
cana-3487	104	4	⇒	⇒	NOUN
cana-3487	104	5	(	(	PUNCT
cana-3487	104	6	2	2	X
cana-3487	104	7	)	)	PUNCT
cana-3487	104	8	assume	assume	VERB
cana-3487	104	9	that	that	SCONJ
cana-3487	104	10	𝑋	𝑋	PROPN
cana-3487	104	11	is	be	AUX
cana-3487	104	12	involutory	involutory	ADJ
cana-3487	104	13	be	be	NOUN
cana-3487	104	14	-	-	PUNCT
cana-3487	104	15	algebra	algebra	NOUN
cana-3487	104	16	.	.	PUNCT
cana-3487	105	1	let	let	VERB
cana-3487	105	2	𝑥	𝑥	PRON
cana-3487	105	3	,	,	PUNCT
cana-3487	105	4	𝑦	𝑦	NOUN
cana-3487	105	5	∈	∈	NOUN
cana-3487	105	6	𝑋	𝑋	NOUN
cana-3487	105	7	be	be	VERB
cana-3487	105	8	such	such	ADJ
cana-3487	105	9	that	that	SCONJ
cana-3487	105	10	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	105	11	=	=	PUNCT
cana-3487	105	12	𝑦𝑁.	𝑦𝑁.	NOUN
cana-3487	105	13	then	then	ADV
cana-3487	105	14	𝑥𝑁𝑁	𝑥𝑁𝑁	ADV
cana-3487	105	15	=	=	PUNCT
cana-3487	105	16	𝑦𝑁𝑁	𝑦𝑁𝑁	NOUN
cana-3487	105	17	and	and	CCONJ
cana-3487	105	18	hence	hence	ADV
cana-3487	105	19	𝑥	𝑥	PROPN
cana-3487	105	20	=	=	PUNCT
cana-3487	105	21	𝑦.	𝑦.	PROPN
cana-3487	105	22	(	(	PUNCT
cana-3487	105	23	2	2	NUM
cana-3487	105	24	)	)	PUNCT
cana-3487	105	25	⇒	⇒	NOUN
cana-3487	105	26	(	(	PUNCT
cana-3487	105	27	3	3	X
cana-3487	105	28	)	)	PUNCT
cana-3487	105	29	assume	assume	VERB
cana-3487	105	30	the	the	DET
cana-3487	105	31	condition	condition	NOUN
cana-3487	105	32	(	(	PUNCT
cana-3487	105	33	2	2	NUM
cana-3487	105	34	)	)	PUNCT
cana-3487	105	35	.	.	PUNCT
cana-3487	106	1	let	let	VERB
cana-3487	106	2	𝑥	𝑥	PRON
cana-3487	106	3	,	,	PUNCT
cana-3487	106	4	𝑦	𝑦	PRON
cana-3487	106	5	∈	∈	NOUN
cana-3487	106	6	𝑋.	𝑋.	PROPN
cana-3487	106	7	since	since	SCONJ
cana-3487	106	8	(	(	PUNCT
cana-3487	106	9	(	(	PUNCT
cana-3487	106	10	𝑥	𝑥	NOUN
cana-3487	106	11	∗	∗	NOUN
cana-3487	106	12	0	0	NUM
cana-3487	106	13	)	)	PUNCT
cana-3487	106	14	∗	∗	NOUN
cana-3487	106	15	0)𝑁	0)𝑁	NUM
cana-3487	107	1	=	=	SYM
cana-3487	107	2	(	(	PUNCT
cana-3487	107	3	𝑥𝑁𝑁)𝑁	𝑥𝑁𝑁)𝑁	NOUN
cana-3487	107	4	=	=	SYM
cana-3487	107	5	𝑥𝑁𝑁𝑁	𝑥𝑁𝑁𝑁	SYM
cana-3487	107	6	=	=	SYM
cana-3487	108	1	𝑥𝑁.	𝑥𝑁.	NOUN
cana-3487	108	2	then	then	ADV
cana-3487	108	3	by	by	ADP
cana-3487	108	4	(	(	PUNCT
cana-3487	108	5	2	2	NUM
cana-3487	108	6	)	)	PUNCT
cana-3487	108	7	,	,	PUNCT
cana-3487	108	8	we	we	PRON
cana-3487	108	9	get	get	VERB
cana-3487	108	10	(	(	PUNCT
cana-3487	108	11	𝑥	𝑥	NOUN
cana-3487	108	12	∗	∗	NOUN
cana-3487	108	13	0	0	NUM
cana-3487	108	14	)	)	PUNCT
cana-3487	108	15	∗	∗	NOUN
cana-3487	108	16	0	0	NUM
cana-3487	109	1	=	=	SYM
cana-3487	109	2	𝑥	𝑥	NOUN
cana-3487	109	3	=	=	SYM
cana-3487	109	4	1	1	NUM
cana-3487	109	5	∗	∗	NOUN
cana-3487	109	6	𝑥	𝑥	NOUN
cana-3487	109	7	=	=	SYM
cana-3487	109	8	(	(	PUNCT
cana-3487	109	9	0	0	NUM
cana-3487	109	10	∗	∗	NUM
cana-3487	109	11	𝑥	𝑥	NOUN
cana-3487	109	12	)	)	PUNCT
cana-3487	109	13	∗	∗	NOUN
cana-3487	109	14	𝑥.	𝑥.	NOUN
cana-3487	109	15	(	(	PUNCT
cana-3487	109	16	3	3	NUM
cana-3487	109	17	)	)	PUNCT
cana-3487	109	18	⇒	⇒	NOUN
cana-3487	109	19	(	(	PUNCT
cana-3487	109	20	1	1	X
cana-3487	109	21	)	)	PUNCT
cana-3487	109	22	assume	assume	VERB
cana-3487	109	23	the	the	DET
cana-3487	109	24	condition	condition	NOUN
cana-3487	109	25	(	(	PUNCT
cana-3487	109	26	3	3	NUM
cana-3487	109	27	)	)	PUNCT
cana-3487	109	28	.	.	PUNCT
cana-3487	110	1	let	let	VERB
cana-3487	110	2	𝑥	𝑥	X
cana-3487	110	3	∈	∈	PROPN
cana-3487	110	4	𝑋.	𝑋.	PROPN
cana-3487	110	5	then	then	ADV
cana-3487	110	6	(	(	PUNCT
cana-3487	110	7	𝑥	𝑥	NOUN
cana-3487	110	8	∗	∗	NOUN
cana-3487	110	9	0	0	NUM
cana-3487	110	10	)	)	PUNCT
cana-3487	110	11	∗	∗	NOUN
cana-3487	110	12	0	0	NUM
cana-3487	111	1	=	=	SYM
cana-3487	111	2	(	(	PUNCT
cana-3487	111	3	0	0	NUM
cana-3487	111	4	∗	∗	NOUN
cana-3487	111	5	𝑥)𝑥.	𝑥)𝑥.	PUNCT
cana-3487	111	6	hence	hence	ADV
cana-3487	111	7	𝑥𝑁𝑁	𝑥𝑁𝑁	ADV
cana-3487	111	8	=	=	PUNCT
cana-3487	111	9	(	(	PUNCT
cana-3487	111	10	𝑥𝑁)𝑁	𝑥𝑁)𝑁	NOUN
cana-3487	111	11	=	=	SYM
cana-3487	111	12	(	(	PUNCT
cana-3487	111	13	𝑥	𝑥	NOUN
cana-3487	111	14	∗	∗	NOUN
cana-3487	111	15	0	0	NUM
cana-3487	111	16	)	)	PUNCT
cana-3487	111	17	∗	∗	NOUN
cana-3487	111	18	0	0	NUM
cana-3487	112	1	=	=	SYM
cana-3487	112	2	(	(	PUNCT
cana-3487	112	3	0	0	NUM
cana-3487	112	4	∗	∗	NOUN
cana-3487	112	5	𝑥)𝑥	𝑥)𝑥	PUNCT
cana-3487	112	6	=	=	SYM
cana-3487	112	7	1	1	NUM
cana-3487	112	8	∗	∗	NOUN
cana-3487	112	9	𝑥	𝑥	NOUN
cana-3487	112	10	=	=	PUNCT
cana-3487	112	11	𝑥.	𝑥.	NOUN
cana-3487	112	12	therefore	therefore	ADV
cana-3487	112	13	𝑋	𝑋	PROPN
cana-3487	112	14	is	be	AUX
cana-3487	112	15	involutory	involutory	NOUN
cana-3487	112	16	.	.	PUNCT
cana-3487	113	1	theorem	theorem	VERB
cana-3487	113	2	2.8	2.8	NUM
cana-3487	113	3	.	.	PUNCT
cana-3487	114	1	a	a	DET
cana-3487	114	2	normal	normal	ADJ
cana-3487	114	3	be	be	NOUN
cana-3487	114	4	-	-	PUNCT
cana-3487	114	5	algebra	algebra	NOUN
cana-3487	114	6	𝑋	𝑋	NOUN
cana-3487	114	7	satisfies	satisfy	VERB
cana-3487	114	8	the	the	DET
cana-3487	114	9	property	property	NOUN
cana-3487	114	10	,	,	PUNCT
cana-3487	114	11	(	(	PUNCT
cana-3487	114	12	𝑥	𝑥	NOUN
cana-3487	114	13	∗	∗	NOUN
cana-3487	114	14	𝑦)𝑁𝑁	𝑦)𝑁𝑁	NOUN
cana-3487	115	1	=	=	SYM
cana-3487	116	1	𝑥𝑁𝑁	𝑥𝑁𝑁	ADJ
cana-3487	116	2	∗	∗	VERB
cana-3487	116	3	𝑦𝑁𝑁	𝑦𝑁𝑁	NUM
cana-3487	116	4	if	if	SCONJ
cana-3487	116	5	and	and	CCONJ
cana-3487	116	6	only	only	ADV
cana-3487	116	7	if	if	SCONJ
cana-3487	116	8	(	(	PUNCT
cana-3487	116	9	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	116	10	∗	∗	VERB
cana-3487	116	11	𝑥𝑁)𝑁𝑁	𝑥𝑁)𝑁𝑁	NOUN
cana-3487	116	12	=	=	PUNCT
cana-3487	116	13	(	(	PUNCT
cana-3487	116	14	𝑥	𝑥	NOUN
cana-3487	116	15	∗	∗	NOUN
cana-3487	116	16	𝑦)𝑁𝑁	𝑦)𝑁𝑁	X
cana-3487	116	17	for	for	ADP
cana-3487	116	18	all	all	PRON
cana-3487	116	19	𝑥	𝑥	PROPN
cana-3487	116	20	,	,	PUNCT
cana-3487	116	21	𝑦	𝑦	NOUN
cana-3487	116	22	∈	∈	NOUN
cana-3487	116	23	𝑋.	𝑋.	NOUN
cana-3487	116	24	proof	proof	NOUN
cana-3487	116	25	.	.	PUNCT
cana-3487	117	1	let	let	VERB
cana-3487	117	2	𝑋	𝑋	NOUN
cana-3487	117	3	be	be	AUX
cana-3487	117	4	a	a	DET
cana-3487	117	5	normal	normal	ADJ
cana-3487	117	6	be	be	NOUN
cana-3487	117	7	-	-	PUNCT
cana-3487	117	8	algebra	algebra	NOUN
cana-3487	117	9	.	.	PUNCT
cana-3487	118	1	assume	assume	VERB
cana-3487	118	2	that	that	SCONJ
cana-3487	118	3	(	(	PUNCT
cana-3487	118	4	𝑥	𝑥	NOUN
cana-3487	118	5	∗	∗	NOUN
cana-3487	118	6	𝑦)𝑁𝑁	𝑦)𝑁𝑁	NOUN
cana-3487	118	7	=	=	SYM
cana-3487	119	1	𝑥𝑁𝑁	𝑥𝑁𝑁	ADJ
cana-3487	119	2	∗	∗	NOUN
cana-3487	119	3	𝑦𝑁𝑁	𝑦𝑁𝑁	NOUN
cana-3487	119	4	for	for	ADP
cana-3487	119	5	all	all	PRON
cana-3487	119	6	𝑥	𝑥	PROPN
cana-3487	119	7	,	,	PUNCT
cana-3487	119	8	𝑦	𝑦	PROPN
cana-3487	119	9	∈	∈	PROPN
cana-3487	119	10	𝑋.	𝑋.	PROPN
cana-3487	119	11	then	then	ADV
cana-3487	119	12	we	we	PRON
cana-3487	119	13	have	have	VERB
cana-3487	119	14	(	(	PUNCT
cana-3487	119	15	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	119	16	∗	∗	VERB
cana-3487	119	17	𝑥𝑁)𝑁𝑁	𝑥𝑁)𝑁𝑁	NOUN
cana-3487	120	1	=	=	PUNCT
cana-3487	121	1	(	(	PUNCT
cana-3487	121	2	𝑥	𝑥	NOUN
cana-3487	121	3	∗	∗	X
cana-3487	121	4	𝑦𝑁𝑁)𝑁𝑁	𝑦𝑁𝑁)𝑁𝑁	NOUN
cana-3487	121	5	=	=	PUNCT
cana-3487	121	6	𝑥	𝑥	NOUN
cana-3487	121	7	∗	∗	NOUN
cana-3487	121	8	𝑦𝑁𝑁	𝑦𝑁𝑁	NUM
cana-3487	121	9	=	=	PUNCT
cana-3487	121	10	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	121	11	∗	∗	VERB
cana-3487	121	12	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	121	13	=	=	PUNCT
cana-3487	121	14	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	121	15	∗	∗	NOUN
cana-3487	121	16	𝑥𝑁𝑁𝑁	𝑥𝑁𝑁𝑁	NUM
cana-3487	121	17	=	=	SYM
cana-3487	122	1	𝑥𝑁𝑁	𝑥𝑁𝑁	NOUN
cana-3487	122	2	∗	∗	VERB
cana-3487	122	3	𝑦𝑁𝑁	𝑦𝑁𝑁	NUM
cana-3487	122	4	=	=	PUNCT
cana-3487	122	5	(	(	PUNCT
cana-3487	122	6	𝑥	𝑥	NOUN
cana-3487	122	7	∗	∗	NOUN
cana-3487	122	8	𝑦)𝑁𝑁.	𝑦)𝑁𝑁.	NOUN
cana-3487	122	9	conversely	conversely	ADV
cana-3487	122	10	,	,	PUNCT
cana-3487	122	11	assume	assume	VERB
cana-3487	122	12	the	the	DET
cana-3487	122	13	condition	condition	NOUN
cana-3487	122	14	(	(	PUNCT
cana-3487	122	15	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	122	16	∗	∗	VERB
cana-3487	122	17	𝑥𝑁)𝑁𝑁	𝑥𝑁)𝑁𝑁	NOUN
cana-3487	122	18	=	=	PUNCT
cana-3487	122	19	(	(	PUNCT
cana-3487	122	20	𝑥	𝑥	NOUN
cana-3487	122	21	∗	∗	NOUN
cana-3487	122	22	𝑦)𝑁𝑁for	𝑦)𝑁𝑁for	ADP
cana-3487	122	23	all	all	DET
cana-3487	122	24	𝑥	𝑥	PROPN
cana-3487	122	25	,	,	PUNCT
cana-3487	122	26	𝑦	𝑦	NOUN
cana-3487	122	27	∈	∈	PROPN
cana-3487	122	28	𝑋.	𝑋.	PROPN
cana-3487	122	29	for	for	ADP
cana-3487	122	30	any	any	DET
cana-3487	122	31	𝑥	𝑥	PROPN
cana-3487	122	32	,	,	PUNCT
cana-3487	122	33	𝑦	𝑦	NOUN
cana-3487	122	34	∈	∈	PROPN
cana-3487	122	35	𝑋	𝑋	PROPN
cana-3487	122	36	,	,	PUNCT
cana-3487	122	37	we	we	PRON
cana-3487	122	38	get	get	VERB
cana-3487	123	1	𝑥𝑁𝑁	𝑥𝑁𝑁	ADJ
cana-3487	123	2	∗	∗	NOUN
cana-3487	123	3	𝑦𝑁𝑁	𝑦𝑁𝑁	NUM
cana-3487	123	4	=	=	PUNCT
cana-3487	123	5	(	(	PUNCT
cana-3487	123	6	𝑥𝑁𝑁	𝑥𝑁𝑁	PROPN
cana-3487	123	7	∗	∗	NOUN
cana-3487	123	8	𝑦𝑁𝑁)𝑁𝑁	𝑦𝑁𝑁)𝑁𝑁	NOUN
cana-3487	123	9	=	=	SYM
cana-3487	123	10	(	(	PUNCT
cana-3487	123	11	𝑦𝑁	𝑦𝑁	PROPN
cana-3487	123	12	∗	∗	VERB
cana-3487	123	13	𝑥𝑁𝑁𝑁)𝑁𝑁	𝑥𝑁𝑁𝑁)𝑁𝑁	PROPN
cana-3487	124	1	=	=	PUNCT
cana-3487	124	2	(	(	PUNCT
cana-3487	124	3	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	124	4	∗	∗	VERB
cana-3487	124	5	𝑥𝑁)𝑁𝑁	𝑥𝑁)𝑁𝑁	NOUN
cana-3487	124	6	=	=	PUNCT
cana-3487	124	7	(	(	PUNCT
cana-3487	124	8	𝑥	𝑥	NOUN
cana-3487	124	9	∗	∗	NOUN
cana-3487	124	10	𝑦)𝑁𝑁.	𝑦)𝑁𝑁.	NOUN
cana-3487	124	11	in	in	ADP
cana-3487	124	12	the	the	DET
cana-3487	124	13	following	following	NOUN
cana-3487	124	14	theorem	theorem	NOUN
cana-3487	124	15	,	,	PUNCT
cana-3487	124	16	we	we	PRON
cana-3487	124	17	derive	derive	VERB
cana-3487	124	18	a	a	DET
cana-3487	124	19	set	set	NOUN
cana-3487	124	20	of	of	ADP
cana-3487	124	21	equivalent	equivalent	ADJ
cana-3487	124	22	conditions	condition	NOUN
cana-3487	124	23	for	for	ADP
cana-3487	124	24	a	a	DET
cana-3487	124	25	transitive	transitive	ADJ
cana-3487	124	26	be	be	NOUN
cana-3487	124	27	-	-	PUNCT
cana-3487	124	28	algebra	algebra	NOUN
cana-3487	124	29	to	to	PART
cana-3487	124	30	become	become	VERB
cana-3487	124	31	normal	normal	ADJ
cana-3487	124	32	.	.	PUNCT
cana-3487	125	1	proposition	proposition	NOUN
cana-3487	125	2	2.9	2.9	NUM
cana-3487	125	3	.	.	PUNCT
cana-3487	126	1	let	let	VERB
cana-3487	126	2	𝑋	𝑋	NOUN
cana-3487	126	3	be	be	AUX
cana-3487	126	4	a	a	DET
cana-3487	126	5	transitive	transitive	ADJ
cana-3487	126	6	be	be	NOUN
cana-3487	126	7	-	-	PUNCT
cana-3487	126	8	algebra	algebra	NOUN
cana-3487	126	9	which	which	PRON
cana-3487	126	10	satisfies	satisfy	VERB
cana-3487	126	11	the	the	DET
cana-3487	126	12	property	property	NOUN
cana-3487	126	13	,	,	PUNCT
cana-3487	126	14	𝑥𝑁𝑁𝑁	𝑥𝑁𝑁𝑁	NOUN
cana-3487	126	15	=	=	PUNCT
cana-3487	126	16	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	126	17	for	for	ADP
cana-3487	126	18	all	all	PRON
cana-3487	126	19	𝑥	𝑥	PROPN
cana-3487	126	20	,	,	PUNCT
cana-3487	126	21	𝑦	𝑦	PROPN
cana-3487	126	22	∈	∈	PROPN
cana-3487	126	23	𝑋.	𝑋.	PROPN
cana-3487	126	24	then	then	ADV
cana-3487	126	25	the	the	DET
cana-3487	126	26	following	follow	VERB
cana-3487	126	27	conditions	condition	NOUN
cana-3487	126	28	are	be	AUX
cana-3487	126	29	equivalent	equivalent	ADJ
cana-3487	126	30	:	:	PUNCT
cana-3487	126	31	(	(	PUNCT
cana-3487	126	32	1	1	X
cana-3487	126	33	)	)	PUNCT
cana-3487	126	34	𝑋	𝑋	NOUN
cana-3487	126	35	is	be	AUX
cana-3487	126	36	a	a	DET
cana-3487	126	37	normal	normal	ADJ
cana-3487	126	38	be	be	NOUN
cana-3487	126	39	-	-	PUNCT
cana-3487	126	40	algebra	algebra	NOUN
cana-3487	126	41	.	.	PUNCT
cana-3487	127	1	(	(	PUNCT
cana-3487	127	2	2	2	X
cana-3487	127	3	)	)	PUNCT
cana-3487	127	4	for	for	ADP
cana-3487	127	5	all	all	PRON
cana-3487	127	6	𝑥	𝑥	PROPN
cana-3487	127	7	,	,	PUNCT
cana-3487	127	8	𝑦	𝑦	NOUN
cana-3487	127	9	∈	∈	PROPN
cana-3487	127	10	𝑋	𝑋	PROPN
cana-3487	127	11	,	,	PUNCT
cana-3487	127	12	(	(	PUNCT
cana-3487	127	13	𝑥𝑁𝑁	𝑥𝑁𝑁	PROPN
cana-3487	127	14	∗	∗	NOUN
cana-3487	127	15	𝑦𝑁𝑁)𝑁𝑁	𝑦𝑁𝑁)𝑁𝑁	NOUN
cana-3487	127	16	=	=	PUNCT
cana-3487	128	1	𝑥𝑁𝑁	𝑥𝑁𝑁	ADJ
cana-3487	128	2	∗	∗	NOUN
cana-3487	128	3	𝑦𝑁𝑁	𝑦𝑁𝑁	NOUN
cana-3487	128	4	;	;	PUNCT
cana-3487	128	5	(	(	PUNCT
cana-3487	128	6	3	3	X
cana-3487	128	7	)	)	PUNCT
cana-3487	128	8	for	for	ADP
cana-3487	128	9	all	all	PRON
cana-3487	128	10	𝑥	𝑥	PROPN
cana-3487	128	11	,	,	PUNCT
cana-3487	128	12	𝑦	𝑦	NOUN
cana-3487	128	13	∈	∈	PROPN
cana-3487	128	14	𝑋	𝑋	PROPN
cana-3487	128	15	,	,	PUNCT
cana-3487	128	16	(	(	PUNCT
cana-3487	128	17	𝑥	𝑥	NOUN
cana-3487	128	18	∗	∗	X
cana-3487	128	19	𝑦𝑁𝑁)𝑁𝑁	𝑦𝑁𝑁)𝑁𝑁	NOUN
cana-3487	128	20	=	=	PUNCT
cana-3487	129	1	𝑥𝑁𝑁	𝑥𝑁𝑁	ADJ
cana-3487	129	2	∗	∗	NOUN
cana-3487	129	3	𝑦𝑁𝑁	𝑦𝑁𝑁	NOUN
cana-3487	129	4	;	;	PUNCT
cana-3487	129	5	(	(	PUNCT
cana-3487	129	6	4	4	X
cana-3487	129	7	)	)	PUNCT
cana-3487	129	8	for	for	ADP
cana-3487	129	9	all	all	PRON
cana-3487	129	10	𝑥	𝑥	PROPN
cana-3487	129	11	,	,	PUNCT
cana-3487	129	12	𝑦	𝑦	NOUN
cana-3487	129	13	∈	∈	PROPN
cana-3487	129	14	𝑋	𝑋	PROPN
cana-3487	129	15	,	,	PUNCT
cana-3487	129	16	(	(	PUNCT
cana-3487	129	17	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	129	18	∗	∗	NOUN
cana-3487	129	19	𝑦𝑁)𝑁𝑁	𝑦𝑁)𝑁𝑁	ADJ
cana-3487	129	20	=	=	PUNCT
cana-3487	129	21	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	129	22	∗	∗	NOUN
cana-3487	129	23	𝑦𝑁.	𝑦𝑁.	NOUN
cana-3487	129	24	proof	proof	NOUN
cana-3487	129	25	.	.	PUNCT
cana-3487	130	1	(	(	PUNCT
cana-3487	130	2	1	1	X
cana-3487	130	3	)	)	PUNCT
cana-3487	130	4	⇔	⇔	X
cana-3487	130	5	(	(	PUNCT
cana-3487	130	6	2	2	NUM
cana-3487	130	7	)	)	PUNCT
cana-3487	130	8	assume	assume	VERB
cana-3487	130	9	that	that	SCONJ
cana-3487	130	10	𝑋	𝑋	PROPN
cana-3487	130	11	is	be	AUX
cana-3487	130	12	a	a	DET
cana-3487	130	13	normal	normal	ADJ
cana-3487	130	14	be	be	NOUN
cana-3487	130	15	-	-	PUNCT
cana-3487	130	16	algebra	algebra	NOUN
cana-3487	130	17	.	.	PUNCT
cana-3487	131	1	let	let	VERB
cana-3487	131	2	𝑥	𝑥	PRON
cana-3487	131	3	,	,	PUNCT
cana-3487	131	4	𝑦	𝑦	PRON
cana-3487	131	5	∈	∈	PROPN
cana-3487	131	6	𝑋.	𝑋.	PROPN
cana-3487	131	7	then	then	ADV
cana-3487	131	8	by	by	ADP
cana-3487	131	9	the	the	DET
cana-3487	131	10	property	property	NOUN
cana-3487	131	11	(	(	PUNCT
cana-3487	131	12	n2	n2	NOUN
cana-3487	131	13	)	)	PUNCT
cana-3487	131	14	,	,	PUNCT
cana-3487	131	15	we	we	PRON
cana-3487	131	16	get	get	VERB
cana-3487	131	17	(	(	PUNCT
cana-3487	131	18	𝑥𝑁𝑁	𝑥𝑁𝑁	ADV
cana-3487	131	19	∗	∗	NOUN
cana-3487	131	20	𝑦𝑁𝑁)𝑁𝑁	𝑦𝑁𝑁)𝑁𝑁	NOUN
cana-3487	131	21	=	=	PUNCT
cana-3487	132	1	𝑥𝑁𝑁	𝑥𝑁𝑁	PROPN
cana-3487	132	2	∗	∗	NOUN
cana-3487	132	3	𝑦𝑁𝑁.	𝑦𝑁𝑁.	X
cana-3487	132	4	conversely	conversely	ADV
cana-3487	132	5	,	,	PUNCT
cana-3487	132	6	assume	assume	VERB
cana-3487	132	7	the	the	DET
cana-3487	132	8	condition	condition	NOUN
cana-3487	132	9	(	(	PUNCT
cana-3487	132	10	2	2	NUM
cana-3487	132	11	)	)	PUNCT
cana-3487	132	12	.	.	PUNCT
cana-3487	133	1	let	let	VERB
cana-3487	133	2	𝑥	𝑥	PRON
cana-3487	133	3	,	,	PUNCT
cana-3487	133	4	𝑦	𝑦	PRON
cana-3487	133	5	∈	∈	PROPN
cana-3487	133	6	𝑋.	𝑋.	PROPN
cana-3487	133	7	then	then	ADV
cana-3487	133	8	(	(	PUNCT
cana-3487	133	9	𝑥	𝑥	NOUN
cana-3487	133	10	∗	∗	X
cana-3487	133	11	𝑦𝑁𝑁)𝑁𝑁	𝑦𝑁𝑁)𝑁𝑁	NOUN
cana-3487	134	1	=	=	SYM
cana-3487	134	2	(	(	PUNCT
cana-3487	134	3	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	134	4	∗	∗	VERB
cana-3487	134	5	𝑥𝑁)𝑁𝑁	𝑥𝑁)𝑁𝑁	NOUN
cana-3487	134	6	=	=	PUNCT
cana-3487	134	7	(	(	PUNCT
cana-3487	134	8	𝑦𝑁	𝑦𝑁	PROPN
cana-3487	134	9	∗	∗	VERB
cana-3487	134	10	𝑥𝑁𝑁𝑁)𝑁𝑁	𝑥𝑁𝑁𝑁)𝑁𝑁	PROPN
cana-3487	134	11	=	=	PUNCT
cana-3487	134	12	(	(	PUNCT
cana-3487	134	13	𝑥𝑁𝑁	𝑥𝑁𝑁	PROPN
cana-3487	134	14	∗	∗	NOUN
cana-3487	134	15	𝑦𝑁𝑁)𝑁𝑁	𝑦𝑁𝑁)𝑁𝑁	NOUN
cana-3487	134	16	=	=	PUNCT
cana-3487	135	1	𝑥𝑁𝑁	𝑥𝑁𝑁	ADP
cana-3487	135	2	∗	∗	VERB
cana-3487	135	3	𝑦𝑁𝑁	𝑦𝑁𝑁	NUM
cana-3487	135	4	=	=	PUNCT
cana-3487	135	5	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	135	6	∗	∗	NOUN
cana-3487	135	7	𝑥𝑁𝑁𝑁	𝑥𝑁𝑁𝑁	NOUN
cana-3487	136	1	=	=	SYM
cana-3487	136	2	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	136	3	∗	∗	VERB
cana-3487	136	4	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	136	5	=	=	PUNCT
cana-3487	136	6	𝑥	𝑥	PROPN
cana-3487	136	7	∗	∗	NOUN
cana-3487	136	8	𝑦𝑁𝑁.	𝑦𝑁𝑁.	PROPN
cana-3487	136	9	communications	communication	NOUN
cana-3487	136	10	on	on	ADP
cana-3487	136	11	applied	apply	VERB
cana-3487	136	12	nonlinear	nonlinear	ADJ
cana-3487	136	13	analysis	analysis	NOUN
cana-3487	136	14	issn	issn	NOUN
cana-3487	136	15	:	:	PUNCT
cana-3487	136	16	1074	1074	NUM
cana-3487	136	17	-	-	PUNCT
cana-3487	136	18	133x	133x	NUM
cana-3487	136	19	vol	vol	NOUN
cana-3487	136	20	32	32	NUM
cana-3487	136	21	no	no	NOUN
cana-3487	136	22	.	.	PUNCT
cana-3487	137	1	7s	7	NOUN
cana-3487	137	2	(	(	PUNCT
cana-3487	137	3	2025	2025	NUM
cana-3487	137	4	)	)	PUNCT
cana-3487	137	5	804	804	NUM
cana-3487	137	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3487	137	7	hence	hence	ADV
cana-3487	137	8	𝑋	𝑋	PROPN
cana-3487	137	9	is	be	AUX
cana-3487	137	10	a	a	DET
cana-3487	137	11	normal	normal	ADJ
cana-3487	137	12	be	be	NOUN
cana-3487	137	13	-	-	PUNCT
cana-3487	137	14	algebra	algebra	NOUN
cana-3487	137	15	,	,	PUNCT
cana-3487	137	16	which	which	PRON
cana-3487	137	17	proves	prove	VERB
cana-3487	137	18	the	the	DET
cana-3487	137	19	condition	condition	NOUN
cana-3487	137	20	(	(	PUNCT
cana-3487	137	21	1	1	NUM
cana-3487	137	22	)	)	PUNCT
cana-3487	137	23	.	.	PUNCT
cana-3487	138	1	(	(	PUNCT
cana-3487	138	2	2	2	X
cana-3487	138	3	)	)	PUNCT
cana-3487	138	4	⇒	⇒	NOUN
cana-3487	138	5	(	(	PUNCT
cana-3487	138	6	3	3	X
cana-3487	138	7	)	)	PUNCT
cana-3487	138	8	assume	assume	VERB
cana-3487	138	9	that	that	SCONJ
cana-3487	138	10	the	the	DET
cana-3487	138	11	condition	condition	NOUN
cana-3487	138	12	(	(	PUNCT
cana-3487	138	13	2	2	X
cana-3487	138	14	)	)	PUNCT
cana-3487	138	15	holds	hold	VERB
cana-3487	138	16	.	.	PUNCT
cana-3487	139	1	let	let	VERB
cana-3487	139	2	𝑥	𝑥	PRON
cana-3487	139	3	,	,	PUNCT
cana-3487	139	4	𝑦	𝑦	PRON
cana-3487	139	5	∈	∈	PROPN
cana-3487	139	6	𝑋.	𝑋.	NOUN
cana-3487	139	7	then	then	ADV
cana-3487	139	8	𝑥𝑁𝑁	𝑥𝑁𝑁	ADV
cana-3487	139	9	∗	∗	VERB
cana-3487	139	10	𝑦𝑁𝑁	𝑦𝑁𝑁	NUM
cana-3487	139	11	=	=	PUNCT
cana-3487	139	12	(	(	PUNCT
cana-3487	139	13	𝑥𝑁𝑁	𝑥𝑁𝑁	PROPN
cana-3487	139	14	∗	∗	NOUN
cana-3487	139	15	𝑦𝑁𝑁)𝑁𝑁	𝑦𝑁𝑁)𝑁𝑁	NOUN
cana-3487	139	16	=	=	SYM
cana-3487	139	17	(	(	PUNCT
cana-3487	139	18	𝑦𝑁	𝑦𝑁	PROPN
cana-3487	139	19	∗	∗	VERB
cana-3487	139	20	𝑥𝑁𝑁𝑁)𝑁𝑁	𝑥𝑁𝑁𝑁)𝑁𝑁	PROPN
cana-3487	140	1	=	=	PUNCT
cana-3487	140	2	(	(	PUNCT
cana-3487	140	3	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	140	4	∗	∗	VERB
cana-3487	140	5	𝑥𝑁)𝑁𝑁	𝑥𝑁)𝑁𝑁	NOUN
cana-3487	140	6	=	=	PUNCT
cana-3487	140	7	(	(	PUNCT
cana-3487	140	8	𝑥	𝑥	PROPN
cana-3487	140	9	∗	∗	X
cana-3487	140	10	𝑦𝑁𝑁)𝑁𝑁	𝑦𝑁𝑁)𝑁𝑁	NOUN
cana-3487	140	11	therefore	therefore	ADV
cana-3487	140	12	(	(	PUNCT
cana-3487	140	13	𝑥	𝑥	NOUN
cana-3487	140	14	∗	∗	X
cana-3487	140	15	𝑦𝑁𝑁)𝑁𝑁	𝑦𝑁𝑁)𝑁𝑁	NOUN
cana-3487	140	16	=	=	PUNCT
cana-3487	141	1	𝑥𝑁𝑁	𝑥𝑁𝑁	ADJ
cana-3487	141	2	∗	∗	NOUN
cana-3487	141	3	𝑦𝑁𝑁	𝑦𝑁𝑁	NOUN
cana-3487	141	4	for	for	ADP
cana-3487	141	5	all	all	PRON
cana-3487	141	6	𝑥	𝑥	PROPN
cana-3487	141	7	,	,	PUNCT
cana-3487	141	8	𝑦	𝑦	PROPN
cana-3487	141	9	∈	∈	PROPN
cana-3487	141	10	𝑋.	𝑋.	PROPN
cana-3487	141	11	(	(	PUNCT
cana-3487	141	12	3	3	NUM
cana-3487	141	13	)	)	PUNCT
cana-3487	141	14	⇒	⇒	NOUN
cana-3487	141	15	(	(	PUNCT
cana-3487	141	16	4	4	X
cana-3487	141	17	)	)	PUNCT
cana-3487	141	18	assume	assume	VERB
cana-3487	141	19	that	that	SCONJ
cana-3487	141	20	the	the	DET
cana-3487	141	21	condition	condition	NOUN
cana-3487	141	22	(	(	PUNCT
cana-3487	141	23	3	3	X
cana-3487	141	24	)	)	PUNCT
cana-3487	141	25	holds	hold	VERB
cana-3487	141	26	.	.	PUNCT
cana-3487	142	1	let	let	VERB
cana-3487	142	2	𝑥	𝑥	PRON
cana-3487	142	3	,	,	PUNCT
cana-3487	142	4	𝑦	𝑦	PRON
cana-3487	142	5	∈	∈	PROPN
cana-3487	142	6	𝑋.	𝑋.	NOUN
cana-3487	142	7	then	then	ADV
cana-3487	142	8	by	by	ADP
cana-3487	142	9	(	(	PUNCT
cana-3487	142	10	3	3	NUM
cana-3487	142	11	)	)	PUNCT
cana-3487	142	12	,	,	PUNCT
cana-3487	142	13	we	we	PRON
cana-3487	142	14	get	get	VERB
cana-3487	142	15	(	(	PUNCT
cana-3487	142	16	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	142	17	∗	∗	NOUN
cana-3487	142	18	𝑦𝑁)𝑁𝑁	𝑦𝑁)𝑁𝑁	PROPN
cana-3487	143	1	=	=	SYM
cana-3487	144	1	(	(	PUNCT
cana-3487	144	2	𝑦	𝑦	NOUN
cana-3487	144	3	∗	∗	X
cana-3487	144	4	𝑥𝑁𝑁)𝑁𝑁	𝑥𝑁𝑁)𝑁𝑁	NOUN
cana-3487	145	1	=	=	X
cana-3487	145	2	𝑦𝑁𝑁	𝑦𝑁𝑁	NUM
cana-3487	145	3	∗	∗	NOUN
cana-3487	145	4	𝑥𝑁𝑁	𝑥𝑁𝑁	PUNCT
cana-3487	146	1	=	=	PUNCT
cana-3487	146	2	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	146	3	∗	∗	NOUN
cana-3487	146	4	𝑦𝑁𝑁𝑁	𝑦𝑁𝑁𝑁	NOUN
cana-3487	147	1	=	=	SYM
cana-3487	147	2	𝑥𝑁	𝑥𝑁	NOUN
cana-3487	147	3	∗	∗	NOUN
cana-3487	147	4	𝑦𝑁.	𝑦𝑁.	NOUN
cana-3487	147	5	(	(	PUNCT
cana-3487	147	6	4	4	NUM
cana-3487	147	7	)	)	PUNCT
cana-3487	147	8	⇒	⇒	NOUN
cana-3487	147	9	(	(	PUNCT
cana-3487	147	10	2	2	X
cana-3487	147	11	)	)	PUNCT
cana-3487	147	12	assume	assume	VERB
cana-3487	147	13	the	the	DET
cana-3487	147	14	condition	condition	NOUN
cana-3487	147	15	(	(	PUNCT
cana-3487	147	16	4	4	X
cana-3487	147	17	)	)	PUNCT
cana-3487	147	18	holds	hold	VERB
cana-3487	147	19	.	.	PUNCT
cana-3487	148	1	let	let	VERB
cana-3487	148	2	𝑥	𝑥	PRON
cana-3487	148	3	,	,	PUNCT
cana-3487	148	4	𝑦	𝑦	PRON
cana-3487	148	5	∈	∈	PROPN
cana-3487	148	6	𝑋.	𝑋.	NOUN
cana-3487	148	7	then	then	ADV
cana-3487	148	8	by	by	ADP
cana-3487	148	9	(	(	PUNCT
cana-3487	148	10	4	4	NUM
cana-3487	148	11	)	)	PUNCT
cana-3487	148	12	,	,	PUNCT
cana-3487	148	13	we	we	PRON
cana-3487	148	14	get	get	VERB
cana-3487	148	15	(	(	PUNCT
cana-3487	148	16	𝑥𝑁𝑁	𝑥𝑁𝑁	ADV
cana-3487	148	17	∗	∗	NOUN
cana-3487	148	18	𝑦𝑁𝑁)𝑁𝑁	𝑦𝑁𝑁)𝑁𝑁	NOUN
cana-3487	148	19	=	=	PUNCT
cana-3487	149	1	𝑥𝑁𝑁	𝑥𝑁𝑁	PROPN
cana-3487	149	2	∗	∗	NOUN
cana-3487	149	3	𝑦𝑁𝑁.	𝑦𝑁𝑁.	ADP
cana-3487	149	4	hence	hence	ADV
cana-3487	149	5	condition	condition	NOUN
cana-3487	149	6	(	(	PUNCT
cana-3487	149	7	2	2	X
cana-3487	149	8	)	)	PUNCT
cana-3487	149	9	is	be	AUX
cana-3487	149	10	derived	derive	VERB
cana-3487	149	11	.	.	PUNCT
cana-3487	150	1	theorem	theorem	VERB
cana-3487	150	2	2.10	2.10	NUM
cana-3487	150	3	.	.	PUNCT
cana-3487	151	1	let	let	VERB
cana-3487	151	2	(	(	PUNCT
cana-3487	151	3	𝑋	𝑋	NOUN
cana-3487	151	4	,	,	PUNCT
cana-3487	151	5	∗	∗	NOUN
cana-3487	151	6	,	,	PUNCT
cana-3487	151	7	0	0	NUM
cana-3487	151	8	,	,	PUNCT
cana-3487	151	9	1	1	NUM
cana-3487	151	10	)	)	PUNCT
cana-3487	151	11	be	be	AUX
cana-3487	151	12	a	a	DET
cana-3487	151	13	normal	normal	ADJ
cana-3487	151	14	be	be	NOUN
cana-3487	151	15	-	-	PUNCT
cana-3487	151	16	algebra	algebra	NOUN
cana-3487	151	17	.	.	PUNCT
cana-3487	152	1	then	then	ADV
cana-3487	152	2	for	for	ADP
cana-3487	152	3	any	any	DET
cana-3487	152	4	𝑥	𝑥	PROPN
cana-3487	152	5	,	,	PUNCT
cana-3487	152	6	𝑦	𝑦	NOUN
cana-3487	152	7	∈	∈	PROPN
cana-3487	152	8	𝑋	𝑋	NOUN
cana-3487	152	9	,	,	PUNCT
cana-3487	152	10	define	define	VERB
cana-3487	152	11	a	a	DET
cana-3487	152	12	binary	binary	ADJ
cana-3487	152	13	relation	relation	NOUN
cana-3487	152	14	𝜃	𝜃	NOUN
cana-3487	152	15	on	on	ADP
cana-3487	152	16	𝑋	𝑋	PROPN
cana-3487	152	17	as	as	ADP
cana-3487	152	18	(	(	PUNCT
cana-3487	152	19	𝑥	𝑥	PROPN
cana-3487	152	20	,	,	PUNCT
cana-3487	152	21	𝑦	𝑦	NOUN
cana-3487	152	22	)	)	PUNCT
cana-3487	152	23	∈	∈	NOUN
cana-3487	152	24	𝜃	𝜃	NOUN
cana-3487	153	1	if	if	SCONJ
cana-3487	153	2	and	and	CCONJ
cana-3487	153	3	only	only	ADV
cana-3487	153	4	if	if	SCONJ
cana-3487	153	5	𝑥𝑁	𝑥𝑁	ADJ
cana-3487	153	6	=	=	SYM
cana-3487	153	7	𝑦𝑁.	𝑦𝑁.	NOUN
cana-3487	153	8	then	then	ADV
cana-3487	153	9	𝜃	𝜃	PRON
cana-3487	153	10	is	be	AUX
cana-3487	153	11	an	an	DET
cana-3487	153	12	equivalence	equivalence	NOUN
cana-3487	153	13	relation	relation	NOUN
cana-3487	153	14	on	on	ADP
cana-3487	153	15	𝑋	𝑋	PROPN
cana-3487	153	16	and	and	CCONJ
cana-3487	153	17	for	for	ADP
cana-3487	153	18	any	any	DET
cana-3487	153	19	𝑎	𝑎	PROPN
cana-3487	153	20	∈	∈	PROPN
cana-3487	153	21	𝑋	𝑋	NOUN
cana-3487	153	22	,	,	PUNCT
cana-3487	153	23	the	the	DET
cana-3487	153	24	following	follow	VERB
cana-3487	153	25	are	be	AUX
cana-3487	153	26	hold	hold	ADJ
cana-3487	153	27	:	:	PUNCT
cana-3487	153	28	(	(	PUNCT
cana-3487	153	29	1	1	X
cana-3487	153	30	)	)	PUNCT
cana-3487	153	31	the	the	DET
cana-3487	153	32	element	element	NOUN
cana-3487	153	33	𝑎𝑁𝑁	𝑎𝑁𝑁	PROPN
cana-3487	153	34	is	be	AUX
cana-3487	153	35	the	the	DET
cana-3487	153	36	greatest	great	ADJ
cana-3487	153	37	element	element	NOUN
cana-3487	153	38	in	in	ADP
cana-3487	153	39	the	the	DET
cana-3487	153	40	class	class	NOUN
cana-3487	154	1	[	[	X
cana-3487	154	2	𝑎]𝜃	𝑎]𝜃	NOUN
cana-3487	154	3	where[𝑎]𝜃	where[𝑎]𝜃	NOUN
cana-3487	154	4	=	=	SYM
cana-3487	154	5	{	{	PUNCT
cana-3487	154	6	𝑏	𝑏	PROPN
cana-3487	154	7	∈	∈	PROPN
cana-3487	154	8	𝑋/(𝑎	𝑋/(𝑎	PROPN
cana-3487	154	9	,	,	PUNCT
cana-3487	154	10	𝑏	𝑏	NOUN
cana-3487	154	11	)	)	PUNCT
cana-3487	154	12	∈	∈	PROPN
cana-3487	154	13	𝜃	𝜃	PROPN
cana-3487	154	14	}	}	PUNCT
cana-3487	154	15	;	;	PUNCT
cana-3487	154	16	(	(	PUNCT
cana-3487	154	17	2	2	X
cana-3487	154	18	)	)	PUNCT
cana-3487	154	19	the	the	DET
cana-3487	154	20	class	class	NOUN
cana-3487	155	1	[	[	X
cana-3487	155	2	𝑎]𝜃	𝑎]𝜃	NOUN
cana-3487	155	3	contains	contain	VERB
cana-3487	155	4	just	just	ADV
cana-3487	155	5	one	one	NUM
cana-3487	155	6	element	element	NOUN
cana-3487	155	7	from	from	ADP
cana-3487	155	8	𝐶(𝑋	𝐶(𝑋	PROPN
cana-3487	155	9	)	)	PUNCT
cana-3487	155	10	which	which	PRON
cana-3487	155	11	is	be	AUX
cana-3487	155	12	𝑎𝑁𝑁.	𝑎𝑁𝑁.	ADP
cana-3487	155	13	proof	proof	NOUN
cana-3487	155	14	.	.	PUNCT
cana-3487	156	1	clearly	clearly	ADV
cana-3487	156	2	𝜃	𝜃	PRON
cana-3487	156	3	is	be	AUX
cana-3487	156	4	an	an	DET
cana-3487	156	5	equivalence	equivalence	NOUN
cana-3487	156	6	relation	relation	NOUN
cana-3487	156	7	on	on	ADP
cana-3487	156	8	𝑋.	𝑋.	PROPN
cana-3487	156	9	(	(	PUNCT
cana-3487	156	10	1	1	NUM
cana-3487	156	11	)	)	PUNCT
cana-3487	156	12	.	.	PUNCT
cana-3487	157	1	let	let	VERB
cana-3487	157	2	𝑎	𝑎	PRON
cana-3487	157	3	∈	∈	NOUN
cana-3487	157	4	𝑋.	𝑋.	NOUN
cana-3487	157	5	since	since	SCONJ
cana-3487	157	6	𝑋	𝑋	PROPN
cana-3487	157	7	is	be	AUX
cana-3487	157	8	normal	normal	ADJ
cana-3487	157	9	,	,	PUNCT
cana-3487	157	10	we	we	PRON
cana-3487	157	11	get	get	VERB
cana-3487	157	12	𝑎𝑁𝑁𝑁	𝑎𝑁𝑁𝑁	NOUN
cana-3487	157	13	=	=	SYM
cana-3487	157	14	𝑎𝑁.	𝑎𝑁.	NOUN
cana-3487	157	15	hence	hence	ADV
cana-3487	157	16	(	(	PUNCT
cana-3487	157	17	𝑎	𝑎	NOUN
cana-3487	157	18	,	,	PUNCT
cana-3487	157	19	𝑎𝑁𝑁	𝑎𝑁𝑁	NOUN
cana-3487	157	20	)	)	PUNCT
cana-3487	157	21	∈	∈	PROPN
cana-3487	157	22	𝜃	𝜃	PROPN
cana-3487	157	23	,	,	PUNCT
cana-3487	157	24	which	which	PRON
cana-3487	157	25	means	mean	VERB
cana-3487	157	26	𝑎𝑁𝑁	𝑎𝑁𝑁	PROPN
cana-3487	157	27	∈	∈	PROPN
cana-3487	158	1	[	[	X
cana-3487	158	2	𝑎]𝜃.	𝑎]𝜃.	NOUN
cana-3487	158	3	let	let	VERB
cana-3487	158	4	𝑥	𝑥	PRON
cana-3487	158	5	∈	∈	PROPN
cana-3487	159	1	[	[	X
cana-3487	159	2	𝑎]𝜃.	𝑎]𝜃.	VERB
cana-3487	159	3	then	then	ADV
cana-3487	159	4	𝑥𝑁	𝑥𝑁	ADJ
cana-3487	159	5	=	=	PUNCT
cana-3487	159	6	𝑎𝑁.	𝑎𝑁.	NOUN
cana-3487	159	7	since	since	SCONJ
cana-3487	159	8	𝑥	𝑥	NOUN
cana-3487	159	9	≤	≤	NUM
cana-3487	159	10	𝑥𝑁𝑁	𝑥𝑁𝑁	ADV
cana-3487	160	1	=	=	PUNCT
cana-3487	160	2	𝑎𝑁𝑁	𝑎𝑁𝑁	PROPN
cana-3487	160	3	,	,	PUNCT
cana-3487	160	4	we	we	PRON
cana-3487	160	5	get	get	VERB
cana-3487	160	6	that	that	PRON
cana-3487	160	7	𝑎𝑁𝑁	𝑎𝑁𝑁	PROPN
cana-3487	160	8	is	be	AUX
cana-3487	160	9	the	the	DET
cana-3487	160	10	greatest	great	ADJ
cana-3487	160	11	element	element	NOUN
cana-3487	160	12	in	in	ADP
cana-3487	160	13	the	the	DET
cana-3487	160	14	class	class	NOUN
cana-3487	161	1	[	[	X
cana-3487	161	2	𝑎]𝜃.	𝑎]𝜃.	NUM
cana-3487	161	3	(	(	PUNCT
cana-3487	161	4	2	2	NUM
cana-3487	161	5	)	)	PUNCT
cana-3487	161	6	.	.	PUNCT
cana-3487	162	1	let	let	VERB
cana-3487	162	2	𝑏	𝑏	DET
cana-3487	162	3	∈	∈	PROPN
cana-3487	162	4	𝐶(𝑋	𝐶(𝑋	PROPN
cana-3487	162	5	)	)	PUNCT
cana-3487	162	6	such	such	ADJ
cana-3487	162	7	that	that	SCONJ
cana-3487	162	8	𝑏	𝑏	PROPN
cana-3487	162	9	∈	∈	PROPN
cana-3487	162	10	[	[	X
cana-3487	162	11	𝑎]𝜃.	𝑎]𝜃.	VERB
cana-3487	162	12	then	then	ADV
cana-3487	162	13	𝑏𝑁	𝑏𝑁	ADV
cana-3487	162	14	=	=	SYM
cana-3487	162	15	𝑎𝑁.	𝑎𝑁.	VERB
cana-3487	162	16	hence	hence	ADV
cana-3487	162	17	𝑏	𝑏	NOUN
cana-3487	162	18	=	=	PUNCT
cana-3487	162	19	𝑏𝑁𝑁	𝑏𝑁𝑁	NOUN
cana-3487	162	20	=	=	SYM
cana-3487	162	21	𝑎𝑁𝑁.	𝑎𝑁𝑁.	ADP
cana-3487	162	22	therefore	therefore	ADV
cana-3487	162	23	,	,	PUNCT
cana-3487	162	24	the	the	DET
cana-3487	162	25	class	class	NOUN
cana-3487	162	26	[	[	X
cana-3487	162	27	𝑎]𝜃	𝑎]𝜃	NOUN
cana-3487	162	28	contains	contain	VERB
cana-3487	162	29	one	one	NUM
cana-3487	162	30	element	element	NOUN
cana-3487	162	31	from	from	ADP
cana-3487	162	32	𝐶(𝑋	𝐶(𝑋	PROPN
cana-3487	162	33	)	)	PUNCT
cana-3487	162	34	which	which	PRON
cana-3487	162	35	is	be	AUX
cana-3487	162	36	𝑎𝑁𝑁.	𝑎𝑁𝑁.	ADP
cana-3487	162	37	theorem	theorem	NOUN
cana-3487	162	38	2.11	2.11	NUM
cana-3487	162	39	.	.	PUNCT
cana-3487	163	1	let	let	VERB
cana-3487	163	2	(	(	PUNCT
cana-3487	163	3	𝑋	𝑋	NOUN
cana-3487	163	4	,	,	PUNCT
cana-3487	163	5	∗	∗	NOUN
cana-3487	163	6	,	,	PUNCT
cana-3487	163	7	0	0	NUM
cana-3487	163	8	,	,	PUNCT
cana-3487	163	9	1	1	NUM
cana-3487	163	10	)	)	PUNCT
cana-3487	163	11	be	be	AUX
cana-3487	163	12	a	a	DET
cana-3487	163	13	normal	normal	ADJ
cana-3487	163	14	be	be	NOUN
cana-3487	163	15	-	-	PUNCT
cana-3487	163	16	algebra	algebra	NOUN
cana-3487	163	17	which	which	PRON
cana-3487	163	18	satisfies	satisfy	VERB
cana-3487	163	19	the	the	DET
cana-3487	163	20	condition	condition	NOUN
cana-3487	163	21	(	(	PUNCT
cana-3487	163	22	𝑥	𝑥	NOUN
cana-3487	163	23	∗	∗	NOUN
cana-3487	163	24	𝑦)𝑁𝑁	𝑦)𝑁𝑁	NOUN
cana-3487	163	25	=	=	SYM
cana-3487	164	1	𝑥𝑁𝑁	𝑥𝑁𝑁	ADJ
cana-3487	164	2	∗	∗	NOUN
cana-3487	164	3	𝑦𝑁𝑁	𝑦𝑁𝑁	NOUN
cana-3487	164	4	for	for	ADP
cana-3487	164	5	all	all	PRON
cana-3487	164	6	𝑥	𝑥	PROPN
cana-3487	164	7	,	,	PUNCT
cana-3487	164	8	𝑦	𝑦	PRON
cana-3487	164	9	∈	∈	NOUN
cana-3487	164	10	𝑋.then	𝑋.then	ADV
cana-3487	164	11	we	we	PRON
cana-3487	164	12	get	get	VERB
cana-3487	164	13	the	the	DET
cana-3487	164	14	following	following	NOUN
cana-3487	164	15	:	:	PUNCT
cana-3487	164	16	(	(	PUNCT
cana-3487	164	17	1	1	X
cana-3487	164	18	)	)	PUNCT
cana-3487	164	19	𝜃	𝜃	PRON
cana-3487	164	20	is	be	AUX
cana-3487	164	21	a	a	DET
cana-3487	164	22	congruence	congruence	NOUN
cana-3487	164	23	on	on	ADP
cana-3487	164	24	𝑋	𝑋	PROPN
cana-3487	164	25	,	,	PUNCT
cana-3487	164	26	(	(	PUNCT
cana-3487	164	27	2	2	NUM
cana-3487	164	28	)	)	PUNCT
cana-3487	164	29	𝐶(𝑋)is	𝐶(𝑋)is	NOUN
cana-3487	164	30	a	a	DET
cana-3487	164	31	retract	retract	NOUN
cana-3487	164	32	of	of	ADP
cana-3487	164	33	𝑋.	𝑋.	PROPN
cana-3487	164	34	proof	proof	NOUN
cana-3487	164	35	.	.	PUNCT
cana-3487	165	1	(	(	PUNCT
cana-3487	165	2	1	1	NUM
cana-3487	165	3	)	)	PUNCT
cana-3487	165	4	.	.	PUNCT
cana-3487	166	1	let	let	VERB
cana-3487	166	2	(	(	PUNCT
cana-3487	166	3	𝑥	𝑥	NOUN
cana-3487	166	4	,	,	PUNCT
cana-3487	166	5	𝑦	𝑦	X
cana-3487	166	6	)	)	PUNCT
cana-3487	166	7	∈	∈	PROPN
cana-3487	166	8	𝜃	𝜃	X
cana-3487	166	9	and	and	CCONJ
cana-3487	166	10	(	(	PUNCT
cana-3487	166	11	𝑧	𝑧	X
cana-3487	166	12	,	,	PUNCT
cana-3487	166	13	𝑤	𝑤	ADJ
cana-3487	166	14	)	)	PUNCT
cana-3487	166	15	∈	∈	PROPN
cana-3487	166	16	𝜃	𝜃	NOUN
cana-3487	166	17	for	for	ADP
cana-3487	166	18	𝑥	𝑥	PROPN
cana-3487	166	19	,	,	PUNCT
cana-3487	166	20	𝑦	𝑦	NOUN
cana-3487	166	21	,	,	PUNCT
cana-3487	166	22	𝑧	𝑧	VERB
cana-3487	166	23	,	,	PUNCT
cana-3487	166	24	𝑤	𝑤	ADP
cana-3487	166	25	∈	∈	PROPN
cana-3487	166	26	𝑋.	𝑋.	PROPN
cana-3487	166	27	then	then	ADV
cana-3487	166	28	we	we	PRON
cana-3487	166	29	get	get	VERB
cana-3487	166	30	𝑥𝑁	𝑥𝑁	ADJ
cana-3487	166	31	=	=	PUNCT
cana-3487	166	32	𝑦𝑁	𝑦𝑁	NOUN
cana-3487	166	33	and	and	CCONJ
cana-3487	166	34	𝑧𝑁	𝑧𝑁	NOUN
cana-3487	166	35	=	=	SYM
cana-3487	167	1	𝑤𝑁.	𝑤𝑁.	NOUN
cana-3487	167	2	now	now	ADV
cana-3487	167	3	(	(	PUNCT
cana-3487	167	4	𝑥	𝑥	NOUN
cana-3487	167	5	∗	∗	NOUN
cana-3487	167	6	𝑧)𝑁𝑁	𝑧)𝑁𝑁	NOUN
cana-3487	168	1	=	=	PUNCT
cana-3487	169	1	𝑥𝑁𝑁	𝑥𝑁𝑁	ADJ
cana-3487	169	2	∗	∗	NOUN
cana-3487	169	3	𝑧𝑁𝑁	𝑧𝑁𝑁	ADJ
cana-3487	169	4	=	=	PUNCT
cana-3487	169	5	𝑦𝑁𝑁	𝑦𝑁𝑁	NOUN
cana-3487	169	6	∗	∗	NOUN
cana-3487	169	7	𝑤𝑁𝑁	𝑤𝑁𝑁	NOUN
cana-3487	169	8	=	=	SYM
cana-3487	169	9	(	(	PUNCT
cana-3487	169	10	𝑦	𝑦	NOUN
cana-3487	169	11	∗	∗	NOUN
cana-3487	169	12	𝑤)𝑁𝑁.	𝑤)𝑁𝑁.	X
cana-3487	169	13	hence	hence	ADV
cana-3487	169	14	(	(	PUNCT
cana-3487	169	15	𝑥	𝑥	PROPN
cana-3487	169	16	∗	∗	DET
cana-3487	169	17	𝑧	𝑧	PROPN
cana-3487	169	18	,	,	PUNCT
cana-3487	169	19	𝑦	𝑦	PRON
cana-3487	169	20	∗	∗	NOUN
cana-3487	169	21	𝑤	𝑤	NOUN
cana-3487	169	22	)	)	PUNCT
cana-3487	169	23	∈	∈	PROPN
cana-3487	169	24	𝜃.	𝜃.	NOUN
cana-3487	169	25	therefore	therefore	ADV
cana-3487	169	26	𝜃	𝜃	X
cana-3487	169	27	is	be	AUX
cana-3487	169	28	a	a	DET
cana-3487	169	29	congruence	congruence	NOUN
cana-3487	169	30	on	on	ADP
cana-3487	169	31	𝑋.	𝑋.	PROPN
cana-3487	169	32	(	(	PUNCT
cana-3487	169	33	2	2	NUM
cana-3487	169	34	)	)	PUNCT
cana-3487	169	35	.	.	PUNCT
cana-3487	170	1	by	by	ADP
cana-3487	170	2	proposition	proposition	NOUN
cana-3487	170	3	2.9(2	2.9(2	NUM
cana-3487	170	4	)	)	PUNCT
cana-3487	170	5	,	,	PUNCT
cana-3487	170	6	𝐶(𝑋	𝐶(𝑋	ADJ
cana-3487	170	7	)	)	PUNCT
cana-3487	170	8	is	be	AUX
cana-3487	170	9	a	a	DET
cana-3487	170	10	subalgebra	subalgebra	NOUN
cana-3487	170	11	of	of	ADP
cana-3487	170	12	𝑋.	𝑋.	PROPN
cana-3487	170	13	define	define	VERB
cana-3487	170	14	𝜑	𝜑	NOUN
cana-3487	170	15	:	:	PUNCT
cana-3487	170	16	𝑋	𝑋	NOUN
cana-3487	170	17	→	→	SYM
cana-3487	170	18	𝑋	𝑋	NOUN
cana-3487	170	19	by	by	ADP
cana-3487	170	20	𝜑(𝑥	𝜑(𝑥	NOUN
cana-3487	170	21	)	)	PUNCT
cana-3487	170	22	=	=	PUNCT
cana-3487	171	1	𝑥𝑁𝑁	𝑥𝑁𝑁	ADJ
cana-3487	171	2	for	for	ADP
cana-3487	171	3	all𝑥	all𝑥	ADJ
cana-3487	171	4	∈	∈	PROPN
cana-3487	171	5	𝑋.	𝑋.	PROPN
cana-3487	171	6	then	then	ADV
cana-3487	171	7	we	we	PRON
cana-3487	171	8	get	get	VERB
cana-3487	171	9	𝜑(𝑥	𝜑(𝑥	NOUN
cana-3487	171	10	)	)	PUNCT
cana-3487	171	11	=	=	SYM
cana-3487	172	1	𝑥	𝑥	PROPN
cana-3487	172	2	for	for	ADP
cana-3487	172	3	all	all	PRON
cana-3487	172	4	𝑥	𝑥	DET
cana-3487	172	5	∈	∈	NOUN
cana-3487	172	6	𝐶(𝑋	𝐶(𝑋	PROPN
cana-3487	172	7	)	)	PUNCT
cana-3487	172	8	.	.	PUNCT
cana-3487	173	1	clearly	clearly	ADV
cana-3487	173	2	𝜑(𝑥	𝜑(𝑥	VERB
cana-3487	173	3	)	)	PUNCT
cana-3487	173	4	=	=	PUNCT
cana-3487	174	1	𝑥𝑁𝑁	𝑥𝑁𝑁	PROPN
cana-3487	174	2	∈	∈	PROPN
cana-3487	174	3	𝐶(𝑋	𝐶(𝑋	PROPN
cana-3487	174	4	)	)	PUNCT
cana-3487	174	5	for	for	ADP
cana-3487	174	6	all	all	DET
cana-3487	174	7	𝑥	𝑥	DET
cana-3487	174	8	∈	∈	NOUN
cana-3487	174	9	𝑋	𝑋	NOUN
cana-3487	174	10	−	−	PROPN
cana-3487	174	11	𝐶(𝑋	𝐶(𝑋	PROPN
cana-3487	174	12	)	)	PUNCT
cana-3487	174	13	.	.	PUNCT
cana-3487	175	1	therefore	therefore	ADV
cana-3487	175	2	𝐶(𝑋	𝐶(𝑋	ADJ
cana-3487	175	3	)	)	PUNCT
cana-3487	175	4	is	be	AUX
cana-3487	175	5	a	a	DET
cana-3487	175	6	retract	retract	NOUN
cana-3487	175	7	of	of	ADP
cana-3487	175	8	𝑋.	𝑋.	PROPN
cana-3487	175	9	acknowledgement	acknowledgement	NOUN
cana-3487	175	10	:	:	PUNCT
cana-3487	175	11	the	the	DET
cana-3487	175	12	authors	author	NOUN
cana-3487	175	13	would	would	AUX
cana-3487	175	14	like	like	VERB
cana-3487	175	15	to	to	PART
cana-3487	175	16	thank	thank	VERB
cana-3487	175	17	the	the	DET
cana-3487	175	18	editor	editor	NOUN
cana-3487	175	19	and	and	CCONJ
cana-3487	175	20	referees	referee	NOUN
cana-3487	175	21	for	for	ADP
cana-3487	175	22	their	their	PRON
cana-3487	175	23	valuable	valuable	ADJ
cana-3487	175	24	suggestions	suggestion	NOUN
cana-3487	175	25	to	to	PART
cana-3487	175	26	improve	improve	VERB
cana-3487	175	27	this	this	DET
cana-3487	175	28	work	work	NOUN
cana-3487	175	29	.	.	PUNCT
cana-3487	176	1	the	the	DET
cana-3487	176	2	authors	author	NOUN
cana-3487	176	3	also	also	ADV
cana-3487	176	4	thank	thank	VERB
cana-3487	176	5	their	their	PRON
cana-3487	176	6	respective	respective	ADJ
cana-3487	176	7	college	college	NOUN
cana-3487	176	8	managements	management	NOUN
cana-3487	176	9	for	for	ADP
cana-3487	176	10	their	their	PRON
cana-3487	176	11	continuous	continuous	ADJ
cana-3487	176	12	support	support	NOUN
cana-3487	176	13	and	and	CCONJ
cana-3487	176	14	constant	constant	ADJ
cana-3487	176	15	encouragement	encouragement	NOUN
cana-3487	176	16	to	to	PART
cana-3487	176	17	carried	carry	VERB
cana-3487	176	18	out	out	ADP
cana-3487	176	19	the	the	DET
cana-3487	176	20	research	research	NOUN
cana-3487	176	21	work	work	NOUN
cana-3487	176	22	.	.	PUNCT
cana-3487	177	1	conflict	conflict	NOUN
cana-3487	177	2	of	of	ADP
cana-3487	177	3	interest	interest	NOUN
cana-3487	177	4	:	:	PUNCT
cana-3487	177	5	the	the	DET
cana-3487	177	6	authors	author	NOUN
cana-3487	177	7	have	have	VERB
cana-3487	177	8	no	no	DET
cana-3487	177	9	conflict	conflict	NOUN
cana-3487	177	10	of	of	ADP
cana-3487	177	11	interest	interest	NOUN
cana-3487	177	12	related	relate	VERB
cana-3487	177	13	to	to	ADP
cana-3487	177	14	this	this	DET
cana-3487	177	15	publication	publication	NOUN
cana-3487	177	16	.	.	PUNCT
cana-3487	178	1	communications	communication	NOUN
cana-3487	178	2	on	on	ADP
cana-3487	178	3	applied	apply	VERB
cana-3487	178	4	nonlinear	nonlinear	ADJ
cana-3487	178	5	analysis	analysis	NOUN
cana-3487	178	6	issn	issn	NOUN
cana-3487	178	7	:	:	PUNCT
cana-3487	178	8	1074	1074	NUM
cana-3487	178	9	-	-	PUNCT
cana-3487	178	10	133x	133x	NUM
cana-3487	178	11	vol	vol	NOUN
cana-3487	178	12	32	32	NUM
cana-3487	178	13	no	no	NOUN
cana-3487	178	14	.	.	PUNCT
cana-3487	179	1	7s	7	NOUN
cana-3487	179	2	(	(	PUNCT
cana-3487	179	3	2025	2025	NUM
cana-3487	179	4	)	)	PUNCT
cana-3487	179	5	805	805	NUM
cana-3487	179	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-3487	179	7	references	reference	NOUN
cana-3487	179	8	:	:	PUNCT
cana-3487	180	1	[	[	X
cana-3487	180	2	1	1	X
cana-3487	180	3	]	]	X
cana-3487	180	4	s.s	s.s	PROPN
cana-3487	180	5	.	.	PROPN
cana-3487	180	6	ahn	ahn	PROPN
cana-3487	180	7	,	,	PUNCT
cana-3487	180	8	y.h	y.h	PROPN
cana-3487	180	9	.	.	PROPN
cana-3487	180	10	kim	kim	PROPN
cana-3487	180	11	and	and	CCONJ
cana-3487	180	12	j.m	j.m	PROPN
cana-3487	180	13	.	.	PROPN
cana-3487	180	14	ko	ko	PROPN
cana-3487	180	15	,	,	PUNCT
cana-3487	180	16	filters	filter	NOUN
cana-3487	180	17	in	in	ADP
cana-3487	180	18	commutative	commutative	ADJ
cana-3487	180	19	be	be	AUX
cana-3487	180	20	-	-	PUNCT
cana-3487	180	21	algebras	algebra	NOUN
cana-3487	180	22	,	,	PUNCT
cana-3487	180	23	commun	commun	PROPN
cana-3487	180	24	.	.	PUNCT
cana-3487	181	1	korean	korean	PROPN
cana-3487	181	2	.	.	PUNCT
cana-3487	181	3	math	math	PROPN
cana-3487	181	4	.	.	PUNCT
cana-3487	182	1	soc	soc	PROPN
cana-3487	182	2	.	.	PROPN
cana-3487	182	3	,	,	PUNCT
cana-3487	182	4	27	27	NUM
cana-3487	182	5	,	,	PUNCT
cana-3487	182	6	no.2	no.2	PROPN
cana-3487	182	7	(	(	PUNCT
cana-3487	182	8	2012	2012	NUM
cana-3487	182	9	)	)	PUNCT
cana-3487	182	10	,	,	PUNCT
cana-3487	182	11	233	233	NUM
cana-3487	182	12	-	-	SYM
cana-3487	182	13	242	242	NUM
cana-3487	182	14	.	.	PUNCT
cana-3487	183	1	[	[	X
cana-3487	183	2	2	2	NUM
cana-3487	183	3	]	]	PUNCT
cana-3487	183	4	m.	m.	NOUN
cana-3487	183	5	bala	bala	PROPN
cana-3487	183	6	prabhakar	prabhakar	PROPN
cana-3487	183	7	,	,	PUNCT
cana-3487	183	8	s.	s.	PROPN
cana-3487	183	9	kalesha	kalesha	PROPN
cana-3487	183	10	vali	vali	PROPN
cana-3487	183	11	and	and	CCONJ
cana-3487	183	12	m.	m.	PROPN
cana-3487	183	13	sambasiva	sambasiva	PROPN
cana-3487	183	14	rao	rao	PROPN
cana-3487	183	15	,	,	PUNCT
cana-3487	183	16	closed	closed	ADJ
cana-3487	183	17	and	and	CCONJ
cana-3487	183	18	dense	dense	ADJ
cana-3487	183	19	elements	element	NOUN
cana-3487	183	20	of	of	ADP
cana-3487	183	21	bealgebras	bealgebras	ADJ
cana-3487	183	22	,	,	PUNCT
cana-3487	183	23	journal	journal	NOUN
cana-3487	183	24	of	of	ADP
cana-3487	183	25	the	the	DET
cana-3487	183	26	chungcheong	chungcheong	PROPN
cana-3487	183	27	mathematical	mathematical	ADJ
cana-3487	183	28	society	society	NOUN
cana-3487	183	29	,	,	PUNCT
cana-3487	183	30	32	32	NUM
cana-3487	183	31	,	,	PUNCT
cana-3487	183	32	no	no	INTJ
cana-3487	183	33	.	.	PUNCT
cana-3487	184	1	1(2019	1(2019	NUM
cana-3487	184	2	)	)	PUNCT
cana-3487	184	3	,	,	PUNCT
cana-3487	184	4	53	53	NUM
cana-3487	184	5	-	-	SYM
cana-3487	184	6	67	67	NUM
cana-3487	184	7	.	.	PUNCT
cana-3487	185	1	[	[	X
cana-3487	185	2	3	3	X
cana-3487	185	3	]	]	PUNCT
cana-3487	185	4	m.	m.	NOUN
cana-3487	185	5	bala	bala	PROPN
cana-3487	185	6	prabhakar	prabhakar	PROPN
cana-3487	185	7	,	,	PUNCT
cana-3487	185	8	s.	s.	PROPN
cana-3487	185	9	kalesha	kalesha	PROPN
cana-3487	185	10	vali	vali	PROPN
cana-3487	185	11	and	and	CCONJ
cana-3487	185	12	m.	m.	PROPN
cana-3487	185	13	sambasiva	sambasiva	PROPN
cana-3487	185	14	rao	rao	PROPN
cana-3487	185	15	,	,	PUNCT
cana-3487	185	16	ideals	ideal	NOUN
cana-3487	185	17	of	of	ADP
cana-3487	185	18	transitive	transitive	ADJ
cana-3487	185	19	be	be	AUX
cana-3487	185	20	-	-	PUNCT
cana-3487	185	21	algebras	algebra	NOUN
cana-3487	185	22	,	,	PUNCT
cana-3487	185	23	palestine	palestine	PROPN
cana-3487	185	24	journal	journal	PROPN
cana-3487	185	25	of	of	ADP
cana-3487	185	26	mathematics	mathematic	NOUN
cana-3487	185	27	,	,	PUNCT
cana-3487	185	28	10	10	NUM
cana-3487	185	29	,	,	PUNCT
cana-3487	185	30	no.2	no.2	PROPN
cana-3487	185	31	(	(	PUNCT
cana-3487	185	32	2021	2021	NUM
cana-3487	185	33	)	)	PUNCT
cana-3487	185	34	,	,	PUNCT
cana-3487	185	35	852	852	NUM
cana-3487	185	36	-	-	SYM
cana-3487	185	37	862	862	NUM
cana-3487	185	38	.	.	PUNCT
cana-3487	186	1	[	[	X
cana-3487	186	2	4	4	X
cana-3487	186	3	]	]	PUNCT
cana-3487	186	4	m.	m.	NOUN
cana-3487	186	5	bala	bala	PROPN
cana-3487	186	6	prabhakar	prabhakar	PROPN
cana-3487	186	7	,	,	PUNCT
cana-3487	186	8	s.	s.	PROPN
cana-3487	186	9	kalesha	kalesha	PROPN
cana-3487	186	10	vali	vali	PROPN
cana-3487	186	11	and	and	CCONJ
cana-3487	186	12	m.	m.	PROPN
cana-3487	186	13	sambasiva	sambasiva	PROPN
cana-3487	186	14	rao	rao	PROPN
cana-3487	186	15	,	,	PUNCT
cana-3487	186	16	semi	semi	ADJ
cana-3487	186	17	maximal	maximal	ADJ
cana-3487	186	18	ideals	ideal	NOUN
cana-3487	186	19	of	of	ADP
cana-3487	186	20	bealgebras	bealgebras	ADJ
cana-3487	186	21	,	,	PUNCT
cana-3487	186	22	i	i	PROPN
cana-3487	186	23	-	-	PUNCT
cana-3487	186	24	manager	manager	NOUN
cana-3487	186	25	’s	’s	PART
cana-3487	186	26	journal	journal	NOUN
cana-3487	186	27	on	on	ADP
cana-3487	186	28	mathematics	mathematic	NOUN
cana-3487	186	29	,	,	PUNCT
cana-3487	186	30	9	9	NUM
cana-3487	186	31	,	,	PUNCT
cana-3487	186	32	no	no	INTJ
cana-3487	186	33	.	.	PUNCT
cana-3487	187	1	2(2021	2(2021	NUM
cana-3487	187	2	)	)	PUNCT
cana-3487	188	1	,	,	PUNCT
cana-3487	188	2	18	18	NUM
cana-3487	188	3	-	-	SYM
cana-3487	188	4	29	29	NUM
cana-3487	188	5	.	.	PUNCT
cana-3487	189	1	[	[	X
cana-3487	189	2	5	5	X
cana-3487	189	3	]	]	PUNCT
cana-3487	189	4	m.	m.	NOUN
cana-3487	189	5	bala	bala	PROPN
cana-3487	189	6	prabhakar	prabhakar	PROPN
cana-3487	189	7	,	,	PUNCT
cana-3487	189	8	s.	s.	PROPN
cana-3487	189	9	kalesha	kalesha	PROPN
cana-3487	189	10	vali	vali	PROPN
cana-3487	189	11	and	and	CCONJ
cana-3487	189	12	m.	m.	PROPN
cana-3487	189	13	sambasiva	sambasiva	PROPN
cana-3487	189	14	rao	rao	PROPN
cana-3487	189	15	,	,	PUNCT
cana-3487	189	16	maximal	maximal	ADJ
cana-3487	189	17	ideals	ideal	NOUN
cana-3487	189	18	of	of	ADP
cana-3487	189	19	transitive	transitive	ADJ
cana-3487	189	20	bealgebras	bealgebra	NOUN
cana-3487	189	21	,	,	PUNCT
cana-3487	189	22	jordan	jordan	PROPN
cana-3487	189	23	journal	journal	PROPN
cana-3487	189	24	of	of	ADP
cana-3487	189	25	mathematics	mathematics	PROPN
cana-3487	189	26	and	and	CCONJ
cana-3487	189	27	statistics,15	statistics,15	PROPN
cana-3487	189	28	,	,	PUNCT
cana-3487	189	29	no	no	INTJ
cana-3487	189	30	.	.	PUNCT
cana-3487	190	1	1(2022	1(2022	NUM
cana-3487	190	2	)	)	PUNCT
cana-3487	190	3	,	,	PUNCT
cana-3487	190	4	1	1	NUM
cana-3487	190	5	-	-	SYM
cana-3487	190	6	14	14	NUM
cana-3487	190	7	.	.	PUNCT
cana-3487	191	1	[	[	X
cana-3487	191	2	6	6	NUM
cana-3487	191	3	]	]	PUNCT
cana-3487	191	4	m.	m.	NOUN
cana-3487	191	5	bala	bala	PROPN
cana-3487	191	6	prabhakar	prabhakar	PROPN
cana-3487	191	7	,	,	PUNCT
cana-3487	191	8	s.	s.	PROPN
cana-3487	191	9	kalesha	kalesha	PROPN
cana-3487	191	10	vali	vali	PROPN
cana-3487	191	11	and	and	CCONJ
cana-3487	191	12	m.	m.	PROPN
cana-3487	191	13	sambasiva	sambasiva	PROPN
cana-3487	191	14	rao	rao	PROPN
cana-3487	191	15	,	,	PUNCT
cana-3487	191	16	prime	prime	ADJ
cana-3487	191	17	ideals	ideal	NOUN
cana-3487	191	18	of	of	ADP
cana-3487	191	19	transitive	transitive	ADJ
cana-3487	191	20	bealgebras	bealgebra	NOUN
cana-3487	191	21	,	,	PUNCT
cana-3487	191	22	dicussiones	dicussione	NOUN
cana-3487	191	23	mathematicae	mathematicae	VERB
cana-3487	191	24	general	general	ADJ
cana-3487	191	25	algebra	algebra	PROPN
cana-3487	191	26	and	and	CCONJ
cana-3487	191	27	applications	application	NOUN
cana-3487	191	28	,	,	PUNCT
cana-3487	191	29	42	42	NUM
cana-3487	191	30	,	,	PUNCT
cana-3487	191	31	no	no	INTJ
cana-3487	191	32	.	.	PUNCT
cana-3487	192	1	1(2022	1(2022	NUM
cana-3487	192	2	)	)	PUNCT
cana-3487	192	3	,	,	PUNCT
cana-3487	192	4	97	97	NUM
cana-3487	192	5	-	-	SYM
cana-3487	192	6	119	119	NUM
cana-3487	192	7	.	.	PUNCT
cana-3487	193	1	[	[	X
cana-3487	193	2	7	7	X
cana-3487	193	3	]	]	PUNCT
cana-3487	193	4	m.	m.	NOUN
cana-3487	193	5	bala	bala	PROPN
cana-3487	193	6	prabhakar	prabhakar	PROPN
cana-3487	193	7	,	,	PUNCT
cana-3487	193	8	s.	s.	PROPN
cana-3487	193	9	kalesha	kalesha	PROPN
cana-3487	193	10	vali	vali	PROPN
cana-3487	193	11	and	and	CCONJ
cana-3487	193	12	m.	m.	PROPN
cana-3487	193	13	sambasiva	sambasiva	PROPN
cana-3487	193	14	rao	rao	PROPN
cana-3487	193	15	,	,	PUNCT
cana-3487	193	16	generalized	generalize	VERB
cana-3487	193	17	lower	low	ADJ
cana-3487	193	18	sets	set	NOUN
cana-3487	193	19	of	of	ADP
cana-3487	193	20	transitive	transitive	ADJ
cana-3487	193	21	be	be	AUX
cana-3487	193	22	-	-	PUNCT
cana-3487	193	23	algebras	algebra	NOUN
cana-3487	193	24	,	,	PUNCT
cana-3487	193	25	palestine	palestine	PROPN
cana-3487	193	26	journal	journal	PROPN
cana-3487	193	27	of	of	ADP
cana-3487	193	28	mathematics	mathematic	NOUN
cana-3487	193	29	,	,	PUNCT
cana-3487	193	30	11	11	NUM
cana-3487	193	31	,	,	PUNCT
cana-3487	193	32	no.1	no.1	NUM
cana-3487	193	33	(	(	PUNCT
cana-3487	193	34	2022	2022	NUM
cana-3487	193	35	)	)	PUNCT
cana-3487	193	36	,	,	PUNCT
cana-3487	193	37	176	176	NUM
cana-3487	193	38	-	-	SYM
cana-3487	193	39	181	181	NUM
cana-3487	193	40	.	.	PUNCT
cana-3487	194	1	[	[	X
cana-3487	194	2	8	8	NUM
cana-3487	194	3	]	]	X
cana-3487	194	4	r.	r.	PROPN
cana-3487	194	5	borzooei	borzooei	PROPN
cana-3487	194	6	,	,	PUNCT
cana-3487	194	7	a.b	a.b	PROPN
cana-3487	194	8	.	.	PROPN
cana-3487	194	9	saeid	saeid	PROPN
cana-3487	194	10	,	,	PUNCT
cana-3487	194	11	r.	r.	PROPN
cana-3487	194	12	ameri	ameri	PROPN
cana-3487	194	13	and	and	CCONJ
cana-3487	194	14	a.	a.	NOUN
cana-3487	194	15	rezaei	rezaei	PROPN
cana-3487	194	16	,	,	PUNCT
cana-3487	194	17	involutory	involutory	NOUN
cana-3487	194	18	be	be	NOUN
cana-3487	194	19	-	-	PUNCT
cana-3487	194	20	algebras	algebra	NOUN
cana-3487	194	21	,	,	PUNCT
cana-3487	194	22	journal	journal	NOUN
cana-3487	194	23	of	of	ADP
cana-3487	194	24	mathematics	mathematic	NOUN
cana-3487	194	25	and	and	CCONJ
cana-3487	194	26	app	app	PROPN
cana-3487	194	27	.	.	PROPN
cana-3487	194	28	,	,	PUNCT
cana-3487	194	29	37(2014	37(2014	NUM
cana-3487	194	30	)	)	PUNCT
cana-3487	194	31	,	,	PUNCT
cana-3487	194	32	13	13	NUM
cana-3487	194	33	-	-	SYM
cana-3487	194	34	26	26	NUM
cana-3487	194	35	.	.	PUNCT
cana-3487	195	1	[	[	X
cana-3487	195	2	9	9	NUM
cana-3487	195	3	]	]	PUNCT
cana-3487	195	4	z.	z.	PROPN
cana-3487	195	5	ciloglu	ciloglu	PROPN
cana-3487	195	6	and	and	CCONJ
cana-3487	195	7	y.	y.	PROPN
cana-3487	195	8	ceven	ceven	PROPN
cana-3487	195	9	,	,	PUNCT
cana-3487	195	10	commutative	commutative	ADJ
cana-3487	195	11	and	and	CCONJ
cana-3487	195	12	bounded	bound	VERB
cana-3487	195	13	be	be	AUX
cana-3487	195	14	-	-	PUNCT
cana-3487	195	15	algebras	algebra	NOUN
cana-3487	195	16	,	,	PUNCT
cana-3487	195	17	algebra	algebra	NOUN
cana-3487	195	18	,	,	PUNCT
cana-3487	195	19	volume	volume	NOUN
cana-3487	195	20	2013(2013	2013(2013	NUM
cana-3487	195	21	)	)	PUNCT
cana-3487	195	22	,	,	PUNCT
cana-3487	195	23	article	article	NOUN
cana-3487	195	24	i	i	PROPN
cana-3487	195	25	d	d	PROPN
cana-3487	195	26	473714	473714	NUM
cana-3487	195	27	,	,	PUNCT
cana-3487	195	28	5	5	NUM
cana-3487	195	29	pages	page	NOUN
cana-3487	195	30	.	.	PUNCT
cana-3487	196	1	[	[	X
cana-3487	196	2	10	10	NUM
cana-3487	196	3	]	]	PUNCT
cana-3487	196	4	k.	k.	PROPN
cana-3487	196	5	iseki	iseki	PROPN
cana-3487	196	6	and	and	CCONJ
cana-3487	196	7	s.	s.	PROPN
cana-3487	196	8	tanaka	tanaka	PROPN
cana-3487	196	9	,	,	PUNCT
cana-3487	196	10	an	an	DET
cana-3487	196	11	introduction	introduction	NOUN
cana-3487	196	12	to	to	ADP
cana-3487	196	13	the	the	DET
cana-3487	196	14	theory	theory	NOUN
cana-3487	196	15	of	of	ADP
cana-3487	196	16	bck	bck	PROPN
cana-3487	196	17	-	-	PUNCT
cana-3487	196	18	algebras	algebras	PROPN
cana-3487	196	19	,	,	PUNCT
cana-3487	196	20	math	math	NOUN
cana-3487	196	21	.	.	PUNCT
cana-3487	197	1	japon	japon	PROPN
cana-3487	197	2	.	.	PROPN
cana-3487	198	1	23	23	NUM
cana-3487	198	2	,	,	PUNCT
cana-3487	198	3	no.1	no.1	X
cana-3487	198	4	(	(	PUNCT
cana-3487	198	5	1979	1979	NUM
cana-3487	198	6	)	)	PUNCT
cana-3487	198	7	,	,	PUNCT
cana-3487	198	8	1	1	NUM
cana-3487	198	9	-	-	SYM
cana-3487	198	10	26	26	NUM
cana-3487	198	11	.	.	PUNCT
cana-3487	199	1	[	[	X
cana-3487	199	2	11	11	NUM
cana-3487	199	3	]	]	X
cana-3487	199	4	h.s	h.s	PROPN
cana-3487	199	5	.	.	PROPN
cana-3487	199	6	kim	kim	PROPN
cana-3487	199	7	and	and	CCONJ
cana-3487	199	8	y.h	y.h	PROPN
cana-3487	199	9	.	.	PROPN
cana-3487	199	10	kim	kim	PROPN
cana-3487	199	11	,	,	PUNCT
cana-3487	199	12	on	on	ADP
cana-3487	199	13	be	be	AUX
cana-3487	199	14	-	-	PUNCT
cana-3487	199	15	algebras	algebra	NOUN
cana-3487	199	16	,	,	PUNCT
cana-3487	199	17	sci	sci	PROPN
cana-3487	199	18	.	.	PROPN
cana-3487	199	19	math	math	PROPN
cana-3487	199	20	.	.	PUNCT
cana-3487	200	1	jpn	jpn	PROPN
cana-3487	200	2	.	.	PROPN
cana-3487	200	3	,	,	PUNCT
cana-3487	200	4	66	66	NUM
cana-3487	200	5	,	,	PUNCT
cana-3487	200	6	no.1	no.1	X
cana-3487	200	7	(	(	PUNCT
cana-3487	200	8	2006	2006	NUM
cana-3487	200	9	)	)	PUNCT
cana-3487	200	10	,	,	PUNCT
cana-3487	200	11	1299	1299	NUM
cana-3487	200	12	-	-	SYM
cana-3487	200	13	1302	1302	NUM
cana-3487	200	14	.	.	PUNCT
cana-3487	201	1	[	[	X
cana-3487	201	2	12	12	NUM
cana-3487	201	3	]	]	X
cana-3487	201	4	b.l	b.l	PROPN
cana-3487	201	5	.	.	PROPN
cana-3487	201	6	meng	meng	PROPN
cana-3487	201	7	,	,	PUNCT
cana-3487	201	8	on	on	ADP
cana-3487	201	9	filters	filter	NOUN
cana-3487	201	10	in	in	ADP
cana-3487	201	11	be	be	NOUN
cana-3487	201	12	-	-	PUNCT
cana-3487	201	13	algebras	algebra	NOUN
cana-3487	201	14	,	,	PUNCT
cana-3487	201	15	sci	sci	PROPN
cana-3487	201	16	.	.	PROPN
cana-3487	201	17	math	math	PROPN
cana-3487	201	18	.	.	PUNCT
cana-3487	202	1	japon	japon	PROPN
cana-3487	202	2	,	,	PUNCT
cana-3487	202	3	online	online	INTJ
cana-3487	202	4	,	,	PUNCT
cana-3487	202	5	e-2010	e-2010	PROPN
cana-3487	202	6	,	,	PUNCT
cana-3487	202	7	105	105	NUM
cana-3487	202	8	-	-	SYM
cana-3487	202	9	111	111	NUM
cana-3487	202	10	.	.	PUNCT
cana-3487	203	1	[	[	X
cana-3487	203	2	13	13	NUM
cana-3487	203	3	]	]	X
cana-3487	203	4	p.	p.	NOUN
cana-3487	203	5	sun	sun	PROPN
cana-3487	203	6	,	,	PUNCT
cana-3487	203	7	homomorphism	homomorphism	NOUN
cana-3487	203	8	theorems	theorem	VERB
cana-3487	203	9	on	on	ADP
cana-3487	203	10	dual	dual	ADJ
cana-3487	203	11	ideals	ideal	NOUN
cana-3487	203	12	in	in	ADP
cana-3487	203	13	bck	bck	NOUN
cana-3487	203	14	-	-	PUNCT
cana-3487	203	15	algebras	algebras	PROPN
cana-3487	203	16	,	,	PUNCT
cana-3487	203	17	soo	soo	PROPN
cana-3487	203	18	.	.	PUNCT
cana-3487	203	19	j.	j.	PROPN
cana-3487	203	20	math	math	PROPN
cana-3487	203	21	.	.	PROPN
cana-3487	203	22	,	,	PUNCT
cana-3487	203	23	26	26	NUM
cana-3487	203	24	,	,	PUNCT
cana-3487	203	25	no.3	no.3	VERB
cana-3487	203	26	(	(	PUNCT
cana-3487	203	27	2000	2000	NUM
cana-3487	203	28	)	)	PUNCT
cana-3487	203	29	,	,	PUNCT
cana-3487	203	30	309	309	NUM
cana-3487	203	31	-	-	SYM
cana-3487	203	32	316	316	NUM
cana-3487	203	33	.	.	PUNCT
cana-3487	204	1	[	[	X
cana-3487	204	2	14	14	NUM
cana-3487	204	3	]	]	PUNCT
cana-3487	204	4	a.	a.	NOUN
cana-3487	204	5	walendziak	walendziak	PROPN
cana-3487	204	6	,	,	PUNCT
cana-3487	204	7	on	on	ADP
cana-3487	204	8	normal	normal	ADJ
cana-3487	204	9	filters	filter	NOUN
cana-3487	204	10	and	and	CCONJ
cana-3487	204	11	congruence	congruence	NOUN
cana-3487	204	12	relations	relation	NOUN
cana-3487	204	13	in	in	ADP
cana-3487	204	14	be	be	NOUN
cana-3487	204	15	-	-	PUNCT
cana-3487	204	16	algebras	algebra	NOUN
cana-3487	204	17	,	,	PUNCT
cana-3487	204	18	commentationes	commentatione	NOUN
cana-3487	204	19	mathematicae	mathematicae	PROPN
cana-3487	204	20	,	,	PUNCT
cana-3487	204	21	52	52	NUM
cana-3487	204	22	(	(	PUNCT
cana-3487	204	23	2012	2012	NUM
cana-3487	204	24	)	)	PUNCT
cana-3487	204	25	,	,	PUNCT
cana-3487	204	26	199	199	NUM
cana-3487	204	27	-	-	SYM
cana-3487	204	28	205	205	NUM
cana-3487	204	29	.	.	PUNCT