id	sid	tid	token	lemma	pos
cana-1395	1	1	communications	communication	NOUN
cana-1395	1	2	on	on	ADP
cana-1395	1	3	applied	apply	VERB
cana-1395	1	4	nonlinear	nonlinear	ADJ
cana-1395	1	5	analysis	analysis	NOUN
cana-1395	1	6	issn	issn	NOUN
cana-1395	1	7	:	:	PUNCT
cana-1395	1	8	1074	1074	NUM
cana-1395	1	9	-	-	PUNCT
cana-1395	1	10	133x	133x	NUM
cana-1395	1	11	vol	vol	NOUN
cana-1395	1	12	31	31	NUM
cana-1395	1	13	no	no	NOUN
cana-1395	1	14	.	.	PUNCT
cana-1395	2	1	7s	7	NOUN
cana-1395	2	2	(	(	PUNCT
cana-1395	2	3	2024	2024	NUM
cana-1395	2	4	)	)	PUNCT
cana-1395	2	5	531	531	NUM
cana-1395	2	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1395	2	7	on	on	ADP
cana-1395	2	8	neutrosophic	neutrosophic	ADJ
cana-1395	2	9	ideal	ideal	NOUN
cana-1395	2	10	of	of	ADP
cana-1395	2	11	kk	kk	PROPN
cana-1395	2	12	-	-	PUNCT
cana-1395	2	13	algebras	algebras	PROPN
cana-1395	2	14	a.	a.	PROPN
cana-1395	2	15	ibrahim	ibrahim	PROPN
cana-1395	2	16	1	1	PROPN
cana-1395	2	17	and	and	CCONJ
cana-1395	2	18	b.	b.	PROPN
cana-1395	2	19	kavitha	kavitha	PROPN
cana-1395	2	20	2	2	NUM
cana-1395	2	21	1	1	NUM
cana-1395	2	22	assistant	assistant	NOUN
cana-1395	2	23	professor	professor	NOUN
cana-1395	2	24	,	,	PUNCT
cana-1395	2	25	p.g	p.g	PROPN
cana-1395	2	26	.	.	PROPN
cana-1395	2	27	and	and	CCONJ
cana-1395	2	28	research	research	PROPN
cana-1395	2	29	department	department	PROPN
cana-1395	2	30	of	of	ADP
cana-1395	2	31	mathematics	mathematics	PROPN
cana-1395	2	32	,	,	PUNCT
cana-1395	2	33	h.h	h.h	PROPN
cana-1395	2	34	.	.	PROPN
cana-1395	2	35	the	the	DET
cana-1395	2	36	rajah	rajah	NOUN
cana-1395	2	37	’s	’s	PART
cana-1395	2	38	college	college	NOUN
cana-1395	2	39	,	,	PUNCT
cana-1395	2	40	pudukkottai	pudukkottai	NOUN
cana-1395	2	41	,	,	PUNCT
cana-1395	2	42	affiliated	affiliate	VERB
cana-1395	2	43	to	to	PART
cana-1395	2	44	bharathidasan	bharathidasan	VERB
cana-1395	2	45	university	university	NOUN
cana-1395	2	46	,	,	PUNCT
cana-1395	2	47	trichirappalli	trichirappalli	PROPN
cana-1395	2	48	,	,	PUNCT
cana-1395	2	49	tamilnadu	tamilnadu	PROPN
cana-1395	2	50	,	,	PUNCT
cana-1395	2	51	india	india	PROPN
cana-1395	2	52	.	.	PUNCT
cana-1395	2	53	email	email	NOUN
cana-1395	2	54	:	:	PUNCT
cana-1395	2	55	dribra@hhrc.ac.in	dribra@hhrc.ac.in	ADV
cana-1395	2	56	;	;	PUNCT
cana-1395	2	57	dribrahimaadhil@gmail.com	dribrahimaadhil@gmail.com	PROPN
cana-1395	2	58	2	2	NUM
cana-1395	2	59	research	research	NOUN
cana-1395	2	60	scholar	scholar	NOUN
cana-1395	2	61	,	,	PUNCT
cana-1395	2	62	p.g	p.g	PROPN
cana-1395	2	63	.	.	PROPN
cana-1395	2	64	and	and	CCONJ
cana-1395	2	65	research	research	PROPN
cana-1395	2	66	department	department	PROPN
cana-1395	2	67	of	of	ADP
cana-1395	2	68	mathematics	mathematics	PROPN
cana-1395	2	69	,	,	PUNCT
cana-1395	2	70	h.h	h.h	PROPN
cana-1395	2	71	.	.	PROPN
cana-1395	2	72	the	the	DET
cana-1395	2	73	rajah	rajah	NOUN
cana-1395	2	74	’s	’s	PART
cana-1395	2	75	college	college	NOUN
cana-1395	2	76	,	,	PUNCT
cana-1395	2	77	pudukkottai	pudukkottai	NOUN
cana-1395	2	78	,	,	PUNCT
cana-1395	2	79	affiliated	affiliate	VERB
cana-1395	2	80	to	to	PART
cana-1395	2	81	bharathidasan	bharathidasan	VERB
cana-1395	2	82	university	university	NOUN
cana-1395	2	83	,	,	PUNCT
cana-1395	2	84	trichirappalli	trichirappalli	PROPN
cana-1395	2	85	,	,	PUNCT
cana-1395	2	86	tamilnadu	tamilnadu	PROPN
cana-1395	2	87	,	,	PUNCT
cana-1395	2	88	india	india	PROPN
cana-1395	2	89	.	.	PUNCT
cana-1395	3	1	email:bkavitha835@gmail.com	email:bkavitha835@gmail.com	PROPN
cana-1395	3	2	article	article	NOUN
cana-1395	3	3	history	history	NOUN
cana-1395	3	4	:	:	PUNCT
cana-1395	3	5	received	receive	VERB
cana-1395	3	6	:	:	PUNCT
cana-1395	3	7	04	04	NUM
cana-1395	3	8	-	-	PUNCT
cana-1395	3	9	06	06	NUM
cana-1395	3	10	-	-	PUNCT
cana-1395	3	11	2024	2024	NUM
cana-1395	3	12	revised	revise	VERB
cana-1395	3	13	:	:	PUNCT
cana-1395	3	14	03	03	NUM
cana-1395	3	15	-	-	PUNCT
cana-1395	3	16	07	07	NUM
cana-1395	3	17	-	-	PUNCT
cana-1395	3	18	2024	2024	NUM
cana-1395	3	19	accepted	accept	VERB
cana-1395	3	20	:	:	PUNCT
cana-1395	3	21	30	30	NUM
cana-1395	3	22	-	-	SYM
cana-1395	3	23	07	07	NUM
cana-1395	3	24	-	-	PUNCT
cana-1395	3	25	2024	2024	NUM
cana-1395	3	26	abstract	abstract	NOUN
cana-1395	3	27	:	:	PUNCT
cana-1395	3	28	in	in	ADP
cana-1395	3	29	this	this	DET
cana-1395	3	30	paper	paper	NOUN
cana-1395	3	31	,	,	PUNCT
cana-1395	3	32	we	we	PRON
cana-1395	3	33	introduce	introduce	VERB
cana-1395	3	34	the	the	DET
cana-1395	3	35	notion	notion	NOUN
cana-1395	3	36	of	of	ADP
cana-1395	3	37	a	a	DET
cana-1395	3	38	neutrosophic	neutrosophic	ADJ
cana-1395	3	39	ideal	ideal	ADJ
cana-1395	3	40	kk	kk	NOUN
cana-1395	3	41	-	-	PUNCT
cana-1395	3	42	algebra	algebra	NOUN
cana-1395	3	43	.	.	PUNCT
cana-1395	4	1	also	also	ADV
cana-1395	4	2	,	,	PUNCT
cana-1395	4	3	we	we	PRON
cana-1395	4	4	investigate	investigate	VERB
cana-1395	4	5	some	some	DET
cana-1395	4	6	properties	property	NOUN
cana-1395	4	7	of	of	ADP
cana-1395	4	8	the	the	DET
cana-1395	4	9	neutrosophic	neutrosophic	ADJ
cana-1395	4	10	ideal	ideal	NOUN
cana-1395	4	11	with	with	ADP
cana-1395	4	12	suitable	suitable	ADJ
cana-1395	4	13	illustrations	illustration	NOUN
cana-1395	4	14	.	.	PUNCT
cana-1395	5	1	further	far	ADV
cana-1395	5	2	,	,	PUNCT
cana-1395	5	3	we	we	PRON
cana-1395	5	4	describe	describe	VERB
cana-1395	5	5	how	how	SCONJ
cana-1395	5	6	to	to	PART
cana-1395	5	7	deal	deal	VERB
cana-1395	5	8	with	with	ADP
cana-1395	5	9	the	the	DET
cana-1395	5	10	homomorphism	homomorphism	NOUN
cana-1395	5	11	of	of	ADP
cana-1395	5	12	the	the	DET
cana-1395	5	13	image	image	NOUN
cana-1395	5	14	and	and	CCONJ
cana-1395	5	15	the	the	DET
cana-1395	5	16	inverse	inverse	ADJ
cana-1395	5	17	image	image	NOUN
cana-1395	5	18	of	of	ADP
cana-1395	5	19	the	the	DET
cana-1395	5	20	neutrosophic	neutrosophic	ADJ
cana-1395	5	21	ideal	ideal	NOUN
cana-1395	5	22	of	of	ADP
cana-1395	5	23	the	the	DET
cana-1395	5	24	kk	kk	NOUN
cana-1395	5	25	-	-	PUNCT
cana-1395	5	26	algebra	algebra	NOUN
cana-1395	5	27	.	.	PUNCT
cana-1395	6	1	ams	am	NOUN
cana-1395	6	2	mathematical	mathematical	ADJ
cana-1395	6	3	subject	subject	NOUN
cana-1395	6	4	classification	classification	NOUN
cana-1395	6	5	(	(	PUNCT
cana-1395	6	6	2020	2020	NUM
cana-1395	6	7	):	):	PUNCT
cana-1395	6	8	06f35	06f35	NUM
cana-1395	6	9	,	,	PUNCT
cana-1395	6	10	03b47	03b47	NOUN
cana-1395	6	11	,	,	PUNCT
cana-1395	6	12	03b52	03b52	NUM
cana-1395	6	13	,	,	PUNCT
cana-1395	6	14	03e70	03e70	NUM
cana-1395	6	15	.	.	PUNCT
cana-1395	7	1	keywords	keyword	NOUN
cana-1395	7	2	:	:	PUNCT
cana-1395	7	3	kk	kk	NOUN
cana-1395	7	4	-	-	PUNCT
cana-1395	7	5	algebra	algebra	NOUN
cana-1395	7	6	;	;	PUNCT
cana-1395	7	7	ideal	ideal	ADJ
cana-1395	7	8	;	;	PUNCT
cana-1395	7	9	neutrosophic	neutrosophic	ADJ
cana-1395	7	10	set	set	NOUN
cana-1395	7	11	;	;	PUNCT
cana-1395	7	12	neutrosophic	neutrosophic	ADJ
cana-1395	7	13	ideal	ideal	NOUN
cana-1395	7	14	;	;	PUNCT
cana-1395	7	15	neutrosophic	neutrosophic	ADJ
cana-1395	7	16	subalgebra	subalgebra	NOUN
cana-1395	7	17	.	.	PUNCT
cana-1395	8	1	1	1	X
cana-1395	8	2	.	.	X
cana-1395	8	3	introduction	introduction	NOUN
cana-1395	8	4	:	:	PUNCT
cana-1395	8	5	the	the	DET
cana-1395	8	6	concept	concept	NOUN
cana-1395	8	7	of	of	ADP
cana-1395	8	8	kk	kk	PROPN
cana-1395	8	9	-	-	PUNCT
cana-1395	8	10	akgebra	akgebra	PROPN
cana-1395	8	11	was	be	AUX
cana-1395	8	12	first	first	ADV
cana-1395	8	13	developed	develop	VERB
cana-1395	8	14	by	by	ADP
cana-1395	8	15	asawasamrit	asawasamrit	NOUN
cana-1395	8	16	and	and	CCONJ
cana-1395	8	17	a.	a.	NOUN
cana-1395	8	18	sudprasert	sudprasert	NOUN
cana-1395	9	1	[	[	X
cana-1395	9	2	3	3	NUM
cana-1395	9	3	]	]	PUNCT
cana-1395	9	4	.	.	PUNCT
cana-1395	10	1	asawasamrit	asawasamrit	NOUN
cana-1395	10	2	and	and	CCONJ
cana-1395	10	3	a.	a.	NOUN
cana-1395	10	4	sudprasert	sudprasert	NOUN
cana-1395	11	1	[	[	X
cana-1395	11	2	4	4	NUM
cana-1395	11	3	-	-	SYM
cana-1395	11	4	6	6	NUM
cana-1395	11	5	]	]	PUNCT
cana-1395	11	6	established	establish	VERB
cana-1395	11	7	the	the	DET
cana-1395	11	8	concepts	concept	NOUN
cana-1395	11	9	of	of	ADP
cana-1395	11	10	ideals	ideal	NOUN
cana-1395	11	11	,	,	PUNCT
cana-1395	11	12	subalgebras	subalgebras	PROPN
cana-1395	11	13	of	of	ADP
cana-1395	11	14	kk	kk	PROPN
cana-1395	11	15	-	-	PUNCT
cana-1395	11	16	algebras	algebras	PROPN
cana-1395	11	17	,	,	PUNCT
cana-1395	11	18	then	then	ADV
cana-1395	11	19	they	they	PRON
cana-1395	11	20	examined	examine	VERB
cana-1395	11	21	the	the	DET
cana-1395	11	22	relationships	relationship	NOUN
cana-1395	11	23	between	between	ADP
cana-1395	11	24	them	they	PRON
cana-1395	11	25	by	by	ADP
cana-1395	11	26	using	use	VERB
cana-1395	11	27	the	the	DET
cana-1395	11	28	idea	idea	NOUN
cana-1395	11	29	of	of	ADP
cana-1395	11	30	kk	kk	NOUN
cana-1395	11	31	-	-	PUNCT
cana-1395	11	32	algebra	algebra	NOUN
cana-1395	11	33	homomorphism	homomorphism	NOUN
cana-1395	11	34	and	and	CCONJ
cana-1395	11	35	looked	look	VERB
cana-1395	11	36	into	into	ADP
cana-1395	11	37	a	a	DET
cana-1395	11	38	few	few	ADJ
cana-1395	11	39	associated	associated	ADJ
cana-1395	11	40	properties	property	NOUN
cana-1395	11	41	.	.	PUNCT
cana-1395	12	1	in	in	ADP
cana-1395	12	2	1965	1965	NUM
cana-1395	12	3	,	,	PUNCT
cana-1395	12	4	zadeh	zadeh	PROPN
cana-1395	12	5	[	[	X
cana-1395	12	6	11	11	NUM
cana-1395	12	7	]	]	PUNCT
cana-1395	12	8	introduced	introduce	VERB
cana-1395	12	9	the	the	DET
cana-1395	12	10	concept	concept	NOUN
cana-1395	12	11	of	of	ADP
cana-1395	12	12	a	a	DET
cana-1395	12	13	fuzzy	fuzzy	ADJ
cana-1395	12	14	set	set	NOUN
cana-1395	12	15	.	.	PUNCT
cana-1395	13	1	u.	u.	PROPN
cana-1395	13	2	leerawat	leerawat	PROPN
cana-1395	13	3	and	and	CCONJ
cana-1395	13	4	c.	c.	PROPN
cana-1395	13	5	prabpayak	prabpayak	NOUN
cana-1395	14	1	[	[	X
cana-1395	14	2	8	8	NUM
cana-1395	14	3	]	]	PUNCT
cana-1395	14	4	introduced	introduce	VERB
cana-1395	14	5	the	the	DET
cana-1395	14	6	idea	idea	NOUN
cana-1395	14	7	of	of	ADP
cana-1395	14	8	ku	ku	PROPN
cana-1395	14	9	-	-	PUNCT
cana-1395	14	10	algebras	algebras	PROPN
cana-1395	14	11	and	and	CCONJ
cana-1395	14	12	examined	examine	VERB
cana-1395	14	13	certain	certain	ADJ
cana-1395	14	14	associated	associate	VERB
cana-1395	14	15	properties	property	NOUN
cana-1395	14	16	and	and	CCONJ
cana-1395	14	17	provided	provide	VERB
cana-1395	14	18	the	the	DET
cana-1395	14	19	homomorphism	homomorphism	NOUN
cana-1395	14	20	of	of	ADP
cana-1395	14	21	ku	ku	PROPN
cana-1395	14	22	-	-	PUNCT
cana-1395	14	23	algebras	algebras	PROPN
cana-1395	14	24	.	.	PUNCT
cana-1395	15	1	fuzzy	fuzzy	ADJ
cana-1395	15	2	ideals	ideal	NOUN
cana-1395	15	3	of	of	ADP
cana-1395	15	4	kk	kk	PROPN
cana-1395	15	5	-	-	PUNCT
cana-1395	15	6	algebras	algebras	PROPN
cana-1395	15	7	have	have	AUX
cana-1395	15	8	been	be	AUX
cana-1395	15	9	introduced	introduce	VERB
cana-1395	15	10	by	by	ADP
cana-1395	15	11	huda	huda	PROPN
cana-1395	15	12	ali	ali	PROPN
cana-1395	15	13	faleh	faleh	PROPN
cana-1395	15	14	,	,	PUNCT
cana-1395	15	15	dr	dr	PROPN
cana-1395	15	16	.	.	PROPN
cana-1395	15	17	ahmed	ahmed	PROPN
cana-1395	15	18	hamzah	hamzah	PROPN
cana-1395	15	19	abed	abed	PROPN
cana-1395	15	20	,	,	PUNCT
cana-1395	15	21	and	and	CCONJ
cana-1395	15	22	dr	dr	PROPN
cana-1395	15	23	.	.	PROPN
cana-1395	15	24	areej	areej	PROPN
cana-1395	15	25	tawfeeq	tawfeeq	PROPN
cana-1395	15	26	hameed	hameed	PROPN
cana-1395	16	1	[	[	X
cana-1395	16	2	2	2	NUM
cana-1395	16	3	]	]	PUNCT
cana-1395	16	4	.	.	PUNCT
cana-1395	17	1	samarandache	samarandache	PROPN
cana-1395	18	1	[	[	X
cana-1395	18	2	10	10	NUM
cana-1395	18	3	]	]	PUNCT
cana-1395	18	4	in	in	ADP
cana-1395	18	5	1998	1998	NUM
cana-1395	18	6	first	first	ADV
cana-1395	18	7	developed	develop	VERB
cana-1395	18	8	the	the	DET
cana-1395	18	9	concept	concept	NOUN
cana-1395	18	10	of	of	ADP
cana-1395	18	11	neutrosophic	neutrosophic	ADJ
cana-1395	18	12	sets	set	NOUN
cana-1395	18	13	.	.	PUNCT
cana-1395	19	1	the	the	DET
cana-1395	19	2	authors	author	NOUN
cana-1395	19	3	[	[	X
cana-1395	19	4	7	7	X
cana-1395	19	5	]	]	PUNCT
cana-1395	19	6	looked	look	VERB
cana-1395	19	7	into	into	ADP
cana-1395	19	8	some	some	DET
cana-1395	19	9	properties	property	NOUN
cana-1395	19	10	and	and	CCONJ
cana-1395	19	11	presented	present	VERB
cana-1395	19	12	the	the	DET
cana-1395	19	13	notation	notation	NOUN
cana-1395	19	14	of	of	ADP
cana-1395	19	15	a	a	DET
cana-1395	19	16	neutrosophic	neutrosophic	ADJ
cana-1395	19	17	ideal	ideal	NOUN
cana-1395	19	18	of	of	ADP
cana-1395	19	19	bnalgebra	bnalgebra	PROPN
cana-1395	19	20	.	.	PUNCT
cana-1395	20	1	in	in	ADP
cana-1395	20	2	this	this	DET
cana-1395	20	3	work	work	NOUN
cana-1395	20	4	,	,	PUNCT
cana-1395	20	5	we	we	PRON
cana-1395	20	6	first	first	ADV
cana-1395	20	7	introduce	introduce	VERB
cana-1395	20	8	the	the	DET
cana-1395	20	9	notion	notion	NOUN
cana-1395	20	10	of	of	ADP
cana-1395	20	11	the	the	DET
cana-1395	20	12	neutrosophic	neutrosophic	ADJ
cana-1395	20	13	ideal	ideal	NOUN
cana-1395	20	14	of	of	ADP
cana-1395	20	15	kk	kk	PROPN
cana-1395	20	16	-	-	PUNCT
cana-1395	20	17	algebra	algebra	NOUN
cana-1395	20	18	's	's	PART
cana-1395	20	19	and	and	CCONJ
cana-1395	20	20	then	then	ADV
cana-1395	20	21	look	look	VERB
cana-1395	20	22	into	into	ADP
cana-1395	20	23	a	a	DET
cana-1395	20	24	number	number	NOUN
cana-1395	20	25	of	of	ADP
cana-1395	20	26	fundamental	fundamental	ADJ
cana-1395	20	27	properties	property	NOUN
cana-1395	20	28	that	that	PRON
cana-1395	20	29	are	be	AUX
cana-1395	20	30	connected	connect	VERB
cana-1395	20	31	to	to	ADP
cana-1395	20	32	it	it	PRON
cana-1395	20	33	.	.	PUNCT
cana-1395	21	1	for	for	ADP
cana-1395	21	2	the	the	DET
cana-1395	21	3	neutrosophic	neutrosophic	ADJ
cana-1395	21	4	ideal	ideal	NOUN
cana-1395	21	5	,	,	PUNCT
cana-1395	21	6	we	we	PRON
cana-1395	21	7	explain	explain	VERB
cana-1395	21	8	how	how	SCONJ
cana-1395	21	9	to	to	PART
cana-1395	21	10	handle	handle	VERB
cana-1395	21	11	the	the	DET
cana-1395	21	12	image	image	NOUN
cana-1395	21	13	and	and	CCONJ
cana-1395	21	14	inverse	inverse	NOUN
cana-1395	21	15	image	image	NOUN
cana-1395	21	16	homomorphism	homomorphism	NOUN
cana-1395	21	17	.	.	PUNCT
cana-1395	22	1	2	2	X
cana-1395	22	2	.	.	X
cana-1395	22	3	preliminaries	preliminary	NOUN
cana-1395	22	4	in	in	ADP
cana-1395	22	5	order	order	NOUN
cana-1395	22	6	to	to	PART
cana-1395	22	7	better	well	ADV
cana-1395	22	8	understand	understand	VERB
cana-1395	22	9	the	the	DET
cana-1395	22	10	primary	primary	ADJ
cana-1395	22	11	findings	finding	NOUN
cana-1395	22	12	,	,	PUNCT
cana-1395	22	13	we	we	PRON
cana-1395	22	14	go	go	VERB
cana-1395	22	15	over	over	ADP
cana-1395	22	16	the	the	DET
cana-1395	22	17	basic	basic	ADJ
cana-1395	22	18	definitions	definition	NOUN
cana-1395	22	19	of	of	ADP
cana-1395	22	20	kk	kk	NOUN
cana-1395	22	21	-	-	PUNCT
cana-1395	22	22	algebra	algebra	NOUN
cana-1395	22	23	,	,	PUNCT
cana-1395	22	24	ideals	ideal	NOUN
cana-1395	22	25	,	,	PUNCT
cana-1395	22	26	and	and	CCONJ
cana-1395	22	27	ideal	ideal	ADJ
cana-1395	22	28	characteristics	characteristic	NOUN
cana-1395	22	29	in	in	ADP
cana-1395	22	30	this	this	DET
cana-1395	22	31	section	section	NOUN
cana-1395	22	32	.	.	PUNCT
cana-1395	23	1	we	we	PRON
cana-1395	23	2	also	also	ADV
cana-1395	23	3	go	go	VERB
cana-1395	23	4	over	over	ADP
cana-1395	23	5	the	the	DET
cana-1395	23	6	concepts	concept	NOUN
cana-1395	23	7	of	of	ADP
cana-1395	23	8	neutrosophic	neutrosophic	ADJ
cana-1395	23	9	sets	set	NOUN
cana-1395	23	10	and	and	CCONJ
cana-1395	23	11	neutrosophic	neutrosophic	ADJ
cana-1395	23	12	ideals	ideal	NOUN
cana-1395	23	13	of	of	ADP
cana-1395	23	14	kk	kk	NOUN
cana-1395	23	15	-	-	PUNCT
cana-1395	23	16	algebra	algebra	NOUN
cana-1395	23	17	.	.	PUNCT
cana-1395	24	1	definition	definition	NOUN
cana-1395	24	2	2.1[2	2.1[2	NUM
cana-1395	24	3	]	]	X
cana-1395	24	4	:	:	PUNCT
cana-1395	24	5	an	an	DET
cana-1395	24	6	algebra	algebra	NOUN
cana-1395	24	7	(	(	PUNCT
cana-1395	24	8	,	,	PUNCT
cana-1395	24	9	,	,	PUNCT
cana-1395	24	10	0	0	NUM
cana-1395	24	11	)	)	PUNCT
cana-1395	24	12	of	of	ADP
cana-1395	24	13	type	type	NOUN
cana-1395	24	14	(	(	PUNCT
cana-1395	24	15	2	2	NUM
cana-1395	24	16	,	,	PUNCT
cana-1395	24	17	0	0	NUM
cana-1395	24	18	)	)	PUNCT
cana-1395	24	19	is	be	AUX
cana-1395	24	20	called	call	VERB
cana-1395	24	21	a	a	DET
cana-1395	24	22	kk	kk	NOUN
cana-1395	24	23	-	-	PUNCT
cana-1395	24	24	algebra	algebra	NOUN
cana-1395	24	25	,	,	PUNCT
cana-1395	24	26	if	if	SCONJ
cana-1395	24	27	it	it	PRON
cana-1395	24	28	satisfied	satisfy	VERB
cana-1395	24	29	the	the	DET
cana-1395	24	30	following	follow	VERB
cana-1395	24	31	axioms	axiom	NOUN
cana-1395	24	32	for	for	SCONJ
cana-1395	24	33	all	all	DET
cana-1395	24	34	communications	communication	NOUN
cana-1395	24	35	on	on	ADP
cana-1395	24	36	applied	apply	VERB
cana-1395	24	37	nonlinear	nonlinear	ADJ
cana-1395	24	38	analysis	analysis	NOUN
cana-1395	24	39	issn	issn	NOUN
cana-1395	24	40	:	:	PUNCT
cana-1395	24	41	1074	1074	NUM
cana-1395	24	42	-	-	PUNCT
cana-1395	24	43	133x	133x	NUM
cana-1395	24	44	vol	vol	NOUN
cana-1395	24	45	31	31	NUM
cana-1395	24	46	no	no	NOUN
cana-1395	24	47	.	.	PUNCT
cana-1395	25	1	7s	7	NOUN
cana-1395	25	2	(	(	PUNCT
cana-1395	25	3	2024	2024	NUM
cana-1395	25	4	)	)	PUNCT
cana-1395	25	5	532	532	NUM
cana-1395	25	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1395	25	7	(	(	PUNCT
cana-1395	25	8	i	i	NOUN
cana-1395	25	9	)	)	PUNCT
cana-1395	25	10	(	(	PUNCT
cana-1395	25	11	ii	ii	NOUN
cana-1395	25	12	)	)	PUNCT
cana-1395	25	13	(	(	PUNCT
cana-1395	25	14	iii	iii	NOUN
cana-1395	25	15	)	)	PUNCT
cana-1395	25	16	.	.	PUNCT
cana-1395	26	1	definition	definition	NOUN
cana-1395	26	2	2.2[2	2.2[2	NUM
cana-1395	26	3	]	]	X
cana-1395	26	4	:	:	PUNCT
cana-1395	26	5	a	a	DET
cana-1395	26	6	binary	binary	ADJ
cana-1395	26	7	relation	relation	NOUN
cana-1395	26	8	on	on	ADP
cana-1395	26	9	kk	kk	NOUN
cana-1395	26	10	-	-	PUNCT
cana-1395	26	11	algebra	algebra	PROPN
cana-1395	26	12	(	(	PUNCT
cana-1395	26	13	,	,	PUNCT
cana-1395	26	14	,	,	PUNCT
cana-1395	26	15	0	0	NUM
cana-1395	26	16	)	)	PUNCT
cana-1395	26	17	,	,	PUNCT
cana-1395	26	18	such	such	ADJ
cana-1395	26	19	that	that	SCONJ
cana-1395	26	20	if	if	SCONJ
cana-1395	26	21	and	and	CCONJ
cana-1395	26	22	only	only	ADV
cana-1395	26	23	if	if	SCONJ
cana-1395	26	24	for	for	ADP
cana-1395	26	25	all	all	DET
cana-1395	26	26	proposition	proposition	NOUN
cana-1395	26	27	2.3[2	2.3[2	NUM
cana-1395	26	28	]	]	X
cana-1395	26	29	:	:	PUNCT
cana-1395	26	30	a	a	DET
cana-1395	26	31	any	any	DET
cana-1395	26	32	kk	kk	NOUN
cana-1395	26	33	-	-	PUNCT
cana-1395	26	34	algebra	algebra	PROPN
cana-1395	26	35	(	(	PUNCT
cana-1395	26	36	,	,	PUNCT
cana-1395	26	37	,	,	PUNCT
cana-1395	26	38	0	0	NUM
cana-1395	26	39	)	)	PUNCT
cana-1395	26	40	,	,	PUNCT
cana-1395	26	41	the	the	DET
cana-1395	26	42	following	follow	VERB
cana-1395	26	43	axioms	axiom	NOUN
cana-1395	26	44	are	be	AUX
cana-1395	26	45	hold	hold	ADJ
cana-1395	26	46	for	for	SCONJ
cana-1395	26	47	all	all	PRON
cana-1395	26	48	(	(	PUNCT
cana-1395	26	49	i	i	NOUN
cana-1395	26	50	)	)	PUNCT
cana-1395	26	51	(	(	PUNCT
cana-1395	26	52	ii	ii	NOUN
cana-1395	26	53	)	)	PUNCT
cana-1395	26	54	(	(	PUNCT
cana-1395	26	55	iii	iii	NOUN
cana-1395	26	56	)	)	PUNCT
cana-1395	26	57	(	(	PUNCT
cana-1395	26	58	iv	iv	X
cana-1395	26	59	)	)	PUNCT
cana-1395	26	60	(	(	PUNCT
cana-1395	26	61	)	)	PUNCT
cana-1395	26	62	(	(	PUNCT
cana-1395	26	63	v	v	NOUN
cana-1395	26	64	)	)	PUNCT
cana-1395	26	65	(	(	PUNCT
cana-1395	26	66	vi	vi	NOUN
cana-1395	26	67	)	)	PUNCT
cana-1395	26	68	(	(	PUNCT
cana-1395	26	69	vii	vii	PROPN
cana-1395	26	70	)	)	PUNCT
cana-1395	26	71	implies	imply	VERB
cana-1395	26	72	(	(	PUNCT
cana-1395	26	73	viii	viii	NOUN
cana-1395	26	74	)	)	PUNCT
cana-1395	26	75	implies	imply	VERB
cana-1395	26	76	.	.	PUNCT
cana-1395	27	1	definition	definition	NOUN
cana-1395	27	2	2.4[2	2.4[2	NUM
cana-1395	27	3	]	]	X
cana-1395	27	4	:	:	PUNCT
cana-1395	27	5	let	let	VERB
cana-1395	27	6	(	(	PUNCT
cana-1395	27	7	,	,	PUNCT
cana-1395	27	8	,	,	PUNCT
cana-1395	27	9	0	0	X
cana-1395	27	10	)	)	PUNCT
cana-1395	27	11	be	be	AUX
cana-1395	27	12	a	a	DET
cana-1395	27	13	kk	kk	NOUN
cana-1395	27	14	-	-	PUNCT
cana-1395	27	15	algebra	algebra	NOUN
cana-1395	27	16	,	,	PUNCT
cana-1395	27	17	and	and	CCONJ
cana-1395	27	18	let	let	VERB
cana-1395	27	19	be	be	AUX
cana-1395	27	20	a	a	DET
cana-1395	27	21	nonempty	nonempty	ADJ
cana-1395	27	22	subset	subset	NOUN
cana-1395	27	23	of	of	ADP
cana-1395	27	24	is	be	AUX
cana-1395	27	25	called	call	VERB
cana-1395	27	26	subalgebra	subalgebra	NOUN
cana-1395	27	27	of	of	ADP
cana-1395	27	28	if	if	SCONJ
cana-1395	27	29	for	for	ADP
cana-1395	27	30	all	all	DET
cana-1395	27	31	definition	definition	NOUN
cana-1395	27	32	2.5[2	2.5[2	NUM
cana-1395	27	33	]	]	X
cana-1395	27	34	:	:	PUNCT
cana-1395	27	35	an	an	DET
cana-1395	27	36	ideal	ideal	NOUN
cana-1395	27	37	of	of	ADP
cana-1395	27	38	is	be	AUX
cana-1395	27	39	a	a	DET
cana-1395	27	40	non	non	ADJ
cana-1395	27	41	-	-	ADJ
cana-1395	27	42	empty	empty	ADJ
cana-1395	27	43	subset	subset	NOUN
cana-1395	27	44	of	of	ADP
cana-1395	27	45	a	a	DET
cana-1395	27	46	kk	kk	NOUN
cana-1395	27	47	-	-	PUNCT
cana-1395	27	48	algebra	algebra	PROPN
cana-1395	27	49	(	(	PUNCT
cana-1395	27	50	,	,	PUNCT
cana-1395	27	51	,	,	PUNCT
cana-1395	27	52	0	0	NUM
cana-1395	27	53	)	)	PUNCT
cana-1395	27	54	that	that	PRON
cana-1395	27	55	satisfies	satisfy	VERB
cana-1395	27	56	the	the	DET
cana-1395	27	57	following	follow	VERB
cana-1395	27	58	for	for	ADP
cana-1395	27	59	all	all	PRON
cana-1395	27	60	(	(	PUNCT
cana-1395	27	61	i	i	NOUN
cana-1395	27	62	)	)	PUNCT
cana-1395	27	63	(	(	PUNCT
cana-1395	27	64	ii	ii	NOUN
cana-1395	27	65	)	)	PUNCT
cana-1395	27	66	and	and	CCONJ
cana-1395	27	67	imply	imply	VERB
cana-1395	27	68	.	.	PUNCT
cana-1395	28	1	proposition	proposition	NOUN
cana-1395	28	2	2.6[2	2.6[2	NUM
cana-1395	28	3	]	]	PUNCT
cana-1395	28	4	:	:	PUNCT
cana-1395	28	5	each	each	DET
cana-1395	28	6	ideal	ideal	NOUN
cana-1395	28	7	in	in	ADP
cana-1395	28	8	the	the	DET
cana-1395	28	9	kk	kk	NOUN
cana-1395	28	10	-	-	PUNCT
cana-1395	28	11	algebra	algebra	PROPN
cana-1395	28	12	(	(	PUNCT
cana-1395	28	13	,	,	PUNCT
cana-1395	28	14	,	,	PUNCT
cana-1395	28	15	0	0	NUM
cana-1395	28	16	)	)	PUNCT
cana-1395	28	17	is	be	AUX
cana-1395	28	18	a	a	DET
cana-1395	28	19	subalgebra	subalgebra	NOUN
cana-1395	28	20	of	of	ADP
cana-1395	28	21	.	.	PUNCT
cana-1395	29	1	proposition	proposition	NOUN
cana-1395	29	2	2.7[2	2.7[2	NUM
cana-1395	29	3	]	]	PUNCT
cana-1395	29	4	:	:	PUNCT
cana-1395	29	5	let	let	VERB
cana-1395	29	6	a	a	DET
cana-1395	29	7	family	family	NOUN
cana-1395	29	8	of	of	ADP
cana-1395	29	9	ideals	ideal	NOUN
cana-1395	29	10	of	of	ADP
cana-1395	29	11	kk	kk	PROPN
cana-1395	29	12	-	-	PUNCT
cana-1395	29	13	algebra	algebra	NOUN
cana-1395	29	14	be	be	AUX
cana-1395	29	15	denoted	denote	VERB
cana-1395	29	16	by	by	ADP
cana-1395	29	17	.the	.the	DET
cana-1395	29	18	intersection	intersection	NOUN
cana-1395	29	19	of	of	ADP
cana-1395	29	20	any	any	DET
cana-1395	29	21	set	set	NOUN
cana-1395	29	22	of	of	ADP
cana-1395	29	23	ideals	ideal	NOUN
cana-1395	29	24	of	of	ADP
cana-1395	29	25	kk	kk	PROPN
cana-1395	29	26	-	-	PUNCT
cana-1395	29	27	algebra	algebra	PROPN
cana-1395	29	28	is	be	AUX
cana-1395	29	29	also	also	ADV
cana-1395	29	30	is	be	AUX
cana-1395	29	31	an	an	DET
cana-1395	29	32	ideal	ideal	NOUN
cana-1395	29	33	.	.	PUNCT
cana-1395	30	1	definition	definition	NOUN
cana-1395	30	2	2.8[2	2.8[2	NUM
cana-1395	30	3	]	]	PUNCT
cana-1395	30	4	:	:	PUNCT
cana-1395	30	5	let	let	VERB
cana-1395	30	6	(	(	PUNCT
cana-1395	30	7	and	and	CCONJ
cana-1395	30	8	(	(	PUNCT
cana-1395	30	9	,	,	PUNCT
cana-1395	30	10	,	,	PUNCT
cana-1395	30	11	)	)	PUNCT
cana-1395	30	12	be	be	AUX
cana-1395	30	13	an	an	DET
cana-1395	30	14	any	any	DET
cana-1395	30	15	two	two	NUM
cana-1395	30	16	kk	kk	NOUN
cana-1395	30	17	-	-	PUNCT
cana-1395	30	18	algebras	algebras	PROPN
cana-1395	30	19	.	.	PUNCT
cana-1395	31	1	then	then	ADV
cana-1395	31	2	,	,	PUNCT
cana-1395	31	3	the	the	DET
cana-1395	31	4	mapping	mapping	NOUN
cana-1395	31	5	is	be	AUX
cana-1395	31	6	called	call	VERB
cana-1395	31	7	homomorphism	homomorphism	NOUN
cana-1395	31	8	,	,	PUNCT
cana-1395	31	9	if	if	SCONJ
cana-1395	31	10	it	it	PRON
cana-1395	31	11	satisfied	satisfied	ADJ
cana-1395	31	12	for	for	ADP
cana-1395	31	13	all	all	DET
cana-1395	31	14	definition	definition	NOUN
cana-1395	31	15	2.9	2.9	NUM
cana-1395	32	1	[	[	NOUN
cana-1395	32	2	10	10	NUM
cana-1395	32	3	]	]	PUNCT
cana-1395	32	4	:	:	PUNCT
cana-1395	32	5	let	let	AUX
cana-1395	32	6	be	be	AUX
cana-1395	32	7	the	the	DET
cana-1395	32	8	discourse	discourse	NOUN
cana-1395	32	9	universe	universe	NOUN
cana-1395	32	10	.	.	PUNCT
cana-1395	33	1	a	a	DET
cana-1395	33	2	neutrosophic	neutrosophic	ADJ
cana-1395	33	3	set	set	NOUN
cana-1395	33	4	of	of	ADP
cana-1395	33	5	is	be	AUX
cana-1395	33	6	characterized	characterize	VERB
cana-1395	33	7	by	by	ADP
cana-1395	33	8	a	a	DET
cana-1395	33	9	truth	truth	NOUN
cana-1395	33	10	membership	membership	NOUN
cana-1395	33	11	function	function	NOUN
cana-1395	33	12	,	,	PUNCT
cana-1395	33	13	an	an	DET
cana-1395	33	14	indeterminacy	indeterminacy	NOUN
cana-1395	33	15	membership	membership	NOUN
cana-1395	33	16	function	function	NOUN
cana-1395	33	17	,	,	PUNCT
cana-1395	33	18	and	and	CCONJ
cana-1395	33	19	a	a	DET
cana-1395	33	20	falsity	falsity	NOUN
cana-1395	33	21	membership	membership	NOUN
cana-1395	33	22	function	function	NOUN
cana-1395	33	23	,	,	PUNCT
cana-1395	33	24	where	where	SCONJ
cana-1395	33	25	,	,	PUNCT
cana-1395	33	26	,	,	PUNCT
cana-1395	33	27	and	and	CCONJ
cana-1395	33	28	are	be	AUX
cana-1395	33	29	real	real	ADJ
cana-1395	33	30	standard	standard	ADJ
cana-1395	33	31	elements	element	NOUN
cana-1395	33	32	of	of	ADP
cana-1395	33	33	[	[	X
cana-1395	33	34	0	0	NUM
cana-1395	33	35	,	,	PUNCT
cana-1395	33	36	1	1	NUM
cana-1395	33	37	]	]	PUNCT
cana-1395	33	38	.	.	PUNCT
cana-1395	34	1	it	it	PRON
cana-1395	34	2	can	can	AUX
cana-1395	34	3	be	be	AUX
cana-1395	34	4	written	write	VERB
cana-1395	34	5	as	as	ADP
cana-1395	34	6	{	{	PUNCT
cana-1395	34	7	)	)	PUNCT
cana-1395	34	8	where	where	SCONJ
cana-1395	34	9	]	]	PUNCT
cana-1395	34	10	[	[	PUNCT
cana-1395	34	11	.	.	PUNCT
cana-1395	35	1	there	there	PRON
cana-1395	35	2	is	be	VERB
cana-1395	35	3	no	no	DET
cana-1395	35	4	restriction	restriction	NOUN
cana-1395	35	5	on	on	ADP
cana-1395	35	6	the	the	DET
cana-1395	35	7	sum	sum	NOUN
cana-1395	35	8	of	of	ADP
cana-1395	35	9	and	and	CCONJ
cana-1395	35	10	so	so	ADV
cana-1395	35	11	+	+	NUM
cana-1395	35	12	definition	definition	NOUN
cana-1395	35	13	2.10	2.10	NUM
cana-1395	35	14	[	[	X
cana-1395	35	15	7	7	NUM
cana-1395	35	16	]	]	X
cana-1395	35	17	:	:	PUNCT
cana-1395	35	18	a	a	DET
cana-1395	35	19	neutrosophic	neutrosophic	ADJ
cana-1395	35	20	set	set	NOUN
cana-1395	35	21	of	of	ADP
cana-1395	35	22	bn	bn	NOUN
cana-1395	35	23	-	-	PUNCT
cana-1395	35	24	algebra	algebra	NOUN
cana-1395	35	25	(	(	PUNCT
cana-1395	35	26	,	,	PUNCT
cana-1395	35	27	,	,	PUNCT
cana-1395	35	28	0	0	NUM
cana-1395	35	29	)	)	PUNCT
cana-1395	35	30	,	,	PUNCT
cana-1395	35	31	if	if	SCONJ
cana-1395	35	32	is	be	AUX
cana-1395	35	33	called	call	VERB
cana-1395	35	34	an	an	DET
cana-1395	35	35	neutrosophic	neutrosophic	ADJ
cana-1395	35	36	ideal	ideal	NOUN
cana-1395	35	37	of	of	ADP
cana-1395	35	38	bn	bn	NOUN
cana-1395	35	39	-	-	PUNCT
cana-1395	35	40	algebra	algebra	NOUN
cana-1395	35	41	then	then	ADV
cana-1395	35	42	it	it	PRON
cana-1395	35	43	satisfies	satisfy	VERB
cana-1395	35	44	the	the	DET
cana-1395	35	45	following	follow	VERB
cana-1395	35	46	for	for	ADP
cana-1395	35	47	all	all	DET
cana-1395	35	48	communications	communication	NOUN
cana-1395	35	49	on	on	ADP
cana-1395	35	50	applied	apply	VERB
cana-1395	35	51	nonlinear	nonlinear	ADJ
cana-1395	35	52	analysis	analysis	NOUN
cana-1395	35	53	issn	issn	NOUN
cana-1395	35	54	:	:	PUNCT
cana-1395	35	55	1074	1074	NUM
cana-1395	35	56	-	-	PUNCT
cana-1395	35	57	133x	133x	NUM
cana-1395	35	58	vol	vol	NOUN
cana-1395	35	59	31	31	NUM
cana-1395	35	60	no	no	NOUN
cana-1395	35	61	.	.	PUNCT
cana-1395	36	1	7s	7	NOUN
cana-1395	36	2	(	(	PUNCT
cana-1395	36	3	2024	2024	NUM
cana-1395	36	4	)	)	PUNCT
cana-1395	36	5	533	533	NUM
cana-1395	36	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1395	36	7	(	(	PUNCT
cana-1395	36	8	i	i	NOUN
cana-1395	36	9	)	)	PUNCT
cana-1395	36	10	(	(	PUNCT
cana-1395	36	11	ii	ii	PROPN
cana-1395	36	12	)	)	PUNCT
cana-1395	36	13	;	;	PUNCT
cana-1395	36	14	definition	definition	NOUN
cana-1395	36	15	2.11[7	2.11[7	NUM
cana-1395	36	16	]	]	PUNCT
cana-1395	36	17	:	:	PUNCT
cana-1395	36	18	let	let	AUX
cana-1395	36	19	be	be	AUX
cana-1395	36	20	any	any	DET
cana-1395	36	21	neutrosophic	neutrosophic	ADJ
cana-1395	36	22	set	set	NOUN
cana-1395	36	23	.	.	PUNCT
cana-1395	37	1	if	if	SCONJ
cana-1395	37	2	for	for	ADP
cana-1395	37	3	any	any	DET
cana-1395	37	4	[	[	NOUN
cana-1395	37	5	0	0	NUM
cana-1395	37	6	,	,	PUNCT
cana-1395	37	7	3	3	NUM
cana-1395	37	8	]	]	PUNCT
cana-1395	37	9	,	,	PUNCT
cana-1395	37	10	then	then	ADV
cana-1395	37	11	an	an	DET
cana-1395	37	12	neutrosophic	neutrosophic	ADJ
cana-1395	37	13	level	level	NOUN
cana-1395	37	14	set	set	NOUN
cana-1395	37	15	of	of	ADP
cana-1395	37	16	is	be	AUX
cana-1395	37	17	defined	define	VERB
cana-1395	37	18	by	by	ADP
cana-1395	37	19	=	=	PROPN
cana-1395	37	20	3	3	X
cana-1395	37	21	.	.	NOUN
cana-1395	37	22	main	main	ADJ
cana-1395	37	23	results	result	NOUN
cana-1395	37	24	this	this	DET
cana-1395	37	25	part	part	NOUN
cana-1395	37	26	presents	present	VERB
cana-1395	37	27	the	the	DET
cana-1395	37	28	key	key	ADJ
cana-1395	37	29	findings	finding	NOUN
cana-1395	37	30	of	of	ADP
cana-1395	37	31	the	the	DET
cana-1395	37	32	study	study	NOUN
cana-1395	37	33	,	,	PUNCT
cana-1395	37	34	starting	start	VERB
cana-1395	37	35	with	with	ADP
cana-1395	37	36	an	an	DET
cana-1395	37	37	idea	idea	NOUN
cana-1395	37	38	of	of	ADP
cana-1395	37	39	neutrosophic	neutrosophic	ADJ
cana-1395	37	40	ideal	ideal	NOUN
cana-1395	37	41	of	of	ADP
cana-1395	37	42	kk	kk	PROPN
cana-1395	37	43	algebra	algebra	PROPN
cana-1395	37	44	,	,	PUNCT
cana-1395	37	45	and	and	CCONJ
cana-1395	37	46	an	an	DET
cana-1395	37	47	explanation	explanation	NOUN
cana-1395	37	48	of	of	ADP
cana-1395	37	49	neutrosophic	neutrosophic	ADJ
cana-1395	37	50	ideal	ideal	NOUN
cana-1395	37	51	.	.	PUNCT
cana-1395	38	1	furthermore	furthermore	ADV
cana-1395	38	2	,	,	PUNCT
cana-1395	38	3	various	various	ADJ
cana-1395	38	4	properties	property	NOUN
cana-1395	38	5	of	of	ADP
cana-1395	38	6	the	the	DET
cana-1395	38	7	neutrosophic	neutrosophic	ADJ
cana-1395	38	8	ideal	ideal	NOUN
cana-1395	38	9	in	in	ADP
cana-1395	38	10	kk	kk	PROPN
cana-1395	38	11	-	-	PUNCT
cana-1395	38	12	algebra	algebra	NOUN
cana-1395	38	13	are	be	AUX
cana-1395	38	14	investigated	investigate	VERB
cana-1395	38	15	.	.	PUNCT
cana-1395	39	1	definition	definition	NOUN
cana-1395	39	2	3.1	3.1	NUM
cana-1395	39	3	:	:	PUNCT
cana-1395	39	4	a	a	DET
cana-1395	39	5	non	non	X
cana-1395	39	6	empty	empty	ADJ
cana-1395	39	7	subset	subset	NOUN
cana-1395	39	8	of	of	ADP
cana-1395	39	9	a	a	DET
cana-1395	39	10	kk	kk	NOUN
cana-1395	39	11	-	-	PUNCT
cana-1395	39	12	algebra	algebra	PROPN
cana-1395	39	13	is	be	AUX
cana-1395	39	14	called	call	VERB
cana-1395	39	15	neutrosophic	neutrosophic	ADJ
cana-1395	39	16	ideal	ideal	NOUN
cana-1395	39	17	of	of	ADP
cana-1395	39	18	kk	kk	PROPN
cana-1395	39	19	-	-	NOUN
cana-1395	39	20	algebra	algebra	NOUN
cana-1395	39	21	,	,	PUNCT
cana-1395	39	22	if	if	SCONJ
cana-1395	39	23	it	it	PRON
cana-1395	39	24	satisfies	satisfy	VERB
cana-1395	39	25	the	the	DET
cana-1395	39	26	following	follow	VERB
cana-1395	39	27	for	for	ADP
cana-1395	39	28	all	all	PRON
cana-1395	39	29	(	(	PUNCT
cana-1395	39	30	i	i	NOUN
cana-1395	39	31	)	)	PUNCT
cana-1395	39	32	,	,	PUNCT
cana-1395	39	33	(	(	PUNCT
cana-1395	39	34	ii	ii	NOUN
cana-1395	39	35	)	)	PUNCT
cana-1395	39	36	;	;	PUNCT
cana-1395	39	37	;	;	PUNCT
cana-1395	39	38	.	.	PUNCT
cana-1395	40	1	example	example	NOUN
cana-1395	40	2	3.2	3.2	NUM
cana-1395	40	3	:	:	PUNCT
cana-1395	40	4	consider	consider	VERB
cana-1395	40	5	a	a	DET
cana-1395	40	6	set	set	NOUN
cana-1395	40	7	define	define	VERB
cana-1395	40	8	a	a	DET
cana-1395	40	9	binary	binary	ADJ
cana-1395	40	10	operation	operation	NOUN
cana-1395	40	11	on	on	ADP
cana-1395	40	12	given	give	VERB
cana-1395	40	13	by	by	ADP
cana-1395	40	14	the	the	DET
cana-1395	40	15	following	follow	VERB
cana-1395	40	16	table	table	NOUN
cana-1395	40	17	3.1	3.1	NUM
cana-1395	40	18	,	,	PUNCT
cana-1395	40	19	and	and	CCONJ
cana-1395	40	20	neutrosophic	neutrosophic	ADJ
cana-1395	40	21	set	set	VERB
cana-1395	40	22	by	by	ADP
cana-1395	40	23	the	the	DET
cana-1395	40	24	table	table	NOUN
cana-1395	40	25	3.2	3.2	NUM
cana-1395	40	26	as	as	SCONJ
cana-1395	40	27	shown	show	VERB
cana-1395	40	28	below	below	ADP
cana-1395	40	29	:	:	PUNCT
cana-1395	40	30	0.7	0.7	NUM
cana-1395	40	31	0.5	0.5	NUM
cana-1395	40	32	0.5	0.5	NUM
cana-1395	40	33	0.5	0.5	NUM
cana-1395	40	34	0.5	0.5	NUM
cana-1395	40	35	0.6	0.6	NUM
cana-1395	40	36	0.4	0.4	NUM
cana-1395	40	37	0.4	0.4	NUM
cana-1395	40	38	0.4	0.4	NUM
cana-1395	40	39	0.4	0.4	NUM
cana-1395	40	40	0.2	0.2	NUM
cana-1395	40	41	0.1	0.1	NUM
cana-1395	40	42	0.1	0.1	NUM
cana-1395	40	43	0.1	0.1	NUM
cana-1395	40	44	0.1	0.1	NUM
cana-1395	40	45	table	table	NOUN
cana-1395	40	46	3.1	3.1	NUM
cana-1395	40	47	:	:	PUNCT
cana-1395	40	48	operation	operation	NOUN
cana-1395	40	49	table	table	NOUN
cana-1395	40	50	3.2	3.2	NUM
cana-1395	40	51	:	:	PUNCT
cana-1395	40	52	neutrosophic	neutrosophic	PROPN
cana-1395	40	53	set	set	NOUN
cana-1395	40	54	it	it	PRON
cana-1395	40	55	is	be	AUX
cana-1395	40	56	easily	easily	ADV
cana-1395	40	57	verified	verify	VERB
cana-1395	40	58	that	that	PRON
cana-1395	40	59	is	be	AUX
cana-1395	40	60	a	a	DET
cana-1395	40	61	neutrosophic	neutrosophic	ADJ
cana-1395	40	62	ideal	ideal	NOUN
cana-1395	40	63	of	of	ADP
cana-1395	40	64	,	,	PUNCT
cana-1395	40	65	and	and	CCONJ
cana-1395	40	66	that	that	SCONJ
cana-1395	40	67	it	it	PRON
cana-1395	40	68	satisfies	satisfy	VERB
cana-1395	40	69	the	the	DET
cana-1395	40	70	conditions	condition	NOUN
cana-1395	40	71	of	of	ADP
cana-1395	40	72	definition	definition	NOUN
cana-1395	40	73	3.1	3.1	NUM
cana-1395	40	74	.	.	PUNCT
cana-1395	41	1	definition	definition	NOUN
cana-1395	41	2	3.3	3.3	NUM
cana-1395	41	3	:	:	PUNCT
cana-1395	41	4	let	let	VERB
cana-1395	41	5	(	(	PUNCT
cana-1395	41	6	,	,	PUNCT
cana-1395	41	7	0	0	NUM
cana-1395	41	8	)	)	PUNCT
cana-1395	41	9	be	be	AUX
cana-1395	41	10	a	a	DET
cana-1395	41	11	kk	kk	NOUN
cana-1395	41	12	-	-	PUNCT
cana-1395	41	13	algebra	algebra	NOUN
cana-1395	41	14	,	,	PUNCT
cana-1395	41	15	and	and	CCONJ
cana-1395	41	16	let	let	VERB
cana-1395	41	17	be	be	AUX
cana-1395	41	18	a	a	DET
cana-1395	41	19	nonempty	nonempty	ADJ
cana-1395	41	20	neutrosophic	neutrosophic	ADJ
cana-1395	41	21	subset	subset	NOUN
cana-1395	41	22	of	of	ADP
cana-1395	41	23	is	be	AUX
cana-1395	41	24	called	call	VERB
cana-1395	41	25	neutrosophic	neutrosophic	ADJ
cana-1395	41	26	sub	sub	NOUN
cana-1395	41	27	algebra	algebra	NOUN
cana-1395	41	28	of	of	ADP
cana-1395	41	29	if	if	SCONJ
cana-1395	41	30	it	it	PRON
cana-1395	41	31	satisfies	satisfy	VERB
cana-1395	41	32	the	the	DET
cana-1395	41	33	following	following	ADJ
cana-1395	41	34	axioms	axiom	NOUN
cana-1395	41	35	for	for	ADP
cana-1395	41	36	all	all	PRON
cana-1395	41	37	(	(	PUNCT
cana-1395	41	38	i	i	NOUN
cana-1395	41	39	)	)	PUNCT
cana-1395	41	40	(	(	PUNCT
cana-1395	41	41	ii	ii	NOUN
cana-1395	41	42	)	)	PUNCT
cana-1395	41	43	(	(	PUNCT
cana-1395	41	44	iii	iii	NOUN
cana-1395	41	45	)	)	PUNCT
cana-1395	41	46	.	.	PUNCT
cana-1395	42	1	communications	communication	NOUN
cana-1395	42	2	on	on	ADP
cana-1395	42	3	applied	apply	VERB
cana-1395	42	4	nonlinear	nonlinear	ADJ
cana-1395	42	5	analysis	analysis	NOUN
cana-1395	42	6	issn	issn	NOUN
cana-1395	42	7	:	:	PUNCT
cana-1395	42	8	1074	1074	NUM
cana-1395	42	9	-	-	PUNCT
cana-1395	42	10	133x	133x	NUM
cana-1395	42	11	vol	vol	NOUN
cana-1395	42	12	31	31	NUM
cana-1395	42	13	no	no	NOUN
cana-1395	42	14	.	.	PUNCT
cana-1395	43	1	7s	7	NOUN
cana-1395	43	2	(	(	PUNCT
cana-1395	43	3	2024	2024	NUM
cana-1395	43	4	)	)	PUNCT
cana-1395	44	1	534	534	NUM
cana-1395	44	2	https://internationalpubls.com	https://internationalpubls.com	X
cana-1395	44	3	example	example	NOUN
cana-1395	44	4	3.4	3.4	NUM
cana-1395	44	5	:	:	PUNCT
cana-1395	44	6	consider	consider	VERB
cana-1395	44	7	a	a	DET
cana-1395	44	8	set	set	NOUN
cana-1395	44	9	be	be	AUX
cana-1395	44	10	a	a	DET
cana-1395	44	11	neutrosophic	neutrosophic	ADJ
cana-1395	44	12	subset	subset	NOUN
cana-1395	44	13	of	of	ADP
cana-1395	44	14	define	define	VERB
cana-1395	44	15	a	a	DET
cana-1395	44	16	binary	binary	ADJ
cana-1395	44	17	operation	operation	NOUN
cana-1395	44	18	on	on	ADP
cana-1395	44	19	given	give	VERB
cana-1395	44	20	by	by	ADP
cana-1395	44	21	the	the	DET
cana-1395	44	22	following	follow	VERB
cana-1395	44	23	table	table	NOUN
cana-1395	44	24	3.3	3.3	NUM
cana-1395	44	25	,	,	PUNCT
cana-1395	44	26	and	and	CCONJ
cana-1395	44	27	neutrosophic	neutrosophic	ADJ
cana-1395	44	28	set	set	VERB
cana-1395	44	29	by	by	ADP
cana-1395	44	30	the	the	DET
cana-1395	44	31	table	table	NOUN
cana-1395	44	32	3.4	3.4	NUM
cana-1395	44	33	as	as	SCONJ
cana-1395	44	34	shown	show	VERB
cana-1395	44	35	below	below	ADP
cana-1395	44	36	:	:	PUNCT
cana-1395	44	37	table	table	NOUN
cana-1395	44	38	3.3	3.3	NUM
cana-1395	44	39	:	:	PUNCT
cana-1395	44	40	operation	operation	NOUN
cana-1395	44	41	0.7	0.7	NUM
cana-1395	44	42	0.5	0.5	NUM
cana-1395	44	43	0.5	0.5	NUM
cana-1395	44	44	0.6	0.6	NUM
cana-1395	44	45	0.4	0.4	NUM
cana-1395	44	46	0.4	0.4	NUM
cana-1395	44	47	0.2	0.2	NUM
cana-1395	44	48	0.1	0.1	NUM
cana-1395	44	49	0.1	0.1	NUM
cana-1395	44	50	table	table	NOUN
cana-1395	44	51	3.4	3.4	NUM
cana-1395	44	52	:	:	PUNCT
cana-1395	44	53	neutrosophic	neutrosophic	PROPN
cana-1395	44	54	set	set	NOUN
cana-1395	44	55	it	it	PRON
cana-1395	44	56	is	be	AUX
cana-1395	44	57	easily	easily	ADV
cana-1395	44	58	verified	verify	VERB
cana-1395	44	59	that	that	PRON
cana-1395	44	60	is	be	AUX
cana-1395	44	61	neutrosophic	neutrosophic	ADJ
cana-1395	44	62	sub	sub	NOUN
cana-1395	44	63	algebra	algebra	PROPN
cana-1395	44	64	of	of	ADP
cana-1395	44	65	,	,	PUNCT
cana-1395	44	66	and	and	CCONJ
cana-1395	44	67	that	that	SCONJ
cana-1395	44	68	it	it	PRON
cana-1395	44	69	satisfies	satisfy	VERB
cana-1395	44	70	the	the	DET
cana-1395	44	71	conditions	condition	NOUN
cana-1395	44	72	of	of	ADP
cana-1395	44	73	definition	definition	NOUN
cana-1395	44	74	3.3	3.3	NUM
cana-1395	44	75	.	.	PUNCT
cana-1395	45	1	proposition	proposition	NOUN
cana-1395	45	2	3.5	3.5	NUM
cana-1395	45	3	:	:	PUNCT
cana-1395	45	4	let	let	AUX
cana-1395	45	5	be	be	AUX
cana-1395	45	6	a	a	DET
cana-1395	45	7	neutrosophic	neutrosophic	ADJ
cana-1395	45	8	ideal	ideal	NOUN
cana-1395	45	9	in	in	ADP
cana-1395	45	10	kk	kk	NOUN
cana-1395	45	11	-	-	NOUN
cana-1395	45	12	algebra	algebra	NOUN
cana-1395	45	13	and	and	CCONJ
cana-1395	45	14	if	if	SCONJ
cana-1395	45	15	then	then	ADV
cana-1395	45	16	for	for	ADP
cana-1395	45	17	all	all	DET
cana-1395	45	18	proof	proof	NOUN
cana-1395	45	19	:	:	PUNCT
cana-1395	45	20	let	let	AUX
cana-1395	45	21	be	be	AUX
cana-1395	45	22	a	a	DET
cana-1395	45	23	neutrosophic	neutrosophic	ADJ
cana-1395	45	24	ideal	ideal	NOUN
cana-1395	45	25	of	of	ADP
cana-1395	45	26	kk	kk	NOUN
cana-1395	45	27	-	-	PUNCT
cana-1395	45	28	algebra	algebra	PROPN
cana-1395	45	29	and	and	CCONJ
cana-1395	45	30	.	.	PUNCT
cana-1395	46	1	then	then	ADV
cana-1395	46	2	from	from	ADP
cana-1395	46	3	the	the	DET
cana-1395	46	4	definition	definition	NOUN
cana-1395	46	5	2.2	2.2	NUM
cana-1395	46	6	,	,	PUNCT
cana-1395	46	7	we	we	PRON
cana-1395	46	8	have	have	VERB
cana-1395	46	9	for	for	ADP
cana-1395	46	10	all	all	ADV
cana-1395	46	11	from	from	ADP
cana-1395	46	12	(	(	PUNCT
cana-1395	46	13	ii	ii	NOUN
cana-1395	46	14	)	)	PUNCT
cana-1395	46	15	of	of	ADP
cana-1395	46	16	definition	definition	NOUN
cana-1395	46	17	3.1we	3.1we	NUM
cana-1395	46	18	have	have	VERB
cana-1395	46	19	,	,	PUNCT
cana-1395	46	20	then	then	ADV
cana-1395	46	21	,	,	PUNCT
cana-1395	46	22	we	we	PRON
cana-1395	46	23	get	get	VERB
cana-1395	46	24	similarly	similarly	ADV
cana-1395	46	25	,	,	PUNCT
cana-1395	46	26	we	we	PRON
cana-1395	46	27	can	can	AUX
cana-1395	46	28	prove	prove	VERB
cana-1395	46	29	for	for	ADP
cana-1395	46	30	next	next	ADJ
cana-1395	46	31	,	,	PUNCT
cana-1395	46	32	from	from	ADP
cana-1395	46	33	(	(	PUNCT
cana-1395	46	34	ii	ii	NOUN
cana-1395	46	35	)	)	PUNCT
cana-1395	46	36	of	of	ADP
cana-1395	46	37	the	the	DET
cana-1395	46	38	definition	definition	NOUN
cana-1395	46	39	3.1	3.1	NUM
cana-1395	46	40	,	,	PUNCT
cana-1395	46	41	we	we	PRON
cana-1395	46	42	have	have	VERB
cana-1395	46	43	hence	hence	ADV
cana-1395	46	44	,	,	PUNCT
cana-1395	46	45	we	we	PRON
cana-1395	46	46	get	get	VERB
cana-1395	46	47	.	.	PUNCT
cana-1395	47	1	∎	∎	NOUN
cana-1395	47	2	definition	definition	NOUN
cana-1395	47	3	3.6	3.6	NUM
cana-1395	47	4	:	:	PUNCT
cana-1395	47	5	let	let	AUX
cana-1395	47	6	be	be	AUX
cana-1395	47	7	a	a	DET
cana-1395	47	8	nonempty	nonempty	ADV
cana-1395	47	9	set	set	VERB
cana-1395	47	10	and	and	CCONJ
cana-1395	47	11	be	be	AUX
cana-1395	47	12	a	a	DET
cana-1395	47	13	neutrosophic	neutrosophic	ADJ
cana-1395	47	14	subset	subset	NOUN
cana-1395	47	15	of	of	ADP
cana-1395	47	16	x	x	PRON
cana-1395	47	17	,	,	PUNCT
cana-1395	47	18	for	for	SCONJ
cana-1395	47	19	[	[	PUNCT
cana-1395	47	20	]	]	X
cana-1395	47	21	the	the	DET
cana-1395	47	22	set	set	NOUN
cana-1395	47	23	is	be	AUX
cana-1395	47	24	called	call	VERB
cana-1395	47	25	a	a	DET
cana-1395	47	26	level	level	NOUN
cana-1395	47	27	subset	subset	NOUN
cana-1395	47	28	of	of	ADP
cana-1395	47	29	n.	n.	PROPN
cana-1395	47	30	theorem	theorem	VERB
cana-1395	47	31	3.7	3.7	NUM
cana-1395	47	32	:	:	PUNCT
cana-1395	47	33	let	let	AUX
cana-1395	47	34	be	be	AUX
cana-1395	47	35	a	a	DET
cana-1395	47	36	neutrosophic	neutrosophic	ADJ
cana-1395	47	37	subset	subset	NOUN
cana-1395	47	38	of	of	ADP
cana-1395	47	39	kk	kk	PROPN
cana-1395	47	40	-	-	NOUN
cana-1395	47	41	algebra	algebra	NOUN
cana-1395	47	42	.	.	PUNCT
cana-1395	48	1	if	if	SCONJ
cana-1395	48	2	n	n	PRON
cana-1395	48	3	is	be	AUX
cana-1395	48	4	a	a	DET
cana-1395	48	5	neutrosophic	neutrosophic	ADJ
cana-1395	48	6	sub	sub	NOUN
cana-1395	48	7	algebra	algebra	NOUN
cana-1395	48	8	of	of	ADP
cana-1395	48	9	if	if	SCONJ
cana-1395	49	1	and	and	CCONJ
cana-1395	49	2	only	only	ADV
cana-1395	49	3	if	if	SCONJ
cana-1395	49	4	the	the	DET
cana-1395	49	5	level	level	NOUN
cana-1395	49	6	set	set	NOUN
cana-1395	49	7	is	be	AUX
cana-1395	49	8	a	a	DET
cana-1395	49	9	subalgebra	subalgebra	NOUN
cana-1395	49	10	of	of	ADP
cana-1395	49	11	for	for	ADP
cana-1395	49	12	every	every	PRON
cana-1395	49	13	[	[	PUNCT
cana-1395	49	14	]	]	X
cana-1395	49	15	.	.	PUNCT
cana-1395	50	1	proof	proof	NOUN
cana-1395	50	2	:	:	PUNCT
cana-1395	50	3	let	let	AUX
cana-1395	50	4	be	be	AUX
cana-1395	50	5	neutrosophic	neutrosophic	ADJ
cana-1395	50	6	subset	subset	NOUN
cana-1395	50	7	of	of	ADP
cana-1395	50	8	kk	kk	PROPN
cana-1395	50	9	-	-	PUNCT
cana-1395	50	10	algebra	algebra	PROPN
cana-1395	50	11	and	and	CCONJ
cana-1395	50	12	also	also	ADV
cana-1395	50	13	neutrosophic	neutrosophic	ADJ
cana-1395	50	14	sub	sub	NOUN
cana-1395	50	15	algebra	algebra	NOUN
cana-1395	50	16	of	of	ADP
cana-1395	50	17	communications	communication	NOUN
cana-1395	50	18	on	on	ADP
cana-1395	50	19	applied	apply	VERB
cana-1395	50	20	nonlinear	nonlinear	ADJ
cana-1395	50	21	analysis	analysis	NOUN
cana-1395	50	22	issn	issn	NOUN
cana-1395	50	23	:	:	PUNCT
cana-1395	50	24	1074	1074	NUM
cana-1395	50	25	-	-	PUNCT
cana-1395	50	26	133x	133x	NUM
cana-1395	50	27	vol	vol	NOUN
cana-1395	50	28	31	31	NUM
cana-1395	50	29	no	no	NOUN
cana-1395	50	30	.	.	PUNCT
cana-1395	51	1	7s	7	NOUN
cana-1395	51	2	(	(	PUNCT
cana-1395	51	3	2024	2024	NUM
cana-1395	51	4	)	)	PUNCT
cana-1395	51	5	535	535	NUM
cana-1395	51	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1395	51	7	kk	kk	PROPN
cana-1395	51	8	-	-	NOUN
cana-1395	51	9	algebra	algebra	PROPN
cana-1395	51	10	.	.	PUNCT
cana-1395	52	1	let	let	AUX
cana-1395	52	2	be	be	AUX
cana-1395	52	3	such	such	ADJ
cana-1395	52	4	that	that	PRON
cana-1395	52	5	and	and	CCONJ
cana-1395	52	6	then	then	ADV
cana-1395	52	7	and	and	CCONJ
cana-1395	52	8	.	.	PUNCT
cana-1395	53	1	since	since	SCONJ
cana-1395	53	2	n	n	NUM
cana-1395	53	3	is	be	AUX
cana-1395	53	4	neutrosophic	neutrosophic	ADJ
cana-1395	53	5	sub	sub	NOUN
cana-1395	53	6	algebra	algebra	NOUN
cana-1395	53	7	.	.	PUNCT
cana-1395	54	1	it	it	PRON
cana-1395	54	2	follows	follow	VERB
cana-1395	54	3	that	that	PRON
cana-1395	54	4	and	and	CCONJ
cana-1395	54	5	that	that	PRON
cana-1395	54	6	hence	hence	ADV
cana-1395	54	7	is	be	AUX
cana-1395	54	8	a	a	DET
cana-1395	54	9	sub	sub	NOUN
cana-1395	54	10	algebra	algebra	NOUN
cana-1395	54	11	of	of	ADP
cana-1395	54	12	similarly	similarly	ADV
cana-1395	54	13	,	,	PUNCT
cana-1395	54	14	we	we	PRON
cana-1395	54	15	can	can	AUX
cana-1395	54	16	prove	prove	VERB
cana-1395	54	17	for	for	ADP
cana-1395	54	18	and	and	CCONJ
cana-1395	54	19	.	.	PUNCT
cana-1395	55	1	conversely	conversely	ADV
cana-1395	55	2	,	,	PUNCT
cana-1395	55	3	if	if	SCONJ
cana-1395	55	4	is	be	AUX
cana-1395	55	5	not	not	PART
cana-1395	55	6	true	true	ADJ
cana-1395	55	7	.	.	PUNCT
cana-1395	56	1	then	then	ADV
cana-1395	56	2	there	there	PRON
cana-1395	56	3	exists	exist	VERB
cana-1395	56	4	such	such	ADJ
cana-1395	56	5	that	that	SCONJ
cana-1395	56	6	putting	put	VERB
cana-1395	56	7	then	then	ADV
cana-1395	56	8	and	and	CCONJ
cana-1395	56	9	.	.	PUNCT
cana-1395	57	1	hence	hence	ADV
cana-1395	57	2	,	,	PUNCT
cana-1395	57	3	and	and	CCONJ
cana-1395	57	4	which	which	PRON
cana-1395	57	5	imply	imply	VERB
cana-1395	57	6	that	that	PRON
cana-1395	57	7	and	and	CCONJ
cana-1395	57	8	since	since	SCONJ
cana-1395	57	9	is	be	AUX
cana-1395	57	10	a	a	DET
cana-1395	57	11	subalgebra	subalgebra	NOUN
cana-1395	57	12	it	it	PRON
cana-1395	57	13	follows	follow	VERB
cana-1395	57	14	that	that	PRON
cana-1395	57	15	,	,	PUNCT
cana-1395	57	16	and	and	CCONJ
cana-1395	57	17	that	that	SCONJ
cana-1395	57	18	this	this	PRON
cana-1395	57	19	is	be	AUX
cana-1395	57	20	contradiction	contradiction	NOUN
cana-1395	57	21	.	.	PUNCT
cana-1395	58	1	therefore	therefore	ADV
cana-1395	58	2	,	,	PUNCT
cana-1395	58	3	is	be	AUX
cana-1395	58	4	neutrosophic	neutrosophic	ADJ
cana-1395	58	5	sub	sub	NOUN
cana-1395	58	6	algebra	algebra	NOUN
cana-1395	58	7	of	of	ADP
cana-1395	58	8	similarly	similarly	ADV
cana-1395	58	9	,	,	PUNCT
cana-1395	58	10	we	we	PRON
cana-1395	58	11	can	can	AUX
cana-1395	58	12	prove	prove	VERB
cana-1395	58	13	for	for	ADP
cana-1395	58	14	and	and	CCONJ
cana-1395	58	15	.∎	.∎	PROPN
cana-1395	58	16	theorem	theorem	VERB
cana-1395	58	17	3.8	3.8	NUM
cana-1395	58	18	:	:	PUNCT
cana-1395	58	19	let	let	AUX
cana-1395	58	20	be	be	AUX
cana-1395	58	21	a	a	DET
cana-1395	58	22	neutrosophic	neutrosophic	ADJ
cana-1395	58	23	ideal	ideal	NOUN
cana-1395	58	24	of	of	ADP
cana-1395	58	25	kk	kk	PROPN
cana-1395	58	26	-	-	PUNCT
cana-1395	58	27	algebra	algebra	NOUN
cana-1395	58	28	be	be	VERB
cana-1395	58	29	a	a	DET
cana-1395	58	30	neutrosophic	neutrosophic	ADJ
cana-1395	58	31	ideal	ideal	NOUN
cana-1395	58	32	of	of	ADP
cana-1395	58	33	if	if	SCONJ
cana-1395	58	34	and	and	CCONJ
cana-1395	58	35	only	only	ADV
cana-1395	58	36	if	if	SCONJ
cana-1395	58	37	for	for	SCONJ
cana-1395	58	38	every	every	PRON
cana-1395	58	39	[	[	PUNCT
cana-1395	58	40	]	]	X
cana-1395	58	41	is	be	AUX
cana-1395	58	42	an	an	DET
cana-1395	58	43	ideal	ideal	NOUN
cana-1395	58	44	of	of	ADP
cana-1395	58	45	proof	proof	NOUN
cana-1395	58	46	:	:	PUNCT
cana-1395	58	47	let	let	AUX
cana-1395	58	48	be	be	AUX
cana-1395	58	49	a	a	DET
cana-1395	58	50	neutrosophic	neutrosophic	ADJ
cana-1395	58	51	ideal	ideal	NOUN
cana-1395	58	52	of	of	ADP
cana-1395	58	53	kk	kk	NOUN
cana-1395	58	54	-	-	NOUN
cana-1395	58	55	algebra	algebra	PROPN
cana-1395	58	56	.	.	PUNCT
cana-1395	59	1	then	then	ADV
cana-1395	59	2	,	,	PUNCT
cana-1395	59	3	from	from	ADP
cana-1395	59	4	the	the	DET
cana-1395	59	5	definition	definition	NOUN
cana-1395	59	6	of	of	ADP
cana-1395	59	7	3.1	3.1	NUM
cana-1395	59	8	we	we	PRON
cana-1395	59	9	have	have	VERB
cana-1395	59	10	for	for	ADP
cana-1395	59	11	all	all	ADV
cana-1395	59	12	therefore	therefore	ADV
cana-1395	59	13	,	,	PUNCT
cana-1395	59	14	for	for	ADP
cana-1395	59	15	and	and	CCONJ
cana-1395	59	16	so	so	ADV
cana-1395	59	17	,	,	PUNCT
cana-1395	59	18	let	let	VERB
cana-1395	59	19	be	be	AUX
cana-1395	59	20	such	such	ADJ
cana-1395	59	21	that	that	PRON
cana-1395	59	22	and	and	CCONJ
cana-1395	59	23	then	then	ADV
cana-1395	59	24	and	and	CCONJ
cana-1395	59	25	since	since	SCONJ
cana-1395	59	26	is	be	AUX
cana-1395	59	27	a	a	DET
cana-1395	59	28	neutrosophic	neutrosophic	ADJ
cana-1395	59	29	ideal	ideal	NOUN
cana-1395	59	30	,	,	PUNCT
cana-1395	59	31	then	then	ADV
cana-1395	59	32	from	from	ADP
cana-1395	59	33	(	(	PUNCT
cana-1395	59	34	ii	ii	NOUN
cana-1395	59	35	)	)	PUNCT
cana-1395	59	36	of	of	ADP
cana-1395	59	37	the	the	DET
cana-1395	59	38	definition	definition	NOUN
cana-1395	59	39	3.1	3.1	NUM
cana-1395	59	40	,	,	PUNCT
cana-1395	59	41	it	it	PRON
cana-1395	59	42	follows	follow	VERB
cana-1395	59	43	that	that	PRON
cana-1395	59	44	and	and	CCONJ
cana-1395	59	45	,	,	PUNCT
cana-1395	59	46	we	we	PRON
cana-1395	59	47	have	have	VERB
cana-1395	59	48	that	that	PRON
cana-1395	59	49	.	.	PUNCT
cana-1395	60	1	hence	hence	ADV
cana-1395	60	2	,	,	PUNCT
cana-1395	60	3	is	be	AUX
cana-1395	60	4	an	an	DET
cana-1395	60	5	ideal	ideal	NOUN
cana-1395	60	6	of	of	ADP
cana-1395	60	7	.	.	PUNCT
cana-1395	61	1	similarly	similarly	ADV
cana-1395	61	2	,	,	PUNCT
cana-1395	61	3	we	we	PRON
cana-1395	61	4	can	can	AUX
cana-1395	61	5	prove	prove	VERB
cana-1395	61	6	for	for	ADP
cana-1395	61	7	and	and	CCONJ
cana-1395	61	8	.	.	PUNCT
cana-1395	62	1	conversely	conversely	ADV
cana-1395	62	2	,	,	PUNCT
cana-1395	62	3	we	we	PRON
cana-1395	62	4	need	need	VERB
cana-1395	62	5	to	to	PART
cana-1395	62	6	show	show	VERB
cana-1395	62	7	that	that	SCONJ
cana-1395	62	8	(	(	PUNCT
cana-1395	62	9	i	i	NOUN
cana-1395	62	10	)	)	PUNCT
cana-1395	62	11	and	and	CCONJ
cana-1395	62	12	(	(	PUNCT
cana-1395	62	13	ii	ii	NOUN
cana-1395	62	14	)	)	PUNCT
cana-1395	62	15	of	of	ADP
cana-1395	62	16	definition	definition	NOUN
cana-1395	62	17	3.1	3.1	NUM
cana-1395	62	18	are	be	AUX
cana-1395	62	19	true	true	ADJ
cana-1395	62	20	.	.	PUNCT
cana-1395	63	1	if	if	SCONJ
cana-1395	63	2	(	(	PUNCT
cana-1395	63	3	i	i	NOUN
cana-1395	63	4	)	)	PUNCT
cana-1395	63	5	of	of	ADP
cana-1395	63	6	definition	definition	NOUN
cana-1395	63	7	3.1	3.1	NUM
cana-1395	63	8	is	be	AUX
cana-1395	63	9	not	not	PART
cana-1395	63	10	satisfied	satisfied	ADJ
cana-1395	63	11	,	,	PUNCT
cana-1395	63	12	then	then	ADV
cana-1395	63	13	there	there	PRON
cana-1395	63	14	exists	exist	VERB
cana-1395	63	15	such	such	ADJ
cana-1395	63	16	that	that	SCONJ
cana-1395	63	17	if	if	SCONJ
cana-1395	63	18	we	we	PRON
cana-1395	63	19	take	take	VERB
cana-1395	63	20	=	=	PUNCT
cana-1395	63	21	then	then	ADV
cana-1395	63	22	and	and	CCONJ
cana-1395	63	23	0	0	NUM
cana-1395	63	24	then	then	ADV
cana-1395	63	25	and	and	CCONJ
cana-1395	63	26	as	as	SCONJ
cana-1395	63	27	is	be	AUX
cana-1395	63	28	an	an	DET
cana-1395	63	29	ideal	ideal	NOUN
cana-1395	63	30	of	of	ADP
cana-1395	63	31	x	x	PUNCT
cana-1395	63	32	we	we	PRON
cana-1395	63	33	have	have	AUX
cana-1395	63	34	and	and	CCONJ
cana-1395	63	35	so	so	ADV
cana-1395	63	36	this	this	PRON
cana-1395	63	37	is	be	AUX
cana-1395	63	38	a	a	DET
cana-1395	63	39	contradiction	contradiction	NOUN
cana-1395	63	40	.	.	PUNCT
cana-1395	64	1	therefore	therefore	ADV
cana-1395	64	2	we	we	PRON
cana-1395	64	3	have	have	VERB
cana-1395	64	4	if	if	SCONJ
cana-1395	64	5	(	(	PUNCT
cana-1395	64	6	ii	ii	NOUN
cana-1395	64	7	)	)	PUNCT
cana-1395	64	8	is	be	AUX
cana-1395	64	9	not	not	PART
cana-1395	64	10	true	true	ADJ
cana-1395	64	11	then	then	ADV
cana-1395	64	12	there	there	PRON
cana-1395	64	13	exists	exist	VERB
cana-1395	64	14	such	such	ADJ
cana-1395	64	15	that	that	SCONJ
cana-1395	64	16	putting	put	VERB
cana-1395	64	17	=	=	PUNCT
cana-1395	64	18	then	then	ADV
cana-1395	64	19	and	and	CCONJ
cana-1395	64	20	}	}	PUNCT
cana-1395	64	21	,	,	PUNCT
cana-1395	64	22	hence	hence	ADV
cana-1395	64	23	,	,	PUNCT
cana-1395	64	24	which	which	PRON
cana-1395	64	25	imply	imply	VERB
cana-1395	64	26	that	that	PRON
cana-1395	64	27	,	,	PUNCT
cana-1395	64	28	and	and	CCONJ
cana-1395	64	29	.	.	PUNCT
cana-1395	65	1	since	since	SCONJ
cana-1395	65	2	is	be	AUX
cana-1395	65	3	an	an	DET
cana-1395	65	4	ideal	ideal	NOUN
cana-1395	65	5	.	.	PUNCT
cana-1395	66	1	it	it	PRON
cana-1395	66	2	follows	follow	VERB
cana-1395	66	3	that	that	PRON
cana-1395	66	4	and	and	CCONJ
cana-1395	66	5	.	.	PUNCT
cana-1395	67	1	this	this	PRON
cana-1395	67	2	is	be	AUX
cana-1395	67	3	a	a	DET
cana-1395	67	4	contradiction	contradiction	NOUN
cana-1395	67	5	.	.	PUNCT
cana-1395	68	1	thus	thus	ADV
cana-1395	68	2	,	,	PUNCT
cana-1395	68	3	is	be	AUX
cana-1395	68	4	neutrosophic	neutrosophic	ADJ
cana-1395	68	5	ideal	ideal	NOUN
cana-1395	68	6	of	of	ADP
cana-1395	68	7	.	.	PUNCT
cana-1395	69	1	communications	communication	NOUN
cana-1395	69	2	on	on	ADP
cana-1395	69	3	applied	apply	VERB
cana-1395	69	4	nonlinear	nonlinear	ADJ
cana-1395	69	5	analysis	analysis	NOUN
cana-1395	69	6	issn	issn	NOUN
cana-1395	69	7	:	:	PUNCT
cana-1395	69	8	1074	1074	NUM
cana-1395	69	9	-	-	PUNCT
cana-1395	69	10	133x	133x	NUM
cana-1395	69	11	vol	vol	NOUN
cana-1395	69	12	31	31	NUM
cana-1395	69	13	no	no	NOUN
cana-1395	69	14	.	.	PUNCT
cana-1395	70	1	7s	7	NOUN
cana-1395	70	2	(	(	PUNCT
cana-1395	70	3	2024	2024	NUM
cana-1395	70	4	)	)	PUNCT
cana-1395	70	5	536	536	NUM
cana-1395	70	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1395	70	7	similarly	similarly	ADV
cana-1395	70	8	,	,	PUNCT
cana-1395	70	9	we	we	PRON
cana-1395	70	10	can	can	AUX
cana-1395	70	11	prove	prove	VERB
cana-1395	70	12	for	for	ADP
cana-1395	70	13	and	and	CCONJ
cana-1395	70	14	.	.	PUNCT
cana-1395	71	1	∎	∎	PROPN
cana-1395	71	2	proposition	proposition	NOUN
cana-1395	71	3	3.9	3.9	NUM
cana-1395	71	4	:	:	PUNCT
cana-1395	71	5	every	every	DET
cana-1395	71	6	neutrosophic	neutrosophic	ADJ
cana-1395	71	7	ideal	ideal	NOUN
cana-1395	71	8	of	of	ADP
cana-1395	71	9	kk	kk	PROPN
cana-1395	71	10	-	-	PUNCT
cana-1395	71	11	algebra	algebra	PROPN
cana-1395	71	12	is	be	AUX
cana-1395	71	13	neutrosophic	neutrosophic	ADJ
cana-1395	71	14	sub	sub	NOUN
cana-1395	71	15	algebra	algebra	NOUN
cana-1395	71	16	of	of	ADP
cana-1395	71	17	proof	proof	NOUN
cana-1395	71	18	:	:	PUNCT
cana-1395	71	19	assume	assume	VERB
cana-1395	71	20	that	that	PRON
cana-1395	71	21	be	be	VERB
cana-1395	71	22	an	an	DET
cana-1395	71	23	neutrosophic	neutrosophic	ADJ
cana-1395	71	24	ideal	ideal	NOUN
cana-1395	71	25	of	of	ADP
cana-1395	71	26	a	a	DET
cana-1395	71	27	kk	kk	NOUN
cana-1395	71	28	-	-	PUNCT
cana-1395	71	29	algebra	algebra	NOUN
cana-1395	71	30	next	next	ADV
cana-1395	71	31	based	base	VERB
cana-1395	71	32	on	on	ADP
cana-1395	71	33	theorem	theorem	NOUN
cana-1395	71	34	3.8	3.8	NUM
cana-1395	71	35	every	every	PRON
cana-1395	71	36	[	[	PUNCT
cana-1395	71	37	]	]	X
cana-1395	71	38	is	be	AUX
cana-1395	71	39	an	an	DET
cana-1395	71	40	ideal	ideal	NOUN
cana-1395	71	41	of	of	ADP
cana-1395	71	42	.	.	PUNCT
cana-1395	72	1	according	accord	VERB
cana-1395	72	2	to	to	ADP
cana-1395	72	3	proposition	proposition	NOUN
cana-1395	72	4	2.6	2.6	NUM
cana-1395	72	5	for	for	ADP
cana-1395	72	6	every	every	DET
cana-1395	72	7	t	t	NOUN
cana-1395	72	8	[	[	PUNCT
cana-1395	72	9	]	]	X
cana-1395	72	10	is	be	AUX
cana-1395	72	11	a	a	DET
cana-1395	72	12	subalgebra	subalgebra	NOUN
cana-1395	72	13	of	of	ADP
cana-1395	72	14	.	.	PUNCT
cana-1395	73	1	thus	thus	ADV
cana-1395	73	2	,	,	PUNCT
cana-1395	73	3	from	from	ADP
cana-1395	73	4	the	the	DET
cana-1395	73	5	theorem	theorem	NOUN
cana-1395	73	6	3.7	3.7	NUM
cana-1395	73	7	,	,	PUNCT
cana-1395	73	8	is	be	AUX
cana-1395	73	9	neutrosophic	neutrosophic	ADJ
cana-1395	73	10	subalgebra	subalgebra	NOUN
cana-1395	73	11	of	of	ADP
cana-1395	73	12	kk	kk	NOUN
cana-1395	73	13	-	-	NOUN
cana-1395	73	14	algebra	algebra	NOUN
cana-1395	73	15	.	.	PUNCT
cana-1395	74	1	∎	∎	PROPN
cana-1395	74	2	theorem	theorem	VERB
cana-1395	74	3	3.10	3.10	NUM
cana-1395	74	4	:	:	PUNCT
cana-1395	74	5	let	let	VERB
cana-1395	74	6	i	i	PRON
cana-1395	74	7	be	be	AUX
cana-1395	74	8	an	an	DET
cana-1395	74	9	ideal	ideal	NOUN
cana-1395	74	10	of	of	ADP
cana-1395	74	11	kk	kk	NOUN
cana-1395	74	12	-	-	NOUN
cana-1395	74	13	algebra	algebra	NOUN
cana-1395	74	14	of	of	ADP
cana-1395	74	15	then	then	ADV
cana-1395	74	16	for	for	ADP
cana-1395	74	17	any	any	DET
cana-1395	74	18	fixed	fixed	ADJ
cana-1395	74	19	number	number	NOUN
cana-1395	74	20	t	t	NOUN
cana-1395	74	21	in	in	ADP
cana-1395	74	22	an	an	DET
cana-1395	74	23	open	open	ADJ
cana-1395	74	24	interval	interval	NOUN
cana-1395	74	25	(	(	PUNCT
cana-1395	74	26	0	0	NUM
cana-1395	74	27	,	,	PUNCT
cana-1395	74	28	1	1	NUM
cana-1395	74	29	)	)	PUNCT
cana-1395	74	30	,	,	PUNCT
cana-1395	74	31	there	there	PRON
cana-1395	74	32	exists	exist	VERB
cana-1395	74	33	neutrosophic	neutrosophic	ADJ
cana-1395	74	34	ideal	ideal	NOUN
cana-1395	74	35	of	of	ADP
cana-1395	74	36	such	such	ADJ
cana-1395	74	37	that	that	DET
cana-1395	74	38	proof	proof	NOUN
cana-1395	74	39	:	:	PUNCT
cana-1395	74	40	define	define	VERB
cana-1395	74	41	[	[	PUNCT
cana-1395	74	42	]	]	X
cana-1395	74	43	by	by	ADP
cana-1395	74	44	)	)	PUNCT
cana-1395	74	45	{	{	PUNCT
cana-1395	74	46	where	where	SCONJ
cana-1395	74	47	is	be	AUX
cana-1395	74	48	a	a	DET
cana-1395	74	49	fixed	fix	VERB
cana-1395	74	50	number	number	NOUN
cana-1395	74	51	in	in	ADP
cana-1395	74	52	(	(	PUNCT
cana-1395	74	53	0	0	NUM
cana-1395	74	54	,	,	PUNCT
cana-1395	74	55	1	1	NUM
cana-1395	74	56	)	)	PUNCT
cana-1395	74	57	clearly	clearly	ADV
cana-1395	74	58	for	for	ADP
cana-1395	74	59	all	all	PRON
cana-1395	74	60	.	.	PUNCT
cana-1395	75	1	let	let	VERB
cana-1395	75	2	,	,	PUNCT
cana-1395	75	3	if	if	SCONJ
cana-1395	75	4	,	,	PUNCT
cana-1395	75	5	then	then	ADV
cana-1395	76	1	and	and	CCONJ
cana-1395	76	2	so	so	ADV
cana-1395	76	3	if	if	SCONJ
cana-1395	76	4	then	then	ADV
cana-1395	76	5	clearly	clearly	ADV
cana-1395	76	6	if	if	SCONJ
cana-1395	76	7	and	and	CCONJ
cana-1395	76	8	then	then	ADV
cana-1395	76	9	since	since	SCONJ
cana-1395	76	10	is	be	AUX
cana-1395	76	11	an	an	DET
cana-1395	76	12	ideal	ideal	NOUN
cana-1395	76	13	,	,	PUNCT
cana-1395	76	14	then	then	ADV
cana-1395	76	15	.	.	PUNCT
cana-1395	77	1	hence	hence	ADV
cana-1395	77	2	,	,	PUNCT
cana-1395	77	3	is	be	AUX
cana-1395	77	4	neutrosophic	neutrosophic	ADJ
cana-1395	77	5	ideal	ideal	NOUN
cana-1395	77	6	of	of	ADP
cana-1395	77	7	.it	.it	PUNCT
cana-1395	77	8	is	be	AUX
cana-1395	77	9	clear	clear	ADJ
cana-1395	77	10	that	that	SCONJ
cana-1395	77	11	.	.	PUNCT
cana-1395	78	1	similarly	similarly	ADV
cana-1395	78	2	,	,	PUNCT
cana-1395	78	3	we	we	PRON
cana-1395	78	4	can	can	AUX
cana-1395	78	5	prove	prove	VERB
cana-1395	78	6	for	for	ADP
cana-1395	78	7	and	and	CCONJ
cana-1395	78	8	.	.	PUNCT
cana-1395	79	1	∎	∎	PROPN
cana-1395	79	2	theorem	theorem	VERB
cana-1395	79	3	3.11	3.11	NUM
cana-1395	79	4	:	:	PUNCT
cana-1395	79	5	let	let	AUX
cana-1395	79	6	be	be	AUX
cana-1395	79	7	neutrosophic	neutrosophic	ADJ
cana-1395	79	8	ideal	ideal	NOUN
cana-1395	79	9	of	of	ADP
cana-1395	79	10	a	a	DET
cana-1395	79	11	kk	kk	NOUN
cana-1395	79	12	-	-	PUNCT
cana-1395	79	13	algebra	algebra	PROPN
cana-1395	79	14	and	and	CCONJ
cana-1395	79	15	assume	assume	VERB
cana-1395	79	16	that	that	SCONJ
cana-1395	79	17	,	,	PUNCT
cana-1395	79	18	be	be	AUX
cana-1395	79	19	level	level	NOUN
cana-1395	79	20	ideals	ideal	NOUN
cana-1395	79	21	of	of	ADP
cana-1395	79	22	,	,	PUNCT
cana-1395	79	23	as	as	ADV
cana-1395	79	24	well	well	ADV
cana-1395	79	25	as	as	ADP
cana-1395	79	26	<	<	X
cana-1395	79	27	then	then	ADV
cana-1395	79	28	the	the	DET
cana-1395	79	29	following	follow	VERB
cana-1395	79	30	are	be	AUX
cana-1395	79	31	equivalent	equivalent	ADJ
cana-1395	79	32	(	(	PUNCT
cana-1395	79	33	i	i	NOUN
cana-1395	79	34	)	)	PUNCT
cana-1395	79	35	=	=	SYM
cana-1395	79	36	(	(	PUNCT
cana-1395	79	37	ii	ii	NOUN
cana-1395	79	38	)	)	PUNCT
cana-1395	79	39	there	there	PRON
cana-1395	79	40	is	be	VERB
cana-1395	79	41	no	no	DET
cana-1395	79	42	such	such	ADJ
cana-1395	79	43	that	that	PRON
cana-1395	79	44	.	.	PUNCT
cana-1395	80	1	proof	proof	NOUN
cana-1395	80	2	:	:	PUNCT
cana-1395	80	3	let	let	AUX
cana-1395	80	4	be	be	AUX
cana-1395	80	5	neutrosophic	neutrosophic	ADJ
cana-1395	80	6	ideal	ideal	NOUN
cana-1395	80	7	of	of	ADP
cana-1395	80	8	a	a	DET
cana-1395	80	9	kk	kk	NOUN
cana-1395	80	10	-	-	PUNCT
cana-1395	80	11	algebra	algebra	PROPN
cana-1395	80	12	.	.	PUNCT
cana-1395	81	1	assume	assume	VERB
cana-1395	81	2	that	that	SCONJ
cana-1395	81	3	for	for	ADP
cana-1395	81	4	and	and	CCONJ
cana-1395	81	5	that	that	SCONJ
cana-1395	81	6	there	there	PRON
cana-1395	81	7	exists	exist	VERB
cana-1395	81	8	such	such	ADJ
cana-1395	81	9	that	that	SCONJ
cana-1395	81	10	then	then	ADV
cana-1395	81	11	is	be	AUX
cana-1395	81	12	proper	proper	ADJ
cana-1395	81	13	subset	subset	NOUN
cana-1395	81	14	of	of	ADP
cana-1395	81	15	this	this	PRON
cana-1395	81	16	is	be	AUX
cana-1395	81	17	a	a	DET
cana-1395	81	18	contradiction	contradiction	NOUN
cana-1395	81	19	.	.	PUNCT
cana-1395	82	1	therefore	therefore	ADV
cana-1395	82	2	,	,	PUNCT
cana-1395	82	3	there	there	PRON
cana-1395	82	4	is	be	VERB
cana-1395	82	5	no	no	DET
cana-1395	82	6	such	such	ADJ
cana-1395	82	7	that	that	SCONJ
cana-1395	82	8	conversely	conversely	ADV
cana-1395	82	9	,	,	PUNCT
cana-1395	82	10	suppose	suppose	VERB
cana-1395	82	11	that	that	SCONJ
cana-1395	82	12	there	there	PRON
cana-1395	82	13	is	be	VERB
cana-1395	82	14	no	no	DET
cana-1395	82	15	such	such	ADJ
cana-1395	82	16	that	that	PRON
cana-1395	82	17	.	.	PUNCT
cana-1395	83	1	it	it	PRON
cana-1395	83	2	follows	follow	VERB
cana-1395	83	3	that	that	SCONJ
cana-1395	83	4	<	<	X
cana-1395	83	5	then	then	ADV
cana-1395	83	6	.	.	PUNCT
cana-1395	84	1	let	let	VERB
cana-1395	84	2	then	then	ADV
cana-1395	84	3	and	and	CCONJ
cana-1395	84	4	because	because	SCONJ
cana-1395	84	5	does	do	AUX
cana-1395	84	6	not	not	PART
cana-1395	84	7	lie	lie	VERB
cana-1395	84	8	between	between	ADP
cana-1395	84	9	and	and	CCONJ
cana-1395	84	10	hence	hence	ADV
cana-1395	84	11	.this	.this	PRON
cana-1395	84	12	implies	imply	VERB
cana-1395	84	13	that	that	SCONJ
cana-1395	84	14	therefore	therefore	ADV
cana-1395	84	15	=	=	X
cana-1395	84	16	.	.	PUNCT
cana-1395	85	1	∎	∎	PROPN
cana-1395	85	2	proposition	proposition	NOUN
cana-1395	85	3	3.12	3.12	NUM
cana-1395	85	4	:	:	PUNCT
cana-1395	85	5	the	the	DET
cana-1395	85	6	intersection	intersection	NOUN
cana-1395	85	7	of	of	ADP
cana-1395	85	8	any	any	DET
cana-1395	85	9	set	set	NOUN
cana-1395	85	10	of	of	ADP
cana-1395	85	11	neutrosophic	neutrosophic	ADJ
cana-1395	85	12	ideals	ideal	NOUN
cana-1395	85	13	of	of	ADP
cana-1395	85	14	kk	kk	NOUN
cana-1395	85	15	-	-	PUNCT
cana-1395	85	16	algebra	algebra	NOUN
cana-1395	85	17	x	x	PUNCT
cana-1395	85	18	is	be	AUX
cana-1395	85	19	also	also	ADV
cana-1395	85	20	neutrosophic	neutrosophic	ADJ
cana-1395	85	21	ideal	ideal	NOUN
cana-1395	85	22	.	.	PUNCT
cana-1395	86	1	proof	proof	NOUN
cana-1395	86	2	:	:	PUNCT
cana-1395	86	3	for	for	ADP
cana-1395	86	4	any	any	PRON
cana-1395	86	5	,	,	PUNCT
cana-1395	86	6	let	let	VERB
cana-1395	86	7	{	{	PUNCT
cana-1395	86	8	be	be	AUX
cana-1395	86	9	a	a	DET
cana-1395	86	10	family	family	NOUN
cana-1395	86	11	of	of	ADP
cana-1395	86	12	neutrosophic	neutrosophic	ADJ
cana-1395	86	13	ideals	ideal	NOUN
cana-1395	86	14	of	of	ADP
cana-1395	86	15	kk	kk	NOUN
cana-1395	86	16	-	-	NOUN
cana-1395	86	17	algebra	algebra	NOUN
cana-1395	86	18	then	then	ADV
cana-1395	86	19	,	,	PUNCT
cana-1395	86	20	,	,	PUNCT
cana-1395	86	21	⋂	⋂	PROPN
cana-1395	86	22	)	)	PUNCT
cana-1395	86	23	(	(	PUNCT
cana-1395	86	24	0	0	X
cana-1395	86	25	)	)	PUNCT
cana-1395	86	26	⋂	⋂	PROPN
cana-1395	86	27	⋂	⋂	PROPN
cana-1395	86	28	communications	communication	NOUN
cana-1395	86	29	on	on	ADP
cana-1395	86	30	applied	apply	VERB
cana-1395	86	31	nonlinear	nonlinear	ADJ
cana-1395	86	32	analysis	analysis	NOUN
cana-1395	86	33	issn	issn	NOUN
cana-1395	86	34	:	:	PUNCT
cana-1395	86	35	1074	1074	NUM
cana-1395	86	36	-	-	PUNCT
cana-1395	86	37	133x	133x	NUM
cana-1395	86	38	vol	vol	NOUN
cana-1395	86	39	31	31	NUM
cana-1395	86	40	no	no	NOUN
cana-1395	86	41	.	.	PUNCT
cana-1395	87	1	7s	7	NOUN
cana-1395	87	2	(	(	PUNCT
cana-1395	87	3	2024	2024	NUM
cana-1395	87	4	)	)	PUNCT
cana-1395	87	5	537	537	NUM
cana-1395	87	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1395	87	7	=	=	PUNCT
cana-1395	87	8	=	=	X
cana-1395	87	9	{	{	PUNCT
cana-1395	87	10	⋂	⋂	PROPN
cana-1395	87	11	⋂	⋂	PROPN
cana-1395	87	12	}	}	PUNCT
cana-1395	87	13	similarly	similarly	ADV
cana-1395	87	14	,	,	PUNCT
cana-1395	87	15	we	we	PRON
cana-1395	87	16	can	can	AUX
cana-1395	87	17	prove	prove	VERB
cana-1395	87	18	for	for	ADP
cana-1395	87	19	and	and	CCONJ
cana-1395	87	20	.∎	.∎	NOUN
cana-1395	87	21	definition	definition	NOUN
cana-1395	87	22	3.13	3.13	NUM
cana-1395	87	23	:	:	PUNCT
cana-1395	87	24	let	let	VERB
cana-1395	87	25	(	(	PUNCT
cana-1395	87	26	,	,	PUNCT
cana-1395	87	27	,	,	PUNCT
cana-1395	87	28	)	)	PUNCT
cana-1395	87	29	and	and	CCONJ
cana-1395	87	30	(	(	PUNCT
cana-1395	87	31	,	,	PUNCT
cana-1395	87	32	,	,	PUNCT
cana-1395	87	33	)	)	PUNCT
cana-1395	87	34	be	be	AUX
cana-1395	87	35	an	an	DET
cana-1395	87	36	any	any	DET
cana-1395	87	37	two	two	NUM
cana-1395	87	38	kk	kk	NOUN
cana-1395	87	39	-	-	PUNCT
cana-1395	87	40	algebras	algebras	PROPN
cana-1395	87	41	and	and	CCONJ
cana-1395	87	42	(	(	PUNCT
cana-1395	87	43	,	,	PUNCT
cana-1395	87	44	,	,	PUNCT
cana-1395	87	45	)	)	PUNCT
cana-1395	87	46	be	be	AUX
cana-1395	87	47	a	a	DET
cana-1395	87	48	mapping	mapping	NOUN
cana-1395	87	49	between	between	ADP
cana-1395	87	50	the	the	DET
cana-1395	87	51	corresponding	correspond	VERB
cana-1395	87	52	nonempty	nonempty	NOUN
cana-1395	87	53	sets	set	NOUN
cana-1395	87	54	and	and	CCONJ
cana-1395	87	55	.	.	PUNCT
cana-1395	88	1	for	for	ADP
cana-1395	88	2	every	every	DET
cana-1395	88	3	neutrosophic	neutrosophic	ADJ
cana-1395	88	4	subset	subset	NOUN
cana-1395	88	5	of	of	ADP
cana-1395	88	6	,	,	PUNCT
cana-1395	88	7	the	the	DET
cana-1395	88	8	neutrosophic	neutrosophic	ADJ
cana-1395	88	9	subset	subset	NOUN
cana-1395	88	10	of	of	ADP
cana-1395	88	11	defined	define	VERB
cana-1395	88	12	"	"	PUNCT
cana-1395	88	13	by	by	ADP
cana-1395	88	14	=	=	PUNCT
cana-1395	88	15	{	{	PUNCT
cana-1395	88	16	is	be	AUX
cana-1395	88	17	called	call	VERB
cana-1395	88	18	as	as	ADP
cana-1395	88	19	the	the	DET
cana-1395	88	20	image	image	NOUN
cana-1395	88	21	of	of	ADP
cana-1395	88	22	under	under	ADP
cana-1395	88	23	f	f	PROPN
cana-1395	88	24	.	.	PUNCT
cana-1395	89	1	similarly	similarly	ADV
cana-1395	89	2	,	,	PUNCT
cana-1395	89	3	is	be	AUX
cana-1395	89	4	the	the	DET
cana-1395	89	5	subset	subset	NOUN
cana-1395	89	6	of	of	ADP
cana-1395	89	7	then	then	ADV
cana-1395	89	8	the	the	DET
cana-1395	89	9	neutrosophic	neutrosophic	ADJ
cana-1395	89	10	subset	subset	NOUN
cana-1395	89	11	=	=	SYM
cana-1395	89	12	(	(	PUNCT
cana-1395	89	13	that	that	PRON
cana-1395	89	14	is	is	ADV
cana-1395	89	15	,	,	PUNCT
cana-1395	89	16	[	[	PUNCT
cana-1395	89	17	]	]	X
cana-1395	89	18	for	for	ADP
cana-1395	89	19	all	all	PRON
cana-1395	89	20	is	be	AUX
cana-1395	89	21	called	call	VERB
cana-1395	89	22	pre	pre	ADJ
cana-1395	89	23	image	image	NOUN
cana-1395	89	24	of	of	ADP
cana-1395	89	25	under	under	ADP
cana-1395	89	26	theorem	theorem	NOUN
cana-1395	89	27	3.14	3.14	NUM
cana-1395	89	28	:	:	PUNCT
cana-1395	89	29	the	the	DET
cana-1395	89	30	neutrosophic	neutrosophic	ADJ
cana-1395	89	31	ideal	ideal	NOUN
cana-1395	89	32	is	be	AUX
cana-1395	89	33	also	also	ADV
cana-1395	89	34	its	its	PRON
cana-1395	89	35	homomorphic	homomorphic	ADJ
cana-1395	89	36	pre	pre	NOUN
cana-1395	89	37	-	-	NOUN
cana-1395	89	38	image	image	NOUN
cana-1395	89	39	.	.	PUNCT
cana-1395	90	1	proof	proof	NOUN
cana-1395	90	2	:	:	PUNCT
cana-1395	90	3	a	a	DET
cana-1395	90	4	homomorphism	homomorphism	NOUN
cana-1395	90	5	of	of	ADP
cana-1395	90	6	kk	kk	PROPN
cana-1395	90	7	-	-	PUNCT
cana-1395	90	8	algebras	algebras	PROPN
cana-1395	90	9	,	,	PUNCT
cana-1395	90	10	denoted	denote	VERB
cana-1395	90	11	by	by	ADP
cana-1395	90	12	where	where	SCONJ
cana-1395	90	13	is	be	AUX
cana-1395	90	14	the	the	DET
cana-1395	90	15	neutrosophic	neutrosophic	ADJ
cana-1395	90	16	ideal	ideal	NOUN
cana-1395	90	17	of	of	ADP
cana-1395	90	18	and	and	CCONJ
cana-1395	90	19	is	be	AUX
cana-1395	90	20	the	the	DET
cana-1395	90	21	pre	pre	ADJ
cana-1395	90	22	image	image	NOUN
cana-1395	90	23	of	of	ADP
cana-1395	90	24	under	under	ADV
cana-1395	90	25	,	,	PUNCT
cana-1395	90	26	then	then	ADV
cana-1395	90	27	[	[	PUNCT
cana-1395	90	28	]	]	X
cana-1395	90	29	for	for	ADP
cana-1395	90	30	every	every	PRON
cana-1395	90	31	.	.	PUNCT
cana-1395	91	1	since	since	SCONJ
cana-1395	91	2	and	and	CCONJ
cana-1395	91	3	is	be	AUX
cana-1395	91	4	neutrosophic	neutrosophic	ADJ
cana-1395	91	5	ideal	ideal	NOUN
cana-1395	91	6	of	of	ADP
cana-1395	91	7	it	it	PRON
cana-1395	91	8	follows	follow	VERB
cana-1395	91	9	that	that	SCONJ
cana-1395	91	10	[	[	PUNCT
cana-1395	91	11	]=	]=	PUNCT
cana-1395	91	12	for	for	ADP
cana-1395	91	13	every	every	PRON
cana-1395	91	14	where	where	SCONJ
cana-1395	91	15	is	be	AUX
cana-1395	91	16	the	the	DET
cana-1395	91	17	zero	zero	NUM
cana-1395	91	18	element	element	NOUN
cana-1395	91	19	of	of	ADP
cana-1395	91	20	but	but	CCONJ
cana-1395	91	21	,	,	PUNCT
cana-1395	91	22	(	(	PUNCT
cana-1395	91	23	)	)	PUNCT
cana-1395	91	24	and	and	CCONJ
cana-1395	91	25	so	so	ADV
cana-1395	91	26	for	for	ADP
cana-1395	91	27	all	all	PRON
cana-1395	91	28	let	let	VERB
cana-1395	91	29	’s	’s	NOUN
cana-1395	91	30	now	now	ADV
cana-1395	91	31	then	then	ADV
cana-1395	91	32	we	we	PRON
cana-1395	91	33	obtain	obtain	VERB
cana-1395	91	34	)	)	PUNCT
cana-1395	91	35	}	}	PUNCT
cana-1395	91	36	}	}	PUNCT
cana-1395	91	37	that	that	PRON
cana-1395	91	38	is	be	AUX
cana-1395	91	39	.	.	PUNCT
cana-1395	92	1	similarly	similarly	ADV
cana-1395	92	2	,	,	PUNCT
cana-1395	92	3	we	we	PRON
cana-1395	92	4	can	can	AUX
cana-1395	92	5	prove	prove	VERB
cana-1395	92	6	for	for	ADP
cana-1395	92	7	and	and	CCONJ
cana-1395	92	8	.∎	.∎	NUM
cana-1395	92	9	definition	definition	NOUN
cana-1395	92	10	3.15	3.15	NUM
cana-1395	92	11	:	:	PUNCT
cana-1395	92	12	a	a	DET
cana-1395	92	13	neutrosophic	neutrosophic	ADJ
cana-1395	92	14	subset	subset	NOUN
cana-1395	92	15	of	of	ADP
cana-1395	92	16	a	a	DET
cana-1395	92	17	set	set	NOUN
cana-1395	92	18	has	have	VERB
cana-1395	92	19	supremum	supremum	ADJ
cana-1395	92	20	property	property	NOUN
cana-1395	92	21	if	if	SCONJ
cana-1395	92	22	for	for	ADP
cana-1395	92	23	any	any	DET
cana-1395	92	24	subset	subset	NOUN
cana-1395	92	25	of	of	ADP
cana-1395	92	26	,	,	PUNCT
cana-1395	92	27	there	there	PRON
cana-1395	92	28	exist	exist	VERB
cana-1395	92	29	such	such	ADJ
cana-1395	92	30	that	that	SCONJ
cana-1395	92	31	(	(	PUNCT
cana-1395	92	32	)	)	PUNCT
cana-1395	92	33	=	=	SYM
cana-1395	92	34	{	{	PUNCT
cana-1395	92	35	(	(	PUNCT
cana-1395	92	36	)	)	PUNCT
cana-1395	92	37	/	/	SYM
cana-1395	92	38	}	}	PUNCT
cana-1395	92	39	.	.	PUNCT
cana-1395	93	1	theorem	theorem	VERB
cana-1395	93	2	3.16	3.16	NUM
cana-1395	93	3	:	:	PUNCT
cana-1395	93	4	let	let	VERB
cana-1395	93	5	(	(	PUNCT
cana-1395	93	6	,	,	PUNCT
cana-1395	93	7	,	,	PUNCT
cana-1395	93	8	)	)	PUNCT
cana-1395	93	9	be	be	AUX
cana-1395	93	10	a	a	DET
cana-1395	93	11	homomorphism	homomorphism	NOUN
cana-1395	93	12	between	between	ADP
cana-1395	93	13	kk	kk	PROPN
cana-1395	93	14	-	-	PUNCT
cana-1395	93	15	algebras	algebras	PROPN
cana-1395	93	16	and	and	CCONJ
cana-1395	93	17	respectively	respectively	ADV
cana-1395	93	18	.	.	PUNCT
cana-1395	94	1	for	for	ADP
cana-1395	94	2	every	every	DET
cana-1395	94	3	neutrosophic	neutrosophic	ADJ
cana-1395	94	4	ideal	ideal	NOUN
cana-1395	94	5	n	n	CCONJ
cana-1395	94	6	in	in	ADP
cana-1395	94	7	x	x	PUNCT
cana-1395	94	8	with	with	ADP
cana-1395	94	9	supremum	supremum	ADJ
cana-1395	94	10	property	property	NOUN
cana-1395	94	11	,	,	PUNCT
cana-1395	94	12	then	then	ADV
cana-1395	94	13	is	be	AUX
cana-1395	94	14	a	a	DET
cana-1395	94	15	neutrosophic	neutrosophic	ADJ
cana-1395	94	16	ideal	ideal	NOUN
cana-1395	94	17	of	of	ADP
cana-1395	94	18	.	.	PUNCT
cana-1395	95	1	proof	proof	NOUN
cana-1395	95	2	:	:	PUNCT
cana-1395	95	3	let	let	VERB
cana-1395	95	4	(	(	PUNCT
cana-1395	95	5	,	,	PUNCT
cana-1395	95	6	,	,	PUNCT
cana-1395	95	7	0	0	NUM
cana-1395	95	8	)	)	PUNCT
cana-1395	95	9	and	and	CCONJ
cana-1395	95	10	(	(	PUNCT
cana-1395	95	11	,	,	PUNCT
cana-1395	95	12	,	,	PUNCT
cana-1395	95	13	)	)	PUNCT
cana-1395	95	14	be	be	AUX
cana-1395	95	15	an	an	DET
cana-1395	95	16	any	any	DET
cana-1395	95	17	two	two	NUM
cana-1395	95	18	kk	kk	NOUN
cana-1395	95	19	-	-	PUNCT
cana-1395	95	20	algebras	algebras	PROPN
cana-1395	95	21	and	and	CCONJ
cana-1395	95	22	(	(	PUNCT
cana-1395	95	23	,	,	PUNCT
cana-1395	95	24	,	,	PUNCT
cana-1395	95	25	)	)	PUNCT
cana-1395	95	26	satisfies	satisfy	VERB
cana-1395	95	27	a	a	DET
cana-1395	95	28	homomorphism	homomorphism	NOUN
cana-1395	95	29	property	property	NOUN
cana-1395	95	30	,	,	PUNCT
cana-1395	95	31	from	from	ADP
cana-1395	95	32	the	the	DET
cana-1395	95	33	definition	definition	NOUN
cana-1395	95	34	of	of	ADP
cana-1395	95	35	3.15	3.15	NUM
cana-1395	95	36	,	,	PUNCT
cana-1395	95	37	=	=	PRON
cana-1395	95	38	{	{	PUNCT
cana-1395	95	39	/	/	SYM
cana-1395	95	40	(	(	PUNCT
cana-1395	95	41	)	)	PUNCT
cana-1395	95	42	}	}	PUNCT
cana-1395	95	43	for	for	ADP
cana-1395	95	44	all	all	PRON
cana-1395	95	45	(	(	PUNCT
cana-1395	95	46	)	)	PUNCT
cana-1395	95	47	we	we	PRON
cana-1395	95	48	have	have	VERB
cana-1395	95	49	to	to	PART
cana-1395	95	50	prove	prove	VERB
cana-1395	95	51	,	,	PUNCT
cana-1395	95	52	assume	assume	VERB
cana-1395	95	53	that	that	SCONJ
cana-1395	95	54	the	the	PRON
cana-1395	95	55	onto	onto	ADP
cana-1395	95	56	homomorphism	homomorphism	NOUN
cana-1395	95	57	of	of	ADP
cana-1395	95	58	the	the	DET
cana-1395	95	59	kk	kk	PROPN
cana-1395	95	60	-	-	PUNCT
cana-1395	95	61	algebra	algebra	PROPN
cana-1395	95	62	is	be	AUX
cana-1395	95	63	,	,	PUNCT
cana-1395	95	64	communications	communication	NOUN
cana-1395	95	65	on	on	ADP
cana-1395	95	66	applied	apply	VERB
cana-1395	95	67	nonlinear	nonlinear	ADJ
cana-1395	95	68	analysis	analysis	NOUN
cana-1395	95	69	issn	issn	NOUN
cana-1395	95	70	:	:	PUNCT
cana-1395	95	71	1074	1074	NUM
cana-1395	95	72	-	-	PUNCT
cana-1395	95	73	133x	133x	NUM
cana-1395	95	74	vol	vol	NOUN
cana-1395	95	75	31	31	NUM
cana-1395	95	76	no	no	NOUN
cana-1395	95	77	.	.	PUNCT
cana-1395	96	1	7s	7	NOUN
cana-1395	96	2	(	(	PUNCT
cana-1395	96	3	2024	2024	NUM
cana-1395	96	4	)	)	PUNCT
cana-1395	96	5	538	538	NUM
cana-1395	96	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1395	96	7	is	be	AUX
cana-1395	96	8	neutrosophic	neutrosophic	ADJ
cana-1395	96	9	ideal	ideal	NOUN
cana-1395	96	10	of	of	ADP
cana-1395	96	11	with	with	ADP
cana-1395	96	12	supremum	supremum	ADJ
cana-1395	96	13	property	property	NOUN
cana-1395	96	14	and	and	CCONJ
cana-1395	96	15	is	be	AUX
cana-1395	96	16	the	the	DET
cana-1395	96	17	image	image	NOUN
cana-1395	96	18	of	of	ADP
cana-1395	96	19	under	under	ADV
cana-1395	96	20	.	.	PUNCT
cana-1395	97	1	since	since	SCONJ
cana-1395	97	2	is	be	AUX
cana-1395	97	3	neutrosophic	neutrosophic	ADJ
cana-1395	97	4	ideal	ideal	NOUN
cana-1395	97	5	of	of	SCONJ
cana-1395	97	6	we	we	PRON
cana-1395	97	7	have	have	VERB
cana-1395	97	8	(	(	PUNCT
cana-1395	97	9	0	0	NUM
cana-1395	97	10	)	)	PUNCT
cana-1395	97	11	(	(	PUNCT
cana-1395	97	12	)	)	PUNCT
cana-1395	97	13	for	for	ADP
cana-1395	97	14	all	all	DET
cana-1395	97	15	note	note	VERB
cana-1395	97	16	that	that	SCONJ
cana-1395	97	17	0	0	NUM
cana-1395	97	18	)	)	PUNCT
cana-1395	97	19	,	,	PUNCT
cana-1395	97	20	where	where	SCONJ
cana-1395	97	21	and	and	CCONJ
cana-1395	97	22	are	be	AUX
cana-1395	97	23	the	the	DET
cana-1395	97	24	zero	zero	NUM
cana-1395	97	25	elements	element	NOUN
cana-1395	97	26	of	of	ADP
cana-1395	97	27	and	and	CCONJ
cana-1395	97	28	respectively	respectively	ADV
cana-1395	97	29	.	.	PUNCT
cana-1395	98	1	thus	thus	ADV
cana-1395	98	2	we	we	PRON
cana-1395	98	3	have	have	VERB
cana-1395	98	4	(	(	PUNCT
cana-1395	98	5	0	0	NUM
cana-1395	98	6	)	)	PUNCT
cana-1395	98	7	.	.	PUNCT
cana-1395	99	1	which	which	PRON
cana-1395	99	2	implies	imply	VERB
cana-1395	99	3	that	that	SCONJ
cana-1395	99	4	for	for	ADP
cana-1395	99	5	any	any	PRON
cana-1395	99	6	.	.	PUNCT
cana-1395	100	1	for	for	ADP
cana-1395	100	2	any	any	PRON
cana-1395	100	3	,	,	PUNCT
cana-1395	100	4	let	let	AUX
cana-1395	100	5	,	,	PUNCT
cana-1395	100	6	be	be	AUX
cana-1395	100	7	such	such	ADJ
cana-1395	100	8	that	that	SCONJ
cana-1395	100	9	(	(	PUNCT
cana-1395	100	10	)	)	PUNCT
cana-1395	101	1	[	[	PUNCT
cana-1395	101	2	]	]	X
cana-1395	101	3	[	[	PUNCT
cana-1395	101	4	[	[	X
cana-1395	101	5	]	]	X
cana-1395	101	6	[	[	PUNCT
cana-1395	101	7	]	]	X
cana-1395	101	8	[	[	PUNCT
cana-1395	101	9	]	]	X
cana-1395	101	10	(	(	PUNCT
cana-1395	101	11	)	)	PUNCT
cana-1395	101	12	.	.	PUNCT
cana-1395	102	1	then	then	ADV
cana-1395	102	2	,	,	PUNCT
cana-1395	102	3	(	(	PUNCT
cana-1395	102	4	)	)	PUNCT
cana-1395	102	5	(	(	PUNCT
cana-1395	102	6	)	)	PUNCT
cana-1395	102	7	(	(	PUNCT
cana-1395	102	8	)	)	PUNCT
cana-1395	102	9	,	,	PUNCT
cana-1395	102	10	(	(	PUNCT
cana-1395	102	11	)	)	PUNCT
cana-1395	102	12	}	}	PUNCT
cana-1395	102	13	{	{	PUNCT
cana-1395	102	14	(	(	PUNCT
cana-1395	102	15	t	t	NOUN
cana-1395	102	16	)	)	PUNCT
cana-1395	102	17	,	,	PUNCT
cana-1395	102	18	(	(	PUNCT
cana-1395	102	19	)	)	PUNCT
cana-1395	102	20	}	}	PUNCT
cana-1395	102	21	{	{	PUNCT
cana-1395	102	22	)	)	PUNCT
cana-1395	102	23	,	,	PUNCT
cana-1395	102	24	)	)	PUNCT
cana-1395	102	25	}	}	PUNCT
cana-1395	102	26	thus	thus	ADV
cana-1395	102	27	,	,	PUNCT
cana-1395	102	28	therefore	therefore	ADV
cana-1395	102	29	,	,	PUNCT
cana-1395	102	30	is	be	AUX
cana-1395	102	31	neutrosophic	neutrosophic	ADJ
cana-1395	102	32	ideal	ideal	NOUN
cana-1395	102	33	of	of	ADP
cana-1395	102	34	.	.	PUNCT
cana-1395	103	1	similarly	similarly	ADV
cana-1395	103	2	,	,	PUNCT
cana-1395	103	3	we	we	PRON
cana-1395	103	4	can	can	AUX
cana-1395	103	5	prove	prove	VERB
cana-1395	103	6	for	for	ADP
cana-1395	103	7	and	and	CCONJ
cana-1395	103	8	.	.	PUNCT
cana-1395	104	1	∎	∎	ADV
cana-1395	104	2	4	4	NUM
cana-1395	104	3	.	.	PUNCT
cana-1395	105	1	conclusions	conclusion	NOUN
cana-1395	105	2	this	this	DET
cana-1395	105	3	paper	paper	NOUN
cana-1395	105	4	starts	start	VERB
cana-1395	105	5	by	by	ADP
cana-1395	105	6	discussing	discuss	VERB
cana-1395	105	7	the	the	DET
cana-1395	105	8	concepts	concept	NOUN
cana-1395	105	9	of	of	ADP
cana-1395	105	10	kk	kk	PROPN
cana-1395	105	11	-	-	PUNCT
cana-1395	105	12	algebra	algebra	PROPN
cana-1395	105	13	and	and	CCONJ
cana-1395	105	14	the	the	DET
cana-1395	105	15	neutrosophic	neutrosophic	ADJ
cana-1395	105	16	ideal	ideal	NOUN
cana-1395	105	17	with	with	ADP
cana-1395	105	18	appropriate	appropriate	ADJ
cana-1395	105	19	examples	example	NOUN
cana-1395	105	20	.	.	PUNCT
cana-1395	106	1	then	then	ADV
cana-1395	106	2	,	,	PUNCT
cana-1395	106	3	we	we	PRON
cana-1395	106	4	look	look	VERB
cana-1395	106	5	into	into	ADP
cana-1395	106	6	various	various	ADJ
cana-1395	106	7	fundamental	fundamental	ADJ
cana-1395	106	8	aspects	aspect	NOUN
cana-1395	106	9	of	of	ADP
cana-1395	106	10	the	the	DET
cana-1395	106	11	neutrosophic	neutrosophic	ADJ
cana-1395	106	12	ideal	ideal	NOUN
cana-1395	106	13	in	in	ADP
cana-1395	106	14	kk	kk	NOUN
cana-1395	106	15	-	-	NOUN
cana-1395	106	16	algebra	algebra	NOUN
cana-1395	106	17	.	.	PUNCT
cana-1395	107	1	we	we	PRON
cana-1395	107	2	also	also	ADV
cana-1395	107	3	need	need	VERB
cana-1395	107	4	to	to	PART
cana-1395	107	5	deal	deal	VERB
cana-1395	107	6	with	with	ADP
cana-1395	107	7	the	the	DET
cana-1395	107	8	homomorphism	homomorphism	NOUN
cana-1395	107	9	of	of	ADP
cana-1395	107	10	image	image	NOUN
cana-1395	107	11	and	and	CCONJ
cana-1395	107	12	inverse	inverse	NOUN
cana-1395	107	13	image	image	NOUN
cana-1395	107	14	of	of	ADP
cana-1395	107	15	neutrosophic	neutrosophic	ADJ
cana-1395	107	16	ideals	ideal	NOUN
cana-1395	107	17	in	in	ADP
cana-1395	107	18	kk	kk	NOUN
cana-1395	107	19	-	-	PUNCT
cana-1395	107	20	algebra	algebra	NOUN
cana-1395	107	21	.	.	PUNCT
cana-1395	108	1	references	reference	NOUN
cana-1395	108	2	[	[	X
cana-1395	108	3	1	1	NUM
cana-1395	108	4	]	]	PUNCT
cana-1395	108	5	andrzej	andrzej	PROPN
cana-1395	108	6	walendziak	walendziak	PROPN
cana-1395	108	7	and	and	CCONJ
cana-1395	108	8	grzegorz	grzegorz	PROPN
cana-1395	108	9	dymek(2015	dymek(2015	PROPN
cana-1395	108	10	)	)	PUNCT
cana-1395	108	11	,	,	PUNCT
cana-1395	108	12	fuzzy	fuzzy	ADJ
cana-1395	108	13	ideals	ideal	NOUN
cana-1395	108	14	of	of	ADP
cana-1395	108	15	bn	bn	NOUN
cana-1395	108	16	-	-	PUNCT
cana-1395	108	17	algebras	algebras	ADJ
cana-1395	108	18	,	,	PUNCT
cana-1395	108	19	hindawi	hindawi	ADJ
cana-1395	108	20	publishing	publishing	NOUN
cana-1395	108	21	corporation	corporation	NOUN
cana-1395	108	22	the	the	DET
cana-1395	108	23	scientific	scientific	ADJ
cana-1395	108	24	world	world	NOUN
cana-1395	108	25	journal	journal	NOUN
cana-1395	108	26	,	,	PUNCT
cana-1395	108	27	volume	volume	NOUN
cana-1395	108	28	2015	2015	NUM
cana-1395	108	29	,	,	PUNCT
cana-1395	108	30	(	(	PUNCT
cana-1395	108	31	9	9	NUM
cana-1395	108	32	pages	page	NOUN
cana-1395	108	33	)	)	PUNCT
cana-1395	108	34	.	.	PUNCT
cana-1395	109	1	[	[	X
cana-1395	109	2	2	2	NUM
cana-1395	109	3	]	]	X
cana-1395	109	4	dr	dr	PROPN
cana-1395	109	5	.	.	PROPN
cana-1395	109	6	areej	areej	PROPN
cana-1395	109	7	tawfeeq	tawfeeq	PROPN
cana-1395	109	8	hameed1	hameed1	PROPN
cana-1395	109	9	,	,	PUNCT
cana-1395	109	10	huda	huda	PROPN
cana-1395	109	11	ali	ali	PROPN
cana-1395	109	12	faleh	faleh	PROPN
cana-1395	109	13	and	and	CCONJ
cana-1395	109	14	dr	dr	PROPN
cana-1395	109	15	.	.	PROPN
cana-1395	109	16	ahmed	ahmed	PROPN
cana-1395	109	17	hamzah	hamzah	PROPN
cana-1395	109	18	abed	abed	PROPN
cana-1395	109	19	,	,	PUNCT
cana-1395	109	20	fuzzy	fuzzy	ADJ
cana-1395	109	21	ideals	ideal	NOUN
cana-1395	109	22	of	of	ADP
cana-1395	109	23	kk	kk	NOUN
cana-1395	109	24	-	-	PUNCT
cana-1395	109	25	algebra	algebra	PROPN
cana-1395	109	26	,	,	PUNCT
cana-1395	109	27	journal	journal	NOUN
cana-1395	109	28	of	of	ADP
cana-1395	109	29	physics	physics	PROPN
cana-1395	109	30	:	:	PUNCT
cana-1395	109	31	conference	conference	NOUN
cana-1395	109	32	series	series	NOUN
cana-1395	109	33	,	,	PUNCT
cana-1395	109	34	1804	1804	NUM
cana-1395	109	35	(	(	PUNCT
cana-1395	109	36	2021	2021	NUM
cana-1395	109	37	)	)	PUNCT
cana-1395	109	38	012066	012066	NUM
cana-1395	109	39	.	.	PUNCT
cana-1395	110	1	[	[	X
cana-1395	110	2	3	3	X
cana-1395	110	3	]	]	X
cana-1395	110	4	s.	s.	PROPN
cana-1395	110	5	asawasamrit	asawasamrit	PROPN
cana-1395	110	6	and	and	CCONJ
cana-1395	110	7	a.	a.	NOUN
cana-1395	110	8	sudprasert	sudprasert	PROPN
cana-1395	110	9	,	,	PUNCT
cana-1395	110	10	a	a	DET
cana-1395	110	11	structure	structure	NOUN
cana-1395	110	12	of	of	ADP
cana-1395	110	13	kk	kk	PROPN
cana-1395	110	14	-	-	PUNCT
cana-1395	110	15	algebras	algebras	PROPN
cana-1395	110	16	and	and	CCONJ
cana-1395	110	17	its	its	PRON
cana-1395	110	18	properties	property	NOUN
cana-1395	110	19	,	,	PUNCT
cana-1395	110	20	international	international	ADJ
cana-1395	110	21	journal	journal	NOUN
cana-1395	110	22	of	of	ADP
cana-1395	110	23	mathematical	mathematical	ADJ
cana-1395	110	24	analysis	analysis	NOUN
cana-1395	110	25	,	,	PUNCT
cana-1395	110	26	volume	volume	NOUN
cana-1395	110	27	6	6	NUM
cana-1395	110	28	,	,	PUNCT
cana-1395	110	29	number	number	NOUN
cana-1395	110	30	21	21	NUM
cana-1395	110	31	(	(	PUNCT
cana-1395	110	32	2012	2012	NUM
cana-1395	110	33	)	)	PUNCT
cana-1395	110	34	,	,	PUNCT
cana-1395	110	35	1035	1035	NUM
cana-1395	110	36	-	-	SYM
cana-1395	110	37	1044	1044	NUM
cana-1395	110	38	.	.	PUNCT
cana-1395	111	1	[	[	X
cana-1395	111	2	4	4	X
cana-1395	111	3	]	]	PUNCT
cana-1395	111	4	s.	s.	PROPN
cana-1395	111	5	asawasamrit	asawasamrit	PROPN
cana-1395	111	6	,	,	PUNCT
cana-1395	111	7	kk	kk	PROPN
cana-1395	111	8	-	-	PUNCT
cana-1395	111	9	isomorphism	isomorphism	PROPN
cana-1395	111	10	and	and	CCONJ
cana-1395	111	11	its	its	PRON
cana-1395	111	12	properties	property	NOUN
cana-1395	111	13	,	,	PUNCT
cana-1395	111	14	international	international	ADJ
cana-1395	111	15	journal	journal	NOUN
cana-1395	111	16	of	of	ADP
cana-1395	111	17	pure	pure	ADJ
cana-1395	111	18	and	and	CCONJ
cana-1395	111	19	applied	applied	ADJ
cana-1395	111	20	mathematics	mathematic	NOUN
cana-1395	111	21	,	,	PUNCT
cana-1395	111	22	volume	volume	NOUN
cana-1395	111	23	78	78	NUM
cana-1395	111	24	,	,	PUNCT
cana-1395	111	25	number	number	NOUN
cana-1395	111	26	1	1	NUM
cana-1395	111	27	(	(	PUNCT
cana-1395	111	28	2012	2012	NUM
cana-1395	111	29	)	)	PUNCT
cana-1395	111	30	,	,	PUNCT
cana-1395	111	31	65	65	NUM
cana-1395	111	32	-	-	SYM
cana-1395	111	33	73	73	NUM
cana-1395	111	34	.	.	PUNCT
cana-1395	112	1	communications	communication	NOUN
cana-1395	112	2	on	on	ADP
cana-1395	112	3	applied	apply	VERB
cana-1395	112	4	nonlinear	nonlinear	ADJ
cana-1395	112	5	analysis	analysis	NOUN
cana-1395	112	6	issn	issn	NOUN
cana-1395	112	7	:	:	PUNCT
cana-1395	112	8	1074	1074	NUM
cana-1395	112	9	-	-	PUNCT
cana-1395	112	10	133x	133x	NUM
cana-1395	112	11	vol	vol	NOUN
cana-1395	112	12	31	31	NUM
cana-1395	112	13	no	no	NOUN
cana-1395	112	14	.	.	PUNCT
cana-1395	113	1	7s	7	NOUN
cana-1395	113	2	(	(	PUNCT
cana-1395	113	3	2024	2024	NUM
cana-1395	113	4	)	)	PUNCT
cana-1395	113	5	539	539	NUM
cana-1395	113	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1395	114	1	[	[	X
cana-1395	114	2	5	5	X
cana-1395	114	3	]	]	PUNCT
cana-1395	114	4	s.	s.	PROPN
cana-1395	114	5	asawasamrit	asawasamrit	PROPN
cana-1395	114	6	and	and	CCONJ
cana-1395	114	7	a.	a.	NOUN
cana-1395	114	8	sudprasert	sudprasert	NOUN
cana-1395	114	9	,	,	PUNCT
cana-1395	114	10	on	on	ADP
cana-1395	114	11	the	the	DET
cana-1395	114	12	special	special	ADJ
cana-1395	114	13	ideals	ideal	NOUN
cana-1395	114	14	in	in	ADP
cana-1395	114	15	kk	kk	PROPN
cana-1395	114	16	-	-	PUNCT
cana-1395	114	17	algebras	algebras	PROPN
cana-1395	114	18	,	,	PUNCT
cana-1395	114	19	international	international	ADJ
cana-1395	114	20	journal	journal	NOUN
cana-1395	114	21	of	of	ADP
cana-1395	114	22	pure	pure	ADJ
cana-1395	114	23	and	and	CCONJ
cana-1395	114	24	applied	applied	ADJ
cana-1395	114	25	mathematics	mathematic	NOUN
cana-1395	114	26	,	,	PUNCT
cana-1395	114	27	volume	volume	NOUN
cana-1395	114	28	82	82	NUM
cana-1395	114	29	,	,	PUNCT
cana-1395	114	30	number	number	NOUN
cana-1395	114	31	4	4	NUM
cana-1395	114	32	(	(	PUNCT
cana-1395	114	33	2013	2013	NUM
cana-1395	114	34	)	)	PUNCT
cana-1395	114	35	,	,	PUNCT
cana-1395	114	36	605	605	NUM
cana-1395	114	37	-	-	SYM
cana-1395	114	38	613	613	NUM
cana-1395	114	39	.	.	PUNCT
cana-1395	115	1	[	[	X
cana-1395	115	2	6	6	NUM
cana-1395	115	3	]	]	PUNCT
cana-1395	115	4	s.asawasamrit	s.asawasamrit	NOUN
cana-1395	115	5	,	,	PUNCT
cana-1395	115	6	a.	a.	NOUN
cana-1395	115	7	sudprasert	sudprasert	NOUN
cana-1395	115	8	,	,	PUNCT
cana-1395	115	9	on	on	ADP
cana-1395	115	10	p	p	NOUN
cana-1395	115	11	-	-	PUNCT
cana-1395	115	12	semisimple	semisimple	NOUN
cana-1395	115	13	in	in	ADP
cana-1395	115	14	kk	kk	PROPN
cana-1395	115	15	-	-	PUNCT
cana-1395	115	16	algebras	algebras	PROPN
cana-1395	115	17	,	,	PUNCT
cana-1395	115	18	international	international	ADJ
cana-1395	115	19	journal	journal	NOUN
cana-1395	115	20	of	of	ADP
cana-1395	115	21	pure	pure	ADJ
cana-1395	115	22	and	and	CCONJ
cana-1395	115	23	applied	applied	ADJ
cana-1395	115	24	mathematics	mathematic	NOUN
cana-1395	115	25	,	,	PUNCT
cana-1395	115	26	volume	volume	NOUN
cana-1395	115	27	98	98	NUM
cana-1395	115	28	,	,	PUNCT
cana-1395	115	29	number.1	number.1	NOUN
cana-1395	115	30	,	,	PUNCT
cana-1395	115	31	(	(	PUNCT
cana-1395	115	32	2015	2015	NUM
cana-1395	115	33	)	)	PUNCT
cana-1395	115	34	,	,	PUNCT
cana-1395	115	35	23	23	NUM
cana-1395	115	36	-	-	SYM
cana-1395	115	37	32	32	NUM
cana-1395	115	38	.	.	PUNCT
cana-1395	116	1	[	[	X
cana-1395	116	2	7	7	NUM
cana-1395	116	3	]	]	PUNCT
cana-1395	116	4	a.	a.	NOUN
cana-1395	116	5	ibrahim	ibrahim	PROPN
cana-1395	116	6	,	,	PUNCT
cana-1395	116	7	and	and	CCONJ
cana-1395	116	8	b.	b.	PROPN
cana-1395	116	9	kavitha	kavitha	PROPN
cana-1395	116	10	,	,	PUNCT
cana-1395	116	11	neutrosophic	neutrosophic	ADJ
cana-1395	116	12	ideals	ideal	NOUN
cana-1395	116	13	of	of	ADP
cana-1395	116	14	bnalgebra	bnalgebra	PROPN
cana-1395	116	15	,	,	PUNCT
cana-1395	116	16	mathematical	mathematical	PROPN
cana-1395	116	17	forum	forum	PROPN
cana-1395	116	18	,	,	PUNCT
cana-1395	116	19	mathematical	mathematical	ADJ
cana-1395	116	20	forum	forum	NOUN
cana-1395	116	21	volume	volume	NOUN
cana-1395	116	22	30	30	NUM
cana-1395	116	23	(	(	PUNCT
cana-1395	116	24	2022	2022	NUM
cana-1395	116	25	)	)	PUNCT
cana-1395	116	26	,	,	PUNCT
cana-1395	116	27	issn	issn	PROPN
cana-1395	116	28	:	:	PUNCT
cana-1395	116	29	0972	0972	NUM
cana-1395	116	30	-	-	SYM
cana-1395	116	31	9852	9852	NUM
cana-1395	116	32	.	.	PUNCT
cana-1395	117	1	[	[	X
cana-1395	117	2	8	8	NUM
cana-1395	117	3	]	]	X
cana-1395	117	4	c.	c.	NOUN
cana-1395	117	5	prabpayak	prabpayak	NOUN
cana-1395	117	6	and	and	CCONJ
cana-1395	117	7	u.	u.	NOUN
cana-1395	117	8	leerawat	leerawat	PROPN
cana-1395	117	9	,	,	PUNCT
cana-1395	117	10	on	on	ADP
cana-1395	117	11	ideals	ideal	NOUN
cana-1395	117	12	and	and	CCONJ
cana-1395	117	13	congurences	congurence	NOUN
cana-1395	117	14	in	in	ADP
cana-1395	117	15	kualgebras	kualgebra	NOUN
cana-1395	117	16	,	,	PUNCT
cana-1395	117	17	scientia	scientia	PROPN
cana-1395	117	18	magna	magna	PROPN
cana-1395	117	19	journal	journal	PROPN
cana-1395	117	20	,	,	PUNCT
cana-1395	117	21	volume	volume	NOUN
cana-1395	117	22	5	5	NUM
cana-1395	117	23	,	,	PUNCT
cana-1395	117	24	number	number	NOUN
cana-1395	117	25	1	1	NUM
cana-1395	117	26	(	(	PUNCT
cana-1395	117	27	2009	2009	NUM
cana-1395	117	28	)	)	PUNCT
cana-1395	117	29	,	,	PUNCT
cana-1395	117	30	54	54	NUM
cana-1395	117	31	-	-	SYM
cana-1395	117	32	57	57	NUM
cana-1395	117	33	.	.	PUNCT
cana-1395	118	1	[	[	X
cana-1395	118	2	9	9	NUM
cana-1395	118	3	]	]	PUNCT
cana-1395	118	4	c.	c.	NOUN
cana-1395	118	5	prabpayak	prabpayak	NOUN
cana-1395	118	6	and	and	CCONJ
cana-1395	118	7	u.	u.	NOUN
cana-1395	118	8	leerawat	leerawat	PROPN
cana-1395	118	9	,	,	PUNCT
cana-1395	118	10	on	on	ADP
cana-1395	118	11	isomorphisms	isomorphism	NOUN
cana-1395	118	12	of	of	ADP
cana-1395	118	13	ku	ku	PROPN
cana-1395	118	14	-	-	PUNCT
cana-1395	118	15	algebras	algebras	PROPN
cana-1395	118	16	,	,	PUNCT
cana-1395	118	17	scientia	scientia	PROPN
cana-1395	118	18	magna	magna	PROPN
cana-1395	118	19	journal	journal	PROPN
cana-1395	118	20	,	,	PUNCT
cana-1395	118	21	volume	volume	NOUN
cana-1395	118	22	5	5	NUM
cana-1395	118	23	,	,	PUNCT
cana-1395	118	24	number	number	NOUN
cana-1395	118	25	3	3	NUM
cana-1395	118	26	,	,	PUNCT
cana-1395	118	27	(	(	PUNCT
cana-1395	118	28	2009	2009	NUM
cana-1395	118	29	)	)	PUNCT
cana-1395	118	30	,	,	PUNCT
cana-1395	118	31	25	25	NUM
cana-1395	118	32	-	-	SYM
cana-1395	118	33	31	31	NUM
cana-1395	118	34	.	.	PUNCT
cana-1395	119	1	[	[	X
cana-1395	119	2	10	10	NUM
cana-1395	119	3	]	]	X
cana-1395	119	4	smarandache	smarandache	NOUN
cana-1395	119	5	florentin	florentin	PROPN
cana-1395	119	6	(	(	PUNCT
cana-1395	119	7	2005	2005	NUM
cana-1395	119	8	)	)	PUNCT
cana-1395	119	9	,	,	PUNCT
cana-1395	119	10	neutrosophic	neutrosophic	ADJ
cana-1395	119	11	set	set	NOUN
cana-1395	119	12	-	-	PUNCT
cana-1395	119	13	a	a	DET
cana-1395	119	14	generalization	generalization	NOUN
cana-1395	119	15	of	of	ADP
cana-1395	119	16	the	the	DET
cana-1395	119	17	intuitionistic	intuitionistic	ADJ
cana-1395	119	18	fuzzy	fuzzy	ADJ
cana-1395	119	19	set	set	NOUN
cana-1395	119	20	,	,	PUNCT
cana-1395	119	21	international	international	ADJ
cana-1395	119	22	journal	journal	NOUN
cana-1395	119	23	of	of	ADP
cana-1395	119	24	pure	pure	ADJ
cana-1395	119	25	and	and	CCONJ
cana-1395	119	26	applied	applied	ADJ
cana-1395	119	27	mathematics	mathematic	NOUN
cana-1395	119	28	,	,	PUNCT
cana-1395	119	29	volume	volume	NOUN
cana-1395	119	30	24	24	NUM
cana-1395	119	31	,	,	PUNCT
cana-1395	119	32	287	287	NUM
cana-1395	119	33	-	-	SYM
cana-1395	119	34	297	297	NUM
cana-1395	119	35	.	.	PUNCT
cana-1395	120	1	[	[	X
cana-1395	120	2	11	11	NUM
cana-1395	120	3	]	]	X
cana-1395	120	4	l.a	l.a	PROPN
cana-1395	120	5	.	.	PROPN
cana-1395	120	6	zadeh	zadeh	PROPN
cana-1395	120	7	,	,	PUNCT
cana-1395	120	8	fuzzy	fuzzy	ADJ
cana-1395	120	9	sets	set	NOUN
cana-1395	120	10	,	,	PUNCT
cana-1395	120	11	information	information	NOUN
cana-1395	120	12	and	and	CCONJ
cana-1395	120	13	control	control	NOUN
cana-1395	120	14	,	,	PUNCT
cana-1395	120	15	volume	volume	NOUN
cana-1395	120	16	8	8	NUM
cana-1395	120	17	(	(	PUNCT
cana-1395	120	18	1965	1965	NUM
cana-1395	120	19	)	)	PUNCT
cana-1395	120	20	,	,	PUNCT
cana-1395	120	21	338353	338353	NUM
cana-1395	120	22	.	.	PUNCT