id	sid	tid	token	lemma	pos
cana-1620	1	1	communications	communication	NOUN
cana-1620	1	2	on	on	ADP
cana-1620	1	3	applied	apply	VERB
cana-1620	1	4	nonlinear	nonlinear	ADJ
cana-1620	1	5	analysis	analysis	NOUN
cana-1620	1	6	issn	issn	NOUN
cana-1620	1	7	:	:	PUNCT
cana-1620	1	8	1074	1074	NUM
cana-1620	1	9	-	-	PUNCT
cana-1620	1	10	133x	133x	NUM
cana-1620	1	11	vol	vol	NOUN
cana-1620	1	12	32	32	NUM
cana-1620	1	13	no	no	NOUN
cana-1620	1	14	.	.	NOUN
cana-1620	1	15	1	1	NUM
cana-1620	1	16	(	(	PUNCT
cana-1620	1	17	2025	2025	NUM
cana-1620	1	18	)	)	PUNCT
cana-1620	1	19	56	56	NUM
cana-1620	1	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1620	1	21	on	on	ADP
cana-1620	1	22	arithmetical	arithmetical	ADJ
cana-1620	1	23	traits	trait	NOUN
cana-1620	1	24	of	of	ADP
cana-1620	1	25	doubt	doubt	NOUN
cana-1620	1	26	fuzzy	fuzzy	ADJ
cana-1620	1	27	t	t	NOUN
cana-1620	1	28	-	-	PUNCT
cana-1620	1	29	ideals	ideal	NOUN
cana-1620	1	30	beneath	beneath	ADP
cana-1620	1	31	the	the	DET
cana-1620	1	32	normalization	normalization	NOUN
cana-1620	1	33	is	be	AUX
cana-1620	1	34	a	a	DET
cana-1620	1	35	t	t	NOUN
cana-1620	1	36	-	-	PUNCT
cana-1620	1	37	algebra	algebra	NOUN
cana-1620	2	1	ct	ct	PROPN
cana-1620	2	2	.	.	PUNCT
cana-1620	3	1	nagaraj(a	nagaraj(a	ADV
cana-1620	3	2	)	)	PUNCT
cana-1620	3	3	,	,	PUNCT
cana-1620	4	1	m.	m.	NOUN
cana-1620	4	2	premkumar	premkumar	PROPN
cana-1620	4	3	,	,	PUNCT
cana-1620	4	4	y.	y.	PROPN
cana-1620	4	5	immanuel(b	immanuel(b	PROPN
cana-1620	4	6	)	)	PUNCT
cana-1620	4	7	,	,	PUNCT
cana-1620	4	8	abdul	abdul	PROPN
cana-1620	4	9	salam(c	salam(c	PROPN
cana-1620	4	10	)	)	PUNCT
cana-1620	4	11	,	,	PUNCT
cana-1620	4	12	m.	m.	NOUN
cana-1620	4	13	s	s	PART
cana-1620	4	14	franklin	franklin	PROPN
cana-1620	4	15	thamil	thamil	PROPN
cana-1620	4	16	selvi(d	selvi(d	PROPN
cana-1620	4	17	)	)	PUNCT
cana-1620	4	18	,	,	PUNCT
cana-1620	4	19	m.	m.	NOUN
cana-1620	4	20	i.	i.	PROPN
cana-1620	4	21	mary	mary	PROPN
cana-1620	4	22	metilda(e	metilda(e	PROPN
cana-1620	4	23	)	)	PUNCT
cana-1620	4	24	and	and	CCONJ
cana-1620	4	25	j.	j.	PROPN
cana-1620	4	26	juliet	juliet	PROPN
cana-1620	4	27	jeyapackiam(f	jeyapackiam(f	PROPN
cana-1620	4	28	)	)	PUNCT
cana-1620	4	29	(	(	PUNCT
cana-1620	4	30	a)department	a)department	NOUN
cana-1620	4	31	of	of	ADP
cana-1620	4	32	mathematics	mathematics	PROPN
cana-1620	4	33	,	,	PUNCT
cana-1620	4	34	sree	sree	PROPN
cana-1620	4	35	sevugan	sevugan	PROPN
cana-1620	4	36	annamalai	annamalai	PROPN
cana-1620	4	37	college	college	PROPN
cana-1620	4	38	,	,	PUNCT
cana-1620	4	39	devakottai-630303	devakottai-630303	ADJ
cana-1620	4	40	,	,	PUNCT
cana-1620	4	41	india*(1a)[0000	india*(1a)[0000	NUM
cana-1620	4	42	-	-	PUNCT
cana-1620	4	43	0002	0002	NUM
cana-1620	4	44	-	-	PUNCT
cana-1620	4	45	8637	8637	NUM
cana-1620	4	46	-	-	PUNCT
cana-1620	4	47	063x	063x	NOUN
cana-1620	4	48	]	]	PUNCT
cana-1620	4	49	(	(	PUNCT
cana-1620	4	50	*	*	NOUN
cana-1620	4	51	1a	1a	X
cana-1620	4	52	,	,	PUNCT
cana-1620	4	53	b	b	X
cana-1620	4	54	,	,	PUNCT
cana-1620	4	55	d	d	PROPN
cana-1620	4	56	&	&	CCONJ
cana-1620	4	57	e	e	NOUN
cana-1620	4	58	)	)	PUNCT
cana-1620	4	59	department	department	NOUN
cana-1620	4	60	of	of	ADP
cana-1620	4	61	mathematics	mathematic	NOUN
cana-1620	4	62	,	,	PUNCT
cana-1620	4	63	sathyabama	sathyabama	PROPN
cana-1620	4	64	institute	institute	PROPN
cana-1620	4	65	of	of	ADP
cana-1620	4	66	science	science	NOUN
cana-1620	4	67	and	and	CCONJ
cana-1620	4	68	technology	technology	NOUN
cana-1620	4	69	(	(	PUNCT
cana-1620	4	70	deemed	deem	VERB
cana-1620	4	71	to	to	PART
cana-1620	4	72	be	be	AUX
cana-1620	4	73	university	university	NOUN
cana-1620	4	74	)	)	PUNCT
cana-1620	4	75	chennai-600119	chennai-600119	ADJ
cana-1620	4	76	,	,	PUNCT
cana-1620	4	77	tamilnadu	tamilnadu	ADJ
cana-1620	4	78	,	,	PUNCT
cana-1620	4	79	india	india	PROPN
cana-1620	4	80	.	.	PUNCT
cana-1620	5	1	(	(	PUNCT
cana-1620	5	2	b)[0000	b)[0000	NUM
cana-1620	5	3	-	-	PUNCT
cana-1620	5	4	0003	0003	NUM
cana-1620	5	5	-	-	PUNCT
cana-1620	5	6	0719	0719	NUM
cana-1620	5	7	-	-	PUNCT
cana-1620	5	8	375x	375x	NOUN
cana-1620	5	9	]	]	PUNCT
cana-1620	5	10	(	(	PUNCT
cana-1620	5	11	c)gulf	c)gulf	PROPN
cana-1620	5	12	asian	asian	PROPN
cana-1620	5	13	english	english	PROPN
cana-1620	5	14	school	school	PROPN
cana-1620	5	15	,	,	PUNCT
cana-1620	5	16	sharjah	sharjah	PROPN
cana-1620	5	17	,	,	PUNCT
cana-1620	5	18	united	united	PROPN
cana-1620	5	19	arab	arab	PROPN
cana-1620	5	20	emirates	emirates	PROPN
cana-1620	5	21	.	.	PUNCT
cana-1620	6	1	(	(	PUNCT
cana-1620	6	2	f)department	f)department	NOUN
cana-1620	6	3	of	of	ADP
cana-1620	6	4	mathematics	mathematic	NOUN
cana-1620	6	5	,	,	PUNCT
cana-1620	6	6	jayaraj	jayaraj	PROPN
cana-1620	6	7	annapackiam	annapackiam	VERB
cana-1620	6	8	csi	csi	PROPN
cana-1620	6	9	college	college	PROPN
cana-1620	6	10	of	of	ADP
cana-1620	6	11	engineering	engineering	PROPN
cana-1620	6	12	nazareth	nazareth	PROPN
cana-1620	6	13	,	,	PUNCT
cana-1620	6	14	tuticorin-628617	tuticorin-628617	NOUN
cana-1620	6	15	,	,	PUNCT
cana-1620	6	16	india	india	PROPN
cana-1620	6	17	.	.	PUNCT
cana-1620	7	1	(	(	PUNCT
cana-1620	7	2	a)mathsnagaraj.ct@gmail.com	a)mathsnagaraj.ct@gmail.com	X
cana-1620	7	3	,	,	PUNCT
cana-1620	7	4	(	(	PUNCT
cana-1620	7	5	*	*	NOUN
cana-1620	7	6	1a	1a	NUM
cana-1620	7	7	)	)	PUNCT
cana-1620	7	8	mprem.maths3033@gmail.com	mprem.maths3033@gmail.com	PROPN
cana-1620	7	9	,	,	PUNCT
cana-1620	7	10	(	(	PUNCT
cana-1620	7	11	b)y_immanuel@yahoo.com	b)y_immanuel@yahoo.com	X
cana-1620	7	12	,	,	PUNCT
cana-1620	7	13	(	(	PUNCT
cana-1620	7	14	c)abdulsalam.maths@gmail.com	c)abdulsalam.maths@gmail.com	X
cana-1620	7	15	,	,	PUNCT
cana-1620	7	16	(	(	PUNCT
cana-1620	7	17	d)thamizanand@gmail.com	d)thamizanand@gmail.com	PROPN
cana-1620	7	18	,	,	PUNCT
cana-1620	7	19	(	(	PUNCT
cana-1620	7	20	e)metilda81@gmail.com	e)metilda81@gmail.com	X
cana-1620	7	21	(	(	PUNCT
cana-1620	7	22	f	f	X
cana-1620	7	23	)	)	PUNCT
cana-1620	7	24	jeyasjjjeyas@gmail.com	jeyasjjjeyas@gmail.com	PROPN
cana-1620	7	25	,	,	PUNCT
cana-1620	7	26	corresponding	correspond	VERB
cana-1620	7	27	author	author	NOUN
cana-1620	7	28	email	email	NOUN
cana-1620	7	29	i	i	PROPN
cana-1620	7	30	d	d	PROPN
cana-1620	7	31	:	:	PUNCT
cana-1620	7	32	(	(	PUNCT
cana-1620	7	33	*	*	NOUN
cana-1620	7	34	1a	1a	X
cana-1620	7	35	)	)	PUNCT
cana-1620	7	36	mprem.maths3033@gmail.com[0000	mprem.maths3033@gmail.com[0000	NUM
cana-1620	7	37	-	-	PUNCT
cana-1620	7	38	0003	0003	NUM
cana-1620	7	39	-	-	PUNCT
cana-1620	7	40	4656	4656	NUM
cana-1620	7	41	-	-	PUNCT
cana-1620	7	42	3370	3370	NUM
cana-1620	7	43	]	]	PUNCT
cana-1620	7	44	article	article	NOUN
cana-1620	7	45	history	history	NOUN
cana-1620	7	46	:	:	PUNCT
cana-1620	7	47	received	receive	VERB
cana-1620	7	48	:	:	PUNCT
cana-1620	7	49	06	06	NUM
cana-1620	7	50	-	-	SYM
cana-1620	7	51	07	07	NUM
cana-1620	7	52	-	-	PUNCT
cana-1620	7	53	2024	2024	NUM
cana-1620	7	54	revised	revise	VERB
cana-1620	7	55	:	:	PUNCT
cana-1620	7	56	21	21	NUM
cana-1620	7	57	-	-	SYM
cana-1620	7	58	08	08	NUM
cana-1620	7	59	-	-	PUNCT
cana-1620	7	60	2024	2024	NUM
cana-1620	7	61	accepted	accept	VERB
cana-1620	7	62	:	:	PUNCT
cana-1620	7	63	03	03	NUM
cana-1620	7	64	-	-	PUNCT
cana-1620	7	65	09	09	NUM
cana-1620	7	66	-	-	PUNCT
cana-1620	7	67	2024	2024	NUM
cana-1620	7	68	abstract	abstract	NOUN
cana-1620	7	69	:	:	PUNCT
cana-1620	7	70	the	the	DET
cana-1620	7	71	normal	normal	ADJ
cana-1620	7	72	doubt	doubt	NOUN
cana-1620	7	73	fuzzy	fuzzy	ADJ
cana-1620	7	74	t	t	NOUN
cana-1620	7	75	-	-	PUNCT
cana-1620	7	76	ideal	ideal	NOUN
cana-1620	7	77	and	and	CCONJ
cana-1620	7	78	poset	poset	VERB
cana-1620	7	79	under	under	ADP
cana-1620	7	80	the	the	DET
cana-1620	7	81	set	set	NOUN
cana-1620	7	82	of	of	ADP
cana-1620	7	83	inclusion	inclusion	NOUN
cana-1620	7	84	principle	principle	NOUN
cana-1620	7	85	in	in	ADP
cana-1620	7	86	t	t	PROPN
cana-1620	7	87	-	-	PUNCT
cana-1620	7	88	algebra	algebra	NOUN
cana-1620	7	89	are	be	AUX
cana-1620	7	90	defined	define	VERB
cana-1620	7	91	in	in	ADP
cana-1620	7	92	this	this	DET
cana-1620	7	93	article	article	NOUN
cana-1620	7	94	,	,	PUNCT
cana-1620	7	95	along	along	ADP
cana-1620	7	96	with	with	ADP
cana-1620	7	97	several	several	ADJ
cana-1620	7	98	algebraic	algebraic	ADJ
cana-1620	7	99	properties	property	NOUN
cana-1620	7	100	and	and	CCONJ
cana-1620	7	101	instances	instance	NOUN
cana-1620	7	102	that	that	PRON
cana-1620	7	103	are	be	AUX
cana-1620	7	104	covered	cover	VERB
cana-1620	7	105	in	in	ADP
cana-1620	7	106	detail	detail	NOUN
cana-1620	7	107	.	.	PUNCT
cana-1620	8	1	keywords	keyword	NOUN
cana-1620	8	2	:	:	PUNCT
cana-1620	8	3	doubt	doubt	ADV
cana-1620	8	4	fuzzy	fuzzy	ADJ
cana-1620	8	5	set	set	NOUN
cana-1620	8	6	(	(	PUNCT
cana-1620	8	7	dfs	dfs	PROPN
cana-1620	8	8	)	)	PUNCT
cana-1620	8	9	,	,	PUNCT
cana-1620	8	10	doubt	doubt	VERB
cana-1620	8	11	fuzzy	fuzzy	ADJ
cana-1620	8	12	subset	subset	NOUN
cana-1620	8	13	(	(	PUNCT
cana-1620	8	14	dfsb),t	dfsb),t	NOUN
cana-1620	8	15	-	-	NOUN
cana-1620	8	16	algebra	algebra	PROPN
cana-1620	8	17	,	,	PUNCT
cana-1620	8	18	t	t	NOUN
cana-1620	8	19	-	-	PUNCT
cana-1620	8	20	ideal	ideal	NOUN
cana-1620	8	21	,	,	PUNCT
cana-1620	8	22	doubt	doubt	VERB
cana-1620	8	23	fuzzy	fuzzy	ADJ
cana-1620	8	24	t	t	NOUN
cana-1620	8	25	-	-	PUNCT
cana-1620	8	26	ideal	ideal	NOUN
cana-1620	8	27	(	(	PUNCT
cana-1620	8	28	dfti	dfti	NOUN
cana-1620	8	29	)	)	PUNCT
cana-1620	8	30	,	,	PUNCT
cana-1620	8	31	normal	normal	ADJ
cana-1620	8	32	doubt	doubt	VERB
cana-1620	8	33	fuzzy	fuzzy	ADJ
cana-1620	8	34	t	t	NOUN
cana-1620	8	35	-	-	PUNCT
cana-1620	8	36	ideal	ideal	NOUN
cana-1620	8	37	(	(	PUNCT
cana-1620	8	38	ndfti	ndfti	NOUN
cana-1620	8	39	)	)	PUNCT
cana-1620	8	40	.	.	PUNCT
cana-1620	9	1	classification	classification	NOUN
cana-1620	9	2	of	of	ADP
cana-1620	9	3	subject	subject	NOUN
cana-1620	9	4	:	:	PUNCT
cana-1620	9	5	msc2020	msc2020	PROPN
cana-1620	9	6	-	-	PUNCT
cana-1620	9	7	zbmath-03b52	zbmath-03b52	PROPN
cana-1620	9	8	i.	i.	PROPN
cana-1620	9	9	introduction	introduction	NOUN
cana-1620	9	10	abu	abu	PROPN
cana-1620	9	11	ayub	ayub	PROPN
cana-1620	9	12	ansari[1	ansari[1	PROPN
cana-1620	9	13	]	]	PUNCT
cana-1620	9	14	introduced	introduce	VERB
cana-1620	9	15	the	the	DET
cana-1620	9	16	novel	novel	ADJ
cana-1620	9	17	idea	idea	NOUN
cana-1620	9	18	of	of	ADP
cana-1620	9	19	t	t	NOUN
cana-1620	9	20	-	-	PUNCT
cana-1620	9	21	fβsa	fβsa	NOUN
cana-1620	9	22	of	of	ADP
cana-1620	9	23	β	β	NOUN
cana-1620	9	24	-	-	PUNCT
cana-1620	9	25	algebras	algebra	VERB
cana-1620	9	26	in	in	ADP
cana-1620	9	27	2014	2014	NUM
cana-1620	9	28	.	.	PUNCT
cana-1620	10	1	prasanna	prasanna	PROPN
cana-1620	10	2	,	,	PUNCT
cana-1620	10	3	a.	a.	PROPN
cana-1620	10	4	,	,	PUNCT
cana-1620	10	5	et	et	PROPN
cana-1620	10	6	al	al	PROPN
cana-1620	10	7	.	.	PUNCT
cana-1620	11	1	[	[	X
cana-1620	11	2	2&3	2&3	X
cana-1620	11	3	]	]	PUNCT
cana-1620	11	4	.	.	PUNCT
cana-1620	12	1	outlined	outline	VERB
cana-1620	12	2	the	the	DET
cana-1620	12	3	new	new	ADJ
cana-1620	12	4	fbi	fbi	PROPN
cana-1620	12	5	normalization	normalization	NOUN
cana-1620	12	6	notation	notation	NOUN
cana-1620	12	7	in	in	ADP
cana-1620	12	8	b	b	NOUN
cana-1620	12	9	-	-	PUNCT
cana-1620	12	10	algebra	algebra	NOUN
cana-1620	12	11	and	and	CCONJ
cana-1620	12	12	presented	present	VERB
cana-1620	12	13	the	the	DET
cana-1620	12	14	idea	idea	NOUN
cana-1620	12	15	of	of	ADP
cana-1620	12	16	fbgi	fbgi	NOUN
cana-1620	12	17	normalization	normalization	NOUN
cana-1620	12	18	in	in	ADP
cana-1620	12	19	bg	bg	NOUN
cana-1620	12	20	-	-	NOUN
cana-1620	12	21	algebra	algebra	NOUN
cana-1620	12	22	in	in	ADP
cana-1620	12	23	2018	2018	NUM
cana-1620	12	24	.	.	PUNCT
cana-1620	13	1	priya	priya	PROPN
cana-1620	13	2	's	's	PART
cana-1620	13	3	fpsis	fpsis	NOUN
cana-1620	13	4	and	and	CCONJ
cana-1620	13	5	fpssas	fpssa	NOUN
cana-1620	13	6	for	for	ADP
cana-1620	13	7	ps	ps	NOUN
cana-1620	13	8	-	-	PUNCT
cana-1620	13	9	algebras	algebras	PROPN
cana-1620	13	10	were	be	AUX
cana-1620	13	11	standardized	standardize	VERB
cana-1620	13	12	in	in	ADP
cana-1620	13	13	2015[4	2015[4	NUM
cana-1620	13	14	]	]	PUNCT
cana-1620	13	15	.	.	PUNCT
cana-1620	14	1	in	in	ADP
cana-1620	14	2	2015	2015	NUM
cana-1620	14	3	,	,	PUNCT
cana-1620	14	4	rajam[5	rajam[5	PROPN
cana-1620	14	5	]	]	PUNCT
cana-1620	14	6	presented	present	VERB
cana-1620	14	7	the	the	DET
cana-1620	14	8	idea	idea	NOUN
cana-1620	14	9	of	of	ADP
cana-1620	14	10	l	l	PROPN
cana-1620	14	11	-	-	NOUN
cana-1620	14	12	fti	fti	PROPN
cana-1620	14	13	in	in	ADP
cana-1620	14	14	β	β	NOUN
cana-1620	14	15	-	-	PUNCT
cana-1620	14	16	algebras	algebras	X
cana-1620	14	17	.	.	PUNCT
cana-1620	15	1	in	in	ADP
cana-1620	15	2	2016	2016	NUM
cana-1620	15	3	,	,	PUNCT
cana-1620	15	4	sithar	sithar	NOUN
cana-1620	15	5	selvam[6	selvam[6	NOUN
cana-1620	15	6	]	]	PUNCT
cana-1620	15	7	learned	learn	VERB
cana-1620	15	8	about	about	ADP
cana-1620	15	9	the	the	DET
cana-1620	15	10	fpmsa	fpmsa	ADJ
cana-1620	15	11	normalization	normalization	NOUN
cana-1620	15	12	study	study	NOUN
cana-1620	15	13	.	.	PUNCT
cana-1620	16	1	tamil	tamil	PROPN
cana-1620	16	2	created	create	VERB
cana-1620	16	3	fsa	fsa	PROPN
cana-1620	16	4	and	and	CCONJ
cana-1620	16	5	fti	fti	PROPN
cana-1620	16	6	in	in	ADP
cana-1620	16	7	tm	tm	PROPN
cana-1620	16	8	-	-	PUNCT
cana-1620	16	9	algebras	algebras	PROPN
cana-1620	16	10	in	in	ADP
cana-1620	16	11	2011[7	2011[7	NUM
cana-1620	16	12	]	]	PUNCT
cana-1620	16	13	.	.	PUNCT
cana-1620	17	1	zadeh[8	zadeh[8	PROPN
cana-1620	17	2	]	]	PUNCT
cana-1620	17	3	introduced	introduce	VERB
cana-1620	17	4	fuzzy	fuzzy	ADJ
cana-1620	17	5	sets	set	NOUN
cana-1620	17	6	for	for	ADP
cana-1620	17	7	the	the	DET
cana-1620	17	8	first	first	ADJ
cana-1620	17	9	time	time	NOUN
cana-1620	17	10	in	in	ADP
cana-1620	17	11	1965	1965	NUM
cana-1620	17	12	.	.	PUNCT
cana-1620	18	1	this	this	DET
cana-1620	18	2	work	work	NOUN
cana-1620	18	3	describes	describe	VERB
cana-1620	18	4	the	the	DET
cana-1620	18	5	normal	normal	ADJ
cana-1620	18	6	fuzzy	fuzzy	ADJ
cana-1620	18	7	t	t	NOUN
cana-1620	18	8	-	-	PUNCT
cana-1620	18	9	ideal	ideal	NOUN
cana-1620	18	10	and	and	CCONJ
cana-1620	18	11	poset	poset	VERB
cana-1620	18	12	under	under	ADP
cana-1620	18	13	the	the	DET
cana-1620	18	14	set	set	NOUN
cana-1620	18	15	of	of	ADP
cana-1620	18	16	inclusion	inclusion	NOUN
cana-1620	18	17	principle	principle	NOUN
cana-1620	18	18	over	over	ADP
cana-1620	18	19	talgebra	talgebra	NOUN
cana-1620	18	20	and	and	CCONJ
cana-1620	18	21	explores	explore	VERB
cana-1620	18	22	some	some	DET
cana-1620	18	23	algebraic	algebraic	ADJ
cana-1620	18	24	characteristics	characteristic	NOUN
cana-1620	18	25	.	.	PUNCT
cana-1620	19	1	ii	ii	PROPN
cana-1620	19	2	.	.	PUNCT
cana-1620	20	1	preliminaries	preliminary	NOUN
cana-1620	20	2	basic	basic	ADJ
cana-1620	20	3	reference	reference	NOUN
cana-1620	20	4	:	:	PUNCT
cana-1620	20	5	2.1	2.1	NUM
cana-1620	21	1	[	[	SYM
cana-1620	21	2	8	8	NUM
cana-1620	21	3	]	]	PUNCT
cana-1620	21	4	let	let	VERB
cana-1620	21	5	𝑋	𝑋	NOUN
cana-1620	21	6	be	be	AUX
cana-1620	21	7	a	a	DET
cana-1620	21	8	non	non	ADJ
cana-1620	21	9	-	-	ADJ
cana-1620	21	10	empty	empty	ADJ
cana-1620	21	11	set	set	NOUN
cana-1620	21	12	.	.	PUNCT
cana-1620	22	1	a	a	DET
cana-1620	22	2	𝐹𝑆𝑏	𝐹𝑆𝑏	NOUN
cana-1620	22	3	of	of	ADP
cana-1620	22	4	the	the	DET
cana-1620	22	5	set	set	NOUN
cana-1620	22	6	x	x	PUNCT
cana-1620	22	7	is	be	AUX
cana-1620	22	8	a	a	DET
cana-1620	22	9	mapping	mapping	NOUN
cana-1620	22	10	𝜇	𝜇	ADP
cana-1620	22	11	:	:	PUNCT
cana-1620	22	12	𝑋→	𝑋→	PROPN
cana-1620	22	13	[	[	X
cana-1620	22	14	0	0	NUM
cana-1620	22	15	,	,	PUNCT
cana-1620	22	16	1	1	NUM
cana-1620	22	17	]	]	PUNCT
cana-1620	22	18	.	.	PUNCT
cana-1620	23	1	basic	basic	ADJ
cana-1620	23	2	reference	reference	NOUN
cana-1620	23	3	:	:	PUNCT
cana-1620	23	4	2.2[7	2.2[7	NUM
cana-1620	23	5	]	]	X
cana-1620	23	6	a	a	DET
cana-1620	23	7	fs	fs	NOUN
cana-1620	23	8	𝜇	𝜇	X
cana-1620	23	9	in	in	ADP
cana-1620	23	10	a	a	DET
cana-1620	23	11	bp	bp	NOUN
cana-1620	23	12	-	-	PUNCT
cana-1620	23	13	algebra	algebra	NOUN
cana-1620	23	14	x	x	PUNCT
cana-1620	23	15	is	be	AUX
cana-1620	23	16	called	call	VERB
cana-1620	23	17	a	a	DET
cana-1620	23	18	𝐹𝑇𝐼	𝐹𝑇𝐼	NOUN
cana-1620	23	19	of	of	ADP
cana-1620	23	20	x	x	NOUN
cana-1620	23	21	if	if	SCONJ
cana-1620	23	22	it	it	PRON
cana-1620	23	23	satisfies	satisfy	VERB
cana-1620	23	24	the	the	DET
cana-1620	23	25	following	follow	VERB
cana-1620	23	26	conditions	condition	NOUN
cana-1620	23	27	:	:	PUNCT
cana-1620	23	28	(	(	PUNCT
cana-1620	23	29	i	i	NOUN
cana-1620	23	30	)	)	PUNCT
cana-1620	23	31	𝜇(0	𝜇(0	PROPN
cana-1620	23	32	)	)	PUNCT
cana-1620	23	33	≥	≥	NOUN
cana-1620	23	34	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1620	23	35	)	)	PUNCT
cana-1620	23	36	mailto:mathsnagaraj.ct@gmail.com	mailto:mathsnagaraj.ct@gmail.com	X
cana-1620	24	1	mailto:*1a)%20mprem.maths3033@gmail.com	mailto:*1a)%20mprem.maths3033@gmail.com	PROPN
cana-1620	25	1	mailto:y_immanuel@yahoo.com	mailto:y_immanuel@yahoo.com	X
cana-1620	25	2	mailto:abdulsalam.maths@gmail.com	mailto:abdulsalam.maths@gmail.com	X
cana-1620	25	3	mailto:thamizanand@gmail.com	mailto:thamizanand@gmail.com	X
cana-1620	25	4	mailto:metilda81@gmail.com	mailto:metilda81@gmail.com	PROPN
cana-1620	25	5	mailto:jeyasjjjeyas@gmail.com	mailto:jeyasjjjeyas@gmail.com	PROPN
cana-1620	25	6	mailto:mprem.maths3033@gmail.com	mailto:mprem.maths3033@gmail.com	X
cana-1620	26	1	communications	communication	NOUN
cana-1620	26	2	on	on	ADP
cana-1620	26	3	applied	apply	VERB
cana-1620	26	4	nonlinear	nonlinear	ADJ
cana-1620	26	5	analysis	analysis	NOUN
cana-1620	26	6	issn	issn	NOUN
cana-1620	26	7	:	:	PUNCT
cana-1620	26	8	1074	1074	NUM
cana-1620	26	9	-	-	PUNCT
cana-1620	26	10	133x	133x	NUM
cana-1620	26	11	vol	vol	NOUN
cana-1620	26	12	32	32	NUM
cana-1620	26	13	no	no	NOUN
cana-1620	26	14	.	.	NOUN
cana-1620	26	15	1	1	NUM
cana-1620	26	16	(	(	PUNCT
cana-1620	26	17	2025	2025	NUM
cana-1620	26	18	)	)	PUNCT
cana-1620	26	19	57	57	NUM
cana-1620	26	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1620	26	21	(	(	PUNCT
cana-1620	26	22	ii	ii	NOUN
cana-1620	26	23	)	)	PUNCT
cana-1620	26	24	𝜇(𝑥	𝜇(𝑥	PROPN
cana-1620	26	25	∗	∗	NOUN
cana-1620	26	26	𝑧	𝑧	NOUN
cana-1620	26	27	)	)	PUNCT
cana-1620	26	28	≥	≥	NOUN
cana-1620	26	29	𝑚𝑖𝑛{𝜇((𝑥	𝑚𝑖𝑛{𝜇((𝑥	NUM
cana-1620	26	30	∗	∗	NOUN
cana-1620	26	31	𝑦	𝑦	NOUN
cana-1620	26	32	)	)	PUNCT
cana-1620	26	33	∗	∗	NOUN
cana-1620	26	34	𝑧	𝑧	NOUN
cana-1620	26	35	)	)	PUNCT
cana-1620	26	36	,	,	PUNCT
cana-1620	26	37	𝜇(𝑦	𝜇(𝑦	PROPN
cana-1620	26	38	)	)	PUNCT
cana-1620	26	39	}	}	PUNCT
cana-1620	26	40	,	,	PUNCT
cana-1620	26	41	∀	∀	X
cana-1620	26	42	𝑥	𝑥	NOUN
cana-1620	26	43	,	,	PUNCT
cana-1620	26	44	𝑦	𝑦	NOUN
cana-1620	26	45	∈	∈	PROPN
cana-1620	26	46	𝑋.	𝑋.	PROPN
cana-1620	26	47	iii	iii	PROPN
cana-1620	26	48	on	on	ADP
cana-1620	26	49	arithmetical	arithmetical	ADJ
cana-1620	26	50	traits	trait	NOUN
cana-1620	26	51	of	of	ADP
cana-1620	26	52	doubt	doubt	NOUN
cana-1620	26	53	fuzzy	fuzzy	ADJ
cana-1620	26	54	t	t	NOUN
cana-1620	26	55	-	-	PUNCT
cana-1620	26	56	ideals	ideal	NOUN
cana-1620	26	57	beneath	beneath	ADP
cana-1620	26	58	the	the	DET
cana-1620	26	59	normalization	normalization	NOUN
cana-1620	26	60	of	of	ADP
cana-1620	26	61	t	t	PROPN
cana-1620	26	62	-	-	PUNCT
cana-1620	26	63	algebra	algebra	NOUN
cana-1620	26	64	definition	definition	NOUN
cana-1620	26	65	:	:	PUNCT
cana-1620	26	66	3.1	3.1	NUM
cana-1620	26	67	let	let	VERB
cana-1620	26	68	𝐷𝐹𝑇𝐼	𝐷𝐹𝑇𝐼	PROPN
cana-1620	26	69	ώ	ώ	PROPN
cana-1620	26	70	of	of	ADP
cana-1620	26	71	ᾆ	ᾆ	PROPN
cana-1620	26	72	is	be	AUX
cana-1620	26	73	called	call	VERB
cana-1620	26	74	to	to	PART
cana-1620	26	75	be	be	AUX
cana-1620	26	76	𝑁𝐷𝐹𝑇𝐼	𝑁𝐷𝐹𝑇𝐼	ADJ
cana-1620	27	1	if	if	SCONJ
cana-1620	27	2	∃	∃	PROPN
cana-1620	27	3	𝜃	𝜃	X
cana-1620	27	4	∈	∈	PROPN
cana-1620	27	5	ᾆ	ᾆ	PROPN
cana-1620	27	6	s.t	s.t	PROPN
cana-1620	27	7	ώ(0	ώ(0	PROPN
cana-1620	27	8	)	)	PUNCT
cana-1620	27	9	=	=	PUNCT
cana-1620	28	1	1	1	X
cana-1620	28	2	.	.	PUNCT
cana-1620	28	3	example	example	NOUN
cana-1620	28	4	:	:	PUNCT
cana-1620	28	5	3.1.1	3.1.1	NUM
cana-1620	28	6	let	let	VERB
cana-1620	28	7	ᾆ	ᾆ	PRON
cana-1620	28	8	=	=	SYM
cana-1620	28	9	{	{	PUNCT
cana-1620	28	10	0	0	NUM
cana-1620	28	11	,	,	PUNCT
cana-1620	28	12	𝑎	𝑎	NOUN
cana-1620	28	13	,	,	PUNCT
cana-1620	28	14	𝑏	𝑏	NOUN
cana-1620	28	15	,	,	PUNCT
cana-1620	28	16	𝑐	𝑐	NOUN
cana-1620	28	17	,	,	PUNCT
cana-1620	28	18	𝑑	𝑑	AUX
cana-1620	28	19	}	}	PUNCT
cana-1620	28	20	be	be	AUX
cana-1620	28	21	a	a	DET
cana-1620	28	22	t	t	NOUN
cana-1620	28	23	-	-	PUNCT
cana-1620	28	24	algebra	algebra	NOUN
cana-1620	28	25	*	*	PUNCT
cana-1620	28	26	𝟎	𝟎	PROPN
cana-1620	28	27	𝒂	𝒂	PRON
cana-1620	28	28	𝒃	𝒃	NOUN
cana-1620	28	29	𝑪	𝑪	NOUN
cana-1620	28	30	𝒅	𝒅	NOUN
cana-1620	28	31	𝟎	𝟎	NUM
cana-1620	28	32	0	0	NUM
cana-1620	28	33	𝑎	𝑎	X
cana-1620	28	34	𝑏	𝑏	PROPN
cana-1620	28	35	𝑐	𝑐	PROPN
cana-1620	28	36	𝑑	𝑑	PROPN
cana-1620	28	37	𝒂	𝒂	SYM
cana-1620	28	38	0	0	NUM
cana-1620	28	39	0	0	NUM
cana-1620	28	40	0	0	NUM
cana-1620	28	41	0	0	NUM
cana-1620	29	1	𝑎	𝑎	NOUN
cana-1620	29	2	𝒃	𝒃	NOUN
cana-1620	29	3	0	0	NUM
cana-1620	29	4	𝑐	𝑐	NOUN
cana-1620	29	5	0	0	NUM
cana-1620	30	1	𝑐	𝑐	PROPN
cana-1620	30	2	𝑑	𝑑	PROPN
cana-1620	30	3	𝒄	𝒄	NOUN
cana-1620	30	4	0	0	NUM
cana-1620	31	1	𝑎	𝑎	PRON
cana-1620	31	2	𝑏	𝑏	NOUN
cana-1620	31	3	0	0	NUM
cana-1620	31	4	𝑎	𝑎	NOUN
cana-1620	31	5	𝒅	𝒅	NOUN
cana-1620	31	6	0	0	NUM
cana-1620	31	7	0	0	NUM
cana-1620	31	8	0	0	NUM
cana-1620	31	9	0	0	NUM
cana-1620	31	10	0	0	NUM
cana-1620	32	1	then	then	ADV
cana-1620	32	2	(	(	PUNCT
cana-1620	32	3	ᾆ,∗	ᾆ,∗	NOUN
cana-1620	32	4	,	,	PUNCT
cana-1620	32	5	0	0	NUM
cana-1620	32	6	)	)	PUNCT
cana-1620	32	7	is	be	AUX
cana-1620	32	8	a	a	DET
cana-1620	32	9	t	t	NOUN
cana-1620	32	10	-	-	PUNCT
cana-1620	32	11	algebra	algebra	NOUN
cana-1620	32	12	.	.	PUNCT
cana-1620	33	1	define	define	VERB
cana-1620	33	2	𝐷𝐹𝑆	𝐷𝐹𝑆	PROPN
cana-1620	33	3	ώ	ώ	NOUN
cana-1620	33	4	in	in	ADP
cana-1620	33	5	ᾆ	ᾆ	NUM
cana-1620	33	6	by	by	ADP
cana-1620	33	7	ώ(0	ώ(0	PROPN
cana-1620	33	8	)	)	PUNCT
cana-1620	33	9	=	=	SYM
cana-1620	33	10	0.9	0.9	NUM
cana-1620	33	11	,	,	PUNCT
cana-1620	33	12	ώ(𝑎	ώ(𝑎	NUM
cana-1620	33	13	)	)	PUNCT
cana-1620	33	14	=	=	SYM
cana-1620	33	15	0.7	0.7	NUM
cana-1620	33	16	,	,	PUNCT
cana-1620	33	17	ώ(𝑏	ώ(𝑏	PROPN
cana-1620	33	18	)	)	PUNCT
cana-1620	33	19	=	=	PUNCT
cana-1620	33	20	0.8	0.8	NUM
cana-1620	33	21	,	,	PUNCT
cana-1620	33	22	ώ(𝑐	ώ(𝑐	PROPN
cana-1620	33	23	)	)	PUNCT
cana-1620	33	24	=	=	SYM
cana-1620	33	25	0.6	0.6	NUM
cana-1620	33	26	𝑎𝑛𝑑	𝑎𝑛𝑑	X
cana-1620	33	27	ώ(𝑑	ώ(𝑑	PROPN
cana-1620	33	28	)	)	PUNCT
cana-1620	33	29	=	=	VERB
cana-1620	33	30	0.5	0.5	NUM
cana-1620	33	31	.	.	PUNCT
cana-1620	34	1	⇒then	⇒then	ADV
cana-1620	34	2	ώ	ώ	PROPN
cana-1620	34	3	is	be	AUX
cana-1620	34	4	a	a	DET
cana-1620	34	5	𝑁𝐷𝐹𝑇𝐼	𝑁𝐷𝐹𝑇𝐼	NOUN
cana-1620	34	6	of	of	ADP
cana-1620	34	7	ᾆ.	ᾆ.	NOUN
cana-1620	34	8	remark	remark	NOUN
cana-1620	34	9	:	:	PUNCT
cana-1620	34	10	3.2	3.2	NUM
cana-1620	34	11	let	let	VERB
cana-1620	34	12	𝑁𝐷𝐹𝑇𝐼	𝑁𝐷𝐹𝑇𝐼	PROPN
cana-1620	34	13	ώ	ώ	PROPN
cana-1620	34	14	of	of	ADP
cana-1620	34	15	ᾆ	ᾆ	PROPN
cana-1620	34	16	if	if	SCONJ
cana-1620	35	1	and	and	CCONJ
cana-1620	35	2	only	only	ADV
cana-1620	35	3	if	if	SCONJ
cana-1620	35	4	ώ(0	ώ(0	PROPN
cana-1620	35	5	)	)	PUNCT
cana-1620	35	6	=	=	SYM
cana-1620	35	7	1	1	X
cana-1620	35	8	.	.	X
cana-1620	35	9	theorem	theorem	VERB
cana-1620	35	10	:	:	PUNCT
cana-1620	35	11	3.3	3.3	NUM
cana-1620	35	12	let	let	VERB
cana-1620	35	13	any	any	DET
cana-1620	35	14	𝐷𝐹𝑇𝐼	𝐷𝐹𝑇𝐼	PROPN
cana-1620	35	15	ώ	ώ	PROPN
cana-1620	35	16	of	of	ADP
cana-1620	35	17	ᾆ	ᾆ	NUM
cana-1620	35	18	,	,	PUNCT
cana-1620	35	19	we	we	PRON
cana-1620	35	20	can	can	AUX
cana-1620	35	21	generate	generate	VERB
cana-1620	35	22	the	the	DET
cana-1620	35	23	𝑁𝐷𝐹𝑇𝐼	𝑁𝐷𝐹𝑇𝐼	NOUN
cana-1620	35	24	of	of	ADP
cana-1620	35	25	ᾆ	ᾆ	PROPN
cana-1620	36	1	⊂	⊂	X
cana-1620	36	2	ώ.	ώ.	ADV
cana-1620	36	3	proof	proof	NOUN
cana-1620	36	4	:	:	PUNCT
cana-1620	36	5	let	let	VERB
cana-1620	36	6	ώ	ώ	PRON
cana-1620	36	7	be	be	AUX
cana-1620	36	8	a	a	DET
cana-1620	36	9	𝐷𝐹𝑇𝐼	𝐷𝐹𝑇𝐼	NOUN
cana-1620	36	10	of	of	ADP
cana-1620	36	11	ᾆ.	ᾆ.	NOUN
cana-1620	36	12	define	define	VERB
cana-1620	36	13	a	a	DET
cana-1620	36	14	𝐷𝐹𝑆	𝐷𝐹𝑆	NOUN
cana-1620	36	15	ώn	ώn	NOUN
cana-1620	36	16	of	of	ADP
cana-1620	36	17	ᾆ	ᾆ	PROPN
cana-1620	36	18	as	as	ADP
cana-1620	36	19	ώ𝑛(ã	ώ𝑛(ã	NUM
cana-1620	36	20	)	)	PUNCT
cana-1620	36	21	=	=	SYM
cana-1620	36	22	ώ(ã	ώ(ã	NOUN
cana-1620	36	23	)	)	PUNCT
cana-1620	36	24	+	+	CCONJ
cana-1620	36	25	ώ𝑐(0	ώ𝑐(0	PROPN
cana-1620	36	26	)	)	PUNCT
cana-1620	36	27	,	,	PUNCT
cana-1620	36	28	∀ã	∀ã	X
cana-1620	36	29	∈	∈	PROPN
cana-1620	36	30	ᾆ.	ᾆ.	NOUN
cana-1620	36	31	let	let	VERB
cana-1620	36	32	ã	ã	NOUN
cana-1620	36	33	,	,	PUNCT
cana-1620	36	34	ɓ	ɓ	PRON
cana-1620	36	35	∈	∈	PROPN
cana-1620	36	36	ᾆ	ᾆ	X
cana-1620	36	37	(	(	PUNCT
cana-1620	36	38	i	i	NOUN
cana-1620	36	39	)	)	PUNCT
cana-1620	36	40	ώ𝑛(0	ώ𝑛(0	PROPN
cana-1620	36	41	)	)	PUNCT
cana-1620	36	42	=	=	PUNCT
cana-1620	36	43	ώ(0	ώ(0	PROPN
cana-1620	36	44	)	)	PUNCT
cana-1620	37	1	+	+	SYM
cana-1620	37	2	ώ𝑐(0	ώ𝑐(0	NOUN
cana-1620	37	3	)	)	PUNCT
cana-1620	37	4	≤	≤	NOUN
cana-1620	37	5	ώ(ã	ώ(ã	NOUN
cana-1620	37	6	)	)	PUNCT
cana-1620	38	1	+	+	SYM
cana-1620	39	1	ώ𝑐(0	ώ𝑐(0	NOUN
cana-1620	39	2	)	)	PUNCT
cana-1620	39	3	=	=	SYM
cana-1620	39	4	ώ𝑛(ã	ώ𝑛(ã	X
cana-1620	39	5	)	)	PUNCT
cana-1620	39	6	⇒	⇒	PROPN
cana-1620	39	7	ώ𝑛(0	ώ𝑛(0	NOUN
cana-1620	39	8	)	)	PUNCT
cana-1620	39	9	≤	≤	NOUN
cana-1620	39	10	ώ𝑛(ã	ώ𝑛(ã	NUM
cana-1620	39	11	)	)	PUNCT
cana-1620	39	12	(	(	PUNCT
cana-1620	39	13	ii	ii	NOUN
cana-1620	39	14	)	)	PUNCT
cana-1620	39	15	ώ𝑛(ã	ώ𝑛(ã	PUNCT
cana-1620	39	16	∗	∗	NOUN
cana-1620	39	17	ĉ	ĉ	PROPN
cana-1620	39	18	)	)	PUNCT
cana-1620	39	19	=	=	SYM
cana-1620	40	1	ώ((ã	ώ((ã	NOUN
cana-1620	40	2	∗	∗	NOUN
cana-1620	40	3	ɓ	ɓ	NOUN
cana-1620	40	4	)	)	PUNCT
cana-1620	40	5	∗	∗	NOUN
cana-1620	40	6	ĉ	ĉ	PROPN
cana-1620	40	7	)	)	PUNCT
cana-1620	40	8	+	+	SYM
cana-1620	40	9	ώ𝑐(0	ώ𝑐(0	NOUN
cana-1620	40	10	)	)	PUNCT
cana-1620	40	11	≤	≤	NOUN
cana-1620	40	12	𝑚𝑎𝑥{ώ((ã	𝑚𝑎𝑥{ώ((ã	NOUN
cana-1620	40	13	∗	∗	NOUN
cana-1620	40	14	ɓ	ɓ	NOUN
cana-1620	40	15	)	)	PUNCT
cana-1620	40	16	∗	∗	NOUN
cana-1620	40	17	ĉ	ĉ	PROPN
cana-1620	40	18	)	)	PUNCT
cana-1620	40	19	,	,	PUNCT
cana-1620	40	20	ώ(ɓ	ώ(ɓ	NUM
cana-1620	40	21	)	)	PUNCT
cana-1620	40	22	}	}	PUNCT
cana-1620	41	1	+	+	CCONJ
cana-1620	41	2	ώ𝑐(0	ώ𝑐(0	NOUN
cana-1620	41	3	)	)	PUNCT
cana-1620	41	4	=	=	PUNCT
cana-1620	41	5	𝑚𝑎𝑥{[ώ((ã	𝑚𝑎𝑥{[ώ((ã	ADJ
cana-1620	41	6	∗	∗	NOUN
cana-1620	41	7	ɓ	ɓ	NOUN
cana-1620	41	8	)	)	PUNCT
cana-1620	41	9	∗	∗	NOUN
cana-1620	41	10	ĉ	ĉ	PROPN
cana-1620	41	11	)	)	PUNCT
cana-1620	41	12	+	+	X
cana-1620	41	13	ώ𝑐(0	ώ𝑐(0	PROPN
cana-1620	41	14	)	)	PUNCT
cana-1620	41	15	]	]	PUNCT
cana-1620	41	16	,	,	PUNCT
cana-1620	41	17	[	[	X
cana-1620	41	18	ώ(ɓ	ώ(ɓ	NUM
cana-1620	41	19	)	)	PUNCT
cana-1620	41	20	+	+	CCONJ
cana-1620	41	21	ώ𝑐(0	ώ𝑐(0	NOUN
cana-1620	41	22	)	)	PUNCT
cana-1620	41	23	]	]	PUNCT
cana-1620	41	24	}	}	PUNCT
cana-1620	41	25	=	=	SYM
cana-1620	41	26	𝑚𝑎𝑥{ώ𝑛((ã	𝑚𝑎𝑥{ώ𝑛((ã	NUM
cana-1620	41	27	∗	∗	NOUN
cana-1620	41	28	ɓ	ɓ	NOUN
cana-1620	41	29	)	)	PUNCT
cana-1620	41	30	∗	∗	NOUN
cana-1620	41	31	ĉ	ĉ	PROPN
cana-1620	41	32	)	)	PUNCT
cana-1620	41	33	,	,	PUNCT
cana-1620	41	34	ώ𝑛	ώ𝑛	PROPN
cana-1620	41	35	(	(	PUNCT
cana-1620	41	36	ɓ	ɓ	NOUN
cana-1620	41	37	)	)	PUNCT
cana-1620	41	38	}	}	PUNCT
cana-1620	41	39	⇒ώ𝑛(ã	⇒ώ𝑛(ã	NOUN
cana-1620	41	40	∗	∗	NOUN
cana-1620	41	41	ĉ	ĉ	PROPN
cana-1620	41	42	)	)	PUNCT
cana-1620	41	43	≤	≤	NUM
cana-1620	41	44	𝑚𝑎𝑥{ώ𝑛((ã	𝑚𝑎𝑥{ώ𝑛((ã	NUM
cana-1620	41	45	∗	∗	NOUN
cana-1620	41	46	ɓ	ɓ	NOUN
cana-1620	41	47	)	)	PUNCT
cana-1620	41	48	∗	∗	NOUN
cana-1620	41	49	ĉ	ĉ	PROPN
cana-1620	41	50	)	)	PUNCT
cana-1620	41	51	,	,	PUNCT
cana-1620	41	52	ώ𝑛	ώ𝑛	PROPN
cana-1620	41	53	(	(	PUNCT
cana-1620	41	54	ɓ	ɓ	NOUN
cana-1620	41	55	)	)	PUNCT
cana-1620	41	56	}	}	PUNCT
cana-1620	41	57	also	also	ADV
cana-1620	41	58	ώ𝑛(0	ώ𝑛(0	X
cana-1620	41	59	)	)	PUNCT
cana-1620	41	60	=	=	PUNCT
cana-1620	42	1	ώ(0	ώ(0	PROPN
cana-1620	42	2	)	)	PUNCT
cana-1620	42	3	+	+	SYM
cana-1620	42	4	ώ𝑐(0	ώ𝑐(0	NOUN
cana-1620	42	5	)	)	PUNCT
cana-1620	42	6	=	=	PUNCT
cana-1620	42	7	ώ(0	ώ(0	PROPN
cana-1620	42	8	)	)	PUNCT
cana-1620	43	1	+	+	CCONJ
cana-1620	43	2	1	1	NUM
cana-1620	43	3	−	−	PROPN
cana-1620	43	4	ώ(0	ώ(0	PROPN
cana-1620	43	5	)	)	PUNCT
cana-1620	43	6	=	=	SYM
cana-1620	44	1	1	1	X
cana-1620	44	2	.	.	X
cana-1620	44	3	∴	∴	PROPN
cana-1620	44	4	ώ𝑛	ώ𝑛	PROPN
cana-1620	44	5	is	be	AUX
cana-1620	44	6	a	a	DET
cana-1620	44	7	𝑁𝐷𝐹𝑇𝐼	𝑁𝐷𝐹𝑇𝐼	NOUN
cana-1620	44	8	of	of	ADP
cana-1620	44	9	ᾆ.	ᾆ.	NOUN
cana-1620	44	10	communications	communication	NOUN
cana-1620	44	11	on	on	ADP
cana-1620	44	12	applied	apply	VERB
cana-1620	44	13	nonlinear	nonlinear	ADJ
cana-1620	44	14	analysis	analysis	NOUN
cana-1620	44	15	issn	issn	NOUN
cana-1620	44	16	:	:	PUNCT
cana-1620	44	17	1074	1074	NUM
cana-1620	44	18	-	-	PUNCT
cana-1620	44	19	133x	133x	NUM
cana-1620	44	20	vol	vol	NOUN
cana-1620	44	21	32	32	NUM
cana-1620	44	22	no	no	NOUN
cana-1620	44	23	.	.	NOUN
cana-1620	44	24	1	1	NUM
cana-1620	44	25	(	(	PUNCT
cana-1620	44	26	2025	2025	NUM
cana-1620	44	27	)	)	PUNCT
cana-1620	44	28	58	58	NUM
cana-1620	44	29	https://internationalpubls.com	https://internationalpubls.com	X
cana-1620	44	30	lemma	lemma	PROPN
cana-1620	44	31	:	:	PUNCT
cana-1620	44	32	3.4	3.4	NUM
cana-1620	44	33	let	let	VERB
cana-1620	44	34	ώ𝑛	ώ𝑛	PRON
cana-1620	44	35	be	be	AUX
cana-1620	44	36	an	an	DET
cana-1620	44	37	𝐷𝐹𝑆	𝐷𝐹𝑆	NOUN
cana-1620	44	38	in	in	ADP
cana-1620	44	39	ᾆ	ᾆ	PROPN
cana-1620	44	40	defined	define	VERB
cana-1620	44	41	by	by	ADP
cana-1620	44	42	ώ𝑛(ã	ώ𝑛(ã	NOUN
cana-1620	44	43	)	)	PUNCT
cana-1620	44	44	=	=	SYM
cana-1620	45	1	ώ(𝜃	ώ(𝜃	NOUN
cana-1620	45	2	)	)	PUNCT
cana-1620	46	1	+	+	CCONJ
cana-1620	47	1	ώc(0	ώc(0	NOUN
cana-1620	47	2	)	)	PUNCT
cana-1620	47	3	,	,	PUNCT
cana-1620	47	4	∀	∀	PUNCT
cana-1620	48	1	ã	ã	PRON
cana-1620	48	2	∈	∈	NOUN
cana-1620	48	3	ᾆ.	ᾆ.	VERB
cana-1620	48	4	if	if	SCONJ
cana-1620	48	5	∃	∃	PROPN
cana-1620	48	6	element	element	NOUN
cana-1620	48	7	ã	ã	PROPN
cana-1620	48	8	∈	∈	PROPN
cana-1620	48	9	ᾆ	ᾆ	PROPN
cana-1620	48	10	in	in	ADP
cana-1620	48	11	s.t	s.t	PROPN
cana-1620	48	12	ώ𝑛(ã	ώ𝑛(ã	PUNCT
cana-1620	48	13	)	)	PUNCT
cana-1620	48	14	=	=	SYM
cana-1620	48	15	0	0	NUM
cana-1620	48	16	,	,	PUNCT
cana-1620	48	17	then	then	ADV
cana-1620	48	18	ώ(ã	ώ(ã	NOUN
cana-1620	48	19	)	)	PUNCT
cana-1620	48	20	=	=	SYM
cana-1620	49	1	0	0	X
cana-1620	49	2	.	.	PUNCT
cana-1620	50	1	lemma	lemma	PROPN
cana-1620	50	2	:	:	PUNCT
cana-1620	50	3	3.5	3.5	NUM
cana-1620	50	4	let	let	VERB
cana-1620	50	5	ώ	ώ	PRON
cana-1620	50	6	be	be	AUX
cana-1620	50	7	𝐷𝐹𝑇𝐼	𝐷𝐹𝑇𝐼	PROPN
cana-1620	50	8	of	of	ADP
cana-1620	50	9	ᾆ.	ᾆ.	NOUN
cana-1620	50	10	then	then	ADV
cana-1620	50	11	,	,	PUNCT
cana-1620	50	12	(	(	PUNCT
cana-1620	50	13	i	i	NOUN
cana-1620	50	14	)	)	PUNCT
cana-1620	50	15	if	if	SCONJ
cana-1620	50	16	ώ	ώ	PRON
cana-1620	50	17	itself	itself	PRON
cana-1620	50	18	is	be	AUX
cana-1620	50	19	𝐷𝐹𝑇𝐼	𝐷𝐹𝑇𝐼	PROPN
cana-1620	50	20	then	then	ADV
cana-1620	50	21	ώ(ã	ώ(ã	NOUN
cana-1620	50	22	)	)	PUNCT
cana-1620	50	23	=	=	PUNCT
cana-1620	50	24	ώ𝑛(ã	ώ𝑛(ã	NUM
cana-1620	50	25	)	)	PUNCT
cana-1620	50	26	.	.	PUNCT
cana-1620	51	1	(	(	PUNCT
cana-1620	51	2	ii	ii	X
cana-1620	51	3	)	)	PUNCT
cana-1620	51	4	if	if	SCONJ
cana-1620	51	5	ώ	ώ	PRON
cana-1620	51	6	is	be	AUX
cana-1620	51	7	a	a	DET
cana-1620	51	8	𝐷𝐹𝑇𝐼	𝐷𝐹𝑇𝐼	NOUN
cana-1620	51	9	of	of	ADP
cana-1620	51	10	𝑋	𝑋	PROPN
cana-1620	51	11	then	then	ADV
cana-1620	51	12	(	(	PUNCT
cana-1620	51	13	ώ𝑛(ã	ώ𝑛(ã	NUM
cana-1620	51	14	)	)	PUNCT
cana-1620	51	15	)	)	PUNCT
cana-1620	52	1	𝑛	𝑛	PROPN
cana-1620	52	2	=	=	PUNCT
cana-1620	52	3	ώ𝑛(ã	ώ𝑛(ã	NUM
cana-1620	52	4	)	)	PUNCT
cana-1620	52	5	.	.	PUNCT
cana-1620	53	1	proposition	proposition	NOUN
cana-1620	53	2	:	:	PUNCT
cana-1620	53	3	3.6	3.6	NUM
cana-1620	53	4	let	let	VERB
cana-1620	53	5	ώ	ώ	PRON
cana-1620	53	6	𝐷𝐹𝑇𝐼	𝐷𝐹𝑇𝐼	PROPN
cana-1620	53	7	ᾆ.	ᾆ.	VERB
cana-1620	53	8	if	if	SCONJ
cana-1620	53	9	ώ	ώ	NOUN
cana-1620	53	10	contains	contain	VERB
cana-1620	53	11	the	the	DET
cana-1620	53	12	𝑁𝐷𝐹𝑇𝐼	𝑁𝐷𝐹𝑇𝐼	PROPN
cana-1620	53	13	of	of	ADP
cana-1620	53	14	ᾆ	ᾆ	NUM
cana-1620	53	15	,	,	PUNCT
cana-1620	53	16	generated	generate	VERB
cana-1620	53	17	by	by	ADP
cana-1620	53	18	any	any	DET
cana-1620	53	19	other	other	ADJ
cana-1620	53	20	𝐷𝐹𝑇𝐼	𝐷𝐹𝑇𝐼	NOUN
cana-1620	53	21	of	of	ADP
cana-1620	53	22	ᾆ	ᾆ	PROPN
cana-1620	53	23	then	then	ADV
cana-1620	53	24	ώ	ώ	NOUN
cana-1620	53	25	is	be	AUX
cana-1620	53	26	normal	normal	ADJ
cana-1620	53	27	.	.	PUNCT
cana-1620	54	1	proof	proof	NOUN
cana-1620	54	2	:	:	PUNCT
cana-1620	54	3	let	let	VERB
cana-1620	54	4	𝛿	𝛿	PROPN
cana-1620	54	5	be	be	AUX
cana-1620	54	6	a	a	DET
cana-1620	54	7	𝐷𝐹𝑇𝐼	𝐷𝐹𝑇𝐼	NOUN
cana-1620	54	8	of	of	ADP
cana-1620	54	9	ᾆ.	ᾆ.	NOUN
cana-1620	54	10	by	by	ADP
cana-1620	54	11	the	the	DET
cana-1620	54	12	.	.	PROPN
cana-1620	54	13	3.3	3.3	NUM
cana-1620	54	14	,	,	PUNCT
cana-1620	54	15	let	let	VERB
cana-1620	54	16	𝛿𝑛	𝛿𝑛	PRON
cana-1620	54	17	is	be	AUX
cana-1620	54	18	a	a	DET
cana-1620	54	19	𝐷𝐹𝑇𝐼	𝐷𝐹𝑇𝐼	NOUN
cana-1620	54	20	of	of	ADP
cana-1620	54	21	ᾆ	ᾆ	PROPN
cana-1620	54	22	∴	∴	PROPN
cana-1620	54	23	𝛿𝑛(0	𝛿𝑛(0	PROPN
cana-1620	54	24	)	)	PUNCT
cana-1620	54	25	=	=	SYM
cana-1620	54	26	1	1	NUM
cana-1620	54	27	(	(	PUNCT
cana-1620	54	28	lem	lem	PROPN
cana-1620	54	29	.	.	PROPN
cana-1620	54	30	3.4	3.4	NUM
cana-1620	54	31	)	)	PUNCT
cana-1620	54	32	let	let	VERB
cana-1620	54	33	ώ	ώ	PRON
cana-1620	54	34	be	be	AUX
cana-1620	54	35	a	a	DET
cana-1620	54	36	𝐷𝐹𝑇𝐼	𝐷𝐹𝑇𝐼	NOUN
cana-1620	54	37	of	of	ADP
cana-1620	54	38	ᾆ	ᾆ	PROPN
cana-1620	54	39	s.t	s.t	PROPN
cana-1620	54	40	𝛿𝑛	𝛿𝑛	PROPN
cana-1620	54	41	⊂	⊂	PROPN
cana-1620	54	42	ώ.	ώ.	ADV
cana-1620	54	43	⇒	⇒	PROPN
cana-1620	54	44	ώ(ã	ώ(ã	PROPN
cana-1620	54	45	)	)	PUNCT
cana-1620	54	46	≤	≤	NOUN
cana-1620	54	47	𝛿𝑛(ã	𝛿𝑛(ã	NUM
cana-1620	54	48	)	)	PUNCT
cana-1620	54	49	,	,	PUNCT
cana-1620	54	50	∀ã	∀ã	X
cana-1620	55	1	∈	∈	PROPN
cana-1620	56	1	ᾆ	ᾆ	X
cana-1620	56	2	put	put	VERB
cana-1620	56	3	ã	ã	X
cana-1620	56	4	=	=	SYM
cana-1620	56	5	0	0	NUM
cana-1620	56	6	⇒	⇒	PROPN
cana-1620	56	7	ώ(0	ώ(0	PROPN
cana-1620	56	8	)	)	PUNCT
cana-1620	56	9	≤	≤	NOUN
cana-1620	56	10	𝛿𝑛(0	𝛿𝑛(0	NOUN
cana-1620	56	11	)	)	PUNCT
cana-1620	56	12	=	=	SYM
cana-1620	56	13	1	1	NUM
cana-1620	56	14	⇒	⇒	NOUN
cana-1620	56	15	ώ(0	ώ(0	PROPN
cana-1620	56	16	)	)	PUNCT
cana-1620	56	17	≤	≤	NOUN
cana-1620	56	18	1	1	NUM
cana-1620	56	19	∴	∴	PROPN
cana-1620	56	20	ώ	ώ	PROPN
cana-1620	56	21	is	be	AUX
cana-1620	56	22	normal	normal	ADJ
cana-1620	56	23	theorem	theorem	NOUN
cana-1620	56	24	:	:	PUNCT
cana-1620	56	25	3.7	3.7	NUM
cana-1620	56	26	a	a	DET
cana-1620	56	27	set	set	NOUN
cana-1620	56	28	𝑁ώ	𝑁ώ	PROPN
cana-1620	56	29	=	=	SYM
cana-1620	56	30	{	{	PUNCT
cana-1620	56	31	ã	ã	X
cana-1620	56	32	∈	∈	PROPN
cana-1620	56	33	𝑋/ώ(ã	𝑋/ώ(ã	PROPN
cana-1620	56	34	)	)	PUNCT
cana-1620	56	35	=	=	PUNCT
cana-1620	56	36	ώ(0	ώ(0	PROPN
cana-1620	56	37	)	)	PUNCT
cana-1620	56	38	}	}	PUNCT
cana-1620	56	39	.	.	PUNCT
cana-1620	57	1	let	let	VERB
cana-1620	57	2	ώ	ώ	PRON
cana-1620	57	3	and	and	CCONJ
cana-1620	57	4	𝛿	𝛿	PRON
cana-1620	57	5	be	be	AUX
cana-1620	57	6	𝑁𝐷𝐹𝑇𝐼	𝑁𝐷𝐹𝑇𝐼	ADJ
cana-1620	57	7	of	of	ADP
cana-1620	57	8	ᾆ.	ᾆ.	NOUN
cana-1620	57	9	if	if	SCONJ
cana-1620	57	10	ώ	ώ	X
cana-1620	57	11	⊂	⊂	PROPN
cana-1620	57	12	𝛿	𝛿	X
cana-1620	57	13	then	then	ADV
cana-1620	57	14	𝑁ώ	𝑁ώ	PROPN
cana-1620	57	15	⊂	⊂	PROPN
cana-1620	57	16	𝑁𝛿	𝑁𝛿	PROPN
cana-1620	57	17	.	.	PUNCT
cana-1620	58	1	proof	proof	NOUN
cana-1620	58	2	:	:	PUNCT
cana-1620	58	3	let	let	VERB
cana-1620	58	4	ã	ã	X
cana-1620	58	5	∈	∈	PROPN
cana-1620	58	6	𝑁ώ	𝑁ώ	PROPN
cana-1620	58	7	since	since	SCONJ
cana-1620	58	8	ώ	ώ	PROPN
cana-1620	58	9	⊂	⊂	PROPN
cana-1620	58	10	𝛿	𝛿	ADJ
cana-1620	58	11	,	,	PUNCT
cana-1620	58	12	𝛿(ã	𝛿(ã	NOUN
cana-1620	58	13	)	)	PUNCT
cana-1620	58	14	≤	≤	NUM
cana-1620	58	15	ώ(ã	ώ(ã	NOUN
cana-1620	58	16	)	)	PUNCT
cana-1620	58	17	=	=	PUNCT
cana-1620	59	1	ώ(0	ώ(0	PROPN
cana-1620	59	2	)	)	PUNCT
cana-1620	59	3	=	=	SYM
cana-1620	59	4	1	1	NUM
cana-1620	59	5	=	=	SYM
cana-1620	59	6	𝛿(0	𝛿(0	PROPN
cana-1620	59	7	)	)	PUNCT
cana-1620	59	8	⇒ã	⇒ã	NOUN
cana-1620	59	9	∈	∈	PROPN
cana-1620	59	10	𝑁𝛿	𝑁𝛿	PROPN
cana-1620	59	11	∴	∴	PROPN
cana-1620	59	12	𝑁ώ	𝑁ώ	PROPN
cana-1620	59	13	⊂	⊂	PROPN
cana-1620	59	14	𝑁𝛿	𝑁𝛿	PROPN
cana-1620	59	15	theorem	theorem	VERB
cana-1620	59	16	:	:	PUNCT
cana-1620	59	17	3.8	3.8	NUM
cana-1620	59	18	let	let	VERB
cana-1620	59	19	ώ	ώ	PRON
cana-1620	59	20	be	be	AUX
cana-1620	59	21	the	the	DET
cana-1620	59	22	𝐷𝐹𝑇𝐼	𝐷𝐹𝑇𝐼	PROPN
cana-1620	59	23	of	of	ADP
cana-1620	59	24	ᾆ.	ᾆ.	NOUN
cana-1620	59	25	let	let	VERB
cana-1620	59	26	𝑓	𝑓	PRON
cana-1620	59	27	:	:	PUNCT
cana-1620	59	28	[	[	X
cana-1620	59	29	0	0	NUM
cana-1620	59	30	,	,	PUNCT
cana-1620	59	31	ώ(0	ώ(0	PROPN
cana-1620	59	32	)	)	PUNCT
cana-1620	59	33	]	]	PUNCT
cana-1620	60	1	→	→	PUNCT
cana-1620	60	2	[	[	X
cana-1620	60	3	0,1	0,1	NUM
cana-1620	60	4	]	]	PUNCT
cana-1620	60	5	be	be	AUX
cana-1620	60	6	an	an	DET
cana-1620	60	7	increasing	increase	VERB
cana-1620	60	8	function	function	NOUN
cana-1620	60	9	.	.	PUNCT
cana-1620	61	1	let	let	VERB
cana-1620	61	2	’s	’s	PRON
cana-1620	61	3	define	define	VERB
cana-1620	61	4	a	a	DET
cana-1620	61	5	𝐷𝐹𝑆	𝐷𝐹𝑆	PROPN
cana-1620	61	6	ώ𝑓	ώ𝑓	NOUN
cana-1620	61	7	:	:	PUNCT
cana-1620	61	8	ᾆ	ᾆ	X
cana-1620	61	9	→	→	SYM
cana-1620	61	10	[	[	X
cana-1620	61	11	0,1	0,1	NUM
cana-1620	61	12	]	]	PUNCT
cana-1620	61	13	by	by	ADP
cana-1620	61	14	ώ𝑓(ã	ώ𝑓(ã	NUM
cana-1620	61	15	)	)	PUNCT
cana-1620	61	16	=	=	SYM
cana-1620	61	17	𝑓(ώ(ã	𝑓(ώ(ã	PROPN
cana-1620	61	18	)	)	PUNCT
cana-1620	61	19	)	)	PUNCT
cana-1620	61	20	,	,	PUNCT
cana-1620	61	21	∀	∀	PUNCT
cana-1620	62	1	ã	ã	PRON
cana-1620	62	2	∈	∈	NOUN
cana-1620	62	3	ᾆ.	ᾆ.	VERB
cana-1620	62	4	therefore	therefore	ADV
cana-1620	62	5	(	(	PUNCT
cana-1620	62	6	i	i	NOUN
cana-1620	62	7	)	)	PUNCT
cana-1620	62	8	if	if	SCONJ
cana-1620	62	9	ώ𝑓	ώ𝑓	PROPN
cana-1620	62	10	is	be	AUX
cana-1620	62	11	a	a	DET
cana-1620	62	12	𝐷𝐹𝑇𝐼	𝐷𝐹𝑇𝐼	NOUN
cana-1620	62	13	of	of	ADP
cana-1620	62	14	ᾆ	ᾆ	PROPN
cana-1620	62	15	(	(	PUNCT
cana-1620	62	16	ii	ii	NOUN
cana-1620	62	17	)	)	PUNCT
cana-1620	62	18	if	if	SCONJ
cana-1620	62	19	𝑓(ώ(0	𝑓(ώ(0	PROPN
cana-1620	62	20	)	)	PUNCT
cana-1620	62	21	)	)	PUNCT
cana-1620	63	1	=	=	SYM
cana-1620	63	2	1	1	NUM
cana-1620	63	3	,	,	PUNCT
cana-1620	63	4	then	then	ADV
cana-1620	63	5	ώ𝑓	ώ𝑓	PROPN
cana-1620	63	6	is	be	AUX
cana-1620	63	7	normal	normal	ADJ
cana-1620	63	8	(	(	PUNCT
cana-1620	63	9	iii	iii	NOUN
cana-1620	63	10	)	)	PUNCT
cana-1620	63	11	if	if	SCONJ
cana-1620	63	12	𝑓(𝑡	𝑓(𝑡	NOUN
cana-1620	63	13	)	)	PUNCT
cana-1620	63	14	≤	≤	NUM
cana-1620	63	15	𝑡	𝑡	NOUN
cana-1620	63	16	,	,	PUNCT
cana-1620	63	17	∀	∀	PUNCT
cana-1620	63	18	𝑡	𝑡	NOUN
cana-1620	63	19	∈	∈	PROPN
cana-1620	64	1	[	[	X
cana-1620	64	2	0	0	NUM
cana-1620	64	3	,	,	PUNCT
cana-1620	64	4	ώ(0	ώ(0	PROPN
cana-1620	64	5	)	)	PUNCT
cana-1620	64	6	]	]	PUNCT
cana-1620	64	7	then	then	ADV
cana-1620	64	8	ώ	ώ	X
cana-1620	64	9	⊂	⊂	X
cana-1620	64	10	ώ𝑓.	ώ𝑓.	X
cana-1620	65	1	proof	proof	NOUN
cana-1620	65	2	:	:	PUNCT
cana-1620	65	3	communications	communication	NOUN
cana-1620	65	4	on	on	ADP
cana-1620	65	5	applied	apply	VERB
cana-1620	65	6	nonlinear	nonlinear	ADJ
cana-1620	65	7	analysis	analysis	NOUN
cana-1620	65	8	issn	issn	NOUN
cana-1620	65	9	:	:	PUNCT
cana-1620	65	10	1074	1074	NUM
cana-1620	65	11	-	-	PUNCT
cana-1620	65	12	133x	133x	NUM
cana-1620	65	13	vol	vol	NOUN
cana-1620	65	14	32	32	NUM
cana-1620	65	15	no	no	NOUN
cana-1620	65	16	.	.	NOUN
cana-1620	65	17	1	1	NUM
cana-1620	65	18	(	(	PUNCT
cana-1620	65	19	2025	2025	NUM
cana-1620	65	20	)	)	PUNCT
cana-1620	65	21	59	59	NUM
cana-1620	65	22	https://internationalpubls.com	https://internationalpubls.com	X
cana-1620	65	23	let	let	VERB
cana-1620	65	24	ώ	ώ	PRON
cana-1620	65	25	be	be	AUX
cana-1620	65	26	𝐷𝐹𝑇𝐼	𝐷𝐹𝑇𝐼	PROPN
cana-1620	65	27	of	of	ADP
cana-1620	65	28	ᾆ.	ᾆ.	NOUN
cana-1620	65	29	let	let	VERB
cana-1620	65	30	𝑓	𝑓	PRON
cana-1620	65	31	:	:	PUNCT
cana-1620	66	1	[	[	X
cana-1620	66	2	0	0	NUM
cana-1620	66	3	,	,	PUNCT
cana-1620	66	4	ώ(0	ώ(0	PROPN
cana-1620	66	5	)	)	PUNCT
cana-1620	66	6	]	]	PUNCT
cana-1620	66	7	→	→	PUNCT
cana-1620	66	8	[	[	X
cana-1620	66	9	0,1	0,1	NUM
cana-1620	66	10	]	]	PUNCT
cana-1620	66	11	be	be	AUX
cana-1620	66	12	an	an	DET
cana-1620	66	13	increasing	increase	VERB
cana-1620	66	14	function	function	NOUN
cana-1620	66	15	.	.	PUNCT
cana-1620	67	1	define	define	VERB
cana-1620	67	2	a	a	DET
cana-1620	67	3	𝐷𝐹𝑆	𝐷𝐹𝑆	PROPN
cana-1620	67	4	ώ𝑓	ώ𝑓	NOUN
cana-1620	67	5	:	:	PUNCT
cana-1620	67	6	ᾆ	ᾆ	X
cana-1620	67	7	→	→	SYM
cana-1620	67	8	[	[	X
cana-1620	67	9	0,1	0,1	NUM
cana-1620	67	10	]	]	PUNCT
cana-1620	67	11	by	by	ADP
cana-1620	67	12	ώ𝑓(ã	ώ𝑓(ã	NUM
cana-1620	67	13	)	)	PUNCT
cana-1620	67	14	=	=	SYM
cana-1620	67	15	𝑓(ώ(ã	𝑓(ώ(ã	PROPN
cana-1620	67	16	)	)	PUNCT
cana-1620	67	17	)	)	PUNCT
cana-1620	67	18	,	,	PUNCT
cana-1620	67	19	∀	∀	PUNCT
cana-1620	67	20	ã	ã	PRON
cana-1620	67	21	∈	∈	NOUN
cana-1620	67	22	ᾆ.	ᾆ.	NOUN
cana-1620	67	23	(	(	PUNCT
cana-1620	67	24	i	i	NOUN
cana-1620	67	25	)	)	PUNCT
cana-1620	67	26	(	(	PUNCT
cana-1620	67	27	a	a	X
cana-1620	67	28	)	)	PUNCT
cana-1620	67	29	ώ𝑓(0	ώ𝑓(0	PROPN
cana-1620	67	30	)	)	PUNCT
cana-1620	68	1	=	=	SYM
cana-1620	68	2	𝑓(ώ	𝑓(ώ	PROPN
cana-1620	68	3	(	(	PUNCT
cana-1620	68	4	0	0	NUM
cana-1620	68	5	)	)	PUNCT
cana-1620	68	6	)	)	PUNCT
cana-1620	68	7	≤	≤	NUM
cana-1620	68	8	𝑓(ώ(ã	𝑓(ώ(ã	NOUN
cana-1620	68	9	)	)	PUNCT
cana-1620	68	10	)	)	PUNCT
cana-1620	69	1	=	=	SYM
cana-1620	69	2	ώ𝑓(ã	ώ𝑓(ã	X
cana-1620	69	3	)	)	PUNCT
cana-1620	69	4	⇒	⇒	VERB
cana-1620	69	5	ώ𝑓(0	ώ𝑓(0	PROPN
cana-1620	69	6	)	)	PUNCT
cana-1620	69	7	≤	≤	NOUN
cana-1620	69	8	ώ𝑓(ã	ώ𝑓(ã	PUNCT
cana-1620	69	9	)	)	PUNCT
cana-1620	69	10	(	(	PUNCT
cana-1620	69	11	b	b	X
cana-1620	69	12	)	)	PUNCT
cana-1620	69	13	ώ𝑓(ã	ώ𝑓(ã	PUNCT
cana-1620	69	14	∗	∗	NOUN
cana-1620	69	15	ĉ	ĉ	PROPN
cana-1620	69	16	)	)	PUNCT
cana-1620	70	1	=	=	SYM
cana-1620	70	2	𝑓(ώ	𝑓(ώ	PROPN
cana-1620	70	3	(	(	PUNCT
cana-1620	70	4	ã	ã	NOUN
cana-1620	70	5	∗	∗	X
cana-1620	70	6	ĉ	ĉ	PROPN
cana-1620	70	7	)	)	PUNCT
cana-1620	70	8	)	)	PUNCT
cana-1620	70	9	≤	≤	NOUN
cana-1620	71	1	𝑓	𝑓	PRON
cana-1620	71	2	{	{	PUNCT
cana-1620	71	3	𝑚𝑎𝑥{ώ((ã	𝑚𝑎𝑥{ώ((ã	NOUN
cana-1620	71	4	∗	∗	NOUN
cana-1620	71	5	ɓ	ɓ	NOUN
cana-1620	71	6	)	)	PUNCT
cana-1620	71	7	∗	∗	NOUN
cana-1620	71	8	ĉ	ĉ	PROPN
cana-1620	71	9	)	)	PUNCT
cana-1620	71	10	,	,	PUNCT
cana-1620	71	11	ώ(ɓ	ώ(ɓ	NUM
cana-1620	71	12	)	)	PUNCT
cana-1620	71	13	}	}	PUNCT
cana-1620	71	14	}	}	PUNCT
cana-1620	71	15	=	=	SYM
cana-1620	71	16	𝑚𝑎𝑥{𝑓(ώ(ã	𝑚𝑎𝑥{𝑓(ώ(ã	X
cana-1620	71	17	∗	∗	X
cana-1620	71	18	ɓ	ɓ	NOUN
cana-1620	71	19	)	)	PUNCT
cana-1620	71	20	∗	∗	NOUN
cana-1620	71	21	ĉ	ĉ	PROPN
cana-1620	71	22	)	)	PUNCT
cana-1620	71	23	,	,	PUNCT
cana-1620	71	24	𝑓(ώ(ɓ	𝑓(ώ(ɓ	PROPN
cana-1620	71	25	)	)	PUNCT
cana-1620	71	26	)	)	PUNCT
cana-1620	71	27	}	}	PUNCT
cana-1620	71	28	=	=	PUNCT
cana-1620	71	29	𝑚𝑎𝑥{ώ𝑓((ã	𝑚𝑎𝑥{ώ𝑓((ã	NOUN
cana-1620	71	30	∗	∗	NOUN
cana-1620	71	31	ɓ	ɓ	NOUN
cana-1620	71	32	)	)	PUNCT
cana-1620	71	33	∗	∗	NOUN
cana-1620	71	34	ĉ	ĉ	PROPN
cana-1620	71	35	)	)	PUNCT
cana-1620	71	36	,	,	PUNCT
cana-1620	71	37	ώ𝑓(ɓ	ώ𝑓(ɓ	NOUN
cana-1620	71	38	)	)	PUNCT
cana-1620	71	39	}	}	PUNCT
cana-1620	71	40	⇒ώ𝑓	⇒ώ𝑓	NOUN
cana-1620	71	41	is	be	AUX
cana-1620	71	42	a	a	DET
cana-1620	71	43	𝐹𝑇𝐼.	𝐹𝑇𝐼.	NUM
cana-1620	71	44	(	(	PUNCT
cana-1620	71	45	ii	ii	NOUN
cana-1620	71	46	)	)	PUNCT
cana-1620	71	47	if	if	SCONJ
cana-1620	71	48	𝑓(ώ	𝑓(ώ	PROPN
cana-1620	71	49	(	(	PUNCT
cana-1620	71	50	0	0	NUM
cana-1620	71	51	)	)	PUNCT
cana-1620	71	52	)	)	PUNCT
cana-1620	72	1	=	=	SYM
cana-1620	72	2	1	1	NUM
cana-1620	72	3	⇒ώ𝑓(0	⇒ώ𝑓(0	NOUN
cana-1620	72	4	)	)	PUNCT
cana-1620	72	5	=	=	SYM
cana-1620	72	6	1	1	NUM
cana-1620	72	7	⇒	⇒	NOUN
cana-1620	72	8	ώ𝑓	ώ𝑓	NOUN
cana-1620	72	9	is	be	AUX
cana-1620	72	10	normal	normal	ADJ
cana-1620	72	11	(	(	PUNCT
cana-1620	72	12	iii	iii	NOUN
cana-1620	72	13	)	)	PUNCT
cana-1620	72	14	let	let	VERB
cana-1620	72	15	𝑓(𝑡	𝑓(𝑡	NOUN
cana-1620	72	16	)	)	PUNCT
cana-1620	72	17	≤	≤	NUM
cana-1620	72	18	𝑡	𝑡	NOUN
cana-1620	72	19	,	,	PUNCT
cana-1620	72	20	∀	∀	PUNCT
cana-1620	72	21	𝑡	𝑡	NOUN
cana-1620	72	22	∈	∈	PROPN
cana-1620	73	1	[	[	X
cana-1620	73	2	0	0	NUM
cana-1620	73	3	,	,	PUNCT
cana-1620	73	4	ώ(0	ώ(0	PROPN
cana-1620	73	5	)	)	PUNCT
cana-1620	73	6	]	]	PUNCT
cana-1620	74	1	then	then	ADV
cana-1620	74	2	ώ𝑓(ã	ώ𝑓(ã	PUNCT
cana-1620	74	3	)	)	PUNCT
cana-1620	74	4	=	=	SYM
cana-1620	74	5	𝑓(ώ(ã	𝑓(ώ(ã	PROPN
cana-1620	74	6	)	)	PUNCT
cana-1620	74	7	≤	≤	NOUN
cana-1620	74	8	ώ(ã	ώ(ã	NOUN
cana-1620	74	9	)	)	PUNCT
cana-1620	74	10	)	)	PUNCT
cana-1620	74	11	,	,	PUNCT
cana-1620	74	12	∀	∀	PUNCT
cana-1620	75	1	ã	ã	PRON
cana-1620	75	2	∈	∈	PROPN
cana-1620	75	3	ᾆ	ᾆ	PROPN
cana-1620	75	4	∴	∴	PROPN
cana-1620	75	5	ώ	ώ	PROPN
cana-1620	75	6	⊆	⊆	NUM
cana-1620	75	7	ώ𝑔.	ώ𝑔.	NOUN
cana-1620	75	8	definition	definition	NOUN
cana-1620	75	9	:	:	PUNCT
cana-1620	75	10	3.9	3.9	NUM
cana-1620	75	11	let	let	VERB
cana-1620	75	12	𝜗	𝜗	NOUN
cana-1620	75	13	=	=	PUNCT
cana-1620	75	14	(	(	PUNCT
cana-1620	75	15	𝜏	𝜏	NOUN
cana-1620	75	16	𝜏	𝜏	NOUN
cana-1620	75	17	is	be	AUX
cana-1620	75	18	the	the	DET
cana-1620	75	19	𝑁𝐷𝐹𝑇𝐼	𝑁𝐷𝐹𝑇𝐼	PROPN
cana-1620	75	20	of	of	ADP
cana-1620	75	21	ᾆ	ᾆ	NUM
cana-1620	75	22	)	)	PUNCT
cana-1620	75	23	then	then	ADV
cana-1620	75	24	the	the	DET
cana-1620	75	25	𝜗	𝜗	NOUN
cana-1620	75	26	is	be	AUX
cana-1620	75	27	called	call	VERB
cana-1620	75	28	a	a	DET
cana-1620	75	29	poset	poset	NOUN
cana-1620	75	30	according	accord	VERB
cana-1620	75	31	to	to	ADP
cana-1620	75	32	the	the	DET
cana-1620	75	33	principle	principle	NOUN
cana-1620	75	34	of	of	ADP
cana-1620	75	35	inclusion	inclusion	NOUN
cana-1620	75	36	.	.	PUNCT
cana-1620	76	1	definition	definition	NOUN
cana-1620	76	2	:	:	PUNCT
cana-1620	76	3	3.10	3.10	NUM
cana-1620	76	4	let	let	VERB
cana-1620	76	5	𝑠	𝑠	PROPN
cana-1620	76	6	>	>	X
cana-1620	76	7	0	0	PUNCT
cana-1620	76	8	be	be	AUX
cana-1620	76	9	a	a	DET
cana-1620	76	10	real	real	ADJ
cana-1620	76	11	number	number	NOUN
cana-1620	76	12	.	.	PUNCT
cana-1620	77	1	if	if	SCONJ
cana-1620	77	2	𝛽	𝛽	PROPN
cana-1620	77	3	∈	∈	PROPN
cana-1620	77	4	[	[	X
cana-1620	77	5	0,1	0,1	NUM
cana-1620	77	6	]	]	PUNCT
cana-1620	77	7	,	,	PUNCT
cana-1620	77	8	𝛽𝑠	𝛽𝑠	PROPN
cana-1620	77	9	be	be	AUX
cana-1620	77	10	the	the	DET
cana-1620	77	11	positive	positive	ADJ
cana-1620	77	12	root	root	NOUN
cana-1620	77	13	in	in	ADP
cana-1620	77	14	case	case	NOUN
cana-1620	77	15	𝑠	𝑠	X
cana-1620	77	16	<	<	X
cana-1620	77	17	1	1	NUM
cana-1620	77	18	.	.	PUNCT
cana-1620	78	1	we	we	PRON
cana-1620	78	2	define	define	VERB
cana-1620	78	3	ώ𝑠	ώ𝑠	X
cana-1620	78	4	:	:	PUNCT
cana-1620	78	5	𝐾	𝐾	NOUN
cana-1620	78	6	→	→	SYM
cana-1620	78	7	[	[	X
cana-1620	78	8	0,1	0,1	NUM
cana-1620	78	9	]	]	PUNCT
cana-1620	78	10	by	by	ADP
cana-1620	78	11	ώ𝑠(ã	ώ𝑠(ã	NOUN
cana-1620	78	12	)	)	PUNCT
cana-1620	78	13	=	=	SYM
cana-1620	78	14	(	(	PUNCT
cana-1620	78	15	ώ(ã	ώ(ã	NOUN
cana-1620	78	16	)	)	PUNCT
cana-1620	78	17	)	)	PUNCT
cana-1620	79	1	𝑠	𝑠	PROPN
cana-1620	79	2	,	,	PUNCT
cana-1620	79	3	∀ã	∀ã	PROPN
cana-1620	79	4	∈	∈	PROPN
cana-1620	79	5	ᾆ.	ᾆ.	NOUN
cana-1620	79	6	theorem	theorem	VERB
cana-1620	79	7	:	:	PUNCT
cana-1620	79	8	3.11	3.11	NUM
cana-1620	79	9	let	let	VERB
cana-1620	79	10	,	,	PUNCT
cana-1620	79	11	ώ∈	ώ∈	VERB
cana-1620	79	12	𝜗	𝜗	AUX
cana-1620	79	13	be	be	AUX
cana-1620	79	14	a	a	DET
cana-1620	79	15	constant	constant	ADJ
cana-1620	79	16	s.t	s.t	PROPN
cana-1620	79	17	it	it	PRON
cana-1620	79	18	is	be	AUX
cana-1620	79	19	a	a	DET
cana-1620	79	20	maximum	maximum	ADJ
cana-1620	79	21	element	element	NOUN
cana-1620	79	22	of	of	ADP
cana-1620	79	23	(	(	PUNCT
cana-1620	79	24	𝜗	𝜗	PROPN
cana-1620	79	25	,	,	PUNCT
cana-1620	79	26	⊆	⊆	NUM
cana-1620	79	27	)	)	PUNCT
cana-1620	79	28	.	.	PUNCT
cana-1620	80	1	then	then	ADV
cana-1620	80	2	,	,	PUNCT
cana-1620	80	3	ώ	ώ	PRON
cana-1620	80	4	only	only	ADV
cana-1620	80	5	accept	accept	VERB
cana-1620	80	6	the	the	DET
cana-1620	80	7	values	value	NOUN
cana-1620	80	8	of	of	ADP
cana-1620	80	9	0	0	NUM
cana-1620	80	10	&	&	CCONJ
cana-1620	80	11	1	1	NUM
cana-1620	80	12	.	.	PUNCT
cana-1620	81	1	proof	proof	NOUN
cana-1620	81	2	:	:	PUNCT
cana-1620	81	3	let	let	VERB
cana-1620	81	4	,	,	PUNCT
cana-1620	81	5	ώ∈	ώ∈	VERB
cana-1620	82	1	𝜗.	𝜗.	PROPN
cana-1620	82	2	then	then	ADV
cana-1620	82	3	ώ(0	ώ(0	PROPN
cana-1620	82	4	)	)	PUNCT
cana-1620	83	1	=	=	SYM
cana-1620	83	2	1	1	NUM
cana-1620	83	3	,	,	PUNCT
cana-1620	83	4	let	let	VERB
cana-1620	83	5	,	,	PUNCT
cana-1620	83	6	ã	ã	PROPN
cana-1620	83	7	∈	∈	PROPN
cana-1620	83	8	ᾆ	ᾆ	PROPN
cana-1620	83	9	s.t	s.t	PROPN
cana-1620	83	10	ώ(ã	ώ(ã	PROPN
cana-1620	83	11	)	)	PUNCT
cana-1620	83	12	≠	≠	PROPN
cana-1620	83	13	1	1	X
cana-1620	83	14	.	.	PUNCT
cana-1620	84	1	we	we	PRON
cana-1620	84	2	claim	claim	VERB
cana-1620	84	3	that	that	SCONJ
cana-1620	84	4	ώ(0	ώ(0	PROPN
cana-1620	84	5	)	)	PUNCT
cana-1620	84	6	=	=	PUNCT
cana-1620	85	1	0	0	X
cana-1620	85	2	.	.	PUNCT
cana-1620	86	1	if	if	SCONJ
cana-1620	86	2	not	not	PART
cana-1620	86	3	,	,	PUNCT
cana-1620	86	4	then	then	ADV
cana-1620	86	5	∃	∃	PROPN
cana-1620	86	6	𝑏	𝑏	PROPN
cana-1620	86	7	∈	∈	PROPN
cana-1620	86	8	𝑋	𝑋	PROPN
cana-1620	86	9	s.t	s.t	PROPN
cana-1620	86	10	0	0	PROPN
cana-1620	86	11	<	<	X
cana-1620	86	12	ώ(𝑏	ώ(𝑏	PROPN
cana-1620	86	13	)	)	PUNCT
cana-1620	86	14	<	<	X
cana-1620	87	1	1	1	X
cana-1620	87	2	.	.	PUNCT
cana-1620	87	3	we	we	PRON
cana-1620	87	4	now	now	ADV
cana-1620	87	5	define	define	VERB
cana-1620	87	6	a	a	DET
cana-1620	87	7	𝐷𝐹𝑆	𝐷𝐹𝑆	PROPN
cana-1620	87	8	,	,	PUNCT
cana-1620	87	9	𝜋	𝜋	NOUN
cana-1620	87	10	:	:	PUNCT
cana-1620	87	11	ᾆ	ᾆ	X
cana-1620	87	12	→	→	SYM
cana-1620	87	13	[	[	X
cana-1620	87	14	0,1	0,1	NUM
cana-1620	87	15	]	]	PUNCT
cana-1620	87	16	by	by	ADP
cana-1620	87	17	𝜋(ã	𝜋(ã	NOUN
cana-1620	87	18	)	)	PUNCT
cana-1620	87	19	=	=	SYM
cana-1620	87	20	1	1	NUM
cana-1620	87	21	2	2	NUM
cana-1620	87	22	{	{	PUNCT
cana-1620	87	23	ώ(ã	ώ(ã	NOUN
cana-1620	87	24	)	)	PUNCT
cana-1620	87	25	+	+	CCONJ
cana-1620	87	26	ώ(𝑏	ώ(𝑏	NOUN
cana-1620	87	27	)	)	PUNCT
cana-1620	87	28	}	}	PUNCT
cana-1620	87	29	,	,	PUNCT
cana-1620	87	30	∀	∀	PUNCT
cana-1620	88	1	ã	ã	PRON
cana-1620	88	2	∈	∈	NOUN
cana-1620	88	3	ᾆ.	ᾆ.	VERB
cana-1620	88	4	then	then	ADV
cana-1620	88	5	ώ	ώ	X
cana-1620	88	6	obviously	obviously	ADV
cana-1620	88	7	is	be	AUX
cana-1620	88	8	well	well	ADV
cana-1620	88	9	defined	define	VERB
cana-1620	88	10	now	now	ADV
cana-1620	88	11	,	,	PUNCT
cana-1620	88	12	(	(	PUNCT
cana-1620	88	13	i	i	NOUN
cana-1620	88	14	)	)	PUNCT
cana-1620	88	15	𝜋(0	𝜋(0	PROPN
cana-1620	88	16	)	)	PUNCT
cana-1620	88	17	=	=	SYM
cana-1620	88	18	1	1	NUM
cana-1620	88	19	2	2	NUM
cana-1620	88	20	{	{	PUNCT
cana-1620	88	21	ώ(0	ώ(0	PROPN
cana-1620	88	22	)	)	PUNCT
cana-1620	88	23	+	+	NUM
cana-1620	88	24	ώ(𝑏	ώ(𝑏	NOUN
cana-1620	88	25	)	)	PUNCT
cana-1620	88	26	}	}	PUNCT
cana-1620	88	27	≤	≤	NUM
cana-1620	88	28	1	1	NUM
cana-1620	88	29	2	2	NUM
cana-1620	88	30	{	{	PUNCT
cana-1620	88	31	ώ(ã	ώ(ã	NOUN
cana-1620	88	32	)	)	PUNCT
cana-1620	88	33	+	+	CCONJ
cana-1620	88	34	ώ(𝑏	ώ(𝑏	NOUN
cana-1620	88	35	)	)	PUNCT
cana-1620	88	36	}	}	PUNCT
cana-1620	88	37	=	=	SYM
cana-1620	88	38	𝜋(ã	𝜋(ã	PROPN
cana-1620	88	39	)	)	PUNCT
cana-1620	88	40	⇒𝜋(0	⇒𝜋(0	PROPN
cana-1620	88	41	)	)	PUNCT
cana-1620	88	42	≤	≤	NOUN
cana-1620	88	43	𝜋(ã	𝜋(ã	NOUN
cana-1620	88	44	)	)	PUNCT
cana-1620	88	45	communications	communication	NOUN
cana-1620	88	46	on	on	ADP
cana-1620	88	47	applied	apply	VERB
cana-1620	88	48	nonlinear	nonlinear	ADJ
cana-1620	88	49	analysis	analysis	NOUN
cana-1620	88	50	issn	issn	NOUN
cana-1620	88	51	:	:	PUNCT
cana-1620	88	52	1074	1074	NUM
cana-1620	88	53	-	-	PUNCT
cana-1620	88	54	133x	133x	NUM
cana-1620	88	55	vol	vol	NOUN
cana-1620	88	56	32	32	NUM
cana-1620	88	57	no	no	NOUN
cana-1620	88	58	.	.	NOUN
cana-1620	88	59	1	1	NUM
cana-1620	88	60	(	(	PUNCT
cana-1620	88	61	2025	2025	NUM
cana-1620	88	62	)	)	PUNCT
cana-1620	88	63	60	60	NUM
cana-1620	88	64	https://internationalpubls.com	https://internationalpubls.com	X
cana-1620	88	65	(	(	PUNCT
cana-1620	88	66	ii	ii	NOUN
cana-1620	88	67	)	)	PUNCT
cana-1620	88	68	𝜋(ã	𝜋(ã	PROPN
cana-1620	88	69	∗	∗	NUM
cana-1620	88	70	ĉ	ĉ	PROPN
cana-1620	88	71	)	)	PUNCT
cana-1620	88	72	=	=	SYM
cana-1620	88	73	1	1	NUM
cana-1620	88	74	2	2	NUM
cana-1620	88	75	{	{	PUNCT
cana-1620	88	76	ώ(ã	ώ(ã	NOUN
cana-1620	88	77	∗	∗	NOUN
cana-1620	88	78	ĉ	ĉ	PROPN
cana-1620	88	79	)	)	PUNCT
cana-1620	88	80	+	+	NUM
cana-1620	88	81	ώ(𝑏	ώ(𝑏	NOUN
cana-1620	88	82	)	)	PUNCT
cana-1620	88	83	}	}	PUNCT
cana-1620	88	84	≤	≤	NUM
cana-1620	88	85	1	1	NUM
cana-1620	88	86	2	2	NUM
cana-1620	88	87	{	{	PUNCT
cana-1620	88	88	𝑚𝑎𝑥{ώ((ã	𝑚𝑎𝑥{ώ((ã	NOUN
cana-1620	88	89	∗	∗	NOUN
cana-1620	88	90	ĉ	ĉ	PROPN
cana-1620	88	91	)	)	PUNCT
cana-1620	88	92	∗	∗	NOUN
cana-1620	88	93	ĉ	ĉ	PROPN
cana-1620	88	94	)	)	PUNCT
cana-1620	88	95	,	,	PUNCT
cana-1620	88	96	ώ(ɓ	ώ(ɓ	NUM
cana-1620	88	97	)	)	PUNCT
cana-1620	88	98	}	}	PUNCT
cana-1620	88	99	+	+	CCONJ
cana-1620	88	100	ώ(𝑏	ώ(𝑏	NOUN
cana-1620	88	101	)	)	PUNCT
cana-1620	88	102	}	}	PUNCT
cana-1620	89	1	=	=	SYM
cana-1620	89	2	1	1	NUM
cana-1620	89	3	2	2	NUM
cana-1620	89	4	{	{	PUNCT
cana-1620	89	5	𝑚𝑎𝑥({ώ((ã	𝑚𝑎𝑥({ώ((ã	NOUN
cana-1620	89	6	∗	∗	NOUN
cana-1620	89	7	ĉ	ĉ	NOUN
cana-1620	89	8	)	)	PUNCT
cana-1620	89	9	∗	∗	NOUN
cana-1620	89	10	ĉ	ĉ	PROPN
cana-1620	89	11	)	)	PUNCT
cana-1620	89	12	+	+	NUM
cana-1620	89	13	ώ(𝑏	ώ(𝑏	NOUN
cana-1620	89	14	)	)	PUNCT
cana-1620	89	15	}	}	PUNCT
cana-1620	89	16	,	,	PUNCT
cana-1620	89	17	{	{	PUNCT
cana-1620	89	18	ώ(ɓ	ώ(ɓ	NUM
cana-1620	89	19	)	)	PUNCT
cana-1620	90	1	+	+	CCONJ
cana-1620	90	2	ώ(𝑏	ώ(𝑏	NOUN
cana-1620	90	3	)	)	PUNCT
cana-1620	90	4	}	}	PUNCT
cana-1620	90	5	)	)	PUNCT
cana-1620	90	6	}	}	PUNCT
cana-1620	91	1	=	=	PUNCT
cana-1620	91	2	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-1620	91	3	{	{	PUNCT
cana-1620	91	4	1	1	NUM
cana-1620	91	5	2	2	NUM
cana-1620	91	6	{	{	PUNCT
cana-1620	91	7	ώ((ã	ώ((ã	NOUN
cana-1620	91	8	∗	∗	NOUN
cana-1620	91	9	ĉ	ĉ	PROPN
cana-1620	91	10	)	)	PUNCT
cana-1620	91	11	∗	∗	NOUN
cana-1620	91	12	ĉ	ĉ	PROPN
cana-1620	91	13	)	)	PUNCT
cana-1620	91	14	+	+	NUM
cana-1620	91	15	ώ(𝑏	ώ(𝑏	NOUN
cana-1620	91	16	)	)	PUNCT
cana-1620	91	17	}	}	PUNCT
cana-1620	91	18	,	,	PUNCT
cana-1620	91	19	1	1	NUM
cana-1620	91	20	2	2	NUM
cana-1620	91	21	{	{	PUNCT
cana-1620	91	22	ώ(ɓ	ώ(ɓ	NUM
cana-1620	91	23	)	)	PUNCT
cana-1620	91	24	+	+	CCONJ
cana-1620	91	25	ώ(𝑏	ώ(𝑏	NOUN
cana-1620	91	26	)	)	PUNCT
cana-1620	91	27	}	}	PUNCT
cana-1620	91	28	}	}	PUNCT
cana-1620	91	29	=	=	PUNCT
cana-1620	91	30	𝑚𝑎𝑥{𝜋((ã	𝑚𝑎𝑥{𝜋((ã	NUM
cana-1620	91	31	∗	∗	NOUN
cana-1620	91	32	ĉ	ĉ	PROPN
cana-1620	91	33	)	)	PUNCT
cana-1620	91	34	∗	∗	NOUN
cana-1620	91	35	ĉ	ĉ	PROPN
cana-1620	91	36	)	)	PUNCT
cana-1620	91	37	,	,	PUNCT
cana-1620	91	38	𝜋(ɓ	𝜋(ɓ	NOUN
cana-1620	91	39	)	)	PUNCT
cana-1620	91	40	}	}	PUNCT
cana-1620	91	41	⇒	⇒	VERB
cana-1620	91	42	𝜋(ã	𝜋(ã	PROPN
cana-1620	91	43	∗	∗	X
cana-1620	91	44	ĉ	ĉ	PROPN
cana-1620	91	45	)	)	PUNCT
cana-1620	91	46	≤	≤	NUM
cana-1620	91	47	𝑚𝑎𝑥{𝜋((ã	𝑚𝑎𝑥{𝜋((ã	NUM
cana-1620	91	48	∗	∗	NOUN
cana-1620	91	49	ĉ	ĉ	PROPN
cana-1620	91	50	)	)	PUNCT
cana-1620	91	51	∗	∗	NOUN
cana-1620	91	52	ĉ	ĉ	PROPN
cana-1620	91	53	)	)	PUNCT
cana-1620	91	54	,	,	PUNCT
cana-1620	91	55	𝜋(ɓ	𝜋(ɓ	NOUN
cana-1620	91	56	)	)	PUNCT
cana-1620	91	57	}	}	PUNCT
cana-1620	91	58	.	.	PUNCT
cana-1620	92	1	⇒𝜋	⇒𝜋	PROPN
cana-1620	92	2	is	be	AUX
cana-1620	92	3	a	a	DET
cana-1620	92	4	𝐹𝑇𝐼.	𝐹𝑇𝐼.	NUM
cana-1620	92	5	⇒	⇒	NOUN
cana-1620	92	6	𝜋𝑛	𝜋𝑛	X
cana-1620	92	7	is	be	AUX
cana-1620	92	8	a	a	DET
cana-1620	92	9	𝑁𝐹𝑇𝐼.	𝑁𝐹𝑇𝐼.	X
cana-1620	92	10	𝜋𝑛(ã	𝜋𝑛(ã	NUM
cana-1620	92	11	)	)	PUNCT
cana-1620	92	12	=	=	SYM
cana-1620	92	13	𝜋(ã	𝜋(ã	PROPN
cana-1620	92	14	)	)	PUNCT
cana-1620	93	1	+	+	PUNCT
cana-1620	93	2	𝜋𝑐(0	𝜋𝑐(0	X
cana-1620	93	3	)	)	PUNCT
cana-1620	93	4	=	=	SYM
cana-1620	93	5	𝜋(ã	𝜋(ã	PROPN
cana-1620	93	6	)	)	PUNCT
cana-1620	94	1	+	+	CCONJ
cana-1620	94	2	(	(	PUNCT
cana-1620	94	3	1	1	NUM
cana-1620	94	4	−	−	PROPN
cana-1620	94	5	𝜋(0	𝜋(0	PROPN
cana-1620	94	6	)	)	PUNCT
cana-1620	94	7	)	)	PUNCT
cana-1620	95	1	=	=	SYM
cana-1620	96	1	1	1	NUM
cana-1620	96	2	2	2	NUM
cana-1620	96	3	{	{	PUNCT
cana-1620	96	4	𝜋(ã	𝜋(ã	X
cana-1620	96	5	)	)	PUNCT
cana-1620	96	6	+	+	CCONJ
cana-1620	96	7	𝜋(𝑏	𝜋(𝑏	NOUN
cana-1620	96	8	)	)	PUNCT
cana-1620	96	9	}	}	PUNCT
cana-1620	97	1	+	+	CCONJ
cana-1620	97	2	(	(	PUNCT
cana-1620	97	3	1	1	NUM
cana-1620	97	4	−	−	NUM
cana-1620	97	5	1	1	NUM
cana-1620	97	6	2	2	NUM
cana-1620	97	7	{	{	PUNCT
cana-1620	97	8	𝜋(0	𝜋(0	PROPN
cana-1620	97	9	)	)	PUNCT
cana-1620	97	10	+	+	NUM
cana-1620	97	11	𝜋(𝑏	𝜋(𝑏	NOUN
cana-1620	97	12	)	)	PUNCT
cana-1620	97	13	}	}	PUNCT
cana-1620	97	14	)	)	PUNCT
cana-1620	97	15	=	=	SYM
cana-1620	97	16	1	1	NUM
cana-1620	97	17	2	2	NUM
cana-1620	97	18	ώ(ã	ώ(ã	NOUN
cana-1620	97	19	)	)	PUNCT
cana-1620	98	1	+	+	CCONJ
cana-1620	98	2	1	1	NUM
cana-1620	98	3	−	−	NUM
cana-1620	98	4	1	1	NUM
cana-1620	98	5	2	2	NUM
cana-1620	98	6	(	(	PUNCT
cana-1620	98	7	1	1	NUM
cana-1620	98	8	)	)	PUNCT
cana-1620	98	9	=	=	SYM
cana-1620	98	10	1	1	NUM
cana-1620	98	11	2	2	NUM
cana-1620	98	12	ώ(ã	ώ(ã	NOUN
cana-1620	98	13	)	)	PUNCT
cana-1620	98	14	+	+	CCONJ
cana-1620	98	15	1	1	NUM
cana-1620	98	16	2	2	NUM
cana-1620	98	17	=	=	SYM
cana-1620	98	18	1	1	NUM
cana-1620	98	19	2	2	NUM
cana-1620	98	20	(	(	PUNCT
cana-1620	98	21	ώ(ã	ώ(ã	NOUN
cana-1620	98	22	)	)	PUNCT
cana-1620	98	23	+	+	CCONJ
cana-1620	98	24	1	1	X
cana-1620	98	25	)	)	PUNCT
cana-1620	98	26	≤	≤	NOUN
cana-1620	98	27	ώ(ã	ώ(ã	NOUN
cana-1620	98	28	)	)	PUNCT
cana-1620	98	29	,	,	PUNCT
cana-1620	98	30	∀ã	∀ã	X
cana-1620	99	1	∈	∈	PROPN
cana-1620	99	2	𝑋	𝑋	PROPN
cana-1620	99	3	∴	∴	PROPN
cana-1620	99	4	𝜋𝑛(0	𝜋𝑛(0	PROPN
cana-1620	99	5	)	)	PUNCT
cana-1620	99	6	=	=	SYM
cana-1620	99	7	1	1	NUM
cana-1620	99	8	2	2	NUM
cana-1620	99	9	(	(	PUNCT
cana-1620	99	10	ώ(0	ώ(0	NOUN
cana-1620	99	11	)	)	PUNCT
cana-1620	99	12	+	+	NOUN
cana-1620	99	13	1	1	X
cana-1620	99	14	)	)	PUNCT
cana-1620	99	15	=	=	SYM
cana-1620	99	16	1	1	NUM
cana-1620	99	17	∴	∴	NOUN
cana-1620	99	18	𝜋𝑛	𝜋𝑛	X
cana-1620	99	19	is	be	AUX
cana-1620	99	20	normal	normal	ADJ
cana-1620	99	21	⇒𝜋𝑛	⇒𝜋𝑛	NOUN
cana-1620	99	22	∈	∈	PROPN
cana-1620	99	23	𝜗	𝜗	X
cana-1620	99	24	also	also	ADV
cana-1620	99	25	𝜋𝑛(ã	𝜋𝑛(ã	X
cana-1620	99	26	)	)	PUNCT
cana-1620	99	27	<	<	X
cana-1620	99	28	ώ(ã	ώ(ã	PROPN
cana-1620	99	29	)	)	PUNCT
cana-1620	99	30	,	,	PUNCT
cana-1620	99	31	∀ã	∀ã	X
cana-1620	99	32	∈	∈	PROPN
cana-1620	99	33	ᾆ.	ᾆ.	VERB
cana-1620	99	34	the	the	DET
cana-1620	99	35	contradiction	contradiction	NOUN
cana-1620	99	36	the	the	DET
cana-1620	99	37	fact	fact	NOUN
cana-1620	99	38	that	that	SCONJ
cana-1620	99	39	of	of	ADP
cana-1620	99	40	ώ	ώ	NOUN
cana-1620	99	41	is	be	AUX
cana-1620	99	42	normal	normal	ADJ
cana-1620	99	43	.	.	PUNCT
cana-1620	100	1	⇒ώ(ã	⇒ώ(ã	PROPN
cana-1620	100	2	)	)	PUNCT
cana-1620	100	3	=	=	SYM
cana-1620	100	4	0	0	NUM
cana-1620	100	5	,	,	PUNCT
cana-1620	100	6	∀ã	∀ã	X
cana-1620	100	7	∈	∈	PROPN
cana-1620	100	8	ᾆ.	ᾆ.	NOUN
cana-1620	100	9	theorem	theorem	VERB
cana-1620	100	10	:	:	PUNCT
cana-1620	100	11	3.12	3.12	NUM
cana-1620	100	12	let	let	VERB
cana-1620	100	13	ώ	ώ	PRON
cana-1620	100	14	is	be	AUX
cana-1620	100	15	a	a	DET
cana-1620	100	16	𝐷𝐹𝑇𝐼	𝐷𝐹𝑇𝐼	NOUN
cana-1620	100	17	of	of	ADP
cana-1620	100	18	𝑋	𝑋	PROPN
cana-1620	100	19	,	,	PUNCT
cana-1620	100	20	then	then	ADV
cana-1620	100	21	so	so	ADV
cana-1620	100	22	is	be	AUX
cana-1620	100	23	ώ𝑠	ώ𝑠	PROPN
cana-1620	100	24	and	and	CCONJ
cana-1620	100	25	𝑁ώ	𝑁ώ	PROPN
cana-1620	100	26	𝑠	𝑠	PROPN
cana-1620	100	27	=	=	SYM
cana-1620	100	28	𝑁ώ.	𝑁ώ.	PROPN
cana-1620	100	29	proof	proof	NOUN
cana-1620	100	30	:	:	PUNCT
cana-1620	100	31	let	let	VERB
cana-1620	100	32	ã	ã	X
cana-1620	100	33	,	,	PUNCT
cana-1620	100	34	ɓ	ɓ	DET
cana-1620	100	35	∈	∈	NOUN
cana-1620	100	36	ᾆ.	ᾆ.	VERB
cana-1620	100	37	now	now	ADV
cana-1620	100	38	,	,	PUNCT
cana-1620	100	39	(	(	PUNCT
cana-1620	100	40	i	i	NOUN
cana-1620	100	41	)	)	PUNCT
cana-1620	100	42	ώ𝑠(0	ώ𝑠(0	PROPN
cana-1620	100	43	)	)	PUNCT
cana-1620	101	1	=	=	PRON
cana-1620	101	2	(	(	PUNCT
cana-1620	101	3	ώ(0	ώ(0	PROPN
cana-1620	101	4	)	)	PUNCT
cana-1620	101	5	)	)	PUNCT
cana-1620	102	1	𝑠	𝑠	X
cana-1620	102	2	≤	≤	NUM
cana-1620	102	3	(	(	PUNCT
cana-1620	102	4	ώ(ã	ώ(ã	NOUN
cana-1620	102	5	)	)	PUNCT
cana-1620	102	6	)	)	PUNCT
cana-1620	103	1	𝑠	𝑠	PROPN
cana-1620	103	2	=	=	PUNCT
cana-1620	103	3	ώ𝑠(ã	ώ𝑠(ã	PROPN
cana-1620	103	4	)	)	PUNCT
cana-1620	103	5	.	.	PUNCT
cana-1620	104	1	communications	communication	NOUN
cana-1620	104	2	on	on	ADP
cana-1620	104	3	applied	apply	VERB
cana-1620	104	4	nonlinear	nonlinear	ADJ
cana-1620	104	5	analysis	analysis	NOUN
cana-1620	104	6	issn	issn	NOUN
cana-1620	104	7	:	:	PUNCT
cana-1620	104	8	1074	1074	NUM
cana-1620	104	9	-	-	PUNCT
cana-1620	104	10	133x	133x	NUM
cana-1620	104	11	vol	vol	NOUN
cana-1620	104	12	32	32	NUM
cana-1620	104	13	no	no	NOUN
cana-1620	104	14	.	.	NOUN
cana-1620	104	15	1	1	NUM
cana-1620	104	16	(	(	PUNCT
cana-1620	104	17	2025	2025	NUM
cana-1620	104	18	)	)	PUNCT
cana-1620	104	19	61	61	NUM
cana-1620	104	20	https://internationalpubls.com	https://internationalpubls.com	X
cana-1620	104	21	⇒	⇒	PROPN
cana-1620	104	22	ώ𝑠(0	ώ𝑠(0	PROPN
cana-1620	104	23	)	)	PUNCT
cana-1620	104	24	≥	≥	NOUN
cana-1620	104	25	ώ𝑠(ã	ώ𝑠(ã	PUNCT
cana-1620	104	26	)	)	PUNCT
cana-1620	104	27	(	(	PUNCT
cana-1620	104	28	ii	ii	NOUN
cana-1620	104	29	)	)	PUNCT
cana-1620	104	30	ώ𝑠(ã	ώ𝑠(ã	PROPN
cana-1620	105	1	∗	∗	PROPN
cana-1620	105	2	ĉ	ĉ	PROPN
cana-1620	105	3	)	)	PUNCT
cana-1620	106	1	=	=	SYM
cana-1620	106	2	(	(	PUNCT
cana-1620	106	3	ώ(ã	ώ(ã	NOUN
cana-1620	106	4	∗	∗	NOUN
cana-1620	106	5	ĉ	ĉ	PROPN
cana-1620	106	6	)	)	PUNCT
cana-1620	106	7	)	)	PUNCT
cana-1620	107	1	𝑠	𝑠	PROPN
cana-1620	107	2	≤	≤	NUM
cana-1620	107	3	(	(	PUNCT
cana-1620	107	4	𝑚𝑎𝑥{ώ((ã	𝑚𝑎𝑥{ώ((ã	NOUN
cana-1620	107	5	∗	∗	X
cana-1620	107	6	ɓ	ɓ	NOUN
cana-1620	107	7	)	)	PUNCT
cana-1620	107	8	∗	∗	NOUN
cana-1620	107	9	ĉ	ĉ	PROPN
cana-1620	107	10	)	)	PUNCT
cana-1620	107	11	,	,	PUNCT
cana-1620	107	12	ώ(ɓ	ώ(ɓ	NUM
cana-1620	107	13	)	)	PUNCT
cana-1620	107	14	}	}	PUNCT
cana-1620	107	15	)	)	PUNCT
cana-1620	108	1	𝑠	𝑠	X
cana-1620	108	2	=	=	PUNCT
cana-1620	108	3	𝑚𝑎𝑥	𝑚𝑎𝑥	X
cana-1620	108	4	{	{	PUNCT
cana-1620	108	5	(	(	PUNCT
cana-1620	108	6	ώ((ã	ώ((ã	PROPN
cana-1620	108	7	∗	∗	NOUN
cana-1620	108	8	ɓ	ɓ	NOUN
cana-1620	108	9	)	)	PUNCT
cana-1620	108	10	∗	∗	NOUN
cana-1620	108	11	ĉ	ĉ	PROPN
cana-1620	108	12	)	)	PUNCT
cana-1620	108	13	)	)	PUNCT
cana-1620	109	1	𝑠	𝑠	PROPN
cana-1620	109	2	,	,	PUNCT
cana-1620	109	3	(	(	PUNCT
cana-1620	109	4	ώ(ɓ))𝑠	ώ(ɓ))𝑠	NOUN
cana-1620	109	5	}	}	PUNCT
cana-1620	109	6	=	=	PUNCT
cana-1620	109	7	𝑚𝑎𝑥{ώ𝑠((ã	𝑚𝑎𝑥{ώ𝑠((ã	NOUN
cana-1620	109	8	∗	∗	X
cana-1620	109	9	ɓ	ɓ	NOUN
cana-1620	109	10	)	)	PUNCT
cana-1620	109	11	∗	∗	NOUN
cana-1620	109	12	ĉ	ĉ	PROPN
cana-1620	109	13	)	)	PUNCT
cana-1620	109	14	,	,	PUNCT
cana-1620	109	15	ώ𝑠(ɓ	ώ𝑠(ɓ	X
cana-1620	109	16	)	)	PUNCT
cana-1620	109	17	}	}	PUNCT
cana-1620	109	18	⇒	⇒	NOUN
cana-1620	109	19	ώ𝑠(ã	ώ𝑠(ã	PROPN
cana-1620	109	20	∗	∗	PROPN
cana-1620	109	21	ĉ	ĉ	PROPN
cana-1620	109	22	)	)	PUNCT
cana-1620	109	23	≤	≤	NUM
cana-1620	109	24	𝑚𝑎𝑥{ώ𝑠((ã	𝑚𝑎𝑥{ώ𝑠((ã	NUM
cana-1620	109	25	∗	∗	X
cana-1620	109	26	ɓ	ɓ	NOUN
cana-1620	109	27	)	)	PUNCT
cana-1620	109	28	∗	∗	NOUN
cana-1620	109	29	ĉ	ĉ	PROPN
cana-1620	109	30	)	)	PUNCT
cana-1620	109	31	,	,	PUNCT
cana-1620	109	32	ώ𝑠(ɓ	ώ𝑠(ɓ	X
cana-1620	109	33	)	)	PUNCT
cana-1620	109	34	}	}	PUNCT
cana-1620	109	35	∴	∴	PROPN
cana-1620	109	36	ώ𝑠	ώ𝑠	PROPN
cana-1620	109	37	is	be	AUX
cana-1620	109	38	a	a	DET
cana-1620	109	39	𝐷𝐹𝑇𝐼.	𝐷𝐹𝑇𝐼.	ADJ
cana-1620	109	40	𝑁ώ	𝑁ώ	PROPN
cana-1620	109	41	𝑠	𝑠	NOUN
cana-1620	109	42	=	=	PUNCT
cana-1620	109	43	{	{	PUNCT
cana-1620	109	44	ã	ã	X
cana-1620	109	45	∈	∈	PROPN
cana-1620	109	46	𝐽/ώ𝑠(ã	𝐽/ώ𝑠(ã	NOUN
cana-1620	109	47	)	)	PUNCT
cana-1620	110	1	=	=	SYM
cana-1620	110	2	ώ𝑠(0	ώ𝑠(0	PROPN
cana-1620	110	3	)	)	PUNCT
cana-1620	110	4	}	}	PUNCT
cana-1620	110	5	=	=	SYM
cana-1620	110	6	{	{	PUNCT
cana-1620	110	7	ã	ã	X
cana-1620	110	8	∈	∈	PROPN
cana-1620	110	9	𝐽/ώ(ã	𝐽/ώ(ã	PROPN
cana-1620	110	10	)	)	PUNCT
cana-1620	110	11	=	=	PUNCT
cana-1620	111	1	ώ(0	ώ(0	PROPN
cana-1620	111	2	)	)	PUNCT
cana-1620	111	3	}	}	PUNCT
cana-1620	111	4	⇒𝑁ώ	⇒𝑁ώ	NOUN
cana-1620	111	5	𝑠	𝑠	NOUN
cana-1620	111	6	=	=	SYM
cana-1620	111	7	𝑁ώ.	𝑁ώ.	PROPN
cana-1620	111	8	iii	iii	PROPN
cana-1620	111	9	.	.	PUNCT
cana-1620	111	10	conclusion	conclusion	NOUN
cana-1620	111	11	hence	hence	ADV
cana-1620	111	12	we	we	PRON
cana-1620	111	13	have	have	VERB
cana-1620	111	14	to	to	ADP
cana-1620	111	15	this	this	DET
cana-1620	111	16	paper	paper	NOUN
cana-1620	111	17	discussed	discuss	VERB
cana-1620	111	18	about	about	ADP
cana-1620	111	19	the	the	DET
cana-1620	111	20	𝑁𝐷𝐹𝑇𝐼	𝑁𝐷𝐹𝑇𝐼	PROPN
cana-1620	111	21	and	and	CCONJ
cana-1620	111	22	poset	poset	VERB
cana-1620	111	23	under	under	ADP
cana-1620	111	24	the	the	DET
cana-1620	111	25	set	set	NOUN
cana-1620	111	26	of	of	ADP
cana-1620	111	27	inclusion	inclusion	NOUN
cana-1620	111	28	principle	principle	NOUN
cana-1620	111	29	over	over	ADP
cana-1620	111	30	t	t	NOUN
cana-1620	111	31	-	-	PUNCT
cana-1620	111	32	algebra	algebra	NOUN
cana-1620	111	33	.	.	PUNCT
cana-1620	112	1	this	this	DET
cana-1620	112	2	idea	idea	NOUN
cana-1620	112	3	can	can	AUX
cana-1620	112	4	be	be	AUX
cana-1620	112	5	further	far	ADV
cana-1620	112	6	extended	extend	VERB
cana-1620	112	7	to	to	ADP
cana-1620	112	8	normalization	normalization	NOUN
cana-1620	112	9	of	of	ADP
cana-1620	112	10	intuitionistic	intuitionistic	ADJ
cana-1620	112	11	fs	f	NOUN
cana-1620	112	12	,	,	PUNCT
cana-1620	112	13	normalization	normalization	NOUN
cana-1620	112	14	of	of	ADP
cana-1620	112	15	interval	interval	NOUN
cana-1620	112	16	valued	value	VERB
cana-1620	112	17	fs	fs	ADP
cana-1620	112	18	,	,	PUNCT
cana-1620	112	19	and	and	CCONJ
cana-1620	112	20	normalization	normalization	NOUN
cana-1620	112	21	of	of	ADP
cana-1620	112	22	bipolar	bipolar	ADJ
cana-1620	112	23	fss	fss	NOUN
cana-1620	112	24	for	for	ADP
cana-1620	112	25	new	new	ADJ
cana-1620	112	26	findings	finding	NOUN
cana-1620	112	27	in	in	ADP
cana-1620	112	28	future	future	ADJ
cana-1620	112	29	studies	study	NOUN
cana-1620	112	30	.	.	PUNCT
cana-1620	113	1	references	reference	NOUN
cana-1620	113	2	[	[	X
cana-1620	113	3	1	1	NUM
cana-1620	113	4	]	]	PUNCT
cana-1620	113	5	m.	m.	NOUN
cana-1620	113	6	abu	abu	PROPN
cana-1620	113	7	ayub	ayub	PROPN
cana-1620	113	8	ansari	ansari	PROPN
cana-1620	113	9	and	and	CCONJ
cana-1620	113	10	m.	m.	NOUN
cana-1620	113	11	chandramouleeswaran	chandramouleeswaran	NOUN
cana-1620	113	12	,	,	PUNCT
cana-1620	113	13	t	t	NOUN
cana-1620	113	14	-	-	PUNCT
cana-1620	113	15	fuzzy	fuzzy	NOUN
cana-1620	113	16	𝛽	𝛽	NOUN
cana-1620	113	17	−subalgebras	−subalgebras	NOUN
cana-1620	113	18	of	of	ADP
cana-1620	113	19	𝛽	𝛽	PROPN
cana-1620	113	20	−algebras	−algebras	X
cana-1620	113	21	,	,	PUNCT
cana-1620	113	22	international	international	ADJ
cana-1620	113	23	j.	j.	PROPN
cana-1620	113	24	of	of	ADP
cana-1620	113	25	maths	maths	PROPN
cana-1620	113	26	.	.	PUNCT
cana-1620	114	1	sci	sci	PROPN
cana-1620	114	2	.	.	PROPN
cana-1620	114	3	and	and	CCONJ
cana-1620	114	4	engg	engg	PROPN
cana-1620	114	5	.	.	PUNCT
cana-1620	115	1	appls	appls	PROPN
cana-1620	115	2	.	.	PUNCT
cana-1620	116	1	(	(	PUNCT
cana-1620	116	2	ijmsea	ijmsea	NOUN
cana-1620	116	3	)	)	PUNCT
cana-1620	116	4	,	,	PUNCT
cana-1620	116	5	8	8	NUM
cana-1620	116	6	(	(	PUNCT
cana-1620	116	7	2014	2014	NUM
cana-1620	116	8	)	)	PUNCT
cana-1620	116	9	,	,	PUNCT
cana-1620	116	10	no	no	INTJ
cana-1620	116	11	.	.	NOUN
cana-1620	116	12	1	1	NUM
cana-1620	116	13	,	,	PUNCT
cana-1620	116	14	177	177	NUM
cana-1620	116	15	-	-	SYM
cana-1620	116	16	187	187	NUM
cana-1620	116	17	.	.	PUNCT
cana-1620	117	1	[	[	X
cana-1620	117	2	2	2	NUM
cana-1620	117	3	]	]	PUNCT
cana-1620	117	4	a.	a.	NOUN
cana-1620	117	5	prasanna	prasanna	PROPN
cana-1620	117	6	,	,	PUNCT
cana-1620	117	7	m.	m.	NOUN
cana-1620	117	8	premkumar	premkumar	PROPN
cana-1620	117	9	and	and	CCONJ
cana-1620	117	10	a.	a.	NOUN
cana-1620	117	11	solairaju	solairaju	PROPN
cana-1620	117	12	,	,	PUNCT
cana-1620	117	13	normalization	normalization	NOUN
cana-1620	117	14	of	of	ADP
cana-1620	117	15	fuzzy	fuzzy	ADJ
cana-1620	117	16	b	b	NOUN
cana-1620	117	17	-	-	PUNCT
cana-1620	117	18	ideals	ideal	NOUN
cana-1620	117	19	in	in	ADP
cana-1620	117	20	b	b	NOUN
cana-1620	117	21	-	-	PUNCT
cana-1620	117	22	algebra	algebra	NOUN
cana-1620	117	23	,	,	PUNCT
cana-1620	117	24	international	international	ADJ
cana-1620	117	25	journal	journal	NOUN
cana-1620	117	26	of	of	ADP
cana-1620	117	27	matheamtics	matheamtic	NOUN
cana-1620	117	28	trends	trend	NOUN
cana-1620	117	29	and	and	CCONJ
cana-1620	117	30	technology	technology	NOUN
cana-1620	117	31	,	,	PUNCT
cana-1620	117	32	53	53	NUM
cana-1620	117	33	(	(	PUNCT
cana-1620	117	34	2018	2018	NUM
cana-1620	117	35	)	)	PUNCT
cana-1620	117	36	,	,	PUNCT
cana-1620	117	37	no.4	no.4	PROPN
cana-1620	117	38	,	,	PUNCT
cana-1620	117	39	277	277	NUM
cana-1620	117	40	-	-	SYM
cana-1620	117	41	283	283	NUM
cana-1620	117	42	.	.	PUNCT
cana-1620	118	1	[	[	X
cana-1620	118	2	3	3	NUM
cana-1620	118	3	]	]	X
cana-1620	118	4	a.	a.	NOUN
cana-1620	118	5	prasanna	prasanna	PROPN
cana-1620	118	6	,	,	PUNCT
cana-1620	118	7	m.	m.	NOUN
cana-1620	118	8	premkumar	premkumar	PROPN
cana-1620	118	9	and	and	CCONJ
cana-1620	118	10	s.	s.	PROPN
cana-1620	118	11	ismail	ismail	PROPN
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cana-1620	118	15	of	of	ADP
cana-1620	118	16	fuzzy	fuzzy	ADJ
cana-1620	118	17	bg	bg	NOUN
cana-1620	118	18	-	-	PUNCT
cana-1620	118	19	ideals	ideal	NOUN
cana-1620	118	20	in	in	ADP
cana-1620	118	21	bg	bg	NOUN
cana-1620	118	22	-	-	PUNCT
cana-1620	118	23	algebra	algebra	PROPN
cana-1620	118	24	,	,	PUNCT
cana-1620	118	25	international	international	ADJ
cana-1620	118	26	journal	journal	NOUN
cana-1620	118	27	of	of	ADP
cana-1620	118	28	matheamtics	matheamtic	NOUN
cana-1620	118	29	trends	trend	NOUN
cana-1620	118	30	and	and	CCONJ
cana-1620	118	31	technology	technology	NOUN
cana-1620	118	32	,	,	PUNCT
cana-1620	118	33	53	53	NUM
cana-1620	118	34	(	(	PUNCT
cana-1620	118	35	2018	2018	NUM
cana-1620	118	36	)	)	PUNCT
cana-1620	118	37	,	,	PUNCT
cana-1620	118	38	no.4	no.4	PROPN
cana-1620	118	39	,	,	PUNCT
cana-1620	118	40	270	270	NUM
cana-1620	118	41	-	-	SYM
cana-1620	118	42	276	276	NUM
cana-1620	118	43	.	.	PUNCT
cana-1620	119	1	[	[	X
cana-1620	119	2	4	4	X
cana-1620	119	3	]	]	SYM
cana-1620	119	4	priyat	priyat	PROPN
cana-1620	119	5	and	and	CCONJ
cana-1620	119	6	ramachandran	ramachandran	PROPN
cana-1620	119	7	t	t	PROPN
cana-1620	119	8	,	,	PUNCT
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cana-1620	119	10	of	of	ADP
cana-1620	119	11	fuzzy	fuzzy	ADJ
cana-1620	119	12	ps	ps	NOUN
cana-1620	119	13	-	-	PUNCT
cana-1620	119	14	ideals	ideal	NOUN
cana-1620	119	15	and	and	CCONJ
cana-1620	119	16	fuzzy	fuzzy	ADJ
cana-1620	119	17	ps	ps	NOUN
cana-1620	119	18	-	-	PUNCT
cana-1620	119	19	sub	sub	NOUN
cana-1620	119	20	algebras	algebra	NOUN
cana-1620	119	21	of	of	ADP
cana-1620	119	22	ps	ps	NOUN
cana-1620	119	23	-	-	PUNCT
cana-1620	119	24	algebras	algebras	PROPN
cana-1620	119	25	,	,	PUNCT
cana-1620	119	26	research	research	NOUN
cana-1620	119	27	journal	journal	NOUN
cana-1620	119	28	’s	’s	PART
cana-1620	119	29	journal	journal	PROPN
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cana-1620	119	31	mathematics	mathematics	PROPN
cana-1620	119	32	,	,	PUNCT
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cana-1620	119	34	-	-	SYM
cana-1620	119	35	12	12	NUM
cana-1620	119	36	.	.	PUNCT
cana-1620	120	1	[	[	X
cana-1620	120	2	5	5	NUM
cana-1620	120	3	]	]	PUNCT
cana-1620	120	4	k.	k.	NOUN
cana-1620	120	5	rajam	rajam	PROPN
cana-1620	120	6	and	and	CCONJ
cana-1620	120	7	m.	m.	NOUN
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cana-1620	120	9	,	,	PUNCT
cana-1620	120	10	l	l	NOUN
cana-1620	120	11	-	-	ADJ
cana-1620	120	12	fuzzy	fuzzy	ADJ
cana-1620	120	13	t	t	NOUN
cana-1620	120	14	-	-	PUNCT
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cana-1620	120	16	in	in	ADP
cana-1620	120	17	𝛽	𝛽	NOUN
cana-1620	120	18	−algebras	−algebras	NUM
cana-1620	120	19	,	,	PUNCT
cana-1620	120	20	applied	apply	VERB
cana-1620	120	21	mathematical	mathematical	ADJ
cana-1620	120	22	sciences	science	NOUN
cana-1620	120	23	,	,	PUNCT
cana-1620	120	24	9	9	NUM
cana-1620	120	25	(	(	PUNCT
cana-1620	120	26	2015	2015	NUM
cana-1620	120	27	)	)	PUNCT
cana-1620	120	28	,	,	PUNCT
cana-1620	120	29	no	no	INTJ
cana-1620	120	30	.	.	NOUN
cana-1620	120	31	145	145	NUM
cana-1620	120	32	,	,	PUNCT
cana-1620	120	33	7221	7221	NUM
cana-1620	120	34	-	-	SYM
cana-1620	120	35	7228	7228	NUM
cana-1620	120	36	.	.	PUNCT
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cana-1620	121	2	.	.	PUNCT
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cana-1620	122	2	6	6	NUM
cana-1620	122	3	]	]	SYM
cana-1620	122	4	p.m.	p.m.	NOUN
cana-1620	122	5	sithar	sithar	PROPN
cana-1620	122	6	selvam	selvam	PROPN
cana-1620	122	7	and	and	CCONJ
cana-1620	122	8	k.t	k.t	PROPN
cana-1620	122	9	.	.	PROPN
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cana-1620	122	11	,	,	PUNCT
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cana-1620	122	16	of	of	ADP
cana-1620	122	17	fuzy	fuzy	ADJ
cana-1620	122	18	pms	pm	NOUN
cana-1620	122	19	-	-	PUNCT
cana-1620	122	20	algebra	algebra	NOUN
cana-1620	122	21	,	,	PUNCT
cana-1620	122	22	international	international	ADJ
cana-1620	122	23	journal	journal	NOUN
cana-1620	122	24	of	of	ADP
cana-1620	122	25	trend	trend	NOUN
cana-1620	122	26	in	in	ADP
cana-1620	122	27	reesrach	reesrach	NOUN
cana-1620	122	28	and	and	CCONJ
cana-1620	122	29	development	development	NOUN
cana-1620	122	30	,	,	PUNCT
cana-1620	122	31	3	3	NUM
cana-1620	122	32	(	(	PUNCT
cana-1620	122	33	2016	2016	NUM
cana-1620	122	34	)	)	PUNCT
cana-1620	122	35	,	,	PUNCT
cana-1620	122	36	no.6	no.6	PROPN
cana-1620	122	37	,	,	PUNCT
cana-1620	122	38	49	49	NUM
cana-1620	122	39	-	-	SYM
cana-1620	122	40	55	55	NUM
cana-1620	122	41	.	.	PUNCT
cana-1620	123	1	[	[	X
cana-1620	123	2	7	7	NUM
cana-1620	123	3	]	]	PUNCT
cana-1620	123	4	a.	a.	NOUN
cana-1620	123	5	tamilarasi	tamilarasi	PROPN
cana-1620	123	6	and	and	CCONJ
cana-1620	123	7	k.	k.	PROPN
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cana-1620	123	9	,	,	PUNCT
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cana-1620	123	13	fuzzy	fuzzy	ADJ
cana-1620	123	14	t	t	NOUN
cana-1620	123	15	-	-	PUNCT
cana-1620	123	16	ideals	ideal	NOUN
cana-1620	123	17	in	in	ADP
cana-1620	123	18	tm	tm	NOUN
cana-1620	123	19	-	-	PUNCT
cana-1620	123	20	algebras	algebra	NOUN
cana-1620	123	21	,	,	PUNCT
cana-1620	123	22	journal	journal	NOUN
cana-1620	123	23	of	of	ADP
cana-1620	123	24	mathematics	mathematic	NOUN
cana-1620	123	25	and	and	CCONJ
cana-1620	123	26	statistics	statistic	NOUN
cana-1620	123	27	,	,	PUNCT
cana-1620	123	28	7	7	NUM
cana-1620	123	29	(	(	PUNCT
cana-1620	123	30	2011	2011	NUM
cana-1620	123	31	)	)	PUNCT
cana-1620	123	32	,	,	PUNCT
cana-1620	123	33	no	no	INTJ
cana-1620	123	34	.	.	NOUN
cana-1620	123	35	2	2	NUM
cana-1620	123	36	,	,	PUNCT
cana-1620	123	37	107	107	NUM
cana-1620	123	38	-	-	SYM
cana-1620	123	39	111	111	NUM
cana-1620	123	40	.	.	PUNCT
cana-1620	124	1	https://doi.org/10.3844/jmssp.2011.107.111	https://doi.org/10.3844/jmssp.2011.107.111	PROPN
cana-1620	124	2	.	.	PUNCT
cana-1620	125	1	[	[	X
cana-1620	125	2	8	8	NUM
cana-1620	125	3	]	]	X
cana-1620	125	4	l.a	l.a	PROPN
cana-1620	125	5	.	.	PROPN
cana-1620	125	6	zadeh	zadeh	PROPN
cana-1620	125	7	,	,	PUNCT
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cana-1620	125	9	sets	set	NOUN
cana-1620	125	10	,	,	PUNCT
cana-1620	125	11	inform	inform	NOUN
cana-1620	125	12	.	.	PUNCT
cana-1620	126	1	and	and	CCONJ
cana-1620	126	2	control	control	NOUN
cana-1620	126	3	,	,	PUNCT
cana-1620	126	4	8	8	NUM
cana-1620	126	5	(	(	PUNCT
cana-1620	126	6	1965	1965	NUM
cana-1620	126	7	)	)	PUNCT
cana-1620	126	8	,	,	PUNCT
cana-1620	126	9	338	338	NUM
cana-1620	126	10	-	-	SYM
cana-1620	126	11	353	353	NUM
cana-1620	126	12	.	.	PUNCT
cana-1620	127	1	https://doi.org/10.1016/s0019-9958(65)90241-x	https://doi.org/10.1016/s0019-9958(65)90241-x	PROPN
cana-1620	127	2	.	.	PUNCT
cana-1620	128	1	https://doi.org/10.12988/ams.2015.59581	https://doi.org/10.12988/ams.2015.59581	PROPN
cana-1620	128	2	https://doi.org/10.3844/jmssp.2011.107.111	https://doi.org/10.3844/jmssp.2011.107.111	PROPN
cana-1620	128	3	https://doi.org/10.1016/s0019-9958(65)90241-x	https://doi.org/10.1016/s0019-9958(65)90241-x	PROPN