id	sid	tid	token	lemma	pos
cana-1172	1	1	communications	communication	NOUN
cana-1172	1	2	on	on	ADP
cana-1172	1	3	applied	apply	VERB
cana-1172	1	4	nonlinear	nonlinear	ADJ
cana-1172	1	5	analysis	analysis	NOUN
cana-1172	1	6	issn	issn	NOUN
cana-1172	1	7	:	:	PUNCT
cana-1172	1	8	1074	1074	NUM
cana-1172	1	9	-	-	PUNCT
cana-1172	1	10	133x	133x	NUM
cana-1172	1	11	vol	vol	NOUN
cana-1172	1	12	31	31	NUM
cana-1172	1	13	no	no	NOUN
cana-1172	1	14	.	.	PUNCT
cana-1172	2	1	6s	6s	NUM
cana-1172	2	2	(	(	PUNCT
cana-1172	2	3	2024	2024	NUM
cana-1172	2	4	)	)	PUNCT
cana-1172	2	5	131	131	NUM
cana-1172	3	1	https://internationalpubls.com	https://internationalpubls.com	X
cana-1172	3	2	lukasiewicz	lukasiewicz	VERB
cana-1172	3	3	fuzzy	fuzzy	ADJ
cana-1172	3	4	bm	bm	PROPN
cana-1172	3	5	-	-	NOUN
cana-1172	3	6	algebra	algebra	PROPN
cana-1172	3	7	and	and	CCONJ
cana-1172	3	8	bm	bm	NOUN
cana-1172	3	9	-	-	ADJ
cana-1172	3	10	ideal	ideal	ADJ
cana-1172	3	11	t.	t.	PROPN
cana-1172	3	12	gokila1	gokila1	NOUN
cana-1172	3	13	,	,	PUNCT
cana-1172	3	14	dr	dr	PROPN
cana-1172	3	15	.	.	PROPN
cana-1172	3	16	m.	m.	PROPN
cana-1172	3	17	mary	mary	PROPN
cana-1172	3	18	jansirani2	jansirani2	PROPN
cana-1172	4	1	1research	1research	NUM
cana-1172	4	2	scholar	scholar	NOUN
cana-1172	4	3	;	;	PUNCT
cana-1172	4	4	school	school	NOUN
cana-1172	4	5	of	of	ADP
cana-1172	4	6	sciences	science	NOUN
cana-1172	4	7	;	;	PUNCT
cana-1172	4	8	division	division	NOUN
cana-1172	4	9	of	of	ADP
cana-1172	4	10	mathematics	mathematic	NOUN
cana-1172	4	11	;	;	PUNCT
cana-1172	4	12	srminstitute	srminstitute	NOUN
cana-1172	4	13	of	of	ADP
cana-1172	4	14	science	science	NOUN
cana-1172	4	15	and	and	CCONJ
cana-1172	4	16	technology	technology	NOUN
cana-1172	4	17	(	(	PUNCT
cana-1172	4	18	deemed	deem	VERB
cana-1172	4	19	to	to	PART
cana-1172	4	20	be	be	AUX
cana-1172	4	21	university	university	NOUN
cana-1172	4	22	)	)	PUNCT
cana-1172	4	23	;	;	PUNCT
cana-1172	4	24	irungalur	irungalur	NOUN
cana-1172	4	25	,	,	PUNCT
cana-1172	4	26	trichy	trichy	NOUN
cana-1172	4	27	–	–	PUNCT
cana-1172	4	28	621105	621105	NUM
cana-1172	4	29	,	,	PUNCT
cana-1172	4	30	tamil	tamil	PROPN
cana-1172	4	31	nadu	nadu	NOUN
cana-1172	4	32	2associate	2associate	NUM
cana-1172	4	33	professor	professor	NOUN
cana-1172	4	34	;	;	PUNCT
cana-1172	4	35	school	school	NOUN
cana-1172	4	36	of	of	ADP
cana-1172	4	37	science	science	NOUN
cana-1172	4	38	;	;	PUNCT
cana-1172	4	39	division	division	NOUN
cana-1172	4	40	of	of	ADP
cana-1172	4	41	mathematics	mathematic	NOUN
cana-1172	4	42	;	;	PUNCT
cana-1172	4	43	srm	srm	PROPN
cana-1172	4	44	-	-	PUNCT
cana-1172	4	45	institute	institute	NOUN
cana-1172	4	46	of	of	ADP
cana-1172	4	47	science	science	NOUN
cana-1172	4	48	and	and	CCONJ
cana-1172	4	49	technology	technology	NOUN
cana-1172	4	50	(	(	PUNCT
cana-1172	4	51	deemed	deem	VERB
cana-1172	4	52	to	to	PART
cana-1172	4	53	be	be	AUX
cana-1172	4	54	university	university	NOUN
cana-1172	4	55	)	)	PUNCT
cana-1172	4	56	;	;	PUNCT
cana-1172	4	57	irungalur	irungalur	NOUN
cana-1172	4	58	,	,	PUNCT
cana-1172	4	59	trichy	trichy	NOUN
cana-1172	4	60	–	–	PUNCT
cana-1172	4	61	621105	621105	NUM
cana-1172	4	62	,	,	PUNCT
cana-1172	4	63	tamil	tamil	PROPN
cana-1172	4	64	nadu	nadu	ADJ
cana-1172	4	65	article	article	NOUN
cana-1172	4	66	history	history	NOUN
cana-1172	4	67	:	:	PUNCT
cana-1172	4	68	received	receive	VERB
cana-1172	4	69	:	:	PUNCT
cana-1172	4	70	24	24	NUM
cana-1172	4	71	-	-	PUNCT
cana-1172	4	72	05	05	NUM
cana-1172	4	73	-	-	PUNCT
cana-1172	4	74	2024	2024	NUM
cana-1172	4	75	revised	revise	VERB
cana-1172	4	76	:	:	PUNCT
cana-1172	4	77	13	13	NUM
cana-1172	4	78	-	-	SYM
cana-1172	4	79	07	07	NUM
cana-1172	4	80	-	-	PUNCT
cana-1172	4	81	2024	2024	NUM
cana-1172	4	82	accepted	accept	VERB
cana-1172	4	83	:	:	PUNCT
cana-1172	4	84	26	26	NUM
cana-1172	4	85	-	-	SYM
cana-1172	4	86	07	07	NUM
cana-1172	4	87	-	-	PUNCT
cana-1172	4	88	2024	2024	NUM
cana-1172	4	89	abstract	abstract	NOUN
cana-1172	4	90	:	:	PUNCT
cana-1172	4	91	introduction	introduction	NOUN
cana-1172	4	92	:	:	PUNCT
cana-1172	4	93	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	NOUN
cana-1172	4	94	sets	set	NOUN
cana-1172	4	95	is	be	AUX
cana-1172	4	96	a	a	DET
cana-1172	4	97	mathematical	mathematical	ADJ
cana-1172	4	98	framework	framework	NOUN
cana-1172	4	99	that	that	PRON
cana-1172	4	100	expands	expand	VERB
cana-1172	4	101	the	the	DET
cana-1172	4	102	traditional	traditional	ADJ
cana-1172	4	103	concept	concept	NOUN
cana-1172	4	104	of	of	ADP
cana-1172	4	105	sets	set	NOUN
cana-1172	4	106	by	by	ADP
cana-1172	4	107	enabling	enable	VERB
cana-1172	4	108	elements	element	NOUN
cana-1172	4	109	to	to	PART
cana-1172	4	110	have	have	VERB
cana-1172	4	111	degrees	degree	NOUN
cana-1172	4	112	of	of	ADP
cana-1172	4	113	membership	membership	NOUN
cana-1172	4	114	.	.	PUNCT
cana-1172	5	1	this	this	PRON
cana-1172	5	2	enables	enable	VERB
cana-1172	5	3	partial	partial	ADJ
cana-1172	5	4	membership	membership	NOUN
cana-1172	5	5	based	base	VERB
cana-1172	5	6	on	on	ADP
cana-1172	5	7	degree	degree	NOUN
cana-1172	5	8	of	of	ADP
cana-1172	5	9	likeness	likeness	NOUN
cana-1172	5	10	.	.	PUNCT
cana-1172	6	1	in	in	ADP
cana-1172	6	2	classical	classical	ADJ
cana-1172	6	3	set	set	NOUN
cana-1172	6	4	theory	theory	NOUN
cana-1172	6	5	,	,	PUNCT
cana-1172	6	6	an	an	DET
cana-1172	6	7	element	element	NOUN
cana-1172	6	8	can	can	AUX
cana-1172	6	9	be	be	AUX
cana-1172	6	10	represented	represent	VERB
cana-1172	6	11	as	as	ADP
cana-1172	6	12	a	a	DET
cana-1172	6	13	crisp	crisp	ADJ
cana-1172	6	14	set	set	NOUN
cana-1172	6	15	,	,	PUNCT
cana-1172	6	16	indicated	indicate	VERB
cana-1172	6	17	by	by	ADP
cana-1172	6	18	𝑥	𝑥	PROPN
cana-1172	6	19	,	,	PUNCT
cana-1172	6	20	which	which	PRON
cana-1172	6	21	either	either	CCONJ
cana-1172	6	22	belongs	belong	VERB
cana-1172	6	23	to	to	ADP
cana-1172	6	24	or	or	CCONJ
cana-1172	6	25	does	do	AUX
cana-1172	6	26	not	not	PART
cana-1172	6	27	belong	belong	VERB
cana-1172	6	28	to	to	ADP
cana-1172	6	29	the	the	DET
cana-1172	6	30	set	set	NOUN
cana-1172	6	31	.	.	PUNCT
cana-1172	7	1	in	in	ADP
cana-1172	7	2	contrast	contrast	NOUN
cana-1172	7	3	,	,	PUNCT
cana-1172	7	4	an	an	DET
cana-1172	7	5	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	7	6	sets	set	NOUN
cana-1172	7	7	allows	allow	VERB
cana-1172	7	8	for	for	ADP
cana-1172	7	9	various	various	ADJ
cana-1172	7	10	levels	level	NOUN
cana-1172	7	11	of	of	ADP
cana-1172	7	12	membership	membership	NOUN
cana-1172	7	13	.	.	PUNCT
cana-1172	8	1	the	the	DET
cana-1172	8	2	level	level	NOUN
cana-1172	8	3	of	of	ADP
cana-1172	8	4	membership	membership	NOUN
cana-1172	8	5	has	have	VERB
cana-1172	8	6	a	a	DET
cana-1172	8	7	value	value	NOUN
cana-1172	8	8	somewhere	somewhere	ADV
cana-1172	8	9	between	between	ADP
cana-1172	8	10	0	0	NUM
cana-1172	8	11	and	and	CCONJ
cana-1172	8	12	1	1	NUM
cana-1172	8	13	,	,	PUNCT
cana-1172	8	14	with	with	ADP
cana-1172	8	15	0	0	NUM
cana-1172	8	16	representing	represent	VERB
cana-1172	8	17	nonparticipation	nonparticipation	NOUN
cana-1172	8	18	and	and	CCONJ
cana-1172	8	19	1	1	NUM
cana-1172	8	20	representing	represent	VERB
cana-1172	8	21	full	full	ADJ
cana-1172	8	22	participation	participation	NOUN
cana-1172	8	23	.	.	PUNCT
cana-1172	9	1	the	the	DET
cana-1172	9	2	shape	shape	NOUN
cana-1172	9	3	of	of	ADP
cana-1172	9	4	the	the	DET
cana-1172	9	5	member	member	NOUN
cana-1172	9	6	function	function	NOUN
cana-1172	9	7	varies	vary	VERB
cana-1172	9	8	according	accord	VERB
cana-1172	9	9	to	to	ADP
cana-1172	9	10	the	the	DET
cana-1172	9	11	application	application	NOUN
cana-1172	9	12	and	and	CCONJ
cana-1172	9	13	intended	intend	VERB
cana-1172	9	14	behaviour	behaviour	NOUN
cana-1172	9	15	.	.	PUNCT
cana-1172	10	1	jan	jan	PROPN
cana-1172	10	2	lukasiewicz	lukasiewicz	PROPN
cana-1172	10	3	was	be	AUX
cana-1172	10	4	a	a	DET
cana-1172	10	5	logical	logical	ADJ
cana-1172	10	6	thinker	thinker	NOUN
cana-1172	10	7	and	and	CCONJ
cana-1172	10	8	philosopher	philosopher	NOUN
cana-1172	10	9	.	.	PUNCT
cana-1172	11	1	he	he	PRON
cana-1172	11	2	contributed	contribute	VERB
cana-1172	11	3	to	to	ADP
cana-1172	11	4	the	the	DET
cana-1172	11	5	advancement	advancement	NOUN
cana-1172	11	6	of	of	ADP
cana-1172	11	7	proportional	proportional	ADJ
cana-1172	11	8	logic	logic	NOUN
cana-1172	11	9	.	.	PUNCT
cana-1172	12	1	lukasiewicz	lukasiewicz	VERB
cana-1172	12	2	or	or	CCONJ
cana-1172	12	3	lukasz	lukasz	PROPN
cana-1172	12	4	logic	logic	NOUN
cana-1172	12	5	is	be	AUX
cana-1172	12	6	an	an	DET
cana-1172	12	7	uncommon	uncommon	ADJ
cana-1172	12	8	and	and	CCONJ
cana-1172	12	9	highly	highly	ADV
cana-1172	12	10	appreciated	appreciated	ADJ
cana-1172	12	11	logic	logic	NOUN
cana-1172	12	12	that	that	PRON
cana-1172	12	13	follows	follow	VERB
cana-1172	12	14	the	the	DET
cana-1172	12	15	lukasz	lukasz	PROPN
cana-1172	12	16	t	t	PROPN
cana-1172	12	17	-	-	PUNCT
cana-1172	12	18	norm	norm	NOUN
cana-1172	12	19	and	and	CCONJ
cana-1172	12	20	t	t	NOUN
cana-1172	12	21	-	-	PUNCT
cana-1172	12	22	conorm	conorm	NOUN
cana-1172	12	23	operations	operation	NOUN
cana-1172	12	24	to	to	PART
cana-1172	12	25	compute	compute	VERB
cana-1172	12	26	the	the	DET
cana-1172	12	27	intersection	intersection	NOUN
cana-1172	12	28	and	and	CCONJ
cana-1172	12	29	union	union	NOUN
cana-1172	12	30	of	of	ADP
cana-1172	12	31	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	12	32	sets	set	NOUN
cana-1172	12	33	.	.	PUNCT
cana-1172	13	1	this	this	DET
cana-1172	13	2	logic	logic	NOUN
cana-1172	13	3	enables	enable	VERB
cana-1172	13	4	reasoning	reason	VERB
cana-1172	13	5	with	with	ADP
cana-1172	13	6	unclear	unclear	ADJ
cana-1172	13	7	or	or	CCONJ
cana-1172	13	8	incomplete	incomplete	ADJ
cana-1172	13	9	knowledge	knowledge	NOUN
cana-1172	13	10	,	,	PUNCT
cana-1172	13	11	making	make	VERB
cana-1172	13	12	it	it	PRON
cana-1172	13	13	appropriate	appropriate	ADJ
cana-1172	13	14	for	for	ADP
cana-1172	13	15	a	a	DET
cana-1172	13	16	variety	variety	NOUN
cana-1172	13	17	of	of	ADP
cana-1172	13	18	applications	application	NOUN
cana-1172	13	19	including	include	VERB
cana-1172	13	20	ambiguity	ambiguity	NOUN
cana-1172	13	21	and	and	CCONJ
cana-1172	13	22	imprecision	imprecision	NOUN
cana-1172	13	23	.	.	PUNCT
cana-1172	14	1	objectives	objective	NOUN
cana-1172	14	2	:	:	PUNCT
cana-1172	14	3	incorporation	incorporation	NOUN
cana-1172	14	4	of	of	ADP
cana-1172	14	5	lukasz	lukasz	PROPN
cana-1172	14	6	logic	logic	NOUN
cana-1172	14	7	theory	theory	NOUN
cana-1172	14	8	to	to	ADP
cana-1172	14	9	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	14	10	set	set	VERB
cana-1172	14	11	in	in	ADP
cana-1172	14	12	𝐵𝑀-algebra	𝐵𝑀-algebra	PROPN
cana-1172	14	13	for	for	ADP
cana-1172	14	14	the	the	DET
cana-1172	14	15	betterment	betterment	NOUN
cana-1172	14	16	of	of	ADP
cana-1172	14	17	algorithms	algorithm	NOUN
cana-1172	14	18	to	to	PART
cana-1172	14	19	address	address	VERB
cana-1172	14	20	a	a	DET
cana-1172	14	21	variety	variety	NOUN
cana-1172	14	22	of	of	ADP
cana-1172	14	23	real	real	ADJ
cana-1172	14	24	-	-	PUNCT
cana-1172	14	25	world	world	NOUN
cana-1172	14	26	issues	issue	NOUN
cana-1172	14	27	,	,	PUNCT
cana-1172	14	28	including	include	VERB
cana-1172	14	29	risk	risk	NOUN
cana-1172	14	30	management	management	NOUN
cana-1172	14	31	,	,	PUNCT
cana-1172	14	32	decision	decision	NOUN
cana-1172	14	33	making	making	NOUN
cana-1172	14	34	,	,	PUNCT
cana-1172	14	35	managing	manage	VERB
cana-1172	14	36	public	public	ADJ
cana-1172	14	37	transit	transit	NOUN
cana-1172	14	38	,	,	PUNCT
cana-1172	14	39	diagnosing	diagnose	VERB
cana-1172	14	40	medical	medical	ADJ
cana-1172	14	41	conditions	condition	NOUN
cana-1172	14	42	and	and	CCONJ
cana-1172	14	43	more	more	ADJ
cana-1172	14	44	.	.	PUNCT
cana-1172	15	1	methods	method	NOUN
cana-1172	15	2	:	:	PUNCT
cana-1172	15	3	applying	apply	VERB
cana-1172	15	4	𝐵𝑀-algebra	𝐵𝑀-algebra	PROPN
cana-1172	15	5	to	to	ADP
cana-1172	15	6	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	15	7	set	set	NOUN
cana-1172	15	8	theory	theory	NOUN
cana-1172	15	9	and	and	CCONJ
cana-1172	15	10	incorporating	incorporate	VERB
cana-1172	15	11	lukasz	lukasz	NOUN
cana-1172	15	12	logic	logic	NOUN
cana-1172	15	13	theory	theory	NOUN
cana-1172	15	14	with	with	ADP
cana-1172	15	15	the	the	DET
cana-1172	15	16	inclusion	inclusion	NOUN
cana-1172	15	17	of	of	ADP
cana-1172	15	18	certain	certain	ADJ
cana-1172	15	19	attributes	attribute	NOUN
cana-1172	15	20	,	,	PUNCT
cana-1172	15	21	in	in	ADP
cana-1172	15	22	order	order	NOUN
cana-1172	15	23	to	to	PART
cana-1172	15	24	facilitate	facilitate	VERB
cana-1172	15	25	the	the	DET
cana-1172	15	26	production	production	NOUN
cana-1172	15	27	of	of	ADP
cana-1172	15	28	lukasz	lukasz	PROPN
cana-1172	15	29	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	15	30	𝐵𝑀-algebra	𝐵𝑀-algebra	PROPN
cana-1172	15	31	and	and	CCONJ
cana-1172	15	32	𝐵𝑀-ideal	𝐵𝑀-ideal	NOUN
cana-1172	15	33	,	,	PUNCT
cana-1172	15	34	wherein	wherein	SCONJ
cana-1172	15	35	the	the	DET
cana-1172	15	36	characteristics	characteristic	NOUN
cana-1172	15	37	and	and	CCONJ
cana-1172	15	38	attributes	attribute	NOUN
cana-1172	15	39	of	of	ADP
cana-1172	15	40	the	the	DET
cana-1172	15	41	lukasz	lukasz	NOUN
cana-1172	15	42	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	15	43	𝐵𝑀-algebra	𝐵𝑀-algebra	PROPN
cana-1172	15	44	and	and	CCONJ
cana-1172	15	45	𝐵𝑀-ideal	𝐵𝑀-ideal	PROPN
cana-1172	15	46	are	be	AUX
cana-1172	15	47	examined	examine	VERB
cana-1172	15	48	,	,	PUNCT
cana-1172	15	49	and	and	CCONJ
cana-1172	15	50	the	the	DET
cana-1172	15	51	relationships	relationship	NOUN
cana-1172	15	52	between	between	ADP
cana-1172	15	53	them	they	PRON
cana-1172	15	54	are	be	AUX
cana-1172	15	55	demonstrated	demonstrate	VERB
cana-1172	15	56	by	by	ADP
cana-1172	15	57	a	a	DET
cana-1172	15	58	few	few	ADJ
cana-1172	15	59	examples	example	NOUN
cana-1172	15	60	.	.	PUNCT
cana-1172	16	1	results	result	NOUN
cana-1172	16	2	:	:	PUNCT
cana-1172	16	3	theorem	theorem	VERB
cana-1172	16	4	3.5	3.5	NUM
cana-1172	16	5	.	.	PUNCT
cana-1172	17	1	every	every	DET
cana-1172	17	2	lukasz	lukasz	NOUN
cana-1172	17	3	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	17	4	set	set	VERB
cana-1172	17	5	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	17	6	𝜀	𝜀	PROPN
cana-1172	17	7	is	be	AUX
cana-1172	17	8	a	a	DET
cana-1172	17	9	lukasz	lukasz	NOUN
cana-1172	17	10	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	17	11	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	PROPN
cana-1172	17	12	of	of	ADP
cana-1172	17	13	𝔊	𝔊	PROPN
cana-1172	17	14	iff	iff	VERB
cana-1172	17	15	it	it	PRON
cana-1172	17	16	satisfies	satisfy	VERB
cana-1172	17	17	:	:	PUNCT
cana-1172	17	18	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	17	19	𝜀	𝜀	PROPN
cana-1172	17	20	(	(	PUNCT
cana-1172	17	21	�	�	PROPN
cana-1172	17	22	̇	̇	PROPN
cana-1172	17	23	�	�	PROPN
cana-1172	17	24	∗	∗	PROPN
cana-1172	17	25	�	�	PROPN
cana-1172	17	26	̇	̇	PROPN
cana-1172	17	27	�	�	PROPN
cana-1172	17	28	)	)	PUNCT
cana-1172	17	29	≥	≥	NOUN
cana-1172	17	30	𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝐿𝑈	NUM
cana-1172	17	31	𝜀	𝜀	X
cana-1172	17	32	(	(	PUNCT
cana-1172	17	33	�	�	PROPN
cana-1172	17	34	̇	̇	PROPN
cana-1172	17	35	�	�	PROPN
cana-1172	17	36	)	)	PUNCT
cana-1172	17	37	,	,	PUNCT
cana-1172	17	38	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	17	39	𝜀	𝜀	PROPN
cana-1172	17	40	(	(	PUNCT
cana-1172	17	41	�	�	PROPN
cana-1172	17	42	̇	̇	PROPN
cana-1172	17	43	�	�	PROPN
cana-1172	17	44	)	)	PUNCT
cana-1172	17	45	}	}	PUNCT
cana-1172	17	46	,	,	PUNCT
cana-1172	17	47	∀	∀	X
cana-1172	17	48	�	�	PROPN
cana-1172	17	49	̇	̇	PROPN
cana-1172	17	50	�	�	PROPN
cana-1172	17	51	,	,	PUNCT
cana-1172	17	52	�	�	PROPN
cana-1172	17	53	̇	̇	VERB
cana-1172	17	54	�	�	PROPN
cana-1172	17	55	∈	∈	PROPN
cana-1172	17	56	𝔊.	𝔊.	PROPN
cana-1172	17	57	theorem	theorem	VERB
cana-1172	17	58	3.6	3.6	NUM
cana-1172	17	59	.	.	PUNCT
cana-1172	18	1	show	show	VERB
cana-1172	18	2	that	that	SCONJ
cana-1172	18	3	𝜀-lukasz	𝜀-lukasz	NOUN
cana-1172	18	4	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	NOUN
cana-1172	18	5	set	set	VERB
cana-1172	18	6	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	18	7	𝜀	𝜀	NOUN
cana-1172	18	8	in	in	ADP
cana-1172	18	9	𝔊	𝔊	PROPN
cana-1172	18	10	is	be	AUX
cana-1172	18	11	an	an	DET
cana-1172	18	12	𝜀-lukasz	𝜀-lukasz	NOUN
cana-1172	18	13	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	18	14	𝐵𝑀𝑎𝑙𝑔𝑒𝑏𝑟𝑎	𝐵𝑀𝑎𝑙𝑔𝑒𝑏𝑟𝑎	PROPN
cana-1172	18	15	of	of	ADP
cana-1172	18	16	𝔊	𝔊	PROPN
cana-1172	18	17	,	,	PUNCT
cana-1172	18	18	if	if	SCONJ
cana-1172	18	19	𝑈	𝑈	PROPN
cana-1172	18	20	is	be	AUX
cana-1172	18	21	a	a	DET
cana-1172	18	22	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	18	23	sub	sub	NOUN
cana-1172	18	24	algebra	algebra	NOUN
cana-1172	18	25	of	of	ADP
cana-1172	18	26	𝔊.	𝔊.	PROPN
cana-1172	18	27	an	an	DET
cana-1172	18	28	example	example	NOUN
cana-1172	18	29	has	have	AUX
cana-1172	18	30	been	be	AUX
cana-1172	18	31	provided	provide	VERB
cana-1172	18	32	to	to	PART
cana-1172	18	33	show	show	VERB
cana-1172	18	34	that	that	SCONJ
cana-1172	18	35	the	the	DET
cana-1172	18	36	converse	converse	NOUN
cana-1172	18	37	is	be	AUX
cana-1172	18	38	not	not	PART
cana-1172	18	39	true	true	ADJ
cana-1172	18	40	.	.	PUNCT
cana-1172	19	1	theorem	theorem	VERB
cana-1172	19	2	4.3	4.3	NUM
cana-1172	19	3	.	.	PUNCT
cana-1172	20	1	every	every	DET
cana-1172	20	2	lukasz	lukasz	NOUN
cana-1172	20	3	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	20	4	set	set	VERB
cana-1172	20	5	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	20	6	𝜀	𝜀	NOUN
cana-1172	20	7	of	of	ADP
cana-1172	20	8	a	a	DET
cana-1172	20	9	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	20	10	set	set	NOUN
cana-1172	20	11	𝑈	𝑈	PROPN
cana-1172	20	12	in	in	ADP
cana-1172	20	13	𝔊	𝔊	PROPN
cana-1172	20	14	is	be	AUX
cana-1172	20	15	a	a	DET
cana-1172	20	16	lukasz	lukasz	NOUN
cana-1172	20	17	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	20	18	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	PROPN
cana-1172	20	19	of	of	ADP
cana-1172	20	20	𝔊	𝔊	PROPN
cana-1172	20	21	if	if	SCONJ
cana-1172	20	22	and	and	CCONJ
cana-1172	20	23	only	only	ADV
cana-1172	20	24	if	if	SCONJ
cana-1172	20	25	it	it	PRON
cana-1172	20	26	satisfies	satisfy	VERB
cana-1172	20	27	(	(	PUNCT
cana-1172	20	28	i	i	NOUN
cana-1172	20	29	)	)	PUNCT
cana-1172	20	30	∀	∀	PUNCT
cana-1172	20	31	�	�	PROPN
cana-1172	20	32	̇	̇	VERB
cana-1172	20	33	�	�	PROPN
cana-1172	20	34	∈	∈	PROPN
cana-1172	20	35	𝔊	𝔊	PROPN
cana-1172	20	36	,	,	PUNCT
cana-1172	20	37	∀	∀	X
cana-1172	20	38	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	20	39	∈	∈	PROPN
cana-1172	20	40	(	(	PUNCT
cana-1172	20	41	0,1	0,1	NOUN
cana-1172	20	42	]	]	PUNCT
cana-1172	20	43	,	,	PUNCT
cana-1172	20	44	[	[	X
cana-1172	20	45	�	�	NOUN
cana-1172	20	46	̇	̇	PROPN
cana-1172	20	47	�	�	PROPN
cana-1172	20	48	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	20	49	]	]	PUNCT
cana-1172	20	50	∈	∈	PROPN
cana-1172	20	51	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	20	52	𝜀	𝜀	PROPN
cana-1172	20	53	⇒	⇒	NOUN
cana-1172	20	54	[	[	X
cana-1172	20	55	0	0	X
cana-1172	20	56	𝑢𝑎⁄	𝑢𝑎⁄	X
cana-1172	20	57	]	]	PUNCT
cana-1172	20	58	∈	∈	PROPN
cana-1172	20	59	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	20	60	𝜀	𝜀	PROPN
cana-1172	20	61	(	(	PUNCT
cana-1172	20	62	ii	ii	NOUN
cana-1172	20	63	)	)	PUNCT
cana-1172	20	64	∀	∀	PUNCT
cana-1172	20	65	�	�	PROPN
cana-1172	20	66	̇	̇	PROPN
cana-1172	20	67	�	�	PROPN
cana-1172	20	68	,	,	PUNCT
cana-1172	20	69	�	�	PROPN
cana-1172	20	70	̇	̇	VERB
cana-1172	20	71	�	�	PROPN
cana-1172	20	72	∈	∈	PROPN
cana-1172	20	73	𝔊	𝔊	PROPN
cana-1172	20	74	,	,	PUNCT
cana-1172	20	75	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	20	76	𝜀	𝜀	PROPN
cana-1172	20	77	(	(	PUNCT
cana-1172	20	78	�	�	PROPN
cana-1172	20	79	̇	̇	PROPN
cana-1172	20	80	�	�	PROPN
cana-1172	20	81	)	)	PUNCT
cana-1172	20	82	≥	≥	NOUN
cana-1172	20	83	𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝐿𝑈	NUM
cana-1172	20	84	𝜀	𝜀	X
cana-1172	20	85	(	(	PUNCT
cana-1172	20	86	�	�	PROPN
cana-1172	20	87	̇	̇	PROPN
cana-1172	20	88	�	�	PROPN
cana-1172	20	89	∗	∗	PROPN
cana-1172	20	90	�	�	PROPN
cana-1172	20	91	̇	̇	PROPN
cana-1172	20	92	�	�	PROPN
cana-1172	20	93	)	)	PUNCT
cana-1172	20	94	,	,	PUNCT
cana-1172	20	95	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	20	96	𝜀	𝜀	PROPN
cana-1172	20	97	(	(	PUNCT
cana-1172	20	98	�	�	PROPN
cana-1172	20	99	̇	̇	PROPN
cana-1172	20	100	�	�	PROPN
cana-1172	20	101	)	)	PUNCT
cana-1172	20	102	}	}	PUNCT
cana-1172	20	103	conclusions	conclusion	NOUN
cana-1172	20	104	:	:	PUNCT
cana-1172	20	105	the	the	DET
cana-1172	20	106	application	application	NOUN
cana-1172	20	107	of	of	ADP
cana-1172	20	108	𝐵𝑀-algebra	𝐵𝑀-algebra	PROPN
cana-1172	20	109	within	within	ADP
cana-1172	20	110	lukasiewicz	lukasiewicz	ADJ
cana-1172	20	111	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	20	112	logic	logic	NOUN
cana-1172	20	113	operation	operation	NOUN
cana-1172	20	114	can	can	AUX
cana-1172	20	115	optimize	optimize	VERB
cana-1172	20	116	public	public	ADJ
cana-1172	20	117	transportation	transportation	NOUN
cana-1172	20	118	system	system	NOUN
cana-1172	20	119	by	by	ADP
cana-1172	20	120	scheduling	scheduling	NOUN
cana-1172	20	121	time	time	NOUN
cana-1172	20	122	and	and	CCONJ
cana-1172	20	123	routing	route	VERB
cana-1172	20	124	communications	communication	NOUN
cana-1172	20	125	on	on	ADP
cana-1172	20	126	applied	apply	VERB
cana-1172	20	127	nonlinear	nonlinear	ADJ
cana-1172	20	128	analysis	analysis	NOUN
cana-1172	20	129	issn	issn	NOUN
cana-1172	20	130	:	:	PUNCT
cana-1172	20	131	1074	1074	NUM
cana-1172	20	132	-	-	PUNCT
cana-1172	20	133	133x	133x	NUM
cana-1172	20	134	vol	vol	NOUN
cana-1172	20	135	31	31	NUM
cana-1172	20	136	no	no	NOUN
cana-1172	20	137	.	.	PUNCT
cana-1172	21	1	6s	6s	NUM
cana-1172	21	2	(	(	PUNCT
cana-1172	21	3	2024	2024	NUM
cana-1172	21	4	)	)	PUNCT
cana-1172	21	5	132	132	NUM
cana-1172	21	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1172	21	7	based	base	VERB
cana-1172	21	8	on	on	ADP
cana-1172	21	9	passengers	passenger	NOUN
cana-1172	21	10	need	need	VERB
cana-1172	21	11	.	.	PUNCT
cana-1172	22	1	it	it	PRON
cana-1172	22	2	also	also	ADV
cana-1172	22	3	improves	improve	VERB
cana-1172	22	4	service	service	NOUN
cana-1172	22	5	reliability	reliability	NOUN
cana-1172	22	6	using	use	VERB
cana-1172	22	7	operational	operational	ADJ
cana-1172	22	8	constraints	constraint	NOUN
cana-1172	22	9	taken	take	VERB
cana-1172	22	10	from	from	ADP
cana-1172	22	11	the	the	DET
cana-1172	22	12	field	field	NOUN
cana-1172	22	13	.	.	PUNCT
cana-1172	23	1	this	this	DET
cana-1172	23	2	study	study	NOUN
cana-1172	23	3	give	give	VERB
cana-1172	23	4	rise	rise	NOUN
cana-1172	23	5	to	to	ADP
cana-1172	23	6	the	the	DET
cana-1172	23	7	notion	notion	NOUN
cana-1172	23	8	of	of	ADP
cana-1172	23	9	lukasz	lukasz	NOUN
cana-1172	23	10	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	23	11	𝐵𝑀-algebra	𝐵𝑀-algebra	PROPN
cana-1172	23	12	and	and	CCONJ
cana-1172	23	13	lukasz	lukasz	VERB
cana-1172	23	14	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	23	15	𝐵𝑀ideal	𝐵𝑀ideal	PROPN
cana-1172	23	16	along	along	ADP
cana-1172	23	17	with	with	ADP
cana-1172	23	18	some	some	PRON
cana-1172	23	19	of	of	ADP
cana-1172	23	20	their	their	PRON
cana-1172	23	21	properties	property	NOUN
cana-1172	23	22	are	be	AUX
cana-1172	23	23	investigated	investigate	VERB
cana-1172	23	24	.	.	PUNCT
cana-1172	24	1	in	in	ADP
cana-1172	24	2	addition	addition	NOUN
cana-1172	24	3	to	to	ADP
cana-1172	24	4	the	the	DET
cana-1172	24	5	characterization	characterization	NOUN
cana-1172	24	6	of	of	ADP
cana-1172	24	7	both	both	PRON
cana-1172	24	8	lukasz	lukasz	VERB
cana-1172	24	9	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	24	10	𝐵𝑀-algebra	𝐵𝑀-algebra	PROPN
cana-1172	24	11	and	and	CCONJ
cana-1172	24	12	𝐵𝑀-ideal	𝐵𝑀-ideal	PROPN
cana-1172	24	13	,	,	PUNCT
cana-1172	24	14	the	the	DET
cana-1172	24	15	relations	relation	NOUN
cana-1172	24	16	of	of	ADP
cana-1172	24	17	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	24	18	subalgebra	subalgebra	NOUN
cana-1172	24	19	,	,	PUNCT
cana-1172	24	20	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	24	21	ideal	ideal	NOUN
cana-1172	24	22	,	,	PUNCT
cana-1172	24	23	lukasz	lukasz	PROPN
cana-1172	24	24	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	24	25	set	set	NOUN
cana-1172	24	26	,	,	PUNCT
cana-1172	24	27	lukasz	lukasz	VERB
cana-1172	24	28	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	24	29	𝐵𝑀-algebra	𝐵𝑀-algebra	PROPN
cana-1172	24	30	and	and	CCONJ
cana-1172	24	31	lukasz	lukasz	VERB
cana-1172	24	32	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	24	33	𝐵𝑀-ideal	𝐵𝑀-ideal	NOUN
cana-1172	24	34	are	be	AUX
cana-1172	24	35	discussed	discuss	VERB
cana-1172	24	36	.	.	PUNCT
cana-1172	25	1	some	some	DET
cana-1172	25	2	examples	example	NOUN
cana-1172	25	3	are	be	AUX
cana-1172	25	4	provided	provide	VERB
cana-1172	25	5	based	base	VERB
cana-1172	25	6	on	on	ADP
cana-1172	25	7	those	those	DET
cana-1172	25	8	relations	relation	NOUN
cana-1172	25	9	.	.	PUNCT
cana-1172	26	1	in	in	ADP
cana-1172	26	2	the	the	DET
cana-1172	26	3	future	future	NOUN
cana-1172	26	4	,	,	PUNCT
cana-1172	26	5	we	we	PRON
cana-1172	26	6	will	will	AUX
cana-1172	26	7	construct	construct	VERB
cana-1172	26	8	an	an	DET
cana-1172	26	9	algorithm	algorithm	NOUN
cana-1172	26	10	for	for	ADP
cana-1172	26	11	the	the	DET
cana-1172	26	12	advancement	advancement	NOUN
cana-1172	26	13	of	of	ADP
cana-1172	26	14	transportation	transportation	NOUN
cana-1172	26	15	,	,	PUNCT
cana-1172	26	16	making	make	VERB
cana-1172	26	17	use	use	NOUN
cana-1172	26	18	of	of	ADP
cana-1172	26	19	the	the	DET
cana-1172	26	20	ideas	idea	NOUN
cana-1172	26	21	and	and	CCONJ
cana-1172	26	22	results	result	NOUN
cana-1172	26	23	of	of	ADP
cana-1172	26	24	this	this	DET
cana-1172	26	25	study	study	NOUN
cana-1172	26	26	.	.	PUNCT
cana-1172	27	1	keywords	keyword	NOUN
cana-1172	27	2	:	:	PUNCT
cana-1172	27	3	bm	bm	PROPN
cana-1172	27	4	-	-	NOUN
cana-1172	27	5	algebra	algebra	PROPN
cana-1172	27	6	,	,	PUNCT
cana-1172	27	7	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	27	8	subalgebra	subalgebra	NOUN
cana-1172	27	9	,	,	PUNCT
cana-1172	27	10	lukasz	lukasz	PROPN
cana-1172	27	11	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	27	12	set	set	NOUN
cana-1172	27	13	,	,	PUNCT
cana-1172	27	14	lukasz	lukasz	PROPN
cana-1172	27	15	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	27	16	bmalgebra	bmalgebra	NOUN
cana-1172	27	17	,	,	PUNCT
cana-1172	27	18	lukasz	lukasz	VERB
cana-1172	27	19	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	27	20	bm	bm	PROPN
cana-1172	27	21	-	-	NOUN
cana-1172	27	22	ideal	ideal	ADJ
cana-1172	27	23	.	.	PUNCT
cana-1172	28	1	1	1	X
cana-1172	28	2	.	.	X
cana-1172	28	3	introduction	introduction	NOUN
cana-1172	28	4	in	in	ADP
cana-1172	28	5	1966	1966	NUM
cana-1172	28	6	,	,	PUNCT
cana-1172	28	7	𝐵𝐶𝐾/𝐵𝐶𝐼-algebra	𝐵𝐶𝐾/𝐵𝐶𝐼-algebra	NOUN
cana-1172	28	8	are	be	AUX
cana-1172	28	9	developed	develop	VERB
cana-1172	28	10	by	by	ADP
cana-1172	28	11	y.	y.	PROPN
cana-1172	28	12	imai	imai	PROPN
cana-1172	28	13	,	,	PUNCT
cana-1172	28	14	k.	k.	PROPN
cana-1172	28	15	iseki	iseki	PROPN
cana-1172	28	16	and	and	CCONJ
cana-1172	28	17	s.	s.	PROPN
cana-1172	28	18	tanaka	tanaka	PROPN
cana-1172	29	1	[	[	X
cana-1172	29	2	4	4	NUM
cana-1172	29	3	]	]	PUNCT
cana-1172	29	4	.	.	PUNCT
cana-1172	30	1	there	there	PRON
cana-1172	30	2	were	be	VERB
cana-1172	30	3	other	other	ADJ
cana-1172	30	4	algebraic	algebraic	ADJ
cana-1172	30	5	structures	structure	NOUN
cana-1172	30	6	besides	besides	SCONJ
cana-1172	30	7	𝐵𝐶𝐼	𝐵𝐶𝐼	PRON
cana-1172	30	8	and	and	CCONJ
cana-1172	30	9	𝐵𝐶𝐾	𝐵𝐶𝐾	NOUN
cana-1172	30	10	algebras	algebra	NOUN
cana-1172	30	11	.	.	PUNCT
cana-1172	31	1	these	these	DET
cana-1172	31	2	structures	structure	NOUN
cana-1172	31	3	belong	belong	VERB
cana-1172	31	4	to	to	ADP
cana-1172	31	5	universal	universal	ADJ
cana-1172	31	6	algebra	algebra	NOUN
cana-1172	31	7	that	that	PRON
cana-1172	31	8	describes	describe	VERB
cana-1172	31	9	fragments	fragment	NOUN
cana-1172	31	10	of	of	ADP
cana-1172	31	11	proportional	proportional	ADJ
cana-1172	31	12	calculus	calculus	NOUN
cana-1172	31	13	.	.	PUNCT
cana-1172	32	1	such	such	ADJ
cana-1172	32	2	algebraic	algebraic	ADJ
cana-1172	32	3	structures	structure	NOUN
cana-1172	32	4	are	be	AUX
cana-1172	32	5	𝐵𝐶𝐶/𝐵𝐶𝐻/𝐵/𝐵𝐸algebras	𝐵𝐶𝐶/𝐵𝐶𝐻/𝐵/𝐵𝐸algebra	NOUN
cana-1172	32	6	,	,	PUNCT
cana-1172	32	7	etc	etc	X
cana-1172	32	8	.	.	X
cana-1172	33	1	these	these	DET
cana-1172	33	2	algebras	algebra	NOUN
cana-1172	33	3	can	can	AUX
cana-1172	33	4	be	be	AUX
cana-1172	33	5	explored	explore	VERB
cana-1172	33	6	both	both	CCONJ
cana-1172	33	7	theoretically	theoretically	ADV
cana-1172	33	8	and	and	CCONJ
cana-1172	33	9	practically	practically	ADV
cana-1172	33	10	in	in	ADP
cana-1172	33	11	mathematics	mathematics	NOUN
cana-1172	33	12	and	and	CCONJ
cana-1172	33	13	computer	computer	NOUN
cana-1172	33	14	science	science	NOUN
cana-1172	33	15	.	.	PUNCT
cana-1172	34	1	in	in	ADP
cana-1172	34	2	2006	2006	NUM
cana-1172	34	3	,	,	PUNCT
cana-1172	34	4	a	a	DET
cana-1172	34	5	specialized	specialized	PROPN
cana-1172	34	6	𝐵-algebra	𝐵-algebra	PROPN
cana-1172	34	7	,	,	PUNCT
cana-1172	34	8	called	call	VERB
cana-1172	34	9	𝐵𝑀-algebras	𝐵𝑀-algebras	PROPN
cana-1172	34	10	was	be	AUX
cana-1172	34	11	delivered	deliver	VERB
cana-1172	34	12	by	by	ADP
cana-1172	34	13	[	[	X
cana-1172	34	14	2	2	NUM
cana-1172	34	15	]	]	PUNCT
cana-1172	34	16	.	.	PUNCT
cana-1172	35	1	the	the	DET
cana-1172	35	2	concept	concept	NOUN
cana-1172	35	3	of	of	ADP
cana-1172	35	4	lukasz	lukasz	NOUN
cana-1172	35	5	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	35	6	subalgebra	subalgebra	NOUN
cana-1172	35	7	in	in	ADP
cana-1172	35	8	𝐵𝐶𝐾/𝐵𝐶𝐼-algebras	𝐵𝐶𝐾/𝐵𝐶𝐼-algebras	PROPN
cana-1172	35	9	was	be	AUX
cana-1172	35	10	built	build	VERB
cana-1172	35	11	by	by	ADP
cana-1172	35	12	jun	jun	PROPN
cana-1172	35	13	using	use	VERB
cana-1172	35	14	the	the	DET
cana-1172	35	15	thoughts	thought	NOUN
cana-1172	35	16	of	of	ADP
cana-1172	35	17	lukasz	lukasz	PROPN
cana-1172	35	18	t	t	PROPN
cana-1172	35	19	-	-	PUNCT
cana-1172	35	20	norm	norm	NOUN
cana-1172	35	21	[	[	X
cana-1172	35	22	8	8	NUM
cana-1172	35	23	]	]	PUNCT
cana-1172	35	24	.	.	PUNCT
cana-1172	36	1	later	later	ADV
cana-1172	36	2	,	,	PUNCT
cana-1172	36	3	he	he	PRON
cana-1172	36	4	extended	extend	VERB
cana-1172	36	5	it	it	PRON
cana-1172	36	6	to	to	PART
cana-1172	36	7	lukasz	lukasz	VERB
cana-1172	36	8	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	36	9	ideal	ideal	NOUN
cana-1172	36	10	in	in	ADP
cana-1172	36	11	𝐵𝐶𝐾/𝐵𝐶𝐼-algebras	𝐵𝐶𝐾/𝐵𝐶𝐼-algebra	NOUN
cana-1172	36	12	in	in	ADP
cana-1172	36	13	2023	2023	NUM
cana-1172	36	14	[	[	X
cana-1172	36	15	7	7	NUM
cana-1172	36	16	]	]	PUNCT
cana-1172	36	17	.	.	PUNCT
cana-1172	37	1	in	in	ADP
cana-1172	37	2	2002	2002	NUM
cana-1172	37	3	,	,	PUNCT
cana-1172	37	4	jun	jun	PROPN
cana-1172	37	5	and	and	CCONJ
cana-1172	37	6	ahn	ahn	PROPN
cana-1172	37	7	designed	design	VERB
cana-1172	37	8	the	the	DET
cana-1172	37	9	concept	concept	NOUN
cana-1172	37	10	of	of	ADP
cana-1172	37	11	lukasz	lukasz	PROPN
cana-1172	37	12	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	37	13	set	set	VERB
cana-1172	37	14	in	in	ADP
cana-1172	37	15	𝐵𝐸-𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑠	𝐵𝐸-𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑠	PROPN
cana-1172	37	16	to	to	PART
cana-1172	37	17	be	be	AUX
cana-1172	37	18	lukasz	lukasz	VERB
cana-1172	37	19	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	37	20	𝐵𝐸-𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑠	𝐵𝐸-𝑎𝑙𝑔𝑒𝑏𝑟𝑎𝑠	NOUN
cana-1172	37	21	and	and	CCONJ
cana-1172	37	22	𝐵𝐸-filters	𝐵𝐸-filter	NOUN
cana-1172	37	23	[	[	X
cana-1172	37	24	10	10	NUM
cana-1172	37	25	]	]	PUNCT
cana-1172	37	26	along	along	ADP
cana-1172	37	27	with	with	ADP
cana-1172	37	28	the	the	DET
cana-1172	37	29	discussion	discussion	NOUN
cana-1172	37	30	of	of	ADP
cana-1172	37	31	relationship	relationship	NOUN
cana-1172	37	32	between	between	ADP
cana-1172	37	33	lukasz	lukasz	PROPN
cana-1172	37	34	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	37	35	𝐵𝐸-algebra	𝐵𝐸-algebra	PROPN
cana-1172	37	36	and	and	CCONJ
cana-1172	37	37	lukasz	lukasz	VERB
cana-1172	37	38	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	37	39	𝐵𝐸-𝑓𝑖𝑙𝑡𝑒𝑟𝑠.	𝐵𝐸-𝑓𝑖𝑙𝑡𝑒𝑟𝑠.	NOUN
cana-1172	37	40	their	their	PRON
cana-1172	37	41	capacity	capacity	NOUN
cana-1172	37	42	to	to	PART
cana-1172	37	43	handle	handle	VERB
cana-1172	37	44	partial	partial	ADJ
cana-1172	37	45	truth	truth	NOUN
cana-1172	37	46	values	value	NOUN
cana-1172	37	47	,	,	PUNCT
cana-1172	37	48	include	include	VERB
cana-1172	37	49	fuzzy	fuzzy	ADJ
cana-1172	37	50	principles	principle	NOUN
cana-1172	37	51	and	and	CCONJ
cana-1172	37	52	integrate	integrate	VERB
cana-1172	37	53	them	they	PRON
cana-1172	37	54	into	into	ADP
cana-1172	37	55	the	the	DET
cana-1172	37	56	decision	decision	NOUN
cana-1172	37	57	-	-	PUNCT
cana-1172	37	58	making	make	VERB
cana-1172	37	59	process	process	NOUN
cana-1172	37	60	leads	lead	VERB
cana-1172	37	61	to	to	ADP
cana-1172	37	62	a	a	DET
cana-1172	37	63	variety	variety	NOUN
cana-1172	37	64	of	of	ADP
cana-1172	37	65	applications	application	NOUN
cana-1172	37	66	.	.	PUNCT
cana-1172	38	1	to	to	PART
cana-1172	38	2	build	build	VERB
cana-1172	38	3	an	an	DET
cana-1172	38	4	advanced	advanced	ADJ
cana-1172	38	5	algorithm	algorithm	NOUN
cana-1172	38	6	for	for	ADP
cana-1172	38	7	the	the	DET
cana-1172	38	8	solution	solution	NOUN
cana-1172	38	9	of	of	ADP
cana-1172	38	10	real	real	ADJ
cana-1172	38	11	-	-	PUNCT
cana-1172	38	12	life	life	NOUN
cana-1172	38	13	problems	problem	NOUN
cana-1172	38	14	,	,	PUNCT
cana-1172	38	15	we	we	PRON
cana-1172	38	16	can	can	AUX
cana-1172	38	17	explore	explore	VERB
cana-1172	38	18	various	various	ADJ
cana-1172	38	19	algebraic	algebraic	ADJ
cana-1172	38	20	structures	structure	NOUN
cana-1172	38	21	.	.	PUNCT
cana-1172	39	1	this	this	DET
cana-1172	39	2	study	study	NOUN
cana-1172	39	3	led	lead	VERB
cana-1172	39	4	to	to	PART
cana-1172	39	5	explore	explore	VERB
cana-1172	39	6	the	the	DET
cana-1172	39	7	concept	concept	NOUN
cana-1172	39	8	of	of	ADP
cana-1172	39	9	lukasz	lukasz	PROPN
cana-1172	39	10	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	39	11	bm	bm	PROPN
cana-1172	39	12	-	-	NOUN
cana-1172	39	13	algebra	algebra	PROPN
cana-1172	39	14	and	and	CCONJ
cana-1172	39	15	bm	bm	NOUN
cana-1172	39	16	-	-	NOUN
cana-1172	39	17	ideal	ideal	ADJ
cana-1172	39	18	using	use	VERB
cana-1172	39	19	the	the	DET
cana-1172	39	20	notion	notion	NOUN
cana-1172	39	21	of	of	ADP
cana-1172	39	22	lukasz	lukasz	PROPN
cana-1172	39	23	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	39	24	set	set	VERB
cana-1172	39	25	to	to	ADP
cana-1172	39	26	the	the	DET
cana-1172	39	27	given	give	VERB
cana-1172	39	28	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	39	29	set	set	NOUN
cana-1172	39	30	in	in	ADP
cana-1172	39	31	bm	bm	NOUN
cana-1172	39	32	-	-	NOUN
cana-1172	39	33	algebra	algebra	PROPN
cana-1172	39	34	and	and	CCONJ
cana-1172	39	35	investigated	investigate	VERB
cana-1172	39	36	some	some	PRON
cana-1172	39	37	of	of	ADP
cana-1172	39	38	their	their	PRON
cana-1172	39	39	properties	property	NOUN
cana-1172	39	40	,	,	PUNCT
cana-1172	39	41	characterizations	characterization	NOUN
cana-1172	39	42	and	and	CCONJ
cana-1172	39	43	relations	relation	NOUN
cana-1172	39	44	with	with	ADP
cana-1172	39	45	some	some	DET
cana-1172	39	46	examples	example	NOUN
cana-1172	39	47	.	.	PUNCT
cana-1172	40	1	2	2	X
cana-1172	40	2	.	.	X
cana-1172	40	3	objectives	objective	NOUN
cana-1172	40	4	definition	definition	NOUN
cana-1172	40	5	2.1	2.1	NUM
cana-1172	40	6	the	the	DET
cana-1172	40	7	set	set	ADJ
cana-1172	40	8	𝔊	𝔊	PROPN
cana-1172	40	9	be	be	AUX
cana-1172	40	10	a	a	DET
cana-1172	40	11	non	non	ADJ
cana-1172	40	12	-	-	ADJ
cana-1172	40	13	empty	empty	ADJ
cana-1172	40	14	set.the	set.the	PROPN
cana-1172	40	15	𝑩𝑴-𝒂𝒍𝒈𝒆𝒃𝒓𝒂	𝑩𝑴-𝒂𝒍𝒈𝒆𝒃𝒓𝒂	PROPN
cana-1172	40	16	satisfies	satisfy	VERB
cana-1172	40	17	the	the	DET
cana-1172	40	18	given	give	VERB
cana-1172	40	19	axioms	axiom	NOUN
cana-1172	40	20	:	:	PUNCT
cana-1172	40	21	(	(	PUNCT
cana-1172	40	22	𝐵𝑀1	𝐵𝑀1	PROPN
cana-1172	40	23	)	)	PUNCT
cana-1172	40	24	�	�	PROPN
cana-1172	40	25	̇	̇	VERB
cana-1172	40	26	�	�	PROPN
cana-1172	40	27	∗	∗	NOUN
cana-1172	40	28	0	0	NUM
cana-1172	40	29	=	=	SYM
cana-1172	40	30	�	�	PROPN
cana-1172	40	31	̇	̇	PROPN
cana-1172	40	32	�	�	PROPN
cana-1172	40	33	(	(	PUNCT
cana-1172	40	34	𝐵𝑀2)(	𝐵𝑀2)(	PROPN
cana-1172	40	35	�	�	PROPN
cana-1172	40	36	̇	̇	PROPN
cana-1172	40	37	�	�	PROPN
cana-1172	40	38	∗	∗	PROPN
cana-1172	40	39	�	�	PROPN
cana-1172	40	40	̇	̇	PROPN
cana-1172	40	41	�	�	PROPN
cana-1172	40	42	)	)	PUNCT
cana-1172	40	43	∗	∗	NOUN
cana-1172	40	44	(	(	PUNCT
cana-1172	40	45	�	�	PROPN
cana-1172	40	46	̇	̇	PROPN
cana-1172	40	47	�	�	PROPN
cana-1172	40	48	∗	∗	PROPN
cana-1172	40	49	�	�	PROPN
cana-1172	40	50	̇	̇	PROPN
cana-1172	40	51	�	�	PROPN
cana-1172	40	52	)	)	PUNCT
cana-1172	40	53	=	=	SYM
cana-1172	40	54	�	�	PROPN
cana-1172	40	55	̇	̇	PROPN
cana-1172	40	56	�	�	PROPN
cana-1172	40	57	∗	∗	PROPN
cana-1172	40	58	�	�	PROPN
cana-1172	40	59	̇	̇	PROPN
cana-1172	40	60	�	�	PROPN
cana-1172	40	61	,	,	PUNCT
cana-1172	40	62	for	for	ADP
cana-1172	40	63	all	all	DET
cana-1172	40	64	�	�	PROPN
cana-1172	40	65	̇	̇	PROPN
cana-1172	40	66	�	�	PROPN
cana-1172	40	67	,	,	PUNCT
cana-1172	40	68	�	�	PROPN
cana-1172	40	69	̇	̇	PROPN
cana-1172	40	70	�	�	PROPN
cana-1172	40	71	,	,	PUNCT
cana-1172	40	72	�	�	PROPN
cana-1172	40	73	̇	̇	VERB
cana-1172	40	74	�	�	PROPN
cana-1172	40	75	∈	∈	PROPN
cana-1172	40	76	𝔊	𝔊	PROPN
cana-1172	40	77	under	under	ADP
cana-1172	40	78	binary	binary	ADJ
cana-1172	40	79	operation	operation	NOUN
cana-1172	40	80	"	"	PUNCT
cana-1172	40	81	∗	∗	NOUN
cana-1172	40	82	"	"	PUNCT
cana-1172	40	83	with	with	ADP
cana-1172	40	84	a	a	DET
cana-1172	40	85	constant	constant	ADJ
cana-1172	40	86	element	element	NOUN
cana-1172	40	87	"	"	PUNCT
cana-1172	40	88	0	0	NUM
cana-1172	40	89	"	"	PUNCT
cana-1172	40	90	remark	remark	NOUN
cana-1172	40	91	2.2	2.2	NUM
cana-1172	40	92	every	every	DET
cana-1172	40	93	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	PROPN
cana-1172	40	94	satisfies	satisfie	NOUN
cana-1172	40	95	(	(	PUNCT
cana-1172	40	96	i	i	NOUN
cana-1172	40	97	)	)	PUNCT
cana-1172	40	98	(	(	PUNCT
cana-1172	40	99	�	�	PROPN
cana-1172	40	100	̇	̇	PROPN
cana-1172	40	101	�	�	PROPN
cana-1172	40	102	∗	∗	NOUN
cana-1172	40	103	𝓅)̇	𝓅)̇	X
cana-1172	40	104	=	=	SYM
cana-1172	40	105	0	0	NUM
cana-1172	40	106	communications	communication	NOUN
cana-1172	40	107	on	on	ADP
cana-1172	40	108	applied	apply	VERB
cana-1172	40	109	nonlinear	nonlinear	ADJ
cana-1172	40	110	analysis	analysis	NOUN
cana-1172	40	111	issn	issn	NOUN
cana-1172	40	112	:	:	PUNCT
cana-1172	40	113	1074	1074	NUM
cana-1172	40	114	-	-	PUNCT
cana-1172	40	115	133x	133x	NUM
cana-1172	40	116	vol	vol	NOUN
cana-1172	40	117	31	31	NUM
cana-1172	40	118	no	no	NOUN
cana-1172	40	119	.	.	PUNCT
cana-1172	41	1	6s	6s	NUM
cana-1172	41	2	(	(	PUNCT
cana-1172	41	3	2024	2024	NUM
cana-1172	41	4	)	)	PUNCT
cana-1172	41	5	133	133	NUM
cana-1172	41	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1172	41	7	(	(	PUNCT
cana-1172	41	8	ii	ii	NOUN
cana-1172	41	9	)	)	PUNCT
cana-1172	41	10	(	(	PUNCT
cana-1172	41	11	0	0	NUM
cana-1172	41	12	∗	∗	NOUN
cana-1172	41	13	(	(	PUNCT
cana-1172	41	14	0	0	NUM
cana-1172	41	15	∗	∗	NOUN
cana-1172	41	16	�	�	PROPN
cana-1172	41	17	̇	̇	PROPN
cana-1172	41	18	�	�	PROPN
cana-1172	41	19	)	)	PUNCT
cana-1172	41	20	)	)	PUNCT
cana-1172	42	1	=	=	PUNCT
cana-1172	42	2	�	�	PROPN
cana-1172	42	3	̇	̇	PROPN
cana-1172	42	4	�	�	PROPN
cana-1172	42	5	(	(	PUNCT
cana-1172	42	6	iii	iii	NOUN
cana-1172	42	7	)	)	PUNCT
cana-1172	42	8	(	(	PUNCT
cana-1172	42	9	0	0	NUM
cana-1172	42	10	∗	∗	NOUN
cana-1172	42	11	(	(	PUNCT
cana-1172	42	12	�	�	PROPN
cana-1172	42	13	̇	̇	PROPN
cana-1172	42	14	�	�	PROPN
cana-1172	42	15	∗	∗	PROPN
cana-1172	42	16	�	�	PROPN
cana-1172	42	17	̇	̇	PROPN
cana-1172	42	18	�	�	PROPN
cana-1172	42	19	)	)	PUNCT
cana-1172	42	20	)	)	PUNCT
cana-1172	43	1	=	=	PUNCT
cana-1172	43	2	�	�	PROPN
cana-1172	43	3	̇	̇	PROPN
cana-1172	43	4	�	�	PROPN
cana-1172	43	5	∗	∗	PROPN
cana-1172	43	6	�	�	PROPN
cana-1172	43	7	̇	̇	PROPN
cana-1172	43	8	�	�	PROPN
cana-1172	43	9	(	(	PUNCT
cana-1172	43	10	iv	iv	NUM
cana-1172	43	11	)	)	PUNCT
cana-1172	43	12	(	(	PUNCT
cana-1172	43	13	�	�	PROPN
cana-1172	43	14	̇	̇	PROPN
cana-1172	43	15	�	�	PROPN
cana-1172	43	16	∗	∗	PROPN
cana-1172	43	17	�	�	PROPN
cana-1172	43	18	̇	̇	PROPN
cana-1172	43	19	�	�	PROPN
cana-1172	43	20	)	)	PUNCT
cana-1172	43	21	∗	∗	NOUN
cana-1172	43	22	(	(	PUNCT
cana-1172	43	23	�	�	PROPN
cana-1172	43	24	̇	̇	PROPN
cana-1172	43	25	�	�	PROPN
cana-1172	43	26	∗	∗	PROPN
cana-1172	43	27	�	�	PROPN
cana-1172	43	28	̇	̇	PROPN
cana-1172	43	29	�	�	PROPN
cana-1172	43	30	)	)	PUNCT
cana-1172	43	31	=	=	SYM
cana-1172	43	32	�	�	PROPN
cana-1172	43	33	̇	̇	PROPN
cana-1172	43	34	�	�	PROPN
cana-1172	43	35	∗	∗	PROPN
cana-1172	43	36	�	�	PROPN
cana-1172	43	37	̇	̇	PROPN
cana-1172	43	38	�	�	PROPN
cana-1172	43	39	(	(	PUNCT
cana-1172	43	40	v	v	NOUN
cana-1172	43	41	)	)	PUNCT
cana-1172	43	42	(	(	PUNCT
cana-1172	43	43	�	�	PROPN
cana-1172	43	44	̇	̇	PROPN
cana-1172	43	45	�	�	PROPN
cana-1172	43	46	∗	∗	NOUN
cana-1172	43	47	𝓆)̇	𝓆)̇	PUNCT
cana-1172	43	48	=	=	SYM
cana-1172	43	49	0	0	NUM
cana-1172	43	50	⇔	⇔	X
cana-1172	43	51	(	(	PUNCT
cana-1172	43	52	�	�	PROPN
cana-1172	43	53	̇	̇	PROPN
cana-1172	43	54	�	�	PROPN
cana-1172	43	55	∗	∗	NOUN
cana-1172	43	56	𝓅)̇	𝓅)̇	X
cana-1172	43	57	=	=	SYM
cana-1172	43	58	0	0	NUM
cana-1172	43	59	for	for	ADP
cana-1172	43	60	all	all	DET
cana-1172	43	61	�	�	PROPN
cana-1172	43	62	̇	̇	PROPN
cana-1172	43	63	�	�	PROPN
cana-1172	43	64	,	,	PUNCT
cana-1172	43	65	�	�	PROPN
cana-1172	43	66	̇	̇	PROPN
cana-1172	43	67	�	�	PROPN
cana-1172	43	68	,	,	PUNCT
cana-1172	43	69	�	�	PROPN
cana-1172	43	70	̇	̇	VERB
cana-1172	43	71	�	�	PROPN
cana-1172	43	72	∈	∈	PROPN
cana-1172	43	73	𝔊.	𝔊.	PROPN
cana-1172	43	74	definition	definition	NOUN
cana-1172	43	75	2.3	2.3	NUM
cana-1172	43	76	a	a	DET
cana-1172	43	77	𝐵𝑀-algebra	𝐵𝑀-algebra	PROPN
cana-1172	43	78	𝔊	𝔊	PROPN
cana-1172	43	79	is	be	AUX
cana-1172	43	80	called	call	VERB
cana-1172	43	81	subalgebra	subalgebra	NOUN
cana-1172	43	82	of	of	ADP
cana-1172	43	83	𝔊	𝔊	PROPN
cana-1172	43	84	if	if	SCONJ
cana-1172	43	85	𝔖	𝔖	PRON
cana-1172	43	86	be	be	VERB
cana-1172	43	87	a	a	DET
cana-1172	43	88	subset	subset	NOUN
cana-1172	43	89	of	of	ADP
cana-1172	43	90	𝔊	𝔊	PROPN
cana-1172	43	91	then	then	ADV
cana-1172	43	92	�	�	PROPN
cana-1172	43	93	̇	̇	PROPN
cana-1172	43	94	�	�	PROPN
cana-1172	43	95	∗	∗	PROPN
cana-1172	43	96	�	�	PROPN
cana-1172	43	97	̇	̇	PROPN
cana-1172	43	98	�	�	PROPN
cana-1172	43	99	∈	∈	PROPN
cana-1172	43	100	𝔖	𝔖	PROPN
cana-1172	43	101	,	,	PUNCT
cana-1172	43	102	∀	∀	X
cana-1172	43	103	�	�	NOUN
cana-1172	43	104	̇	̇	PROPN
cana-1172	43	105	�	�	PROPN
cana-1172	43	106	,	,	PUNCT
cana-1172	43	107	�	�	PROPN
cana-1172	43	108	̇	̇	VERB
cana-1172	43	109	�	�	PROPN
cana-1172	43	110	∈	∈	PROPN
cana-1172	43	111	𝔖	𝔖	PROPN
cana-1172	43	112	definition	definition	NOUN
cana-1172	43	113	2.4	2.4	NUM
cana-1172	43	114	a	a	DET
cana-1172	43	115	𝐵𝑀-algebra	𝐵𝑀-algebra	PROPN
cana-1172	43	116	𝔊	𝔊	PROPN
cana-1172	43	117	is	be	AUX
cana-1172	43	118	called	call	VERB
cana-1172	43	119	ideal	ideal	NOUN
cana-1172	43	120	of	of	ADP
cana-1172	43	121	𝔊	𝔊	PROPN
cana-1172	43	122	if	if	SCONJ
cana-1172	43	123	𝔖	𝔖	PRON
cana-1172	43	124	be	be	VERB
cana-1172	43	125	a	a	DET
cana-1172	43	126	subset	subset	NOUN
cana-1172	43	127	of	of	ADP
cana-1172	43	128	𝔊	𝔊	PROPN
cana-1172	43	129	then	then	ADV
cana-1172	43	130	0	0	NUM
cana-1172	43	131	∈	∈	PROPN
cana-1172	43	132	𝔊	𝔊	PROPN
cana-1172	43	133	and	and	CCONJ
cana-1172	43	134	�	�	PROPN
cana-1172	43	135	̇	̇	PROPN
cana-1172	43	136	�	�	PROPN
cana-1172	43	137	∗	∗	PROPN
cana-1172	43	138	�	�	PROPN
cana-1172	43	139	̇	̇	PROPN
cana-1172	43	140	�	�	PROPN
cana-1172	43	141	∈	∈	PROPN
cana-1172	43	142	𝔖	𝔖	PROPN
cana-1172	43	143	,	,	PUNCT
cana-1172	43	144	�	�	PROPN
cana-1172	43	145	̇	̇	VERB
cana-1172	43	146	�	�	PROPN
cana-1172	43	147	∈	∈	PROPN
cana-1172	43	148	𝔖	𝔖	PROPN
cana-1172	43	149	⟹	⟹	PRON
cana-1172	43	150	�	�	PROPN
cana-1172	43	151	̇	̇	VERB
cana-1172	43	152	�	�	PROPN
cana-1172	43	153	∈	∈	PROPN
cana-1172	43	154	𝔖	𝔖	PROPN
cana-1172	43	155	,	,	PUNCT
cana-1172	43	156	∀	∀	X
cana-1172	43	157	�	�	NOUN
cana-1172	43	158	̇	̇	PROPN
cana-1172	43	159	�	�	PROPN
cana-1172	43	160	,	,	PUNCT
cana-1172	43	161	�	�	PROPN
cana-1172	43	162	̇	̇	VERB
cana-1172	43	163	�	�	PROPN
cana-1172	43	164	∈	∈	PROPN
cana-1172	43	165	𝔊	𝔊	PROPN
cana-1172	43	166	definition	definition	NOUN
cana-1172	43	167	2.5	2.5	NUM
cana-1172	43	168	zadeh	zadeh	PROPN
cana-1172	43	169	’s	’s	PART
cana-1172	43	170	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	43	171	set	set	VERB
cana-1172	43	172	𝑈	𝑈	PROPN
cana-1172	43	173	in	in	ADP
cana-1172	43	174	𝔊	𝔊	PROPN
cana-1172	43	175	takes	take	VERB
cana-1172	43	176	the	the	DET
cana-1172	43	177	form	form	NOUN
cana-1172	43	178	𝑈(	𝑈(	PROPN
cana-1172	43	179	�	�	PROPN
cana-1172	43	180	̇	̇	PROPN
cana-1172	43	181	�	�	PROPN
cana-1172	43	182	)	)	PUNCT
cana-1172	44	1	=	=	PRON
cana-1172	44	2	{	{	PUNCT
cana-1172	44	3	𝑢	𝑢	PART
cana-1172	44	4	∈	∈	PROPN
cana-1172	44	5	(	(	PUNCT
cana-1172	44	6	0,1	0,1	NOUN
cana-1172	44	7	]	]	PUNCT
cana-1172	44	8	𝑖𝑓	𝑖𝑓	PART
cana-1172	44	9	�	�	PROPN
cana-1172	44	10	̇	̇	PROPN
cana-1172	44	11	�	�	PROPN
cana-1172	44	12	=	=	SYM
cana-1172	44	13	�	�	PROPN
cana-1172	44	14	̇	̇	PROPN
cana-1172	44	15	�	�	PROPN
cana-1172	44	16	0	0	NUM
cana-1172	44	17	𝑖𝑓	𝑖𝑓	PRON
cana-1172	44	18	�	�	PROPN
cana-1172	44	19	̇	̇	PROPN
cana-1172	44	20	�	�	PROPN
cana-1172	44	21	≠	≠	PROPN
cana-1172	44	22	�	�	PROPN
cana-1172	44	23	̇	̇	NOUN
cana-1172	44	24	�	�	PROPN
cana-1172	44	25	is	be	AUX
cana-1172	44	26	regarded	regard	VERB
cana-1172	44	27	as	as	ADP
cana-1172	44	28	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	44	29	point	point	NOUN
cana-1172	44	30	with	with	ADP
cana-1172	44	31	support	support	NOUN
cana-1172	44	32	�	�	PROPN
cana-1172	44	33	̇	̇	PROPN
cana-1172	44	34	�	�	PROPN
cana-1172	44	35	and	and	CCONJ
cana-1172	44	36	value	value	NOUN
cana-1172	44	37	𝑢.	𝑢.	NOUN
cana-1172	45	1	it	it	PRON
cana-1172	45	2	is	be	AUX
cana-1172	45	3	viewed	view	VERB
cana-1172	45	4	by	by	ADP
cana-1172	45	5	[	[	PUNCT
cana-1172	45	6	�	�	PROPN
cana-1172	45	7	̇	̇	NOUN
cana-1172	45	8	�	�	PROPN
cana-1172	45	9	𝑢⁄	𝑢⁄	PROPN
cana-1172	45	10	]	]	PUNCT
cana-1172	45	11	.	.	PUNCT
cana-1172	46	1	definition	definition	NOUN
cana-1172	46	2	2.6	2.6	NUM
cana-1172	46	3	a	a	DET
cana-1172	46	4	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	46	5	point	point	NOUN
cana-1172	46	6	[	[	X
cana-1172	46	7	�	�	NOUN
cana-1172	46	8	̇	̇	NOUN
cana-1172	46	9	�	�	PROPN
cana-1172	46	10	𝑢⁄	𝑢⁄	PROPN
cana-1172	46	11	]	]	PUNCT
cana-1172	46	12	in	in	ADP
cana-1172	46	13	every	every	DET
cana-1172	46	14	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	46	15	set	set	VERB
cana-1172	46	16	𝑈	𝑈	PROPN
cana-1172	46	17	in	in	ADP
cana-1172	46	18	𝔊	𝔊	PROPN
cana-1172	46	19	is	be	AUX
cana-1172	46	20	(	(	PUNCT
cana-1172	46	21	i	i	NOUN
cana-1172	46	22	)	)	PUNCT
cana-1172	46	23	contained	contain	VERB
cana-1172	46	24	in	in	ADP
cana-1172	46	25	𝑈	𝑈	PROPN
cana-1172	46	26	,	,	PUNCT
cana-1172	46	27	noted	note	VERB
cana-1172	46	28	by	by	ADP
cana-1172	46	29	[	[	PUNCT
cana-1172	46	30	�	�	PROPN
cana-1172	46	31	̇	̇	VERB
cana-1172	46	32	�	�	PROPN
cana-1172	46	33	𝑢⁄	𝑢⁄	PROPN
cana-1172	46	34	]	]	PUNCT
cana-1172	46	35	∈	∈	PROPN
cana-1172	46	36	𝑈	𝑈	PROPN
cana-1172	46	37	if	if	SCONJ
cana-1172	46	38	𝑈(	𝑈(	NOUN
cana-1172	46	39	�	�	PROPN
cana-1172	46	40	̇	̇	PROPN
cana-1172	46	41	�	�	PROPN
cana-1172	46	42	)	)	PUNCT
cana-1172	46	43	≥	≥	PROPN
cana-1172	47	1	𝑢.	𝑢.	NOUN
cana-1172	47	2	(	(	PUNCT
cana-1172	47	3	ii	ii	NOUN
cana-1172	47	4	)	)	PUNCT
cana-1172	47	5	quasi	quasi	NOUN
cana-1172	47	6	-	-	VERB
cana-1172	47	7	coincident	coincident	ADJ
cana-1172	47	8	with	with	ADP
cana-1172	47	9	𝑈	𝑈	PROPN
cana-1172	47	10	,	,	PUNCT
cana-1172	47	11	noted	note	VERB
cana-1172	47	12	by	by	ADP
cana-1172	47	13	[	[	PUNCT
cana-1172	47	14	�	�	PROPN
cana-1172	47	15	̇	̇	VERB
cana-1172	47	16	�	�	PROPN
cana-1172	47	17	𝑢⁄	𝑢⁄	PROPN
cana-1172	47	18	]	]	PUNCT
cana-1172	47	19	𝔮𝑈	𝔮𝑈	NOUN
cana-1172	47	20	if	if	SCONJ
cana-1172	47	21	𝑈(	𝑈(	PROPN
cana-1172	47	22	�	�	PROPN
cana-1172	47	23	̇	̇	PROPN
cana-1172	47	24	�	�	PROPN
cana-1172	47	25	)	)	PUNCT
cana-1172	47	26	+	+	NUM
cana-1172	47	27	𝑢	𝑢	X
cana-1172	47	28	>	>	SYM
cana-1172	47	29	1	1	NUM
cana-1172	47	30	definition	definition	NOUN
cana-1172	47	31	2.7	2.7	NUM
cana-1172	47	32	a	a	DET
cana-1172	47	33	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	47	34	set	set	NOUN
cana-1172	47	35	𝑈	𝑈	PROPN
cana-1172	47	36	is	be	AUX
cana-1172	47	37	called	call	VERB
cana-1172	47	38	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	47	39	subalgebra	subalgebra	NOUN
cana-1172	47	40	of	of	ADP
cana-1172	47	41	a	a	DET
cana-1172	47	42	𝐵𝑀-algebra	𝐵𝑀-algebra	PROPN
cana-1172	47	43	𝔊	𝔊	PROPN
cana-1172	47	44	if	if	SCONJ
cana-1172	47	45	it	it	PRON
cana-1172	47	46	satisfies	satisfy	VERB
cana-1172	47	47	(	(	PUNCT
cana-1172	47	48	𝐹𝐴1	𝐹𝐴1	PROPN
cana-1172	47	49	)	)	PUNCT
cana-1172	47	50	𝑈(	𝑈(	PROPN
cana-1172	47	51	�	�	PROPN
cana-1172	47	52	̇	̇	PROPN
cana-1172	47	53	�	�	PROPN
cana-1172	47	54	∗	∗	PROPN
cana-1172	47	55	�	�	PROPN
cana-1172	47	56	̇	̇	PROPN
cana-1172	47	57	�	�	PROPN
cana-1172	47	58	)	)	PUNCT
cana-1172	47	59	≥	≥	NOUN
cana-1172	47	60	𝑚𝑖𝑛{𝑈(	𝑚𝑖𝑛{𝑈(	PROPN
cana-1172	47	61	�	�	PROPN
cana-1172	47	62	̇	̇	PROPN
cana-1172	47	63	�	�	PROPN
cana-1172	47	64	)	)	PUNCT
cana-1172	47	65	,	,	PUNCT
cana-1172	47	66	𝑈(	𝑈(	PROPN
cana-1172	47	67	�	�	PROPN
cana-1172	47	68	̇	̇	PROPN
cana-1172	47	69	�	�	PROPN
cana-1172	47	70	)	)	PUNCT
cana-1172	47	71	}	}	PUNCT
cana-1172	47	72	,	,	PUNCT
cana-1172	47	73	∀	∀	X
cana-1172	47	74	�	�	PROPN
cana-1172	47	75	̇	̇	PROPN
cana-1172	47	76	�	�	PROPN
cana-1172	47	77	,	,	PUNCT
cana-1172	47	78	�	�	PROPN
cana-1172	47	79	̇	̇	VERB
cana-1172	47	80	�	�	PROPN
cana-1172	47	81	∈	∈	PROPN
cana-1172	47	82	𝔊	𝔊	PROPN
cana-1172	47	83	definition	definition	NOUN
cana-1172	47	84	2.8	2.8	NUM
cana-1172	47	85	a	a	DET
cana-1172	47	86	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	47	87	set	set	NOUN
cana-1172	47	88	𝑈	𝑈	PROPN
cana-1172	47	89	is	be	AUX
cana-1172	47	90	called	call	VERB
cana-1172	47	91	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	47	92	ideal	ideal	NOUN
cana-1172	47	93	of	of	ADP
cana-1172	47	94	a	a	DET
cana-1172	47	95	𝐵𝑀-algebra	𝐵𝑀-algebra	PROPN
cana-1172	47	96	𝔊	𝔊	PROPN
cana-1172	47	97	if	if	SCONJ
cana-1172	47	98	it	it	PRON
cana-1172	47	99	satisfies	satisfy	VERB
cana-1172	47	100	(	(	PUNCT
cana-1172	47	101	𝐹𝐼1	𝐹𝐼1	NOUN
cana-1172	47	102	)	)	PUNCT
cana-1172	47	103	𝑈(0	𝑈(0	NUM
cana-1172	47	104	)	)	PUNCT
cana-1172	47	105	≥	≥	NOUN
cana-1172	47	106	𝑈(	𝑈(	PROPN
cana-1172	47	107	�	�	PROPN
cana-1172	47	108	̇	̇	PROPN
cana-1172	47	109	�	�	PROPN
cana-1172	47	110	)	)	PUNCT
cana-1172	47	111	(	(	PUNCT
cana-1172	47	112	𝐹𝐼2	𝐹𝐼2	PROPN
cana-1172	47	113	)	)	PUNCT
cana-1172	47	114	𝑈(	𝑈(	PROPN
cana-1172	47	115	�	�	PROPN
cana-1172	47	116	̇	̇	PROPN
cana-1172	47	117	�	�	PROPN
cana-1172	47	118	)	)	PUNCT
cana-1172	47	119	≥	≥	NOUN
cana-1172	47	120	𝑚𝑖𝑛{𝑈(	𝑚𝑖𝑛{𝑈(	PROPN
cana-1172	47	121	�	�	PROPN
cana-1172	47	122	̇	̇	PROPN
cana-1172	47	123	�	�	PROPN
cana-1172	47	124	∗	∗	PROPN
cana-1172	47	125	�	�	PROPN
cana-1172	47	126	̇	̇	PROPN
cana-1172	47	127	�	�	PROPN
cana-1172	47	128	)	)	PUNCT
cana-1172	47	129	,	,	PUNCT
cana-1172	47	130	𝑈(	𝑈(	PROPN
cana-1172	47	131	�	�	PROPN
cana-1172	47	132	̇	̇	PROPN
cana-1172	47	133	�	�	PROPN
cana-1172	47	134	)	)	PUNCT
cana-1172	47	135	}	}	PUNCT
cana-1172	47	136	,	,	PUNCT
cana-1172	47	137	∀	∀	X
cana-1172	47	138	�	�	PROPN
cana-1172	47	139	̇	̇	PROPN
cana-1172	47	140	�	�	PROPN
cana-1172	47	141	,	,	PUNCT
cana-1172	47	142	�	�	PROPN
cana-1172	47	143	̇	̇	VERB
cana-1172	47	144	�	�	PROPN
cana-1172	47	145	∈	∈	PROPN
cana-1172	47	146	𝔊	𝔊	PROPN
cana-1172	47	147	3	3	NUM
cana-1172	47	148	.	.	PUNCT
cana-1172	47	149	methods	method	NOUN
cana-1172	47	150	definition	definition	NOUN
cana-1172	47	151	3.1	3.1	NUM
cana-1172	47	152	an	an	DET
cana-1172	47	153	𝜺	𝜺	NOUN
cana-1172	47	154	−	−	PROPN
cana-1172	47	155	𝑳𝒖𝒌𝒂𝒔𝒛	𝑳𝒖𝒌𝒂𝒔𝒛	PROPN
cana-1172	47	156	𝓕𝐮𝐳𝐳𝐲	𝓕𝐮𝐳𝐳𝐲	PROPN
cana-1172	47	157	𝑺𝒆𝒕	𝑺𝒆𝒕	PROPN
cana-1172	47	158	of	of	ADP
cana-1172	47	159	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	47	160	set	set	VERB
cana-1172	47	161	𝑈	𝑈	PROPN
cana-1172	47	162	in	in	ADP
cana-1172	47	163	𝔊	𝔊	PROPN
cana-1172	47	164	is	be	AUX
cana-1172	47	165	a	a	DET
cana-1172	47	166	function	function	NOUN
cana-1172	47	167	from	from	ADP
cana-1172	47	168	the	the	DET
cana-1172	47	169	bm	bm	PROPN
cana-1172	47	170	-	-	PROPN
cana-1172	47	171	algebra	algebra	ADJ
cana-1172	47	172	𝔊	𝔊	PROPN
cana-1172	47	173	to	to	ADP
cana-1172	47	174	[	[	X
cana-1172	47	175	0,1	0,1	NUM
cana-1172	47	176	]	]	PUNCT
cana-1172	47	177	and	and	CCONJ
cana-1172	47	178	𝜀	𝜀	X
cana-1172	47	179	∈	∈	PROPN
cana-1172	47	180	[	[	X
cana-1172	47	181	0,1	0,1	NUM
cana-1172	47	182	]	]	X
cana-1172	47	183	.	.	PUNCT
cana-1172	48	1	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	48	2	𝜀	𝜀	NOUN
cana-1172	48	3	:	:	PUNCT
cana-1172	48	4	𝔊	𝔊	PROPN
cana-1172	48	5	→	→	SYM
cana-1172	48	6	[	[	X
cana-1172	48	7	0,1	0,1	NUM
cana-1172	48	8	]	]	PUNCT
cana-1172	48	9	,	,	PUNCT
cana-1172	48	10	�	�	PROPN
cana-1172	48	11	̇	̇	PROPN
cana-1172	48	12	�	�	PROPN
cana-1172	48	13	↦	↦	PRON
cana-1172	48	14	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
cana-1172	48	15	{	{	PUNCT
cana-1172	48	16	0	0	NUM
cana-1172	48	17	,	,	PUNCT
cana-1172	48	18	𝑈(	𝑈(	PROPN
cana-1172	48	19	�	�	PROPN
cana-1172	48	20	̇	̇	PROPN
cana-1172	48	21	�	�	PROPN
cana-1172	48	22	)	)	PUNCT
cana-1172	48	23	+	+	NUM
cana-1172	48	24	𝜀	𝜀	X
cana-1172	48	25	−	−	NUM
cana-1172	48	26	1	1	NUM
cana-1172	48	27	}	}	PUNCT
cana-1172	48	28	(	(	PUNCT
cana-1172	48	29	3.1	3.1	NUM
cana-1172	48	30	)	)	PUNCT
cana-1172	48	31	remark	remark	NOUN
cana-1172	48	32	3.2	3.2	NUM
cana-1172	48	33	if	if	SCONJ
cana-1172	48	34	𝑈	𝑈	PROPN
cana-1172	48	35	is	be	AUX
cana-1172	48	36	a	a	DET
cana-1172	48	37	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	48	38	set	set	NOUN
cana-1172	48	39	in	in	ADP
cana-1172	48	40	𝔊	𝔊	PROPN
cana-1172	48	41	,	,	PUNCT
cana-1172	48	42	then	then	ADV
cana-1172	48	43	its	its	PRON
cana-1172	48	44	𝜀-lukasz	𝜀-lukasz	NOUN
cana-1172	48	45	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	48	46	set	set	VERB
cana-1172	48	47	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	48	48	𝜀	𝜀	PROPN
cana-1172	48	49	satisfies	satisfie	NOUN
cana-1172	48	50	communications	communication	NOUN
cana-1172	48	51	on	on	ADP
cana-1172	48	52	applied	apply	VERB
cana-1172	48	53	nonlinear	nonlinear	ADJ
cana-1172	48	54	analysis	analysis	NOUN
cana-1172	48	55	issn	issn	NOUN
cana-1172	48	56	:	:	PUNCT
cana-1172	48	57	1074	1074	NUM
cana-1172	48	58	-	-	PUNCT
cana-1172	48	59	133x	133x	NUM
cana-1172	48	60	vol	vol	NOUN
cana-1172	48	61	31	31	NUM
cana-1172	48	62	no	no	NOUN
cana-1172	48	63	.	.	PUNCT
cana-1172	49	1	6s	6s	NUM
cana-1172	49	2	(	(	PUNCT
cana-1172	49	3	2024	2024	NUM
cana-1172	49	4	)	)	PUNCT
cana-1172	49	5	134	134	NUM
cana-1172	49	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1172	49	7	𝑈(	𝑈(	PROPN
cana-1172	49	8	�	�	PROPN
cana-1172	49	9	̇	̇	PROPN
cana-1172	49	10	�	�	PROPN
cana-1172	49	11	)	)	PUNCT
cana-1172	49	12	≥	≥	NOUN
cana-1172	49	13	𝑈(	𝑈(	PROPN
cana-1172	49	14	�	�	PROPN
cana-1172	49	15	̇	̇	PROPN
cana-1172	49	16	�	�	PROPN
cana-1172	49	17	)	)	PUNCT
cana-1172	49	18	⇒	⇒	NOUN
cana-1172	49	19	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	49	20	𝜀	𝜀	PROPN
cana-1172	49	21	(	(	PUNCT
cana-1172	49	22	�	�	PROPN
cana-1172	49	23	̇	̇	PROPN
cana-1172	49	24	�	�	PROPN
cana-1172	49	25	)	)	PUNCT
cana-1172	49	26	≥	≥	NOUN
cana-1172	49	27	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	49	28	𝜀	𝜀	PROPN
cana-1172	49	29	(	(	PUNCT
cana-1172	49	30	�	�	PROPN
cana-1172	49	31	̇	̇	PROPN
cana-1172	49	32	�	�	PROPN
cana-1172	49	33	)	)	PUNCT
cana-1172	49	34	,	,	PUNCT
cana-1172	49	35	∀	∀	X
cana-1172	49	36	�	�	PROPN
cana-1172	49	37	̇	̇	PROPN
cana-1172	49	38	�	�	PROPN
cana-1172	49	39	,	,	PUNCT
cana-1172	49	40	�	�	PROPN
cana-1172	49	41	̇	̇	VERB
cana-1172	49	42	�	�	PROPN
cana-1172	49	43	∈	∈	PROPN
cana-1172	49	44	𝔊	𝔊	PROPN
cana-1172	49	45	(	(	PUNCT
cana-1172	49	46	3.2	3.2	NUM
cana-1172	49	47	)	)	PUNCT
cana-1172	49	48	proof	proof	NOUN
cana-1172	49	49	suppose	suppose	VERB
cana-1172	49	50	𝑈	𝑈	PROPN
cana-1172	49	51	be	be	AUX
cana-1172	49	52	a	a	DET
cana-1172	49	53	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	49	54	set	set	NOUN
cana-1172	49	55	in	in	ADP
cana-1172	49	56	𝔊	𝔊	PROPN
cana-1172	49	57	and	and	CCONJ
cana-1172	49	58	𝑈(	𝑈(	PROPN
cana-1172	49	59	�	�	PROPN
cana-1172	49	60	̇	̇	PROPN
cana-1172	49	61	�	�	PROPN
cana-1172	49	62	)	)	PUNCT
cana-1172	49	63	≥	≥	NOUN
cana-1172	49	64	𝑈(	𝑈(	PROPN
cana-1172	49	65	�	�	PROPN
cana-1172	49	66	̇	̇	PROPN
cana-1172	49	67	�	�	PROPN
cana-1172	49	68	)	)	PUNCT
cana-1172	49	69	then	then	ADV
cana-1172	49	70	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	49	71	𝜀	𝜀	PROPN
cana-1172	49	72	(	(	PUNCT
cana-1172	49	73	�	�	PROPN
cana-1172	49	74	̇	̇	PROPN
cana-1172	49	75	�	�	PROPN
cana-1172	49	76	)	)	PUNCT
cana-1172	49	77	=	=	SYM
cana-1172	49	78	𝑚𝑎𝑥{0	𝑚𝑎𝑥{0	PROPN
cana-1172	49	79	,	,	PUNCT
cana-1172	49	80	𝑈(	𝑈(	PROPN
cana-1172	49	81	�	�	PROPN
cana-1172	49	82	̇	̇	PROPN
cana-1172	49	83	�	�	PROPN
cana-1172	49	84	)	)	PUNCT
cana-1172	49	85	+	+	NUM
cana-1172	49	86	𝜀	𝜀	X
cana-1172	49	87	−	−	NUM
cana-1172	49	88	1	1	NUM
cana-1172	49	89	}	}	PUNCT
cana-1172	49	90	≥	≥	NUM
cana-1172	49	91	𝑚𝑎𝑥{0	𝑚𝑎𝑥{0	NOUN
cana-1172	49	92	,	,	PUNCT
cana-1172	49	93	𝑈(	𝑈(	PROPN
cana-1172	49	94	�	�	PROPN
cana-1172	49	95	̇	̇	PROPN
cana-1172	49	96	�	�	PROPN
cana-1172	49	97	)	)	PUNCT
cana-1172	49	98	+	+	NUM
cana-1172	49	99	𝜀	𝜀	X
cana-1172	49	100	−	−	NUM
cana-1172	49	101	1	1	NUM
cana-1172	49	102	}	}	PUNCT
cana-1172	49	103	=	=	SYM
cana-1172	49	104	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	49	105	𝜀	𝜀	PROPN
cana-1172	49	106	(	(	PUNCT
cana-1172	49	107	�	�	PROPN
cana-1172	49	108	̇	̇	PROPN
cana-1172	49	109	�	�	PROPN
cana-1172	49	110	)	)	PUNCT
cana-1172	49	111	.	.	PUNCT
cana-1172	50	1	thus	thus	ADV
cana-1172	50	2	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	50	3	𝜀	𝜀	PROPN
cana-1172	50	4	(	(	PUNCT
cana-1172	50	5	�	�	PROPN
cana-1172	50	6	̇	̇	PROPN
cana-1172	50	7	�	�	PROPN
cana-1172	50	8	)	)	PUNCT
cana-1172	50	9	≥	≥	NOUN
cana-1172	50	10	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	50	11	𝜀	𝜀	PROPN
cana-1172	50	12	(	(	PUNCT
cana-1172	50	13	�	�	PROPN
cana-1172	50	14	̇	̇	PROPN
cana-1172	50	15	�	�	PROPN
cana-1172	50	16	)	)	PUNCT
cana-1172	50	17	.	.	PUNCT
cana-1172	51	1	definition	definition	NOUN
cana-1172	51	2	3.3	3.3	NUM
cana-1172	51	3	an	an	DET
cana-1172	51	4	𝜀-lukasz	𝜀-lukasz	NOUN
cana-1172	51	5	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	51	6	set	set	VERB
cana-1172	51	7	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	51	8	𝜀	𝜀	NOUN
cana-1172	51	9	in	in	ADP
cana-1172	51	10	𝔊	𝔊	PROPN
cana-1172	51	11	is	be	AUX
cana-1172	51	12	called	call	VERB
cana-1172	51	13	an	an	DET
cana-1172	51	14	𝜺lukasz	𝜺lukasz	NOUN
cana-1172	51	15	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	NOUN
cana-1172	51	16	subalgebra	subalgebra	NOUN
cana-1172	51	17	of	of	ADP
cana-1172	51	18	𝕲	𝕲	PROPN
cana-1172	51	19	or	or	CCONJ
cana-1172	51	20	𝜺lukasz	𝜺lukasz	VERB
cana-1172	51	21	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	51	22	𝑩𝑴	𝑩𝑴	PROPN
cana-1172	51	23	−	−	PROPN
cana-1172	51	24	𝒂𝒍𝒈𝒆𝒃𝒓𝒂	𝒂𝒍𝒈𝒆𝒃𝒓𝒂	NOUN
cana-1172	51	25	of	of	ADP
cana-1172	51	26	𝕲	𝕲	PROPN
cana-1172	51	27	if	if	SCONJ
cana-1172	51	28	it	it	PRON
cana-1172	51	29	satisfies	satisfy	VERB
cana-1172	51	30	(	(	PUNCT
cana-1172	51	31	𝐿𝐹𝐴1	𝐿𝐹𝐴1	NOUN
cana-1172	51	32	)	)	PUNCT
cana-1172	51	33	[	[	X
cana-1172	51	34	�	�	NOUN
cana-1172	51	35	̇	̇	PROPN
cana-1172	51	36	�	�	PROPN
cana-1172	51	37	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	51	38	]	]	PUNCT
cana-1172	51	39	,	,	PUNCT
cana-1172	51	40	[	[	X
cana-1172	51	41	�	�	NOUN
cana-1172	51	42	̇	̇	PROPN
cana-1172	51	43	�	�	PROPN
cana-1172	51	44	𝑢𝑏⁄	𝑢𝑏⁄	NOUN
cana-1172	51	45	]	]	PUNCT
cana-1172	51	46	∈	∈	PROPN
cana-1172	51	47	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	51	48	𝜀	𝜀	PROPN
cana-1172	51	49	⇒	⇒	NOUN
cana-1172	51	50	[	[	X
cana-1172	51	51	(	(	PUNCT
cana-1172	51	52	�	�	PROPN
cana-1172	51	53	̇	̇	PROPN
cana-1172	51	54	�	�	PROPN
cana-1172	51	55	∗	∗	PROPN
cana-1172	51	56	�	�	PROPN
cana-1172	51	57	̇	̇	PROPN
cana-1172	51	58	�	�	PROPN
cana-1172	51	59	)	)	PUNCT
cana-1172	51	60	𝑚𝑖𝑛{𝑢𝑎	𝑚𝑖𝑛{𝑢𝑎	VERB
cana-1172	51	61	,	,	PUNCT
cana-1172	51	62	𝑢𝑏}⁄	𝑢𝑏}⁄	PRON
cana-1172	51	63	]	]	PUNCT
cana-1172	51	64	∈	∈	PROPN
cana-1172	51	65	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	51	66	𝜀	𝜀	PROPN
cana-1172	51	67	(	(	PUNCT
cana-1172	51	68	3.3	3.3	NUM
cana-1172	51	69	)	)	PUNCT
cana-1172	51	70	for	for	ADP
cana-1172	51	71	all	all	DET
cana-1172	51	72	�	�	PROPN
cana-1172	51	73	̇	̇	PROPN
cana-1172	51	74	�	�	PROPN
cana-1172	51	75	,	,	PUNCT
cana-1172	51	76	�	�	PROPN
cana-1172	51	77	̇	̇	VERB
cana-1172	51	78	�	�	PROPN
cana-1172	51	79	∈	∈	PROPN
cana-1172	51	80	𝐺,𝜀	𝐺,𝜀	X
cana-1172	51	81	∈	∈	PROPN
cana-1172	51	82	(	(	PUNCT
cana-1172	51	83	0,1	0,1	NUM
cana-1172	51	84	)	)	PUNCT
cana-1172	51	85	and	and	CCONJ
cana-1172	51	86	𝑢𝑎	𝑢𝑎	ADJ
cana-1172	51	87	,	,	PUNCT
cana-1172	51	88	𝑢𝑏	𝑢𝑏	ADP
cana-1172	51	89	∈	∈	PROPN
cana-1172	51	90	(	(	PUNCT
cana-1172	51	91	0,1	0,1	NOUN
cana-1172	51	92	]	]	PUNCT
cana-1172	51	93	.	.	PUNCT
cana-1172	52	1	example	example	NOUN
cana-1172	52	2	3.4	3.4	NUM
cana-1172	52	3	let	let	VERB
cana-1172	52	4	𝔊	𝔊	PROPN
cana-1172	52	5	=	=	SYM
cana-1172	52	6	{	{	PUNCT
cana-1172	52	7	0	0	NUM
cana-1172	52	8	,	,	PUNCT
cana-1172	52	9	𝓅1̇	𝓅1̇	PROPN
cana-1172	52	10	,	,	PUNCT
cana-1172	52	11	𝓅2̇	𝓅2̇	PROPN
cana-1172	52	12	,	,	PUNCT
cana-1172	52	13	𝓅3̇	𝓅3̇	NOUN
cana-1172	52	14	}	}	PUNCT
cana-1172	52	15	be	be	VERB
cana-1172	52	16	a	a	DET
cana-1172	52	17	set	set	NOUN
cana-1172	52	18	and	and	CCONJ
cana-1172	52	19	table	table	NOUN
cana-1172	52	20	3.1	3.1	NUM
cana-1172	52	21	shows	show	VERB
cana-1172	52	22	the	the	DET
cana-1172	52	23	cayley	cayley	ADJ
cana-1172	52	24	table	table	NOUN
cana-1172	52	25	of	of	ADP
cana-1172	52	26	𝔊	𝔊	PROPN
cana-1172	52	27	under	under	ADP
cana-1172	52	28	"	"	PUNCT
cana-1172	52	29	∗	∗	NOUN
cana-1172	52	30	"	"	PUNCT
cana-1172	52	31	∗	∗	X
cana-1172	52	32	0	0	NUM
cana-1172	52	33	𝓅1̇	𝓅1̇	PROPN
cana-1172	52	34	𝓅2̇	𝓅2̇	ADJ
cana-1172	52	35	𝓅3̇	𝓅3̇	NOUN
cana-1172	52	36	0	0	NUM
cana-1172	52	37	0	0	NUM
cana-1172	52	38	𝓅1̇	𝓅1̇	PROPN
cana-1172	52	39	𝓅2̇	𝓅2̇	ADJ
cana-1172	52	40	𝓅3̇	𝓅3̇	NOUN
cana-1172	52	41	𝓅1̇	𝓅1̇	VERB
cana-1172	52	42	𝓅1̇	𝓅1̇	ADJ
cana-1172	52	43	0	0	NUM
cana-1172	52	44	0	0	NUM
cana-1172	52	45	𝓅2̇	𝓅2̇	PROPN
cana-1172	52	46	𝓅2̇	𝓅2̇	PROPN
cana-1172	52	47	𝓅2̇	𝓅2̇	PROPN
cana-1172	52	48	0	0	NUM
cana-1172	52	49	0	0	NUM
cana-1172	52	50	𝓅1̇	𝓅1̇	ADJ
cana-1172	52	51	𝓅3̇	𝓅3̇	NOUN
cana-1172	52	52	𝓅3̇	𝓅3̇	PROPN
cana-1172	52	53	𝓅2̇	𝓅2̇	PROPN
cana-1172	52	54	𝓅1̇	𝓅1̇	VERB
cana-1172	52	55	0	0	NUM
cana-1172	52	56	table	table	NOUN
cana-1172	52	57	3.1	3.1	NUM
cana-1172	52	58	cayley	cayley	NOUN
cana-1172	52	59	table	table	NOUN
cana-1172	52	60	with	with	ADP
cana-1172	52	61	respect	respect	NOUN
cana-1172	52	62	to	to	ADP
cana-1172	52	63	"	"	PUNCT
cana-1172	52	64	∗	∗	NOUN
cana-1172	52	65	"	"	PUNCT
cana-1172	52	66	then	then	ADV
cana-1172	52	67	𝔊	𝔊	PROPN
cana-1172	52	68	is	be	AUX
cana-1172	52	69	a	a	DET
cana-1172	52	70	𝐵𝑀	𝐵𝑀	NOUN
cana-1172	52	71	−	−	NOUN
cana-1172	52	72	𝑎𝑙𝑔𝑒𝑏𝑟𝑎.	𝑎𝑙𝑔𝑒𝑏𝑟𝑎.	NOUN
cana-1172	52	73	defining	define	VERB
cana-1172	52	74	a	a	DET
cana-1172	52	75	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	52	76	set	set	VERB
cana-1172	52	77	𝑈	𝑈	PROPN
cana-1172	52	78	in	in	ADP
cana-1172	52	79	𝔊	𝔊	PROPN
cana-1172	52	80	as	as	SCONJ
cana-1172	52	81	follows	follow	VERB
cana-1172	52	82	:	:	PUNCT
cana-1172	52	83	𝑈	𝑈	ADJ
cana-1172	52	84	:	:	PUNCT
cana-1172	52	85	𝔊	𝔊	PROPN
cana-1172	52	86	→	→	SYM
cana-1172	52	87	[	[	X
cana-1172	52	88	0,1	0,1	NUM
cana-1172	52	89	]	]	PUNCT
cana-1172	52	90	,	,	PUNCT
cana-1172	52	91	�	�	PROPN
cana-1172	52	92	̇	̇	PROPN
cana-1172	52	93	�	�	PROPN
cana-1172	52	94	↦	↦	PROPN
cana-1172	52	95	{	{	PUNCT
cana-1172	52	96	0.88	0.88	NUM
cana-1172	52	97	𝑖𝑓	𝑖𝑓	NUM
cana-1172	52	98	�	�	PROPN
cana-1172	52	99	̇	̇	NOUN
cana-1172	52	100	�	�	PROPN
cana-1172	52	101	=	=	SYM
cana-1172	52	102	0	0	NUM
cana-1172	52	103	0.69	0.69	NUM
cana-1172	52	104	𝑖𝑓	𝑖𝑓	PRON
cana-1172	52	105	�	�	PROPN
cana-1172	52	106	̇	̇	PROPN
cana-1172	52	107	�	�	PROPN
cana-1172	52	108	=	=	PUNCT
cana-1172	52	109	{	{	PUNCT
cana-1172	52	110	𝓅1̇	𝓅1̇	PROPN
cana-1172	52	111	,	,	PUNCT
cana-1172	52	112	𝓅2̇	𝓅2̇	PROPN
cana-1172	52	113	}	}	PUNCT
cana-1172	52	114	0.77	0.77	NUM
cana-1172	52	115	𝑖𝑓	𝑖𝑓	PRON
cana-1172	52	116	�	�	PROPN
cana-1172	52	117	̇	̇	PROPN
cana-1172	52	118	�	�	PROPN
cana-1172	52	119	=	=	PUNCT
cana-1172	52	120	𝓅3	𝓅3	PROPN
cana-1172	52	121	̇	̇	VERB
cana-1172	52	122	.	.	PUNCT
cana-1172	53	1	if	if	SCONJ
cana-1172	53	2	it	it	PRON
cana-1172	53	3	is	be	AUX
cana-1172	53	4	taken	take	VERB
cana-1172	53	5	that	that	DET
cana-1172	53	6	𝜀	𝜀	NOUN
cana-1172	53	7	=	=	SYM
cana-1172	53	8	0.61	0.61	NUM
cana-1172	53	9	,	,	PUNCT
cana-1172	53	10	then	then	ADV
cana-1172	53	11	the	the	DET
cana-1172	53	12	lukasz	lukasz	PROPN
cana-1172	53	13	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	53	14	set	set	VERB
cana-1172	53	15	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	53	16	𝜀	𝜀	NOUN
cana-1172	53	17	of	of	ADP
cana-1172	53	18	𝑈	𝑈	PROPN
cana-1172	53	19	in	in	ADP
cana-1172	53	20	𝔊	𝔊	PROPN
cana-1172	53	21	is	be	AUX
cana-1172	53	22	provided	provide	VERB
cana-1172	53	23	as	as	SCONJ
cana-1172	53	24	follows	follow	VERB
cana-1172	53	25	:	:	PUNCT
cana-1172	53	26	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	53	27	𝜀	𝜀	NOUN
cana-1172	53	28	:	:	PUNCT
cana-1172	53	29	𝔊	𝔊	PROPN
cana-1172	53	30	→	→	SYM
cana-1172	53	31	[	[	X
cana-1172	53	32	0,1	0,1	NUM
cana-1172	53	33	]	]	PUNCT
cana-1172	53	34	,	,	PUNCT
cana-1172	53	35	�	�	PROPN
cana-1172	53	36	̇	̇	PROPN
cana-1172	53	37	�	�	PROPN
cana-1172	53	38	↦	↦	PROPN
cana-1172	53	39	{	{	PUNCT
cana-1172	53	40	0.49	0.49	NUM
cana-1172	53	41	𝑖𝑓	𝑖𝑓	NUM
cana-1172	53	42	�	�	PROPN
cana-1172	53	43	̇	̇	NOUN
cana-1172	53	44	�	�	PROPN
cana-1172	53	45	=	=	SYM
cana-1172	53	46	0	0	NUM
cana-1172	53	47	0.3	0.3	NUM
cana-1172	53	48	𝑖𝑓	𝑖𝑓	PRON
cana-1172	53	49	�	�	PROPN
cana-1172	53	50	̇	̇	PROPN
cana-1172	53	51	�	�	PROPN
cana-1172	53	52	=	=	PUNCT
cana-1172	53	53	{	{	PUNCT
cana-1172	53	54	𝓅1̇	𝓅1̇	PROPN
cana-1172	53	55	,	,	PUNCT
cana-1172	53	56	𝓅2̇	𝓅2̇	PROPN
cana-1172	53	57	}	}	PUNCT
cana-1172	53	58	0.36	0.36	NUM
cana-1172	53	59	𝑖𝑓	𝑖𝑓	ADP
cana-1172	53	60	𝓅	𝓅	NOUN
cana-1172	53	61	=	=	PUNCT
cana-1172	53	62	𝓅3̇	𝓅3̇	NOUN
cana-1172	53	63	typically	typically	ADV
cana-1172	53	64	,	,	PUNCT
cana-1172	53	65	it	it	PRON
cana-1172	53	66	is	be	AUX
cana-1172	53	67	verified	verify	VERB
cana-1172	53	68	that	that	SCONJ
cana-1172	53	69	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	53	70	𝜀	𝜀	PROPN
cana-1172	53	71	is	be	AUX
cana-1172	53	72	a	a	DET
cana-1172	53	73	lukasz	lukasz	NOUN
cana-1172	53	74	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	53	75	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	PROPN
cana-1172	53	76	of	of	ADP
cana-1172	53	77	𝔊.	𝔊.	PROPN
cana-1172	53	78	definition	definition	NOUN
cana-1172	53	79	3.5	3.5	NUM
cana-1172	53	80	a	a	DET
cana-1172	53	81	lukasz	lukasz	NOUN
cana-1172	53	82	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	53	83	set	set	VERB
cana-1172	53	84	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	53	85	𝜀	𝜀	NOUN
cana-1172	53	86	in	in	ADP
cana-1172	53	87	𝔊	𝔊	PROPN
cana-1172	53	88	is	be	AUX
cana-1172	53	89	called	call	VERB
cana-1172	53	90	lukasz	lukasz	PROPN
cana-1172	53	91	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	53	92	𝑩𝑴-𝒊𝒅𝒆𝒂𝒍	𝑩𝑴-𝒊𝒅𝒆𝒂𝒍	PROPN
cana-1172	53	93	of	of	ADP
cana-1172	53	94	𝔊	𝔊	PROPN
cana-1172	53	95	if	if	SCONJ
cana-1172	53	96	it	it	PRON
cana-1172	53	97	satisfies	satisfy	VERB
cana-1172	53	98	(	(	PUNCT
cana-1172	53	99	𝐿𝐹𝐼1	𝐿𝐹𝐼1	NUM
cana-1172	53	100	)	)	PUNCT
cana-1172	53	101	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	53	102	𝜀	𝜀	NOUN
cana-1172	53	103	(	(	PUNCT
cana-1172	53	104	0	0	NUM
cana-1172	53	105	)	)	PUNCT
cana-1172	53	106	is	be	AUX
cana-1172	53	107	an	an	DET
cana-1172	53	108	upper	upper	ADJ
cana-1172	53	109	bound	bind	VERB
cana-1172	53	110	of	of	ADP
cana-1172	53	111	{	{	PUNCT
cana-1172	53	112	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	53	113	𝜀	𝜀	PROPN
cana-1172	53	114	(	(	PUNCT
cana-1172	53	115	�	�	PROPN
cana-1172	53	116	̇	̇	PROPN
cana-1172	53	117	�	�	PROPN
cana-1172	53	118	)|	)|	PROPN
cana-1172	53	119	�	�	PROPN
cana-1172	53	120	̇	̇	PROPN
cana-1172	53	121	�	�	PROPN
cana-1172	53	122	∈	∈	PROPN
cana-1172	53	123	𝔊	𝔊	PROPN
cana-1172	53	124	}	}	PUNCT
cana-1172	53	125	(	(	PUNCT
cana-1172	53	126	3.4	3.4	NUM
cana-1172	53	127	)	)	PUNCT
cana-1172	53	128	communications	communication	NOUN
cana-1172	53	129	on	on	ADP
cana-1172	53	130	applied	apply	VERB
cana-1172	53	131	nonlinear	nonlinear	ADJ
cana-1172	53	132	analysis	analysis	NOUN
cana-1172	53	133	issn	issn	NOUN
cana-1172	53	134	:	:	PUNCT
cana-1172	53	135	1074	1074	NUM
cana-1172	53	136	-	-	PUNCT
cana-1172	53	137	133x	133x	NUM
cana-1172	53	138	vol	vol	NOUN
cana-1172	53	139	31	31	NUM
cana-1172	53	140	no	no	NOUN
cana-1172	53	141	.	.	PUNCT
cana-1172	54	1	6s	6s	NUM
cana-1172	54	2	(	(	PUNCT
cana-1172	54	3	2024	2024	NUM
cana-1172	54	4	)	)	PUNCT
cana-1172	54	5	135	135	NUM
cana-1172	54	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1172	54	7	(	(	PUNCT
cana-1172	54	8	𝐿𝐹𝐼2	𝐿𝐹𝐼2	PROPN
cana-1172	54	9	)	)	PUNCT
cana-1172	54	10	[	[	X
cana-1172	54	11	(	(	PUNCT
cana-1172	54	12	�	�	PROPN
cana-1172	54	13	̇	̇	PROPN
cana-1172	54	14	�	�	PROPN
cana-1172	54	15	∗	∗	PROPN
cana-1172	54	16	�	�	PROPN
cana-1172	54	17	̇	̇	PROPN
cana-1172	54	18	�	�	PROPN
cana-1172	54	19	)	)	PUNCT
cana-1172	54	20	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	54	21	]	]	PUNCT
cana-1172	54	22	∈	∈	PROPN
cana-1172	54	23	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	54	24	𝜀	𝜀	NOUN
cana-1172	54	25	,	,	PUNCT
cana-1172	54	26	[	[	X
cana-1172	54	27	�	�	NOUN
cana-1172	54	28	̇	̇	PROPN
cana-1172	54	29	�	�	PROPN
cana-1172	54	30	𝑢𝑏⁄	𝑢𝑏⁄	NOUN
cana-1172	54	31	]	]	PUNCT
cana-1172	54	32	∈	∈	PROPN
cana-1172	54	33	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	54	34	𝜀	𝜀	PROPN
cana-1172	54	35	⇒	⇒	NOUN
cana-1172	54	36	[	[	PUNCT
cana-1172	54	37	�	�	PROPN
cana-1172	54	38	̇	̇	PROPN
cana-1172	54	39	�	�	PROPN
cana-1172	54	40	𝑚𝑖𝑛{𝑢𝑎	𝑚𝑖𝑛{𝑢𝑎	VERB
cana-1172	54	41	,	,	PUNCT
cana-1172	54	42	𝑢𝑏}⁄	𝑢𝑏}⁄	PRON
cana-1172	54	43	]	]	PUNCT
cana-1172	55	1	∈	∈	PROPN
cana-1172	55	2	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	55	3	𝜀	𝜀	PROPN
cana-1172	55	4	(	(	PUNCT
cana-1172	55	5	3.5	3.5	NUM
cana-1172	55	6	)	)	PUNCT
cana-1172	55	7	for	for	ADP
cana-1172	55	8	all	all	DET
cana-1172	55	9	�	�	PROPN
cana-1172	55	10	̇	̇	PROPN
cana-1172	55	11	�	�	PROPN
cana-1172	55	12	,	,	PUNCT
cana-1172	55	13	�	�	PROPN
cana-1172	55	14	̇	̇	VERB
cana-1172	55	15	�	�	PROPN
cana-1172	55	16	∈	∈	PROPN
cana-1172	55	17	𝔊	𝔊	PROPN
cana-1172	55	18	and	and	CCONJ
cana-1172	55	19	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	55	20	,	,	PUNCT
cana-1172	55	21	𝑢𝑏	𝑢𝑏	ADP
cana-1172	55	22	∈	∈	PROPN
cana-1172	55	23	(	(	PUNCT
cana-1172	55	24	0,1	0,1	NOUN
cana-1172	55	25	]	]	PUNCT
cana-1172	55	26	.	.	PUNCT
cana-1172	56	1	example	example	NOUN
cana-1172	56	2	3.6	3.6	NUM
cana-1172	56	3	suppose	suppose	VERB
cana-1172	56	4	the	the	DET
cana-1172	56	5	set	set	NOUN
cana-1172	56	6	𝔊	𝔊	PROPN
cana-1172	56	7	=	=	SYM
cana-1172	56	8	{	{	PUNCT
cana-1172	56	9	0	0	NUM
cana-1172	56	10	,	,	PUNCT
cana-1172	56	11	𝓅1̇	𝓅1̇	PROPN
cana-1172	56	12	,	,	PUNCT
cana-1172	56	13	𝓅2̇	𝓅2̇	PROPN
cana-1172	56	14	,	,	PUNCT
cana-1172	56	15	𝓅3̇	𝓅3̇	NOUN
cana-1172	56	16	}	}	PUNCT
cana-1172	56	17	be	be	VERB
cana-1172	56	18	a	a	DET
cana-1172	56	19	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	PROPN
cana-1172	56	20	with	with	ADP
cana-1172	56	21	respect	respect	NOUN
cana-1172	56	22	to	to	ADP
cana-1172	56	23	a	a	DET
cana-1172	56	24	binary	binary	ADJ
cana-1172	56	25	operation	operation	NOUN
cana-1172	56	26	"	"	PUNCT
cana-1172	56	27	∗	∗	NOUN
cana-1172	56	28	"	"	PUNCT
cana-1172	56	29	given	give	VERB
cana-1172	56	30	by	by	ADP
cana-1172	56	31	table	table	NOUN
cana-1172	56	32	3.2	3.2	NUM
cana-1172	56	33	∗	∗	NOUN
cana-1172	56	34	0	0	NUM
cana-1172	56	35	𝓅1̇	𝓅1̇	PROPN
cana-1172	56	36	𝓅2̇	𝓅2̇	ADJ
cana-1172	56	37	𝓅3̇	𝓅3̇	NOUN
cana-1172	56	38	0	0	NUM
cana-1172	56	39	0	0	NUM
cana-1172	56	40	𝓅1̇	𝓅1̇	PROPN
cana-1172	56	41	𝓅2̇	𝓅2̇	ADJ
cana-1172	56	42	𝓅3̇	𝓅3̇	NOUN
cana-1172	56	43	𝓅1̇	𝓅1̇	VERB
cana-1172	56	44	𝓅1̇	𝓅1̇	X
cana-1172	56	45	0	0	NUM
cana-1172	56	46	𝓅3̇	𝓅3̇	PROPN
cana-1172	56	47	𝓅2̇	𝓅2̇	PROPN
cana-1172	56	48	𝓅2̇	𝓅2̇	PROPN
cana-1172	56	49	𝓅2̇	𝓅2̇	PROPN
cana-1172	56	50	𝓅3̇	𝓅3̇	PROPN
cana-1172	56	51	0	0	NUM
cana-1172	56	52	𝓅1̇	𝓅1̇	VERB
cana-1172	56	53	𝓅3̇	𝓅3̇	NOUN
cana-1172	56	54	𝓅3̇	𝓅3̇	PROPN
cana-1172	56	55	𝓅2̇	𝓅2̇	PROPN
cana-1172	56	56	𝓅1̇	𝓅1̇	VERB
cana-1172	56	57	0	0	NUM
cana-1172	56	58	table	table	NOUN
cana-1172	56	59	3.2	3.2	NUM
cana-1172	56	60	cayley	cayley	NOUN
cana-1172	56	61	table	table	NOUN
cana-1172	56	62	with	with	ADP
cana-1172	56	63	respect	respect	NOUN
cana-1172	56	64	to	to	ADP
cana-1172	56	65	"	"	PUNCT
cana-1172	56	66	∗	∗	NOUN
cana-1172	56	67	"	"	PUNCT
cana-1172	56	68	then	then	ADV
cana-1172	56	69	𝔊	𝔊	PROPN
cana-1172	56	70	is	be	AUX
cana-1172	56	71	a	a	DET
cana-1172	56	72	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎.	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎.	ADJ
cana-1172	56	73	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	NOUN
cana-1172	56	74	set	set	VERB
cana-1172	56	75	𝑈	𝑈	PROPN
cana-1172	56	76	in	in	ADP
cana-1172	56	77	𝔊	𝔊	PROPN
cana-1172	56	78	is	be	AUX
cana-1172	56	79	defined	define	VERB
cana-1172	56	80	as	as	SCONJ
cana-1172	56	81	follows	follow	VERB
cana-1172	56	82	:	:	PUNCT
cana-1172	56	83	𝑈	𝑈	ADJ
cana-1172	56	84	:	:	PUNCT
cana-1172	56	85	𝔊	𝔊	PROPN
cana-1172	56	86	→	→	SYM
cana-1172	56	87	[	[	X
cana-1172	56	88	0,1	0,1	NUM
cana-1172	56	89	]	]	PUNCT
cana-1172	56	90	,	,	PUNCT
cana-1172	56	91	�	�	PROPN
cana-1172	56	92	̇	̇	PROPN
cana-1172	56	93	�	�	PROPN
cana-1172	56	94	↦	↦	PROPN
cana-1172	56	95	{	{	PUNCT
cana-1172	56	96	0.91	0.91	NUM
cana-1172	56	97	𝑖𝑓	𝑖𝑓	PRON
cana-1172	56	98	�	�	PROPN
cana-1172	56	99	̇	̇	PROPN
cana-1172	56	100	�	�	PROPN
cana-1172	56	101	=	=	SYM
cana-1172	56	102	0	0	NUM
cana-1172	56	103	0.78	0.78	NUM
cana-1172	56	104	𝑖𝑓	𝑖𝑓	NOUN
cana-1172	56	105	�	�	PROPN
cana-1172	56	106	̇	̇	PROPN
cana-1172	56	107	�	�	PROPN
cana-1172	56	108	=	=	PUNCT
cana-1172	56	109	{	{	PUNCT
cana-1172	56	110	𝓅1̇	𝓅1̇	PROPN
cana-1172	56	111	,	,	PUNCT
cana-1172	56	112	𝓅2̇	𝓅2̇	PROPN
cana-1172	56	113	}	}	PUNCT
cana-1172	56	114	0.83	0.83	NUM
cana-1172	56	115	𝑖𝑓	𝑖𝑓	PRON
cana-1172	56	116	�	�	PROPN
cana-1172	56	117	̇	̇	PROPN
cana-1172	56	118	�	�	PROPN
cana-1172	56	119	=	=	PUNCT
cana-1172	56	120	𝓅3	𝓅3	PROPN
cana-1172	56	121	̇	̇	VERB
cana-1172	56	122	.	.	PUNCT
cana-1172	57	1	if	if	SCONJ
cana-1172	57	2	it	it	PRON
cana-1172	57	3	is	be	AUX
cana-1172	57	4	taken	take	VERB
cana-1172	57	5	that	that	DET
cana-1172	57	6	𝜀	𝜀	VERB
cana-1172	57	7	=	=	SYM
cana-1172	57	8	0.54	0.54	NUM
cana-1172	57	9	,	,	PUNCT
cana-1172	57	10	then	then	ADV
cana-1172	57	11	the	the	DET
cana-1172	57	12	lukasz	lukasz	PROPN
cana-1172	57	13	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	57	14	set	set	VERB
cana-1172	57	15	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	57	16	𝜀	𝜀	NOUN
cana-1172	57	17	of	of	ADP
cana-1172	57	18	𝑈	𝑈	PROPN
cana-1172	57	19	in	in	ADP
cana-1172	57	20	𝔊	𝔊	PROPN
cana-1172	57	21	is	be	AUX
cana-1172	57	22	provided	provide	VERB
cana-1172	57	23	as	as	ADP
cana-1172	57	24	below	below	ADP
cana-1172	57	25	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	57	26	𝜀	𝜀	NOUN
cana-1172	57	27	:	:	PUNCT
cana-1172	57	28	𝔊	𝔊	PROPN
cana-1172	57	29	→	→	SYM
cana-1172	57	30	[	[	X
cana-1172	57	31	0,1	0,1	NUM
cana-1172	57	32	]	]	PUNCT
cana-1172	57	33	,	,	PUNCT
cana-1172	57	34	�	�	PROPN
cana-1172	57	35	̇	̇	PROPN
cana-1172	57	36	�	�	PROPN
cana-1172	57	37	↦	↦	PROPN
cana-1172	57	38	{	{	PUNCT
cana-1172	57	39	0.45	0.45	NUM
cana-1172	57	40	𝑖𝑓	𝑖𝑓	PRON
cana-1172	57	41	�	�	PROPN
cana-1172	57	42	̇	̇	NOUN
cana-1172	57	43	�	�	PROPN
cana-1172	57	44	=	=	SYM
cana-1172	57	45	0	0	NUM
cana-1172	57	46	0.32	0.32	NUM
cana-1172	57	47	𝑖𝑓	𝑖𝑓	PRON
cana-1172	57	48	�	�	PROPN
cana-1172	57	49	̇	̇	PROPN
cana-1172	57	50	�	�	PROPN
cana-1172	57	51	=	=	PUNCT
cana-1172	57	52	{	{	PUNCT
cana-1172	57	53	𝓅1̇	𝓅1̇	PROPN
cana-1172	57	54	,	,	PUNCT
cana-1172	57	55	𝓅2̇	𝓅2̇	PROPN
cana-1172	57	56	}	}	PUNCT
cana-1172	57	57	0.37	0.37	NUM
cana-1172	57	58	𝑖𝑓	𝑖𝑓	PRON
cana-1172	57	59	�	�	PROPN
cana-1172	57	60	̇	̇	NOUN
cana-1172	57	61	�	�	PROPN
cana-1172	57	62	=	=	PUNCT
cana-1172	57	63	𝓅3̇	𝓅3̇	NOUN
cana-1172	57	64	typically	typically	ADV
cana-1172	57	65	,	,	PUNCT
cana-1172	57	66	it	it	PRON
cana-1172	57	67	is	be	AUX
cana-1172	57	68	verified	verify	VERB
cana-1172	57	69	that	that	SCONJ
cana-1172	57	70	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	57	71	𝜀	𝜀	PROPN
cana-1172	57	72	is	be	AUX
cana-1172	57	73	a	a	DET
cana-1172	57	74	lukasz	lukasz	NOUN
cana-1172	57	75	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	57	76	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	PROPN
cana-1172	57	77	of	of	ADP
cana-1172	57	78	𝔊.	𝔊.	PROPN
cana-1172	57	79	4	4	NUM
cana-1172	57	80	.	.	PUNCT
cana-1172	58	1	results	result	NOUN
cana-1172	58	2	theorem	theorem	VERB
cana-1172	58	3	4.1	4.1	NUM
cana-1172	58	4	every	every	DET
cana-1172	58	5	lukasz	lukasz	NOUN
cana-1172	58	6	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	58	7	set	set	VERB
cana-1172	58	8	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	58	9	𝜀	𝜀	PROPN
cana-1172	58	10	is	be	AUX
cana-1172	58	11	a	a	DET
cana-1172	58	12	lukasz	lukasz	NOUN
cana-1172	58	13	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	58	14	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	PROPN
cana-1172	58	15	of	of	ADP
cana-1172	58	16	𝔊	𝔊	PROPN
cana-1172	58	17	iff	iff	VERB
cana-1172	58	18	it	it	PRON
cana-1172	58	19	satisfies	satisfy	VERB
cana-1172	58	20	:	:	PUNCT
cana-1172	58	21	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	58	22	𝜀	𝜀	PROPN
cana-1172	58	23	(	(	PUNCT
cana-1172	58	24	�	�	PROPN
cana-1172	58	25	̇	̇	PROPN
cana-1172	58	26	�	�	PROPN
cana-1172	58	27	∗	∗	PROPN
cana-1172	58	28	�	�	PROPN
cana-1172	58	29	̇	̇	PROPN
cana-1172	58	30	�	�	PROPN
cana-1172	58	31	)	)	PUNCT
cana-1172	58	32	≥	≥	NOUN
cana-1172	58	33	𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝐿𝑈	NUM
cana-1172	58	34	𝜀	𝜀	X
cana-1172	58	35	(	(	PUNCT
cana-1172	58	36	�	�	PROPN
cana-1172	58	37	̇	̇	PROPN
cana-1172	58	38	�	�	PROPN
cana-1172	58	39	)	)	PUNCT
cana-1172	58	40	,	,	PUNCT
cana-1172	58	41	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	58	42	𝜀	𝜀	PROPN
cana-1172	58	43	(	(	PUNCT
cana-1172	58	44	�	�	PROPN
cana-1172	58	45	̇	̇	PROPN
cana-1172	58	46	�	�	PROPN
cana-1172	58	47	)	)	PUNCT
cana-1172	58	48	}	}	PUNCT
cana-1172	58	49	,	,	PUNCT
cana-1172	58	50	∀	∀	X
cana-1172	58	51	�	�	PROPN
cana-1172	58	52	̇	̇	PROPN
cana-1172	58	53	�	�	PROPN
cana-1172	58	54	,	,	PUNCT
cana-1172	58	55	�	�	PROPN
cana-1172	58	56	̇	̇	VERB
cana-1172	58	57	�	�	PROPN
cana-1172	58	58	∈	∈	PROPN
cana-1172	58	59	𝔊	𝔊	PROPN
cana-1172	58	60	(	(	PUNCT
cana-1172	58	61	4.1	4.1	NUM
cana-1172	58	62	)	)	PUNCT
cana-1172	58	63	proof	proof	NOUN
cana-1172	58	64	suppose	suppose	VERB
cana-1172	58	65	𝑈	𝑈	PROPN
cana-1172	58	66	be	be	AUX
cana-1172	58	67	a	a	DET
cana-1172	58	68	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	58	69	set	set	NOUN
cana-1172	58	70	in	in	ADP
cana-1172	58	71	𝔊.	𝔊.	PROPN
cana-1172	58	72	for	for	ADP
cana-1172	58	73	instance	instance	NOUN
cana-1172	58	74	,	,	PUNCT
cana-1172	58	75	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	58	76	𝜀	𝜀	PROPN
cana-1172	58	77	is	be	AUX
cana-1172	58	78	a	a	DET
cana-1172	58	79	lukasz	lukasz	NOUN
cana-1172	58	80	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	58	81	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	PROPN
cana-1172	58	82	of	of	ADP
cana-1172	58	83	𝔊.	𝔊.	PROPN
cana-1172	58	84	let	let	VERB
cana-1172	58	85	�	�	SYM
cana-1172	58	86	̇	̇	PROPN
cana-1172	58	87	�	�	PROPN
cana-1172	58	88	,	,	PUNCT
cana-1172	58	89	�	�	PROPN
cana-1172	58	90	̇	̇	VERB
cana-1172	58	91	�	�	PROPN
cana-1172	58	92	∈	∈	PROPN
cana-1172	58	93	𝔊	𝔊	PROPN
cana-1172	59	1	and	and	CCONJ
cana-1172	59	2	it	it	PRON
cana-1172	59	3	is	be	AUX
cana-1172	59	4	clear	clear	ADJ
cana-1172	59	5	that	that	SCONJ
cana-1172	59	6	[	[	PUNCT
cana-1172	59	7	�	�	NOUN
cana-1172	59	8	̇	̇	VERB
cana-1172	59	9	�	�	PROPN
cana-1172	59	10	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	59	11	𝜀	𝜀	PROPN
cana-1172	59	12	(	(	PUNCT
cana-1172	59	13	�	�	PROPN
cana-1172	59	14	̇	̇	NOUN
cana-1172	59	15	�	�	NOUN
cana-1172	59	16	)⁄	)⁄	PRON
cana-1172	59	17	]	]	PUNCT
cana-1172	59	18	∈	∈	PROPN
cana-1172	59	19	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	59	20	𝜀	𝜀	PROPN
cana-1172	59	21	and	and	CCONJ
cana-1172	59	22	[	[	PUNCT
cana-1172	59	23	�	�	PROPN
cana-1172	59	24	̇	̇	VERB
cana-1172	59	25	�	�	PROPN
cana-1172	59	26	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	59	27	𝜀	𝜀	PROPN
cana-1172	59	28	(	(	PUNCT
cana-1172	59	29	�	�	PROPN
cana-1172	59	30	̇	̇	NOUN
cana-1172	59	31	�	�	NOUN
cana-1172	59	32	)⁄	)⁄	PRON
cana-1172	59	33	]	]	PUNCT
cana-1172	59	34	∈	∈	PROPN
cana-1172	59	35	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	59	36	𝜀	𝜀	PROPN
cana-1172	59	37	.	.	PUNCT
cana-1172	60	1	from	from	ADP
cana-1172	60	2	(	(	PUNCT
cana-1172	60	3	3.3	3.3	NUM
cana-1172	60	4	)	)	PUNCT
cana-1172	60	5	,	,	PUNCT
cana-1172	60	6	it	it	PRON
cana-1172	60	7	is	be	AUX
cana-1172	60	8	evident	evident	ADJ
cana-1172	60	9	that	that	SCONJ
cana-1172	60	10	[	[	X
cana-1172	60	11	(	(	PUNCT
cana-1172	60	12	�	�	PROPN
cana-1172	60	13	̇	̇	PROPN
cana-1172	60	14	�	�	PROPN
cana-1172	60	15	∗	∗	PROPN
cana-1172	60	16	�	�	PROPN
cana-1172	60	17	̇	̇	PROPN
cana-1172	60	18	�	�	PROPN
cana-1172	60	19	)	)	PUNCT
cana-1172	60	20	𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝐿𝑈	PROPN
cana-1172	60	21	𝜀	𝜀	X
cana-1172	60	22	(	(	PUNCT
cana-1172	60	23	�	�	PROPN
cana-1172	60	24	̇	̇	PROPN
cana-1172	60	25	�	�	PROPN
cana-1172	60	26	)	)	PUNCT
cana-1172	60	27	,	,	PUNCT
cana-1172	60	28	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	60	29	𝜀	𝜀	PROPN
cana-1172	60	30	(	(	PUNCT
cana-1172	60	31	�	�	PROPN
cana-1172	60	32	̇	̇	PROPN
cana-1172	60	33	�	�	PROPN
cana-1172	60	34	)}⁄	)}⁄	PUNCT
cana-1172	60	35	]	]	PUNCT
cana-1172	60	36	∈	∈	PROPN
cana-1172	60	37	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	60	38	𝜀	𝜀	NOUN
cana-1172	60	39	,	,	PUNCT
cana-1172	60	40	and	and	CCONJ
cana-1172	60	41	hence	hence	ADV
cana-1172	60	42	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	60	43	𝜀	𝜀	PROPN
cana-1172	60	44	(	(	PUNCT
cana-1172	60	45	�	�	PROPN
cana-1172	60	46	̇	̇	PROPN
cana-1172	60	47	�	�	PROPN
cana-1172	60	48	∗	∗	PROPN
cana-1172	60	49	�	�	PROPN
cana-1172	60	50	̇	̇	PROPN
cana-1172	60	51	�	�	PROPN
cana-1172	60	52	)	)	PUNCT
cana-1172	60	53	≥	≥	NOUN
cana-1172	60	54	𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝐿𝑈	NUM
cana-1172	60	55	𝜀	𝜀	X
cana-1172	60	56	(	(	PUNCT
cana-1172	60	57	�	�	PROPN
cana-1172	60	58	̇	̇	PROPN
cana-1172	60	59	�	�	PROPN
cana-1172	60	60	)	)	PUNCT
cana-1172	60	61	,	,	PUNCT
cana-1172	60	62	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	60	63	𝜀	𝜀	PROPN
cana-1172	60	64	(	(	PUNCT
cana-1172	60	65	�	�	PROPN
cana-1172	60	66	̇	̇	PROPN
cana-1172	60	67	�	�	PROPN
cana-1172	60	68	)	)	PUNCT
cana-1172	60	69	}	}	PUNCT
cana-1172	60	70	for	for	SCONJ
cana-1172	60	71	all	all	DET
cana-1172	60	72	�	�	PROPN
cana-1172	60	73	̇	̇	PROPN
cana-1172	60	74	�	�	PROPN
cana-1172	60	75	,	,	PUNCT
cana-1172	60	76	�	�	PROPN
cana-1172	60	77	̇	̇	VERB
cana-1172	60	78	�	�	PROPN
cana-1172	60	79	∈	∈	PROPN
cana-1172	60	80	𝔊.	𝔊.	PROPN
cana-1172	60	81	communications	communication	NOUN
cana-1172	60	82	on	on	ADP
cana-1172	60	83	applied	apply	VERB
cana-1172	60	84	nonlinear	nonlinear	ADJ
cana-1172	60	85	analysis	analysis	NOUN
cana-1172	60	86	issn	issn	NOUN
cana-1172	60	87	:	:	PUNCT
cana-1172	60	88	1074	1074	NUM
cana-1172	60	89	-	-	PUNCT
cana-1172	60	90	133x	133x	NUM
cana-1172	60	91	vol	vol	NOUN
cana-1172	60	92	31	31	NUM
cana-1172	60	93	no	no	NOUN
cana-1172	60	94	.	.	PUNCT
cana-1172	61	1	6s	6s	NUM
cana-1172	61	2	(	(	PUNCT
cana-1172	61	3	2024	2024	NUM
cana-1172	61	4	)	)	PUNCT
cana-1172	61	5	136	136	NUM
cana-1172	61	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1172	61	7	conversely	conversely	ADV
cana-1172	61	8	,	,	PUNCT
cana-1172	61	9	suppose	suppose	VERB
cana-1172	61	10	that	that	SCONJ
cana-1172	61	11	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	61	12	𝜀	𝜀	X
cana-1172	61	13	satisfies	satisfie	NOUN
cana-1172	61	14	(	(	PUNCT
cana-1172	61	15	4.1	4.1	NUM
cana-1172	61	16	)	)	PUNCT
cana-1172	61	17	.	.	PUNCT
cana-1172	62	1	also	also	ADV
cana-1172	62	2	let	let	VERB
cana-1172	62	3	�	�	SYM
cana-1172	62	4	̇	̇	PROPN
cana-1172	62	5	�	�	PROPN
cana-1172	62	6	,	,	PUNCT
cana-1172	62	7	�	�	PROPN
cana-1172	62	8	̇	̇	VERB
cana-1172	62	9	�	�	PROPN
cana-1172	62	10	∈	∈	PROPN
cana-1172	62	11	𝔊	𝔊	PROPN
cana-1172	62	12	and	and	CCONJ
cana-1172	62	13	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	62	14	,	,	PUNCT
cana-1172	62	15	𝑢𝑏	𝑢𝑏	ADP
cana-1172	62	16	∈	∈	PROPN
cana-1172	62	17	(	(	PUNCT
cana-1172	62	18	0,1	0,1	NOUN
cana-1172	62	19	]	]	PUNCT
cana-1172	62	20	be	be	VERB
cana-1172	62	21	such	such	ADJ
cana-1172	62	22	that	that	SCONJ
cana-1172	62	23	[	[	X
cana-1172	62	24	�	�	NOUN
cana-1172	62	25	̇	̇	VERB
cana-1172	62	26	�	�	PROPN
cana-1172	62	27	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	62	28	]	]	PUNCT
cana-1172	62	29	∈	∈	PROPN
cana-1172	62	30	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	62	31	𝜀	𝜀	NOUN
cana-1172	62	32	,	,	PUNCT
cana-1172	62	33	[	[	X
cana-1172	62	34	�	�	NOUN
cana-1172	62	35	̇	̇	PROPN
cana-1172	62	36	�	�	PROPN
cana-1172	62	37	𝑢𝑏⁄	𝑢𝑏⁄	NOUN
cana-1172	62	38	]	]	PUNCT
cana-1172	62	39	∈	∈	PROPN
cana-1172	62	40	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	62	41	𝜀	𝜀	PROPN
cana-1172	62	42	.	.	PUNCT
cana-1172	63	1	then	then	ADV
cana-1172	63	2	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	63	3	𝜀	𝜀	PROPN
cana-1172	63	4	(	(	PUNCT
cana-1172	63	5	�	�	PROPN
cana-1172	63	6	̇	̇	PROPN
cana-1172	63	7	�	�	PROPN
cana-1172	63	8	)	)	PUNCT
cana-1172	63	9	≥	≥	NOUN
cana-1172	63	10	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	63	11	and	and	CCONJ
cana-1172	63	12	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	63	13	𝜀	𝜀	PROPN
cana-1172	63	14	(	(	PUNCT
cana-1172	63	15	�	�	PROPN
cana-1172	63	16	̇	̇	PROPN
cana-1172	63	17	�	�	PROPN
cana-1172	63	18	)	)	PUNCT
cana-1172	63	19	≥	≥	NOUN
cana-1172	63	20	𝑢𝑏	𝑢𝑏	PROPN
cana-1172	63	21	,	,	PUNCT
cana-1172	63	22	which	which	PRON
cana-1172	63	23	imply	imply	VERB
cana-1172	63	24	from	from	ADP
cana-1172	63	25	(	(	PUNCT
cana-1172	63	26	4.1	4.1	NUM
cana-1172	63	27	)	)	PUNCT
cana-1172	63	28	that	that	SCONJ
cana-1172	63	29	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	63	30	𝜀	𝜀	PROPN
cana-1172	63	31	(	(	PUNCT
cana-1172	63	32	�	�	PROPN
cana-1172	63	33	̇	̇	PROPN
cana-1172	63	34	�	�	PROPN
cana-1172	63	35	∗	∗	PROPN
cana-1172	63	36	�	�	PROPN
cana-1172	63	37	̇	̇	PROPN
cana-1172	63	38	�	�	PROPN
cana-1172	63	39	)	)	PUNCT
cana-1172	63	40	≥	≥	NOUN
cana-1172	63	41	𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝐿𝑈	NUM
cana-1172	63	42	𝜀	𝜀	X
cana-1172	63	43	(	(	PUNCT
cana-1172	63	44	�	�	PROPN
cana-1172	63	45	̇	̇	PROPN
cana-1172	63	46	�	�	PROPN
cana-1172	63	47	)	)	PUNCT
cana-1172	63	48	,	,	PUNCT
cana-1172	63	49	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	63	50	𝜀	𝜀	PROPN
cana-1172	63	51	(	(	PUNCT
cana-1172	63	52	�	�	PROPN
cana-1172	63	53	̇	̇	PROPN
cana-1172	63	54	�	�	PROPN
cana-1172	63	55	)	)	PUNCT
cana-1172	63	56	}	}	PUNCT
cana-1172	63	57	≥	≥	X
cana-1172	63	58	𝑚𝑖𝑛{𝑢𝑎	𝑚𝑖𝑛{𝑢𝑎	VERB
cana-1172	63	59	,	,	PUNCT
cana-1172	63	60	𝑢𝑏	𝑢𝑏	ADP
cana-1172	63	61	}	}	PUNCT
cana-1172	63	62	.	.	PUNCT
cana-1172	64	1	thus	thus	ADV
cana-1172	64	2	[	[	X
cana-1172	64	3	(	(	PUNCT
cana-1172	64	4	�	�	PROPN
cana-1172	64	5	̇	̇	PROPN
cana-1172	64	6	�	�	PROPN
cana-1172	64	7	∗	∗	PROPN
cana-1172	64	8	�	�	PROPN
cana-1172	64	9	̇	̇	PROPN
cana-1172	64	10	�	�	PROPN
cana-1172	64	11	)	)	PUNCT
cana-1172	64	12	𝑚𝑖𝑛{𝑢𝑎	𝑚𝑖𝑛{𝑢𝑎	PUNCT
cana-1172	64	13	,	,	PUNCT
cana-1172	64	14	𝑢𝑏}⁄	𝑢𝑏}⁄	PRON
cana-1172	64	15	]	]	PUNCT
cana-1172	64	16	∈	∈	PROPN
cana-1172	64	17	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	64	18	𝜀	𝜀	PROPN
cana-1172	64	19	.	.	PUNCT
cana-1172	65	1	therefore	therefore	ADV
cana-1172	65	2	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	65	3	𝜀	𝜀	PROPN
cana-1172	65	4	is	be	AUX
cana-1172	65	5	a	a	DET
cana-1172	65	6	lukasz	lukasz	NOUN
cana-1172	65	7	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	65	8	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	PROPN
cana-1172	65	9	of	of	ADP
cana-1172	65	10	𝔊.	𝔊.	PROPN
cana-1172	65	11	theorem	theorem	VERB
cana-1172	65	12	4.2	4.2	NUM
cana-1172	65	13	show	show	VERB
cana-1172	65	14	that	that	SCONJ
cana-1172	65	15	𝜀-lukasz	𝜀-lukasz	NOUN
cana-1172	65	16	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	NOUN
cana-1172	65	17	set	set	VERB
cana-1172	65	18	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	65	19	𝜀	𝜀	NOUN
cana-1172	65	20	in	in	ADP
cana-1172	65	21	𝔊	𝔊	PROPN
cana-1172	65	22	is	be	AUX
cana-1172	65	23	an	an	DET
cana-1172	65	24	𝜀-lukasz	𝜀-lukasz	NOUN
cana-1172	65	25	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	65	26	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	PROPN
cana-1172	65	27	of	of	ADP
cana-1172	65	28	𝔊	𝔊	PROPN
cana-1172	65	29	,	,	PUNCT
cana-1172	65	30	if	if	SCONJ
cana-1172	65	31	𝑈	𝑈	PROPN
cana-1172	65	32	is	be	AUX
cana-1172	65	33	a	a	DET
cana-1172	65	34	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	65	35	sub	sub	NOUN
cana-1172	65	36	algebra	algebra	NOUN
cana-1172	65	37	of	of	ADP
cana-1172	65	38	𝔊.	𝔊.	PROPN
cana-1172	65	39	proof	proof	NOUN
cana-1172	65	40	for	for	ADP
cana-1172	65	41	instance	instance	NOUN
cana-1172	65	42	,	,	PUNCT
cana-1172	65	43	𝑈	𝑈	PROPN
cana-1172	65	44	is	be	AUX
cana-1172	65	45	a	a	DET
cana-1172	65	46	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	65	47	subalgebra	subalgebra	NOUN
cana-1172	65	48	of	of	ADP
cana-1172	65	49	𝔊.	𝔊.	PROPN
cana-1172	65	50	let	let	VERB
cana-1172	65	51	�	�	SYM
cana-1172	65	52	̇	̇	PROPN
cana-1172	65	53	�	�	PROPN
cana-1172	65	54	,	,	PUNCT
cana-1172	65	55	�	�	PROPN
cana-1172	65	56	̇	̇	VERB
cana-1172	65	57	�	�	PROPN
cana-1172	65	58	∈	∈	PROPN
cana-1172	65	59	𝔊	𝔊	PROPN
cana-1172	65	60	and	and	CCONJ
cana-1172	65	61	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	65	62	,	,	PUNCT
cana-1172	65	63	𝑢𝑏	𝑢𝑏	ADP
cana-1172	65	64	∈	∈	PROPN
cana-1172	65	65	(	(	PUNCT
cana-1172	65	66	0,1	0,1	NOUN
cana-1172	65	67	]	]	PUNCT
cana-1172	65	68	be	be	VERB
cana-1172	65	69	such	such	ADJ
cana-1172	65	70	that	that	SCONJ
cana-1172	65	71	[	[	X
cana-1172	65	72	�	�	NOUN
cana-1172	65	73	̇	̇	VERB
cana-1172	65	74	�	�	PROPN
cana-1172	65	75	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	65	76	]	]	PUNCT
cana-1172	65	77	∈	∈	PROPN
cana-1172	65	78	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	65	79	𝜀	𝜀	NOUN
cana-1172	65	80	,	,	PUNCT
cana-1172	65	81	[	[	X
cana-1172	65	82	�	�	NOUN
cana-1172	65	83	̇	̇	PROPN
cana-1172	65	84	�	�	PROPN
cana-1172	65	85	𝑢𝑏⁄	𝑢𝑏⁄	NOUN
cana-1172	65	86	]	]	PUNCT
cana-1172	65	87	∈	∈	PROPN
cana-1172	65	88	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	65	89	𝜀	𝜀	PROPN
cana-1172	65	90	.	.	PUNCT
cana-1172	66	1	then	then	ADV
cana-1172	66	2	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	66	3	𝜀	𝜀	PROPN
cana-1172	66	4	(	(	PUNCT
cana-1172	66	5	�	�	PROPN
cana-1172	66	6	̇	̇	PROPN
cana-1172	66	7	�	�	PROPN
cana-1172	66	8	)	)	PUNCT
cana-1172	66	9	≥	≥	NOUN
cana-1172	66	10	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	66	11	and	and	CCONJ
cana-1172	66	12	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	66	13	𝜀	𝜀	PROPN
cana-1172	66	14	(	(	PUNCT
cana-1172	66	15	�	�	PROPN
cana-1172	66	16	̇	̇	PROPN
cana-1172	66	17	�	�	PROPN
cana-1172	66	18	)	)	PUNCT
cana-1172	66	19	≥	≥	NOUN
cana-1172	66	20	𝑢𝑏.	𝑢𝑏.	VERB
cana-1172	66	21	thus	thus	ADV
cana-1172	66	22	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	66	23	𝜀	𝜀	PROPN
cana-1172	66	24	(	(	PUNCT
cana-1172	66	25	�	�	PROPN
cana-1172	66	26	̇	̇	PROPN
cana-1172	66	27	�	�	PROPN
cana-1172	66	28	∗	∗	PROPN
cana-1172	66	29	�	�	PROPN
cana-1172	66	30	̇	̇	PROPN
cana-1172	66	31	�	�	PROPN
cana-1172	66	32	)	)	PUNCT
cana-1172	66	33	=	=	SYM
cana-1172	66	34	𝑚𝑎𝑥{0	𝑚𝑎𝑥{0	PROPN
cana-1172	66	35	,	,	PUNCT
cana-1172	66	36	𝑈(	𝑈(	PROPN
cana-1172	66	37	�	�	PROPN
cana-1172	66	38	̇	̇	PROPN
cana-1172	66	39	�	�	PROPN
cana-1172	66	40	∗	∗	PROPN
cana-1172	66	41	�	�	PROPN
cana-1172	66	42	̇	̇	PROPN
cana-1172	66	43	�	�	PROPN
cana-1172	66	44	)	)	PUNCT
cana-1172	66	45	+	+	NUM
cana-1172	66	46	𝜀	𝜀	X
cana-1172	66	47	−	−	NUM
cana-1172	66	48	1	1	NUM
cana-1172	66	49	}	}	PUNCT
cana-1172	66	50	[	[	X
cana-1172	66	51	∵	∵	X
cana-1172	66	52	(	(	PUNCT
cana-1172	66	53	3.1	3.1	NUM
cana-1172	66	54	)	)	PUNCT
cana-1172	66	55	]	]	PUNCT
cana-1172	66	56	≥	≥	X
cana-1172	66	57	𝑚𝑎𝑥{0	𝑚𝑎𝑥{0	PROPN
cana-1172	66	58	,	,	PUNCT
cana-1172	66	59	𝑚𝑖𝑛{𝑈(	𝑚𝑖𝑛{𝑈(	PROPN
cana-1172	66	60	�	�	PROPN
cana-1172	66	61	̇	̇	PROPN
cana-1172	66	62	�	�	PROPN
cana-1172	66	63	)	)	PUNCT
cana-1172	66	64	,	,	PUNCT
cana-1172	66	65	𝑈(	𝑈(	PROPN
cana-1172	66	66	�	�	PROPN
cana-1172	66	67	̇	̇	PROPN
cana-1172	66	68	�	�	PROPN
cana-1172	66	69	)	)	PUNCT
cana-1172	66	70	}	}	PUNCT
cana-1172	66	71	+	+	NUM
cana-1172	66	72	𝜀	𝜀	X
cana-1172	66	73	−	−	NUM
cana-1172	66	74	1	1	NUM
cana-1172	66	75	}	}	PUNCT
cana-1172	66	76	[	[	X
cana-1172	66	77	∵	∵	X
cana-1172	66	78	(	(	PUNCT
cana-1172	66	79	4.1	4.1	NUM
cana-1172	66	80	)	)	PUNCT
cana-1172	66	81	]	]	PUNCT
cana-1172	66	82	=	=	PUNCT
cana-1172	66	83	𝑚𝑎𝑥{0	𝑚𝑎𝑥{0	PROPN
cana-1172	66	84	,	,	PUNCT
cana-1172	66	85	𝑚𝑖𝑛{𝑈(	𝑚𝑖𝑛{𝑈(	PROPN
cana-1172	66	86	�	�	PROPN
cana-1172	66	87	̇	̇	PROPN
cana-1172	66	88	�	�	PROPN
cana-1172	66	89	)	)	PUNCT
cana-1172	67	1	+	+	NUM
cana-1172	67	2	𝜀	𝜀	X
cana-1172	67	3	−	−	NUM
cana-1172	67	4	1	1	NUM
cana-1172	67	5	,	,	PUNCT
cana-1172	67	6	𝑈(	𝑈(	PROPN
cana-1172	67	7	�	�	PROPN
cana-1172	67	8	̇	̇	PROPN
cana-1172	67	9	�	�	PROPN
cana-1172	67	10	)	)	PUNCT
cana-1172	68	1	+	+	NUM
cana-1172	68	2	𝜀	𝜀	X
cana-1172	68	3	−	−	NUM
cana-1172	68	4	1	1	NUM
cana-1172	68	5	}	}	PUNCT
cana-1172	68	6	}	}	PUNCT
cana-1172	68	7	=	=	SYM
cana-1172	68	8	𝑚𝑖𝑛{𝑚𝑎𝑥{0	𝑚𝑖𝑛{𝑚𝑎𝑥{0	NOUN
cana-1172	68	9	,	,	PUNCT
cana-1172	68	10	𝑈(	𝑈(	PROPN
cana-1172	68	11	�	�	PROPN
cana-1172	68	12	̇	̇	PROPN
cana-1172	68	13	�	�	PROPN
cana-1172	68	14	)	)	PUNCT
cana-1172	69	1	+	+	NUM
cana-1172	69	2	𝜀	𝜀	X
cana-1172	69	3	−	−	NUM
cana-1172	69	4	1	1	NUM
cana-1172	69	5	}	}	PUNCT
cana-1172	69	6	,	,	PUNCT
cana-1172	69	7	𝑚𝑎𝑥{0	𝑚𝑎𝑥{0	PROPN
cana-1172	69	8	,	,	PUNCT
cana-1172	69	9	𝑈(	𝑈(	PROPN
cana-1172	69	10	�	�	PROPN
cana-1172	69	11	̇	̇	PROPN
cana-1172	69	12	�	�	PROPN
cana-1172	69	13	)	)	PUNCT
cana-1172	70	1	+	+	NUM
cana-1172	70	2	𝜀	𝜀	X
cana-1172	70	3	−	−	NUM
cana-1172	70	4	1	1	NUM
cana-1172	70	5	}	}	PUNCT
cana-1172	70	6	}	}	PUNCT
cana-1172	70	7	=	=	SYM
cana-1172	70	8	𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝐿𝑈	NUM
cana-1172	70	9	𝜀	𝜀	X
cana-1172	70	10	(	(	PUNCT
cana-1172	70	11	�	�	PROPN
cana-1172	70	12	̇	̇	PROPN
cana-1172	70	13	�	�	PROPN
cana-1172	70	14	)	)	PUNCT
cana-1172	70	15	,	,	PUNCT
cana-1172	70	16	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	70	17	𝜀	𝜀	PROPN
cana-1172	70	18	(	(	PUNCT
cana-1172	70	19	�	�	PROPN
cana-1172	70	20	̇	̇	PROPN
cana-1172	70	21	�	�	PROPN
cana-1172	70	22	)	)	PUNCT
cana-1172	70	23	}	}	PUNCT
cana-1172	71	1	[	[	X
cana-1172	71	2	∵	∵	X
cana-1172	71	3	(	(	PUNCT
cana-1172	71	4	3.1	3.1	NUM
cana-1172	71	5	)	)	PUNCT
cana-1172	71	6	]	]	PUNCT
cana-1172	71	7	≥	≥	X
cana-1172	71	8	𝑚𝑖𝑛{𝑢𝑎	𝑚𝑖𝑛{𝑢𝑎	VERB
cana-1172	71	9	,	,	PUNCT
cana-1172	71	10	𝑢𝑏	𝑢𝑏	ADP
cana-1172	71	11	}	}	PUNCT
cana-1172	71	12	.	.	PUNCT
cana-1172	72	1	so	so	ADV
cana-1172	72	2	,	,	PUNCT
cana-1172	72	3	[	[	X
cana-1172	72	4	(	(	PUNCT
cana-1172	72	5	�	�	PROPN
cana-1172	72	6	̇	̇	PROPN
cana-1172	72	7	�	�	PROPN
cana-1172	72	8	∗	∗	PROPN
cana-1172	72	9	�	�	PROPN
cana-1172	72	10	̇	̇	PROPN
cana-1172	72	11	�	�	PROPN
cana-1172	72	12	)	)	PUNCT
cana-1172	72	13	𝑚𝑖𝑛{𝑢𝑎	𝑚𝑖𝑛{𝑢𝑎	VERB
cana-1172	72	14	,	,	PUNCT
cana-1172	72	15	𝑢𝑏}⁄	𝑢𝑏}⁄	PRON
cana-1172	72	16	]	]	PUNCT
cana-1172	72	17	∈	∈	PROPN
cana-1172	72	18	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	72	19	𝜀	𝜀	NOUN
cana-1172	72	20	.	.	PUNCT
cana-1172	73	1	hence	hence	ADV
cana-1172	73	2	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	73	3	𝜀	𝜀	PROPN
cana-1172	73	4	is	be	AUX
cana-1172	73	5	a	a	DET
cana-1172	73	6	𝜀-lukasz	𝜀-lukasz	NOUN
cana-1172	73	7	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	73	8	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	PROPN
cana-1172	73	9	of	of	ADP
cana-1172	73	10	𝔊.	𝔊.	PROPN
cana-1172	73	11	the	the	DET
cana-1172	73	12	subsequent	subsequent	ADJ
cana-1172	73	13	example	example	NOUN
cana-1172	73	14	demonstrates	demonstrate	VERB
cana-1172	73	15	why	why	SCONJ
cana-1172	73	16	the	the	DET
cana-1172	73	17	reverse	reverse	ADJ
cana-1172	73	18	portion	portion	NOUN
cana-1172	73	19	of	of	ADP
cana-1172	73	20	theorem	theorem	ADJ
cana-1172	73	21	4.2	4.2	NUM
cana-1172	73	22	is	be	AUX
cana-1172	73	23	false	false	ADJ
cana-1172	73	24	.	.	PUNCT
cana-1172	73	25	example	example	NOUN
cana-1172	73	26	4.3	4.3	NUM
cana-1172	73	27	suppose	suppose	VERB
cana-1172	73	28	the	the	DET
cana-1172	73	29	set	set	NOUN
cana-1172	73	30	𝔊	𝔊	PROPN
cana-1172	73	31	=	=	SYM
cana-1172	73	32	{	{	PUNCT
cana-1172	73	33	0	0	NUM
cana-1172	73	34	,	,	PUNCT
cana-1172	73	35	𝓅1̇	𝓅1̇	NOUN
cana-1172	73	36	,	,	PUNCT
cana-1172	73	37	𝓅2̇	𝓅2̇	PROPN
cana-1172	73	38	}	}	PUNCT
cana-1172	73	39	be	be	VERB
cana-1172	73	40	a	a	DET
cana-1172	73	41	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	PROPN
cana-1172	73	42	and	and	CCONJ
cana-1172	73	43	table	table	NOUN
cana-1172	73	44	4.1	4.1	NUM
cana-1172	73	45	shows	show	VERB
cana-1172	73	46	binary	binary	ADJ
cana-1172	73	47	operation	operation	NOUN
cana-1172	73	48	"	"	PUNCT
cana-1172	73	49	∗	∗	NOUN
cana-1172	73	50	"	"	PUNCT
cana-1172	73	51	in	in	ADP
cana-1172	73	52	𝔊	𝔊	PROPN
cana-1172	73	53	∗	∗	NOUN
cana-1172	73	54	0	0	NUM
cana-1172	73	55	𝓅1̇	𝓅1̇	PROPN
cana-1172	73	56	𝓅2̇	𝓅2̇	PROPN
cana-1172	73	57	0	0	NUM
cana-1172	73	58	0	0	NUM
cana-1172	73	59	𝓅2̇	𝓅2̇	PROPN
cana-1172	73	60	𝓅1̇	𝓅1̇	NOUN
cana-1172	73	61	𝓅1̇	𝓅1̇	NOUN
cana-1172	73	62	𝓅1̇	𝓅1̇	X
cana-1172	73	63	0	0	NUM
cana-1172	73	64	𝓅2̇	𝓅2̇	PROPN
cana-1172	73	65	𝓅2̇	𝓅2̇	PROPN
cana-1172	73	66	𝓅2̇	𝓅2̇	PROPN
cana-1172	73	67	𝓅1̇	𝓅1̇	VERB
cana-1172	73	68	0	0	NUM
cana-1172	73	69	communications	communication	NOUN
cana-1172	73	70	on	on	ADP
cana-1172	73	71	applied	apply	VERB
cana-1172	73	72	nonlinear	nonlinear	ADJ
cana-1172	73	73	analysis	analysis	NOUN
cana-1172	73	74	issn	issn	NOUN
cana-1172	73	75	:	:	PUNCT
cana-1172	73	76	1074	1074	NUM
cana-1172	73	77	-	-	PUNCT
cana-1172	73	78	133x	133x	NUM
cana-1172	73	79	vol	vol	NOUN
cana-1172	73	80	31	31	NUM
cana-1172	73	81	no	no	NOUN
cana-1172	73	82	.	.	PUNCT
cana-1172	74	1	6s	6s	NUM
cana-1172	74	2	(	(	PUNCT
cana-1172	74	3	2024	2024	NUM
cana-1172	74	4	)	)	PUNCT
cana-1172	74	5	137	137	NUM
cana-1172	74	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1172	74	7	table	table	NOUN
cana-1172	74	8	4.1	4.1	NUM
cana-1172	74	9	cayley	cayley	NOUN
cana-1172	74	10	table	table	NOUN
cana-1172	74	11	with	with	ADP
cana-1172	74	12	respect	respect	NOUN
cana-1172	74	13	to	to	ADP
cana-1172	74	14	"	"	PUNCT
cana-1172	74	15	∗	∗	NOUN
cana-1172	74	16	"	"	PUNCT
cana-1172	74	17	defining	define	VERB
cana-1172	74	18	a	a	DET
cana-1172	74	19	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	74	20	set	set	VERB
cana-1172	74	21	𝑈	𝑈	PROPN
cana-1172	74	22	in	in	ADP
cana-1172	74	23	𝔊	𝔊	PROPN
cana-1172	74	24	as	as	SCONJ
cana-1172	74	25	follows	follow	VERB
cana-1172	74	26	:	:	PUNCT
cana-1172	74	27	𝑈	𝑈	ADJ
cana-1172	74	28	:	:	PUNCT
cana-1172	74	29	𝔊	𝔊	PROPN
cana-1172	74	30	→	→	SYM
cana-1172	74	31	[	[	X
cana-1172	74	32	0,1	0,1	NUM
cana-1172	74	33	]	]	PUNCT
cana-1172	74	34	,	,	PUNCT
cana-1172	74	35	�	�	PROPN
cana-1172	74	36	̇	̇	PROPN
cana-1172	74	37	�	�	PROPN
cana-1172	74	38	↦	↦	PROPN
cana-1172	74	39	{	{	PUNCT
cana-1172	74	40	0.72	0.72	NUM
cana-1172	74	41	𝑖𝑓	𝑖𝑓	PRON
cana-1172	74	42	�	�	PROPN
cana-1172	74	43	̇	̇	PROPN
cana-1172	74	44	�	�	PROPN
cana-1172	74	45	=	=	SYM
cana-1172	74	46	0	0	NUM
cana-1172	75	1	0.51	0.51	NUM
cana-1172	75	2	𝑖𝑓	𝑖𝑓	PRON
cana-1172	75	3	�	�	PROPN
cana-1172	75	4	̇	̇	PROPN
cana-1172	75	5	�	�	PROPN
cana-1172	75	6	=	=	PRON
cana-1172	75	7	𝓅1̇	𝓅1̇	VERB
cana-1172	75	8	0.43	0.43	NUM
cana-1172	75	9	𝑖𝑓	𝑖𝑓	SYM
cana-1172	75	10	�	�	PROPN
cana-1172	75	11	̇	̇	PROPN
cana-1172	75	12	�	�	PROPN
cana-1172	75	13	=	=	SYM
cana-1172	75	14	𝓅2̇	𝓅2̇	PROPN
cana-1172	75	15	.	.	PUNCT
cana-1172	76	1	provided	provide	VERB
cana-1172	76	2	that	that	DET
cana-1172	76	3	𝜀	𝜀	NOUN
cana-1172	76	4	=	=	SYM
cana-1172	76	5	0.49	0.49	NUM
cana-1172	76	6	,	,	PUNCT
cana-1172	76	7	then	then	ADV
cana-1172	76	8	the	the	DET
cana-1172	76	9	𝜀-lukasz	𝜀-lukasz	NOUN
cana-1172	76	10	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	76	11	set	set	VERB
cana-1172	76	12	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	76	13	𝜀	𝜀	NOUN
cana-1172	76	14	of	of	ADP
cana-1172	76	15	𝑈	𝑈	PROPN
cana-1172	76	16	in	in	ADP
cana-1172	76	17	𝔊	𝔊	PROPN
cana-1172	76	18	is	be	AUX
cana-1172	76	19	formed	form	VERB
cana-1172	76	20	as	as	SCONJ
cana-1172	76	21	follows	follow	VERB
cana-1172	76	22	:	:	PUNCT
cana-1172	76	23	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	76	24	𝜀	𝜀	NOUN
cana-1172	76	25	:	:	PUNCT
cana-1172	76	26	𝔊	𝔊	PROPN
cana-1172	76	27	→	→	SYM
cana-1172	76	28	[	[	X
cana-1172	76	29	0,1	0,1	NUM
cana-1172	76	30	]	]	PUNCT
cana-1172	76	31	,	,	PUNCT
cana-1172	76	32	�	�	PROPN
cana-1172	76	33	̇	̇	PROPN
cana-1172	76	34	�	�	PROPN
cana-1172	76	35	↦	↦	PROPN
cana-1172	76	36	{	{	PUNCT
cana-1172	76	37	0.21	0.21	NUM
cana-1172	76	38	𝑖𝑓	𝑖𝑓	NUM
cana-1172	76	39	�	�	PROPN
cana-1172	76	40	̇	̇	NOUN
cana-1172	76	41	�	�	PROPN
cana-1172	76	42	=	=	SYM
cana-1172	76	43	0	0	NUM
cana-1172	76	44	0	0	NUM
cana-1172	76	45	𝑖𝑓	𝑖𝑓	PRON
cana-1172	76	46	�	�	PROPN
cana-1172	76	47	̇	̇	PROPN
cana-1172	76	48	�	�	PROPN
cana-1172	76	49	=	=	PUNCT
cana-1172	76	50	𝓅1̇	𝓅1̇	VERB
cana-1172	76	51	0	0	NUM
cana-1172	76	52	𝑖𝑓	𝑖𝑓	SYM
cana-1172	76	53	�	�	PROPN
cana-1172	76	54	̇	̇	PROPN
cana-1172	76	55	�	�	PROPN
cana-1172	76	56	=	=	SYM
cana-1172	76	57	𝓅2̇	𝓅2̇	PROPN
cana-1172	76	58	typically	typically	ADV
cana-1172	76	59	,	,	PUNCT
cana-1172	76	60	it	it	PRON
cana-1172	76	61	is	be	AUX
cana-1172	76	62	verified	verify	VERB
cana-1172	76	63	that	that	SCONJ
cana-1172	76	64	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	76	65	𝜀	𝜀	PROPN
cana-1172	76	66	is	be	AUX
cana-1172	76	67	an	an	DET
cana-1172	76	68	𝜀-lukasz	𝜀-lukasz	NOUN
cana-1172	76	69	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	76	70	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	PROPN
cana-1172	76	71	of	of	ADP
cana-1172	76	72	𝔊.	𝔊.	PROPN
cana-1172	76	73	but	but	CCONJ
cana-1172	76	74	𝑈	𝑈	PROPN
cana-1172	76	75	is	be	AUX
cana-1172	76	76	not	not	PART
cana-1172	76	77	a	a	DET
cana-1172	76	78	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	76	79	subalgebra	subalgebra	NOUN
cana-1172	76	80	of	of	ADP
cana-1172	76	81	𝔊	𝔊	PROPN
cana-1172	76	82	because	because	SCONJ
cana-1172	76	83	of	of	ADP
cana-1172	76	84	𝑈(0	𝑈(0	NUM
cana-1172	76	85	∗	∗	NOUN
cana-1172	76	86	𝓅1̇	𝓅1̇	NOUN
cana-1172	76	87	)	)	PUNCT
cana-1172	76	88	=	=	SYM
cana-1172	76	89	𝑈(𝓅2̇	𝑈(𝓅2̇	NOUN
cana-1172	76	90	)	)	PUNCT
cana-1172	76	91	=	=	SYM
cana-1172	76	92	0.43	0.43	NUM
cana-1172	76	93	≱	≱	PROPN
cana-1172	76	94	0.51	0.51	NUM
cana-1172	76	95	=	=	SYM
cana-1172	76	96	𝑚𝑖𝑛{𝑈(0	𝑚𝑖𝑛{𝑈(0	PROPN
cana-1172	76	97	)	)	PUNCT
cana-1172	76	98	,	,	PUNCT
cana-1172	76	99	𝑈(𝓅1̇	𝑈(𝓅1̇	NOUN
cana-1172	76	100	)	)	PUNCT
cana-1172	76	101	}	}	PUNCT
cana-1172	76	102	.	.	PUNCT
cana-1172	77	1	theorem	theorem	VERB
cana-1172	77	2	4.4	4.4	NUM
cana-1172	77	3	every	every	DET
cana-1172	77	4	lukasz	lukasz	NOUN
cana-1172	77	5	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	77	6	set	set	VERB
cana-1172	77	7	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	77	8	𝜀	𝜀	NOUN
cana-1172	77	9	of	of	ADP
cana-1172	77	10	a	a	DET
cana-1172	77	11	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	77	12	set	set	NOUN
cana-1172	77	13	𝑈	𝑈	PROPN
cana-1172	77	14	in	in	ADP
cana-1172	77	15	𝔊	𝔊	PROPN
cana-1172	77	16	is	be	AUX
cana-1172	77	17	a	a	DET
cana-1172	77	18	lukasz	lukasz	NOUN
cana-1172	77	19	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	77	20	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	PROPN
cana-1172	77	21	of	of	ADP
cana-1172	77	22	𝔊	𝔊	PROPN
cana-1172	77	23	if	if	SCONJ
cana-1172	77	24	and	and	CCONJ
cana-1172	77	25	only	only	ADV
cana-1172	77	26	if	if	SCONJ
cana-1172	77	27	it	it	PRON
cana-1172	77	28	satisfies	satisfy	VERB
cana-1172	77	29	(	(	PUNCT
cana-1172	77	30	i	i	NOUN
cana-1172	77	31	)	)	PUNCT
cana-1172	77	32	∀	∀	PUNCT
cana-1172	77	33	�	�	PROPN
cana-1172	77	34	̇	̇	VERB
cana-1172	77	35	�	�	PROPN
cana-1172	77	36	∈	∈	PROPN
cana-1172	77	37	𝔊	𝔊	PROPN
cana-1172	77	38	,	,	PUNCT
cana-1172	77	39	∀	∀	X
cana-1172	77	40	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	77	41	∈	∈	PROPN
cana-1172	77	42	(	(	PUNCT
cana-1172	77	43	0,1	0,1	NOUN
cana-1172	77	44	]	]	PUNCT
cana-1172	77	45	,	,	PUNCT
cana-1172	77	46	[	[	X
cana-1172	77	47	�	�	NOUN
cana-1172	77	48	̇	̇	PROPN
cana-1172	77	49	�	�	PROPN
cana-1172	77	50	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	77	51	]	]	PUNCT
cana-1172	77	52	∈	∈	PROPN
cana-1172	77	53	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	77	54	𝜀	𝜀	PROPN
cana-1172	77	55	⇒	⇒	NOUN
cana-1172	77	56	[	[	X
cana-1172	77	57	0	0	X
cana-1172	77	58	𝑢𝑎⁄	𝑢𝑎⁄	X
cana-1172	77	59	]	]	PUNCT
cana-1172	77	60	∈	∈	PROPN
cana-1172	77	61	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	77	62	𝜀	𝜀	PROPN
cana-1172	77	63	(	(	PUNCT
cana-1172	77	64	4.2	4.2	NUM
cana-1172	77	65	)	)	PUNCT
cana-1172	77	66	(	(	PUNCT
cana-1172	77	67	ii	ii	NOUN
cana-1172	77	68	)	)	PUNCT
cana-1172	77	69	∀	∀	PUNCT
cana-1172	77	70	�	�	PROPN
cana-1172	77	71	̇	̇	PROPN
cana-1172	77	72	�	�	PROPN
cana-1172	77	73	,	,	PUNCT
cana-1172	77	74	�	�	PROPN
cana-1172	77	75	̇	̇	VERB
cana-1172	77	76	�	�	PROPN
cana-1172	77	77	∈	∈	PROPN
cana-1172	77	78	𝔊	𝔊	PROPN
cana-1172	77	79	,	,	PUNCT
cana-1172	77	80	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	77	81	𝜀	𝜀	PROPN
cana-1172	77	82	(	(	PUNCT
cana-1172	77	83	�	�	PROPN
cana-1172	77	84	̇	̇	PROPN
cana-1172	77	85	�	�	PROPN
cana-1172	77	86	)	)	PUNCT
cana-1172	77	87	≥	≥	NOUN
cana-1172	77	88	𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝐿𝑈	NUM
cana-1172	77	89	𝜀	𝜀	X
cana-1172	77	90	(	(	PUNCT
cana-1172	77	91	�	�	PROPN
cana-1172	77	92	̇	̇	PROPN
cana-1172	77	93	�	�	PROPN
cana-1172	77	94	∗	∗	PROPN
cana-1172	77	95	�	�	PROPN
cana-1172	77	96	̇	̇	PROPN
cana-1172	77	97	�	�	PROPN
cana-1172	77	98	)	)	PUNCT
cana-1172	77	99	,	,	PUNCT
cana-1172	77	100	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	77	101	𝜀	𝜀	PROPN
cana-1172	77	102	(	(	PUNCT
cana-1172	77	103	�	�	PROPN
cana-1172	77	104	̇	̇	PROPN
cana-1172	77	105	�	�	PROPN
cana-1172	77	106	)	)	PUNCT
cana-1172	77	107	}	}	PUNCT
cana-1172	77	108	(	(	PUNCT
cana-1172	77	109	4.3	4.3	NUM
cana-1172	77	110	)	)	PUNCT
cana-1172	77	111	proof	proof	NOUN
cana-1172	77	112	for	for	ADP
cana-1172	77	113	instance	instance	NOUN
cana-1172	77	114	,	,	PUNCT
cana-1172	77	115	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	77	116	𝜀	𝜀	PROPN
cana-1172	77	117	is	be	AUX
cana-1172	77	118	a	a	DET
cana-1172	77	119	lukasz	lukasz	NOUN
cana-1172	77	120	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	77	121	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	PROPN
cana-1172	77	122	of	of	ADP
cana-1172	77	123	𝔊.	𝔊.	PROPN
cana-1172	77	124	let	let	VERB
cana-1172	77	125	�	�	SYM
cana-1172	77	126	̇	̇	VERB
cana-1172	77	127	�	�	PROPN
cana-1172	77	128	∈	∈	PROPN
cana-1172	77	129	𝔊	𝔊	PROPN
cana-1172	77	130	and	and	CCONJ
cana-1172	77	131	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	77	132	∈	∈	PROPN
cana-1172	77	133	(	(	PUNCT
cana-1172	77	134	0,1	0,1	NOUN
cana-1172	77	135	]	]	PUNCT
cana-1172	77	136	be	be	VERB
cana-1172	77	137	such	such	ADJ
cana-1172	77	138	that	that	SCONJ
cana-1172	77	139	[	[	X
cana-1172	77	140	�	�	NOUN
cana-1172	77	141	̇	̇	VERB
cana-1172	77	142	�	�	PROPN
cana-1172	77	143	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	77	144	]	]	PUNCT
cana-1172	77	145	∈	∈	PROPN
cana-1172	77	146	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	77	147	𝜀	𝜀	NOUN
cana-1172	77	148	.	.	PUNCT
cana-1172	78	1	utilising	utilise	VERB
cana-1172	78	2	(	(	PUNCT
cana-1172	78	3	3.4	3.4	NUM
cana-1172	78	4	)	)	PUNCT
cana-1172	78	5	,	,	PUNCT
cana-1172	78	6	leads	lead	VERB
cana-1172	78	7	to	to	ADP
cana-1172	78	8	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	78	9	𝜀	𝜀	PROPN
cana-1172	78	10	(	(	PUNCT
cana-1172	78	11	0	0	NUM
cana-1172	78	12	)	)	PUNCT
cana-1172	78	13	≥	≥	NOUN
cana-1172	78	14	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	78	15	𝜀	𝜀	PROPN
cana-1172	78	16	(	(	PUNCT
cana-1172	78	17	�	�	PROPN
cana-1172	78	18	̇	̇	PROPN
cana-1172	78	19	�	�	PROPN
cana-1172	78	20	)	)	PUNCT
cana-1172	78	21	≥	≥	NOUN
cana-1172	78	22	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	78	23	,	,	PUNCT
cana-1172	78	24	and	and	CCONJ
cana-1172	79	1	so	so	ADV
cana-1172	80	1	[	[	X
cana-1172	80	2	0	0	X
cana-1172	80	3	𝑢𝑎⁄	𝑢𝑎⁄	X
cana-1172	80	4	]	]	PUNCT
cana-1172	80	5	∈	∈	PROPN
cana-1172	80	6	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	80	7	𝜀	𝜀	PROPN
cana-1172	80	8	.	.	PUNCT
cana-1172	81	1	note	note	VERB
cana-1172	81	2	that	that	SCONJ
cana-1172	81	3	[	[	X
cana-1172	81	4	(	(	PUNCT
cana-1172	81	5	�	�	PROPN
cana-1172	81	6	̇	̇	PROPN
cana-1172	81	7	�	�	PROPN
cana-1172	81	8	∗	∗	PROPN
cana-1172	81	9	�	�	PROPN
cana-1172	81	10	̇	̇	PROPN
cana-1172	81	11	�	�	PROPN
cana-1172	81	12	)	)	PUNCT
cana-1172	81	13	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	81	14	𝜀	𝜀	PROPN
cana-1172	81	15	(	(	PUNCT
cana-1172	81	16	�	�	PROPN
cana-1172	81	17	̇	̇	PROPN
cana-1172	81	18	�	�	PROPN
cana-1172	81	19	∗	∗	PROPN
cana-1172	81	20	�	�	PROPN
cana-1172	81	21	̇	̇	PROPN
cana-1172	81	22	�	�	NOUN
cana-1172	81	23	)⁄	)⁄	PRON
cana-1172	81	24	]	]	PUNCT
cana-1172	81	25	∈	∈	PROPN
cana-1172	81	26	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	81	27	𝜀	𝜀	NOUN
cana-1172	81	28	,	,	PUNCT
cana-1172	81	29	[	[	X
cana-1172	81	30	�	�	NOUN
cana-1172	81	31	̇	̇	VERB
cana-1172	81	32	�	�	PROPN
cana-1172	81	33	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	81	34	𝜀	𝜀	PROPN
cana-1172	81	35	(	(	PUNCT
cana-1172	81	36	�	�	PROPN
cana-1172	81	37	̇	̇	NOUN
cana-1172	81	38	�	�	NOUN
cana-1172	81	39	)⁄	)⁄	PRON
cana-1172	81	40	]	]	PUNCT
cana-1172	81	41	∈	∈	PROPN
cana-1172	81	42	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	81	43	𝜀	𝜀	NOUN
cana-1172	81	44	for	for	ADP
cana-1172	81	45	all	all	DET
cana-1172	81	46	�	�	PROPN
cana-1172	81	47	̇	̇	PROPN
cana-1172	81	48	�	�	PROPN
cana-1172	81	49	,	,	PUNCT
cana-1172	81	50	�	�	PROPN
cana-1172	81	51	̇	̇	VERB
cana-1172	81	52	�	�	PROPN
cana-1172	81	53	∈	∈	PROPN
cana-1172	81	54	𝔊.	𝔊.	PROPN
cana-1172	81	55	from	from	ADP
cana-1172	81	56	(	(	PUNCT
cana-1172	81	57	3.5	3.5	NUM
cana-1172	81	58	)	)	PUNCT
cana-1172	81	59	,	,	PUNCT
cana-1172	81	60	it	it	PRON
cana-1172	81	61	is	be	AUX
cana-1172	81	62	evident	evident	ADJ
cana-1172	81	63	that	that	SCONJ
cana-1172	81	64	[	[	PUNCT
cana-1172	81	65	�	�	NOUN
cana-1172	81	66	̇	̇	VERB
cana-1172	81	67	�	�	PROPN
cana-1172	81	68	𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝐿𝑈	PROPN
cana-1172	81	69	𝜀	𝜀	PROPN
cana-1172	81	70	(	(	PUNCT
cana-1172	81	71	�	�	PROPN
cana-1172	81	72	̇	̇	PROPN
cana-1172	81	73	�	�	PROPN
cana-1172	81	74	∗	∗	PROPN
cana-1172	81	75	�	�	PROPN
cana-1172	81	76	̇	̇	PROPN
cana-1172	81	77	�	�	PROPN
cana-1172	81	78	)	)	PUNCT
cana-1172	81	79	,	,	PUNCT
cana-1172	81	80	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	81	81	𝜀	𝜀	PROPN
cana-1172	81	82	(	(	PUNCT
cana-1172	81	83	�	�	PROPN
cana-1172	81	84	̇	̇	PROPN
cana-1172	81	85	�	�	PROPN
cana-1172	81	86	)}⁄	)}⁄	PUNCT
cana-1172	81	87	]	]	PUNCT
cana-1172	81	88	∈	∈	PROPN
cana-1172	81	89	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	81	90	𝜀	𝜀	NOUN
cana-1172	81	91	,	,	PUNCT
cana-1172	81	92	and	and	CCONJ
cana-1172	81	93	hence	hence	ADV
cana-1172	81	94	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	81	95	𝜀	𝜀	PROPN
cana-1172	81	96	(	(	PUNCT
cana-1172	81	97	�	�	PROPN
cana-1172	81	98	̇	̇	PROPN
cana-1172	81	99	�	�	PROPN
cana-1172	81	100	)	)	PUNCT
cana-1172	81	101	≥	≥	NOUN
cana-1172	81	102	𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝐿𝑈	NUM
cana-1172	81	103	𝜀	𝜀	X
cana-1172	81	104	(	(	PUNCT
cana-1172	81	105	�	�	PROPN
cana-1172	81	106	̇	̇	PROPN
cana-1172	81	107	�	�	PROPN
cana-1172	81	108	∗	∗	PROPN
cana-1172	81	109	�	�	PROPN
cana-1172	81	110	̇	̇	PROPN
cana-1172	81	111	�	�	PROPN
cana-1172	81	112	)	)	PUNCT
cana-1172	81	113	,	,	PUNCT
cana-1172	81	114	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	81	115	𝜀	𝜀	PROPN
cana-1172	81	116	(	(	PUNCT
cana-1172	81	117	�	�	PROPN
cana-1172	81	118	̇	̇	PROPN
cana-1172	81	119	�	�	PROPN
cana-1172	81	120	)	)	PUNCT
cana-1172	81	121	}	}	PUNCT
cana-1172	81	122	for	for	ADP
cana-1172	81	123	all	all	DET
cana-1172	81	124	�	�	PROPN
cana-1172	81	125	̇	̇	PROPN
cana-1172	81	126	�	�	PROPN
cana-1172	81	127	,	,	PUNCT
cana-1172	81	128	�	�	PROPN
cana-1172	81	129	̇	̇	VERB
cana-1172	81	130	�	�	PROPN
cana-1172	81	131	∈	∈	PROPN
cana-1172	81	132	𝔊.	𝔊.	PROPN
cana-1172	81	133	conversely	conversely	ADV
cana-1172	81	134	,	,	PUNCT
cana-1172	81	135	let	let	VERB
cana-1172	81	136	us	we	PRON
cana-1172	81	137	consider	consider	VERB
cana-1172	81	138	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	81	139	𝜀	𝜀	PROPN
cana-1172	81	140	satisfies	satisfie	NOUN
cana-1172	81	141	(	(	PUNCT
cana-1172	81	142	4.2	4.2	NUM
cana-1172	81	143	)	)	PUNCT
cana-1172	81	144	and	and	CCONJ
cana-1172	81	145	(	(	PUNCT
cana-1172	81	146	4.3	4.3	NUM
cana-1172	81	147	)	)	PUNCT
cana-1172	81	148	.	.	PUNCT
cana-1172	82	1	since	since	SCONJ
cana-1172	82	2	[	[	X
cana-1172	82	3	�	�	PROPN
cana-1172	82	4	̇	̇	VERB
cana-1172	82	5	�	�	PROPN
cana-1172	82	6	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	82	7	𝜀	𝜀	PROPN
cana-1172	82	8	(	(	PUNCT
cana-1172	82	9	�	�	PROPN
cana-1172	82	10	̇	̇	NOUN
cana-1172	82	11	�	�	NOUN
cana-1172	82	12	)⁄	)⁄	PRON
cana-1172	82	13	]	]	PUNCT
cana-1172	82	14	∈	∈	PROPN
cana-1172	82	15	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	82	16	𝜀	𝜀	NOUN
cana-1172	82	17	for	for	ADP
cana-1172	82	18	all	all	DET
cana-1172	82	19	�	�	NOUN
cana-1172	82	20	̇	̇	VERB
cana-1172	82	21	�	�	PROPN
cana-1172	82	22	∈	∈	PROPN
cana-1172	82	23	𝔊	𝔊	PROPN
cana-1172	82	24	,	,	PUNCT
cana-1172	82	25	we	we	PRON
cana-1172	82	26	have	have	VERB
cana-1172	82	27	[	[	X
cana-1172	82	28	0	0	NUM
cana-1172	82	29	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	82	30	𝜀	𝜀	PROPN
cana-1172	82	31	(	(	PUNCT
cana-1172	82	32	�	�	PROPN
cana-1172	82	33	̇	̇	NOUN
cana-1172	82	34	�	�	NOUN
cana-1172	82	35	)⁄	)⁄	PRON
cana-1172	82	36	]	]	PUNCT
cana-1172	82	37	∈	∈	PROPN
cana-1172	82	38	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	82	39	𝜀	𝜀	PROPN
cana-1172	82	40	and	and	CCONJ
cana-1172	82	41	so	so	ADV
cana-1172	82	42	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	82	43	𝜀	𝜀	PROPN
cana-1172	82	44	(	(	PUNCT
cana-1172	82	45	0	0	NUM
cana-1172	82	46	)	)	PUNCT
cana-1172	82	47	≥	≥	NOUN
cana-1172	82	48	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	82	49	𝜀	𝜀	PROPN
cana-1172	82	50	(	(	PUNCT
cana-1172	82	51	�	�	PROPN
cana-1172	82	52	̇	̇	PROPN
cana-1172	82	53	�	�	PROPN
cana-1172	82	54	)	)	PUNCT
cana-1172	82	55	for	for	ADP
cana-1172	82	56	all	all	DET
cana-1172	82	57	�	�	PROPN
cana-1172	82	58	̇	̇	VERB
cana-1172	82	59	�	�	PROPN
cana-1172	82	60	∈	∈	PROPN
cana-1172	82	61	𝔊	𝔊	PROPN
cana-1172	82	62	by	by	ADP
cana-1172	82	63	(	(	PUNCT
cana-1172	82	64	4.2	4.2	NUM
cana-1172	82	65	)	)	PUNCT
cana-1172	82	66	.	.	PUNCT
cana-1172	83	1	hence	hence	ADV
cana-1172	83	2	,	,	PUNCT
cana-1172	83	3	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	83	4	𝜀	𝜀	PROPN
cana-1172	83	5	(	(	PUNCT
cana-1172	83	6	0	0	NUM
cana-1172	83	7	)	)	PUNCT
cana-1172	83	8	is	be	AUX
cana-1172	83	9	an	an	DET
cana-1172	83	10	upper	upper	ADJ
cana-1172	83	11	bound	bind	VERB
cana-1172	83	12	of	of	ADP
cana-1172	83	13	{	{	PUNCT
cana-1172	83	14	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	83	15	𝜀	𝜀	PROPN
cana-1172	83	16	(	(	PUNCT
cana-1172	83	17	�	�	PROPN
cana-1172	83	18	̇	̇	PROPN
cana-1172	83	19	�	�	PROPN
cana-1172	83	20	)|	)|	PROPN
cana-1172	83	21	�	�	PROPN
cana-1172	83	22	̇	̇	PROPN
cana-1172	83	23	�	�	PROPN
cana-1172	83	24	∈	∈	PROPN
cana-1172	83	25	𝔊	𝔊	PROPN
cana-1172	83	26	}	}	PUNCT
cana-1172	83	27	.	.	PUNCT
cana-1172	84	1	also	also	ADV
cana-1172	84	2	let	let	VERB
cana-1172	84	3	�	�	SYM
cana-1172	84	4	̇	̇	PROPN
cana-1172	84	5	�	�	PROPN
cana-1172	84	6	,	,	PUNCT
cana-1172	84	7	�	�	PROPN
cana-1172	84	8	̇	̇	VERB
cana-1172	84	9	�	�	PROPN
cana-1172	84	10	∈	∈	PROPN
cana-1172	84	11	𝔊	𝔊	PROPN
cana-1172	84	12	and	and	CCONJ
cana-1172	84	13	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	84	14	,	,	PUNCT
cana-1172	84	15	𝑢𝑏	𝑢𝑏	ADP
cana-1172	84	16	∈	∈	PROPN
cana-1172	84	17	(	(	PUNCT
cana-1172	84	18	0,1	0,1	NOUN
cana-1172	84	19	]	]	PUNCT
cana-1172	84	20	be	be	VERB
cana-1172	84	21	such	such	ADJ
cana-1172	84	22	that	that	SCONJ
cana-1172	84	23	[	[	X
cana-1172	84	24	(	(	PUNCT
cana-1172	84	25	�	�	PROPN
cana-1172	84	26	̇	̇	PROPN
cana-1172	84	27	�	�	PROPN
cana-1172	84	28	∗	∗	PROPN
cana-1172	84	29	�	�	PROPN
cana-1172	84	30	̇	̇	PROPN
cana-1172	84	31	�	�	PROPN
cana-1172	84	32	)	)	PUNCT
cana-1172	84	33	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	84	34	]	]	PUNCT
cana-1172	84	35	∈	∈	PROPN
cana-1172	84	36	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	84	37	𝜀	𝜀	NOUN
cana-1172	84	38	,	,	PUNCT
cana-1172	84	39	[	[	X
cana-1172	84	40	�	�	NOUN
cana-1172	84	41	̇	̇	PROPN
cana-1172	84	42	�	�	PROPN
cana-1172	84	43	𝑢𝑏⁄	𝑢𝑏⁄	NOUN
cana-1172	84	44	]	]	PUNCT
cana-1172	84	45	∈	∈	PROPN
cana-1172	84	46	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	84	47	𝜀	𝜀	PROPN
cana-1172	84	48	.	.	PUNCT
cana-1172	85	1	then	then	ADV
cana-1172	85	2	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	85	3	𝜀	𝜀	PROPN
cana-1172	85	4	(	(	PUNCT
cana-1172	85	5	�	�	PROPN
cana-1172	85	6	̇	̇	PROPN
cana-1172	85	7	�	�	PROPN
cana-1172	85	8	∗	∗	PROPN
cana-1172	85	9	�	�	PROPN
cana-1172	85	10	̇	̇	PROPN
cana-1172	85	11	�	�	PROPN
cana-1172	85	12	)	)	PUNCT
cana-1172	85	13	≥	≥	NOUN
cana-1172	85	14	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	85	15	and	and	CCONJ
cana-1172	85	16	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	85	17	𝜀	𝜀	PROPN
cana-1172	85	18	(	(	PUNCT
cana-1172	85	19	�	�	PROPN
cana-1172	85	20	̇	̇	PROPN
cana-1172	85	21	�	�	PROPN
cana-1172	85	22	)	)	PUNCT
cana-1172	85	23	≥	≥	NOUN
cana-1172	85	24	𝑢𝑏	𝑢𝑏	PROPN
cana-1172	85	25	,	,	PUNCT
cana-1172	85	26	which	which	PRON
cana-1172	85	27	imply	imply	VERB
cana-1172	85	28	from	from	ADP
cana-1172	85	29	(	(	PUNCT
cana-1172	85	30	4.3	4.3	NUM
cana-1172	85	31	)	)	PUNCT
cana-1172	85	32	that	that	PRON
cana-1172	85	33	communications	communication	NOUN
cana-1172	85	34	on	on	ADP
cana-1172	85	35	applied	apply	VERB
cana-1172	85	36	nonlinear	nonlinear	ADJ
cana-1172	85	37	analysis	analysis	NOUN
cana-1172	85	38	issn	issn	NOUN
cana-1172	85	39	:	:	PUNCT
cana-1172	85	40	1074	1074	NUM
cana-1172	85	41	-	-	PUNCT
cana-1172	85	42	133x	133x	NUM
cana-1172	85	43	vol	vol	NOUN
cana-1172	85	44	31	31	NUM
cana-1172	85	45	no	no	NOUN
cana-1172	85	46	.	.	PUNCT
cana-1172	86	1	6s	6s	NUM
cana-1172	86	2	(	(	PUNCT
cana-1172	86	3	2024	2024	NUM
cana-1172	86	4	)	)	PUNCT
cana-1172	86	5	138	138	NUM
cana-1172	86	6	https://internationalpubls.com	https://internationalpubls.com	NUM
cana-1172	86	7	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	86	8	𝜀	𝜀	PROPN
cana-1172	86	9	(	(	PUNCT
cana-1172	86	10	�	�	PROPN
cana-1172	86	11	̇	̇	PROPN
cana-1172	86	12	�	�	PROPN
cana-1172	86	13	)	)	PUNCT
cana-1172	86	14	≥	≥	NOUN
cana-1172	86	15	𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝐿𝑈	NUM
cana-1172	86	16	𝜀	𝜀	X
cana-1172	86	17	(	(	PUNCT
cana-1172	86	18	�	�	PROPN
cana-1172	86	19	̇	̇	PROPN
cana-1172	86	20	�	�	PROPN
cana-1172	86	21	∗	∗	PROPN
cana-1172	86	22	�	�	PROPN
cana-1172	86	23	̇	̇	PROPN
cana-1172	86	24	�	�	PROPN
cana-1172	86	25	)	)	PUNCT
cana-1172	86	26	,	,	PUNCT
cana-1172	86	27	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	86	28	𝜀	𝜀	PROPN
cana-1172	86	29	(	(	PUNCT
cana-1172	86	30	�	�	PROPN
cana-1172	86	31	̇	̇	PROPN
cana-1172	86	32	�	�	PROPN
cana-1172	86	33	)	)	PUNCT
cana-1172	86	34	}	}	PUNCT
cana-1172	86	35	≥	≥	X
cana-1172	86	36	𝑚𝑖𝑛{𝑢𝑎	𝑚𝑖𝑛{𝑢𝑎	VERB
cana-1172	86	37	,	,	PUNCT
cana-1172	86	38	𝑢𝑏	𝑢𝑏	ADP
cana-1172	86	39	}	}	PUNCT
cana-1172	86	40	.	.	PUNCT
cana-1172	87	1	thus	thus	ADV
cana-1172	87	2	[	[	X
cana-1172	87	3	�	�	NOUN
cana-1172	87	4	̇	̇	PROPN
cana-1172	87	5	�	�	PROPN
cana-1172	87	6	𝑚𝑖𝑛{𝑢𝑎	𝑚𝑖𝑛{𝑢𝑎	VERB
cana-1172	87	7	,	,	PUNCT
cana-1172	87	8	𝑢𝑏}⁄	𝑢𝑏}⁄	PRON
cana-1172	87	9	]	]	PUNCT
cana-1172	87	10	∈	∈	PROPN
cana-1172	87	11	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	87	12	𝜀	𝜀	PROPN
cana-1172	87	13	.	.	PUNCT
cana-1172	88	1	therefore	therefore	ADV
cana-1172	88	2	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	88	3	𝜀	𝜀	PROPN
cana-1172	88	4	is	be	AUX
cana-1172	88	5	a	a	DET
cana-1172	88	6	lukasz	lukasz	NOUN
cana-1172	88	7	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	88	8	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	PROPN
cana-1172	88	9	of	of	ADP
cana-1172	88	10	𝔊.	𝔊.	PROPN
cana-1172	88	11	5	5	NUM
cana-1172	88	12	.	.	PUNCT
cana-1172	88	13	discussion	discussion	NOUN
cana-1172	88	14	remark	remark	VERB
cana-1172	88	15	5.1	5.1	NUM
cana-1172	88	16	prove	prove	VERB
cana-1172	88	17	that	that	SCONJ
cana-1172	88	18	𝜀	𝜀	ADP
cana-1172	88	19	-lukasz	-lukasz	ADJ
cana-1172	88	20	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	88	21	set	set	VERB
cana-1172	88	22	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	88	23	𝜀	𝜀	PROPN
cana-1172	88	24	satisfies	satisfie	NOUN
cana-1172	88	25	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	88	26	𝜀	𝜀	PROPN
cana-1172	88	27	(	(	PUNCT
cana-1172	88	28	0	0	NUM
cana-1172	88	29	)	)	PUNCT
cana-1172	88	30	≥	≥	NOUN
cana-1172	88	31	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	88	32	𝜀	𝜀	PROPN
cana-1172	88	33	(	(	PUNCT
cana-1172	88	34	�	�	PROPN
cana-1172	88	35	̇	̇	PROPN
cana-1172	88	36	�	�	PROPN
cana-1172	88	37	)	)	PUNCT
cana-1172	88	38	,	,	PUNCT
cana-1172	88	39	∀	∀	X
cana-1172	88	40	�	�	NOUN
cana-1172	88	41	̇	̇	PROPN
cana-1172	88	42	�	�	PROPN
cana-1172	88	43	∈	∈	PROPN
cana-1172	88	44	𝔊	𝔊	PROPN
cana-1172	88	45	,	,	PUNCT
cana-1172	88	46	if	if	SCONJ
cana-1172	88	47	𝑈	𝑈	PROPN
cana-1172	88	48	is	be	AUX
cana-1172	88	49	a	a	DET
cana-1172	88	50	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	88	51	sub	sub	NOUN
cana-1172	88	52	algebra	algebra	NOUN
cana-1172	88	53	of	of	ADP
cana-1172	88	54	𝔊	𝔊	PROPN
cana-1172	88	55	,	,	PUNCT
cana-1172	88	56	then	then	ADV
cana-1172	88	57	it	it	PRON
cana-1172	88	58	proof	proof	NOUN
cana-1172	88	59	let	let	VERB
cana-1172	88	60	𝑈	𝑈	PROPN
cana-1172	88	61	be	be	AUX
cana-1172	88	62	a	a	DET
cana-1172	88	63	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	88	64	subalgebra	subalgebra	NOUN
cana-1172	88	65	of	of	ADP
cana-1172	88	66	𝔊.	𝔊.	PROPN
cana-1172	88	67	then	then	ADV
cana-1172	88	68	,	,	PUNCT
cana-1172	88	69	𝑈(0	𝑈(0	NUM
cana-1172	88	70	)	)	PUNCT
cana-1172	88	71	=	=	PRON
cana-1172	88	72	𝑈(	𝑈(	PROPN
cana-1172	88	73	�	�	PROPN
cana-1172	88	74	̇	̇	PROPN
cana-1172	88	75	�	�	PROPN
cana-1172	88	76	∗	∗	PROPN
cana-1172	88	77	�	�	PROPN
cana-1172	88	78	̇	̇	PROPN
cana-1172	88	79	�	�	PROPN
cana-1172	88	80	)	)	PUNCT
cana-1172	88	81	≥	≥	NOUN
cana-1172	88	82	𝑚𝑖𝑛{𝑈(	𝑚𝑖𝑛{𝑈(	PROPN
cana-1172	88	83	�	�	PROPN
cana-1172	88	84	̇	̇	PROPN
cana-1172	88	85	�	�	PROPN
cana-1172	88	86	)	)	PUNCT
cana-1172	88	87	,	,	PUNCT
cana-1172	88	88	𝑈(	𝑈(	PROPN
cana-1172	88	89	�	�	PROPN
cana-1172	88	90	̇	̇	PROPN
cana-1172	88	91	�	�	PROPN
cana-1172	88	92	)	)	PUNCT
cana-1172	88	93	}	}	PUNCT
cana-1172	88	94	=	=	SYM
cana-1172	88	95	𝑈(	𝑈(	PROPN
cana-1172	88	96	�	�	PROPN
cana-1172	88	97	̇	̇	PROPN
cana-1172	88	98	�	�	PROPN
cana-1172	88	99	)	)	PUNCT
cana-1172	88	100	.	.	PUNCT
cana-1172	89	1	therefore	therefore	ADV
cana-1172	89	2	𝑈(0	𝑈(0	NUM
cana-1172	89	3	)	)	PUNCT
cana-1172	89	4	≥	≥	NOUN
cana-1172	89	5	𝑈(	𝑈(	PROPN
cana-1172	89	6	�	�	PROPN
cana-1172	89	7	̇	̇	PROPN
cana-1172	89	8	�	�	PROPN
cana-1172	89	9	)	)	PUNCT
cana-1172	89	10	for	for	ADP
cana-1172	89	11	all	all	DET
cana-1172	89	12	�	�	PROPN
cana-1172	89	13	̇	̇	VERB
cana-1172	89	14	�	�	PROPN
cana-1172	89	15	∈	∈	PROPN
cana-1172	89	16	𝔊.	𝔊.	PROPN
cana-1172	89	17	from	from	ADP
cana-1172	89	18	(	(	PUNCT
cana-1172	89	19	3.2	3.2	NUM
cana-1172	89	20	)	)	PUNCT
cana-1172	89	21	,	,	PUNCT
cana-1172	89	22	it	it	PRON
cana-1172	89	23	is	be	AUX
cana-1172	89	24	evident	evident	ADJ
cana-1172	89	25	that	that	SCONJ
cana-1172	89	26	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	89	27	𝜀	𝜀	PROPN
cana-1172	89	28	(	(	PUNCT
cana-1172	89	29	0	0	NUM
cana-1172	89	30	)	)	PUNCT
cana-1172	89	31	≥	≥	NOUN
cana-1172	89	32	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	89	33	𝜀	𝜀	PROPN
cana-1172	89	34	(	(	PUNCT
cana-1172	89	35	�	�	PROPN
cana-1172	89	36	̇	̇	PROPN
cana-1172	89	37	�	�	PROPN
cana-1172	89	38	)	)	PUNCT
cana-1172	89	39	for	for	ADP
cana-1172	89	40	all	all	DET
cana-1172	89	41	�	�	PROPN
cana-1172	89	42	̇	̇	VERB
cana-1172	89	43	�	�	PROPN
cana-1172	89	44	∈	∈	PROPN
cana-1172	89	45	𝔊.	𝔊.	PROPN
cana-1172	89	46	remark	remark	VERB
cana-1172	89	47	5.2	5.2	NUM
cana-1172	89	48	every	every	DET
cana-1172	89	49	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	89	50	subalgebra	subalgebra	NOUN
cana-1172	89	51	𝑈	𝑈	PROPN
cana-1172	89	52	of	of	ADP
cana-1172	89	53	𝔊	𝔊	PROPN
cana-1172	89	54	is	be	AUX
cana-1172	89	55	said	say	VERB
cana-1172	89	56	to	to	PART
cana-1172	89	57	be	be	AUX
cana-1172	89	58	𝜀-lukasz	𝜀-lukasz	NOUN
cana-1172	89	59	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	89	60	set	set	VERB
cana-1172	89	61	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	89	62	𝜀	𝜀	NOUN
cana-1172	89	63	if	if	SCONJ
cana-1172	89	64	[	[	X
cana-1172	89	65	∀	∀	X
cana-1172	89	66	�	�	NOUN
cana-1172	89	67	̇	̇	PROPN
cana-1172	89	68	�	�	PROPN
cana-1172	89	69	,	,	PUNCT
cana-1172	89	70	�	�	PROPN
cana-1172	89	71	̇	̇	VERB
cana-1172	89	72	�	�	PROPN
cana-1172	89	73	∈	∈	PROPN
cana-1172	89	74	𝔊	𝔊	PROPN
cana-1172	89	75	]	]	X
cana-1172	89	76	[	[	X
cana-1172	89	77	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	89	78	𝜀	𝜀	PROPN
cana-1172	89	79	(	(	PUNCT
cana-1172	89	80	�	�	PROPN
cana-1172	89	81	̇	̇	PROPN
cana-1172	89	82	�	�	PROPN
cana-1172	89	83	)	)	PUNCT
cana-1172	89	84	=	=	SYM
cana-1172	89	85	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	89	86	𝜀	𝜀	PROPN
cana-1172	89	87	(	(	PUNCT
cana-1172	89	88	0	0	NUM
cana-1172	89	89	)	)	PUNCT
cana-1172	89	90	⇔	⇔	PROPN
cana-1172	89	91	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	89	92	𝜀	𝜀	PROPN
cana-1172	89	93	(	(	PUNCT
cana-1172	89	94	�	�	PROPN
cana-1172	89	95	̇	̇	PROPN
cana-1172	89	96	�	�	PROPN
cana-1172	89	97	∗	∗	PROPN
cana-1172	89	98	�	�	PROPN
cana-1172	89	99	̇	̇	PROPN
cana-1172	89	100	�	�	PROPN
cana-1172	89	101	)	)	PUNCT
cana-1172	89	102	≥	≥	NOUN
cana-1172	89	103	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	89	104	𝜀	𝜀	PROPN
cana-1172	89	105	(	(	PUNCT
cana-1172	89	106	�	�	PROPN
cana-1172	89	107	̇	̇	PROPN
cana-1172	89	108	�	�	PROPN
cana-1172	89	109	)	)	PUNCT
cana-1172	89	110	]	]	PUNCT
cana-1172	89	111	.	.	PUNCT
cana-1172	90	1	proof	proof	NOUN
cana-1172	90	2	let	let	VERB
cana-1172	90	3	𝑈	𝑈	PROPN
cana-1172	90	4	be	be	AUX
cana-1172	90	5	a	a	DET
cana-1172	90	6	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	90	7	subalgebra	subalgebra	NOUN
cana-1172	90	8	of	of	ADP
cana-1172	90	9	𝔊.	𝔊.	PROPN
cana-1172	90	10	for	for	ADP
cana-1172	90	11	instance	instance	NOUN
cana-1172	90	12	,	,	PUNCT
cana-1172	90	13	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	90	14	𝜀	𝜀	PROPN
cana-1172	90	15	(	(	PUNCT
cana-1172	90	16	�	�	PROPN
cana-1172	90	17	̇	̇	PROPN
cana-1172	90	18	�	�	PROPN
cana-1172	90	19	)	)	PUNCT
cana-1172	90	20	=	=	SYM
cana-1172	90	21	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	90	22	𝜀	𝜀	PROPN
cana-1172	90	23	(	(	PUNCT
cana-1172	90	24	0	0	NUM
cana-1172	90	25	)	)	PUNCT
cana-1172	90	26	for	for	ADP
cana-1172	90	27	all	all	DET
cana-1172	90	28	�	�	PROPN
cana-1172	90	29	̇	̇	VERB
cana-1172	90	30	�	�	PROPN
cana-1172	90	31	∈	∈	PROPN
cana-1172	90	32	𝔊.	𝔊.	PROPN
cana-1172	90	33	then	then	ADV
cana-1172	90	34	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	90	35	𝜀	𝜀	PROPN
cana-1172	90	36	(	(	PUNCT
cana-1172	90	37	�	�	PROPN
cana-1172	90	38	̇	̇	PROPN
cana-1172	90	39	�	�	PROPN
cana-1172	90	40	∗	∗	PROPN
cana-1172	90	41	�	�	PROPN
cana-1172	90	42	̇	̇	PROPN
cana-1172	90	43	�	�	PROPN
cana-1172	90	44	)	)	PUNCT
cana-1172	90	45	≥	≥	NOUN
cana-1172	90	46	𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝐿𝑈	NUM
cana-1172	90	47	𝜀	𝜀	X
cana-1172	90	48	(	(	PUNCT
cana-1172	90	49	�	�	PROPN
cana-1172	90	50	̇	̇	PROPN
cana-1172	90	51	�	�	PROPN
cana-1172	90	52	)	)	PUNCT
cana-1172	90	53	,	,	PUNCT
cana-1172	90	54	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	90	55	𝜀	𝜀	PROPN
cana-1172	90	56	(	(	PUNCT
cana-1172	90	57	�	�	PROPN
cana-1172	90	58	̇	̇	PROPN
cana-1172	90	59	�	�	PROPN
cana-1172	90	60	)	)	PUNCT
cana-1172	90	61	}	}	PUNCT
cana-1172	90	62	=	=	PUNCT
cana-1172	90	63	𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝐿𝑈	NUM
cana-1172	90	64	𝜀	𝜀	X
cana-1172	90	65	(	(	PUNCT
cana-1172	90	66	0	0	NUM
cana-1172	90	67	)	)	PUNCT
cana-1172	90	68	,	,	PUNCT
cana-1172	90	69	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	90	70	𝜀	𝜀	PROPN
cana-1172	90	71	(	(	PUNCT
cana-1172	90	72	�	�	PROPN
cana-1172	90	73	̇	̇	PROPN
cana-1172	90	74	�	�	PROPN
cana-1172	90	75	)	)	PUNCT
cana-1172	90	76	}	}	PUNCT
cana-1172	90	77	=	=	SYM
cana-1172	90	78	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	90	79	𝜀	𝜀	PROPN
cana-1172	90	80	(	(	PUNCT
cana-1172	90	81	�	�	PROPN
cana-1172	90	82	̇	̇	PROPN
cana-1172	90	83	�	�	PROPN
cana-1172	90	84	)	)	PUNCT
cana-1172	90	85	.	.	PUNCT
cana-1172	91	1	combining	combine	VERB
cana-1172	91	2	the	the	DET
cana-1172	91	3	results	result	NOUN
cana-1172	91	4	of	of	ADP
cana-1172	91	5	theorem	theorem	ADJ
cana-1172	91	6	4.2	4.2	NUM
cana-1172	91	7	and	and	CCONJ
cana-1172	91	8	remark	remark	NOUN
cana-1172	91	9	3.2	3.2	NUM
cana-1172	91	10	leads	lead	VERB
cana-1172	91	11	to	to	ADP
cana-1172	91	12	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	91	13	𝜀	𝜀	PROPN
cana-1172	91	14	(	(	PUNCT
cana-1172	91	15	�	�	PROPN
cana-1172	91	16	̇	̇	PROPN
cana-1172	91	17	�	�	PROPN
cana-1172	91	18	∗	∗	PROPN
cana-1172	91	19	�	�	PROPN
cana-1172	91	20	̇	̇	PROPN
cana-1172	91	21	�	�	PROPN
cana-1172	91	22	)	)	PUNCT
cana-1172	91	23	≥	≥	NOUN
cana-1172	91	24	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	91	25	𝜀	𝜀	PROPN
cana-1172	91	26	(	(	PUNCT
cana-1172	91	27	�	�	PROPN
cana-1172	91	28	̇	̇	PROPN
cana-1172	91	29	�	�	PROPN
cana-1172	91	30	)	)	PUNCT
cana-1172	91	31	for	for	ADP
cana-1172	91	32	all	all	DET
cana-1172	91	33	�	�	PROPN
cana-1172	91	34	̇	̇	PROPN
cana-1172	91	35	�	�	PROPN
cana-1172	91	36	,	,	PUNCT
cana-1172	91	37	�	�	PROPN
cana-1172	91	38	̇	̇	VERB
cana-1172	91	39	�	�	PROPN
cana-1172	91	40	∈	∈	PROPN
cana-1172	91	41	𝔊.	𝔊.	PROPN
cana-1172	91	42	conversely	conversely	ADV
cana-1172	91	43	,	,	PUNCT
cana-1172	91	44	suppose	suppose	VERB
cana-1172	91	45	that	that	SCONJ
cana-1172	91	46	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	91	47	𝜀	𝜀	PROPN
cana-1172	91	48	(	(	PUNCT
cana-1172	91	49	�	�	PROPN
cana-1172	91	50	̇	̇	PROPN
cana-1172	91	51	�	�	PROPN
cana-1172	91	52	∗	∗	PROPN
cana-1172	91	53	�	�	PROPN
cana-1172	91	54	̇	̇	PROPN
cana-1172	91	55	�	�	PROPN
cana-1172	91	56	)	)	PUNCT
cana-1172	91	57	≥	≥	NOUN
cana-1172	91	58	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	91	59	𝜀	𝜀	PROPN
cana-1172	91	60	(	(	PUNCT
cana-1172	91	61	�	�	PROPN
cana-1172	91	62	̇	̇	PROPN
cana-1172	91	63	�	�	PROPN
cana-1172	91	64	)	)	PUNCT
cana-1172	91	65	for	for	ADP
cana-1172	91	66	all	all	DET
cana-1172	91	67	�	�	PROPN
cana-1172	91	68	̇	̇	PROPN
cana-1172	91	69	�	�	PROPN
cana-1172	91	70	,	,	PUNCT
cana-1172	91	71	�	�	PROPN
cana-1172	91	72	̇	̇	VERB
cana-1172	91	73	�	�	PROPN
cana-1172	91	74	∈	∈	PROPN
cana-1172	91	75	𝔊.	𝔊.	PROPN
cana-1172	91	76	utilising	utilise	VERB
cana-1172	91	77	(	(	PUNCT
cana-1172	91	78	𝐵𝑀1	𝐵𝑀1	PROPN
cana-1172	91	79	)	)	PUNCT
cana-1172	91	80	leads	lead	VERB
cana-1172	91	81	to	to	ADP
cana-1172	91	82	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	91	83	𝜀	𝜀	PROPN
cana-1172	91	84	(	(	PUNCT
cana-1172	91	85	�	�	PROPN
cana-1172	91	86	̇	̇	PROPN
cana-1172	91	87	�	�	PROPN
cana-1172	91	88	)	)	PUNCT
cana-1172	91	89	=	=	SYM
cana-1172	91	90	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	91	91	𝜀	𝜀	PROPN
cana-1172	91	92	(	(	PUNCT
cana-1172	91	93	�	�	PROPN
cana-1172	91	94	̇	̇	PROPN
cana-1172	91	95	�	�	PROPN
cana-1172	91	96	∗	∗	NOUN
cana-1172	91	97	0	0	NUM
cana-1172	91	98	)	)	PUNCT
cana-1172	91	99	≥	≥	NOUN
cana-1172	91	100	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	91	101	𝜀	𝜀	PROPN
cana-1172	91	102	(	(	PUNCT
cana-1172	91	103	0	0	NUM
cana-1172	91	104	)	)	PUNCT
cana-1172	91	105	.	.	PUNCT
cana-1172	92	1	combining	combine	VERB
cana-1172	92	2	the	the	DET
cana-1172	92	3	results	result	NOUN
cana-1172	92	4	of	of	ADP
cana-1172	92	5	above	above	ADP
cana-1172	92	6	inequality	inequality	NOUN
cana-1172	92	7	and	and	CCONJ
cana-1172	92	8	remark	remark	VERB
cana-1172	92	9	5.1	5.1	NUM
cana-1172	92	10	leads	lead	NOUN
cana-1172	92	11	to	to	ADP
cana-1172	92	12	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	92	13	𝜀	𝜀	PROPN
cana-1172	92	14	(	(	PUNCT
cana-1172	92	15	�	�	PROPN
cana-1172	92	16	̇	̇	PROPN
cana-1172	92	17	�	�	PROPN
cana-1172	92	18	)	)	PUNCT
cana-1172	92	19	=	=	SYM
cana-1172	92	20	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	92	21	𝜀	𝜀	PROPN
cana-1172	92	22	(	(	PUNCT
cana-1172	92	23	0	0	NUM
cana-1172	92	24	)	)	PUNCT
cana-1172	92	25	for	for	SCONJ
cana-1172	92	26	all	all	DET
cana-1172	92	27	�	�	PROPN
cana-1172	92	28	̇	̇	VERB
cana-1172	92	29	�	�	PROPN
cana-1172	92	30	∈	∈	PROPN
cana-1172	92	31	𝔊.	𝔊.	PROPN
cana-1172	92	32	remark	remark	VERB
cana-1172	92	33	5.3	5.3	NUM
cana-1172	92	34	every	every	DET
cana-1172	92	35	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	92	36	subalgebra	subalgebra	NOUN
cana-1172	92	37	𝑈	𝑈	PROPN
cana-1172	92	38	of	of	ADP
cana-1172	92	39	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	PROPN
cana-1172	92	40	𝔊	𝔊	PROPN
cana-1172	92	41	is	be	AUX
cana-1172	92	42	said	say	VERB
cana-1172	92	43	to	to	PART
cana-1172	92	44	be	be	AUX
cana-1172	92	45	an	an	DET
cana-1172	92	46	𝜀-lukasz	𝜀-lukasz	NOUN
cana-1172	92	47	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	NOUN
cana-1172	92	48	set	set	VERB
cana-1172	92	49	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	92	50	𝜀	𝜀	PROPN
cana-1172	92	51	if	if	SCONJ
cana-1172	92	52	communications	communication	NOUN
cana-1172	92	53	on	on	ADP
cana-1172	92	54	applied	apply	VERB
cana-1172	92	55	nonlinear	nonlinear	ADJ
cana-1172	92	56	analysis	analysis	NOUN
cana-1172	92	57	issn	issn	NOUN
cana-1172	92	58	:	:	PUNCT
cana-1172	92	59	1074	1074	NUM
cana-1172	92	60	-	-	PUNCT
cana-1172	92	61	133x	133x	NUM
cana-1172	92	62	vol	vol	NOUN
cana-1172	92	63	31	31	NUM
cana-1172	92	64	no	no	NOUN
cana-1172	92	65	.	.	PUNCT
cana-1172	93	1	6s	6s	NUM
cana-1172	93	2	(	(	PUNCT
cana-1172	93	3	2024	2024	NUM
cana-1172	93	4	)	)	PUNCT
cana-1172	93	5	139	139	NUM
cana-1172	93	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1172	94	1	[	[	X
cana-1172	94	2	∀	∀	X
cana-1172	94	3	�	�	NOUN
cana-1172	94	4	̇	̇	VERB
cana-1172	94	5	�	�	PROPN
cana-1172	94	6	∈	∈	PROPN
cana-1172	94	7	𝔊	𝔊	PROPN
cana-1172	94	8	]	]	X
cana-1172	94	9	[	[	X
cana-1172	94	10	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	94	11	𝜀	𝜀	PROPN
cana-1172	94	12	(	(	PUNCT
cana-1172	94	13	0	0	NUM
cana-1172	94	14	∗	∗	PROPN
cana-1172	94	15	�	�	PROPN
cana-1172	94	16	̇	̇	PROPN
cana-1172	94	17	�	�	PROPN
cana-1172	94	18	)	)	PUNCT
cana-1172	94	19	≥	≥	NOUN
cana-1172	94	20	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	94	21	𝜀	𝜀	PROPN
cana-1172	94	22	(	(	PUNCT
cana-1172	94	23	�	�	PROPN
cana-1172	94	24	̇	̇	PROPN
cana-1172	94	25	�	�	PROPN
cana-1172	94	26	)	)	PUNCT
cana-1172	94	27	]	]	PUNCT
cana-1172	94	28	.	.	PUNCT
cana-1172	95	1	proof	proof	NOUN
cana-1172	95	2	let	let	VERB
cana-1172	95	3	𝑈	𝑈	PROPN
cana-1172	95	4	be	be	AUX
cana-1172	95	5	a	a	DET
cana-1172	95	6	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	95	7	subalgebra	subalgebra	NOUN
cana-1172	95	8	of	of	ADP
cana-1172	95	9	𝐵𝑀-algebra	𝐵𝑀-algebra	PROPN
cana-1172	95	10	𝔊.	𝔊.	PROPN
cana-1172	95	11	then	then	ADV
cana-1172	95	12	,	,	PUNCT
cana-1172	95	13	𝑈(0	𝑈(0	NUM
cana-1172	95	14	∗	∗	X
cana-1172	95	15	�	�	PROPN
cana-1172	95	16	̇	̇	PROPN
cana-1172	95	17	�	�	PROPN
cana-1172	95	18	)	)	PUNCT
cana-1172	95	19	≥	≥	NOUN
cana-1172	95	20	𝑚𝑖𝑛{𝑈(0	𝑚𝑖𝑛{𝑈(0	NUM
cana-1172	95	21	)	)	PUNCT
cana-1172	95	22	,	,	PUNCT
cana-1172	95	23	𝑈(	𝑈(	PROPN
cana-1172	95	24	�	�	PROPN
cana-1172	95	25	̇	̇	PROPN
cana-1172	95	26	�	�	PROPN
cana-1172	95	27	)	)	PUNCT
cana-1172	95	28	}	}	PUNCT
cana-1172	95	29	=	=	SYM
cana-1172	95	30	𝑈(	𝑈(	PROPN
cana-1172	95	31	�	�	PROPN
cana-1172	95	32	̇	̇	PROPN
cana-1172	95	33	�	�	PROPN
cana-1172	95	34	)	)	PUNCT
cana-1172	95	35	for	for	ADP
cana-1172	95	36	all	all	DET
cana-1172	95	37	�	�	PROPN
cana-1172	95	38	̇	̇	VERB
cana-1172	95	39	�	�	PROPN
cana-1172	95	40	∈	∈	PROPN
cana-1172	95	41	𝔊.	𝔊.	PROPN
cana-1172	95	42	from	from	ADP
cana-1172	95	43	(	(	PUNCT
cana-1172	95	44	3.2	3.2	NUM
cana-1172	95	45	)	)	PUNCT
cana-1172	95	46	,	,	PUNCT
cana-1172	95	47	it	it	PRON
cana-1172	95	48	is	be	AUX
cana-1172	95	49	evident	evident	ADJ
cana-1172	95	50	that	that	SCONJ
cana-1172	95	51	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	95	52	𝜀	𝜀	PROPN
cana-1172	95	53	(	(	PUNCT
cana-1172	95	54	0	0	NUM
cana-1172	95	55	∗	∗	PROPN
cana-1172	95	56	�	�	PROPN
cana-1172	95	57	̇	̇	PROPN
cana-1172	95	58	�	�	PROPN
cana-1172	95	59	)	)	PUNCT
cana-1172	95	60	≥	≥	NOUN
cana-1172	95	61	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	95	62	𝜀	𝜀	PROPN
cana-1172	95	63	(	(	PUNCT
cana-1172	95	64	�	�	PROPN
cana-1172	95	65	̇	̇	PROPN
cana-1172	95	66	�	�	PROPN
cana-1172	95	67	)	)	PUNCT
cana-1172	95	68	for	for	ADP
cana-1172	95	69	all	all	DET
cana-1172	95	70	�	�	PROPN
cana-1172	95	71	̇	̇	VERB
cana-1172	95	72	�	�	PROPN
cana-1172	95	73	∈	∈	PROPN
cana-1172	95	74	𝔊.	𝔊.	PROPN
cana-1172	95	75	remark	remark	NOUN
cana-1172	95	76	5.4	5.4	NUM
cana-1172	95	77	if	if	SCONJ
cana-1172	95	78	𝑈	𝑈	PROPN
cana-1172	95	79	is	be	AUX
cana-1172	95	80	a	a	DET
cana-1172	95	81	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	95	82	sub	sub	NOUN
cana-1172	95	83	algebra	algebra	NOUN
cana-1172	95	84	of	of	ADP
cana-1172	95	85	𝔊	𝔊	PROPN
cana-1172	95	86	,	,	PUNCT
cana-1172	95	87	then	then	ADV
cana-1172	95	88	prove	prove	VERB
cana-1172	95	89	that	that	SCONJ
cana-1172	95	90	𝜀-lukasz	𝜀-lukasz	NOUN
cana-1172	95	91	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	NOUN
cana-1172	95	92	set	set	VERB
cana-1172	95	93	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	95	94	𝜀	𝜀	PROPN
cana-1172	95	95	satisfies	satisfie	NOUN
cana-1172	95	96	[	[	PUNCT
cana-1172	95	97	�	�	NOUN
cana-1172	95	98	̇	̇	NOUN
cana-1172	95	99	�	�	PROPN
cana-1172	95	100	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	95	101	]	]	PUNCT
cana-1172	95	102	∈	∈	PROPN
cana-1172	95	103	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	95	104	𝜀	𝜀	NOUN
cana-1172	95	105	,	,	PUNCT
cana-1172	95	106	[	[	X
cana-1172	95	107	�	�	NOUN
cana-1172	95	108	̇	̇	PROPN
cana-1172	95	109	�	�	PROPN
cana-1172	95	110	𝑢𝑏⁄	𝑢𝑏⁄	NOUN
cana-1172	95	111	]	]	PUNCT
cana-1172	95	112	∈	∈	PROPN
cana-1172	95	113	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	95	114	𝜀	𝜀	PROPN
cana-1172	95	115	⇒	⇒	NOUN
cana-1172	95	116	[	[	X
cana-1172	95	117	(	(	PUNCT
cana-1172	95	118	0	0	NUM
cana-1172	95	119	∗	∗	NOUN
cana-1172	95	120	(	(	PUNCT
cana-1172	95	121	�	�	PROPN
cana-1172	95	122	̇	̇	PROPN
cana-1172	95	123	�	�	PROPN
cana-1172	95	124	∗	∗	PROPN
cana-1172	95	125	�	�	PROPN
cana-1172	95	126	̇	̇	PROPN
cana-1172	95	127	�	�	PROPN
cana-1172	95	128	))/𝑚𝑖𝑛{𝑢𝑎	))/𝑚𝑖𝑛{𝑢𝑎	PROPN
cana-1172	95	129	,	,	PUNCT
cana-1172	95	130	𝑢𝑏	𝑢𝑏	ADP
cana-1172	95	131	}	}	PUNCT
cana-1172	95	132	]	]	PUNCT
cana-1172	95	133	∈	∈	PROPN
cana-1172	95	134	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	95	135	𝜀	𝜀	PROPN
cana-1172	95	136	,	,	PUNCT
cana-1172	95	137	∀	∀	X
cana-1172	95	138	�	�	PROPN
cana-1172	95	139	̇	̇	PROPN
cana-1172	95	140	�	�	PROPN
cana-1172	95	141	,	,	PUNCT
cana-1172	95	142	�	�	PROPN
cana-1172	95	143	̇	̇	VERB
cana-1172	95	144	�	�	PROPN
cana-1172	95	145	∈	∈	PROPN
cana-1172	95	146	𝔊	𝔊	PROPN
cana-1172	95	147	and	and	CCONJ
cana-1172	95	148	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	95	149	,	,	PUNCT
cana-1172	95	150	𝑢𝑏	𝑢𝑏	ADP
cana-1172	95	151	∈	∈	PROPN
cana-1172	95	152	(	(	PUNCT
cana-1172	95	153	0,1	0,1	NOUN
cana-1172	95	154	]	]	PUNCT
cana-1172	95	155	.	.	PUNCT
cana-1172	96	1	proof	proof	NOUN
cana-1172	96	2	let	let	VERB
cana-1172	96	3	𝑈	𝑈	PROPN
cana-1172	96	4	be	be	AUX
cana-1172	96	5	a	a	DET
cana-1172	96	6	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	96	7	subalgebra	subalgebra	NOUN
cana-1172	96	8	of	of	ADP
cana-1172	96	9	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	PROPN
cana-1172	96	10	𝔊.	𝔊.	PROPN
cana-1172	96	11	it	it	PRON
cana-1172	96	12	is	be	AUX
cana-1172	96	13	given	give	VERB
cana-1172	96	14	that	that	SCONJ
cana-1172	96	15	�	�	PROPN
cana-1172	96	16	̇	̇	PROPN
cana-1172	96	17	�	�	PROPN
cana-1172	96	18	,	,	PUNCT
cana-1172	96	19	�	�	PROPN
cana-1172	96	20	̇	̇	VERB
cana-1172	96	21	�	�	PROPN
cana-1172	96	22	∈	∈	PROPN
cana-1172	96	23	𝔊	𝔊	PROPN
cana-1172	96	24	and	and	CCONJ
cana-1172	96	25	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	96	26	,	,	PUNCT
cana-1172	96	27	𝑢𝑏	𝑢𝑏	ADP
cana-1172	96	28	∈	∈	PROPN
cana-1172	96	29	(	(	PUNCT
cana-1172	96	30	0,1	0,1	NOUN
cana-1172	96	31	]	]	PUNCT
cana-1172	96	32	which	which	PRON
cana-1172	96	33	implies	imply	VERB
cana-1172	96	34	[	[	PUNCT
cana-1172	96	35	�	�	NOUN
cana-1172	96	36	̇	̇	VERB
cana-1172	96	37	�	�	PROPN
cana-1172	96	38	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	96	39	]	]	PUNCT
cana-1172	96	40	∈	∈	PROPN
cana-1172	96	41	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	96	42	𝜀	𝜀	NOUN
cana-1172	96	43	,	,	PUNCT
cana-1172	96	44	[	[	X
cana-1172	96	45	�	�	NOUN
cana-1172	96	46	̇	̇	PROPN
cana-1172	96	47	�	�	PROPN
cana-1172	96	48	𝑢𝑏⁄	𝑢𝑏⁄	NOUN
cana-1172	96	49	]	]	PUNCT
cana-1172	96	50	∈	∈	PROPN
cana-1172	96	51	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	96	52	𝜀	𝜀	PROPN
cana-1172	96	53	.	.	PUNCT
cana-1172	97	1	then	then	ADV
cana-1172	97	2	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	97	3	𝜀	𝜀	PROPN
cana-1172	97	4	(	(	PUNCT
cana-1172	97	5	�	�	PROPN
cana-1172	97	6	̇	̇	PROPN
cana-1172	97	7	�	�	PROPN
cana-1172	97	8	)	)	PUNCT
cana-1172	97	9	≥	≥	NOUN
cana-1172	97	10	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	97	11	and	and	CCONJ
cana-1172	97	12	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	97	13	𝜀	𝜀	PROPN
cana-1172	97	14	(	(	PUNCT
cana-1172	97	15	�	�	PROPN
cana-1172	97	16	̇	̇	PROPN
cana-1172	97	17	�	�	PROPN
cana-1172	97	18	)	)	PUNCT
cana-1172	97	19	≥	≥	NOUN
cana-1172	97	20	𝑢𝑏.	𝑢𝑏.	VERB
cana-1172	97	21	thus	thus	ADV
cana-1172	97	22	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	97	23	𝜀	𝜀	NOUN
cana-1172	97	24	(	(	PUNCT
cana-1172	97	25	0	0	NUM
cana-1172	97	26	∗	∗	NOUN
cana-1172	97	27	(	(	PUNCT
cana-1172	97	28	�	�	PROPN
cana-1172	97	29	̇	̇	PROPN
cana-1172	97	30	�	�	PROPN
cana-1172	97	31	∗	∗	PROPN
cana-1172	97	32	�	�	PROPN
cana-1172	97	33	̇	̇	PROPN
cana-1172	97	34	�	�	PROPN
cana-1172	97	35	)	)	PUNCT
cana-1172	97	36	)	)	PUNCT
cana-1172	98	1	=	=	SYM
cana-1172	98	2	𝑚𝑎𝑥{0	𝑚𝑎𝑥{0	PROPN
cana-1172	98	3	,	,	PUNCT
cana-1172	98	4	𝑈(0	𝑈(0	NUM
cana-1172	98	5	∗	∗	NOUN
cana-1172	98	6	(	(	PUNCT
cana-1172	98	7	�	�	PROPN
cana-1172	98	8	̇	̇	PROPN
cana-1172	98	9	�	�	PROPN
cana-1172	98	10	∗	∗	PROPN
cana-1172	98	11	�	�	PROPN
cana-1172	98	12	̇	̇	PROPN
cana-1172	98	13	�	�	PROPN
cana-1172	98	14	)	)	PUNCT
cana-1172	98	15	)	)	PUNCT
cana-1172	99	1	+	+	CCONJ
cana-1172	99	2	𝜀	𝜀	X
cana-1172	99	3	−	−	NUM
cana-1172	99	4	1	1	NUM
cana-1172	99	5	}	}	PUNCT
cana-1172	99	6	=	=	SYM
cana-1172	99	7	𝑚𝑎𝑥{0	𝑚𝑎𝑥{0	NOUN
cana-1172	99	8	,	,	PUNCT
cana-1172	99	9	𝑈(	𝑈(	PROPN
cana-1172	99	10	�	�	PROPN
cana-1172	99	11	̇	̇	PROPN
cana-1172	99	12	�	�	PROPN
cana-1172	99	13	∗	∗	PROPN
cana-1172	99	14	�	�	PROPN
cana-1172	99	15	̇	̇	PROPN
cana-1172	99	16	�	�	PROPN
cana-1172	99	17	)	)	PUNCT
cana-1172	100	1	+	+	NUM
cana-1172	100	2	𝜀	𝜀	X
cana-1172	100	3	−	−	NUM
cana-1172	100	4	1	1	NUM
cana-1172	100	5	}	}	PUNCT
cana-1172	100	6	≥	≥	NUM
cana-1172	100	7	𝑚𝑎𝑥{0	𝑚𝑎𝑥{0	PROPN
cana-1172	100	8	,	,	PUNCT
cana-1172	100	9	𝑚𝑖𝑛{𝑈(	𝑚𝑖𝑛{𝑈(	PROPN
cana-1172	100	10	�	�	PROPN
cana-1172	100	11	̇	̇	PROPN
cana-1172	100	12	�	�	PROPN
cana-1172	100	13	)	)	PUNCT
cana-1172	100	14	,	,	PUNCT
cana-1172	100	15	𝑈(	𝑈(	PROPN
cana-1172	100	16	�	�	PROPN
cana-1172	100	17	̇	̇	PROPN
cana-1172	100	18	�	�	PROPN
cana-1172	100	19	)	)	PUNCT
cana-1172	100	20	}	}	PUNCT
cana-1172	101	1	+	+	NUM
cana-1172	101	2	𝜀	𝜀	VERB
cana-1172	101	3	−	−	NUM
cana-1172	101	4	1	1	NUM
cana-1172	101	5	}	}	PUNCT
cana-1172	101	6	≥	≥	NUM
cana-1172	101	7	𝑚𝑎𝑥{0	𝑚𝑎𝑥{0	PROPN
cana-1172	101	8	,	,	PUNCT
cana-1172	101	9	𝑚𝑖𝑛{𝑈(	𝑚𝑖𝑛{𝑈(	PROPN
cana-1172	101	10	�	�	PROPN
cana-1172	101	11	̇	̇	PROPN
cana-1172	101	12	�	�	PROPN
cana-1172	101	13	)	)	PUNCT
cana-1172	102	1	+	+	NUM
cana-1172	102	2	𝜀	𝜀	X
cana-1172	102	3	−	−	NUM
cana-1172	102	4	1	1	NUM
cana-1172	102	5	,	,	PUNCT
cana-1172	102	6	𝑈(	𝑈(	PROPN
cana-1172	102	7	�	�	PROPN
cana-1172	102	8	̇	̇	PROPN
cana-1172	102	9	�	�	PROPN
cana-1172	102	10	)	)	PUNCT
cana-1172	103	1	+	+	NUM
cana-1172	103	2	𝜀	𝜀	X
cana-1172	103	3	−	−	NOUN
cana-1172	103	4	1	1	NUM
cana-1172	103	5	}	}	PUNCT
cana-1172	103	6	}	}	PUNCT
cana-1172	103	7	≥	≥	PROPN
cana-1172	103	8	𝑚𝑖𝑛{𝑚𝑎𝑥{0	𝑚𝑖𝑛{𝑚𝑎𝑥{0	NOUN
cana-1172	103	9	,	,	PUNCT
cana-1172	103	10	𝑈(	𝑈(	PROPN
cana-1172	103	11	�	�	PROPN
cana-1172	103	12	̇	̇	PROPN
cana-1172	103	13	�	�	PROPN
cana-1172	103	14	)	)	PUNCT
cana-1172	104	1	+	+	NUM
cana-1172	104	2	𝜀	𝜀	X
cana-1172	104	3	−	−	NUM
cana-1172	104	4	1	1	NUM
cana-1172	104	5	}	}	PUNCT
cana-1172	104	6	,	,	PUNCT
cana-1172	104	7	𝑚𝑎𝑥{0	𝑚𝑎𝑥{0	PROPN
cana-1172	104	8	,	,	PUNCT
cana-1172	104	9	𝑈(	𝑈(	PROPN
cana-1172	104	10	�	�	PROPN
cana-1172	104	11	̇	̇	PROPN
cana-1172	104	12	�	�	PROPN
cana-1172	104	13	)	)	PUNCT
cana-1172	105	1	+	+	NUM
cana-1172	105	2	𝜀	𝜀	X
cana-1172	105	3	−	−	NOUN
cana-1172	105	4	1	1	NUM
cana-1172	105	5	}	}	PUNCT
cana-1172	105	6	}	}	PUNCT
cana-1172	105	7	≥	≥	PROPN
cana-1172	105	8	𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝐿𝑈	NUM
cana-1172	105	9	𝜀	𝜀	X
cana-1172	105	10	(	(	PUNCT
cana-1172	105	11	�	�	PROPN
cana-1172	105	12	̇	̇	PROPN
cana-1172	105	13	�	�	PROPN
cana-1172	105	14	)	)	PUNCT
cana-1172	105	15	,	,	PUNCT
cana-1172	105	16	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	105	17	𝜀	𝜀	PROPN
cana-1172	105	18	(	(	PUNCT
cana-1172	105	19	�	�	PROPN
cana-1172	105	20	̇	̇	PROPN
cana-1172	105	21	�	�	PROPN
cana-1172	105	22	)	)	PUNCT
cana-1172	105	23	}	}	PUNCT
cana-1172	105	24	≥	≥	X
cana-1172	105	25	𝑚𝑖𝑛{𝑢𝑏	𝑚𝑖𝑛{𝑢𝑏	NOUN
cana-1172	105	26	,	,	PUNCT
cana-1172	105	27	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	105	28	}	}	PUNCT
cana-1172	105	29	.	.	PUNCT
cana-1172	106	1	so	so	ADV
cana-1172	106	2	[	[	X
cana-1172	106	3	(	(	PUNCT
cana-1172	106	4	0	0	NUM
cana-1172	106	5	∗	∗	NOUN
cana-1172	106	6	(	(	PUNCT
cana-1172	106	7	�	�	PROPN
cana-1172	106	8	̇	̇	PROPN
cana-1172	106	9	�	�	PROPN
cana-1172	106	10	∗	∗	PROPN
cana-1172	106	11	�	�	PROPN
cana-1172	106	12	̇	̇	PROPN
cana-1172	106	13	�	�	PROPN
cana-1172	106	14	))/𝑚𝑖𝑛{𝑢𝑎	))/𝑚𝑖𝑛{𝑢𝑎	PROPN
cana-1172	106	15	,	,	PUNCT
cana-1172	106	16	𝑢𝑏	𝑢𝑏	ADP
cana-1172	106	17	}	}	PUNCT
cana-1172	106	18	]	]	PUNCT
cana-1172	106	19	∈	∈	PROPN
cana-1172	106	20	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	106	21	𝜀	𝜀	PROPN
cana-1172	106	22	.	.	PUNCT
cana-1172	107	1	lemma	lemma	PROPN
cana-1172	107	2	5.5	5.5	NUM
cana-1172	107	3	every	every	DET
cana-1172	107	4	lukasz	lukasz	NOUN
cana-1172	107	5	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	107	6	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	PROPN
cana-1172	107	7	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	107	8	𝜀	𝜀	NOUN
cana-1172	107	9	of	of	ADP
cana-1172	107	10	𝔊	𝔊	PROPN
cana-1172	107	11	satisfies	satisfy	VERB
cana-1172	107	12	the	the	DET
cana-1172	107	13	condition	condition	NOUN
cana-1172	107	14	if	if	SCONJ
cana-1172	107	15	�	�	NOUN
cana-1172	107	16	̇	̇	VERB
cana-1172	107	17	�	�	PROPN
cana-1172	107	18	≤	≤	PROPN
cana-1172	107	19	�	�	PROPN
cana-1172	107	20	̇	̇	VERB
cana-1172	107	21	�	�	PROPN
cana-1172	107	22	and	and	CCONJ
cana-1172	107	23	�	�	PROPN
cana-1172	107	24	̇	̇	PROPN
cana-1172	107	25	�	�	PROPN
cana-1172	107	26	∗	∗	PROPN
cana-1172	107	27	�	�	PROPN
cana-1172	107	28	̇	̇	PROPN
cana-1172	107	29	�	�	PROPN
cana-1172	107	30	=	=	SYM
cana-1172	107	31	0	0	NUM
cana-1172	107	32	,	,	PUNCT
cana-1172	107	33	then	then	ADV
cana-1172	107	34	[	[	X
cana-1172	107	35	�	�	NOUN
cana-1172	107	36	̇	̇	PROPN
cana-1172	107	37	�	�	PROPN
cana-1172	107	38	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	107	39	]	]	PUNCT
cana-1172	107	40	∈	∈	PROPN
cana-1172	107	41	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	107	42	𝜀	𝜀	PROPN
cana-1172	107	43	⇒	⇒	NOUN
cana-1172	107	44	[	[	PUNCT
cana-1172	107	45	�	�	NOUN
cana-1172	107	46	̇	̇	VERB
cana-1172	107	47	�	�	PROPN
cana-1172	107	48	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	107	49	]	]	PUNCT
cana-1172	107	50	∈	∈	PROPN
cana-1172	107	51	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	107	52	𝜀	𝜀	PROPN
cana-1172	107	53	,	,	PUNCT
cana-1172	107	54	∀	∀	X
cana-1172	107	55	�	�	PROPN
cana-1172	107	56	̇	̇	PROPN
cana-1172	107	57	�	�	PROPN
cana-1172	107	58	,	,	PUNCT
cana-1172	107	59	�	�	PROPN
cana-1172	107	60	̇	̇	VERB
cana-1172	107	61	�	�	PROPN
cana-1172	107	62	∈	∈	PROPN
cana-1172	107	63	𝔊	𝔊	PROPN
cana-1172	107	64	,	,	PUNCT
cana-1172	107	65	∀	∀	X
cana-1172	107	66	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	107	67	∈	∈	PROPN
cana-1172	107	68	(	(	PUNCT
cana-1172	107	69	0,1	0,1	NUM
cana-1172	107	70	]	]	PUNCT
cana-1172	107	71	(	(	PUNCT
cana-1172	107	72	5.1	5.1	NUM
cana-1172	107	73	)	)	PUNCT
cana-1172	107	74	proof	proof	NOUN
cana-1172	107	75	let	let	VERB
cana-1172	107	76	�	�	PROPN
cana-1172	107	77	̇	̇	PROPN
cana-1172	107	78	�	�	PROPN
cana-1172	107	79	,	,	PUNCT
cana-1172	107	80	�	�	PROPN
cana-1172	107	81	̇	̇	VERB
cana-1172	107	82	�	�	PROPN
cana-1172	107	83	∈	∈	PROPN
cana-1172	107	84	𝔊	𝔊	PROPN
cana-1172	107	85	and	and	CCONJ
cana-1172	107	86	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	107	87	∈	∈	PROPN
cana-1172	107	88	(	(	PUNCT
cana-1172	107	89	0,1	0,1	NOUN
cana-1172	107	90	]	]	PUNCT
cana-1172	107	91	be	be	VERB
cana-1172	107	92	such	such	ADJ
cana-1172	107	93	that	that	SCONJ
cana-1172	107	94	�	�	NOUN
cana-1172	107	95	̇	̇	PROPN
cana-1172	107	96	�	�	PROPN
cana-1172	107	97	≤	≤	PROPN
cana-1172	107	98	�	�	PROPN
cana-1172	107	99	̇	̇	PROPN
cana-1172	107	100	�	�	PROPN
cana-1172	107	101	,	,	PUNCT
cana-1172	107	102	�	�	PROPN
cana-1172	107	103	̇	̇	PROPN
cana-1172	107	104	�	�	PROPN
cana-1172	107	105	∗	∗	PROPN
cana-1172	107	106	�	�	PROPN
cana-1172	107	107	̇	̇	PROPN
cana-1172	107	108	�	�	PROPN
cana-1172	107	109	=	=	SYM
cana-1172	107	110	0	0	NUM
cana-1172	108	1	and	and	CCONJ
cana-1172	108	2	[	[	X
cana-1172	108	3	�	�	PROPN
cana-1172	108	4	̇	̇	PROPN
cana-1172	108	5	�	�	PROPN
cana-1172	108	6	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	108	7	]	]	PUNCT
cana-1172	108	8	∈	∈	PROPN
cana-1172	108	9	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	108	10	𝜀	𝜀	PROPN
cana-1172	108	11	then	then	ADV
cana-1172	108	12	(	(	PUNCT
cana-1172	108	13	�	�	PROPN
cana-1172	108	14	̇	̇	PROPN
cana-1172	108	15	�	�	PROPN
cana-1172	108	16	∗	∗	PROPN
cana-1172	108	17	�	�	PROPN
cana-1172	108	18	̇	̇	PROPN
cana-1172	108	19	�	�	PROPN
cana-1172	108	20	)	)	PUNCT
cana-1172	108	21	=	=	SYM
cana-1172	108	22	0	0	NUM
cana-1172	108	23	and	and	CCONJ
cana-1172	108	24	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	108	25	𝜀	𝜀	PROPN
cana-1172	108	26	(	(	PUNCT
cana-1172	108	27	�	�	PROPN
cana-1172	108	28	̇	̇	PROPN
cana-1172	108	29	�	�	PROPN
cana-1172	108	30	)	)	PUNCT
cana-1172	108	31	≥	≥	NOUN
cana-1172	108	32	𝑢𝑎	𝑢𝑎	VERB
cana-1172	108	33	so	so	ADV
cana-1172	108	34	,	,	PUNCT
cana-1172	108	35	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	108	36	𝜀	𝜀	PROPN
cana-1172	108	37	(	(	PUNCT
cana-1172	108	38	�	�	PROPN
cana-1172	108	39	̇	̇	PROPN
cana-1172	108	40	�	�	PROPN
cana-1172	108	41	)	)	PUNCT
cana-1172	108	42	≥	≥	NOUN
cana-1172	108	43	min{𝐿𝑈	min{𝐿𝑈	NOUN
cana-1172	108	44	𝜀	𝜀	PROPN
cana-1172	108	45	(	(	PUNCT
cana-1172	108	46	�	�	PROPN
cana-1172	108	47	̇	̇	PROPN
cana-1172	108	48	�	�	PROPN
cana-1172	108	49	∗	∗	PROPN
cana-1172	108	50	�	�	PROPN
cana-1172	108	51	̇	̇	PROPN
cana-1172	108	52	�	�	PROPN
cana-1172	108	53	)	)	PUNCT
cana-1172	108	54	,	,	PUNCT
cana-1172	108	55	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	108	56	𝜀	𝜀	PROPN
cana-1172	108	57	(	(	PUNCT
cana-1172	108	58	�	�	PROPN
cana-1172	108	59	̇	̇	PROPN
cana-1172	108	60	�	�	PROPN
cana-1172	108	61	)	)	PUNCT
cana-1172	108	62	}	}	PUNCT
cana-1172	108	63	=	=	PUNCT
cana-1172	108	64	𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝐿𝑈	NUM
cana-1172	108	65	𝜀	𝜀	X
cana-1172	108	66	(	(	PUNCT
cana-1172	108	67	0	0	NUM
cana-1172	108	68	)	)	PUNCT
cana-1172	108	69	,	,	PUNCT
cana-1172	108	70	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	108	71	𝜀	𝜀	PROPN
cana-1172	108	72	(	(	PUNCT
cana-1172	108	73	�	�	PROPN
cana-1172	108	74	̇	̇	PROPN
cana-1172	108	75	�	�	PROPN
cana-1172	108	76	)	)	PUNCT
cana-1172	108	77	}	}	PUNCT
cana-1172	108	78	=	=	SYM
cana-1172	108	79	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	108	80	𝜀	𝜀	PROPN
cana-1172	108	81	(	(	PUNCT
cana-1172	108	82	�	�	PROPN
cana-1172	108	83	̇	̇	PROPN
cana-1172	108	84	�	�	PROPN
cana-1172	108	85	)	)	PUNCT
cana-1172	108	86	≥	≥	NUM
cana-1172	108	87	𝑢𝑎.	𝑢𝑎.	VERB
cana-1172	108	88	hence	hence	ADV
cana-1172	108	89	[	[	X
cana-1172	108	90	�	�	NOUN
cana-1172	108	91	̇	̇	PROPN
cana-1172	108	92	�	�	PROPN
cana-1172	108	93	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	108	94	]	]	PUNCT
cana-1172	108	95	∈	∈	PROPN
cana-1172	108	96	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	108	97	𝜀	𝜀	PROPN
cana-1172	108	98	.	.	PUNCT
cana-1172	109	1	communications	communication	NOUN
cana-1172	109	2	on	on	ADP
cana-1172	109	3	applied	apply	VERB
cana-1172	109	4	nonlinear	nonlinear	ADJ
cana-1172	109	5	analysis	analysis	NOUN
cana-1172	109	6	issn	issn	NOUN
cana-1172	109	7	:	:	PUNCT
cana-1172	109	8	1074	1074	NUM
cana-1172	109	9	-	-	PUNCT
cana-1172	109	10	133x	133x	NUM
cana-1172	109	11	vol	vol	NOUN
cana-1172	109	12	31	31	NUM
cana-1172	109	13	no	no	NOUN
cana-1172	109	14	.	.	PUNCT
cana-1172	110	1	6s	6s	NUM
cana-1172	110	2	(	(	PUNCT
cana-1172	110	3	2024	2024	NUM
cana-1172	110	4	)	)	PUNCT
cana-1172	110	5	140	140	NUM
cana-1172	110	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1172	110	7	lemma	lemma	PROPN
cana-1172	110	8	5.6	5.6	NUM
cana-1172	110	9	every	every	DET
cana-1172	110	10	lukasz	lukasz	NOUN
cana-1172	110	11	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	110	12	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	PROPN
cana-1172	110	13	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	110	14	𝜀	𝜀	NOUN
cana-1172	110	15	of	of	ADP
cana-1172	110	16	𝔊	𝔊	PROPN
cana-1172	110	17	fulfils	fulfil	VERB
cana-1172	110	18	the	the	DET
cana-1172	110	19	condition	condition	NOUN
cana-1172	110	20	if	if	SCONJ
cana-1172	110	21	�	�	NOUN
cana-1172	110	22	̇	̇	PROPN
cana-1172	110	23	�	�	PROPN
cana-1172	110	24	∗	∗	PROPN
cana-1172	110	25	�	�	PROPN
cana-1172	110	26	̇	̇	PROPN
cana-1172	110	27	�	�	PROPN
cana-1172	110	28	≤	≤	PROPN
cana-1172	110	29	�	�	PROPN
cana-1172	110	30	̇	̇	VERB
cana-1172	110	31	�	�	PROPN
cana-1172	110	32	and	and	CCONJ
cana-1172	110	33	�	�	PROPN
cana-1172	110	34	̇	̇	PROPN
cana-1172	110	35	�	�	PROPN
cana-1172	110	36	∗	∗	NOUN
cana-1172	110	37	(	(	PUNCT
cana-1172	110	38	�	�	PROPN
cana-1172	110	39	̇	̇	PROPN
cana-1172	110	40	�	�	PROPN
cana-1172	110	41	∗	∗	PROPN
cana-1172	110	42	�	�	PROPN
cana-1172	110	43	̇	̇	PROPN
cana-1172	110	44	�	�	PROPN
cana-1172	110	45	)	)	PUNCT
cana-1172	110	46	=	=	SYM
cana-1172	111	1	0	0	NUM
cana-1172	111	2	,	,	PUNCT
cana-1172	111	3	then	then	ADV
cana-1172	111	4	[	[	X
cana-1172	111	5	�	�	NOUN
cana-1172	111	6	̇	̇	PROPN
cana-1172	111	7	�	�	PROPN
cana-1172	111	8	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	111	9	]	]	PUNCT
cana-1172	111	10	∈	∈	PROPN
cana-1172	111	11	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	111	12	𝜀	𝜀	NOUN
cana-1172	111	13	,	,	PUNCT
cana-1172	111	14	[	[	X
cana-1172	111	15	�	�	NOUN
cana-1172	111	16	̇	̇	PROPN
cana-1172	111	17	�	�	PROPN
cana-1172	111	18	𝑢𝑏⁄	𝑢𝑏⁄	NOUN
cana-1172	111	19	]	]	PUNCT
cana-1172	111	20	∈	∈	PROPN
cana-1172	111	21	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	111	22	𝜀	𝜀	PROPN
cana-1172	111	23	⇒	⇒	NOUN
cana-1172	111	24	[	[	PUNCT
cana-1172	111	25	�	�	PROPN
cana-1172	111	26	̇	̇	PROPN
cana-1172	111	27	�	�	PROPN
cana-1172	111	28	𝑚𝑖𝑛{𝑢𝑎	𝑚𝑖𝑛{𝑢𝑎	VERB
cana-1172	111	29	,	,	PUNCT
cana-1172	111	30	𝑢𝑏}⁄	𝑢𝑏}⁄	PRON
cana-1172	111	31	]	]	PUNCT
cana-1172	111	32	∈	∈	PROPN
cana-1172	111	33	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	111	34	𝜀	𝜀	PROPN
cana-1172	111	35	(	(	PUNCT
cana-1172	111	36	5.2	5.2	NUM
cana-1172	111	37	)	)	PUNCT
cana-1172	111	38	for	for	ADP
cana-1172	111	39	all	all	DET
cana-1172	111	40	�	�	PROPN
cana-1172	111	41	̇	̇	PROPN
cana-1172	111	42	�	�	PROPN
cana-1172	111	43	,	,	PUNCT
cana-1172	111	44	�	�	PROPN
cana-1172	111	45	̇	̇	PROPN
cana-1172	111	46	�	�	PROPN
cana-1172	111	47	,	,	PUNCT
cana-1172	111	48	�	�	PROPN
cana-1172	111	49	̇	̇	VERB
cana-1172	111	50	�	�	PROPN
cana-1172	111	51	∈	∈	PROPN
cana-1172	111	52	𝔊	𝔊	PROPN
cana-1172	111	53	,	,	PUNCT
cana-1172	111	54	and	and	CCONJ
cana-1172	111	55	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	111	56	,	,	PUNCT
cana-1172	111	57	𝑢𝑏	𝑢𝑏	ADP
cana-1172	111	58	∈	∈	PROPN
cana-1172	111	59	(	(	PUNCT
cana-1172	111	60	0,1	0,1	NUM
cana-1172	111	61	]	]	PUNCT
cana-1172	111	62	proof	proof	NOUN
cana-1172	111	63	let	let	VERB
cana-1172	111	64	�	�	PROPN
cana-1172	111	65	̇	̇	PROPN
cana-1172	111	66	�	�	PROPN
cana-1172	111	67	,	,	PUNCT
cana-1172	111	68	�	�	PROPN
cana-1172	111	69	̇	̇	PROPN
cana-1172	111	70	�	�	PROPN
cana-1172	111	71	,	,	PUNCT
cana-1172	111	72	�	�	PROPN
cana-1172	111	73	̇	̇	VERB
cana-1172	111	74	�	�	PROPN
cana-1172	111	75	∈	∈	PROPN
cana-1172	111	76	𝔊	𝔊	PROPN
cana-1172	111	77	and	and	CCONJ
cana-1172	111	78	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	111	79	,	,	PUNCT
cana-1172	111	80	𝑢𝑏	𝑢𝑏	ADP
cana-1172	111	81	∈	∈	PROPN
cana-1172	111	82	(	(	PUNCT
cana-1172	111	83	0,1	0,1	NOUN
cana-1172	111	84	]	]	PUNCT
cana-1172	111	85	be	be	VERB
cana-1172	111	86	such	such	ADJ
cana-1172	111	87	that	that	SCONJ
cana-1172	111	88	�	�	NOUN
cana-1172	111	89	̇	̇	PROPN
cana-1172	111	90	�	�	PROPN
cana-1172	111	91	∗	∗	PROPN
cana-1172	111	92	�	�	PROPN
cana-1172	111	93	̇	̇	PROPN
cana-1172	111	94	�	�	PROPN
cana-1172	111	95	≤	≤	PROPN
cana-1172	111	96	�	�	PROPN
cana-1172	111	97	̇	̇	PROPN
cana-1172	111	98	�	�	PROPN
cana-1172	111	99	,	,	PUNCT
cana-1172	111	100	�	�	PROPN
cana-1172	111	101	̇	̇	PROPN
cana-1172	111	102	�	�	PROPN
cana-1172	111	103	∗	∗	NOUN
cana-1172	111	104	(	(	PUNCT
cana-1172	111	105	�	�	PROPN
cana-1172	111	106	̇	̇	PROPN
cana-1172	111	107	�	�	PROPN
cana-1172	111	108	∗	∗	PROPN
cana-1172	111	109	�	�	PROPN
cana-1172	111	110	̇	̇	PROPN
cana-1172	111	111	�	�	PROPN
cana-1172	111	112	)	)	PUNCT
cana-1172	111	113	=	=	SYM
cana-1172	111	114	0	0	NUM
cana-1172	111	115	,	,	PUNCT
cana-1172	111	116	[	[	PUNCT
cana-1172	111	117	�	�	NOUN
cana-1172	111	118	̇	̇	ADJ
cana-1172	111	119	�	�	PROPN
cana-1172	111	120	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	111	121	]	]	PUNCT
cana-1172	111	122	∈	∈	PROPN
cana-1172	111	123	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	111	124	𝜀	𝜀	PROPN
cana-1172	111	125	and	and	CCONJ
cana-1172	111	126	[	[	PUNCT
cana-1172	111	127	�	�	PROPN
cana-1172	111	128	̇	̇	PROPN
cana-1172	111	129	�	�	PROPN
cana-1172	111	130	𝑢𝑏⁄	𝑢𝑏⁄	NOUN
cana-1172	111	131	]	]	PUNCT
cana-1172	111	132	∈	∈	PROPN
cana-1172	111	133	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	111	134	𝜀	𝜀	PROPN
cana-1172	111	135	.	.	PUNCT
cana-1172	112	1	then	then	ADV
cana-1172	112	2	,	,	PUNCT
cana-1172	112	3	(	(	PUNCT
cana-1172	112	4	�	�	PROPN
cana-1172	112	5	̇	̇	PROPN
cana-1172	112	6	�	�	PROPN
cana-1172	112	7	∗	∗	PROPN
cana-1172	112	8	�	�	PROPN
cana-1172	112	9	̇	̇	PROPN
cana-1172	112	10	�	�	PROPN
cana-1172	112	11	)	)	PUNCT
cana-1172	112	12	∗	∗	PROPN
cana-1172	112	13	�	�	PROPN
cana-1172	112	14	̇	̇	PROPN
cana-1172	112	15	�	�	PROPN
cana-1172	112	16	=	=	SYM
cana-1172	112	17	0	0	NUM
cana-1172	112	18	,	,	PUNCT
cana-1172	112	19	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	112	20	𝜀	𝜀	PROPN
cana-1172	112	21	(	(	PUNCT
cana-1172	112	22	�	�	PROPN
cana-1172	112	23	̇	̇	PROPN
cana-1172	112	24	�	�	PROPN
cana-1172	112	25	)	)	PUNCT
cana-1172	112	26	≥	≥	NOUN
cana-1172	112	27	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	112	28	and	and	CCONJ
cana-1172	112	29	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	112	30	𝜀	𝜀	PROPN
cana-1172	112	31	(	(	PUNCT
cana-1172	112	32	�	�	PROPN
cana-1172	112	33	̇	̇	PROPN
cana-1172	112	34	�	�	PROPN
cana-1172	112	35	)	)	PUNCT
cana-1172	112	36	≥	≥	NOUN
cana-1172	112	37	𝑢𝑏.	𝑢𝑏.	NOUN
cana-1172	112	38	hence	hence	ADV
cana-1172	112	39	,	,	PUNCT
cana-1172	112	40	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	112	41	𝜀	𝜀	PROPN
cana-1172	112	42	(	(	PUNCT
cana-1172	112	43	�	�	PROPN
cana-1172	112	44	̇	̇	PROPN
cana-1172	112	45	�	�	PROPN
cana-1172	112	46	)	)	PUNCT
cana-1172	112	47	≥	≥	NOUN
cana-1172	112	48	min{𝐿𝑈	min{𝐿𝑈	NOUN
cana-1172	112	49	𝜀	𝜀	PROPN
cana-1172	112	50	(	(	PUNCT
cana-1172	112	51	�	�	PROPN
cana-1172	112	52	̇	̇	PROPN
cana-1172	112	53	�	�	PROPN
cana-1172	112	54	∗	∗	PROPN
cana-1172	112	55	�	�	PROPN
cana-1172	112	56	̇	̇	PROPN
cana-1172	112	57	�	�	PROPN
cana-1172	112	58	)	)	PUNCT
cana-1172	112	59	,	,	PUNCT
cana-1172	112	60	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	112	61	𝜀	𝜀	PROPN
cana-1172	112	62	(	(	PUNCT
cana-1172	112	63	�	�	PROPN
cana-1172	112	64	̇	̇	PROPN
cana-1172	112	65	�	�	PROPN
cana-1172	112	66	)	)	PUNCT
cana-1172	112	67	}	}	PUNCT
cana-1172	112	68	≥	≥	NUM
cana-1172	112	69	𝑚𝑖𝑛{𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝑚𝑖𝑛{𝐿𝑈	NOUN
cana-1172	113	1	𝜀	𝜀	X
cana-1172	113	2	(	(	PUNCT
cana-1172	113	3	(	(	PUNCT
cana-1172	113	4	�	�	PROPN
cana-1172	113	5	̇	̇	PROPN
cana-1172	113	6	�	�	PROPN
cana-1172	113	7	∗	∗	PROPN
cana-1172	113	8	�	�	PROPN
cana-1172	113	9	̇	̇	PROPN
cana-1172	113	10	�	�	PROPN
cana-1172	113	11	)	)	PUNCT
cana-1172	113	12	∗	∗	PROPN
cana-1172	113	13	�	�	PROPN
cana-1172	113	14	̇	̇	PROPN
cana-1172	113	15	�	�	PROPN
cana-1172	113	16	)	)	PUNCT
cana-1172	113	17	,	,	PUNCT
cana-1172	113	18	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	113	19	𝜀	𝜀	PROPN
cana-1172	113	20	(	(	PUNCT
cana-1172	113	21	�	�	PROPN
cana-1172	113	22	̇	̇	PROPN
cana-1172	113	23	�	�	PROPN
cana-1172	113	24	)	)	PUNCT
cana-1172	113	25	}	}	PUNCT
cana-1172	113	26	,	,	PUNCT
cana-1172	113	27	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	113	28	𝜀	𝜀	PROPN
cana-1172	113	29	(	(	PUNCT
cana-1172	113	30	�	�	PROPN
cana-1172	113	31	̇	̇	PROPN
cana-1172	113	32	�	�	PROPN
cana-1172	113	33	)	)	PUNCT
cana-1172	113	34	}	}	PUNCT
cana-1172	113	35	=	=	SYM
cana-1172	113	36	𝑚𝑖𝑛{𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝑚𝑖𝑛{𝐿𝑈	NOUN
cana-1172	113	37	𝜀	𝜀	X
cana-1172	113	38	(	(	PUNCT
cana-1172	113	39	0	0	NUM
cana-1172	113	40	)	)	PUNCT
cana-1172	113	41	,	,	PUNCT
cana-1172	113	42	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	113	43	𝜀	𝜀	PROPN
cana-1172	113	44	(	(	PUNCT
cana-1172	113	45	�	�	PROPN
cana-1172	113	46	̇	̇	PROPN
cana-1172	113	47	�	�	PROPN
cana-1172	113	48	)	)	PUNCT
cana-1172	113	49	}	}	PUNCT
cana-1172	113	50	,	,	PUNCT
cana-1172	113	51	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	113	52	𝜀	𝜀	PROPN
cana-1172	113	53	(	(	PUNCT
cana-1172	113	54	�	�	PROPN
cana-1172	113	55	̇	̇	PROPN
cana-1172	113	56	�	�	PROPN
cana-1172	113	57	)	)	PUNCT
cana-1172	113	58	}	}	PUNCT
cana-1172	113	59	=	=	SYM
cana-1172	113	60	min{𝐿𝑈	min{𝐿𝑈	PROPN
cana-1172	113	61	𝜀	𝜀	PROPN
cana-1172	113	62	(	(	PUNCT
cana-1172	113	63	�	�	PROPN
cana-1172	113	64	̇	̇	PROPN
cana-1172	113	65	�	�	PROPN
cana-1172	113	66	)	)	PUNCT
cana-1172	113	67	,	,	PUNCT
cana-1172	113	68	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	113	69	𝜀	𝜀	PROPN
cana-1172	113	70	(	(	PUNCT
cana-1172	113	71	�	�	PROPN
cana-1172	113	72	̇	̇	PROPN
cana-1172	113	73	�	�	PROPN
cana-1172	113	74	)	)	PUNCT
cana-1172	113	75	}	}	PUNCT
cana-1172	113	76	≥	≥	X
cana-1172	113	77	𝑚𝑖𝑛{𝑢𝑏	𝑚𝑖𝑛{𝑢𝑏	NOUN
cana-1172	113	78	,	,	PUNCT
cana-1172	113	79	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	113	80	}	}	PUNCT
cana-1172	113	81	and	and	CCONJ
cana-1172	113	82	so	so	ADV
cana-1172	113	83	[	[	X
cana-1172	113	84	�	�	PROPN
cana-1172	113	85	̇	̇	PROPN
cana-1172	113	86	�	�	PROPN
cana-1172	113	87	𝑚𝑖𝑛{𝑢𝑎	𝑚𝑖𝑛{𝑢𝑎	VERB
cana-1172	113	88	,	,	PUNCT
cana-1172	113	89	𝑢𝑏}⁄	𝑢𝑏}⁄	PRON
cana-1172	113	90	]	]	PUNCT
cana-1172	114	1	∈	∈	PROPN
cana-1172	114	2	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	114	3	𝜀	𝜀	PROPN
cana-1172	114	4	.	.	PUNCT
cana-1172	115	1	remark	remark	VERB
cana-1172	115	2	5.7	5.7	NUM
cana-1172	115	3	if	if	SCONJ
cana-1172	115	4	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	115	5	𝜀	𝜀	PROPN
cana-1172	115	6	is	be	AUX
cana-1172	115	7	a	a	DET
cana-1172	115	8	lukasz	lukasz	NOUN
cana-1172	115	9	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	115	10	ideal	ideal	NOUN
cana-1172	115	11	of	of	ADP
cana-1172	115	12	𝔊	𝔊	PROPN
cana-1172	115	13	,	,	PUNCT
cana-1172	115	14	then	then	ADV
cana-1172	115	15	it	it	PRON
cana-1172	115	16	satisfies	satisfy	VERB
cana-1172	115	17	the	the	DET
cana-1172	115	18	following	follow	VERB
cana-1172	115	19	inequalities	inequality	NOUN
cana-1172	115	20	(	(	PUNCT
cana-1172	115	21	i	i	NOUN
cana-1172	115	22	)	)	PUNCT
cana-1172	116	1	[	[	X
cana-1172	116	2	�	�	NOUN
cana-1172	116	3	̇	̇	PROPN
cana-1172	116	4	�	�	PROPN
cana-1172	116	5	≤	≤	PROPN
cana-1172	116	6	�	�	PROPN
cana-1172	116	7	̇	̇	PROPN
cana-1172	116	8	�	�	PROPN
cana-1172	116	9	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-1172	116	10	�	�	PROPN
cana-1172	116	11	̇	̇	PROPN
cana-1172	116	12	�	�	PROPN
cana-1172	116	13	∗	∗	PROPN
cana-1172	116	14	�	�	PROPN
cana-1172	116	15	̇	̇	PROPN
cana-1172	116	16	�	�	PROPN
cana-1172	116	17	=	=	SYM
cana-1172	116	18	0	0	NUM
cana-1172	116	19	⇒	⇒	PROPN
cana-1172	116	20	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	116	21	𝜀	𝜀	PROPN
cana-1172	116	22	(	(	PUNCT
cana-1172	116	23	�	�	PROPN
cana-1172	116	24	̇	̇	PROPN
cana-1172	116	25	�	�	PROPN
cana-1172	116	26	)	)	PUNCT
cana-1172	116	27	≥	≥	NOUN
cana-1172	116	28	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	116	29	𝜀	𝜀	PROPN
cana-1172	116	30	(	(	PUNCT
cana-1172	116	31	�	�	PROPN
cana-1172	116	32	̇	̇	PROPN
cana-1172	116	33	�	�	PROPN
cana-1172	116	34	)	)	PUNCT
cana-1172	116	35	]	]	PUNCT
cana-1172	116	36	(	(	PUNCT
cana-1172	116	37	5.3	5.3	NUM
cana-1172	116	38	)	)	PUNCT
cana-1172	116	39	(	(	PUNCT
cana-1172	116	40	ii	ii	NOUN
cana-1172	116	41	)	)	PUNCT
cana-1172	117	1	[	[	X
cana-1172	117	2	�	�	PROPN
cana-1172	117	3	̇	̇	PROPN
cana-1172	117	4	�	�	PROPN
cana-1172	117	5	∗	∗	PROPN
cana-1172	117	6	�	�	PROPN
cana-1172	117	7	̇	̇	PROPN
cana-1172	117	8	�	�	PROPN
cana-1172	117	9	≤	≤	PROPN
cana-1172	117	10	�	�	PROPN
cana-1172	117	11	̇	̇	PROPN
cana-1172	117	12	�	�	PROPN
cana-1172	117	13	𝑎𝑛𝑑	𝑎𝑛𝑑	PROPN
cana-1172	117	14	�	�	PROPN
cana-1172	117	15	̇	̇	PROPN
cana-1172	117	16	�	�	PROPN
cana-1172	117	17	∗	∗	NOUN
cana-1172	117	18	(	(	PUNCT
cana-1172	117	19	�	�	PROPN
cana-1172	117	20	̇	̇	PROPN
cana-1172	117	21	�	�	PROPN
cana-1172	117	22	∗	∗	PROPN
cana-1172	117	23	�	�	PROPN
cana-1172	117	24	̇	̇	PROPN
cana-1172	117	25	�	�	PROPN
cana-1172	117	26	)	)	PUNCT
cana-1172	117	27	=	=	SYM
cana-1172	117	28	0	0	NUM
cana-1172	117	29	⇒	⇒	PROPN
cana-1172	117	30	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	117	31	𝜀	𝜀	PROPN
cana-1172	117	32	(	(	PUNCT
cana-1172	117	33	�	�	PROPN
cana-1172	117	34	̇	̇	PROPN
cana-1172	117	35	�	�	PROPN
cana-1172	117	36	)	)	PUNCT
cana-1172	117	37	≥	≥	NOUN
cana-1172	117	38	𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝐿𝑈	NUM
cana-1172	117	39	𝜀	𝜀	X
cana-1172	117	40	(	(	PUNCT
cana-1172	117	41	�	�	PROPN
cana-1172	117	42	̇	̇	PROPN
cana-1172	117	43	�	�	PROPN
cana-1172	117	44	)	)	PUNCT
cana-1172	117	45	,	,	PUNCT
cana-1172	117	46	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	117	47	𝜀	𝜀	PROPN
cana-1172	117	48	(	(	PUNCT
cana-1172	117	49	�	�	PROPN
cana-1172	117	50	̇	̇	PROPN
cana-1172	117	51	�	�	PROPN
cana-1172	117	52	)	)	PUNCT
cana-1172	117	53	}	}	PUNCT
cana-1172	117	54	]	]	PUNCT
cana-1172	117	55	(	(	PUNCT
cana-1172	117	56	5.4	5.4	NUM
cana-1172	117	57	)	)	PUNCT
cana-1172	117	58	for	for	ADP
cana-1172	117	59	all	all	DET
cana-1172	117	60	�	�	PROPN
cana-1172	117	61	̇	̇	PROPN
cana-1172	117	62	�	�	PROPN
cana-1172	117	63	,	,	PUNCT
cana-1172	117	64	�	�	PROPN
cana-1172	117	65	̇	̇	PROPN
cana-1172	117	66	�	�	PROPN
cana-1172	117	67	,	,	PUNCT
cana-1172	117	68	�	�	PROPN
cana-1172	117	69	̇	̇	VERB
cana-1172	117	70	�	�	PROPN
cana-1172	117	71	∈	∈	PROPN
cana-1172	117	72	𝔊	𝔊	PROPN
cana-1172	117	73	theorem	theorem	VERB
cana-1172	117	74	5.8	5.8	NUM
cana-1172	117	75	every	every	DET
cana-1172	117	76	lukasz	lukasz	NOUN
cana-1172	117	77	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	117	78	set	set	VERB
cana-1172	117	79	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	117	80	𝜀	𝜀	NOUN
cana-1172	117	81	in	in	ADP
cana-1172	117	82	𝔊	𝔊	PROPN
cana-1172	117	83	is	be	AUX
cana-1172	117	84	a	a	DET
cana-1172	117	85	lukasz	lukasz	NOUN
cana-1172	117	86	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	117	87	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	PROPN
cana-1172	117	88	of	of	ADP
cana-1172	117	89	𝔊	𝔊	PROPN
cana-1172	117	90	if	if	SCONJ
cana-1172	117	91	𝑈	𝑈	PROPN
cana-1172	117	92	is	be	AUX
cana-1172	117	93	a	a	DET
cana-1172	117	94	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	117	95	ideal	ideal	NOUN
cana-1172	117	96	of	of	ADP
cana-1172	117	97	𝐵𝑀-algebra	𝐵𝑀-algebra	PROPN
cana-1172	117	98	𝔊.	𝔊.	PROPN
cana-1172	117	99	proof	proof	NOUN
cana-1172	117	100	for	for	ADP
cana-1172	117	101	instance	instance	NOUN
cana-1172	117	102	,	,	PUNCT
cana-1172	117	103	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	117	104	𝜀	𝜀	PROPN
cana-1172	117	105	is	be	AUX
cana-1172	117	106	a	a	DET
cana-1172	117	107	lukasz	lukasz	NOUN
cana-1172	117	108	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	117	109	set	set	NOUN
cana-1172	117	110	of	of	ADP
cana-1172	117	111	a	a	DET
cana-1172	117	112	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	117	113	ideal	ideal	NOUN
cana-1172	117	114	𝑈	𝑈	PROPN
cana-1172	117	115	in	in	ADP
cana-1172	117	116	𝔊.	𝔊.	PROPN
cana-1172	117	117	let	let	VERB
cana-1172	117	118	�	�	SYM
cana-1172	117	119	̇	̇	PROPN
cana-1172	117	120	�	�	PROPN
cana-1172	117	121	,	,	PUNCT
cana-1172	117	122	�	�	PROPN
cana-1172	117	123	̇	̇	VERB
cana-1172	117	124	�	�	PROPN
cana-1172	117	125	∈	∈	PROPN
cana-1172	117	126	𝔊	𝔊	PROPN
cana-1172	117	127	and	and	CCONJ
cana-1172	117	128	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	117	129	,	,	PUNCT
cana-1172	117	130	𝑢𝑏	𝑢𝑏	ADP
cana-1172	117	131	∈	∈	PROPN
cana-1172	117	132	(	(	PUNCT
cana-1172	117	133	0,1	0,1	NOUN
cana-1172	117	134	]	]	PUNCT
cana-1172	117	135	be	be	VERB
cana-1172	117	136	such	such	ADJ
cana-1172	117	137	that	that	SCONJ
cana-1172	117	138	[	[	X
cana-1172	117	139	(	(	PUNCT
cana-1172	117	140	�	�	PROPN
cana-1172	117	141	̇	̇	PROPN
cana-1172	117	142	�	�	PROPN
cana-1172	117	143	∗	∗	PROPN
cana-1172	117	144	�	�	PROPN
cana-1172	117	145	̇	̇	PROPN
cana-1172	117	146	�	�	PROPN
cana-1172	117	147	)	)	PUNCT
cana-1172	117	148	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	117	149	]	]	PUNCT
cana-1172	117	150	∈	∈	PROPN
cana-1172	117	151	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	117	152	𝜀	𝜀	NOUN
cana-1172	117	153	,	,	PUNCT
cana-1172	117	154	[	[	X
cana-1172	117	155	�	�	NOUN
cana-1172	117	156	̇	̇	PROPN
cana-1172	117	157	�	�	PROPN
cana-1172	117	158	𝑢𝑏⁄	𝑢𝑏⁄	NOUN
cana-1172	117	159	]	]	PUNCT
cana-1172	117	160	∈	∈	PROPN
cana-1172	117	161	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	117	162	𝜀	𝜀	PROPN
cana-1172	117	163	.	.	PUNCT
cana-1172	118	1	then	then	ADV
cana-1172	118	2	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	118	3	𝜀	𝜀	PROPN
cana-1172	118	4	(	(	PUNCT
cana-1172	118	5	�	�	PROPN
cana-1172	118	6	̇	̇	PROPN
cana-1172	118	7	�	�	PROPN
cana-1172	118	8	∗	∗	PROPN
cana-1172	118	9	�	�	PROPN
cana-1172	118	10	̇	̇	PROPN
cana-1172	118	11	�	�	PROPN
cana-1172	118	12	)	)	PUNCT
cana-1172	118	13	≥	≥	NOUN
cana-1172	118	14	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	118	15	and	and	CCONJ
cana-1172	118	16	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	118	17	𝜀	𝜀	PROPN
cana-1172	118	18	(	(	PUNCT
cana-1172	118	19	�	�	PROPN
cana-1172	118	20	̇	̇	PROPN
cana-1172	118	21	�	�	PROPN
cana-1172	118	22	)	)	PUNCT
cana-1172	118	23	≥	≥	NOUN
cana-1172	118	24	𝑢𝑏.	𝑢𝑏.	VERB
cana-1172	118	25	thus	thus	ADV
cana-1172	118	26	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	118	27	𝜀	𝜀	PROPN
cana-1172	118	28	(	(	PUNCT
cana-1172	118	29	�	�	PROPN
cana-1172	118	30	̇	̇	PROPN
cana-1172	118	31	�	�	PROPN
cana-1172	118	32	)	)	PUNCT
cana-1172	118	33	=	=	SYM
cana-1172	118	34	𝑚𝑎𝑥{0	𝑚𝑎𝑥{0	PROPN
cana-1172	118	35	,	,	PUNCT
cana-1172	118	36	𝑈(	𝑈(	PROPN
cana-1172	118	37	�	�	PROPN
cana-1172	118	38	̇	̇	PROPN
cana-1172	118	39	�	�	PROPN
cana-1172	118	40	)	)	PUNCT
cana-1172	118	41	+	+	NUM
cana-1172	118	42	𝜀	𝜀	X
cana-1172	118	43	−	−	NUM
cana-1172	118	44	1	1	NUM
cana-1172	118	45	}	}	PUNCT
cana-1172	118	46	[	[	X
cana-1172	118	47	∵	∵	X
cana-1172	118	48	(	(	PUNCT
cana-1172	118	49	3.1	3.1	NUM
cana-1172	118	50	)	)	PUNCT
cana-1172	118	51	]	]	PUNCT
cana-1172	118	52	≥	≥	X
cana-1172	118	53	𝑚𝑎𝑥{0	𝑚𝑎𝑥{0	PROPN
cana-1172	118	54	,	,	PUNCT
cana-1172	118	55	𝑚𝑖𝑛{𝑈(	𝑚𝑖𝑛{𝑈(	PROPN
cana-1172	118	56	�	�	PROPN
cana-1172	118	57	̇	̇	PROPN
cana-1172	118	58	�	�	PROPN
cana-1172	118	59	∗	∗	PROPN
cana-1172	118	60	�	�	PROPN
cana-1172	118	61	̇	̇	PROPN
cana-1172	118	62	�	�	PROPN
cana-1172	118	63	)	)	PUNCT
cana-1172	118	64	,	,	PUNCT
cana-1172	118	65	𝑈(	𝑈(	PROPN
cana-1172	118	66	�	�	PROPN
cana-1172	118	67	̇	̇	PROPN
cana-1172	118	68	�	�	PROPN
cana-1172	118	69	)	)	PUNCT
cana-1172	118	70	}	}	PUNCT
cana-1172	118	71	+	+	NUM
cana-1172	118	72	𝜀	𝜀	X
cana-1172	118	73	−	−	NUM
cana-1172	118	74	1	1	NUM
cana-1172	118	75	}	}	PUNCT
cana-1172	118	76	[	[	X
cana-1172	118	77	∵	∵	X
cana-1172	118	78	(	(	PUNCT
cana-1172	118	79	4.3	4.3	NUM
cana-1172	118	80	)	)	PUNCT
cana-1172	118	81	]	]	PUNCT
cana-1172	118	82	communications	communication	NOUN
cana-1172	118	83	on	on	ADP
cana-1172	118	84	applied	apply	VERB
cana-1172	118	85	nonlinear	nonlinear	ADJ
cana-1172	118	86	analysis	analysis	NOUN
cana-1172	118	87	issn	issn	NOUN
cana-1172	118	88	:	:	PUNCT
cana-1172	118	89	1074	1074	NUM
cana-1172	118	90	-	-	PUNCT
cana-1172	118	91	133x	133x	NUM
cana-1172	118	92	vol	vol	NOUN
cana-1172	118	93	31	31	NUM
cana-1172	118	94	no	no	NOUN
cana-1172	118	95	.	.	PUNCT
cana-1172	119	1	6s	6s	NUM
cana-1172	119	2	(	(	PUNCT
cana-1172	119	3	2024	2024	NUM
cana-1172	119	4	)	)	PUNCT
cana-1172	119	5	141	141	NUM
cana-1172	119	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1172	119	7	=	=	SYM
cana-1172	119	8	𝑚𝑎𝑥{0	𝑚𝑎𝑥{0	PROPN
cana-1172	119	9	,	,	PUNCT
cana-1172	119	10	𝑚𝑖𝑛{𝑈(	𝑚𝑖𝑛{𝑈(	PROPN
cana-1172	119	11	�	�	PROPN
cana-1172	119	12	̇	̇	PROPN
cana-1172	119	13	�	�	PROPN
cana-1172	119	14	∗	∗	PROPN
cana-1172	119	15	�	�	PROPN
cana-1172	119	16	̇	̇	PROPN
cana-1172	119	17	�	�	PROPN
cana-1172	119	18	)	)	PUNCT
cana-1172	119	19	+	+	NUM
cana-1172	119	20	𝜀	𝜀	X
cana-1172	119	21	−	−	NUM
cana-1172	119	22	1	1	NUM
cana-1172	119	23	,	,	PUNCT
cana-1172	119	24	𝑈(	𝑈(	PROPN
cana-1172	119	25	�	�	PROPN
cana-1172	119	26	̇	̇	PROPN
cana-1172	119	27	�	�	PROPN
cana-1172	119	28	)	)	PUNCT
cana-1172	119	29	+	+	NUM
cana-1172	119	30	𝜀	𝜀	X
cana-1172	119	31	−	−	NUM
cana-1172	119	32	1	1	NUM
cana-1172	119	33	}	}	PUNCT
cana-1172	119	34	}	}	PUNCT
cana-1172	119	35	=	=	SYM
cana-1172	119	36	𝑚𝑖𝑛{𝑚𝑎𝑥{0	𝑚𝑖𝑛{𝑚𝑎𝑥{0	NOUN
cana-1172	119	37	,	,	PUNCT
cana-1172	119	38	𝑈(	𝑈(	PROPN
cana-1172	119	39	�	�	PROPN
cana-1172	119	40	̇	̇	PROPN
cana-1172	119	41	�	�	PROPN
cana-1172	119	42	∗	∗	PROPN
cana-1172	119	43	�	�	PROPN
cana-1172	119	44	̇	̇	PROPN
cana-1172	119	45	�	�	PROPN
cana-1172	119	46	)	)	PUNCT
cana-1172	119	47	+	+	NUM
cana-1172	119	48	𝜀	𝜀	X
cana-1172	119	49	−	−	NUM
cana-1172	119	50	1	1	NUM
cana-1172	119	51	}	}	PUNCT
cana-1172	119	52	,	,	PUNCT
cana-1172	119	53	𝑚𝑎𝑥{0	𝑚𝑎𝑥{0	PROPN
cana-1172	119	54	,	,	PUNCT
cana-1172	119	55	𝑈(	𝑈(	PROPN
cana-1172	119	56	�	�	PROPN
cana-1172	119	57	̇	̇	PROPN
cana-1172	119	58	�	�	PROPN
cana-1172	119	59	)	)	PUNCT
cana-1172	119	60	+	+	NUM
cana-1172	119	61	𝜀	𝜀	X
cana-1172	119	62	−	−	NUM
cana-1172	119	63	1	1	NUM
cana-1172	119	64	}	}	PUNCT
cana-1172	119	65	}	}	PUNCT
cana-1172	119	66	=	=	SYM
cana-1172	119	67	𝑚𝑖𝑛{𝐿𝑈	𝑚𝑖𝑛{𝐿𝑈	NUM
cana-1172	119	68	𝜀	𝜀	X
cana-1172	119	69	(	(	PUNCT
cana-1172	119	70	�	�	PROPN
cana-1172	119	71	̇	̇	PROPN
cana-1172	119	72	�	�	PROPN
cana-1172	119	73	∗	∗	PROPN
cana-1172	119	74	�	�	PROPN
cana-1172	119	75	̇	̇	PROPN
cana-1172	119	76	�	�	PROPN
cana-1172	119	77	)	)	PUNCT
cana-1172	119	78	,	,	PUNCT
cana-1172	119	79	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	119	80	𝜀	𝜀	PROPN
cana-1172	119	81	(	(	PUNCT
cana-1172	119	82	�	�	PROPN
cana-1172	119	83	̇	̇	PROPN
cana-1172	119	84	�	�	PROPN
cana-1172	119	85	)	)	PUNCT
cana-1172	119	86	}	}	PUNCT
cana-1172	120	1	[	[	X
cana-1172	120	2	∵	∵	X
cana-1172	120	3	(	(	PUNCT
cana-1172	120	4	3.1	3.1	NUM
cana-1172	120	5	)	)	PUNCT
cana-1172	120	6	]	]	PUNCT
cana-1172	120	7	≥	≥	X
cana-1172	120	8	𝑚𝑖𝑛{𝑢𝑎	𝑚𝑖𝑛{𝑢𝑎	VERB
cana-1172	120	9	,	,	PUNCT
cana-1172	120	10	𝑢𝑏	𝑢𝑏	ADP
cana-1172	120	11	}	}	PUNCT
cana-1172	120	12	.	.	PUNCT
cana-1172	121	1	so	so	ADV
cana-1172	121	2	,	,	PUNCT
cana-1172	121	3	[	[	X
cana-1172	121	4	�	�	NOUN
cana-1172	121	5	̇	̇	PROPN
cana-1172	121	6	�	�	PROPN
cana-1172	121	7	𝑚𝑖𝑛{𝑢𝑎	𝑚𝑖𝑛{𝑢𝑎	VERB
cana-1172	121	8	,	,	PUNCT
cana-1172	121	9	𝑢𝑏}⁄	𝑢𝑏}⁄	PRON
cana-1172	121	10	]	]	PUNCT
cana-1172	121	11	∈	∈	PROPN
cana-1172	121	12	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	121	13	𝜀	𝜀	NOUN
cana-1172	121	14	.	.	PUNCT
cana-1172	122	1	hence	hence	ADV
cana-1172	122	2	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	122	3	𝜀	𝜀	PROPN
cana-1172	122	4	is	be	AUX
cana-1172	122	5	a	a	DET
cana-1172	122	6	lukasz	lukasz	NOUN
cana-1172	122	7	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	122	8	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	PROPN
cana-1172	122	9	of	of	ADP
cana-1172	122	10	𝔊.	𝔊.	NOUN
cana-1172	122	11	the	the	DET
cana-1172	122	12	subsequent	subsequent	ADJ
cana-1172	122	13	example	example	NOUN
cana-1172	122	14	demonstrates	demonstrate	VERB
cana-1172	122	15	why	why	SCONJ
cana-1172	122	16	the	the	DET
cana-1172	122	17	reverse	reverse	ADJ
cana-1172	122	18	portion	portion	NOUN
cana-1172	122	19	of	of	ADP
cana-1172	122	20	theorem	theorem	ADJ
cana-1172	122	21	5.8	5.8	NUM
cana-1172	122	22	is	be	AUX
cana-1172	122	23	false	false	ADJ
cana-1172	122	24	.	.	PUNCT
cana-1172	122	25	example	example	NOUN
cana-1172	122	26	5.9	5.9	NUM
cana-1172	122	27	consider	consider	VERB
cana-1172	122	28	the	the	DET
cana-1172	122	29	𝐵𝑀-algebra	𝐵𝑀-algebra	PROPN
cana-1172	122	30	set	set	VERB
cana-1172	122	31	𝔊	𝔊	PROPN
cana-1172	122	32	in	in	ADP
cana-1172	122	33	example	example	NOUN
cana-1172	122	34	3.4	3.4	NUM
cana-1172	122	35	and	and	CCONJ
cana-1172	122	36	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	122	37	set	set	VERB
cana-1172	122	38	𝑈	𝑈	PROPN
cana-1172	122	39	in	in	ADP
cana-1172	122	40	𝔊	𝔊	PROPN
cana-1172	122	41	defined	define	VERB
cana-1172	122	42	by	by	ADP
cana-1172	122	43	𝑈	𝑈	PROPN
cana-1172	122	44	:	:	PUNCT
cana-1172	122	45	𝔊	𝔊	PROPN
cana-1172	122	46	→	→	SYM
cana-1172	122	47	[	[	X
cana-1172	122	48	0,1	0,1	NUM
cana-1172	122	49	]	]	PUNCT
cana-1172	122	50	,	,	PUNCT
cana-1172	122	51	�	�	PROPN
cana-1172	122	52	̇	̇	PROPN
cana-1172	122	53	�	�	PROPN
cana-1172	122	54	↦	↦	PROPN
cana-1172	122	55	{	{	PUNCT
cana-1172	122	56	0.81	0.81	NUM
cana-1172	122	57	𝑖𝑓	𝑖𝑓	PRON
cana-1172	122	58	�	�	PROPN
cana-1172	122	59	̇	̇	NOUN
cana-1172	122	60	�	�	PROPN
cana-1172	122	61	=	=	SYM
cana-1172	122	62	0	0	NUM
cana-1172	122	63	0.42	0.42	NUM
cana-1172	122	64	𝑖𝑓	𝑖𝑓	PRON
cana-1172	122	65	�	�	PROPN
cana-1172	122	66	̇	̇	PROPN
cana-1172	122	67	�	�	PROPN
cana-1172	122	68	=	=	PRON
cana-1172	122	69	𝓅1̇	𝓅1̇	VERB
cana-1172	122	70	0.57	0.57	NUM
cana-1172	122	71	𝑖𝑓	𝑖𝑓	SYM
cana-1172	122	72	�	�	PROPN
cana-1172	122	73	̇	̇	PROPN
cana-1172	122	74	�	�	PROPN
cana-1172	122	75	=	=	SYM
cana-1172	122	76	𝓅2̇	𝓅2̇	PROPN
cana-1172	122	77	0.31	0.31	NUM
cana-1172	122	78	𝑖𝑓	𝑖𝑓	SYM
cana-1172	122	79	�	�	PROPN
cana-1172	122	80	̇	̇	PROPN
cana-1172	122	81	�	�	PROPN
cana-1172	122	82	=	=	PUNCT
cana-1172	122	83	𝓅3̇	𝓅3̇	NOUN
cana-1172	122	84	.	.	PUNCT
cana-1172	123	1	then	then	ADV
cana-1172	123	2	𝑈	𝑈	PROPN
cana-1172	123	3	is	be	AUX
cana-1172	123	4	not	not	PART
cana-1172	123	5	a	a	DET
cana-1172	123	6	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	123	7	ideal	ideal	NOUN
cana-1172	123	8	of	of	ADP
cana-1172	123	9	𝔊.	𝔊.	PROPN
cana-1172	123	10	since	since	SCONJ
cana-1172	123	11	𝑈(𝓅1̇	𝑈(𝓅1̇	PROPN
cana-1172	123	12	)	)	PUNCT
cana-1172	123	13	=	=	PUNCT
cana-1172	123	14	0.42	0.42	NUM
cana-1172	123	15	≱	≱	PROPN
cana-1172	123	16	0.57	0.57	NUM
cana-1172	123	17	=	=	PUNCT
cana-1172	123	18	𝑚𝑖𝑛{𝑈(𝓅1̇	𝑚𝑖𝑛{𝑈(𝓅1̇	ADJ
cana-1172	123	19	∗	∗	X
cana-1172	123	20	𝓅2̇	𝓅2̇	PROPN
cana-1172	123	21	)	)	PUNCT
cana-1172	123	22	,	,	PUNCT
cana-1172	123	23	𝑈(𝓅2̇	𝑈(𝓅2̇	PROPN
cana-1172	123	24	)	)	PUNCT
cana-1172	123	25	}	}	PUNCT
cana-1172	123	26	given	give	VERB
cana-1172	123	27	that	that	DET
cana-1172	123	28	𝜀	𝜀	NOUN
cana-1172	123	29	=	=	SYM
cana-1172	123	30	0.55	0.55	NUM
cana-1172	123	31	,	,	PUNCT
cana-1172	123	32	then	then	ADV
cana-1172	123	33	the	the	DET
cana-1172	123	34	lukasz	lukasz	PROPN
cana-1172	123	35	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	123	36	set	set	VERB
cana-1172	123	37	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	123	38	𝜀	𝜀	NOUN
cana-1172	123	39	of	of	ADP
cana-1172	123	40	𝑈	𝑈	PROPN
cana-1172	123	41	in	in	ADP
cana-1172	123	42	𝔊	𝔊	PROPN
cana-1172	123	43	is	be	AUX
cana-1172	123	44	provided	provide	VERB
cana-1172	123	45	as	as	ADP
cana-1172	123	46	below	below	ADV
cana-1172	123	47	:	:	PUNCT
cana-1172	123	48	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	123	49	𝜀	𝜀	NOUN
cana-1172	123	50	:	:	PUNCT
cana-1172	123	51	𝔊	𝔊	PROPN
cana-1172	123	52	→	→	SYM
cana-1172	123	53	[	[	X
cana-1172	123	54	0,1	0,1	NUM
cana-1172	123	55	]	]	PUNCT
cana-1172	123	56	,	,	PUNCT
cana-1172	123	57	�	�	PROPN
cana-1172	123	58	̇	̇	PROPN
cana-1172	123	59	�	�	PROPN
cana-1172	123	60	↦	↦	PROPN
cana-1172	123	61	{	{	PUNCT
cana-1172	123	62	0.36	0.36	NUM
cana-1172	123	63	𝑖𝑓	𝑖𝑓	PRON
cana-1172	123	64	�	�	PROPN
cana-1172	123	65	̇	̇	NOUN
cana-1172	123	66	�	�	PROPN
cana-1172	123	67	=	=	SYM
cana-1172	123	68	0	0	NUM
cana-1172	123	69	0	0	NUM
cana-1172	123	70	𝑖𝑓	𝑖𝑓	PRON
cana-1172	123	71	�	�	PROPN
cana-1172	123	72	̇	̇	PROPN
cana-1172	123	73	�	�	PROPN
cana-1172	123	74	=	=	PRON
cana-1172	123	75	𝓅1̇	𝓅1̇	VERB
cana-1172	123	76	0.12	0.12	NUM
cana-1172	123	77	𝑖𝑓	𝑖𝑓	PRON
cana-1172	123	78	�	�	PROPN
cana-1172	123	79	̇	̇	PROPN
cana-1172	123	80	�	�	PROPN
cana-1172	123	81	=	=	SYM
cana-1172	123	82	𝓅2̇	𝓅2̇	PROPN
cana-1172	123	83	0	0	NUM
cana-1172	123	84	𝑖𝑓	𝑖𝑓	SYM
cana-1172	123	85	�	�	PROPN
cana-1172	123	86	̇	̇	PROPN
cana-1172	123	87	�	�	PROPN
cana-1172	123	88	=	=	PUNCT
cana-1172	123	89	𝓅3̇	𝓅3̇	NOUN
cana-1172	123	90	.	.	PUNCT
cana-1172	124	1	and	and	CCONJ
cana-1172	124	2	it	it	PRON
cana-1172	124	3	is	be	AUX
cana-1172	124	4	a	a	DET
cana-1172	124	5	lukasz	lukasz	NOUN
cana-1172	124	6	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	124	7	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	PROPN
cana-1172	124	8	of	of	ADP
cana-1172	124	9	𝔊.	𝔊.	PROPN
cana-1172	124	10	theorem	theorem	VERB
cana-1172	124	11	5.10	5.10	NUM
cana-1172	124	12	every	every	DET
cana-1172	124	13	lukasz	lukasz	NOUN
cana-1172	124	14	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	124	15	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	PROPN
cana-1172	124	16	of	of	ADP
cana-1172	124	17	𝔊	𝔊	PROPN
cana-1172	124	18	is	be	AUX
cana-1172	124	19	a	a	DET
cana-1172	124	20	lukasz	lukasz	NOUN
cana-1172	124	21	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	124	22	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	PROPN
cana-1172	124	23	of	of	ADP
cana-1172	124	24	𝔊.	𝔊.	PROPN
cana-1172	124	25	proof	proof	NOUN
cana-1172	124	26	for	for	ADP
cana-1172	124	27	instance	instance	NOUN
cana-1172	124	28	,	,	PUNCT
cana-1172	124	29	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	124	30	𝜀	𝜀	PROPN
cana-1172	124	31	is	be	AUX
cana-1172	124	32	a	a	DET
cana-1172	124	33	lukasz	lukasz	NOUN
cana-1172	124	34	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	124	35	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	PROPN
cana-1172	124	36	of	of	ADP
cana-1172	124	37	𝔊.	𝔊.	PROPN
cana-1172	124	38	let	let	VERB
cana-1172	124	39	�	�	SYM
cana-1172	124	40	̇	̇	PROPN
cana-1172	124	41	�	�	PROPN
cana-1172	124	42	,	,	PUNCT
cana-1172	124	43	�	�	PROPN
cana-1172	124	44	̇	̇	VERB
cana-1172	124	45	�	�	PROPN
cana-1172	124	46	∈	∈	PROPN
cana-1172	124	47	𝔊	𝔊	PROPN
cana-1172	124	48	and	and	CCONJ
cana-1172	124	49	𝑢𝑎	𝑢𝑎	NOUN
cana-1172	124	50	,	,	PUNCT
cana-1172	124	51	𝑢𝑏	𝑢𝑏	ADP
cana-1172	124	52	∈	∈	PROPN
cana-1172	124	53	(	(	PUNCT
cana-1172	124	54	0,1	0,1	NOUN
cana-1172	124	55	]	]	PUNCT
cana-1172	124	56	be	be	VERB
cana-1172	124	57	such	such	ADJ
cana-1172	124	58	that	that	SCONJ
cana-1172	124	59	[	[	X
cana-1172	124	60	�	�	NOUN
cana-1172	124	61	̇	̇	VERB
cana-1172	124	62	�	�	PROPN
cana-1172	124	63	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	124	64	]	]	PUNCT
cana-1172	124	65	∈	∈	PROPN
cana-1172	124	66	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	124	67	𝜀	𝜀	NOUN
cana-1172	124	68	,	,	PUNCT
cana-1172	124	69	[	[	X
cana-1172	124	70	�	�	NOUN
cana-1172	124	71	̇	̇	PROPN
cana-1172	124	72	�	�	PROPN
cana-1172	124	73	𝑢𝑏⁄	𝑢𝑏⁄	NOUN
cana-1172	124	74	]	]	PUNCT
cana-1172	124	75	∈	∈	PROPN
cana-1172	124	76	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	124	77	𝜀	𝜀	NOUN
cana-1172	124	78	.	.	PUNCT
cana-1172	125	1	since	since	SCONJ
cana-1172	125	2	�	�	PROPN
cana-1172	125	3	̇	̇	PROPN
cana-1172	125	4	�	�	PROPN
cana-1172	125	5	∗	∗	PROPN
cana-1172	125	6	�	�	PROPN
cana-1172	125	7	̇	̇	PROPN
cana-1172	125	8	�	�	PROPN
cana-1172	125	9	≤	≤	PROPN
cana-1172	125	10	�	�	PROPN
cana-1172	125	11	̇	̇	VERB
cana-1172	125	12	�	�	PROPN
cana-1172	125	13	and	and	CCONJ
cana-1172	125	14	�	�	PROPN
cana-1172	125	15	̇	̇	PROPN
cana-1172	125	16	�	�	PROPN
cana-1172	125	17	∗	∗	NOUN
cana-1172	125	18	(	(	PUNCT
cana-1172	125	19	�	�	PROPN
cana-1172	125	20	̇	̇	PROPN
cana-1172	125	21	�	�	PROPN
cana-1172	125	22	∗	∗	PROPN
cana-1172	125	23	�	�	PROPN
cana-1172	125	24	̇	̇	PROPN
cana-1172	125	25	�	�	PROPN
cana-1172	125	26	)	)	PUNCT
cana-1172	125	27	=	=	SYM
cana-1172	126	1	0	0	NUM
cana-1172	126	2	we	we	PRON
cana-1172	126	3	have	have	AUX
cana-1172	126	4	[	[	X
cana-1172	126	5	(	(	PUNCT
cana-1172	126	6	�	�	PROPN
cana-1172	126	7	̇	̇	PROPN
cana-1172	126	8	�	�	PROPN
cana-1172	126	9	∗	∗	PROPN
cana-1172	126	10	�	�	PROPN
cana-1172	126	11	̇	̇	PROPN
cana-1172	126	12	�	�	PROPN
cana-1172	126	13	)	)	PUNCT
cana-1172	126	14	𝑢𝑎⁄	𝑢𝑎⁄	VERB
cana-1172	126	15	]	]	PUNCT
cana-1172	126	16	∈	∈	PROPN
cana-1172	126	17	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	126	18	𝜀	𝜀	NOUN
cana-1172	126	19	by	by	ADP
cana-1172	126	20	(	(	PUNCT
cana-1172	126	21	5.1	5.1	NUM
cana-1172	126	22	)	)	PUNCT
cana-1172	126	23	.	.	PUNCT
cana-1172	127	1	hence	hence	ADV
cana-1172	127	2	[	[	X
cana-1172	127	3	�	�	PROPN
cana-1172	127	4	̇	̇	PROPN
cana-1172	127	5	�	�	PROPN
cana-1172	127	6	𝑚𝑖𝑛{𝑢𝑎	𝑚𝑖𝑛{𝑢𝑎	VERB
cana-1172	127	7	,	,	PUNCT
cana-1172	127	8	𝑢𝑏}⁄	𝑢𝑏}⁄	PRON
cana-1172	127	9	]	]	PUNCT
cana-1172	128	1	∈	∈	PROPN
cana-1172	128	2	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	128	3	𝜀	𝜀	NOUN
cana-1172	128	4	by	by	ADP
cana-1172	128	5	(	(	PUNCT
cana-1172	128	6	3.5	3.5	NUM
cana-1172	128	7	)	)	PUNCT
cana-1172	128	8	,	,	PUNCT
cana-1172	128	9	and	and	CCONJ
cana-1172	128	10	so	so	ADV
cana-1172	128	11	[	[	X
cana-1172	128	12	(	(	PUNCT
cana-1172	128	13	�	�	PROPN
cana-1172	128	14	̇	̇	PROPN
cana-1172	128	15	�	�	PROPN
cana-1172	128	16	∗	∗	PROPN
cana-1172	128	17	�	�	PROPN
cana-1172	128	18	̇	̇	PROPN
cana-1172	128	19	�	�	PROPN
cana-1172	128	20	)	)	PUNCT
cana-1172	128	21	𝑚𝑖𝑛{𝑢𝑎	𝑚𝑖𝑛{𝑢𝑎	VERB
cana-1172	128	22	,	,	PUNCT
cana-1172	128	23	𝑢𝑏}⁄	𝑢𝑏}⁄	PRON
cana-1172	128	24	]	]	PUNCT
cana-1172	128	25	∈	∈	PROPN
cana-1172	128	26	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	128	27	𝜀	𝜀	NOUN
cana-1172	128	28	by	by	ADP
cana-1172	128	29	(	(	PUNCT
cana-1172	128	30	5.1	5.1	NUM
cana-1172	128	31	)	)	PUNCT
cana-1172	128	32	.	.	PUNCT
cana-1172	129	1	therefore	therefore	ADV
cana-1172	129	2	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	129	3	𝜀	𝜀	PROPN
cana-1172	129	4	is	be	AUX
cana-1172	129	5	a	a	DET
cana-1172	129	6	lukasz	lukasz	NOUN
cana-1172	129	7	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	129	8	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	PROPN
cana-1172	129	9	of	of	ADP
cana-1172	129	10	𝔊.	𝔊.	PROPN
cana-1172	129	11	the	the	DET
cana-1172	129	12	subsequent	subsequent	ADJ
cana-1172	129	13	example	example	NOUN
cana-1172	129	14	demonstrates	demonstrate	VERB
cana-1172	129	15	why	why	SCONJ
cana-1172	129	16	the	the	DET
cana-1172	129	17	reverse	reverse	ADJ
cana-1172	129	18	portion	portion	NOUN
cana-1172	129	19	of	of	ADP
cana-1172	129	20	theorem	theorem	ADJ
cana-1172	129	21	5.10	5.10	NUM
cana-1172	129	22	is	be	AUX
cana-1172	129	23	false	false	ADJ
cana-1172	129	24	.	.	PUNCT
cana-1172	130	1	communications	communication	NOUN
cana-1172	130	2	on	on	ADP
cana-1172	130	3	applied	apply	VERB
cana-1172	130	4	nonlinear	nonlinear	ADJ
cana-1172	130	5	analysis	analysis	NOUN
cana-1172	130	6	issn	issn	NOUN
cana-1172	130	7	:	:	PUNCT
cana-1172	130	8	1074	1074	NUM
cana-1172	130	9	-	-	PUNCT
cana-1172	130	10	133x	133x	NUM
cana-1172	130	11	vol	vol	NOUN
cana-1172	130	12	31	31	NUM
cana-1172	130	13	no	no	NOUN
cana-1172	130	14	.	.	PUNCT
cana-1172	131	1	6s	6s	NUM
cana-1172	131	2	(	(	PUNCT
cana-1172	131	3	2024	2024	NUM
cana-1172	131	4	)	)	PUNCT
cana-1172	131	5	142	142	NUM
cana-1172	131	6	https://internationalpubls.com	https://internationalpubls.com	X
cana-1172	131	7	example	example	NOUN
cana-1172	131	8	5.11	5.11	NUM
cana-1172	131	9	a	a	DET
cana-1172	131	10	set	set	NOUN
cana-1172	131	11	in	in	ADP
cana-1172	131	12	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	PROPN
cana-1172	131	13	𝔊	𝔊	PROPN
cana-1172	131	14	=	=	SYM
cana-1172	131	15	{	{	PUNCT
cana-1172	131	16	0	0	NUM
cana-1172	131	17	,	,	PUNCT
cana-1172	131	18	𝓅1̇	𝓅1̇	NOUN
cana-1172	131	19	,	,	PUNCT
cana-1172	131	20	𝓅2̇	𝓅2̇	PROPN
cana-1172	131	21	}	}	PUNCT
cana-1172	131	22	be	be	AUX
cana-1172	131	23	considered	consider	VERB
cana-1172	131	24	and	and	CCONJ
cana-1172	131	25	the	the	DET
cana-1172	131	26	table	table	NOUN
cana-1172	131	27	5.1	5.1	NUM
cana-1172	131	28	is	be	AUX
cana-1172	131	29	built	build	VERB
cana-1172	131	30	under	under	ADP
cana-1172	131	31	the	the	DET
cana-1172	131	32	"	"	PUNCT
cana-1172	131	33	∗	∗	NOUN
cana-1172	131	34	"	"	PUNCT
cana-1172	131	35	operation	operation	NOUN
cana-1172	131	36	∗	∗	NOUN
cana-1172	131	37	0	0	NUM
cana-1172	131	38	𝓅1̇	𝓅1̇	PROPN
cana-1172	131	39	𝓅2̇	𝓅2̇	PROPN
cana-1172	131	40	0	0	NUM
cana-1172	131	41	0	0	NUM
cana-1172	131	42	𝓅1̇	𝓅1̇	PROPN
cana-1172	131	43	𝓅2̇	𝓅2̇	PROPN
cana-1172	131	44	𝓅1̇	𝓅1̇	NOUN
cana-1172	131	45	𝓅1̇	𝓅1̇	X
cana-1172	131	46	0	0	NUM
cana-1172	131	47	𝓅1̇	𝓅1̇	PROPN
cana-1172	131	48	𝓅2̇	𝓅2̇	PROPN
cana-1172	131	49	𝓅2̇	𝓅2̇	PROPN
cana-1172	131	50	𝓅1̇	𝓅1̇	VERB
cana-1172	131	51	0	0	NUM
cana-1172	131	52	table	table	NOUN
cana-1172	131	53	5.1	5.1	NUM
cana-1172	131	54	cayley	cayley	NOUN
cana-1172	131	55	table	table	NOUN
cana-1172	131	56	with	with	ADP
cana-1172	131	57	respect	respect	NOUN
cana-1172	131	58	to	to	ADP
cana-1172	131	59	"	"	PUNCT
cana-1172	131	60	∗	∗	NOUN
cana-1172	131	61	"	"	PUNCT
cana-1172	131	62	defining	define	VERB
cana-1172	131	63	a	a	DET
cana-1172	131	64	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	131	65	set	set	VERB
cana-1172	131	66	𝑈	𝑈	PROPN
cana-1172	131	67	in	in	ADP
cana-1172	131	68	𝔊	𝔊	PROPN
cana-1172	131	69	as	as	SCONJ
cana-1172	131	70	follows	follow	VERB
cana-1172	131	71	𝑈	𝑈	PROPN
cana-1172	131	72	:	:	PUNCT
cana-1172	131	73	𝔊	𝔊	PROPN
cana-1172	131	74	→	→	SYM
cana-1172	131	75	[	[	X
cana-1172	131	76	0,1	0,1	NUM
cana-1172	131	77	]	]	PUNCT
cana-1172	131	78	,	,	PUNCT
cana-1172	131	79	�	�	PROPN
cana-1172	131	80	̇	̇	PROPN
cana-1172	131	81	�	�	PROPN
cana-1172	131	82	↦	↦	PROPN
cana-1172	131	83	{	{	PUNCT
cana-1172	131	84	0.84	0.84	NUM
cana-1172	131	85	𝑖𝑓	𝑖𝑓	PRON
cana-1172	131	86	�	�	PROPN
cana-1172	131	87	̇	̇	PROPN
cana-1172	131	88	�	�	PROPN
cana-1172	131	89	=	=	SYM
cana-1172	131	90	0	0	NUM
cana-1172	131	91	0.72	0.72	NUM
cana-1172	131	92	𝑖𝑓	𝑖𝑓	PRON
cana-1172	131	93	�	�	PROPN
cana-1172	131	94	̇	̇	PROPN
cana-1172	131	95	�	�	PROPN
cana-1172	131	96	=	=	PRON
cana-1172	131	97	𝓅1̇	𝓅1̇	VERB
cana-1172	131	98	0.51	0.51	NUM
cana-1172	131	99	𝑖𝑓	𝑖𝑓	SYM
cana-1172	131	100	�	�	PROPN
cana-1172	131	101	̇	̇	PROPN
cana-1172	131	102	�	�	PROPN
cana-1172	131	103	=	=	SYM
cana-1172	131	104	𝓅2̇	𝓅2̇	PROPN
cana-1172	131	105	.	.	PUNCT
cana-1172	132	1	given	give	VERB
cana-1172	132	2	that	that	DET
cana-1172	132	3	𝜀	𝜀	NOUN
cana-1172	132	4	=	=	SYM
cana-1172	132	5	0.58	0.58	NUM
cana-1172	132	6	,	,	PUNCT
cana-1172	132	7	the	the	DET
cana-1172	132	8	𝜀-lukasz	𝜀-lukasz	NOUN
cana-1172	132	9	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	132	10	set	set	VERB
cana-1172	132	11	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	132	12	𝜀	𝜀	NOUN
cana-1172	132	13	of	of	ADP
cana-1172	132	14	𝑈	𝑈	PROPN
cana-1172	132	15	in	in	ADP
cana-1172	132	16	𝔊	𝔊	PROPN
cana-1172	132	17	is	be	AUX
cana-1172	132	18	provided	provide	VERB
cana-1172	132	19	as	as	ADP
cana-1172	132	20	below	below	ADP
cana-1172	132	21	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	132	22	𝜀	𝜀	NOUN
cana-1172	132	23	:	:	PUNCT
cana-1172	132	24	𝔊	𝔊	PROPN
cana-1172	132	25	→	→	SYM
cana-1172	132	26	[	[	X
cana-1172	132	27	0,1	0,1	NUM
cana-1172	132	28	]	]	PUNCT
cana-1172	132	29	,	,	PUNCT
cana-1172	132	30	�	�	PROPN
cana-1172	132	31	̇	̇	PROPN
cana-1172	132	32	�	�	PROPN
cana-1172	132	33	↦	↦	PROPN
cana-1172	132	34	{	{	PUNCT
cana-1172	132	35	0.42	0.42	NUM
cana-1172	132	36	𝑖𝑓	𝑖𝑓	NUM
cana-1172	132	37	�	�	PROPN
cana-1172	132	38	̇	̇	NOUN
cana-1172	132	39	�	�	PROPN
cana-1172	132	40	=	=	SYM
cana-1172	132	41	0	0	NUM
cana-1172	132	42	0.3	0.3	NUM
cana-1172	132	43	𝑖𝑓	𝑖𝑓	PRON
cana-1172	132	44	�	�	PROPN
cana-1172	132	45	̇	̇	PROPN
cana-1172	132	46	�	�	PROPN
cana-1172	132	47	=	=	PRON
cana-1172	132	48	𝓅1̇	𝓅1̇	VERB
cana-1172	132	49	0.09	0.09	NUM
cana-1172	132	50	𝑖𝑓	𝑖𝑓	NUM
cana-1172	132	51	�	�	PROPN
cana-1172	132	52	̇	̇	PROPN
cana-1172	132	53	�	�	PROPN
cana-1172	132	54	=	=	SYM
cana-1172	132	55	𝓅2̇	𝓅2̇	PROPN
cana-1172	132	56	typically	typically	ADV
cana-1172	132	57	,	,	PUNCT
cana-1172	132	58	it	it	PRON
cana-1172	132	59	is	be	AUX
cana-1172	132	60	verified	verify	VERB
cana-1172	132	61	that	that	SCONJ
cana-1172	132	62	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	132	63	𝜀	𝜀	PROPN
cana-1172	132	64	is	be	AUX
cana-1172	132	65	an	an	DET
cana-1172	132	66	𝜀-lukasz	𝜀-lukasz	NOUN
cana-1172	132	67	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	132	68	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	𝐵𝑀-𝑎𝑙𝑔𝑒𝑏𝑟𝑎	PROPN
cana-1172	132	69	of	of	ADP
cana-1172	132	70	𝔊.	𝔊.	PROPN
cana-1172	132	71	but	but	CCONJ
cana-1172	132	72	𝐿𝑈	𝐿𝑈	PROPN
cana-1172	132	73	𝜀	𝜀	PROPN
cana-1172	132	74	is	be	AUX
cana-1172	132	75	not	not	PART
cana-1172	132	76	a	a	DET
cana-1172	132	77	lukasz	lukasz	NOUN
cana-1172	132	78	ℱ𝑢𝑧𝑧𝑦	ℱ𝑢𝑧𝑧𝑦	PROPN
cana-1172	132	79	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	𝐵𝑀-𝑖𝑑𝑒𝑎𝑙	PROPN
cana-1172	132	80	of	of	ADP
cana-1172	132	81	𝔊	𝔊	PROPN
cana-1172	132	82	because	because	SCONJ
cana-1172	132	83	of	of	ADP
cana-1172	132	84	𝑈(𝓅2̇	𝑈(𝓅2̇	NOUN
cana-1172	132	85	)	)	PUNCT
cana-1172	132	86	=	=	SYM
cana-1172	132	87	0.09	0.09	NUM
cana-1172	132	88	≱	≱	PROPN
cana-1172	132	89	0.3	0.3	NUM
cana-1172	132	90	=	=	PUNCT
cana-1172	132	91	𝑚𝑖𝑛{𝑈(𝓅2̇	𝑚𝑖𝑛{𝑈(𝓅2̇	PROPN
cana-1172	132	92	∗	∗	NOUN
cana-1172	132	93	𝓅1̇	𝓅1̇	NOUN
cana-1172	132	94	)	)	PUNCT
cana-1172	132	95	,	,	PUNCT
cana-1172	132	96	𝑈(𝓅1̇	𝑈(𝓅1̇	NOUN
cana-1172	132	97	)	)	PUNCT
cana-1172	132	98	}	}	PUNCT
cana-1172	132	99	.	.	PUNCT
cana-1172	133	1	references	reference	NOUN
cana-1172	133	2	[	[	X
cana-1172	133	3	1	1	NUM
cana-1172	133	4	]	]	PUNCT
cana-1172	133	5	a.	a.	NOUN
cana-1172	133	6	paad	paad	NOUN
cana-1172	133	7	and	and	CCONJ
cana-1172	133	8	a.	a.	NOUN
cana-1172	133	9	jafari	jafari	PROPN
cana-1172	133	10	,	,	PUNCT
cana-1172	133	11	n	n	CCONJ
cana-1172	133	12	-	-	ADJ
cana-1172	133	13	fold	fold	ADJ
cana-1172	133	14	obstinate	obstinate	NOUN
cana-1172	133	15	and	and	CCONJ
cana-1172	133	16	n	n	CCONJ
cana-1172	133	17	-	-	ADJ
cana-1172	133	18	fold	fold	ADJ
cana-1172	133	19	fantastic	fantastic	ADJ
cana-1172	133	20	(	(	PUNCT
cana-1172	133	21	pre)filters	pre)filter	NOUN
cana-1172	133	22	of	of	ADP
cana-1172	133	23	eq	eq	NOUN
cana-1172	133	24	-	-	PUNCT
cana-1172	133	25	algebras	algebras	PROPN
cana-1172	133	26	,	,	PUNCT
cana-1172	133	27	j.	j.	PROPN
cana-1172	133	28	algebra	algebra	PROPN
cana-1172	133	29	relat	relat	PROPN
cana-1172	133	30	.	.	PUNCT
cana-1172	134	1	topics	topic	NOUN
cana-1172	134	2	,	,	PUNCT
cana-1172	134	3	(	(	PUNCT
cana-1172	134	4	1	1	X
cana-1172	134	5	)	)	PUNCT
cana-1172	134	6	9	9	NUM
cana-1172	134	7	(	(	PUNCT
cana-1172	134	8	2021	2021	NUM
cana-1172	134	9	)	)	PUNCT
cana-1172	134	10	,	,	PUNCT
cana-1172	134	11	31–50	31–50	NUM
cana-1172	134	12	.	.	PUNCT
cana-1172	135	1	[	[	X
cana-1172	135	2	2	2	NUM
cana-1172	135	3	]	]	PUNCT
cana-1172	135	4	c.	c.	PROPN
cana-1172	135	5	b.	b.	PROPN
cana-1172	135	6	kim	kim	PROPN
cana-1172	135	7	,	,	PUNCT
cana-1172	135	8	h.	h.	PROPN
cana-1172	135	9	s.	s.	PROPN
cana-1172	135	10	kim	kim	PROPN
cana-1172	135	11	,	,	PUNCT
cana-1172	135	12	on	on	ADP
cana-1172	135	13	bm	bm	PROPN
cana-1172	135	14	-	-	NOUN
cana-1172	135	15	algebra	algebra	PROPN
cana-1172	135	16	,	,	PUNCT
cana-1172	135	17	sci	sci	PROPN
cana-1172	135	18	.	.	PROPN
cana-1172	135	19	math	math	PROPN
cana-1172	135	20	,	,	PUNCT
cana-1172	135	21	japan	japan	PROPN
cana-1172	135	22	,	,	PUNCT
cana-1172	135	23	63	63	NUM
cana-1172	135	24	(	(	PUNCT
cana-1172	135	25	2006	2006	NUM
cana-1172	135	26	)	)	PUNCT
cana-1172	135	27	,	,	PUNCT
cana-1172	135	28	421	421	NUM
cana-1172	135	29	-	-	SYM
cana-1172	135	30	427	427	NUM
cana-1172	135	31	.	.	PUNCT
cana-1172	136	1	[	[	X
cana-1172	136	2	3	3	X
cana-1172	136	3	]	]	PUNCT
cana-1172	136	4	h.	h.	PROPN
cana-1172	136	5	s.	s.	PROPN
cana-1172	136	6	kim	kim	PROPN
cana-1172	136	7	and	and	CCONJ
cana-1172	136	8	y.	y.	PROPN
cana-1172	136	9	h.	h.	PROPN
cana-1172	136	10	kim	kim	PROPN
cana-1172	136	11	,	,	PUNCT
cana-1172	136	12	on	on	ADP
cana-1172	136	13	be	be	AUX
cana-1172	136	14	-	-	PUNCT
cana-1172	136	15	algebras	algebra	NOUN
cana-1172	136	16	,	,	PUNCT
cana-1172	136	17	sci	sci	PROPN
cana-1172	136	18	.	.	PROPN
cana-1172	136	19	math	math	PROPN
cana-1172	136	20	.	.	PUNCT
cana-1172	137	1	japan	japan	PROPN
cana-1172	137	2	,	,	PUNCT
cana-1172	137	3	66	66	NUM
cana-1172	137	4	(	(	PUNCT
cana-1172	137	5	2007	2007	NUM
cana-1172	137	6	)	)	PUNCT
cana-1172	137	7	,	,	PUNCT
cana-1172	137	8	113	113	NUM
cana-1172	137	9	-	-	SYM
cana-1172	137	10	116	116	NUM
cana-1172	137	11	.	.	PUNCT
cana-1172	138	1	[	[	X
cana-1172	138	2	4	4	X
cana-1172	138	3	]	]	PUNCT
cana-1172	138	4	k.	k.	PROPN
cana-1172	138	5	iseki	iseki	PROPN
cana-1172	138	6	and	and	CCONJ
cana-1172	138	7	s.	s.	PROPN
cana-1172	138	8	tanaka	tanaka	PROPN
cana-1172	138	9	,	,	PUNCT
cana-1172	138	10	an	an	DET
cana-1172	138	11	introduction	introduction	NOUN
cana-1172	138	12	to	to	ADP
cana-1172	138	13	the	the	DET
cana-1172	138	14	theory	theory	NOUN
cana-1172	138	15	of	of	ADP
cana-1172	138	16	bck	bck	PROPN
cana-1172	138	17	-	-	PUNCT
cana-1172	138	18	algebras	algebras	PROPN
cana-1172	138	19	,	,	PUNCT
cana-1172	138	20	math	math	NOUN
cana-1172	138	21	.	.	PUNCT
cana-1172	139	1	japon	japon	PROPN
cana-1172	139	2	.	.	PUNCT
cana-1172	140	1	23	23	NUM
cana-1172	140	2	(	(	PUNCT
cana-1172	140	3	1978	1978	NUM
cana-1172	140	4	)	)	PUNCT
cana-1172	140	5	,	,	PUNCT
cana-1172	140	6	1–26	1–26	NOUN
cana-1172	140	7	.	.	PUNCT
cana-1172	141	1	[	[	X
cana-1172	141	2	5	5	X
cana-1172	141	3	]	]	PUNCT
cana-1172	141	4	k.	k.	PROPN
cana-1172	141	5	iseki	iseki	PROPN
cana-1172	141	6	,	,	PUNCT
cana-1172	141	7	on	on	ADP
cana-1172	141	8	bci	bci	NOUN
cana-1172	141	9	-	-	PUNCT
cana-1172	141	10	algebras	algebra	NOUN
cana-1172	141	11	,	,	PUNCT
cana-1172	141	12	math	math	NOUN
cana-1172	141	13	.	.	PUNCT
cana-1172	142	1	seminar	seminar	NOUN
cana-1172	142	2	notes	note	NOUN
cana-1172	142	3	,	,	PUNCT
cana-1172	142	4	8	8	NUM
cana-1172	142	5	(	(	PUNCT
cana-1172	142	6	1980	1980	NUM
cana-1172	142	7	)	)	PUNCT
cana-1172	142	8	,	,	PUNCT
cana-1172	142	9	125–130	125–130	NUM
cana-1172	142	10	.	.	PUNCT
cana-1172	143	1	[	[	X
cana-1172	143	2	6	6	NUM
cana-1172	143	3	]	]	PUNCT
cana-1172	143	4	l.	l.	PROPN
cana-1172	143	5	a.	a.	PROPN
cana-1172	143	6	zadeh	zadeh	PROPN
cana-1172	143	7	,	,	PUNCT
cana-1172	143	8	f𝑢𝑧𝑧𝑦	f𝑢𝑧𝑧𝑦	NOUN
cana-1172	143	9	sets	set	NOUN
cana-1172	143	10	,	,	PUNCT
cana-1172	143	11	information	information	NOUN
cana-1172	143	12	and	and	CCONJ
cana-1172	143	13	control	control	NOUN
cana-1172	143	14	,	,	PUNCT
cana-1172	143	15	(	(	PUNCT
cana-1172	143	16	3	3	X
cana-1172	143	17	)	)	PUNCT
cana-1172	143	18	8	8	NUM
cana-1172	143	19	(	(	PUNCT
cana-1172	143	20	1965	1965	NUM
cana-1172	143	21	)	)	PUNCT
cana-1172	143	22	,	,	PUNCT
cana-1172	143	23	338–353	338–353	NUM
cana-1172	143	24	.	.	PUNCT
cana-1172	144	1	[	[	X
cana-1172	144	2	7	7	X
cana-1172	144	3	]	]	X
cana-1172	144	4	y.b	y.b	PROPN
cana-1172	144	5	.	.	PROPN
cana-1172	144	6	jun	jun	PROPN
cana-1172	144	7	,	,	PUNCT
cana-1172	144	8	lukasiewicz	lukasiewicz	VERB
cana-1172	144	9	f𝑢𝑧𝑧𝑦	f𝑢𝑧𝑧𝑦	ADJ
cana-1172	144	10	ideals	ideal	NOUN
cana-1172	144	11	in	in	ADP
cana-1172	144	12	bck	bck	NOUN
cana-1172	144	13	-	-	PUNCT
cana-1172	144	14	algebras	algebras	PROPN
cana-1172	144	15	and	and	CCONJ
cana-1172	144	16	bci	bci	NOUN
cana-1172	144	17	-	-	PUNCT
cana-1172	144	18	algebras	algebra	NOUN
cana-1172	144	19	,	,	PUNCT
cana-1172	144	20	journal	journal	NOUN
cana-1172	144	21	of	of	ADP
cana-1172	144	22	algebra	algebra	PROPN
cana-1172	144	23	and	and	CCONJ
cana-1172	144	24	related	related	ADJ
cana-1172	144	25	topics	topic	NOUN
cana-1172	144	26	,	,	PUNCT
cana-1172	144	27	vol	vol	NOUN
cana-1172	144	28	.	.	PROPN
cana-1172	145	1	11	11	NUM
cana-1172	145	2	,	,	PUNCT
cana-1172	145	3	no	no	DET
cana-1172	145	4	1	1	NUM
cana-1172	145	5	,	,	PUNCT
cana-1172	145	6	(	(	PUNCT
cana-1172	145	7	2023	2023	NUM
cana-1172	145	8	)	)	PUNCT
cana-1172	145	9	,	,	PUNCT
cana-1172	145	10	pp	pp	ADP
cana-1172	145	11	1	1	NUM
cana-1172	145	12	-	-	SYM
cana-1172	145	13	14	14	NUM
cana-1172	145	14	.	.	PUNCT
cana-1172	146	1	[	[	X
cana-1172	146	2	8	8	NUM
cana-1172	146	3	]	]	X
cana-1172	146	4	y.	y.	PROPN
cana-1172	146	5	b.	b.	PROPN
cana-1172	146	6	jun	jun	PROPN
cana-1172	146	7	,	,	PUNCT
cana-1172	146	8	lukasiewicz	lukasiewicz	VERB
cana-1172	146	9	f𝑢𝑧𝑧𝑦	f𝑢𝑧𝑧𝑦	ADJ
cana-1172	146	10	sub	sub	NOUN
cana-1172	146	11	algebras	algebras	PROPN
cana-1172	146	12	in	in	ADP
cana-1172	146	13	bck	bck	PROPN
cana-1172	146	14	-	-	PUNCT
cana-1172	146	15	algebras	algebras	PROPN
cana-1172	146	16	and	and	CCONJ
cana-1172	146	17	bci	bci	NOUN
cana-1172	146	18	-	-	PUNCT
cana-1172	146	19	algebras	algebras	PROPN
cana-1172	146	20	,	,	PUNCT
cana-1172	146	21	ann	ann	PROPN
cana-1172	146	22	.	.	PROPN
cana-1172	146	23	f𝑢𝑧𝑧𝑦	f𝑢𝑧𝑧𝑦	PROPN
cana-1172	146	24	math	math	PROPN
cana-1172	146	25	.	.	PUNCT
cana-1172	147	1	inform	inform	NOUN
cana-1172	147	2	.	.	PUNCT
cana-1172	148	1	(	(	PUNCT
cana-1172	148	2	2	2	NUM
cana-1172	148	3	)	)	PUNCT
cana-1172	148	4	23	23	NUM
cana-1172	148	5	(	(	PUNCT
cana-1172	148	6	2022	2022	NUM
cana-1172	148	7	)	)	PUNCT
cana-1172	148	8	,	,	PUNCT
cana-1172	148	9	213–223	213–223	NUM
cana-1172	148	10	.	.	PUNCT
cana-1172	149	1	[	[	X
cana-1172	149	2	9	9	NUM
cana-1172	149	3	]	]	X
cana-1172	149	4	y.	y.	PROPN
cana-1172	149	5	b.	b.	PROPN
cana-1172	149	6	jun	jun	PROPN
cana-1172	149	7	,	,	PUNCT
cana-1172	149	8	s.	s.	PROPN
cana-1172	149	9	m.	m.	PROPN
cana-1172	149	10	hong	hong	PROPN
cana-1172	149	11	,	,	PUNCT
cana-1172	149	12	s.	s.	PROPN
cana-1172	149	13	j.	j.	PROPN
cana-1172	149	14	kim	kim	PROPN
cana-1172	149	15	and	and	CCONJ
cana-1172	149	16	s.	s.	PROPN
cana-1172	149	17	z.	z.	PROPN
cana-1172	149	18	song	song	PROPN
cana-1172	149	19	,	,	PUNCT
cana-1172	149	20	f𝑢𝑧𝑧𝑦	f𝑢𝑧𝑧𝑦	ADJ
cana-1172	149	21	ideals	ideal	NOUN
cana-1172	149	22	and	and	CCONJ
cana-1172	149	23	f𝑢𝑧𝑧𝑦	f𝑢𝑧𝑧𝑦	ADJ
cana-1172	149	24	sub	sub	NOUN
cana-1172	149	25	algebras	algebra	NOUN
cana-1172	149	26	of	of	ADP
cana-1172	149	27	bck	bck	PROPN
cana-1172	149	28	-	-	PUNCT
cana-1172	149	29	algebras	algebras	PROPN
cana-1172	149	30	,	,	PUNCT
cana-1172	149	31	j.	j.	PROPN
cana-1172	149	32	f𝑢𝑧𝑧𝑦	f𝑢𝑧𝑧𝑦	PROPN
cana-1172	149	33	math	math	PROPN
cana-1172	149	34	.	.	PUNCT
cana-1172	150	1	7	7	NUM
cana-1172	150	2	(	(	PUNCT
cana-1172	150	3	1999	1999	NUM
cana-1172	150	4	)	)	PUNCT
cana-1172	150	5	,	,	PUNCT
cana-1172	150	6	411–418	411–418	NUM
cana-1172	150	7	.	.	PUNCT
cana-1172	151	1	[	[	X
cana-1172	151	2	10	10	NUM
cana-1172	151	3	]	]	X
cana-1172	151	4	y.	y.	PROPN
cana-1172	151	5	b.	b.	PROPN
cana-1172	151	6	jun	jun	PROPN
cana-1172	151	7	,	,	PUNCT
cana-1172	151	8	s.	s.	PROPN
cana-1172	151	9	s.	s.	PROPN
cana-1172	151	10	ahn	ahn	PROPN
cana-1172	151	11	,	,	PUNCT
cana-1172	151	12	lukasiewicz	lukasiewicz	VERB
cana-1172	151	13	f𝑢𝑧𝑧𝑦	f𝑢𝑧𝑧𝑦	NOUN
cana-1172	151	14	be	be	AUX
cana-1172	151	15	-	-	PUNCT
cana-1172	151	16	algebras	algebra	VERB
cana-1172	151	17	and	and	CCONJ
cana-1172	151	18	be	be	NOUN
cana-1172	151	19	-	-	PUNCT
cana-1172	151	20	filters	filter	NOUN
cana-1172	151	21	,	,	PUNCT
cana-1172	151	22	european	european	ADJ
cana-1172	151	23	journal	journal	PROPN
cana-1172	151	24	of	of	ADP
cana-1172	151	25	pure	pure	ADJ
cana-1172	151	26	and	and	CCONJ
cana-1172	151	27	applied	applied	ADJ
cana-1172	151	28	mathematics,15(2002	mathematics,15(2002	NOUN
cana-1172	151	29	)	)	PUNCT
cana-1172	151	30	,	,	PUNCT
cana-1172	151	31	924	924	NUM
cana-1172	151	32	-	-	SYM
cana-1172	151	33	937	937	NUM
cana-1172	151	34	.	.	PUNCT