id	sid	tid	token	lemma	pos
iajs-2724	1	1	47	47	NUM
iajs-2724	1	2	this	this	DET
iajs-2724	1	3	work	work	NOUN
iajs-2724	1	4	is	be	AUX
iajs-2724	1	5	licensed	license	VERB
iajs-2724	1	6	under	under	ADP
iajs-2724	1	7	a	a	DET
iajs-2724	1	8	creative	creative	ADJ
iajs-2724	1	9	commons	common	NOUN
iajs-2724	1	10	attribution	attribution	NOUN
iajs-2724	1	11	4.0	4.0	NUM
iajs-2724	1	12	international	international	ADJ
iajs-2724	1	13	license	license	NOUN
iajs-2724	1	14	.	.	PUNCT
iajs-2724	2	1	cubic	cubic	ADJ
iajs-2724	2	2	bipolar	bipolar	ADJ
iajs-2724	2	3	fuzzy	fuzzy	ADJ
iajs-2724	2	4	ideals	ideal	NOUN
iajs-2724	2	5	with	with	ADP
iajs-2724	2	6	thresholds	threshold	NOUN
iajs-2724	2	7	(	(	PUNCT
iajs-2724	2	8	α	α	X
iajs-2724	2	9	,	,	PUNCT
iajs-2724	2	10	β	β	NOUN
iajs-2724	2	11	)	)	PUNCT
iajs-2724	2	12	,	,	PUNCT
iajs-2724	2	13	(	(	PUNCT
iajs-2724	2	14	𝛚	𝛚	NOUN
iajs-2724	2	15	,	,	PUNCT
iajs-2724	2	16	𝛝	𝛝	PROPN
iajs-2724	2	17	)	)	PUNCT
iajs-2724	2	18	of	of	ADP
iajs-2724	2	19	a	a	DET
iajs-2724	2	20	semigroup	semigroup	NOUN
iajs-2724	2	21	in	in	ADP
iajs-2724	2	22	ku	ku	PROPN
iajs-2724	2	23	-	-	PUNCT
iajs-2724	2	24	algebra	algebra	PROPN
iajs-2724	2	25	abstract	abstract	NOUN
iajs-2724	2	26	in	in	ADP
iajs-2724	2	27	this	this	DET
iajs-2724	2	28	paper	paper	NOUN
iajs-2724	2	29	,	,	PUNCT
iajs-2724	2	30	we	we	PRON
iajs-2724	2	31	introduce	introduce	VERB
iajs-2724	2	32	the	the	DET
iajs-2724	2	33	concept	concept	NOUN
iajs-2724	2	34	of	of	ADP
iajs-2724	2	35	cubic	cubic	ADJ
iajs-2724	2	36	bipolar	bipolar	ADJ
iajs-2724	2	37	-	-	PUNCT
iajs-2724	2	38	fuzzy	fuzzy	ADJ
iajs-2724	2	39	ideals	ideal	NOUN
iajs-2724	2	40	with	with	ADP
iajs-2724	2	41	thresholds	threshold	NOUN
iajs-2724	2	42	(	(	PUNCT
iajs-2724	2	43	α	α	NOUN
iajs-2724	2	44	,	,	PUNCT
iajs-2724	2	45	β),(ω,ϑ	β),(ω,ϑ	NUM
iajs-2724	2	46	)	)	PUNCT
iajs-2724	2	47	of	of	ADP
iajs-2724	2	48	a	a	DET
iajs-2724	2	49	semigroup	semigroup	NOUN
iajs-2724	2	50	in	in	ADP
iajs-2724	2	51	ku	ku	PROPN
iajs-2724	2	52	-	-	PUNCT
iajs-2724	2	53	algebra	algebra	PROPN
iajs-2724	2	54	as	as	ADP
iajs-2724	2	55	a	a	DET
iajs-2724	2	56	generalization	generalization	NOUN
iajs-2724	2	57	of	of	ADP
iajs-2724	2	58	sets	set	NOUN
iajs-2724	2	59	and	and	CCONJ
iajs-2724	2	60	in	in	ADP
iajs-2724	2	61	short	short	ADJ
iajs-2724	2	62	(	(	PUNCT
iajs-2724	2	63	cbf	cbf	PROPN
iajs-2724	2	64	)	)	PUNCT
iajs-2724	2	65	.	.	PUNCT
iajs-2724	3	1	firstly	firstly	ADV
iajs-2724	3	2	,	,	PUNCT
iajs-2724	3	3	a	a	DET
iajs-2724	3	4	(	(	PUNCT
iajs-2724	3	5	cbf	cbf	PROPN
iajs-2724	3	6	)	)	PUNCT
iajs-2724	3	7	subku	subku	NOUN
iajs-2724	3	8	-	-	PUNCT
iajs-2724	3	9	semigroup	semigroup	NOUN
iajs-2724	3	10	with	with	ADP
iajs-2724	3	11	a	a	DET
iajs-2724	3	12	threshold	threshold	NOUN
iajs-2724	3	13	(	(	PUNCT
iajs-2724	3	14	α	α	NOUN
iajs-2724	3	15	,	,	PUNCT
iajs-2724	3	16	β),(ω,ϑ	β),(ω,ϑ	PUNCT
iajs-2724	3	17	)	)	PUNCT
iajs-2724	3	18	and	and	CCONJ
iajs-2724	3	19	some	some	DET
iajs-2724	3	20	results	result	NOUN
iajs-2724	3	21	in	in	ADP
iajs-2724	3	22	this	this	DET
iajs-2724	3	23	notion	notion	NOUN
iajs-2724	3	24	are	be	AUX
iajs-2724	3	25	achieved	achieve	VERB
iajs-2724	3	26	.	.	PUNCT
iajs-2724	4	1	also	also	ADV
iajs-2724	4	2	,	,	PUNCT
iajs-2724	4	3	(	(	PUNCT
iajs-2724	4	4	cubic	cubic	ADJ
iajs-2724	4	5	bipolar	bipolar	ADJ
iajs-2724	4	6	fuzzy	fuzzy	ADJ
iajs-2724	4	7	ideals	ideal	NOUN
iajs-2724	4	8	and	and	CCONJ
iajs-2724	4	9	cubic	cubic	ADJ
iajs-2724	4	10	bipolar	bipolar	ADJ
iajs-2724	4	11	fuzzy	fuzzy	ADJ
iajs-2724	4	12	k	k	NOUN
iajs-2724	4	13	-	-	NOUN
iajs-2724	4	14	ideals	ideal	NOUN
iajs-2724	4	15	)	)	PUNCT
iajs-2724	4	16	with	with	ADP
iajs-2724	4	17	thresholds	threshold	NOUN
iajs-2724	4	18	(	(	PUNCT
iajs-2724	4	19	α	α	NOUN
iajs-2724	4	20	,	,	PUNCT
iajs-2724	4	21	β),(ω	β),(ω	NOUN
iajs-2724	4	22	,	,	PUNCT
iajs-2724	4	23	ϑ	ϑ	X
iajs-2724	4	24	)	)	PUNCT
iajs-2724	4	25	are	be	AUX
iajs-2724	4	26	defined	define	VERB
iajs-2724	4	27	and	and	CCONJ
iajs-2724	4	28	some	some	DET
iajs-2724	4	29	properties	property	NOUN
iajs-2724	4	30	of	of	ADP
iajs-2724	4	31	these	these	DET
iajs-2724	4	32	ideals	ideal	NOUN
iajs-2724	4	33	are	be	AUX
iajs-2724	4	34	given	give	VERB
iajs-2724	4	35	.	.	PUNCT
iajs-2724	5	1	relations	relation	NOUN
iajs-2724	5	2	between	between	ADP
iajs-2724	5	3	a	a	PRON
iajs-2724	5	4	(	(	PUNCT
iajs-2724	5	5	cbf).sub	cbf).sub	ADP
iajs-2724	5	6	algebra	algebra	NOUN
iajs-2724	5	7	and	and	CCONJ
iajs-2724	5	8	-	-	PUNCT
iajs-2724	5	9	a	a	DET
iajs-2724	5	10	(	(	PUNCT
iajs-2724	5	11	cbf	cbf	PROPN
iajs-2724	5	12	)	)	PUNCT
iajs-2724	5	13	ideal	ideal	NOUN
iajs-2724	5	14	are	be	AUX
iajs-2724	5	15	proved	prove	VERB
iajs-2724	5	16	.	.	PUNCT
iajs-2724	6	1	a	a	DET
iajs-2724	6	2	few	few	ADJ
iajs-2724	6	3	characterizations	characterization	NOUN
iajs-2724	6	4	of	of	ADP
iajs-2724	6	5	a	a	DET
iajs-2724	6	6	(	(	PUNCT
iajs-2724	6	7	cbf	cbf	PROPN
iajs-2724	6	8	)	)	PUNCT
iajs-2724	6	9	k	k	NOUN
iajs-2724	6	10	-	-	NOUN
iajs-2724	6	11	ideal	ideal	NOUN
iajs-2724	6	12	with	with	ADP
iajs-2724	6	13	thresholds	threshold	NOUN
iajs-2724	6	14	(	(	PUNCT
iajs-2724	6	15	α	α	X
iajs-2724	6	16	,	,	PUNCT
iajs-2724	6	17	β	β	NOUN
iajs-2724	6	18	)	)	PUNCT
iajs-2724	6	19	,	,	PUNCT
iajs-2724	6	20	(	(	PUNCT
iajs-2724	6	21	ω,ϑ	ω,ϑ	NOUN
iajs-2724	6	22	)	)	PUNCT
iajs-2724	6	23	are	be	AUX
iajs-2724	6	24	discussed	discuss	VERB
iajs-2724	6	25	.	.	PUNCT
iajs-2724	7	1	finally	finally	ADV
iajs-2724	7	2	,	,	PUNCT
iajs-2724	7	3	we	we	PRON
iajs-2724	7	4	proved	prove	VERB
iajs-2724	7	5	that	that	SCONJ
iajs-2724	7	6	a	a	DET
iajs-2724	7	7	(	(	PUNCT
iajs-2724	7	8	cbf	cbf	PROPN
iajs-2724	7	9	)	)	PUNCT
iajs-2724	7	10	k	k	NOUN
iajs-2724	7	11	-	-	PUNCT
iajs-2724	7	12	ideal	ideal	NOUN
iajs-2724	7	13	and	and	CCONJ
iajs-2724	7	14	a	a	DET
iajs-2724	7	15	(	(	PUNCT
iajs-2724	7	16	cbf	cbf	PROPN
iajs-2724	7	17	)	)	PUNCT
iajs-2724	7	18	ideal	ideal	NOUN
iajs-2724	7	19	with	with	ADP
iajs-2724	7	20	thresholds	threshold	NOUN
iajs-2724	7	21	(	(	PUNCT
iajs-2724	7	22	α	α	X
iajs-2724	7	23	,	,	PUNCT
iajs-2724	7	24	β	β	NOUN
iajs-2724	7	25	)	)	PUNCT
iajs-2724	7	26	,	,	PUNCT
iajs-2724	7	27	(	(	PUNCT
iajs-2724	7	28	ω,ϑ	ω,ϑ	NOUN
iajs-2724	7	29	)	)	PUNCT
iajs-2724	7	30	of	of	ADP
iajs-2724	7	31	a	a	DET
iajs-2724	7	32	ku	ku	NOUN
iajs-2724	7	33	-	-	PUNCT
iajs-2724	7	34	semi	semi	NOUN
iajs-2724	7	35	group	group	NOUN
iajs-2724	7	36	are	be	AUX
iajs-2724	7	37	equivalent	equivalent	ADJ
iajs-2724	7	38	relations	relation	NOUN
iajs-2724	7	39	.	.	PUNCT
iajs-2724	8	1	keywords	keyword	NOUN
iajs-2724	8	2	:	:	PUNCT
iajs-2724	8	3	a	a	DET
iajs-2724	8	4	ku	ku	PROPN
iajs-2724	8	5	-	-	PUNCT
iajs-2724	8	6	semigroup	semigroup	PROPN
iajs-2724	8	7	,	,	PUNCT
iajs-2724	8	8	cubic	cubic	ADJ
iajs-2724	8	9	k	k	NOUN
iajs-2724	8	10	-	-	PUNCT
iajs-2724	8	11	ideal	ideal	ADJ
iajs-2724	8	12	,	,	PUNCT
iajs-2724	8	13	cubic	cubic	ADJ
iajs-2724	8	14	bipolar	bipolar	ADJ
iajs-2724	8	15	fuzzy	fuzzy	ADJ
iajs-2724	8	16	k	k	NOUN
iajs-2724	8	17	-	-	NOUN
iajs-2724	8	18	ideal	ideal	NOUN
iajs-2724	8	19	with	with	ADP
iajs-2724	8	20	thresholds	threshold	NOUN
iajs-2724	8	21	(	(	PUNCT
iajs-2724	8	22	α	α	X
iajs-2724	8	23	,	,	PUNCT
iajs-2724	8	24	β	β	NOUN
iajs-2724	8	25	)	)	PUNCT
iajs-2724	8	26	,	,	PUNCT
iajs-2724	8	27	(	(	PUNCT
iajs-2724	8	28	ω,ϑ	ω,ϑ	NOUN
iajs-2724	8	29	)	)	PUNCT
iajs-2724	8	30	.	.	PUNCT
iajs-2724	9	1	1	1	X
iajs-2724	9	2	.	.	X
iajs-2724	9	3	introduction	introduction	NOUN
iajs-2724	9	4	the	the	DET
iajs-2724	9	5	fuzzy	fuzzy	ADJ
iajs-2724	9	6	sets	set	NOUN
iajs-2724	9	7	were	be	AUX
iajs-2724	9	8	introduced	introduce	VERB
iajs-2724	9	9	by	by	ADP
iajs-2724	9	10	zadeh	zadeh	PROPN
iajs-2724	10	1	[	[	X
iajs-2724	10	2	1	1	X
iajs-2724	10	3	]	]	PUNCT
iajs-2724	10	4	in	in	ADP
iajs-2724	10	5	1956	1956	NUM
iajs-2724	10	6	;	;	PUNCT
iajs-2724	10	7	after	after	ADP
iajs-2724	10	8	that	that	PRON
iajs-2724	10	9	,	,	PUNCT
iajs-2724	10	10	many	many	ADJ
iajs-2724	10	11	authors	author	NOUN
iajs-2724	10	12	applied	apply	VERB
iajs-2724	10	13	this	this	DET
iajs-2724	10	14	concept	concept	NOUN
iajs-2724	10	15	in	in	ADP
iajs-2724	10	16	different	different	ADJ
iajs-2724	10	17	mathematics	mathematic	NOUN
iajs-2724	10	18	fields	field	NOUN
iajs-2724	10	19	.	.	PUNCT
iajs-2724	11	1	mostafa	mostafa	PROPN
iajs-2724	12	1	[	[	X
iajs-2724	12	2	2	2	NUM
iajs-2724	12	3	,	,	PUNCT
iajs-2724	12	4	3	3	NUM
iajs-2724	12	5	]	]	PUNCT
iajs-2724	12	6	studied	study	VERB
iajs-2724	12	7	the	the	DET
iajs-2724	12	8	notion	notion	NOUN
iajs-2724	12	9	of	of	ADP
iajs-2724	12	10	fuzzy	fuzzy	ADJ
iajs-2724	12	11	ku	ku	NOUN
iajs-2724	12	12	-	-	PUNCT
iajs-2724	12	13	ideals	ideal	NOUN
iajs-2724	12	14	of	of	ADP
iajs-2724	12	15	ku	ku	PROPN
iajs-2724	12	16	-	-	PUNCT
iajs-2724	12	17	algebras	algebras	PROPN
iajs-2724	12	18	and	and	CCONJ
iajs-2724	12	19	generalizations	generalization	NOUN
iajs-2724	12	20	of	of	ADP
iajs-2724	12	21	fuzzy	fuzzy	ADJ
iajs-2724	12	22	sets	set	NOUN
iajs-2724	12	23	,	,	PUNCT
iajs-2724	12	24	which	which	PRON
iajs-2724	12	25	are	be	AUX
iajs-2724	12	26	called	call	VERB
iajs-2724	12	27	bipolarfuzzy	bipolarfuzzy	ADJ
iajs-2724	12	28	n	n	CCONJ
iajs-2724	12	29	-	-	ADJ
iajs-2724	12	30	fold	fold	ADJ
iajs-2724	12	31	ku	ku	NOUN
iajs-2724	12	32	-	-	PUNCT
iajs-2724	12	33	ideals	ideal	NOUN
iajs-2724	12	34	.	.	PUNCT
iajs-2724	13	1	jun	jun	PROPN
iajs-2724	14	1	[	[	X
iajs-2724	14	2	46	46	NUM
iajs-2724	14	3	]	]	PUNCT
iajs-2724	14	4	studied	study	VERB
iajs-2724	14	5	the	the	DET
iajs-2724	14	6	notion	notion	NOUN
iajs-2724	14	7	of	of	ADP
iajs-2724	14	8	a	a	DET
iajs-2724	14	9	cubic	cubic	ADJ
iajs-2724	14	10	set	set	NOUN
iajs-2724	14	11	as	as	ADP
iajs-2724	14	12	a	a	DET
iajs-2724	14	13	generalization	generalization	NOUN
iajs-2724	14	14	of	of	ADP
iajs-2724	14	15	fuzzy	fuzzy	ADJ
iajs-2724	14	16	set	set	NOUN
iajs-2724	14	17	and	and	CCONJ
iajs-2724	14	18	interval	interval	NOUN
iajs-2724	14	19	-	-	PUNCT
iajs-2724	14	20	valued	value	VERB
iajs-2724	14	21	fuzzy	fuzzy	ADJ
iajs-2724	14	22	set	set	NOUN
iajs-2724	14	23	.	.	PUNCT
iajs-2724	15	1	kareem	kareem	PROPN
iajs-2724	15	2	and	and	CCONJ
iajs-2724	15	3	hasan[7,8	hasan[7,8	PRON
iajs-2724	15	4	]	]	PUNCT
iajs-2724	15	5	defined	define	VERB
iajs-2724	15	6	the	the	DET
iajs-2724	15	7	cubic	cubic	ADJ
iajs-2724	15	8	ideals	ideal	NOUN
iajs-2724	15	9	of	of	ADP
iajs-2724	15	10	a	a	DET
iajs-2724	15	11	ku	ku	PROPN
iajs-2724	15	12	-	-	PUNCT
iajs-2724	15	13	semigroup	semigroup	PROPN
iajs-2724	15	14	and	and	CCONJ
iajs-2724	15	15	a	a	DET
iajs-2724	15	16	homomorphism	homomorphism	NOUN
iajs-2724	15	17	of	of	ADP
iajs-2724	15	18	a	a	DET
iajs-2724	15	19	cubic	cubic	ADJ
iajs-2724	15	20	set	set	NOUN
iajs-2724	15	21	in	in	ADP
iajs-2724	15	22	this	this	DET
iajs-2724	15	23	structure	structure	NOUN
iajs-2724	15	24	.	.	PUNCT
iajs-2724	16	1	bipolar	bipolar	ADJ
iajs-2724	16	2	–	–	PUNCT
iajs-2724	16	3	valued	value	VERB
iajs-2724	16	4	fuzzy	fuzzy	ADJ
iajs-2724	16	5	sets	set	NOUN
iajs-2724	16	6	are	be	AUX
iajs-2724	16	7	extensions	extension	NOUN
iajs-2724	16	8	of	of	ADP
iajs-2724	16	9	fuzzy	fuzzy	ADJ
iajs-2724	16	10	sets	set	NOUN
iajs-2724	16	11	whose	whose	DET
iajs-2724	16	12	membership	membership	NOUN
iajs-2724	16	13	degree	degree	NOUN
iajs-2724	16	14	range	range	NOUN
iajs-2724	16	15	is	be	AUX
iajs-2724	16	16	enlarged	enlarge	VERB
iajs-2724	16	17	from	from	ADP
iajs-2724	16	18	the	the	DET
iajs-2724	16	19	interval	interval	NOUN
iajs-2724	16	20	[	[	X
iajs-2724	16	21	0,1	0,1	NUM
iajs-2724	16	22	]	]	PUNCT
iajs-2724	16	23	to	to	ADP
iajs-2724	16	24	[	[	X
iajs-2724	16	25	-1,1	-1,1	NOUN
iajs-2724	16	26	]	]	PUNCT
iajs-2724	16	27	.	.	PUNCT
iajs-2724	17	1	kareem	kareem	PROPN
iajs-2724	17	2	and	and	CCONJ
iajs-2724	17	3	article	article	PROPN
iajs-2724	17	4	history	history	NOUN
iajs-2724	17	5	:	:	PUNCT
iajs-2724	17	6	received,21	received,21	NOUN
iajs-2724	17	7	,	,	PUNCT
iajs-2724	17	8	november,2021	november,2021	NOUN
iajs-2724	17	9	,	,	PUNCT
iajs-2724	17	10	accepted,25	accepted,25	PROPN
iajs-2724	17	11	,	,	PUNCT
iajs-2724	17	12	january	january	PROPN
iajs-2724	17	13	,	,	PUNCT
iajs-2724	17	14	2022	2022	NUM
iajs-2724	17	15	,	,	PUNCT
iajs-2724	17	16	published	publish	VERB
iajs-2724	17	17	in	in	ADP
iajs-2724	17	18	april	april	PROPN
iajs-2724	17	19	2022	2022	NUM
iajs-2724	17	20	.	.	PUNCT
iajs-2724	18	1	ibn	ibn	PROPN
iajs-2724	18	2	al	al	PROPN
iajs-2724	18	3	haitham	haitham	PROPN
iajs-2724	18	4	journal	journal	PROPN
iajs-2724	18	5	for	for	ADP
iajs-2724	18	6	pure	pure	ADJ
iajs-2724	18	7	and	and	CCONJ
iajs-2724	18	8	applied	applied	ADJ
iajs-2724	18	9	sciences	sciences	PROPN
iajs-2724	18	10	journal	journal	PROPN
iajs-2724	18	11	homepage	homepage	NOUN
iajs-2724	18	12	:	:	PUNCT
iajs-2724	18	13	http://jih.uobaghdad.edu.iq/index.php/j/index	http://jih.uobaghdad.edu.iq/index.php/j/index	NOUN
iajs-2724	18	14	doi	doi	NOUN
iajs-2724	18	15	:	:	PUNCT
iajs-2724	18	16	10.30526/35.2.2724	10.30526/35.2.2724	PROPN
iajs-2724	18	17	omniat	omniat	PROPN
iajs-2724	18	18	adnan	adnan	PROPN
iajs-2724	18	19	hasan	hasan	PROPN
iajs-2724	18	20	department	department	PROPN
iajs-2724	18	21	of	of	ADP
iajs-2724	18	22	mathematics	mathematics	PROPN
iajs-2724	18	23	,	,	PUNCT
iajs-2724	18	24	college	college	NOUN
iajs-2724	18	25	of	of	ADP
iajs-2724	18	26	education	education	NOUN
iajs-2724	18	27	for	for	ADP
iajs-2724	18	28	pure	pure	ADJ
iajs-2724	18	29	sciences	science	NOUN
iajs-2724	18	30	,	,	PUNCT
iajs-2724	18	31	ibn	ibn	PROPN
iajs-2724	18	32	al	al	PROPN
iajs-2724	18	33	haitham	haitham	PROPN
iajs-2724	18	34	,	,	PUNCT
iajs-2724	18	35	university	university	PROPN
iajs-2724	18	36	of	of	ADP
iajs-2724	18	37	baghdad	baghdad	PROPN
iajs-2724	18	38	,	,	PUNCT
iajs-2724	18	39	iraq	iraq	PROPN
iajs-2724	18	40	umniyatadnan@gmail.com	umniyatadnan@gmail.com	X
iajs-2724	19	1	fatema	fatema	PROPN
iajs-2724	19	2	f.	f.	PROPN
iajs-2724	19	3	kareem	kareem	PROPN
iajs-2724	19	4	department	department	PROPN
iajs-2724	19	5	of	of	ADP
iajs-2724	19	6	mathematics	mathematics	PROPN
iajs-2724	19	7	,	,	PUNCT
iajs-2724	19	8	college	college	NOUN
iajs-2724	19	9	of	of	ADP
iajs-2724	19	10	education	education	NOUN
iajs-2724	19	11	for	for	ADP
iajs-2724	19	12	pure	pure	ADJ
iajs-2724	19	13	sciences	science	NOUN
iajs-2724	19	14	,	,	PUNCT
iajs-2724	19	15	ibn	ibn	PROPN
iajs-2724	19	16	al	al	PROPN
iajs-2724	19	17	haitham	haitham	PROPN
iajs-2724	19	18	,	,	PUNCT
iajs-2724	19	19	university	university	PROPN
iajs-2724	19	20	of	of	ADP
iajs-2724	19	21	baghdad	baghdad	PROPN
iajs-2724	19	22	,	,	PUNCT
iajs-2724	19	23	iraq	iraq	PROPN
iajs-2724	19	24	fatma.f.k@ihcoedu.uobaghdad.edu.iq	fatma.f.k@ihcoedu.uobaghdad.edu.iq	PROPN
iajs-2724	19	25	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-2724	19	26	mailto:umniyatadnan@gmail.com	mailto:umniyatadnan@gmail.com	PROPN
iajs-2724	19	27	mailto:fatma.f.k@ihcoedu.uobaghdad.edu.iq	mailto:fatma.f.k@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-2724	19	28	ibn	ibn	PROPN
iajs-2724	19	29	al	al	PROPN
iajs-2724	19	30	-	-	PUNCT
iajs-2724	19	31	haitham	haitham	PROPN
iajs-2724	19	32	jour	jour	X
iajs-2724	19	33	.	.	PROPN
iajs-2724	20	1	for	for	ADP
iajs-2724	20	2	pure	pure	ADJ
iajs-2724	20	3	&	&	CCONJ
iajs-2724	20	4	appl	appl	PROPN
iajs-2724	20	5	.	.	PUNCT
iajs-2724	21	1	sci	sci	PROPN
iajs-2724	21	2	.	.	PROPN
iajs-2724	22	1	53	53	NUM
iajs-2724	22	2	(	(	PUNCT
iajs-2724	22	3	2)2022	2)2022	VERB
iajs-2724	22	4	48	48	NUM
iajs-2724	22	5	hassan[9	hassan[9	NUM
iajs-2724	22	6	]	]	PUNCT
iajs-2724	22	7	and	and	CCONJ
iajs-2724	22	8	kareem	kareem	PROPN
iajs-2724	22	9	and	and	CCONJ
iajs-2724	22	10	awad	awad	PROPN
iajs-2724	23	1	[	[	X
iajs-2724	23	2	10	10	NUM
iajs-2724	23	3	]	]	PUNCT
iajs-2724	23	4	defined	define	VERB
iajs-2724	23	5	the	the	DET
iajs-2724	23	6	concepts	concept	NOUN
iajs-2724	23	7	of	of	ADP
iajs-2724	23	8	bipolar	bipolar	ADJ
iajs-2724	23	9	fuzzy	fuzzy	ADJ
iajs-2724	23	10	k	k	NOUN
iajs-2724	23	11	-	-	NOUN
iajs-2724	23	12	ideals	ideal	NOUN
iajs-2724	23	13	and	and	CCONJ
iajs-2724	23	14	cubic	cubic	ADJ
iajs-2724	23	15	bipolar	bipolar	ADJ
iajs-2724	23	16	ideals	ideal	NOUN
iajs-2724	23	17	in	in	ADP
iajs-2724	23	18	ku	ku	PROPN
iajs-2724	23	19	-	-	PUNCT
iajs-2724	23	20	semigroup	semigroup	PROPN
iajs-2724	23	21	respectively	respectively	ADV
iajs-2724	23	22	,	,	PUNCT
iajs-2724	23	23	also	also	ADV
iajs-2724	23	24	kareem	kareem	PROPN
iajs-2724	23	25	and	and	CCONJ
iajs-2724	23	26	abed	abe	VERB
iajs-2724	23	27	[	[	PUNCT
iajs-2724	23	28	11	11	NUM
iajs-2724	23	29	]	]	PUNCT
iajs-2724	23	30	presented	present	VERB
iajs-2724	23	31	the	the	DET
iajs-2724	23	32	idea	idea	NOUN
iajs-2724	23	33	of	of	ADP
iajs-2724	23	34	bipolar	bipolar	ADJ
iajs-2724	23	35	fuzzy	fuzzy	ADJ
iajs-2724	23	36	-	-	PUNCT
iajs-2724	23	37	k	k	NOUN
iajs-2724	23	38	-	-	NOUN
iajs-2724	23	39	ideals	ideal	NOUN
iajs-2724	23	40	with	with	ADP
iajs-2724	23	41	a	a	DET
iajs-2724	23	42	threshold	threshold	NOUN
iajs-2724	23	43	of	of	ADP
iajs-2724	23	44	ku	ku	PROPN
iajs-2724	23	45	--	--	PUNCT
iajs-2724	23	46	semigroup	semigroup	PROPN
iajs-2724	23	47	.	.	PUNCT
iajs-2724	24	1	the	the	DET
iajs-2724	24	2	paper	paper	NOUN
iajs-2724	24	3	aims	aim	VERB
iajs-2724	24	4	to	to	PART
iajs-2724	24	5	introduce	introduce	VERB
iajs-2724	24	6	a	a	DET
iajs-2724	24	7	cubic	cubic	ADJ
iajs-2724	24	8	bipolar	bipolar	ADJ
iajs-2724	24	9	fuzzy	fuzzy	ADJ
iajs-2724	24	10	k	k	NOUN
iajs-2724	24	11	-	-	NOUN
iajs-2724	24	12	ideals	ideal	NOUN
iajs-2724	24	13	with	with	ADP
iajs-2724	24	14	thresholds	threshold	NOUN
iajs-2724	24	15	(	(	PUNCT
iajs-2724	24	16	α	α	NOUN
iajs-2724	24	17	,	,	PUNCT
iajs-2724	24	18	β),(ω,ϑ	β),(ω,ϑ	NUM
iajs-2724	24	19	)	)	PUNCT
iajs-2724	24	20	of	of	ADP
iajs-2724	24	21	ku	ku	PROPN
iajs-2724	24	22	-	-	PUNCT
iajs-2724	24	23	semi	semi	NOUN
iajs-2724	24	24	group	group	NOUN
iajs-2724	24	25	and	and	CCONJ
iajs-2724	24	26	discuss	discuss	VERB
iajs-2724	24	27	some	some	DET
iajs-2724	24	28	relations	relation	NOUN
iajs-2724	24	29	between	between	ADP
iajs-2724	24	30	a	a	DET
iajs-2724	24	31	cubic	cubic	ADJ
iajs-2724	24	32	bipolar	bipolar	ADJ
iajs-2724	24	33	fuzzy	fuzzy	ADJ
iajs-2724	24	34	k	k	NOUN
iajs-2724	24	35	-	-	NOUN
iajs-2724	24	36	ideal	ideal	NOUN
iajs-2724	24	37	with	with	ADP
iajs-2724	24	38	thresholds	threshold	NOUN
iajs-2724	24	39	(	(	PUNCT
iajs-2724	24	40	α	α	X
iajs-2724	24	41	,	,	PUNCT
iajs-2724	24	42	β	β	NOUN
iajs-2724	24	43	)	)	PUNCT
iajs-2724	24	44	,	,	PUNCT
iajs-2724	24	45	(	(	PUNCT
iajs-2724	24	46	ω,ϑ	ω,ϑ	NOUN
iajs-2724	24	47	)	)	PUNCT
iajs-2724	24	48	and	and	CCONJ
iajs-2724	24	49	a	a	DET
iajs-2724	24	50	bipolar	bipolar	ADJ
iajs-2724	24	51	fuzzy	fuzzy	ADJ
iajs-2724	24	52	k	k	NOUN
iajs-2724	24	53	-	-	NOUN
iajs-2724	24	54	ideal	ideal	ADJ
iajs-2724	24	55	.	.	PUNCT
iajs-2724	25	1	2	2	X
iajs-2724	25	2	.	.	NUM
iajs-2724	25	3	basic	basic	ADJ
iajs-2724	25	4	concepts	concept	NOUN
iajs-2724	25	5	definition(1)[12	definition(1)[12	PROPN
iajs-2724	25	6	]	]	X
iajs-2724	25	7	.	.	PUNCT
iajs-2724	26	1	algebra(ℵ,∗	algebra(ℵ,∗	ADV
iajs-2724	26	2	,	,	PUNCT
iajs-2724	26	3	0	0	NUM
iajs-2724	26	4	)	)	PUNCT
iajs-2724	26	5	is	be	AUX
iajs-2724	26	6	a	a	DET
iajs-2724	26	7	set	set	NOUN
iajs-2724	26	8	ℵ	ℵ	NOUN
iajs-2724	26	9	,	,	PUNCT
iajs-2724	26	10	and	and	CCONJ
iajs-2724	26	11	a	a	DET
iajs-2724	26	12	binary	binary	ADJ
iajs-2724	26	13	operation	operation	NOUN
iajs-2724	26	14	∗	∗	NOUN
iajs-2724	26	15	which	which	PRON
iajs-2724	26	16	is	be	AUX
iajs-2724	26	17	satisfies	satisfie	NOUN
iajs-2724	26	18	the	the	DET
iajs-2724	26	19	following	following	NOUN
iajs-2724	26	20	,	,	PUNCT
iajs-2724	26	21	for	for	ADP
iajs-2724	26	22	all	all	DET
iajs-2724	26	23	χ	χ	ADJ
iajs-2724	26	24	,	,	PUNCT
iajs-2724	26	25	𝛾	𝛾	PROPN
iajs-2724	26	26	,	,	PUNCT
iajs-2724	26	27	τ	τ	PROPN
iajs-2724	26	28	∈	∈	PROPN
iajs-2724	26	29	ℵ	ℵ	NOUN
iajs-2724	26	30	(	(	PUNCT
iajs-2724	26	31	ku1)(χ	ku1)(χ	PROPN
iajs-2724	26	32	∗	∗	NOUN
iajs-2724	26	33	𝛾	𝛾	NOUN
iajs-2724	26	34	)	)	PUNCT
iajs-2724	26	35	∗	∗	NOUN
iajs-2724	26	36	[	[	X
iajs-2724	26	37	(	(	PUNCT
iajs-2724	26	38	𝛾	𝛾	PROPN
iajs-2724	26	39	∗	∗	NOUN
iajs-2724	26	40	τ	τ	NOUN
iajs-2724	26	41	)	)	PUNCT
iajs-2724	26	42	∗	∗	NOUN
iajs-2724	26	43	(	(	PUNCT
iajs-2724	26	44	χ	χ	PROPN
iajs-2724	26	45	∗	∗	NOUN
iajs-2724	26	46	τ	τ	PROPN
iajs-2724	26	47	)	)	PUNCT
iajs-2724	26	48	]	]	PUNCT
iajs-2724	27	1	=	=	SYM
iajs-2724	27	2	0	0	NUM
iajs-2724	27	3	(	(	PUNCT
iajs-2724	27	4	ku2	ku2	NOUN
iajs-2724	27	5	)	)	PUNCT
iajs-2724	27	6	χ	χ	NOUN
iajs-2724	27	7	∗0	∗0	PROPN
iajs-2724	27	8	=	=	SYM
iajs-2724	27	9	0	0	NUM
iajs-2724	27	10	(	(	PUNCT
iajs-2724	27	11	ku3	ku3	X
iajs-2724	27	12	)	)	PUNCT
iajs-2724	27	13	0∗	0∗	PUNCT
iajs-2724	28	1	χ	χ	X
iajs-2724	28	2	=	=	SYM
iajs-2724	28	3	χ	χ	X
iajs-2724	28	4	(	(	PUNCT
iajs-2724	28	5	ku4	ku4	PROPN
iajs-2724	28	6	)	)	PUNCT
iajs-2724	29	1	χ	χ	DET
iajs-2724	29	2	∗	∗	NOUN
iajs-2724	29	3	𝛾	𝛾	ADP
iajs-2724	29	4	=	=	SYM
iajs-2724	29	5	𝛾	𝛾	NOUN
iajs-2724	29	6	∗	∗	NOUN
iajs-2724	29	7	χ	χ	X
iajs-2724	29	8	=	=	SYM
iajs-2724	29	9	0	0	NUM
iajs-2724	29	10	and	and	CCONJ
iajs-2724	29	11	𝛾	𝛾	PROPN
iajs-2724	29	12	∗	∗	NOUN
iajs-2724	29	13	χ	χ	PROPN
iajs-2724	29	14	implies	imply	VERB
iajs-2724	29	15	χ	χ	X
iajs-2724	29	16	=	=	SYM
iajs-2724	29	17	𝛾	𝛾	PROPN
iajs-2724	29	18	(	(	PUNCT
iajs-2724	29	19	ku5	ku5	NOUN
iajs-2724	29	20	)	)	PUNCT
iajs-2724	29	21	χ	χ	NOUN
iajs-2724	29	22	∗	∗	NOUN
iajs-2724	29	23	χ	χ	X
iajs-2724	29	24	=	=	NOUN
iajs-2724	29	25	0	0	X
iajs-2724	29	26	.	.	PUNCT
iajs-2724	30	1	we	we	PRON
iajs-2724	30	2	can	can	AUX
iajs-2724	30	3	define	define	VERB
iajs-2724	30	4	a	a	DET
iajs-2724	30	5	binary	binary	ADJ
iajs-2724	30	6	operation	operation	NOUN
iajs-2724	30	7	≤	≤	NOUN
iajs-2724	30	8	on	on	ADP
iajs-2724	30	9	ℵ	ℵ	NOUN
iajs-2724	30	10	is	be	AUX
iajs-2724	30	11	defined	define	VERB
iajs-2724	30	12	by	by	ADP
iajs-2724	30	13	χ	χ	PRON
iajs-2724	30	14	≤	≤	PROPN
iajs-2724	30	15	𝛾	𝛾	ADP
iajs-2724	30	16	⟺	⟺	PRON
iajs-2724	30	17	𝛾	𝛾	NOUN
iajs-2724	30	18	∗	∗	NOUN
iajs-2724	30	19	χ	χ	X
iajs-2724	30	20	=	=	SYM
iajs-2724	30	21	0	0	X
iajs-2724	30	22	.	.	PUNCT
iajs-2724	31	1	it	it	PRON
iajs-2724	31	2	follows	follow	VERB
iajs-2724	31	3	that	that	SCONJ
iajs-2724	31	4	(	(	PUNCT
iajs-2724	31	5	ℵ,≤)is	ℵ,≤)is	PUNCT
iajs-2724	31	6	a~	a~	PROPN
iajs-2724	31	7	partially	partially	ADV
iajs-2724	31	8	ordered	order	VERB
iajs-2724	31	9	set~.	set~.	VERB
iajs-2724	31	10	theorem(2)[2	theorem(2)[2	NOUN
iajs-2724	31	11	]	]	PUNCT
iajs-2724	31	12	.	.	PUNCT
iajs-2724	32	1	in	in	ADP
iajs-2724	32	2	a	a	DET
iajs-2724	32	3	ku	ku	NOUN
iajs-2724	32	4	-	-	PUNCT
iajs-2724	32	5	algebra	algebra	PROPN
iajs-2724	32	6	(	(	PUNCT
iajs-2724	32	7	ℵ,∗	ℵ,∗	PROPN
iajs-2724	32	8	,	,	PUNCT
iajs-2724	32	9	0	0	NUM
iajs-2724	32	10	)	)	PUNCT
iajs-2724	32	11	∀	∀	PUNCT
iajs-2724	33	1	χ	χ	NOUN
iajs-2724	33	2	,	,	PUNCT
iajs-2724	33	3	𝛾	𝛾	PROPN
iajs-2724	33	4	,	,	PUNCT
iajs-2724	33	5	τ	τ	PROPN
iajs-2724	33	6	∈	∈	PROPN
iajs-2724	33	7	ℵ	ℵ	NOUN
iajs-2724	33	8	,	,	PUNCT
iajs-2724	33	9	then	then	ADV
iajs-2724	33	10	the	the	DET
iajs-2724	33	11	following	follow	VERB
iajs-2724	33	12	holds	hold	VERB
iajs-2724	33	13	(	(	PUNCT
iajs-2724	33	14	1	1	NUM
iajs-2724	33	15	)	)	PUNCT
iajs-2724	33	16	χ	χ	PRON
iajs-2724	33	17	≤	≤	PROPN
iajs-2724	33	18	𝛾	𝛾	AUX
iajs-2724	33	19	𝑖𝑚𝑝𝑙𝑦	𝑖𝑚𝑝𝑙𝑦	NOUN
iajs-2724	33	20	𝛾	𝛾	NOUN
iajs-2724	33	21	∗	∗	NOUN
iajs-2724	33	22	τ	τ	X
iajs-2724	33	23	≤	≤	NUM
iajs-2724	33	24	χ	χ	DET
iajs-2724	33	25	∗	∗	X
iajs-2724	33	26	τ	τ	X
iajs-2724	33	27	(	(	PUNCT
iajs-2724	33	28	2	2	NUM
iajs-2724	33	29	)	)	PUNCT
iajs-2724	33	30	χ	χ	NOUN
iajs-2724	33	31	∗	∗	NOUN
iajs-2724	33	32	(	(	PUNCT
iajs-2724	33	33	𝛾	𝛾	PROPN
iajs-2724	33	34	∗	∗	NOUN
iajs-2724	33	35	τ	τ	X
iajs-2724	33	36	)	)	PUNCT
iajs-2724	33	37	=	=	SYM
iajs-2724	33	38	𝛾	𝛾	NOUN
iajs-2724	33	39	∗	∗	NOUN
iajs-2724	33	40	(	(	PUNCT
iajs-2724	33	41	χ	χ	PROPN
iajs-2724	33	42	∗	∗	NOUN
iajs-2724	33	43	τ	τ	PROPN
iajs-2724	33	44	)	)	PUNCT
iajs-2724	33	45	(	(	PUNCT
iajs-2724	33	46	3	3	X
iajs-2724	33	47	)	)	PUNCT
iajs-2724	33	48	𝛾	𝛾	NOUN
iajs-2724	33	49	∗	∗	NOUN
iajs-2724	33	50	χ	χ	PRON
iajs-2724	33	51	≤	≤	NUM
iajs-2724	33	52	χ	χ	PRON
iajs-2724	33	53	,	,	PUNCT
iajs-2724	33	54	also	also	ADV
iajs-2724	33	55	(	(	PUNCT
iajs-2724	33	56	𝛾	𝛾	PROPN
iajs-2724	33	57	∗	∗	NOUN
iajs-2724	33	58	χ	χ	NOUN
iajs-2724	33	59	)	)	PUNCT
iajs-2724	33	60	∗	∗	NOUN
iajs-2724	33	61	χ	χ	PRON
iajs-2724	33	62	≤	≤	NOUN
iajs-2724	33	63	𝛾	𝛾	ADP
iajs-2724	33	64	definition(3)[2	definition(3)[2	NOUN
iajs-2724	33	65	]	]	PUNCT
iajs-2724	33	66	.	.	PUNCT
iajs-2724	34	1	a	a	DET
iajs-2724	34	2	non	non	ADJ
iajs-2724	34	3	-	-	ADJ
iajs-2724	34	4	empty	empty	ADJ
iajs-2724	34	5	subset	subset	NOUN
iajs-2724	34	6	i	i	PRON
iajs-2724	34	7	of	of	ADP
iajs-2724	34	8	a	a	DET
iajs-2724	34	9	ku	ku	PROPN
iajs-2724	34	10	-	-	PUNCT
iajs-2724	34	11	algebra	algebra	PROPN
iajs-2724	34	12	ℵ	ℵ	NOUN
iajs-2724	34	13	is	be	AUX
iajs-2724	34	14	namedˑˑan	namedˑˑan	ADJ
iajs-2724	34	15	ˑideal	ˑideal	NOUN
iajs-2724	34	16	if	if	SCONJ
iajs-2724	34	17	ˑfor	ˑfor	PROPN
iajs-2724	34	18	anyχ	anyχ	ADV
iajs-2724	34	19	,	,	PUNCT
iajs-2724	34	20	𝛾	𝛾	PROPN
iajs-2724	34	21	∈	∈	NOUN
iajs-2724	34	22	ℵ	ℵ	NOUN
iajs-2724	34	23	,	,	PUNCT
iajs-2724	34	24	then	then	ADV
iajs-2724	34	25	(	(	PUNCT
iajs-2724	34	26	1	1	NUM
iajs-2724	34	27	)	)	PUNCT
iajs-2724	34	28	0	0	NUM
iajs-2724	35	1	∈	∈	PROPN
iajs-2724	35	2	𝐼	𝐼	PROPN
iajs-2724	35	3	(	(	PUNCT
iajs-2724	35	4	2	2	NUM
iajs-2724	35	5	)	)	PUNCT
iajs-2724	35	6	if	if	SCONJ
iajs-2724	35	7	χ	χ	PRON
iajs-2724	35	8	∗	∗	NOUN
iajs-2724	35	9	𝛾	𝛾	ADP
iajs-2724	35	10	∈	∈	NOUN
iajs-2724	35	11	𝐼	𝐼	PROPN
iajs-2724	35	12	implies	imply	VERB
iajs-2724	35	13	that	that	SCONJ
iajs-2724	35	14	𝛾	𝛾	ADP
iajs-2724	35	15	∈	∈	NOUN
iajs-2724	35	16	𝐼	𝐼	ADV
iajs-2724	35	17	.	.	PUNCT
iajs-2724	36	1	definition(4)[2	definition(4)[2	PROPN
iajs-2724	36	2	]	]	PUNCT
iajs-2724	36	3	.	.	PUNCT
iajs-2724	37	1	a	a	DET
iajs-2724	37	2	non	non	ADJ
iajs-2724	37	3	-	-	ADJ
iajs-2724	37	4	empty	empty	ADJ
iajs-2724	37	5	subset	subset	NOUN
iajs-2724	37	6	i	i	PRON
iajs-2724	37	7	of	of	ADP
iajs-2724	37	8	a	a	DET
iajs-2724	37	9	ku	ku	PROPN
iajs-2724	37	10	-	-	PUNCT
iajs-2724	37	11	algebra	algebra	PROPN
iajs-2724	37	12	ℵ	ℵ	NOUN
iajs-2724	37	13	is	be	AUX
iajs-2724	37	14	namedˑˑa	namedˑˑa	NOUN
iajs-2724	37	15	ku	ku	NOUN
iajs-2724	37	16	-	-	PUNCT
iajs-2724	37	17	ideal	ideal	NOUN
iajs-2724	37	18	if	if	SCONJ
iajs-2724	37	19	(	(	PUNCT
iajs-2724	37	20	1	1	NUM
iajs-2724	37	21	)	)	PUNCT
iajs-2724	37	22	0	0	NUM
iajs-2724	37	23	∈	∈	PROPN
iajs-2724	37	24	𝐼	𝐼	PROPN
iajs-2724	37	25	(	(	PUNCT
iajs-2724	37	26	2	2	NUM
iajs-2724	37	27	)	)	PUNCT
iajs-2724	37	28	if	if	SCONJ
iajs-2724	37	29	χ	χ	PRON
iajs-2724	37	30	∗	∗	NOUN
iajs-2724	37	31	(	(	PUNCT
iajs-2724	37	32	𝛾	𝛾	NOUN
iajs-2724	37	33	∗	∗	NOUN
iajs-2724	37	34	𝜏	𝜏	NOUN
iajs-2724	37	35	)	)	PUNCT
iajs-2724	37	36	∈	∈	NOUN
iajs-2724	37	37	𝐼	𝐼	PROPN
iajs-2724	37	38	,	,	PUNCT
iajs-2724	37	39	and	and	CCONJ
iajs-2724	37	40	𝛾	𝛾	AUX
iajs-2724	37	41	∈	∈	NOUN
iajs-2724	37	42	𝐼	𝐼	ADP
iajs-2724	37	43	imply	imply	VERB
iajs-2724	37	44	that	that	SCONJ
iajs-2724	37	45	χ	χ	DET
iajs-2724	37	46	∗	∗	NOUN
iajs-2724	37	47	𝜏	𝜏	NOUN
iajs-2724	37	48	∈	∈	NOUN
iajs-2724	37	49	𝐼	𝐼	NOUN
iajs-2724	37	50	.	.	PUNCT
iajs-2724	38	1	definition(5)[13	definition(5)[13	PROPN
iajs-2724	38	2	]	]	PUNCT
iajs-2724	38	3	.	.	PUNCT
iajs-2724	39	1	an	an	DET
iajs-2724	39	2	algebra	algebra	PROPN
iajs-2724	39	3	-	-	PUNCT
iajs-2724	39	4	ku	ku	NOUN
iajs-2724	39	5	--	--	PUNCT
iajs-2724	39	6	semi	semi	ADJ
iajs-2724	39	7	group	group	NOUN
iajs-2724	39	8	is	be	AUX
iajs-2724	39	9	-	-	PUNCT
iajs-2724	39	10	a	a	DET
iajs-2724	39	11	structure	structure	NOUN
iajs-2724	39	12	containssa	containssa	NOUN
iajs-2724	39	13	nonemptyset	nonemptyset	VERB
iajs-2724	39	14	ℵ	ℵ	PRON
iajs-2724	39	15	-with	-with	PROPN
iajs-2724	39	16	twobinary	twobinary	NOUN
iajs-2724	39	17	operations	operation	NOUN
iajs-2724	39	18	∗,∘	∗,∘	PROPN
iajs-2724	39	19	and	and	CCONJ
iajs-2724	39	20	a	a	PRON
iajs-2724	39	21	-	-	PUNCT
iajs-2724	39	22	constant	constant	ADJ
iajs-2724	39	23	0	0	NUM
iajs-2724	39	24	satisfying	satisfy	VERB
iajs-2724	39	25	the	the	DET
iajs-2724	39	26	following	follow	VERB
iajs-2724	39	27	(	(	PUNCT
iajs-2724	39	28	i	i	NOUN
iajs-2724	39	29	)	)	PUNCT
iajs-2724	39	30	the	the	DET
iajs-2724	39	31	set	set	NOUN
iajs-2724	39	32	ℵ	ℵ	NOUN
iajs-2724	39	33	with	with	ADP
iajs-2724	39	34	operation	operation	NOUN
iajs-2724	39	35	∗	∗	NOUN
iajs-2724	39	36	and	and	CCONJ
iajs-2724	39	37	constant	constant	ADJ
iajs-2724	39	38	0	0	NUM
iajs-2724	39	39	isˑ	isˑ	NOUN
iajs-2724	39	40	ku	ku	NOUN
iajs-2724	39	41	-	-	PUNCT
iajs-2724	39	42	algebra	algebra	PROPN
iajs-2724	39	43	(	(	PUNCT
iajs-2724	39	44	ii	ii	NOUN
iajs-2724	39	45	)	)	PUNCT
iajs-2724	39	46	the	the	DET
iajs-2724	39	47	set	set	NOUN
iajs-2724	39	48	ℵ	ℵ	NOUN
iajs-2724	39	49	with	with	ADP
iajs-2724	39	50	operation	operation	NOUN
iajs-2724	39	51	∘	∘	PROPN
iajs-2724	39	52	is	be	AUX
iajs-2724	39	53	semigroup	semigroup	NOUN
iajs-2724	39	54	.	.	PUNCT
iajs-2724	40	1	(	(	PUNCT
iajs-2724	40	2	iii	iii	X
iajs-2724	40	3	)	)	PUNCT
iajs-2724	41	1	χ	χ	NOUN
iajs-2724	41	2	∘	∘	PROPN
iajs-2724	41	3	(	(	PUNCT
iajs-2724	41	4	𝛾	𝛾	PROPN
iajs-2724	41	5	∗	∗	NOUN
iajs-2724	41	6	τ	τ	X
iajs-2724	41	7	)	)	PUNCT
iajs-2724	41	8	=	=	PUNCT
iajs-2724	42	1	(	(	PUNCT
iajs-2724	42	2	χ	χ	DET
iajs-2724	42	3	∘	∘	PROPN
iajs-2724	42	4	𝛾	𝛾	NOUN
iajs-2724	42	5	)	)	PUNCT
iajs-2724	42	6	∗	∗	NOUN
iajs-2724	42	7	(	(	PUNCT
iajs-2724	42	8	χ	χ	DET
iajs-2724	42	9	∘	∘	NUM
iajs-2724	42	10	τ	τ	X
iajs-2724	42	11	)	)	PUNCT
iajs-2724	42	12	,	,	PUNCT
iajs-2724	42	13	and(χ	and(χ	PROPN
iajs-2724	42	14	∗	∗	NOUN
iajs-2724	42	15	𝛾	𝛾	NOUN
iajs-2724	42	16	)	)	PUNCT
iajs-2724	42	17	∘	∘	NOUN
iajs-2724	42	18	τ	τ	X
iajs-2724	42	19	=	=	SYM
iajs-2724	42	20	(	(	PUNCT
iajs-2724	42	21	χ	χ	PRON
iajs-2724	42	22	∘	∘	NUM
iajs-2724	42	23	τ	τ	X
iajs-2724	42	24	)	)	PUNCT
iajs-2724	42	25	∗	∗	NOUN
iajs-2724	42	26	(	(	PUNCT
iajs-2724	42	27	𝛾	𝛾	PROPN
iajs-2724	42	28	∘	∘	X
iajs-2724	42	29	τ	τ	X
iajs-2724	42	30	)	)	PUNCT
iajs-2724	42	31	,	,	PUNCT
iajs-2724	42	32	for	for	ADP
iajs-2724	42	33	all	all	DET
iajs-2724	42	34	χ	χ	ADJ
iajs-2724	42	35	,	,	PUNCT
iajs-2724	42	36	𝛾	𝛾	PROPN
iajs-2724	42	37	,	,	PUNCT
iajs-2724	42	38	τ	τ	PROPN
iajs-2724	42	39	∈	∈	PROPN
iajs-2724	42	40	ℵ.	ℵ.	PROPN
iajs-2724	43	1	ibn	ibn	PROPN
iajs-2724	43	2	al	al	PROPN
iajs-2724	43	3	-	-	PUNCT
iajs-2724	43	4	haitham	haitham	PROPN
iajs-2724	43	5	jour	jour	X
iajs-2724	43	6	.	.	PROPN
iajs-2724	43	7	for	for	ADP
iajs-2724	43	8	pure	pure	ADJ
iajs-2724	43	9	&	&	CCONJ
iajs-2724	43	10	appl	appl	PROPN
iajs-2724	43	11	.	.	PUNCT
iajs-2724	44	1	sci	sci	PROPN
iajs-2724	44	2	.	.	PROPN
iajs-2724	45	1	53	53	NUM
iajs-2724	45	2	(	(	PUNCT
iajs-2724	45	3	2)2022	2)2022	NOUN
iajs-2724	45	4	49	49	NUM
iajs-2724	45	5	definition(6)[13	definition(6)[13	NOUN
iajs-2724	45	6	]	]	PUNCT
iajs-2724	45	7	.	.	PUNCT
iajs-2724	46	1	a	a	DET
iajs-2724	46	2	non	non	ADJ
iajs-2724	46	3	-	-	ADJ
iajs-2724	46	4	empty	empty	ADJ
iajs-2724	46	5	subset	subset	NOUN
iajs-2724	46	6	a	a	PRON
iajs-2724	46	7	of	of	ADP
iajs-2724	46	8	ℵ	ℵ	NOUN
iajs-2724	46	9	is	be	AUX
iajs-2724	46	10	called	call	VERB
iajs-2724	46	11	a	a	DET
iajs-2724	46	12	sub	sub	ADJ
iajs-2724	46	13	-	-	ADJ
iajs-2724	46	14	ku	ku	ADJ
iajs-2724	46	15	-	-	PUNCT
iajs-2724	46	16	semi	semi	NOUN
iajs-2724	46	17	group	group	NOUN
iajs-2724	46	18	of	of	ADP
iajs-2724	46	19	ℵ	ℵ	NOUN
iajs-2724	46	20	if	if	SCONJ
iajs-2724	46	21	χ	χ	DET
iajs-2724	46	22	∗	∗	NOUN
iajs-2724	46	23	𝛾	𝛾	ADP
iajs-2724	46	24	∈	∈	PROPN
iajs-2724	46	25	𝑨	𝑨	NOUN
iajs-2724	46	26	,	,	PUNCT
iajs-2724	46	27	and	and	CCONJ
iajs-2724	46	28	χ	χ	DET
iajs-2724	46	29	∘	∘	NOUN
iajs-2724	46	30	𝛾	𝛾	ADP
iajs-2724	46	31	∈	∈	PROPN
iajs-2724	46	32	𝑨	𝑨	NOUN
iajs-2724	46	33	,	,	PUNCT
iajs-2724	46	34	for	for	ADP
iajs-2724	46	35	all	all	DET
iajs-2724	46	36	χ	χ	NOUN
iajs-2724	46	37	,	,	PUNCT
iajs-2724	46	38	𝛾	𝛾	AUX
iajs-2724	46	39	∈	∈	PROPN
iajs-2724	46	40	𝑨	𝑨	NOUN
iajs-2724	46	41	definition(7)[13	definition(7)[13	PROPN
iajs-2724	46	42	]	]	PUNCT
iajs-2724	46	43	.	.	PUNCT
iajs-2724	47	1	in	in	ADP
iajs-2724	47	2	a	a	DET
iajs-2724	47	3	ku	ku	NOUN
iajs-2724	47	4	-	-	PUNCT
iajs-2724	47	5	semi	semi	NOUN
iajs-2724	47	6	group	group	NOUN
iajs-2724	47	7	(	(	PUNCT
iajs-2724	47	8	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2724	47	9	,	,	PUNCT
iajs-2724	47	10	0	0	NUM
iajs-2724	47	11	)	)	PUNCT
iajs-2724	47	12	,	,	PUNCT
iajs-2724	47	13	the	the	DET
iajs-2724	47	14	subset	subset	NOUN
iajs-2724	47	15	𝜑	𝜑	X
iajs-2724	47	16	≠	≠	PROPN
iajs-2724	47	17	𝐼	𝐼	PROPN
iajs-2724	47	18	of	of	ADP
iajs-2724	47	19	ℵ	ℵ	NOUN
iajs-2724	47	20	is	be	AUX
iajs-2724	47	21	said	say	VERB
iajs-2724	47	22	to	to	PART
iajs-2724	47	23	be	be	AUX
iajs-2724	47	24	s	s	NOUN
iajs-2724	47	25	ideal	ideal	ADJ
iajs-2724	47	26	-	-	PUNCT
iajs-2724	47	27	,	,	PUNCT
iajs-2724	47	28	if	if	SCONJ
iajs-2724	47	29	(	(	PUNCT
iajs-2724	47	30	i	i	NOUN
iajs-2724	47	31	)	)	PUNCT
iajs-2724	47	32	it	it	PRON
iajs-2724	47	33	is	be	AUX
iajs-2724	47	34	an	an	DET
iajs-2724	47	35	ideal	ideal	NOUN
iajs-2724	47	36	in	in	ADP
iajs-2724	47	37	a	a	DET
iajs-2724	47	38	ku	ku	NOUN
iajs-2724	47	39	-	-	PUNCT
iajs-2724	47	40	algebra	algebra	PROPN
iajs-2724	47	41	(	(	PUNCT
iajs-2724	47	42	ii	ii	NOUN
iajs-2724	47	43	)	)	PUNCT
iajs-2724	48	1	χ	χ	X
iajs-2724	48	2	∘	∘	NUM
iajs-2724	48	3	𝑎	𝑎	PRON
iajs-2724	48	4	∈	∈	PROPN
iajs-2724	48	5	𝑰	𝑰	PROPN
iajs-2724	48	6	,	,	PUNCT
iajs-2724	48	7	and	and	CCONJ
iajs-2724	48	8	𝑎	𝑎	DET
iajs-2724	48	9	∘	∘	NOUN
iajs-2724	48	10	χ	χ	DET
iajs-2724	48	11	∈	∈	PROPN
iajs-2724	48	12	𝑰	𝑰	PROPN
iajs-2724	48	13	,	,	PUNCT
iajs-2724	48	14	∀	∀	X
iajs-2724	48	15	χ	χ	DET
iajs-2724	48	16	∈	∈	PROPN
iajs-2724	48	17	ℵ	ℵ	NOUN
iajs-2724	48	18	,	,	PUNCT
iajs-2724	48	19	𝑎	𝑎	PROPN
iajs-2724	48	20	∈	∈	NOUN
iajs-2724	48	21	𝑰	𝑰	PROPN
iajs-2724	48	22	definition(8)[13	definition(8)[13	PROPN
iajs-2724	48	23	]	]	PUNCT
iajs-2724	48	24	.	.	PUNCT
iajs-2724	49	1	in	in	ADP
iajs-2724	49	2	ku	ku	PROPN
iajs-2724	49	3	-	-	PUNCT
iajs-2724	49	4	semigroup	semigroup	PROPN
iajs-2724	49	5	(	(	PUNCT
iajs-2724	49	6	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2724	49	7	,	,	PUNCT
iajs-2724	49	8	0	0	NUM
iajs-2724	49	9	)	)	PUNCT
iajs-2724	49	10	,	,	PUNCT
iajs-2724	49	11	the	the	DET
iajs-2724	49	12	subset	subset	NOUN
iajs-2724	49	13	𝜑	𝜑	ADP
iajs-2724	49	14	≠	≠	PROPN
iajs-2724	49	15	𝐴	𝐴	PROPN
iajs-2724	49	16	of	of	ADP
iajs-2724	49	17	ℵ	ℵ	PROPN
iajs-2724	49	18	is	be	AUX
iajs-2724	49	19	named	name	VERB
iajs-2724	49	20	a	a	DET
iajs-2724	49	21	-k	-k	PUNCT
iajs-2724	49	22	--	--	PUNCT
iajs-2724	49	23	ideal-	ideal-	NOUN
iajs-2724	49	24	,	,	PUNCT
iajs-2724	49	25	if	if	SCONJ
iajs-2724	49	26	(	(	PUNCT
iajs-2724	49	27	i	i	NOUN
iajs-2724	49	28	)	)	PUNCT
iajs-2724	49	29	it	it	PRON
iajs-2724	49	30	is	be	AUX
iajs-2724	49	31	a	a	DET
iajs-2724	49	32	ku	ku	NOUN
iajs-2724	49	33	-	-	PUNCT
iajs-2724	49	34	ideal	ideal	NOUN
iajs-2724	49	35	of	of	ADP
iajs-2724	49	36	ℵ	ℵ	PROPN
iajs-2724	49	37	(	(	PUNCT
iajs-2724	49	38	ii	ii	NOUN
iajs-2724	49	39	)	)	PUNCT
iajs-2724	50	1	χ	χ	X
iajs-2724	50	2	∘	∘	NUM
iajs-2724	50	3	𝑎	𝑎	PRON
iajs-2724	50	4	∈	∈	PROPN
iajs-2724	50	5	𝑰	𝑰	PROPN
iajs-2724	50	6	,	,	PUNCT
iajs-2724	50	7	and	and	CCONJ
iajs-2724	50	8	𝑎	𝑎	DET
iajs-2724	50	9	∘	∘	NOUN
iajs-2724	50	10	χ	χ	DET
iajs-2724	50	11	∈	∈	PROPN
iajs-2724	50	12	𝑰	𝑰	PROPN
iajs-2724	50	13	,	,	PUNCT
iajs-2724	50	14	∀	∀	X
iajs-2724	50	15	χ	χ	DET
iajs-2724	50	16	∈	∈	PROPN
iajs-2724	50	17	ℵ	ℵ	NOUN
iajs-2724	50	18	,	,	PUNCT
iajs-2724	50	19	𝑎	𝑎	PROPN
iajs-2724	50	20	∈	∈	ADJ
iajs-2724	50	21	𝑰	𝑰	PROPN
iajs-2724	50	22	in	in	ADP
iajs-2724	50	23	this	this	DET
iajs-2724	50	24	part	part	NOUN
iajs-2724	50	25	,	,	PUNCT
iajs-2724	50	26	we	we	PRON
iajs-2724	50	27	recall	recall	VERB
iajs-2724	50	28	some	some	DET
iajs-2724	50	29	concepts	concept	NOUN
iajs-2724	50	30	of	of	ADP
iajs-2724	50	31	fuzzy	fuzzy	ADJ
iajs-2724	50	32	logic	logic	NOUN
iajs-2724	50	33	a	a	DET
iajs-2724	50	34	function	function	NOUN
iajs-2724	50	35	μ	μ	NOUN
iajs-2724	50	36	:	:	PUNCT
iajs-2724	50	37	ℵ	ℵ	NOUN
iajs-2724	50	38	⟶	⟶	NOUN
iajs-2724	50	39	[	[	X
iajs-2724	50	40	0,1	0,1	NUM
iajs-2724	50	41	]	]	PUNCT
iajs-2724	50	42	is	be	AUX
iajs-2724	50	43	said	say	VERB
iajs-2724	50	44	to	to	PART
iajs-2724	50	45	be	be	AUX
iajs-2724	50	46	a	a	DET
iajs-2724	50	47	fuzzy	fuzzy	ADJ
iajs-2724	50	48	set	set	NOUN
iajs-2724	50	49	of	of	ADP
iajs-2724	50	50	a	a	DET
iajs-2724	50	51	set	set	NOUN
iajs-2724	50	52	ℵ,and	ℵ,and	PUNCT
iajs-2724	50	53	the	the	DET
iajs-2724	50	54	set	set	NOUN
iajs-2724	50	55	𝑈(𝜇	𝑈(𝜇	NOUN
iajs-2724	50	56	,	,	PUNCT
iajs-2724	50	57	𝑡	𝑡	X
iajs-2724	50	58	)	)	PUNCT
iajs-2724	50	59	=	=	SYM
iajs-2724	50	60	{	{	PUNCT
iajs-2724	50	61	𝜒	𝜒	X
iajs-2724	50	62	∈	∈	NOUN
iajs-2724	50	63	ℵ	ℵ	X
iajs-2724	50	64	∶	∶	NOUN
iajs-2724	50	65	𝜇(𝜒	𝜇(𝜒	NOUN
iajs-2724	50	66	)	)	PUNCT
iajs-2724	50	67	≥	≥	PROPN
iajs-2724	51	1	𝑡}ˑis	𝑡}ˑis	PROPN
iajs-2724	51	2	said	say	VERB
iajs-2724	51	3	to	to	PART
iajs-2724	51	4	be	be	AUX
iajs-2724	51	5	a	a	DET
iajs-2724	51	6	level	level	NOUN
iajs-2724	51	7	set	set	NOUN
iajs-2724	51	8	of	of	ADP
iajs-2724	51	9	𝜇	𝜇	ADP
iajs-2724	51	10	,	,	PUNCT
iajs-2724	51	11	for	for	ADP
iajs-2724	51	12	t	t	PROPN
iajs-2724	51	13	,	,	PUNCT
iajs-2724	51	14	where	where	SCONJ
iajs-2724	51	15	1	1	NUM
iajs-2724	51	16	≥	≥	NOUN
iajs-2724	51	17	𝑡	𝑡	X
iajs-2724	51	18	≥	≥	NOUN
iajs-2724	51	19	0	0	PUNCT
iajs-2724	51	20	now	now	ADV
iajs-2724	51	21	,	,	PUNCT
iajs-2724	51	22	an	an	DET
iajs-2724	51	23	interval	interval	NOUN
iajs-2724	51	24	valued	value	VERB
iajs-2724	51	25	fuzzy	fuzzy	ADJ
iajs-2724	51	26	set	set	VERB
iajs-2724	51	27	𝜇	𝜇	ADP
iajs-2724	51	28	of	of	ADP
iajs-2724	51	29	ℵ	ℵ	NOUN
iajs-2724	51	30	is	be	AUX
iajs-2724	51	31	defined	define	VERB
iajs-2724	51	32	as	as	SCONJ
iajs-2724	51	33	follows	follow	VERB
iajs-2724	51	34	:	:	PUNCT
iajs-2724	51	35	remark(9)[7	remark(9)[7	NOUN
iajs-2724	51	36	-	-	PUNCT
iajs-2724	51	37	8].a	8].a	NUM
iajs-2724	51	38	function	function	NOUN
iajs-2724	51	39	𝜇	𝜇	ADP
iajs-2724	51	40	:	:	PUNCT
iajs-2724	51	41	ℵ	ℵ	DET
iajs-2724	51	42	⟶	⟶	NOUN
iajs-2724	51	43	𝐷[0,1	𝐷[0,1	NOUN
iajs-2724	51	44	]	]	X
iajs-2724	51	45	,	,	PUNCT
iajs-2724	51	46	where	where	SCONJ
iajs-2724	51	47	d[0,1	d[0,1	NOUN
iajs-2724	51	48	]	]	PUNCT
iajs-2724	51	49	is	be	AUX
iajs-2724	51	50	a	a	DET
iajs-2724	51	51	family	family	NOUN
iajs-2724	51	52	of	of	ADP
iajs-2724	51	53	the	the	DET
iajs-2724	51	54	closed	closed	ADJ
iajs-2724	51	55	sub	sub	NOUN
iajs-2724	51	56	̂intervals	̂interval	NOUN
iajs-2724	51	57	̂of[0	̂of[0	PROPN
iajs-2724	51	58	,	,	PUNCT
iajs-2724	51	59	1	1	NUM
iajs-2724	51	60	]	]	PUNCT
iajs-2724	51	61	.	.	PUNCT
iajs-2724	52	1	the	the	DET
iajs-2724	52	2	level	level	NOUN
iajs-2724	52	3	subset	subset	NOUN
iajs-2724	52	4	of	of	ADP
iajs-2724	52	5	𝜇~is	𝜇~is	PROPN
iajs-2724	52	6	~denoted	~denote	VERB
iajs-2724	52	7	~by	~by	PROPN
iajs-2724	52	8	𝜇	𝜇	ADP
iajs-2724	52	9	�	�	PROPN
iajs-2724	52	10	̃	̃	NOUN
iajs-2724	52	11	�	�	PROPN
iajs-2724	52	12	and	and	CCONJ
iajs-2724	52	13	it	it	PRON
iajs-2724	52	14	is	be	AUX
iajs-2724	52	15	~defined	~define	VERB
iajs-2724	52	16	~by	~by	PROPN
iajs-2724	52	17	𝜇	𝜇	ADP
iajs-2724	52	18	�	�	PROPN
iajs-2724	52	19	̃	̃	NOUN
iajs-2724	52	20	�	�	NOUN
iajs-2724	52	21	=	=	PRON
iajs-2724	52	22	{	{	PUNCT
iajs-2724	52	23	𝜒	𝜒	NOUN
iajs-2724	52	24	∈	∈	ADJ
iajs-2724	52	25	ℵ	ℵ	NOUN
iajs-2724	52	26	:	:	PUNCT
iajs-2724	52	27	𝜇(𝜒	𝜇(𝜒	NUM
iajs-2724	52	28	)	)	PUNCT
iajs-2724	52	29	≥	≥	NOUN
iajs-2724	52	30	�	�	PROPN
iajs-2724	52	31	̃	̃	PROPN
iajs-2724	52	32	�	�	PROPN
iajs-2724	52	33	}	}	PUNCT
iajs-2724	52	34	,	,	PUNCT
iajs-2724	52	35	for	for	ADP
iajs-2724	52	36	every	every	DET
iajs-2724	52	37	~[0,0	~[0,0	NOUN
iajs-2724	52	38	]	]	X
iajs-2724	53	1	≤	≤	NUM
iajs-2724	53	2	�	�	PROPN
iajs-2724	53	3	̃	̃	PROPN
iajs-2724	53	4	�	�	PROPN
iajs-2724	53	5	≤	≤	NOUN
iajs-2724	53	6	[	[	X
iajs-2724	53	7	1,1	1,1	NUM
iajs-2724	53	8	]	]	PUNCT
iajs-2724	53	9	.	.	PUNCT
iajs-2724	54	1	o.	o.	PROPN
iajs-2724	54	2	hasan	hasan	PROPN
iajs-2724	55	1	and	and	CCONJ
iajs-2724	55	2	f.kareem	f.kareem	VERB
iajs-2724	56	1	[	[	X
iajs-2724	56	2	7	7	NUM
iajs-2724	56	3	-	-	SYM
iajs-2724	56	4	8	8	NUM
iajs-2724	56	5	]	]	PUNCT
iajs-2724	56	6	introduced	introduce	VERB
iajs-2724	56	7	the	the	DET
iajs-2724	56	8	cubic	cubic	ADJ
iajs-2724	56	9	ideals	ideal	NOUN
iajs-2724	56	10	of	of	ADP
iajs-2724	56	11	the	the	DET
iajs-2724	56	12	ku	ku	PROPN
iajs-2724	56	13	-	-	PUNCT
iajs-2724	56	14	semigroup	semigroup	PROPN
iajs-2724	56	15	as	as	SCONJ
iajs-2724	56	16	follows	follow	VERB
iajs-2724	56	17	:	:	PUNCT
iajs-2724	56	18	definition(10)[7	definition(10)[7	NOUN
iajs-2724	56	19	-	-	PUNCT
iajs-2724	56	20	8	8	NUM
iajs-2724	56	21	]	]	PUNCT
iajs-2724	56	22	.	.	PUNCT
iajs-2724	57	1	in	in	ADP
iajs-2724	57	2	the	the	DET
iajs-2724	57	3	ku	ku	PROPN
iajs-2724	57	4	-	-	PUNCT
iajs-2724	57	5	semigroup(ℵ,∗,∘	semigroup(ℵ,∗,∘	PROPN
iajs-2724	57	6	,	,	PUNCT
iajs-2724	57	7	0	0	NUM
iajs-2724	57	8	)	)	PUNCT
iajs-2724	57	9	,	,	PUNCT
iajs-2724	57	10	a	a	DET
iajs-2724	57	11	cubic	cubic	ADJ
iajs-2724	57	12	set	set	NOUN
iajs-2724	57	13	θ	θ	PROPN
iajs-2724	57	14	is	be	AUX
iajs-2724	57	15	the	the	DET
iajs-2724	57	16	form	form	NOUN
iajs-2724	57	17	θ	θ	NOUN
iajs-2724	57	18	=	=	PUNCT
iajs-2724	57	19	{	{	PUNCT
iajs-2724	57	20	〈	〈	NOUN
iajs-2724	57	21	𝜒	𝜒	NOUN
iajs-2724	57	22	,	,	PUNCT
iajs-2724	57	23	𝜇θ(𝜒	𝜇θ(𝜒	NUM
iajs-2724	57	24	)	)	PUNCT
iajs-2724	57	25	,	,	PUNCT
iajs-2724	57	26	𝜆θ(𝜒	𝜆θ(𝜒	X
iajs-2724	57	27	)	)	PUNCT
iajs-2724	57	28	〉	〉	NOUN
iajs-2724	57	29	:	:	PUNCT
iajs-2724	57	30	𝜒	𝜒	NUM
iajs-2724	57	31	∈	∈	ADJ
iajs-2724	57	32	ℵ	ℵ	NOUN
iajs-2724	57	33	}	}	PUNCT
iajs-2724	57	34	,	,	PUNCT
iajs-2724	57	35	such	such	ADJ
iajs-2724	57	36	that	that	SCONJ
iajs-2724	57	37	𝜆θ(𝜒)"is	𝜆θ(𝜒)"i	VERB
iajs-2724	57	38	a	a	DET
iajs-2724	57	39	fuzzy	fuzzy	ADJ
iajs-2724	57	40	"	"	PUNCT
iajs-2724	57	41	set	set	NOUN
iajs-2724	57	42	and	and	CCONJ
iajs-2724	57	43	𝜇θ	𝜇θ	ADP
iajs-2724	57	44	:	:	PUNCT
iajs-2724	57	45	ℵ	ℵ	PROPN
iajs-2724	57	46	→	→	SYM
iajs-2724	57	47	𝐷[0,1	𝐷[0,1	NOUN
iajs-2724	57	48	]	]	X
iajs-2724	57	49	"	"	PUNCT
iajs-2724	57	50	is	be	AUX
iajs-2724	57	51	an	an	PRON
iajs-2724	57	52	"	"	PUNCT
iajs-2724	57	53	interval	interval	NOUN
iajs-2724	57	54	-	-	PUNCT
iajs-2724	57	55	valued	value	VERB
iajs-2724	57	56	"	"	PUNCT
iajs-2724	57	57	,	,	PUNCT
iajs-2724	57	58	briefly	briefly	NOUN
iajs-2724	57	59	θ	θ	X
iajs-2724	57	60	=	=	PUNCT
iajs-2724	58	1	〈	〈	PROPN
iajs-2724	58	2	𝜇θ	𝜇θ	PROPN
iajs-2724	58	3	,	,	PUNCT
iajs-2724	58	4	𝜆θ	𝜆θ	ADP
iajs-2724	58	5	〉	〉	NOUN
iajs-2724	58	6	.	.	PUNCT
iajs-2724	59	1	definition(11)[7	definition(11)[7	NOUN
iajs-2724	59	2	-	-	PUNCT
iajs-2724	59	3	8	8	NUM
iajs-2724	59	4	]	]	PUNCT
iajs-2724	59	5	.	.	PUNCT
iajs-2724	60	1	in	in	ADP
iajs-2724	60	2	the	the	DET
iajs-2724	60	3	ku	ku	PROPN
iajs-2724	60	4	-	-	PUNCT
iajs-2724	60	5	semigroup	semigroup	PROPN
iajs-2724	60	6	(	(	PUNCT
iajs-2724	60	7	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2724	60	8	,	,	PUNCT
iajs-2724	60	9	0	0	NUM
iajs-2724	60	10	)	)	PUNCT
iajs-2724	60	11	a	a	DET
iajs-2724	60	12	cubic	cubic	ADJ
iajs-2724	60	13	set	set	VERB
iajs-2724	60	14	θ	θ	PROPN
iajs-2724	60	15	=	=	PUNCT
iajs-2724	60	16	〈	〈	PROPN
iajs-2724	60	17	𝜇θ	𝜇θ	PROPN
iajs-2724	60	18	,	,	PUNCT
iajs-2724	60	19	𝜆θ〉in	𝜆θ〉in	NOUN
iajs-2724	60	20	ℵ	ℵ	NOUN
iajs-2724	60	21	is	be	AUX
iajs-2724	60	22	named	name	VERB
iajs-2724	60	23	a	a	DET
iajs-2724	60	24	cubic	cubic	ADJ
iajs-2724	60	25	sub	sub	ADJ
iajs-2724	60	26	-	-	ADJ
iajs-2724	60	27	ku	ku	NOUN
iajs-2724	60	28	-	-	PUNCT
iajs-2724	60	29	semigroup	semigroup	PROPN
iajs-2724	60	30	if	if	SCONJ
iajs-2724	60	31	:	:	PUNCT
iajs-2724	60	32	for	for	ADP
iajs-2724	60	33	all	all	DET
iajs-2724	60	34	𝜒	𝜒	NOUN
iajs-2724	60	35	,	,	PUNCT
iajs-2724	60	36	𝛾	𝛾	ADP
iajs-2724	60	37	∈	∈	NOUN
iajs-2724	60	38	ℵ	ℵ	NOUN
iajs-2724	60	39	,	,	PUNCT
iajs-2724	60	40	(	(	PUNCT
iajs-2724	60	41	1	1	X
iajs-2724	60	42	)	)	PUNCT
iajs-2724	60	43	�	�	PROPN
iajs-2724	60	44	̃	̃	PROPN
iajs-2724	60	45	�	�	PROPN
iajs-2724	60	46	θ(𝜒	θ(𝜒	PROPN
iajs-2724	60	47	∗	∗	PROPN
iajs-2724	60	48	𝛾	𝛾	NOUN
iajs-2724	60	49	)	)	PUNCT
iajs-2724	60	50	≥	≥	PROPN
iajs-2724	60	51	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2724	60	52	{	{	PUNCT
iajs-2724	60	53	�	�	PROPN
iajs-2724	60	54	̃	̃	PROPN
iajs-2724	60	55	�	�	NOUN
iajs-2724	60	56	θ(𝜒)ˑ	θ(𝜒)ˑ	NOUN
iajs-2724	60	57	,	,	PUNCT
iajs-2724	60	58	�	�	PROPN
iajs-2724	60	59	̃	̃	PROPN
iajs-2724	60	60	�	�	PROPN
iajs-2724	60	61	θ(𝛾)},𝜆θ(𝜒	θ(𝛾)},𝜆θ(𝜒	PROPN
iajs-2724	60	62	∗	∗	NOUN
iajs-2724	60	63	𝛾	𝛾	NOUN
iajs-2724	60	64	)	)	PUNCT
iajs-2724	60	65	≤	≤	NUM
iajs-2724	60	66	𝑚𝑎𝑥	𝑚𝑎𝑥	NOUN
iajs-2724	60	67	{	{	PUNCT
iajs-2724	60	68	𝜆θ(𝜒)ˑ	𝜆θ(𝜒)ˑ	NOUN
iajs-2724	60	69	,	,	PUNCT
iajs-2724	60	70	𝜆θ(𝛾	𝜆θ(𝛾	X
iajs-2724	60	71	)	)	PUNCT
iajs-2724	60	72	}	}	PUNCT
iajs-2724	60	73	(	(	PUNCT
iajs-2724	60	74	2	2	X
iajs-2724	60	75	)	)	PUNCT
iajs-2724	60	76	�	�	PROPN
iajs-2724	60	77	̃	̃	PROPN
iajs-2724	60	78	�	�	PROPN
iajs-2724	60	79	θ(𝜒	θ(𝜒	PROPN
iajs-2724	60	80	∘	∘	PROPN
iajs-2724	60	81	𝛾	𝛾	PROPN
iajs-2724	60	82	)	)	PUNCT
iajs-2724	60	83	≥	≥	PROPN
iajs-2724	60	84	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2724	60	85	{	{	PUNCT
iajs-2724	60	86	�	�	PROPN
iajs-2724	60	87	̃	̃	PROPN
iajs-2724	60	88	�	�	NOUN
iajs-2724	60	89	θ(𝜒)ˑ	θ(𝜒)ˑ	NOUN
iajs-2724	60	90	,	,	PUNCT
iajs-2724	60	91	ˑ	ˑ	PROPN
iajs-2724	60	92	�	�	PROPN
iajs-2724	60	93	̃	̃	NOUN
iajs-2724	60	94	�	�	NOUN
iajs-2724	60	95	θ(𝛾	θ(𝛾	NOUN
iajs-2724	60	96	)	)	PUNCT
iajs-2724	60	97	}	}	PUNCT
iajs-2724	60	98	,	,	PUNCT
iajs-2724	60	99	𝜆θ(𝜒	𝜆θ(𝜒	PUNCT
iajs-2724	60	100	∘	∘	PROPN
iajs-2724	60	101	𝛾	𝛾	NOUN
iajs-2724	60	102	)	)	PUNCT
iajs-2724	60	103	≤	≤	NUM
iajs-2724	60	104	𝑚𝑎𝑥	𝑚𝑎𝑥	NOUN
iajs-2724	60	105	{	{	PUNCT
iajs-2724	60	106	𝜆θ(𝜒)ˑ	𝜆θ(𝜒)ˑ	PROPN
iajs-2724	60	107	,	,	PUNCT
iajs-2724	60	108	ˑ𝜆θ(𝛾	ˑ𝜆θ(𝛾	PROPN
iajs-2724	60	109	)	)	PUNCT
iajs-2724	60	110	}	}	PUNCT
iajs-2724	60	111	.	.	PUNCT
iajs-2724	61	1	definition(12)[7	definition(12)[7	NOUN
iajs-2724	61	2	-	-	PUNCT
iajs-2724	61	3	8	8	NUM
iajs-2724	61	4	]	]	PUNCT
iajs-2724	61	5	.	.	PUNCT
iajs-2724	62	1	the	the	DET
iajs-2724	62	2	set	set	VERB
iajs-2724	62	3	θ	θ	PROPN
iajs-2724	62	4	in	in	ADP
iajs-2724	62	5	ℵ	ℵ	PROPN
iajs-2724	62	6	is	be	AUX
iajs-2724	62	7	named	name	VERB
iajs-2724	62	8	a	a	DET
iajs-2724	62	9	cubic	cubic	ADJ
iajs-2724	62	10	ideal	ideal	NOUN
iajs-2724	62	11	of	of	ADP
iajs-2724	62	12	a	a	DET
iajs-2724	62	13	ku	ku	PROPN
iajs-2724	62	14	-	-	PUNCT
iajs-2724	62	15	semigroup	semigroup	PROPN
iajs-2724	62	16	(	(	PUNCT
iajs-2724	62	17	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2724	62	18	,	,	PUNCT
iajs-2724	62	19	0	0	NUM
iajs-2724	62	20	)	)	PUNCT
iajs-2724	62	21	if	if	SCONJ
iajs-2724	62	22	,	,	PUNCT
iajs-2724	62	23	∀	∀	X
iajs-2724	62	24	𝜒	𝜒	NOUN
iajs-2724	62	25	,	,	PUNCT
iajs-2724	62	26	𝛾	𝛾	PROPN
iajs-2724	62	27	∈	∈	PROPN
iajs-2724	62	28	ℵ	ℵ	X
iajs-2724	62	29	(	(	PUNCT
iajs-2724	62	30	ci1	ci1	NOUN
iajs-2724	62	31	)	)	PUNCT
iajs-2724	62	32	𝜇𝛩(0	𝜇𝛩(0	PROPN
iajs-2724	62	33	)	)	PUNCT
iajs-2724	62	34	≥	≥	NOUN
iajs-2724	62	35	𝜇𝛩(𝜒	𝜇𝛩(𝜒	NOUN
iajs-2724	62	36	)	)	PUNCT
iajs-2724	62	37	and	and	CCONJ
iajs-2724	62	38	𝜆𝛩(0	𝜆𝛩(0	PROPN
iajs-2724	62	39	)	)	PUNCT
iajs-2724	62	40	≤	≤	NOUN
iajs-2724	62	41	𝜆𝛩(𝜒	𝜆𝛩(𝜒	NUM
iajs-2724	62	42	)	)	PUNCT
iajs-2724	62	43	,	,	PUNCT
iajs-2724	62	44	(	(	PUNCT
iajs-2724	62	45	ci2	ci2	NOUN
iajs-2724	62	46	)	)	PUNCT
iajs-2724	62	47	μ̃θ(γ	μ̃θ(γ	NUM
iajs-2724	62	48	)	)	PUNCT
iajs-2724	62	49	≥	≥	PROPN
iajs-2724	62	50	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	62	51	�	�	PROPN
iajs-2724	62	52	̃	̃	PROPN
iajs-2724	62	53	�	�	PROPN
iajs-2724	62	54	𝛩(𝜒	𝛩(𝜒	X
iajs-2724	62	55	∗	∗	NOUN
iajs-2724	62	56	𝛾	𝛾	NOUN
iajs-2724	62	57	)	)	PUNCT
iajs-2724	62	58	,	,	PUNCT
iajs-2724	62	59	𝜇𝛩(𝜒	𝜇𝛩(𝜒	PROPN
iajs-2724	62	60	)	)	PUNCT
iajs-2724	62	61	}	}	PUNCT
iajs-2724	62	62	,	,	PUNCT
iajs-2724	62	63	𝜆𝛩(𝛾	𝜆𝛩(𝛾	PROPN
iajs-2724	62	64	)	)	PUNCT
iajs-2724	62	65	≤	≤	NUM
iajs-2724	62	66	𝑚𝑎𝑥	𝑚𝑎𝑥	X
iajs-2724	62	67	{	{	PUNCT
iajs-2724	62	68	λθ(χ	λθ(χ	X
iajs-2724	62	69	∗	∗	NOUN
iajs-2724	62	70	γ)ˑ	γ)ˑ	NOUN
iajs-2724	62	71	,	,	PUNCT
iajs-2724	62	72	λθ(χ	λθ(χ	NOUN
iajs-2724	62	73	)	)	PUNCT
iajs-2724	62	74	}	}	PUNCT
iajs-2724	62	75	(	(	PUNCT
iajs-2724	62	76	ci3	ci3	ADJ
iajs-2724	62	77	)	)	PUNCT
iajs-2724	62	78	μ̃θ(𝜒	μ̃θ(𝜒	ADJ
iajs-2724	62	79	∘	∘	PROPN
iajs-2724	62	80	𝛾	𝛾	NOUN
iajs-2724	62	81	)	)	PUNCT
iajs-2724	62	82	≥	≥	PROPN
iajs-2724	62	83	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2724	62	84	{	{	PUNCT
iajs-2724	62	85	�	�	PROPN
iajs-2724	62	86	̃	̃	PROPN
iajs-2724	62	87	�	�	NOUN
iajs-2724	62	88	θ(𝜒	θ(𝜒	NOUN
iajs-2724	62	89	)	)	PUNCT
iajs-2724	62	90	,	,	PUNCT
iajs-2724	62	91	ˑ𝜇θ(𝛾	ˑ𝜇θ(𝛾	NUM
iajs-2724	62	92	)	)	PUNCT
iajs-2724	62	93	}	}	PUNCT
iajs-2724	62	94	,	,	PUNCT
iajs-2724	62	95	𝜆θ(𝜒	𝜆θ(𝜒	PUNCT
iajs-2724	62	96	∘	∘	PROPN
iajs-2724	62	97	𝛾	𝛾	NOUN
iajs-2724	62	98	)	)	PUNCT
iajs-2724	62	99	≤	≤	NUM
iajs-2724	62	100	𝑚𝑎𝑥	𝑚𝑎𝑥	NOUN
iajs-2724	62	101	{	{	PUNCT
iajs-2724	62	102	𝜆θ(𝜒)ˑ	𝜆θ(𝜒)ˑ	PROPN
iajs-2724	62	103	,	,	PUNCT
iajs-2724	62	104	ˑ𝜆θ(𝛾	ˑ𝜆θ(𝛾	PROPN
iajs-2724	62	105	)	)	PUNCT
iajs-2724	62	106	}	}	PUNCT
iajs-2724	62	107	.	.	PUNCT
iajs-2724	63	1	ibn	ibn	PROPN
iajs-2724	63	2	al	al	PROPN
iajs-2724	63	3	-	-	PUNCT
iajs-2724	63	4	haitham	haitham	PROPN
iajs-2724	63	5	jour	jour	X
iajs-2724	63	6	.	.	PROPN
iajs-2724	63	7	for	for	ADP
iajs-2724	63	8	pure	pure	ADJ
iajs-2724	63	9	&	&	CCONJ
iajs-2724	63	10	appl	appl	PROPN
iajs-2724	63	11	.	.	PUNCT
iajs-2724	64	1	sci	sci	PROPN
iajs-2724	64	2	.	.	PROPN
iajs-2724	65	1	53	53	NUM
iajs-2724	65	2	(	(	PUNCT
iajs-2724	65	3	2)2022	2)2022	VERB
iajs-2724	65	4	50	50	NUM
iajs-2724	65	5	example(13)[7	example(13)[7	NOUN
iajs-2724	65	6	-	-	PUNCT
iajs-2724	65	7	8	8	NUM
iajs-2724	65	8	]	]	PUNCT
iajs-2724	65	9	.	.	PUNCT
iajs-2724	66	1	let	let	VERB
iajs-2724	66	2	ℵ	ℵ	NOUN
iajs-2724	66	3	=	=	X
iajs-2724	66	4	{	{	PUNCT
iajs-2724	66	5	0,1,2	0,1,2	NOUN
iajs-2724	66	6	}	}	PUNCT
iajs-2724	66	7	be	be	AUX
iajs-2724	66	8	a	a	DET
iajs-2724	66	9	set	set	NOUN
iajs-2724	66	10	.	.	PUNCT
iajs-2724	67	1	define	define	VERB
iajs-2724	67	2	the	the	DET
iajs-2724	67	3	operations	operation	NOUN
iajs-2724	67	4	∗,∘	∗,∘	PROPN
iajs-2724	67	5	by	by	ADP
iajs-2724	67	6	the	the	DET
iajs-2724	67	7	following	follow	VERB
iajs-2724	67	8	tables	table	NOUN
iajs-2724	67	9	.	.	PUNCT
iajs-2724	68	1	then	then	ADV
iajs-2724	68	2	the	the	DET
iajs-2724	68	3	structure	structure	NOUN
iajs-2724	68	4	(	(	PUNCT
iajs-2724	68	5	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2724	68	6	,	,	PUNCT
iajs-2724	68	7	0)ˑis	0)ˑis	NUM
iajs-2724	68	8	a	a	DET
iajs-2724	68	9	ku	ku	PROPN
iajs-2724	68	10	-	-	PUNCT
iajs-2724	68	11	semi	semi	NOUN
iajs-2724	68	12	group	group	NOUN
iajs-2724	68	13	.	.	PUNCT
iajs-2724	69	1	a	a	DET
iajs-2724	69	2	cubic	cubic	ADJ
iajs-2724	69	3	set	set	VERB
iajs-2724	69	4	θ	θ	PROPN
iajs-2724	69	5	=	=	PUNCT
iajs-2724	69	6	〈	〈	PROPN
iajs-2724	69	7	𝜇θ	𝜇θ	PROPN
iajs-2724	69	8	,	,	PUNCT
iajs-2724	69	9	𝜆θ	𝜆θ	SCONJ
iajs-2724	69	10	〉	〉	NOUN
iajs-2724	69	11	is	be	AUX
iajs-2724	69	12	defined	define	VERB
iajs-2724	69	13	by	by	ADP
iajs-2724	69	14	:	:	PUNCT
iajs-2724	69	15	𝜇θ(𝑥	𝜇θ(𝑥	NUM
iajs-2724	69	16	)	)	PUNCT
iajs-2724	69	17	=	=	SYM
iajs-2724	69	18	{	{	PUNCT
iajs-2724	70	1	[	[	X
iajs-2724	70	2	0.4,0.8	0.4,0.8	X
iajs-2724	70	3	]	]	X
iajs-2724	70	4	𝑖𝑓	𝑖𝑓	X
iajs-2724	70	5	𝜒ϵ	𝜒ϵ	ADP
iajs-2724	70	6	{	{	PUNCT
iajs-2724	70	7	0,2	0,2	NUM
iajs-2724	70	8	}	}	PUNCT
iajs-2724	71	1	[	[	X
iajs-2724	71	2	0.1,0.3	0.1,0.3	X
iajs-2724	71	3	]	]	X
iajs-2724	71	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2724	71	5	𝜒	𝜒	X
iajs-2724	71	6	=	=	SYM
iajs-2724	71	7	1	1	NUM
iajs-2724	71	8	and	and	CCONJ
iajs-2724	71	9	𝜆θ(𝑥	𝜆θ(𝑥	NUM
iajs-2724	71	10	)	)	PUNCT
iajs-2724	71	11	=	=	PRON
iajs-2724	71	12	{	{	PUNCT
iajs-2724	71	13	0.1	0.1	NUM
iajs-2724	71	14	𝑖𝑓	𝑖𝑓	NUM
iajs-2724	71	15	𝜒ϵ{0,2	𝜒ϵ{0,2	PROPN
iajs-2724	71	16	}	}	PUNCT
iajs-2724	71	17	0.3	0.3	NUM
iajs-2724	71	18	𝑖𝑓	𝑖𝑓	NOUN
iajs-2724	72	1	𝜒	𝜒	NOUN
iajs-2724	72	2	=	=	NOUN
iajs-2724	72	3	1	1	NUM
iajs-2724	72	4	then	then	ADV
iajs-2724	72	5	θ	θ	PROPN
iajs-2724	72	6	=	=	PUNCT
iajs-2724	73	1	〈	〈	PROPN
iajs-2724	73	2	𝜇θ	𝜇θ	PROPN
iajs-2724	73	3	,	,	PUNCT
iajs-2724	73	4	𝜆θ	𝜆θ	ADP
iajs-2724	73	5	〉	〉	NOUN
iajs-2724	73	6	is	be	AUX
iajs-2724	73	7	a	a	DET
iajs-2724	73	8	cubic	cubic	ADJ
iajs-2724	73	9	ideal	ideal	NOUN
iajs-2724	73	10	of	of	ADP
iajs-2724	73	11	ℵ	ℵ	NOUN
iajs-2724	73	12	.	.	PUNCT
iajs-2724	74	1	definition(14)[7	definition(14)[7	PROPN
iajs-2724	74	2	-	-	PUNCT
iajs-2724	74	3	8	8	NUM
iajs-2724	74	4	]	]	PUNCT
iajs-2724	74	5	.	.	PUNCT
iajs-2724	75	1	in	in	ADP
iajs-2724	75	2	a	a	DET
iajs-2724	75	3	ku	ku	PROPN
iajs-2724	75	4	-	-	PUNCT
iajs-2724	75	5	semigroup	semigroup	PROPN
iajs-2724	75	6	(	(	PUNCT
iajs-2724	75	7	ℵ,∗,∘	ℵ,∗,∘	NUM
iajs-2724	75	8	,	,	PUNCT
iajs-2724	75	9	0	0	NUM
iajs-2724	75	10	)	)	PUNCT
iajs-2724	75	11	,	,	PUNCT
iajs-2724	75	12	a	a	DET
iajs-2724	75	13	cubic	cubic	ADJ
iajs-2724	75	14	set	set	VERB
iajs-2724	75	15	θ	θ	PROPN
iajs-2724	75	16	=	=	PUNCT
iajs-2724	75	17	〈	〈	PROPN
iajs-2724	75	18	𝜇θ	𝜇θ	PROPN
iajs-2724	75	19	,	,	PUNCT
iajs-2724	75	20	𝜆θ	𝜆θ	SCONJ
iajs-2724	75	21	〉	〉	NOUN
iajs-2724	75	22	in	in	ADP
iajs-2724	75	23	ℵ	ℵ	NOUN
iajs-2724	75	24	is	be	AUX
iajs-2724	75	25	named	name	VERB
iajs-2724	75	26	a	a	DET
iajs-2724	75	27	cubic	cubic	ADJ
iajs-2724	75	28	k	k	NOUN
iajs-2724	75	29	-	-	NOUN
iajs-2724	75	30	ideal	ideal	NOUN
iajs-2724	75	31	if	if	SCONJ
iajs-2724	75	32	∀	∀	X
iajs-2724	75	33	𝜒	𝜒	X
iajs-2724	75	34	,	,	PUNCT
iajs-2724	75	35	𝛾	𝛾	PROPN
iajs-2724	75	36	,	,	PUNCT
iajs-2724	75	37	τ	τ	PROPN
iajs-2724	75	38	∈	∈	PROPN
iajs-2724	75	39	ℵ	ℵ	X
iajs-2724	75	40	(	(	PUNCT
iajs-2724	75	41	𝑪𝒌𝟏	𝑪𝒌𝟏	ADJ
iajs-2724	75	42	)	)	PUNCT
iajs-2724	75	43	�	�	PROPN
iajs-2724	75	44	̃	̃	PROPN
iajs-2724	75	45	�	�	PROPN
iajs-2724	75	46	θ(0	θ(0	PROPN
iajs-2724	75	47	)	)	PUNCT
iajs-2724	75	48	)	)	PUNCT
iajs-2724	75	49	≥	≥	NOUN
iajs-2724	75	50	𝜇θ(𝜒	𝜇θ(𝜒	NUM
iajs-2724	75	51	)	)	PUNCT
iajs-2724	75	52	,	,	PUNCT
iajs-2724	75	53	and	and	CCONJ
iajs-2724	75	54	𝜆θ(0	𝜆θ(0	PROPN
iajs-2724	75	55	)	)	PUNCT
iajs-2724	75	56	≤	≤	NOUN
iajs-2724	75	57	𝜆θ(𝑥	𝜆θ(𝑥	NUM
iajs-2724	75	58	)	)	PUNCT
iajs-2724	75	59	(	(	PUNCT
iajs-2724	75	60	𝑪𝒌𝟐)	𝑪𝒌𝟐)	PROPN
iajs-2724	75	61	�	�	PROPN
iajs-2724	75	62	̃	̃	PROPN
iajs-2724	75	63	�	�	PROPN
iajs-2724	75	64	θ(𝜒	θ(𝜒	PROPN
iajs-2724	75	65	∗	∗	PROPN
iajs-2724	75	66	τ	τ	PROPN
iajs-2724	75	67	)	)	PUNCT
iajs-2724	75	68	≥	≥	PROPN
iajs-2724	75	69	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	75	70	�	�	PROPN
iajs-2724	75	71	̃	̃	PROPN
iajs-2724	75	72	�	�	PROPN
iajs-2724	75	73	θ(𝜒	θ(𝜒	ADJ
iajs-2724	75	74	∗	∗	NOUN
iajs-2724	75	75	(	(	PUNCT
iajs-2724	75	76	𝛾	𝛾	PROPN
iajs-2724	75	77	∗	∗	NOUN
iajs-2724	75	78	τ	τ	PROPN
iajs-2724	75	79	)	)	PUNCT
iajs-2724	75	80	)	)	PUNCT
iajs-2724	75	81	,	,	PUNCT
iajs-2724	75	82	𝜇θ(𝛾	𝜇θ(𝛾	NOUN
iajs-2724	75	83	)	)	PUNCT
iajs-2724	75	84	}	}	PUNCT
iajs-2724	75	85	,	,	PUNCT
iajs-2724	75	86	𝜆θ(𝜒	𝜆θ(𝜒	PUNCT
iajs-2724	75	87	∗	∗	X
iajs-2724	75	88	τ	τ	NOUN
iajs-2724	75	89	)	)	PUNCT
iajs-2724	75	90	≤	≤	NUM
iajs-2724	75	91	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-2724	75	92	{	{	PUNCT
iajs-2724	75	93	𝜆θ(𝜒	𝜆θ(𝜒	NOUN
iajs-2724	75	94	∗	∗	NOUN
iajs-2724	75	95	(	(	PUNCT
iajs-2724	75	96	𝛾	𝛾	NOUN
iajs-2724	75	97	∗	∗	NOUN
iajs-2724	75	98	τ	τ	PROPN
iajs-2724	75	99	)	)	PUNCT
iajs-2724	75	100	)	)	PUNCT
iajs-2724	75	101	,	,	PUNCT
iajs-2724	75	102	𝜆θ(𝛾	𝜆θ(𝛾	X
iajs-2724	75	103	)	)	PUNCT
iajs-2724	75	104	}	}	PUNCT
iajs-2724	75	105	(	(	PUNCT
iajs-2724	75	106	𝑪𝒌𝟑)	𝑪𝒌𝟑)	PROPN
iajs-2724	75	107	�	�	PROPN
iajs-2724	75	108	̃	̃	PROPN
iajs-2724	75	109	�	�	NOUN
iajs-2724	75	110	θ(𝜒	θ(𝜒	PROPN
iajs-2724	75	111	∘	∘	PROPN
iajs-2724	75	112	𝛾	𝛾	PROPN
iajs-2724	75	113	)	)	PUNCT
iajs-2724	75	114	≥	≥	PROPN
iajs-2724	75	115	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2724	75	116	{	{	PUNCT
iajs-2724	75	117	�	�	PROPN
iajs-2724	75	118	̃	̃	PROPN
iajs-2724	75	119	�	�	NOUN
iajs-2724	75	120	θ(𝜒)ˑ	θ(𝜒)ˑ	NOUN
iajs-2724	75	121	,	,	PUNCT
iajs-2724	75	122	𝜇θ(𝛾	𝜇θ(𝛾	NOUN
iajs-2724	75	123	)	)	PUNCT
iajs-2724	75	124	}	}	PUNCT
iajs-2724	75	125	,	,	PUNCT
iajs-2724	75	126	𝜆θ(𝜒	𝜆θ(𝜒	PUNCT
iajs-2724	75	127	∘	∘	PROPN
iajs-2724	75	128	𝛾	𝛾	NOUN
iajs-2724	75	129	)	)	PUNCT
iajs-2724	75	130	≤	≤	NUM
iajs-2724	75	131	𝑚𝑎𝑥	𝑚𝑎𝑥	NOUN
iajs-2724	75	132	{	{	PUNCT
iajs-2724	75	133	𝜆θ(𝜒)ˑ	𝜆θ(𝜒)ˑ	PROPN
iajs-2724	75	134	,	,	PUNCT
iajs-2724	75	135	ˑ𝜆θ(𝛾	ˑ𝜆θ(𝛾	PROPN
iajs-2724	75	136	)	)	PUNCT
iajs-2724	75	137	}	}	PUNCT
iajs-2724	75	138	.	.	PUNCT
iajs-2724	76	1	in	in	ADP
iajs-2724	76	2	the	the	DET
iajs-2724	76	3	following	following	NOUN
iajs-2724	76	4	,	,	PUNCT
iajs-2724	76	5	we	we	PRON
iajs-2724	76	6	recall	recall	VERB
iajs-2724	76	7	some	some	DET
iajs-2724	76	8	basic	basic	ADJ
iajs-2724	76	9	concepts	concept	NOUN
iajs-2724	76	10	of	of	ADP
iajs-2724	76	11	a	a	DET
iajs-2724	76	12	bipolar	bipolar	ADJ
iajs-2724	76	13	fuzzy	fuzzy	ADJ
iajs-2724	76	14	set	set	NOUN
iajs-2724	76	15	.	.	PUNCT
iajs-2724	77	1	definition(15)[9	definition(15)[9	PROPN
iajs-2724	77	2	]	]	X
iajs-2724	77	3	.	.	PUNCT
iajs-2724	78	1	a	a	DET
iajs-2724	78	2	bipolar	bipolar	ADJ
iajs-2724	78	3	fuzzy	fuzzy	ADJ
iajs-2724	78	4	set	set	NOUN
iajs-2724	78	5	in	in	ADP
iajs-2724	78	6	a	a	DET
iajs-2724	78	7	set	set	NOUN
iajs-2724	78	8	ℵ	ℵ	NOUN
iajs-2724	78	9	is	be	AUX
iajs-2724	78	10	a	a	DET
iajs-2724	78	11	form	form	NOUN
iajs-2724	78	12	}	}	PUNCT
iajs-2724	78	13	:))	:))	PROPN
iajs-2724	78	14	(	(	PUNCT
iajs-2724	78	15	)	)	PUNCT
iajs-2724	78	16	,	,	PUNCT
iajs-2724	78	17	(	(	PUNCT
iajs-2724	78	18	,	,	PUNCT
iajs-2724	78	19	{	{	PUNCT
iajs-2724	78	20	(	(	PUNCT
iajs-2724	78	21			PROPN
iajs-2724	78	22			PROPN
iajs-2724	78	23			PROPN
iajs-2724	78	24	,	,	PUNCT
iajs-2724	78	25	where	where	SCONJ
iajs-2724	78	26	]	]	X
iajs-2724	78	27	0,1	0,1	NUM
iajs-2724	78	28	[:	[:	X
iajs-2724	78	29	)	)	PUNCT
iajs-2724	78	30	(	(	PUNCT
iajs-2724	78	31			PUNCT
iajs-2724	78	32			NOUN
iajs-2724	78	33	and	and	CCONJ
iajs-2724	78	34	]	]	X
iajs-2724	78	35	1,0	1,0	NUM
iajs-2724	78	36	[:	[:	NOUN
iajs-2724	78	37	)	)	PUNCT
iajs-2724	78	38	(	(	PUNCT
iajs-2724	78	39			NOUN
iajs-2724	78	40			NOUN
iajs-2724	78	41	are	be	AUX
iajs-2724	78	42	two	two	NUM
iajs-2724	78	43	fuzzy	fuzzy	ADJ
iajs-2724	78	44	mappings	mapping	NOUN
iajs-2724	78	45	.	.	PUNCT
iajs-2724	79	1	the	the	DET
iajs-2724	79	2	two	two	NUM
iajs-2724	79	3	membershipdegrees	membershipdegree	NOUN
iajs-2724	79	4	)	)	PUNCT
iajs-2724	79	5	(	(	PUNCT
iajs-2724	79	6			NOUN
iajs-2724	79	7			VERB
iajs-2724	79	8	and	and	CCONJ
iajs-2724	79	9	)	)	PUNCT
iajs-2724	79	10	(	(	PUNCT
iajs-2724	79	11			NOUN
iajs-2724	79	12			VERB
iajs-2724	79	13	denotethe	denotethe	PRON
iajs-2724	79	14	fulfillment	fulfillment	NOUN
iajs-2724	79	15	-degree	-degree	NOUN
iajs-2724	79	16	of-	of-	NOUN
iajs-2724	79	17	to	to	ADP
iajs-2724	79	18	the	the	DET
iajs-2724	79	19	property	property	NOUN
iajs-2724	79	20	corresponding	corresponding	NOUN
iajs-2724	79	21	of	of	ADP
iajs-2724	79	22			NOUN
iajs-2724	79	23	and	and	CCONJ
iajs-2724	79	24	-	-	PUNCT
iajs-2724	79	25	the	the	DET
iajs-2724	79	26	fulfillment	fulfillment	NOUN
iajs-2724	79	27	-	-	PUNCT
iajs-2724	79	28	degree	degree	NOUN
iajs-2724	79	29	of	of	ADP
iajs-2724	79	30			NOUN
iajs-2724	79	31	to	to	ADP
iajs-2724	79	32	some	some	DET
iajs-2724	79	33	implicitcounter	implicitcounter	NOUN
iajs-2724	79	34	-	-	PUNCT
iajs-2724	79	35	property	property	NOUN
iajs-2724	79	36	of	of	PROPN
iajs-2724	79	37	,	,	PUNCT
iajs-2724	79	38	respectively	respectively	ADV
iajs-2724	79	39	.	.	PUNCT
iajs-2724	80	1	kareem	kareem	PROPN
iajs-2724	80	2	and	and	CCONJ
iajs-2724	80	3	awad[10	awad[10	PROPN
iajs-2724	80	4	]	]	PUNCT
iajs-2724	80	5	introduced	introduce	VERB
iajs-2724	80	6	the	the	DET
iajs-2724	80	7	cubic	cubic	ADJ
iajs-2724	80	8	bipolar	bipolar	ADJ
iajs-2724	80	9	-idealsof	-idealsof	PROPN
iajs-2724	80	10	a	a	DET
iajs-2724	80	11	ku	ku	PROPN
iajs-2724	80	12	--	--	PUNCT
iajs-2724	80	13	semigroupin	semigroupin	PROPN
iajs-2724	80	14	-ku	-ku	ADJ
iajs-2724	80	15	-	-	PUNCT
iajs-2724	80	16	algebra	algebra	NOUN
iajs-2724	80	17	asfollows	asfollow	VERB
iajs-2724	80	18	:	:	PUNCT
iajs-2724	80	19	-	-	PUNCT
iajs-2724	80	20	definition(16)[10	definition(16)[10	NUM
iajs-2724	80	21	]	]	PUNCT
iajs-2724	80	22	.	.	PUNCT
iajs-2724	81	1	let	let	VERB
iajs-2724	81	2	ℵ	ℵ	PRON
iajs-2724	81	3	be	be	AUX
iajs-2724	81	4	a	a	DET
iajs-2724	81	5	non	non	ADJ
iajs-2724	81	6	-	-	ADJ
iajs-2724	81	7	empty	empty	ADJ
iajs-2724	81	8	set	set	NOUN
iajs-2724	81	9	.	.	PUNCT
iajs-2724	82	1	a	a	DET
iajs-2724	82	2	cubic	cubic	ADJ
iajs-2724	82	3	bipolar	bipolar	ADJ
iajs-2724	82	4	set	set	VERB
iajs-2724	82	5	in	in	ADP
iajs-2724	82	6	a	a	DET
iajs-2724	82	7	set	set	NOUN
iajs-2724	82	8	ℵ	ℵ	NOUN
iajs-2724	82	9	is	be	AUX
iajs-2724	82	10	the	the	DET
iajs-2724	82	11	structure	structure	NOUN
iajs-2724	82	12	θ	θ	NOUN
iajs-2724	82	13	=	=	PUNCT
iajs-2724	82	14	{	{	PUNCT
iajs-2724	82	15	〈	〈	NOUN
iajs-2724	82	16	χ	χ	NOUN
iajs-2724	82	17	,	,	PUNCT
iajs-2724	82	18	𝜇θ	𝜇θ	PROPN
iajs-2724	83	1	+	+	ADJ
iajs-2724	83	2	(	(	PUNCT
iajs-2724	83	3	χ	χ	NOUN
iajs-2724	83	4	)	)	PUNCT
iajs-2724	83	5	,	,	PUNCT
iajs-2724	83	6	𝜇θ	𝜇θ	ADP
iajs-2724	83	7	−(χ	−(χ	NOUN
iajs-2724	83	8	)	)	PUNCT
iajs-2724	83	9	,	,	PUNCT
iajs-2724	83	10	𝜆θ	𝜆θ	ADP
iajs-2724	83	11	+	+	ADJ
iajs-2724	83	12	(	(	PUNCT
iajs-2724	83	13	χ	χ	NOUN
iajs-2724	83	14	)	)	PUNCT
iajs-2724	83	15	,	,	PUNCT
iajs-2724	83	16	𝜆θ	𝜆θ	ADP
iajs-2724	83	17	−(χ	−(χ	NOUN
iajs-2724	83	18	):	):	PUNCT
iajs-2724	83	19	χ	χ	PROPN
iajs-2724	83	20	∈	∈	PROPN
iajs-2724	83	21	ℵ	ℵ	NOUN
iajs-2724	83	22	〉	〉	NOUN
iajs-2724	83	23	}	}	PUNCT
iajs-2724	83	24	is	be	AUX
iajs-2724	83	25	denoted	denote	VERB
iajs-2724	83	26	as	as	ADP
iajs-2724	83	27	θ	θ	X
iajs-2724	83	28	=	=	SYM
iajs-2724	83	29	〈	〈	PROPN
iajs-2724	83	30	𝑁,𝐾	𝑁,𝐾	NOUN
iajs-2724	83	31	〉	〉	NOUN
iajs-2724	83	32	,	,	PUNCT
iajs-2724	83	33	where	where	SCONJ
iajs-2724	83	34	𝑁(χ	𝑁(χ	NOUN
iajs-2724	83	35	)	)	PUNCT
iajs-2724	83	36	=	=	PRON
iajs-2724	83	37	{	{	PUNCT
iajs-2724	83	38	�	�	PROPN
iajs-2724	83	39	̃	̃	NOUN
iajs-2724	83	40	�	�	NOUN
iajs-2724	83	41	θ	θ	NOUN
iajs-2724	83	42	+	+	PROPN
iajs-2724	83	43	(	(	PUNCT
iajs-2724	83	44	χ	χ	NOUN
iajs-2724	83	45	)	)	PUNCT
iajs-2724	83	46	,	,	PUNCT
iajs-2724	83	47	𝜇θ	𝜇θ	ADP
iajs-2724	83	48	−(χ	−(χ	NOUN
iajs-2724	83	49	)	)	PUNCT
iajs-2724	83	50	}	}	PUNCT
iajs-2724	83	51	is	be	AUX
iajs-2724	83	52	called	call	VERB
iajs-2724	83	53	interval	interval	NOUN
iajs-2724	83	54	-	-	PUNCT
iajs-2724	83	55	valued	value	VERB
iajs-2724	83	56	bipolar	bipolar	ADJ
iajs-2724	83	57	fuzzy	fuzzy	ADJ
iajs-2724	83	58	set	set	NOUN
iajs-2724	83	59	and	and	CCONJ
iajs-2724	83	60	𝐾(χ	𝐾(χ	PRON
iajs-2724	83	61	)	)	PUNCT
iajs-2724	84	1	=	=	NOUN
iajs-2724	84	2	{	{	PUNCT
iajs-2724	84	3	𝜆θ	𝜆θ	ADP
iajs-2724	84	4	+	+	ADJ
iajs-2724	84	5	(	(	PUNCT
iajs-2724	84	6	χ	χ	NOUN
iajs-2724	84	7	)	)	PUNCT
iajs-2724	84	8	,	,	PUNCT
iajs-2724	84	9	𝜆θ	𝜆θ	ADP
iajs-2724	84	10	−(χ	−(χ	NOUN
iajs-2724	84	11	)	)	PUNCT
iajs-2724	84	12	}	}	PUNCT
iajs-2724	84	13	is	be	AUX
iajs-2724	84	14	a	a	DET
iajs-2724	84	15	bipolar	bipolar	ADJ
iajs-2724	84	16	fuzzy	fuzzy	ADJ
iajs-2724	84	17	set	set	NOUN
iajs-2724	84	18	.	.	PUNCT
iajs-2724	85	1	consider	consider	VERB
iajs-2724	85	2	𝜇θ	𝜇θ	ADP
iajs-2724	85	3	+	+	ADJ
iajs-2724	85	4	:	:	PUNCT
iajs-2724	85	5	ℵ	ℵ	PROPN
iajs-2724	85	6	→	→	SYM
iajs-2724	85	7	𝐷[0,1	𝐷[0,1	NOUN
iajs-2724	85	8	]	]	X
iajs-2724	85	9	such	such	ADJ
iajs-2724	85	10	that	that	SCONJ
iajs-2724	85	11	𝜇θ	𝜇θ	PROPN
iajs-2724	85	12	+	+	ADJ
iajs-2724	85	13	(	(	PUNCT
iajs-2724	85	14	χ	χ	NOUN
iajs-2724	85	15	)	)	PUNCT
iajs-2724	85	16	=	=	NOUN
iajs-2724	86	1	[	[	X
iajs-2724	86	2	𝜉θl	𝜉θl	NOUN
iajs-2724	86	3	+	+	CCONJ
iajs-2724	86	4	(	(	PUNCT
iajs-2724	86	5	χ	χ	X
iajs-2724	86	6	)	)	PUNCT
iajs-2724	86	7	,	,	PUNCT
iajs-2724	86	8	𝜉θu	𝜉θu	NOUN
iajs-2724	86	9	+	+	CCONJ
iajs-2724	86	10	(	(	PUNCT
iajs-2724	86	11	χ	χ	NOUN
iajs-2724	86	12	)	)	PUNCT
iajs-2724	86	13	]	]	PUNCT
iajs-2724	86	14	and	and	CCONJ
iajs-2724	86	15	𝜇θ	𝜇θ	ADP
iajs-2724	86	16	−	−	NOUN
iajs-2724	86	17	:	:	PUNCT
iajs-2724	86	18	ℵ	ℵ	X
iajs-2724	86	19	→	→	SYM
iajs-2724	86	20	𝐷[−1,0	𝐷[−1,0	NOUN
iajs-2724	86	21	]	]	PUNCT
iajs-2724	86	22	such	such	ADJ
iajs-2724	86	23	that	that	SCONJ
iajs-2724	86	24	𝜇θ	𝜇θ	ADP
iajs-2724	86	25	−(χ	−(χ	NOUN
iajs-2724	86	26	)	)	PUNCT
iajs-2724	86	27	=	=	PUNCT
iajs-2724	87	1	[	[	X
iajs-2724	87	2	𝜉θl	𝜉θl	NOUN
iajs-2724	87	3	−	−	PROPN
iajs-2724	87	4	(	(	PUNCT
iajs-2724	87	5	χ	χ	NOUN
iajs-2724	87	6	)	)	PUNCT
iajs-2724	87	7	,	,	PUNCT
iajs-2724	87	8	𝜉θu	𝜉θu	NOUN
iajs-2724	87	9	−	−	PROPN
iajs-2724	87	10	(	(	PUNCT
iajs-2724	87	11	χ	χ	NOUN
iajs-2724	87	12	)	)	PUNCT
iajs-2724	87	13	]	]	PUNCT
iajs-2724	87	14	,	,	PUNCT
iajs-2724	87	15	also	also	ADV
iajs-2724	87	16	𝜆θ	𝜆θ	ADP
iajs-2724	87	17	+	+	ADJ
iajs-2724	87	18	:	:	PUNCT
iajs-2724	87	19	ℵ	ℵ	ADJ
iajs-2724	87	20	→	→	SYM
iajs-2724	87	21	[	[	X
iajs-2724	87	22	0,1	0,1	NUM
iajs-2724	87	23	]	]	PUNCT
iajs-2724	87	24	and	and	CCONJ
iajs-2724	87	25	𝜆θ	𝜆θ	ADP
iajs-2724	87	26	−	−	NOUN
iajs-2724	87	27	:	:	PUNCT
iajs-2724	87	28	ℵ	ℵ	X
iajs-2724	87	29	→	→	SYM
iajs-2724	87	30	[	[	X
iajs-2724	87	31	−1,0	−1,0	X
iajs-2724	87	32	]	]	X
iajs-2724	87	33	it	it	PRON
iajs-2724	87	34	follows	follow	VERB
iajs-2724	87	35	that	that	SCONJ
iajs-2724	87	36	∗	∗	NOUN
iajs-2724	87	37	0	0	NUM
iajs-2724	88	1	1	1	NUM
iajs-2724	88	2	2	2	NUM
iajs-2724	88	3	0	0	NUM
iajs-2724	88	4	0	0	NUM
iajs-2724	88	5	1	1	NUM
iajs-2724	88	6	2	2	NUM
iajs-2724	88	7	1	1	NUM
iajs-2724	88	8	0	0	NUM
iajs-2724	88	9	0	0	NUM
iajs-2724	88	10	1	1	NUM
iajs-2724	88	11	2	2	NUM
iajs-2724	88	12	0	0	NUM
iajs-2724	88	13	1	1	NUM
iajs-2724	88	14	0	0	NUM
iajs-2724	88	15	∘	∘	NUM
iajs-2724	88	16	0	0	NUM
iajs-2724	88	17	1	1	NUM
iajs-2724	88	18	2	2	NUM
iajs-2724	88	19	0	0	NUM
iajs-2724	88	20	0	0	NUM
iajs-2724	88	21	0	0	NUM
iajs-2724	88	22	0	0	NUM
iajs-2724	88	23	1	1	NUM
iajs-2724	88	24	0	0	NUM
iajs-2724	88	25	1	1	NUM
iajs-2724	88	26	0	0	NUM
iajs-2724	88	27	2	2	NUM
iajs-2724	88	28	0	0	NUM
iajs-2724	88	29	0	0	NUM
iajs-2724	88	30	2	2	NUM
iajs-2724	88	31	ibn	ibn	PROPN
iajs-2724	88	32	al	al	PROPN
iajs-2724	88	33	-	-	PUNCT
iajs-2724	88	34	haitham	haitham	PROPN
iajs-2724	88	35	jour	jour	X
iajs-2724	88	36	.	.	PROPN
iajs-2724	89	1	for	for	ADP
iajs-2724	89	2	pure	pure	ADJ
iajs-2724	89	3	&	&	CCONJ
iajs-2724	89	4	appl	appl	PROPN
iajs-2724	89	5	.	.	PUNCT
iajs-2724	90	1	sci	sci	PROPN
iajs-2724	90	2	.	.	PROPN
iajs-2724	91	1	53	53	NUM
iajs-2724	91	2	(	(	PUNCT
iajs-2724	91	3	2)2022	2)2022	VERB
iajs-2724	91	4	51	51	NUM
iajs-2724	91	5	θ	θ	NOUN
iajs-2724	91	6	=	=	PUNCT
iajs-2724	91	7	{	{	PUNCT
iajs-2724	91	8	<	<	X
iajs-2724	91	9	χ	χ	NOUN
iajs-2724	91	10	,	,	PUNCT
iajs-2724	91	11	{	{	PUNCT
iajs-2724	91	12	[	[	X
iajs-2724	91	13	𝜉θl	𝜉θl	NOUN
iajs-2724	91	14	+	+	CCONJ
iajs-2724	91	15	(	(	PUNCT
iajs-2724	91	16	χ	χ	X
iajs-2724	91	17	)	)	PUNCT
iajs-2724	91	18	,	,	PUNCT
iajs-2724	91	19	𝜉θu	𝜉θu	NOUN
iajs-2724	91	20	+	+	CCONJ
iajs-2724	91	21	(	(	PUNCT
iajs-2724	91	22	χ	χ	NOUN
iajs-2724	91	23	)	)	PUNCT
iajs-2724	91	24	]	]	PUNCT
iajs-2724	91	25	,	,	PUNCT
iajs-2724	92	1	[	[	X
iajs-2724	92	2	𝜉θl	𝜉θl	NOUN
iajs-2724	92	3	−	−	PROPN
iajs-2724	92	4	(	(	PUNCT
iajs-2724	92	5	χ	χ	NOUN
iajs-2724	92	6	)	)	PUNCT
iajs-2724	92	7	,	,	PUNCT
iajs-2724	92	8	𝜉θu	𝜉θu	NOUN
iajs-2724	92	9	−	−	PROPN
iajs-2724	92	10	(	(	PUNCT
iajs-2724	92	11	χ	χ	NOUN
iajs-2724	92	12	)	)	PUNCT
iajs-2724	92	13	]	]	PUNCT
iajs-2724	92	14	}	}	PUNCT
iajs-2724	92	15	,	,	PUNCT
iajs-2724	92	16	𝜆θ	𝜆θ	ADP
iajs-2724	92	17	+	+	ADJ
iajs-2724	92	18	(	(	PUNCT
iajs-2724	92	19	χ	χ	NOUN
iajs-2724	92	20	)	)	PUNCT
iajs-2724	92	21	,	,	PUNCT
iajs-2724	92	22	𝜆θ	𝜆θ	ADP
iajs-2724	92	23	−(χ	−(χ	NOUN
iajs-2724	92	24	)	)	PUNCT
iajs-2724	92	25	}	}	PUNCT
iajs-2724	92	26	>	>	PUNCT
iajs-2724	92	27	:	:	PUNCT
iajs-2724	92	28	χ	χ	X
iajs-2724	92	29	∈	∈	ADJ
iajs-2724	92	30	ℵ	ℵ	AUX
iajs-2724	92	31	}	}	PUNCT
iajs-2724	92	32	definition(17)[10	definition(17)[10	PROPN
iajs-2724	92	33	]	]	PUNCT
iajs-2724	92	34	.	.	PUNCT
iajs-2724	93	1	a	a	DET
iajs-2724	93	2	(	(	PUNCT
iajs-2724	93	3	cb	cb	PROPN
iajs-2724	93	4	)	)	PUNCT
iajs-2724	93	5	θ	θ	PROPN
iajs-2724	93	6	=	=	PUNCT
iajs-2724	93	7	〈	〈	PROPN
iajs-2724	93	8	𝑁	𝑁	PROPN
iajs-2724	93	9	,	,	PUNCT
iajs-2724	93	10	𝐾	𝐾	NOUN
iajs-2724	93	11	〉	〉	NOUN
iajs-2724	93	12	in	in	ADP
iajs-2724	93	13	ℵ	ℵ	NOUN
iajs-2724	93	14	is	be	AUX
iajs-2724	93	15	named	name	VERB
iajs-2724	93	16	a	a	DET
iajs-2724	93	17	(	(	PUNCT
iajs-2724	93	18	cb	cb	PROPN
iajs-2724	93	19	)	)	PUNCT
iajs-2724	93	20	sub	sub	PROPN
iajs-2724	93	21	-	-	ADJ
iajs-2724	93	22	ku	ku	NOUN
iajs-2724	93	23	-	-	PUNCT
iajs-2724	93	24	semigroup	semigroup	PROPN
iajs-2724	93	25	if	if	SCONJ
iajs-2724	93	26	:	:	PUNCT
iajs-2724	93	27	∀𝜒	∀𝜒	PROPN
iajs-2724	93	28	,	,	PUNCT
iajs-2724	93	29	𝛾	𝛾	PROPN
iajs-2724	93	30	∈	∈	NOUN
iajs-2724	93	31	ℵ	ℵ	NOUN
iajs-2724	93	32	,	,	PUNCT
iajs-2724	93	33	(	(	PUNCT
iajs-2724	93	34	1	1	NUM
iajs-2724	93	35	)	)	PUNCT
iajs-2724	93	36	𝜇θ	𝜇θ	ADP
iajs-2724	94	1	+	+	ADJ
iajs-2724	94	2	(	(	PUNCT
iajs-2724	94	3	𝜒	𝜒	X
iajs-2724	94	4	∗	∗	NOUN
iajs-2724	94	5	𝛾	𝛾	NOUN
iajs-2724	94	6	)	)	PUNCT
iajs-2724	94	7	≥	≥	PROPN
iajs-2724	94	8	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2724	94	9	{	{	PUNCT
iajs-2724	94	10	𝜇θ	𝜇θ	PROPN
iajs-2724	94	11	+	+	ADJ
iajs-2724	94	12	(	(	PUNCT
iajs-2724	94	13	𝜒)ˑ	𝜒)ˑ	ADJ
iajs-2724	94	14	,	,	PUNCT
iajs-2724	94	15	𝜇θ	𝜇θ	ADP
iajs-2724	94	16	+	+	ADJ
iajs-2724	94	17	(	(	PUNCT
iajs-2724	94	18	𝛾	𝛾	NOUN
iajs-2724	94	19	)	)	PUNCT
iajs-2724	94	20	}	}	PUNCT
iajs-2724	94	21	,	,	PUNCT
iajs-2724	94	22	𝜇θ	𝜇θ	ADP
iajs-2724	94	23	−(𝜒	−(𝜒	VERB
iajs-2724	94	24	∗	∗	NOUN
iajs-2724	94	25	𝛾	𝛾	NOUN
iajs-2724	94	26	)	)	PUNCT
iajs-2724	94	27	≤	≤	NOUN
iajs-2724	94	28	𝑟𝑚𝑎𝑥	𝑟𝑚𝑎𝑥	NOUN
iajs-2724	94	29	{	{	PUNCT
iajs-2724	94	30	𝜇θ	𝜇θ	ADP
iajs-2724	94	31	−(𝜒)ˑ	−(𝜒)ˑ	PROPN
iajs-2724	94	32	,	,	PUNCT
iajs-2724	94	33	𝜇θ	𝜇θ	ADV
iajs-2724	94	34	−(𝛾	−(𝛾	NOUN
iajs-2724	94	35	)	)	PUNCT
iajs-2724	94	36	}	}	PUNCT
iajs-2724	94	37	𝜆θ	𝜆θ	ADP
iajs-2724	95	1	+	+	ADJ
iajs-2724	95	2	(	(	PUNCT
iajs-2724	95	3	𝜒	𝜒	NOUN
iajs-2724	95	4	∗	∗	NOUN
iajs-2724	95	5	𝛾	𝛾	NOUN
iajs-2724	95	6	)	)	PUNCT
iajs-2724	95	7	≥	≥	NOUN
iajs-2724	95	8	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
iajs-2724	95	9	{	{	PUNCT
iajs-2724	95	10	𝜆θ	𝜆θ	ADP
iajs-2724	95	11	+	+	ADJ
iajs-2724	95	12	(	(	PUNCT
iajs-2724	95	13	𝜒)ˑ	𝜒)ˑ	ADJ
iajs-2724	95	14	,	,	PUNCT
iajs-2724	95	15	𝜆θ	𝜆θ	ADP
iajs-2724	95	16	+	+	ADJ
iajs-2724	95	17	(	(	PUNCT
iajs-2724	95	18	𝛾	𝛾	NOUN
iajs-2724	95	19	)	)	PUNCT
iajs-2724	95	20	}	}	PUNCT
iajs-2724	95	21	,	,	PUNCT
iajs-2724	95	22	𝜆θ	𝜆θ	ADP
iajs-2724	95	23	−(𝜒	−(𝜒	ADJ
iajs-2724	95	24	∗	∗	NOUN
iajs-2724	95	25	𝛾	𝛾	NOUN
iajs-2724	95	26	)	)	PUNCT
iajs-2724	95	27	≤	≤	NUM
iajs-2724	95	28	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-2724	95	29	{	{	PUNCT
iajs-2724	95	30	𝜆θ	𝜆θ	ADP
iajs-2724	95	31	−(𝜒)ˑ	−(𝜒)ˑ	PROPN
iajs-2724	95	32	,	,	PUNCT
iajs-2724	95	33	𝜆θ	𝜆θ	ADP
iajs-2724	95	34	−(𝛾	−(𝛾	NOUN
iajs-2724	95	35	)	)	PUNCT
iajs-2724	95	36	}	}	PUNCT
iajs-2724	95	37	,	,	PUNCT
iajs-2724	95	38	(	(	PUNCT
iajs-2724	95	39	2	2	X
iajs-2724	95	40	)	)	PUNCT
iajs-2724	95	41	𝜇θ	𝜇θ	ADP
iajs-2724	96	1	+	+	ADJ
iajs-2724	96	2	(	(	PUNCT
iajs-2724	96	3	𝜒	𝜒	PRON
iajs-2724	96	4	∘	∘	NUM
iajs-2724	96	5	𝛾	𝛾	NOUN
iajs-2724	96	6	)	)	PUNCT
iajs-2724	96	7	≥	≥	NOUN
iajs-2724	96	8	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-2724	96	9	{	{	PUNCT
iajs-2724	96	10	𝜇θ	𝜇θ	PROPN
iajs-2724	96	11	+	+	ADJ
iajs-2724	96	12	(	(	PUNCT
iajs-2724	96	13	𝜒)ˑ	𝜒)ˑ	ADJ
iajs-2724	96	14	,	,	PUNCT
iajs-2724	96	15	𝜇θ	𝜇θ	ADP
iajs-2724	96	16	+	+	ADJ
iajs-2724	96	17	(	(	PUNCT
iajs-2724	96	18	𝛾	𝛾	NOUN
iajs-2724	96	19	)	)	PUNCT
iajs-2724	96	20	}	}	PUNCT
iajs-2724	96	21	,	,	PUNCT
iajs-2724	96	22	𝜇θ	𝜇θ	ADP
iajs-2724	96	23	−(𝜒	−(𝜒	VERB
iajs-2724	96	24	∘	∘	NUM
iajs-2724	96	25	𝛾	𝛾	NOUN
iajs-2724	96	26	)	)	PUNCT
iajs-2724	96	27	≤	≤	NOUN
iajs-2724	96	28	𝑟𝑚𝑎𝑥	𝑟𝑚𝑎𝑥	NOUN
iajs-2724	96	29	{	{	PUNCT
iajs-2724	96	30	𝜇θ	𝜇θ	ADP
iajs-2724	96	31	−(𝜒)ˑ	−(𝜒)ˑ	PROPN
iajs-2724	96	32	,	,	PUNCT
iajs-2724	96	33	𝜇θ	𝜇θ	ADV
iajs-2724	96	34	−(𝛾	−(𝛾	NOUN
iajs-2724	96	35	)	)	PUNCT
iajs-2724	96	36	}	}	PUNCT
iajs-2724	96	37	𝜆θ	𝜆θ	ADP
iajs-2724	97	1	+	+	ADJ
iajs-2724	97	2	(	(	PUNCT
iajs-2724	97	3	𝜒	𝜒	PRON
iajs-2724	97	4	∘	∘	NUM
iajs-2724	97	5	𝛾	𝛾	NOUN
iajs-2724	97	6	)	)	PUNCT
iajs-2724	97	7	≥	≥	NOUN
iajs-2724	97	8	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
iajs-2724	97	9	{	{	PUNCT
iajs-2724	97	10	𝜆θ	𝜆θ	ADP
iajs-2724	97	11	+	+	ADJ
iajs-2724	97	12	(	(	PUNCT
iajs-2724	97	13	𝜒)ˑ	𝜒)ˑ	ADJ
iajs-2724	97	14	,	,	PUNCT
iajs-2724	97	15	𝜆θ	𝜆θ	ADP
iajs-2724	97	16	+	+	ADJ
iajs-2724	97	17	(	(	PUNCT
iajs-2724	97	18	𝛾	𝛾	NOUN
iajs-2724	97	19	)	)	PUNCT
iajs-2724	97	20	}	}	PUNCT
iajs-2724	97	21	,	,	PUNCT
iajs-2724	97	22	𝜆θ	𝜆θ	SCONJ
iajs-2724	97	23	−(𝜒	−(𝜒	ADJ
iajs-2724	97	24	∘	∘	PROPN
iajs-2724	97	25	𝛾	𝛾	NOUN
iajs-2724	97	26	)	)	PUNCT
iajs-2724	97	27	≤	≤	NUM
iajs-2724	97	28	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-2724	97	29	{	{	PUNCT
iajs-2724	97	30	𝜆θ	𝜆θ	ADP
iajs-2724	97	31	−(𝜒)ˑ	−(𝜒)ˑ	PROPN
iajs-2724	97	32	,	,	PUNCT
iajs-2724	97	33	𝜆θ	𝜆θ	ADP
iajs-2724	97	34	−(𝛾	−(𝛾	PROPN
iajs-2724	97	35	)	)	PUNCT
iajs-2724	97	36	}	}	PUNCT
iajs-2724	97	37	,	,	PUNCT
iajs-2724	97	38	example(18)[10	example(18)[10	PROPN
iajs-2724	97	39	]	]	PUNCT
iajs-2724	97	40	:	:	PUNCT
iajs-2724	97	41	the	the	DET
iajs-2724	97	42	following	follow	VERB
iajs-2724	97	43	table	table	NOUN
iajs-2724	97	44	is	be	AUX
iajs-2724	97	45	illustrates	illustrate	VERB
iajs-2724	97	46	that	that	SCONJ
iajs-2724	97	47	the	the	DET
iajs-2724	97	48	set	set	NOUN
iajs-2724	97	49	ˑℵ	ˑℵ	X
iajs-2724	97	50	=	=	SYM
iajs-2724	97	51	{	{	PUNCT
iajs-2724	97	52	0,1,2,3	0,1,2,3	NOUN
iajs-2724	97	53	}	}	PUNCT
iajs-2724	97	54	with	with	ADP
iajs-2724	97	55	binary	binary	ADJ
iajs-2724	97	56	operations	operation	NOUN
iajs-2724	97	57	∗	∗	NOUN
iajs-2724	97	58	and	and	CCONJ
iajs-2724	97	59	∘	∘	PROPN
iajs-2724	97	60	then(ℵ,∗,∘	then(ℵ,∗,∘	PROPN
iajs-2724	97	61	,	,	PUNCT
iajs-2724	97	62	0)ˑis	0)ˑis	NUM
iajs-2724	97	63	a	a	DET
iajs-2724	97	64	ku	ku	PROPN
iajs-2724	97	65	-	-	PUNCT
iajs-2724	97	66	semigroup	semigroup	PROPN
iajs-2724	97	67	.	.	PUNCT
iajs-2724	98	1	define	define	VERB
iajs-2724	98	2	θ	θ	PROPN
iajs-2724	99	1	=	=	PUNCT
iajs-2724	99	2	〈	〈	PROPN
iajs-2724	99	3	𝑁,𝐾	𝑁,𝐾	NOUN
iajs-2724	99	4	〉	〉	NOUN
iajs-2724	99	5	as	as	SCONJ
iajs-2724	99	6	follows	follow	VERB
iajs-2724	99	7	𝑀(𝑥	𝑀(𝑥	NOUN
iajs-2724	99	8	)	)	PUNCT
iajs-2724	99	9	=	=	PRON
iajs-2724	99	10	{	{	PUNCT
iajs-2724	99	11	{	{	PUNCT
iajs-2724	99	12	[	[	X
iajs-2724	99	13	−0.2	−0.2	PROPN
iajs-2724	99	14	,	,	PUNCT
iajs-2724	99	15	−0.5	−0.5	PROPN
iajs-2724	99	16	]	]	PUNCT
iajs-2724	99	17	,	,	PUNCT
iajs-2724	100	1	[	[	X
iajs-2724	100	2	0.1,0.9	0.1,0.9	X
iajs-2724	100	3	]	]	X
iajs-2724	100	4	}	}	PUNCT
iajs-2724	100	5	𝑖𝑓	𝑖𝑓	ADP
iajs-2724	100	6	𝜒	𝜒	X
iajs-2724	100	7	=	=	SYM
iajs-2724	100	8	{	{	PUNCT
iajs-2724	100	9	0,1	0,1	NOUN
iajs-2724	100	10	}	}	PUNCT
iajs-2724	100	11	{	{	PUNCT
iajs-2724	100	12	[	[	X
iajs-2724	100	13	−0.6	−0.6	PROPN
iajs-2724	100	14	,	,	PUNCT
iajs-2724	100	15	−0.2	−0.2	PROPN
iajs-2724	100	16	]	]	PUNCT
iajs-2724	100	17	,	,	PUNCT
iajs-2724	100	18	[	[	X
iajs-2724	100	19	0.2,0.5	0.2,0.5	NUM
iajs-2724	100	20	]	]	X
iajs-2724	100	21	}	}	PUNCT
iajs-2724	100	22	𝑖𝑓	𝑖𝑓	ADP
iajs-2724	100	23	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2724	100	24	,	,	PUNCT
iajs-2724	100	25	𝜆θ	𝜆θ	INTJ
iajs-2724	100	26	+	+	ADJ
iajs-2724	100	27	(	(	PUNCT
iajs-2724	100	28	𝑥	𝑥	NOUN
iajs-2724	100	29	)	)	PUNCT
iajs-2724	100	30	=	=	VERB
iajs-2724	100	31	{	{	PUNCT
iajs-2724	100	32	0.5	0.5	NUM
iajs-2724	100	33	𝑖𝑓	𝑖𝑓	NOUN
iajs-2724	100	34	𝜒	𝜒	X
iajs-2724	100	35	=	=	SYM
iajs-2724	100	36	{	{	PUNCT
iajs-2724	100	37	0,1	0,1	NOUN
iajs-2724	100	38	}	}	PUNCT
iajs-2724	100	39	0.3	0.3	NUM
iajs-2724	100	40	𝑖𝑓	𝑖𝑓	NOUN
iajs-2724	100	41	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2724	100	42	𝜆θ	𝜆θ	ADP
iajs-2724	100	43	−(𝑥	−(𝑥	NOUN
iajs-2724	100	44	)	)	PUNCT
iajs-2724	101	1	=	=	PRON
iajs-2724	101	2	{	{	PUNCT
iajs-2724	101	3	−0.6	−0.6	PROPN
iajs-2724	101	4	𝑖𝑓	𝑖𝑓	ADP
iajs-2724	101	5	𝜒	𝜒	X
iajs-2724	101	6	=	=	SYM
iajs-2724	101	7	{	{	PUNCT
iajs-2724	101	8	0,1	0,1	NOUN
iajs-2724	101	9	}	}	PUNCT
iajs-2724	101	10	−0.3	−0.3	PROPN
iajs-2724	101	11	𝑖𝑓	𝑖𝑓	ADP
iajs-2724	101	12	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒	NOUN
iajs-2724	101	13	and	and	CCONJ
iajs-2724	101	14	by	by	ADP
iajs-2724	101	15	applying	apply	VERB
iajs-2724	101	16	definition	definition	NOUN
iajs-2724	101	17	2.17	2.17	NUM
iajs-2724	101	18	,	,	PUNCT
iajs-2724	101	19	we	we	PRON
iajs-2724	101	20	can	can	AUX
iajs-2724	101	21	easily	easily	ADV
iajs-2724	101	22	prove	prove	VERB
iajs-2724	101	23	that	that	SCONJ
iajs-2724	101	24	θ	θ	PROPN
iajs-2724	101	25	=	=	PUNCT
iajs-2724	101	26	〈	〈	PROPN
iajs-2724	101	27	𝑁	𝑁	PROPN
iajs-2724	101	28	,	,	PUNCT
iajs-2724	101	29	𝐾	𝐾	NOUN
iajs-2724	101	30	〉	〉	NOUN
iajs-2724	101	31	is	be	AUX
iajs-2724	101	32	a	a	DET
iajs-2724	101	33	cubic	cubic	ADJ
iajs-2724	101	34	bipolar	bipolar	ADJ
iajs-2724	101	35	sub	sub	NOUN
iajs-2724	101	36	kusemigroup	kusemigroup	NOUN
iajs-2724	101	37	of	of	ADP
iajs-2724	101	38	ℵˑ.	ℵˑ.	PRON
iajs-2724	101	39	3	3	X
iajs-2724	101	40	.	.	PUNCT
iajs-2724	101	41	cubic	cubic	ADJ
iajs-2724	101	42	bipolar	bipolar	ADJ
iajs-2724	101	43	ideals	ideal	NOUN
iajs-2724	101	44	of	of	ADP
iajs-2724	101	45	a	a	DET
iajs-2724	101	46	ku	ku	NOUN
iajs-2724	101	47	-	-	PUNCT
iajs-2724	101	48	semi	semi	NOUN
iajs-2724	101	49	group	group	NOUN
iajs-2724	101	50	with	with	ADP
iajs-2724	101	51	thresholds	threshold	NOUN
iajs-2724	101	52	(	(	PUNCT
iajs-2724	101	53	𝜶	𝜶	NOUN
iajs-2724	101	54	,	,	PUNCT
iajs-2724	101	55	𝜷	𝜷	NOUN
iajs-2724	101	56	)	)	PUNCT
iajs-2724	101	57	,	,	PUNCT
iajs-2724	101	58	(	(	PUNCT
iajs-2724	101	59	𝝎	𝝎	X
iajs-2724	101	60	,	,	PUNCT
iajs-2724	101	61	𝝑	𝝑	NOUN
iajs-2724	101	62	)	)	PUNCT
iajs-2724	101	63	in	in	ADP
iajs-2724	101	64	this	this	DET
iajs-2724	101	65	part	part	NOUN
iajs-2724	101	66	,	,	PUNCT
iajs-2724	101	67	the	the	DET
iajs-2724	101	68	notion	notion	NOUN
iajs-2724	101	69	of	of	ADP
iajs-2724	101	70	cubic	cubic	ADJ
iajs-2724	101	71	bipolar	bipolar	ADJ
iajs-2724	101	72	k	k	NOUN
iajs-2724	101	73	-	-	NOUN
iajs-2724	101	74	ideals	ideal	NOUN
iajs-2724	101	75	with	with	ADP
iajs-2724	101	76	thresholds	threshold	NOUN
iajs-2724	101	77	(	(	PUNCT
iajs-2724	101	78	𝛼	𝛼	X
iajs-2724	101	79	,	,	PUNCT
iajs-2724	101	80	𝛽	𝛽	NOUN
iajs-2724	101	81	)	)	PUNCT
iajs-2724	101	82	,	,	PUNCT
iajs-2724	101	83	(	(	PUNCT
iajs-2724	101	84	𝜔	𝜔	NOUN
iajs-2724	101	85	,	,	PUNCT
iajs-2724	101	86	𝜗	𝜗	NOUN
iajs-2724	101	87	)	)	PUNCT
iajs-2724	101	88	of	of	ADP
iajs-2724	101	89	a	a	DET
iajs-2724	101	90	ku	ku	NOUN
iajs-2724	101	91	-	-	PUNCT
iajs-2724	101	92	semi	semi	NOUN
iajs-2724	101	93	group	group	NOUN
iajs-2724	101	94	and	and	CCONJ
iajs-2724	101	95	some	some	DET
iajs-2724	101	96	properties	property	NOUN
iajs-2724	101	97	are	be	AUX
iajs-2724	101	98	defined	define	VERB
iajs-2724	101	99	.	.	PUNCT
iajs-2724	102	1	in	in	ADP
iajs-2724	102	2	the	the	DET
iajs-2724	102	3	following	following	NOUN
iajs-2724	102	4	,	,	PUNCT
iajs-2724	102	5	we	we	PRON
iajs-2724	102	6	denote	denote	VERB
iajs-2724	102	7	a	a	DET
iajs-2724	102	8	cubic	cubic	ADJ
iajs-2724	102	9	bipolar	bipolar	ADJ
iajs-2724	102	10	fuzzy	fuzzy	ADJ
iajs-2724	102	11	set	set	VERB
iajs-2724	102	12	by	by	ADP
iajs-2724	102	13	(	(	PUNCT
iajs-2724	102	14	cbf	cbf	PROPN
iajs-2724	102	15	)	)	PUNCT
iajs-2724	102	16	,	,	PUNCT
iajs-2724	102	17	and	and	CCONJ
iajs-2724	102	18	let	let	VERB
iajs-2724	102	19	𝛼	𝛼	X
iajs-2724	102	20	,	,	PUNCT
iajs-2724	102	21	𝛽	𝛽	PROPN
iajs-2724	102	22	∈	∈	PROPN
iajs-2724	102	23	𝐷[0,1	𝐷[0,1	NOUN
iajs-2724	102	24	]	]	X
iajs-2724	102	25	,	,	PUNCT
iajs-2724	102	26	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-2724	102	27	,	,	PUNCT
iajs-2724	102	28	𝜔	𝜔	X
iajs-2724	102	29	,	,	PUNCT
iajs-2724	102	30	𝜗	𝜗	PROPN
iajs-2724	102	31	∈	∈	NOUN
iajs-2724	103	1	[	[	X
iajs-2724	103	2	0,1	0,1	NUM
iajs-2724	103	3	]	]	PUNCT
iajs-2724	103	4	,	,	PUNCT
iajs-2724	103	5	𝑠𝑢𝑐ℎ	𝑠𝑢𝑐ℎ	X
iajs-2724	104	1	𝑡ℎ𝑎𝑡	𝑡ℎ𝑎𝑡	ADJ
iajs-2724	105	1	[	[	X
iajs-2724	105	2	0,0	0,0	NOUN
iajs-2724	105	3	]	]	PUNCT
iajs-2724	105	4	<	<	X
iajs-2724	105	5	𝛼	𝛼	X
iajs-2724	105	6	<	<	X
iajs-2724	105	7	𝛽	𝛽	X
iajs-2724	105	8	<	<	X
iajs-2724	105	9	[	[	X
iajs-2724	105	10	1,1	1,1	NUM
iajs-2724	105	11	]	]	PUNCT
iajs-2724	105	12	,	,	PUNCT
iajs-2724	105	13	0	0	PUNCT
iajs-2724	105	14	<	<	X
iajs-2724	105	15	𝜔	𝜔	X
iajs-2724	105	16	<	<	X
iajs-2724	105	17	𝜗	𝜗	X
iajs-2724	105	18	<	<	X
iajs-2724	105	19	1	1	NUM
iajs-2724	105	20	,	,	PUNCT
iajs-2724	105	21	where	where	SCONJ
iajs-2724	105	22	𝜔	𝜔	PRON
iajs-2724	105	23	,	,	PUNCT
iajs-2724	105	24	𝜗	𝜗	NOUN
iajs-2724	105	25	are	be	AUX
iajs-2724	105	26	arbitrary	arbitrary	ADJ
iajs-2724	105	27	values	value	NOUN
iajs-2724	105	28	,	,	PUNCT
iajs-2724	105	29	and	and	CCONJ
iajs-2724	105	30	𝛼	𝛼	X
iajs-2724	105	31	,	,	PUNCT
iajs-2724	105	32	𝛽	𝛽	NOUN
iajs-2724	105	33	,	,	PUNCT
iajs-2724	105	34	are	be	AUX
iajs-2724	105	35	arbitrary	arbitrary	ADJ
iajs-2724	105	36	closed	closed	ADJ
iajs-2724	105	37	sub	sub	NOUN
iajs-2724	105	38	-	-	NOUN
iajs-2724	105	39	intervals	interval	NOUN
iajs-2724	105	40	definition(19	definition(19	NOUN
iajs-2724	105	41	)	)	PUNCT
iajs-2724	105	42	.	.	PUNCT
iajs-2724	106	1	a	a	PRON
iajs-2724	106	2	(	(	PUNCT
iajs-2724	106	3	cbf	cbf	PROPN
iajs-2724	106	4	)	)	PUNCT
iajs-2724	106	5	set	set	VERB
iajs-2724	106	6	θ	θ	PROPN
iajs-2724	106	7	=	=	SYM
iajs-2724	106	8	〈	〈	PROPN
iajs-2724	106	9	𝑀	𝑀	PROPN
iajs-2724	106	10	,	,	PUNCT
iajs-2724	106	11	𝐿	𝐿	PROPN
iajs-2724	106	12	〉	〉	NOUN
iajs-2724	106	13	is	be	AUX
iajs-2724	106	14	named	name	VERB
iajs-2724	106	15	a	a	DET
iajs-2724	106	16	(	(	PUNCT
iajs-2724	106	17	cbf	cbf	PROPN
iajs-2724	106	18	)	)	PUNCT
iajs-2724	106	19	sub	sub	PROPN
iajs-2724	106	20	-	-	ADJ
iajs-2724	106	21	ku	ku	ADJ
iajs-2724	106	22	-	-	PUNCT
iajs-2724	106	23	semi	semi	NOUN
iajs-2724	106	24	group	group	NOUN
iajs-2724	106	25	with	with	ADP
iajs-2724	106	26	thresholds	threshold	NOUN
iajs-2724	106	27	(	(	PUNCT
iajs-2724	106	28	𝛼	𝛼	X
iajs-2724	106	29	,	,	PUNCT
iajs-2724	106	30	𝛽	𝛽	NOUN
iajs-2724	106	31	)	)	PUNCT
iajs-2724	106	32	,	,	PUNCT
iajs-2724	106	33	(	(	PUNCT
iajs-2724	106	34	𝜔	𝜔	NOUN
iajs-2724	106	35	,	,	PUNCT
iajs-2724	106	36	𝜗	𝜗	NOUN
iajs-2724	106	37	)	)	PUNCT
iajs-2724	106	38	if	if	SCONJ
iajs-2724	106	39	∀	∀	X
iajs-2724	106	40	𝜒	𝜒	VERB
iajs-2724	106	41	,	,	PUNCT
iajs-2724	106	42	𝛾	𝛾	ADP
iajs-2724	106	43	∈	∈	PROPN
iajs-2724	106	44	ℵ	ℵ	NOUN
iajs-2724	106	45	(	(	PUNCT
iajs-2724	106	46	1)𝑚𝑖𝑛{𝜇θ	1)𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	106	47	−(𝜒	−(𝜒	PROPN
iajs-2724	106	48	∗	∗	NOUN
iajs-2724	106	49	𝛾),−𝛼	𝛾),−𝛼	NOUN
iajs-2724	106	50	}	}	PUNCT
iajs-2724	106	51	≤	≤	NUM
iajs-2724	107	1	𝑟𝑚𝑎𝑥{	𝑟𝑚𝑎𝑥{	NOUN
iajs-2724	107	2	�	�	NOUN
iajs-2724	107	3	̃	̃	NOUN
iajs-2724	107	4	�	�	NOUN
iajs-2724	107	5	θ	θ	NOUN
iajs-2724	107	6	−(𝜒	−(𝜒	ADJ
iajs-2724	107	7	)	)	PUNCT
iajs-2724	107	8	,	,	PUNCT
iajs-2724	107	9	𝜇θ	𝜇θ	ADV
iajs-2724	107	10	−(𝛾),−𝛽	−(𝛾),−𝛽	PUNCT
iajs-2724	107	11	}	}	PUNCT
iajs-2724	107	12	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	X
iajs-2724	108	1	+	+	NOUN
iajs-2724	108	2	(	(	PUNCT
iajs-2724	108	3	𝜒	𝜒	NOUN
iajs-2724	108	4	∗	∗	NOUN
iajs-2724	108	5	𝛾	𝛾	NOUN
iajs-2724	108	6	)	)	PUNCT
iajs-2724	108	7	,	,	PUNCT
iajs-2724	108	8	𝛼	𝛼	X
iajs-2724	108	9	}	}	PUNCT
iajs-2724	108	10	≥	≥	NOUN
iajs-2724	108	11	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	108	12	+	+	ADJ
iajs-2724	108	13	(	(	PUNCT
iajs-2724	108	14	𝜒	𝜒	NOUN
iajs-2724	108	15	)	)	PUNCT
iajs-2724	108	16	,	,	PUNCT
iajs-2724	108	17	𝜇θ	𝜇θ	PROPN
iajs-2724	108	18	+	+	ADJ
iajs-2724	108	19	(	(	PUNCT
iajs-2724	108	20	𝛾	𝛾	NOUN
iajs-2724	108	21	)	)	PUNCT
iajs-2724	108	22	,	,	PUNCT
iajs-2724	108	23	𝛽	𝛽	AUX
iajs-2724	108	24	}	}	PUNCT
iajs-2724	108	25	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	108	26	−(𝜒	−(𝜒	ADJ
iajs-2724	108	27	∗	∗	NOUN
iajs-2724	108	28	𝛾),−𝜔	𝛾),−𝜔	NOUN
iajs-2724	108	29	}	}	PUNCT
iajs-2724	108	30	≤	≤	NUM
iajs-2724	108	31	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	ADJ
iajs-2724	108	32	−(𝜒	−(𝜒	ADJ
iajs-2724	108	33	)	)	PUNCT
iajs-2724	108	34	,	,	PUNCT
iajs-2724	108	35	𝜆θ	𝜆θ	ADP
iajs-2724	108	36	−(𝛾),−𝜗	−(𝛾),−𝜗	VERB
iajs-2724	108	37	}	}	PUNCT
iajs-2724	108	38	ibn	ibn	PROPN
iajs-2724	108	39	al	al	PROPN
iajs-2724	108	40	-	-	PUNCT
iajs-2724	108	41	haitham	haitham	PROPN
iajs-2724	108	42	jour	jour	X
iajs-2724	108	43	.	.	PROPN
iajs-2724	109	1	for	for	ADP
iajs-2724	109	2	pure	pure	ADJ
iajs-2724	109	3	&	&	CCONJ
iajs-2724	109	4	appl	appl	PROPN
iajs-2724	109	5	.	.	PUNCT
iajs-2724	110	1	sci	sci	PROPN
iajs-2724	110	2	.	.	PROPN
iajs-2724	111	1	53	53	NUM
iajs-2724	111	2	(	(	PUNCT
iajs-2724	111	3	2)2022	2)2022	VERB
iajs-2724	111	4	52	52	NUM
iajs-2724	111	5	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	ADJ
iajs-2724	112	1	+	+	PROPN
iajs-2724	112	2	(	(	PUNCT
iajs-2724	112	3	𝜒	𝜒	X
iajs-2724	112	4	∗	∗	X
iajs-2724	112	5	𝛾),𝜔	𝛾),𝜔	PROPN
iajs-2724	112	6	}	}	PUNCT
iajs-2724	112	7	≥	≥	NOUN
iajs-2724	112	8	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	112	9	+	+	ADJ
iajs-2724	112	10	(	(	PUNCT
iajs-2724	112	11	𝜒	𝜒	NOUN
iajs-2724	112	12	)	)	PUNCT
iajs-2724	112	13	,	,	PUNCT
iajs-2724	112	14	𝜆θ	𝜆θ	ADP
iajs-2724	112	15	+	+	ADJ
iajs-2724	112	16	(	(	PUNCT
iajs-2724	112	17	𝛾	𝛾	NOUN
iajs-2724	112	18	)	)	PUNCT
iajs-2724	112	19	,	,	PUNCT
iajs-2724	112	20	𝜗	𝜗	NOUN
iajs-2724	112	21	}	}	PUNCT
iajs-2724	112	22	(	(	PUNCT
iajs-2724	112	23	2)𝑟𝑚𝑖𝑛{𝜇θ	2)𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	112	24	−(𝜒	−(𝜒	ADJ
iajs-2724	112	25	∘	∘	NOUN
iajs-2724	112	26	𝛾),−𝛼	𝛾),−𝛼	PRON
iajs-2724	112	27	}	}	PUNCT
iajs-2724	112	28	≤	≤	NUM
iajs-2724	112	29	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NUM
iajs-2724	112	30	−(𝜒	−(𝜒	ADJ
iajs-2724	112	31	)	)	PUNCT
iajs-2724	112	32	,	,	PUNCT
iajs-2724	112	33	𝜇θ	𝜇θ	ADV
iajs-2724	112	34	−(𝛾),−𝛽	−(𝛾),−𝛽	PUNCT
iajs-2724	112	35	}	}	PUNCT
iajs-2724	112	36	𝑟𝑚𝑎𝑥{	𝑟𝑚𝑎𝑥{	ADJ
iajs-2724	112	37	�	�	NOUN
iajs-2724	112	38	̃	̃	NOUN
iajs-2724	112	39	�	�	NOUN
iajs-2724	112	40	θ	θ	NOUN
iajs-2724	112	41	+	+	PROPN
iajs-2724	112	42	(	(	PUNCT
iajs-2724	112	43	𝜒	𝜒	PRON
iajs-2724	112	44	∘	∘	NUM
iajs-2724	112	45	𝛾	𝛾	NOUN
iajs-2724	112	46	)	)	PUNCT
iajs-2724	112	47	,	,	PUNCT
iajs-2724	112	48	𝛼	𝛼	X
iajs-2724	112	49	}	}	PUNCT
iajs-2724	112	50	≥	≥	NOUN
iajs-2724	112	51	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	112	52	+	+	ADJ
iajs-2724	112	53	(	(	PUNCT
iajs-2724	112	54	𝜒	𝜒	NOUN
iajs-2724	112	55	)	)	PUNCT
iajs-2724	112	56	,	,	PUNCT
iajs-2724	112	57	𝜇θ	𝜇θ	PROPN
iajs-2724	112	58	+	+	ADJ
iajs-2724	112	59	(	(	PUNCT
iajs-2724	112	60	𝛾	𝛾	NOUN
iajs-2724	112	61	)	)	PUNCT
iajs-2724	112	62	,	,	PUNCT
iajs-2724	112	63	𝛽	𝛽	AUX
iajs-2724	112	64	}	}	PUNCT
iajs-2724	112	65	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	112	66	−(𝜒	−(𝜒	VERB
iajs-2724	112	67	∘	∘	PROPN
iajs-2724	112	68	𝛾),−𝜔	𝛾),−𝜔	NOUN
iajs-2724	112	69	}	}	PUNCT
iajs-2724	112	70	≤	≤	NUM
iajs-2724	112	71	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	ADJ
iajs-2724	112	72	−(𝜒	−(𝜒	ADJ
iajs-2724	112	73	)	)	PUNCT
iajs-2724	112	74	,	,	PUNCT
iajs-2724	112	75	𝜆θ	𝜆θ	INTJ
iajs-2724	112	76	−(𝛾),−𝜗	−(𝛾),−𝜗	VERB
iajs-2724	112	77	}	}	PUNCT
iajs-2724	112	78	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	113	1	+	+	PROPN
iajs-2724	113	2	(	(	PUNCT
iajs-2724	113	3	𝜒	𝜒	X
iajs-2724	113	4	∘	∘	PROPN
iajs-2724	113	5	𝛾),𝜔	𝛾),𝜔	PROPN
iajs-2724	113	6	}	}	PUNCT
iajs-2724	113	7	≥	≥	NOUN
iajs-2724	113	8	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	113	9	+	+	ADJ
iajs-2724	113	10	(	(	PUNCT
iajs-2724	113	11	𝜒	𝜒	NOUN
iajs-2724	113	12	)	)	PUNCT
iajs-2724	113	13	,	,	PUNCT
iajs-2724	113	14	𝜆θ	𝜆θ	ADP
iajs-2724	113	15	+	+	ADJ
iajs-2724	113	16	(	(	PUNCT
iajs-2724	113	17	𝛾	𝛾	NOUN
iajs-2724	113	18	)	)	PUNCT
iajs-2724	113	19	,	,	PUNCT
iajs-2724	113	20	𝜗	𝜗	NOUN
iajs-2724	113	21	}	}	PUNCT
iajs-2724	113	22	remark(20	remark(20	NOUN
iajs-2724	113	23	)	)	PUNCT
iajs-2724	113	24	.	.	PUNCT
iajs-2724	114	1	every	every	DET
iajs-2724	114	2	(	(	PUNCT
iajs-2724	114	3	cbf	cbf	PROPN
iajs-2724	114	4	)	)	PUNCT
iajs-2724	114	5	sub	sub	PROPN
iajs-2724	114	6	-	-	ADJ
iajs-2724	114	7	ku	ku	ADJ
iajs-2724	114	8	-	-	PUNCT
iajs-2724	114	9	semi	semi	NOUN
iajs-2724	114	10	group	group	NOUN
iajs-2724	114	11	of	of	ADP
iajs-2724	114	12	ℵ	ℵ	PROPN
iajs-2724	114	13	is	be	AUX
iajs-2724	114	14	a	a	DET
iajs-2724	114	15	(	(	PUNCT
iajs-2724	114	16	cbf	cbf	PROPN
iajs-2724	114	17	)	)	PUNCT
iajs-2724	114	18	sub	sub	PROPN
iajs-2724	114	19	-	-	PROPN
iajs-2724	114	20	ku	ku	NOUN
iajs-2724	114	21	-	-	PUNCT
iajs-2724	114	22	semigroup	semigroup	PROPN
iajs-2724	114	23	with	with	ADP
iajs-2724	114	24	thresholds	threshold	NOUN
iajs-2724	114	25	(	(	PUNCT
iajs-2724	114	26	𝜶	𝜶	NOUN
iajs-2724	114	27	,	,	PUNCT
iajs-2724	114	28	𝜷	𝜷	NOUN
iajs-2724	114	29	)	)	PUNCT
iajs-2724	114	30	,	,	PUNCT
iajs-2724	114	31	(	(	PUNCT
iajs-2724	114	32	𝝎	𝝎	X
iajs-2724	114	33	,	,	PUNCT
iajs-2724	114	34	𝝑	𝝑	NOUN
iajs-2724	114	35	)	)	PUNCT
iajs-2724	114	36	,	,	PUNCT
iajs-2724	114	37	but	but	CCONJ
iajs-2724	114	38	not	not	PART
iajs-2724	114	39	converse	converse	VERB
iajs-2724	114	40	as	as	SCONJ
iajs-2724	114	41	it	it	PRON
iajs-2724	114	42	is	be	AUX
iajs-2724	114	43	shown	show	VERB
iajs-2724	114	44	in	in	ADP
iajs-2724	114	45	the	the	DET
iajs-2724	114	46	following	follow	VERB
iajs-2724	114	47	example	example	NOUN
iajs-2724	114	48	example(21).let	example(21).let	PROPN
iajs-2724	114	49	ℵ	ℵ	NOUN
iajs-2724	114	50	=	=	SYM
iajs-2724	114	51	{	{	PUNCT
iajs-2724	114	52	0,1,2,3	0,1,2,3	NOUN
iajs-2724	114	53	}	}	PUNCT
iajs-2724	114	54	be	be	AUX
iajs-2724	114	55	a	a	DET
iajs-2724	114	56	set	set	NOUN
iajs-2724	114	57	with	with	ADP
iajs-2724	114	58	two	two	NUM
iajs-2724	114	59	operations	operation	NOUN
iajs-2724	114	60	∗	∗	NOUN
iajs-2724	114	61	and	and	CCONJ
iajs-2724	114	62	which	which	NOUN
iajs-2724	114	63	are	be	AUX
iajs-2724	114	64	defined	define	VERB
iajs-2724	114	65	by	by	ADP
iajs-2724	114	66	the	the	DET
iajs-2724	114	67	following	follow	VERB
iajs-2724	114	68	tables	table	NOUN
iajs-2724	114	69	.	.	PUNCT
iajs-2724	115	1	then(ℵ,∗,∘	then(ℵ,∗,∘	NOUN
iajs-2724	115	2	,	,	PUNCT
iajs-2724	115	3	0)ˑis	0)ˑis	NUM
iajs-2724	115	4	a	a	DET
iajs-2724	115	5	ku	ku	NOUN
iajs-2724	115	6	-	-	PUNCT
iajs-2724	115	7	semi	semi	NOUN
iajs-2724	115	8	group.now	group.now	ADV
iajs-2724	115	9	,	,	PUNCT
iajs-2724	115	10	we	we	PRON
iajs-2724	115	11	define	define	VERB
iajs-2724	115	12	θ	θ	PROPN
iajs-2724	115	13	=	=	SYM
iajs-2724	115	14	〈	〈	PROPN
iajs-2724	115	15	𝑀	𝑀	PROPN
iajs-2724	115	16	,	,	PUNCT
iajs-2724	115	17	𝐿	𝐿	PROPN
iajs-2724	115	18	〉	〉	NOUN
iajs-2724	115	19	by	by	ADP
iajs-2724	115	20	the	the	DET
iajs-2724	115	21	next	next	ADJ
iajs-2724	115	22	𝑀(𝑥	𝑀(𝑥	NUM
iajs-2724	115	23	)	)	PUNCT
iajs-2724	115	24	=	=	PRON
iajs-2724	115	25	{	{	PUNCT
iajs-2724	115	26	[	[	PUNCT
iajs-2724	115	27	−0.9	−0.9	PROPN
iajs-2724	115	28	,	,	PUNCT
iajs-2724	115	29	−0.8	−0.8	X
iajs-2724	115	30	]	]	PUNCT
iajs-2724	115	31	,	,	PUNCT
iajs-2724	115	32	[	[	X
iajs-2724	115	33	0.8	0.8	NUM
iajs-2724	115	34	,	,	PUNCT
iajs-2724	115	35	0.9	0.9	NUM
iajs-2724	115	36	]	]	PUNCT
iajs-2724	115	37	𝑖𝑓	𝑖𝑓	ADP
iajs-2724	115	38	𝜒	𝜒	X
iajs-2724	115	39	=	=	SYM
iajs-2724	115	40	0	0	PUNCT
iajs-2724	116	1	[	[	X
iajs-2724	116	2	−0.8	−0.8	PROPN
iajs-2724	116	3	,	,	PUNCT
iajs-2724	116	4	−0.7	−0.7	PROPN
iajs-2724	116	5	]	]	PUNCT
iajs-2724	116	6	,	,	PUNCT
iajs-2724	116	7	[	[	X
iajs-2724	116	8	0.7	0.7	NUM
iajs-2724	116	9	,	,	PUNCT
iajs-2724	116	10	0.8	0.8	NUM
iajs-2724	116	11	]	]	PUNCT
iajs-2724	116	12	𝑖𝑓	𝑖𝑓	ADP
iajs-2724	116	13	𝜒	𝜒	X
iajs-2724	116	14	=	=	SYM
iajs-2724	116	15	1	1	NUM
iajs-2724	117	1	[	[	X
iajs-2724	117	2	−0.6	−0.6	PROPN
iajs-2724	117	3	,	,	PUNCT
iajs-2724	117	4	−0.5	−0.5	PROPN
iajs-2724	117	5	]	]	PUNCT
iajs-2724	117	6	,	,	PUNCT
iajs-2724	117	7	[	[	X
iajs-2724	117	8	0.5	0.5	NUM
iajs-2724	117	9	,	,	PUNCT
iajs-2724	117	10	0.6	0.6	NUM
iajs-2724	117	11	]	]	PUNCT
iajs-2724	117	12	𝑖𝑓	𝑖𝑓	ADP
iajs-2724	117	13	𝜒	𝜒	X
iajs-2724	117	14	=	=	SYM
iajs-2724	117	15	3	3	NUM
iajs-2724	118	1	[	[	X
iajs-2724	118	2	−0.3	−0.3	PROPN
iajs-2724	118	3	,	,	PUNCT
iajs-2724	118	4	−0.2	−0.2	PROPN
iajs-2724	118	5	]	]	PUNCT
iajs-2724	118	6	,	,	PUNCT
iajs-2724	118	7	[	[	X
iajs-2724	118	8	0.2	0.2	NUM
iajs-2724	118	9	,	,	PUNCT
iajs-2724	118	10	0.3	0.3	NUM
iajs-2724	118	11	]	]	PUNCT
iajs-2724	118	12	𝑖𝑓	𝑖𝑓	ADP
iajs-2724	118	13	𝜒	𝜒	X
iajs-2724	118	14	=	=	SYM
iajs-2724	118	15	2	2	NUM
iajs-2724	118	16	𝐿(𝑥	𝐿(𝑥	NOUN
iajs-2724	118	17	)	)	PUNCT
iajs-2724	118	18	=	=	PRON
iajs-2724	118	19	{	{	PUNCT
iajs-2724	118	20	−0.9	−0.9	PROPN
iajs-2724	118	21	,	,	PUNCT
iajs-2724	118	22	0.9	0.9	NUM
iajs-2724	118	23	𝑖𝑓	𝑖𝑓	NUM
iajs-2724	118	24	𝜒	𝜒	X
iajs-2724	118	25	=	=	SYM
iajs-2724	118	26	0	0	SYM
iajs-2724	119	1	−0.5	−0.5	NOUN
iajs-2724	119	2	,	,	PUNCT
iajs-2724	119	3	0.6	0.6	NUM
iajs-2724	119	4	𝑖𝑓	𝑖𝑓	NOUN
iajs-2724	119	5	𝜒	𝜒	X
iajs-2724	119	6	=	=	SYM
iajs-2724	119	7	1	1	NUM
iajs-2724	119	8	−0.4	−0.4	NUM
iajs-2724	119	9	,	,	PUNCT
iajs-2724	119	10	0.5	0.5	NUM
iajs-2724	119	11	𝑖𝑓	𝑖𝑓	NOUN
iajs-2724	119	12	𝜒	𝜒	X
iajs-2724	119	13	=	=	SYM
iajs-2724	119	14	3	3	NUM
iajs-2724	119	15	−0.2	−0.2	PROPN
iajs-2724	119	16	,	,	PUNCT
iajs-2724	119	17	0.2	0.2	NUM
iajs-2724	119	18	𝑖𝑓	𝑖𝑓	NUM
iajs-2724	119	19	𝜒	𝜒	X
iajs-2724	119	20	=	=	SYM
iajs-2724	119	21	2	2	NUM
iajs-2724	119	22	and	and	CCONJ
iajs-2724	119	23	by	by	ADP
iajs-2724	119	24	applying	apply	VERB
iajs-2724	119	25	definition	definition	NOUN
iajs-2724	119	26	(	(	PUNCT
iajs-2724	119	27	19	19	NUM
iajs-2724	119	28	)	)	PUNCT
iajs-2724	119	29	,	,	PUNCT
iajs-2724	119	30	we	we	PRON
iajs-2724	119	31	can	can	AUX
iajs-2724	119	32	easily	easily	ADV
iajs-2724	119	33	prove	prove	VERB
iajs-2724	119	34	that	that	SCONJ
iajs-2724	119	35	θ	θ	PROPN
iajs-2724	119	36	=	=	SYM
iajs-2724	119	37	〈	〈	PROPN
iajs-2724	119	38	𝑀	𝑀	PROPN
iajs-2724	119	39	,	,	PUNCT
iajs-2724	119	40	𝐿	𝐿	PROPN
iajs-2724	119	41	〉	〉	NOUN
iajs-2724	119	42	is	be	AUX
iajs-2724	119	43	a(cbf)sub	a(cbf)sub	NOUN
iajs-2724	119	44	ku	ku	NOUN
iajs-2724	119	45	-	-	PUNCT
iajs-2724	119	46	semi	semi	NOUN
iajs-2724	119	47	group	group	NOUN
iajs-2724	119	48	with	with	ADP
iajs-2724	119	49	thresholds	threshold	NOUN
iajs-2724	119	50	(	(	PUNCT
iajs-2724	119	51	𝛼	𝛼	X
iajs-2724	119	52	,	,	PUNCT
iajs-2724	119	53	𝛽	𝛽	NOUN
iajs-2724	119	54	)	)	PUNCT
iajs-2724	119	55	=	=	PUNCT
iajs-2724	120	1	(	(	PUNCT
iajs-2724	120	2	[	[	X
iajs-2724	120	3	0.1,0.2	0.1,0.2	X
iajs-2724	120	4	]	]	X
iajs-2724	120	5	,	,	PUNCT
iajs-2724	120	6	[	[	X
iajs-2724	120	7	0.2	0.2	NUM
iajs-2724	120	8	,	,	PUNCT
iajs-2724	120	9	0.2	0.2	NUM
iajs-2724	120	10	]	]	PUNCT
iajs-2724	120	11	)	)	PUNCT
iajs-2724	120	12	,	,	PUNCT
iajs-2724	120	13	and	and	CCONJ
iajs-2724	120	14	(	(	PUNCT
iajs-2724	120	15	𝜔	𝜔	NOUN
iajs-2724	120	16	,	,	PUNCT
iajs-2724	120	17	𝜗)=(0.1,0.2	𝜗)=(0.1,0.2	NOUN
iajs-2724	120	18	)	)	PUNCT
iajs-2724	120	19	,	,	PUNCT
iajs-2724	120	20	but	but	CCONJ
iajs-2724	120	21	not	not	PART
iajs-2724	120	22	a	a	DET
iajs-2724	120	23	(	(	PUNCT
iajs-2724	120	24	cbf)sub	cbf)sub	PROPN
iajs-2724	120	25	ku	ku	PROPN
iajs-2724	120	26	-	-	PUNCT
iajs-2724	120	27	semi	semi	NOUN
iajs-2724	120	28	group	group	NOUN
iajs-2724	120	29	since	since	SCONJ
iajs-2724	120	30	𝜇θ	𝜇θ	PROPN
iajs-2724	120	31	+	+	NOUN
iajs-2724	120	32	(	(	PUNCT
iajs-2724	120	33	1	1	NUM
iajs-2724	120	34	∗	∗	NOUN
iajs-2724	120	35	3	3	NUM
iajs-2724	120	36	)	)	PUNCT
iajs-2724	120	37	≥	≥	NOUN
iajs-2724	120	38	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	X
iajs-2724	121	1	+	+	ADJ
iajs-2724	121	2	(	(	PUNCT
iajs-2724	121	3	1	1	NUM
iajs-2724	121	4	)	)	PUNCT
iajs-2724	121	5	,	,	PUNCT
iajs-2724	121	6	𝜇θ	𝜇θ	PROPN
iajs-2724	121	7	+	+	ADJ
iajs-2724	121	8	(	(	PUNCT
iajs-2724	121	9	3	3	NUM
iajs-2724	121	10	)	)	PUNCT
iajs-2724	121	11	}	}	PUNCT
iajs-2724	121	12	{	{	PUNCT
iajs-2724	121	13	𝜇θ	𝜇θ	PROPN
iajs-2724	121	14	+	+	ADJ
iajs-2724	121	15	(	(	PUNCT
iajs-2724	121	16	2	2	NUM
iajs-2724	121	17	)	)	PUNCT
iajs-2724	121	18	}	}	PUNCT
iajs-2724	121	19	≥	≥	PROPN
iajs-2724	121	20	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	121	21	�	�	PROPN
iajs-2724	121	22	̃	̃	PROPN
iajs-2724	121	23	�	�	PROPN
iajs-2724	121	24	θ	θ	NOUN
iajs-2724	121	25	+	+	PROPN
iajs-2724	121	26	(	(	PUNCT
iajs-2724	121	27	1	1	NUM
iajs-2724	121	28	)	)	PUNCT
iajs-2724	121	29	,	,	PUNCT
iajs-2724	121	30	𝜇θ	𝜇θ	PROPN
iajs-2724	121	31	+	+	ADJ
iajs-2724	121	32	(	(	PUNCT
iajs-2724	121	33	3	3	NUM
iajs-2724	121	34	)	)	PUNCT
iajs-2724	121	35	}	}	PUNCT
iajs-2724	122	1	[	[	X
iajs-2724	122	2	0.2	0.2	NUM
iajs-2724	122	3	,	,	PUNCT
iajs-2724	122	4	0.3	0.3	NUM
iajs-2724	122	5	]	]	PUNCT
iajs-2724	122	6	≥	≥	X
iajs-2724	122	7	𝑟𝑚𝑖𝑛{[0.7	𝑟𝑚𝑖𝑛{[0.7	NUM
iajs-2724	122	8	,	,	PUNCT
iajs-2724	122	9	0.8	0.8	NUM
iajs-2724	122	10	]	]	PUNCT
iajs-2724	122	11	,	,	PUNCT
iajs-2724	122	12	[	[	X
iajs-2724	122	13	0.5	0.5	NUM
iajs-2724	122	14	,	,	PUNCT
iajs-2724	122	15	0.6	0.6	NUM
iajs-2724	122	16	]	]	PUNCT
iajs-2724	122	17	}	}	PUNCT
iajs-2724	123	1	[	[	X
iajs-2724	123	2	0.2	0.2	NUM
iajs-2724	123	3	,	,	PUNCT
iajs-2724	123	4	0.1	0.1	NUM
iajs-2724	123	5	]	]	PUNCT
iajs-2724	123	6	≥	≥	NOUN
iajs-2724	124	1	[	[	X
iajs-2724	124	2	0.5	0.5	NUM
iajs-2724	124	3	,	,	PUNCT
iajs-2724	124	4	0.6	0.6	NUM
iajs-2724	124	5	]	]	PUNCT
iajs-2724	124	6	,	,	PUNCT
iajs-2724	124	7	which	which	PRON
iajs-2724	124	8	is	be	AUX
iajs-2724	124	9	incorrect	incorrect	ADJ
iajs-2724	124	10	phrase	phrase	NOUN
iajs-2724	124	11	𝜇θ	𝜇θ	ADP
iajs-2724	124	12	−(1	−(1	NOUN
iajs-2724	124	13	∗	∗	NOUN
iajs-2724	124	14	3	3	NUM
iajs-2724	124	15	)	)	PUNCT
iajs-2724	124	16	≤	≤	NUM
iajs-2724	124	17	𝑟𝑚𝑎𝑥{	𝑟𝑚𝑎𝑥{	NOUN
iajs-2724	124	18	�	�	NOUN
iajs-2724	124	19	̃	̃	PROPN
iajs-2724	124	20	�	�	PROPN
iajs-2724	124	21	θ	θ	NOUN
iajs-2724	124	22	−(1)ˑ	−(1)ˑ	NOUN
iajs-2724	124	23	,	,	PUNCT
iajs-2724	124	24	𝜇θ	𝜇θ	ADV
iajs-2724	124	25	−(3	−(3	NOUN
iajs-2724	124	26	)	)	PUNCT
iajs-2724	124	27	}	}	PUNCT
iajs-2724	124	28	ibn	ibn	PROPN
iajs-2724	124	29	al	al	PROPN
iajs-2724	124	30	-	-	PUNCT
iajs-2724	124	31	haitham	haitham	PROPN
iajs-2724	124	32	jour	jour	X
iajs-2724	124	33	.	.	PROPN
iajs-2724	125	1	for	for	ADP
iajs-2724	125	2	pure	pure	ADJ
iajs-2724	125	3	&	&	CCONJ
iajs-2724	125	4	appl	appl	PROPN
iajs-2724	125	5	.	.	PUNCT
iajs-2724	126	1	sci	sci	PROPN
iajs-2724	126	2	.	.	PROPN
iajs-2724	127	1	53	53	NUM
iajs-2724	127	2	(	(	PUNCT
iajs-2724	127	3	2)2022	2)2022	VERB
iajs-2724	127	4	53	53	NUM
iajs-2724	127	5	𝜇θ	𝜇θ	ADV
iajs-2724	127	6	−(2	−(2	PROPN
iajs-2724	127	7	)	)	PUNCT
iajs-2724	127	8	≤	≤	PUNCT
iajs-2724	128	1	𝑟𝑚𝑎𝑥{[−0.8	𝑟𝑚𝑎𝑥{[−0.8	ADV
iajs-2724	128	2	,	,	PUNCT
iajs-2724	128	3	−0.7	−0.7	PROPN
iajs-2724	128	4	]	]	PUNCT
iajs-2724	128	5	,	,	PUNCT
iajs-2724	129	1	[	[	X
iajs-2724	129	2	−0.6	−0.6	X
iajs-2724	129	3	,	,	PUNCT
iajs-2724	129	4	−0.5	−0.5	PROPN
iajs-2724	129	5	]	]	X
iajs-2724	129	6	}	}	PUNCT
iajs-2724	130	1	[	[	X
iajs-2724	130	2	−0.3	−0.3	PROPN
iajs-2724	130	3	,	,	PUNCT
iajs-2724	130	4	−0.2≤	−0.2≤	PUNCT
iajs-2724	130	5	[	[	X
iajs-2724	130	6	−0.6	−0.6	PROPN
iajs-2724	130	7	,	,	PUNCT
iajs-2724	130	8	−0.5	−0.5	PROPN
iajs-2724	130	9	]	]	X
iajs-2724	130	10	,	,	PUNCT
iajs-2724	130	11	which	which	PRON
iajs-2724	130	12	is	be	AUX
iajs-2724	130	13	the	the	DET
iajs-2724	130	14	incorrect	incorrect	ADJ
iajs-2724	130	15	phrase	phrase	NOUN
iajs-2724	130	16	,	,	PUNCT
iajs-2724	130	17	and	and	CCONJ
iajs-2724	130	18	𝜆θ	𝜆θ	ADP
iajs-2724	130	19	+	+	ADJ
iajs-2724	130	20	(	(	PUNCT
iajs-2724	130	21	1	1	NUM
iajs-2724	130	22	∗	∗	NOUN
iajs-2724	130	23	3	3	NUM
iajs-2724	130	24	)	)	PUNCT
iajs-2724	130	25	≥	≥	NOUN
iajs-2724	130	26	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	130	27	+	+	PROPN
iajs-2724	130	28	(	(	PUNCT
iajs-2724	130	29	1)ˑ	1)ˑ	NOUN
iajs-2724	130	30	,	,	PUNCT
iajs-2724	130	31	𝜆θ	𝜆θ	ADP
iajs-2724	130	32	+	+	ADJ
iajs-2724	130	33	(	(	PUNCT
iajs-2724	130	34	3	3	NUM
iajs-2724	130	35	)	)	PUNCT
iajs-2724	130	36	}	}	PUNCT
iajs-2724	130	37	𝜆θ	𝜆θ	ADP
iajs-2724	130	38	+	+	ADJ
iajs-2724	130	39	(	(	PUNCT
iajs-2724	130	40	2	2	NUM
iajs-2724	130	41	)	)	PUNCT
iajs-2724	130	42	≥	≥	NOUN
iajs-2724	130	43	𝑚𝑖𝑛{0.6ˑ	𝑚𝑖𝑛{0.6ˑ	NUM
iajs-2724	130	44	,	,	PUNCT
iajs-2724	130	45	0.5	0.5	NUM
iajs-2724	130	46	}	}	SYM
iajs-2724	130	47	0.2	0.2	NUM
iajs-2724	130	48	≥	≥	NOUN
iajs-2724	130	49	0.5	0.5	NUM
iajs-2724	130	50	,	,	PUNCT
iajs-2724	130	51	it	it	PRON
iajs-2724	130	52	is	be	AUX
iajs-2724	130	53	wrong	wrong	ADJ
iajs-2724	130	54	𝜆θ	𝜆θ	ADP
iajs-2724	130	55	−(1	−(1	ADJ
iajs-2724	130	56	∗	∗	NOUN
iajs-2724	130	57	3	3	NUM
iajs-2724	130	58	)	)	PUNCT
iajs-2724	130	59	≤	≤	NUM
iajs-2724	130	60	𝑚𝑎𝑥	𝑚𝑎𝑥	NOUN
iajs-2724	130	61	{	{	PUNCT
iajs-2724	130	62	𝜆θ	𝜆θ	ADP
iajs-2724	130	63	−(1)ˑ	−(1)ˑ	PROPN
iajs-2724	130	64	,	,	PUNCT
iajs-2724	130	65	𝜆θ	𝜆θ	ADP
iajs-2724	130	66	−(3	−(3	NOUN
iajs-2724	130	67	)	)	PUNCT
iajs-2724	130	68	}	}	PUNCT
iajs-2724	130	69	𝜆θ	𝜆θ	ADP
iajs-2724	130	70	−(2	−(2	PROPN
iajs-2724	130	71	)	)	PUNCT
iajs-2724	130	72	≤	≤	NUM
iajs-2724	130	73	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-2724	130	74	{	{	PUNCT
iajs-2724	130	75	−0.5ˑ	−0.5ˑ	ADJ
iajs-2724	130	76	,	,	PUNCT
iajs-2724	130	77	−0.4	−0.4	NUM
iajs-2724	130	78	}	}	PUNCT
iajs-2724	130	79	−0.2	−0.2	PROPN
iajs-2724	130	80	≤	≤	PROPN
iajs-2724	130	81	−0.4	−0.4	PUNCT
iajs-2724	130	82	,	,	PUNCT
iajs-2724	130	83	which	which	PRON
iajs-2724	130	84	is	be	AUX
iajs-2724	130	85	also	also	ADV
iajs-2724	130	86	wrong	wrong	ADJ
iajs-2724	130	87	.	.	PUNCT
iajs-2724	131	1	remark(22	remark(22	NOUN
iajs-2724	131	2	)	)	PUNCT
iajs-2724	131	3	.	.	PUNCT
iajs-2724	132	1	if	if	SCONJ
iajs-2724	132	2	θ	θ	PROPN
iajs-2724	132	3	=	=	SYM
iajs-2724	132	4	〈	〈	PROPN
iajs-2724	132	5	𝑀	𝑀	PROPN
iajs-2724	132	6	,	,	PUNCT
iajs-2724	132	7	𝐿	𝐿	PROPN
iajs-2724	132	8	〉	〉	NOUN
iajs-2724	132	9	is	be	AUX
iajs-2724	132	10	a	a	DET
iajs-2724	132	11	(	(	PUNCT
iajs-2724	132	12	cbf	cbf	PROPN
iajs-2724	132	13	)	)	PUNCT
iajs-2724	132	14	sub	sub	PROPN
iajs-2724	132	15	ku	ku	PROPN
iajs-2724	132	16	-	-	PUNCT
iajs-2724	132	17	semi	semi	NOUN
iajs-2724	132	18	group	group	NOUN
iajs-2724	132	19	with	with	ADP
iajs-2724	132	20	thresholds	threshold	NOUN
iajs-2724	132	21	(	(	PUNCT
iajs-2724	132	22	α	α	X
iajs-2724	132	23	,	,	PUNCT
iajs-2724	132	24	β	β	NOUN
iajs-2724	132	25	)	)	PUNCT
iajs-2724	132	26	,	,	PUNCT
iajs-2724	132	27	(	(	PUNCT
iajs-2724	132	28	ω	ω	X
iajs-2724	132	29	,	,	PUNCT
iajs-2724	132	30	ϑ	ϑ	NOUN
iajs-2724	132	31	)	)	PUNCT
iajs-2724	132	32	such	such	ADJ
iajs-2724	132	33	that	that	SCONJ
iajs-2724	132	34	α	α	NOUN
iajs-2724	132	35	=	=	PUNCT
iajs-2724	133	1	[	[	X
iajs-2724	133	2	0,0	0,0	NOUN
iajs-2724	133	3	]	]	X
iajs-2724	133	4	,	,	PUNCT
iajs-2724	133	5	β	β	X
iajs-2724	133	6	=	=	PUNCT
iajs-2724	134	1	[	[	X
iajs-2724	134	2	1,1	1,1	NUM
iajs-2724	134	3	,	,	PUNCT
iajs-2724	134	4	]	]	PUNCT
iajs-2724	134	5	,	,	PUNCT
iajs-2724	134	6	ω	ω	X
iajs-2724	134	7	=	=	SYM
iajs-2724	134	8	0	0	NUM
iajs-2724	134	9	,	,	PUNCT
iajs-2724	134	10	and	and	CCONJ
iajs-2724	134	11	ϑ	ϑ	X
iajs-2724	134	12	=	=	SYM
iajs-2724	134	13	1	1	NUM
iajs-2724	134	14	,	,	PUNCT
iajs-2724	134	15	then	then	ADV
iajs-2724	134	16	θ	θ	PROPN
iajs-2724	134	17	=	=	SYM
iajs-2724	134	18	〈	〈	PROPN
iajs-2724	134	19	𝑀	𝑀	PROPN
iajs-2724	134	20	,	,	PUNCT
iajs-2724	134	21	𝐿	𝐿	PROPN
iajs-2724	134	22	〉	〉	NOUN
iajs-2724	134	23	is	be	AUX
iajs-2724	134	24	a	a	DET
iajs-2724	134	25	(	(	PUNCT
iajs-2724	134	26	cbf	cbf	PROPN
iajs-2724	134	27	)	)	PUNCT
iajs-2724	134	28	sub	sub	PROPN
iajs-2724	134	29	-	-	ADJ
iajs-2724	134	30	ku	ku	ADJ
iajs-2724	134	31	-	-	PUNCT
iajs-2724	134	32	semi	semi	NOUN
iajs-2724	134	33	group	group	NOUN
iajs-2724	134	34	of	of	ADP
iajs-2724	134	35	ℵ.	ℵ.	PROPN
iajs-2724	134	36	proposition(23).if	proposition(23).if	PROPN
iajs-2724	134	37	θ	θ	PROPN
iajs-2724	134	38	=	=	PUNCT
iajs-2724	134	39	〈	〈	PROPN
iajs-2724	134	40	𝑀	𝑀	PROPN
iajs-2724	134	41	,	,	PUNCT
iajs-2724	134	42	𝐿	𝐿	PROPN
iajs-2724	134	43	〉	〉	NOUN
iajs-2724	134	44	is	be	AUX
iajs-2724	134	45	a	a	DET
iajs-2724	134	46	cubic	cubic	ADJ
iajs-2724	134	47	bipolar	bipolar	ADJ
iajs-2724	134	48	sub	sub	ADJ
iajs-2724	134	49	-	-	ADJ
iajs-2724	134	50	ku	ku	ADJ
iajs-2724	134	51	-	-	PUNCT
iajs-2724	134	52	semi	semi	NOUN
iajs-2724	134	53	group	group	NOUN
iajs-2724	134	54	with	with	ADP
iajs-2724	134	55	thresholds	threshold	NOUN
iajs-2724	134	56	(	(	PUNCT
iajs-2724	134	57	𝛼	𝛼	X
iajs-2724	134	58	,	,	PUNCT
iajs-2724	134	59	𝛽	𝛽	NOUN
iajs-2724	134	60	)	)	PUNCT
iajs-2724	134	61	,	,	PUNCT
iajs-2724	134	62	(	(	PUNCT
iajs-2724	134	63	𝜔	𝜔	NOUN
iajs-2724	134	64	,	,	PUNCT
iajs-2724	134	65	𝜗	𝜗	NOUN
iajs-2724	134	66	)	)	PUNCT
iajs-2724	134	67	of	of	ADP
iajs-2724	134	68	ℵ	ℵ	NOUN
iajs-2724	134	69	,	,	PUNCT
iajs-2724	134	70	then	then	ADV
iajs-2724	134	71	for	for	ADP
iajs-2724	134	72	all	all	DET
iajs-2724	134	73	𝜒	𝜒	NUM
iajs-2724	134	74	∈	∈	ADJ
iajs-2724	134	75	ℵ	ℵ	X
iajs-2724	134	76	(	(	PUNCT
iajs-2724	134	77	1	1	NUM
iajs-2724	134	78	)	)	PUNCT
iajs-2724	134	79	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NOUN
iajs-2724	135	1	+	+	ADJ
iajs-2724	135	2	(	(	PUNCT
iajs-2724	135	3	0	0	NUM
iajs-2724	135	4	)	)	PUNCT
iajs-2724	135	5	,	,	PUNCT
iajs-2724	135	6	𝛼	𝛼	X
iajs-2724	135	7	}	}	PUNCT
iajs-2724	135	8	≥	≥	NOUN
iajs-2724	135	9	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	135	10	+	+	ADJ
iajs-2724	135	11	(	(	PUNCT
iajs-2724	135	12	𝜒	𝜒	NOUN
iajs-2724	135	13	)	)	PUNCT
iajs-2724	135	14	,	,	PUNCT
iajs-2724	135	15	𝛽	𝛽	NOUN
iajs-2724	135	16	}	}	PUNCT
iajs-2724	135	17	(	(	PUNCT
iajs-2724	135	18	2	2	NUM
iajs-2724	135	19	)	)	PUNCT
iajs-2724	135	20	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	135	21	−(0),−𝛼	−(0),−𝛼	PROPN
iajs-2724	135	22	}	}	PUNCT
iajs-2724	135	23	≤	≤	NUM
iajs-2724	135	24	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NUM
iajs-2724	135	25	−(𝜒	−(𝜒	ADJ
iajs-2724	135	26	)	)	PUNCT
iajs-2724	135	27	,	,	PUNCT
iajs-2724	135	28	−𝛽	−𝛽	PROPN
iajs-2724	135	29	}	}	PUNCT
iajs-2724	135	30	(	(	PUNCT
iajs-2724	135	31	3	3	X
iajs-2724	135	32	)	)	PUNCT
iajs-2724	135	33	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	136	1	+	+	PROPN
iajs-2724	136	2	(	(	PUNCT
iajs-2724	136	3	0	0	NUM
iajs-2724	136	4	)	)	PUNCT
iajs-2724	136	5	,	,	PUNCT
iajs-2724	136	6	𝜔	𝜔	X
iajs-2724	136	7	}	}	PUNCT
iajs-2724	136	8	≥	≥	NOUN
iajs-2724	136	9	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	136	10	+	+	ADJ
iajs-2724	136	11	(	(	PUNCT
iajs-2724	136	12	𝜒	𝜒	NOUN
iajs-2724	136	13	)	)	PUNCT
iajs-2724	136	14	,	,	PUNCT
iajs-2724	136	15	𝜗	𝜗	NOUN
iajs-2724	136	16	}	}	PUNCT
iajs-2724	136	17	(	(	PUNCT
iajs-2724	136	18	4	4	NUM
iajs-2724	136	19	)	)	PUNCT
iajs-2724	136	20	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	X
iajs-2724	136	21	−(0),−𝜔	−(0),−𝜔	NUM
iajs-2724	136	22	}	}	PUNCT
iajs-2724	136	23	≤	≤	NUM
iajs-2724	136	24	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	ADJ
iajs-2724	136	25	−(𝜒	−(𝜒	ADJ
iajs-2724	136	26	)	)	PUNCT
iajs-2724	136	27	,	,	PUNCT
iajs-2724	136	28	−𝜗	−𝜗	ADJ
iajs-2724	136	29	}	}	PUNCT
iajs-2724	136	30	semi	semi	ADV
iajs-2724	136	31	group	group	NOUN
iajs-2724	136	32	-	-	PUNCT
iajs-2724	136	33	ku	ku	NOUN
iajs-2724	136	34	-	-	PUNCT
iajs-2724	136	35	sub	sub	NOUN
iajs-2724	136	36	bipolar	bipolar	ADJ
iajs-2724	136	37	is	be	AUX
iajs-2724	136	38	a	a	DET
iajs-2724	136	39	cubicθ	cubicθ	NOUN
iajs-2724	136	40	=	=	PUNCT
iajs-2724	136	41	〈	〈	PROPN
iajs-2724	136	42	𝑀	𝑀	PROPN
iajs-2724	136	43	,	,	PUNCT
iajs-2724	136	44	𝐿	𝐿	NOUN
iajs-2724	136	45	〉	〉	NOUN
iajs-2724	136	46	and	and	CCONJ
iajs-2724	136	47	since	since	ADV
iajs-2724	136	48	,	,	PUNCT
iajs-2724	136	49	𝜒	𝜒	X
iajs-2724	136	50	∗	∗	NOUN
iajs-2724	136	51	𝜒	𝜒	X
iajs-2724	136	52	=	=	NOUN
iajs-2724	136	53	0)5ku(by	0)5ku(by	NOUN
iajs-2724	136	54	proof	proof	NOUN
iajs-2724	136	55	:	:	PUNCT
iajs-2724	136	56	with	with	ADP
iajs-2724	136	57	thresholds	threshold	NOUN
iajs-2724	136	58	(	(	PUNCT
iajs-2724	136	59	𝛼	𝛼	X
iajs-2724	136	60	,	,	PUNCT
iajs-2724	136	61	𝛽	𝛽	NOUN
iajs-2724	136	62	)	)	PUNCT
iajs-2724	136	63	,	,	PUNCT
iajs-2724	136	64	(	(	PUNCT
iajs-2724	136	65	𝜔	𝜔	NOUN
iajs-2724	136	66	,	,	PUNCT
iajs-2724	136	67	𝜗	𝜗	NOUN
iajs-2724	136	68	)	)	PUNCT
iajs-2724	136	69	of	of	ADP
iajs-2724	136	70	ℵ	ℵ	NOUN
iajs-2724	136	71	,	,	PUNCT
iajs-2724	136	72	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	PROPN
iajs-2724	136	73	+	+	ADJ
iajs-2724	136	74	(	(	PUNCT
iajs-2724	136	75	0	0	NUM
iajs-2724	136	76	)	)	PUNCT
iajs-2724	136	77	,	,	PUNCT
iajs-2724	136	78	𝛼	𝛼	NOUN
iajs-2724	136	79	}	}	PUNCT
iajs-2724	136	80	=	=	SYM
iajs-2724	136	81	𝑟𝑚𝑎𝑥{	𝑟𝑚𝑎𝑥{	NOUN
iajs-2724	136	82	�	�	NOUN
iajs-2724	136	83	̃	̃	PROPN
iajs-2724	136	84	�	�	NOUN
iajs-2724	136	85	θ	θ	NOUN
iajs-2724	136	86	+	+	PROPN
iajs-2724	136	87	(	(	PUNCT
iajs-2724	136	88	𝜒	𝜒	X
iajs-2724	136	89	∗	∗	NOUN
iajs-2724	136	90	𝜒	𝜒	NOUN
iajs-2724	136	91	)	)	PUNCT
iajs-2724	136	92	,	,	PUNCT
iajs-2724	136	93	𝛼	𝛼	X
iajs-2724	136	94	}	}	PUNCT
iajs-2724	136	95	≥	≥	NOUN
iajs-2724	136	96	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	136	97	+	+	ADJ
iajs-2724	136	98	(	(	PUNCT
iajs-2724	136	99	𝜒	𝜒	NOUN
iajs-2724	136	100	)	)	PUNCT
iajs-2724	136	101	,	,	PUNCT
iajs-2724	136	102	𝜇θ	𝜇θ	ADP
iajs-2724	137	1	+	+	ADJ
iajs-2724	137	2	(	(	PUNCT
iajs-2724	137	3	𝜒	𝜒	NOUN
iajs-2724	137	4	)	)	PUNCT
iajs-2724	137	5	,	,	PUNCT
iajs-2724	137	6	𝛽	𝛽	NOUN
iajs-2724	137	7	}	}	PUNCT
iajs-2724	137	8	=	=	SYM
iajs-2724	137	9	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	137	10	�	�	PROPN
iajs-2724	137	11	̃	̃	PROPN
iajs-2724	137	12	�	�	PROPN
iajs-2724	137	13	θ	θ	NOUN
iajs-2724	137	14	+	+	NOUN
iajs-2724	137	15	(	(	PUNCT
iajs-2724	137	16	𝜒	𝜒	NOUN
iajs-2724	137	17	)	)	PUNCT
iajs-2724	137	18	,	,	PUNCT
iajs-2724	138	1	𝛽	𝛽	NOUN
iajs-2724	138	2	}	}	PUNCT
iajs-2724	138	3	,	,	PUNCT
iajs-2724	138	4	that	that	ADV
iajs-2724	138	5	is	is	ADV
iajs-2724	138	6	(	(	PUNCT
iajs-2724	138	7	1	1	NUM
iajs-2724	138	8	)	)	PUNCT
iajs-2724	138	9	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	138	10	−(0	−(0	NOUN
iajs-2724	138	11	)	)	PUNCT
iajs-2724	138	12	,	,	PUNCT
iajs-2724	138	13	−𝛼	−𝛼	ADJ
iajs-2724	138	14	}	}	PUNCT
iajs-2724	138	15	=	=	SYM
iajs-2724	138	16	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	138	17	−(𝜒	−(𝜒	ADJ
iajs-2724	138	18	∗	∗	NOUN
iajs-2724	138	19	𝜒),−𝛼	𝜒),−𝛼	X
iajs-2724	138	20	}	}	PUNCT
iajs-2724	138	21	≤	≤	NUM
iajs-2724	138	22	𝑟𝑚𝑎𝑥{	𝑟𝑚𝑎𝑥{	NOUN
iajs-2724	138	23	�	�	NOUN
iajs-2724	138	24	̃	̃	NOUN
iajs-2724	138	25	�	�	NOUN
iajs-2724	138	26	θ	θ	NOUN
iajs-2724	138	27	−(𝜒	−(𝜒	ADJ
iajs-2724	138	28	)	)	PUNCT
iajs-2724	138	29	,	,	PUNCT
iajs-2724	138	30	𝜇θ	𝜇θ	ADP
iajs-2724	138	31	−(𝜒),−𝛽	−(𝜒),−𝛽	ADP
iajs-2724	138	32	}	}	PUNCT
iajs-2724	138	33	=	=	SYM
iajs-2724	138	34	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NOUN
iajs-2724	138	35	−(𝜒),−𝛽	−(𝜒),−𝛽	ADP
iajs-2724	138	36	}	}	PUNCT
iajs-2724	138	37	,	,	PUNCT
iajs-2724	138	38	that	that	ADV
iajs-2724	138	39	is	is	ADV
iajs-2724	138	40	(	(	PUNCT
iajs-2724	138	41	2	2	NUM
iajs-2724	138	42	)	)	PUNCT
iajs-2724	138	43	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	139	1	+	+	PROPN
iajs-2724	139	2	(	(	PUNCT
iajs-2724	139	3	0	0	NUM
iajs-2724	139	4	)	)	PUNCT
iajs-2724	139	5	,	,	PUNCT
iajs-2724	139	6	𝜔	𝜔	X
iajs-2724	139	7	}	}	PUNCT
iajs-2724	139	8	=	=	SYM
iajs-2724	139	9	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	139	10	+	+	ADJ
iajs-2724	139	11	(	(	PUNCT
iajs-2724	139	12	𝜒	𝜒	X
iajs-2724	139	13	∗	∗	NOUN
iajs-2724	139	14	𝜒	𝜒	NOUN
iajs-2724	139	15	)	)	PUNCT
iajs-2724	139	16	,	,	PUNCT
iajs-2724	139	17	𝜔	𝜔	X
iajs-2724	139	18	}	}	PUNCT
iajs-2724	139	19	≥	≥	NUM
iajs-2724	139	20	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
iajs-2724	139	21	{	{	PUNCT
iajs-2724	139	22	𝜆θ	𝜆θ	ADP
iajs-2724	139	23	+	+	ADJ
iajs-2724	139	24	(	(	PUNCT
iajs-2724	139	25	𝜒	𝜒	NOUN
iajs-2724	139	26	)	)	PUNCT
iajs-2724	139	27	,	,	PUNCT
iajs-2724	139	28	𝜆θ	𝜆θ	ADP
iajs-2724	139	29	+	+	ADJ
iajs-2724	139	30	(	(	PUNCT
iajs-2724	139	31	𝜒	𝜒	NOUN
iajs-2724	139	32	)	)	PUNCT
iajs-2724	139	33	,	,	PUNCT
iajs-2724	139	34	𝜗	𝜗	NOUN
iajs-2724	139	35	}	}	PUNCT
iajs-2724	139	36	=	=	SYM
iajs-2724	139	37	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
iajs-2724	139	38	{	{	PUNCT
iajs-2724	139	39	𝜆θ	𝜆θ	ADP
iajs-2724	139	40	+	+	ADJ
iajs-2724	139	41	(	(	PUNCT
iajs-2724	139	42	𝜒	𝜒	NOUN
iajs-2724	139	43	)	)	PUNCT
iajs-2724	139	44	,	,	PUNCT
iajs-2724	139	45	𝜗	𝜗	NOUN
iajs-2724	139	46	}	}	PUNCT
iajs-2724	139	47	,	,	PUNCT
iajs-2724	139	48	that	that	ADV
iajs-2724	139	49	is	is	ADV
iajs-2724	139	50	(	(	PUNCT
iajs-2724	139	51	3	3	NUM
iajs-2724	139	52	)	)	PUNCT
iajs-2724	139	53	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	X
iajs-2724	139	54	−(0),−𝜔	−(0),−𝜔	NUM
iajs-2724	139	55	}	}	PUNCT
iajs-2724	139	56	=	=	PUNCT
iajs-2724	139	57	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	X
iajs-2724	139	58	−(𝜒	−(𝜒	VERB
iajs-2724	139	59	∗	∗	NOUN
iajs-2724	139	60	𝜒	𝜒	NOUN
iajs-2724	139	61	)	)	PUNCT
iajs-2724	139	62	,	,	PUNCT
iajs-2724	139	63	−𝜔	−𝜔	ADP
iajs-2724	139	64	}	}	PUNCT
iajs-2724	139	65	≤	≤	NUM
iajs-2724	139	66	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	ADJ
iajs-2724	139	67	−(𝜒	−(𝜒	ADJ
iajs-2724	139	68	)	)	PUNCT
iajs-2724	139	69	,	,	PUNCT
iajs-2724	139	70	𝜆θ	𝜆θ	ADP
iajs-2724	139	71	−(𝜒),−𝜗	−(𝜒),−𝜗	VERB
iajs-2724	139	72	}	}	PUNCT
iajs-2724	139	73	=	=	SYM
iajs-2724	139	74	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	ADJ
iajs-2724	139	75	−(𝜒	−(𝜒	ADJ
iajs-2724	139	76	)	)	PUNCT
iajs-2724	139	77	,	,	PUNCT
iajs-2724	139	78	−𝜗},that	−𝜗},that	PRON
iajs-2724	139	79	is	be	AUX
iajs-2724	139	80	(	(	PUNCT
iajs-2724	139	81	4	4	NUM
iajs-2724	139	82	)	)	PUNCT
iajs-2724	139	83	proposition(24).if	proposition(24).if	NOUN
iajs-2724	139	84	θ	θ	NOUN
iajs-2724	139	85	=	=	SYM
iajs-2724	139	86	〈	〈	PROPN
iajs-2724	139	87	𝑀	𝑀	PROPN
iajs-2724	139	88	,	,	PUNCT
iajs-2724	139	89	𝐿	𝐿	PROPN
iajs-2724	139	90	〉	〉	NOUN
iajs-2724	139	91	is	be	AUX
iajs-2724	139	92	a	a	DET
iajs-2724	139	93	(	(	PUNCT
iajs-2724	139	94	cbf	cbf	PROPN
iajs-2724	139	95	)	)	PUNCT
iajs-2724	139	96	sub	sub	PROPN
iajs-2724	139	97	-	-	ADJ
iajs-2724	139	98	ku	ku	ADJ
iajs-2724	139	99	-	-	PUNCT
iajs-2724	139	100	semi	semi	NOUN
iajs-2724	139	101	group	group	NOUN
iajs-2724	139	102	with	with	ADP
iajs-2724	139	103	thresholds	threshold	NOUN
iajs-2724	139	104	(	(	PUNCT
iajs-2724	139	105	𝛼	𝛼	X
iajs-2724	139	106	,	,	PUNCT
iajs-2724	139	107	𝛽	𝛽	NOUN
iajs-2724	139	108	)	)	PUNCT
iajs-2724	139	109	,	,	PUNCT
iajs-2724	139	110	(	(	PUNCT
iajs-2724	139	111	𝜔	𝜔	NOUN
iajs-2724	139	112	,	,	PUNCT
iajs-2724	139	113	𝜗	𝜗	NOUN
iajs-2724	139	114	)	)	PUNCT
iajs-2724	139	115	of	of	ADP
iajs-2724	139	116	ℵ	ℵ	NOUN
iajs-2724	139	117	,	,	PUNCT
iajs-2724	139	118	then	then	ADV
iajs-2724	139	119	for	for	ADP
iajs-2724	139	120	all	all	DET
iajs-2724	139	121	𝜒	𝜒	NUM
iajs-2724	139	122	∈	∈	ADJ
iajs-2724	139	123	ℵ	ℵ	X
iajs-2724	139	124	(	(	PUNCT
iajs-2724	139	125	1	1	NUM
iajs-2724	139	126	)	)	PUNCT
iajs-2724	139	127	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NOUN
iajs-2724	140	1	+	+	ADJ
iajs-2724	140	2	(	(	PUNCT
iajs-2724	140	3	0	0	NUM
iajs-2724	140	4	∘	∘	NUM
iajs-2724	140	5	𝜒	𝜒	NUM
iajs-2724	140	6	)	)	PUNCT
iajs-2724	140	7	,	,	PUNCT
iajs-2724	140	8	𝛼	𝛼	X
iajs-2724	140	9	}	}	PUNCT
iajs-2724	140	10	≥	≥	NOUN
iajs-2724	140	11	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	140	12	+	+	ADJ
iajs-2724	140	13	(	(	PUNCT
iajs-2724	140	14	𝜒	𝜒	NOUN
iajs-2724	140	15	)	)	PUNCT
iajs-2724	140	16	,	,	PUNCT
iajs-2724	140	17	𝛽	𝛽	NOUN
iajs-2724	140	18	}	}	PUNCT
iajs-2724	140	19	(	(	PUNCT
iajs-2724	140	20	2	2	X
iajs-2724	140	21	)	)	PUNCT
iajs-2724	140	22	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	140	23	−(0	−(0	NOUN
iajs-2724	140	24	∘	∘	NUM
iajs-2724	140	25	𝜒	𝜒	NUM
iajs-2724	140	26	)	)	PUNCT
iajs-2724	140	27	,	,	PUNCT
iajs-2724	140	28	−𝛼	−𝛼	ADJ
iajs-2724	140	29	}	}	PUNCT
iajs-2724	140	30	≤	≤	NUM
iajs-2724	140	31	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NUM
iajs-2724	140	32	−(𝜒	−(𝜒	ADJ
iajs-2724	140	33	)	)	PUNCT
iajs-2724	140	34	,	,	PUNCT
iajs-2724	140	35	−𝛽	−𝛽	PROPN
iajs-2724	140	36	}	}	PUNCT
iajs-2724	140	37	(	(	PUNCT
iajs-2724	140	38	3	3	X
iajs-2724	140	39	)	)	PUNCT
iajs-2724	140	40	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	141	1	+	+	PROPN
iajs-2724	141	2	(	(	PUNCT
iajs-2724	141	3	0	0	NUM
iajs-2724	141	4	∘	∘	NUM
iajs-2724	141	5	𝜒	𝜒	NUM
iajs-2724	141	6	)	)	PUNCT
iajs-2724	141	7	,	,	PUNCT
iajs-2724	141	8	𝜔	𝜔	X
iajs-2724	141	9	}	}	PUNCT
iajs-2724	141	10	≥	≥	NOUN
iajs-2724	141	11	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	141	12	+	+	ADJ
iajs-2724	141	13	(	(	PUNCT
iajs-2724	141	14	𝜒	𝜒	NOUN
iajs-2724	141	15	)	)	PUNCT
iajs-2724	141	16	,	,	PUNCT
iajs-2724	141	17	𝜗	𝜗	NOUN
iajs-2724	141	18	}	}	PUNCT
iajs-2724	141	19	(	(	PUNCT
iajs-2724	141	20	4	4	NUM
iajs-2724	141	21	)	)	PUNCT
iajs-2724	141	22	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	X
iajs-2724	141	23	−(0	−(0	VERB
iajs-2724	141	24	∘	∘	NUM
iajs-2724	141	25	𝜒	𝜒	NUM
iajs-2724	141	26	)	)	PUNCT
iajs-2724	141	27	,	,	PUNCT
iajs-2724	141	28	−𝜔	−𝜔	ADP
iajs-2724	141	29	}	}	PUNCT
iajs-2724	141	30	≤	≤	NUM
iajs-2724	141	31	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	ADJ
iajs-2724	141	32	−(𝜒	−(𝜒	ADJ
iajs-2724	141	33	)	)	PUNCT
iajs-2724	141	34	,	,	PUNCT
iajs-2724	141	35	−𝜗	−𝜗	ADV
iajs-2724	141	36	}	}	PUNCT
iajs-2724	141	37	ibn	ibn	PROPN
iajs-2724	141	38	al	al	PROPN
iajs-2724	141	39	-	-	PUNCT
iajs-2724	141	40	haitham	haitham	PROPN
iajs-2724	141	41	jour	jour	X
iajs-2724	141	42	.	.	PROPN
iajs-2724	142	1	for	for	ADP
iajs-2724	142	2	pure	pure	ADJ
iajs-2724	142	3	&	&	CCONJ
iajs-2724	142	4	appl	appl	PROPN
iajs-2724	142	5	.	.	PUNCT
iajs-2724	143	1	sci	sci	PROPN
iajs-2724	143	2	.	.	PROPN
iajs-2724	144	1	53	53	NUM
iajs-2724	144	2	(	(	PUNCT
iajs-2724	144	3	2)2022	2)2022	VERB
iajs-2724	144	4	54	54	NUM
iajs-2724	144	5	proof	proof	NOUN
iajs-2724	144	6	:	:	PUNCT
iajs-2724	144	7	since	since	SCONJ
iajs-2724	144	8	θ	θ	PROPN
iajs-2724	144	9	=	=	SYM
iajs-2724	144	10	〈	〈	PROPN
iajs-2724	144	11	𝑀	𝑀	PROPN
iajs-2724	144	12	,	,	PUNCT
iajs-2724	144	13	𝐿	𝐿	PROPN
iajs-2724	144	14	〉	〉	NOUN
iajs-2724	144	15	is	be	AUX
iajs-2724	144	16	a	a	DET
iajs-2724	144	17	(	(	PUNCT
iajs-2724	144	18	cbf	cbf	PROPN
iajs-2724	144	19	)	)	PUNCT
iajs-2724	144	20	sub	sub	PROPN
iajs-2724	144	21	-	-	ADJ
iajs-2724	144	22	ku	ku	ADJ
iajs-2724	144	23	-	-	PUNCT
iajs-2724	144	24	semi	semi	NOUN
iajs-2724	144	25	group	group	NOUN
iajs-2724	144	26	with	with	ADP
iajs-2724	144	27	thresholds	threshold	NOUN
iajs-2724	144	28	(	(	PUNCT
iajs-2724	144	29	𝛼	𝛼	X
iajs-2724	144	30	,	,	PUNCT
iajs-2724	144	31	𝛽	𝛽	NOUN
iajs-2724	144	32	)	)	PUNCT
iajs-2724	144	33	,	,	PUNCT
iajs-2724	144	34	(	(	PUNCT
iajs-2724	144	35	𝜔	𝜔	NOUN
iajs-2724	144	36	,	,	PUNCT
iajs-2724	144	37	𝜗	𝜗	NOUN
iajs-2724	144	38	)	)	PUNCT
iajs-2724	144	39	of	of	ADP
iajs-2724	144	40	ℵ	ℵ	NOUN
iajs-2724	144	41	,	,	PUNCT
iajs-2724	144	42	we	we	PRON
iajs-2724	144	43	have	have	VERB
iajs-2724	144	44	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NOUN
iajs-2724	144	45	+	+	NOUN
iajs-2724	144	46	(	(	PUNCT
iajs-2724	144	47	0	0	NUM
iajs-2724	144	48	∘	∘	NUM
iajs-2724	144	49	𝜒	𝜒	NUM
iajs-2724	144	50	)	)	PUNCT
iajs-2724	144	51	,	,	PUNCT
iajs-2724	144	52	𝛼	𝛼	PROPN
iajs-2724	144	53	}	}	PUNCT
iajs-2724	144	54	≥	≥	NUM
iajs-2724	144	55	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	144	56	�	�	PROPN
iajs-2724	144	57	̃	̃	PROPN
iajs-2724	144	58	�	�	PROPN
iajs-2724	144	59	θ	θ	NOUN
iajs-2724	144	60	+	+	PROPN
iajs-2724	144	61	(	(	PUNCT
iajs-2724	144	62	0	0	NUM
iajs-2724	144	63	)	)	PUNCT
iajs-2724	144	64	,	,	PUNCT
iajs-2724	144	65	𝜇θ	𝜇θ	ADP
iajs-2724	145	1	+	+	ADJ
iajs-2724	145	2	(	(	PUNCT
iajs-2724	145	3	𝜒	𝜒	NOUN
iajs-2724	145	4	)	)	PUNCT
iajs-2724	145	5	,	,	PUNCT
iajs-2724	145	6	𝛽	𝛽	NOUN
iajs-2724	145	7	}	}	PUNCT
iajs-2724	145	8	=	=	SYM
iajs-2724	145	9	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	145	10	+	+	ADJ
iajs-2724	145	11	(	(	PUNCT
iajs-2724	145	12	𝜒	𝜒	NOUN
iajs-2724	145	13	)	)	PUNCT
iajs-2724	145	14	,	,	PUNCT
iajs-2724	145	15	𝛽	𝛽	NOUN
iajs-2724	145	16	}	}	PUNCT
iajs-2724	145	17	,	,	PUNCT
iajs-2724	145	18	which	which	PRON
iajs-2724	145	19	is	be	AUX
iajs-2724	145	20	(	(	PUNCT
iajs-2724	145	21	1	1	NUM
iajs-2724	145	22	)	)	PUNCT
iajs-2724	145	23	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	145	24	−(0	−(0	NOUN
iajs-2724	145	25	∘	∘	NUM
iajs-2724	145	26	𝜒	𝜒	NUM
iajs-2724	145	27	)	)	PUNCT
iajs-2724	145	28	,	,	PUNCT
iajs-2724	145	29	−𝛼	−𝛼	ADJ
iajs-2724	145	30	}	}	PUNCT
iajs-2724	145	31	≤	≤	NUM
iajs-2724	145	32	𝑟𝑚𝑎𝑥{	𝑟𝑚𝑎𝑥{	NOUN
iajs-2724	145	33	�	�	NOUN
iajs-2724	145	34	̃	̃	PROPN
iajs-2724	145	35	�	�	PROPN
iajs-2724	145	36	θ	θ	NOUN
iajs-2724	145	37	−(0	−(0	NOUN
iajs-2724	145	38	)	)	PUNCT
iajs-2724	145	39	,	,	PUNCT
iajs-2724	145	40	𝜇θ	𝜇θ	ADP
iajs-2724	145	41	−(𝜒	−(𝜒	ADJ
iajs-2724	145	42	)	)	PUNCT
iajs-2724	146	1	−	−	ADP
iajs-2724	146	2	𝛽	𝛽	NOUN
iajs-2724	146	3	}	}	PUNCT
iajs-2724	146	4	=	=	SYM
iajs-2724	146	5	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NOUN
iajs-2724	146	6	−(𝜒),−𝛽	−(𝜒),−𝛽	ADP
iajs-2724	146	7	}	}	PUNCT
iajs-2724	146	8	,	,	PUNCT
iajs-2724	146	9	which	which	PRON
iajs-2724	146	10	is	be	AUX
iajs-2724	146	11	(	(	PUNCT
iajs-2724	146	12	2	2	NUM
iajs-2724	146	13	)	)	PUNCT
iajs-2724	146	14	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	147	1	+	+	PROPN
iajs-2724	147	2	(	(	PUNCT
iajs-2724	147	3	0	0	NUM
iajs-2724	147	4	∘	∘	NUM
iajs-2724	147	5	𝜒	𝜒	NUM
iajs-2724	147	6	)	)	PUNCT
iajs-2724	147	7	,	,	PUNCT
iajs-2724	147	8	𝜔	𝜔	X
iajs-2724	147	9	}	}	PUNCT
iajs-2724	147	10	≥	≥	NOUN
iajs-2724	147	11	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	147	12	+	+	ADJ
iajs-2724	147	13	(	(	PUNCT
iajs-2724	147	14	0	0	NUM
iajs-2724	147	15	)	)	PUNCT
iajs-2724	147	16	,	,	PUNCT
iajs-2724	147	17	𝜆θ	𝜆θ	ADP
iajs-2724	147	18	+	+	ADJ
iajs-2724	147	19	(	(	PUNCT
iajs-2724	147	20	𝜒	𝜒	NOUN
iajs-2724	147	21	)	)	PUNCT
iajs-2724	147	22	,	,	PUNCT
iajs-2724	147	23	𝜗	𝜗	NOUN
iajs-2724	147	24	}	}	PUNCT
iajs-2724	147	25	=	=	SYM
iajs-2724	147	26	𝑚𝑖𝑛	𝑚𝑖𝑛	NOUN
iajs-2724	147	27	{	{	PUNCT
iajs-2724	147	28	𝜆θ	𝜆θ	ADP
iajs-2724	147	29	+	+	ADJ
iajs-2724	147	30	(	(	PUNCT
iajs-2724	147	31	𝜒	𝜒	NOUN
iajs-2724	147	32	)	)	PUNCT
iajs-2724	147	33	,	,	PUNCT
iajs-2724	147	34	𝜗	𝜗	PROPN
iajs-2724	147	35	}	}	PUNCT
iajs-2724	147	36	,	,	PUNCT
iajs-2724	147	37	which	which	PRON
iajs-2724	147	38	is	be	AUX
iajs-2724	147	39	(	(	PUNCT
iajs-2724	147	40	3	3	NUM
iajs-2724	147	41	)	)	PUNCT
iajs-2724	147	42	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	X
iajs-2724	147	43	−(0	−(0	VERB
iajs-2724	147	44	∘	∘	NUM
iajs-2724	147	45	𝜒),−𝜔	𝜒),−𝜔	NOUN
iajs-2724	147	46	}	}	PUNCT
iajs-2724	147	47	≤	≤	NUM
iajs-2724	147	48	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	NUM
iajs-2724	147	49	−(0	−(0	NOUN
iajs-2724	147	50	)	)	PUNCT
iajs-2724	147	51	,	,	PUNCT
iajs-2724	147	52	𝜆θ	𝜆θ	ADP
iajs-2724	147	53	−(𝜒),−𝜗	−(𝜒),−𝜗	VERB
iajs-2724	147	54	}	}	PUNCT
iajs-2724	147	55	=	=	SYM
iajs-2724	147	56	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PROPN
iajs-2724	147	57	−(𝜒),−𝜗},which	−(𝜒),−𝜗},which	VERB
iajs-2724	147	58	is	be	AUX
iajs-2724	147	59	(	(	PUNCT
iajs-2724	147	60	4	4	NUM
iajs-2724	147	61	)	)	PUNCT
iajs-2724	147	62	definition(25	definition(25	NOUN
iajs-2724	147	63	)	)	PUNCT
iajs-2724	147	64	.	.	PUNCT
iajs-2724	148	1	a	a	PRON
iajs-2724	148	2	(	(	PUNCT
iajs-2724	148	3	cbf	cbf	PROPN
iajs-2724	148	4	)	)	PUNCT
iajs-2724	148	5	set	set	VERB
iajs-2724	148	6	θ	θ	PROPN
iajs-2724	148	7	=	=	SYM
iajs-2724	148	8	〈	〈	PROPN
iajs-2724	148	9	𝑀	𝑀	PROPN
iajs-2724	148	10	,	,	PUNCT
iajs-2724	148	11	𝐿	𝐿	PROPN
iajs-2724	148	12	〉	〉	NOUN
iajs-2724	148	13	is	be	AUX
iajs-2724	148	14	named	name	VERB
iajs-2724	148	15	a	a	DET
iajs-2724	148	16	(	(	PUNCT
iajs-2724	148	17	cbf	cbf	PROPN
iajs-2724	148	18	)	)	PUNCT
iajs-2724	148	19	ideal	ideal	NOUN
iajs-2724	148	20	of	of	ADP
iajs-2724	148	21	the	the	DET
iajs-2724	148	22	ku	ku	PROPN
iajs-2724	148	23	-	-	PUNCT
iajs-2724	148	24	semi	semi	NOUN
iajs-2724	148	25	group	group	NOUN
iajs-2724	148	26	with	with	ADP
iajs-2724	148	27	thresholds	threshold	NOUN
iajs-2724	148	28	(	(	PUNCT
iajs-2724	148	29	𝛼	𝛼	X
iajs-2724	148	30	,	,	PUNCT
iajs-2724	148	31	𝛽	𝛽	NOUN
iajs-2724	148	32	)	)	PUNCT
iajs-2724	148	33	,	,	PUNCT
iajs-2724	148	34	(	(	PUNCT
iajs-2724	148	35	𝜔	𝜔	NOUN
iajs-2724	148	36	,	,	PUNCT
iajs-2724	148	37	𝜗	𝜗	NOUN
iajs-2724	148	38	)	)	PUNCT
iajs-2724	148	39	if	if	SCONJ
iajs-2724	148	40	∀	∀	X
iajs-2724	148	41	𝜒	𝜒	VERB
iajs-2724	148	42	,	,	PUNCT
iajs-2724	148	43	𝛾	𝛾	ADP
iajs-2724	148	44	∈	∈	PROPN
iajs-2724	148	45	ℵ	ℵ	NOUN
iajs-2724	148	46	(	(	PUNCT
iajs-2724	148	47	cbt1	cbt1	PROPN
iajs-2724	148	48	)	)	PUNCT
iajs-2724	148	49	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	148	50	�	�	PROPN
iajs-2724	148	51	̃	̃	PROPN
iajs-2724	148	52	�	�	PROPN
iajs-2724	148	53	θ	θ	ADJ
iajs-2724	148	54	−(0),−𝛼	−(0),−𝛼	PROPN
iajs-2724	148	55	}	}	PUNCT
iajs-2724	148	56	≤	≤	NUM
iajs-2724	148	57	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NUM
iajs-2724	148	58	−(𝜒	−(𝜒	ADJ
iajs-2724	148	59	)	)	PUNCT
iajs-2724	148	60	,	,	PUNCT
iajs-2724	148	61	−𝛽	−𝛽	PROPN
iajs-2724	148	62	}	}	PUNCT
iajs-2724	148	63	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	PROPN
iajs-2724	149	1	+	+	ADJ
iajs-2724	149	2	(	(	PUNCT
iajs-2724	149	3	0	0	NUM
iajs-2724	149	4	)	)	PUNCT
iajs-2724	149	5	,	,	PUNCT
iajs-2724	149	6	𝛼	𝛼	PROPN
iajs-2724	149	7	}	}	PUNCT
iajs-2724	149	8	≥	≥	NUM
iajs-2724	149	9	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	149	10	�	�	PROPN
iajs-2724	149	11	̃	̃	PROPN
iajs-2724	149	12	�	�	PROPN
iajs-2724	149	13	θ	θ	NOUN
iajs-2724	149	14	+	+	NOUN
iajs-2724	149	15	(	(	PUNCT
iajs-2724	149	16	𝜒	𝜒	NOUN
iajs-2724	149	17	)	)	PUNCT
iajs-2724	149	18	,	,	PUNCT
iajs-2724	149	19	𝛽},and	𝛽},and	ADP
iajs-2724	149	20	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	ADJ
iajs-2724	149	21	−(0),−𝜔	−(0),−𝜔	NUM
iajs-2724	149	22	}	}	PUNCT
iajs-2724	149	23	≤	≤	NUM
iajs-2724	149	24	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	ADV
iajs-2724	150	1	−(𝜒),−𝜗	−(𝜒),−𝜗	PUNCT
iajs-2724	150	2	}	}	PUNCT
iajs-2724	150	3	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	150	4	+	+	PROPN
iajs-2724	150	5	(	(	PUNCT
iajs-2724	150	6	0	0	NUM
iajs-2724	150	7	)	)	PUNCT
iajs-2724	150	8	,	,	PUNCT
iajs-2724	150	9	𝜔	𝜔	X
iajs-2724	150	10	}	}	PUNCT
iajs-2724	150	11	≥	≥	NOUN
iajs-2724	150	12	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	150	13	+	+	ADJ
iajs-2724	150	14	(	(	PUNCT
iajs-2724	150	15	𝜒	𝜒	NOUN
iajs-2724	150	16	)	)	PUNCT
iajs-2724	150	17	,	,	PUNCT
iajs-2724	150	18	𝜗	𝜗	NOUN
iajs-2724	150	19	}	}	PUNCT
iajs-2724	150	20	(	(	PUNCT
iajs-2724	150	21	cbt2	cbt2	PROPN
iajs-2724	150	22	)	)	PUNCT
iajs-2724	150	23	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	150	24	�	�	PROPN
iajs-2724	150	25	̃	̃	PROPN
iajs-2724	150	26	�	�	PROPN
iajs-2724	150	27	θ	θ	ADJ
iajs-2724	150	28	−(𝛾),−𝛼	−(𝛾),−𝛼	NOUN
iajs-2724	150	29	}	}	PUNCT
iajs-2724	150	30	≤	≤	NUM
iajs-2724	150	31	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NUM
iajs-2724	150	32	−(𝜒	−(𝜒	ADJ
iajs-2724	150	33	∗	∗	NOUN
iajs-2724	150	34	𝛾	𝛾	NOUN
iajs-2724	150	35	)	)	PUNCT
iajs-2724	150	36	,	,	PUNCT
iajs-2724	150	37	𝜇θ	𝜇θ	ADP
iajs-2724	150	38	−(𝜒),−𝛽	−(𝜒),−𝛽	ADP
iajs-2724	150	39	}	}	PUNCT
iajs-2724	150	40	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	X
iajs-2724	151	1	+	+	ADJ
iajs-2724	151	2	(	(	PUNCT
iajs-2724	151	3	𝛾	𝛾	NOUN
iajs-2724	151	4	)	)	PUNCT
iajs-2724	151	5	,	,	PUNCT
iajs-2724	151	6	𝛼	𝛼	X
iajs-2724	151	7	}	}	PUNCT
iajs-2724	151	8	≥	≥	NOUN
iajs-2724	151	9	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	151	10	+	+	NOUN
iajs-2724	151	11	(	(	PUNCT
iajs-2724	151	12	𝜒	𝜒	NOUN
iajs-2724	151	13	∗	∗	NOUN
iajs-2724	151	14	𝛾	𝛾	NOUN
iajs-2724	151	15	)	)	PUNCT
iajs-2724	151	16	,	,	PUNCT
iajs-2724	152	1	𝜇θ	𝜇θ	ADP
iajs-2724	152	2	+	+	ADJ
iajs-2724	152	3	(	(	PUNCT
iajs-2724	152	4	𝜒	𝜒	NOUN
iajs-2724	152	5	)	)	PUNCT
iajs-2724	152	6	,	,	PUNCT
iajs-2724	152	7	𝛽	𝛽	NOUN
iajs-2724	152	8	}	}	PUNCT
iajs-2724	152	9	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	152	10	−(𝛾),−𝜔	−(𝛾),−𝜔	ADV
iajs-2724	152	11	}	}	PUNCT
iajs-2724	152	12	≤	≤	NUM
iajs-2724	152	13	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	152	14	−(𝜒	−(𝜒	ADJ
iajs-2724	152	15	∗	∗	NOUN
iajs-2724	152	16	𝛾	𝛾	NOUN
iajs-2724	152	17	)	)	PUNCT
iajs-2724	152	18	,	,	PUNCT
iajs-2724	152	19	𝜆θ	𝜆θ	ADP
iajs-2724	152	20	−(𝜒),−𝜗	−(𝜒),−𝜗	VERB
iajs-2724	152	21	}	}	PUNCT
iajs-2724	152	22	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	152	23	+	+	PROPN
iajs-2724	152	24	(	(	PUNCT
iajs-2724	152	25	𝛾	𝛾	NOUN
iajs-2724	152	26	)	)	PUNCT
iajs-2724	152	27	,	,	PUNCT
iajs-2724	152	28	𝜔	𝜔	X
iajs-2724	152	29	}	}	PUNCT
iajs-2724	152	30	≥	≥	NOUN
iajs-2724	152	31	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	152	32	+	+	PROPN
iajs-2724	152	33	(	(	PUNCT
iajs-2724	152	34	𝜒	𝜒	NOUN
iajs-2724	152	35	∗	∗	NOUN
iajs-2724	152	36	𝛾	𝛾	NOUN
iajs-2724	152	37	)	)	PUNCT
iajs-2724	152	38	,	,	PUNCT
iajs-2724	152	39	𝜆θ	𝜆θ	ADP
iajs-2724	152	40	+	+	ADJ
iajs-2724	152	41	(	(	PUNCT
iajs-2724	152	42	𝜒	𝜒	NOUN
iajs-2724	152	43	)	)	PUNCT
iajs-2724	152	44	,	,	PUNCT
iajs-2724	152	45	𝜗	𝜗	NOUN
iajs-2724	152	46	}	}	PUNCT
iajs-2724	152	47	(	(	PUNCT
iajs-2724	152	48	cbt3)𝑟𝑚𝑖𝑛{	cbt3)𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	152	49	�	�	PROPN
iajs-2724	152	50	̃	̃	PROPN
iajs-2724	152	51	�	�	NOUN
iajs-2724	152	52	θ	θ	NOUN
iajs-2724	152	53	−(𝜒	−(𝜒	ADJ
iajs-2724	152	54	∘	∘	NOUN
iajs-2724	152	55	𝛾),−𝛼	𝛾),−𝛼	PRON
iajs-2724	152	56	}	}	PUNCT
iajs-2724	152	57	≤	≤	NUM
iajs-2724	152	58	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NUM
iajs-2724	152	59	−(𝜒	−(𝜒	ADJ
iajs-2724	152	60	)	)	PUNCT
iajs-2724	152	61	,	,	PUNCT
iajs-2724	152	62	𝜇θ	𝜇θ	ADV
iajs-2724	152	63	−(𝛾),−𝛽	−(𝛾),−𝛽	PUNCT
iajs-2724	152	64	}	}	PUNCT
iajs-2724	152	65	𝑟𝑚𝑎𝑥{	𝑟𝑚𝑎𝑥{	ADJ
iajs-2724	152	66	�	�	NOUN
iajs-2724	152	67	̃	̃	NOUN
iajs-2724	152	68	�	�	NOUN
iajs-2724	152	69	θ	θ	NOUN
iajs-2724	152	70	+	+	PROPN
iajs-2724	152	71	(	(	PUNCT
iajs-2724	152	72	𝜒	𝜒	PRON
iajs-2724	152	73	∘	∘	NUM
iajs-2724	152	74	𝛾	𝛾	NOUN
iajs-2724	152	75	)	)	PUNCT
iajs-2724	152	76	,	,	PUNCT
iajs-2724	152	77	𝛼	𝛼	X
iajs-2724	152	78	}	}	PUNCT
iajs-2724	152	79	≥	≥	NOUN
iajs-2724	152	80	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	152	81	+	+	ADJ
iajs-2724	152	82	(	(	PUNCT
iajs-2724	152	83	𝜒	𝜒	NOUN
iajs-2724	152	84	)	)	PUNCT
iajs-2724	152	85	,	,	PUNCT
iajs-2724	152	86	𝜇θ	𝜇θ	PROPN
iajs-2724	153	1	+	+	ADJ
iajs-2724	153	2	(	(	PUNCT
iajs-2724	153	3	𝛾	𝛾	NOUN
iajs-2724	153	4	)	)	PUNCT
iajs-2724	153	5	,	,	PUNCT
iajs-2724	153	6	𝛽	𝛽	NOUN
iajs-2724	153	7	}	}	PUNCT
iajs-2724	153	8	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	153	9	−(𝜒	−(𝜒	VERB
iajs-2724	153	10	∘	∘	PROPN
iajs-2724	153	11	𝛾),−𝜔	𝛾),−𝜔	NOUN
iajs-2724	153	12	}	}	PUNCT
iajs-2724	153	13	≤	≤	NUM
iajs-2724	153	14	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	ADJ
iajs-2724	153	15	−(𝜒	−(𝜒	ADJ
iajs-2724	153	16	)	)	PUNCT
iajs-2724	153	17	,	,	PUNCT
iajs-2724	153	18	𝜆θ	𝜆θ	INTJ
iajs-2724	153	19	−(𝛾),−𝜗	−(𝛾),−𝜗	VERB
iajs-2724	153	20	}	}	PUNCT
iajs-2724	153	21	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	154	1	+	+	PROPN
iajs-2724	154	2	(	(	PUNCT
iajs-2724	154	3	𝜒	𝜒	PRON
iajs-2724	154	4	∘	∘	NUM
iajs-2724	154	5	𝛾	𝛾	NOUN
iajs-2724	154	6	)	)	PUNCT
iajs-2724	154	7	,	,	PUNCT
iajs-2724	154	8	𝜔	𝜔	X
iajs-2724	154	9	}	}	PUNCT
iajs-2724	154	10	≥	≥	NOUN
iajs-2724	154	11	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	154	12	+	+	ADJ
iajs-2724	154	13	(	(	PUNCT
iajs-2724	154	14	𝜒	𝜒	NOUN
iajs-2724	154	15	)	)	PUNCT
iajs-2724	154	16	,	,	PUNCT
iajs-2724	154	17	𝜆θ	𝜆θ	ADP
iajs-2724	154	18	+	+	ADJ
iajs-2724	154	19	(	(	PUNCT
iajs-2724	154	20	𝛾	𝛾	NOUN
iajs-2724	154	21	)	)	PUNCT
iajs-2724	154	22	,	,	PUNCT
iajs-2724	154	23	𝜗	𝜗	NOUN
iajs-2724	154	24	}	}	PUNCT
iajs-2724	154	25	example(26).the	example(26).the	PRON
iajs-2724	154	26	following	follow	VERB
iajs-2724	154	27	table	table	NOUN
iajs-2724	154	28	illustrates	illustrate	VERB
iajs-2724	154	29	the	the	DET
iajs-2724	154	30	set	set	NOUN
iajs-2724	154	31	ˑℵ	ˑℵ	NOUN
iajs-2724	154	32	=	=	SYM
iajs-2724	154	33	{	{	PUNCT
iajs-2724	154	34	0,1,2	0,1,2	NOUN
iajs-2724	154	35	}	}	PUNCT
iajs-2724	154	36	with	with	ADP
iajs-2724	154	37	binary	binary	ADJ
iajs-2724	154	38	operations	operation	NOUN
iajs-2724	154	39	∗	∗	NOUN
iajs-2724	154	40	and	and	CCONJ
iajs-2724	154	41	∘	∘	PROPN
iajs-2724	154	42	then(ℵ,∗,∘	then(ℵ,∗,∘	PROPN
iajs-2724	154	43	,	,	PUNCT
iajs-2724	154	44	0)ˑis	0)ˑis	NUM
iajs-2724	154	45	a	a	DET
iajs-2724	154	46	ku	ku	PROPN
iajs-2724	154	47	-	-	PUNCT
iajs-2724	154	48	semigroup	semigroup	PROPN
iajs-2724	154	49	.	.	PUNCT
iajs-2724	155	1	define	define	VERB
iajs-2724	155	2	θ	θ	PROPN
iajs-2724	155	3	=	=	SYM
iajs-2724	155	4	〈	〈	PROPN
iajs-2724	155	5	𝑀	𝑀	PROPN
iajs-2724	155	6	,	,	PUNCT
iajs-2724	155	7	𝐿	𝐿	PROPN
iajs-2724	155	8	〉	〉	NOUN
iajs-2724	155	9	as	as	SCONJ
iajs-2724	155	10	follows	follow	VERB
iajs-2724	155	11	:	:	PUNCT
iajs-2724	155	12	𝑀(𝜒	𝑀(𝜒	NUM
iajs-2724	155	13	)	)	PUNCT
iajs-2724	155	14	=	=	PRON
iajs-2724	155	15	{	{	PUNCT
iajs-2724	156	1	[	[	X
iajs-2724	156	2	−0.8	−0.8	ADJ
iajs-2724	156	3	,	,	PUNCT
iajs-2724	156	4	−0.7	−0.7	PROPN
iajs-2724	156	5	]	]	PUNCT
iajs-2724	156	6	,	,	PUNCT
iajs-2724	157	1	[	[	X
iajs-2724	157	2	0.6	0.6	NUM
iajs-2724	157	3	,	,	PUNCT
iajs-2724	157	4	0.8	0.8	NUM
iajs-2724	157	5	]	]	PUNCT
iajs-2724	157	6	𝑖𝑓	𝑖𝑓	ADP
iajs-2724	157	7	𝜒	𝜒	X
iajs-2724	157	8	=	=	SYM
iajs-2724	157	9	0	0	PUNCT
iajs-2724	158	1	[	[	X
iajs-2724	158	2	−0.6	−0.6	PROPN
iajs-2724	158	3	,	,	PUNCT
iajs-2724	158	4	−0.5	−0.5	PROPN
iajs-2724	158	5	]	]	PUNCT
iajs-2724	158	6	,	,	PUNCT
iajs-2724	158	7	[	[	X
iajs-2724	158	8	0.4	0.4	NUM
iajs-2724	158	9	,	,	PUNCT
iajs-2724	158	10	0.6	0.6	NUM
iajs-2724	158	11	]	]	PUNCT
iajs-2724	158	12	𝑖𝑓	𝑖𝑓	ADP
iajs-2724	158	13	𝜒	𝜒	X
iajs-2724	158	14	=	=	SYM
iajs-2724	158	15	1	1	NUM
iajs-2724	158	16	[	[	X
iajs-2724	158	17	−0.4	−0.4	X
iajs-2724	158	18	,	,	PUNCT
iajs-2724	158	19	−0.3	−0.3	PROPN
iajs-2724	158	20	]	]	PUNCT
iajs-2724	158	21	,	,	PUNCT
iajs-2724	158	22	[	[	X
iajs-2724	158	23	0.3	0.3	NUM
iajs-2724	158	24	,	,	PUNCT
iajs-2724	158	25	0.2	0.2	NUM
iajs-2724	158	26	]	]	PUNCT
iajs-2724	158	27	𝑖𝑓	𝑖𝑓	ADP
iajs-2724	158	28	𝜒	𝜒	X
iajs-2724	158	29	=	=	SYM
iajs-2724	158	30	2	2	NUM
iajs-2724	158	31	∗	∗	NOUN
iajs-2724	158	32	0	0	NUM
iajs-2724	158	33	1	1	NUM
iajs-2724	158	34	2	2	NUM
iajs-2724	158	35	0	0	NUM
iajs-2724	158	36	0	0	NUM
iajs-2724	158	37	1	1	NUM
iajs-2724	158	38	2	2	NUM
iajs-2724	158	39	1	1	NUM
iajs-2724	158	40	0	0	NUM
iajs-2724	158	41	0	0	NUM
iajs-2724	158	42	1	1	NUM
iajs-2724	158	43	2	2	NUM
iajs-2724	158	44	0	0	NUM
iajs-2724	158	45	1	1	NUM
iajs-2724	158	46	0	0	NUM
iajs-2724	158	47	∘	∘	NUM
iajs-2724	158	48	0	0	NUM
iajs-2724	158	49	1	1	NUM
iajs-2724	158	50	2	2	NUM
iajs-2724	158	51	0	0	NUM
iajs-2724	158	52	0	0	NUM
iajs-2724	158	53	0	0	NUM
iajs-2724	158	54	0	0	NUM
iajs-2724	158	55	1	1	NUM
iajs-2724	158	56	0	0	NUM
iajs-2724	158	57	1	1	NUM
iajs-2724	158	58	0	0	NUM
iajs-2724	158	59	2	2	NUM
iajs-2724	158	60	0	0	NUM
iajs-2724	158	61	0	0	NUM
iajs-2724	158	62	2	2	NUM
iajs-2724	158	63	ibn	ibn	PROPN
iajs-2724	158	64	al	al	PROPN
iajs-2724	158	65	-	-	PUNCT
iajs-2724	158	66	haitham	haitham	PROPN
iajs-2724	158	67	jour	jour	X
iajs-2724	158	68	.	.	PROPN
iajs-2724	158	69	for	for	ADP
iajs-2724	158	70	pure	pure	ADJ
iajs-2724	158	71	&	&	CCONJ
iajs-2724	158	72	appl	appl	PROPN
iajs-2724	158	73	.	.	PUNCT
iajs-2724	159	1	sci	sci	PROPN
iajs-2724	159	2	.	.	PROPN
iajs-2724	160	1	53	53	NUM
iajs-2724	160	2	(	(	PUNCT
iajs-2724	160	3	2)2022	2)2022	VERB
iajs-2724	160	4	55	55	NUM
iajs-2724	160	5	𝐿(𝜒	𝐿(𝜒	NUM
iajs-2724	160	6	)	)	PUNCT
iajs-2724	160	7	{	{	PUNCT
iajs-2724	160	8	−0.6	−0.6	PROPN
iajs-2724	160	9	,	,	PUNCT
iajs-2724	160	10	0.8	0.8	NUM
iajs-2724	160	11	𝑖𝑓	𝑖𝑓	NUM
iajs-2724	160	12	𝜒	𝜒	X
iajs-2724	160	13	=	=	SYM
iajs-2724	160	14	0	0	X
iajs-2724	160	15	−0.5	−0.5	NOUN
iajs-2724	160	16	,	,	PUNCT
iajs-2724	160	17	0.6	0.6	NUM
iajs-2724	160	18	𝑖𝑓	𝑖𝑓	NUM
iajs-2724	160	19	𝜒	𝜒	X
iajs-2724	160	20	=	=	SYM
iajs-2724	160	21	1	1	NUM
iajs-2724	160	22	−0.3	−0.3	PROPN
iajs-2724	160	23	,	,	PUNCT
iajs-2724	160	24	0.3	0.3	NUM
iajs-2724	160	25	𝑖𝑓	𝑖𝑓	NOUN
iajs-2724	160	26	𝜒	𝜒	NOUN
iajs-2724	160	27	=	=	SYM
iajs-2724	160	28	2	2	NUM
iajs-2724	160	29	we	we	PRON
iajs-2724	160	30	can	can	AUX
iajs-2724	160	31	show	show	VERB
iajs-2724	160	32	that	that	SCONJ
iajs-2724	160	33	θ	θ	PROPN
iajs-2724	160	34	=	=	SYM
iajs-2724	160	35	〈	〈	PROPN
iajs-2724	160	36	𝑀	𝑀	PROPN
iajs-2724	160	37	,	,	PUNCT
iajs-2724	160	38	𝐿	𝐿	PROPN
iajs-2724	160	39	〉	〉	NOUN
iajs-2724	160	40	is	be	AUX
iajs-2724	160	41	a	a	DET
iajs-2724	160	42	(	(	PUNCT
iajs-2724	160	43	cbf	cbf	PROPN
iajs-2724	160	44	)	)	PUNCT
iajs-2724	160	45	ideal	ideal	NOUN
iajs-2724	160	46	with	with	ADP
iajs-2724	160	47	thresholds	threshold	NOUN
iajs-2724	160	48	(	(	PUNCT
iajs-2724	160	49	[	[	X
iajs-2724	160	50	0.1	0.1	NUM
iajs-2724	160	51	,	,	PUNCT
iajs-2724	160	52	0.1	0.1	NUM
iajs-2724	160	53	]	]	PUNCT
iajs-2724	160	54	,	,	PUNCT
iajs-2724	161	1	[	[	X
iajs-2724	161	2	0.3,0.2	0.3,0.2	NOUN
iajs-2724	161	3	]	]	PUNCT
iajs-2724	161	4	)	)	PUNCT
iajs-2724	161	5	and	and	CCONJ
iajs-2724	161	6	(	(	PUNCT
iajs-2724	161	7	0.4	0.4	NUM
iajs-2724	161	8	,	,	PUNCT
iajs-2724	161	9	0.2	0.2	NUM
iajs-2724	161	10	)	)	PUNCT
iajs-2724	161	11	of	of	ADP
iajs-2724	161	12	ℵ	ℵ	NOUN
iajs-2724	161	13	definition(27	definition(27	NOUN
iajs-2724	161	14	)	)	PUNCT
iajs-2724	161	15	.	.	PUNCT
iajs-2724	162	1	a	a	DET
iajs-2724	162	2	(	(	PUNCT
iajs-2724	162	3	cbf)set	cbf)set	NOUN
iajs-2724	162	4	θ	θ	X
iajs-2724	162	5	=	=	SYM
iajs-2724	162	6	〈	〈	PROPN
iajs-2724	162	7	𝑀	𝑀	PROPN
iajs-2724	162	8	,	,	PUNCT
iajs-2724	162	9	𝐿	𝐿	PROPN
iajs-2724	162	10	〉	〉	NOUN
iajs-2724	162	11	is	be	AUX
iajs-2724	162	12	named	name	VERB
iajs-2724	162	13	a	a	DET
iajs-2724	162	14	(	(	PUNCT
iajs-2724	162	15	cbf)k	cbf)k	NOUN
iajs-2724	162	16	-	-	PUNCT
iajs-2724	162	17	ideal	ideal	NOUN
iajs-2724	162	18	of	of	ADP
iajs-2724	162	19	ku	ku	PROPN
iajs-2724	162	20	-	-	PUNCT
iajs-2724	162	21	semigroup	semigroup	PROPN
iajs-2724	162	22	with	with	ADP
iajs-2724	162	23	thresholds	threshold	NOUN
iajs-2724	162	24	(	(	PUNCT
iajs-2724	162	25	𝛼	𝛼	X
iajs-2724	162	26	,	,	PUNCT
iajs-2724	162	27	𝛽	𝛽	NOUN
iajs-2724	162	28	)	)	PUNCT
iajs-2724	162	29	,	,	PUNCT
iajs-2724	162	30	(	(	PUNCT
iajs-2724	162	31	𝜔	𝜔	NOUN
iajs-2724	162	32	,	,	PUNCT
iajs-2724	162	33	𝜗	𝜗	NOUN
iajs-2724	162	34	)	)	PUNCT
iajs-2724	162	35	if	if	SCONJ
iajs-2724	162	36	∀	∀	X
iajs-2724	162	37	𝜒	𝜒	X
iajs-2724	162	38	,	,	PUNCT
iajs-2724	162	39	𝛾	𝛾	NOUN
iajs-2724	162	40	,	,	PUNCT
iajs-2724	162	41	𝜏	𝜏	PROPN
iajs-2724	162	42	∈	∈	PROPN
iajs-2724	162	43	ℵ	ℵ	NOUN
iajs-2724	162	44	(	(	PUNCT
iajs-2724	162	45	cbҡ1)𝑟𝑚𝑖𝑛{	cbҡ1)𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	162	46	�	�	PROPN
iajs-2724	162	47	̃	̃	PROPN
iajs-2724	162	48	�	�	PROPN
iajs-2724	162	49	θ	θ	ADJ
iajs-2724	162	50	−(0),−𝛼	−(0),−𝛼	ADP
iajs-2724	162	51	}	}	PUNCT
iajs-2724	162	52	≤	≤	NUM
iajs-2724	162	53	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NOUN
iajs-2724	162	54	−(𝜒),−𝛽	−(𝜒),−𝛽	ADP
iajs-2724	162	55	}	}	PUNCT
iajs-2724	162	56	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	X
iajs-2724	162	57	+	+	ADJ
iajs-2724	162	58	(	(	PUNCT
iajs-2724	162	59	0	0	NUM
iajs-2724	162	60	)	)	PUNCT
iajs-2724	162	61	,	,	PUNCT
iajs-2724	162	62	𝛼	𝛼	PROPN
iajs-2724	162	63	}	}	PUNCT
iajs-2724	162	64	≥	≥	NUM
iajs-2724	162	65	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	162	66	�	�	PROPN
iajs-2724	162	67	̃	̃	PROPN
iajs-2724	162	68	�	�	PROPN
iajs-2724	162	69	θ	θ	NOUN
iajs-2724	162	70	+	+	NOUN
iajs-2724	162	71	(	(	PUNCT
iajs-2724	162	72	𝜒	𝜒	NOUN
iajs-2724	162	73	)	)	PUNCT
iajs-2724	162	74	,	,	PUNCT
iajs-2724	162	75	𝛽	𝛽	NOUN
iajs-2724	162	76	}	}	PUNCT
iajs-2724	162	77	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	ADV
iajs-2724	162	78	−(0),−𝜔	−(0),−𝜔	NUM
iajs-2724	162	79	}	}	PUNCT
iajs-2724	162	80	≤	≤	NUM
iajs-2724	162	81	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	ADV
iajs-2724	162	82	−(𝜒),−𝜗	−(𝜒),−𝜗	PUNCT
iajs-2724	162	83	}	}	PUNCT
iajs-2724	162	84	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	163	1	+	+	PROPN
iajs-2724	163	2	(	(	PUNCT
iajs-2724	163	3	0	0	NUM
iajs-2724	163	4	)	)	PUNCT
iajs-2724	163	5	,	,	PUNCT
iajs-2724	163	6	𝜔	𝜔	X
iajs-2724	163	7	}	}	PUNCT
iajs-2724	163	8	≥	≥	NOUN
iajs-2724	163	9	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	163	10	+	+	ADJ
iajs-2724	163	11	(	(	PUNCT
iajs-2724	163	12	𝜒	𝜒	NOUN
iajs-2724	163	13	)	)	PUNCT
iajs-2724	163	14	,	,	PUNCT
iajs-2724	163	15	𝜗	𝜗	NOUN
iajs-2724	163	16	}	}	PUNCT
iajs-2724	163	17	(	(	PUNCT
iajs-2724	163	18	cbҡ2	cbҡ2	PROPN
iajs-2724	163	19	)	)	PUNCT
iajs-2724	163	20	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	163	21	�	�	PROPN
iajs-2724	163	22	̃	̃	PROPN
iajs-2724	163	23	�	�	NOUN
iajs-2724	163	24	θ	θ	NOUN
iajs-2724	163	25	−(𝜒	−(𝜒	ADJ
iajs-2724	163	26	∗	∗	NOUN
iajs-2724	163	27	𝜏	𝜏	NOUN
iajs-2724	163	28	)	)	PUNCT
iajs-2724	163	29	,	,	PUNCT
iajs-2724	163	30	−𝛼	−𝛼	ADJ
iajs-2724	163	31	}	}	PUNCT
iajs-2724	163	32	≤	≤	NUM
iajs-2724	163	33	𝑟𝑚𝑎𝑥{	𝑟𝑚𝑎𝑥{	NOUN
iajs-2724	163	34	�	�	NOUN
iajs-2724	163	35	̃	̃	PROPN
iajs-2724	163	36	�	�	NOUN
iajs-2724	163	37	θ	θ	NOUN
iajs-2724	163	38	−(𝜒	−(𝜒	ADJ
iajs-2724	163	39	∗	∗	NOUN
iajs-2724	163	40	(	(	PUNCT
iajs-2724	163	41	𝛾	𝛾	NOUN
iajs-2724	163	42	∗	∗	NOUN
iajs-2724	163	43	𝜏	𝜏	NOUN
iajs-2724	163	44	)	)	PUNCT
iajs-2724	163	45	)	)	PUNCT
iajs-2724	163	46	,	,	PUNCT
iajs-2724	163	47	𝜇θ	𝜇θ	ADV
iajs-2724	163	48	−(𝛾),−𝛽	−(𝛾),−𝛽	PUNCT
iajs-2724	163	49	}	}	PUNCT
iajs-2724	163	50	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	X
iajs-2724	163	51	+	+	NOUN
iajs-2724	163	52	(	(	PUNCT
iajs-2724	163	53	𝜒	𝜒	X
iajs-2724	163	54	∗	∗	NOUN
iajs-2724	163	55	𝜏	𝜏	NOUN
iajs-2724	163	56	)	)	PUNCT
iajs-2724	163	57	,	,	PUNCT
iajs-2724	163	58	𝛼	𝛼	PROPN
iajs-2724	163	59	}	}	PUNCT
iajs-2724	163	60	≥	≥	NOUN
iajs-2724	163	61	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	163	62	+	+	ADJ
iajs-2724	163	63	(	(	PUNCT
iajs-2724	163	64	𝜒	𝜒	X
iajs-2724	163	65	∗	∗	NOUN
iajs-2724	163	66	(	(	PUNCT
iajs-2724	163	67	𝛾	𝛾	NOUN
iajs-2724	163	68	∗	∗	NOUN
iajs-2724	163	69	𝜏	𝜏	NOUN
iajs-2724	163	70	)	)	PUNCT
iajs-2724	163	71	,	,	PUNCT
iajs-2724	163	72	𝜇θ	𝜇θ	PROPN
iajs-2724	163	73	+	+	ADJ
iajs-2724	163	74	(	(	PUNCT
iajs-2724	163	75	𝛾	𝛾	NOUN
iajs-2724	163	76	)	)	PUNCT
iajs-2724	163	77	,	,	PUNCT
iajs-2724	163	78	𝛽	𝛽	AUX
iajs-2724	163	79	}	}	PUNCT
iajs-2724	163	80	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	163	81	−(𝜒	−(𝜒	VERB
iajs-2724	163	82	∗	∗	NOUN
iajs-2724	163	83	𝜏	𝜏	NOUN
iajs-2724	163	84	)	)	PUNCT
iajs-2724	163	85	,	,	PUNCT
iajs-2724	163	86	−𝜔	−𝜔	ADP
iajs-2724	163	87	}	}	PUNCT
iajs-2724	163	88	≤	≤	NUM
iajs-2724	163	89	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	163	90	−(𝜒	−(𝜒	ADJ
iajs-2724	163	91	∗	∗	NOUN
iajs-2724	163	92	(	(	PUNCT
iajs-2724	163	93	𝛾	𝛾	NOUN
iajs-2724	163	94	∗	∗	NOUN
iajs-2724	163	95	𝜏	𝜏	NOUN
iajs-2724	163	96	)	)	PUNCT
iajs-2724	163	97	)	)	PUNCT
iajs-2724	163	98	,	,	PUNCT
iajs-2724	163	99	𝜆θ	𝜆θ	INTJ
iajs-2724	163	100	−(𝛾),−𝜗	−(𝛾),−𝜗	VERB
iajs-2724	163	101	}	}	PUNCT
iajs-2724	163	102	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	164	1	+	+	PROPN
iajs-2724	164	2	(	(	PUNCT
iajs-2724	164	3	𝜒	𝜒	X
iajs-2724	164	4	∗	∗	NOUN
iajs-2724	164	5	𝜏	𝜏	NOUN
iajs-2724	164	6	)	)	PUNCT
iajs-2724	164	7	,	,	PUNCT
iajs-2724	164	8	𝜔	𝜔	X
iajs-2724	164	9	}	}	PUNCT
iajs-2724	164	10	≥	≥	NOUN
iajs-2724	164	11	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	164	12	+	+	PROPN
iajs-2724	164	13	(	(	PUNCT
iajs-2724	164	14	𝜒	𝜒	NOUN
iajs-2724	164	15	∗	∗	NOUN
iajs-2724	164	16	(	(	PUNCT
iajs-2724	164	17	𝛾	𝛾	NOUN
iajs-2724	164	18	∗	∗	NOUN
iajs-2724	164	19	𝜏	𝜏	NOUN
iajs-2724	164	20	)	)	PUNCT
iajs-2724	164	21	)	)	PUNCT
iajs-2724	164	22	,	,	PUNCT
iajs-2724	164	23	𝜆θ	𝜆θ	ADP
iajs-2724	164	24	+	+	ADJ
iajs-2724	164	25	(	(	PUNCT
iajs-2724	164	26	𝛾	𝛾	NOUN
iajs-2724	164	27	)	)	PUNCT
iajs-2724	164	28	,	,	PUNCT
iajs-2724	164	29	𝜗	𝜗	NOUN
iajs-2724	164	30	}	}	PUNCT
iajs-2724	164	31	(	(	PUNCT
iajs-2724	164	32	cbҡ3)𝑟𝑚𝑖𝑛{	cbҡ3)𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	164	33	�	�	PROPN
iajs-2724	164	34	̃	̃	PROPN
iajs-2724	164	35	�	�	NOUN
iajs-2724	164	36	θ	θ	NOUN
iajs-2724	164	37	−(𝜒	−(𝜒	ADJ
iajs-2724	164	38	∘	∘	NOUN
iajs-2724	164	39	𝛾),−𝛼	𝛾),−𝛼	PRON
iajs-2724	164	40	}	}	PUNCT
iajs-2724	164	41	≤	≤	NUM
iajs-2724	164	42	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NUM
iajs-2724	164	43	−(𝜒	−(𝜒	ADJ
iajs-2724	164	44	)	)	PUNCT
iajs-2724	164	45	,	,	PUNCT
iajs-2724	164	46	𝜇θ	𝜇θ	ADV
iajs-2724	164	47	−(𝛾),−𝛽	−(𝛾),−𝛽	PUNCT
iajs-2724	164	48	}	}	PUNCT
iajs-2724	164	49	𝑟𝑚𝑎𝑥{	𝑟𝑚𝑎𝑥{	ADJ
iajs-2724	164	50	�	�	NOUN
iajs-2724	164	51	̃	̃	NOUN
iajs-2724	164	52	�	�	NOUN
iajs-2724	164	53	θ	θ	NOUN
iajs-2724	164	54	+	+	PROPN
iajs-2724	164	55	(	(	PUNCT
iajs-2724	164	56	𝜒	𝜒	PRON
iajs-2724	164	57	∘	∘	NUM
iajs-2724	164	58	𝛾	𝛾	NOUN
iajs-2724	164	59	)	)	PUNCT
iajs-2724	164	60	,	,	PUNCT
iajs-2724	164	61	𝛼	𝛼	X
iajs-2724	164	62	}	}	PUNCT
iajs-2724	164	63	≥	≥	NOUN
iajs-2724	164	64	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	164	65	+	+	ADJ
iajs-2724	164	66	(	(	PUNCT
iajs-2724	164	67	𝜒	𝜒	NOUN
iajs-2724	164	68	)	)	PUNCT
iajs-2724	164	69	,	,	PUNCT
iajs-2724	164	70	𝜇θ	𝜇θ	PROPN
iajs-2724	165	1	+	+	ADJ
iajs-2724	165	2	(	(	PUNCT
iajs-2724	165	3	𝛾	𝛾	NOUN
iajs-2724	165	4	)	)	PUNCT
iajs-2724	165	5	,	,	PUNCT
iajs-2724	165	6	𝛽	𝛽	NOUN
iajs-2724	165	7	}	}	PUNCT
iajs-2724	165	8	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	165	9	−(𝜒	−(𝜒	VERB
iajs-2724	165	10	∘	∘	PROPN
iajs-2724	165	11	𝛾),−𝜔	𝛾),−𝜔	NOUN
iajs-2724	165	12	}	}	PUNCT
iajs-2724	165	13	≤	≤	NUM
iajs-2724	165	14	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	ADJ
iajs-2724	165	15	−(𝜒	−(𝜒	ADJ
iajs-2724	165	16	)	)	PUNCT
iajs-2724	165	17	,	,	PUNCT
iajs-2724	165	18	𝜆θ	𝜆θ	INTJ
iajs-2724	165	19	−(𝛾),−𝜗	−(𝛾),−𝜗	VERB
iajs-2724	165	20	}	}	PUNCT
iajs-2724	165	21	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	166	1	+	+	PROPN
iajs-2724	166	2	(	(	PUNCT
iajs-2724	166	3	𝜒	𝜒	PRON
iajs-2724	166	4	∘	∘	NUM
iajs-2724	166	5	𝛾	𝛾	NOUN
iajs-2724	166	6	)	)	PUNCT
iajs-2724	166	7	,	,	PUNCT
iajs-2724	166	8	𝜔	𝜔	X
iajs-2724	166	9	}	}	PUNCT
iajs-2724	166	10	≥	≥	NOUN
iajs-2724	166	11	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	166	12	+	+	ADJ
iajs-2724	166	13	(	(	PUNCT
iajs-2724	166	14	𝜒	𝜒	NOUN
iajs-2724	166	15	)	)	PUNCT
iajs-2724	166	16	,	,	PUNCT
iajs-2724	166	17	𝜆θ	𝜆θ	ADP
iajs-2724	166	18	+	+	ADJ
iajs-2724	166	19	(	(	PUNCT
iajs-2724	166	20	𝛾	𝛾	NOUN
iajs-2724	166	21	)	)	PUNCT
iajs-2724	166	22	,	,	PUNCT
iajs-2724	166	23	𝜗	𝜗	NOUN
iajs-2724	166	24	}	}	PUNCT
iajs-2724	166	25	lemma(28	lemma(28	NOUN
iajs-2724	166	26	)	)	PUNCT
iajs-2724	166	27	.	.	PUNCT
iajs-2724	167	1	every	every	DET
iajs-2724	167	2	(	(	PUNCT
iajs-2724	167	3	cbf	cbf	PROPN
iajs-2724	167	4	)	)	PUNCT
iajs-2724	167	5	k	k	NOUN
iajs-2724	167	6	-	-	PUNCT
iajs-2724	167	7	ideal	ideal	NOUN
iajs-2724	167	8	of	of	ADP
iajs-2724	167	9	ℵ	ℵ	NOUN
iajs-2724	167	10	is	be	AUX
iajs-2724	167	11	a	a	DET
iajs-2724	167	12	(	(	PUNCT
iajs-2724	167	13	cbf	cbf	PROPN
iajs-2724	167	14	)	)	PUNCT
iajs-2724	167	15	k	k	NOUN
iajs-2724	167	16	-	-	NOUN
iajs-2724	167	17	ideal	ideal	NOUN
iajs-2724	167	18	with	with	ADP
iajs-2724	167	19	thresholds	threshold	NOUN
iajs-2724	167	20	(	(	PUNCT
iajs-2724	167	21	𝛼	𝛼	X
iajs-2724	167	22	,	,	PUNCT
iajs-2724	167	23	𝛽	𝛽	NOUN
iajs-2724	167	24	)	)	PUNCT
iajs-2724	167	25	,	,	PUNCT
iajs-2724	167	26	(	(	PUNCT
iajs-2724	167	27	𝜔	𝜔	NOUN
iajs-2724	167	28	,	,	PUNCT
iajs-2724	167	29	𝜗	𝜗	NOUN
iajs-2724	167	30	)	)	PUNCT
iajs-2724	167	31	of	of	ADP
iajs-2724	167	32	ℵ	ℵ	PRON
iajs-2724	167	33	then	then	ADV
iajs-2724	167	34	let	let	VERB
iajs-2724	167	35	,	,	PUNCT
iajs-2724	167	36	ℵideal	ℵideal	NOUN
iajs-2724	167	37	of	of	ADP
iajs-2724	167	38	-	-	PUNCT
iajs-2724	167	39	k	k	PROPN
iajs-2724	167	40	)	)	PUNCT
iajs-2724	167	41	cbf	cbf	PROPN
iajs-2724	167	42	(	(	PUNCT
iajs-2724	167	43	is	be	AUX
iajs-2724	167	44	a	a	DET
iajs-2724	167	45	θ	θ	NOUN
iajs-2724	167	46	=	=	SYM
iajs-2724	167	47	〈	〈	PROPN
iajs-2724	167	48	𝑀	𝑀	PROPN
iajs-2724	167	49	,	,	PUNCT
iajs-2724	167	50	𝐿〉suppose	𝐿〉suppose	ADP
iajs-2724	167	51	that	that	DET
iajs-2724	167	52	proof	proof	NOUN
iajs-2724	167	53	:	:	PUNCT
iajs-2724	167	54	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	PROPN
iajs-2724	167	55	+	+	ADJ
iajs-2724	167	56	(	(	PUNCT
iajs-2724	167	57	0	0	NUM
iajs-2724	167	58	)	)	PUNCT
iajs-2724	167	59	,	,	PUNCT
iajs-2724	167	60	𝛼	𝛼	X
iajs-2724	167	61	}	}	PUNCT
iajs-2724	167	62	<	<	X
iajs-2724	167	63	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	X
iajs-2724	167	64	+	+	ADJ
iajs-2724	167	65	(	(	PUNCT
iajs-2724	167	66	𝜒	𝜒	NOUN
iajs-2724	167	67	)	)	PUNCT
iajs-2724	167	68	,	,	PUNCT
iajs-2724	167	69	𝛽},and	𝛽},and	NOUN
iajs-2724	167	70	𝛼	𝛼	PART
iajs-2724	167	71	<	<	X
iajs-2724	167	72	𝛽	𝛽	NOUN
iajs-2724	167	73	it	it	PRON
iajs-2724	167	74	follows	follow	VERB
iajs-2724	167	75	that	that	SCONJ
iajs-2724	167	76	𝜇θ	𝜇θ	ADP
iajs-2724	168	1	+	+	ADJ
iajs-2724	168	2	(	(	PUNCT
iajs-2724	168	3	0	0	NUM
iajs-2724	168	4	)	)	PUNCT
iajs-2724	169	1	<	<	X
iajs-2724	169	2	𝜇θ	𝜇θ	X
iajs-2724	169	3	+	+	ADJ
iajs-2724	169	4	(	(	PUNCT
iajs-2724	169	5	𝜒	𝜒	NOUN
iajs-2724	169	6	)	)	PUNCT
iajs-2724	169	7	.	.	PUNCT
iajs-2724	170	1	but	but	CCONJ
iajs-2724	170	2	that	that	PRON
iajs-2724	170	3	is	be	AUX
iajs-2724	170	4	a	a	DET
iajs-2724	170	5	contradiction	contradiction	NOUN
iajs-2724	170	6	,	,	PUNCT
iajs-2724	170	7	since	since	SCONJ
iajs-2724	170	8	θ	θ	PROPN
iajs-2724	170	9	is	be	AUX
iajs-2724	170	10	a(cbf	a(cbf	NOUN
iajs-2724	170	11	)	)	PUNCT
iajs-2724	171	1	k	k	X
iajs-2724	171	2	-	-	NOUN
iajs-2724	171	3	ideal	ideal	NOUN
iajs-2724	171	4	of	of	ADP
iajs-2724	171	5	ℵ	ℵ	NOUN
iajs-2724	171	6	,	,	PUNCT
iajs-2724	171	7	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	PROPN
iajs-2724	171	8	+	+	ADJ
iajs-2724	171	9	(	(	PUNCT
iajs-2724	171	10	0	0	NUM
iajs-2724	171	11	)	)	PUNCT
iajs-2724	171	12	,	,	PUNCT
iajs-2724	171	13	𝛼	𝛼	X
iajs-2724	171	14	}	}	PUNCT
iajs-2724	171	15	≥	≥	NOUN
iajs-2724	171	16	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	171	17	+	+	ADJ
iajs-2724	171	18	(	(	PUNCT
iajs-2724	171	19	𝜒	𝜒	NOUN
iajs-2724	171	20	)	)	PUNCT
iajs-2724	171	21	,	,	PUNCT
iajs-2724	171	22	𝛽	𝛽	NOUN
iajs-2724	171	23	}	}	PUNCT
iajs-2724	171	24	,	,	PUNCT
iajs-2724	171	25	also	also	ADV
iajs-2724	171	26	let	let	VERB
iajs-2724	171	27	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	ADJ
iajs-2724	171	28	−(0),−𝜔	−(0),−𝜔	NUM
iajs-2724	171	29	}	}	PUNCT
iajs-2724	171	30	>	>	SYM
iajs-2724	171	31	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PROPN
iajs-2724	171	32	−(𝜒	−(𝜒	PROPN
iajs-2724	171	33	)	)	PUNCT
iajs-2724	171	34	,	,	PUNCT
iajs-2724	171	35	−𝜗	−𝜗	ADV
iajs-2724	171	36	}	}	PUNCT
iajs-2724	171	37	,	,	PUNCT
iajs-2724	171	38	and	and	CCONJ
iajs-2724	171	39	𝜔	𝜔	ADP
iajs-2724	171	40	<	<	X
iajs-2724	171	41	𝜗	𝜗	NOUN
iajs-2724	171	42	,	,	PUNCT
iajs-2724	171	43	it	it	PRON
iajs-2724	171	44	follows	follow	VERB
iajs-2724	171	45	that	that	SCONJ
iajs-2724	171	46	𝜆θ	𝜆θ	ADP
iajs-2724	171	47	−(0	−(0	NOUN
iajs-2724	171	48	)	)	PUNCT
iajs-2724	171	49	>	>	X
iajs-2724	172	1	𝜆θ	𝜆θ	ADP
iajs-2724	172	2	−(𝜒	−(𝜒	PROPN
iajs-2724	172	3	)	)	PUNCT
iajs-2724	172	4	;	;	PUNCT
iajs-2724	172	5	this	this	PRON
iajs-2724	172	6	is	be	AUX
iajs-2724	172	7	a	a	DET
iajs-2724	172	8	contradiction	contradiction	NOUN
iajs-2724	172	9	since	since	SCONJ
iajs-2724	172	10	θ	θ	PROPN
iajs-2724	172	11	is	be	AUX
iajs-2724	172	12	a(cbf	a(cbf	NOUN
iajs-2724	172	13	)	)	PUNCT
iajs-2724	173	1	k	k	X
iajs-2724	173	2	-	-	NOUN
iajs-2724	173	3	ideal	ideal	NOUN
iajs-2724	173	4	of	of	ADP
iajs-2724	173	5	ℵ	ℵ	NOUN
iajs-2724	173	6	.	.	PUNCT
iajs-2724	174	1	this	this	PRON
iajs-2724	174	2	means	mean	VERB
iajs-2724	174	3	that	that	SCONJ
iajs-2724	174	4	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	ADV
iajs-2724	174	5	−(0),−𝜔	−(0),−𝜔	NUM
iajs-2724	174	6	}	}	PUNCT
iajs-2724	174	7	≤	≤	NUM
iajs-2724	174	8	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	ADV
iajs-2724	174	9	−(𝜒),−𝜗	−(𝜒),−𝜗	NOUN
iajs-2724	174	10	}	}	PUNCT
iajs-2724	174	11	,	,	PUNCT
iajs-2724	174	12	in	in	ADP
iajs-2724	174	13	the	the	DET
iajs-2724	174	14	same	same	ADJ
iajs-2724	174	15	way	way	NOUN
iajs-2724	174	16	,	,	PUNCT
iajs-2724	174	17	we	we	PRON
iajs-2724	174	18	can	can	AUX
iajs-2724	174	19	prove	prove	VERB
iajs-2724	174	20	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NOUN
iajs-2724	174	21	−(0	−(0	NOUN
iajs-2724	174	22	)	)	PUNCT
iajs-2724	174	23	,	,	PUNCT
iajs-2724	174	24	−𝛼	−𝛼	ADJ
iajs-2724	174	25	}	}	PUNCT
iajs-2724	174	26	≤	≤	NUM
iajs-2724	174	27	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NUM
iajs-2724	174	28	−(𝜒	−(𝜒	ADJ
iajs-2724	174	29	)	)	PUNCT
iajs-2724	174	30	,	,	PUNCT
iajs-2724	174	31	−𝛽	−𝛽	PROPN
iajs-2724	174	32	}	}	PUNCT
iajs-2724	174	33	,	,	PUNCT
iajs-2724	174	34	and	and	CCONJ
iajs-2724	174	35	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PROPN
iajs-2724	174	36	+	+	ADJ
iajs-2724	174	37	(	(	PUNCT
iajs-2724	174	38	0),𝜔	0),𝜔	ADJ
iajs-2724	174	39	}	}	PUNCT
iajs-2724	174	40	≥	≥	NOUN
iajs-2724	174	41	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	174	42	+	+	ADJ
iajs-2724	174	43	(	(	PUNCT
iajs-2724	174	44	𝜒	𝜒	NOUN
iajs-2724	174	45	)	)	PUNCT
iajs-2724	174	46	,	,	PUNCT
iajs-2724	174	47	𝜗	𝜗	NOUN
iajs-2724	174	48	}	}	PUNCT
iajs-2724	174	49	again	again	ADV
iajs-2724	174	50	,	,	PUNCT
iajs-2724	174	51	assume	assume	VERB
iajs-2724	174	52	that	that	SCONJ
iajs-2724	174	53	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	PRON
iajs-2724	174	54	+	+	NOUN
iajs-2724	174	55	(	(	PUNCT
iajs-2724	174	56	𝜒	𝜒	X
iajs-2724	174	57	∗	∗	NOUN
iajs-2724	174	58	𝜏	𝜏	NOUN
iajs-2724	174	59	)	)	PUNCT
iajs-2724	174	60	,	,	PUNCT
iajs-2724	174	61	𝛼	𝛼	X
iajs-2724	174	62	}	}	PUNCT
iajs-2724	174	63	<	<	X
iajs-2724	174	64	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	174	65	�	�	PROPN
iajs-2724	174	66	̃	̃	PROPN
iajs-2724	174	67	�	�	PROPN
iajs-2724	174	68	θ	θ	NOUN
iajs-2724	174	69	+	+	PROPN
iajs-2724	174	70	(	(	PUNCT
iajs-2724	174	71	𝜒	𝜒	NOUN
iajs-2724	174	72	∗	∗	NOUN
iajs-2724	174	73	(	(	PUNCT
iajs-2724	174	74	𝛾	𝛾	NOUN
iajs-2724	174	75	∗	∗	NOUN
iajs-2724	174	76	𝜏	𝜏	NOUN
iajs-2724	174	77	)	)	PUNCT
iajs-2724	174	78	,	,	PUNCT
iajs-2724	174	79	𝜇θ	𝜇θ	PROPN
iajs-2724	174	80	+	+	ADJ
iajs-2724	174	81	(	(	PUNCT
iajs-2724	174	82	𝛾	𝛾	NOUN
iajs-2724	174	83	)	)	PUNCT
iajs-2724	174	84	,	,	PUNCT
iajs-2724	174	85	𝛽	𝛽	NOUN
iajs-2724	174	86	}	}	PUNCT
iajs-2724	174	87	,	,	PUNCT
iajs-2724	174	88	and	and	CCONJ
iajs-2724	174	89	𝛼	𝛼	ADJ
iajs-2724	174	90	<	<	X
iajs-2724	174	91	𝛽	𝛽	NOUN
iajs-2724	174	92	it	it	PRON
iajs-2724	174	93	follows	follow	VERB
iajs-2724	174	94	that	that	SCONJ
iajs-2724	174	95	𝜇θ	𝜇θ	PROPN
iajs-2724	174	96	+	+	ADJ
iajs-2724	174	97	(	(	PUNCT
iajs-2724	174	98	𝜒	𝜒	X
iajs-2724	174	99	∗	∗	NOUN
iajs-2724	174	100	𝜏	𝜏	NOUN
iajs-2724	174	101	)	)	PUNCT
iajs-2724	174	102	<	<	X
iajs-2724	174	103	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	174	104	�	�	PROPN
iajs-2724	174	105	̃	̃	PROPN
iajs-2724	174	106	�	�	PROPN
iajs-2724	174	107	θ	θ	NOUN
iajs-2724	174	108	+	+	PROPN
iajs-2724	174	109	(	(	PUNCT
iajs-2724	174	110	𝜒	𝜒	NOUN
iajs-2724	174	111	∗	∗	NOUN
iajs-2724	174	112	(	(	PUNCT
iajs-2724	174	113	𝛾	𝛾	NOUN
iajs-2724	174	114	∗	∗	NOUN
iajs-2724	174	115	𝜏	𝜏	NOUN
iajs-2724	174	116	)	)	PUNCT
iajs-2724	174	117	,	,	PUNCT
iajs-2724	174	118	𝜇θ	𝜇θ	PROPN
iajs-2724	174	119	+	+	ADJ
iajs-2724	174	120	(	(	PUNCT
iajs-2724	174	121	𝛾	𝛾	NOUN
iajs-2724	174	122	)	)	PUNCT
iajs-2724	174	123	}	}	PUNCT
iajs-2724	174	124	,	,	PUNCT
iajs-2724	174	125	which	which	PRON
iajs-2724	174	126	is	be	AUX
iajs-2724	174	127	a	a	DET
iajs-2724	174	128	contradiction	contradiction	NOUN
iajs-2724	174	129	,	,	PUNCT
iajs-2724	174	130	so	so	ADV
iajs-2724	174	131	ibn	ibn	PROPN
iajs-2724	174	132	al	al	PROPN
iajs-2724	174	133	-	-	PUNCT
iajs-2724	174	134	haitham	haitham	PROPN
iajs-2724	174	135	jour	jour	X
iajs-2724	174	136	.	.	PROPN
iajs-2724	175	1	for	for	ADP
iajs-2724	175	2	pure	pure	ADJ
iajs-2724	175	3	&	&	CCONJ
iajs-2724	175	4	appl	appl	PROPN
iajs-2724	175	5	.	.	PUNCT
iajs-2724	176	1	sci	sci	PROPN
iajs-2724	176	2	.	.	PROPN
iajs-2724	177	1	53	53	NUM
iajs-2724	177	2	(	(	PUNCT
iajs-2724	177	3	2)2022	2)2022	VERB
iajs-2724	177	4	56	56	NUM
iajs-2724	177	5	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NOUN
iajs-2724	177	6	+	+	NOUN
iajs-2724	177	7	(	(	PUNCT
iajs-2724	177	8	𝜒	𝜒	X
iajs-2724	177	9	∗	∗	NOUN
iajs-2724	177	10	𝜏	𝜏	NOUN
iajs-2724	177	11	)	)	PUNCT
iajs-2724	177	12	,	,	PUNCT
iajs-2724	177	13	𝛼	𝛼	PROPN
iajs-2724	177	14	}	}	PUNCT
iajs-2724	177	15	≥	≥	NUM
iajs-2724	177	16	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	177	17	�	�	PROPN
iajs-2724	177	18	̃	̃	PROPN
iajs-2724	177	19	�	�	PROPN
iajs-2724	177	20	θ	θ	NOUN
iajs-2724	177	21	+	+	PROPN
iajs-2724	177	22	(	(	PUNCT
iajs-2724	177	23	𝜒	𝜒	NOUN
iajs-2724	177	24	∗	∗	NOUN
iajs-2724	177	25	(	(	PUNCT
iajs-2724	177	26	𝛾	𝛾	NOUN
iajs-2724	177	27	∗	∗	NOUN
iajs-2724	177	28	𝜏	𝜏	NOUN
iajs-2724	177	29	)	)	PUNCT
iajs-2724	177	30	,	,	PUNCT
iajs-2724	177	31	𝜇θ	𝜇θ	PROPN
iajs-2724	177	32	+	+	ADJ
iajs-2724	177	33	(	(	PUNCT
iajs-2724	177	34	𝛾	𝛾	NOUN
iajs-2724	177	35	)	)	PUNCT
iajs-2724	177	36	,	,	PUNCT
iajs-2724	177	37	𝛽	𝛽	NOUN
iajs-2724	177	38	}	}	PUNCT
iajs-2724	177	39	,	,	PUNCT
iajs-2724	177	40	also	also	ADV
iajs-2724	177	41	let	let	VERB
iajs-2724	177	42	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	177	43	−(𝜒	−(𝜒	VERB
iajs-2724	177	44	∗	∗	NOUN
iajs-2724	177	45	𝜏	𝜏	NOUN
iajs-2724	177	46	)	)	PUNCT
iajs-2724	177	47	,	,	PUNCT
iajs-2724	177	48	−𝜔	−𝜔	ADP
iajs-2724	177	49	}	}	PUNCT
iajs-2724	177	50	>	>	PUNCT
iajs-2724	177	51	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	177	52	−(𝜒	−(𝜒	ADJ
iajs-2724	177	53	∗	∗	NOUN
iajs-2724	177	54	(	(	PUNCT
iajs-2724	177	55	𝛾	𝛾	NOUN
iajs-2724	177	56	∗	∗	NOUN
iajs-2724	177	57	𝜏	𝜏	NOUN
iajs-2724	177	58	)	)	PUNCT
iajs-2724	177	59	)	)	PUNCT
iajs-2724	177	60	,	,	PUNCT
iajs-2724	177	61	𝜆θ	𝜆θ	INTJ
iajs-2724	177	62	−(𝛾),−𝜗	−(𝛾),−𝜗	VERB
iajs-2724	177	63	}	}	PUNCT
iajs-2724	177	64	,	,	PUNCT
iajs-2724	177	65	and	and	CCONJ
iajs-2724	177	66	𝜔	𝜔	ADP
iajs-2724	177	67	<	<	X
iajs-2724	177	68	𝜗	𝜗	NOUN
iajs-2724	177	69	,	,	PUNCT
iajs-2724	177	70	so	so	ADV
iajs-2724	177	71	𝜆θ	𝜆θ	ADP
iajs-2724	178	1	−(𝜒	−(𝜒	ADJ
iajs-2724	178	2	∗	∗	NOUN
iajs-2724	178	3	𝜏	𝜏	NOUN
iajs-2724	178	4	)	)	PUNCT
iajs-2724	178	5	>	>	PUNCT
iajs-2724	178	6	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PROPN
iajs-2724	178	7	−(𝜒	−(𝜒	ADJ
iajs-2724	178	8	∗	∗	NOUN
iajs-2724	178	9	(	(	PUNCT
iajs-2724	178	10	𝛾	𝛾	NOUN
iajs-2724	178	11	∗	∗	NOUN
iajs-2724	178	12	𝜏	𝜏	NOUN
iajs-2724	178	13	)	)	PUNCT
iajs-2724	178	14	)	)	PUNCT
iajs-2724	178	15	,	,	PUNCT
iajs-2724	178	16	𝜆θ	𝜆θ	INTJ
iajs-2724	178	17	−(𝛾	−(𝛾	PROPN
iajs-2724	178	18	)	)	PUNCT
iajs-2724	178	19	}	}	PUNCT
iajs-2724	178	20	,	,	PUNCT
iajs-2724	178	21	which	which	PRON
iajs-2724	178	22	is	be	AUX
iajs-2724	178	23	a	a	DET
iajs-2724	178	24	contradiction	contradiction	NOUN
iajs-2724	178	25	.	.	PUNCT
iajs-2724	179	1	that	that	PRON
iajs-2724	179	2	is	be	AUX
iajs-2724	179	3	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	ADV
iajs-2724	179	4	−(𝜒	−(𝜒	VERB
iajs-2724	179	5	∗	∗	NOUN
iajs-2724	179	6	𝜏	𝜏	NOUN
iajs-2724	179	7	)	)	PUNCT
iajs-2724	179	8	,	,	PUNCT
iajs-2724	179	9	−𝜔	−𝜔	ADP
iajs-2724	179	10	}	}	PUNCT
iajs-2724	179	11	≤	≤	NUM
iajs-2724	179	12	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	179	13	−(𝜒	−(𝜒	ADJ
iajs-2724	179	14	∗	∗	NOUN
iajs-2724	179	15	(	(	PUNCT
iajs-2724	179	16	𝛾	𝛾	NOUN
iajs-2724	179	17	∗	∗	NOUN
iajs-2724	179	18	𝜏	𝜏	NOUN
iajs-2724	179	19	)	)	PUNCT
iajs-2724	179	20	)	)	PUNCT
iajs-2724	179	21	,	,	PUNCT
iajs-2724	179	22	𝜆θ	𝜆θ	INTJ
iajs-2724	179	23	−(𝛾),−𝜗	−(𝛾),−𝜗	VERB
iajs-2724	179	24	}	}	PUNCT
iajs-2724	179	25	in	in	ADP
iajs-2724	179	26	the	the	DET
iajs-2724	179	27	same	same	ADJ
iajs-2724	179	28	way	way	NOUN
iajs-2724	179	29	,	,	PUNCT
iajs-2724	179	30	we	we	PRON
iajs-2724	179	31	get	get	VERB
iajs-2724	179	32	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	179	33	−(𝜒	−(𝜒	ADJ
iajs-2724	179	34	∗	∗	NOUN
iajs-2724	179	35	𝜏	𝜏	NOUN
iajs-2724	179	36	)	)	PUNCT
iajs-2724	179	37	,	,	PUNCT
iajs-2724	179	38	−𝛼	−𝛼	ADJ
iajs-2724	179	39	}	}	PUNCT
iajs-2724	179	40	≤	≤	NUM
iajs-2724	179	41	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NUM
iajs-2724	179	42	−(𝜒	−(𝜒	ADJ
iajs-2724	179	43	∗	∗	NOUN
iajs-2724	179	44	(	(	PUNCT
iajs-2724	179	45	𝛾	𝛾	NOUN
iajs-2724	179	46	∗	∗	NOUN
iajs-2724	179	47	𝜏	𝜏	NOUN
iajs-2724	179	48	)	)	PUNCT
iajs-2724	179	49	)	)	PUNCT
iajs-2724	179	50	,	,	PUNCT
iajs-2724	179	51	𝜇θ	𝜇θ	PROPN
iajs-2724	179	52	−(𝛾	−(𝛾	PROPN
iajs-2724	179	53	)	)	PUNCT
iajs-2724	179	54	,	,	PUNCT
iajs-2724	179	55	−𝛽	−𝛽	PROPN
iajs-2724	179	56	}	}	PUNCT
iajs-2724	179	57	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	180	1	+	+	PROPN
iajs-2724	180	2	(	(	PUNCT
iajs-2724	180	3	𝜒	𝜒	X
iajs-2724	180	4	∗	∗	NOUN
iajs-2724	180	5	𝜏	𝜏	NOUN
iajs-2724	180	6	)	)	PUNCT
iajs-2724	180	7	,	,	PUNCT
iajs-2724	180	8	𝜔	𝜔	X
iajs-2724	180	9	}	}	PUNCT
iajs-2724	180	10	≥	≥	NOUN
iajs-2724	180	11	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	180	12	+	+	PROPN
iajs-2724	180	13	(	(	PUNCT
iajs-2724	180	14	𝜒	𝜒	NOUN
iajs-2724	180	15	∗	∗	NOUN
iajs-2724	180	16	(	(	PUNCT
iajs-2724	180	17	𝛾	𝛾	NOUN
iajs-2724	180	18	∗	∗	NOUN
iajs-2724	180	19	𝜏	𝜏	NOUN
iajs-2724	180	20	)	)	PUNCT
iajs-2724	180	21	)	)	PUNCT
iajs-2724	180	22	,	,	PUNCT
iajs-2724	180	23	𝜆θ	𝜆θ	ADP
iajs-2724	180	24	+	+	ADJ
iajs-2724	180	25	(	(	PUNCT
iajs-2724	180	26	𝛾	𝛾	NOUN
iajs-2724	180	27	)	)	PUNCT
iajs-2724	180	28	,	,	PUNCT
iajs-2724	180	29	𝜗},and	𝜗},and	CCONJ
iajs-2724	180	30	the	the	DET
iajs-2724	180	31	condition	condition	NOUN
iajs-2724	180	32	(	(	PUNCT
iajs-2724	180	33	cbҡ3	cbҡ3	NOUN
iajs-2724	180	34	)	)	PUNCT
iajs-2724	180	35	then	then	ADV
iajs-2724	180	36	,	,	PUNCT
iajs-2724	180	37	θ	θ	PROPN
iajs-2724	180	38	=	=	SYM
iajs-2724	180	39	〈	〈	PROPN
iajs-2724	180	40	𝑀	𝑀	PROPN
iajs-2724	180	41	,	,	PUNCT
iajs-2724	180	42	𝐿〉is	𝐿〉is	PROPN
iajs-2724	180	43	a	a	PRON
iajs-2724	180	44	(	(	PUNCT
iajs-2724	180	45	cbf	cbf	PROPN
iajs-2724	180	46	)	)	PUNCT
iajs-2724	180	47	k	k	NOUN
iajs-2724	180	48	-	-	NOUN
iajs-2724	180	49	ideal	ideal	NOUN
iajs-2724	180	50	with	with	ADP
iajs-2724	180	51	thresholds	threshold	NOUN
iajs-2724	180	52	(	(	PUNCT
iajs-2724	180	53	𝛼	𝛼	X
iajs-2724	180	54	,	,	PUNCT
iajs-2724	180	55	𝛽),(𝜔	𝛽),(𝜔	NOUN
iajs-2724	180	56	,	,	PUNCT
iajs-2724	180	57	𝜗	𝜗	NOUN
iajs-2724	180	58	)	)	PUNCT
iajs-2724	180	59	of	of	ADP
iajs-2724	180	60	ℵ.	ℵ.	PROPN
iajs-2724	180	61	proposition(29	proposition(29	PROPN
iajs-2724	180	62	)	)	PUNCT
iajs-2724	180	63	.	.	PUNCT
iajs-2724	181	1	let	let	VERB
iajs-2724	181	2	θ	θ	NOUN
iajs-2724	181	3	=	=	SYM
iajs-2724	181	4	〈	〈	PROPN
iajs-2724	181	5	𝑀	𝑀	PROPN
iajs-2724	181	6	,	,	PUNCT
iajs-2724	181	7	𝐿	𝐿	PROPN
iajs-2724	181	8	〉	〉	NOUN
iajs-2724	181	9	be	be	VERB
iajs-2724	181	10	a	a	DET
iajs-2724	181	11	cubic	cubic	ADJ
iajs-2724	181	12	bipolar	bipolar	ADJ
iajs-2724	181	13	k	k	NOUN
iajs-2724	181	14	-	-	NOUN
iajs-2724	181	15	ideal	ideal	NOUN
iajs-2724	181	16	with	with	ADP
iajs-2724	181	17	thresholds	threshold	NOUN
iajs-2724	181	18	(	(	PUNCT
iajs-2724	181	19	𝛼	𝛼	X
iajs-2724	181	20	,	,	PUNCT
iajs-2724	181	21	𝛽	𝛽	NOUN
iajs-2724	181	22	)	)	PUNCT
iajs-2724	181	23	,	,	PUNCT
iajs-2724	181	24	(	(	PUNCT
iajs-2724	181	25	𝜔	𝜔	NOUN
iajs-2724	181	26	,	,	PUNCT
iajs-2724	181	27	𝜗	𝜗	NOUN
iajs-2724	181	28	)	)	PUNCT
iajs-2724	181	29	of	of	ADP
iajs-2724	181	30	ℵ	ℵ	NOUN
iajs-2724	181	31	if	if	SCONJ
iajs-2724	181	32	𝜒	𝜒	PRON
iajs-2724	181	33	≤	≤	NOUN
iajs-2724	181	34	𝛾	𝛾	NOUN
iajs-2724	181	35	,	,	PUNCT
iajs-2724	181	36	then	then	ADV
iajs-2724	181	37	(	(	PUNCT
iajs-2724	181	38	a	a	X
iajs-2724	181	39	)	)	PUNCT
iajs-2724	181	40	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	181	41	−(𝜒),−𝛼	−(𝜒),−𝛼	NOUN
iajs-2724	181	42	}	}	PUNCT
iajs-2724	181	43	≤	≤	NUM
iajs-2724	181	44	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	PROPN
iajs-2724	181	45	−(𝛾	−(𝛾	NOUN
iajs-2724	181	46	)	)	PUNCT
iajs-2724	181	47	,	,	PUNCT
iajs-2724	181	48	−𝛽	−𝛽	PROPN
iajs-2724	181	49	}	}	PUNCT
iajs-2724	181	50	,	,	PUNCT
iajs-2724	181	51	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	PROPN
iajs-2724	181	52	+	+	ADJ
iajs-2724	181	53	(	(	PUNCT
iajs-2724	181	54	𝜒	𝜒	NOUN
iajs-2724	181	55	)	)	PUNCT
iajs-2724	181	56	,	,	PUNCT
iajs-2724	181	57	𝛼	𝛼	X
iajs-2724	181	58	}	}	PUNCT
iajs-2724	181	59	≥	≥	NOUN
iajs-2724	181	60	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	181	61	+	+	ADJ
iajs-2724	181	62	(	(	PUNCT
iajs-2724	181	63	𝛾	𝛾	NOUN
iajs-2724	181	64	)	)	PUNCT
iajs-2724	181	65	,	,	PUNCT
iajs-2724	181	66	𝛽	𝛽	NOUN
iajs-2724	181	67	}	}	PUNCT
iajs-2724	181	68	(	(	PUNCT
iajs-2724	181	69	b	b	NOUN
iajs-2724	181	70	)	)	PUNCT
iajs-2724	181	71	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	X
iajs-2724	181	72	−(𝜒),−𝜔	−(𝜒),−𝜔	ADP
iajs-2724	181	73	}	}	PUNCT
iajs-2724	181	74	≤	≤	NUM
iajs-2724	181	75	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PROPN
iajs-2724	181	76	−(𝛾	−(𝛾	PROPN
iajs-2724	181	77	)	)	PUNCT
iajs-2724	181	78	,	,	PUNCT
iajs-2724	181	79	−𝜗	−𝜗	ADV
iajs-2724	181	80	}	}	PUNCT
iajs-2724	181	81	,	,	PUNCT
iajs-2724	181	82	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PROPN
iajs-2724	182	1	+	+	ADJ
iajs-2724	182	2	(	(	PUNCT
iajs-2724	182	3	𝜒	𝜒	NOUN
iajs-2724	182	4	)	)	PUNCT
iajs-2724	182	5	,	,	PUNCT
iajs-2724	182	6	𝜔	𝜔	X
iajs-2724	182	7	}	}	PUNCT
iajs-2724	182	8	≥	≥	NOUN
iajs-2724	182	9	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	182	10	+	+	ADJ
iajs-2724	182	11	(	(	PUNCT
iajs-2724	182	12	𝛾	𝛾	NOUN
iajs-2724	182	13	)	)	PUNCT
iajs-2724	182	14	,	,	PUNCT
iajs-2724	182	15	𝜗	𝜗	NOUN
iajs-2724	182	16	}	}	PUNCT
iajs-2724	182	17	proof	proof	NOUN
iajs-2724	182	18	:	:	PUNCT
iajs-2724	182	19	since	since	SCONJ
iajs-2724	182	20	𝜒	𝜒	X
iajs-2724	182	21	≤	≤	NOUN
iajs-2724	182	22	𝛾	𝛾	NOUN
iajs-2724	182	23	,	,	PUNCT
iajs-2724	182	24	then	then	ADV
iajs-2724	182	25	𝛾	𝛾	ADP
iajs-2724	182	26	∗	∗	X
iajs-2724	182	27	𝜒	𝜒	X
iajs-2724	182	28	=	=	SYM
iajs-2724	182	29	0	0	NUM
iajs-2724	182	30	,	,	PUNCT
iajs-2724	182	31	and	and	CCONJ
iajs-2724	182	32	by	by	ADP
iajs-2724	182	33	(	(	PUNCT
iajs-2724	182	34	ku3	ku3	X
iajs-2724	182	35	)	)	PUNCT
iajs-2724	182	36	0	0	NUM
iajs-2724	182	37	∗	∗	NOUN
iajs-2724	182	38	𝜒	𝜒	X
iajs-2724	182	39	=	=	SYM
iajs-2724	182	40	𝜒	𝜒	NOUN
iajs-2724	182	41	since	since	SCONJ
iajs-2724	182	42	θ	θ	PROPN
iajs-2724	182	43	=	=	SYM
iajs-2724	182	44	〈	〈	PROPN
iajs-2724	182	45	𝑀	𝑀	PROPN
iajs-2724	182	46	,	,	PUNCT
iajs-2724	182	47	𝐿	𝐿	PROPN
iajs-2724	182	48	〉	〉	NOUN
iajs-2724	182	49	is	be	AUX
iajs-2724	182	50	a	a	DET
iajs-2724	182	51	(	(	PUNCT
iajs-2724	182	52	cb	cb	PROPN
iajs-2724	182	53	)	)	PUNCT
iajs-2724	182	54	k	k	NOUN
iajs-2724	182	55	-	-	NOUN
iajs-2724	182	56	ideal	ideal	NOUN
iajs-2724	182	57	with	with	ADP
iajs-2724	182	58	thresholds	threshold	NOUN
iajs-2724	182	59	(	(	PUNCT
iajs-2724	182	60	𝛼	𝛼	X
iajs-2724	182	61	,	,	PUNCT
iajs-2724	182	62	𝛽	𝛽	NOUN
iajs-2724	182	63	)	)	PUNCT
iajs-2724	182	64	,	,	PUNCT
iajs-2724	182	65	(	(	PUNCT
iajs-2724	182	66	𝜔	𝜔	NOUN
iajs-2724	182	67	,	,	PUNCT
iajs-2724	182	68	𝜗	𝜗	NOUN
iajs-2724	182	69	)	)	PUNCT
iajs-2724	182	70	of	of	ADP
iajs-2724	182	71	ℵ	ℵ	NOUN
iajs-2724	182	72	,	,	PUNCT
iajs-2724	182	73	we	we	PRON
iajs-2724	182	74	get	get	VERB
iajs-2724	182	75	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	182	76	−(𝜒),−𝛼	−(𝜒),−𝛼	NOUN
iajs-2724	182	77	}	}	PUNCT
iajs-2724	182	78	=	=	SYM
iajs-2724	182	79	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	182	80	−(0	−(0	NOUN
iajs-2724	182	81	∗	∗	NOUN
iajs-2724	182	82	𝜒),−𝛼	𝜒),−𝛼	X
iajs-2724	182	83	}	}	PUNCT
iajs-2724	182	84	≤	≤	NUM
iajs-2724	182	85	𝑟𝑚𝑎𝑥{	𝑟𝑚𝑎𝑥{	NOUN
iajs-2724	182	86	�	�	NOUN
iajs-2724	182	87	̃	̃	PROPN
iajs-2724	182	88	�	�	NOUN
iajs-2724	182	89	θ	θ	NOUN
iajs-2724	182	90	−(0	−(0	NOUN
iajs-2724	182	91	∗	∗	NOUN
iajs-2724	182	92	(	(	PUNCT
iajs-2724	182	93	𝛾	𝛾	NOUN
iajs-2724	182	94	∗	∗	NOUN
iajs-2724	182	95	𝜒	𝜒	NOUN
iajs-2724	182	96	)	)	PUNCT
iajs-2724	182	97	)	)	PUNCT
iajs-2724	182	98	,	,	PUNCT
iajs-2724	182	99	𝜇θ	𝜇θ	ADV
iajs-2724	182	100	−(𝛾),−𝛽	−(𝛾),−𝛽	PUNCT
iajs-2724	182	101	}	}	PUNCT
iajs-2724	182	102	=	=	SYM
iajs-2724	182	103	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NUM
iajs-2724	182	104	−(0	−(0	NOUN
iajs-2724	182	105	∗	∗	NOUN
iajs-2724	182	106	0	0	NUM
iajs-2724	182	107	)	)	PUNCT
iajs-2724	182	108	,	,	PUNCT
iajs-2724	182	109	𝜇θ	𝜇θ	ADV
iajs-2724	182	110	−(𝛾),−𝛽	−(𝛾),−𝛽	PUNCT
iajs-2724	182	111	}	}	PUNCT
iajs-2724	182	112	=	=	SYM
iajs-2724	182	113	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NUM
iajs-2724	182	114	−(0	−(0	NOUN
iajs-2724	182	115	)	)	PUNCT
iajs-2724	182	116	,	,	PUNCT
iajs-2724	182	117	𝜇θ	𝜇θ	ADV
iajs-2724	182	118	−(𝛾),−𝛽	−(𝛾),−𝛽	PUNCT
iajs-2724	182	119	}	}	PUNCT
iajs-2724	182	120	=	=	SYM
iajs-2724	182	121	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NUM
iajs-2724	182	122	−(𝛾),−𝛽	−(𝛾),−𝛽	PUNCT
iajs-2724	182	123	}	}	PUNCT
iajs-2724	182	124	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	X
iajs-2724	183	1	+	+	ADJ
iajs-2724	183	2	(	(	PUNCT
iajs-2724	183	3	𝜒	𝜒	NOUN
iajs-2724	183	4	)	)	PUNCT
iajs-2724	183	5	,	,	PUNCT
iajs-2724	183	6	𝛼	𝛼	X
iajs-2724	183	7	}	}	PUNCT
iajs-2724	183	8	=	=	SYM
iajs-2724	183	9	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	X
iajs-2724	183	10	+	+	ADJ
iajs-2724	183	11	(	(	PUNCT
iajs-2724	183	12	0	0	NUM
iajs-2724	183	13	∗	∗	NOUN
iajs-2724	183	14	𝜒	𝜒	NOUN
iajs-2724	183	15	)	)	PUNCT
iajs-2724	183	16	,	,	PUNCT
iajs-2724	184	1	𝛼	𝛼	PROPN
iajs-2724	184	2	}	}	PUNCT
iajs-2724	184	3	≥	≥	NUM
iajs-2724	184	4	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	184	5	�	�	PROPN
iajs-2724	184	6	̃	̃	PROPN
iajs-2724	184	7	�	�	PROPN
iajs-2724	184	8	θ	θ	NOUN
iajs-2724	184	9	+	+	PROPN
iajs-2724	184	10	(	(	PUNCT
iajs-2724	184	11	0	0	NUM
iajs-2724	184	12	∗	∗	NOUN
iajs-2724	184	13	(	(	PUNCT
iajs-2724	184	14	𝛾	𝛾	NOUN
iajs-2724	184	15	∗	∗	NOUN
iajs-2724	184	16	𝜒	𝜒	NOUN
iajs-2724	184	17	)	)	PUNCT
iajs-2724	184	18	)	)	PUNCT
iajs-2724	184	19	,	,	PUNCT
iajs-2724	184	20	𝜇θ	𝜇θ	PROPN
iajs-2724	184	21	+	+	ADJ
iajs-2724	184	22	(	(	PUNCT
iajs-2724	184	23	𝛾	𝛾	NOUN
iajs-2724	184	24	)	)	PUNCT
iajs-2724	184	25	,	,	PUNCT
iajs-2724	184	26	𝛽	𝛽	NOUN
iajs-2724	184	27	}	}	PUNCT
iajs-2724	184	28	=	=	SYM
iajs-2724	184	29	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	184	30	�	�	PROPN
iajs-2724	184	31	̃	̃	PROPN
iajs-2724	184	32	�	�	PROPN
iajs-2724	184	33	θ	θ	NOUN
iajs-2724	184	34	+	+	PROPN
iajs-2724	184	35	(	(	PUNCT
iajs-2724	184	36	0	0	NUM
iajs-2724	184	37	∗	∗	NOUN
iajs-2724	184	38	0	0	NUM
iajs-2724	184	39	)	)	PUNCT
iajs-2724	184	40	,	,	PUNCT
iajs-2724	184	41	𝜇θ	𝜇θ	PROPN
iajs-2724	184	42	+	+	ADJ
iajs-2724	184	43	(	(	PUNCT
iajs-2724	184	44	𝛾	𝛾	NOUN
iajs-2724	184	45	)	)	PUNCT
iajs-2724	184	46	,	,	PUNCT
iajs-2724	184	47	𝛽	𝛽	NOUN
iajs-2724	184	48	}	}	PUNCT
iajs-2724	184	49	=	=	SYM
iajs-2724	184	50	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	184	51	�	�	PROPN
iajs-2724	184	52	̃	̃	PROPN
iajs-2724	184	53	�	�	PROPN
iajs-2724	184	54	θ	θ	NOUN
iajs-2724	184	55	+	+	PROPN
iajs-2724	184	56	(	(	PUNCT
iajs-2724	184	57	0	0	NUM
iajs-2724	184	58	)	)	PUNCT
iajs-2724	184	59	,	,	PUNCT
iajs-2724	184	60	𝜇θ	𝜇θ	PROPN
iajs-2724	184	61	+	+	ADJ
iajs-2724	184	62	(	(	PUNCT
iajs-2724	184	63	𝛾	𝛾	NOUN
iajs-2724	184	64	)	)	PUNCT
iajs-2724	184	65	,	,	PUNCT
iajs-2724	184	66	𝛽	𝛽	NOUN
iajs-2724	184	67	}	}	PUNCT
iajs-2724	184	68	=	=	SYM
iajs-2724	184	69	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	184	70	�	�	PROPN
iajs-2724	184	71	̃	̃	PROPN
iajs-2724	184	72	�	�	PROPN
iajs-2724	184	73	θ	θ	NOUN
iajs-2724	184	74	+	+	PROPN
iajs-2724	184	75	(	(	PUNCT
iajs-2724	184	76	𝛾	𝛾	NOUN
iajs-2724	184	77	)	)	PUNCT
iajs-2724	184	78	,	,	PUNCT
iajs-2724	184	79	𝛽},which	𝛽},which	PROPN
iajs-2724	184	80	is	be	AUX
iajs-2724	184	81	(	(	PUNCT
iajs-2724	184	82	a	a	NOUN
iajs-2724	184	83	)	)	PUNCT
iajs-2724	184	84	,	,	PUNCT
iajs-2724	184	85	and	and	CCONJ
iajs-2724	184	86	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	ADV
iajs-2724	184	87	−(𝜒),−𝜔	−(𝜒),−𝜔	ADP
iajs-2724	185	1	}	}	PUNCT
iajs-2724	185	2	=	=	PUNCT
iajs-2724	185	3	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PUNCT
iajs-2724	185	4	−(0	−(0	NOUN
iajs-2724	185	5	∗	∗	NOUN
iajs-2724	185	6	𝜒	𝜒	NOUN
iajs-2724	185	7	)	)	PUNCT
iajs-2724	185	8	,	,	PUNCT
iajs-2724	185	9	−𝜔	−𝜔	ADP
iajs-2724	185	10	}	}	PUNCT
iajs-2724	185	11	≤	≤	NUM
iajs-2724	185	12	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	185	13	−(0	−(0	NOUN
iajs-2724	185	14	∗	∗	NOUN
iajs-2724	185	15	(	(	PUNCT
iajs-2724	185	16	𝛾	𝛾	NOUN
iajs-2724	185	17	∗	∗	NOUN
iajs-2724	185	18	𝜒	𝜒	NOUN
iajs-2724	185	19	)	)	PUNCT
iajs-2724	185	20	)	)	PUNCT
iajs-2724	185	21	,	,	PUNCT
iajs-2724	185	22	𝜆θ	𝜆θ	ADP
iajs-2724	185	23	−(𝛾),−𝜗	−(𝛾),−𝜗	VERB
iajs-2724	185	24	}	}	PUNCT
iajs-2724	185	25	=	=	SYM
iajs-2724	185	26	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	ADJ
iajs-2724	185	27	−(0	−(0	NOUN
iajs-2724	185	28	∗	∗	NOUN
iajs-2724	185	29	0	0	NUM
iajs-2724	185	30	)	)	PUNCT
iajs-2724	185	31	,	,	PUNCT
iajs-2724	185	32	𝜆θ	𝜆θ	ADP
iajs-2724	185	33	−(𝛾),−𝜗	−(𝛾),−𝜗	VERB
iajs-2724	185	34	}	}	PUNCT
iajs-2724	185	35	=	=	SYM
iajs-2724	185	36	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	X
iajs-2724	185	37	−(0	−(0	NOUN
iajs-2724	185	38	)	)	PUNCT
iajs-2724	185	39	,	,	PUNCT
iajs-2724	185	40	𝜆θ	𝜆θ	ADP
iajs-2724	185	41	−(𝛾),−𝜗	−(𝛾),−𝜗	VERB
iajs-2724	185	42	}	}	PUNCT
iajs-2724	185	43	=	=	SYM
iajs-2724	185	44	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	ADJ
iajs-2724	185	45	−(𝛾),−𝜗	−(𝛾),−𝜗	PUNCT
iajs-2724	185	46	}	}	PUNCT
iajs-2724	185	47	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	186	1	+	+	PROPN
iajs-2724	186	2	(	(	PUNCT
iajs-2724	186	3	𝜒),𝜔	𝜒),𝜔	NOUN
iajs-2724	186	4	}	}	PUNCT
iajs-2724	186	5	=	=	SYM
iajs-2724	186	6	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	187	1	+	+	ADJ
iajs-2724	187	2	(	(	PUNCT
iajs-2724	187	3	0	0	NUM
iajs-2724	187	4	∗	∗	NOUN
iajs-2724	187	5	𝜒	𝜒	NOUN
iajs-2724	187	6	)	)	PUNCT
iajs-2724	187	7	,	,	PUNCT
iajs-2724	187	8	𝜔	𝜔	X
iajs-2724	187	9	}	}	PUNCT
iajs-2724	187	10	≥	≥	NOUN
iajs-2724	187	11	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	187	12	+	+	ADJ
iajs-2724	187	13	(	(	PUNCT
iajs-2724	187	14	0	0	NUM
iajs-2724	187	15	∗	∗	NOUN
iajs-2724	187	16	(	(	PUNCT
iajs-2724	187	17	𝛾	𝛾	NOUN
iajs-2724	187	18	∗	∗	NOUN
iajs-2724	187	19	𝜒	𝜒	NOUN
iajs-2724	187	20	)	)	PUNCT
iajs-2724	187	21	)	)	PUNCT
iajs-2724	187	22	,	,	PUNCT
iajs-2724	187	23	𝜆θ	𝜆θ	ADP
iajs-2724	187	24	+	+	ADJ
iajs-2724	187	25	(	(	PUNCT
iajs-2724	187	26	𝛾	𝛾	NOUN
iajs-2724	187	27	)	)	PUNCT
iajs-2724	187	28	,	,	PUNCT
iajs-2724	187	29	𝜗	𝜗	NOUN
iajs-2724	187	30	}	}	PUNCT
iajs-2724	187	31	=	=	PUNCT
iajs-2724	187	32	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	X
iajs-2724	187	33	+	+	ADJ
iajs-2724	187	34	(	(	PUNCT
iajs-2724	187	35	0	0	NUM
iajs-2724	187	36	∗	∗	NOUN
iajs-2724	187	37	0	0	NUM
iajs-2724	187	38	)	)	PUNCT
iajs-2724	187	39	,	,	PUNCT
iajs-2724	187	40	𝜆θ	𝜆θ	ADP
iajs-2724	187	41	+	+	ADJ
iajs-2724	187	42	(	(	PUNCT
iajs-2724	187	43	𝛾	𝛾	NOUN
iajs-2724	187	44	)	)	PUNCT
iajs-2724	187	45	,	,	PUNCT
iajs-2724	187	46	𝜗	𝜗	NOUN
iajs-2724	187	47	}	}	PUNCT
iajs-2724	187	48	=	=	PUNCT
iajs-2724	187	49	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	X
iajs-2724	187	50	+	+	ADJ
iajs-2724	187	51	(	(	PUNCT
iajs-2724	187	52	0	0	NUM
iajs-2724	187	53	)	)	PUNCT
iajs-2724	187	54	,	,	PUNCT
iajs-2724	187	55	𝜆θ	𝜆θ	ADP
iajs-2724	187	56	+	+	ADJ
iajs-2724	187	57	(	(	PUNCT
iajs-2724	187	58	𝛾	𝛾	NOUN
iajs-2724	187	59	)	)	PUNCT
iajs-2724	187	60	,	,	PUNCT
iajs-2724	187	61	𝜗	𝜗	NOUN
iajs-2724	187	62	}	}	PUNCT
iajs-2724	187	63	ibn	ibn	PROPN
iajs-2724	187	64	al	al	PROPN
iajs-2724	187	65	-	-	PUNCT
iajs-2724	187	66	haitham	haitham	PROPN
iajs-2724	187	67	jour	jour	X
iajs-2724	187	68	.	.	PROPN
iajs-2724	188	1	for	for	ADP
iajs-2724	188	2	pure	pure	ADJ
iajs-2724	188	3	&	&	CCONJ
iajs-2724	188	4	appl	appl	PROPN
iajs-2724	188	5	.	.	PUNCT
iajs-2724	189	1	sci	sci	PROPN
iajs-2724	189	2	.	.	PROPN
iajs-2724	190	1	53	53	NUM
iajs-2724	190	2	(	(	PUNCT
iajs-2724	190	3	2)2022	2)2022	VERB
iajs-2724	190	4	57	57	NUM
iajs-2724	190	5	=	=	SYM
iajs-2724	190	6	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PUNCT
iajs-2724	190	7	+	+	ADJ
iajs-2724	190	8	(	(	PUNCT
iajs-2724	190	9	𝛾	𝛾	NOUN
iajs-2724	190	10	)	)	PUNCT
iajs-2724	190	11	,	,	PUNCT
iajs-2724	190	12	𝜗},which	𝜗},which	ADP
iajs-2724	190	13	is(b	is(b	NOUN
iajs-2724	190	14	)	)	PUNCT
iajs-2724	190	15	theorem(30).let	theorem(30).let	ADP
iajs-2724	190	16	θ	θ	NOUN
iajs-2724	190	17	=	=	SYM
iajs-2724	190	18	〈	〈	PROPN
iajs-2724	190	19	𝑀	𝑀	PROPN
iajs-2724	190	20	,	,	PUNCT
iajs-2724	190	21	𝐿	𝐿	PROPN
iajs-2724	190	22	〉	〉	NOUN
iajs-2724	190	23	be	be	VERB
iajs-2724	190	24	a	a	DET
iajs-2724	190	25	cubic	cubic	ADJ
iajs-2724	190	26	bipolar	bipolar	ADJ
iajs-2724	190	27	fuzzy	fuzzy	ADJ
iajs-2724	190	28	set	set	NOUN
iajs-2724	190	29	of	of	ADP
iajs-2724	190	30	a	a	DET
iajs-2724	190	31	kusemigroup(ℵ,∗,∘	kusemigroup(ℵ,∗,∘	PROPN
iajs-2724	190	32	,	,	PUNCT
iajs-2724	190	33	0	0	NUM
iajs-2724	190	34	)	)	PUNCT
iajs-2724	190	35	then	then	ADV
iajs-2724	190	36	,	,	PUNCT
iajs-2724	190	37	θ	θ	PROPN
iajs-2724	190	38	is	be	AUX
iajs-2724	190	39	a	a	DET
iajs-2724	190	40	(	(	PUNCT
iajs-2724	190	41	cbf	cbf	PROPN
iajs-2724	190	42	)	)	PUNCT
iajs-2724	190	43	k	k	NOUN
iajs-2724	190	44	-	-	NOUN
iajs-2724	190	45	ideal	ideal	NOUN
iajs-2724	190	46	with	with	ADP
iajs-2724	190	47	thresholds	threshold	NOUN
iajs-2724	190	48	(	(	PUNCT
iajs-2724	190	49	𝛼	𝛼	X
iajs-2724	190	50	,	,	PUNCT
iajs-2724	190	51	𝛽	𝛽	NOUN
iajs-2724	190	52	)	)	PUNCT
iajs-2724	190	53	,	,	PUNCT
iajs-2724	190	54	(	(	PUNCT
iajs-2724	190	55	𝜔	𝜔	NOUN
iajs-2724	190	56	,	,	PUNCT
iajs-2724	190	57	𝜗	𝜗	NOUN
iajs-2724	190	58	)	)	PUNCT
iajs-2724	190	59	of	of	ADP
iajs-2724	190	60	ℵ	ℵ	NOUN
iajs-2724	190	61	if	if	SCONJ
iajs-2724	190	62	and	and	CCONJ
iajs-2724	190	63	only	only	ADV
iajs-2724	190	64	if	if	SCONJ
iajs-2724	190	65	it	it	PRON
iajs-2724	190	66	is	be	AUX
iajs-2724	190	67	a	a	DET
iajs-2724	190	68	(	(	PUNCT
iajs-2724	190	69	cbf)-ideal	cbf)-ideal	ADJ
iajs-2724	190	70	with	with	ADP
iajs-2724	190	71	thresholds	threshold	NOUN
iajs-2724	190	72	(	(	PUNCT
iajs-2724	190	73	𝛼	𝛼	X
iajs-2724	190	74	,	,	PUNCT
iajs-2724	190	75	𝛽	𝛽	NOUN
iajs-2724	190	76	)	)	PUNCT
iajs-2724	190	77	,	,	PUNCT
iajs-2724	190	78	(	(	PUNCT
iajs-2724	190	79	𝜔	𝜔	NOUN
iajs-2724	190	80	,	,	PUNCT
iajs-2724	190	81	𝜗	𝜗	NOUN
iajs-2724	190	82	)	)	PUNCT
iajs-2724	190	83	of	of	ADP
iajs-2724	190	84	ℵ	ℵ	NOUN
iajs-2724	190	85	.	.	PUNCT
iajs-2724	191	1	proof	proof	NOUN
iajs-2724	191	2	:	:	PUNCT
iajs-2724	191	3	⇒	⇒	NOUN
iajs-2724	191	4	let	let	VERB
iajs-2724	191	5	θ	θ	PROPN
iajs-2724	191	6	=	=	SYM
iajs-2724	191	7	〈	〈	PROPN
iajs-2724	191	8	𝑀	𝑀	PROPN
iajs-2724	191	9	,	,	PUNCT
iajs-2724	191	10	𝐿	𝐿	PROPN
iajs-2724	191	11	〉	〉	NOUN
iajs-2724	191	12	be	be	VERB
iajs-2724	191	13	a	a	DET
iajs-2724	191	14	cubic	cubic	ADJ
iajs-2724	191	15	bipolar	bipolar	ADJ
iajs-2724	191	16	k	k	NOUN
iajs-2724	191	17	-	-	NOUN
iajs-2724	191	18	ideal	ideal	NOUN
iajs-2724	191	19	with	with	ADP
iajs-2724	191	20	thresholds	threshold	NOUN
iajs-2724	191	21	(	(	PUNCT
iajs-2724	191	22	𝛼	𝛼	X
iajs-2724	191	23	,	,	PUNCT
iajs-2724	191	24	𝛽	𝛽	NOUN
iajs-2724	191	25	)	)	PUNCT
iajs-2724	191	26	,	,	PUNCT
iajs-2724	191	27	(	(	PUNCT
iajs-2724	191	28	𝜔	𝜔	NOUN
iajs-2724	191	29	,	,	PUNCT
iajs-2724	191	30	𝜗	𝜗	NOUN
iajs-2724	191	31	)	)	PUNCT
iajs-2724	191	32	of	of	ADP
iajs-2724	191	33	ℵ	ℵ	NOUN
iajs-2724	191	34	,	,	PUNCT
iajs-2724	191	35	if	if	SCONJ
iajs-2724	191	36	we	we	PRON
iajs-2724	191	37	put	put	VERB
iajs-2724	191	38	𝜒	𝜒	NOUN
iajs-2724	191	39	=	=	SYM
iajs-2724	191	40	0	0	NUM
iajs-2724	191	41	in	in	ADP
iajs-2724	191	42	(	(	PUNCT
iajs-2724	191	43	cbҡ2	cbҡ2	PROPN
iajs-2724	191	44	)	)	PUNCT
iajs-2724	191	45	,	,	PUNCT
iajs-2724	191	46	we	we	PRON
iajs-2724	191	47	get	get	VERB
iajs-2724	191	48	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	191	49	−(0	−(0	NOUN
iajs-2724	191	50	∗	∗	NOUN
iajs-2724	191	51	𝜏),−𝛼	𝜏),−𝛼	NUM
iajs-2724	191	52	}	}	PUNCT
iajs-2724	191	53	≤	≤	NUM
iajs-2724	191	54	𝑟𝑚𝑎𝑥{	𝑟𝑚𝑎𝑥{	NOUN
iajs-2724	191	55	�	�	NOUN
iajs-2724	191	56	̃	̃	PROPN
iajs-2724	191	57	�	�	NOUN
iajs-2724	191	58	θ	θ	NOUN
iajs-2724	191	59	−(0	−(0	NOUN
iajs-2724	191	60	∗	∗	NOUN
iajs-2724	191	61	(	(	PUNCT
iajs-2724	192	1	𝛾	𝛾	NOUN
iajs-2724	192	2	∗	∗	NOUN
iajs-2724	192	3	𝜏	𝜏	NOUN
iajs-2724	192	4	)	)	PUNCT
iajs-2724	192	5	)	)	PUNCT
iajs-2724	193	1	,	,	PUNCT
iajs-2724	193	2	𝜇θ	𝜇θ	ADV
iajs-2724	193	3	−(𝛾),−𝛽	−(𝛾),−𝛽	PUNCT
iajs-2724	193	4	}	}	PUNCT
iajs-2724	193	5	is	be	AUX
iajs-2724	193	6	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	193	7	−(𝜏),−𝛼	−(𝜏),−𝛼	NOUN
iajs-2724	193	8	}	}	PUNCT
iajs-2724	193	9	≤	≤	NUM
iajs-2724	193	10	𝑟𝑚𝑎𝑥{	𝑟𝑚𝑎𝑥{	NOUN
iajs-2724	193	11	�	�	NOUN
iajs-2724	193	12	̃	̃	PROPN
iajs-2724	193	13	�	�	NOUN
iajs-2724	193	14	θ	θ	PROPN
iajs-2724	193	15	−(𝛾	−(𝛾	NOUN
iajs-2724	193	16	∗	∗	NOUN
iajs-2724	193	17	𝜏	𝜏	NOUN
iajs-2724	193	18	)	)	PUNCT
iajs-2724	193	19	,	,	PUNCT
iajs-2724	193	20	𝜇θ	𝜇θ	ADV
iajs-2724	193	21	−(𝛾),−𝛽	−(𝛾),−𝛽	PUNCT
iajs-2724	193	22	}	}	PUNCT
iajs-2724	193	23	,	,	PUNCT
iajs-2724	193	24	also	also	ADV
iajs-2724	193	25	𝑟𝑚𝑎𝑥{	𝑟𝑚𝑎𝑥{	VERB
iajs-2724	193	26	�	�	NOUN
iajs-2724	193	27	̃	̃	NOUN
iajs-2724	193	28	�	�	NOUN
iajs-2724	193	29	θ	θ	NOUN
iajs-2724	193	30	+	+	PROPN
iajs-2724	193	31	(	(	PUNCT
iajs-2724	193	32	0	0	NUM
iajs-2724	193	33	∗	∗	NOUN
iajs-2724	193	34	𝜏	𝜏	NOUN
iajs-2724	193	35	)	)	PUNCT
iajs-2724	193	36	,	,	PUNCT
iajs-2724	193	37	𝛼	𝛼	PROPN
iajs-2724	193	38	}	}	PUNCT
iajs-2724	193	39	≥	≥	NOUN
iajs-2724	193	40	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	193	41	+	+	ADJ
iajs-2724	193	42	(	(	PUNCT
iajs-2724	193	43	0	0	NUM
iajs-2724	193	44	∗	∗	NOUN
iajs-2724	193	45	(	(	PUNCT
iajs-2724	193	46	𝛾	𝛾	NOUN
iajs-2724	193	47	∗	∗	NOUN
iajs-2724	193	48	𝜏	𝜏	NOUN
iajs-2724	193	49	)	)	PUNCT
iajs-2724	193	50	,	,	PUNCT
iajs-2724	193	51	𝜇θ	𝜇θ	PROPN
iajs-2724	193	52	−(𝛾	−(𝛾	PROPN
iajs-2724	193	53	)	)	PUNCT
iajs-2724	193	54	,	,	PUNCT
iajs-2724	193	55	𝛽	𝛽	PROPN
iajs-2724	193	56	}	}	PUNCT
iajs-2724	193	57	is	be	AUX
iajs-2724	193	58	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NOUN
iajs-2724	194	1	+	+	ADJ
iajs-2724	194	2	(	(	PUNCT
iajs-2724	194	3	𝜏	𝜏	NOUN
iajs-2724	194	4	)	)	PUNCT
iajs-2724	194	5	,	,	PUNCT
iajs-2724	194	6	𝛼	𝛼	PROPN
iajs-2724	194	7	}	}	PUNCT
iajs-2724	194	8	≥	≥	NOUN
iajs-2724	194	9	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	194	10	+	+	NOUN
iajs-2724	194	11	(	(	PUNCT
iajs-2724	194	12	𝛾	𝛾	NOUN
iajs-2724	194	13	∗	∗	NOUN
iajs-2724	194	14	𝜏	𝜏	NOUN
iajs-2724	194	15	)	)	PUNCT
iajs-2724	194	16	,	,	PUNCT
iajs-2724	194	17	𝜇θ	𝜇θ	PROPN
iajs-2724	194	18	−(𝛾	−(𝛾	PROPN
iajs-2724	194	19	)	)	PUNCT
iajs-2724	194	20	,	,	PUNCT
iajs-2724	194	21	𝛽	𝛽	NOUN
iajs-2724	194	22	}	}	PUNCT
iajs-2724	194	23	,	,	PUNCT
iajs-2724	194	24	and	and	CCONJ
iajs-2724	194	25	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	ADJ
iajs-2724	194	26	−(0	−(0	NOUN
iajs-2724	194	27	∗	∗	NOUN
iajs-2724	194	28	𝜏),−𝜔	𝜏),−𝜔	NOUN
iajs-2724	194	29	}	}	PUNCT
iajs-2724	194	30	≤	≤	NUM
iajs-2724	194	31	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	194	32	−(0	−(0	NOUN
iajs-2724	194	33	∗	∗	NOUN
iajs-2724	194	34	(	(	PUNCT
iajs-2724	194	35	𝛾	𝛾	NOUN
iajs-2724	194	36	∗	∗	NOUN
iajs-2724	194	37	𝜏	𝜏	NOUN
iajs-2724	194	38	)	)	PUNCT
iajs-2724	194	39	)	)	PUNCT
iajs-2724	194	40	,	,	PUNCT
iajs-2724	194	41	𝜆θ	𝜆θ	ADP
iajs-2724	194	42	−(𝛾	−(𝛾	PROPN
iajs-2724	194	43	)	)	PUNCT
iajs-2724	194	44	,	,	PUNCT
iajs-2724	194	45	−𝜗	−𝜗	ADV
iajs-2724	194	46	}	}	PUNCT
iajs-2724	194	47	is	be	AUX
iajs-2724	194	48	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	ADJ
iajs-2724	194	49	−(𝜏),−𝜔	−(𝜏),−𝜔	PROPN
iajs-2724	194	50	}	}	PUNCT
iajs-2724	194	51	≤	≤	NUM
iajs-2724	194	52	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PROPN
iajs-2724	194	53	−(𝛾	−(𝛾	VERB
iajs-2724	194	54	∗	∗	PROPN
iajs-2724	194	55	𝜏	𝜏	NOUN
iajs-2724	194	56	)	)	PUNCT
iajs-2724	194	57	)	)	PUNCT
iajs-2724	194	58	,	,	PUNCT
iajs-2724	194	59	𝜆θ	𝜆θ	INTJ
iajs-2724	194	60	−(𝛾),−𝜗	−(𝛾),−𝜗	VERB
iajs-2724	194	61	}	}	PUNCT
iajs-2724	194	62	,	,	PUNCT
iajs-2724	194	63	also	also	ADV
iajs-2724	194	64	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PROPN
iajs-2724	195	1	+	+	ADJ
iajs-2724	195	2	(	(	PUNCT
iajs-2724	195	3	0	0	NUM
iajs-2724	195	4	∗	∗	NOUN
iajs-2724	195	5	𝜏	𝜏	NOUN
iajs-2724	195	6	)	)	PUNCT
iajs-2724	195	7	,	,	PUNCT
iajs-2724	195	8	𝜔	𝜔	X
iajs-2724	195	9	}	}	PUNCT
iajs-2724	195	10	≥	≥	NOUN
iajs-2724	195	11	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	195	12	+	+	ADJ
iajs-2724	195	13	(	(	PUNCT
iajs-2724	195	14	0	0	NUM
iajs-2724	195	15	∗	∗	NOUN
iajs-2724	195	16	(	(	PUNCT
iajs-2724	195	17	𝛾	𝛾	NOUN
iajs-2724	195	18	∗	∗	NOUN
iajs-2724	195	19	𝜏	𝜏	NOUN
iajs-2724	195	20	)	)	PUNCT
iajs-2724	195	21	)	)	PUNCT
iajs-2724	195	22	,	,	PUNCT
iajs-2724	195	23	𝜆θ	𝜆θ	ADP
iajs-2724	195	24	+	+	ADJ
iajs-2724	195	25	(	(	PUNCT
iajs-2724	195	26	𝛾	𝛾	NOUN
iajs-2724	195	27	)	)	PUNCT
iajs-2724	195	28	,	,	PUNCT
iajs-2724	195	29	𝜗	𝜗	PROPN
iajs-2724	195	30	}	}	PUNCT
iajs-2724	195	31	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PROPN
iajs-2724	196	1	+	+	ADJ
iajs-2724	196	2	(	(	PUNCT
iajs-2724	196	3	𝜏	𝜏	NOUN
iajs-2724	196	4	)	)	PUNCT
iajs-2724	196	5	,	,	PUNCT
iajs-2724	196	6	𝜔	𝜔	X
iajs-2724	196	7	}	}	PUNCT
iajs-2724	196	8	≥	≥	NOUN
iajs-2724	196	9	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	196	10	+	+	PROPN
iajs-2724	196	11	(	(	PUNCT
iajs-2724	196	12	𝛾	𝛾	NOUN
iajs-2724	196	13	∗	∗	NOUN
iajs-2724	196	14	𝜏	𝜏	NOUN
iajs-2724	196	15	)	)	PUNCT
iajs-2724	196	16	,	,	PUNCT
iajs-2724	196	17	𝜆θ	𝜆θ	ADP
iajs-2724	196	18	+	+	ADJ
iajs-2724	196	19	(	(	PUNCT
iajs-2724	196	20	𝛾	𝛾	NOUN
iajs-2724	196	21	)	)	PUNCT
iajs-2724	196	22	,	,	PUNCT
iajs-2724	196	23	𝜗},the	𝜗},the	PRON
iajs-2724	196	24	other	other	ADJ
iajs-2724	196	25	conditions	condition	NOUN
iajs-2724	196	26	(	(	PUNCT
iajs-2724	196	27	cbt1	cbt1	PROPN
iajs-2724	196	28	)	)	PUNCT
iajs-2724	196	29	,	,	PUNCT
iajs-2724	196	30	(	(	PUNCT
iajs-2724	196	31	cbt3	cbt3	PROPN
iajs-2724	196	32	)	)	PUNCT
iajs-2724	196	33	are	be	AUX
iajs-2724	196	34	holds	hold	NOUN
iajs-2724	196	35	from	from	ADP
iajs-2724	196	36	the	the	DET
iajs-2724	196	37	definition	definition	NOUN
iajs-2724	196	38	of	of	ADP
iajs-2724	196	39	(	(	PUNCT
iajs-2724	196	40	cbf)k	cbf)k	NOUN
iajs-2724	196	41	-	-	PUNCT
iajs-2724	196	42	ideal	ideal	NOUN
iajs-2724	196	43	;	;	PUNCT
iajs-2724	196	44	therefore	therefore	ADV
iajs-2724	196	45	θ	θ	PROPN
iajs-2724	196	46	=	=	SYM
iajs-2724	196	47	〈	〈	PROPN
iajs-2724	196	48	𝑀	𝑀	PROPN
iajs-2724	196	49	,	,	PUNCT
iajs-2724	196	50	𝐿〉is	𝐿〉is	PROPN
iajs-2724	196	51	a	a	DET
iajs-2724	196	52	(	(	PUNCT
iajs-2724	196	53	cb)-ideal	cb)-ideal	NOUN
iajs-2724	196	54	with	with	ADP
iajs-2724	196	55	thresholds	threshold	NOUN
iajs-2724	196	56	(	(	PUNCT
iajs-2724	196	57	𝛼	𝛼	X
iajs-2724	196	58	,	,	PUNCT
iajs-2724	196	59	𝛽	𝛽	NOUN
iajs-2724	196	60	)	)	PUNCT
iajs-2724	196	61	,	,	PUNCT
iajs-2724	196	62	(	(	PUNCT
iajs-2724	196	63	𝜔	𝜔	NOUN
iajs-2724	196	64	,	,	PUNCT
iajs-2724	196	65	𝜗	𝜗	NOUN
iajs-2724	196	66	)	)	PUNCT
iajs-2724	196	67	of	of	ADP
iajs-2724	196	68	ℵ	ℵ	PRON
iajs-2724	196	69	⇐	⇐	NOUN
iajs-2724	196	70	let	let	VERB
iajs-2724	196	71	θ	θ	PROPN
iajs-2724	196	72	=	=	SYM
iajs-2724	196	73	〈	〈	PROPN
iajs-2724	196	74	𝑀	𝑀	PROPN
iajs-2724	196	75	,	,	PUNCT
iajs-2724	196	76	𝐿	𝐿	PROPN
iajs-2724	196	77	〉	〉	NOUN
iajs-2724	196	78	be	be	VERB
iajs-2724	196	79	a	a	DET
iajs-2724	196	80	cubic	cubic	ADJ
iajs-2724	196	81	bipolar	bipolar	ADJ
iajs-2724	196	82	ideal	ideal	NOUN
iajs-2724	196	83	with	with	ADP
iajs-2724	196	84	thresholds	threshold	NOUN
iajs-2724	196	85	(	(	PUNCT
iajs-2724	196	86	𝛼	𝛼	X
iajs-2724	196	87	,	,	PUNCT
iajs-2724	196	88	𝛽	𝛽	NOUN
iajs-2724	196	89	)	)	PUNCT
iajs-2724	196	90	,	,	PUNCT
iajs-2724	196	91	(	(	PUNCT
iajs-2724	196	92	𝜔	𝜔	NOUN
iajs-2724	196	93	,	,	PUNCT
iajs-2724	196	94	𝜗	𝜗	NOUN
iajs-2724	196	95	)	)	PUNCT
iajs-2724	196	96	of	of	ADP
iajs-2724	196	97	ℵ	ℵ	NOUN
iajs-2724	196	98	,	,	PUNCT
iajs-2724	196	99	by	by	ADP
iajs-2724	196	100	(	(	PUNCT
iajs-2724	196	101	cbt2	cbt2	PROPN
iajs-2724	196	102	)	)	PUNCT
iajs-2724	196	103	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	196	104	−(𝜒	−(𝜒	ADJ
iajs-2724	196	105	∗	∗	NOUN
iajs-2724	196	106	𝜏),−𝛼	𝜏),−𝛼	NUM
iajs-2724	196	107	}	}	PUNCT
iajs-2724	196	108	≤	≤	NUM
iajs-2724	196	109	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	PROPN
iajs-2724	196	110	−(𝛾	−(𝛾	VERB
iajs-2724	196	111	∗	∗	NOUN
iajs-2724	196	112	(	(	PUNCT
iajs-2724	196	113	𝜒	𝜒	NOUN
iajs-2724	196	114	∗	∗	NOUN
iajs-2724	196	115	𝜏	𝜏	NOUN
iajs-2724	196	116	)	)	PUNCT
iajs-2724	196	117	,	,	PUNCT
iajs-2724	196	118	𝜇θ	𝜇θ	ADV
iajs-2724	196	119	−(𝛾),−𝛽	−(𝛾),−𝛽	PUNCT
iajs-2724	196	120	}	}	PUNCT
iajs-2724	196	121	,	,	PUNCT
iajs-2724	196	122	also	also	ADV
iajs-2724	196	123	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	PRON
iajs-2724	196	124	+	+	NOUN
iajs-2724	196	125	(	(	PUNCT
iajs-2724	196	126	𝜒	𝜒	X
iajs-2724	196	127	∗	∗	NOUN
iajs-2724	196	128	𝜏	𝜏	NOUN
iajs-2724	196	129	)	)	PUNCT
iajs-2724	196	130	,	,	PUNCT
iajs-2724	196	131	𝛼	𝛼	NOUN
iajs-2724	196	132	}	}	PUNCT
iajs-2724	196	133	≤	≤	NUM
iajs-2724	196	134	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	197	1	+	+	NOUN
iajs-2724	197	2	(	(	PUNCT
iajs-2724	197	3	𝛾	𝛾	NOUN
iajs-2724	197	4	∗	∗	NOUN
iajs-2724	197	5	(	(	PUNCT
iajs-2724	197	6	𝜒	𝜒	NOUN
iajs-2724	197	7	∗	∗	NOUN
iajs-2724	197	8	𝜏	𝜏	NOUN
iajs-2724	197	9	)	)	PUNCT
iajs-2724	197	10	,	,	PUNCT
iajs-2724	197	11	𝜇θ	𝜇θ	PROPN
iajs-2724	197	12	+	+	ADJ
iajs-2724	197	13	(	(	PUNCT
iajs-2724	197	14	𝛾	𝛾	NOUN
iajs-2724	197	15	)	)	PUNCT
iajs-2724	197	16	,	,	PUNCT
iajs-2724	197	17	𝛽	𝛽	NOUN
iajs-2724	197	18	}	}	PUNCT
iajs-2724	197	19	,	,	PUNCT
iajs-2724	197	20	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	ADV
iajs-2724	197	21	−(𝜒	−(𝜒	VERB
iajs-2724	197	22	∗	∗	NOUN
iajs-2724	197	23	𝜏	𝜏	NOUN
iajs-2724	197	24	)	)	PUNCT
iajs-2724	197	25	,	,	PUNCT
iajs-2724	197	26	−𝜔	−𝜔	ADP
iajs-2724	197	27	}	}	PUNCT
iajs-2724	197	28	≤	≤	NUM
iajs-2724	197	29	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PROPN
iajs-2724	197	30	−(𝛾	−(𝛾	PROPN
iajs-2724	197	31	∗	∗	PROPN
iajs-2724	197	32	(	(	PUNCT
iajs-2724	197	33	𝜒	𝜒	NOUN
iajs-2724	197	34	∗	∗	NOUN
iajs-2724	197	35	𝜏	𝜏	NOUN
iajs-2724	197	36	)	)	PUNCT
iajs-2724	197	37	,	,	PUNCT
iajs-2724	197	38	𝜆θ	𝜆θ	INTJ
iajs-2724	197	39	−(𝛾),−𝜗	−(𝛾),−𝜗	VERB
iajs-2724	197	40	}	}	PUNCT
iajs-2724	197	41	,	,	PUNCT
iajs-2724	197	42	also	also	ADV
iajs-2724	197	43	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PROPN
iajs-2724	198	1	+	+	ADJ
iajs-2724	198	2	(	(	PUNCT
iajs-2724	198	3	𝜒	𝜒	X
iajs-2724	198	4	∗	∗	NOUN
iajs-2724	198	5	𝜏	𝜏	NOUN
iajs-2724	198	6	)	)	PUNCT
iajs-2724	198	7	,	,	PUNCT
iajs-2724	198	8	𝜔	𝜔	X
iajs-2724	198	9	}	}	PUNCT
iajs-2724	198	10	≥	≥	NOUN
iajs-2724	198	11	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	198	12	+	+	PROPN
iajs-2724	198	13	(	(	PUNCT
iajs-2724	198	14	𝛾	𝛾	NOUN
iajs-2724	198	15	∗	∗	NOUN
iajs-2724	198	16	(	(	PUNCT
iajs-2724	198	17	𝜒	𝜒	NOUN
iajs-2724	198	18	∗	∗	NOUN
iajs-2724	198	19	𝜏	𝜏	NOUN
iajs-2724	198	20	)	)	PUNCT
iajs-2724	198	21	,	,	PUNCT
iajs-2724	198	22	𝜆θ	𝜆θ	ADP
iajs-2724	198	23	+	+	ADJ
iajs-2724	198	24	(	(	PUNCT
iajs-2724	198	25	𝛾	𝛾	NOUN
iajs-2724	198	26	)	)	PUNCT
iajs-2724	198	27	,	,	PUNCT
iajs-2724	198	28	𝜗	𝜗	AUX
iajs-2724	198	29	}	}	PUNCT
iajs-2724	198	30	applying	apply	VERB
iajs-2724	198	31	theorem	theorem	ADJ
iajs-2724	198	32	2	2	NUM
iajs-2724	198	33	(	(	PUNCT
iajs-2724	198	34	2	2	NUM
iajs-2724	198	35	)	)	PUNCT
iajs-2724	198	36	to	to	ADP
iajs-2724	198	37	the	the	DET
iajs-2724	198	38	previous	previous	ADJ
iajs-2724	198	39	four	four	NUM
iajs-2724	198	40	steps	step	NOUN
iajs-2724	198	41	,	,	PUNCT
iajs-2724	198	42	we	we	PRON
iajs-2724	198	43	obtain	obtain	VERB
iajs-2724	198	44	𝑟𝑚𝑖𝑛{𝜇θ	𝑟𝑚𝑖𝑛{𝜇θ	NUM
iajs-2724	198	45	−(𝜒	−(𝜒	ADJ
iajs-2724	198	46	∗	∗	NOUN
iajs-2724	198	47	𝜏	𝜏	NOUN
iajs-2724	198	48	)	)	PUNCT
iajs-2724	198	49	,	,	PUNCT
iajs-2724	198	50	−𝛼	−𝛼	ADJ
iajs-2724	198	51	}	}	PUNCT
iajs-2724	198	52	≤	≤	NUM
iajs-2724	198	53	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	NUM
iajs-2724	198	54	−(𝜒	−(𝜒	ADJ
iajs-2724	198	55	∗	∗	NOUN
iajs-2724	198	56	(	(	PUNCT
iajs-2724	198	57	𝛾	𝛾	NOUN
iajs-2724	198	58	∗	∗	NOUN
iajs-2724	198	59	𝜏	𝜏	NOUN
iajs-2724	198	60	)	)	PUNCT
iajs-2724	198	61	,	,	PUNCT
iajs-2724	198	62	𝜇θ	𝜇θ	ADV
iajs-2724	198	63	−(𝛾),−𝛽	−(𝛾),−𝛽	PUNCT
iajs-2724	198	64	}	}	PUNCT
iajs-2724	198	65	,	,	PUNCT
iajs-2724	198	66	𝑟𝑚𝑎𝑥{𝜇θ	𝑟𝑚𝑎𝑥{𝜇θ	PROPN
iajs-2724	198	67	+	+	NOUN
iajs-2724	198	68	(	(	PUNCT
iajs-2724	198	69	𝜒	𝜒	X
iajs-2724	198	70	∗	∗	NOUN
iajs-2724	198	71	𝜏	𝜏	NOUN
iajs-2724	198	72	)	)	PUNCT
iajs-2724	198	73	,	,	PUNCT
iajs-2724	198	74	𝛼	𝛼	PROPN
iajs-2724	198	75	}	}	PUNCT
iajs-2724	198	76	≤	≤	NUM
iajs-2724	198	77	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-2724	198	78	�	�	PROPN
iajs-2724	198	79	̃	̃	PROPN
iajs-2724	198	80	�	�	PROPN
iajs-2724	198	81	θ	θ	NOUN
iajs-2724	198	82	+	+	PROPN
iajs-2724	198	83	(	(	PUNCT
iajs-2724	198	84	𝜒	𝜒	NOUN
iajs-2724	198	85	∗	∗	NOUN
iajs-2724	198	86	(	(	PUNCT
iajs-2724	198	87	𝛾	𝛾	NOUN
iajs-2724	198	88	∗	∗	NOUN
iajs-2724	198	89	𝜏	𝜏	NOUN
iajs-2724	198	90	)	)	PUNCT
iajs-2724	198	91	,	,	PUNCT
iajs-2724	198	92	𝜇θ	𝜇θ	PROPN
iajs-2724	199	1	+	+	ADJ
iajs-2724	199	2	(	(	PUNCT
iajs-2724	199	3	𝛾	𝛾	NOUN
iajs-2724	199	4	)	)	PUNCT
iajs-2724	199	5	,	,	PUNCT
iajs-2724	199	6	𝛽	𝛽	NOUN
iajs-2724	199	7	}	}	PUNCT
iajs-2724	199	8	,	,	PUNCT
iajs-2724	199	9	and	and	CCONJ
iajs-2724	199	10	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	ADV
iajs-2724	199	11	−(𝜒	−(𝜒	ADJ
iajs-2724	199	12	∗	∗	NOUN
iajs-2724	199	13	𝜏	𝜏	NOUN
iajs-2724	199	14	)	)	PUNCT
iajs-2724	199	15	,	,	PUNCT
iajs-2724	199	16	−𝜔	−𝜔	ADP
iajs-2724	199	17	}	}	PUNCT
iajs-2724	199	18	≤	≤	NUM
iajs-2724	199	19	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PUNCT
iajs-2724	199	20	−(𝜒	−(𝜒	ADJ
iajs-2724	199	21	∗	∗	NOUN
iajs-2724	199	22	(	(	PUNCT
iajs-2724	200	1	𝛾	𝛾	NOUN
iajs-2724	200	2	∗	∗	NOUN
iajs-2724	200	3	𝜏	𝜏	NOUN
iajs-2724	200	4	)	)	PUNCT
iajs-2724	200	5	,	,	PUNCT
iajs-2724	200	6	𝜆θ	𝜆θ	INTJ
iajs-2724	200	7	−(𝛾),−𝜗	−(𝛾),−𝜗	VERB
iajs-2724	200	8	}	}	PUNCT
iajs-2724	200	9	,	,	PUNCT
iajs-2724	200	10	𝑚𝑎𝑥{𝜆θ	𝑚𝑎𝑥{𝜆θ	PROPN
iajs-2724	201	1	+	+	PROPN
iajs-2724	201	2	(	(	PUNCT
iajs-2724	201	3	𝜒	𝜒	X
iajs-2724	201	4	∗	∗	NOUN
iajs-2724	201	5	𝜏	𝜏	NOUN
iajs-2724	201	6	)	)	PUNCT
iajs-2724	201	7	,	,	PUNCT
iajs-2724	201	8	𝜔	𝜔	X
iajs-2724	201	9	}	}	PUNCT
iajs-2724	201	10	≥	≥	NOUN
iajs-2724	201	11	𝑚𝑖𝑛{𝜆θ	𝑚𝑖𝑛{𝜆θ	PRON
iajs-2724	201	12	+	+	PROPN
iajs-2724	201	13	(	(	PUNCT
iajs-2724	201	14	𝜒	𝜒	NOUN
iajs-2724	201	15	∗	∗	NOUN
iajs-2724	201	16	(	(	PUNCT
iajs-2724	201	17	𝛾	𝛾	NOUN
iajs-2724	201	18	∗	∗	NOUN
iajs-2724	201	19	𝜏	𝜏	NOUN
iajs-2724	201	20	)	)	PUNCT
iajs-2724	201	21	,	,	PUNCT
iajs-2724	201	22	𝜆θ	𝜆θ	ADP
iajs-2724	201	23	+	+	ADJ
iajs-2724	201	24	(	(	PUNCT
iajs-2724	201	25	𝛾	𝛾	NOUN
iajs-2724	201	26	)	)	PUNCT
iajs-2724	201	27	,	,	PUNCT
iajs-2724	201	28	𝜗	𝜗	PROPN
iajs-2724	201	29	}	}	PUNCT
iajs-2724	201	30	,	,	PUNCT
iajs-2724	201	31	which	which	PRON
iajs-2724	201	32	is	be	AUX
iajs-2724	201	33	a	a	DET
iajs-2724	201	34	(	(	PUNCT
iajs-2724	201	35	cbf	cbf	PROPN
iajs-2724	201	36	)	)	PUNCT
iajs-2724	201	37	k	k	NOUN
iajs-2724	201	38	-	-	PUNCT
iajs-2724	201	39	ideal	ideal	ADJ
iajs-2724	201	40	,	,	PUNCT
iajs-2724	201	41	the	the	DET
iajs-2724	201	42	remaining	remain	VERB
iajs-2724	201	43	two	two	NUM
iajs-2724	201	44	conditions	condition	NOUN
iajs-2724	201	45	(	(	PUNCT
iajs-2724	201	46	cbҡ1),(cbҡ3	cbҡ1),(cbҡ3	X
iajs-2724	201	47	)	)	PUNCT
iajs-2724	201	48	are	be	AUX
iajs-2724	201	49	holds	hold	NOUN
iajs-2724	201	50	from	from	ADP
iajs-2724	201	51	the	the	DET
iajs-2724	201	52	definition	definition	NOUN
iajs-2724	201	53	of	of	ADP
iajs-2724	201	54	(	(	PUNCT
iajs-2724	201	55	cbf)-ideal	cbf)-ideal	NOUN
iajs-2724	201	56	.	.	PUNCT
iajs-2724	202	1	4.conclusion	4.conclusion	NUM
iajs-2724	202	2	during	during	ADP
iajs-2724	202	3	this	this	DET
iajs-2724	202	4	work	work	NOUN
iajs-2724	202	5	,	,	PUNCT
iajs-2724	202	6	we	we	PRON
iajs-2724	202	7	present	present	VERB
iajs-2724	202	8	the	the	DET
iajs-2724	202	9	definitions	definition	NOUN
iajs-2724	202	10	of	of	ADP
iajs-2724	202	11	the	the	DET
iajs-2724	202	12	cubic	cubic	ADJ
iajs-2724	202	13	bipolar	bipolar	ADJ
iajs-2724	202	14	sub	sub	ADJ
iajs-2724	202	15	-	-	ADJ
iajs-2724	202	16	ku	ku	NOUN
iajs-2724	202	17	-	-	PUNCT
iajs-2724	202	18	semigroup	semigroup	PROPN
iajs-2724	202	19	with	with	ADP
iajs-2724	202	20	thresholds	threshold	NOUN
iajs-2724	202	21	(	(	PUNCT
iajs-2724	202	22	𝛼	𝛼	X
iajs-2724	202	23	,	,	PUNCT
iajs-2724	202	24	𝛽	𝛽	NOUN
iajs-2724	202	25	)	)	PUNCT
iajs-2724	202	26	,	,	PUNCT
iajs-2724	202	27	(	(	PUNCT
iajs-2724	202	28	𝜔	𝜔	NOUN
iajs-2724	202	29	,	,	PUNCT
iajs-2724	202	30	𝜗	𝜗	NOUN
iajs-2724	202	31	)	)	PUNCT
iajs-2724	202	32	and	and	CCONJ
iajs-2724	202	33	cubic	cubic	ADJ
iajs-2724	202	34	bipolar	bipolar	ADJ
iajs-2724	202	35	k	k	NOUN
iajs-2724	202	36	-	-	NOUN
iajs-2724	202	37	ideal	ideal	NOUN
iajs-2724	202	38	with	with	ADP
iajs-2724	202	39	thresholds	threshold	NOUN
iajs-2724	202	40	(	(	PUNCT
iajs-2724	202	41	𝛼	𝛼	X
iajs-2724	202	42	,	,	PUNCT
iajs-2724	202	43	𝛽	𝛽	NOUN
iajs-2724	202	44	)	)	PUNCT
iajs-2724	202	45	,	,	PUNCT
iajs-2724	202	46	(	(	PUNCT
iajs-2724	202	47	𝜔	𝜔	NOUN
iajs-2724	202	48	,	,	PUNCT
iajs-2724	202	49	𝜗	𝜗	NOUN
iajs-2724	202	50	)	)	PUNCT
iajs-2724	202	51	of	of	ADP
iajs-2724	202	52	ℵ.	ℵ.	PROPN
iajs-2724	203	1	the	the	DET
iajs-2724	203	2	relationship	relationship	NOUN
iajs-2724	203	3	among	among	ADP
iajs-2724	203	4	these	these	DET
iajs-2724	203	5	types	type	NOUN
iajs-2724	203	6	of	of	ADP
iajs-2724	203	7	ideals	ideal	NOUN
iajs-2724	203	8	and	and	CCONJ
iajs-2724	203	9	some	some	DET
iajs-2724	203	10	properties	property	NOUN
iajs-2724	203	11	are	be	AUX
iajs-2724	203	12	studied	study	VERB
iajs-2724	203	13	,	,	PUNCT
iajs-2724	203	14	we	we	PRON
iajs-2724	203	15	obtained	obtain	VERB
iajs-2724	203	16	the	the	DET
iajs-2724	203	17	following	following	ADJ
iajs-2724	203	18	result	result	NOUN
iajs-2724	203	19	:	:	PUNCT
iajs-2724	203	20	every	every	DET
iajs-2724	203	21	(	(	PUNCT
iajs-2724	203	22	cbf	cbf	PROPN
iajs-2724	203	23	)	)	PUNCT
iajs-2724	203	24	sub	sub	PROPN
iajs-2724	203	25	-	-	ADJ
iajs-2724	203	26	ku	ku	ADJ
iajs-2724	203	27	-	-	PUNCT
iajs-2724	203	28	semi	semi	NOUN
iajs-2724	203	29	group	group	NOUN
iajs-2724	203	30	of	of	ADP
iajs-2724	203	31	ℵ	ℵ	PROPN
iajs-2724	203	32	is	be	AUX
iajs-2724	203	33	a	a	DET
iajs-2724	203	34	(	(	PUNCT
iajs-2724	203	35	cbf	cbf	PROPN
iajs-2724	203	36	)	)	PUNCT
iajs-2724	203	37	sub	sub	PROPN
iajs-2724	203	38	-	-	ADJ
iajs-2724	203	39	ku	ku	ADJ
iajs-2724	203	40	-	-	PUNCT
iajs-2724	203	41	semi	semi	NOUN
iajs-2724	203	42	group	group	NOUN
iajs-2724	203	43	with	with	ADP
iajs-2724	203	44	thresholds	threshold	NOUN
iajs-2724	203	45	(	(	PUNCT
iajs-2724	203	46	𝛼	𝛼	X
iajs-2724	203	47	,	,	PUNCT
iajs-2724	203	48	𝛽	𝛽	NOUN
iajs-2724	203	49	)	)	PUNCT
iajs-2724	203	50	,	,	PUNCT
iajs-2724	203	51	(	(	PUNCT
iajs-2724	203	52	𝜔	𝜔	NOUN
iajs-2724	203	53	,	,	PUNCT
iajs-2724	203	54	𝜗	𝜗	NOUN
iajs-2724	203	55	)	)	PUNCT
iajs-2724	203	56	of	of	ADP
iajs-2724	203	57	ℵ	ℵ	NOUN
iajs-2724	203	58	,	,	PUNCT
iajs-2724	203	59	but	but	CCONJ
iajs-2724	203	60	the	the	DET
iajs-2724	203	61	converse	converse	NOUN
iajs-2724	203	62	is	be	AUX
iajs-2724	203	63	not	not	PART
iajs-2724	203	64	true	true	ADJ
iajs-2724	203	65	.	.	PUNCT
iajs-2724	204	1	finally	finally	ADV
iajs-2724	204	2	,	,	PUNCT
iajs-2724	204	3	we	we	PRON
iajs-2724	204	4	proved	prove	VERB
iajs-2724	204	5	that	that	SCONJ
iajs-2724	204	6	a	a	DET
iajs-2724	204	7	cubic	cubic	ADJ
iajs-2724	204	8	bipolar	bipolar	ADJ
iajs-2724	204	9	fuzzy	fuzzy	ADJ
iajs-2724	204	10	k	k	NOUN
iajs-2724	204	11	-	-	NOUN
iajs-2724	204	12	ideal	ideal	NOUN
iajs-2724	204	13	with	with	ADP
iajs-2724	204	14	thresholds	threshold	NOUN
iajs-2724	204	15	(	(	PUNCT
iajs-2724	204	16	𝛼	𝛼	X
iajs-2724	204	17	,	,	PUNCT
iajs-2724	204	18	𝛽	𝛽	NOUN
iajs-2724	204	19	)	)	PUNCT
iajs-2724	204	20	,	,	PUNCT
iajs-2724	204	21	(	(	PUNCT
iajs-2724	204	22	𝜔	𝜔	NOUN
iajs-2724	204	23	,	,	PUNCT
iajs-2724	204	24	𝜗	𝜗	NOUN
iajs-2724	204	25	)	)	PUNCT
iajs-2724	204	26	and	and	CCONJ
iajs-2724	204	27	a	a	DET
iajs-2724	204	28	cubic	cubic	ADJ
iajs-2724	204	29	bipolar	bipolar	ADJ
iajs-2724	204	30	fuzzy	fuzzy	ADJ
iajs-2724	204	31	ideal	ideal	NOUN
iajs-2724	204	32	with	with	ADP
iajs-2724	204	33	thresholds	threshold	NOUN
iajs-2724	204	34	(	(	PUNCT
iajs-2724	204	35	𝛼	𝛼	X
iajs-2724	204	36	,	,	PUNCT
iajs-2724	204	37	𝛽	𝛽	NOUN
iajs-2724	204	38	)	)	PUNCT
iajs-2724	204	39	,	,	PUNCT
iajs-2724	204	40	(	(	PUNCT
iajs-2724	204	41	𝜔	𝜔	NOUN
iajs-2724	204	42	,	,	PUNCT
iajs-2724	204	43	𝜗	𝜗	NOUN
iajs-2724	204	44	)	)	PUNCT
iajs-2724	204	45	of	of	ADP
iajs-2724	204	46	a	a	DET
iajs-2724	204	47	ku	ku	NOUN
iajs-2724	204	48	-	-	PUNCT
iajs-2724	204	49	semi	semi	NOUN
iajs-2724	204	50	group	group	NOUN
iajs-2724	204	51	are	be	AUX
iajs-2724	204	52	equivalents	equivalent	NOUN
iajs-2724	204	53	.	.	PUNCT
iajs-2724	205	1	ibn	ibn	PROPN
iajs-2724	205	2	al	al	PROPN
iajs-2724	205	3	-	-	PUNCT
iajs-2724	205	4	haitham	haitham	PROPN
iajs-2724	205	5	jour	jour	X
iajs-2724	205	6	.	.	PROPN
iajs-2724	205	7	for	for	ADP
iajs-2724	205	8	pure	pure	ADJ
iajs-2724	205	9	&	&	CCONJ
iajs-2724	205	10	appl	appl	PROPN
iajs-2724	205	11	.	.	PUNCT
iajs-2724	206	1	sci	sci	PROPN
iajs-2724	206	2	.	.	PROPN
iajs-2724	207	1	53	53	NUM
iajs-2724	207	2	(	(	PUNCT
iajs-2724	207	3	2)2022	2)2022	NOUN
iajs-2724	207	4	58	58	NUM
iajs-2724	207	5	references	reference	NOUN
iajs-2724	207	6	1	1	NUM
iajs-2724	207	7	.	.	PUNCT
iajs-2724	208	1	zadeh	zadeh	PROPN
iajs-2724	208	2	,	,	PUNCT
iajs-2724	208	3	l.a	l.a	PROPN
iajs-2724	208	4	.	.	PROPN
iajs-2724	208	5	fuzzy	fuzzy	ADJ
iajs-2724	208	6	sets	set	NOUN
iajs-2724	208	7	,	,	PUNCT
iajs-2724	208	8	inform	inform	NOUN
iajs-2724	208	9	and	and	CCONJ
iajs-2724	208	10	control	control	NOUN
iajs-2724	208	11	,	,	PUNCT
iajs-2724	208	12	1965	1965	NUM
iajs-2724	208	13	,	,	PUNCT
iajs-2724	208	14	8	8	NUM
iajs-2724	208	15	,	,	PUNCT
iajs-2724	208	16	338	338	NUM
iajs-2724	208	17	-	-	SYM
iajs-2724	208	18	353	353	NUM
iajs-2724	208	19	.	.	NOUN
iajs-2724	208	20	2	2	NUM
iajs-2724	208	21	.	.	X
iajs-2724	208	22	mostafa	mostafa	PROPN
iajs-2724	208	23	,	,	PUNCT
iajs-2724	208	24	s.m	s.m	PROPN
iajs-2724	208	25	.	.	PROPN
iajs-2724	208	26	;	;	PUNCT
iajs-2724	208	27	abd	abd	PROPN
iajs-2724	208	28	-	-	PUNCT
iajs-2724	208	29	elnaby	elnaby	PROPN
iajs-2724	208	30	,	,	PUNCT
iajs-2724	208	31	m.a	m.a	PROPN
iajs-2724	208	32	.	.	PROPN
iajs-2724	208	33	;	;	PUNCT
iajs-2724	208	34	yousef	yousef	PROPN
iajs-2724	208	35	,	,	PUNCT
iajs-2724	208	36	m.m.m	m.m.m	INTJ
iajs-2724	208	37	.	.	PUNCT
iajs-2724	208	38	fuzzy	fuzzy	ADJ
iajs-2724	208	39	ideals	ideal	NOUN
iajs-2724	208	40	of	of	ADP
iajs-2724	208	41	ku	ku	PROPN
iajs-2724	208	42	-	-	PUNCT
iajs-2724	208	43	algebras	algebras	PROPN
iajs-2724	208	44	,	,	PUNCT
iajs-2724	208	45	int	int	PROPN
iajs-2724	208	46	.	.	PUNCT
iajs-2724	208	47	math	math	PROPN
iajs-2724	208	48	,	,	PUNCT
iajs-2724	208	49	forum	forum	PROPN
iajs-2724	208	50	,	,	PUNCT
iajs-2724	208	51	2011	2011	NUM
iajs-2724	208	52	,	,	PUNCT
iajs-2724	208	53	6(63	6(63	NUM
iajs-2724	208	54	)	)	PUNCT
iajs-2724	208	55	,	,	PUNCT
iajs-2724	208	56	3139	3139	NUM
iajs-2724	208	57	-	-	SYM
iajs-2724	208	58	3149	3149	NUM
iajs-2724	208	59	.	.	PUNCT
iajs-2724	209	1	3	3	X
iajs-2724	209	2	.	.	X
iajs-2724	209	3	mostafa	mostafa	PROPN
iajs-2724	209	4	,	,	PUNCT
iajs-2724	209	5	s.	s.	PROPN
iajs-2724	209	6	m.	m.	PROPN
iajs-2724	209	7	;	;	PUNCT
iajs-2724	209	8	kareem	kareem	PROPN
iajs-2724	209	9	,	,	PUNCT
iajs-2724	209	10	f.	f.	PROPN
iajs-2724	209	11	f.bipolar	f.bipolar	ADJ
iajs-2724	209	12	fuzzy	fuzzy	ADJ
iajs-2724	209	13	n	n	CCONJ
iajs-2724	209	14	-	-	ADJ
iajs-2724	209	15	fold	fold	ADJ
iajs-2724	209	16	ku	ku	NOUN
iajs-2724	209	17	-	-	PUNCT
iajs-2724	209	18	ideals	ideal	NOUN
iajs-2724	209	19	of	of	ADP
iajs-2724	209	20	ku	ku	PROPN
iajs-2724	209	21	-	-	PUNCT
iajs-2724	209	22	algebras	algebras	PROPN
iajs-2724	209	23	,	,	PUNCT
iajs-2724	209	24	mathematica	mathematica	PROPN
iajs-2724	209	25	aeterna	aeterna	PROPN
iajs-2724	209	26	,	,	PUNCT
iajs-2724	209	27	2014	2014	NUM
iajs-2724	209	28	,	,	PUNCT
iajs-2724	209	29	4	4	NUM
iajs-2724	209	30	,	,	PUNCT
iajs-2724	209	31	633	633	NUM
iajs-2724	209	32	-	-	SYM
iajs-2724	209	33	650	650	NUM
iajs-2724	209	34	.	.	NOUN
iajs-2724	210	1	4	4	NUM
iajs-2724	210	2	.	.	X
iajs-2724	211	1	jun	jun	PROPN
iajs-2724	211	2	,	,	PUNCT
iajs-2724	211	3	y.	y.	PROPN
iajs-2724	211	4	b.	b.	PROPN
iajs-2724	211	5	;	;	PUNCT
iajs-2724	211	6	kim	kim	PROPN
iajs-2724	211	7	,	,	PUNCT
iajs-2724	211	8	c.	c.	PROPN
iajs-2724	211	9	s.	s.	PROPN
iajs-2724	211	10	;	;	PUNCT
iajs-2724	211	11	kang	kang	PROPN
iajs-2724	211	12	,	,	PUNCT
iajs-2724	211	13	m.	m.	PROPN
iajs-2724	211	14	s.	s.	PROPN
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iajs-2724	211	16	subalgebras	subalgebras	PROPN
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iajs-2724	211	20	bck	bck	PROPN
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iajs-2724	211	22	bci	bci	NOUN
iajs-2724	211	23	-	-	PUNCT
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iajs-2724	211	25	,	,	PUNCT
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iajs-2724	211	31	sciences	science	NOUN
iajs-2724	211	32	,	,	PUNCT
iajs-2724	211	33	2010	2010	NUM
iajs-2724	211	34	,	,	PUNCT
iajs-2724	211	35	2	2	NUM
iajs-2724	211	36	(	(	PUNCT
iajs-2724	211	37	44	44	NUM
iajs-2724	211	38	)	)	PUNCT
iajs-2724	211	39	,	,	PUNCT
iajs-2724	211	40	239–250	239–250	NUM
iajs-2724	211	41	.	.	PUNCT
iajs-2724	212	1	5	5	NUM
iajs-2724	212	2	.	.	X
iajs-2724	212	3	jun	jun	PROPN
iajs-2724	212	4	,	,	PUNCT
iajs-2724	212	5	y.	y.	PROPN
iajs-2724	212	6	b.	b.	PROPN
iajs-2724	212	7	;	;	PUNCT
iajs-2724	212	8	kim	kim	PROPN
iajs-2724	212	9	,	,	PUNCT
iajs-2724	212	10	c.	c.	PROPN
iajs-2724	212	11	s.	s.	PROPN
iajs-2724	212	12	;	;	PUNCT
iajs-2724	212	13	kang	kang	PROPN
iajs-2724	212	14	,	,	PUNCT
iajs-2724	212	15	j.	j.	PROPN
iajs-2724	212	16	g.	g.	PROPN
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iajs-2724	213	1	q	q	NOUN
iajs-2724	213	2	-	-	PUNCT
iajs-2724	213	3	ideals	ideal	NOUN
iajs-2724	213	4	of	of	ADP
iajs-2724	213	5	bcialgebras	bcialgebra	NOUN
iajs-2724	213	6	,	,	PUNCT
iajs-2724	213	7	ann	ann	PROPN
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iajs-2724	213	9	fuzzy	fuzzy	PROPN
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iajs-2724	213	11	.	.	PROPN
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iajs-2724	213	13	,	,	PUNCT
iajs-2724	213	14	1	1	NUM
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iajs-2724	213	16	1	1	NUM
iajs-2724	213	17	)	)	PUNCT
iajs-2724	213	18	,	,	PUNCT
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iajs-2724	213	23	6	6	NUM
iajs-2724	213	24	.	.	X
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iajs-2724	213	26	,	,	PUNCT
iajs-2724	213	27	y.	y.	PROPN
iajs-2724	213	28	b.	b.	PROPN
iajs-2724	213	29	;	;	PUNCT
iajs-2724	213	30	kim	kim	PROPN
iajs-2724	213	31	,	,	PUNCT
iajs-2724	213	32	c.	c.	PROPN
iajs-2724	213	33	s.	s.	PROPN
iajs-2724	213	34	;	;	PUNCT
iajs-2724	213	35	yang	yang	PROPN
iajs-2724	213	36	,	,	PUNCT
iajs-2724	213	37	k.	k.	PROPN
iajs-2724	213	38	o.	o.	PROPN
iajs-2724	213	39	cubic	cubic	PROPN
iajs-2724	213	40	sets	set	NOUN
iajs-2724	213	41	,	,	PUNCT
iajs-2724	213	42	ann	ann	PROPN
iajs-2724	213	43	.	.	PROPN
iajs-2724	213	44	fuzzy	fuzzy	ADJ
iajs-2724	213	45	math	math	NOUN
iajs-2724	213	46	.	.	PUNCT
iajs-2724	214	1	inform	inform	NOUN
iajs-2724	214	2	.	.	PUNCT
iajs-2724	215	1	2012,4	2012,4	NUM
iajs-2724	215	2	(	(	PUNCT
iajs-2724	215	3	1	1	NUM
iajs-2724	215	4	)	)	PUNCT
iajs-2724	215	5	8398	8398	NUM
iajs-2724	215	6	.	.	PUNCT
iajs-2724	216	1	7	7	X
iajs-2724	216	2	.	.	X
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iajs-2724	216	4	,	,	PUNCT
iajs-2724	216	5	f.	f.	PROPN
iajs-2724	216	6	f.	f.	PROPN
iajs-2724	216	7	;	;	PUNCT
iajs-2724	216	8	hasan	hasan	PROPN
iajs-2724	216	9	,	,	PUNCT
iajs-2724	216	10	o.	o.	PROPN
iajs-2724	216	11	a.	a.	PROPN
iajs-2724	216	12	cubic	cubic	PROPN
iajs-2724	216	13	ideals	ideal	NOUN
iajs-2724	216	14	of	of	ADP
iajs-2724	216	15	semigroup	semigroup	NOUN
iajs-2724	216	16	in	in	ADP
iajs-2724	216	17	ku	ku	PROPN
iajs-2724	216	18	-	-	PUNCT
iajs-2724	216	19	algebra	algebra	PROPN
iajs-2724	216	20	,	,	PUNCT
iajs-2724	216	21	journal	journal	NOUN
iajs-2724	216	22	of	of	ADP
iajs-2724	216	23	physics	physics	PROPN
iajs-2724	216	24	:	:	PUNCT
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iajs-2724	216	26	series	series	NOUN
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iajs-2724	216	28	(	(	PUNCT
iajs-2724	216	29	2021	2021	NUM
iajs-2724	216	30	)	)	PUNCT
iajs-2724	216	31	012018	012018	NUM
iajs-2724	216	32	iop	iop	PROPN
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iajs-2724	216	34	8	8	NUM
iajs-2724	216	35	.	.	PUNCT
iajs-2724	216	36	kareem	kareem	PROPN
iajs-2724	216	37	,	,	PUNCT
iajs-2724	216	38	f.	f.	PROPN
iajs-2724	216	39	f.	f.	PROPN
iajs-2724	216	40	;	;	PUNCT
iajs-2724	216	41	hasan	hasan	PROPN
iajs-2724	216	42	,	,	PUNCT
iajs-2724	216	43	o.	o.	PROPN
iajs-2724	216	44	a.	a.	PROPN
iajs-2724	216	45	the	the	DET
iajs-2724	216	46	homomorphism	homomorphism	NOUN
iajs-2724	216	47	of	of	ADP
iajs-2724	216	48	a	a	DET
iajs-2724	216	49	cubic	cubic	ADJ
iajs-2724	216	50	set	set	NOUN
iajs-2724	216	51	of	of	ADP
iajs-2724	216	52	a	a	DET
iajs-2724	216	53	semigroup	semigroup	NOUN
iajs-2724	216	54	in	in	ADP
iajs-2724	216	55	a	a	DET
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iajs-2724	216	57	,	,	PUNCT
iajs-2724	216	58	journal	journal	NOUN
iajs-2724	216	59	of	of	ADP
iajs-2724	216	60	physics	physics	PROPN
iajs-2724	216	61	,	,	PUNCT
iajs-2724	216	62	1879	1879	NUM
iajs-2724	216	63	(	(	PUNCT
iajs-2724	216	64	2021	2021	NUM
iajs-2724	216	65	)	)	PUNCT
iajs-2724	216	66	022119	022119	NUM
iajs-2724	217	1	iop	iop	NOUN
iajs-2724	217	2	publishing	publishing	NOUN
iajs-2724	217	3	.	.	PUNCT
iajs-2724	218	1	9	9	X
iajs-2724	218	2	.	.	X
iajs-2724	218	3	kareem	kareem	PROPN
iajs-2724	218	4	,	,	PUNCT
iajs-2724	218	5	f.f	f.f	PROPN
iajs-2724	218	6	.	.	PROPN
iajs-2724	218	7	;	;	PUNCT
iajs-2724	218	8	hasan	hasan	PROPN
iajs-2724	218	9	,	,	PUNCT
iajs-2724	218	10	e.	e.	PROPN
iajs-2724	218	11	r.	r.	PROPN
iajs-2724	218	12	bipolar	bipolar	PROPN
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iajs-2724	218	14	k	k	NOUN
iajs-2724	218	15	-	-	NOUN
iajs-2724	218	16	ideals	ideal	NOUN
iajs-2724	218	17	in	in	ADP
iajs-2724	218	18	ku	ku	PROPN
iajs-2724	218	19	-	-	PUNCT
iajs-2724	218	20	semigroups	semigroup	NOUN
iajs-2724	218	21	,	,	PUNCT
iajs-2724	218	22	journal	journal	NOUN
iajs-2724	218	23	of	of	ADP
iajs-2724	218	24	new	new	ADJ
iajs-2724	218	25	theory	theory	NOUN
iajs-2724	218	26	,	,	PUNCT
iajs-2724	218	27	2019,9	2019,9	NUM
iajs-2724	218	28	,	,	PUNCT
iajs-2724	218	29	71	71	NUM
iajs-2724	218	30	-	-	SYM
iajs-2724	218	31	78	78	NUM
iajs-2724	218	32	.	.	PUNCT
iajs-2724	219	1	10	10	NUM
iajs-2724	219	2	.	.	PUNCT
iajs-2724	220	1	k	k	PROPN
iajs-2724	220	2	awad	awad	PROPN
iajs-2724	220	3	,	,	PUNCT
iajs-2724	220	4	wisam	wisam	NOUN
iajs-2724	220	5	k.	k.	PROPN
iajs-2724	220	6	,	,	PUNCT
iajs-2724	220	7	and	and	CCONJ
iajs-2724	220	8	fatema	fatema	PROPN
iajs-2724	220	9	f.	f.	PROPN
iajs-2724	220	10	kareem	kareem	PROPN
iajs-2724	220	11	.	.	PUNCT
iajs-2724	221	1	the	the	DET
iajs-2724	221	2	homomorphism	homomorphism	NOUN
iajs-2724	221	3	of	of	ADP
iajs-2724	221	4	cubic	cubic	ADJ
iajs-2724	221	5	bipolar	bipolar	ADJ
iajs-2724	221	6	ideals	ideal	NOUN
iajs-2724	221	7	of	of	ADP
iajs-2724	221	8	a	a	DET
iajs-2724	221	9	ku	ku	PROPN
iajs-2724	221	10	-	-	PUNCT
iajs-2724	221	11	semigroup	semigroup	NOUN
iajs-2724	221	12	.	.	PUNCT
iajs-2724	222	1	ibn	ibn	PROPN
iajs-2724	222	2	al	al	PROPN
iajs-2724	222	3	-	-	PUNCT
iajs-2724	222	4	haitham	haitham	PROPN
iajs-2724	222	5	journal	journal	PROPN
iajs-2724	222	6	for	for	ADP
iajs-2724	222	7	pure	pure	ADJ
iajs-2724	222	8	and	and	CCONJ
iajs-2724	222	9	applied	applied	ADJ
iajs-2724	222	10	sciences	science	NOUN
iajs-2724	222	11	,	,	PUNCT
iajs-2724	222	12	2022	2022	NUM
iajs-2724	222	13	,	,	PUNCT
iajs-2724	222	14	35(1	35(1	NUM
iajs-2724	222	15	)	)	PUNCT
iajs-2724	222	16	.	.	PUNCT
iajs-2724	222	17	,	,	PUNCT
iajs-2724	222	18	73	73	NUM
iajs-2724	222	19	-	-	SYM
iajs-2724	222	20	83	83	NUM
iajs-2724	222	21	.	.	PUNCT
iajs-2724	223	1	11	11	NUM
iajs-2724	223	2	.	.	X
iajs-2724	223	3	kareem	kareem	PROPN
iajs-2724	223	4	,	,	PUNCT
iajs-2724	223	5	f.f	f.f	PROPN
iajs-2724	223	6	.	.	PROPN
iajs-2724	223	7	;	;	PUNCT
iajs-2724	223	8	abed	abe	VERB
iajs-2724	223	9	,	,	PUNCT
iajs-2724	223	10	m.	m.	NOUN
iajs-2724	223	11	m.	m.	NOUN
iajs-2724	223	12	generalizations	generalization	NOUN
iajs-2724	223	13	of	of	ADP
iajs-2724	223	14	fuzzy	fuzzy	ADJ
iajs-2724	223	15	k	k	NOUN
iajs-2724	223	16	-	-	NOUN
iajs-2724	223	17	ideals	ideal	NOUN
iajs-2724	223	18	in	in	ADP
iajs-2724	223	19	a	a	DET
iajs-2724	223	20	ku	ku	NOUN
iajs-2724	223	21	-	-	PUNCT
iajs-2724	223	22	algebra	algebra	PROPN
iajs-2724	223	23	with	with	ADP
iajs-2724	223	24	semigroup	semigroup	PROPN
iajs-2724	223	25	,	,	PUNCT
iajs-2724	223	26	journal	journal	NOUN
iajs-2724	223	27	of	of	ADP
iajs-2724	223	28	physics	physics	PROPN
iajs-2724	223	29	,	,	PUNCT
iajs-2724	223	30	2021	2021	NUM
iajs-2724	223	31	,	,	PUNCT
iajs-2724	223	32	1879,022108	1879,022108	NUM
iajs-2724	223	33	iop	iop	NOUN
iajs-2724	223	34	publishing	publishing	NOUN
iajs-2724	223	35	.	.	PUNCT
iajs-2724	224	1	12	12	NUM
iajs-2724	224	2	.	.	PUNCT
iajs-2724	224	3	prabpayak	prabpayak	NOUN
iajs-2724	224	4	,	,	PUNCT
iajs-2724	224	5	c.	c.	PROPN
iajs-2724	224	6	leerawat	leerawat	PROPN
iajs-2724	224	7	,	,	PUNCT
iajs-2724	224	8	u.	u.	PROPN
iajs-2724	224	9	on	on	ADP
iajs-2724	224	10	ideals	ideal	NOUN
iajs-2724	224	11	and	and	CCONJ
iajs-2724	224	12	congruence	congruence	NOUN
iajs-2724	224	13	in	in	ADP
iajs-2724	224	14	ku	ku	PROPN
iajs-2724	224	15	-	-	PUNCT
iajs-2724	224	16	algebras	algebras	PROPN
iajs-2724	224	17	,	,	PUNCT
iajs-2724	224	18	scientia	scientia	PROPN
iajs-2724	224	19	magna	magna	PROPN
iajs-2724	224	20	,	,	PUNCT
iajs-2724	224	21	2009	2009	NUM
iajs-2724	224	22	,	,	PUNCT
iajs-2724	224	23	5(1	5(1	NUM
iajs-2724	224	24	)	)	PUNCT
iajs-2724	224	25	,	,	PUNCT
iajs-2724	224	26	54	54	NUM
iajs-2724	224	27	-	-	SYM
iajs-2724	224	28	57	57	NUM
iajs-2724	224	29	.	.	PUNCT
iajs-2724	225	1	13	13	NUM
iajs-2724	225	2	.	.	X
iajs-2724	225	3	kareem	kareem	PROPN
iajs-2724	225	4	,	,	PUNCT
iajs-2724	225	5	f.f	f.f	PROPN
iajs-2724	225	6	.	.	PROPN
iajs-2724	225	7	;	;	PUNCT
iajs-2724	225	8	hasan	hasan	PROPN
iajs-2724	225	9	,	,	PUNCT
iajs-2724	225	10	e.	e.	PROPN
iajs-2724	225	11	r.	r.	PROPN
iajs-2724	225	12	on	on	ADP
iajs-2724	225	13	ku	ku	PROPN
iajs-2724	225	14	-	-	PUNCT
iajs-2724	225	15	semigroups	semigroup	NOUN
iajs-2724	225	16	,	,	PUNCT
iajs-2724	225	17	international	international	ADJ
iajs-2724	225	18	journal	journal	NOUN
iajs-2724	225	19	of	of	ADP
iajs-2724	225	20	science	science	NOUN
iajs-2724	225	21	and	and	CCONJ
iajs-2724	225	22	nature	nature	NOUN
iajs-2724	225	23	.	.	PUNCT
iajs-2724	226	1	2018	2018	NUM
iajs-2724	226	2	,	,	PUNCT
iajs-2724	226	3	9	9	NUM
iajs-2724	226	4	(	(	PUNCT
iajs-2724	226	5	1	1	NUM
iajs-2724	226	6	)	)	PUNCT
iajs-2724	226	7	,	,	PUNCT
iajs-2724	226	8	79	79	NUM
iajs-2724	226	9	-	-	SYM
iajs-2724	226	10	84	84	NUM
iajs-2724	226	11	.	.	PUNCT