id	sid	tid	token	lemma	pos
iajs-3112	1	1	454	454	NUM
iajs-3112	1	2	this	this	DET
iajs-3112	1	3	work	work	NOUN
iajs-3112	1	4	is	be	AUX
iajs-3112	1	5	licensed	license	VERB
iajs-3112	1	6	under	under	ADP
iajs-3112	1	7	a	a	DET
iajs-3112	1	8	creative	creative	ADJ
iajs-3112	1	9	commons	common	NOUN
iajs-3112	1	10	attribution	attribution	NOUN
iajs-3112	1	11	4.0	4.0	NUM
iajs-3112	1	12	international	international	ADJ
iajs-3112	1	13	license	license	NOUN
iajs-3112	1	14	ihjpas	ihjpa	NOUN
iajs-3112	1	15	.	.	PUNCT
iajs-3112	2	1	37	37	NUM
iajs-3112	2	2	(	(	PUNCT
iajs-3112	2	3	1	1	NUM
iajs-3112	2	4	)	)	PUNCT
iajs-3112	2	5	2024	2024	NUM
iajs-3112	2	6	ibn	ibn	PROPN
iajs-3112	2	7	al	al	PROPN
iajs-3112	2	8	-	-	PUNCT
iajs-3112	2	9	haitham	haitham	PROPN
iajs-3112	2	10	journal	journal	PROPN
iajs-3112	2	11	for	for	ADP
iajs-3112	2	12	pure	pure	ADJ
iajs-3112	2	13	and	and	CCONJ
iajs-3112	2	14	applied	applied	ADJ
iajs-3112	2	15	sciences	sciences	PROPN
iajs-3112	2	16	journal	journal	PROPN
iajs-3112	2	17	homepage	homepage	NOUN
iajs-3112	2	18	:	:	PUNCT
iajs-3112	2	19	jih.uobaghdad.edu.iq	jih.uobaghdad.edu.iq	NOUN
iajs-3112	2	20	pissn	pissn	ADJ
iajs-3112	2	21	:	:	PUNCT
iajs-3112	2	22	1609	1609	NUM
iajs-3112	2	23	-	-	SYM
iajs-3112	2	24	4042	4042	NUM
iajs-3112	2	25	,	,	PUNCT
iajs-3112	2	26	eissn	eissn	NOUN
iajs-3112	2	27	:	:	PUNCT
iajs-3112	2	28	2521	2521	NUM
iajs-3112	2	29	-	-	SYM
iajs-3112	2	30	3407	3407	NUM
iajs-3112	2	31	1	1	NUM
iajs-3112	2	32	samy	samy	NOUN
iajs-3112	2	33	m.	m.	NOUN
iajs-3112	2	34	mostafa	mostafa	PROPN
iajs-3112	2	35	2fatema	2fatema	AUX
iajs-3112	2	36	.	.	PUNCT
iajs-3112	3	1	f.	f.	PROPN
iajs-3112	3	2	kareem	kareem	PROPN
iajs-3112	3	3	*	*	PROPN
iajs-3112	4	1	3omniat	3omniat	PROPN
iajs-3112	4	2	.	.	PUNCT
iajs-3112	4	3	a.	a.	PROPN
iajs-3112	4	4	hasan	hasan	PROPN
iajs-3112	4	5	1	1	NUM
iajs-3112	4	6	department	department	PROPN
iajs-3112	4	7	of	of	ADP
iajs-3112	4	8	mathematics	mathematic	NOUN
iajs-3112	4	9	,	,	PUNCT
iajs-3112	4	10	faculty	faculty	NOUN
iajs-3112	4	11	of	of	ADP
iajs-3112	4	12	education	education	NOUN
iajs-3112	4	13	,	,	PUNCT
iajs-3112	4	14	ain	ain	PROPN
iajs-3112	4	15	shams	shams	PROPN
iajs-3112	4	16	university	university	PROPN
iajs-3112	4	17	,	,	PUNCT
iajs-3112	4	18	roxy	roxy	PROPN
iajs-3112	4	19	,	,	PUNCT
iajs-3112	4	20	cairo	cairo	PROPN
iajs-3112	4	21	,	,	PUNCT
iajs-3112	4	22	egypt	egypt	PROPN
iajs-3112	4	23	.	.	PUNCT
iajs-3112	5	1	2,3	2,3	NUM
iajs-3112	5	2	department	department	NOUN
iajs-3112	5	3	of	of	ADP
iajs-3112	5	4	mathematics	mathematic	NOUN
iajs-3112	5	5	,	,	PUNCT
iajs-3112	5	6	college	college	NOUN
iajs-3112	5	7	of	of	ADP
iajs-3112	5	8	education	education	NOUN
iajs-3112	5	9	for	for	ADP
iajs-3112	5	10	pure	pure	ADJ
iajs-3112	5	11	sciences	science	NOUN
iajs-3112	5	12	,	,	PUNCT
iajs-3112	5	13	ibn	ibn	PROPN
iajs-3112	5	14	al	al	PROPN
iajs-3112	5	15	–	–	PUNCT
iajs-3112	5	16	haitham	haitham	PROPN
iajs-3112	5	17	,	,	PUNCT
iajs-3112	5	18	university	university	PROPN
iajs-3112	5	19	of	of	ADP
iajs-3112	5	20	baghdad	baghdad	PROPN
iajs-3112	5	21	,	,	PUNCT
iajs-3112	5	22	baghdad	baghdad	PROPN
iajs-3112	5	23	,	,	PUNCT
iajs-3112	5	24	iraq	iraq	PROPN
iajs-3112	5	25	.	.	PUNCT
iajs-3112	6	1	*	*	PUNCT
iajs-3112	6	2	corresponding	correspond	VERB
iajs-3112	6	3	author	author	NOUN
iajs-3112	6	4	.	.	PUNCT
iajs-3112	7	1	fatma.f.k@ihcoedu.uobaghdad.edu.iq	fatma.f.k@ihcoedu.uobaghdad.edu.iq	NOUN
iajs-3112	7	2	abstract	abstract	NOUN
iajs-3112	7	3	in	in	ADP
iajs-3112	7	4	this	this	DET
iajs-3112	7	5	paper	paper	NOUN
iajs-3112	7	6	,	,	PUNCT
iajs-3112	7	7	we	we	PRON
iajs-3112	7	8	define	define	VERB
iajs-3112	7	9	a	a	DET
iajs-3112	7	10	cubic	cubic	ADJ
iajs-3112	7	11	positive	positive	ADJ
iajs-3112	7	12	implicative	implicative	ADJ
iajs-3112	7	13	-	-	PUNCT
iajs-3112	7	14	ideal	ideal	NOUN
iajs-3112	7	15	,	,	PUNCT
iajs-3112	7	16	a	a	DET
iajs-3112	7	17	cubic	cubic	ADJ
iajs-3112	7	18	implicative	implicative	ADJ
iajs-3112	7	19	-	-	PUNCT
iajs-3112	7	20	ideal	ideal	NOUN
iajs-3112	7	21	and	and	CCONJ
iajs-3112	7	22	a	a	DET
iajs-3112	7	23	cubic	cubic	ADJ
iajs-3112	7	24	commutative	commutative	ADJ
iajs-3112	7	25	-	-	PUNCT
iajs-3112	7	26	ideal	ideal	NOUN
iajs-3112	7	27	of	of	ADP
iajs-3112	7	28	a	a	DET
iajs-3112	7	29	semigroup	semigroup	NOUN
iajs-3112	7	30	in	in	ADP
iajs-3112	7	31	ku	ku	PROPN
iajs-3112	7	32	-	-	PUNCT
iajs-3112	7	33	algebra	algebra	PROPN
iajs-3112	7	34	as	as	ADP
iajs-3112	7	35	a	a	DET
iajs-3112	7	36	generalization	generalization	NOUN
iajs-3112	7	37	of	of	ADP
iajs-3112	7	38	a	a	DET
iajs-3112	7	39	fuzzy	fuzzy	ADJ
iajs-3112	7	40	(	(	PUNCT
iajs-3112	7	41	positive	positive	ADJ
iajs-3112	7	42	implicative	implicative	ADJ
iajs-3112	7	43	-	-	PUNCT
iajs-3112	7	44	ideal	ideal	NOUN
iajs-3112	7	45	,	,	PUNCT
iajs-3112	7	46	an	an	DET
iajs-3112	7	47	implicative	implicative	ADJ
iajs-3112	7	48	-	-	PUNCT
iajs-3112	7	49	ideal	ideal	NOUN
iajs-3112	7	50	and	and	CCONJ
iajs-3112	7	51	a	a	DET
iajs-3112	7	52	commutative	commutative	ADJ
iajs-3112	7	53	-	-	PUNCT
iajs-3112	7	54	ideal	ideal	NOUN
iajs-3112	7	55	)	)	PUNCT
iajs-3112	7	56	of	of	ADP
iajs-3112	7	57	a	a	DET
iajs-3112	7	58	semigroup	semigroup	NOUN
iajs-3112	7	59	in	in	ADP
iajs-3112	7	60	ku	ku	PROPN
iajs-3112	7	61	-	-	PUNCT
iajs-3112	7	62	algebra	algebra	PROPN
iajs-3112	7	63	.	.	PUNCT
iajs-3112	8	1	some	some	DET
iajs-3112	8	2	relations	relation	NOUN
iajs-3112	8	3	between	between	ADP
iajs-3112	8	4	these	these	DET
iajs-3112	8	5	types	type	NOUN
iajs-3112	8	6	of	of	ADP
iajs-3112	8	7	cubic	cubic	ADJ
iajs-3112	8	8	ideals	ideal	NOUN
iajs-3112	8	9	are	be	AUX
iajs-3112	8	10	discussed	discuss	VERB
iajs-3112	8	11	.	.	PUNCT
iajs-3112	9	1	also	also	ADV
iajs-3112	9	2	,	,	PUNCT
iajs-3112	9	3	some	some	DET
iajs-3112	9	4	important	important	ADJ
iajs-3112	9	5	properties	property	NOUN
iajs-3112	9	6	of	of	ADP
iajs-3112	9	7	these	these	DET
iajs-3112	9	8	ideals	ideal	NOUN
iajs-3112	9	9	are	be	AUX
iajs-3112	9	10	studied	study	VERB
iajs-3112	9	11	.	.	PUNCT
iajs-3112	10	1	finally	finally	ADV
iajs-3112	10	2	,	,	PUNCT
iajs-3112	10	3	some	some	DET
iajs-3112	10	4	important	important	ADJ
iajs-3112	10	5	theories	theory	NOUN
iajs-3112	10	6	are	be	AUX
iajs-3112	10	7	discussed	discuss	VERB
iajs-3112	10	8	.	.	PUNCT
iajs-3112	11	1	it	it	PRON
iajs-3112	11	2	is	be	AUX
iajs-3112	11	3	proved	prove	VERB
iajs-3112	11	4	that	that	SCONJ
iajs-3112	11	5	every	every	DET
iajs-3112	11	6	cubic	cubic	ADJ
iajs-3112	11	7	commutative	commutative	ADJ
iajs-3112	11	8	-	-	PUNCT
iajs-3112	11	9	ideal	ideal	ADJ
iajs-3112	11	10	,	,	PUNCT
iajs-3112	11	11	cubic	cubic	ADJ
iajs-3112	11	12	positive	positive	ADJ
iajs-3112	11	13	implicative	implicative	ADJ
iajs-3112	11	14	-	-	PUNCT
iajs-3112	11	15	ideal	ideal	NOUN
iajs-3112	11	16	,	,	PUNCT
iajs-3112	11	17	and	and	CCONJ
iajs-3112	11	18	cubic	cubic	ADJ
iajs-3112	11	19	implicative	implicative	ADJ
iajs-3112	11	20	-	-	PUNCT
iajs-3112	11	21	ideal	ideal	NOUN
iajs-3112	11	22	are	be	AUX
iajs-3112	11	23	a	a	DET
iajs-3112	11	24	cubic	cubic	ADJ
iajs-3112	11	25	ideal	ideal	NOUN
iajs-3112	11	26	,	,	PUNCT
iajs-3112	11	27	but	but	CCONJ
iajs-3112	11	28	not	not	PART
iajs-3112	11	29	conversely	conversely	ADV
iajs-3112	11	30	.	.	PUNCT
iajs-3112	12	1	also	also	ADV
iajs-3112	12	2	,	,	PUNCT
iajs-3112	12	3	we	we	PRON
iajs-3112	12	4	show	show	VERB
iajs-3112	12	5	that	that	SCONJ
iajs-3112	12	6	if	if	SCONJ
iajs-3112	12	7	θ	θ	PROPN
iajs-3112	12	8	is	be	AUX
iajs-3112	12	9	a	a	DET
iajs-3112	12	10	cubic	cubic	ADJ
iajs-3112	12	11	positive	positive	ADJ
iajs-3112	12	12	implicative	implicative	ADJ
iajs-3112	12	13	-	-	PUNCT
iajs-3112	12	14	ideal	ideal	NOUN
iajs-3112	12	15	and	and	CCONJ
iajs-3112	12	16	a	a	DET
iajs-3112	12	17	cubic	cubic	ADJ
iajs-3112	12	18	commutative	commutative	ADJ
iajs-3112	12	19	-	-	PUNCT
iajs-3112	12	20	ideal	ideal	NOUN
iajs-3112	12	21	then	then	ADV
iajs-3112	12	22	θ	θ	PROPN
iajs-3112	12	23	is	be	AUX
iajs-3112	12	24	a	a	DET
iajs-3112	12	25	cubic	cubic	ADJ
iajs-3112	12	26	implicative	implicative	ADJ
iajs-3112	12	27	-	-	PUNCT
iajs-3112	12	28	ideal	ideal	NOUN
iajs-3112	12	29	.	.	PUNCT
iajs-3112	13	1	some	some	DET
iajs-3112	13	2	examples	example	NOUN
iajs-3112	13	3	of	of	ADP
iajs-3112	13	4	the	the	DET
iajs-3112	13	5	opposite	opposite	ADJ
iajs-3112	13	6	direction	direction	NOUN
iajs-3112	13	7	of	of	ADP
iajs-3112	13	8	the	the	DET
iajs-3112	13	9	previous	previous	ADJ
iajs-3112	13	10	theories	theory	NOUN
iajs-3112	13	11	are	be	AUX
iajs-3112	13	12	obtained	obtain	VERB
iajs-3112	13	13	.	.	PUNCT
iajs-3112	14	1	keywords	keyword	VERB
iajs-3112	14	2	a	a	DET
iajs-3112	14	3	ku	ku	PROPN
iajs-3112	14	4	-	-	PUNCT
iajs-3112	14	5	semigroup	semigroup	PROPN
iajs-3112	14	6	,	,	PUNCT
iajs-3112	14	7	cubic	cubic	ADJ
iajs-3112	14	8	ideal	ideal	NOUN
iajs-3112	14	9	,	,	PUNCT
iajs-3112	14	10	cubic	cubic	ADJ
iajs-3112	14	11	k	k	NOUN
iajs-3112	14	12	-	-	PUNCT
iajs-3112	14	13	ideal	ideal	ADJ
iajs-3112	14	14	,	,	PUNCT
iajs-3112	14	15	cubic	cubic	ADJ
iajs-3112	14	16	positive	positive	ADJ
iajs-3112	14	17	implicative	implicative	ADJ
iajs-3112	14	18	ideal	ideal	NOUN
iajs-3112	14	19	,	,	PUNCT
iajs-3112	14	20	cubic	cubic	ADJ
iajs-3112	14	21	implicative	implicative	ADJ
iajs-3112	14	22	ideal	ideal	NOUN
iajs-3112	14	23	,	,	PUNCT
iajs-3112	14	24	cubic	cubic	ADJ
iajs-3112	14	25	commutative	commutative	ADJ
iajs-3112	14	26	ideal	ideal	NOUN
iajs-3112	14	27	.	.	PUNCT
iajs-3112	15	1	1	1	X
iajs-3112	15	2	.	.	X
iajs-3112	15	3	introduction	introduction	NOUN
iajs-3112	15	4	the	the	DET
iajs-3112	15	5	structure	structure	NOUN
iajs-3112	15	6	of	of	ADP
iajs-3112	15	7	ku	ku	PROPN
iajs-3112	15	8	-	-	PUNCT
iajs-3112	15	9	algebra	algebra	PROPN
iajs-3112	15	10	was	be	AUX
iajs-3112	15	11	studied	study	VERB
iajs-3112	15	12	by	by	ADP
iajs-3112	15	13	prabpayak	prabpayak	NOUN
iajs-3112	15	14	and	and	CCONJ
iajs-3112	15	15	leerawat	leerawat	VERB
iajs-3112	16	1	[	[	X
iajs-3112	16	2	1,2	1,2	NUM
iajs-3112	16	3	]	]	PUNCT
iajs-3112	16	4	.	.	PUNCT
iajs-3112	17	1	they	they	PRON
iajs-3112	17	2	gave	give	VERB
iajs-3112	17	3	a	a	DET
iajs-3112	17	4	homomorphism	homomorphism	NOUN
iajs-3112	17	5	of	of	ADP
iajs-3112	17	6	ku	ku	PROPN
iajs-3112	17	7	-	-	PUNCT
iajs-3112	17	8	algebras	algebras	PROPN
iajs-3112	17	9	and	and	CCONJ
iajs-3112	17	10	proved	prove	VERB
iajs-3112	17	11	some	some	DET
iajs-3112	17	12	important	important	ADJ
iajs-3112	17	13	theories	theory	NOUN
iajs-3112	17	14	.	.	PUNCT
iajs-3112	18	1	the	the	DET
iajs-3112	18	2	idea	idea	NOUN
iajs-3112	18	3	of	of	ADP
iajs-3112	18	4	a	a	DET
iajs-3112	18	5	fuzzy	fuzzy	ADJ
iajs-3112	18	6	set	set	NOUN
iajs-3112	18	7	was	be	AUX
iajs-3112	18	8	initialed	initial	VERB
iajs-3112	18	9	in	in	ADP
iajs-3112	18	10	1965	1965	NUM
iajs-3112	18	11	by	by	ADP
iajs-3112	18	12	the	the	DET
iajs-3112	18	13	author	author	NOUN
iajs-3112	18	14	zadeh	zadeh	PROPN
iajs-3112	19	1	[	[	X
iajs-3112	19	2	3	3	NUM
iajs-3112	19	3	]	]	PUNCT
iajs-3112	19	4	.	.	PUNCT
iajs-3112	20	1	since	since	SCONJ
iajs-3112	20	2	then	then	ADV
iajs-3112	20	3	this	this	DET
iajs-3112	20	4	concept	concept	NOUN
iajs-3112	20	5	has	have	AUX
iajs-3112	20	6	been	be	AUX
iajs-3112	20	7	applied	apply	VERB
iajs-3112	20	8	in	in	ADP
iajs-3112	20	9	many	many	ADJ
iajs-3112	20	10	different	different	ADJ
iajs-3112	20	11	branches	branch	NOUN
iajs-3112	20	12	of	of	ADP
iajs-3112	20	13	mathematic	mathematic	NOUN
iajs-3112	20	14	.	.	PUNCT
iajs-3112	21	1	the	the	DET
iajs-3112	21	2	study	study	NOUN
iajs-3112	21	3	of	of	ADP
iajs-3112	21	4	fuzzy	fuzzy	ADJ
iajs-3112	21	5	algebraic	algebraic	ADJ
iajs-3112	21	6	structures	structure	NOUN
iajs-3112	21	7	was	be	AUX
iajs-3112	21	8	started	start	VERB
iajs-3112	21	9	with	with	ADP
iajs-3112	21	10	the	the	DET
iajs-3112	21	11	introduction	introduction	NOUN
iajs-3112	21	12	of	of	ADP
iajs-3112	21	13	the	the	DET
iajs-3112	21	14	concept	concept	NOUN
iajs-3112	21	15	of	of	ADP
iajs-3112	21	16	fuzzy	fuzzy	ADJ
iajs-3112	21	17	groups	group	NOUN
iajs-3112	21	18	by	by	ADP
iajs-3112	21	19	rosenfeld	rosenfeld	PROPN
iajs-3112	21	20	[	[	X
iajs-3112	21	21	4	4	NUM
iajs-3112	21	22	]	]	PUNCT
iajs-3112	21	23	.	.	PUNCT
iajs-3112	22	1	fuzzy	fuzzy	ADJ
iajs-3112	22	2	p	p	NOUN
iajs-3112	22	3	-	-	PUNCT
iajs-3112	22	4	ideals	ideal	NOUN
iajs-3112	22	5	and	and	CCONJ
iajs-3112	22	6	fuzzy	fuzzy	ADJ
iajs-3112	22	7	h	h	NOUN
iajs-3112	22	8	-	-	PUNCT
iajs-3112	22	9	ideals	ideal	NOUN
iajs-3112	22	10	in	in	ADP
iajs-3112	22	11	bci	bci	NOUN
iajs-3112	22	12	-	-	PUNCT
iajs-3112	22	13	algebras	algebra	NOUN
iajs-3112	22	14	are	be	AUX
iajs-3112	22	15	introduced	introduce	VERB
iajs-3112	22	16	in	in	ADP
iajs-3112	22	17	[	[	X
iajs-3112	22	18	5	5	NUM
iajs-3112	22	19	]	]	PUNCT
iajs-3112	22	20	.	.	PUNCT
iajs-3112	23	1	jun	jun	PROPN
iajs-3112	23	2	et	et	PROPN
iajs-3112	23	3	al	al	PROPN
iajs-3112	23	4	,	,	PUNCT
iajs-3112	23	5	in	in	ADP
iajs-3112	23	6	[	[	X
iajs-3112	23	7	6	6	NUM
iajs-3112	23	8	]	]	PUNCT
iajs-3112	23	9	studied	study	VERB
iajs-3112	23	10	of	of	ADP
iajs-3112	23	11	the	the	DET
iajs-3112	23	12	fuzzy	fuzzy	ADJ
iajs-3112	23	13	implicative	implicative	ADJ
iajs-3112	23	14	ideals	ideal	NOUN
iajs-3112	23	15	of	of	ADP
iajs-3112	23	16	bck	bck	PROPN
iajs-3112	23	17	algebras	algebra	NOUN
iajs-3112	23	18	.	.	PUNCT
iajs-3112	24	1	also	also	ADV
iajs-3112	24	2	,	,	PUNCT
iajs-3112	24	3	mostafa	mostafa	PROPN
iajs-3112	24	4	et	et	PROPN
iajs-3112	24	5	al	al	PROPN
iajs-3112	24	6	[	[	X
iajs-3112	24	7	7	7	X
iajs-3112	24	8	]	]	PUNCT
iajs-3112	24	9	introduced	introduce	VERB
iajs-3112	24	10	the	the	DET
iajs-3112	24	11	notion	notion	NOUN
iajs-3112	24	12	of	of	ADP
iajs-3112	24	13	fuzzy	fuzzy	ADJ
iajs-3112	24	14	ku	ku	NOUN
iajs-3112	24	15	-	-	PUNCT
iajs-3112	24	16	ideals	ideal	NOUN
iajs-3112	24	17	of	of	ADP
iajs-3112	24	18	ku	ku	PROPN
iajs-3112	24	19	-	-	PUNCT
iajs-3112	24	20	algebras	algebras	PROPN
iajs-3112	24	21	and	and	CCONJ
iajs-3112	24	22	they	they	PRON
iajs-3112	24	23	investigated	investigate	VERB
iajs-3112	24	24	several	several	ADJ
iajs-3112	24	25	basic	basic	ADJ
iajs-3112	24	26	properties	property	NOUN
iajs-3112	24	27	which	which	PRON
iajs-3112	24	28	are	be	AUX
iajs-3112	24	29	associated	associate	VERB
iajs-3112	24	30	to	to	ADP
iajs-3112	24	31	fuzzy	fuzzy	ADJ
iajs-3112	24	32	ku	ku	NOUN
iajs-3112	24	33	-	-	PUNCT
iajs-3112	24	34	ideals	ideal	NOUN
iajs-3112	24	35	.	.	PUNCT
iajs-3112	25	1	many	many	ADJ
iajs-3112	25	2	mathematicians	mathematician	NOUN
iajs-3112	25	3	have	have	AUX
iajs-3112	25	4	studied	study	VERB
iajs-3112	25	5	“	"	PUNCT
iajs-3112	25	6	a	a	DET
iajs-3112	25	7	fuzzy	fuzzy	NOUN
iajs-3112	25	8	”	"	PUNCT
iajs-3112	25	9	for	for	ADP
iajs-3112	25	10	some	some	DET
iajs-3112	25	11	algebraic	algebraic	ADJ
iajs-3112	25	12	structures	structure	NOUN
iajs-3112	25	13	,	,	PUNCT
iajs-3112	25	14	seeˑ[8	seeˑ[8	NOUN
iajs-3112	25	15	-	-	SYM
iajs-3112	25	16	12	12	NUM
iajs-3112	25	17	]	]	PUNCT
iajs-3112	25	18	.	.	PUNCT
iajs-3112	26	1	there	there	PRON
iajs-3112	26	2	are	be	VERB
iajs-3112	26	3	several	several	ADJ
iajs-3112	26	4	kinds	kind	NOUN
iajs-3112	26	5	of	of	ADP
iajs-3112	26	6	fuzzy	fuzzy	ADJ
iajs-3112	26	7	set	set	NOUN
iajs-3112	26	8	extensions	extension	NOUN
iajs-3112	26	9	in	in	ADP
iajs-3112	26	10	the	the	DET
iajs-3112	26	11	fuzzy	fuzzy	ADJ
iajs-3112	26	12	set	set	NOUN
iajs-3112	26	13	theory	theory	NOUN
iajs-3112	26	14	,	,	PUNCT
iajs-3112	26	15	for	for	ADP
iajs-3112	26	16	example	example	NOUN
iajs-3112	26	17	,	,	PUNCT
iajs-3112	26	18	interval	interval	NOUN
iajs-3112	26	19	-	-	PUNCT
iajs-3112	26	20	valued	value	VERB
iajs-3112	26	21	fuzzy	fuzzy	ADJ
iajs-3112	26	22	sets	set	NOUN
iajs-3112	26	23	and	and	CCONJ
iajs-3112	26	24	bipolar	bipolar	ADV
iajs-3112	26	25	-	-	PUNCT
iajs-3112	26	26	valued	value	VERB
iajs-3112	26	27	fuzzy	fuzzy	ADJ
iajs-3112	26	28	sets	set	NOUN
iajs-3112	26	29	,	,	PUNCT
iajs-3112	26	30	which	which	PRON
iajs-3112	26	31	were	be	AUX
iajs-3112	26	32	introduced	introduce	VERB
iajs-3112	26	33	in	in	ADP
iajs-3112	26	34	[	[	PUNCT
iajs-3112	26	35	13	13	NUM
iajs-3112	26	36	-	-	SYM
iajs-3112	26	37	17	17	NUM
iajs-3112	26	38	]	]	PUNCT
iajs-3112	26	39	.	.	PUNCT
iajs-3112	27	1	the	the	DET
iajs-3112	27	2	concept	concept	NOUN
iajs-3112	27	3	of	of	ADP
iajs-3112	27	4	cubic	cubic	ADJ
iajs-3112	27	5	subalgebras	subalgebras	PROPN
iajs-3112	27	6	/ideals	/ideal	NOUN
iajs-3112	27	7	in	in	ADP
iajs-3112	27	8	bck	bck	PROPN
iajs-3112	27	9	/	/	SYM
iajs-3112	27	10	bci	bci	NOUN
iajs-3112	27	11	-	-	PUNCT
iajs-3112	27	12	algebras	algebras	PROPN
iajs-3112	27	13	was	be	AUX
iajs-3112	27	14	introduced	introduce	VERB
iajs-3112	27	15	by	by	ADP
iajs-3112	27	16	jun	jun	PROPN
iajs-3112	27	17	et	et	PROPN
iajs-3112	27	18	al	al	PROPN
iajs-3112	27	19	.	.	PUNCT
iajs-3112	28	1	[	[	X
iajs-3112	28	2	18	18	NUM
iajs-3112	28	3	,	,	PUNCT
iajs-3112	28	4	19	19	NUM
iajs-3112	28	5	]	]	PUNCT
iajs-3112	28	6	.	.	PUNCT
iajs-3112	29	1	they	they	PRON
iajs-3112	29	2	discussed	discuss	VERB
iajs-3112	29	3	the	the	DET
iajs-3112	29	4	relationship	relationship	NOUN
iajs-3112	29	5	between	between	ADP
iajs-3112	29	6	a	a	DET
iajs-3112	29	7	cubic	cubic	ADJ
iajs-3112	29	8	subalgebra	subalgebra	NOUN
iajs-3112	29	9	and	and	CCONJ
iajs-3112	29	10	a	a	DET
iajs-3112	29	11	cubic	cubic	ADJ
iajs-3112	29	12	ideal	ideal	NOUN
iajs-3112	29	13	.	.	PUNCT
iajs-3112	30	1	and	and	CCONJ
iajs-3112	30	2	then	then	ADV
iajs-3112	30	3	,	,	PUNCT
iajs-3112	30	4	yaqoob	yaqoob	NOUN
iajs-3112	30	5	et	et	NOUN
iajs-3112	30	6	al	al	PROPN
iajs-3112	31	1	[	[	X
iajs-3112	31	2	20	20	NUM
iajs-3112	31	3	]	]	PUNCT
iajs-3112	31	4	introduced	introduce	VERB
iajs-3112	31	5	the	the	DET
iajs-3112	31	6	notion	notion	NOUN
iajs-3112	31	7	of	of	ADP
iajs-3112	31	8	cubic	cubic	ADJ
iajs-3112	31	9	ku	ku	PROPN
iajs-3112	31	10	-	-	PUNCT
iajs-3112	31	11	algebra	algebra	PROPN
iajs-3112	31	12	which	which	PRON
iajs-3112	31	13	is	be	AUX
iajs-3112	31	14	a	a	DET
iajs-3112	31	15	generalization	generalization	NOUN
iajs-3112	31	16	of	of	ADP
iajs-3112	31	17	the	the	DET
iajs-3112	31	18	concept	concept	NOUN
iajs-3112	31	19	of	of	ADP
iajs-3112	31	20	fuzzy	fuzzy	ADJ
iajs-3112	31	21	ku	ku	NOUN
iajs-3112	31	22	-	-	PUNCT
iajs-3112	31	23	ideals	ideal	NOUN
iajs-3112	31	24	of	of	ADP
iajs-3112	31	25	ku	ku	PROPN
iajs-3112	31	26	-	-	PUNCT
iajs-3112	31	27	algebras	algebras	PROPN
iajs-3112	31	28	.	.	PUNCT
iajs-3112	32	1	after	after	ADP
iajs-3112	32	2	that	that	PRON
iajs-3112	32	3	,	,	PUNCT
iajs-3112	32	4	kareem	kareem	PROPN
iajs-3112	32	5	and	and	CCONJ
iajs-3112	32	6	hasan[21	hasan[21	PROPN
iajs-3112	32	7	]	]	PUNCT
iajs-3112	32	8	introduced	introduce	VERB
iajs-3112	32	9	the	the	PRON
iajs-3112	32	10	received	receive	VERB
iajs-3112	32	11	25	25	NUM
iajs-3112	32	12	november	november	PROPN
iajs-3112	32	13	2022	2022	NUM
iajs-3112	32	14	,	,	PUNCT
iajs-3112	32	15	received	receive	VERB
iajs-3112	32	16	18	18	NUM
iajs-3112	32	17	may	may	PROPN
iajs-3112	32	18	2023	2023	NUM
iajs-3112	32	19	,	,	PUNCT
iajs-3112	32	20	accepted	accept	VERB
iajs-3112	32	21	22	22	NUM
iajs-3112	32	22	may	may	PROPN
iajs-3112	32	23	2023	2023	NUM
iajs-3112	32	24	,	,	PUNCT
iajs-3112	32	25	published	publish	VERB
iajs-3112	32	26	20	20	NUM
iajs-3112	32	27	january	january	PROPN
iajs-3112	32	28	2024	2024	NUM
iajs-3112	32	29	cubic	cubic	NOUN
iajs-3112	32	30	of	of	ADP
iajs-3112	32	31	positive	positive	ADJ
iajs-3112	32	32	implicative	implicative	ADJ
iajs-3112	32	33	ideals	ideal	NOUN
iajs-3112	32	34	in	in	ADP
iajs-3112	32	35	ku	ku	PROPN
iajs-3112	32	36	-	-	PUNCT
iajs-3112	32	37	semigroup	semigroup	PROPN
iajs-3112	32	38	doi.org/10.30526/37.1.3112	doi.org/10.30526/37.1.3112	PROPN
iajs-3112	32	39	https://creativecommons.org/licenses/by/4.0/	https://creativecommons.org/licenses/by/4.0/	PROPN
iajs-3112	32	40	https://jih.uobaghdad.edu.iq/index.php/j/index#1609-4042	https://jih.uobaghdad.edu.iq/index.php/j/index#1609-4042	NOUN
iajs-3112	32	41	https://jih.uobaghdad.edu.iq/index.php/j/index#2521-3407	https://jih.uobaghdad.edu.iq/index.php/j/index#2521-3407	VERB
iajs-3112	32	42	mailto:fatma.f.k@ihcoedu.uobaghdad.edu.iq	mailto:fatma.f.k@ihcoedu.uobaghdad.edu.iq	ADJ
iajs-3112	32	43	https://orcid.org/0000-0002-5069-2434	https://orcid.org/0000-0002-5069-2434	PROPN
iajs-3112	32	44	mailto:%20%20%20%20%20%20%20samymostafa@yahoo.com	mailto:%20%20%20%20%20%20%20samymostafa@yahoo.com	PROPN
iajs-3112	32	45	https://orcid.org/0000-0002-6141-8721	https://orcid.org/0000-0002-6141-8721	PROPN
iajs-3112	32	46	mailto:fatma.f.k@ihcoedu.uobaghdad.edu.iq	mailto:fatma.f.k@ihcoedu.uobaghdad.edu.iq	ADJ
iajs-3112	32	47	https://orcid.org/0009-0005-7019-2655	https://orcid.org/0009-0005-7019-2655	NOUN
iajs-3112	32	48	mailto:umniyatadnan@gmail.com	mailto:umniyatadnan@gmail.com	NOUN
iajs-3112	32	49	ihjpas	ihjpa	VERB
iajs-3112	32	50	.	.	PUNCT
iajs-3112	33	1	37	37	NUM
iajs-3112	33	2	(	(	PUNCT
iajs-3112	33	3	1	1	NUM
iajs-3112	33	4	)	)	PUNCT
iajs-3112	33	5	2024	2024	NUM
iajs-3112	33	6	455	455	NUM
iajs-3112	33	7	notion	notion	NOUN
iajs-3112	33	8	of	of	ADP
iajs-3112	33	9	a	a	DET
iajs-3112	33	10	ku	ku	NOUN
iajs-3112	33	11	-	-	PUNCT
iajs-3112	33	12	algebra	algebra	PROPN
iajs-3112	33	13	with	with	ADP
iajs-3112	33	14	semigroup	semigroup	PROPN
iajs-3112	33	15	which	which	PRON
iajs-3112	33	16	is	be	AUX
iajs-3112	33	17	called	call	VERB
iajs-3112	33	18	a	a	DET
iajs-3112	33	19	ku	ku	PROPN
iajs-3112	33	20	-	-	PUNCT
iajs-3112	33	21	semigroup	semigroup	PROPN
iajs-3112	33	22	and	and	CCONJ
iajs-3112	33	23	defined	define	VERB
iajs-3112	33	24	some	some	DET
iajs-3112	33	25	types	type	NOUN
iajs-3112	33	26	of	of	ADP
iajs-3112	33	27	ideals	ideal	NOUN
iajs-3112	33	28	in	in	ADP
iajs-3112	33	29	this	this	DET
iajs-3112	33	30	concept	concept	NOUN
iajs-3112	33	31	.	.	PUNCT
iajs-3112	34	1	also	also	ADV
iajs-3112	34	2	,	,	PUNCT
iajs-3112	34	3	they	they	PRON
iajs-3112	34	4	studied	study	VERB
iajs-3112	34	5	the	the	DET
iajs-3112	34	6	fuzzy	fuzzy	ADJ
iajs-3112	34	7	ideals	ideal	NOUN
iajs-3112	34	8	of	of	ADP
iajs-3112	34	9	a	a	DET
iajs-3112	34	10	ku	ku	PROPN
iajs-3112	34	11	-	-	PUNCT
iajs-3112	34	12	semigroup	semigroup	PROPN
iajs-3112	34	13	.	.	PUNCT
iajs-3112	35	1	after	after	ADP
iajs-3112	35	2	that	that	PRON
iajs-3112	35	3	,	,	PUNCT
iajs-3112	35	4	kareem	kareem	PROPN
iajs-3112	35	5	and	and	CCONJ
iajs-3112	35	6	hasan	hasan	PROPN
iajs-3112	36	1	[	[	X
iajs-3112	36	2	22	22	NUM
iajs-3112	36	3	]	]	PUNCT
iajs-3112	36	4	introduced	introduce	VERB
iajs-3112	36	5	the	the	DET
iajs-3112	36	6	cubic	cubic	ADJ
iajs-3112	36	7	ideals	ideal	NOUN
iajs-3112	36	8	of	of	ADP
iajs-3112	36	9	a	a	DET
iajs-3112	36	10	semigroup	semigroup	NOUN
iajs-3112	36	11	in	in	ADP
iajs-3112	36	12	ku	ku	PROPN
iajs-3112	36	13	-	-	PUNCT
iajs-3112	36	14	algebra	algebra	PROPN
iajs-3112	36	15	and	and	CCONJ
iajs-3112	36	16	defined	define	VERB
iajs-3112	36	17	some	some	DET
iajs-3112	36	18	types	type	NOUN
iajs-3112	36	19	of	of	ADP
iajs-3112	36	20	ideals	ideal	NOUN
iajs-3112	36	21	in	in	ADP
iajs-3112	36	22	this	this	DET
iajs-3112	36	23	concept	concept	NOUN
iajs-3112	36	24	.	.	PUNCT
iajs-3112	37	1	senapati	senapati	PROPN
iajs-3112	37	2	et	et	PROPN
iajs-3112	37	3	al[23,24	al[23,24	PROPN
iajs-3112	37	4	]	]	PUNCT
iajs-3112	37	5	introduced	introduce	VERB
iajs-3112	37	6	the	the	DET
iajs-3112	37	7	two	two	NUM
iajs-3112	37	8	concepts	concept	NOUN
iajs-3112	37	9	which	which	PRON
iajs-3112	37	10	are	be	AUX
iajs-3112	37	11	cubic	cubic	ADJ
iajs-3112	37	12	ideal	ideal	ADJ
iajs-3112	37	13	and	and	CCONJ
iajs-3112	37	14	implicative	implicative	ADJ
iajs-3112	37	15	ideal	ideal	NOUN
iajs-3112	37	16	.	.	PUNCT
iajs-3112	38	1	some	some	DET
iajs-3112	38	2	authors	author	NOUN
iajs-3112	38	3	introduced	introduce	VERB
iajs-3112	38	4	a	a	DET
iajs-3112	38	5	cubic	cubic	ADJ
iajs-3112	38	6	set	set	NOUN
iajs-3112	38	7	of	of	ADP
iajs-3112	38	8	different	different	ADJ
iajs-3112	38	9	structures	structure	NOUN
iajs-3112	38	10	.	.	PUNCT
iajs-3112	39	1	see	see	VERB
iajs-3112	39	2	[	[	X
iajs-3112	39	3	25	25	NUM
iajs-3112	39	4	-	-	SYM
iajs-3112	39	5	30	30	NUM
iajs-3112	39	6	]	]	PUNCT
iajs-3112	39	7	.	.	PUNCT
iajs-3112	40	1	in	in	ADP
iajs-3112	40	2	this	this	DET
iajs-3112	40	3	work	work	NOUN
iajs-3112	40	4	,	,	PUNCT
iajs-3112	40	5	the	the	DET
iajs-3112	40	6	notion	notion	NOUN
iajs-3112	40	7	of	of	ADP
iajs-3112	40	8	cubic	cubic	ADJ
iajs-3112	40	9	(	(	PUNCT
iajs-3112	40	10	positive	positive	ADJ
iajs-3112	40	11	implicative	implicative	ADJ
iajs-3112	40	12	,	,	PUNCT
iajs-3112	40	13	implicative	implicative	ADJ
iajs-3112	40	14	and	and	CCONJ
iajs-3112	40	15	commutative)-ideal	commutative)-ideal	PROPN
iajs-3112	40	16	are	be	AUX
iajs-3112	40	17	discussed	discuss	VERB
iajs-3112	40	18	and	and	CCONJ
iajs-3112	40	19	the	the	DET
iajs-3112	40	20	relationship	relationship	NOUN
iajs-3112	40	21	among	among	ADP
iajs-3112	40	22	these	these	DET
iajs-3112	40	23	types	type	NOUN
iajs-3112	40	24	are	be	AUX
iajs-3112	40	25	studied	study	VERB
iajs-3112	40	26	.	.	PUNCT
iajs-3112	41	1	2	2	X
iajs-3112	41	2	.	.	NUM
iajs-3112	41	3	basic	basic	ADJ
iajs-3112	41	4	concepts	concept	NOUN
iajs-3112	41	5	definition	definition	NOUN
iajs-3112	41	6	(	(	PUNCT
iajs-3112	41	7	1	1	X
iajs-3112	41	8	)	)	PUNCT
iajs-3112	42	1	[	[	X
iajs-3112	42	2	1	1	NUM
iajs-3112	42	3	]	]	PUNCT
iajs-3112	42	4	.	.	PUNCT
iajs-3112	43	1	an	an	DET
iajs-3112	43	2	algebra(ν,∗	algebra(ν,∗	PROPN
iajs-3112	43	3	,	,	PUNCT
iajs-3112	43	4	0	0	NUM
iajs-3112	43	5	)	)	PUNCT
iajs-3112	43	6	is	be	AUX
iajs-3112	43	7	named	name	VERB
iajs-3112	43	8	a	a	DET
iajs-3112	43	9	ku	ku	NOUN
iajs-3112	43	10	-	-	PUNCT
iajs-3112	43	11	algebra	algebra	PROPN
iajs-3112	43	12	if	if	SCONJ
iajs-3112	43	13	,	,	PUNCT
iajs-3112	43	14	for	for	ADP
iajs-3112	43	15	all	all	DET
iajs-3112	43	16	ς	ς	PROPN
iajs-3112	43	17	,	,	PUNCT
iajs-3112	43	18	ω	ω	PROPN
iajs-3112	43	19	,	,	PUNCT
iajs-3112	43	20	κ	κ	PROPN
iajs-3112	43	21	∈	∈	PROPN
iajs-3112	43	22	ν	ν	PROPN
iajs-3112	43	23	,	,	PUNCT
iajs-3112	43	24	(	(	PUNCT
iajs-3112	43	25	ku1	ku1	NOUN
iajs-3112	43	26	)	)	PUNCT
iajs-3112	43	27	(	(	PUNCT
iajs-3112	43	28	ς	ς	PROPN
iajs-3112	43	29	∗	∗	NOUN
iajs-3112	43	30	ω	ω	NOUN
iajs-3112	43	31	)	)	PUNCT
iajs-3112	43	32	∗	∗	NOUN
iajs-3112	44	1	[	[	X
iajs-3112	44	2	(	(	PUNCT
iajs-3112	44	3	ω	ω	PROPN
iajs-3112	44	4	∗	∗	PROPN
iajs-3112	44	5	κ	κ	NOUN
iajs-3112	44	6	)	)	PUNCT
iajs-3112	44	7	∗	∗	NOUN
iajs-3112	44	8	(	(	PUNCT
iajs-3112	44	9	ς	ς	PROPN
iajs-3112	44	10	∗	∗	NOUN
iajs-3112	44	11	κ	κ	NOUN
iajs-3112	44	12	)	)	PUNCT
iajs-3112	44	13	]	]	PUNCT
iajs-3112	45	1	=	=	PUNCT
iajs-3112	45	2	0	0	NUM
iajs-3112	45	3	,	,	PUNCT
iajs-3112	45	4	(	(	PUNCT
iajs-3112	45	5	ku2	ku2	NOUN
iajs-3112	45	6	)	)	PUNCT
iajs-3112	45	7	ς	ς	PROPN
iajs-3112	45	8	∗0	∗0	PROPN
iajs-3112	45	9	=	=	SYM
iajs-3112	45	10	0	0	NUM
iajs-3112	45	11	,	,	PUNCT
iajs-3112	45	12	(	(	PUNCT
iajs-3112	45	13	ku3	ku3	X
iajs-3112	45	14	)	)	PUNCT
iajs-3112	45	15	0∗	0∗	PUNCT
iajs-3112	46	1	ς	ς	PROPN
iajs-3112	46	2	=	=	SYM
iajs-3112	46	3	ς	ς	PROPN
iajs-3112	46	4	,	,	PUNCT
iajs-3112	46	5	(	(	PUNCT
iajs-3112	46	6	ku4	ku4	NOUN
iajs-3112	46	7	)	)	PUNCT
iajs-3112	47	1	ς	ς	PROPN
iajs-3112	47	2	∗	∗	X
iajs-3112	47	3	ω	ω	X
iajs-3112	47	4	=	=	SYM
iajs-3112	47	5	0	0	NUM
iajs-3112	47	6	and	and	CCONJ
iajs-3112	47	7	ω	ω	NUM
iajs-3112	47	8	∗	∗	NOUN
iajs-3112	47	9	ς	ς	PROPN
iajs-3112	47	10	=	=	SYM
iajs-3112	47	11	0	0	NUM
iajs-3112	47	12	implies	imply	VERB
iajs-3112	47	13	ς	ς	PROPN
iajs-3112	47	14	=	=	SYM
iajs-3112	47	15	ω	ω	PROPN
iajs-3112	47	16	,	,	PUNCT
iajs-3112	47	17	(	(	PUNCT
iajs-3112	47	18	ku5	ku5	NOUN
iajs-3112	47	19	)	)	PUNCT
iajs-3112	47	20	ς	ς	PROPN
iajs-3112	47	21	∗	∗	NOUN
iajs-3112	47	22	ς	ς	PROPN
iajs-3112	47	23	=	=	SYM
iajs-3112	47	24	0	0	PROPN
iajs-3112	47	25	.	.	PUNCT
iajs-3112	48	1	the	the	DET
iajs-3112	48	2	binary	binary	PROPN
iajs-3112	48	3	relation	relation	NOUN
iajs-3112	48	4	≤	≤	PUNCT
iajs-3112	48	5	on	on	ADP
iajs-3112	48	6	ν	ν	NOUN
iajs-3112	48	7	is	be	AUX
iajs-3112	48	8	define	define	VERB
iajs-3112	48	9	by	by	ADP
iajs-3112	48	10	ς	ς	PROPN
iajs-3112	48	11	≤	≤	PROPN
iajs-3112	48	12	ω	ω	NUM
iajs-3112	49	1	⟺	⟺	PROPN
iajs-3112	49	2	ω	ω	PROPN
iajs-3112	49	3	∗	∗	NOUN
iajs-3112	49	4	ς	ς	PROPN
iajs-3112	49	5	=	=	SYM
iajs-3112	49	6	0	0	NUM
iajs-3112	49	7	.	.	NUM
iajs-3112	49	8	theorem(2)[2].ˑlet(ν,∗	theorem(2)[2].ˑlet(ν,∗	ADV
iajs-3112	49	9	,	,	PUNCT
iajs-3112	49	10	0	0	NUM
iajs-3112	49	11	)	)	PUNCT
iajs-3112	49	12	be	be	AUX
iajs-3112	49	13	a	a	DET
iajs-3112	49	14	ku	ku	NOUN
iajs-3112	49	15	-	-	PUNCT
iajs-3112	49	16	algebra	algebra	PROPN
iajs-3112	49	17	.	.	PUNCT
iajs-3112	50	1	then	then	ADV
iajs-3112	50	2	for	for	ADP
iajs-3112	50	3	all	all	DET
iajs-3112	50	4	ς	ς	PROPN
iajs-3112	50	5	,	,	PUNCT
iajs-3112	50	6	ω	ω	PROPN
iajs-3112	50	7	,	,	PUNCT
iajs-3112	50	8	κ	κ	PROPN
iajs-3112	50	9	∈	∈	PROPN
iajs-3112	50	10	ν	ν	NOUN
iajs-3112	50	11	:	:	PUNCT
iajs-3112	50	12	(	(	PUNCT
iajs-3112	50	13	1	1	X
iajs-3112	50	14	)	)	PUNCT
iajs-3112	50	15	if	if	SCONJ
iajs-3112	50	16	ς	ς	PROPN
iajs-3112	50	17	≤	≤	X
iajs-3112	50	18	ω	ω	PROPN
iajs-3112	50	19	implyω	implyω	PROPN
iajs-3112	50	20	∗	∗	NOUN
iajs-3112	50	21	κ	κ	PROPN
iajs-3112	50	22	≤	≤	NUM
iajs-3112	50	23	ς	ς	PROPN
iajs-3112	50	24	∗	∗	NOUN
iajs-3112	50	25	κ	κ	NOUN
iajs-3112	50	26	.	.	PUNCT
iajs-3112	51	1	(	(	PUNCT
iajs-3112	51	2	2	2	X
iajs-3112	51	3	)	)	PUNCT
iajs-3112	51	4	ς	ς	PROPN
iajs-3112	51	5	∗	∗	NOUN
iajs-3112	51	6	(	(	PUNCT
iajs-3112	51	7	ω	ω	PROPN
iajs-3112	51	8	∗	∗	X
iajs-3112	51	9	κ	κ	NOUN
iajs-3112	51	10	)	)	PUNCT
iajs-3112	51	11	=	=	SYM
iajs-3112	51	12	ω	ω	PROPN
iajs-3112	51	13	∗	∗	NOUN
iajs-3112	51	14	(	(	PUNCT
iajs-3112	51	15	ς	ς	PROPN
iajs-3112	51	16	∗	∗	NOUN
iajs-3112	51	17	κ	κ	NOUN
iajs-3112	51	18	)	)	PUNCT
iajs-3112	51	19	.	.	PUNCT
iajs-3112	52	1	(	(	PUNCT
iajs-3112	52	2	3	3	X
iajs-3112	52	3	)	)	PUNCT
iajs-3112	52	4	(	(	PUNCT
iajs-3112	52	5	ω	ω	PROPN
iajs-3112	52	6	∗	∗	PROPN
iajs-3112	52	7	ς	ς	NOUN
iajs-3112	52	8	)	)	PUNCT
iajs-3112	52	9	∗	∗	NOUN
iajs-3112	52	10	ς	ς	PROPN
iajs-3112	52	11	≤	≤	NUM
iajs-3112	52	12	ω	ω	PROPN
iajs-3112	52	13	.	.	PUNCT
iajs-3112	53	1	(	(	PUNCT
iajs-3112	53	2	4)((ω	4)((ω	NOUN
iajs-3112	53	3	∗	∗	PROPN
iajs-3112	53	4	ς	ς	NOUN
iajs-3112	53	5	)	)	PUNCT
iajs-3112	53	6	∗	∗	NOUN
iajs-3112	53	7	ς	ς	NOUN
iajs-3112	53	8	)	)	PUNCT
iajs-3112	53	9	∗	∗	NOUN
iajs-3112	53	10	ς	ς	NOUN
iajs-3112	53	11	)	)	PUNCT
iajs-3112	53	12	)	)	PUNCT
iajs-3112	54	1	=	=	SYM
iajs-3112	54	2	ω	ω	PROPN
iajs-3112	54	3	∗	∗	NOUN
iajs-3112	54	4	ς	ς	PROPN
iajs-3112	54	5	.	.	PUNCT
iajs-3112	54	6	definition(3	definition(3	X
iajs-3112	54	7	)	)	PUNCT
iajs-3112	55	1	[	[	X
iajs-3112	55	2	21	21	NUM
iajs-3112	55	3	]	]	PUNCT
iajs-3112	55	4	.	.	PUNCT
iajs-3112	56	1	a	a	DET
iajs-3112	56	2	nonempty	nonempty	ADV
iajs-3112	56	3	set	set	VERB
iajs-3112	56	4	ν	ν	NOUN
iajs-3112	56	5	with~∗,∘	with~∗,∘	NOUN
iajs-3112	56	6	and	and	CCONJ
iajs-3112	56	7	0	0	NUM
iajs-3112	56	8	is	be	AUX
iajs-3112	56	9	called	call	VERB
iajs-3112	56	10	a	a	DET
iajs-3112	56	11	ku	ku	PROPN
iajs-3112	56	12	-	-	PUNCT
iajs-3112	56	13	semigroup	semigroup	PROPN
iajs-3112	56	14	if~~	if~~	NOUN
iajs-3112	56	15	(	(	PUNCT
iajs-3112	56	16	i)the	i)the	DET
iajs-3112	56	17	set	set	NOUN
iajs-3112	56	18	ν	ν	NOUN
iajs-3112	56	19	with	with	ADP
iajs-3112	56	20	∗	∗	NOUN
iajs-3112	56	21	and	and	CCONJ
iajs-3112	56	22	0	0	NUM
iajs-3112	56	23	is	be	AUX
iajs-3112	56	24	ˑa	ˑa	ADJ
iajs-3112	56	25	ku	ku	NOUN
iajs-3112	56	26	-	-	PUNCT
iajs-3112	56	27	algebra	algebra	PROPN
iajs-3112	56	28	.	.	PUNCT
iajs-3112	57	1	(	(	PUNCT
iajs-3112	57	2	ii	ii	NOUN
iajs-3112	57	3	)	)	PUNCT
iajs-3112	57	4	the	the	DET
iajs-3112	57	5	set	set	NOUN
iajs-3112	57	6	νwith	νwith	ADJ
iajs-3112	57	7	∘	∘	NOUN
iajs-3112	57	8	and	and	CCONJ
iajs-3112	57	9	0	0	NUM
iajs-3112	57	10	is	be	AUX
iajs-3112	57	11	aˑsemigroup	aˑsemigroup	NOUN
iajs-3112	57	12	.	.	PUNCT
iajs-3112	58	1	(	(	PUNCT
iajs-3112	58	2	iii	iii	NOUN
iajs-3112	58	3	)	)	PUNCT
iajs-3112	58	4	ς	ς	PROPN
iajs-3112	58	5	∘	∘	X
iajs-3112	58	6	(	(	PUNCT
iajs-3112	58	7	ω	ω	PROPN
iajs-3112	58	8	∗	∗	PROPN
iajs-3112	58	9	κ	κ	NOUN
iajs-3112	58	10	)	)	PUNCT
iajs-3112	58	11	=	=	PUNCT
iajs-3112	58	12	(	(	PUNCT
iajs-3112	58	13	ς	ς	PROPN
iajs-3112	58	14	∘	∘	PROPN
iajs-3112	58	15	ω	ω	NUM
iajs-3112	58	16	)	)	PUNCT
iajs-3112	58	17	∗	∗	NOUN
iajs-3112	58	18	(	(	PUNCT
iajs-3112	58	19	ς	ς	PROPN
iajs-3112	58	20	∘	∘	PROPN
iajs-3112	58	21	κ	κ	NOUN
iajs-3112	58	22	)	)	PUNCT
iajs-3112	58	23	and(ς	and(ς	PROPN
iajs-3112	58	24	∗	∗	X
iajs-3112	58	25	ω	ω	NOUN
iajs-3112	58	26	)	)	PUNCT
iajs-3112	58	27	∘	∘	PROPN
iajs-3112	58	28	κ	κ	X
iajs-3112	58	29	=	=	PUNCT
iajs-3112	58	30	(	(	PUNCT
iajs-3112	58	31	ς	ς	PROPN
iajs-3112	58	32	∘	∘	PROPN
iajs-3112	58	33	κ	κ	NOUN
iajs-3112	58	34	)	)	PUNCT
iajs-3112	58	35	∗	∗	NOUN
iajs-3112	58	36	(	(	PUNCT
iajs-3112	58	37	ω	ω	PROPN
iajs-3112	58	38	∘	∘	PROPN
iajs-3112	58	39	κ	κ	NOUN
iajs-3112	58	40	)	)	PUNCT
iajs-3112	58	41	,	,	PUNCT
iajs-3112	58	42	for	for	ADP
iajs-3112	58	43	all	all	DET
iajs-3112	58	44	ς	ς	PROPN
iajs-3112	58	45	,	,	PUNCT
iajs-3112	58	46	ω	ω	PROPN
iajs-3112	58	47	,	,	PUNCT
iajs-3112	58	48	κ	κ	PROPN
iajs-3112	58	49	∈	∈	PROPN
iajs-3112	58	50	ν	ν	PROPN
iajs-3112	58	51	.	.	PROPN
iajs-3112	58	52	example	example	NOUN
iajs-3112	58	53	(	(	PUNCT
iajs-3112	58	54	4	4	X
iajs-3112	58	55	)	)	PUNCT
iajs-3112	59	1	[	[	X
iajs-3112	59	2	21	21	NUM
iajs-3112	59	3	]	]	PUNCT
iajs-3112	59	4	.	.	PUNCT
iajs-3112	60	1	letˑν	letˑν	PROPN
iajs-3112	60	2	=	=	PRON
iajs-3112	60	3	{	{	PUNCT
iajs-3112	60	4	0,1,2,3	0,1,2,3	NOUN
iajs-3112	60	5	}	}	PUNCT
iajs-3112	60	6	with	with	ADP
iajs-3112	60	7	two	two	NUM
iajs-3112	60	8	operations	operation	NOUN
iajs-3112	60	9	∗	∗	NOUN
iajs-3112	60	10	and	and	CCONJ
iajs-3112	60	11	∘	∘	NOUN
iajs-3112	60	12	defined	define	VERB
iajs-3112	60	13	by	by	ADP
iajs-3112	60	14	table	table	NOUN
iajs-3112	60	15	1	1	NUM
iajs-3112	60	16	:	:	PUNCT
iajs-3112	60	17	table	table	NOUN
iajs-3112	60	18	1	1	NUM
iajs-3112	60	19	.	.	PUNCT
iajs-3112	61	1	a	a	DET
iajs-3112	61	2	ku	ku	PROPN
iajs-3112	61	3	-	-	PUNCT
iajs-3112	61	4	semigroup	semigroup	PROPN
iajs-3112	61	5	it	it	PRON
iajs-3112	61	6	follows	follow	VERB
iajs-3112	61	7	that	that	SCONJ
iajs-3112	61	8	ˑ(ν,∗,∘	ˑ(ν,∗,∘	NOUN
iajs-3112	61	9	,	,	PUNCT
iajs-3112	61	10	0	0	NUM
iajs-3112	61	11	)	)	PUNCT
iajs-3112	61	12	is	be	AUX
iajs-3112	61	13	aˑku	aˑku	NOUN
iajs-3112	61	14	-	-	PUNCT
iajs-3112	61	15	semigroup	semigroup	NOUN
iajs-3112	61	16	.	.	PUNCT
iajs-3112	62	1	*	*	PUNCT
iajs-3112	62	2	0	0	NUM
iajs-3112	62	3	1	1	NUM
iajs-3112	62	4	2	2	NUM
iajs-3112	62	5	3	3	NUM
iajs-3112	62	6	0	0	NUM
iajs-3112	62	7	0	0	NUM
iajs-3112	62	8	1	1	NUM
iajs-3112	62	9	3	3	NUM
iajs-3112	62	10	2	2	NUM
iajs-3112	62	11	1	1	NUM
iajs-3112	62	12	0	0	NUM
iajs-3112	62	13	0	0	NUM
iajs-3112	62	14	0	0	NUM
iajs-3112	62	15	2	2	NUM
iajs-3112	62	16	2	2	NUM
iajs-3112	62	17	2	2	NUM
iajs-3112	62	18	0	0	NUM
iajs-3112	62	19	0	0	NUM
iajs-3112	62	20	1	1	NUM
iajs-3112	62	21	3	3	NUM
iajs-3112	62	22	0	0	NUM
iajs-3112	62	23	0	0	NUM
iajs-3112	62	24	0	0	NUM
iajs-3112	62	25	0	0	NUM
iajs-3112	63	1	∘	∘	NUM
iajs-3112	63	2	0	0	NUM
iajs-3112	63	3	1	1	NUM
iajs-3112	63	4	2	2	NUM
iajs-3112	63	5	3	3	NUM
iajs-3112	63	6	0	0	NUM
iajs-3112	63	7	0	0	NUM
iajs-3112	63	8	0	0	NUM
iajs-3112	63	9	0	0	NUM
iajs-3112	63	10	0	0	NUM
iajs-3112	63	11	1	1	NUM
iajs-3112	63	12	0	0	NUM
iajs-3112	63	13	1	1	NUM
iajs-3112	63	14	0	0	NUM
iajs-3112	63	15	1	1	NUM
iajs-3112	63	16	2	2	NUM
iajs-3112	63	17	0	0	NUM
iajs-3112	63	18	0	0	NUM
iajs-3112	63	19	2	2	NUM
iajs-3112	63	20	2	2	NUM
iajs-3112	63	21	3	3	NUM
iajs-3112	63	22	0	0	NUM
iajs-3112	63	23	1	1	NUM
iajs-3112	63	24	2	2	NUM
iajs-3112	63	25	3	3	NUM
iajs-3112	63	26	ihjpas	ihjpa	NOUN
iajs-3112	63	27	.	.	PUNCT
iajs-3112	64	1	37	37	NUM
iajs-3112	64	2	(	(	PUNCT
iajs-3112	64	3	1	1	NUM
iajs-3112	64	4	)	)	PUNCT
iajs-3112	64	5	2024	2024	NUM
iajs-3112	64	6	456	456	NUM
iajs-3112	64	7	we	we	PRON
iajs-3112	64	8	recall	recall	VERB
iajs-3112	64	9	,	,	PUNCT
iajs-3112	64	10	an	an	DET
iajs-3112	64	11	interval	interval	NOUN
iajs-3112	64	12	valued	value	VERB
iajs-3112	64	13	fuzzy	fuzzy	ADJ
iajs-3112	64	14	set	set	VERB
iajs-3112	64	15	μ̃	μ̃	PROPN
iajs-3112	64	16	in	in	ADP
iajs-3112	64	17	ν	ν	NOUN
iajs-3112	64	18	is	be	AUX
iajs-3112	64	19	defined	define	VERB
iajs-3112	64	20	as	as	ADP
iajs-3112	64	21	μ̃	μ̃	PROPN
iajs-3112	64	22	=	=	SYM
iajs-3112	64	23	{	{	PUNCT
iajs-3112	64	24	〈	〈	NOUN
iajs-3112	64	25	𝜍	𝜍	PROPN
iajs-3112	64	26	,	,	PUNCT
iajs-3112	64	27	[	[	X
iajs-3112	64	28	𝜇𝐿(𝜍	𝜇𝐿(𝜍	NOUN
iajs-3112	64	29	)	)	PUNCT
iajs-3112	64	30	,	,	PUNCT
iajs-3112	64	31	𝜇𝑈(𝜍	𝜇𝑈(𝜍	NOUN
iajs-3112	64	32	)	)	PUNCT
iajs-3112	64	33	]	]	PUNCT
iajs-3112	64	34	,	,	PUNCT
iajs-3112	64	35	𝜍	𝜍	PROPN
iajs-3112	64	36	∈	∈	PROPN
iajs-3112	64	37	𝑁	𝑁	PROPN
iajs-3112	64	38	〉	〉	NOUN
iajs-3112	64	39	}	}	PUNCT
iajs-3112	64	40	,	,	PUNCT
iajs-3112	64	41	where	where	SCONJ
iajs-3112	64	42	the	the	DET
iajs-3112	64	43	ordinary	ordinary	ADJ
iajs-3112	64	44	fuzzy	fuzzy	ADJ
iajs-3112	64	45	sets	set	NOUN
iajs-3112	64	46	𝜇𝐿	𝜇𝐿	NOUN
iajs-3112	64	47	:	:	PUNCT
iajs-3112	64	48	𝑁	𝑁	PROPN
iajs-3112	64	49	→	→	SYM
iajs-3112	64	50	[	[	X
iajs-3112	64	51	0,1	0,1	NUM
iajs-3112	64	52	]	]	PUNCT
iajs-3112	64	53	and	and	CCONJ
iajs-3112	64	54	𝜇	𝜇	ADP
iajs-3112	64	55	𝑈	𝑈	NOUN
iajs-3112	64	56	:	:	PUNCT
iajs-3112	64	57	𝑁	𝑁	PROPN
iajs-3112	64	58	→	→	SYM
iajs-3112	64	59	[	[	X
iajs-3112	64	60	0,1	0,1	NUM
iajs-3112	64	61	]	]	PUNCT
iajs-3112	64	62	are	be	AUX
iajs-3112	64	63	called	call	VERB
iajs-3112	64	64	a	a	DET
iajs-3112	64	65	lower	low	ADJ
iajs-3112	64	66	fuzzy	fuzzy	ADJ
iajs-3112	64	67	set	set	NOUN
iajs-3112	64	68	and	and	CCONJ
iajs-3112	64	69	an	an	DET
iajs-3112	64	70	upper	upper	ADJ
iajs-3112	64	71	fuzzy	fuzzy	ADJ
iajs-3112	64	72	set	set	NOUN
iajs-3112	64	73	of	of	ADP
iajs-3112	64	74	μ̃	μ̃	PROPN
iajs-3112	64	75	respectively	respectively	ADV
iajs-3112	64	76	.	.	PUNCT
iajs-3112	65	1	also	also	ADV
iajs-3112	65	2	,	,	PUNCT
iajs-3112	65	3	we	we	PRON
iajs-3112	65	4	recall	recall	VERB
iajs-3112	65	5	some	some	DET
iajs-3112	65	6	definitions	definition	NOUN
iajs-3112	65	7	of	of	ADP
iajs-3112	65	8	a	a	DET
iajs-3112	65	9	cubic	cubic	ADJ
iajs-3112	65	10	subset	subset	NOUN
iajs-3112	65	11	of	of	ADP
iajs-3112	65	12	aˑku	aˑku	NOUN
iajs-3112	65	13	-	-	PUNCT
iajs-3112	65	14	semigroup	semigroup	NOUN
iajs-3112	65	15	from	from	ADP
iajs-3112	65	16	[	[	X
iajs-3112	65	17	22	22	NUM
iajs-3112	65	18	]	]	PUNCT
iajs-3112	65	19	.	.	PUNCT
iajs-3112	66	1	definition(5	definition(5	NOUN
iajs-3112	66	2	)	)	PUNCT
iajs-3112	67	1	[	[	X
iajs-3112	67	2	18	18	NUM
iajs-3112	67	3	]	]	PUNCT
iajs-3112	67	4	.	.	PUNCT
iajs-3112	68	1	the	the	DET
iajs-3112	68	2	cubic	cubic	ADJ
iajs-3112	68	3	set	set	VERB
iajs-3112	68	4	θ	θ	PROPN
iajs-3112	68	5	of	of	ADP
iajs-3112	68	6	a	a	DET
iajs-3112	68	7	non	non	ADJ
iajs-3112	68	8	-	-	ADJ
iajs-3112	68	9	empty	empty	ADJ
iajs-3112	68	10	set	set	ADJ
iajs-3112	68	11	ν	ν	NOUN
iajs-3112	68	12	is	be	AUX
iajs-3112	68	13	θ	θ	NOUN
iajs-3112	68	14	=	=	PUNCT
iajs-3112	68	15	{	{	PUNCT
iajs-3112	68	16	〈	〈	PROPN
iajs-3112	68	17	ς	ς	PROPN
iajs-3112	68	18	,	,	PUNCT
iajs-3112	68	19	𝜇θ(ς	𝜇θ(ς	ADJ
iajs-3112	68	20	)	)	PUNCT
iajs-3112	68	21	,	,	PUNCT
iajs-3112	68	22	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	68	23	)	)	PUNCT
iajs-3112	68	24	〉	〉	NOUN
iajs-3112	68	25	:	:	PUNCT
iajs-3112	68	26	ς	ς	PROPN
iajs-3112	68	27	∈	∈	PROPN
iajs-3112	68	28	ν	ν	NOUN
iajs-3112	68	29	}	}	PUNCT
iajs-3112	68	30	,	,	PUNCT
iajs-3112	68	31	which	which	PRON
iajs-3112	68	32	is	be	AUX
iajs-3112	68	33	briefly	briefly	ADV
iajs-3112	68	34	indicated	indicate	VERB
iajs-3112	68	35	by	by	ADP
iajs-3112	68	36	θ	θ	X
iajs-3112	68	37	=	=	PUNCT
iajs-3112	68	38	〈	〈	PROPN
iajs-3112	68	39	𝜇θ	𝜇θ	PROPN
iajs-3112	68	40	,	,	PUNCT
iajs-3112	68	41	𝜆θ	𝜆θ	ADP
iajs-3112	68	42	〉	〉	NOUN
iajs-3112	68	43	where	where	SCONJ
iajs-3112	68	44	𝜇θ(ς	𝜇θ(ς	VERB
iajs-3112	68	45	)	)	PUNCT
iajs-3112	68	46	=	=	PUNCT
iajs-3112	69	1	[	[	X
iajs-3112	69	2	𝜇θ	𝜇θ	NOUN
iajs-3112	69	3	𝐿(ς	𝐿(ς	NOUN
iajs-3112	69	4	)	)	PUNCT
iajs-3112	69	5	,	,	PUNCT
iajs-3112	69	6	𝜇θ	𝜇θ	ADV
iajs-3112	69	7	𝑈(ς	𝑈(ς	NOUN
iajs-3112	69	8	)	)	PUNCT
iajs-3112	69	9	]	]	PUNCT
iajs-3112	69	10	is	be	AUX
iajs-3112	69	11	an	an	DET
iajs-3112	69	12	interval	interval	NOUN
iajs-3112	69	13	valued	value	VERB
iajs-3112	69	14	fuzzy	fuzzy	ADJ
iajs-3112	69	15	set	set	VERB
iajs-3112	69	16	in	in	ADP
iajs-3112	69	17	ν	ν	NOUN
iajs-3112	69	18	and	and	CCONJ
iajs-3112	69	19	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	69	20	)	)	PUNCT
iajs-3112	69	21	is	be	AUX
iajs-3112	69	22	a	a	DET
iajs-3112	69	23	fuzzy	fuzzy	ADJ
iajs-3112	69	24	set	set	NOUN
iajs-3112	69	25	in	in	ADP
iajs-3112	69	26	ν.the	ν.the	DET
iajs-3112	69	27	set	set	NOUN
iajs-3112	69	28	{	{	PUNCT
iajs-3112	69	29	ς	ς	PROPN
iajs-3112	69	30	∈	∈	PROPN
iajs-3112	69	31	ν	ν	NOUN
iajs-3112	69	32	∶	∶	NOUN
iajs-3112	69	33	𝜇θ(ς	𝜇θ(ς	PUNCT
iajs-3112	69	34	)	)	PUNCT
iajs-3112	69	35	≥	≥	PROPN
iajs-3112	69	36	�	�	PROPN
iajs-3112	69	37	̃	̃	PROPN
iajs-3112	69	38	�	�	PROPN
iajs-3112	69	39	,	,	PUNCT
iajs-3112	69	40	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	69	41	)	)	PUNCT
iajs-3112	69	42	≤	≤	NUM
iajs-3112	69	43	𝛼	𝛼	X
iajs-3112	69	44	}	}	PUNCT
iajs-3112	69	45	is	be	AUX
iajs-3112	69	46	called	call	VERB
iajs-3112	69	47	a	a	DET
iajs-3112	69	48	cubic	cubic	ADJ
iajs-3112	69	49	level	level	NOUN
iajs-3112	69	50	set	set	NOUN
iajs-3112	69	51	of	of	ADP
iajs-3112	69	52	θ	θ	PROPN
iajs-3112	69	53	=	=	PUNCT
iajs-3112	70	1	〈	〈	PROPN
iajs-3112	70	2	𝜇θ	𝜇θ	PROPN
iajs-3112	70	3	,	,	PUNCT
iajs-3112	70	4	𝜆θ	𝜆θ	ADP
iajs-3112	70	5	〉	〉	NOUN
iajs-3112	70	6	,	,	PUNCT
iajs-3112	70	7	where	where	SCONJ
iajs-3112	70	8	[	[	X
iajs-3112	70	9	0,0]≤	0,0]≤	NOUN
iajs-3112	70	10	�	�	PROPN
iajs-3112	70	11	̃	̃	PROPN
iajs-3112	70	12	�	�	NOUN
iajs-3112	70	13	≤	≤	NOUN
iajs-3112	71	1	[	[	X
iajs-3112	71	2	1,1	1,1	NUM
iajs-3112	71	3	]	]	PUNCT
iajs-3112	71	4	and	and	CCONJ
iajs-3112	71	5	𝛼	𝛼	PRON
iajs-3112	71	6	∈	∈	NOUN
iajs-3112	71	7	[	[	X
iajs-3112	71	8	0,1	0,1	NUM
iajs-3112	71	9	]	]	PUNCT
iajs-3112	71	10	.	.	PUNCT
iajs-3112	72	1	definition(6	definition(6	PROPN
iajs-3112	72	2	)	)	PUNCT
iajs-3112	73	1	[	[	X
iajs-3112	73	2	22	22	NUM
iajs-3112	73	3	]	]	PUNCT
iajs-3112	73	4	.	.	PUNCT
iajs-3112	74	1	the	the	DET
iajs-3112	74	2	cubic	cubic	ADJ
iajs-3112	74	3	set	set	VERB
iajs-3112	74	4	θ	θ	PROPN
iajs-3112	74	5	of	of	ADP
iajs-3112	74	6	ν	ν	PROPN
iajs-3112	74	7	is	be	AUX
iajs-3112	74	8	named	name	VERB
iajs-3112	74	9	a	a	DET
iajs-3112	74	10	cubic	cubic	ADJ
iajs-3112	74	11	sub	sub	NOUN
iajs-3112	74	12	ku	ku	PROPN
iajs-3112	74	13	-	-	PUNCT
iajs-3112	74	14	semigroup	semigroup	PROPN
iajs-3112	74	15	if	if	SCONJ
iajs-3112	74	16	for	for	ADP
iajs-3112	74	17	all	all	DET
iajs-3112	74	18	ς	ς	PROPN
iajs-3112	74	19	,	,	PUNCT
iajs-3112	74	20	ω	ω	PROPN
iajs-3112	74	21	∈	∈	PROPN
iajs-3112	74	22	ν	ν	NOUN
iajs-3112	74	23	,	,	PUNCT
iajs-3112	74	24	1	1	NUM
iajs-3112	74	25	.	.	NUM
iajs-3112	74	26	𝜇θ(ς	𝜇θ(ς	NOUN
iajs-3112	74	27	∗	∗	PROPN
iajs-3112	74	28	ω	ω	NUM
iajs-3112	74	29	)	)	PUNCT
iajs-3112	74	30	≥	≥	NOUN
iajs-3112	74	31	𝑟𝑚𝑖𝑛{𝜇θ(ς)ˑ	𝑟𝑚𝑖𝑛{𝜇θ(ς)ˑ	NOUN
iajs-3112	74	32	,	,	PUNCT
iajs-3112	74	33	𝜇θ(ω)},𝜆θ(ς	𝜇θ(ω)},𝜆θ(ς	NUM
iajs-3112	74	34	∗	∗	NOUN
iajs-3112	74	35	ω	ω	NOUN
iajs-3112	74	36	)	)	PUNCT
iajs-3112	74	37	≤	≤	NOUN
iajs-3112	74	38	𝑚𝑎𝑥{𝜆θ(ς)ˑ	𝑚𝑎𝑥{𝜆θ(ς)ˑ	NOUN
iajs-3112	74	39	,	,	PUNCT
iajs-3112	74	40	𝜆θ(ω	𝜆θ(ω	NOUN
iajs-3112	74	41	)	)	PUNCT
iajs-3112	74	42	}	}	PUNCT
iajs-3112	74	43	.	.	PUNCT
iajs-3112	75	1	2	2	X
iajs-3112	75	2	.	.	NUM
iajs-3112	75	3	𝜇θ(ς	𝜇θ(ς	VERB
iajs-3112	75	4	∘	∘	PROPN
iajs-3112	75	5	ω	ω	NUM
iajs-3112	75	6	)	)	PUNCT
iajs-3112	75	7	≥	≥	NUM
iajs-3112	75	8	𝑟𝑚𝑖	𝑟𝑚𝑖	X
iajs-3112	75	9	𝑛{𝜇𝛩(𝜍)ˑ	𝑛{𝜇𝛩(𝜍)ˑ	NOUN
iajs-3112	75	10	,	,	PUNCT
iajs-3112	75	11	ˑ𝜇𝛩(𝜔	ˑ𝜇𝛩(𝜔	NOUN
iajs-3112	75	12	)	)	PUNCT
iajs-3112	75	13	}	}	PUNCT
iajs-3112	75	14	,	,	PUNCT
iajs-3112	75	15	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	75	16	∘	∘	PROPN
iajs-3112	75	17	ω	ω	NOUN
iajs-3112	75	18	)	)	PUNCT
iajs-3112	75	19	≤	≤	NUM
iajs-3112	75	20	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	75	21	{	{	PUNCT
iajs-3112	75	22	𝜆θ(ς)ˑ	𝜆θ(ς)ˑ	NOUN
iajs-3112	75	23	,	,	PUNCT
iajs-3112	75	24	ˑ𝜆θ(ω	ˑ𝜆θ(ω	PROPN
iajs-3112	75	25	)	)	PUNCT
iajs-3112	75	26	}	}	PUNCT
iajs-3112	75	27	.	.	PUNCT
iajs-3112	76	1	definition(7	definition(7	PROPN
iajs-3112	76	2	)	)	PUNCT
iajs-3112	77	1	[	[	X
iajs-3112	77	2	22	22	NUM
iajs-3112	77	3	]	]	PUNCT
iajs-3112	77	4	.	.	PUNCT
iajs-3112	78	1	the	the	DET
iajs-3112	78	2	cubic	cubic	ADJ
iajs-3112	78	3	set	set	VERB
iajs-3112	78	4	θ	θ	PROPN
iajs-3112	78	5	of	of	ADP
iajs-3112	78	6	ν	ν	PROPN
iajs-3112	78	7	is	be	AUX
iajs-3112	78	8	named	name	VERB
iajs-3112	78	9	a	a	DET
iajs-3112	78	10	cubic	cubic	ADJ
iajs-3112	78	11	ideal	ideal	NOUN
iajs-3112	78	12	,	,	PUNCT
iajs-3112	78	13	if	if	SCONJ
iajs-3112	78	14	for	for	ADP
iajs-3112	78	15	all	all	DET
iajs-3112	78	16	ς	ς	PROPN
iajs-3112	78	17	,	,	PUNCT
iajs-3112	78	18	ω	ω	PROPN
iajs-3112	78	19	∈	∈	PROPN
iajs-3112	78	20	ν	ν	X
iajs-3112	78	21	.	.	PUNCT
iajs-3112	79	1	(	(	PUNCT
iajs-3112	79	2	ci1	ci1	PROPN
iajs-3112	79	3	)	)	PUNCT
iajs-3112	79	4	𝜇θ(0	𝜇θ(0	PROPN
iajs-3112	79	5	)	)	PUNCT
iajs-3112	79	6	≥	≥	NOUN
iajs-3112	79	7	𝜇θ(ς)and	𝜇θ(ς)and	NOUN
iajs-3112	79	8	𝜆θ(0	𝜆θ(0	PROPN
iajs-3112	79	9	)	)	PUNCT
iajs-3112	79	10	≤	≤	NOUN
iajs-3112	79	11	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	79	12	)	)	PUNCT
iajs-3112	79	13	.	.	PUNCT
iajs-3112	80	1	(	(	PUNCT
iajs-3112	80	2	ci2	ci2	NOUN
iajs-3112	80	3	)	)	PUNCT
iajs-3112	80	4	𝜇θ(ω	𝜇θ(ω	NOUN
iajs-3112	80	5	)	)	PUNCT
iajs-3112	80	6	≥	≥	NOUN
iajs-3112	80	7	𝑟𝑚𝑖𝑛{𝜇θ(ς	𝑟𝑚𝑖𝑛{𝜇θ(ς	NOUN
iajs-3112	80	8	∗	∗	NOUN
iajs-3112	80	9	ω	ω	NOUN
iajs-3112	80	10	)	)	PUNCT
iajs-3112	80	11	,	,	PUNCT
iajs-3112	80	12	𝜇θ(ς	𝜇θ(ς	NOUN
iajs-3112	80	13	)	)	PUNCT
iajs-3112	80	14	}	}	PUNCT
iajs-3112	80	15	,	,	PUNCT
iajs-3112	80	16	𝜆θ(ω	𝜆θ(ω	NOUN
iajs-3112	80	17	)	)	PUNCT
iajs-3112	80	18	≤	≤	NUM
iajs-3112	80	19	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	80	20	{	{	PUNCT
iajs-3112	80	21	𝜆θ(ς	𝜆θ(ς	NOUN
iajs-3112	80	22	∗	∗	NOUN
iajs-3112	80	23	ω)ˑ	ω)ˑ	NOUN
iajs-3112	80	24	,	,	PUNCT
iajs-3112	80	25	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	80	26	)	)	PUNCT
iajs-3112	80	27	}	}	PUNCT
iajs-3112	80	28	.	.	PUNCT
iajs-3112	81	1	(	(	PUNCT
iajs-3112	81	2	ci3	ci3	ADJ
iajs-3112	81	3	)	)	PUNCT
iajs-3112	81	4	𝜇θ(ς	𝜇θ(ς	VERB
iajs-3112	81	5	∘	∘	PROPN
iajs-3112	81	6	ω	ω	NUM
iajs-3112	81	7	)	)	PUNCT
iajs-3112	81	8	≥	≥	NOUN
iajs-3112	81	9	𝑟𝑚𝑖𝑛{𝜇θ(ς	𝑟𝑚𝑖𝑛{𝜇θ(ς	NOUN
iajs-3112	81	10	)	)	PUNCT
iajs-3112	81	11	,	,	PUNCT
iajs-3112	81	12	ˑ𝜇θ(ω	ˑ𝜇θ(ω	PROPN
iajs-3112	81	13	)	)	PUNCT
iajs-3112	81	14	}	}	PUNCT
iajs-3112	81	15	,	,	PUNCT
iajs-3112	81	16	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	81	17	∘	∘	PROPN
iajs-3112	81	18	ω	ω	NOUN
iajs-3112	81	19	)	)	PUNCT
iajs-3112	81	20	≤	≤	NUM
iajs-3112	81	21	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	81	22	{	{	PUNCT
iajs-3112	81	23	𝜆θ(ς)ˑ	𝜆θ(ς)ˑ	NOUN
iajs-3112	81	24	,	,	PUNCT
iajs-3112	81	25	ˑ𝜆θ(ω	ˑ𝜆θ(ω	PROPN
iajs-3112	81	26	)	)	PUNCT
iajs-3112	81	27	}	}	PUNCT
iajs-3112	81	28	.	.	PUNCT
iajs-3112	82	1	definition(8	definition(8	NOUN
iajs-3112	82	2	)	)	PUNCT
iajs-3112	83	1	[	[	X
iajs-3112	83	2	22	22	NUM
iajs-3112	83	3	]	]	PUNCT
iajs-3112	83	4	.	.	PUNCT
iajs-3112	84	1	the	the	DET
iajs-3112	84	2	cubic	cubic	ADJ
iajs-3112	84	3	set	set	VERB
iajs-3112	84	4	θ	θ	PROPN
iajs-3112	84	5	of	of	ADP
iajs-3112	84	6	ν	ν	PROPN
iajs-3112	84	7	is	be	AUX
iajs-3112	84	8	named	name	VERB
iajs-3112	84	9	a	a	DET
iajs-3112	84	10	cubic	cubic	ADJ
iajs-3112	84	11	k	k	NOUN
iajs-3112	84	12	-	-	NOUN
iajs-3112	84	13	ideal	ideal	NOUN
iajs-3112	84	14	if	if	SCONJ
iajs-3112	84	15	for	for	ADP
iajs-3112	84	16	all	all	DET
iajs-3112	84	17	ς	ς	PROPN
iajs-3112	84	18	,	,	PUNCT
iajs-3112	84	19	ω	ω	PROPN
iajs-3112	84	20	,	,	PUNCT
iajs-3112	84	21	κ	κ	PROPN
iajs-3112	84	22	∈	∈	PROPN
iajs-3112	84	23	ν	ν	NOUN
iajs-3112	84	24	(	(	PUNCT
iajs-3112	84	25	𝑪𝒌𝟏	𝑪𝒌𝟏	ADJ
iajs-3112	84	26	)	)	PUNCT
iajs-3112	84	27	�	�	PROPN
iajs-3112	84	28	̃	̃	PROPN
iajs-3112	84	29	�	�	PROPN
iajs-3112	84	30	θ(0	θ(0	PROPN
iajs-3112	84	31	)	)	PUNCT
iajs-3112	84	32	)	)	PUNCT
iajs-3112	84	33	≥	≥	NOUN
iajs-3112	84	34	𝜇θ(ς	𝜇θ(ς	NOUN
iajs-3112	84	35	)	)	PUNCT
iajs-3112	84	36	,	,	PUNCT
iajs-3112	84	37	and	and	CCONJ
iajs-3112	84	38	𝜆θ(0	𝜆θ(0	PROPN
iajs-3112	84	39	)	)	PUNCT
iajs-3112	84	40	≤	≤	NOUN
iajs-3112	84	41	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	84	42	)	)	PUNCT
iajs-3112	84	43	.	.	PUNCT
iajs-3112	85	1	(	(	PUNCT
iajs-3112	85	2	𝑪𝒌𝟐	𝑪𝒌𝟐	NOUN
iajs-3112	85	3	)	)	PUNCT
iajs-3112	85	4	�	�	PROPN
iajs-3112	85	5	̃	̃	PROPN
iajs-3112	85	6	�	�	NOUN
iajs-3112	85	7	θ(ς	θ(ς	ADJ
iajs-3112	85	8	∗	∗	NOUN
iajs-3112	85	9	κ	κ	NOUN
iajs-3112	85	10	)	)	PUNCT
iajs-3112	85	11	≥	≥	NOUN
iajs-3112	85	12	𝑟𝑚𝑖𝑛{𝜇θ(ς	𝑟𝑚𝑖𝑛{𝜇θ(ς	ADJ
iajs-3112	85	13	∗	∗	NOUN
iajs-3112	85	14	(	(	PUNCT
iajs-3112	85	15	ω	ω	NOUN
iajs-3112	85	16	∗	∗	PROPN
iajs-3112	85	17	κ	κ	NOUN
iajs-3112	85	18	)	)	PUNCT
iajs-3112	85	19	)	)	PUNCT
iajs-3112	85	20	,	,	PUNCT
iajs-3112	85	21	𝜇θ(ω	𝜇θ(ω	NOUN
iajs-3112	85	22	)	)	PUNCT
iajs-3112	85	23	}	}	PUNCT
iajs-3112	85	24	.	.	PUNCT
iajs-3112	85	25	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	86	1	∗	∗	PROPN
iajs-3112	86	2	κ	κ	NOUN
iajs-3112	86	3	)	)	PUNCT
iajs-3112	86	4	≤	≤	NUM
iajs-3112	86	5	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	86	6	{	{	PUNCT
iajs-3112	86	7	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	86	8	∗	∗	NOUN
iajs-3112	86	9	(	(	PUNCT
iajs-3112	86	10	ω	ω	NOUN
iajs-3112	86	11	∗	∗	PROPN
iajs-3112	86	12	κ	κ	NOUN
iajs-3112	86	13	)	)	PUNCT
iajs-3112	86	14	)	)	PUNCT
iajs-3112	86	15	,	,	PUNCT
iajs-3112	86	16	𝜆θ(ω	𝜆θ(ω	NOUN
iajs-3112	86	17	)	)	PUNCT
iajs-3112	86	18	}	}	PUNCT
iajs-3112	86	19	.	.	PUNCT
iajs-3112	87	1	(	(	PUNCT
iajs-3112	87	2	𝑪𝒌𝟑	𝑪𝒌𝟑	NOUN
iajs-3112	87	3	)	)	PUNCT
iajs-3112	87	4	�	�	PROPN
iajs-3112	87	5	̃	̃	PROPN
iajs-3112	87	6	�	�	PROPN
iajs-3112	87	7	θ(ς	θ(ς	PROPN
iajs-3112	87	8	∘	∘	PROPN
iajs-3112	87	9	ω	ω	NUM
iajs-3112	87	10	)	)	PUNCT
iajs-3112	87	11	≥	≥	PROPN
iajs-3112	87	12	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-3112	87	13	{	{	PUNCT
iajs-3112	87	14	�	�	PROPN
iajs-3112	87	15	̃	̃	PROPN
iajs-3112	87	16	�	�	NOUN
iajs-3112	87	17	θ(ς)ˑ	θ(ς)ˑ	NOUN
iajs-3112	87	18	,	,	PUNCT
iajs-3112	87	19	𝜇θ(ω	𝜇θ(ω	NOUN
iajs-3112	87	20	)	)	PUNCT
iajs-3112	87	21	}	}	PUNCT
iajs-3112	87	22	,	,	PUNCT
iajs-3112	87	23	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	87	24	∘	∘	PROPN
iajs-3112	87	25	ω	ω	NOUN
iajs-3112	87	26	)	)	PUNCT
iajs-3112	87	27	≤	≤	NUM
iajs-3112	87	28	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	87	29	{	{	PUNCT
iajs-3112	87	30	𝜆θ(ς)ˑ	𝜆θ(ς)ˑ	NOUN
iajs-3112	87	31	,	,	PUNCT
iajs-3112	87	32	ˑ𝜆θ(ω	ˑ𝜆θ(ω	PROPN
iajs-3112	87	33	)	)	PUNCT
iajs-3112	87	34	}	}	PUNCT
iajs-3112	87	35	.	.	PUNCT
iajs-3112	88	1	theorem(9	theorem(9	PROPN
iajs-3112	88	2	)	)	PUNCT
iajs-3112	89	1	[	[	X
iajs-3112	89	2	22	22	NUM
iajs-3112	89	3	]	]	PUNCT
iajs-3112	89	4	.	.	PUNCT
iajs-3112	90	1	let	let	AUX
iajs-3112	90	2	(	(	PUNCT
iajs-3112	90	3	ν,∗,∘	ν,∗,∘	PROPN
iajs-3112	90	4	,	,	PUNCT
iajs-3112	90	5	0	0	NUM
iajs-3112	90	6	)	)	PUNCT
iajs-3112	90	7	be	be	AUX
iajs-3112	90	8	a	a	DET
iajs-3112	90	9	ku	ku	PROPN
iajs-3112	90	10	-	-	PUNCT
iajs-3112	90	11	semigroup	semigroup	PROPN
iajs-3112	90	12	.	.	PUNCT
iajs-3112	91	1	a	a	DET
iajs-3112	91	2	non	non	ADJ
iajs-3112	91	3	-	-	ADJ
iajs-3112	91	4	empty	empty	ADJ
iajs-3112	91	5	subset	subset	NOUN
iajs-3112	91	6	θ	θ	NOUN
iajs-3112	91	7	is	be	AUX
iajs-3112	91	8	a	a	DET
iajs-3112	91	9	cubic	cubic	ADJ
iajs-3112	91	10	k	k	NOUN
iajs-3112	91	11	-	-	NOUN
iajs-3112	91	12	ideal	ideal	NOUN
iajs-3112	91	13	if	if	SCONJ
iajs-3112	91	14	and	and	CCONJ
iajs-3112	91	15	only	only	ADV
iajs-3112	91	16	if	if	SCONJ
iajs-3112	91	17	it	it	PRON
iajs-3112	91	18	is	be	AUX
iajs-3112	91	19	a	a	DET
iajs-3112	91	20	cubic	cubic	ADJ
iajs-3112	91	21	ideal	ideal	NOUN
iajs-3112	91	22	of	of	ADP
iajs-3112	91	23	ν	ν	NOUN
iajs-3112	91	24	.	.	PUNCT
iajs-3112	92	1	3	3	X
iajs-3112	92	2	.	.	X
iajs-3112	92	3	cubic	cubic	ADJ
iajs-3112	92	4	positive	positive	ADJ
iajs-3112	92	5	implicative	implicative	ADJ
iajs-3112	92	6	-	-	PUNCT
iajs-3112	92	7	ideals	ideal	NOUN
iajs-3112	92	8	of	of	ADP
iajs-3112	92	9	𝚴	𝚴	PROPN
iajs-3112	92	10	definition(10	definition(10	NOUN
iajs-3112	92	11	)	)	PUNCT
iajs-3112	92	12	.	.	PUNCT
iajs-3112	93	1	the	the	DET
iajs-3112	93	2	cubic	cubic	ADJ
iajs-3112	93	3	set	set	VERB
iajs-3112	93	4	θ	θ	PROPN
iajs-3112	93	5	of	of	ADP
iajs-3112	93	6	ν	ν	PROPN
iajs-3112	93	7	is	be	AUX
iajs-3112	93	8	named	name	VERB
iajs-3112	93	9	a	a	DET
iajs-3112	93	10	cubic	cubic	ADJ
iajs-3112	93	11	positive	positive	ADJ
iajs-3112	93	12	implicative	implicative	ADJ
iajs-3112	93	13	-	-	PUNCT
iajs-3112	93	14	ideal	ideal	NOUN
iajs-3112	93	15	if	if	SCONJ
iajs-3112	93	16	for	for	ADP
iajs-3112	93	17	all	all	DET
iajs-3112	93	18	ς	ς	PROPN
iajs-3112	93	19	,	,	PUNCT
iajs-3112	93	20	ω	ω	PROPN
iajs-3112	93	21	,	,	PUNCT
iajs-3112	93	22	κ	κ	PROPN
iajs-3112	93	23	∈	∈	PROPN
iajs-3112	93	24	ν	ν	NOUN
iajs-3112	93	25	(	(	PUNCT
iajs-3112	93	26	𝑪𝒑𝟏	𝑪𝒑𝟏	NOUN
iajs-3112	93	27	)	)	PUNCT
iajs-3112	93	28	�	�	PROPN
iajs-3112	93	29	̃	̃	PROPN
iajs-3112	93	30	�	�	PROPN
iajs-3112	93	31	θ(0	θ(0	PROPN
iajs-3112	93	32	)	)	PUNCT
iajs-3112	93	33	)	)	PUNCT
iajs-3112	93	34	≥	≥	NOUN
iajs-3112	93	35	𝜇θ(ς	𝜇θ(ς	NOUN
iajs-3112	93	36	)	)	PUNCT
iajs-3112	93	37	,	,	PUNCT
iajs-3112	93	38	and	and	CCONJ
iajs-3112	93	39	𝜆θ(0	𝜆θ(0	PROPN
iajs-3112	93	40	)	)	PUNCT
iajs-3112	93	41	≤	≤	NOUN
iajs-3112	93	42	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	93	43	)	)	PUNCT
iajs-3112	93	44	.	.	PUNCT
iajs-3112	94	1	(	(	PUNCT
iajs-3112	94	2	𝑪𝒑𝟐)	𝑪𝒑𝟐)	PROPN
iajs-3112	94	3	�	�	PROPN
iajs-3112	94	4	̃	̃	NOUN
iajs-3112	94	5	�	�	NOUN
iajs-3112	94	6	θ(κ	θ(κ	PROPN
iajs-3112	94	7	∗	∗	NOUN
iajs-3112	94	8	ω	ω	NOUN
iajs-3112	94	9	)	)	PUNCT
iajs-3112	94	10	≥	≥	PROPN
iajs-3112	94	11	𝑟𝑚𝑖𝑛{𝜇θ(κ	𝑟𝑚𝑖𝑛{𝜇θ(κ	PROPN
iajs-3112	94	12	∗	∗	X
iajs-3112	94	13	(	(	PUNCT
iajs-3112	94	14	ς	ς	PROPN
iajs-3112	94	15	∗	∗	NOUN
iajs-3112	94	16	ω	ω	NOUN
iajs-3112	94	17	)	)	PUNCT
iajs-3112	94	18	)	)	PUNCT
iajs-3112	94	19	,	,	PUNCT
iajs-3112	94	20	𝜇θ(κ	𝜇θ(κ	PUNCT
iajs-3112	94	21	∗	∗	X
iajs-3112	94	22	ς	ς	PROPN
iajs-3112	94	23	)	)	PUNCT
iajs-3112	94	24	}	}	PUNCT
iajs-3112	94	25	,	,	PUNCT
iajs-3112	94	26	𝜆θ(κ	𝜆θ(κ	PUNCT
iajs-3112	94	27	∗	∗	NOUN
iajs-3112	94	28	ω	ω	NOUN
iajs-3112	94	29	)	)	PUNCT
iajs-3112	94	30	≤	≤	NUM
iajs-3112	94	31	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	94	32	{	{	PUNCT
iajs-3112	94	33	𝜆θ(κ	𝜆θ(κ	ADP
iajs-3112	94	34	∗	∗	NOUN
iajs-3112	94	35	(	(	PUNCT
iajs-3112	94	36	ς	ς	PROPN
iajs-3112	94	37	∗	∗	NOUN
iajs-3112	94	38	ω	ω	NOUN
iajs-3112	94	39	)	)	PUNCT
iajs-3112	94	40	)	)	PUNCT
iajs-3112	94	41	,	,	PUNCT
iajs-3112	94	42	𝜆θ(κ	𝜆θ(κ	PUNCT
iajs-3112	94	43	∗	∗	PROPN
iajs-3112	94	44	ς	ς	PROPN
iajs-3112	94	45	)	)	PUNCT
iajs-3112	94	46	}	}	PUNCT
iajs-3112	94	47	.	.	PUNCT
iajs-3112	95	1	(	(	PUNCT
iajs-3112	95	2	𝑪𝒑𝟑)	𝑪𝒑𝟑)	PROPN
iajs-3112	95	3	�	�	PROPN
iajs-3112	95	4	̃	̃	PROPN
iajs-3112	95	5	�	�	NOUN
iajs-3112	95	6	θ(ς	θ(ς	PROPN
iajs-3112	95	7	∘	∘	PROPN
iajs-3112	95	8	ω	ω	NUM
iajs-3112	95	9	)	)	PUNCT
iajs-3112	95	10	≥	≥	PROPN
iajs-3112	95	11	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-3112	95	12	{	{	PUNCT
iajs-3112	95	13	𝜇θ(ς)ˑ	𝜇θ(ς)ˑ	NOUN
iajs-3112	95	14	,	,	PUNCT
iajs-3112	95	15	𝜇θ(ω	𝜇θ(ω	NOUN
iajs-3112	95	16	)	)	PUNCT
iajs-3112	95	17	}	}	PUNCT
iajs-3112	95	18	,	,	PUNCT
iajs-3112	95	19	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	95	20	∘	∘	PROPN
iajs-3112	95	21	ω	ω	NOUN
iajs-3112	95	22	)	)	PUNCT
iajs-3112	95	23	≤	≤	NUM
iajs-3112	95	24	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	95	25	{	{	PUNCT
iajs-3112	95	26	𝜆θ(ς)ˑ	𝜆θ(ς)ˑ	NOUN
iajs-3112	95	27	,	,	PUNCT
iajs-3112	95	28	ˑ𝜆θ(ω	ˑ𝜆θ(ω	PROPN
iajs-3112	95	29	)	)	PUNCT
iajs-3112	95	30	}	}	PUNCT
iajs-3112	95	31	.	.	PUNCT
iajs-3112	96	1	definition(11	definition(11	PROPN
iajs-3112	96	2	)	)	PUNCT
iajs-3112	96	3	.	.	PUNCT
iajs-3112	97	1	the	the	DET
iajs-3112	97	2	cubic	cubic	ADJ
iajs-3112	97	3	set	set	VERB
iajs-3112	97	4	θ	θ	PROPN
iajs-3112	97	5	of	of	ADP
iajs-3112	97	6	ν	ν	PROPN
iajs-3112	97	7	is	be	AUX
iajs-3112	97	8	named	name	VERB
iajs-3112	97	9	a	a	DET
iajs-3112	97	10	cubic	cubic	ADJ
iajs-3112	97	11	implicative	implicative	ADJ
iajs-3112	97	12	-	-	PUNCT
iajs-3112	97	13	ideal	ideal	NOUN
iajs-3112	97	14	if	if	SCONJ
iajs-3112	97	15	for	for	ADP
iajs-3112	97	16	all	all	DET
iajs-3112	97	17	ς	ς	PROPN
iajs-3112	97	18	,	,	PUNCT
iajs-3112	97	19	ω	ω	PROPN
iajs-3112	97	20	,	,	PUNCT
iajs-3112	97	21	κ	κ	PROPN
iajs-3112	97	22	∈	∈	PROPN
iajs-3112	97	23	ν	ν	NOUN
iajs-3112	97	24	(	(	PUNCT
iajs-3112	97	25	𝑪𝑽𝟏	𝑪𝑽𝟏	PROPN
iajs-3112	97	26	)	)	PUNCT
iajs-3112	97	27	�	�	PROPN
iajs-3112	97	28	̃	̃	PROPN
iajs-3112	97	29	�	�	PROPN
iajs-3112	97	30	θ(0	θ(0	PROPN
iajs-3112	97	31	)	)	PUNCT
iajs-3112	97	32	)	)	PUNCT
iajs-3112	97	33	≥	≥	NOUN
iajs-3112	97	34	𝜇θ(ς	𝜇θ(ς	NOUN
iajs-3112	97	35	)	)	PUNCT
iajs-3112	97	36	,	,	PUNCT
iajs-3112	97	37	and	and	CCONJ
iajs-3112	97	38	𝜆θ(0	𝜆θ(0	PROPN
iajs-3112	97	39	)	)	PUNCT
iajs-3112	97	40	≤	≤	NOUN
iajs-3112	97	41	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	97	42	)	)	PUNCT
iajs-3112	97	43	.	.	PUNCT
iajs-3112	98	1	(	(	PUNCT
iajs-3112	98	2	𝑪𝑽𝟐)𝜇θ((ς	𝑪𝑽𝟐)𝜇θ((ς	VERB
iajs-3112	98	3	∗	∗	NOUN
iajs-3112	98	4	ω	ω	NOUN
iajs-3112	98	5	)	)	PUNCT
iajs-3112	98	6	∗	∗	PROPN
iajs-3112	98	7	ς	ς	PROPN
iajs-3112	98	8	)	)	PUNCT
iajs-3112	98	9	≥	≥	PROPN
iajs-3112	98	10	𝑟𝑚𝑖𝑛{𝜇θ(κ	𝑟𝑚𝑖𝑛{𝜇θ(κ	PROPN
iajs-3112	98	11	∗	∗	X
iajs-3112	98	12	(	(	PUNCT
iajs-3112	98	13	(	(	PUNCT
iajs-3112	98	14	ς	ς	PROPN
iajs-3112	98	15	∗	∗	NOUN
iajs-3112	98	16	ω	ω	NOUN
iajs-3112	98	17	)	)	PUNCT
iajs-3112	98	18	∗	∗	NOUN
iajs-3112	98	19	ς	ς	NOUN
iajs-3112	98	20	)	)	PUNCT
iajs-3112	98	21	)	)	PUNCT
iajs-3112	98	22	,	,	PUNCT
iajs-3112	98	23	𝜇θ(κ	𝜇θ(κ	NOUN
iajs-3112	98	24	)	)	PUNCT
iajs-3112	98	25	}	}	PUNCT
iajs-3112	98	26	,	,	PUNCT
iajs-3112	98	27	ihjpas	ihjpa	VERB
iajs-3112	98	28	.	.	PUNCT
iajs-3112	99	1	37	37	NUM
iajs-3112	99	2	(	(	PUNCT
iajs-3112	99	3	1	1	NUM
iajs-3112	99	4	)	)	PUNCT
iajs-3112	99	5	2024	2024	NUM
iajs-3112	99	6	457	457	NUM
iajs-3112	99	7	𝜆θ((ς	𝜆θ((ς	NOUN
iajs-3112	99	8	∗	∗	X
iajs-3112	99	9	ω	ω	NOUN
iajs-3112	99	10	)	)	PUNCT
iajs-3112	99	11	∗	∗	PROPN
iajs-3112	99	12	ς	ς	NOUN
iajs-3112	99	13	)	)	PUNCT
iajs-3112	99	14	≤	≤	NUM
iajs-3112	99	15	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	99	16	{	{	PUNCT
iajs-3112	99	17	𝜆θ(κ	𝜆θ(κ	ADP
iajs-3112	99	18	∗	∗	NOUN
iajs-3112	99	19	(	(	PUNCT
iajs-3112	99	20	(	(	PUNCT
iajs-3112	99	21	ς	ς	PROPN
iajs-3112	99	22	∗	∗	NOUN
iajs-3112	99	23	ω	ω	NOUN
iajs-3112	99	24	)	)	PUNCT
iajs-3112	99	25	∗	∗	NOUN
iajs-3112	99	26	ς	ς	NOUN
iajs-3112	99	27	)	)	PUNCT
iajs-3112	99	28	,	,	PUNCT
iajs-3112	99	29	𝜆θ(κ	𝜆θ(κ	NOUN
iajs-3112	99	30	)	)	PUNCT
iajs-3112	99	31	}	}	PUNCT
iajs-3112	99	32	.	.	PUNCT
iajs-3112	100	1	(	(	PUNCT
iajs-3112	100	2	𝑪𝑽𝟑)𝜇θ(ς	𝑪𝑽𝟑)𝜇θ(ς	PROPN
iajs-3112	100	3	∘	∘	PROPN
iajs-3112	100	4	ω	ω	PROPN
iajs-3112	100	5	)	)	PUNCT
iajs-3112	100	6	≥	≥	PROPN
iajs-3112	100	7	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-3112	100	8	{	{	PUNCT
iajs-3112	100	9	𝜇θ(ς)ˑ	𝜇θ(ς)ˑ	NOUN
iajs-3112	100	10	,	,	PUNCT
iajs-3112	100	11	𝜇θ(ω	𝜇θ(ω	NOUN
iajs-3112	100	12	)	)	PUNCT
iajs-3112	100	13	}	}	PUNCT
iajs-3112	100	14	,	,	PUNCT
iajs-3112	100	15	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	100	16	∘	∘	PROPN
iajs-3112	100	17	ω	ω	NOUN
iajs-3112	100	18	)	)	PUNCT
iajs-3112	100	19	≤	≤	NOUN
iajs-3112	100	20	𝑚𝑎	𝑚𝑎	ADP
iajs-3112	100	21	𝑥{𝜆θ(ς)ˑ	𝑥{𝜆θ(ς)ˑ	NOUN
iajs-3112	100	22	,	,	PUNCT
iajs-3112	100	23	ˑ𝜆θ(ω	ˑ𝜆θ(ω	PROPN
iajs-3112	100	24	)	)	PUNCT
iajs-3112	100	25	}	}	PUNCT
iajs-3112	100	26	.	.	PUNCT
iajs-3112	101	1	definition(12	definition(12	NOUN
iajs-3112	101	2	)	)	PUNCT
iajs-3112	101	3	.	.	PUNCT
iajs-3112	102	1	the	the	DET
iajs-3112	102	2	cubic	cubic	ADJ
iajs-3112	102	3	set	set	VERB
iajs-3112	102	4	θ	θ	PROPN
iajs-3112	102	5	of	of	ADP
iajs-3112	102	6	ν	ν	PROPN
iajs-3112	102	7	is	be	AUX
iajs-3112	102	8	named	name	VERB
iajs-3112	102	9	a	a	DET
iajs-3112	102	10	cubic	cubic	ADJ
iajs-3112	102	11	commutative	commutative	ADJ
iajs-3112	102	12	-	-	PUNCT
iajs-3112	102	13	ideal	ideal	NOUN
iajs-3112	102	14	if	if	SCONJ
iajs-3112	102	15	for	for	ADP
iajs-3112	102	16	all	all	DET
iajs-3112	102	17	ς	ς	PROPN
iajs-3112	102	18	,	,	PUNCT
iajs-3112	102	19	ω	ω	PROPN
iajs-3112	102	20	,	,	PUNCT
iajs-3112	102	21	κ	κ	PROPN
iajs-3112	102	22	∈	∈	PROPN
iajs-3112	102	23	ν	ν	X
iajs-3112	102	24	(	(	PUNCT
iajs-3112	102	25	𝑪𝑪𝟏	𝑪𝑪𝟏	PROPN
iajs-3112	102	26	)	)	PUNCT
iajs-3112	102	27	�	�	PROPN
iajs-3112	102	28	̃	̃	PROPN
iajs-3112	102	29	�	�	PROPN
iajs-3112	102	30	θ(0	θ(0	PROPN
iajs-3112	102	31	)	)	PUNCT
iajs-3112	102	32	)	)	PUNCT
iajs-3112	102	33	≥	≥	NOUN
iajs-3112	102	34	𝜇θ(ς	𝜇θ(ς	NOUN
iajs-3112	102	35	)	)	PUNCT
iajs-3112	102	36	,	,	PUNCT
iajs-3112	102	37	and	and	CCONJ
iajs-3112	102	38	𝜆θ(0	𝜆θ(0	PROPN
iajs-3112	102	39	)	)	PUNCT
iajs-3112	102	40	≤	≤	NOUN
iajs-3112	102	41	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	102	42	)	)	PUNCT
iajs-3112	102	43	.	.	PUNCT
iajs-3112	103	1	(	(	PUNCT
iajs-3112	103	2	𝑪𝑪𝟐	𝑪𝑪𝟐	PROPN
iajs-3112	103	3	)	)	PUNCT
iajs-3112	103	4	𝜇θ((ς	𝜇θ((ς	NOUN
iajs-3112	103	5	∗	∗	PROPN
iajs-3112	103	6	ω	ω	NOUN
iajs-3112	103	7	)	)	PUNCT
iajs-3112	103	8	∗	∗	PROPN
iajs-3112	103	9	ω	ω	NOUN
iajs-3112	103	10	)	)	PUNCT
iajs-3112	103	11	∗	∗	PROPN
iajs-3112	103	12	ς	ς	NOUN
iajs-3112	103	13	)	)	PUNCT
iajs-3112	103	14	≥	≥	NOUN
iajs-3112	103	15	𝑟𝑚𝑖𝑛{𝜇θ(ω	𝑟𝑚𝑖𝑛{𝜇θ(ω	NOUN
iajs-3112	103	16	∗	∗	NOUN
iajs-3112	103	17	(	(	PUNCT
iajs-3112	103	18	κ	κ	NOUN
iajs-3112	103	19	∗	∗	NOUN
iajs-3112	103	20	ς	ς	NOUN
iajs-3112	103	21	)	)	PUNCT
iajs-3112	103	22	)	)	PUNCT
iajs-3112	103	23	,	,	PUNCT
iajs-3112	103	24	�	�	PROPN
iajs-3112	103	25	̃	̃	NOUN
iajs-3112	103	26	�	�	NOUN
iajs-3112	103	27	θ(κ	θ(κ	NOUN
iajs-3112	103	28	)	)	PUNCT
iajs-3112	103	29	}	}	PUNCT
iajs-3112	103	30	,	,	PUNCT
iajs-3112	103	31	𝜆θ((ς	𝜆θ((ς	NOUN
iajs-3112	103	32	∗	∗	NOUN
iajs-3112	103	33	ω	ω	NOUN
iajs-3112	103	34	)	)	PUNCT
iajs-3112	103	35	∗	∗	PROPN
iajs-3112	103	36	ω	ω	NOUN
iajs-3112	103	37	)	)	PUNCT
iajs-3112	103	38	∗	∗	PROPN
iajs-3112	103	39	ς	ς	NOUN
iajs-3112	103	40	)	)	PUNCT
iajs-3112	103	41	≤	≤	NUM
iajs-3112	103	42	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	103	43	{	{	PUNCT
iajs-3112	103	44	𝜆θ(ω	𝜆θ(ω	NOUN
iajs-3112	103	45	∗	∗	NOUN
iajs-3112	103	46	(	(	PUNCT
iajs-3112	103	47	κ	κ	NOUN
iajs-3112	103	48	∗	∗	NOUN
iajs-3112	103	49	ς	ς	NOUN
iajs-3112	103	50	)	)	PUNCT
iajs-3112	103	51	)	)	PUNCT
iajs-3112	103	52	,	,	PUNCT
iajs-3112	103	53	𝜆θ(κ	𝜆θ(κ	NOUN
iajs-3112	103	54	)	)	PUNCT
iajs-3112	103	55	}	}	PUNCT
iajs-3112	103	56	.	.	PUNCT
iajs-3112	104	1	(	(	PUNCT
iajs-3112	104	2	𝑪𝑪𝟑	𝑪𝑪𝟑	PROPN
iajs-3112	104	3	)	)	PUNCT
iajs-3112	104	4	𝜇θ(ς	𝜇θ(ς	VERB
iajs-3112	104	5	∘	∘	PROPN
iajs-3112	104	6	ω	ω	NUM
iajs-3112	104	7	)	)	PUNCT
iajs-3112	104	8	≥	≥	PROPN
iajs-3112	104	9	𝑟𝑚𝑖𝑛	𝑟𝑚𝑖𝑛	NOUN
iajs-3112	104	10	{	{	PUNCT
iajs-3112	104	11	𝜇θ(ς)ˑ	𝜇θ(ς)ˑ	NOUN
iajs-3112	104	12	,	,	PUNCT
iajs-3112	104	13	𝜇θ(ω	𝜇θ(ω	NOUN
iajs-3112	104	14	)	)	PUNCT
iajs-3112	104	15	}	}	PUNCT
iajs-3112	104	16	,	,	PUNCT
iajs-3112	104	17	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	104	18	∘	∘	PROPN
iajs-3112	104	19	ω	ω	NOUN
iajs-3112	104	20	)	)	PUNCT
iajs-3112	104	21	≤	≤	NUM
iajs-3112	104	22	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	104	23	{	{	PUNCT
iajs-3112	104	24	𝜆θ(ς)ˑ	𝜆θ(ς)ˑ	NOUN
iajs-3112	104	25	,	,	PUNCT
iajs-3112	104	26	ˑ𝜆θ(ω	ˑ𝜆θ(ω	PROPN
iajs-3112	104	27	)	)	PUNCT
iajs-3112	104	28	}	}	PUNCT
iajs-3112	104	29	.	.	PUNCT
iajs-3112	105	1	lemma(13	lemma(13	ADJ
iajs-3112	105	2	)	)	PUNCT
iajs-3112	105	3	.	.	PUNCT
iajs-3112	106	1	in	in	ADP
iajs-3112	106	2	a	a	DET
iajs-3112	106	3	cubic	cubic	ADJ
iajs-3112	106	4	positive	positive	ADJ
iajs-3112	106	5	implicative	implicative	ADJ
iajs-3112	106	6	-	-	PUNCT
iajs-3112	106	7	ideal	ideal	NOUN
iajs-3112	106	8	θ	θ	PROPN
iajs-3112	106	9	of	of	ADP
iajs-3112	106	10	ν	ν	NOUN
iajs-3112	106	11	,	,	PUNCT
iajs-3112	106	12	if	if	SCONJ
iajs-3112	106	13	ς	ς	PROPN
iajs-3112	106	14	≤	≤	PROPN
iajs-3112	106	15	ω	ω	NUM
iajs-3112	106	16	,	,	PUNCT
iajs-3112	106	17	then	then	ADV
iajs-3112	106	18	𝜇θ(ς	𝜇θ(ς	PUNCT
iajs-3112	106	19	)	)	PUNCT
iajs-3112	106	20	≥	≥	NUM
iajs-3112	106	21	𝜇θ(ω	𝜇θ(ω	NOUN
iajs-3112	106	22	)	)	PUNCT
iajs-3112	106	23	and	and	CCONJ
iajs-3112	106	24	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	106	25	)	)	PUNCT
iajs-3112	106	26	≤	≤	NUM
iajs-3112	106	27	𝜆θ(ω	𝜆θ(ω	NOUN
iajs-3112	106	28	)	)	PUNCT
iajs-3112	106	29	,	,	PUNCT
iajs-3112	106	30	for	for	ADP
iajs-3112	106	31	all	all	DET
iajs-3112	106	32	ς	ς	PROPN
iajs-3112	106	33	,	,	PUNCT
iajs-3112	106	34	ω	ω	PROPN
iajs-3112	106	35	∈	∈	PROPN
iajs-3112	106	36	ν	ν	NOUN
iajs-3112	106	37	.	.	PUNCT
iajs-3112	106	38	proof	proof	NOUN
iajs-3112	106	39	.	.	PUNCT
iajs-3112	107	1	since	since	SCONJ
iajs-3112	107	2	ς	ς	PROPN
iajs-3112	107	3	≤	≤	PROPN
iajs-3112	107	4	ω	ω	PROPN
iajs-3112	107	5	⇒	⇒	PROPN
iajs-3112	107	6	ω	ω	PROPN
iajs-3112	107	7	∗	∗	NOUN
iajs-3112	107	8	ς	ς	PROPN
iajs-3112	107	9	=	=	SYM
iajs-3112	107	10	0	0	NUM
iajs-3112	107	11	,	,	PUNCT
iajs-3112	107	12	since	since	SCONJ
iajs-3112	107	13	θ	θ	PROPN
iajs-3112	107	14	is	be	AUX
iajs-3112	107	15	a	a	DET
iajs-3112	107	16	cubic	cubic	ADJ
iajs-3112	107	17	positive	positive	ADJ
iajs-3112	107	18	implicative	implicative	ADJ
iajs-3112	107	19	-	-	PUNCT
iajs-3112	107	20	ideal	ideal	NOUN
iajs-3112	107	21	,	,	PUNCT
iajs-3112	107	22	then	then	ADV
iajs-3112	107	23	𝜇θ(κ	𝜇θ(κ	PUNCT
iajs-3112	107	24	∗	∗	PROPN
iajs-3112	107	25	ς	ς	PROPN
iajs-3112	107	26	)	)	PUNCT
iajs-3112	107	27	≥	≥	PROPN
iajs-3112	107	28	𝑟𝑚𝑖𝑛{𝜇θ(κ	𝑟𝑚𝑖𝑛{𝜇θ(κ	PROPN
iajs-3112	107	29	∗	∗	X
iajs-3112	107	30	(	(	PUNCT
iajs-3112	107	31	ω	ω	PROPN
iajs-3112	107	32	∗	∗	PROPN
iajs-3112	107	33	ς	ς	NOUN
iajs-3112	107	34	)	)	PUNCT
iajs-3112	107	35	)	)	PUNCT
iajs-3112	107	36	,	,	PUNCT
iajs-3112	107	37	𝜇θ(κ	𝜇θ(κ	PUNCT
iajs-3112	107	38	∗	∗	X
iajs-3112	107	39	ω	ω	NOUN
iajs-3112	107	40	)	)	PUNCT
iajs-3112	107	41	}	}	PUNCT
iajs-3112	107	42	,	,	PUNCT
iajs-3112	107	43	put	put	VERB
iajs-3112	107	44	κ=0	κ=0	ADJ
iajs-3112	107	45	𝜇θ(0	𝜇θ(0	PROPN
iajs-3112	107	46	∗	∗	NOUN
iajs-3112	107	47	ς	ς	NOUN
iajs-3112	107	48	)	)	PUNCT
iajs-3112	107	49	≥	≥	NOUN
iajs-3112	107	50	𝑟	𝑟	NOUN
iajs-3112	107	51	𝑚𝑖𝑛{	𝑚𝑖𝑛{	PROPN
iajs-3112	107	52	�	�	PROPN
iajs-3112	107	53	̃	̃	PROPN
iajs-3112	107	54	�	�	PROPN
iajs-3112	107	55	θ(0	θ(0	PROPN
iajs-3112	107	56	∗	∗	NOUN
iajs-3112	107	57	(	(	PUNCT
iajs-3112	107	58	ω	ω	NOUN
iajs-3112	107	59	∗	∗	PROPN
iajs-3112	107	60	ς	ς	NOUN
iajs-3112	107	61	)	)	PUNCT
iajs-3112	107	62	)	)	PUNCT
iajs-3112	107	63	,	,	PUNCT
iajs-3112	107	64	𝜇θ(0	𝜇θ(0	PROPN
iajs-3112	107	65	∗	∗	NOUN
iajs-3112	107	66	ω	ω	NOUN
iajs-3112	107	67	)	)	PUNCT
iajs-3112	107	68	}	}	PUNCT
iajs-3112	107	69	,	,	PUNCT
iajs-3112	107	70	𝜇θ(ς	𝜇θ(ς	NOUN
iajs-3112	107	71	)	)	PUNCT
iajs-3112	107	72	≥	≥	PROPN
iajs-3112	107	73	𝑟	𝑟	NOUN
iajs-3112	107	74	𝑚𝑖𝑛{	𝑚𝑖𝑛{	PROPN
iajs-3112	107	75	�	�	PROPN
iajs-3112	107	76	̃	̃	PROPN
iajs-3112	107	77	�	�	NOUN
iajs-3112	107	78	θ(0	θ(0	PROPN
iajs-3112	107	79	∗	∗	NOUN
iajs-3112	107	80	0	0	NUM
iajs-3112	107	81	)	)	PUNCT
iajs-3112	107	82	,	,	PUNCT
iajs-3112	107	83	𝜇θ(ω	𝜇θ(ω	NOUN
iajs-3112	107	84	)	)	PUNCT
iajs-3112	107	85	}	}	PUNCT
iajs-3112	107	86	=	=	SYM
iajs-3112	107	87	𝜇θ(ω	𝜇θ(ω	NOUN
iajs-3112	107	88	)	)	PUNCT
iajs-3112	107	89	,	,	PUNCT
iajs-3112	107	90	and	and	CCONJ
iajs-3112	107	91	𝜆θ(κ	𝜆θ(κ	ADP
iajs-3112	107	92	∗	∗	NOUN
iajs-3112	107	93	ς	ς	NOUN
iajs-3112	107	94	)	)	PUNCT
iajs-3112	107	95	≤	≤	NUM
iajs-3112	107	96	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	107	97	{	{	PUNCT
iajs-3112	107	98	𝜆θ(κ	𝜆θ(κ	ADP
iajs-3112	107	99	∗	∗	NOUN
iajs-3112	107	100	(	(	PUNCT
iajs-3112	107	101	ω	ω	NOUN
iajs-3112	107	102	∗	∗	PROPN
iajs-3112	107	103	ς	ς	NOUN
iajs-3112	107	104	)	)	PUNCT
iajs-3112	107	105	)	)	PUNCT
iajs-3112	107	106	,	,	PUNCT
iajs-3112	107	107	𝜆θ(κ	𝜆θ(κ	PUNCT
iajs-3112	107	108	∗	∗	NOUN
iajs-3112	107	109	ω	ω	NOUN
iajs-3112	107	110	)	)	PUNCT
iajs-3112	107	111	}	}	PUNCT
iajs-3112	107	112	,	,	PUNCT
iajs-3112	107	113	put	put	VERB
iajs-3112	107	114	κ=0	κ=0	PROPN
iajs-3112	107	115	𝜆θ(0	𝜆θ(0	PROPN
iajs-3112	107	116	∗	∗	PROPN
iajs-3112	107	117	ς	ς	NOUN
iajs-3112	107	118	)	)	PUNCT
iajs-3112	107	119	≤	≤	NUM
iajs-3112	107	120	𝑚𝑎𝑥	𝑚𝑎𝑥	NUM
iajs-3112	107	121	{	{	PUNCT
iajs-3112	107	122	𝜆θ(0	𝜆θ(0	PROPN
iajs-3112	107	123	∗	∗	NOUN
iajs-3112	107	124	(	(	PUNCT
iajs-3112	107	125	ω	ω	PROPN
iajs-3112	107	126	∗	∗	PROPN
iajs-3112	107	127	ς	ς	NOUN
iajs-3112	107	128	)	)	PUNCT
iajs-3112	107	129	,	,	PUNCT
iajs-3112	107	130	𝜆θ(0	𝜆θ(0	PROPN
iajs-3112	107	131	∗	∗	NOUN
iajs-3112	107	132	ω	ω	NOUN
iajs-3112	107	133	)	)	PUNCT
iajs-3112	107	134	}	}	PUNCT
iajs-3112	107	135	,	,	PUNCT
iajs-3112	107	136	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	107	137	)	)	PUNCT
iajs-3112	107	138	≤	≤	NUM
iajs-3112	107	139	𝑚𝑎𝑥	𝑚𝑎𝑥	NUM
iajs-3112	107	140	{	{	PUNCT
iajs-3112	107	141	𝜆θ(0	𝜆θ(0	PROPN
iajs-3112	107	142	∗	∗	NOUN
iajs-3112	107	143	0	0	NUM
iajs-3112	107	144	)	)	PUNCT
iajs-3112	107	145	,	,	PUNCT
iajs-3112	107	146	𝜆θ(ω	𝜆θ(ω	NOUN
iajs-3112	107	147	)	)	PUNCT
iajs-3112	107	148	}	}	PUNCT
iajs-3112	107	149	=	=	SYM
iajs-3112	107	150	𝜆θ(ω	𝜆θ(ω	NUM
iajs-3112	107	151	)	)	PUNCT
iajs-3112	107	152	.	.	PUNCT
iajs-3112	108	1	■	■	PUNCT
iajs-3112	108	2	theorem(14	theorem(14	NOUN
iajs-3112	108	3	)	)	PUNCT
iajs-3112	108	4	.	.	PUNCT
iajs-3112	109	1	in	in	ADP
iajs-3112	109	2	a	a	DET
iajs-3112	109	3	cubic	cubic	ADJ
iajs-3112	109	4	positive	positive	ADJ
iajs-3112	109	5	implicative	implicative	ADJ
iajs-3112	109	6	-	-	PUNCT
iajs-3112	109	7	ideal	ideal	NOUN
iajs-3112	109	8	θ	θ	PROPN
iajs-3112	109	9	of	of	ADP
iajs-3112	109	10	ν	ν	NOUN
iajs-3112	109	11	,	,	PUNCT
iajs-3112	109	12	if	if	SCONJ
iajs-3112	109	13	ς	ς	PROPN
iajs-3112	109	14	∗	∗	X
iajs-3112	109	15	ω	ω	NOUN
iajs-3112	109	16	≤	≤	NUM
iajs-3112	109	17	κ	κ	NOUN
iajs-3112	109	18	,	,	PUNCT
iajs-3112	109	19	then	then	ADV
iajs-3112	109	20	𝜇θ(ω	𝜇θ(ω	NOUN
iajs-3112	109	21	)	)	PUNCT
iajs-3112	109	22	≥	≥	PROPN
iajs-3112	109	23	𝑟𝑚𝑖𝑛{𝜇θ(κ	𝑟𝑚𝑖𝑛{𝜇θ(κ	NOUN
iajs-3112	109	24	)	)	PUNCT
iajs-3112	109	25	,	,	PUNCT
iajs-3112	109	26	𝜇θ(ς	𝜇θ(ς	ADJ
iajs-3112	109	27	)	)	PUNCT
iajs-3112	109	28	}	}	PUNCT
iajs-3112	109	29	,	,	PUNCT
iajs-3112	109	30	and	and	CCONJ
iajs-3112	109	31	𝜆θ(ω	𝜆θ(ω	NOUN
iajs-3112	109	32	)	)	PUNCT
iajs-3112	109	33	≤	≤	NUM
iajs-3112	109	34	max	max	PROPN
iajs-3112	109	35	{	{	PUNCT
iajs-3112	109	36	𝜆θ(κ	𝜆θ(κ	NOUN
iajs-3112	109	37	)	)	PUNCT
iajs-3112	109	38	,	,	PUNCT
iajs-3112	109	39	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	109	40	)	)	PUNCT
iajs-3112	109	41	}	}	PUNCT
iajs-3112	109	42	,	,	PUNCT
iajs-3112	109	43	for	for	ADP
iajs-3112	109	44	all	all	DET
iajs-3112	109	45	ς	ς	PROPN
iajs-3112	109	46	,	,	PUNCT
iajs-3112	109	47	ω	ω	PROPN
iajs-3112	109	48	,	,	PUNCT
iajs-3112	109	49	κ	κ	PROPN
iajs-3112	109	50	∈	∈	PROPN
iajs-3112	109	51	ν	ν	X
iajs-3112	109	52	.	.	PUNCT
iajs-3112	109	53	proof	proof	NOUN
iajs-3112	109	54	.	.	PUNCT
iajs-3112	109	55	suppose	suppose	VERB
iajs-3112	109	56	ς	ς	PROPN
iajs-3112	109	57	∗	∗	PROPN
iajs-3112	109	58	ω	ω	PROPN
iajs-3112	109	59	≤	≤	NUM
iajs-3112	109	60	κholds	κhold	NOUN
iajs-3112	109	61	,	,	PUNCT
iajs-3112	109	62	then	then	ADV
iajs-3112	109	63	by	by	ADP
iajs-3112	109	64	lemma(13	lemma(13	ADJ
iajs-3112	109	65	)	)	PUNCT
iajs-3112	109	66	we	we	PRON
iajs-3112	109	67	get	get	VERB
iajs-3112	109	68	𝜇θ(ς	𝜇θ(ς	NOUN
iajs-3112	109	69	∗	∗	NOUN
iajs-3112	109	70	ω	ω	NOUN
iajs-3112	109	71	)	)	PUNCT
iajs-3112	109	72	≥	≥	NOUN
iajs-3112	109	73	𝜇θ(κ	𝜇θ(κ	NOUN
iajs-3112	109	74	)	)	PUNCT
iajs-3112	109	75	,	,	PUNCT
iajs-3112	109	76	and	and	CCONJ
iajs-3112	109	77	𝜆θ(ς	𝜆θ(ς	X
iajs-3112	109	78	∗	∗	X
iajs-3112	109	79	ω	ω	NOUN
iajs-3112	109	80	)	)	PUNCT
iajs-3112	109	81	≤	≤	NOUN
iajs-3112	109	82	𝜆θ(κ	𝜆θ(κ	NOUN
iajs-3112	109	83	)	)	PUNCT
iajs-3112	109	84	.	.	PUNCT
iajs-3112	110	1	since	since	SCONJ
iajs-3112	110	2	θ	θ	PROPN
iajs-3112	110	3	is	be	AUX
iajs-3112	110	4	a	a	DET
iajs-3112	110	5	cubic	cubic	ADJ
iajs-3112	110	6	positive	positive	ADJ
iajs-3112	110	7	implicative	implicative	ADJ
iajs-3112	110	8	-	-	PUNCT
iajs-3112	110	9	ideal	ideal	NOUN
iajs-3112	110	10	,	,	PUNCT
iajs-3112	110	11	that	that	ADV
iajs-3112	110	12	is	is	ADV
iajs-3112	110	13	𝜇θ(κ	𝜇θ(κ	PUNCT
iajs-3112	110	14	∗	∗	NOUN
iajs-3112	110	15	ω	ω	NOUN
iajs-3112	110	16	)	)	PUNCT
iajs-3112	110	17	≥	≥	PROPN
iajs-3112	110	18	𝑟𝑚𝑖𝑛{𝜇θ(κ	𝑟𝑚𝑖𝑛{𝜇θ(κ	PROPN
iajs-3112	110	19	∗	∗	X
iajs-3112	110	20	(	(	PUNCT
iajs-3112	110	21	ς	ς	PROPN
iajs-3112	110	22	∗	∗	NOUN
iajs-3112	110	23	ω	ω	NOUN
iajs-3112	110	24	)	)	PUNCT
iajs-3112	110	25	)	)	PUNCT
iajs-3112	110	26	,	,	PUNCT
iajs-3112	111	1	𝜇θ(κ	𝜇θ(κ	PUNCT
iajs-3112	111	2	∗	∗	X
iajs-3112	111	3	ς	ς	PROPN
iajs-3112	111	4	)	)	PUNCT
iajs-3112	111	5	}	}	PUNCT
iajs-3112	111	6	,	,	PUNCT
iajs-3112	111	7	put	put	VERB
iajs-3112	111	8	κ	κ	NOUN
iajs-3112	111	9	=	=	SYM
iajs-3112	111	10	0	0	PUNCT
iajs-3112	111	11	𝜇θ(0	𝜇θ(0	PROPN
iajs-3112	111	12	∗	∗	NOUN
iajs-3112	111	13	ω	ω	NOUN
iajs-3112	111	14	)	)	PUNCT
iajs-3112	111	15	≥	≥	NOUN
iajs-3112	111	16	𝑟𝑚𝑖𝑛{𝜇θ(0	𝑟𝑚𝑖𝑛{𝜇θ(0	VERB
iajs-3112	111	17	∗	∗	NOUN
iajs-3112	111	18	(	(	PUNCT
iajs-3112	111	19	ς	ς	PROPN
iajs-3112	111	20	∗	∗	NOUN
iajs-3112	111	21	ω	ω	NOUN
iajs-3112	111	22	)	)	PUNCT
iajs-3112	111	23	)	)	PUNCT
iajs-3112	111	24	,	,	PUNCT
iajs-3112	111	25	𝜇θ(0	𝜇θ(0	PROPN
iajs-3112	111	26	∗	∗	NOUN
iajs-3112	111	27	ς	ς	NOUN
iajs-3112	111	28	)	)	PUNCT
iajs-3112	111	29	}	}	PUNCT
iajs-3112	111	30	,	,	PUNCT
iajs-3112	111	31	𝜇θ(ω	𝜇θ(ω	NOUN
iajs-3112	111	32	)	)	PUNCT
iajs-3112	111	33	≥	≥	NOUN
iajs-3112	111	34	𝑟𝑚𝑖𝑛{𝜇θ(ς	𝑟𝑚𝑖𝑛{𝜇θ(ς	NOUN
iajs-3112	111	35	∗	∗	NOUN
iajs-3112	111	36	ω	ω	NOUN
iajs-3112	111	37	)	)	PUNCT
iajs-3112	111	38	,	,	PUNCT
iajs-3112	111	39	𝜇θ(ς	𝜇θ(ς	NOUN
iajs-3112	111	40	)	)	PUNCT
iajs-3112	111	41	}	}	PUNCT
iajs-3112	111	42	,	,	PUNCT
iajs-3112	111	43	but	but	CCONJ
iajs-3112	111	44	𝜇θ(ς	𝜇θ(ς	VERB
iajs-3112	111	45	∗	∗	X
iajs-3112	111	46	ω	ω	NOUN
iajs-3112	111	47	)	)	PUNCT
iajs-3112	111	48	≥	≥	NOUN
iajs-3112	111	49	𝜇θ(κ	𝜇θ(κ	NOUN
iajs-3112	111	50	)	)	PUNCT
iajs-3112	111	51	,	,	PUNCT
iajs-3112	111	52	then	then	ADV
iajs-3112	111	53	𝜇θ(ω	𝜇θ(ω	VERB
iajs-3112	111	54	)	)	PUNCT
iajs-3112	111	55	≥	≥	PROPN
iajs-3112	111	56	𝑟𝑚𝑖𝑛{𝜇θ(κ	𝑟𝑚𝑖𝑛{𝜇θ(κ	NOUN
iajs-3112	111	57	)	)	PUNCT
iajs-3112	111	58	,	,	PUNCT
iajs-3112	111	59	𝜇θ(ς	𝜇θ(ς	NOUN
iajs-3112	111	60	)	)	PUNCT
iajs-3112	111	61	}	}	PUNCT
iajs-3112	111	62	,	,	PUNCT
iajs-3112	111	63	and	and	CCONJ
iajs-3112	111	64	𝜆θ(κ	𝜆θ(κ	X
iajs-3112	111	65	∗	∗	NOUN
iajs-3112	111	66	ω	ω	NOUN
iajs-3112	111	67	)	)	PUNCT
iajs-3112	111	68	≤	≤	NOUN
iajs-3112	111	69	𝑚𝑎𝑥{𝜆θ(κ	𝑚𝑎𝑥{𝜆θ(κ	PROPN
iajs-3112	111	70	∗	∗	NOUN
iajs-3112	111	71	(	(	PUNCT
iajs-3112	111	72	ς	ς	PROPN
iajs-3112	111	73	∗	∗	NOUN
iajs-3112	111	74	ω	ω	NOUN
iajs-3112	111	75	)	)	PUNCT
iajs-3112	111	76	)	)	PUNCT
iajs-3112	111	77	,	,	PUNCT
iajs-3112	111	78	𝜆θ(κ	𝜆θ(κ	PUNCT
iajs-3112	111	79	∗	∗	PROPN
iajs-3112	111	80	ς	ς	PROPN
iajs-3112	111	81	)	)	PUNCT
iajs-3112	111	82	}	}	PUNCT
iajs-3112	111	83	,	,	PUNCT
iajs-3112	111	84	put	put	VERB
iajs-3112	111	85	κ	κ	NOUN
iajs-3112	111	86	=	=	SYM
iajs-3112	111	87	0	0	NUM
iajs-3112	111	88	,	,	PUNCT
iajs-3112	111	89	we	we	PRON
iajs-3112	111	90	obtain	obtain	VERB
iajs-3112	111	91	𝜆θ(ω	𝜆θ(ω	NOUN
iajs-3112	111	92	)	)	PUNCT
iajs-3112	111	93	≤	≤	NOUN
iajs-3112	111	94	𝑚𝑎𝑥{𝜆θ((ς	𝑚𝑎𝑥{𝜆θ((ς	NUM
iajs-3112	111	95	∗	∗	NOUN
iajs-3112	111	96	ω	ω	NOUN
iajs-3112	111	97	)	)	PUNCT
iajs-3112	111	98	)	)	PUNCT
iajs-3112	111	99	,	,	PUNCT
iajs-3112	111	100	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	111	101	)	)	PUNCT
iajs-3112	111	102	}	}	PUNCT
iajs-3112	111	103	,	,	PUNCT
iajs-3112	111	104	but	but	CCONJ
iajs-3112	111	105	𝜆θ(ς	𝜆θ(ς	X
iajs-3112	111	106	∗	∗	X
iajs-3112	111	107	ω	ω	NOUN
iajs-3112	111	108	)	)	PUNCT
iajs-3112	111	109	≤	≤	NOUN
iajs-3112	111	110	𝜆θ(κ	𝜆θ(κ	NOUN
iajs-3112	111	111	)	)	PUNCT
iajs-3112	111	112	,	,	PUNCT
iajs-3112	111	113	then	then	ADV
iajs-3112	111	114	𝜆θ(ω	𝜆θ(ω	NOUN
iajs-3112	111	115	)	)	PUNCT
iajs-3112	111	116	≤	≤	NUM
iajs-3112	111	117	𝑚𝑎𝑥{𝜆θ(κ	𝑚𝑎𝑥{𝜆θ(κ	NOUN
iajs-3112	111	118	)	)	PUNCT
iajs-3112	111	119	,	,	PUNCT
iajs-3112	111	120	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	111	121	)	)	PUNCT
iajs-3112	111	122	}	}	PUNCT
iajs-3112	111	123	,	,	PUNCT
iajs-3112	111	124	which	which	PRON
iajs-3112	111	125	is	be	AUX
iajs-3112	111	126	the	the	DET
iajs-3112	111	127	required	require	VERB
iajs-3112	111	128	.	.	PUNCT
iajs-3112	112	1	■	■	PUNCT
iajs-3112	112	2	theorem(15	theorem(15	NUM
iajs-3112	112	3	)	)	PUNCT
iajs-3112	112	4	.	.	PUNCT
iajs-3112	113	1	every	every	DET
iajs-3112	113	2	cubic	cubic	ADJ
iajs-3112	113	3	positive	positive	ADJ
iajs-3112	113	4	implicative	implicative	ADJ
iajs-3112	113	5	-	-	PUNCT
iajs-3112	113	6	ideal	ideal	NOUN
iajs-3112	113	7	θ	θ	PROPN
iajs-3112	113	8	of	of	ADP
iajs-3112	113	9	ν	ν	PROPN
iajs-3112	113	10	is	be	AUX
iajs-3112	113	11	a	a	DET
iajs-3112	113	12	cubic	cubic	ADJ
iajs-3112	113	13	ideal	ideal	NOUN
iajs-3112	113	14	.	.	PUNCT
iajs-3112	114	1	proof	proof	NOUN
iajs-3112	114	2	.	.	PUNCT
iajs-3112	115	1	let	let	VERB
iajs-3112	115	2	θ	θ	NOUN
iajs-3112	115	3	be	be	AUX
iajs-3112	115	4	a	a	DET
iajs-3112	115	5	cubic	cubic	ADJ
iajs-3112	115	6	positive	positive	ADJ
iajs-3112	115	7	implicative	implicative	ADJ
iajs-3112	115	8	-	-	PUNCT
iajs-3112	115	9	ideal	ideal	NOUN
iajs-3112	115	10	,	,	PUNCT
iajs-3112	115	11	then	then	ADV
iajs-3112	115	12	𝜇θ(κ	𝜇θ(κ	PUNCT
iajs-3112	115	13	∗	∗	PROPN
iajs-3112	115	14	ω	ω	NOUN
iajs-3112	115	15	)	)	PUNCT
iajs-3112	115	16	≥	≥	PROPN
iajs-3112	115	17	𝑟𝑚𝑖𝑛{𝜇θ(κ	𝑟𝑚𝑖𝑛{𝜇θ(κ	PROPN
iajs-3112	115	18	∗	∗	X
iajs-3112	115	19	(	(	PUNCT
iajs-3112	115	20	ς	ς	PROPN
iajs-3112	115	21	∗	∗	NOUN
iajs-3112	115	22	ω	ω	NOUN
iajs-3112	115	23	)	)	PUNCT
iajs-3112	115	24	)	)	PUNCT
iajs-3112	115	25	,	,	PUNCT
iajs-3112	116	1	𝜇θ(κ	𝜇θ(κ	PUNCT
iajs-3112	116	2	∗	∗	X
iajs-3112	116	3	ς	ς	PROPN
iajs-3112	116	4	)	)	PUNCT
iajs-3112	116	5	}	}	PUNCT
iajs-3112	116	6	,	,	PUNCT
iajs-3112	116	7	𝜆θ(κ	𝜆θ(κ	PUNCT
iajs-3112	116	8	∗	∗	NOUN
iajs-3112	116	9	ω	ω	NOUN
iajs-3112	116	10	)	)	PUNCT
iajs-3112	116	11	≤	≤	NUM
iajs-3112	116	12	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	116	13	{	{	PUNCT
iajs-3112	116	14	𝜆θ(κ	𝜆θ(κ	ADP
iajs-3112	116	15	∗	∗	NOUN
iajs-3112	116	16	(	(	PUNCT
iajs-3112	116	17	ς	ς	PROPN
iajs-3112	116	18	∗	∗	NOUN
iajs-3112	116	19	ω	ω	NOUN
iajs-3112	116	20	)	)	PUNCT
iajs-3112	116	21	)	)	PUNCT
iajs-3112	116	22	,	,	PUNCT
iajs-3112	116	23	𝜆θ(κ	𝜆θ(κ	PUNCT
iajs-3112	116	24	∗	∗	PROPN
iajs-3112	116	25	ς	ς	PROPN
iajs-3112	116	26	)	)	PUNCT
iajs-3112	116	27	}	}	PUNCT
iajs-3112	116	28	,	,	PUNCT
iajs-3112	116	29	put	put	VERB
iajs-3112	116	30	κ	κ	NOUN
iajs-3112	116	31	=	=	SYM
iajs-3112	116	32	0	0	NUM
iajs-3112	116	33	we	we	PRON
iajs-3112	116	34	get	get	VERB
iajs-3112	116	35	𝜇θ(0	𝜇θ(0	PROPN
iajs-3112	116	36	∗	∗	NOUN
iajs-3112	116	37	ω	ω	NOUN
iajs-3112	116	38	)	)	PUNCT
iajs-3112	116	39	≥	≥	NOUN
iajs-3112	116	40	𝑟𝑚𝑖𝑛{𝜇θ(0	𝑟𝑚𝑖𝑛{𝜇θ(0	VERB
iajs-3112	116	41	∗	∗	NOUN
iajs-3112	116	42	(	(	PUNCT
iajs-3112	116	43	ς	ς	PROPN
iajs-3112	116	44	∗	∗	NOUN
iajs-3112	116	45	ω	ω	NOUN
iajs-3112	116	46	)	)	PUNCT
iajs-3112	116	47	)	)	PUNCT
iajs-3112	116	48	,	,	PUNCT
iajs-3112	117	1	𝜇θ(0	𝜇θ(0	PROPN
iajs-3112	117	2	∗	∗	NOUN
iajs-3112	117	3	ς	ς	NOUN
iajs-3112	117	4	)	)	PUNCT
iajs-3112	117	5	}	}	PUNCT
iajs-3112	117	6	,	,	PUNCT
iajs-3112	117	7	ihjpas	ihjpas	PROPN
iajs-3112	117	8	.	.	PUNCT
iajs-3112	118	1	37	37	NUM
iajs-3112	118	2	(	(	PUNCT
iajs-3112	118	3	1	1	NUM
iajs-3112	118	4	)	)	PUNCT
iajs-3112	118	5	2024	2024	NUM
iajs-3112	118	6	458	458	NUM
iajs-3112	118	7	𝜇θ(ω	𝜇θ(ω	NOUN
iajs-3112	118	8	)	)	PUNCT
iajs-3112	118	9	≥	≥	NOUN
iajs-3112	118	10	𝑟𝑚𝑖𝑛{𝜇θ(ς	𝑟𝑚𝑖𝑛{𝜇θ(ς	NOUN
iajs-3112	118	11	∗	∗	NOUN
iajs-3112	118	12	ω	ω	NOUN
iajs-3112	118	13	)	)	PUNCT
iajs-3112	118	14	,	,	PUNCT
iajs-3112	118	15	𝜇θ(ς	𝜇θ(ς	NOUN
iajs-3112	118	16	)	)	PUNCT
iajs-3112	118	17	}	}	PUNCT
iajs-3112	118	18	…	…	PUNCT
iajs-3112	118	19	…	…	PUNCT
iajs-3112	118	20	.	.	PUNCT
iajs-3112	118	21	.	.	PUNCT
iajs-3112	119	1	(	(	PUNCT
iajs-3112	119	2	1	1	NUM
iajs-3112	119	3	)	)	PUNCT
iajs-3112	119	4	,	,	PUNCT
iajs-3112	119	5	and	and	CCONJ
iajs-3112	119	6	𝜆θ(0	𝜆θ(0	PROPN
iajs-3112	119	7	∗	∗	PROPN
iajs-3112	119	8	ω	ω	NOUN
iajs-3112	119	9	)	)	PUNCT
iajs-3112	119	10	≤	≤	NUM
iajs-3112	120	1	𝑚𝑎𝑥	𝑚𝑎𝑥	NOUN
iajs-3112	121	1	{	{	PUNCT
iajs-3112	121	2	𝜆θ(0	𝜆θ(0	PROPN
iajs-3112	121	3	∗	∗	NOUN
iajs-3112	121	4	(	(	PUNCT
iajs-3112	121	5	ς	ς	PROPN
iajs-3112	121	6	∗	∗	NOUN
iajs-3112	121	7	ω	ω	NOUN
iajs-3112	121	8	)	)	PUNCT
iajs-3112	121	9	)	)	PUNCT
iajs-3112	121	10	,	,	PUNCT
iajs-3112	121	11	𝜆θ(0	𝜆θ(0	PROPN
iajs-3112	121	12	∗	∗	NOUN
iajs-3112	121	13	ς	ς	NOUN
iajs-3112	121	14	)	)	PUNCT
iajs-3112	121	15	}	}	PUNCT
iajs-3112	121	16	,	,	PUNCT
iajs-3112	121	17	𝜆θ(ω	𝜆θ(ω	NOUN
iajs-3112	121	18	)	)	PUNCT
iajs-3112	121	19	≤	≤	NUM
iajs-3112	121	20	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	121	21	{	{	PUNCT
iajs-3112	121	22	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	121	23	∗	∗	NOUN
iajs-3112	121	24	ω	ω	NOUN
iajs-3112	121	25	)	)	PUNCT
iajs-3112	121	26	,	,	PUNCT
iajs-3112	121	27	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	121	28	)	)	PUNCT
iajs-3112	121	29	}	}	PUNCT
iajs-3112	121	30	…	…	PUNCT
iajs-3112	121	31	…	…	PUNCT
iajs-3112	121	32	…	…	PUNCT
iajs-3112	121	33	(	(	PUNCT
iajs-3112	121	34	2	2	NUM
iajs-3112	121	35	)	)	PUNCT
iajs-3112	121	36	from	from	ADP
iajs-3112	121	37	(	(	PUNCT
iajs-3112	121	38	1	1	NUM
iajs-3112	121	39	)	)	PUNCT
iajs-3112	121	40	and	and	CCONJ
iajs-3112	121	41	(	(	PUNCT
iajs-3112	121	42	2	2	NUM
iajs-3112	121	43	)	)	PUNCT
iajs-3112	121	44	,	,	PUNCT
iajs-3112	121	45	θ	θ	PROPN
iajs-3112	121	46	is	be	AUX
iajs-3112	121	47	a	a	DET
iajs-3112	121	48	cubic	cubic	ADJ
iajs-3112	121	49	ideal	ideal	NOUN
iajs-3112	121	50	.	.	PUNCT
iajs-3112	122	1	the	the	DET
iajs-3112	122	2	following	follow	VERB
iajs-3112	122	3	example	example	NOUN
iajs-3112	122	4	shows	show	VERB
iajs-3112	122	5	the	the	DET
iajs-3112	122	6	converse	converse	NOUN
iajs-3112	122	7	of	of	ADP
iajs-3112	122	8	this	this	DET
iajs-3112	122	9	theorem	theorem	NOUN
iajs-3112	122	10	is	be	AUX
iajs-3112	122	11	not	not	PART
iajs-3112	122	12	true	true	ADJ
iajs-3112	122	13	,	,	PUNCT
iajs-3112	122	14	in	in	ADP
iajs-3112	122	15	general	general	ADJ
iajs-3112	122	16	.	.	PUNCT
iajs-3112	122	17	example(16	example(16	PROPN
iajs-3112	122	18	)	)	PUNCT
iajs-3112	122	19	.	.	PUNCT
iajs-3112	123	1	let	let	VERB
iajs-3112	123	2	ν	ν	X
iajs-3112	123	3	=	=	PRON
iajs-3112	123	4	{	{	PUNCT
iajs-3112	123	5	0	0	NUM
iajs-3112	123	6	,	,	PUNCT
iajs-3112	123	7	𝑎	𝑎	NOUN
iajs-3112	123	8	,	,	PUNCT
iajs-3112	123	9	𝑏	𝑏	NOUN
iajs-3112	123	10	}	}	PUNCT
iajs-3112	123	11	with	with	ADP
iajs-3112	123	12	two	two	NUM
iajs-3112	123	13	operations	operation	NOUN
iajs-3112	123	14	∗	∗	NOUN
iajs-3112	123	15	and	and	CCONJ
iajs-3112	123	16	∘	∘	NOUN
iajs-3112	123	17	defined	define	VERB
iajs-3112	123	18	by	by	ADP
iajs-3112	123	19	table	table	NOUN
iajs-3112	123	20	2	2	NUM
iajs-3112	123	21	:	:	PUNCT
iajs-3112	123	22	table	table	NOUN
iajs-3112	123	23	2	2	NUM
iajs-3112	123	24	.	.	PUNCT
iajs-3112	123	25	a	a	DET
iajs-3112	123	26	cubic	cubic	ADJ
iajs-3112	123	27	ideal	ideal	NOUN
iajs-3112	123	28	it	it	PRON
iajs-3112	123	29	follows	follow	VERB
iajs-3112	123	30	that	that	SCONJ
iajs-3112	123	31	(	(	PUNCT
iajs-3112	123	32	ν,∗,∘	ν,∗,∘	PROPN
iajs-3112	123	33	,	,	PUNCT
iajs-3112	123	34	0	0	NUM
iajs-3112	123	35	)	)	PUNCT
iajs-3112	123	36	is	be	AUX
iajs-3112	123	37	a	a	DET
iajs-3112	123	38	ku	ku	PROPN
iajs-3112	123	39	-	-	PUNCT
iajs-3112	123	40	semigroup	semigroup	PROPN
iajs-3112	123	41	and	and	CCONJ
iajs-3112	123	42	define	define	VERB
iajs-3112	123	43	θ	θ	PROPN
iajs-3112	123	44	by	by	ADP
iajs-3112	123	45	:	:	PUNCT
iajs-3112	123	46	𝜇θ(ς	𝜇θ(ς	PUNCT
iajs-3112	123	47	)	)	PUNCT
iajs-3112	123	48	=	=	PRON
iajs-3112	123	49	{	{	PUNCT
iajs-3112	124	1	[	[	X
iajs-3112	124	2	0.5,0.8	0.5,0.8	X
iajs-3112	124	3	]	]	PUNCT
iajs-3112	124	4	,	,	PUNCT
iajs-3112	124	5	𝑖𝑓	𝑖𝑓	ADP
iajs-3112	124	6	ς	ς	PROPN
iajs-3112	125	1	=	=	SYM
iajs-3112	125	2	0	0	PUNCT
iajs-3112	126	1	[	[	X
iajs-3112	126	2	0.1,0.2	0.1,0.2	X
iajs-3112	126	3	]	]	X
iajs-3112	126	4	,	,	PUNCT
iajs-3112	126	5	𝑖𝑓	𝑖𝑓	ADP
iajs-3112	126	6	ς	ς	PROPN
iajs-3112	127	1	=	=	PUNCT
iajs-3112	127	2	𝑎	𝑎	X
iajs-3112	127	3	[	[	X
iajs-3112	127	4	0.1,0.3	0.1,0.3	X
iajs-3112	127	5	]	]	PUNCT
iajs-3112	127	6	,	,	PUNCT
iajs-3112	127	7	𝑖𝑓	𝑖𝑓	ADP
iajs-3112	127	8	ς	ς	PROPN
iajs-3112	127	9	=	=	SYM
iajs-3112	127	10	𝑏	𝑏	PROPN
iajs-3112	127	11	and	and	CCONJ
iajs-3112	127	12	λθ(ς	λθ(ς	NUM
iajs-3112	127	13	)	)	PUNCT
iajs-3112	128	1	=	=	NOUN
iajs-3112	128	2	{	{	PUNCT
iajs-3112	128	3	0.1	0.1	NUM
iajs-3112	128	4	,	,	PUNCT
iajs-3112	128	5	𝑖𝑓	𝑖𝑓	ADP
iajs-3112	128	6	ς	ς	PROPN
iajs-3112	128	7	=	=	NOUN
iajs-3112	128	8	0	0	NUM
iajs-3112	128	9	0.5	0.5	NUM
iajs-3112	128	10	,	,	PUNCT
iajs-3112	128	11	𝑖𝑓	𝑖𝑓	ADP
iajs-3112	128	12	ς	ς	NOUN
iajs-3112	128	13	=	=	PUNCT
iajs-3112	129	1	𝑎	𝑎	X
iajs-3112	129	2	0.3	0.3	NUM
iajs-3112	129	3	,	,	PUNCT
iajs-3112	129	4	𝑖𝑓	𝑖𝑓	AUX
iajs-3112	129	5	ς	ς	PROPN
iajs-3112	129	6	=	=	SYM
iajs-3112	129	7	𝑏	𝑏	PROPN
iajs-3112	130	1	and	and	CCONJ
iajs-3112	130	2	then	then	ADV
iajs-3112	130	3	we	we	PRON
iajs-3112	130	4	can	can	AUX
iajs-3112	130	5	prove	prove	VERB
iajs-3112	130	6	that	that	SCONJ
iajs-3112	130	7	{	{	PUNCT
iajs-3112	130	8	0,a	0,a	NOUN
iajs-3112	130	9	,	,	PUNCT
iajs-3112	130	10	b	b	AUX
iajs-3112	130	11	}	}	PUNCT
iajs-3112	130	12	is	be	AUX
iajs-3112	130	13	a	a	DET
iajs-3112	130	14	cubic	cubic	ADJ
iajs-3112	130	15	ideal	ideal	NOUN
iajs-3112	130	16	but	but	CCONJ
iajs-3112	130	17	not	not	PART
iajs-3112	130	18	a	a	DET
iajs-3112	130	19	cubic	cubic	ADJ
iajs-3112	130	20	positive	positive	ADJ
iajs-3112	130	21	implicative	implicative	ADJ
iajs-3112	130	22	-	-	PUNCT
iajs-3112	130	23	ideal	ideal	NOUN
iajs-3112	130	24	,	,	PUNCT
iajs-3112	130	25	since	since	SCONJ
iajs-3112	130	26	𝜇θ(0	𝜇θ(0	PROPN
iajs-3112	130	27	∗	∗	NOUN
iajs-3112	130	28	𝑎	𝑎	NOUN
iajs-3112	130	29	)	)	PUNCT
iajs-3112	130	30	=	=	PUNCT
iajs-3112	131	1	[	[	X
iajs-3112	131	2	0.1	0.1	NUM
iajs-3112	131	3	,	,	PUNCT
iajs-3112	131	4	0.2	0.2	NUM
iajs-3112	131	5	}	}	PUNCT
iajs-3112	131	6	≤	≤	PUNCT
iajs-3112	131	7	𝑟𝑚𝑖𝑛{𝜇θ(0	𝑟𝑚𝑖𝑛{𝜇θ(0	VERB
iajs-3112	131	8	∗	∗	NOUN
iajs-3112	131	9	(	(	PUNCT
iajs-3112	131	10	𝑏	𝑏	PROPN
iajs-3112	131	11	∗	∗	NOUN
iajs-3112	131	12	𝑎	𝑎	NOUN
iajs-3112	131	13	)	)	PUNCT
iajs-3112	131	14	)	)	PUNCT
iajs-3112	131	15	,	,	PUNCT
iajs-3112	132	1	𝜇θ(0	𝜇θ(0	PROPN
iajs-3112	132	2	∗	∗	NOUN
iajs-3112	132	3	𝑏	𝑏	NOUN
iajs-3112	132	4	)	)	PUNCT
iajs-3112	132	5	}	}	PUNCT
iajs-3112	132	6	=	=	PUNCT
iajs-3112	133	1	[	[	X
iajs-3112	133	2	0.1,0.3	0.1,0.3	PROPN
iajs-3112	133	3	]	]	PUNCT
iajs-3112	133	4	.	.	PUNCT
iajs-3112	133	5	corollary(17	corollary(17	NOUN
iajs-3112	133	6	)	)	PUNCT
iajs-3112	133	7	.	.	PUNCT
iajs-3112	134	1	every	every	DET
iajs-3112	134	2	cubic	cubic	ADJ
iajs-3112	134	3	positive	positive	ADJ
iajs-3112	134	4	implicative	implicative	ADJ
iajs-3112	134	5	-	-	PUNCT
iajs-3112	134	6	ideal	ideal	NOUN
iajs-3112	134	7	in	in	ADP
iajs-3112	134	8	a	a	DET
iajs-3112	134	9	ku	ku	PROPN
iajs-3112	134	10	-	-	PUNCT
iajs-3112	134	11	semigroup	semigroup	PROPN
iajs-3112	134	12	is	be	AUX
iajs-3112	134	13	a	a	DET
iajs-3112	134	14	cubic	cubic	ADJ
iajs-3112	134	15	k	k	NOUN
iajs-3112	134	16	-	-	PUNCT
iajs-3112	134	17	ideal	ideal	ADJ
iajs-3112	134	18	.	.	PUNCT
iajs-3112	135	1	proof	proof	NOUN
iajs-3112	135	2	.	.	PUNCT
iajs-3112	136	1	by	by	ADP
iajs-3112	136	2	referring	refer	VERB
iajs-3112	136	3	to	to	ADP
iajs-3112	136	4	theorem	theorem	NOUN
iajs-3112	136	5	(	(	PUNCT
iajs-3112	136	6	15	15	NUM
iajs-3112	136	7	)	)	PUNCT
iajs-3112	136	8	,	,	PUNCT
iajs-3112	136	9	we	we	PRON
iajs-3112	136	10	get	get	VERB
iajs-3112	136	11	θ	θ	PROPN
iajs-3112	136	12	is	be	AUX
iajs-3112	136	13	a	a	DET
iajs-3112	136	14	cubic	cubic	ADJ
iajs-3112	136	15	ideal	ideal	NOUN
iajs-3112	136	16	and	and	CCONJ
iajs-3112	136	17	by	by	ADP
iajs-3112	136	18	referring	refer	VERB
iajs-3112	136	19	to	to	ADP
iajs-3112	136	20	theorem	theorem	NOUN
iajs-3112	136	21	(	(	PUNCT
iajs-3112	136	22	9	9	NUM
iajs-3112	136	23	)	)	PUNCT
iajs-3112	136	24	and	and	CCONJ
iajs-3112	136	25	upon	upon	SCONJ
iajs-3112	136	26	it	it	PRON
iajs-3112	136	27	θ	θ	PROPN
iajs-3112	136	28	is	be	AUX
iajs-3112	136	29	achieved	achieve	VERB
iajs-3112	136	30	the	the	DET
iajs-3112	136	31	required	require	VERB
iajs-3112	136	32	.	.	PUNCT
iajs-3112	137	1	lemma(18	lemma(18	NOUN
iajs-3112	137	2	)	)	PUNCT
iajs-3112	137	3	.	.	PUNCT
iajs-3112	138	1	if	if	SCONJ
iajs-3112	138	2	θ	θ	PROPN
iajs-3112	138	3	is	be	AUX
iajs-3112	138	4	a	a	DET
iajs-3112	138	5	cubic	cubic	ADJ
iajs-3112	138	6	ideal	ideal	NOUN
iajs-3112	138	7	of	of	ADP
iajs-3112	138	8	ν	ν	NOUN
iajs-3112	138	9	,	,	PUNCT
iajs-3112	138	10	then	then	ADV
iajs-3112	138	11	θ	θ	PROPN
iajs-3112	138	12	is	be	AUX
iajs-3112	138	13	a	a	DET
iajs-3112	138	14	cubic	cubic	ADJ
iajs-3112	138	15	positive	positive	ADJ
iajs-3112	138	16	implicative	implicative	ADJ
iajs-3112	138	17	-	-	PUNCT
iajs-3112	138	18	ideal	ideal	NOUN
iajs-3112	138	19	if	if	SCONJ
iajs-3112	138	20	and	and	CCONJ
iajs-3112	138	21	only	only	ADV
iajs-3112	138	22	if	if	SCONJ
iajs-3112	138	23	the	the	DET
iajs-3112	138	24	following	follow	VERB
iajs-3112	138	25	conditions	condition	NOUN
iajs-3112	138	26	hold	hold	VERB
iajs-3112	138	27	(	(	PUNCT
iajs-3112	138	28	a	a	X
iajs-3112	138	29	)	)	PUNCT
iajs-3112	138	30	μ̃θ((κ	μ̃θ((κ	NUM
iajs-3112	138	31	∗	∗	NOUN
iajs-3112	138	32	ς	ς	NOUN
iajs-3112	138	33	)	)	PUNCT
iajs-3112	138	34	∗	∗	NOUN
iajs-3112	138	35	(	(	PUNCT
iajs-3112	138	36	κ	κ	NOUN
iajs-3112	138	37	∗	∗	NOUN
iajs-3112	138	38	ω	ω	NOUN
iajs-3112	138	39	)	)	PUNCT
iajs-3112	138	40	)	)	PUNCT
iajs-3112	138	41	≥	≥	NOUN
iajs-3112	138	42	μ̃θ(κ	μ̃θ(κ	X
iajs-3112	139	1	∗	∗	NOUN
iajs-3112	139	2	(	(	PUNCT
iajs-3112	139	3	ς	ς	PROPN
iajs-3112	139	4	∗	∗	NOUN
iajs-3112	139	5	ω	ω	NOUN
iajs-3112	139	6	)	)	PUNCT
iajs-3112	139	7	)	)	PUNCT
iajs-3112	139	8	and	and	CCONJ
iajs-3112	139	9	λθ((κ	λθ((κ	NUM
iajs-3112	139	10	∗	∗	PROPN
iajs-3112	139	11	ς	ς	NOUN
iajs-3112	139	12	)	)	PUNCT
iajs-3112	139	13	∗	∗	NOUN
iajs-3112	139	14	(	(	PUNCT
iajs-3112	139	15	κ	κ	NOUN
iajs-3112	139	16	∗	∗	NOUN
iajs-3112	139	17	ω	ω	NOUN
iajs-3112	139	18	)	)	PUNCT
iajs-3112	139	19	)	)	PUNCT
iajs-3112	139	20	≤	≤	NOUN
iajs-3112	139	21	λθ(κ	λθ(κ	PART
iajs-3112	139	22	∗	∗	NOUN
iajs-3112	139	23	(	(	PUNCT
iajs-3112	139	24	ς	ς	PROPN
iajs-3112	139	25	∗	∗	NOUN
iajs-3112	139	26	ω	ω	NOUN
iajs-3112	139	27	)	)	PUNCT
iajs-3112	139	28	)	)	PUNCT
iajs-3112	139	29	.	.	PUNCT
iajs-3112	140	1	proof	proof	NOUN
iajs-3112	140	2	.	.	PUNCT
iajs-3112	141	1	since	since	SCONJ
iajs-3112	141	2	θ	θ	PROPN
iajs-3112	141	3	is	be	AUX
iajs-3112	141	4	a	a	DET
iajs-3112	141	5	cubic	cubic	ADJ
iajs-3112	141	6	ideal	ideal	NOUN
iajs-3112	141	7	of	of	ADP
iajs-3112	141	8	ν	ν	NOUN
iajs-3112	141	9	,	,	PUNCT
iajs-3112	141	10	then	then	ADV
iajs-3112	141	11	𝜇θ(κ	𝜇θ(κ	PUNCT
iajs-3112	141	12	∗	∗	PROPN
iajs-3112	141	13	ω	ω	NOUN
iajs-3112	141	14	)	)	PUNCT
iajs-3112	141	15	≥	≥	NOUN
iajs-3112	141	16	𝑟𝑚𝑖𝑛{𝜇θ((κ	𝑟𝑚𝑖𝑛{𝜇θ((κ	ADV
iajs-3112	141	17	∗	∗	PROPN
iajs-3112	141	18	ς	ς	NOUN
iajs-3112	141	19	)	)	PUNCT
iajs-3112	141	20	∗	∗	NOUN
iajs-3112	141	21	(	(	PUNCT
iajs-3112	141	22	κ	κ	NOUN
iajs-3112	141	23	∗	∗	NOUN
iajs-3112	141	24	ω	ω	NOUN
iajs-3112	141	25	)	)	PUNCT
iajs-3112	141	26	)	)	PUNCT
iajs-3112	141	27	,	,	PUNCT
iajs-3112	141	28	𝜇θ(κ	𝜇θ(κ	PUNCT
iajs-3112	141	29	∗	∗	X
iajs-3112	141	30	ς	ς	PROPN
iajs-3112	141	31	)	)	PUNCT
iajs-3112	141	32	}	}	PUNCT
iajs-3112	141	33	and	and	CCONJ
iajs-3112	141	34	by	by	ADP
iajs-3112	141	35	above	above	ADP
iajs-3112	141	36	condition	condition	NOUN
iajs-3112	141	37	,	,	PUNCT
iajs-3112	141	38	we	we	PRON
iajs-3112	141	39	have	have	VERB
iajs-3112	141	40	𝜇θ(κ	𝜇θ(κ	NOUN
iajs-3112	141	41	∗	∗	NOUN
iajs-3112	141	42	ω	ω	NUM
iajs-3112	141	43	)	)	PUNCT
iajs-3112	141	44	≥	≥	PROPN
iajs-3112	141	45	𝑟𝑚𝑖𝑛{𝜇θ(κ	𝑟𝑚𝑖𝑛{𝜇θ(κ	PROPN
iajs-3112	141	46	∗	∗	X
iajs-3112	141	47	(	(	PUNCT
iajs-3112	141	48	ς	ς	PROPN
iajs-3112	141	49	∗	∗	NOUN
iajs-3112	141	50	ω	ω	NOUN
iajs-3112	141	51	)	)	PUNCT
iajs-3112	141	52	)	)	PUNCT
iajs-3112	141	53	,	,	PUNCT
iajs-3112	141	54	𝜇θ(κ	𝜇θ(κ	PUNCT
iajs-3112	141	55	∗	∗	X
iajs-3112	141	56	ς	ς	PROPN
iajs-3112	141	57	)	)	PUNCT
iajs-3112	141	58	}	}	PUNCT
iajs-3112	141	59	,	,	PUNCT
iajs-3112	141	60	also	also	ADV
iajs-3112	141	61	𝜆θ(κ	𝜆θ(κ	VERB
iajs-3112	141	62	∗	∗	NOUN
iajs-3112	141	63	ω	ω	NOUN
iajs-3112	141	64	)	)	PUNCT
iajs-3112	141	65	≤	≤	NOUN
iajs-3112	141	66	𝑚𝑎𝑥{𝜆θ((κ	𝑚𝑎𝑥{𝜆θ((κ	NUM
iajs-3112	141	67	∗	∗	NOUN
iajs-3112	141	68	ς	ς	NOUN
iajs-3112	141	69	)	)	PUNCT
iajs-3112	141	70	∗	∗	NOUN
iajs-3112	141	71	(	(	PUNCT
iajs-3112	141	72	κ	κ	NOUN
iajs-3112	141	73	∗	∗	NOUN
iajs-3112	141	74	ω	ω	NOUN
iajs-3112	141	75	)	)	PUNCT
iajs-3112	141	76	)	)	PUNCT
iajs-3112	141	77	,	,	PUNCT
iajs-3112	141	78	𝜆θ(κ	𝜆θ(κ	PUNCT
iajs-3112	141	79	∗	∗	PROPN
iajs-3112	141	80	ς	ς	NOUN
iajs-3112	141	81	)	)	PUNCT
iajs-3112	141	82	}	}	PUNCT
iajs-3112	141	83	and	and	CCONJ
iajs-3112	141	84	by	by	ADP
iajs-3112	141	85	above	above	ADP
iajs-3112	141	86	condition	condition	NOUN
iajs-3112	141	87	,	,	PUNCT
iajs-3112	141	88	we	we	PRON
iajs-3112	141	89	have	have	VERB
iajs-3112	141	90	𝜆θ(κ	𝜆θ(κ	NOUN
iajs-3112	141	91	∗	∗	NOUN
iajs-3112	141	92	ω	ω	NOUN
iajs-3112	141	93	)	)	PUNCT
iajs-3112	141	94	≤	≤	NOUN
iajs-3112	141	95	𝑚𝑎𝑥{𝜆θ(κ	𝑚𝑎𝑥{𝜆θ(κ	PROPN
iajs-3112	141	96	∗	∗	NOUN
iajs-3112	141	97	(	(	PUNCT
iajs-3112	141	98	ς	ς	PROPN
iajs-3112	141	99	∗	∗	NOUN
iajs-3112	141	100	ω	ω	NOUN
iajs-3112	141	101	)	)	PUNCT
iajs-3112	141	102	)	)	PUNCT
iajs-3112	141	103	,	,	PUNCT
iajs-3112	141	104	𝜆θ(κ	𝜆θ(κ	PUNCT
iajs-3112	141	105	∗	∗	PROPN
iajs-3112	141	106	ς	ς	PROPN
iajs-3112	141	107	)	)	PUNCT
iajs-3112	141	108	}	}	PUNCT
iajs-3112	141	109	,	,	PUNCT
iajs-3112	141	110	which	which	PRON
iajs-3112	141	111	is	be	AUX
iajs-3112	141	112	𝑪𝒑𝟐	𝑪𝒑𝟐	NOUN
iajs-3112	141	113	,	,	PUNCT
iajs-3112	141	114	the	the	DET
iajs-3112	141	115	conditions	condition	NOUN
iajs-3112	141	116	𝑪𝒑𝟏	𝑪𝒑𝟏	VERB
iajs-3112	141	117	and	and	CCONJ
iajs-3112	141	118	𝑪𝒑𝟑	𝑪𝒑𝟑	NOUN
iajs-3112	141	119	are	be	AUX
iajs-3112	141	120	verify	verify	VERB
iajs-3112	141	121	from	from	ADP
iajs-3112	141	122	definition(10	definition(10	NOUN
iajs-3112	141	123	)	)	PUNCT
iajs-3112	141	124	.	.	PUNCT
iajs-3112	142	1	on	on	ADP
iajs-3112	142	2	the	the	DET
iajs-3112	142	3	contrary	contrary	NOUN
iajs-3112	142	4	,	,	PUNCT
iajs-3112	142	5	by	by	ADP
iajs-3112	142	6	theorem	theorem	NOUN
iajs-3112	142	7	(	(	PUNCT
iajs-3112	142	8	15),θ	15),θ	NUM
iajs-3112	142	9	is	be	AUX
iajs-3112	142	10	a	a	DET
iajs-3112	142	11	cubic	cubic	ADJ
iajs-3112	142	12	ideal	ideal	NOUN
iajs-3112	142	13	.	.	PUNCT
iajs-3112	143	1	*	*	PUNCT
iajs-3112	143	2	0	0	PUNCT
iajs-3112	144	1	a	a	DET
iajs-3112	144	2	b	b	NOUN
iajs-3112	144	3	0	0	NUM
iajs-3112	144	4	0	0	NUM
iajs-3112	144	5	0	0	NUM
iajs-3112	144	6	0	0	NUM
iajs-3112	144	7	a	a	DET
iajs-3112	144	8	0	0	NUM
iajs-3112	144	9	a	a	DET
iajs-3112	144	10	0	0	NUM
iajs-3112	144	11	b	b	NOUN
iajs-3112	144	12	0	0	NUM
iajs-3112	144	13	0	0	NUM
iajs-3112	144	14	b	b	X
iajs-3112	144	15	∘	∘	NOUN
iajs-3112	144	16	0	0	NUM
iajs-3112	144	17	a	a	DET
iajs-3112	144	18	b	b	NOUN
iajs-3112	144	19	0	0	NUM
iajs-3112	144	20	0	0	NUM
iajs-3112	145	1	a	a	DET
iajs-3112	145	2	b	b	NOUN
iajs-3112	145	3	a	a	DET
iajs-3112	145	4	0	0	NUM
iajs-3112	145	5	0	0	NUM
iajs-3112	146	1	a	a	DET
iajs-3112	146	2	b	b	NOUN
iajs-3112	146	3	0	0	NUM
iajs-3112	146	4	0	0	NUM
iajs-3112	146	5	0	0	NUM
iajs-3112	146	6	ihjpas	ihjpas	PROPN
iajs-3112	146	7	.	.	PUNCT
iajs-3112	147	1	37	37	NUM
iajs-3112	147	2	(	(	PUNCT
iajs-3112	147	3	1	1	NUM
iajs-3112	147	4	)	)	PUNCT
iajs-3112	147	5	2024	2024	NUM
iajs-3112	147	6	459	459	NUM
iajs-3112	147	7	let	let	VERB
iajs-3112	147	8	α	α	PRON
iajs-3112	147	9	=	=	SYM
iajs-3112	147	10	ς	ς	PROPN
iajs-3112	147	11	∗	∗	NOUN
iajs-3112	147	12	ω	ω	NUM
iajs-3112	147	13	𝑎𝑛𝑑	𝑎𝑛𝑑	X
iajs-3112	147	14	β	β	X
iajs-3112	147	15	=	=	SYM
iajs-3112	147	16	(	(	PUNCT
iajs-3112	147	17	κ	κ	NOUN
iajs-3112	147	18	∗	∗	NOUN
iajs-3112	147	19	ς	ς	NOUN
iajs-3112	147	20	)	)	PUNCT
iajs-3112	147	21	∗	∗	NOUN
iajs-3112	147	22	ω	ω	PROPN
iajs-3112	147	23	,	,	PUNCT
iajs-3112	147	24	then	then	ADV
iajs-3112	147	25	𝜇θ(κ	𝜇θ(κ	PUNCT
iajs-3112	147	26	∗	∗	NOUN
iajs-3112	147	27	(	(	PUNCT
iajs-3112	147	28	α	α	X
iajs-3112	147	29	∗	∗	NOUN
iajs-3112	147	30	β	β	NOUN
iajs-3112	147	31	)	)	PUNCT
iajs-3112	147	32	)	)	PUNCT
iajs-3112	148	1	=	=	SYM
iajs-3112	148	2	𝜇θ(κ	𝜇θ(κ	NOUN
iajs-3112	148	3	∗	∗	NOUN
iajs-3112	148	4	(	(	PUNCT
iajs-3112	148	5	(	(	PUNCT
iajs-3112	148	6	ς	ς	PROPN
iajs-3112	148	7	∗	∗	NOUN
iajs-3112	148	8	ω	ω	NOUN
iajs-3112	148	9	)	)	PUNCT
iajs-3112	148	10	∗	∗	NOUN
iajs-3112	148	11	(	(	PUNCT
iajs-3112	148	12	(	(	PUNCT
iajs-3112	148	13	κ	κ	NOUN
iajs-3112	148	14	∗	∗	NOUN
iajs-3112	148	15	ς	ς	NOUN
iajs-3112	148	16	)	)	PUNCT
iajs-3112	148	17	∗	∗	NOUN
iajs-3112	148	18	ω	ω	NUM
iajs-3112	148	19	)	)	PUNCT
iajs-3112	148	20	)	)	PUNCT
iajs-3112	148	21	≥	≥	NOUN
iajs-3112	148	22	𝜇θ(κ	𝜇θ(κ	PUNCT
iajs-3112	148	23	∗	∗	NOUN
iajs-3112	148	24	(	(	PUNCT
iajs-3112	148	25	(	(	PUNCT
iajs-3112	148	26	κ	κ	NOUN
iajs-3112	148	27	∗	∗	NOUN
iajs-3112	148	28	ς	ς	NOUN
iajs-3112	148	29	)	)	PUNCT
iajs-3112	148	30	∗	∗	NOUN
iajs-3112	148	31	ς	ς	NOUN
iajs-3112	148	32	)	)	PUNCT
iajs-3112	148	33	)	)	PUNCT
iajs-3112	148	34	by	by	ADP
iajs-3112	148	35	ku1	ku1	NOUN
iajs-3112	148	36	=	=	SYM
iajs-3112	148	37	𝜇θ(0	𝜇θ(0	NOUN
iajs-3112	148	38	)	)	PUNCT
iajs-3112	148	39	by	by	ADP
iajs-3112	148	40	theorem(2	theorem(2	PRON
iajs-3112	148	41	)	)	PUNCT
iajs-3112	148	42	.	.	PUNCT
iajs-3112	149	1	so	so	ADV
iajs-3112	149	2	𝜇θ(κ	𝜇θ(κ	PUNCT
iajs-3112	149	3	∗	∗	NOUN
iajs-3112	149	4	(	(	PUNCT
iajs-3112	149	5	α	α	X
iajs-3112	149	6	∗	∗	NOUN
iajs-3112	149	7	β	β	NOUN
iajs-3112	149	8	)	)	PUNCT
iajs-3112	149	9	)	)	PUNCT
iajs-3112	150	1	=	=	SYM
iajs-3112	150	2	𝜇θ(0	𝜇θ(0	PROPN
iajs-3112	150	3	)	)	PUNCT
iajs-3112	150	4	and	and	CCONJ
iajs-3112	150	5	𝜆θ(κ	𝜆θ(κ	ADP
iajs-3112	150	6	∗	∗	NOUN
iajs-3112	150	7	(	(	PUNCT
iajs-3112	150	8	α	α	X
iajs-3112	150	9	∗	∗	NOUN
iajs-3112	150	10	β	β	NOUN
iajs-3112	150	11	)	)	PUNCT
iajs-3112	150	12	)	)	PUNCT
iajs-3112	151	1	=	=	SYM
iajs-3112	151	2	𝜆θ(κ	𝜆θ(κ	ADP
iajs-3112	151	3	∗	∗	NOUN
iajs-3112	151	4	(	(	PUNCT
iajs-3112	151	5	(	(	PUNCT
iajs-3112	151	6	ς	ς	PROPN
iajs-3112	151	7	∗	∗	NOUN
iajs-3112	151	8	ω	ω	NOUN
iajs-3112	151	9	)	)	PUNCT
iajs-3112	151	10	∗	∗	NOUN
iajs-3112	151	11	(	(	PUNCT
iajs-3112	151	12	κ	κ	NOUN
iajs-3112	151	13	∗	∗	NOUN
iajs-3112	151	14	ς	ς	NOUN
iajs-3112	151	15	)	)	PUNCT
iajs-3112	151	16	∗	∗	NOUN
iajs-3112	151	17	ω	ω	NUM
iajs-3112	151	18	)	)	PUNCT
iajs-3112	151	19	)	)	PUNCT
iajs-3112	151	20	≥	≥	NOUN
iajs-3112	151	21	𝜆θ(κ	𝜆θ(κ	PART
iajs-3112	151	22	∗	∗	NOUN
iajs-3112	151	23	(	(	PUNCT
iajs-3112	151	24	(	(	PUNCT
iajs-3112	151	25	κ	κ	NOUN
iajs-3112	151	26	∗	∗	NOUN
iajs-3112	151	27	ς	ς	NOUN
iajs-3112	151	28	)	)	PUNCT
iajs-3112	151	29	∗	∗	NOUN
iajs-3112	151	30	ς	ς	NOUN
iajs-3112	151	31	)	)	PUNCT
iajs-3112	151	32	)	)	PUNCT
iajs-3112	151	33	by	by	ADP
iajs-3112	151	34	ku1	ku1	NOUN
iajs-3112	151	35	=	=	SYM
iajs-3112	151	36	𝜆θ(0	𝜆θ(0	PROPN
iajs-3112	151	37	)	)	PUNCT
iajs-3112	151	38	by	by	ADP
iajs-3112	151	39	theorem	theorem	NOUN
iajs-3112	151	40	(	(	PUNCT
iajs-3112	151	41	2	2	NUM
iajs-3112	151	42	)	)	PUNCT
iajs-3112	151	43	.	.	PUNCT
iajs-3112	152	1	so	so	ADV
iajs-3112	152	2	𝜆θ(κ	𝜆θ(κ	PUNCT
iajs-3112	152	3	∗	∗	NOUN
iajs-3112	152	4	(	(	PUNCT
iajs-3112	152	5	α	α	X
iajs-3112	152	6	∗	∗	NOUN
iajs-3112	152	7	β	β	NOUN
iajs-3112	152	8	)	)	PUNCT
iajs-3112	152	9	)	)	PUNCT
iajs-3112	153	1	=	=	PUNCT
iajs-3112	153	2	𝜆θ(0	𝜆θ(0	PROPN
iajs-3112	153	3	)	)	PUNCT
iajs-3112	153	4	.	.	PUNCT
iajs-3112	154	1	now	now	ADV
iajs-3112	154	2	by	by	ADP
iajs-3112	154	3	using	use	VERB
iajs-3112	154	4	theorem	theorem	NOUN
iajs-3112	154	5	(	(	PUNCT
iajs-3112	154	6	2	2	NUM
iajs-3112	154	7	)	)	PUNCT
iajs-3112	154	8	and	and	CCONJ
iajs-3112	154	9	the	the	DET
iajs-3112	154	10	condition	condition	NOUN
iajs-3112	154	11	𝑪𝒑𝟐	𝑪𝒑𝟐	NOUN
iajs-3112	154	12	from	from	ADP
iajs-3112	154	13	definition	definition	NOUN
iajs-3112	154	14	of	of	ADP
iajs-3112	154	15	a	a	DET
iajs-3112	154	16	cubic	cubic	ADJ
iajs-3112	154	17	positive	positive	ADJ
iajs-3112	154	18	implicative	implicative	ADJ
iajs-3112	154	19	-	-	PUNCT
iajs-3112	154	20	ideal	ideal	NOUN
iajs-3112	154	21	we	we	PRON
iajs-3112	154	22	obtain	obtain	VERB
iajs-3112	154	23	:	:	PUNCT
iajs-3112	154	24	𝜇θ((κ	𝜇θ((κ	PROPN
iajs-3112	154	25	∗	∗	PROPN
iajs-3112	154	26	ς	ς	PROPN
iajs-3112	154	27	)	)	PUNCT
iajs-3112	154	28	∗	∗	NOUN
iajs-3112	154	29	(	(	PUNCT
iajs-3112	154	30	κ	κ	NOUN
iajs-3112	154	31	∗	∗	NOUN
iajs-3112	154	32	ω	ω	NOUN
iajs-3112	154	33	)	)	PUNCT
iajs-3112	154	34	)	)	PUNCT
iajs-3112	155	1	=	=	SYM
iajs-3112	155	2	𝜇θ(κ	𝜇θ(κ	NOUN
iajs-3112	155	3	∗	∗	NOUN
iajs-3112	155	4	(	(	PUNCT
iajs-3112	155	5	κ	κ	NOUN
iajs-3112	155	6	∗	∗	NOUN
iajs-3112	155	7	ς	ς	NOUN
iajs-3112	155	8	)	)	PUNCT
iajs-3112	155	9	∗	∗	NOUN
iajs-3112	155	10	ω	ω	NOUN
iajs-3112	155	11	)	)	PUNCT
iajs-3112	155	12	=	=	PUNCT
iajs-3112	155	13	𝜇θ(κ	𝜇θ(κ	NOUN
iajs-3112	155	14	∗	∗	X
iajs-3112	155	15	β	β	X
iajs-3112	155	16	)	)	PUNCT
iajs-3112	155	17	≥	≥	PROPN
iajs-3112	155	18	𝑟𝑚𝑖𝑛{𝜇θ(κ	𝑟𝑚𝑖𝑛{𝜇θ(κ	PROPN
iajs-3112	155	19	∗	∗	PROPN
iajs-3112	155	20	(	(	PUNCT
iajs-3112	155	21	α	α	X
iajs-3112	155	22	∗	∗	NOUN
iajs-3112	155	23	β	β	NOUN
iajs-3112	155	24	)	)	PUNCT
iajs-3112	155	25	)	)	PUNCT
iajs-3112	155	26	,	,	PUNCT
iajs-3112	155	27	𝜇θ(κ	𝜇θ(κ	PUNCT
iajs-3112	155	28	∗	∗	X
iajs-3112	155	29	α	α	NOUN
iajs-3112	155	30	)	)	PUNCT
iajs-3112	155	31	}	}	PUNCT
iajs-3112	155	32	=	=	SYM
iajs-3112	155	33	𝑟𝑚𝑖𝑛{𝜇θ(0	𝑟𝑚𝑖𝑛{𝜇θ(0	PROPN
iajs-3112	155	34	)	)	PUNCT
iajs-3112	155	35	,	,	PUNCT
iajs-3112	155	36	𝜇θ(κ	𝜇θ(κ	PUNCT
iajs-3112	155	37	∗	∗	X
iajs-3112	155	38	α	α	NOUN
iajs-3112	155	39	)	)	PUNCT
iajs-3112	155	40	}	}	PUNCT
iajs-3112	155	41	=	=	SYM
iajs-3112	155	42	𝜇θ(κ	𝜇θ(κ	NOUN
iajs-3112	155	43	∗	∗	NOUN
iajs-3112	155	44	α	α	NOUN
iajs-3112	155	45	)	)	PUNCT
iajs-3112	155	46	=	=	NOUN
iajs-3112	155	47	𝜇θ(κ	𝜇θ(κ	NOUN
iajs-3112	155	48	∗	∗	NOUN
iajs-3112	155	49	(	(	PUNCT
iajs-3112	155	50	ς	ς	PROPN
iajs-3112	155	51	∗	∗	NOUN
iajs-3112	155	52	ω	ω	NOUN
iajs-3112	155	53	)	)	PUNCT
iajs-3112	155	54	)	)	PUNCT
iajs-3112	155	55	,	,	PUNCT
iajs-3112	155	56	and	and	CCONJ
iajs-3112	155	57	𝜆θ((κ	𝜆θ((κ	NOUN
iajs-3112	155	58	∗	∗	NOUN
iajs-3112	155	59	ς	ς	NOUN
iajs-3112	155	60	)	)	PUNCT
iajs-3112	155	61	∗	∗	NOUN
iajs-3112	155	62	(	(	PUNCT
iajs-3112	155	63	κ	κ	NOUN
iajs-3112	155	64	∗	∗	NOUN
iajs-3112	155	65	ω	ω	NOUN
iajs-3112	155	66	)	)	PUNCT
iajs-3112	155	67	)	)	PUNCT
iajs-3112	156	1	=	=	SYM
iajs-3112	156	2	𝜆θ(κ	𝜆θ(κ	ADP
iajs-3112	156	3	∗	∗	NOUN
iajs-3112	156	4	(	(	PUNCT
iajs-3112	156	5	κ	κ	NOUN
iajs-3112	156	6	∗	∗	NOUN
iajs-3112	156	7	ς	ς	NOUN
iajs-3112	156	8	)	)	PUNCT
iajs-3112	156	9	∗	∗	NOUN
iajs-3112	156	10	ω	ω	NOUN
iajs-3112	156	11	)	)	PUNCT
iajs-3112	156	12	=	=	NOUN
iajs-3112	156	13	𝜆θ(κ	𝜆θ(κ	ADP
iajs-3112	156	14	∗	∗	NOUN
iajs-3112	156	15	β	β	NOUN
iajs-3112	156	16	)	)	PUNCT
iajs-3112	156	17	≤	≤	NOUN
iajs-3112	156	18	𝑚𝑎𝑥{𝜆θ(κ	𝑚𝑎𝑥{𝜆θ(κ	PROPN
iajs-3112	156	19	∗	∗	NOUN
iajs-3112	156	20	(	(	PUNCT
iajs-3112	156	21	α	α	X
iajs-3112	156	22	∗	∗	NOUN
iajs-3112	156	23	β	β	NOUN
iajs-3112	156	24	)	)	PUNCT
iajs-3112	156	25	)	)	PUNCT
iajs-3112	156	26	,	,	PUNCT
iajs-3112	156	27	𝜆θ(κ	𝜆θ(κ	PUNCT
iajs-3112	156	28	∗	∗	NOUN
iajs-3112	156	29	α	α	NOUN
iajs-3112	156	30	)	)	PUNCT
iajs-3112	156	31	}	}	PUNCT
iajs-3112	156	32	=	=	SYM
iajs-3112	156	33	𝑚𝑎𝑥{𝜆θ(0	𝑚𝑎𝑥{𝜆θ(0	NUM
iajs-3112	156	34	)	)	PUNCT
iajs-3112	156	35	,	,	PUNCT
iajs-3112	156	36	𝜆θ(κ	𝜆θ(κ	PUNCT
iajs-3112	157	1	∗	∗	NOUN
iajs-3112	157	2	α	α	NOUN
iajs-3112	157	3	)	)	PUNCT
iajs-3112	157	4	}	}	PUNCT
iajs-3112	157	5	=	=	SYM
iajs-3112	157	6	𝜆θ(κ	𝜆θ(κ	ADP
iajs-3112	157	7	∗	∗	NOUN
iajs-3112	157	8	α	α	NOUN
iajs-3112	157	9	)	)	PUNCT
iajs-3112	158	1	=	=	NOUN
iajs-3112	158	2	𝜆θ(κ	𝜆θ(κ	NOUN
iajs-3112	158	3	∗	∗	NOUN
iajs-3112	158	4	(	(	PUNCT
iajs-3112	158	5	ς	ς	PROPN
iajs-3112	158	6	∗	∗	NOUN
iajs-3112	158	7	ω	ω	NOUN
iajs-3112	158	8	)	)	PUNCT
iajs-3112	158	9	)	)	PUNCT
iajs-3112	158	10	,	,	PUNCT
iajs-3112	158	11	which	which	PRON
iajs-3112	158	12	is	be	AUX
iajs-3112	158	13	the	the	DET
iajs-3112	158	14	condition	condition	NOUN
iajs-3112	158	15	(	(	PUNCT
iajs-3112	158	16	a	a	NOUN
iajs-3112	158	17	)	)	PUNCT
iajs-3112	158	18	.	.	PUNCT
iajs-3112	159	1	■	■	PUNCT
iajs-3112	159	2	theorem(19	theorem(19	NUM
iajs-3112	159	3	)	)	PUNCT
iajs-3112	159	4	.	.	PUNCT
iajs-3112	160	1	every	every	DET
iajs-3112	160	2	cubic	cubic	ADJ
iajs-3112	160	3	commutative	commutative	ADJ
iajs-3112	160	4	-	-	PUNCT
iajs-3112	160	5	ideal	ideal	ADJ
iajs-3112	160	6	θ	θ	PROPN
iajs-3112	160	7	of	of	ADP
iajs-3112	160	8	ν	ν	PROPN
iajs-3112	160	9	is	be	AUX
iajs-3112	160	10	a	a	DET
iajs-3112	160	11	cubic	cubic	ADJ
iajs-3112	160	12	ideal	ideal	NOUN
iajs-3112	160	13	.	.	PUNCT
iajs-3112	161	1	proof	proof	NOUN
iajs-3112	161	2	.	.	PUNCT
iajs-3112	162	1	by	by	ADP
iajs-3112	162	2	definition	definition	NOUN
iajs-3112	162	3	of	of	ADP
iajs-3112	162	4	a	a	DET
iajs-3112	162	5	cubic	cubic	ADJ
iajs-3112	162	6	commutative	commutative	ADJ
iajs-3112	162	7	-	-	PUNCT
iajs-3112	162	8	ideal	ideal	NOUN
iajs-3112	162	9	,	,	PUNCT
iajs-3112	162	10	we	we	PRON
iajs-3112	162	11	have	have	VERB
iajs-3112	162	12	ci1	ci1	PROPN
iajs-3112	162	13	and	and	CCONJ
iajs-3112	162	14	ci3	ci3	NOUN
iajs-3112	162	15	are	be	AUX
iajs-3112	162	16	fulfilled	fulfil	VERB
iajs-3112	162	17	(	(	PUNCT
iajs-3112	162	18	𝐂𝑪𝟐	𝐂𝑪𝟐	PROPN
iajs-3112	162	19	)	)	PUNCT
iajs-3112	162	20	𝜇θ((ς	𝜇θ((ς	NOUN
iajs-3112	162	21	∗	∗	PROPN
iajs-3112	162	22	ω	ω	NOUN
iajs-3112	162	23	)	)	PUNCT
iajs-3112	162	24	∗	∗	PROPN
iajs-3112	162	25	ω	ω	NOUN
iajs-3112	162	26	)	)	PUNCT
iajs-3112	162	27	∗	∗	PROPN
iajs-3112	162	28	ς	ς	NOUN
iajs-3112	162	29	)	)	PUNCT
iajs-3112	162	30	≥	≥	NOUN
iajs-3112	162	31	𝑟𝑚𝑖𝑛{𝜇θ(ω	𝑟𝑚𝑖𝑛{𝜇θ(ω	NOUN
iajs-3112	162	32	∗	∗	NOUN
iajs-3112	162	33	(	(	PUNCT
iajs-3112	162	34	κ	κ	NOUN
iajs-3112	162	35	∗	∗	NOUN
iajs-3112	162	36	ς	ς	NOUN
iajs-3112	162	37	)	)	PUNCT
iajs-3112	162	38	)	)	PUNCT
iajs-3112	162	39	,	,	PUNCT
iajs-3112	162	40	�	�	PROPN
iajs-3112	162	41	̃	̃	NOUN
iajs-3112	162	42	�	�	NOUN
iajs-3112	162	43	θ(κ	θ(κ	NOUN
iajs-3112	162	44	)	)	PUNCT
iajs-3112	162	45	}	}	PUNCT
iajs-3112	162	46	,	,	PUNCT
iajs-3112	162	47	put	put	VERB
iajs-3112	162	48	ω	ω	PROPN
iajs-3112	162	49	=	=	SYM
iajs-3112	162	50	0	0	NUM
iajs-3112	162	51	,	,	PUNCT
iajs-3112	162	52	we	we	PRON
iajs-3112	162	53	get	get	VERB
iajs-3112	162	54	𝜇θ((ς	𝜇θ((ς	NOUN
iajs-3112	162	55	∗	∗	NOUN
iajs-3112	162	56	0	0	NUM
iajs-3112	162	57	)	)	PUNCT
iajs-3112	162	58	∗	∗	NOUN
iajs-3112	162	59	0	0	NUM
iajs-3112	162	60	)	)	PUNCT
iajs-3112	162	61	∗	∗	NOUN
iajs-3112	162	62	ς	ς	NOUN
iajs-3112	162	63	)	)	PUNCT
iajs-3112	162	64	≥	≥	NOUN
iajs-3112	162	65	𝑟𝑚𝑖𝑛{𝜇θ(0	𝑟𝑚𝑖𝑛{𝜇θ(0	VERB
iajs-3112	162	66	∗	∗	NOUN
iajs-3112	162	67	(	(	PUNCT
iajs-3112	162	68	κ	κ	NOUN
iajs-3112	162	69	∗	∗	NOUN
iajs-3112	162	70	ς	ς	NOUN
iajs-3112	162	71	)	)	PUNCT
iajs-3112	162	72	)	)	PUNCT
iajs-3112	162	73	,	,	PUNCT
iajs-3112	162	74	�	�	PROPN
iajs-3112	162	75	̃	̃	NOUN
iajs-3112	162	76	�	�	NOUN
iajs-3112	162	77	θ(κ	θ(κ	NOUN
iajs-3112	162	78	)	)	PUNCT
iajs-3112	162	79	}	}	PUNCT
iajs-3112	162	80	,	,	PUNCT
iajs-3112	162	81	𝜇θ(ς	𝜇θ(ς	NOUN
iajs-3112	162	82	)	)	PUNCT
iajs-3112	162	83	≥	≥	PROPN
iajs-3112	162	84	𝑟𝑚𝑖𝑛{𝜇θ(κ	𝑟𝑚𝑖𝑛{𝜇θ(κ	PROPN
iajs-3112	162	85	∗	∗	PROPN
iajs-3112	162	86	ς	ς	PROPN
iajs-3112	162	87	)	)	PUNCT
iajs-3112	162	88	,	,	PUNCT
iajs-3112	162	89	�	�	PROPN
iajs-3112	162	90	̃	̃	NOUN
iajs-3112	162	91	�	�	NOUN
iajs-3112	162	92	θ(κ	θ(κ	NOUN
iajs-3112	162	93	)	)	PUNCT
iajs-3112	162	94	}	}	PUNCT
iajs-3112	162	95	,	,	PUNCT
iajs-3112	162	96	and	and	CCONJ
iajs-3112	162	97	𝜆θ((ς	𝜆θ((ς	NOUN
iajs-3112	162	98	∗	∗	NOUN
iajs-3112	162	99	ω	ω	NOUN
iajs-3112	162	100	)	)	PUNCT
iajs-3112	162	101	∗	∗	PROPN
iajs-3112	162	102	ω	ω	NOUN
iajs-3112	162	103	)	)	PUNCT
iajs-3112	162	104	∗	∗	PROPN
iajs-3112	162	105	ς	ς	NOUN
iajs-3112	162	106	)	)	PUNCT
iajs-3112	162	107	≤	≤	NUM
iajs-3112	162	108	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	162	109	{	{	PUNCT
iajs-3112	162	110	𝜆θ(ω	𝜆θ(ω	NOUN
iajs-3112	162	111	∗	∗	NOUN
iajs-3112	162	112	(	(	PUNCT
iajs-3112	162	113	κ	κ	NOUN
iajs-3112	162	114	∗	∗	NOUN
iajs-3112	162	115	ς	ς	NOUN
iajs-3112	162	116	)	)	PUNCT
iajs-3112	162	117	)	)	PUNCT
iajs-3112	162	118	,	,	PUNCT
iajs-3112	162	119	𝜆θ(κ	𝜆θ(κ	NOUN
iajs-3112	162	120	)	)	PUNCT
iajs-3112	162	121	}	}	PUNCT
iajs-3112	162	122	,	,	PUNCT
iajs-3112	162	123	put	put	VERB
iajs-3112	162	124	ω	ω	NOUN
iajs-3112	162	125	=	=	SYM
iajs-3112	162	126	0	0	NUM
iajs-3112	162	127	𝜆θ((ς	𝜆θ((ς	NOUN
iajs-3112	162	128	∗	∗	NOUN
iajs-3112	162	129	0	0	NUM
iajs-3112	162	130	)	)	PUNCT
iajs-3112	162	131	∗	∗	NOUN
iajs-3112	162	132	0	0	NUM
iajs-3112	162	133	)	)	PUNCT
iajs-3112	162	134	∗	∗	NOUN
iajs-3112	162	135	ς	ς	NOUN
iajs-3112	162	136	)	)	PUNCT
iajs-3112	162	137	≤	≤	NUM
iajs-3112	162	138	𝑚𝑎𝑥	𝑚𝑎𝑥	NOUN
iajs-3112	162	139	{	{	PUNCT
iajs-3112	162	140	𝜆θ(0	𝜆θ(0	PROPN
iajs-3112	162	141	∗	∗	X
iajs-3112	162	142	(	(	PUNCT
iajs-3112	162	143	κ	κ	NOUN
iajs-3112	162	144	∗	∗	NOUN
iajs-3112	162	145	ς	ς	NOUN
iajs-3112	162	146	)	)	PUNCT
iajs-3112	162	147	)	)	PUNCT
iajs-3112	162	148	,	,	PUNCT
iajs-3112	162	149	𝜆θ(κ	𝜆θ(κ	NOUN
iajs-3112	162	150	)	)	PUNCT
iajs-3112	162	151	}	}	PUNCT
iajs-3112	162	152	,	,	PUNCT
iajs-3112	162	153	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	162	154	)	)	PUNCT
iajs-3112	162	155	≤	≤	NUM
iajs-3112	162	156	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	162	157	{	{	PUNCT
iajs-3112	162	158	𝜆θ(κ	𝜆θ(κ	ADP
iajs-3112	162	159	∗	∗	NOUN
iajs-3112	162	160	ς	ς	NOUN
iajs-3112	162	161	)	)	PUNCT
iajs-3112	162	162	,	,	PUNCT
iajs-3112	162	163	𝜆θ(κ	𝜆θ(κ	NOUN
iajs-3112	162	164	)	)	PUNCT
iajs-3112	162	165	}	}	PUNCT
iajs-3112	162	166	,	,	PUNCT
iajs-3112	162	167	which	which	PRON
iajs-3112	162	168	is	be	AUX
iajs-3112	162	169	a	a	DET
iajs-3112	162	170	ci2	ci2	NOUN
iajs-3112	162	171	condition	condition	NOUN
iajs-3112	162	172	,	,	PUNCT
iajs-3112	162	173	therefore	therefore	ADV
iajs-3112	162	174	θ	θ	PROPN
iajs-3112	162	175	is	be	AUX
iajs-3112	162	176	a	a	DET
iajs-3112	162	177	cubic	cubic	ADJ
iajs-3112	162	178	ideal	ideal	NOUN
iajs-3112	162	179	.	.	PUNCT
iajs-3112	163	1	■	■	PUNCT
iajs-3112	163	2	example	example	NOUN
iajs-3112	163	3	(	(	PUNCT
iajs-3112	163	4	20	20	NUM
iajs-3112	163	5	)	)	PUNCT
iajs-3112	163	6	.	.	PUNCT
iajs-3112	164	1	let	let	VERB
iajs-3112	164	2	ν	ν	X
iajs-3112	164	3	=	=	PRON
iajs-3112	164	4	{	{	PUNCT
iajs-3112	164	5	0	0	NUM
iajs-3112	164	6	,	,	PUNCT
iajs-3112	164	7	𝑎	𝑎	NOUN
iajs-3112	164	8	,	,	PUNCT
iajs-3112	164	9	𝑏	𝑏	NOUN
iajs-3112	164	10	}	}	PUNCT
iajs-3112	164	11	with	with	ADP
iajs-3112	164	12	two	two	NUM
iajs-3112	164	13	operations	operation	NOUN
iajs-3112	164	14	∗	∗	NOUN
iajs-3112	164	15	and	and	CCONJ
iajs-3112	164	16	∘	∘	NOUN
iajs-3112	164	17	defined	define	VERB
iajs-3112	164	18	by	by	ADP
iajs-3112	164	19	the	the	DET
iajs-3112	164	20	tables	table	NOUN
iajs-3112	164	21	in	in	ADP
iajs-3112	164	22	example	example	NOUN
iajs-3112	164	23	(	(	PUNCT
iajs-3112	164	24	16	16	NUM
iajs-3112	164	25	)	)	PUNCT
iajs-3112	164	26	.	.	PUNCT
iajs-3112	165	1	define	define	VERB
iajs-3112	165	2	𝜇θ(ν	𝜇θ(ν	NOUN
iajs-3112	165	3	)	)	PUNCT
iajs-3112	165	4	and	and	CCONJ
iajs-3112	165	5	𝜆θ(ν	𝜆θ(ν	NUM
iajs-3112	165	6	)	)	PUNCT
iajs-3112	165	7	as	as	SCONJ
iajs-3112	165	8	follows	follow	VERB
iajs-3112	165	9	:	:	PUNCT
iajs-3112	165	10	𝜇θ(ς	𝜇θ(ς	PUNCT
iajs-3112	165	11	)	)	PUNCT
iajs-3112	166	1	=	=	PRON
iajs-3112	166	2	{	{	PUNCT
iajs-3112	167	1	[	[	X
iajs-3112	167	2	0.5	0.5	NUM
iajs-3112	167	3	,	,	PUNCT
iajs-3112	167	4	0.9	0.9	NUM
iajs-3112	167	5	]	]	PUNCT
iajs-3112	167	6	𝑖𝑓	𝑖𝑓	ADP
iajs-3112	167	7	ς	ς	PROPN
iajs-3112	167	8	=	=	SYM
iajs-3112	167	9	0	0	NUM
iajs-3112	167	10	,	,	PUNCT
iajs-3112	167	11	𝑎	𝑎	X
iajs-3112	167	12	[	[	X
iajs-3112	167	13	0.2	0.2	NUM
iajs-3112	167	14	,	,	PUNCT
iajs-3112	167	15	0.4	0.4	NUM
iajs-3112	167	16	]	]	PUNCT
iajs-3112	167	17	𝑖𝑓	𝑖𝑓	ADP
iajs-3112	167	18	ς	ς	PROPN
iajs-3112	167	19	=	=	SYM
iajs-3112	167	20	𝑏	𝑏	PROPN
iajs-3112	167	21	𝑎𝑛𝑑	𝑎𝑛𝑑	ADJ
iajs-3112	167	22	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	167	23	)	)	PUNCT
iajs-3112	167	24	=	=	PRON
iajs-3112	167	25	{	{	PUNCT
iajs-3112	167	26	0.2	0.2	NUM
iajs-3112	167	27	𝑖𝑓	𝑖𝑓	NOUN
iajs-3112	167	28	ς	ς	PROPN
iajs-3112	167	29	=	=	SYM
iajs-3112	167	30	0	0	NUM
iajs-3112	167	31	,	,	PUNCT
iajs-3112	168	1	𝑎	𝑎	PRON
iajs-3112	168	2	0.5	0.5	NUM
iajs-3112	168	3	𝑖𝑓	𝑖𝑓	ADP
iajs-3112	168	4	ς	ς	PROPN
iajs-3112	168	5	=	=	SYM
iajs-3112	168	6	𝑏	𝑏	PROPN
iajs-3112	168	7	then	then	ADV
iajs-3112	168	8	we	we	PRON
iajs-3112	168	9	can	can	AUX
iajs-3112	168	10	prove	prove	VERB
iajs-3112	168	11	that	that	SCONJ
iajs-3112	168	12	θ	θ	PROPN
iajs-3112	168	13	is	be	AUX
iajs-3112	168	14	a	a	DET
iajs-3112	168	15	cubic	cubic	ADJ
iajs-3112	168	16	ideal	ideal	NOUN
iajs-3112	168	17	but	but	CCONJ
iajs-3112	168	18	not	not	PART
iajs-3112	168	19	a	a	DET
iajs-3112	168	20	cubic	cubic	ADJ
iajs-3112	168	21	commutative	commutative	ADJ
iajs-3112	168	22	-	-	PUNCT
iajs-3112	168	23	ideal	ideal	NOUN
iajs-3112	168	24	,	,	PUNCT
iajs-3112	168	25	since	since	SCONJ
iajs-3112	168	26	𝜇θ(((𝑏	𝜇θ(((𝑏	X
iajs-3112	168	27	∗	∗	NOUN
iajs-3112	168	28	0	0	NUM
iajs-3112	168	29	)	)	PUNCT
iajs-3112	168	30	∗	∗	NOUN
iajs-3112	168	31	0	0	NUM
iajs-3112	168	32	)	)	PUNCT
iajs-3112	168	33	∗	∗	NOUN
iajs-3112	168	34	𝑏	𝑏	NOUN
iajs-3112	168	35	)	)	PUNCT
iajs-3112	168	36	)	)	PUNCT
iajs-3112	168	37	≤	≤	PUNCT
iajs-3112	168	38	𝑟𝑚𝑖𝑛{𝜇θ(0	𝑟𝑚𝑖𝑛{𝜇θ(0	VERB
iajs-3112	168	39	∗	∗	NOUN
iajs-3112	168	40	(	(	PUNCT
iajs-3112	168	41	𝑎	𝑎	NOUN
iajs-3112	168	42	∗	∗	NOUN
iajs-3112	168	43	𝑏	𝑏	NOUN
iajs-3112	168	44	)	)	PUNCT
iajs-3112	168	45	)	)	PUNCT
iajs-3112	168	46	,	,	PUNCT
iajs-3112	168	47	𝜇θ(𝑎	𝜇θ(𝑎	NOUN
iajs-3112	168	48	)	)	PUNCT
iajs-3112	168	49	}	}	PUNCT
iajs-3112	168	50	,	,	PUNCT
iajs-3112	168	51	𝜇θ(𝑏	𝜇θ(𝑏	X
iajs-3112	168	52	)	)	PUNCT
iajs-3112	168	53	=	=	PUNCT
iajs-3112	169	1	[	[	X
iajs-3112	169	2	0.2	0.2	NUM
iajs-3112	169	3	,	,	PUNCT
iajs-3112	169	4	0.4	0.4	NUM
iajs-3112	169	5	]	]	PUNCT
iajs-3112	169	6	≤	≤	NUM
iajs-3112	169	7	𝜇θ(𝑎	𝜇θ(𝑎	NOUN
iajs-3112	169	8	)	)	PUNCT
iajs-3112	169	9	=	=	PUNCT
iajs-3112	170	1	[	[	X
iajs-3112	170	2	0.5	0.5	NUM
iajs-3112	170	3	,	,	PUNCT
iajs-3112	170	4	0.9	0.9	NUM
iajs-3112	170	5	]	]	PUNCT
iajs-3112	170	6	and	and	CCONJ
iajs-3112	170	7	𝜆θ(((𝑏	𝜆θ(((𝑏	NUM
iajs-3112	170	8	∗	∗	NOUN
iajs-3112	170	9	0	0	NUM
iajs-3112	170	10	)	)	PUNCT
iajs-3112	170	11	∗	∗	NOUN
iajs-3112	170	12	0	0	NUM
iajs-3112	170	13	)	)	PUNCT
iajs-3112	170	14	∗	∗	NOUN
iajs-3112	170	15	𝑏	𝑏	NOUN
iajs-3112	170	16	)	)	PUNCT
iajs-3112	170	17	)	)	PUNCT
iajs-3112	170	18	≥	≥	X
iajs-3112	171	1	𝑚𝑎𝑥{𝜆θ(0	𝑚𝑎𝑥{𝜆θ(0	ADP
iajs-3112	171	2	∗	∗	NOUN
iajs-3112	171	3	(	(	PUNCT
iajs-3112	171	4	𝑎	𝑎	NOUN
iajs-3112	171	5	∗	∗	NOUN
iajs-3112	171	6	𝑏	𝑏	NOUN
iajs-3112	171	7	)	)	PUNCT
iajs-3112	171	8	)	)	PUNCT
iajs-3112	171	9	,	,	PUNCT
iajs-3112	171	10	𝜆θ(𝑎	𝜆θ(𝑎	ADP
iajs-3112	171	11	)	)	PUNCT
iajs-3112	171	12	}	}	PUNCT
iajs-3112	171	13	,	,	PUNCT
iajs-3112	171	14	so	so	SCONJ
iajs-3112	171	15	𝜆θ(𝑏	𝜆θ(𝑏	PRON
iajs-3112	171	16	)	)	PUNCT
iajs-3112	171	17	=	=	SYM
iajs-3112	171	18	0.5	0.5	NUM
iajs-3112	171	19	≥	≥	NOUN
iajs-3112	171	20	𝜆θ(𝑎	𝜆θ(𝑎	PUNCT
iajs-3112	171	21	)	)	PUNCT
iajs-3112	171	22	=	=	SYM
iajs-3112	171	23	0.2	0.2	NUM
iajs-3112	171	24	,	,	PUNCT
iajs-3112	171	25	this	this	PRON
iajs-3112	171	26	is	be	AUX
iajs-3112	171	27	also	also	ADV
iajs-3112	171	28	a	a	DET
iajs-3112	171	29	wrong	wrong	ADJ
iajs-3112	171	30	phrase	phrase	NOUN
iajs-3112	171	31	.	.	PUNCT
iajs-3112	172	1	so	so	ADV
iajs-3112	172	2	we	we	PRON
iajs-3112	172	3	conclude	conclude	VERB
iajs-3112	172	4	that	that	SCONJ
iajs-3112	172	5	the	the	DET
iajs-3112	172	6	converse	converse	NOUN
iajs-3112	172	7	of	of	ADP
iajs-3112	172	8	the	the	DET
iajs-3112	172	9	previous	previous	ADJ
iajs-3112	172	10	theorem	theorem	NOUN
iajs-3112	172	11	is	be	AUX
iajs-3112	172	12	not	not	PART
iajs-3112	172	13	true	true	ADJ
iajs-3112	172	14	in	in	ADP
iajs-3112	172	15	general	general	ADJ
iajs-3112	172	16	,	,	PUNCT
iajs-3112	172	17	as	as	SCONJ
iajs-3112	172	18	we	we	PRON
iajs-3112	172	19	will	will	AUX
iajs-3112	172	20	clarify	clarify	VERB
iajs-3112	172	21	in	in	ADP
iajs-3112	172	22	the	the	DET
iajs-3112	172	23	following	following	ADJ
iajs-3112	172	24	theorem	theorem	ADJ
iajs-3112	172	25	ihjpas	ihjpa	NOUN
iajs-3112	172	26	.	.	PUNCT
iajs-3112	173	1	37	37	NUM
iajs-3112	173	2	(	(	PUNCT
iajs-3112	173	3	1	1	NUM
iajs-3112	173	4	)	)	PUNCT
iajs-3112	173	5	2024	2024	NUM
iajs-3112	173	6	460	460	NUM
iajs-3112	173	7	theorem(21	theorem(21	NUM
iajs-3112	173	8	)	)	PUNCT
iajs-3112	173	9	.	.	PUNCT
iajs-3112	174	1	a	a	DET
iajs-3112	174	2	cubic	cubic	ADJ
iajs-3112	174	3	ideal	ideal	ADJ
iajs-3112	174	4	θ	θ	PROPN
iajs-3112	174	5	of	of	ADP
iajs-3112	174	6	ν	ν	PROPN
iajs-3112	174	7	is	be	AUX
iajs-3112	174	8	a	a	DET
iajs-3112	174	9	cubic	cubic	ADJ
iajs-3112	174	10	commutative	commutative	ADJ
iajs-3112	174	11	-	-	PUNCT
iajs-3112	174	12	ideal	ideal	NOUN
iajs-3112	174	13	if	if	SCONJ
iajs-3112	174	14	and	and	CCONJ
iajs-3112	174	15	only	only	ADV
iajs-3112	174	16	if	if	SCONJ
iajs-3112	174	17	it	it	PRON
iajs-3112	174	18	is	be	AUX
iajs-3112	174	19	satisfies	satisfie	NOUN
iajs-3112	174	20	the	the	DET
iajs-3112	174	21	following	follow	VERB
iajs-3112	174	22	inequalities	inequality	NOUN
iajs-3112	174	23	:	:	PUNCT
iajs-3112	174	24	(	(	PUNCT
iajs-3112	174	25	a	a	X
iajs-3112	174	26	)	)	PUNCT
iajs-3112	174	27	μ̃θ(ω	μ̃θ(ω	PROPN
iajs-3112	174	28	∗	∗	PROPN
iajs-3112	174	29	ς	ς	PROPN
iajs-3112	174	30	)	)	PUNCT
iajs-3112	174	31	≤	≤	NOUN
iajs-3112	174	32	𝜇θ((ς	𝜇θ((ς	NOUN
iajs-3112	174	33	∗	∗	NOUN
iajs-3112	174	34	ω	ω	NOUN
iajs-3112	174	35	)	)	PUNCT
iajs-3112	174	36	∗	∗	PROPN
iajs-3112	174	37	ω	ω	NOUN
iajs-3112	174	38	)	)	PUNCT
iajs-3112	174	39	∗	∗	NOUN
iajs-3112	174	40	ς	ς	NOUN
iajs-3112	174	41	)	)	PUNCT
iajs-3112	174	42	.	.	PUNCT
iajs-3112	175	1	(	(	PUNCT
iajs-3112	175	2	b	b	X
iajs-3112	175	3	)	)	PUNCT
iajs-3112	175	4	λθ(ω	λθ(ω	X
iajs-3112	175	5	∗	∗	NOUN
iajs-3112	175	6	ς	ς	NOUN
iajs-3112	175	7	)	)	PUNCT
iajs-3112	175	8	≥	≥	NOUN
iajs-3112	175	9	𝜆θ((ς	𝜆θ((ς	NOUN
iajs-3112	175	10	∗	∗	NOUN
iajs-3112	175	11	ω	ω	NOUN
iajs-3112	175	12	)	)	PUNCT
iajs-3112	175	13	∗	∗	PROPN
iajs-3112	175	14	ω	ω	NOUN
iajs-3112	175	15	)	)	PUNCT
iajs-3112	175	16	∗	∗	NOUN
iajs-3112	175	17	ς	ς	PROPN
iajs-3112	175	18	)	)	PUNCT
iajs-3112	175	19	,	,	PUNCT
iajs-3112	175	20	for	for	ADP
iajs-3112	175	21	all	all	DET
iajs-3112	175	22	ς	ς	PROPN
iajs-3112	175	23	,	,	PUNCT
iajs-3112	175	24	ω	ω	PROPN
iajs-3112	175	25	∈	∈	PROPN
iajs-3112	175	26	ν	ν	NOUN
iajs-3112	175	27	.	.	PUNCT
iajs-3112	175	28	proof	proof	NOUN
iajs-3112	175	29	.	.	PUNCT
iajs-3112	176	1	let	let	VERB
iajs-3112	176	2	θ	θ	PROPN
iajs-3112	176	3	is	be	AUX
iajs-3112	176	4	a	a	DET
iajs-3112	176	5	cubic	cubic	ADJ
iajs-3112	176	6	ideal	ideal	NOUN
iajs-3112	176	7	which	which	PRON
iajs-3112	176	8	satisfies	satisfy	VERB
iajs-3112	176	9	(	(	PUNCT
iajs-3112	176	10	a	a	X
iajs-3112	176	11	)	)	PUNCT
iajs-3112	176	12	and	and	CCONJ
iajs-3112	176	13	(	(	PUNCT
iajs-3112	176	14	b	b	NOUN
iajs-3112	176	15	)	)	PUNCT
iajs-3112	176	16	,	,	PUNCT
iajs-3112	176	17	then	then	ADV
iajs-3112	176	18	𝜇θ(ω	𝜇θ(ω	VERB
iajs-3112	176	19	∗	∗	PROPN
iajs-3112	176	20	ς	ς	NOUN
iajs-3112	176	21	)	)	PUNCT
iajs-3112	176	22	≥	≥	PROPN
iajs-3112	176	23	𝑟𝑚𝑖𝑛{𝜇θ(κ	𝑟𝑚𝑖𝑛{𝜇θ(κ	PROPN
iajs-3112	176	24	∗	∗	X
iajs-3112	176	25	(	(	PUNCT
iajs-3112	176	26	ω	ω	PROPN
iajs-3112	176	27	∗	∗	PROPN
iajs-3112	176	28	ς	ς	NOUN
iajs-3112	176	29	)	)	PUNCT
iajs-3112	176	30	)	)	PUNCT
iajs-3112	176	31	,	,	PUNCT
iajs-3112	176	32	𝜇θ(κ	𝜇θ(κ	NOUN
iajs-3112	176	33	)	)	PUNCT
iajs-3112	176	34	}	}	PUNCT
iajs-3112	176	35	,	,	PUNCT
iajs-3112	176	36	by	by	ADP
iajs-3112	176	37	substituting	substitute	VERB
iajs-3112	176	38	(	(	PUNCT
iajs-3112	176	39	a	a	NOUN
iajs-3112	176	40	)	)	PUNCT
iajs-3112	176	41	and	and	CCONJ
iajs-3112	176	42	using	use	VERB
iajs-3112	176	43	theorem(2	theorem(2	NUM
iajs-3112	176	44	)	)	PUNCT
iajs-3112	176	45	,	,	PUNCT
iajs-3112	176	46	we	we	PRON
iajs-3112	176	47	obtain	obtain	VERB
iajs-3112	176	48	𝜇θ((ς	𝜇θ((ς	NOUN
iajs-3112	176	49	∗	∗	NOUN
iajs-3112	176	50	ω	ω	NOUN
iajs-3112	176	51	)	)	PUNCT
iajs-3112	176	52	∗	∗	PROPN
iajs-3112	176	53	ω	ω	NOUN
iajs-3112	176	54	)	)	PUNCT
iajs-3112	176	55	∗	∗	PROPN
iajs-3112	176	56	ς	ς	NOUN
iajs-3112	176	57	)	)	PUNCT
iajs-3112	176	58	≥	≥	NOUN
iajs-3112	176	59	𝑟𝑚𝑖𝑛{𝜇θ(ω	𝑟𝑚𝑖𝑛{𝜇θ(ω	NOUN
iajs-3112	176	60	∗	∗	NOUN
iajs-3112	176	61	(	(	PUNCT
iajs-3112	176	62	κ	κ	NOUN
iajs-3112	176	63	∗	∗	NOUN
iajs-3112	176	64	ς	ς	NOUN
iajs-3112	176	65	)	)	PUNCT
iajs-3112	176	66	)	)	PUNCT
iajs-3112	176	67	,	,	PUNCT
iajs-3112	176	68	�	�	PROPN
iajs-3112	176	69	̃	̃	NOUN
iajs-3112	176	70	�	�	NOUN
iajs-3112	176	71	θ(κ	θ(κ	NOUN
iajs-3112	176	72	)	)	PUNCT
iajs-3112	176	73	}	}	PUNCT
iajs-3112	176	74	and	and	CCONJ
iajs-3112	176	75	𝜆θ(ω	𝜆θ(ω	NOUN
iajs-3112	176	76	∗	∗	NOUN
iajs-3112	176	77	ς	ς	NOUN
iajs-3112	176	78	)	)	PUNCT
iajs-3112	176	79	≤	≤	NUM
iajs-3112	176	80	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	176	81	{	{	PUNCT
iajs-3112	176	82	𝜆θ(κ	𝜆θ(κ	ADP
iajs-3112	176	83	∗	∗	NOUN
iajs-3112	176	84	(	(	PUNCT
iajs-3112	176	85	ω	ω	NOUN
iajs-3112	176	86	∗	∗	PROPN
iajs-3112	176	87	ς	ς	NOUN
iajs-3112	176	88	)	)	PUNCT
iajs-3112	176	89	)	)	PUNCT
iajs-3112	176	90	,	,	PUNCT
iajs-3112	176	91	𝜆θ(κ	𝜆θ(κ	NOUN
iajs-3112	176	92	)	)	PUNCT
iajs-3112	176	93	}	}	PUNCT
iajs-3112	176	94	,	,	PUNCT
iajs-3112	176	95	substituting	substitute	VERB
iajs-3112	176	96	(	(	PUNCT
iajs-3112	176	97	b	b	NOUN
iajs-3112	176	98	)	)	PUNCT
iajs-3112	176	99	and	and	CCONJ
iajs-3112	176	100	using	use	VERB
iajs-3112	176	101	theorem	theorem	NOUN
iajs-3112	176	102	(	(	PUNCT
iajs-3112	176	103	2	2	X
iajs-3112	176	104	)	)	PUNCT
iajs-3112	176	105	we	we	PRON
iajs-3112	176	106	get	get	VERB
iajs-3112	176	107	𝜆θ((ς	𝜆θ((ς	NOUN
iajs-3112	176	108	∗	∗	NOUN
iajs-3112	176	109	ω	ω	NOUN
iajs-3112	176	110	)	)	PUNCT
iajs-3112	176	111	∗	∗	PROPN
iajs-3112	176	112	ω	ω	NOUN
iajs-3112	176	113	)	)	PUNCT
iajs-3112	176	114	∗	∗	PROPN
iajs-3112	176	115	ς	ς	NOUN
iajs-3112	176	116	)	)	PUNCT
iajs-3112	176	117	≤	≤	NUM
iajs-3112	176	118	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	176	119	{	{	PUNCT
iajs-3112	176	120	𝜆θ(ω	𝜆θ(ω	NOUN
iajs-3112	176	121	∗	∗	NOUN
iajs-3112	176	122	(	(	PUNCT
iajs-3112	176	123	κ	κ	NOUN
iajs-3112	176	124	∗	∗	NOUN
iajs-3112	176	125	ς	ς	NOUN
iajs-3112	176	126	)	)	PUNCT
iajs-3112	176	127	)	)	PUNCT
iajs-3112	176	128	,	,	PUNCT
iajs-3112	176	129	𝜆θ(κ	𝜆θ(κ	NOUN
iajs-3112	176	130	)	)	PUNCT
iajs-3112	176	131	}	}	PUNCT
iajs-3112	176	132	,	,	PUNCT
iajs-3112	176	133	then	then	ADV
iajs-3112	176	134	θ	θ	PROPN
iajs-3112	176	135	is	be	AUX
iajs-3112	176	136	a	a	DET
iajs-3112	176	137	cubic	cubic	ADJ
iajs-3112	176	138	commutative	commutative	ADJ
iajs-3112	176	139	-	-	PUNCT
iajs-3112	176	140	ideal	ideal	NOUN
iajs-3112	176	141	.	.	PUNCT
iajs-3112	177	1	on	on	ADP
iajs-3112	177	2	the	the	DET
iajs-3112	177	3	contrary	contrary	NOUN
iajs-3112	177	4	,	,	PUNCT
iajs-3112	177	5	suppose	suppose	VERB
iajs-3112	177	6	θ	θ	PROPN
iajs-3112	177	7	=	=	PUNCT
iajs-3112	177	8	〈	〈	PROPN
iajs-3112	177	9	𝜇θ	𝜇θ	PROPN
iajs-3112	177	10	,	,	PUNCT
iajs-3112	177	11	𝜆θ	𝜆θ	SCONJ
iajs-3112	177	12	〉	〉	NOUN
iajs-3112	177	13	is	be	AUX
iajs-3112	177	14	a	a	DET
iajs-3112	177	15	cubic	cubic	ADJ
iajs-3112	177	16	commutative	commutative	ADJ
iajs-3112	177	17	ideal	ideal	NOUN
iajs-3112	177	18	,	,	PUNCT
iajs-3112	177	19	so	so	SCONJ
iajs-3112	177	20	𝜇θ((ς	𝜇θ((ς	NOUN
iajs-3112	177	21	∗	∗	PROPN
iajs-3112	177	22	ω	ω	NOUN
iajs-3112	177	23	)	)	PUNCT
iajs-3112	177	24	∗	∗	PROPN
iajs-3112	177	25	ω	ω	NOUN
iajs-3112	177	26	)	)	PUNCT
iajs-3112	177	27	∗	∗	PROPN
iajs-3112	177	28	ς	ς	NOUN
iajs-3112	177	29	)	)	PUNCT
iajs-3112	177	30	≥	≥	NOUN
iajs-3112	177	31	𝑟𝑚𝑖𝑛{𝜇θ(ω	𝑟𝑚𝑖𝑛{𝜇θ(ω	NOUN
iajs-3112	177	32	∗	∗	NOUN
iajs-3112	177	33	(	(	PUNCT
iajs-3112	177	34	κ	κ	NOUN
iajs-3112	177	35	∗	∗	NOUN
iajs-3112	177	36	ς	ς	NOUN
iajs-3112	177	37	)	)	PUNCT
iajs-3112	177	38	)	)	PUNCT
iajs-3112	177	39	,	,	PUNCT
iajs-3112	177	40	𝜇θ(κ	𝜇θ(κ	NOUN
iajs-3112	177	41	)	)	PUNCT
iajs-3112	177	42	}	}	PUNCT
iajs-3112	177	43	,	,	PUNCT
iajs-3112	177	44	put	put	VERB
iajs-3112	177	45	κ	κ	NOUN
iajs-3112	177	46	=	=	SYM
iajs-3112	177	47	0	0	NUM
iajs-3112	177	48	𝜇θ((ς	𝜇θ((ς	NOUN
iajs-3112	177	49	∗	∗	NOUN
iajs-3112	177	50	ω	ω	NOUN
iajs-3112	177	51	)	)	PUNCT
iajs-3112	177	52	∗	∗	PROPN
iajs-3112	177	53	ω	ω	NOUN
iajs-3112	177	54	)	)	PUNCT
iajs-3112	177	55	∗	∗	PROPN
iajs-3112	177	56	ς	ς	NOUN
iajs-3112	177	57	)	)	PUNCT
iajs-3112	177	58	≥	≥	NOUN
iajs-3112	177	59	𝑟𝑚𝑖𝑛{𝜇θ(ω	𝑟𝑚𝑖𝑛{𝜇θ(ω	NOUN
iajs-3112	177	60	∗	∗	NOUN
iajs-3112	177	61	(	(	PUNCT
iajs-3112	177	62	0	0	NUM
iajs-3112	177	63	∗	∗	NOUN
iajs-3112	177	64	ς	ς	NOUN
iajs-3112	177	65	)	)	PUNCT
iajs-3112	177	66	)	)	PUNCT
iajs-3112	177	67	,	,	PUNCT
iajs-3112	177	68	𝜇θ(0	𝜇θ(0	PROPN
iajs-3112	177	69	)	)	PUNCT
iajs-3112	177	70	}	}	PUNCT
iajs-3112	177	71	,	,	PUNCT
iajs-3112	177	72	𝜇θ((ς	𝜇θ((ς	NOUN
iajs-3112	177	73	∗	∗	PROPN
iajs-3112	177	74	ω	ω	NOUN
iajs-3112	177	75	)	)	PUNCT
iajs-3112	177	76	∗	∗	PROPN
iajs-3112	177	77	ω	ω	NOUN
iajs-3112	177	78	)	)	PUNCT
iajs-3112	177	79	∗	∗	PROPN
iajs-3112	177	80	ς	ς	NOUN
iajs-3112	177	81	)	)	PUNCT
iajs-3112	177	82	≥	≥	PROPN
iajs-3112	177	83	𝑟𝑚𝑖𝑛{𝜇θ(ω	𝑟𝑚𝑖𝑛{𝜇θ(ω	PROPN
iajs-3112	177	84	∗	∗	PROPN
iajs-3112	177	85	ς	ς	PROPN
iajs-3112	177	86	)	)	PUNCT
iajs-3112	177	87	,	,	PUNCT
iajs-3112	177	88	𝜇θ(0	𝜇θ(0	PROPN
iajs-3112	177	89	)	)	PUNCT
iajs-3112	177	90	}	}	PUNCT
iajs-3112	177	91	,	,	PUNCT
iajs-3112	177	92	𝜇θ((ς	𝜇θ((ς	NOUN
iajs-3112	177	93	∗	∗	PROPN
iajs-3112	177	94	ω	ω	NOUN
iajs-3112	177	95	)	)	PUNCT
iajs-3112	177	96	∗	∗	PROPN
iajs-3112	177	97	ω	ω	NOUN
iajs-3112	177	98	)	)	PUNCT
iajs-3112	177	99	∗	∗	PROPN
iajs-3112	177	100	ς	ς	PROPN
iajs-3112	177	101	)	)	PUNCT
iajs-3112	177	102	≥	≥	NOUN
iajs-3112	177	103	𝜇θ(ω	𝜇θ(ω	NOUN
iajs-3112	177	104	∗	∗	NOUN
iajs-3112	177	105	ς	ς	NOUN
iajs-3112	177	106	)	)	PUNCT
iajs-3112	177	107	,	,	PUNCT
iajs-3112	177	108	which	which	PRON
iajs-3112	177	109	is	be	AUX
iajs-3112	177	110	(	(	PUNCT
iajs-3112	177	111	a	a	X
iajs-3112	177	112	)	)	PUNCT
iajs-3112	177	113	likewise	likewise	ADV
iajs-3112	177	114	put	put	VERB
iajs-3112	177	115	κ	κ	NOUN
iajs-3112	177	116	=	=	SYM
iajs-3112	177	117	0	0	NUM
iajs-3112	177	118	in	in	ADP
iajs-3112	177	119	𝜆θ((ς	𝜆θ((ς	NOUN
iajs-3112	177	120	∗	∗	X
iajs-3112	177	121	ω	ω	NOUN
iajs-3112	177	122	)	)	PUNCT
iajs-3112	177	123	∗	∗	PROPN
iajs-3112	177	124	ω	ω	NOUN
iajs-3112	177	125	)	)	PUNCT
iajs-3112	177	126	∗	∗	PROPN
iajs-3112	177	127	ς	ς	NOUN
iajs-3112	177	128	)	)	PUNCT
iajs-3112	177	129	≤	≤	NOUN
iajs-3112	177	130	𝑚𝑎	𝑚𝑎	ADP
iajs-3112	177	131	𝑥{𝜆θ(ω	𝑥{𝜆θ(ω	NUM
iajs-3112	177	132	∗	∗	NOUN
iajs-3112	177	133	(	(	PUNCT
iajs-3112	177	134	κ	κ	NOUN
iajs-3112	177	135	∗	∗	NOUN
iajs-3112	177	136	ς	ς	NOUN
iajs-3112	177	137	)	)	PUNCT
iajs-3112	177	138	)	)	PUNCT
iajs-3112	177	139	,	,	PUNCT
iajs-3112	177	140	𝜆θ(κ	𝜆θ(κ	NOUN
iajs-3112	177	141	)	)	PUNCT
iajs-3112	177	142	}	}	PUNCT
iajs-3112	177	143	,	,	PUNCT
iajs-3112	177	144	so	so	SCONJ
iajs-3112	177	145	we	we	PRON
iajs-3112	177	146	get	get	VERB
iajs-3112	177	147	𝜆θ((ς	𝜆θ((ς	NOUN
iajs-3112	177	148	∗	∗	NOUN
iajs-3112	177	149	ω	ω	NOUN
iajs-3112	177	150	)	)	PUNCT
iajs-3112	177	151	∗	∗	PROPN
iajs-3112	177	152	ω	ω	NOUN
iajs-3112	177	153	)	)	PUNCT
iajs-3112	177	154	∗	∗	PROPN
iajs-3112	177	155	ς	ς	PROPN
iajs-3112	177	156	)	)	PUNCT
iajs-3112	177	157	≥	≥	NOUN
iajs-3112	177	158	𝜆θ(ω	𝜆θ(ω	NOUN
iajs-3112	177	159	∗	∗	NOUN
iajs-3112	177	160	ς	ς	NOUN
iajs-3112	177	161	)	)	PUNCT
iajs-3112	177	162	.	.	PUNCT
iajs-3112	178	1	■	■	PUNCT
iajs-3112	178	2	theorem(22	theorem(22	NUM
iajs-3112	178	3	)	)	PUNCT
iajs-3112	178	4	.	.	PUNCT
iajs-3112	179	1	every	every	DET
iajs-3112	179	2	cubic	cubic	ADJ
iajs-3112	179	3	implicative	implicative	ADJ
iajs-3112	179	4	-	-	PUNCT
iajs-3112	179	5	ideal	ideal	NOUN
iajs-3112	179	6	of	of	ADP
iajs-3112	179	7	ν	ν	PROPN
iajs-3112	179	8	is	be	AUX
iajs-3112	179	9	a	a	DET
iajs-3112	179	10	cubic	cubic	ADJ
iajs-3112	179	11	ideal	ideal	NOUN
iajs-3112	179	12	.	.	PUNCT
iajs-3112	180	1	proof	proof	NOUN
iajs-3112	180	2	.	.	PUNCT
iajs-3112	181	1	by	by	ADP
iajs-3112	181	2	definition(11	definition(11	PROPN
iajs-3112	181	3	)	)	PUNCT
iajs-3112	181	4	,	,	PUNCT
iajs-3112	181	5	we	we	PRON
iajs-3112	181	6	have	have	VERB
iajs-3112	181	7	ci1	ci1	PROPN
iajs-3112	181	8	and	and	CCONJ
iajs-3112	181	9	ci3	ci3	NOUN
iajs-3112	181	10	are	be	AUX
iajs-3112	181	11	hold	hold	NOUN
iajs-3112	181	12	,	,	PUNCT
iajs-3112	181	13	then	then	ADV
iajs-3112	181	14	by	by	ADP
iajs-3112	181	15	(	(	PUNCT
iajs-3112	181	16	𝐂𝐕𝟐)𝜇θ((ς	𝐂𝐕𝟐)𝜇θ((ς	NOUN
iajs-3112	181	17	∗	∗	X
iajs-3112	181	18	ω	ω	NOUN
iajs-3112	181	19	)	)	PUNCT
iajs-3112	181	20	∗	∗	PROPN
iajs-3112	181	21	ς	ς	PROPN
iajs-3112	181	22	)	)	PUNCT
iajs-3112	181	23	≥	≥	PROPN
iajs-3112	181	24	𝑟𝑚𝑖𝑛{𝜇θ(κ	𝑟𝑚𝑖𝑛{𝜇θ(κ	PROPN
iajs-3112	181	25	∗	∗	X
iajs-3112	181	26	(	(	PUNCT
iajs-3112	181	27	(	(	PUNCT
iajs-3112	181	28	ς	ς	PROPN
iajs-3112	181	29	∗	∗	NOUN
iajs-3112	181	30	ω	ω	NOUN
iajs-3112	181	31	)	)	PUNCT
iajs-3112	181	32	∗	∗	NOUN
iajs-3112	181	33	ς	ς	NOUN
iajs-3112	181	34	)	)	PUNCT
iajs-3112	181	35	)	)	PUNCT
iajs-3112	181	36	,	,	PUNCT
iajs-3112	181	37	𝜇θ(κ	𝜇θ(κ	NOUN
iajs-3112	181	38	)	)	PUNCT
iajs-3112	181	39	}	}	PUNCT
iajs-3112	181	40	,	,	PUNCT
iajs-3112	181	41	take	take	VERB
iajs-3112	181	42	ω=0	ω=0	PRON
iajs-3112	181	43	𝜇θ((ς	𝜇θ((ς	NOUN
iajs-3112	181	44	∗	∗	NOUN
iajs-3112	181	45	0	0	NUM
iajs-3112	181	46	)	)	PUNCT
iajs-3112	181	47	∗	∗	NOUN
iajs-3112	181	48	ς	ς	PROPN
iajs-3112	181	49	)	)	PUNCT
iajs-3112	181	50	≥	≥	PROPN
iajs-3112	181	51	𝑟𝑚𝑖𝑛{𝜇θ(κ	𝑟𝑚𝑖𝑛{𝜇θ(κ	PROPN
iajs-3112	181	52	∗	∗	X
iajs-3112	181	53	(	(	PUNCT
iajs-3112	181	54	(	(	PUNCT
iajs-3112	181	55	ς	ς	PROPN
iajs-3112	181	56	∗	∗	NOUN
iajs-3112	181	57	0	0	NUM
iajs-3112	181	58	)	)	PUNCT
iajs-3112	181	59	∗	∗	NOUN
iajs-3112	181	60	ς	ς	PROPN
iajs-3112	181	61	)	)	PUNCT
iajs-3112	181	62	)	)	PUNCT
iajs-3112	181	63	,	,	PUNCT
iajs-3112	181	64	𝜇θ(κ	𝜇θ(κ	NOUN
iajs-3112	181	65	)	)	PUNCT
iajs-3112	181	66	}	}	PUNCT
iajs-3112	181	67	,	,	PUNCT
iajs-3112	181	68	𝜇θ(ς	𝜇θ(ς	NOUN
iajs-3112	181	69	)	)	PUNCT
iajs-3112	181	70	≥	≥	PROPN
iajs-3112	181	71	𝑟𝑚𝑖𝑛{𝜇θ(κ	𝑟𝑚𝑖𝑛{𝜇θ(κ	PROPN
iajs-3112	181	72	∗	∗	PROPN
iajs-3112	181	73	ς	ς	PROPN
iajs-3112	181	74	)	)	PUNCT
iajs-3112	181	75	,	,	PUNCT
iajs-3112	181	76	𝜇θ(κ	𝜇θ(κ	NOUN
iajs-3112	181	77	)	)	PUNCT
iajs-3112	181	78	}	}	PUNCT
iajs-3112	181	79	,	,	PUNCT
iajs-3112	181	80	also	also	ADV
iajs-3112	181	81	take	take	VERB
iajs-3112	181	82	ω=0	ω=0	NOUN
iajs-3112	181	83	in	in	ADP
iajs-3112	181	84	𝜆θ((ς	𝜆θ((ς	NOUN
iajs-3112	181	85	∗	∗	X
iajs-3112	181	86	ω	ω	NOUN
iajs-3112	181	87	)	)	PUNCT
iajs-3112	181	88	∗	∗	PROPN
iajs-3112	181	89	ς	ς	NOUN
iajs-3112	181	90	)	)	PUNCT
iajs-3112	181	91	≤	≤	NUM
iajs-3112	181	92	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	181	93	{	{	PUNCT
iajs-3112	181	94	𝜆θ(κ	𝜆θ(κ	ADP
iajs-3112	181	95	∗	∗	NOUN
iajs-3112	181	96	(	(	PUNCT
iajs-3112	181	97	(	(	PUNCT
iajs-3112	181	98	ς	ς	PROPN
iajs-3112	181	99	∗	∗	NOUN
iajs-3112	181	100	ω	ω	NOUN
iajs-3112	181	101	)	)	PUNCT
iajs-3112	181	102	∗	∗	NOUN
iajs-3112	181	103	ς	ς	NOUN
iajs-3112	181	104	)	)	PUNCT
iajs-3112	181	105	,	,	PUNCT
iajs-3112	181	106	𝜆θ(κ	𝜆θ(κ	NOUN
iajs-3112	181	107	)	)	PUNCT
iajs-3112	181	108	}	}	PUNCT
iajs-3112	181	109	,	,	PUNCT
iajs-3112	181	110	we	we	PRON
iajs-3112	181	111	get	get	VERB
iajs-3112	181	112	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	181	113	)	)	PUNCT
iajs-3112	181	114	≤	≤	NUM
iajs-3112	181	115	𝑚𝑎𝑥	𝑚𝑎𝑥	ADJ
iajs-3112	181	116	{	{	PUNCT
iajs-3112	181	117	𝜆θ(κ	𝜆θ(κ	ADP
iajs-3112	181	118	∗	∗	NOUN
iajs-3112	181	119	ς	ς	NOUN
iajs-3112	181	120	)	)	PUNCT
iajs-3112	181	121	,	,	PUNCT
iajs-3112	181	122	𝜆θ(κ	𝜆θ(κ	NOUN
iajs-3112	181	123	)	)	PUNCT
iajs-3112	181	124	}	}	PUNCT
iajs-3112	181	125	.	.	PUNCT
iajs-3112	182	1	■	■	ADJ
iajs-3112	182	2	theorem(23	theorem(23	NOUN
iajs-3112	182	3	)	)	PUNCT
iajs-3112	182	4	.	.	PUNCT
iajs-3112	183	1	a	a	DET
iajs-3112	183	2	cubic	cubic	ADJ
iajs-3112	183	3	ideal	ideal	NOUN
iajs-3112	183	4	of	of	ADP
iajs-3112	183	5	ν	ν	PROPN
iajs-3112	183	6	is	be	AUX
iajs-3112	183	7	a	a	DET
iajs-3112	183	8	cubic	cubic	ADJ
iajs-3112	183	9	implicative	implicative	NOUN
iajs-3112	183	10	if	if	SCONJ
iajs-3112	183	11	it	it	PRON
iajs-3112	183	12	satisfies	satisfy	VERB
iajs-3112	183	13	the	the	DET
iajs-3112	183	14	following	following	NOUN
iajs-3112	183	15	:	:	PUNCT
iajs-3112	183	16	μ̃θ(ς	μ̃θ(ς	NUM
iajs-3112	183	17	)	)	PUNCT
iajs-3112	183	18	≥	≥	NOUN
iajs-3112	183	19	𝜇θ((ς	𝜇θ((ς	NOUN
iajs-3112	183	20	∗	∗	PROPN
iajs-3112	183	21	ω	ω	NOUN
iajs-3112	183	22	)	)	PUNCT
iajs-3112	183	23	∗	∗	NOUN
iajs-3112	183	24	ς	ς	NOUN
iajs-3112	183	25	)	)	PUNCT
iajs-3112	183	26	and	and	CCONJ
iajs-3112	183	27	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	183	28	)	)	PUNCT
iajs-3112	183	29	≤	≤	NUM
iajs-3112	183	30	𝜆θ((ς	𝜆θ((ς	NOUN
iajs-3112	183	31	∗	∗	NOUN
iajs-3112	183	32	ω	ω	NOUN
iajs-3112	183	33	)	)	PUNCT
iajs-3112	183	34	∗	∗	NOUN
iajs-3112	183	35	ς	ς	NOUN
iajs-3112	183	36	)	)	PUNCT
iajs-3112	183	37	,	,	PUNCT
iajs-3112	183	38	𝑓𝑜𝑟	𝑓𝑜𝑟	ADV
iajs-3112	183	39	𝑎𝑙𝑙	𝑎𝑙𝑙	VERB
iajs-3112	183	40	ς	ς	PROPN
iajs-3112	183	41	,	,	PUNCT
iajs-3112	183	42	ω	ω	PROPN
iajs-3112	183	43	∈	∈	NOUN
iajs-3112	183	44	ν	ν	NOUN
iajs-3112	183	45	proof	proof	NOUN
iajs-3112	183	46	.	.	PUNCT
iajs-3112	184	1	suppose	suppose	VERB
iajs-3112	184	2	θ	θ	NOUN
iajs-3112	184	3	is	be	AUX
iajs-3112	184	4	a	a	DET
iajs-3112	184	5	cubic	cubic	ADJ
iajs-3112	184	6	ideal	ideal	NOUN
iajs-3112	184	7	satisfies	satisfy	VERB
iajs-3112	184	8	the	the	DET
iajs-3112	184	9	above	above	ADJ
iajs-3112	184	10	two	two	NUM
iajs-3112	184	11	inequalities	inequality	NOUN
iajs-3112	184	12	,	,	PUNCT
iajs-3112	184	13	hence	hence	ADV
iajs-3112	184	14	𝜇θ(ς	𝜇θ(ς	PUNCT
iajs-3112	184	15	)	)	PUNCT
iajs-3112	185	1	≥	≥	PROPN
iajs-3112	185	2	𝜇θ((ς	𝜇θ((ς	NOUN
iajs-3112	185	3	∗	∗	PROPN
iajs-3112	185	4	ω	ω	NOUN
iajs-3112	185	5	)	)	PUNCT
iajs-3112	185	6	∗	∗	PROPN
iajs-3112	185	7	ς	ς	PROPN
iajs-3112	185	8	)	)	PUNCT
iajs-3112	185	9	≥	≥	PROPN
iajs-3112	185	10	𝑟𝑚𝑖𝑛{	𝑟𝑚𝑖𝑛{	PROPN
iajs-3112	185	11	�	�	PROPN
iajs-3112	185	12	̃	̃	PROPN
iajs-3112	185	13	�	�	NOUN
iajs-3112	185	14	θ(κ	θ(κ	NOUN
iajs-3112	185	15	∗	∗	NOUN
iajs-3112	185	16	(	(	PUNCT
iajs-3112	185	17	(	(	PUNCT
iajs-3112	185	18	ς	ς	PROPN
iajs-3112	185	19	∗	∗	NOUN
iajs-3112	185	20	ω	ω	NOUN
iajs-3112	185	21	)	)	PUNCT
iajs-3112	185	22	∗	∗	NOUN
iajs-3112	185	23	ς	ς	NOUN
iajs-3112	185	24	)	)	PUNCT
iajs-3112	185	25	)	)	PUNCT
iajs-3112	185	26	,	,	PUNCT
iajs-3112	185	27	𝜇θ(κ	𝜇θ(κ	NOUN
iajs-3112	185	28	)	)	PUNCT
iajs-3112	185	29	}	}	PUNCT
iajs-3112	185	30	and	and	CCONJ
iajs-3112	185	31	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	185	32	)	)	PUNCT
iajs-3112	185	33	≤	≤	NUM
iajs-3112	185	34	𝜆θ((ς	𝜆θ((ς	NOUN
iajs-3112	185	35	∗	∗	NOUN
iajs-3112	185	36	ω	ω	NOUN
iajs-3112	185	37	)	)	PUNCT
iajs-3112	185	38	∗	∗	PROPN
iajs-3112	185	39	ς	ς	NOUN
iajs-3112	185	40	)	)	PUNCT
iajs-3112	185	41	≤	≤	NOUN
iajs-3112	185	42	𝑚𝑎𝑥{𝜆𝛩(κ	𝑚𝑎𝑥{𝜆𝛩(κ	X
iajs-3112	185	43	∗	∗	NOUN
iajs-3112	185	44	(	(	PUNCT
iajs-3112	185	45	(	(	PUNCT
iajs-3112	185	46	ς	ς	PROPN
iajs-3112	185	47	∗	∗	NOUN
iajs-3112	185	48	ω	ω	NOUN
iajs-3112	185	49	)	)	PUNCT
iajs-3112	185	50	∗	∗	NOUN
iajs-3112	185	51	ς	ς	NOUN
iajs-3112	185	52	)	)	PUNCT
iajs-3112	185	53	)	)	PUNCT
iajs-3112	185	54	,	,	PUNCT
iajs-3112	185	55	𝜆𝛩(κ	𝜆𝛩(κ	NUM
iajs-3112	185	56	)	)	PUNCT
iajs-3112	185	57	}	}	PUNCT
iajs-3112	185	58	.	.	PUNCT
iajs-3112	186	1	■	■	PUNCT
iajs-3112	186	2	theorem(24	theorem(24	NUM
iajs-3112	186	3	)	)	PUNCT
iajs-3112	186	4	.	.	PUNCT
iajs-3112	187	1	if	if	SCONJ
iajs-3112	187	2	θ	θ	PROPN
iajs-3112	187	3	of	of	ADP
iajs-3112	187	4	ν	ν	PROPN
iajs-3112	187	5	is	be	AUX
iajs-3112	187	6	a	a	DET
iajs-3112	187	7	cubic	cubic	ADJ
iajs-3112	187	8	positive	positive	ADJ
iajs-3112	187	9	implicative	implicative	ADJ
iajs-3112	187	10	-	-	PUNCT
iajs-3112	187	11	ideal	ideal	NOUN
iajs-3112	187	12	and	and	CCONJ
iajs-3112	187	13	a	a	DET
iajs-3112	187	14	cubic	cubic	ADJ
iajs-3112	187	15	commutative	commutative	ADJ
iajs-3112	187	16	-	-	PUNCT
iajs-3112	187	17	ideal	ideal	NOUN
iajs-3112	187	18	,	,	PUNCT
iajs-3112	187	19	then	then	ADV
iajs-3112	187	20	θ	θ	PROPN
iajs-3112	187	21	is	be	AUX
iajs-3112	187	22	a	a	DET
iajs-3112	187	23	cubic	cubic	ADJ
iajs-3112	187	24	implicative	implicative	ADJ
iajs-3112	187	25	-	-	PUNCT
iajs-3112	187	26	ideal	ideal	NOUN
iajs-3112	187	27	.	.	PUNCT
iajs-3112	188	1	ihjpas	ihjpas	PROPN
iajs-3112	188	2	.	.	PUNCT
iajs-3112	189	1	37	37	NUM
iajs-3112	189	2	(	(	PUNCT
iajs-3112	189	3	1	1	NUM
iajs-3112	189	4	)	)	PUNCT
iajs-3112	189	5	2024	2024	NUM
iajs-3112	189	6	461	461	NUM
iajs-3112	189	7	proof	proof	NOUN
iajs-3112	189	8	.	.	PUNCT
iajs-3112	190	1	by	by	ADP
iajs-3112	190	2	theorem	theorem	NOUN
iajs-3112	190	3	(	(	PUNCT
iajs-3112	190	4	21	21	NUM
iajs-3112	190	5	)	)	PUNCT
iajs-3112	190	6	and	and	CCONJ
iajs-3112	190	7	lemma	lemma	PROPN
iajs-3112	190	8	(	(	PUNCT
iajs-3112	190	9	18	18	NUM
iajs-3112	190	10	)	)	PUNCT
iajs-3112	190	11	,	,	PUNCT
iajs-3112	190	12	we	we	PRON
iajs-3112	190	13	have	have	VERB
iajs-3112	190	14	:	:	PUNCT
iajs-3112	190	15	𝜇θ((ς	𝜇θ((ς	NOUN
iajs-3112	190	16	∗	∗	PROPN
iajs-3112	190	17	ω	ω	NOUN
iajs-3112	190	18	)	)	PUNCT
iajs-3112	190	19	∗	∗	PROPN
iajs-3112	190	20	ς	ς	NOUN
iajs-3112	190	21	)	)	PUNCT
iajs-3112	190	22	∗	∗	NOUN
iajs-3112	190	23	ς	ς	NOUN
iajs-3112	190	24	)	)	PUNCT
iajs-3112	190	25	≥	≥	NOUN
iajs-3112	190	26	𝜇θ((ς	𝜇θ((ς	NOUN
iajs-3112	190	27	∗	∗	NOUN
iajs-3112	190	28	(	(	PUNCT
iajs-3112	190	29	ς	ς	PROPN
iajs-3112	190	30	∗	∗	NOUN
iajs-3112	190	31	ω	ω	NOUN
iajs-3112	190	32	)	)	PUNCT
iajs-3112	190	33	)	)	PUNCT
iajs-3112	190	34	∗	∗	NOUN
iajs-3112	190	35	(	(	PUNCT
iajs-3112	190	36	ς	ς	PROPN
iajs-3112	190	37	∗	∗	NOUN
iajs-3112	190	38	ω	ω	NOUN
iajs-3112	190	39	)	)	PUNCT
iajs-3112	190	40	)	)	PUNCT
iajs-3112	191	1	≥	≥	X
iajs-3112	191	2	𝜇θ(ς	𝜇θ(ς	VERB
iajs-3112	191	3	∗	∗	NOUN
iajs-3112	191	4	(	(	PUNCT
iajs-3112	191	5	(	(	PUNCT
iajs-3112	191	6	ς	ς	PROPN
iajs-3112	191	7	∗	∗	NOUN
iajs-3112	191	8	ω	ω	NOUN
iajs-3112	191	9	)	)	PUNCT
iajs-3112	191	10	∗	∗	PROPN
iajs-3112	191	11	ω	ω	NUM
iajs-3112	191	12	)	)	PUNCT
iajs-3112	191	13	)	)	PUNCT
iajs-3112	192	1	=	=	NOUN
iajs-3112	192	2	𝜇θ((ς	𝜇θ((ς	NOUN
iajs-3112	192	3	∗	∗	NOUN
iajs-3112	192	4	ω	ω	NOUN
iajs-3112	192	5	)	)	PUNCT
iajs-3112	192	6	∗	∗	NOUN
iajs-3112	192	7	(	(	PUNCT
iajs-3112	192	8	ς	ς	PROPN
iajs-3112	192	9	∗	∗	NOUN
iajs-3112	192	10	ω	ω	NOUN
iajs-3112	192	11	)	)	PUNCT
iajs-3112	192	12	)	)	PUNCT
iajs-3112	192	13	and	and	CCONJ
iajs-3112	192	14	by	by	ADP
iajs-3112	192	15	theorem(2	theorem(2	PRON
iajs-3112	192	16	)	)	PUNCT
iajs-3112	192	17	=	=	SYM
iajs-3112	192	18	𝜇θ(0	𝜇θ(0	PROPN
iajs-3112	192	19	)	)	PUNCT
iajs-3112	192	20	.	.	PUNCT
iajs-3112	193	1	also	also	ADV
iajs-3112	193	2	,	,	PUNCT
iajs-3112	193	3	𝜆θ((ς	𝜆θ((ς	NOUN
iajs-3112	193	4	∗	∗	NOUN
iajs-3112	193	5	ω	ω	NOUN
iajs-3112	193	6	)	)	PUNCT
iajs-3112	193	7	∗	∗	PROPN
iajs-3112	193	8	ς	ς	NOUN
iajs-3112	193	9	)	)	PUNCT
iajs-3112	193	10	∗	∗	NOUN
iajs-3112	193	11	ς	ς	NOUN
iajs-3112	193	12	)	)	PUNCT
iajs-3112	193	13	≤	≤	NUM
iajs-3112	193	14	𝜆θ((ς	𝜆θ((ς	NOUN
iajs-3112	193	15	∗	∗	NOUN
iajs-3112	193	16	(	(	PUNCT
iajs-3112	193	17	ς	ς	PROPN
iajs-3112	193	18	∗	∗	NOUN
iajs-3112	193	19	ω	ω	NOUN
iajs-3112	193	20	)	)	PUNCT
iajs-3112	193	21	)	)	PUNCT
iajs-3112	194	1	∗	∗	NOUN
iajs-3112	194	2	(	(	PUNCT
iajs-3112	194	3	ς	ς	PROPN
iajs-3112	194	4	∗	∗	NOUN
iajs-3112	194	5	ω	ω	NOUN
iajs-3112	194	6	)	)	PUNCT
iajs-3112	194	7	)	)	PUNCT
iajs-3112	194	8	≤	≤	NOUN
iajs-3112	194	9	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	194	10	∗	∗	NOUN
iajs-3112	194	11	(	(	PUNCT
iajs-3112	194	12	ω	ω	NOUN
iajs-3112	194	13	∗	∗	X
iajs-3112	194	14	(	(	PUNCT
iajs-3112	194	15	ς	ς	PROPN
iajs-3112	194	16	∗	∗	NOUN
iajs-3112	194	17	ω	ω	NOUN
iajs-3112	194	18	)	)	PUNCT
iajs-3112	194	19	)	)	PUNCT
iajs-3112	194	20	)	)	PUNCT
iajs-3112	195	1	=	=	PRON
iajs-3112	195	2	𝜆θ((ς	𝜆θ((ς	NOUN
iajs-3112	195	3	∗	∗	NOUN
iajs-3112	195	4	ω	ω	NOUN
iajs-3112	195	5	)	)	PUNCT
iajs-3112	195	6	∗	∗	NOUN
iajs-3112	195	7	(	(	PUNCT
iajs-3112	195	8	ς	ς	PROPN
iajs-3112	195	9	∗	∗	NOUN
iajs-3112	195	10	ω	ω	NOUN
iajs-3112	195	11	)	)	PUNCT
iajs-3112	195	12	)	)	PUNCT
iajs-3112	195	13	and	and	CCONJ
iajs-3112	195	14	by	by	ADP
iajs-3112	195	15	theorem(2	theorem(2	PRON
iajs-3112	195	16	)	)	PUNCT
iajs-3112	195	17	=	=	SYM
iajs-3112	195	18	𝜆θ(0	𝜆θ(0	PROPN
iajs-3112	195	19	)	)	PUNCT
iajs-3112	195	20	by	by	ADP
iajs-3112	195	21	two	two	NUM
iajs-3112	195	22	theorems	theorem	NOUN
iajs-3112	195	23	(	(	PUNCT
iajs-3112	195	24	15	15	NUM
iajs-3112	195	25	)	)	PUNCT
iajs-3112	195	26	and	and	CCONJ
iajs-3112	195	27	(	(	PUNCT
iajs-3112	195	28	19	19	NUM
iajs-3112	195	29	)	)	PUNCT
iajs-3112	195	30	and	and	CCONJ
iajs-3112	195	31	θ	θ	PROPN
iajs-3112	195	32	is	be	AUX
iajs-3112	195	33	a	a	DET
iajs-3112	195	34	cubic	cubic	ADJ
iajs-3112	195	35	-	-	PUNCT
iajs-3112	195	36	ideal	ideal	NOUN
iajs-3112	195	37	,	,	PUNCT
iajs-3112	195	38	then	then	ADV
iajs-3112	195	39	𝜇θ(ς	𝜇θ(ς	PUNCT
iajs-3112	195	40	)	)	PUNCT
iajs-3112	195	41	≥	≥	PROPN
iajs-3112	195	42	𝑟𝑚𝑖𝑛{𝜇θ(((ς	𝑟𝑚𝑖𝑛{𝜇θ(((ς	X
iajs-3112	195	43	∗	∗	PROPN
iajs-3112	195	44	ω	ω	NOUN
iajs-3112	195	45	)	)	PUNCT
iajs-3112	195	46	∗	∗	PROPN
iajs-3112	195	47	ς	ς	NOUN
iajs-3112	195	48	)	)	PUNCT
iajs-3112	195	49	∗	∗	NOUN
iajs-3112	195	50	ς	ς	NOUN
iajs-3112	195	51	)	)	PUNCT
iajs-3112	195	52	,	,	PUNCT
iajs-3112	195	53	𝜇θ((ς	𝜇θ((ς	NOUN
iajs-3112	195	54	∗	∗	PROPN
iajs-3112	195	55	ω	ω	NOUN
iajs-3112	195	56	)	)	PUNCT
iajs-3112	195	57	∗	∗	NOUN
iajs-3112	195	58	ς	ς	NOUN
iajs-3112	195	59	)	)	PUNCT
iajs-3112	195	60	}	}	PUNCT
iajs-3112	195	61	≥	≥	NOUN
iajs-3112	195	62	𝑟𝑚𝑖𝑛{𝜇θ(0	𝑟𝑚𝑖𝑛{𝜇θ(0	VERB
iajs-3112	195	63	)	)	PUNCT
iajs-3112	195	64	,	,	PUNCT
iajs-3112	195	65	𝜇θ((ς	𝜇θ((ς	NOUN
iajs-3112	195	66	∗	∗	PROPN
iajs-3112	195	67	ω	ω	NOUN
iajs-3112	195	68	)	)	PUNCT
iajs-3112	195	69	∗	∗	NOUN
iajs-3112	195	70	ς	ς	NOUN
iajs-3112	195	71	)	)	PUNCT
iajs-3112	195	72	}	}	PUNCT
iajs-3112	195	73	=	=	NOUN
iajs-3112	195	74	𝜇θ((ς	𝜇θ((ς	NOUN
iajs-3112	195	75	∗	∗	NOUN
iajs-3112	195	76	ω	ω	NOUN
iajs-3112	195	77	)	)	PUNCT
iajs-3112	195	78	∗	∗	NOUN
iajs-3112	195	79	ς	ς	NOUN
iajs-3112	195	80	)	)	PUNCT
iajs-3112	195	81	,	,	PUNCT
iajs-3112	195	82	and	and	CCONJ
iajs-3112	195	83	𝜆θ(ς	𝜆θ(ς	PUNCT
iajs-3112	195	84	)	)	PUNCT
iajs-3112	195	85	≤	≤	NOUN
iajs-3112	195	86	𝑚𝑎𝑥{𝜆θ((ς	𝑚𝑎𝑥{𝜆θ((ς	NUM
iajs-3112	195	87	∗	∗	NOUN
iajs-3112	195	88	ω	ω	NOUN
iajs-3112	195	89	)	)	PUNCT
iajs-3112	195	90	∗	∗	PROPN
iajs-3112	195	91	ς	ς	NOUN
iajs-3112	195	92	)	)	PUNCT
iajs-3112	195	93	∗	∗	NOUN
iajs-3112	195	94	ς	ς	NOUN
iajs-3112	195	95	)	)	PUNCT
iajs-3112	195	96	,	,	PUNCT
iajs-3112	195	97	𝜆θ((ς	𝜆θ((ς	VERB
iajs-3112	195	98	∗	∗	NOUN
iajs-3112	195	99	ω	ω	NOUN
iajs-3112	195	100	)	)	PUNCT
iajs-3112	195	101	∗	∗	NOUN
iajs-3112	195	102	ς	ς	NOUN
iajs-3112	195	103	)	)	PUNCT
iajs-3112	195	104	}	}	PUNCT
iajs-3112	195	105	≤	≤	NUM
iajs-3112	195	106	𝑚𝑎𝑥{𝜆θ(0	𝑚𝑎𝑥{𝜆θ(0	NUM
iajs-3112	195	107	)	)	PUNCT
iajs-3112	195	108	,	,	PUNCT
iajs-3112	195	109	𝜆θ((ς	𝜆θ((ς	NOUN
iajs-3112	195	110	∗	∗	NOUN
iajs-3112	195	111	ω	ω	NOUN
iajs-3112	195	112	)	)	PUNCT
iajs-3112	195	113	∗	∗	NOUN
iajs-3112	195	114	ς	ς	NOUN
iajs-3112	195	115	)	)	PUNCT
iajs-3112	195	116	}	}	PUNCT
iajs-3112	195	117	=	=	SYM
iajs-3112	195	118	𝜆θ((ς	𝜆θ((ς	NOUN
iajs-3112	195	119	∗	∗	NOUN
iajs-3112	195	120	ω	ω	NOUN
iajs-3112	195	121	)	)	PUNCT
iajs-3112	195	122	∗	∗	NOUN
iajs-3112	195	123	ς	ς	NOUN
iajs-3112	195	124	)	)	PUNCT
iajs-3112	195	125	so	so	ADV
iajs-3112	195	126	from	from	ADP
iajs-3112	195	127	theorem(23	theorem(23	NOUN
iajs-3112	195	128	)	)	PUNCT
iajs-3112	195	129	,	,	PUNCT
iajs-3112	195	130	we	we	PRON
iajs-3112	195	131	get	get	VERB
iajs-3112	195	132	the	the	DET
iajs-3112	195	133	required	require	VERB
iajs-3112	195	134	.	.	PUNCT
iajs-3112	196	1	■	■	PUNCT
iajs-3112	196	2	4	4	X
iajs-3112	196	3	.	.	X
iajs-3112	196	4	conclusion	conclusion	NOUN
iajs-3112	196	5	through	through	ADP
iajs-3112	196	6	this	this	DET
iajs-3112	196	7	work	work	NOUN
iajs-3112	196	8	,	,	PUNCT
iajs-3112	196	9	we	we	PRON
iajs-3112	196	10	present	present	VERB
iajs-3112	196	11	the	the	DET
iajs-3112	196	12	definitions	definition	NOUN
iajs-3112	196	13	of	of	ADP
iajs-3112	196	14	the	the	DET
iajs-3112	196	15	cubic	cubic	ADJ
iajs-3112	196	16	(	(	PUNCT
iajs-3112	196	17	commutative	commutative	ADJ
iajs-3112	196	18	,	,	PUNCT
iajs-3112	196	19	implicative	implicative	ADJ
iajs-3112	196	20	,	,	PUNCT
iajs-3112	196	21	positive	positive	ADJ
iajs-3112	196	22	implicative)-ideals	implicative)-ideal	NOUN
iajs-3112	196	23	and	and	CCONJ
iajs-3112	196	24	study	study	VERB
iajs-3112	196	25	some	some	DET
iajs-3112	196	26	relationships	relationship	NOUN
iajs-3112	196	27	among	among	ADP
iajs-3112	196	28	these	these	DET
iajs-3112	196	29	types	type	NOUN
iajs-3112	196	30	.	.	PUNCT
iajs-3112	197	1	also	also	ADV
iajs-3112	197	2	,	,	PUNCT
iajs-3112	197	3	some	some	DET
iajs-3112	197	4	important	important	ADJ
iajs-3112	197	5	theories	theory	NOUN
iajs-3112	197	6	are	be	AUX
iajs-3112	197	7	discussed	discuss	VERB
iajs-3112	197	8	.	.	PUNCT
iajs-3112	198	1	the	the	DET
iajs-3112	198	2	main	main	ADJ
iajs-3112	198	3	purpose	purpose	NOUN
iajs-3112	198	4	of	of	ADP
iajs-3112	198	5	our	our	PRON
iajs-3112	198	6	future	future	ADJ
iajs-3112	198	7	work	work	NOUN
iajs-3112	198	8	is	be	AUX
iajs-3112	198	9	to	to	PART
iajs-3112	198	10	study	study	VERB
iajs-3112	198	11	some	some	DET
iajs-3112	198	12	concepts	concept	NOUN
iajs-3112	198	13	such	such	ADJ
iajs-3112	198	14	as	as	ADP
iajs-3112	198	15	hyper	hyper	ADJ
iajs-3112	198	16	cubicideal	cubicideal	NOUN
iajs-3112	198	17	,	,	PUNCT
iajs-3112	198	18	filter	filter	NOUN
iajs-3112	198	19	cubic	cubic	ADJ
iajs-3112	198	20	-	-	PUNCT
iajs-3112	198	21	ideal	ideal	ADJ
iajs-3112	198	22	and	and	CCONJ
iajs-3112	198	23	intuitionistic	intuitionistic	ADJ
iajs-3112	198	24	cubic	cubic	ADJ
iajs-3112	198	25	-	-	PUNCT
iajs-3112	198	26	ideals	ideal	NOUN
iajs-3112	198	27	.	.	PUNCT
iajs-3112	199	1	also	also	ADV
iajs-3112	199	2	,	,	PUNCT
iajs-3112	199	3	we	we	PRON
iajs-3112	199	4	can	can	AUX
iajs-3112	199	5	study	study	VERB
iajs-3112	199	6	the	the	DET
iajs-3112	199	7	notion	notion	NOUN
iajs-3112	199	8	of	of	ADP
iajs-3112	199	9	the	the	DET
iajs-3112	199	10	graph	graph	NOUN
iajs-3112	199	11	theory	theory	NOUN
iajs-3112	199	12	of	of	ADP
iajs-3112	199	13	cubic	cubic	ADJ
iajs-3112	199	14	-	-	PUNCT
iajs-3112	199	15	ideal	ideal	NOUN
iajs-3112	199	16	in	in	ADP
iajs-3112	199	17	a	a	DET
iajs-3112	199	18	ku	ku	PROPN
iajs-3112	199	19	-	-	PUNCT
iajs-3112	199	20	semigroup	semigroup	PROPN
iajs-3112	199	21	.	.	PUNCT
iajs-3112	200	1	acknowledgment	acknowledgment	NOUN
iajs-3112	200	2	the	the	DET
iajs-3112	200	3	authors	author	NOUN
iajs-3112	200	4	are	be	AUX
iajs-3112	200	5	greatly	greatly	ADV
iajs-3112	200	6	appreciated	appreciate	VERB
iajs-3112	200	7	the	the	DET
iajs-3112	200	8	referees	referee	NOUN
iajs-3112	200	9	for	for	ADP
iajs-3112	200	10	their	their	PRON
iajs-3112	200	11	valuable	valuable	ADJ
iajs-3112	200	12	comments	comment	NOUN
iajs-3112	200	13	and	and	CCONJ
iajs-3112	200	14	suggestions	suggestion	NOUN
iajs-3112	200	15	for	for	ADP
iajs-3112	200	16	improving	improve	VERB
iajs-3112	200	17	the	the	DET
iajs-3112	200	18	paper	paper	NOUN
iajs-3112	200	19	.	.	PUNCT
iajs-3112	201	1	conflict	conflict	NOUN
iajs-3112	201	2	of	of	ADP
iajs-3112	201	3	interest	interest	NOUN
iajs-3112	201	4	the	the	DET
iajs-3112	201	5	authors	author	NOUN
iajs-3112	201	6	declare	declare	VERB
iajs-3112	201	7	that	that	SCONJ
iajs-3112	201	8	they	they	PRON
iajs-3112	201	9	have	have	VERB
iajs-3112	201	10	no	no	DET
iajs-3112	201	11	conflicts	conflict	NOUN
iajs-3112	201	12	of	of	ADP
iajs-3112	201	13	interest	interest	NOUN
iajs-3112	201	14	funding	funding	NOUN
iajs-3112	201	15	there	there	PRON
iajs-3112	201	16	is	be	VERB
iajs-3112	201	17	no	no	DET
iajs-3112	201	18	financial	financial	ADJ
iajs-3112	201	19	support	support	NOUN
iajs-3112	201	20	in	in	ADP
iajs-3112	201	21	preparation	preparation	NOUN
iajs-3112	201	22	for	for	ADP
iajs-3112	201	23	the	the	DET
iajs-3112	201	24	publication	publication	NOUN
iajs-3112	201	25	.	.	PUNCT
iajs-3112	202	1	references	reference	NOUN
iajs-3112	202	2	1	1	NUM
iajs-3112	202	3	.	.	PUNCT
iajs-3112	202	4	prabpayak	prabpayak	NOUN
iajs-3112	202	5	,	,	PUNCT
iajs-3112	202	6	c.	c.	NOUN
iajs-3112	202	7	;	;	PUNCT
iajs-3112	202	8	leerawat	leerawat	NOUN
iajs-3112	202	9	,	,	PUNCT
iajs-3112	202	10	u.	u.	VERB
iajs-3112	202	11	on	on	ADP
iajs-3112	202	12	ideals	ideal	NOUN
iajs-3112	202	13	and	and	CCONJ
iajs-3112	202	14	congruence	congruence	NOUN
iajs-3112	202	15	in	in	ADP
iajs-3112	202	16	ku	ku	PROPN
iajs-3112	202	17	-	-	PUNCT
iajs-3112	202	18	algebras	algebras	PROPN
iajs-3112	202	19	.	.	PUNCT
iajs-3112	203	1	scientia	scientia	PROPN
iajs-3112	203	2	magna(international	magna(international	ADJ
iajs-3112	203	3	book	book	NOUN
iajs-3112	203	4	series	series	NOUN
iajs-3112	203	5	)	)	PUNCT
iajs-3112	203	6	.	.	PUNCT
iajs-3112	204	1	2009,5,1	2009,5,1	NUM
iajs-3112	204	2	,	,	PUNCT
iajs-3112	204	3	54	54	NUM
iajs-3112	204	4	-	-	SYM
iajs-3112	204	5	57	57	NUM
iajs-3112	204	6	.	.	X
iajs-3112	205	1	2	2	NUM
iajs-3112	205	2	.	.	X
iajs-3112	205	3	prabpayak	prabpayak	NOUN
iajs-3112	205	4	,	,	PUNCT
iajs-3112	205	5	c.	c.	NOUN
iajs-3112	205	6	;	;	PUNCT
iajs-3112	205	7	leerawat	leerawat	NOUN
iajs-3112	205	8	,	,	PUNCT
iajs-3112	205	9	on	on	ADP
iajs-3112	205	10	isomorphisms	isomorphism	NOUN
iajs-3112	205	11	of	of	ADP
iajs-3112	205	12	ku	ku	PROPN
iajs-3112	205	13	-	-	PUNCT
iajs-3112	205	14	algebras	algebras	PROPN
iajs-3112	205	15	,	,	PUNCT
iajs-3112	205	16	scientia	scientia	PROPN
iajs-3112	205	17	magna	magna	PROPN
iajs-3112	205	18	(	(	PUNCT
iajs-3112	205	19	international	international	ADJ
iajs-3112	205	20	book	book	NOUN
iajs-3112	205	21	series	series	NOUN
iajs-3112	205	22	)	)	PUNCT
iajs-3112	205	23	.	.	PUNCT
iajs-3112	206	1	2009,5,3	2009,5,3	ADP
iajs-3112	206	2	,	,	PUNCT
iajs-3112	206	3	25	25	NUM
iajs-3112	206	4	-	-	SYM
iajs-3112	206	5	31	31	NUM
iajs-3112	206	6	.	.	PUNCT
iajs-3112	207	1	3	3	X
iajs-3112	207	2	.	.	X
iajs-3112	207	3	zadeh	zadeh	PROPN
iajs-3112	207	4	,	,	PUNCT
iajs-3112	207	5	l.	l.	PROPN
iajs-3112	207	6	a.	a.	PROPN
iajs-3112	207	7	the	the	DET
iajs-3112	207	8	concept	concept	NOUN
iajs-3112	207	9	of	of	ADP
iajs-3112	207	10	a	a	DET
iajs-3112	207	11	linguistic	linguistic	ADJ
iajs-3112	207	12	variable	variable	NOUN
iajs-3112	207	13	and	and	CCONJ
iajs-3112	207	14	its	its	PRON
iajs-3112	207	15	application	application	NOUN
iajs-3112	207	16	to	to	PART
iajs-3112	207	17	approximate	approximate	ADJ
iajs-3112	207	18	reasoning	reasoning	NOUN
iajs-3112	207	19	.	.	PUNCT
iajs-3112	208	1	i	i	PRON
iajs-3112	208	2	,	,	PUNCT
iajs-3112	208	3	information	information	NOUN
iajs-3112	208	4	sci	sci	PROPN
iajs-3112	208	5	.	.	PROPN
iajs-3112	208	6	and	and	CCONJ
iajs-3112	208	7	control	control	NOUN
iajs-3112	208	8	.	.	PUNCT
iajs-3112	209	1	1975	1975	NUM
iajs-3112	209	2	,	,	PUNCT
iajs-3112	209	3	8	8	NUM
iajs-3112	209	4	,	,	PUNCT
iajs-3112	209	5	199	199	NUM
iajs-3112	209	6	-	-	SYM
iajs-3112	209	7	249	249	NUM
iajs-3112	209	8	.	.	PUNCT
iajs-3112	210	1	4	4	X
iajs-3112	210	2	.	.	X
iajs-3112	210	3	rosenfeld	rosenfeld	PROPN
iajs-3112	210	4	,	,	PUNCT
iajs-3112	210	5	a.	a.	NOUN
iajs-3112	210	6	fuzzy	fuzzy	ADJ
iajs-3112	210	7	groups	group	NOUN
iajs-3112	210	8	,	,	PUNCT
iajs-3112	210	9	j.	j.	PROPN
iajs-3112	210	10	math	math	PROPN
iajs-3112	210	11	.	.	PUNCT
iajs-3112	211	1	anal	anal	NOUN
iajs-3112	211	2	.	.	PUNCT
iajs-3112	212	1	appl.1971	appl.1971	X
iajs-3112	212	2	,	,	PUNCT
iajs-3112	212	3	35	35	NUM
iajs-3112	212	4	,	,	PUNCT
iajs-3112	212	5	512	512	NUM
iajs-3112	212	6	-	-	SYM
iajs-3112	212	7	517	517	NUM
iajs-3112	212	8	.	.	NOUN
iajs-3112	213	1	5	5	NUM
iajs-3112	213	2	.	.	X
iajs-3112	213	3	jun	jun	PROPN
iajs-3112	213	4	,	,	PUNCT
iajs-3112	213	5	y.	y.	PROPN
iajs-3112	213	6	b.	b.	PROPN
iajs-3112	213	7	;	;	PUNCT
iajs-3112	214	1	meng	meng	PROPN
iajs-3112	214	2	j.	j.	PROPN
iajs-3112	214	3	fuzzy	fuzzy	PROPN
iajs-3112	214	4	p	p	NOUN
iajs-3112	214	5	-	-	PUNCT
iajs-3112	214	6	ideals	ideal	NOUN
iajs-3112	214	7	in	in	ADP
iajs-3112	214	8	bci	bci	NOUN
iajs-3112	214	9	-	-	PUNCT
iajs-3112	214	10	algebras	algebra	NOUN
iajs-3112	214	11	.	.	PUNCT
iajs-3112	214	12	math	math	NOUN
iajs-3112	214	13	.	.	PUNCT
iajs-3112	215	1	japonica	japonica	PROPN
iajs-3112	215	2	.	.	PUNCT
iajs-3112	216	1	1994	1994	NUM
iajs-3112	216	2	,	,	PUNCT
iajs-3112	216	3	40	40	NUM
iajs-3112	216	4	,	,	PUNCT
iajs-3112	216	5	2	2	NUM
iajs-3112	216	6	,	,	PUNCT
iajs-3112	216	7	271–282	271–282	NUM
iajs-3112	216	8	.	.	PUNCT
iajs-3112	217	1	6	6	NUM
iajs-3112	217	2	.	.	X
iajs-3112	217	3	jun	jun	PROPN
iajs-3112	217	4	,	,	PUNCT
iajs-3112	217	5	y.	y.	PROPN
iajs-3112	217	6	b.	b.	PROPN
iajs-3112	217	7	;	;	PUNCT
iajs-3112	217	8	meng	meng	PROPN
iajs-3112	217	9	j	j	PROPN
iajs-3112	217	10	;	;	PUNCT
iajs-3112	217	11	mostafa	mostafa	PROPN
iajs-3112	217	12	,	,	PUNCT
iajs-3112	217	13	s.m	s.m	PROPN
iajs-3112	217	14	.	.	PROPN
iajs-3112	217	15	on	on	ADP
iajs-3112	217	16	fuzzy	fuzzy	ADJ
iajs-3112	217	17	implicative	implicative	ADJ
iajs-3112	217	18	ideals	ideal	NOUN
iajs-3112	217	19	of	of	ADP
iajs-3112	217	20	bckalgebras	bckalgebras	PROPN
iajs-3112	217	21	.	.	PUNCT
iajs-3112	218	1	soochw	soochw	PROPN
iajs-3112	218	2	journal	journal	PROPN
iajs-3112	218	3	of	of	ADP
iajs-3112	218	4	mathematics	mathematics	PROPN
iajs-3112	218	5	.	.	PUNCT
iajs-3112	219	1	1999	1999	NUM
iajs-3112	219	2	,	,	PUNCT
iajs-3112	219	3	25	25	NUM
iajs-3112	219	4	,	,	PUNCT
iajs-3112	219	5	1	1	NUM
iajs-3112	219	6	,	,	PUNCT
iajs-3112	219	7	57	57	NUM
iajs-3112	219	8	-	-	SYM
iajs-3112	219	9	70	70	NUM
iajs-3112	219	10	.	.	PUNCT
iajs-3112	220	1	ihjpas	ihjpas	PROPN
iajs-3112	220	2	.	.	PUNCT
iajs-3112	221	1	37	37	NUM
iajs-3112	221	2	(	(	PUNCT
iajs-3112	221	3	1	1	NUM
iajs-3112	221	4	)	)	PUNCT
iajs-3112	221	5	2024	2024	NUM
iajs-3112	221	6	462	462	NUM
iajs-3112	221	7	7	7	NUM
iajs-3112	221	8	.	.	X
iajs-3112	221	9	mostafa	mostafa	PROPN
iajs-3112	221	10	,	,	PUNCT
iajs-3112	221	11	s.m	s.m	PROPN
iajs-3112	221	12	.	.	PROPN
iajs-3112	221	13	;	;	PUNCT
iajs-3112	221	14	abd	abd	PROPN
iajs-3112	221	15	-	-	PUNCT
iajs-3112	221	16	elnaby	elnaby	PROPN
iajs-3112	221	17	,	,	PUNCT
iajs-3112	221	18	m.a	m.a	PROPN
iajs-3112	221	19	.	.	PROPN
iajs-3112	221	20	;	;	PUNCT
iajs-3112	221	21	yousef	yousef	PROPN
iajs-3112	221	22	,	,	PUNCT
iajs-3112	221	23	m.m.m	m.m.m	INTJ
iajs-3112	221	24	.	.	PUNCT
iajs-3112	221	25	fuzzy	fuzzy	ADJ
iajs-3112	221	26	ideals	ideal	NOUN
iajs-3112	221	27	of	of	ADP
iajs-3112	221	28	ku	ku	PROPN
iajs-3112	221	29	-	-	PUNCT
iajs-3112	221	30	algebras	algebras	PROPN
iajs-3112	221	31	.	.	PUNCT
iajs-3112	222	1	int	int	NOUN
iajs-3112	222	2	.	.	PUNCT
iajs-3112	223	1	math	math	PROPN
iajs-3112	223	2	,	,	PUNCT
iajs-3112	223	3	forum	forum	PROPN
iajs-3112	223	4	.	.	PROPN
iajs-3112	223	5	2011	2011	NUM
iajs-3112	223	6	,	,	PUNCT
iajs-3112	223	7	6,63	6,63	NUM
iajs-3112	223	8	,	,	PUNCT
iajs-3112	223	9	3139	3139	NUM
iajs-3112	223	10	-	-	SYM
iajs-3112	223	11	3149	3149	NUM
iajs-3112	223	12	.	.	PUNCT
iajs-3112	224	1	8	8	NUM
iajs-3112	224	2	.	.	X
iajs-3112	225	1	xin	xin	PROPN
iajs-3112	225	2	,	,	PUNCT
iajs-3112	225	3	x.	x.	PROPN
iajs-3112	225	4	l.	l.	PROPN
iajs-3112	225	5	;	;	PUNCT
iajs-3112	225	6	ji	ji	PROPN
iajs-3112	225	7	,	,	PUNCT
iajs-3112	225	8	w.	w.	PROPN
iajs-3112	225	9	;	;	PUNCT
iajs-3112	225	10	hua	hua	PROPN
iajs-3112	225	11	,	,	PUNCT
iajs-3112	225	12	x.	x.	PROPN
iajs-3112	225	13	j	j	PROPN
iajs-3112	225	14	.fuzzy	.fuzzy	PUNCT
iajs-3112	225	15	filter	filter	NOUN
iajs-3112	225	16	spectrum	spectrum	NOUN
iajs-3112	225	17	of	of	ADP
iajs-3112	225	18	a	a	DET
iajs-3112	225	19	bck	bck	NOUN
iajs-3112	225	20	algebra	algebra	NOUN
iajs-3112	225	21	.	.	PUNCT
iajs-3112	226	1	international	international	ADJ
iajs-3112	226	2	journal	journal	PROPN
iajs-3112	226	3	of	of	ADP
iajs-3112	226	4	mathematics	mathematics	PROPN
iajs-3112	226	5	and	and	CCONJ
iajs-3112	226	6	mathematical	mathematical	ADJ
iajs-3112	226	7	sciences	science	NOUN
iajs-3112	226	8	2011	2011	NUM
iajs-3112	226	9	,	,	PUNCT
iajs-3112	226	10	13	13	NUM
iajs-3112	226	11	pages	page	NOUN
iajs-3112	226	12	.	.	PUNCT
iajs-3112	227	1	https://doi.org/10.1155/2011/795934	https://doi.org/10.1155/2011/795934	PROPN
iajs-3112	227	2	9	9	NUM
iajs-3112	227	3	.	.	PUNCT
iajs-3112	228	1	satyanarayana	satyanarayana	PROPN
iajs-3112	228	2	,	,	PUNCT
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iajs-3112	230	6	computer	computer	NOUN
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iajs-3112	230	8	.	.	PUNCT
iajs-3112	231	1	2016	2016	NUM
iajs-3112	231	2	,	,	PUNCT
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iajs-3112	231	4	,	,	PUNCT
iajs-3112	231	5	1	1	NUM
iajs-3112	231	6	-	-	SYM
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iajs-3112	231	10	.	.	X
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iajs-3112	232	7	,	,	PUNCT
iajs-3112	232	8	m.	m.	NOUN
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iajs-3112	232	10	on	on	ADP
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iajs-3112	232	14	group	group	NOUN
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iajs-3112	232	16	.	.	PUNCT
iajs-3112	233	1	ibn	ibn	PROPN
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iajs-3112	233	4	for	for	ADP
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iajs-3112	233	10	,	,	PUNCT
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iajs-3112	233	14	-	-	SYM
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iajs-3112	233	16	.	.	PUNCT
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iajs-3112	234	3	.	.	PUNCT
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iajs-3112	235	2	,	,	PUNCT
iajs-3112	235	3	s.m	s.m	PROPN
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iajs-3112	235	5	;	;	PUNCT
iajs-3112	235	6	abd	abd	PROPN
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iajs-3112	235	8	-	-	PUNCT
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iajs-3112	235	10	,	,	PUNCT
iajs-3112	235	11	o.w	o.w	PROPN
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iajs-3112	235	16	in	in	ADP
iajs-3112	235	17	ku	ku	PROPN
iajs-3112	235	18	-	-	PUNCT
iajs-3112	235	19	algebra	algebra	PROPN
iajs-3112	235	20	.	.	PUNCT
iajs-3112	236	1	journal	journal	NOUN
iajs-3112	236	2	of	of	ADP
iajs-3112	236	3	new	new	ADJ
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iajs-3112	236	5	.	.	PUNCT
iajs-3112	237	1	2018	2018	NUM
iajs-3112	237	2	,	,	PUNCT
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iajs-3112	237	4	,	,	PUNCT
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iajs-3112	237	6	-	-	SYM
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iajs-3112	237	8	.	.	PUNCT
iajs-3112	238	1	12	12	NUM
iajs-3112	238	2	.	.	PUNCT
iajs-3112	239	1	megalai	megalai	PROPN
iajs-3112	239	2	,	,	PUNCT
iajs-3112	239	3	k.	k.	PROPN
iajs-3112	239	4	;	;	PUNCT
iajs-3112	239	5	tamilarasi	tamilarasi	NOUN
iajs-3112	239	6	,	,	PUNCT
iajs-3112	239	7	a.	a.	NOUN
iajs-3112	239	8	fuzzy	fuzzy	ADJ
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iajs-3112	239	10	and	and	CCONJ
iajs-3112	239	11	fuzzy	fuzzy	ADJ
iajs-3112	239	12	t	t	NOUN
iajs-3112	239	13	-	-	PUNCT
iajs-3112	239	14	ideals	ideal	NOUN
iajs-3112	239	15	in	in	ADP
iajs-3112	239	16	tm	tm	NOUN
iajs-3112	239	17	-	-	PUNCT
iajs-3112	239	18	algebras	algebras	PROPN
iajs-3112	239	19	.	.	PUNCT
iajs-3112	240	1	j	j	PROPN
iajs-3112	240	2	math	math	PROPN
iajs-3112	240	3	stat	stat	PROPN
iajs-3112	240	4	.	.	PUNCT
iajs-3112	241	1	2011	2011	NUM
iajs-3112	241	2	,	,	PUNCT
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iajs-3112	241	4	,	,	PUNCT
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iajs-3112	241	6	-	-	SYM
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iajs-3112	241	8	.	.	PUNCT
iajs-3112	242	1	https://doi.org/10.3844/jmssp.2011.107.111	https://doi.org/10.3844/jmssp.2011.107.111	PROPN
iajs-3112	242	2	13	13	NUM
iajs-3112	242	3	.	.	PUNCT
iajs-3112	243	1	lee	lee	PROPN
iajs-3112	243	2	,	,	PUNCT
iajs-3112	243	3	k.m	k.m	PROPN
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iajs-3112	243	6	-	-	PUNCT
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iajs-3112	243	11	their	their	PRON
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iajs-3112	243	13	.	.	PUNCT
iajs-3112	244	1	proc	proc	NOUN
iajs-3112	244	2	.	.	PUNCT
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iajs-3112	245	2	.	.	PUNCT
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iajs-3112	245	4	.	.	PUNCT
iajs-3112	246	1	on	on	ADP
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iajs-3112	246	4	.	.	PUNCT
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iajs-3112	246	6	,	,	PUNCT
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iajs-3112	246	8	-	-	SYM
iajs-3112	246	9	312	312	NUM
iajs-3112	246	10	.	.	PUNCT
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iajs-3112	246	12	.	.	PUNCT
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iajs-3112	246	20	-	-	PUNCT
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iajs-3112	246	24	,	,	PUNCT
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iajs-3112	246	32	.	.	PUNCT
iajs-3112	247	1	j.	j.	PROPN
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iajs-3112	247	6	.	.	PUNCT
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iajs-3112	248	2	,	,	PUNCT
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iajs-3112	248	6	-	-	SYM
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iajs-3112	248	8	.	.	PUNCT
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iajs-3112	249	2	.	.	X
iajs-3112	250	1	yaqoob	yaqoob	NOUN
iajs-3112	250	2	,	,	PUNCT
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iajs-3112	250	6	,	,	PUNCT
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iajs-3112	250	11	𝜆	𝜆	X
iajs-3112	250	12	,	,	PUNCT
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iajs-3112	250	18	.	.	PUNCT
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iajs-3112	252	1	j.	j.	PROPN
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iajs-3112	252	3	.	.	PUNCT
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iajs-3112	253	2	.	.	PUNCT
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iajs-3112	254	2	,	,	PUNCT
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iajs-3112	254	4	-	-	SYM
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iajs-3112	254	6	.	.	PUNCT
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iajs-3112	255	2	.	.	PUNCT
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iajs-3112	268	24	-	-	PUNCT
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iajs-3112	268	26	.	.	PUNCT
iajs-3112	269	1	comput	comput	PROPN
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iajs-3112	270	8	.	.	PUNCT
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iajs-3112	271	3	.	.	PUNCT
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iajs-3112	272	4	;	;	PUNCT
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iajs-3112	272	6	,	,	PUNCT
iajs-3112	272	7	s.m	s.m	PROPN
iajs-3112	272	8	.	.	PROPN
iajs-3112	272	9	;	;	PUNCT
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iajs-3112	272	15	cubic	cubic	PROPN
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iajs-3112	272	17	-	-	PUNCT
iajs-3112	272	18	ideals	ideal	NOUN
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iajs-3112	272	20	ku	ku	PROPN
iajs-3112	272	21	-	-	PUNCT
iajs-3112	272	22	algebras	algebras	PROPN
iajs-3112	272	23	.	.	PUNCT
iajs-3112	273	1	isrn	isrn	PROPN
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iajs-3112	274	7	.	.	PUNCT
iajs-3112	275	1	21	21	NUM
iajs-3112	275	2	.	.	PUNCT
iajs-3112	276	1	hasan	hasan	PROPN
iajs-3112	276	2	,	,	PUNCT
iajs-3112	276	3	e.	e.	PROPN
iajs-3112	276	4	r.	r.	PROPN
iajs-3112	276	5	;	;	PUNCT
iajs-3112	276	6	kareem	kareem	PROPN
iajs-3112	276	7	,	,	PUNCT
iajs-3112	276	8	f.f	f.f	PROPN
iajs-3112	276	9	.	.	PROPN
iajs-3112	276	10	fuzzy	fuzzy	PROPN
iajs-3112	276	11	ku	ku	PROPN
iajs-3112	276	12	-	-	PUNCT
iajs-3112	276	13	semi	semi	NOUN
iajs-3112	276	14	-	-	NOUN
iajs-3112	276	15	groups	group	NOUN
iajs-3112	276	16	and	and	CCONJ
iajs-3112	276	17	investigate	investigate	VERB
iajs-3112	276	18	some	some	DET
iajs-3112	276	19	basic	basic	ADJ
iajs-3112	276	20	properties	property	NOUN
iajs-3112	276	21	.	.	PUNCT
iajs-3112	277	1	journal	journal	NOUN
iajs-3112	277	2	of	of	ADP
iajs-3112	277	3	engineering	engineering	NOUN
iajs-3112	277	4	and	and	CCONJ
iajs-3112	277	5	applied	apply	VERB
iajs-3112	277	6	science	science	NOUN
iajs-3112	277	7	.	.	PUNCT
iajs-3112	278	1	2018,13,18	2018,13,18	NUM
iajs-3112	278	2	,	,	PUNCT
iajs-3112	278	3	7739	7739	NUM
iajs-3112	278	4	-	-	SYM
iajs-3112	278	5	7744	7744	NUM
iajs-3112	278	6	.	.	PUNCT
iajs-3112	279	1	22	22	NUM
iajs-3112	279	2	.	.	PUNCT
iajs-3112	279	3	kareem	kareem	PROPN
iajs-3112	279	4	,	,	PUNCT
iajs-3112	279	5	f.f	f.f	PROPN
iajs-3112	279	6	.	.	PROPN
iajs-3112	279	7	;	;	PUNCT
iajs-3112	279	8	hasan	hasan	PROPN
iajs-3112	279	9	,	,	PUNCT
iajs-3112	279	10	o.a	o.a	PROPN
iajs-3112	279	11	.	.	PROPN
iajs-3112	279	12	cubic	cubic	ADJ
iajs-3112	279	13	ideals	ideal	NOUN
iajs-3112	279	14	of	of	ADP
iajs-3112	279	15	semigroup	semigroup	NOUN
iajs-3112	279	16	in	in	ADP
iajs-3112	279	17	ku	ku	PROPN
iajs-3112	279	18	-	-	PUNCT
iajs-3112	279	19	algebra	algebra	PROPN
iajs-3112	279	20	.	.	PUNCT
iajs-3112	280	1	j.	j.	PROPN
iajs-3112	280	2	phys	phys	PROPN
iajs-3112	280	3	.	.	PUNCT
iajs-3112	280	4	:	:	PUNCT
iajs-3112	281	1	conf	conf	PROPN
iajs-3112	281	2	.	.	PUNCT
iajs-3112	281	3	ser	ser	PROPN
iajs-3112	281	4	.	.	PROPN
iajs-3112	281	5	1804	1804	NUM
iajs-3112	281	6	.	.	PUNCT
iajs-3112	282	1	2021	2021	NUM
iajs-3112	282	2	,	,	PUNCT
iajs-3112	282	3	012018	012018	NUM
iajs-3112	282	4	.	.	PUNCT
iajs-3112	283	1	23	23	NUM
iajs-3112	283	2	.	.	PUNCT
iajs-3112	284	1	senapati	senapati	PROPN
iajs-3112	284	2	,	,	PUNCT
iajs-3112	284	3	t.	t.	PROPN
iajs-3112	284	4	;	;	PUNCT
iajs-3112	284	5	shum	shum	X
iajs-3112	284	6	,	,	PUNCT
iajs-3112	284	7	k.p	k.p	PROPN
iajs-3112	284	8	.	.	PROPN
iajs-3112	284	9	cubic	cubic	ADJ
iajs-3112	284	10	implicative	implicative	ADJ
iajs-3112	284	11	ideals	ideal	NOUN
iajs-3112	284	12	of	of	ADP
iajs-3112	284	13	bck	bck	NOUN
iajs-3112	284	14	-	-	PUNCT
iajs-3112	284	15	algebras	algebras	PROPN
iajs-3112	284	16	.	.	PUNCT
iajs-3112	285	1	missouri	missouri	PROPN
iajs-3112	285	2	j.	j.	PROPN
iajs-3112	285	3	of	of	ADP
iajs-3112	285	4	math	math	PROPN
iajs-3112	285	5	.	.	PUNCT
iajs-3112	286	1	sci	sci	PROPN
iajs-3112	286	2	.	.	PROPN
iajs-3112	286	3	2017	2017	NUM
iajs-3112	286	4	,	,	PUNCT
iajs-3112	286	5	29	29	NUM
iajs-3112	286	6	,	,	PUNCT
iajs-3112	286	7	2	2	NUM
iajs-3112	286	8	,	,	PUNCT
iajs-3112	286	9	125	125	NUM
iajs-3112	286	10	-	-	SYM
iajs-3112	286	11	138	138	NUM
iajs-3112	286	12	.	.	PUNCT
iajs-3112	286	13	24	24	NUM
iajs-3112	286	14	.	.	PUNCT
iajs-3112	287	1	senapati	senapati	PROPN
iajs-3112	287	2	,	,	PUNCT
iajs-3112	287	3	t.	t.	PROPN
iajs-3112	287	4	;	;	PUNCT
iajs-3112	287	5	jun	jun	PROPN
iajs-3112	287	6	,	,	PUNCT
iajs-3112	287	7	y.	y.	PROPN
iajs-3112	287	8	b.	b.	PROPN
iajs-3112	287	9	;	;	PUNCT
iajs-3112	287	10	shum	shum	PROPN
iajs-3112	287	11	,	,	PUNCT
iajs-3112	287	12	k.p	k.p	PROPN
iajs-3112	287	13	.	.	PROPN
iajs-3112	287	14	cubic	cubic	PROPN
iajs-3112	287	15	intuitionistic	intuitionistic	ADJ
iajs-3112	287	16	implicative	implicative	ADJ
iajs-3112	287	17	ideals	ideal	NOUN
iajs-3112	287	18	of	of	ADP
iajs-3112	287	19	bck	bck	NOUN
iajs-3112	287	20	-	-	PUNCT
iajs-3112	287	21	algebras	algebras	PROPN
iajs-3112	287	22	.	.	PUNCT
iajs-3112	288	1	proc	proc	PROPN
iajs-3112	288	2	.	.	PUNCT
iajs-3112	289	1	natl	natl	PROPN
iajs-3112	289	2	.	.	PUNCT
iajs-3112	290	1	acad	acad	PROPN
iajs-3112	290	2	.	.	PUNCT
iajs-3112	291	1	sci	sci	PROPN
iajs-3112	291	2	.	.	PROPN
iajs-3112	291	3	,	,	PUNCT
iajs-3112	291	4	india	india	PROPN
iajs-3112	291	5	,	,	PUNCT
iajs-3112	291	6	sect	sect	NOUN
iajs-3112	291	7	.	.	PUNCT
iajs-3112	292	1	a	a	DET
iajs-3112	292	2	phys	phy	NOUN
iajs-3112	292	3	.	.	PUNCT
iajs-3112	293	1	sci	sci	PROPN
iajs-3112	293	2	.	.	PROPN
iajs-3112	293	3	2021	2021	NUM
iajs-3112	293	4	,	,	PUNCT
iajs-3112	293	5	91	91	NUM
iajs-3112	293	6	,	,	PUNCT
iajs-3112	293	7	2	2	NUM
iajs-3112	293	8	,	,	PUNCT
iajs-3112	293	9	273	273	NUM
iajs-3112	293	10	-	-	SYM
iajs-3112	293	11	282	282	NUM
iajs-3112	293	12	.	.	PUNCT
iajs-3112	294	1	25	25	NUM
iajs-3112	294	2	.	.	PUNCT
iajs-3112	295	1	jun	jun	PROPN
iajs-3112	295	2	,	,	PUNCT
iajs-3112	295	3	y.	y.	PROPN
iajs-3112	295	4	b.	b.	PROPN
iajs-3112	295	5	;	;	PUNCT
iajs-3112	295	6	kim	kim	PROPN
iajs-3112	295	7	,	,	PUNCT
iajs-3112	295	8	c.	c.	PROPN
iajs-3112	295	9	s.	s.	PROPN
iajs-3112	295	10	;	;	PUNCT
iajs-3112	295	11	kang	kang	PROPN
iajs-3112	295	12	;	;	PUNCT
iajs-3112	295	13	kang	kang	PROPN
iajs-3112	295	14	,	,	PUNCT
iajs-3112	295	15	j.	j.	PROPN
iajs-3112	295	16	g.	g.	PROPN
iajs-3112	295	17	cubic	cubic	PROPN
iajs-3112	296	1	q	q	PROPN
iajs-3112	296	2	-ideals	-ideal	NOUN
iajs-3112	296	3	of	of	ADP
iajs-3112	296	4	bci	bci	NOUN
iajs-3112	296	5	-	-	PUNCT
iajs-3112	296	6	algebras	algebra	NOUN
iajs-3112	296	7	.	.	PUNCT
iajs-3112	297	1	annals	annal	NOUN
iajs-3112	297	2	of	of	ADP
iajs-3112	297	3	fuzzy	fuzzy	ADJ
iajs-3112	297	4	mathematics	mathematic	NOUN
iajs-3112	297	5	and	and	CCONJ
iajs-3112	297	6	informatics	informatic	NOUN
iajs-3112	297	7	.	.	PUNCT
iajs-3112	298	1	2011	2011	NUM
iajs-3112	298	2	,	,	PUNCT
iajs-3112	298	3	1	1	NUM
iajs-3112	298	4	,	,	PUNCT
iajs-3112	298	5	1	1	NUM
iajs-3112	298	6	,	,	PUNCT
iajs-3112	298	7	2534	2534	NUM
iajs-3112	298	8	.	.	PUNCT
iajs-3112	299	1	26	26	NUM
iajs-3112	299	2	.	.	X
iajs-3112	299	3	akram	akram	PROPN
iajs-3112	299	4	,	,	PUNCT
iajs-3112	299	5	m.	m.	NOUN
iajs-3112	299	6	;	;	PUNCT
iajs-3112	299	7	yaqoob	yaqoob	NOUN
iajs-3112	299	8	,	,	PUNCT
iajs-3112	299	9	n.	n.	PROPN
iajs-3112	299	10	;	;	PUNCT
iajs-3112	299	11	gulistan	gulistan	PROPN
iajs-3112	299	12	,	,	PUNCT
iajs-3112	299	13	m.	m.	NOUN
iajs-3112	299	14	cubic	cubic	PROPN
iajs-3112	299	15	ku	ku	PROPN
iajs-3112	299	16	-	-	PUNCT
iajs-3112	299	17	subalgebras	subalgebras	PROPN
iajs-3112	299	18	.	.	PUNCT
iajs-3112	300	1	int	int	NOUN
iajs-3112	300	2	.	.	PUNCT
iajs-3112	301	1	j.	j.	PROPN
iajs-3112	301	2	pure	pure	PROPN
iajs-3112	301	3	appl	appl	PROPN
iajs-3112	301	4	.	.	PUNCT
iajs-3112	301	5	math	math	PROPN
iajs-3112	301	6	.	.	PUNCT
iajs-3112	302	1	2013	2013	NUM
iajs-3112	302	2	,	,	PUNCT
iajs-3112	302	3	89	89	NUM
iajs-3112	302	4	,	,	PUNCT
iajs-3112	302	5	5	5	NUM
iajs-3112	302	6	,	,	PUNCT
iajs-3112	302	7	659665	659665	NUM
iajs-3112	302	8	.	.	PUNCT
iajs-3112	303	1	27	27	NUM
iajs-3112	303	2	.	.	X
iajs-3112	304	1	muhiuddin	muhiuddin	PROPN
iajs-3112	304	2	,	,	PUNCT
iajs-3112	304	3	g.	g.	PROPN
iajs-3112	304	4	;	;	PUNCT
iajs-3112	304	5	al	al	PROPN
iajs-3112	304	6	-	-	PUNCT
iajs-3112	304	7	roqi	roqi	ADV
iajs-3112	304	8	,	,	PUNCT
iajs-3112	304	9	a.	a.	NOUN
iajs-3112	304	10	m.	m.	NOUN
iajs-3112	304	11	cubic	cubic	ADJ
iajs-3112	304	12	soft	soft	ADJ
iajs-3112	304	13	sets	set	NOUN
iajs-3112	304	14	with	with	ADP
iajs-3112	304	15	applications	application	NOUN
iajs-3112	304	16	in	in	ADP
iajs-3112	304	17	bck/	bck/	NUM
iajs-3112	304	18	bci	bci	NOUN
iajs-3112	304	19	-	-	PUNCT
iajs-3112	304	20	algebras	algebra	NOUN
iajs-3112	304	21	.	.	PUNCT
iajs-3112	305	1	annals	annal	NOUN
iajs-3112	305	2	of	of	ADP
iajs-3112	305	3	fuzzy	fuzzy	ADJ
iajs-3112	305	4	mathematics	mathematic	NOUN
iajs-3112	305	5	and	and	CCONJ
iajs-3112	305	6	informatics	informatic	NOUN
iajs-3112	305	7	.	.	PUNCT
iajs-3112	306	1	2014	2014	NUM
iajs-3112	306	2	,	,	PUNCT
iajs-3112	306	3	8	8	NUM
iajs-3112	306	4	,	,	PUNCT
iajs-3112	306	5	291	291	NUM
iajs-3112	306	6	-	-	SYM
iajs-3112	306	7	304	304	NUM
iajs-3112	306	8	.	.	PUNCT
iajs-3112	307	1	https://doi.org/10.1155/2011/795934	https://doi.org/10.1155/2011/795934	PROPN
iajs-3112	307	2	https://jih.uobaghdad.edu.iq/index.php/j/article/view/641	https://jih.uobaghdad.edu.iq/index.php/j/article/view/641	PROPN
iajs-3112	307	3	https://doi.org/10.3844/jmssp.2011.107.111	https://doi.org/10.3844/jmssp.2011.107.111	PROPN
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iajs-3112	307	5	ihjpas	ihjpas	PROPN
iajs-3112	307	6	.	.	PUNCT
iajs-3112	308	1	37	37	NUM
iajs-3112	308	2	(	(	PUNCT
iajs-3112	308	3	1	1	NUM
iajs-3112	308	4	)	)	PUNCT
iajs-3112	308	5	2024	2024	NUM
iajs-3112	308	6	463	463	NUM
iajs-3112	308	7	28	28	NUM
iajs-3112	308	8	.	.	PUNCT
iajs-3112	309	1	janaa	janaa	PROPN
iajs-3112	309	2	,	,	PUNCT
iajs-3112	309	3	c.	c.	PROPN
iajs-3112	309	4	;	;	PUNCT
iajs-3112	309	5	senapati	senapati	PROPN
iajs-3112	309	6	,	,	PUNCT
iajs-3112	309	7	t.	t.	PROPN
iajs-3112	309	8	cubic	cubic	PROPN
iajs-3112	309	9	g	g	PROPN
iajs-3112	309	10	-	-	PUNCT
iajs-3112	309	11	subalgebras	subalgebras	NOUN
iajs-3112	309	12	of	of	ADP
iajs-3112	309	13	g	g	NOUN
iajs-3112	309	14	-	-	PUNCT
iajs-3112	309	15	algebras	algebras	NOUN
iajs-3112	309	16	.	.	PUNCT
iajs-3112	310	1	annals	annal	NOUN
iajs-3112	310	2	of	of	ADP
iajs-3112	310	3	pure	pure	ADJ
iajs-3112	310	4	and	and	CCONJ
iajs-3112	310	5	applied	applied	ADJ
iajs-3112	310	6	mathematics	mathematic	NOUN
iajs-3112	310	7	.	.	PUNCT
iajs-3112	311	1	2015	2015	NUM
iajs-3112	311	2	,	,	PUNCT
iajs-3112	311	3	10	10	NUM
iajs-3112	311	4	,	,	PUNCT
iajs-3112	311	5	1	1	NUM
iajs-3112	311	6	,	,	PUNCT
iajs-3112	311	7	105	105	NUM
iajs-3112	311	8	-	-	SYM
iajs-3112	311	9	115	115	NUM
iajs-3112	311	10	.	.	PUNCT
iajs-3112	312	1	29	29	NUM
iajs-3112	312	2	.	.	X
iajs-3112	313	1	muhiuddin	muhiuddin	PROPN
iajs-3112	313	2	,	,	PUNCT
iajs-3112	313	3	g.	g.	PROPN
iajs-3112	313	4	;	;	PUNCT
iajs-3112	313	5	ahn	ahn	PROPN
iajs-3112	313	6	,	,	PUNCT
iajs-3112	313	7	s.	s.	PROPN
iajs-3112	313	8	s.	s.	PROPN
iajs-3112	313	9	;	;	PUNCT
iajs-3112	313	10	kim	kim	PROPN
iajs-3112	313	11	,	,	PUNCT
iajs-3112	313	12	c.	c.	PROPN
iajs-3112	313	13	s.	s.	PROPN
iajs-3112	313	14	;	;	PUNCT
iajs-3112	313	15	jun	jun	PROPN
iajs-3112	313	16	,	,	PUNCT
iajs-3112	313	17	y	y	PROPN
iajs-3112	313	18	.b	.b	PROPN
iajs-3112	313	19	.	.	PUNCT
iajs-3112	314	1	stable	stable	ADJ
iajs-3112	314	2	cubic	cubic	ADJ
iajs-3112	314	3	sets	set	NOUN
iajs-3112	314	4	.	.	PUNCT
iajs-3112	315	1	journal	journal	NOUN
iajs-3112	315	2	of	of	ADP
iajs-3112	315	3	computational	computational	ADJ
iajs-3112	315	4	analysis	analysis	NOUN
iajs-3112	315	5	and	and	CCONJ
iajs-3112	315	6	applications	application	NOUN
iajs-3112	315	7	.	.	PUNCT
iajs-3112	316	1	2017	2017	NUM
iajs-3112	316	2	,	,	PUNCT
iajs-3112	316	3	23,5	23,5	NOUN
iajs-3112	316	4	,	,	PUNCT
iajs-3112	316	5	802–819	802–819	NUM
iajs-3112	316	6	.	.	PUNCT
iajs-3112	317	1	30	30	NUM
iajs-3112	317	2	.	.	PUNCT
iajs-3112	317	3	lee	lee	PROPN
iajs-3112	317	4	,	,	PUNCT
iajs-3112	317	5	j.	j.	PROPN
iajs-3112	317	6	g.	g.	PROPN
iajs-3112	317	7	;	;	PUNCT
iajs-3112	317	8	hur	hur	PROPN
iajs-3112	317	9	,	,	PUNCT
iajs-3112	317	10	k.	k.	PROPN
iajs-3112	317	11	;	;	PUNCT
iajs-3112	317	12	mostafa	mostafa	PROPN
iajs-3112	317	13	,	,	PUNCT
iajs-3112	317	14	s.	s.	PROPN
iajs-3112	317	15	m.	m.	PROPN
iajs-3112	317	16	cubic	cubic	ADJ
iajs-3112	317	17	bipolar	bipolar	ADJ
iajs-3112	317	18	structures	structure	NOUN
iajs-3112	317	19	of	of	ADP
iajs-3112	317	20	bcc	bcc	PROPN
iajs-3112	317	21	-	-	PUNCT
iajs-3112	317	22	ideal	ideal	NOUN
iajs-3112	317	23	on	on	ADP
iajs-3112	317	24	bcc	bcc	PROPN
iajs-3112	317	25	-	-	PUNCT
iajs-3112	317	26	algebras	algebras	PROPN
iajs-3112	317	27	.	.	PUNCT
iajs-3112	318	1	annals	annal	NOUN
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iajs-3112	318	5	and	and	CCONJ
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iajs-3112	318	7	,	,	PUNCT
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iajs-3112	318	9	,	,	PUNCT
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iajs-3112	318	11	-	-	SYM
iajs-3112	318	12	103	103	NUM
iajs-3112	318	13	.	.	PUNCT