id	sid	tid	token	lemma	pos
ejpam-4788	1	1	european	european	PROPN
ejpam-4788	1	2	journal	journal	PROPN
ejpam-4788	1	3	of	of	ADP
ejpam-4788	1	4	pure	pure	ADJ
ejpam-4788	1	5	and	and	CCONJ
ejpam-4788	1	6	applied	apply	VERB
ejpam-4788	1	7	mathematics	mathematic	NOUN
ejpam-4788	1	8	vol	vol	NOUN
ejpam-4788	1	9	.	.	PUNCT
ejpam-4788	2	1	16	16	NUM
ejpam-4788	2	2	,	,	PUNCT
ejpam-4788	2	3	no	no	INTJ
ejpam-4788	2	4	.	.	NOUN
ejpam-4788	2	5	3	3	NUM
ejpam-4788	2	6	,	,	PUNCT
ejpam-4788	2	7	2023	2023	NUM
ejpam-4788	2	8	,	,	PUNCT
ejpam-4788	2	9	1592	1592	NUM
ejpam-4788	2	10	-	-	SYM
ejpam-4788	2	11	1607	1607	NUM
ejpam-4788	3	1	issn	issn	PROPN
ejpam-4788	3	2	1307	1307	NUM
ejpam-4788	3	3	-	-	SYM
ejpam-4788	3	4	5543	5543	NUM
ejpam-4788	3	5	–	–	PUNCT
ejpam-4788	3	6	ejpam.com	ejpam.com	X
ejpam-4788	3	7	published	publish	VERB
ejpam-4788	3	8	by	by	ADP
ejpam-4788	3	9	new	new	PROPN
ejpam-4788	3	10	york	york	PROPN
ejpam-4788	3	11	business	business	PROPN
ejpam-4788	3	12	global	global	ADJ
ejpam-4788	3	13	application	application	NOUN
ejpam-4788	3	14	of	of	ADP
ejpam-4788	3	15	bipolar	bipolar	ADJ
ejpam-4788	3	16	fuzzy	fuzzy	ADJ
ejpam-4788	3	17	set	set	VERB
ejpam-4788	3	18	to	to	ADP
ejpam-4788	3	19	a	a	DET
ejpam-4788	3	20	novel	novel	NOUN
ejpam-4788	3	21	of	of	ADP
ejpam-4788	3	22	fuzzy	fuzzy	ADJ
ejpam-4788	3	23	ideal	ideal	NOUN
ejpam-4788	3	24	in	in	ADP
ejpam-4788	3	25	γ	γ	NOUN
ejpam-4788	3	26	-	-	PUNCT
ejpam-4788	3	27	semigroups	semigroup	NOUN
ejpam-4788	3	28	p.	p.	PROPN
ejpam-4788	3	29	khamrot1	khamrot1	PROPN
ejpam-4788	3	30	,	,	PUNCT
ejpam-4788	3	31	t.	t.	PROPN
ejpam-4788	3	32	gaketem2,∗	gaketem2,∗	PROPN
ejpam-4788	3	33	1	1	NUM
ejpam-4788	3	34	department	department	NOUN
ejpam-4788	3	35	of	of	ADP
ejpam-4788	3	36	mathematics	mathematic	NOUN
ejpam-4788	3	37	,	,	PUNCT
ejpam-4788	3	38	faculty	faculty	NOUN
ejpam-4788	3	39	of	of	ADP
ejpam-4788	3	40	science	science	NOUN
ejpam-4788	3	41	and	and	CCONJ
ejpam-4788	3	42	agricultural	agricultural	ADJ
ejpam-4788	3	43	technology	technology	NOUN
ejpam-4788	3	44	,	,	PUNCT
ejpam-4788	3	45	rajamangala	rajamangala	PROPN
ejpam-4788	3	46	university	university	PROPN
ejpam-4788	3	47	of	of	ADP
ejpam-4788	3	48	technology	technology	PROPN
ejpam-4788	3	49	lanna	lanna	PROPN
ejpam-4788	3	50	of	of	ADP
ejpam-4788	3	51	phitsanulok	phitsanulok	PROPN
ejpam-4788	3	52	,	,	PUNCT
ejpam-4788	3	53	phitsanulok	phitsanulok	PROPN
ejpam-4788	3	54	,	,	PUNCT
ejpam-4788	3	55	thailand	thailand	PROPN
ejpam-4788	3	56	2	2	NUM
ejpam-4788	3	57	fuzzy	fuzzy	ADJ
ejpam-4788	3	58	algebras	algebra	NOUN
ejpam-4788	3	59	and	and	CCONJ
ejpam-4788	3	60	decision	decision	NOUN
ejpam-4788	3	61	-	-	PUNCT
ejpam-4788	3	62	making	make	VERB
ejpam-4788	3	63	problems	problem	NOUN
ejpam-4788	3	64	research	research	NOUN
ejpam-4788	3	65	unit	unit	NOUN
ejpam-4788	3	66	,	,	PUNCT
ejpam-4788	3	67	department	department	NOUN
ejpam-4788	3	68	of	of	ADP
ejpam-4788	3	69	mathematics	mathematic	NOUN
ejpam-4788	3	70	,	,	PUNCT
ejpam-4788	3	71	school	school	NOUN
ejpam-4788	3	72	of	of	ADP
ejpam-4788	3	73	science	science	NOUN
ejpam-4788	3	74	,	,	PUNCT
ejpam-4788	3	75	university	university	NOUN
ejpam-4788	3	76	of	of	ADP
ejpam-4788	3	77	phayao	phayao	NOUN
ejpam-4788	3	78	,	,	PUNCT
ejpam-4788	3	79	phayao	phayao	NOUN
ejpam-4788	3	80	56000	56000	NUM
ejpam-4788	3	81	,	,	PUNCT
ejpam-4788	3	82	thailand	thailand	PROPN
ejpam-4788	3	83	abstract	abstract	NOUN
ejpam-4788	3	84	.	.	PUNCT
ejpam-4788	4	1	in	in	ADP
ejpam-4788	4	2	this	this	DET
ejpam-4788	4	3	paper	paper	NOUN
ejpam-4788	4	4	we	we	PRON
ejpam-4788	4	5	define	define	VERB
ejpam-4788	4	6	new	new	ADJ
ejpam-4788	4	7	types	type	NOUN
ejpam-4788	4	8	of	of	ADP
ejpam-4788	4	9	bipolar	bipolar	ADJ
ejpam-4788	4	10	fuzzy	fuzzy	ADJ
ejpam-4788	4	11	ideals	ideal	NOUN
ejpam-4788	4	12	,	,	PUNCT
ejpam-4788	4	13	bipolar	bipolar	ADJ
ejpam-4788	4	14	fuzzy	fuzzy	ADJ
ejpam-4788	4	15	almost	almost	ADV
ejpam-4788	4	16	ideals	ideal	NOUN
ejpam-4788	4	17	in	in	ADP
ejpam-4788	4	18	γ	γ	NOUN
ejpam-4788	4	19	-	-	PUNCT
ejpam-4788	4	20	semigroups	semigroup	NOUN
ejpam-4788	4	21	.	.	PUNCT
ejpam-4788	5	1	we	we	PRON
ejpam-4788	5	2	discussed	discuss	VERB
ejpam-4788	5	3	properties	property	NOUN
ejpam-4788	5	4	of	of	ADP
ejpam-4788	5	5	bipolar	bipolar	ADJ
ejpam-4788	5	6	fuzzy	fuzzy	ADJ
ejpam-4788	5	7	ideals	ideal	NOUN
ejpam-4788	5	8	,	,	PUNCT
ejpam-4788	5	9	bipolar	bipolar	ADJ
ejpam-4788	5	10	fuzzy	fuzzy	ADJ
ejpam-4788	5	11	almost	almost	ADV
ejpam-4788	5	12	ideals	ideal	NOUN
ejpam-4788	5	13	,	,	PUNCT
ejpam-4788	5	14	minimal	minimal	ADJ
ejpam-4788	5	15	bipolar	bipolar	ADJ
ejpam-4788	5	16	fuzzy	fuzzy	ADJ
ejpam-4788	5	17	ideal	ideal	NOUN
ejpam-4788	5	18	,	,	PUNCT
ejpam-4788	5	19	minimal	minimal	ADJ
ejpam-4788	5	20	bipolar	bipolar	ADJ
ejpam-4788	5	21	almost	almost	ADV
ejpam-4788	5	22	ideal	ideal	ADJ
ejpam-4788	5	23	in	in	ADP
ejpam-4788	5	24	γ	γ	NOUN
ejpam-4788	5	25	-	-	PUNCT
ejpam-4788	5	26	semigroups	semigroup	NOUN
ejpam-4788	5	27	.	.	PUNCT
ejpam-4788	6	1	moreover	moreover	ADV
ejpam-4788	6	2	,	,	PUNCT
ejpam-4788	6	3	we	we	PRON
ejpam-4788	6	4	prove	prove	VERB
ejpam-4788	6	5	connection	connection	NOUN
ejpam-4788	6	6	between	between	ADP
ejpam-4788	6	7	almost	almost	ADV
ejpam-4788	6	8	ideals	ideal	NOUN
ejpam-4788	6	9	and	and	CCONJ
ejpam-4788	6	10	bipolar	bipolar	ADJ
ejpam-4788	6	11	fuzzy	fuzzy	ADJ
ejpam-4788	6	12	almost	almost	ADV
ejpam-4788	6	13	ideals	ideal	NOUN
ejpam-4788	6	14	of	of	ADP
ejpam-4788	6	15	γ	γ	NOUN
ejpam-4788	6	16	-	-	PUNCT
ejpam-4788	6	17	semigroups	semigroup	NOUN
ejpam-4788	6	18	.	.	PUNCT
ejpam-4788	7	1	2020	2020	NUM
ejpam-4788	7	2	mathematics	mathematic	NOUN
ejpam-4788	7	3	subject	subject	NOUN
ejpam-4788	7	4	classifications	classification	NOUN
ejpam-4788	7	5	:	:	PUNCT
ejpam-4788	7	6	20m12	20m12	NUM
ejpam-4788	7	7	,	,	PUNCT
ejpam-4788	7	8	06f05	06f05	PRON
ejpam-4788	7	9	key	key	ADJ
ejpam-4788	7	10	words	word	NOUN
ejpam-4788	7	11	and	and	CCONJ
ejpam-4788	7	12	phrases	phrase	NOUN
ejpam-4788	7	13	:	:	PUNCT
ejpam-4788	7	14	bipolar	bipolar	ADJ
ejpam-4788	7	15	fuzzy	fuzzy	ADJ
ejpam-4788	7	16	ideals	ideal	NOUN
ejpam-4788	7	17	,	,	PUNCT
ejpam-4788	7	18	bipolar	bipolar	ADJ
ejpam-4788	7	19	fuzzy	fuzzy	ADJ
ejpam-4788	7	20	almost	almost	ADV
ejpam-4788	7	21	ideals	ideal	NOUN
ejpam-4788	7	22	,	,	PUNCT
ejpam-4788	7	23	minimal	minimal	ADJ
ejpam-4788	7	24	bipolar	bipolar	ADJ
ejpam-4788	7	25	fuzzy	fuzzy	ADJ
ejpam-4788	7	26	ideals	ideal	NOUN
ejpam-4788	7	27	,	,	PUNCT
ejpam-4788	7	28	minimal	minimal	ADJ
ejpam-4788	7	29	biploar	biploar	NOUN
ejpam-4788	7	30	fuzzy	fuzzy	ADJ
ejpam-4788	7	31	almost	almost	ADV
ejpam-4788	7	32	ideals	ideal	VERB
ejpam-4788	7	33	1	1	NUM
ejpam-4788	7	34	.	.	PUNCT
ejpam-4788	8	1	introduction	introduction	NOUN
ejpam-4788	8	2	as	as	ADP
ejpam-4788	8	3	a	a	DET
ejpam-4788	8	4	theory	theory	NOUN
ejpam-4788	8	5	that	that	SCONJ
ejpam-4788	8	6	deals	deal	VERB
ejpam-4788	8	7	with	with	ADP
ejpam-4788	8	8	uncertainty	uncertainty	NOUN
ejpam-4788	8	9	,	,	PUNCT
ejpam-4788	8	10	the	the	DET
ejpam-4788	8	11	fuzzy	fuzzy	ADJ
ejpam-4788	8	12	set	set	NOUN
ejpam-4788	8	13	theory	theory	NOUN
ejpam-4788	8	14	was	be	AUX
ejpam-4788	8	15	studied	study	VERB
ejpam-4788	8	16	by	by	ADP
ejpam-4788	8	17	zadeh	zadeh	PROPN
ejpam-4788	8	18	in	in	ADP
ejpam-4788	8	19	1965	1965	NUM
ejpam-4788	9	1	[	[	X
ejpam-4788	9	2	12	12	NUM
ejpam-4788	9	3	]	]	PUNCT
ejpam-4788	9	4	.	.	PUNCT
ejpam-4788	10	1	it	it	PRON
ejpam-4788	10	2	has	have	AUX
ejpam-4788	10	3	been	be	AUX
ejpam-4788	10	4	applied	apply	VERB
ejpam-4788	10	5	to	to	ADP
ejpam-4788	10	6	many	many	ADJ
ejpam-4788	10	7	areas	area	NOUN
ejpam-4788	10	8	,	,	PUNCT
ejpam-4788	10	9	such	such	ADJ
ejpam-4788	10	10	as	as	ADP
ejpam-4788	10	11	medical	medical	ADJ
ejpam-4788	10	12	science	science	NOUN
ejpam-4788	10	13	,	,	PUNCT
ejpam-4788	10	14	robotics	robotic	NOUN
ejpam-4788	10	15	,	,	PUNCT
ejpam-4788	10	16	computer	computer	NOUN
ejpam-4788	10	17	science	science	NOUN
ejpam-4788	10	18	,	,	PUNCT
ejpam-4788	10	19	information	information	NOUN
ejpam-4788	10	20	science	science	NOUN
ejpam-4788	10	21	,	,	PUNCT
ejpam-4788	10	22	control	control	NOUN
ejpam-4788	10	23	engineering	engineering	NOUN
ejpam-4788	10	24	,	,	PUNCT
ejpam-4788	10	25	measure	measure	NOUN
ejpam-4788	10	26	theory	theory	NOUN
ejpam-4788	10	27	,	,	PUNCT
ejpam-4788	10	28	logic	logic	NOUN
ejpam-4788	10	29	,	,	PUNCT
ejpam-4788	10	30	set	set	ADJ
ejpam-4788	10	31	theory	theory	NOUN
ejpam-4788	10	32	,	,	PUNCT
ejpam-4788	10	33	topology	topology	NOUN
ejpam-4788	10	34	and	and	CCONJ
ejpam-4788	10	35	others	other	NOUN
ejpam-4788	10	36	.	.	PUNCT
ejpam-4788	11	1	in	in	ADP
ejpam-4788	11	2	1994	1994	NUM
ejpam-4788	11	3	,	,	PUNCT
ejpam-4788	11	4	zhang	zhang	PROPN
ejpam-4788	12	1	[	[	X
ejpam-4788	12	2	13	13	NUM
ejpam-4788	12	3	]	]	PUNCT
ejpam-4788	12	4	introduced	introduce	VERB
ejpam-4788	12	5	the	the	DET
ejpam-4788	12	6	concept	concept	NOUN
ejpam-4788	12	7	of	of	ADP
ejpam-4788	12	8	bipolar	bipolar	ADJ
ejpam-4788	12	9	fuzzy	fuzzy	ADJ
ejpam-4788	12	10	sets	set	NOUN
ejpam-4788	12	11	.	.	PUNCT
ejpam-4788	13	1	in	in	ADP
ejpam-4788	13	2	2000	2000	NUM
ejpam-4788	13	3	,	,	PUNCT
ejpam-4788	13	4	lee	lee	PROPN
ejpam-4788	14	1	[	[	X
ejpam-4788	14	2	8	8	NUM
ejpam-4788	14	3	]	]	PUNCT
ejpam-4788	14	4	extended	extend	VERB
ejpam-4788	14	5	a	a	DET
ejpam-4788	14	6	fuzzy	fuzzy	ADJ
ejpam-4788	14	7	set	set	NOUN
ejpam-4788	14	8	to	to	ADP
ejpam-4788	14	9	theory	theory	NOUN
ejpam-4788	14	10	of	of	ADP
ejpam-4788	14	11	a	a	DET
ejpam-4788	14	12	bipolar	bipolar	ADJ
ejpam-4788	14	13	fuzzy	fuzzy	NOUN
ejpam-4788	14	14	set	set	VERB
ejpam-4788	14	15	whose	whose	DET
ejpam-4788	14	16	function	function	NOUN
ejpam-4788	14	17	ranges	range	VERB
ejpam-4788	14	18	from	from	ADP
ejpam-4788	14	19	the	the	DET
ejpam-4788	14	20	interval	interval	NOUN
ejpam-4788	14	21	[	[	X
ejpam-4788	14	22	−1	−1	NOUN
ejpam-4788	14	23	,	,	PUNCT
ejpam-4788	14	24	0	0	NUM
ejpam-4788	14	25	]	]	PUNCT
ejpam-4788	14	26	∪	∪	ADP
ejpam-4788	14	27	[	[	X
ejpam-4788	14	28	0	0	NUM
ejpam-4788	14	29	,	,	PUNCT
ejpam-4788	14	30	1	1	NUM
ejpam-4788	14	31	]	]	PUNCT
ejpam-4788	14	32	.	.	PUNCT
ejpam-4788	15	1	as	as	SCONJ
ejpam-4788	15	2	application	application	NOUN
ejpam-4788	15	3	of	of	ADP
ejpam-4788	15	4	the	the	DET
ejpam-4788	15	5	bipolr	bipolr	NOUN
ejpam-4788	15	6	set	set	NOUN
ejpam-4788	15	7	theory	theory	NOUN
ejpam-4788	15	8	affects	affect	VERB
ejpam-4788	15	9	and	and	CCONJ
ejpam-4788	15	10	effectiveness	effectiveness	NOUN
ejpam-4788	15	11	and	and	CCONJ
ejpam-4788	15	12	efficiency	efficiency	NOUN
ejpam-4788	15	13	of	of	ADP
ejpam-4788	15	14	decision	decision	NOUN
ejpam-4788	15	15	making	making	NOUN
ejpam-4788	15	16	.	.	PUNCT
ejpam-4788	16	1	therefore	therefore	ADV
ejpam-4788	16	2	,	,	PUNCT
ejpam-4788	16	3	it	it	PRON
ejpam-4788	16	4	is	be	AUX
ejpam-4788	16	5	used	use	VERB
ejpam-4788	16	6	to	to	PART
ejpam-4788	16	7	solve	solve	VERB
ejpam-4788	16	8	problem	problem	NOUN
ejpam-4788	16	9	related	relate	VERB
ejpam-4788	16	10	to	to	ADP
ejpam-4788	16	11	decision	decision	NOUN
ejpam-4788	16	12	-	-	PUNCT
ejpam-4788	16	13	making	making	NOUN
ejpam-4788	16	14	,	,	PUNCT
ejpam-4788	16	15	organization	organization	NOUN
ejpam-4788	16	16	problems	problem	NOUN
ejpam-4788	16	17	,	,	PUNCT
ejpam-4788	16	18	economic	economic	ADJ
ejpam-4788	16	19	problems	problem	NOUN
ejpam-4788	16	20	,	,	PUNCT
ejpam-4788	16	21	and	and	CCONJ
ejpam-4788	16	22	evaluation	evaluation	NOUN
ejpam-4788	16	23	,	,	PUNCT
ejpam-4788	16	24	risk	risk	NOUN
ejpam-4788	16	25	management	management	NOUN
ejpam-4788	16	26	,	,	PUNCT
ejpam-4788	16	27	environmental	environmental	ADJ
ejpam-4788	16	28	and	and	CCONJ
ejpam-4788	16	29	social	social	ADJ
ejpam-4788	16	30	impact	impact	NOUN
ejpam-4788	16	31	assessments	assessment	NOUN
ejpam-4788	16	32	.	.	PUNCT
ejpam-4788	17	1	later	later	ADV
ejpam-4788	17	2	in	in	ADP
ejpam-4788	17	3	2012	2012	NUM
ejpam-4788	17	4	,	,	PUNCT
ejpam-4788	17	5	s.k	s.k	PROPN
ejpam-4788	17	6	.	.	PROPN
ejpam-4788	17	7	majumder	majumder	NOUN
ejpam-4788	18	1	[	[	X
ejpam-4788	18	2	9	9	NUM
ejpam-4788	18	3	]	]	PUNCT
ejpam-4788	18	4	studied	study	VERB
ejpam-4788	18	5	the	the	DET
ejpam-4788	18	6	bipolar	bipolar	ADJ
ejpam-4788	18	7	fuzzy	fuzzy	ADJ
ejpam-4788	18	8	set	set	NOUN
ejpam-4788	18	9	in	in	ADP
ejpam-4788	18	10	γ	γ	NOUN
ejpam-4788	18	11	-	-	PUNCT
ejpam-4788	18	12	semigroups	semigroup	NOUN
ejpam-4788	18	13	and	and	CCONJ
ejpam-4788	18	14	integration	integration	NOUN
ejpam-4788	18	15	properties	property	NOUN
ejpam-4788	18	16	of	of	ADP
ejpam-4788	18	17	bipolar	bipolar	ADJ
ejpam-4788	18	18	fuzzy	fuzzy	ADJ
ejpam-4788	18	19	ideals	ideal	NOUN
ejpam-4788	18	20	in	in	ADP
ejpam-4788	18	21	γ	γ	NOUN
ejpam-4788	18	22	-	-	PUNCT
ejpam-4788	18	23	semigroups	semigroup	NOUN
ejpam-4788	18	24	.	.	PUNCT
ejpam-4788	19	1	the	the	DET
ejpam-4788	19	2	ideal	ideal	ADJ
ejpam-4788	19	3	theory	theory	NOUN
ejpam-4788	19	4	is	be	AUX
ejpam-4788	19	5	an	an	DET
ejpam-4788	19	6	essential	essential	ADJ
ejpam-4788	19	7	structures	structure	NOUN
ejpam-4788	19	8	in	in	ADP
ejpam-4788	19	9	semigroups	semigroup	NOUN
ejpam-4788	19	10	and	and	CCONJ
ejpam-4788	19	11	many	many	ADJ
ejpam-4788	19	12	researchers	researcher	NOUN
ejpam-4788	19	13	have	have	AUX
ejpam-4788	19	14	applied	apply	VERB
ejpam-4788	19	15	the	the	DET
ejpam-4788	19	16	knowledge	knowledge	NOUN
ejpam-4788	19	17	of	of	ADP
ejpam-4788	19	18	ideals	ideal	NOUN
ejpam-4788	19	19	in	in	ADP
ejpam-4788	19	20	γ	γ	NOUN
ejpam-4788	19	21	-	-	PUNCT
ejpam-4788	19	22	semigroups	semigroup	NOUN
ejpam-4788	19	23	studies	study	NOUN
ejpam-4788	19	24	in	in	ADP
ejpam-4788	19	25	a	a	DET
ejpam-4788	19	26	fuzzy	fuzzy	ADJ
ejpam-4788	19	27	semigroup	semigroup	NOUN
ejpam-4788	19	28	.	.	PUNCT
ejpam-4788	20	1	for	for	ADP
ejpam-4788	20	2	example	example	NOUN
ejpam-4788	20	3	,	,	PUNCT
ejpam-4788	20	4	chinram	chinram	PROPN
ejpam-4788	20	5	et	et	PROPN
ejpam-4788	20	6	al	al	PROPN
ejpam-4788	20	7	.	.	PUNCT
ejpam-4788	21	1	[	[	X
ejpam-4788	21	2	1	1	X
ejpam-4788	21	3	]	]	PUNCT
ejpam-4788	21	4	studied	study	VERB
ejpam-4788	21	5	almost	almost	ADV
ejpam-4788	21	6	quasi	quasi	ADJ
ejpam-4788	21	7	-	-	ADJ
ejpam-4788	21	8	γ	γ	NOUN
ejpam-4788	21	9	-	-	ADJ
ejpam-4788	21	10	ideal	ideal	ADJ
ejpam-4788	21	11	and	and	CCONJ
ejpam-4788	21	12	fuzzy	fuzzy	ADJ
ejpam-4788	21	13	almost	almost	ADV
ejpam-4788	21	14	quasi	quasi	ADJ
ejpam-4788	21	15	-	-	ADJ
ejpam-4788	21	16	γ	γ	NOUN
ejpam-4788	21	17	-	-	PUNCT
ejpam-4788	21	18	ideals	ideal	NOUN
ejpam-4788	21	19	in	in	ADP
ejpam-4788	21	20	γ	γ	NOUN
ejpam-4788	21	21	-	-	PUNCT
ejpam-4788	21	22	semigroup	semigroup	NOUN
ejpam-4788	21	23	,	,	PUNCT
ejpam-4788	21	24	m.	m.	PROPN
ejpam-4788	21	25	k.	k.	PROPN
ejpam-4788	21	26	r.	r.	PROPN
ejpam-4788	21	27	marapureddy	marapureddy	PROPN
ejpam-4788	21	28	and	and	CCONJ
ejpam-4788	21	29	prv	prv	PROPN
ejpam-4788	21	30	s.	s.	PROPN
ejpam-4788	21	31	r.	r.	PROPN
ejpam-4788	21	32	doradla	doradla	NOUN
ejpam-4788	22	1	[	[	X
ejpam-4788	22	2	7	7	X
ejpam-4788	22	3	]	]	PUNCT
ejpam-4788	22	4	investigated	investigate	VERB
ejpam-4788	22	5	weak	weak	ADJ
ejpam-4788	22	6	interior	interior	ADJ
ejpam-4788	22	7	ideals	ideal	NOUN
ejpam-4788	22	8	of	of	ADP
ejpam-4788	22	9	γ	γ	NOUN
ejpam-4788	22	10	-	-	PUNCT
ejpam-4788	22	11	semigroups	semigroup	NOUN
ejpam-4788	22	12	,	,	PUNCT
ejpam-4788	22	13	s.k	s.k	PROPN
ejpam-4788	22	14	.	.	PROPN
ejpam-4788	22	15	majumder	majumder	PROPN
ejpam-4788	22	16	and	and	CCONJ
ejpam-4788	22	17	m.	m.	PROPN
ejpam-4788	22	18	mandal	mandal	PROPN
ejpam-4788	23	1	[	[	X
ejpam-4788	23	2	4	4	X
ejpam-4788	23	3	]	]	PUNCT
ejpam-4788	23	4	examimed	examime	VERB
ejpam-4788	23	5	∗corresponding	∗corresponde	VERB
ejpam-4788	23	6	author	author	NOUN
ejpam-4788	23	7	.	.	PUNCT
ejpam-4788	24	1	doi	doi	NOUN
ejpam-4788	24	2	:	:	PUNCT
ejpam-4788	24	3	https://doi.org/10.29020/nybg.ejpam.v16i3.4788	https://doi.org/10.29020/nybg.ejpam.v16i3.4788	ADP
ejpam-4788	24	4	email	email	NOUN
ejpam-4788	24	5	addresses	address	VERB
ejpam-4788	24	6	:	:	PUNCT
ejpam-4788	24	7	thiti.ga@up.ac.th	thiti.ga@up.ac.th	PROPN
ejpam-4788	24	8	(	(	PUNCT
ejpam-4788	24	9	t.	t.	NOUN
ejpam-4788	24	10	gaketem	gaketem	PROPN
ejpam-4788	24	11	)	)	PUNCT
ejpam-4788	24	12	,	,	PUNCT
ejpam-4788	24	13	pk	pk	NOUN
ejpam-4788	24	14	g@rmutl.ac.th	g@rmutl.ac.th	PROPN
ejpam-4788	24	15	(	(	PUNCT
ejpam-4788	24	16	p.	p.	NOUN
ejpam-4788	24	17	khamrot	khamrot	PROPN
ejpam-4788	24	18	)	)	PUNCT
ejpam-4788	24	19	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4788	24	20	1592	1592	NUM
ejpam-4788	25	1	©	©	PROPN
ejpam-4788	25	2	2023	2023	NUM
ejpam-4788	25	3	ejpam	ejpam	NOUN
ejpam-4788	25	4	all	all	DET
ejpam-4788	25	5	rights	right	NOUN
ejpam-4788	25	6	reserved	reserve	VERB
ejpam-4788	25	7	.	.	PUNCT
ejpam-4788	26	1	p.	p.	NOUN
ejpam-4788	26	2	khamrot	khamrot	PROPN
ejpam-4788	26	3	,	,	PUNCT
ejpam-4788	26	4	t.	t.	PROPN
ejpam-4788	26	5	gaketem	gaketem	PROPN
ejpam-4788	26	6	/	/	SYM
ejpam-4788	26	7	eur	eur	PROPN
ejpam-4788	26	8	.	.	PUNCT
ejpam-4788	27	1	j.	j.	PROPN
ejpam-4788	27	2	pure	pure	PROPN
ejpam-4788	27	3	appl	appl	PROPN
ejpam-4788	27	4	.	.	PROPN
ejpam-4788	27	5	math	math	PROPN
ejpam-4788	27	6	,	,	PUNCT
ejpam-4788	27	7	16	16	NUM
ejpam-4788	27	8	(	(	PUNCT
ejpam-4788	27	9	3	3	NUM
ejpam-4788	27	10	)	)	PUNCT
ejpam-4788	27	11	(	(	PUNCT
ejpam-4788	27	12	2023	2023	NUM
ejpam-4788	27	13	)	)	PUNCT
ejpam-4788	27	14	,	,	PUNCT
ejpam-4788	27	15	1592	1592	NUM
ejpam-4788	27	16	-	-	SYM
ejpam-4788	27	17	1607	1607	NUM
ejpam-4788	27	18	1593	1593	NUM
ejpam-4788	27	19	a	a	DET
ejpam-4788	27	20	fuzzy	fuzzy	ADJ
ejpam-4788	27	21	generalized	generalize	VERB
ejpam-4788	27	22	bi	bi	NOUN
ejpam-4788	27	23	-	-	NOUN
ejpam-4788	27	24	ideal	ideal	NOUN
ejpam-4788	27	25	in	in	ADP
ejpam-4788	27	26	γ	γ	NOUN
ejpam-4788	27	27	-	-	PUNCT
ejpam-4788	27	28	semigroups	semigroup	NOUN
ejpam-4788	27	29	.	.	PUNCT
ejpam-4788	28	1	in	in	ADP
ejpam-4788	28	2	2021	2021	NUM
ejpam-4788	28	3	,	,	PUNCT
ejpam-4788	28	4	t.	t.	PROPN
ejpam-4788	28	5	gaketem	gaketem	PROPN
ejpam-4788	28	6	and	and	CCONJ
ejpam-4788	28	7	p.	p.	NOUN
ejpam-4788	28	8	khamrot	khamrot	NOUN
ejpam-4788	29	1	[	[	X
ejpam-4788	29	2	2	2	X
ejpam-4788	29	3	]	]	PUNCT
ejpam-4788	29	4	discussed	discuss	VERB
ejpam-4788	29	5	bipolar	bipolar	ADJ
ejpam-4788	29	6	fuzzy	fuzzy	ADJ
ejpam-4788	29	7	weakly	weakly	ADJ
ejpam-4788	29	8	interior	interior	ADJ
ejpam-4788	29	9	ideals	ideal	NOUN
ejpam-4788	29	10	in	in	ADP
ejpam-4788	29	11	semigroups	semigroup	NOUN
ejpam-4788	29	12	.	.	PUNCT
ejpam-4788	30	1	in	in	ADP
ejpam-4788	30	2	the	the	DET
ejpam-4788	30	3	same	same	ADJ
ejpam-4788	30	4	year	year	NOUN
ejpam-4788	30	5	,	,	PUNCT
ejpam-4788	30	6	[	[	X
ejpam-4788	30	7	11	11	NUM
ejpam-4788	30	8	]	]	PUNCT
ejpam-4788	30	9	a.	a.	NOUN
ejpam-4788	30	10	simuen	simuen	PROPN
ejpam-4788	30	11	et	et	PROPN
ejpam-4788	30	12	al	al	PROPN
ejpam-4788	30	13	.	.	PROPN
ejpam-4788	30	14	studied	study	VERB
ejpam-4788	30	15	a	a	DET
ejpam-4788	30	16	novel	novel	NOUN
ejpam-4788	30	17	of	of	ADP
ejpam-4788	30	18	ideals	ideal	NOUN
ejpam-4788	30	19	and	and	CCONJ
ejpam-4788	30	20	fuzzy	fuzzy	ADJ
ejpam-4788	30	21	ideals	ideal	NOUN
ejpam-4788	30	22	of	of	ADP
ejpam-4788	30	23	γ	γ	NOUN
ejpam-4788	30	24	-	-	PUNCT
ejpam-4788	30	25	semigroups	semigroup	NOUN
ejpam-4788	30	26	.	.	PUNCT
ejpam-4788	31	1	recently	recently	ADV
ejpam-4788	31	2	,	,	PUNCT
ejpam-4788	31	3	in	in	ADP
ejpam-4788	31	4	2022	2022	NUM
ejpam-4788	31	5	-	-	SYM
ejpam-4788	31	6	2023	2023	NUM
ejpam-4788	31	7	,	,	PUNCT
ejpam-4788	31	8	t.	t.	PROPN
ejpam-4788	31	9	gaketem	gaketem	PROPN
ejpam-4788	31	10	and	and	CCONJ
ejpam-4788	31	11	p.	p.	NOUN
ejpam-4788	31	12	khamrot	khamrot	NOUN
ejpam-4788	32	1	[	[	X
ejpam-4788	32	2	3	3	NUM
ejpam-4788	32	3	,	,	PUNCT
ejpam-4788	32	4	5	5	NUM
ejpam-4788	32	5	,	,	PUNCT
ejpam-4788	32	6	6	6	NUM
ejpam-4788	32	7	]	]	PUNCT
ejpam-4788	32	8	investigated	investigate	VERB
ejpam-4788	32	9	propertes	properte	NOUN
ejpam-4788	32	10	of	of	ADP
ejpam-4788	32	11	a	a	DET
ejpam-4788	32	12	novel	novel	NOUN
ejpam-4788	32	13	of	of	ADP
ejpam-4788	32	14	ideals	ideal	NOUN
ejpam-4788	32	15	on	on	ADP
ejpam-4788	32	16	intuitionistic	intuitionistic	ADJ
ejpam-4788	32	17	fuzzy	fuzzy	ADJ
ejpam-4788	32	18	ideals	ideal	NOUN
ejpam-4788	32	19	,	,	PUNCT
ejpam-4788	32	20	cubic	cubic	ADJ
ejpam-4788	32	21	ideals	ideal	NOUN
ejpam-4788	32	22	and	and	CCONJ
ejpam-4788	32	23	interval	interval	NOUN
ejpam-4788	32	24	valued	value	VERB
ejpam-4788	32	25	fuzzy	fuzzy	ADJ
ejpam-4788	32	26	ideals	ideal	NOUN
ejpam-4788	32	27	of	of	ADP
ejpam-4788	32	28	γ	γ	NOUN
ejpam-4788	32	29	-	-	PUNCT
ejpam-4788	32	30	semigroups	semigroup	NOUN
ejpam-4788	32	31	.	.	PUNCT
ejpam-4788	33	1	in	in	ADP
ejpam-4788	33	2	this	this	DET
ejpam-4788	33	3	paper	paper	NOUN
ejpam-4788	33	4	,	,	PUNCT
ejpam-4788	33	5	we	we	PRON
ejpam-4788	33	6	extend	extend	VERB
ejpam-4788	33	7	the	the	DET
ejpam-4788	33	8	new	new	ADJ
ejpam-4788	33	9	fuzzy	fuzzy	ADJ
ejpam-4788	33	10	ideal	ideal	NOUN
ejpam-4788	33	11	to	to	ADP
ejpam-4788	33	12	the	the	DET
ejpam-4788	33	13	bipolar	bipolar	ADJ
ejpam-4788	33	14	fuzzy	fuzzy	ADJ
ejpam-4788	33	15	ideal	ideal	NOUN
ejpam-4788	33	16	of	of	ADP
ejpam-4788	33	17	γ	γ	NOUN
ejpam-4788	33	18	-	-	PUNCT
ejpam-4788	33	19	semigroups	semigroup	NOUN
ejpam-4788	33	20	and	and	CCONJ
ejpam-4788	33	21	investigate	investigate	VERB
ejpam-4788	33	22	their	their	PRON
ejpam-4788	33	23	properties	property	NOUN
ejpam-4788	33	24	.	.	PUNCT
ejpam-4788	34	1	we	we	PRON
ejpam-4788	34	2	prove	prove	VERB
ejpam-4788	34	3	connection	connection	NOUN
ejpam-4788	34	4	between	between	ADP
ejpam-4788	34	5	almost	almost	ADV
ejpam-4788	34	6	ideals	ideal	NOUN
ejpam-4788	34	7	and	and	CCONJ
ejpam-4788	34	8	bipolar	bipolar	ADJ
ejpam-4788	34	9	fuzzy	fuzzy	ADJ
ejpam-4788	34	10	almost	almost	ADV
ejpam-4788	34	11	ideals	ideal	NOUN
ejpam-4788	34	12	of	of	ADP
ejpam-4788	34	13	γ	γ	NOUN
ejpam-4788	34	14	-	-	PUNCT
ejpam-4788	34	15	semigroups	semigroup	NOUN
ejpam-4788	34	16	.	.	PUNCT
ejpam-4788	35	1	2	2	X
ejpam-4788	35	2	.	.	NUM
ejpam-4788	35	3	preliminaries	preliminary	NOUN
ejpam-4788	35	4	in	in	ADP
ejpam-4788	35	5	this	this	DET
ejpam-4788	35	6	section	section	NOUN
ejpam-4788	35	7	,	,	PUNCT
ejpam-4788	35	8	we	we	PRON
ejpam-4788	35	9	review	review	VERB
ejpam-4788	35	10	some	some	DET
ejpam-4788	35	11	basic	basic	ADJ
ejpam-4788	35	12	concepts	concept	NOUN
ejpam-4788	35	13	that	that	PRON
ejpam-4788	35	14	are	be	AUX
ejpam-4788	35	15	necessary	necessary	ADJ
ejpam-4788	35	16	to	to	PART
ejpam-4788	35	17	understand	understand	VERB
ejpam-4788	35	18	our	our	PRON
ejpam-4788	35	19	section	section	NOUN
ejpam-4788	35	20	.	.	PUNCT
ejpam-4788	36	1	a	a	DET
ejpam-4788	36	2	sub	sub	ADJ
ejpam-4788	36	3	-	-	ADJ
ejpam-4788	36	4	γ	γ	NOUN
ejpam-4788	36	5	-	-	PUNCT
ejpam-4788	36	6	semigroup	semigroup	NOUN
ejpam-4788	36	7	of	of	ADP
ejpam-4788	36	8	a	a	DET
ejpam-4788	36	9	γ	γ	NOUN
ejpam-4788	36	10	-	-	PUNCT
ejpam-4788	36	11	semigroup	semigroup	NOUN
ejpam-4788	36	12	s	s	VERB
ejpam-4788	36	13	is	be	AUX
ejpam-4788	36	14	a	a	DET
ejpam-4788	36	15	non	non	ADJ
ejpam-4788	36	16	-	-	ADJ
ejpam-4788	36	17	empty	empty	ADJ
ejpam-4788	36	18	set	set	NOUN
ejpam-4788	37	1	k	k	PROPN
ejpam-4788	37	2	of	of	ADP
ejpam-4788	37	3	s	s	PRON
ejpam-4788	37	4	such	such	ADJ
ejpam-4788	37	5	that	that	DET
ejpam-4788	37	6	kγk	kγk	NOUN
ejpam-4788	37	7	⊆	⊆	NUM
ejpam-4788	37	8	k.	k.	NOUN
ejpam-4788	37	9	a	a	DET
ejpam-4788	37	10	left	left	ADJ
ejpam-4788	37	11	(	(	PUNCT
ejpam-4788	37	12	right	right	ADJ
ejpam-4788	37	13	)	)	PUNCT
ejpam-4788	37	14	ideal	ideal	NOUN
ejpam-4788	37	15	of	of	ADP
ejpam-4788	37	16	a	a	DET
ejpam-4788	37	17	γ	γ	NOUN
ejpam-4788	37	18	-	-	PUNCT
ejpam-4788	37	19	semigroup	semigroup	NOUN
ejpam-4788	37	20	s	s	VERB
ejpam-4788	37	21	is	be	AUX
ejpam-4788	37	22	a	a	DET
ejpam-4788	37	23	non	non	ADJ
ejpam-4788	37	24	-	-	ADJ
ejpam-4788	37	25	empty	empty	ADJ
ejpam-4788	37	26	set	set	NOUN
ejpam-4788	37	27	k	k	PROPN
ejpam-4788	37	28	of	of	ADP
ejpam-4788	37	29	s	s	PRON
ejpam-4788	37	30	such	such	ADJ
ejpam-4788	37	31	that	that	DET
ejpam-4788	37	32	sγk	sγk	NOUN
ejpam-4788	37	33	⊆	⊆	NUM
ejpam-4788	37	34	k	k	PROPN
ejpam-4788	37	35	(	(	PUNCT
ejpam-4788	37	36	kγs	kγs	NOUN
ejpam-4788	37	37	⊆	⊆	NUM
ejpam-4788	37	38	k	k	NOUN
ejpam-4788	37	39	)	)	PUNCT
ejpam-4788	37	40	.	.	PUNCT
ejpam-4788	38	1	by	by	ADP
ejpam-4788	38	2	an	an	DET
ejpam-4788	38	3	ideal	ideal	NOUN
ejpam-4788	38	4	of	of	ADP
ejpam-4788	38	5	a	a	DET
ejpam-4788	38	6	γ	γ	NOUN
ejpam-4788	38	7	-	-	PUNCT
ejpam-4788	38	8	semigroup	semigroup	NOUN
ejpam-4788	38	9	s	s	PROPN
ejpam-4788	38	10	,	,	PUNCT
ejpam-4788	38	11	we	we	PRON
ejpam-4788	38	12	mean	mean	VERB
ejpam-4788	38	13	a	a	DET
ejpam-4788	38	14	non	non	ADJ
ejpam-4788	38	15	-	-	ADJ
ejpam-4788	38	16	empty	empty	ADJ
ejpam-4788	38	17	set	set	NOUN
ejpam-4788	38	18	of	of	ADP
ejpam-4788	38	19	s	s	PRON
ejpam-4788	38	20	which	which	PRON
ejpam-4788	38	21	is	be	AUX
ejpam-4788	38	22	both	both	CCONJ
ejpam-4788	38	23	a	a	DET
ejpam-4788	38	24	left	left	NOUN
ejpam-4788	38	25	and	and	CCONJ
ejpam-4788	38	26	a	a	DET
ejpam-4788	38	27	right	right	ADJ
ejpam-4788	38	28	ideal	ideal	NOUN
ejpam-4788	38	29	of	of	ADP
ejpam-4788	38	30	s.	s.	PROPN
ejpam-4788	38	31	a	a	DET
ejpam-4788	38	32	quasi	quasi	NOUN
ejpam-4788	38	33	-	-	NOUN
ejpam-4788	38	34	ideal	ideal	ADJ
ejpam-4788	38	35	of	of	ADP
ejpam-4788	38	36	a	a	DET
ejpam-4788	38	37	γ	γ	NOUN
ejpam-4788	38	38	-	-	PUNCT
ejpam-4788	38	39	semigroup	semigroup	NOUN
ejpam-4788	38	40	s	s	VERB
ejpam-4788	38	41	is	be	AUX
ejpam-4788	38	42	a	a	DET
ejpam-4788	38	43	non	non	ADJ
ejpam-4788	38	44	-	-	ADJ
ejpam-4788	38	45	empty	empty	ADJ
ejpam-4788	38	46	set	set	NOUN
ejpam-4788	38	47	k	k	PROPN
ejpam-4788	38	48	of	of	ADP
ejpam-4788	38	49	s	s	PRON
ejpam-4788	38	50	such	such	ADJ
ejpam-4788	38	51	that	that	DET
ejpam-4788	38	52	kγs	kγs	NOUN
ejpam-4788	38	53	∩	∩	NOUN
ejpam-4788	38	54	sγk	sγk	VERB
ejpam-4788	38	55	⊆	⊆	NUM
ejpam-4788	38	56	k.	k.	NOUN
ejpam-4788	38	57	a	a	DET
ejpam-4788	38	58	sub	sub	ADJ
ejpam-4788	38	59	-	-	ADJ
ejpam-4788	38	60	γ	γ	ADJ
ejpam-4788	38	61	-	-	PUNCT
ejpam-4788	38	62	semigroup	semigroup	NOUN
ejpam-4788	38	63	k	k	PROPN
ejpam-4788	38	64	of	of	ADP
ejpam-4788	38	65	a	a	DET
ejpam-4788	38	66	γ	γ	PROPN
ejpam-4788	38	67	-	-	PUNCT
ejpam-4788	38	68	semigroup	semigroup	NOUN
ejpam-4788	38	69	s	s	PART
ejpam-4788	38	70	is	be	AUX
ejpam-4788	38	71	called	call	VERB
ejpam-4788	38	72	a	a	DET
ejpam-4788	38	73	bi	bi	NOUN
ejpam-4788	38	74	-	-	NOUN
ejpam-4788	38	75	ideal	ideal	NOUN
ejpam-4788	38	76	of	of	ADP
ejpam-4788	38	77	s	s	PRON
ejpam-4788	38	78	if	if	SCONJ
ejpam-4788	38	79	kγsγk	kγsγk	VERB
ejpam-4788	38	80	⊆	⊆	NUM
ejpam-4788	38	81	k.	k.	NOUN
ejpam-4788	38	82	definition	definition	NOUN
ejpam-4788	38	83	1	1	NUM
ejpam-4788	38	84	.	.	PUNCT
ejpam-4788	39	1	[	[	X
ejpam-4788	39	2	11	11	NUM
ejpam-4788	39	3	]	]	PUNCT
ejpam-4788	39	4	let	let	VERB
ejpam-4788	39	5	s	s	PRON
ejpam-4788	39	6	be	be	AUX
ejpam-4788	39	7	a	a	DET
ejpam-4788	39	8	γ	γ	NOUN
ejpam-4788	39	9	-	-	PUNCT
ejpam-4788	39	10	semigroup	semigroup	NOUN
ejpam-4788	39	11	,	,	PUNCT
ejpam-4788	39	12	k	k	X
ejpam-4788	39	13	be	be	AUX
ejpam-4788	39	14	a	a	DET
ejpam-4788	39	15	non	non	ADJ
ejpam-4788	39	16	-	-	ADJ
ejpam-4788	39	17	empty	empty	ADJ
ejpam-4788	39	18	subset	subset	NOUN
ejpam-4788	39	19	of	of	ADP
ejpam-4788	39	20	s	s	PROPN
ejpam-4788	39	21	and	and	CCONJ
ejpam-4788	39	22	α	α	NOUN
ejpam-4788	39	23	,	,	PUNCT
ejpam-4788	39	24	β	β	PROPN
ejpam-4788	39	25	∈	∈	PROPN
ejpam-4788	39	26	γ	γ	X
ejpam-4788	39	27	.	.	PROPN
ejpam-4788	40	1	then	then	ADV
ejpam-4788	40	2	k	k	PROPN
ejpam-4788	40	3	is	be	AUX
ejpam-4788	40	4	said	say	VERB
ejpam-4788	40	5	to	to	PART
ejpam-4788	40	6	be	be	AUX
ejpam-4788	40	7	[	[	PUNCT
ejpam-4788	40	8	(	(	PUNCT
ejpam-4788	40	9	i	i	NOUN
ejpam-4788	40	10	)	)	PUNCT
ejpam-4788	40	11	]	]	PUNCT
ejpam-4788	41	1	(	(	PUNCT
ejpam-4788	41	2	i	i	NOUN
ejpam-4788	41	3	)	)	PUNCT
ejpam-4788	41	4	a	a	DET
ejpam-4788	41	5	left	left	ADJ
ejpam-4788	41	6	(	(	PUNCT
ejpam-4788	41	7	right	right	ADJ
ejpam-4788	41	8	)	)	PUNCT
ejpam-4788	41	9	almost	almost	ADV
ejpam-4788	41	10	ideal	ideal	ADJ
ejpam-4788	41	11	of	of	ADP
ejpam-4788	41	12	γ	γ	PROPN
ejpam-4788	41	13	-	-	PUNCT
ejpam-4788	41	14	semigroup	semigroup	NOUN
ejpam-4788	41	15	s	s	X
ejpam-4788	41	16	which	which	PRON
ejpam-4788	41	17	is	be	AUX
ejpam-4788	41	18	a	a	DET
ejpam-4788	41	19	non	non	ADJ
ejpam-4788	41	20	-	-	ADJ
ejpam-4788	41	21	empty	empty	ADJ
ejpam-4788	41	22	set	set	NOUN
ejpam-4788	41	23	k	k	ADP
ejpam-4788	41	24	such	such	ADJ
ejpam-4788	41	25	that	that	DET
ejpam-4788	41	26	(	(	PUNCT
ejpam-4788	41	27	sγk	sγk	NOUN
ejpam-4788	41	28	)	)	PUNCT
ejpam-4788	41	29	∩k	∩k	NOUN
ejpam-4788	41	30	̸=	̸=	PROPN
ejpam-4788	41	31	∅	∅	NOUN
ejpam-4788	41	32	(	(	PUNCT
ejpam-4788	41	33	(	(	PUNCT
ejpam-4788	41	34	kγs	kγs	NOUN
ejpam-4788	41	35	)	)	PUNCT
ejpam-4788	41	36	∩k	∩k	NOUN
ejpam-4788	41	37	̸=	̸=	PROPN
ejpam-4788	41	38	∅	∅	NOUN
ejpam-4788	41	39	)	)	PUNCT
ejpam-4788	41	40	for	for	ADP
ejpam-4788	41	41	all	all	PRON
ejpam-4788	41	42	s	s	PROPN
ejpam-4788	41	43	∈	∈	PROPN
ejpam-4788	41	44	s.	s.	PROPN
ejpam-4788	41	45	(	(	PUNCT
ejpam-4788	41	46	ii	ii	PROPN
ejpam-4788	41	47	)	)	PUNCT
ejpam-4788	41	48	an	an	DET
ejpam-4788	41	49	almost	almost	ADV
ejpam-4788	41	50	bi	bi	NOUN
ejpam-4788	41	51	-	-	NOUN
ejpam-4788	41	52	ideal	ideal	NOUN
ejpam-4788	41	53	of	of	ADP
ejpam-4788	41	54	γ	γ	PROPN
ejpam-4788	41	55	-	-	PUNCT
ejpam-4788	41	56	semigroup	semigroup	NOUN
ejpam-4788	41	57	s	s	X
ejpam-4788	41	58	which	which	PRON
ejpam-4788	41	59	is	be	AUX
ejpam-4788	41	60	a	a	DET
ejpam-4788	41	61	non	non	ADJ
ejpam-4788	41	62	-	-	ADJ
ejpam-4788	41	63	empty	empty	ADJ
ejpam-4788	41	64	set	set	NOUN
ejpam-4788	41	65	k	k	ADP
ejpam-4788	42	1	such	such	ADJ
ejpam-4788	42	2	that	that	DET
ejpam-4788	42	3	(	(	PUNCT
ejpam-4788	42	4	kγsγk)∩	kγsγk)∩	PROPN
ejpam-4788	42	5	k	k	PROPN
ejpam-4788	42	6	̸=	̸=	PROPN
ejpam-4788	42	7	∅	∅	NOUN
ejpam-4788	42	8	for	for	ADP
ejpam-4788	42	9	all	all	PRON
ejpam-4788	42	10	s	s	PART
ejpam-4788	42	11	∈	∈	PROPN
ejpam-4788	42	12	s.	s.	PROPN
ejpam-4788	42	13	(	(	PUNCT
ejpam-4788	42	14	iii	iii	PROPN
ejpam-4788	42	15	)	)	PUNCT
ejpam-4788	42	16	an	an	DET
ejpam-4788	42	17	almost	almost	ADV
ejpam-4788	42	18	quasi	quasi	NOUN
ejpam-4788	42	19	-	-	NOUN
ejpam-4788	42	20	ideal	ideal	ADJ
ejpam-4788	42	21	of	of	ADP
ejpam-4788	42	22	γ	γ	PROPN
ejpam-4788	42	23	-	-	PUNCT
ejpam-4788	42	24	semigroup	semigroup	NOUN
ejpam-4788	42	25	s	s	X
ejpam-4788	42	26	which	which	PRON
ejpam-4788	42	27	is	be	AUX
ejpam-4788	42	28	a	a	DET
ejpam-4788	42	29	non	non	ADJ
ejpam-4788	42	30	-	-	ADJ
ejpam-4788	42	31	empty	empty	ADJ
ejpam-4788	42	32	set	set	NOUN
ejpam-4788	42	33	k	k	ADP
ejpam-4788	42	34	such	such	ADJ
ejpam-4788	42	35	that	that	PRON
ejpam-4788	42	36	(	(	PUNCT
ejpam-4788	42	37	sγk∩	sγk∩	NOUN
ejpam-4788	42	38	kγs	kγs	NOUN
ejpam-4788	42	39	)	)	PUNCT
ejpam-4788	42	40	∩k	∩k	PROPN
ejpam-4788	42	41	̸=	̸=	PROPN
ejpam-4788	42	42	∅	∅	NOUN
ejpam-4788	42	43	for	for	ADP
ejpam-4788	42	44	all	all	PRON
ejpam-4788	42	45	s	s	PART
ejpam-4788	42	46	∈	∈	PROPN
ejpam-4788	42	47	s.	s.	PROPN
ejpam-4788	42	48	(	(	PUNCT
ejpam-4788	42	49	iv	iv	X
ejpam-4788	42	50	)	)	PUNCT
ejpam-4788	42	51	a	a	DET
ejpam-4788	42	52	left	left	ADJ
ejpam-4788	42	53	α	α	NOUN
ejpam-4788	42	54	-	-	NOUN
ejpam-4788	42	55	ideal	ideal	NOUN
ejpam-4788	42	56	of	of	ADP
ejpam-4788	42	57	a	a	DET
ejpam-4788	42	58	γ	γ	NOUN
ejpam-4788	42	59	-	-	PUNCT
ejpam-4788	42	60	semigroup	semigroup	NOUN
ejpam-4788	42	61	s	s	X
ejpam-4788	42	62	which	which	PRON
ejpam-4788	42	63	is	be	AUX
ejpam-4788	42	64	a	a	DET
ejpam-4788	42	65	non	non	ADJ
ejpam-4788	42	66	-	-	ADJ
ejpam-4788	42	67	empty	empty	ADJ
ejpam-4788	42	68	set	set	NOUN
ejpam-4788	42	69	k	k	ADP
ejpam-4788	42	70	such	such	ADJ
ejpam-4788	42	71	that	that	DET
ejpam-4788	42	72	sαk	sαk	NOUN
ejpam-4788	42	73	⊆	⊆	NUM
ejpam-4788	42	74	k.	k.	NOUN
ejpam-4788	42	75	a	a	DET
ejpam-4788	42	76	right	right	ADJ
ejpam-4788	42	77	α	α	NOUN
ejpam-4788	42	78	-	-	NOUN
ejpam-4788	42	79	ideal	ideal	NOUN
ejpam-4788	42	80	of	of	ADP
ejpam-4788	42	81	a	a	DET
ejpam-4788	42	82	γ	γ	NOUN
ejpam-4788	42	83	-	-	PUNCT
ejpam-4788	42	84	semigroup	semigroup	NOUN
ejpam-4788	42	85	s	s	VERB
ejpam-4788	42	86	is	be	AUX
ejpam-4788	42	87	a	a	DET
ejpam-4788	42	88	non	non	ADJ
ejpam-4788	42	89	-	-	ADJ
ejpam-4788	42	90	empty	empty	ADJ
ejpam-4788	42	91	set	set	NOUN
ejpam-4788	42	92	k	k	SCONJ
ejpam-4788	42	93	such	such	ADJ
ejpam-4788	42	94	that	that	PRON
ejpam-4788	42	95	kβs	kβs	VERB
ejpam-4788	42	96	⊆	⊆	NUM
ejpam-4788	42	97	k.	k.	PROPN
ejpam-4788	42	98	(	(	PUNCT
ejpam-4788	42	99	v	v	NOUN
ejpam-4788	42	100	)	)	PUNCT
ejpam-4788	42	101	an	an	DET
ejpam-4788	42	102	(	(	PUNCT
ejpam-4788	42	103	α	α	NOUN
ejpam-4788	42	104	,	,	PUNCT
ejpam-4788	42	105	β)-ideal	β)-ideal	NOUN
ejpam-4788	42	106	of	of	ADP
ejpam-4788	42	107	a	a	DET
ejpam-4788	42	108	γ	γ	NOUN
ejpam-4788	42	109	-	-	PUNCT
ejpam-4788	42	110	semigroup	semigroup	NOUN
ejpam-4788	42	111	s	s	X
ejpam-4788	42	112	which	which	PRON
ejpam-4788	42	113	is	be	AUX
ejpam-4788	42	114	a	a	DET
ejpam-4788	42	115	non	non	ADJ
ejpam-4788	42	116	-	-	ADJ
ejpam-4788	42	117	empty	empty	ADJ
ejpam-4788	42	118	set	set	NOUN
ejpam-4788	42	119	k	k	ADP
ejpam-4788	42	120	such	such	ADJ
ejpam-4788	42	121	that	that	SCONJ
ejpam-4788	42	122	it	it	PRON
ejpam-4788	42	123	is	be	AUX
ejpam-4788	42	124	both	both	CCONJ
ejpam-4788	42	125	a	a	DET
ejpam-4788	42	126	left	left	ADJ
ejpam-4788	42	127	α	α	NOUN
ejpam-4788	42	128	-	-	NOUN
ejpam-4788	42	129	ideal	ideal	NOUN
ejpam-4788	42	130	and	and	CCONJ
ejpam-4788	42	131	a	a	DET
ejpam-4788	42	132	right	right	ADJ
ejpam-4788	42	133	β	β	NOUN
ejpam-4788	42	134	-	-	NOUN
ejpam-4788	42	135	ideal	ideal	NOUN
ejpam-4788	42	136	of	of	ADP
ejpam-4788	42	137	s.	s.	PROPN
ejpam-4788	42	138	for	for	ADP
ejpam-4788	42	139	any	any	DET
ejpam-4788	42	140	mi	mi	PROPN
ejpam-4788	42	141	∈	∈	PROPN
ejpam-4788	43	1	[	[	X
ejpam-4788	43	2	0	0	NUM
ejpam-4788	43	3	,	,	PUNCT
ejpam-4788	43	4	1	1	NUM
ejpam-4788	43	5	]	]	PUNCT
ejpam-4788	43	6	,	,	PUNCT
ejpam-4788	43	7	i	i	PRON
ejpam-4788	43	8	∈	∈	VERB
ejpam-4788	43	9	a	a	PRON
ejpam-4788	43	10	,	,	PUNCT
ejpam-4788	43	11	define	define	VERB
ejpam-4788	43	12	∨	∨	NUM
ejpam-4788	43	13	i∈a	i∈a	ADJ
ejpam-4788	43	14	mi	mi	NOUN
ejpam-4788	43	15	:	:	PUNCT
ejpam-4788	43	16	=	=	SYM
ejpam-4788	43	17	sup	sup	NOUN
ejpam-4788	43	18	i∈a	i∈a	ADJ
ejpam-4788	43	19	{	{	PUNCT
ejpam-4788	43	20	mi	mi	NOUN
ejpam-4788	43	21	}	}	PUNCT
ejpam-4788	43	22	and	and	CCONJ
ejpam-4788	43	23	∧	∧	PROPN
ejpam-4788	43	24	i∈a	i∈a	ADJ
ejpam-4788	43	25	mi	mi	NOUN
ejpam-4788	43	26	:	:	PUNCT
ejpam-4788	43	27	=	=	SYM
ejpam-4788	43	28	inf	inf	PROPN
ejpam-4788	43	29	i∈a	i∈a	PROPN
ejpam-4788	43	30	{	{	PUNCT
ejpam-4788	43	31	mi	mi	NOUN
ejpam-4788	43	32	}	}	PUNCT
ejpam-4788	43	33	.	.	PUNCT
ejpam-4788	44	1	we	we	PRON
ejpam-4788	44	2	see	see	VERB
ejpam-4788	44	3	that	that	PRON
ejpam-4788	44	4	for	for	ADP
ejpam-4788	44	5	any	any	DET
ejpam-4788	44	6	m	m	NOUN
ejpam-4788	44	7	,	,	PUNCT
ejpam-4788	44	8	n	n	PRON
ejpam-4788	44	9	∈	∈	NOUN
ejpam-4788	45	1	[	[	X
ejpam-4788	45	2	0	0	NUM
ejpam-4788	45	3	,	,	PUNCT
ejpam-4788	45	4	1	1	NUM
ejpam-4788	45	5	]	]	PUNCT
ejpam-4788	45	6	,	,	PUNCT
ejpam-4788	45	7	we	we	PRON
ejpam-4788	45	8	have	have	VERB
ejpam-4788	45	9	m	m	PROPN
ejpam-4788	45	10	∨	∨	NUM
ejpam-4788	45	11	n	n	CCONJ
ejpam-4788	45	12	=	=	SYM
ejpam-4788	45	13	max{m	max{m	NOUN
ejpam-4788	45	14	,	,	PUNCT
ejpam-4788	45	15	n	n	CCONJ
ejpam-4788	45	16	}	}	PUNCT
ejpam-4788	45	17	and	and	CCONJ
ejpam-4788	45	18	m	m	PROPN
ejpam-4788	45	19	∧	∧	PROPN
ejpam-4788	45	20	n	n	PROPN
ejpam-4788	45	21	=	=	SYM
ejpam-4788	45	22	min{m	min{m	PROPN
ejpam-4788	45	23	,	,	PUNCT
ejpam-4788	45	24	n	n	CCONJ
ejpam-4788	45	25	}	}	PUNCT
ejpam-4788	45	26	.	.	PUNCT
ejpam-4788	46	1	p.	p.	NOUN
ejpam-4788	46	2	khamrot	khamrot	PROPN
ejpam-4788	46	3	,	,	PUNCT
ejpam-4788	46	4	t.	t.	PROPN
ejpam-4788	46	5	gaketem	gaketem	PROPN
ejpam-4788	46	6	/	/	SYM
ejpam-4788	46	7	eur	eur	PROPN
ejpam-4788	46	8	.	.	PUNCT
ejpam-4788	47	1	j.	j.	PROPN
ejpam-4788	47	2	pure	pure	PROPN
ejpam-4788	47	3	appl	appl	PROPN
ejpam-4788	47	4	.	.	PROPN
ejpam-4788	47	5	math	math	PROPN
ejpam-4788	47	6	,	,	PUNCT
ejpam-4788	47	7	16	16	NUM
ejpam-4788	47	8	(	(	PUNCT
ejpam-4788	47	9	3	3	NUM
ejpam-4788	47	10	)	)	PUNCT
ejpam-4788	47	11	(	(	PUNCT
ejpam-4788	47	12	2023	2023	NUM
ejpam-4788	47	13	)	)	PUNCT
ejpam-4788	47	14	,	,	PUNCT
ejpam-4788	47	15	1592	1592	NUM
ejpam-4788	47	16	-	-	SYM
ejpam-4788	47	17	1607	1607	NUM
ejpam-4788	47	18	1594	1594	NUM
ejpam-4788	47	19	definition	definition	NOUN
ejpam-4788	47	20	2	2	NUM
ejpam-4788	47	21	.	.	PUNCT
ejpam-4788	48	1	[	[	X
ejpam-4788	48	2	12	12	NUM
ejpam-4788	48	3	]	]	PUNCT
ejpam-4788	48	4	a	a	DET
ejpam-4788	48	5	fuzzy	fuzzy	ADJ
ejpam-4788	48	6	set	set	VERB
ejpam-4788	48	7	ξ	ξ	PROPN
ejpam-4788	48	8	of	of	ADP
ejpam-4788	48	9	a	a	DET
ejpam-4788	48	10	non	non	ADJ
ejpam-4788	48	11	-	-	ADJ
ejpam-4788	48	12	empty	empty	ADJ
ejpam-4788	48	13	set	set	ADJ
ejpam-4788	48	14	t	t	PROPN
ejpam-4788	48	15	is	be	AUX
ejpam-4788	48	16	a	a	DET
ejpam-4788	48	17	function	function	NOUN
ejpam-4788	48	18	ξ	ξ	NOUN
ejpam-4788	48	19	:	:	PUNCT
ejpam-4788	48	20	t	t	X
ejpam-4788	48	21	→	→	PUNCT
ejpam-4788	49	1	[	[	X
ejpam-4788	49	2	0	0	NUM
ejpam-4788	49	3	,	,	PUNCT
ejpam-4788	49	4	1	1	NUM
ejpam-4788	49	5	]	]	PUNCT
ejpam-4788	49	6	.	.	PUNCT
ejpam-4788	50	1	for	for	ADP
ejpam-4788	50	2	any	any	DET
ejpam-4788	50	3	two	two	NUM
ejpam-4788	50	4	fuzzy	fuzzy	ADJ
ejpam-4788	50	5	sets	set	NOUN
ejpam-4788	50	6	ξ	ξ	PROPN
ejpam-4788	50	7	and	and	CCONJ
ejpam-4788	50	8	ς	ς	PROPN
ejpam-4788	50	9	of	of	ADP
ejpam-4788	50	10	a	a	DET
ejpam-4788	50	11	non	non	ADJ
ejpam-4788	50	12	-	-	ADJ
ejpam-4788	50	13	empty	empty	ADJ
ejpam-4788	50	14	set	set	ADJ
ejpam-4788	50	15	t	t	NOUN
ejpam-4788	50	16	,	,	PUNCT
ejpam-4788	50	17	define	define	VERB
ejpam-4788	50	18	≥,=,∧	≥,=,∧	NUM
ejpam-4788	50	19	,	,	PUNCT
ejpam-4788	50	20	and	and	CCONJ
ejpam-4788	50	21	∨	∨	NUM
ejpam-4788	50	22	as	as	SCONJ
ejpam-4788	50	23	follows	follow	VERB
ejpam-4788	50	24	:	:	PUNCT
ejpam-4788	50	25	[	[	X
ejpam-4788	50	26	(	(	PUNCT
ejpam-4788	50	27	i)]ξ	i)]ξ	ADJ
ejpam-4788	50	28	≥	≥	NOUN
ejpam-4788	50	29	ς	ς	PROPN
ejpam-4788	50	30	⇔	⇔	X
ejpam-4788	50	31	ξ(k	ξ(k	PROPN
ejpam-4788	50	32	)	)	PUNCT
ejpam-4788	50	33	≥	≥	NOUN
ejpam-4788	50	34	ς(k	ς(k	NOUN
ejpam-4788	50	35	)	)	PUNCT
ejpam-4788	50	36	for	for	ADP
ejpam-4788	50	37	all	all	DET
ejpam-4788	50	38	k	k	PROPN
ejpam-4788	50	39	∈	∈	PROPN
ejpam-4788	50	40	t	t	PROPN
ejpam-4788	50	41	,	,	PUNCT
ejpam-4788	50	42	ξ	ξ	PROPN
ejpam-4788	50	43	=	=	SYM
ejpam-4788	50	44	ς	ς	PROPN
ejpam-4788	50	45	⇔	⇔	PROPN
ejpam-4788	50	46	ξ	ξ	PROPN
ejpam-4788	50	47	≥	≥	NOUN
ejpam-4788	50	48	ς	ς	PROPN
ejpam-4788	50	49	and	and	CCONJ
ejpam-4788	50	50	ς	ς	PROPN
ejpam-4788	50	51	≥	≥	NUM
ejpam-4788	50	52	ξ	ξ	PROPN
ejpam-4788	50	53	,	,	PUNCT
ejpam-4788	50	54	(	(	PUNCT
ejpam-4788	50	55	ξ	ξ	X
ejpam-4788	50	56	∧	∧	PROPN
ejpam-4788	50	57	ς)(k	ς)(k	NOUN
ejpam-4788	50	58	)	)	PUNCT
ejpam-4788	50	59	=	=	SYM
ejpam-4788	50	60	min{ξ(k	min{ξ(k	NOUN
ejpam-4788	50	61	)	)	PUNCT
ejpam-4788	50	62	,	,	PUNCT
ejpam-4788	50	63	ς(k	ς(k	NOUN
ejpam-4788	50	64	)	)	PUNCT
ejpam-4788	50	65	}	}	PUNCT
ejpam-4788	50	66	=	=	SYM
ejpam-4788	50	67	ξ(k)∧ς(k	ξ(k)∧ς(k	PRON
ejpam-4788	50	68	)	)	PUNCT
ejpam-4788	50	69	for	for	ADP
ejpam-4788	50	70	all	all	DET
ejpam-4788	50	71	k	k	PROPN
ejpam-4788	50	72	∈	∈	PROPN
ejpam-4788	50	73	t	t	PROPN
ejpam-4788	50	74	,	,	PUNCT
ejpam-4788	50	75	(	(	PUNCT
ejpam-4788	50	76	ξ∨ς)(k	ξ∨ς)(k	NOUN
ejpam-4788	50	77	)	)	PUNCT
ejpam-4788	50	78	=	=	SYM
ejpam-4788	50	79	max{ξ(k	max{ξ(k	NOUN
ejpam-4788	50	80	)	)	PUNCT
ejpam-4788	50	81	,	,	PUNCT
ejpam-4788	50	82	ς(k	ς(k	NOUN
ejpam-4788	50	83	)	)	PUNCT
ejpam-4788	50	84	}	}	PUNCT
ejpam-4788	50	85	=	=	SYM
ejpam-4788	50	86	ξ(k)∨ς(k	ξ(k)∨ς(k	NOUN
ejpam-4788	50	87	)	)	PUNCT
ejpam-4788	50	88	for	for	ADP
ejpam-4788	50	89	all	all	DET
ejpam-4788	50	90	k	k	PROPN
ejpam-4788	50	91	∈	∈	PROPN
ejpam-4788	50	92	t.	t.	NOUN
ejpam-4788	50	93	for	for	ADP
ejpam-4788	50	94	the	the	DET
ejpam-4788	50	95	symbol	symbol	NOUN
ejpam-4788	50	96	ξ	ξ	X
ejpam-4788	50	97	≤	≤	NUM
ejpam-4788	51	1	ς	ς	NOUN
ejpam-4788	51	2	,	,	PUNCT
ejpam-4788	51	3	we	we	PRON
ejpam-4788	51	4	mean	mean	VERB
ejpam-4788	51	5	ς	ς	PROPN
ejpam-4788	51	6	≥	≥	NUM
ejpam-4788	51	7	ξ	ξ	X
ejpam-4788	51	8	.	.	PUNCT
ejpam-4788	52	1	for	for	ADP
ejpam-4788	52	2	any	any	DET
ejpam-4788	52	3	two	two	NUM
ejpam-4788	52	4	fuzzy	fuzzy	ADJ
ejpam-4788	52	5	sets	set	NOUN
ejpam-4788	52	6	of	of	ADP
ejpam-4788	52	7	ξ	ξ	PROPN
ejpam-4788	52	8	and	and	CCONJ
ejpam-4788	52	9	ς	ς	PROPN
ejpam-4788	52	10	of	of	ADP
ejpam-4788	52	11	a	a	DET
ejpam-4788	52	12	non	non	ADJ
ejpam-4788	52	13	-	-	ADJ
ejpam-4788	52	14	empty	empty	ADJ
ejpam-4788	52	15	of	of	ADP
ejpam-4788	52	16	t	t	PROPN
ejpam-4788	52	17	,	,	PUNCT
ejpam-4788	52	18	we	we	PRON
ejpam-4788	52	19	define	define	VERB
ejpam-4788	52	20	the	the	DET
ejpam-4788	52	21	support	support	NOUN
ejpam-4788	52	22	of	of	ADP
ejpam-4788	52	23	ξ	ξ	PROPN
ejpam-4788	52	24	instead	instead	ADV
ejpam-4788	52	25	of	of	ADP
ejpam-4788	52	26	supp(ξ	supp(ξ	PROPN
ejpam-4788	52	27	)	)	PUNCT
ejpam-4788	52	28	=	=	PRON
ejpam-4788	52	29	{	{	PUNCT
ejpam-4788	52	30	k	k	PROPN
ejpam-4788	52	31	∈	∈	PROPN
ejpam-4788	52	32	t	t	PROPN
ejpam-4788	52	33	|	|	ADV
ejpam-4788	52	34	ξ(k	ξ(k	PROPN
ejpam-4788	52	35	)	)	PUNCT
ejpam-4788	52	36	̸=	̸=	PROPN
ejpam-4788	52	37	0	0	NUM
ejpam-4788	52	38	}	}	PUNCT
ejpam-4788	52	39	,	,	PUNCT
ejpam-4788	52	40	ξ	ξ	PROPN
ejpam-4788	52	41	⊆	⊆	NUM
ejpam-4788	52	42	ς	ς	NOUN
ejpam-4788	52	43	if	if	SCONJ
ejpam-4788	52	44	ξ(k	ξ(k	NOUN
ejpam-4788	52	45	)	)	PUNCT
ejpam-4788	52	46	≤	≤	NOUN
ejpam-4788	52	47	ς(k	ς(k	NOUN
ejpam-4788	52	48	)	)	PUNCT
ejpam-4788	52	49	,	,	PUNCT
ejpam-4788	52	50	(	(	PUNCT
ejpam-4788	52	51	ξ	ξ	X
ejpam-4788	52	52	∪	∪	ADP
ejpam-4788	52	53	ς)(k	ς)(k	NOUN
ejpam-4788	52	54	)	)	PUNCT
ejpam-4788	52	55	=	=	SYM
ejpam-4788	52	56	max{ξ(k	max{ξ(k	NOUN
ejpam-4788	52	57	)	)	PUNCT
ejpam-4788	52	58	,	,	PUNCT
ejpam-4788	52	59	ς(k	ς(k	NOUN
ejpam-4788	52	60	)	)	PUNCT
ejpam-4788	52	61	}	}	PUNCT
ejpam-4788	52	62	and	and	CCONJ
ejpam-4788	52	63	(	(	PUNCT
ejpam-4788	52	64	ξ	ξ	X
ejpam-4788	52	65	∩	∩	NOUN
ejpam-4788	52	66	ς)(k	ς)(k	NOUN
ejpam-4788	52	67	)	)	PUNCT
ejpam-4788	52	68	=	=	SYM
ejpam-4788	52	69	min{ξ(k	min{ξ(k	NOUN
ejpam-4788	52	70	)	)	PUNCT
ejpam-4788	52	71	,	,	PUNCT
ejpam-4788	52	72	ς(k	ς(k	NOUN
ejpam-4788	52	73	)	)	PUNCT
ejpam-4788	52	74	}	}	PUNCT
ejpam-4788	52	75	for	for	ADP
ejpam-4788	52	76	all	all	DET
ejpam-4788	52	77	k	k	PROPN
ejpam-4788	52	78	∈	∈	PROPN
ejpam-4788	52	79	t.	t.	NOUN
ejpam-4788	52	80	for	for	ADP
ejpam-4788	52	81	any	any	DET
ejpam-4788	52	82	element	element	NOUN
ejpam-4788	52	83	k	k	PROPN
ejpam-4788	52	84	in	in	ADP
ejpam-4788	52	85	a	a	DET
ejpam-4788	52	86	semigroup	semigroup	NOUN
ejpam-4788	52	87	s	s	NOUN
ejpam-4788	52	88	,	,	PUNCT
ejpam-4788	52	89	define	define	VERB
ejpam-4788	52	90	the	the	DET
ejpam-4788	52	91	set	set	NOUN
ejpam-4788	52	92	fk	fk	INTJ
ejpam-4788	52	93	by	by	ADP
ejpam-4788	52	94	fk	fk	INTJ
ejpam-4788	52	95	:	:	PUNCT
ejpam-4788	53	1	=	=	SYM
ejpam-4788	53	2	{	{	PUNCT
ejpam-4788	53	3	(	(	PUNCT
ejpam-4788	53	4	y	y	PROPN
ejpam-4788	53	5	,	,	PUNCT
ejpam-4788	53	6	z	z	NOUN
ejpam-4788	53	7	)	)	PUNCT
ejpam-4788	53	8	∈	∈	PROPN
ejpam-4788	53	9	s×s	s×s	PROPN
ejpam-4788	54	1	|	|	ADV
ejpam-4788	54	2	k	k	PROPN
ejpam-4788	54	3	=	=	SYM
ejpam-4788	54	4	yz	yz	PROPN
ejpam-4788	54	5	}	}	PUNCT
ejpam-4788	54	6	.	.	PUNCT
ejpam-4788	55	1	for	for	ADP
ejpam-4788	55	2	two	two	NUM
ejpam-4788	55	3	fuzzy	fuzzy	ADJ
ejpam-4788	55	4	sets	set	NOUN
ejpam-4788	55	5	ξ	ξ	PROPN
ejpam-4788	55	6	and	and	CCONJ
ejpam-4788	55	7	ς	ς	PROPN
ejpam-4788	55	8	on	on	ADP
ejpam-4788	55	9	a	a	DET
ejpam-4788	55	10	semigroup	semigroup	NOUN
ejpam-4788	55	11	s	s	NOUN
ejpam-4788	55	12	,	,	PUNCT
ejpam-4788	55	13	define	define	VERB
ejpam-4788	55	14	the	the	DET
ejpam-4788	55	15	product	product	NOUN
ejpam-4788	55	16	ξ	ξ	PROPN
ejpam-4788	55	17	◦	◦	NOUN
ejpam-4788	55	18	ς	ς	PROPN
ejpam-4788	55	19	as	as	SCONJ
ejpam-4788	55	20	follows	follow	VERB
ejpam-4788	55	21	:	:	PUNCT
ejpam-4788	55	22	for	for	ADP
ejpam-4788	55	23	all	all	DET
ejpam-4788	55	24	k	k	PROPN
ejpam-4788	55	25	∈	∈	PROPN
ejpam-4788	55	26	s	s	NOUN
ejpam-4788	55	27	,	,	PUNCT
ejpam-4788	55	28	(	(	PUNCT
ejpam-4788	55	29	ξ	ξ	X
ejpam-4788	55	30	◦	◦	NOUN
ejpam-4788	55	31	ς)(k	ς)(k	NOUN
ejpam-4788	55	32	)	)	PUNCT
ejpam-4788	55	33	=	=	SYM
ejpam-4788	56	1			PROPN
ejpam-4788	56	2	∨	∨	NOUN
ejpam-4788	56	3	(	(	PUNCT
ejpam-4788	56	4	y	y	PROPN
ejpam-4788	56	5	,	,	PUNCT
ejpam-4788	56	6	z)∈fk	z)∈fk	PROPN
ejpam-4788	56	7	{	{	PUNCT
ejpam-4788	56	8	ξ(y	ξ(y	PROPN
ejpam-4788	56	9	)	)	PUNCT
ejpam-4788	56	10	∧	∧	PROPN
ejpam-4788	56	11	ς(z	ς(z	PROPN
ejpam-4788	56	12	)	)	PUNCT
ejpam-4788	56	13	}	}	PUNCT
ejpam-4788	56	14	if	if	SCONJ
ejpam-4788	56	15	fk	fk	INTJ
ejpam-4788	56	16	̸=	̸=	PROPN
ejpam-4788	56	17	∅	∅	NOUN
ejpam-4788	56	18	,	,	PUNCT
ejpam-4788	56	19	0	0	PUNCT
ejpam-4788	56	20	if	if	SCONJ
ejpam-4788	56	21	fk	fk	INTJ
ejpam-4788	56	22	=	=	PUNCT
ejpam-4788	56	23	∅.	∅.	PRON
ejpam-4788	56	24	definition	definition	NOUN
ejpam-4788	56	25	3	3	NUM
ejpam-4788	56	26	.	.	PUNCT
ejpam-4788	57	1	let	let	VERB
ejpam-4788	57	2	i	i	PRON
ejpam-4788	57	3	be	be	AUX
ejpam-4788	57	4	a	a	DET
ejpam-4788	57	5	non	non	ADJ
ejpam-4788	57	6	-	-	ADJ
ejpam-4788	57	7	empty	empty	ADJ
ejpam-4788	57	8	set	set	NOUN
ejpam-4788	57	9	of	of	ADP
ejpam-4788	57	10	a	a	DET
ejpam-4788	57	11	semigroup	semigroup	PROPN
ejpam-4788	57	12	s.	s.	PROPN
ejpam-4788	57	13	a	a	DET
ejpam-4788	57	14	positive	positive	ADJ
ejpam-4788	57	15	characteristic	characteristic	ADJ
ejpam-4788	57	16	function	function	NOUN
ejpam-4788	57	17	and	and	CCONJ
ejpam-4788	57	18	a	a	DET
ejpam-4788	57	19	negative	negative	ADJ
ejpam-4788	57	20	characteristic	characteristic	ADJ
ejpam-4788	57	21	function	function	NOUN
ejpam-4788	57	22	are	be	AUX
ejpam-4788	57	23	respectively	respectively	ADV
ejpam-4788	57	24	defined	define	VERB
ejpam-4788	57	25	by	by	ADP
ejpam-4788	57	26	λi	λi	ADP
ejpam-4788	57	27	:	:	PUNCT
ejpam-4788	57	28	s	s	X
ejpam-4788	57	29	→	→	SYM
ejpam-4788	57	30	[	[	X
ejpam-4788	57	31	0	0	NUM
ejpam-4788	57	32	,	,	PUNCT
ejpam-4788	57	33	1	1	NUM
ejpam-4788	57	34	]	]	PUNCT
ejpam-4788	57	35	,	,	PUNCT
ejpam-4788	57	36	k	k	PROPN
ejpam-4788	57	37	7→	7→	NUM
ejpam-4788	57	38	λi(u	λi(u	NUM
ejpam-4788	57	39	)	)	PUNCT
ejpam-4788	58	1	:	:	PUNCT
ejpam-4788	58	2	=	=	SYM
ejpam-4788	58	3	{	{	PUNCT
ejpam-4788	58	4	1	1	NUM
ejpam-4788	58	5	k	k	X
ejpam-4788	58	6	∈	∈	PROPN
ejpam-4788	58	7	i	i	PRON
ejpam-4788	58	8	,	,	PUNCT
ejpam-4788	58	9	0	0	PUNCT
ejpam-4788	59	1	k	k	NOUN
ejpam-4788	59	2	/∈	/∈	PUNCT
ejpam-4788	60	1	i	i	PRON
ejpam-4788	60	2	,	,	PUNCT
ejpam-4788	60	3	the	the	DET
ejpam-4788	60	4	following	follow	VERB
ejpam-4788	60	5	definitions	definition	NOUN
ejpam-4788	60	6	are	be	AUX
ejpam-4788	60	7	types	type	NOUN
ejpam-4788	60	8	of	of	ADP
ejpam-4788	60	9	fuzzy	fuzzy	ADJ
ejpam-4788	60	10	almost	almost	ADV
ejpam-4788	60	11	ideal	ideal	ADJ
ejpam-4788	60	12	on	on	ADP
ejpam-4788	60	13	semigroups	semigroup	NOUN
ejpam-4788	60	14	.	.	PUNCT
ejpam-4788	61	1	definition	definition	NOUN
ejpam-4788	61	2	4	4	NUM
ejpam-4788	61	3	.	.	PUNCT
ejpam-4788	62	1	[	[	X
ejpam-4788	62	2	11	11	NUM
ejpam-4788	62	3	]	]	PUNCT
ejpam-4788	62	4	a	a	DET
ejpam-4788	62	5	fuzzy	fuzzy	ADJ
ejpam-4788	62	6	set	set	VERB
ejpam-4788	62	7	ξ	ξ	PROPN
ejpam-4788	62	8	of	of	ADP
ejpam-4788	62	9	a	a	DET
ejpam-4788	62	10	semigroup	semigroup	NOUN
ejpam-4788	62	11	s	s	NOUN
ejpam-4788	62	12	and	and	CCONJ
ejpam-4788	62	13	s	s	PROPN
ejpam-4788	62	14	∈	∈	PROPN
ejpam-4788	62	15	s	s	NOUN
ejpam-4788	62	16	,	,	PUNCT
ejpam-4788	62	17	is	be	AUX
ejpam-4788	62	18	said	say	VERB
ejpam-4788	62	19	to	to	PART
ejpam-4788	62	20	be	be	AUX
ejpam-4788	62	21	[	[	PUNCT
ejpam-4788	62	22	(	(	PUNCT
ejpam-4788	62	23	i	i	NOUN
ejpam-4788	62	24	)	)	PUNCT
ejpam-4788	62	25	]	]	PUNCT
ejpam-4788	63	1	(	(	PUNCT
ejpam-4788	63	2	i	i	NOUN
ejpam-4788	63	3	)	)	PUNCT
ejpam-4788	63	4	a	a	DET
ejpam-4788	63	5	fuzzy	fuzzy	ADJ
ejpam-4788	63	6	almost	almost	ADV
ejpam-4788	63	7	left	leave	VERB
ejpam-4788	63	8	(	(	PUNCT
ejpam-4788	63	9	right	right	ADJ
ejpam-4788	63	10	)	)	PUNCT
ejpam-4788	63	11	ideal	ideal	NOUN
ejpam-4788	63	12	of	of	ADP
ejpam-4788	63	13	s	s	PRON
ejpam-4788	63	14	if	if	SCONJ
ejpam-4788	63	15	λs	λs	ADP
ejpam-4788	63	16	◦	◦	NOUN
ejpam-4788	63	17	ξ	ξ	X
ejpam-4788	63	18	∩	∩	NOUN
ejpam-4788	63	19	ξ	ξ	PROPN
ejpam-4788	63	20	̸=	̸=	PROPN
ejpam-4788	63	21	0	0	NUM
ejpam-4788	63	22	(	(	PUNCT
ejpam-4788	63	23	ξ	ξ	PROPN
ejpam-4788	63	24	◦	◦	NOUN
ejpam-4788	63	25	λs	λs	ADP
ejpam-4788	63	26	∩	∩	NOUN
ejpam-4788	63	27	ξ	ξ	PROPN
ejpam-4788	63	28	̸=	̸=	PROPN
ejpam-4788	63	29	0	0	NUM
ejpam-4788	63	30	)	)	PUNCT
ejpam-4788	63	31	for	for	ADP
ejpam-4788	63	32	all	all	PRON
ejpam-4788	63	33	s	s	PART
ejpam-4788	63	34	∈	∈	PROPN
ejpam-4788	63	35	s	s	NOUN
ejpam-4788	63	36	,	,	PUNCT
ejpam-4788	63	37	(	(	PUNCT
ejpam-4788	63	38	ii	ii	NOUN
ejpam-4788	63	39	)	)	PUNCT
ejpam-4788	63	40	a	a	DET
ejpam-4788	63	41	fuzzy	fuzzy	ADJ
ejpam-4788	63	42	almost	almost	ADV
ejpam-4788	63	43	ideal	ideal	ADJ
ejpam-4788	63	44	of	of	ADP
ejpam-4788	63	45	s	s	PRON
ejpam-4788	63	46	if	if	SCONJ
ejpam-4788	63	47	it	it	PRON
ejpam-4788	63	48	is	be	AUX
ejpam-4788	63	49	both	both	CCONJ
ejpam-4788	63	50	a	a	DET
ejpam-4788	63	51	fuzzy	fuzzy	ADJ
ejpam-4788	63	52	almost	almost	ADV
ejpam-4788	63	53	left	leave	VERB
ejpam-4788	63	54	ideal	ideal	NOUN
ejpam-4788	63	55	and	and	CCONJ
ejpam-4788	63	56	a	a	DET
ejpam-4788	63	57	fuzzy	fuzzy	ADJ
ejpam-4788	63	58	almost	almost	ADV
ejpam-4788	63	59	right	right	ADJ
ejpam-4788	63	60	ideal	ideal	NOUN
ejpam-4788	63	61	of	of	ADP
ejpam-4788	63	62	s	s	PRON
ejpam-4788	63	63	for	for	ADP
ejpam-4788	63	64	all	all	PRON
ejpam-4788	63	65	s	s	PART
ejpam-4788	63	66	∈	∈	PROPN
ejpam-4788	63	67	s	s	NOUN
ejpam-4788	63	68	,	,	PUNCT
ejpam-4788	63	69	(	(	PUNCT
ejpam-4788	63	70	iii	iii	NOUN
ejpam-4788	63	71	)	)	PUNCT
ejpam-4788	63	72	a	a	DET
ejpam-4788	63	73	fuzzy	fuzzy	ADJ
ejpam-4788	63	74	almost	almost	ADV
ejpam-4788	63	75	bi	bi	NOUN
ejpam-4788	63	76	-	-	NOUN
ejpam-4788	63	77	ideal	ideal	NOUN
ejpam-4788	63	78	of	of	ADP
ejpam-4788	63	79	s	s	PRON
ejpam-4788	63	80	if	if	SCONJ
ejpam-4788	63	81	ξ	ξ	X
ejpam-4788	63	82	◦	◦	NOUN
ejpam-4788	63	83	λs	λs	ADP
ejpam-4788	63	84	◦	◦	NOUN
ejpam-4788	63	85	ξ	ξ	X
ejpam-4788	63	86	∩	∩	NOUN
ejpam-4788	63	87	ξ	ξ	X
ejpam-4788	63	88	̸=	̸=	PROPN
ejpam-4788	63	89	0	0	NUM
ejpam-4788	63	90	for	for	ADP
ejpam-4788	63	91	all	all	PRON
ejpam-4788	63	92	s	s	PART
ejpam-4788	63	93	∈	∈	PROPN
ejpam-4788	63	94	s	s	NOUN
ejpam-4788	63	95	,	,	PUNCT
ejpam-4788	63	96	(	(	PUNCT
ejpam-4788	63	97	iv	iv	X
ejpam-4788	63	98	)	)	PUNCT
ejpam-4788	63	99	a	a	DET
ejpam-4788	63	100	fuzzy	fuzzy	ADJ
ejpam-4788	63	101	almost	almost	ADV
ejpam-4788	63	102	quasi	quasi	ADJ
ejpam-4788	63	103	-	-	NOUN
ejpam-4788	63	104	ideal	ideal	ADJ
ejpam-4788	63	105	of	of	ADP
ejpam-4788	63	106	s	s	PRON
ejpam-4788	63	107	if	if	SCONJ
ejpam-4788	63	108	λs	λs	ADP
ejpam-4788	63	109	◦	◦	NOUN
ejpam-4788	63	110	ξ	ξ	X
ejpam-4788	63	111	∩	∩	X
ejpam-4788	63	112	ξ	ξ	X
ejpam-4788	63	113	◦	◦	NOUN
ejpam-4788	63	114	λs	λs	ADP
ejpam-4788	63	115	∩	∩	NOUN
ejpam-4788	63	116	ξ	ξ	PROPN
ejpam-4788	63	117	̸=	̸=	PROPN
ejpam-4788	63	118	0	0	NUM
ejpam-4788	63	119	.	.	PUNCT
ejpam-4788	64	1	theorem	theorem	NOUN
ejpam-4788	64	2	1	1	NUM
ejpam-4788	64	3	.	.	PUNCT
ejpam-4788	65	1	[	[	X
ejpam-4788	65	2	10	10	NUM
ejpam-4788	65	3	]	]	PUNCT
ejpam-4788	65	4	let	let	VERB
ejpam-4788	65	5	ξ	ξ	X
ejpam-4788	65	6	be	be	AUX
ejpam-4788	65	7	a	a	DET
ejpam-4788	65	8	nonzero	nonzero	ADJ
ejpam-4788	65	9	fuzzy	fuzzy	ADJ
ejpam-4788	65	10	set	set	NOUN
ejpam-4788	65	11	of	of	ADP
ejpam-4788	65	12	a	a	DET
ejpam-4788	65	13	semigroup	semigroup	PROPN
ejpam-4788	65	14	s.	s.	PROPN
ejpam-4788	65	15	then	then	ADV
ejpam-4788	65	16	ξ	ξ	PROPN
ejpam-4788	65	17	is	be	AUX
ejpam-4788	65	18	a	a	DET
ejpam-4788	65	19	fuzzy	fuzzy	ADJ
ejpam-4788	65	20	subsemigroup	subsemigroup	NOUN
ejpam-4788	65	21	(	(	PUNCT
ejpam-4788	65	22	ideal	ideal	NOUN
ejpam-4788	65	23	)	)	PUNCT
ejpam-4788	65	24	of	of	ADP
ejpam-4788	65	25	s	s	PRON
ejpam-4788	65	26	if	if	SCONJ
ejpam-4788	65	27	and	and	CCONJ
ejpam-4788	65	28	only	only	ADV
ejpam-4788	65	29	if	if	SCONJ
ejpam-4788	65	30	supp(ξ	supp(ξ	PROPN
ejpam-4788	65	31	)	)	PUNCT
ejpam-4788	65	32	is	be	AUX
ejpam-4788	65	33	a	a	DET
ejpam-4788	65	34	subsemigroup	subsemigroup	NOUN
ejpam-4788	65	35	(	(	PUNCT
ejpam-4788	65	36	ideal	ideal	NOUN
ejpam-4788	65	37	)	)	PUNCT
ejpam-4788	65	38	of	of	ADP
ejpam-4788	65	39	s.	s.	PROPN
ejpam-4788	65	40	the	the	DET
ejpam-4788	65	41	following	follow	VERB
ejpam-4788	65	42	definitions	definition	NOUN
ejpam-4788	65	43	are	be	AUX
ejpam-4788	65	44	types	type	NOUN
ejpam-4788	65	45	of	of	ADP
ejpam-4788	65	46	fuzzy	fuzzy	ADJ
ejpam-4788	65	47	subsemigroups	subsemigroup	NOUN
ejpam-4788	65	48	on	on	ADP
ejpam-4788	65	49	γ	γ	NOUN
ejpam-4788	65	50	-	-	PUNCT
ejpam-4788	65	51	semigroups	semigroup	NOUN
ejpam-4788	65	52	.	.	PUNCT
ejpam-4788	66	1	definition	definition	NOUN
ejpam-4788	66	2	5	5	NUM
ejpam-4788	66	3	.	.	PUNCT
ejpam-4788	67	1	[	[	X
ejpam-4788	67	2	10	10	NUM
ejpam-4788	67	3	]	]	X
ejpam-4788	67	4	a	a	DET
ejpam-4788	67	5	fuzzy	fuzzy	ADJ
ejpam-4788	67	6	set	set	VERB
ejpam-4788	67	7	ξ	ξ	PROPN
ejpam-4788	67	8	of	of	ADP
ejpam-4788	67	9	a	a	DET
ejpam-4788	67	10	γ	γ	NOUN
ejpam-4788	67	11	-	-	PUNCT
ejpam-4788	67	12	semigroup	semigroup	NOUN
ejpam-4788	67	13	s	s	VERB
ejpam-4788	67	14	is	be	AUX
ejpam-4788	67	15	said	say	VERB
ejpam-4788	67	16	to	to	PART
ejpam-4788	67	17	be	be	AUX
ejpam-4788	67	18	[	[	PUNCT
ejpam-4788	67	19	(	(	PUNCT
ejpam-4788	67	20	i	i	NOUN
ejpam-4788	67	21	)	)	PUNCT
ejpam-4788	67	22	]	]	PUNCT
ejpam-4788	68	1	(	(	PUNCT
ejpam-4788	68	2	i	i	NOUN
ejpam-4788	68	3	)	)	PUNCT
ejpam-4788	68	4	a	a	DET
ejpam-4788	68	5	fuzzy	fuzzy	ADJ
ejpam-4788	68	6	subsemigroup	subsemigroup	NOUN
ejpam-4788	68	7	of	of	ADP
ejpam-4788	68	8	s	s	PRON
ejpam-4788	68	9	if	if	SCONJ
ejpam-4788	68	10	ξ(uγv	ξ(uγv	VERB
ejpam-4788	68	11	)	)	PUNCT
ejpam-4788	68	12	≥	≥	NOUN
ejpam-4788	68	13	ξ(u	ξ(u	NOUN
ejpam-4788	68	14	)	)	PUNCT
ejpam-4788	68	15	∧	∧	PROPN
ejpam-4788	68	16	ξ(v	ξ(v	NOUN
ejpam-4788	68	17	)	)	PUNCT
ejpam-4788	68	18	for	for	ADP
ejpam-4788	68	19	all	all	DET
ejpam-4788	68	20	u	u	NOUN
ejpam-4788	68	21	,	,	PUNCT
ejpam-4788	68	22	v	v	ADP
ejpam-4788	68	23	∈	∈	PROPN
ejpam-4788	68	24	s	s	NOUN
ejpam-4788	68	25	and	and	CCONJ
ejpam-4788	68	26	γ	γ	PROPN
ejpam-4788	68	27	∈	∈	PROPN
ejpam-4788	68	28	γ	γ	X
ejpam-4788	68	29	.	.	PUNCT
ejpam-4788	69	1	p.	p.	NOUN
ejpam-4788	69	2	khamrot	khamrot	PROPN
ejpam-4788	69	3	,	,	PUNCT
ejpam-4788	69	4	t.	t.	PROPN
ejpam-4788	69	5	gaketem	gaketem	PROPN
ejpam-4788	69	6	/	/	SYM
ejpam-4788	69	7	eur	eur	PROPN
ejpam-4788	69	8	.	.	PUNCT
ejpam-4788	70	1	j.	j.	PROPN
ejpam-4788	70	2	pure	pure	PROPN
ejpam-4788	70	3	appl	appl	PROPN
ejpam-4788	70	4	.	.	PROPN
ejpam-4788	70	5	math	math	PROPN
ejpam-4788	70	6	,	,	PUNCT
ejpam-4788	70	7	16	16	NUM
ejpam-4788	70	8	(	(	PUNCT
ejpam-4788	70	9	3	3	NUM
ejpam-4788	70	10	)	)	PUNCT
ejpam-4788	70	11	(	(	PUNCT
ejpam-4788	70	12	2023	2023	NUM
ejpam-4788	70	13	)	)	PUNCT
ejpam-4788	70	14	,	,	PUNCT
ejpam-4788	70	15	1592	1592	NUM
ejpam-4788	70	16	-	-	SYM
ejpam-4788	70	17	1607	1607	NUM
ejpam-4788	70	18	1595	1595	NUM
ejpam-4788	70	19	(	(	PUNCT
ejpam-4788	70	20	ii	ii	NOUN
ejpam-4788	70	21	)	)	PUNCT
ejpam-4788	70	22	a	a	DET
ejpam-4788	70	23	fuzzy	fuzzy	ADJ
ejpam-4788	70	24	left	left	ADJ
ejpam-4788	70	25	(	(	PUNCT
ejpam-4788	70	26	right	right	ADJ
ejpam-4788	70	27	)	)	PUNCT
ejpam-4788	70	28	ideal	ideal	NOUN
ejpam-4788	70	29	of	of	ADP
ejpam-4788	70	30	s	s	PRON
ejpam-4788	70	31	if	if	SCONJ
ejpam-4788	70	32	ξ(uγv	ξ(uγv	ADP
ejpam-4788	70	33	)	)	PUNCT
ejpam-4788	70	34	≥	≥	NUM
ejpam-4788	70	35	ξ(v	ξ(v	PROPN
ejpam-4788	70	36	)	)	PUNCT
ejpam-4788	70	37	(	(	PUNCT
ejpam-4788	70	38	ξ(uγv	ξ(uγv	VERB
ejpam-4788	70	39	)	)	PUNCT
ejpam-4788	70	40	≥	≥	NOUN
ejpam-4788	70	41	ξ(u	ξ(u	NOUN
ejpam-4788	70	42	)	)	PUNCT
ejpam-4788	70	43	)	)	PUNCT
ejpam-4788	70	44	for	for	ADP
ejpam-4788	70	45	all	all	DET
ejpam-4788	70	46	u	u	NOUN
ejpam-4788	70	47	,	,	PUNCT
ejpam-4788	70	48	v	v	ADP
ejpam-4788	70	49	∈	∈	PROPN
ejpam-4788	70	50	s	s	NOUN
ejpam-4788	70	51	and	and	CCONJ
ejpam-4788	70	52	γ	γ	PROPN
ejpam-4788	70	53	∈	∈	PROPN
ejpam-4788	70	54	γ	γ	X
ejpam-4788	70	55	.	.	PUNCT
ejpam-4788	70	56	(	(	PUNCT
ejpam-4788	70	57	iii	iii	X
ejpam-4788	70	58	)	)	PUNCT
ejpam-4788	70	59	a	a	DET
ejpam-4788	70	60	fuzzy	fuzzy	ADJ
ejpam-4788	70	61	ideal	ideal	NOUN
ejpam-4788	70	62	of	of	ADP
ejpam-4788	70	63	s	s	PRON
ejpam-4788	70	64	if	if	SCONJ
ejpam-4788	70	65	it	it	PRON
ejpam-4788	70	66	is	be	AUX
ejpam-4788	70	67	both	both	CCONJ
ejpam-4788	70	68	a	a	DET
ejpam-4788	70	69	fuzzy	fuzzy	ADJ
ejpam-4788	70	70	left	leave	VERB
ejpam-4788	70	71	ideal	ideal	NOUN
ejpam-4788	70	72	and	and	CCONJ
ejpam-4788	70	73	a	a	DET
ejpam-4788	70	74	fuzzy	fuzzy	ADJ
ejpam-4788	70	75	right	right	ADJ
ejpam-4788	70	76	ideal	ideal	NOUN
ejpam-4788	70	77	of	of	ADP
ejpam-4788	70	78	s.	s.	PROPN
ejpam-4788	70	79	(	(	PUNCT
ejpam-4788	70	80	iv	iv	X
ejpam-4788	70	81	)	)	PUNCT
ejpam-4788	70	82	a	a	DET
ejpam-4788	70	83	fuzzy	fuzzy	ADJ
ejpam-4788	70	84	bi	bi	NOUN
ejpam-4788	70	85	-	-	NOUN
ejpam-4788	70	86	ideal	ideal	NOUN
ejpam-4788	70	87	of	of	ADP
ejpam-4788	70	88	s	s	PRON
ejpam-4788	70	89	if	if	SCONJ
ejpam-4788	70	90	ξ	ξ	PROPN
ejpam-4788	70	91	is	be	AUX
ejpam-4788	70	92	a	a	DET
ejpam-4788	70	93	fuzzy	fuzzy	ADJ
ejpam-4788	70	94	subsemigroup	subsemigroup	NOUN
ejpam-4788	70	95	of	of	ADP
ejpam-4788	70	96	s	s	NOUN
ejpam-4788	70	97	and	and	CCONJ
ejpam-4788	70	98	ξ(uγvβw	ξ(uγvβw	NOUN
ejpam-4788	70	99	)	)	PUNCT
ejpam-4788	70	100	≥	≥	NOUN
ejpam-4788	70	101	ξ(u	ξ(u	NOUN
ejpam-4788	70	102	)	)	PUNCT
ejpam-4788	70	103	∧	∧	PROPN
ejpam-4788	70	104	ξ(w	ξ(w	PROPN
ejpam-4788	70	105	)	)	PUNCT
ejpam-4788	70	106	for	for	ADP
ejpam-4788	70	107	all	all	DET
ejpam-4788	70	108	u	u	NOUN
ejpam-4788	70	109	,	,	PUNCT
ejpam-4788	70	110	v	v	NOUN
ejpam-4788	70	111	,	,	PUNCT
ejpam-4788	70	112	w	w	PROPN
ejpam-4788	70	113	∈	∈	PROPN
ejpam-4788	70	114	s	s	NOUN
ejpam-4788	70	115	and	and	CCONJ
ejpam-4788	70	116	γ	γ	PROPN
ejpam-4788	70	117	,	,	PUNCT
ejpam-4788	70	118	β	β	PROPN
ejpam-4788	70	119	∈	∈	PROPN
ejpam-4788	70	120	γ	γ	X
ejpam-4788	70	121	.	.	PUNCT
ejpam-4788	71	1	now	now	ADV
ejpam-4788	71	2	,	,	PUNCT
ejpam-4788	71	3	we	we	PRON
ejpam-4788	71	4	review	review	VERB
ejpam-4788	71	5	the	the	DET
ejpam-4788	71	6	definition	definition	NOUN
ejpam-4788	71	7	of	of	ADP
ejpam-4788	71	8	a	a	DET
ejpam-4788	71	9	bipolar	bipolar	ADJ
ejpam-4788	71	10	valued	value	VERB
ejpam-4788	71	11	fuzzy	fuzzy	ADJ
ejpam-4788	71	12	set	set	ADJ
ejpam-4788	71	13	and	and	CCONJ
ejpam-4788	71	14	basic	basic	ADJ
ejpam-4788	71	15	properties	property	NOUN
ejpam-4788	71	16	used	use	VERB
ejpam-4788	71	17	in	in	ADP
ejpam-4788	71	18	the	the	DET
ejpam-4788	71	19	next	next	ADJ
ejpam-4788	71	20	section	section	NOUN
ejpam-4788	71	21	.	.	PUNCT
ejpam-4788	72	1	definition	definition	NOUN
ejpam-4788	72	2	6	6	NUM
ejpam-4788	72	3	.	.	PUNCT
ejpam-4788	73	1	[	[	X
ejpam-4788	73	2	8	8	NUM
ejpam-4788	73	3	]	]	PUNCT
ejpam-4788	73	4	let	let	VERB
ejpam-4788	73	5	s	s	PRON
ejpam-4788	73	6	be	be	AUX
ejpam-4788	73	7	a	a	DET
ejpam-4788	73	8	non	non	ADJ
ejpam-4788	73	9	-	-	ADJ
ejpam-4788	73	10	empty	empty	ADJ
ejpam-4788	73	11	set	set	NOUN
ejpam-4788	73	12	.	.	PUNCT
ejpam-4788	74	1	a	a	DET
ejpam-4788	74	2	bipolar	bipolar	ADJ
ejpam-4788	74	3	fuzzy	fuzzy	ADJ
ejpam-4788	74	4	set	set	NOUN
ejpam-4788	74	5	(	(	PUNCT
ejpam-4788	74	6	bf	bf	NOUN
ejpam-4788	74	7	set	set	NOUN
ejpam-4788	74	8	)	)	PUNCT
ejpam-4788	74	9	ξ	ξ	PROPN
ejpam-4788	74	10	on	on	ADP
ejpam-4788	74	11	s	s	PROPN
ejpam-4788	74	12	is	be	AUX
ejpam-4788	74	13	an	an	DET
ejpam-4788	74	14	object	object	NOUN
ejpam-4788	74	15	having	have	VERB
ejpam-4788	74	16	the	the	DET
ejpam-4788	74	17	form	form	NOUN
ejpam-4788	74	18	ξ	ξ	NOUN
ejpam-4788	74	19	:	:	PUNCT
ejpam-4788	74	20	=	=	SYM
ejpam-4788	74	21	{	{	PUNCT
ejpam-4788	74	22	(	(	PUNCT
ejpam-4788	74	23	k	k	NOUN
ejpam-4788	74	24	,	,	PUNCT
ejpam-4788	74	25	ξp(k	ξp(k	NOUN
ejpam-4788	74	26	)	)	PUNCT
ejpam-4788	74	27	,	,	PUNCT
ejpam-4788	74	28	ξn(k	ξn(k	NUM
ejpam-4788	74	29	)	)	PUNCT
ejpam-4788	74	30	)	)	PUNCT
ejpam-4788	75	1	|	|	ADV
ejpam-4788	75	2	k	k	PROPN
ejpam-4788	75	3	∈	∈	PROPN
ejpam-4788	75	4	s	s	PART
ejpam-4788	75	5	}	}	PUNCT
ejpam-4788	75	6	,	,	PUNCT
ejpam-4788	75	7	where	where	SCONJ
ejpam-4788	75	8	ξp	ξp	X
ejpam-4788	75	9	:	:	PUNCT
ejpam-4788	75	10	s	s	X
ejpam-4788	75	11	→	→	SYM
ejpam-4788	75	12	[	[	X
ejpam-4788	75	13	0	0	NUM
ejpam-4788	75	14	,	,	PUNCT
ejpam-4788	75	15	1	1	NUM
ejpam-4788	75	16	]	]	PUNCT
ejpam-4788	75	17	and	and	CCONJ
ejpam-4788	75	18	ξn	ξn	INTJ
ejpam-4788	75	19	:	:	PUNCT
ejpam-4788	75	20	s	s	X
ejpam-4788	75	21	→	→	X
ejpam-4788	75	22	[	[	X
ejpam-4788	75	23	−1	−1	NOUN
ejpam-4788	75	24	,	,	PUNCT
ejpam-4788	75	25	0	0	NUM
ejpam-4788	75	26	]	]	PUNCT
ejpam-4788	75	27	.	.	PUNCT
ejpam-4788	76	1	remark	remark	PROPN
ejpam-4788	76	2	1	1	NUM
ejpam-4788	76	3	.	.	PUNCT
ejpam-4788	77	1	for	for	ADP
ejpam-4788	77	2	the	the	DET
ejpam-4788	77	3	sake	sake	NOUN
ejpam-4788	77	4	of	of	ADP
ejpam-4788	77	5	simplicity	simplicity	NOUN
ejpam-4788	77	6	,	,	PUNCT
ejpam-4788	77	7	we	we	PRON
ejpam-4788	77	8	shall	shall	AUX
ejpam-4788	77	9	use	use	VERB
ejpam-4788	77	10	the	the	DET
ejpam-4788	77	11	symbol	symbol	NOUN
ejpam-4788	77	12	ξ	ξ	X
ejpam-4788	77	13	=	=	SYM
ejpam-4788	77	14	(	(	PUNCT
ejpam-4788	77	15	s	s	PROPN
ejpam-4788	77	16	;	;	PUNCT
ejpam-4788	77	17	ξp	ξp	NUM
ejpam-4788	77	18	,	,	PUNCT
ejpam-4788	77	19	ξn	ξn	NOUN
ejpam-4788	77	20	)	)	PUNCT
ejpam-4788	77	21	for	for	ADP
ejpam-4788	77	22	the	the	DET
ejpam-4788	77	23	bf	bf	NOUN
ejpam-4788	77	24	set	set	VERB
ejpam-4788	77	25	ξ	ξ	X
ejpam-4788	77	26	=	=	SYM
ejpam-4788	77	27	{	{	PUNCT
ejpam-4788	77	28	(	(	PUNCT
ejpam-4788	77	29	k	k	NOUN
ejpam-4788	77	30	,	,	PUNCT
ejpam-4788	77	31	ξp(k	ξp(k	NOUN
ejpam-4788	77	32	)	)	PUNCT
ejpam-4788	77	33	,	,	PUNCT
ejpam-4788	77	34	ξn(k	ξn(k	NUM
ejpam-4788	77	35	)	)	PUNCT
ejpam-4788	77	36	)	)	PUNCT
ejpam-4788	78	1	|	|	ADV
ejpam-4788	78	2	k	k	PROPN
ejpam-4788	78	3	∈	∈	PROPN
ejpam-4788	78	4	s	s	PART
ejpam-4788	78	5	}	}	PUNCT
ejpam-4788	78	6	.	.	PUNCT
ejpam-4788	79	1	the	the	DET
ejpam-4788	79	2	following	follow	VERB
ejpam-4788	79	3	presents	present	VERB
ejpam-4788	79	4	an	an	DET
ejpam-4788	79	5	example	example	NOUN
ejpam-4788	79	6	of	of	ADP
ejpam-4788	79	7	a	a	DET
ejpam-4788	79	8	bf	bf	NOUN
ejpam-4788	79	9	set	set	NOUN
ejpam-4788	79	10	.	.	PUNCT
ejpam-4788	79	11	example	example	NOUN
ejpam-4788	80	1	1	1	NUM
ejpam-4788	80	2	.	.	PUNCT
ejpam-4788	81	1	let	let	VERB
ejpam-4788	81	2	s	s	VERB
ejpam-4788	81	3	=	=	PUNCT
ejpam-4788	81	4	{	{	PUNCT
ejpam-4788	81	5	21	21	NUM
ejpam-4788	81	6	,	,	PUNCT
ejpam-4788	81	7	22	22	NUM
ejpam-4788	81	8	,	,	PUNCT
ejpam-4788	81	9	23	23	NUM
ejpam-4788	81	10	...	...	PUNCT
ejpam-4788	81	11	}	}	PUNCT
ejpam-4788	81	12	.	.	PUNCT
ejpam-4788	82	1	define	define	VERB
ejpam-4788	82	2	ξp	ξp	ADP
ejpam-4788	82	3	:	:	PUNCT
ejpam-4788	82	4	s	s	X
ejpam-4788	82	5	→	→	SYM
ejpam-4788	82	6	[	[	X
ejpam-4788	82	7	0	0	NUM
ejpam-4788	82	8	,	,	PUNCT
ejpam-4788	82	9	1	1	NUM
ejpam-4788	82	10	]	]	PUNCT
ejpam-4788	82	11	as	as	ADP
ejpam-4788	82	12	a	a	DET
ejpam-4788	82	13	function	function	NOUN
ejpam-4788	82	14	ξp(u	ξp(u	NOUN
ejpam-4788	82	15	)	)	PUNCT
ejpam-4788	83	1	=	=	PRON
ejpam-4788	83	2	{	{	PUNCT
ejpam-4788	83	3	0	0	NUM
ejpam-4788	83	4	if	if	SCONJ
ejpam-4788	83	5	u	u	NOUN
ejpam-4788	83	6	is	be	AUX
ejpam-4788	83	7	an	an	DET
ejpam-4788	83	8	old	old	ADJ
ejpam-4788	83	9	number	number	NOUN
ejpam-4788	83	10	1	1	NUM
ejpam-4788	83	11	if	if	SCONJ
ejpam-4788	83	12	u	u	NOUN
ejpam-4788	83	13	is	be	AUX
ejpam-4788	83	14	an	an	DET
ejpam-4788	83	15	even	even	ADJ
ejpam-4788	83	16	number	number	NOUN
ejpam-4788	83	17	and	and	CCONJ
ejpam-4788	83	18	ξn	ξn	X
ejpam-4788	83	19	:	:	PUNCT
ejpam-4788	83	20	s	s	X
ejpam-4788	83	21	→	→	X
ejpam-4788	83	22	[	[	X
ejpam-4788	83	23	−1	−1	NOUN
ejpam-4788	83	24	,	,	PUNCT
ejpam-4788	83	25	0	0	NUM
ejpam-4788	83	26	]	]	PUNCT
ejpam-4788	83	27	as	as	ADP
ejpam-4788	83	28	a	a	DET
ejpam-4788	83	29	function	function	NOUN
ejpam-4788	83	30	ξn(u	ξn(u	NOUN
ejpam-4788	83	31	)	)	PUNCT
ejpam-4788	83	32	=	=	PRON
ejpam-4788	83	33	{	{	PUNCT
ejpam-4788	83	34	−1	−1	NOUN
ejpam-4788	83	35	if	if	SCONJ
ejpam-4788	83	36	u	u	NOUN
ejpam-4788	83	37	is	be	AUX
ejpam-4788	83	38	an	an	DET
ejpam-4788	83	39	old	old	ADJ
ejpam-4788	83	40	number	number	NOUN
ejpam-4788	83	41	0	0	NUM
ejpam-4788	84	1	if	if	SCONJ
ejpam-4788	84	2	u	u	NOUN
ejpam-4788	84	3	is	be	AUX
ejpam-4788	84	4	an	an	DET
ejpam-4788	84	5	even	even	ADJ
ejpam-4788	84	6	number	number	NOUN
ejpam-4788	84	7	.	.	PUNCT
ejpam-4788	85	1	then	then	ADV
ejpam-4788	85	2	ξ	ξ	X
ejpam-4788	85	3	=	=	SYM
ejpam-4788	85	4	(	(	PUNCT
ejpam-4788	85	5	s	s	PROPN
ejpam-4788	85	6	;	;	PUNCT
ejpam-4788	85	7	ξp	ξp	NUM
ejpam-4788	85	8	,	,	PUNCT
ejpam-4788	85	9	ξn	ξn	NOUN
ejpam-4788	85	10	)	)	PUNCT
ejpam-4788	85	11	is	be	AUX
ejpam-4788	85	12	a	a	DET
ejpam-4788	85	13	bf	bf	NOUN
ejpam-4788	85	14	set	set	NOUN
ejpam-4788	85	15	.	.	PUNCT
ejpam-4788	86	1	for	for	ADP
ejpam-4788	86	2	bipolar	bipolar	ADJ
ejpam-4788	86	3	fuzzy	fuzzy	ADJ
ejpam-4788	86	4	sets	set	NOUN
ejpam-4788	86	5	ξ	ξ	NOUN
ejpam-4788	86	6	=	=	SYM
ejpam-4788	86	7	(	(	PUNCT
ejpam-4788	86	8	s	s	PROPN
ejpam-4788	86	9	;	;	PUNCT
ejpam-4788	86	10	ξp	ξp	NUM
ejpam-4788	86	11	,	,	PUNCT
ejpam-4788	86	12	ξn	ξn	NOUN
ejpam-4788	86	13	)	)	PUNCT
ejpam-4788	86	14	and	and	CCONJ
ejpam-4788	86	15	ς	ς	PROPN
ejpam-4788	86	16	=	=	PUNCT
ejpam-4788	86	17	(	(	PUNCT
ejpam-4788	86	18	s	s	NOUN
ejpam-4788	86	19	;	;	PUNCT
ejpam-4788	86	20	ςp	ςp	NUM
ejpam-4788	86	21	,	,	PUNCT
ejpam-4788	86	22	ςn	ςn	NOUN
ejpam-4788	86	23	)	)	PUNCT
ejpam-4788	86	24	,	,	PUNCT
ejpam-4788	86	25	define	define	VERB
ejpam-4788	86	26	the	the	DET
ejpam-4788	86	27	products	product	NOUN
ejpam-4788	86	28	ξp	ξp	PART
ejpam-4788	86	29	◦	◦	VERB
ejpam-4788	86	30	ςp	ςp	NOUN
ejpam-4788	86	31	and	and	CCONJ
ejpam-4788	86	32	ξn	ξn	PROPN
ejpam-4788	86	33	◦	◦	NOUN
ejpam-4788	86	34	ςn	ςn	NOUN
ejpam-4788	86	35	as	as	SCONJ
ejpam-4788	86	36	follows	follow	VERB
ejpam-4788	86	37	:	:	PUNCT
ejpam-4788	86	38	for	for	SCONJ
ejpam-4788	86	39	u	u	PROPN
ejpam-4788	86	40	∈	∈	PROPN
ejpam-4788	86	41	s	s	X
ejpam-4788	86	42	(	(	PUNCT
ejpam-4788	86	43	ξp	ξp	PART
ejpam-4788	86	44	◦	◦	VERB
ejpam-4788	86	45	ςp)(k	ςp)(k	NOUN
ejpam-4788	86	46	)	)	PUNCT
ejpam-4788	86	47	=	=	PUNCT
ejpam-4788	87	1			PUNCT
ejpam-4788	87	2	∨	∨	X
ejpam-4788	87	3	(	(	PUNCT
ejpam-4788	87	4	y	y	PROPN
ejpam-4788	87	5	,	,	PUNCT
ejpam-4788	87	6	z)∈fk	z)∈fk	PROPN
ejpam-4788	87	7	{	{	PUNCT
ejpam-4788	87	8	ξp(y	ξp(y	NOUN
ejpam-4788	87	9	)	)	PUNCT
ejpam-4788	87	10	∧	∧	PROPN
ejpam-4788	87	11	ςp(z	ςp(z	NUM
ejpam-4788	87	12	)	)	PUNCT
ejpam-4788	87	13	}	}	PUNCT
ejpam-4788	87	14	if	if	SCONJ
ejpam-4788	87	15	k	k	PROPN
ejpam-4788	87	16	=	=	PUNCT
ejpam-4788	87	17	yz	yz	PROPN
ejpam-4788	87	18	0	0	PUNCT
ejpam-4788	88	1	if	if	SCONJ
ejpam-4788	88	2	otherwise	otherwise	ADV
ejpam-4788	88	3	.	.	PUNCT
ejpam-4788	89	1	and	and	CCONJ
ejpam-4788	89	2	(	(	PUNCT
ejpam-4788	89	3	ξn	ξn	NOUN
ejpam-4788	89	4	◦	◦	NOUN
ejpam-4788	89	5	ςn)(k	ςn)(k	NUM
ejpam-4788	89	6	)	)	PUNCT
ejpam-4788	90	1	=	=	PUNCT
ejpam-4788	90	2			PUNCT
ejpam-4788	90	3	∧	∧	PROPN
ejpam-4788	90	4	(	(	PUNCT
ejpam-4788	90	5	y	y	PROPN
ejpam-4788	90	6	,	,	PUNCT
ejpam-4788	90	7	z)∈fk	z)∈fk	PROPN
ejpam-4788	90	8	{	{	PUNCT
ejpam-4788	90	9	ξn(y	ξn(y	NOUN
ejpam-4788	90	10	)	)	PUNCT
ejpam-4788	90	11	∨	∨	NUM
ejpam-4788	90	12	ςn(z	ςn(z	NOUN
ejpam-4788	90	13	)	)	PUNCT
ejpam-4788	90	14	}	}	PUNCT
ejpam-4788	90	15	if	if	SCONJ
ejpam-4788	90	16	k	k	PROPN
ejpam-4788	90	17	=	=	PUNCT
ejpam-4788	90	18	yz	yz	PROPN
ejpam-4788	90	19	0	0	PUNCT
ejpam-4788	90	20	if	if	SCONJ
ejpam-4788	90	21	otherwise	otherwise	ADV
ejpam-4788	90	22	.	.	PUNCT
ejpam-4788	91	1	definition	definition	NOUN
ejpam-4788	91	2	7	7	NUM
ejpam-4788	91	3	.	.	PUNCT
ejpam-4788	92	1	let	let	VERB
ejpam-4788	92	2	i	i	PRON
ejpam-4788	92	3	be	be	AUX
ejpam-4788	92	4	a	a	DET
ejpam-4788	92	5	non	non	ADJ
ejpam-4788	92	6	-	-	ADJ
ejpam-4788	92	7	empty	empty	ADJ
ejpam-4788	92	8	set	set	NOUN
ejpam-4788	92	9	of	of	ADP
ejpam-4788	92	10	a	a	DET
ejpam-4788	92	11	semigroup	semigroup	PROPN
ejpam-4788	92	12	s.	s.	PROPN
ejpam-4788	92	13	a	a	DET
ejpam-4788	92	14	positive	positive	ADJ
ejpam-4788	92	15	characteristic	characteristic	ADJ
ejpam-4788	92	16	function	function	NOUN
ejpam-4788	92	17	and	and	CCONJ
ejpam-4788	92	18	a	a	DET
ejpam-4788	92	19	negative	negative	ADJ
ejpam-4788	92	20	characteristic	characteristic	ADJ
ejpam-4788	92	21	function	function	NOUN
ejpam-4788	92	22	are	be	AUX
ejpam-4788	92	23	respectively	respectively	ADV
ejpam-4788	92	24	defined	define	VERB
ejpam-4788	92	25	by	by	ADP
ejpam-4788	92	26	p.	p.	PROPN
ejpam-4788	92	27	khamrot	khamrot	PROPN
ejpam-4788	92	28	,	,	PUNCT
ejpam-4788	92	29	t.	t.	PROPN
ejpam-4788	92	30	gaketem	gaketem	PROPN
ejpam-4788	92	31	/	/	SYM
ejpam-4788	92	32	eur	eur	PROPN
ejpam-4788	92	33	.	.	PUNCT
ejpam-4788	93	1	j.	j.	PROPN
ejpam-4788	93	2	pure	pure	PROPN
ejpam-4788	93	3	appl	appl	PROPN
ejpam-4788	93	4	.	.	PROPN
ejpam-4788	93	5	math	math	PROPN
ejpam-4788	93	6	,	,	PUNCT
ejpam-4788	93	7	16	16	NUM
ejpam-4788	93	8	(	(	PUNCT
ejpam-4788	93	9	3	3	NUM
ejpam-4788	93	10	)	)	PUNCT
ejpam-4788	93	11	(	(	PUNCT
ejpam-4788	93	12	2023	2023	NUM
ejpam-4788	93	13	)	)	PUNCT
ejpam-4788	93	14	,	,	PUNCT
ejpam-4788	93	15	1592	1592	NUM
ejpam-4788	93	16	-	-	SYM
ejpam-4788	93	17	1607	1607	NUM
ejpam-4788	93	18	1596	1596	NUM
ejpam-4788	93	19	λp	λp	ADP
ejpam-4788	94	1	i	i	PRON
ejpam-4788	94	2	:	:	PUNCT
ejpam-4788	94	3	s	s	X
ejpam-4788	94	4	→	→	SYM
ejpam-4788	94	5	[	[	X
ejpam-4788	94	6	0	0	NUM
ejpam-4788	94	7	,	,	PUNCT
ejpam-4788	94	8	1	1	NUM
ejpam-4788	94	9	]	]	PUNCT
ejpam-4788	94	10	,	,	PUNCT
ejpam-4788	94	11	k	k	PROPN
ejpam-4788	94	12	7→	7→	NUM
ejpam-4788	94	13	λp	λp	X
ejpam-4788	94	14	i(u	i(u	PROPN
ejpam-4788	94	15	)	)	PUNCT
ejpam-4788	94	16	:	:	PUNCT
ejpam-4788	95	1	=	=	SYM
ejpam-4788	95	2	{	{	PUNCT
ejpam-4788	95	3	1	1	NUM
ejpam-4788	95	4	k	k	X
ejpam-4788	95	5	∈	∈	PROPN
ejpam-4788	95	6	i	i	PRON
ejpam-4788	95	7	,	,	PUNCT
ejpam-4788	95	8	0	0	PUNCT
ejpam-4788	96	1	k	k	NOUN
ejpam-4788	96	2	/∈	/∈	PUNCT
ejpam-4788	97	1	i	i	PRON
ejpam-4788	97	2	,	,	PUNCT
ejpam-4788	97	3	and	and	CCONJ
ejpam-4788	97	4	λn	λn	INTJ
ejpam-4788	98	1	i	i	PRON
ejpam-4788	98	2	:	:	PUNCT
ejpam-4788	98	3	s	s	X
ejpam-4788	98	4	→	→	X
ejpam-4788	98	5	[	[	X
ejpam-4788	98	6	−1	−1	NOUN
ejpam-4788	98	7	,	,	PUNCT
ejpam-4788	98	8	0	0	NUM
ejpam-4788	98	9	]	]	PUNCT
ejpam-4788	98	10	,	,	PUNCT
ejpam-4788	98	11	k	k	PROPN
ejpam-4788	99	1	7→	7→	NUM
ejpam-4788	99	2	λn	λn	VERB
ejpam-4788	100	1	i	i	PRON
ejpam-4788	100	2	(	(	PUNCT
ejpam-4788	100	3	k	k	NOUN
ejpam-4788	100	4	)	)	PUNCT
ejpam-4788	100	5	:	:	PUNCT
ejpam-4788	100	6	=	=	PRON
ejpam-4788	100	7	{	{	PUNCT
ejpam-4788	100	8	−1	−1	NOUN
ejpam-4788	100	9	k	k	PROPN
ejpam-4788	100	10	∈	∈	PROPN
ejpam-4788	101	1	i	i	PRON
ejpam-4788	101	2	,	,	PUNCT
ejpam-4788	101	3	0	0	PUNCT
ejpam-4788	101	4	k	k	PROPN
ejpam-4788	101	5	/∈	/∈	PROPN
ejpam-4788	101	6	i.	i.	PROPN
ejpam-4788	101	7	remark	remark	PROPN
ejpam-4788	101	8	2	2	NUM
ejpam-4788	101	9	.	.	PUNCT
ejpam-4788	102	1	for	for	ADP
ejpam-4788	102	2	the	the	DET
ejpam-4788	102	3	sake	sake	NOUN
ejpam-4788	102	4	of	of	ADP
ejpam-4788	102	5	simplicity	simplicity	NOUN
ejpam-4788	102	6	,	,	PUNCT
ejpam-4788	102	7	we	we	PRON
ejpam-4788	102	8	shall	shall	AUX
ejpam-4788	102	9	use	use	VERB
ejpam-4788	102	10	the	the	DET
ejpam-4788	102	11	symbol	symbol	NOUN
ejpam-4788	102	12	λi	λi	NOUN
ejpam-4788	102	13	=	=	PUNCT
ejpam-4788	102	14	(	(	PUNCT
ejpam-4788	102	15	s;λp	s;λp	NOUN
ejpam-4788	102	16	i	i	PRON
ejpam-4788	102	17	,	,	PUNCT
ejpam-4788	102	18	λ	λ	PROPN
ejpam-4788	102	19	n	n	X
ejpam-4788	102	20	i	i	NOUN
ejpam-4788	102	21	)	)	PUNCT
ejpam-4788	102	22	for	for	SCONJ
ejpam-4788	102	23	the	the	DET
ejpam-4788	102	24	bf	bf	NOUN
ejpam-4788	102	25	set	set	VERB
ejpam-4788	102	26	λi	λi	INTJ
ejpam-4788	102	27	:	:	PUNCT
ejpam-4788	102	28	=	=	SYM
ejpam-4788	102	29	{	{	PUNCT
ejpam-4788	102	30	(	(	PUNCT
ejpam-4788	102	31	k	k	X
ejpam-4788	102	32	,	,	PUNCT
ejpam-4788	102	33	λp	λp	X
ejpam-4788	102	34	i(k	i(k	PROPN
ejpam-4788	102	35	)	)	PUNCT
ejpam-4788	102	36	,	,	PUNCT
ejpam-4788	102	37	λ	λ	PROPN
ejpam-4788	102	38	n	n	VERB
ejpam-4788	102	39	i	i	PRON
ejpam-4788	102	40	(	(	PUNCT
ejpam-4788	102	41	k	k	NOUN
ejpam-4788	102	42	)	)	PUNCT
ejpam-4788	102	43	)	)	PUNCT
ejpam-4788	103	1	|	|	ADV
ejpam-4788	103	2	k	k	PROPN
ejpam-4788	103	3	∈	∈	PROPN
ejpam-4788	103	4	i	i	X
ejpam-4788	103	5	}	}	PUNCT
ejpam-4788	103	6	.	.	PUNCT
ejpam-4788	104	1	for	for	ADP
ejpam-4788	104	2	u	u	PROPN
ejpam-4788	104	3	∈	∈	PROPN
ejpam-4788	104	4	s	s	X
ejpam-4788	104	5	and	and	CCONJ
ejpam-4788	104	6	(	(	PUNCT
ejpam-4788	104	7	t	t	PROPN
ejpam-4788	104	8	,	,	PUNCT
ejpam-4788	104	9	s	s	PART
ejpam-4788	104	10	)	)	PUNCT
ejpam-4788	104	11	∈	∈	PROPN
ejpam-4788	105	1	[	[	X
ejpam-4788	105	2	0	0	NUM
ejpam-4788	105	3	,	,	PUNCT
ejpam-4788	105	4	1]×	1]×	NUM
ejpam-4788	105	5	[	[	X
ejpam-4788	105	6	−1	−1	NOUN
ejpam-4788	105	7	,	,	PUNCT
ejpam-4788	105	8	0	0	NUM
ejpam-4788	105	9	]	]	PUNCT
ejpam-4788	105	10	,	,	PUNCT
ejpam-4788	105	11	a	a	DET
ejpam-4788	105	12	bipolar	bipolar	ADJ
ejpam-4788	105	13	fuzzy	fuzzy	ADJ
ejpam-4788	105	14	point	point	NOUN
ejpam-4788	105	15	x(t	x(t	PROPN
ejpam-4788	105	16	,	,	PUNCT
ejpam-4788	105	17	s	s	NOUN
ejpam-4788	105	18	)	)	PUNCT
ejpam-4788	105	19	=	=	SYM
ejpam-4788	105	20	(	(	PUNCT
ejpam-4788	105	21	s;xpt	s;xpt	NOUN
ejpam-4788	105	22	,	,	PUNCT
ejpam-4788	105	23	x	x	PUNCT
ejpam-4788	105	24	n	n	NOUN
ejpam-4788	105	25	s	s	NOUN
ejpam-4788	105	26	)	)	PUNCT
ejpam-4788	105	27	of	of	ADP
ejpam-4788	105	28	a	a	DET
ejpam-4788	105	29	set	set	NOUN
ejpam-4788	105	30	s	s	PART
ejpam-4788	105	31	is	be	AUX
ejpam-4788	105	32	a	a	DET
ejpam-4788	105	33	bipolar	bipolar	ADJ
ejpam-4788	105	34	set	set	NOUN
ejpam-4788	105	35	of	of	ADP
ejpam-4788	105	36	s	s	PRON
ejpam-4788	105	37	defined	define	VERB
ejpam-4788	105	38	by	by	ADP
ejpam-4788	105	39	xpt	xpt	PROPN
ejpam-4788	105	40	(	(	PUNCT
ejpam-4788	105	41	u	u	NOUN
ejpam-4788	105	42	)	)	PUNCT
ejpam-4788	105	43	=	=	SYM
ejpam-4788	105	44	{	{	PUNCT
ejpam-4788	105	45	t	t	X
ejpam-4788	105	46	if	if	SCONJ
ejpam-4788	105	47	u	u	PROPN
ejpam-4788	105	48	=	=	NOUN
ejpam-4788	105	49	x	x	SYM
ejpam-4788	105	50	0	0	PUNCT
ejpam-4788	105	51	if	if	SCONJ
ejpam-4788	105	52	u	u	PRON
ejpam-4788	105	53	̸=	̸=	PROPN
ejpam-4788	105	54	x	x	X
ejpam-4788	105	55	and	and	CCONJ
ejpam-4788	105	56	xns	xns	PROPN
ejpam-4788	105	57	(	(	PUNCT
ejpam-4788	105	58	u	u	NOUN
ejpam-4788	105	59	)	)	PUNCT
ejpam-4788	105	60	=	=	PRON
ejpam-4788	106	1	{	{	PUNCT
ejpam-4788	106	2	s	s	X
ejpam-4788	106	3	if	if	SCONJ
ejpam-4788	106	4	u	u	X
ejpam-4788	106	5	=	=	NOUN
ejpam-4788	106	6	x	x	SYM
ejpam-4788	106	7	0	0	PUNCT
ejpam-4788	106	8	if	if	SCONJ
ejpam-4788	106	9	u	u	PROPN
ejpam-4788	106	10	̸=	̸=	PROPN
ejpam-4788	106	11	x.	x.	NOUN
ejpam-4788	106	12	next	next	ADV
ejpam-4788	106	13	,	,	PUNCT
ejpam-4788	106	14	we	we	PRON
ejpam-4788	106	15	study	study	VERB
ejpam-4788	106	16	the	the	DET
ejpam-4788	106	17	intersection	intersection	NOUN
ejpam-4788	106	18	of	of	ADP
ejpam-4788	106	19	the	the	DET
ejpam-4788	106	20	bf	bf	NOUN
ejpam-4788	106	21	set	set	VERB
ejpam-4788	106	22	as	as	SCONJ
ejpam-4788	106	23	defined	define	VERB
ejpam-4788	106	24	.	.	PUNCT
ejpam-4788	107	1	let	let	VERB
ejpam-4788	107	2	ξ	ξ	X
ejpam-4788	107	3	=	=	SYM
ejpam-4788	107	4	(	(	PUNCT
ejpam-4788	107	5	s	s	PROPN
ejpam-4788	107	6	;	;	PUNCT
ejpam-4788	107	7	ξp	ξp	NUM
ejpam-4788	107	8	,	,	PUNCT
ejpam-4788	107	9	ξn	ξn	NOUN
ejpam-4788	107	10	)	)	PUNCT
ejpam-4788	107	11	and	and	CCONJ
ejpam-4788	107	12	ς	ς	PROPN
ejpam-4788	107	13	=	=	PUNCT
ejpam-4788	107	14	(	(	PUNCT
ejpam-4788	107	15	s	s	NOUN
ejpam-4788	107	16	;	;	PUNCT
ejpam-4788	107	17	ςp	ςp	NUM
ejpam-4788	107	18	,	,	PUNCT
ejpam-4788	107	19	ςn	ςn	NOUN
ejpam-4788	107	20	)	)	PUNCT
ejpam-4788	107	21	be	be	VERB
ejpam-4788	107	22	bf	bf	NOUN
ejpam-4788	107	23	sets	set	NOUN
ejpam-4788	107	24	of	of	ADP
ejpam-4788	107	25	a	a	DET
ejpam-4788	107	26	semigroup	semigroup	PROPN
ejpam-4788	107	27	s.	s.	PROPN
ejpam-4788	107	28	define	define	VERB
ejpam-4788	107	29	ξ	ξ	PROPN
ejpam-4788	107	30	∩	∩	NOUN
ejpam-4788	107	31	ς	ς	X
ejpam-4788	107	32	=	=	PUNCT
ejpam-4788	107	33	(	(	PUNCT
ejpam-4788	107	34	ξp	ξp	ADP
ejpam-4788	107	35	∩	∩	ADJ
ejpam-4788	107	36	ςp	ςp	NOUN
ejpam-4788	107	37	,	,	PUNCT
ejpam-4788	107	38	ξn	ξn	NOUN
ejpam-4788	107	39	∩	∩	NOUN
ejpam-4788	107	40	ςn	ςn	NOUN
ejpam-4788	107	41	)	)	PUNCT
ejpam-4788	108	1	where	where	SCONJ
ejpam-4788	108	2	(	(	PUNCT
ejpam-4788	108	3	ξp	ξp	ADP
ejpam-4788	108	4	∩	∩	ADJ
ejpam-4788	108	5	ςp)(k	ςp)(k	NOUN
ejpam-4788	108	6	)	)	PUNCT
ejpam-4788	108	7	=	=	SYM
ejpam-4788	108	8	ξp(k	ξp(k	NOUN
ejpam-4788	108	9	)	)	PUNCT
ejpam-4788	108	10	∧	∧	NOUN
ejpam-4788	108	11	ςp(k	ςp(k	NOUN
ejpam-4788	108	12	)	)	PUNCT
ejpam-4788	108	13	and	and	CCONJ
ejpam-4788	108	14	(	(	PUNCT
ejpam-4788	108	15	ξn	ξn	NOUN
ejpam-4788	108	16	∩	∩	NOUN
ejpam-4788	108	17	ςn)(k	ςn)(k	NOUN
ejpam-4788	108	18	)	)	PUNCT
ejpam-4788	108	19	=	=	SYM
ejpam-4788	108	20	ξn(k	ξn(k	NUM
ejpam-4788	108	21	)	)	PUNCT
ejpam-4788	108	22	∨	∨	NUM
ejpam-4788	108	23	ςn(k	ςn(k	NUM
ejpam-4788	108	24	)	)	PUNCT
ejpam-4788	108	25	for	for	ADP
ejpam-4788	108	26	all	all	DET
ejpam-4788	108	27	k	k	PROPN
ejpam-4788	108	28	∈	∈	PROPN
ejpam-4788	108	29	s.	s.	PROPN
ejpam-4788	108	30	definition	definition	NOUN
ejpam-4788	108	31	8	8	NUM
ejpam-4788	109	1	.	.	PUNCT
ejpam-4788	110	1	[	[	X
ejpam-4788	110	2	9	9	NUM
ejpam-4788	110	3	]	]	PUNCT
ejpam-4788	110	4	a	a	DET
ejpam-4788	110	5	bf	bf	NOUN
ejpam-4788	110	6	set	set	VERB
ejpam-4788	110	7	ξ	ξ	X
ejpam-4788	110	8	=	=	SYM
ejpam-4788	110	9	(	(	PUNCT
ejpam-4788	110	10	s	s	PROPN
ejpam-4788	110	11	;	;	PUNCT
ejpam-4788	110	12	ξp	ξp	NUM
ejpam-4788	110	13	,	,	PUNCT
ejpam-4788	110	14	ξn	ξn	NOUN
ejpam-4788	110	15	)	)	PUNCT
ejpam-4788	110	16	on	on	ADP
ejpam-4788	110	17	a	a	DET
ejpam-4788	110	18	γ	γ	NOUN
ejpam-4788	110	19	-	-	PUNCT
ejpam-4788	110	20	semigroup	semigroup	NOUN
ejpam-4788	110	21	s	s	PART
ejpam-4788	110	22	is	be	AUX
ejpam-4788	110	23	called	call	VERB
ejpam-4788	110	24	a	a	DET
ejpam-4788	110	25	bf	bf	NOUN
ejpam-4788	110	26	subsemigroup	subsemigroup	NOUN
ejpam-4788	110	27	on	on	ADP
ejpam-4788	110	28	s	s	PRON
ejpam-4788	110	29	if	if	SCONJ
ejpam-4788	110	30	it	it	PRON
ejpam-4788	110	31	satisfies	satisfy	VERB
ejpam-4788	110	32	the	the	DET
ejpam-4788	110	33	following	follow	VERB
ejpam-4788	110	34	conditions	condition	NOUN
ejpam-4788	110	35	:	:	PUNCT
ejpam-4788	110	36	ξp(uαv	ξp(uαv	NOUN
ejpam-4788	110	37	)	)	PUNCT
ejpam-4788	110	38	≥	≥	NOUN
ejpam-4788	110	39	ξp(u	ξp(u	NOUN
ejpam-4788	110	40	)	)	PUNCT
ejpam-4788	111	1	∧	∧	NOUN
ejpam-4788	111	2	ξp(v	ξp(v	NOUN
ejpam-4788	111	3	)	)	PUNCT
ejpam-4788	111	4	and	and	CCONJ
ejpam-4788	111	5	ξn(uαv	ξn(uαv	PROPN
ejpam-4788	111	6	)	)	PUNCT
ejpam-4788	111	7	≤	≤	NOUN
ejpam-4788	111	8	ξn(u	ξn(u	NOUN
ejpam-4788	111	9	)	)	PUNCT
ejpam-4788	111	10	∨	∨	NUM
ejpam-4788	111	11	ξn(v	ξn(v	NUM
ejpam-4788	111	12	)	)	PUNCT
ejpam-4788	111	13	for	for	ADP
ejpam-4788	111	14	all	all	DET
ejpam-4788	111	15	u	u	NOUN
ejpam-4788	111	16	,	,	PUNCT
ejpam-4788	111	17	v	v	ADP
ejpam-4788	111	18	∈	∈	PROPN
ejpam-4788	111	19	s	s	PART
ejpam-4788	111	20	and	and	CCONJ
ejpam-4788	111	21	α	α	PRON
ejpam-4788	111	22	∈	∈	PROPN
ejpam-4788	111	23	γ	γ	X
ejpam-4788	111	24	.	.	PUNCT
ejpam-4788	112	1	the	the	DET
ejpam-4788	112	2	following	follow	VERB
ejpam-4788	112	3	presents	present	VERB
ejpam-4788	112	4	an	an	DET
ejpam-4788	112	5	example	example	NOUN
ejpam-4788	112	6	of	of	ADP
ejpam-4788	112	7	a	a	DET
ejpam-4788	112	8	bf	bf	NOUN
ejpam-4788	112	9	subsemigroup	subsemigroup	NOUN
ejpam-4788	112	10	.	.	PUNCT
ejpam-4788	113	1	example	example	NOUN
ejpam-4788	114	1	2	2	NUM
ejpam-4788	114	2	.	.	PUNCT
ejpam-4788	114	3	let	let	VERB
ejpam-4788	114	4	s	s	PRON
ejpam-4788	114	5	be	be	AUX
ejpam-4788	114	6	the	the	DET
ejpam-4788	114	7	set	set	NOUN
ejpam-4788	114	8	of	of	ADP
ejpam-4788	114	9	all	all	DET
ejpam-4788	114	10	negative	negative	ADJ
ejpam-4788	114	11	integers	integer	NOUN
ejpam-4788	114	12	and	and	CCONJ
ejpam-4788	114	13	γ	γ	NOUN
ejpam-4788	114	14	be	be	AUX
ejpam-4788	114	15	the	the	DET
ejpam-4788	114	16	set	set	NOUN
ejpam-4788	114	17	of	of	ADP
ejpam-4788	114	18	all	all	DET
ejpam-4788	114	19	non	non	ADJ
ejpam-4788	114	20	positive	positive	ADJ
ejpam-4788	114	21	even	even	ADJ
ejpam-4788	114	22	intergers	interger	NOUN
ejpam-4788	114	23	.	.	PUNCT
ejpam-4788	115	1	then	then	ADV
ejpam-4788	115	2	s	s	VERB
ejpam-4788	115	3	is	be	AUX
ejpam-4788	115	4	a	a	DET
ejpam-4788	115	5	γ	γ	NOUN
ejpam-4788	115	6	-	-	PUNCT
ejpam-4788	115	7	semigroup	semigroup	NOUN
ejpam-4788	115	8	by	by	ADP
ejpam-4788	115	9	usual	usual	ADJ
ejpam-4788	115	10	multiplication	multiplication	NOUN
ejpam-4788	115	11	of	of	ADP
ejpam-4788	115	12	integers	integer	NOUN
ejpam-4788	115	13	.	.	PUNCT
ejpam-4788	116	1	let	let	VERB
ejpam-4788	116	2	ξ	ξ	X
ejpam-4788	116	3	=	=	SYM
ejpam-4788	116	4	(	(	PUNCT
ejpam-4788	116	5	s	s	PROPN
ejpam-4788	116	6	;	;	PUNCT
ejpam-4788	116	7	ξp	ξp	NUM
ejpam-4788	116	8	,	,	PUNCT
ejpam-4788	116	9	ξn	ξn	NOUN
ejpam-4788	116	10	)	)	PUNCT
ejpam-4788	116	11	be	be	VERB
ejpam-4788	116	12	a	a	DET
ejpam-4788	116	13	bf	bf	NOUN
ejpam-4788	116	14	set	set	NOUN
ejpam-4788	116	15	of	of	ADP
ejpam-4788	116	16	s	s	PRON
ejpam-4788	116	17	,	,	PUNCT
ejpam-4788	116	18	as	as	SCONJ
ejpam-4788	116	19	defined	define	VERB
ejpam-4788	116	20	follows	follow	VERB
ejpam-4788	116	21	ξp(u	ξp(u	NOUN
ejpam-4788	116	22	)	)	PUNCT
ejpam-4788	116	23	=	=	PUNCT
ejpam-4788	117	1			PUNCT
ejpam-4788	117	2	1	1	NUM
ejpam-4788	117	3	if	if	SCONJ
ejpam-4788	117	4	u	u	NOUN
ejpam-4788	117	5	=	=	NOUN
ejpam-4788	117	6	0	0	NUM
ejpam-4788	117	7	−	−	PROPN
ejpam-4788	117	8	1	1	NUM
ejpam-4788	117	9	u	u	NOUN
ejpam-4788	117	10	if	if	SCONJ
ejpam-4788	117	11	u	u	NOUN
ejpam-4788	117	12	=	=	PUNCT
ejpam-4788	117	13	−1,−2	−1,−2	VERB
ejpam-4788	117	14	−	−	PROPN
ejpam-4788	117	15	1	1	NUM
ejpam-4788	118	1	u−2	u−2	INTJ
ejpam-4788	118	2	if	if	SCONJ
ejpam-4788	118	3	u	u	NOUN
ejpam-4788	118	4	<	<	X
ejpam-4788	118	5	−2	−2	NOUN
ejpam-4788	118	6	and	and	CCONJ
ejpam-4788	118	7	ξn(u	ξn(u	NOUN
ejpam-4788	118	8	)	)	PUNCT
ejpam-4788	119	1	=	=	PUNCT
ejpam-4788	119	2			PUNCT
ejpam-4788	119	3	−1	−1	NOUN
ejpam-4788	119	4	if	if	SCONJ
ejpam-4788	119	5	u	u	NOUN
ejpam-4788	119	6	=	=	NOUN
ejpam-4788	119	7	0	0	NUM
ejpam-4788	119	8	1	1	NUM
ejpam-4788	119	9	u	u	NOUN
ejpam-4788	119	10	if	if	SCONJ
ejpam-4788	119	11	u	u	NOUN
ejpam-4788	119	12	=	=	PUNCT
ejpam-4788	119	13	−1,−2	−1,−2	VERB
ejpam-4788	119	14	1	1	NUM
ejpam-4788	119	15	u−2	u−2	INTJ
ejpam-4788	119	16	if	if	SCONJ
ejpam-4788	119	17	u	u	NOUN
ejpam-4788	119	18	<	<	X
ejpam-4788	119	19	−2	−2	X
ejpam-4788	119	20	then	then	ADV
ejpam-4788	119	21	ξ	ξ	X
ejpam-4788	119	22	=	=	SYM
ejpam-4788	119	23	(	(	PUNCT
ejpam-4788	119	24	s	s	PROPN
ejpam-4788	119	25	;	;	PUNCT
ejpam-4788	119	26	ξp	ξp	NUM
ejpam-4788	119	27	,	,	PUNCT
ejpam-4788	119	28	ξn	ξn	NOUN
ejpam-4788	119	29	)	)	PUNCT
ejpam-4788	119	30	is	be	AUX
ejpam-4788	119	31	a	a	DET
ejpam-4788	119	32	bf	bf	NOUN
ejpam-4788	119	33	subsemigroup	subsemigroup	NOUN
ejpam-4788	119	34	of	of	ADP
ejpam-4788	119	35	s.	s.	PROPN
ejpam-4788	119	36	definition	definition	NOUN
ejpam-4788	119	37	9	9	NUM
ejpam-4788	119	38	.	.	PUNCT
ejpam-4788	120	1	[	[	X
ejpam-4788	120	2	9	9	NUM
ejpam-4788	120	3	]	]	PUNCT
ejpam-4788	120	4	a	a	DET
ejpam-4788	120	5	bf	bf	NOUN
ejpam-4788	120	6	set	set	VERB
ejpam-4788	120	7	ξ	ξ	X
ejpam-4788	120	8	=	=	SYM
ejpam-4788	120	9	(	(	PUNCT
ejpam-4788	120	10	s	s	PROPN
ejpam-4788	120	11	;	;	PUNCT
ejpam-4788	120	12	ξp	ξp	NUM
ejpam-4788	120	13	,	,	PUNCT
ejpam-4788	120	14	ξn	ξn	NOUN
ejpam-4788	120	15	)	)	PUNCT
ejpam-4788	120	16	on	on	ADP
ejpam-4788	120	17	a	a	DET
ejpam-4788	120	18	γ	γ	NOUN
ejpam-4788	120	19	-	-	PUNCT
ejpam-4788	120	20	semigroup	semigroup	NOUN
ejpam-4788	120	21	s	s	PART
ejpam-4788	120	22	is	be	AUX
ejpam-4788	120	23	called	call	VERB
ejpam-4788	120	24	a	a	DET
ejpam-4788	120	25	bf	bf	NOUN
ejpam-4788	120	26	left	left	ADJ
ejpam-4788	120	27	(	(	PUNCT
ejpam-4788	120	28	right	right	ADJ
ejpam-4788	120	29	)	)	PUNCT
ejpam-4788	120	30	ideal	ideal	NOUN
ejpam-4788	120	31	on	on	ADP
ejpam-4788	120	32	s	s	PRON
ejpam-4788	120	33	if	if	SCONJ
ejpam-4788	120	34	it	it	PRON
ejpam-4788	120	35	satisfies	satisfy	VERB
ejpam-4788	120	36	the	the	DET
ejpam-4788	120	37	following	follow	VERB
ejpam-4788	120	38	conditions	condition	NOUN
ejpam-4788	120	39	:	:	PUNCT
ejpam-4788	120	40	ξp(uαv	ξp(uαv	NOUN
ejpam-4788	120	41	)	)	PUNCT
ejpam-4788	120	42	≥	≥	NOUN
ejpam-4788	120	43	ξp(v	ξp(v	PUNCT
ejpam-4788	120	44	)	)	PUNCT
ejpam-4788	120	45	(	(	PUNCT
ejpam-4788	120	46	ξp(uαv	ξp(uαv	NOUN
ejpam-4788	120	47	)	)	PUNCT
ejpam-4788	120	48	≥	≥	NOUN
ejpam-4788	120	49	ξp(u	ξp(u	NOUN
ejpam-4788	120	50	)	)	PUNCT
ejpam-4788	120	51	)	)	PUNCT
ejpam-4788	120	52	and	and	CCONJ
ejpam-4788	120	53	ξn(uαv	ξn(uαv	PROPN
ejpam-4788	120	54	)	)	PUNCT
ejpam-4788	120	55	≤	≤	NOUN
ejpam-4788	120	56	ξn(v	ξn(v	NUM
ejpam-4788	120	57	)	)	PUNCT
ejpam-4788	120	58	(	(	PUNCT
ejpam-4788	120	59	ξn(uαv	ξn(uαv	PROPN
ejpam-4788	120	60	)	)	PUNCT
ejpam-4788	120	61	≤	≤	NOUN
ejpam-4788	120	62	ξn(u	ξn(u	NOUN
ejpam-4788	120	63	)	)	PUNCT
ejpam-4788	120	64	)	)	PUNCT
ejpam-4788	120	65	for	for	ADP
ejpam-4788	120	66	all	all	DET
ejpam-4788	120	67	u	u	NOUN
ejpam-4788	120	68	,	,	PUNCT
ejpam-4788	120	69	v	v	ADP
ejpam-4788	120	70	∈	∈	PROPN
ejpam-4788	120	71	s	s	PART
ejpam-4788	120	72	and	and	CCONJ
ejpam-4788	120	73	α	α	PRON
ejpam-4788	120	74	∈	∈	PROPN
ejpam-4788	120	75	γ	γ	X
ejpam-4788	120	76	.	.	PUNCT
ejpam-4788	121	1	p.	p.	NOUN
ejpam-4788	121	2	khamrot	khamrot	PROPN
ejpam-4788	121	3	,	,	PUNCT
ejpam-4788	121	4	t.	t.	PROPN
ejpam-4788	121	5	gaketem	gaketem	PROPN
ejpam-4788	121	6	/	/	SYM
ejpam-4788	121	7	eur	eur	PROPN
ejpam-4788	121	8	.	.	PUNCT
ejpam-4788	122	1	j.	j.	PROPN
ejpam-4788	122	2	pure	pure	PROPN
ejpam-4788	122	3	appl	appl	PROPN
ejpam-4788	122	4	.	.	PROPN
ejpam-4788	122	5	math	math	PROPN
ejpam-4788	122	6	,	,	PUNCT
ejpam-4788	122	7	16	16	NUM
ejpam-4788	122	8	(	(	PUNCT
ejpam-4788	122	9	3	3	NUM
ejpam-4788	122	10	)	)	PUNCT
ejpam-4788	122	11	(	(	PUNCT
ejpam-4788	122	12	2023	2023	NUM
ejpam-4788	122	13	)	)	PUNCT
ejpam-4788	122	14	,	,	PUNCT
ejpam-4788	122	15	1592	1592	NUM
ejpam-4788	122	16	-	-	SYM
ejpam-4788	122	17	1607	1607	NUM
ejpam-4788	122	18	1597	1597	NUM
ejpam-4788	122	19	3	3	NUM
ejpam-4788	122	20	.	.	PUNCT
ejpam-4788	123	1	new	new	ADJ
ejpam-4788	123	2	types	type	NOUN
ejpam-4788	123	3	of	of	ADP
ejpam-4788	123	4	bipolar	bipolar	ADJ
ejpam-4788	123	5	valued	value	VERB
ejpam-4788	123	6	fuzzy	fuzzy	ADJ
ejpam-4788	123	7	ideals	ideal	NOUN
ejpam-4788	123	8	in	in	ADP
ejpam-4788	123	9	this	this	DET
ejpam-4788	123	10	section	section	NOUN
ejpam-4788	123	11	,	,	PUNCT
ejpam-4788	123	12	we	we	PRON
ejpam-4788	123	13	define	define	VERB
ejpam-4788	123	14	the	the	DET
ejpam-4788	123	15	bipolar	bipolar	ADJ
ejpam-4788	123	16	fuzzy	fuzzy	NOUN
ejpam-4788	123	17	(	(	PUNCT
ejpam-4788	123	18	α	α	NOUN
ejpam-4788	123	19	,	,	PUNCT
ejpam-4788	123	20	β)-ideal	β)-ideal	PUNCT
ejpam-4788	123	21	and	and	CCONJ
ejpam-4788	123	22	study	study	VERB
ejpam-4788	123	23	its	its	PRON
ejpam-4788	123	24	basic	basic	ADJ
ejpam-4788	123	25	properties	property	NOUN
ejpam-4788	123	26	.	.	PUNCT
ejpam-4788	124	1	definition	definition	NOUN
ejpam-4788	124	2	10	10	NUM
ejpam-4788	124	3	.	.	PUNCT
ejpam-4788	125	1	let	let	VERB
ejpam-4788	125	2	ξ	ξ	X
ejpam-4788	125	3	=	=	SYM
ejpam-4788	125	4	(	(	PUNCT
ejpam-4788	125	5	s	s	PROPN
ejpam-4788	125	6	;	;	PUNCT
ejpam-4788	125	7	ξp	ξp	NUM
ejpam-4788	125	8	,	,	PUNCT
ejpam-4788	125	9	ξn	ξn	NOUN
ejpam-4788	125	10	)	)	PUNCT
ejpam-4788	125	11	be	be	VERB
ejpam-4788	125	12	a	a	DET
ejpam-4788	125	13	bf	bf	NOUN
ejpam-4788	125	14	set	set	NOUN
ejpam-4788	125	15	of	of	ADP
ejpam-4788	125	16	a	a	DET
ejpam-4788	125	17	γ	γ	NOUN
ejpam-4788	125	18	-	-	PUNCT
ejpam-4788	125	19	semigroup	semigroup	NOUN
ejpam-4788	125	20	s	s	X
ejpam-4788	125	21	and	and	CCONJ
ejpam-4788	125	22	α	α	NOUN
ejpam-4788	125	23	,	,	PUNCT
ejpam-4788	125	24	β	β	PROPN
ejpam-4788	125	25	∈	∈	PROPN
ejpam-4788	125	26	γ	γ	X
ejpam-4788	125	27	.	.	PROPN
ejpam-4788	126	1	then	then	ADV
ejpam-4788	126	2	ξ	ξ	X
ejpam-4788	126	3	=	=	SYM
ejpam-4788	126	4	(	(	PUNCT
ejpam-4788	126	5	s	s	PROPN
ejpam-4788	126	6	;	;	PUNCT
ejpam-4788	126	7	ξp	ξp	NUM
ejpam-4788	126	8	,	,	PUNCT
ejpam-4788	126	9	ξn	ξn	NOUN
ejpam-4788	126	10	)	)	PUNCT
ejpam-4788	126	11	is	be	AUX
ejpam-4788	126	12	called	call	VERB
ejpam-4788	126	13	[	[	X
ejpam-4788	126	14	(	(	PUNCT
ejpam-4788	126	15	i)]a	i)]a	NOUN
ejpam-4788	126	16	bf	bf	NOUN
ejpam-4788	126	17	left	leave	VERB
ejpam-4788	126	18	α	α	NOUN
ejpam-4788	126	19	-	-	NOUN
ejpam-4788	126	20	ideal	ideal	NOUN
ejpam-4788	126	21	of	of	ADP
ejpam-4788	126	22	s	s	PRON
ejpam-4788	126	23	if	if	SCONJ
ejpam-4788	126	24	ξp(uαv	ξp(uαv	NOUN
ejpam-4788	126	25	)	)	PUNCT
ejpam-4788	126	26	≥	≥	NOUN
ejpam-4788	126	27	ξp(v	ξp(v	PUNCT
ejpam-4788	126	28	)	)	PUNCT
ejpam-4788	126	29	and	and	CCONJ
ejpam-4788	126	30	ξn(uαv	ξn(uαv	PROPN
ejpam-4788	126	31	)	)	PUNCT
ejpam-4788	126	32	≤	≤	NOUN
ejpam-4788	126	33	ξn(v	ξn(v	NUM
ejpam-4788	126	34	)	)	PUNCT
ejpam-4788	126	35	for	for	ADP
ejpam-4788	126	36	all	all	DET
ejpam-4788	126	37	u	u	NOUN
ejpam-4788	126	38	,	,	PUNCT
ejpam-4788	126	39	v	v	ADP
ejpam-4788	126	40	∈	∈	PROPN
ejpam-4788	126	41	s.	s.	PROPN
ejpam-4788	126	42	a	a	DET
ejpam-4788	126	43	bf	bf	NOUN
ejpam-4788	126	44	right	right	ADJ
ejpam-4788	127	1	β	β	NOUN
ejpam-4788	127	2	-	-	NOUN
ejpam-4788	127	3	ideal	ideal	NOUN
ejpam-4788	127	4	of	of	ADP
ejpam-4788	127	5	s	s	PRON
ejpam-4788	127	6	if	if	SCONJ
ejpam-4788	127	7	ξp(uβv	ξp(uβv	NOUN
ejpam-4788	127	8	)	)	PUNCT
ejpam-4788	127	9	≥	≥	NOUN
ejpam-4788	127	10	ξ(u	ξ(u	NOUN
ejpam-4788	127	11	)	)	PUNCT
ejpam-4788	127	12	and	and	CCONJ
ejpam-4788	127	13	ξn(uβv	ξn(uβv	NOUN
ejpam-4788	127	14	)	)	PUNCT
ejpam-4788	127	15	≤	≤	NOUN
ejpam-4788	127	16	ξn(u	ξn(u	NOUN
ejpam-4788	127	17	)	)	PUNCT
ejpam-4788	127	18	for	for	ADP
ejpam-4788	127	19	all	all	DET
ejpam-4788	127	20	u	u	NOUN
ejpam-4788	127	21	,	,	PUNCT
ejpam-4788	127	22	v	v	ADP
ejpam-4788	127	23	∈	∈	PROPN
ejpam-4788	127	24	s.	s.	PROPN
ejpam-4788	127	25	a	a	DET
ejpam-4788	127	26	bf	bf	NOUN
ejpam-4788	127	27	(	(	PUNCT
ejpam-4788	127	28	α	α	NOUN
ejpam-4788	127	29	,	,	PUNCT
ejpam-4788	127	30	β)-ideal	β)-ideal	PUNCT
ejpam-4788	127	31	of	of	ADP
ejpam-4788	127	32	s	s	PRON
ejpam-4788	127	33	if	if	SCONJ
ejpam-4788	127	34	it	it	PRON
ejpam-4788	127	35	is	be	AUX
ejpam-4788	127	36	both	both	PRON
ejpam-4788	127	37	a	a	DET
ejpam-4788	127	38	bf	bf	NOUN
ejpam-4788	127	39	left	leave	VERB
ejpam-4788	127	40	α	α	NOUN
ejpam-4788	127	41	-	-	NOUN
ejpam-4788	127	42	ideal	ideal	NOUN
ejpam-4788	127	43	and	and	CCONJ
ejpam-4788	127	44	a	a	DET
ejpam-4788	127	45	bf	bf	NOUN
ejpam-4788	128	1	right	right	ADJ
ejpam-4788	128	2	β	β	NOUN
ejpam-4788	128	3	-	-	NOUN
ejpam-4788	128	4	ideal	ideal	NOUN
ejpam-4788	128	5	of	of	ADP
ejpam-4788	128	6	s.	s.	PROPN
ejpam-4788	128	7	a	a	DET
ejpam-4788	128	8	bf	bf	NOUN
ejpam-4788	128	9	α	α	NOUN
ejpam-4788	128	10	-	-	NOUN
ejpam-4788	128	11	ideal	ideal	NOUN
ejpam-4788	128	12	of	of	ADP
ejpam-4788	128	13	s	s	PRON
ejpam-4788	128	14	if	if	SCONJ
ejpam-4788	128	15	it	it	PRON
ejpam-4788	128	16	is	be	AUX
ejpam-4788	128	17	a	a	DET
ejpam-4788	128	18	bf	bf	NOUN
ejpam-4788	128	19	(	(	PUNCT
ejpam-4788	128	20	α	α	NOUN
ejpam-4788	128	21	,	,	PUNCT
ejpam-4788	128	22	α)-ideal	α)-ideal	NUM
ejpam-4788	128	23	of	of	ADP
ejpam-4788	128	24	s.	s.	PROPN
ejpam-4788	128	25	theorem	theorem	VERB
ejpam-4788	128	26	2	2	X
ejpam-4788	128	27	.	.	PUNCT
ejpam-4788	129	1	let	let	VERB
ejpam-4788	129	2	k	k	PRON
ejpam-4788	129	3	be	be	AUX
ejpam-4788	129	4	a	a	DET
ejpam-4788	129	5	non	non	ADJ
ejpam-4788	129	6	-	-	ADJ
ejpam-4788	129	7	empty	empty	ADJ
ejpam-4788	129	8	subset	subset	NOUN
ejpam-4788	129	9	of	of	ADP
ejpam-4788	129	10	γ	γ	PROPN
ejpam-4788	129	11	-	-	PUNCT
ejpam-4788	129	12	semigroup	semigroup	PROPN
ejpam-4788	129	13	s.	s.	PROPN
ejpam-4788	130	1	then	then	ADV
ejpam-4788	130	2	k	k	PROPN
ejpam-4788	130	3	is	be	AUX
ejpam-4788	130	4	a	a	DET
ejpam-4788	130	5	left	left	ADJ
ejpam-4788	130	6	α	α	NOUN
ejpam-4788	130	7	-	-	NOUN
ejpam-4788	130	8	ideal	ideal	ADJ
ejpam-4788	130	9	(	(	PUNCT
ejpam-4788	130	10	right	right	ADJ
ejpam-4788	130	11	β	β	NOUN
ejpam-4788	130	12	-	-	NOUN
ejpam-4788	130	13	ideal	ideal	ADJ
ejpam-4788	130	14	,	,	PUNCT
ejpam-4788	130	15	(	(	PUNCT
ejpam-4788	130	16	α	α	NOUN
ejpam-4788	130	17	,	,	PUNCT
ejpam-4788	130	18	β)-ideal	β)-ideal	NUM
ejpam-4788	130	19	)	)	PUNCT
ejpam-4788	130	20	of	of	ADP
ejpam-4788	130	21	s	s	PRON
ejpam-4788	130	22	if	if	SCONJ
ejpam-4788	131	1	and	and	CCONJ
ejpam-4788	131	2	only	only	ADV
ejpam-4788	131	3	if	if	SCONJ
ejpam-4788	131	4	λk	λk	X
ejpam-4788	131	5	=	=	SYM
ejpam-4788	131	6	(	(	PUNCT
ejpam-4788	131	7	s;λp	s;λp	PROPN
ejpam-4788	131	8	k	k	PROPN
ejpam-4788	131	9	,	,	PUNCT
ejpam-4788	131	10	λn	λn	PROPN
ejpam-4788	131	11	k	k	X
ejpam-4788	131	12	)	)	PUNCT
ejpam-4788	131	13	is	be	AUX
ejpam-4788	131	14	a	a	DET
ejpam-4788	131	15	bf	bf	NOUN
ejpam-4788	131	16	left	leave	VERB
ejpam-4788	131	17	α	α	NOUN
ejpam-4788	131	18	-	-	NOUN
ejpam-4788	131	19	ideal	ideal	ADJ
ejpam-4788	131	20	(	(	PUNCT
ejpam-4788	131	21	right	right	ADJ
ejpam-4788	131	22	β	β	NOUN
ejpam-4788	131	23	-	-	NOUN
ejpam-4788	131	24	ideal	ideal	ADJ
ejpam-4788	131	25	,	,	PUNCT
ejpam-4788	131	26	(	(	PUNCT
ejpam-4788	131	27	α	α	NOUN
ejpam-4788	131	28	,	,	PUNCT
ejpam-4788	131	29	β)-ideal	β)-ideal	NUM
ejpam-4788	131	30	)	)	PUNCT
ejpam-4788	131	31	of	of	ADP
ejpam-4788	131	32	s.	s.	PROPN
ejpam-4788	131	33	proof	proof	PROPN
ejpam-4788	131	34	.	.	PUNCT
ejpam-4788	132	1	suppose	suppose	VERB
ejpam-4788	132	2	that	that	SCONJ
ejpam-4788	132	3	k	k	PROPN
ejpam-4788	132	4	is	be	AUX
ejpam-4788	132	5	a	a	DET
ejpam-4788	132	6	left	left	ADJ
ejpam-4788	132	7	α	α	NOUN
ejpam-4788	132	8	-	-	NOUN
ejpam-4788	132	9	ideal	ideal	NOUN
ejpam-4788	132	10	of	of	ADP
ejpam-4788	132	11	s	s	NOUN
ejpam-4788	132	12	and	and	CCONJ
ejpam-4788	132	13	u	u	NOUN
ejpam-4788	132	14	,	,	PUNCT
ejpam-4788	132	15	v	v	ADP
ejpam-4788	132	16	∈	∈	NOUN
ejpam-4788	132	17	s.	s.	PROPN
ejpam-4788	132	18	if	if	SCONJ
ejpam-4788	132	19	v	v	NUM
ejpam-4788	133	1	∈	∈	PROPN
ejpam-4788	133	2	k	k	NOUN
ejpam-4788	133	3	,	,	PUNCT
ejpam-4788	133	4	then	then	ADV
ejpam-4788	133	5	uαv	uαv	PROPN
ejpam-4788	133	6	∈	∈	PROPN
ejpam-4788	133	7	k.	k.	PROPN
ejpam-4788	134	1	thus	thus	ADV
ejpam-4788	134	2	,	,	PUNCT
ejpam-4788	134	3	λp	λp	PROPN
ejpam-4788	134	4	k(v	k(v	PROPN
ejpam-4788	134	5	)	)	PUNCT
ejpam-4788	134	6	=	=	PRON
ejpam-4788	135	1	λp	λp	DET
ejpam-4788	135	2	k(uαv	k(uαv	PROPN
ejpam-4788	135	3	)	)	PUNCT
ejpam-4788	135	4	=	=	SYM
ejpam-4788	135	5	1	1	NUM
ejpam-4788	135	6	and	and	CCONJ
ejpam-4788	135	7	λn	λn	PROPN
ejpam-4788	135	8	k(v	k(v	PROPN
ejpam-4788	135	9	)	)	PUNCT
ejpam-4788	136	1	=	=	SYM
ejpam-4788	136	2	λn	λn	PROPN
ejpam-4788	136	3	k(uαv	k(uαv	NOUN
ejpam-4788	136	4	)	)	PUNCT
ejpam-4788	136	5	=	=	PUNCT
ejpam-4788	136	6	−1	−1	NOUN
ejpam-4788	136	7	.	.	PUNCT
ejpam-4788	137	1	hence	hence	ADV
ejpam-4788	137	2	,	,	PUNCT
ejpam-4788	137	3	λp	λp	PRON
ejpam-4788	137	4	k(uαv	k(uαv	PROPN
ejpam-4788	137	5	)	)	PUNCT
ejpam-4788	137	6	≥	≥	NOUN
ejpam-4788	137	7	λp	λp	X
ejpam-4788	137	8	k(v	k(v	PROPN
ejpam-4788	137	9	)	)	PUNCT
ejpam-4788	137	10	and	and	CCONJ
ejpam-4788	137	11	λn	λn	PROPN
ejpam-4788	137	12	k(uαv	k(uαv	PROPN
ejpam-4788	137	13	)	)	PUNCT
ejpam-4788	137	14	≤	≤	NUM
ejpam-4788	137	15	λn	λn	ADP
ejpam-4788	137	16	k(v	k(v	PROPN
ejpam-4788	137	17	)	)	PUNCT
ejpam-4788	137	18	.	.	PUNCT
ejpam-4788	138	1	if	if	SCONJ
ejpam-4788	138	2	v	v	NUM
ejpam-4788	138	3	/∈	/∈	PUNCT
ejpam-4788	139	1	k	k	NOUN
ejpam-4788	139	2	,	,	PUNCT
ejpam-4788	139	3	then	then	ADV
ejpam-4788	139	4	uαv	uαv	PROPN
ejpam-4788	139	5	∈	∈	PROPN
ejpam-4788	139	6	k.	k.	PROPN
ejpam-4788	140	1	thus	thus	ADV
ejpam-4788	140	2	,	,	PUNCT
ejpam-4788	140	3	λp	λp	PROPN
ejpam-4788	140	4	k(v	k(v	PROPN
ejpam-4788	140	5	)	)	PUNCT
ejpam-4788	141	1	=	=	PRON
ejpam-4788	141	2	λn	λn	PROPN
ejpam-4788	141	3	k(v	k(v	PROPN
ejpam-4788	141	4	)	)	PUNCT
ejpam-4788	142	1	=	=	SYM
ejpam-4788	142	2	0	0	NUM
ejpam-4788	142	3	and	and	CCONJ
ejpam-4788	142	4	λp	λp	X
ejpam-4788	142	5	k(uαv	k(uαv	PROPN
ejpam-4788	142	6	)	)	PUNCT
ejpam-4788	142	7	=	=	SYM
ejpam-4788	142	8	1	1	NUM
ejpam-4788	142	9	,	,	PUNCT
ejpam-4788	142	10	λn	λn	PROPN
ejpam-4788	142	11	k(uαv	k(uαv	PROPN
ejpam-4788	142	12	)	)	PUNCT
ejpam-4788	142	13	=	=	PUNCT
ejpam-4788	142	14	−1	−1	NOUN
ejpam-4788	142	15	.	.	PUNCT
ejpam-4788	143	1	hence	hence	ADV
ejpam-4788	143	2	,	,	PUNCT
ejpam-4788	143	3	λp	λp	PRON
ejpam-4788	143	4	k(uαv	k(uαv	PROPN
ejpam-4788	143	5	)	)	PUNCT
ejpam-4788	143	6	≥	≥	NOUN
ejpam-4788	143	7	λp	λp	X
ejpam-4788	143	8	k(v	k(v	PROPN
ejpam-4788	143	9	)	)	PUNCT
ejpam-4788	143	10	and	and	CCONJ
ejpam-4788	143	11	λn	λn	PROPN
ejpam-4788	143	12	k(uαv	k(uαv	PROPN
ejpam-4788	143	13	)	)	PUNCT
ejpam-4788	143	14	≤	≤	NUM
ejpam-4788	143	15	λn	λn	ADP
ejpam-4788	143	16	k(v	k(v	PROPN
ejpam-4788	143	17	)	)	PUNCT
ejpam-4788	143	18	.	.	PUNCT
ejpam-4788	144	1	therefore	therefore	ADV
ejpam-4788	144	2	,	,	PUNCT
ejpam-4788	144	3	λk	λk	X
ejpam-4788	144	4	=	=	PUNCT
ejpam-4788	144	5	(	(	PUNCT
ejpam-4788	144	6	s;λp	s;λp	PROPN
ejpam-4788	144	7	k	k	PROPN
ejpam-4788	144	8	,	,	PUNCT
ejpam-4788	144	9	λn	λn	PROPN
ejpam-4788	144	10	k	k	X
ejpam-4788	144	11	)	)	PUNCT
ejpam-4788	144	12	is	be	AUX
ejpam-4788	144	13	a	a	DET
ejpam-4788	144	14	bf	bf	NOUN
ejpam-4788	144	15	left	leave	VERB
ejpam-4788	144	16	α	α	NOUN
ejpam-4788	144	17	-	-	NOUN
ejpam-4788	144	18	ideal	ideal	NOUN
ejpam-4788	144	19	of	of	ADP
ejpam-4788	144	20	s.	s.	PROPN
ejpam-4788	144	21	conversely	conversely	ADV
ejpam-4788	144	22	,	,	PUNCT
ejpam-4788	144	23	assume	assume	VERB
ejpam-4788	144	24	that	that	SCONJ
ejpam-4788	144	25	λk	λk	ADV
ejpam-4788	144	26	=	=	SYM
ejpam-4788	144	27	(	(	PUNCT
ejpam-4788	144	28	s;λp	s;λp	PROPN
ejpam-4788	144	29	k	k	PROPN
ejpam-4788	144	30	,	,	PUNCT
ejpam-4788	144	31	λn	λn	PROPN
ejpam-4788	144	32	k	k	X
ejpam-4788	144	33	)	)	PUNCT
ejpam-4788	144	34	is	be	AUX
ejpam-4788	144	35	a	a	DET
ejpam-4788	144	36	bf	bf	NOUN
ejpam-4788	144	37	left	leave	VERB
ejpam-4788	144	38	α	α	NOUN
ejpam-4788	144	39	-	-	NOUN
ejpam-4788	144	40	ideal	ideal	NOUN
ejpam-4788	144	41	of	of	ADP
ejpam-4788	144	42	s	s	NOUN
ejpam-4788	144	43	and	and	CCONJ
ejpam-4788	144	44	u	u	NOUN
ejpam-4788	144	45	,	,	PUNCT
ejpam-4788	144	46	v	v	PROPN
ejpam-4788	144	47	∈	∈	NOUN
ejpam-4788	144	48	s	s	NOUN
ejpam-4788	144	49	with	with	ADP
ejpam-4788	144	50	v	v	PROPN
ejpam-4788	144	51	∈	∈	PROPN
ejpam-4788	145	1	k.	k.	NOUN
ejpam-4788	146	1	then	then	ADV
ejpam-4788	146	2	λp	λp	PROPN
ejpam-4788	146	3	k(v	k(v	PROPN
ejpam-4788	146	4	)	)	PUNCT
ejpam-4788	146	5	=	=	SYM
ejpam-4788	147	1	1	1	NUM
ejpam-4788	147	2	and	and	CCONJ
ejpam-4788	147	3	λp	λp	ADP
ejpam-4788	147	4	i(v	i(v	NOUN
ejpam-4788	147	5	)	)	PUNCT
ejpam-4788	147	6	=	=	SYM
ejpam-4788	147	7	−1	−1	NOUN
ejpam-4788	147	8	.	.	PUNCT
ejpam-4788	148	1	by	by	ADP
ejpam-4788	148	2	assumption	assumption	NOUN
ejpam-4788	148	3	,	,	PUNCT
ejpam-4788	148	4	λp	λp	PRON
ejpam-4788	148	5	k(uαv	k(uαv	PROPN
ejpam-4788	148	6	)	)	PUNCT
ejpam-4788	148	7	≥	≥	NOUN
ejpam-4788	148	8	λp	λp	X
ejpam-4788	148	9	k(v	k(v	PROPN
ejpam-4788	148	10	)	)	PUNCT
ejpam-4788	148	11	and	and	CCONJ
ejpam-4788	148	12	λn	λn	PROPN
ejpam-4788	148	13	k(uαv	k(uαv	PROPN
ejpam-4788	148	14	)	)	PUNCT
ejpam-4788	148	15	≤	≤	NUM
ejpam-4788	148	16	λn	λn	ADP
ejpam-4788	148	17	k(v	k(v	PROPN
ejpam-4788	148	18	)	)	PUNCT
ejpam-4788	148	19	thus	thus	ADV
ejpam-4788	148	20	,	,	PUNCT
ejpam-4788	148	21	uαv	uαv	PROPN
ejpam-4788	148	22	∈	∈	PROPN
ejpam-4788	148	23	k.	k.	PROPN
ejpam-4788	148	24	hence	hence	ADV
ejpam-4788	148	25	,	,	PUNCT
ejpam-4788	148	26	k	k	PROPN
ejpam-4788	148	27	is	be	AUX
ejpam-4788	148	28	a	a	DET
ejpam-4788	148	29	left	left	ADJ
ejpam-4788	148	30	α	α	NOUN
ejpam-4788	148	31	-	-	NOUN
ejpam-4788	148	32	ideal	ideal	NOUN
ejpam-4788	148	33	of	of	ADP
ejpam-4788	148	34	s.	s.	PROPN
ejpam-4788	148	35	theorem	theorem	VERB
ejpam-4788	148	36	3	3	X
ejpam-4788	148	37	.	.	PUNCT
ejpam-4788	149	1	the	the	DET
ejpam-4788	149	2	positive	positive	ADJ
ejpam-4788	149	3	and	and	CCONJ
ejpam-4788	149	4	negative	negative	ADJ
ejpam-4788	149	5	of	of	ADP
ejpam-4788	149	6	the	the	DET
ejpam-4788	149	7	intersection	intersection	NOUN
ejpam-4788	149	8	and	and	CCONJ
ejpam-4788	149	9	union	union	NOUN
ejpam-4788	149	10	of	of	ADP
ejpam-4788	149	11	any	any	DET
ejpam-4788	149	12	two	two	NUM
ejpam-4788	149	13	bf	bf	NOUN
ejpam-4788	149	14	left	leave	VERB
ejpam-4788	149	15	αideals	αideal	NOUN
ejpam-4788	149	16	(	(	PUNCT
ejpam-4788	149	17	right	right	ADJ
ejpam-4788	149	18	β	β	NOUN
ejpam-4788	149	19	-	-	NOUN
ejpam-4788	149	20	ideals	ideal	NOUN
ejpam-4788	149	21	,	,	PUNCT
ejpam-4788	149	22	(	(	PUNCT
ejpam-4788	149	23	α	α	X
ejpam-4788	149	24	,	,	PUNCT
ejpam-4788	149	25	β)-ideals	β)-ideal	NOUN
ejpam-4788	149	26	)	)	PUNCT
ejpam-4788	149	27	of	of	ADP
ejpam-4788	149	28	a	a	DET
ejpam-4788	149	29	γ	γ	PROPN
ejpam-4788	149	30	-	-	PUNCT
ejpam-4788	149	31	semigroup	semigroup	NOUN
ejpam-4788	149	32	s	s	VERB
ejpam-4788	149	33	is	be	AUX
ejpam-4788	149	34	a	a	DET
ejpam-4788	149	35	bf	bf	NOUN
ejpam-4788	149	36	left	leave	VERB
ejpam-4788	149	37	α	α	NOUN
ejpam-4788	149	38	-	-	NOUN
ejpam-4788	149	39	ideal	ideal	ADJ
ejpam-4788	149	40	(	(	PUNCT
ejpam-4788	149	41	right	right	ADJ
ejpam-4788	149	42	β	β	NOUN
ejpam-4788	149	43	-	-	NOUN
ejpam-4788	149	44	ideal	ideal	ADJ
ejpam-4788	149	45	,	,	PUNCT
ejpam-4788	149	46	(	(	PUNCT
ejpam-4788	149	47	α	α	NOUN
ejpam-4788	149	48	,	,	PUNCT
ejpam-4788	149	49	β)-ideal	β)-ideal	NUM
ejpam-4788	149	50	)	)	PUNCT
ejpam-4788	149	51	of	of	ADP
ejpam-4788	149	52	s.	s.	PROPN
ejpam-4788	149	53	proof	proof	PROPN
ejpam-4788	149	54	.	.	PUNCT
ejpam-4788	150	1	let	let	VERB
ejpam-4788	150	2	ξ	ξ	X
ejpam-4788	150	3	=	=	SYM
ejpam-4788	150	4	(	(	PUNCT
ejpam-4788	150	5	s	s	PROPN
ejpam-4788	150	6	;	;	PUNCT
ejpam-4788	150	7	ξp	ξp	NUM
ejpam-4788	150	8	,	,	PUNCT
ejpam-4788	150	9	ξn	ξn	NOUN
ejpam-4788	150	10	)	)	PUNCT
ejpam-4788	150	11	and	and	CCONJ
ejpam-4788	150	12	ς	ς	PROPN
ejpam-4788	150	13	=	=	PUNCT
ejpam-4788	150	14	(	(	PUNCT
ejpam-4788	150	15	s	s	NOUN
ejpam-4788	150	16	;	;	PUNCT
ejpam-4788	150	17	ςp	ςp	NUM
ejpam-4788	150	18	,	,	PUNCT
ejpam-4788	150	19	ςn	ςn	NOUN
ejpam-4788	150	20	)	)	PUNCT
ejpam-4788	150	21	be	be	VERB
ejpam-4788	150	22	bf	bf	NOUN
ejpam-4788	150	23	left	leave	VERB
ejpam-4788	150	24	α	α	NOUN
ejpam-4788	150	25	-	-	NOUN
ejpam-4788	150	26	ideals	ideal	NOUN
ejpam-4788	150	27	of	of	ADP
ejpam-4788	150	28	s	s	PRON
ejpam-4788	150	29	and	and	CCONJ
ejpam-4788	150	30	let	let	VERB
ejpam-4788	150	31	u	u	NOUN
ejpam-4788	150	32	,	,	PUNCT
ejpam-4788	150	33	v	v	PROPN
ejpam-4788	150	34	∈	∈	NOUN
ejpam-4788	150	35	s.	s.	PROPN
ejpam-4788	150	36	then	then	ADV
ejpam-4788	150	37	(	(	PUNCT
ejpam-4788	150	38	ξp	ξp	ADP
ejpam-4788	150	39	∩	∩	NOUN
ejpam-4788	150	40	ςp)(uαv	ςp)(uαv	X
ejpam-4788	150	41	)	)	PUNCT
ejpam-4788	150	42	=	=	SYM
ejpam-4788	150	43	ξp(uαv	ξp(uαv	NOUN
ejpam-4788	150	44	)	)	PUNCT
ejpam-4788	150	45	∧	∧	PROPN
ejpam-4788	150	46	ςp(uαv	ςp(uαv	PROPN
ejpam-4788	150	47	)	)	PUNCT
ejpam-4788	150	48	≥	≥	NOUN
ejpam-4788	150	49	ξp(v	ξp(v	NOUN
ejpam-4788	150	50	)	)	PUNCT
ejpam-4788	150	51	∧	∧	NOUN
ejpam-4788	150	52	ςp(v	ςp(v	NOUN
ejpam-4788	150	53	)	)	PUNCT
ejpam-4788	150	54	=	=	SYM
ejpam-4788	150	55	(	(	PUNCT
ejpam-4788	150	56	ξp	ξp	NUM
ejpam-4788	150	57	∩	∩	NOUN
ejpam-4788	150	58	ςp)(v	ςp)(v	NOUN
ejpam-4788	150	59	)	)	PUNCT
ejpam-4788	150	60	and	and	CCONJ
ejpam-4788	150	61	(	(	PUNCT
ejpam-4788	150	62	ξn	ξn	PROPN
ejpam-4788	150	63	∩	∩	NOUN
ejpam-4788	150	64	ςn)(uαv	ςn)(uαv	NUM
ejpam-4788	150	65	)	)	PUNCT
ejpam-4788	150	66	=	=	SYM
ejpam-4788	150	67	ξn(uαv	ξn(uαv	PROPN
ejpam-4788	150	68	)	)	PUNCT
ejpam-4788	150	69	∨	∨	NUM
ejpam-4788	150	70	ςn(uαv	ςn(uαv	PROPN
ejpam-4788	150	71	)	)	PUNCT
ejpam-4788	150	72	≤	≤	NOUN
ejpam-4788	150	73	ξn(v	ξn(v	NUM
ejpam-4788	150	74	)	)	PUNCT
ejpam-4788	150	75	∨	∨	NUM
ejpam-4788	150	76	ςn(v	ςn(v	NUM
ejpam-4788	150	77	)	)	PUNCT
ejpam-4788	150	78	=	=	PUNCT
ejpam-4788	150	79	(	(	PUNCT
ejpam-4788	150	80	ξn	ξn	PROPN
ejpam-4788	150	81	∩	∩	NOUN
ejpam-4788	150	82	ςn)(v	ςn)(v	NOUN
ejpam-4788	150	83	)	)	PUNCT
ejpam-4788	150	84	.	.	PUNCT
ejpam-4788	151	1	similarly	similarly	ADV
ejpam-4788	151	2	,	,	PUNCT
ejpam-4788	151	3	(	(	PUNCT
ejpam-4788	151	4	ξp	ξp	ADP
ejpam-4788	151	5	∪	∪	VERB
ejpam-4788	151	6	ςp)(uαv	ςp)(uαv	NOUN
ejpam-4788	151	7	)	)	PUNCT
ejpam-4788	151	8	=	=	SYM
ejpam-4788	151	9	ξp(uαv	ξp(uαv	NOUN
ejpam-4788	151	10	)	)	PUNCT
ejpam-4788	151	11	∨	∨	NUM
ejpam-4788	151	12	ςp(uαv	ςp(uαv	PROPN
ejpam-4788	151	13	)	)	PUNCT
ejpam-4788	151	14	≥	≥	NOUN
ejpam-4788	151	15	ξp(v	ξp(v	NOUN
ejpam-4788	151	16	)	)	PUNCT
ejpam-4788	151	17	∨	∨	NUM
ejpam-4788	151	18	ςp(v	ςp(v	NOUN
ejpam-4788	151	19	)	)	PUNCT
ejpam-4788	151	20	=	=	SYM
ejpam-4788	151	21	(	(	PUNCT
ejpam-4788	151	22	ξp	ξp	NOUN
ejpam-4788	151	23	∪	∪	VERB
ejpam-4788	151	24	ςp)(v	ςp)(v	NUM
ejpam-4788	151	25	)	)	PUNCT
ejpam-4788	151	26	and	and	CCONJ
ejpam-4788	151	27	(	(	PUNCT
ejpam-4788	151	28	ξn	ξn	PROPN
ejpam-4788	151	29	∪	∪	ADP
ejpam-4788	151	30	ςn)(uαv	ςn)(uαv	NOUN
ejpam-4788	151	31	)	)	PUNCT
ejpam-4788	151	32	=	=	SYM
ejpam-4788	151	33	ξn(uαv	ξn(uαv	PROPN
ejpam-4788	151	34	)	)	PUNCT
ejpam-4788	151	35	∧	∧	PROPN
ejpam-4788	151	36	ςn(uαv	ςn(uαv	PROPN
ejpam-4788	151	37	)	)	PUNCT
ejpam-4788	151	38	≤	≤	NOUN
ejpam-4788	151	39	ξn(v	ξn(v	X
ejpam-4788	151	40	)	)	PUNCT
ejpam-4788	151	41	∧	∧	NOUN
ejpam-4788	151	42	ςn(v	ςn(v	NUM
ejpam-4788	151	43	)	)	PUNCT
ejpam-4788	151	44	=	=	PUNCT
ejpam-4788	151	45	(	(	PUNCT
ejpam-4788	151	46	ξn	ξn	PROPN
ejpam-4788	151	47	∩	∩	NOUN
ejpam-4788	151	48	ςn)(v	ςn)(v	NOUN
ejpam-4788	151	49	)	)	PUNCT
ejpam-4788	151	50	.	.	PUNCT
ejpam-4788	152	1	thus	thus	ADV
ejpam-4788	152	2	,	,	PUNCT
ejpam-4788	152	3	ξ	ξ	PROPN
ejpam-4788	152	4	∩	∩	NOUN
ejpam-4788	152	5	ς	ς	PROPN
ejpam-4788	152	6	and	and	CCONJ
ejpam-4788	152	7	ξ	ξ	PROPN
ejpam-4788	152	8	∪	∪	X
ejpam-4788	152	9	ς	ς	PROPN
ejpam-4788	152	10	are	be	AUX
ejpam-4788	152	11	bf	bf	NOUN
ejpam-4788	152	12	left	leave	VERB
ejpam-4788	152	13	α	α	NOUN
ejpam-4788	152	14	-	-	NOUN
ejpam-4788	152	15	ideals	ideal	NOUN
ejpam-4788	152	16	of	of	ADP
ejpam-4788	152	17	s.	s.	PROPN
ejpam-4788	152	18	p.	p.	PROPN
ejpam-4788	152	19	khamrot	khamrot	PROPN
ejpam-4788	152	20	,	,	PUNCT
ejpam-4788	152	21	t.	t.	PROPN
ejpam-4788	152	22	gaketem	gaketem	PROPN
ejpam-4788	152	23	/	/	SYM
ejpam-4788	152	24	eur	eur	PROPN
ejpam-4788	152	25	.	.	PUNCT
ejpam-4788	153	1	j.	j.	PROPN
ejpam-4788	153	2	pure	pure	PROPN
ejpam-4788	153	3	appl	appl	PROPN
ejpam-4788	153	4	.	.	PROPN
ejpam-4788	153	5	math	math	PROPN
ejpam-4788	153	6	,	,	PUNCT
ejpam-4788	153	7	16	16	NUM
ejpam-4788	153	8	(	(	PUNCT
ejpam-4788	153	9	3	3	NUM
ejpam-4788	153	10	)	)	PUNCT
ejpam-4788	153	11	(	(	PUNCT
ejpam-4788	153	12	2023	2023	NUM
ejpam-4788	153	13	)	)	PUNCT
ejpam-4788	153	14	,	,	PUNCT
ejpam-4788	153	15	1592	1592	NUM
ejpam-4788	153	16	-	-	SYM
ejpam-4788	153	17	1607	1607	NUM
ejpam-4788	153	18	1598	1598	NUM
ejpam-4788	153	19	theorem	theorem	NOUN
ejpam-4788	153	20	4	4	NUM
ejpam-4788	153	21	.	.	PUNCT
ejpam-4788	154	1	let	let	VERB
ejpam-4788	154	2	ξ	ξ	X
ejpam-4788	154	3	=	=	SYM
ejpam-4788	154	4	(	(	PUNCT
ejpam-4788	154	5	s	s	PROPN
ejpam-4788	154	6	;	;	PUNCT
ejpam-4788	154	7	ξp	ξp	NUM
ejpam-4788	154	8	,	,	PUNCT
ejpam-4788	154	9	ξn	ξn	NOUN
ejpam-4788	154	10	)	)	PUNCT
ejpam-4788	154	11	be	be	VERB
ejpam-4788	154	12	a	a	DET
ejpam-4788	154	13	bf	bf	NOUN
ejpam-4788	154	14	set	set	NOUN
ejpam-4788	154	15	of	of	ADP
ejpam-4788	154	16	a	a	DET
ejpam-4788	154	17	γ	γ	NOUN
ejpam-4788	154	18	-	-	PUNCT
ejpam-4788	154	19	semigroup	semigroup	ADJ
ejpam-4788	154	20	s	s	NOUN
ejpam-4788	154	21	and	and	CCONJ
ejpam-4788	154	22	ξ(l	ξ(l	PROPN
ejpam-4788	154	23	,	,	PUNCT
ejpam-4788	154	24	m	m	NOUN
ejpam-4788	154	25	)	)	PUNCT
ejpam-4788	155	1	=	=	SYM
ejpam-4788	155	2	(	(	PUNCT
ejpam-4788	155	3	s	s	X
ejpam-4788	155	4	;	;	PUNCT
ejpam-4788	155	5	ξpl	ξpl	X
ejpam-4788	155	6	,	,	PUNCT
ejpam-4788	155	7	ξ	ξ	PROPN
ejpam-4788	155	8	n	n	PRON
ejpam-4788	155	9	m	m	PRON
ejpam-4788	155	10	)	)	PUNCT
ejpam-4788	155	11	be	be	VERB
ejpam-4788	155	12	bf	bf	NOUN
ejpam-4788	155	13	point	point	NOUN
ejpam-4788	155	14	with	with	ADP
ejpam-4788	155	15	ξpl	ξpl	NOUN
ejpam-4788	155	16	=	=	SYM
ejpam-4788	155	17	{	{	PUNCT
ejpam-4788	155	18	x	x	PUNCT
ejpam-4788	155	19	∈	∈	PROPN
ejpam-4788	155	20	s	s	PART
ejpam-4788	155	21	|	|	ADV
ejpam-4788	155	22	ξpl	ξpl	NOUN
ejpam-4788	155	23	(	(	PUNCT
ejpam-4788	155	24	x	x	NOUN
ejpam-4788	155	25	)	)	PUNCT
ejpam-4788	155	26	≥	≥	NOUN
ejpam-4788	155	27	l	l	NOUN
ejpam-4788	155	28	}	}	PUNCT
ejpam-4788	155	29	and	and	CCONJ
ejpam-4788	155	30	ξnm	ξnm	PROPN
ejpam-4788	155	31	=	=	SYM
ejpam-4788	155	32	{	{	PUNCT
ejpam-4788	155	33	x	x	PUNCT
ejpam-4788	155	34	∈	∈	PROPN
ejpam-4788	155	35	s	s	VERB
ejpam-4788	155	36	|	|	ADV
ejpam-4788	155	37	ξnm(x	ξnm(x	NOUN
ejpam-4788	155	38	)	)	PUNCT
ejpam-4788	155	39	≤	≤	NUM
ejpam-4788	155	40	m	m	ADP
ejpam-4788	155	41	}	}	PUNCT
ejpam-4788	155	42	.	.	PUNCT
ejpam-4788	156	1	then	then	ADV
ejpam-4788	156	2	ξ	ξ	X
ejpam-4788	156	3	=	=	SYM
ejpam-4788	156	4	(	(	PUNCT
ejpam-4788	156	5	s	s	PROPN
ejpam-4788	156	6	;	;	PUNCT
ejpam-4788	156	7	ξp	ξp	NUM
ejpam-4788	156	8	,	,	PUNCT
ejpam-4788	156	9	ξn	ξn	NOUN
ejpam-4788	156	10	)	)	PUNCT
ejpam-4788	156	11	is	be	AUX
ejpam-4788	156	12	a	a	DET
ejpam-4788	156	13	bf	bf	NOUN
ejpam-4788	156	14	left	leave	VERB
ejpam-4788	156	15	α	α	NOUN
ejpam-4788	156	16	-	-	NOUN
ejpam-4788	156	17	ideal	ideal	ADJ
ejpam-4788	156	18	(	(	PUNCT
ejpam-4788	156	19	right	right	ADJ
ejpam-4788	156	20	β	β	NOUN
ejpam-4788	156	21	-	-	NOUN
ejpam-4788	156	22	ideal	ideal	ADJ
ejpam-4788	156	23	,	,	PUNCT
ejpam-4788	156	24	(	(	PUNCT
ejpam-4788	156	25	α	α	NOUN
ejpam-4788	156	26	,	,	PUNCT
ejpam-4788	156	27	β)-ideal	β)-ideal	NUM
ejpam-4788	156	28	)	)	PUNCT
ejpam-4788	156	29	of	of	ADP
ejpam-4788	156	30	s	s	PRON
ejpam-4788	156	31	if	if	SCONJ
ejpam-4788	157	1	and	and	CCONJ
ejpam-4788	157	2	only	only	ADV
ejpam-4788	157	3	if	if	SCONJ
ejpam-4788	157	4	ξ(l	ξ(l	NOUN
ejpam-4788	157	5	,	,	PUNCT
ejpam-4788	157	6	m	m	NOUN
ejpam-4788	157	7	)	)	PUNCT
ejpam-4788	157	8	=	=	SYM
ejpam-4788	157	9	(	(	PUNCT
ejpam-4788	157	10	s	s	X
ejpam-4788	157	11	;	;	PUNCT
ejpam-4788	157	12	ξpl	ξpl	X
ejpam-4788	157	13	,	,	PUNCT
ejpam-4788	157	14	ξ	ξ	PROPN
ejpam-4788	157	15	n	n	PRON
ejpam-4788	157	16	m	m	PRON
ejpam-4788	157	17	)	)	PUNCT
ejpam-4788	157	18	is	be	AUX
ejpam-4788	157	19	a	a	DET
ejpam-4788	157	20	non	non	ADJ
ejpam-4788	157	21	-	-	ADJ
ejpam-4788	157	22	empty	empty	ADJ
ejpam-4788	157	23	set	set	NOUN
ejpam-4788	157	24	and	and	CCONJ
ejpam-4788	157	25	ξ(l	ξ(l	NOUN
ejpam-4788	157	26	,	,	PUNCT
ejpam-4788	157	27	m	m	PRON
ejpam-4788	157	28	)	)	PUNCT
ejpam-4788	157	29	is	be	AUX
ejpam-4788	157	30	a	a	DET
ejpam-4788	157	31	left	left	ADJ
ejpam-4788	157	32	α	α	NOUN
ejpam-4788	157	33	-	-	NOUN
ejpam-4788	157	34	ideal	ideal	ADJ
ejpam-4788	157	35	(	(	PUNCT
ejpam-4788	157	36	right	right	ADJ
ejpam-4788	157	37	β	β	NOUN
ejpam-4788	157	38	-	-	NOUN
ejpam-4788	157	39	ideal	ideal	ADJ
ejpam-4788	157	40	,	,	PUNCT
ejpam-4788	157	41	(	(	PUNCT
ejpam-4788	157	42	α	α	NOUN
ejpam-4788	157	43	,	,	PUNCT
ejpam-4788	157	44	β)-ideal	β)-ideal	NUM
ejpam-4788	157	45	)	)	PUNCT
ejpam-4788	157	46	of	of	ADP
ejpam-4788	157	47	s	s	PRON
ejpam-4788	157	48	for	for	ADP
ejpam-4788	157	49	all	all	DET
ejpam-4788	157	50	(	(	PUNCT
ejpam-4788	157	51	l	l	NOUN
ejpam-4788	157	52	,	,	PUNCT
ejpam-4788	157	53	m	m	NOUN
ejpam-4788	157	54	)	)	PUNCT
ejpam-4788	157	55	∈	∈	PROPN
ejpam-4788	157	56	(	(	PUNCT
ejpam-4788	157	57	0	0	NUM
ejpam-4788	157	58	,	,	PUNCT
ejpam-4788	157	59	1]×	1]×	NUM
ejpam-4788	157	60	[	[	X
ejpam-4788	157	61	−1	−1	NOUN
ejpam-4788	157	62	,	,	PUNCT
ejpam-4788	157	63	0	0	NUM
ejpam-4788	157	64	)	)	PUNCT
ejpam-4788	157	65	.	.	PUNCT
ejpam-4788	158	1	proof	proof	NOUN
ejpam-4788	158	2	.	.	PUNCT
ejpam-4788	159	1	suppose	suppose	VERB
ejpam-4788	159	2	that	that	SCONJ
ejpam-4788	159	3	ξ	ξ	PROPN
ejpam-4788	159	4	=	=	SYM
ejpam-4788	159	5	(	(	PUNCT
ejpam-4788	159	6	s	s	PROPN
ejpam-4788	159	7	;	;	PUNCT
ejpam-4788	159	8	ξp	ξp	NUM
ejpam-4788	159	9	,	,	PUNCT
ejpam-4788	159	10	ξn	ξn	NOUN
ejpam-4788	159	11	)	)	PUNCT
ejpam-4788	159	12	is	be	AUX
ejpam-4788	159	13	a	a	DET
ejpam-4788	159	14	bf	bf	NOUN
ejpam-4788	159	15	left	leave	VERB
ejpam-4788	159	16	α	α	NOUN
ejpam-4788	159	17	-	-	NOUN
ejpam-4788	159	18	ideal	ideal	NOUN
ejpam-4788	159	19	of	of	ADP
ejpam-4788	159	20	s.	s.	PROPN
ejpam-4788	159	21	then	then	ADV
ejpam-4788	159	22	ξp(uαr	ξp(uαr	PROPN
ejpam-4788	159	23	)	)	PUNCT
ejpam-4788	159	24	≥	≥	NOUN
ejpam-4788	159	25	ξp(r	ξp(r	NOUN
ejpam-4788	159	26	)	)	PUNCT
ejpam-4788	159	27	and	and	CCONJ
ejpam-4788	159	28	ξn(uαr	ξn(uαr	PROPN
ejpam-4788	159	29	)	)	PUNCT
ejpam-4788	159	30	≤	≤	NOUN
ejpam-4788	159	31	ξn(r	ξn(r	NOUN
ejpam-4788	159	32	)	)	PUNCT
ejpam-4788	159	33	for	for	ADP
ejpam-4788	159	34	all	all	DET
ejpam-4788	159	35	u	u	NOUN
ejpam-4788	159	36	,	,	PUNCT
ejpam-4788	159	37	r	r	PROPN
ejpam-4788	159	38	∈	∈	PROPN
ejpam-4788	159	39	s.	s.	PROPN
ejpam-4788	159	40	let	let	VERB
ejpam-4788	159	41	(	(	PUNCT
ejpam-4788	159	42	l	l	NOUN
ejpam-4788	159	43	,	,	PUNCT
ejpam-4788	159	44	m	m	NOUN
ejpam-4788	159	45	)	)	PUNCT
ejpam-4788	159	46	∈	∈	PROPN
ejpam-4788	159	47	(	(	PUNCT
ejpam-4788	159	48	0	0	NUM
ejpam-4788	159	49	,	,	PUNCT
ejpam-4788	159	50	1	1	NUM
ejpam-4788	159	51	]	]	SYM
ejpam-4788	159	52	×	×	NOUN
ejpam-4788	160	1	[	[	X
ejpam-4788	160	2	−1	−1	NOUN
ejpam-4788	160	3	,	,	PUNCT
ejpam-4788	160	4	0	0	NUM
ejpam-4788	160	5	)	)	PUNCT
ejpam-4788	160	6	be	be	AUX
ejpam-4788	160	7	such	such	ADJ
ejpam-4788	160	8	that	that	SCONJ
ejpam-4788	160	9	ξ(l	ξ(l	NOUN
ejpam-4788	160	10	,	,	PUNCT
ejpam-4788	160	11	m	m	NOUN
ejpam-4788	160	12	)	)	PUNCT
ejpam-4788	160	13	̸=	̸=	PROPN
ejpam-4788	160	14	∅.	∅.	ADV
ejpam-4788	160	15	let	let	VERB
ejpam-4788	160	16	r	r	NOUN
ejpam-4788	160	17	∈	∈	PROPN
ejpam-4788	160	18	ξ(l	ξ(l	PROPN
ejpam-4788	160	19	,	,	PUNCT
ejpam-4788	160	20	m	m	NOUN
ejpam-4788	160	21	)	)	PUNCT
ejpam-4788	160	22	and	and	CCONJ
ejpam-4788	160	23	u	u	PROPN
ejpam-4788	160	24	∈	∈	PROPN
ejpam-4788	160	25	s.	s.	PROPN
ejpam-4788	160	26	then	then	ADV
ejpam-4788	160	27	ξp(r	ξp(r	PUNCT
ejpam-4788	160	28	)	)	PUNCT
ejpam-4788	160	29	≥	≥	X
ejpam-4788	160	30	l	l	NOUN
ejpam-4788	160	31	and	and	CCONJ
ejpam-4788	160	32	ξn(r	ξn(r	NUM
ejpam-4788	160	33	)	)	PUNCT
ejpam-4788	161	1	≤	≤	NUM
ejpam-4788	161	2	m.	m.	NOUN
ejpam-4788	161	3	thus	thus	ADV
ejpam-4788	161	4	,	,	PUNCT
ejpam-4788	161	5	ξp(uαr	ξp(uαr	NOUN
ejpam-4788	161	6	)	)	PUNCT
ejpam-4788	161	7	≥	≥	NOUN
ejpam-4788	161	8	ξp(r	ξp(r	NOUN
ejpam-4788	161	9	)	)	PUNCT
ejpam-4788	161	10	≥	≥	X
ejpam-4788	161	11	l	l	NOUN
ejpam-4788	161	12	and	and	CCONJ
ejpam-4788	161	13	ξp(uαr	ξp(uαr	NOUN
ejpam-4788	161	14	)	)	PUNCT
ejpam-4788	161	15	≤	≤	NOUN
ejpam-4788	161	16	ξn(r	ξn(r	NOUN
ejpam-4788	161	17	)	)	PUNCT
ejpam-4788	161	18	≤	≤	NUM
ejpam-4788	161	19	m.	m.	NOUN
ejpam-4788	161	20	so	so	ADV
ejpam-4788	161	21	,	,	PUNCT
ejpam-4788	161	22	uαr	uαr	NOUN
ejpam-4788	161	23	∈	∈	PROPN
ejpam-4788	161	24	ξ(l	ξ(l	PROPN
ejpam-4788	161	25	,	,	PUNCT
ejpam-4788	161	26	m	m	NOUN
ejpam-4788	161	27	)	)	PUNCT
ejpam-4788	161	28	.	.	PUNCT
ejpam-4788	162	1	hence	hence	ADV
ejpam-4788	162	2	,	,	PUNCT
ejpam-4788	162	3	ξ(l	ξ(l	PROPN
ejpam-4788	162	4	,	,	PUNCT
ejpam-4788	162	5	m	m	NOUN
ejpam-4788	162	6	)	)	PUNCT
ejpam-4788	162	7	=	=	SYM
ejpam-4788	162	8	(	(	PUNCT
ejpam-4788	162	9	s	s	X
ejpam-4788	162	10	;	;	PUNCT
ejpam-4788	162	11	ξpl	ξpl	X
ejpam-4788	162	12	,	,	PUNCT
ejpam-4788	162	13	ξ	ξ	PROPN
ejpam-4788	162	14	n	n	PRON
ejpam-4788	162	15	m	m	PRON
ejpam-4788	162	16	)	)	PUNCT
ejpam-4788	162	17	is	be	AUX
ejpam-4788	162	18	a	a	DET
ejpam-4788	162	19	left	left	ADJ
ejpam-4788	162	20	α	α	NOUN
ejpam-4788	162	21	-	-	NOUN
ejpam-4788	162	22	ideal	ideal	NOUN
ejpam-4788	162	23	of	of	ADP
ejpam-4788	162	24	s.	s.	PROPN
ejpam-4788	162	25	conversely	conversely	ADV
ejpam-4788	162	26	,	,	PUNCT
ejpam-4788	162	27	assume	assume	VERB
ejpam-4788	162	28	that	that	SCONJ
ejpam-4788	162	29	ξ(l	ξ(l	PROPN
ejpam-4788	162	30	,	,	PUNCT
ejpam-4788	162	31	m	m	NOUN
ejpam-4788	162	32	)	)	PUNCT
ejpam-4788	162	33	=	=	SYM
ejpam-4788	162	34	(	(	PUNCT
ejpam-4788	162	35	s	s	X
ejpam-4788	162	36	;	;	PUNCT
ejpam-4788	162	37	ξpl	ξpl	X
ejpam-4788	162	38	,	,	PUNCT
ejpam-4788	162	39	ξ	ξ	PROPN
ejpam-4788	162	40	n	n	PRON
ejpam-4788	162	41	m	m	PRON
ejpam-4788	162	42	)	)	PUNCT
ejpam-4788	162	43	is	be	AUX
ejpam-4788	162	44	a	a	DET
ejpam-4788	162	45	left	left	ADJ
ejpam-4788	162	46	α	α	NOUN
ejpam-4788	162	47	-	-	NOUN
ejpam-4788	162	48	ideal	ideal	NOUN
ejpam-4788	162	49	of	of	ADP
ejpam-4788	162	50	s	s	PRON
ejpam-4788	162	51	if	if	SCONJ
ejpam-4788	162	52	(	(	PUNCT
ejpam-4788	162	53	l	l	NOUN
ejpam-4788	162	54	,	,	PUNCT
ejpam-4788	162	55	m	m	NOUN
ejpam-4788	162	56	)	)	PUNCT
ejpam-4788	162	57	∈	∈	PROPN
ejpam-4788	162	58	(	(	PUNCT
ejpam-4788	162	59	0	0	NUM
ejpam-4788	162	60	,	,	PUNCT
ejpam-4788	162	61	1	1	NUM
ejpam-4788	162	62	]	]	SYM
ejpam-4788	162	63	×	×	NOUN
ejpam-4788	163	1	[	[	X
ejpam-4788	163	2	−1	−1	NOUN
ejpam-4788	163	3	,	,	PUNCT
ejpam-4788	163	4	0	0	NUM
ejpam-4788	163	5	)	)	PUNCT
ejpam-4788	163	6	and	and	CCONJ
ejpam-4788	163	7	ξ(l	ξ(l	PROPN
ejpam-4788	163	8	,	,	PUNCT
ejpam-4788	163	9	m	m	NOUN
ejpam-4788	163	10	)	)	PUNCT
ejpam-4788	163	11	̸=	̸=	PROPN
ejpam-4788	163	12	∅.	∅.	ADV
ejpam-4788	163	13	let	let	VERB
ejpam-4788	163	14	u	u	NOUN
ejpam-4788	163	15	,	,	PUNCT
ejpam-4788	163	16	v	v	ADP
ejpam-4788	163	17	∈	∈	PROPN
ejpam-4788	163	18	s	s	PART
ejpam-4788	163	19	and	and	CCONJ
ejpam-4788	163	20	l	l	NOUN
ejpam-4788	163	21	=	=	NOUN
ejpam-4788	163	22	ξp(v),m	ξp(v),m	NOUN
ejpam-4788	163	23	=	=	SYM
ejpam-4788	163	24	ξn(v	ξn(v	NUM
ejpam-4788	163	25	)	)	PUNCT
ejpam-4788	163	26	.	.	PUNCT
ejpam-4788	164	1	by	by	ADP
ejpam-4788	164	2	assumption	assumption	NOUN
ejpam-4788	164	3	,	,	PUNCT
ejpam-4788	164	4	ξp(v	ξp(v	NUM
ejpam-4788	164	5	)	)	PUNCT
ejpam-4788	164	6	≥	≥	NOUN
ejpam-4788	164	7	l	l	NOUN
ejpam-4788	164	8	and	and	CCONJ
ejpam-4788	164	9	ξn(v	ξn(v	NUM
ejpam-4788	164	10	)	)	PUNCT
ejpam-4788	164	11	≤	≤	NOUN
ejpam-4788	164	12	m.	m.	NOUN
ejpam-4788	164	13	then	then	ADV
ejpam-4788	164	14	v	v	NUM
ejpam-4788	164	15	∈	∈	PROPN
ejpam-4788	164	16	ξ(l	ξ(l	PROPN
ejpam-4788	164	17	,	,	PUNCT
ejpam-4788	164	18	m	m	NOUN
ejpam-4788	164	19	)	)	PUNCT
ejpam-4788	164	20	.	.	PUNCT
ejpam-4788	165	1	thus	thus	ADV
ejpam-4788	165	2	,	,	PUNCT
ejpam-4788	165	3	ξl	ξl	X
ejpam-4788	165	4	̸=	̸=	PROPN
ejpam-4788	165	5	∅.	∅.	PRON
ejpam-4788	165	6	hence	hence	ADV
ejpam-4788	165	7	,	,	PUNCT
ejpam-4788	165	8	ξl	ξl	NOUN
ejpam-4788	165	9	is	be	AUX
ejpam-4788	165	10	a	a	DET
ejpam-4788	165	11	left	left	ADJ
ejpam-4788	165	12	α	α	NOUN
ejpam-4788	165	13	-	-	NOUN
ejpam-4788	165	14	ideal	ideal	NOUN
ejpam-4788	165	15	of	of	ADP
ejpam-4788	165	16	s.	s.	PROPN
ejpam-4788	165	17	since	since	SCONJ
ejpam-4788	165	18	v	v	NUM
ejpam-4788	165	19	∈	∈	PROPN
ejpam-4788	165	20	ξ(l	ξ(l	PROPN
ejpam-4788	165	21	,	,	PUNCT
ejpam-4788	165	22	m	m	NOUN
ejpam-4788	165	23	)	)	PUNCT
ejpam-4788	165	24	and	and	CCONJ
ejpam-4788	165	25	u	u	PROPN
ejpam-4788	165	26	∈	∈	PROPN
ejpam-4788	165	27	s	s	X
ejpam-4788	165	28	,	,	PUNCT
ejpam-4788	165	29	we	we	PRON
ejpam-4788	165	30	have	have	VERB
ejpam-4788	165	31	xαv	xαv	PROPN
ejpam-4788	165	32	∈	∈	PROPN
ejpam-4788	165	33	ξ(l	ξ(l	PROPN
ejpam-4788	165	34	,	,	PUNCT
ejpam-4788	165	35	m	m	NOUN
ejpam-4788	165	36	)	)	PUNCT
ejpam-4788	165	37	.	.	PUNCT
ejpam-4788	166	1	thus	thus	ADV
ejpam-4788	166	2	,	,	PUNCT
ejpam-4788	166	3	ξp(uαv	ξp(uαv	NOUN
ejpam-4788	166	4	)	)	PUNCT
ejpam-4788	166	5	≥	≥	X
ejpam-4788	166	6	l	l	NOUN
ejpam-4788	166	7	=	=	PUNCT
ejpam-4788	166	8	ξp(v	ξp(v	X
ejpam-4788	166	9	)	)	PUNCT
ejpam-4788	166	10	and	and	CCONJ
ejpam-4788	166	11	ξn(uαv	ξn(uαv	PROPN
ejpam-4788	166	12	)	)	PUNCT
ejpam-4788	166	13	≤	≤	NUM
ejpam-4788	166	14	m	m	NOUN
ejpam-4788	166	15	=	=	SYM
ejpam-4788	166	16	ξn(v	ξn(v	NUM
ejpam-4788	166	17	)	)	PUNCT
ejpam-4788	166	18	.	.	PUNCT
ejpam-4788	167	1	hence	hence	ADV
ejpam-4788	167	2	,	,	PUNCT
ejpam-4788	167	3	ξ	ξ	X
ejpam-4788	167	4	=	=	SYM
ejpam-4788	167	5	(	(	PUNCT
ejpam-4788	167	6	s	s	PROPN
ejpam-4788	167	7	;	;	PUNCT
ejpam-4788	167	8	ξp	ξp	NUM
ejpam-4788	167	9	,	,	PUNCT
ejpam-4788	167	10	ξn	ξn	NOUN
ejpam-4788	167	11	)	)	PUNCT
ejpam-4788	167	12	is	be	AUX
ejpam-4788	167	13	a	a	DET
ejpam-4788	167	14	bf	bf	NOUN
ejpam-4788	167	15	left	leave	VERB
ejpam-4788	167	16	α	α	NOUN
ejpam-4788	167	17	-	-	NOUN
ejpam-4788	167	18	ideal	ideal	NOUN
ejpam-4788	167	19	of	of	ADP
ejpam-4788	167	20	s.	s.	PROPN
ejpam-4788	167	21	next	next	ADV
ejpam-4788	167	22	,	,	PUNCT
ejpam-4788	167	23	we	we	PRON
ejpam-4788	167	24	will	will	AUX
ejpam-4788	167	25	define	define	VERB
ejpam-4788	167	26	the	the	DET
ejpam-4788	167	27	(	(	PUNCT
ejpam-4788	167	28	α	α	NOUN
ejpam-4788	167	29	,	,	PUNCT
ejpam-4788	167	30	β)-product	β)-product	NOUN
ejpam-4788	167	31	.	.	PUNCT
ejpam-4788	168	1	for	for	ADP
ejpam-4788	168	2	bf	bf	NOUN
ejpam-4788	168	3	sets	set	VERB
ejpam-4788	168	4	ξ	ξ	X
ejpam-4788	168	5	=	=	SYM
ejpam-4788	168	6	(	(	PUNCT
ejpam-4788	168	7	s	s	PROPN
ejpam-4788	168	8	;	;	PUNCT
ejpam-4788	168	9	ξp	ξp	NUM
ejpam-4788	168	10	,	,	PUNCT
ejpam-4788	168	11	ξn	ξn	NOUN
ejpam-4788	168	12	)	)	PUNCT
ejpam-4788	168	13	and	and	CCONJ
ejpam-4788	168	14	ς	ς	PROPN
ejpam-4788	168	15	=	=	PUNCT
ejpam-4788	168	16	(	(	PUNCT
ejpam-4788	168	17	s	s	NOUN
ejpam-4788	168	18	;	;	PUNCT
ejpam-4788	168	19	ςp	ςp	NUM
ejpam-4788	168	20	,	,	PUNCT
ejpam-4788	168	21	ςn	ςn	NOUN
ejpam-4788	168	22	)	)	PUNCT
ejpam-4788	168	23	,	,	PUNCT
ejpam-4788	168	24	define	define	VERB
ejpam-4788	168	25	the	the	DET
ejpam-4788	168	26	product	product	NOUN
ejpam-4788	168	27	ξp	ξp	ADP
ejpam-4788	168	28	◦	◦	NOUN
ejpam-4788	168	29	α	α	NOUN
ejpam-4788	168	30	ςp	ςp	NOUN
ejpam-4788	168	31	and	and	CCONJ
ejpam-4788	168	32	ξn	ξn	PROPN
ejpam-4788	168	33	◦	◦	NOUN
ejpam-4788	168	34	α	α	NOUN
ejpam-4788	168	35	ςn	ςn	NOUN
ejpam-4788	168	36	as	as	SCONJ
ejpam-4788	168	37	follows	follow	VERB
ejpam-4788	168	38	:	:	PUNCT
ejpam-4788	168	39	for	for	ADP
ejpam-4788	168	40	u	u	PROPN
ejpam-4788	168	41	∈	∈	PROPN
ejpam-4788	168	42	s	s	X
ejpam-4788	168	43	(	(	PUNCT
ejpam-4788	168	44	ξp	ξp	AUX
ejpam-4788	168	45	◦	◦	NOUN
ejpam-4788	168	46	α	α	DET
ejpam-4788	168	47	ςp)(u	ςp)(u	NOUN
ejpam-4788	168	48	)	)	PUNCT
ejpam-4788	168	49	=	=	PUNCT
ejpam-4788	168	50			PUNCT
ejpam-4788	168	51	∨	∨	X
ejpam-4788	168	52	(	(	PUNCT
ejpam-4788	168	53	y	y	PROPN
ejpam-4788	168	54	,	,	PUNCT
ejpam-4788	168	55	α	α	PROPN
ejpam-4788	168	56	,	,	PUNCT
ejpam-4788	168	57	z)∈fuα	z)∈fuα	PROPN
ejpam-4788	168	58	{	{	PUNCT
ejpam-4788	168	59	ξp(y	ξp(y	NOUN
ejpam-4788	168	60	)	)	PUNCT
ejpam-4788	168	61	∧	∧	PROPN
ejpam-4788	168	62	ςp(z	ςp(z	NUM
ejpam-4788	168	63	)	)	PUNCT
ejpam-4788	168	64	}	}	PUNCT
ejpam-4788	168	65	if	if	SCONJ
ejpam-4788	168	66	u	u	NOUN
ejpam-4788	168	67	=	=	X
ejpam-4788	168	68	yαz	yαz	X
ejpam-4788	168	69	0	0	PUNCT
ejpam-4788	169	1	if	if	SCONJ
ejpam-4788	169	2	otherwise	otherwise	ADV
ejpam-4788	169	3	.	.	PUNCT
ejpam-4788	170	1	and	and	CCONJ
ejpam-4788	170	2	(	(	PUNCT
ejpam-4788	170	3	ξn	ξn	PROPN
ejpam-4788	170	4	◦	◦	NOUN
ejpam-4788	170	5	α	α	NOUN
ejpam-4788	170	6	ςn)(u	ςn)(u	NOUN
ejpam-4788	170	7	)	)	PUNCT
ejpam-4788	171	1	=	=	SYM
ejpam-4788	171	2			PUNCT
ejpam-4788	171	3	∧	∧	PROPN
ejpam-4788	171	4	(	(	PUNCT
ejpam-4788	171	5	y	y	PROPN
ejpam-4788	171	6	,	,	PUNCT
ejpam-4788	171	7	α	α	PROPN
ejpam-4788	171	8	,	,	PUNCT
ejpam-4788	171	9	z)∈fuα	z)∈fuα	PROPN
ejpam-4788	171	10	{	{	PUNCT
ejpam-4788	171	11	ξn(y	ξn(y	NOUN
ejpam-4788	171	12	)	)	PUNCT
ejpam-4788	171	13	∨	∨	NUM
ejpam-4788	171	14	ςn(z	ςn(z	NOUN
ejpam-4788	171	15	)	)	PUNCT
ejpam-4788	171	16	}	}	PUNCT
ejpam-4788	171	17	if	if	SCONJ
ejpam-4788	171	18	u	u	NOUN
ejpam-4788	171	19	=	=	X
ejpam-4788	171	20	yαz	yαz	X
ejpam-4788	171	21	0	0	PUNCT
ejpam-4788	172	1	if	if	SCONJ
ejpam-4788	172	2	otherwise	otherwise	ADV
ejpam-4788	172	3	,	,	PUNCT
ejpam-4788	172	4	where	where	SCONJ
ejpam-4788	172	5	fuα	fuα	NOUN
ejpam-4788	172	6	=	=	PUNCT
ejpam-4788	172	7	{	{	PUNCT
ejpam-4788	172	8	(	(	PUNCT
ejpam-4788	172	9	y	y	PROPN
ejpam-4788	172	10	,	,	PUNCT
ejpam-4788	172	11	z	z	NOUN
ejpam-4788	172	12	)	)	PUNCT
ejpam-4788	172	13	∈	∈	PROPN
ejpam-4788	172	14	s	s	PART
ejpam-4788	172	15	×	×	NOUN
ejpam-4788	172	16	γ×	γ×	PROPN
ejpam-4788	172	17	s	s	PART
ejpam-4788	172	18	|	|	ADV
ejpam-4788	172	19	u	u	NOUN
ejpam-4788	172	20	=	=	PROPN
ejpam-4788	172	21	yαz	yαz	PROPN
ejpam-4788	172	22	}	}	PUNCT
ejpam-4788	172	23	,	,	PUNCT
ejpam-4788	172	24	for	for	ADP
ejpam-4788	172	25	u	u	PROPN
ejpam-4788	172	26	∈	∈	PROPN
ejpam-4788	172	27	s	s	PART
ejpam-4788	172	28	and	and	CCONJ
ejpam-4788	172	29	α	α	PRON
ejpam-4788	172	30	∈	∈	PROPN
ejpam-4788	172	31	γ	γ	PROPN
ejpam-4788	172	32	.	.	PROPN
ejpam-4788	173	1	next	next	ADV
ejpam-4788	173	2	,	,	PUNCT
ejpam-4788	173	3	we	we	PRON
ejpam-4788	173	4	define	define	VERB
ejpam-4788	173	5	bf	bf	NOUN
ejpam-4788	173	6	(	(	PUNCT
ejpam-4788	173	7	α	α	NOUN
ejpam-4788	173	8	,	,	PUNCT
ejpam-4788	173	9	β)-bi	β)-bi	ADJ
ejpam-4788	173	10	-	-	PUNCT
ejpam-4788	173	11	ideal	ideal	NOUN
ejpam-4788	173	12	and	and	CCONJ
ejpam-4788	173	13	study	study	VERB
ejpam-4788	173	14	its	its	PRON
ejpam-4788	173	15	basic	basic	ADJ
ejpam-4788	173	16	properties	property	NOUN
ejpam-4788	173	17	.	.	PUNCT
ejpam-4788	174	1	definition	definition	NOUN
ejpam-4788	174	2	11	11	NUM
ejpam-4788	174	3	.	.	PUNCT
ejpam-4788	175	1	let	let	VERB
ejpam-4788	175	2	ξ	ξ	X
ejpam-4788	175	3	=	=	SYM
ejpam-4788	175	4	(	(	PUNCT
ejpam-4788	175	5	s	s	PROPN
ejpam-4788	175	6	;	;	PUNCT
ejpam-4788	175	7	ξp	ξp	NUM
ejpam-4788	175	8	,	,	PUNCT
ejpam-4788	175	9	ξn	ξn	NOUN
ejpam-4788	175	10	)	)	PUNCT
ejpam-4788	175	11	be	be	VERB
ejpam-4788	175	12	a	a	DET
ejpam-4788	175	13	bf	bf	NOUN
ejpam-4788	175	14	set	set	NOUN
ejpam-4788	175	15	of	of	ADP
ejpam-4788	175	16	a	a	DET
ejpam-4788	175	17	γ	γ	NOUN
ejpam-4788	175	18	-	-	PUNCT
ejpam-4788	175	19	semigroup	semigroup	NOUN
ejpam-4788	175	20	s	s	X
ejpam-4788	175	21	and	and	CCONJ
ejpam-4788	175	22	α	α	NOUN
ejpam-4788	175	23	,	,	PUNCT
ejpam-4788	175	24	β	β	PROPN
ejpam-4788	175	25	∈	∈	PROPN
ejpam-4788	175	26	γ	γ	X
ejpam-4788	175	27	.	.	PROPN
ejpam-4788	176	1	then	then	ADV
ejpam-4788	176	2	ξ	ξ	X
ejpam-4788	176	3	=	=	SYM
ejpam-4788	176	4	(	(	PUNCT
ejpam-4788	176	5	s	s	PROPN
ejpam-4788	176	6	;	;	PUNCT
ejpam-4788	176	7	ξp	ξp	NUM
ejpam-4788	176	8	,	,	PUNCT
ejpam-4788	176	9	ξn	ξn	NOUN
ejpam-4788	176	10	)	)	PUNCT
ejpam-4788	176	11	is	be	AUX
ejpam-4788	176	12	called	call	VERB
ejpam-4788	176	13	a	a	DET
ejpam-4788	176	14	bf	bf	NOUN
ejpam-4788	176	15	(	(	PUNCT
ejpam-4788	176	16	α	α	NOUN
ejpam-4788	176	17	,	,	PUNCT
ejpam-4788	176	18	β)-bi	β)-bi	NOUN
ejpam-4788	176	19	-	-	PUNCT
ejpam-4788	176	20	ideal	ideal	NOUN
ejpam-4788	176	21	of	of	ADP
ejpam-4788	176	22	s	s	PRON
ejpam-4788	176	23	if	if	SCONJ
ejpam-4788	176	24	ξp	ξp	PART
ejpam-4788	176	25	◦	◦	NOUN
ejpam-4788	176	26	αλp	αλp	NOUN
ejpam-4788	176	27	s	s	NOUN
ejpam-4788	176	28	◦	◦	NOUN
ejpam-4788	176	29	βξp	βξp	NOUN
ejpam-4788	176	30	≥	≥	NOUN
ejpam-4788	176	31	ξp	ξp	NOUN
ejpam-4788	176	32	and	and	CCONJ
ejpam-4788	176	33	ξn	ξn	PRON
ejpam-4788	176	34	◦	◦	NOUN
ejpam-4788	176	35	αλn	αλn	NOUN
ejpam-4788	176	36	s	s	SYM
ejpam-4788	176	37	◦	◦	NOUN
ejpam-4788	176	38	βξn	βξn	X
ejpam-4788	176	39	≤	≤	NOUN
ejpam-4788	176	40	ξn	ξn	NOUN
ejpam-4788	176	41	where	where	SCONJ
ejpam-4788	176	42	λs	λs	NOUN
ejpam-4788	176	43	=	=	SYM
ejpam-4788	176	44	(	(	PUNCT
ejpam-4788	176	45	s;λp	s;λp	NOUN
ejpam-4788	176	46	s	s	PART
ejpam-4788	176	47	,	,	PUNCT
ejpam-4788	176	48	λ	λ	PROPN
ejpam-4788	176	49	n	n	NOUN
ejpam-4788	176	50	s	s	NOUN
ejpam-4788	176	51	)	)	PUNCT
ejpam-4788	176	52	is	be	AUX
ejpam-4788	176	53	a	a	DET
ejpam-4788	176	54	bf	bf	NOUN
ejpam-4788	176	55	set	set	VERB
ejpam-4788	176	56	mapping	map	VERB
ejpam-4788	176	57	every	every	DET
ejpam-4788	176	58	element	element	NOUN
ejpam-4788	176	59	of	of	ADP
ejpam-4788	176	60	s	s	NOUN
ejpam-4788	176	61	to	to	ADP
ejpam-4788	176	62	[	[	X
ejpam-4788	176	63	−1	−1	NOUN
ejpam-4788	176	64	,	,	PUNCT
ejpam-4788	176	65	1	1	NUM
ejpam-4788	176	66	]	]	PUNCT
ejpam-4788	176	67	.	.	PUNCT
ejpam-4788	177	1	theorem	theorem	NOUN
ejpam-4788	177	2	5	5	NUM
ejpam-4788	177	3	.	.	PUNCT
ejpam-4788	178	1	let	let	VERB
ejpam-4788	178	2	k	k	PRON
ejpam-4788	178	3	be	be	AUX
ejpam-4788	178	4	a	a	DET
ejpam-4788	178	5	non	non	ADJ
ejpam-4788	178	6	-	-	ADJ
ejpam-4788	178	7	empty	empty	ADJ
ejpam-4788	178	8	subset	subset	NOUN
ejpam-4788	178	9	of	of	ADP
ejpam-4788	178	10	γ	γ	PROPN
ejpam-4788	178	11	-	-	PUNCT
ejpam-4788	178	12	semigroup	semigroup	PROPN
ejpam-4788	178	13	s.	s.	PROPN
ejpam-4788	179	1	then	then	ADV
ejpam-4788	179	2	k	k	PROPN
ejpam-4788	179	3	is	be	AUX
ejpam-4788	179	4	an	an	DET
ejpam-4788	179	5	(	(	PUNCT
ejpam-4788	179	6	α	α	NOUN
ejpam-4788	179	7	,	,	PUNCT
ejpam-4788	179	8	β)-bi	β)-bi	NOUN
ejpam-4788	179	9	-	-	PUNCT
ejpam-4788	179	10	ideal	ideal	NOUN
ejpam-4788	179	11	of	of	ADP
ejpam-4788	179	12	s	s	PRON
ejpam-4788	179	13	if	if	SCONJ
ejpam-4788	179	14	and	and	CCONJ
ejpam-4788	179	15	only	only	ADV
ejpam-4788	179	16	if	if	SCONJ
ejpam-4788	179	17	the	the	DET
ejpam-4788	179	18	characteristic	characteristic	ADJ
ejpam-4788	179	19	function	function	NOUN
ejpam-4788	179	20	λk	λk	X
ejpam-4788	179	21	=	=	PUNCT
ejpam-4788	179	22	(	(	PUNCT
ejpam-4788	179	23	s;λp	s;λp	PROPN
ejpam-4788	179	24	k	k	PROPN
ejpam-4788	179	25	,	,	PUNCT
ejpam-4788	179	26	λn	λn	PROPN
ejpam-4788	179	27	k	k	X
ejpam-4788	179	28	)	)	PUNCT
ejpam-4788	179	29	is	be	AUX
ejpam-4788	179	30	a	a	DET
ejpam-4788	179	31	bf	bf	NOUN
ejpam-4788	179	32	(	(	PUNCT
ejpam-4788	179	33	α	α	NOUN
ejpam-4788	179	34	,	,	PUNCT
ejpam-4788	179	35	β)-bi	β)-bi	NOUN
ejpam-4788	179	36	-	-	PUNCT
ejpam-4788	179	37	ideal	ideal	NOUN
ejpam-4788	179	38	of	of	ADP
ejpam-4788	179	39	s.	s.	PROPN
ejpam-4788	179	40	proof	proof	PROPN
ejpam-4788	179	41	.	.	PUNCT
ejpam-4788	180	1	suppose	suppose	VERB
ejpam-4788	180	2	that	that	SCONJ
ejpam-4788	180	3	k	k	PROPN
ejpam-4788	180	4	is	be	AUX
ejpam-4788	180	5	an	an	DET
ejpam-4788	180	6	(	(	PUNCT
ejpam-4788	180	7	α	α	NOUN
ejpam-4788	180	8	,	,	PUNCT
ejpam-4788	180	9	β)-bi	β)-bi	NOUN
ejpam-4788	180	10	-	-	PUNCT
ejpam-4788	180	11	ideal	ideal	NOUN
ejpam-4788	180	12	of	of	ADP
ejpam-4788	180	13	s	s	PRON
ejpam-4788	180	14	and	and	CCONJ
ejpam-4788	180	15	kαsβk	kαsβk	VERB
ejpam-4788	180	16	⊆	⊆	NUM
ejpam-4788	180	17	k.	k.	NOUN
ejpam-4788	180	18	if	if	SCONJ
ejpam-4788	180	19	u	u	PROPN
ejpam-4788	180	20	∈	∈	PROPN
ejpam-4788	180	21	kαsβk	kαsβk	NOUN
ejpam-4788	180	22	,	,	PUNCT
ejpam-4788	180	23	then	then	ADV
ejpam-4788	180	24	λp	λp	X
ejpam-4788	180	25	k(u	k(u	X
ejpam-4788	180	26	)	)	PUNCT
ejpam-4788	180	27	=	=	SYM
ejpam-4788	180	28	(	(	PUNCT
ejpam-4788	180	29	ξp	ξp	PART
ejpam-4788	180	30	◦	◦	VERB
ejpam-4788	180	31	α	α	NOUN
ejpam-4788	180	32	λp	λp	X
ejpam-4788	180	33	s	s	PROPN
ejpam-4788	180	34	◦	◦	NOUN
ejpam-4788	180	35	β	β	NOUN
ejpam-4788	180	36	ξp)(u	ξp)(u	ADJ
ejpam-4788	180	37	)	)	PUNCT
ejpam-4788	180	38	=	=	SYM
ejpam-4788	180	39	1	1	NUM
ejpam-4788	180	40	and	and	CCONJ
ejpam-4788	180	41	λn	λn	ADP
ejpam-4788	180	42	k(u	k(u	NOUN
ejpam-4788	180	43	)	)	PUNCT
ejpam-4788	180	44	=	=	SYM
ejpam-4788	180	45	(	(	PUNCT
ejpam-4788	180	46	ξn	ξn	PROPN
ejpam-4788	180	47	◦	◦	VERB
ejpam-4788	180	48	α	α	NOUN
ejpam-4788	180	49	λn	λn	NOUN
ejpam-4788	180	50	s	s	PART
ejpam-4788	180	51	◦	◦	NOUN
ejpam-4788	180	52	β	β	X
ejpam-4788	180	53	ξn)(u	ξn)(u	ADJ
ejpam-4788	180	54	)	)	PUNCT
ejpam-4788	181	1	=	=	SYM
ejpam-4788	181	2	−1	−1	NOUN
ejpam-4788	181	3	.	.	PUNCT
ejpam-4788	182	1	hence	hence	ADV
ejpam-4788	182	2	,	,	PUNCT
ejpam-4788	182	3	(	(	PUNCT
ejpam-4788	182	4	ξp	ξp	PART
ejpam-4788	182	5	◦	◦	VERB
ejpam-4788	182	6	α	α	NOUN
ejpam-4788	182	7	λp	λp	X
ejpam-4788	182	8	s	s	PROPN
ejpam-4788	182	9	◦	◦	NOUN
ejpam-4788	182	10	β	β	NOUN
ejpam-4788	182	11	ξp)(u	ξp)(u	ADJ
ejpam-4788	182	12	)	)	PUNCT
ejpam-4788	182	13	≥	≥	NOUN
ejpam-4788	182	14	ξp(u	ξp(u	NOUN
ejpam-4788	182	15	)	)	PUNCT
ejpam-4788	182	16	and	and	CCONJ
ejpam-4788	182	17	(	(	PUNCT
ejpam-4788	182	18	ξn	ξn	PROPN
ejpam-4788	182	19	◦	◦	NOUN
ejpam-4788	182	20	α	α	NOUN
ejpam-4788	182	21	λn	λn	NOUN
ejpam-4788	182	22	s	s	PART
ejpam-4788	182	23	◦	◦	NOUN
ejpam-4788	182	24	β	β	X
ejpam-4788	182	25	ξn)(u	ξn)(u	ADJ
ejpam-4788	182	26	)	)	PUNCT
ejpam-4788	182	27	≤	≤	NOUN
ejpam-4788	182	28	ξn(u	ξn(u	NOUN
ejpam-4788	182	29	)	)	PUNCT
ejpam-4788	182	30	if	if	SCONJ
ejpam-4788	182	31	u	u	PROPN
ejpam-4788	182	32	/∈	/∈	VERB
ejpam-4788	182	33	kαsβk	kαsβk	PROPN
ejpam-4788	182	34	,	,	PUNCT
ejpam-4788	182	35	then	then	ADV
ejpam-4788	182	36	λp	λp	X
ejpam-4788	182	37	k(u	k(u	X
ejpam-4788	182	38	)	)	PUNCT
ejpam-4788	183	1	=	=	SYM
ejpam-4788	183	2	(	(	PUNCT
ejpam-4788	183	3	ξp	ξp	PART
ejpam-4788	183	4	◦	◦	VERB
ejpam-4788	183	5	α	α	NOUN
ejpam-4788	183	6	λp	λp	X
ejpam-4788	183	7	s	s	PROPN
ejpam-4788	183	8	◦	◦	NOUN
ejpam-4788	183	9	β	β	NOUN
ejpam-4788	183	10	ξp)(u	ξp)(u	ADJ
ejpam-4788	183	11	)	)	PUNCT
ejpam-4788	183	12	=	=	SYM
ejpam-4788	183	13	0	0	PUNCT
ejpam-4788	183	14	and	and	CCONJ
ejpam-4788	183	15	λn	λn	ADP
ejpam-4788	183	16	k(u	k(u	NOUN
ejpam-4788	183	17	)	)	PUNCT
ejpam-4788	183	18	=	=	SYM
ejpam-4788	184	1	(	(	PUNCT
ejpam-4788	184	2	ξn	ξn	PROPN
ejpam-4788	184	3	◦	◦	VERB
ejpam-4788	184	4	α	α	NOUN
ejpam-4788	184	5	λn	λn	NOUN
ejpam-4788	184	6	s	s	PART
ejpam-4788	184	7	◦	◦	NOUN
ejpam-4788	184	8	β	β	X
ejpam-4788	184	9	ξn)(u	ξn)(u	ADJ
ejpam-4788	184	10	)	)	PUNCT
ejpam-4788	184	11	=	=	SYM
ejpam-4788	185	1	0	0	X
ejpam-4788	185	2	.	.	PUNCT
ejpam-4788	186	1	hence	hence	ADV
ejpam-4788	186	2	,	,	PUNCT
ejpam-4788	186	3	(	(	PUNCT
ejpam-4788	186	4	ξp	ξp	PART
ejpam-4788	186	5	◦	◦	VERB
ejpam-4788	186	6	α	α	NOUN
ejpam-4788	186	7	λp	λp	X
ejpam-4788	186	8	s	s	PROPN
ejpam-4788	186	9	◦	◦	NOUN
ejpam-4788	186	10	β	β	NOUN
ejpam-4788	186	11	ξp)(u	ξp)(u	ADJ
ejpam-4788	186	12	)	)	PUNCT
ejpam-4788	186	13	≥	≥	NOUN
ejpam-4788	186	14	ξp(u	ξp(u	NOUN
ejpam-4788	186	15	)	)	PUNCT
ejpam-4788	186	16	and	and	CCONJ
ejpam-4788	186	17	(	(	PUNCT
ejpam-4788	186	18	ξn	ξn	PROPN
ejpam-4788	186	19	◦	◦	NOUN
ejpam-4788	186	20	α	α	NOUN
ejpam-4788	186	21	λn	λn	NOUN
ejpam-4788	186	22	s	s	PART
ejpam-4788	186	23	◦	◦	NOUN
ejpam-4788	186	24	β	β	X
ejpam-4788	186	25	ξn)(u	ξn)(u	ADJ
ejpam-4788	186	26	)	)	PUNCT
ejpam-4788	186	27	≤	≤	NOUN
ejpam-4788	186	28	ξn(u	ξn(u	NOUN
ejpam-4788	186	29	)	)	PUNCT
ejpam-4788	187	1	p.	p.	NOUN
ejpam-4788	187	2	khamrot	khamrot	PROPN
ejpam-4788	187	3	,	,	PUNCT
ejpam-4788	187	4	t.	t.	PROPN
ejpam-4788	187	5	gaketem	gaketem	PROPN
ejpam-4788	187	6	/	/	SYM
ejpam-4788	187	7	eur	eur	PROPN
ejpam-4788	187	8	.	.	PUNCT
ejpam-4788	188	1	j.	j.	PROPN
ejpam-4788	188	2	pure	pure	PROPN
ejpam-4788	188	3	appl	appl	PROPN
ejpam-4788	188	4	.	.	PROPN
ejpam-4788	188	5	math	math	PROPN
ejpam-4788	188	6	,	,	PUNCT
ejpam-4788	188	7	16	16	NUM
ejpam-4788	188	8	(	(	PUNCT
ejpam-4788	188	9	3	3	NUM
ejpam-4788	188	10	)	)	PUNCT
ejpam-4788	188	11	(	(	PUNCT
ejpam-4788	188	12	2023	2023	NUM
ejpam-4788	188	13	)	)	PUNCT
ejpam-4788	188	14	,	,	PUNCT
ejpam-4788	188	15	1592	1592	NUM
ejpam-4788	188	16	-	-	SYM
ejpam-4788	188	17	1607	1607	NUM
ejpam-4788	188	18	1599	1599	NUM
ejpam-4788	188	19	therefore	therefore	ADV
ejpam-4788	188	20	,	,	PUNCT
ejpam-4788	188	21	λk	λk	X
ejpam-4788	188	22	=	=	SYM
ejpam-4788	188	23	(	(	PUNCT
ejpam-4788	188	24	s;λp	s;λp	PROPN
ejpam-4788	188	25	k	k	PROPN
ejpam-4788	188	26	,	,	PUNCT
ejpam-4788	188	27	λn	λn	PROPN
ejpam-4788	188	28	k	k	X
ejpam-4788	188	29	)	)	PUNCT
ejpam-4788	188	30	is	be	AUX
ejpam-4788	188	31	a	a	DET
ejpam-4788	188	32	bf	bf	NOUN
ejpam-4788	188	33	(	(	PUNCT
ejpam-4788	188	34	α	α	NOUN
ejpam-4788	188	35	,	,	PUNCT
ejpam-4788	188	36	β)-bi	β)-bi	NOUN
ejpam-4788	188	37	-	-	PUNCT
ejpam-4788	188	38	ideal	ideal	NOUN
ejpam-4788	188	39	of	of	ADP
ejpam-4788	188	40	s.	s.	PROPN
ejpam-4788	188	41	conversely	conversely	ADV
ejpam-4788	188	42	,	,	PUNCT
ejpam-4788	188	43	assume	assume	VERB
ejpam-4788	188	44	that	that	SCONJ
ejpam-4788	188	45	λk	λk	ADV
ejpam-4788	188	46	=	=	SYM
ejpam-4788	188	47	(	(	PUNCT
ejpam-4788	188	48	s;λp	s;λp	PROPN
ejpam-4788	188	49	k	k	PROPN
ejpam-4788	188	50	,	,	PUNCT
ejpam-4788	188	51	λn	λn	PROPN
ejpam-4788	188	52	k	k	X
ejpam-4788	188	53	)	)	PUNCT
ejpam-4788	188	54	is	be	AUX
ejpam-4788	188	55	a	a	DET
ejpam-4788	188	56	bf	bf	NOUN
ejpam-4788	188	57	(	(	PUNCT
ejpam-4788	188	58	α	α	NOUN
ejpam-4788	188	59	,	,	PUNCT
ejpam-4788	188	60	β)-bi	β)-bi	NOUN
ejpam-4788	188	61	-	-	PUNCT
ejpam-4788	188	62	ideal	ideal	NOUN
ejpam-4788	188	63	of	of	ADP
ejpam-4788	188	64	s	s	PRON
ejpam-4788	188	65	and	and	CCONJ
ejpam-4788	188	66	u	u	PROPN
ejpam-4788	188	67	∈	∈	PROPN
ejpam-4788	188	68	kαsβk	kαsβk	NOUN
ejpam-4788	188	69	.	.	PUNCT
ejpam-4788	189	1	then	then	ADV
ejpam-4788	189	2	(	(	PUNCT
ejpam-4788	189	3	ξp	ξp	AUX
ejpam-4788	189	4	◦	◦	VERB
ejpam-4788	189	5	α	α	NOUN
ejpam-4788	189	6	λp	λp	X
ejpam-4788	189	7	s	s	PROPN
ejpam-4788	189	8	◦	◦	NOUN
ejpam-4788	189	9	β	β	NOUN
ejpam-4788	189	10	ξp)(u	ξp)(u	ADJ
ejpam-4788	189	11	)	)	PUNCT
ejpam-4788	189	12	=	=	SYM
ejpam-4788	189	13	1	1	NUM
ejpam-4788	189	14	and	and	CCONJ
ejpam-4788	189	15	(	(	PUNCT
ejpam-4788	189	16	ξn	ξn	PROPN
ejpam-4788	189	17	◦	◦	NOUN
ejpam-4788	189	18	α	α	NOUN
ejpam-4788	189	19	λn	λn	NOUN
ejpam-4788	189	20	s	s	PART
ejpam-4788	189	21	◦	◦	NOUN
ejpam-4788	189	22	β	β	X
ejpam-4788	189	23	ξn)(u	ξn)(u	ADJ
ejpam-4788	189	24	)	)	PUNCT
ejpam-4788	189	25	=	=	SYM
ejpam-4788	189	26	−1	−1	NOUN
ejpam-4788	189	27	.	.	PUNCT
ejpam-4788	190	1	by	by	ADP
ejpam-4788	190	2	assumption	assumption	NOUN
ejpam-4788	190	3	,	,	PUNCT
ejpam-4788	190	4	(	(	PUNCT
ejpam-4788	190	5	ξp	ξp	PART
ejpam-4788	190	6	◦	◦	VERB
ejpam-4788	190	7	α	α	NOUN
ejpam-4788	190	8	λp	λp	X
ejpam-4788	190	9	s	s	PROPN
ejpam-4788	190	10	◦	◦	NOUN
ejpam-4788	190	11	β	β	NOUN
ejpam-4788	190	12	ξp)(u	ξp)(u	ADJ
ejpam-4788	190	13	)	)	PUNCT
ejpam-4788	190	14	≥	≥	NOUN
ejpam-4788	190	15	ξp(u	ξp(u	NOUN
ejpam-4788	190	16	)	)	PUNCT
ejpam-4788	190	17	and	and	CCONJ
ejpam-4788	190	18	(	(	PUNCT
ejpam-4788	190	19	ξn	ξn	PROPN
ejpam-4788	190	20	◦	◦	NOUN
ejpam-4788	190	21	α	α	NOUN
ejpam-4788	190	22	λn	λn	NOUN
ejpam-4788	190	23	s	s	PART
ejpam-4788	190	24	◦	◦	NOUN
ejpam-4788	190	25	β	β	X
ejpam-4788	190	26	ξn)(u	ξn)(u	ADJ
ejpam-4788	190	27	)	)	PUNCT
ejpam-4788	190	28	≤	≤	NOUN
ejpam-4788	190	29	ξn(u	ξn(u	NOUN
ejpam-4788	190	30	)	)	PUNCT
ejpam-4788	190	31	.	.	PUNCT
ejpam-4788	191	1	thus	thus	ADV
ejpam-4788	191	2	,	,	PUNCT
ejpam-4788	191	3	u	u	PROPN
ejpam-4788	191	4	∈	∈	PROPN
ejpam-4788	191	5	k.	k.	NOUN
ejpam-4788	191	6	hence	hence	ADV
ejpam-4788	191	7	,	,	PUNCT
ejpam-4788	191	8	k	k	PROPN
ejpam-4788	191	9	is	be	AUX
ejpam-4788	191	10	an	an	DET
ejpam-4788	191	11	(	(	PUNCT
ejpam-4788	191	12	α	α	NOUN
ejpam-4788	191	13	,	,	PUNCT
ejpam-4788	191	14	β)-bi	β)-bi	NOUN
ejpam-4788	191	15	-	-	PUNCT
ejpam-4788	191	16	ideal	ideal	NOUN
ejpam-4788	191	17	of	of	ADP
ejpam-4788	191	18	s.	s.	PROPN
ejpam-4788	191	19	theorem	theorem	VERB
ejpam-4788	191	20	6	6	NUM
ejpam-4788	191	21	.	.	PUNCT
ejpam-4788	192	1	the	the	DET
ejpam-4788	192	2	positive	positive	ADJ
ejpam-4788	192	3	and	and	CCONJ
ejpam-4788	192	4	negative	negative	ADJ
ejpam-4788	192	5	of	of	ADP
ejpam-4788	192	6	intersection	intersection	NOUN
ejpam-4788	192	7	of	of	ADP
ejpam-4788	192	8	any	any	DET
ejpam-4788	192	9	two	two	NUM
ejpam-4788	192	10	bf	bf	NOUN
ejpam-4788	192	11	(	(	PUNCT
ejpam-4788	192	12	α	α	NOUN
ejpam-4788	192	13	,	,	PUNCT
ejpam-4788	192	14	β)-bi	β)-bi	NOUN
ejpam-4788	192	15	-	-	PUNCT
ejpam-4788	192	16	ideals	ideal	NOUN
ejpam-4788	192	17	of	of	ADP
ejpam-4788	192	18	a	a	DET
ejpam-4788	192	19	γ	γ	NOUN
ejpam-4788	192	20	-	-	PUNCT
ejpam-4788	192	21	semigroup	semigroup	NOUN
ejpam-4788	192	22	s	s	VERB
ejpam-4788	192	23	are	be	AUX
ejpam-4788	192	24	a	a	DET
ejpam-4788	192	25	bf	bf	NOUN
ejpam-4788	192	26	(	(	PUNCT
ejpam-4788	192	27	α	α	NOUN
ejpam-4788	192	28	,	,	PUNCT
ejpam-4788	192	29	β)-bi	β)-bi	NOUN
ejpam-4788	192	30	-	-	PUNCT
ejpam-4788	192	31	ideal	ideal	NOUN
ejpam-4788	192	32	of	of	ADP
ejpam-4788	192	33	s.	s.	PROPN
ejpam-4788	192	34	proof	proof	PROPN
ejpam-4788	192	35	.	.	PUNCT
ejpam-4788	193	1	let	let	VERB
ejpam-4788	193	2	ξ	ξ	X
ejpam-4788	193	3	=	=	SYM
ejpam-4788	193	4	(	(	PUNCT
ejpam-4788	193	5	s	s	PROPN
ejpam-4788	193	6	;	;	PUNCT
ejpam-4788	193	7	ξp	ξp	NUM
ejpam-4788	193	8	,	,	PUNCT
ejpam-4788	193	9	ξn	ξn	NOUN
ejpam-4788	193	10	)	)	PUNCT
ejpam-4788	193	11	and	and	CCONJ
ejpam-4788	193	12	ς	ς	PROPN
ejpam-4788	193	13	=	=	PUNCT
ejpam-4788	193	14	(	(	PUNCT
ejpam-4788	193	15	s	s	NOUN
ejpam-4788	193	16	;	;	PUNCT
ejpam-4788	193	17	ςp	ςp	NUM
ejpam-4788	193	18	,	,	PUNCT
ejpam-4788	193	19	ςn	ςn	NOUN
ejpam-4788	193	20	)	)	PUNCT
ejpam-4788	193	21	be	be	VERB
ejpam-4788	193	22	bf	bf	NOUN
ejpam-4788	193	23	(	(	PUNCT
ejpam-4788	193	24	α	α	NOUN
ejpam-4788	193	25	,	,	PUNCT
ejpam-4788	193	26	β)-bi	β)-bi	NOUN
ejpam-4788	193	27	-	-	PUNCT
ejpam-4788	193	28	ideals	ideal	NOUN
ejpam-4788	193	29	of	of	ADP
ejpam-4788	193	30	s	s	NOUN
ejpam-4788	193	31	and	and	CCONJ
ejpam-4788	193	32	u	u	PROPN
ejpam-4788	193	33	∈	∈	PROPN
ejpam-4788	193	34	s.	s.	PROPN
ejpam-4788	193	35	then	then	ADV
ejpam-4788	193	36	(	(	PUNCT
ejpam-4788	193	37	(	(	PUNCT
ejpam-4788	193	38	ξp	ξp	ADP
ejpam-4788	193	39	∩	∩	ADJ
ejpam-4788	193	40	ςp	ςp	NUM
ejpam-4788	193	41	)	)	PUNCT
ejpam-4788	193	42	◦	◦	NOUN
ejpam-4788	193	43	α	α	NOUN
ejpam-4788	193	44	λp	λp	X
ejpam-4788	193	45	s	s	PROPN
ejpam-4788	193	46	◦	◦	NOUN
ejpam-4788	193	47	β	β	X
ejpam-4788	193	48	(	(	PUNCT
ejpam-4788	193	49	ξp	ξp	ADP
ejpam-4788	193	50	∩	∩	NOUN
ejpam-4788	193	51	ςp))(u	ςp))(u	NUM
ejpam-4788	193	52	)	)	PUNCT
ejpam-4788	193	53	≥	≥	NOUN
ejpam-4788	193	54	(	(	PUNCT
ejpam-4788	193	55	ξp	ξp	PART
ejpam-4788	193	56	◦	◦	VERB
ejpam-4788	193	57	α	α	NOUN
ejpam-4788	193	58	λp	λp	X
ejpam-4788	193	59	s	s	PROPN
ejpam-4788	193	60	◦	◦	NOUN
ejpam-4788	193	61	β	β	NOUN
ejpam-4788	193	62	ξp)(u	ξp)(u	ADJ
ejpam-4788	193	63	)	)	PUNCT
ejpam-4788	193	64	∧	∧	NOUN
ejpam-4788	193	65	(	(	PUNCT
ejpam-4788	193	66	ςp	ςp	ADP
ejpam-4788	193	67	◦	◦	NOUN
ejpam-4788	193	68	α	α	NOUN
ejpam-4788	193	69	λp	λp	X
ejpam-4788	193	70	s	s	PROPN
ejpam-4788	193	71	◦	◦	NOUN
ejpam-4788	193	72	β	β	NOUN
ejpam-4788	193	73	ςp)(u	ςp)(u	NOUN
ejpam-4788	193	74	)	)	PUNCT
ejpam-4788	193	75	≥	≥	NOUN
ejpam-4788	193	76	(	(	PUNCT
ejpam-4788	193	77	ξp	ξp	ADP
ejpam-4788	193	78	∩	∩	NOUN
ejpam-4788	193	79	ςp)(u	ςp)(u	NUM
ejpam-4788	193	80	)	)	PUNCT
ejpam-4788	193	81	and	and	CCONJ
ejpam-4788	193	82	(	(	PUNCT
ejpam-4788	193	83	(	(	PUNCT
ejpam-4788	193	84	ξn	ξn	PROPN
ejpam-4788	193	85	∩	∩	NOUN
ejpam-4788	193	86	ςn	ςn	NOUN
ejpam-4788	193	87	)	)	PUNCT
ejpam-4788	193	88	◦	◦	NOUN
ejpam-4788	193	89	α	α	NOUN
ejpam-4788	193	90	λn	λn	NOUN
ejpam-4788	193	91	s	s	PART
ejpam-4788	193	92	◦	◦	NOUN
ejpam-4788	193	93	β	β	X
ejpam-4788	193	94	(	(	PUNCT
ejpam-4788	193	95	ξn	ξn	PROPN
ejpam-4788	193	96	∩	∩	NOUN
ejpam-4788	193	97	ςn))(u	ςn))(u	NUM
ejpam-4788	193	98	)	)	PUNCT
ejpam-4788	193	99	≤	≤	NOUN
ejpam-4788	193	100	(	(	PUNCT
ejpam-4788	193	101	ξn	ξn	PROPN
ejpam-4788	193	102	◦	◦	VERB
ejpam-4788	193	103	α	α	NOUN
ejpam-4788	193	104	λn	λn	NOUN
ejpam-4788	193	105	s	s	PART
ejpam-4788	193	106	◦	◦	NOUN
ejpam-4788	193	107	β	β	X
ejpam-4788	193	108	ξn)(u)∨	ξn)(u)∨	NOUN
ejpam-4788	193	109	(	(	PUNCT
ejpam-4788	193	110	ςn	ςn	NOUN
ejpam-4788	193	111	◦	◦	NOUN
ejpam-4788	193	112	α	α	NOUN
ejpam-4788	193	113	λn	λn	NOUN
ejpam-4788	193	114	s	s	PART
ejpam-4788	193	115	◦	◦	NOUN
ejpam-4788	193	116	β	β	X
ejpam-4788	193	117	ςn)(u	ςn)(u	ADJ
ejpam-4788	193	118	)	)	PUNCT
ejpam-4788	193	119	≤	≤	NOUN
ejpam-4788	193	120	(	(	PUNCT
ejpam-4788	193	121	ξn	ξn	PROPN
ejpam-4788	193	122	∩	∩	NOUN
ejpam-4788	193	123	ςn)(u	ςn)(u	ADJ
ejpam-4788	193	124	)	)	PUNCT
ejpam-4788	193	125	.	.	PUNCT
ejpam-4788	194	1	thus	thus	ADV
ejpam-4788	194	2	,	,	PUNCT
ejpam-4788	194	3	ξ	ξ	PROPN
ejpam-4788	194	4	∩	∩	NOUN
ejpam-4788	194	5	ς	ς	PROPN
ejpam-4788	194	6	is	be	AUX
ejpam-4788	194	7	a	a	DET
ejpam-4788	194	8	bf	bf	NOUN
ejpam-4788	194	9	(	(	PUNCT
ejpam-4788	194	10	α	α	NOUN
ejpam-4788	194	11	,	,	PUNCT
ejpam-4788	194	12	β)-bi	β)-bi	NOUN
ejpam-4788	194	13	-	-	PUNCT
ejpam-4788	194	14	ideal	ideal	NOUN
ejpam-4788	194	15	of	of	ADP
ejpam-4788	194	16	s.	s.	PROPN
ejpam-4788	194	17	theorem	theorem	VERB
ejpam-4788	194	18	7	7	NUM
ejpam-4788	194	19	.	.	PUNCT
ejpam-4788	195	1	let	let	VERB
ejpam-4788	195	2	ξ	ξ	X
ejpam-4788	195	3	=	=	SYM
ejpam-4788	195	4	(	(	PUNCT
ejpam-4788	195	5	s	s	PROPN
ejpam-4788	195	6	;	;	PUNCT
ejpam-4788	195	7	ξp	ξp	NUM
ejpam-4788	195	8	,	,	PUNCT
ejpam-4788	195	9	ξn	ξn	NOUN
ejpam-4788	195	10	)	)	PUNCT
ejpam-4788	195	11	be	be	VERB
ejpam-4788	195	12	a	a	DET
ejpam-4788	195	13	bf	bf	NOUN
ejpam-4788	195	14	set	set	NOUN
ejpam-4788	195	15	of	of	ADP
ejpam-4788	195	16	a	a	DET
ejpam-4788	195	17	γ	γ	NOUN
ejpam-4788	195	18	-	-	PUNCT
ejpam-4788	195	19	semigroup	semigroup	ADJ
ejpam-4788	195	20	s	s	NOUN
ejpam-4788	195	21	and	and	CCONJ
ejpam-4788	195	22	ξ(l	ξ(l	PROPN
ejpam-4788	195	23	,	,	PUNCT
ejpam-4788	195	24	m	m	NOUN
ejpam-4788	195	25	)	)	PUNCT
ejpam-4788	196	1	=	=	SYM
ejpam-4788	196	2	(	(	PUNCT
ejpam-4788	196	3	s	s	X
ejpam-4788	196	4	;	;	PUNCT
ejpam-4788	196	5	ξpl	ξpl	X
ejpam-4788	196	6	,	,	PUNCT
ejpam-4788	196	7	ξ	ξ	PROPN
ejpam-4788	196	8	n	n	PRON
ejpam-4788	196	9	m	m	PRON
ejpam-4788	196	10	)	)	PUNCT
ejpam-4788	196	11	be	be	VERB
ejpam-4788	196	12	bf	bf	NOUN
ejpam-4788	196	13	point	point	NOUN
ejpam-4788	196	14	with	with	ADP
ejpam-4788	196	15	ξpl	ξpl	NOUN
ejpam-4788	196	16	=	=	SYM
ejpam-4788	196	17	{	{	PUNCT
ejpam-4788	196	18	x	x	PUNCT
ejpam-4788	196	19	∈	∈	PROPN
ejpam-4788	196	20	s	s	PART
ejpam-4788	196	21	|	|	ADV
ejpam-4788	196	22	ξpl	ξpl	NOUN
ejpam-4788	196	23	(	(	PUNCT
ejpam-4788	196	24	x	x	NOUN
ejpam-4788	196	25	)	)	PUNCT
ejpam-4788	196	26	≥	≥	NOUN
ejpam-4788	196	27	l	l	NOUN
ejpam-4788	196	28	}	}	PUNCT
ejpam-4788	196	29	and	and	CCONJ
ejpam-4788	196	30	ξnm	ξnm	PROPN
ejpam-4788	196	31	=	=	SYM
ejpam-4788	196	32	{	{	PUNCT
ejpam-4788	196	33	x	x	PUNCT
ejpam-4788	196	34	∈	∈	PROPN
ejpam-4788	196	35	s	s	VERB
ejpam-4788	196	36	|	|	ADV
ejpam-4788	196	37	ξnm(x	ξnm(x	NOUN
ejpam-4788	196	38	)	)	PUNCT
ejpam-4788	196	39	≤	≤	NUM
ejpam-4788	196	40	m	m	ADP
ejpam-4788	196	41	}	}	PUNCT
ejpam-4788	196	42	.	.	PUNCT
ejpam-4788	197	1	then	then	ADV
ejpam-4788	197	2	ξ	ξ	X
ejpam-4788	197	3	=	=	SYM
ejpam-4788	197	4	(	(	PUNCT
ejpam-4788	197	5	s	s	PROPN
ejpam-4788	197	6	;	;	PUNCT
ejpam-4788	197	7	ξp	ξp	NUM
ejpam-4788	197	8	,	,	PUNCT
ejpam-4788	197	9	ξn	ξn	NOUN
ejpam-4788	197	10	)	)	PUNCT
ejpam-4788	197	11	is	be	AUX
ejpam-4788	197	12	a	a	DET
ejpam-4788	197	13	bf	bf	NOUN
ejpam-4788	197	14	(	(	PUNCT
ejpam-4788	197	15	α	α	NOUN
ejpam-4788	197	16	,	,	PUNCT
ejpam-4788	197	17	β)-bi	β)-bi	NOUN
ejpam-4788	197	18	-	-	PUNCT
ejpam-4788	197	19	ideal	ideal	NOUN
ejpam-4788	197	20	of	of	ADP
ejpam-4788	197	21	s	s	PRON
ejpam-4788	197	22	if	if	SCONJ
ejpam-4788	198	1	and	and	CCONJ
ejpam-4788	198	2	only	only	ADV
ejpam-4788	198	3	if	if	SCONJ
ejpam-4788	198	4	ξ(l	ξ(l	NOUN
ejpam-4788	198	5	,	,	PUNCT
ejpam-4788	198	6	m	m	NOUN
ejpam-4788	198	7	)	)	PUNCT
ejpam-4788	198	8	=	=	SYM
ejpam-4788	198	9	(	(	PUNCT
ejpam-4788	198	10	s	s	X
ejpam-4788	198	11	;	;	PUNCT
ejpam-4788	198	12	ξpl	ξpl	X
ejpam-4788	198	13	,	,	PUNCT
ejpam-4788	198	14	ξ	ξ	PROPN
ejpam-4788	198	15	n	n	PRON
ejpam-4788	198	16	m	m	PRON
ejpam-4788	198	17	)	)	PUNCT
ejpam-4788	198	18	is	be	AUX
ejpam-4788	198	19	a	a	DET
ejpam-4788	198	20	non	non	ADJ
ejpam-4788	198	21	-	-	ADJ
ejpam-4788	198	22	empty	empty	ADJ
ejpam-4788	198	23	set	set	NOUN
ejpam-4788	198	24	and	and	CCONJ
ejpam-4788	198	25	ξ(l	ξ(l	NOUN
ejpam-4788	198	26	,	,	PUNCT
ejpam-4788	198	27	m	m	PRON
ejpam-4788	198	28	)	)	PUNCT
ejpam-4788	198	29	is	be	AUX
ejpam-4788	198	30	an	an	DET
ejpam-4788	198	31	(	(	PUNCT
ejpam-4788	198	32	α	α	NOUN
ejpam-4788	198	33	,	,	PUNCT
ejpam-4788	198	34	β)-bi	β)-bi	NOUN
ejpam-4788	198	35	-	-	PUNCT
ejpam-4788	198	36	ideal	ideal	NOUN
ejpam-4788	198	37	of	of	ADP
ejpam-4788	198	38	s	s	PRON
ejpam-4788	198	39	for	for	ADP
ejpam-4788	198	40	all	all	DET
ejpam-4788	198	41	(	(	PUNCT
ejpam-4788	198	42	l	l	NOUN
ejpam-4788	198	43	,	,	PUNCT
ejpam-4788	198	44	m	m	NOUN
ejpam-4788	198	45	)	)	PUNCT
ejpam-4788	198	46	∈	∈	PROPN
ejpam-4788	198	47	(	(	PUNCT
ejpam-4788	198	48	0	0	NUM
ejpam-4788	198	49	,	,	PUNCT
ejpam-4788	198	50	1]×	1]×	NUM
ejpam-4788	198	51	[	[	X
ejpam-4788	198	52	−1	−1	NOUN
ejpam-4788	198	53	,	,	PUNCT
ejpam-4788	198	54	0	0	NUM
ejpam-4788	198	55	)	)	PUNCT
ejpam-4788	198	56	.	.	PUNCT
ejpam-4788	199	1	proof	proof	NOUN
ejpam-4788	199	2	.	.	PUNCT
ejpam-4788	200	1	suppose	suppose	VERB
ejpam-4788	200	2	that	that	SCONJ
ejpam-4788	200	3	ξ	ξ	PROPN
ejpam-4788	200	4	=	=	SYM
ejpam-4788	200	5	(	(	PUNCT
ejpam-4788	200	6	s	s	PROPN
ejpam-4788	200	7	;	;	PUNCT
ejpam-4788	200	8	ξp	ξp	NUM
ejpam-4788	200	9	,	,	PUNCT
ejpam-4788	200	10	ξn	ξn	NOUN
ejpam-4788	200	11	)	)	PUNCT
ejpam-4788	200	12	is	be	AUX
ejpam-4788	200	13	a	a	DET
ejpam-4788	200	14	bf	bf	NOUN
ejpam-4788	200	15	(	(	PUNCT
ejpam-4788	200	16	α	α	NOUN
ejpam-4788	200	17	,	,	PUNCT
ejpam-4788	200	18	β)-bi	β)-bi	NOUN
ejpam-4788	200	19	-	-	PUNCT
ejpam-4788	200	20	ideal	ideal	NOUN
ejpam-4788	200	21	of	of	ADP
ejpam-4788	200	22	s.	s.	PROPN
ejpam-4788	200	23	then	then	ADV
ejpam-4788	200	24	ξp(uαvβw	ξp(uαvβw	PROPN
ejpam-4788	200	25	)	)	PUNCT
ejpam-4788	200	26	≥	≥	NOUN
ejpam-4788	200	27	ξp(u)∧ξp(w	ξp(u)∧ξp(w	PROPN
ejpam-4788	200	28	)	)	PUNCT
ejpam-4788	200	29	and	and	CCONJ
ejpam-4788	200	30	ξn(uαvβw	ξn(uαvβw	NOUN
ejpam-4788	200	31	)	)	PUNCT
ejpam-4788	200	32	≤	≤	NOUN
ejpam-4788	200	33	ξn(u)∨ξn(w	ξn(u)∨ξn(w	NUM
ejpam-4788	200	34	)	)	PUNCT
ejpam-4788	200	35	for	for	ADP
ejpam-4788	200	36	all	all	DET
ejpam-4788	200	37	u	u	NOUN
ejpam-4788	200	38	,	,	PUNCT
ejpam-4788	200	39	v	v	NOUN
ejpam-4788	200	40	,	,	PUNCT
ejpam-4788	200	41	w	w	PROPN
ejpam-4788	200	42	∈	∈	PROPN
ejpam-4788	200	43	s.	s.	PROPN
ejpam-4788	200	44	let	let	VERB
ejpam-4788	200	45	(	(	PUNCT
ejpam-4788	200	46	l	l	NOUN
ejpam-4788	200	47	,	,	PUNCT
ejpam-4788	200	48	m	m	NOUN
ejpam-4788	200	49	)	)	PUNCT
ejpam-4788	200	50	∈	∈	PROPN
ejpam-4788	200	51	(	(	PUNCT
ejpam-4788	200	52	0	0	NUM
ejpam-4788	200	53	,	,	PUNCT
ejpam-4788	200	54	1]×[−1	1]×[−1	NUM
ejpam-4788	200	55	,	,	PUNCT
ejpam-4788	200	56	0	0	NUM
ejpam-4788	200	57	)	)	PUNCT
ejpam-4788	200	58	be	be	AUX
ejpam-4788	200	59	such	such	ADJ
ejpam-4788	200	60	that	that	SCONJ
ejpam-4788	200	61	ξ(l	ξ(l	NOUN
ejpam-4788	200	62	,	,	PUNCT
ejpam-4788	200	63	m	m	NOUN
ejpam-4788	200	64	)	)	PUNCT
ejpam-4788	200	65	̸=	̸=	PROPN
ejpam-4788	200	66	∅.	∅.	ADV
ejpam-4788	200	67	let	let	VERB
ejpam-4788	200	68	u	u	NOUN
ejpam-4788	200	69	,	,	PUNCT
ejpam-4788	200	70	w	w	PROPN
ejpam-4788	200	71	∈	∈	PROPN
ejpam-4788	200	72	ξ(l	ξ(l	PROPN
ejpam-4788	200	73	,	,	PUNCT
ejpam-4788	200	74	m	m	NOUN
ejpam-4788	200	75	)	)	PUNCT
ejpam-4788	200	76	and	and	CCONJ
ejpam-4788	201	1	v	v	X
ejpam-4788	201	2	∈	∈	PROPN
ejpam-4788	201	3	s.	s.	PROPN
ejpam-4788	201	4	then	then	ADV
ejpam-4788	201	5	ξp(u	ξp(u	NOUN
ejpam-4788	201	6	)	)	PUNCT
ejpam-4788	201	7	≥	≥	PROPN
ejpam-4788	201	8	l	l	NOUN
ejpam-4788	201	9	,	,	PUNCT
ejpam-4788	201	10	ξp(w	ξp(w	NUM
ejpam-4788	201	11	)	)	PUNCT
ejpam-4788	201	12	and	and	CCONJ
ejpam-4788	201	13	ξn(u	ξn(u	NOUN
ejpam-4788	201	14	)	)	PUNCT
ejpam-4788	201	15	≤	≤	NUM
ejpam-4788	201	16	m	m	PROPN
ejpam-4788	201	17	,	,	PUNCT
ejpam-4788	201	18	ξn(w	ξn(w	NOUN
ejpam-4788	201	19	)	)	PUNCT
ejpam-4788	201	20	≤	≤	NUM
ejpam-4788	201	21	m.	m.	NOUN
ejpam-4788	201	22	thus	thus	ADV
ejpam-4788	201	23	,	,	PUNCT
ejpam-4788	201	24	ξp(uαvβw	ξp(uαvβw	NOUN
ejpam-4788	201	25	)	)	PUNCT
ejpam-4788	201	26	≥	≥	NOUN
ejpam-4788	201	27	ξp(u	ξp(u	NOUN
ejpam-4788	201	28	)	)	PUNCT
ejpam-4788	201	29	∧	∧	PROPN
ejpam-4788	201	30	ξp(w	ξp(w	NOUN
ejpam-4788	201	31	)	)	PUNCT
ejpam-4788	201	32	≥	≥	NOUN
ejpam-4788	201	33	l	l	NOUN
ejpam-4788	201	34	and	and	CCONJ
ejpam-4788	201	35	ξp(uαvβw	ξp(uαvβw	NOUN
ejpam-4788	201	36	)	)	PUNCT
ejpam-4788	201	37	≤	≤	NOUN
ejpam-4788	201	38	ξn(u	ξn(u	NOUN
ejpam-4788	201	39	)	)	PUNCT
ejpam-4788	201	40	∨	∨	NUM
ejpam-4788	201	41	ξn(w	ξn(w	NOUN
ejpam-4788	201	42	)	)	PUNCT
ejpam-4788	201	43	≤	≤	NUM
ejpam-4788	201	44	m.	m.	NOUN
ejpam-4788	201	45	so	so	ADV
ejpam-4788	201	46	,	,	PUNCT
ejpam-4788	201	47	uαvβw	uαvβw	ADJ
ejpam-4788	201	48	∈	∈	PROPN
ejpam-4788	201	49	ξ(l	ξ(l	PROPN
ejpam-4788	201	50	,	,	PUNCT
ejpam-4788	201	51	m	m	NOUN
ejpam-4788	201	52	)	)	PUNCT
ejpam-4788	201	53	.	.	PUNCT
ejpam-4788	202	1	hence	hence	ADV
ejpam-4788	202	2	,	,	PUNCT
ejpam-4788	202	3	ξ(l	ξ(l	PROPN
ejpam-4788	202	4	,	,	PUNCT
ejpam-4788	202	5	m	m	NOUN
ejpam-4788	202	6	)	)	PUNCT
ejpam-4788	202	7	=	=	SYM
ejpam-4788	202	8	(	(	PUNCT
ejpam-4788	202	9	s	s	X
ejpam-4788	202	10	;	;	PUNCT
ejpam-4788	202	11	ξpl	ξpl	X
ejpam-4788	202	12	,	,	PUNCT
ejpam-4788	202	13	ξ	ξ	PROPN
ejpam-4788	202	14	n	n	PRON
ejpam-4788	202	15	m	m	PRON
ejpam-4788	202	16	)	)	PUNCT
ejpam-4788	202	17	is	be	AUX
ejpam-4788	202	18	an	an	DET
ejpam-4788	202	19	(	(	PUNCT
ejpam-4788	202	20	α	α	NOUN
ejpam-4788	202	21	,	,	PUNCT
ejpam-4788	202	22	β)-bi	β)-bi	NOUN
ejpam-4788	202	23	-	-	PUNCT
ejpam-4788	202	24	ideal	ideal	NOUN
ejpam-4788	202	25	of	of	ADP
ejpam-4788	202	26	s.	s.	PROPN
ejpam-4788	202	27	conversely	conversely	ADV
ejpam-4788	202	28	,	,	PUNCT
ejpam-4788	202	29	assume	assume	VERB
ejpam-4788	202	30	that	that	SCONJ
ejpam-4788	202	31	ξ(l	ξ(l	PROPN
ejpam-4788	202	32	,	,	PUNCT
ejpam-4788	202	33	m	m	NOUN
ejpam-4788	202	34	)	)	PUNCT
ejpam-4788	202	35	=	=	SYM
ejpam-4788	202	36	(	(	PUNCT
ejpam-4788	202	37	s	s	X
ejpam-4788	202	38	;	;	PUNCT
ejpam-4788	202	39	ξpl	ξpl	X
ejpam-4788	202	40	,	,	PUNCT
ejpam-4788	202	41	ξ	ξ	PROPN
ejpam-4788	202	42	n	n	PRON
ejpam-4788	202	43	m	m	PRON
ejpam-4788	202	44	)	)	PUNCT
ejpam-4788	202	45	is	be	AUX
ejpam-4788	202	46	an	an	DET
ejpam-4788	202	47	(	(	PUNCT
ejpam-4788	202	48	α	α	NOUN
ejpam-4788	202	49	,	,	PUNCT
ejpam-4788	202	50	β)-bi	β)-bi	NOUN
ejpam-4788	202	51	-	-	PUNCT
ejpam-4788	202	52	ideal	ideal	NOUN
ejpam-4788	202	53	of	of	ADP
ejpam-4788	202	54	s	s	PRON
ejpam-4788	202	55	if	if	SCONJ
ejpam-4788	202	56	(	(	PUNCT
ejpam-4788	202	57	l	l	NOUN
ejpam-4788	202	58	,	,	PUNCT
ejpam-4788	202	59	m	m	NOUN
ejpam-4788	202	60	)	)	PUNCT
ejpam-4788	202	61	∈	∈	PROPN
ejpam-4788	202	62	(	(	PUNCT
ejpam-4788	202	63	0	0	NUM
ejpam-4788	202	64	,	,	PUNCT
ejpam-4788	202	65	1]×	1]×	NUM
ejpam-4788	203	1	[	[	X
ejpam-4788	203	2	−1	−1	NOUN
ejpam-4788	203	3	,	,	PUNCT
ejpam-4788	203	4	0	0	NUM
ejpam-4788	203	5	)	)	PUNCT
ejpam-4788	203	6	and	and	CCONJ
ejpam-4788	203	7	ξ(l	ξ(l	PROPN
ejpam-4788	203	8	,	,	PUNCT
ejpam-4788	203	9	m	m	NOUN
ejpam-4788	203	10	)	)	PUNCT
ejpam-4788	203	11	̸=	̸=	PROPN
ejpam-4788	203	12	∅.	∅.	ADV
ejpam-4788	203	13	let	let	VERB
ejpam-4788	203	14	u	u	NOUN
ejpam-4788	203	15	,	,	PUNCT
ejpam-4788	203	16	v	v	NOUN
ejpam-4788	203	17	,	,	PUNCT
ejpam-4788	203	18	w	w	PROPN
ejpam-4788	203	19	∈	∈	PROPN
ejpam-4788	203	20	s	s	PART
ejpam-4788	203	21	and	and	CCONJ
ejpam-4788	203	22	l	l	NOUN
ejpam-4788	203	23	=	=	PUNCT
ejpam-4788	203	24	ξp(u	ξp(u	PROPN
ejpam-4788	203	25	)	)	PUNCT
ejpam-4788	203	26	,	,	PUNCT
ejpam-4788	203	27	l	l	NOUN
ejpam-4788	203	28	=	=	PUNCT
ejpam-4788	203	29	ξp(u	ξp(u	PROPN
ejpam-4788	203	30	)	)	PUNCT
ejpam-4788	203	31	,	,	PUNCT
ejpam-4788	203	32	m	m	NOUN
ejpam-4788	203	33	=	=	SYM
ejpam-4788	203	34	ξn(u	ξn(u	NOUN
ejpam-4788	203	35	)	)	PUNCT
ejpam-4788	203	36	,	,	PUNCT
ejpam-4788	203	37	m	m	VERB
ejpam-4788	203	38	=	=	SYM
ejpam-4788	203	39	ξn(w	ξn(w	NUM
ejpam-4788	203	40	)	)	PUNCT
ejpam-4788	203	41	.	.	PUNCT
ejpam-4788	204	1	by	by	ADP
ejpam-4788	204	2	assumption	assumption	NOUN
ejpam-4788	204	3	,	,	PUNCT
ejpam-4788	204	4	ξp(u)∧ξp(w	ξp(u)∧ξp(w	PROPN
ejpam-4788	204	5	)	)	PUNCT
ejpam-4788	204	6	≥	≥	NOUN
ejpam-4788	204	7	l	l	NOUN
ejpam-4788	204	8	and	and	CCONJ
ejpam-4788	204	9	ξn(u)∨ξn(w	ξn(u)∨ξn(w	NUM
ejpam-4788	204	10	)	)	PUNCT
ejpam-4788	204	11	≤	≤	NUM
ejpam-4788	204	12	m.	m.	NOUN
ejpam-4788	204	13	then	then	ADV
ejpam-4788	204	14	u	u	NOUN
ejpam-4788	204	15	,	,	PUNCT
ejpam-4788	204	16	w	w	PROPN
ejpam-4788	204	17	∈	∈	PROPN
ejpam-4788	204	18	ξ(l	ξ(l	PROPN
ejpam-4788	204	19	,	,	PUNCT
ejpam-4788	204	20	m	m	NOUN
ejpam-4788	204	21	)	)	PUNCT
ejpam-4788	204	22	.	.	PUNCT
ejpam-4788	205	1	thus	thus	ADV
ejpam-4788	205	2	,	,	PUNCT
ejpam-4788	205	3	ξl	ξl	X
ejpam-4788	205	4	̸=	̸=	PROPN
ejpam-4788	205	5	∅.	∅.	PRON
ejpam-4788	205	6	hence	hence	ADV
ejpam-4788	205	7	,	,	PUNCT
ejpam-4788	205	8	ξl	ξl	NOUN
ejpam-4788	205	9	is	be	AUX
ejpam-4788	205	10	an	an	DET
ejpam-4788	205	11	(	(	PUNCT
ejpam-4788	205	12	α	α	NOUN
ejpam-4788	205	13	,	,	PUNCT
ejpam-4788	205	14	β)-bi	β)-bi	NOUN
ejpam-4788	205	15	-	-	PUNCT
ejpam-4788	205	16	ideal	ideal	NOUN
ejpam-4788	205	17	of	of	ADP
ejpam-4788	205	18	s.	s.	PROPN
ejpam-4788	205	19	since	since	SCONJ
ejpam-4788	205	20	u	u	PROPN
ejpam-4788	205	21	,	,	PUNCT
ejpam-4788	205	22	w	w	PROPN
ejpam-4788	205	23	∈	∈	PROPN
ejpam-4788	205	24	ξ(l	ξ(l	PROPN
ejpam-4788	205	25	,	,	PUNCT
ejpam-4788	205	26	m	m	NOUN
ejpam-4788	205	27	)	)	PUNCT
ejpam-4788	205	28	and	and	CCONJ
ejpam-4788	205	29	v	v	ADP
ejpam-4788	205	30	∈	∈	NOUN
ejpam-4788	205	31	s	s	NOUN
ejpam-4788	205	32	,	,	PUNCT
ejpam-4788	205	33	we	we	PRON
ejpam-4788	205	34	have	have	VERB
ejpam-4788	205	35	uαvβw	uαvβw	ADJ
ejpam-4788	205	36	∈	∈	PROPN
ejpam-4788	205	37	ξ(l	ξ(l	PROPN
ejpam-4788	205	38	,	,	PUNCT
ejpam-4788	205	39	m	m	NOUN
ejpam-4788	205	40	)	)	PUNCT
ejpam-4788	205	41	.	.	PUNCT
ejpam-4788	206	1	thus	thus	ADV
ejpam-4788	206	2	,	,	PUNCT
ejpam-4788	206	3	ξp(uαvβw	ξp(uαvβw	NOUN
ejpam-4788	206	4	)	)	PUNCT
ejpam-4788	206	5	≥	≥	NOUN
ejpam-4788	206	6	l	l	NOUN
ejpam-4788	206	7	=	=	PUNCT
ejpam-4788	206	8	ξp(u	ξp(u	PROPN
ejpam-4788	206	9	)	)	PUNCT
ejpam-4788	206	10	∧	∧	NOUN
ejpam-4788	206	11	ξ(p)(w	ξ(p)(w	NOUN
ejpam-4788	206	12	)	)	PUNCT
ejpam-4788	206	13	and	and	CCONJ
ejpam-4788	206	14	ξn(uαvβw	ξn(uαvβw	NOUN
ejpam-4788	206	15	)	)	PUNCT
ejpam-4788	206	16	≤	≤	NUM
ejpam-4788	206	17	m	m	NOUN
ejpam-4788	206	18	=	=	SYM
ejpam-4788	206	19	ξn(u	ξn(u	NOUN
ejpam-4788	206	20	)	)	PUNCT
ejpam-4788	206	21	∨	∨	NUM
ejpam-4788	206	22	ξn(w	ξn(w	NUM
ejpam-4788	206	23	)	)	PUNCT
ejpam-4788	206	24	.	.	PUNCT
ejpam-4788	207	1	hence	hence	ADV
ejpam-4788	207	2	,	,	PUNCT
ejpam-4788	207	3	ξ	ξ	X
ejpam-4788	207	4	=	=	SYM
ejpam-4788	207	5	(	(	PUNCT
ejpam-4788	207	6	s	s	PROPN
ejpam-4788	207	7	;	;	PUNCT
ejpam-4788	207	8	ξp	ξp	NUM
ejpam-4788	207	9	,	,	PUNCT
ejpam-4788	207	10	ξn	ξn	NOUN
ejpam-4788	207	11	)	)	PUNCT
ejpam-4788	207	12	is	be	AUX
ejpam-4788	207	13	a	a	DET
ejpam-4788	207	14	bf	bf	NOUN
ejpam-4788	207	15	(	(	PUNCT
ejpam-4788	207	16	α	α	NOUN
ejpam-4788	207	17	,	,	PUNCT
ejpam-4788	207	18	β)-bi	β)-bi	NOUN
ejpam-4788	207	19	-	-	PUNCT
ejpam-4788	207	20	ideal	ideal	NOUN
ejpam-4788	207	21	of	of	ADP
ejpam-4788	207	22	s.	s.	PROPN
ejpam-4788	207	23	next	next	ADV
ejpam-4788	207	24	,	,	PUNCT
ejpam-4788	207	25	we	we	PRON
ejpam-4788	207	26	define	define	VERB
ejpam-4788	207	27	a	a	DET
ejpam-4788	207	28	bf	bf	NOUN
ejpam-4788	207	29	(	(	PUNCT
ejpam-4788	207	30	α	α	NOUN
ejpam-4788	207	31	,	,	PUNCT
ejpam-4788	207	32	β)-quasi	β)-quasi	ADJ
ejpam-4788	207	33	-	-	PUNCT
ejpam-4788	207	34	ideal	ideal	NOUN
ejpam-4788	207	35	and	and	CCONJ
ejpam-4788	207	36	study	study	VERB
ejpam-4788	207	37	its	its	PRON
ejpam-4788	207	38	basic	basic	ADJ
ejpam-4788	207	39	properties	property	NOUN
ejpam-4788	207	40	.	.	PUNCT
ejpam-4788	208	1	definition	definition	NOUN
ejpam-4788	208	2	12	12	NUM
ejpam-4788	208	3	.	.	PUNCT
ejpam-4788	209	1	let	let	VERB
ejpam-4788	209	2	ξ	ξ	X
ejpam-4788	209	3	=	=	SYM
ejpam-4788	209	4	(	(	PUNCT
ejpam-4788	209	5	s	s	PROPN
ejpam-4788	209	6	;	;	PUNCT
ejpam-4788	209	7	ξp	ξp	NUM
ejpam-4788	209	8	,	,	PUNCT
ejpam-4788	209	9	ξn	ξn	NOUN
ejpam-4788	209	10	)	)	PUNCT
ejpam-4788	209	11	be	be	VERB
ejpam-4788	209	12	a	a	DET
ejpam-4788	209	13	bf	bf	NOUN
ejpam-4788	209	14	set	set	NOUN
ejpam-4788	209	15	of	of	ADP
ejpam-4788	209	16	a	a	DET
ejpam-4788	209	17	γ	γ	NOUN
ejpam-4788	209	18	-	-	PUNCT
ejpam-4788	209	19	semigroup	semigroup	NOUN
ejpam-4788	209	20	s	s	X
ejpam-4788	209	21	and	and	CCONJ
ejpam-4788	209	22	α	α	NOUN
ejpam-4788	209	23	,	,	PUNCT
ejpam-4788	209	24	β	β	PROPN
ejpam-4788	209	25	∈	∈	PROPN
ejpam-4788	209	26	γ	γ	X
ejpam-4788	209	27	.	.	PROPN
ejpam-4788	210	1	then	then	ADV
ejpam-4788	210	2	ξ	ξ	X
ejpam-4788	210	3	=	=	SYM
ejpam-4788	210	4	(	(	PUNCT
ejpam-4788	210	5	s	s	PROPN
ejpam-4788	210	6	;	;	PUNCT
ejpam-4788	210	7	ξp	ξp	NUM
ejpam-4788	210	8	,	,	PUNCT
ejpam-4788	210	9	ξn	ξn	NOUN
ejpam-4788	210	10	)	)	PUNCT
ejpam-4788	210	11	is	be	AUX
ejpam-4788	210	12	called	call	VERB
ejpam-4788	210	13	a	a	DET
ejpam-4788	210	14	bf	bf	NOUN
ejpam-4788	210	15	(	(	PUNCT
ejpam-4788	210	16	α	α	NOUN
ejpam-4788	210	17	,	,	PUNCT
ejpam-4788	210	18	β)-quasi	β)-quasi	NOUN
ejpam-4788	210	19	-	-	NOUN
ejpam-4788	210	20	ideal	ideal	NOUN
ejpam-4788	210	21	of	of	ADP
ejpam-4788	210	22	s	s	PRON
ejpam-4788	210	23	if	if	SCONJ
ejpam-4788	210	24	λp	λp	PRON
ejpam-4788	210	25	s	s	VERB
ejpam-4788	210	26	◦	◦	NOUN
ejpam-4788	210	27	α	α	NOUN
ejpam-4788	210	28	ξp	ξp	NOUN
ejpam-4788	210	29	∩	∩	NOUN
ejpam-4788	210	30	ξp	ξp	AUX
ejpam-4788	210	31	◦	◦	VERB
ejpam-4788	210	32	β	β	X
ejpam-4788	210	33	λp	λp	X
ejpam-4788	210	34	s	s	ADP
ejpam-4788	210	35	⊆	⊆	NUM
ejpam-4788	210	36	ξp	ξp	NOUN
ejpam-4788	211	1	and	and	CCONJ
ejpam-4788	211	2	λn	λn	PROPN
ejpam-4788	211	3	s	s	PART
ejpam-4788	211	4	◦	◦	NOUN
ejpam-4788	211	5	α	α	NOUN
ejpam-4788	211	6	ξn	ξn	NOUN
ejpam-4788	211	7	∪	∪	ADP
ejpam-4788	211	8	ξn	ξn	PROPN
ejpam-4788	211	9	◦	◦	NOUN
ejpam-4788	211	10	β	β	X
ejpam-4788	211	11	λn	λn	NOUN
ejpam-4788	211	12	s	s	PROPN
ejpam-4788	211	13	⊇	⊇	PROPN
ejpam-4788	211	14	ξn	ξn	PROPN
ejpam-4788	211	15	.	.	PROPN
ejpam-4788	211	16	theorem	theorem	VERB
ejpam-4788	211	17	8	8	NUM
ejpam-4788	211	18	.	.	PUNCT
ejpam-4788	212	1	if	if	SCONJ
ejpam-4788	212	2	ξ	ξ	X
ejpam-4788	212	3	=	=	SYM
ejpam-4788	212	4	(	(	PUNCT
ejpam-4788	212	5	s	s	PROPN
ejpam-4788	212	6	;	;	PUNCT
ejpam-4788	212	7	ξp	ξp	NUM
ejpam-4788	212	8	,	,	PUNCT
ejpam-4788	212	9	ξn	ξn	NOUN
ejpam-4788	212	10	)	)	PUNCT
ejpam-4788	212	11	and	and	CCONJ
ejpam-4788	212	12	ς	ς	PROPN
ejpam-4788	212	13	=	=	PUNCT
ejpam-4788	212	14	(	(	PUNCT
ejpam-4788	212	15	s	s	NOUN
ejpam-4788	212	16	;	;	PUNCT
ejpam-4788	212	17	ςp	ςp	NUM
ejpam-4788	212	18	,	,	PUNCT
ejpam-4788	212	19	ςn	ςn	NOUN
ejpam-4788	212	20	)	)	PUNCT
ejpam-4788	212	21	is	be	AUX
ejpam-4788	212	22	a	a	DET
ejpam-4788	212	23	bf	bf	NOUN
ejpam-4788	212	24	left	leave	VERB
ejpam-4788	212	25	α	α	NOUN
ejpam-4788	212	26	-	-	NOUN
ejpam-4788	212	27	ideal	ideal	NOUN
ejpam-4788	212	28	and	and	CCONJ
ejpam-4788	212	29	a	a	DET
ejpam-4788	212	30	bf	bf	NOUN
ejpam-4788	213	1	right	right	NOUN
ejpam-4788	214	1	α	α	NOUN
ejpam-4788	214	2	-	-	NOUN
ejpam-4788	214	3	ideal	ideal	NOUN
ejpam-4788	214	4	of	of	ADP
ejpam-4788	214	5	a	a	DET
ejpam-4788	214	6	γ	γ	NOUN
ejpam-4788	214	7	-	-	PUNCT
ejpam-4788	214	8	semigroup	semigroup	NOUN
ejpam-4788	214	9	s	s	NOUN
ejpam-4788	214	10	,	,	PUNCT
ejpam-4788	214	11	respectively	respectively	ADV
ejpam-4788	214	12	,	,	PUNCT
ejpam-4788	214	13	then	then	ADV
ejpam-4788	214	14	ξ	ξ	PROPN
ejpam-4788	214	15	∩	∩	NOUN
ejpam-4788	214	16	ς	ς	PROPN
ejpam-4788	214	17	is	be	AUX
ejpam-4788	214	18	a	a	DET
ejpam-4788	214	19	bf	bf	NOUN
ejpam-4788	215	1	α	α	NOUN
ejpam-4788	215	2	-	-	PUNCT
ejpam-4788	215	3	quasi	quasi	NOUN
ejpam-4788	215	4	-	-	NOUN
ejpam-4788	215	5	ideal	ideal	NOUN
ejpam-4788	215	6	of	of	ADP
ejpam-4788	215	7	s.	s.	PROPN
ejpam-4788	215	8	p.	p.	PROPN
ejpam-4788	215	9	khamrot	khamrot	PROPN
ejpam-4788	215	10	,	,	PUNCT
ejpam-4788	215	11	t.	t.	PROPN
ejpam-4788	215	12	gaketem	gaketem	PROPN
ejpam-4788	215	13	/	/	SYM
ejpam-4788	215	14	eur	eur	PROPN
ejpam-4788	215	15	.	.	PUNCT
ejpam-4788	216	1	j.	j.	PROPN
ejpam-4788	216	2	pure	pure	PROPN
ejpam-4788	216	3	appl	appl	PROPN
ejpam-4788	216	4	.	.	PROPN
ejpam-4788	216	5	math	math	PROPN
ejpam-4788	216	6	,	,	PUNCT
ejpam-4788	216	7	16	16	NUM
ejpam-4788	216	8	(	(	PUNCT
ejpam-4788	216	9	3	3	NUM
ejpam-4788	216	10	)	)	PUNCT
ejpam-4788	216	11	(	(	PUNCT
ejpam-4788	216	12	2023	2023	NUM
ejpam-4788	216	13	)	)	PUNCT
ejpam-4788	216	14	,	,	PUNCT
ejpam-4788	216	15	1592	1592	NUM
ejpam-4788	216	16	-	-	SYM
ejpam-4788	216	17	1607	1607	NUM
ejpam-4788	216	18	1600	1600	NUM
ejpam-4788	216	19	proof	proof	NOUN
ejpam-4788	216	20	.	.	PUNCT
ejpam-4788	217	1	let	let	VERB
ejpam-4788	217	2	ξ	ξ	X
ejpam-4788	217	3	=	=	SYM
ejpam-4788	217	4	(	(	PUNCT
ejpam-4788	217	5	s	s	PROPN
ejpam-4788	217	6	;	;	PUNCT
ejpam-4788	217	7	ξp	ξp	NUM
ejpam-4788	217	8	,	,	PUNCT
ejpam-4788	217	9	ξn	ξn	NOUN
ejpam-4788	217	10	)	)	PUNCT
ejpam-4788	217	11	and	and	CCONJ
ejpam-4788	217	12	ς	ς	PROPN
ejpam-4788	217	13	=	=	PUNCT
ejpam-4788	217	14	(	(	PUNCT
ejpam-4788	217	15	s	s	NOUN
ejpam-4788	217	16	;	;	PUNCT
ejpam-4788	217	17	ςp	ςp	NUM
ejpam-4788	217	18	,	,	PUNCT
ejpam-4788	217	19	ςn	ςn	NOUN
ejpam-4788	217	20	)	)	PUNCT
ejpam-4788	217	21	is	be	AUX
ejpam-4788	217	22	a	a	DET
ejpam-4788	217	23	bf	bf	NOUN
ejpam-4788	217	24	left	leave	VERB
ejpam-4788	217	25	α	α	NOUN
ejpam-4788	217	26	-	-	NOUN
ejpam-4788	217	27	ideal	ideal	NOUN
ejpam-4788	217	28	and	and	CCONJ
ejpam-4788	217	29	a	a	DET
ejpam-4788	217	30	bf	bf	NOUN
ejpam-4788	218	1	right	right	NOUN
ejpam-4788	219	1	α	α	NOUN
ejpam-4788	219	2	-	-	NOUN
ejpam-4788	219	3	ideal	ideal	NOUN
ejpam-4788	219	4	of	of	ADP
ejpam-4788	219	5	s	s	PROPN
ejpam-4788	219	6	,	,	PUNCT
ejpam-4788	219	7	respectively	respectively	ADV
ejpam-4788	219	8	.	.	PUNCT
ejpam-4788	220	1	then	then	ADV
ejpam-4788	220	2	ςp	ςp	DET
ejpam-4788	220	3	◦	◦	NOUN
ejpam-4788	220	4	α	α	NOUN
ejpam-4788	220	5	ξp	ξp	ADP
ejpam-4788	220	6	⊆	⊆	NUM
ejpam-4788	220	7	λp	λp	SYM
ejpam-4788	220	8	s	s	NOUN
ejpam-4788	220	9	◦	◦	NOUN
ejpam-4788	220	10	α	α	NOUN
ejpam-4788	220	11	ξp	ξp	ADP
ejpam-4788	220	12	⊆	⊆	NUM
ejpam-4788	220	13	ξp	ξp	NOUN
ejpam-4788	220	14	and	and	CCONJ
ejpam-4788	220	15	ςp	ςp	ADP
ejpam-4788	220	16	◦	◦	NOUN
ejpam-4788	220	17	α	α	NOUN
ejpam-4788	220	18	ξp	ξp	ADP
ejpam-4788	220	19	⊆	⊆	NUM
ejpam-4788	220	20	ςp	ςp	NOUN
ejpam-4788	220	21	◦	◦	NOUN
ejpam-4788	220	22	α	α	NOUN
ejpam-4788	220	23	λp	λp	X
ejpam-4788	220	24	s	s	PROPN
ejpam-4788	220	25	⊆	⊆	NUM
ejpam-4788	220	26	ςp	ςp	NOUN
ejpam-4788	220	27	.	.	PUNCT
ejpam-4788	221	1	thus	thus	ADV
ejpam-4788	221	2	,	,	PUNCT
ejpam-4788	221	3	ςp	ςp	ADP
ejpam-4788	221	4	◦	◦	NOUN
ejpam-4788	221	5	α	α	NOUN
ejpam-4788	221	6	ξp	ξp	ADP
ejpam-4788	221	7	⊆	⊆	NUM
ejpam-4788	221	8	ξp	ξp	ADP
ejpam-4788	221	9	∩	∩	ADJ
ejpam-4788	221	10	ςp	ςp	X
ejpam-4788	221	11	.	.	PUNCT
ejpam-4788	222	1	so	so	ADV
ejpam-4788	222	2	,	,	PUNCT
ejpam-4788	222	3	λp	λp	PRON
ejpam-4788	222	4	s	s	VERB
ejpam-4788	222	5	◦	◦	NOUN
ejpam-4788	222	6	α	α	NOUN
ejpam-4788	222	7	(	(	PUNCT
ejpam-4788	222	8	ξp	ξp	ADP
ejpam-4788	222	9	∩	∩	ADJ
ejpam-4788	222	10	ςp	ςp	NUM
ejpam-4788	222	11	)	)	PUNCT
ejpam-4788	222	12	∩	∩	NOUN
ejpam-4788	222	13	(	(	PUNCT
ejpam-4788	222	14	ξp	ξp	NUM
ejpam-4788	222	15	∩	∩	ADJ
ejpam-4788	222	16	ςp	ςp	NUM
ejpam-4788	222	17	)	)	PUNCT
ejpam-4788	222	18	◦	◦	NOUN
ejpam-4788	222	19	α	α	NOUN
ejpam-4788	222	20	λp	λp	X
ejpam-4788	222	21	s	s	PROPN
ejpam-4788	222	22	⊆	⊆	NUM
ejpam-4788	222	23	λp	λp	SYM
ejpam-4788	222	24	s	s	NOUN
ejpam-4788	222	25	◦	◦	NOUN
ejpam-4788	222	26	α	α	NOUN
ejpam-4788	222	27	(	(	PUNCT
ejpam-4788	222	28	ξp	ξp	ADP
ejpam-4788	222	29	∩	∩	ADJ
ejpam-4788	222	30	ςp	ςp	NUM
ejpam-4788	222	31	)	)	PUNCT
ejpam-4788	222	32	◦	◦	NOUN
ejpam-4788	222	33	α	α	NOUN
ejpam-4788	222	34	λp	λp	X
ejpam-4788	222	35	s	s	PROPN
ejpam-4788	222	36	⊆	⊆	NUM
ejpam-4788	222	37	ξp	ξp	NOUN
ejpam-4788	222	38	∩	∩	ADJ
ejpam-4788	222	39	ςp	ςp	X
ejpam-4788	222	40	.	.	PUNCT
ejpam-4788	223	1	thus	thus	ADV
ejpam-4788	223	2	,	,	PUNCT
ejpam-4788	223	3	ξp	ξp	ADP
ejpam-4788	223	4	∩	∩	ADJ
ejpam-4788	223	5	ςp	ςp	PROPN
ejpam-4788	223	6	is	be	AUX
ejpam-4788	223	7	a	a	DET
ejpam-4788	223	8	bf	bf	NOUN
ejpam-4788	223	9	α	α	NOUN
ejpam-4788	223	10	-	-	PUNCT
ejpam-4788	223	11	quasi	quasi	NOUN
ejpam-4788	223	12	-	-	NOUN
ejpam-4788	223	13	ideal	ideal	NOUN
ejpam-4788	223	14	of	of	ADP
ejpam-4788	223	15	s.	s.	PROPN
ejpam-4788	223	16	similarly	similarly	ADV
ejpam-4788	223	17	,	,	PUNCT
ejpam-4788	223	18	we	we	PRON
ejpam-4788	223	19	can	can	AUX
ejpam-4788	223	20	show	show	VERB
ejpam-4788	223	21	that	that	SCONJ
ejpam-4788	223	22	ξn	ξn	PROPN
ejpam-4788	223	23	∩	∩	NOUN
ejpam-4788	223	24	ςn	ςn	PRON
ejpam-4788	223	25	is	be	AUX
ejpam-4788	223	26	a	a	DET
ejpam-4788	223	27	bf	bf	NOUN
ejpam-4788	224	1	α	α	NOUN
ejpam-4788	224	2	-	-	PUNCT
ejpam-4788	224	3	quasi	quasi	NOUN
ejpam-4788	224	4	-	-	NOUN
ejpam-4788	224	5	ideal	ideal	NOUN
ejpam-4788	224	6	of	of	ADP
ejpam-4788	224	7	s.	s.	PROPN
ejpam-4788	224	8	hence	hence	PROPN
ejpam-4788	225	1	,	,	PUNCT
ejpam-4788	225	2	ξ	ξ	PROPN
ejpam-4788	225	3	∩	∩	NOUN
ejpam-4788	225	4	ς	ς	PROPN
ejpam-4788	225	5	is	be	AUX
ejpam-4788	225	6	a	a	DET
ejpam-4788	225	7	bf	bf	NOUN
ejpam-4788	225	8	α	α	NOUN
ejpam-4788	225	9	-	-	PUNCT
ejpam-4788	225	10	quasi	quasi	NOUN
ejpam-4788	225	11	-	-	NOUN
ejpam-4788	225	12	ideal	ideal	NOUN
ejpam-4788	225	13	of	of	ADP
ejpam-4788	225	14	s.	s.	PROPN
ejpam-4788	225	15	theorem	theorem	VERB
ejpam-4788	225	16	9	9	NUM
ejpam-4788	225	17	.	.	PUNCT
ejpam-4788	226	1	every	every	DET
ejpam-4788	226	2	bf	bf	NOUN
ejpam-4788	226	3	(	(	PUNCT
ejpam-4788	226	4	α	α	NOUN
ejpam-4788	226	5	,	,	PUNCT
ejpam-4788	226	6	β)-quasi	β)-quasi	NOUN
ejpam-4788	226	7	-	-	PUNCT
ejpam-4788	226	8	ideal	ideal	NOUN
ejpam-4788	226	9	of	of	ADP
ejpam-4788	226	10	γ	γ	PROPN
ejpam-4788	226	11	-	-	PUNCT
ejpam-4788	226	12	semigroup	semigroup	NOUN
ejpam-4788	226	13	s	s	VERB
ejpam-4788	226	14	is	be	AUX
ejpam-4788	226	15	the	the	DET
ejpam-4788	226	16	intersection	intersection	NOUN
ejpam-4788	226	17	of	of	ADP
ejpam-4788	226	18	a	a	DET
ejpam-4788	226	19	bf	bf	NOUN
ejpam-4788	226	20	left	leave	VERB
ejpam-4788	226	21	α	α	NOUN
ejpam-4788	226	22	-	-	NOUN
ejpam-4788	226	23	ideal	ideal	NOUN
ejpam-4788	226	24	and	and	CCONJ
ejpam-4788	226	25	a	a	DET
ejpam-4788	226	26	bf	bf	NOUN
ejpam-4788	226	27	right	right	ADJ
ejpam-4788	226	28	β	β	NOUN
ejpam-4788	226	29	-	-	NOUN
ejpam-4788	226	30	ideal	ideal	NOUN
ejpam-4788	226	31	of	of	ADP
ejpam-4788	226	32	s	s	NOUN
ejpam-4788	226	33	proof	proof	NOUN
ejpam-4788	226	34	.	.	PUNCT
ejpam-4788	227	1	let	let	VERB
ejpam-4788	227	2	ξ	ξ	X
ejpam-4788	227	3	=	=	SYM
ejpam-4788	227	4	(	(	PUNCT
ejpam-4788	227	5	s	s	PROPN
ejpam-4788	227	6	;	;	PUNCT
ejpam-4788	227	7	ξp	ξp	NUM
ejpam-4788	227	8	,	,	PUNCT
ejpam-4788	227	9	ξn	ξn	NOUN
ejpam-4788	227	10	)	)	PUNCT
ejpam-4788	227	11	be	be	VERB
ejpam-4788	227	12	a	a	DET
ejpam-4788	227	13	bf	bf	NOUN
ejpam-4788	227	14	(	(	PUNCT
ejpam-4788	227	15	α	α	NOUN
ejpam-4788	227	16	,	,	PUNCT
ejpam-4788	227	17	β)-quasi	β)-quasi	NOUN
ejpam-4788	227	18	-	-	PUNCT
ejpam-4788	227	19	ideal	ideal	NOUN
ejpam-4788	227	20	of	of	ADP
ejpam-4788	227	21	s.	s.	PROPN
ejpam-4788	227	22	consider	consider	VERB
ejpam-4788	227	23	ςp	ςp	NOUN
ejpam-4788	227	24	=	=	PUNCT
ejpam-4788	227	25	ξp	ξp	NOUN
ejpam-4788	227	26	∪	∪	ADV
ejpam-4788	227	27	(	(	PUNCT
ejpam-4788	227	28	λp	λp	X
ejpam-4788	227	29	s	s	NOUN
ejpam-4788	227	30	◦	◦	NOUN
ejpam-4788	227	31	α	α	NOUN
ejpam-4788	227	32	ξp	ξp	NOUN
ejpam-4788	227	33	)	)	PUNCT
ejpam-4788	227	34	and	and	CCONJ
ejpam-4788	227	35	ςn	ςn	X
ejpam-4788	227	36	=	=	NOUN
ejpam-4788	227	37	ξn	ξn	PROPN
ejpam-4788	227	38	∪	∪	ADV
ejpam-4788	227	39	(	(	PUNCT
ejpam-4788	227	40	λn	λn	NOUN
ejpam-4788	227	41	s	s	PART
ejpam-4788	227	42	◦	◦	NOUN
ejpam-4788	227	43	α	α	NOUN
ejpam-4788	227	44	ξn	ξn	NOUN
ejpam-4788	227	45	)	)	PUNCT
ejpam-4788	228	1	where	where	SCONJ
ejpam-4788	228	2	ς	ς	PROPN
ejpam-4788	228	3	=	=	PUNCT
ejpam-4788	228	4	(	(	PUNCT
ejpam-4788	228	5	s	s	NOUN
ejpam-4788	228	6	;	;	PUNCT
ejpam-4788	228	7	ςp	ςp	NUM
ejpam-4788	228	8	,	,	PUNCT
ejpam-4788	228	9	ςn	ςn	PROPN
ejpam-4788	228	10	)	)	PUNCT
ejpam-4788	228	11	,	,	PUNCT
ejpam-4788	228	12	κp	κp	NOUN
ejpam-4788	228	13	=	=	PUNCT
ejpam-4788	228	14	ξp	ξp	AUX
ejpam-4788	228	15	∪	∪	ADJ
ejpam-4788	228	16	(	(	PUNCT
ejpam-4788	228	17	ξp	ξp	ADP
ejpam-4788	228	18	◦	◦	VERB
ejpam-4788	228	19	β	β	X
ejpam-4788	228	20	λp	λp	X
ejpam-4788	228	21	s	s	PROPN
ejpam-4788	228	22	)	)	PUNCT
ejpam-4788	228	23	and	and	CCONJ
ejpam-4788	228	24	κn	κn	NOUN
ejpam-4788	228	25	=	=	PUNCT
ejpam-4788	228	26	ξn	ξn	PROPN
ejpam-4788	228	27	∪	∪	X
ejpam-4788	228	28	(	(	PUNCT
ejpam-4788	228	29	ξn	ξn	PROPN
ejpam-4788	228	30	◦	◦	NOUN
ejpam-4788	228	31	β	β	X
ejpam-4788	228	32	λn	λn	NOUN
ejpam-4788	228	33	s	s	NOUN
ejpam-4788	228	34	)	)	PUNCT
ejpam-4788	228	35	where	where	SCONJ
ejpam-4788	228	36	κ	κ	NOUN
ejpam-4788	228	37	=	=	PUNCT
ejpam-4788	228	38	(	(	PUNCT
ejpam-4788	228	39	s;κp	s;κp	NOUN
ejpam-4788	228	40	,	,	PUNCT
ejpam-4788	228	41	κn	κn	NOUN
ejpam-4788	228	42	)	)	PUNCT
ejpam-4788	228	43	.	.	PUNCT
ejpam-4788	229	1	then	then	ADV
ejpam-4788	229	2	λp	λp	X
ejpam-4788	229	3	s	s	VERB
ejpam-4788	229	4	◦	◦	NOUN
ejpam-4788	229	5	α	α	NOUN
ejpam-4788	229	6	ςp	ςp	X
ejpam-4788	229	7	=	=	PUNCT
ejpam-4788	230	1	λp	λp	X
ejpam-4788	230	2	s	s	PROPN
ejpam-4788	230	3	◦	◦	NOUN
ejpam-4788	230	4	α	α	X
ejpam-4788	230	5	(	(	PUNCT
ejpam-4788	230	6	ξp	ξp	ADP
ejpam-4788	230	7	∪	∪	ADV
ejpam-4788	230	8	(	(	PUNCT
ejpam-4788	230	9	λp	λp	X
ejpam-4788	230	10	s	s	NOUN
ejpam-4788	230	11	◦	◦	NOUN
ejpam-4788	230	12	α	α	NOUN
ejpam-4788	230	13	ξp	ξp	NOUN
ejpam-4788	230	14	)	)	PUNCT
ejpam-4788	230	15	)	)	PUNCT
ejpam-4788	231	1	=	=	PUNCT
ejpam-4788	231	2	(	(	PUNCT
ejpam-4788	231	3	λp	λp	X
ejpam-4788	231	4	s	s	NOUN
ejpam-4788	231	5	◦	◦	NOUN
ejpam-4788	231	6	α	α	NOUN
ejpam-4788	231	7	ξp	ξp	NOUN
ejpam-4788	231	8	)	)	PUNCT
ejpam-4788	231	9	∪	∪	NOUN
ejpam-4788	231	10	(	(	PUNCT
ejpam-4788	231	11	λp	λp	X
ejpam-4788	231	12	s	s	NOUN
ejpam-4788	231	13	◦	◦	NOUN
ejpam-4788	231	14	α	α	NOUN
ejpam-4788	231	15	(	(	PUNCT
ejpam-4788	231	16	λp	λp	X
ejpam-4788	231	17	s	s	NOUN
ejpam-4788	231	18	◦	◦	NOUN
ejpam-4788	231	19	α	α	NOUN
ejpam-4788	231	20	ξp	ξp	NOUN
ejpam-4788	231	21	)	)	PUNCT
ejpam-4788	231	22	)	)	PUNCT
ejpam-4788	232	1	=	=	PUNCT
ejpam-4788	232	2	(	(	PUNCT
ejpam-4788	232	3	λp	λp	X
ejpam-4788	232	4	s	s	NOUN
ejpam-4788	232	5	◦	◦	NOUN
ejpam-4788	232	6	α	α	NOUN
ejpam-4788	232	7	ξp	ξp	NOUN
ejpam-4788	232	8	)	)	PUNCT
ejpam-4788	232	9	∪	∪	NOUN
ejpam-4788	232	10	(	(	PUNCT
ejpam-4788	232	11	(	(	PUNCT
ejpam-4788	232	12	λp	λp	X
ejpam-4788	232	13	s	s	NOUN
ejpam-4788	232	14	◦	◦	NOUN
ejpam-4788	232	15	α	α	NOUN
ejpam-4788	232	16	λp	λp	X
ejpam-4788	232	17	s	s	PROPN
ejpam-4788	232	18	)	)	PUNCT
ejpam-4788	232	19	◦	◦	NOUN
ejpam-4788	232	20	α	α	PRON
ejpam-4788	232	21	ξp	ξp	NOUN
ejpam-4788	232	22	)	)	PUNCT
ejpam-4788	232	23	=	=	SYM
ejpam-4788	233	1	(	(	PUNCT
ejpam-4788	233	2	λp	λp	X
ejpam-4788	233	3	s	s	NOUN
ejpam-4788	233	4	◦	◦	NOUN
ejpam-4788	233	5	α	α	NOUN
ejpam-4788	233	6	ξp	ξp	NOUN
ejpam-4788	233	7	)	)	PUNCT
ejpam-4788	233	8	∪	∪	NOUN
ejpam-4788	233	9	(	(	PUNCT
ejpam-4788	233	10	λp	λp	X
ejpam-4788	233	11	s	s	NOUN
ejpam-4788	233	12	◦	◦	NOUN
ejpam-4788	233	13	α	α	NOUN
ejpam-4788	233	14	ξp	ξp	NOUN
ejpam-4788	233	15	)	)	PUNCT
ejpam-4788	233	16	⊆	⊆	NUM
ejpam-4788	233	17	ξp	ξp	NOUN
ejpam-4788	233	18	∪	∪	ADV
ejpam-4788	233	19	(	(	PUNCT
ejpam-4788	233	20	λp	λp	X
ejpam-4788	233	21	s	s	NOUN
ejpam-4788	233	22	◦	◦	NOUN
ejpam-4788	233	23	α	α	NOUN
ejpam-4788	233	24	ξp	ξp	NOUN
ejpam-4788	233	25	)	)	PUNCT
ejpam-4788	233	26	=	=	SYM
ejpam-4788	234	1	ςp	ςp	X
ejpam-4788	234	2	.	.	PUNCT
ejpam-4788	235	1	and	and	CCONJ
ejpam-4788	235	2	κp	κp	PROPN
ejpam-4788	235	3	◦	◦	PROPN
ejpam-4788	235	4	β	β	X
ejpam-4788	235	5	λp	λp	X
ejpam-4788	235	6	s	s	NOUN
ejpam-4788	235	7	=	=	PUNCT
ejpam-4788	235	8	(	(	PUNCT
ejpam-4788	235	9	ξp	ξp	NOUN
ejpam-4788	235	10	∪	∪	ADJ
ejpam-4788	235	11	(	(	PUNCT
ejpam-4788	235	12	ξp	ξp	AUX
ejpam-4788	235	13	◦	◦	VERB
ejpam-4788	235	14	β	β	X
ejpam-4788	235	15	λp	λp	X
ejpam-4788	235	16	s	s	PROPN
ejpam-4788	235	17	)	)	PUNCT
ejpam-4788	235	18	)	)	PUNCT
ejpam-4788	236	1	◦	◦	NOUN
ejpam-4788	236	2	α	α	PRON
ejpam-4788	236	3	λp	λp	X
ejpam-4788	236	4	s	s	PROPN
ejpam-4788	236	5	=	=	PUNCT
ejpam-4788	236	6	(	(	PUNCT
ejpam-4788	236	7	ξp	ξp	PART
ejpam-4788	236	8	◦	◦	VERB
ejpam-4788	236	9	α	α	NOUN
ejpam-4788	236	10	λp	λp	X
ejpam-4788	236	11	s	s	PROPN
ejpam-4788	236	12	)	)	PUNCT
ejpam-4788	236	13	∪	∪	NOUN
ejpam-4788	236	14	(	(	PUNCT
ejpam-4788	236	15	ξp	ξp	PART
ejpam-4788	236	16	◦	◦	VERB
ejpam-4788	236	17	β	β	X
ejpam-4788	236	18	λp	λp	X
ejpam-4788	236	19	s	s	PROPN
ejpam-4788	236	20	◦	◦	NOUN
ejpam-4788	236	21	α	α	NOUN
ejpam-4788	236	22	λp	λp	X
ejpam-4788	236	23	s	s	PROPN
ejpam-4788	236	24	)	)	PUNCT
ejpam-4788	236	25	=	=	SYM
ejpam-4788	236	26	(	(	PUNCT
ejpam-4788	236	27	ξp	ξp	PART
ejpam-4788	236	28	◦	◦	VERB
ejpam-4788	236	29	α	α	NOUN
ejpam-4788	236	30	λp	λp	X
ejpam-4788	236	31	s	s	PROPN
ejpam-4788	236	32	)	)	PUNCT
ejpam-4788	236	33	∪	∪	NOUN
ejpam-4788	236	34	ξp	ξp	ADP
ejpam-4788	236	35	◦	◦	NOUN
ejpam-4788	236	36	β	β	X
ejpam-4788	236	37	(	(	PUNCT
ejpam-4788	236	38	λp	λp	X
ejpam-4788	236	39	s	s	PROPN
ejpam-4788	236	40	◦	◦	NOUN
ejpam-4788	236	41	α	α	NOUN
ejpam-4788	236	42	λp	λp	X
ejpam-4788	236	43	s	s	PROPN
ejpam-4788	236	44	)	)	PUNCT
ejpam-4788	236	45	=	=	SYM
ejpam-4788	236	46	(	(	PUNCT
ejpam-4788	236	47	ξp	ξp	PART
ejpam-4788	236	48	◦	◦	VERB
ejpam-4788	236	49	α	α	NOUN
ejpam-4788	236	50	λp	λp	X
ejpam-4788	236	51	s	s	PROPN
ejpam-4788	236	52	)	)	PUNCT
ejpam-4788	236	53	∪	∪	NOUN
ejpam-4788	236	54	(	(	PUNCT
ejpam-4788	236	55	ξp	ξp	ADP
ejpam-4788	236	56	◦	◦	VERB
ejpam-4788	236	57	β	β	X
ejpam-4788	236	58	λp	λp	X
ejpam-4788	236	59	s	s	PROPN
ejpam-4788	236	60	)	)	PUNCT
ejpam-4788	236	61	⊆	⊆	NUM
ejpam-4788	236	62	ξp	ξp	NOUN
ejpam-4788	236	63	∪	∪	ADV
ejpam-4788	236	64	(	(	PUNCT
ejpam-4788	236	65	ξp	ξp	ADP
ejpam-4788	236	66	◦	◦	VERB
ejpam-4788	236	67	β	β	X
ejpam-4788	236	68	λp	λp	X
ejpam-4788	236	69	s	s	PROPN
ejpam-4788	236	70	)	)	PUNCT
ejpam-4788	236	71	=	=	SYM
ejpam-4788	236	72	κp	κp	PROPN
ejpam-4788	236	73	.	.	PUNCT
ejpam-4788	237	1	similarly	similarly	ADV
ejpam-4788	237	2	,	,	PUNCT
ejpam-4788	237	3	we	we	PRON
ejpam-4788	237	4	can	can	AUX
ejpam-4788	237	5	show	show	VERB
ejpam-4788	237	6	that	that	SCONJ
ejpam-4788	237	7	λn	λn	PROPN
ejpam-4788	237	8	s	s	VERB
ejpam-4788	237	9	◦	◦	NOUN
ejpam-4788	237	10	α	α	X
ejpam-4788	237	11	ςn	ςn	NOUN
ejpam-4788	237	12	⊇	⊇	PROPN
ejpam-4788	237	13	ςn	ςn	PROPN
ejpam-4788	237	14	and	and	CCONJ
ejpam-4788	237	15	κn	κn	PROPN
ejpam-4788	237	16	◦	◦	PROPN
ejpam-4788	237	17	β	β	NOUN
ejpam-4788	237	18	λn	λn	NOUN
ejpam-4788	237	19	s	s	PART
ejpam-4788	237	20	⊇	⊇	PROPN
ejpam-4788	237	21	κn	κn	NOUN
ejpam-4788	237	22	.	.	PUNCT
ejpam-4788	238	1	thus	thus	ADV
ejpam-4788	238	2	,	,	PUNCT
ejpam-4788	238	3	ς	ς	PROPN
ejpam-4788	238	4	=	=	PUNCT
ejpam-4788	238	5	(	(	PUNCT
ejpam-4788	238	6	s	s	NOUN
ejpam-4788	238	7	;	;	PUNCT
ejpam-4788	238	8	ςp	ςp	NUM
ejpam-4788	238	9	,	,	PUNCT
ejpam-4788	238	10	ςn	ςn	NOUN
ejpam-4788	238	11	)	)	PUNCT
ejpam-4788	238	12	and	and	CCONJ
ejpam-4788	238	13	κ	κ	X
ejpam-4788	238	14	=	=	SYM
ejpam-4788	238	15	(	(	PUNCT
ejpam-4788	238	16	s;κp	s;κp	NOUN
ejpam-4788	238	17	,	,	PUNCT
ejpam-4788	238	18	κn	κn	NOUN
ejpam-4788	238	19	)	)	PUNCT
ejpam-4788	238	20	is	be	AUX
ejpam-4788	238	21	a	a	DET
ejpam-4788	238	22	bf	bf	NOUN
ejpam-4788	238	23	left	leave	VERB
ejpam-4788	238	24	α	α	NOUN
ejpam-4788	238	25	-	-	NOUN
ejpam-4788	238	26	ideal	ideal	NOUN
ejpam-4788	238	27	and	and	CCONJ
ejpam-4788	238	28	a	a	DET
ejpam-4788	238	29	bf	bf	NOUN
ejpam-4788	238	30	right	right	ADJ
ejpam-4788	238	31	β	β	NOUN
ejpam-4788	238	32	-	-	NOUN
ejpam-4788	238	33	ideal	ideal	NOUN
ejpam-4788	238	34	of	of	ADP
ejpam-4788	238	35	s.	s.	PROPN
ejpam-4788	238	36	now	now	ADV
ejpam-4788	238	37	,	,	PUNCT
ejpam-4788	238	38	ξp	ξp	ADP
ejpam-4788	238	39	⊆	⊆	NUM
ejpam-4788	238	40	(	(	PUNCT
ejpam-4788	238	41	ξp	ξp	NOUN
ejpam-4788	238	42	∪	∪	ADV
ejpam-4788	238	43	(	(	PUNCT
ejpam-4788	238	44	λp	λp	X
ejpam-4788	238	45	s	s	NOUN
ejpam-4788	238	46	◦	◦	NOUN
ejpam-4788	238	47	α	α	NOUN
ejpam-4788	238	48	ξp	ξp	NOUN
ejpam-4788	238	49	)	)	PUNCT
ejpam-4788	238	50	)	)	PUNCT
ejpam-4788	238	51	∩	∩	NOUN
ejpam-4788	238	52	(	(	PUNCT
ejpam-4788	238	53	ξp	ξp	ADP
ejpam-4788	238	54	∪	∪	ADJ
ejpam-4788	238	55	(	(	PUNCT
ejpam-4788	238	56	ξp	ξp	AUX
ejpam-4788	238	57	◦	◦	VERB
ejpam-4788	238	58	β	β	X
ejpam-4788	238	59	λp	λp	X
ejpam-4788	238	60	s	s	PROPN
ejpam-4788	238	61	)	)	PUNCT
ejpam-4788	238	62	)	)	PUNCT
ejpam-4788	239	1	=	=	SYM
ejpam-4788	239	2	ςp	ςp	NOUN
ejpam-4788	239	3	∩	∩	PROPN
ejpam-4788	239	4	κp	κp	PROPN
ejpam-4788	239	5	and	and	CCONJ
ejpam-4788	239	6	ςp	ςp	ADP
ejpam-4788	239	7	∩κp	∩κp	NOUN
ejpam-4788	239	8	=	=	SYM
ejpam-4788	239	9	(	(	PUNCT
ejpam-4788	239	10	ξp	ξp	NOUN
ejpam-4788	239	11	∪	∪	ADV
ejpam-4788	239	12	(	(	PUNCT
ejpam-4788	239	13	λp	λp	X
ejpam-4788	239	14	s	s	NOUN
ejpam-4788	239	15	◦	◦	NOUN
ejpam-4788	239	16	α	α	NOUN
ejpam-4788	239	17	ξp))∩	ξp))∩	NOUN
ejpam-4788	239	18	(	(	PUNCT
ejpam-4788	239	19	ξp	ξp	ADP
ejpam-4788	239	20	∪	∪	ADJ
ejpam-4788	239	21	(	(	PUNCT
ejpam-4788	239	22	ξp	ξp	AUX
ejpam-4788	239	23	◦	◦	VERB
ejpam-4788	239	24	β	β	X
ejpam-4788	239	25	λp	λp	X
ejpam-4788	239	26	s	s	PROPN
ejpam-4788	239	27	)	)	PUNCT
ejpam-4788	239	28	)	)	PUNCT
ejpam-4788	239	29	=	=	SYM
ejpam-4788	240	1	ξp	ξp	ADP
ejpam-4788	240	2	∩	∩	NOUN
ejpam-4788	240	3	(	(	PUNCT
ejpam-4788	240	4	(	(	PUNCT
ejpam-4788	240	5	λp	λp	X
ejpam-4788	240	6	s	s	NOUN
ejpam-4788	240	7	◦	◦	NOUN
ejpam-4788	240	8	α	α	X
ejpam-4788	240	9	ξ	ξ	X
ejpam-4788	240	10	p)∪	p)∪	NOUN
ejpam-4788	240	11	(	(	PUNCT
ejpam-4788	240	12	ξp	ξp	AUX
ejpam-4788	240	13	◦	◦	VERB
ejpam-4788	240	14	β	β	X
ejpam-4788	240	15	λp	λp	X
ejpam-4788	240	16	s	s	PROPN
ejpam-4788	240	17	)	)	PUNCT
ejpam-4788	240	18	)	)	PUNCT
ejpam-4788	241	1	⊆	⊆	NUM
ejpam-4788	241	2	ξp	ξp	ADP
ejpam-4788	241	3	∩	∩	NOUN
ejpam-4788	241	4	ξp	ξp	ADP
ejpam-4788	241	5	=	=	PUNCT
ejpam-4788	241	6	ξp	ξp	NOUN
ejpam-4788	241	7	.	.	PUNCT
ejpam-4788	242	1	hence	hence	ADV
ejpam-4788	242	2	,	,	PUNCT
ejpam-4788	242	3	ξp	ξp	ADP
ejpam-4788	242	4	=	=	SYM
ejpam-4788	242	5	ςp	ςp	NOUN
ejpam-4788	242	6	∩	∩	PROPN
ejpam-4788	242	7	κp	κp	PROPN
ejpam-4788	242	8	.	.	PUNCT
ejpam-4788	243	1	simlarly	simlarly	ADV
ejpam-4788	243	2	,	,	PUNCT
ejpam-4788	243	3	we	we	PRON
ejpam-4788	243	4	can	can	AUX
ejpam-4788	243	5	show	show	VERB
ejpam-4788	243	6	that	that	SCONJ
ejpam-4788	243	7	ξn	ξn	NOUN
ejpam-4788	243	8	=	=	SYM
ejpam-4788	243	9	ςn	ςn	PROPN
ejpam-4788	243	10	∩	∩	PROPN
ejpam-4788	243	11	κn	κn	NOUN
ejpam-4788	243	12	.	.	PUNCT
ejpam-4788	244	1	theorem	theorem	NOUN
ejpam-4788	244	2	10	10	NUM
ejpam-4788	244	3	.	.	PUNCT
ejpam-4788	245	1	let	let	VERB
ejpam-4788	245	2	k	k	PRON
ejpam-4788	245	3	be	be	AUX
ejpam-4788	245	4	a	a	DET
ejpam-4788	245	5	non	non	ADJ
ejpam-4788	245	6	-	-	ADJ
ejpam-4788	245	7	empty	empty	ADJ
ejpam-4788	245	8	subset	subset	NOUN
ejpam-4788	245	9	of	of	ADP
ejpam-4788	245	10	γ	γ	PROPN
ejpam-4788	245	11	-	-	PUNCT
ejpam-4788	245	12	semigroup	semigroup	PROPN
ejpam-4788	245	13	s.	s.	PROPN
ejpam-4788	246	1	then	then	ADV
ejpam-4788	246	2	k	k	PROPN
ejpam-4788	246	3	is	be	AUX
ejpam-4788	246	4	a	a	DET
ejpam-4788	246	5	(	(	PUNCT
ejpam-4788	246	6	α	α	NOUN
ejpam-4788	246	7	,	,	PUNCT
ejpam-4788	246	8	β)-quasiideal	β)-quasiideal	PUNCT
ejpam-4788	246	9	of	of	ADP
ejpam-4788	246	10	s	s	PRON
ejpam-4788	246	11	if	if	SCONJ
ejpam-4788	247	1	and	and	CCONJ
ejpam-4788	247	2	only	only	ADV
ejpam-4788	247	3	if	if	SCONJ
ejpam-4788	247	4	the	the	DET
ejpam-4788	247	5	characteristic	characteristic	ADJ
ejpam-4788	247	6	function	function	NOUN
ejpam-4788	247	7	λk	λk	X
ejpam-4788	247	8	=	=	PUNCT
ejpam-4788	247	9	(	(	PUNCT
ejpam-4788	247	10	s;λp	s;λp	PROPN
ejpam-4788	247	11	k	k	PROPN
ejpam-4788	247	12	,	,	PUNCT
ejpam-4788	247	13	λn	λn	PROPN
ejpam-4788	247	14	k	k	X
ejpam-4788	247	15	)	)	PUNCT
ejpam-4788	247	16	is	be	AUX
ejpam-4788	247	17	a	a	DET
ejpam-4788	247	18	bf	bf	NOUN
ejpam-4788	247	19	(	(	PUNCT
ejpam-4788	247	20	α	α	NOUN
ejpam-4788	247	21	,	,	PUNCT
ejpam-4788	247	22	β)-quasiideal	β)-quasiideal	PUNCT
ejpam-4788	247	23	of	of	ADP
ejpam-4788	247	24	s.	s.	PROPN
ejpam-4788	247	25	proof	proof	PROPN
ejpam-4788	247	26	.	.	PUNCT
ejpam-4788	248	1	suppose	suppose	VERB
ejpam-4788	248	2	that	that	SCONJ
ejpam-4788	248	3	k	k	PROPN
ejpam-4788	248	4	is	be	AUX
ejpam-4788	248	5	a	a	DET
ejpam-4788	248	6	(	(	PUNCT
ejpam-4788	248	7	α	α	NOUN
ejpam-4788	248	8	,	,	PUNCT
ejpam-4788	248	9	β)-quasi	β)-quasi	NOUN
ejpam-4788	248	10	-	-	PUNCT
ejpam-4788	248	11	ideal	ideal	NOUN
ejpam-4788	248	12	of	of	ADP
ejpam-4788	248	13	s	s	NOUN
ejpam-4788	248	14	and	and	CCONJ
ejpam-4788	248	15	u	u	PROPN
ejpam-4788	248	16	∈	∈	PROPN
ejpam-4788	248	17	s.	s.	PROPN
ejpam-4788	248	18	if	if	SCONJ
ejpam-4788	248	19	u	u	PROPN
ejpam-4788	248	20	∈	∈	PROPN
ejpam-4788	248	21	(	(	PUNCT
ejpam-4788	248	22	sαk	sαk	NOUN
ejpam-4788	248	23	)	)	PUNCT
ejpam-4788	248	24	∩	∩	NOUN
ejpam-4788	248	25	(	(	PUNCT
ejpam-4788	248	26	kβs	kβs	PROPN
ejpam-4788	248	27	)	)	PUNCT
ejpam-4788	248	28	,	,	PUNCT
ejpam-4788	248	29	then	then	ADV
ejpam-4788	248	30	u	u	PROPN
ejpam-4788	248	31	∈	∈	PROPN
ejpam-4788	248	32	k.	k.	PROPN
ejpam-4788	248	33	thus	thus	ADV
ejpam-4788	248	34	,	,	PUNCT
ejpam-4788	248	35	λp	λp	X
ejpam-4788	248	36	k(u	k(u	X
ejpam-4788	248	37	)	)	PUNCT
ejpam-4788	248	38	=	=	SYM
ejpam-4788	248	39	1	1	NUM
ejpam-4788	248	40	and	and	CCONJ
ejpam-4788	248	41	λn	λn	ADP
ejpam-4788	248	42	k(u	k(u	NOUN
ejpam-4788	248	43	)	)	PUNCT
ejpam-4788	248	44	=	=	SYM
ejpam-4788	248	45	−1	−1	NOUN
ejpam-4788	248	46	.	.	PUNCT
ejpam-4788	249	1	hence	hence	ADV
ejpam-4788	249	2	,	,	PUNCT
ejpam-4788	249	3	(	(	PUNCT
ejpam-4788	249	4	(	(	PUNCT
ejpam-4788	249	5	λp	λp	X
ejpam-4788	249	6	k	k	PROPN
ejpam-4788	249	7	◦	◦	PROPN
ejpam-4788	249	8	α	α	X
ejpam-4788	249	9	λp	λp	X
ejpam-4788	249	10	s	s	PROPN
ejpam-4788	249	11	)	)	PUNCT
ejpam-4788	249	12	∩	∩	NOUN
ejpam-4788	249	13	(	(	PUNCT
ejpam-4788	249	14	λp	λp	X
ejpam-4788	249	15	s	s	PROPN
ejpam-4788	249	16	◦	◦	NOUN
ejpam-4788	249	17	β	β	X
ejpam-4788	249	18	λp	λp	NOUN
ejpam-4788	249	19	k))(u	k))(u	NOUN
ejpam-4788	249	20	)	)	PUNCT
ejpam-4788	249	21	≤	≤	NOUN
ejpam-4788	249	22	λp	λp	X
ejpam-4788	249	23	k(u	k(u	NOUN
ejpam-4788	249	24	)	)	PUNCT
ejpam-4788	249	25	and	and	CCONJ
ejpam-4788	249	26	(	(	PUNCT
ejpam-4788	249	27	(	(	PUNCT
ejpam-4788	249	28	λn	λn	X
ejpam-4788	249	29	k	k	PROPN
ejpam-4788	249	30	◦	◦	NOUN
ejpam-4788	249	31	α	α	NOUN
ejpam-4788	249	32	λn	λn	NOUN
ejpam-4788	249	33	s	s	NOUN
ejpam-4788	249	34	)	)	PUNCT
ejpam-4788	249	35	∪	∪	NOUN
ejpam-4788	249	36	(	(	PUNCT
ejpam-4788	249	37	λn	λn	NOUN
ejpam-4788	249	38	s	s	PART
ejpam-4788	249	39	◦	◦	NOUN
ejpam-4788	249	40	β	β	X
ejpam-4788	249	41	λn	λn	NOUN
ejpam-4788	249	42	k))(u	k))(u	NOUN
ejpam-4788	249	43	)	)	PUNCT
ejpam-4788	249	44	≥	≥	NOUN
ejpam-4788	249	45	λn	λn	NOUN
ejpam-4788	249	46	k(u	k(u	NOUN
ejpam-4788	249	47	)	)	PUNCT
ejpam-4788	249	48	if	if	SCONJ
ejpam-4788	249	49	u	u	PRON
ejpam-4788	249	50	/∈	/∈	PROPN
ejpam-4788	249	51	(	(	PUNCT
ejpam-4788	249	52	sαk	sαk	PROPN
ejpam-4788	249	53	)	)	PUNCT
ejpam-4788	249	54	∩	∩	NOUN
ejpam-4788	249	55	(	(	PUNCT
ejpam-4788	249	56	kβs	kβs	PROPN
ejpam-4788	249	57	)	)	PUNCT
ejpam-4788	249	58	,	,	PUNCT
ejpam-4788	249	59	then	then	ADV
ejpam-4788	249	60	λp	λp	X
ejpam-4788	249	61	k(u	k(u	X
ejpam-4788	249	62	)	)	PUNCT
ejpam-4788	249	63	=	=	SYM
ejpam-4788	249	64	0	0	NUM
ejpam-4788	249	65	and	and	CCONJ
ejpam-4788	249	66	λp	λp	X
ejpam-4788	249	67	k(u	k(u	X
ejpam-4788	249	68	)	)	PUNCT
ejpam-4788	249	69	=	=	SYM
ejpam-4788	250	1	0	0	X
ejpam-4788	250	2	.	.	PUNCT
ejpam-4788	251	1	hence	hence	ADV
ejpam-4788	251	2	,	,	PUNCT
ejpam-4788	251	3	(	(	PUNCT
ejpam-4788	251	4	(	(	PUNCT
ejpam-4788	251	5	λp	λp	X
ejpam-4788	251	6	k	k	PROPN
ejpam-4788	251	7	◦	◦	PROPN
ejpam-4788	251	8	α	α	X
ejpam-4788	251	9	λp	λp	X
ejpam-4788	251	10	s	s	PROPN
ejpam-4788	251	11	)	)	PUNCT
ejpam-4788	251	12	∩	∩	NOUN
ejpam-4788	251	13	(	(	PUNCT
ejpam-4788	251	14	λp	λp	X
ejpam-4788	251	15	s	s	PROPN
ejpam-4788	251	16	◦	◦	NOUN
ejpam-4788	251	17	β	β	X
ejpam-4788	251	18	λp	λp	NOUN
ejpam-4788	251	19	k))(u	k))(u	NOUN
ejpam-4788	251	20	)	)	PUNCT
ejpam-4788	251	21	≤	≤	NOUN
ejpam-4788	251	22	λp	λp	X
ejpam-4788	251	23	k(u	k(u	NOUN
ejpam-4788	251	24	)	)	PUNCT
ejpam-4788	251	25	and	and	CCONJ
ejpam-4788	251	26	(	(	PUNCT
ejpam-4788	251	27	(	(	PUNCT
ejpam-4788	251	28	λn	λn	X
ejpam-4788	251	29	k	k	PROPN
ejpam-4788	251	30	◦	◦	NOUN
ejpam-4788	251	31	α	α	NOUN
ejpam-4788	251	32	λn	λn	NOUN
ejpam-4788	251	33	s	s	NOUN
ejpam-4788	251	34	)	)	PUNCT
ejpam-4788	251	35	∪	∪	NOUN
ejpam-4788	251	36	(	(	PUNCT
ejpam-4788	251	37	λn	λn	NOUN
ejpam-4788	251	38	s	s	PART
ejpam-4788	251	39	◦	◦	NOUN
ejpam-4788	251	40	β	β	X
ejpam-4788	251	41	λn	λn	NOUN
ejpam-4788	251	42	k))(u	k))(u	NOUN
ejpam-4788	251	43	)	)	PUNCT
ejpam-4788	251	44	≥	≥	NOUN
ejpam-4788	251	45	λn	λn	NOUN
ejpam-4788	251	46	k(u	k(u	NOUN
ejpam-4788	251	47	)	)	PUNCT
ejpam-4788	251	48	.	.	PUNCT
ejpam-4788	252	1	therefore	therefore	ADV
ejpam-4788	252	2	,	,	PUNCT
ejpam-4788	252	3	λk	λk	X
ejpam-4788	252	4	=	=	PUNCT
ejpam-4788	252	5	(	(	PUNCT
ejpam-4788	252	6	s;λp	s;λp	PROPN
ejpam-4788	252	7	k	k	PROPN
ejpam-4788	252	8	,	,	PUNCT
ejpam-4788	252	9	λn	λn	PROPN
ejpam-4788	252	10	k	k	X
ejpam-4788	252	11	)	)	PUNCT
ejpam-4788	252	12	is	be	AUX
ejpam-4788	252	13	a	a	DET
ejpam-4788	252	14	bf	bf	NOUN
ejpam-4788	252	15	(	(	PUNCT
ejpam-4788	252	16	α	α	NOUN
ejpam-4788	252	17	,	,	PUNCT
ejpam-4788	252	18	β)-quasi	β)-quasi	NOUN
ejpam-4788	252	19	-	-	NOUN
ejpam-4788	252	20	ideal	ideal	NOUN
ejpam-4788	252	21	of	of	ADP
ejpam-4788	252	22	s.	s.	PROPN
ejpam-4788	252	23	p.	p.	PROPN
ejpam-4788	252	24	khamrot	khamrot	PROPN
ejpam-4788	252	25	,	,	PUNCT
ejpam-4788	252	26	t.	t.	PROPN
ejpam-4788	252	27	gaketem	gaketem	PROPN
ejpam-4788	252	28	/	/	SYM
ejpam-4788	252	29	eur	eur	PROPN
ejpam-4788	252	30	.	.	PUNCT
ejpam-4788	253	1	j.	j.	PROPN
ejpam-4788	253	2	pure	pure	PROPN
ejpam-4788	253	3	appl	appl	PROPN
ejpam-4788	253	4	.	.	PROPN
ejpam-4788	253	5	math	math	PROPN
ejpam-4788	253	6	,	,	PUNCT
ejpam-4788	253	7	16	16	NUM
ejpam-4788	253	8	(	(	PUNCT
ejpam-4788	253	9	3	3	NUM
ejpam-4788	253	10	)	)	PUNCT
ejpam-4788	253	11	(	(	PUNCT
ejpam-4788	253	12	2023	2023	NUM
ejpam-4788	253	13	)	)	PUNCT
ejpam-4788	253	14	,	,	PUNCT
ejpam-4788	253	15	1592	1592	NUM
ejpam-4788	253	16	-	-	SYM
ejpam-4788	253	17	1607	1607	NUM
ejpam-4788	253	18	1601	1601	NUM
ejpam-4788	253	19	conversely	conversely	ADV
ejpam-4788	253	20	,	,	PUNCT
ejpam-4788	253	21	assume	assume	VERB
ejpam-4788	253	22	that	that	SCONJ
ejpam-4788	253	23	λk	λk	ADV
ejpam-4788	253	24	=	=	SYM
ejpam-4788	253	25	(	(	PUNCT
ejpam-4788	253	26	s;λp	s;λp	PROPN
ejpam-4788	253	27	k	k	PROPN
ejpam-4788	253	28	,	,	PUNCT
ejpam-4788	253	29	λn	λn	PROPN
ejpam-4788	253	30	k	k	X
ejpam-4788	253	31	)	)	PUNCT
ejpam-4788	253	32	is	be	AUX
ejpam-4788	253	33	a	a	DET
ejpam-4788	253	34	bf	bf	NOUN
ejpam-4788	253	35	(	(	PUNCT
ejpam-4788	253	36	α	α	NOUN
ejpam-4788	253	37	,	,	PUNCT
ejpam-4788	253	38	β)-quasi	β)-quasi	NOUN
ejpam-4788	253	39	-	-	PUNCT
ejpam-4788	253	40	ideal	ideal	NOUN
ejpam-4788	253	41	of	of	ADP
ejpam-4788	253	42	s	s	NOUN
ejpam-4788	253	43	and	and	CCONJ
ejpam-4788	253	44	u	u	PROPN
ejpam-4788	253	45	∈	∈	PROPN
ejpam-4788	253	46	(	(	PUNCT
ejpam-4788	253	47	sαk	sαk	NOUN
ejpam-4788	253	48	)	)	PUNCT
ejpam-4788	253	49	∩	∩	NOUN
ejpam-4788	253	50	(	(	PUNCT
ejpam-4788	253	51	kβs	kβs	PROPN
ejpam-4788	253	52	)	)	PUNCT
ejpam-4788	253	53	.	.	PUNCT
ejpam-4788	254	1	then	then	ADV
ejpam-4788	254	2	(	(	PUNCT
ejpam-4788	254	3	(	(	PUNCT
ejpam-4788	254	4	λp	λp	X
ejpam-4788	254	5	k	k	PROPN
ejpam-4788	254	6	◦	◦	PROPN
ejpam-4788	254	7	α	α	X
ejpam-4788	254	8	λp	λp	X
ejpam-4788	254	9	s	s	PROPN
ejpam-4788	254	10	)	)	PUNCT
ejpam-4788	254	11	∩	∩	NOUN
ejpam-4788	254	12	(	(	PUNCT
ejpam-4788	254	13	λp	λp	X
ejpam-4788	254	14	s	s	PROPN
ejpam-4788	254	15	◦	◦	NOUN
ejpam-4788	254	16	β	β	X
ejpam-4788	254	17	λp	λp	NOUN
ejpam-4788	254	18	k))(u	k))(u	NOUN
ejpam-4788	254	19	)	)	PUNCT
ejpam-4788	254	20	=	=	SYM
ejpam-4788	254	21	1	1	NUM
ejpam-4788	254	22	and	and	CCONJ
ejpam-4788	254	23	(	(	PUNCT
ejpam-4788	254	24	(	(	PUNCT
ejpam-4788	254	25	λn	λn	X
ejpam-4788	254	26	k	k	PROPN
ejpam-4788	254	27	◦	◦	NOUN
ejpam-4788	254	28	α	α	X
ejpam-4788	254	29	λn	λn	NOUN
ejpam-4788	254	30	s)∩	s)∩	NOUN
ejpam-4788	254	31	(	(	PUNCT
ejpam-4788	254	32	λn	λn	NOUN
ejpam-4788	254	33	s	s	PART
ejpam-4788	254	34	◦	◦	NOUN
ejpam-4788	254	35	β	β	X
ejpam-4788	254	36	λn	λn	NOUN
ejpam-4788	254	37	k))(u	k))(u	NOUN
ejpam-4788	254	38	)	)	PUNCT
ejpam-4788	254	39	=	=	SYM
ejpam-4788	254	40	−1	−1	NOUN
ejpam-4788	254	41	.	.	PUNCT
ejpam-4788	255	1	by	by	ADP
ejpam-4788	255	2	assumption	assumption	NOUN
ejpam-4788	255	3	,	,	PUNCT
ejpam-4788	255	4	(	(	PUNCT
ejpam-4788	255	5	(	(	PUNCT
ejpam-4788	255	6	λp	λp	X
ejpam-4788	255	7	k	k	PROPN
ejpam-4788	255	8	◦	◦	PROPN
ejpam-4788	255	9	α	α	NOUN
ejpam-4788	255	10	λp	λp	NOUN
ejpam-4788	255	11	s)∩	s)∩	NOUN
ejpam-4788	255	12	(	(	PUNCT
ejpam-4788	255	13	λp	λp	X
ejpam-4788	255	14	s	s	PROPN
ejpam-4788	255	15	◦	◦	NOUN
ejpam-4788	255	16	β	β	X
ejpam-4788	255	17	λp	λp	NOUN
ejpam-4788	255	18	k))(u	k))(u	NOUN
ejpam-4788	255	19	)	)	PUNCT
ejpam-4788	255	20	≤	≤	NOUN
ejpam-4788	255	21	λp	λp	X
ejpam-4788	255	22	k(u	k(u	NOUN
ejpam-4788	255	23	)	)	PUNCT
ejpam-4788	255	24	and	and	CCONJ
ejpam-4788	255	25	(	(	PUNCT
ejpam-4788	255	26	(	(	PUNCT
ejpam-4788	255	27	λn	λn	PART
ejpam-4788	255	28	k	k	PROPN
ejpam-4788	255	29	◦	◦	PROPN
ejpam-4788	255	30	αλn	αλn	PROPN
ejpam-4788	255	31	s)∪(λn	s)∪(λn	PROPN
ejpam-4788	255	32	s	s	PART
ejpam-4788	255	33	◦	◦	NOUN
ejpam-4788	255	34	β	β	X
ejpam-4788	255	35	λn	λn	NOUN
ejpam-4788	255	36	k))(u	k))(u	NOUN
ejpam-4788	255	37	)	)	PUNCT
ejpam-4788	255	38	≥	≥	NOUN
ejpam-4788	255	39	λn	λn	NOUN
ejpam-4788	255	40	k(u	k(u	NOUN
ejpam-4788	255	41	)	)	PUNCT
ejpam-4788	255	42	.	.	PUNCT
ejpam-4788	256	1	thus	thus	ADV
ejpam-4788	256	2	,	,	PUNCT
ejpam-4788	256	3	u	u	PROPN
ejpam-4788	256	4	∈	∈	PROPN
ejpam-4788	256	5	k.	k.	NOUN
ejpam-4788	256	6	hence	hence	ADV
ejpam-4788	256	7	,	,	PUNCT
ejpam-4788	256	8	k	k	PROPN
ejpam-4788	256	9	is	be	AUX
ejpam-4788	256	10	an	an	DET
ejpam-4788	256	11	(	(	PUNCT
ejpam-4788	256	12	α	α	NOUN
ejpam-4788	256	13	,	,	PUNCT
ejpam-4788	256	14	β)-quasi	β)-quasi	NOUN
ejpam-4788	256	15	-	-	NOUN
ejpam-4788	256	16	ideal	ideal	NOUN
ejpam-4788	256	17	of	of	ADP
ejpam-4788	256	18	s.	s.	PROPN
ejpam-4788	256	19	4	4	NUM
ejpam-4788	256	20	.	.	PUNCT
ejpam-4788	257	1	new	new	ADJ
ejpam-4788	257	2	types	type	NOUN
ejpam-4788	257	3	of	of	ADP
ejpam-4788	257	4	bipolar	bipolar	ADJ
ejpam-4788	257	5	fuzzy	fuzzy	ADJ
ejpam-4788	257	6	almost	almost	ADV
ejpam-4788	257	7	ideals	ideal	VERB
ejpam-4788	257	8	definition	definition	NOUN
ejpam-4788	257	9	13	13	NUM
ejpam-4788	257	10	.	.	PUNCT
ejpam-4788	258	1	let	let	VERB
ejpam-4788	258	2	ξ	ξ	X
ejpam-4788	258	3	=	=	SYM
ejpam-4788	258	4	(	(	PUNCT
ejpam-4788	258	5	s	s	PROPN
ejpam-4788	258	6	;	;	PUNCT
ejpam-4788	258	7	ξp	ξp	NUM
ejpam-4788	258	8	,	,	PUNCT
ejpam-4788	258	9	ξn	ξn	NOUN
ejpam-4788	258	10	)	)	PUNCT
ejpam-4788	258	11	be	be	VERB
ejpam-4788	258	12	a	a	DET
ejpam-4788	258	13	bf	bf	NOUN
ejpam-4788	258	14	set	set	NOUN
ejpam-4788	258	15	of	of	ADP
ejpam-4788	258	16	a	a	DET
ejpam-4788	258	17	γ	γ	NOUN
ejpam-4788	258	18	-	-	PUNCT
ejpam-4788	258	19	semigroup	semigroup	NOUN
ejpam-4788	258	20	s	s	PROPN
ejpam-4788	258	21	,	,	PUNCT
ejpam-4788	258	22	and	and	CCONJ
ejpam-4788	258	23	α	α	NOUN
ejpam-4788	258	24	,	,	PUNCT
ejpam-4788	258	25	β	β	PROPN
ejpam-4788	258	26	∈	∈	NOUN
ejpam-4788	258	27	γ	γ	NOUN
ejpam-4788	258	28	is	be	AUX
ejpam-4788	258	29	said	say	VERB
ejpam-4788	258	30	to	to	PART
ejpam-4788	258	31	be	be	AUX
ejpam-4788	258	32	[	[	PUNCT
ejpam-4788	258	33	(	(	PUNCT
ejpam-4788	258	34	i	i	NOUN
ejpam-4788	258	35	)	)	PUNCT
ejpam-4788	258	36	]	]	PUNCT
ejpam-4788	259	1	(	(	PUNCT
ejpam-4788	259	2	i	i	NOUN
ejpam-4788	259	3	)	)	PUNCT
ejpam-4788	259	4	a	a	DET
ejpam-4788	259	5	bf	bf	NOUN
ejpam-4788	259	6	almost	almost	ADV
ejpam-4788	259	7	left	leave	VERB
ejpam-4788	259	8	α	α	NOUN
ejpam-4788	259	9	-	-	NOUN
ejpam-4788	259	10	ideal	ideal	NOUN
ejpam-4788	259	11	of	of	ADP
ejpam-4788	259	12	s	s	PRON
ejpam-4788	259	13	if	if	SCONJ
ejpam-4788	259	14	(	(	PUNCT
ejpam-4788	259	15	xpt	xpt	PROPN
ejpam-4788	259	16	◦	◦	PROPN
ejpam-4788	259	17	α	α	PROPN
ejpam-4788	259	18	ξp	ξp	NOUN
ejpam-4788	259	19	)	)	PUNCT
ejpam-4788	259	20	∧	∧	PROPN
ejpam-4788	259	21	ξp	ξp	ADP
ejpam-4788	259	22	̸=	̸=	PROPN
ejpam-4788	259	23	0	0	NUM
ejpam-4788	260	1	and	and	CCONJ
ejpam-4788	260	2	(	(	PUNCT
ejpam-4788	260	3	xns	xns	PROPN
ejpam-4788	260	4	◦	◦	PROPN
ejpam-4788	260	5	α	α	PROPN
ejpam-4788	260	6	ξn	ξn	NOUN
ejpam-4788	260	7	)	)	PUNCT
ejpam-4788	260	8	∨	∨	PROPN
ejpam-4788	260	9	ξn	ξn	PROPN
ejpam-4788	260	10	̸=	̸=	PROPN
ejpam-4788	260	11	0	0	NUM
ejpam-4788	260	12	.	.	PUNCT
ejpam-4788	261	1	(	(	PUNCT
ejpam-4788	261	2	ii	ii	NOUN
ejpam-4788	261	3	)	)	PUNCT
ejpam-4788	261	4	a	a	DET
ejpam-4788	261	5	bf	bf	NOUN
ejpam-4788	261	6	almost	almost	ADV
ejpam-4788	261	7	right	right	ADJ
ejpam-4788	261	8	β	β	NOUN
ejpam-4788	261	9	-	-	NOUN
ejpam-4788	261	10	ideal	ideal	NOUN
ejpam-4788	261	11	of	of	ADP
ejpam-4788	261	12	s	s	PRON
ejpam-4788	261	13	if	if	SCONJ
ejpam-4788	261	14	ξp	ξp	PART
ejpam-4788	261	15	◦	◦	VERB
ejpam-4788	261	16	β	β	X
ejpam-4788	261	17	(	(	PUNCT
ejpam-4788	261	18	xpt	xpt	PROPN
ejpam-4788	261	19	)	)	PUNCT
ejpam-4788	261	20	∧	∧	PROPN
ejpam-4788	261	21	ξp	ξp	ADP
ejpam-4788	261	22	̸=	̸=	PROPN
ejpam-4788	261	23	0	0	NUM
ejpam-4788	262	1	and	and	CCONJ
ejpam-4788	262	2	(	(	PUNCT
ejpam-4788	262	3	ξn	ξn	PROPN
ejpam-4788	262	4	◦	◦	PROPN
ejpam-4788	262	5	β	β	X
ejpam-4788	262	6	xns	xns	PROPN
ejpam-4788	262	7	)	)	PUNCT
ejpam-4788	262	8	∨	∨	PROPN
ejpam-4788	262	9	ξn	ξn	PROPN
ejpam-4788	262	10	̸=	̸=	PROPN
ejpam-4788	262	11	0	0	NUM
ejpam-4788	262	12	.	.	PUNCT
ejpam-4788	263	1	(	(	PUNCT
ejpam-4788	263	2	iii	iii	X
ejpam-4788	263	3	)	)	PUNCT
ejpam-4788	263	4	a	a	DET
ejpam-4788	263	5	bf	bf	NOUN
ejpam-4788	263	6	almost	almost	ADV
ejpam-4788	263	7	(	(	PUNCT
ejpam-4788	263	8	α	α	NOUN
ejpam-4788	263	9	,	,	PUNCT
ejpam-4788	263	10	β)-ideal	β)-ideal	PUNCT
ejpam-4788	263	11	of	of	ADP
ejpam-4788	263	12	s	s	PRON
ejpam-4788	263	13	if	if	SCONJ
ejpam-4788	263	14	it	it	PRON
ejpam-4788	263	15	is	be	AUX
ejpam-4788	263	16	both	both	CCONJ
ejpam-4788	263	17	a	a	DET
ejpam-4788	263	18	bf	bf	NOUN
ejpam-4788	263	19	almost	almost	ADV
ejpam-4788	263	20	left	leave	VERB
ejpam-4788	263	21	α	α	PRON
ejpam-4788	263	22	-	-	NOUN
ejpam-4788	263	23	ideal	ideal	NOUN
ejpam-4788	263	24	and	and	CCONJ
ejpam-4788	263	25	a	a	DET
ejpam-4788	263	26	bf	bf	NOUN
ejpam-4788	263	27	almost	almost	ADV
ejpam-4788	263	28	right	right	ADJ
ejpam-4788	263	29	β	β	NOUN
ejpam-4788	263	30	-	-	NOUN
ejpam-4788	263	31	ideal	ideal	NOUN
ejpam-4788	263	32	of	of	ADP
ejpam-4788	263	33	s.	s.	PROPN
ejpam-4788	263	34	theorem	theorem	VERB
ejpam-4788	263	35	11	11	NUM
ejpam-4788	263	36	.	.	PUNCT
ejpam-4788	264	1	if	if	SCONJ
ejpam-4788	264	2	ξ	ξ	X
ejpam-4788	264	3	=	=	SYM
ejpam-4788	264	4	(	(	PUNCT
ejpam-4788	264	5	s	s	PROPN
ejpam-4788	264	6	;	;	PUNCT
ejpam-4788	264	7	ξp	ξp	NUM
ejpam-4788	264	8	,	,	PUNCT
ejpam-4788	264	9	ξn	ξn	NOUN
ejpam-4788	264	10	)	)	PUNCT
ejpam-4788	264	11	is	be	AUX
ejpam-4788	264	12	a	a	DET
ejpam-4788	264	13	bf	bf	NOUN
ejpam-4788	264	14	almost	almost	ADV
ejpam-4788	264	15	left	leave	VERB
ejpam-4788	264	16	α	α	NOUN
ejpam-4788	264	17	-	-	NOUN
ejpam-4788	264	18	ideal	ideal	ADJ
ejpam-4788	264	19	(	(	PUNCT
ejpam-4788	264	20	right	right	ADJ
ejpam-4788	264	21	β	β	NOUN
ejpam-4788	264	22	-	-	NOUN
ejpam-4788	264	23	ideal	ideal	ADJ
ejpam-4788	264	24	,	,	PUNCT
ejpam-4788	264	25	(	(	PUNCT
ejpam-4788	264	26	α	α	NOUN
ejpam-4788	264	27	,	,	PUNCT
ejpam-4788	264	28	β)-ideal	β)-ideal	NUM
ejpam-4788	264	29	)	)	PUNCT
ejpam-4788	264	30	of	of	ADP
ejpam-4788	264	31	a	a	DET
ejpam-4788	264	32	γ	γ	NOUN
ejpam-4788	264	33	-	-	PUNCT
ejpam-4788	264	34	semigroup	semigroup	NOUN
ejpam-4788	264	35	s	s	PROPN
ejpam-4788	264	36	,	,	PUNCT
ejpam-4788	264	37	and	and	CCONJ
ejpam-4788	264	38	ς	ς	PROPN
ejpam-4788	264	39	=	=	PUNCT
ejpam-4788	264	40	(	(	PUNCT
ejpam-4788	264	41	s	s	NOUN
ejpam-4788	264	42	;	;	PUNCT
ejpam-4788	264	43	ςp	ςp	NUM
ejpam-4788	264	44	,	,	PUNCT
ejpam-4788	264	45	ςn	ςn	NOUN
ejpam-4788	264	46	)	)	PUNCT
ejpam-4788	264	47	is	be	AUX
ejpam-4788	264	48	a	a	DET
ejpam-4788	264	49	bf	bf	NOUN
ejpam-4788	264	50	set	set	NOUN
ejpam-4788	264	51	of	of	ADP
ejpam-4788	264	52	s	s	PRON
ejpam-4788	264	53	such	such	ADJ
ejpam-4788	264	54	that	that	SCONJ
ejpam-4788	264	55	ξ	ξ	PROPN
ejpam-4788	264	56	⊆	⊆	NUM
ejpam-4788	264	57	ς	ς	NOUN
ejpam-4788	264	58	,	,	PUNCT
ejpam-4788	264	59	then	then	ADV
ejpam-4788	264	60	ς	ς	PROPN
ejpam-4788	264	61	=	=	PUNCT
ejpam-4788	264	62	(	(	PUNCT
ejpam-4788	264	63	s	s	NOUN
ejpam-4788	264	64	;	;	PUNCT
ejpam-4788	264	65	ςp	ςp	NUM
ejpam-4788	264	66	,	,	PUNCT
ejpam-4788	264	67	ςn	ςn	NOUN
ejpam-4788	264	68	)	)	PUNCT
ejpam-4788	264	69	is	be	AUX
ejpam-4788	264	70	a	a	DET
ejpam-4788	264	71	bf	bf	NOUN
ejpam-4788	264	72	left	leave	VERB
ejpam-4788	264	73	almost	almost	ADV
ejpam-4788	264	74	α	α	NOUN
ejpam-4788	264	75	-	-	NOUN
ejpam-4788	264	76	ideal	ideal	ADJ
ejpam-4788	264	77	(	(	PUNCT
ejpam-4788	264	78	right	right	ADJ
ejpam-4788	264	79	β	β	NOUN
ejpam-4788	264	80	-	-	NOUN
ejpam-4788	264	81	ideal	ideal	ADJ
ejpam-4788	264	82	,	,	PUNCT
ejpam-4788	264	83	(	(	PUNCT
ejpam-4788	264	84	α	α	NOUN
ejpam-4788	264	85	,	,	PUNCT
ejpam-4788	264	86	β)-ideal	β)-ideal	NUM
ejpam-4788	264	87	)	)	PUNCT
ejpam-4788	264	88	of	of	ADP
ejpam-4788	264	89	s.	s.	PROPN
ejpam-4788	264	90	proof	proof	PROPN
ejpam-4788	264	91	.	.	PUNCT
ejpam-4788	265	1	suppose	suppose	VERB
ejpam-4788	265	2	that	that	SCONJ
ejpam-4788	265	3	ξ	ξ	PROPN
ejpam-4788	265	4	=	=	SYM
ejpam-4788	265	5	(	(	PUNCT
ejpam-4788	265	6	s	s	PROPN
ejpam-4788	265	7	;	;	PUNCT
ejpam-4788	265	8	ξp	ξp	NUM
ejpam-4788	265	9	,	,	PUNCT
ejpam-4788	265	10	ξn	ξn	NOUN
ejpam-4788	265	11	)	)	PUNCT
ejpam-4788	265	12	is	be	AUX
ejpam-4788	265	13	a	a	DET
ejpam-4788	265	14	bf	bf	NOUN
ejpam-4788	265	15	almost	almost	ADV
ejpam-4788	265	16	left	leave	VERB
ejpam-4788	265	17	α	α	NOUN
ejpam-4788	265	18	-	-	NOUN
ejpam-4788	265	19	ideal	ideal	NOUN
ejpam-4788	265	20	of	of	ADP
ejpam-4788	265	21	s	s	PROPN
ejpam-4788	265	22	,	,	PUNCT
ejpam-4788	265	23	and	and	CCONJ
ejpam-4788	265	24	ς	ς	PROPN
ejpam-4788	265	25	=	=	PUNCT
ejpam-4788	265	26	(	(	PUNCT
ejpam-4788	265	27	s	s	NOUN
ejpam-4788	265	28	;	;	PUNCT
ejpam-4788	265	29	ςp	ςp	NUM
ejpam-4788	265	30	,	,	PUNCT
ejpam-4788	265	31	ςn	ςn	NOUN
ejpam-4788	265	32	)	)	PUNCT
ejpam-4788	265	33	is	be	AUX
ejpam-4788	265	34	a	a	DET
ejpam-4788	265	35	bf	bf	NOUN
ejpam-4788	265	36	set	set	NOUN
ejpam-4788	265	37	of	of	ADP
ejpam-4788	265	38	s	s	PRON
ejpam-4788	265	39	such	such	ADJ
ejpam-4788	265	40	that	that	SCONJ
ejpam-4788	265	41	ξ	ξ	PROPN
ejpam-4788	265	42	⊆	⊆	NUM
ejpam-4788	265	43	ς	ς	X
ejpam-4788	265	44	.	.	PUNCT
ejpam-4788	266	1	then	then	ADV
ejpam-4788	266	2	(	(	PUNCT
ejpam-4788	266	3	xpt	xpt	PROPN
ejpam-4788	266	4	◦	◦	PROPN
ejpam-4788	266	5	α	α	PROPN
ejpam-4788	266	6	ξp	ξp	NOUN
ejpam-4788	266	7	)	)	PUNCT
ejpam-4788	266	8	∧	∧	PROPN
ejpam-4788	266	9	ξp	ξp	ADP
ejpam-4788	266	10	̸=	̸=	PROPN
ejpam-4788	266	11	0	0	NUM
ejpam-4788	267	1	and	and	CCONJ
ejpam-4788	267	2	(	(	PUNCT
ejpam-4788	267	3	xns	xns	PROPN
ejpam-4788	267	4	◦	◦	PROPN
ejpam-4788	267	5	α	α	PROPN
ejpam-4788	267	6	ξn	ξn	NOUN
ejpam-4788	267	7	)	)	PUNCT
ejpam-4788	267	8	∨	∨	PROPN
ejpam-4788	267	9	ξn	ξn	PROPN
ejpam-4788	267	10	̸=	̸=	PROPN
ejpam-4788	267	11	0	0	NUM
ejpam-4788	267	12	.	.	PUNCT
ejpam-4788	268	1	thus	thus	ADV
ejpam-4788	268	2	,	,	PUNCT
ejpam-4788	268	3	(	(	PUNCT
ejpam-4788	268	4	xpt	xpt	PROPN
ejpam-4788	268	5	◦	◦	PROPN
ejpam-4788	268	6	α	α	PROPN
ejpam-4788	268	7	ξp	ξp	NOUN
ejpam-4788	268	8	)	)	PUNCT
ejpam-4788	268	9	∧	∧	NOUN
ejpam-4788	268	10	ξp	ξp	ADP
ejpam-4788	268	11	⊆	⊆	NUM
ejpam-4788	268	12	(	(	PUNCT
ejpam-4788	268	13	xpt	xpt	PROPN
ejpam-4788	268	14	◦	◦	NOUN
ejpam-4788	268	15	α	α	NOUN
ejpam-4788	268	16	ςp	ςp	NOUN
ejpam-4788	268	17	)	)	PUNCT
ejpam-4788	268	18	∧	∧	NOUN
ejpam-4788	268	19	ςp	ςp	ADP
ejpam-4788	268	20	̸=	̸=	PROPN
ejpam-4788	268	21	0	0	NUM
ejpam-4788	269	1	and	and	CCONJ
ejpam-4788	269	2	(	(	PUNCT
ejpam-4788	269	3	xns	xns	PROPN
ejpam-4788	269	4	◦	◦	PROPN
ejpam-4788	269	5	α	α	PROPN
ejpam-4788	269	6	ξn	ξn	NOUN
ejpam-4788	269	7	)	)	PUNCT
ejpam-4788	269	8	∨	∨	NOUN
ejpam-4788	269	9	ξn	ξn	PROPN
ejpam-4788	269	10	⊆	⊆	NUM
ejpam-4788	269	11	(	(	PUNCT
ejpam-4788	269	12	xns	xns	PROPN
ejpam-4788	269	13	◦	◦	PROPN
ejpam-4788	269	14	α	α	PROPN
ejpam-4788	269	15	ςn	ςn	NOUN
ejpam-4788	269	16	)	)	PUNCT
ejpam-4788	269	17	∨	∨	NOUN
ejpam-4788	269	18	ςn	ςn	ADP
ejpam-4788	269	19	̸=	̸=	PROPN
ejpam-4788	269	20	0	0	NUM
ejpam-4788	269	21	.	.	PUNCT
ejpam-4788	270	1	hence	hence	ADV
ejpam-4788	270	2	,	,	PUNCT
ejpam-4788	270	3	ς	ς	PROPN
ejpam-4788	270	4	=	=	PUNCT
ejpam-4788	270	5	(	(	PUNCT
ejpam-4788	270	6	s	s	NOUN
ejpam-4788	270	7	;	;	PUNCT
ejpam-4788	270	8	ςp	ςp	NUM
ejpam-4788	270	9	,	,	PUNCT
ejpam-4788	270	10	ςn	ςn	NOUN
ejpam-4788	270	11	)	)	PUNCT
ejpam-4788	270	12	is	be	AUX
ejpam-4788	270	13	a	a	DET
ejpam-4788	270	14	bf	bf	NOUN
ejpam-4788	270	15	left	leave	VERB
ejpam-4788	270	16	almost	almost	ADV
ejpam-4788	270	17	α	α	NOUN
ejpam-4788	270	18	-	-	NOUN
ejpam-4788	270	19	ideal	ideal	NOUN
ejpam-4788	270	20	of	of	ADP
ejpam-4788	270	21	s.	s.	PROPN
ejpam-4788	270	22	theorem	theorem	VERB
ejpam-4788	270	23	12	12	NUM
ejpam-4788	270	24	.	.	PUNCT
ejpam-4788	271	1	let	let	VERB
ejpam-4788	271	2	k	k	PRON
ejpam-4788	271	3	be	be	AUX
ejpam-4788	271	4	a	a	DET
ejpam-4788	271	5	non	non	ADJ
ejpam-4788	271	6	-	-	ADJ
ejpam-4788	271	7	empty	empty	ADJ
ejpam-4788	271	8	subset	subset	NOUN
ejpam-4788	271	9	of	of	ADP
ejpam-4788	271	10	γ	γ	PROPN
ejpam-4788	271	11	-	-	PUNCT
ejpam-4788	271	12	semigroup	semigroup	PROPN
ejpam-4788	271	13	s.	s.	PROPN
ejpam-4788	272	1	then	then	ADV
ejpam-4788	272	2	k	k	PROPN
ejpam-4788	272	3	is	be	AUX
ejpam-4788	272	4	an	an	DET
ejpam-4788	272	5	almost	almost	ADV
ejpam-4788	272	6	left	leave	VERB
ejpam-4788	272	7	α	α	NOUN
ejpam-4788	272	8	-	-	NOUN
ejpam-4788	272	9	ideal	ideal	ADJ
ejpam-4788	272	10	(	(	PUNCT
ejpam-4788	272	11	right	right	ADJ
ejpam-4788	272	12	β	β	NOUN
ejpam-4788	272	13	-	-	NOUN
ejpam-4788	272	14	ideal	ideal	ADJ
ejpam-4788	272	15	,	,	PUNCT
ejpam-4788	272	16	(	(	PUNCT
ejpam-4788	272	17	α	α	NOUN
ejpam-4788	272	18	,	,	PUNCT
ejpam-4788	272	19	β)-ideal	β)-ideal	NUM
ejpam-4788	272	20	)	)	PUNCT
ejpam-4788	272	21	of	of	ADP
ejpam-4788	272	22	s	s	PRON
ejpam-4788	272	23	if	if	SCONJ
ejpam-4788	272	24	and	and	CCONJ
ejpam-4788	272	25	only	only	ADV
ejpam-4788	272	26	if	if	SCONJ
ejpam-4788	272	27	the	the	DET
ejpam-4788	272	28	characteristic	characteristic	ADJ
ejpam-4788	272	29	function	function	NOUN
ejpam-4788	272	30	λk	λk	X
ejpam-4788	272	31	=	=	PUNCT
ejpam-4788	272	32	(	(	PUNCT
ejpam-4788	272	33	s;λp	s;λp	PROPN
ejpam-4788	272	34	k	k	PROPN
ejpam-4788	272	35	,	,	PUNCT
ejpam-4788	272	36	λn	λn	PROPN
ejpam-4788	272	37	k	k	X
ejpam-4788	272	38	)	)	PUNCT
ejpam-4788	272	39	is	be	AUX
ejpam-4788	272	40	a	a	DET
ejpam-4788	272	41	bf	bf	NOUN
ejpam-4788	272	42	almost	almost	ADV
ejpam-4788	272	43	left	leave	VERB
ejpam-4788	272	44	α	α	NOUN
ejpam-4788	272	45	-	-	NOUN
ejpam-4788	272	46	ideal	ideal	ADJ
ejpam-4788	272	47	(	(	PUNCT
ejpam-4788	272	48	right	right	ADJ
ejpam-4788	272	49	β	β	NOUN
ejpam-4788	272	50	-	-	NOUN
ejpam-4788	272	51	ideal	ideal	ADJ
ejpam-4788	272	52	,	,	PUNCT
ejpam-4788	272	53	(	(	PUNCT
ejpam-4788	272	54	α	α	NOUN
ejpam-4788	272	55	,	,	PUNCT
ejpam-4788	272	56	β)-ideal	β)-ideal	NUM
ejpam-4788	272	57	)	)	PUNCT
ejpam-4788	272	58	of	of	ADP
ejpam-4788	272	59	s.	s.	PROPN
ejpam-4788	272	60	proof	proof	PROPN
ejpam-4788	272	61	.	.	PUNCT
ejpam-4788	273	1	suppose	suppose	VERB
ejpam-4788	273	2	that	that	SCONJ
ejpam-4788	273	3	k	k	PROPN
ejpam-4788	273	4	is	be	AUX
ejpam-4788	273	5	an	an	DET
ejpam-4788	273	6	almost	almost	ADV
ejpam-4788	273	7	left	leave	VERB
ejpam-4788	273	8	α	α	NOUN
ejpam-4788	273	9	-	-	NOUN
ejpam-4788	273	10	ideal	ideal	NOUN
ejpam-4788	273	11	of	of	ADP
ejpam-4788	273	12	s.	s.	PROPN
ejpam-4788	273	13	then	then	ADV
ejpam-4788	273	14	uαk	uαk	ADP
ejpam-4788	273	15	∩k	∩k	PROPN
ejpam-4788	273	16	̸=	̸=	PROPN
ejpam-4788	273	17	∅	∅	NOUN
ejpam-4788	273	18	for	for	ADP
ejpam-4788	273	19	all	all	PRON
ejpam-4788	273	20	u	u	NOUN
ejpam-4788	273	21	∈	∈	PROPN
ejpam-4788	273	22	s.	s.	PROPN
ejpam-4788	273	23	thus	thus	ADV
ejpam-4788	273	24	,	,	PUNCT
ejpam-4788	273	25	there	there	PRON
ejpam-4788	273	26	exists	exist	VERB
ejpam-4788	273	27	v	v	ADP
ejpam-4788	273	28	∈	∈	PROPN
ejpam-4788	273	29	uαk	uαk	NOUN
ejpam-4788	273	30	and	and	CCONJ
ejpam-4788	273	31	v	v	ADP
ejpam-4788	273	32	∈	∈	PROPN
ejpam-4788	273	33	k.	k.	NOUN
ejpam-4788	274	1	so	so	ADV
ejpam-4788	274	2	,	,	PUNCT
ejpam-4788	274	3	(	(	PUNCT
ejpam-4788	274	4	xpt	xpt	PROPN
ejpam-4788	274	5	◦	◦	PROPN
ejpam-4788	274	6	α	α	PROPN
ejpam-4788	274	7	λp	λp	X
ejpam-4788	274	8	k)(v	k)(v	PROPN
ejpam-4788	274	9	)	)	PUNCT
ejpam-4788	275	1	=	=	SYM
ejpam-4788	275	2	λp	λp	PROPN
ejpam-4788	275	3	k(v	k(v	PROPN
ejpam-4788	275	4	)	)	PUNCT
ejpam-4788	275	5	=	=	SYM
ejpam-4788	275	6	1	1	NUM
ejpam-4788	275	7	and	and	CCONJ
ejpam-4788	275	8	(	(	PUNCT
ejpam-4788	275	9	xns	xns	PROPN
ejpam-4788	275	10	◦	◦	PROPN
ejpam-4788	275	11	α	α	NOUN
ejpam-4788	275	12	λn	λn	X
ejpam-4788	275	13	k)(v	k)(v	X
ejpam-4788	275	14	)	)	PUNCT
ejpam-4788	276	1	=	=	SYM
ejpam-4788	276	2	λn	λn	PROPN
ejpam-4788	276	3	k(v	k(v	PROPN
ejpam-4788	276	4	)	)	PUNCT
ejpam-4788	276	5	=	=	PUNCT
ejpam-4788	276	6	−1	−1	NOUN
ejpam-4788	276	7	.	.	PUNCT
ejpam-4788	277	1	hence	hence	ADV
ejpam-4788	277	2	,	,	PUNCT
ejpam-4788	277	3	(	(	PUNCT
ejpam-4788	277	4	xpt	xpt	PROPN
ejpam-4788	277	5	◦	◦	PROPN
ejpam-4788	277	6	α	α	PROPN
ejpam-4788	277	7	λp	λp	X
ejpam-4788	277	8	k	k	NOUN
ejpam-4788	277	9	)	)	PUNCT
ejpam-4788	277	10	∧	∧	PROPN
ejpam-4788	277	11	λp	λp	ADP
ejpam-4788	277	12	k	k	PROPN
ejpam-4788	277	13	̸=	̸=	PROPN
ejpam-4788	277	14	0	0	PUNCT
ejpam-4788	277	15	and	and	CCONJ
ejpam-4788	277	16	(	(	PUNCT
ejpam-4788	277	17	xns	xns	PROPN
ejpam-4788	277	18	◦	◦	PROPN
ejpam-4788	277	19	α	α	NOUN
ejpam-4788	277	20	λn	λn	PROPN
ejpam-4788	277	21	k	k	X
ejpam-4788	277	22	)	)	PUNCT
ejpam-4788	277	23	∨	∨	NUM
ejpam-4788	277	24	λn	λn	PROPN
ejpam-4788	277	25	k	k	PROPN
ejpam-4788	277	26	̸=	̸=	PROPN
ejpam-4788	277	27	0	0	NUM
ejpam-4788	277	28	.	.	PUNCT
ejpam-4788	278	1	therefore	therefore	ADV
ejpam-4788	278	2	,	,	PUNCT
ejpam-4788	278	3	λk	λk	X
ejpam-4788	278	4	=	=	PUNCT
ejpam-4788	278	5	(	(	PUNCT
ejpam-4788	278	6	s;λp	s;λp	PROPN
ejpam-4788	278	7	k	k	PROPN
ejpam-4788	278	8	,	,	PUNCT
ejpam-4788	278	9	λn	λn	PROPN
ejpam-4788	278	10	k	k	X
ejpam-4788	278	11	)	)	PUNCT
ejpam-4788	278	12	is	be	AUX
ejpam-4788	278	13	a	a	DET
ejpam-4788	278	14	bf	bf	NOUN
ejpam-4788	278	15	almost	almost	ADV
ejpam-4788	278	16	left	leave	VERB
ejpam-4788	278	17	α	α	NOUN
ejpam-4788	278	18	-	-	NOUN
ejpam-4788	278	19	ideal	ideal	NOUN
ejpam-4788	278	20	of	of	ADP
ejpam-4788	278	21	s.	s.	PROPN
ejpam-4788	278	22	conversely	conversely	ADV
ejpam-4788	278	23	,	,	PUNCT
ejpam-4788	278	24	assume	assume	VERB
ejpam-4788	278	25	that	that	SCONJ
ejpam-4788	278	26	λk	λk	ADV
ejpam-4788	278	27	=	=	SYM
ejpam-4788	278	28	(	(	PUNCT
ejpam-4788	278	29	s;λp	s;λp	PROPN
ejpam-4788	278	30	k	k	PROPN
ejpam-4788	278	31	,	,	PUNCT
ejpam-4788	278	32	λn	λn	PROPN
ejpam-4788	278	33	k	k	X
ejpam-4788	278	34	)	)	PUNCT
ejpam-4788	278	35	is	be	AUX
ejpam-4788	278	36	a	a	DET
ejpam-4788	278	37	bf	bf	NOUN
ejpam-4788	278	38	almost	almost	ADV
ejpam-4788	278	39	left	leave	VERB
ejpam-4788	278	40	α	α	NOUN
ejpam-4788	278	41	-	-	NOUN
ejpam-4788	278	42	ideal	ideal	NOUN
ejpam-4788	278	43	of	of	ADP
ejpam-4788	278	44	s	s	PRON
ejpam-4788	278	45	and	and	CCONJ
ejpam-4788	278	46	u	u	PROPN
ejpam-4788	278	47	∈	∈	PROPN
ejpam-4788	278	48	s.	s.	PROPN
ejpam-4788	279	1	then	then	ADV
ejpam-4788	279	2	(	(	PUNCT
ejpam-4788	279	3	xpt	xpt	PROPN
ejpam-4788	279	4	◦	◦	PROPN
ejpam-4788	279	5	α	α	PROPN
ejpam-4788	279	6	λp	λp	X
ejpam-4788	279	7	k	k	NOUN
ejpam-4788	279	8	)	)	PUNCT
ejpam-4788	279	9	∧	∧	PROPN
ejpam-4788	279	10	λp	λp	ADP
ejpam-4788	279	11	k	k	PROPN
ejpam-4788	279	12	̸=	̸=	PROPN
ejpam-4788	279	13	0	0	PUNCT
ejpam-4788	280	1	and	and	CCONJ
ejpam-4788	280	2	(	(	PUNCT
ejpam-4788	280	3	xns	xns	PROPN
ejpam-4788	280	4	◦	◦	PROPN
ejpam-4788	280	5	α	α	NOUN
ejpam-4788	280	6	λn	λn	PROPN
ejpam-4788	280	7	k	k	X
ejpam-4788	280	8	)	)	PUNCT
ejpam-4788	280	9	∨	∨	NUM
ejpam-4788	280	10	λn	λn	PROPN
ejpam-4788	280	11	k	k	PROPN
ejpam-4788	280	12	̸=	̸=	PROPN
ejpam-4788	280	13	0	0	NUM
ejpam-4788	280	14	.	.	PUNCT
ejpam-4788	281	1	thus	thus	ADV
ejpam-4788	281	2	,	,	PUNCT
ejpam-4788	281	3	there	there	PRON
ejpam-4788	281	4	exists	exist	VERB
ejpam-4788	281	5	r	r	NOUN
ejpam-4788	281	6	∈	∈	PROPN
ejpam-4788	281	7	s	s	VERB
ejpam-4788	281	8	such	such	ADJ
ejpam-4788	281	9	that	that	SCONJ
ejpam-4788	281	10	(	(	PUNCT
ejpam-4788	281	11	(	(	PUNCT
ejpam-4788	281	12	xpt	xpt	PROPN
ejpam-4788	281	13	◦	◦	NOUN
ejpam-4788	281	14	α	α	PROPN
ejpam-4788	281	15	λp	λp	X
ejpam-4788	281	16	k	k	NOUN
ejpam-4788	281	17	)	)	PUNCT
ejpam-4788	281	18	∧	∧	NOUN
ejpam-4788	281	19	λp	λp	PRON
ejpam-4788	281	20	k)(r	k)(r	PROPN
ejpam-4788	281	21	)	)	PUNCT
ejpam-4788	281	22	̸=	̸=	PROPN
ejpam-4788	281	23	0	0	NUM
ejpam-4788	282	1	and	and	CCONJ
ejpam-4788	282	2	(	(	PUNCT
ejpam-4788	282	3	(	(	PUNCT
ejpam-4788	282	4	xns	xns	PROPN
ejpam-4788	282	5	◦	◦	NOUN
ejpam-4788	282	6	α	α	NOUN
ejpam-4788	282	7	λn	λn	PROPN
ejpam-4788	282	8	k	k	X
ejpam-4788	282	9	)	)	PUNCT
ejpam-4788	282	10	∨	∨	NOUN
ejpam-4788	282	11	λn	λn	PROPN
ejpam-4788	282	12	k)(r	k)(r	NOUN
ejpam-4788	282	13	)	)	PUNCT
ejpam-4788	282	14	̸=	̸=	PROPN
ejpam-4788	282	15	0	0	NUM
ejpam-4788	282	16	.	.	PUNCT
ejpam-4788	283	1	hence	hence	ADV
ejpam-4788	283	2	,	,	PUNCT
ejpam-4788	283	3	r	r	PROPN
ejpam-4788	283	4	∈	∈	PROPN
ejpam-4788	283	5	uαk	uαk	PROPN
ejpam-4788	283	6	∩	∩	PROPN
ejpam-4788	283	7	k	k	PROPN
ejpam-4788	283	8	implies	imply	VERB
ejpam-4788	283	9	uαk	uαk	ADP
ejpam-4788	283	10	∩k	∩k	PROPN
ejpam-4788	283	11	̸=	̸=	PROPN
ejpam-4788	283	12	∅.	∅.	VERB
ejpam-4788	283	13	therefore	therefore	ADV
ejpam-4788	283	14	,	,	PUNCT
ejpam-4788	283	15	k	k	PROPN
ejpam-4788	283	16	is	be	AUX
ejpam-4788	283	17	an	an	DET
ejpam-4788	283	18	almost	almost	ADV
ejpam-4788	283	19	left	leave	VERB
ejpam-4788	283	20	α	α	NOUN
ejpam-4788	283	21	-	-	NOUN
ejpam-4788	283	22	ideal	ideal	NOUN
ejpam-4788	283	23	of	of	ADP
ejpam-4788	283	24	s.	s.	PROPN
ejpam-4788	283	25	next	next	ADV
ejpam-4788	283	26	,	,	PUNCT
ejpam-4788	283	27	we	we	PRON
ejpam-4788	283	28	review	review	VERB
ejpam-4788	283	29	the	the	DET
ejpam-4788	283	30	definition	definition	NOUN
ejpam-4788	283	31	of	of	ADP
ejpam-4788	283	32	supp(ξ	supp(ξ	PROPN
ejpam-4788	283	33	)	)	PUNCT
ejpam-4788	283	34	,	,	PUNCT
ejpam-4788	283	35	and	and	CCONJ
ejpam-4788	283	36	we	we	PRON
ejpam-4788	283	37	study	study	VERB
ejpam-4788	283	38	the	the	DET
ejpam-4788	283	39	properties	property	NOUN
ejpam-4788	283	40	between	between	ADP
ejpam-4788	283	41	supp(ξ	supp(ξ	PROPN
ejpam-4788	283	42	)	)	PUNCT
ejpam-4788	283	43	and	and	CCONJ
ejpam-4788	283	44	bf	bf	NOUN
ejpam-4788	283	45	almost	almost	ADV
ejpam-4788	283	46	left	leave	VERB
ejpam-4788	283	47	α	α	NOUN
ejpam-4788	283	48	-	-	NOUN
ejpam-4788	283	49	ideal	ideal	ADJ
ejpam-4788	283	50	(	(	PUNCT
ejpam-4788	283	51	right	right	ADJ
ejpam-4788	283	52	β	β	NOUN
ejpam-4788	283	53	-	-	NOUN
ejpam-4788	283	54	ideal	ideal	ADJ
ejpam-4788	283	55	,	,	PUNCT
ejpam-4788	283	56	(	(	PUNCT
ejpam-4788	283	57	α	α	NOUN
ejpam-4788	283	58	,	,	PUNCT
ejpam-4788	283	59	β)-ideal	β)-ideal	NUM
ejpam-4788	283	60	)	)	PUNCT
ejpam-4788	283	61	of	of	ADP
ejpam-4788	283	62	γ	γ	NOUN
ejpam-4788	283	63	-	-	PUNCT
ejpam-4788	283	64	semigroups	semigroup	NOUN
ejpam-4788	283	65	.	.	PUNCT
ejpam-4788	284	1	let	let	VERB
ejpam-4788	284	2	ξ	ξ	X
ejpam-4788	284	3	=	=	SYM
ejpam-4788	284	4	(	(	PUNCT
ejpam-4788	284	5	s	s	PROPN
ejpam-4788	284	6	;	;	PUNCT
ejpam-4788	284	7	ξp	ξp	NUM
ejpam-4788	284	8	,	,	PUNCT
ejpam-4788	284	9	ξn	ξn	NOUN
ejpam-4788	284	10	)	)	PUNCT
ejpam-4788	284	11	be	be	VERB
ejpam-4788	284	12	a	a	DET
ejpam-4788	284	13	bf	bf	NOUN
ejpam-4788	284	14	set	set	NOUN
ejpam-4788	284	15	of	of	ADP
ejpam-4788	284	16	a	a	DET
ejpam-4788	284	17	non	non	ADJ
ejpam-4788	284	18	-	-	ADJ
ejpam-4788	284	19	empty	empty	ADJ
ejpam-4788	284	20	of	of	ADP
ejpam-4788	284	21	s.	s.	PROPN
ejpam-4788	284	22	then	then	ADV
ejpam-4788	284	23	the	the	DET
ejpam-4788	284	24	support	support	NOUN
ejpam-4788	284	25	of	of	ADP
ejpam-4788	284	26	ξ	ξ	PROPN
ejpam-4788	284	27	instead	instead	ADV
ejpam-4788	284	28	of	of	ADP
ejpam-4788	284	29	supp(ξ	supp(ξ	PROPN
ejpam-4788	284	30	)	)	PUNCT
ejpam-4788	284	31	=	=	PRON
ejpam-4788	284	32	{	{	PUNCT
ejpam-4788	284	33	u	u	NOUN
ejpam-4788	284	34	∈	∈	PROPN
ejpam-4788	284	35	s	s	PART
ejpam-4788	284	36	|	|	ADV
ejpam-4788	284	37	ξ(u	ξ(u	NOUN
ejpam-4788	284	38	)	)	PUNCT
ejpam-4788	284	39	̸=	̸=	NOUN
ejpam-4788	284	40	0	0	NUM
ejpam-4788	284	41	}	}	PUNCT
ejpam-4788	284	42	where	where	SCONJ
ejpam-4788	284	43	ξp(u	ξp(u	NOUN
ejpam-4788	284	44	)	)	PUNCT
ejpam-4788	284	45	̸=	̸=	PROPN
ejpam-4788	284	46	0	0	NUM
ejpam-4788	284	47	and	and	CCONJ
ejpam-4788	284	48	ξn(u	ξn(u	NOUN
ejpam-4788	284	49	)	)	PUNCT
ejpam-4788	284	50	̸=	̸=	NOUN
ejpam-4788	284	51	0	0	NUM
ejpam-4788	284	52	for	for	ADP
ejpam-4788	284	53	all	all	PRON
ejpam-4788	284	54	u	u	PROPN
ejpam-4788	284	55	∈	∈	PROPN
ejpam-4788	284	56	s.	s.	PROPN
ejpam-4788	284	57	p.	p.	PROPN
ejpam-4788	284	58	khamrot	khamrot	PROPN
ejpam-4788	284	59	,	,	PUNCT
ejpam-4788	284	60	t.	t.	PROPN
ejpam-4788	284	61	gaketem	gaketem	PROPN
ejpam-4788	284	62	/	/	SYM
ejpam-4788	284	63	eur	eur	PROPN
ejpam-4788	284	64	.	.	PUNCT
ejpam-4788	285	1	j.	j.	PROPN
ejpam-4788	285	2	pure	pure	PROPN
ejpam-4788	285	3	appl	appl	PROPN
ejpam-4788	285	4	.	.	PROPN
ejpam-4788	285	5	math	math	PROPN
ejpam-4788	285	6	,	,	PUNCT
ejpam-4788	285	7	16	16	NUM
ejpam-4788	285	8	(	(	PUNCT
ejpam-4788	285	9	3	3	NUM
ejpam-4788	285	10	)	)	PUNCT
ejpam-4788	285	11	(	(	PUNCT
ejpam-4788	285	12	2023	2023	NUM
ejpam-4788	285	13	)	)	PUNCT
ejpam-4788	285	14	,	,	PUNCT
ejpam-4788	285	15	1592	1592	NUM
ejpam-4788	285	16	-	-	SYM
ejpam-4788	285	17	1607	1607	NUM
ejpam-4788	285	18	1602	1602	NUM
ejpam-4788	285	19	theorem	theorem	VERB
ejpam-4788	285	20	13	13	NUM
ejpam-4788	285	21	.	.	PUNCT
ejpam-4788	286	1	let	let	VERB
ejpam-4788	286	2	ξ	ξ	X
ejpam-4788	286	3	=	=	SYM
ejpam-4788	286	4	(	(	PUNCT
ejpam-4788	286	5	s	s	PROPN
ejpam-4788	286	6	;	;	PUNCT
ejpam-4788	286	7	ξp	ξp	NUM
ejpam-4788	286	8	,	,	PUNCT
ejpam-4788	286	9	ξn	ξn	NOUN
ejpam-4788	286	10	)	)	PUNCT
ejpam-4788	286	11	be	be	VERB
ejpam-4788	286	12	a	a	DET
ejpam-4788	286	13	bf	bf	NOUN
ejpam-4788	286	14	set	set	NOUN
ejpam-4788	286	15	of	of	ADP
ejpam-4788	286	16	a	a	DET
ejpam-4788	286	17	non	non	ADJ
ejpam-4788	286	18	-	-	ADJ
ejpam-4788	286	19	empty	empty	ADJ
ejpam-4788	286	20	of	of	ADP
ejpam-4788	286	21	a	a	DET
ejpam-4788	286	22	γ	γ	PROPN
ejpam-4788	286	23	-	-	PUNCT
ejpam-4788	286	24	semigroup	semigroup	PROPN
ejpam-4788	286	25	s.	s.	PROPN
ejpam-4788	286	26	then	then	ADV
ejpam-4788	286	27	ξ	ξ	X
ejpam-4788	286	28	=	=	SYM
ejpam-4788	286	29	(	(	PUNCT
ejpam-4788	286	30	s	s	PROPN
ejpam-4788	286	31	;	;	PUNCT
ejpam-4788	286	32	ξp	ξp	NUM
ejpam-4788	286	33	,	,	PUNCT
ejpam-4788	286	34	ξn	ξn	NOUN
ejpam-4788	286	35	)	)	PUNCT
ejpam-4788	286	36	is	be	AUX
ejpam-4788	286	37	a	a	DET
ejpam-4788	286	38	bf	bf	NOUN
ejpam-4788	286	39	almost	almost	ADV
ejpam-4788	286	40	left	leave	VERB
ejpam-4788	286	41	α	α	NOUN
ejpam-4788	286	42	-	-	NOUN
ejpam-4788	286	43	ideal	ideal	ADJ
ejpam-4788	286	44	(	(	PUNCT
ejpam-4788	286	45	right	right	ADJ
ejpam-4788	286	46	β	β	NOUN
ejpam-4788	286	47	-	-	NOUN
ejpam-4788	286	48	ideal	ideal	ADJ
ejpam-4788	286	49	,	,	PUNCT
ejpam-4788	286	50	(	(	PUNCT
ejpam-4788	286	51	α	α	NOUN
ejpam-4788	286	52	,	,	PUNCT
ejpam-4788	286	53	β)-ideal	β)-ideal	NUM
ejpam-4788	286	54	)	)	PUNCT
ejpam-4788	286	55	of	of	ADP
ejpam-4788	286	56	s	s	PRON
ejpam-4788	286	57	if	if	SCONJ
ejpam-4788	287	1	and	and	CCONJ
ejpam-4788	287	2	only	only	ADV
ejpam-4788	287	3	if	if	SCONJ
ejpam-4788	287	4	supp(ξ	supp(ξ	PROPN
ejpam-4788	287	5	)	)	PUNCT
ejpam-4788	287	6	is	be	AUX
ejpam-4788	287	7	an	an	DET
ejpam-4788	287	8	almost	almost	ADV
ejpam-4788	287	9	left	leave	VERB
ejpam-4788	287	10	α	α	NOUN
ejpam-4788	287	11	-	-	NOUN
ejpam-4788	287	12	ideal	ideal	ADJ
ejpam-4788	287	13	(	(	PUNCT
ejpam-4788	287	14	right	right	ADJ
ejpam-4788	287	15	β	β	NOUN
ejpam-4788	287	16	-	-	NOUN
ejpam-4788	287	17	ideal	ideal	ADJ
ejpam-4788	287	18	,	,	PUNCT
ejpam-4788	287	19	(	(	PUNCT
ejpam-4788	287	20	α	α	NOUN
ejpam-4788	287	21	,	,	PUNCT
ejpam-4788	287	22	β)-ideal	β)-ideal	NUM
ejpam-4788	287	23	)	)	PUNCT
ejpam-4788	287	24	of	of	ADP
ejpam-4788	287	25	s.	s.	PROPN
ejpam-4788	287	26	proof	proof	PROPN
ejpam-4788	287	27	.	.	PUNCT
ejpam-4788	288	1	let	let	VERB
ejpam-4788	288	2	ξ	ξ	X
ejpam-4788	288	3	=	=	SYM
ejpam-4788	288	4	(	(	PUNCT
ejpam-4788	288	5	s	s	PROPN
ejpam-4788	288	6	;	;	PUNCT
ejpam-4788	288	7	ξp	ξp	NUM
ejpam-4788	288	8	,	,	PUNCT
ejpam-4788	288	9	ξn	ξn	NOUN
ejpam-4788	288	10	)	)	PUNCT
ejpam-4788	288	11	be	be	VERB
ejpam-4788	288	12	a	a	DET
ejpam-4788	288	13	bf	bf	NOUN
ejpam-4788	288	14	almost	almost	ADV
ejpam-4788	288	15	left	leave	VERB
ejpam-4788	288	16	α	α	NOUN
ejpam-4788	288	17	-	-	NOUN
ejpam-4788	288	18	ideal	ideal	NOUN
ejpam-4788	288	19	of	of	ADP
ejpam-4788	288	20	s	s	PRON
ejpam-4788	288	21	and	and	CCONJ
ejpam-4788	288	22	u	u	PROPN
ejpam-4788	288	23	∈	∈	PROPN
ejpam-4788	288	24	s.	s.	PROPN
ejpam-4788	288	25	then	then	ADV
ejpam-4788	288	26	(	(	PUNCT
ejpam-4788	288	27	xpt	xpt	PROPN
ejpam-4788	288	28	◦	◦	PROPN
ejpam-4788	288	29	α	α	PROPN
ejpam-4788	288	30	ξp	ξp	NOUN
ejpam-4788	288	31	)	)	PUNCT
ejpam-4788	288	32	∧	∧	PROPN
ejpam-4788	288	33	ξp	ξp	ADP
ejpam-4788	288	34	̸=	̸=	PROPN
ejpam-4788	288	35	0	0	NUM
ejpam-4788	289	1	and	and	CCONJ
ejpam-4788	289	2	(	(	PUNCT
ejpam-4788	289	3	xns	xns	PROPN
ejpam-4788	289	4	◦	◦	PROPN
ejpam-4788	289	5	α	α	PROPN
ejpam-4788	289	6	ξn	ξn	NOUN
ejpam-4788	289	7	)	)	PUNCT
ejpam-4788	289	8	∨	∨	PROPN
ejpam-4788	289	9	ξn	ξn	PROPN
ejpam-4788	289	10	̸=	̸=	PROPN
ejpam-4788	289	11	0	0	NUM
ejpam-4788	289	12	.	.	PUNCT
ejpam-4788	290	1	thus	thus	ADV
ejpam-4788	290	2	,	,	PUNCT
ejpam-4788	290	3	there	there	PRON
ejpam-4788	290	4	exists	exist	VERB
ejpam-4788	290	5	r	r	NOUN
ejpam-4788	290	6	∈	∈	PROPN
ejpam-4788	290	7	s	s	VERB
ejpam-4788	290	8	such	such	ADJ
ejpam-4788	290	9	that	that	SCONJ
ejpam-4788	290	10	(	(	PUNCT
ejpam-4788	290	11	(	(	PUNCT
ejpam-4788	290	12	xpt	xpt	PROPN
ejpam-4788	290	13	◦	◦	PROPN
ejpam-4788	290	14	α	α	PROPN
ejpam-4788	290	15	ξp	ξp	NOUN
ejpam-4788	290	16	)	)	PUNCT
ejpam-4788	290	17	∧	∧	PROPN
ejpam-4788	290	18	ξp)(r	ξp)(r	PROPN
ejpam-4788	290	19	)	)	PUNCT
ejpam-4788	290	20	̸=	̸=	PROPN
ejpam-4788	290	21	0	0	NUM
ejpam-4788	290	22	and	and	CCONJ
ejpam-4788	290	23	(	(	PUNCT
ejpam-4788	290	24	(	(	PUNCT
ejpam-4788	290	25	xns	xns	PROPN
ejpam-4788	290	26	◦	◦	PROPN
ejpam-4788	290	27	α	α	PROPN
ejpam-4788	290	28	ξn	ξn	PROPN
ejpam-4788	290	29	)	)	PUNCT
ejpam-4788	290	30	∨	∨	PROPN
ejpam-4788	290	31	ξn)(r	ξn)(r	PROPN
ejpam-4788	290	32	)	)	PUNCT
ejpam-4788	291	1	̸=	̸=	PROPN
ejpam-4788	291	2	0	0	NUM
ejpam-4788	291	3	.	.	PUNCT
ejpam-4788	292	1	so	so	ADV
ejpam-4788	292	2	,	,	PUNCT
ejpam-4788	292	3	there	there	PRON
ejpam-4788	292	4	exists	exist	VERB
ejpam-4788	292	5	k	k	PROPN
ejpam-4788	292	6	∈	∈	PROPN
ejpam-4788	292	7	s	s	VERB
ejpam-4788	292	8	such	such	ADJ
ejpam-4788	292	9	that	that	DET
ejpam-4788	292	10	r	r	NOUN
ejpam-4788	292	11	=	=	SYM
ejpam-4788	292	12	uαk	uαk	PROPN
ejpam-4788	292	13	,	,	PUNCT
ejpam-4788	292	14	xpt	xpt	PROPN
ejpam-4788	292	15	(	(	PUNCT
ejpam-4788	292	16	r	r	NOUN
ejpam-4788	292	17	)	)	PUNCT
ejpam-4788	292	18	̸=	̸=	PROPN
ejpam-4788	292	19	0	0	NUM
ejpam-4788	292	20	,	,	PUNCT
ejpam-4788	292	21	xns	xns	X
ejpam-4788	292	22	(	(	PUNCT
ejpam-4788	292	23	r	r	NOUN
ejpam-4788	292	24	)	)	PUNCT
ejpam-4788	292	25	̸=	̸=	PROPN
ejpam-4788	292	26	0	0	NUM
ejpam-4788	292	27	and	and	CCONJ
ejpam-4788	292	28	xpt	xpt	PROPN
ejpam-4788	292	29	(	(	PUNCT
ejpam-4788	292	30	k	k	X
ejpam-4788	292	31	)	)	PUNCT
ejpam-4788	292	32	̸=	̸=	PROPN
ejpam-4788	292	33	0	0	NUM
ejpam-4788	292	34	,	,	PUNCT
ejpam-4788	292	35	xns	xns	PROPN
ejpam-4788	292	36	(	(	PUNCT
ejpam-4788	292	37	k	k	NOUN
ejpam-4788	292	38	)	)	PUNCT
ejpam-4788	292	39	̸=	̸=	PROPN
ejpam-4788	292	40	0	0	NUM
ejpam-4788	292	41	.	.	PUNCT
ejpam-4788	293	1	it	it	PRON
ejpam-4788	293	2	implies	imply	VERB
ejpam-4788	293	3	that	that	SCONJ
ejpam-4788	293	4	r	r	NOUN
ejpam-4788	293	5	,	,	PUNCT
ejpam-4788	293	6	k	k	PROPN
ejpam-4788	293	7	∈	∈	PROPN
ejpam-4788	293	8	supp(ξ	supp(ξ	PROPN
ejpam-4788	293	9	)	)	PUNCT
ejpam-4788	293	10	.	.	PUNCT
ejpam-4788	294	1	thus	thus	ADV
ejpam-4788	294	2	,	,	PUNCT
ejpam-4788	294	3	(	(	PUNCT
ejpam-4788	294	4	xpt	xpt	PROPN
ejpam-4788	294	5	◦	◦	PROPN
ejpam-4788	294	6	α	α	PROPN
ejpam-4788	294	7	λ	λ	X
ejpam-4788	294	8	p	p	PROPN
ejpam-4788	294	9	supp(ξ))(r	supp(ξ))(r	PROPN
ejpam-4788	294	10	)	)	PUNCT
ejpam-4788	294	11	̸=	̸=	PROPN
ejpam-4788	294	12	0	0	NUM
ejpam-4788	294	13	,	,	PUNCT
ejpam-4788	294	14	(	(	PUNCT
ejpam-4788	294	15	xns	xns	PROPN
ejpam-4788	294	16	◦	◦	PROPN
ejpam-4788	294	17	α	α	PROPN
ejpam-4788	294	18	λn	λn	PROPN
ejpam-4788	294	19	supp(ξ))(r	supp(ξ))(r	NOUN
ejpam-4788	294	20	)	)	PUNCT
ejpam-4788	294	21	̸=	̸=	PROPN
ejpam-4788	294	22	0	0	NUM
ejpam-4788	294	23	,	,	PUNCT
ejpam-4788	294	24	and	and	CCONJ
ejpam-4788	294	25	λp	λp	ADP
ejpam-4788	294	26	supp(ξ	supp(ξ	PROPN
ejpam-4788	294	27	)	)	PUNCT
ejpam-4788	294	28	̸=	̸=	PROPN
ejpam-4788	294	29	0	0	NUM
ejpam-4788	294	30	,	,	PUNCT
ejpam-4788	294	31	λn	λn	PROPN
ejpam-4788	294	32	supp(ξ	supp(ξ	PROPN
ejpam-4788	294	33	)	)	PUNCT
ejpam-4788	294	34	̸=	̸=	PROPN
ejpam-4788	294	35	0	0	NUM
ejpam-4788	294	36	.	.	PUNCT
ejpam-4788	295	1	hence	hence	ADV
ejpam-4788	295	2	,	,	PUNCT
ejpam-4788	295	3	(	(	PUNCT
ejpam-4788	295	4	xpt	xpt	PROPN
ejpam-4788	295	5	◦	◦	PROPN
ejpam-4788	295	6	α	α	X
ejpam-4788	295	7	λp	λp	X
ejpam-4788	295	8	supp(ξ	supp(ξ	PROPN
ejpam-4788	295	9	)	)	PUNCT
ejpam-4788	295	10	)	)	PUNCT
ejpam-4788	296	1	∧	∧	PROPN
ejpam-4788	296	2	λp	λp	PRON
ejpam-4788	296	3	supp(ξ	supp(ξ	PROPN
ejpam-4788	296	4	)	)	PUNCT
ejpam-4788	296	5	̸=	̸=	PROPN
ejpam-4788	296	6	0	0	NUM
ejpam-4788	296	7	and	and	CCONJ
ejpam-4788	296	8	(	(	PUNCT
ejpam-4788	296	9	xns	xns	PROPN
ejpam-4788	296	10	◦	◦	PROPN
ejpam-4788	296	11	α	α	X
ejpam-4788	296	12	λn	λn	PROPN
ejpam-4788	296	13	supp(ξ	supp(ξ	PROPN
ejpam-4788	296	14	)	)	PUNCT
ejpam-4788	296	15	)	)	PUNCT
ejpam-4788	296	16	∨	∨	NUM
ejpam-4788	296	17	λn	λn	PROPN
ejpam-4788	296	18	supp(ξ	supp(ξ	PROPN
ejpam-4788	296	19	)	)	PUNCT
ejpam-4788	296	20	̸=	̸=	PROPN
ejpam-4788	296	21	0	0	NUM
ejpam-4788	296	22	.	.	PUNCT
ejpam-4788	297	1	therefore	therefore	ADV
ejpam-4788	297	2	,	,	PUNCT
ejpam-4788	297	3	λsupp(ξ	λsupp(ξ	NOUN
ejpam-4788	297	4	)	)	PUNCT
ejpam-4788	297	5	is	be	AUX
ejpam-4788	297	6	a	a	DET
ejpam-4788	297	7	bf	bf	NOUN
ejpam-4788	297	8	almost	almost	ADV
ejpam-4788	297	9	left	leave	VERB
ejpam-4788	297	10	α	α	NOUN
ejpam-4788	297	11	-	-	NOUN
ejpam-4788	297	12	ideal	ideal	NOUN
ejpam-4788	297	13	of	of	ADP
ejpam-4788	297	14	s.	s.	PROPN
ejpam-4788	297	15	this	this	PRON
ejpam-4788	297	16	shows	show	VERB
ejpam-4788	297	17	that	that	SCONJ
ejpam-4788	297	18	supp(ξ	supp(ξ	NOUN
ejpam-4788	297	19	)	)	PUNCT
ejpam-4788	297	20	is	be	AUX
ejpam-4788	297	21	an	an	DET
ejpam-4788	297	22	almost	almost	ADV
ejpam-4788	297	23	left	leave	VERB
ejpam-4788	297	24	α	α	NOUN
ejpam-4788	297	25	-	-	NOUN
ejpam-4788	297	26	ideal	ideal	NOUN
ejpam-4788	297	27	of	of	ADP
ejpam-4788	297	28	s.	s.	PROPN
ejpam-4788	297	29	conversely	conversely	ADV
ejpam-4788	297	30	,	,	PUNCT
ejpam-4788	297	31	let	let	VERB
ejpam-4788	297	32	supp(ξ	supp(ξ	PROPN
ejpam-4788	297	33	)	)	PUNCT
ejpam-4788	297	34	be	be	AUX
ejpam-4788	297	35	an	an	DET
ejpam-4788	297	36	almost	almost	ADV
ejpam-4788	297	37	left	leave	VERB
ejpam-4788	297	38	α	α	NOUN
ejpam-4788	297	39	-	-	NOUN
ejpam-4788	297	40	ideal	ideal	NOUN
ejpam-4788	297	41	of	of	ADP
ejpam-4788	297	42	s.	s.	PROPN
ejpam-4788	297	43	then	then	ADV
ejpam-4788	297	44	,	,	PUNCT
ejpam-4788	297	45	by	by	ADP
ejpam-4788	297	46	theorem	theorem	NOUN
ejpam-4788	297	47	12	12	NUM
ejpam-4788	297	48	,	,	PUNCT
ejpam-4788	297	49	λsupp(ξ	λsupp(ξ	NOUN
ejpam-4788	297	50	)	)	PUNCT
ejpam-4788	297	51	is	be	AUX
ejpam-4788	297	52	a	a	DET
ejpam-4788	297	53	bf	bf	NOUN
ejpam-4788	297	54	almost	almost	ADV
ejpam-4788	297	55	left	leave	VERB
ejpam-4788	297	56	α	α	NOUN
ejpam-4788	297	57	-	-	NOUN
ejpam-4788	297	58	ideal	ideal	NOUN
ejpam-4788	297	59	of	of	ADP
ejpam-4788	297	60	s.	s.	PROPN
ejpam-4788	297	61	thus	thus	ADV
ejpam-4788	297	62	,	,	PUNCT
ejpam-4788	297	63	(	(	PUNCT
ejpam-4788	297	64	xpt	xpt	PROPN
ejpam-4788	297	65	◦	◦	PROPN
ejpam-4788	297	66	α	α	PROPN
ejpam-4788	297	67	λp	λp	X
ejpam-4788	297	68	supp(ξ))∧λp	supp(ξ))∧λp	X
ejpam-4788	297	69	supp(ξ	supp(ξ	PROPN
ejpam-4788	297	70	)	)	PUNCT
ejpam-4788	297	71	̸=	̸=	PROPN
ejpam-4788	297	72	0	0	NUM
ejpam-4788	297	73	and	and	CCONJ
ejpam-4788	297	74	(	(	PUNCT
ejpam-4788	297	75	xns	xns	PROPN
ejpam-4788	297	76	◦	◦	PROPN
ejpam-4788	297	77	α	α	NOUN
ejpam-4788	297	78	λn	λn	NOUN
ejpam-4788	297	79	supp(ξ))∨	supp(ξ))∨	PROPN
ejpam-4788	297	80	λn	λn	PROPN
ejpam-4788	297	81	supp(ξ	supp(ξ	PROPN
ejpam-4788	297	82	)	)	PUNCT
ejpam-4788	297	83	̸=	̸=	PROPN
ejpam-4788	297	84	0	0	NUM
ejpam-4788	297	85	.	.	PUNCT
ejpam-4788	298	1	so	so	ADV
ejpam-4788	298	2	,	,	PUNCT
ejpam-4788	298	3	there	there	PRON
ejpam-4788	298	4	exists	exist	VERB
ejpam-4788	298	5	r	r	NOUN
ejpam-4788	298	6	∈	∈	PROPN
ejpam-4788	298	7	s	s	VERB
ejpam-4788	298	8	such	such	ADJ
ejpam-4788	298	9	that	that	SCONJ
ejpam-4788	298	10	(	(	PUNCT
ejpam-4788	298	11	(	(	PUNCT
ejpam-4788	298	12	xpt	xpt	PROPN
ejpam-4788	298	13	◦	◦	NOUN
ejpam-4788	298	14	α	α	X
ejpam-4788	298	15	λp	λp	X
ejpam-4788	298	16	supp(ξ	supp(ξ	PROPN
ejpam-4788	298	17	)	)	PUNCT
ejpam-4788	298	18	)	)	PUNCT
ejpam-4788	299	1	∧	∧	PROPN
ejpam-4788	299	2	λp	λp	ADP
ejpam-4788	299	3	supp(ξ))(r	supp(ξ))(r	PROPN
ejpam-4788	299	4	)	)	PUNCT
ejpam-4788	299	5	̸=	̸=	PROPN
ejpam-4788	299	6	0	0	NUM
ejpam-4788	300	1	and	and	CCONJ
ejpam-4788	300	2	(	(	PUNCT
ejpam-4788	300	3	(	(	PUNCT
ejpam-4788	300	4	xns	xns	PROPN
ejpam-4788	300	5	◦	◦	NOUN
ejpam-4788	300	6	αλn	αλn	PROPN
ejpam-4788	300	7	supp(ξ))∨λ	supp(ξ))∨λ	PROPN
ejpam-4788	300	8	n	n	PRON
ejpam-4788	300	9	supp(ξ))(r	supp(ξ))(r	PROPN
ejpam-4788	300	10	)	)	PUNCT
ejpam-4788	300	11	̸=	̸=	PROPN
ejpam-4788	300	12	0	0	NUM
ejpam-4788	300	13	.	.	PUNCT
ejpam-4788	301	1	it	it	PRON
ejpam-4788	301	2	implies	imply	VERB
ejpam-4788	301	3	that	that	SCONJ
ejpam-4788	301	4	(	(	PUNCT
ejpam-4788	301	5	xpt	xpt	PROPN
ejpam-4788	301	6	◦	◦	VERB
ejpam-4788	301	7	αλ	αλ	NUM
ejpam-4788	301	8	p	p	PROPN
ejpam-4788	301	9	supp(ξ))(r	supp(ξ))(r	PROPN
ejpam-4788	301	10	)	)	PUNCT
ejpam-4788	301	11	̸=	̸=	PROPN
ejpam-4788	301	12	0	0	NUM
ejpam-4788	301	13	,	,	PUNCT
ejpam-4788	301	14	(	(	PUNCT
ejpam-4788	301	15	xnt	xnt	NOUN
ejpam-4788	301	16	◦	◦	NOUN
ejpam-4788	301	17	αλn	αλn	PROPN
ejpam-4788	301	18	supp(ξ))(r	supp(ξ))(r	PROPN
ejpam-4788	301	19	)	)	PUNCT
ejpam-4788	301	20	̸=	̸=	PROPN
ejpam-4788	301	21	0	0	NUM
ejpam-4788	301	22	and	and	CCONJ
ejpam-4788	301	23	λp	λp	ADJ
ejpam-4788	301	24	k(r	k(r	PROPN
ejpam-4788	301	25	)	)	PUNCT
ejpam-4788	301	26	̸=	̸=	PROPN
ejpam-4788	301	27	0	0	NUM
ejpam-4788	301	28	,	,	PUNCT
ejpam-4788	301	29	λn	λn	PROPN
ejpam-4788	301	30	k(r	k(r	PROPN
ejpam-4788	301	31	)	)	PUNCT
ejpam-4788	301	32	̸=	̸=	PROPN
ejpam-4788	301	33	0	0	NUM
ejpam-4788	301	34	.	.	PUNCT
ejpam-4788	302	1	thus	thus	ADV
ejpam-4788	302	2	,	,	PUNCT
ejpam-4788	302	3	there	there	PRON
ejpam-4788	302	4	exists	exist	VERB
ejpam-4788	302	5	k	k	PROPN
ejpam-4788	302	6	∈	∈	PROPN
ejpam-4788	302	7	s	s	VERB
ejpam-4788	302	8	such	such	ADJ
ejpam-4788	302	9	that	that	DET
ejpam-4788	302	10	r	r	NOUN
ejpam-4788	302	11	=	=	SYM
ejpam-4788	302	12	uαk	uαk	PROPN
ejpam-4788	302	13	,	,	PUNCT
ejpam-4788	302	14	xpt	xpt	PROPN
ejpam-4788	302	15	(	(	PUNCT
ejpam-4788	302	16	r	r	NOUN
ejpam-4788	302	17	)	)	PUNCT
ejpam-4788	302	18	̸=	̸=	PROPN
ejpam-4788	302	19	0	0	NUM
ejpam-4788	302	20	,	,	PUNCT
ejpam-4788	302	21	xns	xns	X
ejpam-4788	302	22	(	(	PUNCT
ejpam-4788	302	23	r	r	NOUN
ejpam-4788	302	24	)	)	PUNCT
ejpam-4788	302	25	̸=	̸=	PROPN
ejpam-4788	302	26	0	0	NUM
ejpam-4788	302	27	and	and	CCONJ
ejpam-4788	302	28	xpt	xpt	PROPN
ejpam-4788	302	29	(	(	PUNCT
ejpam-4788	302	30	k	k	X
ejpam-4788	302	31	)	)	PUNCT
ejpam-4788	302	32	̸=	̸=	PROPN
ejpam-4788	302	33	0	0	NUM
ejpam-4788	302	34	,	,	PUNCT
ejpam-4788	302	35	xns	xns	PROPN
ejpam-4788	302	36	(	(	PUNCT
ejpam-4788	302	37	k	k	NOUN
ejpam-4788	302	38	)	)	PUNCT
ejpam-4788	302	39	̸=	̸=	PROPN
ejpam-4788	302	40	0	0	NUM
ejpam-4788	302	41	.	.	PUNCT
ejpam-4788	303	1	hence	hence	ADV
ejpam-4788	303	2	,	,	PUNCT
ejpam-4788	303	3	(	(	PUNCT
ejpam-4788	303	4	xpt	xpt	PROPN
ejpam-4788	303	5	◦	◦	PROPN
ejpam-4788	303	6	α	α	PROPN
ejpam-4788	303	7	ξp	ξp	NOUN
ejpam-4788	303	8	)	)	PUNCT
ejpam-4788	303	9	∧	∧	PROPN
ejpam-4788	303	10	ξp	ξp	ADP
ejpam-4788	303	11	̸=	̸=	PROPN
ejpam-4788	303	12	0	0	NUM
ejpam-4788	304	1	and	and	CCONJ
ejpam-4788	304	2	(	(	PUNCT
ejpam-4788	304	3	xns	xns	PROPN
ejpam-4788	304	4	◦	◦	PROPN
ejpam-4788	304	5	α	α	PROPN
ejpam-4788	304	6	ξn	ξn	NOUN
ejpam-4788	304	7	)	)	PUNCT
ejpam-4788	304	8	∨	∨	PROPN
ejpam-4788	304	9	ξn	ξn	PROPN
ejpam-4788	304	10	̸=	̸=	PROPN
ejpam-4788	304	11	0	0	NUM
ejpam-4788	304	12	.	.	PUNCT
ejpam-4788	305	1	therefore	therefore	ADV
ejpam-4788	305	2	,	,	PUNCT
ejpam-4788	305	3	ξ	ξ	X
ejpam-4788	305	4	=	=	SYM
ejpam-4788	305	5	(	(	PUNCT
ejpam-4788	305	6	s	s	PROPN
ejpam-4788	305	7	;	;	PUNCT
ejpam-4788	305	8	ξp	ξp	NUM
ejpam-4788	305	9	,	,	PUNCT
ejpam-4788	305	10	ξn	ξn	NOUN
ejpam-4788	305	11	)	)	PUNCT
ejpam-4788	305	12	is	be	AUX
ejpam-4788	305	13	a	a	DET
ejpam-4788	305	14	bf	bf	NOUN
ejpam-4788	305	15	almost	almost	ADV
ejpam-4788	305	16	left	leave	VERB
ejpam-4788	305	17	α	α	NOUN
ejpam-4788	305	18	-	-	NOUN
ejpam-4788	305	19	ideal	ideal	NOUN
ejpam-4788	305	20	of	of	ADP
ejpam-4788	305	21	s.	s.	PROPN
ejpam-4788	305	22	definition	definition	PROPN
ejpam-4788	305	23	14	14	NUM
ejpam-4788	305	24	.	.	PUNCT
ejpam-4788	306	1	an	an	DET
ejpam-4788	306	2	almost	almost	ADV
ejpam-4788	306	3	ideal	ideal	ADJ
ejpam-4788	306	4	i	i	PRON
ejpam-4788	306	5	of	of	ADP
ejpam-4788	306	6	a	a	DET
ejpam-4788	306	7	γ	γ	PROPN
ejpam-4788	306	8	-	-	PUNCT
ejpam-4788	306	9	semigroup	semigroup	NOUN
ejpam-4788	306	10	s	s	VERB
ejpam-4788	306	11	is	be	AUX
ejpam-4788	306	12	called	call	VERB
ejpam-4788	306	13	minimal	minimal	ADJ
ejpam-4788	306	14	if	if	SCONJ
ejpam-4788	306	15	for	for	ADP
ejpam-4788	306	16	every	every	DET
ejpam-4788	306	17	almost	almost	ADV
ejpam-4788	306	18	ideal	ideal	NOUN
ejpam-4788	306	19	of	of	ADP
ejpam-4788	306	20	j	j	PROPN
ejpam-4788	306	21	of	of	ADP
ejpam-4788	306	22	s	s	PRON
ejpam-4788	306	23	such	such	ADJ
ejpam-4788	306	24	that	that	SCONJ
ejpam-4788	306	25	j	j	PROPN
ejpam-4788	306	26	⊆	⊆	NUM
ejpam-4788	306	27	i	i	PROPN
ejpam-4788	306	28	,	,	PUNCT
ejpam-4788	306	29	we	we	PRON
ejpam-4788	306	30	have	have	VERB
ejpam-4788	306	31	j	j	PROPN
ejpam-4788	306	32	=	=	PROPN
ejpam-4788	306	33	i.	i.	PROPN
ejpam-4788	306	34	definition	definition	NOUN
ejpam-4788	306	35	15	15	NUM
ejpam-4788	306	36	.	.	PUNCT
ejpam-4788	307	1	a	a	DET
ejpam-4788	307	2	bf	bf	NOUN
ejpam-4788	307	3	almost	almost	ADV
ejpam-4788	307	4	left	leave	VERB
ejpam-4788	307	5	α	α	NOUN
ejpam-4788	307	6	-	-	NOUN
ejpam-4788	307	7	ideal	ideal	ADJ
ejpam-4788	307	8	(	(	PUNCT
ejpam-4788	307	9	right	right	ADJ
ejpam-4788	307	10	β	β	NOUN
ejpam-4788	307	11	-	-	NOUN
ejpam-4788	307	12	ideal	ideal	ADJ
ejpam-4788	307	13	,	,	PUNCT
ejpam-4788	307	14	(	(	PUNCT
ejpam-4788	307	15	α	α	NOUN
ejpam-4788	307	16	,	,	PUNCT
ejpam-4788	307	17	β)-ideal	β)-ideal	ADJ
ejpam-4788	307	18	)	)	PUNCT
ejpam-4788	307	19	ξ	ξ	X
ejpam-4788	307	20	=	=	SYM
ejpam-4788	307	21	(	(	PUNCT
ejpam-4788	307	22	s	s	PROPN
ejpam-4788	307	23	;	;	PUNCT
ejpam-4788	307	24	ξp	ξp	NUM
ejpam-4788	307	25	,	,	PUNCT
ejpam-4788	307	26	ξn	ξn	NOUN
ejpam-4788	307	27	)	)	PUNCT
ejpam-4788	307	28	of	of	ADP
ejpam-4788	307	29	a	a	DET
ejpam-4788	307	30	γ	γ	PROPN
ejpam-4788	307	31	-	-	PUNCT
ejpam-4788	307	32	semigroup	semigroup	NOUN
ejpam-4788	307	33	s	s	VERB
ejpam-4788	307	34	is	be	AUX
ejpam-4788	307	35	minimal	minimal	ADJ
ejpam-4788	307	36	if	if	SCONJ
ejpam-4788	307	37	for	for	SCONJ
ejpam-4788	307	38	all	all	DET
ejpam-4788	307	39	bf	bf	NOUN
ejpam-4788	307	40	almost	almost	ADV
ejpam-4788	307	41	left	leave	VERB
ejpam-4788	307	42	α	α	PRON
ejpam-4788	307	43	-	-	NOUN
ejpam-4788	307	44	ideal	ideal	ADJ
ejpam-4788	307	45	(	(	PUNCT
ejpam-4788	307	46	right	right	ADJ
ejpam-4788	307	47	β	β	NOUN
ejpam-4788	307	48	-	-	NOUN
ejpam-4788	307	49	ideal	ideal	ADJ
ejpam-4788	307	50	,	,	PUNCT
ejpam-4788	307	51	(	(	PUNCT
ejpam-4788	307	52	α	α	NOUN
ejpam-4788	307	53	,	,	PUNCT
ejpam-4788	307	54	β)-ideal	β)-ideal	ADJ
ejpam-4788	307	55	)	)	PUNCT
ejpam-4788	307	56	ς	ς	PROPN
ejpam-4788	308	1	=	=	PUNCT
ejpam-4788	308	2	(	(	PUNCT
ejpam-4788	308	3	s	s	NOUN
ejpam-4788	308	4	;	;	PUNCT
ejpam-4788	308	5	ςp	ςp	NUM
ejpam-4788	308	6	,	,	PUNCT
ejpam-4788	308	7	ςn	ςn	NOUN
ejpam-4788	308	8	)	)	PUNCT
ejpam-4788	308	9	of	of	ADP
ejpam-4788	308	10	s	s	PRON
ejpam-4788	308	11	such	such	ADJ
ejpam-4788	308	12	that	that	SCONJ
ejpam-4788	308	13	ς	ς	PROPN
ejpam-4788	308	14	⊆	⊆	NUM
ejpam-4788	308	15	ξ	ξ	NUM
ejpam-4788	308	16	,	,	PUNCT
ejpam-4788	308	17	then	then	ADV
ejpam-4788	308	18	supp(ς	supp(ς	NOUN
ejpam-4788	308	19	)	)	PUNCT
ejpam-4788	308	20	=	=	SYM
ejpam-4788	308	21	supp(ξ	supp(ξ	PROPN
ejpam-4788	308	22	)	)	PUNCT
ejpam-4788	308	23	.	.	PUNCT
ejpam-4788	309	1	theorem	theorem	PROPN
ejpam-4788	309	2	14	14	NUM
ejpam-4788	309	3	.	.	PUNCT
ejpam-4788	310	1	let	let	VERB
ejpam-4788	310	2	k	k	PRON
ejpam-4788	310	3	be	be	AUX
ejpam-4788	310	4	a	a	DET
ejpam-4788	310	5	non	non	ADJ
ejpam-4788	310	6	-	-	ADJ
ejpam-4788	310	7	empty	empty	ADJ
ejpam-4788	310	8	subset	subset	NOUN
ejpam-4788	310	9	of	of	ADP
ejpam-4788	310	10	a	a	DET
ejpam-4788	310	11	γ	γ	PROPN
ejpam-4788	310	12	-	-	PUNCT
ejpam-4788	310	13	semigroup	semigroup	PROPN
ejpam-4788	310	14	s.	s.	PROPN
ejpam-4788	311	1	then	then	ADV
ejpam-4788	311	2	k	k	PROPN
ejpam-4788	311	3	is	be	AUX
ejpam-4788	311	4	a	a	DET
ejpam-4788	311	5	minimal	minimal	ADJ
ejpam-4788	311	6	almost	almost	ADV
ejpam-4788	311	7	left	leave	VERB
ejpam-4788	311	8	α	α	NOUN
ejpam-4788	311	9	-	-	NOUN
ejpam-4788	311	10	ideal	ideal	ADJ
ejpam-4788	311	11	(	(	PUNCT
ejpam-4788	311	12	right	right	ADJ
ejpam-4788	311	13	β	β	NOUN
ejpam-4788	311	14	-	-	NOUN
ejpam-4788	311	15	ideal	ideal	ADJ
ejpam-4788	311	16	,	,	PUNCT
ejpam-4788	311	17	(	(	PUNCT
ejpam-4788	311	18	α	α	NOUN
ejpam-4788	311	19	,	,	PUNCT
ejpam-4788	311	20	β)-ideal	β)-ideal	PUNCT
ejpam-4788	311	21	)	)	PUNCT
ejpam-4788	311	22	if	if	SCONJ
ejpam-4788	311	23	and	and	CCONJ
ejpam-4788	311	24	only	only	ADV
ejpam-4788	311	25	if	if	SCONJ
ejpam-4788	311	26	λk	λk	X
ejpam-4788	311	27	=	=	SYM
ejpam-4788	311	28	(	(	PUNCT
ejpam-4788	311	29	s;λp	s;λp	PROPN
ejpam-4788	311	30	k	k	PROPN
ejpam-4788	311	31	,	,	PUNCT
ejpam-4788	311	32	λn	λn	PROPN
ejpam-4788	311	33	k	k	X
ejpam-4788	311	34	)	)	PUNCT
ejpam-4788	311	35	is	be	AUX
ejpam-4788	311	36	a	a	DET
ejpam-4788	311	37	minimal	minimal	ADJ
ejpam-4788	311	38	bf	bf	NOUN
ejpam-4788	311	39	almost	almost	ADV
ejpam-4788	311	40	left	leave	VERB
ejpam-4788	311	41	α	α	NOUN
ejpam-4788	311	42	-	-	NOUN
ejpam-4788	311	43	ideal	ideal	ADJ
ejpam-4788	311	44	(	(	PUNCT
ejpam-4788	311	45	right	right	ADJ
ejpam-4788	311	46	β	β	NOUN
ejpam-4788	311	47	-	-	NOUN
ejpam-4788	311	48	ideal	ideal	ADJ
ejpam-4788	311	49	,	,	PUNCT
ejpam-4788	311	50	(	(	PUNCT
ejpam-4788	311	51	α	α	NOUN
ejpam-4788	311	52	,	,	PUNCT
ejpam-4788	311	53	β)-ideal	β)-ideal	NUM
ejpam-4788	311	54	)	)	PUNCT
ejpam-4788	311	55	of	of	ADP
ejpam-4788	311	56	s.	s.	PROPN
ejpam-4788	311	57	proof	proof	PROPN
ejpam-4788	311	58	.	.	PUNCT
ejpam-4788	312	1	suppose	suppose	VERB
ejpam-4788	312	2	that	that	SCONJ
ejpam-4788	312	3	k	k	PROPN
ejpam-4788	312	4	is	be	AUX
ejpam-4788	312	5	a	a	DET
ejpam-4788	312	6	minimal	minimal	ADJ
ejpam-4788	312	7	almost	almost	ADV
ejpam-4788	312	8	left	leave	VERB
ejpam-4788	312	9	α	α	NOUN
ejpam-4788	312	10	-	-	NOUN
ejpam-4788	312	11	ideal	ideal	NOUN
ejpam-4788	312	12	of	of	ADP
ejpam-4788	312	13	s.	s.	PROPN
ejpam-4788	312	14	then	then	ADV
ejpam-4788	312	15	k	k	PROPN
ejpam-4788	312	16	is	be	AUX
ejpam-4788	312	17	an	an	DET
ejpam-4788	312	18	almost	almost	ADV
ejpam-4788	312	19	left	leave	VERB
ejpam-4788	312	20	α	α	NOUN
ejpam-4788	312	21	-	-	NOUN
ejpam-4788	312	22	ideal	ideal	NOUN
ejpam-4788	312	23	of	of	ADP
ejpam-4788	312	24	s.	s.	PROPN
ejpam-4788	312	25	thus	thus	ADV
ejpam-4788	312	26	,	,	PUNCT
ejpam-4788	312	27	by	by	ADP
ejpam-4788	312	28	theorem	theorem	NOUN
ejpam-4788	312	29	12	12	NUM
ejpam-4788	312	30	,	,	PUNCT
ejpam-4788	312	31	λk	λk	NOUN
ejpam-4788	312	32	=	=	SYM
ejpam-4788	312	33	(	(	PUNCT
ejpam-4788	312	34	s;λp	s;λp	PROPN
ejpam-4788	312	35	k	k	PROPN
ejpam-4788	312	36	,	,	PUNCT
ejpam-4788	312	37	λn	λn	PROPN
ejpam-4788	312	38	k	k	X
ejpam-4788	312	39	)	)	PUNCT
ejpam-4788	312	40	is	be	AUX
ejpam-4788	312	41	a	a	DET
ejpam-4788	312	42	bf	bf	NOUN
ejpam-4788	312	43	left	leave	VERB
ejpam-4788	312	44	α	α	NOUN
ejpam-4788	312	45	-	-	NOUN
ejpam-4788	312	46	ideal	ideal	NOUN
ejpam-4788	312	47	of	of	ADP
ejpam-4788	312	48	s.	s.	PROPN
ejpam-4788	312	49	let	let	VERB
ejpam-4788	312	50	ς	ς	PROPN
ejpam-4788	312	51	=	=	PUNCT
ejpam-4788	312	52	(	(	PUNCT
ejpam-4788	312	53	s	s	NOUN
ejpam-4788	312	54	;	;	PUNCT
ejpam-4788	312	55	ςp	ςp	NUM
ejpam-4788	312	56	,	,	PUNCT
ejpam-4788	312	57	ςn	ςn	NOUN
ejpam-4788	312	58	)	)	PUNCT
ejpam-4788	312	59	be	be	VERB
ejpam-4788	312	60	a	a	DET
ejpam-4788	312	61	bf	bf	NOUN
ejpam-4788	312	62	left	leave	VERB
ejpam-4788	312	63	α	α	NOUN
ejpam-4788	312	64	-	-	NOUN
ejpam-4788	312	65	ideal	ideal	NOUN
ejpam-4788	312	66	of	of	ADP
ejpam-4788	312	67	s	s	PRON
ejpam-4788	312	68	such	such	ADJ
ejpam-4788	312	69	that	that	SCONJ
ejpam-4788	312	70	ς	ς	PROPN
ejpam-4788	312	71	⊆	⊆	NUM
ejpam-4788	312	72	ξ	ξ	PROPN
ejpam-4788	312	73	.	.	PUNCT
ejpam-4788	313	1	then	then	ADV
ejpam-4788	313	2	,	,	PUNCT
ejpam-4788	313	3	by	by	ADP
ejpam-4788	313	4	theorem	theorem	NOUN
ejpam-4788	313	5	13	13	NUM
ejpam-4788	313	6	,	,	PUNCT
ejpam-4788	313	7	supp(ς	supp(ς	NOUN
ejpam-4788	313	8	)	)	PUNCT
ejpam-4788	313	9	is	be	AUX
ejpam-4788	313	10	an	an	DET
ejpam-4788	313	11	almost	almost	ADV
ejpam-4788	313	12	left	leave	VERB
ejpam-4788	313	13	α	α	NOUN
ejpam-4788	313	14	-	-	NOUN
ejpam-4788	313	15	ideal	ideal	NOUN
ejpam-4788	313	16	of	of	ADP
ejpam-4788	313	17	s.	s.	PROPN
ejpam-4788	313	18	thus	thus	ADV
ejpam-4788	313	19	,	,	PUNCT
ejpam-4788	313	20	supp(ς	supp(ς	NOUN
ejpam-4788	313	21	)	)	PUNCT
ejpam-4788	313	22	⊆	⊆	NUM
ejpam-4788	313	23	supp(λk	supp(λk	NOUN
ejpam-4788	313	24	)	)	PUNCT
ejpam-4788	313	25	=	=	SYM
ejpam-4788	313	26	k.	k.	NOUN
ejpam-4788	313	27	by	by	ADP
ejpam-4788	313	28	assumption	assumption	NOUN
ejpam-4788	313	29	,	,	PUNCT
ejpam-4788	313	30	supp(ς	supp(ς	NOUN
ejpam-4788	313	31	)	)	PUNCT
ejpam-4788	314	1	=	=	SYM
ejpam-4788	314	2	k	k	NOUN
ejpam-4788	314	3	=	=	PUNCT
ejpam-4788	314	4	supp(λk	supp(λk	NOUN
ejpam-4788	314	5	)	)	PUNCT
ejpam-4788	314	6	.	.	PUNCT
ejpam-4788	315	1	thus	thus	ADV
ejpam-4788	315	2	,	,	PUNCT
ejpam-4788	315	3	λk	λk	X
ejpam-4788	315	4	=	=	SYM
ejpam-4788	315	5	(	(	PUNCT
ejpam-4788	315	6	s;λp	s;λp	PROPN
ejpam-4788	315	7	k	k	PROPN
ejpam-4788	315	8	,	,	PUNCT
ejpam-4788	315	9	λn	λn	PROPN
ejpam-4788	315	10	k	k	X
ejpam-4788	315	11	)	)	PUNCT
ejpam-4788	315	12	is	be	AUX
ejpam-4788	315	13	a	a	DET
ejpam-4788	315	14	minimal	minimal	ADJ
ejpam-4788	315	15	bf	bf	NOUN
ejpam-4788	315	16	almost	almost	ADV
ejpam-4788	315	17	left	leave	VERB
ejpam-4788	315	18	α	α	NOUN
ejpam-4788	315	19	-	-	NOUN
ejpam-4788	315	20	ideal	ideal	NOUN
ejpam-4788	315	21	of	of	ADP
ejpam-4788	315	22	s.	s.	PROPN
ejpam-4788	315	23	conversely	conversely	ADV
ejpam-4788	315	24	,	,	PUNCT
ejpam-4788	315	25	suppose	suppose	VERB
ejpam-4788	315	26	that	that	SCONJ
ejpam-4788	315	27	λk	λk	PROPN
ejpam-4788	315	28	=	=	SYM
ejpam-4788	315	29	(	(	PUNCT
ejpam-4788	315	30	s;λp	s;λp	PROPN
ejpam-4788	315	31	k	k	PROPN
ejpam-4788	315	32	,	,	PUNCT
ejpam-4788	315	33	λn	λn	PROPN
ejpam-4788	315	34	k	k	X
ejpam-4788	315	35	)	)	PUNCT
ejpam-4788	315	36	is	be	AUX
ejpam-4788	315	37	a	a	DET
ejpam-4788	315	38	minimal	minimal	ADJ
ejpam-4788	315	39	bf	bf	NOUN
ejpam-4788	315	40	almost	almost	ADV
ejpam-4788	315	41	left	leave	VERB
ejpam-4788	315	42	α	α	NOUN
ejpam-4788	315	43	-	-	NOUN
ejpam-4788	315	44	ideal	ideal	NOUN
ejpam-4788	315	45	of	of	ADP
ejpam-4788	315	46	s.	s.	PROPN
ejpam-4788	315	47	then	then	ADV
ejpam-4788	315	48	,	,	PUNCT
ejpam-4788	315	49	by	by	ADP
ejpam-4788	315	50	theorem	theorem	NOUN
ejpam-4788	315	51	12	12	NUM
ejpam-4788	315	52	,	,	PUNCT
ejpam-4788	315	53	k	k	PROPN
ejpam-4788	315	54	is	be	AUX
ejpam-4788	315	55	an	an	DET
ejpam-4788	315	56	almost	almost	ADV
ejpam-4788	315	57	left	leave	VERB
ejpam-4788	315	58	α	α	NOUN
ejpam-4788	315	59	-	-	NOUN
ejpam-4788	315	60	ideal	ideal	NOUN
ejpam-4788	315	61	of	of	ADP
ejpam-4788	315	62	s.	s.	PROPN
ejpam-4788	315	63	let	let	VERB
ejpam-4788	315	64	j	j	PROPN
ejpam-4788	315	65	be	be	AUX
ejpam-4788	315	66	an	an	DET
ejpam-4788	315	67	almost	almost	ADV
ejpam-4788	315	68	left	leave	VERB
ejpam-4788	315	69	α	α	NOUN
ejpam-4788	315	70	-	-	NOUN
ejpam-4788	315	71	ideal	ideal	NOUN
ejpam-4788	315	72	of	of	ADP
ejpam-4788	315	73	s	s	PRON
ejpam-4788	316	1	such	such	ADJ
ejpam-4788	316	2	that	that	SCONJ
ejpam-4788	316	3	j	j	PROPN
ejpam-4788	316	4	⊆	⊆	NUM
ejpam-4788	316	5	k.	k.	PROPN
ejpam-4788	316	6	then	then	ADV
ejpam-4788	316	7	by	by	ADP
ejpam-4788	316	8	theorem	theorem	NOUN
ejpam-4788	316	9	12	12	NUM
ejpam-4788	316	10	,	,	PUNCT
ejpam-4788	316	11	λj	λj	PROPN
ejpam-4788	316	12	=	=	SYM
ejpam-4788	316	13	(	(	PUNCT
ejpam-4788	316	14	s;λp	s;λp	PROPN
ejpam-4788	316	15	j	j	PROPN
ejpam-4788	316	16	,	,	PUNCT
ejpam-4788	316	17	λ	λ	PROPN
ejpam-4788	316	18	n	n	CCONJ
ejpam-4788	316	19	j	j	NOUN
ejpam-4788	316	20	)	)	PUNCT
ejpam-4788	316	21	is	be	AUX
ejpam-4788	316	22	a	a	DET
ejpam-4788	316	23	bf	bf	NOUN
ejpam-4788	316	24	left	leave	VERB
ejpam-4788	316	25	α	α	NOUN
ejpam-4788	316	26	-	-	NOUN
ejpam-4788	316	27	ideal	ideal	NOUN
ejpam-4788	316	28	of	of	ADP
ejpam-4788	316	29	s	s	PRON
ejpam-4788	316	30	such	such	ADJ
ejpam-4788	316	31	that	that	SCONJ
ejpam-4788	316	32	λj	λj	PROPN
ejpam-4788	316	33	⊆	⊆	NUM
ejpam-4788	316	34	λk	λk	NOUN
ejpam-4788	316	35	.	.	PUNCT
ejpam-4788	317	1	thus	thus	ADV
ejpam-4788	317	2	,	,	PUNCT
ejpam-4788	317	3	j	j	PROPN
ejpam-4788	317	4	=	=	PUNCT
ejpam-4788	317	5	supp(λj	supp(λj	PROPN
ejpam-4788	317	6	)	)	PUNCT
ejpam-4788	317	7	=	=	SYM
ejpam-4788	317	8	supp(λk	supp(λk	NOUN
ejpam-4788	317	9	)	)	PUNCT
ejpam-4788	317	10	=	=	SYM
ejpam-4788	317	11	k.	k.	PROPN
ejpam-4788	318	1	hence	hence	ADV
ejpam-4788	318	2	,	,	PUNCT
ejpam-4788	318	3	k	k	PROPN
ejpam-4788	318	4	is	be	AUX
ejpam-4788	318	5	a	a	DET
ejpam-4788	318	6	minimal	minimal	ADJ
ejpam-4788	318	7	almost	almost	ADV
ejpam-4788	318	8	left	leave	VERB
ejpam-4788	318	9	α	α	NOUN
ejpam-4788	318	10	-	-	NOUN
ejpam-4788	318	11	ideal	ideal	NOUN
ejpam-4788	318	12	of	of	ADP
ejpam-4788	318	13	s.	s.	PROPN
ejpam-4788	318	14	corollary	corollary	PROPN
ejpam-4788	318	15	1	1	NUM
ejpam-4788	318	16	.	.	PUNCT
ejpam-4788	319	1	let	let	VERB
ejpam-4788	319	2	s	s	PRON
ejpam-4788	319	3	be	be	AUX
ejpam-4788	319	4	a	a	DET
ejpam-4788	319	5	γ	γ	NOUN
ejpam-4788	319	6	-	-	PUNCT
ejpam-4788	319	7	semigroup	semigroup	ADJ
ejpam-4788	319	8	s.	s.	PROPN
ejpam-4788	320	1	then	then	ADV
ejpam-4788	320	2	s	s	AUX
ejpam-4788	320	3	has	have	VERB
ejpam-4788	320	4	no	no	DET
ejpam-4788	320	5	proper	proper	ADJ
ejpam-4788	320	6	almost	almost	ADV
ejpam-4788	320	7	left	leave	VERB
ejpam-4788	320	8	α	α	NOUN
ejpam-4788	320	9	-	-	NOUN
ejpam-4788	320	10	ideal	ideal	ADJ
ejpam-4788	320	11	(	(	PUNCT
ejpam-4788	320	12	right	right	ADJ
ejpam-4788	320	13	β	β	NOUN
ejpam-4788	320	14	-	-	NOUN
ejpam-4788	320	15	ideal	ideal	ADJ
ejpam-4788	320	16	,	,	PUNCT
ejpam-4788	320	17	(	(	PUNCT
ejpam-4788	320	18	α	α	NOUN
ejpam-4788	320	19	,	,	PUNCT
ejpam-4788	320	20	β)-ideal	β)-ideal	NUM
ejpam-4788	320	21	)	)	PUNCT
ejpam-4788	320	22	of	of	ADP
ejpam-4788	320	23	s	s	PRON
ejpam-4788	320	24	if	if	SCONJ
ejpam-4788	320	25	and	and	CCONJ
ejpam-4788	320	26	only	only	ADV
ejpam-4788	320	27	if	if	SCONJ
ejpam-4788	320	28	for	for	ADP
ejpam-4788	320	29	any	any	DET
ejpam-4788	320	30	bf	bf	NOUN
ejpam-4788	320	31	almost	almost	ADV
ejpam-4788	320	32	left	leave	VERB
ejpam-4788	320	33	α	α	NOUN
ejpam-4788	320	34	-	-	NOUN
ejpam-4788	320	35	ideal	ideal	ADJ
ejpam-4788	320	36	(	(	PUNCT
ejpam-4788	320	37	right	right	ADJ
ejpam-4788	320	38	β	β	NOUN
ejpam-4788	320	39	-	-	NOUN
ejpam-4788	320	40	ideal	ideal	ADJ
ejpam-4788	320	41	,	,	PUNCT
ejpam-4788	320	42	(	(	PUNCT
ejpam-4788	320	43	α	α	NOUN
ejpam-4788	320	44	,	,	PUNCT
ejpam-4788	320	45	β)-ideal	β)-ideal	ADJ
ejpam-4788	320	46	)	)	PUNCT
ejpam-4788	321	1	ξ	ξ	X
ejpam-4788	321	2	=	=	SYM
ejpam-4788	321	3	(	(	PUNCT
ejpam-4788	321	4	s	s	PROPN
ejpam-4788	321	5	;	;	PUNCT
ejpam-4788	321	6	ξp	ξp	NUM
ejpam-4788	321	7	,	,	PUNCT
ejpam-4788	321	8	ξn	ξn	NOUN
ejpam-4788	321	9	)	)	PUNCT
ejpam-4788	321	10	of	of	ADP
ejpam-4788	321	11	s	s	PROPN
ejpam-4788	321	12	,	,	PUNCT
ejpam-4788	321	13	supp(ξ	supp(ξ	PROPN
ejpam-4788	321	14	)	)	PUNCT
ejpam-4788	321	15	=	=	PUNCT
ejpam-4788	322	1	s.	s.	PROPN
ejpam-4788	322	2	p.	p.	PROPN
ejpam-4788	322	3	khamrot	khamrot	PROPN
ejpam-4788	322	4	,	,	PUNCT
ejpam-4788	322	5	t.	t.	PROPN
ejpam-4788	322	6	gaketem	gaketem	PROPN
ejpam-4788	322	7	/	/	SYM
ejpam-4788	322	8	eur	eur	PROPN
ejpam-4788	322	9	.	.	PUNCT
ejpam-4788	323	1	j.	j.	PROPN
ejpam-4788	323	2	pure	pure	PROPN
ejpam-4788	323	3	appl	appl	PROPN
ejpam-4788	323	4	.	.	PROPN
ejpam-4788	323	5	math	math	PROPN
ejpam-4788	323	6	,	,	PUNCT
ejpam-4788	323	7	16	16	NUM
ejpam-4788	323	8	(	(	PUNCT
ejpam-4788	323	9	3	3	NUM
ejpam-4788	323	10	)	)	PUNCT
ejpam-4788	323	11	(	(	PUNCT
ejpam-4788	323	12	2023	2023	NUM
ejpam-4788	323	13	)	)	PUNCT
ejpam-4788	323	14	,	,	PUNCT
ejpam-4788	323	15	1592	1592	NUM
ejpam-4788	323	16	-	-	SYM
ejpam-4788	323	17	1607	1607	NUM
ejpam-4788	323	18	1603	1603	NUM
ejpam-4788	323	19	next	next	ADV
ejpam-4788	323	20	,	,	PUNCT
ejpam-4788	323	21	we	we	PRON
ejpam-4788	323	22	define	define	VERB
ejpam-4788	323	23	the	the	DET
ejpam-4788	323	24	bf	bf	NOUN
ejpam-4788	323	25	almost	almost	ADV
ejpam-4788	323	26	(	(	PUNCT
ejpam-4788	323	27	α	α	NOUN
ejpam-4788	323	28	,	,	PUNCT
ejpam-4788	323	29	β)-quasi	β)-quasi	NOUN
ejpam-4788	323	30	-	-	NOUN
ejpam-4788	323	31	ideals	ideal	NOUN
ejpam-4788	323	32	,	,	PUNCT
ejpam-4788	323	33	and	and	CCONJ
ejpam-4788	323	34	we	we	PRON
ejpam-4788	323	35	study	study	VERB
ejpam-4788	323	36	their	their	PRON
ejpam-4788	323	37	properties	property	NOUN
ejpam-4788	323	38	.	.	PUNCT
ejpam-4788	324	1	definition	definition	NOUN
ejpam-4788	324	2	16	16	NUM
ejpam-4788	324	3	.	.	PUNCT
ejpam-4788	325	1	let	let	VERB
ejpam-4788	325	2	ξ	ξ	X
ejpam-4788	325	3	=	=	SYM
ejpam-4788	325	4	(	(	PUNCT
ejpam-4788	325	5	s	s	PROPN
ejpam-4788	325	6	;	;	PUNCT
ejpam-4788	325	7	ξp	ξp	NUM
ejpam-4788	325	8	,	,	PUNCT
ejpam-4788	325	9	ξn	ξn	NOUN
ejpam-4788	325	10	)	)	PUNCT
ejpam-4788	325	11	be	be	VERB
ejpam-4788	325	12	a	a	DET
ejpam-4788	325	13	bf	bf	NOUN
ejpam-4788	325	14	set	set	NOUN
ejpam-4788	325	15	of	of	ADP
ejpam-4788	325	16	a	a	DET
ejpam-4788	325	17	γ	γ	NOUN
ejpam-4788	325	18	-	-	PUNCT
ejpam-4788	325	19	semigroup	semigroup	NOUN
ejpam-4788	325	20	s	s	PROPN
ejpam-4788	325	21	,	,	PUNCT
ejpam-4788	325	22	and	and	CCONJ
ejpam-4788	325	23	α	α	NOUN
ejpam-4788	325	24	,	,	PUNCT
ejpam-4788	325	25	β	β	PROPN
ejpam-4788	325	26	∈	∈	NOUN
ejpam-4788	325	27	γ	γ	NOUN
ejpam-4788	325	28	is	be	AUX
ejpam-4788	325	29	said	say	VERB
ejpam-4788	325	30	to	to	PART
ejpam-4788	325	31	be	be	AUX
ejpam-4788	325	32	a	a	DET
ejpam-4788	325	33	bf	bf	NOUN
ejpam-4788	325	34	almost	almost	ADV
ejpam-4788	325	35	(	(	PUNCT
ejpam-4788	325	36	α	α	NOUN
ejpam-4788	325	37	,	,	PUNCT
ejpam-4788	325	38	β)-quasi	β)-quasi	NOUN
ejpam-4788	325	39	-	-	NOUN
ejpam-4788	325	40	ideal	ideal	NOUN
ejpam-4788	325	41	of	of	ADP
ejpam-4788	325	42	s	s	PRON
ejpam-4788	325	43	if	if	SCONJ
ejpam-4788	325	44	(	(	PUNCT
ejpam-4788	325	45	ξ	ξ	X
ejpam-4788	325	46	◦	◦	NOUN
ejpam-4788	325	47	α	α	NOUN
ejpam-4788	325	48	x(t	x(t	PROPN
ejpam-4788	325	49	,	,	PUNCT
ejpam-4788	325	50	s	s	NOUN
ejpam-4788	325	51	)	)	PUNCT
ejpam-4788	325	52	)	)	PUNCT
ejpam-4788	325	53	∩	∩	NOUN
ejpam-4788	325	54	(	(	PUNCT
ejpam-4788	325	55	x(t	x(t	PROPN
ejpam-4788	325	56	,	,	PUNCT
ejpam-4788	325	57	s	s	NOUN
ejpam-4788	325	58	)	)	PUNCT
ejpam-4788	325	59	◦	◦	NOUN
ejpam-4788	325	60	β	β	X
ejpam-4788	325	61	ξ	ξ	X
ejpam-4788	325	62	)	)	PUNCT
ejpam-4788	325	63	̸=	̸=	PROPN
ejpam-4788	325	64	0	0	NUM
ejpam-4788	325	65	.	.	PUNCT
ejpam-4788	326	1	theorem	theorem	VERB
ejpam-4788	326	2	15	15	NUM
ejpam-4788	326	3	.	.	PUNCT
ejpam-4788	327	1	if	if	SCONJ
ejpam-4788	327	2	ξ	ξ	X
ejpam-4788	327	3	=	=	SYM
ejpam-4788	327	4	(	(	PUNCT
ejpam-4788	327	5	s	s	PROPN
ejpam-4788	327	6	;	;	PUNCT
ejpam-4788	327	7	ξp	ξp	NUM
ejpam-4788	327	8	,	,	PUNCT
ejpam-4788	327	9	ξn	ξn	NOUN
ejpam-4788	327	10	)	)	PUNCT
ejpam-4788	327	11	is	be	AUX
ejpam-4788	327	12	a	a	DET
ejpam-4788	327	13	bf	bf	NOUN
ejpam-4788	327	14	almost	almost	ADV
ejpam-4788	327	15	(	(	PUNCT
ejpam-4788	327	16	α	α	NOUN
ejpam-4788	327	17	,	,	PUNCT
ejpam-4788	327	18	β)-quasi	β)-quasi	NOUN
ejpam-4788	327	19	-	-	NOUN
ejpam-4788	327	20	ideal	ideal	NOUN
ejpam-4788	327	21	of	of	ADP
ejpam-4788	327	22	a	a	DET
ejpam-4788	327	23	γ	γ	NOUN
ejpam-4788	327	24	-	-	PUNCT
ejpam-4788	327	25	semigroup	semigroup	NOUN
ejpam-4788	327	26	s	s	PROPN
ejpam-4788	327	27	,	,	PUNCT
ejpam-4788	327	28	and	and	CCONJ
ejpam-4788	327	29	ς	ς	PROPN
ejpam-4788	327	30	=	=	PUNCT
ejpam-4788	327	31	(	(	PUNCT
ejpam-4788	327	32	s	s	NOUN
ejpam-4788	327	33	;	;	PUNCT
ejpam-4788	327	34	ςp	ςp	NUM
ejpam-4788	327	35	,	,	PUNCT
ejpam-4788	327	36	ςn	ςn	NOUN
ejpam-4788	327	37	)	)	PUNCT
ejpam-4788	327	38	is	be	AUX
ejpam-4788	327	39	a	a	DET
ejpam-4788	327	40	bf	bf	NOUN
ejpam-4788	327	41	set	set	NOUN
ejpam-4788	327	42	of	of	ADP
ejpam-4788	327	43	s	s	PRON
ejpam-4788	327	44	such	such	ADJ
ejpam-4788	327	45	that	that	SCONJ
ejpam-4788	327	46	ξ	ξ	PROPN
ejpam-4788	327	47	⊆	⊆	NUM
ejpam-4788	327	48	ς	ς	NOUN
ejpam-4788	327	49	,	,	PUNCT
ejpam-4788	327	50	then	then	ADV
ejpam-4788	327	51	ς	ς	PROPN
ejpam-4788	327	52	=	=	PUNCT
ejpam-4788	327	53	(	(	PUNCT
ejpam-4788	327	54	s	s	NOUN
ejpam-4788	327	55	;	;	PUNCT
ejpam-4788	327	56	ςp	ςp	NUM
ejpam-4788	327	57	,	,	PUNCT
ejpam-4788	327	58	ςn	ςn	NOUN
ejpam-4788	327	59	)	)	PUNCT
ejpam-4788	327	60	is	be	AUX
ejpam-4788	327	61	a	a	DET
ejpam-4788	327	62	bf	bf	NOUN
ejpam-4788	327	63	(	(	PUNCT
ejpam-4788	327	64	α	α	NOUN
ejpam-4788	327	65	,	,	PUNCT
ejpam-4788	327	66	β)-quasiideal	β)-quasiideal	PUNCT
ejpam-4788	327	67	of	of	ADP
ejpam-4788	327	68	s.	s.	PROPN
ejpam-4788	327	69	proof	proof	PROPN
ejpam-4788	327	70	.	.	PUNCT
ejpam-4788	328	1	suppose	suppose	VERB
ejpam-4788	328	2	that	that	SCONJ
ejpam-4788	328	3	ξ	ξ	PROPN
ejpam-4788	328	4	=	=	SYM
ejpam-4788	328	5	(	(	PUNCT
ejpam-4788	328	6	s	s	PROPN
ejpam-4788	328	7	;	;	PUNCT
ejpam-4788	328	8	ξp	ξp	NUM
ejpam-4788	328	9	,	,	PUNCT
ejpam-4788	328	10	ξn	ξn	NOUN
ejpam-4788	328	11	)	)	PUNCT
ejpam-4788	328	12	is	be	AUX
ejpam-4788	328	13	a	a	DET
ejpam-4788	328	14	bf	bf	NOUN
ejpam-4788	328	15	almost	almost	ADV
ejpam-4788	328	16	(	(	PUNCT
ejpam-4788	328	17	α	α	NOUN
ejpam-4788	328	18	,	,	PUNCT
ejpam-4788	328	19	β)-quasi	β)-quasi	NOUN
ejpam-4788	328	20	-	-	NOUN
ejpam-4788	328	21	ideal	ideal	NOUN
ejpam-4788	328	22	of	of	ADP
ejpam-4788	328	23	s	s	PROPN
ejpam-4788	328	24	,	,	PUNCT
ejpam-4788	328	25	and	and	CCONJ
ejpam-4788	328	26	ς	ς	PROPN
ejpam-4788	328	27	=	=	PUNCT
ejpam-4788	328	28	(	(	PUNCT
ejpam-4788	328	29	s	s	NOUN
ejpam-4788	328	30	;	;	PUNCT
ejpam-4788	328	31	ςp	ςp	NUM
ejpam-4788	328	32	,	,	PUNCT
ejpam-4788	328	33	ςn	ςn	NOUN
ejpam-4788	328	34	)	)	PUNCT
ejpam-4788	328	35	is	be	AUX
ejpam-4788	328	36	a	a	DET
ejpam-4788	328	37	bf	bf	NOUN
ejpam-4788	328	38	set	set	NOUN
ejpam-4788	328	39	of	of	ADP
ejpam-4788	328	40	s	s	PRON
ejpam-4788	328	41	such	such	ADJ
ejpam-4788	328	42	that	that	SCONJ
ejpam-4788	328	43	ξ	ξ	PROPN
ejpam-4788	328	44	⊆	⊆	NUM
ejpam-4788	328	45	ς	ς	X
ejpam-4788	328	46	.	.	PUNCT
ejpam-4788	329	1	then	then	ADV
ejpam-4788	329	2	(	(	PUNCT
ejpam-4788	329	3	xpt	xpt	PROPN
ejpam-4788	329	4	◦	◦	PROPN
ejpam-4788	329	5	α	α	PROPN
ejpam-4788	329	6	ξp	ξp	NOUN
ejpam-4788	329	7	)	)	PUNCT
ejpam-4788	329	8	∧	∧	PROPN
ejpam-4788	329	9	(	(	PUNCT
ejpam-4788	329	10	ξp	ξp	AUX
ejpam-4788	329	11	◦	◦	VERB
ejpam-4788	329	12	β	β	X
ejpam-4788	329	13	xpt	xpt	X
ejpam-4788	329	14	)	)	PUNCT
ejpam-4788	330	1	̸=	̸=	PROPN
ejpam-4788	330	2	0	0	NUM
ejpam-4788	331	1	and	and	CCONJ
ejpam-4788	331	2	(	(	PUNCT
ejpam-4788	331	3	xns	xns	PROPN
ejpam-4788	331	4	◦	◦	PROPN
ejpam-4788	331	5	α	α	PROPN
ejpam-4788	331	6	ξn	ξn	PROPN
ejpam-4788	331	7	)	)	PUNCT
ejpam-4788	331	8	∨	∨	PROPN
ejpam-4788	331	9	(	(	PUNCT
ejpam-4788	331	10	ξn	ξn	PROPN
ejpam-4788	331	11	◦	◦	PROPN
ejpam-4788	331	12	β	β	X
ejpam-4788	331	13	xns	xns	X
ejpam-4788	331	14	)	)	PUNCT
ejpam-4788	331	15	̸=	̸=	PROPN
ejpam-4788	331	16	0	0	NUM
ejpam-4788	331	17	.	.	PUNCT
ejpam-4788	332	1	thus	thus	ADV
ejpam-4788	332	2	,	,	PUNCT
ejpam-4788	332	3	(	(	PUNCT
ejpam-4788	332	4	xpt	xpt	PROPN
ejpam-4788	332	5	◦	◦	PROPN
ejpam-4788	332	6	α	α	PROPN
ejpam-4788	332	7	ξp	ξp	NOUN
ejpam-4788	332	8	)	)	PUNCT
ejpam-4788	332	9	∧	∧	PROPN
ejpam-4788	332	10	(	(	PUNCT
ejpam-4788	332	11	ξp	ξp	AUX
ejpam-4788	332	12	◦	◦	VERB
ejpam-4788	332	13	β	β	X
ejpam-4788	332	14	xpt	xpt	PROPN
ejpam-4788	332	15	)	)	PUNCT
ejpam-4788	333	1	⊆	⊆	X
ejpam-4788	333	2	(	(	PUNCT
ejpam-4788	333	3	xpt	xpt	PROPN
ejpam-4788	333	4	◦	◦	NOUN
ejpam-4788	333	5	α	α	NOUN
ejpam-4788	333	6	ςp	ςp	NOUN
ejpam-4788	333	7	)	)	PUNCT
ejpam-4788	333	8	∧	∧	PROPN
ejpam-4788	333	9	(	(	PUNCT
ejpam-4788	333	10	ςp	ςp	ADP
ejpam-4788	333	11	◦	◦	NOUN
ejpam-4788	333	12	β	β	X
ejpam-4788	333	13	xpt	xpt	PROPN
ejpam-4788	333	14	)	)	PUNCT
ejpam-4788	334	1	̸=	̸=	PROPN
ejpam-4788	334	2	0	0	NUM
ejpam-4788	334	3	,	,	PUNCT
ejpam-4788	334	4	and	and	CCONJ
ejpam-4788	334	5	(	(	PUNCT
ejpam-4788	334	6	xns	xns	PROPN
ejpam-4788	334	7	◦	◦	PROPN
ejpam-4788	334	8	α	α	PROPN
ejpam-4788	334	9	ξn	ξn	PROPN
ejpam-4788	334	10	)	)	PUNCT
ejpam-4788	334	11	∨	∨	PROPN
ejpam-4788	334	12	(	(	PUNCT
ejpam-4788	334	13	ξn	ξn	PROPN
ejpam-4788	334	14	◦	◦	PROPN
ejpam-4788	334	15	β	β	X
ejpam-4788	334	16	xns	xns	X
ejpam-4788	334	17	)	)	PUNCT
ejpam-4788	334	18	⊆	⊆	NUM
ejpam-4788	334	19	(	(	PUNCT
ejpam-4788	334	20	xns	xns	PROPN
ejpam-4788	334	21	◦	◦	NOUN
ejpam-4788	334	22	α	α	NOUN
ejpam-4788	334	23	ςn	ςn	NOUN
ejpam-4788	334	24	)	)	PUNCT
ejpam-4788	334	25	∧	∧	PROPN
ejpam-4788	334	26	(	(	PUNCT
ejpam-4788	334	27	ςn	ςn	NOUN
ejpam-4788	334	28	◦	◦	PROPN
ejpam-4788	334	29	β	β	X
ejpam-4788	334	30	xns	xns	X
ejpam-4788	334	31	)	)	PUNCT
ejpam-4788	334	32	̸=	̸=	PROPN
ejpam-4788	334	33	0	0	NUM
ejpam-4788	334	34	.	.	PUNCT
ejpam-4788	335	1	hence	hence	ADV
ejpam-4788	335	2	,	,	PUNCT
ejpam-4788	335	3	ς	ς	PROPN
ejpam-4788	335	4	=	=	PUNCT
ejpam-4788	335	5	(	(	PUNCT
ejpam-4788	335	6	s	s	NOUN
ejpam-4788	335	7	;	;	PUNCT
ejpam-4788	335	8	ςp	ςp	NUM
ejpam-4788	335	9	,	,	PUNCT
ejpam-4788	335	10	ςn	ςn	NOUN
ejpam-4788	335	11	)	)	PUNCT
ejpam-4788	335	12	is	be	AUX
ejpam-4788	335	13	a	a	DET
ejpam-4788	335	14	bf	bf	NOUN
ejpam-4788	335	15	(	(	PUNCT
ejpam-4788	335	16	α	α	NOUN
ejpam-4788	335	17	,	,	PUNCT
ejpam-4788	335	18	β)-quasi	β)-quasi	NOUN
ejpam-4788	335	19	-	-	PUNCT
ejpam-4788	335	20	ideal	ideal	NOUN
ejpam-4788	335	21	of	of	ADP
ejpam-4788	335	22	s.	s.	PROPN
ejpam-4788	335	23	theorem	theorem	VERB
ejpam-4788	335	24	16	16	NUM
ejpam-4788	335	25	.	.	PUNCT
ejpam-4788	336	1	let	let	VERB
ejpam-4788	336	2	k	k	PRON
ejpam-4788	336	3	be	be	AUX
ejpam-4788	336	4	a	a	DET
ejpam-4788	336	5	non	non	ADJ
ejpam-4788	336	6	-	-	ADJ
ejpam-4788	336	7	empty	empty	ADJ
ejpam-4788	336	8	subset	subset	NOUN
ejpam-4788	336	9	of	of	ADP
ejpam-4788	336	10	γ	γ	PROPN
ejpam-4788	336	11	-	-	PUNCT
ejpam-4788	336	12	semigroup	semigroup	PROPN
ejpam-4788	336	13	s.	s.	PROPN
ejpam-4788	337	1	then	then	ADV
ejpam-4788	337	2	k	k	PROPN
ejpam-4788	337	3	is	be	AUX
ejpam-4788	337	4	an	an	DET
ejpam-4788	337	5	almost	almost	ADV
ejpam-4788	337	6	(	(	PUNCT
ejpam-4788	337	7	α	α	NOUN
ejpam-4788	337	8	,	,	PUNCT
ejpam-4788	337	9	β)-quasi	β)-quasi	NOUN
ejpam-4788	337	10	-	-	NOUN
ejpam-4788	337	11	ideal	ideal	NOUN
ejpam-4788	337	12	of	of	ADP
ejpam-4788	337	13	s	s	PRON
ejpam-4788	337	14	if	if	SCONJ
ejpam-4788	337	15	and	and	CCONJ
ejpam-4788	337	16	only	only	ADV
ejpam-4788	337	17	if	if	SCONJ
ejpam-4788	337	18	the	the	DET
ejpam-4788	337	19	characteristic	characteristic	ADJ
ejpam-4788	337	20	function	function	NOUN
ejpam-4788	337	21	λk	λk	X
ejpam-4788	337	22	=	=	PUNCT
ejpam-4788	337	23	(	(	PUNCT
ejpam-4788	337	24	s;λp	s;λp	PROPN
ejpam-4788	337	25	k	k	PROPN
ejpam-4788	337	26	,	,	PUNCT
ejpam-4788	337	27	λn	λn	PROPN
ejpam-4788	337	28	k	k	X
ejpam-4788	337	29	)	)	PUNCT
ejpam-4788	337	30	is	be	AUX
ejpam-4788	337	31	a	a	DET
ejpam-4788	337	32	bf	bf	NOUN
ejpam-4788	337	33	almost	almost	ADV
ejpam-4788	337	34	(	(	PUNCT
ejpam-4788	337	35	α	α	NOUN
ejpam-4788	337	36	,	,	PUNCT
ejpam-4788	337	37	β)-quasi	β)-quasi	NOUN
ejpam-4788	337	38	-	-	NOUN
ejpam-4788	337	39	ideal	ideal	NOUN
ejpam-4788	337	40	of	of	ADP
ejpam-4788	337	41	s.	s.	PROPN
ejpam-4788	337	42	proof	proof	PROPN
ejpam-4788	337	43	.	.	PUNCT
ejpam-4788	338	1	suppose	suppose	VERB
ejpam-4788	338	2	that	that	SCONJ
ejpam-4788	338	3	k	k	PROPN
ejpam-4788	338	4	is	be	AUX
ejpam-4788	338	5	an	an	DET
ejpam-4788	338	6	almost	almost	ADV
ejpam-4788	338	7	(	(	PUNCT
ejpam-4788	338	8	α	α	NOUN
ejpam-4788	338	9	,	,	PUNCT
ejpam-4788	338	10	β)-quasi	β)-quasi	NOUN
ejpam-4788	338	11	-	-	PUNCT
ejpam-4788	338	12	ideal	ideal	NOUN
ejpam-4788	338	13	of	of	ADP
ejpam-4788	338	14	s.	s.	PROPN
ejpam-4788	338	15	then	then	ADV
ejpam-4788	338	16	(	(	PUNCT
ejpam-4788	338	17	kαu	kαu	NOUN
ejpam-4788	338	18	)	)	PUNCT
ejpam-4788	338	19	∩	∩	NOUN
ejpam-4788	338	20	(	(	PUNCT
ejpam-4788	338	21	uβk	uβk	PROPN
ejpam-4788	338	22	)	)	PUNCT
ejpam-4788	338	23	∩	∩	NOUN
ejpam-4788	338	24	k	k	PROPN
ejpam-4788	338	25	̸=	̸=	PROPN
ejpam-4788	338	26	∅	∅	NOUN
ejpam-4788	338	27	for	for	ADP
ejpam-4788	338	28	all	all	PRON
ejpam-4788	338	29	u	u	NOUN
ejpam-4788	338	30	∈	∈	PROPN
ejpam-4788	338	31	s.	s.	PROPN
ejpam-4788	338	32	thus	thus	ADV
ejpam-4788	338	33	,	,	PUNCT
ejpam-4788	338	34	there	there	PRON
ejpam-4788	338	35	exists	exist	VERB
ejpam-4788	338	36	v	v	ADP
ejpam-4788	338	37	∈	∈	PROPN
ejpam-4788	338	38	(	(	PUNCT
ejpam-4788	338	39	kαu	kαu	NOUN
ejpam-4788	338	40	)	)	PUNCT
ejpam-4788	338	41	∩	∩	NOUN
ejpam-4788	338	42	(	(	PUNCT
ejpam-4788	338	43	uβk	uβk	PROPN
ejpam-4788	338	44	)	)	PUNCT
ejpam-4788	338	45	and	and	CCONJ
ejpam-4788	338	46	v	v	ADP
ejpam-4788	338	47	∈	∈	PROPN
ejpam-4788	338	48	k.	k.	NOUN
ejpam-4788	339	1	so	so	ADV
ejpam-4788	339	2	,	,	PUNCT
ejpam-4788	339	3	(	(	PUNCT
ejpam-4788	339	4	(	(	PUNCT
ejpam-4788	339	5	xpt	xpt	PROPN
ejpam-4788	339	6	◦	◦	NOUN
ejpam-4788	339	7	α	α	PROPN
ejpam-4788	339	8	λp	λp	X
ejpam-4788	339	9	k	k	NOUN
ejpam-4788	339	10	)	)	PUNCT
ejpam-4788	339	11	∧	∧	PROPN
ejpam-4788	339	12	(	(	PUNCT
ejpam-4788	339	13	λp	λp	X
ejpam-4788	339	14	k	k	PROPN
ejpam-4788	339	15	◦	◦	PROPN
ejpam-4788	339	16	β	β	X
ejpam-4788	339	17	xpt	xpt	PROPN
ejpam-4788	339	18	)	)	PUNCT
ejpam-4788	339	19	)	)	PUNCT
ejpam-4788	340	1	(	(	PUNCT
ejpam-4788	340	2	v	v	NOUN
ejpam-4788	340	3	)	)	PUNCT
ejpam-4788	340	4	̸=	̸=	PROPN
ejpam-4788	340	5	0	0	NUM
ejpam-4788	341	1	and	and	CCONJ
ejpam-4788	341	2	(	(	PUNCT
ejpam-4788	341	3	(	(	PUNCT
ejpam-4788	341	4	xns	xns	PROPN
ejpam-4788	341	5	◦	◦	NOUN
ejpam-4788	341	6	α	α	NOUN
ejpam-4788	341	7	λn	λn	PROPN
ejpam-4788	341	8	k	k	X
ejpam-4788	341	9	)	)	PUNCT
ejpam-4788	341	10	∨	∨	PROPN
ejpam-4788	341	11	(	(	PUNCT
ejpam-4788	341	12	λn	λn	PROPN
ejpam-4788	341	13	k	k	PROPN
ejpam-4788	341	14	◦	◦	PROPN
ejpam-4788	341	15	β	β	X
ejpam-4788	341	16	xns	xns	NOUN
ejpam-4788	341	17	)	)	PUNCT
ejpam-4788	341	18	)	)	PUNCT
ejpam-4788	341	19	(	(	PUNCT
ejpam-4788	341	20	v	v	NOUN
ejpam-4788	341	21	)	)	PUNCT
ejpam-4788	341	22	̸=	̸=	PROPN
ejpam-4788	341	23	0	0	NUM
ejpam-4788	341	24	.	.	PUNCT
ejpam-4788	342	1	hence	hence	ADV
ejpam-4788	342	2	,	,	PUNCT
ejpam-4788	342	3	(	(	PUNCT
ejpam-4788	342	4	λk	λk	AUX
ejpam-4788	342	5	◦	◦	NOUN
ejpam-4788	342	6	α	α	NOUN
ejpam-4788	342	7	x(t	x(t	PROPN
ejpam-4788	342	8	,	,	PUNCT
ejpam-4788	342	9	s	s	NOUN
ejpam-4788	342	10	)	)	PUNCT
ejpam-4788	342	11	)	)	PUNCT
ejpam-4788	342	12	∩	∩	NOUN
ejpam-4788	342	13	(	(	PUNCT
ejpam-4788	342	14	x(t	x(t	PROPN
ejpam-4788	342	15	,	,	PUNCT
ejpam-4788	342	16	s	s	NOUN
ejpam-4788	342	17	)	)	PUNCT
ejpam-4788	342	18	◦	◦	NOUN
ejpam-4788	342	19	β	β	X
ejpam-4788	342	20	λk	λk	NOUN
ejpam-4788	342	21	)	)	PUNCT
ejpam-4788	342	22	̸=	̸=	PROPN
ejpam-4788	342	23	0	0	NUM
ejpam-4788	342	24	.	.	PUNCT
ejpam-4788	343	1	therefore	therefore	ADV
ejpam-4788	343	2	,	,	PUNCT
ejpam-4788	343	3	λk	λk	X
ejpam-4788	343	4	=	=	PUNCT
ejpam-4788	343	5	(	(	PUNCT
ejpam-4788	343	6	s;λp	s;λp	PROPN
ejpam-4788	343	7	k	k	PROPN
ejpam-4788	343	8	,	,	PUNCT
ejpam-4788	343	9	λn	λn	PROPN
ejpam-4788	343	10	k	k	X
ejpam-4788	343	11	)	)	PUNCT
ejpam-4788	343	12	is	be	AUX
ejpam-4788	343	13	a	a	DET
ejpam-4788	343	14	bf	bf	NOUN
ejpam-4788	343	15	almost	almost	ADV
ejpam-4788	343	16	(	(	PUNCT
ejpam-4788	343	17	α	α	NOUN
ejpam-4788	343	18	,	,	PUNCT
ejpam-4788	343	19	β)-quasi	β)-quasi	NOUN
ejpam-4788	343	20	-	-	NOUN
ejpam-4788	343	21	ideal	ideal	NOUN
ejpam-4788	343	22	of	of	ADP
ejpam-4788	343	23	s.	s.	PROPN
ejpam-4788	343	24	conversely	conversely	ADV
ejpam-4788	343	25	,	,	PUNCT
ejpam-4788	343	26	assume	assume	VERB
ejpam-4788	343	27	that	that	SCONJ
ejpam-4788	343	28	λk	λk	ADV
ejpam-4788	343	29	=	=	SYM
ejpam-4788	343	30	(	(	PUNCT
ejpam-4788	343	31	s;λp	s;λp	PROPN
ejpam-4788	343	32	k	k	PROPN
ejpam-4788	343	33	,	,	PUNCT
ejpam-4788	343	34	λn	λn	PROPN
ejpam-4788	343	35	k	k	X
ejpam-4788	343	36	)	)	PUNCT
ejpam-4788	343	37	is	be	AUX
ejpam-4788	343	38	a	a	DET
ejpam-4788	343	39	bf	bf	NOUN
ejpam-4788	343	40	almost	almost	ADV
ejpam-4788	343	41	(	(	PUNCT
ejpam-4788	343	42	α	α	NOUN
ejpam-4788	343	43	,	,	PUNCT
ejpam-4788	343	44	β)-quasi	β)-quasi	NOUN
ejpam-4788	343	45	-	-	PUNCT
ejpam-4788	343	46	ideal	ideal	NOUN
ejpam-4788	343	47	of	of	ADP
ejpam-4788	343	48	s	s	NOUN
ejpam-4788	343	49	and	and	CCONJ
ejpam-4788	343	50	u	u	PROPN
ejpam-4788	343	51	∈	∈	PROPN
ejpam-4788	343	52	s.	s.	PROPN
ejpam-4788	344	1	then	then	ADV
ejpam-4788	344	2	(	(	PUNCT
ejpam-4788	344	3	λk	λk	ADP
ejpam-4788	344	4	◦	◦	NOUN
ejpam-4788	344	5	α	α	NOUN
ejpam-4788	344	6	x(t	x(t	PROPN
ejpam-4788	344	7	,	,	PUNCT
ejpam-4788	344	8	s	s	NOUN
ejpam-4788	344	9	)	)	PUNCT
ejpam-4788	344	10	)	)	PUNCT
ejpam-4788	344	11	∩	∩	NOUN
ejpam-4788	344	12	(	(	PUNCT
ejpam-4788	344	13	x(t	x(t	PROPN
ejpam-4788	344	14	,	,	PUNCT
ejpam-4788	344	15	s	s	NOUN
ejpam-4788	344	16	)	)	PUNCT
ejpam-4788	344	17	◦	◦	NOUN
ejpam-4788	344	18	β	β	X
ejpam-4788	344	19	λk	λk	NOUN
ejpam-4788	344	20	)	)	PUNCT
ejpam-4788	344	21	̸=	̸=	PROPN
ejpam-4788	344	22	0	0	NUM
ejpam-4788	344	23	.	.	PUNCT
ejpam-4788	345	1	thus	thus	ADV
ejpam-4788	345	2	,	,	PUNCT
ejpam-4788	345	3	there	there	PRON
ejpam-4788	345	4	exists	exist	VERB
ejpam-4788	345	5	r	r	NOUN
ejpam-4788	345	6	∈	∈	PROPN
ejpam-4788	345	7	s	s	VERB
ejpam-4788	345	8	such	such	ADJ
ejpam-4788	345	9	that	that	SCONJ
ejpam-4788	345	10	(	(	PUNCT
ejpam-4788	345	11	(	(	PUNCT
ejpam-4788	345	12	xpt	xpt	PROPN
ejpam-4788	345	13	◦	◦	NOUN
ejpam-4788	345	14	α	α	PROPN
ejpam-4788	345	15	λp	λp	X
ejpam-4788	345	16	k	k	NOUN
ejpam-4788	345	17	)	)	PUNCT
ejpam-4788	345	18	∧	∧	PROPN
ejpam-4788	345	19	(	(	PUNCT
ejpam-4788	345	20	λp	λp	X
ejpam-4788	345	21	k	k	PROPN
ejpam-4788	345	22	◦	◦	PROPN
ejpam-4788	345	23	β	β	X
ejpam-4788	345	24	xpt	xpt	PROPN
ejpam-4788	345	25	)	)	PUNCT
ejpam-4788	345	26	)	)	PUNCT
ejpam-4788	345	27	(	(	PUNCT
ejpam-4788	345	28	r	r	X
ejpam-4788	345	29	)	)	PUNCT
ejpam-4788	345	30	̸=	̸=	NOUN
ejpam-4788	345	31	0	0	NUM
ejpam-4788	346	1	and	and	CCONJ
ejpam-4788	346	2	(	(	PUNCT
ejpam-4788	346	3	(	(	PUNCT
ejpam-4788	346	4	xns	xns	PROPN
ejpam-4788	346	5	◦	◦	NOUN
ejpam-4788	346	6	α	α	NOUN
ejpam-4788	346	7	λn	λn	PROPN
ejpam-4788	346	8	k	k	X
ejpam-4788	346	9	)	)	PUNCT
ejpam-4788	346	10	∨	∨	PROPN
ejpam-4788	346	11	(	(	PUNCT
ejpam-4788	346	12	λn	λn	PROPN
ejpam-4788	346	13	k	k	PROPN
ejpam-4788	346	14	◦	◦	PROPN
ejpam-4788	346	15	β	β	X
ejpam-4788	346	16	xns	xns	NOUN
ejpam-4788	346	17	)	)	PUNCT
ejpam-4788	346	18	)	)	PUNCT
ejpam-4788	347	1	(	(	PUNCT
ejpam-4788	347	2	r	r	X
ejpam-4788	347	3	)	)	PUNCT
ejpam-4788	347	4	̸=	̸=	PROPN
ejpam-4788	347	5	0	0	NUM
ejpam-4788	347	6	.	.	PUNCT
ejpam-4788	348	1	hence	hence	ADV
ejpam-4788	348	2	,	,	PUNCT
ejpam-4788	348	3	r	r	NOUN
ejpam-4788	348	4	∈	∈	PROPN
ejpam-4788	348	5	(	(	PUNCT
ejpam-4788	348	6	kαu	kαu	NOUN
ejpam-4788	348	7	)	)	PUNCT
ejpam-4788	348	8	∩	∩	NOUN
ejpam-4788	348	9	(	(	PUNCT
ejpam-4788	348	10	uβk	uβk	PROPN
ejpam-4788	348	11	)	)	PUNCT
ejpam-4788	348	12	∩	∩	NOUN
ejpam-4788	348	13	k	k	PROPN
ejpam-4788	348	14	implies	imply	VERB
ejpam-4788	348	15	(	(	PUNCT
ejpam-4788	348	16	kαu	kαu	NOUN
ejpam-4788	348	17	)	)	PUNCT
ejpam-4788	348	18	∩	∩	NOUN
ejpam-4788	348	19	(	(	PUNCT
ejpam-4788	348	20	uβk	uβk	PROPN
ejpam-4788	348	21	)	)	PUNCT
ejpam-4788	348	22	∩	∩	NOUN
ejpam-4788	348	23	k	k	PROPN
ejpam-4788	348	24	̸=	̸=	PROPN
ejpam-4788	348	25	∅.	∅.	VERB
ejpam-4788	348	26	therefore	therefore	ADV
ejpam-4788	348	27	,	,	PUNCT
ejpam-4788	348	28	k	k	PROPN
ejpam-4788	348	29	is	be	AUX
ejpam-4788	348	30	an	an	DET
ejpam-4788	348	31	almost	almost	ADV
ejpam-4788	348	32	(	(	PUNCT
ejpam-4788	348	33	α	α	NOUN
ejpam-4788	348	34	,	,	PUNCT
ejpam-4788	348	35	β)-quasi	β)-quasi	NOUN
ejpam-4788	348	36	-	-	NOUN
ejpam-4788	348	37	ideal	ideal	NOUN
ejpam-4788	348	38	of	of	ADP
ejpam-4788	348	39	s.	s.	PROPN
ejpam-4788	348	40	next	next	ADV
ejpam-4788	348	41	,	,	PUNCT
ejpam-4788	348	42	we	we	PRON
ejpam-4788	348	43	study	study	VERB
ejpam-4788	348	44	the	the	DET
ejpam-4788	348	45	properties	property	NOUN
ejpam-4788	348	46	between	between	ADP
ejpam-4788	348	47	supp(ξ	supp(ξ	PROPN
ejpam-4788	348	48	)	)	PUNCT
ejpam-4788	348	49	and	and	CCONJ
ejpam-4788	348	50	a	a	DET
ejpam-4788	348	51	bf	bf	NOUN
ejpam-4788	348	52	almost	almost	ADV
ejpam-4788	348	53	(	(	PUNCT
ejpam-4788	348	54	α	α	NOUN
ejpam-4788	348	55	,	,	PUNCT
ejpam-4788	348	56	β)-quasi	β)-quasi	NOUN
ejpam-4788	348	57	-	-	PUNCT
ejpam-4788	348	58	ideal	ideal	NOUN
ejpam-4788	348	59	of	of	ADP
ejpam-4788	348	60	γ	γ	NOUN
ejpam-4788	348	61	-	-	PUNCT
ejpam-4788	348	62	semigroups	semigroup	NOUN
ejpam-4788	348	63	.	.	PUNCT
ejpam-4788	349	1	theorem	theorem	NOUN
ejpam-4788	349	2	17	17	NUM
ejpam-4788	349	3	.	.	PUNCT
ejpam-4788	350	1	let	let	VERB
ejpam-4788	350	2	ξ	ξ	X
ejpam-4788	350	3	=	=	SYM
ejpam-4788	350	4	(	(	PUNCT
ejpam-4788	350	5	s	s	PROPN
ejpam-4788	350	6	;	;	PUNCT
ejpam-4788	350	7	ξp	ξp	NUM
ejpam-4788	350	8	,	,	PUNCT
ejpam-4788	350	9	ξn	ξn	NOUN
ejpam-4788	350	10	)	)	PUNCT
ejpam-4788	350	11	be	be	VERB
ejpam-4788	350	12	a	a	DET
ejpam-4788	350	13	bf	bf	NOUN
ejpam-4788	350	14	set	set	NOUN
ejpam-4788	350	15	of	of	ADP
ejpam-4788	350	16	a	a	DET
ejpam-4788	350	17	non	non	ADJ
ejpam-4788	350	18	-	-	ADJ
ejpam-4788	350	19	empty	empty	ADJ
ejpam-4788	350	20	of	of	ADP
ejpam-4788	350	21	a	a	DET
ejpam-4788	350	22	γ	γ	PROPN
ejpam-4788	350	23	-	-	PUNCT
ejpam-4788	350	24	semigroup	semigroup	PROPN
ejpam-4788	350	25	s.	s.	PROPN
ejpam-4788	350	26	then	then	ADV
ejpam-4788	350	27	ξ	ξ	X
ejpam-4788	350	28	=	=	SYM
ejpam-4788	350	29	(	(	PUNCT
ejpam-4788	350	30	s	s	PROPN
ejpam-4788	350	31	;	;	PUNCT
ejpam-4788	350	32	ξp	ξp	NUM
ejpam-4788	350	33	,	,	PUNCT
ejpam-4788	350	34	ξn	ξn	NOUN
ejpam-4788	350	35	)	)	PUNCT
ejpam-4788	350	36	is	be	AUX
ejpam-4788	350	37	a	a	DET
ejpam-4788	350	38	bf	bf	NOUN
ejpam-4788	350	39	almost	almost	ADV
ejpam-4788	350	40	(	(	PUNCT
ejpam-4788	350	41	α	α	NOUN
ejpam-4788	350	42	,	,	PUNCT
ejpam-4788	350	43	β)-quasi	β)-quasi	NOUN
ejpam-4788	350	44	-	-	NOUN
ejpam-4788	350	45	ideal	ideal	NOUN
ejpam-4788	350	46	of	of	ADP
ejpam-4788	350	47	s	s	PRON
ejpam-4788	350	48	if	if	SCONJ
ejpam-4788	350	49	and	and	CCONJ
ejpam-4788	350	50	only	only	ADV
ejpam-4788	350	51	if	if	SCONJ
ejpam-4788	350	52	supp(ξ	supp(ξ	PROPN
ejpam-4788	350	53	)	)	PUNCT
ejpam-4788	350	54	is	be	AUX
ejpam-4788	350	55	an	an	PRON
ejpam-4788	350	56	almost	almost	ADV
ejpam-4788	350	57	(	(	PUNCT
ejpam-4788	350	58	α	α	NOUN
ejpam-4788	350	59	,	,	PUNCT
ejpam-4788	350	60	β)-quasi	β)-quasi	NOUN
ejpam-4788	350	61	-	-	NOUN
ejpam-4788	350	62	ideal	ideal	NOUN
ejpam-4788	350	63	of	of	ADP
ejpam-4788	350	64	s.	s.	PROPN
ejpam-4788	350	65	proof	proof	PROPN
ejpam-4788	350	66	.	.	PUNCT
ejpam-4788	351	1	let	let	VERB
ejpam-4788	351	2	ξ	ξ	X
ejpam-4788	351	3	=	=	SYM
ejpam-4788	351	4	(	(	PUNCT
ejpam-4788	351	5	s	s	PROPN
ejpam-4788	351	6	;	;	PUNCT
ejpam-4788	351	7	ξp	ξp	NUM
ejpam-4788	351	8	,	,	PUNCT
ejpam-4788	351	9	ξn	ξn	NOUN
ejpam-4788	351	10	)	)	PUNCT
ejpam-4788	351	11	be	be	VERB
ejpam-4788	351	12	a	a	DET
ejpam-4788	351	13	bf	bf	NOUN
ejpam-4788	351	14	almost	almost	ADV
ejpam-4788	351	15	(	(	PUNCT
ejpam-4788	351	16	α	α	NOUN
ejpam-4788	351	17	,	,	PUNCT
ejpam-4788	351	18	β)-quasi	β)-quasi	NOUN
ejpam-4788	351	19	-	-	PUNCT
ejpam-4788	351	20	ideal	ideal	NOUN
ejpam-4788	351	21	of	of	ADP
ejpam-4788	351	22	s	s	NOUN
ejpam-4788	351	23	and	and	CCONJ
ejpam-4788	351	24	u	u	PROPN
ejpam-4788	351	25	∈	∈	PROPN
ejpam-4788	351	26	s.	s.	PROPN
ejpam-4788	351	27	then	then	ADV
ejpam-4788	351	28	(	(	PUNCT
ejpam-4788	351	29	λk	λk	ADP
ejpam-4788	351	30	◦	◦	NOUN
ejpam-4788	351	31	αx(t	αx(t	NOUN
ejpam-4788	351	32	,	,	PUNCT
ejpam-4788	351	33	s))∩(x(t	s))∩(x(t	PROPN
ejpam-4788	351	34	,	,	PUNCT
ejpam-4788	351	35	s)	s)	X
ejpam-4788	351	36	◦	◦	NOUN
ejpam-4788	351	37	βλk	βλk	NOUN
ejpam-4788	351	38	)	)	PUNCT
ejpam-4788	351	39	̸=	̸=	PROPN
ejpam-4788	351	40	0	0	NUM
ejpam-4788	351	41	.	.	PUNCT
ejpam-4788	352	1	thus	thus	ADV
ejpam-4788	352	2	,	,	PUNCT
ejpam-4788	352	3	there	there	PRON
ejpam-4788	352	4	exists	exist	VERB
ejpam-4788	352	5	r	r	NOUN
ejpam-4788	352	6	∈	∈	PROPN
ejpam-4788	352	7	s	s	VERB
ejpam-4788	352	8	such	such	ADJ
ejpam-4788	352	9	that	that	SCONJ
ejpam-4788	352	10	(	(	PUNCT
ejpam-4788	352	11	ξp	ξp	NUM
ejpam-4788	352	12	◦	◦	NOUN
ejpam-4788	352	13	αxt)∩(xt	αxt)∩(xt	NOUN
ejpam-4788	352	14	◦	◦	NOUN
ejpam-4788	352	15	β	β	NOUN
ejpam-4788	352	16	ξp	ξp	NOUN
ejpam-4788	352	17	)	)	PUNCT
ejpam-4788	352	18	̸=	̸=	PROPN
ejpam-4788	352	19	0	0	NUM
ejpam-4788	353	1	and	and	CCONJ
ejpam-4788	353	2	(	(	PUNCT
ejpam-4788	353	3	ξn	ξn	PROPN
ejpam-4788	353	4	◦	◦	PROPN
ejpam-4788	353	5	α	α	PROPN
ejpam-4788	353	6	xs	xs	NOUN
ejpam-4788	353	7	)	)	PUNCT
ejpam-4788	354	1	∪	∪	NOUN
ejpam-4788	354	2	(	(	PUNCT
ejpam-4788	354	3	xs	xs	PROPN
ejpam-4788	354	4	◦	◦	PROPN
ejpam-4788	354	5	β	β	X
ejpam-4788	354	6	ξn	ξn	NOUN
ejpam-4788	354	7	)	)	PUNCT
ejpam-4788	354	8	̸=	̸=	PROPN
ejpam-4788	354	9	0	0	NUM
ejpam-4788	355	1	so	so	CCONJ
ejpam-4788	355	2	,	,	PUNCT
ejpam-4788	355	3	there	there	PRON
ejpam-4788	355	4	exists	exist	VERB
ejpam-4788	355	5	k1	k1	NOUN
ejpam-4788	355	6	,	,	PUNCT
ejpam-4788	356	1	k2	k2	PROPN
ejpam-4788	356	2	∈	∈	PROPN
ejpam-4788	356	3	s	s	VERB
ejpam-4788	356	4	such	such	ADJ
ejpam-4788	356	5	that	that	DET
ejpam-4788	356	6	r	r	NOUN
ejpam-4788	356	7	=	=	SYM
ejpam-4788	356	8	k1αu	k1αu	PROPN
ejpam-4788	356	9	=	=	SYM
ejpam-4788	356	10	uβk2	uβk2	PROPN
ejpam-4788	356	11	,	,	PUNCT
ejpam-4788	356	12	xpt	xpt	PROPN
ejpam-4788	356	13	(	(	PUNCT
ejpam-4788	356	14	r	r	NOUN
ejpam-4788	356	15	)	)	PUNCT
ejpam-4788	356	16	̸=	̸=	PROPN
ejpam-4788	356	17	0	0	NUM
ejpam-4788	356	18	,	,	PUNCT
ejpam-4788	356	19	xns	xns	X
ejpam-4788	356	20	(	(	PUNCT
ejpam-4788	356	21	r	r	NOUN
ejpam-4788	356	22	)	)	PUNCT
ejpam-4788	356	23	̸=	̸=	PROPN
ejpam-4788	356	24	0	0	NUM
ejpam-4788	356	25	and	and	CCONJ
ejpam-4788	356	26	xpt	xpt	PROPN
ejpam-4788	356	27	(	(	PUNCT
ejpam-4788	356	28	k1	k1	PROPN
ejpam-4788	356	29	)	)	PUNCT
ejpam-4788	356	30	̸=	̸=	PROPN
ejpam-4788	356	31	0	0	NUM
ejpam-4788	356	32	,	,	PUNCT
ejpam-4788	356	33	xns	xns	PROPN
ejpam-4788	356	34	(	(	PUNCT
ejpam-4788	356	35	k1	k1	PROPN
ejpam-4788	356	36	)	)	PUNCT
ejpam-4788	356	37	̸=	̸=	PROPN
ejpam-4788	356	38	0	0	NUM
ejpam-4788	356	39	.	.	PUNCT
ejpam-4788	357	1	it	it	PRON
ejpam-4788	357	2	implies	imply	VERB
ejpam-4788	357	3	that	that	SCONJ
ejpam-4788	357	4	r	r	NOUN
ejpam-4788	357	5	,	,	PUNCT
ejpam-4788	357	6	k1	k1	NOUN
ejpam-4788	357	7	,	,	PUNCT
ejpam-4788	357	8	k2	k2	PROPN
ejpam-4788	357	9	∈	∈	PROPN
ejpam-4788	357	10	supp(ξ	supp(ξ	PROPN
ejpam-4788	357	11	)	)	PUNCT
ejpam-4788	357	12	.	.	PUNCT
ejpam-4788	358	1	thus	thus	ADV
ejpam-4788	358	2	(	(	PUNCT
ejpam-4788	358	3	(	(	PUNCT
ejpam-4788	358	4	ξp	ξp	PART
ejpam-4788	358	5	◦	◦	VERB
ejpam-4788	358	6	α	α	NOUN
ejpam-4788	358	7	xt	xt	NOUN
ejpam-4788	358	8	)	)	PUNCT
ejpam-4788	358	9	∩	∩	NOUN
ejpam-4788	358	10	(	(	PUNCT
ejpam-4788	358	11	xt	xt	X
ejpam-4788	358	12	◦	◦	NOUN
ejpam-4788	358	13	β	β	X
ejpam-4788	358	14	ξp))(r	ξp))(r	NUM
ejpam-4788	358	15	)	)	PUNCT
ejpam-4788	358	16	̸=	̸=	PROPN
ejpam-4788	358	17	0	0	NUM
ejpam-4788	358	18	and	and	CCONJ
ejpam-4788	358	19	λp	λp	ADP
ejpam-4788	358	20	supp(ξ	supp(ξ	PROPN
ejpam-4788	358	21	)	)	PUNCT
ejpam-4788	358	22	̸=	̸=	PROPN
ejpam-4788	358	23	0	0	NUM
ejpam-4788	358	24	.	.	PUNCT
ejpam-4788	359	1	similalry	similalry	ADJ
ejpam-4788	359	2	,	,	PUNCT
ejpam-4788	359	3	(	(	PUNCT
ejpam-4788	359	4	ξn	ξn	PROPN
ejpam-4788	359	5	◦	◦	PROPN
ejpam-4788	359	6	α	α	PROPN
ejpam-4788	359	7	xs	xs	NOUN
ejpam-4788	359	8	)	)	PUNCT
ejpam-4788	359	9	∪	∪	NOUN
ejpam-4788	359	10	(	(	PUNCT
ejpam-4788	359	11	xs	xs	PROPN
ejpam-4788	359	12	◦	◦	PROPN
ejpam-4788	359	13	β	β	X
ejpam-4788	359	14	ξn)(r	ξn)(r	PROPN
ejpam-4788	359	15	)	)	PUNCT
ejpam-4788	360	1	̸=	̸=	PROPN
ejpam-4788	360	2	0	0	NUM
ejpam-4788	360	3	and	and	CCONJ
ejpam-4788	360	4	λn	λn	PROPN
ejpam-4788	360	5	supp(ξ	supp(ξ	PROPN
ejpam-4788	360	6	)	)	PUNCT
ejpam-4788	360	7	̸=	̸=	PROPN
ejpam-4788	360	8	0	0	NUM
ejpam-4788	360	9	.	.	PUNCT
ejpam-4788	361	1	hence	hence	ADV
ejpam-4788	361	2	,	,	PUNCT
ejpam-4788	361	3	(	(	PUNCT
ejpam-4788	361	4	λp	λp	X
ejpam-4788	361	5	supp(ξ	supp(ξ	PROPN
ejpam-4788	361	6	)	)	PUNCT
ejpam-4788	361	7	◦	◦	NOUN
ejpam-4788	361	8	α	α	X
ejpam-4788	361	9	xpt	xpt	PROPN
ejpam-4788	361	10	)	)	PUNCT
ejpam-4788	361	11	∩	∩	NOUN
ejpam-4788	361	12	(	(	PUNCT
ejpam-4788	361	13	xpt	xpt	PROPN
ejpam-4788	361	14	◦	◦	PROPN
ejpam-4788	361	15	β	β	X
ejpam-4788	361	16	λp	λp	X
ejpam-4788	361	17	supp(ξ	supp(ξ	PROPN
ejpam-4788	361	18	)	)	PUNCT
ejpam-4788	361	19	)	)	PUNCT
ejpam-4788	361	20	∩	∩	PROPN
ejpam-4788	361	21	λp	λp	X
ejpam-4788	361	22	supp(ξ	supp(ξ	PROPN
ejpam-4788	361	23	)	)	PUNCT
ejpam-4788	361	24	̸=	̸=	PROPN
ejpam-4788	361	25	0	0	NUM
ejpam-4788	362	1	and	and	CCONJ
ejpam-4788	362	2	(	(	PUNCT
ejpam-4788	362	3	λn	λn	PROPN
ejpam-4788	362	4	supp(ξ	supp(ξ	PROPN
ejpam-4788	362	5	)	)	PUNCT
ejpam-4788	362	6	◦	◦	NOUN
ejpam-4788	362	7	α	α	X
ejpam-4788	362	8	xns	xns	NOUN
ejpam-4788	362	9	)	)	PUNCT
ejpam-4788	362	10	∪	∪	PROPN
ejpam-4788	362	11	(	(	PUNCT
ejpam-4788	362	12	xns	xns	PROPN
ejpam-4788	362	13	◦	◦	PROPN
ejpam-4788	362	14	β	β	X
ejpam-4788	362	15	λn	λn	NOUN
ejpam-4788	362	16	supp(ξ	supp(ξ	PROPN
ejpam-4788	362	17	)	)	PUNCT
ejpam-4788	362	18	)	)	PUNCT
ejpam-4788	362	19	∪	∪	ADP
ejpam-4788	362	20	λn	λn	PROPN
ejpam-4788	362	21	supp(ξ	supp(ξ	PROPN
ejpam-4788	362	22	)	)	PUNCT
ejpam-4788	362	23	̸=	̸=	PROPN
ejpam-4788	362	24	0	0	NUM
ejpam-4788	362	25	.	.	PUNCT
ejpam-4788	363	1	therefore	therefore	ADV
ejpam-4788	363	2	,	,	PUNCT
ejpam-4788	363	3	λsupp(ξ	λsupp(ξ	NOUN
ejpam-4788	363	4	)	)	PUNCT
ejpam-4788	363	5	is	be	AUX
ejpam-4788	363	6	a	a	DET
ejpam-4788	363	7	bf	bf	NOUN
ejpam-4788	363	8	almost	almost	ADV
ejpam-4788	363	9	(	(	PUNCT
ejpam-4788	363	10	α	α	NOUN
ejpam-4788	363	11	,	,	PUNCT
ejpam-4788	363	12	β)-quasi	β)-quasi	NOUN
ejpam-4788	363	13	-	-	NOUN
ejpam-4788	363	14	ideal	ideal	NOUN
ejpam-4788	363	15	of	of	ADP
ejpam-4788	363	16	s.	s.	PROPN
ejpam-4788	363	17	this	this	PRON
ejpam-4788	363	18	shows	show	VERB
ejpam-4788	363	19	that	that	SCONJ
ejpam-4788	363	20	supp(ξ	supp(ξ	NOUN
ejpam-4788	363	21	)	)	PUNCT
ejpam-4788	363	22	is	be	AUX
ejpam-4788	363	23	an	an	DET
ejpam-4788	363	24	almost	almost	ADV
ejpam-4788	363	25	(	(	PUNCT
ejpam-4788	363	26	α	α	NOUN
ejpam-4788	363	27	,	,	PUNCT
ejpam-4788	363	28	β)-quasi	β)-quasi	NOUN
ejpam-4788	363	29	-	-	NOUN
ejpam-4788	363	30	ideal	ideal	NOUN
ejpam-4788	363	31	of	of	ADP
ejpam-4788	363	32	s.	s.	PROPN
ejpam-4788	363	33	p.	p.	PROPN
ejpam-4788	363	34	khamrot	khamrot	PROPN
ejpam-4788	363	35	,	,	PUNCT
ejpam-4788	363	36	t.	t.	PROPN
ejpam-4788	363	37	gaketem	gaketem	PROPN
ejpam-4788	363	38	/	/	SYM
ejpam-4788	363	39	eur	eur	PROPN
ejpam-4788	363	40	.	.	PUNCT
ejpam-4788	364	1	j.	j.	PROPN
ejpam-4788	364	2	pure	pure	PROPN
ejpam-4788	364	3	appl	appl	PROPN
ejpam-4788	364	4	.	.	PROPN
ejpam-4788	364	5	math	math	PROPN
ejpam-4788	364	6	,	,	PUNCT
ejpam-4788	364	7	16	16	NUM
ejpam-4788	364	8	(	(	PUNCT
ejpam-4788	364	9	3	3	NUM
ejpam-4788	364	10	)	)	PUNCT
ejpam-4788	364	11	(	(	PUNCT
ejpam-4788	364	12	2023	2023	NUM
ejpam-4788	364	13	)	)	PUNCT
ejpam-4788	364	14	,	,	PUNCT
ejpam-4788	364	15	1592	1592	NUM
ejpam-4788	364	16	-	-	SYM
ejpam-4788	364	17	1607	1607	NUM
ejpam-4788	364	18	1604	1604	NUM
ejpam-4788	364	19	conversely	conversely	ADV
ejpam-4788	364	20	,	,	PUNCT
ejpam-4788	364	21	let	let	VERB
ejpam-4788	364	22	supp(ξ	supp(ξ	PROPN
ejpam-4788	364	23	)	)	PUNCT
ejpam-4788	364	24	be	be	AUX
ejpam-4788	365	1	an	an	DET
ejpam-4788	365	2	almost	almost	ADV
ejpam-4788	365	3	(	(	PUNCT
ejpam-4788	365	4	α	α	NOUN
ejpam-4788	365	5	,	,	PUNCT
ejpam-4788	365	6	β)-quasi	β)-quasi	NOUN
ejpam-4788	365	7	-	-	PUNCT
ejpam-4788	365	8	ideal	ideal	NOUN
ejpam-4788	365	9	of	of	ADP
ejpam-4788	365	10	s.	s.	PROPN
ejpam-4788	365	11	then	then	ADV
ejpam-4788	365	12	,	,	PUNCT
ejpam-4788	365	13	by	by	ADP
ejpam-4788	365	14	theorem	theorem	NOUN
ejpam-4788	365	15	16	16	NUM
ejpam-4788	365	16	,	,	PUNCT
ejpam-4788	365	17	χsupp(ξ	χsupp(ξ	PROPN
ejpam-4788	365	18	)	)	PUNCT
ejpam-4788	365	19	is	be	AUX
ejpam-4788	365	20	a	a	DET
ejpam-4788	365	21	bf	bf	NOUN
ejpam-4788	365	22	(	(	PUNCT
ejpam-4788	365	23	α	α	NOUN
ejpam-4788	365	24	,	,	PUNCT
ejpam-4788	365	25	β)-quasi	β)-quasi	NOUN
ejpam-4788	365	26	-	-	PUNCT
ejpam-4788	365	27	ideal	ideal	NOUN
ejpam-4788	365	28	of	of	ADP
ejpam-4788	365	29	s.	s.	PROPN
ejpam-4788	365	30	thus	thus	ADV
ejpam-4788	365	31	,	,	PUNCT
ejpam-4788	365	32	[	[	X
ejpam-4788	365	33	(	(	PUNCT
ejpam-4788	365	34	λp	λp	X
ejpam-4788	365	35	supp(ξ)	supp(ξ)	NOUN
ejpam-4788	365	36	◦	◦	VERB
ejpam-4788	365	37	αx	αx	PROPN
ejpam-4788	365	38	p	p	PROPN
ejpam-4788	365	39	t	t	PROPN
ejpam-4788	365	40	)	)	PUNCT
ejpam-4788	365	41	∩(x	∩(x	PROPN
ejpam-4788	366	1	p	p	PROPN
ejpam-4788	366	2	t	t	PROPN
ejpam-4788	366	3	◦	◦	NOUN
ejpam-4788	366	4	βλ	βλ	ADV
ejpam-4788	366	5	p	p	X
ejpam-4788	366	6	supp(ξ))]∩λ	supp(ξ))]∩λ	X
ejpam-4788	366	7	p	p	PROPN
ejpam-4788	366	8	supp(ξ	supp(ξ	PROPN
ejpam-4788	366	9	)	)	PUNCT
ejpam-4788	366	10	̸=	̸=	PROPN
ejpam-4788	366	11	0	0	NUM
ejpam-4788	367	1	and	and	CCONJ
ejpam-4788	367	2	[	[	X
ejpam-4788	367	3	(	(	PUNCT
ejpam-4788	367	4	λn	λn	PROPN
ejpam-4788	367	5	supp(ξ	supp(ξ	PROPN
ejpam-4788	367	6	)	)	PUNCT
ejpam-4788	367	7	◦	◦	NOUN
ejpam-4788	367	8	α	α	X
ejpam-4788	367	9	xns	xns	NOUN
ejpam-4788	367	10	)	)	PUNCT
ejpam-4788	367	11	∩	∩	NOUN
ejpam-4788	367	12	(	(	PUNCT
ejpam-4788	367	13	xns	xns	PROPN
ejpam-4788	367	14	◦	◦	PROPN
ejpam-4788	367	15	β	β	X
ejpam-4788	367	16	λn	λn	NOUN
ejpam-4788	367	17	supp(ξ	supp(ξ	PROPN
ejpam-4788	367	18	)	)	PUNCT
ejpam-4788	367	19	)	)	PUNCT
ejpam-4788	367	20	]	]	PUNCT
ejpam-4788	367	21	∩	∩	PROPN
ejpam-4788	367	22	λn	λn	PROPN
ejpam-4788	367	23	supp(ξ	supp(ξ	PROPN
ejpam-4788	367	24	)	)	PUNCT
ejpam-4788	367	25	̸=	̸=	PROPN
ejpam-4788	367	26	0	0	NUM
ejpam-4788	367	27	.	.	PUNCT
ejpam-4788	368	1	so	so	ADV
ejpam-4788	368	2	,	,	PUNCT
ejpam-4788	368	3	there	there	PRON
ejpam-4788	368	4	exists	exist	VERB
ejpam-4788	368	5	r	r	NOUN
ejpam-4788	368	6	∈	∈	PROPN
ejpam-4788	368	7	s	s	VERB
ejpam-4788	368	8	such	such	ADJ
ejpam-4788	368	9	that	that	SCONJ
ejpam-4788	369	1	[	[	X
ejpam-4788	369	2	(	(	PUNCT
ejpam-4788	369	3	λp	λp	X
ejpam-4788	369	4	supp(ξ	supp(ξ	PROPN
ejpam-4788	369	5	)	)	PUNCT
ejpam-4788	369	6	◦	◦	NOUN
ejpam-4788	369	7	α	α	NOUN
ejpam-4788	369	8	x	x	X
ejpam-4788	369	9	p	p	X
ejpam-4788	369	10	t	t	NOUN
ejpam-4788	369	11	)	)	PUNCT
ejpam-4788	369	12	∩	∩	NOUN
ejpam-4788	369	13	(	(	PUNCT
ejpam-4788	369	14	xpt	xpt	PROPN
ejpam-4788	369	15	◦	◦	PROPN
ejpam-4788	369	16	β	β	X
ejpam-4788	369	17	λ	λ	X
ejpam-4788	369	18	p	p	NOUN
ejpam-4788	369	19	supp(ξ))]∩λp	supp(ξ))]∩λp	PROPN
ejpam-4788	369	20	supp(ξ)(r	supp(ξ)(r	NOUN
ejpam-4788	369	21	)	)	PUNCT
ejpam-4788	369	22	̸=	̸=	PROPN
ejpam-4788	369	23	0	0	NUM
ejpam-4788	370	1	and	and	CCONJ
ejpam-4788	370	2	(	(	PUNCT
ejpam-4788	370	3	[	[	X
ejpam-4788	370	4	(	(	PUNCT
ejpam-4788	370	5	λn	λn	PROPN
ejpam-4788	370	6	supp(ξ	supp(ξ	PROPN
ejpam-4788	370	7	)	)	PUNCT
ejpam-4788	370	8	◦	◦	NOUN
ejpam-4788	370	9	α	α	NOUN
ejpam-4788	370	10	x	x	SYM
ejpam-4788	370	11	n	n	PROPN
ejpam-4788	370	12	s	s	PART
ejpam-4788	370	13	)	)	PUNCT
ejpam-4788	370	14	∩	∩	NOUN
ejpam-4788	370	15	(	(	PUNCT
ejpam-4788	370	16	xns	xns	PROPN
ejpam-4788	370	17	◦	◦	PROPN
ejpam-4788	370	18	β	β	NOUN
ejpam-4788	370	19	λn	λn	NOUN
ejpam-4788	370	20	supp(ξ))]∩	supp(ξ))]∩	PROPN
ejpam-4788	370	21	λn	λn	NOUN
ejpam-4788	370	22	supp(ξ))(r	supp(ξ))(r	VERB
ejpam-4788	370	23	)	)	PUNCT
ejpam-4788	370	24	̸=	̸=	PROPN
ejpam-4788	370	25	0	0	NUM
ejpam-4788	370	26	.	.	PUNCT
ejpam-4788	371	1	it	it	PRON
ejpam-4788	371	2	implies	imply	VERB
ejpam-4788	371	3	that	that	SCONJ
ejpam-4788	371	4	(	(	PUNCT
ejpam-4788	371	5	(	(	PUNCT
ejpam-4788	371	6	λsupp(ξ	λsupp(ξ	NOUN
ejpam-4788	371	7	)	)	PUNCT
ejpam-4788	371	8	◦	◦	NOUN
ejpam-4788	371	9	α	α	PRON
ejpam-4788	371	10	xt	xt	NOUN
ejpam-4788	371	11	)	)	PUNCT
ejpam-4788	371	12	∩	∩	NOUN
ejpam-4788	371	13	(	(	PUNCT
ejpam-4788	371	14	xt	xt	ADP
ejpam-4788	371	15	◦	◦	PROPN
ejpam-4788	371	16	β	β	X
ejpam-4788	371	17	λsupp(ξ)))(r	λsupp(ξ)))(r	PROPN
ejpam-4788	371	18	)	)	PUNCT
ejpam-4788	371	19	̸=	̸=	PROPN
ejpam-4788	371	20	0	0	NUM
ejpam-4788	371	21	and	and	CCONJ
ejpam-4788	371	22	,	,	PUNCT
ejpam-4788	371	23	λk(r	λk(r	PROPN
ejpam-4788	371	24	)	)	PUNCT
ejpam-4788	371	25	̸=	̸=	PROPN
ejpam-4788	371	26	0	0	NUM
ejpam-4788	371	27	.	.	PUNCT
ejpam-4788	372	1	thus	thus	ADV
ejpam-4788	372	2	,	,	PUNCT
ejpam-4788	372	3	there	there	PRON
ejpam-4788	372	4	exist	exist	VERB
ejpam-4788	372	5	k1	k1	NOUN
ejpam-4788	372	6	,	,	PUNCT
ejpam-4788	372	7	k2	k2	PROPN
ejpam-4788	372	8	∈	∈	PROPN
ejpam-4788	372	9	s	s	VERB
ejpam-4788	372	10	such	such	ADJ
ejpam-4788	372	11	that	that	DET
ejpam-4788	372	12	r	r	NOUN
ejpam-4788	372	13	=	=	SYM
ejpam-4788	372	14	k1αu	k1αu	PROPN
ejpam-4788	372	15	=	=	SYM
ejpam-4788	372	16	uβk2	uβk2	NOUN
ejpam-4788	372	17	,	,	PUNCT
ejpam-4788	372	18	ξp(r	ξp(r	NOUN
ejpam-4788	372	19	)	)	PUNCT
ejpam-4788	372	20	̸=	̸=	PROPN
ejpam-4788	372	21	0	0	NUM
ejpam-4788	372	22	,	,	PUNCT
ejpam-4788	372	23	ξn(r	ξn(r	NUM
ejpam-4788	372	24	)	)	PUNCT
ejpam-4788	372	25	̸=	̸=	PROPN
ejpam-4788	372	26	0	0	NUM
ejpam-4788	372	27	,	,	PUNCT
ejpam-4788	372	28	and	and	CCONJ
ejpam-4788	372	29	ξp(k	ξp(k	NOUN
ejpam-4788	372	30	)	)	PUNCT
ejpam-4788	372	31	̸=	̸=	PROPN
ejpam-4788	372	32	0	0	NUM
ejpam-4788	372	33	,	,	PUNCT
ejpam-4788	372	34	ξn(k	ξn(k	NUM
ejpam-4788	372	35	)	)	PUNCT
ejpam-4788	372	36	̸=	̸=	PROPN
ejpam-4788	372	37	0	0	NUM
ejpam-4788	372	38	.	.	PUNCT
ejpam-4788	373	1	hence	hence	ADV
ejpam-4788	373	2	,	,	PUNCT
ejpam-4788	373	3	(	(	PUNCT
ejpam-4788	373	4	ξp	ξp	PART
ejpam-4788	373	5	◦	◦	NOUN
ejpam-4788	373	6	α	α	NOUN
ejpam-4788	373	7	xt	xt	NOUN
ejpam-4788	373	8	)	)	PUNCT
ejpam-4788	373	9	∩	∩	NOUN
ejpam-4788	373	10	(	(	PUNCT
ejpam-4788	373	11	xt	xt	ADP
ejpam-4788	373	12	◦	◦	PROPN
ejpam-4788	373	13	β	β	NOUN
ejpam-4788	373	14	ξp	ξp	NOUN
ejpam-4788	373	15	)	)	PUNCT
ejpam-4788	373	16	̸=	̸=	PROPN
ejpam-4788	373	17	0	0	NUM
ejpam-4788	373	18	,	,	PUNCT
ejpam-4788	373	19	and	and	CCONJ
ejpam-4788	373	20	(	(	PUNCT
ejpam-4788	373	21	ξn	ξn	PROPN
ejpam-4788	373	22	◦	◦	PROPN
ejpam-4788	373	23	α	α	PROPN
ejpam-4788	373	24	xs	xs	NOUN
ejpam-4788	373	25	)	)	PUNCT
ejpam-4788	373	26	∪	∪	NOUN
ejpam-4788	373	27	(	(	PUNCT
ejpam-4788	373	28	xs	xs	PROPN
ejpam-4788	373	29	◦	◦	PROPN
ejpam-4788	373	30	β	β	X
ejpam-4788	373	31	ξn	ξn	NOUN
ejpam-4788	373	32	)	)	PUNCT
ejpam-4788	373	33	̸=	̸=	PROPN
ejpam-4788	373	34	0	0	NUM
ejpam-4788	373	35	.	.	PUNCT
ejpam-4788	374	1	so	so	ADV
ejpam-4788	374	2	,	,	PUNCT
ejpam-4788	374	3	(	(	PUNCT
ejpam-4788	374	4	ξ	ξ	X
ejpam-4788	374	5	◦	◦	NOUN
ejpam-4788	374	6	α	α	NOUN
ejpam-4788	374	7	x(t	x(t	PROPN
ejpam-4788	374	8	,	,	PUNCT
ejpam-4788	374	9	s	s	NOUN
ejpam-4788	374	10	)	)	PUNCT
ejpam-4788	374	11	)	)	PUNCT
ejpam-4788	374	12	∩	∩	NOUN
ejpam-4788	374	13	(	(	PUNCT
ejpam-4788	374	14	x(t	x(t	PROPN
ejpam-4788	374	15	,	,	PUNCT
ejpam-4788	374	16	s	s	NOUN
ejpam-4788	374	17	)	)	PUNCT
ejpam-4788	374	18	◦	◦	NOUN
ejpam-4788	374	19	β	β	X
ejpam-4788	374	20	ξ	ξ	X
ejpam-4788	374	21	)	)	PUNCT
ejpam-4788	374	22	̸=	̸=	PROPN
ejpam-4788	374	23	0	0	NUM
ejpam-4788	374	24	.	.	PUNCT
ejpam-4788	375	1	therefore	therefore	ADV
ejpam-4788	375	2	,	,	PUNCT
ejpam-4788	375	3	ξ	ξ	PROPN
ejpam-4788	375	4	is	be	AUX
ejpam-4788	375	5	a	a	DET
ejpam-4788	375	6	bf	bf	NOUN
ejpam-4788	375	7	almost	almost	ADV
ejpam-4788	375	8	(	(	PUNCT
ejpam-4788	375	9	α	α	NOUN
ejpam-4788	375	10	,	,	PUNCT
ejpam-4788	375	11	β)-quasi	β)-quasi	NOUN
ejpam-4788	375	12	-	-	NOUN
ejpam-4788	375	13	ideal	ideal	NOUN
ejpam-4788	375	14	of	of	ADP
ejpam-4788	375	15	s.	s.	PROPN
ejpam-4788	375	16	definition	definition	PROPN
ejpam-4788	375	17	17	17	NUM
ejpam-4788	375	18	.	.	PUNCT
ejpam-4788	376	1	a	a	DET
ejpam-4788	376	2	bf	bf	NOUN
ejpam-4788	376	3	almost	almost	ADV
ejpam-4788	376	4	(	(	PUNCT
ejpam-4788	376	5	α	α	NOUN
ejpam-4788	376	6	,	,	PUNCT
ejpam-4788	376	7	β)-quasi	β)-quasi	NOUN
ejpam-4788	376	8	-	-	PUNCT
ejpam-4788	376	9	ideal	ideal	NOUN
ejpam-4788	376	10	ξ	ξ	NOUN
ejpam-4788	377	1	=	=	SYM
ejpam-4788	378	1	(	(	PUNCT
ejpam-4788	378	2	s	s	PROPN
ejpam-4788	378	3	;	;	PUNCT
ejpam-4788	378	4	ξp	ξp	NUM
ejpam-4788	378	5	,	,	PUNCT
ejpam-4788	378	6	ξn	ξn	NOUN
ejpam-4788	378	7	)	)	PUNCT
ejpam-4788	378	8	of	of	ADP
ejpam-4788	378	9	a	a	DET
ejpam-4788	378	10	γ	γ	PROPN
ejpam-4788	378	11	-	-	PUNCT
ejpam-4788	378	12	semigroup	semigroup	NOUN
ejpam-4788	378	13	s	s	VERB
ejpam-4788	378	14	is	be	AUX
ejpam-4788	378	15	minimal	minimal	ADJ
ejpam-4788	378	16	if	if	SCONJ
ejpam-4788	378	17	for	for	ADP
ejpam-4788	378	18	all	all	DET
ejpam-4788	378	19	bf	bf	NOUN
ejpam-4788	378	20	almost	almost	ADV
ejpam-4788	378	21	(	(	PUNCT
ejpam-4788	378	22	α	α	NOUN
ejpam-4788	378	23	,	,	PUNCT
ejpam-4788	378	24	β)-quasi	β)-quasi	NOUN
ejpam-4788	378	25	-	-	PUNCT
ejpam-4788	378	26	ideal	ideal	ADJ
ejpam-4788	378	27	ς	ς	X
ejpam-4788	378	28	=	=	PUNCT
ejpam-4788	378	29	(	(	PUNCT
ejpam-4788	378	30	s	s	NOUN
ejpam-4788	378	31	;	;	PUNCT
ejpam-4788	378	32	ςp	ςp	NUM
ejpam-4788	378	33	,	,	PUNCT
ejpam-4788	378	34	ςn	ςn	NOUN
ejpam-4788	378	35	)	)	PUNCT
ejpam-4788	378	36	of	of	ADP
ejpam-4788	378	37	s	s	PRON
ejpam-4788	378	38	such	such	ADJ
ejpam-4788	378	39	that	that	SCONJ
ejpam-4788	378	40	ς	ς	PROPN
ejpam-4788	378	41	⊆	⊆	NUM
ejpam-4788	378	42	ξ	ξ	NUM
ejpam-4788	378	43	,	,	PUNCT
ejpam-4788	378	44	then	then	ADV
ejpam-4788	378	45	supp(ς	supp(ς	NOUN
ejpam-4788	378	46	)	)	PUNCT
ejpam-4788	378	47	=	=	SYM
ejpam-4788	378	48	supp(ξ	supp(ξ	PROPN
ejpam-4788	378	49	)	)	PUNCT
ejpam-4788	378	50	.	.	PUNCT
ejpam-4788	379	1	theorem	theorem	NOUN
ejpam-4788	379	2	18	18	NUM
ejpam-4788	379	3	.	.	PUNCT
ejpam-4788	380	1	let	let	VERB
ejpam-4788	380	2	k	k	PRON
ejpam-4788	380	3	be	be	AUX
ejpam-4788	380	4	a	a	DET
ejpam-4788	380	5	non	non	ADJ
ejpam-4788	380	6	-	-	ADJ
ejpam-4788	380	7	empty	empty	ADJ
ejpam-4788	380	8	subset	subset	NOUN
ejpam-4788	380	9	of	of	ADP
ejpam-4788	380	10	a	a	DET
ejpam-4788	380	11	γ	γ	PROPN
ejpam-4788	380	12	-	-	PUNCT
ejpam-4788	380	13	semigroup	semigroup	PROPN
ejpam-4788	380	14	s.	s.	PROPN
ejpam-4788	381	1	then	then	ADV
ejpam-4788	381	2	k	k	PROPN
ejpam-4788	381	3	is	be	AUX
ejpam-4788	381	4	a	a	DET
ejpam-4788	381	5	minimal	minimal	ADJ
ejpam-4788	381	6	almost	almost	ADV
ejpam-4788	381	7	(	(	PUNCT
ejpam-4788	381	8	α	α	NOUN
ejpam-4788	381	9	,	,	PUNCT
ejpam-4788	381	10	β)-quasi	β)-quasi	NOUN
ejpam-4788	381	11	-	-	PUNCT
ejpam-4788	381	12	ideal	ideal	NOUN
ejpam-4788	381	13	if	if	SCONJ
ejpam-4788	381	14	and	and	CCONJ
ejpam-4788	381	15	only	only	ADV
ejpam-4788	381	16	if	if	SCONJ
ejpam-4788	381	17	λk	λk	X
ejpam-4788	381	18	=	=	SYM
ejpam-4788	381	19	(	(	PUNCT
ejpam-4788	381	20	s;λp	s;λp	PROPN
ejpam-4788	381	21	k	k	PROPN
ejpam-4788	381	22	,	,	PUNCT
ejpam-4788	381	23	λn	λn	PROPN
ejpam-4788	381	24	k	k	X
ejpam-4788	381	25	)	)	PUNCT
ejpam-4788	381	26	is	be	AUX
ejpam-4788	381	27	a	a	DET
ejpam-4788	381	28	minimal	minimal	ADJ
ejpam-4788	381	29	bf	bf	NOUN
ejpam-4788	381	30	almost	almost	ADV
ejpam-4788	381	31	(	(	PUNCT
ejpam-4788	381	32	α	α	NOUN
ejpam-4788	381	33	,	,	PUNCT
ejpam-4788	381	34	β)quasi	β)quasi	ADJ
ejpam-4788	381	35	-	-	PUNCT
ejpam-4788	381	36	ideal	ideal	NOUN
ejpam-4788	381	37	of	of	ADP
ejpam-4788	381	38	s.	s.	PROPN
ejpam-4788	381	39	proof	proof	PROPN
ejpam-4788	381	40	.	.	PUNCT
ejpam-4788	382	1	suppose	suppose	VERB
ejpam-4788	382	2	thatk	thatk	PROPN
ejpam-4788	382	3	is	be	AUX
ejpam-4788	382	4	a	a	DET
ejpam-4788	382	5	minimal	minimal	ADJ
ejpam-4788	382	6	almost	almost	ADV
ejpam-4788	382	7	(	(	PUNCT
ejpam-4788	382	8	α	α	NOUN
ejpam-4788	382	9	,	,	PUNCT
ejpam-4788	382	10	β)-quasi	β)-quasi	NOUN
ejpam-4788	382	11	-	-	NOUN
ejpam-4788	382	12	ideal	ideal	NOUN
ejpam-4788	382	13	of	of	ADP
ejpam-4788	382	14	s.	s.	PROPN
ejpam-4788	382	15	thenk	thenk	PROPN
ejpam-4788	382	16	is	be	AUX
ejpam-4788	382	17	an	an	DET
ejpam-4788	382	18	almost	almost	ADV
ejpam-4788	382	19	(	(	PUNCT
ejpam-4788	382	20	α	α	NOUN
ejpam-4788	382	21	,	,	PUNCT
ejpam-4788	382	22	β)-quasi	β)-quasi	NOUN
ejpam-4788	382	23	-	-	PUNCT
ejpam-4788	382	24	ideal	ideal	NOUN
ejpam-4788	382	25	of	of	ADP
ejpam-4788	382	26	s.	s.	PROPN
ejpam-4788	382	27	thus	thus	ADV
ejpam-4788	382	28	,	,	PUNCT
ejpam-4788	382	29	by	by	ADP
ejpam-4788	382	30	theorem	theorem	NOUN
ejpam-4788	382	31	16	16	NUM
ejpam-4788	382	32	,	,	PUNCT
ejpam-4788	382	33	λk	λk	X
ejpam-4788	382	34	=	=	SYM
ejpam-4788	382	35	(	(	PUNCT
ejpam-4788	382	36	s;λp	s;λp	PROPN
ejpam-4788	382	37	k	k	PROPN
ejpam-4788	382	38	,	,	PUNCT
ejpam-4788	382	39	λn	λn	PROPN
ejpam-4788	382	40	k	k	X
ejpam-4788	382	41	)	)	PUNCT
ejpam-4788	382	42	is	be	AUX
ejpam-4788	382	43	a	a	DET
ejpam-4788	382	44	bf	bf	NOUN
ejpam-4788	382	45	(	(	PUNCT
ejpam-4788	382	46	α	α	NOUN
ejpam-4788	382	47	,	,	PUNCT
ejpam-4788	382	48	β)-quasi	β)-quasi	NOUN
ejpam-4788	382	49	-	-	PUNCT
ejpam-4788	382	50	ideal	ideal	NOUN
ejpam-4788	382	51	of	of	ADP
ejpam-4788	382	52	s.	s.	PROPN
ejpam-4788	382	53	let	let	VERB
ejpam-4788	382	54	ς	ς	PROPN
ejpam-4788	382	55	=	=	PUNCT
ejpam-4788	382	56	(	(	PUNCT
ejpam-4788	382	57	s	s	NOUN
ejpam-4788	382	58	;	;	PUNCT
ejpam-4788	382	59	ςp	ςp	NUM
ejpam-4788	382	60	,	,	PUNCT
ejpam-4788	382	61	ςn	ςn	NOUN
ejpam-4788	382	62	)	)	PUNCT
ejpam-4788	382	63	be	be	VERB
ejpam-4788	382	64	a	a	DET
ejpam-4788	382	65	bf	bf	NOUN
ejpam-4788	382	66	(	(	PUNCT
ejpam-4788	382	67	α	α	NOUN
ejpam-4788	382	68	,	,	PUNCT
ejpam-4788	382	69	β)-quasi	β)-quasi	NOUN
ejpam-4788	382	70	-	-	NOUN
ejpam-4788	382	71	ideal	ideal	NOUN
ejpam-4788	382	72	of	of	ADP
ejpam-4788	382	73	s	s	PRON
ejpam-4788	382	74	such	such	ADJ
ejpam-4788	382	75	that	that	SCONJ
ejpam-4788	382	76	ς	ς	PROPN
ejpam-4788	382	77	⊆	⊆	NUM
ejpam-4788	382	78	λk	λk	NOUN
ejpam-4788	382	79	.	.	PUNCT
ejpam-4788	383	1	then	then	ADV
ejpam-4788	383	2	,	,	PUNCT
ejpam-4788	383	3	by	by	ADP
ejpam-4788	383	4	theorem	theorem	NOUN
ejpam-4788	383	5	15	15	NUM
ejpam-4788	383	6	,	,	PUNCT
ejpam-4788	383	7	supp(ς	supp(ς	NOUN
ejpam-4788	383	8	)	)	PUNCT
ejpam-4788	383	9	is	be	AUX
ejpam-4788	383	10	an	an	DET
ejpam-4788	383	11	almost	almost	ADV
ejpam-4788	383	12	(	(	PUNCT
ejpam-4788	383	13	α	α	NOUN
ejpam-4788	383	14	,	,	PUNCT
ejpam-4788	383	15	β)-quasi	β)-quasi	NOUN
ejpam-4788	383	16	-	-	PUNCT
ejpam-4788	383	17	ideal	ideal	NOUN
ejpam-4788	383	18	of	of	ADP
ejpam-4788	383	19	s.	s.	PROPN
ejpam-4788	383	20	thus	thus	ADV
ejpam-4788	383	21	,	,	PUNCT
ejpam-4788	383	22	supp(ς	supp(ς	NOUN
ejpam-4788	383	23	)	)	PUNCT
ejpam-4788	383	24	⊆	⊆	NUM
ejpam-4788	383	25	supp(λk	supp(λk	NOUN
ejpam-4788	383	26	)	)	PUNCT
ejpam-4788	383	27	=	=	SYM
ejpam-4788	383	28	k.	k.	NOUN
ejpam-4788	383	29	by	by	ADP
ejpam-4788	383	30	assumption	assumption	NOUN
ejpam-4788	383	31	,	,	PUNCT
ejpam-4788	383	32	supp(ς	supp(ς	NOUN
ejpam-4788	383	33	)	)	PUNCT
ejpam-4788	384	1	=	=	SYM
ejpam-4788	384	2	k	k	NOUN
ejpam-4788	384	3	=	=	PUNCT
ejpam-4788	384	4	supp(λk	supp(λk	NOUN
ejpam-4788	384	5	)	)	PUNCT
ejpam-4788	384	6	.	.	PUNCT
ejpam-4788	385	1	thus	thus	ADV
ejpam-4788	385	2	,	,	PUNCT
ejpam-4788	385	3	λk	λk	X
ejpam-4788	385	4	=	=	SYM
ejpam-4788	385	5	(	(	PUNCT
ejpam-4788	385	6	s;λp	s;λp	PROPN
ejpam-4788	385	7	k	k	PROPN
ejpam-4788	385	8	,	,	PUNCT
ejpam-4788	385	9	λn	λn	PROPN
ejpam-4788	385	10	k	k	X
ejpam-4788	385	11	)	)	PUNCT
ejpam-4788	385	12	is	be	AUX
ejpam-4788	385	13	a	a	DET
ejpam-4788	385	14	minimal	minimal	ADJ
ejpam-4788	385	15	bf	bf	NOUN
ejpam-4788	385	16	almost	almost	ADV
ejpam-4788	385	17	(	(	PUNCT
ejpam-4788	385	18	α	α	NOUN
ejpam-4788	385	19	,	,	PUNCT
ejpam-4788	385	20	β)-quasi	β)-quasi	NOUN
ejpam-4788	385	21	-	-	NOUN
ejpam-4788	385	22	ideal	ideal	NOUN
ejpam-4788	385	23	of	of	ADP
ejpam-4788	385	24	s.	s.	PROPN
ejpam-4788	385	25	conversely	conversely	ADV
ejpam-4788	385	26	,	,	PUNCT
ejpam-4788	385	27	suppose	suppose	VERB
ejpam-4788	385	28	that	that	SCONJ
ejpam-4788	385	29	λk	λk	PROPN
ejpam-4788	385	30	=	=	SYM
ejpam-4788	385	31	(	(	PUNCT
ejpam-4788	385	32	s;λp	s;λp	PROPN
ejpam-4788	385	33	k	k	PROPN
ejpam-4788	385	34	,	,	PUNCT
ejpam-4788	385	35	λn	λn	PROPN
ejpam-4788	385	36	k	k	X
ejpam-4788	385	37	)	)	PUNCT
ejpam-4788	385	38	is	be	AUX
ejpam-4788	385	39	a	a	DET
ejpam-4788	385	40	minimal	minimal	ADJ
ejpam-4788	385	41	bf	bf	NOUN
ejpam-4788	385	42	almost	almost	ADV
ejpam-4788	385	43	(	(	PUNCT
ejpam-4788	385	44	α	α	NOUN
ejpam-4788	385	45	,	,	PUNCT
ejpam-4788	385	46	β)-quasi	β)-quasi	NOUN
ejpam-4788	385	47	-	-	PUNCT
ejpam-4788	385	48	ideal	ideal	NOUN
ejpam-4788	385	49	of	of	ADP
ejpam-4788	385	50	s.	s.	PROPN
ejpam-4788	385	51	then	then	ADV
ejpam-4788	385	52	,	,	PUNCT
ejpam-4788	385	53	by	by	ADP
ejpam-4788	385	54	theorem	theorem	NOUN
ejpam-4788	385	55	17	17	NUM
ejpam-4788	385	56	,	,	PUNCT
ejpam-4788	385	57	k	k	PROPN
ejpam-4788	385	58	is	be	AUX
ejpam-4788	385	59	an	an	DET
ejpam-4788	385	60	almost	almost	ADV
ejpam-4788	385	61	(	(	PUNCT
ejpam-4788	385	62	α	α	NOUN
ejpam-4788	385	63	,	,	PUNCT
ejpam-4788	385	64	β)-quasi	β)-quasi	NOUN
ejpam-4788	385	65	-	-	PUNCT
ejpam-4788	385	66	ideal	ideal	NOUN
ejpam-4788	385	67	of	of	ADP
ejpam-4788	385	68	s.	s.	PROPN
ejpam-4788	385	69	let	let	VERB
ejpam-4788	385	70	j	j	PROPN
ejpam-4788	385	71	be	be	AUX
ejpam-4788	385	72	an	an	PRON
ejpam-4788	385	73	almost	almost	ADV
ejpam-4788	385	74	(	(	PUNCT
ejpam-4788	385	75	α	α	NOUN
ejpam-4788	385	76	,	,	PUNCT
ejpam-4788	385	77	β)-quasi	β)-quasi	NOUN
ejpam-4788	385	78	-	-	NOUN
ejpam-4788	385	79	ideal	ideal	NOUN
ejpam-4788	385	80	of	of	ADP
ejpam-4788	385	81	s	s	PRON
ejpam-4788	385	82	such	such	ADJ
ejpam-4788	385	83	that	that	SCONJ
ejpam-4788	385	84	j	j	PROPN
ejpam-4788	385	85	⊆	⊆	NUM
ejpam-4788	385	86	k.	k.	PROPN
ejpam-4788	385	87	then	then	ADV
ejpam-4788	385	88	,	,	PUNCT
ejpam-4788	385	89	by	by	ADP
ejpam-4788	385	90	theorem	theorem	NOUN
ejpam-4788	385	91	17	17	NUM
ejpam-4788	385	92	,	,	PUNCT
ejpam-4788	385	93	λj	λj	PROPN
ejpam-4788	385	94	=	=	SYM
ejpam-4788	385	95	(	(	PUNCT
ejpam-4788	385	96	s;λp	s;λp	PROPN
ejpam-4788	385	97	j	j	PROPN
ejpam-4788	385	98	,	,	PUNCT
ejpam-4788	385	99	λ	λ	PROPN
ejpam-4788	385	100	n	n	CCONJ
ejpam-4788	385	101	j	j	NOUN
ejpam-4788	385	102	)	)	PUNCT
ejpam-4788	385	103	is	be	AUX
ejpam-4788	385	104	a	a	DET
ejpam-4788	385	105	bf	bf	NOUN
ejpam-4788	385	106	(	(	PUNCT
ejpam-4788	385	107	α	α	NOUN
ejpam-4788	385	108	,	,	PUNCT
ejpam-4788	385	109	β)-quasi	β)-quasi	NOUN
ejpam-4788	385	110	-	-	NOUN
ejpam-4788	385	111	ideal	ideal	NOUN
ejpam-4788	385	112	of	of	ADP
ejpam-4788	385	113	s	s	PRON
ejpam-4788	385	114	such	such	ADJ
ejpam-4788	385	115	that	that	SCONJ
ejpam-4788	385	116	λj	λj	PROPN
ejpam-4788	385	117	⊆	⊆	NUM
ejpam-4788	385	118	λk	λk	NOUN
ejpam-4788	385	119	.	.	PUNCT
ejpam-4788	386	1	thus	thus	ADV
ejpam-4788	386	2	,	,	PUNCT
ejpam-4788	386	3	j	j	PROPN
ejpam-4788	386	4	=	=	PUNCT
ejpam-4788	386	5	supp(λj	supp(λj	PROPN
ejpam-4788	386	6	)	)	PUNCT
ejpam-4788	386	7	=	=	SYM
ejpam-4788	386	8	supp(λk	supp(λk	NOUN
ejpam-4788	386	9	)	)	PUNCT
ejpam-4788	386	10	=	=	SYM
ejpam-4788	386	11	k.	k.	PROPN
ejpam-4788	387	1	hence	hence	ADV
ejpam-4788	387	2	,	,	PUNCT
ejpam-4788	387	3	k	k	PROPN
ejpam-4788	387	4	is	be	AUX
ejpam-4788	387	5	a	a	DET
ejpam-4788	387	6	minimal	minimal	ADJ
ejpam-4788	387	7	almost	almost	ADV
ejpam-4788	387	8	(	(	PUNCT
ejpam-4788	387	9	α	α	NOUN
ejpam-4788	387	10	,	,	PUNCT
ejpam-4788	387	11	β)-quasi	β)-quasi	NOUN
ejpam-4788	387	12	-	-	NOUN
ejpam-4788	387	13	ideal	ideal	NOUN
ejpam-4788	387	14	of	of	ADP
ejpam-4788	387	15	s.	s.	PROPN
ejpam-4788	387	16	corollary	corollary	PROPN
ejpam-4788	387	17	2	2	PROPN
ejpam-4788	387	18	.	.	PUNCT
ejpam-4788	388	1	let	let	VERB
ejpam-4788	388	2	s	s	PRON
ejpam-4788	388	3	be	be	AUX
ejpam-4788	388	4	a	a	DET
ejpam-4788	388	5	γ	γ	NOUN
ejpam-4788	388	6	-	-	PUNCT
ejpam-4788	388	7	semigroup	semigroup	NOUN
ejpam-4788	388	8	.	.	PUNCT
ejpam-4788	389	1	then	then	ADV
ejpam-4788	389	2	s	s	VERB
ejpam-4788	389	3	has	have	VERB
ejpam-4788	389	4	no	no	DET
ejpam-4788	389	5	proper	proper	ADJ
ejpam-4788	389	6	almost	almost	ADV
ejpam-4788	389	7	(	(	PUNCT
ejpam-4788	389	8	α	α	NOUN
ejpam-4788	389	9	,	,	PUNCT
ejpam-4788	389	10	β)-quasi	β)-quasi	NOUN
ejpam-4788	389	11	-	-	NOUN
ejpam-4788	389	12	ideal	ideal	NOUN
ejpam-4788	389	13	of	of	ADP
ejpam-4788	389	14	s	s	PRON
ejpam-4788	389	15	if	if	SCONJ
ejpam-4788	389	16	and	and	CCONJ
ejpam-4788	389	17	only	only	ADV
ejpam-4788	389	18	if	if	SCONJ
ejpam-4788	389	19	for	for	ADP
ejpam-4788	389	20	any	any	DET
ejpam-4788	389	21	bf	bf	NOUN
ejpam-4788	389	22	almost	almost	ADV
ejpam-4788	389	23	(	(	PUNCT
ejpam-4788	389	24	α	α	NOUN
ejpam-4788	389	25	,	,	PUNCT
ejpam-4788	389	26	β)-quasi	β)-quasi	NOUN
ejpam-4788	389	27	-	-	PUNCT
ejpam-4788	389	28	ideal	ideal	NOUN
ejpam-4788	389	29	ξ	ξ	NOUN
ejpam-4788	389	30	=	=	SYM
ejpam-4788	389	31	(	(	PUNCT
ejpam-4788	389	32	s	s	PROPN
ejpam-4788	389	33	;	;	PUNCT
ejpam-4788	389	34	ξp	ξp	NUM
ejpam-4788	389	35	,	,	PUNCT
ejpam-4788	389	36	ξn	ξn	NOUN
ejpam-4788	389	37	)	)	PUNCT
ejpam-4788	389	38	of	of	ADP
ejpam-4788	389	39	s	s	PROPN
ejpam-4788	389	40	,	,	PUNCT
ejpam-4788	389	41	supp(ξ	supp(ξ	PROPN
ejpam-4788	389	42	)	)	PUNCT
ejpam-4788	389	43	=	=	VERB
ejpam-4788	390	1	k.	k.	PROPN
ejpam-4788	391	1	next	next	ADV
ejpam-4788	391	2	,	,	PUNCT
ejpam-4788	391	3	we	we	PRON
ejpam-4788	391	4	define	define	VERB
ejpam-4788	391	5	the	the	DET
ejpam-4788	391	6	bf	bf	NOUN
ejpam-4788	391	7	almost	almost	ADV
ejpam-4788	391	8	(	(	PUNCT
ejpam-4788	391	9	α	α	NOUN
ejpam-4788	391	10	,	,	PUNCT
ejpam-4788	391	11	β)-bi	β)-bi	ADJ
ejpam-4788	391	12	-	-	PUNCT
ejpam-4788	391	13	ideals	ideal	NOUN
ejpam-4788	392	1	and	and	CCONJ
ejpam-4788	392	2	we	we	PRON
ejpam-4788	392	3	study	study	VERB
ejpam-4788	392	4	their	their	PRON
ejpam-4788	392	5	properties	property	NOUN
ejpam-4788	392	6	.	.	PUNCT
ejpam-4788	393	1	definition	definition	NOUN
ejpam-4788	393	2	18	18	NUM
ejpam-4788	393	3	.	.	PUNCT
ejpam-4788	394	1	let	let	VERB
ejpam-4788	394	2	ξ	ξ	X
ejpam-4788	394	3	=	=	SYM
ejpam-4788	394	4	(	(	PUNCT
ejpam-4788	394	5	s	s	PROPN
ejpam-4788	394	6	;	;	PUNCT
ejpam-4788	394	7	ξp	ξp	NUM
ejpam-4788	394	8	,	,	PUNCT
ejpam-4788	394	9	ξn	ξn	NOUN
ejpam-4788	394	10	)	)	PUNCT
ejpam-4788	394	11	be	be	VERB
ejpam-4788	394	12	a	a	DET
ejpam-4788	394	13	bf	bf	NOUN
ejpam-4788	394	14	set	set	NOUN
ejpam-4788	394	15	of	of	ADP
ejpam-4788	394	16	a	a	DET
ejpam-4788	394	17	γ	γ	NOUN
ejpam-4788	394	18	-	-	PUNCT
ejpam-4788	394	19	semigroup	semigroup	NOUN
ejpam-4788	394	20	s	s	PROPN
ejpam-4788	394	21	,	,	PUNCT
ejpam-4788	394	22	and	and	CCONJ
ejpam-4788	394	23	α	α	NOUN
ejpam-4788	394	24	,	,	PUNCT
ejpam-4788	394	25	β	β	PROPN
ejpam-4788	394	26	∈	∈	NOUN
ejpam-4788	394	27	γ	γ	NOUN
ejpam-4788	394	28	is	be	AUX
ejpam-4788	394	29	said	say	VERB
ejpam-4788	394	30	to	to	PART
ejpam-4788	394	31	be	be	AUX
ejpam-4788	394	32	bf	bf	NOUN
ejpam-4788	394	33	almost	almost	ADV
ejpam-4788	394	34	(	(	PUNCT
ejpam-4788	394	35	α	α	NOUN
ejpam-4788	394	36	,	,	PUNCT
ejpam-4788	394	37	β)-bi	β)-bi	NOUN
ejpam-4788	394	38	-	-	PUNCT
ejpam-4788	394	39	ideal	ideal	NOUN
ejpam-4788	394	40	of	of	ADP
ejpam-4788	394	41	s	s	PRON
ejpam-4788	394	42	if	if	SCONJ
ejpam-4788	394	43	(	(	PUNCT
ejpam-4788	394	44	ξp	ξp	PART
ejpam-4788	394	45	◦	◦	VERB
ejpam-4788	394	46	αxpt	αxpt	NOUN
ejpam-4788	394	47	◦	◦	NOUN
ejpam-4788	394	48	β	β	NOUN
ejpam-4788	394	49	ξp)∧ξp	ξp)∧ξp	NUM
ejpam-4788	394	50	̸=	̸=	PROPN
ejpam-4788	394	51	0	0	NUM
ejpam-4788	394	52	and	and	CCONJ
ejpam-4788	394	53	(	(	PUNCT
ejpam-4788	394	54	ξn	ξn	NOUN
ejpam-4788	394	55	◦	◦	NOUN
ejpam-4788	394	56	αxns	αxns	NOUN
ejpam-4788	394	57	◦	◦	NOUN
ejpam-4788	394	58	β	β	NOUN
ejpam-4788	394	59	ξn)∨ξn	ξn)∨ξn	NOUN
ejpam-4788	394	60	̸=	̸=	PROPN
ejpam-4788	394	61	0	0	NUM
ejpam-4788	394	62	.	.	PUNCT
ejpam-4788	395	1	theorem	theorem	NOUN
ejpam-4788	395	2	19	19	NUM
ejpam-4788	395	3	.	.	PUNCT
ejpam-4788	396	1	if	if	SCONJ
ejpam-4788	396	2	ξ	ξ	X
ejpam-4788	396	3	=	=	SYM
ejpam-4788	396	4	(	(	PUNCT
ejpam-4788	396	5	s	s	PROPN
ejpam-4788	396	6	;	;	PUNCT
ejpam-4788	396	7	ξp	ξp	NUM
ejpam-4788	396	8	,	,	PUNCT
ejpam-4788	396	9	ξn	ξn	NOUN
ejpam-4788	396	10	)	)	PUNCT
ejpam-4788	396	11	is	be	AUX
ejpam-4788	396	12	a	a	DET
ejpam-4788	396	13	bf	bf	NOUN
ejpam-4788	396	14	almost	almost	ADV
ejpam-4788	396	15	(	(	PUNCT
ejpam-4788	396	16	α	α	NOUN
ejpam-4788	396	17	,	,	PUNCT
ejpam-4788	396	18	β)-bi	β)-bi	NOUN
ejpam-4788	396	19	-	-	PUNCT
ejpam-4788	396	20	ideal	ideal	NOUN
ejpam-4788	396	21	of	of	ADP
ejpam-4788	396	22	a	a	DET
ejpam-4788	396	23	γ	γ	NOUN
ejpam-4788	396	24	-	-	PUNCT
ejpam-4788	396	25	semigroup	semigroup	NOUN
ejpam-4788	396	26	s	s	PROPN
ejpam-4788	396	27	,	,	PUNCT
ejpam-4788	396	28	and	and	CCONJ
ejpam-4788	396	29	ς	ς	PROPN
ejpam-4788	396	30	=	=	PUNCT
ejpam-4788	396	31	(	(	PUNCT
ejpam-4788	396	32	s	s	NOUN
ejpam-4788	396	33	;	;	PUNCT
ejpam-4788	396	34	ςp	ςp	NUM
ejpam-4788	396	35	,	,	PUNCT
ejpam-4788	396	36	ςn	ςn	NOUN
ejpam-4788	396	37	)	)	PUNCT
ejpam-4788	396	38	is	be	AUX
ejpam-4788	396	39	a	a	DET
ejpam-4788	396	40	bf	bf	NOUN
ejpam-4788	396	41	set	set	NOUN
ejpam-4788	396	42	of	of	ADP
ejpam-4788	396	43	s	s	PRON
ejpam-4788	396	44	such	such	ADJ
ejpam-4788	396	45	that	that	SCONJ
ejpam-4788	396	46	ξ	ξ	PROPN
ejpam-4788	396	47	⊆	⊆	NUM
ejpam-4788	396	48	ς	ς	NOUN
ejpam-4788	396	49	,	,	PUNCT
ejpam-4788	396	50	then	then	ADV
ejpam-4788	396	51	ς	ς	PROPN
ejpam-4788	396	52	=	=	PUNCT
ejpam-4788	396	53	(	(	PUNCT
ejpam-4788	396	54	s	s	NOUN
ejpam-4788	396	55	;	;	PUNCT
ejpam-4788	396	56	ςp	ςp	NUM
ejpam-4788	396	57	,	,	PUNCT
ejpam-4788	396	58	ςn	ςn	NOUN
ejpam-4788	396	59	)	)	PUNCT
ejpam-4788	396	60	is	be	AUX
ejpam-4788	396	61	a	a	DET
ejpam-4788	396	62	bf	bf	NOUN
ejpam-4788	396	63	(	(	PUNCT
ejpam-4788	396	64	α	α	NOUN
ejpam-4788	396	65	,	,	PUNCT
ejpam-4788	396	66	β)-bi	β)-bi	NOUN
ejpam-4788	396	67	-	-	PUNCT
ejpam-4788	396	68	ideal	ideal	NOUN
ejpam-4788	396	69	of	of	ADP
ejpam-4788	396	70	s.	s.	PROPN
ejpam-4788	396	71	proof	proof	PROPN
ejpam-4788	396	72	.	.	PUNCT
ejpam-4788	397	1	suppose	suppose	VERB
ejpam-4788	397	2	that	that	SCONJ
ejpam-4788	397	3	ξ	ξ	PROPN
ejpam-4788	397	4	=	=	SYM
ejpam-4788	397	5	(	(	PUNCT
ejpam-4788	397	6	s	s	PROPN
ejpam-4788	397	7	;	;	PUNCT
ejpam-4788	397	8	ξp	ξp	NUM
ejpam-4788	397	9	,	,	PUNCT
ejpam-4788	397	10	ξn	ξn	NOUN
ejpam-4788	397	11	)	)	PUNCT
ejpam-4788	397	12	is	be	AUX
ejpam-4788	397	13	a	a	DET
ejpam-4788	397	14	bf	bf	NOUN
ejpam-4788	397	15	almost	almost	ADV
ejpam-4788	397	16	(	(	PUNCT
ejpam-4788	397	17	α	α	NOUN
ejpam-4788	397	18	,	,	PUNCT
ejpam-4788	397	19	β)-bi	β)-bi	NOUN
ejpam-4788	397	20	-	-	PUNCT
ejpam-4788	397	21	ideal	ideal	NOUN
ejpam-4788	397	22	of	of	ADP
ejpam-4788	397	23	s	s	PROPN
ejpam-4788	397	24	,	,	PUNCT
ejpam-4788	397	25	and	and	CCONJ
ejpam-4788	397	26	ς	ς	PROPN
ejpam-4788	397	27	=	=	PUNCT
ejpam-4788	397	28	(	(	PUNCT
ejpam-4788	397	29	s	s	NOUN
ejpam-4788	397	30	;	;	PUNCT
ejpam-4788	397	31	ςp	ςp	NUM
ejpam-4788	397	32	,	,	PUNCT
ejpam-4788	397	33	ςn	ςn	NOUN
ejpam-4788	397	34	)	)	PUNCT
ejpam-4788	397	35	is	be	AUX
ejpam-4788	397	36	a	a	DET
ejpam-4788	397	37	bf	bf	NOUN
ejpam-4788	397	38	set	set	NOUN
ejpam-4788	397	39	of	of	ADP
ejpam-4788	397	40	s	s	PRON
ejpam-4788	397	41	such	such	ADJ
ejpam-4788	397	42	that	that	SCONJ
ejpam-4788	397	43	ξ	ξ	PROPN
ejpam-4788	397	44	⊆	⊆	NUM
ejpam-4788	397	45	ς	ς	X
ejpam-4788	397	46	.	.	PUNCT
ejpam-4788	398	1	then	then	ADV
ejpam-4788	398	2	(	(	PUNCT
ejpam-4788	398	3	ξp	ξp	AUX
ejpam-4788	398	4	◦	◦	NOUN
ejpam-4788	398	5	α	α	NOUN
ejpam-4788	398	6	xpt	xpt	PROPN
ejpam-4788	398	7	◦	◦	PROPN
ejpam-4788	398	8	β	β	X
ejpam-4788	398	9	ξp)∧	ξp)∧	NOUN
ejpam-4788	398	10	ξp	ξp	ADP
ejpam-4788	398	11	̸=	̸=	PROPN
ejpam-4788	398	12	0	0	NUM
ejpam-4788	398	13	,	,	PUNCT
ejpam-4788	398	14	and	and	CCONJ
ejpam-4788	398	15	(	(	PUNCT
ejpam-4788	398	16	ξn	ξn	PROPN
ejpam-4788	398	17	◦	◦	NOUN
ejpam-4788	398	18	α	α	PROPN
ejpam-4788	398	19	xns	xns	PROPN
ejpam-4788	398	20	◦	◦	PROPN
ejpam-4788	398	21	β	β	NOUN
ejpam-4788	398	22	ξn)∨ξn	ξn)∨ξn	PROPN
ejpam-4788	398	23	̸=	̸=	PROPN
ejpam-4788	398	24	0	0	NUM
ejpam-4788	398	25	.	.	PUNCT
ejpam-4788	399	1	thus	thus	ADV
ejpam-4788	399	2	,	,	PUNCT
ejpam-4788	399	3	(	(	PUNCT
ejpam-4788	399	4	ξp	ξp	PART
ejpam-4788	399	5	◦	◦	NOUN
ejpam-4788	399	6	αxpt	αxpt	NOUN
ejpam-4788	399	7	◦	◦	NOUN
ejpam-4788	399	8	β	β	NOUN
ejpam-4788	399	9	ξp)∧ξp	ξp)∧ξp	NOUN
ejpam-4788	399	10	⊆	⊆	NUM
ejpam-4788	399	11	(	(	PUNCT
ejpam-4788	399	12	ςp	ςp	NUM
ejpam-4788	399	13	◦	◦	NOUN
ejpam-4788	399	14	αxpt	αxpt	NOUN
ejpam-4788	399	15	◦	◦	NOUN
ejpam-4788	399	16	β	β	NOUN
ejpam-4788	399	17	ςp)∧ςp	ςp)∧ςp	NUM
ejpam-4788	399	18	̸=	̸=	PROPN
ejpam-4788	399	19	0	0	NUM
ejpam-4788	399	20	,	,	PUNCT
ejpam-4788	399	21	and	and	CCONJ
ejpam-4788	399	22	(	(	PUNCT
ejpam-4788	399	23	ξn	ξn	NOUN
ejpam-4788	399	24	◦	◦	NOUN
ejpam-4788	399	25	αxns	αxns	NOUN
ejpam-4788	399	26	◦	◦	NOUN
ejpam-4788	399	27	β	β	NOUN
ejpam-4788	399	28	ξn)∨ξn	ξn)∨ξn	NOUN
ejpam-4788	399	29	⊆	⊆	NUM
ejpam-4788	399	30	(	(	PUNCT
ejpam-4788	399	31	ςn	ςn	NOUN
ejpam-4788	399	32	◦	◦	NOUN
ejpam-4788	399	33	α	α	X
ejpam-4788	399	34	xns	xns	PROPN
ejpam-4788	399	35	◦	◦	PROPN
ejpam-4788	399	36	β	β	X
ejpam-4788	399	37	ςn	ςn	NOUN
ejpam-4788	399	38	)	)	PUNCT
ejpam-4788	399	39	∨	∨	NOUN
ejpam-4788	399	40	ςn	ςn	ADP
ejpam-4788	399	41	̸=	̸=	PROPN
ejpam-4788	399	42	0	0	NUM
ejpam-4788	399	43	.	.	PUNCT
ejpam-4788	400	1	hence	hence	ADV
ejpam-4788	400	2	,	,	PUNCT
ejpam-4788	400	3	ς	ς	PROPN
ejpam-4788	400	4	=	=	PUNCT
ejpam-4788	400	5	(	(	PUNCT
ejpam-4788	400	6	s	s	NOUN
ejpam-4788	400	7	;	;	PUNCT
ejpam-4788	400	8	ςp	ςp	NUM
ejpam-4788	400	9	,	,	PUNCT
ejpam-4788	400	10	ςn	ςn	NOUN
ejpam-4788	400	11	)	)	PUNCT
ejpam-4788	400	12	is	be	AUX
ejpam-4788	400	13	a	a	DET
ejpam-4788	400	14	bf	bf	NOUN
ejpam-4788	400	15	(	(	PUNCT
ejpam-4788	400	16	α	α	NOUN
ejpam-4788	400	17	,	,	PUNCT
ejpam-4788	400	18	β)-bi	β)-bi	NOUN
ejpam-4788	400	19	-	-	PUNCT
ejpam-4788	400	20	ideal	ideal	NOUN
ejpam-4788	400	21	of	of	ADP
ejpam-4788	400	22	s.	s.	PROPN
ejpam-4788	400	23	p.	p.	PROPN
ejpam-4788	400	24	khamrot	khamrot	PROPN
ejpam-4788	400	25	,	,	PUNCT
ejpam-4788	400	26	t.	t.	PROPN
ejpam-4788	400	27	gaketem	gaketem	PROPN
ejpam-4788	400	28	/	/	SYM
ejpam-4788	400	29	eur	eur	PROPN
ejpam-4788	400	30	.	.	PUNCT
ejpam-4788	401	1	j.	j.	PROPN
ejpam-4788	401	2	pure	pure	PROPN
ejpam-4788	401	3	appl	appl	PROPN
ejpam-4788	401	4	.	.	PROPN
ejpam-4788	401	5	math	math	PROPN
ejpam-4788	401	6	,	,	PUNCT
ejpam-4788	401	7	16	16	NUM
ejpam-4788	401	8	(	(	PUNCT
ejpam-4788	401	9	3	3	NUM
ejpam-4788	401	10	)	)	PUNCT
ejpam-4788	401	11	(	(	PUNCT
ejpam-4788	401	12	2023	2023	NUM
ejpam-4788	401	13	)	)	PUNCT
ejpam-4788	401	14	,	,	PUNCT
ejpam-4788	401	15	1592	1592	NUM
ejpam-4788	401	16	-	-	SYM
ejpam-4788	401	17	1607	1607	NUM
ejpam-4788	401	18	1605	1605	NUM
ejpam-4788	401	19	theorem	theorem	VERB
ejpam-4788	401	20	20	20	NUM
ejpam-4788	401	21	.	.	PUNCT
ejpam-4788	402	1	let	let	VERB
ejpam-4788	402	2	k	k	PRON
ejpam-4788	402	3	be	be	AUX
ejpam-4788	402	4	a	a	DET
ejpam-4788	402	5	non	non	ADJ
ejpam-4788	402	6	-	-	ADJ
ejpam-4788	402	7	empty	empty	ADJ
ejpam-4788	402	8	subset	subset	NOUN
ejpam-4788	402	9	of	of	ADP
ejpam-4788	402	10	γ	γ	PROPN
ejpam-4788	402	11	-	-	PUNCT
ejpam-4788	402	12	semigroup	semigroup	PROPN
ejpam-4788	402	13	s.	s.	PROPN
ejpam-4788	403	1	then	then	ADV
ejpam-4788	403	2	k	k	PROPN
ejpam-4788	403	3	is	be	AUX
ejpam-4788	403	4	an	an	DET
ejpam-4788	403	5	almost	almost	ADV
ejpam-4788	403	6	(	(	PUNCT
ejpam-4788	403	7	α	α	NOUN
ejpam-4788	403	8	,	,	PUNCT
ejpam-4788	403	9	β)-bi	β)-bi	NOUN
ejpam-4788	403	10	-	-	PUNCT
ejpam-4788	403	11	ideal	ideal	NOUN
ejpam-4788	403	12	of	of	ADP
ejpam-4788	403	13	s	s	PRON
ejpam-4788	403	14	if	if	SCONJ
ejpam-4788	403	15	and	and	CCONJ
ejpam-4788	403	16	only	only	ADV
ejpam-4788	403	17	if	if	SCONJ
ejpam-4788	403	18	the	the	DET
ejpam-4788	403	19	characteristic	characteristic	ADJ
ejpam-4788	403	20	function	function	NOUN
ejpam-4788	403	21	λk	λk	X
ejpam-4788	403	22	=	=	PUNCT
ejpam-4788	403	23	(	(	PUNCT
ejpam-4788	403	24	s;λp	s;λp	PROPN
ejpam-4788	403	25	k	k	PROPN
ejpam-4788	403	26	,	,	PUNCT
ejpam-4788	403	27	λn	λn	PROPN
ejpam-4788	403	28	k	k	X
ejpam-4788	403	29	)	)	PUNCT
ejpam-4788	403	30	is	be	AUX
ejpam-4788	403	31	a	a	DET
ejpam-4788	403	32	bf	bf	NOUN
ejpam-4788	403	33	almost	almost	ADV
ejpam-4788	403	34	(	(	PUNCT
ejpam-4788	403	35	α	α	NOUN
ejpam-4788	403	36	,	,	PUNCT
ejpam-4788	403	37	β)-bi	β)-bi	NOUN
ejpam-4788	403	38	-	-	PUNCT
ejpam-4788	403	39	ideal	ideal	NOUN
ejpam-4788	403	40	of	of	ADP
ejpam-4788	403	41	s.	s.	PROPN
ejpam-4788	403	42	proof	proof	PROPN
ejpam-4788	403	43	.	.	PUNCT
ejpam-4788	404	1	suppose	suppose	VERB
ejpam-4788	404	2	that	that	SCONJ
ejpam-4788	404	3	k	k	PROPN
ejpam-4788	404	4	is	be	AUX
ejpam-4788	404	5	an	an	DET
ejpam-4788	404	6	almost	almost	ADV
ejpam-4788	404	7	(	(	PUNCT
ejpam-4788	404	8	α	α	NOUN
ejpam-4788	404	9	,	,	PUNCT
ejpam-4788	404	10	β)-bi	β)-bi	NOUN
ejpam-4788	404	11	-	-	PUNCT
ejpam-4788	404	12	ideal	ideal	NOUN
ejpam-4788	404	13	of	of	ADP
ejpam-4788	404	14	s.	s.	PROPN
ejpam-4788	404	15	then	then	ADV
ejpam-4788	404	16	kαuβk	kαuβk	PROPN
ejpam-4788	404	17	∩k	∩k	PROPN
ejpam-4788	404	18	̸=	̸=	PROPN
ejpam-4788	404	19	∅	∅	NOUN
ejpam-4788	404	20	for	for	ADP
ejpam-4788	404	21	all	all	PRON
ejpam-4788	404	22	u	u	NOUN
ejpam-4788	404	23	∈	∈	PROPN
ejpam-4788	404	24	s.	s.	PROPN
ejpam-4788	404	25	thus	thus	ADV
ejpam-4788	404	26	,	,	PUNCT
ejpam-4788	404	27	there	there	PRON
ejpam-4788	404	28	exists	exist	VERB
ejpam-4788	404	29	v	v	ADP
ejpam-4788	404	30	∈	∈	PROPN
ejpam-4788	404	31	kαuβk	kαuβk	NOUN
ejpam-4788	404	32	and	and	CCONJ
ejpam-4788	404	33	v	v	ADP
ejpam-4788	404	34	∈	∈	PROPN
ejpam-4788	404	35	k.	k.	NOUN
ejpam-4788	405	1	so	so	ADV
ejpam-4788	405	2	,	,	PUNCT
ejpam-4788	405	3	(	(	PUNCT
ejpam-4788	405	4	(	(	PUNCT
ejpam-4788	405	5	λp	λp	X
ejpam-4788	405	6	k	k	PROPN
ejpam-4788	405	7	◦	◦	PROPN
ejpam-4788	405	8	α	α	X
ejpam-4788	405	9	xpt	xpt	PROPN
ejpam-4788	405	10	◦	◦	PROPN
ejpam-4788	405	11	β	β	NOUN
ejpam-4788	405	12	λ	λ	X
ejpam-4788	405	13	p	p	X
ejpam-4788	405	14	k))(v	k))(v	PROPN
ejpam-4788	405	15	)	)	PUNCT
ejpam-4788	406	1	=	=	SYM
ejpam-4788	407	1	λp	λp	PROPN
ejpam-4788	407	2	k(v	k(v	PROPN
ejpam-4788	407	3	)	)	PUNCT
ejpam-4788	407	4	=	=	SYM
ejpam-4788	407	5	1	1	NUM
ejpam-4788	407	6	,	,	PUNCT
ejpam-4788	407	7	and	and	CCONJ
ejpam-4788	407	8	(	(	PUNCT
ejpam-4788	407	9	(	(	PUNCT
ejpam-4788	407	10	λn	λn	X
ejpam-4788	407	11	k	k	PROPN
ejpam-4788	407	12	◦	◦	NOUN
ejpam-4788	407	13	α	α	PROPN
ejpam-4788	407	14	xns	xns	PROPN
ejpam-4788	407	15	◦	◦	PROPN
ejpam-4788	407	16	β	β	X
ejpam-4788	407	17	λn	λn	NOUN
ejpam-4788	407	18	k))(v	k))(v	NOUN
ejpam-4788	407	19	)	)	PUNCT
ejpam-4788	408	1	=	=	PROPN
ejpam-4788	408	2	λn	λn	PROPN
ejpam-4788	408	3	k(v	k(v	PROPN
ejpam-4788	408	4	)	)	PUNCT
ejpam-4788	408	5	=	=	PUNCT
ejpam-4788	408	6	−1	−1	NOUN
ejpam-4788	408	7	.	.	PUNCT
ejpam-4788	409	1	hence	hence	ADV
ejpam-4788	409	2	,	,	PUNCT
ejpam-4788	409	3	(	(	PUNCT
ejpam-4788	409	4	λp	λp	X
ejpam-4788	409	5	k	k	PROPN
ejpam-4788	409	6	◦	◦	PROPN
ejpam-4788	409	7	α	α	X
ejpam-4788	409	8	xpt	xpt	PROPN
ejpam-4788	409	9	◦	◦	PROPN
ejpam-4788	409	10	β	β	X
ejpam-4788	409	11	λp	λp	X
ejpam-4788	409	12	k	k	NOUN
ejpam-4788	409	13	)	)	PUNCT
ejpam-4788	409	14	∧	∧	PROPN
ejpam-4788	409	15	λp	λp	ADP
ejpam-4788	409	16	k	k	PROPN
ejpam-4788	409	17	̸=	̸=	PROPN
ejpam-4788	409	18	0	0	PUNCT
ejpam-4788	410	1	and	and	CCONJ
ejpam-4788	410	2	(	(	PUNCT
ejpam-4788	410	3	λn	λn	X
ejpam-4788	410	4	k	k	PROPN
ejpam-4788	410	5	◦	◦	NOUN
ejpam-4788	410	6	α	α	PROPN
ejpam-4788	410	7	xns	xns	PROPN
ejpam-4788	410	8	◦	◦	PROPN
ejpam-4788	410	9	β	β	X
ejpam-4788	410	10	λn	λn	NOUN
ejpam-4788	410	11	k	k	X
ejpam-4788	410	12	)	)	PUNCT
ejpam-4788	410	13	∨	∨	NUM
ejpam-4788	410	14	λn	λn	PROPN
ejpam-4788	410	15	k	k	PROPN
ejpam-4788	410	16	̸=	̸=	PROPN
ejpam-4788	410	17	0	0	NUM
ejpam-4788	410	18	.	.	PUNCT
ejpam-4788	411	1	therefore	therefore	ADV
ejpam-4788	411	2	,	,	PUNCT
ejpam-4788	411	3	λk	λk	X
ejpam-4788	411	4	=	=	PUNCT
ejpam-4788	411	5	(	(	PUNCT
ejpam-4788	411	6	s;λp	s;λp	PROPN
ejpam-4788	411	7	k	k	PROPN
ejpam-4788	411	8	,	,	PUNCT
ejpam-4788	411	9	λn	λn	PROPN
ejpam-4788	411	10	k	k	X
ejpam-4788	411	11	)	)	PUNCT
ejpam-4788	411	12	is	be	AUX
ejpam-4788	411	13	a	a	DET
ejpam-4788	411	14	bf	bf	NOUN
ejpam-4788	411	15	almost	almost	ADV
ejpam-4788	411	16	(	(	PUNCT
ejpam-4788	411	17	α	α	NOUN
ejpam-4788	411	18	,	,	PUNCT
ejpam-4788	411	19	β)-bi	β)-bi	NOUN
ejpam-4788	411	20	-	-	PUNCT
ejpam-4788	411	21	ideal	ideal	NOUN
ejpam-4788	411	22	of	of	ADP
ejpam-4788	411	23	s.	s.	PROPN
ejpam-4788	411	24	conversely	conversely	ADV
ejpam-4788	411	25	,	,	PUNCT
ejpam-4788	411	26	assume	assume	VERB
ejpam-4788	411	27	that	that	SCONJ
ejpam-4788	411	28	λk	λk	ADV
ejpam-4788	411	29	=	=	SYM
ejpam-4788	411	30	(	(	PUNCT
ejpam-4788	411	31	s;λp	s;λp	PROPN
ejpam-4788	411	32	k	k	PROPN
ejpam-4788	411	33	,	,	PUNCT
ejpam-4788	411	34	λn	λn	PROPN
ejpam-4788	411	35	k	k	X
ejpam-4788	411	36	)	)	PUNCT
ejpam-4788	411	37	is	be	AUX
ejpam-4788	411	38	a	a	DET
ejpam-4788	411	39	bf	bf	NOUN
ejpam-4788	411	40	almost	almost	ADV
ejpam-4788	411	41	(	(	PUNCT
ejpam-4788	411	42	α	α	NOUN
ejpam-4788	411	43	,	,	PUNCT
ejpam-4788	411	44	β)-bi	β)-bi	NOUN
ejpam-4788	411	45	-	-	PUNCT
ejpam-4788	411	46	ideal	ideal	NOUN
ejpam-4788	411	47	of	of	ADP
ejpam-4788	411	48	s	s	PROPN
ejpam-4788	411	49	,	,	PUNCT
ejpam-4788	411	50	and	and	CCONJ
ejpam-4788	411	51	u	u	PROPN
ejpam-4788	411	52	∈	∈	PROPN
ejpam-4788	411	53	s.	s.	PROPN
ejpam-4788	411	54	then	then	ADV
ejpam-4788	412	1	(	(	PUNCT
ejpam-4788	412	2	λp	λp	X
ejpam-4788	412	3	k	k	PROPN
ejpam-4788	412	4	◦	◦	PROPN
ejpam-4788	412	5	α	α	X
ejpam-4788	412	6	xpt	xpt	PROPN
ejpam-4788	412	7	◦	◦	PROPN
ejpam-4788	412	8	β	β	NOUN
ejpam-4788	412	9	λ	λ	X
ejpam-4788	412	10	p	p	X
ejpam-4788	412	11	k)∧λp	k)∧λp	PROPN
ejpam-4788	412	12	k	k	PROPN
ejpam-4788	412	13	̸=	̸=	PROPN
ejpam-4788	412	14	0	0	NUM
ejpam-4788	412	15	,	,	PUNCT
ejpam-4788	412	16	and	and	CCONJ
ejpam-4788	412	17	(	(	PUNCT
ejpam-4788	412	18	λn	λn	NOUN
ejpam-4788	412	19	k	k	PROPN
ejpam-4788	412	20	◦	◦	NOUN
ejpam-4788	412	21	α	α	PROPN
ejpam-4788	412	22	xns	xns	PROPN
ejpam-4788	412	23	◦	◦	PROPN
ejpam-4788	412	24	β	β	X
ejpam-4788	412	25	λn	λn	X
ejpam-4788	412	26	k)∨λn	k)∨λn	PROPN
ejpam-4788	412	27	k	k	PROPN
ejpam-4788	412	28	̸=	̸=	PROPN
ejpam-4788	412	29	0	0	NUM
ejpam-4788	412	30	.	.	PUNCT
ejpam-4788	413	1	thus	thus	ADV
ejpam-4788	413	2	,	,	PUNCT
ejpam-4788	413	3	there	there	PRON
ejpam-4788	413	4	exists	exist	VERB
ejpam-4788	413	5	r	r	NOUN
ejpam-4788	413	6	∈	∈	PROPN
ejpam-4788	413	7	s	s	VERB
ejpam-4788	413	8	such	such	ADJ
ejpam-4788	413	9	that	that	SCONJ
ejpam-4788	413	10	(	(	PUNCT
ejpam-4788	413	11	(	(	PUNCT
ejpam-4788	413	12	λp	λp	X
ejpam-4788	413	13	k	k	PROPN
ejpam-4788	413	14	◦	◦	PROPN
ejpam-4788	413	15	αxpt	αxpt	PROPN
ejpam-4788	413	16	◦	◦	VERB
ejpam-4788	413	17	β	β	NOUN
ejpam-4788	413	18	λ	λ	X
ejpam-4788	413	19	p	p	X
ejpam-4788	413	20	k)∧λp	k)∧λp	PROPN
ejpam-4788	413	21	k)(r	k)(r	NOUN
ejpam-4788	413	22	)	)	PUNCT
ejpam-4788	413	23	̸=	̸=	PROPN
ejpam-4788	413	24	0	0	NUM
ejpam-4788	413	25	,	,	PUNCT
ejpam-4788	413	26	and	and	CCONJ
ejpam-4788	413	27	(	(	PUNCT
ejpam-4788	413	28	(	(	PUNCT
ejpam-4788	413	29	λn	λn	NOUN
ejpam-4788	413	30	k	k	PROPN
ejpam-4788	413	31	◦	◦	NOUN
ejpam-4788	413	32	αxns	αxns	NOUN
ejpam-4788	413	33	◦	◦	NOUN
ejpam-4788	413	34	β	β	X
ejpam-4788	413	35	λn	λn	NOUN
ejpam-4788	413	36	k)∨λn	k)∨λn	PROPN
ejpam-4788	413	37	k)(r	k)(r	PROPN
ejpam-4788	413	38	)	)	PUNCT
ejpam-4788	413	39	̸=	̸=	PROPN
ejpam-4788	413	40	0	0	NUM
ejpam-4788	413	41	.	.	PUNCT
ejpam-4788	414	1	hence	hence	ADV
ejpam-4788	414	2	,	,	PUNCT
ejpam-4788	414	3	r	r	NOUN
ejpam-4788	414	4	∈	∈	PROPN
ejpam-4788	414	5	kαuβk	kαuβk	NOUN
ejpam-4788	414	6	∩k	∩k	PROPN
ejpam-4788	414	7	implies	imply	VERB
ejpam-4788	414	8	that	that	SCONJ
ejpam-4788	414	9	kαuβk	kαuβk	PROPN
ejpam-4788	414	10	∩k	∩k	PROPN
ejpam-4788	414	11	̸=	̸=	PROPN
ejpam-4788	414	12	∅.	∅.	VERB
ejpam-4788	414	13	therefore	therefore	ADV
ejpam-4788	414	14	,	,	PUNCT
ejpam-4788	414	15	k	k	PROPN
ejpam-4788	414	16	is	be	AUX
ejpam-4788	414	17	an	an	DET
ejpam-4788	414	18	almost	almost	ADV
ejpam-4788	414	19	(	(	PUNCT
ejpam-4788	414	20	α	α	NOUN
ejpam-4788	414	21	,	,	PUNCT
ejpam-4788	414	22	β)-bi	β)-bi	NOUN
ejpam-4788	414	23	-	-	PUNCT
ejpam-4788	414	24	ideal	ideal	NOUN
ejpam-4788	414	25	of	of	ADP
ejpam-4788	414	26	s.	s.	PROPN
ejpam-4788	414	27	next	next	ADV
ejpam-4788	414	28	,	,	PUNCT
ejpam-4788	414	29	we	we	PRON
ejpam-4788	414	30	study	study	VERB
ejpam-4788	414	31	the	the	DET
ejpam-4788	414	32	properties	property	NOUN
ejpam-4788	414	33	between	between	ADP
ejpam-4788	414	34	supp(ξ	supp(ξ	PROPN
ejpam-4788	414	35	)	)	PUNCT
ejpam-4788	414	36	and	and	CCONJ
ejpam-4788	414	37	a	a	DET
ejpam-4788	414	38	bf	bf	NOUN
ejpam-4788	414	39	almost	almost	ADV
ejpam-4788	414	40	(	(	PUNCT
ejpam-4788	414	41	α	α	NOUN
ejpam-4788	414	42	,	,	PUNCT
ejpam-4788	414	43	β)-bi	β)-bi	ADJ
ejpam-4788	414	44	-	-	PUNCT
ejpam-4788	414	45	ideal	ideal	NOUN
ejpam-4788	414	46	of	of	ADP
ejpam-4788	414	47	γ	γ	NOUN
ejpam-4788	414	48	-	-	PUNCT
ejpam-4788	414	49	semigroups	semigroup	NOUN
ejpam-4788	414	50	.	.	PUNCT
ejpam-4788	415	1	theorem	theorem	NOUN
ejpam-4788	415	2	21	21	NUM
ejpam-4788	415	3	.	.	PUNCT
ejpam-4788	416	1	let	let	VERB
ejpam-4788	416	2	ξ	ξ	X
ejpam-4788	416	3	=	=	SYM
ejpam-4788	416	4	(	(	PUNCT
ejpam-4788	416	5	s	s	PROPN
ejpam-4788	416	6	;	;	PUNCT
ejpam-4788	416	7	ξp	ξp	NUM
ejpam-4788	416	8	,	,	PUNCT
ejpam-4788	416	9	ξn	ξn	NOUN
ejpam-4788	416	10	)	)	PUNCT
ejpam-4788	416	11	be	be	AUX
ejpam-4788	416	12	a	a	DET
ejpam-4788	416	13	fuzzy	fuzzy	ADJ
ejpam-4788	416	14	set	set	NOUN
ejpam-4788	416	15	of	of	ADP
ejpam-4788	416	16	a	a	DET
ejpam-4788	416	17	non	non	ADJ
ejpam-4788	416	18	-	-	ADJ
ejpam-4788	416	19	empty	empty	ADJ
ejpam-4788	416	20	of	of	ADP
ejpam-4788	416	21	a	a	DET
ejpam-4788	416	22	γ	γ	PROPN
ejpam-4788	416	23	-	-	PUNCT
ejpam-4788	416	24	semigroup	semigroup	PROPN
ejpam-4788	416	25	s.	s.	PROPN
ejpam-4788	416	26	then	then	ADV
ejpam-4788	416	27	ξ	ξ	X
ejpam-4788	416	28	=	=	SYM
ejpam-4788	416	29	(	(	PUNCT
ejpam-4788	416	30	s	s	PROPN
ejpam-4788	416	31	;	;	PUNCT
ejpam-4788	416	32	ξp	ξp	NUM
ejpam-4788	416	33	,	,	PUNCT
ejpam-4788	416	34	ξn	ξn	NOUN
ejpam-4788	416	35	)	)	PUNCT
ejpam-4788	416	36	is	be	AUX
ejpam-4788	416	37	a	a	DET
ejpam-4788	416	38	bf	bf	NOUN
ejpam-4788	416	39	almost	almost	ADV
ejpam-4788	416	40	(	(	PUNCT
ejpam-4788	416	41	α	α	NOUN
ejpam-4788	416	42	,	,	PUNCT
ejpam-4788	416	43	β)-bi	β)-bi	NOUN
ejpam-4788	416	44	-	-	PUNCT
ejpam-4788	416	45	ideal	ideal	NOUN
ejpam-4788	416	46	of	of	ADP
ejpam-4788	416	47	s	s	PRON
ejpam-4788	416	48	if	if	SCONJ
ejpam-4788	417	1	and	and	CCONJ
ejpam-4788	417	2	only	only	ADV
ejpam-4788	417	3	if	if	SCONJ
ejpam-4788	417	4	supp(ξ	supp(ξ	PROPN
ejpam-4788	417	5	)	)	PUNCT
ejpam-4788	417	6	is	be	AUX
ejpam-4788	417	7	an	an	DET
ejpam-4788	417	8	almost	almost	ADV
ejpam-4788	417	9	(	(	PUNCT
ejpam-4788	417	10	α	α	NOUN
ejpam-4788	417	11	,	,	PUNCT
ejpam-4788	417	12	β)-bi	β)-bi	NOUN
ejpam-4788	417	13	-	-	PUNCT
ejpam-4788	417	14	ideal	ideal	NOUN
ejpam-4788	417	15	of	of	ADP
ejpam-4788	417	16	s.	s.	PROPN
ejpam-4788	417	17	proof	proof	PROPN
ejpam-4788	417	18	.	.	PUNCT
ejpam-4788	418	1	let	let	VERB
ejpam-4788	418	2	ξ	ξ	X
ejpam-4788	418	3	=	=	SYM
ejpam-4788	418	4	(	(	PUNCT
ejpam-4788	418	5	s	s	PROPN
ejpam-4788	418	6	;	;	PUNCT
ejpam-4788	418	7	ξp	ξp	NUM
ejpam-4788	418	8	,	,	PUNCT
ejpam-4788	418	9	ξn	ξn	NOUN
ejpam-4788	418	10	)	)	PUNCT
ejpam-4788	418	11	be	be	VERB
ejpam-4788	418	12	a	a	DET
ejpam-4788	418	13	bf	bf	NOUN
ejpam-4788	418	14	almost	almost	ADV
ejpam-4788	418	15	(	(	PUNCT
ejpam-4788	418	16	α	α	NOUN
ejpam-4788	418	17	,	,	PUNCT
ejpam-4788	418	18	β)-bi	β)-bi	NOUN
ejpam-4788	418	19	-	-	PUNCT
ejpam-4788	418	20	ideal	ideal	NOUN
ejpam-4788	418	21	of	of	ADP
ejpam-4788	418	22	s	s	PROPN
ejpam-4788	418	23	,	,	PUNCT
ejpam-4788	418	24	and	and	CCONJ
ejpam-4788	418	25	u	u	PROPN
ejpam-4788	418	26	∈	∈	PROPN
ejpam-4788	418	27	s.	s.	PROPN
ejpam-4788	418	28	then	then	ADV
ejpam-4788	418	29	(	(	PUNCT
ejpam-4788	418	30	ξp	ξp	AUX
ejpam-4788	418	31	◦	◦	NOUN
ejpam-4788	418	32	α	α	NOUN
ejpam-4788	418	33	xpt	xpt	PROPN
ejpam-4788	418	34	◦	◦	PROPN
ejpam-4788	418	35	β	β	NOUN
ejpam-4788	418	36	ξp	ξp	NOUN
ejpam-4788	418	37	)	)	PUNCT
ejpam-4788	418	38	∧	∧	PROPN
ejpam-4788	418	39	ξp	ξp	ADP
ejpam-4788	418	40	̸=	̸=	PROPN
ejpam-4788	418	41	0	0	NUM
ejpam-4788	419	1	and	and	CCONJ
ejpam-4788	419	2	(	(	PUNCT
ejpam-4788	419	3	ξn	ξn	PROPN
ejpam-4788	419	4	◦	◦	NOUN
ejpam-4788	419	5	α	α	X
ejpam-4788	419	6	xns	xns	PROPN
ejpam-4788	419	7	◦	◦	PROPN
ejpam-4788	419	8	β	β	X
ejpam-4788	419	9	ξn	ξn	NOUN
ejpam-4788	419	10	)	)	PUNCT
ejpam-4788	419	11	∨	∨	PROPN
ejpam-4788	419	12	ξn	ξn	PROPN
ejpam-4788	419	13	̸=	̸=	PROPN
ejpam-4788	419	14	0	0	NUM
ejpam-4788	419	15	.	.	PUNCT
ejpam-4788	420	1	thus	thus	ADV
ejpam-4788	420	2	,	,	PUNCT
ejpam-4788	420	3	there	there	PRON
ejpam-4788	420	4	exists	exist	VERB
ejpam-4788	420	5	r	r	NOUN
ejpam-4788	420	6	∈	∈	PROPN
ejpam-4788	420	7	s	s	VERB
ejpam-4788	420	8	such	such	ADJ
ejpam-4788	420	9	that	that	SCONJ
ejpam-4788	420	10	(	(	PUNCT
ejpam-4788	420	11	(	(	PUNCT
ejpam-4788	420	12	ξp	ξp	PART
ejpam-4788	420	13	◦	◦	VERB
ejpam-4788	420	14	α	α	NOUN
ejpam-4788	420	15	xpt	xpt	PROPN
ejpam-4788	420	16	◦	◦	PROPN
ejpam-4788	420	17	β	β	X
ejpam-4788	420	18	ξp)∧	ξp)∧	ADJ
ejpam-4788	420	19	ξp)(r	ξp)(r	PROPN
ejpam-4788	420	20	)	)	PUNCT
ejpam-4788	420	21	̸=	̸=	PROPN
ejpam-4788	420	22	0	0	NUM
ejpam-4788	420	23	,	,	PUNCT
ejpam-4788	420	24	and	and	CCONJ
ejpam-4788	420	25	(	(	PUNCT
ejpam-4788	420	26	(	(	PUNCT
ejpam-4788	420	27	ξn	ξn	PROPN
ejpam-4788	420	28	◦	◦	NOUN
ejpam-4788	420	29	α	α	X
ejpam-4788	420	30	xns	xns	PROPN
ejpam-4788	420	31	◦	◦	PROPN
ejpam-4788	420	32	β	β	X
ejpam-4788	420	33	ξn)∨	ξn)∨	PROPN
ejpam-4788	420	34	ξn)(r	ξn)(r	PROPN
ejpam-4788	420	35	)	)	PUNCT
ejpam-4788	420	36	̸=	̸=	PROPN
ejpam-4788	420	37	0	0	NUM
ejpam-4788	420	38	.	.	PUNCT
ejpam-4788	421	1	so	so	ADV
ejpam-4788	421	2	,	,	PUNCT
ejpam-4788	421	3	there	there	PRON
ejpam-4788	421	4	exists	exist	VERB
ejpam-4788	421	5	k1	k1	NOUN
ejpam-4788	421	6	,	,	PUNCT
ejpam-4788	421	7	k2	k2	PROPN
ejpam-4788	421	8	∈	∈	PROPN
ejpam-4788	421	9	s	s	VERB
ejpam-4788	421	10	such	such	ADJ
ejpam-4788	421	11	that	that	DET
ejpam-4788	421	12	r	r	NOUN
ejpam-4788	421	13	=	=	SYM
ejpam-4788	421	14	k1αβk2	k1αβk2	NOUN
ejpam-4788	421	15	,	,	PUNCT
ejpam-4788	421	16	ξ	ξ	PROPN
ejpam-4788	421	17	p(r	p(r	PROPN
ejpam-4788	421	18	)	)	PUNCT
ejpam-4788	421	19	̸=	̸=	PROPN
ejpam-4788	421	20	0	0	NUM
ejpam-4788	421	21	,	,	PUNCT
ejpam-4788	421	22	ξn(r	ξn(r	NUM
ejpam-4788	421	23	)	)	PUNCT
ejpam-4788	421	24	̸=	̸=	PROPN
ejpam-4788	421	25	0	0	NUM
ejpam-4788	421	26	,	,	PUNCT
ejpam-4788	421	27	and	and	CCONJ
ejpam-4788	421	28	ξp(k	ξp(k	NOUN
ejpam-4788	421	29	)	)	PUNCT
ejpam-4788	421	30	̸=	̸=	PROPN
ejpam-4788	421	31	0	0	NUM
ejpam-4788	421	32	,	,	PUNCT
ejpam-4788	421	33	ξn(k	ξn(k	NUM
ejpam-4788	421	34	)	)	PUNCT
ejpam-4788	421	35	̸=	̸=	PROPN
ejpam-4788	421	36	0	0	NUM
ejpam-4788	421	37	.	.	PUNCT
ejpam-4788	422	1	it	it	PRON
ejpam-4788	422	2	implies	imply	VERB
ejpam-4788	422	3	that	that	SCONJ
ejpam-4788	422	4	r	r	NOUN
ejpam-4788	422	5	,	,	PUNCT
ejpam-4788	422	6	k1	k1	NOUN
ejpam-4788	422	7	,	,	PUNCT
ejpam-4788	422	8	k2	k2	PROPN
ejpam-4788	422	9	∈	∈	PROPN
ejpam-4788	422	10	supp(ξ	supp(ξ	PROPN
ejpam-4788	422	11	)	)	PUNCT
ejpam-4788	422	12	.	.	PUNCT
ejpam-4788	423	1	thus	thus	ADV
ejpam-4788	423	2	,	,	PUNCT
ejpam-4788	423	3	(	(	PUNCT
ejpam-4788	423	4	λp	λp	X
ejpam-4788	423	5	supp(ξ	supp(ξ	PROPN
ejpam-4788	423	6	)	)	PUNCT
ejpam-4788	423	7	◦	◦	NOUN
ejpam-4788	423	8	α	α	NOUN
ejpam-4788	423	9	xpt	xpt	PROPN
ejpam-4788	423	10	◦	◦	PROPN
ejpam-4788	423	11	β	β	X
ejpam-4788	423	12	λp	λp	X
ejpam-4788	423	13	supp(ξ))(r	supp(ξ))(r	PROPN
ejpam-4788	423	14	)	)	PUNCT
ejpam-4788	423	15	̸=	̸=	PROPN
ejpam-4788	423	16	0	0	NUM
ejpam-4788	423	17	and	and	CCONJ
ejpam-4788	423	18	λp	λp	ADP
ejpam-4788	423	19	supp(ξ	supp(ξ	PROPN
ejpam-4788	423	20	)	)	PUNCT
ejpam-4788	423	21	̸=	̸=	PROPN
ejpam-4788	423	22	0	0	NUM
ejpam-4788	423	23	.	.	PUNCT
ejpam-4788	424	1	similarly	similarly	ADV
ejpam-4788	424	2	,	,	PUNCT
ejpam-4788	424	3	(	(	PUNCT
ejpam-4788	424	4	λn	λn	PROPN
ejpam-4788	424	5	supp(ξ	supp(ξ	PROPN
ejpam-4788	424	6	)	)	PUNCT
ejpam-4788	424	7	◦	◦	NOUN
ejpam-4788	424	8	α	α	PROPN
ejpam-4788	424	9	xns	xns	PROPN
ejpam-4788	424	10	◦	◦	PROPN
ejpam-4788	424	11	β	β	X
ejpam-4788	424	12	λn	λn	NOUN
ejpam-4788	424	13	supp(ξ))(r	supp(ξ))(r	PROPN
ejpam-4788	424	14	)	)	PUNCT
ejpam-4788	424	15	̸=	̸=	PROPN
ejpam-4788	424	16	0	0	NUM
ejpam-4788	424	17	,	,	PUNCT
ejpam-4788	424	18	and	and	CCONJ
ejpam-4788	424	19	λn	λn	PROPN
ejpam-4788	424	20	supp(ξ	supp(ξ	PROPN
ejpam-4788	424	21	)	)	PUNCT
ejpam-4788	424	22	̸=	̸=	PROPN
ejpam-4788	424	23	0	0	NUM
ejpam-4788	424	24	.	.	PUNCT
ejpam-4788	425	1	hence	hence	ADV
ejpam-4788	425	2	,	,	PUNCT
ejpam-4788	425	3	[	[	X
ejpam-4788	425	4	(	(	PUNCT
ejpam-4788	425	5	λp	λp	X
ejpam-4788	425	6	supp(ξ	supp(ξ	PROPN
ejpam-4788	425	7	)	)	PUNCT
ejpam-4788	425	8	◦	◦	NOUN
ejpam-4788	425	9	α	α	NOUN
ejpam-4788	425	10	xpt	xpt	PROPN
ejpam-4788	425	11	◦	◦	PROPN
ejpam-4788	425	12	β	β	X
ejpam-4788	425	13	λp	λp	X
ejpam-4788	425	14	supp(ξ	supp(ξ	PROPN
ejpam-4788	425	15	)	)	PUNCT
ejpam-4788	425	16	)	)	PUNCT
ejpam-4788	425	17	]	]	PUNCT
ejpam-4788	426	1	∧	∧	PROPN
ejpam-4788	426	2	λp	λp	X
ejpam-4788	426	3	supp(ξ	supp(ξ	PROPN
ejpam-4788	426	4	)	)	PUNCT
ejpam-4788	426	5	̸=	̸=	PROPN
ejpam-4788	426	6	0	0	NUM
ejpam-4788	426	7	,	,	PUNCT
ejpam-4788	426	8	and	and	CCONJ
ejpam-4788	426	9	[	[	X
ejpam-4788	426	10	(	(	PUNCT
ejpam-4788	426	11	λn	λn	PROPN
ejpam-4788	426	12	supp(ξ	supp(ξ	PROPN
ejpam-4788	426	13	)	)	PUNCT
ejpam-4788	426	14	◦	◦	NOUN
ejpam-4788	426	15	α	α	PROPN
ejpam-4788	426	16	xns	xns	PROPN
ejpam-4788	426	17	◦	◦	PROPN
ejpam-4788	426	18	β	β	X
ejpam-4788	426	19	λn	λn	NOUN
ejpam-4788	426	20	supp(ξ	supp(ξ	PROPN
ejpam-4788	426	21	)	)	PUNCT
ejpam-4788	426	22	)	)	PUNCT
ejpam-4788	426	23	]	]	PUNCT
ejpam-4788	427	1	∨	∨	NUM
ejpam-4788	427	2	λn	λn	PROPN
ejpam-4788	427	3	supp(ξ	supp(ξ	PROPN
ejpam-4788	427	4	)	)	PUNCT
ejpam-4788	427	5	̸=	̸=	PROPN
ejpam-4788	427	6	0	0	NUM
ejpam-4788	427	7	.	.	PUNCT
ejpam-4788	428	1	therefore	therefore	ADV
ejpam-4788	428	2	,	,	PUNCT
ejpam-4788	428	3	λsupp(ξ	λsupp(ξ	NOUN
ejpam-4788	428	4	)	)	PUNCT
ejpam-4788	428	5	is	be	AUX
ejpam-4788	428	6	a	a	DET
ejpam-4788	428	7	bf	bf	NOUN
ejpam-4788	428	8	almost	almost	ADV
ejpam-4788	428	9	(	(	PUNCT
ejpam-4788	428	10	α	α	NOUN
ejpam-4788	428	11	,	,	PUNCT
ejpam-4788	428	12	β)-bi	β)-bi	NOUN
ejpam-4788	428	13	-	-	PUNCT
ejpam-4788	428	14	ideal	ideal	NOUN
ejpam-4788	428	15	of	of	ADP
ejpam-4788	428	16	s.	s.	PROPN
ejpam-4788	428	17	this	this	PRON
ejpam-4788	428	18	shows	show	VERB
ejpam-4788	428	19	that	that	SCONJ
ejpam-4788	428	20	supp(ξ	supp(ξ	NOUN
ejpam-4788	428	21	)	)	PUNCT
ejpam-4788	428	22	is	be	AUX
ejpam-4788	428	23	an	an	DET
ejpam-4788	428	24	almost	almost	ADV
ejpam-4788	428	25	(	(	PUNCT
ejpam-4788	428	26	α	α	NOUN
ejpam-4788	428	27	,	,	PUNCT
ejpam-4788	428	28	β)-bi	β)-bi	NOUN
ejpam-4788	428	29	-	-	PUNCT
ejpam-4788	428	30	ideal	ideal	NOUN
ejpam-4788	428	31	of	of	ADP
ejpam-4788	428	32	s.	s.	PROPN
ejpam-4788	428	33	conversely	conversely	ADV
ejpam-4788	428	34	,	,	PUNCT
ejpam-4788	428	35	let	let	VERB
ejpam-4788	428	36	supp(ξ	supp(ξ	PROPN
ejpam-4788	428	37	)	)	PUNCT
ejpam-4788	428	38	be	be	AUX
ejpam-4788	428	39	an	an	DET
ejpam-4788	428	40	almost	almost	ADV
ejpam-4788	428	41	(	(	PUNCT
ejpam-4788	428	42	α	α	NOUN
ejpam-4788	428	43	,	,	PUNCT
ejpam-4788	428	44	β)-bi	β)-bi	NOUN
ejpam-4788	428	45	-	-	PUNCT
ejpam-4788	428	46	ideal	ideal	NOUN
ejpam-4788	428	47	of	of	ADP
ejpam-4788	428	48	s.	s.	PROPN
ejpam-4788	428	49	then	then	ADV
ejpam-4788	428	50	,	,	PUNCT
ejpam-4788	428	51	by	by	ADP
ejpam-4788	428	52	theorem	theorem	NOUN
ejpam-4788	428	53	20	20	NUM
ejpam-4788	428	54	,	,	PUNCT
ejpam-4788	428	55	λsupp(ξ	λsupp(ξ	NOUN
ejpam-4788	428	56	)	)	PUNCT
ejpam-4788	428	57	is	be	AUX
ejpam-4788	428	58	a	a	DET
ejpam-4788	428	59	bf	bf	NOUN
ejpam-4788	428	60	almost	almost	ADV
ejpam-4788	428	61	(	(	PUNCT
ejpam-4788	428	62	α	α	NOUN
ejpam-4788	428	63	,	,	PUNCT
ejpam-4788	428	64	β)-bi	β)-bi	NOUN
ejpam-4788	428	65	-	-	PUNCT
ejpam-4788	428	66	ideal	ideal	NOUN
ejpam-4788	428	67	of	of	ADP
ejpam-4788	428	68	s.	s.	PROPN
ejpam-4788	428	69	thus	thus	ADV
ejpam-4788	428	70	,	,	PUNCT
ejpam-4788	428	71	[	[	X
ejpam-4788	428	72	(	(	PUNCT
ejpam-4788	428	73	λp	λp	X
ejpam-4788	428	74	supp(ξ	supp(ξ	PROPN
ejpam-4788	428	75	)	)	PUNCT
ejpam-4788	428	76	◦	◦	NOUN
ejpam-4788	428	77	α	α	NOUN
ejpam-4788	428	78	xpt	xpt	PROPN
ejpam-4788	428	79	◦	◦	PROPN
ejpam-4788	428	80	β	β	X
ejpam-4788	428	81	λp	λp	X
ejpam-4788	428	82	supp(ξ	supp(ξ	PROPN
ejpam-4788	428	83	)	)	PUNCT
ejpam-4788	428	84	)	)	PUNCT
ejpam-4788	428	85	]	]	PUNCT
ejpam-4788	429	1	∧	∧	PROPN
ejpam-4788	429	2	λp	λp	X
ejpam-4788	429	3	supp(ξ	supp(ξ	PROPN
ejpam-4788	429	4	)	)	PUNCT
ejpam-4788	429	5	̸=	̸=	PROPN
ejpam-4788	429	6	0	0	NUM
ejpam-4788	429	7	,	,	PUNCT
ejpam-4788	429	8	and	and	CCONJ
ejpam-4788	429	9	[	[	X
ejpam-4788	429	10	(	(	PUNCT
ejpam-4788	429	11	λn	λn	PROPN
ejpam-4788	429	12	supp(ξ	supp(ξ	PROPN
ejpam-4788	429	13	)	)	PUNCT
ejpam-4788	429	14	◦	◦	NOUN
ejpam-4788	429	15	α	α	PROPN
ejpam-4788	429	16	xns	xns	PROPN
ejpam-4788	429	17	◦	◦	PROPN
ejpam-4788	429	18	β	β	X
ejpam-4788	429	19	λn	λn	NOUN
ejpam-4788	429	20	supp(ξ	supp(ξ	PROPN
ejpam-4788	429	21	)	)	PUNCT
ejpam-4788	429	22	)	)	PUNCT
ejpam-4788	429	23	]	]	PUNCT
ejpam-4788	430	1	∨	∨	NUM
ejpam-4788	430	2	λn	λn	PROPN
ejpam-4788	430	3	supp(ξ	supp(ξ	PROPN
ejpam-4788	430	4	)	)	PUNCT
ejpam-4788	430	5	̸=	̸=	PROPN
ejpam-4788	430	6	0	0	NUM
ejpam-4788	430	7	.	.	PUNCT
ejpam-4788	431	1	so	so	ADV
ejpam-4788	431	2	,	,	PUNCT
ejpam-4788	431	3	there	there	PRON
ejpam-4788	431	4	exists	exist	VERB
ejpam-4788	431	5	r	r	NOUN
ejpam-4788	431	6	∈	∈	PROPN
ejpam-4788	431	7	s	s	VERB
ejpam-4788	431	8	such	such	ADJ
ejpam-4788	431	9	that	that	SCONJ
ejpam-4788	431	10	(	(	PUNCT
ejpam-4788	431	11	(	(	PUNCT
ejpam-4788	431	12	λp	λp	X
ejpam-4788	431	13	supp(ξ)	supp(ξ)	NOUN
ejpam-4788	431	14	◦	◦	VERB
ejpam-4788	431	15	αx	αx	PROPN
ejpam-4788	431	16	p	p	PROPN
ejpam-4788	431	17	t	t	PROPN
ejpam-4788	431	18	◦	◦	NOUN
ejpam-4788	431	19	βλ	βλ	ADV
ejpam-4788	431	20	p	p	X
ejpam-4788	431	21	supp(ξ))∧λ	supp(ξ))∧λ	PRON
ejpam-4788	431	22	p	p	PROPN
ejpam-4788	431	23	supp(ξ))(r	supp(ξ))(r	PROPN
ejpam-4788	431	24	)	)	PUNCT
ejpam-4788	431	25	̸=	̸=	PROPN
ejpam-4788	431	26	0	0	NUM
ejpam-4788	431	27	,	,	PUNCT
ejpam-4788	431	28	and	and	CCONJ
ejpam-4788	431	29	(	(	PUNCT
ejpam-4788	431	30	(	(	PUNCT
ejpam-4788	431	31	λn	λn	NOUN
ejpam-4788	431	32	supp(ξ)	supp(ξ)	PROPN
ejpam-4788	431	33	◦	◦	NOUN
ejpam-4788	431	34	αx	αx	ADV
ejpam-4788	431	35	n	n	PROPN
ejpam-4788	431	36	s	s	PART
ejpam-4788	431	37	◦	◦	NOUN
ejpam-4788	431	38	βλn	βλn	ADJ
ejpam-4788	431	39	supp(ξ))∨λ	supp(ξ))∨λ	PROPN
ejpam-4788	431	40	n	n	CCONJ
ejpam-4788	431	41	supp(ξ))(r	supp(ξ))(r	PROPN
ejpam-4788	431	42	)	)	PUNCT
ejpam-4788	431	43	̸=	̸=	PROPN
ejpam-4788	431	44	0	0	NUM
ejpam-4788	431	45	.	.	PUNCT
ejpam-4788	432	1	it	it	PRON
ejpam-4788	432	2	implies	imply	VERB
ejpam-4788	432	3	that	that	SCONJ
ejpam-4788	432	4	(	(	PUNCT
ejpam-4788	432	5	λp	λp	X
ejpam-4788	432	6	supp(ξ	supp(ξ	PROPN
ejpam-4788	432	7	)	)	PUNCT
ejpam-4788	432	8	◦	◦	NOUN
ejpam-4788	432	9	α	α	NOUN
ejpam-4788	432	10	xpt	xpt	PROPN
ejpam-4788	432	11	◦	◦	PROPN
ejpam-4788	432	12	β	β	X
ejpam-4788	432	13	λp	λp	X
ejpam-4788	432	14	supp(ξ))(r	supp(ξ))(r	PROPN
ejpam-4788	432	15	)	)	PUNCT
ejpam-4788	432	16	̸=	̸=	PROPN
ejpam-4788	432	17	0	0	NUM
ejpam-4788	432	18	,	,	PUNCT
ejpam-4788	432	19	and	and	CCONJ
ejpam-4788	432	20	λp	λp	ADP
ejpam-4788	432	21	supp(ξ)(r	supp(ξ)(r	PROPN
ejpam-4788	432	22	)	)	PUNCT
ejpam-4788	432	23	̸=	̸=	PROPN
ejpam-4788	432	24	0	0	NUM
ejpam-4788	432	25	.	.	PUNCT
ejpam-4788	433	1	similarly	similarly	ADV
ejpam-4788	433	2	,	,	PUNCT
ejpam-4788	433	3	(	(	PUNCT
ejpam-4788	433	4	λn	λn	PROPN
ejpam-4788	433	5	supp(ξ	supp(ξ	PROPN
ejpam-4788	433	6	)	)	PUNCT
ejpam-4788	433	7	◦	◦	NOUN
ejpam-4788	433	8	α	α	PROPN
ejpam-4788	433	9	xns	xns	PROPN
ejpam-4788	433	10	◦	◦	PROPN
ejpam-4788	433	11	β	β	X
ejpam-4788	433	12	λn	λn	NOUN
ejpam-4788	433	13	supp(ξ))(r	supp(ξ))(r	PROPN
ejpam-4788	433	14	)	)	PUNCT
ejpam-4788	433	15	̸=	̸=	PROPN
ejpam-4788	433	16	0	0	NUM
ejpam-4788	433	17	,	,	PUNCT
ejpam-4788	433	18	and	and	CCONJ
ejpam-4788	433	19	λn	λn	ADP
ejpam-4788	433	20	supp(ξ)(r	supp(ξ)(r	NOUN
ejpam-4788	433	21	)	)	PUNCT
ejpam-4788	433	22	̸=	̸=	PROPN
ejpam-4788	433	23	0	0	NUM
ejpam-4788	433	24	.	.	PUNCT
ejpam-4788	434	1	thus	thus	ADV
ejpam-4788	434	2	,	,	PUNCT
ejpam-4788	434	3	there	there	PRON
ejpam-4788	434	4	exist	exist	VERB
ejpam-4788	434	5	k1	k1	NOUN
ejpam-4788	434	6	,	,	PUNCT
ejpam-4788	434	7	k2	k2	PROPN
ejpam-4788	434	8	∈	∈	PROPN
ejpam-4788	434	9	s	s	VERB
ejpam-4788	434	10	such	such	ADJ
ejpam-4788	434	11	that	that	DET
ejpam-4788	434	12	r	r	NOUN
ejpam-4788	434	13	=	=	SYM
ejpam-4788	434	14	k1αuβk2	k1αuβk2	PROPN
ejpam-4788	434	15	,	,	PUNCT
ejpam-4788	434	16	ξ	ξ	PROPN
ejpam-4788	434	17	p(r	p(r	PROPN
ejpam-4788	434	18	)	)	PUNCT
ejpam-4788	434	19	̸=	̸=	PROPN
ejpam-4788	434	20	0	0	NUM
ejpam-4788	434	21	,	,	PUNCT
ejpam-4788	434	22	ξn(r	ξn(r	NUM
ejpam-4788	434	23	)	)	PUNCT
ejpam-4788	434	24	̸=	̸=	PROPN
ejpam-4788	434	25	0	0	NUM
ejpam-4788	434	26	,	,	PUNCT
ejpam-4788	434	27	and	and	CCONJ
ejpam-4788	434	28	ξp(k	ξp(k	NOUN
ejpam-4788	434	29	)	)	PUNCT
ejpam-4788	434	30	̸=	̸=	PROPN
ejpam-4788	434	31	0	0	NUM
ejpam-4788	434	32	,	,	PUNCT
ejpam-4788	434	33	ξn(k	ξn(k	NUM
ejpam-4788	434	34	)	)	PUNCT
ejpam-4788	434	35	̸=	̸=	PROPN
ejpam-4788	434	36	0	0	NUM
ejpam-4788	434	37	.	.	PUNCT
ejpam-4788	435	1	hence	hence	ADV
ejpam-4788	435	2	,	,	PUNCT
ejpam-4788	435	3	(	(	PUNCT
ejpam-4788	435	4	ξp	ξp	PART
ejpam-4788	435	5	◦	◦	NOUN
ejpam-4788	435	6	α	α	NOUN
ejpam-4788	435	7	xpt	xpt	PROPN
ejpam-4788	435	8	◦	◦	PROPN
ejpam-4788	435	9	β	β	X
ejpam-4788	435	10	ξp)∧	ξp)∧	NOUN
ejpam-4788	435	11	ξp	ξp	ADP
ejpam-4788	435	12	̸=	̸=	PROPN
ejpam-4788	435	13	0	0	NUM
ejpam-4788	435	14	,	,	PUNCT
ejpam-4788	435	15	and	and	CCONJ
ejpam-4788	435	16	(	(	PUNCT
ejpam-4788	435	17	ξn	ξn	PROPN
ejpam-4788	435	18	◦	◦	NOUN
ejpam-4788	435	19	α	α	PROPN
ejpam-4788	435	20	xns	xns	PROPN
ejpam-4788	435	21	◦	◦	PROPN
ejpam-4788	435	22	β	β	X
ejpam-4788	435	23	ξn)∨	ξn)∨	PROPN
ejpam-4788	435	24	ξn	ξn	PROPN
ejpam-4788	435	25	̸=	̸=	PROPN
ejpam-4788	435	26	0	0	NUM
ejpam-4788	435	27	.	.	PUNCT
ejpam-4788	436	1	therefore	therefore	ADV
ejpam-4788	436	2	,	,	PUNCT
ejpam-4788	436	3	ξ	ξ	PROPN
ejpam-4788	436	4	is	be	AUX
ejpam-4788	436	5	a	a	DET
ejpam-4788	436	6	bf	bf	NOUN
ejpam-4788	436	7	almost	almost	ADV
ejpam-4788	436	8	(	(	PUNCT
ejpam-4788	436	9	α	α	NOUN
ejpam-4788	436	10	,	,	PUNCT
ejpam-4788	436	11	β)-bi	β)-bi	NOUN
ejpam-4788	436	12	-	-	PUNCT
ejpam-4788	436	13	ideal	ideal	NOUN
ejpam-4788	436	14	of	of	ADP
ejpam-4788	436	15	s.	s.	PROPN
ejpam-4788	436	16	definition	definition	NOUN
ejpam-4788	436	17	19	19	NUM
ejpam-4788	436	18	.	.	PUNCT
ejpam-4788	437	1	a	a	DET
ejpam-4788	437	2	bf	bf	NOUN
ejpam-4788	437	3	almost	almost	ADV
ejpam-4788	437	4	(	(	PUNCT
ejpam-4788	437	5	α	α	NOUN
ejpam-4788	437	6	,	,	PUNCT
ejpam-4788	437	7	β)-bi	β)-bi	NOUN
ejpam-4788	437	8	-	-	PUNCT
ejpam-4788	437	9	ideal	ideal	NOUN
ejpam-4788	437	10	ξ	ξ	X
ejpam-4788	437	11	=	=	SYM
ejpam-4788	437	12	(	(	PUNCT
ejpam-4788	437	13	s	s	PROPN
ejpam-4788	437	14	;	;	PUNCT
ejpam-4788	437	15	ξp	ξp	NUM
ejpam-4788	437	16	,	,	PUNCT
ejpam-4788	437	17	ξn	ξn	NOUN
ejpam-4788	437	18	)	)	PUNCT
ejpam-4788	437	19	of	of	ADP
ejpam-4788	437	20	a	a	DET
ejpam-4788	437	21	γ	γ	PROPN
ejpam-4788	437	22	-	-	PUNCT
ejpam-4788	437	23	semigroup	semigroup	NOUN
ejpam-4788	437	24	s	s	VERB
ejpam-4788	437	25	is	be	AUX
ejpam-4788	437	26	minimal	minimal	ADJ
ejpam-4788	437	27	if	if	SCONJ
ejpam-4788	437	28	for	for	ADP
ejpam-4788	437	29	all	all	DET
ejpam-4788	437	30	bf	bf	NOUN
ejpam-4788	437	31	almost	almost	ADV
ejpam-4788	437	32	(	(	PUNCT
ejpam-4788	437	33	α	α	NOUN
ejpam-4788	437	34	,	,	PUNCT
ejpam-4788	437	35	β)-bi	β)-bi	ADJ
ejpam-4788	437	36	-	-	PUNCT
ejpam-4788	437	37	ideal	ideal	NOUN
ejpam-4788	437	38	ς	ς	PROPN
ejpam-4788	437	39	=	=	PUNCT
ejpam-4788	437	40	(	(	PUNCT
ejpam-4788	437	41	s	s	NOUN
ejpam-4788	437	42	;	;	PUNCT
ejpam-4788	437	43	ςp	ςp	NUM
ejpam-4788	437	44	,	,	PUNCT
ejpam-4788	437	45	ςn	ςn	NOUN
ejpam-4788	437	46	)	)	PUNCT
ejpam-4788	437	47	of	of	ADP
ejpam-4788	437	48	s	s	PRON
ejpam-4788	437	49	such	such	ADJ
ejpam-4788	437	50	that	that	SCONJ
ejpam-4788	437	51	ς	ς	PROPN
ejpam-4788	437	52	⊆	⊆	NUM
ejpam-4788	437	53	ξ	ξ	NUM
ejpam-4788	437	54	,	,	PUNCT
ejpam-4788	437	55	then	then	ADV
ejpam-4788	437	56	supp(ς	supp(ς	NOUN
ejpam-4788	437	57	)	)	PUNCT
ejpam-4788	437	58	=	=	SYM
ejpam-4788	437	59	supp(ξ	supp(ξ	PROPN
ejpam-4788	437	60	)	)	PUNCT
ejpam-4788	437	61	.	.	PUNCT
ejpam-4788	438	1	references	reference	NOUN
ejpam-4788	438	2	1606	1606	NUM
ejpam-4788	438	3	theorem	theorem	VERB
ejpam-4788	438	4	22	22	NUM
ejpam-4788	438	5	.	.	PUNCT
ejpam-4788	439	1	let	let	VERB
ejpam-4788	439	2	k	k	PRON
ejpam-4788	439	3	be	be	AUX
ejpam-4788	439	4	a	a	DET
ejpam-4788	439	5	non	non	ADJ
ejpam-4788	439	6	-	-	ADJ
ejpam-4788	439	7	empty	empty	ADJ
ejpam-4788	439	8	subset	subset	NOUN
ejpam-4788	439	9	of	of	ADP
ejpam-4788	439	10	a	a	DET
ejpam-4788	439	11	γ	γ	PROPN
ejpam-4788	439	12	-	-	PUNCT
ejpam-4788	439	13	semigroup	semigroup	PROPN
ejpam-4788	439	14	s.	s.	PROPN
ejpam-4788	440	1	then	then	ADV
ejpam-4788	440	2	k	k	PROPN
ejpam-4788	440	3	is	be	AUX
ejpam-4788	440	4	a	a	DET
ejpam-4788	440	5	minimal	minimal	ADJ
ejpam-4788	440	6	almost	almost	ADV
ejpam-4788	440	7	(	(	PUNCT
ejpam-4788	440	8	α	α	NOUN
ejpam-4788	440	9	,	,	PUNCT
ejpam-4788	440	10	β)-bi	β)-bi	ADJ
ejpam-4788	440	11	-	-	PUNCT
ejpam-4788	440	12	ideal	ideal	NOUN
ejpam-4788	440	13	if	if	SCONJ
ejpam-4788	440	14	and	and	CCONJ
ejpam-4788	440	15	only	only	ADV
ejpam-4788	440	16	if	if	SCONJ
ejpam-4788	440	17	λk	λk	X
ejpam-4788	440	18	=	=	SYM
ejpam-4788	440	19	(	(	PUNCT
ejpam-4788	440	20	s;λp	s;λp	PROPN
ejpam-4788	440	21	k	k	PROPN
ejpam-4788	440	22	,	,	PUNCT
ejpam-4788	440	23	λn	λn	PROPN
ejpam-4788	440	24	k	k	X
ejpam-4788	440	25	)	)	PUNCT
ejpam-4788	440	26	is	be	AUX
ejpam-4788	440	27	a	a	DET
ejpam-4788	440	28	minimal	minimal	ADJ
ejpam-4788	440	29	bf	bf	NOUN
ejpam-4788	440	30	almost	almost	ADV
ejpam-4788	440	31	(	(	PUNCT
ejpam-4788	440	32	α	α	NOUN
ejpam-4788	440	33	,	,	PUNCT
ejpam-4788	440	34	β)-biideal	β)-biideal	PUNCT
ejpam-4788	440	35	of	of	ADP
ejpam-4788	440	36	s.	s.	PROPN
ejpam-4788	440	37	proof	proof	PROPN
ejpam-4788	440	38	.	.	PUNCT
ejpam-4788	441	1	suppose	suppose	VERB
ejpam-4788	441	2	that	that	SCONJ
ejpam-4788	441	3	k	k	PROPN
ejpam-4788	441	4	is	be	AUX
ejpam-4788	441	5	a	a	DET
ejpam-4788	441	6	minimal	minimal	ADJ
ejpam-4788	441	7	almost	almost	ADV
ejpam-4788	441	8	(	(	PUNCT
ejpam-4788	441	9	α	α	NOUN
ejpam-4788	441	10	,	,	PUNCT
ejpam-4788	441	11	β)-bi	β)-bi	NOUN
ejpam-4788	441	12	-	-	PUNCT
ejpam-4788	441	13	ideal	ideal	NOUN
ejpam-4788	441	14	of	of	ADP
ejpam-4788	441	15	s.	s.	PROPN
ejpam-4788	441	16	then	then	ADV
ejpam-4788	441	17	k	k	PROPN
ejpam-4788	441	18	is	be	AUX
ejpam-4788	441	19	an	an	DET
ejpam-4788	441	20	almost	almost	ADV
ejpam-4788	441	21	(	(	PUNCT
ejpam-4788	441	22	α	α	NOUN
ejpam-4788	441	23	,	,	PUNCT
ejpam-4788	441	24	β)-quasi	β)-quasi	NOUN
ejpam-4788	441	25	-	-	PUNCT
ejpam-4788	441	26	ideal	ideal	NOUN
ejpam-4788	441	27	of	of	ADP
ejpam-4788	441	28	s.	s.	PROPN
ejpam-4788	441	29	thus	thus	ADV
ejpam-4788	441	30	,	,	PUNCT
ejpam-4788	441	31	by	by	ADP
ejpam-4788	441	32	theorem	theorem	NOUN
ejpam-4788	441	33	20	20	NUM
ejpam-4788	441	34	,	,	PUNCT
ejpam-4788	441	35	λk	λk	NOUN
ejpam-4788	441	36	=	=	SYM
ejpam-4788	441	37	(	(	PUNCT
ejpam-4788	441	38	s;λp	s;λp	PROPN
ejpam-4788	441	39	k	k	PROPN
ejpam-4788	441	40	,	,	PUNCT
ejpam-4788	441	41	λn	λn	PROPN
ejpam-4788	441	42	k	k	X
ejpam-4788	441	43	)	)	PUNCT
ejpam-4788	441	44	is	be	AUX
ejpam-4788	441	45	a	a	DET
ejpam-4788	441	46	bf	bf	NOUN
ejpam-4788	441	47	(	(	PUNCT
ejpam-4788	441	48	α	α	NOUN
ejpam-4788	441	49	,	,	PUNCT
ejpam-4788	441	50	β)-quasiideal	β)-quasiideal	PUNCT
ejpam-4788	441	51	of	of	ADP
ejpam-4788	441	52	s.	s.	PROPN
ejpam-4788	441	53	let	let	VERB
ejpam-4788	441	54	ς	ς	PROPN
ejpam-4788	441	55	=	=	PUNCT
ejpam-4788	441	56	(	(	PUNCT
ejpam-4788	441	57	s	s	NOUN
ejpam-4788	441	58	;	;	PUNCT
ejpam-4788	441	59	ςp	ςp	NUM
ejpam-4788	441	60	,	,	PUNCT
ejpam-4788	441	61	ςn	ςn	NOUN
ejpam-4788	441	62	)	)	PUNCT
ejpam-4788	441	63	be	be	VERB
ejpam-4788	441	64	a	a	DET
ejpam-4788	441	65	bf	bf	NOUN
ejpam-4788	441	66	(	(	PUNCT
ejpam-4788	441	67	α	α	NOUN
ejpam-4788	441	68	,	,	PUNCT
ejpam-4788	441	69	β)-bi	β)-bi	NOUN
ejpam-4788	441	70	-	-	PUNCT
ejpam-4788	441	71	ideal	ideal	NOUN
ejpam-4788	441	72	of	of	ADP
ejpam-4788	441	73	s	s	PRON
ejpam-4788	441	74	such	such	ADJ
ejpam-4788	441	75	that	that	SCONJ
ejpam-4788	441	76	ς	ς	PROPN
ejpam-4788	441	77	⊆	⊆	NUM
ejpam-4788	441	78	λk	λk	NOUN
ejpam-4788	441	79	.	.	PUNCT
ejpam-4788	442	1	then	then	ADV
ejpam-4788	442	2	,	,	PUNCT
ejpam-4788	442	3	by	by	ADP
ejpam-4788	442	4	theorem	theorem	NOUN
ejpam-4788	442	5	21	21	NUM
ejpam-4788	442	6	,	,	PUNCT
ejpam-4788	442	7	supp(ς	supp(ς	NOUN
ejpam-4788	442	8	)	)	PUNCT
ejpam-4788	442	9	is	be	AUX
ejpam-4788	442	10	an	an	DET
ejpam-4788	442	11	almost	almost	ADV
ejpam-4788	442	12	(	(	PUNCT
ejpam-4788	442	13	α	α	NOUN
ejpam-4788	442	14	,	,	PUNCT
ejpam-4788	442	15	β)-bi	β)-bi	NOUN
ejpam-4788	442	16	-	-	PUNCT
ejpam-4788	442	17	ideal	ideal	NOUN
ejpam-4788	442	18	of	of	ADP
ejpam-4788	442	19	s.	s.	PROPN
ejpam-4788	442	20	thus	thus	ADV
ejpam-4788	442	21	,	,	PUNCT
ejpam-4788	442	22	supp(ς	supp(ς	NOUN
ejpam-4788	442	23	)	)	PUNCT
ejpam-4788	442	24	⊆	⊆	NUM
ejpam-4788	442	25	supp(λk	supp(λk	NOUN
ejpam-4788	442	26	)	)	PUNCT
ejpam-4788	442	27	=	=	SYM
ejpam-4788	442	28	k.	k.	NOUN
ejpam-4788	442	29	by	by	ADP
ejpam-4788	442	30	assumption	assumption	NOUN
ejpam-4788	442	31	,	,	PUNCT
ejpam-4788	442	32	supp(ς	supp(ς	NOUN
ejpam-4788	442	33	)	)	PUNCT
ejpam-4788	443	1	=	=	SYM
ejpam-4788	443	2	k	k	NOUN
ejpam-4788	443	3	=	=	PUNCT
ejpam-4788	443	4	supp(λk	supp(λk	NOUN
ejpam-4788	443	5	)	)	PUNCT
ejpam-4788	443	6	.	.	PUNCT
ejpam-4788	444	1	thus	thus	ADV
ejpam-4788	444	2	,	,	PUNCT
ejpam-4788	444	3	λk	λk	X
ejpam-4788	444	4	=	=	SYM
ejpam-4788	444	5	(	(	PUNCT
ejpam-4788	444	6	s;λp	s;λp	PROPN
ejpam-4788	444	7	k	k	PROPN
ejpam-4788	444	8	,	,	PUNCT
ejpam-4788	444	9	λn	λn	PROPN
ejpam-4788	444	10	k	k	X
ejpam-4788	444	11	)	)	PUNCT
ejpam-4788	444	12	is	be	AUX
ejpam-4788	444	13	a	a	DET
ejpam-4788	444	14	minimal	minimal	ADJ
ejpam-4788	444	15	bf	bf	NOUN
ejpam-4788	444	16	almost	almost	ADV
ejpam-4788	444	17	(	(	PUNCT
ejpam-4788	444	18	α	α	NOUN
ejpam-4788	444	19	,	,	PUNCT
ejpam-4788	444	20	β)-bi	β)-bi	NOUN
ejpam-4788	444	21	-	-	PUNCT
ejpam-4788	444	22	ideal	ideal	NOUN
ejpam-4788	444	23	of	of	ADP
ejpam-4788	444	24	s.	s.	PROPN
ejpam-4788	444	25	conversely	conversely	ADV
ejpam-4788	444	26	,	,	PUNCT
ejpam-4788	444	27	let	let	VERB
ejpam-4788	444	28	λk	λk	X
ejpam-4788	444	29	=	=	PUNCT
ejpam-4788	444	30	(	(	PUNCT
ejpam-4788	444	31	s;λp	s;λp	PROPN
ejpam-4788	444	32	k	k	PROPN
ejpam-4788	444	33	,	,	PUNCT
ejpam-4788	444	34	λn	λn	PROPN
ejpam-4788	444	35	k	k	X
ejpam-4788	444	36	)	)	PUNCT
ejpam-4788	444	37	be	be	AUX
ejpam-4788	444	38	a	a	DET
ejpam-4788	444	39	minimal	minimal	ADJ
ejpam-4788	444	40	bf	bf	NOUN
ejpam-4788	444	41	almost	almost	ADV
ejpam-4788	444	42	(	(	PUNCT
ejpam-4788	444	43	α	α	NOUN
ejpam-4788	444	44	,	,	PUNCT
ejpam-4788	444	45	β)-bi	β)-bi	NOUN
ejpam-4788	444	46	-	-	PUNCT
ejpam-4788	444	47	ideal	ideal	NOUN
ejpam-4788	444	48	of	of	ADP
ejpam-4788	444	49	s.	s.	PROPN
ejpam-4788	444	50	then	then	ADV
ejpam-4788	444	51	,	,	PUNCT
ejpam-4788	444	52	by	by	ADP
ejpam-4788	444	53	theorem	theorem	NOUN
ejpam-4788	444	54	20	20	NUM
ejpam-4788	444	55	,	,	PUNCT
ejpam-4788	444	56	k	k	PROPN
ejpam-4788	444	57	is	be	AUX
ejpam-4788	444	58	an	an	DET
ejpam-4788	444	59	almost	almost	ADV
ejpam-4788	444	60	(	(	PUNCT
ejpam-4788	444	61	α	α	NOUN
ejpam-4788	444	62	,	,	PUNCT
ejpam-4788	444	63	β)-bi	β)-bi	NOUN
ejpam-4788	444	64	-	-	PUNCT
ejpam-4788	444	65	ideal	ideal	NOUN
ejpam-4788	444	66	of	of	ADP
ejpam-4788	444	67	s.	s.	PROPN
ejpam-4788	444	68	let	let	VERB
ejpam-4788	444	69	j	j	PROPN
ejpam-4788	444	70	be	be	AUX
ejpam-4788	444	71	an	an	DET
ejpam-4788	444	72	almost	almost	ADV
ejpam-4788	444	73	(	(	PUNCT
ejpam-4788	444	74	α	α	NOUN
ejpam-4788	444	75	,	,	PUNCT
ejpam-4788	444	76	β)-bi	β)-bi	NOUN
ejpam-4788	444	77	-	-	PUNCT
ejpam-4788	444	78	ideal	ideal	NOUN
ejpam-4788	444	79	of	of	ADP
ejpam-4788	444	80	s	s	PRON
ejpam-4788	444	81	such	such	ADJ
ejpam-4788	444	82	that	that	SCONJ
ejpam-4788	444	83	j	j	PROPN
ejpam-4788	444	84	⊆	⊆	NUM
ejpam-4788	444	85	k.	k.	PROPN
ejpam-4788	444	86	then	then	ADV
ejpam-4788	444	87	by	by	ADP
ejpam-4788	444	88	theorem	theorem	NOUN
ejpam-4788	444	89	20	20	NUM
ejpam-4788	444	90	,	,	PUNCT
ejpam-4788	444	91	λj	λj	PROPN
ejpam-4788	444	92	is	be	AUX
ejpam-4788	444	93	a	a	DET
ejpam-4788	444	94	bf	bf	NOUN
ejpam-4788	444	95	(	(	PUNCT
ejpam-4788	444	96	α	α	NOUN
ejpam-4788	444	97	,	,	PUNCT
ejpam-4788	444	98	β)-bi	β)-bi	NOUN
ejpam-4788	444	99	-	-	PUNCT
ejpam-4788	444	100	ideal	ideal	NOUN
ejpam-4788	444	101	of	of	ADP
ejpam-4788	444	102	s	s	PRON
ejpam-4788	444	103	such	such	ADJ
ejpam-4788	444	104	that	that	SCONJ
ejpam-4788	444	105	λj	λj	PROPN
ejpam-4788	444	106	⊆	⊆	NUM
ejpam-4788	444	107	λk	λk	NOUN
ejpam-4788	444	108	.	.	PUNCT
ejpam-4788	445	1	thus	thus	ADV
ejpam-4788	445	2	,	,	PUNCT
ejpam-4788	445	3	j	j	PROPN
ejpam-4788	445	4	=	=	PUNCT
ejpam-4788	445	5	supp(λj	supp(λj	PROPN
ejpam-4788	445	6	)	)	PUNCT
ejpam-4788	445	7	=	=	SYM
ejpam-4788	445	8	supp(λk	supp(λk	NOUN
ejpam-4788	445	9	)	)	PUNCT
ejpam-4788	445	10	=	=	SYM
ejpam-4788	445	11	k.	k.	PROPN
ejpam-4788	446	1	hence	hence	ADV
ejpam-4788	446	2	,	,	PUNCT
ejpam-4788	446	3	k	k	PROPN
ejpam-4788	446	4	is	be	AUX
ejpam-4788	446	5	a	a	DET
ejpam-4788	446	6	minimal	minimal	ADJ
ejpam-4788	446	7	almost	almost	ADV
ejpam-4788	446	8	(	(	PUNCT
ejpam-4788	446	9	α	α	NOUN
ejpam-4788	446	10	,	,	PUNCT
ejpam-4788	446	11	β)-bi	β)-bi	NOUN
ejpam-4788	446	12	-	-	PUNCT
ejpam-4788	446	13	ideal	ideal	NOUN
ejpam-4788	446	14	of	of	ADP
ejpam-4788	446	15	s.	s.	PROPN
ejpam-4788	446	16	corollary	corollary	PROPN
ejpam-4788	446	17	3	3	X
ejpam-4788	446	18	.	.	PUNCT
ejpam-4788	447	1	let	let	VERB
ejpam-4788	447	2	s	s	PRON
ejpam-4788	447	3	be	be	AUX
ejpam-4788	447	4	a	a	DET
ejpam-4788	447	5	γ	γ	NOUN
ejpam-4788	447	6	-	-	PUNCT
ejpam-4788	447	7	semigroup	semigroup	ADJ
ejpam-4788	447	8	s.	s.	PROPN
ejpam-4788	448	1	then	then	ADV
ejpam-4788	448	2	s	s	AUX
ejpam-4788	448	3	has	have	VERB
ejpam-4788	448	4	no	no	DET
ejpam-4788	448	5	proper	proper	ADJ
ejpam-4788	448	6	almost	almost	ADV
ejpam-4788	448	7	(	(	PUNCT
ejpam-4788	448	8	α	α	NOUN
ejpam-4788	448	9	,	,	PUNCT
ejpam-4788	448	10	β)-bi	β)-bi	NOUN
ejpam-4788	448	11	-	-	PUNCT
ejpam-4788	448	12	ideal	ideal	NOUN
ejpam-4788	448	13	of	of	ADP
ejpam-4788	448	14	s	s	PRON
ejpam-4788	448	15	if	if	SCONJ
ejpam-4788	448	16	and	and	CCONJ
ejpam-4788	448	17	only	only	ADV
ejpam-4788	448	18	if	if	SCONJ
ejpam-4788	448	19	for	for	ADP
ejpam-4788	448	20	any	any	DET
ejpam-4788	448	21	bf	bf	NOUN
ejpam-4788	448	22	almost	almost	ADV
ejpam-4788	448	23	(	(	PUNCT
ejpam-4788	448	24	α	α	NOUN
ejpam-4788	448	25	,	,	PUNCT
ejpam-4788	448	26	β)-bi	β)-bi	NOUN
ejpam-4788	448	27	-	-	PUNCT
ejpam-4788	448	28	ideal	ideal	NOUN
ejpam-4788	448	29	ξ	ξ	X
ejpam-4788	448	30	=	=	SYM
ejpam-4788	448	31	(	(	PUNCT
ejpam-4788	448	32	s	s	PROPN
ejpam-4788	448	33	;	;	PUNCT
ejpam-4788	448	34	ξp	ξp	NUM
ejpam-4788	448	35	,	,	PUNCT
ejpam-4788	448	36	ξn	ξn	NOUN
ejpam-4788	448	37	)	)	PUNCT
ejpam-4788	448	38	of	of	ADP
ejpam-4788	448	39	s	s	PROPN
ejpam-4788	448	40	,	,	PUNCT
ejpam-4788	448	41	supp(ξ	supp(ξ	PROPN
ejpam-4788	448	42	)	)	PUNCT
ejpam-4788	448	43	=	=	PUNCT
ejpam-4788	448	44	s.	s.	PROPN
ejpam-4788	448	45	5	5	NUM
ejpam-4788	448	46	.	.	PUNCT
ejpam-4788	448	47	conclusion	conclusion	NOUN
ejpam-4788	448	48	in	in	ADP
ejpam-4788	448	49	this	this	DET
ejpam-4788	448	50	article	article	NOUN
ejpam-4788	448	51	,	,	PUNCT
ejpam-4788	448	52	we	we	PRON
ejpam-4788	448	53	introduce	introduce	VERB
ejpam-4788	448	54	the	the	DET
ejpam-4788	448	55	concept	concept	NOUN
ejpam-4788	448	56	of	of	ADP
ejpam-4788	448	57	a	a	DET
ejpam-4788	448	58	new	new	ADJ
ejpam-4788	448	59	bipolar	bipolar	ADJ
ejpam-4788	448	60	fuzzy	fuzzy	ADJ
ejpam-4788	448	61	ideal	ideal	NOUN
ejpam-4788	448	62	and	and	CCONJ
ejpam-4788	448	63	bipolar	bipolar	ADJ
ejpam-4788	448	64	fuzzy	fuzzy	ADJ
ejpam-4788	448	65	almost	almost	ADV
ejpam-4788	448	66	ideals	ideal	NOUN
ejpam-4788	448	67	in	in	ADP
ejpam-4788	448	68	γ	γ	NOUN
ejpam-4788	448	69	-	-	PUNCT
ejpam-4788	448	70	semigroups	semigroup	NOUN
ejpam-4788	448	71	.	.	PUNCT
ejpam-4788	449	1	we	we	PRON
ejpam-4788	449	2	also	also	ADV
ejpam-4788	449	3	study	study	VERB
ejpam-4788	449	4	properties	property	NOUN
ejpam-4788	449	5	of	of	ADP
ejpam-4788	449	6	new	new	ADJ
ejpam-4788	449	7	bipolar	bipolar	ADJ
ejpam-4788	449	8	fuzzy	fuzzy	ADJ
ejpam-4788	449	9	ideals	ideal	NOUN
ejpam-4788	449	10	and	and	CCONJ
ejpam-4788	449	11	bipolar	bipolar	ADJ
ejpam-4788	449	12	fuzzy	fuzzy	ADJ
ejpam-4788	449	13	almost	almost	ADV
ejpam-4788	449	14	ideals	ideal	NOUN
ejpam-4788	449	15	.	.	PUNCT
ejpam-4788	450	1	we	we	PRON
ejpam-4788	450	2	hope	hope	VERB
ejpam-4788	450	3	that	that	SCONJ
ejpam-4788	450	4	the	the	DET
ejpam-4788	450	5	present	present	ADJ
ejpam-4788	450	6	study	study	NOUN
ejpam-4788	450	7	will	will	AUX
ejpam-4788	450	8	be	be	AUX
ejpam-4788	450	9	useful	useful	ADJ
ejpam-4788	450	10	mathematical	mathematical	ADJ
ejpam-4788	450	11	tools	tool	NOUN
ejpam-4788	450	12	.	.	PUNCT
ejpam-4788	451	1	in	in	ADP
ejpam-4788	451	2	further	far	ADV
ejpam-4788	451	3	,	,	PUNCT
ejpam-4788	451	4	we	we	PRON
ejpam-4788	451	5	extend	extend	VERB
ejpam-4788	451	6	to	to	ADP
ejpam-4788	451	7	hesitant	hesitant	ADJ
ejpam-4788	451	8	fuzzy	fuzzy	ADJ
ejpam-4788	451	9	almost	almost	ADV
ejpam-4788	451	10	ideals	ideal	NOUN
ejpam-4788	451	11	and	and	CCONJ
ejpam-4788	451	12	algebraic	algebraic	ADJ
ejpam-4788	451	13	systems	system	NOUN
ejpam-4788	451	14	.	.	PUNCT
ejpam-4788	452	1	acknowledgements	acknowledgement	NOUN
ejpam-4788	452	2	this	this	DET
ejpam-4788	452	3	research	research	NOUN
ejpam-4788	452	4	project	project	NOUN
ejpam-4788	452	5	was	be	AUX
ejpam-4788	452	6	supported	support	VERB
ejpam-4788	452	7	by	by	ADP
ejpam-4788	452	8	the	the	DET
ejpam-4788	452	9	thailand	thailand	PROPN
ejpam-4788	452	10	science	science	PROPN
ejpam-4788	452	11	research	research	PROPN
ejpam-4788	452	12	and	and	CCONJ
ejpam-4788	452	13	innovation	innovation	NOUN
ejpam-4788	452	14	fund	fund	NOUN
ejpam-4788	452	15	and	and	CCONJ
ejpam-4788	452	16	the	the	DET
ejpam-4788	452	17	university	university	NOUN
ejpam-4788	452	18	of	of	ADP
ejpam-4788	452	19	phayao	phayao	NOUN
ejpam-4788	452	20	(	(	PUNCT
ejpam-4788	452	21	grant	grant	VERB
ejpam-4788	452	22	no	no	INTJ
ejpam-4788	452	23	.	.	PUNCT
ejpam-4788	453	1	ff66	ff66	PROPN
ejpam-4788	453	2	-	-	PUNCT
ejpam-4788	453	3	uoe017	uoe017	ADJ
ejpam-4788	453	4	)	)	PUNCT
ejpam-4788	453	5	fuzzy	fuzzy	ADJ
ejpam-4788	453	6	algebras	algebra	NOUN
ejpam-4788	453	7	and	and	CCONJ
ejpam-4788	453	8	decision	decision	NOUN
ejpam-4788	453	9	-	-	PUNCT
ejpam-4788	453	10	making	make	VERB
ejpam-4788	453	11	problems	problem	NOUN
ejpam-4788	453	12	research	research	NOUN
ejpam-4788	453	13	unit	unit	NOUN
ejpam-4788	453	14	,	,	PUNCT
ejpam-4788	453	15	department	department	NOUN
ejpam-4788	453	16	of	of	ADP
ejpam-4788	453	17	mathematics	mathematic	NOUN
ejpam-4788	453	18	,	,	PUNCT
ejpam-4788	453	19	school	school	NOUN
ejpam-4788	453	20	of	of	ADP
ejpam-4788	453	21	science	science	NOUN
ejpam-4788	453	22	,	,	PUNCT
ejpam-4788	453	23	university	university	NOUN
ejpam-4788	453	24	of	of	ADP
ejpam-4788	453	25	phayao	phayao	NOUN
ejpam-4788	453	26	,	,	PUNCT
ejpam-4788	453	27	phayao	phayao	NOUN
ejpam-4788	453	28	56000	56000	NUM
ejpam-4788	453	29	,	,	PUNCT
ejpam-4788	453	30	thailand	thailand	PROPN
ejpam-4788	453	31	.	.	PUNCT
ejpam-4788	454	1	references	reference	NOUN
ejpam-4788	454	2	[	[	X
ejpam-4788	454	3	1	1	NUM
ejpam-4788	454	4	]	]	PUNCT
ejpam-4788	454	5	r.	r.	PROPN
ejpam-4788	454	6	chinram	chinram	PROPN
ejpam-4788	454	7	.	.	PUNCT
ejpam-4788	455	1	on	on	ADP
ejpam-4788	455	2	quasi	quasi	ADJ
ejpam-4788	455	3	-	-	ADJ
ejpam-4788	455	4	gamma	gamma	ADJ
ejpam-4788	455	5	-	-	PUNCT
ejpam-4788	455	6	ideals	ideal	NOUN
ejpam-4788	455	7	in	in	ADP
ejpam-4788	455	8	γ	γ	NOUN
ejpam-4788	455	9	-	-	PUNCT
ejpam-4788	455	10	semigroups	semigroup	NOUN
ejpam-4788	455	11	.	.	PUNCT
ejpam-4788	456	1	science	science	PROPN
ejpam-4788	456	2	asia	asia	PROPN
ejpam-4788	456	3	,	,	PUNCT
ejpam-4788	456	4	32:351–353	32:351–353	PROPN
ejpam-4788	456	5	,	,	PUNCT
ejpam-4788	456	6	2016	2016	NUM
ejpam-4788	456	7	.	.	PUNCT
ejpam-4788	457	1	[	[	X
ejpam-4788	457	2	2	2	X
ejpam-4788	457	3	]	]	PUNCT
ejpam-4788	457	4	t.	t.	PROPN
ejpam-4788	457	5	gaketem	gaketem	PROPN
ejpam-4788	457	6	and	and	CCONJ
ejpam-4788	457	7	p.	p.	PROPN
ejpam-4788	457	8	khamrot	khamrot	PROPN
ejpam-4788	457	9	.	.	PUNCT
ejpam-4788	458	1	on	on	ADP
ejpam-4788	458	2	some	some	DET
ejpam-4788	458	3	semigroups	semigroup	NOUN
ejpam-4788	458	4	characterized	characterize	VERB
ejpam-4788	458	5	in	in	ADP
ejpam-4788	458	6	terms	term	NOUN
ejpam-4788	458	7	of	of	ADP
ejpam-4788	458	8	biplar	biplar	ADJ
ejpam-4788	458	9	fuzzy	fuzzy	ADJ
ejpam-4788	458	10	weakly	weakly	ADJ
ejpam-4788	458	11	interior	interior	ADJ
ejpam-4788	458	12	ideals	ideal	NOUN
ejpam-4788	458	13	.	.	PUNCT
ejpam-4788	459	1	iaeng	iaeng	PROPN
ejpam-4788	459	2	international	international	PROPN
ejpam-4788	459	3	journal	journal	PROPN
ejpam-4788	459	4	of	of	ADP
ejpam-4788	459	5	computer	computer	NOUN
ejpam-4788	459	6	science	science	NOUN
ejpam-4788	459	7	,	,	PUNCT
ejpam-4788	459	8	48(2):250–256	48(2):250–256	PROPN
ejpam-4788	459	9	,	,	PUNCT
ejpam-4788	459	10	2021	2021	NUM
ejpam-4788	459	11	.	.	PUNCT
ejpam-4788	460	1	[	[	X
ejpam-4788	460	2	3	3	X
ejpam-4788	460	3	]	]	PUNCT
ejpam-4788	460	4	t.	t.	PROPN
ejpam-4788	460	5	gaketem	gaketem	PROPN
ejpam-4788	460	6	and	and	CCONJ
ejpam-4788	460	7	p.	p.	PROPN
ejpam-4788	460	8	khamrot	khamrot	PROPN
ejpam-4788	460	9	.	.	PUNCT
ejpam-4788	461	1	new	new	ADJ
ejpam-4788	461	2	types	type	NOUN
ejpam-4788	461	3	of	of	ADP
ejpam-4788	461	4	intuitionistic	intuitionistic	ADJ
ejpam-4788	461	5	fuzzy	fuzzy	ADJ
ejpam-4788	461	6	ideals	ideal	NOUN
ejpam-4788	461	7	in	in	ADP
ejpam-4788	461	8	γ	γ	NOUN
ejpam-4788	461	9	-	-	PUNCT
ejpam-4788	461	10	semigroups	semigroup	NOUN
ejpam-4788	461	11	.	.	PUNCT
ejpam-4788	462	1	international	international	ADJ
ejpam-4788	462	2	journal	journal	NOUN
ejpam-4788	462	3	of	of	ADP
ejpam-4788	462	4	fuzzy	fuzzy	ADJ
ejpam-4788	462	5	logic	logic	NOUN
ejpam-4788	462	6	and	and	CCONJ
ejpam-4788	462	7	intelligent	intelligent	ADJ
ejpam-4788	462	8	systems	system	NOUN
ejpam-4788	462	9	,	,	PUNCT
ejpam-4788	462	10	22(2):135–143	22(2):135–143	PROPN
ejpam-4788	462	11	,	,	PUNCT
ejpam-4788	462	12	2022	2022	NUM
ejpam-4788	462	13	.	.	PUNCT
ejpam-4788	463	1	references	reference	NOUN
ejpam-4788	463	2	1607	1607	PROPN
ejpam-4788	463	3	[	[	X
ejpam-4788	463	4	4	4	NUM
ejpam-4788	463	5	]	]	PUNCT
ejpam-4788	463	6	a.	a.	NOUN
ejpam-4788	463	7	iampan	iampan	PROPN
ejpam-4788	463	8	.	.	PUNCT
ejpam-4788	464	1	note	note	NOUN
ejpam-4788	464	2	on	on	ADP
ejpam-4788	464	3	bi	bi	NOUN
ejpam-4788	464	4	-	-	NOUN
ejpam-4788	464	5	ideal	ideal	NOUN
ejpam-4788	464	6	in	in	ADP
ejpam-4788	464	7	γ	γ	NOUN
ejpam-4788	464	8	-	-	PUNCT
ejpam-4788	464	9	semigroups	semigroup	NOUN
ejpam-4788	464	10	.	.	PUNCT
ejpam-4788	465	1	international	international	ADJ
ejpam-4788	465	2	journal	journal	NOUN
ejpam-4788	465	3	of	of	ADP
ejpam-4788	465	4	algebra	algebra	PROPN
ejpam-4788	465	5	,	,	PUNCT
ejpam-4788	465	6	3(4):181–188	3(4):181–188	NUM
ejpam-4788	465	7	,	,	PUNCT
ejpam-4788	465	8	2009	2009	NUM
ejpam-4788	465	9	.	.	PUNCT
ejpam-4788	466	1	[	[	X
ejpam-4788	466	2	5	5	X
ejpam-4788	466	3	]	]	PUNCT
ejpam-4788	466	4	p.	p.	NOUN
ejpam-4788	466	5	khamrot	khamrot	PROPN
ejpam-4788	466	6	and	and	CCONJ
ejpam-4788	466	7	t.	t.	PROPN
ejpam-4788	466	8	gaketem	gaketem	PROPN
ejpam-4788	466	9	.	.	PUNCT
ejpam-4788	467	1	a	a	DET
ejpam-4788	467	2	novel	novel	NOUN
ejpam-4788	467	3	of	of	ADP
ejpam-4788	467	4	cubic	cubic	ADJ
ejpam-4788	467	5	ideals	ideal	NOUN
ejpam-4788	467	6	in	in	ADP
ejpam-4788	467	7	γ	γ	NOUN
ejpam-4788	467	8	-	-	PUNCT
ejpam-4788	467	9	semigroups	semigroup	NOUN
ejpam-4788	467	10	.	.	PUNCT
ejpam-4788	468	1	international	international	ADJ
ejpam-4788	468	2	journal	journal	NOUN
ejpam-4788	468	3	of	of	ADP
ejpam-4788	468	4	analysis	analysis	NOUN
ejpam-4788	468	5	and	and	CCONJ
ejpam-4788	468	6	applications	application	NOUN
ejpam-4788	468	7	,	,	PUNCT
ejpam-4788	468	8	20:1–15	20:1–15	NUM
ejpam-4788	468	9	,	,	PUNCT
ejpam-4788	468	10	2012	2012	NUM
ejpam-4788	468	11	.	.	PUNCT
ejpam-4788	469	1	[	[	X
ejpam-4788	469	2	6	6	NUM
ejpam-4788	469	3	]	]	PUNCT
ejpam-4788	469	4	p.	p.	NOUN
ejpam-4788	469	5	khamrot	khamrot	PROPN
ejpam-4788	469	6	and	and	CCONJ
ejpam-4788	469	7	t.	t.	PROPN
ejpam-4788	469	8	gaketem	gaketem	PROPN
ejpam-4788	469	9	.	.	PUNCT
ejpam-4788	470	1	a	a	DET
ejpam-4788	470	2	new	new	ADJ
ejpam-4788	470	3	types	type	NOUN
ejpam-4788	470	4	interval	interval	NOUN
ejpam-4788	470	5	valued	value	VERB
ejpam-4788	470	6	fuzzy	fuzzy	ADJ
ejpam-4788	470	7	ideals	ideal	NOUN
ejpam-4788	470	8	in	in	ADP
ejpam-4788	470	9	γsemigroups	γsemigroup	NOUN
ejpam-4788	470	10	.	.	PUNCT
ejpam-4788	471	1	iaeng	iaeng	PROPN
ejpam-4788	471	2	international	international	PROPN
ejpam-4788	471	3	journal	journal	PROPN
ejpam-4788	471	4	of	of	ADP
ejpam-4788	471	5	computer	computer	NOUN
ejpam-4788	471	6	sciences	science	NOUN
ejpam-4788	471	7	,	,	PUNCT
ejpam-4788	471	8	50(2	50(2	NUM
ejpam-4788	471	9	)	)	PUNCT
ejpam-4788	471	10	,	,	PUNCT
ejpam-4788	471	11	2023	2023	NUM
ejpam-4788	471	12	.	.	PUNCT
ejpam-4788	472	1	[	[	X
ejpam-4788	472	2	7	7	X
ejpam-4788	472	3	]	]	X
ejpam-4788	472	4	p.	p.	NOUN
ejpam-4788	472	5	kummoon	kummoon	NOUN
ejpam-4788	472	6	and	and	CCONJ
ejpam-4788	472	7	t.	t.	PROPN
ejpam-4788	472	8	changphas	changphas	PROPN
ejpam-4788	472	9	.	.	PUNCT
ejpam-4788	473	1	bi	bi	NOUN
ejpam-4788	473	2	-	-	NOUN
ejpam-4788	473	3	bases	basis	NOUN
ejpam-4788	473	4	of	of	ADP
ejpam-4788	473	5	γ	γ	NOUN
ejpam-4788	473	6	-	-	PUNCT
ejpam-4788	473	7	semigroups	semigroup	NOUN
ejpam-4788	473	8	.	.	PUNCT
ejpam-4788	474	1	thai	thai	PROPN
ejpam-4788	474	2	journal	journal	PROPN
ejpam-4788	474	3	of	of	ADP
ejpam-4788	474	4	mathematics	mathematic	NOUN
ejpam-4788	474	5	,	,	PUNCT
ejpam-4788	474	6	pages	page	NOUN
ejpam-4788	474	7	75–86	75–86	NUM
ejpam-4788	474	8	,	,	PUNCT
ejpam-4788	474	9	2017	2017	NUM
ejpam-4788	474	10	.	.	PUNCT
ejpam-4788	475	1	[	[	X
ejpam-4788	475	2	8	8	X
ejpam-4788	475	3	]	]	PUNCT
ejpam-4788	475	4	k.	k.	PROPN
ejpam-4788	475	5	lee	lee	PROPN
ejpam-4788	475	6	.	.	PUNCT
ejpam-4788	476	1	bipolar	bipolar	ADJ
ejpam-4788	476	2	-	-	PUNCT
ejpam-4788	476	3	valued	value	VERB
ejpam-4788	476	4	fuzzy	fuzzy	ADJ
ejpam-4788	476	5	sets	set	NOUN
ejpam-4788	476	6	and	and	CCONJ
ejpam-4788	476	7	their	their	PRON
ejpam-4788	476	8	operations	operation	NOUN
ejpam-4788	476	9	.	.	PUNCT
ejpam-4788	477	1	in	in	ADP
ejpam-4788	477	2	proceeding	proceed	VERB
ejpam-4788	477	3	international	international	ADJ
ejpam-4788	477	4	conference	conference	NOUN
ejpam-4788	477	5	on	on	ADP
ejpam-4788	477	6	intelligent	intelligent	ADJ
ejpam-4788	477	7	technologies	technology	NOUN
ejpam-4788	477	8	bangkok	bangkok	PROPN
ejpam-4788	477	9	,	,	PUNCT
ejpam-4788	477	10	thailand	thailand	PROPN
ejpam-4788	477	11	,	,	PUNCT
ejpam-4788	477	12	pages	page	NOUN
ejpam-4788	477	13	307–312	307–312	NUM
ejpam-4788	477	14	,	,	PUNCT
ejpam-4788	477	15	2000	2000	NUM
ejpam-4788	477	16	.	.	PUNCT
ejpam-4788	478	1	[	[	X
ejpam-4788	478	2	9	9	NUM
ejpam-4788	478	3	]	]	PUNCT
ejpam-4788	478	4	sk	sk	PROPN
ejpam-4788	478	5	.	.	PROPN
ejpam-4788	478	6	majumder	majumder	NOUN
ejpam-4788	478	7	.	.	PUNCT
ejpam-4788	479	1	bipolar	bipolar	PROPN
ejpam-4788	479	2	valued	value	VERB
ejpam-4788	479	3	fuzzy	fuzzy	ADJ
ejpam-4788	479	4	sets	set	NOUN
ejpam-4788	479	5	in	in	ADP
ejpam-4788	479	6	γ	γ	NOUN
ejpam-4788	479	7	-	-	PUNCT
ejpam-4788	479	8	semigroups	semigroup	NOUN
ejpam-4788	479	9	.	.	PUNCT
ejpam-4788	480	1	mathematica	mathematica	PROPN
ejpam-4788	480	2	aeterna	aeterna	PROPN
ejpam-4788	480	3	,	,	PUNCT
ejpam-4788	480	4	2(3):203–213	2(3):203–213	NUM
ejpam-4788	480	5	,	,	PUNCT
ejpam-4788	480	6	2013	2013	NUM
ejpam-4788	480	7	.	.	PUNCT
ejpam-4788	481	1	[	[	X
ejpam-4788	481	2	10	10	NUM
ejpam-4788	481	3	]	]	X
ejpam-4788	481	4	j.n	j.n	PROPN
ejpam-4788	481	5	.	.	PROPN
ejpam-4788	481	6	mordeson	mordeson	PROPN
ejpam-4788	481	7	,	,	PUNCT
ejpam-4788	481	8	d.	d.	PROPN
ejpam-4788	481	9	s.	s.	PROPN
ejpam-4788	481	10	malik	malik	PROPN
ejpam-4788	481	11	,	,	PUNCT
ejpam-4788	481	12	and	and	CCONJ
ejpam-4788	481	13	n.	n.	PROPN
ejpam-4788	481	14	kuroki	kuroki	PROPN
ejpam-4788	481	15	.	.	PUNCT
ejpam-4788	482	1	fuzzy	fuzzy	PROPN
ejpam-4788	482	2	semigroup	semigroup	PROPN
ejpam-4788	482	3	.	.	PUNCT
ejpam-4788	483	1	springer	springer	NOUN
ejpam-4788	483	2	science	science	PROPN
ejpam-4788	483	3	and	and	CCONJ
ejpam-4788	483	4	business	business	NOUN
ejpam-4788	483	5	media	medium	NOUN
ejpam-4788	483	6	,	,	PUNCT
ejpam-4788	483	7	2003	2003	NUM
ejpam-4788	483	8	.	.	PUNCT
ejpam-4788	484	1	[	[	X
ejpam-4788	484	2	11	11	NUM
ejpam-4788	484	3	]	]	PUNCT
ejpam-4788	484	4	a.	a.	NOUN
ejpam-4788	484	5	simuen	simuen	PROPN
ejpam-4788	484	6	,	,	PUNCT
ejpam-4788	484	7	a.	a.	NOUN
ejpam-4788	484	8	iampan	iampan	PROPN
ejpam-4788	484	9	,	,	PUNCT
ejpam-4788	484	10	and	and	CCONJ
ejpam-4788	484	11	r.	r.	PROPN
ejpam-4788	484	12	chinram	chinram	PROPN
ejpam-4788	484	13	.	.	PUNCT
ejpam-4788	485	1	a	a	DET
ejpam-4788	485	2	novel	novel	NOUN
ejpam-4788	485	3	of	of	ADP
ejpam-4788	485	4	ideals	ideal	NOUN
ejpam-4788	485	5	and	and	CCONJ
ejpam-4788	485	6	fuzzy	fuzzy	ADJ
ejpam-4788	485	7	ideals	ideal	NOUN
ejpam-4788	485	8	of	of	ADP
ejpam-4788	485	9	γsemigroups	γsemigroup	NOUN
ejpam-4788	485	10	.	.	PUNCT
ejpam-4788	486	1	journal	journal	PROPN
ejpam-4788	486	2	of	of	ADP
ejpam-4788	486	3	mathematics	mathematic	NOUN
ejpam-4788	486	4	,	,	PUNCT
ejpam-4788	486	5	pages	page	NOUN
ejpam-4788	486	6	1–14	1–14	PROPN
ejpam-4788	486	7	,	,	PUNCT
ejpam-4788	486	8	2021	2021	NUM
ejpam-4788	486	9	.	.	PUNCT
ejpam-4788	487	1	[	[	X
ejpam-4788	487	2	12	12	NUM
ejpam-4788	487	3	]	]	X
ejpam-4788	487	4	l.a	l.a	PROPN
ejpam-4788	487	5	.	.	PROPN
ejpam-4788	487	6	zadeh	zadeh	PROPN
ejpam-4788	487	7	.	.	PUNCT
ejpam-4788	487	8	fuzzy	fuzzy	ADJ
ejpam-4788	487	9	sets	set	NOUN
ejpam-4788	487	10	.	.	PUNCT
ejpam-4788	488	1	information	information	NOUN
ejpam-4788	488	2	and	and	CCONJ
ejpam-4788	488	3	control	control	NOUN
ejpam-4788	488	4	,	,	PUNCT
ejpam-4788	488	5	8:338–353	8:338–353	NUM
ejpam-4788	488	6	,	,	PUNCT
ejpam-4788	488	7	1965	1965	NUM
ejpam-4788	488	8	.	.	PUNCT
ejpam-4788	489	1	[	[	X
ejpam-4788	489	2	13	13	NUM
ejpam-4788	489	3	]	]	PUNCT
ejpam-4788	489	4	w.	w.	PROPN
ejpam-4788	489	5	zhang	zhang	PROPN
ejpam-4788	489	6	.	.	PUNCT
ejpam-4788	490	1	bipolar	bipolar	ADJ
ejpam-4788	490	2	fuzzy	fuzzy	ADJ
ejpam-4788	490	3	sets	set	NOUN
ejpam-4788	490	4	and	and	CCONJ
ejpam-4788	490	5	relations	relation	NOUN
ejpam-4788	490	6	:	:	PUNCT
ejpam-4788	490	7	a	a	DET
ejpam-4788	490	8	computational	computational	ADJ
ejpam-4788	490	9	framework	framework	NOUN
ejpam-4788	490	10	forcognitive	forcognitive	ADJ
ejpam-4788	490	11	modeling	modeling	NOUN
ejpam-4788	490	12	and	and	CCONJ
ejpam-4788	490	13	multiagent	multiagent	ADJ
ejpam-4788	490	14	decision	decision	NOUN
ejpam-4788	490	15	analysis	analysis	NOUN
ejpam-4788	490	16	.	.	PUNCT
ejpam-4788	491	1	in	in	ADP
ejpam-4788	491	2	proceedings	proceeding	NOUN
ejpam-4788	491	3	of	of	ADP
ejpam-4788	491	4	ieee	ieee	NOUN
ejpam-4788	491	5	conference	conference	NOUN
ejpam-4788	491	6	,	,	PUNCT
ejpam-4788	491	7	pages	page	NOUN
ejpam-4788	491	8	305–309	305–309	NUM
ejpam-4788	491	9	,	,	PUNCT
ejpam-4788	491	10	1994	1994	NUM
ejpam-4788	491	11	.	.	PUNCT