id	sid	tid	token	lemma	pos
ejpam-4616	1	1	european	european	PROPN
ejpam-4616	1	2	journal	journal	PROPN
ejpam-4616	1	3	of	of	ADP
ejpam-4616	1	4	pure	pure	ADJ
ejpam-4616	1	5	and	and	CCONJ
ejpam-4616	1	6	applied	apply	VERB
ejpam-4616	1	7	mathematics	mathematic	NOUN
ejpam-4616	1	8	vol	vol	NOUN
ejpam-4616	1	9	.	.	PUNCT
ejpam-4616	2	1	16	16	NUM
ejpam-4616	2	2	,	,	PUNCT
ejpam-4616	2	3	no	no	INTJ
ejpam-4616	2	4	.	.	NOUN
ejpam-4616	2	5	1	1	NUM
ejpam-4616	2	6	,	,	PUNCT
ejpam-4616	2	7	2023	2023	NUM
ejpam-4616	2	8	,	,	PUNCT
ejpam-4616	2	9	121	121	NUM
ejpam-4616	2	10	-	-	SYM
ejpam-4616	2	11	130	130	NUM
ejpam-4616	2	12	issn	issn	PROPN
ejpam-4616	2	13	1307	1307	NUM
ejpam-4616	2	14	-	-	SYM
ejpam-4616	2	15	5543	5543	NUM
ejpam-4616	2	16	–	–	PUNCT
ejpam-4616	2	17	ejpam.com	ejpam.com	X
ejpam-4616	2	18	published	publish	VERB
ejpam-4616	2	19	by	by	ADP
ejpam-4616	2	20	new	new	PROPN
ejpam-4616	2	21	york	york	PROPN
ejpam-4616	2	22	business	business	PROPN
ejpam-4616	2	23	global	global	PROPN
ejpam-4616	2	24	on	on	ADP
ejpam-4616	2	25	interval	interval	NOUN
ejpam-4616	2	26	valued	value	VERB
ejpam-4616	2	27	fuzzy	fuzzy	ADJ
ejpam-4616	2	28	bi	bi	ADJ
ejpam-4616	2	29	-	-	ADJ
ejpam-4616	2	30	interior	interior	ADJ
ejpam-4616	2	31	ideals	ideal	NOUN
ejpam-4616	2	32	in	in	ADP
ejpam-4616	2	33	semigroups	semigroup	NOUN
ejpam-4616	2	34	thiti	thiti	PROPN
ejpam-4616	2	35	gaketem1	gaketem1	PROPN
ejpam-4616	2	36	,	,	PUNCT
ejpam-4616	2	37	tanaphong	tanaphong	ADJ
ejpam-4616	2	38	prommai1,∗	prommai1,∗	NOUN
ejpam-4616	2	39	1	1	NUM
ejpam-4616	2	40	fuzzy	fuzzy	ADJ
ejpam-4616	2	41	algebras	algebra	NOUN
ejpam-4616	2	42	and	and	CCONJ
ejpam-4616	2	43	decision	decision	NOUN
ejpam-4616	2	44	-	-	PUNCT
ejpam-4616	2	45	making	make	VERB
ejpam-4616	2	46	problems	problem	NOUN
ejpam-4616	2	47	research	research	NOUN
ejpam-4616	2	48	unit	unit	NOUN
ejpam-4616	2	49	,	,	PUNCT
ejpam-4616	2	50	department	department	NOUN
ejpam-4616	2	51	of	of	ADP
ejpam-4616	2	52	mathematics	mathematic	NOUN
ejpam-4616	2	53	,	,	PUNCT
ejpam-4616	2	54	school	school	NOUN
ejpam-4616	2	55	of	of	ADP
ejpam-4616	2	56	science	science	NOUN
ejpam-4616	2	57	,	,	PUNCT
ejpam-4616	2	58	university	university	NOUN
ejpam-4616	2	59	of	of	ADP
ejpam-4616	2	60	phayao	phayao	NOUN
ejpam-4616	2	61	,	,	PUNCT
ejpam-4616	2	62	phayao	phayao	NOUN
ejpam-4616	2	63	56000	56000	NUM
ejpam-4616	2	64	,	,	PUNCT
ejpam-4616	2	65	thailand	thailand	PROPN
ejpam-4616	2	66	abstract	abstract	NOUN
ejpam-4616	2	67	.	.	PUNCT
ejpam-4616	3	1	in	in	ADP
ejpam-4616	3	2	this	this	DET
ejpam-4616	3	3	paper	paper	NOUN
ejpam-4616	3	4	,	,	PUNCT
ejpam-4616	3	5	we	we	PRON
ejpam-4616	3	6	study	study	VERB
ejpam-4616	3	7	the	the	DET
ejpam-4616	3	8	concept	concept	NOUN
ejpam-4616	3	9	of	of	ADP
ejpam-4616	3	10	an	an	DET
ejpam-4616	3	11	interval	interval	NOUN
ejpam-4616	3	12	valued	value	VERB
ejpam-4616	3	13	fuzzy	fuzzy	ADJ
ejpam-4616	3	14	bi	bi	ADJ
ejpam-4616	3	15	-	-	ADJ
ejpam-4616	3	16	interior	interior	ADJ
ejpam-4616	3	17	ideal	ideal	NOUN
ejpam-4616	3	18	.	.	PUNCT
ejpam-4616	4	1	we	we	PRON
ejpam-4616	4	2	investigate	investigate	VERB
ejpam-4616	4	3	the	the	DET
ejpam-4616	4	4	properties	property	NOUN
ejpam-4616	4	5	of	of	ADP
ejpam-4616	4	6	an	an	DET
ejpam-4616	4	7	interval	interval	NOUN
ejpam-4616	4	8	valued	value	VERB
ejpam-4616	4	9	fuzzy	fuzzy	ADJ
ejpam-4616	4	10	bi	bi	ADJ
ejpam-4616	4	11	-	-	ADJ
ejpam-4616	4	12	interior	interior	ADJ
ejpam-4616	4	13	ideal	ideal	NOUN
ejpam-4616	4	14	in	in	ADP
ejpam-4616	4	15	semigroups	semigroup	NOUN
ejpam-4616	4	16	.	.	PUNCT
ejpam-4616	5	1	we	we	PRON
ejpam-4616	5	2	characterize	characterize	VERB
ejpam-4616	5	3	a	a	DET
ejpam-4616	5	4	regular	regular	ADJ
ejpam-4616	5	5	semigroup	semigroup	NOUN
ejpam-4616	5	6	in	in	ADP
ejpam-4616	5	7	terms	term	NOUN
ejpam-4616	5	8	of	of	ADP
ejpam-4616	5	9	an	an	DET
ejpam-4616	5	10	interval	interval	NOUN
ejpam-4616	5	11	valued	value	VERB
ejpam-4616	5	12	fuzzy	fuzzy	ADJ
ejpam-4616	5	13	bi	bi	ADJ
ejpam-4616	5	14	-	-	ADJ
ejpam-4616	5	15	interior	interior	ADJ
ejpam-4616	5	16	ideal	ideal	NOUN
ejpam-4616	5	17	.	.	PUNCT
ejpam-4616	6	1	2020	2020	NUM
ejpam-4616	6	2	mathematics	mathematic	NOUN
ejpam-4616	6	3	subject	subject	NOUN
ejpam-4616	6	4	classifications	classification	NOUN
ejpam-4616	6	5	:	:	PUNCT
ejpam-4616	6	6	03e72	03e72	NUM
ejpam-4616	6	7	,	,	PUNCT
ejpam-4616	6	8	18b40	18b40	NUM
ejpam-4616	6	9	key	key	ADJ
ejpam-4616	6	10	words	word	NOUN
ejpam-4616	6	11	and	and	CCONJ
ejpam-4616	6	12	phrases	phrase	NOUN
ejpam-4616	6	13	:	:	PUNCT
ejpam-4616	6	14	bi	bi	ADJ
ejpam-4616	6	15	-	-	ADJ
ejpam-4616	6	16	interior	interior	ADJ
ejpam-4616	6	17	ideals	ideal	NOUN
ejpam-4616	6	18	,	,	PUNCT
ejpam-4616	6	19	interval	interval	NOUN
ejpam-4616	6	20	valued	value	VERB
ejpam-4616	6	21	fuzzy	fuzzy	ADJ
ejpam-4616	6	22	bi	bi	ADJ
ejpam-4616	6	23	-	-	ADJ
ejpam-4616	6	24	interior	interior	ADJ
ejpam-4616	6	25	ideals	ideal	NOUN
ejpam-4616	6	26	1	1	NUM
ejpam-4616	6	27	.	.	PUNCT
ejpam-4616	7	1	introduction	introduction	NOUN
ejpam-4616	7	2	uncertainties	uncertainty	NOUN
ejpam-4616	7	3	can	can	AUX
ejpam-4616	7	4	not	not	PART
ejpam-4616	7	5	be	be	AUX
ejpam-4616	7	6	handled	handle	VERB
ejpam-4616	7	7	using	use	VERB
ejpam-4616	7	8	traditional	traditional	ADJ
ejpam-4616	7	9	mathematical	mathematical	ADJ
ejpam-4616	7	10	tools	tool	NOUN
ejpam-4616	7	11	but	but	CCONJ
ejpam-4616	7	12	maybe	maybe	ADV
ejpam-4616	7	13	deal	deal	VERB
ejpam-4616	7	14	with	with	ADP
ejpam-4616	7	15	using	use	VERB
ejpam-4616	7	16	a	a	DET
ejpam-4616	7	17	wide	wide	ADJ
ejpam-4616	7	18	range	range	NOUN
ejpam-4616	7	19	of	of	ADP
ejpam-4616	7	20	existing	exist	VERB
ejpam-4616	7	21	theories	theory	NOUN
ejpam-4616	7	22	such	such	ADJ
ejpam-4616	7	23	as	as	ADP
ejpam-4616	7	24	probability	probability	NOUN
ejpam-4616	7	25	theory	theory	NOUN
ejpam-4616	7	26	,	,	PUNCT
ejpam-4616	7	27	theory	theory	NOUN
ejpam-4616	7	28	of	of	ADP
ejpam-4616	7	29	fuzzy	fuzzy	ADJ
ejpam-4616	7	30	sets	set	NOUN
ejpam-4616	7	31	,	,	PUNCT
ejpam-4616	7	32	interval	interval	NOUN
ejpam-4616	7	33	valued	value	VERB
ejpam-4616	7	34	fuzzy	fuzzy	ADJ
ejpam-4616	7	35	sets	set	NOUN
ejpam-4616	7	36	.	.	PUNCT
ejpam-4616	8	1	in	in	ADP
ejpam-4616	8	2	1975	1975	NUM
ejpam-4616	8	3	,	,	PUNCT
ejpam-4616	8	4	zadeh	zadeh	PROPN
ejpam-4616	8	5	[	[	X
ejpam-4616	8	6	11	11	NUM
ejpam-4616	8	7	]	]	PUNCT
ejpam-4616	8	8	introduced	introduce	VERB
ejpam-4616	8	9	the	the	DET
ejpam-4616	8	10	theory	theory	NOUN
ejpam-4616	8	11	of	of	ADP
ejpam-4616	8	12	interval	interval	NOUN
ejpam-4616	8	13	valued	value	VERB
ejpam-4616	8	14	fuzzy	fuzzy	ADJ
ejpam-4616	8	15	sets	set	NOUN
ejpam-4616	8	16	as	as	ADP
ejpam-4616	8	17	a	a	DET
ejpam-4616	8	18	generalization	generalization	NOUN
ejpam-4616	8	19	of	of	ADP
ejpam-4616	8	20	the	the	DET
ejpam-4616	8	21	notion	notion	NOUN
ejpam-4616	8	22	of	of	ADP
ejpam-4616	8	23	fuzzy	fuzzy	ADJ
ejpam-4616	8	24	sets	set	NOUN
ejpam-4616	8	25	.	.	PUNCT
ejpam-4616	9	1	interval	interval	NOUN
ejpam-4616	9	2	valued	value	VERB
ejpam-4616	9	3	fuzzy	fuzzy	ADJ
ejpam-4616	9	4	sets	set	NOUN
ejpam-4616	9	5	have	have	VERB
ejpam-4616	9	6	various	various	ADJ
ejpam-4616	9	7	applications	application	NOUN
ejpam-4616	9	8	in	in	ADP
ejpam-4616	9	9	several	several	ADJ
ejpam-4616	9	10	areas	area	NOUN
ejpam-4616	9	11	like	like	ADP
ejpam-4616	9	12	medical	medical	ADJ
ejpam-4616	9	13	science	science	NOUN
ejpam-4616	9	14	[	[	X
ejpam-4616	9	15	2	2	NUM
ejpam-4616	9	16	]	]	PUNCT
ejpam-4616	9	17	,	,	PUNCT
ejpam-4616	9	18	image	image	NOUN
ejpam-4616	9	19	processing	processing	NOUN
ejpam-4616	9	20	[	[	X
ejpam-4616	9	21	1	1	NUM
ejpam-4616	9	22	]	]	PUNCT
ejpam-4616	9	23	,	,	PUNCT
ejpam-4616	9	24	decision	decision	NOUN
ejpam-4616	9	25	making	make	VERB
ejpam-4616	9	26	[	[	X
ejpam-4616	9	27	12	12	NUM
ejpam-4616	9	28	]	]	PUNCT
ejpam-4616	9	29	,	,	PUNCT
ejpam-4616	9	30	etc	etc	X
ejpam-4616	9	31	.	.	X
ejpam-4616	9	32	in	in	ADP
ejpam-4616	9	33	2006	2006	NUM
ejpam-4616	9	34	,	,	PUNCT
ejpam-4616	9	35	narayanan	narayanan	PROPN
ejpam-4616	9	36	and	and	CCONJ
ejpam-4616	9	37	manikantan	manikantan	PROPN
ejpam-4616	10	1	[	[	X
ejpam-4616	10	2	9	9	NUM
ejpam-4616	10	3	]	]	PUNCT
ejpam-4616	10	4	for	for	ADP
ejpam-4616	10	5	the	the	DET
ejpam-4616	10	6	first	first	ADJ
ejpam-4616	10	7	time	time	NOUN
ejpam-4616	10	8	employed	employ	VERB
ejpam-4616	10	9	the	the	DET
ejpam-4616	10	10	theory	theory	NOUN
ejpam-4616	10	11	of	of	ADP
ejpam-4616	10	12	interval	interval	NOUN
ejpam-4616	10	13	valued	value	VERB
ejpam-4616	10	14	fuzzy	fuzzy	ADJ
ejpam-4616	10	15	subsemigroup	subsemigroup	ADV
ejpam-4616	10	16	and	and	CCONJ
ejpam-4616	10	17	studied	studied	ADJ
ejpam-4616	10	18	types	type	NOUN
ejpam-4616	10	19	of	of	ADP
ejpam-4616	10	20	interval	interval	NOUN
ejpam-4616	10	21	valued	value	VERB
ejpam-4616	10	22	fuzzy	fuzzy	ADJ
ejpam-4616	10	23	ideals	ideal	NOUN
ejpam-4616	10	24	in	in	ADP
ejpam-4616	10	25	semigroups	semigroup	NOUN
ejpam-4616	10	26	.	.	PUNCT
ejpam-4616	11	1	in	in	ADP
ejpam-4616	11	2	2018	2018	NUM
ejpam-4616	11	3	,	,	PUNCT
ejpam-4616	11	4	mk	mk	PROPN
ejpam-4616	11	5	.	.	PUNCT
ejpam-4616	11	6	rao	rao	PROPN
ejpam-4616	11	7	[	[	X
ejpam-4616	11	8	5	5	NUM
ejpam-4616	11	9	]	]	PUNCT
ejpam-4616	11	10	introduced	introduce	VERB
ejpam-4616	11	11	and	and	CCONJ
ejpam-4616	11	12	studied	study	VERB
ejpam-4616	11	13	the	the	DET
ejpam-4616	11	14	definition	definition	NOUN
ejpam-4616	11	15	and	and	CCONJ
ejpam-4616	11	16	properties	property	NOUN
ejpam-4616	11	17	of	of	ADP
ejpam-4616	11	18	the	the	DET
ejpam-4616	11	19	bi	bi	ADJ
ejpam-4616	11	20	-	-	ADJ
ejpam-4616	11	21	interior	interior	ADJ
ejpam-4616	11	22	ideal	ideal	NOUN
ejpam-4616	11	23	in	in	ADP
ejpam-4616	11	24	the	the	DET
ejpam-4616	11	25	semigroup	semigroup	NOUN
ejpam-4616	11	26	.	.	PUNCT
ejpam-4616	12	1	in	in	ADP
ejpam-4616	12	2	2019	2019	NUM
ejpam-4616	12	3	,	,	PUNCT
ejpam-4616	12	4	a.	a.	PROPN
ejpam-4616	12	5	mahboob	mahboob	PROPN
ejpam-4616	12	6	et	et	PROPN
ejpam-4616	12	7	al	al	PROPN
ejpam-4616	12	8	.	.	PUNCT
ejpam-4616	13	1	[	[	X
ejpam-4616	13	2	8	8	NUM
ejpam-4616	13	3	]	]	PUNCT
ejpam-4616	13	4	characterizations	characterization	NOUN
ejpam-4616	13	5	of	of	ADP
ejpam-4616	13	6	regular	regular	ADJ
ejpam-4616	13	7	ordered	order	VERB
ejpam-4616	13	8	semigroups	semigroup	NOUN
ejpam-4616	13	9	by	by	ADP
ejpam-4616	13	10	(	(	PUNCT
ejpam-4616	13	11	ε	ε	PROPN
ejpam-4616	13	12	,	,	PUNCT
ejpam-4616	13	13	ε	ε	PROPN
ejpam-4616	13	14	∨	∨	PROPN
ejpam-4616	13	15	(	(	PUNCT
ejpam-4616	13	16	k	k	NOUN
ejpam-4616	13	17	,	,	PUNCT
ejpam-4616	13	18	qk))-fuzzy	qk))-fuzzy	ADJ
ejpam-4616	13	19	quasi	quasi	ADJ
ejpam-4616	13	20	ideals	ideal	NOUN
ejpam-4616	13	21	.	.	PUNCT
ejpam-4616	14	1	g	g	PROPN
ejpam-4616	14	2	muhiuddin	muhiuddin	PROPN
ejpam-4616	14	3	et	et	PROPN
ejpam-4616	14	4	al	al	PROPN
ejpam-4616	14	5	.	.	PROPN
ejpam-4616	14	6	discussed	discuss	VERB
ejpam-4616	14	7	a	a	DET
ejpam-4616	14	8	new	new	ADJ
ejpam-4616	14	9	type	type	NOUN
ejpam-4616	14	10	of	of	ADP
ejpam-4616	14	11	fuzzy	fuzzy	ADJ
ejpam-4616	14	12	semiprime	semiprime	NOUN
ejpam-4616	14	13	subsets	subset	NOUN
ejpam-4616	14	14	in	in	ADP
ejpam-4616	14	15	ordered	order	VERB
ejpam-4616	14	16	semigroups	semigroup	NOUN
ejpam-4616	14	17	.	.	PUNCT
ejpam-4616	15	1	many	many	ADJ
ejpam-4616	15	2	researchers	researcher	NOUN
ejpam-4616	15	3	studied	study	VERB
ejpam-4616	15	4	in	in	ADP
ejpam-4616	15	5	interval	interval	NOUN
ejpam-4616	15	6	valued	value	VERB
ejpam-4616	15	7	fuzzy	fuzzy	ADJ
ejpam-4616	15	8	semigroup	semigroup	NOUN
ejpam-4616	15	9	such	such	ADJ
ejpam-4616	15	10	that	that	SCONJ
ejpam-4616	15	11	in	in	ADP
ejpam-4616	15	12	2020	2020	NUM
ejpam-4616	15	13	ahsan	ahsan	PROPN
ejpam-4616	15	14	et	et	PROPN
ejpam-4616	15	15	al	al	PROPN
ejpam-4616	15	16	.	.	PUNCT
ejpam-4616	16	1	[	[	X
ejpam-4616	16	2	6	6	NUM
ejpam-4616	16	3	]	]	PUNCT
ejpam-4616	16	4	extend	extend	VERB
ejpam-4616	16	5	the	the	DET
ejpam-4616	16	6	ideals	ideal	NOUN
ejpam-4616	16	7	of	of	ADP
ejpam-4616	16	8	(	(	PUNCT
ejpam-4616	16	9	m	m	PROPN
ejpam-4616	16	10	,	,	PUNCT
ejpam-4616	16	11	n)-ideals	n)-ideal	NOUN
ejpam-4616	16	12	in	in	ADP
ejpam-4616	16	13	semigroups	semigroup	NOUN
ejpam-4616	16	14	to	to	ADP
ejpam-4616	16	15	fuzzy	fuzzy	ADJ
ejpam-4616	16	16	sets	set	NOUN
ejpam-4616	16	17	in	in	ADP
ejpam-4616	16	18	semigroup	semigroup	PROPN
ejpam-4616	17	1	and	and	CCONJ
ejpam-4616	17	2	they	they	PRON
ejpam-4616	17	3	characterize	characterize	VERB
ejpam-4616	17	4	the	the	DET
ejpam-4616	17	5	regular	regular	ADJ
ejpam-4616	17	6	semigroup	semigroup	NOUN
ejpam-4616	17	7	by	by	ADP
ejpam-4616	17	8	using	use	VERB
ejpam-4616	17	9	fuzzy	fuzzy	ADJ
ejpam-4616	17	10	(	(	PUNCT
ejpam-4616	17	11	m	m	NOUN
ejpam-4616	17	12	,	,	PUNCT
ejpam-4616	17	13	n)-ideals	n)-ideal	NOUN
ejpam-4616	17	14	.	.	PUNCT
ejpam-4616	17	15	i.	i.	PROPN
ejpam-4616	17	16	crista	crista	PROPN
ejpam-4616	17	17	et	et	PROPN
ejpam-4616	17	18	al	al	PROPN
ejpam-4616	17	19	.	.	PUNCT
ejpam-4616	18	1	[	[	X
ejpam-4616	18	2	3	3	NUM
ejpam-4616	18	3	]	]	PUNCT
ejpam-4616	18	4	studied	study	VERB
ejpam-4616	18	5	a	a	DET
ejpam-4616	18	6	new	new	ADJ
ejpam-4616	18	7	type	type	NOUN
ejpam-4616	18	8	fuzzy	fuzzy	ADJ
ejpam-4616	18	9	quasi	quasi	NOUN
ejpam-4616	18	10	-	-	NOUN
ejpam-4616	18	11	ideal	ideal	ADJ
ejpam-4616	18	12	in	in	ADP
ejpam-4616	18	13	ordered	order	VERB
ejpam-4616	18	14	semigroups	semigroup	NOUN
ejpam-4616	18	15	.	.	PUNCT
ejpam-4616	19	1	in	in	ADP
ejpam-4616	19	2	2021	2021	NUM
ejpam-4616	19	3	t.	t.	NOUN
ejpam-4616	19	4	gaketem	gaketem	NOUN
ejpam-4616	20	1	[	[	X
ejpam-4616	20	2	4	4	NUM
ejpam-4616	20	3	]	]	PUNCT
ejpam-4616	20	4	,	,	PUNCT
ejpam-4616	20	5	studied	study	VERB
ejpam-4616	20	6	interval	interval	NOUN
ejpam-4616	20	7	valued	value	VERB
ejpam-4616	20	8	fuzzy	fuzzy	ADJ
ejpam-4616	20	9	almost	almost	ADV
ejpam-4616	20	10	(	(	PUNCT
ejpam-4616	20	11	m	m	NOUN
ejpam-4616	20	12	,	,	PUNCT
ejpam-4616	20	13	n)-bi	n)-bi	NOUN
ejpam-4616	20	14	-	-	PUNCT
ejpam-4616	20	15	ideal	ideal	NOUN
ejpam-4616	20	16	in	in	ADP
ejpam-4616	20	17	semigroups	semigroup	NOUN
ejpam-4616	20	18	.	.	PUNCT
ejpam-4616	21	1	a.	a.	NOUN
ejpam-4616	21	2	mahboob	mahboob	PROPN
ejpam-4616	21	3	and	and	CCONJ
ejpam-4616	21	4	g.	g.	PROPN
ejpam-4616	21	5	muhiuddin	muhiuddin	PROPN
ejpam-4616	22	1	[	[	X
ejpam-4616	22	2	7	7	NUM
ejpam-4616	22	3	]	]	PUNCT
ejpam-4616	22	4	studied	study	VERB
ejpam-4616	22	5	a	a	DET
ejpam-4616	22	6	fuzzy	fuzzy	ADJ
ejpam-4616	22	7	prime	prime	NOUN
ejpam-4616	22	8	subset	subset	NOUN
ejpam-4616	22	9	in	in	ADP
ejpam-4616	22	10	ordered	order	VERB
ejpam-4616	22	11	semigroups	semigroup	NOUN
ejpam-4616	22	12	.	.	PUNCT
ejpam-4616	23	1	∗corresponding	∗corresponde	VERB
ejpam-4616	23	2	author	author	NOUN
ejpam-4616	23	3	.	.	PUNCT
ejpam-4616	24	1	doi	doi	NOUN
ejpam-4616	24	2	:	:	PUNCT
ejpam-4616	24	3	https://doi.org/10.29020/nybg.ejpam.v16i1.4616	https://doi.org/10.29020/nybg.ejpam.v16i1.4616	PRON
ejpam-4616	24	4	email	email	NOUN
ejpam-4616	24	5	addresses	address	VERB
ejpam-4616	24	6	:	:	PUNCT
ejpam-4616	24	7	thiti.ga@up.ac.th	thiti.ga@up.ac.th	PROPN
ejpam-4616	24	8	(	(	PUNCT
ejpam-4616	24	9	t.	t.	NOUN
ejpam-4616	24	10	gaketem	gaketem	PROPN
ejpam-4616	24	11	)	)	PUNCT
ejpam-4616	24	12	,	,	PUNCT
ejpam-4616	24	13	,	,	PUNCT
ejpam-4616	24	14	tanaphong.pr@up.ac.th	tanaphong.pr@up.ac.th	PRON
ejpam-4616	24	15	(	(	PUNCT
ejpam-4616	24	16	t.	t.	PROPN
ejpam-4616	24	17	prommai	prommai	PROPN
ejpam-4616	24	18	)	)	PUNCT
ejpam-4616	24	19	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4616	25	1	121	121	NUM
ejpam-4616	25	2	©	©	PROPN
ejpam-4616	25	3	2023	2023	NUM
ejpam-4616	25	4	ejpam	ejpam	NOUN
ejpam-4616	25	5	all	all	DET
ejpam-4616	25	6	rights	right	NOUN
ejpam-4616	25	7	reserved	reserve	VERB
ejpam-4616	25	8	.	.	PUNCT
ejpam-4616	26	1	t.	t.	PROPN
ejpam-4616	26	2	gaketem	gaketem	PROPN
ejpam-4616	26	3	,	,	PUNCT
ejpam-4616	26	4	t.	t.	PROPN
ejpam-4616	26	5	prommai	prommai	PROPN
ejpam-4616	26	6	/	/	SYM
ejpam-4616	26	7	eur	eur	PROPN
ejpam-4616	26	8	.	.	PUNCT
ejpam-4616	27	1	j.	j.	PROPN
ejpam-4616	27	2	pure	pure	PROPN
ejpam-4616	27	3	appl	appl	PROPN
ejpam-4616	27	4	.	.	PROPN
ejpam-4616	27	5	math	math	PROPN
ejpam-4616	27	6	,	,	PUNCT
ejpam-4616	27	7	16	16	NUM
ejpam-4616	27	8	(	(	PUNCT
ejpam-4616	27	9	1	1	NUM
ejpam-4616	27	10	)	)	PUNCT
ejpam-4616	27	11	(	(	PUNCT
ejpam-4616	27	12	2023	2023	NUM
ejpam-4616	27	13	)	)	PUNCT
ejpam-4616	27	14	,	,	PUNCT
ejpam-4616	27	15	121	121	NUM
ejpam-4616	27	16	-	-	SYM
ejpam-4616	27	17	130	130	NUM
ejpam-4616	27	18	122	122	NUM
ejpam-4616	27	19	in	in	ADP
ejpam-4616	27	20	this	this	DET
ejpam-4616	27	21	work	work	NOUN
ejpam-4616	27	22	,	,	PUNCT
ejpam-4616	27	23	we	we	PRON
ejpam-4616	27	24	establish	establish	VERB
ejpam-4616	27	25	the	the	DET
ejpam-4616	27	26	concept	concept	NOUN
ejpam-4616	27	27	of	of	ADP
ejpam-4616	27	28	an	an	DET
ejpam-4616	27	29	interval	interval	NOUN
ejpam-4616	27	30	valued	value	VERB
ejpam-4616	27	31	fuzzy	fuzzy	ADJ
ejpam-4616	27	32	bi	bi	ADJ
ejpam-4616	27	33	-	-	ADJ
ejpam-4616	27	34	interior	interior	ADJ
ejpam-4616	27	35	ideal	ideal	NOUN
ejpam-4616	27	36	.	.	PUNCT
ejpam-4616	28	1	we	we	PRON
ejpam-4616	28	2	investigate	investigate	VERB
ejpam-4616	28	3	the	the	DET
ejpam-4616	28	4	properties	property	NOUN
ejpam-4616	28	5	of	of	ADP
ejpam-4616	28	6	an	an	DET
ejpam-4616	28	7	interval	interval	NOUN
ejpam-4616	28	8	valued	value	VERB
ejpam-4616	28	9	fuzzy	fuzzy	ADJ
ejpam-4616	28	10	bi	bi	ADJ
ejpam-4616	28	11	-	-	ADJ
ejpam-4616	28	12	interior	interior	ADJ
ejpam-4616	28	13	ideal	ideal	NOUN
ejpam-4616	28	14	in	in	ADP
ejpam-4616	28	15	semigroups	semigroup	NOUN
ejpam-4616	28	16	.	.	PUNCT
ejpam-4616	29	1	finaly	finaly	VERB
ejpam-4616	29	2	,	,	PUNCT
ejpam-4616	29	3	we	we	PRON
ejpam-4616	29	4	characterize	characterize	VERB
ejpam-4616	29	5	a	a	DET
ejpam-4616	29	6	regular	regular	ADJ
ejpam-4616	29	7	semigroup	semigroup	NOUN
ejpam-4616	29	8	in	in	ADP
ejpam-4616	29	9	terms	term	NOUN
ejpam-4616	29	10	of	of	ADP
ejpam-4616	29	11	an	an	DET
ejpam-4616	29	12	interval	interval	NOUN
ejpam-4616	29	13	valued	value	VERB
ejpam-4616	29	14	fuzzy	fuzzy	ADJ
ejpam-4616	29	15	bi	bi	ADJ
ejpam-4616	29	16	-	-	ADJ
ejpam-4616	29	17	interior	interior	ADJ
ejpam-4616	29	18	ideal	ideal	NOUN
ejpam-4616	29	19	in	in	ADP
ejpam-4616	29	20	semigroups	semigroup	NOUN
ejpam-4616	29	21	.	.	PUNCT
ejpam-4616	30	1	2	2	X
ejpam-4616	30	2	.	.	NUM
ejpam-4616	30	3	preliminaries	preliminary	NOUN
ejpam-4616	30	4	in	in	ADP
ejpam-4616	30	5	this	this	DET
ejpam-4616	30	6	section	section	NOUN
ejpam-4616	30	7	,	,	PUNCT
ejpam-4616	30	8	we	we	PRON
ejpam-4616	30	9	begin	begin	VERB
ejpam-4616	30	10	with	with	ADP
ejpam-4616	30	11	elementary	elementary	ADJ
ejpam-4616	30	12	some	some	DET
ejpam-4616	30	13	fundamental	fundamental	ADJ
ejpam-4616	30	14	concepts	concept	NOUN
ejpam-4616	30	15	about	about	ADP
ejpam-4616	30	16	semigroups	semigroup	NOUN
ejpam-4616	30	17	,	,	PUNCT
ejpam-4616	30	18	fuzzy	fuzzy	ADJ
ejpam-4616	30	19	sets	set	NOUN
ejpam-4616	30	20	,	,	PUNCT
ejpam-4616	30	21	and	and	CCONJ
ejpam-4616	30	22	interval	interval	NOUN
ejpam-4616	30	23	valued	value	VERB
ejpam-4616	30	24	fuzzy	fuzzy	ADJ
ejpam-4616	30	25	sets	set	NOUN
ejpam-4616	30	26	that	that	PRON
ejpam-4616	30	27	are	be	AUX
ejpam-4616	30	28	necessary	necessary	ADJ
ejpam-4616	30	29	for	for	ADP
ejpam-4616	30	30	this	this	DET
ejpam-4616	30	31	paper	paper	NOUN
ejpam-4616	30	32	.	.	PUNCT
ejpam-4616	31	1	by	by	ADP
ejpam-4616	31	2	a	a	DET
ejpam-4616	31	3	subsemigroup	subsemigroup	NOUN
ejpam-4616	31	4	of	of	ADP
ejpam-4616	31	5	a	a	DET
ejpam-4616	31	6	semigroup	semigroup	NOUN
ejpam-4616	31	7	s	s	VERB
ejpam-4616	31	8	we	we	PRON
ejpam-4616	31	9	mean	mean	VERB
ejpam-4616	31	10	a	a	DET
ejpam-4616	31	11	non	non	ADJ
ejpam-4616	31	12	-	-	ADJ
ejpam-4616	31	13	empty	empty	ADJ
ejpam-4616	31	14	subset	subset	NOUN
ejpam-4616	31	15	m	m	VERB
ejpam-4616	31	16	of	of	ADP
ejpam-4616	31	17	s	s	PRON
ejpam-4616	31	18	such	such	ADJ
ejpam-4616	31	19	that	that	SCONJ
ejpam-4616	31	20	m2	m2	PROPN
ejpam-4616	31	21	⊆	⊆	NUM
ejpam-4616	31	22	m	m	NOUN
ejpam-4616	31	23	,	,	PUNCT
ejpam-4616	31	24	and	and	CCONJ
ejpam-4616	31	25	by	by	ADP
ejpam-4616	31	26	a	a	DET
ejpam-4616	31	27	left	left	ADJ
ejpam-4616	31	28	(	(	PUNCT
ejpam-4616	31	29	right	right	ADJ
ejpam-4616	31	30	)	)	PUNCT
ejpam-4616	31	31	ideal	ideal	NOUN
ejpam-4616	31	32	of	of	ADP
ejpam-4616	31	33	s	s	PRON
ejpam-4616	31	34	we	we	PRON
ejpam-4616	31	35	mean	mean	VERB
ejpam-4616	31	36	a	a	DET
ejpam-4616	31	37	non	non	ADJ
ejpam-4616	31	38	-	-	ADJ
ejpam-4616	31	39	empty	empty	ADJ
ejpam-4616	31	40	subset	subset	NOUN
ejpam-4616	31	41	m	m	VERB
ejpam-4616	31	42	of	of	ADP
ejpam-4616	31	43	s	s	PRON
ejpam-4616	31	44	such	such	ADJ
ejpam-4616	31	45	that	that	SCONJ
ejpam-4616	31	46	sm	sm	PROPN
ejpam-4616	31	47	⊆	⊆	NUM
ejpam-4616	31	48	m(ms	m(ms	PROPN
ejpam-4616	31	49	⊆	⊆	NUM
ejpam-4616	31	50	m	m	NOUN
ejpam-4616	31	51	)	)	PUNCT
ejpam-4616	31	52	.	.	PUNCT
ejpam-4616	32	1	by	by	ADP
ejpam-4616	32	2	a	a	DET
ejpam-4616	32	3	two	two	NUM
ejpam-4616	32	4	-	-	PUNCT
ejpam-4616	32	5	sided	sided	ADJ
ejpam-4616	32	6	ideal	ideal	NOUN
ejpam-4616	32	7	or	or	CCONJ
ejpam-4616	32	8	simply	simply	ADV
ejpam-4616	32	9	an	an	DET
ejpam-4616	32	10	ideal	ideal	NOUN
ejpam-4616	32	11	,	,	PUNCT
ejpam-4616	32	12	we	we	PRON
ejpam-4616	32	13	mean	mean	VERB
ejpam-4616	32	14	a	a	DET
ejpam-4616	32	15	non	non	ADJ
ejpam-4616	32	16	-	-	ADJ
ejpam-4616	32	17	empty	empty	ADJ
ejpam-4616	32	18	subset	subset	NOUN
ejpam-4616	32	19	of	of	ADP
ejpam-4616	32	20	a	a	DET
ejpam-4616	32	21	semigroup	semigroup	NOUN
ejpam-4616	32	22	s	s	NOUN
ejpam-4616	32	23	that	that	PRON
ejpam-4616	32	24	is	be	AUX
ejpam-4616	32	25	both	both	CCONJ
ejpam-4616	32	26	a	a	DET
ejpam-4616	32	27	left	left	NOUN
ejpam-4616	32	28	and	and	CCONJ
ejpam-4616	32	29	a	a	DET
ejpam-4616	32	30	right	right	ADJ
ejpam-4616	32	31	ideal	ideal	NOUN
ejpam-4616	32	32	of	of	ADP
ejpam-4616	32	33	s.	s.	PROPN
ejpam-4616	32	34	a	a	DET
ejpam-4616	32	35	non	non	ADJ
ejpam-4616	32	36	-	-	ADJ
ejpam-4616	32	37	empty	empty	ADJ
ejpam-4616	32	38	subset	subset	NOUN
ejpam-4616	32	39	m	m	NOUN
ejpam-4616	32	40	of	of	ADP
ejpam-4616	32	41	s	s	PRON
ejpam-4616	32	42	is	be	AUX
ejpam-4616	32	43	called	call	VERB
ejpam-4616	32	44	a	a	DET
ejpam-4616	32	45	quasi	quasi	NOUN
ejpam-4616	32	46	-	-	NOUN
ejpam-4616	32	47	ideal	ideal	ADJ
ejpam-4616	32	48	of	of	ADP
ejpam-4616	32	49	s	s	PRON
ejpam-4616	32	50	if	if	SCONJ
ejpam-4616	32	51	ms	ms	PROPN
ejpam-4616	32	52	∩	∩	PROPN
ejpam-4616	32	53	sm	sm	VERB
ejpam-4616	32	54	⊆	⊆	NUM
ejpam-4616	32	55	m	m	NOUN
ejpam-4616	32	56	.	.	PUNCT
ejpam-4616	33	1	a	a	DET
ejpam-4616	33	2	subsemigroup	subsemigroup	NOUN
ejpam-4616	33	3	m	m	NOUN
ejpam-4616	33	4	of	of	ADP
ejpam-4616	33	5	s	s	PROPN
ejpam-4616	33	6	is	be	AUX
ejpam-4616	33	7	called	call	VERB
ejpam-4616	33	8	a	a	DET
ejpam-4616	33	9	bi	bi	NOUN
ejpam-4616	33	10	-	-	NOUN
ejpam-4616	33	11	ideal	ideal	NOUN
ejpam-4616	33	12	of	of	ADP
ejpam-4616	33	13	s	s	PRON
ejpam-4616	33	14	if	if	SCONJ
ejpam-4616	33	15	msm	msm	NOUN
ejpam-4616	33	16	⊆	⊆	NUM
ejpam-4616	33	17	m	m	NOUN
ejpam-4616	33	18	.	.	PUNCT
ejpam-4616	34	1	a	a	DET
ejpam-4616	34	2	subsemigroup	subsemigroup	NOUN
ejpam-4616	34	3	m	m	PROPN
ejpam-4616	34	4	of	of	ADP
ejpam-4616	34	5	a	a	DET
ejpam-4616	34	6	semigroup	semigroup	NOUN
ejpam-4616	34	7	s	s	PART
ejpam-4616	34	8	is	be	AUX
ejpam-4616	34	9	called	call	VERB
ejpam-4616	34	10	an	an	DET
ejpam-4616	34	11	interior	interior	ADJ
ejpam-4616	34	12	ideal	ideal	NOUN
ejpam-4616	34	13	of	of	ADP
ejpam-4616	34	14	s	s	PRON
ejpam-4616	34	15	if	if	SCONJ
ejpam-4616	34	16	sms	sm	VERB
ejpam-4616	34	17	⊆	⊆	NUM
ejpam-4616	34	18	m	m	NOUN
ejpam-4616	34	19	.	.	PUNCT
ejpam-4616	35	1	a	a	DET
ejpam-4616	35	2	subsemigroup	subsemigroup	NOUN
ejpam-4616	35	3	m	m	PROPN
ejpam-4616	35	4	of	of	ADP
ejpam-4616	35	5	a	a	DET
ejpam-4616	35	6	semigroup	semigroup	NOUN
ejpam-4616	35	7	s	s	NOUN
ejpam-4616	35	8	is	be	AUX
ejpam-4616	35	9	said	say	VERB
ejpam-4616	35	10	to	to	PART
ejpam-4616	35	11	be	be	AUX
ejpam-4616	35	12	a	a	DET
ejpam-4616	35	13	bi	bi	ADJ
ejpam-4616	35	14	-	-	ADJ
ejpam-4616	35	15	interior	interior	ADJ
ejpam-4616	35	16	ideal	ideal	NOUN
ejpam-4616	35	17	of	of	ADP
ejpam-4616	35	18	s	s	PRON
ejpam-4616	35	19	if	if	SCONJ
ejpam-4616	35	20	m	m	NOUN
ejpam-4616	35	21	is	be	AUX
ejpam-4616	35	22	a	a	DET
ejpam-4616	35	23	subsemigroup	subsemigroup	NOUN
ejpam-4616	35	24	of	of	ADP
ejpam-4616	35	25	s	s	NOUN
ejpam-4616	35	26	and	and	CCONJ
ejpam-4616	35	27	sms	sm	VERB
ejpam-4616	35	28	∩msm	∩msm	NOUN
ejpam-4616	36	1	⊆	⊆	NUM
ejpam-4616	36	2	m	m	NOUN
ejpam-4616	36	3	.[5	.[5	X
ejpam-4616	36	4	]	]	PUNCT
ejpam-4616	36	5	.	.	PUNCT
ejpam-4616	37	1	we	we	PRON
ejpam-4616	37	2	note	note	VERB
ejpam-4616	37	3	here	here	ADV
ejpam-4616	37	4	that	that	SCONJ
ejpam-4616	37	5	the	the	DET
ejpam-4616	37	6	properties	property	NOUN
ejpam-4616	37	7	is	be	AUX
ejpam-4616	37	8	hold	hold	ADJ
ejpam-4616	37	9	:	:	PUNCT
ejpam-4616	37	10	(	(	PUNCT
ejpam-4616	37	11	1	1	X
ejpam-4616	37	12	)	)	PUNCT
ejpam-4616	37	13	every	every	DET
ejpam-4616	37	14	left	leave	VERB
ejpam-4616	37	15	ideal	ideal	NOUN
ejpam-4616	37	16	is	be	AUX
ejpam-4616	37	17	a	a	DET
ejpam-4616	37	18	bi	bi	ADJ
ejpam-4616	37	19	-	-	ADJ
ejpam-4616	37	20	interior	interior	ADJ
ejpam-4616	37	21	ideal	ideal	NOUN
ejpam-4616	37	22	of	of	ADP
ejpam-4616	37	23	s.	s.	PROPN
ejpam-4616	37	24	(	(	PUNCT
ejpam-4616	37	25	2	2	X
ejpam-4616	37	26	)	)	PUNCT
ejpam-4616	37	27	every	every	DET
ejpam-4616	37	28	right	right	ADJ
ejpam-4616	37	29	ideal	ideal	NOUN
ejpam-4616	37	30	is	be	AUX
ejpam-4616	37	31	a	a	DET
ejpam-4616	37	32	bi	bi	ADJ
ejpam-4616	37	33	-	-	ADJ
ejpam-4616	37	34	interior	interior	ADJ
ejpam-4616	37	35	ideal	ideal	NOUN
ejpam-4616	37	36	of	of	ADP
ejpam-4616	37	37	s.	s.	PROPN
ejpam-4616	37	38	(	(	PUNCT
ejpam-4616	37	39	3	3	X
ejpam-4616	37	40	)	)	PUNCT
ejpam-4616	37	41	every	every	DET
ejpam-4616	37	42	ideal	ideal	NOUN
ejpam-4616	37	43	is	be	AUX
ejpam-4616	37	44	a	a	DET
ejpam-4616	37	45	bi	bi	ADJ
ejpam-4616	37	46	-	-	ADJ
ejpam-4616	37	47	interior	interior	ADJ
ejpam-4616	37	48	ideal	ideal	NOUN
ejpam-4616	37	49	of	of	ADP
ejpam-4616	37	50	s.	s.	PROPN
ejpam-4616	37	51	(	(	PUNCT
ejpam-4616	37	52	4	4	X
ejpam-4616	37	53	)	)	PUNCT
ejpam-4616	37	54	every	every	DET
ejpam-4616	37	55	quasi	quasi	ADJ
ejpam-4616	37	56	ideal	ideal	NOUN
ejpam-4616	37	57	is	be	AUX
ejpam-4616	37	58	a	a	DET
ejpam-4616	37	59	bi	bi	ADJ
ejpam-4616	37	60	-	-	ADJ
ejpam-4616	37	61	interior	interior	ADJ
ejpam-4616	37	62	ideal	ideal	NOUN
ejpam-4616	37	63	of	of	ADP
ejpam-4616	37	64	s.	s.	PROPN
ejpam-4616	37	65	(	(	PUNCT
ejpam-4616	37	66	5	5	X
ejpam-4616	37	67	)	)	PUNCT
ejpam-4616	37	68	the	the	DET
ejpam-4616	37	69	arbitrary	arbitrary	ADJ
ejpam-4616	37	70	intersection	intersection	NOUN
ejpam-4616	37	71	of	of	ADP
ejpam-4616	37	72	bi	bi	ADJ
ejpam-4616	37	73	-	-	NOUN
ejpam-4616	37	74	interior	interior	ADJ
ejpam-4616	37	75	of	of	ADP
ejpam-4616	37	76	s	s	PROPN
ejpam-4616	37	77	is	be	AUX
ejpam-4616	37	78	also	also	ADV
ejpam-4616	37	79	bi	bi	ADJ
ejpam-4616	37	80	-	-	ADJ
ejpam-4616	37	81	interior	interior	ADJ
ejpam-4616	37	82	ideal	ideal	NOUN
ejpam-4616	37	83	of	of	ADP
ejpam-4616	37	84	s.	s.	PROPN
ejpam-4616	37	85	(	(	PUNCT
ejpam-4616	37	86	6	6	NUM
ejpam-4616	37	87	)	)	PUNCT
ejpam-4616	37	88	if	if	SCONJ
ejpam-4616	37	89	m	m	NOUN
ejpam-4616	37	90	is	be	AUX
ejpam-4616	37	91	a	a	DET
ejpam-4616	37	92	bi	bi	ADJ
ejpam-4616	37	93	-	-	ADJ
ejpam-4616	37	94	interior	interior	ADJ
ejpam-4616	37	95	ideal	ideal	NOUN
ejpam-4616	37	96	of	of	ADP
ejpam-4616	37	97	s	s	PRON
ejpam-4616	37	98	then	then	ADV
ejpam-4616	37	99	ms	ms	PROPN
ejpam-4616	37	100	and	and	CCONJ
ejpam-4616	37	101	sm	sm	PROPN
ejpam-4616	37	102	are	be	AUX
ejpam-4616	37	103	bi	bi	ADJ
ejpam-4616	37	104	-	-	ADJ
ejpam-4616	37	105	interior	interior	ADJ
ejpam-4616	37	106	ideals	ideal	NOUN
ejpam-4616	37	107	of	of	ADP
ejpam-4616	37	108	s	s	NOUN
ejpam-4616	37	109	[	[	X
ejpam-4616	37	110	5	5	NUM
ejpam-4616	37	111	]	]	PUNCT
ejpam-4616	37	112	.	.	PUNCT
ejpam-4616	38	1	definition	definition	NOUN
ejpam-4616	38	2	1	1	NUM
ejpam-4616	38	3	.	.	PUNCT
ejpam-4616	39	1	[	[	X
ejpam-4616	39	2	10	10	NUM
ejpam-4616	39	3	]	]	X
ejpam-4616	39	4	a	a	DET
ejpam-4616	39	5	fuzzy	fuzzy	ADJ
ejpam-4616	39	6	subset	subset	PROPN
ejpam-4616	39	7	η	η	PROPN
ejpam-4616	39	8	of	of	ADP
ejpam-4616	39	9	a	a	DET
ejpam-4616	39	10	non	non	ADJ
ejpam-4616	39	11	-	-	ADJ
ejpam-4616	39	12	empty	empty	ADJ
ejpam-4616	39	13	set	set	NOUN
ejpam-4616	39	14	x	x	PUNCT
ejpam-4616	39	15	is	be	AUX
ejpam-4616	39	16	a	a	DET
ejpam-4616	39	17	function	function	NOUN
ejpam-4616	39	18	η	η	NOUN
ejpam-4616	39	19	:	:	PUNCT
ejpam-4616	39	20	x	x	SYM
ejpam-4616	39	21	→	→	SYM
ejpam-4616	40	1	[	[	X
ejpam-4616	40	2	0	0	NUM
ejpam-4616	40	3	,	,	PUNCT
ejpam-4616	40	4	1	1	NUM
ejpam-4616	40	5	]	]	PUNCT
ejpam-4616	40	6	.	.	PUNCT
ejpam-4616	41	1	for	for	ADP
ejpam-4616	41	2	any	any	DET
ejpam-4616	41	3	ηi	ηi	PROPN
ejpam-4616	41	4	∈	∈	PROPN
ejpam-4616	42	1	[	[	X
ejpam-4616	42	2	0	0	NUM
ejpam-4616	42	3	,	,	PUNCT
ejpam-4616	42	4	1	1	NUM
ejpam-4616	42	5	]	]	PUNCT
ejpam-4616	42	6	where	where	SCONJ
ejpam-4616	42	7	i	i	PRON
ejpam-4616	42	8	∈	∈	VERB
ejpam-4616	42	9	a	a	DET
ejpam-4616	42	10	define	define	ADJ
ejpam-4616	42	11	∨	∨	NUM
ejpam-4616	42	12	i∈a	i∈a	ADJ
ejpam-4616	42	13	ηi	ηi	NOUN
ejpam-4616	42	14	:	:	PUNCT
ejpam-4616	42	15	=	=	SYM
ejpam-4616	42	16	sup	sup	NOUN
ejpam-4616	42	17	i∈a	i∈a	ADJ
ejpam-4616	42	18	{	{	PUNCT
ejpam-4616	42	19	ηi	ηi	NOUN
ejpam-4616	42	20	}	}	PUNCT
ejpam-4616	42	21	and	and	CCONJ
ejpam-4616	42	22	∧	∧	PROPN
ejpam-4616	42	23	i∈a	i∈a	ADJ
ejpam-4616	42	24	ηi	ηi	NOUN
ejpam-4616	42	25	:	:	PUNCT
ejpam-4616	42	26	=	=	SYM
ejpam-4616	42	27	inf	inf	PROPN
ejpam-4616	42	28	i∈a	i∈a	PROPN
ejpam-4616	42	29	{	{	PUNCT
ejpam-4616	42	30	ηi	ηi	NOUN
ejpam-4616	42	31	}	}	PUNCT
ejpam-4616	42	32	.	.	PUNCT
ejpam-4616	43	1	we	we	PRON
ejpam-4616	43	2	see	see	VERB
ejpam-4616	43	3	that	that	PRON
ejpam-4616	43	4	for	for	ADP
ejpam-4616	43	5	any	any	DET
ejpam-4616	43	6	η1	η1	NOUN
ejpam-4616	43	7	,	,	PUNCT
ejpam-4616	43	8	η2	η2	X
ejpam-4616	43	9	∈	∈	PROPN
ejpam-4616	44	1	[	[	X
ejpam-4616	44	2	0	0	NUM
ejpam-4616	44	3	,	,	PUNCT
ejpam-4616	44	4	1	1	NUM
ejpam-4616	44	5	]	]	PUNCT
ejpam-4616	44	6	,	,	PUNCT
ejpam-4616	44	7	we	we	PRON
ejpam-4616	44	8	have	have	VERB
ejpam-4616	44	9	η1	η1	NOUN
ejpam-4616	44	10	∨	∨	NUM
ejpam-4616	44	11	η2	η2	PROPN
ejpam-4616	44	12	=	=	PUNCT
ejpam-4616	44	13	max{η1	max{η1	NOUN
ejpam-4616	44	14	,	,	PUNCT
ejpam-4616	44	15	η2	η2	X
ejpam-4616	44	16	}	}	PUNCT
ejpam-4616	44	17	and	and	CCONJ
ejpam-4616	44	18	η1	η1	PROPN
ejpam-4616	44	19	∧	∧	PROPN
ejpam-4616	44	20	η2	η2	VERB
ejpam-4616	44	21	=	=	PUNCT
ejpam-4616	44	22	min{η1	min{η1	NOUN
ejpam-4616	44	23	,	,	PUNCT
ejpam-4616	44	24	η2	η2	PROPN
ejpam-4616	44	25	}	}	PUNCT
ejpam-4616	44	26	.	.	PUNCT
ejpam-4616	45	1	let	let	VERB
ejpam-4616	45	2	ω[0	ω[0	PROPN
ejpam-4616	45	3	,	,	PUNCT
ejpam-4616	45	4	1	1	NUM
ejpam-4616	45	5	]	]	PUNCT
ejpam-4616	45	6	be	be	AUX
ejpam-4616	45	7	the	the	DET
ejpam-4616	45	8	set	set	NOUN
ejpam-4616	45	9	of	of	ADP
ejpam-4616	45	10	all	all	DET
ejpam-4616	45	11	closed	closed	ADJ
ejpam-4616	45	12	subintervals	subinterval	NOUN
ejpam-4616	45	13	of	of	ADP
ejpam-4616	45	14	[	[	X
ejpam-4616	45	15	0	0	NUM
ejpam-4616	45	16	,	,	PUNCT
ejpam-4616	45	17	1	1	NUM
ejpam-4616	45	18	]	]	PUNCT
ejpam-4616	45	19	,	,	PUNCT
ejpam-4616	45	20	i.e.	i.e.	X
ejpam-4616	45	21	,	,	PUNCT
ejpam-4616	45	22	ω[0	ω[0	PROPN
ejpam-4616	45	23	,	,	PUNCT
ejpam-4616	45	24	1	1	NUM
ejpam-4616	45	25	]	]	PUNCT
ejpam-4616	45	26	=	=	SYM
ejpam-4616	45	27	{	{	PUNCT
ejpam-4616	46	1	p̃	p̃	PROPN
ejpam-4616	46	2	=	=	PUNCT
ejpam-4616	47	1	[	[	X
ejpam-4616	47	2	p−	p−	X
ejpam-4616	47	3	,	,	PUNCT
ejpam-4616	47	4	p+	p+	NOUN
ejpam-4616	47	5	]	]	X
ejpam-4616	47	6	|	|	ADV
ejpam-4616	47	7	0	0	X
ejpam-4616	47	8	≤	≤	NOUN
ejpam-4616	47	9	p−	p−	NOUN
ejpam-4616	47	10	≤	≤	NOUN
ejpam-4616	47	11	p+	p+	VERB
ejpam-4616	47	12	≤	≤	NOUN
ejpam-4616	47	13	1	1	NUM
ejpam-4616	47	14	}	}	PUNCT
ejpam-4616	47	15	.	.	PUNCT
ejpam-4616	48	1	let	let	VERB
ejpam-4616	49	1	p̃	p̃	PROPN
ejpam-4616	49	2	=	=	PUNCT
ejpam-4616	50	1	[	[	X
ejpam-4616	50	2	p−	p−	X
ejpam-4616	50	3	,	,	PUNCT
ejpam-4616	50	4	p+	p+	NOUN
ejpam-4616	50	5	]	]	PUNCT
ejpam-4616	50	6	and	and	CCONJ
ejpam-4616	50	7	q̃	q̃	PROPN
ejpam-4616	51	1	=	=	PUNCT
ejpam-4616	52	1	[	[	X
ejpam-4616	52	2	q−	q−	PROPN
ejpam-4616	52	3	,	,	PUNCT
ejpam-4616	52	4	q+	q+	ADP
ejpam-4616	52	5	]	]	PUNCT
ejpam-4616	52	6	∈	∈	PROPN
ejpam-4616	52	7	ω[0	ω[0	PROPN
ejpam-4616	52	8	,	,	PUNCT
ejpam-4616	52	9	1	1	NUM
ejpam-4616	52	10	]	]	PUNCT
ejpam-4616	52	11	.	.	PUNCT
ejpam-4616	53	1	define	define	VERB
ejpam-4616	53	2	the	the	DET
ejpam-4616	53	3	operations	operation	NOUN
ejpam-4616	53	4	⪯	⪯	NOUN
ejpam-4616	53	5	,	,	PUNCT
ejpam-4616	53	6	=	=	NOUN
ejpam-4616	53	7	,	,	PUNCT
ejpam-4616	53	8	⋏	⋏	PROPN
ejpam-4616	53	9	and	and	CCONJ
ejpam-4616	53	10	⋎	⋎	NOUN
ejpam-4616	53	11	as	as	SCONJ
ejpam-4616	53	12	follows	follow	VERB
ejpam-4616	53	13	:	:	PUNCT
ejpam-4616	53	14	t.	t.	NOUN
ejpam-4616	53	15	gaketem	gaketem	PROPN
ejpam-4616	53	16	,	,	PUNCT
ejpam-4616	53	17	t.	t.	PROPN
ejpam-4616	53	18	prommai	prommai	PROPN
ejpam-4616	53	19	/	/	SYM
ejpam-4616	53	20	eur	eur	PROPN
ejpam-4616	53	21	.	.	PUNCT
ejpam-4616	54	1	j.	j.	PROPN
ejpam-4616	54	2	pure	pure	PROPN
ejpam-4616	54	3	appl	appl	PROPN
ejpam-4616	54	4	.	.	PROPN
ejpam-4616	54	5	math	math	PROPN
ejpam-4616	54	6	,	,	PUNCT
ejpam-4616	54	7	16	16	NUM
ejpam-4616	54	8	(	(	PUNCT
ejpam-4616	54	9	1	1	NUM
ejpam-4616	54	10	)	)	PUNCT
ejpam-4616	54	11	(	(	PUNCT
ejpam-4616	54	12	2023	2023	NUM
ejpam-4616	54	13	)	)	PUNCT
ejpam-4616	54	14	,	,	PUNCT
ejpam-4616	54	15	121	121	NUM
ejpam-4616	54	16	-	-	SYM
ejpam-4616	54	17	130	130	NUM
ejpam-4616	54	18	123	123	NUM
ejpam-4616	54	19	(	(	PUNCT
ejpam-4616	54	20	1	1	X
ejpam-4616	54	21	)	)	PUNCT
ejpam-4616	54	22	p̃	p̃	PROPN
ejpam-4616	54	23	⪯	⪯	NOUN
ejpam-4616	54	24	q̃	q̃	PROPN
ejpam-4616	54	25	if	if	SCONJ
ejpam-4616	54	26	and	and	CCONJ
ejpam-4616	54	27	only	only	ADV
ejpam-4616	54	28	if	if	SCONJ
ejpam-4616	54	29	p−	p−	NOUN
ejpam-4616	54	30	≤	≤	NOUN
ejpam-4616	54	31	q−	q−	PROPN
ejpam-4616	54	32	and	and	CCONJ
ejpam-4616	54	33	p+	p+	PROPN
ejpam-4616	54	34	≤	≤	NOUN
ejpam-4616	54	35	q+	q+	ADV
ejpam-4616	54	36	(	(	PUNCT
ejpam-4616	54	37	2	2	X
ejpam-4616	54	38	)	)	PUNCT
ejpam-4616	54	39	p̃	p̃	PROPN
ejpam-4616	54	40	=	=	SYM
ejpam-4616	54	41	q̃	q̃	PROPN
ejpam-4616	54	42	if	if	SCONJ
ejpam-4616	54	43	and	and	CCONJ
ejpam-4616	54	44	only	only	ADV
ejpam-4616	54	45	if	if	SCONJ
ejpam-4616	54	46	p−	p−	NOUN
ejpam-4616	54	47	=	=	SYM
ejpam-4616	54	48	q−	q−	PROPN
ejpam-4616	54	49	and	and	CCONJ
ejpam-4616	54	50	p+	p+	NOUN
ejpam-4616	54	51	=	=	SYM
ejpam-4616	54	52	q+	q+	X
ejpam-4616	54	53	(	(	PUNCT
ejpam-4616	54	54	3	3	X
ejpam-4616	54	55	)	)	PUNCT
ejpam-4616	54	56	p̃⋏	p̃⋏	NOUN
ejpam-4616	54	57	q̃	q̃	PROPN
ejpam-4616	54	58	=	=	SYM
ejpam-4616	55	1	[	[	X
ejpam-4616	55	2	(	(	PUNCT
ejpam-4616	55	3	p−	p−	NOUN
ejpam-4616	55	4	∧	∧	PROPN
ejpam-4616	55	5	q−	q−	PROPN
ejpam-4616	55	6	)	)	PUNCT
ejpam-4616	55	7	,	,	PUNCT
ejpam-4616	55	8	(	(	PUNCT
ejpam-4616	55	9	p+	p+	VERB
ejpam-4616	55	10	∧	∧	NOUN
ejpam-4616	55	11	q+	q+	ADP
ejpam-4616	55	12	)	)	PUNCT
ejpam-4616	55	13	]	]	PUNCT
ejpam-4616	55	14	(	(	PUNCT
ejpam-4616	55	15	4	4	X
ejpam-4616	55	16	)	)	PUNCT
ejpam-4616	55	17	p̃⋎	p̃⋎	NOUN
ejpam-4616	55	18	q̃	q̃	PROPN
ejpam-4616	55	19	=	=	SYM
ejpam-4616	56	1	[	[	X
ejpam-4616	56	2	(	(	PUNCT
ejpam-4616	56	3	p−	p−	PROPN
ejpam-4616	56	4	∨	∨	NUM
ejpam-4616	56	5	q−	q−	PROPN
ejpam-4616	56	6	)	)	PUNCT
ejpam-4616	56	7	,	,	PUNCT
ejpam-4616	56	8	(	(	PUNCT
ejpam-4616	56	9	p+	p+	NOUN
ejpam-4616	56	10	∨	∨	NUM
ejpam-4616	56	11	q+	q+	ADV
ejpam-4616	56	12	)	)	PUNCT
ejpam-4616	56	13	]	]	PUNCT
ejpam-4616	56	14	.	.	PUNCT
ejpam-4616	57	1	if	if	SCONJ
ejpam-4616	57	2	p̃	p̃	PROPN
ejpam-4616	57	3	⪰	⪰	NOUN
ejpam-4616	57	4	q̃	q̃	PROPN
ejpam-4616	57	5	,	,	PUNCT
ejpam-4616	57	6	we	we	PRON
ejpam-4616	57	7	mean	mean	VERB
ejpam-4616	57	8	q̃	q̃	PROPN
ejpam-4616	57	9	⪯	⪯	AUX
ejpam-4616	57	10	p̃.	p̃.	VERB
ejpam-4616	57	11	for	for	ADP
ejpam-4616	57	12	each	each	DET
ejpam-4616	57	13	interval	interval	NOUN
ejpam-4616	57	14	p̃i	p̃i	NOUN
ejpam-4616	57	15	=	=	PUNCT
ejpam-4616	58	1	[	[	X
ejpam-4616	58	2	p−i	p−i	X
ejpam-4616	58	3	,	,	PUNCT
ejpam-4616	58	4	p	p	X
ejpam-4616	59	1	+	+	CCONJ
ejpam-4616	59	2	i	i	PRON
ejpam-4616	59	3	]	]	PUNCT
ejpam-4616	59	4	∈	∈	PROPN
ejpam-4616	59	5	µ[0	µ[0	PROPN
ejpam-4616	59	6	,	,	PUNCT
ejpam-4616	59	7	1	1	NUM
ejpam-4616	59	8	]	]	PUNCT
ejpam-4616	59	9	,	,	PUNCT
ejpam-4616	59	10	i	i	PRON
ejpam-4616	59	11	∈	∈	VERB
ejpam-4616	59	12	a	a	PRON
ejpam-4616	59	13	where	where	SCONJ
ejpam-4616	59	14	a	a	PRON
ejpam-4616	59	15	is	be	AUX
ejpam-4616	59	16	an	an	DET
ejpam-4616	59	17	index	index	NOUN
ejpam-4616	59	18	set	set	NOUN
ejpam-4616	59	19	,	,	PUNCT
ejpam-4616	59	20	we	we	PRON
ejpam-4616	59	21	define	define	VERB
ejpam-4616	59	22	⋏	⋏	PROPN
ejpam-4616	59	23	i∈a	i∈a	VERB
ejpam-4616	60	1	p̃i	p̃i	NOUN
ejpam-4616	60	2	=	=	PUNCT
ejpam-4616	61	1	[	[	PUNCT
ejpam-4616	61	2	∧	∧	PROPN
ejpam-4616	61	3	i∈a	i∈a	VERB
ejpam-4616	61	4	p−i	p−i	NOUN
ejpam-4616	61	5	,	,	PUNCT
ejpam-4616	61	6	∧	∧	PROPN
ejpam-4616	61	7	i∈a	i∈a	VERB
ejpam-4616	61	8	p+i	p+i	NUM
ejpam-4616	61	9	]	]	PUNCT
ejpam-4616	61	10	and	and	CCONJ
ejpam-4616	61	11	⋎	⋎	PROPN
ejpam-4616	61	12	i∈a	i∈a	VERB
ejpam-4616	61	13	p̃i	p̃i	X
ejpam-4616	61	14	=	=	PUNCT
ejpam-4616	61	15	[	[	PUNCT
ejpam-4616	61	16	∨	∨	NUM
ejpam-4616	61	17	i∈a	i∈a	ADJ
ejpam-4616	61	18	p−i	p−i	PROPN
ejpam-4616	61	19	,	,	PUNCT
ejpam-4616	61	20	∨	∨	NOUN
ejpam-4616	61	21	i∈a	i∈a	VERB
ejpam-4616	61	22	p+i	p+i	NUM
ejpam-4616	61	23	]	]	PUNCT
ejpam-4616	61	24	.	.	PUNCT
ejpam-4616	62	1	definition	definition	NOUN
ejpam-4616	62	2	2	2	NUM
ejpam-4616	62	3	.	.	PUNCT
ejpam-4616	63	1	[	[	X
ejpam-4616	63	2	9	9	NUM
ejpam-4616	63	3	]	]	PUNCT
ejpam-4616	63	4	let	let	VERB
ejpam-4616	63	5	t	t	NOUN
ejpam-4616	63	6	be	be	AUX
ejpam-4616	63	7	a	a	DET
ejpam-4616	63	8	non	non	ADJ
ejpam-4616	63	9	-	-	ADJ
ejpam-4616	63	10	empty	empty	ADJ
ejpam-4616	63	11	set	set	NOUN
ejpam-4616	63	12	.	.	PUNCT
ejpam-4616	64	1	then	then	ADV
ejpam-4616	64	2	the	the	DET
ejpam-4616	64	3	function	function	NOUN
ejpam-4616	64	4	µ̃	µ̃	PROPN
ejpam-4616	64	5	:	:	PUNCT
ejpam-4616	64	6	t	t	PROPN
ejpam-4616	64	7	→	→	SYM
ejpam-4616	64	8	ω[0	ω[0	PROPN
ejpam-4616	64	9	,	,	PUNCT
ejpam-4616	64	10	1	1	NUM
ejpam-4616	64	11	]	]	PUNCT
ejpam-4616	64	12	is	be	AUX
ejpam-4616	64	13	called	call	VERB
ejpam-4616	64	14	an	an	DET
ejpam-4616	64	15	interval	interval	NOUN
ejpam-4616	64	16	valued	value	VERB
ejpam-4616	64	17	fuzzy	fuzzy	ADJ
ejpam-4616	64	18	set	set	NOUN
ejpam-4616	64	19	(	(	PUNCT
ejpam-4616	64	20	shortly	shortly	ADV
ejpam-4616	64	21	,	,	PUNCT
ejpam-4616	64	22	ivf	ivf	NOUN
ejpam-4616	64	23	set	set	NOUN
ejpam-4616	64	24	)	)	PUNCT
ejpam-4616	64	25	of	of	ADP
ejpam-4616	64	26	t	t	PROPN
ejpam-4616	64	27	.	.	PUNCT
ejpam-4616	65	1	definition	definition	NOUN
ejpam-4616	65	2	3	3	NUM
ejpam-4616	65	3	.	.	PUNCT
ejpam-4616	66	1	[	[	X
ejpam-4616	66	2	9	9	NUM
ejpam-4616	66	3	]	]	PUNCT
ejpam-4616	66	4	let	let	VERB
ejpam-4616	66	5	m	m	PRON
ejpam-4616	66	6	be	be	AUX
ejpam-4616	66	7	a	a	DET
ejpam-4616	66	8	subset	subset	NOUN
ejpam-4616	66	9	of	of	ADP
ejpam-4616	66	10	a	a	DET
ejpam-4616	66	11	non	non	ADJ
ejpam-4616	66	12	-	-	ADJ
ejpam-4616	66	13	empty	empty	ADJ
ejpam-4616	66	14	set	set	ADJ
ejpam-4616	66	15	t	t	PROPN
ejpam-4616	66	16	.	.	PUNCT
ejpam-4616	67	1	an	an	DET
ejpam-4616	67	2	interval	interval	NOUN
ejpam-4616	67	3	valued	value	VERB
ejpam-4616	67	4	characteristic	characteristic	ADJ
ejpam-4616	67	5	function	function	NOUN
ejpam-4616	67	6	of	of	ADP
ejpam-4616	67	7	t	t	PROPN
ejpam-4616	67	8	is	be	AUX
ejpam-4616	67	9	defined	define	VERB
ejpam-4616	67	10	to	to	PART
ejpam-4616	67	11	be	be	AUX
ejpam-4616	67	12	a	a	DET
ejpam-4616	67	13	function	function	NOUN
ejpam-4616	67	14	χ̃m	χ̃m	NOUN
ejpam-4616	67	15	:	:	PUNCT
ejpam-4616	67	16	t	t	PROPN
ejpam-4616	67	17	→	→	SYM
ejpam-4616	67	18	ω[0	ω[0	PROPN
ejpam-4616	67	19	,	,	PUNCT
ejpam-4616	67	20	1	1	NUM
ejpam-4616	67	21	]	]	PUNCT
ejpam-4616	67	22	by	by	ADP
ejpam-4616	67	23	χ̃m	χ̃m	PROPN
ejpam-4616	67	24	(	(	PUNCT
ejpam-4616	67	25	e	e	NOUN
ejpam-4616	67	26	)	)	PUNCT
ejpam-4616	67	27	=	=	SYM
ejpam-4616	67	28	{	{	PUNCT
ejpam-4616	68	1	[	[	X
ejpam-4616	68	2	1	1	NUM
ejpam-4616	68	3	,	,	PUNCT
ejpam-4616	68	4	1	1	NUM
ejpam-4616	68	5	]	]	PUNCT
ejpam-4616	68	6	if	if	SCONJ
ejpam-4616	68	7	e	e	PROPN
ejpam-4616	68	8	∈	∈	PROPN
ejpam-4616	68	9	m	m	NOUN
ejpam-4616	68	10	,	,	PUNCT
ejpam-4616	69	1	[	[	X
ejpam-4616	69	2	0	0	NUM
ejpam-4616	69	3	,	,	PUNCT
ejpam-4616	69	4	0	0	NUM
ejpam-4616	69	5	]	]	PUNCT
ejpam-4616	69	6	if	if	SCONJ
ejpam-4616	69	7	e	e	PROPN
ejpam-4616	69	8	/∈	/∈	NOUN
ejpam-4616	69	9	m	m	VERB
ejpam-4616	69	10	for	for	ADP
ejpam-4616	69	11	all	all	DET
ejpam-4616	69	12	e	e	PROPN
ejpam-4616	69	13	∈	∈	PROPN
ejpam-4616	69	14	t	t	PROPN
ejpam-4616	69	15	.	.	PUNCT
ejpam-4616	70	1	for	for	ADP
ejpam-4616	70	2	two	two	NUM
ejpam-4616	70	3	ivf	ivf	NOUN
ejpam-4616	70	4	sets	set	NOUN
ejpam-4616	70	5	µ̃	µ̃	PROPN
ejpam-4616	70	6	and	and	CCONJ
ejpam-4616	70	7	ϖ̃	ϖ̃	PROPN
ejpam-4616	70	8	of	of	ADP
ejpam-4616	70	9	a	a	DET
ejpam-4616	70	10	non	non	ADJ
ejpam-4616	70	11	-	-	ADJ
ejpam-4616	70	12	empty	empty	ADJ
ejpam-4616	70	13	set	set	ADJ
ejpam-4616	70	14	t	t	NOUN
ejpam-4616	70	15	,	,	PUNCT
ejpam-4616	70	16	define	define	VERB
ejpam-4616	70	17	(	(	PUNCT
ejpam-4616	70	18	1	1	X
ejpam-4616	70	19	)	)	PUNCT
ejpam-4616	70	20	µ̃	µ̃	PROPN
ejpam-4616	70	21	⊑	⊑	DET
ejpam-4616	70	22	ϖ̃	ϖ̃	PROPN
ejpam-4616	70	23	⇔	⇔	PROPN
ejpam-4616	70	24	µ̃(e	µ̃(e	PROPN
ejpam-4616	70	25	)	)	PUNCT
ejpam-4616	70	26	⪯	⪯	NOUN
ejpam-4616	70	27	ϖ̃(e	ϖ̃(e	NOUN
ejpam-4616	70	28	)	)	PUNCT
ejpam-4616	70	29	for	for	ADP
ejpam-4616	70	30	all	all	DET
ejpam-4616	70	31	e	e	PROPN
ejpam-4616	70	32	∈	∈	PROPN
ejpam-4616	70	33	t	t	NOUN
ejpam-4616	70	34	,	,	PUNCT
ejpam-4616	70	35	(	(	PUNCT
ejpam-4616	70	36	2	2	X
ejpam-4616	70	37	)	)	PUNCT
ejpam-4616	70	38	µ̃	µ̃	PROPN
ejpam-4616	70	39	=	=	SYM
ejpam-4616	70	40	ϖ̃	ϖ̃	PROPN
ejpam-4616	70	41	⇔	⇔	NOUN
ejpam-4616	70	42	µ̃	µ̃	PROPN
ejpam-4616	70	43	⊑	⊑	X
ejpam-4616	70	44	ϖ̃	ϖ̃	PROPN
ejpam-4616	70	45	and	and	CCONJ
ejpam-4616	70	46	ϖ̃	ϖ̃	PROPN
ejpam-4616	70	47	⊑	⊑	X
ejpam-4616	70	48	µ̃	µ̃	PROPN
ejpam-4616	70	49	,	,	PUNCT
ejpam-4616	70	50	(	(	PUNCT
ejpam-4616	70	51	3	3	NUM
ejpam-4616	70	52	)	)	PUNCT
ejpam-4616	70	53	(	(	PUNCT
ejpam-4616	70	54	µ̃	µ̃	PROPN
ejpam-4616	70	55	⊓	⊓	PROPN
ejpam-4616	70	56	ϖ̃)(e	ϖ̃)(e	X
ejpam-4616	70	57	)	)	PUNCT
ejpam-4616	70	58	=	=	NOUN
ejpam-4616	70	59	µ̃(e)⋏	µ̃(e)⋏	ADP
ejpam-4616	70	60	ϖ̃(e	ϖ̃(e	NOUN
ejpam-4616	70	61	)	)	PUNCT
ejpam-4616	70	62	for	for	ADP
ejpam-4616	70	63	all	all	DET
ejpam-4616	70	64	e	e	PROPN
ejpam-4616	70	65	∈	∈	PROPN
ejpam-4616	70	66	t	t	NOUN
ejpam-4616	70	67	,	,	PUNCT
ejpam-4616	70	68	(	(	PUNCT
ejpam-4616	70	69	4	4	NUM
ejpam-4616	70	70	)	)	PUNCT
ejpam-4616	70	71	(	(	PUNCT
ejpam-4616	70	72	µ̃	µ̃	PROPN
ejpam-4616	70	73	⊔	⊔	PROPN
ejpam-4616	70	74	ϖ̃)(e	ϖ̃)(e	NOUN
ejpam-4616	70	75	)	)	PUNCT
ejpam-4616	71	1	=	=	PUNCT
ejpam-4616	72	1	µ̃(e)⋎	µ̃(e)⋎	X
ejpam-4616	72	2	ϖ̃(e	ϖ̃(e	NOUN
ejpam-4616	72	3	)	)	PUNCT
ejpam-4616	72	4	for	for	ADP
ejpam-4616	72	5	all	all	DET
ejpam-4616	72	6	e	e	PROPN
ejpam-4616	72	7	∈	∈	PROPN
ejpam-4616	72	8	t	t	PROPN
ejpam-4616	72	9	.	.	PUNCT
ejpam-4616	73	1	for	for	ADP
ejpam-4616	73	2	two	two	NUM
ejpam-4616	73	3	ivf	ivf	NOUN
ejpam-4616	73	4	sets	set	NOUN
ejpam-4616	73	5	µ̃	µ̃	PROPN
ejpam-4616	73	6	and	and	CCONJ
ejpam-4616	73	7	ϖ̃	ϖ̃	PROPN
ejpam-4616	73	8	in	in	ADP
ejpam-4616	73	9	a	a	DET
ejpam-4616	73	10	semigroup	semigroup	NOUN
ejpam-4616	73	11	s	s	NOUN
ejpam-4616	73	12	,	,	PUNCT
ejpam-4616	73	13	define	define	VERB
ejpam-4616	73	14	the	the	DET
ejpam-4616	73	15	product	product	NOUN
ejpam-4616	73	16	µ̃	µ̃	PROPN
ejpam-4616	73	17	◦	◦	NOUN
ejpam-4616	73	18	ϖ̃	ϖ̃	PROPN
ejpam-4616	73	19	as	as	SCONJ
ejpam-4616	73	20	follows	follow	VERB
ejpam-4616	73	21	:	:	PUNCT
ejpam-4616	73	22	for	for	ADP
ejpam-4616	73	23	all	all	DET
ejpam-4616	73	24	e	e	PROPN
ejpam-4616	73	25	∈	∈	PROPN
ejpam-4616	73	26	s	s	NOUN
ejpam-4616	73	27	,	,	PUNCT
ejpam-4616	73	28	(	(	PUNCT
ejpam-4616	73	29	µ̃	µ̃	PROPN
ejpam-4616	73	30	◦	◦	NOUN
ejpam-4616	73	31	ϖ̃)(e	ϖ̃)(e	NOUN
ejpam-4616	73	32	)	)	PUNCT
ejpam-4616	74	1	=	=	PUNCT
ejpam-4616	74	2			PUNCT
ejpam-4616	74	3	⋃	⋃	NOUN
ejpam-4616	74	4	e	e	NOUN
ejpam-4616	74	5	=	=	NOUN
ejpam-4616	74	6	th	th	X
ejpam-4616	74	7	{	{	PUNCT
ejpam-4616	74	8	µ̃(t)⋏	µ̃(t)⋏	ADV
ejpam-4616	74	9	ϖ̃(h	ϖ̃(h	PROPN
ejpam-4616	74	10	)	)	PUNCT
ejpam-4616	74	11	}	}	PUNCT
ejpam-4616	74	12	,	,	PUNCT
ejpam-4616	74	13	[	[	X
ejpam-4616	74	14	0	0	NUM
ejpam-4616	74	15	,	,	PUNCT
ejpam-4616	74	16	0	0	NUM
ejpam-4616	74	17	]	]	PUNCT
ejpam-4616	74	18	.	.	PUNCT
ejpam-4616	75	1	definition	definition	NOUN
ejpam-4616	75	2	4	4	NUM
ejpam-4616	75	3	.	.	PUNCT
ejpam-4616	76	1	[	[	X
ejpam-4616	76	2	9	9	NUM
ejpam-4616	76	3	]	]	X
ejpam-4616	76	4	an	an	DET
ejpam-4616	76	5	ivf	ivf	NOUN
ejpam-4616	76	6	subset	subset	VERB
ejpam-4616	76	7	µ̃	µ̃	PROPN
ejpam-4616	76	8	of	of	ADP
ejpam-4616	76	9	a	a	DET
ejpam-4616	76	10	semigroup	semigroup	NOUN
ejpam-4616	76	11	s	s	NOUN
ejpam-4616	76	12	is	be	AUX
ejpam-4616	76	13	said	say	VERB
ejpam-4616	76	14	to	to	PART
ejpam-4616	76	15	be	be	AUX
ejpam-4616	76	16	(	(	PUNCT
ejpam-4616	76	17	1	1	X
ejpam-4616	76	18	)	)	PUNCT
ejpam-4616	76	19	an	an	DET
ejpam-4616	76	20	ivf	ivf	NOUN
ejpam-4616	76	21	subsemigroup	subsemigroup	NOUN
ejpam-4616	76	22	of	of	ADP
ejpam-4616	76	23	s	s	PRON
ejpam-4616	76	24	if	if	SCONJ
ejpam-4616	76	25	µ̃(uv	µ̃(uv	NOUN
ejpam-4616	76	26	)	)	PUNCT
ejpam-4616	76	27	⪰	⪰	NOUN
ejpam-4616	76	28	µ̃(u)⋏	µ̃(u)⋏	PROPN
ejpam-4616	76	29	µ̃(v	µ̃(v	PROPN
ejpam-4616	76	30	)	)	PUNCT
ejpam-4616	76	31	for	for	ADP
ejpam-4616	76	32	all	all	DET
ejpam-4616	76	33	u	u	NOUN
ejpam-4616	76	34	,	,	PUNCT
ejpam-4616	76	35	v	v	ADP
ejpam-4616	76	36	∈	∈	PROPN
ejpam-4616	76	37	s	s	NOUN
ejpam-4616	76	38	,	,	PUNCT
ejpam-4616	76	39	(	(	PUNCT
ejpam-4616	76	40	2	2	X
ejpam-4616	76	41	)	)	PUNCT
ejpam-4616	76	42	an	an	DET
ejpam-4616	76	43	ivf	ivf	NOUN
ejpam-4616	76	44	left	left	NOUN
ejpam-4616	76	45	(	(	PUNCT
ejpam-4616	76	46	right	right	ADJ
ejpam-4616	76	47	)	)	PUNCT
ejpam-4616	76	48	ideal	ideal	NOUN
ejpam-4616	76	49	of	of	ADP
ejpam-4616	76	50	s	s	PRON
ejpam-4616	76	51	if	if	SCONJ
ejpam-4616	76	52	µ̃(uv	µ̃(uv	NOUN
ejpam-4616	76	53	)	)	PUNCT
ejpam-4616	76	54	⪰	⪰	NOUN
ejpam-4616	76	55	µ̃(v)(µ̃(uv	µ̃(v)(µ̃(uv	NOUN
ejpam-4616	76	56	)	)	PUNCT
ejpam-4616	76	57	⪰	⪰	NOUN
ejpam-4616	76	58	µ̃(u	µ̃(u	VERB
ejpam-4616	76	59	)	)	PUNCT
ejpam-4616	76	60	)	)	PUNCT
ejpam-4616	76	61	for	for	ADP
ejpam-4616	76	62	all	all	DET
ejpam-4616	76	63	u	u	NOUN
ejpam-4616	76	64	,	,	PUNCT
ejpam-4616	76	65	v	v	ADP
ejpam-4616	76	66	∈	∈	PROPN
ejpam-4616	76	67	s.	s.	PROPN
ejpam-4616	76	68	an	an	DET
ejpam-4616	76	69	ivf	ivf	NOUN
ejpam-4616	76	70	subset	subset	VERB
ejpam-4616	76	71	µ̃	µ̃	PROPN
ejpam-4616	76	72	of	of	ADP
ejpam-4616	76	73	s	s	PROPN
ejpam-4616	76	74	is	be	AUX
ejpam-4616	76	75	called	call	VERB
ejpam-4616	76	76	an	an	DET
ejpam-4616	76	77	ivf	ivf	ADJ
ejpam-4616	76	78	ideal	ideal	NOUN
ejpam-4616	76	79	of	of	ADP
ejpam-4616	76	80	s	s	PRON
ejpam-4616	76	81	if	if	SCONJ
ejpam-4616	76	82	it	it	PRON
ejpam-4616	76	83	is	be	AUX
ejpam-4616	76	84	both	both	CCONJ
ejpam-4616	76	85	an	an	DET
ejpam-4616	76	86	ivf	ivf	NOUN
ejpam-4616	76	87	left	leave	VERB
ejpam-4616	76	88	ideal	ideal	NOUN
ejpam-4616	76	89	and	and	CCONJ
ejpam-4616	76	90	an	an	DET
ejpam-4616	76	91	ivf	ivf	ADJ
ejpam-4616	76	92	right	right	ADJ
ejpam-4616	76	93	ideal	ideal	NOUN
ejpam-4616	76	94	of	of	ADP
ejpam-4616	76	95	s	s	PROPN
ejpam-4616	76	96	,	,	PUNCT
ejpam-4616	76	97	t.	t.	PROPN
ejpam-4616	76	98	gaketem	gaketem	NOUN
ejpam-4616	76	99	,	,	PUNCT
ejpam-4616	76	100	t.	t.	PROPN
ejpam-4616	76	101	prommai	prommai	PROPN
ejpam-4616	76	102	/	/	SYM
ejpam-4616	76	103	eur	eur	PROPN
ejpam-4616	76	104	.	.	PUNCT
ejpam-4616	77	1	j.	j.	PROPN
ejpam-4616	77	2	pure	pure	PROPN
ejpam-4616	77	3	appl	appl	PROPN
ejpam-4616	77	4	.	.	PROPN
ejpam-4616	77	5	math	math	PROPN
ejpam-4616	77	6	,	,	PUNCT
ejpam-4616	77	7	16	16	NUM
ejpam-4616	77	8	(	(	PUNCT
ejpam-4616	77	9	1	1	NUM
ejpam-4616	77	10	)	)	PUNCT
ejpam-4616	77	11	(	(	PUNCT
ejpam-4616	77	12	2023	2023	NUM
ejpam-4616	77	13	)	)	PUNCT
ejpam-4616	77	14	,	,	PUNCT
ejpam-4616	77	15	121	121	NUM
ejpam-4616	77	16	-	-	SYM
ejpam-4616	77	17	130	130	NUM
ejpam-4616	77	18	124	124	NUM
ejpam-4616	77	19	(	(	PUNCT
ejpam-4616	77	20	3	3	X
ejpam-4616	77	21	)	)	PUNCT
ejpam-4616	77	22	an	an	DET
ejpam-4616	77	23	ivf	ivf	ADJ
ejpam-4616	77	24	bi	bi	NOUN
ejpam-4616	77	25	-	-	NOUN
ejpam-4616	77	26	ideal	ideal	NOUN
ejpam-4616	77	27	of	of	ADP
ejpam-4616	77	28	s	s	PRON
ejpam-4616	77	29	if	if	SCONJ
ejpam-4616	77	30	µ̃	µ̃	PROPN
ejpam-4616	77	31	is	be	AUX
ejpam-4616	77	32	an	an	DET
ejpam-4616	77	33	ivf	ivf	ADJ
ejpam-4616	77	34	subsemigroup	subsemigroup	NOUN
ejpam-4616	77	35	and	and	CCONJ
ejpam-4616	77	36	µ̃(uvw	µ̃(uvw	NOUN
ejpam-4616	77	37	)	)	PUNCT
ejpam-4616	77	38	⪰	⪰	NOUN
ejpam-4616	77	39	µ̃(u	µ̃(u	VERB
ejpam-4616	77	40	)	)	PUNCT
ejpam-4616	77	41	⋏	⋏	PROPN
ejpam-4616	77	42	µ̃(w	µ̃(w	NOUN
ejpam-4616	77	43	)	)	PUNCT
ejpam-4616	77	44	for	for	ADP
ejpam-4616	77	45	all	all	DET
ejpam-4616	77	46	u	u	NOUN
ejpam-4616	77	47	,	,	PUNCT
ejpam-4616	77	48	v	v	NOUN
ejpam-4616	77	49	,	,	PUNCT
ejpam-4616	77	50	w	w	PROPN
ejpam-4616	77	51	∈	∈	PROPN
ejpam-4616	77	52	s	s	NOUN
ejpam-4616	77	53	,	,	PUNCT
ejpam-4616	77	54	(	(	PUNCT
ejpam-4616	77	55	4	4	X
ejpam-4616	77	56	)	)	PUNCT
ejpam-4616	77	57	an	an	DET
ejpam-4616	77	58	ivf	ivf	ADJ
ejpam-4616	77	59	interior	interior	ADJ
ejpam-4616	77	60	ideal	ideal	NOUN
ejpam-4616	77	61	of	of	ADP
ejpam-4616	77	62	s	s	PRON
ejpam-4616	77	63	if	if	SCONJ
ejpam-4616	77	64	µ̃	µ̃	PROPN
ejpam-4616	77	65	is	be	AUX
ejpam-4616	77	66	an	an	DET
ejpam-4616	77	67	ivf	ivf	ADJ
ejpam-4616	77	68	subsemigroup	subsemigroup	NOUN
ejpam-4616	77	69	and	and	CCONJ
ejpam-4616	77	70	µ̃(uav	µ̃(uav	NOUN
ejpam-4616	77	71	)	)	PUNCT
ejpam-4616	77	72	⪰	⪰	NOUN
ejpam-4616	77	73	µ̃(a	µ̃(a	NOUN
ejpam-4616	77	74	)	)	PUNCT
ejpam-4616	77	75	for	for	ADP
ejpam-4616	77	76	all	all	DET
ejpam-4616	77	77	a	a	DET
ejpam-4616	77	78	,	,	PUNCT
ejpam-4616	77	79	u	u	NOUN
ejpam-4616	77	80	,	,	PUNCT
ejpam-4616	77	81	v	v	ADP
ejpam-4616	77	82	∈	∈	PROPN
ejpam-4616	77	83	s	s	NOUN
ejpam-4616	77	84	,	,	PUNCT
ejpam-4616	77	85	(	(	PUNCT
ejpam-4616	77	86	5	5	X
ejpam-4616	77	87	)	)	PUNCT
ejpam-4616	77	88	an	an	DET
ejpam-4616	77	89	ivf	ivf	NOUN
ejpam-4616	77	90	quasi	quasi	NOUN
ejpam-4616	77	91	-	-	NOUN
ejpam-4616	77	92	ideal	ideal	ADJ
ejpam-4616	77	93	of	of	ADP
ejpam-4616	77	94	s	s	PRON
ejpam-4616	77	95	if	if	SCONJ
ejpam-4616	77	96	(	(	PUNCT
ejpam-4616	77	97	s̃	s̃	NOUN
ejpam-4616	77	98	◦	◦	NOUN
ejpam-4616	77	99	µ̃)(u)⋏	µ̃)(u)⋏	X
ejpam-4616	77	100	(	(	PUNCT
ejpam-4616	77	101	µ̃	µ̃	PROPN
ejpam-4616	77	102	◦	◦	NOUN
ejpam-4616	77	103	s̃)(u	s̃)(u	ADV
ejpam-4616	77	104	)	)	PUNCT
ejpam-4616	77	105	⪯	⪯	NOUN
ejpam-4616	77	106	µ̃(u	µ̃(u	VERB
ejpam-4616	77	107	)	)	PUNCT
ejpam-4616	77	108	for	for	ADP
ejpam-4616	77	109	all	all	DET
ejpam-4616	77	110	u	u	PROPN
ejpam-4616	77	111	∈	∈	NOUN
ejpam-4616	77	112	s	s	VERB
ejpam-4616	77	113	where	where	SCONJ
ejpam-4616	77	114	s̃	s̃	PROPN
ejpam-4616	77	115	is	be	AUX
ejpam-4616	77	116	an	an	DET
ejpam-4616	77	117	ivf	ivf	NOUN
ejpam-4616	77	118	subset	subset	NOUN
ejpam-4616	77	119	of	of	ADP
ejpam-4616	77	120	s	s	PRON
ejpam-4616	77	121	mapping	map	VERB
ejpam-4616	77	122	every	every	DET
ejpam-4616	77	123	element	element	NOUN
ejpam-4616	77	124	of	of	ADP
ejpam-4616	77	125	s	s	PRON
ejpam-4616	77	126	on	on	ADP
ejpam-4616	77	127	[	[	X
ejpam-4616	77	128	1	1	NUM
ejpam-4616	77	129	,	,	PUNCT
ejpam-4616	77	130	1	1	NUM
ejpam-4616	77	131	]	]	PUNCT
ejpam-4616	77	132	.	.	PUNCT
ejpam-4616	78	1	theorem	theorem	NOUN
ejpam-4616	78	2	1	1	NUM
ejpam-4616	78	3	.	.	PUNCT
ejpam-4616	79	1	[	[	X
ejpam-4616	79	2	9	9	NUM
ejpam-4616	79	3	]	]	PUNCT
ejpam-4616	79	4	let	let	VERB
ejpam-4616	79	5	s	s	PRON
ejpam-4616	79	6	be	be	AUX
ejpam-4616	79	7	a	a	DET
ejpam-4616	79	8	semigroup	semigroup	NOUN
ejpam-4616	79	9	and	and	CCONJ
ejpam-4616	79	10	let	let	VERB
ejpam-4616	79	11	m	m	PRON
ejpam-4616	79	12	be	be	AUX
ejpam-4616	79	13	non	non	ADJ
ejpam-4616	79	14	-	-	ADJ
ejpam-4616	79	15	empty	empty	ADJ
ejpam-4616	79	16	subset	subset	NOUN
ejpam-4616	79	17	of	of	ADP
ejpam-4616	79	18	s.	s.	PROPN
ejpam-4616	79	19	then	then	ADV
ejpam-4616	79	20	m	m	PROPN
ejpam-4616	79	21	is	be	AUX
ejpam-4616	79	22	a	a	DET
ejpam-4616	79	23	subsemigroup	subsemigroup	NOUN
ejpam-4616	79	24	(	(	PUNCT
ejpam-4616	79	25	left	leave	VERB
ejpam-4616	79	26	ideals	ideal	NOUN
ejpam-4616	79	27	,	,	PUNCT
ejpam-4616	79	28	right	right	ADJ
ejpam-4616	79	29	ideals	ideal	NOUN
ejpam-4616	79	30	,	,	PUNCT
ejpam-4616	79	31	interior	interior	ADJ
ejpam-4616	79	32	ideals	ideal	NOUN
ejpam-4616	79	33	,	,	PUNCT
ejpam-4616	79	34	bi	bi	NOUN
ejpam-4616	79	35	-	-	NOUN
ejpam-4616	79	36	ideals	ideal	NOUN
ejpam-4616	79	37	,	,	PUNCT
ejpam-4616	79	38	quasi	quasi	NOUN
ejpam-4616	79	39	-	-	NOUN
ejpam-4616	79	40	ideals	ideal	NOUN
ejpam-4616	79	41	)	)	PUNCT
ejpam-4616	79	42	of	of	ADP
ejpam-4616	79	43	s	s	PRON
ejpam-4616	79	44	if	if	SCONJ
ejpam-4616	79	45	and	and	CCONJ
ejpam-4616	79	46	only	only	ADV
ejpam-4616	79	47	if	if	SCONJ
ejpam-4616	79	48	the	the	DET
ejpam-4616	79	49	characteristic	characteristic	ADJ
ejpam-4616	79	50	set	set	NOUN
ejpam-4616	79	51	χ̃m	χ̃m	PROPN
ejpam-4616	79	52	is	be	AUX
ejpam-4616	79	53	an	an	DET
ejpam-4616	79	54	ivf	ivf	ADJ
ejpam-4616	79	55	subsemigroup	subsemigroup	NOUN
ejpam-4616	79	56	(	(	PUNCT
ejpam-4616	79	57	left	leave	VERB
ejpam-4616	79	58	ideals	ideal	NOUN
ejpam-4616	79	59	,	,	PUNCT
ejpam-4616	79	60	right	right	ADJ
ejpam-4616	79	61	ideals	ideal	NOUN
ejpam-4616	79	62	,	,	PUNCT
ejpam-4616	79	63	interior	interior	ADJ
ejpam-4616	79	64	ideals	ideal	NOUN
ejpam-4616	79	65	,	,	PUNCT
ejpam-4616	79	66	bi	bi	NOUN
ejpam-4616	79	67	-	-	NOUN
ejpam-4616	79	68	ideals	ideal	NOUN
ejpam-4616	79	69	,	,	PUNCT
ejpam-4616	79	70	quasi	quasi	NOUN
ejpam-4616	79	71	-	-	NOUN
ejpam-4616	79	72	ideals	ideal	NOUN
ejpam-4616	79	73	)	)	PUNCT
ejpam-4616	79	74	of	of	ADP
ejpam-4616	79	75	s.	s.	PROPN
ejpam-4616	79	76	3	3	NUM
ejpam-4616	80	1	.	.	PUNCT
ejpam-4616	80	2	interval	interval	NOUN
ejpam-4616	80	3	valued	value	VERB
ejpam-4616	80	4	fuzzy	fuzzy	ADJ
ejpam-4616	80	5	bi	bi	ADJ
ejpam-4616	80	6	-	-	ADJ
ejpam-4616	80	7	interior	interior	ADJ
ejpam-4616	80	8	ideals	ideal	NOUN
ejpam-4616	80	9	of	of	ADP
ejpam-4616	80	10	semigroups	semigroup	NOUN
ejpam-4616	80	11	in	in	ADP
ejpam-4616	80	12	this	this	DET
ejpam-4616	80	13	section	section	NOUN
ejpam-4616	80	14	,	,	PUNCT
ejpam-4616	80	15	we	we	PRON
ejpam-4616	80	16	introduce	introduce	VERB
ejpam-4616	80	17	the	the	DET
ejpam-4616	80	18	notion	notion	NOUN
ejpam-4616	80	19	of	of	ADP
ejpam-4616	80	20	an	an	DET
ejpam-4616	80	21	interval	interval	NOUN
ejpam-4616	80	22	valued	value	VERB
ejpam-4616	80	23	fuzzy	fuzzy	ADJ
ejpam-4616	80	24	bi	bi	ADJ
ejpam-4616	80	25	-	-	ADJ
ejpam-4616	80	26	interior	interior	ADJ
ejpam-4616	80	27	ideal	ideal	NOUN
ejpam-4616	80	28	and	and	CCONJ
ejpam-4616	80	29	study	study	VERB
ejpam-4616	80	30	the	the	DET
ejpam-4616	80	31	properties	property	NOUN
ejpam-4616	80	32	of	of	ADP
ejpam-4616	80	33	interval	interval	NOUN
ejpam-4616	80	34	valued	value	VERB
ejpam-4616	80	35	fuzzy	fuzzy	ADJ
ejpam-4616	80	36	bi	bi	ADJ
ejpam-4616	80	37	-	-	ADJ
ejpam-4616	80	38	interior	interior	ADJ
ejpam-4616	80	39	ideals	ideal	NOUN
ejpam-4616	80	40	of	of	ADP
ejpam-4616	80	41	semigroups	semigroup	NOUN
ejpam-4616	80	42	.	.	PUNCT
ejpam-4616	81	1	definition	definition	NOUN
ejpam-4616	81	2	5	5	NUM
ejpam-4616	81	3	.	.	PUNCT
ejpam-4616	82	1	an	an	DET
ejpam-4616	82	2	ivf	ivf	NOUN
ejpam-4616	82	3	subsemigroup	subsemigroup	NOUN
ejpam-4616	82	4	µ̃	µ̃	PROPN
ejpam-4616	82	5	of	of	ADP
ejpam-4616	82	6	a	a	DET
ejpam-4616	82	7	semigroup	semigroup	NOUN
ejpam-4616	82	8	s	s	PART
ejpam-4616	82	9	is	be	AUX
ejpam-4616	82	10	called	call	VERB
ejpam-4616	82	11	an	an	DET
ejpam-4616	82	12	ivf	ivf	ADJ
ejpam-4616	82	13	bi	bi	ADJ
ejpam-4616	82	14	-	-	ADJ
ejpam-4616	82	15	interior	interior	ADJ
ejpam-4616	82	16	ideal	ideal	NOUN
ejpam-4616	82	17	of	of	ADP
ejpam-4616	82	18	s	s	PRON
ejpam-4616	82	19	if	if	SCONJ
ejpam-4616	82	20	it	it	PRON
ejpam-4616	82	21	satisfies	satisfy	VERB
ejpam-4616	82	22	the	the	DET
ejpam-4616	82	23	following	follow	VERB
ejpam-4616	82	24	condition	condition	NOUN
ejpam-4616	82	25	:	:	PUNCT
ejpam-4616	82	26	χ̃s	χ̃s	PROPN
ejpam-4616	83	1	◦	◦	VERB
ejpam-4616	83	2	µ̃	µ̃	PROPN
ejpam-4616	83	3	◦	◦	NOUN
ejpam-4616	83	4	χ̃s	χ̃s	PUNCT
ejpam-4616	83	5	⊓	⊓	PROPN
ejpam-4616	83	6	µ̃	µ̃	PROPN
ejpam-4616	83	7	◦	◦	VERB
ejpam-4616	83	8	χ̃s	χ̃s	NOUN
ejpam-4616	83	9	◦	◦	NOUN
ejpam-4616	83	10	µ̃⊆̃µ̃	µ̃⊆̃µ̃	PROPN
ejpam-4616	83	11	,	,	PUNCT
ejpam-4616	83	12	example	example	NOUN
ejpam-4616	83	13	1	1	NUM
ejpam-4616	83	14	.	.	X
ejpam-4616	83	15	define	define	VERB
ejpam-4616	83	16	µ̃	µ̃	PROPN
ejpam-4616	83	17	:	:	PUNCT
ejpam-4616	83	18	t	t	PROPN
ejpam-4616	83	19	→	→	SYM
ejpam-4616	83	20	ω[0	ω[0	PROPN
ejpam-4616	83	21	,	,	PUNCT
ejpam-4616	83	22	1	1	NUM
ejpam-4616	83	23	]	]	PUNCT
ejpam-4616	83	24	by	by	ADP
ejpam-4616	83	25	µ̃(e	µ̃(e	PROPN
ejpam-4616	83	26	)	)	PUNCT
ejpam-4616	84	1	=	=	PRON
ejpam-4616	84	2	{	{	PUNCT
ejpam-4616	85	1	[	[	X
ejpam-4616	85	2	1	1	NUM
ejpam-4616	85	3	,	,	PUNCT
ejpam-4616	85	4	1	1	NUM
ejpam-4616	85	5	]	]	PUNCT
ejpam-4616	85	6	if	if	SCONJ
ejpam-4616	85	7	e	e	PROPN
ejpam-4616	85	8	∈	∈	PROPN
ejpam-4616	85	9	t	t	PROPN
ejpam-4616	85	10	,	,	PUNCT
ejpam-4616	85	11	[	[	X
ejpam-4616	85	12	0	0	NUM
ejpam-4616	85	13	,	,	PUNCT
ejpam-4616	85	14	0	0	NUM
ejpam-4616	85	15	]	]	PUNCT
ejpam-4616	86	1	if	if	SCONJ
ejpam-4616	86	2	e	e	PROPN
ejpam-4616	86	3	/∈	/∈	PROPN
ejpam-4616	86	4	t	t	PROPN
ejpam-4616	86	5	then	then	ADV
ejpam-4616	86	6	µ̃	µ̃	PROPN
ejpam-4616	86	7	is	be	AUX
ejpam-4616	86	8	an	an	DET
ejpam-4616	86	9	ivf	ivf	ADJ
ejpam-4616	86	10	bi	bi	ADJ
ejpam-4616	86	11	-	-	ADJ
ejpam-4616	86	12	interior	interior	ADJ
ejpam-4616	86	13	ideal	ideal	NOUN
ejpam-4616	86	14	of	of	ADP
ejpam-4616	86	15	t	t	PROPN
ejpam-4616	86	16	.	.	PUNCT
ejpam-4616	87	1	the	the	DET
ejpam-4616	87	2	next	next	ADJ
ejpam-4616	87	3	theorems	theorem	NOUN
ejpam-4616	87	4	are	be	AUX
ejpam-4616	87	5	studies	study	NOUN
ejpam-4616	87	6	ivf	ivf	ADJ
ejpam-4616	87	7	ideals	ideal	NOUN
ejpam-4616	87	8	in	in	ADP
ejpam-4616	87	9	semigroup	semigroup	PROPN
ejpam-4616	87	10	are	be	AUX
ejpam-4616	87	11	ivf	ivf	ADJ
ejpam-4616	87	12	bi	bi	ADJ
ejpam-4616	87	13	-	-	ADJ
ejpam-4616	87	14	interior	interior	ADJ
ejpam-4616	87	15	ideals	ideal	NOUN
ejpam-4616	87	16	of	of	ADP
ejpam-4616	87	17	semigroups	semigroup	NOUN
ejpam-4616	87	18	theorem	theorem	VERB
ejpam-4616	87	19	2	2	NUM
ejpam-4616	87	20	.	.	X
ejpam-4616	88	1	every	every	DET
ejpam-4616	88	2	ivf	ivf	NOUN
ejpam-4616	88	3	left	leave	VERB
ejpam-4616	88	4	ideal	ideal	NOUN
ejpam-4616	88	5	of	of	ADP
ejpam-4616	88	6	a	a	DET
ejpam-4616	88	7	semigroup	semigroup	NOUN
ejpam-4616	88	8	s	s	PART
ejpam-4616	88	9	is	be	AUX
ejpam-4616	88	10	an	an	DET
ejpam-4616	88	11	ivf	ivf	ADJ
ejpam-4616	88	12	bi	bi	ADJ
ejpam-4616	88	13	-	-	ADJ
ejpam-4616	88	14	interior	interior	ADJ
ejpam-4616	88	15	ideal	ideal	NOUN
ejpam-4616	88	16	of	of	ADP
ejpam-4616	88	17	s.	s.	PROPN
ejpam-4616	88	18	proof	proof	PROPN
ejpam-4616	88	19	.	.	PUNCT
ejpam-4616	89	1	let	let	VERB
ejpam-4616	89	2	µ̃	µ̃	PROPN
ejpam-4616	89	3	be	be	AUX
ejpam-4616	89	4	an	an	DET
ejpam-4616	89	5	ivf	ivf	NOUN
ejpam-4616	89	6	left	leave	VERB
ejpam-4616	89	7	ideal	ideal	NOUN
ejpam-4616	89	8	of	of	ADP
ejpam-4616	89	9	s.	s.	PROPN
ejpam-4616	89	10	let	let	VERB
ejpam-4616	89	11	x	x	PROPN
ejpam-4616	89	12	∈	∈	PROPN
ejpam-4616	89	13	s.	s.	PROPN
ejpam-4616	89	14	then	then	ADV
ejpam-4616	89	15	(	(	PUNCT
ejpam-4616	89	16	χ̃s	χ̃s	PROPN
ejpam-4616	89	17	◦	◦	PROPN
ejpam-4616	89	18	µ̃)(x	µ̃)(x	NOUN
ejpam-4616	89	19	)	)	PUNCT
ejpam-4616	90	1	=	=	PUNCT
ejpam-4616	90	2	⋃	⋃	NOUN
ejpam-4616	90	3	x	x	X
ejpam-4616	90	4	=	=	NOUN
ejpam-4616	90	5	yz	yz	X
ejpam-4616	90	6	{	{	PUNCT
ejpam-4616	90	7	χ̃s(y)⋏	χ̃s(y)⋏	PROPN
ejpam-4616	90	8	µ̃(z	µ̃(z	PROPN
ejpam-4616	90	9	)	)	PUNCT
ejpam-4616	90	10	}	}	PUNCT
ejpam-4616	90	11	=	=	SYM
ejpam-4616	91	1	⋃	⋃	NOUN
ejpam-4616	91	2	x	x	X
ejpam-4616	91	3	=	=	NOUN
ejpam-4616	91	4	yz	yz	X
ejpam-4616	91	5	{	{	PUNCT
ejpam-4616	91	6	µ̃(z	µ̃(z	PROPN
ejpam-4616	91	7	)	)	PUNCT
ejpam-4616	91	8	}	}	PUNCT
ejpam-4616	91	9	⊆	⊆	NUM
ejpam-4616	91	10	⋃	⋃	NOUN
ejpam-4616	91	11	x	x	SYM
ejpam-4616	91	12	=	=	NOUN
ejpam-4616	91	13	yz	yz	X
ejpam-4616	91	14	{	{	PUNCT
ejpam-4616	91	15	µ̃(yz	µ̃(yz	NOUN
ejpam-4616	91	16	)	)	PUNCT
ejpam-4616	91	17	}	}	PUNCT
ejpam-4616	92	1	=	=	SYM
ejpam-4616	92	2	⋃	⋃	NOUN
ejpam-4616	92	3	x	x	X
ejpam-4616	92	4	=	=	PROPN
ejpam-4616	92	5	yz	yz	X
ejpam-4616	92	6	{	{	PUNCT
ejpam-4616	92	7	µ̃(x	µ̃(x	PROPN
ejpam-4616	92	8	)	)	PUNCT
ejpam-4616	92	9	}	}	PUNCT
ejpam-4616	92	10	=	=	SYM
ejpam-4616	92	11	µ̃(x	µ̃(x	PROPN
ejpam-4616	92	12	)	)	PUNCT
ejpam-4616	92	13	.	.	PUNCT
ejpam-4616	93	1	we	we	PRON
ejpam-4616	93	2	have	have	AUX
ejpam-4616	93	3	,	,	PUNCT
ejpam-4616	93	4	(	(	PUNCT
ejpam-4616	93	5	µ̃	µ̃	PROPN
ejpam-4616	93	6	◦	◦	VERB
ejpam-4616	93	7	χ̃s	χ̃s	NOUN
ejpam-4616	93	8	◦	◦	NOUN
ejpam-4616	93	9	µ̃)(x	µ̃)(x	NOUN
ejpam-4616	93	10	)	)	PUNCT
ejpam-4616	94	1	=	=	PUNCT
ejpam-4616	94	2	⋃	⋃	NOUN
ejpam-4616	94	3	x	x	SYM
ejpam-4616	94	4	=	=	NOUN
ejpam-4616	94	5	abc{µ̃(a	abc{µ̃(a	NOUN
ejpam-4616	94	6	)	)	PUNCT
ejpam-4616	94	7	⋏	⋏	PROPN
ejpam-4616	94	8	(	(	PUNCT
ejpam-4616	94	9	χ̃s	χ̃s	PROPN
ejpam-4616	94	10	◦	◦	NOUN
ejpam-4616	94	11	µ̃)(bc	µ̃)(bc	ADJ
ejpam-4616	94	12	)	)	PUNCT
ejpam-4616	94	13	}	}	PUNCT
ejpam-4616	94	14	⊆	⊆	NUM
ejpam-4616	94	15	⋃	⋃	NOUN
ejpam-4616	94	16	x	x	SYM
ejpam-4616	94	17	=	=	NOUN
ejpam-4616	94	18	abc{µ̃(a	abc{µ̃(a	NOUN
ejpam-4616	94	19	)	)	PUNCT
ejpam-4616	94	20	⋏	⋏	PROPN
ejpam-4616	94	21	µ̃(bc	µ̃(bc	NOUN
ejpam-4616	94	22	)	)	PUNCT
ejpam-4616	94	23	}	}	PUNCT
ejpam-4616	94	24	=	=	SYM
ejpam-4616	94	25	µ̃(x	µ̃(x	PROPN
ejpam-4616	94	26	)	)	PUNCT
ejpam-4616	94	27	.	.	PUNCT
ejpam-4616	95	1	now	now	ADV
ejpam-4616	95	2	(	(	PUNCT
ejpam-4616	95	3	χ̃s	χ̃s	PROPN
ejpam-4616	95	4	◦	◦	VERB
ejpam-4616	95	5	µ̃	µ̃	PROPN
ejpam-4616	95	6	◦	◦	NOUN
ejpam-4616	95	7	χ̃s	χ̃s	PUNCT
ejpam-4616	95	8	⊓	⊓	PROPN
ejpam-4616	95	9	µ̃	µ̃	PROPN
ejpam-4616	95	10	◦	◦	VERB
ejpam-4616	95	11	χ̃s	χ̃s	NOUN
ejpam-4616	95	12	◦	◦	NOUN
ejpam-4616	95	13	µ̃)(x	µ̃)(x	NOUN
ejpam-4616	95	14	)	)	PUNCT
ejpam-4616	96	1	=	=	PRON
ejpam-4616	96	2	(	(	PUNCT
ejpam-4616	96	3	χ̃s	χ̃s	PROPN
ejpam-4616	96	4	◦	◦	NOUN
ejpam-4616	96	5	µ̃	µ̃	PROPN
ejpam-4616	96	6	◦	◦	PROPN
ejpam-4616	96	7	χ̃s)(x	χ̃s)(x	NOUN
ejpam-4616	96	8	)	)	PUNCT
ejpam-4616	96	9	⊓	⊓	PROPN
ejpam-4616	96	10	(	(	PUNCT
ejpam-4616	96	11	µ̃	µ̃	PROPN
ejpam-4616	96	12	◦	◦	VERB
ejpam-4616	96	13	χ̃s	χ̃s	NOUN
ejpam-4616	96	14	◦	◦	PROPN
ejpam-4616	96	15	µ̃)(x	µ̃)(x	NOUN
ejpam-4616	96	16	)	)	PUNCT
ejpam-4616	96	17	⪯	⪯	NOUN
ejpam-4616	96	18	(	(	PUNCT
ejpam-4616	96	19	χ̃s	χ̃s	PROPN
ejpam-4616	96	20	◦	◦	VERB
ejpam-4616	96	21	µ̃	µ̃	PROPN
ejpam-4616	96	22	◦	◦	PROPN
ejpam-4616	96	23	χ̃s)(x	χ̃s)(x	NOUN
ejpam-4616	96	24	)	)	PUNCT
ejpam-4616	96	25	⊓	⊓	PROPN
ejpam-4616	96	26	µ̃(x	µ̃(x	PROPN
ejpam-4616	96	27	)	)	PUNCT
ejpam-4616	96	28	⪯	⪯	PROPN
ejpam-4616	96	29	µ̃(x	µ̃(x	PROPN
ejpam-4616	96	30	)	)	PUNCT
ejpam-4616	96	31	.	.	PUNCT
ejpam-4616	97	1	therefor	therefor	ADP
ejpam-4616	97	2	χ̃s	χ̃s	PROPN
ejpam-4616	97	3	◦	◦	PROPN
ejpam-4616	97	4	µ̃	µ̃	PROPN
ejpam-4616	97	5	◦	◦	NOUN
ejpam-4616	97	6	χ̃s	χ̃s	PUNCT
ejpam-4616	97	7	⊓	⊓	PROPN
ejpam-4616	97	8	µ̃	µ̃	PROPN
ejpam-4616	97	9	◦	◦	VERB
ejpam-4616	97	10	χ̃s	χ̃s	NUM
ejpam-4616	97	11	◦	◦	VERB
ejpam-4616	97	12	µ̃	µ̃	PROPN
ejpam-4616	97	13	⊑	⊑	PRON
ejpam-4616	97	14	µ̃.	µ̃.	PROPN
ejpam-4616	97	15	hence	hence	ADV
ejpam-4616	97	16	µ̃	µ̃	PROPN
ejpam-4616	97	17	is	be	AUX
ejpam-4616	97	18	an	an	DET
ejpam-4616	97	19	ivf	ivf	ADJ
ejpam-4616	97	20	bi	bi	ADJ
ejpam-4616	97	21	-	-	ADJ
ejpam-4616	97	22	interior	interior	ADJ
ejpam-4616	97	23	ideal	ideal	NOUN
ejpam-4616	97	24	of	of	ADP
ejpam-4616	97	25	s.	s.	PROPN
ejpam-4616	97	26	t.	t.	PROPN
ejpam-4616	97	27	gaketem	gaketem	PROPN
ejpam-4616	97	28	,	,	PUNCT
ejpam-4616	97	29	t.	t.	PROPN
ejpam-4616	97	30	prommai	prommai	PROPN
ejpam-4616	97	31	/	/	SYM
ejpam-4616	97	32	eur	eur	PROPN
ejpam-4616	97	33	.	.	PUNCT
ejpam-4616	98	1	j.	j.	PROPN
ejpam-4616	98	2	pure	pure	PROPN
ejpam-4616	98	3	appl	appl	PROPN
ejpam-4616	98	4	.	.	PROPN
ejpam-4616	98	5	math	math	PROPN
ejpam-4616	98	6	,	,	PUNCT
ejpam-4616	98	7	16	16	NUM
ejpam-4616	98	8	(	(	PUNCT
ejpam-4616	98	9	1	1	NUM
ejpam-4616	98	10	)	)	PUNCT
ejpam-4616	98	11	(	(	PUNCT
ejpam-4616	98	12	2023	2023	NUM
ejpam-4616	98	13	)	)	PUNCT
ejpam-4616	98	14	,	,	PUNCT
ejpam-4616	98	15	121	121	NUM
ejpam-4616	98	16	-	-	SYM
ejpam-4616	98	17	130	130	NUM
ejpam-4616	98	18	125	125	NUM
ejpam-4616	98	19	theorem	theorem	NOUN
ejpam-4616	98	20	3	3	NUM
ejpam-4616	98	21	.	.	PUNCT
ejpam-4616	99	1	every	every	DET
ejpam-4616	99	2	ivf	ivf	ADJ
ejpam-4616	99	3	right	right	ADJ
ejpam-4616	99	4	ideal	ideal	NOUN
ejpam-4616	99	5	of	of	ADP
ejpam-4616	99	6	a	a	DET
ejpam-4616	99	7	semigroup	semigroup	NOUN
ejpam-4616	99	8	s	s	PART
ejpam-4616	99	9	is	be	AUX
ejpam-4616	99	10	an	an	DET
ejpam-4616	99	11	ivf	ivf	ADJ
ejpam-4616	99	12	bi	bi	ADJ
ejpam-4616	99	13	-	-	ADJ
ejpam-4616	99	14	interior	interior	ADJ
ejpam-4616	99	15	ideal	ideal	NOUN
ejpam-4616	99	16	of	of	ADP
ejpam-4616	99	17	s.	s.	PROPN
ejpam-4616	99	18	proof	proof	PROPN
ejpam-4616	99	19	.	.	PUNCT
ejpam-4616	100	1	it	it	PRON
ejpam-4616	100	2	follows	follow	VERB
ejpam-4616	100	3	theorem	theorem	ADJ
ejpam-4616	100	4	2	2	NUM
ejpam-4616	100	5	.	.	PUNCT
ejpam-4616	100	6	corollary	corollary	ADJ
ejpam-4616	100	7	1	1	NUM
ejpam-4616	100	8	.	.	PUNCT
ejpam-4616	101	1	every	every	DET
ejpam-4616	101	2	ivf	ivf	ADJ
ejpam-4616	101	3	ideal	ideal	NOUN
ejpam-4616	101	4	of	of	ADP
ejpam-4616	101	5	a	a	DET
ejpam-4616	101	6	semigroup	semigroup	NOUN
ejpam-4616	101	7	s	s	PART
ejpam-4616	101	8	is	be	AUX
ejpam-4616	101	9	an	an	DET
ejpam-4616	101	10	ivf	ivf	ADJ
ejpam-4616	101	11	bi	bi	ADJ
ejpam-4616	101	12	-	-	ADJ
ejpam-4616	101	13	interior	interior	ADJ
ejpam-4616	101	14	ideal	ideal	NOUN
ejpam-4616	101	15	of	of	ADP
ejpam-4616	101	16	s.	s.	PROPN
ejpam-4616	101	17	definition	definition	NOUN
ejpam-4616	101	18	6	6	NUM
ejpam-4616	101	19	.	.	PUNCT
ejpam-4616	102	1	let	let	VERB
ejpam-4616	102	2	µ̃	µ̃	PROPN
ejpam-4616	102	3	be	be	AUX
ejpam-4616	102	4	an	an	DET
ejpam-4616	102	5	ivf	ivf	NOUN
ejpam-4616	102	6	set	set	NOUN
ejpam-4616	102	7	in	in	ADP
ejpam-4616	102	8	a	a	DET
ejpam-4616	102	9	non	non	ADJ
ejpam-4616	102	10	-	-	ADJ
ejpam-4616	102	11	empty	empty	ADJ
ejpam-4616	102	12	set	set	NOUN
ejpam-4616	102	13	x.	x.	NOUN
ejpam-4616	102	14	define	define	VERB
ejpam-4616	102	15	u(µ̃	u(µ̃	PROPN
ejpam-4616	102	16	;	;	PUNCT
ejpam-4616	102	17	t̃	t̃	PROPN
ejpam-4616	102	18	)	)	PUNCT
ejpam-4616	102	19	=	=	PRON
ejpam-4616	103	1	{	{	PUNCT
ejpam-4616	103	2	x	x	SYM
ejpam-4616	103	3	∈	∈	PROPN
ejpam-4616	103	4	x|t̃	x|t̃	NOUN
ejpam-4616	103	5	⊆	⊆	NUM
ejpam-4616	103	6	µ̃(x	µ̃(x	PROPN
ejpam-4616	103	7	)	)	PUNCT
ejpam-4616	103	8	}	}	PUNCT
ejpam-4616	103	9	where	where	SCONJ
ejpam-4616	103	10	t̄	t̄	PROPN
ejpam-4616	103	11	∈	∈	PROPN
ejpam-4616	103	12	ω[0	ω[0	PROPN
ejpam-4616	103	13	,	,	PUNCT
ejpam-4616	103	14	1	1	NUM
ejpam-4616	103	15	]	]	PUNCT
ejpam-4616	103	16	is	be	AUX
ejpam-4616	103	17	called	call	VERB
ejpam-4616	103	18	the	the	DET
ejpam-4616	103	19	ivf	ivf	ADJ
ejpam-4616	103	20	level	level	NOUN
ejpam-4616	103	21	set	set	NOUN
ejpam-4616	103	22	of	of	ADP
ejpam-4616	103	23	µ̃.	µ̃.	NOUN
ejpam-4616	103	24	theorem	theorem	VERB
ejpam-4616	103	25	4	4	NUM
ejpam-4616	103	26	.	.	PUNCT
ejpam-4616	104	1	let	let	VERB
ejpam-4616	104	2	s	s	PRON
ejpam-4616	104	3	be	be	AUX
ejpam-4616	104	4	a	a	DET
ejpam-4616	104	5	semigroup	semigroup	NOUN
ejpam-4616	104	6	and	and	CCONJ
ejpam-4616	104	7	µ̃	µ̃	PROPN
ejpam-4616	104	8	be	be	VERB
ejpam-4616	104	9	a	a	DET
ejpam-4616	104	10	non	non	ADJ
ejpam-4616	104	11	-	-	ADJ
ejpam-4616	104	12	empty	empty	ADJ
ejpam-4616	104	13	ivf	ivf	NOUN
ejpam-4616	104	14	set	set	NOUN
ejpam-4616	104	15	of	of	ADP
ejpam-4616	104	16	s.	s.	PROPN
ejpam-4616	104	17	an	an	DET
ejpam-4616	104	18	ivf	ivf	NOUN
ejpam-4616	104	19	set	set	VERB
ejpam-4616	104	20	µ̃	µ̃	PROPN
ejpam-4616	104	21	is	be	AUX
ejpam-4616	104	22	an	an	DET
ejpam-4616	104	23	ivf	ivf	ADJ
ejpam-4616	104	24	bi	bi	ADJ
ejpam-4616	104	25	-	-	ADJ
ejpam-4616	104	26	interior	interior	ADJ
ejpam-4616	104	27	ideal	ideal	NOUN
ejpam-4616	104	28	of	of	ADP
ejpam-4616	104	29	a	a	DET
ejpam-4616	104	30	semigroup	semigroup	NOUN
ejpam-4616	104	31	s	s	X
ejpam-4616	104	32	if	if	SCONJ
ejpam-4616	105	1	and	and	CCONJ
ejpam-4616	105	2	only	only	ADV
ejpam-4616	105	3	if	if	SCONJ
ejpam-4616	105	4	the	the	DET
ejpam-4616	105	5	ivf	ivf	ADJ
ejpam-4616	105	6	level	level	NOUN
ejpam-4616	105	7	set	set	VERB
ejpam-4616	105	8	u(µ̃	u(µ̃	PROPN
ejpam-4616	105	9	;	;	PUNCT
ejpam-4616	105	10	t̃	t̃	PROPN
ejpam-4616	105	11	)	)	PUNCT
ejpam-4616	105	12	of	of	ADP
ejpam-4616	105	13	s	s	PROPN
ejpam-4616	105	14	is	be	AUX
ejpam-4616	105	15	a	a	DET
ejpam-4616	105	16	bi	bi	ADJ
ejpam-4616	105	17	-	-	ADJ
ejpam-4616	105	18	interior	interior	ADJ
ejpam-4616	105	19	ideal	ideal	NOUN
ejpam-4616	105	20	of	of	ADP
ejpam-4616	105	21	a	a	DET
ejpam-4616	105	22	semigroup	semigroup	NOUN
ejpam-4616	105	23	s	s	NOUN
ejpam-4616	105	24	for	for	ADP
ejpam-4616	105	25	every	every	DET
ejpam-4616	105	26	t̃	t̃	PROPN
ejpam-4616	105	27	∈	∈	PROPN
ejpam-4616	105	28	ω[0	ω[0	PROPN
ejpam-4616	105	29	,	,	PUNCT
ejpam-4616	105	30	1	1	NUM
ejpam-4616	105	31	]	]	PUNCT
ejpam-4616	105	32	,	,	PUNCT
ejpam-4616	105	33	where	where	SCONJ
ejpam-4616	105	34	u(µ̃	u(µ̃	PROPN
ejpam-4616	105	35	;	;	PUNCT
ejpam-4616	105	36	t̃	t̃	PROPN
ejpam-4616	105	37	)	)	PUNCT
ejpam-4616	105	38	̸=	̸=	PROPN
ejpam-4616	105	39	∅.	∅.	ADP
ejpam-4616	105	40	proof	proof	NOUN
ejpam-4616	105	41	.	.	PUNCT
ejpam-4616	106	1	assume	assume	VERB
ejpam-4616	106	2	that	that	SCONJ
ejpam-4616	106	3	µ̃	µ̃	PROPN
ejpam-4616	106	4	is	be	AUX
ejpam-4616	106	5	an	an	DET
ejpam-4616	106	6	ivf	ivf	ADJ
ejpam-4616	106	7	bi	bi	ADJ
ejpam-4616	106	8	-	-	ADJ
ejpam-4616	106	9	interior	interior	ADJ
ejpam-4616	106	10	ideal	ideal	NOUN
ejpam-4616	106	11	of	of	ADP
ejpam-4616	106	12	s	s	PRON
ejpam-4616	106	13	and	and	CCONJ
ejpam-4616	106	14	let	let	VERB
ejpam-4616	106	15	x	x	X
ejpam-4616	106	16	∈	∈	PROPN
ejpam-4616	106	17	su(µ̃	su(µ̃	PROPN
ejpam-4616	106	18	;	;	PUNCT
ejpam-4616	106	19	t̃)s	t̃)s	NOUN
ejpam-4616	106	20	∩	∩	NOUN
ejpam-4616	106	21	u(µ̃	u(µ̃	PROPN
ejpam-4616	106	22	;	;	PUNCT
ejpam-4616	106	23	t̃)su(µ̃	t̃)su(µ̃	NOUN
ejpam-4616	106	24	;	;	PUNCT
ejpam-4616	106	25	t̃	t̃	PROPN
ejpam-4616	106	26	)	)	PUNCT
ejpam-4616	106	27	.	.	PUNCT
ejpam-4616	107	1	then	then	ADV
ejpam-4616	107	2	x	x	X
ejpam-4616	107	3	=	=	PUNCT
ejpam-4616	107	4	bau	bau	X
ejpam-4616	107	5	=	=	SYM
ejpam-4616	107	6	cde	cde	PROPN
ejpam-4616	107	7	where	where	SCONJ
ejpam-4616	107	8	b	b	NOUN
ejpam-4616	107	9	,	,	PUNCT
ejpam-4616	107	10	u	u	NOUN
ejpam-4616	107	11	,	,	PUNCT
ejpam-4616	107	12	d	d	PROPN
ejpam-4616	107	13	∈	∈	PROPN
ejpam-4616	107	14	s	s	X
ejpam-4616	107	15	and	and	CCONJ
ejpam-4616	107	16	a	a	DET
ejpam-4616	107	17	,	,	PUNCT
ejpam-4616	107	18	c	c	NOUN
ejpam-4616	107	19	,	,	PUNCT
ejpam-4616	107	20	e	e	PROPN
ejpam-4616	107	21	∈	∈	PROPN
ejpam-4616	107	22	u(µ̃	u(µ̃	PROPN
ejpam-4616	107	23	;	;	PUNCT
ejpam-4616	107	24	t̃	t̃	PROPN
ejpam-4616	107	25	)	)	PUNCT
ejpam-4616	107	26	.	.	PUNCT
ejpam-4616	108	1	then	then	ADV
ejpam-4616	108	2	t̃	t̃	PROPN
ejpam-4616	108	3	⪯	⪯	NOUN
ejpam-4616	108	4	(	(	PUNCT
ejpam-4616	108	5	χ̃s	χ̃s	PROPN
ejpam-4616	108	6	◦	◦	VERB
ejpam-4616	108	7	µ̃	µ̃	PROPN
ejpam-4616	108	8	◦	◦	PROPN
ejpam-4616	108	9	χ̃s)(x	χ̃s)(x	NOUN
ejpam-4616	108	10	)	)	PUNCT
ejpam-4616	108	11	and	and	CCONJ
ejpam-4616	108	12	t̃	t̃	PROPN
ejpam-4616	108	13	⪯	⪯	NOUN
ejpam-4616	108	14	(	(	PUNCT
ejpam-4616	108	15	µ̃	µ̃	PROPN
ejpam-4616	108	16	◦	◦	VERB
ejpam-4616	108	17	χ̃s	χ̃s	NOUN
ejpam-4616	108	18	◦	◦	PROPN
ejpam-4616	108	19	µ̃)(x	µ̃)(x	NOUN
ejpam-4616	108	20	)	)	PUNCT
ejpam-4616	108	21	implies	imply	VERB
ejpam-4616	108	22	that	that	SCONJ
ejpam-4616	108	23	t̃	t̃	PROPN
ejpam-4616	108	24	⪯	⪯	PROPN
ejpam-4616	108	25	µ̃(x	µ̃(x	PROPN
ejpam-4616	108	26	)	)	PUNCT
ejpam-4616	108	27	then	then	ADV
ejpam-4616	108	28	x	x	SYM
ejpam-4616	108	29	∈	∈	PROPN
ejpam-4616	108	30	u(µ̃	u(µ̃	PROPN
ejpam-4616	108	31	;	;	PUNCT
ejpam-4616	108	32	t̃	t̃	PROPN
ejpam-4616	108	33	)	)	PUNCT
ejpam-4616	108	34	.	.	PUNCT
ejpam-4616	109	1	therefore	therefore	ADV
ejpam-4616	109	2	u(µ̃	u(µ̃	PROPN
ejpam-4616	109	3	;	;	PUNCT
ejpam-4616	109	4	t̃	t̃	PROPN
ejpam-4616	109	5	)	)	PUNCT
ejpam-4616	109	6	is	be	AUX
ejpam-4616	109	7	a	a	DET
ejpam-4616	109	8	bi	bi	ADJ
ejpam-4616	109	9	-	-	ADJ
ejpam-4616	109	10	interior	interior	ADJ
ejpam-4616	109	11	ideal	ideal	NOUN
ejpam-4616	109	12	of	of	ADP
ejpam-4616	109	13	s.	s.	PROPN
ejpam-4616	109	14	conversely	conversely	ADV
ejpam-4616	109	15	suppose	suppose	VERB
ejpam-4616	109	16	that	that	SCONJ
ejpam-4616	109	17	u(µ̃	u(µ̃	PROPN
ejpam-4616	109	18	;	;	PUNCT
ejpam-4616	109	19	t̃	t̃	PROPN
ejpam-4616	109	20	)	)	PUNCT
ejpam-4616	109	21	is	be	AUX
ejpam-4616	109	22	a	a	DET
ejpam-4616	109	23	bi	bi	ADJ
ejpam-4616	109	24	-	-	ADJ
ejpam-4616	109	25	interior	interior	ADJ
ejpam-4616	109	26	ideal	ideal	NOUN
ejpam-4616	109	27	of	of	ADP
ejpam-4616	109	28	s	s	PROPN
ejpam-4616	109	29	,	,	PUNCT
ejpam-4616	109	30	for	for	ADP
ejpam-4616	109	31	all	all	DET
ejpam-4616	109	32	t̃	t̃	PROPN
ejpam-4616	109	33	∈	∈	PROPN
ejpam-4616	109	34	im(µ̃	im(µ̃	NOUN
ejpam-4616	109	35	)	)	PUNCT
ejpam-4616	109	36	.	.	PUNCT
ejpam-4616	110	1	let	let	VERB
ejpam-4616	110	2	x	x	PRON
ejpam-4616	110	3	,	,	PUNCT
ejpam-4616	110	4	y	y	PROPN
ejpam-4616	110	5	∈	∈	PROPN
ejpam-4616	110	6	s.	s.	PROPN
ejpam-4616	110	7	then	then	ADV
ejpam-4616	110	8	µ̃(x	µ̃(x	PROPN
ejpam-4616	110	9	)	)	PUNCT
ejpam-4616	110	10	=	=	SYM
ejpam-4616	110	11	t̃1	t̃1	PROPN
ejpam-4616	110	12	,	,	PUNCT
ejpam-4616	110	13	µ̃(y	µ̃(y	NOUN
ejpam-4616	110	14	)	)	PUNCT
ejpam-4616	110	15	=	=	SYM
ejpam-4616	110	16	t̃2	t̃2	PROPN
ejpam-4616	110	17	,	,	PUNCT
ejpam-4616	110	18	t̃1	t̃1	NOUN
ejpam-4616	110	19	⪰	⪰	NOUN
ejpam-4616	110	20	t̃2	t̃2	PROPN
ejpam-4616	110	21	.	.	PUNCT
ejpam-4616	111	1	then	then	ADV
ejpam-4616	111	2	x	x	X
ejpam-4616	111	3	,	,	PUNCT
ejpam-4616	111	4	y	y	PROPN
ejpam-4616	111	5	∈	∈	PROPN
ejpam-4616	111	6	u(µ̃	u(µ̃	PROPN
ejpam-4616	111	7	;	;	PUNCT
ejpam-4616	111	8	t̃	t̃	PROPN
ejpam-4616	111	9	)	)	PUNCT
ejpam-4616	111	10	.	.	PUNCT
ejpam-4616	112	1	thus	thus	ADV
ejpam-4616	112	2	su(µ̃	su(µ̃	ADJ
ejpam-4616	112	3	;	;	PUNCT
ejpam-4616	112	4	l̃)s	l̃)s	PROPN
ejpam-4616	112	5	⊓	⊓	PROPN
ejpam-4616	112	6	u(µ̃	u(µ̃	PROPN
ejpam-4616	112	7	;	;	PUNCT
ejpam-4616	112	8	l̃)su(µ̃	l̃)su(µ̃	PROPN
ejpam-4616	112	9	;	;	PUNCT
ejpam-4616	112	10	l̃	l̃	PROPN
ejpam-4616	112	11	)	)	PUNCT
ejpam-4616	112	12	⪯	⪯	NOUN
ejpam-4616	112	13	u(µ̃	u(µ̃	PROPN
ejpam-4616	112	14	;	;	PUNCT
ejpam-4616	112	15	l̃	l̃	PROPN
ejpam-4616	112	16	)	)	PUNCT
ejpam-4616	112	17	,	,	PUNCT
ejpam-4616	112	18	for	for	ADP
ejpam-4616	112	19	all	all	DET
ejpam-4616	112	20	l̃	l̃	PROPN
ejpam-4616	112	21	∈	∈	PROPN
ejpam-4616	112	22	im(µ̃	im(µ̃	NOUN
ejpam-4616	112	23	)	)	PUNCT
ejpam-4616	112	24	.	.	PUNCT
ejpam-4616	113	1	suppose	suppose	VERB
ejpam-4616	113	2	t̃	t̃	PROPN
ejpam-4616	113	3	=	=	PUNCT
ejpam-4616	113	4	min{im(µ̃	min{im(µ̃	PROPN
ejpam-4616	113	5	)	)	PUNCT
ejpam-4616	113	6	}	}	PUNCT
ejpam-4616	113	7	.	.	PUNCT
ejpam-4616	114	1	then	then	ADV
ejpam-4616	114	2	su(µ̃	su(µ̃	ADJ
ejpam-4616	114	3	;	;	PUNCT
ejpam-4616	114	4	t̃)s	t̃)s	NOUN
ejpam-4616	114	5	⊓	⊓	PROPN
ejpam-4616	114	6	u(µ̃	u(µ̃	PROPN
ejpam-4616	114	7	;	;	PUNCT
ejpam-4616	114	8	t̃)su(µ̃	t̃)su(µ̃	NOUN
ejpam-4616	114	9	;	;	PUNCT
ejpam-4616	114	10	t̃	t̃	PROPN
ejpam-4616	114	11	)	)	PUNCT
ejpam-4616	114	12	⪯	⪯	NOUN
ejpam-4616	114	13	u(µ̃	u(µ̃	PROPN
ejpam-4616	114	14	;	;	PUNCT
ejpam-4616	114	15	t̃	t̃	PROPN
ejpam-4616	114	16	)	)	PUNCT
ejpam-4616	114	17	.	.	PUNCT
ejpam-4616	115	1	therefor	therefor	ADP
ejpam-4616	115	2	χ̃s	χ̃s	PROPN
ejpam-4616	115	3	◦	◦	PROPN
ejpam-4616	115	4	µ̃	µ̃	PROPN
ejpam-4616	115	5	◦	◦	NOUN
ejpam-4616	115	6	χ̃s	χ̃s	PUNCT
ejpam-4616	115	7	⊓	⊓	PROPN
ejpam-4616	115	8	µ̃	µ̃	PROPN
ejpam-4616	115	9	◦	◦	VERB
ejpam-4616	115	10	χ̃s	χ̃s	NUM
ejpam-4616	115	11	◦	◦	VERB
ejpam-4616	115	12	µ̃	µ̃	PROPN
ejpam-4616	115	13	⊑	⊑	PRON
ejpam-4616	115	14	µ̃.	µ̃.	PROPN
ejpam-4616	115	15	hence	hence	ADV
ejpam-4616	115	16	µ̃	µ̃	PROPN
ejpam-4616	115	17	is	be	AUX
ejpam-4616	115	18	an	an	DET
ejpam-4616	115	19	ivf	ivf	ADJ
ejpam-4616	115	20	bi	bi	ADJ
ejpam-4616	115	21	-	-	ADJ
ejpam-4616	115	22	interior	interior	ADJ
ejpam-4616	115	23	ideal	ideal	NOUN
ejpam-4616	115	24	of	of	ADP
ejpam-4616	115	25	s.	s.	PROPN
ejpam-4616	115	26	theorem	theorem	VERB
ejpam-4616	115	27	5	5	X
ejpam-4616	115	28	.	.	PUNCT
ejpam-4616	116	1	let	let	VERB
ejpam-4616	116	2	m	m	PRON
ejpam-4616	116	3	be	be	AUX
ejpam-4616	116	4	a	a	DET
ejpam-4616	116	5	non	non	ADJ
ejpam-4616	116	6	-	-	ADJ
ejpam-4616	116	7	empty	empty	ADJ
ejpam-4616	116	8	subset	subset	NOUN
ejpam-4616	116	9	of	of	ADP
ejpam-4616	116	10	a	a	DET
ejpam-4616	116	11	semigroup	semigroup	NOUN
ejpam-4616	116	12	s	s	PART
ejpam-4616	116	13	and	and	CCONJ
ejpam-4616	116	14	χ̃m	χ̃m	AUX
ejpam-4616	116	15	be	be	AUX
ejpam-4616	116	16	the	the	DET
ejpam-4616	116	17	characteristic	characteristic	ADJ
ejpam-4616	116	18	ivf	ivf	NOUN
ejpam-4616	116	19	set	set	NOUN
ejpam-4616	116	20	of	of	ADP
ejpam-4616	116	21	m	m	PROPN
ejpam-4616	116	22	.	.	PUNCT
ejpam-4616	117	1	then	then	ADV
ejpam-4616	117	2	m	m	PROPN
ejpam-4616	117	3	is	be	AUX
ejpam-4616	117	4	a	a	DET
ejpam-4616	117	5	bi	bi	ADJ
ejpam-4616	117	6	-	-	ADJ
ejpam-4616	117	7	interior	interior	ADJ
ejpam-4616	117	8	ideal	ideal	NOUN
ejpam-4616	117	9	of	of	ADP
ejpam-4616	117	10	a	a	DET
ejpam-4616	117	11	semigroup	semigroup	NOUN
ejpam-4616	117	12	s	s	X
ejpam-4616	117	13	if	if	SCONJ
ejpam-4616	118	1	and	and	CCONJ
ejpam-4616	118	2	only	only	ADV
ejpam-4616	118	3	if	if	SCONJ
ejpam-4616	118	4	χ̃m	χ̃m	PROPN
ejpam-4616	118	5	is	be	AUX
ejpam-4616	118	6	an	an	DET
ejpam-4616	118	7	ivf	ivf	ADJ
ejpam-4616	118	8	bi	bi	ADJ
ejpam-4616	118	9	-	-	ADJ
ejpam-4616	118	10	interior	interior	ADJ
ejpam-4616	118	11	ideal	ideal	NOUN
ejpam-4616	118	12	of	of	ADP
ejpam-4616	118	13	a	a	DET
ejpam-4616	118	14	semigroup	semigroup	PROPN
ejpam-4616	118	15	s.	s.	PROPN
ejpam-4616	118	16	proof	proof	PROPN
ejpam-4616	118	17	.	.	PUNCT
ejpam-4616	119	1	suppose	suppose	VERB
ejpam-4616	119	2	m	m	PRON
ejpam-4616	119	3	is	be	AUX
ejpam-4616	119	4	a	a	DET
ejpam-4616	119	5	bi	bi	ADJ
ejpam-4616	119	6	-	-	ADJ
ejpam-4616	119	7	interior	interior	ADJ
ejpam-4616	119	8	ideal	ideal	NOUN
ejpam-4616	119	9	of	of	ADP
ejpam-4616	119	10	s.	s.	PROPN
ejpam-4616	119	11	then	then	ADV
ejpam-4616	119	12	m	m	PROPN
ejpam-4616	119	13	is	be	AUX
ejpam-4616	119	14	a	a	DET
ejpam-4616	119	15	subsemigroup	subsemigroup	NOUN
ejpam-4616	119	16	of	of	ADP
ejpam-4616	119	17	s.	s.	PROPN
ejpam-4616	119	18	thus	thus	ADV
ejpam-4616	119	19	by	by	ADP
ejpam-4616	119	20	theorem	theorem	NOUN
ejpam-4616	119	21	1	1	NUM
ejpam-4616	119	22	,	,	PUNCT
ejpam-4616	119	23	χ̃m	χ̃m	PROPN
ejpam-4616	119	24	is	be	AUX
ejpam-4616	119	25	an	an	DET
ejpam-4616	119	26	ivf	ivf	ADJ
ejpam-4616	119	27	subsemigroup	subsemigroup	NOUN
ejpam-4616	119	28	of	of	ADP
ejpam-4616	119	29	s.	s.	PROPN
ejpam-4616	119	30	let	let	VERB
ejpam-4616	119	31	x	x	PROPN
ejpam-4616	119	32	∈	∈	PROPN
ejpam-4616	119	33	s.	s.	PROPN
ejpam-4616	119	34	since	since	SCONJ
ejpam-4616	119	35	m	m	PROPN
ejpam-4616	119	36	is	be	AUX
ejpam-4616	119	37	a	a	DET
ejpam-4616	119	38	bi	bi	ADJ
ejpam-4616	119	39	-	-	ADJ
ejpam-4616	119	40	interior	interior	ADJ
ejpam-4616	119	41	ideal	ideal	NOUN
ejpam-4616	119	42	of	of	ADP
ejpam-4616	119	43	s	s	PROPN
ejpam-4616	119	44	,	,	PUNCT
ejpam-4616	119	45	we	we	PRON
ejpam-4616	119	46	have	have	VERB
ejpam-4616	119	47	sms	sm	NOUN
ejpam-4616	119	48	∩msm	∩msm	NOUN
ejpam-4616	119	49	⊆	⊆	NUM
ejpam-4616	119	50	m	m	NOUN
ejpam-4616	119	51	.	.	PUNCT
ejpam-4616	120	1	thus	thus	ADV
ejpam-4616	120	2	(	(	PUNCT
ejpam-4616	120	3	χ̃s	χ̃s	PROPN
ejpam-4616	120	4	◦	◦	NOUN
ejpam-4616	120	5	χ̃m	χ̃m	VERB
ejpam-4616	120	6	◦	◦	NOUN
ejpam-4616	120	7	χ̃s	χ̃s	PUNCT
ejpam-4616	121	1	⊓	⊓	PROPN
ejpam-4616	121	2	χ̃m	χ̃m	VERB
ejpam-4616	121	3	◦	◦	NOUN
ejpam-4616	121	4	χ̃s	χ̃s	NUM
ejpam-4616	121	5	◦	◦	NOUN
ejpam-4616	121	6	χ̃m	χ̃m	PROPN
ejpam-4616	121	7	)	)	PUNCT
ejpam-4616	121	8	(	(	PUNCT
ejpam-4616	121	9	x	x	X
ejpam-4616	121	10	)	)	PUNCT
ejpam-4616	121	11	=	=	SYM
ejpam-4616	121	12	(	(	PUNCT
ejpam-4616	121	13	χ̃s	χ̃s	PROPN
ejpam-4616	121	14	◦	◦	NOUN
ejpam-4616	121	15	χ̃m	χ̃m	VERB
ejpam-4616	121	16	◦	◦	VERB
ejpam-4616	121	17	χ̃s)(x)⋏	χ̃s)(x)⋏	PROPN
ejpam-4616	121	18	(	(	PUNCT
ejpam-4616	121	19	χ̃m	χ̃m	VERB
ejpam-4616	121	20	◦	◦	VERB
ejpam-4616	121	21	χ̃s	χ̃s	PROPN
ejpam-4616	121	22	◦	◦	NOUN
ejpam-4616	121	23	χ̃m	χ̃m	PROPN
ejpam-4616	121	24	)	)	PUNCT
ejpam-4616	121	25	(	(	PUNCT
ejpam-4616	121	26	x	x	X
ejpam-4616	121	27	)	)	PUNCT
ejpam-4616	121	28	=	=	SYM
ejpam-4616	121	29	χ̃sms(x)⋏	χ̃sms(x)⋏	PROPN
ejpam-4616	121	30	˜χmsm	˜χmsm	PROPN
ejpam-4616	121	31	(	(	PUNCT
ejpam-4616	121	32	x	x	X
ejpam-4616	121	33	)	)	PUNCT
ejpam-4616	121	34	=	=	SYM
ejpam-4616	121	35	χ̃sis∩msm	χ̃sis∩msm	PROPN
ejpam-4616	121	36	(	(	PUNCT
ejpam-4616	121	37	x	x	X
ejpam-4616	121	38	)	)	PUNCT
ejpam-4616	121	39	⪯	⪯	PROPN
ejpam-4616	121	40	χ̃m	χ̃m	PROPN
ejpam-4616	121	41	(	(	PUNCT
ejpam-4616	121	42	x	x	NOUN
ejpam-4616	121	43	)	)	PUNCT
ejpam-4616	121	44	.	.	PUNCT
ejpam-4616	122	1	therefore	therefore	ADV
ejpam-4616	122	2	χ̃s	χ̃s	PROPN
ejpam-4616	122	3	◦	◦	NOUN
ejpam-4616	122	4	χ̃m	χ̃m	VERB
ejpam-4616	122	5	◦	◦	NOUN
ejpam-4616	122	6	χ̃s	χ̃s	PUNCT
ejpam-4616	123	1	⊓	⊓	PROPN
ejpam-4616	123	2	χ̃m	χ̃m	AUX
ejpam-4616	123	3	◦	◦	NOUN
ejpam-4616	123	4	χ̃s	χ̃s	PROPN
ejpam-4616	123	5	◦	◦	NOUN
ejpam-4616	123	6	χ̃m	χ̃m	VERB
ejpam-4616	123	7	⊑	⊑	PRON
ejpam-4616	123	8	χ̃m	χ̃m	PROPN
ejpam-4616	123	9	.	.	PUNCT
ejpam-4616	124	1	hence	hence	ADV
ejpam-4616	124	2	χ̃m	χ̃m	PROPN
ejpam-4616	124	3	is	be	AUX
ejpam-4616	124	4	an	an	DET
ejpam-4616	124	5	ivf	ivf	ADJ
ejpam-4616	124	6	bi	bi	ADJ
ejpam-4616	124	7	-	-	ADJ
ejpam-4616	124	8	interior	interior	ADJ
ejpam-4616	124	9	ideal	ideal	NOUN
ejpam-4616	124	10	of	of	ADP
ejpam-4616	124	11	s.	s.	PROPN
ejpam-4616	124	12	conversely	conversely	ADV
ejpam-4616	124	13	,	,	PUNCT
ejpam-4616	124	14	suppose	suppose	VERB
ejpam-4616	124	15	that	that	SCONJ
ejpam-4616	124	16	χ̃m	χ̃m	PROPN
ejpam-4616	124	17	is	be	AUX
ejpam-4616	124	18	an	an	DET
ejpam-4616	124	19	ivf	ivf	ADJ
ejpam-4616	124	20	bi	bi	ADJ
ejpam-4616	124	21	-	-	ADJ
ejpam-4616	124	22	interior	interior	ADJ
ejpam-4616	124	23	ideal	ideal	NOUN
ejpam-4616	124	24	of	of	ADP
ejpam-4616	124	25	s.	s.	PROPN
ejpam-4616	124	26	then	then	ADV
ejpam-4616	124	27	χ̃m	χ̃m	VERB
ejpam-4616	124	28	is	be	AUX
ejpam-4616	124	29	an	an	DET
ejpam-4616	124	30	ivf	ivf	ADJ
ejpam-4616	124	31	subsemigroup	subsemigroup	NOUN
ejpam-4616	124	32	of	of	ADP
ejpam-4616	124	33	s.	s.	PROPN
ejpam-4616	124	34	thus	thus	ADV
ejpam-4616	124	35	by	by	ADP
ejpam-4616	124	36	theorem	theorem	NOUN
ejpam-4616	124	37	1	1	NUM
ejpam-4616	124	38	,	,	PUNCT
ejpam-4616	124	39	m	m	VERB
ejpam-4616	124	40	is	be	AUX
ejpam-4616	124	41	a	a	DET
ejpam-4616	124	42	subsemigroup	subsemigroup	NOUN
ejpam-4616	124	43	of	of	ADP
ejpam-4616	124	44	s.	s.	PROPN
ejpam-4616	124	45	let	let	VERB
ejpam-4616	124	46	x	x	X
ejpam-4616	124	47	∈	∈	PROPN
ejpam-4616	124	48	m	m	VERB
ejpam-4616	124	49	.	.	PUNCT
ejpam-4616	125	1	we	we	PRON
ejpam-4616	125	2	have	have	AUX
ejpam-4616	125	3	(	(	PUNCT
ejpam-4616	125	4	χ̃s	χ̃s	PROPN
ejpam-4616	125	5	◦	◦	NOUN
ejpam-4616	125	6	χ̃m	χ̃m	VERB
ejpam-4616	125	7	◦	◦	VERB
ejpam-4616	125	8	χ̃s)(x)⋏	χ̃s)(x)⋏	PROPN
ejpam-4616	125	9	(	(	PUNCT
ejpam-4616	125	10	χ̃m	χ̃m	VERB
ejpam-4616	125	11	◦	◦	VERB
ejpam-4616	125	12	χ̃s	χ̃s	PROPN
ejpam-4616	125	13	◦	◦	NOUN
ejpam-4616	125	14	χ̃m	χ̃m	PROPN
ejpam-4616	125	15	)	)	PUNCT
ejpam-4616	125	16	(	(	PUNCT
ejpam-4616	125	17	x	x	X
ejpam-4616	125	18	)	)	PUNCT
ejpam-4616	125	19	⪯	⪯	PROPN
ejpam-4616	125	20	χ̃m	χ̃m	PROPN
ejpam-4616	125	21	(	(	PUNCT
ejpam-4616	125	22	x	x	NOUN
ejpam-4616	125	23	)	)	PUNCT
ejpam-4616	125	24	⇒	⇒	NOUN
ejpam-4616	125	25	χ̃sms(x)⋏	χ̃sms(x)⋏	NOUN
ejpam-4616	125	26	χ̃msm	χ̃msm	NOUN
ejpam-4616	125	27	(	(	PUNCT
ejpam-4616	125	28	x	x	X
ejpam-4616	125	29	)	)	PUNCT
ejpam-4616	125	30	⪯	⪯	PROPN
ejpam-4616	125	31	χ̃m	χ̃m	PROPN
ejpam-4616	125	32	(	(	PUNCT
ejpam-4616	125	33	x	x	NOUN
ejpam-4616	125	34	)	)	PUNCT
ejpam-4616	125	35	⇒	⇒	NOUN
ejpam-4616	125	36	χ̃sms∩msm	χ̃sms∩msm	NUM
ejpam-4616	125	37	(	(	PUNCT
ejpam-4616	125	38	x	x	X
ejpam-4616	125	39	)	)	PUNCT
ejpam-4616	125	40	⪯	⪯	PROPN
ejpam-4616	125	41	χ̃m	χ̃m	PROPN
ejpam-4616	125	42	(	(	PUNCT
ejpam-4616	125	43	x	x	NOUN
ejpam-4616	125	44	)	)	PUNCT
ejpam-4616	125	45	.	.	PUNCT
ejpam-4616	126	1	therefore	therefore	ADV
ejpam-4616	126	2	sms	sms	VERB
ejpam-4616	126	3	∩msm	∩msm	NOUN
ejpam-4616	126	4	⊆	⊆	NUM
ejpam-4616	126	5	m	m	NOUN
ejpam-4616	126	6	.	.	PUNCT
ejpam-4616	127	1	hence	hence	ADV
ejpam-4616	127	2	m	m	PROPN
ejpam-4616	127	3	is	be	AUX
ejpam-4616	127	4	a	a	DET
ejpam-4616	127	5	bi	bi	ADJ
ejpam-4616	127	6	-	-	ADJ
ejpam-4616	127	7	interior	interior	ADJ
ejpam-4616	127	8	ideal	ideal	NOUN
ejpam-4616	127	9	of	of	ADP
ejpam-4616	127	10	s.	s.	PROPN
ejpam-4616	127	11	t.	t.	PROPN
ejpam-4616	127	12	gaketem	gaketem	PROPN
ejpam-4616	127	13	,	,	PUNCT
ejpam-4616	127	14	t.	t.	PROPN
ejpam-4616	127	15	prommai	prommai	PROPN
ejpam-4616	127	16	/	/	SYM
ejpam-4616	127	17	eur	eur	PROPN
ejpam-4616	127	18	.	.	PUNCT
ejpam-4616	128	1	j.	j.	PROPN
ejpam-4616	128	2	pure	pure	PROPN
ejpam-4616	128	3	appl	appl	PROPN
ejpam-4616	128	4	.	.	PROPN
ejpam-4616	128	5	math	math	PROPN
ejpam-4616	128	6	,	,	PUNCT
ejpam-4616	128	7	16	16	NUM
ejpam-4616	128	8	(	(	PUNCT
ejpam-4616	128	9	1	1	NUM
ejpam-4616	128	10	)	)	PUNCT
ejpam-4616	128	11	(	(	PUNCT
ejpam-4616	128	12	2023	2023	NUM
ejpam-4616	128	13	)	)	PUNCT
ejpam-4616	128	14	,	,	PUNCT
ejpam-4616	128	15	121	121	NUM
ejpam-4616	128	16	-	-	SYM
ejpam-4616	128	17	130	130	NUM
ejpam-4616	128	18	126	126	NUM
ejpam-4616	128	19	theorem	theorem	NOUN
ejpam-4616	128	20	6	6	NUM
ejpam-4616	128	21	.	.	PUNCT
ejpam-4616	129	1	if	if	SCONJ
ejpam-4616	129	2	µ̃	µ̃	PROPN
ejpam-4616	129	3	and	and	CCONJ
ejpam-4616	129	4	λ̃	λ̃	PROPN
ejpam-4616	129	5	are	be	AUX
ejpam-4616	129	6	ivf	ivf	ADJ
ejpam-4616	129	7	bi	bi	ADJ
ejpam-4616	129	8	-	-	ADJ
ejpam-4616	129	9	interior	interior	ADJ
ejpam-4616	129	10	ideals	ideal	NOUN
ejpam-4616	129	11	of	of	ADP
ejpam-4616	129	12	a	a	DET
ejpam-4616	129	13	semigroup	semigroup	NOUN
ejpam-4616	129	14	s	s	NOUN
ejpam-4616	129	15	,	,	PUNCT
ejpam-4616	129	16	then	then	ADV
ejpam-4616	129	17	µ̃⊓	µ̃⊓	PROPN
ejpam-4616	129	18	λ̃	λ̃	PROPN
ejpam-4616	129	19	is	be	AUX
ejpam-4616	129	20	an	an	DET
ejpam-4616	129	21	ivf	ivf	ADJ
ejpam-4616	129	22	bi	bi	ADJ
ejpam-4616	129	23	-	-	ADJ
ejpam-4616	129	24	interior	interior	ADJ
ejpam-4616	129	25	ideal	ideal	NOUN
ejpam-4616	129	26	of	of	ADP
ejpam-4616	129	27	s.	s.	PROPN
ejpam-4616	129	28	proof	proof	PROPN
ejpam-4616	129	29	.	.	PUNCT
ejpam-4616	130	1	let	let	VERB
ejpam-4616	130	2	µ̃	µ̃	PROPN
ejpam-4616	130	3	and	and	CCONJ
ejpam-4616	130	4	λ̃	λ̃	PROPN
ejpam-4616	130	5	be	be	AUX
ejpam-4616	130	6	ivf	ivf	ADJ
ejpam-4616	130	7	bi	bi	ADJ
ejpam-4616	130	8	-	-	ADJ
ejpam-4616	130	9	interior	interior	ADJ
ejpam-4616	130	10	ideals	ideal	NOUN
ejpam-4616	130	11	of	of	ADP
ejpam-4616	130	12	s.	s.	PROPN
ejpam-4616	130	13	then	then	ADV
ejpam-4616	130	14	(	(	PUNCT
ejpam-4616	130	15	χ̃s	χ̃s	PROPN
ejpam-4616	130	16	◦	◦	PROPN
ejpam-4616	130	17	µ̃∩̃λ̃)(x	µ̃∩̃λ̃)(x	PROPN
ejpam-4616	130	18	)	)	PUNCT
ejpam-4616	131	1	=	=	PUNCT
ejpam-4616	131	2	⋃	⋃	NOUN
ejpam-4616	131	3	x	x	X
ejpam-4616	131	4	=	=	NOUN
ejpam-4616	131	5	ab	ab	X
ejpam-4616	131	6	{	{	PUNCT
ejpam-4616	131	7	χ̃s(a)⋏	χ̃s(a)⋏	PROPN
ejpam-4616	131	8	(	(	PUNCT
ejpam-4616	131	9	µ̃∩̃λ̃)(b	µ̃∩̃λ̃)(b	PROPN
ejpam-4616	131	10	)	)	PUNCT
ejpam-4616	131	11	}	}	PUNCT
ejpam-4616	131	12	=	=	SYM
ejpam-4616	132	1	⋃	⋃	NOUN
ejpam-4616	132	2	x	x	X
ejpam-4616	132	3	=	=	NOUN
ejpam-4616	132	4	ab	ab	X
ejpam-4616	132	5	{	{	PUNCT
ejpam-4616	132	6	χ̃s(a)⋏	χ̃s(a)⋏	DET
ejpam-4616	132	7	µ̃(b)⋏	µ̃(b)⋏	PROPN
ejpam-4616	132	8	λ̃(b	λ̃(b	NOUN
ejpam-4616	132	9	)	)	PUNCT
ejpam-4616	132	10	}	}	PUNCT
ejpam-4616	132	11	=	=	SYM
ejpam-4616	132	12	⋃	⋃	NOUN
ejpam-4616	132	13	x	x	X
ejpam-4616	132	14	=	=	NOUN
ejpam-4616	132	15	ab	ab	X
ejpam-4616	132	16	{	{	PUNCT
ejpam-4616	132	17	{	{	PUNCT
ejpam-4616	132	18	χ̃s(a)⋏	χ̃s(a)⋏	PROPN
ejpam-4616	132	19	µ̃(b)}⋏	µ̃(b)}⋏	PROPN
ejpam-4616	132	20	{	{	PUNCT
ejpam-4616	132	21	χ̃s(a	χ̃s(a	PROPN
ejpam-4616	132	22	)	)	PUNCT
ejpam-4616	132	23	∩	∩	NOUN
ejpam-4616	132	24	λ̃(b	λ̃(b	PROPN
ejpam-4616	132	25	)	)	PUNCT
ejpam-4616	132	26	}	}	PUNCT
ejpam-4616	132	27	}	}	PUNCT
ejpam-4616	132	28	=	=	SYM
ejpam-4616	132	29	⋃	⋃	NOUN
ejpam-4616	132	30	x	x	X
ejpam-4616	132	31	=	=	NOUN
ejpam-4616	132	32	ab	ab	X
ejpam-4616	132	33	{	{	PUNCT
ejpam-4616	132	34	χ̃s(a)⋏	χ̃s(a)⋏	X
ejpam-4616	132	35	µ̃(b	µ̃(b	PROPN
ejpam-4616	132	36	)	)	PUNCT
ejpam-4616	132	37	}	}	PUNCT
ejpam-4616	132	38	∩	∩	NOUN
ejpam-4616	132	39	⋃	⋃	NOUN
ejpam-4616	132	40	x	x	SYM
ejpam-4616	132	41	=	=	PROPN
ejpam-4616	132	42	ab	ab	PROPN
ejpam-4616	132	43	{	{	PUNCT
ejpam-4616	132	44	χ̃s(a	χ̃s(a	PROPN
ejpam-4616	132	45	)	)	PUNCT
ejpam-4616	132	46	∩	∩	NOUN
ejpam-4616	132	47	λ̃(b	λ̃(b	PROPN
ejpam-4616	132	48	)	)	PUNCT
ejpam-4616	132	49	}	}	PUNCT
ejpam-4616	132	50	=	=	SYM
ejpam-4616	132	51	(	(	PUNCT
ejpam-4616	132	52	χ̃s	χ̃s	PROPN
ejpam-4616	132	53	◦	◦	VERB
ejpam-4616	132	54	µ̃)(x)⋏	µ̃)(x)⋏	ADP
ejpam-4616	132	55	(	(	PUNCT
ejpam-4616	132	56	χ̃s	χ̃s	PROPN
ejpam-4616	132	57	◦	◦	NOUN
ejpam-4616	132	58	λ̃)(x	λ̃)(x	NOUN
ejpam-4616	132	59	)	)	PUNCT
ejpam-4616	133	1	=	=	PRON
ejpam-4616	133	2	(	(	PUNCT
ejpam-4616	133	3	χ̃s	χ̃s	PROPN
ejpam-4616	133	4	◦	◦	NOUN
ejpam-4616	133	5	µ̃∩̃χ̃s	µ̃∩̃χ̃s	NOUN
ejpam-4616	134	1	◦	◦	NOUN
ejpam-4616	134	2	λ̃)(x	λ̃)(x	NOUN
ejpam-4616	134	3	)	)	PUNCT
ejpam-4616	134	4	.	.	PUNCT
ejpam-4616	135	1	therefore	therefore	ADV
ejpam-4616	135	2	χ̃s	χ̃s	PROPN
ejpam-4616	135	3	◦	◦	VERB
ejpam-4616	135	4	µ̃	µ̃	PROPN
ejpam-4616	135	5	⊓	⊓	PROPN
ejpam-4616	135	6	λ̃	λ̃	PROPN
ejpam-4616	135	7	=	=	PUNCT
ejpam-4616	135	8	χ̃s	χ̃s	PROPN
ejpam-4616	136	1	◦	◦	VERB
ejpam-4616	136	2	µ̃	µ̃	PROPN
ejpam-4616	136	3	⊓	⊓	PROPN
ejpam-4616	136	4	χ̃s	χ̃s	PROPN
ejpam-4616	137	1	◦	◦	NOUN
ejpam-4616	137	2	λ̃.	λ̃.	ADV
ejpam-4616	137	3	(	(	PUNCT
ejpam-4616	137	4	µ̃∩̃λ̃	µ̃∩̃λ̃	NOUN
ejpam-4616	137	5	◦	◦	NOUN
ejpam-4616	137	6	χ̃s	χ̃s	NOUN
ejpam-4616	137	7	◦	◦	VERB
ejpam-4616	137	8	µ̃	µ̃	PROPN
ejpam-4616	137	9	⊓	⊓	PROPN
ejpam-4616	137	10	λ̃)(x	λ̃)(x	NOUN
ejpam-4616	137	11	)	)	PUNCT
ejpam-4616	138	1	=	=	PUNCT
ejpam-4616	138	2	⋃	⋃	NOUN
ejpam-4616	138	3	x	x	X
ejpam-4616	138	4	=	=	PRON
ejpam-4616	138	5	abc	abc	X
ejpam-4616	138	6	{	{	PUNCT
ejpam-4616	138	7	(	(	PUNCT
ejpam-4616	138	8	µ̃∩̃λ̃)(a)⋏	µ̃∩̃λ̃)(a)⋏	X
ejpam-4616	138	9	(	(	PUNCT
ejpam-4616	138	10	χ̃s	χ̃s	PROPN
ejpam-4616	138	11	◦	◦	NOUN
ejpam-4616	138	12	µ̃∩̃λ̃)(bc	µ̃∩̃λ̃)(bc	ADJ
ejpam-4616	138	13	)	)	PUNCT
ejpam-4616	138	14	}	}	PUNCT
ejpam-4616	138	15	=	=	PUNCT
ejpam-4616	138	16	⋃	⋃	NOUN
ejpam-4616	138	17	x	x	X
ejpam-4616	138	18	=	=	PRON
ejpam-4616	138	19	abc	abc	X
ejpam-4616	138	20	{	{	PUNCT
ejpam-4616	138	21	(	(	PUNCT
ejpam-4616	138	22	µ̃	µ̃	PROPN
ejpam-4616	138	23	⊓	⊓	PROPN
ejpam-4616	138	24	λ̃)(a)⋏	λ̃)(a)⋏	PRON
ejpam-4616	138	25	{	{	PUNCT
ejpam-4616	138	26	(	(	PUNCT
ejpam-4616	138	27	χ̃s	χ̃s	PROPN
ejpam-4616	138	28	◦	◦	NOUN
ejpam-4616	138	29	µ̃∩̃χ̃s	µ̃∩̃χ̃s	NOUN
ejpam-4616	138	30	◦	◦	NOUN
ejpam-4616	138	31	λ̃)(bc	λ̃)(bc	VERB
ejpam-4616	138	32	)	)	PUNCT
ejpam-4616	138	33	}	}	PUNCT
ejpam-4616	138	34	}	}	PUNCT
ejpam-4616	138	35	=	=	SYM
ejpam-4616	138	36	⋃	⋃	NOUN
ejpam-4616	138	37	x	x	X
ejpam-4616	138	38	=	=	PRON
ejpam-4616	138	39	abc	abc	X
ejpam-4616	138	40	{	{	PUNCT
ejpam-4616	138	41	(	(	PUNCT
ejpam-4616	138	42	µ̃	µ̃	PROPN
ejpam-4616	138	43	⊓	⊓	PROPN
ejpam-4616	138	44	λ̃)(a)⋏	λ̃)(a)⋏	PRON
ejpam-4616	138	45	{	{	PUNCT
ejpam-4616	138	46	(	(	PUNCT
ejpam-4616	138	47	χ̃s	χ̃s	PROPN
ejpam-4616	138	48	◦	◦	NOUN
ejpam-4616	138	49	µ̃)(bc)⋏	µ̃)(bc)⋏	ADP
ejpam-4616	138	50	(	(	PUNCT
ejpam-4616	138	51	χ̃s	χ̃s	PROPN
ejpam-4616	138	52	◦	◦	NOUN
ejpam-4616	138	53	λ̃)(bc	λ̃)(bc	VERB
ejpam-4616	138	54	)	)	PUNCT
ejpam-4616	138	55	}	}	PUNCT
ejpam-4616	138	56	}	}	PUNCT
ejpam-4616	138	57	=	=	SYM
ejpam-4616	138	58	⋃	⋃	NOUN
ejpam-4616	138	59	x	x	X
ejpam-4616	138	60	=	=	PRON
ejpam-4616	138	61	abc	abc	PROPN
ejpam-4616	138	62	{	{	PUNCT
ejpam-4616	138	63	{	{	PUNCT
ejpam-4616	138	64	µ̃(a)⋏	µ̃(a)⋏	PROPN
ejpam-4616	138	65	(	(	PUNCT
ejpam-4616	138	66	χ̃s	χ̃s	PROPN
ejpam-4616	138	67	◦	◦	NOUN
ejpam-4616	138	68	µ̃)(bc	µ̃)(bc	ADJ
ejpam-4616	138	69	)	)	PUNCT
ejpam-4616	138	70	}	}	PUNCT
ejpam-4616	138	71	∩	∩	NOUN
ejpam-4616	138	72	{	{	PUNCT
ejpam-4616	138	73	λ̃(a)⋏	λ̃(a)⋏	NOUN
ejpam-4616	138	74	(	(	PUNCT
ejpam-4616	138	75	χ̃s	χ̃s	PROPN
ejpam-4616	138	76	◦	◦	NOUN
ejpam-4616	138	77	λ̃)(bc	λ̃)(bc	VERB
ejpam-4616	138	78	)	)	PUNCT
ejpam-4616	138	79	}	}	PUNCT
ejpam-4616	138	80	}	}	PUNCT
ejpam-4616	138	81	=	=	SYM
ejpam-4616	138	82	(	(	PUNCT
ejpam-4616	138	83	µ̃	µ̃	PROPN
ejpam-4616	138	84	◦	◦	VERB
ejpam-4616	138	85	χ̃s	χ̃s	NOUN
ejpam-4616	138	86	◦	◦	NOUN
ejpam-4616	138	87	µ̃)(x)⋏	µ̃)(x)⋏	ADP
ejpam-4616	138	88	(	(	PUNCT
ejpam-4616	138	89	λ̃	λ̃	PROPN
ejpam-4616	138	90	◦	◦	NOUN
ejpam-4616	138	91	χ̃s	χ̃s	NUM
ejpam-4616	138	92	◦	◦	NOUN
ejpam-4616	138	93	λ̃)(x	λ̃)(x	NOUN
ejpam-4616	138	94	)	)	PUNCT
ejpam-4616	139	1	=	=	PUNCT
ejpam-4616	139	2	(	(	PUNCT
ejpam-4616	139	3	µ̃	µ̃	PROPN
ejpam-4616	139	4	◦	◦	VERB
ejpam-4616	139	5	χ̃s	χ̃s	NUM
ejpam-4616	140	1	◦	◦	NOUN
ejpam-4616	140	2	µ̃∩̃λ̃	µ̃∩̃λ̃	NOUN
ejpam-4616	140	3	◦	◦	NOUN
ejpam-4616	140	4	χ̃s	χ̃s	NOUN
ejpam-4616	140	5	◦	◦	NOUN
ejpam-4616	140	6	λ̃)(x	λ̃)(x	NOUN
ejpam-4616	140	7	)	)	PUNCT
ejpam-4616	141	1	.	.	PUNCT
ejpam-4616	142	1	therefore	therefore	ADV
ejpam-4616	142	2	µ̃	µ̃	PROPN
ejpam-4616	142	3	⊓	⊓	PROPN
ejpam-4616	142	4	λ̃	λ̃	PROPN
ejpam-4616	142	5	◦	◦	NOUN
ejpam-4616	142	6	χ̃s	χ̃s	NOUN
ejpam-4616	142	7	◦	◦	VERB
ejpam-4616	143	1	µ̃	µ̃	PROPN
ejpam-4616	143	2	⊓	⊓	PROPN
ejpam-4616	143	3	λ̃	λ̃	PROPN
ejpam-4616	143	4	=	=	SYM
ejpam-4616	143	5	µ̃	µ̃	PROPN
ejpam-4616	143	6	◦	◦	NOUN
ejpam-4616	143	7	χ̃s	χ̃s	NOUN
ejpam-4616	144	1	◦	◦	VERB
ejpam-4616	145	1	µ̃	µ̃	PROPN
ejpam-4616	145	2	⊓	⊓	PROPN
ejpam-4616	145	3	λ̃	λ̃	PROPN
ejpam-4616	145	4	◦	◦	NOUN
ejpam-4616	145	5	χ̃s	χ̃s	NUM
ejpam-4616	146	1	◦	◦	NOUN
ejpam-4616	146	2	λ̃.	λ̃.	ADV
ejpam-4616	146	3	then	then	ADV
ejpam-4616	146	4	(	(	PUNCT
ejpam-4616	146	5	χ̃s	χ̃s	PROPN
ejpam-4616	146	6	◦	◦	VERB
ejpam-4616	146	7	µ̃	µ̃	PROPN
ejpam-4616	146	8	⊓	⊓	PROPN
ejpam-4616	146	9	λ̃	λ̃	PROPN
ejpam-4616	146	10	◦	◦	PROPN
ejpam-4616	146	11	χ̃s)(x)⋏	χ̃s)(x)⋏	PROPN
ejpam-4616	146	12	(	(	PUNCT
ejpam-4616	146	13	µ̃	µ̃	PROPN
ejpam-4616	146	14	⊓	⊓	PROPN
ejpam-4616	146	15	λ̃	λ̃	PROPN
ejpam-4616	146	16	◦	◦	NOUN
ejpam-4616	146	17	χ̃s	χ̃s	NOUN
ejpam-4616	146	18	◦	◦	VERB
ejpam-4616	146	19	µ̃	µ̃	PROPN
ejpam-4616	146	20	⊓	⊓	PROPN
ejpam-4616	146	21	λ̃)(x	λ̃)(x	NOUN
ejpam-4616	146	22	)	)	PUNCT
ejpam-4616	147	1	=	=	PRON
ejpam-4616	147	2	(	(	PUNCT
ejpam-4616	147	3	χ̃s	χ̃s	PROPN
ejpam-4616	147	4	◦	◦	PROPN
ejpam-4616	147	5	µ̃	µ̃	PROPN
ejpam-4616	147	6	◦	◦	PROPN
ejpam-4616	147	7	χ̃s)(x)⋏	χ̃s)(x)⋏	PROPN
ejpam-4616	147	8	(	(	PUNCT
ejpam-4616	147	9	µ̃	µ̃	PROPN
ejpam-4616	147	10	◦	◦	VERB
ejpam-4616	147	11	χ̃s	χ̃s	NOUN
ejpam-4616	147	12	◦	◦	NOUN
ejpam-4616	147	13	µ̃)(x)⋏	µ̃)(x)⋏	ADP
ejpam-4616	147	14	(	(	PUNCT
ejpam-4616	147	15	χ̃s	χ̃s	PROPN
ejpam-4616	147	16	◦	◦	PROPN
ejpam-4616	147	17	λ̃	λ̃	PROPN
ejpam-4616	147	18	◦	◦	NOUN
ejpam-4616	147	19	µ̃χs	µ̃χs	PROPN
ejpam-4616	147	20	)	)	PUNCT
ejpam-4616	147	21	(	(	PUNCT
ejpam-4616	147	22	x)⋏	x)⋏	PROPN
ejpam-4616	147	23	(	(	PUNCT
ejpam-4616	147	24	λ̃	λ̃	PROPN
ejpam-4616	147	25	◦	◦	VERB
ejpam-4616	147	26	χ̃s	χ̃s	NUM
ejpam-4616	147	27	◦	◦	NOUN
ejpam-4616	147	28	λ̃)(x	λ̃)(x	NOUN
ejpam-4616	147	29	)	)	PUNCT
ejpam-4616	147	30	⪯	⪯	NOUN
ejpam-4616	147	31	(	(	PUNCT
ejpam-4616	147	32	µ̃	µ̃	PROPN
ejpam-4616	147	33	⊓	⊓	PROPN
ejpam-4616	147	34	λ̃)(x	λ̃)(x	NOUN
ejpam-4616	147	35	)	)	PUNCT
ejpam-4616	147	36	.	.	PUNCT
ejpam-4616	148	1	therefore	therefore	ADV
ejpam-4616	148	2	(	(	PUNCT
ejpam-4616	148	3	χ̃s	χ̃s	PROPN
ejpam-4616	148	4	◦	◦	VERB
ejpam-4616	148	5	µ̃	µ̃	PROPN
ejpam-4616	148	6	⊓	⊓	PROPN
ejpam-4616	148	7	λ̃	λ̃	PROPN
ejpam-4616	148	8	◦	◦	NOUN
ejpam-4616	148	9	χ̃s	χ̃s	NOUN
ejpam-4616	148	10	)	)	PUNCT
ejpam-4616	149	1	⊓	⊓	PROPN
ejpam-4616	149	2	(	(	PUNCT
ejpam-4616	149	3	µ̃	µ̃	PROPN
ejpam-4616	149	4	⊓	⊓	PROPN
ejpam-4616	149	5	λ̃	λ̃	PROPN
ejpam-4616	149	6	◦	◦	NOUN
ejpam-4616	149	7	χ̃s	χ̃s	NOUN
ejpam-4616	149	8	◦	◦	VERB
ejpam-4616	149	9	µ̃	µ̃	PROPN
ejpam-4616	149	10	⊓	⊓	PROPN
ejpam-4616	149	11	λ̃	λ̃	PROPN
ejpam-4616	149	12	)	)	PUNCT
ejpam-4616	149	13	⊑	⊑	X
ejpam-4616	150	1	µ̃	µ̃	PROPN
ejpam-4616	150	2	⊓	⊓	PROPN
ejpam-4616	150	3	λ̃.	λ̃.	ADJ
ejpam-4616	150	4	hence	hence	ADV
ejpam-4616	150	5	µ̃∩̃λ̃	µ̃∩̃λ̃	NOUN
ejpam-4616	150	6	is	be	AUX
ejpam-4616	150	7	an	an	DET
ejpam-4616	150	8	ivf	ivf	ADJ
ejpam-4616	150	9	bi	bi	ADJ
ejpam-4616	150	10	-	-	ADJ
ejpam-4616	150	11	interior	interior	ADJ
ejpam-4616	150	12	ideal	ideal	NOUN
ejpam-4616	150	13	of	of	ADP
ejpam-4616	150	14	a	a	DET
ejpam-4616	150	15	semigroup	semigroup	PROPN
ejpam-4616	150	16	s.	s.	PROPN
ejpam-4616	150	17	we	we	PRON
ejpam-4616	150	18	know	know	VERB
ejpam-4616	150	19	that	that	SCONJ
ejpam-4616	150	20	every	every	DET
ejpam-4616	150	21	ivf	ivf	NOUN
ejpam-4616	150	22	ideal	ideal	NOUN
ejpam-4616	150	23	is	be	AUX
ejpam-4616	150	24	an	an	DET
ejpam-4616	150	25	ivf	ivf	ADJ
ejpam-4616	150	26	bi	bi	ADJ
ejpam-4616	150	27	-	-	ADJ
ejpam-4616	150	28	interior	interior	ADJ
ejpam-4616	150	29	ideal	ideal	NOUN
ejpam-4616	150	30	then	then	ADV
ejpam-4616	150	31	the	the	DET
ejpam-4616	150	32	following	follow	VERB
ejpam-4616	150	33	theorem	theorem	ADJ
ejpam-4616	150	34	holds	hold	NOUN
ejpam-4616	150	35	.	.	PUNCT
ejpam-4616	151	1	theorem	theorem	NOUN
ejpam-4616	151	2	7	7	NUM
ejpam-4616	151	3	.	.	PUNCT
ejpam-4616	152	1	if	if	SCONJ
ejpam-4616	152	2	µ̃	µ̃	PROPN
ejpam-4616	152	3	and	and	CCONJ
ejpam-4616	152	4	λ̃	λ̃	PROPN
ejpam-4616	152	5	are	be	AUX
ejpam-4616	152	6	ivf	ivf	ADJ
ejpam-4616	152	7	right	right	ADJ
ejpam-4616	152	8	ideals	ideal	NOUN
ejpam-4616	152	9	and	and	CCONJ
ejpam-4616	152	10	an	an	DET
ejpam-4616	152	11	ivf	ivf	NOUN
ejpam-4616	152	12	left	leave	VERB
ejpam-4616	152	13	ideal	ideal	NOUN
ejpam-4616	152	14	of	of	ADP
ejpam-4616	152	15	a	a	DET
ejpam-4616	152	16	semigroup	semigroup	NOUN
ejpam-4616	152	17	s	s	VERB
ejpam-4616	152	18	respectively	respectively	ADV
ejpam-4616	152	19	.	.	PUNCT
ejpam-4616	153	1	then	then	ADV
ejpam-4616	153	2	µ̃	µ̃	PROPN
ejpam-4616	153	3	⊓	⊓	PROPN
ejpam-4616	153	4	λ̃	λ̃	PROPN
ejpam-4616	153	5	is	be	AUX
ejpam-4616	153	6	an	an	DET
ejpam-4616	153	7	ivf	ivf	ADJ
ejpam-4616	153	8	bi	bi	ADJ
ejpam-4616	153	9	-	-	ADJ
ejpam-4616	153	10	interior	interior	ADJ
ejpam-4616	153	11	ideal	ideal	NOUN
ejpam-4616	153	12	of	of	ADP
ejpam-4616	153	13	s.	s.	PROPN
ejpam-4616	153	14	t.	t.	PROPN
ejpam-4616	153	15	gaketem	gaketem	PROPN
ejpam-4616	153	16	,	,	PUNCT
ejpam-4616	153	17	t.	t.	PROPN
ejpam-4616	153	18	prommai	prommai	PROPN
ejpam-4616	153	19	/	/	SYM
ejpam-4616	153	20	eur	eur	PROPN
ejpam-4616	153	21	.	.	PUNCT
ejpam-4616	154	1	j.	j.	PROPN
ejpam-4616	154	2	pure	pure	PROPN
ejpam-4616	154	3	appl	appl	PROPN
ejpam-4616	154	4	.	.	PROPN
ejpam-4616	154	5	math	math	PROPN
ejpam-4616	154	6	,	,	PUNCT
ejpam-4616	154	7	16	16	NUM
ejpam-4616	154	8	(	(	PUNCT
ejpam-4616	154	9	1	1	NUM
ejpam-4616	154	10	)	)	PUNCT
ejpam-4616	154	11	(	(	PUNCT
ejpam-4616	154	12	2023	2023	NUM
ejpam-4616	154	13	)	)	PUNCT
ejpam-4616	154	14	,	,	PUNCT
ejpam-4616	154	15	121	121	NUM
ejpam-4616	154	16	-	-	SYM
ejpam-4616	154	17	130	130	NUM
ejpam-4616	154	18	127	127	NUM
ejpam-4616	154	19	proof	proof	NOUN
ejpam-4616	154	20	.	.	PUNCT
ejpam-4616	155	1	assume	assume	VERB
ejpam-4616	155	2	that	that	SCONJ
ejpam-4616	155	3	µ̃	µ̃	PROPN
ejpam-4616	155	4	and	and	CCONJ
ejpam-4616	155	5	λ̃	λ̃	PROPN
ejpam-4616	155	6	are	be	AUX
ejpam-4616	155	7	ivf	ivf	ADJ
ejpam-4616	155	8	right	right	ADJ
ejpam-4616	155	9	ideals	ideal	NOUN
ejpam-4616	155	10	and	and	CCONJ
ejpam-4616	155	11	an	an	DET
ejpam-4616	155	12	ivf	ivf	NOUN
ejpam-4616	155	13	left	leave	VERB
ejpam-4616	155	14	ideal	ideal	NOUN
ejpam-4616	155	15	of	of	ADP
ejpam-4616	155	16	s	s	PRON
ejpam-4616	155	17	respectively	respectively	ADV
ejpam-4616	155	18	.	.	PUNCT
ejpam-4616	156	1	then	then	ADV
ejpam-4616	156	2	by	by	ADP
ejpam-4616	156	3	theorems	theorem	NOUN
ejpam-4616	156	4	2	2	NUM
ejpam-4616	156	5	and	and	CCONJ
ejpam-4616	156	6	3	3	NUM
ejpam-4616	156	7	,	,	PUNCT
ejpam-4616	156	8	we	we	PRON
ejpam-4616	156	9	have	have	VERB
ejpam-4616	156	10	µ̃	µ̃	PROPN
ejpam-4616	156	11	and	and	CCONJ
ejpam-4616	156	12	λ̃	λ̃	PROPN
ejpam-4616	156	13	are	be	AUX
ejpam-4616	156	14	ivf	ivf	ADJ
ejpam-4616	156	15	bi	bi	ADJ
ejpam-4616	156	16	-	-	ADJ
ejpam-4616	156	17	interior	interior	ADJ
ejpam-4616	156	18	ideals	ideal	NOUN
ejpam-4616	156	19	of	of	ADP
ejpam-4616	156	20	s.	s.	PROPN
ejpam-4616	156	21	by	by	ADP
ejpam-4616	156	22	theorem	theorem	NOUN
ejpam-4616	156	23	6	6	NUM
ejpam-4616	156	24	we	we	PRON
ejpam-4616	156	25	have	have	VERB
ejpam-4616	156	26	µ̃	µ̃	PROPN
ejpam-4616	156	27	⊓	⊓	PROPN
ejpam-4616	156	28	λ̃	λ̃	PROPN
ejpam-4616	156	29	is	be	AUX
ejpam-4616	156	30	an	an	DET
ejpam-4616	156	31	ivf	ivf	ADJ
ejpam-4616	156	32	bi	bi	ADJ
ejpam-4616	156	33	-	-	ADJ
ejpam-4616	156	34	interior	interior	ADJ
ejpam-4616	156	35	ideal	ideal	NOUN
ejpam-4616	156	36	of	of	ADP
ejpam-4616	156	37	s.	s.	PROPN
ejpam-4616	156	38	the	the	DET
ejpam-4616	156	39	following	follow	VERB
ejpam-4616	156	40	are	be	AUX
ejpam-4616	156	41	tools	tool	NOUN
ejpam-4616	156	42	the	the	DET
ejpam-4616	156	43	converse	converse	NOUN
ejpam-4616	156	44	of	of	ADP
ejpam-4616	156	45	an	an	DET
ejpam-4616	156	46	ivf	ivf	ADJ
ejpam-4616	156	47	bi	bi	ADJ
ejpam-4616	156	48	-	-	ADJ
ejpam-4616	156	49	interior	interior	ADJ
ejpam-4616	156	50	ideals	ideal	NOUN
ejpam-4616	156	51	is	be	AUX
ejpam-4616	156	52	ivf	ivf	ADJ
ejpam-4616	156	53	ideals	ideal	NOUN
ejpam-4616	156	54	on	on	ADP
ejpam-4616	156	55	semigroups	semigroup	NOUN
ejpam-4616	156	56	.	.	PUNCT
ejpam-4616	157	1	definition	definition	NOUN
ejpam-4616	157	2	7	7	NUM
ejpam-4616	157	3	.	.	PUNCT
ejpam-4616	158	1	a	a	DET
ejpam-4616	158	2	semigroup	semigroup	NOUN
ejpam-4616	158	3	s	s	VERB
ejpam-4616	158	4	is	be	AUX
ejpam-4616	158	5	called	call	VERB
ejpam-4616	158	6	regular	regular	ADJ
ejpam-4616	158	7	if	if	SCONJ
ejpam-4616	158	8	for	for	ADP
ejpam-4616	158	9	all	all	DET
ejpam-4616	158	10	a	a	DET
ejpam-4616	158	11	∈	∈	NOUN
ejpam-4616	158	12	s	s	VERB
ejpam-4616	158	13	there	there	PRON
ejpam-4616	158	14	exists	exist	VERB
ejpam-4616	158	15	x	x	X
ejpam-4616	158	16	∈	∈	NOUN
ejpam-4616	158	17	s	s	VERB
ejpam-4616	158	18	such	such	ADJ
ejpam-4616	158	19	that	that	SCONJ
ejpam-4616	158	20	a	a	DET
ejpam-4616	158	21	=	=	PUNCT
ejpam-4616	158	22	axa	axa	NOUN
ejpam-4616	158	23	.	.	PUNCT
ejpam-4616	159	1	theorem	theorem	VERB
ejpam-4616	159	2	8	8	NUM
ejpam-4616	159	3	.	.	PUNCT
ejpam-4616	160	1	if	if	SCONJ
ejpam-4616	160	2	µ̃	µ̃	PROPN
ejpam-4616	160	3	be	be	VERB
ejpam-4616	160	4	an	an	DET
ejpam-4616	160	5	ivf	ivf	ADJ
ejpam-4616	160	6	quasi	quasi	NOUN
ejpam-4616	160	7	-	-	NOUN
ejpam-4616	160	8	ideal	ideal	ADJ
ejpam-4616	160	9	of	of	ADP
ejpam-4616	160	10	a	a	DET
ejpam-4616	160	11	regular	regular	ADJ
ejpam-4616	160	12	semigroup	semigroup	NOUN
ejpam-4616	160	13	s.then	s.then	ADP
ejpam-4616	160	14	µ̃	µ̃	PROPN
ejpam-4616	160	15	is	be	AUX
ejpam-4616	160	16	an	an	DET
ejpam-4616	160	17	ivf	ivf	ADJ
ejpam-4616	160	18	ideal	ideal	NOUN
ejpam-4616	160	19	of	of	ADP
ejpam-4616	160	20	a	a	DET
ejpam-4616	160	21	semigroup	semigroup	PROPN
ejpam-4616	160	22	s.	s.	PROPN
ejpam-4616	160	23	proof	proof	PROPN
ejpam-4616	160	24	.	.	PUNCT
ejpam-4616	161	1	assume	assume	VERB
ejpam-4616	161	2	that	that	SCONJ
ejpam-4616	161	3	µ̃	µ̃	PROPN
ejpam-4616	161	4	is	be	AUX
ejpam-4616	161	5	an	an	DET
ejpam-4616	161	6	ivf	ivf	ADJ
ejpam-4616	161	7	quasi	quasi	NOUN
ejpam-4616	161	8	-	-	NOUN
ejpam-4616	161	9	ideal	ideal	ADJ
ejpam-4616	161	10	of	of	ADP
ejpam-4616	161	11	s	s	PRON
ejpam-4616	161	12	and	and	CCONJ
ejpam-4616	161	13	let	let	VERB
ejpam-4616	161	14	x	x	PRON
ejpam-4616	161	15	,	,	PUNCT
ejpam-4616	161	16	y	y	PROPN
ejpam-4616	161	17	∈	∈	PROPN
ejpam-4616	161	18	s.	s.	PROPN
ejpam-4616	161	19	then	then	ADV
ejpam-4616	161	20	µ̃(xy	µ̃(xy	ADJ
ejpam-4616	161	21	)	)	PUNCT
ejpam-4616	161	22	⪰	⪰	NOUN
ejpam-4616	161	23	(	(	PUNCT
ejpam-4616	161	24	µ̃	µ̃	PROPN
ejpam-4616	161	25	◦	◦	NOUN
ejpam-4616	161	26	χ̃s)(xy)⋏	χ̃s)(xy)⋏	NOUN
ejpam-4616	161	27	(	(	PUNCT
ejpam-4616	161	28	χ̃s	χ̃s	PROPN
ejpam-4616	161	29	◦	◦	NOUN
ejpam-4616	161	30	µ̃)(xy	µ̃)(xy	NUM
ejpam-4616	161	31	)	)	PUNCT
ejpam-4616	162	1	=	=	PUNCT
ejpam-4616	162	2	⋃	⋃	NOUN
ejpam-4616	162	3	xy	xy	NOUN
ejpam-4616	162	4	=	=	ADJ
ejpam-4616	162	5	ab	ab	X
ejpam-4616	162	6	{	{	PUNCT
ejpam-4616	162	7	µ̃(a)⋏	µ̃(a)⋏	NOUN
ejpam-4616	162	8	χ̃s(b)}⋏	χ̃s(b)}⋏	NOUN
ejpam-4616	162	9	⋃	⋃	NOUN
ejpam-4616	162	10	xy	xy	NOUN
ejpam-4616	162	11	=	=	ADJ
ejpam-4616	162	12	ij	ij	X
ejpam-4616	162	13	{	{	PUNCT
ejpam-4616	162	14	χ̃s(i)⋏	χ̃s(i)⋏	PROPN
ejpam-4616	162	15	µ̃(j	µ̃(j	PROPN
ejpam-4616	162	16	)	)	PUNCT
ejpam-4616	162	17	}	}	PUNCT
ejpam-4616	162	18	⪰	⪰	NOUN
ejpam-4616	162	19	µ̃(x	µ̃(x	ADJ
ejpam-4616	162	20	)	)	PUNCT
ejpam-4616	162	21	∩	∩	NOUN
ejpam-4616	162	22	χ̃s(y)⋏	χ̃s(y)⋏	X
ejpam-4616	162	23	χ̃s(x)⋏	χ̃s(x)⋏	X
ejpam-4616	162	24	µ̃(y	µ̃(y	NOUN
ejpam-4616	162	25	)	)	PUNCT
ejpam-4616	162	26	=	=	SYM
ejpam-4616	162	27	(	(	PUNCT
ejpam-4616	162	28	µ̃(x)⋏	µ̃(x)⋏	ADP
ejpam-4616	162	29	[	[	X
ejpam-4616	162	30	1	1	NUM
ejpam-4616	162	31	,	,	PUNCT
ejpam-4616	162	32	1])⋏	1])⋏	NUM
ejpam-4616	162	33	(	(	PUNCT
ejpam-4616	162	34	[	[	X
ejpam-4616	162	35	1	1	NUM
ejpam-4616	162	36	,	,	PUNCT
ejpam-4616	162	37	1]⋏	1]⋏	ADJ
ejpam-4616	162	38	µ̃(y	µ̃(y	NOUN
ejpam-4616	162	39	)	)	PUNCT
ejpam-4616	162	40	)	)	PUNCT
ejpam-4616	163	1	=	=	SYM
ejpam-4616	163	2	µ̃(x	µ̃(x	ADJ
ejpam-4616	163	3	)	)	PUNCT
ejpam-4616	163	4	∩	∩	NOUN
ejpam-4616	163	5	µ̃(y	µ̃(y	NOUN
ejpam-4616	163	6	)	)	PUNCT
ejpam-4616	163	7	.	.	PUNCT
ejpam-4616	164	1	thus	thus	ADV
ejpam-4616	164	2	µ̃(xy	µ̃(xy	ADJ
ejpam-4616	164	3	)	)	PUNCT
ejpam-4616	164	4	⪰	⪰	NOUN
ejpam-4616	164	5	µ̃(x	µ̃(x	PROPN
ejpam-4616	164	6	)	)	PUNCT
ejpam-4616	164	7	∩	∩	NOUN
ejpam-4616	164	8	µ̃(y	µ̃(y	NOUN
ejpam-4616	164	9	)	)	PUNCT
ejpam-4616	164	10	.	.	PUNCT
ejpam-4616	165	1	hence	hence	ADV
ejpam-4616	165	2	µ̃	µ̃	PROPN
ejpam-4616	165	3	is	be	AUX
ejpam-4616	165	4	an	an	DET
ejpam-4616	165	5	ivf	ivf	ADJ
ejpam-4616	165	6	subsemigroup	subsemigroup	NOUN
ejpam-4616	165	7	of	of	ADP
ejpam-4616	165	8	s.	s.	PROPN
ejpam-4616	165	9	let	let	VERB
ejpam-4616	165	10	x	x	PRON
ejpam-4616	165	11	,	,	PUNCT
ejpam-4616	165	12	y	y	PROPN
ejpam-4616	165	13	,	,	PUNCT
ejpam-4616	165	14	z	z	PROPN
ejpam-4616	165	15	∈	∈	PROPN
ejpam-4616	165	16	s.	s.	PROPN
ejpam-4616	165	17	then	then	ADV
ejpam-4616	165	18	µ̃(xyz	µ̃(xyz	PROPN
ejpam-4616	165	19	)	)	PUNCT
ejpam-4616	165	20	⪰	⪰	NOUN
ejpam-4616	165	21	(	(	PUNCT
ejpam-4616	165	22	µ̃	µ̃	PROPN
ejpam-4616	165	23	◦	◦	PROPN
ejpam-4616	165	24	χ̃s)(xyz)⋏	χ̃s)(xyz)⋏	NOUN
ejpam-4616	165	25	(	(	PUNCT
ejpam-4616	165	26	χ̃s	χ̃s	PROPN
ejpam-4616	165	27	◦	◦	NOUN
ejpam-4616	165	28	µ̃)(xyz	µ̃)(xyz	PROPN
ejpam-4616	165	29	)	)	PUNCT
ejpam-4616	165	30	=	=	SYM
ejpam-4616	165	31	⋃	⋃	NOUN
ejpam-4616	165	32	xyz	xyz	SYM
ejpam-4616	165	33	=	=	NOUN
ejpam-4616	165	34	ab	ab	PROPN
ejpam-4616	165	35	{	{	PUNCT
ejpam-4616	165	36	µ̃(a)⋏	µ̃(a)⋏	NOUN
ejpam-4616	165	37	χ̃s(b)}⋏	χ̃s(b)}⋏	NOUN
ejpam-4616	165	38	⋃	⋃	NOUN
ejpam-4616	165	39	xyz	xyz	SYM
ejpam-4616	165	40	=	=	NOUN
ejpam-4616	165	41	ij	ij	X
ejpam-4616	165	42	{	{	PUNCT
ejpam-4616	165	43	χ̃s(i)⋏	χ̃s(i)⋏	PROPN
ejpam-4616	165	44	µ̃(j	µ̃(j	PROPN
ejpam-4616	165	45	)	)	PUNCT
ejpam-4616	165	46	}	}	PUNCT
ejpam-4616	165	47	⪰	⪰	VERB
ejpam-4616	165	48	µ̃(x)⋏	µ̃(x)⋏	ADP
ejpam-4616	165	49	χ̃s(yz)⋏	χ̃s(yz)⋏	PROPN
ejpam-4616	165	50	χ̃s(xy)⋏	χ̃s(xy)⋏	PROPN
ejpam-4616	165	51	µ̃(z	µ̃(z	PROPN
ejpam-4616	165	52	)	)	PUNCT
ejpam-4616	165	53	=	=	NOUN
ejpam-4616	165	54	(	(	PUNCT
ejpam-4616	165	55	µ̃(x)⋏	µ̃(x)⋏	ADP
ejpam-4616	165	56	[	[	X
ejpam-4616	165	57	1	1	NUM
ejpam-4616	165	58	,	,	PUNCT
ejpam-4616	165	59	1])⋏	1])⋏	NUM
ejpam-4616	165	60	(	(	PUNCT
ejpam-4616	165	61	[	[	X
ejpam-4616	165	62	1	1	NUM
ejpam-4616	165	63	,	,	PUNCT
ejpam-4616	165	64	1]⋏	1]⋏	PROPN
ejpam-4616	165	65	µ̃(z	µ̃(z	PROPN
ejpam-4616	165	66	)	)	PUNCT
ejpam-4616	165	67	)	)	PUNCT
ejpam-4616	166	1	=	=	PUNCT
ejpam-4616	166	2	µ̃(x)⋏	µ̃(x)⋏	ADP
ejpam-4616	166	3	µ̃(z	µ̃(z	PROPN
ejpam-4616	166	4	)	)	PUNCT
ejpam-4616	166	5	.	.	PUNCT
ejpam-4616	167	1	thus	thus	ADV
ejpam-4616	167	2	µ̃(xyz	µ̃(xyz	NOUN
ejpam-4616	167	3	)	)	PUNCT
ejpam-4616	167	4	⪰	⪰	PROPN
ejpam-4616	167	5	µ̃(x	µ̃(x	PROPN
ejpam-4616	167	6	)	)	PUNCT
ejpam-4616	167	7	⋏	⋏	PROPN
ejpam-4616	167	8	µ̃(z	µ̃(z	PROPN
ejpam-4616	167	9	)	)	PUNCT
ejpam-4616	167	10	.	.	PUNCT
ejpam-4616	168	1	hence	hence	ADV
ejpam-4616	168	2	µ̃	µ̃	PROPN
ejpam-4616	168	3	is	be	AUX
ejpam-4616	168	4	an	an	DET
ejpam-4616	168	5	ivf	ivf	ADJ
ejpam-4616	168	6	bi	bi	NOUN
ejpam-4616	168	7	-	-	NOUN
ejpam-4616	168	8	ideal	ideal	NOUN
ejpam-4616	168	9	of	of	ADP
ejpam-4616	168	10	s.	s.	PROPN
ejpam-4616	168	11	since	since	SCONJ
ejpam-4616	168	12	s	s	PROPN
ejpam-4616	168	13	is	be	AUX
ejpam-4616	168	14	regular	regular	ADJ
ejpam-4616	168	15	,	,	PUNCT
ejpam-4616	168	16	µ̃	µ̃	PROPN
ejpam-4616	168	17	is	be	AUX
ejpam-4616	168	18	an	an	DET
ejpam-4616	168	19	ivf	ivf	ADJ
ejpam-4616	168	20	bi	bi	NOUN
ejpam-4616	168	21	-	-	NOUN
ejpam-4616	168	22	ideal	ideal	NOUN
ejpam-4616	168	23	of	of	ADP
ejpam-4616	168	24	s	s	PRON
ejpam-4616	168	25	and	and	CCONJ
ejpam-4616	168	26	x	x	NOUN
ejpam-4616	168	27	,	,	PUNCT
ejpam-4616	168	28	y	y	PROPN
ejpam-4616	168	29	∈	∈	PROPN
ejpam-4616	168	30	s	s	VERB
ejpam-4616	168	31	we	we	PRON
ejpam-4616	168	32	have	have	VERB
ejpam-4616	168	33	xy	xy	PROPN
ejpam-4616	168	34	∈	∈	PROPN
ejpam-4616	168	35	(	(	PUNCT
ejpam-4616	168	36	xsx)s	xsx)s	PROPN
ejpam-4616	168	37	⊆	⊆	NUM
ejpam-4616	168	38	xsx	xsx	PROPN
ejpam-4616	168	39	.	.	PUNCT
ejpam-4616	169	1	thus	thus	ADV
ejpam-4616	169	2	there	there	PRON
ejpam-4616	169	3	exists	exist	VERB
ejpam-4616	169	4	k	k	PROPN
ejpam-4616	169	5	∈	∈	PROPN
ejpam-4616	169	6	s	s	VERB
ejpam-4616	169	7	such	such	ADJ
ejpam-4616	169	8	that	that	SCONJ
ejpam-4616	169	9	xy	xy	PROPN
ejpam-4616	169	10	=	=	PUNCT
ejpam-4616	169	11	xkx	xkx	PROPN
ejpam-4616	169	12	.	.	PUNCT
ejpam-4616	170	1	so	so	ADV
ejpam-4616	170	2	µ̃(xy	µ̃(xy	ADJ
ejpam-4616	170	3	)	)	PUNCT
ejpam-4616	170	4	=	=	SYM
ejpam-4616	170	5	µ̃(xkx	µ̃(xkx	PROPN
ejpam-4616	170	6	)	)	PUNCT
ejpam-4616	170	7	⪰	⪰	PROPN
ejpam-4616	170	8	µ̃(x	µ̃(x	PROPN
ejpam-4616	170	9	)	)	PUNCT
ejpam-4616	170	10	⋏	⋏	PROPN
ejpam-4616	170	11	µ̃(x	µ̃(x	PROPN
ejpam-4616	170	12	)	)	PUNCT
ejpam-4616	170	13	=	=	SYM
ejpam-4616	170	14	µ̃(x	µ̃(x	PROPN
ejpam-4616	170	15	)	)	PUNCT
ejpam-4616	170	16	.	.	PUNCT
ejpam-4616	171	1	similarly	similarly	ADV
ejpam-4616	171	2	,	,	PUNCT
ejpam-4616	171	3	we	we	PRON
ejpam-4616	171	4	can	can	AUX
ejpam-4616	171	5	show	show	VERB
ejpam-4616	171	6	that	that	SCONJ
ejpam-4616	171	7	µ̃(xy	µ̃(xy	ADJ
ejpam-4616	171	8	)	)	PUNCT
ejpam-4616	171	9	⪰	⪰	NOUN
ejpam-4616	171	10	µ̃(y	µ̃(y	NOUN
ejpam-4616	171	11	)	)	PUNCT
ejpam-4616	171	12	.	.	PUNCT
ejpam-4616	172	1	thus	thus	ADV
ejpam-4616	172	2	µ̃	µ̃	PROPN
ejpam-4616	172	3	is	be	AUX
ejpam-4616	172	4	an	an	DET
ejpam-4616	172	5	ivf	ivf	NOUN
ejpam-4616	172	6	left	leave	VERB
ejpam-4616	172	7	ideal	ideal	NOUN
ejpam-4616	172	8	of	of	ADP
ejpam-4616	172	9	s.	s.	PROPN
ejpam-4616	172	10	hence	hence	ADV
ejpam-4616	172	11	µ̃	µ̃	PROPN
ejpam-4616	172	12	is	be	AUX
ejpam-4616	172	13	an	an	DET
ejpam-4616	172	14	ivf	ivf	ADJ
ejpam-4616	172	15	ideal	ideal	NOUN
ejpam-4616	172	16	of	of	ADP
ejpam-4616	172	17	s.	s.	PROPN
ejpam-4616	172	18	theorem	theorem	VERB
ejpam-4616	172	19	9	9	NUM
ejpam-4616	172	20	.	.	PUNCT
ejpam-4616	173	1	let	let	VERB
ejpam-4616	173	2	s	s	PRON
ejpam-4616	173	3	be	be	AUX
ejpam-4616	173	4	a	a	DET
ejpam-4616	173	5	regular	regular	ADJ
ejpam-4616	173	6	semigroup	semigroup	NOUN
ejpam-4616	173	7	.	.	PUNCT
ejpam-4616	174	1	then	then	ADV
ejpam-4616	174	2	µ̃	µ̃	PROPN
ejpam-4616	174	3	is	be	AUX
ejpam-4616	174	4	an	an	DET
ejpam-4616	174	5	ivf	ivf	ADJ
ejpam-4616	174	6	bi	bi	ADJ
ejpam-4616	174	7	-	-	ADJ
ejpam-4616	174	8	interior	interior	ADJ
ejpam-4616	174	9	ideal	ideal	NOUN
ejpam-4616	174	10	of	of	ADP
ejpam-4616	174	11	s	s	PRON
ejpam-4616	174	12	if	if	SCONJ
ejpam-4616	175	1	and	and	CCONJ
ejpam-4616	175	2	only	only	ADV
ejpam-4616	175	3	if	if	SCONJ
ejpam-4616	175	4	µ̃	µ̃	PROPN
ejpam-4616	175	5	is	be	AUX
ejpam-4616	175	6	an	an	DET
ejpam-4616	175	7	ivf	ivf	ADJ
ejpam-4616	175	8	quasi	quasi	NOUN
ejpam-4616	175	9	-	-	NOUN
ejpam-4616	175	10	ideal	ideal	NOUN
ejpam-4616	175	11	of	of	ADP
ejpam-4616	175	12	s.	s.	PROPN
ejpam-4616	175	13	proof	proof	PROPN
ejpam-4616	175	14	.	.	PUNCT
ejpam-4616	176	1	let	let	VERB
ejpam-4616	176	2	µ̃	µ̃	PROPN
ejpam-4616	176	3	be	be	AUX
ejpam-4616	176	4	an	an	DET
ejpam-4616	176	5	ivf	ivf	ADJ
ejpam-4616	176	6	bi	bi	ADJ
ejpam-4616	176	7	-	-	ADJ
ejpam-4616	176	8	interior	interior	ADJ
ejpam-4616	176	9	ideal	ideal	NOUN
ejpam-4616	176	10	of	of	ADP
ejpam-4616	176	11	s	s	PRON
ejpam-4616	176	12	and	and	CCONJ
ejpam-4616	176	13	x	x	SYM
ejpam-4616	176	14	∈	∈	PROPN
ejpam-4616	176	15	s.	s.	PROPN
ejpam-4616	176	16	then	then	ADV
ejpam-4616	176	17	(	(	PUNCT
ejpam-4616	176	18	χ̃s	χ̃s	PROPN
ejpam-4616	176	19	◦	◦	PROPN
ejpam-4616	176	20	µ̃	µ̃	PROPN
ejpam-4616	176	21	◦	◦	PROPN
ejpam-4616	176	22	χ̃s)(x)⋏	χ̃s)(x)⋏	PROPN
ejpam-4616	176	23	(	(	PUNCT
ejpam-4616	176	24	µ̃	µ̃	PROPN
ejpam-4616	176	25	◦	◦	NOUN
ejpam-4616	176	26	µ̃χs	µ̃χs	PROPN
ejpam-4616	176	27	◦	◦	NOUN
ejpam-4616	176	28	µ̃)(x	µ̃)(x	NOUN
ejpam-4616	176	29	)	)	PUNCT
ejpam-4616	176	30	⪰	⪰	PROPN
ejpam-4616	176	31	µ̃(x	µ̃(x	PROPN
ejpam-4616	176	32	)	)	PUNCT
ejpam-4616	176	33	.	.	PUNCT
ejpam-4616	177	1	suppose	suppose	VERB
ejpam-4616	177	2	(	(	PUNCT
ejpam-4616	177	3	χ̃s	χ̃s	PROPN
ejpam-4616	177	4	◦	◦	PROPN
ejpam-4616	177	5	µ̃)(x	µ̃)(x	NOUN
ejpam-4616	177	6	)	)	PUNCT
ejpam-4616	177	7	⪰	⪰	PROPN
ejpam-4616	177	8	µ̃(x	µ̃(x	PROPN
ejpam-4616	177	9	)	)	PUNCT
ejpam-4616	177	10	.	.	PUNCT
ejpam-4616	178	1	since	since	SCONJ
ejpam-4616	178	2	s	s	PROPN
ejpam-4616	178	3	is	be	AUX
ejpam-4616	178	4	regular	regular	ADJ
ejpam-4616	178	5	,	,	PUNCT
ejpam-4616	178	6	there	there	PRON
ejpam-4616	178	7	exists	exist	VERB
ejpam-4616	178	8	y	y	PROPN
ejpam-4616	178	9	∈	∈	PROPN
ejpam-4616	178	10	s	s	VERB
ejpam-4616	179	1	such	such	ADJ
ejpam-4616	179	2	that	that	SCONJ
ejpam-4616	179	3	x	x	PROPN
ejpam-4616	179	4	=	=	SYM
ejpam-4616	179	5	xyx	xyx	PROPN
ejpam-4616	179	6	.	.	PUNCT
ejpam-4616	180	1	then	then	ADV
ejpam-4616	180	2	(	(	PUNCT
ejpam-4616	180	3	µ̃	µ̃	PROPN
ejpam-4616	180	4	◦	◦	VERB
ejpam-4616	180	5	χ̃s	χ̃s	NOUN
ejpam-4616	180	6	◦	◦	NOUN
ejpam-4616	180	7	µ̃)(x	µ̃)(x	NOUN
ejpam-4616	180	8	)	)	PUNCT
ejpam-4616	181	1	=	=	PUNCT
ejpam-4616	181	2	⋃	⋃	NOUN
ejpam-4616	181	3	x	x	SYM
ejpam-4616	181	4	=	=	NOUN
ejpam-4616	181	5	xyx	xyx	X
ejpam-4616	181	6	{	{	PUNCT
ejpam-4616	181	7	µ̃(xy)⋏	µ̃(xy)⋏	PROPN
ejpam-4616	181	8	(	(	PUNCT
ejpam-4616	181	9	χ̃s	χ̃s	PROPN
ejpam-4616	181	10	◦	◦	PROPN
ejpam-4616	181	11	µ̃)(x	µ̃)(x	NOUN
ejpam-4616	181	12	)	)	PUNCT
ejpam-4616	181	13	}	}	PUNCT
ejpam-4616	181	14	⊇	⊇	NOUN
ejpam-4616	181	15	⋃	⋃	NOUN
ejpam-4616	181	16	x	x	SYM
ejpam-4616	181	17	=	=	NOUN
ejpam-4616	181	18	xyx	xyx	X
ejpam-4616	181	19	{	{	PUNCT
ejpam-4616	181	20	µ̃(x)⋏	µ̃(x)⋏	ADP
ejpam-4616	181	21	µ̃(x	µ̃(x	PROPN
ejpam-4616	181	22	)	)	PUNCT
ejpam-4616	181	23	}	}	PUNCT
ejpam-4616	181	24	=	=	SYM
ejpam-4616	181	25	µ̃(x	µ̃(x	PROPN
ejpam-4616	181	26	)	)	PUNCT
ejpam-4616	181	27	.	.	PUNCT
ejpam-4616	182	1	which	which	PRON
ejpam-4616	182	2	is	be	AUX
ejpam-4616	182	3	a	a	DET
ejpam-4616	182	4	contradiction	contradiction	NOUN
ejpam-4616	182	5	.	.	PUNCT
ejpam-4616	183	1	therefore	therefore	ADV
ejpam-4616	183	2	µ̃	µ̃	PROPN
ejpam-4616	183	3	is	be	AUX
ejpam-4616	183	4	an	an	DET
ejpam-4616	183	5	ivf	ivf	ADJ
ejpam-4616	183	6	quasi	quasi	NOUN
ejpam-4616	183	7	-	-	NOUN
ejpam-4616	183	8	ideal	ideal	NOUN
ejpam-4616	183	9	of	of	ADP
ejpam-4616	183	10	s.	s.	PROPN
ejpam-4616	183	11	by	by	ADP
ejpam-4616	183	12	theorem	theorem	NOUN
ejpam-4616	183	13	8	8	NUM
ejpam-4616	183	14	,	,	PUNCT
ejpam-4616	183	15	converse	converse	NOUN
ejpam-4616	183	16	is	be	AUX
ejpam-4616	183	17	true	true	ADJ
ejpam-4616	183	18	.	.	PUNCT
ejpam-4616	184	1	t.	t.	PROPN
ejpam-4616	184	2	gaketem	gaketem	PROPN
ejpam-4616	184	3	,	,	PUNCT
ejpam-4616	184	4	t.	t.	PROPN
ejpam-4616	184	5	prommai	prommai	PROPN
ejpam-4616	184	6	/	/	SYM
ejpam-4616	184	7	eur	eur	PROPN
ejpam-4616	184	8	.	.	PUNCT
ejpam-4616	185	1	j.	j.	PROPN
ejpam-4616	185	2	pure	pure	PROPN
ejpam-4616	185	3	appl	appl	PROPN
ejpam-4616	185	4	.	.	PROPN
ejpam-4616	185	5	math	math	PROPN
ejpam-4616	185	6	,	,	PUNCT
ejpam-4616	185	7	16	16	NUM
ejpam-4616	185	8	(	(	PUNCT
ejpam-4616	185	9	1	1	NUM
ejpam-4616	185	10	)	)	PUNCT
ejpam-4616	185	11	(	(	PUNCT
ejpam-4616	185	12	2023	2023	NUM
ejpam-4616	185	13	)	)	PUNCT
ejpam-4616	185	14	,	,	PUNCT
ejpam-4616	185	15	121	121	NUM
ejpam-4616	185	16	-	-	SYM
ejpam-4616	185	17	130	130	NUM
ejpam-4616	185	18	128	128	NUM
ejpam-4616	185	19	theorem	theorem	NOUN
ejpam-4616	185	20	10	10	NUM
ejpam-4616	185	21	.	.	PUNCT
ejpam-4616	186	1	if	if	SCONJ
ejpam-4616	186	2	µ̃	µ̃	PROPN
ejpam-4616	186	3	be	be	VERB
ejpam-4616	186	4	an	an	DET
ejpam-4616	186	5	ivf	ivf	ADJ
ejpam-4616	186	6	bi	bi	ADJ
ejpam-4616	186	7	-	-	ADJ
ejpam-4616	186	8	interior	interior	ADJ
ejpam-4616	186	9	ideal	ideal	NOUN
ejpam-4616	186	10	of	of	ADP
ejpam-4616	186	11	a	a	DET
ejpam-4616	186	12	regular	regular	ADJ
ejpam-4616	186	13	semigroup	semigroup	NOUN
ejpam-4616	186	14	s.	s.	PROPN
ejpam-4616	186	15	then	then	ADV
ejpam-4616	186	16	µ̃	µ̃	PROPN
ejpam-4616	186	17	is	be	AUX
ejpam-4616	186	18	an	an	DET
ejpam-4616	186	19	ivf	ivf	ADJ
ejpam-4616	186	20	ideal	ideal	NOUN
ejpam-4616	186	21	of	of	ADP
ejpam-4616	186	22	a	a	DET
ejpam-4616	186	23	semigroup	semigroup	PROPN
ejpam-4616	186	24	s.	s.	PROPN
ejpam-4616	186	25	proof	proof	PROPN
ejpam-4616	186	26	.	.	PUNCT
ejpam-4616	187	1	suppose	suppose	VERB
ejpam-4616	187	2	that	that	SCONJ
ejpam-4616	187	3	µ̃	µ̃	PROPN
ejpam-4616	187	4	is	be	AUX
ejpam-4616	187	5	an	an	DET
ejpam-4616	187	6	ivf	ivf	ADJ
ejpam-4616	187	7	bi	bi	ADJ
ejpam-4616	187	8	-	-	ADJ
ejpam-4616	187	9	interior	interior	ADJ
ejpam-4616	187	10	ideal	ideal	NOUN
ejpam-4616	187	11	of	of	ADP
ejpam-4616	187	12	s.	s.	PROPN
ejpam-4616	187	13	then	then	ADV
ejpam-4616	187	14	by	by	ADP
ejpam-4616	187	15	theorem	theorem	NOUN
ejpam-4616	187	16	9	9	NUM
ejpam-4616	187	17	,	,	PUNCT
ejpam-4616	187	18	µ̃	µ̃	PROPN
ejpam-4616	187	19	is	be	AUX
ejpam-4616	187	20	an	an	DET
ejpam-4616	187	21	ivf	ivf	ADJ
ejpam-4616	187	22	quasi	quasi	NOUN
ejpam-4616	187	23	-	-	NOUN
ejpam-4616	187	24	ideal	ideal	NOUN
ejpam-4616	187	25	of	of	ADP
ejpam-4616	187	26	s.	s.	PROPN
ejpam-4616	187	27	thus	thus	ADV
ejpam-4616	187	28	by	by	ADP
ejpam-4616	187	29	theorem	theorem	NOUN
ejpam-4616	187	30	8	8	NUM
ejpam-4616	187	31	,	,	PUNCT
ejpam-4616	187	32	µ̃	µ̃	PROPN
ejpam-4616	187	33	is	be	AUX
ejpam-4616	187	34	an	an	DET
ejpam-4616	187	35	ivf	ivf	ADJ
ejpam-4616	187	36	ideal	ideal	NOUN
ejpam-4616	187	37	of	of	ADP
ejpam-4616	187	38	s.	s.	PROPN
ejpam-4616	187	39	hence	hence	ADV
ejpam-4616	187	40	the	the	DET
ejpam-4616	187	41	theorem	theorem	NOUN
ejpam-4616	187	42	is	be	AUX
ejpam-4616	187	43	complete	complete	ADJ
ejpam-4616	187	44	.	.	PUNCT
ejpam-4616	188	1	the	the	DET
ejpam-4616	188	2	following	follow	VERB
ejpam-4616	188	3	theorems	theorem	NOUN
ejpam-4616	188	4	are	be	AUX
ejpam-4616	188	5	a	a	DET
ejpam-4616	188	6	tool	tool	NOUN
ejpam-4616	188	7	in	in	ADP
ejpam-4616	188	8	characterization	characterization	NOUN
ejpam-4616	188	9	regular	regular	ADJ
ejpam-4616	188	10	semigroup	semigroup	NOUN
ejpam-4616	188	11	in	in	ADP
ejpam-4616	188	12	terms	term	NOUN
ejpam-4616	188	13	of	of	ADP
ejpam-4616	188	14	ivf	ivf	ADJ
ejpam-4616	188	15	bi	bi	ADJ
ejpam-4616	188	16	-	-	ADJ
ejpam-4616	188	17	interior	interior	ADJ
ejpam-4616	188	18	ideals	ideal	NOUN
ejpam-4616	188	19	on	on	ADP
ejpam-4616	188	20	semigroups	semigroup	NOUN
ejpam-4616	188	21	.	.	PUNCT
ejpam-4616	189	1	theorem	theorem	VERB
ejpam-4616	189	2	11	11	NUM
ejpam-4616	189	3	.	.	PUNCT
ejpam-4616	190	1	[	[	X
ejpam-4616	190	2	4	4	X
ejpam-4616	190	3	]	]	PUNCT
ejpam-4616	190	4	for	for	ADP
ejpam-4616	190	5	non	non	ADJ
ejpam-4616	190	6	-	-	ADJ
ejpam-4616	190	7	empty	empty	ADJ
ejpam-4616	190	8	subsets	subset	NOUN
ejpam-4616	190	9	g	g	PROPN
ejpam-4616	190	10	and	and	CCONJ
ejpam-4616	190	11	h	h	NOUN
ejpam-4616	190	12	of	of	ADP
ejpam-4616	190	13	a	a	DET
ejpam-4616	190	14	semigroup	semigroup	NOUN
ejpam-4616	190	15	s	s	NOUN
ejpam-4616	190	16	,	,	PUNCT
ejpam-4616	190	17	we	we	PRON
ejpam-4616	190	18	have	have	VERB
ejpam-4616	190	19	(	(	PUNCT
ejpam-4616	190	20	1	1	X
ejpam-4616	190	21	)	)	PUNCT
ejpam-4616	190	22	χ̃g	χ̃g	PROPN
ejpam-4616	190	23	◦	◦	NOUN
ejpam-4616	190	24	χ̃h	χ̃h	NOUN
ejpam-4616	190	25	=	=	SYM
ejpam-4616	190	26	˜χgh	˜χgh	X
ejpam-4616	190	27	,	,	PUNCT
ejpam-4616	190	28	(	(	PUNCT
ejpam-4616	190	29	2	2	X
ejpam-4616	190	30	)	)	PUNCT
ejpam-4616	191	1	χ̃g	χ̃g	PRON
ejpam-4616	191	2	⊓	⊓	PROPN
ejpam-4616	191	3	χ̃h	χ̃h	NOUN
ejpam-4616	191	4	=	=	NOUN
ejpam-4616	191	5	˜χg∩h	˜χg∩h	NUM
ejpam-4616	191	6	.	.	PUNCT
ejpam-4616	192	1	proof	proof	NOUN
ejpam-4616	192	2	.	.	PUNCT
ejpam-4616	193	1	it	it	PRON
ejpam-4616	193	2	is	be	AUX
ejpam-4616	193	3	straightforward	straightforward	ADJ
ejpam-4616	193	4	.	.	PUNCT
ejpam-4616	194	1	theorem	theorem	NOUN
ejpam-4616	194	2	12	12	NUM
ejpam-4616	194	3	.	.	PUNCT
ejpam-4616	195	1	let	let	VERB
ejpam-4616	195	2	s	s	PRON
ejpam-4616	195	3	be	be	AUX
ejpam-4616	195	4	a	a	DET
ejpam-4616	195	5	semigroup	semigroup	NOUN
ejpam-4616	195	6	.	.	PUNCT
ejpam-4616	196	1	then	then	ADV
ejpam-4616	196	2	s	s	VERB
ejpam-4616	196	3	is	be	AUX
ejpam-4616	196	4	a	a	DET
ejpam-4616	196	5	regular	regular	ADJ
ejpam-4616	196	6	semigroup	semigroup	NOUN
ejpam-4616	196	7	if	if	SCONJ
ejpam-4616	196	8	and	and	CCONJ
ejpam-4616	196	9	only	only	ADV
ejpam-4616	196	10	if	if	SCONJ
ejpam-4616	196	11	b	b	NOUN
ejpam-4616	196	12	=	=	SYM
ejpam-4616	196	13	sbs	sbs	ADJ
ejpam-4616	196	14	∩	∩	ADJ
ejpam-4616	196	15	sbs	sbs	NOUN
ejpam-4616	196	16	,	,	PUNCT
ejpam-4616	196	17	for	for	ADP
ejpam-4616	196	18	every	every	DET
ejpam-4616	196	19	bi	bi	ADJ
ejpam-4616	196	20	-	-	ADJ
ejpam-4616	196	21	interior	interior	ADJ
ejpam-4616	196	22	ideal	ideal	NOUN
ejpam-4616	196	23	of	of	ADP
ejpam-4616	196	24	s.	s.	PROPN
ejpam-4616	196	25	the	the	DET
ejpam-4616	196	26	following	follow	VERB
ejpam-4616	196	27	theorems	theorem	NOUN
ejpam-4616	196	28	are	be	AUX
ejpam-4616	196	29	characterization	characterization	NOUN
ejpam-4616	196	30	regular	regular	ADJ
ejpam-4616	196	31	semigroup	semigroup	NOUN
ejpam-4616	196	32	in	in	ADP
ejpam-4616	196	33	terms	term	NOUN
ejpam-4616	196	34	of	of	ADP
ejpam-4616	196	35	ivf	ivf	ADJ
ejpam-4616	196	36	biinterior	biinterior	PROPN
ejpam-4616	196	37	ideals	ideal	NOUN
ejpam-4616	196	38	in	in	ADP
ejpam-4616	196	39	semigroups	semigroup	NOUN
ejpam-4616	196	40	.	.	PUNCT
ejpam-4616	197	1	theorem	theorem	NOUN
ejpam-4616	197	2	13	13	NUM
ejpam-4616	197	3	.	.	PUNCT
ejpam-4616	198	1	let	let	VERB
ejpam-4616	198	2	s	s	PRON
ejpam-4616	198	3	be	be	AUX
ejpam-4616	198	4	a	a	DET
ejpam-4616	198	5	semigroup	semigroup	NOUN
ejpam-4616	198	6	.	.	PUNCT
ejpam-4616	199	1	then	then	ADV
ejpam-4616	199	2	s	s	VERB
ejpam-4616	199	3	is	be	AUX
ejpam-4616	199	4	a	a	DET
ejpam-4616	199	5	regular	regular	ADJ
ejpam-4616	199	6	if	if	SCONJ
ejpam-4616	199	7	and	and	CCONJ
ejpam-4616	199	8	only	only	ADV
ejpam-4616	199	9	if	if	SCONJ
ejpam-4616	199	10	µ̃	µ̃	PROPN
ejpam-4616	199	11	=	=	SYM
ejpam-4616	199	12	χ̃s	χ̃s	PROPN
ejpam-4616	199	13	◦	◦	VERB
ejpam-4616	199	14	µ̃	µ̃	PROPN
ejpam-4616	199	15	◦	◦	NOUN
ejpam-4616	199	16	χ̃s	χ̃s	PUNCT
ejpam-4616	199	17	⊓	⊓	PROPN
ejpam-4616	199	18	µ̃	µ̃	PROPN
ejpam-4616	199	19	◦	◦	VERB
ejpam-4616	199	20	χ̃s	χ̃s	NUM
ejpam-4616	199	21	◦	◦	VERB
ejpam-4616	199	22	µ̃	µ̃	PROPN
ejpam-4616	199	23	for	for	ADP
ejpam-4616	199	24	every	every	DET
ejpam-4616	199	25	ivf	ivf	ADJ
ejpam-4616	199	26	bi	bi	ADJ
ejpam-4616	199	27	-	-	ADJ
ejpam-4616	199	28	interior	interior	ADJ
ejpam-4616	199	29	ideal	ideal	NOUN
ejpam-4616	199	30	of	of	ADP
ejpam-4616	199	31	a	a	DET
ejpam-4616	199	32	semigroup	semigroup	PROPN
ejpam-4616	199	33	s.	s.	PROPN
ejpam-4616	199	34	proof	proof	PROPN
ejpam-4616	199	35	.	.	PUNCT
ejpam-4616	200	1	let	let	VERB
ejpam-4616	200	2	µ̃	µ̃	PROPN
ejpam-4616	200	3	be	be	AUX
ejpam-4616	200	4	an	an	DET
ejpam-4616	200	5	ivf	ivf	ADJ
ejpam-4616	200	6	bi	bi	ADJ
ejpam-4616	200	7	-	-	ADJ
ejpam-4616	200	8	interior	interior	ADJ
ejpam-4616	200	9	ideal	ideal	NOUN
ejpam-4616	200	10	of	of	ADP
ejpam-4616	200	11	the	the	DET
ejpam-4616	200	12	regular	regular	ADJ
ejpam-4616	200	13	semigroup	semigroup	NOUN
ejpam-4616	200	14	s	s	PART
ejpam-4616	200	15	and	and	CCONJ
ejpam-4616	200	16	let	let	VERB
ejpam-4616	200	17	x	x	PROPN
ejpam-4616	200	18	∈	∈	PROPN
ejpam-4616	200	19	s.	s.	PROPN
ejpam-4616	200	20	since	since	SCONJ
ejpam-4616	200	21	s	s	PROPN
ejpam-4616	200	22	is	be	AUX
ejpam-4616	200	23	regular	regular	ADJ
ejpam-4616	200	24	,	,	PUNCT
ejpam-4616	200	25	there	there	PRON
ejpam-4616	200	26	exists	exist	VERB
ejpam-4616	200	27	a	a	DET
ejpam-4616	200	28	∈	∈	NOUN
ejpam-4616	200	29	s	s	VERB
ejpam-4616	200	30	such	such	ADJ
ejpam-4616	200	31	that	that	SCONJ
ejpam-4616	200	32	x	x	NOUN
ejpam-4616	200	33	=	=	PUNCT
ejpam-4616	200	34	xax	xax	PROPN
ejpam-4616	200	35	.	.	PUNCT
ejpam-4616	201	1	thus	thus	ADV
ejpam-4616	201	2	(	(	PUNCT
ejpam-4616	201	3	µ̃	µ̃	PROPN
ejpam-4616	201	4	◦	◦	VERB
ejpam-4616	201	5	χ̃s	χ̃s	NOUN
ejpam-4616	201	6	◦	◦	NOUN
ejpam-4616	201	7	µ̃)(x	µ̃)(x	NOUN
ejpam-4616	201	8	)	)	PUNCT
ejpam-4616	202	1	=	=	PUNCT
ejpam-4616	202	2	⋃	⋃	NOUN
ejpam-4616	202	3	x	x	X
ejpam-4616	202	4	=	=	NOUN
ejpam-4616	202	5	xax	xax	X
ejpam-4616	202	6	{	{	PUNCT
ejpam-4616	202	7	µ̃(x)⋏	µ̃(x)⋏	ADV
ejpam-4616	202	8	(	(	PUNCT
ejpam-4616	202	9	χ̃s	χ̃s	PROPN
ejpam-4616	202	10	◦	◦	PROPN
ejpam-4616	202	11	µ̃)(ax	µ̃)(ax	PROPN
ejpam-4616	202	12	)	)	PUNCT
ejpam-4616	202	13	}	}	PUNCT
ejpam-4616	203	1	=	=	SYM
ejpam-4616	203	2	⋃	⋃	NOUN
ejpam-4616	203	3	x	x	X
ejpam-4616	203	4	=	=	NOUN
ejpam-4616	203	5	xax	xax	PROPN
ejpam-4616	203	6	{	{	PUNCT
ejpam-4616	203	7	µ̃(x)⋏	µ̃(x)⋏	ADP
ejpam-4616	203	8	⋃	⋃	NOUN
ejpam-4616	203	9	ax	ax	NOUN
ejpam-4616	203	10	=	=	SYM
ejpam-4616	203	11	yz	yz	X
ejpam-4616	203	12	{	{	PUNCT
ejpam-4616	203	13	χ̃s(y)⋏	χ̃s(y)⋏	PROPN
ejpam-4616	203	14	µ̃(z	µ̃(z	PROPN
ejpam-4616	203	15	)	)	PUNCT
ejpam-4616	203	16	}	}	PUNCT
ejpam-4616	203	17	}	}	PUNCT
ejpam-4616	203	18	⊇	⊇	NOUN
ejpam-4616	203	19	⋃	⋃	NOUN
ejpam-4616	203	20	x	x	SYM
ejpam-4616	203	21	=	=	NOUN
ejpam-4616	203	22	xax	xax	PROPN
ejpam-4616	203	23	{	{	PUNCT
ejpam-4616	203	24	µ̃(x)⋏	µ̃(x)⋏	ADP
ejpam-4616	203	25	µ̃(x	µ̃(x	PROPN
ejpam-4616	203	26	)	)	PUNCT
ejpam-4616	203	27	}	}	PUNCT
ejpam-4616	203	28	=	=	SYM
ejpam-4616	203	29	µ̃(x	µ̃(x	PROPN
ejpam-4616	203	30	)	)	PUNCT
ejpam-4616	203	31	.	.	PUNCT
ejpam-4616	204	1	similarly	similarly	ADV
ejpam-4616	204	2	,	,	PUNCT
ejpam-4616	204	3	(	(	PUNCT
ejpam-4616	204	4	χ̃s	χ̃s	PROPN
ejpam-4616	204	5	◦	◦	PROPN
ejpam-4616	204	6	µ̃	µ̃	PROPN
ejpam-4616	204	7	◦	◦	PROPN
ejpam-4616	204	8	χ̃s)(x	χ̃s)(x	NOUN
ejpam-4616	204	9	)	)	PUNCT
ejpam-4616	204	10	⪯	⪯	PROPN
ejpam-4616	204	11	µ̃(x	µ̃(x	PROPN
ejpam-4616	204	12	)	)	PUNCT
ejpam-4616	204	13	.	.	PUNCT
ejpam-4616	205	1	therefore	therefore	ADV
ejpam-4616	205	2	µ̃	µ̃	PROPN
ejpam-4616	205	3	=	=	SYM
ejpam-4616	205	4	χ̃s	χ̃s	PROPN
ejpam-4616	205	5	◦	◦	VERB
ejpam-4616	205	6	µ̃	µ̃	PROPN
ejpam-4616	205	7	◦	◦	NOUN
ejpam-4616	205	8	χ̃s	χ̃s	PUNCT
ejpam-4616	205	9	⊓	⊓	PROPN
ejpam-4616	205	10	µ̃	µ̃	PROPN
ejpam-4616	205	11	◦	◦	VERB
ejpam-4616	205	12	χ̃s	χ̃s	NOUN
ejpam-4616	205	13	◦	◦	NOUN
ejpam-4616	205	14	µ̃.	µ̃.	NOUN
ejpam-4616	205	15	conversely	conversely	ADV
ejpam-4616	205	16	,	,	PUNCT
ejpam-4616	205	17	suppose	suppose	VERB
ejpam-4616	205	18	that	that	SCONJ
ejpam-4616	205	19	b	b	PROPN
ejpam-4616	205	20	is	be	AUX
ejpam-4616	205	21	a	a	DET
ejpam-4616	205	22	bi	bi	ADJ
ejpam-4616	205	23	-	-	ADJ
ejpam-4616	205	24	interior	interior	ADJ
ejpam-4616	205	25	ideal	ideal	NOUN
ejpam-4616	205	26	of	of	ADP
ejpam-4616	205	27	a	a	DET
ejpam-4616	205	28	semigroup	semigroup	PROPN
ejpam-4616	205	29	s.	s.	PROPN
ejpam-4616	205	30	then	then	ADV
ejpam-4616	205	31	by	by	ADP
ejpam-4616	205	32	theorem	theorem	NOUN
ejpam-4616	205	33	5	5	NUM
ejpam-4616	205	34	,	,	PUNCT
ejpam-4616	205	35	χ̃b	χ̃b	PROPN
ejpam-4616	205	36	is	be	AUX
ejpam-4616	205	37	an	an	DET
ejpam-4616	205	38	ivf	ivf	ADJ
ejpam-4616	205	39	bi	bi	ADJ
ejpam-4616	205	40	-	-	ADJ
ejpam-4616	205	41	interior	interior	ADJ
ejpam-4616	205	42	ideal	ideal	NOUN
ejpam-4616	205	43	of	of	ADP
ejpam-4616	205	44	the	the	DET
ejpam-4616	205	45	semigroup	semigroup	PROPN
ejpam-4616	205	46	s.	s.	PROPN
ejpam-4616	205	47	thus	thus	ADV
ejpam-4616	205	48	by	by	ADP
ejpam-4616	205	49	theorem	theorem	ADJ
ejpam-4616	205	50	11	11	NUM
ejpam-4616	205	51	,	,	PUNCT
ejpam-4616	205	52	µ̃χb	µ̃χb	NOUN
ejpam-4616	205	53	(	(	PUNCT
ejpam-4616	205	54	x	x	NOUN
ejpam-4616	205	55	)	)	PUNCT
ejpam-4616	205	56	=	=	SYM
ejpam-4616	205	57	χ̃s	χ̃s	PROPN
ejpam-4616	205	58	◦	◦	NOUN
ejpam-4616	205	59	µ̃χb	µ̃χb	NOUN
ejpam-4616	205	60	◦	◦	NOUN
ejpam-4616	205	61	χ̃s(x)⋏	χ̃s(x)⋏	VERB
ejpam-4616	205	62	µ̃χb	µ̃χb	NOUN
ejpam-4616	205	63	◦	◦	NOUN
ejpam-4616	205	64	χ̃s	χ̃s	NOUN
ejpam-4616	205	65	◦	◦	NOUN
ejpam-4616	205	66	µ̃χb	µ̃χb	NOUN
ejpam-4616	205	67	(	(	PUNCT
ejpam-4616	205	68	x	x	X
ejpam-4616	205	69	)	)	PUNCT
ejpam-4616	205	70	=	=	VERB
ejpam-4616	205	71	µ̃χsbs	µ̃χsbs	NOUN
ejpam-4616	205	72	(	(	PUNCT
ejpam-4616	205	73	x)⋏	x)⋏	PUNCT
ejpam-4616	205	74	µ̃χbsb	µ̃χbsb	PROPN
ejpam-4616	205	75	(	(	PUNCT
ejpam-4616	205	76	x	x	X
ejpam-4616	205	77	)	)	PUNCT
ejpam-4616	205	78	=	=	SYM
ejpam-4616	206	1	µ̃χsbs∩bsb	µ̃χsbs∩bsb	PROPN
ejpam-4616	206	2	(	(	PUNCT
ejpam-4616	206	3	x	x	NOUN
ejpam-4616	206	4	)	)	PUNCT
ejpam-4616	206	5	.	.	PUNCT
ejpam-4616	207	1	therefore	therefore	ADV
ejpam-4616	207	2	b	b	X
ejpam-4616	207	3	=	=	SYM
ejpam-4616	207	4	sbs	sbs	PROPN
ejpam-4616	207	5	∩bsb	∩bsb	NOUN
ejpam-4616	207	6	.	.	PUNCT
ejpam-4616	208	1	by	by	ADP
ejpam-4616	208	2	theorem	theorem	NOUN
ejpam-4616	208	3	12	12	NUM
ejpam-4616	208	4	,	,	PUNCT
ejpam-4616	208	5	s	s	PART
ejpam-4616	208	6	is	be	AUX
ejpam-4616	208	7	a	a	DET
ejpam-4616	208	8	regular	regular	ADJ
ejpam-4616	208	9	semigroup	semigroup	NOUN
ejpam-4616	208	10	.	.	PUNCT
ejpam-4616	209	1	theorem	theorem	PROPN
ejpam-4616	209	2	14	14	NUM
ejpam-4616	209	3	.	.	PUNCT
ejpam-4616	210	1	let	let	VERB
ejpam-4616	210	2	s	s	PRON
ejpam-4616	210	3	be	be	AUX
ejpam-4616	210	4	a	a	DET
ejpam-4616	210	5	semigroup	semigroup	NOUN
ejpam-4616	210	6	.	.	PUNCT
ejpam-4616	211	1	then	then	ADV
ejpam-4616	211	2	s	s	VERB
ejpam-4616	211	3	is	be	AUX
ejpam-4616	211	4	regular	regular	ADJ
ejpam-4616	211	5	if	if	SCONJ
ejpam-4616	211	6	and	and	CCONJ
ejpam-4616	211	7	only	only	ADV
ejpam-4616	211	8	if	if	SCONJ
ejpam-4616	211	9	µ̃	µ̃	PROPN
ejpam-4616	211	10	⊓	⊓	PROPN
ejpam-4616	211	11	λ̃⊆̃λ̃	λ̃⊆̃λ̃	PUNCT
ejpam-4616	211	12	◦	◦	NOUN
ejpam-4616	211	13	µ̃	µ̃	PROPN
ejpam-4616	211	14	◦	◦	NOUN
ejpam-4616	211	15	λ̃	λ̃	PROPN
ejpam-4616	211	16	⊓	⊓	PROPN
ejpam-4616	211	17	µ̃	µ̃	PROPN
ejpam-4616	211	18	◦	◦	NOUN
ejpam-4616	211	19	λ̃	λ̃	PUNCT
ejpam-4616	211	20	◦	◦	VERB
ejpam-4616	211	21	µ̃	µ̃	PROPN
ejpam-4616	211	22	for	for	ADP
ejpam-4616	211	23	every	every	DET
ejpam-4616	211	24	ivf	ivf	ADJ
ejpam-4616	211	25	bi	bi	ADJ
ejpam-4616	211	26	-	-	ADJ
ejpam-4616	211	27	interior	interior	ADJ
ejpam-4616	211	28	ideal	ideal	NOUN
ejpam-4616	211	29	µ̃	µ̃	PROPN
ejpam-4616	211	30	and	and	CCONJ
ejpam-4616	211	31	every	every	DET
ejpam-4616	211	32	ivf	ivf	NOUN
ejpam-4616	211	33	ideal	ideal	NOUN
ejpam-4616	211	34	λ̃	λ̃	PROPN
ejpam-4616	211	35	of	of	ADP
ejpam-4616	211	36	s.	s.	PROPN
ejpam-4616	211	37	references	reference	VERB
ejpam-4616	211	38	129	129	NUM
ejpam-4616	211	39	proof	proof	NOUN
ejpam-4616	211	40	.	.	PUNCT
ejpam-4616	212	1	let	let	VERB
ejpam-4616	212	2	µ̃	µ̃	PROPN
ejpam-4616	212	3	be	be	AUX
ejpam-4616	212	4	an	an	DET
ejpam-4616	212	5	ivf	ivf	ADJ
ejpam-4616	212	6	bi	bi	ADJ
ejpam-4616	212	7	-	-	ADJ
ejpam-4616	212	8	interior	interior	ADJ
ejpam-4616	212	9	ideal	ideal	NOUN
ejpam-4616	212	10	and	and	CCONJ
ejpam-4616	212	11	λ̃	λ̃	PROPN
ejpam-4616	212	12	be	be	VERB
ejpam-4616	212	13	an	an	DET
ejpam-4616	212	14	ivf	ivf	ADJ
ejpam-4616	212	15	ideal	ideal	NOUN
ejpam-4616	212	16	of	of	ADP
ejpam-4616	212	17	a	a	DET
ejpam-4616	212	18	regular	regular	ADJ
ejpam-4616	212	19	semigroup	semigroup	NOUN
ejpam-4616	212	20	s	s	PART
ejpam-4616	212	21	and	and	CCONJ
ejpam-4616	212	22	let	let	VERB
ejpam-4616	212	23	x	x	PROPN
ejpam-4616	212	24	∈	∈	PROPN
ejpam-4616	212	25	s.	s.	PROPN
ejpam-4616	212	26	then	then	ADV
ejpam-4616	212	27	there	there	PRON
ejpam-4616	212	28	exists	exist	VERB
ejpam-4616	212	29	y	y	PROPN
ejpam-4616	212	30	∈	∈	PROPN
ejpam-4616	212	31	s	s	VERB
ejpam-4616	213	1	such	such	ADJ
ejpam-4616	213	2	that	that	SCONJ
ejpam-4616	213	3	x	x	PROPN
ejpam-4616	213	4	=	=	SYM
ejpam-4616	213	5	xyx	xyx	PROPN
ejpam-4616	213	6	.	.	PUNCT
ejpam-4616	214	1	(	(	PUNCT
ejpam-4616	214	2	µ̃	µ̃	PROPN
ejpam-4616	214	3	◦	◦	VERB
ejpam-4616	214	4	λ̃	λ̃	PROPN
ejpam-4616	214	5	◦	◦	NOUN
ejpam-4616	214	6	µ̃)(x	µ̃)(x	NOUN
ejpam-4616	214	7	)	)	PUNCT
ejpam-4616	215	1	=	=	PUNCT
ejpam-4616	215	2	⋃	⋃	NOUN
ejpam-4616	215	3	x	x	SYM
ejpam-4616	215	4	=	=	NOUN
ejpam-4616	215	5	xyx	xyx	X
ejpam-4616	215	6	{	{	PUNCT
ejpam-4616	215	7	(	(	PUNCT
ejpam-4616	215	8	µ̃	µ̃	PROPN
ejpam-4616	215	9	◦	◦	NOUN
ejpam-4616	215	10	λ̃)(xy)⋏	λ̃)(xy)⋏	PRON
ejpam-4616	215	11	µ̃(x	µ̃(x	PROPN
ejpam-4616	215	12	)	)	PUNCT
ejpam-4616	215	13	}	}	PUNCT
ejpam-4616	216	1	=	=	SYM
ejpam-4616	216	2	⋃	⋃	NOUN
ejpam-4616	216	3	x	x	SYM
ejpam-4616	216	4	=	=	NOUN
ejpam-4616	216	5	xyx	xyx	X
ejpam-4616	216	6	{	{	PUNCT
ejpam-4616	216	7	⋃	⋃	NOUN
ejpam-4616	216	8	xy	xy	PROPN
ejpam-4616	216	9	=	=	PRON
ejpam-4616	216	10	xyxy	xyxy	PROPN
ejpam-4616	216	11	{	{	PUNCT
ejpam-4616	216	12	µ̃(x	µ̃(x	PROPN
ejpam-4616	216	13	)	)	PUNCT
ejpam-4616	216	14	∩	∩	PROPN
ejpam-4616	216	15	λ̃(yxy)}⋏	λ̃(yxy)}⋏	PROPN
ejpam-4616	216	16	µ̃(x	µ̃(x	PROPN
ejpam-4616	216	17	)	)	PUNCT
ejpam-4616	216	18	}	}	PUNCT
ejpam-4616	216	19	⪰	⪰	NOUN
ejpam-4616	216	20	{	{	PUNCT
ejpam-4616	216	21	µ̃(x)⋏	µ̃(x)⋏	ADP
ejpam-4616	216	22	λ̃(x)}⋏	λ̃(x)}⋏	PROPN
ejpam-4616	216	23	µ̃(x	µ̃(x	PROPN
ejpam-4616	216	24	)	)	PUNCT
ejpam-4616	216	25	=	=	VERB
ejpam-4616	216	26	µ̃(x)⋏	µ̃(x)⋏	ADP
ejpam-4616	216	27	λ̃(x	λ̃(x	NOUN
ejpam-4616	216	28	)	)	PUNCT
ejpam-4616	216	29	=	=	PUNCT
ejpam-4616	216	30	(	(	PUNCT
ejpam-4616	216	31	µ̃	µ̃	PROPN
ejpam-4616	216	32	⊓	⊓	PROPN
ejpam-4616	216	33	λ̃)(x	λ̃)(x	NOUN
ejpam-4616	216	34	)	)	PUNCT
ejpam-4616	216	35	.	.	PUNCT
ejpam-4616	217	1	(	(	PUNCT
ejpam-4616	217	2	λ̃	λ̃	PROPN
ejpam-4616	217	3	◦	◦	NOUN
ejpam-4616	217	4	µ̃)(x	µ̃)(x	NOUN
ejpam-4616	217	5	)	)	PUNCT
ejpam-4616	218	1	=	=	PUNCT
ejpam-4616	218	2	⋃	⋃	NOUN
ejpam-4616	218	3	x	x	SYM
ejpam-4616	218	4	=	=	NOUN
ejpam-4616	218	5	xyx	xyx	X
ejpam-4616	218	6	{	{	PUNCT
ejpam-4616	218	7	λ̃(xy)⋏	λ̃(xy)⋏	PROPN
ejpam-4616	218	8	µ̃(x	µ̃(x	PROPN
ejpam-4616	218	9	)	)	PUNCT
ejpam-4616	218	10	}	}	PUNCT
ejpam-4616	218	11	⪰	⪰	NOUN
ejpam-4616	218	12	{	{	PUNCT
ejpam-4616	218	13	λ̃(x)⋏	λ̃(x)⋏	NOUN
ejpam-4616	218	14	µ̃(x	µ̃(x	PROPN
ejpam-4616	218	15	)	)	PUNCT
ejpam-4616	218	16	}	}	PUNCT
ejpam-4616	218	17	=	=	SYM
ejpam-4616	218	18	(	(	PUNCT
ejpam-4616	218	19	µ̃	µ̃	PROPN
ejpam-4616	218	20	⊓	⊓	PROPN
ejpam-4616	218	21	λ̃)(x	λ̃)(x	NOUN
ejpam-4616	218	22	)	)	PUNCT
ejpam-4616	218	23	.	.	PUNCT
ejpam-4616	219	1	therefore	therefore	ADV
ejpam-4616	219	2	µ̃	µ̃	PROPN
ejpam-4616	219	3	◦	◦	PROPN
ejpam-4616	219	4	λ̃	λ̃	PUNCT
ejpam-4616	219	5	◦	◦	VERB
ejpam-4616	219	6	µ̃	µ̃	PROPN
ejpam-4616	219	7	⊑	⊑	DET
ejpam-4616	219	8	µ̃	µ̃	PROPN
ejpam-4616	219	9	⊓	⊓	PROPN
ejpam-4616	219	10	λ̃.	λ̃.	PROPN
ejpam-4616	219	11	similary	similary	NOUN
ejpam-4616	219	12	,	,	PUNCT
ejpam-4616	219	13	we	we	PRON
ejpam-4616	219	14	can	can	AUX
ejpam-4616	219	15	prove	prove	VERB
ejpam-4616	219	16	λ̃	λ̃	PRON
ejpam-4616	219	17	◦	◦	VERB
ejpam-4616	219	18	µ̃	µ̃	PROPN
ejpam-4616	219	19	◦	◦	NOUN
ejpam-4616	219	20	λ̃	λ̃	PROPN
ejpam-4616	219	21	⊒	⊒	PUNCT
ejpam-4616	219	22	µ̃	µ̃	PROPN
ejpam-4616	219	23	⊓	⊓	PROPN
ejpam-4616	219	24	λ̃.	λ̃.	ADV
ejpam-4616	219	25	hence	hence	ADV
ejpam-4616	219	26	µ̃	µ̃	PROPN
ejpam-4616	219	27	⊓	⊓	PROPN
ejpam-4616	219	28	λ̃	λ̃	PROPN
ejpam-4616	219	29	⊑	⊑	X
ejpam-4616	219	30	λ̃	λ̃	PROPN
ejpam-4616	219	31	◦	◦	VERB
ejpam-4616	219	32	µ̃	µ̃	PROPN
ejpam-4616	219	33	◦	◦	NOUN
ejpam-4616	219	34	λ̃	λ̃	PROPN
ejpam-4616	219	35	⊓	⊓	PROPN
ejpam-4616	219	36	µ̃	µ̃	PROPN
ejpam-4616	219	37	◦	◦	NOUN
ejpam-4616	219	38	λ̃	λ̃	PROPN
ejpam-4616	219	39	◦	◦	NOUN
ejpam-4616	219	40	µ̃.	µ̃.	NOUN
ejpam-4616	219	41	conversely	conversely	ADV
ejpam-4616	219	42	,	,	PUNCT
ejpam-4616	219	43	suppose	suppose	VERB
ejpam-4616	219	44	that	that	SCONJ
ejpam-4616	219	45	the	the	DET
ejpam-4616	219	46	condition	condition	NOUN
ejpam-4616	219	47	holds	hold	VERB
ejpam-4616	219	48	.	.	PUNCT
ejpam-4616	220	1	let	let	VERB
ejpam-4616	220	2	µ̃	µ̃	PROPN
ejpam-4616	220	3	be	be	AUX
ejpam-4616	220	4	an	an	DET
ejpam-4616	220	5	ivf	ivf	ADJ
ejpam-4616	220	6	bi	bi	ADJ
ejpam-4616	220	7	-	-	ADJ
ejpam-4616	220	8	interior	interior	ADJ
ejpam-4616	220	9	ideal	ideal	NOUN
ejpam-4616	220	10	.	.	PUNCT
ejpam-4616	221	1	we	we	PRON
ejpam-4616	221	2	have	have	VERB
ejpam-4616	221	3	µ̃	µ̃	PROPN
ejpam-4616	221	4	⊓	⊓	PROPN
ejpam-4616	221	5	χ̃s	χ̃s	PROPN
ejpam-4616	221	6	⊑	⊑	PRON
ejpam-4616	221	7	χ̃s	χ̃s	PROPN
ejpam-4616	222	1	◦	◦	PROPN
ejpam-4616	222	2	µ̃	µ̃	PROPN
ejpam-4616	222	3	◦	◦	NOUN
ejpam-4616	222	4	χ̃s	χ̃s	PUNCT
ejpam-4616	222	5	⊓	⊓	PROPN
ejpam-4616	222	6	µ̃	µ̃	PROPN
ejpam-4616	222	7	◦	◦	VERB
ejpam-4616	222	8	χ̃s	χ̃s	NUM
ejpam-4616	222	9	◦	◦	NOUN
ejpam-4616	222	10	µ̃	µ̃	PROPN
ejpam-4616	222	11	and	and	CCONJ
ejpam-4616	222	12	implies	imply	VERB
ejpam-4616	222	13	that	that	SCONJ
ejpam-4616	222	14	µ̃	µ̃	PROPN
ejpam-4616	223	1	⊑	⊑	DET
ejpam-4616	223	2	χ̃s	χ̃s	PROPN
ejpam-4616	223	3	◦	◦	PROPN
ejpam-4616	223	4	µ̃	µ̃	PROPN
ejpam-4616	223	5	◦	◦	NOUN
ejpam-4616	223	6	χ̃s	χ̃s	PUNCT
ejpam-4616	223	7	⊓	⊓	PROPN
ejpam-4616	223	8	µ̃	µ̃	PROPN
ejpam-4616	223	9	◦	◦	VERB
ejpam-4616	223	10	χ̃s	χ̃s	NUM
ejpam-4616	223	11	◦	◦	NOUN
ejpam-4616	223	12	µ̃.	µ̃.	NOUN
ejpam-4616	223	13	by	by	ADP
ejpam-4616	223	14	theorem	theorem	ADJ
ejpam-4616	223	15	13	13	NUM
ejpam-4616	223	16	,	,	PUNCT
ejpam-4616	223	17	s	s	VERB
ejpam-4616	223	18	is	be	AUX
ejpam-4616	223	19	a	a	DET
ejpam-4616	223	20	regular	regular	ADJ
ejpam-4616	223	21	semigroup	semigroup	NOUN
ejpam-4616	223	22	.	.	PUNCT
ejpam-4616	224	1	4	4	X
ejpam-4616	224	2	.	.	X
ejpam-4616	224	3	conclusion	conclusion	NOUN
ejpam-4616	224	4	in	in	ADP
ejpam-4616	224	5	this	this	DET
ejpam-4616	224	6	paper	paper	NOUN
ejpam-4616	224	7	,	,	PUNCT
ejpam-4616	224	8	we	we	PRON
ejpam-4616	224	9	give	give	VERB
ejpam-4616	224	10	the	the	DET
ejpam-4616	224	11	concept	concept	NOUN
ejpam-4616	224	12	of	of	ADP
ejpam-4616	224	13	ivf	ivf	ADJ
ejpam-4616	224	14	bi	bi	ADJ
ejpam-4616	224	15	-	-	ADJ
ejpam-4616	224	16	interior	interior	ADJ
ejpam-4616	224	17	ideals	ideal	NOUN
ejpam-4616	224	18	in	in	ADP
ejpam-4616	224	19	semigroups	semigroup	NOUN
ejpam-4616	224	20	and	and	CCONJ
ejpam-4616	224	21	we	we	PRON
ejpam-4616	224	22	study	study	VERB
ejpam-4616	224	23	properties	property	NOUN
ejpam-4616	224	24	of	of	ADP
ejpam-4616	224	25	ivf	ivf	ADJ
ejpam-4616	224	26	bi	bi	ADJ
ejpam-4616	224	27	-	-	ADJ
ejpam-4616	224	28	interior	interior	ADJ
ejpam-4616	224	29	ideals	ideal	NOUN
ejpam-4616	224	30	in	in	ADP
ejpam-4616	224	31	semigroups	semigroup	NOUN
ejpam-4616	224	32	.	.	PUNCT
ejpam-4616	225	1	moreover	moreover	ADV
ejpam-4616	225	2	,	,	PUNCT
ejpam-4616	225	3	we	we	PRON
ejpam-4616	225	4	prove	prove	VERB
ejpam-4616	225	5	relationship	relationship	NOUN
ejpam-4616	225	6	between	between	ADP
ejpam-4616	225	7	ivf	ivf	ADJ
ejpam-4616	225	8	bi	bi	ADJ
ejpam-4616	225	9	-	-	ADJ
ejpam-4616	225	10	interior	interior	ADJ
ejpam-4616	225	11	ideals	ideal	NOUN
ejpam-4616	225	12	and	and	CCONJ
ejpam-4616	225	13	bi	bi	ADJ
ejpam-4616	225	14	-	-	ADJ
ejpam-4616	225	15	interior	interior	ADJ
ejpam-4616	225	16	ideals	ideal	NOUN
ejpam-4616	225	17	.	.	PUNCT
ejpam-4616	226	1	in	in	ADP
ejpam-4616	226	2	the	the	DET
ejpam-4616	226	3	future	future	NOUN
ejpam-4616	226	4	we	we	PRON
ejpam-4616	226	5	study	study	VERB
ejpam-4616	226	6	other	other	ADJ
ejpam-4616	226	7	kinds	kind	NOUN
ejpam-4616	226	8	of	of	ADP
ejpam-4616	226	9	ivf	ivf	ADJ
ejpam-4616	226	10	bi	bi	ADJ
ejpam-4616	226	11	-	-	ADJ
ejpam-4616	226	12	quasi	quasi	ADJ
ejpam-4616	226	13	interior	interior	ADJ
ejpam-4616	226	14	ideals	ideal	NOUN
ejpam-4616	226	15	in	in	ADP
ejpam-4616	226	16	semigroup	semigroup	ADJ
ejpam-4616	226	17	or	or	CCONJ
ejpam-4616	226	18	algebric	algebric	ADJ
ejpam-4616	226	19	system	system	NOUN
ejpam-4616	226	20	.	.	PUNCT
ejpam-4616	227	1	acknowledgements	acknowledgement	NOUN
ejpam-4616	227	2	the	the	DET
ejpam-4616	227	3	authors	author	NOUN
ejpam-4616	227	4	are	be	AUX
ejpam-4616	227	5	grateful	grateful	ADJ
ejpam-4616	227	6	to	to	ADP
ejpam-4616	227	7	the	the	DET
ejpam-4616	227	8	school	school	NOUN
ejpam-4616	227	9	of	of	ADP
ejpam-4616	227	10	science	science	NOUN
ejpam-4616	227	11	,	,	PUNCT
ejpam-4616	227	12	university	university	NOUN
ejpam-4616	227	13	of	of	ADP
ejpam-4616	227	14	phayao	phayao	NOUN
ejpam-4616	227	15	for	for	ADP
ejpam-4616	227	16	grant	grant	NOUN
ejpam-4616	227	17	support	support	NOUN
ejpam-4616	227	18	.	.	PUNCT
ejpam-4616	228	1	references	reference	NOUN
ejpam-4616	228	2	[	[	X
ejpam-4616	228	3	1	1	X
ejpam-4616	228	4	]	]	PUNCT
ejpam-4616	228	5	j.	j.	PROPN
ejpam-4616	228	6	aranzazu	aranzazu	PROPN
ejpam-4616	228	7	,	,	PUNCT
ejpam-4616	228	8	s.	s.	PROPN
ejpam-4616	228	9	antonio	antonio	PROPN
ejpam-4616	228	10	,	,	PUNCT
ejpam-4616	228	11	p.	p.	PROPN
ejpam-4616	228	12	daniel	daniel	PROPN
ejpam-4616	228	13	,	,	PUNCT
ejpam-4616	228	14	f.	f.	PROPN
ejpam-4616	228	15	javier	javier	PROPN
ejpam-4616	228	16	,	,	PUNCT
ejpam-4616	228	17	and	and	CCONJ
ejpam-4616	228	18	b.	b.	PROPN
ejpam-4616	228	19	humberto	humberto	PROPN
ejpam-4616	228	20	.	.	PUNCT
ejpam-4616	229	1	interval	interval	NOUN
ejpam-4616	229	2	valued	value	VERB
ejpam-4616	229	3	fuzzy	fuzzy	ADJ
ejpam-4616	229	4	sets	set	NOUN
ejpam-4616	229	5	for	for	ADP
ejpam-4616	229	6	color	color	NOUN
ejpam-4616	229	7	image	image	NOUN
ejpam-4616	229	8	super	super	NOUN
ejpam-4616	229	9	-	-	NOUN
ejpam-4616	229	10	resolution	resolution	NOUN
ejpam-4616	229	11	.	.	PUNCT
ejpam-4616	230	1	advances	advance	NOUN
ejpam-4616	230	2	in	in	ADP
ejpam-4616	230	3	artificial	artificial	ADJ
ejpam-4616	230	4	intelligence	intelligence	NOUN
ejpam-4616	230	5	,	,	PUNCT
ejpam-4616	230	6	pages	page	NOUN
ejpam-4616	230	7	373–382	373–382	NUM
ejpam-4616	230	8	,	,	PUNCT
ejpam-4616	230	9	2011	2011	NUM
ejpam-4616	230	10	.	.	PUNCT
ejpam-4616	231	1	[	[	X
ejpam-4616	231	2	2	2	X
ejpam-4616	231	3	]	]	PUNCT
ejpam-4616	231	4	h.	h.	NOUN
ejpam-4616	231	5	bustince	bustince	NOUN
ejpam-4616	231	6	.	.	PUNCT
ejpam-4616	232	1	indicator	indicator	NOUN
ejpam-4616	232	2	of	of	ADP
ejpam-4616	232	3	inclusion	inclusion	NOUN
ejpam-4616	232	4	grade	grade	NOUN
ejpam-4616	232	5	for	for	ADP
ejpam-4616	232	6	interval	interval	NOUN
ejpam-4616	232	7	valued	value	VERB
ejpam-4616	232	8	fuzzy	fuzzy	ADJ
ejpam-4616	232	9	sets	set	NOUN
ejpam-4616	232	10	.	.	PUNCT
ejpam-4616	233	1	application	application	NOUN
ejpam-4616	233	2	to	to	PART
ejpam-4616	233	3	approximate	approximate	ADJ
ejpam-4616	233	4	reasoning	reasoning	NOUN
ejpam-4616	233	5	based	base	VERB
ejpam-4616	233	6	on	on	ADP
ejpam-4616	233	7	interval	interval	NOUN
ejpam-4616	233	8	valued	value	VERB
ejpam-4616	233	9	fuzzy	fuzzy	ADJ
ejpam-4616	233	10	sets	set	NOUN
ejpam-4616	233	11	.	.	PUNCT
ejpam-4616	234	1	international	international	ADJ
ejpam-4616	234	2	journal	journal	PROPN
ejpam-4616	234	3	of	of	ADP
ejpam-4616	234	4	approximate	approximate	ADJ
ejpam-4616	234	5	reasoning	reasoning	NOUN
ejpam-4616	234	6	,	,	PUNCT
ejpam-4616	234	7	23:137–209	23:137–209	PROPN
ejpam-4616	234	8	,	,	PUNCT
ejpam-4616	234	9	1998	1998	NUM
ejpam-4616	234	10	.	.	PUNCT
ejpam-4616	235	1	[	[	X
ejpam-4616	235	2	3	3	NUM
ejpam-4616	235	3	]	]	X
ejpam-4616	235	4	i.	i.	PROPN
ejpam-4616	235	5	cristea	cristea	PROPN
ejpam-4616	235	6	,	,	PUNCT
ejpam-4616	235	7	a.	a.	NOUN
ejpam-4616	235	8	mahboob	mahboob	PROPN
ejpam-4616	235	9	,	,	PUNCT
ejpam-4616	235	10	and	and	CCONJ
ejpam-4616	235	11	n.	n.	PROPN
ejpam-4616	235	12	mohammad	mohammad	PROPN
ejpam-4616	235	13	khan	khan	PROPN
ejpam-4616	235	14	.	.	PUNCT
ejpam-4616	236	1	a	a	DET
ejpam-4616	236	2	new	new	ADJ
ejpam-4616	236	3	type	type	NOUN
ejpam-4616	236	4	fuzzy	fuzzy	ADJ
ejpam-4616	236	5	quasi	quasi	NOUN
ejpam-4616	236	6	-	-	NOUN
ejpam-4616	236	7	ideals	ideal	NOUN
ejpam-4616	236	8	of	of	ADP
ejpam-4616	236	9	ordered	order	VERB
ejpam-4616	236	10	semigrouops	semigrouop	NOUN
ejpam-4616	236	11	.	.	PUNCT
ejpam-4616	237	1	journal	journal	PROPN
ejpam-4616	237	2	of	of	ADP
ejpam-4616	237	3	multiple	multiple	ADV
ejpam-4616	237	4	-	-	PUNCT
ejpam-4616	237	5	valued	value	VERB
ejpam-4616	237	6	logic	logic	NOUN
ejpam-4616	237	7	and	and	CCONJ
ejpam-4616	237	8	soft	soft	ADJ
ejpam-4616	237	9	computing	computing	NOUN
ejpam-4616	237	10	,	,	PUNCT
ejpam-4616	237	11	17(3):739	17(3):739	NUM
ejpam-4616	237	12	–	–	PUNCT
ejpam-4616	237	13	752	752	NUM
ejpam-4616	237	14	,	,	PUNCT
ejpam-4616	237	15	2020	2020	NUM
ejpam-4616	237	16	.	.	PUNCT
ejpam-4616	238	1	references	reference	NOUN
ejpam-4616	238	2	130	130	NUM
ejpam-4616	238	3	[	[	SYM
ejpam-4616	238	4	4	4	NUM
ejpam-4616	238	5	]	]	PUNCT
ejpam-4616	238	6	t.	t.	NOUN
ejpam-4616	238	7	gaketem	gaketem	NOUN
ejpam-4616	238	8	.	.	PUNCT
ejpam-4616	239	1	on	on	ADP
ejpam-4616	239	2	interval	interval	NOUN
ejpam-4616	239	3	valued	value	VERB
ejpam-4616	239	4	fuzzy	fuzzy	ADJ
ejpam-4616	239	5	almost	almost	ADV
ejpam-4616	239	6	(	(	PUNCT
ejpam-4616	239	7	m	m	NOUN
ejpam-4616	239	8	,	,	PUNCT
ejpam-4616	239	9	n)-bi	n)-bi	NOUN
ejpam-4616	239	10	-	-	PUNCT
ejpam-4616	239	11	ideal	ideal	NOUN
ejpam-4616	239	12	in	in	ADP
ejpam-4616	239	13	semigroups	semigroup	NOUN
ejpam-4616	239	14	.	.	PUNCT
ejpam-4616	240	1	j.	j.	PROPN
ejpam-4616	240	2	math	math	PROPN
ejpam-4616	240	3	.	.	PUNCT
ejpam-4616	241	1	comput	comput	NOUN
ejpam-4616	241	2	.	.	PUNCT
ejpam-4616	242	1	sci	sci	PROPN
ejpam-4616	242	2	.	.	PROPN
ejpam-4616	242	3	,	,	PUNCT
ejpam-4616	242	4	11(6):6657–6665	11(6):6657–6665	NUM
ejpam-4616	242	5	,	,	PUNCT
ejpam-4616	242	6	2021	2021	NUM
ejpam-4616	242	7	.	.	PUNCT
ejpam-4616	243	1	[	[	X
ejpam-4616	243	2	5	5	NUM
ejpam-4616	243	3	]	]	PUNCT
ejpam-4616	243	4	m.	m.	NOUN
ejpam-4616	243	5	krishna	krishna	PROPN
ejpam-4616	243	6	and	and	CCONJ
ejpam-4616	243	7	m.	m.	PROPN
ejpam-4616	243	8	rao	rao	PROPN
ejpam-4616	243	9	.	.	PUNCT
ejpam-4616	244	1	bi	bi	ADJ
ejpam-4616	244	2	-	-	ADJ
ejpam-4616	244	3	interior	interior	ADJ
ejpam-4616	244	4	ideals	ideal	NOUN
ejpam-4616	244	5	of	of	ADP
ejpam-4616	244	6	semigroups	semigroup	NOUN
ejpam-4616	244	7	.	.	PUNCT
ejpam-4616	245	1	discussiones	discussione	NOUN
ejpam-4616	245	2	mathematicae	mathematicae	VERB
ejpam-4616	245	3	general	general	ADJ
ejpam-4616	245	4	algebra	algebra	PROPN
ejpam-4616	245	5	and	and	CCONJ
ejpam-4616	245	6	applications	application	NOUN
ejpam-4616	245	7	,	,	PUNCT
ejpam-4616	245	8	38:69–78	38:69–78	NUM
ejpam-4616	245	9	,	,	PUNCT
ejpam-4616	245	10	2018	2018	NUM
ejpam-4616	245	11	.	.	PUNCT
ejpam-4616	246	1	[	[	X
ejpam-4616	246	2	6	6	NUM
ejpam-4616	246	3	]	]	PUNCT
ejpam-4616	246	4	a.	a.	NOUN
ejpam-4616	246	5	mahboob	mahboob	PROPN
ejpam-4616	246	6	,	,	PUNCT
ejpam-4616	246	7	b.	b.	PROPN
ejpam-4616	246	8	davvaz	davvaz	PROPN
ejpam-4616	246	9	,	,	PUNCT
ejpam-4616	246	10	and	and	CCONJ
ejpam-4616	246	11	n.	n.	PROPN
ejpam-4616	246	12	m.	m.	PROPN
ejpam-4616	246	13	khan	khan	PROPN
ejpam-4616	246	14	.	.	PUNCT
ejpam-4616	247	1	fuzzy	fuzzy	ADJ
ejpam-4616	247	2	(	(	PUNCT
ejpam-4616	247	3	m	m	PROPN
ejpam-4616	247	4	,	,	PUNCT
ejpam-4616	247	5	n)-ideals	n)-ideal	NOUN
ejpam-4616	247	6	in	in	ADP
ejpam-4616	247	7	semigroups	semigroup	NOUN
ejpam-4616	247	8	.	.	PUNCT
ejpam-4616	248	1	computational	computational	ADJ
ejpam-4616	248	2	and	and	CCONJ
ejpam-4616	248	3	applied	applied	ADJ
ejpam-4616	248	4	mathematics	mathematic	NOUN
ejpam-4616	248	5	,	,	PUNCT
ejpam-4616	248	6	38(189):1–18	38(189):1–18	PROPN
ejpam-4616	248	7	,	,	PUNCT
ejpam-4616	248	8	2019	2019	NUM
ejpam-4616	248	9	.	.	PUNCT
ejpam-4616	249	1	[	[	X
ejpam-4616	249	2	7	7	X
ejpam-4616	249	3	]	]	PUNCT
ejpam-4616	249	4	a.	a.	NOUN
ejpam-4616	249	5	mahboob	mahboob	PROPN
ejpam-4616	249	6	and	and	CCONJ
ejpam-4616	249	7	g.	g.	PROPN
ejpam-4616	249	8	muhiuddin	muhiuddin	PROPN
ejpam-4616	249	9	.	.	PUNCT
ejpam-4616	250	1	a	a	DET
ejpam-4616	250	2	new	new	ADJ
ejpam-4616	250	3	type	type	NOUN
ejpam-4616	250	4	of	of	ADP
ejpam-4616	250	5	fuzzy	fuzzy	ADJ
ejpam-4616	250	6	prime	prime	NOUN
ejpam-4616	250	7	subset	subset	NOUN
ejpam-4616	250	8	in	in	ADP
ejpam-4616	250	9	ordered	order	VERB
ejpam-4616	250	10	semigroups	semigroup	NOUN
ejpam-4616	250	11	.	.	PUNCT
ejpam-4616	251	1	new	new	ADJ
ejpam-4616	251	2	mathematics	mathematic	NOUN
ejpam-4616	251	3	and	and	CCONJ
ejpam-4616	251	4	natural	natural	ADJ
ejpam-4616	251	5	computing	computing	NOUN
ejpam-4616	251	6	,	,	PUNCT
ejpam-4616	251	7	17(3):739–752	17(3):739–752	NUM
ejpam-4616	251	8	,	,	PUNCT
ejpam-4616	251	9	2021	2021	NUM
ejpam-4616	251	10	.	.	PUNCT
ejpam-4616	252	1	[	[	X
ejpam-4616	252	2	8	8	NUM
ejpam-4616	252	3	]	]	PUNCT
ejpam-4616	252	4	a.	a.	NOUN
ejpam-4616	252	5	mahboob	mahboob	PROPN
ejpam-4616	252	6	,	,	PUNCT
ejpam-4616	252	7	a.	a.	PROPN
ejpam-4616	252	8	salam	salam	PROPN
ejpam-4616	252	9	,	,	PUNCT
ejpam-4616	252	10	md	md	PROPN
ejpam-4616	252	11	.	.	PROPN
ejpam-4616	252	12	f.	f.	PROPN
ejpam-4616	252	13	ali	ali	PROPN
ejpam-4616	252	14	,	,	PUNCT
ejpam-4616	252	15	and	and	CCONJ
ejpam-4616	252	16	n.	n.	PROPN
ejpam-4616	252	17	m.	m.	PROPN
ejpam-4616	252	18	khan	khan	PROPN
ejpam-4616	252	19	.	.	PUNCT
ejpam-4616	253	1	characterizations	characterization	NOUN
ejpam-4616	253	2	of	of	ADP
ejpam-4616	253	3	regular	regular	ADJ
ejpam-4616	253	4	ordered	order	VERB
ejpam-4616	253	5	semigroups	semigroup	NOUN
ejpam-4616	253	6	by	by	ADP
ejpam-4616	253	7	(	(	PUNCT
ejpam-4616	253	8	ε	ε	PROPN
ejpam-4616	253	9	,	,	PUNCT
ejpam-4616	253	10	ε	ε	PROPN
ejpam-4616	253	11	∨	∨	PROPN
ejpam-4616	253	12	(	(	PUNCT
ejpam-4616	253	13	k	k	NOUN
ejpam-4616	253	14	,	,	PUNCT
ejpam-4616	253	15	qk))-fuzzy	qk))-fuzzy	ADJ
ejpam-4616	253	16	quasi	quasi	ADJ
ejpam-4616	253	17	ideals	ideal	NOUN
ejpam-4616	253	18	.	.	PUNCT
ejpam-4616	254	1	mathematics	mathematic	NOUN
ejpam-4616	254	2	,	,	PUNCT
ejpam-4616	254	3	7:1–16	7:1–16	NUM
ejpam-4616	254	4	,	,	PUNCT
ejpam-4616	254	5	2019	2019	NUM
ejpam-4616	254	6	.	.	PUNCT
ejpam-4616	255	1	[	[	X
ejpam-4616	255	2	9	9	NUM
ejpam-4616	255	3	]	]	X
ejpam-4616	255	4	al	al	PROPN
ejpam-4616	255	5	.	.	PROPN
ejpam-4616	255	6	narayanan	narayanan	PROPN
ejpam-4616	255	7	and	and	CCONJ
ejpam-4616	255	8	t.	t.	PROPN
ejpam-4616	255	9	manikantan	manikantan	PROPN
ejpam-4616	255	10	.	.	PUNCT
ejpam-4616	256	1	interval	interval	NOUN
ejpam-4616	256	2	valued	value	VERB
ejpam-4616	256	3	fuzzy	fuzzy	ADJ
ejpam-4616	256	4	ideals	ideal	NOUN
ejpam-4616	256	5	generated	generate	VERB
ejpam-4616	256	6	by	by	ADP
ejpam-4616	256	7	an	an	DET
ejpam-4616	256	8	interval	interval	NOUN
ejpam-4616	256	9	valued	value	VERB
ejpam-4616	256	10	fuzzy	fuzzy	ADJ
ejpam-4616	256	11	subset	subset	NOUN
ejpam-4616	256	12	in	in	ADP
ejpam-4616	256	13	semigroups	semigroup	NOUN
ejpam-4616	256	14	.	.	PUNCT
ejpam-4616	257	1	journal	journal	NOUN
ejpam-4616	257	2	of	of	ADP
ejpam-4616	257	3	applied	apply	VERB
ejpam-4616	257	4	mathematics	mathematic	NOUN
ejpam-4616	257	5	and	and	CCONJ
ejpam-4616	257	6	computing	computing	NOUN
ejpam-4616	257	7	,	,	PUNCT
ejpam-4616	257	8	20(1	20(1	NUM
ejpam-4616	257	9	-	-	PUNCT
ejpam-4616	257	10	2):455–464	2):455–464	NUM
ejpam-4616	257	11	,	,	PUNCT
ejpam-4616	257	12	2006	2006	NUM
ejpam-4616	257	13	.	.	PUNCT
ejpam-4616	258	1	[	[	X
ejpam-4616	258	2	10	10	NUM
ejpam-4616	258	3	]	]	X
ejpam-4616	258	4	l.a	l.a	PROPN
ejpam-4616	258	5	.	.	PROPN
ejpam-4616	258	6	zadeh	zadeh	PROPN
ejpam-4616	258	7	.	.	PUNCT
ejpam-4616	258	8	fuzzy	fuzzy	ADJ
ejpam-4616	258	9	sets	set	NOUN
ejpam-4616	258	10	.	.	PUNCT
ejpam-4616	259	1	information	information	NOUN
ejpam-4616	259	2	and	and	CCONJ
ejpam-4616	259	3	control	control	NOUN
ejpam-4616	259	4	,	,	PUNCT
ejpam-4616	259	5	8:338–353	8:338–353	NUM
ejpam-4616	259	6	,	,	PUNCT
ejpam-4616	259	7	1965	1965	NUM
ejpam-4616	259	8	.	.	PUNCT
ejpam-4616	260	1	[	[	X
ejpam-4616	260	2	11	11	NUM
ejpam-4616	260	3	]	]	X
ejpam-4616	260	4	l.a	l.a	PROPN
ejpam-4616	260	5	.	.	PROPN
ejpam-4616	260	6	zadeh	zadeh	PROPN
ejpam-4616	260	7	.	.	PUNCT
ejpam-4616	261	1	the	the	DET
ejpam-4616	261	2	concept	concept	NOUN
ejpam-4616	261	3	of	of	ADP
ejpam-4616	261	4	a	a	DET
ejpam-4616	261	5	linguistic	linguistic	ADJ
ejpam-4616	261	6	variable	variable	NOUN
ejpam-4616	261	7	and	and	CCONJ
ejpam-4616	261	8	its	its	PRON
ejpam-4616	261	9	application	application	NOUN
ejpam-4616	261	10	to	to	PART
ejpam-4616	261	11	approximate	approximate	ADJ
ejpam-4616	261	12	reasoning	reasoning	NOUN
ejpam-4616	261	13	.	.	PUNCT
ejpam-4616	262	1	information	information	NOUN
ejpam-4616	262	2	sciences	sciences	PROPN
ejpam-4616	262	3	,	,	PUNCT
ejpam-4616	262	4	8:199–249	8:199–249	NUM
ejpam-4616	262	5	,	,	PUNCT
ejpam-4616	262	6	1975	1975	NUM
ejpam-4616	262	7	.	.	PUNCT
ejpam-4616	263	1	[	[	X
ejpam-4616	263	2	12	12	NUM
ejpam-4616	263	3	]	]	PUNCT
ejpam-4616	263	4	m.	m.	NOUN
ejpam-4616	263	5	zulquanain	zulquanain	NOUN
ejpam-4616	263	6	and	and	CCONJ
ejpam-4616	263	7	m.	m.	PROPN
ejpam-4616	263	8	saeed	saeed	PROPN
ejpam-4616	263	9	.	.	PUNCT
ejpam-4616	264	1	a	a	DET
ejpam-4616	264	2	new	new	ADJ
ejpam-4616	264	3	decision	decision	NOUN
ejpam-4616	264	4	making	make	VERB
ejpam-4616	264	5	method	method	NOUN
ejpam-4616	264	6	on	on	ADP
ejpam-4616	264	7	interval	interval	NOUN
ejpam-4616	264	8	value	value	NOUN
ejpam-4616	264	9	fuzzy	fuzzy	ADJ
ejpam-4616	264	10	soft	soft	ADJ
ejpam-4616	264	11	matrix	matrix	NOUN
ejpam-4616	264	12	.	.	PUNCT
ejpam-4616	265	1	british	british	ADJ
ejpam-4616	265	2	journal	journal	PROPN
ejpam-4616	265	3	of	of	ADP
ejpam-4616	265	4	mathematics	mathematics	PROPN
ejpam-4616	265	5	and	and	CCONJ
ejpam-4616	265	6	computer	computer	NOUN
ejpam-4616	265	7	science	science	NOUN
ejpam-4616	265	8	,	,	PUNCT
ejpam-4616	265	9	20(5):1–17	20(5):1–17	NUM
ejpam-4616	265	10	,	,	PUNCT
ejpam-4616	265	11	2017	2017	NUM
ejpam-4616	265	12	.	.	PUNCT