id	sid	tid	token	lemma	pos
ejpam-4279	1	1	european	european	PROPN
ejpam-4279	1	2	journal	journal	PROPN
ejpam-4279	1	3	of	of	ADP
ejpam-4279	1	4	pure	pure	ADJ
ejpam-4279	1	5	and	and	CCONJ
ejpam-4279	1	6	applied	apply	VERB
ejpam-4279	1	7	mathematics	mathematic	NOUN
ejpam-4279	1	8	vol	vol	NOUN
ejpam-4279	1	9	.	.	PROPN
ejpam-4279	2	1	15	15	NUM
ejpam-4279	2	2	,	,	PUNCT
ejpam-4279	2	3	no	no	INTJ
ejpam-4279	2	4	.	.	NOUN
ejpam-4279	2	5	1	1	NUM
ejpam-4279	2	6	,	,	PUNCT
ejpam-4279	2	7	2022	2022	NUM
ejpam-4279	2	8	,	,	PUNCT
ejpam-4279	2	9	281	281	NUM
ejpam-4279	2	10	-	-	SYM
ejpam-4279	2	11	289	289	NUM
ejpam-4279	2	12	issn	issn	PROPN
ejpam-4279	2	13	1307	1307	NUM
ejpam-4279	2	14	-	-	SYM
ejpam-4279	2	15	5543	5543	NUM
ejpam-4279	2	16	–	–	PUNCT
ejpam-4279	3	1	ejpam.com	ejpam.com	X
ejpam-4279	3	2	published	publish	VERB
ejpam-4279	3	3	by	by	ADP
ejpam-4279	3	4	new	new	PROPN
ejpam-4279	3	5	york	york	PROPN
ejpam-4279	3	6	business	business	PROPN
ejpam-4279	3	7	global	global	ADJ
ejpam-4279	3	8	almost	almost	ADV
ejpam-4279	3	9	bi	bi	ADJ
ejpam-4279	3	10	interior	interior	ADJ
ejpam-4279	3	11	ideal	ideal	NOUN
ejpam-4279	3	12	in	in	ADP
ejpam-4279	3	13	semigroups	semigroup	NOUN
ejpam-4279	3	14	and	and	CCONJ
ejpam-4279	3	15	their	their	PRON
ejpam-4279	3	16	fuzzifications	fuzzification	NOUN
ejpam-4279	3	17	t.	t.	NOUN
ejpam-4279	3	18	gaketem	gaketem	PROPN
ejpam-4279	3	19	fuzzy	fuzzy	ADJ
ejpam-4279	3	20	algebras	algebra	NOUN
ejpam-4279	3	21	and	and	CCONJ
ejpam-4279	3	22	decision	decision	NOUN
ejpam-4279	3	23	-	-	PUNCT
ejpam-4279	3	24	making	make	VERB
ejpam-4279	3	25	problems	problem	NOUN
ejpam-4279	3	26	research	research	NOUN
ejpam-4279	3	27	unit	unit	NOUN
ejpam-4279	3	28	,	,	PUNCT
ejpam-4279	3	29	department	department	NOUN
ejpam-4279	3	30	of	of	ADP
ejpam-4279	3	31	mathematics	mathematics	PROPN
ejpam-4279	3	32	/school	/school	PUNCT
ejpam-4279	3	33	of	of	ADP
ejpam-4279	3	34	science	science	NOUN
ejpam-4279	3	35	,	,	PUNCT
ejpam-4279	3	36	university	university	NOUN
ejpam-4279	3	37	of	of	ADP
ejpam-4279	3	38	phayao	phayao	NOUN
ejpam-4279	3	39	,	,	PUNCT
ejpam-4279	3	40	phayao	phayao	NOUN
ejpam-4279	3	41	56000	56000	NUM
ejpam-4279	3	42	,	,	PUNCT
ejpam-4279	3	43	thailand	thailand	PROPN
ejpam-4279	3	44	abstract	abstract	NOUN
ejpam-4279	3	45	.	.	PUNCT
ejpam-4279	4	1	in	in	ADP
ejpam-4279	4	2	this	this	DET
ejpam-4279	4	3	paper	paper	NOUN
ejpam-4279	4	4	,	,	PUNCT
ejpam-4279	4	5	we	we	PRON
ejpam-4279	4	6	define	define	VERB
ejpam-4279	4	7	the	the	DET
ejpam-4279	4	8	concepts	concept	NOUN
ejpam-4279	4	9	of	of	ADP
ejpam-4279	4	10	almost	almost	ADV
ejpam-4279	4	11	bi	bi	ADJ
ejpam-4279	4	12	-	-	ADJ
ejpam-4279	4	13	interior	interior	ADJ
ejpam-4279	4	14	ideal	ideal	NOUN
ejpam-4279	4	15	and	and	CCONJ
ejpam-4279	4	16	fuzzy	fuzzy	ADJ
ejpam-4279	4	17	almost	almost	ADV
ejpam-4279	4	18	biinterior	biinterior	ADJ
ejpam-4279	4	19	ideal	ideal	NOUN
ejpam-4279	4	20	in	in	ADP
ejpam-4279	4	21	semigroups	semigroup	NOUN
ejpam-4279	4	22	.	.	PUNCT
ejpam-4279	5	1	moreover	moreover	ADV
ejpam-4279	5	2	,	,	PUNCT
ejpam-4279	5	3	we	we	PRON
ejpam-4279	5	4	prove	prove	VERB
ejpam-4279	5	5	that	that	SCONJ
ejpam-4279	5	6	relation	relation	NOUN
ejpam-4279	5	7	between	between	ADP
ejpam-4279	5	8	almost	almost	ADV
ejpam-4279	5	9	bi	bi	ADJ
ejpam-4279	5	10	-	-	ADJ
ejpam-4279	5	11	interior	interior	ADJ
ejpam-4279	5	12	ideal	ideal	NOUN
ejpam-4279	5	13	and	and	CCONJ
ejpam-4279	5	14	fuzzy	fuzzy	ADJ
ejpam-4279	5	15	almost	almost	ADV
ejpam-4279	5	16	bi	bi	ADJ
ejpam-4279	5	17	-	-	ADJ
ejpam-4279	5	18	interior	interior	ADJ
ejpam-4279	5	19	ideal	ideal	NOUN
ejpam-4279	5	20	.	.	PUNCT
ejpam-4279	6	1	2020	2020	NUM
ejpam-4279	6	2	mathematics	mathematic	NOUN
ejpam-4279	6	3	subject	subject	NOUN
ejpam-4279	6	4	classifications	classification	NOUN
ejpam-4279	6	5	:	:	PUNCT
ejpam-4279	6	6	20m12	20m12	NUM
ejpam-4279	6	7	,	,	PUNCT
ejpam-4279	6	8	06f05	06f05	PRON
ejpam-4279	6	9	key	key	ADJ
ejpam-4279	6	10	words	word	NOUN
ejpam-4279	6	11	and	and	CCONJ
ejpam-4279	6	12	phrases	phrase	NOUN
ejpam-4279	6	13	:	:	PUNCT
ejpam-4279	6	14	bi	bi	ADJ
ejpam-4279	6	15	-	-	ADJ
ejpam-4279	6	16	interior	interior	ADJ
ejpam-4279	6	17	ideal	ideal	NOUN
ejpam-4279	6	18	,	,	PUNCT
ejpam-4279	6	19	fuzzy	fuzzy	ADJ
ejpam-4279	6	20	bi	bi	ADJ
ejpam-4279	6	21	-	-	ADJ
ejpam-4279	6	22	interior	interior	ADJ
ejpam-4279	6	23	ideal	ideal	NOUN
ejpam-4279	6	24	,	,	PUNCT
ejpam-4279	6	25	weakly	weakly	ADJ
ejpam-4279	6	26	bi	bi	ADJ
ejpam-4279	6	27	-	-	ADJ
ejpam-4279	6	28	interior	interior	ADJ
ejpam-4279	6	29	ideal	ideal	NOUN
ejpam-4279	6	30	,	,	PUNCT
ejpam-4279	6	31	weakly	weakly	ADV
ejpam-4279	6	32	fuzzy	fuzzy	ADJ
ejpam-4279	6	33	bi	bi	ADJ
ejpam-4279	6	34	-	-	ADJ
ejpam-4279	6	35	interior	interior	ADJ
ejpam-4279	6	36	ideal	ideal	NOUN
ejpam-4279	6	37	1	1	NUM
ejpam-4279	6	38	.	.	PUNCT
ejpam-4279	7	1	introduction	introduction	NOUN
ejpam-4279	7	2	fuzzy	fuzzy	ADJ
ejpam-4279	7	3	sets	set	NOUN
ejpam-4279	7	4	are	be	AUX
ejpam-4279	7	5	a	a	DET
ejpam-4279	7	6	kind	kind	NOUN
ejpam-4279	7	7	of	of	ADP
ejpam-4279	7	8	useful	useful	ADJ
ejpam-4279	7	9	mathematical	mathematical	ADJ
ejpam-4279	7	10	structure	structure	NOUN
ejpam-4279	7	11	to	to	PART
ejpam-4279	7	12	represent	represent	VERB
ejpam-4279	7	13	a	a	DET
ejpam-4279	7	14	collection	collection	NOUN
ejpam-4279	7	15	of	of	ADP
ejpam-4279	7	16	objects	object	NOUN
ejpam-4279	7	17	whose	whose	DET
ejpam-4279	7	18	boundary	boundary	NOUN
ejpam-4279	7	19	is	be	AUX
ejpam-4279	7	20	vague	vague	ADJ
ejpam-4279	7	21	,	,	PUNCT
ejpam-4279	7	22	which	which	PRON
ejpam-4279	7	23	introduced	introduce	VERB
ejpam-4279	7	24	by	by	ADP
ejpam-4279	7	25	zadeh	zadeh	PROPN
ejpam-4279	7	26	in	in	ADP
ejpam-4279	7	27	1965	1965	NUM
ejpam-4279	7	28	[	[	X
ejpam-4279	7	29	9	9	NUM
ejpam-4279	7	30	]	]	PUNCT
ejpam-4279	7	31	.	.	PUNCT
ejpam-4279	8	1	fuzzy	fuzzy	ADJ
ejpam-4279	8	2	set	set	PROPN
ejpam-4279	8	3	theory	theory	NOUN
ejpam-4279	8	4	became	become	VERB
ejpam-4279	8	5	a	a	DET
ejpam-4279	8	6	phenomenon	phenomenon	NOUN
ejpam-4279	8	7	since	since	SCONJ
ejpam-4279	8	8	its	its	PRON
ejpam-4279	8	9	logic	logic	NOUN
ejpam-4279	8	10	can	can	AUX
ejpam-4279	8	11	deal	deal	VERB
ejpam-4279	8	12	with	with	ADP
ejpam-4279	8	13	information	information	NOUN
ejpam-4279	8	14	that	that	PRON
ejpam-4279	8	15	is	be	AUX
ejpam-4279	8	16	imprecise	imprecise	ADV
ejpam-4279	8	17	,	,	PUNCT
ejpam-4279	8	18	vague	vague	ADJ
ejpam-4279	8	19	,	,	PUNCT
ejpam-4279	8	20	partially	partially	ADV
ejpam-4279	8	21	true	true	ADJ
ejpam-4279	8	22	,	,	PUNCT
ejpam-4279	8	23	or	or	CCONJ
ejpam-4279	8	24	without	without	ADP
ejpam-4279	8	25	sharp	sharp	ADJ
ejpam-4279	8	26	boundaries	boundary	NOUN
ejpam-4279	8	27	.	.	PUNCT
ejpam-4279	9	1	the	the	DET
ejpam-4279	9	2	reader	reader	NOUN
ejpam-4279	9	3	may	may	AUX
ejpam-4279	9	4	for	for	ADP
ejpam-4279	9	5	a	a	DET
ejpam-4279	9	6	compilation	compilation	NOUN
ejpam-4279	9	7	of	of	ADP
ejpam-4279	9	8	articles	article	NOUN
ejpam-4279	9	9	on	on	ADP
ejpam-4279	9	10	fuzzy	fuzzy	ADJ
ejpam-4279	9	11	sets	set	NOUN
ejpam-4279	9	12	,	,	PUNCT
ejpam-4279	9	13	fuzzy	fuzzy	ADJ
ejpam-4279	9	14	logic	logic	NOUN
ejpam-4279	9	15	and	and	CCONJ
ejpam-4279	9	16	their	their	PRON
ejpam-4279	9	17	applications	application	NOUN
ejpam-4279	9	18	.	.	PUNCT
ejpam-4279	10	1	in	in	ADP
ejpam-4279	10	2	1979	1979	NUM
ejpam-4279	10	3	kuroki	kuroki	NOUN
ejpam-4279	10	4	used	use	VERB
ejpam-4279	10	5	fuzzy	fuzzy	ADJ
ejpam-4279	10	6	set	set	VERB
ejpam-4279	10	7	in	in	ADP
ejpam-4279	10	8	semigroup	semigroup	NOUN
ejpam-4279	10	9	and	and	CCONJ
ejpam-4279	10	10	chracterizations	chracterization	NOUN
ejpam-4279	10	11	properties	property	NOUN
ejpam-4279	10	12	of	of	ADP
ejpam-4279	10	13	fuzzy	fuzzy	ADJ
ejpam-4279	10	14	semigroup	semigroup	NOUN
ejpam-4279	10	15	.	.	PUNCT
ejpam-4279	11	1	the	the	DET
ejpam-4279	11	2	almost	almost	ADV
ejpam-4279	11	3	ideal	ideal	ADJ
ejpam-4279	11	4	theory	theory	NOUN
ejpam-4279	11	5	in	in	ADP
ejpam-4279	11	6	semigroups	semigroup	NOUN
ejpam-4279	11	7	studied	study	VERB
ejpam-4279	11	8	by	by	ADP
ejpam-4279	11	9	grosek	grosek	NOUN
ejpam-4279	11	10	and	and	CCONJ
ejpam-4279	11	11	satko	satko	NOUN
ejpam-4279	11	12	in	in	ADP
ejpam-4279	11	13	1980	1980	NUM
ejpam-4279	11	14	[	[	X
ejpam-4279	11	15	3	3	NUM
ejpam-4279	11	16	]	]	PUNCT
ejpam-4279	11	17	.	.	PUNCT
ejpam-4279	12	1	in	in	ADP
ejpam-4279	12	2	1981	1981	NUM
ejpam-4279	12	3	,	,	PUNCT
ejpam-4279	12	4	bogdanvic	bogdanvic	VERB
ejpam-4279	12	5	,	,	PUNCT
ejpam-4279	12	6	[	[	X
ejpam-4279	12	7	4	4	X
ejpam-4279	12	8	]	]	PUNCT
ejpam-4279	12	9	established	establish	VERB
ejpam-4279	12	10	definitions	definition	NOUN
ejpam-4279	12	11	of	of	ADP
ejpam-4279	12	12	almost	almost	ADV
ejpam-4279	12	13	bi	bi	NOUN
ejpam-4279	12	14	-	-	NOUN
ejpam-4279	12	15	ideals	ideal	NOUN
ejpam-4279	12	16	in	in	ADP
ejpam-4279	12	17	semigroups	semigroup	NOUN
ejpam-4279	12	18	and	and	CCONJ
ejpam-4279	12	19	studies	study	NOUN
ejpam-4279	12	20	properties	property	NOUN
ejpam-4279	12	21	of	of	ADP
ejpam-4279	12	22	almost	almost	ADV
ejpam-4279	12	23	bi	bi	NOUN
ejpam-4279	12	24	-	-	NOUN
ejpam-4279	12	25	ideals	ideal	NOUN
ejpam-4279	12	26	in	in	ADP
ejpam-4279	12	27	semigroups	semigroup	NOUN
ejpam-4279	12	28	.	.	PUNCT
ejpam-4279	13	1	later	later	ADV
ejpam-4279	13	2	,	,	PUNCT
ejpam-4279	13	3	chinram	chinram	PROPN
ejpam-4279	13	4	give	give	VERB
ejpam-4279	13	5	definition	definition	NOUN
ejpam-4279	13	6	the	the	DET
ejpam-4279	13	7	definitions	definition	NOUN
ejpam-4279	13	8	of	of	ADP
ejpam-4279	13	9	types	type	NOUN
ejpam-4279	13	10	of	of	ADP
ejpam-4279	13	11	alomst	alomst	ADJ
ejpam-4279	13	12	ideals	ideal	NOUN
ejpam-4279	13	13	in	in	ADP
ejpam-4279	13	14	semigroups	semigroup	NOUN
ejpam-4279	13	15	such	such	ADJ
ejpam-4279	13	16	that	that	SCONJ
ejpam-4279	13	17	almost	almost	ADV
ejpam-4279	13	18	quasi	quasi	ADJ
ejpam-4279	13	19	-	-	NOUN
ejpam-4279	13	20	ideal	ideal	ADJ
ejpam-4279	13	21	[	[	X
ejpam-4279	13	22	7	7	NUM
ejpam-4279	13	23	]	]	PUNCT
ejpam-4279	13	24	.	.	PUNCT
ejpam-4279	14	1	in	in	ADP
ejpam-4279	14	2	2020	2020	NUM
ejpam-4279	14	3	,	,	PUNCT
ejpam-4279	14	4	n.	n.	PROPN
ejpam-4279	14	5	kaopusek	kaopusek	PROPN
ejpam-4279	14	6	et	et	PROPN
ejpam-4279	14	7	al	al	PROPN
ejpam-4279	14	8	.	.	PUNCT
ejpam-4279	15	1	[	[	X
ejpam-4279	15	2	6	6	NUM
ejpam-4279	15	3	]	]	PUNCT
ejpam-4279	15	4	discussed	discuss	VERB
ejpam-4279	15	5	almost	almost	ADV
ejpam-4279	15	6	interior	interior	ADJ
ejpam-4279	15	7	ideals	ideal	NOUN
ejpam-4279	15	8	and	and	CCONJ
ejpam-4279	15	9	weakly	weakly	ADJ
ejpam-4279	15	10	almost	almost	ADV
ejpam-4279	15	11	interior	interior	ADJ
ejpam-4279	15	12	ideals	ideal	NOUN
ejpam-4279	15	13	of	of	ADP
ejpam-4279	15	14	semigroups	semigroup	NOUN
ejpam-4279	15	15	by	by	ADP
ejpam-4279	15	16	using	use	VERB
ejpam-4279	15	17	the	the	DET
ejpam-4279	15	18	concepts	concept	NOUN
ejpam-4279	15	19	of	of	ADP
ejpam-4279	15	20	almost	almost	ADV
ejpam-4279	15	21	ideals	ideal	NOUN
ejpam-4279	15	22	and	and	CCONJ
ejpam-4279	15	23	interior	interior	ADJ
ejpam-4279	15	24	ideals	ideal	NOUN
ejpam-4279	15	25	of	of	ADP
ejpam-4279	15	26	semigroups	semigroup	NOUN
ejpam-4279	15	27	and	and	CCONJ
ejpam-4279	15	28	investigated	investigate	VERB
ejpam-4279	15	29	their	their	PRON
ejpam-4279	15	30	properties	property	NOUN
ejpam-4279	15	31	.	.	PUNCT
ejpam-4279	16	1	recently	recently	ADV
ejpam-4279	16	2	,	,	PUNCT
ejpam-4279	16	3	r.	r.	PROPN
ejpam-4279	16	4	chinram	chinram	PROPN
ejpam-4279	16	5	and	and	CCONJ
ejpam-4279	16	6	w.	w.	PROPN
ejpam-4279	16	7	nakkhasen	nakkhasen	PROPN
ejpam-4279	17	1	[	[	X
ejpam-4279	17	2	2	2	NUM
ejpam-4279	17	3	]	]	PUNCT
ejpam-4279	17	4	sutdied	sutdie	VERB
ejpam-4279	17	5	properteis	properteis	NOUN
ejpam-4279	17	6	of	of	ADP
ejpam-4279	17	7	almost	almost	ADV
ejpam-4279	17	8	bi	bi	ADJ
ejpam-4279	17	9	-	-	ADJ
ejpam-4279	17	10	quasi	quasi	ADJ
ejpam-4279	17	11	-	-	ADJ
ejpam-4279	17	12	interior	interior	ADJ
ejpam-4279	17	13	ideals	ideal	NOUN
ejpam-4279	17	14	and	and	CCONJ
ejpam-4279	17	15	their	their	PRON
ejpam-4279	17	16	fuzzy	fuzzy	ADJ
ejpam-4279	17	17	bi	bi	ADJ
ejpam-4279	17	18	-	-	ADJ
ejpam-4279	17	19	interior	interior	ADJ
ejpam-4279	17	20	ideals	ideal	NOUN
ejpam-4279	17	21	in	in	ADP
ejpam-4279	17	22	semigroups	semigroup	NOUN
ejpam-4279	17	23	.	.	PUNCT
ejpam-4279	18	1	moreover	moreover	ADV
ejpam-4279	18	2	,	,	PUNCT
ejpam-4279	18	3	the	the	DET
ejpam-4279	18	4	concept	concept	NOUN
ejpam-4279	18	5	of	of	ADP
ejpam-4279	18	6	almost	almost	ADV
ejpam-4279	18	7	interior	interior	ADJ
ejpam-4279	18	8	ideal	ideal	NOUN
ejpam-4279	18	9	has	have	AUX
ejpam-4279	18	10	been	be	AUX
ejpam-4279	18	11	discussed	discuss	VERB
ejpam-4279	18	12	in	in	ADP
ejpam-4279	18	13	other	other	ADJ
ejpam-4279	18	14	research	research	NOUN
ejpam-4279	18	15	sucht	sucht	NOUN
ejpam-4279	18	16	that	that	SCONJ
ejpam-4279	19	1	[	[	X
ejpam-4279	19	2	1	1	NUM
ejpam-4279	19	3	]	]	PUNCT
ejpam-4279	19	4	,	,	PUNCT
ejpam-4279	19	5	[	[	X
ejpam-4279	19	6	8	8	NUM
ejpam-4279	19	7	]	]	PUNCT
ejpam-4279	19	8	.	.	PUNCT
ejpam-4279	20	1	krishna	krishna	PROPN
ejpam-4279	20	2	and	and	CCONJ
ejpam-4279	20	3	rao	rao	PROPN
ejpam-4279	20	4	gave	give	VERB
ejpam-4279	20	5	definition	definition	NOUN
ejpam-4279	20	6	of	of	ADP
ejpam-4279	20	7	bi	bi	ADJ
ejpam-4279	20	8	-	-	ADJ
ejpam-4279	20	9	interior	interior	ADJ
ejpam-4279	20	10	ideal	ideal	NOUN
ejpam-4279	20	11	in	in	ADP
ejpam-4279	20	12	semigroups	semigroup	NOUN
ejpam-4279	20	13	in	in	ADP
ejpam-4279	20	14	2018	2018	NUM
ejpam-4279	20	15	[	[	X
ejpam-4279	20	16	5	5	NUM
ejpam-4279	20	17	]	]	PUNCT
ejpam-4279	20	18	.	.	PUNCT
ejpam-4279	21	1	in	in	ADP
ejpam-4279	21	2	this	this	DET
ejpam-4279	21	3	paper	paper	NOUN
ejpam-4279	21	4	,	,	PUNCT
ejpam-4279	21	5	we	we	PRON
ejpam-4279	21	6	give	give	VERB
ejpam-4279	21	7	definition	definition	NOUN
ejpam-4279	21	8	of	of	ADP
ejpam-4279	21	9	almost	almost	ADV
ejpam-4279	21	10	bi	bi	ADJ
ejpam-4279	21	11	-	-	ADJ
ejpam-4279	21	12	interior	interior	ADJ
ejpam-4279	21	13	ideal	ideal	NOUN
ejpam-4279	21	14	and	and	CCONJ
ejpam-4279	21	15	fuzzy	fuzzy	ADJ
ejpam-4279	21	16	almost	almost	ADV
ejpam-4279	21	17	bi	bi	ADJ
ejpam-4279	21	18	-	-	ADJ
ejpam-4279	21	19	interior	interior	ADJ
ejpam-4279	21	20	ideal	ideal	NOUN
ejpam-4279	21	21	in	in	ADP
ejpam-4279	21	22	semigroups	semigroup	NOUN
ejpam-4279	21	23	.	.	PUNCT
ejpam-4279	22	1	moreover	moreover	ADV
ejpam-4279	22	2	,	,	PUNCT
ejpam-4279	22	3	we	we	PRON
ejpam-4279	22	4	prove	prove	VERB
ejpam-4279	22	5	that	that	SCONJ
ejpam-4279	22	6	relation	relation	NOUN
ejpam-4279	22	7	between	between	ADP
ejpam-4279	22	8	almost	almost	ADV
ejpam-4279	22	9	bi	bi	ADJ
ejpam-4279	22	10	-	-	ADJ
ejpam-4279	22	11	interior	interior	ADJ
ejpam-4279	22	12	ideal	ideal	NOUN
ejpam-4279	22	13	and	and	CCONJ
ejpam-4279	22	14	fuzzy	fuzzy	ADJ
ejpam-4279	22	15	almost	almost	ADV
ejpam-4279	22	16	bi	bi	ADJ
ejpam-4279	22	17	-	-	ADJ
ejpam-4279	22	18	interior	interior	ADJ
ejpam-4279	22	19	ideal	ideal	NOUN
ejpam-4279	22	20	.	.	PUNCT
ejpam-4279	23	1	doi	doi	NOUN
ejpam-4279	23	2	:	:	PUNCT
ejpam-4279	23	3	https://doi.org/10.29020/nybg.ejpam.v15i1.4279	https://doi.org/10.29020/nybg.ejpam.v15i1.4279	ADJ
ejpam-4279	23	4	email	email	NOUN
ejpam-4279	23	5	address	address	NOUN
ejpam-4279	23	6	:	:	PUNCT
ejpam-4279	23	7	thiti.ga@up.ac.th	thiti.ga@up.ac.th	PROPN
ejpam-4279	23	8	(	(	PUNCT
ejpam-4279	23	9	t.	t.	NOUN
ejpam-4279	23	10	gaketem	gaketem	PROPN
ejpam-4279	23	11	)	)	PUNCT
ejpam-4279	23	12	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-4279	24	1	281	281	NUM
ejpam-4279	24	2	©	©	ADP
ejpam-4279	24	3	2022	2022	NUM
ejpam-4279	24	4	ejpam	ejpam	VERB
ejpam-4279	24	5	all	all	DET
ejpam-4279	24	6	rights	right	NOUN
ejpam-4279	24	7	reserved	reserve	VERB
ejpam-4279	24	8	.	.	PUNCT
ejpam-4279	25	1	t.	t.	PROPN
ejpam-4279	25	2	gaketem	gaketem	PROPN
ejpam-4279	25	3	/	/	SYM
ejpam-4279	25	4	eur	eur	PROPN
ejpam-4279	25	5	.	.	PUNCT
ejpam-4279	26	1	j.	j.	PROPN
ejpam-4279	26	2	pure	pure	PROPN
ejpam-4279	26	3	appl	appl	PROPN
ejpam-4279	26	4	.	.	PROPN
ejpam-4279	26	5	math	math	PROPN
ejpam-4279	26	6	,	,	PUNCT
ejpam-4279	26	7	15	15	NUM
ejpam-4279	26	8	(	(	PUNCT
ejpam-4279	26	9	1	1	NUM
ejpam-4279	26	10	)	)	PUNCT
ejpam-4279	26	11	(	(	PUNCT
ejpam-4279	26	12	2022	2022	NUM
ejpam-4279	26	13	)	)	PUNCT
ejpam-4279	26	14	,	,	PUNCT
ejpam-4279	26	15	281	281	NUM
ejpam-4279	26	16	-	-	SYM
ejpam-4279	26	17	289	289	NUM
ejpam-4279	26	18	282	282	NUM
ejpam-4279	26	19	2	2	NUM
ejpam-4279	26	20	.	.	PUNCT
ejpam-4279	26	21	preliminaries	preliminary	NOUN
ejpam-4279	26	22	in	in	ADP
ejpam-4279	26	23	this	this	DET
ejpam-4279	26	24	section	section	NOUN
ejpam-4279	26	25	we	we	PRON
ejpam-4279	26	26	give	give	VERB
ejpam-4279	26	27	some	some	DET
ejpam-4279	26	28	concepts	concept	NOUN
ejpam-4279	26	29	and	and	CCONJ
ejpam-4279	26	30	results	result	NOUN
ejpam-4279	26	31	,	,	PUNCT
ejpam-4279	26	32	which	which	PRON
ejpam-4279	26	33	will	will	AUX
ejpam-4279	26	34	be	be	AUX
ejpam-4279	26	35	helpful	helpful	ADJ
ejpam-4279	26	36	in	in	ADP
ejpam-4279	26	37	later	later	ADJ
ejpam-4279	26	38	sections	section	NOUN
ejpam-4279	26	39	.	.	PUNCT
ejpam-4279	27	1	definition	definition	NOUN
ejpam-4279	27	2	1	1	NUM
ejpam-4279	27	3	.	.	PUNCT
ejpam-4279	28	1	[	[	X
ejpam-4279	28	2	5	5	NUM
ejpam-4279	28	3	]	]	PUNCT
ejpam-4279	28	4	a	a	DET
ejpam-4279	28	5	non	non	ADJ
ejpam-4279	28	6	-	-	ADJ
ejpam-4279	28	7	empty	empty	ADJ
ejpam-4279	28	8	subset	subset	NOUN
ejpam-4279	28	9	m	m	PROPN
ejpam-4279	28	10	of	of	ADP
ejpam-4279	28	11	semigroup	semigroup	PROPN
ejpam-4279	28	12	s	s	PART
ejpam-4279	28	13	is	be	AUX
ejpam-4279	28	14	called	call	VERB
ejpam-4279	28	15	(	(	PUNCT
ejpam-4279	28	16	1	1	NUM
ejpam-4279	28	17	)	)	PUNCT
ejpam-4279	28	18	a	a	DET
ejpam-4279	28	19	subsemigroup	subsemigroup	NOUN
ejpam-4279	28	20	of	of	ADP
ejpam-4279	28	21	s	s	PRON
ejpam-4279	28	22	if	if	SCONJ
ejpam-4279	28	23	m2	m2	PROPN
ejpam-4279	28	24	⊆	⊆	NUM
ejpam-4279	28	25	m	m	NOUN
ejpam-4279	28	26	,	,	PUNCT
ejpam-4279	28	27	(	(	PUNCT
ejpam-4279	28	28	2	2	X
ejpam-4279	28	29	)	)	PUNCT
ejpam-4279	28	30	a	a	DET
ejpam-4279	28	31	left	left	ADJ
ejpam-4279	28	32	(	(	PUNCT
ejpam-4279	28	33	right	right	ADJ
ejpam-4279	28	34	)	)	PUNCT
ejpam-4279	28	35	ideal	ideal	NOUN
ejpam-4279	28	36	of	of	ADP
ejpam-4279	28	37	s	s	PRON
ejpam-4279	28	38	if	if	SCONJ
ejpam-4279	28	39	sm	sm	PROPN
ejpam-4279	28	40	⊆	⊆	NUM
ejpam-4279	28	41	m	m	VERB
ejpam-4279	28	42	(	(	PUNCT
ejpam-4279	28	43	ms	ms	PROPN
ejpam-4279	28	44	⊆	⊆	NUM
ejpam-4279	28	45	m	m	NOUN
ejpam-4279	28	46	)	)	PUNCT
ejpam-4279	28	47	.	.	PUNCT
ejpam-4279	29	1	by	by	ADP
ejpam-4279	29	2	an	an	DET
ejpam-4279	29	3	ideal	ideal	ADJ
ejpam-4279	29	4	m	m	NOUN
ejpam-4279	29	5	of	of	ADP
ejpam-4279	29	6	a	a	DET
ejpam-4279	29	7	semigroup	semigroup	NOUN
ejpam-4279	29	8	s	s	VERB
ejpam-4279	29	9	we	we	PRON
ejpam-4279	29	10	mean	mean	VERB
ejpam-4279	29	11	a	a	DET
ejpam-4279	29	12	left	left	ADJ
ejpam-4279	29	13	ideal	ideal	NOUN
ejpam-4279	29	14	and	and	CCONJ
ejpam-4279	29	15	a	a	DET
ejpam-4279	29	16	right	right	ADJ
ejpam-4279	29	17	ideal	ideal	NOUN
ejpam-4279	29	18	of	of	ADP
ejpam-4279	29	19	s	s	PROPN
ejpam-4279	29	20	,	,	PUNCT
ejpam-4279	29	21	(	(	PUNCT
ejpam-4279	29	22	3	3	X
ejpam-4279	29	23	)	)	PUNCT
ejpam-4279	29	24	a	a	DET
ejpam-4279	29	25	bi	bi	NOUN
ejpam-4279	29	26	-	-	NOUN
ejpam-4279	29	27	ideal	ideal	NOUN
ejpam-4279	29	28	of	of	ADP
ejpam-4279	29	29	s	s	PRON
ejpam-4279	29	30	if	if	SCONJ
ejpam-4279	29	31	m	m	NOUN
ejpam-4279	29	32	is	be	AUX
ejpam-4279	29	33	a	a	DET
ejpam-4279	29	34	subsemigroup	subsemigroup	NOUN
ejpam-4279	29	35	of	of	ADP
ejpam-4279	29	36	s	s	PRON
ejpam-4279	29	37	and	and	CCONJ
ejpam-4279	29	38	msm	msm	VERB
ejpam-4279	29	39	⊆	⊆	NUM
ejpam-4279	29	40	m	m	NOUN
ejpam-4279	29	41	,	,	PUNCT
ejpam-4279	29	42	(	(	PUNCT
ejpam-4279	29	43	4	4	X
ejpam-4279	29	44	)	)	PUNCT
ejpam-4279	29	45	an	an	DET
ejpam-4279	29	46	interior	interior	ADJ
ejpam-4279	29	47	ideal	ideal	NOUN
ejpam-4279	29	48	of	of	ADP
ejpam-4279	29	49	s	s	PRON
ejpam-4279	29	50	if	if	SCONJ
ejpam-4279	29	51	m	m	NOUN
ejpam-4279	29	52	is	be	AUX
ejpam-4279	29	53	a	a	DET
ejpam-4279	29	54	subsemigroup	subsemigroup	NOUN
ejpam-4279	29	55	of	of	ADP
ejpam-4279	29	56	s	s	NOUN
ejpam-4279	29	57	and	and	CCONJ
ejpam-4279	29	58	sms	sm	VERB
ejpam-4279	29	59	⊆	⊆	NUM
ejpam-4279	29	60	m	m	NOUN
ejpam-4279	29	61	,	,	PUNCT
ejpam-4279	29	62	(	(	PUNCT
ejpam-4279	29	63	5	5	X
ejpam-4279	29	64	)	)	PUNCT
ejpam-4279	29	65	a	a	DET
ejpam-4279	29	66	quasi	quasi	NOUN
ejpam-4279	29	67	-	-	NOUN
ejpam-4279	29	68	ideal	ideal	ADJ
ejpam-4279	29	69	of	of	ADP
ejpam-4279	29	70	s	s	PRON
ejpam-4279	29	71	if	if	SCONJ
ejpam-4279	29	72	ms	ms	PROPN
ejpam-4279	29	73	∩	∩	PROPN
ejpam-4279	29	74	sm	sm	VERB
ejpam-4279	29	75	⊆	⊆	NUM
ejpam-4279	29	76	m	m	NOUN
ejpam-4279	29	77	,	,	PUNCT
ejpam-4279	29	78	(	(	PUNCT
ejpam-4279	29	79	6	6	X
ejpam-4279	29	80	)	)	PUNCT
ejpam-4279	29	81	a	a	DET
ejpam-4279	29	82	left	left	ADJ
ejpam-4279	29	83	(	(	PUNCT
ejpam-4279	29	84	right	right	ADJ
ejpam-4279	29	85	)	)	PUNCT
ejpam-4279	29	86	almost	almost	ADV
ejpam-4279	29	87	ideal	ideal	ADJ
ejpam-4279	29	88	of	of	ADP
ejpam-4279	29	89	s	s	PRON
ejpam-4279	29	90	if	if	SCONJ
ejpam-4279	29	91	am	be	AUX
ejpam-4279	29	92	∩m	∩m	PROPN
ejpam-4279	29	93	̸=	̸=	PROPN
ejpam-4279	29	94	∅	∅	NOUN
ejpam-4279	29	95	(	(	PUNCT
ejpam-4279	29	96	ma	ma	PROPN
ejpam-4279	29	97	∩m	∩m	PROPN
ejpam-4279	29	98	̸=	̸=	PROPN
ejpam-4279	29	99	∅	∅	NOUN
ejpam-4279	29	100	)	)	PUNCT
ejpam-4279	29	101	for	for	ADP
ejpam-4279	29	102	all	all	DET
ejpam-4279	29	103	a	a	DET
ejpam-4279	29	104	∈	∈	NOUN
ejpam-4279	29	105	s.	s.	PROPN
ejpam-4279	29	106	by	by	ADP
ejpam-4279	29	107	an	an	DET
ejpam-4279	29	108	almost	almost	ADV
ejpam-4279	29	109	ideal	ideal	ADJ
ejpam-4279	29	110	m	m	NOUN
ejpam-4279	29	111	of	of	ADP
ejpam-4279	29	112	a	a	DET
ejpam-4279	29	113	semigroup	semigroup	NOUN
ejpam-4279	29	114	s	s	VERB
ejpam-4279	29	115	we	we	PRON
ejpam-4279	29	116	mean	mean	VERB
ejpam-4279	29	117	a	a	DET
ejpam-4279	29	118	left	left	NOUN
ejpam-4279	29	119	almost	almost	ADV
ejpam-4279	29	120	ideal	ideal	ADJ
ejpam-4279	29	121	and	and	CCONJ
ejpam-4279	29	122	a	a	DET
ejpam-4279	29	123	right	right	NOUN
ejpam-4279	29	124	almost	almost	ADV
ejpam-4279	29	125	ideal	ideal	ADJ
ejpam-4279	29	126	of	of	ADP
ejpam-4279	29	127	s	s	PROPN
ejpam-4279	29	128	,	,	PUNCT
ejpam-4279	29	129	(	(	PUNCT
ejpam-4279	29	130	7	7	X
ejpam-4279	29	131	)	)	PUNCT
ejpam-4279	29	132	a	a	DET
ejpam-4279	29	133	almost	almost	ADV
ejpam-4279	29	134	bi	bi	NOUN
ejpam-4279	29	135	-	-	NOUN
ejpam-4279	29	136	ideal	ideal	NOUN
ejpam-4279	29	137	of	of	ADP
ejpam-4279	29	138	s	s	PRON
ejpam-4279	29	139	if	if	SCONJ
ejpam-4279	29	140	mam	mam	PROPN
ejpam-4279	29	141	∩m	∩m	PROPN
ejpam-4279	29	142	̸=	̸=	PROPN
ejpam-4279	29	143	∅	∅	NOUN
ejpam-4279	29	144	for	for	ADP
ejpam-4279	29	145	all	all	DET
ejpam-4279	29	146	a	a	DET
ejpam-4279	29	147	∈	∈	ADJ
ejpam-4279	29	148	s	s	NOUN
ejpam-4279	29	149	,	,	PUNCT
ejpam-4279	29	150	(	(	PUNCT
ejpam-4279	29	151	8)	8)	NUM
ejpam-4279	29	152	a	a	DET
ejpam-4279	29	153	almost	almost	ADV
ejpam-4279	29	154	interior	interior	ADJ
ejpam-4279	29	155	ideal	ideal	NOUN
ejpam-4279	29	156	of	of	ADP
ejpam-4279	29	157	s	s	PRON
ejpam-4279	29	158	if	if	SCONJ
ejpam-4279	29	159	amb	amb	PROPN
ejpam-4279	29	160	∩m	∩m	PROPN
ejpam-4279	29	161	̸=	̸=	PROPN
ejpam-4279	29	162	∅	∅	NOUN
ejpam-4279	29	163	for	for	ADP
ejpam-4279	29	164	all	all	DET
ejpam-4279	29	165	a	a	PRON
ejpam-4279	29	166	,	,	PUNCT
ejpam-4279	29	167	b	b	PROPN
ejpam-4279	29	168	∈	∈	PROPN
ejpam-4279	29	169	s.	s.	PROPN
ejpam-4279	29	170	(	(	PUNCT
ejpam-4279	29	171	9	9	X
ejpam-4279	29	172	)	)	PUNCT
ejpam-4279	29	173	a	a	DET
ejpam-4279	29	174	almost	almost	ADV
ejpam-4279	29	175	quasi	quasi	ADJ
ejpam-4279	29	176	ideal	ideal	NOUN
ejpam-4279	29	177	of	of	ADP
ejpam-4279	29	178	s	s	PRON
ejpam-4279	29	179	if	if	SCONJ
ejpam-4279	29	180	(	(	PUNCT
ejpam-4279	29	181	am	be	AUX
ejpam-4279	29	182	∩ma	∩ma	ADJ
ejpam-4279	29	183	)	)	PUNCT
ejpam-4279	30	1	∩m	∩m	PROPN
ejpam-4279	30	2	̸=	̸=	PROPN
ejpam-4279	30	3	∅	∅	NOUN
ejpam-4279	30	4	for	for	ADP
ejpam-4279	30	5	all	all	DET
ejpam-4279	30	6	a	a	PRON
ejpam-4279	30	7	,	,	PUNCT
ejpam-4279	30	8	b	b	X
ejpam-4279	30	9	∈	∈	PROPN
ejpam-4279	30	10	s.	s.	PROPN
ejpam-4279	31	1	a	a	DET
ejpam-4279	31	2	subsemigroup	subsemigroup	PROPN
ejpam-4279	31	3	m	m	PROPN
ejpam-4279	31	4	of	of	ADP
ejpam-4279	31	5	a	a	DET
ejpam-4279	31	6	semigroup	semigroup	NOUN
ejpam-4279	31	7	s	s	NOUN
ejpam-4279	31	8	is	be	AUX
ejpam-4279	31	9	said	say	VERB
ejpam-4279	31	10	to	to	PART
ejpam-4279	31	11	be	be	AUX
ejpam-4279	31	12	left	leave	VERB
ejpam-4279	31	13	(	(	PUNCT
ejpam-4279	31	14	right	right	ADJ
ejpam-4279	31	15	)	)	PUNCT
ejpam-4279	31	16	bi	bi	ADJ
ejpam-4279	31	17	-	-	ADJ
ejpam-4279	31	18	quasi	quasi	ADJ
ejpam-4279	31	19	ideal	ideal	NOUN
ejpam-4279	31	20	of	of	ADP
ejpam-4279	31	21	s	s	PRON
ejpam-4279	31	22	if	if	SCONJ
ejpam-4279	31	23	sm	sm	PROPN
ejpam-4279	31	24	∩msm	∩msm	NOUN
ejpam-4279	31	25	⊆	⊆	NUM
ejpam-4279	31	26	m(ms	m(ms	PROPN
ejpam-4279	31	27	∩msm	∩msm	NOUN
ejpam-4279	31	28	⊆	⊆	NUM
ejpam-4279	31	29	m	m	NOUN
ejpam-4279	31	30	)	)	PUNCT
ejpam-4279	31	31	.	.	PUNCT
ejpam-4279	32	1	a	a	DET
ejpam-4279	32	2	subsemigroup	subsemigroup	NOUN
ejpam-4279	32	3	m	m	PROPN
ejpam-4279	32	4	of	of	ADP
ejpam-4279	32	5	a	a	DET
ejpam-4279	32	6	semigroup	semigroup	NOUN
ejpam-4279	32	7	s	s	NOUN
ejpam-4279	32	8	is	be	AUX
ejpam-4279	32	9	said	say	VERB
ejpam-4279	32	10	to	to	PART
ejpam-4279	32	11	be	be	AUX
ejpam-4279	32	12	bi	bi	ADJ
ejpam-4279	32	13	-	-	ADJ
ejpam-4279	32	14	quasi	quasi	ADJ
ejpam-4279	32	15	ideal	ideal	NOUN
ejpam-4279	32	16	of	of	ADP
ejpam-4279	32	17	s	s	PRON
ejpam-4279	32	18	if	if	SCONJ
ejpam-4279	32	19	it	it	PRON
ejpam-4279	32	20	is	be	AUX
ejpam-4279	32	21	both	both	CCONJ
ejpam-4279	32	22	a	a	DET
ejpam-4279	32	23	left	left	ADJ
ejpam-4279	32	24	bi	bi	NOUN
ejpam-4279	32	25	-	-	NOUN
ejpam-4279	32	26	quasi	quasi	ADJ
ejpam-4279	32	27	and	and	CCONJ
ejpam-4279	32	28	right	right	ADJ
ejpam-4279	32	29	bi	bi	ADJ
ejpam-4279	32	30	-	-	ADJ
ejpam-4279	32	31	quasi	quasi	ADJ
ejpam-4279	32	32	ideal	ideal	NOUN
ejpam-4279	32	33	of	of	ADP
ejpam-4279	32	34	s.	s.	PROPN
ejpam-4279	32	35	a	a	DET
ejpam-4279	32	36	subsemigroup	subsemigroup	NOUN
ejpam-4279	32	37	m	m	PROPN
ejpam-4279	32	38	of	of	ADP
ejpam-4279	32	39	a	a	DET
ejpam-4279	32	40	semigroup	semigroup	NOUN
ejpam-4279	32	41	s	s	NOUN
ejpam-4279	32	42	is	be	AUX
ejpam-4279	32	43	said	say	VERB
ejpam-4279	32	44	to	to	PART
ejpam-4279	32	45	be	be	AUX
ejpam-4279	32	46	bi	bi	ADJ
ejpam-4279	32	47	-	-	ADJ
ejpam-4279	32	48	interior	interior	ADJ
ejpam-4279	32	49	ideal	ideal	NOUN
ejpam-4279	32	50	of	of	ADP
ejpam-4279	32	51	s	s	PRON
ejpam-4279	32	52	if	if	SCONJ
ejpam-4279	32	53	m	m	NOUN
ejpam-4279	32	54	is	be	AUX
ejpam-4279	32	55	a	a	DET
ejpam-4279	32	56	subsemigroup	subsemigroup	NOUN
ejpam-4279	32	57	of	of	ADP
ejpam-4279	32	58	s	s	NOUN
ejpam-4279	32	59	and	and	CCONJ
ejpam-4279	32	60	sms	sm	VERB
ejpam-4279	32	61	∩msm	∩msm	NOUN
ejpam-4279	32	62	⊆	⊆	NUM
ejpam-4279	32	63	m	m	NOUN
ejpam-4279	32	64	.	.	PUNCT
ejpam-4279	33	1	[	[	X
ejpam-4279	33	2	5	5	NUM
ejpam-4279	33	3	]	]	PUNCT
ejpam-4279	33	4	.	.	PUNCT
ejpam-4279	34	1	we	we	PRON
ejpam-4279	34	2	note	note	VERB
ejpam-4279	34	3	here	here	ADV
ejpam-4279	34	4	that	that	SCONJ
ejpam-4279	34	5	the	the	DET
ejpam-4279	34	6	properties	property	NOUN
ejpam-4279	34	7	is	be	AUX
ejpam-4279	34	8	hold	hold	ADJ
ejpam-4279	34	9	:	:	PUNCT
ejpam-4279	34	10	(	(	PUNCT
ejpam-4279	34	11	1	1	X
ejpam-4279	34	12	)	)	PUNCT
ejpam-4279	34	13	every	every	DET
ejpam-4279	34	14	left	leave	VERB
ejpam-4279	34	15	ideal	ideal	NOUN
ejpam-4279	34	16	is	be	AUX
ejpam-4279	34	17	a	a	DET
ejpam-4279	34	18	bi	bi	ADJ
ejpam-4279	34	19	-	-	ADJ
ejpam-4279	34	20	interior	interior	ADJ
ejpam-4279	34	21	ideal	ideal	NOUN
ejpam-4279	34	22	of	of	ADP
ejpam-4279	34	23	s.	s.	PROPN
ejpam-4279	34	24	(	(	PUNCT
ejpam-4279	34	25	2	2	X
ejpam-4279	34	26	)	)	PUNCT
ejpam-4279	34	27	every	every	DET
ejpam-4279	34	28	right	right	ADJ
ejpam-4279	34	29	ideal	ideal	NOUN
ejpam-4279	34	30	is	be	AUX
ejpam-4279	34	31	a	a	DET
ejpam-4279	34	32	bi	bi	ADJ
ejpam-4279	34	33	-	-	ADJ
ejpam-4279	34	34	interior	interior	ADJ
ejpam-4279	34	35	ideal	ideal	NOUN
ejpam-4279	34	36	of	of	ADP
ejpam-4279	34	37	s.	s.	PROPN
ejpam-4279	34	38	(	(	PUNCT
ejpam-4279	34	39	3	3	X
ejpam-4279	34	40	)	)	PUNCT
ejpam-4279	34	41	every	every	DET
ejpam-4279	34	42	ideal	ideal	NOUN
ejpam-4279	34	43	is	be	AUX
ejpam-4279	34	44	a	a	DET
ejpam-4279	34	45	bi	bi	ADJ
ejpam-4279	34	46	-	-	ADJ
ejpam-4279	34	47	interior	interior	ADJ
ejpam-4279	34	48	ideal	ideal	NOUN
ejpam-4279	34	49	of	of	ADP
ejpam-4279	34	50	s.	s.	PROPN
ejpam-4279	34	51	(	(	PUNCT
ejpam-4279	34	52	4	4	X
ejpam-4279	34	53	)	)	PUNCT
ejpam-4279	34	54	every	every	DET
ejpam-4279	34	55	quasi	quasi	ADJ
ejpam-4279	34	56	ideal	ideal	NOUN
ejpam-4279	34	57	is	be	AUX
ejpam-4279	34	58	a	a	DET
ejpam-4279	34	59	bi	bi	ADJ
ejpam-4279	34	60	-	-	ADJ
ejpam-4279	34	61	interior	interior	ADJ
ejpam-4279	34	62	ideal	ideal	NOUN
ejpam-4279	34	63	of	of	ADP
ejpam-4279	34	64	s.	s.	PROPN
ejpam-4279	34	65	(	(	PUNCT
ejpam-4279	34	66	5	5	X
ejpam-4279	34	67	)	)	PUNCT
ejpam-4279	34	68	the	the	DET
ejpam-4279	34	69	arbitrary	arbitrary	ADJ
ejpam-4279	34	70	intersection	intersection	NOUN
ejpam-4279	34	71	of	of	ADP
ejpam-4279	34	72	bi	bi	ADJ
ejpam-4279	34	73	-	-	NOUN
ejpam-4279	34	74	interior	interior	ADJ
ejpam-4279	34	75	of	of	ADP
ejpam-4279	34	76	s	s	PROPN
ejpam-4279	34	77	is	be	AUX
ejpam-4279	34	78	also	also	ADV
ejpam-4279	34	79	bi	bi	ADJ
ejpam-4279	34	80	-	-	ADJ
ejpam-4279	34	81	interior	interior	ADJ
ejpam-4279	34	82	ideal	ideal	NOUN
ejpam-4279	34	83	of	of	ADP
ejpam-4279	34	84	s.	s.	PROPN
ejpam-4279	34	85	(	(	PUNCT
ejpam-4279	34	86	6	6	NUM
ejpam-4279	34	87	)	)	PUNCT
ejpam-4279	34	88	if	if	SCONJ
ejpam-4279	34	89	m	m	VERB
ejpam-4279	34	90	a	a	DET
ejpam-4279	34	91	bi	bi	ADJ
ejpam-4279	34	92	-	-	ADJ
ejpam-4279	34	93	interior	interior	ADJ
ejpam-4279	34	94	ideal	ideal	NOUN
ejpam-4279	34	95	of	of	ADP
ejpam-4279	34	96	s	s	PROPN
ejpam-4279	34	97	,	,	PUNCT
ejpam-4279	34	98	then	then	ADV
ejpam-4279	34	99	ms	ms	PROPN
ejpam-4279	34	100	and	and	CCONJ
ejpam-4279	34	101	sm	sm	PROPN
ejpam-4279	34	102	are	be	AUX
ejpam-4279	34	103	bi	bi	ADJ
ejpam-4279	34	104	-	-	ADJ
ejpam-4279	34	105	interior	interior	ADJ
ejpam-4279	34	106	ideals	ideal	NOUN
ejpam-4279	34	107	of	of	ADP
ejpam-4279	34	108	s.	s.	PROPN
ejpam-4279	34	109	for	for	ADP
ejpam-4279	34	110	any	any	DET
ejpam-4279	34	111	hi	hi	NOUN
ejpam-4279	34	112	∈	∈	PROPN
ejpam-4279	35	1	[	[	X
ejpam-4279	35	2	0	0	NUM
ejpam-4279	35	3	,	,	PUNCT
ejpam-4279	35	4	1	1	NUM
ejpam-4279	35	5	]	]	PUNCT
ejpam-4279	35	6	,	,	PUNCT
ejpam-4279	35	7	i	i	PROPN
ejpam-4279	35	8	∈	∈	PROPN
ejpam-4279	35	9	f	f	PROPN
ejpam-4279	35	10	,	,	PUNCT
ejpam-4279	35	11	define	define	VERB
ejpam-4279	35	12	∨	∨	NUM
ejpam-4279	35	13	i∈f	i∈f	VERB
ejpam-4279	35	14	hi	hi	INTJ
ejpam-4279	35	15	:	:	PUNCT
ejpam-4279	35	16	=	=	NOUN
ejpam-4279	35	17	sup	sup	NOUN
ejpam-4279	35	18	i∈f	i∈f	VERB
ejpam-4279	35	19	{	{	PUNCT
ejpam-4279	35	20	hi	hi	INTJ
ejpam-4279	35	21	}	}	PUNCT
ejpam-4279	35	22	and	and	CCONJ
ejpam-4279	35	23	∧	∧	PROPN
ejpam-4279	35	24	i∈f	i∈f	VERB
ejpam-4279	35	25	hi	hi	INTJ
ejpam-4279	35	26	:	:	PUNCT
ejpam-4279	35	27	=	=	SYM
ejpam-4279	35	28	inf	inf	PROPN
ejpam-4279	35	29	i∈f	i∈f	VERB
ejpam-4279	35	30	{	{	PUNCT
ejpam-4279	35	31	hi	hi	INTJ
ejpam-4279	35	32	}	}	PUNCT
ejpam-4279	35	33	.	.	PUNCT
ejpam-4279	36	1	we	we	PRON
ejpam-4279	36	2	see	see	VERB
ejpam-4279	36	3	that	that	PRON
ejpam-4279	36	4	for	for	ADP
ejpam-4279	36	5	any	any	DET
ejpam-4279	36	6	h	h	NOUN
ejpam-4279	36	7	,	,	PUNCT
ejpam-4279	36	8	r	r	NOUN
ejpam-4279	36	9	∈	∈	PROPN
ejpam-4279	37	1	[	[	X
ejpam-4279	37	2	0	0	NUM
ejpam-4279	37	3	,	,	PUNCT
ejpam-4279	37	4	1	1	NUM
ejpam-4279	37	5	]	]	PUNCT
ejpam-4279	37	6	,	,	PUNCT
ejpam-4279	37	7	we	we	PRON
ejpam-4279	37	8	have	have	VERB
ejpam-4279	37	9	t.	t.	PROPN
ejpam-4279	37	10	gaketem	gaketem	PROPN
ejpam-4279	37	11	/	/	SYM
ejpam-4279	37	12	eur	eur	PROPN
ejpam-4279	37	13	.	.	PUNCT
ejpam-4279	38	1	j.	j.	PROPN
ejpam-4279	38	2	pure	pure	PROPN
ejpam-4279	38	3	appl	appl	PROPN
ejpam-4279	38	4	.	.	PROPN
ejpam-4279	38	5	math	math	PROPN
ejpam-4279	38	6	,	,	PUNCT
ejpam-4279	38	7	15	15	NUM
ejpam-4279	38	8	(	(	PUNCT
ejpam-4279	38	9	1	1	NUM
ejpam-4279	38	10	)	)	PUNCT
ejpam-4279	38	11	(	(	PUNCT
ejpam-4279	38	12	2022	2022	NUM
ejpam-4279	38	13	)	)	PUNCT
ejpam-4279	38	14	,	,	PUNCT
ejpam-4279	38	15	281	281	NUM
ejpam-4279	38	16	-	-	SYM
ejpam-4279	38	17	289	289	NUM
ejpam-4279	38	18	283	283	NUM
ejpam-4279	38	19	h	h	NOUN
ejpam-4279	38	20	∨	∨	NOUN
ejpam-4279	38	21	r	r	NOUN
ejpam-4279	38	22	=	=	SYM
ejpam-4279	38	23	max{h	max{h	PROPN
ejpam-4279	38	24	,	,	PUNCT
ejpam-4279	38	25	r	r	NOUN
ejpam-4279	38	26	}	}	PUNCT
ejpam-4279	38	27	and	and	CCONJ
ejpam-4279	38	28	h	h	NOUN
ejpam-4279	38	29	∧	∧	NOUN
ejpam-4279	38	30	r	r	NOUN
ejpam-4279	38	31	=	=	SYM
ejpam-4279	38	32	min{h	min{h	ADJ
ejpam-4279	38	33	,	,	PUNCT
ejpam-4279	38	34	r	r	NOUN
ejpam-4279	38	35	}	}	PUNCT
ejpam-4279	38	36	.	.	PUNCT
ejpam-4279	39	1	a	a	DET
ejpam-4279	39	2	fuzzy	fuzzy	ADJ
ejpam-4279	39	3	set	set	NOUN
ejpam-4279	39	4	(	(	PUNCT
ejpam-4279	39	5	fuzzy	fuzzy	ADJ
ejpam-4279	39	6	subset	subset	NOUN
ejpam-4279	39	7	)	)	PUNCT
ejpam-4279	39	8	of	of	ADP
ejpam-4279	39	9	a	a	DET
ejpam-4279	39	10	non	non	ADJ
ejpam-4279	39	11	-	-	ADJ
ejpam-4279	39	12	empty	empty	ADJ
ejpam-4279	39	13	set	set	NOUN
ejpam-4279	39	14	e	e	NOUN
ejpam-4279	39	15	is	be	AUX
ejpam-4279	39	16	a	a	DET
ejpam-4279	39	17	function	function	NOUN
ejpam-4279	39	18	φ	φ	NOUN
ejpam-4279	39	19	:	:	PUNCT
ejpam-4279	40	1	e	e	X
ejpam-4279	40	2	→	→	PUNCT
ejpam-4279	40	3	[	[	X
ejpam-4279	40	4	0	0	NUM
ejpam-4279	40	5	,	,	PUNCT
ejpam-4279	40	6	1	1	NUM
ejpam-4279	40	7	]	]	PUNCT
ejpam-4279	40	8	.	.	PUNCT
ejpam-4279	41	1	for	for	ADP
ejpam-4279	41	2	any	any	DET
ejpam-4279	41	3	two	two	NUM
ejpam-4279	41	4	fuzzy	fuzzy	ADJ
ejpam-4279	41	5	sets	set	NOUN
ejpam-4279	41	6	φ	φ	PROPN
ejpam-4279	41	7	and	and	CCONJ
ejpam-4279	41	8	ξ	ξ	PROPN
ejpam-4279	41	9	of	of	ADP
ejpam-4279	41	10	a	a	DET
ejpam-4279	41	11	non	non	ADJ
ejpam-4279	41	12	-	-	ADJ
ejpam-4279	41	13	empty	empty	ADJ
ejpam-4279	41	14	set	set	ADJ
ejpam-4279	41	15	e	e	NOUN
ejpam-4279	41	16	,	,	PUNCT
ejpam-4279	41	17	define	define	VERB
ejpam-4279	41	18	the	the	DET
ejpam-4279	41	19	symbol	symbol	NOUN
ejpam-4279	41	20	as	as	SCONJ
ejpam-4279	41	21	follows	follow	VERB
ejpam-4279	41	22	:	:	PUNCT
ejpam-4279	41	23	(	(	PUNCT
ejpam-4279	41	24	1	1	X
ejpam-4279	41	25	)	)	PUNCT
ejpam-4279	41	26	φ	φ	PROPN
ejpam-4279	41	27	≥	≥	PROPN
ejpam-4279	41	28	ξ	ξ	X
ejpam-4279	41	29	⇔	⇔	X
ejpam-4279	41	30	φ(h	φ(h	PROPN
ejpam-4279	41	31	)	)	PUNCT
ejpam-4279	41	32	≥	≥	NUM
ejpam-4279	41	33	ξ(h	ξ(h	NOUN
ejpam-4279	41	34	)	)	PUNCT
ejpam-4279	41	35	for	for	ADP
ejpam-4279	41	36	all	all	DET
ejpam-4279	41	37	h	h	NOUN
ejpam-4279	41	38	∈	∈	PROPN
ejpam-4279	41	39	e	e	NOUN
ejpam-4279	41	40	,	,	PUNCT
ejpam-4279	41	41	(	(	PUNCT
ejpam-4279	41	42	2	2	X
ejpam-4279	41	43	)	)	PUNCT
ejpam-4279	41	44	φ	φ	NOUN
ejpam-4279	41	45	=	=	SYM
ejpam-4279	41	46	ξ	ξ	PROPN
ejpam-4279	41	47	⇔	⇔	PROPN
ejpam-4279	41	48	φ	φ	PROPN
ejpam-4279	41	49	≥	≥	PROPN
ejpam-4279	41	50	ξ	ξ	PROPN
ejpam-4279	41	51	and	and	CCONJ
ejpam-4279	41	52	ξ	ξ	PROPN
ejpam-4279	41	53	≥	≥	X
ejpam-4279	41	54	φ	φ	NUM
ejpam-4279	41	55	,	,	PUNCT
ejpam-4279	41	56	(	(	PUNCT
ejpam-4279	41	57	3	3	NUM
ejpam-4279	41	58	)	)	PUNCT
ejpam-4279	41	59	(	(	PUNCT
ejpam-4279	41	60	φ	φ	PROPN
ejpam-4279	41	61	∧	∧	PROPN
ejpam-4279	41	62	ξ)(h	ξ)(h	NUM
ejpam-4279	41	63	)	)	PUNCT
ejpam-4279	41	64	=	=	SYM
ejpam-4279	42	1	min{φ(h	min{φ(h	PROPN
ejpam-4279	42	2	)	)	PUNCT
ejpam-4279	42	3	,	,	PUNCT
ejpam-4279	42	4	ξ(h	ξ(h	NOUN
ejpam-4279	42	5	)	)	PUNCT
ejpam-4279	42	6	}	}	PUNCT
ejpam-4279	42	7	=	=	SYM
ejpam-4279	42	8	φ(h	φ(h	ADJ
ejpam-4279	42	9	)	)	PUNCT
ejpam-4279	42	10	∧	∧	PROPN
ejpam-4279	42	11	ξ(h	ξ(h	PROPN
ejpam-4279	42	12	)	)	PUNCT
ejpam-4279	42	13	and	and	CCONJ
ejpam-4279	42	14	(	(	PUNCT
ejpam-4279	42	15	φ	φ	PROPN
ejpam-4279	42	16	∨	∨	NUM
ejpam-4279	42	17	ξ)(h	ξ)(h	NUM
ejpam-4279	42	18	)	)	PUNCT
ejpam-4279	42	19	=	=	SYM
ejpam-4279	42	20	max{φ(h	max{φ(h	PROPN
ejpam-4279	42	21	)	)	PUNCT
ejpam-4279	42	22	,	,	PUNCT
ejpam-4279	42	23	ξ(h	ξ(h	NOUN
ejpam-4279	42	24	)	)	PUNCT
ejpam-4279	42	25	}	}	PUNCT
ejpam-4279	42	26	=	=	SYM
ejpam-4279	42	27	φ(h	φ(h	NOUN
ejpam-4279	42	28	)	)	PUNCT
ejpam-4279	42	29	∨	∨	NUM
ejpam-4279	42	30	ξ(h	ξ(h	PROPN
ejpam-4279	42	31	)	)	PUNCT
ejpam-4279	42	32	for	for	ADP
ejpam-4279	42	33	all	all	DET
ejpam-4279	42	34	h	h	NOUN
ejpam-4279	42	35	∈	∈	PROPN
ejpam-4279	42	36	e	e	NOUN
ejpam-4279	42	37	,	,	PUNCT
ejpam-4279	42	38	(	(	PUNCT
ejpam-4279	42	39	4	4	X
ejpam-4279	42	40	)	)	PUNCT
ejpam-4279	42	41	φ	φ	NOUN
ejpam-4279	42	42	⊆	⊆	NUM
ejpam-4279	42	43	ξ	ξ	X
ejpam-4279	42	44	if	if	SCONJ
ejpam-4279	42	45	φ(h	φ(h	NOUN
ejpam-4279	42	46	)	)	PUNCT
ejpam-4279	42	47	≤	≤	NUM
ejpam-4279	42	48	ξ(h	ξ(h	NOUN
ejpam-4279	42	49	)	)	PUNCT
ejpam-4279	42	50	,	,	PUNCT
ejpam-4279	42	51	(	(	PUNCT
ejpam-4279	42	52	5	5	X
ejpam-4279	42	53	)	)	PUNCT
ejpam-4279	42	54	(	(	PUNCT
ejpam-4279	42	55	φ	φ	X
ejpam-4279	42	56	∪	∪	ADP
ejpam-4279	42	57	ξ)(h	ξ)(h	PROPN
ejpam-4279	42	58	)	)	PUNCT
ejpam-4279	42	59	=	=	SYM
ejpam-4279	42	60	max{φ(h	max{φ(h	PROPN
ejpam-4279	42	61	)	)	PUNCT
ejpam-4279	42	62	,	,	PUNCT
ejpam-4279	42	63	ξ(h	ξ(h	NOUN
ejpam-4279	42	64	)	)	PUNCT
ejpam-4279	42	65	}	}	PUNCT
ejpam-4279	42	66	and	and	CCONJ
ejpam-4279	42	67	(	(	PUNCT
ejpam-4279	42	68	φ	φ	PROPN
ejpam-4279	42	69	∩	∩	ADJ
ejpam-4279	42	70	ξ)(h	ξ)(h	NUM
ejpam-4279	42	71	)	)	PUNCT
ejpam-4279	42	72	=	=	SYM
ejpam-4279	43	1	min{φ(h	min{φ(h	PROPN
ejpam-4279	43	2	)	)	PUNCT
ejpam-4279	43	3	,	,	PUNCT
ejpam-4279	43	4	ξ(h	ξ(h	NOUN
ejpam-4279	43	5	)	)	PUNCT
ejpam-4279	43	6	}	}	PUNCT
ejpam-4279	43	7	for	for	ADP
ejpam-4279	43	8	all	all	DET
ejpam-4279	43	9	h	h	NOUN
ejpam-4279	43	10	∈	∈	PROPN
ejpam-4279	43	11	e.	e.	PROPN
ejpam-4279	43	12	(	(	PUNCT
ejpam-4279	43	13	6	6	NUM
ejpam-4279	43	14	)	)	PUNCT
ejpam-4279	43	15	the	the	DET
ejpam-4279	43	16	support	support	NOUN
ejpam-4279	43	17	of	of	ADP
ejpam-4279	43	18	φ	φ	PROPN
ejpam-4279	43	19	instead	instead	ADV
ejpam-4279	43	20	of	of	ADP
ejpam-4279	43	21	supp(φ	supp(φ	PROPN
ejpam-4279	43	22	)	)	PUNCT
ejpam-4279	43	23	=	=	PRON
ejpam-4279	43	24	{	{	PUNCT
ejpam-4279	43	25	h	h	NOUN
ejpam-4279	43	26	∈	∈	PROPN
ejpam-4279	43	27	e	e	NOUN
ejpam-4279	43	28	|	|	ADV
ejpam-4279	43	29	φ(h	φ(h	NOUN
ejpam-4279	43	30	)	)	PUNCT
ejpam-4279	43	31	̸=	̸=	PROPN
ejpam-4279	43	32	0	0	NUM
ejpam-4279	43	33	}	}	PUNCT
ejpam-4279	43	34	.	.	PUNCT
ejpam-4279	44	1	for	for	ADP
ejpam-4279	44	2	the	the	DET
ejpam-4279	44	3	symbol	symbol	NOUN
ejpam-4279	44	4	φ	φ	NOUN
ejpam-4279	44	5	≤	≤	PROPN
ejpam-4279	44	6	ξ	ξ	PUNCT
ejpam-4279	44	7	,	,	PUNCT
ejpam-4279	44	8	we	we	PRON
ejpam-4279	44	9	mean	mean	VERB
ejpam-4279	44	10	ξ	ξ	PRON
ejpam-4279	44	11	≥	≥	X
ejpam-4279	44	12	φ	φ	NOUN
ejpam-4279	44	13	.	.	PUNCT
ejpam-4279	45	1	for	for	ADP
ejpam-4279	45	2	any	any	DET
ejpam-4279	45	3	two	two	NUM
ejpam-4279	45	4	fuzzy	fuzzy	ADJ
ejpam-4279	45	5	sets	set	NOUN
ejpam-4279	45	6	φ	φ	PROPN
ejpam-4279	45	7	and	and	CCONJ
ejpam-4279	45	8	ξ	ξ	PROPN
ejpam-4279	45	9	of	of	ADP
ejpam-4279	45	10	a	a	DET
ejpam-4279	45	11	semigroup	semigroup	PROPN
ejpam-4279	45	12	s.	s.	PROPN
ejpam-4279	45	13	the	the	DET
ejpam-4279	45	14	product	product	NOUN
ejpam-4279	45	15	of	of	ADP
ejpam-4279	45	16	fuzzy	fuzzy	ADJ
ejpam-4279	45	17	subsets	subset	NOUN
ejpam-4279	45	18	φ	φ	PROPN
ejpam-4279	45	19	and	and	CCONJ
ejpam-4279	45	20	ξ	ξ	PROPN
ejpam-4279	45	21	of	of	ADP
ejpam-4279	45	22	s	s	PRON
ejpam-4279	45	23	is	be	AUX
ejpam-4279	45	24	defined	define	VERB
ejpam-4279	45	25	as	as	ADP
ejpam-4279	45	26	follow	follow	NOUN
ejpam-4279	45	27	,	,	PUNCT
ejpam-4279	45	28	for	for	ADP
ejpam-4279	45	29	all	all	DET
ejpam-4279	45	30	h	h	NOUN
ejpam-4279	45	31	∈	∈	NOUN
ejpam-4279	45	32	s	s	X
ejpam-4279	45	33	(	(	PUNCT
ejpam-4279	45	34	φ	φ	X
ejpam-4279	45	35	◦	◦	NOUN
ejpam-4279	45	36	ξ)(h	ξ)(h	NUM
ejpam-4279	45	37	)	)	PUNCT
ejpam-4279	46	1	=	=	PUNCT
ejpam-4279	47	1			PROPN
ejpam-4279	47	2	∨	∨	NOUN
ejpam-4279	47	3	h	h	NOUN
ejpam-4279	47	4	=	=	SYM
ejpam-4279	47	5	yz	yz	X
ejpam-4279	47	6	{	{	PUNCT
ejpam-4279	47	7	φ(y	φ(y	PROPN
ejpam-4279	47	8	)	)	PUNCT
ejpam-4279	47	9	∧	∧	PROPN
ejpam-4279	47	10	ξ(z	ξ(z	PROPN
ejpam-4279	47	11	)	)	PUNCT
ejpam-4279	47	12	}	}	PUNCT
ejpam-4279	47	13	if	if	SCONJ
ejpam-4279	47	14	h	h	NOUN
ejpam-4279	47	15	=	=	SYM
ejpam-4279	47	16	yz	yz	PROPN
ejpam-4279	47	17	,	,	PUNCT
ejpam-4279	47	18	0	0	NUM
ejpam-4279	47	19	otherwise	otherwise	ADV
ejpam-4279	47	20	.	.	PUNCT
ejpam-4279	48	1	the	the	DET
ejpam-4279	48	2	characteristic	characteristic	ADJ
ejpam-4279	48	3	function	function	NOUN
ejpam-4279	48	4	of	of	ADP
ejpam-4279	48	5	a	a	DET
ejpam-4279	48	6	subset	subset	NOUN
ejpam-4279	48	7	m	m	NOUN
ejpam-4279	48	8	of	of	ADP
ejpam-4279	48	9	a	a	DET
ejpam-4279	48	10	nonempty	nonempty	ADJ
ejpam-4279	48	11	set	set	NOUN
ejpam-4279	48	12	s	s	NOUN
ejpam-4279	48	13	is	be	AUX
ejpam-4279	48	14	a	a	DET
ejpam-4279	48	15	fuzzy	fuzzy	ADJ
ejpam-4279	48	16	set	set	NOUN
ejpam-4279	48	17	of	of	ADP
ejpam-4279	48	18	s	s	PRON
ejpam-4279	48	19	λm	λm	X
ejpam-4279	48	20	(	(	PUNCT
ejpam-4279	48	21	h	h	NOUN
ejpam-4279	48	22	)	)	PUNCT
ejpam-4279	48	23	=	=	PRON
ejpam-4279	48	24	{	{	PUNCT
ejpam-4279	48	25	1	1	NUM
ejpam-4279	48	26	if	if	SCONJ
ejpam-4279	48	27	h	h	NOUN
ejpam-4279	48	28	∈	∈	PROPN
ejpam-4279	48	29	m	m	VERB
ejpam-4279	48	30	0	0	NUM
ejpam-4279	49	1	if	if	SCONJ
ejpam-4279	49	2	h	h	NOUN
ejpam-4279	49	3	/∈	/∈	PUNCT
ejpam-4279	49	4	m.	m.	NOUN
ejpam-4279	49	5	for	for	ADP
ejpam-4279	49	6	all	all	DET
ejpam-4279	49	7	h	h	NOUN
ejpam-4279	49	8	∈	∈	PROPN
ejpam-4279	49	9	s.	s.	PROPN
ejpam-4279	49	10	definition	definition	NOUN
ejpam-4279	49	11	2	2	NUM
ejpam-4279	49	12	.	.	PUNCT
ejpam-4279	50	1	[	[	X
ejpam-4279	50	2	5	5	NUM
ejpam-4279	50	3	]	]	PUNCT
ejpam-4279	50	4	a	a	DET
ejpam-4279	50	5	fuzzy	fuzzy	ADJ
ejpam-4279	50	6	set	set	VERB
ejpam-4279	50	7	φ	φ	PROPN
ejpam-4279	50	8	of	of	ADP
ejpam-4279	50	9	a	a	DET
ejpam-4279	50	10	semigroup	semigroup	NOUN
ejpam-4279	50	11	s	s	NOUN
ejpam-4279	50	12	is	be	AUX
ejpam-4279	50	13	said	say	VERB
ejpam-4279	50	14	to	to	PART
ejpam-4279	50	15	be	be	AUX
ejpam-4279	50	16	(	(	PUNCT
ejpam-4279	50	17	1	1	X
ejpam-4279	50	18	)	)	PUNCT
ejpam-4279	50	19	a	a	DET
ejpam-4279	50	20	fuzzy	fuzzy	ADJ
ejpam-4279	50	21	subsemigroup	subsemigroup	NOUN
ejpam-4279	50	22	of	of	ADP
ejpam-4279	50	23	s	s	PRON
ejpam-4279	50	24	if	if	SCONJ
ejpam-4279	50	25	φ(hr	φ(hr	NOUN
ejpam-4279	50	26	)	)	PUNCT
ejpam-4279	50	27	≥	≥	NOUN
ejpam-4279	50	28	φ(h	φ(h	NOUN
ejpam-4279	50	29	)	)	PUNCT
ejpam-4279	50	30	∧	∧	PROPN
ejpam-4279	50	31	φ(r	φ(r	PROPN
ejpam-4279	50	32	)	)	PUNCT
ejpam-4279	50	33	,	,	PUNCT
ejpam-4279	50	34	for	for	ADP
ejpam-4279	50	35	all	all	DET
ejpam-4279	50	36	h	h	NOUN
ejpam-4279	50	37	,	,	PUNCT
ejpam-4279	50	38	r	r	NOUN
ejpam-4279	50	39	∈	∈	PROPN
ejpam-4279	50	40	s	s	PART
ejpam-4279	50	41	,	,	PUNCT
ejpam-4279	50	42	(	(	PUNCT
ejpam-4279	50	43	2	2	X
ejpam-4279	50	44	)	)	PUNCT
ejpam-4279	50	45	a	a	DET
ejpam-4279	50	46	fuzzy	fuzzy	ADJ
ejpam-4279	50	47	left	left	NOUN
ejpam-4279	50	48	(	(	PUNCT
ejpam-4279	50	49	right	right	ADJ
ejpam-4279	50	50	)	)	PUNCT
ejpam-4279	50	51	ideal	ideal	NOUN
ejpam-4279	50	52	of	of	ADP
ejpam-4279	50	53	s	s	PRON
ejpam-4279	50	54	if	if	SCONJ
ejpam-4279	50	55	φ(hr	φ(hr	NOUN
ejpam-4279	50	56	)	)	PUNCT
ejpam-4279	50	57	≥	≥	NOUN
ejpam-4279	50	58	φ(r	φ(r	ADJ
ejpam-4279	50	59	)	)	PUNCT
ejpam-4279	50	60	(	(	PUNCT
ejpam-4279	50	61	φ(hr	φ(hr	NOUN
ejpam-4279	50	62	)	)	PUNCT
ejpam-4279	50	63	≥	≥	NOUN
ejpam-4279	50	64	φ(h	φ(h	NOUN
ejpam-4279	50	65	)	)	PUNCT
ejpam-4279	50	66	)	)	PUNCT
ejpam-4279	50	67	,	,	PUNCT
ejpam-4279	50	68	for	for	ADP
ejpam-4279	50	69	all	all	DET
ejpam-4279	50	70	h	h	NOUN
ejpam-4279	50	71	,	,	PUNCT
ejpam-4279	50	72	r	r	NOUN
ejpam-4279	50	73	∈	∈	PROPN
ejpam-4279	50	74	s.	s.	PROPN
ejpam-4279	50	75	a	a	DET
ejpam-4279	50	76	fuzzy	fuzzy	ADJ
ejpam-4279	50	77	ideal	ideal	NOUN
ejpam-4279	50	78	of	of	ADP
ejpam-4279	50	79	s	s	PRON
ejpam-4279	50	80	if	if	SCONJ
ejpam-4279	50	81	it	it	PRON
ejpam-4279	50	82	is	be	AUX
ejpam-4279	50	83	both	both	CCONJ
ejpam-4279	50	84	a	a	DET
ejpam-4279	50	85	fuzzy	fuzzy	ADJ
ejpam-4279	50	86	left	leave	VERB
ejpam-4279	50	87	ideal	ideal	NOUN
ejpam-4279	50	88	and	and	CCONJ
ejpam-4279	50	89	a	a	DET
ejpam-4279	50	90	fuzzy	fuzzy	ADJ
ejpam-4279	50	91	right	right	ADJ
ejpam-4279	50	92	ideal	ideal	NOUN
ejpam-4279	50	93	of	of	ADP
ejpam-4279	50	94	s	s	PROPN
ejpam-4279	50	95	,	,	PUNCT
ejpam-4279	50	96	(	(	PUNCT
ejpam-4279	50	97	3	3	X
ejpam-4279	50	98	)	)	PUNCT
ejpam-4279	50	99	a	a	DET
ejpam-4279	50	100	fuzzy	fuzzy	ADJ
ejpam-4279	50	101	bi	bi	NOUN
ejpam-4279	50	102	-	-	NOUN
ejpam-4279	50	103	ideal	ideal	NOUN
ejpam-4279	50	104	of	of	ADP
ejpam-4279	50	105	s	s	PRON
ejpam-4279	50	106	if	if	SCONJ
ejpam-4279	50	107	φ	φ	PROPN
ejpam-4279	50	108	is	be	AUX
ejpam-4279	50	109	a	a	DET
ejpam-4279	50	110	fuzzy	fuzzy	ADJ
ejpam-4279	50	111	subsemigroup	subsemigroup	NOUN
ejpam-4279	50	112	of	of	ADP
ejpam-4279	50	113	s	s	NOUN
ejpam-4279	50	114	and	and	CCONJ
ejpam-4279	50	115	φ(hrk	φ(hrk	NOUN
ejpam-4279	50	116	)	)	PUNCT
ejpam-4279	50	117	≥	≥	NOUN
ejpam-4279	50	118	φ(h	φ(h	NOUN
ejpam-4279	50	119	)	)	PUNCT
ejpam-4279	50	120	∧	∧	PROPN
ejpam-4279	50	121	φ(k	φ(k	PROPN
ejpam-4279	50	122	)	)	PUNCT
ejpam-4279	50	123	for	for	ADP
ejpam-4279	50	124	all	all	DET
ejpam-4279	50	125	h	h	NOUN
ejpam-4279	50	126	,	,	PUNCT
ejpam-4279	50	127	r	r	NOUN
ejpam-4279	50	128	,	,	PUNCT
ejpam-4279	50	129	k	k	PROPN
ejpam-4279	50	130	∈	∈	PROPN
ejpam-4279	50	131	s	s	PART
ejpam-4279	50	132	,	,	PUNCT
ejpam-4279	50	133	(	(	PUNCT
ejpam-4279	50	134	4	4	X
ejpam-4279	50	135	)	)	PUNCT
ejpam-4279	50	136	a	a	DET
ejpam-4279	50	137	fuzzy	fuzzy	ADJ
ejpam-4279	50	138	interior	interior	ADJ
ejpam-4279	50	139	ideal	ideal	NOUN
ejpam-4279	50	140	of	of	ADP
ejpam-4279	50	141	s	s	PRON
ejpam-4279	50	142	if	if	SCONJ
ejpam-4279	50	143	φ	φ	PROPN
ejpam-4279	50	144	is	be	AUX
ejpam-4279	50	145	a	a	DET
ejpam-4279	50	146	fuzzy	fuzzy	ADJ
ejpam-4279	50	147	subsemigroup	subsemigroup	NOUN
ejpam-4279	50	148	of	of	ADP
ejpam-4279	50	149	s	s	NOUN
ejpam-4279	50	150	and	and	CCONJ
ejpam-4279	50	151	φ(hrk	φ(hrk	NOUN
ejpam-4279	50	152	)	)	PUNCT
ejpam-4279	50	153	≥	≥	NOUN
ejpam-4279	50	154	φ(r	φ(r	ADJ
ejpam-4279	50	155	)	)	PUNCT
ejpam-4279	50	156	for	for	ADP
ejpam-4279	50	157	all	all	DET
ejpam-4279	50	158	h	h	NOUN
ejpam-4279	50	159	,	,	PUNCT
ejpam-4279	50	160	r	r	NOUN
ejpam-4279	50	161	,	,	PUNCT
ejpam-4279	50	162	k	k	PROPN
ejpam-4279	50	163	∈	∈	PROPN
ejpam-4279	50	164	s	s	PART
ejpam-4279	50	165	,	,	PUNCT
ejpam-4279	50	166	(	(	PUNCT
ejpam-4279	50	167	5	5	X
ejpam-4279	50	168	)	)	PUNCT
ejpam-4279	50	169	a	a	DET
ejpam-4279	50	170	fuzzy	fuzzy	ADJ
ejpam-4279	50	171	quasi	quasi	NOUN
ejpam-4279	50	172	-	-	NOUN
ejpam-4279	50	173	ideal	ideal	ADJ
ejpam-4279	50	174	of	of	ADP
ejpam-4279	50	175	s	s	PRON
ejpam-4279	50	176	if	if	SCONJ
ejpam-4279	50	177	φ(h	φ(h	NOUN
ejpam-4279	50	178	)	)	PUNCT
ejpam-4279	50	179	≥	≥	NUM
ejpam-4279	50	180	(	(	PUNCT
ejpam-4279	50	181	s	s	AUX
ejpam-4279	50	182	◦	◦	NOUN
ejpam-4279	50	183	ϑ)(h	ϑ)(h	NOUN
ejpam-4279	50	184	)	)	PUNCT
ejpam-4279	50	185	∧	∧	PROPN
ejpam-4279	50	186	(	(	PUNCT
ejpam-4279	50	187	φ	φ	PROPN
ejpam-4279	50	188	◦	◦	PROPN
ejpam-4279	50	189	s)(h	s)(h	NOUN
ejpam-4279	50	190	)	)	PUNCT
ejpam-4279	50	191	for	for	ADP
ejpam-4279	50	192	all	all	DET
ejpam-4279	50	193	h	h	NOUN
ejpam-4279	50	194	∈	∈	NOUN
ejpam-4279	50	195	s	s	VERB
ejpam-4279	50	196	where	where	SCONJ
ejpam-4279	50	197	s	s	NOUN
ejpam-4279	50	198	is	be	AUX
ejpam-4279	50	199	a	a	DET
ejpam-4279	50	200	fuzzy	fuzzy	ADJ
ejpam-4279	50	201	subset	subset	NOUN
ejpam-4279	50	202	of	of	ADP
ejpam-4279	50	203	s	s	PRON
ejpam-4279	50	204	mapping	map	VERB
ejpam-4279	50	205	every	every	DET
ejpam-4279	50	206	element	element	NOUN
ejpam-4279	50	207	of	of	ADP
ejpam-4279	50	208	s	s	PRON
ejpam-4279	50	209	to	to	PART
ejpam-4279	50	210	1	1	NUM
ejpam-4279	50	211	,	,	PUNCT
ejpam-4279	50	212	t.	t.	PROPN
ejpam-4279	50	213	gaketem	gaketem	PROPN
ejpam-4279	50	214	/	/	SYM
ejpam-4279	50	215	eur	eur	PROPN
ejpam-4279	50	216	.	.	PUNCT
ejpam-4279	51	1	j.	j.	PROPN
ejpam-4279	51	2	pure	pure	PROPN
ejpam-4279	51	3	appl	appl	PROPN
ejpam-4279	51	4	.	.	PROPN
ejpam-4279	51	5	math	math	PROPN
ejpam-4279	51	6	,	,	PUNCT
ejpam-4279	51	7	15	15	NUM
ejpam-4279	51	8	(	(	PUNCT
ejpam-4279	51	9	1	1	NUM
ejpam-4279	51	10	)	)	PUNCT
ejpam-4279	51	11	(	(	PUNCT
ejpam-4279	51	12	2022	2022	NUM
ejpam-4279	51	13	)	)	PUNCT
ejpam-4279	51	14	,	,	PUNCT
ejpam-4279	51	15	281	281	NUM
ejpam-4279	51	16	-	-	SYM
ejpam-4279	51	17	289	289	NUM
ejpam-4279	51	18	284	284	NUM
ejpam-4279	51	19	(	(	PUNCT
ejpam-4279	51	20	6	6	NUM
ejpam-4279	51	21	)	)	PUNCT
ejpam-4279	51	22	a	a	DET
ejpam-4279	51	23	fuzzy	fuzzy	ADJ
ejpam-4279	51	24	bi	bi	ADJ
ejpam-4279	51	25	-	-	ADJ
ejpam-4279	51	26	interior	interior	ADJ
ejpam-4279	51	27	ideal	ideal	NOUN
ejpam-4279	51	28	of	of	ADP
ejpam-4279	51	29	s	s	PRON
ejpam-4279	51	30	if	if	SCONJ
ejpam-4279	51	31	(	(	PUNCT
ejpam-4279	51	32	λm	λm	ADP
ejpam-4279	51	33	◦	◦	NOUN
ejpam-4279	51	34	φ	φ	NUM
ejpam-4279	51	35	◦	◦	NOUN
ejpam-4279	51	36	λm	λm	NOUN
ejpam-4279	51	37	)	)	PUNCT
ejpam-4279	51	38	∧	∧	PROPN
ejpam-4279	51	39	(	(	PUNCT
ejpam-4279	51	40	φ	φ	PROPN
ejpam-4279	51	41	◦	◦	NOUN
ejpam-4279	51	42	λm	λm	ADP
ejpam-4279	51	43	◦	◦	NOUN
ejpam-4279	51	44	φ	φ	NUM
ejpam-4279	51	45	)	)	PUNCT
ejpam-4279	51	46	⊆	⊆	NUM
ejpam-4279	51	47	φ	φ	X
ejpam-4279	51	48	.	.	PUNCT
ejpam-4279	52	1	the	the	DET
ejpam-4279	52	2	definition	definition	NOUN
ejpam-4279	52	3	of	of	ADP
ejpam-4279	52	4	fuzzy	fuzzy	ADJ
ejpam-4279	52	5	point	point	NOUN
ejpam-4279	52	6	of	of	ADP
ejpam-4279	52	7	a	a	DET
ejpam-4279	52	8	set	set	NOUN
ejpam-4279	52	9	.	.	PUNCT
ejpam-4279	53	1	for	for	ADP
ejpam-4279	53	2	h	h	PRON
ejpam-4279	53	3	∈	∈	PROPN
ejpam-4279	53	4	s	s	PART
ejpam-4279	53	5	and	and	CCONJ
ejpam-4279	53	6	t	t	PROPN
ejpam-4279	53	7	∈	∈	PROPN
ejpam-4279	53	8	(	(	PUNCT
ejpam-4279	53	9	0	0	NUM
ejpam-4279	53	10	,	,	PUNCT
ejpam-4279	53	11	1	1	NUM
ejpam-4279	53	12	]	]	PUNCT
ejpam-4279	53	13	,	,	PUNCT
ejpam-4279	53	14	a	a	DET
ejpam-4279	53	15	fuzzy	fuzzy	ADJ
ejpam-4279	53	16	point	point	NOUN
ejpam-4279	53	17	pδ	pδ	INTJ
ejpam-4279	53	18	of	of	ADP
ejpam-4279	53	19	a	a	DET
ejpam-4279	53	20	set	set	NOUN
ejpam-4279	53	21	s	s	PART
ejpam-4279	53	22	is	be	AUX
ejpam-4279	53	23	a	a	DET
ejpam-4279	53	24	fuzzy	fuzzy	ADJ
ejpam-4279	53	25	subset	subset	NOUN
ejpam-4279	53	26	of	of	ADP
ejpam-4279	53	27	s	s	PRON
ejpam-4279	53	28	defined	define	VERB
ejpam-4279	53	29	by	by	ADP
ejpam-4279	53	30	pδ(h	pδ(h	NOUN
ejpam-4279	53	31	)	)	PUNCT
ejpam-4279	54	1	=	=	PRON
ejpam-4279	54	2	{	{	PUNCT
ejpam-4279	54	3	δ	δ	NOUN
ejpam-4279	54	4	if	if	SCONJ
ejpam-4279	54	5	h	h	NOUN
ejpam-4279	55	1	=	=	NOUN
ejpam-4279	55	2	r	r	NOUN
ejpam-4279	55	3	0	0	PUNCT
ejpam-4279	56	1	if	if	SCONJ
ejpam-4279	56	2	r	r	NOUN
ejpam-4279	56	3	̸=	̸=	PROPN
ejpam-4279	56	4	h.	h.	PROPN
ejpam-4279	56	5	definition	definition	NOUN
ejpam-4279	56	6	3	3	NUM
ejpam-4279	57	1	.	.	PUNCT
ejpam-4279	58	1	[	[	X
ejpam-4279	58	2	2	2	X
ejpam-4279	58	3	]	]	PUNCT
ejpam-4279	58	4	a	a	DET
ejpam-4279	58	5	fuzzy	fuzzy	ADJ
ejpam-4279	58	6	set	set	VERB
ejpam-4279	58	7	φ	φ	PROPN
ejpam-4279	58	8	of	of	ADP
ejpam-4279	58	9	a	a	DET
ejpam-4279	58	10	semigroup	semigroup	NOUN
ejpam-4279	58	11	s	s	NOUN
ejpam-4279	58	12	is	be	AUX
ejpam-4279	58	13	said	say	VERB
ejpam-4279	58	14	to	to	PART
ejpam-4279	58	15	be	be	AUX
ejpam-4279	58	16	(	(	PUNCT
ejpam-4279	58	17	1	1	X
ejpam-4279	58	18	)	)	PUNCT
ejpam-4279	58	19	a	a	DET
ejpam-4279	58	20	fuzzy	fuzzy	ADJ
ejpam-4279	58	21	left	left	NOUN
ejpam-4279	58	22	(	(	PUNCT
ejpam-4279	58	23	right	right	ADJ
ejpam-4279	58	24	)	)	PUNCT
ejpam-4279	58	25	almost	almost	ADV
ejpam-4279	58	26	ideal	ideal	ADJ
ejpam-4279	58	27	of	of	ADP
ejpam-4279	58	28	s	s	PRON
ejpam-4279	58	29	if	if	SCONJ
ejpam-4279	58	30	(	(	PUNCT
ejpam-4279	58	31	pδ	pδ	ADP
ejpam-4279	58	32	◦	◦	NOUN
ejpam-4279	58	33	φ	φ	NUM
ejpam-4279	58	34	)	)	PUNCT
ejpam-4279	58	35	∧	∧	PROPN
ejpam-4279	58	36	φ	φ	NUM
ejpam-4279	58	37	̸=	̸=	PROPN
ejpam-4279	58	38	0	0	NUM
ejpam-4279	58	39	(	(	PUNCT
ejpam-4279	58	40	(	(	PUNCT
ejpam-4279	58	41	φ	φ	PROPN
ejpam-4279	58	42	◦	◦	PROPN
ejpam-4279	58	43	pδ	pδ	NOUN
ejpam-4279	58	44	)	)	PUNCT
ejpam-4279	58	45	∧	∧	PROPN
ejpam-4279	58	46	φ	φ	NUM
ejpam-4279	58	47	̸=	̸=	PROPN
ejpam-4279	58	48	0	0	NUM
ejpam-4279	58	49	)	)	PUNCT
ejpam-4279	58	50	for	for	ADP
ejpam-4279	58	51	all	all	DET
ejpam-4279	58	52	fuzzy	fuzzy	ADJ
ejpam-4279	58	53	point	point	NOUN
ejpam-4279	58	54	pδ	pδ	PROPN
ejpam-4279	58	55	.	.	PUNCT
ejpam-4279	59	1	a	a	DET
ejpam-4279	59	2	fuzzy	fuzzy	ADJ
ejpam-4279	59	3	almost	almost	ADV
ejpam-4279	59	4	ideal	ideal	ADJ
ejpam-4279	59	5	of	of	ADP
ejpam-4279	59	6	s	s	PRON
ejpam-4279	59	7	if	if	SCONJ
ejpam-4279	59	8	it	it	PRON
ejpam-4279	59	9	is	be	AUX
ejpam-4279	59	10	both	both	CCONJ
ejpam-4279	59	11	a	a	DET
ejpam-4279	59	12	fuzzy	fuzzy	ADJ
ejpam-4279	59	13	left	leave	VERB
ejpam-4279	59	14	almost	almost	ADV
ejpam-4279	59	15	ideal	ideal	ADJ
ejpam-4279	59	16	and	and	CCONJ
ejpam-4279	59	17	a	a	DET
ejpam-4279	59	18	fuzzy	fuzzy	ADJ
ejpam-4279	59	19	right	right	NOUN
ejpam-4279	59	20	almost	almost	ADV
ejpam-4279	59	21	ideal	ideal	ADJ
ejpam-4279	59	22	of	of	ADP
ejpam-4279	59	23	s	s	PROPN
ejpam-4279	59	24	,	,	PUNCT
ejpam-4279	59	25	(	(	PUNCT
ejpam-4279	59	26	2	2	X
ejpam-4279	59	27	)	)	PUNCT
ejpam-4279	59	28	a	a	DET
ejpam-4279	59	29	fuzzy	fuzzy	ADJ
ejpam-4279	59	30	almost	almost	ADV
ejpam-4279	59	31	bi	bi	NOUN
ejpam-4279	59	32	-	-	NOUN
ejpam-4279	59	33	ideal	ideal	NOUN
ejpam-4279	59	34	of	of	ADP
ejpam-4279	59	35	s	s	PRON
ejpam-4279	59	36	if	if	SCONJ
ejpam-4279	59	37	(	(	PUNCT
ejpam-4279	59	38	pδ	pδ	ADP
ejpam-4279	59	39	◦	◦	NOUN
ejpam-4279	59	40	φ	φ	NUM
ejpam-4279	59	41	◦	◦	NOUN
ejpam-4279	59	42	pδ	pδ	NOUN
ejpam-4279	59	43	)	)	PUNCT
ejpam-4279	59	44	∧	∧	PROPN
ejpam-4279	59	45	φ	φ	NUM
ejpam-4279	59	46	̸=	̸=	PROPN
ejpam-4279	59	47	0	0	NUM
ejpam-4279	59	48	for	for	ADP
ejpam-4279	59	49	all	all	DET
ejpam-4279	59	50	fuzzy	fuzzy	ADJ
ejpam-4279	59	51	point	point	NOUN
ejpam-4279	59	52	pδ	pδ	NOUN
ejpam-4279	59	53	.	.	PUNCT
ejpam-4279	60	1	(	(	PUNCT
ejpam-4279	60	2	3	3	X
ejpam-4279	60	3	)	)	PUNCT
ejpam-4279	60	4	a	a	DET
ejpam-4279	60	5	fuzzy	fuzzy	ADJ
ejpam-4279	60	6	almost	almost	ADV
ejpam-4279	60	7	interior	interior	ADJ
ejpam-4279	60	8	ideal	ideal	NOUN
ejpam-4279	60	9	of	of	ADP
ejpam-4279	60	10	s	s	PRON
ejpam-4279	60	11	if	if	SCONJ
ejpam-4279	60	12	(	(	PUNCT
ejpam-4279	60	13	φ	φ	NUM
ejpam-4279	60	14	◦	◦	NOUN
ejpam-4279	60	15	pδ	pδ	ADP
ejpam-4279	60	16	◦	◦	NOUN
ejpam-4279	60	17	φ	φ	NUM
ejpam-4279	60	18	)	)	PUNCT
ejpam-4279	60	19	∧	∧	PROPN
ejpam-4279	60	20	φ	φ	NUM
ejpam-4279	60	21	̸=	̸=	PROPN
ejpam-4279	60	22	0	0	NUM
ejpam-4279	60	23	for	for	ADP
ejpam-4279	60	24	all	all	DET
ejpam-4279	60	25	fuzzy	fuzzy	ADJ
ejpam-4279	60	26	point	point	NOUN
ejpam-4279	60	27	pδ	pδ	PROPN
ejpam-4279	60	28	.	.	PUNCT
ejpam-4279	61	1	(	(	PUNCT
ejpam-4279	61	2	4	4	X
ejpam-4279	61	3	)	)	PUNCT
ejpam-4279	61	4	a	a	DET
ejpam-4279	61	5	fuzzy	fuzzy	ADJ
ejpam-4279	61	6	almost	almost	ADV
ejpam-4279	61	7	quasi	quasi	ADJ
ejpam-4279	61	8	-	-	NOUN
ejpam-4279	61	9	ideal	ideal	ADJ
ejpam-4279	61	10	of	of	ADP
ejpam-4279	61	11	s	s	PRON
ejpam-4279	61	12	if	if	SCONJ
ejpam-4279	61	13	[	[	X
ejpam-4279	61	14	(	(	PUNCT
ejpam-4279	61	15	pδ	pδ	ADP
ejpam-4279	61	16	◦	◦	NOUN
ejpam-4279	61	17	φ	φ	NUM
ejpam-4279	61	18	)	)	PUNCT
ejpam-4279	62	1	∧	∧	PROPN
ejpam-4279	62	2	(	(	PUNCT
ejpam-4279	62	3	φ	φ	PROPN
ejpam-4279	62	4	◦	◦	PROPN
ejpam-4279	62	5	pδ	pδ	PROPN
ejpam-4279	62	6	)	)	PUNCT
ejpam-4279	62	7	]	]	PUNCT
ejpam-4279	63	1	∧	∧	PROPN
ejpam-4279	63	2	φ	φ	NUM
ejpam-4279	63	3	̸=	̸=	PROPN
ejpam-4279	63	4	0	0	NUM
ejpam-4279	63	5	for	for	ADP
ejpam-4279	63	6	all	all	DET
ejpam-4279	63	7	fuzzy	fuzzy	ADJ
ejpam-4279	63	8	point	point	NOUN
ejpam-4279	63	9	pδ	pδ	PROPN
ejpam-4279	63	10	.	.	PROPN
ejpam-4279	63	11	3	3	X
ejpam-4279	63	12	.	.	X
ejpam-4279	63	13	almost	almost	ADV
ejpam-4279	63	14	bi	bi	ADJ
ejpam-4279	63	15	-	-	ADJ
ejpam-4279	63	16	interior	interior	ADJ
ejpam-4279	63	17	ideals	ideal	NOUN
ejpam-4279	63	18	in	in	ADP
ejpam-4279	63	19	semigroups	semigroup	NOUN
ejpam-4279	63	20	.	.	PUNCT
ejpam-4279	64	1	in	in	ADP
ejpam-4279	64	2	this	this	DET
ejpam-4279	64	3	section	section	NOUN
ejpam-4279	64	4	,	,	PUNCT
ejpam-4279	64	5	we	we	PRON
ejpam-4279	64	6	define	define	VERB
ejpam-4279	64	7	the	the	DET
ejpam-4279	64	8	notions	notion	NOUN
ejpam-4279	64	9	of	of	ADP
ejpam-4279	64	10	almost	almost	ADV
ejpam-4279	64	11	bi	bi	ADJ
ejpam-4279	64	12	-	-	ADJ
ejpam-4279	64	13	interior	interior	ADJ
ejpam-4279	64	14	ideals	ideal	NOUN
ejpam-4279	64	15	,	,	PUNCT
ejpam-4279	64	16	weakly	weakly	ADJ
ejpam-4279	64	17	almost	almost	ADV
ejpam-4279	64	18	biinterior	biinterior	ADJ
ejpam-4279	64	19	ideals	ideal	NOUN
ejpam-4279	64	20	in	in	ADP
ejpam-4279	64	21	ordered	order	VERB
ejpam-4279	64	22	semigroups	semigroup	NOUN
ejpam-4279	64	23	and	and	CCONJ
ejpam-4279	64	24	some	some	DET
ejpam-4279	64	25	properties	property	NOUN
ejpam-4279	64	26	of	of	ADP
ejpam-4279	64	27	them	they	PRON
ejpam-4279	64	28	are	be	AUX
ejpam-4279	64	29	investigated	investigate	VERB
ejpam-4279	64	30	.	.	PUNCT
ejpam-4279	65	1	definition	definition	NOUN
ejpam-4279	65	2	4	4	NUM
ejpam-4279	65	3	.	.	PUNCT
ejpam-4279	66	1	a	a	DET
ejpam-4279	66	2	nonempty	nonempty	ADV
ejpam-4279	66	3	set	set	VERB
ejpam-4279	66	4	m	m	PROPN
ejpam-4279	66	5	of	of	ADP
ejpam-4279	66	6	a	a	DET
ejpam-4279	66	7	semigroup	semigroup	NOUN
ejpam-4279	66	8	s	s	PART
ejpam-4279	66	9	is	be	AUX
ejpam-4279	66	10	called	call	VERB
ejpam-4279	66	11	a	a	DET
ejpam-4279	66	12	(	(	PUNCT
ejpam-4279	66	13	1	1	NUM
ejpam-4279	66	14	)	)	PUNCT
ejpam-4279	66	15	almost	almost	ADV
ejpam-4279	66	16	bi	bi	ADJ
ejpam-4279	66	17	-	-	ADJ
ejpam-4279	66	18	interior	interior	ADJ
ejpam-4279	66	19	ideal	ideal	NOUN
ejpam-4279	66	20	of	of	ADP
ejpam-4279	66	21	s	s	PRON
ejpam-4279	66	22	if	if	SCONJ
ejpam-4279	66	23	(	(	PUNCT
ejpam-4279	66	24	hmr	hmr	NOUN
ejpam-4279	66	25	∩mnm	∩mnm	PROPN
ejpam-4279	66	26	)	)	PUNCT
ejpam-4279	67	1	∩m	∩m	PROPN
ejpam-4279	67	2	̸=	̸=	PROPN
ejpam-4279	67	3	∅	∅	NOUN
ejpam-4279	67	4	,	,	PUNCT
ejpam-4279	67	5	for	for	ADP
ejpam-4279	67	6	all	all	DET
ejpam-4279	67	7	h	h	NOUN
ejpam-4279	67	8	,	,	PUNCT
ejpam-4279	67	9	r	r	NOUN
ejpam-4279	67	10	,	,	PUNCT
ejpam-4279	67	11	n	n	PROPN
ejpam-4279	67	12	∈	∈	NOUN
ejpam-4279	67	13	s	s	X
ejpam-4279	67	14	(	(	PUNCT
ejpam-4279	67	15	2	2	NUM
ejpam-4279	67	16	)	)	PUNCT
ejpam-4279	67	17	weakly	weakly	ADV
ejpam-4279	67	18	almost	almost	ADV
ejpam-4279	67	19	bi	bi	ADJ
ejpam-4279	67	20	-	-	ADJ
ejpam-4279	67	21	interior	interior	ADJ
ejpam-4279	67	22	ideal	ideal	NOUN
ejpam-4279	67	23	of	of	ADP
ejpam-4279	67	24	s	s	PRON
ejpam-4279	67	25	if	if	SCONJ
ejpam-4279	67	26	(	(	PUNCT
ejpam-4279	67	27	hmh	hmh	NOUN
ejpam-4279	67	28	∩mhm	∩mhm	NOUN
ejpam-4279	67	29	)	)	PUNCT
ejpam-4279	68	1	∩m	∩m	PROPN
ejpam-4279	68	2	̸=	̸=	PROPN
ejpam-4279	68	3	∅	∅	NOUN
ejpam-4279	68	4	,	,	PUNCT
ejpam-4279	68	5	for	for	ADP
ejpam-4279	68	6	all	all	DET
ejpam-4279	68	7	h	h	NOUN
ejpam-4279	68	8	∈	∈	PROPN
ejpam-4279	68	9	s.	s.	PROPN
ejpam-4279	68	10	theorem	theorem	VERB
ejpam-4279	68	11	1	1	X
ejpam-4279	68	12	.	.	PUNCT
ejpam-4279	69	1	let	let	VERB
ejpam-4279	69	2	s	s	PRON
ejpam-4279	69	3	be	be	AUX
ejpam-4279	69	4	a	a	DET
ejpam-4279	69	5	semigroup	semigroup	NOUN
ejpam-4279	69	6	.	.	PUNCT
ejpam-4279	70	1	then	then	ADV
ejpam-4279	70	2	the	the	DET
ejpam-4279	70	3	following	follow	VERB
ejpam-4279	70	4	statement	statement	NOUN
ejpam-4279	70	5	hold	hold	NOUN
ejpam-4279	70	6	.	.	PUNCT
ejpam-4279	71	1	(	(	PUNCT
ejpam-4279	71	2	1	1	X
ejpam-4279	71	3	)	)	PUNCT
ejpam-4279	71	4	every	every	DET
ejpam-4279	71	5	bi	bi	ADJ
ejpam-4279	71	6	-	-	ADJ
ejpam-4279	71	7	interior	interior	ADJ
ejpam-4279	71	8	ideal	ideal	NOUN
ejpam-4279	71	9	of	of	ADP
ejpam-4279	71	10	s	s	PROPN
ejpam-4279	71	11	is	be	AUX
ejpam-4279	71	12	an	an	DET
ejpam-4279	71	13	almost	almost	ADV
ejpam-4279	71	14	bi	bi	ADJ
ejpam-4279	71	15	-	-	ADJ
ejpam-4279	71	16	interior	interior	ADJ
ejpam-4279	71	17	ideal	ideal	NOUN
ejpam-4279	71	18	of	of	ADP
ejpam-4279	71	19	s.	s.	PROPN
ejpam-4279	71	20	(	(	PUNCT
ejpam-4279	71	21	2	2	X
ejpam-4279	71	22	)	)	PUNCT
ejpam-4279	71	23	every	every	DET
ejpam-4279	71	24	weak	weak	ADJ
ejpam-4279	71	25	bi	bi	ADJ
ejpam-4279	71	26	-	-	ADJ
ejpam-4279	71	27	interior	interior	ADJ
ejpam-4279	71	28	ideal	ideal	NOUN
ejpam-4279	71	29	of	of	ADP
ejpam-4279	71	30	s	s	PROPN
ejpam-4279	71	31	is	be	AUX
ejpam-4279	71	32	an	an	DET
ejpam-4279	71	33	weak	weak	ADJ
ejpam-4279	71	34	almost	almost	ADV
ejpam-4279	71	35	bi	bi	ADJ
ejpam-4279	71	36	-	-	ADJ
ejpam-4279	71	37	interior	interior	ADJ
ejpam-4279	71	38	ideal	ideal	NOUN
ejpam-4279	71	39	of	of	ADP
ejpam-4279	71	40	s.	s.	PROPN
ejpam-4279	71	41	proof	proof	PROPN
ejpam-4279	71	42	.	.	PUNCT
ejpam-4279	72	1	suppose	suppose	VERB
ejpam-4279	72	2	that	that	SCONJ
ejpam-4279	72	3	m	m	PROPN
ejpam-4279	72	4	is	be	AUX
ejpam-4279	72	5	a	a	DET
ejpam-4279	72	6	bi	bi	ADJ
ejpam-4279	72	7	-	-	ADJ
ejpam-4279	72	8	interior	interior	ADJ
ejpam-4279	72	9	ideal	ideal	NOUN
ejpam-4279	72	10	of	of	ADP
ejpam-4279	72	11	s	s	PRON
ejpam-4279	72	12	and	and	CCONJ
ejpam-4279	72	13	let	let	VERB
ejpam-4279	72	14	h	h	NOUN
ejpam-4279	72	15	,	,	PUNCT
ejpam-4279	72	16	r	r	NOUN
ejpam-4279	72	17	,	,	PUNCT
ejpam-4279	72	18	n	n	PROPN
ejpam-4279	72	19	∈	∈	PROPN
ejpam-4279	72	20	s.	s.	PROPN
ejpam-4279	72	21	then	then	ADV
ejpam-4279	72	22	(	(	PUNCT
ejpam-4279	72	23	hmr	hmr	PROPN
ejpam-4279	72	24	∩	∩	PROPN
ejpam-4279	72	25	mnm	mnm	ADJ
ejpam-4279	72	26	)	)	PUNCT
ejpam-4279	72	27	̸=	̸=	PROPN
ejpam-4279	72	28	∅.	∅.	ADP
ejpam-4279	72	29	thus	thus	ADV
ejpam-4279	72	30	(	(	PUNCT
ejpam-4279	72	31	hmr	hmr	PROPN
ejpam-4279	72	32	∩	∩	PROPN
ejpam-4279	72	33	mnm	mnm	PROPN
ejpam-4279	72	34	)	)	PUNCT
ejpam-4279	72	35	⊆	⊆	NUM
ejpam-4279	72	36	(	(	PUNCT
ejpam-4279	72	37	sms	sms	NOUN
ejpam-4279	72	38	∩	∩	NOUN
ejpam-4279	72	39	msm	msm	NOUN
ejpam-4279	72	40	)	)	PUNCT
ejpam-4279	72	41	⊆	⊆	NUM
ejpam-4279	72	42	m	m	NOUN
ejpam-4279	72	43	.	.	PUNCT
ejpam-4279	73	1	it	it	PRON
ejpam-4279	73	2	implies	imply	VERB
ejpam-4279	73	3	that	that	SCONJ
ejpam-4279	73	4	(	(	PUNCT
ejpam-4279	73	5	hmr	hmr	PROPN
ejpam-4279	73	6	∩	∩	PROPN
ejpam-4279	73	7	mnm	mnm	PROPN
ejpam-4279	73	8	)	)	PUNCT
ejpam-4279	73	9	∩m	∩m	PROPN
ejpam-4279	74	1	⊆	⊆	NUM
ejpam-4279	74	2	(	(	PUNCT
ejpam-4279	74	3	sms	sms	NOUN
ejpam-4279	74	4	∩msm	∩msm	NOUN
ejpam-4279	74	5	)	)	PUNCT
ejpam-4279	75	1	̸=	̸=	PROPN
ejpam-4279	75	2	∅.	∅.	PRON
ejpam-4279	75	3	hence	hence	ADV
ejpam-4279	75	4	m	m	VERB
ejpam-4279	75	5	is	be	AUX
ejpam-4279	75	6	an	an	DET
ejpam-4279	75	7	almost	almost	ADV
ejpam-4279	75	8	bi	bi	ADJ
ejpam-4279	75	9	-	-	ADJ
ejpam-4279	75	10	interior	interior	ADJ
ejpam-4279	75	11	ideal	ideal	NOUN
ejpam-4279	75	12	of	of	ADP
ejpam-4279	75	13	s.	s.	PROPN
ejpam-4279	75	14	the	the	DET
ejpam-4279	75	15	proof	proof	NOUN
ejpam-4279	75	16	of	of	ADP
ejpam-4279	75	17	the	the	DET
ejpam-4279	75	18	other	other	ADJ
ejpam-4279	75	19	similar	similar	ADJ
ejpam-4279	75	20	to	to	ADP
ejpam-4279	75	21	the	the	DET
ejpam-4279	75	22	proof	proof	NOUN
ejpam-4279	75	23	(	(	PUNCT
ejpam-4279	75	24	1	1	NUM
ejpam-4279	75	25	)	)	PUNCT
ejpam-4279	75	26	.	.	PUNCT
ejpam-4279	76	1	theorem	theorem	NOUN
ejpam-4279	76	2	2	2	NUM
ejpam-4279	76	3	.	.	PUNCT
ejpam-4279	77	1	let	let	AUX
ejpam-4279	77	2	m	m	PRON
ejpam-4279	77	3	and	and	CCONJ
ejpam-4279	77	4	l	l	NOUN
ejpam-4279	77	5	be	be	AUX
ejpam-4279	77	6	nonempty	nonempty	X
ejpam-4279	77	7	subsets	subset	NOUN
ejpam-4279	77	8	a	a	DET
ejpam-4279	77	9	semigroup	semigroup	NOUN
ejpam-4279	77	10	of	of	ADP
ejpam-4279	77	11	s	s	PRON
ejpam-4279	77	12	with	with	ADP
ejpam-4279	77	13	m	m	PROPN
ejpam-4279	77	14	⊆	⊆	NUM
ejpam-4279	77	15	l.	l.	NOUN
ejpam-4279	77	16	then	then	ADV
ejpam-4279	77	17	the	the	DET
ejpam-4279	77	18	following	follow	VERB
ejpam-4279	77	19	statement	statement	NOUN
ejpam-4279	77	20	hold	hold	NOUN
ejpam-4279	77	21	.	.	PUNCT
ejpam-4279	78	1	(	(	PUNCT
ejpam-4279	78	2	1	1	X
ejpam-4279	78	3	)	)	PUNCT
ejpam-4279	78	4	if	if	SCONJ
ejpam-4279	78	5	m	m	NOUN
ejpam-4279	78	6	is	be	AUX
ejpam-4279	78	7	an	an	DET
ejpam-4279	78	8	almost	almost	ADV
ejpam-4279	78	9	bi	bi	ADJ
ejpam-4279	78	10	-	-	ADJ
ejpam-4279	78	11	interior	interior	ADJ
ejpam-4279	78	12	ideal	ideal	NOUN
ejpam-4279	78	13	of	of	ADP
ejpam-4279	78	14	s	s	PROPN
ejpam-4279	78	15	,	,	PUNCT
ejpam-4279	78	16	then	then	ADV
ejpam-4279	78	17	l	l	NOUN
ejpam-4279	78	18	is	be	AUX
ejpam-4279	78	19	an	an	DET
ejpam-4279	78	20	almost	almost	ADV
ejpam-4279	78	21	bi	bi	ADJ
ejpam-4279	78	22	-	-	ADJ
ejpam-4279	78	23	interior	interior	ADJ
ejpam-4279	78	24	ideal	ideal	NOUN
ejpam-4279	78	25	of	of	ADP
ejpam-4279	78	26	s.	s.	PROPN
ejpam-4279	78	27	(	(	PUNCT
ejpam-4279	78	28	2	2	X
ejpam-4279	78	29	)	)	PUNCT
ejpam-4279	78	30	if	if	SCONJ
ejpam-4279	78	31	m	m	NOUN
ejpam-4279	78	32	is	be	AUX
ejpam-4279	78	33	a	a	DET
ejpam-4279	78	34	weak	weak	ADJ
ejpam-4279	78	35	almost	almost	ADV
ejpam-4279	78	36	bi	bi	ADJ
ejpam-4279	78	37	-	-	ADJ
ejpam-4279	78	38	interior	interior	ADJ
ejpam-4279	78	39	ideal	ideal	NOUN
ejpam-4279	78	40	of	of	ADP
ejpam-4279	78	41	s	s	PROPN
ejpam-4279	78	42	,	,	PUNCT
ejpam-4279	78	43	then	then	ADV
ejpam-4279	78	44	l	l	NOUN
ejpam-4279	78	45	is	be	AUX
ejpam-4279	78	46	a	a	DET
ejpam-4279	78	47	weak	weak	ADJ
ejpam-4279	78	48	almost	almost	ADV
ejpam-4279	78	49	bi	bi	ADJ
ejpam-4279	78	50	-	-	ADJ
ejpam-4279	78	51	interior	interior	ADJ
ejpam-4279	78	52	ideal	ideal	NOUN
ejpam-4279	78	53	of	of	ADP
ejpam-4279	78	54	s.	s.	PROPN
ejpam-4279	78	55	t.	t.	PROPN
ejpam-4279	78	56	gaketem	gaketem	PROPN
ejpam-4279	78	57	/	/	SYM
ejpam-4279	78	58	eur	eur	PROPN
ejpam-4279	78	59	.	.	PUNCT
ejpam-4279	79	1	j.	j.	PROPN
ejpam-4279	79	2	pure	pure	PROPN
ejpam-4279	79	3	appl	appl	PROPN
ejpam-4279	79	4	.	.	PROPN
ejpam-4279	79	5	math	math	PROPN
ejpam-4279	79	6	,	,	PUNCT
ejpam-4279	79	7	15	15	NUM
ejpam-4279	79	8	(	(	PUNCT
ejpam-4279	79	9	1	1	NUM
ejpam-4279	79	10	)	)	PUNCT
ejpam-4279	79	11	(	(	PUNCT
ejpam-4279	79	12	2022	2022	NUM
ejpam-4279	79	13	)	)	PUNCT
ejpam-4279	79	14	,	,	PUNCT
ejpam-4279	79	15	281	281	NUM
ejpam-4279	79	16	-	-	SYM
ejpam-4279	79	17	289	289	NUM
ejpam-4279	79	18	285	285	NUM
ejpam-4279	79	19	proof	proof	NOUN
ejpam-4279	79	20	.	.	PUNCT
ejpam-4279	79	21	suppose	suppose	VERB
ejpam-4279	79	22	that	that	SCONJ
ejpam-4279	79	23	m	m	PROPN
ejpam-4279	79	24	is	be	AUX
ejpam-4279	79	25	an	an	DET
ejpam-4279	79	26	almost	almost	ADV
ejpam-4279	79	27	bi	bi	ADJ
ejpam-4279	79	28	-	-	ADJ
ejpam-4279	79	29	interior	interior	ADJ
ejpam-4279	79	30	ideal	ideal	NOUN
ejpam-4279	79	31	of	of	ADP
ejpam-4279	79	32	s	s	NOUN
ejpam-4279	79	33	and	and	CCONJ
ejpam-4279	79	34	h	h	NOUN
ejpam-4279	79	35	,	,	PUNCT
ejpam-4279	79	36	n	n	CCONJ
ejpam-4279	79	37	,	,	PUNCT
ejpam-4279	79	38	r	r	PROPN
ejpam-4279	79	39	∈	∈	PROPN
ejpam-4279	79	40	s.	s.	PROPN
ejpam-4279	79	41	then	then	ADV
ejpam-4279	79	42	(	(	PUNCT
ejpam-4279	79	43	hmr	hmr	NOUN
ejpam-4279	79	44	∩mnm	∩mnm	PROPN
ejpam-4279	79	45	)	)	PUNCT
ejpam-4279	80	1	̸=	̸=	PROPN
ejpam-4279	80	2	∅.	∅.	ADP
ejpam-4279	80	3	thus	thus	ADV
ejpam-4279	80	4	(	(	PUNCT
ejpam-4279	80	5	hmr	hmr	NOUN
ejpam-4279	80	6	∩mnm	∩mnm	PROPN
ejpam-4279	80	7	)	)	PUNCT
ejpam-4279	81	1	⊆	⊆	NUM
ejpam-4279	81	2	(	(	PUNCT
ejpam-4279	81	3	klr	klr	PROPN
ejpam-4279	81	4	∩	∩	X
ejpam-4279	81	5	lnl	lnl	PROPN
ejpam-4279	81	6	)	)	PUNCT
ejpam-4279	81	7	̸=	̸=	PROPN
ejpam-4279	81	8	∅.	∅.	NOUN
ejpam-4279	81	9	by	by	ADP
ejpam-4279	81	10	assumption	assumption	NOUN
ejpam-4279	81	11	,	,	PUNCT
ejpam-4279	81	12	(	(	PUNCT
ejpam-4279	81	13	hmr	hmr	PROPN
ejpam-4279	81	14	∩	∩	PROPN
ejpam-4279	81	15	mnm	mnm	ADJ
ejpam-4279	81	16	)	)	PUNCT
ejpam-4279	81	17	∩	∩	NOUN
ejpam-4279	81	18	m	m	VERB
ejpam-4279	81	19	̸=	̸=	PROPN
ejpam-4279	81	20	∅.	∅.	ADP
ejpam-4279	81	21	it	it	PRON
ejpam-4279	81	22	implies	imply	VERB
ejpam-4279	81	23	that	that	SCONJ
ejpam-4279	81	24	∅	∅	NOUN
ejpam-4279	81	25	=	=	NOUN
ejpam-4279	81	26	̸	̸	NUM
ejpam-4279	81	27	(	(	PUNCT
ejpam-4279	81	28	hmr	hmr	PROPN
ejpam-4279	81	29	∩	∩	PROPN
ejpam-4279	81	30	mnm	mnm	ADJ
ejpam-4279	81	31	)	)	PUNCT
ejpam-4279	81	32	∩	∩	PROPN
ejpam-4279	81	33	m	m	VERB
ejpam-4279	81	34	⊆	⊆	NUM
ejpam-4279	81	35	(	(	PUNCT
ejpam-4279	81	36	hlr	hlr	PROPN
ejpam-4279	81	37	∩	∩	PROPN
ejpam-4279	81	38	lnl	lnl	PROPN
ejpam-4279	81	39	)	)	PUNCT
ejpam-4279	81	40	∩	∩	PROPN
ejpam-4279	81	41	l.	l.	PROPN
ejpam-4279	81	42	thus	thus	ADV
ejpam-4279	81	43	(	(	PUNCT
ejpam-4279	81	44	hlr	hlr	PROPN
ejpam-4279	81	45	∩	∩	PROPN
ejpam-4279	81	46	lnl	lnl	PROPN
ejpam-4279	81	47	)	)	PUNCT
ejpam-4279	81	48	∩	∩	NOUN
ejpam-4279	81	49	l	l	PROPN
ejpam-4279	81	50	̸=	̸=	PROPN
ejpam-4279	81	51	∅.	∅.	PRON
ejpam-4279	81	52	hence	hence	ADV
ejpam-4279	81	53	l	l	NOUN
ejpam-4279	81	54	is	be	AUX
ejpam-4279	81	55	an	an	DET
ejpam-4279	81	56	almost	almost	ADV
ejpam-4279	81	57	bi	bi	ADJ
ejpam-4279	81	58	-	-	ADJ
ejpam-4279	81	59	interior	interior	ADJ
ejpam-4279	81	60	ideal	ideal	NOUN
ejpam-4279	81	61	of	of	ADP
ejpam-4279	81	62	s.	s.	PROPN
ejpam-4279	81	63	the	the	DET
ejpam-4279	81	64	proof	proof	NOUN
ejpam-4279	81	65	of	of	ADP
ejpam-4279	81	66	the	the	DET
ejpam-4279	81	67	other	other	ADJ
ejpam-4279	81	68	similar	similar	ADJ
ejpam-4279	81	69	to	to	ADP
ejpam-4279	81	70	the	the	DET
ejpam-4279	81	71	proof	proof	NOUN
ejpam-4279	81	72	(	(	PUNCT
ejpam-4279	81	73	1	1	NUM
ejpam-4279	81	74	)	)	PUNCT
ejpam-4279	81	75	.	.	PUNCT
ejpam-4279	82	1	corollary	corollary	ADJ
ejpam-4279	82	2	1	1	NUM
ejpam-4279	82	3	.	.	PUNCT
ejpam-4279	83	1	let	let	VERB
ejpam-4279	83	2	s	s	PRON
ejpam-4279	83	3	be	be	AUX
ejpam-4279	83	4	a	a	DET
ejpam-4279	83	5	semigroup	semigroup	NOUN
ejpam-4279	83	6	.	.	PUNCT
ejpam-4279	84	1	then	then	ADV
ejpam-4279	84	2	the	the	DET
ejpam-4279	84	3	following	follow	VERB
ejpam-4279	84	4	statement	statement	NOUN
ejpam-4279	84	5	hold	hold	NOUN
ejpam-4279	84	6	.	.	PUNCT
ejpam-4279	85	1	(	(	PUNCT
ejpam-4279	85	2	1	1	X
ejpam-4279	85	3	)	)	PUNCT
ejpam-4279	85	4	the	the	DET
ejpam-4279	85	5	finite	finite	PROPN
ejpam-4279	85	6	union	union	PROPN
ejpam-4279	85	7	almost	almost	ADV
ejpam-4279	85	8	bi	bi	ADJ
ejpam-4279	85	9	-	-	ADJ
ejpam-4279	85	10	interior	interior	ADJ
ejpam-4279	85	11	ideal	ideal	NOUN
ejpam-4279	85	12	of	of	ADP
ejpam-4279	85	13	s	s	PROPN
ejpam-4279	85	14	is	be	AUX
ejpam-4279	85	15	an	an	DET
ejpam-4279	85	16	almost	almost	ADV
ejpam-4279	85	17	bi	bi	ADJ
ejpam-4279	85	18	-	-	ADJ
ejpam-4279	85	19	interior	interior	ADJ
ejpam-4279	85	20	ideal	ideal	NOUN
ejpam-4279	85	21	of	of	ADP
ejpam-4279	85	22	s.	s.	PROPN
ejpam-4279	85	23	(	(	PUNCT
ejpam-4279	85	24	2	2	X
ejpam-4279	85	25	)	)	PUNCT
ejpam-4279	85	26	the	the	DET
ejpam-4279	85	27	finite	finite	PROPN
ejpam-4279	85	28	union	union	NOUN
ejpam-4279	85	29	weak	weak	ADJ
ejpam-4279	85	30	almost	almost	ADV
ejpam-4279	85	31	bi	bi	ADJ
ejpam-4279	85	32	-	-	ADJ
ejpam-4279	85	33	interior	interior	ADJ
ejpam-4279	85	34	ideal	ideal	NOUN
ejpam-4279	85	35	of	of	ADP
ejpam-4279	85	36	s	s	PROPN
ejpam-4279	85	37	is	be	AUX
ejpam-4279	85	38	an	an	DET
ejpam-4279	85	39	weak	weak	ADJ
ejpam-4279	85	40	almost	almost	ADV
ejpam-4279	85	41	bi	bi	ADJ
ejpam-4279	85	42	-	-	ADJ
ejpam-4279	85	43	interior	interior	ADJ
ejpam-4279	85	44	ideal	ideal	NOUN
ejpam-4279	85	45	of	of	ADP
ejpam-4279	85	46	s.	s.	PROPN
ejpam-4279	85	47	4	4	NUM
ejpam-4279	85	48	.	.	PUNCT
ejpam-4279	85	49	fuzzy	fuzzy	ADJ
ejpam-4279	85	50	almost	almost	ADV
ejpam-4279	85	51	bi	bi	ADJ
ejpam-4279	85	52	-	-	ADJ
ejpam-4279	85	53	interior	interior	ADJ
ejpam-4279	85	54	ideals	ideal	NOUN
ejpam-4279	85	55	and	and	CCONJ
ejpam-4279	85	56	weak	weak	ADJ
ejpam-4279	85	57	fuzzy	fuzzy	ADJ
ejpam-4279	85	58	almost	almost	ADV
ejpam-4279	85	59	bi	bi	ADJ
ejpam-4279	85	60	-	-	ADJ
ejpam-4279	85	61	interior	interior	ADJ
ejpam-4279	85	62	ideals	ideal	NOUN
ejpam-4279	85	63	.	.	PUNCT
ejpam-4279	86	1	in	in	ADP
ejpam-4279	86	2	this	this	DET
ejpam-4279	86	3	section	section	NOUN
ejpam-4279	86	4	,	,	PUNCT
ejpam-4279	86	5	we	we	PRON
ejpam-4279	86	6	define	define	VERB
ejpam-4279	86	7	the	the	DET
ejpam-4279	86	8	notions	notion	NOUN
ejpam-4279	86	9	of	of	ADP
ejpam-4279	86	10	fuzzy	fuzzy	ADJ
ejpam-4279	86	11	almost	almost	ADV
ejpam-4279	86	12	bi	bi	ADJ
ejpam-4279	86	13	-	-	ADJ
ejpam-4279	86	14	interior	interior	ADJ
ejpam-4279	86	15	ideals	ideal	NOUN
ejpam-4279	86	16	and	and	CCONJ
ejpam-4279	86	17	weakly	weakly	ADJ
ejpam-4279	86	18	fuzzy	fuzzy	ADJ
ejpam-4279	86	19	bi	bi	ADJ
ejpam-4279	86	20	-	-	ADJ
ejpam-4279	86	21	interior	interior	ADJ
ejpam-4279	86	22	ideal	ideal	NOUN
ejpam-4279	86	23	in	in	ADP
ejpam-4279	86	24	semigroups	semigroup	NOUN
ejpam-4279	86	25	and	and	CCONJ
ejpam-4279	86	26	some	some	DET
ejpam-4279	86	27	properties	property	NOUN
ejpam-4279	86	28	of	of	ADP
ejpam-4279	86	29	them	they	PRON
ejpam-4279	86	30	are	be	AUX
ejpam-4279	86	31	investigated	investigate	VERB
ejpam-4279	86	32	.	.	PUNCT
ejpam-4279	87	1	definition	definition	NOUN
ejpam-4279	87	2	5	5	NUM
ejpam-4279	87	3	.	.	PUNCT
ejpam-4279	88	1	a	a	DET
ejpam-4279	88	2	nonzero	nonzero	ADJ
ejpam-4279	88	3	fuzzy	fuzzy	ADJ
ejpam-4279	88	4	set	set	VERB
ejpam-4279	88	5	φ	φ	PROPN
ejpam-4279	88	6	of	of	ADP
ejpam-4279	88	7	a	a	DET
ejpam-4279	88	8	semigroup	semigroup	NOUN
ejpam-4279	88	9	s	s	PART
ejpam-4279	88	10	is	be	AUX
ejpam-4279	88	11	called	call	VERB
ejpam-4279	88	12	a	a	DET
ejpam-4279	88	13	(	(	PUNCT
ejpam-4279	88	14	1	1	NUM
ejpam-4279	88	15	)	)	PUNCT
ejpam-4279	88	16	fuzzy	fuzzy	ADJ
ejpam-4279	88	17	almost	almost	ADV
ejpam-4279	88	18	bi	bi	ADJ
ejpam-4279	88	19	-	-	ADJ
ejpam-4279	88	20	interior	interior	ADJ
ejpam-4279	88	21	ideal	ideal	NOUN
ejpam-4279	88	22	of	of	ADP
ejpam-4279	88	23	s	s	PRON
ejpam-4279	88	24	if	if	SCONJ
ejpam-4279	88	25	(	(	PUNCT
ejpam-4279	88	26	(	(	PUNCT
ejpam-4279	88	27	pδ	pδ	ADP
ejpam-4279	88	28	◦	◦	NOUN
ejpam-4279	88	29	φ	φ	NUM
ejpam-4279	88	30	◦	◦	NOUN
ejpam-4279	88	31	qδ1	qδ1	NOUN
ejpam-4279	88	32	)	)	PUNCT
ejpam-4279	88	33	∧	∧	PROPN
ejpam-4279	88	34	(	(	PUNCT
ejpam-4279	88	35	φ	φ	NOUN
ejpam-4279	88	36	◦	◦	PROPN
ejpam-4279	88	37	gδ2	gδ2	PROPN
ejpam-4279	88	38	◦	◦	NOUN
ejpam-4279	88	39	φ	φ	NUM
ejpam-4279	88	40	)	)	PUNCT
ejpam-4279	88	41	)	)	PUNCT
ejpam-4279	89	1	∧	∧	PROPN
ejpam-4279	89	2	φ	φ	NUM
ejpam-4279	89	3	̸=	̸=	PROPN
ejpam-4279	89	4	0	0	NUM
ejpam-4279	89	5	,	,	PUNCT
ejpam-4279	89	6	for	for	ADP
ejpam-4279	89	7	fuzzy	fuzzy	ADJ
ejpam-4279	89	8	point	point	NOUN
ejpam-4279	89	9	pδ	pδ	PROPN
ejpam-4279	89	10	,	,	PUNCT
ejpam-4279	89	11	qδ1	qδ1	NOUN
ejpam-4279	89	12	and	and	CCONJ
ejpam-4279	89	13	gδ2	gδ2	PROPN
ejpam-4279	89	14	of	of	ADP
ejpam-4279	89	15	s.	s.	PROPN
ejpam-4279	89	16	(	(	PUNCT
ejpam-4279	89	17	2	2	X
ejpam-4279	89	18	)	)	PUNCT
ejpam-4279	89	19	weakly	weakly	ADJ
ejpam-4279	89	20	almost	almost	ADV
ejpam-4279	89	21	interior	interior	ADJ
ejpam-4279	89	22	ideal	ideal	NOUN
ejpam-4279	89	23	of	of	ADP
ejpam-4279	89	24	s	s	PRON
ejpam-4279	89	25	if	if	SCONJ
ejpam-4279	89	26	(	(	PUNCT
ejpam-4279	89	27	(	(	PUNCT
ejpam-4279	89	28	pδ	pδ	ADP
ejpam-4279	89	29	◦	◦	NOUN
ejpam-4279	89	30	φ	φ	NUM
ejpam-4279	89	31	◦	◦	NOUN
ejpam-4279	89	32	pδ1	pδ1	ADV
ejpam-4279	89	33	)	)	PUNCT
ejpam-4279	89	34	∧	∧	PROPN
ejpam-4279	89	35	(	(	PUNCT
ejpam-4279	89	36	φ	φ	X
ejpam-4279	89	37	◦	◦	PROPN
ejpam-4279	89	38	pδ2	pδ2	PROPN
ejpam-4279	89	39	◦	◦	NOUN
ejpam-4279	89	40	φ	φ	NUM
ejpam-4279	89	41	)	)	PUNCT
ejpam-4279	89	42	)	)	PUNCT
ejpam-4279	90	1	∧	∧	PROPN
ejpam-4279	90	2	φ	φ	NUM
ejpam-4279	90	3	̸=	̸=	PROPN
ejpam-4279	90	4	0	0	NUM
ejpam-4279	90	5	,	,	PUNCT
ejpam-4279	90	6	for	for	ADP
ejpam-4279	90	7	fuzzy	fuzzy	ADJ
ejpam-4279	90	8	point	point	NOUN
ejpam-4279	90	9	pδ	pδ	PROPN
ejpam-4279	90	10	,	,	PUNCT
ejpam-4279	90	11	pδ1	pδ1	ADJ
ejpam-4279	90	12	and	and	CCONJ
ejpam-4279	90	13	pδ2	pδ2	PROPN
ejpam-4279	90	14	of	of	ADP
ejpam-4279	90	15	s.	s.	PROPN
ejpam-4279	90	16	the	the	DET
ejpam-4279	90	17	following	follow	VERB
ejpam-4279	90	18	theorem	theorem	VERB
ejpam-4279	90	19	easily	easily	ADV
ejpam-4279	90	20	to	to	PART
ejpam-4279	90	21	prove	prove	VERB
ejpam-4279	90	22	.	.	PUNCT
ejpam-4279	91	1	theorem	theorem	NOUN
ejpam-4279	91	2	3	3	X
ejpam-4279	91	3	.	.	PUNCT
ejpam-4279	92	1	let	let	VERB
ejpam-4279	92	2	φ	φ	PROPN
ejpam-4279	92	3	be	be	AUX
ejpam-4279	92	4	a	a	DET
ejpam-4279	92	5	nonzero	nonzero	ADJ
ejpam-4279	92	6	fuzzy	fuzzy	ADJ
ejpam-4279	92	7	subset	subset	NOUN
ejpam-4279	92	8	of	of	ADP
ejpam-4279	92	9	a	a	DET
ejpam-4279	92	10	semigroup	semigroup	PROPN
ejpam-4279	92	11	s.	s.	PROPN
ejpam-4279	92	12	then	then	ADV
ejpam-4279	92	13	the	the	DET
ejpam-4279	92	14	following	follow	VERB
ejpam-4279	92	15	statement	statement	NOUN
ejpam-4279	92	16	hold	hold	NOUN
ejpam-4279	92	17	.	.	PUNCT
ejpam-4279	93	1	(	(	PUNCT
ejpam-4279	93	2	1	1	X
ejpam-4279	93	3	)	)	PUNCT
ejpam-4279	93	4	every	every	DET
ejpam-4279	93	5	fuzzy	fuzzy	ADJ
ejpam-4279	93	6	bi	bi	ADJ
ejpam-4279	93	7	-	-	ADJ
ejpam-4279	93	8	interior	interior	ADJ
ejpam-4279	93	9	ideal	ideal	NOUN
ejpam-4279	93	10	of	of	ADP
ejpam-4279	93	11	s	s	PROPN
ejpam-4279	93	12	is	be	AUX
ejpam-4279	93	13	a	a	DET
ejpam-4279	93	14	fuzzy	fuzzy	ADJ
ejpam-4279	93	15	almost	almost	ADV
ejpam-4279	93	16	bi	bi	ADJ
ejpam-4279	93	17	-	-	ADJ
ejpam-4279	93	18	interior	interior	ADJ
ejpam-4279	93	19	ideal	ideal	NOUN
ejpam-4279	93	20	of	of	ADP
ejpam-4279	93	21	s.	s.	PROPN
ejpam-4279	93	22	(	(	PUNCT
ejpam-4279	93	23	2	2	X
ejpam-4279	93	24	)	)	PUNCT
ejpam-4279	93	25	every	every	DET
ejpam-4279	93	26	weak	weak	ADJ
ejpam-4279	93	27	fuzzy	fuzzy	ADJ
ejpam-4279	93	28	bi	bi	ADJ
ejpam-4279	93	29	-	-	ADJ
ejpam-4279	93	30	interior	interior	ADJ
ejpam-4279	93	31	ideal	ideal	NOUN
ejpam-4279	93	32	of	of	ADP
ejpam-4279	93	33	s	s	PROPN
ejpam-4279	93	34	is	be	AUX
ejpam-4279	93	35	a	a	DET
ejpam-4279	93	36	weak	weak	ADJ
ejpam-4279	93	37	fuzzy	fuzzy	ADJ
ejpam-4279	93	38	almost	almost	ADV
ejpam-4279	93	39	bi	bi	ADJ
ejpam-4279	93	40	-	-	ADJ
ejpam-4279	93	41	interior	interior	ADJ
ejpam-4279	93	42	ideal	ideal	NOUN
ejpam-4279	93	43	of	of	ADP
ejpam-4279	93	44	s.	s.	PROPN
ejpam-4279	93	45	theorem	theorem	VERB
ejpam-4279	93	46	4	4	X
ejpam-4279	93	47	.	.	PUNCT
ejpam-4279	94	1	let	let	VERB
ejpam-4279	94	2	φ	φ	NOUN
ejpam-4279	94	3	and	and	CCONJ
ejpam-4279	94	4	ν	ν	PROPN
ejpam-4279	94	5	be	be	AUX
ejpam-4279	94	6	a	a	DET
ejpam-4279	94	7	nonzero	nonzero	ADJ
ejpam-4279	94	8	fuzzy	fuzzy	ADJ
ejpam-4279	94	9	sets	set	NOUN
ejpam-4279	94	10	of	of	ADP
ejpam-4279	94	11	a	a	DET
ejpam-4279	94	12	semigroup	semigroup	NOUN
ejpam-4279	94	13	s	s	PROPN
ejpam-4279	94	14	with	with	ADP
ejpam-4279	94	15	φ	φ	PROPN
ejpam-4279	94	16	⪯	⪯	PROPN
ejpam-4279	94	17	ξ	ξ	PROPN
ejpam-4279	94	18	.	.	PUNCT
ejpam-4279	95	1	then	then	ADV
ejpam-4279	95	2	the	the	DET
ejpam-4279	95	3	following	follow	VERB
ejpam-4279	95	4	statement	statement	NOUN
ejpam-4279	95	5	hold	hold	NOUN
ejpam-4279	95	6	.	.	PUNCT
ejpam-4279	96	1	(	(	PUNCT
ejpam-4279	96	2	1	1	X
ejpam-4279	96	3	)	)	PUNCT
ejpam-4279	96	4	if	if	SCONJ
ejpam-4279	96	5	φ	φ	PROPN
ejpam-4279	96	6	is	be	AUX
ejpam-4279	96	7	a	a	DET
ejpam-4279	96	8	fuzzy	fuzzy	ADJ
ejpam-4279	96	9	almost	almost	ADV
ejpam-4279	96	10	bi	bi	ADJ
ejpam-4279	96	11	-	-	ADJ
ejpam-4279	96	12	interior	interior	ADJ
ejpam-4279	96	13	ideal	ideal	NOUN
ejpam-4279	96	14	of	of	ADP
ejpam-4279	96	15	s	s	PROPN
ejpam-4279	96	16	,	,	PUNCT
ejpam-4279	96	17	then	then	ADV
ejpam-4279	96	18	ξ	ξ	PROPN
ejpam-4279	96	19	is	be	AUX
ejpam-4279	96	20	a	a	DET
ejpam-4279	96	21	fuzzy	fuzzy	ADJ
ejpam-4279	96	22	almost	almost	ADV
ejpam-4279	96	23	bi	bi	ADJ
ejpam-4279	96	24	-	-	ADJ
ejpam-4279	96	25	interior	interior	ADJ
ejpam-4279	96	26	ideal	ideal	NOUN
ejpam-4279	96	27	of	of	ADP
ejpam-4279	96	28	s.	s.	PROPN
ejpam-4279	96	29	(	(	PUNCT
ejpam-4279	96	30	2	2	X
ejpam-4279	96	31	)	)	PUNCT
ejpam-4279	96	32	if	if	SCONJ
ejpam-4279	96	33	φ	φ	PROPN
ejpam-4279	96	34	is	be	AUX
ejpam-4279	96	35	a	a	DET
ejpam-4279	96	36	weakly	weakly	ADJ
ejpam-4279	96	37	fuzzy	fuzzy	ADJ
ejpam-4279	96	38	almost	almost	ADV
ejpam-4279	96	39	bi	bi	ADJ
ejpam-4279	96	40	-	-	ADJ
ejpam-4279	96	41	interior	interior	ADJ
ejpam-4279	96	42	ideal	ideal	NOUN
ejpam-4279	96	43	of	of	ADP
ejpam-4279	96	44	s	s	PROPN
ejpam-4279	96	45	,	,	PUNCT
ejpam-4279	96	46	then	then	ADV
ejpam-4279	96	47	ξ	ξ	PROPN
ejpam-4279	96	48	is	be	AUX
ejpam-4279	96	49	a	a	DET
ejpam-4279	96	50	weakly	weakly	ADJ
ejpam-4279	96	51	almost	almost	ADV
ejpam-4279	96	52	bi	bi	NOUN
ejpam-4279	96	53	-	-	NOUN
ejpam-4279	96	54	ideal	ideal	NOUN
ejpam-4279	96	55	of	of	ADP
ejpam-4279	96	56	s.	s.	PROPN
ejpam-4279	96	57	proof	proof	PROPN
ejpam-4279	96	58	.	.	PUNCT
ejpam-4279	97	1	suppose	suppose	VERB
ejpam-4279	97	2	that	that	SCONJ
ejpam-4279	97	3	φ	φ	PROPN
ejpam-4279	97	4	is	be	AUX
ejpam-4279	97	5	a	a	DET
ejpam-4279	97	6	fuzzy	fuzzy	ADJ
ejpam-4279	97	7	almost	almost	ADV
ejpam-4279	97	8	bi	bi	ADJ
ejpam-4279	97	9	-	-	ADJ
ejpam-4279	97	10	interior	interior	ADJ
ejpam-4279	97	11	ideal	ideal	NOUN
ejpam-4279	97	12	of	of	ADP
ejpam-4279	97	13	s	s	NOUN
ejpam-4279	97	14	and	and	CCONJ
ejpam-4279	97	15	pδ	pδ	PROPN
ejpam-4279	97	16	,	,	PUNCT
ejpam-4279	97	17	qδ1	qδ1	INTJ
ejpam-4279	97	18	,	,	PUNCT
ejpam-4279	97	19	gδ2	gδ2	PROPN
ejpam-4279	97	20	∈	∈	PROPN
ejpam-4279	97	21	(	(	PUNCT
ejpam-4279	97	22	0	0	NUM
ejpam-4279	97	23	,	,	PUNCT
ejpam-4279	97	24	1	1	NUM
ejpam-4279	97	25	]	]	PUNCT
ejpam-4279	97	26	.	.	PUNCT
ejpam-4279	98	1	then	then	ADV
ejpam-4279	98	2	(	(	PUNCT
ejpam-4279	98	3	(	(	PUNCT
ejpam-4279	98	4	(	(	PUNCT
ejpam-4279	98	5	pδ	pδ	ADP
ejpam-4279	98	6	◦	◦	NOUN
ejpam-4279	98	7	φ	φ	NUM
ejpam-4279	98	8	◦	◦	NOUN
ejpam-4279	98	9	qδ1	qδ1	NOUN
ejpam-4279	98	10	)	)	PUNCT
ejpam-4279	98	11	∧	∧	PROPN
ejpam-4279	98	12	(	(	PUNCT
ejpam-4279	98	13	φ	φ	NOUN
ejpam-4279	98	14	◦	◦	PROPN
ejpam-4279	98	15	gδ2	gδ2	PROPN
ejpam-4279	98	16	◦	◦	NOUN
ejpam-4279	98	17	φ	φ	NUM
ejpam-4279	98	18	)	)	PUNCT
ejpam-4279	98	19	)	)	PUNCT
ejpam-4279	99	1	∧	∧	PROPN
ejpam-4279	99	2	φ	φ	NUM
ejpam-4279	99	3	̸=	̸=	PROPN
ejpam-4279	99	4	0	0	NUM
ejpam-4279	99	5	.	.	PUNCT
ejpam-4279	100	1	by	by	ADP
ejpam-4279	100	2	assumption	assumption	NOUN
ejpam-4279	100	3	,	,	PUNCT
ejpam-4279	100	4	(	(	PUNCT
ejpam-4279	100	5	(	(	PUNCT
ejpam-4279	100	6	(	(	PUNCT
ejpam-4279	100	7	pδ	pδ	ADP
ejpam-4279	100	8	◦	◦	NOUN
ejpam-4279	100	9	φ	φ	NUM
ejpam-4279	100	10	◦	◦	NOUN
ejpam-4279	100	11	qδ1	qδ1	NOUN
ejpam-4279	100	12	)	)	PUNCT
ejpam-4279	100	13	∧	∧	PROPN
ejpam-4279	100	14	(	(	PUNCT
ejpam-4279	100	15	φ	φ	NOUN
ejpam-4279	100	16	◦	◦	PROPN
ejpam-4279	100	17	gδ2	gδ2	PROPN
ejpam-4279	100	18	◦	◦	NOUN
ejpam-4279	100	19	φ	φ	NUM
ejpam-4279	100	20	)	)	PUNCT
ejpam-4279	100	21	)	)	PUNCT
ejpam-4279	100	22	∧	∧	PROPN
ejpam-4279	100	23	φ	φ	X
ejpam-4279	100	24	⪯	⪯	PROPN
ejpam-4279	100	25	(	(	PUNCT
ejpam-4279	100	26	(	(	PUNCT
ejpam-4279	100	27	(	(	PUNCT
ejpam-4279	100	28	pδ	pδ	ADP
ejpam-4279	100	29	◦	◦	NOUN
ejpam-4279	100	30	ξ	ξ	X
ejpam-4279	100	31	◦	◦	NOUN
ejpam-4279	100	32	qδ1	qδ1	NOUN
ejpam-4279	100	33	)	)	PUNCT
ejpam-4279	100	34	∧	∧	NOUN
ejpam-4279	100	35	(	(	PUNCT
ejpam-4279	100	36	ξ	ξ	X
ejpam-4279	100	37	◦	◦	NOUN
ejpam-4279	100	38	gδ2	gδ2	PROPN
ejpam-4279	100	39	◦	◦	NOUN
ejpam-4279	100	40	ξ	ξ	X
ejpam-4279	100	41	)	)	PUNCT
ejpam-4279	100	42	)	)	PUNCT
ejpam-4279	100	43	∧	∧	PROPN
ejpam-4279	100	44	ξ	ξ	X
ejpam-4279	100	45	̸=	̸=	PROPN
ejpam-4279	100	46	0	0	NUM
ejpam-4279	100	47	.	.	PUNCT
ejpam-4279	101	1	thus	thus	ADV
ejpam-4279	101	2	(	(	PUNCT
ejpam-4279	101	3	(	(	PUNCT
ejpam-4279	101	4	(	(	PUNCT
ejpam-4279	101	5	pδ	pδ	ADP
ejpam-4279	101	6	◦	◦	NOUN
ejpam-4279	101	7	ξ	ξ	X
ejpam-4279	101	8	◦	◦	NOUN
ejpam-4279	101	9	qδ1	qδ1	NOUN
ejpam-4279	101	10	)	)	PUNCT
ejpam-4279	101	11	∧	∧	NOUN
ejpam-4279	101	12	(	(	PUNCT
ejpam-4279	101	13	ξ	ξ	X
ejpam-4279	101	14	◦	◦	NOUN
ejpam-4279	101	15	gδ2	gδ2	PROPN
ejpam-4279	101	16	◦	◦	NOUN
ejpam-4279	101	17	ξ	ξ	X
ejpam-4279	101	18	)	)	PUNCT
ejpam-4279	101	19	)	)	PUNCT
ejpam-4279	102	1	∧	∧	PROPN
ejpam-4279	102	2	ξ	ξ	X
ejpam-4279	102	3	̸=	̸=	PROPN
ejpam-4279	102	4	0	0	NUM
ejpam-4279	102	5	.	.	PUNCT
ejpam-4279	103	1	hence	hence	ADV
ejpam-4279	103	2	ξ	ξ	PROPN
ejpam-4279	103	3	is	be	AUX
ejpam-4279	103	4	a	a	DET
ejpam-4279	103	5	fuzzy	fuzzy	ADJ
ejpam-4279	103	6	almost	almost	ADV
ejpam-4279	103	7	bi	bi	ADJ
ejpam-4279	103	8	-	-	ADJ
ejpam-4279	103	9	interior	interior	ADJ
ejpam-4279	103	10	ideal	ideal	NOUN
ejpam-4279	103	11	of	of	ADP
ejpam-4279	103	12	s.	s.	PROPN
ejpam-4279	103	13	the	the	DET
ejpam-4279	103	14	proof	proof	NOUN
ejpam-4279	103	15	of	of	ADP
ejpam-4279	103	16	the	the	DET
ejpam-4279	103	17	other	other	ADJ
ejpam-4279	103	18	similar	similar	ADJ
ejpam-4279	103	19	to	to	ADP
ejpam-4279	103	20	the	the	DET
ejpam-4279	103	21	proof	proof	NOUN
ejpam-4279	103	22	(	(	PUNCT
ejpam-4279	103	23	1	1	NUM
ejpam-4279	103	24	)	)	PUNCT
ejpam-4279	103	25	.	.	PUNCT
ejpam-4279	104	1	t.	t.	PROPN
ejpam-4279	104	2	gaketem	gaketem	PROPN
ejpam-4279	104	3	/	/	SYM
ejpam-4279	104	4	eur	eur	PROPN
ejpam-4279	104	5	.	.	PUNCT
ejpam-4279	105	1	j.	j.	PROPN
ejpam-4279	105	2	pure	pure	PROPN
ejpam-4279	105	3	appl	appl	PROPN
ejpam-4279	105	4	.	.	PROPN
ejpam-4279	105	5	math	math	PROPN
ejpam-4279	105	6	,	,	PUNCT
ejpam-4279	105	7	15	15	NUM
ejpam-4279	105	8	(	(	PUNCT
ejpam-4279	105	9	1	1	NUM
ejpam-4279	105	10	)	)	PUNCT
ejpam-4279	105	11	(	(	PUNCT
ejpam-4279	105	12	2022	2022	NUM
ejpam-4279	105	13	)	)	PUNCT
ejpam-4279	105	14	,	,	PUNCT
ejpam-4279	105	15	281	281	NUM
ejpam-4279	105	16	-	-	SYM
ejpam-4279	105	17	289	289	NUM
ejpam-4279	105	18	286	286	NUM
ejpam-4279	105	19	corollary	corollary	ADJ
ejpam-4279	105	20	2	2	NUM
ejpam-4279	105	21	.	.	PUNCT
ejpam-4279	106	1	let	let	VERB
ejpam-4279	106	2	s	s	PRON
ejpam-4279	106	3	be	be	AUX
ejpam-4279	106	4	an	an	DET
ejpam-4279	106	5	ordered	order	VERB
ejpam-4279	106	6	semigroup	semigroup	NOUN
ejpam-4279	106	7	.	.	PUNCT
ejpam-4279	107	1	then	then	ADV
ejpam-4279	107	2	the	the	DET
ejpam-4279	107	3	following	follow	VERB
ejpam-4279	107	4	statement	statement	NOUN
ejpam-4279	107	5	hold	hold	NOUN
ejpam-4279	107	6	.	.	PUNCT
ejpam-4279	108	1	(	(	PUNCT
ejpam-4279	108	2	1	1	X
ejpam-4279	108	3	)	)	PUNCT
ejpam-4279	108	4	the	the	DET
ejpam-4279	108	5	finite	finite	PROPN
ejpam-4279	108	6	union	union	NOUN
ejpam-4279	108	7	fuzzy	fuzzy	ADJ
ejpam-4279	108	8	almost	almost	ADV
ejpam-4279	108	9	bi	bi	ADJ
ejpam-4279	108	10	-	-	ADJ
ejpam-4279	108	11	interior	interior	ADJ
ejpam-4279	108	12	ideal	ideal	NOUN
ejpam-4279	108	13	of	of	ADP
ejpam-4279	108	14	s	s	PROPN
ejpam-4279	108	15	is	be	AUX
ejpam-4279	108	16	a	a	DET
ejpam-4279	108	17	fuzzy	fuzzy	ADJ
ejpam-4279	108	18	almost	almost	ADV
ejpam-4279	108	19	bi	bi	ADJ
ejpam-4279	108	20	-	-	ADJ
ejpam-4279	108	21	interior	interior	ADJ
ejpam-4279	108	22	ideal	ideal	NOUN
ejpam-4279	108	23	of	of	ADP
ejpam-4279	108	24	s.	s.	PROPN
ejpam-4279	108	25	(	(	PUNCT
ejpam-4279	108	26	2	2	X
ejpam-4279	108	27	)	)	PUNCT
ejpam-4279	108	28	the	the	DET
ejpam-4279	108	29	finite	finite	PROPN
ejpam-4279	108	30	union	union	PROPN
ejpam-4279	108	31	weakly	weakly	ADV
ejpam-4279	108	32	fuzzy	fuzzy	ADJ
ejpam-4279	108	33	almost	almost	ADV
ejpam-4279	108	34	bi	bi	ADJ
ejpam-4279	108	35	-	-	ADJ
ejpam-4279	108	36	interior	interior	ADJ
ejpam-4279	108	37	ideal	ideal	NOUN
ejpam-4279	108	38	of	of	ADP
ejpam-4279	108	39	s	s	PROPN
ejpam-4279	108	40	is	be	AUX
ejpam-4279	108	41	a	a	DET
ejpam-4279	108	42	weakly	weakly	ADJ
ejpam-4279	108	43	fuzzy	fuzzy	ADJ
ejpam-4279	108	44	almost	almost	ADV
ejpam-4279	108	45	bi	bi	ADJ
ejpam-4279	108	46	-	-	ADJ
ejpam-4279	108	47	interior	interior	ADJ
ejpam-4279	108	48	ideal	ideal	NOUN
ejpam-4279	108	49	of	of	ADP
ejpam-4279	108	50	s.	s.	PROPN
ejpam-4279	108	51	theorem	theorem	VERB
ejpam-4279	108	52	5	5	X
ejpam-4279	108	53	.	.	PUNCT
ejpam-4279	109	1	let	let	VERB
ejpam-4279	109	2	m	m	PRON
ejpam-4279	109	3	be	be	AUX
ejpam-4279	109	4	a	a	DET
ejpam-4279	109	5	nonempty	nonempty	ADJ
ejpam-4279	109	6	subset	subset	NOUN
ejpam-4279	109	7	of	of	ADP
ejpam-4279	109	8	an	an	DET
ejpam-4279	109	9	ordered	order	VERB
ejpam-4279	109	10	semigroup	semigroup	PROPN
ejpam-4279	109	11	s.	s.	PROPN
ejpam-4279	109	12	then	then	ADV
ejpam-4279	109	13	the	the	DET
ejpam-4279	109	14	following	follow	VERB
ejpam-4279	109	15	statement	statement	NOUN
ejpam-4279	109	16	hold	hold	NOUN
ejpam-4279	109	17	.	.	PUNCT
ejpam-4279	110	1	(	(	PUNCT
ejpam-4279	110	2	1	1	X
ejpam-4279	110	3	)	)	PUNCT
ejpam-4279	110	4	m	m	VERB
ejpam-4279	110	5	is	be	AUX
ejpam-4279	110	6	an	an	DET
ejpam-4279	110	7	almost	almost	ADV
ejpam-4279	110	8	bi	bi	ADJ
ejpam-4279	110	9	-	-	ADJ
ejpam-4279	110	10	interior	interior	ADJ
ejpam-4279	110	11	ideal	ideal	NOUN
ejpam-4279	110	12	of	of	ADP
ejpam-4279	110	13	s	s	PRON
ejpam-4279	110	14	if	if	SCONJ
ejpam-4279	110	15	and	and	CCONJ
ejpam-4279	110	16	only	only	ADV
ejpam-4279	110	17	if	if	SCONJ
ejpam-4279	110	18	λm	λm	ADV
ejpam-4279	110	19	is	be	AUX
ejpam-4279	110	20	a	a	DET
ejpam-4279	110	21	fuzzy	fuzzy	ADJ
ejpam-4279	110	22	almost	almost	ADV
ejpam-4279	110	23	bi	bi	ADJ
ejpam-4279	110	24	-	-	ADJ
ejpam-4279	110	25	interior	interior	ADJ
ejpam-4279	110	26	ideal	ideal	NOUN
ejpam-4279	110	27	of	of	ADP
ejpam-4279	110	28	s.	s.	PROPN
ejpam-4279	110	29	(	(	PUNCT
ejpam-4279	110	30	2	2	X
ejpam-4279	110	31	)	)	PUNCT
ejpam-4279	110	32	m	m	VERB
ejpam-4279	110	33	is	be	AUX
ejpam-4279	110	34	a	a	DET
ejpam-4279	110	35	weakly	weakly	ADJ
ejpam-4279	110	36	almost	almost	ADV
ejpam-4279	110	37	interior	interior	ADJ
ejpam-4279	110	38	ideal	ideal	NOUN
ejpam-4279	110	39	of	of	ADP
ejpam-4279	110	40	s	s	PRON
ejpam-4279	110	41	if	if	SCONJ
ejpam-4279	110	42	and	and	CCONJ
ejpam-4279	110	43	only	only	ADV
ejpam-4279	110	44	if	if	SCONJ
ejpam-4279	110	45	λm	λm	ADV
ejpam-4279	110	46	is	be	AUX
ejpam-4279	110	47	a	a	DET
ejpam-4279	110	48	weakly	weakly	ADJ
ejpam-4279	110	49	fuzzy	fuzzy	ADJ
ejpam-4279	110	50	almost	almost	ADV
ejpam-4279	110	51	bi	bi	ADJ
ejpam-4279	110	52	-	-	ADJ
ejpam-4279	110	53	interior	interior	ADJ
ejpam-4279	110	54	ideal	ideal	NOUN
ejpam-4279	110	55	of	of	ADP
ejpam-4279	110	56	s.	s.	PROPN
ejpam-4279	110	57	proof	proof	PROPN
ejpam-4279	110	58	.	.	PUNCT
ejpam-4279	111	1	suppose	suppose	VERB
ejpam-4279	111	2	that	that	SCONJ
ejpam-4279	111	3	m	m	PROPN
ejpam-4279	111	4	is	be	AUX
ejpam-4279	111	5	an	an	DET
ejpam-4279	111	6	almost	almost	ADV
ejpam-4279	111	7	bi	bi	ADJ
ejpam-4279	111	8	-	-	ADJ
ejpam-4279	111	9	interior	interior	ADJ
ejpam-4279	111	10	ideal	ideal	NOUN
ejpam-4279	111	11	of	of	ADP
ejpam-4279	111	12	s	s	PROPN
ejpam-4279	111	13	,	,	PUNCT
ejpam-4279	111	14	h	h	NOUN
ejpam-4279	111	15	,	,	PUNCT
ejpam-4279	111	16	r	r	NOUN
ejpam-4279	111	17	,	,	PUNCT
ejpam-4279	111	18	n	n	PROPN
ejpam-4279	111	19	∈	∈	PROPN
ejpam-4279	111	20	s	s	X
ejpam-4279	111	21	and	and	CCONJ
ejpam-4279	111	22	δ	δ	PROPN
ejpam-4279	111	23	,	,	PUNCT
ejpam-4279	111	24	δ1	δ1	NOUN
ejpam-4279	111	25	,	,	PUNCT
ejpam-4279	111	26	δ2	δ2	VERB
ejpam-4279	111	27	∈	∈	PROPN
ejpam-4279	111	28	(	(	PUNCT
ejpam-4279	111	29	0	0	NUM
ejpam-4279	111	30	,	,	PUNCT
ejpam-4279	111	31	1	1	NUM
ejpam-4279	111	32	]	]	PUNCT
ejpam-4279	111	33	.	.	PUNCT
ejpam-4279	112	1	then	then	ADV
ejpam-4279	112	2	(	(	PUNCT
ejpam-4279	112	3	hmr∩mnm)∩m	hmr∩mnm)∩m	NOUN
ejpam-4279	112	4	̸=	̸=	PROPN
ejpam-4279	112	5	∅.	∅.	VERB
ejpam-4279	112	6	thus	thus	ADV
ejpam-4279	112	7	there	there	PRON
ejpam-4279	112	8	exists	exist	VERB
ejpam-4279	112	9	c	c	PROPN
ejpam-4279	112	10	∈	∈	PROPN
ejpam-4279	112	11	s	s	VERB
ejpam-4279	112	12	such	such	ADJ
ejpam-4279	112	13	that	that	SCONJ
ejpam-4279	112	14	c	c	PROPN
ejpam-4279	112	15	∈	∈	PROPN
ejpam-4279	112	16	(	(	PUNCT
ejpam-4279	112	17	hmr∩mnm	hmr∩mnm	NOUN
ejpam-4279	112	18	)	)	PUNCT
ejpam-4279	112	19	and	and	CCONJ
ejpam-4279	112	20	c	c	X
ejpam-4279	112	21	∈	∈	PROPN
ejpam-4279	112	22	m	m	VERB
ejpam-4279	113	1	so	so	ADV
ejpam-4279	113	2	c	c	NOUN
ejpam-4279	113	3	=	=	PUNCT
ejpam-4279	114	1	hm1r	hm1r	PROPN
ejpam-4279	114	2	and	and	CCONJ
ejpam-4279	114	3	c	c	NOUN
ejpam-4279	114	4	=	=	SYM
ejpam-4279	114	5	m2nm3	m2nm3	NOUN
ejpam-4279	114	6	for	for	ADP
ejpam-4279	114	7	some	some	DET
ejpam-4279	114	8	m1,m2,m3	m1,m2,m3	ADJ
ejpam-4279	114	9	∈	∈	PROPN
ejpam-4279	114	10	m	m	VERB
ejpam-4279	114	11	.	.	PUNCT
ejpam-4279	115	1	it	it	PRON
ejpam-4279	115	2	follows	follow	VERB
ejpam-4279	115	3	that	that	SCONJ
ejpam-4279	115	4	(	(	PUNCT
ejpam-4279	115	5	pδ	pδ	ADP
ejpam-4279	115	6	◦	◦	NOUN
ejpam-4279	115	7	λm	λm	ADP
ejpam-4279	115	8	◦	◦	NOUN
ejpam-4279	115	9	qδ1	qδ1	NOUN
ejpam-4279	115	10	)	)	PUNCT
ejpam-4279	115	11	∧	∧	NOUN
ejpam-4279	115	12	(	(	PUNCT
ejpam-4279	115	13	λm	λm	ADP
ejpam-4279	115	14	◦	◦	NOUN
ejpam-4279	115	15	gδ2	gδ2	INTJ
ejpam-4279	116	1	◦	◦	NOUN
ejpam-4279	116	2	λm	λm	X
ejpam-4279	116	3	)	)	PUNCT
ejpam-4279	116	4	(	(	PUNCT
ejpam-4279	116	5	c	c	X
ejpam-4279	116	6	)	)	PUNCT
ejpam-4279	116	7	=	=	SYM
ejpam-4279	116	8	(	(	PUNCT
ejpam-4279	116	9	∨	∨	X
ejpam-4279	116	10	c	c	X
ejpam-4279	116	11	=	=	X
ejpam-4279	116	12	hm1r	hm1r	X
ejpam-4279	116	13	{	{	PUNCT
ejpam-4279	116	14	pδ(h	pδ(h	NOUN
ejpam-4279	116	15	)	)	PUNCT
ejpam-4279	116	16	∧	∧	NOUN
ejpam-4279	116	17	λm	λm	ADP
ejpam-4279	116	18	(	(	PUNCT
ejpam-4279	116	19	m1	m1	NOUN
ejpam-4279	116	20	)	)	PUNCT
ejpam-4279	116	21	∧	∧	PROPN
ejpam-4279	116	22	qδ1(r	qδ1(r	NOUN
ejpam-4279	116	23	)	)	PUNCT
ejpam-4279	116	24	}	}	PUNCT
ejpam-4279	116	25	)	)	PUNCT
ejpam-4279	117	1	∧	∧	NOUN
ejpam-4279	117	2	(	(	PUNCT
ejpam-4279	117	3	∨	∨	PROPN
ejpam-4279	117	4	c	c	NOUN
ejpam-4279	117	5	=	=	X
ejpam-4279	117	6	m2nm3	m2nm3	NOUN
ejpam-4279	117	7	{	{	PUNCT
ejpam-4279	117	8	λm	λm	X
ejpam-4279	117	9	(	(	PUNCT
ejpam-4279	117	10	m2	m2	PROPN
ejpam-4279	117	11	)	)	PUNCT
ejpam-4279	117	12	∧	∧	PROPN
ejpam-4279	117	13	gδ2(n	gδ2(n	PROPN
ejpam-4279	117	14	)	)	PUNCT
ejpam-4279	117	15	∧	∧	NOUN
ejpam-4279	117	16	λm	λm	ADP
ejpam-4279	117	17	(	(	PUNCT
ejpam-4279	117	18	m3	m3	PROPN
ejpam-4279	117	19	)	)	PUNCT
ejpam-4279	117	20	}	}	PUNCT
ejpam-4279	117	21	)	)	PUNCT
ejpam-4279	117	22	̸=	̸=	PROPN
ejpam-4279	117	23	0	0	NUM
ejpam-4279	117	24	,	,	PUNCT
ejpam-4279	117	25	and	and	CCONJ
ejpam-4279	117	26	λm	λm	X
ejpam-4279	117	27	(	(	PUNCT
ejpam-4279	117	28	c	c	X
ejpam-4279	117	29	)	)	PUNCT
ejpam-4279	117	30	=	=	SYM
ejpam-4279	117	31	1	1	X
ejpam-4279	117	32	.	.	PUNCT
ejpam-4279	118	1	thus	thus	ADV
ejpam-4279	118	2	(	(	PUNCT
ejpam-4279	118	3	pδ	pδ	ADP
ejpam-4279	118	4	◦	◦	NOUN
ejpam-4279	118	5	λm	λm	ADP
ejpam-4279	118	6	◦	◦	NOUN
ejpam-4279	118	7	qδ1	qδ1	NOUN
ejpam-4279	118	8	)	)	PUNCT
ejpam-4279	118	9	∧	∧	NOUN
ejpam-4279	118	10	(	(	PUNCT
ejpam-4279	118	11	λm	λm	ADP
ejpam-4279	118	12	◦	◦	NOUN
ejpam-4279	118	13	gδ2	gδ2	INTJ
ejpam-4279	118	14	◦	◦	NOUN
ejpam-4279	118	15	λm	λm	NOUN
ejpam-4279	118	16	)	)	PUNCT
ejpam-4279	118	17	∧	∧	NOUN
ejpam-4279	118	18	λm	λm	ADP
ejpam-4279	118	19	̸=	̸=	PROPN
ejpam-4279	118	20	0	0	NUM
ejpam-4279	118	21	.	.	PUNCT
ejpam-4279	119	1	hence	hence	ADV
ejpam-4279	119	2	λm	λm	ADV
ejpam-4279	119	3	is	be	AUX
ejpam-4279	119	4	a	a	DET
ejpam-4279	119	5	fuzzy	fuzzy	ADJ
ejpam-4279	119	6	almost	almost	ADV
ejpam-4279	119	7	bi	bi	ADJ
ejpam-4279	119	8	-	-	ADJ
ejpam-4279	119	9	interior	interior	ADJ
ejpam-4279	119	10	ideal	ideal	NOUN
ejpam-4279	119	11	of	of	ADP
ejpam-4279	119	12	s.	s.	PROPN
ejpam-4279	119	13	for	for	ADP
ejpam-4279	119	14	the	the	DET
ejpam-4279	119	15	converse	converse	NOUN
ejpam-4279	119	16	,	,	PUNCT
ejpam-4279	119	17	assume	assume	VERB
ejpam-4279	119	18	that	that	SCONJ
ejpam-4279	119	19	λm	λm	SCONJ
ejpam-4279	119	20	is	be	AUX
ejpam-4279	119	21	a	a	DET
ejpam-4279	119	22	fuzzy	fuzzy	ADJ
ejpam-4279	119	23	almost	almost	ADV
ejpam-4279	119	24	bi	bi	ADJ
ejpam-4279	119	25	-	-	ADJ
ejpam-4279	119	26	interior	interior	ADJ
ejpam-4279	119	27	ideal	ideal	NOUN
ejpam-4279	119	28	of	of	ADP
ejpam-4279	119	29	s	s	PRON
ejpam-4279	119	30	,	,	PUNCT
ejpam-4279	119	31	let	let	VERB
ejpam-4279	119	32	h	h	NOUN
ejpam-4279	119	33	,	,	PUNCT
ejpam-4279	119	34	r	r	NOUN
ejpam-4279	119	35	,	,	PUNCT
ejpam-4279	119	36	n	n	PROPN
ejpam-4279	119	37	∈	∈	PROPN
ejpam-4279	119	38	s	s	X
ejpam-4279	119	39	and	and	CCONJ
ejpam-4279	119	40	δ	δ	PROPN
ejpam-4279	119	41	,	,	PUNCT
ejpam-4279	119	42	δ1	δ1	NOUN
ejpam-4279	119	43	,	,	PUNCT
ejpam-4279	119	44	δ2	δ2	VERB
ejpam-4279	119	45	∈	∈	PROPN
ejpam-4279	119	46	(	(	PUNCT
ejpam-4279	119	47	0	0	NUM
ejpam-4279	119	48	,	,	PUNCT
ejpam-4279	119	49	1	1	NUM
ejpam-4279	119	50	]	]	PUNCT
ejpam-4279	119	51	.	.	PUNCT
ejpam-4279	120	1	then	then	ADV
ejpam-4279	120	2	(	(	PUNCT
ejpam-4279	120	3	pδ	pδ	ADP
ejpam-4279	120	4	◦	◦	NOUN
ejpam-4279	120	5	λm	λm	ADP
ejpam-4279	120	6	◦	◦	NOUN
ejpam-4279	120	7	qδ1)∧(λm	qδ1)∧(λm	NOUN
ejpam-4279	120	8	◦	◦	NOUN
ejpam-4279	120	9	gδ2	gδ2	NOUN
ejpam-4279	120	10	◦	◦	NOUN
ejpam-4279	120	11	λm	λm	NOUN
ejpam-4279	120	12	)	)	PUNCT
ejpam-4279	120	13	∧λm	∧λm	PROPN
ejpam-4279	120	14	̸=	̸=	PROPN
ejpam-4279	120	15	0	0	NUM
ejpam-4279	120	16	.	.	PUNCT
ejpam-4279	121	1	thus	thus	ADV
ejpam-4279	121	2	there	there	PRON
ejpam-4279	121	3	exists	exist	VERB
ejpam-4279	121	4	c	c	PROPN
ejpam-4279	121	5	∈	∈	PROPN
ejpam-4279	121	6	s	s	VERB
ejpam-4279	121	7	such	such	ADJ
ejpam-4279	121	8	that	that	SCONJ
ejpam-4279	121	9	(	(	PUNCT
ejpam-4279	121	10	pδ	pδ	ADP
ejpam-4279	121	11	◦	◦	NOUN
ejpam-4279	121	12	λm	λm	ADP
ejpam-4279	121	13	◦	◦	NOUN
ejpam-4279	121	14	qδ1	qδ1	NOUN
ejpam-4279	121	15	)	)	PUNCT
ejpam-4279	121	16	∧	∧	NOUN
ejpam-4279	121	17	(	(	PUNCT
ejpam-4279	121	18	λm	λm	ADP
ejpam-4279	121	19	◦	◦	NOUN
ejpam-4279	121	20	gδ2	gδ2	INTJ
ejpam-4279	121	21	◦	◦	NOUN
ejpam-4279	121	22	λm	λm	X
ejpam-4279	121	23	)	)	PUNCT
ejpam-4279	121	24	(	(	PUNCT
ejpam-4279	121	25	c	c	X
ejpam-4279	121	26	)	)	PUNCT
ejpam-4279	121	27	̸=	̸=	NOUN
ejpam-4279	121	28	0	0	NUM
ejpam-4279	121	29	and	and	CCONJ
ejpam-4279	121	30	λm	λm	X
ejpam-4279	121	31	(	(	PUNCT
ejpam-4279	121	32	c	c	X
ejpam-4279	121	33	)	)	PUNCT
ejpam-4279	121	34	̸=	̸=	PROPN
ejpam-4279	121	35	0	0	NUM
ejpam-4279	121	36	.	.	PUNCT
ejpam-4279	122	1	so	so	ADV
ejpam-4279	122	2	c	c	PROPN
ejpam-4279	122	3	∈	∈	PROPN
ejpam-4279	122	4	(	(	PUNCT
ejpam-4279	122	5	hmr	hmr	NOUN
ejpam-4279	122	6	∩mnm	∩mnm	PROPN
ejpam-4279	122	7	)	)	PUNCT
ejpam-4279	122	8	and	and	CCONJ
ejpam-4279	122	9	c	c	X
ejpam-4279	122	10	∈	∈	PROPN
ejpam-4279	122	11	m	m	VERB
ejpam-4279	122	12	implies	imply	VERB
ejpam-4279	122	13	that	that	SCONJ
ejpam-4279	122	14	(	(	PUNCT
ejpam-4279	122	15	hmr	hmr	PROPN
ejpam-4279	122	16	∩	∩	PROPN
ejpam-4279	122	17	mnm	mnm	ADJ
ejpam-4279	122	18	)	)	PUNCT
ejpam-4279	122	19	∩	∩	NOUN
ejpam-4279	122	20	m	m	VERB
ejpam-4279	122	21	̸=	̸=	PROPN
ejpam-4279	122	22	∅.	∅.	PRON
ejpam-4279	122	23	therefore	therefore	ADV
ejpam-4279	122	24	m	m	PROPN
ejpam-4279	122	25	is	be	AUX
ejpam-4279	122	26	an	an	DET
ejpam-4279	122	27	almost	almost	ADV
ejpam-4279	122	28	bi	bi	ADJ
ejpam-4279	122	29	-	-	ADJ
ejpam-4279	122	30	interior	interior	ADJ
ejpam-4279	122	31	ideal	ideal	NOUN
ejpam-4279	122	32	of	of	ADP
ejpam-4279	122	33	s.	s.	PROPN
ejpam-4279	122	34	the	the	DET
ejpam-4279	122	35	proof	proof	NOUN
ejpam-4279	122	36	of	of	ADP
ejpam-4279	122	37	the	the	DET
ejpam-4279	122	38	other	other	ADJ
ejpam-4279	122	39	similar	similar	ADJ
ejpam-4279	122	40	to	to	ADP
ejpam-4279	122	41	the	the	DET
ejpam-4279	122	42	proof	proof	NOUN
ejpam-4279	122	43	(	(	PUNCT
ejpam-4279	122	44	1	1	NUM
ejpam-4279	122	45	)	)	PUNCT
ejpam-4279	122	46	.	.	PUNCT
ejpam-4279	123	1	theorem	theorem	ADJ
ejpam-4279	123	2	6	6	NUM
ejpam-4279	123	3	.	.	PUNCT
ejpam-4279	124	1	let	let	VERB
ejpam-4279	124	2	φ	φ	PROPN
ejpam-4279	124	3	be	be	AUX
ejpam-4279	124	4	a	a	DET
ejpam-4279	124	5	nonzero	nonzero	ADJ
ejpam-4279	124	6	fuzzy	fuzzy	ADJ
ejpam-4279	124	7	set	set	NOUN
ejpam-4279	124	8	of	of	ADP
ejpam-4279	124	9	a	a	DET
ejpam-4279	124	10	semigroup	semigroup	PROPN
ejpam-4279	124	11	s.	s.	PROPN
ejpam-4279	124	12	then	then	ADV
ejpam-4279	124	13	the	the	DET
ejpam-4279	124	14	following	follow	VERB
ejpam-4279	124	15	statement	statement	NOUN
ejpam-4279	124	16	hold	hold	NOUN
ejpam-4279	124	17	.	.	PUNCT
ejpam-4279	125	1	(	(	PUNCT
ejpam-4279	125	2	1	1	X
ejpam-4279	125	3	)	)	PUNCT
ejpam-4279	125	4	φ	φ	PROPN
ejpam-4279	125	5	is	be	AUX
ejpam-4279	125	6	a	a	DET
ejpam-4279	125	7	fuzzy	fuzzy	ADJ
ejpam-4279	125	8	almost	almost	ADV
ejpam-4279	125	9	bi	bi	ADJ
ejpam-4279	125	10	-	-	ADJ
ejpam-4279	125	11	interior	interior	ADJ
ejpam-4279	125	12	ideal	ideal	NOUN
ejpam-4279	125	13	of	of	ADP
ejpam-4279	125	14	s	s	PRON
ejpam-4279	125	15	if	if	SCONJ
ejpam-4279	126	1	and	and	CCONJ
ejpam-4279	126	2	only	only	ADV
ejpam-4279	126	3	if	if	SCONJ
ejpam-4279	126	4	supp(φ	supp(φ	VERB
ejpam-4279	126	5	)	)	PUNCT
ejpam-4279	126	6	is	be	AUX
ejpam-4279	126	7	an	an	DET
ejpam-4279	126	8	almost	almost	ADV
ejpam-4279	126	9	bi	bi	ADJ
ejpam-4279	126	10	-	-	ADJ
ejpam-4279	126	11	interior	interior	ADJ
ejpam-4279	126	12	ideal	ideal	NOUN
ejpam-4279	126	13	of	of	ADP
ejpam-4279	126	14	s.	s.	PROPN
ejpam-4279	126	15	(	(	PUNCT
ejpam-4279	126	16	2	2	X
ejpam-4279	126	17	)	)	PUNCT
ejpam-4279	126	18	φ	φ	PROPN
ejpam-4279	126	19	is	be	AUX
ejpam-4279	126	20	a	a	DET
ejpam-4279	126	21	weakly	weakly	ADJ
ejpam-4279	126	22	fuzzy	fuzzy	ADJ
ejpam-4279	126	23	almost	almost	ADV
ejpam-4279	126	24	bi	bi	ADJ
ejpam-4279	126	25	-	-	ADJ
ejpam-4279	126	26	interior	interior	ADJ
ejpam-4279	126	27	ideal	ideal	NOUN
ejpam-4279	126	28	of	of	ADP
ejpam-4279	126	29	s	s	PRON
ejpam-4279	126	30	if	if	SCONJ
ejpam-4279	126	31	and	and	CCONJ
ejpam-4279	126	32	only	only	ADV
ejpam-4279	127	1	if	if	SCONJ
ejpam-4279	127	2	supp(φ	supp(φ	VERB
ejpam-4279	127	3	)	)	PUNCT
ejpam-4279	127	4	is	be	AUX
ejpam-4279	127	5	a	a	DET
ejpam-4279	127	6	weakly	weakly	ADJ
ejpam-4279	127	7	almost	almost	ADV
ejpam-4279	127	8	bi	bi	ADJ
ejpam-4279	127	9	-	-	ADJ
ejpam-4279	127	10	interior	interior	ADJ
ejpam-4279	127	11	ideal	ideal	NOUN
ejpam-4279	127	12	of	of	ADP
ejpam-4279	127	13	s.	s.	PROPN
ejpam-4279	127	14	proof	proof	PROPN
ejpam-4279	127	15	.	.	PUNCT
ejpam-4279	128	1	suppose	suppose	VERB
ejpam-4279	128	2	that	that	SCONJ
ejpam-4279	128	3	φ	φ	PROPN
ejpam-4279	128	4	is	be	AUX
ejpam-4279	128	5	a	a	DET
ejpam-4279	128	6	fuzzy	fuzzy	ADJ
ejpam-4279	128	7	almost	almost	ADV
ejpam-4279	128	8	bi	bi	ADJ
ejpam-4279	128	9	-	-	ADJ
ejpam-4279	128	10	interior	interior	ADJ
ejpam-4279	128	11	ideal	ideal	NOUN
ejpam-4279	128	12	of	of	ADP
ejpam-4279	128	13	s	s	PROPN
ejpam-4279	128	14	,	,	PUNCT
ejpam-4279	128	15	p	p	X
ejpam-4279	128	16	,	,	PUNCT
ejpam-4279	128	17	q	q	X
ejpam-4279	128	18	,	,	PUNCT
ejpam-4279	128	19	g	g	PROPN
ejpam-4279	128	20	∈	∈	PROPN
ejpam-4279	128	21	s	s	PART
ejpam-4279	128	22	and	and	CCONJ
ejpam-4279	128	23	δ	δ	PROPN
ejpam-4279	128	24	,	,	PUNCT
ejpam-4279	128	25	δ1	δ1	NOUN
ejpam-4279	128	26	,	,	PUNCT
ejpam-4279	128	27	δ2	δ2	VERB
ejpam-4279	128	28	∈	∈	PROPN
ejpam-4279	128	29	(	(	PUNCT
ejpam-4279	128	30	0	0	NUM
ejpam-4279	128	31	,	,	PUNCT
ejpam-4279	128	32	1	1	NUM
ejpam-4279	128	33	]	]	PUNCT
ejpam-4279	128	34	.	.	PUNCT
ejpam-4279	129	1	then	then	ADV
ejpam-4279	129	2	(	(	PUNCT
ejpam-4279	129	3	pδ	pδ	ADP
ejpam-4279	129	4	◦	◦	NOUN
ejpam-4279	129	5	φ	φ	NUM
ejpam-4279	129	6	◦	◦	NOUN
ejpam-4279	129	7	qδ1	qδ1	NOUN
ejpam-4279	129	8	)	)	PUNCT
ejpam-4279	129	9	∧	∧	PROPN
ejpam-4279	129	10	(	(	PUNCT
ejpam-4279	129	11	φ	φ	NOUN
ejpam-4279	129	12	◦	◦	PROPN
ejpam-4279	129	13	gδ2	gδ2	PROPN
ejpam-4279	129	14	◦	◦	NOUN
ejpam-4279	129	15	φ	φ	NOUN
ejpam-4279	129	16	)	)	PUNCT
ejpam-4279	129	17	∧	∧	PROPN
ejpam-4279	129	18	φ	φ	NUM
ejpam-4279	129	19	̸=	̸=	PROPN
ejpam-4279	129	20	0	0	NUM
ejpam-4279	129	21	.	.	PUNCT
ejpam-4279	130	1	thus	thus	ADV
ejpam-4279	130	2	there	there	PRON
ejpam-4279	130	3	exist	exist	VERB
ejpam-4279	130	4	c	c	PROPN
ejpam-4279	130	5	∈	∈	PROPN
ejpam-4279	130	6	s	s	VERB
ejpam-4279	130	7	such	such	ADJ
ejpam-4279	130	8	that	that	SCONJ
ejpam-4279	130	9	(	(	PUNCT
ejpam-4279	130	10	pδ	pδ	ADP
ejpam-4279	130	11	◦	◦	NOUN
ejpam-4279	130	12	φ	φ	NUM
ejpam-4279	130	13	◦	◦	NOUN
ejpam-4279	130	14	qδ1	qδ1	NOUN
ejpam-4279	130	15	)	)	PUNCT
ejpam-4279	130	16	∧	∧	PROPN
ejpam-4279	130	17	(	(	PUNCT
ejpam-4279	130	18	φ	φ	NOUN
ejpam-4279	130	19	◦	◦	PROPN
ejpam-4279	130	20	gδ2	gδ2	PROPN
ejpam-4279	130	21	◦	◦	NOUN
ejpam-4279	130	22	φ)(c	φ)(c	NOUN
ejpam-4279	130	23	)	)	PUNCT
ejpam-4279	130	24	̸=	̸=	PROPN
ejpam-4279	130	25	0	0	NUM
ejpam-4279	130	26	and	and	CCONJ
ejpam-4279	130	27	φ(c	φ(c	NOUN
ejpam-4279	130	28	)	)	PUNCT
ejpam-4279	130	29	̸=	̸=	PROPN
ejpam-4279	130	30	0	0	NUM
ejpam-4279	130	31	.	.	PUNCT
ejpam-4279	131	1	so	so	ADV
ejpam-4279	131	2	there	there	PRON
ejpam-4279	131	3	exist	exist	VERB
ejpam-4279	131	4	h	h	NOUN
ejpam-4279	131	5	,	,	PUNCT
ejpam-4279	131	6	r	r	NOUN
ejpam-4279	131	7	,	,	PUNCT
ejpam-4279	131	8	n	n	PRON
ejpam-4279	131	9	such	such	ADJ
ejpam-4279	131	10	that	that	DET
ejpam-4279	131	11	c	c	NOUN
ejpam-4279	132	1	=	=	PUNCT
ejpam-4279	132	2	hm1r	hm1r	PROPN
ejpam-4279	132	3	and	and	CCONJ
ejpam-4279	132	4	c	c	NOUN
ejpam-4279	132	5	=	=	SYM
ejpam-4279	132	6	m2nm3	m2nm3	NOUN
ejpam-4279	132	7	it	it	PRON
ejpam-4279	132	8	follows	follow	VERB
ejpam-4279	132	9	that	that	SCONJ
ejpam-4279	132	10	t.	t.	PROPN
ejpam-4279	132	11	gaketem	gaketem	PROPN
ejpam-4279	132	12	/	/	SYM
ejpam-4279	132	13	eur	eur	PROPN
ejpam-4279	132	14	.	.	PUNCT
ejpam-4279	133	1	j.	j.	PROPN
ejpam-4279	133	2	pure	pure	PROPN
ejpam-4279	133	3	appl	appl	PROPN
ejpam-4279	133	4	.	.	PROPN
ejpam-4279	133	5	math	math	PROPN
ejpam-4279	133	6	,	,	PUNCT
ejpam-4279	133	7	15	15	NUM
ejpam-4279	133	8	(	(	PUNCT
ejpam-4279	133	9	1	1	NUM
ejpam-4279	133	10	)	)	PUNCT
ejpam-4279	133	11	(	(	PUNCT
ejpam-4279	133	12	2022	2022	NUM
ejpam-4279	133	13	)	)	PUNCT
ejpam-4279	133	14	,	,	PUNCT
ejpam-4279	133	15	281	281	NUM
ejpam-4279	133	16	-	-	SYM
ejpam-4279	133	17	289	289	NUM
ejpam-4279	133	18	287	287	NUM
ejpam-4279	133	19	(	(	PUNCT
ejpam-4279	133	20	∨	∨	NOUN
ejpam-4279	133	21	c	c	X
ejpam-4279	133	22	=	=	X
ejpam-4279	133	23	hm1r	hm1r	X
ejpam-4279	133	24	{	{	PUNCT
ejpam-4279	133	25	pδ(h	pδ(h	NOUN
ejpam-4279	133	26	)	)	PUNCT
ejpam-4279	133	27	∧	∧	PROPN
ejpam-4279	133	28	φ(m1	φ(m1	NOUN
ejpam-4279	133	29	)	)	PUNCT
ejpam-4279	133	30	∧	∧	PROPN
ejpam-4279	133	31	qδ1(r	qδ1(r	NOUN
ejpam-4279	133	32	)	)	PUNCT
ejpam-4279	133	33	}	}	PUNCT
ejpam-4279	133	34	)	)	PUNCT
ejpam-4279	134	1	∧	∧	NOUN
ejpam-4279	134	2	(	(	PUNCT
ejpam-4279	134	3	∨	∨	PROPN
ejpam-4279	134	4	c	c	NOUN
ejpam-4279	134	5	=	=	X
ejpam-4279	134	6	m2nm3	m2nm3	NOUN
ejpam-4279	134	7	{	{	PUNCT
ejpam-4279	134	8	φ(m2	φ(m2	NOUN
ejpam-4279	134	9	)	)	PUNCT
ejpam-4279	134	10	∧	∧	PROPN
ejpam-4279	134	11	gδ2(n	gδ2(n	PROPN
ejpam-4279	134	12	)	)	PUNCT
ejpam-4279	134	13	∧	∧	PROPN
ejpam-4279	134	14	φ(m3	φ(m3	ADJ
ejpam-4279	134	15	)	)	PUNCT
ejpam-4279	134	16	}	}	PUNCT
ejpam-4279	134	17	)	)	PUNCT
ejpam-4279	135	1	=	=	SYM
ejpam-4279	135	2	(	(	PUNCT
ejpam-4279	135	3	pδ	pδ	ADP
ejpam-4279	135	4	◦	◦	NOUN
ejpam-4279	135	5	φ	φ	NUM
ejpam-4279	135	6	◦	◦	NOUN
ejpam-4279	135	7	qδ1	qδ1	NOUN
ejpam-4279	135	8	)	)	PUNCT
ejpam-4279	135	9	∧	∧	PROPN
ejpam-4279	135	10	(	(	PUNCT
ejpam-4279	135	11	φ	φ	X
ejpam-4279	135	12	◦	◦	NOUN
ejpam-4279	135	13	gδ2	gδ2	PROPN
ejpam-4279	135	14	∧	∧	PROPN
ejpam-4279	135	15	φ	φ	NOUN
ejpam-4279	135	16	)	)	PUNCT
ejpam-4279	135	17	̸=	̸=	PROPN
ejpam-4279	135	18	0	0	NUM
ejpam-4279	135	19	.	.	PUNCT
ejpam-4279	136	1	thus	thus	ADV
ejpam-4279	136	2	φ(m1	φ(m1	NOUN
ejpam-4279	136	3	)	)	PUNCT
ejpam-4279	136	4	̸=	̸=	PROPN
ejpam-4279	136	5	0	0	NUM
ejpam-4279	136	6	,	,	PUNCT
ejpam-4279	136	7	φ(m2	φ(m2	NUM
ejpam-4279	136	8	)	)	PUNCT
ejpam-4279	136	9	̸=	̸=	PROPN
ejpam-4279	136	10	0	0	NUM
ejpam-4279	136	11	,	,	PUNCT
ejpam-4279	136	12	φ(m3	φ(m3	ADJ
ejpam-4279	136	13	)	)	PUNCT
ejpam-4279	136	14	̸=	̸=	PROPN
ejpam-4279	136	15	0	0	NUM
ejpam-4279	137	1	so	so	CCONJ
ejpam-4279	137	2	m1,m2,m3	m1,m2,m3	PROPN
ejpam-4279	137	3	∈	∈	PROPN
ejpam-4279	137	4	supp(φ	supp(φ	NOUN
ejpam-4279	137	5	)	)	PUNCT
ejpam-4279	137	6	implies	imply	VERB
ejpam-4279	137	7	that	that	SCONJ
ejpam-4279	137	8	(	(	PUNCT
ejpam-4279	137	9	pδ	pδ	ADP
ejpam-4279	137	10	◦	◦	NOUN
ejpam-4279	137	11	λsupp(φ	λsupp(φ	PROPN
ejpam-4279	137	12	)	)	PUNCT
ejpam-4279	137	13	◦	◦	NOUN
ejpam-4279	137	14	qδ1)∧	qδ1)∧	NOUN
ejpam-4279	137	15	(	(	PUNCT
ejpam-4279	137	16	λsupp(φ	λsupp(φ	PROPN
ejpam-4279	137	17	)	)	PUNCT
ejpam-4279	137	18	◦	◦	NOUN
ejpam-4279	137	19	gδ2	gδ2	NOUN
ejpam-4279	137	20	∧λsupp(φ))(c	∧λsupp(φ))(c	NOUN
ejpam-4279	137	21	)	)	PUNCT
ejpam-4279	137	22	̸=	̸=	PROPN
ejpam-4279	137	23	0	0	NUM
ejpam-4279	137	24	.	.	PUNCT
ejpam-4279	138	1	thus	thus	ADV
ejpam-4279	138	2	λsupp(φ	λsupp(φ	VERB
ejpam-4279	138	3	)	)	PUNCT
ejpam-4279	138	4	is	be	AUX
ejpam-4279	138	5	a	a	DET
ejpam-4279	138	6	fuzzy	fuzzy	ADJ
ejpam-4279	138	7	almost	almost	ADV
ejpam-4279	138	8	bi	bi	ADJ
ejpam-4279	138	9	-	-	ADJ
ejpam-4279	138	10	interior	interior	ADJ
ejpam-4279	138	11	ideal	ideal	NOUN
ejpam-4279	138	12	of	of	ADP
ejpam-4279	138	13	s.	s.	PROPN
ejpam-4279	138	14	by	by	ADP
ejpam-4279	138	15	theorem	theorem	NOUN
ejpam-4279	138	16	5	5	NUM
ejpam-4279	138	17	,	,	PUNCT
ejpam-4279	138	18	supp(φ	supp(φ	NOUN
ejpam-4279	138	19	)	)	PUNCT
ejpam-4279	138	20	is	be	AUX
ejpam-4279	138	21	an	an	DET
ejpam-4279	138	22	almost	almost	ADV
ejpam-4279	138	23	bi	bi	ADJ
ejpam-4279	138	24	-	-	ADJ
ejpam-4279	138	25	interior	interior	ADJ
ejpam-4279	138	26	ideal	ideal	NOUN
ejpam-4279	138	27	of	of	ADP
ejpam-4279	138	28	s.	s.	PROPN
ejpam-4279	138	29	for	for	ADP
ejpam-4279	138	30	the	the	DET
ejpam-4279	138	31	converse	converse	NOUN
ejpam-4279	138	32	,	,	PUNCT
ejpam-4279	138	33	assume	assume	VERB
ejpam-4279	138	34	that	that	SCONJ
ejpam-4279	138	35	supp(φ	supp(φ	NOUN
ejpam-4279	138	36	)	)	PUNCT
ejpam-4279	138	37	is	be	AUX
ejpam-4279	138	38	an	an	DET
ejpam-4279	138	39	almost	almost	ADV
ejpam-4279	138	40	bi	bi	ADJ
ejpam-4279	138	41	-	-	ADJ
ejpam-4279	138	42	interior	interior	ADJ
ejpam-4279	138	43	ideal	ideal	NOUN
ejpam-4279	138	44	of	of	ADP
ejpam-4279	138	45	s.	s.	PROPN
ejpam-4279	138	46	by	by	ADP
ejpam-4279	138	47	theorem	theorem	ADJ
ejpam-4279	138	48	5	5	NUM
ejpam-4279	138	49	,	,	PUNCT
ejpam-4279	138	50	λsupp(φ	λsupp(φ	PROPN
ejpam-4279	138	51	)	)	PUNCT
ejpam-4279	138	52	is	be	AUX
ejpam-4279	138	53	a	a	DET
ejpam-4279	138	54	fuzzy	fuzzy	ADJ
ejpam-4279	138	55	almost	almost	ADV
ejpam-4279	138	56	bi	bi	ADJ
ejpam-4279	138	57	-	-	ADJ
ejpam-4279	138	58	interior	interior	ADJ
ejpam-4279	138	59	ideal	ideal	NOUN
ejpam-4279	138	60	of	of	ADP
ejpam-4279	138	61	s.	s.	PROPN
ejpam-4279	138	62	thus	thus	ADV
ejpam-4279	138	63	for	for	ADP
ejpam-4279	138	64	any	any	DET
ejpam-4279	138	65	fuzzy	fuzzy	ADJ
ejpam-4279	138	66	point	point	NOUN
ejpam-4279	138	67	pδ	pδ	PROPN
ejpam-4279	138	68	,	,	PUNCT
ejpam-4279	138	69	qδ1,gδ2	qδ1,gδ2	PROPN
ejpam-4279	138	70	of	of	ADP
ejpam-4279	138	71	s	s	PRON
ejpam-4279	138	72	such	such	ADJ
ejpam-4279	138	73	that	that	SCONJ
ejpam-4279	138	74	(	(	PUNCT
ejpam-4279	138	75	pδ	pδ	ADP
ejpam-4279	138	76	◦	◦	NOUN
ejpam-4279	138	77	λsupp(φ	λsupp(φ	PROPN
ejpam-4279	138	78	)	)	PUNCT
ejpam-4279	138	79	◦	◦	NOUN
ejpam-4279	138	80	qδ1	qδ1	NOUN
ejpam-4279	138	81	)	)	PUNCT
ejpam-4279	138	82	∧	∧	PROPN
ejpam-4279	138	83	(	(	PUNCT
ejpam-4279	138	84	λsupp(φ	λsupp(φ	PROPN
ejpam-4279	138	85	)	)	PUNCT
ejpam-4279	138	86	◦	◦	NOUN
ejpam-4279	138	87	gδ2	gδ2	PROPN
ejpam-4279	138	88	◦	◦	NOUN
ejpam-4279	138	89	λsupp(φ	λsupp(φ	PROPN
ejpam-4279	138	90	)	)	PUNCT
ejpam-4279	138	91	)	)	PUNCT
ejpam-4279	139	1	̸=	̸=	PROPN
ejpam-4279	139	2	0	0	NUM
ejpam-4279	139	3	.	.	PUNCT
ejpam-4279	140	1	so	so	ADV
ejpam-4279	140	2	there	there	PRON
ejpam-4279	140	3	exists	exist	VERB
ejpam-4279	140	4	c	c	PROPN
ejpam-4279	140	5	∈	∈	PROPN
ejpam-4279	140	6	s	s	VERB
ejpam-4279	140	7	such	such	ADJ
ejpam-4279	140	8	that	that	SCONJ
ejpam-4279	140	9	(	(	PUNCT
ejpam-4279	140	10	pδ	pδ	NOUN
ejpam-4279	140	11	◦	◦	NOUN
ejpam-4279	140	12	λsupp(φ)	λsupp(φ)	NOUN
ejpam-4279	140	13	◦	◦	NOUN
ejpam-4279	140	14	qδ1)∧(λsupp(φ)	qδ1)∧(λsupp(φ)	NOUN
ejpam-4279	140	15	◦	◦	ADJ
ejpam-4279	140	16	gδ2)(c	gδ2)(c	NOUN
ejpam-4279	140	17	)	)	PUNCT
ejpam-4279	140	18	̸=	̸=	PROPN
ejpam-4279	140	19	0	0	NUM
ejpam-4279	140	20	and	and	CCONJ
ejpam-4279	140	21	λsupp(φ)(c	λsupp(φ)(c	NOUN
ejpam-4279	140	22	)	)	PUNCT
ejpam-4279	140	23	̸=	̸=	PROPN
ejpam-4279	140	24	0	0	NUM
ejpam-4279	140	25	.	.	PUNCT
ejpam-4279	141	1	thus	thus	ADV
ejpam-4279	141	2	there	there	PRON
ejpam-4279	141	3	exist	exist	VERB
ejpam-4279	141	4	h	h	NOUN
ejpam-4279	141	5	,	,	PUNCT
ejpam-4279	141	6	r	r	NOUN
ejpam-4279	141	7	,	,	PUNCT
ejpam-4279	141	8	n	n	PRON
ejpam-4279	141	9	∈	∈	NOUN
ejpam-4279	141	10	s	s	VERB
ejpam-4279	141	11	such	such	ADJ
ejpam-4279	141	12	that	that	PRON
ejpam-4279	141	13	c	c	NOUN
ejpam-4279	141	14	=	=	PUNCT
ejpam-4279	141	15	hm1r	hm1r	PROPN
ejpam-4279	141	16	and	and	CCONJ
ejpam-4279	141	17	c	c	NOUN
ejpam-4279	141	18	=	=	SYM
ejpam-4279	141	19	m2nm3	m2nm3	NOUN
ejpam-4279	141	20	it	it	PRON
ejpam-4279	141	21	follows	follow	VERB
ejpam-4279	141	22	that	that	SCONJ
ejpam-4279	141	23	(	(	PUNCT
ejpam-4279	141	24	∨	∨	NOUN
ejpam-4279	141	25	c	c	X
ejpam-4279	141	26	=	=	X
ejpam-4279	141	27	hm1r	hm1r	X
ejpam-4279	141	28	{	{	PUNCT
ejpam-4279	141	29	pδ(h	pδ(h	NOUN
ejpam-4279	141	30	)	)	PUNCT
ejpam-4279	141	31	∧	∧	PROPN
ejpam-4279	141	32	λsupp(φ)(m1	λsupp(φ)(m1	NOUN
ejpam-4279	141	33	)	)	PUNCT
ejpam-4279	141	34	∧	∧	PROPN
ejpam-4279	141	35	qδ1(r)})∧	qδ1(r)})∧	PROPN
ejpam-4279	141	36	(	(	PUNCT
ejpam-4279	141	37	∨	∨	PROPN
ejpam-4279	141	38	c	c	NOUN
ejpam-4279	141	39	=	=	X
ejpam-4279	141	40	m2nm3	m2nm3	NOUN
ejpam-4279	141	41	{	{	PUNCT
ejpam-4279	141	42	λsupp(φ)(m2	λsupp(φ)(m2	NOUN
ejpam-4279	141	43	)	)	PUNCT
ejpam-4279	141	44	∧	∧	PROPN
ejpam-4279	141	45	gδ2(n	gδ2(n	PROPN
ejpam-4279	141	46	)	)	PUNCT
ejpam-4279	141	47	∧	∧	NOUN
ejpam-4279	141	48	λsupp(φ)(m3	λsupp(φ)(m3	NOUN
ejpam-4279	141	49	)	)	PUNCT
ejpam-4279	141	50	}	}	PUNCT
ejpam-4279	141	51	)	)	PUNCT
ejpam-4279	141	52	̸=	̸=	PROPN
ejpam-4279	141	53	0	0	NUM
ejpam-4279	141	54	.	.	PUNCT
ejpam-4279	142	1	so	so	ADV
ejpam-4279	142	2	λsupp(φ)(m1	λsupp(φ)(m1	NOUN
ejpam-4279	142	3	)	)	PUNCT
ejpam-4279	142	4	̸=	̸=	PROPN
ejpam-4279	142	5	0	0	NUM
ejpam-4279	142	6	,	,	PUNCT
ejpam-4279	142	7	λsupp(φ)(m2	λsupp(φ)(m2	X
ejpam-4279	142	8	)	)	PUNCT
ejpam-4279	142	9	̸=	̸=	PROPN
ejpam-4279	142	10	0	0	NUM
ejpam-4279	142	11	,	,	PUNCT
ejpam-4279	142	12	λsupp(φ)(m3	λsupp(φ)(m3	ADJ
ejpam-4279	142	13	)	)	PUNCT
ejpam-4279	142	14	̸=	̸=	PROPN
ejpam-4279	142	15	0	0	NUM
ejpam-4279	142	16	implies	imply	VERB
ejpam-4279	142	17	that	that	SCONJ
ejpam-4279	142	18	φ(m1	φ(m1	NOUN
ejpam-4279	142	19	)	)	PUNCT
ejpam-4279	142	20	̸=	̸=	PROPN
ejpam-4279	142	21	0	0	NUM
ejpam-4279	142	22	,	,	PUNCT
ejpam-4279	142	23	φ(m2	φ(m2	NUM
ejpam-4279	142	24	)	)	PUNCT
ejpam-4279	142	25	̸=	̸=	PROPN
ejpam-4279	142	26	0	0	NUM
ejpam-4279	142	27	,	,	PUNCT
ejpam-4279	142	28	φ(m3	φ(m3	ADJ
ejpam-4279	142	29	)	)	PUNCT
ejpam-4279	142	30	̸=	̸=	PROPN
ejpam-4279	142	31	0	0	NUM
ejpam-4279	142	32	.	.	PUNCT
ejpam-4279	143	1	hence	hence	ADV
ejpam-4279	143	2	(	(	PUNCT
ejpam-4279	143	3	pδ	pδ	ADP
ejpam-4279	143	4	◦	◦	NOUN
ejpam-4279	143	5	λsupp(φ	λsupp(φ	PROPN
ejpam-4279	143	6	)	)	PUNCT
ejpam-4279	143	7	◦	◦	NOUN
ejpam-4279	143	8	qδ1	qδ1	NOUN
ejpam-4279	143	9	)	)	PUNCT
ejpam-4279	143	10	∧	∧	PROPN
ejpam-4279	143	11	(	(	PUNCT
ejpam-4279	143	12	λsupp(φ	λsupp(φ	PROPN
ejpam-4279	143	13	)	)	PUNCT
ejpam-4279	143	14	◦	◦	NOUN
ejpam-4279	143	15	gδ2)(c	gδ2)(c	PROPN
ejpam-4279	143	16	)	)	PUNCT
ejpam-4279	143	17	=	=	PUNCT
ejpam-4279	143	18	(	(	PUNCT
ejpam-4279	143	19	∨	∨	X
ejpam-4279	143	20	c	c	X
ejpam-4279	143	21	=	=	X
ejpam-4279	143	22	hm1r	hm1r	X
ejpam-4279	143	23	{	{	PUNCT
ejpam-4279	143	24	pδ(h	pδ(h	NOUN
ejpam-4279	143	25	)	)	PUNCT
ejpam-4279	143	26	∧	∧	PROPN
ejpam-4279	143	27	φ(m1	φ(m1	NOUN
ejpam-4279	143	28	)	)	PUNCT
ejpam-4279	143	29	∧	∧	PROPN
ejpam-4279	143	30	qδ1(r	qδ1(r	NOUN
ejpam-4279	143	31	)	)	PUNCT
ejpam-4279	143	32	}	}	PUNCT
ejpam-4279	143	33	)	)	PUNCT
ejpam-4279	143	34	∧	∧	NOUN
ejpam-4279	143	35	(	(	PUNCT
ejpam-4279	143	36	∨	∨	PROPN
ejpam-4279	143	37	c	c	NOUN
ejpam-4279	143	38	=	=	X
ejpam-4279	143	39	m2nm3	m2nm3	NOUN
ejpam-4279	143	40	{	{	PUNCT
ejpam-4279	143	41	φ(m2	φ(m2	NOUN
ejpam-4279	143	42	)	)	PUNCT
ejpam-4279	143	43	∧	∧	PROPN
ejpam-4279	143	44	gδ2(n	gδ2(n	PROPN
ejpam-4279	143	45	)	)	PUNCT
ejpam-4279	143	46	∧	∧	PROPN
ejpam-4279	143	47	φ(m3	φ(m3	ADJ
ejpam-4279	143	48	)	)	PUNCT
ejpam-4279	143	49	}	}	PUNCT
ejpam-4279	143	50	)	)	PUNCT
ejpam-4279	144	1	̸=	̸=	PROPN
ejpam-4279	144	2	0	0	NUM
ejpam-4279	144	3	.	.	PUNCT
ejpam-4279	145	1	it	it	PRON
ejpam-4279	145	2	follows	follow	VERB
ejpam-4279	145	3	that	that	SCONJ
ejpam-4279	145	4	(	(	PUNCT
ejpam-4279	145	5	pδ	pδ	ADP
ejpam-4279	145	6	◦	◦	NOUN
ejpam-4279	145	7	φ	φ	NUM
ejpam-4279	145	8	◦	◦	NOUN
ejpam-4279	145	9	qδ1	qδ1	NOUN
ejpam-4279	145	10	)	)	PUNCT
ejpam-4279	145	11	∧	∧	PROPN
ejpam-4279	145	12	(	(	PUNCT
ejpam-4279	145	13	φ	φ	NOUN
ejpam-4279	145	14	◦	◦	PROPN
ejpam-4279	145	15	gδ2	gδ2	PROPN
ejpam-4279	145	16	◦	◦	NOUN
ejpam-4279	145	17	φ	φ	NOUN
ejpam-4279	145	18	)	)	PUNCT
ejpam-4279	145	19	∧	∧	PROPN
ejpam-4279	145	20	φ	φ	NUM
ejpam-4279	145	21	̸=	̸=	PROPN
ejpam-4279	145	22	0	0	NUM
ejpam-4279	145	23	for	for	ADP
ejpam-4279	145	24	any	any	DET
ejpam-4279	145	25	fuzzy	fuzzy	ADJ
ejpam-4279	145	26	point	point	NOUN
ejpam-4279	145	27	pδ	pδ	PROPN
ejpam-4279	145	28	,	,	PUNCT
ejpam-4279	145	29	qδ1,gδ2	qδ1,gδ2	PROPN
ejpam-4279	145	30	of	of	ADP
ejpam-4279	145	31	s	s	PRON
ejpam-4279	145	32	therefore	therefore	ADV
ejpam-4279	145	33	φ	φ	PROPN
ejpam-4279	145	34	is	be	AUX
ejpam-4279	145	35	a	a	DET
ejpam-4279	145	36	fuzzy	fuzzy	ADJ
ejpam-4279	145	37	almost	almost	ADV
ejpam-4279	145	38	bi	bi	ADJ
ejpam-4279	145	39	-	-	ADJ
ejpam-4279	145	40	interior	interior	ADJ
ejpam-4279	145	41	ideal	ideal	NOUN
ejpam-4279	145	42	of	of	ADP
ejpam-4279	145	43	s.	s.	PROPN
ejpam-4279	145	44	the	the	DET
ejpam-4279	145	45	proof	proof	NOUN
ejpam-4279	145	46	of	of	ADP
ejpam-4279	145	47	the	the	DET
ejpam-4279	145	48	other	other	ADJ
ejpam-4279	145	49	similar	similar	ADJ
ejpam-4279	145	50	to	to	ADP
ejpam-4279	145	51	the	the	DET
ejpam-4279	145	52	proof	proof	NOUN
ejpam-4279	145	53	(	(	PUNCT
ejpam-4279	145	54	1	1	NUM
ejpam-4279	145	55	)	)	PUNCT
ejpam-4279	145	56	.	.	PUNCT
ejpam-4279	146	1	next	next	ADV
ejpam-4279	146	2	,	,	PUNCT
ejpam-4279	146	3	we	we	PRON
ejpam-4279	146	4	define	define	VERB
ejpam-4279	146	5	minimal	minimal	ADJ
ejpam-4279	146	6	fuzzy	fuzzy	ADJ
ejpam-4279	146	7	almost	almost	ADV
ejpam-4279	146	8	bi	bi	ADJ
ejpam-4279	146	9	-	-	ADJ
ejpam-4279	146	10	interior	interior	ADJ
ejpam-4279	146	11	ideals	ideal	NOUN
ejpam-4279	146	12	in	in	ADP
ejpam-4279	146	13	semigroups	semigroup	NOUN
ejpam-4279	146	14	and	and	CCONJ
ejpam-4279	146	15	study	study	VERB
ejpam-4279	146	16	the	the	PRON
ejpam-4279	146	17	between	between	ADP
ejpam-4279	146	18	minimal	minimal	ADJ
ejpam-4279	146	19	almost	almost	ADV
ejpam-4279	146	20	bi	bi	ADJ
ejpam-4279	146	21	-	-	ADJ
ejpam-4279	146	22	interior	interior	ADJ
ejpam-4279	146	23	ideals	ideal	NOUN
ejpam-4279	146	24	and	and	CCONJ
ejpam-4279	146	25	minimal	minimal	ADJ
ejpam-4279	146	26	fuzzy	fuzzy	ADJ
ejpam-4279	146	27	almost	almost	ADV
ejpam-4279	146	28	bi	bi	ADJ
ejpam-4279	146	29	-	-	ADJ
ejpam-4279	146	30	interior	interior	ADJ
ejpam-4279	146	31	ideals	ideal	NOUN
ejpam-4279	146	32	of	of	ADP
ejpam-4279	146	33	semigroups	semigroup	NOUN
ejpam-4279	146	34	.	.	PUNCT
ejpam-4279	147	1	definition	definition	NOUN
ejpam-4279	147	2	6	6	NUM
ejpam-4279	147	3	.	.	PUNCT
ejpam-4279	148	1	a	a	DET
ejpam-4279	148	2	fuzzy	fuzzy	ADJ
ejpam-4279	148	3	almost	almost	ADV
ejpam-4279	148	4	bi	bi	ADJ
ejpam-4279	148	5	-	-	ADJ
ejpam-4279	148	6	interior	interior	ADJ
ejpam-4279	148	7	ideal	ideal	NOUN
ejpam-4279	148	8	(	(	PUNCT
ejpam-4279	148	9	weak	weak	ADJ
ejpam-4279	148	10	bi	bi	ADJ
ejpam-4279	148	11	-	-	ADJ
ejpam-4279	148	12	interior	interior	ADJ
ejpam-4279	148	13	ideal	ideal	NOUN
ejpam-4279	148	14	)	)	PUNCT
ejpam-4279	148	15	φ	φ	NOUN
ejpam-4279	148	16	of	of	ADP
ejpam-4279	148	17	a	a	DET
ejpam-4279	148	18	semigroup	semigroup	NOUN
ejpam-4279	148	19	s	s	PART
ejpam-4279	148	20	is	be	AUX
ejpam-4279	148	21	called	call	VERB
ejpam-4279	148	22	minimal	minimal	ADJ
ejpam-4279	148	23	if	if	SCONJ
ejpam-4279	148	24	for	for	ADP
ejpam-4279	148	25	any	any	DET
ejpam-4279	148	26	fuzzy	fuzzy	ADJ
ejpam-4279	148	27	almost	almost	ADV
ejpam-4279	148	28	bi	bi	ADJ
ejpam-4279	148	29	-	-	ADJ
ejpam-4279	148	30	interior	interior	ADJ
ejpam-4279	148	31	ideal	ideal	NOUN
ejpam-4279	148	32	(	(	PUNCT
ejpam-4279	148	33	weak	weak	ADJ
ejpam-4279	148	34	bi	bi	ADJ
ejpam-4279	148	35	-	-	ADJ
ejpam-4279	148	36	interior	interior	ADJ
ejpam-4279	148	37	ideal	ideal	NOUN
ejpam-4279	148	38	)	)	PUNCT
ejpam-4279	149	1	ξ	ξ	PROPN
ejpam-4279	149	2	of	of	ADP
ejpam-4279	149	3	s	s	PRON
ejpam-4279	149	4	if	if	SCONJ
ejpam-4279	149	5	whenever	whenever	SCONJ
ejpam-4279	149	6	ξ	ξ	PROPN
ejpam-4279	149	7	⊆	⊆	NUM
ejpam-4279	149	8	φ	φ	NUM
ejpam-4279	149	9	,	,	PUNCT
ejpam-4279	149	10	then	then	ADV
ejpam-4279	149	11	sup(ξ	sup(ξ	PROPN
ejpam-4279	149	12	)	)	PUNCT
ejpam-4279	149	13	=	=	SYM
ejpam-4279	149	14	sup(φ	sup(φ	PROPN
ejpam-4279	149	15	)	)	PUNCT
ejpam-4279	149	16	.	.	PUNCT
ejpam-4279	150	1	theorem	theorem	ADJ
ejpam-4279	150	2	7	7	NUM
ejpam-4279	150	3	.	.	PUNCT
ejpam-4279	151	1	let	let	VERB
ejpam-4279	151	2	m	m	PRON
ejpam-4279	151	3	be	be	AUX
ejpam-4279	151	4	a	a	DET
ejpam-4279	151	5	nonempty	nonempty	ADJ
ejpam-4279	151	6	subset	subset	NOUN
ejpam-4279	151	7	of	of	ADP
ejpam-4279	151	8	a	a	DET
ejpam-4279	151	9	semigroup	semigroup	PROPN
ejpam-4279	151	10	s.	s.	PROPN
ejpam-4279	151	11	then	then	ADV
ejpam-4279	151	12	(	(	PUNCT
ejpam-4279	151	13	1	1	X
ejpam-4279	151	14	)	)	PUNCT
ejpam-4279	152	1	m	m	VERB
ejpam-4279	152	2	is	be	AUX
ejpam-4279	152	3	a	a	DET
ejpam-4279	152	4	minimal	minimal	ADJ
ejpam-4279	152	5	almost	almost	ADV
ejpam-4279	152	6	bi	bi	ADJ
ejpam-4279	152	7	-	-	ADJ
ejpam-4279	152	8	interior	interior	ADJ
ejpam-4279	152	9	ideal	ideal	NOUN
ejpam-4279	152	10	of	of	ADP
ejpam-4279	152	11	s	s	PRON
ejpam-4279	152	12	if	if	SCONJ
ejpam-4279	152	13	and	and	CCONJ
ejpam-4279	152	14	only	only	ADV
ejpam-4279	152	15	if	if	SCONJ
ejpam-4279	152	16	λm	λm	ADV
ejpam-4279	152	17	is	be	AUX
ejpam-4279	152	18	a	a	DET
ejpam-4279	152	19	minimal	minimal	ADJ
ejpam-4279	152	20	fuzzy	fuzzy	ADJ
ejpam-4279	152	21	almost	almost	ADV
ejpam-4279	152	22	bi	bi	ADJ
ejpam-4279	152	23	-	-	ADJ
ejpam-4279	152	24	interior	interior	ADJ
ejpam-4279	152	25	ideal	ideal	NOUN
ejpam-4279	152	26	of	of	ADP
ejpam-4279	152	27	s.	s.	PROPN
ejpam-4279	152	28	(	(	PUNCT
ejpam-4279	152	29	2	2	X
ejpam-4279	152	30	)	)	PUNCT
ejpam-4279	153	1	m	m	VERB
ejpam-4279	153	2	is	be	AUX
ejpam-4279	153	3	a	a	DET
ejpam-4279	153	4	minimal	minimal	ADJ
ejpam-4279	153	5	weak	weak	ADJ
ejpam-4279	153	6	almost	almost	ADV
ejpam-4279	153	7	bi	bi	ADJ
ejpam-4279	153	8	-	-	ADJ
ejpam-4279	153	9	interior	interior	ADJ
ejpam-4279	153	10	ideal	ideal	NOUN
ejpam-4279	153	11	of	of	ADP
ejpam-4279	153	12	s	s	PRON
ejpam-4279	153	13	if	if	SCONJ
ejpam-4279	153	14	and	and	CCONJ
ejpam-4279	153	15	only	only	ADV
ejpam-4279	153	16	if	if	SCONJ
ejpam-4279	153	17	λm	λm	ADV
ejpam-4279	153	18	is	be	AUX
ejpam-4279	153	19	a	a	DET
ejpam-4279	153	20	minimal	minimal	ADJ
ejpam-4279	153	21	fuzzy	fuzzy	ADJ
ejpam-4279	153	22	weak	weak	ADJ
ejpam-4279	153	23	almost	almost	ADV
ejpam-4279	153	24	bi	bi	ADJ
ejpam-4279	153	25	-	-	ADJ
ejpam-4279	153	26	interior	interior	ADJ
ejpam-4279	153	27	ideal	ideal	NOUN
ejpam-4279	153	28	of	of	ADP
ejpam-4279	153	29	s.	s.	PROPN
ejpam-4279	153	30	references	reference	VERB
ejpam-4279	153	31	288	288	NUM
ejpam-4279	153	32	proof	proof	NOUN
ejpam-4279	153	33	.	.	PUNCT
ejpam-4279	154	1	assume	assume	VERB
ejpam-4279	154	2	that	that	SCONJ
ejpam-4279	154	3	m	m	PROPN
ejpam-4279	154	4	is	be	AUX
ejpam-4279	154	5	a	a	DET
ejpam-4279	154	6	minimal	minimal	ADJ
ejpam-4279	154	7	almost	almost	ADV
ejpam-4279	154	8	bi	bi	ADJ
ejpam-4279	154	9	-	-	ADJ
ejpam-4279	154	10	interior	interior	ADJ
ejpam-4279	154	11	ideal	ideal	NOUN
ejpam-4279	154	12	of	of	ADP
ejpam-4279	154	13	s.	s.	PROPN
ejpam-4279	154	14	by	by	ADP
ejpam-4279	154	15	theorem	theorem	NOUN
ejpam-4279	154	16	5	5	NUM
ejpam-4279	154	17	,	,	PUNCT
ejpam-4279	154	18	λm	λm	SCONJ
ejpam-4279	154	19	is	be	AUX
ejpam-4279	154	20	a	a	DET
ejpam-4279	154	21	fuzzy	fuzzy	ADJ
ejpam-4279	154	22	almost	almost	ADV
ejpam-4279	154	23	bi	bi	ADJ
ejpam-4279	154	24	-	-	ADJ
ejpam-4279	154	25	interior	interior	ADJ
ejpam-4279	154	26	ideal	ideal	NOUN
ejpam-4279	154	27	of	of	ADP
ejpam-4279	154	28	s.	s.	PROPN
ejpam-4279	154	29	let	let	VERB
ejpam-4279	154	30	ξ	ξ	X
ejpam-4279	154	31	be	be	AUX
ejpam-4279	154	32	a	a	DET
ejpam-4279	154	33	fuzzy	fuzzy	ADJ
ejpam-4279	154	34	almost	almost	ADV
ejpam-4279	154	35	bi	bi	ADJ
ejpam-4279	154	36	-	-	ADJ
ejpam-4279	154	37	interior	interior	ADJ
ejpam-4279	154	38	ideal	ideal	NOUN
ejpam-4279	154	39	of	of	ADP
ejpam-4279	154	40	s	s	PRON
ejpam-4279	154	41	such	such	ADJ
ejpam-4279	154	42	that	that	SCONJ
ejpam-4279	154	43	ξ	ξ	PROPN
ejpam-4279	154	44	⊆	⊆	NUM
ejpam-4279	154	45	λm	λm	ADP
ejpam-4279	154	46	then	then	ADV
ejpam-4279	154	47	supp(ξ	supp(ξ	PROPN
ejpam-4279	154	48	)	)	PUNCT
ejpam-4279	154	49	⊆	⊆	NUM
ejpam-4279	154	50	supp(λm	supp(λm	NOUN
ejpam-4279	154	51	)	)	PUNCT
ejpam-4279	154	52	=	=	PUNCT
ejpam-4279	154	53	m.	m.	NOUN
ejpam-4279	154	54	by	by	ADP
ejpam-4279	154	55	theorem	theorem	ADJ
ejpam-4279	154	56	6	6	NUM
ejpam-4279	154	57	,	,	PUNCT
ejpam-4279	154	58	supp(ξ	supp(ξ	PROPN
ejpam-4279	154	59	)	)	PUNCT
ejpam-4279	154	60	is	be	AUX
ejpam-4279	154	61	an	an	DET
ejpam-4279	154	62	almost	almost	ADV
ejpam-4279	154	63	bi	bi	ADJ
ejpam-4279	154	64	-	-	ADJ
ejpam-4279	154	65	interior	interior	ADJ
ejpam-4279	154	66	ideal	ideal	NOUN
ejpam-4279	154	67	of	of	ADP
ejpam-4279	154	68	s.	s.	PROPN
ejpam-4279	154	69	since	since	SCONJ
ejpam-4279	154	70	k	k	PROPN
ejpam-4279	154	71	is	be	AUX
ejpam-4279	154	72	minimal	minimal	ADJ
ejpam-4279	154	73	we	we	PRON
ejpam-4279	154	74	have	have	AUX
ejpam-4279	154	75	supp(ξ	supp(ξ	NOUN
ejpam-4279	154	76	)	)	PUNCT
ejpam-4279	155	1	=	=	SYM
ejpam-4279	155	2	k	k	NOUN
ejpam-4279	155	3	=	=	PUNCT
ejpam-4279	155	4	supp(λk	supp(λk	NOUN
ejpam-4279	155	5	)	)	PUNCT
ejpam-4279	155	6	.	.	PUNCT
ejpam-4279	156	1	therefore	therefore	ADV
ejpam-4279	156	2	,	,	PUNCT
ejpam-4279	156	3	λk	λk	X
ejpam-4279	156	4	is	be	AUX
ejpam-4279	156	5	minimal	minimal	ADJ
ejpam-4279	156	6	of	of	ADP
ejpam-4279	156	7	s.	s.	PROPN
ejpam-4279	156	8	conversely	conversely	ADV
ejpam-4279	156	9	,	,	PUNCT
ejpam-4279	156	10	suppose	suppose	VERB
ejpam-4279	156	11	that	that	SCONJ
ejpam-4279	156	12	λk	λk	PROPN
ejpam-4279	156	13	is	be	AUX
ejpam-4279	156	14	a	a	DET
ejpam-4279	156	15	minimal	minimal	ADJ
ejpam-4279	156	16	bf	bf	NOUN
ejpam-4279	156	17	almost	almost	ADV
ejpam-4279	156	18	interior	interior	ADJ
ejpam-4279	156	19	ideal	ideal	NOUN
ejpam-4279	156	20	of	of	ADP
ejpam-4279	156	21	s.	s.	PROPN
ejpam-4279	156	22	by	by	ADP
ejpam-4279	156	23	theorem	theorem	NOUN
ejpam-4279	156	24	5	5	NUM
ejpam-4279	156	25	,	,	PUNCT
ejpam-4279	156	26	k	k	PROPN
ejpam-4279	156	27	is	be	AUX
ejpam-4279	156	28	an	an	DET
ejpam-4279	156	29	almost	almost	ADV
ejpam-4279	156	30	bi	bi	ADJ
ejpam-4279	156	31	-	-	ADJ
ejpam-4279	156	32	interior	interior	ADJ
ejpam-4279	156	33	ideal	ideal	NOUN
ejpam-4279	156	34	of	of	ADP
ejpam-4279	156	35	s.	s.	PROPN
ejpam-4279	156	36	let	let	VERB
ejpam-4279	156	37	m	m	PRON
ejpam-4279	156	38	be	be	AUX
ejpam-4279	156	39	an	an	DET
ejpam-4279	156	40	almost	almost	ADV
ejpam-4279	156	41	bi	bi	ADJ
ejpam-4279	156	42	-	-	ADJ
ejpam-4279	156	43	interior	interior	ADJ
ejpam-4279	156	44	ideal	ideal	NOUN
ejpam-4279	156	45	of	of	ADP
ejpam-4279	156	46	s	s	PRON
ejpam-4279	156	47	such	such	ADJ
ejpam-4279	156	48	that	that	SCONJ
ejpam-4279	156	49	m	m	PROPN
ejpam-4279	156	50	⊆	⊆	NUM
ejpam-4279	156	51	k.	k.	NOUN
ejpam-4279	156	52	then	then	ADV
ejpam-4279	156	53	λk	λk	X
ejpam-4279	156	54	is	be	AUX
ejpam-4279	156	55	a	a	DET
ejpam-4279	156	56	fuzzy	fuzzy	ADJ
ejpam-4279	156	57	almost	almost	ADV
ejpam-4279	156	58	interior	interior	ADJ
ejpam-4279	156	59	ideal	ideal	NOUN
ejpam-4279	156	60	of	of	ADP
ejpam-4279	156	61	s	s	PRON
ejpam-4279	156	62	such	such	ADJ
ejpam-4279	156	63	that	that	PRON
ejpam-4279	156	64	λm	λm	ADP
ejpam-4279	156	65	⊆	⊆	NUM
ejpam-4279	156	66	λk	λk	NOUN
ejpam-4279	156	67	.	.	PUNCT
ejpam-4279	157	1	hence	hence	ADV
ejpam-4279	157	2	,	,	PUNCT
ejpam-4279	157	3	m	m	VERB
ejpam-4279	157	4	=	=	NOUN
ejpam-4279	157	5	supp(λm	supp(λm	ADJ
ejpam-4279	157	6	)	)	PUNCT
ejpam-4279	157	7	=	=	SYM
ejpam-4279	157	8	supp(λk	supp(λk	NOUN
ejpam-4279	157	9	)	)	PUNCT
ejpam-4279	157	10	=	=	SYM
ejpam-4279	157	11	k.	k.	PROPN
ejpam-4279	158	1	therefore	therefore	ADV
ejpam-4279	158	2	,	,	PUNCT
ejpam-4279	158	3	k	k	PROPN
ejpam-4279	158	4	is	be	AUX
ejpam-4279	158	5	minimal	minimal	ADJ
ejpam-4279	158	6	of	of	ADP
ejpam-4279	158	7	s.	s.	PROPN
ejpam-4279	158	8	the	the	DET
ejpam-4279	158	9	proof	proof	NOUN
ejpam-4279	158	10	of	of	ADP
ejpam-4279	158	11	the	the	DET
ejpam-4279	158	12	other	other	ADJ
ejpam-4279	158	13	similar	similar	ADJ
ejpam-4279	158	14	to	to	ADP
ejpam-4279	158	15	the	the	DET
ejpam-4279	158	16	proof	proof	NOUN
ejpam-4279	158	17	(	(	PUNCT
ejpam-4279	158	18	1	1	NUM
ejpam-4279	158	19	)	)	PUNCT
ejpam-4279	158	20	.	.	PUNCT
ejpam-4279	159	1	5	5	X
ejpam-4279	159	2	.	.	X
ejpam-4279	159	3	conclusion	conclusion	VERB
ejpam-4279	159	4	the	the	DET
ejpam-4279	159	5	union	union	NOUN
ejpam-4279	159	6	of	of	ADP
ejpam-4279	159	7	two	two	NUM
ejpam-4279	159	8	almost	almost	ADV
ejpam-4279	159	9	bi	bi	ADJ
ejpam-4279	159	10	-	-	ADJ
ejpam-4279	159	11	interior	interior	ADJ
ejpam-4279	159	12	ideal	ideal	NOUN
ejpam-4279	159	13	,	,	PUNCT
ejpam-4279	159	14	and	and	CCONJ
ejpam-4279	159	15	weakly	weakly	ADJ
ejpam-4279	159	16	bi	bi	ADJ
ejpam-4279	159	17	-	-	ADJ
ejpam-4279	159	18	interior	interior	ADJ
ejpam-4279	159	19	ideal	ideal	NOUN
ejpam-4279	159	20	is	be	AUX
ejpam-4279	159	21	also	also	ADV
ejpam-4279	159	22	an	an	DET
ejpam-4279	159	23	almost	almost	ADV
ejpam-4279	159	24	bi	bi	ADJ
ejpam-4279	159	25	-	-	ADJ
ejpam-4279	159	26	interior	interior	ADJ
ejpam-4279	159	27	ideal	ideal	NOUN
ejpam-4279	159	28	and	and	CCONJ
ejpam-4279	159	29	weakly	weakly	ADJ
ejpam-4279	159	30	bi	bi	ADJ
ejpam-4279	159	31	-	-	ADJ
ejpam-4279	159	32	interior	interior	ADJ
ejpam-4279	159	33	ideal	ideal	NOUN
ejpam-4279	159	34	respectively	respectively	ADV
ejpam-4279	159	35	in	in	ADP
ejpam-4279	159	36	semigroups	semigroup	NOUN
ejpam-4279	159	37	and	and	CCONJ
ejpam-4279	159	38	results	result	NOUN
ejpam-4279	159	39	in	in	ADP
ejpam-4279	159	40	class	class	NOUN
ejpam-4279	159	41	fuzzifications	fuzzification	NOUN
ejpam-4279	159	42	is	be	AUX
ejpam-4279	159	43	the	the	DET
ejpam-4279	159	44	same	same	ADJ
ejpam-4279	159	45	.	.	PUNCT
ejpam-4279	160	1	we	we	PRON
ejpam-4279	160	2	prove	prove	VERB
ejpam-4279	160	3	relationship	relationship	NOUN
ejpam-4279	160	4	between	between	ADP
ejpam-4279	160	5	almost	almost	ADV
ejpam-4279	160	6	bi	bi	ADJ
ejpam-4279	160	7	-	-	ADJ
ejpam-4279	160	8	interior	interior	ADJ
ejpam-4279	160	9	ideal	ideal	NOUN
ejpam-4279	160	10	,	,	PUNCT
ejpam-4279	160	11	weakly	weakly	ADJ
ejpam-4279	160	12	bi	bi	ADJ
ejpam-4279	160	13	-	-	ADJ
ejpam-4279	160	14	interior	interior	ADJ
ejpam-4279	160	15	ideal	ideal	NOUN
ejpam-4279	160	16	and	and	CCONJ
ejpam-4279	160	17	class	class	NOUN
ejpam-4279	160	18	fuzzifications	fuzzification	NOUN
ejpam-4279	160	19	.	.	PUNCT
ejpam-4279	161	1	in	in	ADP
ejpam-4279	161	2	the	the	DET
ejpam-4279	161	3	future	future	ADJ
ejpam-4279	161	4	work	work	NOUN
ejpam-4279	161	5	,	,	PUNCT
ejpam-4279	161	6	we	we	PRON
ejpam-4279	161	7	can	can	AUX
ejpam-4279	161	8	study	study	VERB
ejpam-4279	161	9	bi	bi	ADJ
ejpam-4279	161	10	-	-	ADJ
ejpam-4279	161	11	interior	interior	ADJ
ejpam-4279	161	12	ideals	ideal	NOUN
ejpam-4279	161	13	and	and	CCONJ
ejpam-4279	161	14	their	their	PRON
ejpam-4279	161	15	fuzzifications	fuzzification	NOUN
ejpam-4279	161	16	in	in	ADP
ejpam-4279	161	17	algebraic	algebraic	ADJ
ejpam-4279	161	18	structures	structure	NOUN
ejpam-4279	161	19	.	.	PUNCT
ejpam-4279	162	1	references	reference	NOUN
ejpam-4279	162	2	[	[	X
ejpam-4279	162	3	1	1	NUM
ejpam-4279	162	4	]	]	PUNCT
ejpam-4279	162	5	w.	w.	PROPN
ejpam-4279	162	6	yonthanthum	yonthanthum	PROPN
ejpam-4279	162	7	a.	a.	PROPN
ejpam-4279	162	8	simuen	simuen	PROPN
ejpam-4279	162	9	and	and	CCONJ
ejpam-4279	162	10	r.	r.	PROPN
ejpam-4279	162	11	chinram	chinram	PROPN
ejpam-4279	162	12	.	.	PUNCT
ejpam-4279	163	1	almost	almost	ADV
ejpam-4279	163	2	interior	interior	ADJ
ejpam-4279	163	3	gamma	gamma	NOUN
ejpam-4279	163	4	ideals	ideal	NOUN
ejpam-4279	163	5	and	and	CCONJ
ejpam-4279	163	6	fuzzy	fuzzy	ADJ
ejpam-4279	163	7	almost	almost	ADV
ejpam-4279	163	8	interior	interior	ADJ
ejpam-4279	163	9	gamma	gamma	NOUN
ejpam-4279	163	10	ideals	ideal	NOUN
ejpam-4279	163	11	in	in	ADP
ejpam-4279	163	12	gamma	gamma	NOUN
ejpam-4279	163	13	semigroups	semigroup	NOUN
ejpam-4279	163	14	.	.	PUNCT
ejpam-4279	164	1	mathematics	mathematic	NOUN
ejpam-4279	164	2	and	and	CCONJ
ejpam-4279	164	3	staistics	staistic	NOUN
ejpam-4279	164	4	,	,	PUNCT
ejpam-4279	164	5	9:302–308	9:302–308	NOUN
ejpam-4279	164	6	,	,	PUNCT
ejpam-4279	164	7	2021	2021	NUM
ejpam-4279	164	8	.	.	PUNCT
ejpam-4279	165	1	[	[	X
ejpam-4279	165	2	2	2	NUM
ejpam-4279	165	3	]	]	PUNCT
ejpam-4279	165	4	r.	r.	PROPN
ejpam-4279	165	5	chinram	chinram	PROPN
ejpam-4279	165	6	and	and	CCONJ
ejpam-4279	165	7	w.	w.	PROPN
ejpam-4279	165	8	nakkhasen	nakkhasen	PROPN
ejpam-4279	165	9	and	and	CCONJ
ejpam-4279	165	10	.	.	PUNCT
ejpam-4279	166	1	almost	almost	ADV
ejpam-4279	166	2	bi	bi	ADJ
ejpam-4279	166	3	-	-	ADJ
ejpam-4279	166	4	quasi	quasi	ADJ
ejpam-4279	166	5	-	-	ADJ
ejpam-4279	166	6	interior	interior	ADJ
ejpam-4279	166	7	ideals	ideal	NOUN
ejpam-4279	166	8	and	and	CCONJ
ejpam-4279	166	9	fuzzy	fuzzy	ADJ
ejpam-4279	166	10	almost	almost	ADV
ejpam-4279	166	11	bi	bi	ADJ
ejpam-4279	166	12	-	-	ADJ
ejpam-4279	166	13	quasiinterior	quasiinterior	ADJ
ejpam-4279	166	14	ideals	ideal	NOUN
ejpam-4279	166	15	of	of	ADP
ejpam-4279	166	16	semigroups	semigroup	NOUN
ejpam-4279	166	17	.	.	PUNCT
ejpam-4279	167	1	journal	journal	NOUN
ejpam-4279	167	2	of	of	ADP
ejpam-4279	167	3	mathematics	mathematic	NOUN
ejpam-4279	167	4	and	and	CCONJ
ejpam-4279	167	5	computer	computer	NOUN
ejpam-4279	167	6	science	science	NOUN
ejpam-4279	167	7	,	,	PUNCT
ejpam-4279	167	8	26:128–136	26:128–136	PROPN
ejpam-4279	167	9	,	,	PUNCT
ejpam-4279	167	10	2022	2022	NUM
ejpam-4279	167	11	.	.	PUNCT
ejpam-4279	168	1	[	[	X
ejpam-4279	168	2	3	3	X
ejpam-4279	168	3	]	]	X
ejpam-4279	168	4	o.	o.	NOUN
ejpam-4279	168	5	grosek	grosek	NOUN
ejpam-4279	168	6	and	and	CCONJ
ejpam-4279	168	7	l.	l.	PROPN
ejpam-4279	168	8	satko	satko	PROPN
ejpam-4279	168	9	.	.	PUNCT
ejpam-4279	169	1	a	a	DET
ejpam-4279	169	2	new	new	ADJ
ejpam-4279	169	3	notion	notion	NOUN
ejpam-4279	169	4	in	in	ADP
ejpam-4279	169	5	the	the	DET
ejpam-4279	169	6	theory	theory	NOUN
ejpam-4279	169	7	of	of	ADP
ejpam-4279	169	8	semigroup	semigroup	PROPN
ejpam-4279	169	9	.	.	PUNCT
ejpam-4279	170	1	semigroup	semigroup	PROPN
ejpam-4279	170	2	forum	forum	PROPN
ejpam-4279	170	3	,	,	PUNCT
ejpam-4279	170	4	20:233–240	20:233–240	NUM
ejpam-4279	170	5	,	,	PUNCT
ejpam-4279	170	6	1980	1980	NUM
ejpam-4279	170	7	.	.	PUNCT
ejpam-4279	171	1	[	[	X
ejpam-4279	171	2	4	4	X
ejpam-4279	171	3	]	]	X
ejpam-4279	171	4	o.	o.	NOUN
ejpam-4279	171	5	grosek	grosek	NOUN
ejpam-4279	171	6	and	and	CCONJ
ejpam-4279	171	7	l.	l.	PROPN
ejpam-4279	171	8	satko	satko	PROPN
ejpam-4279	171	9	.	.	PUNCT
ejpam-4279	172	1	on	on	ADP
ejpam-4279	172	2	minimal	minimal	ADJ
ejpam-4279	172	3	a	a	DET
ejpam-4279	172	4	-	-	PUNCT
ejpam-4279	172	5	ideals	ideal	NOUN
ejpam-4279	172	6	of	of	ADP
ejpam-4279	172	7	semigroups	semigroup	NOUN
ejpam-4279	172	8	.	.	PUNCT
ejpam-4279	173	1	semigroup	semigroup	PROPN
ejpam-4279	173	2	forum	forum	PROPN
ejpam-4279	173	3	,	,	PUNCT
ejpam-4279	173	4	23:283–295	23:283–295	PROPN
ejpam-4279	173	5	,	,	PUNCT
ejpam-4279	173	6	1981	1981	NUM
ejpam-4279	173	7	.	.	PUNCT
ejpam-4279	174	1	[	[	X
ejpam-4279	174	2	5	5	NUM
ejpam-4279	174	3	]	]	PUNCT
ejpam-4279	174	4	m.	m.	NOUN
ejpam-4279	174	5	krishna	krishna	PROPN
ejpam-4279	174	6	and	and	CCONJ
ejpam-4279	174	7	m.	m.	PROPN
ejpam-4279	174	8	rao	rao	PROPN
ejpam-4279	174	9	.	.	PUNCT
ejpam-4279	175	1	bi	bi	ADJ
ejpam-4279	175	2	-	-	ADJ
ejpam-4279	175	3	interior	interior	ADJ
ejpam-4279	175	4	ideals	ideal	NOUN
ejpam-4279	175	5	of	of	ADP
ejpam-4279	175	6	semigroups	semigroup	NOUN
ejpam-4279	175	7	.	.	PUNCT
ejpam-4279	176	1	discussiones	discussione	NOUN
ejpam-4279	176	2	mathematicae	mathematicae	VERB
ejpam-4279	176	3	general	general	ADJ
ejpam-4279	176	4	algebra	algebra	PROPN
ejpam-4279	176	5	and	and	CCONJ
ejpam-4279	176	6	applications	application	NOUN
ejpam-4279	176	7	,	,	PUNCT
ejpam-4279	176	8	38:69–78	38:69–78	NUM
ejpam-4279	176	9	,	,	PUNCT
ejpam-4279	176	10	2018	2018	NUM
ejpam-4279	176	11	.	.	PUNCT
ejpam-4279	177	1	[	[	X
ejpam-4279	177	2	6	6	NUM
ejpam-4279	177	3	]	]	PUNCT
ejpam-4279	177	4	t.	t.	PROPN
ejpam-4279	177	5	kaewnoi	kaewnoi	PROPN
ejpam-4279	177	6	n.	n.	PROPN
ejpam-4279	177	7	kaopusek	kaopusek	PROPN
ejpam-4279	177	8	and	and	CCONJ
ejpam-4279	177	9	r.	r.	PROPN
ejpam-4279	177	10	chinram	chinram	PROPN
ejpam-4279	177	11	.	.	PUNCT
ejpam-4279	178	1	on	on	ADP
ejpam-4279	178	2	almost	almost	ADV
ejpam-4279	178	3	interior	interior	ADJ
ejpam-4279	178	4	ideals	ideal	NOUN
ejpam-4279	178	5	and	and	CCONJ
ejpam-4279	178	6	weakly	weakly	ADJ
ejpam-4279	178	7	almost	almost	ADV
ejpam-4279	178	8	interior	interior	ADJ
ejpam-4279	178	9	ideals	ideal	NOUN
ejpam-4279	178	10	in	in	ADP
ejpam-4279	178	11	semigroups	semigroup	NOUN
ejpam-4279	178	12	.	.	PUNCT
ejpam-4279	179	1	journal	journal	PROPN
ejpam-4279	179	2	of	of	ADP
ejpam-4279	179	3	discrecte	discrecte	PROPN
ejpam-4279	179	4	mathematical	mathematical	ADJ
ejpam-4279	179	5	science	science	NOUN
ejpam-4279	179	6	and	and	CCONJ
ejpam-4279	179	7	cryptography	cryptography	NOUN
ejpam-4279	179	8	,	,	PUNCT
ejpam-4279	179	9	2020	2020	NUM
ejpam-4279	179	10	.	.	PUNCT
ejpam-4279	180	1	[	[	X
ejpam-4279	180	2	7	7	X
ejpam-4279	180	3	]	]	PUNCT
ejpam-4279	180	4	k.	k.	PROPN
ejpam-4279	180	5	wattanatripop	wattanatripop	PROPN
ejpam-4279	180	6	s.	s.	PROPN
ejpam-4279	180	7	suebsung	suebsung	PROPN
ejpam-4279	180	8	and	and	CCONJ
ejpam-4279	180	9	r.	r.	PROPN
ejpam-4279	180	10	chinram	chinram	PROPN
ejpam-4279	180	11	.	.	PUNCT
ejpam-4279	181	1	characterizing	characterize	VERB
ejpam-4279	181	2	almost	almost	ADV
ejpam-4279	181	3	quasi	quasi	ADJ
ejpam-4279	181	4	-	-	ADJ
ejpam-4279	181	5	γ	γ	NOUN
ejpam-4279	181	6	-	-	PUNCT
ejpam-4279	181	7	ideals	ideal	NOUN
ejpam-4279	181	8	and	and	CCONJ
ejpam-4279	181	9	fuzzy	fuzzy	ADJ
ejpam-4279	181	10	almost	almost	ADV
ejpam-4279	181	11	quasi	quasi	ADJ
ejpam-4279	181	12	-	-	ADJ
ejpam-4279	181	13	γ	γ	NOUN
ejpam-4279	181	14	-	-	PUNCT
ejpam-4279	181	15	ideals	ideal	NOUN
ejpam-4279	181	16	of	of	ADP
ejpam-4279	181	17	γ	γ	NOUN
ejpam-4279	181	18	-	-	PUNCT
ejpam-4279	181	19	semigroups	semigroup	NOUN
ejpam-4279	181	20	.	.	PUNCT
ejpam-4279	182	1	communications	communication	NOUN
ejpam-4279	182	2	in	in	ADP
ejpam-4279	182	3	mathematics	mathematic	NOUN
ejpam-4279	182	4	and	and	CCONJ
ejpam-4279	182	5	applications	application	NOUN
ejpam-4279	182	6	,	,	PUNCT
ejpam-4279	182	7	11:233–240	11:233–240	NUM
ejpam-4279	182	8	,	,	PUNCT
ejpam-4279	182	9	2020	2020	NUM
ejpam-4279	182	10	.	.	PUNCT
ejpam-4279	183	1	references	reference	NOUN
ejpam-4279	183	2	289	289	NUM
ejpam-4279	184	1	[	[	SYM
ejpam-4279	184	2	8	8	NUM
ejpam-4279	184	3	]	]	PUNCT
ejpam-4279	184	4	a.	a.	NOUN
ejpam-4279	184	5	simuen	simuen	PROPN
ejpam-4279	184	6	r.	r.	PROPN
ejpam-4279	184	7	chinram	chinram	PROPN
ejpam-4279	184	8	w.	w.	PROPN
ejpam-4279	184	9	krailoet	krailoet	PROPN
ejpam-4279	184	10	and	and	CCONJ
ejpam-4279	184	11	p.	p.	NOUN
ejpam-4279	184	12	petchkaew	petchkaew	NOUN
ejpam-4279	184	13	.	.	PUNCT
ejpam-4279	185	1	a	a	DET
ejpam-4279	185	2	note	note	NOUN
ejpam-4279	185	3	on	on	ADP
ejpam-4279	185	4	fuzzy	fuzzy	ADJ
ejpam-4279	185	5	almost	almost	ADV
ejpam-4279	185	6	interior	interior	ADJ
ejpam-4279	185	7	ideals	ideal	NOUN
ejpam-4279	185	8	in	in	ADP
ejpam-4279	185	9	semigroups	semigroup	NOUN
ejpam-4279	185	10	.	.	PUNCT
ejpam-4279	186	1	mathematics	mathematic	NOUN
ejpam-4279	186	2	and	and	CCONJ
ejpam-4279	186	3	staistics	staistic	NOUN
ejpam-4279	186	4	,	,	PUNCT
ejpam-4279	186	5	9:302–308	9:302–308	NOUN
ejpam-4279	186	6	,	,	PUNCT
ejpam-4279	186	7	2021	2021	NUM
ejpam-4279	186	8	.	.	PUNCT
ejpam-4279	187	1	[	[	X
ejpam-4279	187	2	9	9	NUM
ejpam-4279	187	3	]	]	X
ejpam-4279	187	4	l.a	l.a	PROPN
ejpam-4279	187	5	.	.	PROPN
ejpam-4279	187	6	zadeh	zadeh	PROPN
ejpam-4279	187	7	.	.	PUNCT
ejpam-4279	187	8	fuzzy	fuzzy	ADJ
ejpam-4279	187	9	sets	set	NOUN
ejpam-4279	187	10	.	.	PUNCT
ejpam-4279	188	1	information	information	NOUN
ejpam-4279	188	2	and	and	CCONJ
ejpam-4279	188	3	control	control	NOUN
ejpam-4279	188	4	,	,	PUNCT
ejpam-4279	188	5	8:338–353	8:338–353	NUM
ejpam-4279	188	6	,	,	PUNCT
ejpam-4279	188	7	1965	1965	NUM
ejpam-4279	188	8	.	.	PUNCT