id	sid	tid	token	lemma	pos
ejpam-5596	1	1	european	european	PROPN
ejpam-5596	1	2	journal	journal	PROPN
ejpam-5596	1	3	of	of	ADP
ejpam-5596	1	4	pure	pure	ADJ
ejpam-5596	1	5	and	and	CCONJ
ejpam-5596	1	6	applied	applied	ADJ
ejpam-5596	1	7	mathematics	mathematic	NOUN
ejpam-5596	1	8	2025	2025	NUM
ejpam-5596	1	9	,	,	PUNCT
ejpam-5596	1	10	vol	vol	NOUN
ejpam-5596	1	11	.	.	PROPN
ejpam-5596	1	12	18	18	NUM
ejpam-5596	1	13	,	,	PUNCT
ejpam-5596	1	14	issue	issue	NOUN
ejpam-5596	1	15	1	1	NUM
ejpam-5596	1	16	,	,	PUNCT
ejpam-5596	1	17	article	article	NOUN
ejpam-5596	1	18	number	number	NOUN
ejpam-5596	1	19	5596	5596	NUM
ejpam-5596	1	20	issn	issn	PROPN
ejpam-5596	1	21	1307	1307	NUM
ejpam-5596	1	22	-	-	SYM
ejpam-5596	1	23	5543	5543	NUM
ejpam-5596	1	24	–	–	PUNCT
ejpam-5596	1	25	ejpam.com	ejpam.com	X
ejpam-5596	1	26	published	publish	VERB
ejpam-5596	1	27	by	by	ADP
ejpam-5596	1	28	new	new	PROPN
ejpam-5596	1	29	york	york	PROPN
ejpam-5596	1	30	business	business	PROPN
ejpam-5596	1	31	global	global	ADJ
ejpam-5596	1	32	fuzzy	fuzzy	NOUN
ejpam-5596	1	33	(	(	PUNCT
ejpam-5596	1	34	m	m	NOUN
ejpam-5596	1	35	,	,	PUNCT
ejpam-5596	1	36	n)-ideals	n)-ideal	NOUN
ejpam-5596	1	37	and	and	CCONJ
ejpam-5596	1	38	n	n	CCONJ
ejpam-5596	1	39	-	-	PUNCT
ejpam-5596	1	40	interior	interior	ADJ
ejpam-5596	1	41	ideals	ideal	NOUN
ejpam-5596	1	42	in	in	ADP
ejpam-5596	1	43	ordered	order	VERB
ejpam-5596	1	44	semigroups	semigroup	NOUN
ejpam-5596	1	45	p.	p.	PROPN
ejpam-5596	1	46	khamrot1	khamrot1	PROPN
ejpam-5596	1	47	,	,	PUNCT
ejpam-5596	1	48	a.	a.	PROPN
ejpam-5596	1	49	iampan2	iampan2	PROPN
ejpam-5596	1	50	,	,	PUNCT
ejpam-5596	1	51	t.	t.	PROPN
ejpam-5596	1	52	gaketem2,∗	gaketem2,∗	PROPN
ejpam-5596	1	53	1	1	NUM
ejpam-5596	1	54	department	department	NOUN
ejpam-5596	1	55	of	of	ADP
ejpam-5596	1	56	mathematics	mathematic	NOUN
ejpam-5596	1	57	,	,	PUNCT
ejpam-5596	1	58	faculty	faculty	NOUN
ejpam-5596	1	59	of	of	ADP
ejpam-5596	1	60	science	science	NOUN
ejpam-5596	1	61	and	and	CCONJ
ejpam-5596	1	62	agricultural	agricultural	ADJ
ejpam-5596	1	63	technology	technology	NOUN
ejpam-5596	1	64	,	,	PUNCT
ejpam-5596	1	65	rajamangala	rajamangala	PROPN
ejpam-5596	1	66	university	university	PROPN
ejpam-5596	1	67	of	of	ADP
ejpam-5596	1	68	technology	technology	PROPN
ejpam-5596	1	69	lanna	lanna	PROPN
ejpam-5596	1	70	of	of	ADP
ejpam-5596	1	71	phitsanulok	phitsanulok	PROPN
ejpam-5596	1	72	,	,	PUNCT
ejpam-5596	1	73	phitsanulok	phitsanulok	PROPN
ejpam-5596	1	74	,	,	PUNCT
ejpam-5596	1	75	thailand	thailand	PROPN
ejpam-5596	1	76	2	2	NUM
ejpam-5596	1	77	department	department	NOUN
ejpam-5596	1	78	of	of	ADP
ejpam-5596	1	79	mathematics	mathematic	NOUN
ejpam-5596	1	80	,	,	PUNCT
ejpam-5596	1	81	school	school	NOUN
ejpam-5596	1	82	of	of	ADP
ejpam-5596	1	83	science	science	NOUN
ejpam-5596	1	84	,	,	PUNCT
ejpam-5596	1	85	university	university	NOUN
ejpam-5596	1	86	of	of	ADP
ejpam-5596	1	87	phayao	phayao	NOUN
ejpam-5596	1	88	,	,	PUNCT
ejpam-5596	1	89	mae	mae	PROPN
ejpam-5596	1	90	ka	ka	PROPN
ejpam-5596	1	91	,	,	PUNCT
ejpam-5596	1	92	mueang	mueang	PROPN
ejpam-5596	1	93	,	,	PUNCT
ejpam-5596	1	94	phayao	phayao	NOUN
ejpam-5596	1	95	56000	56000	NUM
ejpam-5596	1	96	,	,	PUNCT
ejpam-5596	1	97	thailand	thailand	PROPN
ejpam-5596	1	98	abstract	abstract	NOUN
ejpam-5596	1	99	.	.	PUNCT
ejpam-5596	2	1	in	in	ADP
ejpam-5596	2	2	2019	2019	NUM
ejpam-5596	2	3	,	,	PUNCT
ejpam-5596	2	4	ahsan	ahsan	PROPN
ejpam-5596	2	5	et	et	PROPN
ejpam-5596	2	6	al	al	PROPN
ejpam-5596	2	7	.	.	PROPN
ejpam-5596	2	8	developed	develop	VERB
ejpam-5596	2	9	the	the	DET
ejpam-5596	2	10	concept	concept	NOUN
ejpam-5596	2	11	of	of	ADP
ejpam-5596	2	12	fuzzy	fuzzy	ADJ
ejpam-5596	2	13	(	(	PUNCT
ejpam-5596	2	14	m	m	PROPN
ejpam-5596	2	15	,	,	PUNCT
ejpam-5596	2	16	n)-ideals	n)-ideal	NOUN
ejpam-5596	2	17	in	in	ADP
ejpam-5596	2	18	semigroups	semigroup	NOUN
ejpam-5596	2	19	.	.	PUNCT
ejpam-5596	3	1	later	later	ADV
ejpam-5596	3	2	,	,	PUNCT
ejpam-5596	3	3	in	in	ADP
ejpam-5596	3	4	2022	2022	NUM
ejpam-5596	3	5	tiprachot	tiprachot	NOUN
ejpam-5596	3	6	studied	study	VERB
ejpam-5596	3	7	n	n	CCONJ
ejpam-5596	3	8	-	-	PUNCT
ejpam-5596	3	9	interior	interior	ADJ
ejpam-5596	3	10	ideals	ideal	NOUN
ejpam-5596	3	11	in	in	ADP
ejpam-5596	3	12	ordered	order	VERB
ejpam-5596	3	13	semigroup	semigroup	NOUN
ejpam-5596	3	14	,	,	PUNCT
ejpam-5596	3	15	which	which	PRON
ejpam-5596	3	16	is	be	AUX
ejpam-5596	3	17	a	a	DET
ejpam-5596	3	18	generalization	generalization	NOUN
ejpam-5596	3	19	of	of	ADP
ejpam-5596	3	20	fuzzy	fuzzy	ADJ
ejpam-5596	3	21	ideals	ideal	NOUN
ejpam-5596	3	22	and	and	CCONJ
ejpam-5596	3	23	interior	interior	ADJ
ejpam-5596	3	24	ideals	ideal	NOUN
ejpam-5596	3	25	in	in	ADP
ejpam-5596	3	26	semigroups	semigroup	NOUN
ejpam-5596	3	27	.	.	PUNCT
ejpam-5596	4	1	the	the	DET
ejpam-5596	4	2	aim	aim	NOUN
ejpam-5596	4	3	of	of	ADP
ejpam-5596	4	4	this	this	DET
ejpam-5596	4	5	paper	paper	NOUN
ejpam-5596	4	6	is	be	AUX
ejpam-5596	4	7	to	to	PART
ejpam-5596	4	8	study	study	VERB
ejpam-5596	4	9	the	the	DET
ejpam-5596	4	10	concept	concept	NOUN
ejpam-5596	4	11	of	of	ADP
ejpam-5596	4	12	fuzzy	fuzzy	ADJ
ejpam-5596	4	13	(	(	PUNCT
ejpam-5596	4	14	m	m	NOUN
ejpam-5596	4	15	,	,	PUNCT
ejpam-5596	4	16	n)-ideals	n)-ideal	NOUN
ejpam-5596	4	17	and	and	CCONJ
ejpam-5596	4	18	n	n	CCONJ
ejpam-5596	4	19	-	-	PUNCT
ejpam-5596	4	20	interior	interior	ADJ
ejpam-5596	4	21	ideals	ideal	NOUN
ejpam-5596	4	22	in	in	ADP
ejpam-5596	4	23	ordered	order	VERB
ejpam-5596	4	24	semigroup	semigroup	NOUN
ejpam-5596	4	25	and	and	CCONJ
ejpam-5596	4	26	investigate	investigate	VERB
ejpam-5596	4	27	the	the	DET
ejpam-5596	4	28	properties	property	NOUN
ejpam-5596	4	29	of	of	ADP
ejpam-5596	4	30	fuzzy	fuzzy	ADJ
ejpam-5596	4	31	(	(	PUNCT
ejpam-5596	4	32	m	m	NOUN
ejpam-5596	4	33	,	,	PUNCT
ejpam-5596	4	34	n)-ideals	n)-ideal	NOUN
ejpam-5596	4	35	and	and	CCONJ
ejpam-5596	4	36	n	n	CCONJ
ejpam-5596	4	37	-	-	ADJ
ejpam-5596	4	38	interior	interior	ADJ
ejpam-5596	4	39	ideals	ideal	NOUN
ejpam-5596	4	40	.	.	PUNCT
ejpam-5596	5	1	2020	2020	NUM
ejpam-5596	5	2	mathematics	mathematic	NOUN
ejpam-5596	5	3	subject	subject	NOUN
ejpam-5596	5	4	classifications	classification	NOUN
ejpam-5596	5	5	:	:	PUNCT
ejpam-5596	5	6	20m12	20m12	NUM
ejpam-5596	5	7	,	,	PUNCT
ejpam-5596	5	8	06f05	06f05	PRON
ejpam-5596	5	9	key	key	ADJ
ejpam-5596	5	10	words	word	NOUN
ejpam-5596	5	11	and	and	CCONJ
ejpam-5596	5	12	phrases	phrase	NOUN
ejpam-5596	5	13	:	:	PUNCT
ejpam-5596	5	14	fuzzy	fuzzy	ADJ
ejpam-5596	5	15	(	(	PUNCT
ejpam-5596	5	16	m	m	NOUN
ejpam-5596	5	17	,	,	PUNCT
ejpam-5596	5	18	n)-ideals	n)-ideal	NOUN
ejpam-5596	5	19	,	,	PUNCT
ejpam-5596	5	20	fuzzy	fuzzy	ADJ
ejpam-5596	5	21	n	n	CCONJ
ejpam-5596	5	22	-	-	PUNCT
ejpam-5596	5	23	interior	interior	ADJ
ejpam-5596	5	24	ideals	ideal	NOUN
ejpam-5596	5	25	,	,	PUNCT
ejpam-5596	5	26	regular	regular	ADJ
ejpam-5596	5	27	ordered	order	VERB
ejpam-5596	5	28	semigroups	semigroup	NOUN
ejpam-5596	5	29	1	1	X
ejpam-5596	5	30	.	.	X
ejpam-5596	5	31	introduction	introduction	NOUN
ejpam-5596	5	32	the	the	DET
ejpam-5596	5	33	theory	theory	NOUN
ejpam-5596	5	34	of	of	ADP
ejpam-5596	5	35	ordered	order	VERB
ejpam-5596	5	36	semigroups	semigroup	NOUN
ejpam-5596	5	37	originated	originate	VERB
ejpam-5596	5	38	as	as	ADP
ejpam-5596	5	39	a	a	DET
ejpam-5596	5	40	generalization	generalization	NOUN
ejpam-5596	5	41	of	of	ADP
ejpam-5596	5	42	the	the	DET
ejpam-5596	5	43	semigroup	semigroup	PROPN
ejpam-5596	5	44	theory	theory	NOUN
ejpam-5596	5	45	.	.	PUNCT
ejpam-5596	6	1	the	the	DET
ejpam-5596	6	2	concept	concept	NOUN
ejpam-5596	6	3	of	of	ADP
ejpam-5596	6	4	(	(	PUNCT
ejpam-5596	6	5	m	m	PROPN
ejpam-5596	6	6	,	,	PUNCT
ejpam-5596	6	7	n)-t	n)-t	PROPN
ejpam-5596	6	8	.	.	PROPN
ejpam-5596	6	9	changphas	changphas	PROPN
ejpam-5596	6	10	gave	give	VERB
ejpam-5596	6	11	ideals	ideal	NOUN
ejpam-5596	6	12	in	in	ADP
ejpam-5596	6	13	ordered	order	VERB
ejpam-5596	6	14	semigroups	semigroup	NOUN
ejpam-5596	6	15	in	in	ADP
ejpam-5596	6	16	[	[	X
ejpam-5596	6	17	3	3	X
ejpam-5596	6	18	]	]	PUNCT
ejpam-5596	6	19	which	which	PRON
ejpam-5596	6	20	was	be	AUX
ejpam-5596	6	21	obtained	obtain	VERB
ejpam-5596	6	22	by	by	ADP
ejpam-5596	6	23	generalizing	generalize	VERB
ejpam-5596	6	24	the	the	DET
ejpam-5596	6	25	idea	idea	NOUN
ejpam-5596	6	26	of	of	ADP
ejpam-5596	6	27	(	(	PUNCT
ejpam-5596	6	28	m	m	PROPN
ejpam-5596	6	29	,	,	PUNCT
ejpam-5596	6	30	n)-ideals	n)-ideal	NOUN
ejpam-5596	6	31	in	in	ADP
ejpam-5596	6	32	semigroups	semigroup	NOUN
ejpam-5596	6	33	.	.	PUNCT
ejpam-5596	7	1	as	as	ADP
ejpam-5596	7	2	a	a	DET
ejpam-5596	7	3	theory	theory	NOUN
ejpam-5596	7	4	of	of	ADP
ejpam-5596	7	5	fuzzy	fuzzy	ADJ
ejpam-5596	7	6	set	set	NOUN
ejpam-5596	7	7	it	it	PRON
ejpam-5596	7	8	is	be	AUX
ejpam-5596	7	9	tool	tool	NOUN
ejpam-5596	7	10	for	for	ADP
ejpam-5596	7	11	dealing	deal	VERB
ejpam-5596	7	12	with	with	ADP
ejpam-5596	7	13	possibilities	possibility	NOUN
ejpam-5596	7	14	of	of	ADP
ejpam-5596	7	15	uncertainty	uncertainty	NOUN
ejpam-5596	7	16	,	,	PUNCT
ejpam-5596	7	17	connected	connect	VERB
ejpam-5596	7	18	with	with	ADP
ejpam-5596	7	19	the	the	DET
ejpam-5596	7	20	imprecision	imprecision	NOUN
ejpam-5596	7	21	of	of	ADP
ejpam-5596	7	22	states	state	NOUN
ejpam-5596	7	23	,	,	PUNCT
ejpam-5596	7	24	perceptions	perception	NOUN
ejpam-5596	7	25	,	,	PUNCT
ejpam-5596	7	26	and	and	CCONJ
ejpam-5596	7	27	preferences	preference	NOUN
ejpam-5596	7	28	,	,	PUNCT
ejpam-5596	7	29	and	and	CCONJ
ejpam-5596	7	30	was	be	AUX
ejpam-5596	7	31	studied	study	VERB
ejpam-5596	7	32	by	by	ADP
ejpam-5596	7	33	zadeh	zadeh	PROPN
ejpam-5596	7	34	in	in	ADP
ejpam-5596	7	35	1965	1965	NUM
ejpam-5596	7	36	[	[	X
ejpam-5596	7	37	14	14	NUM
ejpam-5596	7	38	]	]	PUNCT
ejpam-5596	7	39	.	.	PUNCT
ejpam-5596	8	1	it	it	PRON
ejpam-5596	8	2	has	have	AUX
ejpam-5596	8	3	been	be	AUX
ejpam-5596	8	4	applied	apply	VERB
ejpam-5596	8	5	to	to	ADP
ejpam-5596	8	6	many	many	ADJ
ejpam-5596	8	7	areas	area	NOUN
ejpam-5596	8	8	,	,	PUNCT
ejpam-5596	8	9	such	such	ADJ
ejpam-5596	8	10	as	as	ADP
ejpam-5596	8	11	medical	medical	ADJ
ejpam-5596	8	12	science	science	NOUN
ejpam-5596	8	13	,	,	PUNCT
ejpam-5596	8	14	robotics	robotic	NOUN
ejpam-5596	8	15	,	,	PUNCT
ejpam-5596	8	16	computer	computer	NOUN
ejpam-5596	8	17	science	science	NOUN
ejpam-5596	8	18	,	,	PUNCT
ejpam-5596	8	19	information	information	NOUN
ejpam-5596	8	20	science	science	NOUN
ejpam-5596	8	21	,	,	PUNCT
ejpam-5596	8	22	control	control	NOUN
ejpam-5596	8	23	engineering	engineering	NOUN
ejpam-5596	8	24	,	,	PUNCT
ejpam-5596	8	25	measure	measure	NOUN
ejpam-5596	8	26	theory	theory	NOUN
ejpam-5596	8	27	,	,	PUNCT
ejpam-5596	8	28	logic	logic	NOUN
ejpam-5596	8	29	,	,	PUNCT
ejpam-5596	8	30	set	set	ADJ
ejpam-5596	8	31	theory	theory	NOUN
ejpam-5596	8	32	,	,	PUNCT
ejpam-5596	8	33	topology	topology	NOUN
ejpam-5596	8	34	and	and	CCONJ
ejpam-5596	8	35	others	other	NOUN
ejpam-5596	8	36	.	.	PUNCT
ejpam-5596	9	1	the	the	DET
ejpam-5596	9	2	study	study	NOUN
ejpam-5596	9	3	of	of	ADP
ejpam-5596	9	4	fuzzy	fuzzy	ADJ
ejpam-5596	9	5	sets	set	NOUN
ejpam-5596	9	6	in	in	ADP
ejpam-5596	9	7	semigroups	semigroup	NOUN
ejpam-5596	9	8	was	be	AUX
ejpam-5596	9	9	introduced	introduce	VERB
ejpam-5596	9	10	by	by	ADP
ejpam-5596	9	11	kuroki	kuroki	NOUN
ejpam-5596	9	12	in	in	ADP
ejpam-5596	9	13	1981	1981	NUM
ejpam-5596	9	14	.	.	PUNCT
ejpam-5596	10	1	in	in	ADP
ejpam-5596	10	2	2020	2020	NUM
ejpam-5596	10	3	kehayopulu	kehayopulu	VERB
ejpam-5596	10	4	and	and	CCONJ
ejpam-5596	10	5	tsingelis	tsingeli	NOUN
ejpam-5596	11	1	[	[	X
ejpam-5596	11	2	5	5	NUM
ejpam-5596	11	3	]	]	PUNCT
ejpam-5596	11	4	extended	extend	VERB
ejpam-5596	11	5	the	the	DET
ejpam-5596	11	6	concept	concept	NOUN
ejpam-5596	11	7	of	of	ADP
ejpam-5596	11	8	fuzzy	fuzzy	ADJ
ejpam-5596	11	9	semigroups	semigroup	NOUN
ejpam-5596	11	10	to	to	ADP
ejpam-5596	11	11	the	the	DET
ejpam-5596	11	12	fuzzy	fuzzy	ADJ
ejpam-5596	11	13	ordered	order	VERB
ejpam-5596	11	14	semigroups	semigroup	NOUN
ejpam-5596	11	15	and	and	CCONJ
ejpam-5596	11	16	studied	study	VERB
ejpam-5596	11	17	some	some	DET
ejpam-5596	11	18	properties	property	NOUN
ejpam-5596	11	19	of	of	ADP
ejpam-5596	11	20	fuzzy	fuzzy	ADJ
ejpam-5596	11	21	left	left	NOUN
ejpam-5596	11	22	(	(	PUNCT
ejpam-5596	11	23	right	right	ADJ
ejpam-5596	11	24	)	)	PUNCT
ejpam-5596	11	25	ideals	ideal	NOUN
ejpam-5596	11	26	and	and	CCONJ
ejpam-5596	11	27	fuzzy	fuzzy	ADJ
ejpam-5596	11	28	filters	filter	NOUN
ejpam-5596	11	29	in	in	ADP
ejpam-5596	11	30	ordered	order	VERB
ejpam-5596	11	31	semigroups	semigroup	NOUN
ejpam-5596	11	32	.	.	PUNCT
ejpam-5596	12	1	in	in	ADP
ejpam-5596	12	2	2019	2019	NUM
ejpam-5596	12	3	,	,	PUNCT
ejpam-5596	12	4	ahsan	ahsan	PROPN
ejpam-5596	12	5	et	et	PROPN
ejpam-5596	12	6	al	al	PROPN
ejpam-5596	12	7	.	.	PUNCT
ejpam-5596	13	1	[	[	X
ejpam-5596	13	2	9	9	NUM
ejpam-5596	13	3	]	]	PUNCT
ejpam-5596	13	4	extended	extend	VERB
ejpam-5596	13	5	the	the	DET
ejpam-5596	13	6	notion	notion	NOUN
ejpam-5596	13	7	of	of	ADP
ejpam-5596	13	8	(	(	PUNCT
ejpam-5596	13	9	m	m	PROPN
ejpam-5596	13	10	,	,	PUNCT
ejpam-5596	13	11	n)-ideals	n)-ideal	NOUN
ejpam-5596	13	12	in	in	ADP
ejpam-5596	13	13	semigroups	semigroup	NOUN
ejpam-5596	13	14	to	to	ADP
ejpam-5596	13	15	the	the	DET
ejpam-5596	13	16	notion	notion	NOUN
ejpam-5596	13	17	of	of	ADP
ejpam-5596	13	18	fuzzy	fuzzy	ADJ
ejpam-5596	13	19	(	(	PUNCT
ejpam-5596	13	20	m	m	PROPN
ejpam-5596	13	21	,	,	PUNCT
ejpam-5596	13	22	n)-ideals	n)-ideal	NOUN
ejpam-5596	13	23	in	in	ADP
ejpam-5596	13	24	semigroups	semigroup	NOUN
ejpam-5596	13	25	and	and	CCONJ
ejpam-5596	13	26	they	they	PRON
ejpam-5596	13	27	characterized	characterize	VERB
ejpam-5596	13	28	(	(	PUNCT
ejpam-5596	13	29	m	m	PROPN
ejpam-5596	13	30	,	,	PUNCT
ejpam-5596	13	31	n)-regular	n)-regular	PRON
ejpam-5596	13	32	semigroups	semigroup	NOUN
ejpam-5596	13	33	by	by	ADP
ejpam-5596	13	34	using	use	VERB
ejpam-5596	13	35	fuzzy	fuzzy	ADJ
ejpam-5596	13	36	(	(	PUNCT
ejpam-5596	13	37	m	m	NOUN
ejpam-5596	13	38	,	,	PUNCT
ejpam-5596	13	39	n)-ideals	n)-ideal	NOUN
ejpam-5596	13	40	.	.	PUNCT
ejpam-5596	14	1	tiprachot	tiprachot	PROPN
ejpam-5596	14	2	et	et	PROPN
ejpam-5596	14	3	al	al	PROPN
ejpam-5596	14	4	.	.	PUNCT
ejpam-5596	15	1	[	[	X
ejpam-5596	15	2	12	12	NUM
ejpam-5596	15	3	]	]	PUNCT
ejpam-5596	15	4	∗corresponding	∗corresponde	VERB
ejpam-5596	15	5	author	author	NOUN
ejpam-5596	15	6	.	.	PUNCT
ejpam-5596	16	1	doi	doi	NOUN
ejpam-5596	16	2	:	:	PUNCT
ejpam-5596	16	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5596	https://doi.org/10.29020/nybg.ejpam.v18i1.5596	NUM
ejpam-5596	16	4	email	email	NOUN
ejpam-5596	16	5	addresses	address	NOUN
ejpam-5596	16	6	:	:	PUNCT
ejpam-5596	16	7	pk	pk	NOUN
ejpam-5596	16	8	g@rmutl.ac.th	g@rmutl.ac.th	PROPN
ejpam-5596	16	9	(	(	PUNCT
ejpam-5596	16	10	p.	p.	NOUN
ejpam-5596	16	11	khamrot	khamrot	PROPN
ejpam-5596	16	12	)	)	PUNCT
ejpam-5596	16	13	,	,	PUNCT
ejpam-5596	16	14	aiyared.ia@up.ac.th	aiyared.ia@up.ac.th	NOUN
ejpam-5596	16	15	(	(	PUNCT
ejpam-5596	16	16	a.	a.	NOUN
ejpam-5596	16	17	iampan	iampan	PROPN
ejpam-5596	16	18	)	)	PUNCT
ejpam-5596	16	19	,	,	PUNCT
ejpam-5596	16	20	thiti.ga@up.ac.th	thiti.ga@up.ac.th	PROPN
ejpam-5596	16	21	(	(	PUNCT
ejpam-5596	16	22	t.	t.	NOUN
ejpam-5596	16	23	gaketem	gaketem	PROPN
ejpam-5596	16	24	)	)	PUNCT
ejpam-5596	16	25	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5596	16	26	1	1	NUM
ejpam-5596	16	27	copyright	copyright	NOUN
ejpam-5596	16	28	:	:	PUNCT
ejpam-5596	17	1	©	©	PROPN
ejpam-5596	17	2	2025	2025	NUM
ejpam-5596	17	3	the	the	DET
ejpam-5596	17	4	author(s	author(s	NOUN
ejpam-5596	17	5	)	)	PUNCT
ejpam-5596	17	6	.	.	PUNCT
ejpam-5596	18	1	(	(	PUNCT
ejpam-5596	18	2	cc	cc	NOUN
ejpam-5596	18	3	by	by	ADP
ejpam-5596	18	4	-	-	PUNCT
ejpam-5596	18	5	nc	nc	PROPN
ejpam-5596	18	6	4.0	4.0	NUM
ejpam-5596	18	7	)	)	PUNCT
ejpam-5596	18	8	p.	p.	NOUN
ejpam-5596	18	9	khamrot	khamrot	PROPN
ejpam-5596	18	10	,	,	PUNCT
ejpam-5596	18	11	a.	a.	NOUN
ejpam-5596	18	12	iampan	iampan	PROPN
ejpam-5596	18	13	,	,	PUNCT
ejpam-5596	18	14	t.	t.	PROPN
ejpam-5596	18	15	gaketem	gaketem	PROPN
ejpam-5596	18	16	/	/	SYM
ejpam-5596	18	17	eur	eur	PROPN
ejpam-5596	18	18	.	.	PUNCT
ejpam-5596	19	1	j.	j.	PROPN
ejpam-5596	19	2	pure	pure	PROPN
ejpam-5596	19	3	appl	appl	PROPN
ejpam-5596	19	4	.	.	PROPN
ejpam-5596	19	5	math	math	PROPN
ejpam-5596	19	6	,	,	PUNCT
ejpam-5596	19	7	18	18	NUM
ejpam-5596	19	8	(	(	PUNCT
ejpam-5596	19	9	1	1	NUM
ejpam-5596	19	10	)	)	PUNCT
ejpam-5596	19	11	(	(	PUNCT
ejpam-5596	19	12	2025	2025	NUM
ejpam-5596	19	13	)	)	PUNCT
ejpam-5596	19	14	,	,	PUNCT
ejpam-5596	19	15	5596	5596	NUM
ejpam-5596	19	16	2	2	NUM
ejpam-5596	19	17	of	of	ADP
ejpam-5596	19	18	12	12	NUM
ejpam-5596	19	19	discussed	discuss	VERB
ejpam-5596	19	20	the	the	DET
ejpam-5596	19	21	notion	notion	NOUN
ejpam-5596	19	22	of	of	ADP
ejpam-5596	19	23	n	n	CCONJ
ejpam-5596	19	24	-	-	PUNCT
ejpam-5596	19	25	interior	interior	ADJ
ejpam-5596	19	26	ideals	ideal	NOUN
ejpam-5596	19	27	as	as	ADP
ejpam-5596	19	28	a	a	DET
ejpam-5596	19	29	generalization	generalization	NOUN
ejpam-5596	19	30	of	of	ADP
ejpam-5596	19	31	interior	interior	ADJ
ejpam-5596	19	32	ideals	ideal	NOUN
ejpam-5596	19	33	and	and	CCONJ
ejpam-5596	19	34	characterized	characterize	VERB
ejpam-5596	19	35	many	many	ADJ
ejpam-5596	19	36	classes	class	NOUN
ejpam-5596	19	37	of	of	ADP
ejpam-5596	19	38	ordered	order	VERB
ejpam-5596	19	39	semigroups	semigroup	NOUN
ejpam-5596	19	40	in	in	ADP
ejpam-5596	19	41	terms	term	NOUN
ejpam-5596	19	42	of	of	ADP
ejpam-5596	19	43	(	(	PUNCT
ejpam-5596	19	44	m	m	PROPN
ejpam-5596	19	45	,	,	PUNCT
ejpam-5596	19	46	n)-ideals	n)-ideal	NOUN
ejpam-5596	19	47	and	and	CCONJ
ejpam-5596	19	48	n	n	CCONJ
ejpam-5596	19	49	-	-	ADJ
ejpam-5596	19	50	interior	interior	ADJ
ejpam-5596	19	51	ideals	ideal	NOUN
ejpam-5596	19	52	.	.	PUNCT
ejpam-5596	20	1	in	in	ADP
ejpam-5596	20	2	2023	2023	NUM
ejpam-5596	20	3	,	,	PUNCT
ejpam-5596	20	4	tiprachot	tiprachot	NOUN
ejpam-5596	20	5	et	et	PROPN
ejpam-5596	20	6	al	al	PROPN
ejpam-5596	20	7	.	.	PUNCT
ejpam-5596	21	1	[	[	X
ejpam-5596	21	2	13	13	NUM
ejpam-5596	21	3	]	]	PUNCT
ejpam-5596	21	4	extend	extend	NOUN
ejpam-5596	21	5	n	n	CCONJ
ejpam-5596	21	6	-	-	PUNCT
ejpam-5596	21	7	interior	interior	ADJ
ejpam-5596	21	8	ideals	ideal	NOUN
ejpam-5596	21	9	and	and	CCONJ
ejpam-5596	21	10	(	(	PUNCT
ejpam-5596	21	11	m	m	PROPN
ejpam-5596	21	12	,	,	PUNCT
ejpam-5596	21	13	n)-ideals	n)-ideal	NOUN
ejpam-5596	21	14	to	to	PART
ejpam-5596	21	15	hybrid	hybrid	VERB
ejpam-5596	21	16	in	in	ADP
ejpam-5596	21	17	ordered	order	VERB
ejpam-5596	21	18	semigroups	semigroup	NOUN
ejpam-5596	21	19	.	.	PUNCT
ejpam-5596	22	1	recently	recently	ADV
ejpam-5596	22	2	t.	t.	PROPN
ejpam-5596	22	3	gaketem	gaketem	PROPN
ejpam-5596	22	4	and	and	CCONJ
ejpam-5596	22	5	p.	p.	NOUN
ejpam-5596	22	6	khamrot	khamrot	NOUN
ejpam-5596	23	1	[	[	X
ejpam-5596	23	2	6	6	NUM
ejpam-5596	23	3	]	]	PUNCT
ejpam-5596	23	4	studied	study	VERB
ejpam-5596	23	5	concepts	concept	NOUN
ejpam-5596	23	6	interval	interval	NOUN
ejpam-5596	23	7	valued	value	VERB
ejpam-5596	23	8	fuzzy	fuzzy	ADJ
ejpam-5596	23	9	(	(	PUNCT
ejpam-5596	23	10	m	m	PROPN
ejpam-5596	23	11	,	,	PUNCT
ejpam-5596	23	12	n)-ideals	n)-ideal	NOUN
ejpam-5596	23	13	in	in	ADP
ejpam-5596	23	14	semigroups	semigroup	NOUN
ejpam-5596	23	15	.	.	PUNCT
ejpam-5596	24	1	in	in	ADP
ejpam-5596	24	2	the	the	DET
ejpam-5596	24	3	same	same	ADJ
ejpam-5596	24	4	year	year	NOUN
ejpam-5596	24	5	a.	a.	NOUN
ejpam-5596	24	6	mahoob	mahoob	PROPN
ejpam-5596	24	7	et	et	PROPN
ejpam-5596	24	8	al	al	PROPN
ejpam-5596	24	9	.	.	PUNCT
ejpam-5596	25	1	[	[	X
ejpam-5596	25	2	8	8	NUM
ejpam-5596	25	3	]	]	PUNCT
ejpam-5596	25	4	gave	give	VERB
ejpam-5596	25	5	the	the	DET
ejpam-5596	25	6	concepts	concept	NOUN
ejpam-5596	25	7	of	of	ADP
ejpam-5596	25	8	structures	structure	NOUN
ejpam-5596	25	9	of	of	ADP
ejpam-5596	25	10	fuzzy	fuzzy	ADJ
ejpam-5596	25	11	(	(	PUNCT
ejpam-5596	25	12	m	m	PROPN
ejpam-5596	25	13	,	,	PUNCT
ejpam-5596	25	14	n)-quasi	n)-quasi	ADJ
ejpam-5596	25	15	ideals	ideal	NOUN
ejpam-5596	25	16	in	in	ADP
ejpam-5596	25	17	ordered	order	VERB
ejpam-5596	25	18	semigroups	semigroup	NOUN
ejpam-5596	25	19	.	.	PUNCT
ejpam-5596	26	1	other	other	ADJ
ejpam-5596	26	2	work	work	NOUN
ejpam-5596	26	3	of	of	ADP
ejpam-5596	26	4	ordered	order	VERB
ejpam-5596	26	5	semigroup	semigroup	NOUN
ejpam-5596	26	6	studied	study	VERB
ejpam-5596	26	7	more	more	ADV
ejpam-5596	26	8	like	like	ADP
ejpam-5596	26	9	fuzzy	fuzzy	ADJ
ejpam-5596	26	10	quasi	quasi	NOUN
ejpam-5596	26	11	-	-	NOUN
ejpam-5596	26	12	ideal	ideal	ADJ
ejpam-5596	26	13	[	[	X
ejpam-5596	26	14	4	4	NUM
ejpam-5596	26	15	]	]	PUNCT
ejpam-5596	26	16	,	,	PUNCT
ejpam-5596	26	17	fuzzy	fuzzy	ADJ
ejpam-5596	26	18	filters	filter	NOUN
ejpam-5596	26	19	[	[	X
ejpam-5596	26	20	1	1	NUM
ejpam-5596	26	21	,	,	PUNCT
ejpam-5596	26	22	2	2	NUM
ejpam-5596	26	23	,	,	PUNCT
ejpam-5596	26	24	7	7	NUM
ejpam-5596	26	25	]	]	PUNCT
ejpam-5596	26	26	and	and	CCONJ
ejpam-5596	26	27	fuzzy	fuzzy	ADJ
ejpam-5596	26	28	prime	prime	NOUN
ejpam-5596	26	29	[	[	X
ejpam-5596	26	30	10	10	NUM
ejpam-5596	26	31	]	]	PUNCT
ejpam-5596	26	32	.	.	PUNCT
ejpam-5596	27	1	the	the	DET
ejpam-5596	27	2	purpose	purpose	NOUN
ejpam-5596	27	3	of	of	ADP
ejpam-5596	27	4	this	this	DET
ejpam-5596	27	5	paper	paper	NOUN
ejpam-5596	27	6	is	be	AUX
ejpam-5596	27	7	to	to	PART
ejpam-5596	27	8	extend	extend	VERB
ejpam-5596	27	9	the	the	DET
ejpam-5596	27	10	study	study	NOUN
ejpam-5596	27	11	of	of	ADP
ejpam-5596	27	12	fuzzy	fuzzy	ADJ
ejpam-5596	27	13	(	(	PUNCT
ejpam-5596	27	14	m	m	PROPN
ejpam-5596	27	15	,	,	PUNCT
ejpam-5596	27	16	n)-ideals	n)-ideal	NOUN
ejpam-5596	27	17	in	in	ADP
ejpam-5596	27	18	semigroups	semigroup	NOUN
ejpam-5596	27	19	to	to	PART
ejpam-5596	27	20	ordred	ordre	VERB
ejpam-5596	27	21	semigroups	semigroup	NOUN
ejpam-5596	27	22	.	.	PUNCT
ejpam-5596	28	1	we	we	PRON
ejpam-5596	28	2	prove	prove	VERB
ejpam-5596	28	3	the	the	DET
ejpam-5596	28	4	properties	property	NOUN
ejpam-5596	28	5	of	of	ADP
ejpam-5596	28	6	fuzzy	fuzzy	ADJ
ejpam-5596	28	7	(	(	PUNCT
ejpam-5596	28	8	m	m	PROPN
ejpam-5596	28	9	,	,	PUNCT
ejpam-5596	28	10	n)-ideals	n)-ideal	NOUN
ejpam-5596	28	11	in	in	ADP
ejpam-5596	28	12	ordered	order	VERB
ejpam-5596	28	13	semigroup	semigroup	NOUN
ejpam-5596	28	14	and	and	CCONJ
ejpam-5596	28	15	investigate	investigate	VERB
ejpam-5596	28	16	the	the	DET
ejpam-5596	28	17	properties	property	NOUN
ejpam-5596	28	18	of	of	ADP
ejpam-5596	28	19	minimal	minimal	ADJ
ejpam-5596	28	20	fuzzy	fuzzy	ADJ
ejpam-5596	28	21	(	(	PUNCT
ejpam-5596	28	22	m	m	NOUN
ejpam-5596	28	23	,	,	PUNCT
ejpam-5596	28	24	n	n	CCONJ
ejpam-5596	28	25	)	)	PUNCT
ejpam-5596	28	26	ideals	ideal	NOUN
ejpam-5596	28	27	,	,	PUNCT
ejpam-5596	28	28	and	and	CCONJ
ejpam-5596	28	29	fuzzy	fuzzy	ADJ
ejpam-5596	28	30	prime	prime	ADJ
ejpam-5596	28	31	(	(	PUNCT
ejpam-5596	28	32	semiprime	semiprime	NOUN
ejpam-5596	28	33	)	)	PUNCT
ejpam-5596	28	34	(	(	PUNCT
ejpam-5596	28	35	m	m	PROPN
ejpam-5596	28	36	,	,	PUNCT
ejpam-5596	28	37	n	n	CCONJ
ejpam-5596	28	38	)	)	PUNCT
ejpam-5596	28	39	ideals	ideal	NOUN
ejpam-5596	28	40	in	in	ADP
ejpam-5596	28	41	semigroups	semigroup	NOUN
ejpam-5596	28	42	.	.	PUNCT
ejpam-5596	29	1	the	the	DET
ejpam-5596	29	2	relationship	relationship	NOUN
ejpam-5596	29	3	between	between	ADP
ejpam-5596	29	4	(	(	PUNCT
ejpam-5596	29	5	m	m	PROPN
ejpam-5596	29	6	,	,	PUNCT
ejpam-5596	29	7	n	n	CCONJ
ejpam-5596	29	8	)	)	PUNCT
ejpam-5596	29	9	ideals	ideal	NOUN
ejpam-5596	29	10	and	and	CCONJ
ejpam-5596	29	11	fuzzy	fuzzy	ADJ
ejpam-5596	29	12	(	(	PUNCT
ejpam-5596	29	13	m	m	NOUN
ejpam-5596	29	14	,	,	PUNCT
ejpam-5596	29	15	n	n	CCONJ
ejpam-5596	29	16	)	)	PUNCT
ejpam-5596	29	17	ideals	ideal	NOUN
ejpam-5596	29	18	in	in	ADP
ejpam-5596	29	19	ordered	order	VERB
ejpam-5596	29	20	semigroups	semigroup	NOUN
ejpam-5596	29	21	.	.	PUNCT
ejpam-5596	30	1	finally	finally	ADV
ejpam-5596	30	2	,	,	PUNCT
ejpam-5596	30	3	we	we	PRON
ejpam-5596	30	4	discuss	discuss	VERB
ejpam-5596	30	5	the	the	DET
ejpam-5596	30	6	properties	property	NOUN
ejpam-5596	30	7	of	of	ADP
ejpam-5596	30	8	fuzzy	fuzzy	ADJ
ejpam-5596	30	9	n	n	CCONJ
ejpam-5596	30	10	-	-	PUNCT
ejpam-5596	30	11	interior	interior	ADJ
ejpam-5596	30	12	ideals	ideal	NOUN
ejpam-5596	30	13	and	and	CCONJ
ejpam-5596	30	14	the	the	DET
ejpam-5596	30	15	relationship	relationship	NOUN
ejpam-5596	30	16	between	between	ADP
ejpam-5596	30	17	n	n	CCONJ
ejpam-5596	30	18	-	-	PUNCT
ejpam-5596	30	19	interior	interior	ADJ
ejpam-5596	30	20	ideals	ideal	NOUN
ejpam-5596	30	21	and	and	CCONJ
ejpam-5596	30	22	fuzzy	fuzzy	ADJ
ejpam-5596	30	23	n	n	CCONJ
ejpam-5596	30	24	-	-	PUNCT
ejpam-5596	30	25	interior	interior	ADJ
ejpam-5596	30	26	ideals	ideal	NOUN
ejpam-5596	30	27	in	in	ADP
ejpam-5596	30	28	ordered	order	VERB
ejpam-5596	30	29	semigroups	semigroup	NOUN
ejpam-5596	30	30	.	.	PUNCT
ejpam-5596	31	1	2	2	X
ejpam-5596	31	2	.	.	NUM
ejpam-5596	31	3	preliminaries	preliminary	NOUN
ejpam-5596	31	4	in	in	ADP
ejpam-5596	31	5	this	this	DET
ejpam-5596	31	6	section	section	NOUN
ejpam-5596	31	7	,	,	PUNCT
ejpam-5596	31	8	we	we	PRON
ejpam-5596	31	9	review	review	VERB
ejpam-5596	31	10	some	some	DET
ejpam-5596	31	11	basic	basic	ADJ
ejpam-5596	31	12	concepts	concept	NOUN
ejpam-5596	31	13	that	that	PRON
ejpam-5596	31	14	are	be	AUX
ejpam-5596	31	15	necessary	necessary	ADJ
ejpam-5596	31	16	to	to	PART
ejpam-5596	31	17	understand	understand	VERB
ejpam-5596	31	18	our	our	PRON
ejpam-5596	31	19	next	next	ADJ
ejpam-5596	31	20	section	section	NOUN
ejpam-5596	31	21	.	.	PUNCT
ejpam-5596	32	1	let	let	VERB
ejpam-5596	32	2	(	(	PUNCT
ejpam-5596	32	3	s	s	X
ejpam-5596	32	4	,	,	PUNCT
ejpam-5596	32	5	·	·	PUNCT
ejpam-5596	32	6	)	)	PUNCT
ejpam-5596	32	7	be	be	AUX
ejpam-5596	32	8	a	a	DET
ejpam-5596	32	9	semigroup	semigroup	NOUN
ejpam-5596	32	10	ordered	order	VERB
ejpam-5596	32	11	semigroup	semigroup	NOUN
ejpam-5596	32	12	and	and	CCONJ
ejpam-5596	32	13	s,≤	s,≤	NOUN
ejpam-5596	32	14	)	)	PUNCT
ejpam-5596	32	15	is	be	AUX
ejpam-5596	32	16	a	a	DET
ejpam-5596	32	17	partially	partially	ADV
ejpam-5596	32	18	ordered	order	VERB
ejpam-5596	32	19	set	set	NOUN
ejpam-5596	32	20	.	.	PUNCT
ejpam-5596	33	1	then	then	ADV
ejpam-5596	33	2	(	(	PUNCT
ejpam-5596	33	3	s	s	X
ejpam-5596	33	4	,	,	PUNCT
ejpam-5596	33	5	·	·	PUNCT
ejpam-5596	33	6	,	,	PUNCT
ejpam-5596	33	7	≤	≤	NUM
ejpam-5596	33	8	)	)	PUNCT
ejpam-5596	33	9	is	be	AUX
ejpam-5596	33	10	an	an	DET
ejpam-5596	33	11	ordered	order	VERB
ejpam-5596	33	12	semigroup	semigroup	NOUN
ejpam-5596	33	13	if	if	SCONJ
ejpam-5596	33	14	for	for	ADP
ejpam-5596	33	15	all	all	DET
ejpam-5596	33	16	a	a	DET
ejpam-5596	33	17	,	,	PUNCT
ejpam-5596	33	18	b	b	NOUN
ejpam-5596	33	19	,	,	PUNCT
ejpam-5596	33	20	c	c	PROPN
ejpam-5596	33	21	∈	∈	PROPN
ejpam-5596	33	22	s	s	X
ejpam-5596	33	23	,	,	PUNCT
ejpam-5596	33	24	we	we	PRON
ejpam-5596	33	25	have	have	VERB
ejpam-5596	33	26	a	a	DET
ejpam-5596	33	27	≤	≤	NUM
ejpam-5596	33	28	b	b	NOUN
ejpam-5596	33	29	then	then	ADV
ejpam-5596	33	30	ac	ac	PROPN
ejpam-5596	33	31	≤	≤	PUNCT
ejpam-5596	33	32	bc	bc	PROPN
ejpam-5596	33	33	and	and	CCONJ
ejpam-5596	33	34	ca	can	AUX
ejpam-5596	33	35	≤	≤	NUM
ejpam-5596	33	36	cb	cb	PROPN
ejpam-5596	33	37	.	.	PROPN
ejpam-5596	34	1	for	for	ADP
ejpam-5596	34	2	a	a	DET
ejpam-5596	34	3	nonempty	nonempty	NOUN
ejpam-5596	34	4	subset	subset	NOUN
ejpam-5596	34	5	x	x	PUNCT
ejpam-5596	34	6	and	and	CCONJ
ejpam-5596	34	7	y	y	PROPN
ejpam-5596	34	8	of	of	ADP
ejpam-5596	34	9	ordered	order	VERB
ejpam-5596	34	10	semigroup	semigroup	PROPN
ejpam-5596	34	11	s	s	PROPN
ejpam-5596	34	12	,	,	PUNCT
ejpam-5596	34	13	we	we	PRON
ejpam-5596	34	14	write	write	VERB
ejpam-5596	34	15	(	(	PUNCT
ejpam-5596	34	16	x	x	SYM
ejpam-5596	34	17	]	]	X
ejpam-5596	34	18	:	:	PUNCT
ejpam-5596	34	19	=	=	SYM
ejpam-5596	34	20	{	{	PUNCT
ejpam-5596	34	21	a	a	DET
ejpam-5596	34	22	∈	∈	NOUN
ejpam-5596	34	23	s	s	VERB
ejpam-5596	34	24	|	|	ADV
ejpam-5596	34	25	a	a	DET
ejpam-5596	34	26	≤	≤	NUM
ejpam-5596	34	27	b	b	NOUN
ejpam-5596	34	28	for	for	ADP
ejpam-5596	34	29	some	some	DET
ejpam-5596	34	30	b	b	NOUN
ejpam-5596	34	31	∈	∈	PROPN
ejpam-5596	34	32	x	x	NOUN
ejpam-5596	34	33	}	}	PUNCT
ejpam-5596	34	34	and	and	CCONJ
ejpam-5596	34	35	xy	xy	INTJ
ejpam-5596	34	36	:	:	PUNCT
ejpam-5596	34	37	=	=	X
ejpam-5596	34	38	{	{	PUNCT
ejpam-5596	35	1	xy	xy	INTJ
ejpam-5596	35	2	|	|	ADV
ejpam-5596	35	3	x	x	SYM
ejpam-5596	35	4	∈	∈	PROPN
ejpam-5596	35	5	x	x	X
ejpam-5596	35	6	and	and	CCONJ
ejpam-5596	35	7	y	y	PROPN
ejpam-5596	35	8	∈	∈	PROPN
ejpam-5596	35	9	y	y	PROPN
ejpam-5596	35	10	}	}	PUNCT
ejpam-5596	35	11	.	.	PUNCT
ejpam-5596	36	1	it	it	PRON
ejpam-5596	36	2	is	be	AUX
ejpam-5596	36	3	observed	observe	VERB
ejpam-5596	36	4	that	that	SCONJ
ejpam-5596	36	5	(	(	PUNCT
ejpam-5596	36	6	1	1	X
ejpam-5596	36	7	)	)	PUNCT
ejpam-5596	36	8	x	x	X
ejpam-5596	37	1	⊆	⊆	X
ejpam-5596	37	2	(	(	PUNCT
ejpam-5596	37	3	x	x	SYM
ejpam-5596	37	4	]	]	X
ejpam-5596	37	5	,	,	PUNCT
ejpam-5596	37	6	(	(	PUNCT
ejpam-5596	37	7	2	2	X
ejpam-5596	37	8	)	)	PUNCT
ejpam-5596	38	1	if	if	SCONJ
ejpam-5596	38	2	x	x	PROPN
ejpam-5596	38	3	⊆	⊆	NUM
ejpam-5596	38	4	y	y	NUM
ejpam-5596	38	5	,	,	PUNCT
ejpam-5596	38	6	then	then	ADV
ejpam-5596	38	7	(	(	PUNCT
ejpam-5596	38	8	x	x	X
ejpam-5596	38	9	]	]	X
ejpam-5596	38	10	⊆	⊆	NUM
ejpam-5596	38	11	(	(	PUNCT
ejpam-5596	38	12	y	y	NOUN
ejpam-5596	38	13	]	]	X
ejpam-5596	38	14	,	,	PUNCT
ejpam-5596	38	15	(	(	PUNCT
ejpam-5596	38	16	3	3	X
ejpam-5596	38	17	)	)	PUNCT
ejpam-5596	38	18	(	(	PUNCT
ejpam-5596	38	19	(	(	PUNCT
ejpam-5596	38	20	x	x	X
ejpam-5596	38	21	]	]	X
ejpam-5596	38	22	]	]	X
ejpam-5596	38	23	=	=	SYM
ejpam-5596	38	24	(	(	PUNCT
ejpam-5596	38	25	x	x	X
ejpam-5596	38	26	]	]	X
ejpam-5596	38	27	,	,	PUNCT
ejpam-5596	38	28	(	(	PUNCT
ejpam-5596	38	29	4	4	NUM
ejpam-5596	38	30	)	)	PUNCT
ejpam-5596	38	31	(	(	PUNCT
ejpam-5596	38	32	x	x	X
ejpam-5596	38	33	]	]	X
ejpam-5596	38	34	(	(	PUNCT
ejpam-5596	38	35	y	y	X
ejpam-5596	38	36	]	]	X
ejpam-5596	38	37	⊆	⊆	NUM
ejpam-5596	38	38	(	(	PUNCT
ejpam-5596	38	39	xy	xy	PROPN
ejpam-5596	38	40	]	]	X
ejpam-5596	38	41	,	,	PUNCT
ejpam-5596	38	42	(	(	PUNCT
ejpam-5596	38	43	5	5	NUM
ejpam-5596	38	44	)	)	PUNCT
ejpam-5596	38	45	(	(	PUNCT
ejpam-5596	38	46	(	(	PUNCT
ejpam-5596	38	47	x	x	X
ejpam-5596	38	48	]	]	X
ejpam-5596	38	49	(	(	PUNCT
ejpam-5596	38	50	y	y	NOUN
ejpam-5596	38	51	]	]	X
ejpam-5596	38	52	]	]	X
ejpam-5596	38	53	=	=	X
ejpam-5596	38	54	(	(	PUNCT
ejpam-5596	38	55	xy	xy	PROPN
ejpam-5596	38	56	]	]	X
ejpam-5596	38	57	,	,	PUNCT
ejpam-5596	38	58	(	(	PUNCT
ejpam-5596	38	59	6	6	NUM
ejpam-5596	38	60	)	)	PUNCT
ejpam-5596	38	61	(	(	PUNCT
ejpam-5596	38	62	x	x	SYM
ejpam-5596	38	63	∪	∪	PROPN
ejpam-5596	38	64	y	y	PROPN
ejpam-5596	38	65	]	]	X
ejpam-5596	38	66	=	=	PUNCT
ejpam-5596	38	67	(	(	PUNCT
ejpam-5596	38	68	x	x	SYM
ejpam-5596	38	69	]	]	X
ejpam-5596	38	70	∪	∪	X
ejpam-5596	38	71	(	(	PUNCT
ejpam-5596	38	72	y	y	NOUN
ejpam-5596	38	73	]	]	X
ejpam-5596	38	74	,	,	PUNCT
ejpam-5596	38	75	(	(	PUNCT
ejpam-5596	38	76	7	7	NUM
ejpam-5596	38	77	)	)	PUNCT
ejpam-5596	38	78	(	(	PUNCT
ejpam-5596	38	79	x	x	X
ejpam-5596	38	80	∩	∩	ADJ
ejpam-5596	38	81	y	y	NOUN
ejpam-5596	38	82	]	]	X
ejpam-5596	38	83	=	=	PUNCT
ejpam-5596	38	84	(	(	PUNCT
ejpam-5596	38	85	x	x	SYM
ejpam-5596	38	86	]	]	X
ejpam-5596	38	87	∩	∩	NOUN
ejpam-5596	38	88	(	(	PUNCT
ejpam-5596	38	89	y	y	NOUN
ejpam-5596	38	90	]	]	PUNCT
ejpam-5596	38	91	.	.	PUNCT
ejpam-5596	39	1	let	let	VERB
ejpam-5596	39	2	(	(	PUNCT
ejpam-5596	39	3	s	s	X
ejpam-5596	39	4	,	,	PUNCT
ejpam-5596	39	5	·	·	PUNCT
ejpam-5596	39	6	,	,	PUNCT
ejpam-5596	39	7	≤	≤	NUM
ejpam-5596	39	8	)	)	PUNCT
ejpam-5596	39	9	be	be	AUX
ejpam-5596	39	10	an	an	DET
ejpam-5596	39	11	ordered	order	VERB
ejpam-5596	39	12	semigroup	semigroup	NOUN
ejpam-5596	39	13	,	,	PUNCT
ejpam-5596	39	14	(	(	PUNCT
ejpam-5596	39	15	∅	∅	NOUN
ejpam-5596	39	16	=	=	VERB
ejpam-5596	39	17	̸)k	̸)k	PROPN
ejpam-5596	39	18	⊆	⊆	NUM
ejpam-5596	39	19	k	k	NOUN
ejpam-5596	39	20	is	be	AUX
ejpam-5596	39	21	called	call	VERB
ejpam-5596	39	22	a	a	DET
ejpam-5596	39	23	subsemigroup	subsemigroup	NOUN
ejpam-5596	39	24	such	such	ADJ
ejpam-5596	39	25	that	that	SCONJ
ejpam-5596	39	26	k2	k2	PROPN
ejpam-5596	39	27	⊆	⊆	NUM
ejpam-5596	39	28	k.	k.	PROPN
ejpam-5596	39	29	a	a	DET
ejpam-5596	39	30	left	left	ADJ
ejpam-5596	39	31	(	(	PUNCT
ejpam-5596	39	32	right	right	ADJ
ejpam-5596	39	33	)	)	PUNCT
ejpam-5596	39	34	ideal	ideal	NOUN
ejpam-5596	39	35	of	of	ADP
ejpam-5596	39	36	a	a	DET
ejpam-5596	39	37	ordered	order	VERB
ejpam-5596	39	38	semigroup	semigroup	NOUN
ejpam-5596	39	39	(	(	PUNCT
ejpam-5596	39	40	s	s	PROPN
ejpam-5596	39	41	,	,	PUNCT
ejpam-5596	39	42	·	·	PUNCT
ejpam-5596	39	43	,	,	PUNCT
ejpam-5596	39	44	≤	≤	NUM
ejpam-5596	39	45	)	)	PUNCT
ejpam-5596	39	46	is	be	AUX
ejpam-5596	39	47	a	a	DET
ejpam-5596	39	48	non	non	ADJ
ejpam-5596	39	49	-	-	ADJ
ejpam-5596	39	50	empty	empty	ADJ
ejpam-5596	39	51	set	set	NOUN
ejpam-5596	39	52	k	k	PROPN
ejpam-5596	39	53	of	of	ADP
ejpam-5596	39	54	k	k	PROPN
ejpam-5596	40	1	such	such	ADJ
ejpam-5596	40	2	that	that	PRON
ejpam-5596	40	3	sk	sk	VERB
ejpam-5596	40	4	⊆	⊆	NUM
ejpam-5596	40	5	k	k	X
ejpam-5596	40	6	(	(	PUNCT
ejpam-5596	40	7	ks	ks	PROPN
ejpam-5596	40	8	⊆	⊆	NUM
ejpam-5596	40	9	k	k	NOUN
ejpam-5596	40	10	)	)	PUNCT
ejpam-5596	40	11	and	and	CCONJ
ejpam-5596	40	12	(	(	PUNCT
ejpam-5596	40	13	k	k	X
ejpam-5596	40	14	]	]	X
ejpam-5596	40	15	.	.	PUNCT
ejpam-5596	41	1	by	by	ADP
ejpam-5596	41	2	an	an	DET
ejpam-5596	41	3	ideal	ideal	NOUN
ejpam-5596	41	4	of	of	ADP
ejpam-5596	41	5	an	an	DET
ejpam-5596	41	6	ordered	order	VERB
ejpam-5596	41	7	semigroup	semigroup	NOUN
ejpam-5596	41	8	(	(	PUNCT
ejpam-5596	41	9	s	s	PROPN
ejpam-5596	41	10	,	,	PUNCT
ejpam-5596	41	11	·	·	PUNCT
ejpam-5596	41	12	,	,	PUNCT
ejpam-5596	41	13	≤	≤	NUM
ejpam-5596	41	14	)	)	PUNCT
ejpam-5596	41	15	,	,	PUNCT
ejpam-5596	41	16	we	we	PRON
ejpam-5596	41	17	mean	mean	VERB
ejpam-5596	41	18	a	a	DET
ejpam-5596	41	19	non	non	ADJ
ejpam-5596	41	20	-	-	ADJ
ejpam-5596	41	21	empty	empty	ADJ
ejpam-5596	41	22	set	set	NOUN
ejpam-5596	41	23	of	of	ADP
ejpam-5596	41	24	s	s	PRON
ejpam-5596	41	25	which	which	PRON
ejpam-5596	41	26	is	be	AUX
ejpam-5596	41	27	both	both	CCONJ
ejpam-5596	41	28	a	a	DET
ejpam-5596	41	29	left	left	NOUN
ejpam-5596	41	30	and	and	CCONJ
ejpam-5596	41	31	a	a	DET
ejpam-5596	41	32	right	right	ADJ
ejpam-5596	41	33	ideal	ideal	NOUN
ejpam-5596	41	34	of	of	ADP
ejpam-5596	41	35	s.	s.	PROPN
ejpam-5596	41	36	definition	definition	NOUN
ejpam-5596	41	37	1	1	NUM
ejpam-5596	41	38	.	.	PUNCT
ejpam-5596	42	1	[	[	X
ejpam-5596	42	2	3	3	X
ejpam-5596	42	3	]	]	X
ejpam-5596	42	4	a	a	DET
ejpam-5596	42	5	subsemigroup	subsemigroup	NOUN
ejpam-5596	42	6	k	k	PROPN
ejpam-5596	42	7	of	of	ADP
ejpam-5596	42	8	an	an	DET
ejpam-5596	42	9	ordered	order	VERB
ejpam-5596	42	10	semigroup	semigroup	NOUN
ejpam-5596	42	11	(	(	PUNCT
ejpam-5596	42	12	s	s	PROPN
ejpam-5596	42	13	,	,	PUNCT
ejpam-5596	42	14	·	·	PUNCT
ejpam-5596	42	15	,	,	PUNCT
ejpam-5596	42	16	≤	≤	NUM
ejpam-5596	42	17	)	)	PUNCT
ejpam-5596	42	18	is	be	AUX
ejpam-5596	42	19	called	call	VERB
ejpam-5596	42	20	an	an	DET
ejpam-5596	42	21	(	(	PUNCT
ejpam-5596	42	22	m	m	PROPN
ejpam-5596	42	23	,	,	PUNCT
ejpam-5596	42	24	n)ideal	n)ideal	NOUN
ejpam-5596	42	25	of	of	ADP
ejpam-5596	42	26	s	s	PRON
ejpam-5596	42	27	if	if	SCONJ
ejpam-5596	42	28	k	k	PROPN
ejpam-5596	42	29	satisfies	satisfy	VERB
ejpam-5596	42	30	the	the	DET
ejpam-5596	42	31	following	follow	VERB
ejpam-5596	42	32	conditions	condition	NOUN
ejpam-5596	42	33	:	:	PUNCT
ejpam-5596	42	34	p.	p.	PROPN
ejpam-5596	42	35	khamrot	khamrot	PROPN
ejpam-5596	42	36	,	,	PUNCT
ejpam-5596	42	37	a.	a.	NOUN
ejpam-5596	42	38	iampan	iampan	PROPN
ejpam-5596	42	39	,	,	PUNCT
ejpam-5596	42	40	t.	t.	PROPN
ejpam-5596	42	41	gaketem	gaketem	PROPN
ejpam-5596	42	42	/	/	SYM
ejpam-5596	42	43	eur	eur	PROPN
ejpam-5596	42	44	.	.	PUNCT
ejpam-5596	43	1	j.	j.	PROPN
ejpam-5596	43	2	pure	pure	PROPN
ejpam-5596	43	3	appl	appl	PROPN
ejpam-5596	43	4	.	.	PROPN
ejpam-5596	43	5	math	math	PROPN
ejpam-5596	43	6	,	,	PUNCT
ejpam-5596	43	7	18	18	NUM
ejpam-5596	43	8	(	(	PUNCT
ejpam-5596	43	9	1	1	NUM
ejpam-5596	43	10	)	)	PUNCT
ejpam-5596	43	11	(	(	PUNCT
ejpam-5596	43	12	2025	2025	NUM
ejpam-5596	43	13	)	)	PUNCT
ejpam-5596	43	14	,	,	PUNCT
ejpam-5596	43	15	5596	5596	NUM
ejpam-5596	43	16	3	3	NUM
ejpam-5596	43	17	of	of	ADP
ejpam-5596	43	18	12	12	NUM
ejpam-5596	43	19	(	(	PUNCT
ejpam-5596	43	20	1	1	NUM
ejpam-5596	43	21	)	)	PUNCT
ejpam-5596	43	22	kmskn	kmskn	NOUN
ejpam-5596	43	23	⊆	⊆	NUM
ejpam-5596	43	24	k.	k.	NOUN
ejpam-5596	43	25	(	(	PUNCT
ejpam-5596	43	26	2	2	NUM
ejpam-5596	43	27	)	)	PUNCT
ejpam-5596	43	28	k	k	NOUN
ejpam-5596	44	1	=	=	SYM
ejpam-5596	44	2	(	(	PUNCT
ejpam-5596	44	3	k	k	X
ejpam-5596	44	4	]	]	X
ejpam-5596	44	5	,	,	PUNCT
ejpam-5596	44	6	that	that	PRON
ejpam-5596	44	7	is	be	AUX
ejpam-5596	44	8	for	for	ADP
ejpam-5596	44	9	x	x	PROPN
ejpam-5596	44	10	∈	∈	PROPN
ejpam-5596	44	11	k	k	PROPN
ejpam-5596	44	12	and	and	CCONJ
ejpam-5596	44	13	y	y	PROPN
ejpam-5596	44	14	∈	∈	PROPN
ejpam-5596	44	15	s	s	PROPN
ejpam-5596	44	16	,	,	PUNCT
ejpam-5596	44	17	y	y	PROPN
ejpam-5596	44	18	≤	≤	NUM
ejpam-5596	44	19	x	x	PUNCT
ejpam-5596	44	20	implies	imply	VERB
ejpam-5596	44	21	y	y	PROPN
ejpam-5596	44	22	∈	∈	PROPN
ejpam-5596	44	23	k.	k.	PROPN
ejpam-5596	44	24	where	where	SCONJ
ejpam-5596	44	25	m	m	PROPN
ejpam-5596	44	26	,	,	PUNCT
ejpam-5596	44	27	n	n	PRON
ejpam-5596	44	28	are	be	AUX
ejpam-5596	44	29	non	non	ADJ
ejpam-5596	44	30	-	-	ADJ
ejpam-5596	44	31	negative	negative	ADJ
ejpam-5596	44	32	integers	integer	NOUN
ejpam-5596	44	33	.	.	PUNCT
ejpam-5596	45	1	we	we	PRON
ejpam-5596	45	2	see	see	VERB
ejpam-5596	45	3	that	that	SCONJ
ejpam-5596	45	4	for	for	ADP
ejpam-5596	45	5	any	any	DET
ejpam-5596	45	6	δ1	δ1	NOUN
ejpam-5596	45	7	,	,	PUNCT
ejpam-5596	45	8	δ2	δ2	VERB
ejpam-5596	45	9	∈	∈	NOUN
ejpam-5596	46	1	[	[	X
ejpam-5596	46	2	0	0	NUM
ejpam-5596	46	3	,	,	PUNCT
ejpam-5596	46	4	1	1	NUM
ejpam-5596	46	5	]	]	PUNCT
ejpam-5596	46	6	,	,	PUNCT
ejpam-5596	46	7	we	we	PRON
ejpam-5596	46	8	have	have	VERB
ejpam-5596	46	9	δ1	δ1	NOUN
ejpam-5596	46	10	∨	∨	NUM
ejpam-5596	46	11	δ2	δ2	VERB
ejpam-5596	46	12	=	=	PUNCT
ejpam-5596	46	13	max{δ1	max{δ1	NOUN
ejpam-5596	46	14	,	,	PUNCT
ejpam-5596	46	15	δ2	δ2	ADJ
ejpam-5596	46	16	}	}	PUNCT
ejpam-5596	46	17	and	and	CCONJ
ejpam-5596	46	18	δ1	δ1	VERB
ejpam-5596	46	19	∧	∧	PROPN
ejpam-5596	46	20	δ2	δ2	VERB
ejpam-5596	46	21	=	=	SYM
ejpam-5596	46	22	min{δ1	min{δ1	NOUN
ejpam-5596	46	23	,	,	PUNCT
ejpam-5596	46	24	δ2	δ2	VERB
ejpam-5596	46	25	}	}	PUNCT
ejpam-5596	46	26	.	.	PUNCT
ejpam-5596	47	1	a	a	DET
ejpam-5596	47	2	fuzzy	fuzzy	ADJ
ejpam-5596	47	3	set	set	VERB
ejpam-5596	47	4	δ	δ	PROPN
ejpam-5596	47	5	of	of	ADP
ejpam-5596	47	6	a	a	DET
ejpam-5596	47	7	non	non	ADJ
ejpam-5596	47	8	-	-	ADJ
ejpam-5596	47	9	empty	empty	ADJ
ejpam-5596	47	10	set	set	ADJ
ejpam-5596	47	11	t	t	PROPN
ejpam-5596	47	12	is	be	AUX
ejpam-5596	47	13	function	function	NOUN
ejpam-5596	47	14	from	from	ADP
ejpam-5596	47	15	t	t	PROPN
ejpam-5596	47	16	into	into	ADP
ejpam-5596	47	17	unit	unit	NOUN
ejpam-5596	47	18	closed	close	VERB
ejpam-5596	47	19	interval	interval	NOUN
ejpam-5596	47	20	[	[	X
ejpam-5596	47	21	0	0	NUM
ejpam-5596	47	22	,	,	PUNCT
ejpam-5596	47	23	1	1	NUM
ejpam-5596	47	24	]	]	PUNCT
ejpam-5596	47	25	of	of	ADP
ejpam-5596	47	26	real	real	ADJ
ejpam-5596	47	27	numbers	number	NOUN
ejpam-5596	47	28	,	,	PUNCT
ejpam-5596	47	29	i.e.	i.e.	X
ejpam-5596	47	30	,	,	PUNCT
ejpam-5596	47	31	δ	δ	PROPN
ejpam-5596	47	32	:	:	PUNCT
ejpam-5596	47	33	t	t	PROPN
ejpam-5596	47	34	→	→	PUNCT
ejpam-5596	48	1	[	[	X
ejpam-5596	48	2	0	0	NUM
ejpam-5596	48	3	,	,	PUNCT
ejpam-5596	48	4	1	1	NUM
ejpam-5596	48	5	]	]	PUNCT
ejpam-5596	48	6	.	.	PUNCT
ejpam-5596	49	1	for	for	ADP
ejpam-5596	49	2	any	any	DET
ejpam-5596	49	3	two	two	NUM
ejpam-5596	49	4	fuzzy	fuzzy	ADJ
ejpam-5596	49	5	sets	set	NOUN
ejpam-5596	49	6	δ	δ	PROPN
ejpam-5596	49	7	and	and	CCONJ
ejpam-5596	49	8	ϑ	ϑ	X
ejpam-5596	49	9	of	of	ADP
ejpam-5596	49	10	a	a	DET
ejpam-5596	49	11	non	non	ADJ
ejpam-5596	49	12	-	-	ADJ
ejpam-5596	49	13	empty	empty	ADJ
ejpam-5596	49	14	set	set	ADJ
ejpam-5596	49	15	t	t	NOUN
ejpam-5596	49	16	,	,	PUNCT
ejpam-5596	49	17	define	define	VERB
ejpam-5596	49	18	≥,=,∧	≥,=,∧	NUM
ejpam-5596	49	19	and	and	CCONJ
ejpam-5596	49	20	∨	∨	NUM
ejpam-5596	49	21	as	as	SCONJ
ejpam-5596	49	22	follows	follow	VERB
ejpam-5596	49	23	:	:	PUNCT
ejpam-5596	49	24	(	(	PUNCT
ejpam-5596	49	25	1	1	X
ejpam-5596	49	26	)	)	PUNCT
ejpam-5596	49	27	δ	δ	PROPN
ejpam-5596	49	28	≥	≥	X
ejpam-5596	49	29	ϑ	ϑ	X
ejpam-5596	49	30	⇔	⇔	X
ejpam-5596	49	31	δ(e	δ(e	NUM
ejpam-5596	49	32	)	)	PUNCT
ejpam-5596	49	33	≥	≥	NOUN
ejpam-5596	49	34	ϑ(e	ϑ(e	PROPN
ejpam-5596	49	35	)	)	PUNCT
ejpam-5596	49	36	for	for	ADP
ejpam-5596	49	37	all	all	DET
ejpam-5596	49	38	e	e	PROPN
ejpam-5596	49	39	∈	∈	PROPN
ejpam-5596	49	40	t	t	NOUN
ejpam-5596	49	41	,	,	PUNCT
ejpam-5596	49	42	(	(	PUNCT
ejpam-5596	49	43	2	2	X
ejpam-5596	49	44	)	)	PUNCT
ejpam-5596	49	45	δ	δ	NOUN
ejpam-5596	49	46	=	=	PROPN
ejpam-5596	49	47	ϑ	ϑ	PROPN
ejpam-5596	49	48	⇔	⇔	PROPN
ejpam-5596	49	49	δ	δ	PROPN
ejpam-5596	49	50	≥	≥	X
ejpam-5596	49	51	ϑ	ϑ	X
ejpam-5596	49	52	and	and	CCONJ
ejpam-5596	49	53	ϑ	ϑ	PRON
ejpam-5596	49	54	≥	≥	X
ejpam-5596	49	55	δ	δ	PROPN
ejpam-5596	49	56	,	,	PUNCT
ejpam-5596	49	57	(	(	PUNCT
ejpam-5596	49	58	3	3	NUM
ejpam-5596	49	59	)	)	PUNCT
ejpam-5596	49	60	(	(	PUNCT
ejpam-5596	49	61	δ	δ	PROPN
ejpam-5596	49	62	∧	∧	PROPN
ejpam-5596	49	63	ϑ)(e	ϑ)(e	NOUN
ejpam-5596	49	64	)	)	PUNCT
ejpam-5596	49	65	=	=	PUNCT
ejpam-5596	50	1	min{δ(e	min{δ(e	NOUN
ejpam-5596	50	2	)	)	PUNCT
ejpam-5596	50	3	,	,	PUNCT
ejpam-5596	50	4	ϑ(e	ϑ(e	PROPN
ejpam-5596	50	5	)	)	PUNCT
ejpam-5596	50	6	}	}	PUNCT
ejpam-5596	50	7	=	=	PUNCT
ejpam-5596	50	8	δ(e	δ(e	X
ejpam-5596	50	9	)	)	PUNCT
ejpam-5596	50	10	∧	∧	PROPN
ejpam-5596	50	11	ϑ(e	ϑ(e	PROPN
ejpam-5596	50	12	)	)	PUNCT
ejpam-5596	51	1	for	for	ADP
ejpam-5596	51	2	all	all	DET
ejpam-5596	51	3	e	e	PROPN
ejpam-5596	51	4	∈	∈	PROPN
ejpam-5596	51	5	t	t	NOUN
ejpam-5596	51	6	,	,	PUNCT
ejpam-5596	51	7	(	(	PUNCT
ejpam-5596	51	8	4	4	NUM
ejpam-5596	51	9	)	)	PUNCT
ejpam-5596	51	10	(	(	PUNCT
ejpam-5596	51	11	δ	δ	PROPN
ejpam-5596	51	12	∨	∨	VERB
ejpam-5596	51	13	ϑ)(e	ϑ)(e	NOUN
ejpam-5596	51	14	)	)	PUNCT
ejpam-5596	51	15	=	=	SYM
ejpam-5596	51	16	max{δ(e	max{δ(e	NOUN
ejpam-5596	51	17	)	)	PUNCT
ejpam-5596	51	18	,	,	PUNCT
ejpam-5596	51	19	ϑ(e	ϑ(e	PROPN
ejpam-5596	51	20	)	)	PUNCT
ejpam-5596	51	21	}	}	PUNCT
ejpam-5596	51	22	=	=	PUNCT
ejpam-5596	51	23	δ(e	δ(e	X
ejpam-5596	51	24	)	)	PUNCT
ejpam-5596	51	25	∨	∨	NUM
ejpam-5596	51	26	ϑ(e	ϑ(e	PROPN
ejpam-5596	51	27	)	)	PUNCT
ejpam-5596	51	28	for	for	ADP
ejpam-5596	51	29	all	all	DET
ejpam-5596	51	30	e	e	PROPN
ejpam-5596	51	31	∈	∈	PROPN
ejpam-5596	51	32	t	t	PROPN
ejpam-5596	51	33	.	.	PUNCT
ejpam-5596	52	1	for	for	ADP
ejpam-5596	52	2	the	the	DET
ejpam-5596	52	3	symbol	symbol	NOUN
ejpam-5596	52	4	δ	δ	PROPN
ejpam-5596	52	5	≤	≤	PROPN
ejpam-5596	52	6	ϑ	ϑ	X
ejpam-5596	52	7	,	,	PUNCT
ejpam-5596	52	8	we	we	PRON
ejpam-5596	52	9	mean	mean	VERB
ejpam-5596	52	10	ϑ	ϑ	PRON
ejpam-5596	52	11	≥	≥	X
ejpam-5596	52	12	δ	δ	PROPN
ejpam-5596	52	13	.	.	PUNCT
ejpam-5596	53	1	the	the	DET
ejpam-5596	53	2	following	follow	VERB
ejpam-5596	53	3	definitions	definition	NOUN
ejpam-5596	53	4	are	be	AUX
ejpam-5596	53	5	types	type	NOUN
ejpam-5596	53	6	of	of	ADP
ejpam-5596	53	7	fuzzy	fuzzy	ADJ
ejpam-5596	53	8	substructures	substructure	NOUN
ejpam-5596	53	9	of	of	ADP
ejpam-5596	53	10	a	a	DET
ejpam-5596	53	11	semigroup	semigroup	NOUN
ejpam-5596	53	12	.	.	PUNCT
ejpam-5596	54	1	definition	definition	NOUN
ejpam-5596	54	2	2	2	NUM
ejpam-5596	54	3	.	.	PUNCT
ejpam-5596	55	1	[	[	X
ejpam-5596	55	2	11	11	NUM
ejpam-5596	55	3	]	]	PUNCT
ejpam-5596	55	4	a	a	DET
ejpam-5596	55	5	fuzzy	fuzzy	ADJ
ejpam-5596	55	6	set	set	VERB
ejpam-5596	55	7	δ	δ	PROPN
ejpam-5596	55	8	of	of	ADP
ejpam-5596	55	9	a	a	DET
ejpam-5596	55	10	semigroup	semigroup	NOUN
ejpam-5596	55	11	s	s	NOUN
ejpam-5596	55	12	is	be	AUX
ejpam-5596	55	13	said	say	VERB
ejpam-5596	55	14	to	to	PART
ejpam-5596	55	15	be	be	AUX
ejpam-5596	55	16	a	a	DET
ejpam-5596	55	17	fuzzy	fuzzy	ADJ
ejpam-5596	55	18	ideal	ideal	NOUN
ejpam-5596	55	19	of	of	ADP
ejpam-5596	55	20	s	s	PRON
ejpam-5596	55	21	if	if	SCONJ
ejpam-5596	55	22	δ(uv	δ(uv	NOUN
ejpam-5596	55	23	)	)	PUNCT
ejpam-5596	55	24	≥	≥	NOUN
ejpam-5596	55	25	δ(u	δ(u	NOUN
ejpam-5596	55	26	)	)	PUNCT
ejpam-5596	55	27	∨	∨	NOUN
ejpam-5596	55	28	δ(v	δ(v	PROPN
ejpam-5596	55	29	)	)	PUNCT
ejpam-5596	55	30	for	for	ADP
ejpam-5596	55	31	all	all	DET
ejpam-5596	55	32	u	u	NOUN
ejpam-5596	55	33	,	,	PUNCT
ejpam-5596	55	34	v	v	PROPN
ejpam-5596	55	35	∈	∈	PROPN
ejpam-5596	55	36	s.	s.	PROPN
ejpam-5596	55	37	definition	definition	NOUN
ejpam-5596	55	38	3	3	NUM
ejpam-5596	55	39	.	.	PUNCT
ejpam-5596	56	1	[	[	X
ejpam-5596	56	2	9	9	NUM
ejpam-5596	56	3	]	]	PUNCT
ejpam-5596	56	4	a	a	DET
ejpam-5596	56	5	fuzzy	fuzzy	ADJ
ejpam-5596	56	6	subsemigroup	subsemigroup	NOUN
ejpam-5596	56	7	of	of	ADP
ejpam-5596	56	8	δ	δ	PROPN
ejpam-5596	56	9	if	if	SCONJ
ejpam-5596	56	10	a	a	DET
ejpam-5596	56	11	semigroup	semigroup	NOUN
ejpam-5596	56	12	s	s	NOUN
ejpam-5596	56	13	is	be	AUX
ejpam-5596	56	14	said	say	VERB
ejpam-5596	56	15	to	to	PART
ejpam-5596	56	16	be	be	AUX
ejpam-5596	56	17	a	a	DET
ejpam-5596	56	18	fuzzy	fuzzy	ADJ
ejpam-5596	56	19	(	(	PUNCT
ejpam-5596	56	20	m	m	NOUN
ejpam-5596	56	21	,	,	PUNCT
ejpam-5596	56	22	n)ideal	n)ideal	NOUN
ejpam-5596	56	23	of	of	ADP
ejpam-5596	56	24	s	s	PRON
ejpam-5596	56	25	if	if	SCONJ
ejpam-5596	56	26	for	for	ADP
ejpam-5596	56	27	all	all	DET
ejpam-5596	56	28	u1	u1	NOUN
ejpam-5596	56	29	,	,	PUNCT
ejpam-5596	56	30	u2	u2	NOUN
ejpam-5596	56	31	,	,	PUNCT
ejpam-5596	56	32	...	...	PUNCT
ejpam-5596	56	33	,	,	PUNCT
ejpam-5596	56	34	um	um	INTJ
ejpam-5596	56	35	,	,	PUNCT
ejpam-5596	56	36	v1	v1	PROPN
ejpam-5596	56	37	,	,	PUNCT
ejpam-5596	56	38	v2	v2	PROPN
ejpam-5596	56	39	,	,	PUNCT
ejpam-5596	56	40	...	...	PUNCT
ejpam-5596	56	41	,	,	PUNCT
ejpam-5596	56	42	vn	vn	INTJ
ejpam-5596	56	43	,	,	PUNCT
ejpam-5596	56	44	z	z	PROPN
ejpam-5596	56	45	∈	∈	PROPN
ejpam-5596	56	46	s	s	X
ejpam-5596	56	47	and	and	CCONJ
ejpam-5596	56	48	m	m	PROPN
ejpam-5596	56	49	,	,	PUNCT
ejpam-5596	56	50	n	n	PROPN
ejpam-5596	56	51	∈	∈	PROPN
ejpam-5596	56	52	n	n	CCONJ
ejpam-5596	56	53	,	,	PUNCT
ejpam-5596	56	54	we	we	PRON
ejpam-5596	56	55	have	have	VERB
ejpam-5596	56	56	δs(u1u2	δs(u1u2	X
ejpam-5596	56	57	·	·	PUNCT
ejpam-5596	56	58	·	·	PUNCT
ejpam-5596	56	59	·	·	PUNCT
ejpam-5596	56	60	umzv1v2	umzv1v2	X
ejpam-5596	56	61	·	·	PUNCT
ejpam-5596	56	62	·	·	PUNCT
ejpam-5596	56	63	·	·	PUNCT
ejpam-5596	56	64	vn	vn	X
ejpam-5596	56	65	)	)	PUNCT
ejpam-5596	56	66	≥	≥	NOUN
ejpam-5596	56	67	δs(u1)∧	δs(u1)∧	NOUN
ejpam-5596	56	68	ϑf(u2)∧	ϑf(u2)∧	INTJ
ejpam-5596	56	69	...	...	PUNCT
ejpam-5596	57	1	∧	∧	PROPN
ejpam-5596	57	2	δs(um)∧	δs(um)∧	X
ejpam-5596	57	3	δs(v1)∧	δs(v1)∧	NOUN
ejpam-5596	57	4	δs(v2)∧	δs(v2)∧	NOUN
ejpam-5596	57	5	...	...	PUNCT
ejpam-5596	57	6	∧	∧	PROPN
ejpam-5596	57	7	δs(vn	δs(vn	PROPN
ejpam-5596	57	8	)	)	PUNCT
ejpam-5596	57	9	.	.	PUNCT
ejpam-5596	58	1	for	for	ADP
ejpam-5596	58	2	any	any	DET
ejpam-5596	58	3	element	element	NOUN
ejpam-5596	58	4	k	k	PROPN
ejpam-5596	58	5	in	in	ADP
ejpam-5596	58	6	an	an	DET
ejpam-5596	58	7	ordered	order	VERB
ejpam-5596	58	8	semigroup	semigroup	NOUN
ejpam-5596	58	9	s	s	PROPN
ejpam-5596	58	10	,	,	PUNCT
ejpam-5596	58	11	define	define	VERB
ejpam-5596	58	12	the	the	DET
ejpam-5596	58	13	set	set	NOUN
ejpam-5596	58	14	fk	fk	INTJ
ejpam-5596	58	15	by	by	ADP
ejpam-5596	58	16	fk	fk	INTJ
ejpam-5596	58	17	:	:	PUNCT
ejpam-5596	59	1	=	=	SYM
ejpam-5596	59	2	{	{	PUNCT
ejpam-5596	59	3	(	(	PUNCT
ejpam-5596	59	4	y	y	PROPN
ejpam-5596	59	5	,	,	PUNCT
ejpam-5596	59	6	z	z	NOUN
ejpam-5596	59	7	)	)	PUNCT
ejpam-5596	59	8	∈	∈	PROPN
ejpam-5596	59	9	s	s	PART
ejpam-5596	59	10	×	×	NOUN
ejpam-5596	59	11	s	s	X
ejpam-5596	59	12	|	|	NOUN
ejpam-5596	59	13	k	k	PROPN
ejpam-5596	59	14	≤	≤	PROPN
ejpam-5596	59	15	yz	yz	PROPN
ejpam-5596	59	16	}	}	PUNCT
ejpam-5596	59	17	.	.	PUNCT
ejpam-5596	60	1	for	for	ADP
ejpam-5596	60	2	two	two	NUM
ejpam-5596	60	3	fuzzy	fuzzy	ADJ
ejpam-5596	60	4	sets	set	NOUN
ejpam-5596	60	5	δ	δ	PROPN
ejpam-5596	60	6	and	and	CCONJ
ejpam-5596	60	7	ϑ	ϑ	X
ejpam-5596	60	8	on	on	ADP
ejpam-5596	60	9	a	a	DET
ejpam-5596	60	10	semigroup	semigroup	NOUN
ejpam-5596	60	11	s	s	NOUN
ejpam-5596	60	12	,	,	PUNCT
ejpam-5596	60	13	define	define	VERB
ejpam-5596	60	14	the	the	DET
ejpam-5596	60	15	product	product	NOUN
ejpam-5596	60	16	δ	δ	NOUN
ejpam-5596	60	17	◦	◦	NOUN
ejpam-5596	60	18	ϑ	ϑ	X
ejpam-5596	60	19	as	as	SCONJ
ejpam-5596	60	20	follows	follow	VERB
ejpam-5596	60	21	:	:	PUNCT
ejpam-5596	60	22	for	for	ADP
ejpam-5596	60	23	all	all	DET
ejpam-5596	60	24	k	k	PROPN
ejpam-5596	60	25	∈	∈	PROPN
ejpam-5596	60	26	s	s	NOUN
ejpam-5596	60	27	,	,	PUNCT
ejpam-5596	60	28	(	(	PUNCT
ejpam-5596	60	29	δ	δ	PROPN
ejpam-5596	60	30	◦	◦	NOUN
ejpam-5596	60	31	ϑ)(k	ϑ)(k	PUNCT
ejpam-5596	60	32	)	)	PUNCT
ejpam-5596	60	33	=	=	SYM
ejpam-5596	61	1			PROPN
ejpam-5596	61	2	∨	∨	NOUN
ejpam-5596	61	3	(	(	PUNCT
ejpam-5596	61	4	y	y	PROPN
ejpam-5596	61	5	,	,	PUNCT
ejpam-5596	61	6	z)∈fk	z)∈fk	PROPN
ejpam-5596	61	7	{	{	PUNCT
ejpam-5596	61	8	δ(y	δ(y	ADV
ejpam-5596	61	9	)	)	PUNCT
ejpam-5596	61	10	∧	∧	PROPN
ejpam-5596	61	11	ϑ(z	ϑ(z	PROPN
ejpam-5596	61	12	)	)	PUNCT
ejpam-5596	61	13	}	}	PUNCT
ejpam-5596	61	14	if	if	SCONJ
ejpam-5596	61	15	fk	fk	INTJ
ejpam-5596	61	16	̸=	̸=	PROPN
ejpam-5596	61	17	∅	∅	NOUN
ejpam-5596	61	18	,	,	PUNCT
ejpam-5596	61	19	0	0	PUNCT
ejpam-5596	61	20	if	if	SCONJ
ejpam-5596	61	21	fk	fk	INTJ
ejpam-5596	61	22	=	=	PUNCT
ejpam-5596	61	23	∅.	∅.	PRON
ejpam-5596	61	24	definition	definition	NOUN
ejpam-5596	61	25	4	4	NUM
ejpam-5596	61	26	.	.	PUNCT
ejpam-5596	62	1	let	let	VERB
ejpam-5596	62	2	i	i	PRON
ejpam-5596	62	3	be	be	AUX
ejpam-5596	62	4	a	a	DET
ejpam-5596	62	5	non	non	ADJ
ejpam-5596	62	6	-	-	ADJ
ejpam-5596	62	7	empty	empty	ADJ
ejpam-5596	62	8	set	set	NOUN
ejpam-5596	62	9	of	of	ADP
ejpam-5596	62	10	an	an	DET
ejpam-5596	62	11	ordered	order	VERB
ejpam-5596	62	12	semigroup	semigroup	PROPN
ejpam-5596	62	13	s.	s.	PROPN
ejpam-5596	62	14	a	a	DET
ejpam-5596	62	15	characteristic	characteristic	ADJ
ejpam-5596	62	16	function	function	NOUN
ejpam-5596	62	17	are	be	AUX
ejpam-5596	62	18	respectively	respectively	ADV
ejpam-5596	62	19	defined	define	VERB
ejpam-5596	62	20	by	by	ADP
ejpam-5596	62	21	λi	λi	ADP
ejpam-5596	62	22	:	:	PUNCT
ejpam-5596	62	23	s	s	X
ejpam-5596	62	24	→	→	SYM
ejpam-5596	62	25	[	[	X
ejpam-5596	62	26	0	0	NUM
ejpam-5596	62	27	,	,	PUNCT
ejpam-5596	62	28	1	1	NUM
ejpam-5596	62	29	]	]	PUNCT
ejpam-5596	62	30	,	,	PUNCT
ejpam-5596	62	31	k	k	PROPN
ejpam-5596	62	32	7→	7→	NUM
ejpam-5596	62	33	λi(u	λi(u	NUM
ejpam-5596	62	34	)	)	PUNCT
ejpam-5596	63	1	:	:	PUNCT
ejpam-5596	63	2	=	=	SYM
ejpam-5596	63	3	{	{	PUNCT
ejpam-5596	63	4	1	1	NUM
ejpam-5596	63	5	k	k	X
ejpam-5596	63	6	∈	∈	PROPN
ejpam-5596	63	7	i	i	PRON
ejpam-5596	63	8	,	,	PUNCT
ejpam-5596	63	9	0	0	PUNCT
ejpam-5596	64	1	k	k	NOUN
ejpam-5596	64	2	/∈	/∈	PUNCT
ejpam-5596	65	1	i	i	PRON
ejpam-5596	65	2	,	,	PUNCT
ejpam-5596	65	3	the	the	DET
ejpam-5596	65	4	following	follow	VERB
ejpam-5596	65	5	definitions	definition	NOUN
ejpam-5596	65	6	are	be	AUX
ejpam-5596	65	7	types	type	NOUN
ejpam-5596	65	8	of	of	ADP
ejpam-5596	65	9	fuzzy	fuzzy	ADJ
ejpam-5596	65	10	subsemigroups	subsemigroup	NOUN
ejpam-5596	65	11	on	on	ADP
ejpam-5596	65	12	ordered	order	VERB
ejpam-5596	65	13	semigroups	semigroup	NOUN
ejpam-5596	65	14	.	.	PUNCT
ejpam-5596	66	1	p.	p.	NOUN
ejpam-5596	66	2	khamrot	khamrot	PROPN
ejpam-5596	66	3	,	,	PUNCT
ejpam-5596	66	4	a.	a.	NOUN
ejpam-5596	66	5	iampan	iampan	PROPN
ejpam-5596	66	6	,	,	PUNCT
ejpam-5596	66	7	t.	t.	PROPN
ejpam-5596	66	8	gaketem	gaketem	PROPN
ejpam-5596	66	9	/	/	SYM
ejpam-5596	66	10	eur	eur	PROPN
ejpam-5596	66	11	.	.	PUNCT
ejpam-5596	67	1	j.	j.	PROPN
ejpam-5596	67	2	pure	pure	PROPN
ejpam-5596	67	3	appl	appl	PROPN
ejpam-5596	67	4	.	.	PROPN
ejpam-5596	67	5	math	math	PROPN
ejpam-5596	67	6	,	,	PUNCT
ejpam-5596	67	7	18	18	NUM
ejpam-5596	67	8	(	(	PUNCT
ejpam-5596	67	9	1	1	NUM
ejpam-5596	67	10	)	)	PUNCT
ejpam-5596	67	11	(	(	PUNCT
ejpam-5596	67	12	2025	2025	NUM
ejpam-5596	67	13	)	)	PUNCT
ejpam-5596	67	14	,	,	PUNCT
ejpam-5596	67	15	5596	5596	NUM
ejpam-5596	67	16	4	4	NUM
ejpam-5596	67	17	of	of	ADP
ejpam-5596	67	18	12	12	NUM
ejpam-5596	67	19	definition	definition	NOUN
ejpam-5596	67	20	5	5	NUM
ejpam-5596	67	21	.	.	PUNCT
ejpam-5596	68	1	[	[	X
ejpam-5596	68	2	11	11	NUM
ejpam-5596	68	3	]	]	PUNCT
ejpam-5596	68	4	a	a	DET
ejpam-5596	68	5	fuzzy	fuzzy	ADJ
ejpam-5596	68	6	set	set	VERB
ejpam-5596	68	7	ξ	ξ	PROPN
ejpam-5596	68	8	of	of	ADP
ejpam-5596	68	9	an	an	DET
ejpam-5596	68	10	ordered	order	VERB
ejpam-5596	68	11	semigroups	semigroup	NOUN
ejpam-5596	68	12	s	s	X
ejpam-5596	68	13	is	be	AUX
ejpam-5596	68	14	said	say	VERB
ejpam-5596	68	15	to	to	PART
ejpam-5596	68	16	be	be	AUX
ejpam-5596	68	17	a	a	DET
ejpam-5596	68	18	fuzzy	fuzzy	ADJ
ejpam-5596	68	19	left	left	NOUN
ejpam-5596	68	20	(	(	PUNCT
ejpam-5596	68	21	right	right	ADJ
ejpam-5596	68	22	)	)	PUNCT
ejpam-5596	68	23	ideal	ideal	NOUN
ejpam-5596	68	24	of	of	ADP
ejpam-5596	68	25	s	s	PRON
ejpam-5596	68	26	if	if	SCONJ
ejpam-5596	68	27	u	u	NOUN
ejpam-5596	68	28	≤	≤	NOUN
ejpam-5596	68	29	v	v	NOUN
ejpam-5596	68	30	implies	imply	VERB
ejpam-5596	68	31	δ(u	δ(u	NOUN
ejpam-5596	68	32	)	)	PUNCT
ejpam-5596	68	33	≥	≥	NOUN
ejpam-5596	68	34	δ(y	δ(y	ADV
ejpam-5596	68	35	)	)	PUNCT
ejpam-5596	68	36	for	for	ADP
ejpam-5596	68	37	all	all	DET
ejpam-5596	68	38	u	u	NOUN
ejpam-5596	68	39	,	,	PUNCT
ejpam-5596	68	40	v	v	ADP
ejpam-5596	68	41	∈	∈	PROPN
ejpam-5596	68	42	s	s	NOUN
ejpam-5596	68	43	and	and	CCONJ
ejpam-5596	68	44	δ(uv	δ(uv	NOUN
ejpam-5596	68	45	)	)	PUNCT
ejpam-5596	68	46	≥	≥	NOUN
ejpam-5596	68	47	δ(v	δ(v	PROPN
ejpam-5596	68	48	)	)	PUNCT
ejpam-5596	68	49	(	(	PUNCT
ejpam-5596	68	50	δ(uv	δ(uv	NOUN
ejpam-5596	68	51	)	)	PUNCT
ejpam-5596	68	52	≥	≥	NOUN
ejpam-5596	68	53	δ(u	δ(u	NOUN
ejpam-5596	68	54	)	)	PUNCT
ejpam-5596	68	55	)	)	PUNCT
ejpam-5596	68	56	for	for	ADP
ejpam-5596	68	57	all	all	DET
ejpam-5596	68	58	u	u	NOUN
ejpam-5596	68	59	,	,	PUNCT
ejpam-5596	68	60	v	v	ADP
ejpam-5596	68	61	∈	∈	PROPN
ejpam-5596	68	62	s.	s.	PROPN
ejpam-5596	68	63	lemma	lemma	PROPN
ejpam-5596	69	1	1	1	X
ejpam-5596	69	2	.	.	PUNCT
ejpam-5596	69	3	let	let	VERB
ejpam-5596	69	4	k	k	PRON
ejpam-5596	69	5	be	be	AUX
ejpam-5596	69	6	a	a	DET
ejpam-5596	69	7	nonempty	nonempty	ADJ
ejpam-5596	69	8	subset	subset	NOUN
ejpam-5596	69	9	of	of	ADP
ejpam-5596	69	10	an	an	DET
ejpam-5596	69	11	ordered	order	VERB
ejpam-5596	69	12	semigroup	semigroup	PROPN
ejpam-5596	69	13	s.	s.	PROPN
ejpam-5596	70	1	then	then	ADV
ejpam-5596	70	2	k	k	PROPN
ejpam-5596	70	3	is	be	AUX
ejpam-5596	70	4	a	a	DET
ejpam-5596	70	5	subsemigroup	subsemigroup	NOUN
ejpam-5596	70	6	of	of	ADP
ejpam-5596	70	7	s	s	PRON
ejpam-5596	70	8	if	if	SCONJ
ejpam-5596	70	9	and	and	CCONJ
ejpam-5596	70	10	only	only	ADV
ejpam-5596	70	11	if	if	SCONJ
ejpam-5596	70	12	the	the	DET
ejpam-5596	70	13	characteristic	characteristic	ADJ
ejpam-5596	70	14	function	function	NOUN
ejpam-5596	70	15	λk	λk	ADV
ejpam-5596	70	16	is	be	AUX
ejpam-5596	70	17	a	a	DET
ejpam-5596	70	18	fuzzy	fuzzy	ADJ
ejpam-5596	70	19	subsemigroup	subsemigroup	NOUN
ejpam-5596	70	20	of	of	ADP
ejpam-5596	70	21	s.	s.	PROPN
ejpam-5596	70	22	3	3	NUM
ejpam-5596	70	23	.	.	X
ejpam-5596	70	24	fuzzy	fuzzy	ADJ
ejpam-5596	70	25	(	(	PUNCT
ejpam-5596	70	26	m	m	NOUN
ejpam-5596	70	27	,	,	PUNCT
ejpam-5596	70	28	n)-ideals	n)-ideal	NOUN
ejpam-5596	70	29	in	in	ADP
ejpam-5596	70	30	this	this	DET
ejpam-5596	70	31	section	section	NOUN
ejpam-5596	70	32	,	,	PUNCT
ejpam-5596	70	33	we	we	PRON
ejpam-5596	70	34	outline	outline	VERB
ejpam-5596	70	35	the	the	DET
ejpam-5596	70	36	concept	concept	NOUN
ejpam-5596	70	37	of	of	ADP
ejpam-5596	70	38	fuzzy	fuzzy	ADJ
ejpam-5596	70	39	(	(	PUNCT
ejpam-5596	70	40	m	m	NOUN
ejpam-5596	70	41	,	,	PUNCT
ejpam-5596	70	42	n)-ideals	n)-ideal	NOUN
ejpam-5596	70	43	and	and	CCONJ
ejpam-5596	70	44	explore	explore	VERB
ejpam-5596	70	45	their	their	PRON
ejpam-5596	70	46	properties	property	NOUN
ejpam-5596	70	47	within	within	ADP
ejpam-5596	70	48	ordered	order	VERB
ejpam-5596	70	49	semigroups	semigroup	NOUN
ejpam-5596	70	50	.	.	PUNCT
ejpam-5596	71	1	definition	definition	NOUN
ejpam-5596	71	2	6	6	NUM
ejpam-5596	71	3	.	.	PUNCT
ejpam-5596	72	1	a	a	DET
ejpam-5596	72	2	fuzzy	fuzzy	ADJ
ejpam-5596	72	3	subsemigroup	subsemigroup	PROPN
ejpam-5596	72	4	δ	δ	PROPN
ejpam-5596	72	5	of	of	ADP
ejpam-5596	72	6	an	an	DET
ejpam-5596	72	7	ordered	order	VERB
ejpam-5596	72	8	semigroup	semigroup	NOUN
ejpam-5596	72	9	s	s	VERB
ejpam-5596	72	10	is	be	AUX
ejpam-5596	72	11	called	call	VERB
ejpam-5596	72	12	a	a	DET
ejpam-5596	72	13	fuzzy	fuzzy	ADJ
ejpam-5596	72	14	(	(	PUNCT
ejpam-5596	72	15	m	m	NOUN
ejpam-5596	72	16	,	,	PUNCT
ejpam-5596	72	17	n)ideal	n)ideal	NOUN
ejpam-5596	72	18	of	of	ADP
ejpam-5596	72	19	s	s	PRON
ejpam-5596	72	20	if	if	SCONJ
ejpam-5596	72	21	(	(	PUNCT
ejpam-5596	72	22	1	1	X
ejpam-5596	72	23	)	)	PUNCT
ejpam-5596	72	24	δ(u1u2	δ(u1u2	X
ejpam-5596	72	25	·	·	PUNCT
ejpam-5596	72	26	·	·	PUNCT
ejpam-5596	73	1	·	·	PUNCT
ejpam-5596	73	2	umkv1v2	umkv1v2	X
ejpam-5596	73	3	·	·	PUNCT
ejpam-5596	73	4	·	·	PUNCT
ejpam-5596	73	5	·	·	PUNCT
ejpam-5596	73	6	vn	vn	X
ejpam-5596	73	7	)	)	PUNCT
ejpam-5596	73	8	≥	≥	PROPN
ejpam-5596	73	9	δ(u1)∧	δ(u1)∧	NOUN
ejpam-5596	73	10	δ(u2)∧	δ(u2)∧	NOUN
ejpam-5596	73	11	·	·	PUNCT
ejpam-5596	73	12	·	·	PUNCT
ejpam-5596	73	13	·	·	PUNCT
ejpam-5596	74	1	∧	∧	NOUN
ejpam-5596	74	2	δ(um)∧	δ(um)∧	PROPN
ejpam-5596	74	3	δ(v1)∧	δ(v1)∧	VERB
ejpam-5596	74	4	δ(v2)∧	δ(v2)∧	X
ejpam-5596	74	5	·	·	PUNCT
ejpam-5596	74	6	·	·	PUNCT
ejpam-5596	74	7	·	·	PUNCT
ejpam-5596	74	8	∧	∧	NOUN
ejpam-5596	74	9	δ(vn	δ(vn	PROPN
ejpam-5596	74	10	)	)	PUNCT
ejpam-5596	74	11	for	for	ADP
ejpam-5596	74	12	all	all	DET
ejpam-5596	74	13	u1	u1	NOUN
ejpam-5596	74	14	,	,	PUNCT
ejpam-5596	74	15	u2	u2	NOUN
ejpam-5596	74	16	,	,	PUNCT
ejpam-5596	74	17	.	.	PUNCT
ejpam-5596	74	18	.	.	PUNCT
ejpam-5596	74	19	.	.	PUNCT
ejpam-5596	75	1	,	,	PUNCT
ejpam-5596	75	2	um	um	INTJ
ejpam-5596	75	3	,	,	PUNCT
ejpam-5596	75	4	k	k	PROPN
ejpam-5596	75	5	,	,	PUNCT
ejpam-5596	75	6	v1	v1	NOUN
ejpam-5596	75	7	,	,	PUNCT
ejpam-5596	75	8	v2	v2	NOUN
ejpam-5596	75	9	,	,	PUNCT
ejpam-5596	75	10	.	.	PUNCT
ejpam-5596	75	11	.	.	PUNCT
ejpam-5596	76	1	.	.	PUNCT
ejpam-5596	77	1	vn	vn	INTJ
ejpam-5596	77	2	of	of	ADP
ejpam-5596	77	3	s	s	PROPN
ejpam-5596	77	4	and	and	CCONJ
ejpam-5596	77	5	m	m	PROPN
ejpam-5596	77	6	,	,	PUNCT
ejpam-5596	77	7	n	n	PROPN
ejpam-5596	77	8	∈	∈	PROPN
ejpam-5596	77	9	n.	n.	NOUN
ejpam-5596	77	10	(	(	PUNCT
ejpam-5596	77	11	2	2	NUM
ejpam-5596	77	12	)	)	PUNCT
ejpam-5596	77	13	u	u	NOUN
ejpam-5596	77	14	≤	≤	NOUN
ejpam-5596	77	15	v	v	NOUN
ejpam-5596	77	16	implies	imply	VERB
ejpam-5596	77	17	δ(u	δ(u	NOUN
ejpam-5596	77	18	)	)	PUNCT
ejpam-5596	77	19	≥	≥	NOUN
ejpam-5596	77	20	δ(v	δ(v	PROPN
ejpam-5596	77	21	)	)	PUNCT
ejpam-5596	77	22	for	for	ADP
ejpam-5596	77	23	all	all	DET
ejpam-5596	77	24	u	u	NOUN
ejpam-5596	77	25	,	,	PUNCT
ejpam-5596	77	26	v	v	PROPN
ejpam-5596	77	27	∈	∈	PROPN
ejpam-5596	77	28	s.	s.	PROPN
ejpam-5596	77	29	example	example	NOUN
ejpam-5596	78	1	1	1	X
ejpam-5596	78	2	.	.	X
ejpam-5596	78	3	consider	consider	VERB
ejpam-5596	78	4	the	the	DET
ejpam-5596	78	5	ordered	order	VERB
ejpam-5596	78	6	semigroup	semigroup	NOUN
ejpam-5596	78	7	s	s	PART
ejpam-5596	78	8	=	=	PUNCT
ejpam-5596	78	9	{	{	PUNCT
ejpam-5596	78	10	w	w	PROPN
ejpam-5596	78	11	,	,	PUNCT
ejpam-5596	78	12	x	x	NOUN
ejpam-5596	78	13	,	,	PUNCT
ejpam-5596	78	14	y	y	PROPN
ejpam-5596	78	15	,	,	PUNCT
ejpam-5596	78	16	z	z	NOUN
ejpam-5596	78	17	}	}	PUNCT
ejpam-5596	78	18	with	with	ADP
ejpam-5596	78	19	the	the	DET
ejpam-5596	78	20	following	follow	VERB
ejpam-5596	78	21	cayley	cayley	ADJ
ejpam-5596	78	22	table	table	NOUN
ejpam-5596	78	23	:	:	PUNCT
ejpam-5596	78	24	·	·	PUNCT
ejpam-5596	78	25	w	w	X
ejpam-5596	78	26	x	x	PUNCT
ejpam-5596	78	27	y	y	PROPN
ejpam-5596	78	28	z	z	PROPN
ejpam-5596	78	29	w	w	PROPN
ejpam-5596	78	30	w	w	PROPN
ejpam-5596	78	31	w	w	PROPN
ejpam-5596	78	32	w	w	PROPN
ejpam-5596	78	33	w	w	PROPN
ejpam-5596	78	34	x	x	PROPN
ejpam-5596	78	35	w	w	PROPN
ejpam-5596	78	36	w	w	PROPN
ejpam-5596	78	37	z	z	PROPN
ejpam-5596	78	38	w	w	PROPN
ejpam-5596	78	39	y	y	PROPN
ejpam-5596	78	40	w	w	PROPN
ejpam-5596	78	41	w	w	PROPN
ejpam-5596	78	42	w	w	PROPN
ejpam-5596	78	43	w	w	PROPN
ejpam-5596	78	44	z	z	PROPN
ejpam-5596	78	45	w	w	PROPN
ejpam-5596	78	46	w	w	PROPN
ejpam-5596	78	47	w	w	PROPN
ejpam-5596	78	48	w	w	PROPN
ejpam-5596	78	49	and	and	CCONJ
ejpam-5596	78	50	≤	≤	NUM
ejpam-5596	78	51	:	:	PUNCT
ejpam-5596	78	52	{	{	PUNCT
ejpam-5596	78	53	(	(	PUNCT
ejpam-5596	78	54	w	w	PROPN
ejpam-5596	78	55	,	,	PUNCT
ejpam-5596	78	56	w	w	NOUN
ejpam-5596	78	57	)	)	PUNCT
ejpam-5596	78	58	,	,	PUNCT
ejpam-5596	78	59	(	(	PUNCT
ejpam-5596	78	60	x	x	X
ejpam-5596	78	61	,	,	PUNCT
ejpam-5596	78	62	x	x	NOUN
ejpam-5596	78	63	)	)	PUNCT
ejpam-5596	78	64	,	,	PUNCT
ejpam-5596	78	65	(	(	PUNCT
ejpam-5596	78	66	y	y	PROPN
ejpam-5596	78	67	,	,	PUNCT
ejpam-5596	78	68	y	y	PROPN
ejpam-5596	78	69	)	)	PUNCT
ejpam-5596	78	70	,	,	PUNCT
ejpam-5596	78	71	(	(	PUNCT
ejpam-5596	78	72	z	z	X
ejpam-5596	78	73	,	,	PUNCT
ejpam-5596	78	74	z	z	NOUN
ejpam-5596	78	75	)	)	PUNCT
ejpam-5596	78	76	}	}	PUNCT
ejpam-5596	78	77	.	.	PUNCT
ejpam-5596	79	1	define	define	VERB
ejpam-5596	79	2	a	a	DET
ejpam-5596	79	3	function	function	NOUN
ejpam-5596	79	4	δ	δ	NOUN
ejpam-5596	79	5	:	:	PUNCT
ejpam-5596	79	6	s	s	X
ejpam-5596	79	7	→	→	SYM
ejpam-5596	79	8	[	[	X
ejpam-5596	79	9	0	0	NUM
ejpam-5596	79	10	,	,	PUNCT
ejpam-5596	79	11	1	1	NUM
ejpam-5596	79	12	]	]	PUNCT
ejpam-5596	79	13	by	by	ADP
ejpam-5596	79	14	δ(w	δ(w	PROPN
ejpam-5596	79	15	)	)	PUNCT
ejpam-5596	79	16	=	=	PUNCT
ejpam-5596	79	17	0.4	0.4	NUM
ejpam-5596	79	18	,	,	PUNCT
ejpam-5596	79	19	δ(x	δ(x	ADJ
ejpam-5596	79	20	)	)	PUNCT
ejpam-5596	79	21	=	=	PUNCT
ejpam-5596	79	22	0.4	0.4	NUM
ejpam-5596	79	23	,	,	PUNCT
ejpam-5596	79	24	δ(y	δ(y	ADV
ejpam-5596	79	25	)	)	PUNCT
ejpam-5596	79	26	=	=	SYM
ejpam-5596	79	27	0	0	NUM
ejpam-5596	79	28	,	,	PUNCT
ejpam-5596	79	29	δ(z	δ(z	NOUN
ejpam-5596	79	30	)	)	PUNCT
ejpam-5596	79	31	=	=	SYM
ejpam-5596	80	1	0	0	X
ejpam-5596	80	2	.	.	PUNCT
ejpam-5596	81	1	then	then	ADV
ejpam-5596	81	2	δ	δ	PROPN
ejpam-5596	81	3	is	be	AUX
ejpam-5596	81	4	a	a	DET
ejpam-5596	81	5	fuzzy	fuzzy	ADJ
ejpam-5596	81	6	(	(	PUNCT
ejpam-5596	81	7	m	m	PROPN
ejpam-5596	81	8	,	,	PUNCT
ejpam-5596	81	9	n)-ideal	n)-ideal	NOUN
ejpam-5596	81	10	of	of	ADP
ejpam-5596	81	11	s.	s.	PROPN
ejpam-5596	81	12	theorem	theorem	VERB
ejpam-5596	81	13	1	1	X
ejpam-5596	81	14	.	.	PUNCT
ejpam-5596	82	1	let	let	VERB
ejpam-5596	82	2	{	{	PUNCT
ejpam-5596	82	3	δi	δi	VERB
ejpam-5596	83	1	|	|	ADV
ejpam-5596	83	2	i	i	PRON
ejpam-5596	83	3	∈	∈	PROPN
ejpam-5596	83	4	j	j	PROPN
ejpam-5596	83	5	}	}	PUNCT
ejpam-5596	83	6	be	be	AUX
ejpam-5596	83	7	a	a	DET
ejpam-5596	83	8	family	family	NOUN
ejpam-5596	83	9	of	of	ADP
ejpam-5596	83	10	fuzzy	fuzzy	ADJ
ejpam-5596	83	11	(	(	PUNCT
ejpam-5596	83	12	m	m	PROPN
ejpam-5596	83	13	,	,	PUNCT
ejpam-5596	83	14	n)-ideals	n)-ideal	NOUN
ejpam-5596	83	15	of	of	ADP
ejpam-5596	83	16	an	an	DET
ejpam-5596	83	17	ordered	order	VERB
ejpam-5596	83	18	semigroup	semigroup	NOUN
ejpam-5596	83	19	s	s	PROPN
ejpam-5596	83	20	with	with	ADP
ejpam-5596	83	21	δ(u	δ(u	PROPN
ejpam-5596	83	22	)	)	PUNCT
ejpam-5596	83	23	≥	≥	NOUN
ejpam-5596	83	24	δ(v	δ(v	PROPN
ejpam-5596	83	25	)	)	PUNCT
ejpam-5596	83	26	whenever	whenever	SCONJ
ejpam-5596	83	27	u	u	NOUN
ejpam-5596	83	28	≤	≤	X
ejpam-5596	84	1	v.	v.	CCONJ
ejpam-5596	84	2	then	then	ADV
ejpam-5596	84	3	∧	∧	PROPN
ejpam-5596	84	4	i∈f	i∈f	VERB
ejpam-5596	84	5	ϑi	ϑi	PRON
ejpam-5596	84	6	is	be	AUX
ejpam-5596	84	7	a	a	DET
ejpam-5596	84	8	fuzzy	fuzzy	ADJ
ejpam-5596	84	9	(	(	PUNCT
ejpam-5596	84	10	m	m	PROPN
ejpam-5596	84	11	,	,	PUNCT
ejpam-5596	84	12	n)-ideal	n)-ideal	NOUN
ejpam-5596	84	13	of	of	ADP
ejpam-5596	84	14	s.	s.	PROPN
ejpam-5596	84	15	proof	proof	PROPN
ejpam-5596	84	16	.	.	PUNCT
ejpam-5596	85	1	let	let	VERB
ejpam-5596	85	2	u	u	NOUN
ejpam-5596	85	3	,	,	PUNCT
ejpam-5596	85	4	v	v	ADP
ejpam-5596	85	5	∈	∈	NOUN
ejpam-5596	85	6	s.	s.	PROPN
ejpam-5596	85	7	then,∧	then,∧	PROPN
ejpam-5596	85	8	i∈j	i∈j	NOUN
ejpam-5596	85	9	δi(uv	δi(uv	NOUN
ejpam-5596	85	10	)	)	PUNCT
ejpam-5596	85	11	≥	≥	NOUN
ejpam-5596	85	12	∧	∧	PROPN
ejpam-5596	85	13	i∈j	i∈j	NOUN
ejpam-5596	85	14	{	{	PUNCT
ejpam-5596	85	15	δi(u	δi(u	NUM
ejpam-5596	85	16	)	)	PUNCT
ejpam-5596	85	17	∧	∧	PROPN
ejpam-5596	85	18	δ(v	δ(v	PROPN
ejpam-5596	85	19	)	)	PUNCT
ejpam-5596	85	20	}	}	PUNCT
ejpam-5596	85	21	=	=	SYM
ejpam-5596	85	22	∧	∧	NOUN
ejpam-5596	85	23	i∈j	i∈j	NOUN
ejpam-5596	85	24	δi(u	δi(u	NOUN
ejpam-5596	85	25	)	)	PUNCT
ejpam-5596	86	1	∧	∧	NOUN
ejpam-5596	86	2	∧	∧	PROPN
ejpam-5596	86	3	i∈j	i∈j	NOUN
ejpam-5596	86	4	δi(v	δi(v	NUM
ejpam-5596	86	5	)	)	PUNCT
ejpam-5596	86	6	.	.	PUNCT
ejpam-5596	87	1	thus	thus	ADV
ejpam-5596	87	2	,	,	PUNCT
ejpam-5596	87	3	∧	∧	NOUN
ejpam-5596	87	4	i∈j	i∈j	NOUN
ejpam-5596	87	5	δi	δi	NOUN
ejpam-5596	87	6	is	be	AUX
ejpam-5596	87	7	a	a	DET
ejpam-5596	87	8	fuzzy	fuzzy	ADJ
ejpam-5596	87	9	subsemigroup	subsemigroup	NOUN
ejpam-5596	87	10	of	of	ADP
ejpam-5596	87	11	s.	s.	PROPN
ejpam-5596	87	12	let	let	VERB
ejpam-5596	87	13	u1	u1	NOUN
ejpam-5596	87	14	,	,	PUNCT
ejpam-5596	87	15	u2	u2	NOUN
ejpam-5596	87	16	,	,	PUNCT
ejpam-5596	87	17	.	.	PUNCT
ejpam-5596	87	18	.	.	PUNCT
ejpam-5596	88	1	.	.	PUNCT
ejpam-5596	89	1	,	,	PUNCT
ejpam-5596	89	2	um	um	INTJ
ejpam-5596	89	3	,	,	PUNCT
ejpam-5596	89	4	k	k	PROPN
ejpam-5596	89	5	,	,	PUNCT
ejpam-5596	89	6	v1	v1	NOUN
ejpam-5596	89	7	,	,	PUNCT
ejpam-5596	89	8	v2	v2	NOUN
ejpam-5596	89	9	,	,	PUNCT
ejpam-5596	89	10	.	.	PUNCT
ejpam-5596	89	11	.	.	PUNCT
ejpam-5596	90	1	.	.	PUNCT
ejpam-5596	91	1	vn	vn	PROPN
ejpam-5596	91	2	∈	∈	PROPN
ejpam-5596	91	3	s.	s.	PROPN
ejpam-5596	91	4	then,∧	then,∧	PROPN
ejpam-5596	91	5	i∈j	i∈j	NOUN
ejpam-5596	91	6	δi(u1u2	δi(u1u2	PROPN
ejpam-5596	91	7	·	·	PUNCT
ejpam-5596	91	8	·	·	PUNCT
ejpam-5596	91	9	·	·	PUNCT
ejpam-5596	91	10	umkv1v2	umkv1v2	X
ejpam-5596	91	11	·	·	PUNCT
ejpam-5596	91	12	·	·	PUNCT
ejpam-5596	91	13	·	·	PUNCT
ejpam-5596	91	14	vn	vn	X
ejpam-5596	91	15	)	)	PUNCT
ejpam-5596	91	16	p.	p.	NOUN
ejpam-5596	91	17	khamrot	khamrot	PROPN
ejpam-5596	91	18	,	,	PUNCT
ejpam-5596	91	19	a.	a.	NOUN
ejpam-5596	91	20	iampan	iampan	PROPN
ejpam-5596	91	21	,	,	PUNCT
ejpam-5596	91	22	t.	t.	PROPN
ejpam-5596	91	23	gaketem	gaketem	PROPN
ejpam-5596	91	24	/	/	SYM
ejpam-5596	91	25	eur	eur	PROPN
ejpam-5596	91	26	.	.	PUNCT
ejpam-5596	92	1	j.	j.	PROPN
ejpam-5596	92	2	pure	pure	PROPN
ejpam-5596	92	3	appl	appl	PROPN
ejpam-5596	92	4	.	.	PROPN
ejpam-5596	92	5	math	math	PROPN
ejpam-5596	92	6	,	,	PUNCT
ejpam-5596	92	7	18	18	NUM
ejpam-5596	92	8	(	(	PUNCT
ejpam-5596	92	9	1	1	NUM
ejpam-5596	92	10	)	)	PUNCT
ejpam-5596	92	11	(	(	PUNCT
ejpam-5596	92	12	2025	2025	NUM
ejpam-5596	92	13	)	)	PUNCT
ejpam-5596	92	14	,	,	PUNCT
ejpam-5596	92	15	5596	5596	NUM
ejpam-5596	92	16	5	5	NUM
ejpam-5596	92	17	of	of	ADP
ejpam-5596	92	18	12	12	NUM
ejpam-5596	92	19	≥	≥	NOUN
ejpam-5596	92	20	∧	∧	PROPN
ejpam-5596	92	21	i∈j	i∈j	NOUN
ejpam-5596	92	22	{	{	PUNCT
ejpam-5596	92	23	δi(u1	δi(u1	NOUN
ejpam-5596	92	24	)	)	PUNCT
ejpam-5596	92	25	∧	∧	PROPN
ejpam-5596	92	26	δi(u2	δi(u2	NOUN
ejpam-5596	92	27	)	)	PUNCT
ejpam-5596	92	28	·	·	PUNCT
ejpam-5596	92	29	·	·	PUNCT
ejpam-5596	92	30	·	·	PUNCT
ejpam-5596	93	1	∧	∧	PROPN
ejpam-5596	93	2	δi(un	δi(un	PROPN
ejpam-5596	93	3	)	)	PUNCT
ejpam-5596	93	4	∧	∧	NOUN
ejpam-5596	93	5	δi(v1	δi(v1	NOUN
ejpam-5596	93	6	)	)	PUNCT
ejpam-5596	93	7	∧	∧	PROPN
ejpam-5596	93	8	δi(v2	δi(v2	NOUN
ejpam-5596	93	9	)	)	PUNCT
ejpam-5596	93	10	.	.	PUNCT
ejpam-5596	93	11	.	.	PUNCT
ejpam-5596	93	12	.	.	PUNCT
ejpam-5596	94	1	δi(vn	δi(vn	PROPN
ejpam-5596	94	2	)	)	PUNCT
ejpam-5596	94	3	}	}	PUNCT
ejpam-5596	95	1	=	=	PUNCT
ejpam-5596	95	2	∧	∧	PROPN
ejpam-5596	95	3	i∈j	i∈j	NOUN
ejpam-5596	95	4	δi(u1	δi(u1	NOUN
ejpam-5596	95	5	)	)	PUNCT
ejpam-5596	95	6	∧	∧	NOUN
ejpam-5596	95	7	∧	∧	PROPN
ejpam-5596	95	8	i∈j	i∈j	NOUN
ejpam-5596	95	9	δi(u2	δi(u2	PROPN
ejpam-5596	95	10	)	)	PUNCT
ejpam-5596	95	11	·	·	PUNCT
ejpam-5596	95	12	·	·	PUNCT
ejpam-5596	95	13	·	·	PUNCT
ejpam-5596	96	1	∧	∧	NOUN
ejpam-5596	96	2	∧	∧	PROPN
ejpam-5596	96	3	i∈j	i∈j	NOUN
ejpam-5596	96	4	δi(un	δi(un	NOUN
ejpam-5596	96	5	)	)	PUNCT
ejpam-5596	96	6	∧	∧	NOUN
ejpam-5596	96	7	∧	∧	PROPN
ejpam-5596	96	8	i∈j	i∈j	NOUN
ejpam-5596	96	9	δi(v1	δi(v1	NOUN
ejpam-5596	96	10	)	)	PUNCT
ejpam-5596	96	11	∧	∧	PROPN
ejpam-5596	96	12	∧	∧	PROPN
ejpam-5596	96	13	i∈j	i∈j	NOUN
ejpam-5596	96	14	δi(v2	δi(v2	NOUN
ejpam-5596	96	15	)	)	PUNCT
ejpam-5596	96	16	.	.	PUNCT
ejpam-5596	96	17	.	.	PUNCT
ejpam-5596	96	18	.	.	PUNCT
ejpam-5596	97	1	∧	∧	NOUN
ejpam-5596	97	2	i∈j	i∈j	NOUN
ejpam-5596	97	3	δi(vn	δi(vn	PROPN
ejpam-5596	97	4	)	)	PUNCT
ejpam-5596	97	5	.	.	PUNCT
ejpam-5596	98	1	thus	thus	ADV
ejpam-5596	98	2	,	,	PUNCT
ejpam-5596	98	3	∧	∧	NOUN
ejpam-5596	98	4	i∈j	i∈j	NOUN
ejpam-5596	98	5	δi	δi	NOUN
ejpam-5596	98	6	is	be	AUX
ejpam-5596	98	7	a	a	DET
ejpam-5596	98	8	fuzzy	fuzzy	ADJ
ejpam-5596	98	9	(	(	PUNCT
ejpam-5596	98	10	m	m	PROPN
ejpam-5596	98	11	,	,	PUNCT
ejpam-5596	98	12	n)-ideal	n)-ideal	NOUN
ejpam-5596	98	13	of	of	ADP
ejpam-5596	98	14	s.	s.	PROPN
ejpam-5596	98	15	theorem	theorem	VERB
ejpam-5596	98	16	2	2	X
ejpam-5596	98	17	.	.	PUNCT
ejpam-5596	99	1	let	let	VERB
ejpam-5596	99	2	k	k	PRON
ejpam-5596	99	3	be	be	AUX
ejpam-5596	99	4	a	a	DET
ejpam-5596	99	5	nonempty	nonempty	ADJ
ejpam-5596	99	6	subset	subset	NOUN
ejpam-5596	99	7	of	of	ADP
ejpam-5596	99	8	an	an	DET
ejpam-5596	99	9	ordered	order	VERB
ejpam-5596	99	10	semigroup	semigroup	PROPN
ejpam-5596	99	11	s	s	PROPN
ejpam-5596	99	12	and	and	CCONJ
ejpam-5596	99	13	m	m	PROPN
ejpam-5596	99	14	,	,	PUNCT
ejpam-5596	99	15	n	n	PRON
ejpam-5596	99	16	are	be	AUX
ejpam-5596	99	17	positive	positive	ADJ
ejpam-5596	99	18	integers	integer	NOUN
ejpam-5596	99	19	.	.	PUNCT
ejpam-5596	100	1	then	then	ADV
ejpam-5596	100	2	k	k	PROPN
ejpam-5596	100	3	is	be	AUX
ejpam-5596	100	4	an	an	DET
ejpam-5596	100	5	(	(	PUNCT
ejpam-5596	100	6	m	m	PROPN
ejpam-5596	100	7	,	,	PUNCT
ejpam-5596	100	8	n)-ideal	n)-ideal	NOUN
ejpam-5596	100	9	of	of	ADP
ejpam-5596	100	10	s	s	PRON
ejpam-5596	100	11	if	if	SCONJ
ejpam-5596	101	1	and	and	CCONJ
ejpam-5596	101	2	only	only	ADV
ejpam-5596	101	3	if	if	SCONJ
ejpam-5596	101	4	the	the	DET
ejpam-5596	101	5	characteristic	characteristic	ADJ
ejpam-5596	101	6	function	function	NOUN
ejpam-5596	101	7	λk	λk	ADV
ejpam-5596	101	8	is	be	AUX
ejpam-5596	101	9	a	a	DET
ejpam-5596	101	10	fuzzy	fuzzy	ADJ
ejpam-5596	101	11	(	(	PUNCT
ejpam-5596	101	12	m	m	PROPN
ejpam-5596	101	13	,	,	PUNCT
ejpam-5596	101	14	n)-ideal	n)-ideal	NOUN
ejpam-5596	101	15	of	of	ADP
ejpam-5596	101	16	s.	s.	PROPN
ejpam-5596	101	17	proof	proof	PROPN
ejpam-5596	101	18	.	.	PUNCT
ejpam-5596	102	1	suppose	suppose	VERB
ejpam-5596	102	2	that	that	SCONJ
ejpam-5596	102	3	k	k	PROPN
ejpam-5596	102	4	is	be	AUX
ejpam-5596	102	5	an	an	DET
ejpam-5596	102	6	(	(	PUNCT
ejpam-5596	102	7	m	m	PROPN
ejpam-5596	102	8	,	,	PUNCT
ejpam-5596	102	9	n)-ideal	n)-ideal	NOUN
ejpam-5596	102	10	of	of	ADP
ejpam-5596	102	11	s.	s.	PROPN
ejpam-5596	102	12	then	then	ADV
ejpam-5596	102	13	,	,	PUNCT
ejpam-5596	102	14	k	k	PROPN
ejpam-5596	102	15	is	be	AUX
ejpam-5596	102	16	a	a	DET
ejpam-5596	102	17	subsemigroup	subsemigroup	NOUN
ejpam-5596	102	18	of	of	ADP
ejpam-5596	102	19	s.	s.	PROPN
ejpam-5596	102	20	by	by	ADP
ejpam-5596	102	21	lemma	lemma	PROPN
ejpam-5596	102	22	1	1	NUM
ejpam-5596	102	23	,	,	PUNCT
ejpam-5596	102	24	λk	λk	PRON
ejpam-5596	102	25	is	be	AUX
ejpam-5596	102	26	a	a	DET
ejpam-5596	102	27	fuzzy	fuzzy	ADJ
ejpam-5596	102	28	subsemigroup	subsemigroup	NOUN
ejpam-5596	102	29	of	of	ADP
ejpam-5596	102	30	s.	s.	PROPN
ejpam-5596	102	31	let	let	VERB
ejpam-5596	102	32	u1	u1	NOUN
ejpam-5596	102	33	,	,	PUNCT
ejpam-5596	102	34	u2	u2	NOUN
ejpam-5596	102	35	,	,	PUNCT
ejpam-5596	102	36	.	.	PUNCT
ejpam-5596	102	37	.	.	PUNCT
ejpam-5596	102	38	.	.	PUNCT
ejpam-5596	103	1	um	um	INTJ
ejpam-5596	103	2	,	,	PUNCT
ejpam-5596	103	3	k	k	PROPN
ejpam-5596	103	4	,	,	PUNCT
ejpam-5596	103	5	v1	v1	NOUN
ejpam-5596	103	6	,	,	PUNCT
ejpam-5596	103	7	v2	v2	NOUN
ejpam-5596	103	8	,	,	PUNCT
ejpam-5596	103	9	.	.	PUNCT
ejpam-5596	103	10	.	.	PUNCT
ejpam-5596	104	1	.	.	PUNCT
ejpam-5596	105	1	,	,	PUNCT
ejpam-5596	105	2	vn	vn	PROPN
ejpam-5596	105	3	∈	∈	PROPN
ejpam-5596	105	4	s.	s.	PROPN
ejpam-5596	105	5	then	then	ADV
ejpam-5596	105	6	the	the	DET
ejpam-5596	105	7	following	follow	VERB
ejpam-5596	105	8	cases	case	NOUN
ejpam-5596	105	9	:	:	PUNCT
ejpam-5596	105	10	case	case	NOUN
ejpam-5596	105	11	1	1	NUM
ejpam-5596	105	12	if	if	SCONJ
ejpam-5596	105	13	ui	ui	PROPN
ejpam-5596	105	14	,	,	PUNCT
ejpam-5596	105	15	vj	vj	PROPN
ejpam-5596	105	16	∈	∈	PROPN
ejpam-5596	105	17	k	k	PROPN
ejpam-5596	105	18	for	for	ADP
ejpam-5596	105	19	all	all	PRON
ejpam-5596	105	20	i	i	PRON
ejpam-5596	105	21	∈	∈	PROPN
ejpam-5596	105	22	{	{	PUNCT
ejpam-5596	105	23	1	1	NUM
ejpam-5596	105	24	,	,	PUNCT
ejpam-5596	105	25	2	2	NUM
ejpam-5596	105	26	,	,	PUNCT
ejpam-5596	105	27	.	.	PUNCT
ejpam-5596	105	28	.	.	PUNCT
ejpam-5596	105	29	.	.	PUNCT
ejpam-5596	106	1	,	,	PUNCT
ejpam-5596	106	2	m	m	VERB
ejpam-5596	106	3	}	}	PUNCT
ejpam-5596	106	4	and	and	CCONJ
ejpam-5596	106	5	j	j	PROPN
ejpam-5596	106	6	∈	∈	PROPN
ejpam-5596	106	7	{	{	PUNCT
ejpam-5596	106	8	1	1	NUM
ejpam-5596	106	9	,	,	PUNCT
ejpam-5596	106	10	2	2	NUM
ejpam-5596	106	11	,	,	PUNCT
ejpam-5596	106	12	.	.	PUNCT
ejpam-5596	106	13	.	.	PUNCT
ejpam-5596	106	14	.	.	PUNCT
ejpam-5596	107	1	,	,	PUNCT
ejpam-5596	107	2	n	n	CCONJ
ejpam-5596	107	3	}	}	PUNCT
ejpam-5596	107	4	,	,	PUNCT
ejpam-5596	107	5	then	then	ADV
ejpam-5596	107	6	u1u2	u1u2	X
ejpam-5596	107	7	·	·	PUNCT
ejpam-5596	107	8	·	·	PUNCT
ejpam-5596	107	9	·	·	PUNCT
ejpam-5596	107	10	umkv1v2	umkv1v2	X
ejpam-5596	107	11	·	·	PUNCT
ejpam-5596	107	12	·	·	PUNCT
ejpam-5596	107	13	·	·	PUNCT
ejpam-5596	107	14	vn	vn	PROPN
ejpam-5596	107	15	∈	∈	PROPN
ejpam-5596	107	16	kmskn	kmskn	PROPN
ejpam-5596	107	17	.	.	PUNCT
ejpam-5596	108	1	thus	thus	ADV
ejpam-5596	108	2	,	,	PUNCT
ejpam-5596	108	3	λk(u1u2	λk(u1u2	X
ejpam-5596	108	4	·	·	PUNCT
ejpam-5596	108	5	·	·	PUNCT
ejpam-5596	108	6	·	·	PUNCT
ejpam-5596	108	7	umkv1v2	umkv1v2	X
ejpam-5596	108	8	·	·	PUNCT
ejpam-5596	108	9	·	·	PUNCT
ejpam-5596	108	10	·	·	PUNCT
ejpam-5596	108	11	vn	vn	X
ejpam-5596	108	12	)	)	PUNCT
ejpam-5596	108	13	=	=	SYM
ejpam-5596	108	14	1	1	NUM
ejpam-5596	108	15	,	,	PUNCT
ejpam-5596	108	16	λk(ui	λk(ui	X
ejpam-5596	108	17	)	)	PUNCT
ejpam-5596	108	18	=	=	SYM
ejpam-5596	108	19	1	1	NUM
ejpam-5596	108	20	for	for	ADP
ejpam-5596	108	21	all	all	PRON
ejpam-5596	108	22	i	i	PRON
ejpam-5596	108	23	∈	∈	PROPN
ejpam-5596	108	24	{	{	PUNCT
ejpam-5596	108	25	1	1	NUM
ejpam-5596	108	26	,	,	PUNCT
ejpam-5596	108	27	2	2	NUM
ejpam-5596	108	28	,	,	PUNCT
ejpam-5596	108	29	.	.	PUNCT
ejpam-5596	108	30	.	.	PUNCT
ejpam-5596	108	31	.	.	PUNCT
ejpam-5596	109	1	,	,	PUNCT
ejpam-5596	109	2	m	m	VERB
ejpam-5596	109	3	}	}	PUNCT
ejpam-5596	109	4	and	and	CCONJ
ejpam-5596	109	5	λk(rj	λk(rj	NOUN
ejpam-5596	109	6	)	)	PUNCT
ejpam-5596	109	7	=	=	SYM
ejpam-5596	109	8	1	1	NUM
ejpam-5596	109	9	for	for	ADP
ejpam-5596	109	10	all	all	DET
ejpam-5596	109	11	j	j	PROPN
ejpam-5596	109	12	∈	∈	PROPN
ejpam-5596	109	13	{	{	PUNCT
ejpam-5596	109	14	1	1	NUM
ejpam-5596	109	15	,	,	PUNCT
ejpam-5596	109	16	2	2	NUM
ejpam-5596	109	17	,	,	PUNCT
ejpam-5596	109	18	.	.	PUNCT
ejpam-5596	109	19	.	.	PUNCT
ejpam-5596	110	1	.	.	PUNCT
ejpam-5596	110	2	,	,	PUNCT
ejpam-5596	110	3	n	n	CCONJ
ejpam-5596	110	4	}	}	PUNCT
ejpam-5596	110	5	.	.	PUNCT
ejpam-5596	111	1	so	so	ADV
ejpam-5596	111	2	,	,	PUNCT
ejpam-5596	111	3	we	we	PRON
ejpam-5596	111	4	have	have	VERB
ejpam-5596	111	5	λk(u1u2	λk(u1u2	PROPN
ejpam-5596	111	6	·	·	PUNCT
ejpam-5596	111	7	·	·	PUNCT
ejpam-5596	112	1	·	·	PUNCT
ejpam-5596	112	2	umkv1v2	umkv1v2	X
ejpam-5596	112	3	·	·	PUNCT
ejpam-5596	112	4	·	·	PUNCT
ejpam-5596	112	5	·	·	PUNCT
ejpam-5596	112	6	vn	vn	X
ejpam-5596	112	7	)	)	PUNCT
ejpam-5596	112	8	≥	≥	NOUN
ejpam-5596	112	9	λk(u1)∧λk(u2)∧	λk(u1)∧λk(u2)∧	NOUN
ejpam-5596	112	10	·	·	PUNCT
ejpam-5596	112	11	·	·	PUNCT
ejpam-5596	112	12	·	·	PUNCT
ejpam-5596	112	13	∧λk(um)∧	∧λk(um)∧	X
ejpam-5596	112	14	·	·	SYM
ejpam-5596	112	15	·	·	PUNCT
ejpam-5596	112	16	·	·	PUNCT
ejpam-5596	112	17	∧λk(v1)∧λk(r2)∧	∧λk(v1)∧λk(r2)∧	X
ejpam-5596	112	18	·	·	SYM
ejpam-5596	112	19	·	·	PUNCT
ejpam-5596	112	20	·	·	SYM
ejpam-5596	112	21	∧λk(rn	∧λk(rn	NUM
ejpam-5596	112	22	)	)	PUNCT
ejpam-5596	112	23	.	.	PUNCT
ejpam-5596	113	1	case	case	NOUN
ejpam-5596	113	2	2	2	NUM
ejpam-5596	113	3	if	if	SCONJ
ejpam-5596	113	4	ei	ei	NOUN
ejpam-5596	113	5	/∈	/∈	PUNCT
ejpam-5596	114	1	k	k	PROPN
ejpam-5596	114	2	or	or	CCONJ
ejpam-5596	114	3	rj	rj	PROPN
ejpam-5596	114	4	/∈	/∈	PUNCT
ejpam-5596	115	1	k	k	PROPN
ejpam-5596	116	1	for	for	ADP
ejpam-5596	116	2	some	some	DET
ejpam-5596	116	3	i	i	PRON
ejpam-5596	116	4	∈	∈	PROPN
ejpam-5596	116	5	{	{	PUNCT
ejpam-5596	116	6	1	1	NUM
ejpam-5596	116	7	,	,	PUNCT
ejpam-5596	116	8	2	2	NUM
ejpam-5596	116	9	,	,	PUNCT
ejpam-5596	116	10	.	.	PUNCT
ejpam-5596	116	11	.	.	PUNCT
ejpam-5596	116	12	.	.	PUNCT
ejpam-5596	117	1	,	,	PUNCT
ejpam-5596	117	2	m	m	VERB
ejpam-5596	117	3	}	}	PUNCT
ejpam-5596	117	4	and	and	CCONJ
ejpam-5596	117	5	j	j	PROPN
ejpam-5596	117	6	∈	∈	PROPN
ejpam-5596	117	7	{	{	PUNCT
ejpam-5596	117	8	1	1	NUM
ejpam-5596	117	9	,	,	PUNCT
ejpam-5596	117	10	2	2	NUM
ejpam-5596	117	11	,	,	PUNCT
ejpam-5596	117	12	.	.	PUNCT
ejpam-5596	117	13	.	.	PUNCT
ejpam-5596	117	14	.	.	PUNCT
ejpam-5596	118	1	,	,	PUNCT
ejpam-5596	118	2	n	n	CCONJ
ejpam-5596	118	3	}	}	PUNCT
ejpam-5596	118	4	,	,	PUNCT
ejpam-5596	118	5	then	then	ADV
ejpam-5596	118	6	λk(u1u2	λk(u1u2	PROPN
ejpam-5596	118	7	·	·	PUNCT
ejpam-5596	118	8	·	·	PUNCT
ejpam-5596	119	1	·	·	PUNCT
ejpam-5596	119	2	umkv1v2	umkv1v2	X
ejpam-5596	119	3	·	·	PUNCT
ejpam-5596	119	4	·	·	PUNCT
ejpam-5596	119	5	·	·	PUNCT
ejpam-5596	119	6	vn	vn	X
ejpam-5596	119	7	)	)	PUNCT
ejpam-5596	119	8	≥	≥	NOUN
ejpam-5596	119	9	λk(u1)∧λk(u2)∧	λk(u1)∧λk(u2)∧	NOUN
ejpam-5596	119	10	·	·	PUNCT
ejpam-5596	119	11	·	·	PUNCT
ejpam-5596	119	12	·	·	PUNCT
ejpam-5596	119	13	∧λk(um)∧	∧λk(um)∧	X
ejpam-5596	119	14	·	·	SYM
ejpam-5596	119	15	·	·	PUNCT
ejpam-5596	119	16	·	·	PUNCT
ejpam-5596	119	17	∧λk(v1)∧λk(r2)∧	∧λk(v1)∧λk(r2)∧	X
ejpam-5596	119	18	·	·	SYM
ejpam-5596	119	19	·	·	PUNCT
ejpam-5596	119	20	·	·	SYM
ejpam-5596	119	21	∧λk(rn	∧λk(rn	NUM
ejpam-5596	119	22	)	)	PUNCT
ejpam-5596	119	23	.	.	PUNCT
ejpam-5596	120	1	let	let	VERB
ejpam-5596	120	2	u	u	NOUN
ejpam-5596	120	3	,	,	PUNCT
ejpam-5596	120	4	v	v	PROPN
ejpam-5596	120	5	∈	∈	NOUN
ejpam-5596	120	6	s	s	VERB
ejpam-5596	121	1	such	such	ADJ
ejpam-5596	121	2	that	that	SCONJ
ejpam-5596	121	3	u	u	PROPN
ejpam-5596	121	4	≤	≤	X
ejpam-5596	121	5	v	v	NOUN
ejpam-5596	121	6	and	and	CCONJ
ejpam-5596	121	7	u	u	PROPN
ejpam-5596	121	8	∈	∈	PROPN
ejpam-5596	121	9	k.	k.	PROPN
ejpam-5596	121	10	then	then	ADV
ejpam-5596	121	11	λk(u	λk(u	PUNCT
ejpam-5596	121	12	)	)	PUNCT
ejpam-5596	121	13	=	=	SYM
ejpam-5596	121	14	1	1	X
ejpam-5596	121	15	.	.	PUNCT
ejpam-5596	121	16	thus	thus	ADV
ejpam-5596	121	17	,	,	PUNCT
ejpam-5596	121	18	λk(u	λk(u	PUNCT
ejpam-5596	121	19	)	)	PUNCT
ejpam-5596	121	20	≥	≥	NOUN
ejpam-5596	121	21	λk(v	λk(v	NOUN
ejpam-5596	121	22	)	)	PUNCT
ejpam-5596	121	23	.	.	PUNCT
ejpam-5596	122	1	therefore	therefore	ADV
ejpam-5596	122	2	,	,	PUNCT
ejpam-5596	122	3	λk	λk	PRON
ejpam-5596	122	4	is	be	AUX
ejpam-5596	122	5	a	a	DET
ejpam-5596	122	6	fuzzy	fuzzy	ADJ
ejpam-5596	122	7	(	(	PUNCT
ejpam-5596	122	8	m	m	PROPN
ejpam-5596	122	9	,	,	PUNCT
ejpam-5596	122	10	n)-ideal	n)-ideal	NOUN
ejpam-5596	122	11	of	of	ADP
ejpam-5596	122	12	s.	s.	PROPN
ejpam-5596	122	13	conversely	conversely	ADV
ejpam-5596	122	14	,	,	PUNCT
ejpam-5596	122	15	suppose	suppose	VERB
ejpam-5596	122	16	that	that	SCONJ
ejpam-5596	122	17	λk	λk	PROPN
ejpam-5596	122	18	is	be	AUX
ejpam-5596	122	19	a	a	DET
ejpam-5596	122	20	fuzzy	fuzzy	ADJ
ejpam-5596	122	21	(	(	PUNCT
ejpam-5596	122	22	m	m	PROPN
ejpam-5596	122	23	,	,	PUNCT
ejpam-5596	122	24	n)-ideal	n)-ideal	NOUN
ejpam-5596	122	25	of	of	ADP
ejpam-5596	122	26	s.	s.	PROPN
ejpam-5596	122	27	then	then	ADV
ejpam-5596	122	28	λk	λk	X
ejpam-5596	122	29	is	be	AUX
ejpam-5596	122	30	a	a	DET
ejpam-5596	122	31	fuzzy	fuzzy	ADJ
ejpam-5596	122	32	subsemigroup	subsemigroup	NOUN
ejpam-5596	122	33	of	of	ADP
ejpam-5596	122	34	s.	s.	PROPN
ejpam-5596	122	35	by	by	ADP
ejpam-5596	122	36	lemma	lemma	PROPN
ejpam-5596	122	37	1	1	NUM
ejpam-5596	122	38	,	,	PUNCT
ejpam-5596	122	39	k	k	PROPN
ejpam-5596	122	40	is	be	AUX
ejpam-5596	122	41	a	a	DET
ejpam-5596	122	42	subsemigroup	subsemigroup	NOUN
ejpam-5596	122	43	of	of	ADP
ejpam-5596	122	44	s.	s.	PROPN
ejpam-5596	122	45	let	let	VERB
ejpam-5596	122	46	u1	u1	NOUN
ejpam-5596	122	47	,	,	PUNCT
ejpam-5596	122	48	u2	u2	NOUN
ejpam-5596	122	49	,	,	PUNCT
ejpam-5596	122	50	.	.	PUNCT
ejpam-5596	122	51	.	.	PUNCT
ejpam-5596	122	52	.	.	PUNCT
ejpam-5596	123	1	um	um	INTJ
ejpam-5596	123	2	,	,	PUNCT
ejpam-5596	123	3	k	k	PROPN
ejpam-5596	123	4	,	,	PUNCT
ejpam-5596	123	5	v1	v1	NOUN
ejpam-5596	123	6	,	,	PUNCT
ejpam-5596	123	7	v2	v2	NOUN
ejpam-5596	123	8	,	,	PUNCT
ejpam-5596	123	9	.	.	PUNCT
ejpam-5596	123	10	.	.	PUNCT
ejpam-5596	124	1	.	.	PUNCT
ejpam-5596	125	1	,	,	PUNCT
ejpam-5596	125	2	vn	vn	PROPN
ejpam-5596	125	3	∈	∈	PROPN
ejpam-5596	125	4	kmskn	kmskn	PROPN
ejpam-5596	125	5	.	.	PUNCT
ejpam-5596	126	1	then	then	ADV
ejpam-5596	126	2	λk(ui	λk(ui	NOUN
ejpam-5596	126	3	)	)	PUNCT
ejpam-5596	127	1	=	=	SYM
ejpam-5596	127	2	1	1	NUM
ejpam-5596	127	3	and	and	CCONJ
ejpam-5596	127	4	λk(vj	λk(vj	VERB
ejpam-5596	127	5	)	)	PUNCT
ejpam-5596	128	1	=	=	SYM
ejpam-5596	128	2	1	1	NUM
ejpam-5596	128	3	for	for	ADP
ejpam-5596	128	4	some	some	DET
ejpam-5596	128	5	i	i	PRON
ejpam-5596	128	6	∈	∈	PROPN
ejpam-5596	128	7	{	{	PUNCT
ejpam-5596	128	8	1	1	NUM
ejpam-5596	128	9	,	,	PUNCT
ejpam-5596	128	10	2	2	NUM
ejpam-5596	128	11	,	,	PUNCT
ejpam-5596	128	12	.	.	PUNCT
ejpam-5596	128	13	.	.	PUNCT
ejpam-5596	129	1	.m	.m	PROPN
ejpam-5596	129	2	}	}	PUNCT
ejpam-5596	130	1	and	and	CCONJ
ejpam-5596	130	2	j	j	PROPN
ejpam-5596	130	3	∈	∈	PROPN
ejpam-5596	130	4	{	{	PUNCT
ejpam-5596	130	5	1	1	NUM
ejpam-5596	130	6	,	,	PUNCT
ejpam-5596	130	7	2	2	NUM
ejpam-5596	130	8	,	,	PUNCT
ejpam-5596	130	9	.	.	PUNCT
ejpam-5596	130	10	.	.	PUNCT
ejpam-5596	130	11	.	.	PUNCT
ejpam-5596	130	12	,	,	PUNCT
ejpam-5596	130	13	n	n	CCONJ
ejpam-5596	130	14	}	}	PUNCT
ejpam-5596	130	15	.	.	PUNCT
ejpam-5596	131	1	by	by	ADP
ejpam-5596	131	2	assumption	assumption	NOUN
ejpam-5596	131	3	,	,	PUNCT
ejpam-5596	131	4	λk(u1u2	λk(u1u2	PROPN
ejpam-5596	131	5	·	·	PUNCT
ejpam-5596	131	6	·	·	PUNCT
ejpam-5596	131	7	·	·	PUNCT
ejpam-5596	131	8	umkv1v2	umkv1v2	X
ejpam-5596	131	9	·	·	PUNCT
ejpam-5596	131	10	·	·	PUNCT
ejpam-5596	131	11	·	·	PUNCT
ejpam-5596	131	12	vn	vn	X
ejpam-5596	131	13	)	)	PUNCT
ejpam-5596	131	14	≥	≥	X
ejpam-5596	131	15	λk(u1	λk(u1	X
ejpam-5596	131	16	)	)	PUNCT
ejpam-5596	131	17	∧	∧	PROPN
ejpam-5596	131	18	λk(u2	λk(u2	NOUN
ejpam-5596	131	19	)	)	PUNCT
ejpam-5596	131	20	∧	∧	PROPN
ejpam-5596	131	21	·	·	PUNCT
ejpam-5596	131	22	·	·	PUNCT
ejpam-5596	131	23	·	·	PUNCT
ejpam-5596	131	24	∧	∧	NOUN
ejpam-5596	131	25	λk(um	λk(um	PROPN
ejpam-5596	131	26	)	)	PUNCT
ejpam-5596	131	27	∧	∧	NOUN
ejpam-5596	131	28	·	·	PUNCT
ejpam-5596	131	29	·	·	PUNCT
ejpam-5596	131	30	·	·	PUNCT
ejpam-5596	131	31	∧	∧	NOUN
ejpam-5596	131	32	λk(v1	λk(v1	NOUN
ejpam-5596	131	33	)	)	PUNCT
ejpam-5596	131	34	∧	∧	PROPN
ejpam-5596	131	35	λk(v2	λk(v2	NOUN
ejpam-5596	131	36	)	)	PUNCT
ejpam-5596	131	37	∧	∧	PROPN
ejpam-5596	131	38	·	·	PUNCT
ejpam-5596	131	39	·	·	PUNCT
ejpam-5596	131	40	·	·	PUNCT
ejpam-5596	131	41	∧	∧	PROPN
ejpam-5596	131	42	λk(vn	λk(vn	PROPN
ejpam-5596	131	43	)	)	PUNCT
ejpam-5596	131	44	.	.	PUNCT
ejpam-5596	132	1	thus	thus	ADV
ejpam-5596	132	2	,	,	PUNCT
ejpam-5596	132	3	λk(u1u2	λk(u1u2	X
ejpam-5596	132	4	·	·	PUNCT
ejpam-5596	132	5	·	·	PUNCT
ejpam-5596	132	6	·	·	PUNCT
ejpam-5596	132	7	umkv1v2	umkv1v2	X
ejpam-5596	132	8	·	·	PUNCT
ejpam-5596	132	9	·	·	PUNCT
ejpam-5596	132	10	·	·	PUNCT
ejpam-5596	132	11	vn	vn	X
ejpam-5596	132	12	)	)	PUNCT
ejpam-5596	132	13	=	=	SYM
ejpam-5596	133	1	1	1	X
ejpam-5596	133	2	.	.	X
ejpam-5596	133	3	it	it	PRON
ejpam-5596	133	4	impiles	impile	VERB
ejpam-5596	133	5	that	that	SCONJ
ejpam-5596	133	6	,	,	PUNCT
ejpam-5596	133	7	e1e2	e1e2	X
ejpam-5596	133	8	·	·	PUNCT
ejpam-5596	133	9	·	·	PUNCT
ejpam-5596	133	10	·	·	PUNCT
ejpam-5596	133	11	emkv1v2	emkv1v2	X
ejpam-5596	133	12	·	·	PUNCT
ejpam-5596	133	13	·	·	PUNCT
ejpam-5596	133	14	·	·	PUNCT
ejpam-5596	133	15	vn	vn	PROPN
ejpam-5596	133	16	∈	∈	PROPN
ejpam-5596	133	17	k.	k.	PROPN
ejpam-5596	133	18	hence	hence	ADV
ejpam-5596	133	19	,	,	PUNCT
ejpam-5596	133	20	kmskn	kmskn	PROPN
ejpam-5596	133	21	⊆	⊆	NUM
ejpam-5596	133	22	k.	k.	NOUN
ejpam-5596	133	23	let	let	VERB
ejpam-5596	133	24	u	u	PRON
ejpam-5596	133	25	∈	∈	PROPN
ejpam-5596	133	26	k	k	ADP
ejpam-5596	133	27	such	such	ADJ
ejpam-5596	133	28	that	that	DET
ejpam-5596	133	29	v	v	NOUN
ejpam-5596	133	30	≤	≤	NOUN
ejpam-5596	133	31	u	u	NOUN
ejpam-5596	133	32	and	and	CCONJ
ejpam-5596	133	33	v	v	ADP
ejpam-5596	133	34	∈	∈	PROPN
ejpam-5596	133	35	s.	s.	PROPN
ejpam-5596	133	36	then	then	ADV
ejpam-5596	133	37	λk(u	λk(u	PUNCT
ejpam-5596	133	38	)	)	PUNCT
ejpam-5596	133	39	≥	≥	NOUN
ejpam-5596	133	40	λk(v	λk(v	PUNCT
ejpam-5596	133	41	)	)	PUNCT
ejpam-5596	133	42	≥	≥	NOUN
ejpam-5596	133	43	1	1	NUM
ejpam-5596	133	44	.	.	PUNCT
ejpam-5596	134	1	thus	thus	ADV
ejpam-5596	134	2	,	,	PUNCT
ejpam-5596	134	3	v	v	PROPN
ejpam-5596	134	4	∈	∈	PROPN
ejpam-5596	134	5	k.	k.	PROPN
ejpam-5596	134	6	therefore	therefore	ADV
ejpam-5596	134	7	,	,	PUNCT
ejpam-5596	134	8	k	k	PROPN
ejpam-5596	134	9	is	be	AUX
ejpam-5596	134	10	an	an	DET
ejpam-5596	134	11	(	(	PUNCT
ejpam-5596	134	12	m	m	PROPN
ejpam-5596	134	13	,	,	PUNCT
ejpam-5596	134	14	n)-ideal	n)-ideal	NOUN
ejpam-5596	134	15	of	of	ADP
ejpam-5596	134	16	s.	s.	PROPN
ejpam-5596	134	17	let	let	VERB
ejpam-5596	134	18	δ	δ	PRON
ejpam-5596	134	19	be	be	AUX
ejpam-5596	134	20	a	a	DET
ejpam-5596	134	21	fuzzy	fuzzy	ADJ
ejpam-5596	134	22	set	set	NOUN
ejpam-5596	134	23	and	and	CCONJ
ejpam-5596	134	24	t	t	NOUN
ejpam-5596	134	25	∈	∈	PROPN
ejpam-5596	135	1	[	[	X
ejpam-5596	135	2	0	0	NUM
ejpam-5596	135	3	,	,	PUNCT
ejpam-5596	135	4	1	1	NUM
ejpam-5596	135	5	]	]	PUNCT
ejpam-5596	135	6	.	.	PUNCT
ejpam-5596	136	1	define	define	VERB
ejpam-5596	136	2	the	the	DET
ejpam-5596	136	3	set	set	NOUN
ejpam-5596	136	4	ut	ut	PROPN
ejpam-5596	136	5	:	:	PUNCT
ejpam-5596	136	6	=	=	SYM
ejpam-5596	136	7	{	{	PUNCT
ejpam-5596	136	8	e	e	X
ejpam-5596	136	9	∈	∈	PROPN
ejpam-5596	136	10	s	s	VERB
ejpam-5596	136	11	|	|	ADV
ejpam-5596	136	12	δ(e	δ(e	NOUN
ejpam-5596	136	13	)	)	PUNCT
ejpam-5596	136	14	≥	≥	NOUN
ejpam-5596	136	15	t	t	PROPN
ejpam-5596	136	16	}	}	PUNCT
ejpam-5596	136	17	is	be	AUX
ejpam-5596	136	18	called	call	VERB
ejpam-5596	136	19	an	an	DET
ejpam-5596	136	20	t	t	NOUN
ejpam-5596	136	21	-	-	PUNCT
ejpam-5596	136	22	level	level	NOUN
ejpam-5596	136	23	subset	subset	NOUN
ejpam-5596	136	24	of	of	ADP
ejpam-5596	136	25	fuzzy	fuzzy	ADJ
ejpam-5596	136	26	set	set	NOUN
ejpam-5596	136	27	of	of	ADP
ejpam-5596	136	28	δ	δ	PROPN
ejpam-5596	136	29	.	.	PUNCT
ejpam-5596	137	1	lemma	lemma	PROPN
ejpam-5596	137	2	2	2	NUM
ejpam-5596	137	3	.	.	PUNCT
ejpam-5596	138	1	a	a	DET
ejpam-5596	138	2	fuzzy	fuzzy	ADJ
ejpam-5596	138	3	set	set	NOUN
ejpam-5596	138	4	δ	δ	PROPN
ejpam-5596	138	5	is	be	AUX
ejpam-5596	138	6	a	a	DET
ejpam-5596	138	7	fuzzy	fuzzy	ADJ
ejpam-5596	138	8	subsemigroup	subsemigroup	NOUN
ejpam-5596	138	9	of	of	ADP
ejpam-5596	138	10	a	a	DET
ejpam-5596	138	11	semigroup	semigroup	NOUN
ejpam-5596	138	12	s	s	X
ejpam-5596	138	13	if	if	SCONJ
ejpam-5596	138	14	and	and	CCONJ
ejpam-5596	138	15	only	only	ADV
ejpam-5596	138	16	if	if	SCONJ
ejpam-5596	138	17	the	the	DET
ejpam-5596	138	18	level	level	NOUN
ejpam-5596	138	19	set	set	VERB
ejpam-5596	138	20	ut	ut	PROPN
ejpam-5596	138	21	is	be	AUX
ejpam-5596	138	22	a	a	DET
ejpam-5596	138	23	subsemigroup	subsemigroup	NOUN
ejpam-5596	138	24	of	of	ADP
ejpam-5596	138	25	s	s	PRON
ejpam-5596	138	26	for	for	ADP
ejpam-5596	138	27	all	all	DET
ejpam-5596	138	28	t	t	NOUN
ejpam-5596	138	29	∈	∈	PROPN
ejpam-5596	139	1	[	[	X
ejpam-5596	139	2	0	0	NUM
ejpam-5596	139	3	,	,	PUNCT
ejpam-5596	139	4	1	1	NUM
ejpam-5596	139	5	]	]	PUNCT
ejpam-5596	139	6	.	.	PUNCT
ejpam-5596	140	1	proof	proof	NOUN
ejpam-5596	140	2	.	.	PUNCT
ejpam-5596	141	1	let	let	VERB
ejpam-5596	141	2	δ	δ	PRON
ejpam-5596	141	3	be	be	AUX
ejpam-5596	141	4	a	a	DET
ejpam-5596	141	5	fuzzy	fuzzy	ADJ
ejpam-5596	141	6	subsemigroup	subsemigroup	NOUN
ejpam-5596	141	7	of	of	ADP
ejpam-5596	141	8	s	s	NOUN
ejpam-5596	141	9	and	and	CCONJ
ejpam-5596	141	10	u	u	NOUN
ejpam-5596	141	11	,	,	PUNCT
ejpam-5596	141	12	v	v	PROPN
ejpam-5596	141	13	∈	∈	PROPN
ejpam-5596	141	14	ut	ut	PROPN
ejpam-5596	141	15	.	.	PROPN
ejpam-5596	141	16	then	then	ADV
ejpam-5596	141	17	δ(u1	δ(u1	VERB
ejpam-5596	141	18	)	)	PUNCT
ejpam-5596	141	19	≥	≥	PROPN
ejpam-5596	141	20	t	t	PROPN
ejpam-5596	141	21	,	,	PUNCT
ejpam-5596	141	22	δ(v	δ(v	PROPN
ejpam-5596	141	23	)	)	PUNCT
ejpam-5596	141	24	≥	≥	NOUN
ejpam-5596	141	25	t.	t.	NOUN
ejpam-5596	141	26	by	by	ADP
ejpam-5596	141	27	assumption	assumption	NOUN
ejpam-5596	141	28	,	,	PUNCT
ejpam-5596	141	29	δ(uv	δ(uv	NOUN
ejpam-5596	141	30	)	)	PUNCT
ejpam-5596	141	31	≥	≥	NOUN
ejpam-5596	141	32	δ(u)∧	δ(u)∧	VERB
ejpam-5596	141	33	δ(ev	δ(ev	PROPN
ejpam-5596	141	34	)	)	PUNCT
ejpam-5596	141	35	.	.	PUNCT
ejpam-5596	142	1	thus	thus	ADV
ejpam-5596	142	2	,	,	PUNCT
ejpam-5596	142	3	δ(uv	δ(uv	NOUN
ejpam-5596	142	4	)	)	PUNCT
ejpam-5596	142	5	≥	≥	NOUN
ejpam-5596	142	6	δ(u)∧	δ(u)∧	PROPN
ejpam-5596	142	7	δ(v	δ(v	PROPN
ejpam-5596	142	8	)	)	PUNCT
ejpam-5596	142	9	≥	≥	NOUN
ejpam-5596	142	10	t.	t.	NOUN
ejpam-5596	143	1	it	it	PRON
ejpam-5596	143	2	impiles	impile	VERB
ejpam-5596	143	3	that	that	SCONJ
ejpam-5596	143	4	,	,	PUNCT
ejpam-5596	143	5	uv	uv	PROPN
ejpam-5596	143	6	∈	∈	PROPN
ejpam-5596	143	7	ut	ut	PROPN
ejpam-5596	143	8	.	.	PROPN
ejpam-5596	144	1	hence	hence	ADV
ejpam-5596	144	2	,	,	PUNCT
ejpam-5596	144	3	ut	ut	PROPN
ejpam-5596	144	4	is	be	AUX
ejpam-5596	144	5	a	a	DET
ejpam-5596	144	6	subsemigroup	subsemigroup	NOUN
ejpam-5596	144	7	of	of	ADP
ejpam-5596	144	8	s.	s.	PROPN
ejpam-5596	144	9	p.	p.	PROPN
ejpam-5596	144	10	khamrot	khamrot	PROPN
ejpam-5596	144	11	,	,	PUNCT
ejpam-5596	144	12	a.	a.	NOUN
ejpam-5596	144	13	iampan	iampan	PROPN
ejpam-5596	144	14	,	,	PUNCT
ejpam-5596	144	15	t.	t.	PROPN
ejpam-5596	144	16	gaketem	gaketem	PROPN
ejpam-5596	144	17	/	/	SYM
ejpam-5596	144	18	eur	eur	PROPN
ejpam-5596	144	19	.	.	PUNCT
ejpam-5596	145	1	j.	j.	PROPN
ejpam-5596	145	2	pure	pure	PROPN
ejpam-5596	145	3	appl	appl	PROPN
ejpam-5596	145	4	.	.	PROPN
ejpam-5596	145	5	math	math	PROPN
ejpam-5596	145	6	,	,	PUNCT
ejpam-5596	145	7	18	18	NUM
ejpam-5596	145	8	(	(	PUNCT
ejpam-5596	145	9	1	1	NUM
ejpam-5596	145	10	)	)	PUNCT
ejpam-5596	145	11	(	(	PUNCT
ejpam-5596	145	12	2025	2025	NUM
ejpam-5596	145	13	)	)	PUNCT
ejpam-5596	145	14	,	,	PUNCT
ejpam-5596	145	15	5596	5596	NUM
ejpam-5596	145	16	6	6	NUM
ejpam-5596	145	17	of	of	ADP
ejpam-5596	145	18	12	12	NUM
ejpam-5596	145	19	conversely	conversely	ADV
ejpam-5596	145	20	,	,	PUNCT
ejpam-5596	145	21	suppose	suppose	VERB
ejpam-5596	145	22	that	that	SCONJ
ejpam-5596	145	23	ut	ut	PROPN
ejpam-5596	145	24	is	be	AUX
ejpam-5596	145	25	a	a	DET
ejpam-5596	145	26	subsemigroup	subsemigroup	NOUN
ejpam-5596	145	27	of	of	ADP
ejpam-5596	145	28	s	s	NOUN
ejpam-5596	145	29	and	and	CCONJ
ejpam-5596	145	30	u	u	NOUN
ejpam-5596	145	31	,	,	PUNCT
ejpam-5596	145	32	v	v	ADP
ejpam-5596	145	33	∈	∈	NOUN
ejpam-5596	145	34	s.	s.	PROPN
ejpam-5596	145	35	if	if	SCONJ
ejpam-5596	145	36	u	u	PROPN
ejpam-5596	145	37	,	,	PUNCT
ejpam-5596	145	38	v	v	PROPN
ejpam-5596	145	39	∈	∈	PROPN
ejpam-5596	145	40	ut	ut	PROPN
ejpam-5596	145	41	,	,	PUNCT
ejpam-5596	145	42	then	then	ADV
ejpam-5596	145	43	δ(u	δ(u	PROPN
ejpam-5596	145	44	)	)	PUNCT
ejpam-5596	145	45	≥	≥	NOUN
ejpam-5596	145	46	t	t	PROPN
ejpam-5596	145	47	,	,	PUNCT
ejpam-5596	145	48	δ(v	δ(v	PROPN
ejpam-5596	145	49	)	)	PUNCT
ejpam-5596	145	50	≥	≥	NOUN
ejpam-5596	145	51	t.	t.	PROPN
ejpam-5596	145	52	thus	thus	ADV
ejpam-5596	145	53	,	,	PUNCT
ejpam-5596	145	54	δ(uv	δ(uv	NOUN
ejpam-5596	145	55	)	)	PUNCT
ejpam-5596	145	56	≥	≥	NOUN
ejpam-5596	145	57	δ(u	δ(u	NOUN
ejpam-5596	145	58	)	)	PUNCT
ejpam-5596	145	59	∧	∧	PROPN
ejpam-5596	145	60	δ(v	δ(v	PROPN
ejpam-5596	145	61	)	)	PUNCT
ejpam-5596	145	62	.	.	PUNCT
ejpam-5596	146	1	if	if	SCONJ
ejpam-5596	146	2	u	u	PROPN
ejpam-5596	146	3	/∈	/∈	PROPN
ejpam-5596	146	4	ut	ut	PROPN
ejpam-5596	146	5	or	or	CCONJ
ejpam-5596	146	6	v	v	ADP
ejpam-5596	146	7	/∈	/∈	INTJ
ejpam-5596	146	8	ut	ut	PROPN
ejpam-5596	146	9	,	,	PUNCT
ejpam-5596	146	10	then	then	ADV
ejpam-5596	146	11	δ(uv	δ(uv	NOUN
ejpam-5596	146	12	)	)	PUNCT
ejpam-5596	146	13	≥	≥	NOUN
ejpam-5596	146	14	δ(u	δ(u	NOUN
ejpam-5596	146	15	)	)	PUNCT
ejpam-5596	146	16	∧	∧	PROPN
ejpam-5596	146	17	δ(v	δ(v	PROPN
ejpam-5596	146	18	)	)	PUNCT
ejpam-5596	146	19	.	.	PUNCT
ejpam-5596	147	1	hence	hence	ADV
ejpam-5596	147	2	,	,	PUNCT
ejpam-5596	147	3	δ	δ	PROPN
ejpam-5596	147	4	be	be	VERB
ejpam-5596	147	5	a	a	DET
ejpam-5596	147	6	fuzzy	fuzzy	ADJ
ejpam-5596	147	7	subsemigroup	subsemigroup	NOUN
ejpam-5596	147	8	of	of	ADP
ejpam-5596	147	9	s.	s.	PROPN
ejpam-5596	147	10	theorem	theorem	VERB
ejpam-5596	147	11	3	3	NUM
ejpam-5596	147	12	.	.	PUNCT
ejpam-5596	148	1	a	a	DET
ejpam-5596	148	2	fuzzy	fuzzy	ADJ
ejpam-5596	148	3	set	set	NOUN
ejpam-5596	148	4	δ	δ	PROPN
ejpam-5596	148	5	is	be	AUX
ejpam-5596	148	6	a	a	DET
ejpam-5596	148	7	fuzzy	fuzzy	ADJ
ejpam-5596	148	8	(	(	PUNCT
ejpam-5596	148	9	m	m	PROPN
ejpam-5596	148	10	,	,	PUNCT
ejpam-5596	148	11	n)-ideal	n)-ideal	NOUN
ejpam-5596	148	12	of	of	ADP
ejpam-5596	148	13	an	an	DET
ejpam-5596	148	14	ordered	order	VERB
ejpam-5596	148	15	semigroup	semigroup	NOUN
ejpam-5596	148	16	s	s	X
ejpam-5596	148	17	if	if	SCONJ
ejpam-5596	148	18	and	and	CCONJ
ejpam-5596	148	19	only	only	ADV
ejpam-5596	148	20	if	if	SCONJ
ejpam-5596	148	21	the	the	DET
ejpam-5596	148	22	level	level	NOUN
ejpam-5596	148	23	set	set	VERB
ejpam-5596	148	24	ut	ut	PROPN
ejpam-5596	148	25	is	be	AUX
ejpam-5596	148	26	an	an	DET
ejpam-5596	148	27	(	(	PUNCT
ejpam-5596	148	28	m	m	PROPN
ejpam-5596	148	29	,	,	PUNCT
ejpam-5596	148	30	n)-ideal	n)-ideal	NOUN
ejpam-5596	148	31	of	of	ADP
ejpam-5596	148	32	s	s	PRON
ejpam-5596	148	33	for	for	ADP
ejpam-5596	148	34	all	all	DET
ejpam-5596	148	35	t	t	NOUN
ejpam-5596	148	36	∈	∈	PROPN
ejpam-5596	149	1	[	[	X
ejpam-5596	149	2	0	0	NUM
ejpam-5596	149	3	,	,	PUNCT
ejpam-5596	149	4	1	1	NUM
ejpam-5596	149	5	]	]	PUNCT
ejpam-5596	149	6	.	.	PUNCT
ejpam-5596	150	1	proof	proof	NOUN
ejpam-5596	150	2	.	.	PUNCT
ejpam-5596	151	1	let	let	VERB
ejpam-5596	151	2	δ	δ	PRON
ejpam-5596	151	3	be	be	AUX
ejpam-5596	151	4	a	a	DET
ejpam-5596	151	5	fuzzy	fuzzy	ADJ
ejpam-5596	151	6	(	(	PUNCT
ejpam-5596	151	7	m	m	PROPN
ejpam-5596	151	8	,	,	PUNCT
ejpam-5596	151	9	n)-ideal	n)-ideal	NOUN
ejpam-5596	151	10	of	of	ADP
ejpam-5596	151	11	s.	s.	PROPN
ejpam-5596	151	12	then	then	ADV
ejpam-5596	151	13	δ	δ	PROPN
ejpam-5596	151	14	is	be	AUX
ejpam-5596	151	15	a	a	DET
ejpam-5596	151	16	fuzzy	fuzzy	ADJ
ejpam-5596	151	17	subsemigroup	subsemigroup	NOUN
ejpam-5596	151	18	of	of	ADP
ejpam-5596	151	19	s.	s.	PROPN
ejpam-5596	151	20	by	by	ADP
ejpam-5596	151	21	lemma	lemma	PROPN
ejpam-5596	151	22	2	2	NUM
ejpam-5596	151	23	,	,	PUNCT
ejpam-5596	151	24	ut	ut	PROPN
ejpam-5596	151	25	is	be	AUX
ejpam-5596	151	26	a	a	DET
ejpam-5596	151	27	subsemigroup	subsemigroup	NOUN
ejpam-5596	151	28	of	of	ADP
ejpam-5596	151	29	s.	s.	PROPN
ejpam-5596	151	30	let	let	VERB
ejpam-5596	151	31	u1	u1	NOUN
ejpam-5596	151	32	,	,	PUNCT
ejpam-5596	151	33	u2	u2	NOUN
ejpam-5596	151	34	,	,	PUNCT
ejpam-5596	151	35	.	.	PUNCT
ejpam-5596	151	36	.	.	PUNCT
ejpam-5596	152	1	.	.	PUNCT
ejpam-5596	153	1	um	um	INTJ
ejpam-5596	153	2	,	,	PUNCT
ejpam-5596	153	3	k	k	PROPN
ejpam-5596	153	4	,	,	PUNCT
ejpam-5596	153	5	v1	v1	NOUN
ejpam-5596	153	6	,	,	PUNCT
ejpam-5596	153	7	v2	v2	NOUN
ejpam-5596	153	8	,	,	PUNCT
ejpam-5596	153	9	.	.	PUNCT
ejpam-5596	153	10	.	.	PUNCT
ejpam-5596	154	1	.	.	PUNCT
ejpam-5596	155	1	,	,	PUNCT
ejpam-5596	155	2	vn	vn	PROPN
ejpam-5596	155	3	∈	∈	PROPN
ejpam-5596	155	4	ut	ut	PROPN
ejpam-5596	155	5	.	.	PROPN
ejpam-5596	155	6	then	then	ADV
ejpam-5596	155	7	δ(ui	δ(ui	NUM
ejpam-5596	155	8	)	)	PUNCT
ejpam-5596	155	9	≥	≥	PROPN
ejpam-5596	155	10	t	t	PROPN
ejpam-5596	155	11	,	,	PUNCT
ejpam-5596	155	12	δ(vj	δ(vj	PROPN
ejpam-5596	155	13	)	)	PUNCT
ejpam-5596	155	14	≥	≥	NOUN
ejpam-5596	155	15	t	t	NOUN
ejpam-5596	155	16	for	for	ADP
ejpam-5596	155	17	some	some	DET
ejpam-5596	155	18	i	i	PRON
ejpam-5596	155	19	∈	∈	PROPN
ejpam-5596	155	20	{	{	PUNCT
ejpam-5596	155	21	1	1	NUM
ejpam-5596	155	22	,	,	PUNCT
ejpam-5596	155	23	2	2	NUM
ejpam-5596	155	24	,	,	PUNCT
ejpam-5596	155	25	.	.	PUNCT
ejpam-5596	155	26	.	.	PUNCT
ejpam-5596	155	27	.	.	PUNCT
ejpam-5596	156	1	,	,	PUNCT
ejpam-5596	156	2	m	m	VERB
ejpam-5596	156	3	}	}	PUNCT
ejpam-5596	156	4	and	and	CCONJ
ejpam-5596	156	5	j	j	PROPN
ejpam-5596	156	6	∈	∈	PROPN
ejpam-5596	156	7	{	{	PUNCT
ejpam-5596	156	8	1	1	NUM
ejpam-5596	156	9	,	,	PUNCT
ejpam-5596	156	10	2	2	NUM
ejpam-5596	156	11	,	,	PUNCT
ejpam-5596	156	12	.	.	PUNCT
ejpam-5596	156	13	.	.	PUNCT
ejpam-5596	156	14	.	.	PUNCT
ejpam-5596	157	1	,	,	PUNCT
ejpam-5596	157	2	m	m	VERB
ejpam-5596	157	3	}	}	PUNCT
ejpam-5596	157	4	.	.	PUNCT
ejpam-5596	158	1	by	by	ADP
ejpam-5596	158	2	assumption	assumption	NOUN
ejpam-5596	158	3	,	,	PUNCT
ejpam-5596	158	4	δ(u1u2	δ(u1u2	PROPN
ejpam-5596	158	5	·	·	PUNCT
ejpam-5596	158	6	·	·	PUNCT
ejpam-5596	158	7	·	·	PUNCT
ejpam-5596	158	8	umkv1v2	umkv1v2	X
ejpam-5596	158	9	·	·	PUNCT
ejpam-5596	158	10	·	·	PUNCT
ejpam-5596	158	11	·	·	PUNCT
ejpam-5596	158	12	vn	vn	X
ejpam-5596	158	13	)	)	PUNCT
ejpam-5596	158	14	≥	≥	NOUN
ejpam-5596	158	15	δ(u1	δ(u1	NOUN
ejpam-5596	158	16	)	)	PUNCT
ejpam-5596	158	17	∧	∧	PROPN
ejpam-5596	158	18	δ(u2	δ(u2	NOUN
ejpam-5596	158	19	)	)	PUNCT
ejpam-5596	158	20	∧	∧	PROPN
ejpam-5596	158	21	·	·	PUNCT
ejpam-5596	158	22	·	·	PUNCT
ejpam-5596	158	23	·	·	PUNCT
ejpam-5596	158	24	∧	∧	NOUN
ejpam-5596	158	25	δ(um	δ(um	NOUN
ejpam-5596	158	26	)	)	PUNCT
ejpam-5596	158	27	∧	∧	PROPN
ejpam-5596	158	28	·	·	PUNCT
ejpam-5596	158	29	·	·	PUNCT
ejpam-5596	158	30	·	·	PUNCT
ejpam-5596	158	31	∧	∧	NOUN
ejpam-5596	158	32	δ(v1	δ(v1	NOUN
ejpam-5596	158	33	)	)	PUNCT
ejpam-5596	158	34	∧	∧	NOUN
ejpam-5596	158	35	δ(v2	δ(v2	NOUN
ejpam-5596	158	36	)	)	PUNCT
ejpam-5596	158	37	∧	∧	PROPN
ejpam-5596	158	38	·	·	PUNCT
ejpam-5596	158	39	·	·	PUNCT
ejpam-5596	158	40	·	·	PUNCT
ejpam-5596	158	41	∧	∧	NOUN
ejpam-5596	158	42	δ(vn	δ(vn	PROPN
ejpam-5596	158	43	)	)	PUNCT
ejpam-5596	158	44	.	.	PUNCT
ejpam-5596	159	1	thus	thus	ADV
ejpam-5596	159	2	,	,	PUNCT
ejpam-5596	159	3	δ(u1u2	δ(u1u2	PROPN
ejpam-5596	159	4	·	·	PUNCT
ejpam-5596	159	5	·	·	PUNCT
ejpam-5596	159	6	·	·	PUNCT
ejpam-5596	159	7	umkv1v2	umkv1v2	X
ejpam-5596	159	8	·	·	PUNCT
ejpam-5596	159	9	·	·	PUNCT
ejpam-5596	159	10	·	·	PUNCT
ejpam-5596	159	11	vn	vn	X
ejpam-5596	159	12	)	)	PUNCT
ejpam-5596	159	13	≥	≥	PROPN
ejpam-5596	159	14	t.	t.	NOUN
ejpam-5596	159	15	it	it	PRON
ejpam-5596	159	16	impiles	impile	VERB
ejpam-5596	159	17	that	that	SCONJ
ejpam-5596	159	18	,	,	PUNCT
ejpam-5596	159	19	u1u2	u1u2	X
ejpam-5596	159	20	·	·	PUNCT
ejpam-5596	159	21	·	·	PUNCT
ejpam-5596	159	22	·	·	PUNCT
ejpam-5596	159	23	umkv1v2	umkv1v2	X
ejpam-5596	159	24	·	·	PUNCT
ejpam-5596	159	25	·	·	PUNCT
ejpam-5596	159	26	·	·	PUNCT
ejpam-5596	159	27	vn	vn	PROPN
ejpam-5596	159	28	∈	∈	PROPN
ejpam-5596	159	29	ut	ut	PROPN
ejpam-5596	159	30	.	.	PROPN
ejpam-5596	159	31	let	let	VERB
ejpam-5596	159	32	u	u	NOUN
ejpam-5596	159	33	,	,	PUNCT
ejpam-5596	159	34	v	v	PROPN
ejpam-5596	159	35	∈	∈	NOUN
ejpam-5596	159	36	s	s	VERB
ejpam-5596	159	37	such	such	ADJ
ejpam-5596	159	38	that	that	SCONJ
ejpam-5596	159	39	u	u	PROPN
ejpam-5596	159	40	≤	≤	X
ejpam-5596	159	41	v	v	NOUN
ejpam-5596	159	42	and	and	CCONJ
ejpam-5596	159	43	v	v	ADP
ejpam-5596	159	44	∈	∈	PROPN
ejpam-5596	160	1	ut	ut	PROPN
ejpam-5596	160	2	.	.	PROPN
ejpam-5596	161	1	then	then	ADV
ejpam-5596	161	2	δ(u	δ(u	PROPN
ejpam-5596	161	3	)	)	PUNCT
ejpam-5596	161	4	≥	≥	NOUN
ejpam-5596	161	5	δ(v	δ(v	PROPN
ejpam-5596	161	6	)	)	PUNCT
ejpam-5596	161	7	≥	≥	NOUN
ejpam-5596	161	8	t.	t.	PROPN
ejpam-5596	161	9	thus	thus	ADV
ejpam-5596	161	10	,	,	PUNCT
ejpam-5596	161	11	u	u	PROPN
ejpam-5596	161	12	∈	∈	PROPN
ejpam-5596	161	13	ut	ut	PROPN
ejpam-5596	161	14	.	.	PROPN
ejpam-5596	162	1	hence	hence	ADV
ejpam-5596	162	2	,	,	PUNCT
ejpam-5596	162	3	ut	ut	PROPN
ejpam-5596	162	4	is	be	AUX
ejpam-5596	162	5	an	an	DET
ejpam-5596	162	6	(	(	PUNCT
ejpam-5596	162	7	m	m	PROPN
ejpam-5596	162	8	,	,	PUNCT
ejpam-5596	162	9	n)-ideal	n)-ideal	NOUN
ejpam-5596	162	10	of	of	ADP
ejpam-5596	162	11	s.	s.	PROPN
ejpam-5596	162	12	conversely	conversely	ADV
ejpam-5596	162	13	,	,	PUNCT
ejpam-5596	162	14	suppose	suppose	VERB
ejpam-5596	162	15	that	that	SCONJ
ejpam-5596	162	16	ut	ut	PROPN
ejpam-5596	162	17	is	be	AUX
ejpam-5596	162	18	an	an	DET
ejpam-5596	162	19	(	(	PUNCT
ejpam-5596	162	20	m	m	PROPN
ejpam-5596	162	21	,	,	PUNCT
ejpam-5596	162	22	n)-ideal	n)-ideal	NOUN
ejpam-5596	162	23	of	of	ADP
ejpam-5596	162	24	s.	s.	PROPN
ejpam-5596	162	25	then	then	ADV
ejpam-5596	162	26	ut	ut	PROPN
ejpam-5596	162	27	is	be	AUX
ejpam-5596	162	28	a	a	DET
ejpam-5596	162	29	subsemigroup	subsemigroup	NOUN
ejpam-5596	162	30	of	of	ADP
ejpam-5596	162	31	s.	s.	PROPN
ejpam-5596	162	32	by	by	ADP
ejpam-5596	162	33	lemma	lemma	PROPN
ejpam-5596	162	34	2	2	NUM
ejpam-5596	162	35	,	,	PUNCT
ejpam-5596	162	36	δ	δ	PROPN
ejpam-5596	162	37	is	be	AUX
ejpam-5596	162	38	a	a	DET
ejpam-5596	162	39	fuzzy	fuzzy	ADJ
ejpam-5596	162	40	subsemigroup	subsemigroup	NOUN
ejpam-5596	162	41	of	of	ADP
ejpam-5596	162	42	a	a	DET
ejpam-5596	162	43	semigroup	semigroup	PROPN
ejpam-5596	162	44	s.	s.	PROPN
ejpam-5596	162	45	let	let	VERB
ejpam-5596	162	46	u	u	NOUN
ejpam-5596	162	47	,	,	PUNCT
ejpam-5596	162	48	v	v	PROPN
ejpam-5596	162	49	∈	∈	NOUN
ejpam-5596	162	50	s	s	VERB
ejpam-5596	162	51	such	such	ADJ
ejpam-5596	162	52	that	that	SCONJ
ejpam-5596	162	53	u	u	PROPN
ejpam-5596	162	54	≤	≤	X
ejpam-5596	163	1	v.	v.	CCONJ
ejpam-5596	163	2	we	we	PRON
ejpam-5596	163	3	choose	choose	VERB
ejpam-5596	163	4	δ(v	δ(v	PROPN
ejpam-5596	163	5	)	)	PUNCT
ejpam-5596	164	1	=	=	PUNCT
ejpam-5596	165	1	t.	t.	PROPN
ejpam-5596	165	2	thus	thus	ADV
ejpam-5596	165	3	,	,	PUNCT
ejpam-5596	165	4	v	v	PROPN
ejpam-5596	165	5	∈	∈	PROPN
ejpam-5596	165	6	ut	ut	PROPN
ejpam-5596	165	7	.	.	PROPN
ejpam-5596	165	8	by	by	ADP
ejpam-5596	165	9	assumption	assumption	NOUN
ejpam-5596	165	10	,	,	PUNCT
ejpam-5596	165	11	u	u	PROPN
ejpam-5596	165	12	∈	∈	PROPN
ejpam-5596	165	13	ut	ut	PROPN
ejpam-5596	165	14	.	.	PROPN
ejpam-5596	166	1	then	then	ADV
ejpam-5596	166	2	δ(u	δ(u	PROPN
ejpam-5596	166	3	)	)	PUNCT
ejpam-5596	166	4	≥	≥	NOUN
ejpam-5596	166	5	t	t	NOUN
ejpam-5596	166	6	=	=	SYM
ejpam-5596	166	7	δ(v	δ(v	PROPN
ejpam-5596	166	8	)	)	PUNCT
ejpam-5596	166	9	.	.	PUNCT
ejpam-5596	167	1	if	if	SCONJ
ejpam-5596	167	2	δ	δ	PROPN
ejpam-5596	167	3	is	be	AUX
ejpam-5596	167	4	not	not	PART
ejpam-5596	167	5	a	a	DET
ejpam-5596	167	6	fuzzy	fuzzy	ADJ
ejpam-5596	167	7	(	(	PUNCT
ejpam-5596	167	8	m	m	PROPN
ejpam-5596	167	9	,	,	PUNCT
ejpam-5596	167	10	n)-ideal	n)-ideal	NOUN
ejpam-5596	167	11	of	of	ADP
ejpam-5596	167	12	s	s	PROPN
ejpam-5596	167	13	,	,	PUNCT
ejpam-5596	167	14	then	then	ADV
ejpam-5596	167	15	there	there	PRON
ejpam-5596	167	16	exists	exist	VERB
ejpam-5596	167	17	ui	ui	PROPN
ejpam-5596	167	18	,	,	PUNCT
ejpam-5596	167	19	k	k	PROPN
ejpam-5596	167	20	,	,	PUNCT
ejpam-5596	167	21	vj	vj	PROPN
ejpam-5596	167	22	∈	∈	PROPN
ejpam-5596	167	23	s	s	VERB
ejpam-5596	167	24	such	such	ADJ
ejpam-5596	167	25	that	that	SCONJ
ejpam-5596	167	26	δ(u1u2	δ(u1u2	PROPN
ejpam-5596	167	27	·	·	PUNCT
ejpam-5596	167	28	·	·	PUNCT
ejpam-5596	167	29	·	·	PUNCT
ejpam-5596	167	30	umkv1v2	umkv1v2	X
ejpam-5596	167	31	·	·	PUNCT
ejpam-5596	167	32	·	·	PUNCT
ejpam-5596	167	33	·	·	PUNCT
ejpam-5596	167	34	vn	vn	X
ejpam-5596	167	35	)	)	PUNCT
ejpam-5596	167	36	<	<	X
ejpam-5596	167	37	δ(u1	δ(u1	NOUN
ejpam-5596	167	38	)	)	PUNCT
ejpam-5596	167	39	∧	∧	PROPN
ejpam-5596	167	40	δ(u2	δ(u2	NOUN
ejpam-5596	167	41	)	)	PUNCT
ejpam-5596	167	42	∧	∧	PROPN
ejpam-5596	167	43	·	·	PUNCT
ejpam-5596	167	44	·	·	PUNCT
ejpam-5596	167	45	·	·	PUNCT
ejpam-5596	167	46	∧	∧	NOUN
ejpam-5596	167	47	δ(um	δ(um	NOUN
ejpam-5596	167	48	)	)	PUNCT
ejpam-5596	167	49	∧	∧	PROPN
ejpam-5596	167	50	·	·	PUNCT
ejpam-5596	167	51	·	·	PUNCT
ejpam-5596	167	52	·	·	PUNCT
ejpam-5596	167	53	∧	∧	NOUN
ejpam-5596	167	54	δ(v1	δ(v1	NOUN
ejpam-5596	167	55	)	)	PUNCT
ejpam-5596	167	56	∧	∧	NOUN
ejpam-5596	167	57	δ(v2	δ(v2	NOUN
ejpam-5596	167	58	)	)	PUNCT
ejpam-5596	167	59	∧	∧	PROPN
ejpam-5596	167	60	·	·	PUNCT
ejpam-5596	167	61	·	·	PUNCT
ejpam-5596	167	62	·	·	PUNCT
ejpam-5596	167	63	∧	∧	NOUN
ejpam-5596	167	64	δ(vn	δ(vn	PROPN
ejpam-5596	167	65	)	)	PUNCT
ejpam-5596	167	66	.	.	PUNCT
ejpam-5596	168	1	by	by	ADP
ejpam-5596	168	2	assumption	assumption	NOUN
ejpam-5596	168	3	,	,	PUNCT
ejpam-5596	168	4	we	we	PRON
ejpam-5596	168	5	have	have	VERB
ejpam-5596	168	6	u1u2	u1u2	X
ejpam-5596	168	7	·	·	PUNCT
ejpam-5596	168	8	·	·	PUNCT
ejpam-5596	168	9	·	·	PUNCT
ejpam-5596	168	10	umkv1v2	umkv1v2	X
ejpam-5596	168	11	·	·	PUNCT
ejpam-5596	168	12	·	·	PUNCT
ejpam-5596	168	13	·	·	PUNCT
ejpam-5596	168	14	vn	vn	PROPN
ejpam-5596	168	15	∈	∈	PROPN
ejpam-5596	168	16	ut	ut	PROPN
ejpam-5596	168	17	.	.	PROPN
ejpam-5596	169	1	thus	thus	ADV
ejpam-5596	169	2	,	,	PUNCT
ejpam-5596	169	3	δ(u1u2	δ(u1u2	PROPN
ejpam-5596	169	4	·	·	PUNCT
ejpam-5596	169	5	·	·	PUNCT
ejpam-5596	169	6	·	·	PUNCT
ejpam-5596	169	7	umkv1v2	umkv1v2	X
ejpam-5596	169	8	·	·	PUNCT
ejpam-5596	169	9	·	·	PUNCT
ejpam-5596	169	10	·	·	PUNCT
ejpam-5596	169	11	vn	vn	X
ejpam-5596	169	12	)	)	PUNCT
ejpam-5596	169	13	≥	≥	NOUN
ejpam-5596	169	14	δ(u1	δ(u1	NOUN
ejpam-5596	169	15	)	)	PUNCT
ejpam-5596	169	16	∧	∧	PROPN
ejpam-5596	169	17	δ(u2	δ(u2	NOUN
ejpam-5596	169	18	)	)	PUNCT
ejpam-5596	169	19	∧	∧	PROPN
ejpam-5596	169	20	·	·	PUNCT
ejpam-5596	169	21	·	·	PUNCT
ejpam-5596	169	22	·	·	PUNCT
ejpam-5596	170	1	∧	∧	NOUN
ejpam-5596	170	2	δ(um	δ(um	NOUN
ejpam-5596	170	3	)	)	PUNCT
ejpam-5596	170	4	∧	∧	PROPN
ejpam-5596	170	5	·	·	PUNCT
ejpam-5596	170	6	·	·	PUNCT
ejpam-5596	170	7	·	·	PUNCT
ejpam-5596	170	8	∧	∧	NOUN
ejpam-5596	170	9	δ(v1	δ(v1	NOUN
ejpam-5596	170	10	)	)	PUNCT
ejpam-5596	170	11	∧	∧	NOUN
ejpam-5596	170	12	δ(v2	δ(v2	NOUN
ejpam-5596	170	13	)	)	PUNCT
ejpam-5596	170	14	∧	∧	PROPN
ejpam-5596	170	15	·	·	PUNCT
ejpam-5596	170	16	·	·	PUNCT
ejpam-5596	170	17	·	·	PUNCT
ejpam-5596	170	18	∧	∧	NOUN
ejpam-5596	170	19	δ(vn	δ(vn	PROPN
ejpam-5596	170	20	)	)	PUNCT
ejpam-5596	170	21	.	.	PUNCT
ejpam-5596	171	1	it	it	PRON
ejpam-5596	171	2	is	be	AUX
ejpam-5596	171	3	a	a	DET
ejpam-5596	171	4	contradiction	contradiction	NOUN
ejpam-5596	171	5	.	.	PUNCT
ejpam-5596	172	1	hence	hence	ADV
ejpam-5596	172	2	,	,	PUNCT
ejpam-5596	172	3	δ	δ	PROPN
ejpam-5596	172	4	be	be	VERB
ejpam-5596	172	5	a	a	DET
ejpam-5596	172	6	fuzzy	fuzzy	ADJ
ejpam-5596	172	7	(	(	PUNCT
ejpam-5596	172	8	m	m	PROPN
ejpam-5596	172	9	,	,	PUNCT
ejpam-5596	172	10	n)-ideal	n)-ideal	NOUN
ejpam-5596	172	11	of	of	ADP
ejpam-5596	172	12	s.	s.	PROPN
ejpam-5596	172	13	definition	definition	PROPN
ejpam-5596	172	14	7	7	NUM
ejpam-5596	172	15	.	.	PUNCT
ejpam-5596	173	1	an	an	DET
ejpam-5596	173	2	(	(	PUNCT
ejpam-5596	173	3	m	m	PROPN
ejpam-5596	173	4	,	,	PUNCT
ejpam-5596	173	5	n)-ideal	n)-ideal	PROPN
ejpam-5596	173	6	k	k	PROPN
ejpam-5596	173	7	of	of	ADP
ejpam-5596	173	8	a	a	DET
ejpam-5596	173	9	ordered	order	VERB
ejpam-5596	173	10	semigroup	semigroup	NOUN
ejpam-5596	173	11	s	s	VERB
ejpam-5596	173	12	is	be	AUX
ejpam-5596	173	13	called	call	VERB
ejpam-5596	173	14	(	(	PUNCT
ejpam-5596	173	15	1	1	NUM
ejpam-5596	173	16	)	)	PUNCT
ejpam-5596	173	17	a	a	DET
ejpam-5596	173	18	minimal	minimal	ADJ
ejpam-5596	173	19	if	if	SCONJ
ejpam-5596	173	20	for	for	ADP
ejpam-5596	173	21	every	every	DET
ejpam-5596	173	22	(	(	PUNCT
ejpam-5596	173	23	m	m	PROPN
ejpam-5596	173	24	,	,	PUNCT
ejpam-5596	173	25	n)-ideal	n)-ideal	NOUN
ejpam-5596	173	26	of	of	ADP
ejpam-5596	173	27	j	j	PROPN
ejpam-5596	173	28	of	of	ADP
ejpam-5596	173	29	s	s	PRON
ejpam-5596	173	30	such	such	ADJ
ejpam-5596	173	31	that	that	SCONJ
ejpam-5596	173	32	j	j	PROPN
ejpam-5596	173	33	⊆	⊆	NUM
ejpam-5596	173	34	k	k	NOUN
ejpam-5596	173	35	,	,	PUNCT
ejpam-5596	173	36	we	we	PRON
ejpam-5596	173	37	have	have	VERB
ejpam-5596	173	38	j	j	PROPN
ejpam-5596	173	39	=	=	PROPN
ejpam-5596	173	40	k.	k.	PROPN
ejpam-5596	173	41	(	(	PUNCT
ejpam-5596	173	42	2	2	X
ejpam-5596	173	43	)	)	PUNCT
ejpam-5596	173	44	a	a	DET
ejpam-5596	173	45	maximal	maximal	ADJ
ejpam-5596	173	46	if	if	SCONJ
ejpam-5596	173	47	for	for	ADP
ejpam-5596	173	48	every	every	DET
ejpam-5596	173	49	(	(	PUNCT
ejpam-5596	173	50	m	m	PROPN
ejpam-5596	173	51	,	,	PUNCT
ejpam-5596	173	52	n)-ideal	n)-ideal	NOUN
ejpam-5596	173	53	of	of	ADP
ejpam-5596	173	54	j	j	PROPN
ejpam-5596	173	55	of	of	ADP
ejpam-5596	173	56	s	s	PRON
ejpam-5596	173	57	such	such	ADJ
ejpam-5596	173	58	that	that	SCONJ
ejpam-5596	173	59	k	k	PROPN
ejpam-5596	173	60	⊆	⊆	NUM
ejpam-5596	173	61	j	j	PROPN
ejpam-5596	173	62	,	,	PUNCT
ejpam-5596	173	63	we	we	PRON
ejpam-5596	173	64	have	have	VERB
ejpam-5596	173	65	j	j	PROPN
ejpam-5596	173	66	=	=	SYM
ejpam-5596	173	67	k.	k.	PROPN
ejpam-5596	173	68	definition	definition	NOUN
ejpam-5596	173	69	8	8	NUM
ejpam-5596	173	70	.	.	PUNCT
ejpam-5596	174	1	a	a	DET
ejpam-5596	174	2	fuzzy	fuzzy	ADJ
ejpam-5596	174	3	(	(	PUNCT
ejpam-5596	174	4	m	m	PROPN
ejpam-5596	174	5	,	,	PUNCT
ejpam-5596	174	6	n)-ideal	n)-ideal	PROPN
ejpam-5596	174	7	δ	δ	PROPN
ejpam-5596	174	8	of	of	ADP
ejpam-5596	174	9	an	an	DET
ejpam-5596	174	10	ordered	order	VERB
ejpam-5596	174	11	semigroup	semigroup	NOUN
ejpam-5596	174	12	s	s	X
ejpam-5596	174	13	is	be	AUX
ejpam-5596	174	14	(	(	PUNCT
ejpam-5596	174	15	1	1	NUM
ejpam-5596	174	16	)	)	PUNCT
ejpam-5596	174	17	a	a	DET
ejpam-5596	174	18	minimal	minimal	ADJ
ejpam-5596	174	19	if	if	SCONJ
ejpam-5596	174	20	for	for	ADP
ejpam-5596	174	21	all	all	PRON
ejpam-5596	174	22	fuzzy	fuzzy	ADJ
ejpam-5596	174	23	(	(	PUNCT
ejpam-5596	174	24	m	m	PROPN
ejpam-5596	174	25	,	,	PUNCT
ejpam-5596	174	26	n)-ideal	n)-ideal	PROPN
ejpam-5596	174	27	ξ	ξ	PROPN
ejpam-5596	174	28	of	of	ADP
ejpam-5596	174	29	s	s	PRON
ejpam-5596	174	30	such	such	ADJ
ejpam-5596	174	31	that	that	SCONJ
ejpam-5596	174	32	ξ	ξ	PROPN
ejpam-5596	174	33	≤	≤	PROPN
ejpam-5596	174	34	δ	δ	PROPN
ejpam-5596	174	35	,	,	PUNCT
ejpam-5596	174	36	then	then	ADV
ejpam-5596	174	37	ξ	ξ	X
ejpam-5596	174	38	=	=	SYM
ejpam-5596	174	39	δ	δ	PROPN
ejpam-5596	174	40	.	.	PUNCT
ejpam-5596	175	1	(	(	PUNCT
ejpam-5596	175	2	2	2	X
ejpam-5596	175	3	)	)	PUNCT
ejpam-5596	175	4	a	a	DET
ejpam-5596	175	5	maximal	maximal	ADJ
ejpam-5596	175	6	if	if	SCONJ
ejpam-5596	175	7	for	for	ADP
ejpam-5596	175	8	all	all	PRON
ejpam-5596	175	9	fuzzy	fuzzy	ADJ
ejpam-5596	175	10	(	(	PUNCT
ejpam-5596	175	11	m	m	PROPN
ejpam-5596	175	12	,	,	PUNCT
ejpam-5596	175	13	n)-ideal	n)-ideal	PROPN
ejpam-5596	175	14	ξ	ξ	PROPN
ejpam-5596	175	15	of	of	ADP
ejpam-5596	175	16	s	s	PRON
ejpam-5596	175	17	such	such	ADJ
ejpam-5596	175	18	that	that	SCONJ
ejpam-5596	175	19	δ	δ	PROPN
ejpam-5596	175	20	≤	≤	PROPN
ejpam-5596	175	21	ξ	ξ	PROPN
ejpam-5596	175	22	,	,	PUNCT
ejpam-5596	175	23	then	then	ADV
ejpam-5596	175	24	ξ	ξ	X
ejpam-5596	175	25	=	=	SYM
ejpam-5596	175	26	δ	δ	PROPN
ejpam-5596	175	27	.	.	PUNCT
ejpam-5596	175	28	theorem	theorem	VERB
ejpam-5596	175	29	4	4	NUM
ejpam-5596	175	30	.	.	PUNCT
ejpam-5596	176	1	a	a	DET
ejpam-5596	176	2	non	non	ADJ
ejpam-5596	176	3	-	-	ADJ
ejpam-5596	176	4	empty	empty	ADJ
ejpam-5596	176	5	subset	subset	NOUN
ejpam-5596	176	6	k	k	PROPN
ejpam-5596	176	7	of	of	ADP
ejpam-5596	176	8	an	an	DET
ejpam-5596	176	9	ordered	order	VERB
ejpam-5596	176	10	semigroup	semigroup	PROPN
ejpam-5596	176	11	s.	s.	PROPN
ejpam-5596	176	12	then	then	ADV
ejpam-5596	176	13	the	the	DET
ejpam-5596	176	14	following	follow	VERB
ejpam-5596	176	15	statements	statement	NOUN
ejpam-5596	176	16	ture	ture	NOUN
ejpam-5596	176	17	(	(	PUNCT
ejpam-5596	176	18	1	1	NUM
ejpam-5596	176	19	)	)	PUNCT
ejpam-5596	176	20	k	k	X
ejpam-5596	176	21	is	be	AUX
ejpam-5596	176	22	a	a	DET
ejpam-5596	176	23	minimal	minimal	ADJ
ejpam-5596	176	24	(	(	PUNCT
ejpam-5596	176	25	m	m	PROPN
ejpam-5596	176	26	,	,	PUNCT
ejpam-5596	176	27	n)-ideal	n)-ideal	NOUN
ejpam-5596	176	28	if	if	SCONJ
ejpam-5596	176	29	and	and	CCONJ
ejpam-5596	176	30	only	only	ADV
ejpam-5596	176	31	if	if	SCONJ
ejpam-5596	176	32	λk	λk	PRON
ejpam-5596	176	33	is	be	AUX
ejpam-5596	176	34	a	a	DET
ejpam-5596	176	35	minimal	minimal	ADJ
ejpam-5596	176	36	fuzzy	fuzzy	ADJ
ejpam-5596	176	37	(	(	PUNCT
ejpam-5596	176	38	m	m	NOUN
ejpam-5596	176	39	,	,	PUNCT
ejpam-5596	176	40	n)-ideal	n)-ideal	NOUN
ejpam-5596	176	41	.	.	PUNCT
ejpam-5596	177	1	(	(	PUNCT
ejpam-5596	177	2	2	2	X
ejpam-5596	177	3	)	)	PUNCT
ejpam-5596	177	4	k	k	X
ejpam-5596	177	5	is	be	AUX
ejpam-5596	177	6	a	a	DET
ejpam-5596	177	7	maximal	maximal	ADJ
ejpam-5596	177	8	(	(	PUNCT
ejpam-5596	177	9	m	m	PROPN
ejpam-5596	177	10	,	,	PUNCT
ejpam-5596	177	11	n)-ideal	n)-ideal	NOUN
ejpam-5596	177	12	if	if	SCONJ
ejpam-5596	177	13	and	and	CCONJ
ejpam-5596	177	14	only	only	ADV
ejpam-5596	177	15	if	if	SCONJ
ejpam-5596	177	16	λk	λk	PRON
ejpam-5596	177	17	is	be	AUX
ejpam-5596	177	18	a	a	DET
ejpam-5596	177	19	maximal	maximal	ADJ
ejpam-5596	177	20	fuzzy	fuzzy	ADJ
ejpam-5596	177	21	(	(	PUNCT
ejpam-5596	177	22	m	m	NOUN
ejpam-5596	177	23	,	,	PUNCT
ejpam-5596	177	24	n)-ideal	n)-ideal	NOUN
ejpam-5596	177	25	.	.	PUNCT
ejpam-5596	178	1	proof	proof	NOUN
ejpam-5596	178	2	.	.	PUNCT
ejpam-5596	179	1	p.	p.	NOUN
ejpam-5596	179	2	khamrot	khamrot	PROPN
ejpam-5596	179	3	,	,	PUNCT
ejpam-5596	179	4	a.	a.	NOUN
ejpam-5596	179	5	iampan	iampan	PROPN
ejpam-5596	179	6	,	,	PUNCT
ejpam-5596	179	7	t.	t.	PROPN
ejpam-5596	179	8	gaketem	gaketem	PROPN
ejpam-5596	179	9	/	/	SYM
ejpam-5596	179	10	eur	eur	PROPN
ejpam-5596	179	11	.	.	PUNCT
ejpam-5596	180	1	j.	j.	PROPN
ejpam-5596	180	2	pure	pure	PROPN
ejpam-5596	180	3	appl	appl	PROPN
ejpam-5596	180	4	.	.	PROPN
ejpam-5596	180	5	math	math	PROPN
ejpam-5596	180	6	,	,	PUNCT
ejpam-5596	180	7	18	18	NUM
ejpam-5596	180	8	(	(	PUNCT
ejpam-5596	180	9	1	1	NUM
ejpam-5596	180	10	)	)	PUNCT
ejpam-5596	180	11	(	(	PUNCT
ejpam-5596	180	12	2025	2025	NUM
ejpam-5596	180	13	)	)	PUNCT
ejpam-5596	180	14	,	,	PUNCT
ejpam-5596	180	15	5596	5596	NUM
ejpam-5596	180	16	7	7	NUM
ejpam-5596	180	17	of	of	ADP
ejpam-5596	180	18	12	12	NUM
ejpam-5596	180	19	(	(	PUNCT
ejpam-5596	180	20	1	1	NUM
ejpam-5596	180	21	)	)	PUNCT
ejpam-5596	180	22	let	let	VERB
ejpam-5596	180	23	k	k	X
ejpam-5596	180	24	be	be	AUX
ejpam-5596	180	25	a	a	DET
ejpam-5596	180	26	minimal	minimal	ADJ
ejpam-5596	180	27	(	(	PUNCT
ejpam-5596	180	28	m	m	PROPN
ejpam-5596	180	29	,	,	PUNCT
ejpam-5596	180	30	n)-ideal	n)-ideal	NOUN
ejpam-5596	180	31	of	of	ADP
ejpam-5596	180	32	s.	s.	PROPN
ejpam-5596	180	33	then	then	ADV
ejpam-5596	180	34	k	k	PROPN
ejpam-5596	180	35	is	be	AUX
ejpam-5596	180	36	an	an	DET
ejpam-5596	180	37	(	(	PUNCT
ejpam-5596	180	38	m	m	PROPN
ejpam-5596	180	39	,	,	PUNCT
ejpam-5596	180	40	n)-ideal	n)-ideal	NOUN
ejpam-5596	180	41	of	of	ADP
ejpam-5596	180	42	s.	s.	PROPN
ejpam-5596	180	43	thus	thus	ADV
ejpam-5596	180	44	,	,	PUNCT
ejpam-5596	180	45	by	by	ADP
ejpam-5596	180	46	theorem	theorem	NOUN
ejpam-5596	180	47	2	2	NUM
ejpam-5596	180	48	,	,	PUNCT
ejpam-5596	180	49	λk	λk	PRON
ejpam-5596	180	50	is	be	AUX
ejpam-5596	180	51	a	a	DET
ejpam-5596	180	52	fuzzy	fuzzy	ADJ
ejpam-5596	180	53	(	(	PUNCT
ejpam-5596	180	54	m	m	PROPN
ejpam-5596	180	55	,	,	PUNCT
ejpam-5596	180	56	n)-ideal	n)-ideal	NOUN
ejpam-5596	180	57	of	of	ADP
ejpam-5596	180	58	s.	s.	PROPN
ejpam-5596	180	59	let	let	VERB
ejpam-5596	180	60	j	j	PROPN
ejpam-5596	180	61	be	be	AUX
ejpam-5596	180	62	an	an	DET
ejpam-5596	180	63	(	(	PUNCT
ejpam-5596	180	64	m	m	PROPN
ejpam-5596	180	65	,	,	PUNCT
ejpam-5596	180	66	n)-ideal	n)-ideal	NOUN
ejpam-5596	180	67	of	of	ADP
ejpam-5596	180	68	s	s	PRON
ejpam-5596	180	69	such	such	ADJ
ejpam-5596	180	70	that	that	SCONJ
ejpam-5596	180	71	j	j	PROPN
ejpam-5596	180	72	⊆	⊆	NUM
ejpam-5596	180	73	k.	k.	PROPN
ejpam-5596	180	74	then	then	ADV
ejpam-5596	180	75	by	by	ADP
ejpam-5596	180	76	theorem	theorem	NOUN
ejpam-5596	180	77	2	2	NUM
ejpam-5596	180	78	,	,	PUNCT
ejpam-5596	180	79	λj	λj	X
ejpam-5596	180	80	is	be	AUX
ejpam-5596	180	81	a	a	DET
ejpam-5596	180	82	fuzzy	fuzzy	ADJ
ejpam-5596	180	83	(	(	PUNCT
ejpam-5596	180	84	m	m	PROPN
ejpam-5596	180	85	,	,	PUNCT
ejpam-5596	180	86	n)-ideal	n)-ideal	NOUN
ejpam-5596	180	87	of	of	ADP
ejpam-5596	180	88	s	s	PRON
ejpam-5596	180	89	and	and	CCONJ
ejpam-5596	180	90	λj	λj	PROPN
ejpam-5596	180	91	≤	≤	PROPN
ejpam-5596	181	1	λk	λk	X
ejpam-5596	181	2	.	.	PUNCT
ejpam-5596	182	1	since	since	SCONJ
ejpam-5596	182	2	k	k	PROPN
ejpam-5596	182	3	is	be	AUX
ejpam-5596	182	4	a	a	PRON
ejpam-5596	182	5	minimal	minimal	ADJ
ejpam-5596	182	6	(	(	PUNCT
ejpam-5596	182	7	m	m	PROPN
ejpam-5596	182	8	,	,	PUNCT
ejpam-5596	182	9	n)-ideal	n)-ideal	NOUN
ejpam-5596	182	10	of	of	ADP
ejpam-5596	182	11	s	s	PRON
ejpam-5596	182	12	we	we	PRON
ejpam-5596	182	13	have	have	VERB
ejpam-5596	182	14	j	j	PROPN
ejpam-5596	182	15	=	=	PROPN
ejpam-5596	182	16	k.	k.	PROPN
ejpam-5596	183	1	thus	thus	ADV
ejpam-5596	183	2	,	,	PUNCT
ejpam-5596	183	3	λj	λj	PROPN
ejpam-5596	183	4	=	=	X
ejpam-5596	183	5	λk	λk	PROPN
ejpam-5596	183	6	.	.	PUNCT
ejpam-5596	184	1	hence	hence	ADV
ejpam-5596	184	2	,	,	PUNCT
ejpam-5596	184	3	λk	λk	ADV
ejpam-5596	184	4	is	be	AUX
ejpam-5596	184	5	minimal	minimal	ADJ
ejpam-5596	184	6	fuzzy	fuzzy	ADJ
ejpam-5596	184	7	(	(	PUNCT
ejpam-5596	184	8	m	m	PROPN
ejpam-5596	184	9	,	,	PUNCT
ejpam-5596	184	10	n)-ideal	n)-ideal	NOUN
ejpam-5596	184	11	of	of	ADP
ejpam-5596	184	12	s.	s.	PROPN
ejpam-5596	184	13	conversely	conversely	ADV
ejpam-5596	184	14	,	,	PUNCT
ejpam-5596	184	15	λk	λk	PRON
ejpam-5596	184	16	is	be	AUX
ejpam-5596	184	17	minimal	minimal	ADJ
ejpam-5596	184	18	fuzzy	fuzzy	ADJ
ejpam-5596	184	19	(	(	PUNCT
ejpam-5596	184	20	m	m	PROPN
ejpam-5596	184	21	,	,	PUNCT
ejpam-5596	184	22	n)-ideal	n)-ideal	NOUN
ejpam-5596	184	23	of	of	ADP
ejpam-5596	184	24	s.	s.	PROPN
ejpam-5596	184	25	then	then	ADV
ejpam-5596	184	26	λk	λk	X
ejpam-5596	184	27	is	be	AUX
ejpam-5596	184	28	a	a	DET
ejpam-5596	184	29	fuzzy	fuzzy	ADJ
ejpam-5596	184	30	(	(	PUNCT
ejpam-5596	184	31	m	m	PROPN
ejpam-5596	184	32	,	,	PUNCT
ejpam-5596	184	33	n)-ideal	n)-ideal	NOUN
ejpam-5596	184	34	of	of	ADP
ejpam-5596	184	35	s.	s.	PROPN
ejpam-5596	184	36	thus	thus	ADV
ejpam-5596	184	37	,	,	PUNCT
ejpam-5596	184	38	by	by	ADP
ejpam-5596	184	39	theorem	theorem	NOUN
ejpam-5596	184	40	2	2	NUM
ejpam-5596	184	41	,	,	PUNCT
ejpam-5596	184	42	k	k	PROPN
ejpam-5596	184	43	is	be	AUX
ejpam-5596	184	44	an	an	DET
ejpam-5596	184	45	(	(	PUNCT
ejpam-5596	184	46	m	m	PROPN
ejpam-5596	184	47	,	,	PUNCT
ejpam-5596	184	48	n)-ideal	n)-ideal	NOUN
ejpam-5596	184	49	of	of	ADP
ejpam-5596	184	50	s.	s.	PROPN
ejpam-5596	184	51	let	let	VERB
ejpam-5596	184	52	λj	λj	PRON
ejpam-5596	184	53	be	be	AUX
ejpam-5596	184	54	a	a	DET
ejpam-5596	184	55	fuzzy	fuzzy	ADJ
ejpam-5596	184	56	(	(	PUNCT
ejpam-5596	184	57	m	m	PROPN
ejpam-5596	184	58	,	,	PUNCT
ejpam-5596	184	59	n)-ideal	n)-ideal	NOUN
ejpam-5596	184	60	of	of	ADP
ejpam-5596	184	61	s	s	PRON
ejpam-5596	184	62	such	such	ADJ
ejpam-5596	184	63	that	that	SCONJ
ejpam-5596	184	64	λj	λj	PROPN
ejpam-5596	184	65	≤	≤	X
ejpam-5596	185	1	λk	λk	X
ejpam-5596	185	2	.	.	PUNCT
ejpam-5596	186	1	then	then	ADV
ejpam-5596	186	2	by	by	ADP
ejpam-5596	186	3	theorem	theorem	NOUN
ejpam-5596	186	4	2	2	NUM
ejpam-5596	186	5	,	,	PUNCT
ejpam-5596	186	6	j	j	PROPN
ejpam-5596	186	7	is	be	AUX
ejpam-5596	186	8	an	an	DET
ejpam-5596	186	9	(	(	PUNCT
ejpam-5596	186	10	m	m	PROPN
ejpam-5596	186	11	,	,	PUNCT
ejpam-5596	186	12	n)-ideal	n)-ideal	NOUN
ejpam-5596	186	13	of	of	ADP
ejpam-5596	186	14	s	s	PRON
ejpam-5596	186	15	such	such	ADJ
ejpam-5596	186	16	that	that	SCONJ
ejpam-5596	186	17	j	j	PROPN
ejpam-5596	186	18	⊆	⊆	NUM
ejpam-5596	186	19	k.	k.	NOUN
ejpam-5596	186	20	since	since	SCONJ
ejpam-5596	186	21	λk	λk	PROPN
ejpam-5596	186	22	is	be	AUX
ejpam-5596	186	23	minimal	minimal	ADJ
ejpam-5596	186	24	fuzzy	fuzzy	ADJ
ejpam-5596	186	25	(	(	PUNCT
ejpam-5596	186	26	m	m	PROPN
ejpam-5596	186	27	,	,	PUNCT
ejpam-5596	186	28	n)-ideal	n)-ideal	NOUN
ejpam-5596	186	29	of	of	ADP
ejpam-5596	186	30	s	s	PRON
ejpam-5596	186	31	we	we	PRON
ejpam-5596	186	32	have	have	VERB
ejpam-5596	186	33	λj	λj	PROPN
ejpam-5596	186	34	=	=	SYM
ejpam-5596	186	35	λk	λk	PROPN
ejpam-5596	186	36	.	.	PUNCT
ejpam-5596	187	1	thus	thus	ADV
ejpam-5596	187	2	,	,	PUNCT
ejpam-5596	187	3	j	j	PROPN
ejpam-5596	187	4	=	=	PROPN
ejpam-5596	187	5	k.	k.	PROPN
ejpam-5596	187	6	hence	hence	ADV
ejpam-5596	187	7	,	,	PUNCT
ejpam-5596	187	8	k	k	PROPN
ejpam-5596	187	9	is	be	AUX
ejpam-5596	187	10	a	a	DET
ejpam-5596	187	11	minimal	minimal	ADJ
ejpam-5596	187	12	(	(	PUNCT
ejpam-5596	187	13	m	m	PROPN
ejpam-5596	187	14	,	,	PUNCT
ejpam-5596	187	15	n)-ideal	n)-ideal	NOUN
ejpam-5596	187	16	of	of	ADP
ejpam-5596	187	17	s.	s.	PROPN
ejpam-5596	187	18	(	(	PUNCT
ejpam-5596	187	19	2	2	X
ejpam-5596	187	20	)	)	PUNCT
ejpam-5596	187	21	if	if	SCONJ
ejpam-5596	187	22	follows	follow	VERB
ejpam-5596	187	23	from	from	ADP
ejpam-5596	187	24	(	(	PUNCT
ejpam-5596	187	25	1	1	NUM
ejpam-5596	187	26	)	)	PUNCT
ejpam-5596	187	27	.	.	PUNCT
ejpam-5596	188	1	next	next	ADV
ejpam-5596	188	2	,	,	PUNCT
ejpam-5596	188	3	we	we	PRON
ejpam-5596	188	4	give	give	VERB
ejpam-5596	188	5	the	the	DET
ejpam-5596	188	6	relationship	relationship	NOUN
ejpam-5596	188	7	between	between	ADP
ejpam-5596	188	8	prime	prime	ADJ
ejpam-5596	188	9	,	,	PUNCT
ejpam-5596	188	10	semiprime	semiprime	NOUN
ejpam-5596	188	11	(	(	PUNCT
ejpam-5596	188	12	m	m	NOUN
ejpam-5596	188	13	,	,	PUNCT
ejpam-5596	188	14	n)-ideals	n)-ideal	NOUN
ejpam-5596	188	15	and	and	CCONJ
ejpam-5596	188	16	prime	prime	ADJ
ejpam-5596	188	17	,	,	PUNCT
ejpam-5596	188	18	semiprime	semiprime	NOUN
ejpam-5596	188	19	fuzzy	fuzzy	ADJ
ejpam-5596	188	20	(	(	PUNCT
ejpam-5596	188	21	m	m	NOUN
ejpam-5596	188	22	,	,	PUNCT
ejpam-5596	188	23	n)-ideals	n)-ideal	NOUN
ejpam-5596	188	24	.	.	PUNCT
ejpam-5596	189	1	definition	definition	NOUN
ejpam-5596	189	2	9	9	NUM
ejpam-5596	189	3	.	.	PUNCT
ejpam-5596	190	1	let	let	VERB
ejpam-5596	190	2	k	k	PRON
ejpam-5596	190	3	be	be	AUX
ejpam-5596	190	4	an	an	DET
ejpam-5596	190	5	(	(	PUNCT
ejpam-5596	190	6	m	m	PROPN
ejpam-5596	190	7	,	,	PUNCT
ejpam-5596	190	8	n)-ideal	n)-ideal	NOUN
ejpam-5596	190	9	of	of	ADP
ejpam-5596	190	10	an	an	DET
ejpam-5596	190	11	ordered	order	VERB
ejpam-5596	190	12	semigroup	semigroup	NOUN
ejpam-5596	190	13	s	s	VERB
ejpam-5596	190	14	is	be	AUX
ejpam-5596	190	15	called	call	VERB
ejpam-5596	190	16	(	(	PUNCT
ejpam-5596	190	17	1	1	NUM
ejpam-5596	190	18	)	)	PUNCT
ejpam-5596	190	19	prime	prime	NOUN
ejpam-5596	190	20	if	if	SCONJ
ejpam-5596	190	21	uv	uv	PROPN
ejpam-5596	190	22	∈	∈	PROPN
ejpam-5596	190	23	k	k	PROPN
ejpam-5596	190	24	implies	imply	VERB
ejpam-5596	190	25	u	u	PROPN
ejpam-5596	190	26	∈	∈	PROPN
ejpam-5596	190	27	k	k	PROPN
ejpam-5596	190	28	or	or	CCONJ
ejpam-5596	190	29	v	v	ADP
ejpam-5596	190	30	∈	∈	PROPN
ejpam-5596	190	31	k	k	NOUN
ejpam-5596	190	32	for	for	ADP
ejpam-5596	190	33	all	all	DET
ejpam-5596	190	34	e	e	NOUN
ejpam-5596	190	35	,	,	PUNCT
ejpam-5596	190	36	h	h	NOUN
ejpam-5596	190	37	∈	∈	PROPN
ejpam-5596	190	38	s	s	PROPN
ejpam-5596	190	39	,	,	PUNCT
ejpam-5596	190	40	(	(	PUNCT
ejpam-5596	190	41	2	2	NUM
ejpam-5596	190	42	)	)	PUNCT
ejpam-5596	190	43	semiprime	semiprime	NOUN
ejpam-5596	190	44	if	if	SCONJ
ejpam-5596	190	45	u2	u2	PROPN
ejpam-5596	190	46	∈	∈	PROPN
ejpam-5596	190	47	k	k	PROPN
ejpam-5596	190	48	implies	imply	VERB
ejpam-5596	190	49	u	u	PROPN
ejpam-5596	190	50	∈	∈	PROPN
ejpam-5596	190	51	k	k	PROPN
ejpam-5596	190	52	for	for	ADP
ejpam-5596	190	53	all	all	DET
ejpam-5596	190	54	u	u	PROPN
ejpam-5596	190	55	∈	∈	PROPN
ejpam-5596	190	56	s.	s.	PROPN
ejpam-5596	190	57	definition	definition	NOUN
ejpam-5596	190	58	10	10	NUM
ejpam-5596	190	59	.	.	PUNCT
ejpam-5596	191	1	let	let	VERB
ejpam-5596	191	2	δ	δ	PRON
ejpam-5596	191	3	be	be	AUX
ejpam-5596	191	4	a	a	DET
ejpam-5596	191	5	fuzzy	fuzzy	ADJ
ejpam-5596	191	6	(	(	PUNCT
ejpam-5596	191	7	m	m	PROPN
ejpam-5596	191	8	,	,	PUNCT
ejpam-5596	191	9	n)-ideal	n)-ideal	NOUN
ejpam-5596	191	10	of	of	ADP
ejpam-5596	191	11	a	a	DET
ejpam-5596	191	12	ordered	order	VERB
ejpam-5596	191	13	semigroup	semigroup	NOUN
ejpam-5596	191	14	is	be	AUX
ejpam-5596	191	15	called	call	VERB
ejpam-5596	191	16	(	(	PUNCT
ejpam-5596	191	17	1	1	NUM
ejpam-5596	191	18	)	)	PUNCT
ejpam-5596	191	19	prime	prime	NOUN
ejpam-5596	191	20	if	if	SCONJ
ejpam-5596	191	21	δ(uv	δ(uv	NOUN
ejpam-5596	191	22	)	)	PUNCT
ejpam-5596	191	23	≤	≤	NOUN
ejpam-5596	191	24	δ(u	δ(u	PROPN
ejpam-5596	191	25	)	)	PUNCT
ejpam-5596	191	26	∨	∨	PROPN
ejpam-5596	191	27	δ(v	δ(v	PROPN
ejpam-5596	191	28	)	)	PUNCT
ejpam-5596	191	29	for	for	ADP
ejpam-5596	191	30	all	all	DET
ejpam-5596	191	31	u	u	NOUN
ejpam-5596	191	32	,	,	PUNCT
ejpam-5596	191	33	v	v	ADP
ejpam-5596	191	34	∈	∈	PROPN
ejpam-5596	191	35	s	s	NOUN
ejpam-5596	191	36	,	,	PUNCT
ejpam-5596	191	37	(	(	PUNCT
ejpam-5596	191	38	2	2	NUM
ejpam-5596	191	39	)	)	PUNCT
ejpam-5596	191	40	semiprime	semiprime	NOUN
ejpam-5596	191	41	if	if	SCONJ
ejpam-5596	191	42	δ(u2	δ(u2	NOUN
ejpam-5596	191	43	)	)	PUNCT
ejpam-5596	191	44	≤	≤	NOUN
ejpam-5596	192	1	δ(u	δ(u	PROPN
ejpam-5596	192	2	)	)	PUNCT
ejpam-5596	192	3	for	for	ADP
ejpam-5596	192	4	all	all	DET
ejpam-5596	192	5	u	u	PROPN
ejpam-5596	192	6	∈	∈	PROPN
ejpam-5596	192	7	s.	s.	PROPN
ejpam-5596	192	8	remark	remark	VERB
ejpam-5596	192	9	1	1	NUM
ejpam-5596	192	10	.	.	PUNCT
ejpam-5596	193	1	every	every	DET
ejpam-5596	193	2	prime	prime	NOUN
ejpam-5596	193	3	(	(	PUNCT
ejpam-5596	193	4	m	m	PROPN
ejpam-5596	193	5	,	,	PUNCT
ejpam-5596	193	6	n)-ideal	n)-ideal	PROPN
ejpam-5596	193	7	is	be	AUX
ejpam-5596	193	8	semiprime	semiprime	NOUN
ejpam-5596	193	9	(	(	PUNCT
ejpam-5596	193	10	m	m	PROPN
ejpam-5596	193	11	,	,	PUNCT
ejpam-5596	193	12	n)-ideal	n)-ideal	NOUN
ejpam-5596	193	13	in	in	ADP
ejpam-5596	193	14	an	an	DET
ejpam-5596	193	15	ordered	order	VERB
ejpam-5596	193	16	semigroup	semigroup	NOUN
ejpam-5596	193	17	.	.	PUNCT
ejpam-5596	194	1	theorem	theorem	NOUN
ejpam-5596	194	2	5	5	NUM
ejpam-5596	194	3	.	.	PUNCT
ejpam-5596	195	1	let	let	VERB
ejpam-5596	195	2	k	k	PRON
ejpam-5596	195	3	be	be	AUX
ejpam-5596	195	4	a	a	DET
ejpam-5596	195	5	non	non	ADJ
ejpam-5596	195	6	-	-	ADJ
ejpam-5596	195	7	empty	empty	ADJ
ejpam-5596	195	8	subset	subset	NOUN
ejpam-5596	195	9	of	of	ADP
ejpam-5596	195	10	an	an	DET
ejpam-5596	195	11	ordered	order	VERB
ejpam-5596	195	12	semigroup	semigroup	PROPN
ejpam-5596	195	13	s.	s.	PROPN
ejpam-5596	196	1	then	then	ADV
ejpam-5596	196	2	k	k	PROPN
ejpam-5596	196	3	is	be	AUX
ejpam-5596	196	4	a	a	DET
ejpam-5596	196	5	prime	prime	ADJ
ejpam-5596	196	6	(	(	PUNCT
ejpam-5596	196	7	m	m	PROPN
ejpam-5596	196	8	,	,	PUNCT
ejpam-5596	196	9	n)-ideal	n)-ideal	NOUN
ejpam-5596	196	10	of	of	ADP
ejpam-5596	196	11	s	s	PRON
ejpam-5596	196	12	if	if	SCONJ
ejpam-5596	196	13	and	and	CCONJ
ejpam-5596	196	14	only	only	ADV
ejpam-5596	196	15	if	if	SCONJ
ejpam-5596	196	16	λk	λk	PRON
ejpam-5596	196	17	is	be	AUX
ejpam-5596	196	18	a	a	DET
ejpam-5596	196	19	prime	prime	ADJ
ejpam-5596	196	20	fuzzy	fuzzy	NOUN
ejpam-5596	196	21	(	(	PUNCT
ejpam-5596	196	22	m	m	PROPN
ejpam-5596	196	23	,	,	PUNCT
ejpam-5596	196	24	n)-ideal	n)-ideal	NOUN
ejpam-5596	196	25	of	of	ADP
ejpam-5596	196	26	s.	s.	PROPN
ejpam-5596	196	27	proof	proof	PROPN
ejpam-5596	196	28	.	.	PUNCT
ejpam-5596	197	1	suppose	suppose	VERB
ejpam-5596	197	2	that	that	SCONJ
ejpam-5596	197	3	k	k	PROPN
ejpam-5596	197	4	is	be	AUX
ejpam-5596	197	5	a	a	DET
ejpam-5596	197	6	prime	prime	ADJ
ejpam-5596	197	7	(	(	PUNCT
ejpam-5596	197	8	m	m	PROPN
ejpam-5596	197	9	,	,	PUNCT
ejpam-5596	197	10	n)-ideal	n)-ideal	NOUN
ejpam-5596	197	11	of	of	ADP
ejpam-5596	197	12	s.	s.	PROPN
ejpam-5596	197	13	then	then	ADV
ejpam-5596	197	14	k	k	PROPN
ejpam-5596	197	15	is	be	AUX
ejpam-5596	197	16	an	an	DET
ejpam-5596	197	17	(	(	PUNCT
ejpam-5596	197	18	m	m	PROPN
ejpam-5596	197	19	,	,	PUNCT
ejpam-5596	197	20	n)-ideal	n)-ideal	NOUN
ejpam-5596	197	21	of	of	ADP
ejpam-5596	197	22	s.	s.	PROPN
ejpam-5596	197	23	thus	thus	ADV
ejpam-5596	197	24	,	,	PUNCT
ejpam-5596	197	25	by	by	SCONJ
ejpam-5596	197	26	theorem	theorem	NOUN
ejpam-5596	197	27	2	2	NUM
ejpam-5596	197	28	λk	λk	NOUN
ejpam-5596	197	29	is	be	AUX
ejpam-5596	197	30	a	a	DET
ejpam-5596	197	31	fuzzy	fuzzy	ADJ
ejpam-5596	197	32	(	(	PUNCT
ejpam-5596	197	33	m	m	PROPN
ejpam-5596	197	34	,	,	PUNCT
ejpam-5596	197	35	n)-ideal	n)-ideal	NOUN
ejpam-5596	197	36	of	of	ADP
ejpam-5596	197	37	s.	s.	PROPN
ejpam-5596	197	38	let	let	VERB
ejpam-5596	197	39	u	u	NOUN
ejpam-5596	197	40	,	,	PUNCT
ejpam-5596	197	41	v	v	PROPN
ejpam-5596	197	42	∈	∈	PROPN
ejpam-5596	197	43	s.	s.	PROPN
ejpam-5596	197	44	case	case	NOUN
ejpam-5596	197	45	1	1	NUM
ejpam-5596	197	46	:	:	PUNCT
ejpam-5596	198	1	if	if	SCONJ
ejpam-5596	198	2	uv	uv	PROPN
ejpam-5596	198	3	∈	∈	PROPN
ejpam-5596	198	4	k	k	NOUN
ejpam-5596	198	5	,	,	PUNCT
ejpam-5596	198	6	then	then	ADV
ejpam-5596	198	7	u	u	PROPN
ejpam-5596	198	8	∈	∈	PROPN
ejpam-5596	198	9	k	k	NOUN
ejpam-5596	198	10	or	or	CCONJ
ejpam-5596	198	11	v	v	ADP
ejpam-5596	198	12	∈	∈	PROPN
ejpam-5596	198	13	k.	k.	PROPN
ejpam-5596	199	1	thus	thus	ADV
ejpam-5596	199	2	,	,	PUNCT
ejpam-5596	199	3	λk(uv	λk(uv	NOUN
ejpam-5596	199	4	)	)	PUNCT
ejpam-5596	199	5	=	=	SYM
ejpam-5596	199	6	1	1	NUM
ejpam-5596	199	7	=	=	PUNCT
ejpam-5596	199	8	λk(u	λk(u	X
ejpam-5596	199	9	)	)	PUNCT
ejpam-5596	199	10	and	and	CCONJ
ejpam-5596	199	11	λk(uv	λk(uv	NOUN
ejpam-5596	199	12	)	)	PUNCT
ejpam-5596	199	13	=	=	SYM
ejpam-5596	199	14	−1	−1	NOUN
ejpam-5596	199	15	=	=	PUNCT
ejpam-5596	199	16	λk(u	λk(u	X
ejpam-5596	199	17	)	)	PUNCT
ejpam-5596	199	18	or	or	CCONJ
ejpam-5596	199	19	λk(v	λk(v	NUM
ejpam-5596	199	20	)	)	PUNCT
ejpam-5596	200	1	=	=	SYM
ejpam-5596	200	2	1	1	NUM
ejpam-5596	200	3	=	=	SYM
ejpam-5596	200	4	λk(uv	λk(uv	NOUN
ejpam-5596	200	5	)	)	PUNCT
ejpam-5596	200	6	.	.	PUNCT
ejpam-5596	201	1	hence	hence	ADV
ejpam-5596	201	2	,	,	PUNCT
ejpam-5596	201	3	λk(uv	λk(uv	NOUN
ejpam-5596	201	4	)	)	PUNCT
ejpam-5596	201	5	≤	≤	NOUN
ejpam-5596	201	6	λk(u	λk(u	PUNCT
ejpam-5596	201	7	)	)	PUNCT
ejpam-5596	201	8	∨	∨	NUM
ejpam-5596	201	9	λk(v	λk(v	NOUN
ejpam-5596	201	10	)	)	PUNCT
ejpam-5596	201	11	.	.	PUNCT
ejpam-5596	202	1	case	case	NOUN
ejpam-5596	202	2	2	2	NUM
ejpam-5596	202	3	:	:	PUNCT
ejpam-5596	202	4	if	if	SCONJ
ejpam-5596	202	5	uv	uv	NOUN
ejpam-5596	202	6	/∈	/∈	PUNCT
ejpam-5596	203	1	k	k	NOUN
ejpam-5596	203	2	,	,	PUNCT
ejpam-5596	203	3	then	then	ADV
ejpam-5596	203	4	λk(uv	λk(uv	NOUN
ejpam-5596	203	5	)	)	PUNCT
ejpam-5596	204	1	=	=	SYM
ejpam-5596	204	2	0	0	X
ejpam-5596	204	3	.	.	PUNCT
ejpam-5596	205	1	thus	thus	ADV
ejpam-5596	205	2	,	,	PUNCT
ejpam-5596	205	3	λk(uv	λk(uv	NOUN
ejpam-5596	205	4	)	)	PUNCT
ejpam-5596	205	5	≤	≤	NOUN
ejpam-5596	205	6	λk(u	λk(u	PUNCT
ejpam-5596	205	7	)	)	PUNCT
ejpam-5596	205	8	∨	∨	NUM
ejpam-5596	205	9	λk(v	λk(v	NOUN
ejpam-5596	205	10	)	)	PUNCT
ejpam-5596	205	11	.	.	PUNCT
ejpam-5596	206	1	therefore	therefore	ADV
ejpam-5596	206	2	,	,	PUNCT
ejpam-5596	206	3	λk	λk	PRON
ejpam-5596	206	4	is	be	AUX
ejpam-5596	206	5	a	a	DET
ejpam-5596	206	6	prime	prime	ADJ
ejpam-5596	206	7	fuzzy	fuzzy	NOUN
ejpam-5596	206	8	(	(	PUNCT
ejpam-5596	206	9	m	m	PROPN
ejpam-5596	206	10	,	,	PUNCT
ejpam-5596	206	11	n)-ideal	n)-ideal	NOUN
ejpam-5596	206	12	of	of	ADP
ejpam-5596	206	13	s.	s.	PROPN
ejpam-5596	206	14	conversely	conversely	ADV
ejpam-5596	206	15	,	,	PUNCT
ejpam-5596	206	16	suppose	suppose	VERB
ejpam-5596	206	17	that	that	SCONJ
ejpam-5596	206	18	λk	λk	PROPN
ejpam-5596	206	19	is	be	AUX
ejpam-5596	206	20	a	a	DET
ejpam-5596	206	21	prime	prime	ADJ
ejpam-5596	206	22	fuzzy	fuzzy	NOUN
ejpam-5596	206	23	(	(	PUNCT
ejpam-5596	206	24	m	m	PROPN
ejpam-5596	206	25	,	,	PUNCT
ejpam-5596	206	26	n)-ideal	n)-ideal	NOUN
ejpam-5596	206	27	of	of	ADP
ejpam-5596	206	28	s.	s.	PROPN
ejpam-5596	206	29	then	then	ADV
ejpam-5596	206	30	λk	λk	X
ejpam-5596	206	31	is	be	AUX
ejpam-5596	206	32	a	a	DET
ejpam-5596	206	33	fuzzy	fuzzy	ADJ
ejpam-5596	206	34	(	(	PUNCT
ejpam-5596	206	35	m	m	PROPN
ejpam-5596	206	36	,	,	PUNCT
ejpam-5596	206	37	n)-ideal	n)-ideal	NOUN
ejpam-5596	206	38	of	of	ADP
ejpam-5596	206	39	s.	s.	PROPN
ejpam-5596	206	40	thus	thus	ADV
ejpam-5596	206	41	,	,	PUNCT
ejpam-5596	206	42	by	by	ADP
ejpam-5596	206	43	theorem	theorem	NOUN
ejpam-5596	206	44	2	2	NUM
ejpam-5596	206	45	,	,	PUNCT
ejpam-5596	206	46	k	k	PROPN
ejpam-5596	206	47	is	be	AUX
ejpam-5596	206	48	an	an	DET
ejpam-5596	206	49	(	(	PUNCT
ejpam-5596	206	50	m	m	PROPN
ejpam-5596	206	51	,	,	PUNCT
ejpam-5596	206	52	n)-ideal	n)-ideal	NOUN
ejpam-5596	206	53	of	of	ADP
ejpam-5596	206	54	s.	s.	PROPN
ejpam-5596	206	55	let	let	VERB
ejpam-5596	206	56	u	u	NOUN
ejpam-5596	206	57	,	,	PUNCT
ejpam-5596	206	58	v	v	PROPN
ejpam-5596	206	59	∈	∈	NOUN
ejpam-5596	206	60	s	s	PART
ejpam-5596	206	61	with	with	ADP
ejpam-5596	206	62	uv	uv	PROPN
ejpam-5596	206	63	∈	∈	PROPN
ejpam-5596	206	64	k.	k.	NOUN
ejpam-5596	207	1	then	then	ADV
ejpam-5596	207	2	,	,	PUNCT
ejpam-5596	207	3	λk(uv	λk(uv	NOUN
ejpam-5596	207	4	)	)	PUNCT
ejpam-5596	207	5	=	=	SYM
ejpam-5596	208	1	1	1	X
ejpam-5596	208	2	.	.	PUNCT
ejpam-5596	209	1	if	if	SCONJ
ejpam-5596	209	2	u	u	PROPN
ejpam-5596	209	3	/∈	/∈	PROPN
ejpam-5596	209	4	k	k	PROPN
ejpam-5596	209	5	and	and	CCONJ
ejpam-5596	209	6	v	v	NOUN
ejpam-5596	209	7	/∈	/∈	PUNCT
ejpam-5596	210	1	k	k	NOUN
ejpam-5596	210	2	,	,	PUNCT
ejpam-5596	210	3	then	then	ADV
ejpam-5596	210	4	λk(u	λk(u	PUNCT
ejpam-5596	210	5	)	)	PUNCT
ejpam-5596	210	6	=	=	SYM
ejpam-5596	210	7	0	0	PUNCT
ejpam-5596	210	8	=	=	SYM
ejpam-5596	210	9	λk(v	λk(v	PROPN
ejpam-5596	210	10	)	)	PUNCT
ejpam-5596	210	11	.	.	PUNCT
ejpam-5596	211	1	by	by	ADP
ejpam-5596	211	2	assumption	assumption	NOUN
ejpam-5596	211	3	,	,	PUNCT
ejpam-5596	211	4	λk(uv	λk(uv	NOUN
ejpam-5596	211	5	)	)	PUNCT
ejpam-5596	211	6	≤	≤	NOUN
ejpam-5596	211	7	λk(u	λk(u	PUNCT
ejpam-5596	211	8	)	)	PUNCT
ejpam-5596	211	9	∨	∨	NUM
ejpam-5596	211	10	λk(v	λk(v	NOUN
ejpam-5596	211	11	)	)	PUNCT
ejpam-5596	211	12	.	.	PUNCT
ejpam-5596	212	1	thus	thus	ADV
ejpam-5596	212	2	,	,	PUNCT
ejpam-5596	212	3	λk(uv	λk(uv	NOUN
ejpam-5596	212	4	)	)	PUNCT
ejpam-5596	212	5	=	=	SYM
ejpam-5596	213	1	0	0	X
ejpam-5596	213	2	.	.	PUNCT
ejpam-5596	214	1	it	it	PRON
ejpam-5596	214	2	is	be	AUX
ejpam-5596	214	3	a	a	DET
ejpam-5596	214	4	contradiction	contradiction	NOUN
ejpam-5596	214	5	,	,	PUNCT
ejpam-5596	214	6	so	so	CCONJ
ejpam-5596	214	7	u	u	PROPN
ejpam-5596	214	8	∈	∈	PROPN
ejpam-5596	214	9	k	k	NOUN
ejpam-5596	214	10	or	or	CCONJ
ejpam-5596	214	11	v	v	ADP
ejpam-5596	214	12	∈	∈	PROPN
ejpam-5596	214	13	k.	k.	NOUN
ejpam-5596	215	1	hence	hence	ADV
ejpam-5596	215	2	,	,	PUNCT
ejpam-5596	215	3	k	k	PROPN
ejpam-5596	215	4	is	be	AUX
ejpam-5596	215	5	a	a	DET
ejpam-5596	215	6	prime	prime	ADJ
ejpam-5596	215	7	(	(	PUNCT
ejpam-5596	215	8	m	m	PROPN
ejpam-5596	215	9	,	,	PUNCT
ejpam-5596	215	10	n)-ideal	n)-ideal	NOUN
ejpam-5596	215	11	of	of	ADP
ejpam-5596	215	12	s.	s.	PROPN
ejpam-5596	215	13	theorem	theorem	VERB
ejpam-5596	215	14	6	6	NUM
ejpam-5596	215	15	.	.	PUNCT
ejpam-5596	216	1	let	let	VERB
ejpam-5596	216	2	k	k	PRON
ejpam-5596	216	3	be	be	AUX
ejpam-5596	216	4	a	a	DET
ejpam-5596	216	5	non	non	ADJ
ejpam-5596	216	6	-	-	ADJ
ejpam-5596	216	7	empty	empty	ADJ
ejpam-5596	216	8	subset	subset	NOUN
ejpam-5596	216	9	of	of	ADP
ejpam-5596	216	10	an	an	DET
ejpam-5596	216	11	ordered	order	VERB
ejpam-5596	216	12	semigroup	semigroup	PROPN
ejpam-5596	216	13	s.	s.	PROPN
ejpam-5596	217	1	then	then	ADV
ejpam-5596	217	2	k	k	PROPN
ejpam-5596	217	3	is	be	AUX
ejpam-5596	217	4	a	a	DET
ejpam-5596	217	5	semiprime	semiprime	NOUN
ejpam-5596	217	6	(	(	PUNCT
ejpam-5596	217	7	m	m	PROPN
ejpam-5596	217	8	,	,	PUNCT
ejpam-5596	217	9	n)-ideal	n)-ideal	NOUN
ejpam-5596	217	10	of	of	ADP
ejpam-5596	217	11	k	k	PROPN
ejpam-5596	217	12	if	if	SCONJ
ejpam-5596	218	1	and	and	CCONJ
ejpam-5596	218	2	only	only	ADV
ejpam-5596	218	3	if	if	SCONJ
ejpam-5596	218	4	λk	λk	PRON
ejpam-5596	218	5	is	be	AUX
ejpam-5596	218	6	a	a	DET
ejpam-5596	218	7	semiprime	semiprime	NOUN
ejpam-5596	218	8	fuzzy	fuzzy	ADJ
ejpam-5596	218	9	(	(	PUNCT
ejpam-5596	218	10	m	m	PROPN
ejpam-5596	218	11	,	,	PUNCT
ejpam-5596	218	12	n)-ideal	n)-ideal	NOUN
ejpam-5596	218	13	of	of	ADP
ejpam-5596	218	14	s.	s.	PROPN
ejpam-5596	218	15	proof	proof	PROPN
ejpam-5596	218	16	.	.	PUNCT
ejpam-5596	219	1	it	it	PRON
ejpam-5596	219	2	follows	follow	VERB
ejpam-5596	219	3	from	from	ADP
ejpam-5596	219	4	theorem	theorem	ADJ
ejpam-5596	219	5	5	5	NUM
ejpam-5596	219	6	.	.	PUNCT
ejpam-5596	220	1	p.	p.	NOUN
ejpam-5596	220	2	khamrot	khamrot	PROPN
ejpam-5596	220	3	,	,	PUNCT
ejpam-5596	220	4	a.	a.	NOUN
ejpam-5596	220	5	iampan	iampan	PROPN
ejpam-5596	220	6	,	,	PUNCT
ejpam-5596	220	7	t.	t.	PROPN
ejpam-5596	220	8	gaketem	gaketem	PROPN
ejpam-5596	220	9	/	/	SYM
ejpam-5596	220	10	eur	eur	PROPN
ejpam-5596	220	11	.	.	PUNCT
ejpam-5596	221	1	j.	j.	PROPN
ejpam-5596	221	2	pure	pure	PROPN
ejpam-5596	221	3	appl	appl	PROPN
ejpam-5596	221	4	.	.	PROPN
ejpam-5596	221	5	math	math	PROPN
ejpam-5596	221	6	,	,	PUNCT
ejpam-5596	221	7	18	18	NUM
ejpam-5596	221	8	(	(	PUNCT
ejpam-5596	221	9	1	1	NUM
ejpam-5596	221	10	)	)	PUNCT
ejpam-5596	221	11	(	(	PUNCT
ejpam-5596	221	12	2025	2025	NUM
ejpam-5596	221	13	)	)	PUNCT
ejpam-5596	221	14	,	,	PUNCT
ejpam-5596	221	15	5596	5596	NUM
ejpam-5596	221	16	8	8	NUM
ejpam-5596	221	17	of	of	ADP
ejpam-5596	221	18	12	12	NUM
ejpam-5596	221	19	4	4	NUM
ejpam-5596	221	20	.	.	NOUN
ejpam-5596	221	21	fuzzy	fuzzy	ADJ
ejpam-5596	221	22	n	n	CCONJ
ejpam-5596	221	23	-	-	PUNCT
ejpam-5596	221	24	interior	interior	ADJ
ejpam-5596	221	25	ideals	ideal	NOUN
ejpam-5596	221	26	before	before	ADV
ejpam-5596	221	27	,	,	PUNCT
ejpam-5596	221	28	we	we	PRON
ejpam-5596	221	29	will	will	AUX
ejpam-5596	221	30	review	review	VERB
ejpam-5596	221	31	the	the	DET
ejpam-5596	221	32	definition	definition	NOUN
ejpam-5596	221	33	of	of	ADP
ejpam-5596	221	34	n	n	CCONJ
ejpam-5596	221	35	-	-	PUNCT
ejpam-5596	221	36	interior	interior	ADJ
ejpam-5596	221	37	ideals	ideal	NOUN
ejpam-5596	221	38	in	in	ADP
ejpam-5596	221	39	ordered	order	VERB
ejpam-5596	221	40	semigroups	semigroup	NOUN
ejpam-5596	221	41	.	.	PUNCT
ejpam-5596	222	1	definition	definition	NOUN
ejpam-5596	222	2	11	11	NUM
ejpam-5596	222	3	.	.	PUNCT
ejpam-5596	223	1	[	[	X
ejpam-5596	223	2	12	12	NUM
ejpam-5596	223	3	]	]	PUNCT
ejpam-5596	223	4	a	a	DET
ejpam-5596	223	5	subsemigroup	subsemigroup	NOUN
ejpam-5596	223	6	k	k	PROPN
ejpam-5596	223	7	of	of	ADP
ejpam-5596	223	8	an	an	DET
ejpam-5596	223	9	ordered	order	VERB
ejpam-5596	223	10	semigroup	semigroup	NOUN
ejpam-5596	223	11	s	s	VERB
ejpam-5596	223	12	is	be	AUX
ejpam-5596	223	13	said	say	VERB
ejpam-5596	223	14	to	to	PART
ejpam-5596	223	15	be	be	AUX
ejpam-5596	223	16	an	an	DET
ejpam-5596	223	17	ninterior	ninterior	ADJ
ejpam-5596	223	18	ideal	ideal	NOUN
ejpam-5596	223	19	of	of	ADP
ejpam-5596	223	20	s	s	PRON
ejpam-5596	223	21	if	if	SCONJ
ejpam-5596	223	22	skns	skn	VERB
ejpam-5596	223	23	⊆	⊆	NUM
ejpam-5596	223	24	k	k	NOUN
ejpam-5596	223	25	where	where	SCONJ
ejpam-5596	223	26	n	n	PRON
ejpam-5596	223	27	is	be	AUX
ejpam-5596	223	28	an	an	DET
ejpam-5596	223	29	integer	integer	NOUN
ejpam-5596	223	30	and	and	CCONJ
ejpam-5596	223	31	k	k	NOUN
ejpam-5596	223	32	=	=	PUNCT
ejpam-5596	223	33	(	(	PUNCT
ejpam-5596	223	34	k	k	X
ejpam-5596	223	35	]	]	X
ejpam-5596	223	36	,	,	PUNCT
ejpam-5596	223	37	that	that	PRON
ejpam-5596	223	38	is	be	AUX
ejpam-5596	223	39	for	for	ADP
ejpam-5596	223	40	x	x	PROPN
ejpam-5596	223	41	∈	∈	PROPN
ejpam-5596	223	42	k	k	PROPN
ejpam-5596	223	43	and	and	CCONJ
ejpam-5596	223	44	y	y	PROPN
ejpam-5596	223	45	∈	∈	PROPN
ejpam-5596	223	46	s	s	PROPN
ejpam-5596	223	47	,	,	PUNCT
ejpam-5596	223	48	y	y	PROPN
ejpam-5596	223	49	≤	≤	NUM
ejpam-5596	223	50	x	x	PUNCT
ejpam-5596	223	51	implies	imply	VERB
ejpam-5596	223	52	y	y	PROPN
ejpam-5596	223	53	∈	∈	PROPN
ejpam-5596	223	54	k.	k.	PROPN
ejpam-5596	223	55	definition	definition	NOUN
ejpam-5596	223	56	12	12	NUM
ejpam-5596	223	57	.	.	PUNCT
ejpam-5596	224	1	a	a	DET
ejpam-5596	224	2	fuzzy	fuzzy	ADJ
ejpam-5596	224	3	subsemigroup	subsemigroup	PROPN
ejpam-5596	224	4	δ	δ	PROPN
ejpam-5596	224	5	in	in	ADP
ejpam-5596	224	6	an	an	DET
ejpam-5596	224	7	ordered	order	VERB
ejpam-5596	224	8	semigroup	semigroup	NOUN
ejpam-5596	224	9	s	s	VERB
ejpam-5596	224	10	is	be	AUX
ejpam-5596	224	11	called	call	VERB
ejpam-5596	224	12	fuzzy	fuzzy	ADJ
ejpam-5596	224	13	ninterior	ninterior	ADJ
ejpam-5596	224	14	ideal	ideal	NOUN
ejpam-5596	224	15	of	of	ADP
ejpam-5596	224	16	s	s	PRON
ejpam-5596	224	17	if	if	SCONJ
ejpam-5596	224	18	(	(	PUNCT
ejpam-5596	224	19	1	1	X
ejpam-5596	224	20	)	)	PUNCT
ejpam-5596	224	21	δ(ukni	δ(ukni	NOUN
ejpam-5596	224	22	v	v	NOUN
ejpam-5596	224	23	)	)	PUNCT
ejpam-5596	224	24	≥	≥	NOUN
ejpam-5596	224	25	δ(k1	δ(k1	NOUN
ejpam-5596	224	26	)	)	PUNCT
ejpam-5596	224	27	∧	∧	PROPN
ejpam-5596	224	28	δ(k2	δ(k2	NOUN
ejpam-5596	224	29	)	)	PUNCT
ejpam-5596	224	30	∧	∧	PROPN
ejpam-5596	224	31	·	·	PUNCT
ejpam-5596	224	32	·	·	PUNCT
ejpam-5596	224	33	·	·	PUNCT
ejpam-5596	225	1	∧	∧	PROPN
ejpam-5596	225	2	δ(kn	δ(kn	NOUN
ejpam-5596	225	3	)	)	PUNCT
ejpam-5596	225	4	(	(	PUNCT
ejpam-5596	225	5	2	2	X
ejpam-5596	225	6	)	)	PUNCT
ejpam-5596	225	7	u	u	NOUN
ejpam-5596	225	8	≤	≤	PROPN
ejpam-5596	225	9	v	v	NOUN
ejpam-5596	225	10	imples	imple	VERB
ejpam-5596	225	11	δ(u	δ(u	PROPN
ejpam-5596	225	12	)	)	PUNCT
ejpam-5596	225	13	≥	≥	NOUN
ejpam-5596	225	14	δ(v	δ(v	PROPN
ejpam-5596	225	15	)	)	PUNCT
ejpam-5596	225	16	for	for	ADP
ejpam-5596	225	17	all	all	DET
ejpam-5596	225	18	u	u	NOUN
ejpam-5596	225	19	,	,	PUNCT
ejpam-5596	225	20	kni	kni	INTJ
ejpam-5596	225	21	,	,	PUNCT
ejpam-5596	225	22	v	v	NUM
ejpam-5596	225	23	∈	∈	NOUN
ejpam-5596	225	24	s	s	X
ejpam-5596	225	25	and	and	CCONJ
ejpam-5596	225	26	where	where	SCONJ
ejpam-5596	225	27	i	i	PRON
ejpam-5596	225	28	∈	∈	PROPN
ejpam-5596	225	29	{	{	PUNCT
ejpam-5596	225	30	1	1	NUM
ejpam-5596	225	31	,	,	PUNCT
ejpam-5596	225	32	2	2	NUM
ejpam-5596	225	33	,	,	PUNCT
ejpam-5596	225	34	.	.	PUNCT
ejpam-5596	225	35	.	.	PUNCT
ejpam-5596	226	1	.	.	PUNCT
ejpam-5596	226	2	,	,	PUNCT
ejpam-5596	226	3	n	n	CCONJ
ejpam-5596	226	4	}	}	PUNCT
ejpam-5596	226	5	.	.	PUNCT
ejpam-5596	227	1	example	example	NOUN
ejpam-5596	227	2	2	2	NUM
ejpam-5596	227	3	.	.	X
ejpam-5596	228	1	consider	consider	VERB
ejpam-5596	228	2	the	the	DET
ejpam-5596	228	3	ordered	order	VERB
ejpam-5596	228	4	semigroup	semigroup	NOUN
ejpam-5596	228	5	s	s	PART
ejpam-5596	228	6	=	=	SYM
ejpam-5596	228	7	{	{	PUNCT
ejpam-5596	228	8	u	u	NOUN
ejpam-5596	228	9	,	,	PUNCT
ejpam-5596	228	10	v	v	NOUN
ejpam-5596	228	11	,	,	PUNCT
ejpam-5596	228	12	w	w	PROPN
ejpam-5596	228	13	,	,	PUNCT
ejpam-5596	228	14	x	x	NOUN
ejpam-5596	228	15	,	,	PUNCT
ejpam-5596	228	16	y	y	PROPN
ejpam-5596	228	17	,	,	PUNCT
ejpam-5596	228	18	z	z	NOUN
ejpam-5596	228	19	}	}	PUNCT
ejpam-5596	228	20	with	with	ADP
ejpam-5596	228	21	the	the	DET
ejpam-5596	228	22	following	follow	VERB
ejpam-5596	228	23	cayley	cayley	ADJ
ejpam-5596	228	24	table	table	NOUN
ejpam-5596	228	25	:	:	PUNCT
ejpam-5596	228	26	·	·	PUNCT
ejpam-5596	228	27	u	u	NOUN
ejpam-5596	228	28	v	v	PROPN
ejpam-5596	228	29	w	w	PROPN
ejpam-5596	228	30	x	x	PUNCT
ejpam-5596	228	31	y	y	PROPN
ejpam-5596	228	32	z	z	PROPN
ejpam-5596	228	33	u	u	NOUN
ejpam-5596	228	34	u	u	X
ejpam-5596	228	35	u	u	NOUN
ejpam-5596	228	36	w	w	PROPN
ejpam-5596	228	37	u	u	X
ejpam-5596	228	38	u	u	NOUN
ejpam-5596	228	39	u	u	NOUN
ejpam-5596	228	40	v	v	ADP
ejpam-5596	228	41	u	u	X
ejpam-5596	228	42	u	u	X
ejpam-5596	228	43	u	u	NOUN
ejpam-5596	228	44	u	u	NOUN
ejpam-5596	228	45	u	u	NOUN
ejpam-5596	228	46	v	v	NUM
ejpam-5596	228	47	w	w	PROPN
ejpam-5596	228	48	u	u	PROPN
ejpam-5596	228	49	u	u	NOUN
ejpam-5596	228	50	u	u	NOUN
ejpam-5596	228	51	u	u	NOUN
ejpam-5596	228	52	u	u	NOUN
ejpam-5596	228	53	v	v	ADP
ejpam-5596	228	54	z	z	PROPN
ejpam-5596	228	55	u	u	NOUN
ejpam-5596	228	56	u	u	NOUN
ejpam-5596	228	57	u	u	NOUN
ejpam-5596	228	58	u	u	NOUN
ejpam-5596	228	59	u	u	NOUN
ejpam-5596	228	60	x	x	VERB
ejpam-5596	228	61	y	y	PROPN
ejpam-5596	228	62	u	u	NOUN
ejpam-5596	228	63	x	x	X
ejpam-5596	228	64	x	x	PUNCT
ejpam-5596	228	65	u	u	X
ejpam-5596	228	66	u	u	NOUN
ejpam-5596	228	67	x	x	PROPN
ejpam-5596	228	68	z	z	NOUN
ejpam-5596	228	69	u	u	NOUN
ejpam-5596	228	70	x	x	NOUN
ejpam-5596	228	71	x	x	PUNCT
ejpam-5596	228	72	x	x	PUNCT
ejpam-5596	228	73	y	y	PROPN
ejpam-5596	228	74	z	z	PROPN
ejpam-5596	228	75	and	and	CCONJ
ejpam-5596	228	76	≤	≤	NUM
ejpam-5596	228	77	:	:	PUNCT
ejpam-5596	228	78	{	{	PUNCT
ejpam-5596	228	79	(	(	PUNCT
ejpam-5596	228	80	u	u	NOUN
ejpam-5596	228	81	,	,	PUNCT
ejpam-5596	228	82	u	u	NOUN
ejpam-5596	228	83	)	)	PUNCT
ejpam-5596	228	84	,	,	PUNCT
ejpam-5596	228	85	(	(	PUNCT
ejpam-5596	228	86	v	v	NOUN
ejpam-5596	228	87	,	,	PUNCT
ejpam-5596	228	88	v	v	NOUN
ejpam-5596	228	89	)	)	PUNCT
ejpam-5596	228	90	,	,	PUNCT
ejpam-5596	228	91	(	(	PUNCT
ejpam-5596	228	92	w	w	NOUN
ejpam-5596	228	93	,	,	PUNCT
ejpam-5596	228	94	w	w	NOUN
ejpam-5596	228	95	)	)	PUNCT
ejpam-5596	228	96	,	,	PUNCT
ejpam-5596	228	97	(	(	PUNCT
ejpam-5596	228	98	x	x	X
ejpam-5596	228	99	,	,	PUNCT
ejpam-5596	228	100	x	x	NOUN
ejpam-5596	228	101	)	)	PUNCT
ejpam-5596	228	102	,	,	PUNCT
ejpam-5596	228	103	(	(	PUNCT
ejpam-5596	228	104	y	y	PROPN
ejpam-5596	228	105	,	,	PUNCT
ejpam-5596	228	106	y	y	PROPN
ejpam-5596	228	107	)	)	PUNCT
ejpam-5596	228	108	,	,	PUNCT
ejpam-5596	228	109	(	(	PUNCT
ejpam-5596	228	110	z	z	X
ejpam-5596	228	111	,	,	PUNCT
ejpam-5596	228	112	z	z	NOUN
ejpam-5596	228	113	)	)	PUNCT
ejpam-5596	228	114	}	}	PUNCT
ejpam-5596	228	115	.	.	PUNCT
ejpam-5596	229	1	define	define	VERB
ejpam-5596	229	2	a	a	DET
ejpam-5596	229	3	function	function	NOUN
ejpam-5596	229	4	δ	δ	NOUN
ejpam-5596	229	5	:	:	PUNCT
ejpam-5596	229	6	s	s	X
ejpam-5596	229	7	→	→	SYM
ejpam-5596	229	8	[	[	X
ejpam-5596	229	9	0	0	NUM
ejpam-5596	229	10	,	,	PUNCT
ejpam-5596	229	11	1	1	NUM
ejpam-5596	229	12	]	]	PUNCT
ejpam-5596	229	13	by	by	ADP
ejpam-5596	229	14	δ(u	δ(u	NOUN
ejpam-5596	229	15	)	)	PUNCT
ejpam-5596	229	16	=	=	SYM
ejpam-5596	229	17	0.6	0.6	NUM
ejpam-5596	229	18	,	,	PUNCT
ejpam-5596	229	19	δ(v	δ(v	PROPN
ejpam-5596	229	20	)	)	PUNCT
ejpam-5596	229	21	=	=	SYM
ejpam-5596	229	22	0.2	0.2	NUM
ejpam-5596	229	23	,	,	PUNCT
ejpam-5596	229	24	δ(w	δ(w	PROPN
ejpam-5596	229	25	)	)	PUNCT
ejpam-5596	229	26	=	=	SYM
ejpam-5596	229	27	0.4	0.4	NUM
ejpam-5596	229	28	,	,	PUNCT
ejpam-5596	229	29	δ(x	δ(x	ADJ
ejpam-5596	229	30	)	)	PUNCT
ejpam-5596	229	31	=	=	SYM
ejpam-5596	229	32	0.5	0.5	NUM
ejpam-5596	229	33	δ(y	δ(y	ADV
ejpam-5596	229	34	)	)	PUNCT
ejpam-5596	229	35	=	=	PUNCT
ejpam-5596	229	36	0.3	0.3	NUM
ejpam-5596	229	37	δ(z	δ(z	NOUN
ejpam-5596	229	38	)	)	PUNCT
ejpam-5596	229	39	=	=	PUNCT
ejpam-5596	229	40	0.1	0.1	NUM
ejpam-5596	229	41	.	.	PUNCT
ejpam-5596	230	1	then	then	ADV
ejpam-5596	230	2	δ	δ	PROPN
ejpam-5596	230	3	is	be	AUX
ejpam-5596	230	4	a	a	DET
ejpam-5596	230	5	fuzzy	fuzzy	ADJ
ejpam-5596	230	6	n	n	CCONJ
ejpam-5596	230	7	-	-	ADJ
ejpam-5596	230	8	interior	interior	ADJ
ejpam-5596	230	9	ideal	ideal	NOUN
ejpam-5596	230	10	of	of	ADP
ejpam-5596	230	11	s.	s.	PROPN
ejpam-5596	230	12	theorem	theorem	VERB
ejpam-5596	230	13	7	7	NUM
ejpam-5596	230	14	.	.	PUNCT
ejpam-5596	231	1	let	let	VERB
ejpam-5596	231	2	{	{	PUNCT
ejpam-5596	231	3	δi	δi	VERB
ejpam-5596	232	1	|	|	ADV
ejpam-5596	232	2	i	i	PRON
ejpam-5596	232	3	∈	∈	PROPN
ejpam-5596	232	4	j	j	PROPN
ejpam-5596	232	5	}	}	PUNCT
ejpam-5596	232	6	be	be	AUX
ejpam-5596	232	7	a	a	DET
ejpam-5596	232	8	family	family	NOUN
ejpam-5596	232	9	of	of	ADP
ejpam-5596	232	10	fuzzy	fuzzy	ADJ
ejpam-5596	232	11	n	n	CCONJ
ejpam-5596	232	12	-	-	PUNCT
ejpam-5596	232	13	interior	interior	ADJ
ejpam-5596	232	14	ideals	ideal	NOUN
ejpam-5596	232	15	of	of	ADP
ejpam-5596	232	16	an	an	DET
ejpam-5596	232	17	ordered	order	VERB
ejpam-5596	232	18	semigroup	semigroup	NOUN
ejpam-5596	232	19	s	s	PROPN
ejpam-5596	232	20	with	with	ADP
ejpam-5596	232	21	δ(u	δ(u	PROPN
ejpam-5596	232	22	)	)	PUNCT
ejpam-5596	232	23	≥	≥	NOUN
ejpam-5596	232	24	δ(v	δ(v	PROPN
ejpam-5596	232	25	)	)	PUNCT
ejpam-5596	232	26	whenever	whenever	SCONJ
ejpam-5596	232	27	u	u	NOUN
ejpam-5596	232	28	≤	≤	X
ejpam-5596	233	1	v.	v.	CCONJ
ejpam-5596	233	2	then	then	ADV
ejpam-5596	233	3	∧	∧	PROPN
ejpam-5596	233	4	i∈f	i∈f	VERB
ejpam-5596	233	5	ϑi	ϑi	PRON
ejpam-5596	233	6	is	be	AUX
ejpam-5596	233	7	a	a	DET
ejpam-5596	233	8	fuzzy	fuzzy	ADJ
ejpam-5596	233	9	n	n	CCONJ
ejpam-5596	233	10	-	-	ADJ
ejpam-5596	233	11	interior	interior	ADJ
ejpam-5596	233	12	ideal	ideal	NOUN
ejpam-5596	233	13	of	of	ADP
ejpam-5596	233	14	s.	s.	PROPN
ejpam-5596	233	15	proof	proof	PROPN
ejpam-5596	233	16	.	.	PUNCT
ejpam-5596	234	1	let	let	VERB
ejpam-5596	234	2	u	u	NOUN
ejpam-5596	234	3	,	,	PUNCT
ejpam-5596	234	4	v	v	ADP
ejpam-5596	234	5	∈	∈	NOUN
ejpam-5596	234	6	s.	s.	PROPN
ejpam-5596	234	7	then,∧	then,∧	PROPN
ejpam-5596	234	8	i∈j	i∈j	NOUN
ejpam-5596	234	9	δi(uv	δi(uv	NOUN
ejpam-5596	234	10	)	)	PUNCT
ejpam-5596	234	11	≥	≥	NOUN
ejpam-5596	234	12	∧	∧	PROPN
ejpam-5596	234	13	i∈j	i∈j	NOUN
ejpam-5596	234	14	{	{	PUNCT
ejpam-5596	234	15	δi(u	δi(u	NUM
ejpam-5596	234	16	)	)	PUNCT
ejpam-5596	234	17	∧	∧	NOUN
ejpam-5596	234	18	δi(v	δi(v	NUM
ejpam-5596	234	19	)	)	PUNCT
ejpam-5596	234	20	}	}	PUNCT
ejpam-5596	235	1	=	=	PUNCT
ejpam-5596	235	2	∧	∧	NOUN
ejpam-5596	235	3	i∈j	i∈j	NOUN
ejpam-5596	235	4	δi(u	δi(u	NOUN
ejpam-5596	235	5	)	)	PUNCT
ejpam-5596	235	6	∧	∧	NOUN
ejpam-5596	235	7	∧	∧	PROPN
ejpam-5596	235	8	i∈j	i∈j	NOUN
ejpam-5596	235	9	δi(v	δi(v	NUM
ejpam-5596	235	10	)	)	PUNCT
ejpam-5596	235	11	.	.	PUNCT
ejpam-5596	236	1	thus	thus	ADV
ejpam-5596	236	2	,	,	PUNCT
ejpam-5596	236	3	∧	∧	NOUN
ejpam-5596	236	4	i∈j	i∈j	NOUN
ejpam-5596	236	5	δi	δi	NOUN
ejpam-5596	236	6	is	be	AUX
ejpam-5596	236	7	a	a	DET
ejpam-5596	236	8	fuzzy	fuzzy	ADJ
ejpam-5596	236	9	subsemigroup	subsemigroup	NOUN
ejpam-5596	236	10	of	of	ADP
ejpam-5596	236	11	s.	s.	PROPN
ejpam-5596	236	12	let	let	VERB
ejpam-5596	236	13	u	u	PRON
ejpam-5596	236	14	,	,	PUNCT
ejpam-5596	236	15	kni	kni	INTJ
ejpam-5596	236	16	,	,	PUNCT
ejpam-5596	236	17	v	v	PROPN
ejpam-5596	236	18	∈	∈	NOUN
ejpam-5596	236	19	s	s	NOUN
ejpam-5596	236	20	for	for	ADP
ejpam-5596	236	21	all	all	PRON
ejpam-5596	236	22	i	i	PRON
ejpam-5596	236	23	∈	∈	PROPN
ejpam-5596	236	24	{	{	PUNCT
ejpam-5596	236	25	1	1	NUM
ejpam-5596	236	26	,	,	PUNCT
ejpam-5596	236	27	2	2	NUM
ejpam-5596	236	28	,	,	PUNCT
ejpam-5596	236	29	.	.	PUNCT
ejpam-5596	236	30	.	.	PUNCT
ejpam-5596	237	1	.	.	PUNCT
ejpam-5596	237	2	,	,	PUNCT
ejpam-5596	238	1	n	n	CCONJ
ejpam-5596	238	2	}	}	PUNCT
ejpam-5596	238	3	.	.	PUNCT
ejpam-5596	239	1	then,∧	then,∧	PROPN
ejpam-5596	239	2	i∈j	i∈j	NOUN
ejpam-5596	239	3	δi(uk	δi(uk	VERB
ejpam-5596	239	4	n	n	CCONJ
ejpam-5596	239	5	i	i	PROPN
ejpam-5596	239	6	v	v	NOUN
ejpam-5596	239	7	)	)	PUNCT
ejpam-5596	239	8	≥	≥	NOUN
ejpam-5596	240	1	∧	∧	PROPN
ejpam-5596	240	2	i∈j	i∈j	NOUN
ejpam-5596	240	3	{	{	PUNCT
ejpam-5596	240	4	δi(k1	δi(k1	NOUN
ejpam-5596	240	5	)	)	PUNCT
ejpam-5596	240	6	∧	∧	NOUN
ejpam-5596	240	7	δi(k2	δi(k2	NOUN
ejpam-5596	240	8	)	)	PUNCT
ejpam-5596	240	9	·	·	PUNCT
ejpam-5596	240	10	·	·	PUNCT
ejpam-5596	240	11	·	·	PUNCT
ejpam-5596	241	1	∧	∧	PROPN
ejpam-5596	241	2	δi(kn	δi(kn	PROPN
ejpam-5596	241	3	)	)	PUNCT
ejpam-5596	241	4	}	}	PUNCT
ejpam-5596	242	1	=	=	PUNCT
ejpam-5596	242	2	∧	∧	PROPN
ejpam-5596	242	3	i∈j	i∈j	NOUN
ejpam-5596	242	4	δi(k1	δi(k1	NOUN
ejpam-5596	242	5	)	)	PUNCT
ejpam-5596	242	6	∧	∧	NOUN
ejpam-5596	242	7	∧	∧	PROPN
ejpam-5596	242	8	i∈j	i∈j	NOUN
ejpam-5596	242	9	δi(k2	δi(k2	NOUN
ejpam-5596	242	10	)	)	PUNCT
ejpam-5596	242	11	·	·	PUNCT
ejpam-5596	242	12	·	·	PUNCT
ejpam-5596	243	1	·	·	PUNCT
ejpam-5596	244	1	∧	∧	NOUN
ejpam-5596	244	2	∧	∧	PROPN
ejpam-5596	244	3	i∈j	i∈j	NOUN
ejpam-5596	244	4	δi(kn	δi(kn	PROPN
ejpam-5596	244	5	)	)	PUNCT
ejpam-5596	244	6	.	.	PUNCT
ejpam-5596	245	1	thus	thus	ADV
ejpam-5596	245	2	,	,	PUNCT
ejpam-5596	245	3	∧	∧	NOUN
ejpam-5596	245	4	i∈j	i∈j	NOUN
ejpam-5596	245	5	δi	δi	NOUN
ejpam-5596	245	6	is	be	AUX
ejpam-5596	245	7	a	a	DET
ejpam-5596	245	8	fuzzy	fuzzy	ADJ
ejpam-5596	245	9	n	n	CCONJ
ejpam-5596	245	10	-	-	ADJ
ejpam-5596	245	11	interior	interior	ADJ
ejpam-5596	245	12	ideal	ideal	NOUN
ejpam-5596	245	13	of	of	ADP
ejpam-5596	245	14	s.	s.	PROPN
ejpam-5596	245	15	p.	p.	PROPN
ejpam-5596	245	16	khamrot	khamrot	PROPN
ejpam-5596	245	17	,	,	PUNCT
ejpam-5596	245	18	a.	a.	NOUN
ejpam-5596	245	19	iampan	iampan	PROPN
ejpam-5596	245	20	,	,	PUNCT
ejpam-5596	245	21	t.	t.	PROPN
ejpam-5596	245	22	gaketem	gaketem	PROPN
ejpam-5596	245	23	/	/	SYM
ejpam-5596	245	24	eur	eur	PROPN
ejpam-5596	245	25	.	.	PUNCT
ejpam-5596	246	1	j.	j.	PROPN
ejpam-5596	246	2	pure	pure	PROPN
ejpam-5596	246	3	appl	appl	PROPN
ejpam-5596	246	4	.	.	PROPN
ejpam-5596	246	5	math	math	PROPN
ejpam-5596	246	6	,	,	PUNCT
ejpam-5596	246	7	18	18	NUM
ejpam-5596	246	8	(	(	PUNCT
ejpam-5596	246	9	1	1	NUM
ejpam-5596	246	10	)	)	PUNCT
ejpam-5596	246	11	(	(	PUNCT
ejpam-5596	246	12	2025	2025	NUM
ejpam-5596	246	13	)	)	PUNCT
ejpam-5596	246	14	,	,	PUNCT
ejpam-5596	246	15	5596	5596	NUM
ejpam-5596	246	16	9	9	NUM
ejpam-5596	246	17	of	of	ADP
ejpam-5596	246	18	12	12	NUM
ejpam-5596	246	19	theorem	theorem	NOUN
ejpam-5596	246	20	8	8	NUM
ejpam-5596	246	21	.	.	PUNCT
ejpam-5596	247	1	let	let	VERB
ejpam-5596	247	2	k	k	PRON
ejpam-5596	247	3	be	be	AUX
ejpam-5596	247	4	an	an	DET
ejpam-5596	247	5	ideal	ideal	NOUN
ejpam-5596	247	6	of	of	ADP
ejpam-5596	247	7	a	a	DET
ejpam-5596	247	8	semigroup	semigroup	NOUN
ejpam-5596	247	9	s	s	X
ejpam-5596	247	10	and	and	CCONJ
ejpam-5596	247	11	m	m	PROPN
ejpam-5596	247	12	,	,	PUNCT
ejpam-5596	247	13	n	n	PRON
ejpam-5596	247	14	are	be	AUX
ejpam-5596	247	15	positive	positive	ADJ
ejpam-5596	247	16	integers	integer	NOUN
ejpam-5596	247	17	.	.	PUNCT
ejpam-5596	248	1	then	then	ADV
ejpam-5596	248	2	k	k	PROPN
ejpam-5596	248	3	is	be	AUX
ejpam-5596	248	4	an	an	DET
ejpam-5596	248	5	n	n	CCONJ
ejpam-5596	248	6	-	-	PUNCT
ejpam-5596	248	7	interior	interior	ADJ
ejpam-5596	248	8	ideal	ideal	NOUN
ejpam-5596	248	9	of	of	ADP
ejpam-5596	248	10	s	s	PRON
ejpam-5596	248	11	if	if	SCONJ
ejpam-5596	248	12	and	and	CCONJ
ejpam-5596	248	13	only	only	ADV
ejpam-5596	248	14	if	if	SCONJ
ejpam-5596	248	15	the	the	DET
ejpam-5596	248	16	characteristic	characteristic	ADJ
ejpam-5596	248	17	function	function	NOUN
ejpam-5596	248	18	λk	λk	INTJ
ejpam-5596	248	19	is	be	AUX
ejpam-5596	248	20	a	a	DET
ejpam-5596	248	21	fuzzy	fuzzy	ADJ
ejpam-5596	248	22	n	n	CCONJ
ejpam-5596	248	23	-	-	ADJ
ejpam-5596	248	24	interior	interior	ADJ
ejpam-5596	248	25	ideal	ideal	NOUN
ejpam-5596	248	26	of	of	ADP
ejpam-5596	248	27	s.	s.	PROPN
ejpam-5596	248	28	proof	proof	PROPN
ejpam-5596	248	29	.	.	PUNCT
ejpam-5596	249	1	suppose	suppose	VERB
ejpam-5596	249	2	that	that	SCONJ
ejpam-5596	249	3	k	k	PROPN
ejpam-5596	249	4	is	be	AUX
ejpam-5596	249	5	an	an	DET
ejpam-5596	249	6	n	n	CCONJ
ejpam-5596	249	7	-	-	PUNCT
ejpam-5596	249	8	interior	interior	ADJ
ejpam-5596	249	9	ideal	ideal	NOUN
ejpam-5596	249	10	of	of	ADP
ejpam-5596	249	11	s.	s.	PROPN
ejpam-5596	249	12	then	then	ADV
ejpam-5596	249	13	k	k	PROPN
ejpam-5596	249	14	is	be	AUX
ejpam-5596	249	15	a	a	DET
ejpam-5596	249	16	subsemigroup	subsemigroup	NOUN
ejpam-5596	249	17	of	of	ADP
ejpam-5596	249	18	s.	s.	PROPN
ejpam-5596	249	19	thus	thus	ADV
ejpam-5596	249	20	,	,	PUNCT
ejpam-5596	249	21	by	by	ADP
ejpam-5596	249	22	theorem	theorem	NOUN
ejpam-5596	249	23	1	1	NUM
ejpam-5596	249	24	,	,	PUNCT
ejpam-5596	249	25	λk	λk	PRON
ejpam-5596	249	26	is	be	AUX
ejpam-5596	249	27	a	a	DET
ejpam-5596	249	28	bf	bf	NOUN
ejpam-5596	249	29	subsemigroup	subsemigroup	NOUN
ejpam-5596	249	30	of	of	ADP
ejpam-5596	249	31	e	e	PROPN
ejpam-5596	249	32	.	.	PUNCT
ejpam-5596	250	1	let	let	VERB
ejpam-5596	250	2	h	h	NOUN
ejpam-5596	250	3	,	,	PUNCT
ejpam-5596	250	4	ri	ri	PROPN
ejpam-5596	250	5	,	,	PUNCT
ejpam-5596	250	6	k	k	PROPN
ejpam-5596	250	7	∈	∈	PROPN
ejpam-5596	250	8	e	e	X
ejpam-5596	250	9	where	where	SCONJ
ejpam-5596	250	10	i	i	PRON
ejpam-5596	250	11	∈	∈	PROPN
ejpam-5596	250	12	{	{	PUNCT
ejpam-5596	250	13	1	1	NUM
ejpam-5596	250	14	,	,	PUNCT
ejpam-5596	250	15	2	2	NUM
ejpam-5596	250	16	,	,	PUNCT
ejpam-5596	250	17	.	.	PUNCT
ejpam-5596	250	18	.	.	PUNCT
ejpam-5596	251	1	.	.	PUNCT
ejpam-5596	251	2	,	,	PUNCT
ejpam-5596	251	3	n	n	CCONJ
ejpam-5596	251	4	}	}	PUNCT
ejpam-5596	251	5	.	.	PUNCT
ejpam-5596	252	1	if	if	SCONJ
ejpam-5596	252	2	ri	ri	PROPN
ejpam-5596	252	3	∈	∈	PROPN
ejpam-5596	252	4	k	k	PROPN
ejpam-5596	252	5	for	for	ADP
ejpam-5596	252	6	all	all	PRON
ejpam-5596	252	7	i	i	PRON
ejpam-5596	252	8	∈	∈	PROPN
ejpam-5596	252	9	{	{	PUNCT
ejpam-5596	252	10	1	1	NUM
ejpam-5596	252	11	,	,	PUNCT
ejpam-5596	252	12	2	2	NUM
ejpam-5596	252	13	,	,	PUNCT
ejpam-5596	252	14	.	.	PUNCT
ejpam-5596	252	15	.	.	PUNCT
ejpam-5596	253	1	.	.	PUNCT
ejpam-5596	254	1	,	,	PUNCT
ejpam-5596	255	1	n	n	CCONJ
ejpam-5596	255	2	}	}	PUNCT
ejpam-5596	255	3	,	,	PUNCT
ejpam-5596	255	4	then	then	ADV
ejpam-5596	255	5	hrni	hrni	VERB
ejpam-5596	255	6	k	k	PROPN
ejpam-5596	255	7	∈	∈	PROPN
ejpam-5596	255	8	k.	k.	PROPN
ejpam-5596	256	1	thus	thus	ADV
ejpam-5596	256	2	,	,	PUNCT
ejpam-5596	256	3	λk(ri	λk(ri	PROPN
ejpam-5596	256	4	)	)	PUNCT
ejpam-5596	257	1	=	=	SYM
ejpam-5596	257	2	λk(hr	λk(hr	PROPN
ejpam-5596	257	3	n	n	CCONJ
ejpam-5596	257	4	i	i	PRON
ejpam-5596	257	5	k	k	PROPN
ejpam-5596	257	6	)	)	PUNCT
ejpam-5596	257	7	=	=	SYM
ejpam-5596	257	8	1	1	NUM
ejpam-5596	257	9	for	for	ADP
ejpam-5596	257	10	all	all	PRON
ejpam-5596	257	11	i	i	PRON
ejpam-5596	257	12	∈	∈	PROPN
ejpam-5596	257	13	{	{	PUNCT
ejpam-5596	257	14	1	1	NUM
ejpam-5596	257	15	,	,	PUNCT
ejpam-5596	257	16	2	2	NUM
ejpam-5596	257	17	,	,	PUNCT
ejpam-5596	257	18	.	.	PUNCT
ejpam-5596	257	19	.	.	PUNCT
ejpam-5596	258	1	.	.	PUNCT
ejpam-5596	258	2	,	,	PUNCT
ejpam-5596	258	3	n	n	CCONJ
ejpam-5596	258	4	}	}	PUNCT
ejpam-5596	258	5	.	.	PUNCT
ejpam-5596	259	1	hence	hence	ADV
ejpam-5596	259	2	,	,	PUNCT
ejpam-5596	259	3	λk(hr	λk(hr	PROPN
ejpam-5596	259	4	n	n	PROPN
ejpam-5596	259	5	i	i	PROPN
ejpam-5596	259	6	k	k	PROPN
ejpam-5596	259	7	)	)	PUNCT
ejpam-5596	259	8	≥	≥	NOUN
ejpam-5596	259	9	λk(r1	λk(r1	NOUN
ejpam-5596	259	10	)	)	PUNCT
ejpam-5596	259	11	∧	∧	NOUN
ejpam-5596	259	12	λk(r2	λk(r2	NOUN
ejpam-5596	259	13	)	)	PUNCT
ejpam-5596	259	14	∧	∧	PROPN
ejpam-5596	259	15	·	·	PUNCT
ejpam-5596	259	16	·	·	PUNCT
ejpam-5596	259	17	·	·	PUNCT
ejpam-5596	260	1	∧	∧	PROPN
ejpam-5596	260	2	λk(rn	λk(rn	PROPN
ejpam-5596	260	3	)	)	PUNCT
ejpam-5596	260	4	.	.	PUNCT
ejpam-5596	261	1	if	if	SCONJ
ejpam-5596	261	2	ri	ri	PROPN
ejpam-5596	261	3	/∈	/∈	PUNCT
ejpam-5596	262	1	k	k	PROPN
ejpam-5596	263	1	for	for	ADP
ejpam-5596	263	2	some	some	DET
ejpam-5596	263	3	i	i	PRON
ejpam-5596	263	4	∈	∈	PROPN
ejpam-5596	263	5	{	{	PUNCT
ejpam-5596	263	6	1	1	NUM
ejpam-5596	263	7	,	,	PUNCT
ejpam-5596	263	8	2	2	NUM
ejpam-5596	263	9	,	,	PUNCT
ejpam-5596	263	10	.	.	PUNCT
ejpam-5596	263	11	.	.	PUNCT
ejpam-5596	263	12	.	.	PUNCT
ejpam-5596	263	13	,	,	PUNCT
ejpam-5596	263	14	n	n	CCONJ
ejpam-5596	263	15	}	}	PUNCT
ejpam-5596	263	16	,	,	PUNCT
ejpam-5596	263	17	then	then	ADV
ejpam-5596	263	18	λk(ri	λk(ri	PROPN
ejpam-5596	263	19	)	)	PUNCT
ejpam-5596	263	20	=	=	SYM
ejpam-5596	263	21	0	0	NUM
ejpam-5596	263	22	for	for	ADP
ejpam-5596	263	23	some	some	DET
ejpam-5596	263	24	i	i	PRON
ejpam-5596	263	25	∈	∈	PROPN
ejpam-5596	263	26	{	{	PUNCT
ejpam-5596	263	27	1	1	NUM
ejpam-5596	263	28	,	,	PUNCT
ejpam-5596	263	29	2	2	NUM
ejpam-5596	263	30	,	,	PUNCT
ejpam-5596	263	31	.	.	PUNCT
ejpam-5596	263	32	.	.	PUNCT
ejpam-5596	263	33	.	.	PUNCT
ejpam-5596	263	34	,	,	PUNCT
ejpam-5596	263	35	n	n	CCONJ
ejpam-5596	263	36	}	}	PUNCT
ejpam-5596	263	37	.	.	PUNCT
ejpam-5596	264	1	thus	thus	ADV
ejpam-5596	264	2	,	,	PUNCT
ejpam-5596	264	3	λk(hr	λk(hr	PROPN
ejpam-5596	264	4	n	n	PROPN
ejpam-5596	264	5	i	i	PROPN
ejpam-5596	264	6	k	k	PROPN
ejpam-5596	264	7	)	)	PUNCT
ejpam-5596	264	8	≥	≥	NOUN
ejpam-5596	264	9	λk(r1	λk(r1	NOUN
ejpam-5596	264	10	)	)	PUNCT
ejpam-5596	264	11	∧	∧	NOUN
ejpam-5596	264	12	λk(r2	λk(r2	NOUN
ejpam-5596	264	13	)	)	PUNCT
ejpam-5596	264	14	∧	∧	PROPN
ejpam-5596	264	15	·	·	PUNCT
ejpam-5596	264	16	·	·	PUNCT
ejpam-5596	264	17	·	·	PUNCT
ejpam-5596	264	18	∧	∧	PROPN
ejpam-5596	264	19	λk(rn	λk(rn	PROPN
ejpam-5596	264	20	)	)	PUNCT
ejpam-5596	264	21	.	.	PUNCT
ejpam-5596	265	1	therefore	therefore	ADV
ejpam-5596	265	2	,	,	PUNCT
ejpam-5596	265	3	λk	λk	PRON
ejpam-5596	265	4	is	be	AUX
ejpam-5596	265	5	a	a	DET
ejpam-5596	265	6	fuzzy	fuzzy	ADJ
ejpam-5596	265	7	n	n	CCONJ
ejpam-5596	265	8	-	-	ADJ
ejpam-5596	265	9	interior	interior	ADJ
ejpam-5596	265	10	ideal	ideal	NOUN
ejpam-5596	265	11	of	of	ADP
ejpam-5596	265	12	s.	s.	PROPN
ejpam-5596	265	13	conversely	conversely	ADV
ejpam-5596	265	14	,	,	PUNCT
ejpam-5596	265	15	suppose	suppose	VERB
ejpam-5596	265	16	that	that	SCONJ
ejpam-5596	265	17	λk	λk	PROPN
ejpam-5596	265	18	is	be	AUX
ejpam-5596	265	19	a	a	DET
ejpam-5596	265	20	fuzzy	fuzzy	ADJ
ejpam-5596	265	21	n	n	CCONJ
ejpam-5596	265	22	-	-	ADJ
ejpam-5596	265	23	interior	interior	ADJ
ejpam-5596	265	24	ideal	ideal	NOUN
ejpam-5596	265	25	of	of	ADP
ejpam-5596	265	26	s.	s.	PROPN
ejpam-5596	265	27	then	then	ADV
ejpam-5596	265	28	λk	λk	X
ejpam-5596	265	29	is	be	AUX
ejpam-5596	265	30	a	a	DET
ejpam-5596	265	31	fuzzy	fuzzy	ADJ
ejpam-5596	265	32	subsemigroup	subsemigroup	NOUN
ejpam-5596	265	33	of	of	ADP
ejpam-5596	265	34	s.	s.	PROPN
ejpam-5596	265	35	thus	thus	ADV
ejpam-5596	265	36	,	,	PUNCT
ejpam-5596	265	37	by	by	ADP
ejpam-5596	265	38	theorem	theorem	NOUN
ejpam-5596	265	39	1	1	NUM
ejpam-5596	265	40	,	,	PUNCT
ejpam-5596	265	41	k	k	PROPN
ejpam-5596	265	42	is	be	AUX
ejpam-5596	265	43	a	a	DET
ejpam-5596	265	44	subsemigroup	subsemigroup	NOUN
ejpam-5596	265	45	of	of	ADP
ejpam-5596	265	46	s.	s.	PROPN
ejpam-5596	265	47	let	let	VERB
ejpam-5596	265	48	rni	rni	NOUN
ejpam-5596	265	49	∈	∈	PROPN
ejpam-5596	265	50	skns	skns	NOUN
ejpam-5596	265	51	where	where	SCONJ
ejpam-5596	265	52	n	n	PRON
ejpam-5596	265	53	is	be	AUX
ejpam-5596	265	54	an	an	DET
ejpam-5596	265	55	integer	integer	NOUN
ejpam-5596	265	56	and	and	CCONJ
ejpam-5596	265	57	for	for	ADP
ejpam-5596	265	58	all	all	PRON
ejpam-5596	265	59	i	i	PRON
ejpam-5596	265	60	∈	∈	PROPN
ejpam-5596	265	61	{	{	PUNCT
ejpam-5596	265	62	1	1	NUM
ejpam-5596	265	63	,	,	PUNCT
ejpam-5596	265	64	2	2	NUM
ejpam-5596	265	65	,	,	PUNCT
ejpam-5596	265	66	.	.	PUNCT
ejpam-5596	265	67	.	.	PUNCT
ejpam-5596	266	1	.	.	PUNCT
ejpam-5596	266	2	,	,	PUNCT
ejpam-5596	266	3	n	n	CCONJ
ejpam-5596	266	4	}	}	PUNCT
ejpam-5596	266	5	.	.	PUNCT
ejpam-5596	267	1	then	then	ADV
ejpam-5596	267	2	λk(r	λk(r	PUNCT
ejpam-5596	267	3	n	n	PROPN
ejpam-5596	267	4	i	i	PRON
ejpam-5596	267	5	)	)	PUNCT
ejpam-5596	268	1	=	=	SYM
ejpam-5596	268	2	1	1	NUM
ejpam-5596	268	3	for	for	ADP
ejpam-5596	268	4	all	all	PRON
ejpam-5596	268	5	i	i	PRON
ejpam-5596	268	6	∈	∈	PROPN
ejpam-5596	268	7	{	{	PUNCT
ejpam-5596	268	8	1	1	NUM
ejpam-5596	268	9	,	,	PUNCT
ejpam-5596	268	10	2	2	NUM
ejpam-5596	268	11	,	,	PUNCT
ejpam-5596	268	12	.	.	PUNCT
ejpam-5596	268	13	.	.	PUNCT
ejpam-5596	269	1	.	.	PUNCT
ejpam-5596	269	2	,	,	PUNCT
ejpam-5596	269	3	n	n	CCONJ
ejpam-5596	269	4	}	}	PUNCT
ejpam-5596	269	5	.	.	PUNCT
ejpam-5596	270	1	by	by	ADP
ejpam-5596	270	2	assumption	assumption	NOUN
ejpam-5596	270	3	,	,	PUNCT
ejpam-5596	270	4	λk(hr	λk(hr	PROPN
ejpam-5596	270	5	n	n	PROPN
ejpam-5596	270	6	i	i	PROPN
ejpam-5596	270	7	k	k	PROPN
ejpam-5596	270	8	)	)	PUNCT
ejpam-5596	270	9	≥	≥	NOUN
ejpam-5596	270	10	λk(r1	λk(r1	NOUN
ejpam-5596	270	11	)	)	PUNCT
ejpam-5596	270	12	∧	∧	NOUN
ejpam-5596	270	13	λk(r2	λk(r2	NOUN
ejpam-5596	270	14	)	)	PUNCT
ejpam-5596	270	15	∧	∧	PROPN
ejpam-5596	270	16	·	·	PUNCT
ejpam-5596	270	17	·	·	PUNCT
ejpam-5596	270	18	·	·	PUNCT
ejpam-5596	270	19	∧	∧	PROPN
ejpam-5596	270	20	λk(rn	λk(rn	PROPN
ejpam-5596	270	21	)	)	PUNCT
ejpam-5596	270	22	.	.	PUNCT
ejpam-5596	271	1	thus	thus	ADV
ejpam-5596	271	2	,	,	PUNCT
ejpam-5596	271	3	λk(hr	λk(hr	PROPN
ejpam-5596	271	4	n	n	PROPN
ejpam-5596	271	5	i	i	PROPN
ejpam-5596	271	6	k	k	PROPN
ejpam-5596	271	7	)	)	PUNCT
ejpam-5596	271	8	=	=	SYM
ejpam-5596	271	9	1	1	NUM
ejpam-5596	271	10	for	for	ADP
ejpam-5596	271	11	all	all	PRON
ejpam-5596	271	12	i	i	PRON
ejpam-5596	271	13	∈	∈	PROPN
ejpam-5596	271	14	{	{	PUNCT
ejpam-5596	271	15	1	1	NUM
ejpam-5596	271	16	,	,	PUNCT
ejpam-5596	271	17	2	2	NUM
ejpam-5596	271	18	,	,	PUNCT
ejpam-5596	271	19	.	.	PUNCT
ejpam-5596	271	20	.	.	PUNCT
ejpam-5596	272	1	.	.	PUNCT
ejpam-5596	272	2	,	,	PUNCT
ejpam-5596	272	3	n	n	CCONJ
ejpam-5596	272	4	}	}	PUNCT
ejpam-5596	272	5	.	.	PUNCT
ejpam-5596	273	1	hence	hence	ADV
ejpam-5596	273	2	,	,	PUNCT
ejpam-5596	273	3	rni	rni	NOUN
ejpam-5596	273	4	∈	∈	PROPN
ejpam-5596	273	5	k	k	PROPN
ejpam-5596	273	6	for	for	ADP
ejpam-5596	273	7	all	all	PRON
ejpam-5596	273	8	i	i	PRON
ejpam-5596	273	9	∈	∈	PROPN
ejpam-5596	273	10	{	{	PUNCT
ejpam-5596	273	11	1	1	NUM
ejpam-5596	273	12	,	,	PUNCT
ejpam-5596	273	13	2	2	NUM
ejpam-5596	273	14	,	,	PUNCT
ejpam-5596	273	15	.	.	PUNCT
ejpam-5596	273	16	.	.	PUNCT
ejpam-5596	273	17	.	.	PUNCT
ejpam-5596	273	18	,	,	PUNCT
ejpam-5596	273	19	n	n	CCONJ
ejpam-5596	273	20	}	}	PUNCT
ejpam-5596	273	21	.	.	PUNCT
ejpam-5596	274	1	therefore	therefore	ADV
ejpam-5596	274	2	,	,	PUNCT
ejpam-5596	274	3	k	k	PROPN
ejpam-5596	274	4	is	be	AUX
ejpam-5596	274	5	an	an	DET
ejpam-5596	274	6	n	n	CCONJ
ejpam-5596	274	7	-	-	PUNCT
ejpam-5596	274	8	interior	interior	ADJ
ejpam-5596	274	9	ideal	ideal	NOUN
ejpam-5596	274	10	of	of	ADP
ejpam-5596	274	11	s.	s.	PROPN
ejpam-5596	274	12	theorem	theorem	VERB
ejpam-5596	274	13	9	9	NUM
ejpam-5596	274	14	.	.	PUNCT
ejpam-5596	275	1	a	a	DET
ejpam-5596	275	2	fuzzy	fuzzy	ADJ
ejpam-5596	275	3	set	set	NOUN
ejpam-5596	275	4	δ	δ	PROPN
ejpam-5596	275	5	is	be	AUX
ejpam-5596	275	6	a	a	DET
ejpam-5596	275	7	fuzzy	fuzzy	ADJ
ejpam-5596	275	8	n	n	CCONJ
ejpam-5596	275	9	-	-	ADJ
ejpam-5596	275	10	interior	interior	ADJ
ejpam-5596	275	11	ideal	ideal	NOUN
ejpam-5596	275	12	of	of	ADP
ejpam-5596	275	13	a	a	DET
ejpam-5596	275	14	semigroup	semigroup	NOUN
ejpam-5596	275	15	s	s	X
ejpam-5596	275	16	if	if	SCONJ
ejpam-5596	275	17	and	and	CCONJ
ejpam-5596	275	18	only	only	ADV
ejpam-5596	275	19	if	if	SCONJ
ejpam-5596	275	20	the	the	DET
ejpam-5596	275	21	level	level	NOUN
ejpam-5596	275	22	set	set	VERB
ejpam-5596	275	23	ut	ut	PROPN
ejpam-5596	275	24	is	be	AUX
ejpam-5596	275	25	an	an	DET
ejpam-5596	275	26	n	n	CCONJ
ejpam-5596	275	27	-	-	PUNCT
ejpam-5596	275	28	interior	interior	ADJ
ejpam-5596	275	29	ideal	ideal	NOUN
ejpam-5596	275	30	of	of	ADP
ejpam-5596	275	31	s	s	PRON
ejpam-5596	275	32	for	for	ADP
ejpam-5596	275	33	all	all	DET
ejpam-5596	275	34	t	t	NOUN
ejpam-5596	275	35	∈	∈	PROPN
ejpam-5596	276	1	[	[	X
ejpam-5596	276	2	0	0	NUM
ejpam-5596	276	3	,	,	PUNCT
ejpam-5596	276	4	1	1	NUM
ejpam-5596	276	5	]	]	PUNCT
ejpam-5596	276	6	.	.	PUNCT
ejpam-5596	277	1	proof	proof	NOUN
ejpam-5596	277	2	.	.	PUNCT
ejpam-5596	278	1	let	let	VERB
ejpam-5596	278	2	δ	δ	PRON
ejpam-5596	278	3	be	be	AUX
ejpam-5596	278	4	a	a	DET
ejpam-5596	278	5	fuzzy	fuzzy	ADJ
ejpam-5596	278	6	n	n	CCONJ
ejpam-5596	278	7	-	-	ADJ
ejpam-5596	278	8	interior	interior	ADJ
ejpam-5596	278	9	ideal	ideal	NOUN
ejpam-5596	278	10	of	of	ADP
ejpam-5596	278	11	s.	s.	PROPN
ejpam-5596	278	12	then	then	ADV
ejpam-5596	278	13	δ	δ	PROPN
ejpam-5596	278	14	is	be	AUX
ejpam-5596	278	15	a	a	DET
ejpam-5596	278	16	fuzzy	fuzzy	ADJ
ejpam-5596	278	17	subsemigroup	subsemigroup	NOUN
ejpam-5596	278	18	of	of	ADP
ejpam-5596	278	19	s.	s.	PROPN
ejpam-5596	278	20	by	by	ADP
ejpam-5596	278	21	lemma	lemma	PROPN
ejpam-5596	278	22	2	2	NUM
ejpam-5596	278	23	,	,	PUNCT
ejpam-5596	278	24	ut	ut	PROPN
ejpam-5596	278	25	is	be	AUX
ejpam-5596	278	26	a	a	DET
ejpam-5596	278	27	subsemigroup	subsemigroup	NOUN
ejpam-5596	278	28	of	of	ADP
ejpam-5596	278	29	s.	s.	PROPN
ejpam-5596	278	30	let	let	VERB
ejpam-5596	278	31	r1	r1	PROPN
ejpam-5596	278	32	,	,	PUNCT
ejpam-5596	278	33	r2	r2	PROPN
ejpam-5596	278	34	,	,	PUNCT
ejpam-5596	278	35	.	.	PUNCT
ejpam-5596	278	36	.	.	PUNCT
ejpam-5596	278	37	.	.	PUNCT
ejpam-5596	279	1	rm	rm	PROPN
ejpam-5596	279	2	,	,	PUNCT
ejpam-5596	279	3	k	k	PROPN
ejpam-5596	279	4	,	,	PUNCT
ejpam-5596	279	5	h	h	NOUN
ejpam-5596	279	6	∈	∈	PROPN
ejpam-5596	279	7	u	u	PROPN
ejpam-5596	279	8	(	(	PUNCT
ejpam-5596	279	9	s	s	PROPN
ejpam-5596	279	10	,	,	PUNCT
ejpam-5596	279	11	t	t	PROPN
ejpam-5596	279	12	)	)	PUNCT
ejpam-5596	279	13	ϑ	ϑ	PROPN
ejpam-5596	279	14	.	.	PUNCT
ejpam-5596	280	1	then	then	ADV
ejpam-5596	280	2	δ(ri	δ(ri	NUM
ejpam-5596	280	3	)	)	PUNCT
ejpam-5596	280	4	≥	≥	PROPN
ejpam-5596	280	5	t	t	NOUN
ejpam-5596	280	6	for	for	ADP
ejpam-5596	280	7	some	some	DET
ejpam-5596	280	8	i	i	PRON
ejpam-5596	280	9	∈	∈	PROPN
ejpam-5596	280	10	{	{	PUNCT
ejpam-5596	280	11	1	1	NUM
ejpam-5596	280	12	,	,	PUNCT
ejpam-5596	280	13	2	2	NUM
ejpam-5596	280	14	,	,	PUNCT
ejpam-5596	280	15	.	.	PUNCT
ejpam-5596	280	16	.	.	PUNCT
ejpam-5596	281	1	.	.	PUNCT
ejpam-5596	281	2	,	,	PUNCT
ejpam-5596	281	3	n	n	CCONJ
ejpam-5596	281	4	}	}	PUNCT
ejpam-5596	281	5	.	.	PUNCT
ejpam-5596	282	1	by	by	ADP
ejpam-5596	282	2	assumption	assumption	NOUN
ejpam-5596	282	3	,	,	PUNCT
ejpam-5596	282	4	δ(hrni	δ(hrni	NOUN
ejpam-5596	282	5	k	k	NOUN
ejpam-5596	282	6	)	)	PUNCT
ejpam-5596	282	7	≥	≥	NOUN
ejpam-5596	282	8	δ(r1	δ(r1	NOUN
ejpam-5596	282	9	)	)	PUNCT
ejpam-5596	282	10	∧	∧	NOUN
ejpam-5596	282	11	δ(r2	δ(r2	NOUN
ejpam-5596	282	12	)	)	PUNCT
ejpam-5596	282	13	∧	∧	PROPN
ejpam-5596	282	14	·	·	PUNCT
ejpam-5596	282	15	·	·	PUNCT
ejpam-5596	282	16	·	·	PUNCT
ejpam-5596	283	1	∧	∧	PROPN
ejpam-5596	283	2	δ(rn	δ(rn	PROPN
ejpam-5596	283	3	)	)	PUNCT
ejpam-5596	283	4	.	.	PUNCT
ejpam-5596	284	1	thus	thus	ADV
ejpam-5596	284	2	,	,	PUNCT
ejpam-5596	284	3	δp	δp	INTJ
ejpam-5596	284	4	(	(	PUNCT
ejpam-5596	284	5	hrni	hrni	PROPN
ejpam-5596	284	6	k	k	PROPN
ejpam-5596	284	7	)	)	PUNCT
ejpam-5596	284	8	≥	≥	NOUN
ejpam-5596	284	9	t.	t.	NOUN
ejpam-5596	284	10	it	it	PRON
ejpam-5596	284	11	impiles	impile	VERB
ejpam-5596	284	12	that	that	SCONJ
ejpam-5596	284	13	,	,	PUNCT
ejpam-5596	284	14	rni	rni	PROPN
ejpam-5596	284	15	∈	∈	PROPN
ejpam-5596	284	16	ut	ut	PROPN
ejpam-5596	284	17	.	.	PROPN
ejpam-5596	284	18	hence	hence	ADV
ejpam-5596	284	19	,	,	PUNCT
ejpam-5596	284	20	ut	ut	PROPN
ejpam-5596	284	21	is	be	AUX
ejpam-5596	284	22	an	an	DET
ejpam-5596	284	23	n	n	CCONJ
ejpam-5596	284	24	-	-	PUNCT
ejpam-5596	284	25	interior	interior	ADJ
ejpam-5596	284	26	ideal	ideal	NOUN
ejpam-5596	284	27	of	of	ADP
ejpam-5596	284	28	s.	s.	PROPN
ejpam-5596	284	29	conversely	conversely	ADV
ejpam-5596	284	30	,	,	PUNCT
ejpam-5596	284	31	suppose	suppose	VERB
ejpam-5596	284	32	that	that	SCONJ
ejpam-5596	284	33	ut	ut	PROPN
ejpam-5596	284	34	is	be	AUX
ejpam-5596	284	35	an	an	DET
ejpam-5596	284	36	n	n	CCONJ
ejpam-5596	284	37	-	-	PUNCT
ejpam-5596	284	38	interior	interior	ADJ
ejpam-5596	284	39	ideal	ideal	NOUN
ejpam-5596	284	40	of	of	ADP
ejpam-5596	284	41	s.	s.	PROPN
ejpam-5596	284	42	then	then	ADV
ejpam-5596	284	43	ut	ut	PROPN
ejpam-5596	284	44	is	be	AUX
ejpam-5596	284	45	a	a	DET
ejpam-5596	284	46	subsemigroup	subsemigroup	NOUN
ejpam-5596	284	47	of	of	ADP
ejpam-5596	284	48	s.	s.	PROPN
ejpam-5596	284	49	by	by	ADP
ejpam-5596	284	50	lemma	lemma	PROPN
ejpam-5596	284	51	2	2	NUM
ejpam-5596	284	52	,	,	PUNCT
ejpam-5596	284	53	δ	δ	PROPN
ejpam-5596	284	54	is	be	AUX
ejpam-5596	284	55	a	a	DET
ejpam-5596	284	56	fuzzy	fuzzy	ADJ
ejpam-5596	284	57	subsemigroup	subsemigroup	NOUN
ejpam-5596	284	58	of	of	ADP
ejpam-5596	284	59	s.	s.	PROPN
ejpam-5596	284	60	if	if	SCONJ
ejpam-5596	284	61	δ	δ	PROPN
ejpam-5596	284	62	is	be	AUX
ejpam-5596	284	63	not	not	PART
ejpam-5596	284	64	a	a	DET
ejpam-5596	284	65	fuzzy	fuzzy	ADJ
ejpam-5596	284	66	ninterior	ninterior	ADJ
ejpam-5596	284	67	ideal	ideal	NOUN
ejpam-5596	284	68	of	of	ADP
ejpam-5596	284	69	e	e	NOUN
ejpam-5596	284	70	,	,	PUNCT
ejpam-5596	284	71	then	then	ADV
ejpam-5596	284	72	there	there	PRON
ejpam-5596	284	73	exists	exist	VERB
ejpam-5596	284	74	ri	ri	PROPN
ejpam-5596	284	75	,	,	PUNCT
ejpam-5596	284	76	k	k	PROPN
ejpam-5596	284	77	,	,	PUNCT
ejpam-5596	284	78	h	h	PROPN
ejpam-5596	284	79	∈	∈	PROPN
ejpam-5596	284	80	s	s	VERB
ejpam-5596	284	81	such	such	ADJ
ejpam-5596	284	82	that	that	DET
ejpam-5596	284	83	δ(hrni	δ(hrni	NOUN
ejpam-5596	284	84	k	k	NOUN
ejpam-5596	284	85	)	)	PUNCT
ejpam-5596	284	86	<	<	X
ejpam-5596	284	87	δ(r1)∧	δ(r1)∧	NOUN
ejpam-5596	284	88	δ(r2)∧	δ(r2)∧	ADP
ejpam-5596	284	89	·	·	PUNCT
ejpam-5596	284	90	·	·	PUNCT
ejpam-5596	284	91	·	·	PUNCT
ejpam-5596	285	1	∧	∧	PROPN
ejpam-5596	285	2	δ(rn	δ(rn	PROPN
ejpam-5596	285	3	)	)	PUNCT
ejpam-5596	285	4	.	.	PUNCT
ejpam-5596	286	1	by	by	ADP
ejpam-5596	286	2	assumption	assumption	NOUN
ejpam-5596	286	3	,	,	PUNCT
ejpam-5596	286	4	we	we	PRON
ejpam-5596	286	5	have	have	VERB
ejpam-5596	286	6	hrni	hrni	PROPN
ejpam-5596	286	7	k	k	PROPN
ejpam-5596	286	8	∈	∈	PROPN
ejpam-5596	286	9	ut	ut	PROPN
ejpam-5596	286	10	.	.	PROPN
ejpam-5596	287	1	thus	thus	ADV
ejpam-5596	287	2	,	,	PUNCT
ejpam-5596	287	3	δ(hr	δ(hr	NOUN
ejpam-5596	287	4	n	n	CCONJ
ejpam-5596	287	5	i	i	PRON
ejpam-5596	287	6	k	k	PROPN
ejpam-5596	287	7	)	)	PUNCT
ejpam-5596	287	8	≥	≥	NOUN
ejpam-5596	287	9	δ(r1	δ(r1	NOUN
ejpam-5596	287	10	)	)	PUNCT
ejpam-5596	287	11	∧	∧	NOUN
ejpam-5596	287	12	δ(r2	δ(r2	NOUN
ejpam-5596	287	13	)	)	PUNCT
ejpam-5596	287	14	∧	∧	PROPN
ejpam-5596	287	15	·	·	PUNCT
ejpam-5596	287	16	·	·	PUNCT
ejpam-5596	287	17	·	·	PUNCT
ejpam-5596	287	18	∧	∧	PROPN
ejpam-5596	287	19	δ(rn	δ(rn	PROPN
ejpam-5596	287	20	)	)	PUNCT
ejpam-5596	287	21	.	.	PUNCT
ejpam-5596	288	1	it	it	PRON
ejpam-5596	288	2	is	be	AUX
ejpam-5596	288	3	a	a	DET
ejpam-5596	288	4	contradiction	contradiction	NOUN
ejpam-5596	288	5	.	.	PUNCT
ejpam-5596	289	1	hence	hence	ADV
ejpam-5596	289	2	,	,	PUNCT
ejpam-5596	289	3	δ	δ	PROPN
ejpam-5596	289	4	is	be	AUX
ejpam-5596	289	5	a	a	DET
ejpam-5596	289	6	fuzzy	fuzzy	ADJ
ejpam-5596	289	7	n	n	CCONJ
ejpam-5596	289	8	-	-	ADJ
ejpam-5596	289	9	interior	interior	ADJ
ejpam-5596	289	10	ideal	ideal	NOUN
ejpam-5596	289	11	of	of	ADP
ejpam-5596	289	12	s.	s.	PROPN
ejpam-5596	289	13	definition	definition	NOUN
ejpam-5596	289	14	13	13	NUM
ejpam-5596	289	15	.	.	PUNCT
ejpam-5596	290	1	an	an	DET
ejpam-5596	290	2	n	n	CCONJ
ejpam-5596	290	3	-	-	PUNCT
ejpam-5596	290	4	interior	interior	ADJ
ejpam-5596	290	5	ideal	ideal	NOUN
ejpam-5596	290	6	k	k	PROPN
ejpam-5596	290	7	of	of	ADP
ejpam-5596	290	8	a	a	DET
ejpam-5596	290	9	semigroup	semigroup	NOUN
ejpam-5596	290	10	s	s	PART
ejpam-5596	290	11	is	be	AUX
ejpam-5596	290	12	called	call	VERB
ejpam-5596	290	13	a	a	DET
ejpam-5596	290	14	minimal	minimal	ADJ
ejpam-5596	290	15	if	if	SCONJ
ejpam-5596	290	16	for	for	ADP
ejpam-5596	290	17	every	every	DET
ejpam-5596	290	18	n	n	CCONJ
ejpam-5596	290	19	-	-	ADJ
ejpam-5596	290	20	interior	interior	ADJ
ejpam-5596	290	21	ideal	ideal	NOUN
ejpam-5596	290	22	of	of	ADP
ejpam-5596	290	23	j	j	PROPN
ejpam-5596	290	24	of	of	ADP
ejpam-5596	290	25	s	s	PRON
ejpam-5596	290	26	such	such	ADJ
ejpam-5596	290	27	that	that	SCONJ
ejpam-5596	290	28	j	j	PROPN
ejpam-5596	290	29	⊆	⊆	NUM
ejpam-5596	290	30	k	k	NOUN
ejpam-5596	290	31	,	,	PUNCT
ejpam-5596	290	32	we	we	PRON
ejpam-5596	290	33	have	have	VERB
ejpam-5596	290	34	j	j	PROPN
ejpam-5596	290	35	=	=	SYM
ejpam-5596	290	36	k.	k.	PROPN
ejpam-5596	290	37	definition	definition	NOUN
ejpam-5596	290	38	14	14	NUM
ejpam-5596	290	39	.	.	PUNCT
ejpam-5596	291	1	a	a	DET
ejpam-5596	291	2	fuzzy	fuzzy	ADJ
ejpam-5596	291	3	n	n	CCONJ
ejpam-5596	291	4	-	-	ADJ
ejpam-5596	291	5	interior	interior	ADJ
ejpam-5596	291	6	ideal	ideal	NOUN
ejpam-5596	291	7	δ	δ	PROPN
ejpam-5596	291	8	of	of	ADP
ejpam-5596	291	9	a	a	DET
ejpam-5596	291	10	semigroup	semigroup	NOUN
ejpam-5596	291	11	s	s	VERB
ejpam-5596	291	12	is	be	AUX
ejpam-5596	291	13	a	a	DET
ejpam-5596	291	14	minimal	minimal	ADJ
ejpam-5596	291	15	if	if	SCONJ
ejpam-5596	291	16	for	for	ADP
ejpam-5596	291	17	all	all	PRON
ejpam-5596	291	18	fuzzy	fuzzy	ADJ
ejpam-5596	291	19	n	n	CCONJ
ejpam-5596	291	20	-	-	ADJ
ejpam-5596	291	21	interior	interior	ADJ
ejpam-5596	291	22	ideal	ideal	NOUN
ejpam-5596	291	23	ξ	ξ	PROPN
ejpam-5596	291	24	of	of	ADP
ejpam-5596	291	25	s	s	PRON
ejpam-5596	291	26	such	such	ADJ
ejpam-5596	291	27	that	that	SCONJ
ejpam-5596	291	28	ξ	ξ	PROPN
ejpam-5596	291	29	≤	≤	PROPN
ejpam-5596	291	30	δ	δ	PROPN
ejpam-5596	291	31	,	,	PUNCT
ejpam-5596	291	32	then	then	ADV
ejpam-5596	291	33	ξ	ξ	X
ejpam-5596	291	34	=	=	SYM
ejpam-5596	291	35	δ	δ	PROPN
ejpam-5596	291	36	.	.	PUNCT
ejpam-5596	291	37	theorem	theorem	VERB
ejpam-5596	291	38	10	10	NUM
ejpam-5596	291	39	.	.	PUNCT
ejpam-5596	292	1	a	a	DET
ejpam-5596	292	2	non	non	ADJ
ejpam-5596	292	3	-	-	ADJ
ejpam-5596	292	4	empty	empty	ADJ
ejpam-5596	292	5	subset	subset	NOUN
ejpam-5596	292	6	k	k	PROPN
ejpam-5596	292	7	of	of	ADP
ejpam-5596	292	8	a	a	DET
ejpam-5596	292	9	semigroup	semigroup	NOUN
ejpam-5596	292	10	s	s	VERB
ejpam-5596	292	11	is	be	AUX
ejpam-5596	292	12	a	a	DET
ejpam-5596	292	13	minimal	minimal	ADJ
ejpam-5596	292	14	n	n	CCONJ
ejpam-5596	292	15	-	-	ADJ
ejpam-5596	292	16	interior	interior	ADJ
ejpam-5596	292	17	ideal	ideal	NOUN
ejpam-5596	292	18	if	if	SCONJ
ejpam-5596	292	19	and	and	CCONJ
ejpam-5596	292	20	only	only	ADV
ejpam-5596	292	21	if	if	SCONJ
ejpam-5596	292	22	λk	λk	PRON
ejpam-5596	292	23	is	be	AUX
ejpam-5596	292	24	a	a	DET
ejpam-5596	292	25	minimal	minimal	ADJ
ejpam-5596	292	26	fuzzy	fuzzy	ADJ
ejpam-5596	292	27	n	n	CCONJ
ejpam-5596	292	28	-	-	PUNCT
ejpam-5596	292	29	interior	interior	ADJ
ejpam-5596	292	30	ideal	ideal	NOUN
ejpam-5596	292	31	s.	s.	PROPN
ejpam-5596	292	32	proof	proof	PROPN
ejpam-5596	292	33	.	.	PUNCT
ejpam-5596	293	1	let	let	VERB
ejpam-5596	293	2	k	k	PRON
ejpam-5596	293	3	be	be	AUX
ejpam-5596	293	4	a	a	DET
ejpam-5596	293	5	minimal	minimal	ADJ
ejpam-5596	293	6	n	n	CCONJ
ejpam-5596	293	7	-	-	ADJ
ejpam-5596	293	8	interior	interior	ADJ
ejpam-5596	293	9	ideal	ideal	NOUN
ejpam-5596	293	10	of	of	ADP
ejpam-5596	293	11	s.	s.	PROPN
ejpam-5596	293	12	then	then	ADV
ejpam-5596	293	13	k	k	PROPN
ejpam-5596	293	14	is	be	AUX
ejpam-5596	293	15	an	an	DET
ejpam-5596	293	16	n	n	CCONJ
ejpam-5596	293	17	-	-	PUNCT
ejpam-5596	293	18	interior	interior	ADJ
ejpam-5596	293	19	ideal	ideal	NOUN
ejpam-5596	293	20	of	of	ADP
ejpam-5596	293	21	s.	s.	PROPN
ejpam-5596	293	22	thus	thus	ADV
ejpam-5596	293	23	,	,	PUNCT
ejpam-5596	293	24	by	by	ADP
ejpam-5596	293	25	theorem	theorem	NOUN
ejpam-5596	293	26	8	8	NUM
ejpam-5596	293	27	,	,	PUNCT
ejpam-5596	293	28	λk	λk	PRON
ejpam-5596	293	29	is	be	AUX
ejpam-5596	293	30	a	a	DET
ejpam-5596	293	31	fuzzy	fuzzy	ADJ
ejpam-5596	293	32	n	n	CCONJ
ejpam-5596	293	33	-	-	ADJ
ejpam-5596	293	34	interior	interior	ADJ
ejpam-5596	293	35	ideal	ideal	NOUN
ejpam-5596	293	36	of	of	ADP
ejpam-5596	293	37	e.	e.	PROPN
ejpam-5596	293	38	let	let	VERB
ejpam-5596	293	39	j	j	PROPN
ejpam-5596	293	40	be	be	AUX
ejpam-5596	293	41	an	an	DET
ejpam-5596	293	42	n	n	CCONJ
ejpam-5596	293	43	-	-	PUNCT
ejpam-5596	293	44	interior	interior	ADJ
ejpam-5596	293	45	ideal	ideal	NOUN
ejpam-5596	293	46	of	of	ADP
ejpam-5596	293	47	e	e	PRON
ejpam-5596	294	1	such	such	ADJ
ejpam-5596	294	2	that	that	SCONJ
ejpam-5596	294	3	j	j	PROPN
ejpam-5596	294	4	⊆	⊆	NUM
ejpam-5596	294	5	k.	k.	PROPN
ejpam-5596	294	6	then	then	ADV
ejpam-5596	294	7	by	by	ADP
ejpam-5596	294	8	theorem	theorem	NOUN
ejpam-5596	294	9	8	8	NUM
ejpam-5596	294	10	,	,	PUNCT
ejpam-5596	294	11	λj	λj	X
ejpam-5596	294	12	is	be	AUX
ejpam-5596	294	13	a	a	DET
ejpam-5596	294	14	fuzzy	fuzzy	ADJ
ejpam-5596	294	15	n	n	CCONJ
ejpam-5596	294	16	-	-	ADJ
ejpam-5596	294	17	interior	interior	ADJ
ejpam-5596	294	18	ideal	ideal	NOUN
ejpam-5596	294	19	of	of	ADP
ejpam-5596	294	20	s	s	PRON
ejpam-5596	294	21	and	and	CCONJ
ejpam-5596	294	22	λj	λj	PROPN
ejpam-5596	294	23	≤	≤	PROPN
ejpam-5596	294	24	λk	λk	X
ejpam-5596	294	25	.	.	PUNCT
ejpam-5596	295	1	since	since	SCONJ
ejpam-5596	295	2	k	k	PROPN
ejpam-5596	295	3	is	be	AUX
ejpam-5596	295	4	a	a	DET
ejpam-5596	295	5	minimal	minimal	ADJ
ejpam-5596	295	6	n	n	CCONJ
ejpam-5596	295	7	-	-	ADJ
ejpam-5596	295	8	interior	interior	ADJ
ejpam-5596	295	9	ideal	ideal	NOUN
ejpam-5596	295	10	of	of	ADP
ejpam-5596	295	11	s	s	PRON
ejpam-5596	295	12	we	we	PRON
ejpam-5596	295	13	have	have	VERB
ejpam-5596	295	14	j	j	PROPN
ejpam-5596	295	15	=	=	PROPN
ejpam-5596	295	16	k.	k.	PROPN
ejpam-5596	295	17	thus	thus	ADV
ejpam-5596	295	18	,	,	PUNCT
ejpam-5596	295	19	λj	λj	PROPN
ejpam-5596	295	20	=	=	X
ejpam-5596	295	21	λk	λk	PROPN
ejpam-5596	295	22	.	.	PUNCT
ejpam-5596	296	1	hence	hence	ADV
ejpam-5596	296	2	,	,	PUNCT
ejpam-5596	296	3	λk	λk	ADV
ejpam-5596	296	4	is	be	AUX
ejpam-5596	296	5	minimal	minimal	ADJ
ejpam-5596	296	6	fuzzy	fuzzy	ADJ
ejpam-5596	296	7	n	n	CCONJ
ejpam-5596	296	8	-	-	ADJ
ejpam-5596	296	9	interior	interior	ADJ
ejpam-5596	296	10	ideal	ideal	NOUN
ejpam-5596	296	11	of	of	ADP
ejpam-5596	296	12	s.	s.	PROPN
ejpam-5596	296	13	p.	p.	PROPN
ejpam-5596	296	14	khamrot	khamrot	PROPN
ejpam-5596	296	15	,	,	PUNCT
ejpam-5596	296	16	a.	a.	NOUN
ejpam-5596	296	17	iampan	iampan	PROPN
ejpam-5596	296	18	,	,	PUNCT
ejpam-5596	296	19	t.	t.	PROPN
ejpam-5596	296	20	gaketem	gaketem	PROPN
ejpam-5596	296	21	/	/	SYM
ejpam-5596	296	22	eur	eur	PROPN
ejpam-5596	296	23	.	.	PUNCT
ejpam-5596	297	1	j.	j.	PROPN
ejpam-5596	297	2	pure	pure	PROPN
ejpam-5596	297	3	appl	appl	PROPN
ejpam-5596	297	4	.	.	PROPN
ejpam-5596	297	5	math	math	PROPN
ejpam-5596	297	6	,	,	PUNCT
ejpam-5596	297	7	18	18	NUM
ejpam-5596	297	8	(	(	PUNCT
ejpam-5596	297	9	1	1	NUM
ejpam-5596	297	10	)	)	PUNCT
ejpam-5596	297	11	(	(	PUNCT
ejpam-5596	297	12	2025	2025	NUM
ejpam-5596	297	13	)	)	PUNCT
ejpam-5596	297	14	,	,	PUNCT
ejpam-5596	297	15	5596	5596	NUM
ejpam-5596	297	16	10	10	NUM
ejpam-5596	297	17	of	of	ADP
ejpam-5596	297	18	12	12	NUM
ejpam-5596	297	19	conversely	conversely	ADV
ejpam-5596	297	20	,	,	PUNCT
ejpam-5596	297	21	λk	λk	PRON
ejpam-5596	297	22	is	be	AUX
ejpam-5596	297	23	minimal	minimal	ADJ
ejpam-5596	297	24	fuzzy	fuzzy	ADJ
ejpam-5596	297	25	n	n	CCONJ
ejpam-5596	297	26	-	-	ADJ
ejpam-5596	297	27	interior	interior	ADJ
ejpam-5596	297	28	ideal	ideal	NOUN
ejpam-5596	297	29	of	of	ADP
ejpam-5596	297	30	s.	s.	PROPN
ejpam-5596	298	1	then	then	ADV
ejpam-5596	298	2	λk	λk	X
ejpam-5596	298	3	is	be	AUX
ejpam-5596	298	4	a	a	DET
ejpam-5596	298	5	fuzzy	fuzzy	ADJ
ejpam-5596	298	6	n	n	CCONJ
ejpam-5596	298	7	-	-	ADJ
ejpam-5596	298	8	interior	interior	ADJ
ejpam-5596	298	9	ideal	ideal	NOUN
ejpam-5596	298	10	of	of	ADP
ejpam-5596	298	11	s.	s.	PROPN
ejpam-5596	298	12	thus	thus	ADV
ejpam-5596	298	13	,	,	PUNCT
ejpam-5596	298	14	by	by	ADP
ejpam-5596	298	15	theorem	theorem	NOUN
ejpam-5596	298	16	8	8	NUM
ejpam-5596	298	17	,	,	PUNCT
ejpam-5596	298	18	k	k	PROPN
ejpam-5596	298	19	is	be	AUX
ejpam-5596	298	20	an	an	DET
ejpam-5596	298	21	n	n	CCONJ
ejpam-5596	298	22	-	-	PUNCT
ejpam-5596	298	23	interior	interior	ADJ
ejpam-5596	298	24	ideal	ideal	NOUN
ejpam-5596	298	25	of	of	ADP
ejpam-5596	298	26	s.	s.	PROPN
ejpam-5596	298	27	let	let	VERB
ejpam-5596	298	28	λj	λj	PRON
ejpam-5596	298	29	be	be	AUX
ejpam-5596	298	30	a	a	DET
ejpam-5596	298	31	fuzzy	fuzzy	ADJ
ejpam-5596	298	32	n	n	CCONJ
ejpam-5596	298	33	-	-	ADJ
ejpam-5596	298	34	interior	interior	ADJ
ejpam-5596	298	35	ideal	ideal	NOUN
ejpam-5596	298	36	of	of	ADP
ejpam-5596	298	37	s	s	PRON
ejpam-5596	298	38	such	such	ADJ
ejpam-5596	298	39	that	that	SCONJ
ejpam-5596	298	40	λj	λj	PROPN
ejpam-5596	298	41	≤	≤	X
ejpam-5596	299	1	λk	λk	X
ejpam-5596	299	2	.	.	PUNCT
ejpam-5596	300	1	then	then	ADV
ejpam-5596	300	2	by	by	ADP
ejpam-5596	300	3	theorem	theorem	NOUN
ejpam-5596	300	4	8	8	NUM
ejpam-5596	300	5	,	,	PUNCT
ejpam-5596	300	6	j	j	PROPN
ejpam-5596	300	7	is	be	AUX
ejpam-5596	300	8	an	an	DET
ejpam-5596	300	9	n	n	CCONJ
ejpam-5596	300	10	-	-	PUNCT
ejpam-5596	300	11	interior	interior	ADJ
ejpam-5596	300	12	ideal	ideal	NOUN
ejpam-5596	300	13	of	of	ADP
ejpam-5596	300	14	s	s	PRON
ejpam-5596	300	15	such	such	ADJ
ejpam-5596	300	16	that	that	SCONJ
ejpam-5596	300	17	j	j	PROPN
ejpam-5596	300	18	⊆	⊆	NUM
ejpam-5596	300	19	k.	k.	NOUN
ejpam-5596	300	20	since	since	SCONJ
ejpam-5596	300	21	λk	λk	PROPN
ejpam-5596	300	22	is	be	AUX
ejpam-5596	300	23	minimal	minimal	ADJ
ejpam-5596	300	24	fuzzy	fuzzy	ADJ
ejpam-5596	300	25	n	n	CCONJ
ejpam-5596	300	26	-	-	ADJ
ejpam-5596	300	27	interior	interior	ADJ
ejpam-5596	300	28	ideal	ideal	NOUN
ejpam-5596	300	29	of	of	ADP
ejpam-5596	300	30	s	s	PRON
ejpam-5596	300	31	we	we	PRON
ejpam-5596	300	32	have	have	VERB
ejpam-5596	300	33	λj	λj	PROPN
ejpam-5596	300	34	=	=	SYM
ejpam-5596	300	35	λk	λk	PROPN
ejpam-5596	300	36	.	.	PUNCT
ejpam-5596	301	1	thus	thus	ADV
ejpam-5596	301	2	,	,	PUNCT
ejpam-5596	301	3	j	j	PROPN
ejpam-5596	301	4	=	=	PROPN
ejpam-5596	301	5	k.	k.	PROPN
ejpam-5596	301	6	hence	hence	ADV
ejpam-5596	301	7	,	,	PUNCT
ejpam-5596	301	8	k	k	PROPN
ejpam-5596	301	9	is	be	AUX
ejpam-5596	301	10	a	a	DET
ejpam-5596	301	11	minimal	minimal	ADJ
ejpam-5596	301	12	n	n	CCONJ
ejpam-5596	301	13	-	-	ADJ
ejpam-5596	301	14	interior	interior	ADJ
ejpam-5596	301	15	ideal	ideal	NOUN
ejpam-5596	301	16	of	of	ADP
ejpam-5596	301	17	s.	s.	PROPN
ejpam-5596	301	18	definition	definition	PROPN
ejpam-5596	301	19	15	15	NUM
ejpam-5596	301	20	.	.	PUNCT
ejpam-5596	302	1	an	an	DET
ejpam-5596	302	2	n	n	CCONJ
ejpam-5596	302	3	-	-	PUNCT
ejpam-5596	302	4	interior	interior	ADJ
ejpam-5596	302	5	ideal	ideal	NOUN
ejpam-5596	302	6	k	k	PROPN
ejpam-5596	302	7	of	of	ADP
ejpam-5596	302	8	a	a	DET
ejpam-5596	302	9	semigroup	semigroup	NOUN
ejpam-5596	302	10	s	s	PART
ejpam-5596	302	11	is	be	AUX
ejpam-5596	302	12	called	call	VERB
ejpam-5596	302	13	a	a	DET
ejpam-5596	302	14	maximalif	maximalif	NOUN
ejpam-5596	302	15	for	for	ADP
ejpam-5596	302	16	every	every	DET
ejpam-5596	302	17	n	n	CCONJ
ejpam-5596	302	18	-	-	ADJ
ejpam-5596	302	19	interior	interior	ADJ
ejpam-5596	302	20	ideal	ideal	NOUN
ejpam-5596	302	21	of	of	ADP
ejpam-5596	302	22	j	j	PROPN
ejpam-5596	302	23	of	of	ADP
ejpam-5596	302	24	e	e	PROPN
ejpam-5596	302	25	such	such	ADJ
ejpam-5596	302	26	that	that	SCONJ
ejpam-5596	302	27	k	k	PROPN
ejpam-5596	302	28	⊆	⊆	NUM
ejpam-5596	302	29	j	j	PROPN
ejpam-5596	302	30	,	,	PUNCT
ejpam-5596	302	31	we	we	PRON
ejpam-5596	302	32	have	have	VERB
ejpam-5596	302	33	j	j	PROPN
ejpam-5596	302	34	=	=	SYM
ejpam-5596	302	35	k.	k.	PROPN
ejpam-5596	302	36	definition	definition	NOUN
ejpam-5596	302	37	16	16	NUM
ejpam-5596	302	38	.	.	PUNCT
ejpam-5596	303	1	a	a	DET
ejpam-5596	303	2	fuzzy	fuzzy	ADJ
ejpam-5596	303	3	n	n	CCONJ
ejpam-5596	303	4	-	-	ADJ
ejpam-5596	303	5	interior	interior	ADJ
ejpam-5596	303	6	ideal	ideal	NOUN
ejpam-5596	303	7	δ	δ	PROPN
ejpam-5596	303	8	of	of	ADP
ejpam-5596	303	9	a	a	DET
ejpam-5596	303	10	semigroup	semigroup	NOUN
ejpam-5596	303	11	s	s	VERB
ejpam-5596	303	12	is	be	AUX
ejpam-5596	303	13	a	a	DET
ejpam-5596	303	14	maximal	maximal	ADJ
ejpam-5596	303	15	if	if	SCONJ
ejpam-5596	303	16	for	for	ADP
ejpam-5596	303	17	all	all	PRON
ejpam-5596	303	18	fuzzy	fuzzy	ADJ
ejpam-5596	303	19	n	n	CCONJ
ejpam-5596	303	20	-	-	ADJ
ejpam-5596	303	21	interior	interior	ADJ
ejpam-5596	303	22	ideal	ideal	NOUN
ejpam-5596	303	23	ξ	ξ	PROPN
ejpam-5596	303	24	of	of	ADP
ejpam-5596	303	25	e	e	PRON
ejpam-5596	303	26	such	such	ADJ
ejpam-5596	303	27	that	that	SCONJ
ejpam-5596	303	28	δ	δ	PROPN
ejpam-5596	303	29	≤	≤	PROPN
ejpam-5596	303	30	ξ	ξ	PROPN
ejpam-5596	303	31	,	,	PUNCT
ejpam-5596	303	32	then	then	ADV
ejpam-5596	303	33	ξ	ξ	X
ejpam-5596	303	34	=	=	SYM
ejpam-5596	303	35	δ	δ	PROPN
ejpam-5596	303	36	.	.	PUNCT
ejpam-5596	303	37	theorem	theorem	VERB
ejpam-5596	303	38	11	11	NUM
ejpam-5596	303	39	.	.	PUNCT
ejpam-5596	304	1	a	a	DET
ejpam-5596	304	2	non	non	ADJ
ejpam-5596	304	3	-	-	ADJ
ejpam-5596	304	4	empty	empty	ADJ
ejpam-5596	304	5	subset	subset	NOUN
ejpam-5596	304	6	k	k	PROPN
ejpam-5596	304	7	of	of	ADP
ejpam-5596	304	8	a	a	DET
ejpam-5596	304	9	semigroup	semigroup	NOUN
ejpam-5596	304	10	s	s	VERB
ejpam-5596	304	11	is	be	AUX
ejpam-5596	304	12	a	a	DET
ejpam-5596	304	13	maximal	maximal	ADJ
ejpam-5596	304	14	n	n	CCONJ
ejpam-5596	304	15	-	-	ADJ
ejpam-5596	304	16	interior	interior	ADJ
ejpam-5596	304	17	ideal	ideal	NOUN
ejpam-5596	304	18	if	if	SCONJ
ejpam-5596	304	19	and	and	CCONJ
ejpam-5596	304	20	only	only	ADV
ejpam-5596	304	21	if	if	SCONJ
ejpam-5596	304	22	λk	λk	PRON
ejpam-5596	304	23	is	be	AUX
ejpam-5596	304	24	a	a	DET
ejpam-5596	304	25	maximal	maximal	ADJ
ejpam-5596	304	26	fuzzy	fuzzy	ADJ
ejpam-5596	304	27	n	n	CCONJ
ejpam-5596	304	28	-	-	PUNCT
ejpam-5596	304	29	interior	interior	ADJ
ejpam-5596	304	30	ideal	ideal	NOUN
ejpam-5596	304	31	s.	s.	PROPN
ejpam-5596	304	32	proof	proof	PROPN
ejpam-5596	304	33	.	.	PUNCT
ejpam-5596	305	1	let	let	VERB
ejpam-5596	305	2	k	k	PRON
ejpam-5596	305	3	be	be	AUX
ejpam-5596	305	4	a	a	DET
ejpam-5596	305	5	maximal	maximal	ADJ
ejpam-5596	305	6	n	n	CCONJ
ejpam-5596	305	7	-	-	ADJ
ejpam-5596	305	8	interior	interior	ADJ
ejpam-5596	305	9	ideal	ideal	NOUN
ejpam-5596	305	10	of	of	ADP
ejpam-5596	305	11	s.	s.	PROPN
ejpam-5596	305	12	then	then	ADV
ejpam-5596	305	13	k	k	PROPN
ejpam-5596	305	14	is	be	AUX
ejpam-5596	305	15	an	an	DET
ejpam-5596	305	16	n	n	CCONJ
ejpam-5596	305	17	-	-	PUNCT
ejpam-5596	305	18	interior	interior	ADJ
ejpam-5596	305	19	ideal	ideal	NOUN
ejpam-5596	305	20	of	of	ADP
ejpam-5596	305	21	s.	s.	PROPN
ejpam-5596	305	22	thus	thus	ADV
ejpam-5596	305	23	,	,	PUNCT
ejpam-5596	305	24	by	by	ADP
ejpam-5596	305	25	theorem	theorem	NOUN
ejpam-5596	305	26	8	8	NUM
ejpam-5596	305	27	,	,	PUNCT
ejpam-5596	305	28	λk	λk	PRON
ejpam-5596	305	29	is	be	AUX
ejpam-5596	305	30	a	a	DET
ejpam-5596	305	31	fuzzy	fuzzy	ADJ
ejpam-5596	305	32	n	n	CCONJ
ejpam-5596	305	33	-	-	ADJ
ejpam-5596	305	34	interior	interior	ADJ
ejpam-5596	305	35	ideal	ideal	NOUN
ejpam-5596	305	36	of	of	ADP
ejpam-5596	305	37	e.	e.	PROPN
ejpam-5596	305	38	let	let	VERB
ejpam-5596	305	39	j	j	PROPN
ejpam-5596	305	40	be	be	AUX
ejpam-5596	305	41	an	an	DET
ejpam-5596	305	42	n	n	CCONJ
ejpam-5596	305	43	-	-	PUNCT
ejpam-5596	305	44	interior	interior	ADJ
ejpam-5596	305	45	ideal	ideal	NOUN
ejpam-5596	305	46	of	of	ADP
ejpam-5596	305	47	e	e	PRON
ejpam-5596	305	48	such	such	ADJ
ejpam-5596	305	49	that	that	SCONJ
ejpam-5596	305	50	k	k	PROPN
ejpam-5596	305	51	⊆	⊆	NUM
ejpam-5596	305	52	j	j	PROPN
ejpam-5596	305	53	.	.	PUNCT
ejpam-5596	306	1	then	then	ADV
ejpam-5596	306	2	by	by	ADP
ejpam-5596	306	3	theorem	theorem	NOUN
ejpam-5596	306	4	8	8	NUM
ejpam-5596	306	5	,	,	PUNCT
ejpam-5596	306	6	λj	λj	X
ejpam-5596	306	7	is	be	AUX
ejpam-5596	306	8	a	a	DET
ejpam-5596	306	9	fuzzy	fuzzy	ADJ
ejpam-5596	306	10	n	n	CCONJ
ejpam-5596	306	11	-	-	ADJ
ejpam-5596	306	12	interior	interior	ADJ
ejpam-5596	306	13	ideal	ideal	NOUN
ejpam-5596	306	14	of	of	ADP
ejpam-5596	306	15	s	s	PRON
ejpam-5596	306	16	and	and	CCONJ
ejpam-5596	306	17	λk	λk	X
ejpam-5596	306	18	≤	≤	NUM
ejpam-5596	306	19	λj	λj	INTJ
ejpam-5596	306	20	.	.	PUNCT
ejpam-5596	307	1	since	since	SCONJ
ejpam-5596	307	2	k	k	PROPN
ejpam-5596	307	3	is	be	AUX
ejpam-5596	307	4	a	a	DET
ejpam-5596	307	5	maximal	maximal	ADJ
ejpam-5596	307	6	n	n	CCONJ
ejpam-5596	307	7	-	-	ADJ
ejpam-5596	307	8	interior	interior	ADJ
ejpam-5596	307	9	ideal	ideal	NOUN
ejpam-5596	307	10	of	of	ADP
ejpam-5596	307	11	s	s	PRON
ejpam-5596	307	12	we	we	PRON
ejpam-5596	307	13	have	have	VERB
ejpam-5596	307	14	j	j	PROPN
ejpam-5596	307	15	=	=	PROPN
ejpam-5596	307	16	k.	k.	PROPN
ejpam-5596	307	17	thus	thus	ADV
ejpam-5596	307	18	,	,	PUNCT
ejpam-5596	307	19	λj	λj	PROPN
ejpam-5596	307	20	=	=	X
ejpam-5596	307	21	λk	λk	PROPN
ejpam-5596	307	22	.	.	PUNCT
ejpam-5596	308	1	hence	hence	ADV
ejpam-5596	308	2	,	,	PUNCT
ejpam-5596	308	3	λk	λk	PROPN
ejpam-5596	308	4	is	be	AUX
ejpam-5596	308	5	maximal	maximal	ADJ
ejpam-5596	308	6	fuzzy	fuzzy	ADJ
ejpam-5596	308	7	n	n	CCONJ
ejpam-5596	308	8	-	-	ADJ
ejpam-5596	308	9	interior	interior	ADJ
ejpam-5596	308	10	ideal	ideal	NOUN
ejpam-5596	308	11	of	of	ADP
ejpam-5596	308	12	s.	s.	PROPN
ejpam-5596	308	13	conversely	conversely	ADV
ejpam-5596	308	14	,	,	PUNCT
ejpam-5596	308	15	λk	λk	PROPN
ejpam-5596	308	16	is	be	AUX
ejpam-5596	308	17	maximal	maximal	ADJ
ejpam-5596	308	18	fuzzy	fuzzy	ADJ
ejpam-5596	308	19	n	n	CCONJ
ejpam-5596	308	20	-	-	ADJ
ejpam-5596	308	21	interior	interior	ADJ
ejpam-5596	308	22	ideal	ideal	NOUN
ejpam-5596	308	23	of	of	ADP
ejpam-5596	308	24	s.	s.	PROPN
ejpam-5596	308	25	then	then	ADV
ejpam-5596	308	26	λk	λk	X
ejpam-5596	308	27	is	be	AUX
ejpam-5596	308	28	a	a	DET
ejpam-5596	308	29	fuzzy	fuzzy	ADJ
ejpam-5596	308	30	n	n	CCONJ
ejpam-5596	308	31	-	-	ADJ
ejpam-5596	308	32	interior	interior	ADJ
ejpam-5596	308	33	ideal	ideal	NOUN
ejpam-5596	308	34	of	of	ADP
ejpam-5596	308	35	s.	s.	PROPN
ejpam-5596	308	36	thus	thus	ADV
ejpam-5596	308	37	,	,	PUNCT
ejpam-5596	308	38	by	by	ADP
ejpam-5596	308	39	theorem	theorem	NOUN
ejpam-5596	308	40	8	8	NUM
ejpam-5596	308	41	,	,	PUNCT
ejpam-5596	308	42	k	k	PROPN
ejpam-5596	308	43	is	be	AUX
ejpam-5596	308	44	an	an	DET
ejpam-5596	308	45	n	n	CCONJ
ejpam-5596	308	46	-	-	PUNCT
ejpam-5596	308	47	interior	interior	ADJ
ejpam-5596	308	48	ideal	ideal	NOUN
ejpam-5596	308	49	of	of	ADP
ejpam-5596	308	50	s.	s.	PROPN
ejpam-5596	308	51	let	let	VERB
ejpam-5596	308	52	λj	λj	PRON
ejpam-5596	308	53	be	be	AUX
ejpam-5596	308	54	a	a	DET
ejpam-5596	308	55	fuzzy	fuzzy	ADJ
ejpam-5596	308	56	n	n	CCONJ
ejpam-5596	308	57	-	-	ADJ
ejpam-5596	308	58	interior	interior	ADJ
ejpam-5596	308	59	ideal	ideal	NOUN
ejpam-5596	308	60	of	of	ADP
ejpam-5596	308	61	s	s	PRON
ejpam-5596	308	62	such	such	ADJ
ejpam-5596	308	63	that	that	SCONJ
ejpam-5596	308	64	λk	λk	ADP
ejpam-5596	308	65	≤	≤	ADV
ejpam-5596	308	66	λj	λj	INTJ
ejpam-5596	308	67	.	.	PUNCT
ejpam-5596	309	1	then	then	ADV
ejpam-5596	309	2	by	by	ADP
ejpam-5596	309	3	theorem	theorem	NOUN
ejpam-5596	309	4	8	8	NUM
ejpam-5596	309	5	,	,	PUNCT
ejpam-5596	309	6	j	j	PROPN
ejpam-5596	309	7	is	be	AUX
ejpam-5596	309	8	an	an	DET
ejpam-5596	309	9	n	n	CCONJ
ejpam-5596	309	10	-	-	PUNCT
ejpam-5596	309	11	interior	interior	ADJ
ejpam-5596	309	12	ideal	ideal	NOUN
ejpam-5596	309	13	of	of	ADP
ejpam-5596	309	14	s	s	PRON
ejpam-5596	309	15	such	such	ADJ
ejpam-5596	309	16	that	that	SCONJ
ejpam-5596	309	17	k	k	PROPN
ejpam-5596	309	18	⊆	⊆	NUM
ejpam-5596	309	19	j	j	PROPN
ejpam-5596	309	20	.	.	PUNCT
ejpam-5596	310	1	since	since	SCONJ
ejpam-5596	310	2	λk	λk	PROPN
ejpam-5596	310	3	is	be	AUX
ejpam-5596	310	4	maximal	maximal	ADJ
ejpam-5596	310	5	fuzzy	fuzzy	ADJ
ejpam-5596	310	6	n	n	CCONJ
ejpam-5596	310	7	-	-	ADJ
ejpam-5596	310	8	interior	interior	ADJ
ejpam-5596	310	9	ideal	ideal	NOUN
ejpam-5596	310	10	of	of	ADP
ejpam-5596	310	11	s	s	PRON
ejpam-5596	310	12	we	we	PRON
ejpam-5596	310	13	have	have	VERB
ejpam-5596	310	14	λj	λj	PROPN
ejpam-5596	310	15	=	=	SYM
ejpam-5596	310	16	λk	λk	PROPN
ejpam-5596	310	17	.	.	PUNCT
ejpam-5596	311	1	thus	thus	ADV
ejpam-5596	311	2	,	,	PUNCT
ejpam-5596	311	3	j	j	PROPN
ejpam-5596	311	4	=	=	PROPN
ejpam-5596	311	5	k.	k.	PROPN
ejpam-5596	311	6	hence	hence	ADV
ejpam-5596	311	7	,	,	PUNCT
ejpam-5596	311	8	k	k	PROPN
ejpam-5596	311	9	is	be	AUX
ejpam-5596	311	10	a	a	DET
ejpam-5596	311	11	maximal	maximal	ADJ
ejpam-5596	311	12	n	n	CCONJ
ejpam-5596	311	13	-	-	ADJ
ejpam-5596	311	14	interior	interior	ADJ
ejpam-5596	311	15	ideal	ideal	NOUN
ejpam-5596	311	16	of	of	ADP
ejpam-5596	311	17	s.	s.	PROPN
ejpam-5596	311	18	next	next	ADV
ejpam-5596	311	19	,	,	PUNCT
ejpam-5596	311	20	we	we	PRON
ejpam-5596	311	21	give	give	VERB
ejpam-5596	311	22	the	the	DET
ejpam-5596	311	23	relationship	relationship	NOUN
ejpam-5596	311	24	between	between	ADP
ejpam-5596	311	25	prime	prime	ADJ
ejpam-5596	311	26	,	,	PUNCT
ejpam-5596	311	27	semiprime	semiprime	NOUN
ejpam-5596	311	28	n	n	CCONJ
ejpam-5596	311	29	-	-	PUNCT
ejpam-5596	311	30	interior	interior	ADJ
ejpam-5596	311	31	ideals	ideal	NOUN
ejpam-5596	311	32	and	and	CCONJ
ejpam-5596	311	33	prime	prime	ADJ
ejpam-5596	311	34	,	,	PUNCT
ejpam-5596	311	35	semiprime	semiprime	NOUN
ejpam-5596	311	36	fuzzy	fuzzy	ADJ
ejpam-5596	311	37	n	n	CCONJ
ejpam-5596	311	38	-	-	ADJ
ejpam-5596	311	39	interior	interior	ADJ
ejpam-5596	311	40	ideals	ideal	NOUN
ejpam-5596	311	41	.	.	PUNCT
ejpam-5596	312	1	definition	definition	NOUN
ejpam-5596	312	2	17	17	NUM
ejpam-5596	312	3	.	.	PUNCT
ejpam-5596	313	1	let	let	VERB
ejpam-5596	313	2	k	k	PRON
ejpam-5596	313	3	be	be	AUX
ejpam-5596	313	4	an	an	DET
ejpam-5596	313	5	n	n	CCONJ
ejpam-5596	313	6	-	-	PUNCT
ejpam-5596	313	7	interior	interior	ADJ
ejpam-5596	313	8	ideal	ideal	NOUN
ejpam-5596	313	9	of	of	ADP
ejpam-5596	313	10	a	a	DET
ejpam-5596	313	11	semigroup	semigroup	NOUN
ejpam-5596	313	12	s	s	PART
ejpam-5596	313	13	is	be	AUX
ejpam-5596	313	14	called	call	VERB
ejpam-5596	313	15	(	(	PUNCT
ejpam-5596	313	16	1	1	NUM
ejpam-5596	313	17	)	)	PUNCT
ejpam-5596	313	18	prime	prime	NOUN
ejpam-5596	313	19	if	if	SCONJ
ejpam-5596	313	20	eh	eh	INTJ
ejpam-5596	313	21	∈	∈	PROPN
ejpam-5596	313	22	k	k	PROPN
ejpam-5596	313	23	implies	imply	VERB
ejpam-5596	313	24	e	e	PROPN
ejpam-5596	313	25	∈	∈	PROPN
ejpam-5596	313	26	k	k	PROPN
ejpam-5596	313	27	or	or	CCONJ
ejpam-5596	313	28	h	h	NOUN
ejpam-5596	313	29	∈	∈	PROPN
ejpam-5596	313	30	k	k	PROPN
ejpam-5596	313	31	for	for	ADP
ejpam-5596	313	32	all	all	DET
ejpam-5596	313	33	e	e	NOUN
ejpam-5596	313	34	,	,	PUNCT
ejpam-5596	313	35	h	h	NOUN
ejpam-5596	313	36	∈	∈	PROPN
ejpam-5596	313	37	s	s	PROPN
ejpam-5596	313	38	,	,	PUNCT
ejpam-5596	313	39	(	(	PUNCT
ejpam-5596	313	40	2	2	NUM
ejpam-5596	313	41	)	)	PUNCT
ejpam-5596	313	42	semiprime	semiprime	NOUN
ejpam-5596	313	43	if	if	SCONJ
ejpam-5596	313	44	e2	e2	PROPN
ejpam-5596	313	45	∈	∈	PROPN
ejpam-5596	313	46	k	k	PROPN
ejpam-5596	313	47	implies	imply	VERB
ejpam-5596	313	48	e	e	PROPN
ejpam-5596	313	49	∈	∈	PROPN
ejpam-5596	313	50	k	k	PROPN
ejpam-5596	313	51	for	for	ADP
ejpam-5596	313	52	all	all	DET
ejpam-5596	313	53	e	e	PROPN
ejpam-5596	313	54	∈	∈	PROPN
ejpam-5596	313	55	s.	s.	PROPN
ejpam-5596	313	56	definition	definition	NOUN
ejpam-5596	313	57	18	18	NUM
ejpam-5596	313	58	.	.	PUNCT
ejpam-5596	314	1	let	let	VERB
ejpam-5596	314	2	δ	δ	PRON
ejpam-5596	314	3	be	be	AUX
ejpam-5596	314	4	a	a	DET
ejpam-5596	314	5	fuzzy	fuzzy	ADJ
ejpam-5596	314	6	n	n	CCONJ
ejpam-5596	314	7	-	-	ADJ
ejpam-5596	314	8	interior	interior	ADJ
ejpam-5596	314	9	ideal	ideal	NOUN
ejpam-5596	314	10	of	of	ADP
ejpam-5596	314	11	a	a	DET
ejpam-5596	314	12	semigroup	semigroup	NOUN
ejpam-5596	314	13	s	s	PART
ejpam-5596	314	14	is	be	AUX
ejpam-5596	314	15	called	call	VERB
ejpam-5596	314	16	(	(	PUNCT
ejpam-5596	314	17	1	1	NUM
ejpam-5596	314	18	)	)	PUNCT
ejpam-5596	314	19	prime	prime	NOUN
ejpam-5596	314	20	if	if	SCONJ
ejpam-5596	314	21	δ(eh	δ(eh	NOUN
ejpam-5596	314	22	)	)	PUNCT
ejpam-5596	314	23	≤	≤	NOUN
ejpam-5596	314	24	δ(e	δ(e	ADJ
ejpam-5596	314	25	)	)	PUNCT
ejpam-5596	314	26	∨	∨	NUM
ejpam-5596	314	27	δ(h	δ(h	PROPN
ejpam-5596	314	28	)	)	PUNCT
ejpam-5596	314	29	for	for	ADP
ejpam-5596	314	30	all	all	DET
ejpam-5596	314	31	e	e	NOUN
ejpam-5596	314	32	,	,	PUNCT
ejpam-5596	314	33	h	h	NOUN
ejpam-5596	314	34	∈	∈	PROPN
ejpam-5596	314	35	s	s	PROPN
ejpam-5596	314	36	,	,	PUNCT
ejpam-5596	314	37	(	(	PUNCT
ejpam-5596	314	38	2	2	NUM
ejpam-5596	314	39	)	)	PUNCT
ejpam-5596	314	40	semiprime	semiprime	NOUN
ejpam-5596	314	41	if	if	SCONJ
ejpam-5596	314	42	δ(e2	δ(e2	NOUN
ejpam-5596	314	43	)	)	PUNCT
ejpam-5596	314	44	≤	≤	NOUN
ejpam-5596	314	45	δ(e	δ(e	NOUN
ejpam-5596	314	46	)	)	PUNCT
ejpam-5596	314	47	for	for	ADP
ejpam-5596	314	48	all	all	DET
ejpam-5596	314	49	e	e	PROPN
ejpam-5596	314	50	∈	∈	PROPN
ejpam-5596	314	51	s.	s.	PROPN
ejpam-5596	314	52	remark	remark	PROPN
ejpam-5596	314	53	2	2	NUM
ejpam-5596	314	54	.	.	PUNCT
ejpam-5596	315	1	every	every	DET
ejpam-5596	315	2	prime	prime	ADJ
ejpam-5596	315	3	n	n	CCONJ
ejpam-5596	315	4	-	-	ADJ
ejpam-5596	315	5	interior	interior	ADJ
ejpam-5596	315	6	ideal	ideal	NOUN
ejpam-5596	315	7	is	be	AUX
ejpam-5596	315	8	semiprime	semiprime	NOUN
ejpam-5596	315	9	n	n	CCONJ
ejpam-5596	315	10	-	-	ADJ
ejpam-5596	315	11	interior	interior	ADJ
ejpam-5596	315	12	ideal	ideal	NOUN
ejpam-5596	315	13	in	in	ADP
ejpam-5596	315	14	a	a	DET
ejpam-5596	315	15	semigroup	semigroup	NOUN
ejpam-5596	315	16	.	.	PUNCT
ejpam-5596	316	1	theorem	theorem	PROPN
ejpam-5596	316	2	12	12	NUM
ejpam-5596	316	3	.	.	PUNCT
ejpam-5596	317	1	let	let	VERB
ejpam-5596	317	2	k	k	PRON
ejpam-5596	317	3	be	be	AUX
ejpam-5596	317	4	a	a	DET
ejpam-5596	317	5	non	non	ADJ
ejpam-5596	317	6	-	-	ADJ
ejpam-5596	317	7	empty	empty	ADJ
ejpam-5596	317	8	subset	subset	NOUN
ejpam-5596	317	9	of	of	ADP
ejpam-5596	317	10	a	a	DET
ejpam-5596	317	11	semigroup	semigroup	PROPN
ejpam-5596	317	12	s.	s.	PROPN
ejpam-5596	317	13	then	then	ADV
ejpam-5596	317	14	the	the	DET
ejpam-5596	317	15	following	follow	VERB
ejpam-5596	317	16	statement	statement	NOUN
ejpam-5596	317	17	holds	hold	VERB
ejpam-5596	317	18	:	:	PUNCT
ejpam-5596	317	19	(	(	PUNCT
ejpam-5596	317	20	1	1	X
ejpam-5596	317	21	)	)	PUNCT
ejpam-5596	317	22	k	k	X
ejpam-5596	317	23	is	be	AUX
ejpam-5596	317	24	a	a	DET
ejpam-5596	317	25	prime	prime	ADJ
ejpam-5596	317	26	n	n	CCONJ
ejpam-5596	317	27	-	-	ADJ
ejpam-5596	317	28	interior	interior	ADJ
ejpam-5596	317	29	ideal	ideal	NOUN
ejpam-5596	317	30	of	of	ADP
ejpam-5596	317	31	s	s	PRON
ejpam-5596	317	32	if	if	SCONJ
ejpam-5596	318	1	and	and	CCONJ
ejpam-5596	318	2	only	only	ADV
ejpam-5596	318	3	if	if	SCONJ
ejpam-5596	318	4	λk	λk	PRON
ejpam-5596	318	5	is	be	AUX
ejpam-5596	318	6	a	a	DET
ejpam-5596	318	7	prime	prime	ADJ
ejpam-5596	318	8	fuzzy	fuzzy	ADJ
ejpam-5596	318	9	n	n	CCONJ
ejpam-5596	318	10	-	-	ADJ
ejpam-5596	318	11	interior	interior	ADJ
ejpam-5596	318	12	ideal	ideal	NOUN
ejpam-5596	318	13	of	of	ADP
ejpam-5596	318	14	s.	s.	PROPN
ejpam-5596	318	15	p.	p.	PROPN
ejpam-5596	318	16	khamrot	khamrot	PROPN
ejpam-5596	318	17	,	,	PUNCT
ejpam-5596	318	18	a.	a.	NOUN
ejpam-5596	318	19	iampan	iampan	PROPN
ejpam-5596	318	20	,	,	PUNCT
ejpam-5596	318	21	t.	t.	PROPN
ejpam-5596	318	22	gaketem	gaketem	PROPN
ejpam-5596	318	23	/	/	SYM
ejpam-5596	318	24	eur	eur	PROPN
ejpam-5596	318	25	.	.	PUNCT
ejpam-5596	319	1	j.	j.	PROPN
ejpam-5596	319	2	pure	pure	PROPN
ejpam-5596	319	3	appl	appl	PROPN
ejpam-5596	319	4	.	.	PROPN
ejpam-5596	319	5	math	math	PROPN
ejpam-5596	319	6	,	,	PUNCT
ejpam-5596	319	7	18	18	NUM
ejpam-5596	319	8	(	(	PUNCT
ejpam-5596	319	9	1	1	NUM
ejpam-5596	319	10	)	)	PUNCT
ejpam-5596	319	11	(	(	PUNCT
ejpam-5596	319	12	2025	2025	NUM
ejpam-5596	319	13	)	)	PUNCT
ejpam-5596	319	14	,	,	PUNCT
ejpam-5596	319	15	5596	5596	NUM
ejpam-5596	319	16	11	11	NUM
ejpam-5596	319	17	of	of	ADP
ejpam-5596	319	18	12	12	NUM
ejpam-5596	319	19	(	(	PUNCT
ejpam-5596	319	20	2	2	NUM
ejpam-5596	319	21	)	)	PUNCT
ejpam-5596	319	22	k	k	X
ejpam-5596	319	23	is	be	AUX
ejpam-5596	319	24	a	a	DET
ejpam-5596	319	25	semiprime	semiprime	NOUN
ejpam-5596	319	26	n	n	CCONJ
ejpam-5596	319	27	-	-	ADJ
ejpam-5596	319	28	interior	interior	ADJ
ejpam-5596	319	29	ideal	ideal	NOUN
ejpam-5596	319	30	of	of	ADP
ejpam-5596	319	31	s	s	PRON
ejpam-5596	319	32	if	if	SCONJ
ejpam-5596	320	1	and	and	CCONJ
ejpam-5596	320	2	only	only	ADV
ejpam-5596	320	3	if	if	SCONJ
ejpam-5596	320	4	λk	λk	PRON
ejpam-5596	320	5	is	be	AUX
ejpam-5596	320	6	a	a	DET
ejpam-5596	320	7	semiprime	semiprime	NOUN
ejpam-5596	320	8	fuzzy	fuzzy	ADJ
ejpam-5596	320	9	n	n	CCONJ
ejpam-5596	320	10	-	-	ADJ
ejpam-5596	320	11	interior	interior	ADJ
ejpam-5596	320	12	ideal	ideal	NOUN
ejpam-5596	320	13	of	of	ADP
ejpam-5596	320	14	s.	s.	PROPN
ejpam-5596	320	15	proof	proof	PROPN
ejpam-5596	320	16	.	.	PUNCT
ejpam-5596	321	1	(	(	PUNCT
ejpam-5596	321	2	1	1	X
ejpam-5596	321	3	)	)	PUNCT
ejpam-5596	321	4	suppose	suppose	VERB
ejpam-5596	321	5	that	that	SCONJ
ejpam-5596	321	6	k	k	PROPN
ejpam-5596	321	7	is	be	AUX
ejpam-5596	321	8	a	a	DET
ejpam-5596	321	9	prime	prime	ADJ
ejpam-5596	321	10	n	n	CCONJ
ejpam-5596	321	11	-	-	ADJ
ejpam-5596	321	12	interior	interior	ADJ
ejpam-5596	321	13	ideal	ideal	NOUN
ejpam-5596	321	14	of	of	ADP
ejpam-5596	321	15	s.	s.	PROPN
ejpam-5596	321	16	then	then	ADV
ejpam-5596	321	17	k	k	PROPN
ejpam-5596	321	18	is	be	AUX
ejpam-5596	321	19	an	an	DET
ejpam-5596	321	20	n	n	CCONJ
ejpam-5596	321	21	-	-	PUNCT
ejpam-5596	321	22	interior	interior	ADJ
ejpam-5596	321	23	ideal	ideal	NOUN
ejpam-5596	321	24	of	of	ADP
ejpam-5596	321	25	s.	s.	PROPN
ejpam-5596	321	26	thus	thus	ADV
ejpam-5596	321	27	,	,	PUNCT
ejpam-5596	321	28	by	by	ADP
ejpam-5596	321	29	theorem	theorem	NOUN
ejpam-5596	321	30	8	8	NUM
ejpam-5596	321	31	λk	λk	NOUN
ejpam-5596	321	32	is	be	AUX
ejpam-5596	321	33	a	a	DET
ejpam-5596	321	34	fuzzy	fuzzy	ADJ
ejpam-5596	321	35	n	n	CCONJ
ejpam-5596	321	36	-	-	ADJ
ejpam-5596	321	37	interior	interior	ADJ
ejpam-5596	321	38	ideal	ideal	NOUN
ejpam-5596	321	39	of	of	ADP
ejpam-5596	321	40	s.	s.	PROPN
ejpam-5596	321	41	let	let	VERB
ejpam-5596	321	42	e	e	NOUN
ejpam-5596	321	43	,	,	PUNCT
ejpam-5596	321	44	h	h	PROPN
ejpam-5596	321	45	∈	∈	PROPN
ejpam-5596	321	46	s.	s.	PROPN
ejpam-5596	321	47	case	case	NOUN
ejpam-5596	321	48	1	1	NUM
ejpam-5596	321	49	:	:	PUNCT
ejpam-5596	321	50	if	if	SCONJ
ejpam-5596	321	51	eh	eh	INTJ
ejpam-5596	321	52	∈	∈	PROPN
ejpam-5596	321	53	k	k	NOUN
ejpam-5596	321	54	,	,	PUNCT
ejpam-5596	321	55	then	then	ADV
ejpam-5596	321	56	e	e	PROPN
ejpam-5596	321	57	∈	∈	PROPN
ejpam-5596	321	58	k	k	PROPN
ejpam-5596	321	59	or	or	CCONJ
ejpam-5596	321	60	h	h	NOUN
ejpam-5596	321	61	∈	∈	PROPN
ejpam-5596	321	62	k.	k.	PROPN
ejpam-5596	322	1	thus	thus	ADV
ejpam-5596	322	2	,	,	PUNCT
ejpam-5596	322	3	λk(eh	λk(eh	PROPN
ejpam-5596	322	4	)	)	PUNCT
ejpam-5596	322	5	=	=	SYM
ejpam-5596	322	6	1	1	NUM
ejpam-5596	322	7	=	=	SYM
ejpam-5596	322	8	λk(e	λk(e	X
ejpam-5596	322	9	)	)	PUNCT
ejpam-5596	322	10	and	and	CCONJ
ejpam-5596	322	11	λk(eh	λk(eh	PROPN
ejpam-5596	322	12	)	)	PUNCT
ejpam-5596	322	13	=	=	SYM
ejpam-5596	323	1	1	1	X
ejpam-5596	323	2	.	.	PUNCT
ejpam-5596	324	1	hence	hence	ADV
ejpam-5596	324	2	,	,	PUNCT
ejpam-5596	324	3	λk(eh	λk(eh	PROPN
ejpam-5596	324	4	)	)	PUNCT
ejpam-5596	324	5	≤	≤	NOUN
ejpam-5596	324	6	λk(e	λk(e	PROPN
ejpam-5596	324	7	)	)	PUNCT
ejpam-5596	324	8	∨	∨	NUM
ejpam-5596	324	9	λk(h	λk(h	ADP
ejpam-5596	324	10	)	)	PUNCT
ejpam-5596	324	11	.	.	PUNCT
ejpam-5596	325	1	case	case	NOUN
ejpam-5596	325	2	2	2	NUM
ejpam-5596	325	3	:	:	PUNCT
ejpam-5596	325	4	if	if	SCONJ
ejpam-5596	325	5	eh	eh	INTJ
ejpam-5596	325	6	/∈	/∈	PUNCT
ejpam-5596	326	1	k	k	NOUN
ejpam-5596	326	2	,	,	PUNCT
ejpam-5596	326	3	then	then	ADV
ejpam-5596	326	4	λk(eh	λk(eh	ADV
ejpam-5596	326	5	)	)	PUNCT
ejpam-5596	327	1	=	=	SYM
ejpam-5596	327	2	0	0	X
ejpam-5596	327	3	.	.	PUNCT
ejpam-5596	328	1	thus	thus	ADV
ejpam-5596	328	2	,	,	PUNCT
ejpam-5596	328	3	λk(eh	λk(eh	PROPN
ejpam-5596	328	4	)	)	PUNCT
ejpam-5596	328	5	≤	≤	NOUN
ejpam-5596	328	6	λk(e	λk(e	PROPN
ejpam-5596	328	7	)	)	PUNCT
ejpam-5596	328	8	∨	∨	NUM
ejpam-5596	328	9	λk(h	λk(h	PRON
ejpam-5596	328	10	)	)	PUNCT
ejpam-5596	328	11	.	.	PUNCT
ejpam-5596	329	1	therefore	therefore	ADV
ejpam-5596	329	2	,	,	PUNCT
ejpam-5596	329	3	λk	λk	PRON
ejpam-5596	329	4	is	be	AUX
ejpam-5596	329	5	a	a	DET
ejpam-5596	329	6	prime	prime	ADJ
ejpam-5596	329	7	fuzzy	fuzzy	ADJ
ejpam-5596	329	8	n	n	CCONJ
ejpam-5596	329	9	-	-	ADJ
ejpam-5596	329	10	interior	interior	ADJ
ejpam-5596	329	11	ideal	ideal	NOUN
ejpam-5596	329	12	of	of	ADP
ejpam-5596	329	13	s.	s.	PROPN
ejpam-5596	329	14	conversely	conversely	ADV
ejpam-5596	329	15	,	,	PUNCT
ejpam-5596	329	16	suppose	suppose	VERB
ejpam-5596	329	17	that	that	SCONJ
ejpam-5596	329	18	λk	λk	PROPN
ejpam-5596	329	19	is	be	AUX
ejpam-5596	329	20	a	a	DET
ejpam-5596	329	21	prime	prime	ADJ
ejpam-5596	329	22	fuzzy	fuzzy	ADJ
ejpam-5596	329	23	n	n	CCONJ
ejpam-5596	329	24	-	-	ADJ
ejpam-5596	329	25	interior	interior	ADJ
ejpam-5596	329	26	ideal	ideal	NOUN
ejpam-5596	329	27	of	of	ADP
ejpam-5596	329	28	s.	s.	PROPN
ejpam-5596	329	29	then	then	ADV
ejpam-5596	329	30	λk	λk	X
ejpam-5596	329	31	is	be	AUX
ejpam-5596	329	32	a	a	DET
ejpam-5596	329	33	fuzzy	fuzzy	ADJ
ejpam-5596	329	34	n	n	CCONJ
ejpam-5596	329	35	-	-	ADJ
ejpam-5596	329	36	interior	interior	ADJ
ejpam-5596	329	37	ideal	ideal	NOUN
ejpam-5596	329	38	of	of	ADP
ejpam-5596	329	39	s.	s.	PROPN
ejpam-5596	329	40	thus	thus	ADV
ejpam-5596	329	41	,	,	PUNCT
ejpam-5596	329	42	by	by	ADP
ejpam-5596	329	43	theorem	theorem	NOUN
ejpam-5596	329	44	8	8	NUM
ejpam-5596	329	45	,	,	PUNCT
ejpam-5596	329	46	k	k	PROPN
ejpam-5596	329	47	is	be	AUX
ejpam-5596	329	48	an	an	DET
ejpam-5596	329	49	n	n	CCONJ
ejpam-5596	329	50	-	-	PUNCT
ejpam-5596	329	51	interior	interior	ADJ
ejpam-5596	329	52	ideal	ideal	NOUN
ejpam-5596	329	53	of	of	ADP
ejpam-5596	329	54	s.	s.	PROPN
ejpam-5596	329	55	let	let	VERB
ejpam-5596	329	56	e	e	NOUN
ejpam-5596	329	57	,	,	PUNCT
ejpam-5596	329	58	h	h	PROPN
ejpam-5596	329	59	∈	∈	PROPN
ejpam-5596	329	60	s	s	VERB
ejpam-5596	329	61	with	with	ADP
ejpam-5596	329	62	eh	eh	INTJ
ejpam-5596	329	63	∈	∈	PROPN
ejpam-5596	329	64	k.	k.	NOUN
ejpam-5596	330	1	then	then	ADV
ejpam-5596	330	2	,	,	PUNCT
ejpam-5596	330	3	λk(eh	λk(eh	PROPN
ejpam-5596	330	4	)	)	PUNCT
ejpam-5596	330	5	=	=	SYM
ejpam-5596	331	1	1	1	X
ejpam-5596	331	2	.	.	PUNCT
ejpam-5596	332	1	if	if	SCONJ
ejpam-5596	332	2	e	e	PROPN
ejpam-5596	332	3	/∈	/∈	PROPN
ejpam-5596	333	1	k	k	PROPN
ejpam-5596	333	2	and	and	CCONJ
ejpam-5596	333	3	h	h	NOUN
ejpam-5596	333	4	/∈	/∈	PUNCT
ejpam-5596	334	1	k	k	NOUN
ejpam-5596	334	2	,	,	PUNCT
ejpam-5596	334	3	then	then	ADV
ejpam-5596	334	4	λk(e	λk(e	PROPN
ejpam-5596	334	5	)	)	PUNCT
ejpam-5596	334	6	=	=	SYM
ejpam-5596	334	7	0	0	PUNCT
ejpam-5596	334	8	=	=	SYM
ejpam-5596	334	9	λk(h	λk(h	NOUN
ejpam-5596	334	10	)	)	PUNCT
ejpam-5596	334	11	.	.	PUNCT
ejpam-5596	335	1	by	by	ADP
ejpam-5596	335	2	assumption	assumption	NOUN
ejpam-5596	335	3	,	,	PUNCT
ejpam-5596	335	4	λk(eh	λk(eh	PROPN
ejpam-5596	335	5	)	)	PUNCT
ejpam-5596	335	6	≤	≤	NOUN
ejpam-5596	335	7	λk(e	λk(e	PROPN
ejpam-5596	335	8	)	)	PUNCT
ejpam-5596	335	9	∨	∨	NUM
ejpam-5596	335	10	λk(h	λk(h	PRON
ejpam-5596	335	11	)	)	PUNCT
ejpam-5596	335	12	.	.	PUNCT
ejpam-5596	336	1	thus	thus	ADV
ejpam-5596	336	2	,	,	PUNCT
ejpam-5596	336	3	λk(eh	λk(eh	PROPN
ejpam-5596	336	4	)	)	PUNCT
ejpam-5596	336	5	=	=	SYM
ejpam-5596	337	1	0	0	X
ejpam-5596	337	2	..	..	PUNCT
ejpam-5596	337	3	it	it	PRON
ejpam-5596	337	4	is	be	AUX
ejpam-5596	337	5	a	a	DET
ejpam-5596	337	6	contradiction	contradiction	NOUN
ejpam-5596	337	7	,	,	PUNCT
ejpam-5596	337	8	so	so	ADV
ejpam-5596	337	9	e	e	PROPN
ejpam-5596	337	10	∈	∈	PROPN
ejpam-5596	337	11	k	k	PROPN
ejpam-5596	337	12	or	or	CCONJ
ejpam-5596	337	13	h	h	NOUN
ejpam-5596	337	14	∈	∈	PROPN
ejpam-5596	337	15	k.	k.	PROPN
ejpam-5596	338	1	hence	hence	ADV
ejpam-5596	338	2	,	,	PUNCT
ejpam-5596	338	3	k	k	PROPN
ejpam-5596	338	4	is	be	AUX
ejpam-5596	338	5	a	a	DET
ejpam-5596	338	6	prime	prime	ADJ
ejpam-5596	338	7	n	n	CCONJ
ejpam-5596	338	8	-	-	ADJ
ejpam-5596	338	9	interior	interior	ADJ
ejpam-5596	338	10	ideal	ideal	NOUN
ejpam-5596	338	11	of	of	ADP
ejpam-5596	338	12	s.	s.	PROPN
ejpam-5596	338	13	(	(	PUNCT
ejpam-5596	338	14	2	2	X
ejpam-5596	338	15	)	)	PUNCT
ejpam-5596	338	16	it	it	PRON
ejpam-5596	338	17	follows	follow	VERB
ejpam-5596	338	18	from	from	ADP
ejpam-5596	338	19	(	(	PUNCT
ejpam-5596	338	20	1	1	NUM
ejpam-5596	338	21	)	)	PUNCT
ejpam-5596	338	22	.	.	PUNCT
ejpam-5596	339	1	5	5	X
ejpam-5596	339	2	.	.	X
ejpam-5596	339	3	conclusion	conclusion	NOUN
ejpam-5596	339	4	in	in	ADP
ejpam-5596	339	5	this	this	DET
ejpam-5596	339	6	paper	paper	NOUN
ejpam-5596	339	7	,	,	PUNCT
ejpam-5596	339	8	we	we	PRON
ejpam-5596	339	9	introduce	introduce	VERB
ejpam-5596	339	10	the	the	DET
ejpam-5596	339	11	concept	concept	NOUN
ejpam-5596	339	12	of	of	ADP
ejpam-5596	339	13	bipolar	bipolar	ADJ
ejpam-5596	339	14	fuzzy	fuzzy	ADJ
ejpam-5596	339	15	(	(	PUNCT
ejpam-5596	339	16	m	m	NOUN
ejpam-5596	339	17	,	,	PUNCT
ejpam-5596	339	18	n)-ideals	n)-ideal	NOUN
ejpam-5596	339	19	in	in	ADP
ejpam-5596	339	20	semigroups	semigroup	NOUN
ejpam-5596	339	21	and	and	CCONJ
ejpam-5596	339	22	investigate	investigate	VERB
ejpam-5596	339	23	their	their	PRON
ejpam-5596	339	24	properties	property	NOUN
ejpam-5596	339	25	.	.	PUNCT
ejpam-5596	340	1	additionally	additionally	ADV
ejpam-5596	340	2	,	,	PUNCT
ejpam-5596	340	3	we	we	PRON
ejpam-5596	340	4	establish	establish	VERB
ejpam-5596	340	5	the	the	DET
ejpam-5596	340	6	relationship	relationship	NOUN
ejpam-5596	340	7	between	between	ADP
ejpam-5596	340	8	(	(	PUNCT
ejpam-5596	340	9	m	m	PROPN
ejpam-5596	340	10	,	,	PUNCT
ejpam-5596	340	11	n)ideals	n)ideal	NOUN
ejpam-5596	340	12	and	and	CCONJ
ejpam-5596	340	13	fuzzy	fuzzy	ADJ
ejpam-5596	340	14	(	(	PUNCT
ejpam-5596	340	15	m	m	NOUN
ejpam-5596	340	16	,	,	PUNCT
ejpam-5596	340	17	n)-ideals	n)-ideal	NOUN
ejpam-5596	340	18	.	.	PUNCT
ejpam-5596	341	1	furthermore	furthermore	ADV
ejpam-5596	341	2	,	,	PUNCT
ejpam-5596	341	3	we	we	PRON
ejpam-5596	341	4	define	define	VERB
ejpam-5596	341	5	bipolar	bipolar	ADJ
ejpam-5596	341	6	fuzzy	fuzzy	ADJ
ejpam-5596	341	7	n	n	CCONJ
ejpam-5596	341	8	-	-	PUNCT
ejpam-5596	341	9	interior	interior	ADJ
ejpam-5596	341	10	ideals	ideal	NOUN
ejpam-5596	341	11	in	in	ADP
ejpam-5596	341	12	semigroup	semigroup	NOUN
ejpam-5596	341	13	and	and	CCONJ
ejpam-5596	341	14	prove	prove	VERB
ejpam-5596	341	15	the	the	DET
ejpam-5596	341	16	relationship	relationship	NOUN
ejpam-5596	341	17	between	between	ADP
ejpam-5596	341	18	n	n	CCONJ
ejpam-5596	341	19	-	-	PUNCT
ejpam-5596	341	20	interior	interior	ADJ
ejpam-5596	341	21	ideals	ideal	NOUN
ejpam-5596	341	22	and	and	CCONJ
ejpam-5596	341	23	fuzzy	fuzzy	ADJ
ejpam-5596	341	24	n	n	CCONJ
ejpam-5596	341	25	-	-	PUNCT
ejpam-5596	341	26	interior	interior	ADJ
ejpam-5596	341	27	ideals	ideal	NOUN
ejpam-5596	341	28	.	.	PUNCT
ejpam-5596	342	1	in	in	ADP
ejpam-5596	342	2	the	the	DET
ejpam-5596	342	3	future	future	NOUN
ejpam-5596	342	4	,	,	PUNCT
ejpam-5596	342	5	we	we	PRON
ejpam-5596	342	6	plan	plan	VERB
ejpam-5596	342	7	to	to	PART
ejpam-5596	342	8	explore	explore	VERB
ejpam-5596	342	9	bipolar	bipolar	ADJ
ejpam-5596	342	10	(	(	PUNCT
ejpam-5596	342	11	m	m	NOUN
ejpam-5596	342	12	,	,	PUNCT
ejpam-5596	342	13	n)-ideals	n)-ideal	NOUN
ejpam-5596	342	14	and	and	CCONJ
ejpam-5596	342	15	n	n	CCONJ
ejpam-5596	342	16	-	-	PUNCT
ejpam-5596	342	17	interior	interior	ADJ
ejpam-5596	342	18	ideals	ideal	NOUN
ejpam-5596	342	19	in	in	ADP
ejpam-5596	342	20	ordered	order	VERB
ejpam-5596	342	21	semigroups	semigroup	NOUN
ejpam-5596	342	22	or	or	CCONJ
ejpam-5596	342	23	within	within	ADP
ejpam-5596	342	24	the	the	DET
ejpam-5596	342	25	algebraic	algebraic	ADJ
ejpam-5596	342	26	context	context	NOUN
ejpam-5596	342	27	.	.	PUNCT
ejpam-5596	343	1	acknowledgements	acknowledgement	NOUN
ejpam-5596	343	2	this	this	DET
ejpam-5596	343	3	research	research	NOUN
ejpam-5596	343	4	was	be	AUX
ejpam-5596	343	5	supported	support	VERB
ejpam-5596	343	6	by	by	ADP
ejpam-5596	343	7	the	the	DET
ejpam-5596	343	8	university	university	NOUN
ejpam-5596	343	9	of	of	ADP
ejpam-5596	343	10	phayao	phayao	NOUN
ejpam-5596	343	11	and	and	CCONJ
ejpam-5596	343	12	the	the	DET
ejpam-5596	343	13	thailand	thailand	PROPN
ejpam-5596	343	14	science	science	PROPN
ejpam-5596	343	15	research	research	PROPN
ejpam-5596	343	16	and	and	CCONJ
ejpam-5596	343	17	innovation	innovation	NOUN
ejpam-5596	343	18	fund	fund	NOUN
ejpam-5596	343	19	(	(	PUNCT
ejpam-5596	343	20	fundamental	fundamental	ADJ
ejpam-5596	343	21	fund	fund	NOUN
ejpam-5596	343	22	2025	2025	NUM
ejpam-5596	343	23	,	,	PUNCT
ejpam-5596	343	24	grant	grant	VERB
ejpam-5596	343	25	no	no	NOUN
ejpam-5596	343	26	.	.	PROPN
ejpam-5596	344	1	5027/2567	5027/2567	NUM
ejpam-5596	344	2	)	)	PUNCT
ejpam-5596	344	3	.	.	PUNCT
ejpam-5596	345	1	references	reference	NOUN
ejpam-5596	345	2	[	[	X
ejpam-5596	345	3	1	1	NUM
ejpam-5596	345	4	]	]	PUNCT
ejpam-5596	345	5	m.	m.	NOUN
ejpam-5596	345	6	al	al	PROPN
ejpam-5596	345	7	-	-	PUNCT
ejpam-5596	345	8	tahan	tahan	PROPN
ejpam-5596	345	9	,	,	PUNCT
ejpam-5596	345	10	b.	b.	PROPN
ejpam-5596	345	11	davvaz	davvaz	PROPN
ejpam-5596	345	12	,	,	PUNCT
ejpam-5596	345	13	a.	a.	NOUN
ejpam-5596	345	14	mahboob	mahboob	PROPN
ejpam-5596	345	15	,	,	PUNCT
ejpam-5596	345	16	and	and	CCONJ
ejpam-5596	345	17	n.	n.	PROPN
ejpam-5596	345	18	m.	m.	PROPN
ejpam-5596	345	19	khan	khan	PROPN
ejpam-5596	345	20	.	.	PUNCT
ejpam-5596	346	1	on	on	ADP
ejpam-5596	346	2	a	a	DET
ejpam-5596	346	3	generalization	generalization	NOUN
ejpam-5596	346	4	of	of	ADP
ejpam-5596	346	5	fuzzy	fuzzy	ADJ
ejpam-5596	346	6	filters	filter	NOUN
ejpam-5596	346	7	for	for	ADP
ejpam-5596	346	8	ordered	order	VERB
ejpam-5596	346	9	semigroups	semigroup	NOUN
ejpam-5596	346	10	.	.	PUNCT
ejpam-5596	347	1	new	new	ADJ
ejpam-5596	347	2	mathematics	mathematic	NOUN
ejpam-5596	347	3	and	and	CCONJ
ejpam-5596	347	4	natural	natural	ADJ
ejpam-5596	347	5	computing	computing	NOUN
ejpam-5596	347	6	,	,	PUNCT
ejpam-5596	347	7	19(20):489	19(20):489	NUM
ejpam-5596	347	8	–	–	PUNCT
ejpam-5596	347	9	502	502	NUM
ejpam-5596	347	10	,	,	PUNCT
ejpam-5596	347	11	2023	2023	NUM
ejpam-5596	347	12	.	.	PUNCT
ejpam-5596	348	1	[	[	X
ejpam-5596	348	2	2	2	NUM
ejpam-5596	348	3	]	]	PUNCT
ejpam-5596	348	4	m.	m.	NOUN
ejpam-5596	348	5	al	al	PROPN
ejpam-5596	348	6	-	-	PUNCT
ejpam-5596	348	7	tahan	tahan	PROPN
ejpam-5596	348	8	,	,	PUNCT
ejpam-5596	348	9	b.	b.	PROPN
ejpam-5596	348	10	davvaz	davvaz	PROPN
ejpam-5596	348	11	,	,	PUNCT
ejpam-5596	348	12	a.	a.	PROPN
ejpam-5596	348	13	mahboob	mahboob	PROPN
ejpam-5596	348	14	,	,	PUNCT
ejpam-5596	348	15	s.	s.	PROPN
ejpam-5596	348	16	h.	h.	PROPN
ejpam-5596	348	17	mayerova	mayerova	PROPN
ejpam-5596	348	18	,	,	PUNCT
ejpam-5596	348	19	and	and	CCONJ
ejpam-5596	348	20	a.	a.	PROPN
ejpam-5596	348	21	vagaská.	vagaská.	PROPN
ejpam-5596	348	22	on	on	ADP
ejpam-5596	348	23	new	new	ADJ
ejpam-5596	348	24	filters	filter	NOUN
ejpam-5596	348	25	in	in	ADP
ejpam-5596	348	26	ordered	order	VERB
ejpam-5596	348	27	semigroups	semigroup	NOUN
ejpam-5596	348	28	.	.	PUNCT
ejpam-5596	348	29	symmetry	symmetry	NOUN
ejpam-5596	348	30	,	,	PUNCT
ejpam-5596	348	31	14(8):1564	14(8):1564	NUM
ejpam-5596	348	32	,	,	PUNCT
ejpam-5596	348	33	2022	2022	NUM
ejpam-5596	348	34	.	.	PUNCT
ejpam-5596	349	1	[	[	X
ejpam-5596	349	2	3	3	X
ejpam-5596	349	3	]	]	X
ejpam-5596	349	4	t.	t.	NOUN
ejpam-5596	349	5	changphas	changphas	PROPN
ejpam-5596	349	6	.	.	PUNCT
ejpam-5596	350	1	on	on	ADP
ejpam-5596	350	2	(	(	PUNCT
ejpam-5596	350	3	m	m	PROPN
ejpam-5596	350	4	,	,	PUNCT
ejpam-5596	350	5	n)-ideals	n)-ideal	NOUN
ejpam-5596	350	6	of	of	ADP
ejpam-5596	350	7	an	an	DET
ejpam-5596	350	8	ordered	order	VERB
ejpam-5596	350	9	semigroups	semigroup	NOUN
ejpam-5596	350	10	.	.	PUNCT
ejpam-5596	351	1	international	international	ADJ
ejpam-5596	351	2	journal	journal	NOUN
ejpam-5596	351	3	of	of	ADP
ejpam-5596	351	4	pure	pure	ADJ
ejpam-5596	351	5	and	and	CCONJ
ejpam-5596	351	6	applied	applied	ADJ
ejpam-5596	351	7	mathematics	mathematic	NOUN
ejpam-5596	351	8	,	,	PUNCT
ejpam-5596	351	9	100(1):1–5	100(1):1–5	NUM
ejpam-5596	351	10	,	,	PUNCT
ejpam-5596	351	11	2015	2015	NUM
ejpam-5596	351	12	.	.	PUNCT
ejpam-5596	352	1	p.	p.	NOUN
ejpam-5596	352	2	khamrot	khamrot	PROPN
ejpam-5596	352	3	,	,	PUNCT
ejpam-5596	352	4	a.	a.	NOUN
ejpam-5596	352	5	iampan	iampan	PROPN
ejpam-5596	352	6	,	,	PUNCT
ejpam-5596	352	7	t.	t.	PROPN
ejpam-5596	352	8	gaketem	gaketem	PROPN
ejpam-5596	352	9	/	/	SYM
ejpam-5596	352	10	eur	eur	PROPN
ejpam-5596	352	11	.	.	PUNCT
ejpam-5596	353	1	j.	j.	PROPN
ejpam-5596	353	2	pure	pure	PROPN
ejpam-5596	353	3	appl	appl	PROPN
ejpam-5596	353	4	.	.	PROPN
ejpam-5596	353	5	math	math	PROPN
ejpam-5596	353	6	,	,	PUNCT
ejpam-5596	353	7	18	18	NUM
ejpam-5596	353	8	(	(	PUNCT
ejpam-5596	353	9	1	1	NUM
ejpam-5596	353	10	)	)	PUNCT
ejpam-5596	353	11	(	(	PUNCT
ejpam-5596	353	12	2025	2025	NUM
ejpam-5596	353	13	)	)	PUNCT
ejpam-5596	353	14	,	,	PUNCT
ejpam-5596	353	15	5596	5596	NUM
ejpam-5596	353	16	12	12	NUM
ejpam-5596	353	17	of	of	ADP
ejpam-5596	353	18	12	12	NUM
ejpam-5596	353	19	[	[	SYM
ejpam-5596	353	20	4	4	NUM
ejpam-5596	353	21	]	]	X
ejpam-5596	353	22	i.	i.	PROPN
ejpam-5596	353	23	cristea	cristea	PROPN
ejpam-5596	353	24	,	,	PUNCT
ejpam-5596	353	25	a.	a.	NOUN
ejpam-5596	353	26	mahboob	mahboob	PROPN
ejpam-5596	353	27	,	,	PUNCT
ejpam-5596	353	28	and	and	CCONJ
ejpam-5596	353	29	m.	m.	NOUN
ejpam-5596	353	30	m.	m.	PROPN
ejpam-5596	353	31	khan	khan	PROPN
ejpam-5596	353	32	.	.	PUNCT
ejpam-5596	354	1	a	a	DET
ejpam-5596	354	2	new	new	ADJ
ejpam-5596	354	3	type	type	NOUN
ejpam-5596	354	4	fuzzy	fuzzy	ADJ
ejpam-5596	354	5	quasi	quasi	NOUN
ejpam-5596	354	6	-	-	NOUN
ejpam-5596	354	7	ideals	ideal	NOUN
ejpam-5596	354	8	of	of	ADP
ejpam-5596	354	9	ordered	order	VERB
ejpam-5596	354	10	semigroups	semigroup	NOUN
ejpam-5596	354	11	.	.	PUNCT
ejpam-5596	355	1	journal	journal	NOUN
ejpam-5596	355	2	of	of	ADP
ejpam-5596	355	3	multiple	multiple	ADV
ejpam-5596	355	4	-	-	PUNCT
ejpam-5596	355	5	valued	value	VERB
ejpam-5596	355	6	logic	logic	NOUN
ejpam-5596	355	7	and	and	CCONJ
ejpam-5596	355	8	soft	soft	ADJ
ejpam-5596	355	9	computing	computing	NOUN
ejpam-5596	355	10	,	,	PUNCT
ejpam-5596	355	11	34:283–304	34:283–304	PROPN
ejpam-5596	355	12	,	,	PUNCT
ejpam-5596	355	13	2020	2020	NUM
ejpam-5596	355	14	.	.	PUNCT
ejpam-5596	356	1	[	[	X
ejpam-5596	356	2	5	5	NUM
ejpam-5596	356	3	]	]	X
ejpam-5596	356	4	n.	n.	NOUN
ejpam-5596	356	5	kehayopulu	kehayopulu	ADJ
ejpam-5596	356	6	and	and	CCONJ
ejpam-5596	356	7	m.	m.	NOUN
ejpam-5596	356	8	tsingelis	tsingelis	PROPN
ejpam-5596	356	9	.	.	PUNCT
ejpam-5596	357	1	fuzzy	fuzzy	ADJ
ejpam-5596	357	2	bi	bi	NOUN
ejpam-5596	357	3	-	-	NOUN
ejpam-5596	357	4	ideal	ideal	NOUN
ejpam-5596	357	5	in	in	ADP
ejpam-5596	357	6	ordered	order	VERB
ejpam-5596	357	7	semigroups	semigroup	NOUN
ejpam-5596	357	8	.	.	PUNCT
ejpam-5596	358	1	information	information	NOUN
ejpam-5596	358	2	science	science	NOUN
ejpam-5596	358	3	,	,	PUNCT
ejpam-5596	358	4	pages	page	NOUN
ejpam-5596	358	5	13–28	13–28	NUM
ejpam-5596	358	6	,	,	PUNCT
ejpam-5596	358	7	2005	2005	NUM
ejpam-5596	358	8	.	.	PUNCT
ejpam-5596	359	1	[	[	X
ejpam-5596	359	2	6	6	NUM
ejpam-5596	359	3	]	]	PUNCT
ejpam-5596	359	4	p.	p.	NOUN
ejpam-5596	359	5	khamrot	khamrot	PROPN
ejpam-5596	359	6	and	and	CCONJ
ejpam-5596	359	7	t.	t.	PROPN
ejpam-5596	359	8	gaketem	gaketem	PROPN
ejpam-5596	359	9	.	.	PUNCT
ejpam-5596	360	1	generalized	generalize	VERB
ejpam-5596	360	2	interval	interval	NOUN
ejpam-5596	360	3	valued	value	VERB
ejpam-5596	360	4	fuzzy	fuzzy	ADJ
ejpam-5596	360	5	ideals	ideal	NOUN
ejpam-5596	360	6	in	in	ADP
ejpam-5596	360	7	semigroups	semigroup	NOUN
ejpam-5596	360	8	.	.	PUNCT
ejpam-5596	361	1	journal	journal	PROPN
ejpam-5596	361	2	of	of	ADP
ejpam-5596	361	3	mathematics	mathematic	NOUN
ejpam-5596	361	4	and	and	CCONJ
ejpam-5596	361	5	computer	computer	NOUN
ejpam-5596	361	6	science	science	NOUN
ejpam-5596	361	7	,	,	PUNCT
ejpam-5596	361	8	34(2):116–127	34(2):116–127	PROPN
ejpam-5596	361	9	,	,	PUNCT
ejpam-5596	361	10	2024	2024	NUM
ejpam-5596	361	11	.	.	PUNCT
ejpam-5596	362	1	[	[	X
ejpam-5596	362	2	7	7	X
ejpam-5596	362	3	]	]	PUNCT
ejpam-5596	362	4	a.	a.	NOUN
ejpam-5596	362	5	mahboob	mahboob	PROPN
ejpam-5596	362	6	,	,	PUNCT
ejpam-5596	362	7	m.	m.	PROPN
ejpam-5596	362	8	al	al	PROPN
ejpam-5596	362	9	-	-	PUNCT
ejpam-5596	362	10	tahan	tahan	PROPN
ejpam-5596	362	11	,	,	PUNCT
ejpam-5596	362	12	and	and	CCONJ
ejpam-5596	362	13	g.	g.	PROPN
ejpam-5596	362	14	muhiuddin	muhiuddin	PROPN
ejpam-5596	362	15	.	.	PUNCT
ejpam-5596	363	1	fuzzy	fuzzy	ADJ
ejpam-5596	363	2	(	(	PUNCT
ejpam-5596	363	3	m	m	PROPN
ejpam-5596	363	4	,	,	PUNCT
ejpam-5596	363	5	n)-filters	n)-filter	NOUN
ejpam-5596	363	6	based	base	VERB
ejpam-5596	363	7	on	on	ADP
ejpam-5596	363	8	fuzzy	fuzzy	ADJ
ejpam-5596	363	9	points	point	NOUN
ejpam-5596	363	10	in	in	ADP
ejpam-5596	363	11	ordered	order	VERB
ejpam-5596	363	12	semigroups	semigroup	NOUN
ejpam-5596	363	13	.	.	PUNCT
ejpam-5596	364	1	computational	computational	ADJ
ejpam-5596	364	2	and	and	CCONJ
ejpam-5596	364	3	applied	applied	ADJ
ejpam-5596	364	4	mathematics	mathematic	NOUN
ejpam-5596	364	5	,	,	PUNCT
ejpam-5596	364	6	42:1–14	42:1–14	NUM
ejpam-5596	364	7	,	,	PUNCT
ejpam-5596	364	8	2023	2023	NUM
ejpam-5596	364	9	.	.	PUNCT
ejpam-5596	365	1	[	[	X
ejpam-5596	365	2	8	8	NUM
ejpam-5596	365	3	]	]	PUNCT
ejpam-5596	365	4	a.	a.	NOUN
ejpam-5596	365	5	mahboob	mahboob	PROPN
ejpam-5596	365	6	,	,	PUNCT
ejpam-5596	365	7	m.	m.	PROPN
ejpam-5596	365	8	al	al	PROPN
ejpam-5596	365	9	-	-	PUNCT
ejpam-5596	365	10	tahan	tahan	PROPN
ejpam-5596	365	11	,	,	PUNCT
ejpam-5596	365	12	and	and	CCONJ
ejpam-5596	365	13	g.	g.	PROPN
ejpam-5596	365	14	muhiuddin	muhiuddin	PROPN
ejpam-5596	365	15	.	.	PUNCT
ejpam-5596	366	1	characterizations	characterization	NOUN
ejpam-5596	366	2	of	of	ADP
ejpam-5596	366	3	ordered	order	VERB
ejpam-5596	366	4	semigroups	semigroup	NOUN
ejpam-5596	366	5	in	in	ADP
ejpam-5596	366	6	terms	term	NOUN
ejpam-5596	366	7	of	of	ADP
ejpam-5596	366	8	fuzzy	fuzzy	ADJ
ejpam-5596	366	9	(	(	PUNCT
ejpam-5596	366	10	m	m	PROPN
ejpam-5596	366	11	,	,	PUNCT
ejpam-5596	366	12	n)-substructures	n)-substructure	NOUN
ejpam-5596	366	13	.	.	NOUN
ejpam-5596	366	14	soft	soft	ADJ
ejpam-5596	366	15	computing	computing	NOUN
ejpam-5596	366	16	,	,	PUNCT
ejpam-5596	366	17	pages	page	NOUN
ejpam-5596	366	18	1–8	1–8	NUM
ejpam-5596	366	19	,	,	PUNCT
ejpam-5596	366	20	2024	2024	NUM
ejpam-5596	366	21	.	.	PUNCT
ejpam-5596	367	1	[	[	X
ejpam-5596	367	2	9	9	NUM
ejpam-5596	367	3	]	]	PUNCT
ejpam-5596	367	4	a.	a.	NOUN
ejpam-5596	367	5	mahboob	mahboob	PROPN
ejpam-5596	367	6	,	,	PUNCT
ejpam-5596	367	7	b.	b.	PROPN
ejpam-5596	367	8	davvaz	davvaz	PROPN
ejpam-5596	367	9	,	,	PUNCT
ejpam-5596	367	10	and	and	CCONJ
ejpam-5596	367	11	n.m	n.m	PROPN
ejpam-5596	367	12	.	.	PROPN
ejpam-5596	367	13	khan	khan	PROPN
ejpam-5596	367	14	.	.	PUNCT
ejpam-5596	368	1	fuzzy	fuzzy	ADJ
ejpam-5596	368	2	(	(	PUNCT
ejpam-5596	368	3	m	m	PROPN
ejpam-5596	368	4	,	,	PUNCT
ejpam-5596	368	5	n)-ideals	n)-ideal	NOUN
ejpam-5596	368	6	in	in	ADP
ejpam-5596	368	7	semigroups	semigroup	NOUN
ejpam-5596	368	8	.	.	PUNCT
ejpam-5596	369	1	computational	computational	ADJ
ejpam-5596	369	2	and	and	CCONJ
ejpam-5596	369	3	applied	applied	ADJ
ejpam-5596	369	4	mathematics	mathematic	NOUN
ejpam-5596	369	5	,	,	PUNCT
ejpam-5596	369	6	38(189):1–18	38(189):1–18	PROPN
ejpam-5596	369	7	,	,	PUNCT
ejpam-5596	369	8	2019	2019	NUM
ejpam-5596	369	9	.	.	PUNCT
ejpam-5596	370	1	[	[	X
ejpam-5596	370	2	10	10	NUM
ejpam-5596	370	3	]	]	PUNCT
ejpam-5596	370	4	a.	a.	NOUN
ejpam-5596	370	5	mahboob	mahboob	PROPN
ejpam-5596	370	6	and	and	CCONJ
ejpam-5596	370	7	g.	g.	PROPN
ejpam-5596	370	8	muhiuddin	muhiuddin	PROPN
ejpam-5596	370	9	.	.	PUNCT
ejpam-5596	371	1	a	a	DET
ejpam-5596	371	2	new	new	ADJ
ejpam-5596	371	3	type	type	NOUN
ejpam-5596	371	4	of	of	ADP
ejpam-5596	371	5	fuzzy	fuzzy	ADJ
ejpam-5596	371	6	prime	prime	NOUN
ejpam-5596	371	7	subset	subset	NOUN
ejpam-5596	371	8	in	in	ADP
ejpam-5596	371	9	ordered	order	VERB
ejpam-5596	371	10	semigroups	semigroup	NOUN
ejpam-5596	371	11	.	.	PUNCT
ejpam-5596	372	1	new	new	ADJ
ejpam-5596	372	2	mathematics	mathematic	NOUN
ejpam-5596	372	3	and	and	CCONJ
ejpam-5596	372	4	natural	natural	ADJ
ejpam-5596	372	5	computing	computing	NOUN
ejpam-5596	372	6	,	,	PUNCT
ejpam-5596	372	7	17(3):739–752	17(3):739–752	NUM
ejpam-5596	372	8	,	,	PUNCT
ejpam-5596	372	9	2021	2021	NUM
ejpam-5596	372	10	.	.	PUNCT
ejpam-5596	373	1	[	[	X
ejpam-5596	373	2	11	11	NUM
ejpam-5596	373	3	]	]	X
ejpam-5596	373	4	j.n	j.n	PROPN
ejpam-5596	373	5	.	.	PROPN
ejpam-5596	373	6	mordeson	mordeson	PROPN
ejpam-5596	373	7	,	,	PUNCT
ejpam-5596	373	8	d.	d.	PROPN
ejpam-5596	373	9	s.	s.	PROPN
ejpam-5596	373	10	malik	malik	PROPN
ejpam-5596	373	11	,	,	PUNCT
ejpam-5596	373	12	and	and	CCONJ
ejpam-5596	373	13	n.	n.	PROPN
ejpam-5596	373	14	kuroki	kuroki	PROPN
ejpam-5596	373	15	.	.	PUNCT
ejpam-5596	374	1	fuzzy	fuzzy	PROPN
ejpam-5596	374	2	semigroup	semigroup	PROPN
ejpam-5596	374	3	.	.	PUNCT
ejpam-5596	375	1	springer	springer	NOUN
ejpam-5596	375	2	science	science	PROPN
ejpam-5596	375	3	and	and	CCONJ
ejpam-5596	375	4	business	business	NOUN
ejpam-5596	375	5	media	medium	NOUN
ejpam-5596	375	6	,	,	PUNCT
ejpam-5596	375	7	2003	2003	NUM
ejpam-5596	375	8	.	.	PUNCT
ejpam-5596	376	1	[	[	X
ejpam-5596	376	2	12	12	NUM
ejpam-5596	376	3	]	]	X
ejpam-5596	376	4	n.	n.	NOUN
ejpam-5596	376	5	tiprachot	tiprachot	NOUN
ejpam-5596	376	6	,	,	PUNCT
ejpam-5596	376	7	n.	n.	PROPN
ejpam-5596	376	8	lekkoksung	lekkoksung	PROPN
ejpam-5596	376	9	,	,	PUNCT
ejpam-5596	376	10	and	and	CCONJ
ejpam-5596	376	11	b.	b.	PROPN
ejpam-5596	376	12	pibaljommee	pibaljommee	PROPN
ejpam-5596	376	13	.	.	PUNCT
ejpam-5596	377	1	regularities	regularity	NOUN
ejpam-5596	377	2	of	of	ADP
ejpam-5596	377	3	ordered	order	VERB
ejpam-5596	377	4	semigroups	semigroup	NOUN
ejpam-5596	377	5	in	in	ADP
ejpam-5596	377	6	terms	term	NOUN
ejpam-5596	377	7	of	of	ADP
ejpam-5596	377	8	(	(	PUNCT
ejpam-5596	377	9	m	m	PROPN
ejpam-5596	377	10	,	,	PUNCT
ejpam-5596	377	11	n)-ideals	n)-ideal	NOUN
ejpam-5596	377	12	and	and	CCONJ
ejpam-5596	377	13	n	n	CCONJ
ejpam-5596	377	14	-	-	ADJ
ejpam-5596	377	15	interior	interior	ADJ
ejpam-5596	377	16	ideal	ideal	NOUN
ejpam-5596	377	17	,	,	PUNCT
ejpam-5596	377	18	.	.	PUNCT
ejpam-5596	378	1	international	international	ADJ
ejpam-5596	378	2	journal	journal	NOUN
ejpam-5596	378	3	of	of	ADP
ejpam-5596	378	4	innovative	innovative	ADJ
ejpam-5596	378	5	computing	computing	NOUN
ejpam-5596	378	6	,	,	PUNCT
ejpam-5596	378	7	information	information	NOUN
ejpam-5596	378	8	and	and	CCONJ
ejpam-5596	378	9	control	control	NOUN
ejpam-5596	378	10	,	,	PUNCT
ejpam-5596	378	11	17(2):723–730	17(2):723–730	NUM
ejpam-5596	378	12	,	,	PUNCT
ejpam-5596	378	13	2022	2022	NUM
ejpam-5596	378	14	.	.	PUNCT
ejpam-5596	379	1	[	[	X
ejpam-5596	379	2	13	13	NUM
ejpam-5596	379	3	]	]	X
ejpam-5596	379	4	n.	n.	NOUN
ejpam-5596	379	5	tiprachot	tiprachot	NOUN
ejpam-5596	379	6	,	,	PUNCT
ejpam-5596	379	7	s.	s.	PROPN
ejpam-5596	379	8	lekkoksung	lekkoksung	PROPN
ejpam-5596	379	9	,	,	PUNCT
ejpam-5596	379	10	b.	b.	PROPN
ejpam-5596	379	11	pibaljommee	pibaljommee	PROPN
ejpam-5596	379	12	,	,	PUNCT
ejpam-5596	379	13	and	and	CCONJ
ejpam-5596	379	14	n.	n.	PROPN
ejpam-5596	379	15	lekkoksung	lekkoksung	PROPN
ejpam-5596	379	16	.	.	PUNCT
ejpam-5596	380	1	hybrid	hybrid	ADJ
ejpam-5596	380	2	n	n	CCONJ
ejpam-5596	380	3	-	-	PUNCT
ejpam-5596	380	4	interior	interior	ADJ
ejpam-5596	380	5	ideal	ideal	NOUN
ejpam-5596	380	6	and	and	CCONJ
ejpam-5596	380	7	hybrid	hybrid	ADJ
ejpam-5596	380	8	(	(	PUNCT
ejpam-5596	380	9	m	m	PROPN
ejpam-5596	380	10	,	,	PUNCT
ejpam-5596	380	11	n)-ideals	n)-ideal	NOUN
ejpam-5596	380	12	in	in	ADP
ejpam-5596	380	13	ordered	order	VERB
ejpam-5596	380	14	semigroups	semigroup	NOUN
ejpam-5596	380	15	.	.	PUNCT
ejpam-5596	381	1	fuzzy	fuzzy	ADJ
ejpam-5596	381	2	information	information	NOUN
ejpam-5596	381	3	and	and	CCONJ
ejpam-5596	381	4	engineering	engineering	NOUN
ejpam-5596	381	5	,	,	PUNCT
ejpam-5596	381	6	15(2):128–148	15(2):128–148	NUM
ejpam-5596	381	7	,	,	PUNCT
ejpam-5596	381	8	2023	2023	NUM
ejpam-5596	381	9	.	.	PUNCT
ejpam-5596	382	1	[	[	X
ejpam-5596	382	2	14	14	NUM
ejpam-5596	382	3	]	]	X
ejpam-5596	382	4	l.a	l.a	PROPN
ejpam-5596	382	5	.	.	PROPN
ejpam-5596	382	6	zadeh	zadeh	PROPN
ejpam-5596	382	7	.	.	PUNCT
ejpam-5596	382	8	fuzzy	fuzzy	ADJ
ejpam-5596	382	9	sets	set	NOUN
ejpam-5596	382	10	.	.	PUNCT
ejpam-5596	383	1	information	information	NOUN
ejpam-5596	383	2	and	and	CCONJ
ejpam-5596	383	3	control	control	NOUN
ejpam-5596	383	4	,	,	PUNCT
ejpam-5596	383	5	8:338–353	8:338–353	NUM
ejpam-5596	383	6	,	,	PUNCT
ejpam-5596	383	7	1965	1965	NUM
ejpam-5596	383	8	.	.	PUNCT