id	sid	tid	token	lemma	pos
ejpam-5460	1	1	european	european	PROPN
ejpam-5460	1	2	journal	journal	PROPN
ejpam-5460	1	3	of	of	ADP
ejpam-5460	1	4	pure	pure	ADJ
ejpam-5460	1	5	and	and	CCONJ
ejpam-5460	1	6	applied	apply	VERB
ejpam-5460	1	7	mathematics	mathematic	NOUN
ejpam-5460	1	8	vol	vol	NOUN
ejpam-5460	1	9	.	.	PROPN
ejpam-5460	2	1	17	17	NUM
ejpam-5460	2	2	,	,	PUNCT
ejpam-5460	2	3	no	no	INTJ
ejpam-5460	2	4	.	.	NOUN
ejpam-5460	2	5	4	4	NUM
ejpam-5460	2	6	,	,	PUNCT
ejpam-5460	2	7	2024	2024	NUM
ejpam-5460	2	8	,	,	PUNCT
ejpam-5460	2	9	2962	2962	NUM
ejpam-5460	2	10	-	-	SYM
ejpam-5460	2	11	2984	2984	NUM
ejpam-5460	2	12	issn	issn	PROPN
ejpam-5460	2	13	1307	1307	NUM
ejpam-5460	2	14	-	-	SYM
ejpam-5460	2	15	5543	5543	NUM
ejpam-5460	2	16	–	–	PUNCT
ejpam-5460	2	17	ejpam.com	ejpam.com	X
ejpam-5460	2	18	published	publish	VERB
ejpam-5460	2	19	by	by	ADP
ejpam-5460	2	20	new	new	PROPN
ejpam-5460	2	21	york	york	PROPN
ejpam-5460	2	22	business	business	PROPN
ejpam-5460	2	23	global	global	PROPN
ejpam-5460	2	24	an	an	DET
ejpam-5460	2	25	application	application	NOUN
ejpam-5460	2	26	of	of	ADP
ejpam-5460	2	27	generalized	generalized	ADJ
ejpam-5460	2	28	fuzzy	fuzzy	ADJ
ejpam-5460	2	29	ideals	ideal	NOUN
ejpam-5460	2	30	in	in	ADP
ejpam-5460	2	31	ordered	order	VERB
ejpam-5460	2	32	semigroups	semigroup	NOUN
ejpam-5460	2	33	somsak	somsak	PROPN
ejpam-5460	2	34	lekkoksung1	lekkoksung1	PROPN
ejpam-5460	2	35	,	,	PUNCT
ejpam-5460	2	36	bijan	bijan	PROPN
ejpam-5460	2	37	davvaz2	davvaz2	PROPN
ejpam-5460	2	38	,	,	PUNCT
ejpam-5460	2	39	nareupanat	nareupanat	ADJ
ejpam-5460	2	40	lekkoksung1,∗	lekkoksung1,∗	NOUN
ejpam-5460	2	41	1	1	NUM
ejpam-5460	2	42	division	division	NOUN
ejpam-5460	2	43	of	of	ADP
ejpam-5460	2	44	mathematics	mathematic	NOUN
ejpam-5460	2	45	,	,	PUNCT
ejpam-5460	2	46	faculty	faculty	NOUN
ejpam-5460	2	47	of	of	ADP
ejpam-5460	2	48	engineering	engineering	NOUN
ejpam-5460	2	49	,	,	PUNCT
ejpam-5460	2	50	rajamangala	rajamangala	PROPN
ejpam-5460	2	51	university	university	PROPN
ejpam-5460	2	52	of	of	ADP
ejpam-5460	2	53	technology	technology	PROPN
ejpam-5460	2	54	isan	isan	PROPN
ejpam-5460	2	55	,	,	PUNCT
ejpam-5460	2	56	khon	khon	PROPN
ejpam-5460	2	57	kaen	kaen	PROPN
ejpam-5460	2	58	campus	campus	PROPN
ejpam-5460	2	59	,	,	PUNCT
ejpam-5460	2	60	khon	khon	PROPN
ejpam-5460	2	61	kaen	kaen	PROPN
ejpam-5460	2	62	40000	40000	NUM
ejpam-5460	2	63	,	,	PUNCT
ejpam-5460	2	64	thailand	thailand	PROPN
ejpam-5460	2	65	2	2	NUM
ejpam-5460	2	66	department	department	NOUN
ejpam-5460	2	67	of	of	ADP
ejpam-5460	2	68	mathematical	mathematical	ADJ
ejpam-5460	2	69	sciences	sciences	PROPN
ejpam-5460	2	70	,	,	PUNCT
ejpam-5460	2	71	yazd	yazd	PROPN
ejpam-5460	2	72	university	university	PROPN
ejpam-5460	2	73	,	,	PUNCT
ejpam-5460	2	74	yazd	yazd	PROPN
ejpam-5460	2	75	89136	89136	NUM
ejpam-5460	2	76	,	,	PUNCT
ejpam-5460	2	77	iran	iran	PROPN
ejpam-5460	2	78	abstract	abstract	NOUN
ejpam-5460	2	79	.	.	PUNCT
ejpam-5460	3	1	ordered	order	VERB
ejpam-5460	3	2	semigroups	semigroup	NOUN
ejpam-5460	3	3	are	be	AUX
ejpam-5460	3	4	algebraic	algebraic	ADJ
ejpam-5460	3	5	systems	system	NOUN
ejpam-5460	3	6	consisting	consist	VERB
ejpam-5460	3	7	of	of	ADP
ejpam-5460	3	8	a	a	DET
ejpam-5460	3	9	nonempty	nonempty	ADJ
ejpam-5460	3	10	set	set	NOUN
ejpam-5460	3	11	,	,	PUNCT
ejpam-5460	3	12	an	an	DET
ejpam-5460	3	13	associative	associative	ADJ
ejpam-5460	3	14	binary	binary	ADJ
ejpam-5460	3	15	operation	operation	NOUN
ejpam-5460	3	16	,	,	PUNCT
ejpam-5460	3	17	and	and	CCONJ
ejpam-5460	3	18	a	a	DET
ejpam-5460	3	19	partial	partial	ADJ
ejpam-5460	3	20	order	order	NOUN
ejpam-5460	3	21	compatible	compatible	ADJ
ejpam-5460	3	22	with	with	ADP
ejpam-5460	3	23	this	this	DET
ejpam-5460	3	24	binary	binary	ADJ
ejpam-5460	3	25	operation	operation	NOUN
ejpam-5460	3	26	.	.	PUNCT
ejpam-5460	4	1	this	this	DET
ejpam-5460	4	2	concept	concept	NOUN
ejpam-5460	4	3	is	be	AUX
ejpam-5460	4	4	a	a	DET
ejpam-5460	4	5	generalization	generalization	NOUN
ejpam-5460	4	6	of	of	ADP
ejpam-5460	4	7	semigroups	semigroup	NOUN
ejpam-5460	4	8	.	.	PUNCT
ejpam-5460	5	1	one	one	NUM
ejpam-5460	5	2	mathematical	mathematical	ADJ
ejpam-5460	5	3	tool	tool	NOUN
ejpam-5460	5	4	used	use	VERB
ejpam-5460	5	5	to	to	PART
ejpam-5460	5	6	study	study	VERB
ejpam-5460	5	7	ordered	order	VERB
ejpam-5460	5	8	semigroups	semigroup	NOUN
ejpam-5460	5	9	is	be	AUX
ejpam-5460	5	10	the	the	DET
ejpam-5460	5	11	concept	concept	NOUN
ejpam-5460	5	12	of	of	ADP
ejpam-5460	5	13	ideals	ideal	NOUN
ejpam-5460	5	14	.	.	PUNCT
ejpam-5460	6	1	it	it	PRON
ejpam-5460	6	2	turns	turn	VERB
ejpam-5460	6	3	out	out	ADP
ejpam-5460	6	4	that	that	SCONJ
ejpam-5460	6	5	ordered	order	VERB
ejpam-5460	6	6	semigroups	semigroup	NOUN
ejpam-5460	6	7	can	can	AUX
ejpam-5460	6	8	be	be	AUX
ejpam-5460	6	9	decomposed	decompose	VERB
ejpam-5460	6	10	based	base	VERB
ejpam-5460	6	11	on	on	ADP
ejpam-5460	6	12	their	their	PRON
ejpam-5460	6	13	regularities	regularity	NOUN
ejpam-5460	6	14	using	use	VERB
ejpam-5460	6	15	various	various	ADJ
ejpam-5460	6	16	kinds	kind	NOUN
ejpam-5460	6	17	of	of	ADP
ejpam-5460	6	18	ideals	ideal	NOUN
ejpam-5460	6	19	.	.	PUNCT
ejpam-5460	7	1	the	the	DET
ejpam-5460	7	2	concept	concept	NOUN
ejpam-5460	7	3	of	of	ADP
ejpam-5460	7	4	fuzzy	fuzzy	ADJ
ejpam-5460	7	5	sets	set	NOUN
ejpam-5460	7	6	is	be	AUX
ejpam-5460	7	7	one	one	NUM
ejpam-5460	7	8	of	of	ADP
ejpam-5460	7	9	the	the	DET
ejpam-5460	7	10	mathematical	mathematical	ADJ
ejpam-5460	7	11	tools	tool	NOUN
ejpam-5460	7	12	used	use	VERB
ejpam-5460	7	13	to	to	PART
ejpam-5460	7	14	investigate	investigate	VERB
ejpam-5460	7	15	ordered	order	VERB
ejpam-5460	7	16	semigroups	semigroup	NOUN
ejpam-5460	7	17	,	,	PUNCT
ejpam-5460	7	18	specifically	specifically	ADV
ejpam-5460	7	19	through	through	ADP
ejpam-5460	7	20	so	so	ADV
ejpam-5460	7	21	-	-	PUNCT
ejpam-5460	7	22	called	call	VERB
ejpam-5460	7	23	fuzzy	fuzzy	ADJ
ejpam-5460	7	24	ideals	ideal	NOUN
ejpam-5460	7	25	,	,	PUNCT
ejpam-5460	7	26	which	which	PRON
ejpam-5460	7	27	are	be	AUX
ejpam-5460	7	28	more	more	ADV
ejpam-5460	7	29	appropriate	appropriate	ADJ
ejpam-5460	7	30	than	than	ADP
ejpam-5460	7	31	set	set	NOUN
ejpam-5460	7	32	-	-	PUNCT
ejpam-5460	7	33	theoretical	theoretical	ADJ
ejpam-5460	7	34	ideals	ideal	NOUN
ejpam-5460	7	35	.	.	PUNCT
ejpam-5460	8	1	it	it	PRON
ejpam-5460	8	2	is	be	AUX
ejpam-5460	8	3	known	know	VERB
ejpam-5460	8	4	that	that	SCONJ
ejpam-5460	8	5	the	the	DET
ejpam-5460	8	6	concepts	concept	NOUN
ejpam-5460	8	7	of	of	ADP
ejpam-5460	8	8	(	(	PUNCT
ejpam-5460	8	9	α	α	NOUN
ejpam-5460	8	10	,	,	PUNCT
ejpam-5460	8	11	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	8	12	(	(	PUNCT
ejpam-5460	8	13	m	m	NOUN
ejpam-5460	8	14	,	,	PUNCT
ejpam-5460	8	15	n)-ideals	n)-ideal	NOUN
ejpam-5460	8	16	and	and	CCONJ
ejpam-5460	8	17	(	(	PUNCT
ejpam-5460	8	18	α	α	NOUN
ejpam-5460	8	19	,	,	PUNCT
ejpam-5460	8	20	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	8	21	n	n	CCONJ
ejpam-5460	8	22	-	-	PUNCT
ejpam-5460	8	23	interior	interior	ADJ
ejpam-5460	8	24	ideals	ideal	NOUN
ejpam-5460	8	25	are	be	AUX
ejpam-5460	8	26	generalizations	generalization	NOUN
ejpam-5460	8	27	of	of	ADP
ejpam-5460	8	28	various	various	ADJ
ejpam-5460	8	29	types	type	NOUN
ejpam-5460	8	30	of	of	ADP
ejpam-5460	8	31	ideals	ideal	NOUN
ejpam-5460	8	32	and	and	CCONJ
ejpam-5460	8	33	many	many	ADJ
ejpam-5460	8	34	kinds	kind	NOUN
ejpam-5460	8	35	of	of	ADP
ejpam-5460	8	36	fuzzy	fuzzy	ADJ
ejpam-5460	8	37	ideals	ideal	NOUN
ejpam-5460	8	38	in	in	ADP
ejpam-5460	8	39	ordered	order	VERB
ejpam-5460	8	40	semigroups	semigroup	NOUN
ejpam-5460	8	41	.	.	PUNCT
ejpam-5460	9	1	in	in	ADP
ejpam-5460	9	2	this	this	DET
ejpam-5460	9	3	paper	paper	NOUN
ejpam-5460	9	4	,	,	PUNCT
ejpam-5460	9	5	we	we	PRON
ejpam-5460	9	6	apply	apply	VERB
ejpam-5460	9	7	the	the	DET
ejpam-5460	9	8	notions	notion	NOUN
ejpam-5460	9	9	of	of	ADP
ejpam-5460	9	10	(	(	PUNCT
ejpam-5460	9	11	α	α	NOUN
ejpam-5460	9	12	,	,	PUNCT
ejpam-5460	9	13	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	9	14	(	(	PUNCT
ejpam-5460	9	15	m	m	X
ejpam-5460	9	16	,	,	PUNCT
ejpam-5460	9	17	n)ideals	n)ideal	NOUN
ejpam-5460	9	18	and	and	CCONJ
ejpam-5460	9	19	(	(	PUNCT
ejpam-5460	9	20	α	α	NOUN
ejpam-5460	9	21	,	,	PUNCT
ejpam-5460	9	22	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	9	23	n	n	CCONJ
ejpam-5460	9	24	-	-	PUNCT
ejpam-5460	9	25	interior	interior	ADJ
ejpam-5460	9	26	ideals	ideal	NOUN
ejpam-5460	9	27	to	to	PART
ejpam-5460	9	28	classify	classify	AUX
ejpam-5460	9	29	ordered	order	VERB
ejpam-5460	9	30	semigroups	semigroup	NOUN
ejpam-5460	9	31	into	into	ADP
ejpam-5460	9	32	classes	class	NOUN
ejpam-5460	9	33	depended	depend	VERB
ejpam-5460	9	34	on	on	ADP
ejpam-5460	9	35	their	their	PRON
ejpam-5460	9	36	regularities	regularity	NOUN
ejpam-5460	9	37	,	,	PUNCT
ejpam-5460	9	38	using	use	VERB
ejpam-5460	9	39	the	the	DET
ejpam-5460	9	40	meaning	meaning	NOUN
ejpam-5460	9	41	of	of	ADP
ejpam-5460	9	42	characteristic	characteristic	ADJ
ejpam-5460	9	43	functions	function	NOUN
ejpam-5460	9	44	.	.	PUNCT
ejpam-5460	10	1	2020	2020	NUM
ejpam-5460	10	2	mathematics	mathematic	NOUN
ejpam-5460	10	3	subject	subject	NOUN
ejpam-5460	10	4	classifications	classification	NOUN
ejpam-5460	10	5	:	:	PUNCT
ejpam-5460	10	6	06f05	06f05	NUM
ejpam-5460	10	7	,	,	PUNCT
ejpam-5460	10	8	08a72	08a72	NUM
ejpam-5460	10	9	,	,	PUNCT
ejpam-5460	10	10	20m12	20m12	NUM
ejpam-5460	10	11	key	key	ADJ
ejpam-5460	10	12	words	word	NOUN
ejpam-5460	10	13	and	and	CCONJ
ejpam-5460	10	14	phrases	phrase	NOUN
ejpam-5460	10	15	:	:	PUNCT
ejpam-5460	10	16	ordered	order	VERB
ejpam-5460	10	17	semigroup	semigroup	PROPN
ejpam-5460	10	18	,	,	PUNCT
ejpam-5460	10	19	fuzzy	fuzzy	ADJ
ejpam-5460	10	20	ordered	order	VERB
ejpam-5460	10	21	semigroup	semigroup	NOUN
ejpam-5460	10	22	,	,	PUNCT
ejpam-5460	10	23	regularity	regularity	NOUN
ejpam-5460	10	24	,	,	PUNCT
ejpam-5460	10	25	(	(	PUNCT
ejpam-5460	10	26	α	α	NOUN
ejpam-5460	10	27	,	,	PUNCT
ejpam-5460	10	28	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	10	29	(	(	PUNCT
ejpam-5460	10	30	m	m	PROPN
ejpam-5460	10	31	,	,	PUNCT
ejpam-5460	10	32	n)-ideal	n)-ideal	NOUN
ejpam-5460	10	33	,	,	PUNCT
ejpam-5460	10	34	(	(	PUNCT
ejpam-5460	10	35	α	α	NOUN
ejpam-5460	10	36	,	,	PUNCT
ejpam-5460	10	37	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	10	38	n	n	CCONJ
ejpam-5460	10	39	-	-	PUNCT
ejpam-5460	10	40	interior	interior	ADJ
ejpam-5460	10	41	ideal	ideal	NOUN
ejpam-5460	10	42	1	1	NUM
ejpam-5460	10	43	.	.	PUNCT
ejpam-5460	11	1	introduction	introduction	NOUN
ejpam-5460	11	2	ordered	order	VERB
ejpam-5460	11	3	semigroups	semigroup	NOUN
ejpam-5460	11	4	constitute	constitute	VERB
ejpam-5460	11	5	an	an	DET
ejpam-5460	11	6	algebraic	algebraic	ADJ
ejpam-5460	11	7	structure	structure	NOUN
ejpam-5460	11	8	comprising	comprise	VERB
ejpam-5460	11	9	a	a	DET
ejpam-5460	11	10	binary	binary	ADJ
ejpam-5460	11	11	operation	operation	NOUN
ejpam-5460	11	12	satisfying	satisfy	VERB
ejpam-5460	11	13	associative	associative	ADJ
ejpam-5460	11	14	property	property	NOUN
ejpam-5460	11	15	and	and	CCONJ
ejpam-5460	11	16	a	a	DET
ejpam-5460	11	17	partial	partial	ADJ
ejpam-5460	11	18	order	order	NOUN
ejpam-5460	11	19	with	with	ADP
ejpam-5460	11	20	the	the	DET
ejpam-5460	11	21	compatibility	compatibility	NOUN
ejpam-5460	11	22	(	(	PUNCT
ejpam-5460	11	23	see	see	VERB
ejpam-5460	11	24	[	[	X
ejpam-5460	11	25	2	2	NUM
ejpam-5460	11	26	,	,	PUNCT
ejpam-5460	11	27	8	8	NUM
ejpam-5460	11	28	]	]	NUM
ejpam-5460	11	29	)	)	PUNCT
ejpam-5460	11	30	.	.	PUNCT
ejpam-5460	12	1	this	this	DET
ejpam-5460	12	2	concept	concept	NOUN
ejpam-5460	12	3	extends	extend	VERB
ejpam-5460	12	4	the	the	DET
ejpam-5460	12	5	notion	notion	NOUN
ejpam-5460	12	6	of	of	ADP
ejpam-5460	12	7	semigroups	semigroup	NOUN
ejpam-5460	12	8	,	,	PUNCT
ejpam-5460	12	9	prompting	prompt	VERB
ejpam-5460	12	10	numerous	numerous	ADJ
ejpam-5460	12	11	researchers	researcher	NOUN
ejpam-5460	12	12	to	to	PART
ejpam-5460	12	13	explore	explore	VERB
ejpam-5460	12	14	various	various	ADJ
ejpam-5460	12	15	properties	property	NOUN
ejpam-5460	12	16	of	of	ADP
ejpam-5460	12	17	semigroups	semigroup	NOUN
ejpam-5460	12	18	in	in	ADP
ejpam-5460	12	19	an	an	DET
ejpam-5460	12	20	ordered	order	VERB
ejpam-5460	12	21	semigroup	semigroup	NOUN
ejpam-5460	12	22	setting	setting	NOUN
ejpam-5460	12	23	.	.	PUNCT
ejpam-5460	13	1	studying	study	VERB
ejpam-5460	13	2	ideals	ideal	NOUN
ejpam-5460	13	3	is	be	AUX
ejpam-5460	13	4	key	key	ADJ
ejpam-5460	13	5	to	to	ADP
ejpam-5460	13	6	examining	examine	VERB
ejpam-5460	13	7	several	several	ADJ
ejpam-5460	13	8	properties	property	NOUN
ejpam-5460	13	9	inherent	inherent	ADJ
ejpam-5460	13	10	to	to	PART
ejpam-5460	13	11	ordered	order	VERB
ejpam-5460	13	12	semigroups	semigroup	NOUN
ejpam-5460	13	13	.	.	PUNCT
ejpam-5460	14	1	let	let	VERB
ejpam-5460	14	2	us	we	PRON
ejpam-5460	14	3	briefly	briefly	ADV
ejpam-5460	14	4	outline	outline	VERB
ejpam-5460	14	5	the	the	DET
ejpam-5460	14	6	historical	historical	ADJ
ejpam-5460	14	7	context	context	NOUN
ejpam-5460	14	8	of	of	ADP
ejpam-5460	14	9	the	the	DET
ejpam-5460	14	10	set	set	NOUN
ejpam-5460	14	11	-	-	PUNCT
ejpam-5460	14	12	theoretical	theoretical	ADJ
ejpam-5460	14	13	ideals	ideal	NOUN
ejpam-5460	14	14	under	under	ADP
ejpam-5460	14	15	consideration	consideration	NOUN
ejpam-5460	14	16	in	in	ADP
ejpam-5460	14	17	this	this	DET
ejpam-5460	14	18	paper	paper	NOUN
ejpam-5460	14	19	.	.	PUNCT
ejpam-5460	15	1	the	the	DET
ejpam-5460	15	2	concept	concept	NOUN
ejpam-5460	15	3	of	of	ADP
ejpam-5460	15	4	bi	bi	NOUN
ejpam-5460	15	5	-	-	NOUN
ejpam-5460	15	6	ideals	ideal	NOUN
ejpam-5460	15	7	extends	extend	VERB
ejpam-5460	15	8	left	leave	VERB
ejpam-5460	15	9	and	and	CCONJ
ejpam-5460	15	10	right	right	ADJ
ejpam-5460	15	11	ideals	ideal	NOUN
ejpam-5460	15	12	in	in	ADP
ejpam-5460	15	13	ordered	order	VERB
ejpam-5460	15	14	semigroups	semigroup	NOUN
ejpam-5460	15	15	,	,	PUNCT
ejpam-5460	15	16	started	start	VERB
ejpam-5460	15	17	by	by	ADP
ejpam-5460	15	18	kehayopulu	kehayopulu	PROPN
ejpam-5460	15	19	.	.	PUNCT
ejpam-5460	16	1	the	the	DET
ejpam-5460	16	2	author	author	NOUN
ejpam-5460	16	3	studied	study	VERB
ejpam-5460	16	4	ordered	order	VERB
ejpam-5460	16	5	semigroups	semigroup	NOUN
ejpam-5460	16	6	that	that	PRON
ejpam-5460	16	7	do	do	AUX
ejpam-5460	16	8	not	not	PART
ejpam-5460	16	9	contain	contain	VERB
ejpam-5460	16	10	proper	proper	ADJ
ejpam-5460	16	11	biideals	biideal	NOUN
ejpam-5460	16	12	(	(	PUNCT
ejpam-5460	16	13	see	see	VERB
ejpam-5460	16	14	[	[	X
ejpam-5460	16	15	14	14	NUM
ejpam-5460	16	16	]	]	NUM
ejpam-5460	16	17	)	)	PUNCT
ejpam-5460	16	18	.	.	PUNCT
ejpam-5460	17	1	this	this	DET
ejpam-5460	17	2	property	property	NOUN
ejpam-5460	17	3	resonates	resonate	VERB
ejpam-5460	17	4	with	with	ADP
ejpam-5460	17	5	group	group	NOUN
ejpam-5460	17	6	theory	theory	NOUN
ejpam-5460	17	7	,	,	PUNCT
ejpam-5460	17	8	emphasizing	emphasize	VERB
ejpam-5460	17	9	the	the	DET
ejpam-5460	17	10	notion	notion	NOUN
ejpam-5460	17	11	of	of	ADP
ejpam-5460	17	12	∗corresponding	∗corresponde	VERB
ejpam-5460	17	13	author	author	NOUN
ejpam-5460	17	14	.	.	PUNCT
ejpam-5460	18	1	doi	doi	NOUN
ejpam-5460	18	2	:	:	PUNCT
ejpam-5460	18	3	https://doi.org/10.29020/nybg.ejpam.v17i4.5460	https://doi.org/10.29020/nybg.ejpam.v17i4.5460	ADJ
ejpam-5460	18	4	email	email	NOUN
ejpam-5460	18	5	addresses	address	NOUN
ejpam-5460	18	6	:	:	PUNCT
ejpam-5460	18	7	lekkoksung	lekkoksung	PROPN
ejpam-5460	18	8	somsak@hotmail.com	somsak@hotmail.com	X
ejpam-5460	18	9	(	(	PUNCT
ejpam-5460	18	10	s.	s.	PROPN
ejpam-5460	18	11	lekkoksung	lekkoksung	PROPN
ejpam-5460	18	12	)	)	PUNCT
ejpam-5460	18	13	,	,	PUNCT
ejpam-5460	18	14	davvaz@yazd.ac.ir	davvaz@yazd.ac.ir	PROPN
ejpam-5460	18	15	(	(	PUNCT
ejpam-5460	18	16	b.	b.	PROPN
ejpam-5460	18	17	davvaz	davvaz	PROPN
ejpam-5460	18	18	)	)	PUNCT
ejpam-5460	18	19	,	,	PUNCT
ejpam-5460	18	20	nareupanat.le@rmuti.ac.th	nareupanat.le@rmuti.ac.th	PROPN
ejpam-5460	18	21	(	(	PUNCT
ejpam-5460	18	22	n.	n.	PROPN
ejpam-5460	18	23	lekkoksung	lekkoksung	PROPN
ejpam-5460	18	24	)	)	PUNCT
ejpam-5460	18	25	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-5460	18	26	2962	2962	NUM
ejpam-5460	19	1	copyright	copyright	NOUN
ejpam-5460	19	2	:	:	PUNCT
ejpam-5460	19	3	©	©	PROPN
ejpam-5460	19	4	2024	2024	NUM
ejpam-5460	19	5	the	the	DET
ejpam-5460	19	6	author(s	author(s	NOUN
ejpam-5460	19	7	)	)	PUNCT
ejpam-5460	19	8	.	.	PUNCT
ejpam-5460	20	1	(	(	PUNCT
ejpam-5460	20	2	cc	cc	NOUN
ejpam-5460	20	3	by	by	ADP
ejpam-5460	20	4	-	-	PUNCT
ejpam-5460	20	5	nc	nc	PROPN
ejpam-5460	20	6	4.0	4.0	NUM
ejpam-5460	20	7	)	)	PUNCT
ejpam-5460	20	8	s.	s.	PROPN
ejpam-5460	20	9	lekkoksung	lekkoksung	PROPN
ejpam-5460	20	10	,	,	PUNCT
ejpam-5460	20	11	b.	b.	PROPN
ejpam-5460	20	12	davvaz	davvaz	PROPN
ejpam-5460	20	13	,	,	PUNCT
ejpam-5460	20	14	n.	n.	PROPN
ejpam-5460	20	15	lekkoksung	lekkoksung	PROPN
ejpam-5460	20	16	/	/	SYM
ejpam-5460	20	17	eur	eur	PROPN
ejpam-5460	20	18	.	.	PUNCT
ejpam-5460	21	1	j.	j.	PROPN
ejpam-5460	21	2	pure	pure	PROPN
ejpam-5460	21	3	appl	appl	PROPN
ejpam-5460	21	4	.	.	PROPN
ejpam-5460	21	5	math	math	PROPN
ejpam-5460	21	6	,	,	PUNCT
ejpam-5460	21	7	17	17	NUM
ejpam-5460	21	8	(	(	PUNCT
ejpam-5460	21	9	4	4	NUM
ejpam-5460	21	10	)	)	PUNCT
ejpam-5460	21	11	(	(	PUNCT
ejpam-5460	21	12	2024	2024	NUM
ejpam-5460	21	13	)	)	PUNCT
ejpam-5460	21	14	,	,	PUNCT
ejpam-5460	21	15	2962	2962	NUM
ejpam-5460	21	16	-	-	SYM
ejpam-5460	21	17	2984	2984	NUM
ejpam-5460	21	18	2963	2963	NUM
ejpam-5460	21	19	bi	bi	ADJ
ejpam-5460	21	20	-	-	PUNCT
ejpam-5460	21	21	ideals	ideal	NOUN
ejpam-5460	21	22	significance	significance	NOUN
ejpam-5460	21	23	in	in	ADP
ejpam-5460	21	24	ordered	order	VERB
ejpam-5460	21	25	semigroups	semigroup	NOUN
ejpam-5460	21	26	.	.	PUNCT
ejpam-5460	22	1	subsequently	subsequently	ADV
ejpam-5460	22	2	,	,	PUNCT
ejpam-5460	22	3	in	in	ADP
ejpam-5460	22	4	2013	2013	NUM
ejpam-5460	22	5	,	,	PUNCT
ejpam-5460	22	6	saritha	saritha	PROPN
ejpam-5460	22	7	[	[	X
ejpam-5460	22	8	45	45	NUM
ejpam-5460	22	9	]	]	PUNCT
ejpam-5460	22	10	considered	consider	VERB
ejpam-5460	22	11	prime	prime	ADJ
ejpam-5460	22	12	and	and	CCONJ
ejpam-5460	22	13	semiprime	semiprime	NOUN
ejpam-5460	22	14	properties	property	NOUN
ejpam-5460	22	15	of	of	ADP
ejpam-5460	22	16	bi	bi	NOUN
ejpam-5460	22	17	-	-	NOUN
ejpam-5460	22	18	ideals	ideal	NOUN
ejpam-5460	22	19	.	.	PUNCT
ejpam-5460	23	1	moreover	moreover	ADV
ejpam-5460	23	2	,	,	PUNCT
ejpam-5460	23	3	bi	bi	NOUN
ejpam-5460	23	4	-	-	NOUN
ejpam-5460	23	5	ideals	ideal	NOUN
ejpam-5460	23	6	satisfying	satisfy	VERB
ejpam-5460	23	7	prime	prime	ADJ
ejpam-5460	23	8	and	and	CCONJ
ejpam-5460	23	9	semiprime	semiprime	NOUN
ejpam-5460	23	10	properties	property	NOUN
ejpam-5460	23	11	were	be	AUX
ejpam-5460	23	12	characterized	characterize	VERB
ejpam-5460	23	13	.	.	PUNCT
ejpam-5460	24	1	building	build	VERB
ejpam-5460	24	2	upon	upon	SCONJ
ejpam-5460	24	3	saritha	saritha	PROPN
ejpam-5460	24	4	’s	’s	PART
ejpam-5460	24	5	insights	insight	NOUN
ejpam-5460	24	6	,	,	PUNCT
ejpam-5460	24	7	gu	gu	X
ejpam-5460	25	1	[	[	X
ejpam-5460	25	2	9	9	NUM
ejpam-5460	25	3	]	]	PUNCT
ejpam-5460	25	4	introduced	introduce	VERB
ejpam-5460	25	5	quasi	quasi	ADJ
ejpam-5460	25	6	-	-	ADJ
ejpam-5460	25	7	prime	prime	ADJ
ejpam-5460	25	8	properties	property	NOUN
ejpam-5460	25	9	for	for	ADP
ejpam-5460	25	10	bi	bi	NOUN
ejpam-5460	25	11	-	-	NOUN
ejpam-5460	25	12	ideals	ideal	NOUN
ejpam-5460	25	13	,	,	PUNCT
ejpam-5460	25	14	which	which	PRON
ejpam-5460	25	15	were	be	AUX
ejpam-5460	25	16	then	then	ADV
ejpam-5460	25	17	utilized	utilize	VERB
ejpam-5460	25	18	to	to	PART
ejpam-5460	25	19	characterize	characterize	VERB
ejpam-5460	25	20	regular	regular	ADJ
ejpam-5460	25	21	and	and	CCONJ
ejpam-5460	25	22	intra	intra	ADJ
ejpam-5460	25	23	-	-	ADJ
ejpam-5460	25	24	regular	regular	ADJ
ejpam-5460	25	25	ordered	order	VERB
ejpam-5460	25	26	semigroups	semigroup	NOUN
ejpam-5460	25	27	.	.	PUNCT
ejpam-5460	26	1	hansda	hansda	NOUN
ejpam-5460	27	1	[	[	X
ejpam-5460	27	2	10	10	NUM
ejpam-5460	27	3	]	]	PUNCT
ejpam-5460	27	4	expanded	expand	VERB
ejpam-5460	27	5	upon	upon	SCONJ
ejpam-5460	27	6	gu	gu	NOUN
ejpam-5460	27	7	’s	’s	PART
ejpam-5460	27	8	results	result	NOUN
ejpam-5460	27	9	,	,	PUNCT
ejpam-5460	27	10	further	far	ADV
ejpam-5460	27	11	characterizing	characterize	VERB
ejpam-5460	27	12	regular	regular	ADJ
ejpam-5460	27	13	and	and	CCONJ
ejpam-5460	27	14	completely	completely	ADV
ejpam-5460	27	15	regular	regular	ADJ
ejpam-5460	27	16	ordered	order	VERB
ejpam-5460	27	17	semigroups	semigroup	NOUN
ejpam-5460	27	18	based	base	VERB
ejpam-5460	27	19	on	on	ADP
ejpam-5460	27	20	the	the	DET
ejpam-5460	27	21	minimality	minimality	NOUN
ejpam-5460	27	22	of	of	ADP
ejpam-5460	27	23	bi	bi	NOUN
ejpam-5460	27	24	-	-	NOUN
ejpam-5460	27	25	ideals	ideal	NOUN
ejpam-5460	27	26	.	.	PUNCT
ejpam-5460	28	1	kehayopulu	kehayopulu	PROPN
ejpam-5460	28	2	introduced	introduce	VERB
ejpam-5460	28	3	the	the	DET
ejpam-5460	28	4	notion	notion	NOUN
ejpam-5460	28	5	of	of	ADP
ejpam-5460	28	6	interior	interior	ADJ
ejpam-5460	28	7	ideals	ideal	NOUN
ejpam-5460	28	8	in	in	ADP
ejpam-5460	28	9	ordered	order	VERB
ejpam-5460	28	10	semigroups	semigroup	NOUN
ejpam-5460	28	11	in	in	ADP
ejpam-5460	28	12	1999	1999	NUM
ejpam-5460	28	13	.	.	PUNCT
ejpam-5460	29	1	it	it	PRON
ejpam-5460	29	2	was	be	AUX
ejpam-5460	29	3	demonstrated	demonstrate	VERB
ejpam-5460	29	4	that	that	SCONJ
ejpam-5460	29	5	interior	interior	ADJ
ejpam-5460	29	6	ideals	ideal	NOUN
ejpam-5460	29	7	coincide	coincide	VERB
ejpam-5460	29	8	with	with	ADP
ejpam-5460	29	9	two	two	NUM
ejpam-5460	29	10	-	-	PUNCT
ejpam-5460	29	11	sided	sided	ADJ
ejpam-5460	29	12	ideals	ideal	NOUN
ejpam-5460	29	13	in	in	ADP
ejpam-5460	29	14	certain	certain	ADJ
ejpam-5460	29	15	classes	class	NOUN
ejpam-5460	29	16	of	of	ADP
ejpam-5460	29	17	ordered	order	VERB
ejpam-5460	29	18	semigroups	semigroup	NOUN
ejpam-5460	29	19	(	(	PUNCT
ejpam-5460	29	20	see	see	VERB
ejpam-5460	29	21	[	[	X
ejpam-5460	29	22	15	15	NUM
ejpam-5460	29	23	]	]	NUM
ejpam-5460	29	24	)	)	PUNCT
ejpam-5460	29	25	.	.	PUNCT
ejpam-5460	30	1	sanborisoot	sanborisoot	PROPN
ejpam-5460	30	2	and	and	CCONJ
ejpam-5460	30	3	changphas	changphas	PROPN
ejpam-5460	30	4	initially	initially	ADV
ejpam-5460	30	5	generalized	generalize	VERB
ejpam-5460	30	6	the	the	DET
ejpam-5460	30	7	concept	concept	NOUN
ejpam-5460	30	8	of	of	ADP
ejpam-5460	30	9	bi	bi	NOUN
ejpam-5460	30	10	-	-	NOUN
ejpam-5460	30	11	ideals	ideal	NOUN
ejpam-5460	30	12	in	in	ADP
ejpam-5460	30	13	ordered	order	VERB
ejpam-5460	30	14	semigroups	semigroup	NOUN
ejpam-5460	30	15	to	to	ADP
ejpam-5460	30	16	(	(	PUNCT
ejpam-5460	30	17	m	m	PROPN
ejpam-5460	30	18	,	,	PUNCT
ejpam-5460	30	19	n)-ideals	n)-ideal	NOUN
ejpam-5460	30	20	.	.	PUNCT
ejpam-5460	31	1	subsequently	subsequently	ADV
ejpam-5460	31	2	,	,	PUNCT
ejpam-5460	31	3	the	the	DET
ejpam-5460	31	4	notion	notion	NOUN
ejpam-5460	31	5	of	of	ADP
ejpam-5460	31	6	(	(	PUNCT
ejpam-5460	31	7	m	m	PROPN
ejpam-5460	31	8	,	,	PUNCT
ejpam-5460	31	9	n)-ideals	n)-ideal	NOUN
ejpam-5460	31	10	was	be	AUX
ejpam-5460	31	11	employed	employ	VERB
ejpam-5460	31	12	to	to	PART
ejpam-5460	31	13	characterize	characterize	VERB
ejpam-5460	31	14	a	a	DET
ejpam-5460	31	15	class	class	NOUN
ejpam-5460	31	16	of	of	ADP
ejpam-5460	31	17	ordered	order	VERB
ejpam-5460	31	18	semigroups	semigroup	NOUN
ejpam-5460	31	19	known	know	VERB
ejpam-5460	31	20	as	as	ADP
ejpam-5460	31	21	(	(	PUNCT
ejpam-5460	31	22	m	m	PROPN
ejpam-5460	31	23	,	,	PUNCT
ejpam-5460	31	24	n)-regular	n)-regular	PRON
ejpam-5460	31	25	ordered	order	VERB
ejpam-5460	31	26	semigroups	semigroup	NOUN
ejpam-5460	31	27	(	(	PUNCT
ejpam-5460	31	28	see	see	VERB
ejpam-5460	31	29	[	[	X
ejpam-5460	31	30	44	44	NUM
ejpam-5460	31	31	]	]	SYM
ejpam-5460	31	32	)	)	PUNCT
ejpam-5460	31	33	.	.	PUNCT
ejpam-5460	32	1	in	in	ADP
ejpam-5460	32	2	2015	2015	NUM
ejpam-5460	32	3	,	,	PUNCT
ejpam-5460	32	4	bussaban	bussaban	NOUN
ejpam-5460	32	5	and	and	CCONJ
ejpam-5460	32	6	changphas	changphas	ADJ
ejpam-5460	32	7	[	[	X
ejpam-5460	32	8	3	3	X
ejpam-5460	32	9	]	]	PUNCT
ejpam-5460	32	10	highlighted	highlight	VERB
ejpam-5460	32	11	that	that	SCONJ
ejpam-5460	32	12	in	in	ADP
ejpam-5460	32	13	regular	regular	ADJ
ejpam-5460	32	14	duo	duo	NOUN
ejpam-5460	32	15	ordered	order	VERB
ejpam-5460	32	16	semigroups	semigroup	NOUN
ejpam-5460	32	17	,	,	PUNCT
ejpam-5460	32	18	the	the	DET
ejpam-5460	32	19	concept	concept	NOUN
ejpam-5460	32	20	of	of	ADP
ejpam-5460	32	21	(	(	PUNCT
ejpam-5460	32	22	m	m	PROPN
ejpam-5460	32	23	,	,	PUNCT
ejpam-5460	32	24	n)-ideals	n)-ideal	VERB
ejpam-5460	32	25	coincides	coincide	NOUN
ejpam-5460	32	26	with	with	ADP
ejpam-5460	32	27	two	two	NUM
ejpam-5460	32	28	-	-	PUNCT
ejpam-5460	32	29	sided	sided	ADJ
ejpam-5460	32	30	ideals	ideal	NOUN
ejpam-5460	32	31	.	.	PUNCT
ejpam-5460	33	1	luangchaisri	luangchaisri	VERB
ejpam-5460	33	2	and	and	CCONJ
ejpam-5460	33	3	changphas	changpha	VERB
ejpam-5460	34	1	[	[	X
ejpam-5460	34	2	37	37	NUM
ejpam-5460	34	3	]	]	PUNCT
ejpam-5460	34	4	further	far	ADV
ejpam-5460	34	5	extended	extend	VERB
ejpam-5460	34	6	bi	bi	NOUN
ejpam-5460	34	7	-	-	ADJ
ejpam-5460	34	8	ideals	ideal	NOUN
ejpam-5460	34	9	results	result	NOUN
ejpam-5460	34	10	to	to	ADP
ejpam-5460	34	11	(	(	PUNCT
ejpam-5460	34	12	m	m	PROPN
ejpam-5460	34	13	,	,	PUNCT
ejpam-5460	34	14	n)-ideals	n)-ideal	NOUN
ejpam-5460	34	15	in	in	ADP
ejpam-5460	34	16	2019	2019	NUM
ejpam-5460	34	17	.	.	PUNCT
ejpam-5460	35	1	following	follow	VERB
ejpam-5460	35	2	this	this	DET
ejpam-5460	35	3	extension	extension	NOUN
ejpam-5460	35	4	,	,	PUNCT
ejpam-5460	35	5	tiprachot	tiprachot	NOUN
ejpam-5460	35	6	et	et	PROPN
ejpam-5460	35	7	al	al	AUX
ejpam-5460	35	8	.	.	PUNCT
ejpam-5460	36	1	[	[	X
ejpam-5460	36	2	49	49	NUM
ejpam-5460	36	3	]	]	PUNCT
ejpam-5460	36	4	generalized	generalize	VERB
ejpam-5460	36	5	the	the	DET
ejpam-5460	36	6	notion	notion	NOUN
ejpam-5460	36	7	of	of	ADP
ejpam-5460	36	8	interior	interior	ADJ
ejpam-5460	36	9	ideals	ideal	NOUN
ejpam-5460	36	10	to	to	ADP
ejpam-5460	36	11	n	n	CCONJ
ejpam-5460	36	12	-	-	PUNCT
ejpam-5460	36	13	interior	interior	ADJ
ejpam-5460	36	14	ideals	ideal	NOUN
ejpam-5460	36	15	in	in	ADP
ejpam-5460	36	16	ordered	order	VERB
ejpam-5460	36	17	semigroups	semigroup	NOUN
ejpam-5460	36	18	in	in	ADP
ejpam-5460	36	19	2022	2022	NUM
ejpam-5460	36	20	.	.	PUNCT
ejpam-5460	37	1	the	the	DET
ejpam-5460	37	2	authors	author	NOUN
ejpam-5460	37	3	characterized	characterize	VERB
ejpam-5460	37	4	several	several	ADJ
ejpam-5460	37	5	classes	class	NOUN
ejpam-5460	37	6	of	of	ADP
ejpam-5460	37	7	ordered	order	VERB
ejpam-5460	37	8	semigroups	semigroup	NOUN
ejpam-5460	37	9	by	by	ADP
ejpam-5460	37	10	combining	combine	VERB
ejpam-5460	37	11	(	(	PUNCT
ejpam-5460	37	12	m	m	NOUN
ejpam-5460	37	13	,	,	PUNCT
ejpam-5460	37	14	n)-ideals	n)-ideal	NOUN
ejpam-5460	37	15	and	and	CCONJ
ejpam-5460	37	16	n	n	CCONJ
ejpam-5460	37	17	-	-	ADJ
ejpam-5460	37	18	interior	interior	ADJ
ejpam-5460	37	19	ideals	ideal	NOUN
ejpam-5460	37	20	.	.	PUNCT
ejpam-5460	38	1	the	the	DET
ejpam-5460	38	2	characterization	characterization	NOUN
ejpam-5460	38	3	of	of	ADP
ejpam-5460	38	4	ordered	order	VERB
ejpam-5460	38	5	semigroups	semigroup	NOUN
ejpam-5460	38	6	using	use	VERB
ejpam-5460	38	7	(	(	PUNCT
ejpam-5460	38	8	m	m	NOUN
ejpam-5460	38	9	,	,	PUNCT
ejpam-5460	38	10	n)-ideals	n)-ideal	NOUN
ejpam-5460	38	11	and	and	CCONJ
ejpam-5460	38	12	n	n	CCONJ
ejpam-5460	38	13	-	-	PUNCT
ejpam-5460	38	14	interior	interior	ADJ
ejpam-5460	38	15	ideals	ideal	NOUN
ejpam-5460	38	16	was	be	AUX
ejpam-5460	38	17	further	far	ADV
ejpam-5460	38	18	extended	extend	VERB
ejpam-5460	38	19	by	by	ADP
ejpam-5460	38	20	(	(	PUNCT
ejpam-5460	38	21	m	m	PROPN
ejpam-5460	38	22	,	,	PUNCT
ejpam-5460	38	23	n)-ideal	n)-ideal	NOUN
ejpam-5460	38	24	elements	element	NOUN
ejpam-5460	38	25	and	and	CCONJ
ejpam-5460	38	26	n	n	CCONJ
ejpam-5460	38	27	-	-	ADJ
ejpam-5460	38	28	interior	interior	ADJ
ejpam-5460	38	29	ideal	ideal	ADJ
ejpam-5460	38	30	elements	element	NOUN
ejpam-5460	38	31	(	(	PUNCT
ejpam-5460	38	32	see	see	VERB
ejpam-5460	38	33	[	[	X
ejpam-5460	38	34	36	36	NUM
ejpam-5460	38	35	]	]	NUM
ejpam-5460	38	36	)	)	PUNCT
ejpam-5460	38	37	.	.	PUNCT
ejpam-5460	39	1	additionally	additionally	ADV
ejpam-5460	39	2	,	,	PUNCT
ejpam-5460	39	3	they	they	PRON
ejpam-5460	39	4	provided	provide	VERB
ejpam-5460	39	5	a	a	DET
ejpam-5460	39	6	comprehensive	comprehensive	ADJ
ejpam-5460	39	7	overview	overview	NOUN
ejpam-5460	39	8	of	of	ADP
ejpam-5460	39	9	the	the	DET
ejpam-5460	39	10	classification	classification	NOUN
ejpam-5460	39	11	of	of	ADP
ejpam-5460	39	12	ordered	order	VERB
ejpam-5460	39	13	semigroups	semigroup	NOUN
ejpam-5460	39	14	through	through	ADP
ejpam-5460	39	15	α	α	NOUN
ejpam-5460	39	16	-	-	PUNCT
ejpam-5460	39	17	ideals	ideal	NOUN
ejpam-5460	39	18	(	(	PUNCT
ejpam-5460	39	19	see	see	VERB
ejpam-5460	39	20	[	[	X
ejpam-5460	39	21	48	48	NUM
ejpam-5460	39	22	]	]	PUNCT
ejpam-5460	39	23	)	)	PUNCT
ejpam-5460	39	24	.	.	PUNCT
ejpam-5460	40	1	in	in	ADP
ejpam-5460	40	2	1965	1965	NUM
ejpam-5460	40	3	,	,	PUNCT
ejpam-5460	40	4	zadeh	zadeh	PROPN
ejpam-5460	40	5	[	[	X
ejpam-5460	40	6	51	51	NUM
ejpam-5460	40	7	]	]	PUNCT
ejpam-5460	40	8	introduced	introduce	VERB
ejpam-5460	40	9	the	the	DET
ejpam-5460	40	10	concept	concept	NOUN
ejpam-5460	40	11	of	of	ADP
ejpam-5460	40	12	fuzzy	fuzzy	ADJ
ejpam-5460	40	13	sets	set	NOUN
ejpam-5460	40	14	that	that	PRON
ejpam-5460	40	15	can	can	AUX
ejpam-5460	40	16	deal	deal	VERB
ejpam-5460	40	17	with	with	ADP
ejpam-5460	40	18	problems	problem	NOUN
ejpam-5460	40	19	involving	involve	VERB
ejpam-5460	40	20	uncertain	uncertain	ADJ
ejpam-5460	40	21	conditions	condition	NOUN
ejpam-5460	40	22	.	.	PUNCT
ejpam-5460	41	1	fuzzy	fuzzy	ADJ
ejpam-5460	41	2	sets	set	NOUN
ejpam-5460	41	3	have	have	AUX
ejpam-5460	41	4	widely	widely	ADV
ejpam-5460	41	5	applied	apply	VERB
ejpam-5460	41	6	across	across	ADP
ejpam-5460	41	7	various	various	ADJ
ejpam-5460	41	8	scientific	scientific	ADJ
ejpam-5460	41	9	and	and	CCONJ
ejpam-5460	41	10	engineering	engineering	NOUN
ejpam-5460	41	11	investigations	investigation	NOUN
ejpam-5460	41	12	.	.	PUNCT
ejpam-5460	42	1	particularly	particularly	ADV
ejpam-5460	42	2	in	in	ADP
ejpam-5460	42	3	mathematics	mathematic	NOUN
ejpam-5460	42	4	,	,	PUNCT
ejpam-5460	42	5	they	they	PRON
ejpam-5460	42	6	play	play	VERB
ejpam-5460	42	7	as	as	ADP
ejpam-5460	42	8	a	a	DET
ejpam-5460	42	9	tool	tool	NOUN
ejpam-5460	42	10	to	to	PART
ejpam-5460	42	11	analyze	analyze	VERB
ejpam-5460	42	12	the	the	DET
ejpam-5460	42	13	properties	property	NOUN
ejpam-5460	42	14	of	of	ADP
ejpam-5460	42	15	several	several	ADJ
ejpam-5460	42	16	algebraic	algebraic	ADJ
ejpam-5460	42	17	systems	system	NOUN
ejpam-5460	42	18	,	,	PUNCT
ejpam-5460	42	19	including	include	VERB
ejpam-5460	42	20	semigroups	semigroup	NOUN
ejpam-5460	42	21	,	,	PUNCT
ejpam-5460	42	22	ordered	order	VERB
ejpam-5460	42	23	semigroups	semigroup	NOUN
ejpam-5460	42	24	,	,	PUNCT
ejpam-5460	42	25	groups	group	NOUN
ejpam-5460	42	26	,	,	PUNCT
ejpam-5460	42	27	semirings	semiring	NOUN
ejpam-5460	42	28	,	,	PUNCT
ejpam-5460	42	29	ordered	order	VERB
ejpam-5460	42	30	semirings	semiring	NOUN
ejpam-5460	42	31	,	,	PUNCT
ejpam-5460	42	32	and	and	CCONJ
ejpam-5460	42	33	rings	ring	NOUN
ejpam-5460	42	34	(	(	PUNCT
ejpam-5460	42	35	see	see	VERB
ejpam-5460	42	36	[	[	X
ejpam-5460	42	37	1	1	NUM
ejpam-5460	42	38	,	,	PUNCT
ejpam-5460	42	39	4	4	NUM
ejpam-5460	42	40	,	,	PUNCT
ejpam-5460	42	41	5	5	NUM
ejpam-5460	42	42	,	,	PUNCT
ejpam-5460	42	43	35	35	NUM
ejpam-5460	42	44	,	,	PUNCT
ejpam-5460	42	45	38–41	38–41	NUM
ejpam-5460	42	46	]	]	PUNCT
ejpam-5460	42	47	)	)	PUNCT
ejpam-5460	42	48	.	.	PUNCT
ejpam-5460	43	1	given	give	VERB
ejpam-5460	43	2	the	the	DET
ejpam-5460	43	3	importance	importance	NOUN
ejpam-5460	43	4	of	of	ADP
ejpam-5460	43	5	ideals	ideal	NOUN
ejpam-5460	43	6	in	in	ADP
ejpam-5460	43	7	exploring	explore	VERB
ejpam-5460	43	8	ordered	order	VERB
ejpam-5460	43	9	semigroups	semigroup	NOUN
ejpam-5460	43	10	,	,	PUNCT
ejpam-5460	43	11	we	we	PRON
ejpam-5460	43	12	offer	offer	VERB
ejpam-5460	43	13	a	a	DET
ejpam-5460	43	14	brief	brief	ADJ
ejpam-5460	43	15	overview	overview	NOUN
ejpam-5460	43	16	of	of	ADP
ejpam-5460	43	17	how	how	SCONJ
ejpam-5460	43	18	their	their	PRON
ejpam-5460	43	19	fuzzy	fuzzy	ADJ
ejpam-5460	43	20	counterparts	counterpart	NOUN
ejpam-5460	43	21	are	be	AUX
ejpam-5460	43	22	utilized	utilize	VERB
ejpam-5460	43	23	in	in	ADP
ejpam-5460	43	24	this	this	DET
ejpam-5460	43	25	process	process	NOUN
ejpam-5460	43	26	.	.	PUNCT
ejpam-5460	44	1	in	in	ADP
ejpam-5460	44	2	the	the	DET
ejpam-5460	44	3	early	early	ADJ
ejpam-5460	44	4	2000s	2000s	NUM
ejpam-5460	44	5	,	,	PUNCT
ejpam-5460	44	6	fuzzy	fuzzy	ADJ
ejpam-5460	44	7	sets	set	NOUN
ejpam-5460	44	8	were	be	AUX
ejpam-5460	44	9	applied	apply	VERB
ejpam-5460	44	10	to	to	ADP
ejpam-5460	44	11	ordered	order	VERB
ejpam-5460	44	12	groupoids	groupoid	NOUN
ejpam-5460	44	13	,	,	PUNCT
ejpam-5460	44	14	which	which	PRON
ejpam-5460	44	15	led	lead	VERB
ejpam-5460	44	16	to	to	ADP
ejpam-5460	44	17	their	their	PRON
ejpam-5460	44	18	use	use	NOUN
ejpam-5460	44	19	in	in	ADP
ejpam-5460	44	20	considering	consider	VERB
ejpam-5460	44	21	ordered	order	VERB
ejpam-5460	44	22	semigroups	semigroup	NOUN
ejpam-5460	44	23	(	(	PUNCT
ejpam-5460	44	24	see	see	VERB
ejpam-5460	44	25	[	[	X
ejpam-5460	44	26	19	19	NUM
ejpam-5460	44	27	]	]	NUM
ejpam-5460	44	28	)	)	PUNCT
ejpam-5460	44	29	.	.	PUNCT
ejpam-5460	45	1	in	in	ADP
ejpam-5460	45	2	2003	2003	NUM
ejpam-5460	45	3	,	,	PUNCT
ejpam-5460	45	4	kehayopulu	kehayopulu	VERB
ejpam-5460	45	5	and	and	CCONJ
ejpam-5460	45	6	tsingelis	tsingeli	NOUN
ejpam-5460	46	1	[	[	X
ejpam-5460	46	2	20	20	NUM
ejpam-5460	46	3	]	]	PUNCT
ejpam-5460	46	4	showed	show	VERB
ejpam-5460	46	5	that	that	SCONJ
ejpam-5460	46	6	any	any	DET
ejpam-5460	46	7	ordered	order	VERB
ejpam-5460	46	8	semigroup	semigroup	NOUN
ejpam-5460	46	9	embeds	embed	VERB
ejpam-5460	46	10	into	into	ADP
ejpam-5460	46	11	a	a	DET
ejpam-5460	46	12	fuzzy	fuzzy	ADJ
ejpam-5460	46	13	ordered	order	VERB
ejpam-5460	46	14	semigroup	semigroup	NOUN
ejpam-5460	46	15	,	,	PUNCT
ejpam-5460	46	16	which	which	PRON
ejpam-5460	46	17	was	be	AUX
ejpam-5460	46	18	a	a	DET
ejpam-5460	46	19	significant	significant	ADJ
ejpam-5460	46	20	result	result	NOUN
ejpam-5460	46	21	.	.	PUNCT
ejpam-5460	47	1	two	two	NUM
ejpam-5460	47	2	years	year	NOUN
ejpam-5460	47	3	later	later	ADV
ejpam-5460	47	4	,	,	PUNCT
ejpam-5460	47	5	they	they	PRON
ejpam-5460	47	6	introduced	introduce	VERB
ejpam-5460	47	7	the	the	DET
ejpam-5460	47	8	concept	concept	NOUN
ejpam-5460	47	9	of	of	ADP
ejpam-5460	47	10	fuzzy	fuzzy	ADJ
ejpam-5460	47	11	bi	bi	NOUN
ejpam-5460	47	12	-	-	NOUN
ejpam-5460	47	13	ideals	ideal	NOUN
ejpam-5460	47	14	in	in	ADP
ejpam-5460	47	15	ordered	order	VERB
ejpam-5460	47	16	semigroups	semigroup	NOUN
ejpam-5460	47	17	.	.	PUNCT
ejpam-5460	48	1	the	the	DET
ejpam-5460	48	2	authors	author	NOUN
ejpam-5460	48	3	[	[	X
ejpam-5460	48	4	21	21	NUM
ejpam-5460	48	5	]	]	PUNCT
ejpam-5460	48	6	used	use	VERB
ejpam-5460	48	7	them	they	PRON
ejpam-5460	48	8	to	to	PART
ejpam-5460	48	9	characterize	characterize	VERB
ejpam-5460	48	10	different	different	ADJ
ejpam-5460	48	11	classes	class	NOUN
ejpam-5460	48	12	of	of	ADP
ejpam-5460	48	13	ordered	order	VERB
ejpam-5460	48	14	semigroups	semigroup	NOUN
ejpam-5460	48	15	,	,	PUNCT
ejpam-5460	48	16	for	for	ADP
ejpam-5460	48	17	example	example	NOUN
ejpam-5460	48	18	,	,	PUNCT
ejpam-5460	48	19	left	leave	VERB
ejpam-5460	48	20	(	(	PUNCT
ejpam-5460	48	21	right	right	ADJ
ejpam-5460	48	22	)	)	PUNCT
ejpam-5460	48	23	simple	simple	ADJ
ejpam-5460	48	24	,	,	PUNCT
ejpam-5460	48	25	completely	completely	ADV
ejpam-5460	48	26	regular	regular	ADJ
ejpam-5460	48	27	,	,	PUNCT
ejpam-5460	48	28	and	and	CCONJ
ejpam-5460	48	29	strongly	strongly	ADV
ejpam-5460	48	30	regular	regular	ADJ
ejpam-5460	48	31	.	.	PUNCT
ejpam-5460	49	1	in	in	ADP
ejpam-5460	49	2	2006	2006	NUM
ejpam-5460	49	3	,	,	PUNCT
ejpam-5460	49	4	fuzzy	fuzzy	ADJ
ejpam-5460	49	5	left	left	ADJ
ejpam-5460	49	6	(	(	PUNCT
ejpam-5460	49	7	right	right	ADJ
ejpam-5460	49	8	)	)	PUNCT
ejpam-5460	49	9	ideals	ideal	NOUN
ejpam-5460	49	10	[	[	X
ejpam-5460	49	11	23	23	NUM
ejpam-5460	49	12	]	]	PUNCT
ejpam-5460	49	13	were	be	AUX
ejpam-5460	49	14	used	use	VERB
ejpam-5460	49	15	to	to	PART
ejpam-5460	49	16	study	study	VERB
ejpam-5460	49	17	left	leave	VERB
ejpam-5460	49	18	(	(	PUNCT
ejpam-5460	49	19	right	right	ADJ
ejpam-5460	49	20	)	)	PUNCT
ejpam-5460	49	21	regular	regular	ADJ
ejpam-5460	49	22	and	and	CCONJ
ejpam-5460	49	23	intra	intra	ADJ
ejpam-5460	49	24	-	-	ADJ
ejpam-5460	49	25	regular	regular	ADJ
ejpam-5460	49	26	ordered	order	VERB
ejpam-5460	49	27	semigroups	semigroup	NOUN
ejpam-5460	49	28	.	.	PUNCT
ejpam-5460	50	1	over	over	ADP
ejpam-5460	50	2	two	two	NUM
ejpam-5460	50	3	decades	decade	NOUN
ejpam-5460	50	4	,	,	PUNCT
ejpam-5460	50	5	several	several	ADJ
ejpam-5460	50	6	studies	study	NOUN
ejpam-5460	50	7	have	have	AUX
ejpam-5460	50	8	characterized	characterize	VERB
ejpam-5460	50	9	special	special	ADJ
ejpam-5460	50	10	classes	class	NOUN
ejpam-5460	50	11	of	of	ADP
ejpam-5460	50	12	ordered	order	VERB
ejpam-5460	50	13	semigroups	semigroup	NOUN
ejpam-5460	50	14	using	use	VERB
ejpam-5460	50	15	fuzzy	fuzzy	ADJ
ejpam-5460	50	16	ideals	ideal	NOUN
ejpam-5460	50	17	in	in	ADP
ejpam-5460	50	18	different	different	ADJ
ejpam-5460	50	19	combinations	combination	NOUN
ejpam-5460	50	20	(	(	PUNCT
ejpam-5460	50	21	see	see	VERB
ejpam-5460	50	22	[	[	X
ejpam-5460	50	23	16	16	NUM
ejpam-5460	50	24	,	,	PUNCT
ejpam-5460	50	25	24	24	NUM
ejpam-5460	50	26	,	,	PUNCT
ejpam-5460	50	27	25	25	NUM
ejpam-5460	50	28	,	,	PUNCT
ejpam-5460	50	29	46	46	NUM
ejpam-5460	50	30	,	,	PUNCT
ejpam-5460	50	31	50	50	NUM
ejpam-5460	50	32	,	,	PUNCT
ejpam-5460	50	33	52	52	NUM
ejpam-5460	50	34	]	]	NUM
ejpam-5460	50	35	)	)	PUNCT
ejpam-5460	50	36	.	.	PUNCT
ejpam-5460	51	1	khan	khan	PROPN
ejpam-5460	51	2	and	and	CCONJ
ejpam-5460	51	3	shabir	shabir	PROPN
ejpam-5460	52	1	[	[	X
ejpam-5460	52	2	31	31	NUM
ejpam-5460	52	3	]	]	PUNCT
ejpam-5460	52	4	developed	develop	VERB
ejpam-5460	52	5	the	the	DET
ejpam-5460	52	6	concept	concept	NOUN
ejpam-5460	52	7	of	of	ADP
ejpam-5460	52	8	fuzzy	fuzzy	ADJ
ejpam-5460	52	9	ideals	ideal	NOUN
ejpam-5460	52	10	in	in	ADP
ejpam-5460	52	11	ordered	order	VERB
ejpam-5460	52	12	semigroups	semigroup	NOUN
ejpam-5460	52	13	by	by	ADP
ejpam-5460	52	14	introducing	introduce	VERB
ejpam-5460	52	15	the	the	DET
ejpam-5460	52	16	“	"	PUNCT
ejpam-5460	52	17	belong	belong	NOUN
ejpam-5460	52	18	to	to	ADP
ejpam-5460	52	19	”	"	PUNCT
ejpam-5460	52	20	(	(	PUNCT
ejpam-5460	52	21	∈	∈	PROPN
ejpam-5460	52	22	)	)	PUNCT
ejpam-5460	52	23	and	and	CCONJ
ejpam-5460	52	24	“	"	PUNCT
ejpam-5460	52	25	quasi	quasi	ADJ
ejpam-5460	52	26	-	-	ADJ
ejpam-5460	52	27	coincident	coincident	ADJ
ejpam-5460	52	28	”	"	PUNCT
ejpam-5460	52	29	(	(	PUNCT
ejpam-5460	52	30	q	q	NOUN
ejpam-5460	52	31	)	)	PUNCT
ejpam-5460	52	32	relations	relation	NOUN
ejpam-5460	52	33	.	.	PUNCT
ejpam-5460	53	1	they	they	PRON
ejpam-5460	53	2	introduced	introduce	VERB
ejpam-5460	53	3	various	various	ADJ
ejpam-5460	53	4	notions	notion	NOUN
ejpam-5460	53	5	of	of	ADP
ejpam-5460	53	6	(	(	PUNCT
ejpam-5460	53	7	∈,∈	∈,∈	X
ejpam-5460	53	8	∨q)-fuzzy	∨q)-fuzzy	ADJ
ejpam-5460	53	9	ideals	ideal	NOUN
ejpam-5460	53	10	in	in	ADP
ejpam-5460	53	11	ordered	order	VERB
ejpam-5460	53	12	semigroups	semigroup	NOUN
ejpam-5460	53	13	,	,	PUNCT
ejpam-5460	53	14	including	include	VERB
ejpam-5460	53	15	(	(	PUNCT
ejpam-5460	53	16	∈,∈	∈,∈	X
ejpam-5460	53	17	∨q)-fuzzy	∨q)-fuzzy	ADJ
ejpam-5460	53	18	left	left	ADJ
ejpam-5460	53	19	(	(	PUNCT
ejpam-5460	53	20	right	right	ADJ
ejpam-5460	53	21	,	,	PUNCT
ejpam-5460	53	22	two	two	NUM
ejpam-5460	53	23	-	-	PUNCT
ejpam-5460	53	24	sided	sided	ADJ
ejpam-5460	53	25	,	,	PUNCT
ejpam-5460	53	26	interior	interior	ADJ
ejpam-5460	53	27	)	)	PUNCT
ejpam-5460	53	28	ideals	ideal	NOUN
ejpam-5460	53	29	.	.	PUNCT
ejpam-5460	54	1	subsequently	subsequently	ADV
ejpam-5460	54	2	,	,	PUNCT
ejpam-5460	54	3	jun	jun	PROPN
ejpam-5460	54	4	et	et	PROPN
ejpam-5460	54	5	al	al	PROPN
ejpam-5460	54	6	.	.	PUNCT
ejpam-5460	55	1	[	[	X
ejpam-5460	55	2	11	11	NUM
ejpam-5460	55	3	]	]	PUNCT
ejpam-5460	55	4	defined	define	VERB
ejpam-5460	55	5	the	the	DET
ejpam-5460	55	6	concept	concept	NOUN
ejpam-5460	55	7	of	of	ADP
ejpam-5460	55	8	(	(	PUNCT
ejpam-5460	55	9	∈,∈	∈,∈	X
ejpam-5460	55	10	∨q)-fuzzy	∨q)-fuzzy	ADJ
ejpam-5460	55	11	bi	bi	NOUN
ejpam-5460	55	12	-	-	NOUN
ejpam-5460	55	13	ideals	ideal	NOUN
ejpam-5460	55	14	,	,	PUNCT
ejpam-5460	55	15	which	which	PRON
ejpam-5460	55	16	generalize	generalize	VERB
ejpam-5460	55	17	(	(	PUNCT
ejpam-5460	55	18	∈,∈	∈,∈	X
ejpam-5460	55	19	∨q)-fuzzy	∨q)-fuzzy	ADJ
ejpam-5460	55	20	left	left	ADJ
ejpam-5460	55	21	and	and	CCONJ
ejpam-5460	55	22	right	right	ADJ
ejpam-5460	55	23	ideals	ideal	NOUN
ejpam-5460	55	24	.	.	PUNCT
ejpam-5460	56	1	these	these	DET
ejpam-5460	56	2	s.	s.	PROPN
ejpam-5460	56	3	lekkoksung	lekkoksung	PROPN
ejpam-5460	56	4	,	,	PUNCT
ejpam-5460	56	5	b.	b.	PROPN
ejpam-5460	56	6	davvaz	davvaz	PROPN
ejpam-5460	56	7	,	,	PUNCT
ejpam-5460	56	8	n.	n.	PROPN
ejpam-5460	56	9	lekkoksung	lekkoksung	PROPN
ejpam-5460	56	10	/	/	SYM
ejpam-5460	56	11	eur	eur	PROPN
ejpam-5460	56	12	.	.	PUNCT
ejpam-5460	57	1	j.	j.	PROPN
ejpam-5460	57	2	pure	pure	PROPN
ejpam-5460	57	3	appl	appl	PROPN
ejpam-5460	57	4	.	.	PROPN
ejpam-5460	57	5	math	math	PROPN
ejpam-5460	57	6	,	,	PUNCT
ejpam-5460	57	7	17	17	NUM
ejpam-5460	57	8	(	(	PUNCT
ejpam-5460	57	9	4	4	NUM
ejpam-5460	57	10	)	)	PUNCT
ejpam-5460	57	11	(	(	PUNCT
ejpam-5460	57	12	2024	2024	NUM
ejpam-5460	57	13	)	)	PUNCT
ejpam-5460	57	14	,	,	PUNCT
ejpam-5460	57	15	2962	2962	NUM
ejpam-5460	57	16	-	-	SYM
ejpam-5460	57	17	2984	2984	NUM
ejpam-5460	57	18	2964	2964	NUM
ejpam-5460	57	19	concepts	concept	NOUN
ejpam-5460	57	20	were	be	AUX
ejpam-5460	57	21	applied	apply	VERB
ejpam-5460	57	22	to	to	PART
ejpam-5460	57	23	characterize	characterize	VERB
ejpam-5460	57	24	ordered	order	VERB
ejpam-5460	57	25	semigroups	semigroup	NOUN
ejpam-5460	57	26	satisfying	satisfy	VERB
ejpam-5460	57	27	regular	regular	ADJ
ejpam-5460	57	28	and	and	CCONJ
ejpam-5460	57	29	intraregular	intraregular	ADJ
ejpam-5460	57	30	conditions	condition	NOUN
ejpam-5460	57	31	.	.	PUNCT
ejpam-5460	58	1	in	in	ADP
ejpam-5460	58	2	2012	2012	NUM
ejpam-5460	58	3	,	,	PUNCT
ejpam-5460	58	4	khan	khan	PROPN
ejpam-5460	58	5	et	et	PROPN
ejpam-5460	58	6	al	al	PROPN
ejpam-5460	58	7	.	.	PUNCT
ejpam-5460	59	1	[	[	X
ejpam-5460	59	2	28	28	NUM
ejpam-5460	59	3	]	]	PUNCT
ejpam-5460	59	4	further	far	ADV
ejpam-5460	59	5	investigated	investigate	VERB
ejpam-5460	59	6	the	the	DET
ejpam-5460	59	7	properties	property	NOUN
ejpam-5460	59	8	of	of	ADP
ejpam-5460	59	9	(	(	PUNCT
ejpam-5460	59	10	∈,∈	∈,∈	X
ejpam-5460	59	11	∨q)-fuzzy	∨q)-fuzzy	ADJ
ejpam-5460	59	12	left	left	ADJ
ejpam-5460	59	13	and	and	CCONJ
ejpam-5460	59	14	right	right	ADJ
ejpam-5460	59	15	ideals	ideal	NOUN
ejpam-5460	59	16	in	in	ADP
ejpam-5460	59	17	-	-	PUNCT
ejpam-5460	59	18	depth	depth	NOUN
ejpam-5460	59	19	and	and	CCONJ
ejpam-5460	59	20	utilized	utilize	VERB
ejpam-5460	59	21	them	they	PRON
ejpam-5460	59	22	to	to	PART
ejpam-5460	59	23	characterize	characterize	VERB
ejpam-5460	59	24	regular	regular	ADJ
ejpam-5460	59	25	ordered	order	VERB
ejpam-5460	59	26	semigroups	semigroup	NOUN
ejpam-5460	59	27	.	.	PUNCT
ejpam-5460	60	1	the	the	DET
ejpam-5460	60	2	concepts	concept	NOUN
ejpam-5460	60	3	of	of	ADP
ejpam-5460	60	4	(	(	PUNCT
ejpam-5460	60	5	∈,∈	∈,∈	X
ejpam-5460	60	6	∨q)-fuzzy	∨q)-fuzzy	ADJ
ejpam-5460	60	7	ideals	ideal	NOUN
ejpam-5460	60	8	in	in	ADP
ejpam-5460	60	9	ordered	order	VERB
ejpam-5460	60	10	semigroups	semigroup	NOUN
ejpam-5460	60	11	were	be	AUX
ejpam-5460	60	12	extended	extend	VERB
ejpam-5460	60	13	to	to	ADP
ejpam-5460	60	14	(	(	PUNCT
ejpam-5460	60	15	∈,∈	∈,∈	X
ejpam-5460	60	16	∨qk)-fuzzy	∨qk)-fuzzy	ADJ
ejpam-5460	60	17	ideals	ideal	NOUN
ejpam-5460	60	18	,	,	PUNCT
ejpam-5460	60	19	where	where	SCONJ
ejpam-5460	60	20	k	k	PROPN
ejpam-5460	60	21	∈	∈	PROPN
ejpam-5460	61	1	[	[	X
ejpam-5460	61	2	0	0	NUM
ejpam-5460	61	3	,	,	PUNCT
ejpam-5460	61	4	1	1	NUM
ejpam-5460	61	5	)	)	PUNCT
ejpam-5460	61	6	initially	initially	ADV
ejpam-5460	61	7	by	by	ADP
ejpam-5460	61	8	khan	khan	PROPN
ejpam-5460	61	9	et	et	PROPN
ejpam-5460	61	10	al	al	PROPN
ejpam-5460	61	11	.	.	PUNCT
ejpam-5460	62	1	they	they	PRON
ejpam-5460	62	2	defined	define	VERB
ejpam-5460	62	3	the	the	DET
ejpam-5460	62	4	notions	notion	NOUN
ejpam-5460	62	5	of	of	ADP
ejpam-5460	62	6	(	(	PUNCT
ejpam-5460	62	7	∈,∈	∈,∈	X
ejpam-5460	62	8	∨q)-fuzzy	∨q)-fuzzy	ADJ
ejpam-5460	62	9	left	leave	VERB
ejpam-5460	62	10	(	(	PUNCT
ejpam-5460	62	11	right	right	ADJ
ejpam-5460	62	12	and	and	CCONJ
ejpam-5460	62	13	generalized	generalized	ADJ
ejpam-5460	62	14	bi-	bi-	NUM
ejpam-5460	62	15	)	)	PUNCT
ejpam-5460	62	16	ideals	ideal	NOUN
ejpam-5460	62	17	in	in	ADP
ejpam-5460	62	18	ordered	order	VERB
ejpam-5460	62	19	semigroups	semigroup	NOUN
ejpam-5460	62	20	and	and	CCONJ
ejpam-5460	62	21	utilized	utilize	VERB
ejpam-5460	62	22	them	they	PRON
ejpam-5460	62	23	to	to	PART
ejpam-5460	62	24	characterize	characterize	VERB
ejpam-5460	62	25	several	several	ADJ
ejpam-5460	62	26	classes	class	NOUN
ejpam-5460	62	27	of	of	ADP
ejpam-5460	62	28	ordered	order	VERB
ejpam-5460	62	29	semigroups	semigroup	NOUN
ejpam-5460	62	30	(	(	PUNCT
ejpam-5460	62	31	see	see	VERB
ejpam-5460	62	32	[	[	X
ejpam-5460	62	33	27	27	NUM
ejpam-5460	62	34	]	]	NUM
ejpam-5460	62	35	)	)	PUNCT
ejpam-5460	62	36	.	.	PUNCT
ejpam-5460	63	1	the	the	DET
ejpam-5460	63	2	applications	application	NOUN
ejpam-5460	63	3	of	of	ADP
ejpam-5460	63	4	such	such	ADJ
ejpam-5460	63	5	fuzzy	fuzzy	ADJ
ejpam-5460	63	6	ideals	ideal	NOUN
ejpam-5460	63	7	for	for	ADP
ejpam-5460	63	8	classifying	classify	VERB
ejpam-5460	63	9	ordered	order	VERB
ejpam-5460	63	10	semigroups	semigroup	NOUN
ejpam-5460	63	11	appeared	appear	VERB
ejpam-5460	63	12	in	in	ADP
ejpam-5460	63	13	at	at	ADV
ejpam-5460	63	14	least	least	ADV
ejpam-5460	63	15	two	two	NUM
ejpam-5460	63	16	papers	paper	NOUN
ejpam-5460	63	17	by	by	ADP
ejpam-5460	63	18	khan	khan	PROPN
ejpam-5460	63	19	et	et	PROPN
ejpam-5460	63	20	al	al	PROPN
ejpam-5460	63	21	.	.	PROPN
ejpam-5460	63	22	and	and	CCONJ
ejpam-5460	63	23	khan	khan	PROPN
ejpam-5460	63	24	et	et	PROPN
ejpam-5460	63	25	al	al	PROPN
ejpam-5460	63	26	.	.	PUNCT
ejpam-5460	64	1	(	(	PUNCT
ejpam-5460	64	2	see	see	VERB
ejpam-5460	64	3	[	[	X
ejpam-5460	64	4	29	29	NUM
ejpam-5460	64	5	,	,	PUNCT
ejpam-5460	64	6	32	32	NUM
ejpam-5460	64	7	]	]	PUNCT
ejpam-5460	64	8	)	)	PUNCT
ejpam-5460	64	9	later	later	ADV
ejpam-5460	64	10	,	,	PUNCT
ejpam-5460	64	11	tang	tang	PROPN
ejpam-5460	64	12	and	and	CCONJ
ejpam-5460	64	13	xie	xie	PROPN
ejpam-5460	65	1	[	[	X
ejpam-5460	65	2	47	47	NUM
ejpam-5460	65	3	]	]	PUNCT
ejpam-5460	65	4	examined	examine	VERB
ejpam-5460	65	5	prime	prime	ADJ
ejpam-5460	65	6	property	property	NOUN
ejpam-5460	65	7	of	of	ADP
ejpam-5460	65	8	(	(	PUNCT
ejpam-5460	65	9	∈,∈	∈,∈	X
ejpam-5460	65	10	∨qk)-fuzzy	∨qk)-fuzzy	ADJ
ejpam-5460	65	11	ideals	ideal	NOUN
ejpam-5460	65	12	.	.	PUNCT
ejpam-5460	66	1	the	the	DET
ejpam-5460	66	2	concepts	concept	NOUN
ejpam-5460	66	3	of	of	ADP
ejpam-5460	66	4	(	(	PUNCT
ejpam-5460	66	5	∈,∈	∈,∈	X
ejpam-5460	66	6	∨qk)-fuzzy	∨qk)-fuzzy	ADJ
ejpam-5460	66	7	ideals	ideal	NOUN
ejpam-5460	66	8	in	in	ADP
ejpam-5460	66	9	ordered	order	VERB
ejpam-5460	66	10	semigroups	semigroup	NOUN
ejpam-5460	66	11	were	be	AUX
ejpam-5460	66	12	independently	independently	ADV
ejpam-5460	66	13	extended	extended	ADJ
ejpam-5460	66	14	to	to	ADP
ejpam-5460	66	15	(	(	PUNCT
ejpam-5460	66	16	∈,∈	∈,∈	X
ejpam-5460	66	17	∨(k∗	∨(k∗	ADJ
ejpam-5460	66	18	,	,	PUNCT
ejpam-5460	66	19	qk))-fuzzy	qk))-fuzzy	ADJ
ejpam-5460	66	20	ideals	ideal	NOUN
ejpam-5460	66	21	and	and	CCONJ
ejpam-5460	66	22	(	(	PUNCT
ejpam-5460	66	23	∈,∈	∈,∈	X
ejpam-5460	66	24	∨qδk)fuzzy	∨qδk)fuzzy	ADJ
ejpam-5460	66	25	ideals	ideal	NOUN
ejpam-5460	66	26	by	by	ADP
ejpam-5460	66	27	khan	khan	PROPN
ejpam-5460	66	28	et	et	PROPN
ejpam-5460	66	29	al	al	PROPN
ejpam-5460	66	30	.	.	PUNCT
ejpam-5460	67	1	[	[	X
ejpam-5460	67	2	34	34	NUM
ejpam-5460	67	3	]	]	PUNCT
ejpam-5460	67	4	and	and	CCONJ
ejpam-5460	67	5	ali	ali	PROPN
ejpam-5460	67	6	khan	khan	PROPN
ejpam-5460	67	7	et	et	PROPN
ejpam-5460	67	8	al	al	PROPN
ejpam-5460	67	9	.	.	PUNCT
ejpam-5460	68	1	[	[	X
ejpam-5460	68	2	33	33	NUM
ejpam-5460	68	3	]	]	PUNCT
ejpam-5460	68	4	,	,	PUNCT
ejpam-5460	68	5	respectively	respectively	ADV
ejpam-5460	68	6	.	.	PUNCT
ejpam-5460	69	1	interestingly	interestingly	ADV
ejpam-5460	69	2	,	,	PUNCT
ejpam-5460	69	3	it	it	PRON
ejpam-5460	69	4	was	be	AUX
ejpam-5460	69	5	discovered	discover	VERB
ejpam-5460	69	6	that	that	SCONJ
ejpam-5460	69	7	these	these	DET
ejpam-5460	69	8	two	two	NUM
ejpam-5460	69	9	concepts	concept	NOUN
ejpam-5460	69	10	coincide	coincide	VERB
ejpam-5460	69	11	.	.	PUNCT
ejpam-5460	70	1	in	in	ADP
ejpam-5460	70	2	a	a	DET
ejpam-5460	70	3	recent	recent	ADJ
ejpam-5460	70	4	development	development	NOUN
ejpam-5460	70	5	,	,	PUNCT
ejpam-5460	70	6	muhiuddin	muhiuddin	NOUN
ejpam-5460	70	7	et	et	PROPN
ejpam-5460	70	8	al	al	PROPN
ejpam-5460	70	9	.	.	PUNCT
ejpam-5460	71	1	[	[	X
ejpam-5460	71	2	42	42	NUM
ejpam-5460	71	3	]	]	PUNCT
ejpam-5460	71	4	generalized	generalize	VERB
ejpam-5460	71	5	the	the	DET
ejpam-5460	71	6	notion	notion	NOUN
ejpam-5460	71	7	of	of	ADP
ejpam-5460	71	8	(	(	PUNCT
ejpam-5460	71	9	∈,∈	∈,∈	X
ejpam-5460	71	10	∨qk)-fuzzy	∨qk)-fuzzy	ADJ
ejpam-5460	71	11	ideals	ideal	NOUN
ejpam-5460	71	12	to	to	AUX
ejpam-5460	71	13	(	(	PUNCT
ejpam-5460	71	14	∈,∈	∈,∈	X
ejpam-5460	71	15	∨qk)-fuzzy	∨qk)-fuzzy	ADJ
ejpam-5460	71	16	(	(	PUNCT
ejpam-5460	71	17	m	m	NOUN
ejpam-5460	71	18	,	,	PUNCT
ejpam-5460	71	19	n)-ideals	n)-ideal	NOUN
ejpam-5460	71	20	.	.	PUNCT
ejpam-5460	72	1	a	a	DET
ejpam-5460	72	2	class	class	NOUN
ejpam-5460	72	3	of	of	ADP
ejpam-5460	72	4	ordered	order	VERB
ejpam-5460	72	5	semigroups	semigroup	NOUN
ejpam-5460	72	6	,	,	PUNCT
ejpam-5460	72	7	(	(	PUNCT
ejpam-5460	72	8	m	m	X
ejpam-5460	72	9	,	,	PUNCT
ejpam-5460	72	10	n)-regular	n)-regular	PRON
ejpam-5460	72	11	ordered	order	VERB
ejpam-5460	72	12	semigroups	semigroup	NOUN
ejpam-5460	72	13	,	,	PUNCT
ejpam-5460	72	14	were	be	AUX
ejpam-5460	72	15	characterized	characterize	VERB
ejpam-5460	72	16	by	by	ADP
ejpam-5460	72	17	this	this	DET
ejpam-5460	72	18	extension	extension	NOUN
ejpam-5460	72	19	.	.	PUNCT
ejpam-5460	73	1	another	another	DET
ejpam-5460	73	2	generalization	generalization	NOUN
ejpam-5460	73	3	of	of	ADP
ejpam-5460	73	4	the	the	DET
ejpam-5460	73	5	above	above	ADV
ejpam-5460	73	6	-	-	PUNCT
ejpam-5460	73	7	mentioned	mention	VERB
ejpam-5460	73	8	fuzzy	fuzzy	ADJ
ejpam-5460	73	9	ideals	ideal	NOUN
ejpam-5460	73	10	in	in	ADP
ejpam-5460	73	11	ordered	order	VERB
ejpam-5460	73	12	semigroups	semigroup	NOUN
ejpam-5460	73	13	is	be	AUX
ejpam-5460	73	14	the	the	DET
ejpam-5460	73	15	notion	notion	NOUN
ejpam-5460	73	16	of	of	ADP
ejpam-5460	73	17	(	(	PUNCT
ejpam-5460	73	18	α	α	NOUN
ejpam-5460	73	19	,	,	PUNCT
ejpam-5460	73	20	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	73	21	ideals	ideal	NOUN
ejpam-5460	73	22	,	,	PUNCT
ejpam-5460	74	1	where	where	SCONJ
ejpam-5460	74	2	0	0	NUM
ejpam-5460	74	3	≤	≤	NUM
ejpam-5460	74	4	α	α	NOUN
ejpam-5460	74	5	<	<	X
ejpam-5460	74	6	β	β	X
ejpam-5460	74	7	≤	≤	NUM
ejpam-5460	74	8	1	1	NUM
ejpam-5460	74	9	.	.	PUNCT
ejpam-5460	75	1	this	this	DET
ejpam-5460	75	2	notion	notion	NOUN
ejpam-5460	75	3	was	be	AUX
ejpam-5460	75	4	initiated	initiate	VERB
ejpam-5460	75	5	by	by	ADP
ejpam-5460	75	6	feng	feng	PROPN
ejpam-5460	75	7	and	and	CCONJ
ejpam-5460	75	8	corsini	corsini	PROPN
ejpam-5460	75	9	in	in	ADP
ejpam-5460	75	10	2012	2012	NUM
ejpam-5460	75	11	.	.	PUNCT
ejpam-5460	76	1	they	they	PRON
ejpam-5460	76	2	defined	define	VERB
ejpam-5460	76	3	the	the	DET
ejpam-5460	76	4	concepts	concept	NOUN
ejpam-5460	76	5	of	of	ADP
ejpam-5460	76	6	(	(	PUNCT
ejpam-5460	76	7	α	α	NOUN
ejpam-5460	76	8	,	,	PUNCT
ejpam-5460	76	9	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	76	10	left	leave	VERB
ejpam-5460	76	11	(	(	PUNCT
ejpam-5460	76	12	right	right	ADJ
ejpam-5460	76	13	,	,	PUNCT
ejpam-5460	76	14	interior	interior	ADJ
ejpam-5460	76	15	,	,	PUNCT
ejpam-5460	76	16	quasi-	quasi-	NUM
ejpam-5460	76	17	,	,	PUNCT
ejpam-5460	76	18	bi-	bi-	NUM
ejpam-5460	76	19	)	)	PUNCT
ejpam-5460	76	20	ideals	ideal	NOUN
ejpam-5460	76	21	and	and	CCONJ
ejpam-5460	76	22	studied	study	VERB
ejpam-5460	76	23	the	the	DET
ejpam-5460	76	24	fundamental	fundamental	ADJ
ejpam-5460	76	25	properties	property	NOUN
ejpam-5460	76	26	of	of	ADP
ejpam-5460	76	27	these	these	DET
ejpam-5460	76	28	ideals	ideal	NOUN
ejpam-5460	76	29	(	(	PUNCT
ejpam-5460	76	30	see	see	VERB
ejpam-5460	76	31	[	[	X
ejpam-5460	76	32	7	7	NUM
ejpam-5460	76	33	]	]	NUM
ejpam-5460	76	34	)	)	PUNCT
ejpam-5460	76	35	.	.	PUNCT
ejpam-5460	77	1	independently	independently	ADV
ejpam-5460	77	2	,	,	PUNCT
ejpam-5460	77	3	khan	khan	PROPN
ejpam-5460	77	4	et	et	PROPN
ejpam-5460	77	5	al	al	PROPN
ejpam-5460	77	6	.	.	PUNCT
ejpam-5460	78	1	[	[	X
ejpam-5460	78	2	30	30	NUM
ejpam-5460	78	3	]	]	PUNCT
ejpam-5460	78	4	introduced	introduce	VERB
ejpam-5460	78	5	the	the	DET
ejpam-5460	78	6	notion	notion	NOUN
ejpam-5460	78	7	of	of	ADP
ejpam-5460	78	8	(	(	PUNCT
ejpam-5460	78	9	α	α	NOUN
ejpam-5460	78	10	,	,	PUNCT
ejpam-5460	78	11	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	78	12	bi	bi	NOUN
ejpam-5460	78	13	-	-	NOUN
ejpam-5460	78	14	ideals	ideal	NOUN
ejpam-5460	78	15	and	and	CCONJ
ejpam-5460	78	16	characterized	characterize	VERB
ejpam-5460	78	17	completely	completely	ADV
ejpam-5460	78	18	regular	regular	ADJ
ejpam-5460	78	19	ordered	order	VERB
ejpam-5460	78	20	semigroups	semigroup	NOUN
ejpam-5460	78	21	using	use	VERB
ejpam-5460	78	22	this	this	DET
ejpam-5460	78	23	notion	notion	NOUN
ejpam-5460	78	24	.	.	PUNCT
ejpam-5460	79	1	later	later	ADV
ejpam-5460	79	2	,	,	PUNCT
ejpam-5460	79	3	feng	feng	PROPN
ejpam-5460	79	4	and	and	CCONJ
ejpam-5460	79	5	corsini	corsini	PROPN
ejpam-5460	80	1	[	[	X
ejpam-5460	80	2	6	6	NUM
ejpam-5460	80	3	]	]	PUNCT
ejpam-5460	80	4	discovered	discover	VERB
ejpam-5460	80	5	that	that	SCONJ
ejpam-5460	80	6	in	in	ADP
ejpam-5460	80	7	certain	certain	ADJ
ejpam-5460	80	8	classes	class	NOUN
ejpam-5460	80	9	of	of	ADP
ejpam-5460	80	10	ordered	order	VERB
ejpam-5460	80	11	semigroups	semigroup	NOUN
ejpam-5460	80	12	,	,	PUNCT
ejpam-5460	80	13	for	for	ADP
ejpam-5460	80	14	example	example	NOUN
ejpam-5460	80	15	regular	regular	ADJ
ejpam-5460	80	16	and	and	CCONJ
ejpam-5460	80	17	intra	intra	ADJ
ejpam-5460	80	18	-	-	ADJ
ejpam-5460	80	19	regular	regular	ADJ
ejpam-5460	80	20	ordered	order	VERB
ejpam-5460	80	21	semigroups	semigroup	NOUN
ejpam-5460	80	22	,	,	PUNCT
ejpam-5460	80	23	the	the	DET
ejpam-5460	80	24	concepts	concept	NOUN
ejpam-5460	80	25	of	of	ADP
ejpam-5460	80	26	(	(	PUNCT
ejpam-5460	80	27	α	α	NOUN
ejpam-5460	80	28	,	,	PUNCT
ejpam-5460	80	29	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	80	30	two	two	NUM
ejpam-5460	80	31	-	-	PUNCT
ejpam-5460	80	32	sided	sided	ADJ
ejpam-5460	80	33	ideals	ideal	NOUN
ejpam-5460	80	34	and	and	CCONJ
ejpam-5460	80	35	(	(	PUNCT
ejpam-5460	80	36	α	α	NOUN
ejpam-5460	80	37	,	,	PUNCT
ejpam-5460	80	38	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	80	39	interior	interior	ADJ
ejpam-5460	80	40	ideals	ideal	NOUN
ejpam-5460	80	41	coincide	coincide	VERB
ejpam-5460	80	42	.	.	PUNCT
ejpam-5460	81	1	jun	jun	PROPN
ejpam-5460	81	2	et	et	PROPN
ejpam-5460	81	3	al	al	PROPN
ejpam-5460	81	4	.	.	PUNCT
ejpam-5460	82	1	[	[	X
ejpam-5460	82	2	12	12	NUM
ejpam-5460	82	3	]	]	PUNCT
ejpam-5460	82	4	also	also	ADV
ejpam-5460	82	5	characterized	characterize	VERB
ejpam-5460	82	6	some	some	DET
ejpam-5460	82	7	particular	particular	ADJ
ejpam-5460	82	8	classes	class	NOUN
ejpam-5460	82	9	of	of	ADP
ejpam-5460	82	10	ordered	order	VERB
ejpam-5460	82	11	semigroups	semigroup	NOUN
ejpam-5460	82	12	using	use	VERB
ejpam-5460	82	13	(	(	PUNCT
ejpam-5460	82	14	α	α	NOUN
ejpam-5460	82	15	,	,	PUNCT
ejpam-5460	82	16	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	82	17	ideals	ideal	NOUN
ejpam-5460	82	18	.	.	PUNCT
ejpam-5460	83	1	directly	directly	ADV
ejpam-5460	83	2	studying	study	VERB
ejpam-5460	83	3	the	the	DET
ejpam-5460	83	4	structural	structural	ADJ
ejpam-5460	83	5	properties	property	NOUN
ejpam-5460	83	6	of	of	ADP
ejpam-5460	83	7	(	(	PUNCT
ejpam-5460	83	8	α	α	NOUN
ejpam-5460	83	9	,	,	PUNCT
ejpam-5460	83	10	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	83	11	ideals	ideal	NOUN
ejpam-5460	83	12	is	be	AUX
ejpam-5460	83	13	complex	complex	ADJ
ejpam-5460	83	14	.	.	PUNCT
ejpam-5460	84	1	by	by	ADP
ejpam-5460	84	2	applying	apply	VERB
ejpam-5460	84	3	the	the	DET
ejpam-5460	84	4	operation	operation	NOUN
ejpam-5460	84	5	defined	define	VERB
ejpam-5460	84	6	in	in	ADP
ejpam-5460	84	7	[	[	X
ejpam-5460	84	8	30	30	NUM
ejpam-5460	84	9	]	]	PUNCT
ejpam-5460	84	10	,	,	PUNCT
ejpam-5460	84	11	lekkoksung	lekkoksung	PROPN
ejpam-5460	84	12	et	et	PROPN
ejpam-5460	84	13	al	al	PROPN
ejpam-5460	84	14	.	.	PROPN
ejpam-5460	84	15	recently	recently	ADV
ejpam-5460	84	16	demonstrated	demonstrate	VERB
ejpam-5460	84	17	that	that	SCONJ
ejpam-5460	84	18	the	the	DET
ejpam-5460	84	19	algebraic	algebraic	ADJ
ejpam-5460	84	20	structure	structure	NOUN
ejpam-5460	84	21	comprising	comprise	VERB
ejpam-5460	84	22	the	the	DET
ejpam-5460	84	23	set	set	NOUN
ejpam-5460	84	24	of	of	ADP
ejpam-5460	84	25	all	all	DET
ejpam-5460	84	26	fuzzy	fuzzy	ADJ
ejpam-5460	84	27	sets	set	NOUN
ejpam-5460	84	28	on	on	ADP
ejpam-5460	84	29	an	an	DET
ejpam-5460	84	30	ordered	order	VERB
ejpam-5460	84	31	semigroup	semigroup	NOUN
ejpam-5460	84	32	,	,	PUNCT
ejpam-5460	84	33	along	along	ADP
ejpam-5460	84	34	with	with	ADP
ejpam-5460	84	35	this	this	DET
ejpam-5460	84	36	operation	operation	NOUN
ejpam-5460	84	37	and	and	CCONJ
ejpam-5460	84	38	a	a	DET
ejpam-5460	84	39	binary	binary	PROPN
ejpam-5460	84	40	relation	relation	NOUN
ejpam-5460	84	41	,	,	PUNCT
ejpam-5460	84	42	forms	form	VERB
ejpam-5460	84	43	a	a	DET
ejpam-5460	84	44	representation	representation	NOUN
ejpam-5460	84	45	of	of	ADP
ejpam-5460	84	46	an	an	DET
ejpam-5460	84	47	ordered	order	VERB
ejpam-5460	84	48	semigroup	semigroup	NOUN
ejpam-5460	84	49	.	.	PUNCT
ejpam-5460	85	1	this	this	DET
ejpam-5460	85	2	result	result	NOUN
ejpam-5460	85	3	emphasizes	emphasize	VERB
ejpam-5460	85	4	the	the	DET
ejpam-5460	85	5	significance	significance	NOUN
ejpam-5460	85	6	of	of	ADP
ejpam-5460	85	7	investigating	investigate	VERB
ejpam-5460	85	8	(	(	PUNCT
ejpam-5460	85	9	α	α	NOUN
ejpam-5460	85	10	,	,	PUNCT
ejpam-5460	85	11	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	85	12	ideals	ideal	NOUN
ejpam-5460	85	13	in	in	ADP
ejpam-5460	85	14	ordered	order	VERB
ejpam-5460	85	15	semigroups	semigroup	NOUN
ejpam-5460	85	16	(	(	PUNCT
ejpam-5460	85	17	see	see	VERB
ejpam-5460	85	18	[	[	X
ejpam-5460	85	19	35	35	NUM
ejpam-5460	85	20	]	]	SYM
ejpam-5460	85	21	)	)	PUNCT
ejpam-5460	85	22	.	.	PUNCT
ejpam-5460	86	1	subsequently	subsequently	ADV
ejpam-5460	86	2	,	,	PUNCT
ejpam-5460	86	3	davvaz	davvaz	NOUN
ejpam-5460	86	4	et	et	PROPN
ejpam-5460	86	5	al	al	PROPN
ejpam-5460	86	6	.	.	PUNCT
ejpam-5460	87	1	[	[	X
ejpam-5460	87	2	4	4	X
ejpam-5460	87	3	]	]	PUNCT
ejpam-5460	87	4	introduced	introduce	VERB
ejpam-5460	87	5	the	the	DET
ejpam-5460	87	6	concepts	concept	NOUN
ejpam-5460	87	7	of	of	ADP
ejpam-5460	87	8	(	(	PUNCT
ejpam-5460	87	9	α	α	NOUN
ejpam-5460	87	10	,	,	PUNCT
ejpam-5460	87	11	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	87	12	(	(	PUNCT
ejpam-5460	87	13	m	m	NOUN
ejpam-5460	87	14	,	,	PUNCT
ejpam-5460	87	15	n)-ideals	n)-ideal	NOUN
ejpam-5460	87	16	and	and	CCONJ
ejpam-5460	87	17	(	(	PUNCT
ejpam-5460	87	18	α	α	NOUN
ejpam-5460	87	19	,	,	PUNCT
ejpam-5460	87	20	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	87	21	n	n	CCONJ
ejpam-5460	87	22	-	-	PUNCT
ejpam-5460	87	23	interior	interior	ADJ
ejpam-5460	87	24	ideals	ideal	NOUN
ejpam-5460	87	25	in	in	ADP
ejpam-5460	87	26	ordered	order	VERB
ejpam-5460	87	27	semigroups	semigroup	NOUN
ejpam-5460	87	28	.	.	PUNCT
ejpam-5460	88	1	it	it	PRON
ejpam-5460	88	2	was	be	AUX
ejpam-5460	88	3	discovered	discover	VERB
ejpam-5460	88	4	that	that	SCONJ
ejpam-5460	88	5	several	several	ADJ
ejpam-5460	88	6	notions	notion	NOUN
ejpam-5460	88	7	of	of	ADP
ejpam-5460	88	8	ideals	ideal	NOUN
ejpam-5460	88	9	mentioned	mention	VERB
ejpam-5460	88	10	earlier	early	ADV
ejpam-5460	88	11	are	be	AUX
ejpam-5460	88	12	a	a	DET
ejpam-5460	88	13	particular	particular	ADJ
ejpam-5460	88	14	case	case	NOUN
ejpam-5460	88	15	of	of	ADP
ejpam-5460	88	16	these	these	DET
ejpam-5460	88	17	concepts	concept	NOUN
ejpam-5460	88	18	.	.	PUNCT
ejpam-5460	89	1	the	the	DET
ejpam-5460	89	2	authors	author	NOUN
ejpam-5460	89	3	also	also	ADV
ejpam-5460	89	4	studied	study	VERB
ejpam-5460	89	5	the	the	DET
ejpam-5460	89	6	properties	property	NOUN
ejpam-5460	89	7	of	of	ADP
ejpam-5460	89	8	(	(	PUNCT
ejpam-5460	89	9	α	α	NOUN
ejpam-5460	89	10	,	,	PUNCT
ejpam-5460	89	11	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	89	12	(	(	PUNCT
ejpam-5460	89	13	m	m	NOUN
ejpam-5460	89	14	,	,	PUNCT
ejpam-5460	89	15	n)(n	n)(n	NOUN
ejpam-5460	89	16	-	-	PUNCT
ejpam-5460	89	17	interior	interior	NOUN
ejpam-5460	89	18	)	)	PUNCT
ejpam-5460	89	19	ideals	ideal	NOUN
ejpam-5460	89	20	.	.	PUNCT
ejpam-5460	90	1	furthermore	furthermore	ADV
ejpam-5460	90	2	,	,	PUNCT
ejpam-5460	90	3	the	the	DET
ejpam-5460	90	4	concepts	concept	NOUN
ejpam-5460	90	5	of	of	ADP
ejpam-5460	90	6	(	(	PUNCT
ejpam-5460	90	7	α	α	NOUN
ejpam-5460	90	8	,	,	PUNCT
ejpam-5460	90	9	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	90	10	(	(	PUNCT
ejpam-5460	90	11	m	m	NOUN
ejpam-5460	90	12	,	,	PUNCT
ejpam-5460	90	13	n)-ideals	n)-ideal	NOUN
ejpam-5460	90	14	and	and	CCONJ
ejpam-5460	90	15	(	(	PUNCT
ejpam-5460	90	16	α	α	NOUN
ejpam-5460	90	17	,	,	PUNCT
ejpam-5460	90	18	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	90	19	n	n	CCONJ
ejpam-5460	90	20	-	-	PUNCT
ejpam-5460	90	21	interior	interior	ADJ
ejpam-5460	90	22	ideals	ideal	NOUN
ejpam-5460	90	23	were	be	AUX
ejpam-5460	90	24	characterized	characterize	VERB
ejpam-5460	90	25	by	by	ADP
ejpam-5460	90	26	the	the	DET
ejpam-5460	90	27	operation	operation	NOUN
ejpam-5460	90	28	◦	◦	NOUN
ejpam-5460	90	29	i	i	PRON
ejpam-5460	90	30	.	.	PUNCT
ejpam-5460	91	1	the	the	DET
ejpam-5460	91	2	above	above	ADJ
ejpam-5460	91	3	motivations	motivation	NOUN
ejpam-5460	91	4	provide	provide	VERB
ejpam-5460	91	5	a	a	DET
ejpam-5460	91	6	brief	brief	ADJ
ejpam-5460	91	7	overview	overview	NOUN
ejpam-5460	91	8	of	of	ADP
ejpam-5460	91	9	the	the	DET
ejpam-5460	91	10	investigation	investigation	NOUN
ejpam-5460	91	11	into	into	ADP
ejpam-5460	91	12	fuzzy	fuzzy	ADJ
ejpam-5460	91	13	ideals	ideal	NOUN
ejpam-5460	91	14	in	in	ADP
ejpam-5460	91	15	ordered	order	VERB
ejpam-5460	91	16	semigroups	semigroup	NOUN
ejpam-5460	91	17	.	.	PUNCT
ejpam-5460	92	1	moreover	moreover	ADV
ejpam-5460	92	2	,	,	PUNCT
ejpam-5460	92	3	the	the	DET
ejpam-5460	92	4	importance	importance	NOUN
ejpam-5460	92	5	of	of	ADP
ejpam-5460	92	6	separating	separate	VERB
ejpam-5460	92	7	ordered	order	VERB
ejpam-5460	92	8	semigroups	semigroup	NOUN
ejpam-5460	92	9	into	into	ADP
ejpam-5460	92	10	classes	class	NOUN
ejpam-5460	92	11	using	use	VERB
ejpam-5460	92	12	these	these	DET
ejpam-5460	92	13	various	various	ADJ
ejpam-5460	92	14	fuzzy	fuzzy	ADJ
ejpam-5460	92	15	ideals	ideal	NOUN
ejpam-5460	92	16	is	be	AUX
ejpam-5460	92	17	shown	show	VERB
ejpam-5460	92	18	.	.	PUNCT
ejpam-5460	93	1	in	in	ADP
ejpam-5460	93	2	this	this	DET
ejpam-5460	93	3	paper	paper	NOUN
ejpam-5460	93	4	,	,	PUNCT
ejpam-5460	93	5	we	we	PRON
ejpam-5460	93	6	build	build	VERB
ejpam-5460	93	7	upon	upon	SCONJ
ejpam-5460	93	8	the	the	DET
ejpam-5460	93	9	investigations	investigation	NOUN
ejpam-5460	93	10	in	in	ADP
ejpam-5460	93	11	[	[	X
ejpam-5460	93	12	4	4	NUM
ejpam-5460	93	13	,	,	PUNCT
ejpam-5460	93	14	35	35	NUM
ejpam-5460	93	15	]	]	PUNCT
ejpam-5460	93	16	.	.	PUNCT
ejpam-5460	94	1	we	we	PRON
ejpam-5460	94	2	apply	apply	VERB
ejpam-5460	94	3	the	the	DET
ejpam-5460	94	4	concepts	concept	NOUN
ejpam-5460	94	5	of	of	ADP
ejpam-5460	94	6	(	(	PUNCT
ejpam-5460	94	7	α	α	NOUN
ejpam-5460	94	8	,	,	PUNCT
ejpam-5460	94	9	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	94	10	(	(	PUNCT
ejpam-5460	94	11	m	m	NOUN
ejpam-5460	94	12	,	,	PUNCT
ejpam-5460	94	13	n)-ideals	n)-ideal	NOUN
ejpam-5460	94	14	and	and	CCONJ
ejpam-5460	94	15	(	(	PUNCT
ejpam-5460	94	16	α	α	NOUN
ejpam-5460	94	17	,	,	PUNCT
ejpam-5460	94	18	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	94	19	n	n	CCONJ
ejpam-5460	94	20	-	-	PUNCT
ejpam-5460	94	21	interior	interior	ADJ
ejpam-5460	94	22	ideals	ideal	NOUN
ejpam-5460	94	23	to	to	PART
ejpam-5460	94	24	characterize	characterize	VERB
ejpam-5460	94	25	all	all	DET
ejpam-5460	94	26	classes	class	NOUN
ejpam-5460	94	27	of	of	ADP
ejpam-5460	94	28	ordered	order	VERB
ejpam-5460	94	29	semigroups	semigroup	NOUN
ejpam-5460	94	30	based	base	VERB
ejpam-5460	94	31	on	on	ADP
ejpam-5460	94	32	their	their	PRON
ejpam-5460	94	33	regularities	regularity	NOUN
ejpam-5460	94	34	through	through	ADP
ejpam-5460	94	35	characteristic	characteristic	ADJ
ejpam-5460	94	36	functions	function	NOUN
ejpam-5460	94	37	.	.	PUNCT
ejpam-5460	95	1	the	the	DET
ejpam-5460	95	2	outline	outline	NOUN
ejpam-5460	95	3	of	of	ADP
ejpam-5460	95	4	the	the	DET
ejpam-5460	95	5	paper	paper	NOUN
ejpam-5460	95	6	is	be	AUX
ejpam-5460	95	7	as	as	SCONJ
ejpam-5460	95	8	follows	follow	VERB
ejpam-5460	95	9	.	.	PUNCT
ejpam-5460	96	1	section	section	NOUN
ejpam-5460	96	2	2	2	NUM
ejpam-5460	96	3	presents	present	VERB
ejpam-5460	96	4	the	the	DET
ejpam-5460	96	5	preliminary	preliminary	ADJ
ejpam-5460	96	6	concepts	concept	NOUN
ejpam-5460	96	7	used	use	VERB
ejpam-5460	96	8	in	in	ADP
ejpam-5460	96	9	this	this	DET
ejpam-5460	96	10	study	study	NOUN
ejpam-5460	96	11	.	.	PUNCT
ejpam-5460	97	1	we	we	PRON
ejpam-5460	97	2	revisit	revisit	VERB
ejpam-5460	97	3	the	the	DET
ejpam-5460	97	4	definitions	definition	NOUN
ejpam-5460	97	5	of	of	ADP
ejpam-5460	97	6	(	(	PUNCT
ejpam-5460	97	7	α	α	NOUN
ejpam-5460	97	8	,	,	PUNCT
ejpam-5460	97	9	β)s	β)s	PUNCT
ejpam-5460	97	10	.	.	PUNCT
ejpam-5460	98	1	lekkoksung	lekkoksung	PROPN
ejpam-5460	98	2	,	,	PUNCT
ejpam-5460	98	3	b.	b.	PROPN
ejpam-5460	98	4	davvaz	davvaz	PROPN
ejpam-5460	98	5	,	,	PUNCT
ejpam-5460	98	6	n.	n.	PROPN
ejpam-5460	98	7	lekkoksung	lekkoksung	PROPN
ejpam-5460	98	8	/	/	SYM
ejpam-5460	98	9	eur	eur	PROPN
ejpam-5460	98	10	.	.	PUNCT
ejpam-5460	99	1	j.	j.	PROPN
ejpam-5460	99	2	pure	pure	PROPN
ejpam-5460	99	3	appl	appl	PROPN
ejpam-5460	99	4	.	.	PROPN
ejpam-5460	99	5	math	math	PROPN
ejpam-5460	99	6	,	,	PUNCT
ejpam-5460	99	7	17	17	NUM
ejpam-5460	99	8	(	(	PUNCT
ejpam-5460	99	9	4	4	NUM
ejpam-5460	99	10	)	)	PUNCT
ejpam-5460	99	11	(	(	PUNCT
ejpam-5460	99	12	2024	2024	NUM
ejpam-5460	99	13	)	)	PUNCT
ejpam-5460	99	14	,	,	PUNCT
ejpam-5460	99	15	2962	2962	NUM
ejpam-5460	99	16	-	-	SYM
ejpam-5460	99	17	2984	2984	NUM
ejpam-5460	99	18	2965	2965	NUM
ejpam-5460	99	19	fuzzy	fuzzy	ADJ
ejpam-5460	99	20	(	(	PUNCT
ejpam-5460	99	21	m	m	NOUN
ejpam-5460	99	22	,	,	PUNCT
ejpam-5460	99	23	n)-ideals	n)-ideal	NOUN
ejpam-5460	99	24	and	and	CCONJ
ejpam-5460	99	25	(	(	PUNCT
ejpam-5460	99	26	α	α	NOUN
ejpam-5460	99	27	,	,	PUNCT
ejpam-5460	99	28	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	99	29	n	n	CCONJ
ejpam-5460	99	30	-	-	PUNCT
ejpam-5460	99	31	interior	interior	ADJ
ejpam-5460	99	32	ideals	ideal	NOUN
ejpam-5460	99	33	,	,	PUNCT
ejpam-5460	99	34	along	along	ADP
ejpam-5460	99	35	with	with	ADP
ejpam-5460	99	36	their	their	PRON
ejpam-5460	99	37	significant	significant	ADJ
ejpam-5460	99	38	results	result	NOUN
ejpam-5460	99	39	in	in	ADP
ejpam-5460	99	40	section	section	NOUN
ejpam-5460	99	41	3	3	NUM
ejpam-5460	99	42	.	.	PUNCT
ejpam-5460	100	1	in	in	ADP
ejpam-5460	100	2	section	section	NOUN
ejpam-5460	100	3	4	4	NUM
ejpam-5460	100	4	,	,	PUNCT
ejpam-5460	100	5	we	we	PRON
ejpam-5460	100	6	apply	apply	VERB
ejpam-5460	100	7	these	these	DET
ejpam-5460	100	8	notions	notion	NOUN
ejpam-5460	100	9	of	of	ADP
ejpam-5460	100	10	fuzzy	fuzzy	ADJ
ejpam-5460	100	11	ideals	ideal	NOUN
ejpam-5460	100	12	to	to	PART
ejpam-5460	100	13	classify	classify	VERB
ejpam-5460	100	14	ordered	order	VERB
ejpam-5460	100	15	semigroups	semigroup	NOUN
ejpam-5460	100	16	.	.	PUNCT
ejpam-5460	101	1	2	2	X
ejpam-5460	101	2	.	.	X
ejpam-5460	101	3	preliminaries	preliminary	NOUN
ejpam-5460	101	4	this	this	DET
ejpam-5460	101	5	section	section	NOUN
ejpam-5460	101	6	recalls	recall	VERB
ejpam-5460	101	7	the	the	DET
ejpam-5460	101	8	concept	concept	NOUN
ejpam-5460	101	9	of	of	ADP
ejpam-5460	101	10	ordered	order	VERB
ejpam-5460	101	11	semigroups	semigroup	NOUN
ejpam-5460	101	12	and	and	CCONJ
ejpam-5460	101	13	fuzzy	fuzzy	ADJ
ejpam-5460	101	14	sets	set	NOUN
ejpam-5460	101	15	.	.	PUNCT
ejpam-5460	102	1	additionally	additionally	ADV
ejpam-5460	102	2	,	,	PUNCT
ejpam-5460	102	3	their	their	PRON
ejpam-5460	102	4	preliminary	preliminary	ADJ
ejpam-5460	102	5	results	result	NOUN
ejpam-5460	102	6	are	be	AUX
ejpam-5460	102	7	provided	provide	VERB
ejpam-5460	102	8	.	.	PUNCT
ejpam-5460	103	1	definition	definition	NOUN
ejpam-5460	103	2	1	1	NUM
ejpam-5460	103	3	.	.	PUNCT
ejpam-5460	104	1	an	an	DET
ejpam-5460	104	2	ordered	order	VERB
ejpam-5460	104	3	semigroup	semigroup	NOUN
ejpam-5460	104	4	⟨s	⟨s	PROPN
ejpam-5460	104	5	;	;	PUNCT
ejpam-5460	104	6	·	·	PUNCT
ejpam-5460	104	7	,	,	PUNCT
ejpam-5460	104	8	≤⟩	≤⟩	X
ejpam-5460	104	9	is	be	AUX
ejpam-5460	104	10	an	an	DET
ejpam-5460	104	11	algebraic	algebraic	ADJ
ejpam-5460	104	12	system	system	NOUN
ejpam-5460	104	13	consisting	consist	VERB
ejpam-5460	104	14	of	of	ADP
ejpam-5460	104	15	a	a	DET
ejpam-5460	104	16	nonempty	nonempty	ADJ
ejpam-5460	104	17	set	set	VERB
ejpam-5460	104	18	s	s	PROPN
ejpam-5460	104	19	,	,	PUNCT
ejpam-5460	104	20	an	an	DET
ejpam-5460	104	21	associative	associative	ADJ
ejpam-5460	104	22	binary	binary	NOUN
ejpam-5460	104	23	operation	operation	NOUN
ejpam-5460	104	24	·	·	PUNCT
ejpam-5460	104	25	on	on	ADP
ejpam-5460	104	26	s	s	PRON
ejpam-5460	104	27	and	and	CCONJ
ejpam-5460	104	28	a	a	DET
ejpam-5460	104	29	partial	partial	ADJ
ejpam-5460	104	30	order	order	NOUN
ejpam-5460	104	31	relation	relation	NOUN
ejpam-5460	104	32	on	on	ADP
ejpam-5460	104	33	s	s	AUX
ejpam-5460	104	34	satisfying	satisfy	VERB
ejpam-5460	104	35	the	the	DET
ejpam-5460	104	36	compatibility	compatibility	NOUN
ejpam-5460	104	37	.	.	PUNCT
ejpam-5460	105	1	that	that	PRON
ejpam-5460	105	2	is	be	AUX
ejpam-5460	105	3	,	,	PUNCT
ejpam-5460	105	4	for	for	ADP
ejpam-5460	105	5	any	any	DET
ejpam-5460	105	6	a	a	NOUN
ejpam-5460	105	7	,	,	PUNCT
ejpam-5460	105	8	b	b	X
ejpam-5460	105	9	∈	∈	PROPN
ejpam-5460	105	10	s	s	VERB
ejpam-5460	105	11	a	a	DET
ejpam-5460	105	12	≤	≤	NUM
ejpam-5460	105	13	b	b	NOUN
ejpam-5460	105	14	implies	imply	VERB
ejpam-5460	105	15	a	a	DET
ejpam-5460	105	16	·	·	PUNCT
ejpam-5460	105	17	c	c	NOUN
ejpam-5460	105	18	≤	≤	PROPN
ejpam-5460	105	19	b	b	X
ejpam-5460	105	20	·	·	PUNCT
ejpam-5460	105	21	c	c	NOUN
ejpam-5460	105	22	and	and	CCONJ
ejpam-5460	105	23	c	c	PROPN
ejpam-5460	105	24	·	·	PUNCT
ejpam-5460	105	25	a	a	DET
ejpam-5460	105	26	≤	≤	X
ejpam-5460	105	27	c	c	X
ejpam-5460	105	28	·	·	PUNCT
ejpam-5460	105	29	b	b	X
ejpam-5460	105	30	for	for	ADP
ejpam-5460	105	31	all	all	PRON
ejpam-5460	106	1	c	c	PROPN
ejpam-5460	106	2	∈	∈	PROPN
ejpam-5460	106	3	s.	s.	PROPN
ejpam-5460	106	4	we	we	PRON
ejpam-5460	106	5	observe	observe	VERB
ejpam-5460	106	6	that	that	SCONJ
ejpam-5460	106	7	any	any	DET
ejpam-5460	106	8	semigroup	semigroup	NOUN
ejpam-5460	106	9	⟨s	⟨s	NOUN
ejpam-5460	106	10	;	;	PUNCT
ejpam-5460	106	11	·	·	PUNCT
ejpam-5460	106	12	⟩	⟩	NOUN
ejpam-5460	106	13	can	can	AUX
ejpam-5460	106	14	be	be	AUX
ejpam-5460	106	15	regarded	regard	VERB
ejpam-5460	106	16	as	as	ADP
ejpam-5460	106	17	an	an	DET
ejpam-5460	106	18	ordered	order	VERB
ejpam-5460	106	19	semigroup	semigroup	NOUN
ejpam-5460	106	20	that	that	PRON
ejpam-5460	106	21	defines	define	VERB
ejpam-5460	106	22	the	the	DET
ejpam-5460	106	23	equality	equality	NOUN
ejpam-5460	106	24	relation	relation	NOUN
ejpam-5460	106	25	as	as	ADP
ejpam-5460	106	26	the	the	DET
ejpam-5460	106	27	partial	partial	ADJ
ejpam-5460	106	28	order	order	NOUN
ejpam-5460	106	29	compatible	compatible	ADJ
ejpam-5460	106	30	with	with	ADP
ejpam-5460	106	31	the	the	DET
ejpam-5460	106	32	operation	operation	NOUN
ejpam-5460	106	33	of	of	ADP
ejpam-5460	106	34	such	such	DET
ejpam-5460	106	35	a	a	DET
ejpam-5460	106	36	semigroup	semigroup	NOUN
ejpam-5460	106	37	.	.	PUNCT
ejpam-5460	107	1	the	the	DET
ejpam-5460	107	2	product	product	NOUN
ejpam-5460	107	3	x	x	PUNCT
ejpam-5460	107	4	·	·	PUNCT
ejpam-5460	107	5	y	y	NOUN
ejpam-5460	107	6	of	of	ADP
ejpam-5460	107	7	any	any	DET
ejpam-5460	107	8	elements	element	NOUN
ejpam-5460	107	9	x	x	PUNCT
ejpam-5460	107	10	and	and	CCONJ
ejpam-5460	107	11	y	y	PROPN
ejpam-5460	107	12	of	of	ADP
ejpam-5460	107	13	s	s	PROPN
ejpam-5460	107	14	is	be	AUX
ejpam-5460	107	15	usually	usually	ADV
ejpam-5460	107	16	denoted	denote	VERB
ejpam-5460	107	17	by	by	ADP
ejpam-5460	107	18	xy	xy	PROPN
ejpam-5460	107	19	,	,	PUNCT
ejpam-5460	107	20	and	and	CCONJ
ejpam-5460	107	21	the	the	DET
ejpam-5460	107	22	n	n	NOUN
ejpam-5460	107	23	-	-	PUNCT
ejpam-5460	107	24	product	product	NOUN
ejpam-5460	107	25	a	a	PRON
ejpam-5460	107	26	·	·	PUNCT
ejpam-5460	107	27	·	·	PUNCT
ejpam-5460	107	28	·	·	PUNCT
ejpam-5460	107	29	a︸	a︸	ADV
ejpam-5460	107	30	︷︷	︷︷	PROPN
ejpam-5460	107	31	︸	︸	SYM
ejpam-5460	107	32	n	n	CCONJ
ejpam-5460	107	33	-	-	PUNCT
ejpam-5460	107	34	time	time	NOUN
ejpam-5460	107	35	of	of	ADP
ejpam-5460	107	36	any	any	DET
ejpam-5460	107	37	element	element	NOUN
ejpam-5460	107	38	a	a	PRON
ejpam-5460	107	39	of	of	ADP
ejpam-5460	107	40	s	s	PRON
ejpam-5460	107	41	is	be	AUX
ejpam-5460	107	42	denoted	denote	VERB
ejpam-5460	107	43	by	by	ADP
ejpam-5460	107	44	an	an	PRON
ejpam-5460	107	45	,	,	PUNCT
ejpam-5460	107	46	where	where	SCONJ
ejpam-5460	107	47	n	n	PRON
ejpam-5460	107	48	is	be	AUX
ejpam-5460	107	49	a	a	DET
ejpam-5460	107	50	positive	positive	ADJ
ejpam-5460	107	51	integer	integer	NOUN
ejpam-5460	107	52	.	.	PUNCT
ejpam-5460	108	1	for	for	ADP
ejpam-5460	108	2	convenience	convenience	NOUN
ejpam-5460	108	3	,	,	PUNCT
ejpam-5460	108	4	we	we	PRON
ejpam-5460	108	5	denote	denote	VERB
ejpam-5460	108	6	an	an	DET
ejpam-5460	108	7	ordered	order	VERB
ejpam-5460	108	8	semigroup	semigroup	NOUN
ejpam-5460	108	9	⟨s	⟨s	PROPN
ejpam-5460	108	10	;	;	PUNCT
ejpam-5460	108	11	·	·	PUNCT
ejpam-5460	108	12	,	,	PUNCT
ejpam-5460	108	13	≤⟩	≤⟩	VERB
ejpam-5460	108	14	by	by	ADP
ejpam-5460	108	15	s	s	PRON
ejpam-5460	108	16	the	the	DET
ejpam-5460	108	17	boldface	boldface	NOUN
ejpam-5460	108	18	letter	letter	NOUN
ejpam-5460	108	19	of	of	ADP
ejpam-5460	108	20	its	its	PRON
ejpam-5460	108	21	underlying	underlie	VERB
ejpam-5460	108	22	set	set	NOUN
ejpam-5460	108	23	.	.	PUNCT
ejpam-5460	109	1	let	let	VERB
ejpam-5460	109	2	s	s	PRON
ejpam-5460	109	3	be	be	AUX
ejpam-5460	109	4	an	an	DET
ejpam-5460	109	5	ordered	order	VERB
ejpam-5460	109	6	semigroup	semigroup	NOUN
ejpam-5460	109	7	,	,	PUNCT
ejpam-5460	109	8	and	and	CCONJ
ejpam-5460	109	9	a	a	DET
ejpam-5460	109	10	,	,	PUNCT
ejpam-5460	109	11	b	b	NOUN
ejpam-5460	109	12	subsets	subset	NOUN
ejpam-5460	109	13	of	of	ADP
ejpam-5460	109	14	s.	s.	PROPN
ejpam-5460	109	15	we	we	PRON
ejpam-5460	109	16	define	define	VERB
ejpam-5460	109	17	the	the	DET
ejpam-5460	109	18	sets	set	NOUN
ejpam-5460	109	19	ab	ab	PROPN
ejpam-5460	109	20	and	and	CCONJ
ejpam-5460	109	21	(	(	PUNCT
ejpam-5460	109	22	a	a	X
ejpam-5460	109	23	]	]	X
ejpam-5460	109	24	by	by	ADP
ejpam-5460	109	25	ab	ab	PROPN
ejpam-5460	109	26	:	:	PUNCT
ejpam-5460	109	27	=	=	SYM
ejpam-5460	109	28	{	{	PUNCT
ejpam-5460	109	29	ab	ab	NOUN
ejpam-5460	109	30	:	:	PUNCT
ejpam-5460	109	31	a	a	DET
ejpam-5460	109	32	∈	∈	PROPN
ejpam-5460	109	33	a	a	PRON
ejpam-5460	109	34	and	and	CCONJ
ejpam-5460	109	35	b	b	NOUN
ejpam-5460	109	36	∈	∈	PROPN
ejpam-5460	109	37	b	b	NOUN
ejpam-5460	109	38	}	}	PUNCT
ejpam-5460	109	39	and	and	CCONJ
ejpam-5460	109	40	(	(	PUNCT
ejpam-5460	109	41	a	a	X
ejpam-5460	109	42	]	]	X
ejpam-5460	109	43	:	:	PUNCT
ejpam-5460	109	44	=	=	SYM
ejpam-5460	109	45	{	{	PUNCT
ejpam-5460	109	46	x	x	PUNCT
ejpam-5460	109	47	∈	∈	NOUN
ejpam-5460	109	48	s	s	PART
ejpam-5460	109	49	:	:	PUNCT
ejpam-5460	109	50	x	x	SYM
ejpam-5460	109	51	≤	≤	ADV
ejpam-5460	109	52	a	a	PRON
ejpam-5460	109	53	for	for	ADP
ejpam-5460	109	54	some	some	DET
ejpam-5460	109	55	a	a	DET
ejpam-5460	109	56	∈	∈	PROPN
ejpam-5460	109	57	a	a	PRON
ejpam-5460	109	58	}	}	PUNCT
ejpam-5460	109	59	.	.	PUNCT
ejpam-5460	110	1	the	the	DET
ejpam-5460	110	2	importance	importance	NOUN
ejpam-5460	110	3	of	of	ADP
ejpam-5460	110	4	the	the	DET
ejpam-5460	110	5	operator	operator	NOUN
ejpam-5460	110	6	(	(	PUNCT
ejpam-5460	110	7	·	·	PUNCT
ejpam-5460	110	8	]	]	X
ejpam-5460	110	9	is	be	AUX
ejpam-5460	110	10	shown	show	VERB
ejpam-5460	110	11	as	as	SCONJ
ejpam-5460	110	12	follows	follow	VERB
ejpam-5460	110	13	.	.	PUNCT
ejpam-5460	111	1	lemma	lemma	PROPN
ejpam-5460	111	2	1	1	NUM
ejpam-5460	111	3	(	(	PUNCT
ejpam-5460	111	4	[	[	X
ejpam-5460	111	5	13	13	NUM
ejpam-5460	111	6	]	]	NUM
ejpam-5460	111	7	)	)	PUNCT
ejpam-5460	111	8	.	.	PUNCT
ejpam-5460	112	1	let	let	VERB
ejpam-5460	112	2	s	s	PRON
ejpam-5460	112	3	be	be	AUX
ejpam-5460	112	4	an	an	DET
ejpam-5460	112	5	ordered	order	VERB
ejpam-5460	112	6	semigroup	semigroup	NOUN
ejpam-5460	112	7	,	,	PUNCT
ejpam-5460	112	8	a	a	DET
ejpam-5460	112	9	,	,	PUNCT
ejpam-5460	112	10	b	b	NOUN
ejpam-5460	112	11	,	,	PUNCT
ejpam-5460	112	12	ai	ai	VERB
ejpam-5460	112	13	subsets	subset	NOUN
ejpam-5460	112	14	of	of	ADP
ejpam-5460	112	15	s	s	NOUN
ejpam-5460	112	16	,	,	PUNCT
ejpam-5460	112	17	where	where	SCONJ
ejpam-5460	112	18	i	i	PRON
ejpam-5460	112	19	∈	∈	PROPN
ejpam-5460	112	20	i.	i.	NOUN
ejpam-5460	112	21	then	then	ADV
ejpam-5460	112	22	,	,	PUNCT
ejpam-5460	112	23	we	we	PRON
ejpam-5460	112	24	obtain	obtain	VERB
ejpam-5460	112	25	the	the	DET
ejpam-5460	112	26	following	following	ADJ
ejpam-5460	112	27	statements	statement	NOUN
ejpam-5460	112	28	.	.	PUNCT
ejpam-5460	113	1	(	(	PUNCT
ejpam-5460	113	2	i	i	NOUN
ejpam-5460	113	3	)	)	PUNCT
ejpam-5460	113	4	a	a	DET
ejpam-5460	113	5	⊆	⊆	NUM
ejpam-5460	113	6	(	(	PUNCT
ejpam-5460	113	7	a	a	PRON
ejpam-5460	113	8	]	]	X
ejpam-5460	113	9	.	.	PUNCT
ejpam-5460	114	1	(	(	PUNCT
ejpam-5460	114	2	ii	ii	NOUN
ejpam-5460	114	3	)	)	PUNCT
ejpam-5460	114	4	a	a	DET
ejpam-5460	114	5	⊆	⊆	NUM
ejpam-5460	114	6	b	b	NOUN
ejpam-5460	114	7	implies	imply	VERB
ejpam-5460	114	8	(	(	PUNCT
ejpam-5460	114	9	a	a	X
ejpam-5460	114	10	]	]	X
ejpam-5460	114	11	⊆	⊆	NUM
ejpam-5460	114	12	(	(	PUNCT
ejpam-5460	114	13	b	b	NOUN
ejpam-5460	114	14	]	]	X
ejpam-5460	114	15	.	.	PUNCT
ejpam-5460	115	1	(	(	PUNCT
ejpam-5460	115	2	iii	iii	X
ejpam-5460	115	3	)	)	PUNCT
ejpam-5460	115	4	(	(	PUNCT
ejpam-5460	115	5	a](b	a](b	NOUN
ejpam-5460	115	6	]	]	PUNCT
ejpam-5460	115	7	⊆	⊆	NUM
ejpam-5460	115	8	(	(	PUNCT
ejpam-5460	115	9	ab	ab	X
ejpam-5460	115	10	]	]	X
ejpam-5460	115	11	.	.	PUNCT
ejpam-5460	116	1	(	(	PUNCT
ejpam-5460	116	2	iv	iv	X
ejpam-5460	116	3	)	)	PUNCT
ejpam-5460	116	4	(	(	PUNCT
ejpam-5460	116	5	⋃	⋃	PROPN
ejpam-5460	116	6	i∈i	i∈i	NOUN
ejpam-5460	116	7	ai	ai	VERB
ejpam-5460	116	8	]	]	PUNCT
ejpam-5460	116	9	=	=	SYM
ejpam-5460	116	10	⋃	⋃	NOUN
ejpam-5460	116	11	i∈i(ai	i∈i(ai	X
ejpam-5460	116	12	]	]	PUNCT
ejpam-5460	116	13	.	.	PUNCT
ejpam-5460	117	1	(	(	PUNCT
ejpam-5460	117	2	v	v	NOUN
ejpam-5460	117	3	)	)	PUNCT
ejpam-5460	117	4	(	(	PUNCT
ejpam-5460	117	5	⋂	⋂	PROPN
ejpam-5460	117	6	i∈i	i∈i	ADV
ejpam-5460	117	7	ai	ai	VERB
ejpam-5460	117	8	]	]	PUNCT
ejpam-5460	117	9	⊆	⊆	NUM
ejpam-5460	117	10	⋂	⋂	PROPN
ejpam-5460	117	11	i∈i(ai	i∈i(ai	PROPN
ejpam-5460	117	12	]	]	X
ejpam-5460	117	13	.	.	PUNCT
ejpam-5460	118	1	(	(	PUNCT
ejpam-5460	118	2	vi	vi	NOUN
ejpam-5460	118	3	)	)	PUNCT
ejpam-5460	118	4	(	(	PUNCT
ejpam-5460	118	5	(	(	PUNCT
ejpam-5460	118	6	a	a	X
ejpam-5460	118	7	]	]	X
ejpam-5460	118	8	]	]	X
ejpam-5460	118	9	=	=	X
ejpam-5460	118	10	(	(	PUNCT
ejpam-5460	118	11	a	a	PRON
ejpam-5460	118	12	]	]	X
ejpam-5460	118	13	.	.	PUNCT
ejpam-5460	119	1	let	let	VERB
ejpam-5460	119	2	s	s	PRON
ejpam-5460	119	3	be	be	AUX
ejpam-5460	119	4	an	an	DET
ejpam-5460	119	5	ordered	order	VERB
ejpam-5460	119	6	semigroup	semigroup	NOUN
ejpam-5460	119	7	,	,	PUNCT
ejpam-5460	119	8	a	a	DET
ejpam-5460	119	9	a	a	DET
ejpam-5460	119	10	nonempty	nonempty	NOUN
ejpam-5460	119	11	subset	subset	NOUN
ejpam-5460	119	12	of	of	ADP
ejpam-5460	119	13	s	s	PRON
ejpam-5460	119	14	such	such	ADJ
ejpam-5460	119	15	that	that	SCONJ
ejpam-5460	119	16	(	(	PUNCT
ejpam-5460	119	17	a	a	X
ejpam-5460	119	18	]	]	X
ejpam-5460	119	19	⊆	⊆	NUM
ejpam-5460	119	20	a	a	DET
ejpam-5460	119	21	,	,	PUNCT
ejpam-5460	119	22	and	and	CCONJ
ejpam-5460	119	23	m	m	PROPN
ejpam-5460	119	24	,	,	PUNCT
ejpam-5460	119	25	n	n	CCONJ
ejpam-5460	119	26	nonnegative	nonnegative	ADJ
ejpam-5460	119	27	integers	integer	NOUN
ejpam-5460	119	28	not	not	PART
ejpam-5460	119	29	all	all	DET
ejpam-5460	119	30	zero	zero	NUM
ejpam-5460	119	31	.	.	PUNCT
ejpam-5460	120	1	then	then	ADV
ejpam-5460	120	2	,	,	PUNCT
ejpam-5460	120	3	a	a	PRON
ejpam-5460	120	4	is	be	AUX
ejpam-5460	120	5	said	say	VERB
ejpam-5460	120	6	to	to	PART
ejpam-5460	120	7	be	be	AUX
ejpam-5460	120	8	:	:	PUNCT
ejpam-5460	120	9	(	(	PUNCT
ejpam-5460	120	10	i	i	NOUN
ejpam-5460	120	11	)	)	PUNCT
ejpam-5460	120	12	a	a	DET
ejpam-5460	120	13	subsemigroup	subsemigroup	NOUN
ejpam-5460	120	14	of	of	ADP
ejpam-5460	120	15	s	s	PRON
ejpam-5460	120	16	if	if	SCONJ
ejpam-5460	120	17	aa	aa	PROPN
ejpam-5460	120	18	⊆	⊆	NUM
ejpam-5460	120	19	a	a	PRON
ejpam-5460	120	20	;	;	PUNCT
ejpam-5460	120	21	(	(	PUNCT
ejpam-5460	120	22	ii	ii	NOUN
ejpam-5460	120	23	)	)	PUNCT
ejpam-5460	120	24	an	an	DET
ejpam-5460	120	25	(	(	PUNCT
ejpam-5460	120	26	m	m	PROPN
ejpam-5460	120	27	,	,	PUNCT
ejpam-5460	120	28	n)-ideal	n)-ideal	NOUN
ejpam-5460	120	29	[	[	X
ejpam-5460	120	30	44	44	NUM
ejpam-5460	120	31	]	]	PUNCT
ejpam-5460	120	32	of	of	ADP
ejpam-5460	120	33	s	s	PRON
ejpam-5460	120	34	if	if	SCONJ
ejpam-5460	120	35	a	a	PRON
ejpam-5460	120	36	is	be	AUX
ejpam-5460	120	37	a	a	DET
ejpam-5460	120	38	subsemigroup	subsemigroup	NOUN
ejpam-5460	120	39	of	of	ADP
ejpam-5460	120	40	s	s	PRON
ejpam-5460	120	41	and	and	CCONJ
ejpam-5460	120	42	amsan	amsan	ADJ
ejpam-5460	120	43	⊆	⊆	NUM
ejpam-5460	120	44	a	a	PRON
ejpam-5460	120	45	;	;	PUNCT
ejpam-5460	120	46	(	(	PUNCT
ejpam-5460	120	47	iii	iii	X
ejpam-5460	120	48	)	)	PUNCT
ejpam-5460	120	49	an	an	DET
ejpam-5460	120	50	n	n	CCONJ
ejpam-5460	120	51	-	-	PUNCT
ejpam-5460	120	52	interior	interior	ADJ
ejpam-5460	120	53	ideal	ideal	NOUN
ejpam-5460	121	1	[	[	X
ejpam-5460	121	2	49	49	NUM
ejpam-5460	121	3	]	]	PUNCT
ejpam-5460	121	4	of	of	ADP
ejpam-5460	121	5	s	s	PRON
ejpam-5460	121	6	if	if	SCONJ
ejpam-5460	121	7	a	a	PRON
ejpam-5460	121	8	is	be	AUX
ejpam-5460	121	9	a	a	DET
ejpam-5460	121	10	subsemigroup	subsemigroup	NOUN
ejpam-5460	121	11	of	of	ADP
ejpam-5460	121	12	s	s	NOUN
ejpam-5460	121	13	and	and	CCONJ
ejpam-5460	121	14	sans	san	NOUN
ejpam-5460	121	15	⊆	⊆	NUM
ejpam-5460	121	16	a.	a.	NOUN
ejpam-5460	121	17	we	we	PRON
ejpam-5460	121	18	note	note	VERB
ejpam-5460	121	19	that	that	SCONJ
ejpam-5460	121	20	a0s	a0s	PROPN
ejpam-5460	121	21	=	=	PUNCT
ejpam-5460	121	22	s	s	PART
ejpam-5460	121	23	=	=	NOUN
ejpam-5460	121	24	sa0	sa0	NOUN
ejpam-5460	121	25	.	.	PUNCT
ejpam-5460	122	1	the	the	DET
ejpam-5460	122	2	above	above	ADJ
ejpam-5460	122	3	definition	definition	NOUN
ejpam-5460	122	4	shows	show	VERB
ejpam-5460	122	5	that	that	SCONJ
ejpam-5460	122	6	any	any	PRON
ejpam-5460	122	7	left	left	ADJ
ejpam-5460	122	8	,	,	PUNCT
ejpam-5460	122	9	right	right	INTJ
ejpam-5460	122	10	,	,	PUNCT
ejpam-5460	122	11	bi-	bi-	NUM
ejpam-5460	122	12	,	,	PUNCT
ejpam-5460	122	13	and	and	CCONJ
ejpam-5460	122	14	interior	interior	ADJ
ejpam-5460	122	15	ideal	ideal	NOUN
ejpam-5460	122	16	can	can	AUX
ejpam-5460	122	17	be	be	AUX
ejpam-5460	122	18	considered	consider	VERB
ejpam-5460	122	19	as	as	ADP
ejpam-5460	122	20	a	a	DET
ejpam-5460	122	21	(	(	PUNCT
ejpam-5460	122	22	0	0	NUM
ejpam-5460	122	23	,	,	PUNCT
ejpam-5460	122	24	1)-	1)-	NUM
ejpam-5460	122	25	,	,	PUNCT
ejpam-5460	122	26	(	(	PUNCT
ejpam-5460	122	27	1	1	NUM
ejpam-5460	122	28	,	,	PUNCT
ejpam-5460	122	29	0)-	0)-	NUM
ejpam-5460	122	30	,	,	PUNCT
ejpam-5460	122	31	(	(	PUNCT
ejpam-5460	122	32	1	1	NUM
ejpam-5460	122	33	,	,	PUNCT
ejpam-5460	122	34	1)-	1)-	NUM
ejpam-5460	122	35	,	,	PUNCT
ejpam-5460	122	36	and	and	CCONJ
ejpam-5460	123	1	1	1	NUM
ejpam-5460	123	2	-	-	ADJ
ejpam-5460	123	3	interior	interior	ADJ
ejpam-5460	123	4	ideal	ideal	NOUN
ejpam-5460	123	5	,	,	PUNCT
ejpam-5460	123	6	respectively	respectively	ADV
ejpam-5460	123	7	.	.	PUNCT
ejpam-5460	124	1	s.	s.	PROPN
ejpam-5460	124	2	lekkoksung	lekkoksung	PROPN
ejpam-5460	124	3	,	,	PUNCT
ejpam-5460	124	4	b.	b.	PROPN
ejpam-5460	124	5	davvaz	davvaz	PROPN
ejpam-5460	124	6	,	,	PUNCT
ejpam-5460	124	7	n.	n.	PROPN
ejpam-5460	124	8	lekkoksung	lekkoksung	PROPN
ejpam-5460	124	9	/	/	SYM
ejpam-5460	124	10	eur	eur	PROPN
ejpam-5460	124	11	.	.	PUNCT
ejpam-5460	125	1	j.	j.	PROPN
ejpam-5460	125	2	pure	pure	PROPN
ejpam-5460	125	3	appl	appl	PROPN
ejpam-5460	125	4	.	.	PROPN
ejpam-5460	125	5	math	math	PROPN
ejpam-5460	125	6	,	,	PUNCT
ejpam-5460	125	7	17	17	NUM
ejpam-5460	125	8	(	(	PUNCT
ejpam-5460	125	9	4	4	NUM
ejpam-5460	125	10	)	)	PUNCT
ejpam-5460	125	11	(	(	PUNCT
ejpam-5460	125	12	2024	2024	NUM
ejpam-5460	125	13	)	)	PUNCT
ejpam-5460	125	14	,	,	PUNCT
ejpam-5460	125	15	2962	2962	NUM
ejpam-5460	125	16	-	-	SYM
ejpam-5460	125	17	2984	2984	NUM
ejpam-5460	125	18	2966	2966	NUM
ejpam-5460	125	19	we	we	PRON
ejpam-5460	125	20	note	note	VERB
ejpam-5460	125	21	that	that	SCONJ
ejpam-5460	125	22	for	for	ADP
ejpam-5460	125	23	any	any	DET
ejpam-5460	125	24	positive	positive	ADJ
ejpam-5460	125	25	integers	integer	NOUN
ejpam-5460	125	26	m	m	VERB
ejpam-5460	125	27	and	and	CCONJ
ejpam-5460	125	28	n	n	PROPN
ejpam-5460	125	29	with	with	ADP
ejpam-5460	125	30	m	m	PROPN
ejpam-5460	125	31	≤	≤	NOUN
ejpam-5460	125	32	n	n	CCONJ
ejpam-5460	125	33	,	,	PUNCT
ejpam-5460	125	34	we	we	PRON
ejpam-5460	125	35	obtain	obtain	VERB
ejpam-5460	125	36	that	that	SCONJ
ejpam-5460	125	37	any	any	DET
ejpam-5460	125	38	m	m	ADJ
ejpam-5460	125	39	-	-	ADJ
ejpam-5460	125	40	interior	interior	ADJ
ejpam-5460	125	41	ideal	ideal	NOUN
ejpam-5460	125	42	is	be	AUX
ejpam-5460	125	43	an	an	DET
ejpam-5460	125	44	n	n	CCONJ
ejpam-5460	125	45	-	-	PUNCT
ejpam-5460	125	46	interior	interior	ADJ
ejpam-5460	125	47	ideal	ideal	NOUN
ejpam-5460	125	48	.	.	PUNCT
ejpam-5460	126	1	the	the	DET
ejpam-5460	126	2	converse	converse	NOUN
ejpam-5460	126	3	of	of	ADP
ejpam-5460	126	4	this	this	DET
ejpam-5460	126	5	statement	statement	NOUN
ejpam-5460	126	6	is	be	AUX
ejpam-5460	126	7	not	not	PART
ejpam-5460	126	8	true	true	ADJ
ejpam-5460	126	9	to	to	ADP
ejpam-5460	126	10	the	the	DET
ejpam-5460	126	11	evidence	evidence	NOUN
ejpam-5460	126	12	provided	provide	VERB
ejpam-5460	126	13	in	in	ADP
ejpam-5460	126	14	[	[	X
ejpam-5460	126	15	49	49	NUM
ejpam-5460	126	16	,	,	PUNCT
ejpam-5460	126	17	example	example	NOUN
ejpam-5460	126	18	3.2	3.2	NUM
ejpam-5460	126	19	]	]	PUNCT
ejpam-5460	126	20	.	.	PUNCT
ejpam-5460	127	1	similarly	similarly	ADV
ejpam-5460	127	2	,	,	PUNCT
ejpam-5460	127	3	for	for	ADP
ejpam-5460	127	4	any	any	DET
ejpam-5460	127	5	positive	positive	ADJ
ejpam-5460	127	6	integers	integer	NOUN
ejpam-5460	127	7	m1,m2	m1,m2	PROPN
ejpam-5460	127	8	,	,	PUNCT
ejpam-5460	127	9	n1	n1	NOUN
ejpam-5460	127	10	,	,	PUNCT
ejpam-5460	127	11	n2	n2	NOUN
ejpam-5460	127	12	with	with	ADP
ejpam-5460	127	13	m1	m1	PROPN
ejpam-5460	127	14	≤	≤	NUM
ejpam-5460	127	15	m2	m2	PROPN
ejpam-5460	127	16	and	and	CCONJ
ejpam-5460	127	17	n1	n1	PROPN
ejpam-5460	127	18	≤	≤	NUM
ejpam-5460	127	19	n2	n2	NOUN
ejpam-5460	127	20	,	,	PUNCT
ejpam-5460	127	21	we	we	PRON
ejpam-5460	127	22	obtain	obtain	VERB
ejpam-5460	127	23	that	that	SCONJ
ejpam-5460	127	24	any	any	DET
ejpam-5460	127	25	(	(	PUNCT
ejpam-5460	127	26	m1	m1	NOUN
ejpam-5460	127	27	,	,	PUNCT
ejpam-5460	127	28	n1)-ideal	n1)-ideal	PROPN
ejpam-5460	127	29	is	be	AUX
ejpam-5460	127	30	an	an	DET
ejpam-5460	127	31	(	(	PUNCT
ejpam-5460	127	32	m2	m2	PROPN
ejpam-5460	127	33	,	,	PUNCT
ejpam-5460	127	34	n2)-ideal	n2)-ideal	NOUN
ejpam-5460	127	35	.	.	PUNCT
ejpam-5460	128	1	the	the	DET
ejpam-5460	128	2	converse	converse	NOUN
ejpam-5460	128	3	of	of	ADP
ejpam-5460	128	4	this	this	DET
ejpam-5460	128	5	statement	statement	NOUN
ejpam-5460	128	6	does	do	AUX
ejpam-5460	128	7	not	not	PART
ejpam-5460	128	8	generally	generally	ADV
ejpam-5460	128	9	hold	hold	VERB
ejpam-5460	128	10	,	,	PUNCT
ejpam-5460	128	11	as	as	SCONJ
ejpam-5460	128	12	shown	show	VERB
ejpam-5460	128	13	by	by	ADP
ejpam-5460	128	14	the	the	DET
ejpam-5460	128	15	following	follow	VERB
ejpam-5460	128	16	example	example	NOUN
ejpam-5460	128	17	.	.	PUNCT
ejpam-5460	129	1	example	example	NOUN
ejpam-5460	130	1	1	1	NUM
ejpam-5460	130	2	.	.	PUNCT
ejpam-5460	130	3	let	let	VERB
ejpam-5460	130	4	s	s	VERB
ejpam-5460	130	5	=	=	X
ejpam-5460	130	6	{	{	PUNCT
ejpam-5460	130	7	0	0	NUM
ejpam-5460	130	8	,	,	PUNCT
ejpam-5460	130	9	1	1	NUM
ejpam-5460	130	10	,	,	PUNCT
ejpam-5460	130	11	2	2	NUM
ejpam-5460	130	12	,	,	PUNCT
ejpam-5460	130	13	3	3	NUM
ejpam-5460	130	14	,	,	PUNCT
ejpam-5460	130	15	4	4	NUM
ejpam-5460	130	16	,	,	PUNCT
ejpam-5460	130	17	5	5	NUM
ejpam-5460	130	18	,	,	PUNCT
ejpam-5460	130	19	6	6	NUM
ejpam-5460	130	20	}	}	PUNCT
ejpam-5460	130	21	.	.	PUNCT
ejpam-5460	131	1	define	define	VERB
ejpam-5460	131	2	a	a	DET
ejpam-5460	131	3	binary	binary	ADJ
ejpam-5460	131	4	operation	operation	NOUN
ejpam-5460	131	5	·	·	PUNCT
ejpam-5460	131	6	and	and	CCONJ
ejpam-5460	131	7	a	a	DET
ejpam-5460	131	8	partial	partial	ADJ
ejpam-5460	131	9	order	order	NOUN
ejpam-5460	131	10	≤	≤	X
ejpam-5460	131	11	on	on	ADP
ejpam-5460	131	12	s	s	PRON
ejpam-5460	131	13	as	as	SCONJ
ejpam-5460	131	14	follows	follow	VERB
ejpam-5460	131	15	.	.	PUNCT
ejpam-5460	132	1	·	·	PUNCT
ejpam-5460	132	2	0	0	NUM
ejpam-5460	133	1	1	1	NUM
ejpam-5460	133	2	2	2	NUM
ejpam-5460	133	3	3	3	NUM
ejpam-5460	133	4	4	4	NUM
ejpam-5460	133	5	5	5	NUM
ejpam-5460	133	6	6	6	NUM
ejpam-5460	133	7	0	0	NUM
ejpam-5460	133	8	0	0	NUM
ejpam-5460	133	9	0	0	NUM
ejpam-5460	133	10	0	0	NUM
ejpam-5460	133	11	0	0	NUM
ejpam-5460	133	12	0	0	NUM
ejpam-5460	133	13	0	0	NUM
ejpam-5460	133	14	0	0	NUM
ejpam-5460	133	15	1	1	NUM
ejpam-5460	133	16	0	0	NUM
ejpam-5460	133	17	0	0	NUM
ejpam-5460	133	18	0	0	NUM
ejpam-5460	133	19	0	0	NUM
ejpam-5460	133	20	0	0	NUM
ejpam-5460	133	21	0	0	NUM
ejpam-5460	133	22	0	0	NUM
ejpam-5460	133	23	2	2	NUM
ejpam-5460	133	24	0	0	NUM
ejpam-5460	133	25	0	0	NUM
ejpam-5460	133	26	0	0	NUM
ejpam-5460	133	27	0	0	NUM
ejpam-5460	133	28	0	0	NUM
ejpam-5460	133	29	0	0	NUM
ejpam-5460	133	30	0	0	NUM
ejpam-5460	133	31	3	3	NUM
ejpam-5460	133	32	0	0	NUM
ejpam-5460	133	33	0	0	NUM
ejpam-5460	133	34	0	0	NUM
ejpam-5460	133	35	0	0	NUM
ejpam-5460	133	36	0	0	NUM
ejpam-5460	133	37	0	0	NUM
ejpam-5460	133	38	0	0	NUM
ejpam-5460	133	39	4	4	NUM
ejpam-5460	133	40	0	0	NUM
ejpam-5460	133	41	0	0	NUM
ejpam-5460	133	42	0	0	NUM
ejpam-5460	133	43	0	0	NUM
ejpam-5460	133	44	0	0	NUM
ejpam-5460	133	45	0	0	NUM
ejpam-5460	133	46	1	1	NUM
ejpam-5460	133	47	5	5	NUM
ejpam-5460	133	48	0	0	NUM
ejpam-5460	133	49	0	0	NUM
ejpam-5460	133	50	0	0	NUM
ejpam-5460	133	51	0	0	NUM
ejpam-5460	133	52	0	0	NUM
ejpam-5460	133	53	0	0	NUM
ejpam-5460	133	54	2	2	NUM
ejpam-5460	133	55	6	6	NUM
ejpam-5460	133	56	0	0	NUM
ejpam-5460	133	57	0	0	NUM
ejpam-5460	133	58	1	1	NUM
ejpam-5460	133	59	0	0	NUM
ejpam-5460	133	60	0	0	NUM
ejpam-5460	133	61	4	4	NUM
ejpam-5460	133	62	0	0	NUM
ejpam-5460	133	63	and	and	CCONJ
ejpam-5460	133	64	≤	≤	NUM
ejpam-5460	133	65	:	:	PUNCT
ejpam-5460	133	66	=	=	PUNCT
ejpam-5460	133	67	∆s	∆s	NOUN
ejpam-5460	133	68	∪	∪	X
ejpam-5460	133	69	{	{	PUNCT
ejpam-5460	133	70	(	(	PUNCT
ejpam-5460	133	71	0	0	NUM
ejpam-5460	133	72	,	,	PUNCT
ejpam-5460	133	73	1	1	NUM
ejpam-5460	133	74	)	)	PUNCT
ejpam-5460	133	75	}	}	PUNCT
ejpam-5460	133	76	,	,	PUNCT
ejpam-5460	133	77	where	where	SCONJ
ejpam-5460	133	78	∆s	∆s	NOUN
ejpam-5460	133	79	is	be	AUX
ejpam-5460	133	80	the	the	DET
ejpam-5460	133	81	equality	equality	NOUN
ejpam-5460	133	82	relation	relation	NOUN
ejpam-5460	133	83	on	on	ADP
ejpam-5460	133	84	s.	s.	PROPN
ejpam-5460	133	85	then	then	ADV
ejpam-5460	133	86	,	,	PUNCT
ejpam-5460	133	87	s	s	X
ejpam-5460	133	88	:	:	PUNCT
ejpam-5460	133	89	=	=	SYM
ejpam-5460	133	90	⟨s	⟨s	NOUN
ejpam-5460	133	91	;	;	PUNCT
ejpam-5460	133	92	·	·	PUNCT
ejpam-5460	133	93	,	,	PUNCT
ejpam-5460	133	94	≤⟩	≤⟩	VERB
ejpam-5460	133	95	is	be	AUX
ejpam-5460	133	96	an	an	DET
ejpam-5460	133	97	ordered	order	VERB
ejpam-5460	133	98	semigroup	semigroup	NOUN
ejpam-5460	133	99	.	.	PUNCT
ejpam-5460	134	1	let	let	VERB
ejpam-5460	134	2	a	a	PRON
ejpam-5460	134	3	=	=	PUNCT
ejpam-5460	134	4	{	{	PUNCT
ejpam-5460	134	5	0	0	NUM
ejpam-5460	134	6	,	,	PUNCT
ejpam-5460	134	7	6	6	NUM
ejpam-5460	134	8	}	}	PUNCT
ejpam-5460	134	9	.	.	PUNCT
ejpam-5460	135	1	we	we	PRON
ejpam-5460	135	2	can	can	AUX
ejpam-5460	135	3	see	see	VERB
ejpam-5460	135	4	that	that	SCONJ
ejpam-5460	135	5	a	a	PRON
ejpam-5460	135	6	is	be	AUX
ejpam-5460	135	7	a	a	DET
ejpam-5460	135	8	(	(	PUNCT
ejpam-5460	135	9	2	2	NUM
ejpam-5460	135	10	,	,	PUNCT
ejpam-5460	135	11	2)-ideal	2)-ideal	NUM
ejpam-5460	135	12	of	of	ADP
ejpam-5460	135	13	s	s	PRON
ejpam-5460	135	14	but	but	CCONJ
ejpam-5460	135	15	not	not	PART
ejpam-5460	135	16	a	a	DET
ejpam-5460	135	17	(	(	PUNCT
ejpam-5460	135	18	1	1	NUM
ejpam-5460	135	19	,	,	PUNCT
ejpam-5460	135	20	1)-ideal	1)-ideal	NUM
ejpam-5460	135	21	of	of	ADP
ejpam-5460	135	22	s.	s.	PROPN
ejpam-5460	135	23	now	now	ADV
ejpam-5460	135	24	,	,	PUNCT
ejpam-5460	135	25	we	we	PRON
ejpam-5460	135	26	recall	recall	VERB
ejpam-5460	135	27	the	the	DET
ejpam-5460	135	28	concept	concept	NOUN
ejpam-5460	135	29	of	of	ADP
ejpam-5460	135	30	fuzzy	fuzzy	ADJ
ejpam-5460	135	31	sets	set	NOUN
ejpam-5460	135	32	.	.	PUNCT
ejpam-5460	136	1	let	let	VERB
ejpam-5460	136	2	x	x	PRON
ejpam-5460	136	3	be	be	AUX
ejpam-5460	136	4	a	a	DET
ejpam-5460	136	5	nonempty	nonempty	ADV
ejpam-5460	136	6	set	set	VERB
ejpam-5460	136	7	.	.	PUNCT
ejpam-5460	137	1	a	a	DET
ejpam-5460	137	2	fuzzy	fuzzy	ADJ
ejpam-5460	137	3	set	set	NOUN
ejpam-5460	137	4	f	f	PROPN
ejpam-5460	137	5	in	in	ADP
ejpam-5460	137	6	x	x	PROPN
ejpam-5460	137	7	is	be	AUX
ejpam-5460	137	8	a	a	DET
ejpam-5460	137	9	mapping	mapping	NOUN
ejpam-5460	137	10	f	f	NOUN
ejpam-5460	137	11	:	:	PUNCT
ejpam-5460	137	12	x	x	X
ejpam-5460	137	13	→	→	PUNCT
ejpam-5460	138	1	[	[	X
ejpam-5460	138	2	0	0	NUM
ejpam-5460	138	3	,	,	PUNCT
ejpam-5460	138	4	1	1	NUM
ejpam-5460	138	5	]	]	PUNCT
ejpam-5460	138	6	.	.	PUNCT
ejpam-5460	139	1	we	we	PRON
ejpam-5460	139	2	denote	denote	VERB
ejpam-5460	139	3	by	by	ADP
ejpam-5460	139	4	f	f	PROPN
ejpam-5460	139	5	(	(	PUNCT
ejpam-5460	139	6	x	x	X
ejpam-5460	139	7	)	)	PUNCT
ejpam-5460	139	8	the	the	DET
ejpam-5460	139	9	set	set	NOUN
ejpam-5460	139	10	of	of	ADP
ejpam-5460	139	11	all	all	DET
ejpam-5460	139	12	fuzzy	fuzzy	ADJ
ejpam-5460	139	13	sets	set	NOUN
ejpam-5460	139	14	in	in	ADP
ejpam-5460	139	15	x.	x.	NOUN
ejpam-5460	139	16	any	any	PRON
ejpam-5460	139	17	subset	subset	VERB
ejpam-5460	139	18	a	a	PRON
ejpam-5460	139	19	of	of	ADP
ejpam-5460	139	20	x	x	PUNCT
ejpam-5460	139	21	can	can	AUX
ejpam-5460	139	22	be	be	AUX
ejpam-5460	139	23	regarded	regard	VERB
ejpam-5460	139	24	as	as	ADP
ejpam-5460	139	25	a	a	DET
ejpam-5460	139	26	fuzzy	fuzzy	ADJ
ejpam-5460	139	27	set	set	VERB
ejpam-5460	139	28	χa	χa	PRON
ejpam-5460	139	29	in	in	ADP
ejpam-5460	139	30	x	x	PUNCT
ejpam-5460	139	31	defined	define	VERB
ejpam-5460	139	32	by	by	ADP
ejpam-5460	139	33	χa(x	χa(x	NOUN
ejpam-5460	139	34	)	)	PUNCT
ejpam-5460	139	35	:	:	PUNCT
ejpam-5460	140	1	=	=	SYM
ejpam-5460	140	2	{	{	PUNCT
ejpam-5460	140	3	1	1	NUM
ejpam-5460	140	4	if	if	SCONJ
ejpam-5460	140	5	x	x	PROPN
ejpam-5460	140	6	∈	∈	PROPN
ejpam-5460	140	7	a	a	PRON
ejpam-5460	140	8	,	,	PUNCT
ejpam-5460	140	9	0	0	PUNCT
ejpam-5460	140	10	if	if	SCONJ
ejpam-5460	140	11	x	x	PROPN
ejpam-5460	140	12	̸∈	̸∈	PROPN
ejpam-5460	140	13	a	a	PROPN
ejpam-5460	140	14	,	,	PUNCT
ejpam-5460	140	15	for	for	ADP
ejpam-5460	140	16	all	all	DET
ejpam-5460	140	17	x	x	SYM
ejpam-5460	140	18	∈	∈	NOUN
ejpam-5460	140	19	a.	a.	NOUN
ejpam-5460	140	20	the	the	DET
ejpam-5460	140	21	fuzzy	fuzzy	ADJ
ejpam-5460	140	22	set	set	VERB
ejpam-5460	140	23	χa	χa	PROPN
ejpam-5460	140	24	is	be	AUX
ejpam-5460	140	25	called	call	VERB
ejpam-5460	140	26	the	the	DET
ejpam-5460	140	27	characteristic	characteristic	ADJ
ejpam-5460	140	28	function	function	NOUN
ejpam-5460	140	29	of	of	ADP
ejpam-5460	140	30	a	a	PRON
ejpam-5460	140	31	in	in	ADP
ejpam-5460	140	32	x.	x.	NOUN
ejpam-5460	140	33	since	since	SCONJ
ejpam-5460	140	34	the	the	DET
ejpam-5460	140	35	codomain	codomain	NOUN
ejpam-5460	140	36	of	of	ADP
ejpam-5460	140	37	a	a	DET
ejpam-5460	140	38	fuzzy	fuzzy	ADJ
ejpam-5460	140	39	set	set	NOUN
ejpam-5460	140	40	lies	lie	VERB
ejpam-5460	140	41	in	in	ADP
ejpam-5460	140	42	the	the	DET
ejpam-5460	140	43	unit	unit	NOUN
ejpam-5460	140	44	closed	close	VERB
ejpam-5460	140	45	interval	interval	NOUN
ejpam-5460	140	46	[	[	X
ejpam-5460	140	47	0	0	NUM
ejpam-5460	140	48	,	,	PUNCT
ejpam-5460	140	49	1	1	NUM
ejpam-5460	140	50	]	]	PUNCT
ejpam-5460	140	51	,	,	PUNCT
ejpam-5460	140	52	for	for	ADP
ejpam-5460	140	53	any	any	DET
ejpam-5460	140	54	α	α	NOUN
ejpam-5460	140	55	∈	∈	PROPN
ejpam-5460	141	1	[	[	X
ejpam-5460	141	2	0	0	NUM
ejpam-5460	141	3	,	,	PUNCT
ejpam-5460	141	4	1	1	NUM
ejpam-5460	141	5	]	]	PUNCT
ejpam-5460	141	6	can	can	AUX
ejpam-5460	141	7	be	be	AUX
ejpam-5460	141	8	considered	consider	VERB
ejpam-5460	141	9	as	as	ADP
ejpam-5460	141	10	a	a	DET
ejpam-5460	141	11	fuzzy	fuzzy	ADJ
ejpam-5460	141	12	set	set	NOUN
ejpam-5460	141	13	in	in	ADP
ejpam-5460	141	14	x	x	PUNCT
ejpam-5460	141	15	defined	define	VERB
ejpam-5460	141	16	by	by	ADP
ejpam-5460	141	17	α(x	α(x	PROPN
ejpam-5460	141	18	)	)	PUNCT
ejpam-5460	141	19	:	:	PUNCT
ejpam-5460	141	20	=	=	SYM
ejpam-5460	141	21	α	α	X
ejpam-5460	141	22	for	for	ADP
ejpam-5460	141	23	all	all	DET
ejpam-5460	141	24	x	x	SYM
ejpam-5460	141	25	∈	∈	ADJ
ejpam-5460	141	26	x.	x.	NOUN
ejpam-5460	141	27	for	for	ADP
ejpam-5460	141	28	any	any	DET
ejpam-5460	141	29	fuzzy	fuzzy	ADJ
ejpam-5460	141	30	sets	set	NOUN
ejpam-5460	141	31	f	f	PROPN
ejpam-5460	141	32	and	and	CCONJ
ejpam-5460	141	33	g	g	PROPN
ejpam-5460	141	34	in	in	ADP
ejpam-5460	141	35	x	x	SYM
ejpam-5460	141	36	,	,	PUNCT
ejpam-5460	141	37	we	we	PRON
ejpam-5460	141	38	define	define	VERB
ejpam-5460	141	39	the	the	DET
ejpam-5460	141	40	operations	operation	NOUN
ejpam-5460	141	41	∪	∪	ADJ
ejpam-5460	141	42	and	and	CCONJ
ejpam-5460	141	43	∩	∩	NOUN
ejpam-5460	141	44	on	on	ADP
ejpam-5460	141	45	f	f	PROPN
ejpam-5460	141	46	(	(	PUNCT
ejpam-5460	141	47	x	x	X
ejpam-5460	141	48	)	)	PUNCT
ejpam-5460	141	49	by	by	ADP
ejpam-5460	141	50	(	(	PUNCT
ejpam-5460	141	51	f	f	PROPN
ejpam-5460	141	52	∪	∪	PROPN
ejpam-5460	141	53	g)(x	g)(x	PROPN
ejpam-5460	141	54	)	)	PUNCT
ejpam-5460	141	55	:	:	PUNCT
ejpam-5460	141	56	=	=	SYM
ejpam-5460	141	57	f(x	f(x	PROPN
ejpam-5460	141	58	)	)	PUNCT
ejpam-5460	141	59	∨	∨	X
ejpam-5460	141	60	g(x	g(x	NOUN
ejpam-5460	141	61	)	)	PUNCT
ejpam-5460	141	62	and	and	CCONJ
ejpam-5460	141	63	(	(	PUNCT
ejpam-5460	141	64	f	f	PROPN
ejpam-5460	141	65	∩	∩	PROPN
ejpam-5460	141	66	g)(x	g)(x	PROPN
ejpam-5460	141	67	)	)	PUNCT
ejpam-5460	141	68	:	:	PUNCT
ejpam-5460	141	69	=	=	SYM
ejpam-5460	141	70	f(x	f(x	PROPN
ejpam-5460	141	71	)	)	PUNCT
ejpam-5460	141	72	∧	∧	PROPN
ejpam-5460	141	73	g(x	g(x	NOUN
ejpam-5460	141	74	)	)	PUNCT
ejpam-5460	141	75	for	for	ADP
ejpam-5460	141	76	all	all	DET
ejpam-5460	141	77	x	x	SYM
ejpam-5460	141	78	∈	∈	PROPN
ejpam-5460	141	79	x	x	NOUN
ejpam-5460	141	80	,	,	PUNCT
ejpam-5460	141	81	where	where	SCONJ
ejpam-5460	141	82	∨	∨	NUM
ejpam-5460	141	83	and	and	CCONJ
ejpam-5460	141	84	∧	∧	PROPN
ejpam-5460	141	85	is	be	AUX
ejpam-5460	141	86	the	the	DET
ejpam-5460	141	87	supremum	supremum	ADJ
ejpam-5460	141	88	operation	operation	NOUN
ejpam-5460	141	89	and	and	CCONJ
ejpam-5460	141	90	infimum	infimum	ADJ
ejpam-5460	141	91	operation	operation	NOUN
ejpam-5460	141	92	,	,	PUNCT
ejpam-5460	141	93	respectively	respectively	ADV
ejpam-5460	141	94	.	.	PUNCT
ejpam-5460	142	1	a	a	DET
ejpam-5460	142	2	relation	relation	NOUN
ejpam-5460	142	3	⊆	⊆	NUM
ejpam-5460	142	4	on	on	ADP
ejpam-5460	142	5	f	f	PROPN
ejpam-5460	142	6	(	(	PUNCT
ejpam-5460	142	7	x	x	X
ejpam-5460	142	8	)	)	PUNCT
ejpam-5460	142	9	is	be	AUX
ejpam-5460	142	10	defined	define	VERB
ejpam-5460	142	11	by	by	ADP
ejpam-5460	142	12	f	f	PROPN
ejpam-5460	142	13	⊆	⊆	NUM
ejpam-5460	142	14	g	g	NOUN
ejpam-5460	142	15	if	if	SCONJ
ejpam-5460	142	16	f(x	f(x	PROPN
ejpam-5460	142	17	)	)	PUNCT
ejpam-5460	142	18	≤	≤	PUNCT
ejpam-5460	142	19	g(x	g(x	NOUN
ejpam-5460	142	20	)	)	PUNCT
ejpam-5460	142	21	for	for	ADP
ejpam-5460	142	22	all	all	PRON
ejpam-5460	142	23	x	x	SYM
ejpam-5460	142	24	∈	∈	NOUN
ejpam-5460	142	25	x.	x.	NOUN
ejpam-5460	142	26	now	now	ADV
ejpam-5460	142	27	,	,	PUNCT
ejpam-5460	142	28	we	we	PRON
ejpam-5460	142	29	combine	combine	VERB
ejpam-5460	142	30	the	the	DET
ejpam-5460	142	31	concepts	concept	NOUN
ejpam-5460	142	32	of	of	ADP
ejpam-5460	142	33	ordered	order	VERB
ejpam-5460	142	34	semigroups	semigroup	NOUN
ejpam-5460	142	35	and	and	CCONJ
ejpam-5460	142	36	fuzzy	fuzzy	ADJ
ejpam-5460	142	37	sets	set	NOUN
ejpam-5460	142	38	.	.	PUNCT
ejpam-5460	143	1	in	in	ADP
ejpam-5460	143	2	what	what	PRON
ejpam-5460	143	3	follows	follow	VERB
ejpam-5460	143	4	,	,	PUNCT
ejpam-5460	143	5	the	the	DET
ejpam-5460	143	6	notation	notation	NOUN
ejpam-5460	143	7	f	f	X
ejpam-5460	143	8	(	(	PUNCT
ejpam-5460	143	9	s	s	X
ejpam-5460	143	10	)	)	PUNCT
ejpam-5460	143	11	stands	stand	VERB
ejpam-5460	143	12	for	for	ADP
ejpam-5460	143	13	the	the	DET
ejpam-5460	143	14	set	set	NOUN
ejpam-5460	143	15	of	of	ADP
ejpam-5460	143	16	all	all	DET
ejpam-5460	143	17	fuzzy	fuzzy	ADJ
ejpam-5460	143	18	sets	set	NOUN
ejpam-5460	143	19	in	in	ADP
ejpam-5460	143	20	s	s	NOUN
ejpam-5460	143	21	,	,	PUNCT
ejpam-5460	143	22	where	where	SCONJ
ejpam-5460	143	23	s	s	NOUN
ejpam-5460	143	24	is	be	AUX
ejpam-5460	143	25	the	the	DET
ejpam-5460	143	26	underlying	underlying	ADJ
ejpam-5460	143	27	set	set	NOUN
ejpam-5460	143	28	of	of	ADP
ejpam-5460	143	29	the	the	DET
ejpam-5460	143	30	ordered	order	VERB
ejpam-5460	143	31	semigroups	semigroup	NOUN
ejpam-5460	143	32	s.	s.	PROPN
ejpam-5460	143	33	let	let	VERB
ejpam-5460	143	34	s	s	PRON
ejpam-5460	143	35	be	be	AUX
ejpam-5460	143	36	an	an	DET
ejpam-5460	143	37	ordered	order	VERB
ejpam-5460	143	38	semigroup	semigroup	NOUN
ejpam-5460	143	39	.	.	PUNCT
ejpam-5460	144	1	for	for	ADP
ejpam-5460	144	2	any	any	DET
ejpam-5460	144	3	a	a	DET
ejpam-5460	144	4	∈	∈	ADJ
ejpam-5460	144	5	s	s	NOUN
ejpam-5460	144	6	,	,	PUNCT
ejpam-5460	144	7	we	we	PRON
ejpam-5460	144	8	define	define	VERB
ejpam-5460	144	9	sa	sa	NOUN
ejpam-5460	144	10	:	:	PUNCT
ejpam-5460	144	11	=	=	SYM
ejpam-5460	144	12	{	{	PUNCT
ejpam-5460	144	13	(	(	PUNCT
ejpam-5460	144	14	u	u	NOUN
ejpam-5460	144	15	,	,	PUNCT
ejpam-5460	144	16	v	v	NOUN
ejpam-5460	144	17	)	)	PUNCT
ejpam-5460	144	18	∈	∈	PROPN
ejpam-5460	144	19	s	s	PART
ejpam-5460	144	20	×	×	NOUN
ejpam-5460	144	21	s	s	PART
ejpam-5460	144	22	:	:	PUNCT
ejpam-5460	144	23	a	a	DET
ejpam-5460	144	24	≤	≤	NUM
ejpam-5460	144	25	uv	uv	NOUN
ejpam-5460	144	26	}	}	PUNCT
ejpam-5460	144	27	.	.	PUNCT
ejpam-5460	145	1	given	give	VERB
ejpam-5460	145	2	fuzzy	fuzzy	ADJ
ejpam-5460	145	3	sets	set	NOUN
ejpam-5460	145	4	f	f	AUX
ejpam-5460	145	5	,	,	PUNCT
ejpam-5460	145	6	g	g	PROPN
ejpam-5460	145	7	∈	∈	PROPN
ejpam-5460	145	8	f	f	X
ejpam-5460	145	9	(	(	PUNCT
ejpam-5460	145	10	s	s	PROPN
ejpam-5460	145	11	)	)	PUNCT
ejpam-5460	145	12	,	,	PUNCT
ejpam-5460	145	13	we	we	PRON
ejpam-5460	145	14	define	define	VERB
ejpam-5460	145	15	(	(	PUNCT
ejpam-5460	145	16	f	f	X
ejpam-5460	145	17	◦	◦	PROPN
ejpam-5460	145	18	g)(x	g)(x	PROPN
ejpam-5460	145	19	)	)	PUNCT
ejpam-5460	145	20	:	:	PUNCT
ejpam-5460	145	21	=	=	PUNCT
ejpam-5460	145	22			PUNCT
ejpam-5460	145	23	∨	∨	X
ejpam-5460	145	24	(	(	PUNCT
ejpam-5460	145	25	u	u	NOUN
ejpam-5460	145	26	,	,	PUNCT
ejpam-5460	145	27	v)∈sa	v)∈sa	PROPN
ejpam-5460	145	28	{	{	PUNCT
ejpam-5460	145	29	f(u	f(u	PROPN
ejpam-5460	145	30	)	)	PUNCT
ejpam-5460	145	31	∧	∧	PROPN
ejpam-5460	145	32	g(v	g(v	PROPN
ejpam-5460	145	33	)	)	PUNCT
ejpam-5460	145	34	}	}	PUNCT
ejpam-5460	145	35	if	if	SCONJ
ejpam-5460	145	36	sa	sa	PROPN
ejpam-5460	145	37	̸=	̸=	PROPN
ejpam-5460	145	38	∅	∅	NOUN
ejpam-5460	145	39	,	,	PUNCT
ejpam-5460	145	40	0	0	NUM
ejpam-5460	145	41	otherwise	otherwise	ADV
ejpam-5460	145	42	,	,	PUNCT
ejpam-5460	145	43	(	(	PUNCT
ejpam-5460	145	44	1	1	X
ejpam-5460	145	45	)	)	PUNCT
ejpam-5460	145	46	s.	s.	PROPN
ejpam-5460	145	47	lekkoksung	lekkoksung	PROPN
ejpam-5460	145	48	,	,	PUNCT
ejpam-5460	145	49	b.	b.	PROPN
ejpam-5460	145	50	davvaz	davvaz	PROPN
ejpam-5460	145	51	,	,	PUNCT
ejpam-5460	145	52	n.	n.	PROPN
ejpam-5460	145	53	lekkoksung	lekkoksung	PROPN
ejpam-5460	145	54	/	/	SYM
ejpam-5460	145	55	eur	eur	PROPN
ejpam-5460	145	56	.	.	PUNCT
ejpam-5460	146	1	j.	j.	PROPN
ejpam-5460	146	2	pure	pure	PROPN
ejpam-5460	146	3	appl	appl	PROPN
ejpam-5460	146	4	.	.	PROPN
ejpam-5460	146	5	math	math	PROPN
ejpam-5460	146	6	,	,	PUNCT
ejpam-5460	146	7	17	17	NUM
ejpam-5460	146	8	(	(	PUNCT
ejpam-5460	146	9	4	4	NUM
ejpam-5460	146	10	)	)	PUNCT
ejpam-5460	146	11	(	(	PUNCT
ejpam-5460	146	12	2024	2024	NUM
ejpam-5460	146	13	)	)	PUNCT
ejpam-5460	146	14	,	,	PUNCT
ejpam-5460	146	15	2962	2962	NUM
ejpam-5460	146	16	-	-	SYM
ejpam-5460	146	17	2984	2984	NUM
ejpam-5460	146	18	2967	2967	NUM
ejpam-5460	146	19	for	for	ADP
ejpam-5460	146	20	all	all	DET
ejpam-5460	146	21	x	x	SYM
ejpam-5460	146	22	∈	∈	PROPN
ejpam-5460	146	23	s.	s.	PROPN
ejpam-5460	146	24	it	it	PRON
ejpam-5460	146	25	was	be	AUX
ejpam-5460	146	26	proved	prove	VERB
ejpam-5460	146	27	in	in	ADP
ejpam-5460	146	28	[	[	X
ejpam-5460	146	29	20	20	NUM
ejpam-5460	146	30	]	]	PUNCT
ejpam-5460	146	31	by	by	ADP
ejpam-5460	146	32	kehayopulu	kehayopulu	ADJ
ejpam-5460	146	33	and	and	CCONJ
ejpam-5460	146	34	tsingelis	tsingeli	NOUN
ejpam-5460	146	35	that	that	SCONJ
ejpam-5460	146	36	⟨f	⟨f	PUNCT
ejpam-5460	146	37	(	(	PUNCT
ejpam-5460	146	38	s	s	NOUN
ejpam-5460	146	39	)	)	PUNCT
ejpam-5460	146	40	;	;	PUNCT
ejpam-5460	146	41	◦	◦	NOUN
ejpam-5460	146	42	,	,	PUNCT
ejpam-5460	146	43	⊆⟩	⊆⟩	X
ejpam-5460	146	44	is	be	AUX
ejpam-5460	146	45	an	an	DET
ejpam-5460	146	46	ordered	order	VERB
ejpam-5460	146	47	semigroup	semigroup	NOUN
ejpam-5460	146	48	.	.	PUNCT
ejpam-5460	147	1	moreover	moreover	ADV
ejpam-5460	147	2	,	,	PUNCT
ejpam-5460	147	3	⟨f	⟨f	X
ejpam-5460	147	4	(	(	PUNCT
ejpam-5460	147	5	s	s	NOUN
ejpam-5460	147	6	)	)	PUNCT
ejpam-5460	147	7	;	;	PUNCT
ejpam-5460	147	8	◦	◦	NOUN
ejpam-5460	147	9	,	,	PUNCT
ejpam-5460	147	10	⊆⟩	⊆⟩	X
ejpam-5460	147	11	is	be	AUX
ejpam-5460	147	12	a	a	DET
ejpam-5460	147	13	representation	representation	NOUN
ejpam-5460	147	14	of	of	ADP
ejpam-5460	147	15	an	an	DET
ejpam-5460	147	16	ordered	order	VERB
ejpam-5460	147	17	semigroup	semigroup	PROPN
ejpam-5460	147	18	s.	s.	PROPN
ejpam-5460	147	19	this	this	PRON
ejpam-5460	147	20	means	mean	VERB
ejpam-5460	147	21	that	that	SCONJ
ejpam-5460	147	22	any	any	DET
ejpam-5460	147	23	ordered	order	VERB
ejpam-5460	147	24	semigroup	semigroup	NOUN
ejpam-5460	147	25	k	k	PROPN
ejpam-5460	147	26	can	can	AUX
ejpam-5460	147	27	be	be	AUX
ejpam-5460	147	28	embedded	embed	VERB
ejpam-5460	147	29	into	into	ADP
ejpam-5460	147	30	⟨f	⟨f	PUNCT
ejpam-5460	147	31	(	(	PUNCT
ejpam-5460	147	32	k	k	NOUN
ejpam-5460	147	33	)	)	PUNCT
ejpam-5460	147	34	;	;	PUNCT
ejpam-5460	147	35	◦	◦	NOUN
ejpam-5460	147	36	,	,	PUNCT
ejpam-5460	147	37	⊆⟩.	⊆⟩.	NOUN
ejpam-5460	147	38	this	this	DET
ejpam-5460	147	39	finding	finding	NOUN
ejpam-5460	147	40	emphasizes	emphasize	VERB
ejpam-5460	147	41	the	the	DET
ejpam-5460	147	42	significance	significance	NOUN
ejpam-5460	147	43	of	of	ADP
ejpam-5460	147	44	ordered	order	VERB
ejpam-5460	147	45	semigroups	semigroup	NOUN
ejpam-5460	147	46	induced	induce	VERB
ejpam-5460	147	47	by	by	ADP
ejpam-5460	147	48	fuzzy	fuzzy	ADJ
ejpam-5460	147	49	sets	set	NOUN
ejpam-5460	147	50	,	,	PUNCT
ejpam-5460	147	51	so	so	ADV
ejpam-5460	147	52	-	-	PUNCT
ejpam-5460	147	53	called	call	VERB
ejpam-5460	147	54	fuzzy	fuzzy	ADJ
ejpam-5460	147	55	ordered	order	VERB
ejpam-5460	147	56	semigroups	semigroup	NOUN
ejpam-5460	147	57	.	.	PUNCT
ejpam-5460	148	1	efforts	effort	NOUN
ejpam-5460	148	2	have	have	AUX
ejpam-5460	148	3	been	be	AUX
ejpam-5460	148	4	made	make	VERB
ejpam-5460	148	5	to	to	PART
ejpam-5460	148	6	construct	construct	VERB
ejpam-5460	148	7	ordered	order	VERB
ejpam-5460	148	8	semigroups	semigroup	NOUN
ejpam-5460	148	9	induced	induce	VERB
ejpam-5460	148	10	by	by	ADP
ejpam-5460	148	11	fuzzy	fuzzy	ADJ
ejpam-5460	148	12	sets	set	NOUN
ejpam-5460	148	13	,	,	PUNCT
ejpam-5460	148	14	which	which	PRON
ejpam-5460	148	15	aim	aim	VERB
ejpam-5460	148	16	to	to	PART
ejpam-5460	148	17	represent	represent	VERB
ejpam-5460	148	18	ordered	order	VERB
ejpam-5460	148	19	semigroups	semigroup	NOUN
ejpam-5460	148	20	.	.	PUNCT
ejpam-5460	149	1	as	as	ADP
ejpam-5460	149	2	a	a	DET
ejpam-5460	149	3	result	result	NOUN
ejpam-5460	149	4	,	,	PUNCT
ejpam-5460	149	5	the	the	DET
ejpam-5460	149	6	operation	operation	NOUN
ejpam-5460	149	7	◦	◦	NOUN
ejpam-5460	149	8	defined	define	VERB
ejpam-5460	149	9	in	in	ADP
ejpam-5460	149	10	(	(	PUNCT
ejpam-5460	149	11	1	1	X
ejpam-5460	149	12	)	)	PUNCT
ejpam-5460	149	13	can	can	AUX
ejpam-5460	149	14	be	be	AUX
ejpam-5460	149	15	extended	extend	VERB
ejpam-5460	149	16	accordingly	accordingly	ADV
ejpam-5460	149	17	.	.	PUNCT
ejpam-5460	150	1	let	let	VERB
ejpam-5460	150	2	s	s	PRON
ejpam-5460	150	3	be	be	AUX
ejpam-5460	150	4	an	an	DET
ejpam-5460	150	5	ordered	order	VERB
ejpam-5460	150	6	semigroup	semigroup	NOUN
ejpam-5460	150	7	and	and	CCONJ
ejpam-5460	150	8	α	α	NOUN
ejpam-5460	150	9	,	,	PUNCT
ejpam-5460	150	10	β	β	X
ejpam-5460	150	11	∈	∈	PROPN
ejpam-5460	151	1	[	[	X
ejpam-5460	151	2	0	0	NUM
ejpam-5460	151	3	,	,	PUNCT
ejpam-5460	151	4	1	1	NUM
ejpam-5460	151	5	]	]	PUNCT
ejpam-5460	151	6	with	with	ADP
ejpam-5460	151	7	α	α	X
ejpam-5460	151	8	<	<	X
ejpam-5460	151	9	β	β	X
ejpam-5460	151	10	.	.	PUNCT
ejpam-5460	152	1	for	for	ADP
ejpam-5460	152	2	any	any	DET
ejpam-5460	152	3	f	f	PROPN
ejpam-5460	152	4	∈	∈	PROPN
ejpam-5460	152	5	f	f	X
ejpam-5460	152	6	(	(	PUNCT
ejpam-5460	152	7	s	s	PROPN
ejpam-5460	152	8	)	)	PUNCT
ejpam-5460	152	9	,	,	PUNCT
ejpam-5460	152	10	the	the	DET
ejpam-5460	152	11	fuzzy	fuzzy	ADJ
ejpam-5460	152	12	set	set	VERB
ejpam-5460	152	13	fi	fi	NOUN
ejpam-5460	152	14	with	with	ADP
ejpam-5460	152	15	restricted	restricted	ADJ
ejpam-5460	152	16	range	range	NOUN
ejpam-5460	152	17	i	i	PRON
ejpam-5460	152	18	in	in	ADP
ejpam-5460	152	19	s	s	PROPN
ejpam-5460	152	20	is	be	AUX
ejpam-5460	152	21	defined	define	VERB
ejpam-5460	152	22	by	by	ADP
ejpam-5460	152	23	fi(x	fi(x	NUM
ejpam-5460	152	24	)	)	PUNCT
ejpam-5460	152	25	:	:	PUNCT
ejpam-5460	153	1	=	=	PUNCT
ejpam-5460	154	1	[	[	X
ejpam-5460	154	2	f(x)∧β]∨α	f(x)∧β]∨α	NOUN
ejpam-5460	154	3	for	for	ADP
ejpam-5460	154	4	all	all	PRON
ejpam-5460	154	5	x	x	SYM
ejpam-5460	154	6	∈	∈	PROPN
ejpam-5460	154	7	s.	s.	PROPN
ejpam-5460	154	8	we	we	PRON
ejpam-5460	154	9	note	note	VERB
ejpam-5460	154	10	here	here	ADV
ejpam-5460	154	11	that	that	SCONJ
ejpam-5460	154	12	,	,	PUNCT
ejpam-5460	154	13	for	for	ADP
ejpam-5460	154	14	any	any	DET
ejpam-5460	154	15	x	x	SYM
ejpam-5460	154	16	∈	∈	PROPN
ejpam-5460	154	17	s	s	NOUN
ejpam-5460	154	18	,	,	PUNCT
ejpam-5460	154	19	fi(x	fi(x	NUM
ejpam-5460	154	20	)	)	PUNCT
ejpam-5460	154	21	∈	∈	PROPN
ejpam-5460	155	1	[	[	X
ejpam-5460	155	2	α	α	X
ejpam-5460	155	3	,	,	PUNCT
ejpam-5460	155	4	β	β	X
ejpam-5460	155	5	]	]	X
ejpam-5460	155	6	.	.	PUNCT
ejpam-5460	156	1	therefore	therefore	ADV
ejpam-5460	156	2	,	,	PUNCT
ejpam-5460	156	3	(	(	PUNCT
ejpam-5460	156	4	fi)i	fi)i	PROPN
ejpam-5460	156	5	=	=	NOUN
ejpam-5460	156	6	fi	fi	NOUN
ejpam-5460	156	7	.	.	PUNCT
ejpam-5460	157	1	it	it	PRON
ejpam-5460	157	2	is	be	AUX
ejpam-5460	157	3	not	not	PART
ejpam-5460	157	4	difficult	difficult	ADJ
ejpam-5460	157	5	to	to	PART
ejpam-5460	157	6	observe	observe	VERB
ejpam-5460	157	7	that	that	SCONJ
ejpam-5460	157	8	,	,	PUNCT
ejpam-5460	157	9	for	for	ADP
ejpam-5460	157	10	any	any	DET
ejpam-5460	157	11	f	f	NOUN
ejpam-5460	157	12	,	,	PUNCT
ejpam-5460	157	13	g	g	PROPN
ejpam-5460	157	14	∈	∈	PROPN
ejpam-5460	157	15	f	f	X
ejpam-5460	157	16	(	(	PUNCT
ejpam-5460	157	17	s	s	X
ejpam-5460	157	18	):	):	PUNCT
ejpam-5460	157	19	(	(	PUNCT
ejpam-5460	157	20	i	i	NOUN
ejpam-5460	157	21	)	)	PUNCT
ejpam-5460	157	22	(	(	PUNCT
ejpam-5460	157	23	f	f	PROPN
ejpam-5460	157	24	∩	∩	X
ejpam-5460	157	25	g)i	g)i	NOUN
ejpam-5460	157	26	=	=	SYM
ejpam-5460	157	27	fi	fi	NOUN
ejpam-5460	157	28	∩	∩	ADJ
ejpam-5460	157	29	gi	gi	X
ejpam-5460	157	30	;	;	PUNCT
ejpam-5460	157	31	(	(	PUNCT
ejpam-5460	157	32	ii	ii	NOUN
ejpam-5460	157	33	)	)	PUNCT
ejpam-5460	157	34	fi(x	fi(x	NUM
ejpam-5460	157	35	)	)	PUNCT
ejpam-5460	157	36	≤	≤	NOUN
ejpam-5460	157	37	fi(y	fi(y	NOUN
ejpam-5460	157	38	)	)	PUNCT
ejpam-5460	157	39	if	if	SCONJ
ejpam-5460	157	40	f(x	f(x	PROPN
ejpam-5460	157	41	)	)	PUNCT
ejpam-5460	157	42	≤	≤	NUM
ejpam-5460	157	43	f(y	f(y	NOUN
ejpam-5460	157	44	)	)	PUNCT
ejpam-5460	157	45	for	for	ADP
ejpam-5460	157	46	all	all	DET
ejpam-5460	157	47	x	x	NOUN
ejpam-5460	157	48	,	,	PUNCT
ejpam-5460	157	49	y	y	PROPN
ejpam-5460	157	50	∈	∈	PROPN
ejpam-5460	157	51	s	s	PART
ejpam-5460	157	52	;	;	PUNCT
ejpam-5460	157	53	(	(	PUNCT
ejpam-5460	157	54	iii	iii	NOUN
ejpam-5460	157	55	)	)	PUNCT
ejpam-5460	157	56	fi(x	fi(x	NUM
ejpam-5460	157	57	)	)	PUNCT
ejpam-5460	157	58	≤	≤	NUM
ejpam-5460	157	59	gi(y	gi(y	VERB
ejpam-5460	157	60	)	)	PUNCT
ejpam-5460	157	61	if	if	SCONJ
ejpam-5460	157	62	f(x	f(x	PROPN
ejpam-5460	157	63	)	)	PUNCT
ejpam-5460	157	64	∨	∨	NUM
ejpam-5460	157	65	α	α	NOUN
ejpam-5460	157	66	≤	≤	PROPN
ejpam-5460	157	67	g(y	g(y	NOUN
ejpam-5460	157	68	)	)	PUNCT
ejpam-5460	157	69	∧	∧	PROPN
ejpam-5460	157	70	β	β	NOUN
ejpam-5460	157	71	for	for	ADP
ejpam-5460	157	72	all	all	DET
ejpam-5460	157	73	x	x	NOUN
ejpam-5460	157	74	,	,	PUNCT
ejpam-5460	157	75	y	y	PROPN
ejpam-5460	157	76	∈	∈	PROPN
ejpam-5460	157	77	s.	s.	PROPN
ejpam-5460	157	78	in	in	ADP
ejpam-5460	157	79	[	[	X
ejpam-5460	157	80	30	30	NUM
ejpam-5460	157	81	]	]	PUNCT
ejpam-5460	157	82	,	,	PUNCT
ejpam-5460	157	83	khan	khan	PROPN
ejpam-5460	157	84	et	et	PROPN
ejpam-5460	157	85	al	al	PROPN
ejpam-5460	157	86	.	.	PROPN
ejpam-5460	158	1	introduced	introduce	VERB
ejpam-5460	158	2	operations	operation	NOUN
ejpam-5460	158	3	∩i	∩i	NOUN
ejpam-5460	158	4	,	,	PUNCT
ejpam-5460	158	5	∪i	∪i	PROPN
ejpam-5460	158	6	and	and	CCONJ
ejpam-5460	158	7	◦	◦	VERB
ejpam-5460	158	8	i	i	PRON
ejpam-5460	158	9	on	on	ADP
ejpam-5460	158	10	f	f	PROPN
ejpam-5460	158	11	(	(	PUNCT
ejpam-5460	158	12	s	s	PROPN
ejpam-5460	158	13	)	)	PUNCT
ejpam-5460	158	14	by	by	ADP
ejpam-5460	158	15	f	f	PROPN
ejpam-5460	158	16	∩i	∩i	PROPN
ejpam-5460	158	17	g	g	PROPN
ejpam-5460	158	18	:	:	PUNCT
ejpam-5460	158	19	=	=	SYM
ejpam-5460	158	20	(	(	PUNCT
ejpam-5460	158	21	f	f	PROPN
ejpam-5460	158	22	∩	∩	NOUN
ejpam-5460	158	23	g)i	g)i	NOUN
ejpam-5460	158	24	,	,	PUNCT
ejpam-5460	158	25	f	f	PROPN
ejpam-5460	158	26	∪i	∪i	PROPN
ejpam-5460	158	27	g	g	PROPN
ejpam-5460	158	28	:	:	PUNCT
ejpam-5460	158	29	=	=	SYM
ejpam-5460	158	30	(	(	PUNCT
ejpam-5460	158	31	f	f	PROPN
ejpam-5460	158	32	∪	∪	VERB
ejpam-5460	158	33	g)i	g)i	NOUN
ejpam-5460	158	34	,	,	PUNCT
ejpam-5460	158	35	and	and	CCONJ
ejpam-5460	158	36	f	f	PROPN
ejpam-5460	158	37	◦	◦	NOUN
ejpam-5460	158	38	i	i	PRON
ejpam-5460	158	39	g	g	VERB
ejpam-5460	158	40	:	:	PUNCT
ejpam-5460	158	41	=	=	SYM
ejpam-5460	158	42	(	(	PUNCT
ejpam-5460	158	43	f	f	X
ejpam-5460	158	44	◦	◦	NOUN
ejpam-5460	158	45	g)i	g)i	NOUN
ejpam-5460	158	46	.	.	PUNCT
ejpam-5460	159	1	it	it	PRON
ejpam-5460	159	2	is	be	AUX
ejpam-5460	159	3	routine	routine	ADJ
ejpam-5460	159	4	to	to	PART
ejpam-5460	159	5	verify	verify	VERB
ejpam-5460	159	6	that	that	SCONJ
ejpam-5460	159	7	the	the	DET
ejpam-5460	159	8	following	follow	VERB
ejpam-5460	159	9	statements	statement	NOUN
ejpam-5460	159	10	hold	hold	VERB
ejpam-5460	159	11	for	for	ADP
ejpam-5460	159	12	all	all	DET
ejpam-5460	159	13	f	f	NOUN
ejpam-5460	159	14	,	,	PUNCT
ejpam-5460	159	15	g	g	PROPN
ejpam-5460	159	16	∈	∈	PROPN
ejpam-5460	159	17	f	f	X
ejpam-5460	159	18	(	(	PUNCT
ejpam-5460	159	19	s	s	NOUN
ejpam-5460	159	20	)	)	PUNCT
ejpam-5460	159	21	.	.	PUNCT
ejpam-5460	160	1	(	(	PUNCT
ejpam-5460	160	2	i	i	NOUN
ejpam-5460	160	3	)	)	PUNCT
ejpam-5460	160	4	f	f	PROPN
ejpam-5460	160	5	∩i	∩i	PROPN
ejpam-5460	160	6	g	g	PROPN
ejpam-5460	160	7	=	=	PROPN
ejpam-5460	160	8	fi	fi	NOUN
ejpam-5460	160	9	∩	∩	ADJ
ejpam-5460	160	10	gi	gi	NOUN
ejpam-5460	160	11	=	=	SYM
ejpam-5460	160	12	fi	fi	NOUN
ejpam-5460	160	13	∩i	∩i	NOUN
ejpam-5460	160	14	gi	gi	INTJ
ejpam-5460	160	15	.	.	PUNCT
ejpam-5460	161	1	(	(	PUNCT
ejpam-5460	161	2	ii	ii	X
ejpam-5460	161	3	)	)	PUNCT
ejpam-5460	161	4	f	f	PROPN
ejpam-5460	161	5	∪i	∪i	PROPN
ejpam-5460	161	6	g	g	PROPN
ejpam-5460	161	7	=	=	PROPN
ejpam-5460	161	8	fi	fi	NOUN
ejpam-5460	161	9	∪	∪	NOUN
ejpam-5460	161	10	gi	gi	NOUN
ejpam-5460	161	11	=	=	SYM
ejpam-5460	161	12	fi	fi	NOUN
ejpam-5460	161	13	∪i	∪i	PROPN
ejpam-5460	161	14	gi	gi	NOUN
ejpam-5460	161	15	.	.	PUNCT
ejpam-5460	162	1	(	(	PUNCT
ejpam-5460	162	2	iii	iii	NOUN
ejpam-5460	162	3	)	)	PUNCT
ejpam-5460	162	4	fi	fi	NOUN
ejpam-5460	162	5	◦	◦	NOUN
ejpam-5460	162	6	i	i	PRON
ejpam-5460	162	7	gi	gi	VERB
ejpam-5460	162	8	=	=	SYM
ejpam-5460	162	9	f	f	PROPN
ejpam-5460	162	10	◦	◦	NOUN
ejpam-5460	163	1	i	i	PRON
ejpam-5460	163	2	g	g	NOUN
ejpam-5460	163	3	=	=	PUNCT
ejpam-5460	163	4	(	(	PUNCT
ejpam-5460	163	5	f	f	NOUN
ejpam-5460	163	6	◦	◦	NOUN
ejpam-5460	163	7	g)i	g)i	NOUN
ejpam-5460	163	8	.	.	PUNCT
ejpam-5460	164	1	(	(	PUNCT
ejpam-5460	164	2	iv	iv	X
ejpam-5460	164	3	)	)	PUNCT
ejpam-5460	164	4	(	(	PUNCT
ejpam-5460	164	5	f	f	X
ejpam-5460	164	6	◦	◦	NOUN
ejpam-5460	164	7	i	i	NOUN
ejpam-5460	164	8	g)(x	g)(x	PROPN
ejpam-5460	164	9	)	)	PUNCT
ejpam-5460	164	10	≥	≥	NOUN
ejpam-5460	164	11	(	(	PUNCT
ejpam-5460	164	12	fi	fi	NOUN
ejpam-5460	164	13	◦	◦	PROPN
ejpam-5460	164	14	gi)(x	gi)(x	PROPN
ejpam-5460	164	15	)	)	PUNCT
ejpam-5460	164	16	for	for	ADP
ejpam-5460	164	17	all	all	DET
ejpam-5460	164	18	x	x	SYM
ejpam-5460	164	19	∈	∈	PROPN
ejpam-5460	164	20	s.	s.	PROPN
ejpam-5460	164	21	the	the	DET
ejpam-5460	164	22	equality	equality	NOUN
ejpam-5460	164	23	is	be	AUX
ejpam-5460	164	24	valid	valid	ADJ
ejpam-5460	164	25	if	if	SCONJ
ejpam-5460	164	26	sx	sx	PROPN
ejpam-5460	164	27	̸=	̸=	PROPN
ejpam-5460	164	28	∅.	∅.	NOUN
ejpam-5460	164	29	by	by	ADP
ejpam-5460	164	30	the	the	DET
ejpam-5460	164	31	product	product	NOUN
ejpam-5460	164	32	of	of	ADP
ejpam-5460	164	33	fuzzy	fuzzy	ADJ
ejpam-5460	164	34	sets	set	NOUN
ejpam-5460	164	35	and	and	CCONJ
ejpam-5460	164	36	the	the	DET
ejpam-5460	164	37	properties	property	NOUN
ejpam-5460	164	38	of	of	ADP
ejpam-5460	164	39	fuzzy	fuzzy	ADJ
ejpam-5460	164	40	sets	set	NOUN
ejpam-5460	164	41	with	with	ADP
ejpam-5460	164	42	restricted	restricted	ADJ
ejpam-5460	164	43	range	range	NOUN
ejpam-5460	164	44	i	i	PRON
ejpam-5460	164	45	,	,	PUNCT
ejpam-5460	164	46	we	we	PRON
ejpam-5460	164	47	obtain	obtain	VERB
ejpam-5460	164	48	the	the	DET
ejpam-5460	164	49	following	follow	VERB
ejpam-5460	164	50	result	result	NOUN
ejpam-5460	164	51	.	.	PUNCT
ejpam-5460	165	1	proposition	proposition	NOUN
ejpam-5460	165	2	1	1	NUM
ejpam-5460	165	3	.	.	PUNCT
ejpam-5460	166	1	let	let	VERB
ejpam-5460	166	2	s	s	PRON
ejpam-5460	166	3	be	be	AUX
ejpam-5460	166	4	an	an	DET
ejpam-5460	166	5	ordered	order	VERB
ejpam-5460	166	6	semigroup	semigroup	NOUN
ejpam-5460	166	7	,	,	PUNCT
ejpam-5460	166	8	f	f	PROPN
ejpam-5460	166	9	,	,	PUNCT
ejpam-5460	166	10	g	g	PROPN
ejpam-5460	166	11	fuzzy	fuzzy	ADJ
ejpam-5460	166	12	sets	set	NOUN
ejpam-5460	166	13	in	in	ADP
ejpam-5460	166	14	s	s	NOUN
ejpam-5460	166	15	,	,	PUNCT
ejpam-5460	166	16	and	and	CCONJ
ejpam-5460	166	17	a	a	DET
ejpam-5460	166	18	∈	∈	PROPN
ejpam-5460	166	19	s.	s.	PROPN
ejpam-5460	166	20	suppose	suppose	VERB
ejpam-5460	166	21	that	that	SCONJ
ejpam-5460	166	22	(	(	PUNCT
ejpam-5460	166	23	u	u	NOUN
ejpam-5460	166	24	,	,	PUNCT
ejpam-5460	166	25	v	v	NOUN
ejpam-5460	166	26	)	)	PUNCT
ejpam-5460	166	27	∈	∈	PROPN
ejpam-5460	166	28	sa	sa	NOUN
ejpam-5460	166	29	for	for	ADP
ejpam-5460	166	30	some	some	DET
ejpam-5460	166	31	u	u	NOUN
ejpam-5460	166	32	,	,	PUNCT
ejpam-5460	166	33	v	v	PROPN
ejpam-5460	166	34	∈	∈	PROPN
ejpam-5460	166	35	s.	s.	PROPN
ejpam-5460	166	36	then	then	ADV
ejpam-5460	166	37	,	,	PUNCT
ejpam-5460	166	38	we	we	PRON
ejpam-5460	166	39	have	have	VERB
ejpam-5460	166	40	(	(	PUNCT
ejpam-5460	166	41	f	f	X
ejpam-5460	166	42	◦	◦	NOUN
ejpam-5460	166	43	i	i	PRON
ejpam-5460	166	44	g)(a	g)(a	VERB
ejpam-5460	166	45	)	)	PUNCT
ejpam-5460	166	46	≥	≥	NUM
ejpam-5460	166	47	fi(u	fi(u	NOUN
ejpam-5460	166	48	)	)	PUNCT
ejpam-5460	166	49	∧	∧	NOUN
ejpam-5460	166	50	gi(v	gi(v	NOUN
ejpam-5460	166	51	)	)	PUNCT
ejpam-5460	166	52	.	.	PUNCT
ejpam-5460	167	1	in	in	ADP
ejpam-5460	167	2	particular	particular	ADJ
ejpam-5460	167	3	,	,	PUNCT
ejpam-5460	167	4	(	(	PUNCT
ejpam-5460	167	5	β	β	X
ejpam-5460	167	6	◦	◦	NOUN
ejpam-5460	167	7	i	i	PRON
ejpam-5460	167	8	f)(a	f)(a	NOUN
ejpam-5460	167	9	)	)	PUNCT
ejpam-5460	167	10	≥	≥	NOUN
ejpam-5460	167	11	fi(v	fi(v	PUNCT
ejpam-5460	167	12	)	)	PUNCT
ejpam-5460	167	13	and	and	CCONJ
ejpam-5460	167	14	(	(	PUNCT
ejpam-5460	167	15	f	f	PROPN
ejpam-5460	167	16	◦	◦	NOUN
ejpam-5460	167	17	i	i	PRON
ejpam-5460	167	18	β)(a	β)(a	NOUN
ejpam-5460	167	19	)	)	PUNCT
ejpam-5460	167	20	≥	≥	NUM
ejpam-5460	167	21	fi(u	fi(u	NOUN
ejpam-5460	167	22	)	)	PUNCT
ejpam-5460	167	23	.	.	PUNCT
ejpam-5460	168	1	by	by	ADP
ejpam-5460	168	2	the	the	DET
ejpam-5460	168	3	associativity	associativity	NOUN
ejpam-5460	168	4	of	of	ADP
ejpam-5460	168	5	◦	◦	NOUN
ejpam-5460	168	6	and	and	CCONJ
ejpam-5460	168	7	◦	◦	NOUN
ejpam-5460	168	8	i	i	PRON
ejpam-5460	168	9	,	,	PUNCT
ejpam-5460	168	10	proposition	proposition	NOUN
ejpam-5460	168	11	1	1	NUM
ejpam-5460	168	12	can	can	AUX
ejpam-5460	168	13	be	be	AUX
ejpam-5460	168	14	naturally	naturally	ADV
ejpam-5460	168	15	extended	extend	VERB
ejpam-5460	168	16	to	to	PART
ejpam-5460	168	17	yield	yield	VERB
ejpam-5460	168	18	the	the	DET
ejpam-5460	168	19	following	follow	VERB
ejpam-5460	168	20	corollary	corollary	NOUN
ejpam-5460	168	21	.	.	PUNCT
ejpam-5460	169	1	corollary	corollary	ADJ
ejpam-5460	169	2	1	1	NUM
ejpam-5460	169	3	.	.	PUNCT
ejpam-5460	170	1	let	let	VERB
ejpam-5460	170	2	s	s	PRON
ejpam-5460	170	3	be	be	AUX
ejpam-5460	170	4	an	an	DET
ejpam-5460	170	5	ordered	order	VERB
ejpam-5460	170	6	semigroup	semigroup	NOUN
ejpam-5460	170	7	,	,	PUNCT
ejpam-5460	170	8	f1	f1	NOUN
ejpam-5460	170	9	,	,	PUNCT
ejpam-5460	170	10	.	.	PUNCT
ejpam-5460	170	11	.	.	PUNCT
ejpam-5460	171	1	.	.	PUNCT
ejpam-5460	172	1	,	,	PUNCT
ejpam-5460	172	2	fn	fn	NOUN
ejpam-5460	172	3	fuzzy	fuzzy	ADJ
ejpam-5460	172	4	sets	set	NOUN
ejpam-5460	172	5	in	in	ADP
ejpam-5460	172	6	s	s	NOUN
ejpam-5460	172	7	,	,	PUNCT
ejpam-5460	172	8	and	and	CCONJ
ejpam-5460	172	9	a	a	DET
ejpam-5460	172	10	∈	∈	PROPN
ejpam-5460	172	11	s.	s.	PROPN
ejpam-5460	172	12	suppose	suppose	VERB
ejpam-5460	172	13	that	that	SCONJ
ejpam-5460	172	14	a	a	DET
ejpam-5460	172	15	≤	≤	NUM
ejpam-5460	172	16	u1	u1	NOUN
ejpam-5460	172	17	·	·	PUNCT
ejpam-5460	172	18	·	·	PUNCT
ejpam-5460	172	19	·	·	PUNCT
ejpam-5460	172	20	un	un	PROPN
ejpam-5460	172	21	for	for	ADP
ejpam-5460	172	22	some	some	DET
ejpam-5460	172	23	u1	u1	NOUN
ejpam-5460	172	24	,	,	PUNCT
ejpam-5460	172	25	.	.	PUNCT
ejpam-5460	172	26	.	.	PUNCT
ejpam-5460	172	27	.	.	PUNCT
ejpam-5460	173	1	,	,	PUNCT
ejpam-5460	173	2	un	un	PROPN
ejpam-5460	173	3	∈	∈	PROPN
ejpam-5460	173	4	s.	s.	PROPN
ejpam-5460	173	5	then	then	ADV
ejpam-5460	173	6	,	,	PUNCT
ejpam-5460	173	7	we	we	PRON
ejpam-5460	173	8	have	have	VERB
ejpam-5460	173	9	(	(	PUNCT
ejpam-5460	173	10	f1	f1	PROPN
ejpam-5460	173	11	◦	◦	NOUN
ejpam-5460	173	12	i	i	NOUN
ejpam-5460	173	13	·	·	PUNCT
ejpam-5460	173	14	·	·	PUNCT
ejpam-5460	173	15	·	·	PUNCT
ejpam-5460	174	1	◦	◦	NOUN
ejpam-5460	174	2	i	i	PRON
ejpam-5460	174	3	fn)(a	fn)(a	PROPN
ejpam-5460	174	4	)	)	PUNCT
ejpam-5460	174	5	≥	≥	NOUN
ejpam-5460	174	6	(	(	PUNCT
ejpam-5460	174	7	f1)i(u1	f1)i(u1	NOUN
ejpam-5460	174	8	)	)	PUNCT
ejpam-5460	174	9	∧	∧	PROPN
ejpam-5460	174	10	·	·	PUNCT
ejpam-5460	174	11	·	·	PUNCT
ejpam-5460	175	1	·	·	PUNCT
ejpam-5460	175	2	∧	∧	NOUN
ejpam-5460	175	3	(	(	PUNCT
ejpam-5460	175	4	fn)i(un	fn)i(un	PROPN
ejpam-5460	175	5	)	)	PUNCT
ejpam-5460	175	6	.	.	PUNCT
ejpam-5460	176	1	s.	s.	PROPN
ejpam-5460	176	2	lekkoksung	lekkoksung	PROPN
ejpam-5460	176	3	,	,	PUNCT
ejpam-5460	176	4	b.	b.	PROPN
ejpam-5460	176	5	davvaz	davvaz	PROPN
ejpam-5460	176	6	,	,	PUNCT
ejpam-5460	176	7	n.	n.	PROPN
ejpam-5460	176	8	lekkoksung	lekkoksung	PROPN
ejpam-5460	176	9	/	/	SYM
ejpam-5460	176	10	eur	eur	PROPN
ejpam-5460	176	11	.	.	PUNCT
ejpam-5460	177	1	j.	j.	PROPN
ejpam-5460	177	2	pure	pure	PROPN
ejpam-5460	177	3	appl	appl	PROPN
ejpam-5460	177	4	.	.	PROPN
ejpam-5460	177	5	math	math	PROPN
ejpam-5460	177	6	,	,	PUNCT
ejpam-5460	177	7	17	17	NUM
ejpam-5460	177	8	(	(	PUNCT
ejpam-5460	177	9	4	4	NUM
ejpam-5460	177	10	)	)	PUNCT
ejpam-5460	177	11	(	(	PUNCT
ejpam-5460	177	12	2024	2024	NUM
ejpam-5460	177	13	)	)	PUNCT
ejpam-5460	177	14	,	,	PUNCT
ejpam-5460	177	15	2962	2962	NUM
ejpam-5460	177	16	-	-	SYM
ejpam-5460	177	17	2984	2984	NUM
ejpam-5460	177	18	2968	2968	NUM
ejpam-5460	177	19	in	in	ADP
ejpam-5460	177	20	2023	2023	NUM
ejpam-5460	177	21	,	,	PUNCT
ejpam-5460	177	22	lekkoksung	lekkoksung	PROPN
ejpam-5460	177	23	et	et	PROPN
ejpam-5460	177	24	al	al	PROPN
ejpam-5460	177	25	.	.	PUNCT
ejpam-5460	178	1	[	[	X
ejpam-5460	178	2	35	35	NUM
ejpam-5460	178	3	]	]	PUNCT
ejpam-5460	178	4	illustrated	illustrate	VERB
ejpam-5460	178	5	that	that	SCONJ
ejpam-5460	178	6	any	any	DET
ejpam-5460	178	7	ordered	order	VERB
ejpam-5460	178	8	semigroup	semigroup	NOUN
ejpam-5460	178	9	s	s	VERB
ejpam-5460	178	10	can	can	AUX
ejpam-5460	178	11	be	be	AUX
ejpam-5460	178	12	embedded	embed	VERB
ejpam-5460	178	13	into	into	ADP
ejpam-5460	178	14	an	an	DET
ejpam-5460	178	15	ordered	order	VERB
ejpam-5460	178	16	semigroup	semigroup	NOUN
ejpam-5460	178	17	fi(s	fi(s	NOUN
ejpam-5460	178	18	)	)	PUNCT
ejpam-5460	179	1	=	=	PRON
ejpam-5460	179	2	⟨f	⟨f	X
ejpam-5460	179	3	(	(	PUNCT
ejpam-5460	179	4	s	s	NOUN
ejpam-5460	179	5	)	)	PUNCT
ejpam-5460	179	6	;	;	PUNCT
ejpam-5460	180	1	◦	◦	NOUN
ejpam-5460	180	2	i	i	PRON
ejpam-5460	180	3	,	,	PUNCT
ejpam-5460	180	4	⊆⟩.	⊆⟩.	PROPN
ejpam-5460	180	5	with	with	ADP
ejpam-5460	180	6	a	a	DET
ejpam-5460	180	7	slight	slight	ADJ
ejpam-5460	180	8	modification	modification	NOUN
ejpam-5460	180	9	of	of	ADP
ejpam-5460	180	10	⊆	⊆	NUM
ejpam-5460	180	11	on	on	ADP
ejpam-5460	180	12	f	f	PROPN
ejpam-5460	180	13	(	(	PUNCT
ejpam-5460	180	14	s	s	PROPN
ejpam-5460	180	15	)	)	PUNCT
ejpam-5460	180	16	,	,	PUNCT
ejpam-5460	180	17	we	we	PRON
ejpam-5460	180	18	obtain	obtain	VERB
ejpam-5460	180	19	a	a	DET
ejpam-5460	180	20	new	new	ADJ
ejpam-5460	180	21	quasi	quasi	ADJ
ejpam-5460	180	22	-	-	NOUN
ejpam-5460	180	23	order	order	NOUN
ejpam-5460	180	24	relation	relation	NOUN
ejpam-5460	180	25	⊆i	⊆i	NOUN
ejpam-5460	180	26	defined	define	VERB
ejpam-5460	180	27	by	by	ADP
ejpam-5460	180	28	f	f	PROPN
ejpam-5460	180	29	⊆i	⊆i	NOUN
ejpam-5460	180	30	g	g	PROPN
ejpam-5460	180	31	if	if	SCONJ
ejpam-5460	181	1	and	and	CCONJ
ejpam-5460	181	2	only	only	ADV
ejpam-5460	181	3	if	if	SCONJ
ejpam-5460	181	4	fi	fi	NOUN
ejpam-5460	181	5	⊆	⊆	NUM
ejpam-5460	181	6	gi	gi	NOUN
ejpam-5460	181	7	.	.	PUNCT
ejpam-5460	182	1	it	it	PRON
ejpam-5460	182	2	is	be	AUX
ejpam-5460	182	3	natural	natural	ADJ
ejpam-5460	182	4	to	to	PART
ejpam-5460	182	5	obtain	obtain	VERB
ejpam-5460	182	6	an	an	DET
ejpam-5460	182	7	equivalence	equivalence	NOUN
ejpam-5460	182	8	relation	relation	NOUN
ejpam-5460	182	9	≡i	≡i	PROPN
ejpam-5460	182	10	on	on	ADP
ejpam-5460	182	11	f	f	PROPN
ejpam-5460	182	12	(	(	PUNCT
ejpam-5460	182	13	s	s	NOUN
ejpam-5460	182	14	)	)	PUNCT
ejpam-5460	182	15	given	give	VERB
ejpam-5460	182	16	by	by	ADP
ejpam-5460	182	17	f	f	PROPN
ejpam-5460	182	18	≡i	≡i	PROPN
ejpam-5460	182	19	g	g	PROPN
ejpam-5460	182	20	if	if	SCONJ
ejpam-5460	183	1	and	and	CCONJ
ejpam-5460	183	2	only	only	ADV
ejpam-5460	183	3	if	if	SCONJ
ejpam-5460	183	4	f	f	PROPN
ejpam-5460	183	5	⊆i	⊆i	VERB
ejpam-5460	183	6	g	g	NOUN
ejpam-5460	183	7	and	and	CCONJ
ejpam-5460	183	8	g	g	NOUN
ejpam-5460	183	9	⊆i	⊆i	NOUN
ejpam-5460	183	10	f	f	PROPN
ejpam-5460	183	11	.	.	PUNCT
ejpam-5460	184	1	with	with	ADP
ejpam-5460	184	2	the	the	DET
ejpam-5460	184	3	help	help	NOUN
ejpam-5460	184	4	of	of	ADP
ejpam-5460	184	5	the	the	DET
ejpam-5460	184	6	equivalence	equivalence	NOUN
ejpam-5460	184	7	relation	relation	NOUN
ejpam-5460	184	8	≡i	≡i	PROPN
ejpam-5460	184	9	,	,	PUNCT
ejpam-5460	184	10	we	we	PRON
ejpam-5460	184	11	obtain	obtain	VERB
ejpam-5460	184	12	an	an	DET
ejpam-5460	184	13	associative	associative	ADJ
ejpam-5460	184	14	operation	operation	NOUN
ejpam-5460	184	15	⋄i	⋄i	PROPN
ejpam-5460	184	16	and	and	CCONJ
ejpam-5460	184	17	a	a	DET
ejpam-5460	184	18	partial	partial	ADJ
ejpam-5460	184	19	order	order	NOUN
ejpam-5460	184	20	⊑i	⊑i	ADV
ejpam-5460	184	21	on	on	ADP
ejpam-5460	184	22	f	f	PROPN
ejpam-5460	184	23	(	(	PUNCT
ejpam-5460	184	24	s)/≡i	s)/≡i	PROPN
ejpam-5460	184	25	defined	define	VERB
ejpam-5460	184	26	by	by	ADP
ejpam-5460	184	27	f/≡i	f/≡i	PROPN
ejpam-5460	184	28	⋄i	⋄i	INTJ
ejpam-5460	184	29	g/≡i	g/≡i	NOUN
ejpam-5460	184	30	:	:	PUNCT
ejpam-5460	185	1	=	=	SYM
ejpam-5460	185	2	(	(	PUNCT
ejpam-5460	185	3	f	f	X
ejpam-5460	185	4	◦	◦	NOUN
ejpam-5460	186	1	i	i	PRON
ejpam-5460	186	2	g)/≡i	g)/≡i	PROPN
ejpam-5460	186	3	and	and	CCONJ
ejpam-5460	186	4	f/≡i	f/≡i	VERB
ejpam-5460	186	5	⊑i	⊑i	ADV
ejpam-5460	186	6	g/≡i	g/≡i	NOUN
ejpam-5460	186	7	if	if	SCONJ
ejpam-5460	186	8	and	and	CCONJ
ejpam-5460	186	9	only	only	ADV
ejpam-5460	186	10	if	if	SCONJ
ejpam-5460	186	11	f	f	PROPN
ejpam-5460	186	12	⊆i	⊆i	NOUN
ejpam-5460	186	13	g.	g.	PROPN
ejpam-5460	186	14	then	then	ADV
ejpam-5460	186	15	,	,	PUNCT
ejpam-5460	186	16	the	the	DET
ejpam-5460	186	17	algebraic	algebraic	ADJ
ejpam-5460	186	18	system	system	NOUN
ejpam-5460	186	19	fi(s	fi(s	NUM
ejpam-5460	186	20	)	)	PUNCT
ejpam-5460	186	21	:	:	PUNCT
ejpam-5460	187	1	=	=	SYM
ejpam-5460	187	2	⟨f	⟨f	PUNCT
ejpam-5460	187	3	(	(	PUNCT
ejpam-5460	187	4	s)/≡i	s)/≡i	PROPN
ejpam-5460	187	5	;	;	PUNCT
ejpam-5460	187	6	⋄i	⋄i	NUM
ejpam-5460	187	7	,	,	PUNCT
ejpam-5460	187	8	⊑i⟩	⊑i⟩	PROPN
ejpam-5460	187	9	is	be	AUX
ejpam-5460	187	10	an	an	DET
ejpam-5460	187	11	ordered	order	VERB
ejpam-5460	187	12	semigroup	semigroup	NOUN
ejpam-5460	187	13	which	which	PRON
ejpam-5460	187	14	is	be	AUX
ejpam-5460	187	15	a	a	DET
ejpam-5460	187	16	representation	representation	NOUN
ejpam-5460	187	17	of	of	ADP
ejpam-5460	187	18	an	an	DET
ejpam-5460	187	19	ordered	order	VERB
ejpam-5460	187	20	semigroup	semigroup	PROPN
ejpam-5460	187	21	s.	s.	PROPN
ejpam-5460	187	22	we	we	PRON
ejpam-5460	187	23	conclude	conclude	VERB
ejpam-5460	187	24	that	that	SCONJ
ejpam-5460	187	25	the	the	DET
ejpam-5460	187	26	algebraic	algebraic	PROPN
ejpam-5460	187	27	systems	system	NOUN
ejpam-5460	187	28	we	we	PRON
ejpam-5460	187	29	obtained	obtain	VERB
ejpam-5460	187	30	above	above	ADV
ejpam-5460	187	31	are	be	AUX
ejpam-5460	187	32	representations	representation	NOUN
ejpam-5460	187	33	of	of	ADP
ejpam-5460	187	34	any	any	DET
ejpam-5460	187	35	ordered	order	VERB
ejpam-5460	187	36	semigroup	semigroup	NOUN
ejpam-5460	187	37	,	,	PUNCT
ejpam-5460	187	38	as	as	SCONJ
ejpam-5460	187	39	presented	present	VERB
ejpam-5460	187	40	by	by	ADP
ejpam-5460	187	41	the	the	DET
ejpam-5460	187	42	following	follow	VERB
ejpam-5460	187	43	theorem	theorem	PROPN
ejpam-5460	187	44	.	.	PUNCT
ejpam-5460	188	1	theorem	theorem	NOUN
ejpam-5460	188	2	1	1	NUM
ejpam-5460	188	3	(	(	PUNCT
ejpam-5460	188	4	[	[	X
ejpam-5460	188	5	35	35	NUM
ejpam-5460	188	6	]	]	NUM
ejpam-5460	188	7	)	)	PUNCT
ejpam-5460	188	8	.	.	PUNCT
ejpam-5460	189	1	any	any	DET
ejpam-5460	189	2	ordered	order	VERB
ejpam-5460	189	3	semigroup	semigroup	NOUN
ejpam-5460	189	4	s	s	VERB
ejpam-5460	189	5	can	can	AUX
ejpam-5460	189	6	be	be	AUX
ejpam-5460	189	7	embedded	embed	VERB
ejpam-5460	189	8	into	into	ADP
ejpam-5460	189	9	the	the	DET
ejpam-5460	189	10	algebraic	algebraic	PROPN
ejpam-5460	189	11	systems	system	NOUN
ejpam-5460	189	12	fi(s	fi(s	X
ejpam-5460	189	13	)	)	PUNCT
ejpam-5460	189	14	and	and	CCONJ
ejpam-5460	189	15	fi(s	fi(s	NUM
ejpam-5460	189	16	)	)	PUNCT
ejpam-5460	189	17	.	.	PUNCT
ejpam-5460	190	1	the	the	DET
ejpam-5460	190	2	above	above	ADJ
ejpam-5460	190	3	result	result	NOUN
ejpam-5460	190	4	demonstrates	demonstrate	VERB
ejpam-5460	190	5	the	the	DET
ejpam-5460	190	6	importance	importance	NOUN
ejpam-5460	190	7	of	of	ADP
ejpam-5460	190	8	such	such	ADJ
ejpam-5460	190	9	ordered	order	VERB
ejpam-5460	190	10	semigroups	semigroup	NOUN
ejpam-5460	190	11	induced	induce	VERB
ejpam-5460	190	12	by	by	ADP
ejpam-5460	190	13	fuzzy	fuzzy	ADJ
ejpam-5460	190	14	sets	set	NOUN
ejpam-5460	190	15	.	.	PUNCT
ejpam-5460	191	1	therefore	therefore	ADV
ejpam-5460	191	2	,	,	PUNCT
ejpam-5460	191	3	it	it	PRON
ejpam-5460	191	4	is	be	AUX
ejpam-5460	191	5	worth	worth	ADJ
ejpam-5460	191	6	studying	study	VERB
ejpam-5460	191	7	the	the	DET
ejpam-5460	191	8	algebraic	algebraic	ADJ
ejpam-5460	191	9	properties	property	NOUN
ejpam-5460	191	10	of	of	ADP
ejpam-5460	191	11	ordered	order	VERB
ejpam-5460	191	12	semigroups	semigroup	NOUN
ejpam-5460	191	13	by	by	ADP
ejpam-5460	191	14	these	these	DET
ejpam-5460	191	15	algebraic	algebraic	ADJ
ejpam-5460	191	16	systems	system	NOUN
ejpam-5460	191	17	.	.	PUNCT
ejpam-5460	192	1	3	3	X
ejpam-5460	192	2	.	.	X
ejpam-5460	192	3	the	the	DET
ejpam-5460	192	4	concepts	concept	NOUN
ejpam-5460	192	5	of	of	ADP
ejpam-5460	192	6	(	(	PUNCT
ejpam-5460	192	7	α	α	NOUN
ejpam-5460	192	8	,	,	PUNCT
ejpam-5460	192	9	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	192	10	(	(	PUNCT
ejpam-5460	192	11	m	m	NOUN
ejpam-5460	192	12	,	,	PUNCT
ejpam-5460	192	13	n)-ideals	n)-ideal	NOUN
ejpam-5460	192	14	and	and	CCONJ
ejpam-5460	192	15	(	(	PUNCT
ejpam-5460	192	16	α	α	NOUN
ejpam-5460	192	17	,	,	PUNCT
ejpam-5460	192	18	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	192	19	n	n	CCONJ
ejpam-5460	192	20	-	-	PUNCT
ejpam-5460	192	21	interior	interior	ADJ
ejpam-5460	192	22	ideals	ideal	NOUN
ejpam-5460	192	23	as	as	SCONJ
ejpam-5460	192	24	we	we	PRON
ejpam-5460	192	25	recall	recall	VERB
ejpam-5460	192	26	the	the	DET
ejpam-5460	192	27	concepts	concept	NOUN
ejpam-5460	192	28	of	of	ADP
ejpam-5460	192	29	(	(	PUNCT
ejpam-5460	192	30	m	m	PROPN
ejpam-5460	192	31	,	,	PUNCT
ejpam-5460	192	32	n)-ideals	n)-ideal	NOUN
ejpam-5460	192	33	and	and	CCONJ
ejpam-5460	192	34	n	n	CCONJ
ejpam-5460	192	35	-	-	PUNCT
ejpam-5460	192	36	interior	interior	ADJ
ejpam-5460	192	37	ideals	ideal	NOUN
ejpam-5460	192	38	in	in	ADP
ejpam-5460	192	39	ordered	order	VERB
ejpam-5460	192	40	semigroups	semigroup	NOUN
ejpam-5460	192	41	in	in	ADP
ejpam-5460	192	42	the	the	DET
ejpam-5460	192	43	previous	previous	ADJ
ejpam-5460	192	44	section	section	NOUN
ejpam-5460	192	45	,	,	PUNCT
ejpam-5460	192	46	we	we	PRON
ejpam-5460	192	47	recall	recall	VERB
ejpam-5460	192	48	the	the	DET
ejpam-5460	192	49	concepts	concept	NOUN
ejpam-5460	192	50	of	of	ADP
ejpam-5460	192	51	such	such	ADJ
ejpam-5460	192	52	ideals	ideal	NOUN
ejpam-5460	192	53	applied	apply	VERB
ejpam-5460	192	54	by	by	ADP
ejpam-5460	192	55	fuzzy	fuzzy	ADJ
ejpam-5460	192	56	sets	set	NOUN
ejpam-5460	192	57	in	in	ADP
ejpam-5460	192	58	this	this	DET
ejpam-5460	192	59	section	section	NOUN
ejpam-5460	192	60	.	.	PUNCT
ejpam-5460	193	1	let	let	VERB
ejpam-5460	193	2	s	s	PRON
ejpam-5460	193	3	be	be	AUX
ejpam-5460	193	4	an	an	DET
ejpam-5460	193	5	ordered	order	VERB
ejpam-5460	193	6	semigroup	semigroup	NOUN
ejpam-5460	193	7	,	,	PUNCT
ejpam-5460	193	8	and	and	CCONJ
ejpam-5460	193	9	x1	x1	NUM
ejpam-5460	193	10	,	,	PUNCT
ejpam-5460	193	11	.	.	PUNCT
ejpam-5460	193	12	.	.	PUNCT
ejpam-5460	194	1	.	.	PUNCT
ejpam-5460	195	1	,	,	PUNCT
ejpam-5460	195	2	xn	xn	PROPN
ejpam-5460	195	3	∈	∈	PROPN
ejpam-5460	195	4	s.	s.	PROPN
ejpam-5460	195	5	we	we	PRON
ejpam-5460	195	6	simply	simply	ADV
ejpam-5460	195	7	denote	denote	VERB
ejpam-5460	195	8	the	the	DET
ejpam-5460	195	9	product	product	NOUN
ejpam-5460	195	10	x1	x1	X
ejpam-5460	195	11	·	·	PUNCT
ejpam-5460	195	12	·	·	PUNCT
ejpam-5460	196	1	·	·	PUNCT
ejpam-5460	196	2	xn	xn	NUM
ejpam-5460	196	3	by	by	ADP
ejpam-5460	196	4	xn1	xn1	PROPN
ejpam-5460	196	5	.	.	PUNCT
ejpam-5460	197	1	a	a	DET
ejpam-5460	197	2	fuzzy	fuzzy	ADJ
ejpam-5460	197	3	set	set	VERB
ejpam-5460	197	4	f	f	PROPN
ejpam-5460	197	5	∈	∈	PROPN
ejpam-5460	197	6	f	f	X
ejpam-5460	197	7	(	(	PUNCT
ejpam-5460	197	8	s	s	X
ejpam-5460	197	9	)	)	PUNCT
ejpam-5460	197	10	is	be	AUX
ejpam-5460	197	11	said	say	VERB
ejpam-5460	197	12	to	to	PART
ejpam-5460	197	13	be	be	AUX
ejpam-5460	197	14	(	(	PUNCT
ejpam-5460	197	15	α	α	NOUN
ejpam-5460	197	16	,	,	PUNCT
ejpam-5460	197	17	β)-strongly	β)-strongly	ADV
ejpam-5460	197	18	convex	convex	VERB
ejpam-5460	197	19	if	if	SCONJ
ejpam-5460	197	20	f(x	f(x	PROPN
ejpam-5460	197	21	)	)	PUNCT
ejpam-5460	197	22	∨	∨	NUM
ejpam-5460	197	23	α	α	PROPN
ejpam-5460	197	24	≥	≥	NOUN
ejpam-5460	197	25	f(y	f(y	NOUN
ejpam-5460	197	26	)	)	PUNCT
ejpam-5460	197	27	∧	∧	NOUN
ejpam-5460	197	28	β	β	X
ejpam-5460	197	29	whenever	whenever	SCONJ
ejpam-5460	197	30	x	x	SYM
ejpam-5460	197	31	≤	≤	ADJ
ejpam-5460	197	32	y	y	NOUN
ejpam-5460	197	33	for	for	ADP
ejpam-5460	197	34	all	all	DET
ejpam-5460	197	35	x	x	NOUN
ejpam-5460	197	36	,	,	PUNCT
ejpam-5460	197	37	y	y	PROPN
ejpam-5460	197	38	∈	∈	PROPN
ejpam-5460	197	39	s.	s.	PROPN
ejpam-5460	197	40	an	an	DET
ejpam-5460	197	41	(	(	PUNCT
ejpam-5460	197	42	α	α	NOUN
ejpam-5460	197	43	,	,	PUNCT
ejpam-5460	197	44	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	197	45	subsemigroup	subsemigroup	NOUN
ejpam-5460	197	46	of	of	ADP
ejpam-5460	197	47	s	s	NOUN
ejpam-5460	197	48	is	be	AUX
ejpam-5460	197	49	a	a	DET
ejpam-5460	197	50	fuzzy	fuzzy	ADJ
ejpam-5460	197	51	set	set	NOUN
ejpam-5460	197	52	f	f	PROPN
ejpam-5460	197	53	∈	∈	PROPN
ejpam-5460	197	54	f	f	X
ejpam-5460	197	55	(	(	PUNCT
ejpam-5460	197	56	s	s	NOUN
ejpam-5460	197	57	)	)	PUNCT
ejpam-5460	197	58	such	such	ADJ
ejpam-5460	197	59	that	that	DET
ejpam-5460	197	60	f(xy	f(xy	NOUN
ejpam-5460	197	61	)	)	PUNCT
ejpam-5460	197	62	∨	∨	NUM
ejpam-5460	197	63	α	α	PROPN
ejpam-5460	197	64	≥	≥	NUM
ejpam-5460	197	65	f(x	f(x	PROPN
ejpam-5460	197	66	)	)	PUNCT
ejpam-5460	197	67	∧	∧	PROPN
ejpam-5460	197	68	f(y	f(y	NOUN
ejpam-5460	197	69	)	)	PUNCT
ejpam-5460	197	70	∧	∧	PROPN
ejpam-5460	197	71	β	β	NOUN
ejpam-5460	197	72	for	for	ADP
ejpam-5460	197	73	all	all	DET
ejpam-5460	197	74	x	x	NOUN
ejpam-5460	197	75	,	,	PUNCT
ejpam-5460	197	76	y	y	PROPN
ejpam-5460	197	77	∈	∈	PROPN
ejpam-5460	197	78	s.	s.	PROPN
ejpam-5460	197	79	definition	definition	NOUN
ejpam-5460	197	80	2	2	NUM
ejpam-5460	197	81	(	(	PUNCT
ejpam-5460	197	82	[	[	X
ejpam-5460	197	83	4	4	NUM
ejpam-5460	197	84	]	]	NUM
ejpam-5460	197	85	)	)	PUNCT
ejpam-5460	197	86	.	.	PUNCT
ejpam-5460	198	1	let	let	VERB
ejpam-5460	198	2	s	s	PRON
ejpam-5460	198	3	be	be	AUX
ejpam-5460	198	4	an	an	DET
ejpam-5460	198	5	ordered	order	VERB
ejpam-5460	198	6	semigroup	semigroup	NOUN
ejpam-5460	198	7	.	.	PUNCT
ejpam-5460	199	1	an	an	DET
ejpam-5460	199	2	(	(	PUNCT
ejpam-5460	199	3	α	α	NOUN
ejpam-5460	199	4	,	,	PUNCT
ejpam-5460	199	5	β)-strongly	β)-strongly	ADV
ejpam-5460	199	6	convex	convex	NOUN
ejpam-5460	199	7	(	(	PUNCT
ejpam-5460	199	8	α	α	NOUN
ejpam-5460	199	9	,	,	PUNCT
ejpam-5460	199	10	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	199	11	subsemigroup	subsemigroup	INTJ
ejpam-5460	199	12	f	f	PROPN
ejpam-5460	199	13	of	of	ADP
ejpam-5460	199	14	s	s	PROPN
ejpam-5460	199	15	is	be	AUX
ejpam-5460	199	16	called	call	VERB
ejpam-5460	199	17	:	:	PUNCT
ejpam-5460	199	18	(	(	PUNCT
ejpam-5460	199	19	i	i	NOUN
ejpam-5460	199	20	)	)	PUNCT
ejpam-5460	199	21	an	an	DET
ejpam-5460	199	22	(	(	PUNCT
ejpam-5460	199	23	α	α	NOUN
ejpam-5460	199	24	,	,	PUNCT
ejpam-5460	199	25	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	199	26	(	(	PUNCT
ejpam-5460	199	27	m	m	PROPN
ejpam-5460	199	28	,	,	PUNCT
ejpam-5460	199	29	n)-ideal	n)-ideal	NOUN
ejpam-5460	199	30	of	of	ADP
ejpam-5460	199	31	s	s	PRON
ejpam-5460	199	32	if	if	SCONJ
ejpam-5460	199	33	f(xm1	f(xm1	ADJ
ejpam-5460	199	34	yzn1	yzn1	NOUN
ejpam-5460	199	35	)	)	PUNCT
ejpam-5460	199	36	∨	∨	PROPN
ejpam-5460	199	37	α	α	X
ejpam-5460	199	38	≥	≥	NUM
ejpam-5460	199	39	(	(	PUNCT
ejpam-5460	199	40	m∧	m∧	PROPN
ejpam-5460	199	41	i=1	i=1	PROPN
ejpam-5460	199	42	f(xi	f(xi	PROPN
ejpam-5460	199	43	)	)	PUNCT
ejpam-5460	199	44	)	)	PUNCT
ejpam-5460	200	1	∧	∧	NOUN
ejpam-5460	200	2	(	(	PUNCT
ejpam-5460	200	3	n∧	n∧	NUM
ejpam-5460	200	4	i=1	i=1	PROPN
ejpam-5460	200	5	f(zi	f(zi	PROPN
ejpam-5460	200	6	)	)	PUNCT
ejpam-5460	200	7	)	)	PUNCT
ejpam-5460	201	1	∧	∧	PROPN
ejpam-5460	201	2	β	β	NOUN
ejpam-5460	201	3	for	for	ADP
ejpam-5460	201	4	any	any	DET
ejpam-5460	201	5	x1	x1	PROPN
ejpam-5460	201	6	,	,	PUNCT
ejpam-5460	201	7	.	.	PUNCT
ejpam-5460	201	8	.	.	PUNCT
ejpam-5460	201	9	.	.	PUNCT
ejpam-5460	202	1	,	,	PUNCT
ejpam-5460	202	2	xm	xm	PROPN
ejpam-5460	202	3	,	,	PUNCT
ejpam-5460	202	4	y	y	PROPN
ejpam-5460	202	5	,	,	PUNCT
ejpam-5460	202	6	z1	z1	PROPN
ejpam-5460	202	7	,	,	PUNCT
ejpam-5460	202	8	.	.	PUNCT
ejpam-5460	202	9	.	.	PUNCT
ejpam-5460	202	10	.	.	PUNCT
ejpam-5460	203	1	,	,	PUNCT
ejpam-5460	203	2	zn	zn	PROPN
ejpam-5460	203	3	∈	∈	PROPN
ejpam-5460	203	4	s	s	PROPN
ejpam-5460	203	5	;	;	PUNCT
ejpam-5460	203	6	s.	s.	PROPN
ejpam-5460	203	7	lekkoksung	lekkoksung	PROPN
ejpam-5460	203	8	,	,	PUNCT
ejpam-5460	203	9	b.	b.	PROPN
ejpam-5460	203	10	davvaz	davvaz	PROPN
ejpam-5460	203	11	,	,	PUNCT
ejpam-5460	203	12	n.	n.	PROPN
ejpam-5460	203	13	lekkoksung	lekkoksung	PROPN
ejpam-5460	203	14	/	/	SYM
ejpam-5460	203	15	eur	eur	PROPN
ejpam-5460	203	16	.	.	PUNCT
ejpam-5460	204	1	j.	j.	PROPN
ejpam-5460	204	2	pure	pure	PROPN
ejpam-5460	204	3	appl	appl	PROPN
ejpam-5460	204	4	.	.	PROPN
ejpam-5460	204	5	math	math	PROPN
ejpam-5460	204	6	,	,	PUNCT
ejpam-5460	204	7	17	17	NUM
ejpam-5460	204	8	(	(	PUNCT
ejpam-5460	204	9	4	4	NUM
ejpam-5460	204	10	)	)	PUNCT
ejpam-5460	204	11	(	(	PUNCT
ejpam-5460	204	12	2024	2024	NUM
ejpam-5460	204	13	)	)	PUNCT
ejpam-5460	204	14	,	,	PUNCT
ejpam-5460	204	15	2962	2962	NUM
ejpam-5460	204	16	-	-	SYM
ejpam-5460	204	17	2984	2984	NUM
ejpam-5460	204	18	2969	2969	NUM
ejpam-5460	204	19	(	(	PUNCT
ejpam-5460	204	20	ii	ii	NOUN
ejpam-5460	204	21	)	)	PUNCT
ejpam-5460	204	22	an	an	DET
ejpam-5460	204	23	(	(	PUNCT
ejpam-5460	204	24	α	α	NOUN
ejpam-5460	204	25	,	,	PUNCT
ejpam-5460	204	26	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	204	27	n	n	CCONJ
ejpam-5460	204	28	-	-	PUNCT
ejpam-5460	204	29	interior	interior	ADJ
ejpam-5460	204	30	ideal	ideal	NOUN
ejpam-5460	204	31	of	of	ADP
ejpam-5460	204	32	s	s	PRON
ejpam-5460	204	33	if	if	SCONJ
ejpam-5460	204	34	f(xyn1	f(xyn1	NOUN
ejpam-5460	204	35	z	z	NOUN
ejpam-5460	204	36	)	)	PUNCT
ejpam-5460	204	37	∨	∨	PROPN
ejpam-5460	204	38	α	α	PROPN
ejpam-5460	204	39	≥	≥	NUM
ejpam-5460	204	40	(	(	PUNCT
ejpam-5460	204	41	n∧	n∧	NUM
ejpam-5460	204	42	i=1	i=1	PROPN
ejpam-5460	204	43	f(yi	f(yi	PROPN
ejpam-5460	204	44	)	)	PUNCT
ejpam-5460	204	45	)	)	PUNCT
ejpam-5460	205	1	∧	∧	PROPN
ejpam-5460	205	2	β	β	NOUN
ejpam-5460	205	3	for	for	ADP
ejpam-5460	205	4	any	any	DET
ejpam-5460	205	5	x	x	NOUN
ejpam-5460	205	6	,	,	PUNCT
ejpam-5460	205	7	y1	y1	NOUN
ejpam-5460	205	8	,	,	PUNCT
ejpam-5460	205	9	.	.	PUNCT
ejpam-5460	205	10	.	.	PUNCT
ejpam-5460	205	11	.	.	PUNCT
ejpam-5460	206	1	,	,	PUNCT
ejpam-5460	206	2	yn	yn	PROPN
ejpam-5460	206	3	,	,	PUNCT
ejpam-5460	206	4	z	z	PROPN
ejpam-5460	206	5	∈	∈	PROPN
ejpam-5460	206	6	s.	s.	PROPN
ejpam-5460	206	7	by	by	ADP
ejpam-5460	206	8	putting	put	VERB
ejpam-5460	206	9	an	an	DET
ejpam-5460	206	10	appropriate	appropriate	ADJ
ejpam-5460	206	11	α	α	NOUN
ejpam-5460	206	12	,	,	PUNCT
ejpam-5460	206	13	β	β	X
ejpam-5460	206	14	,	,	PUNCT
ejpam-5460	206	15	m	m	PROPN
ejpam-5460	206	16	,	,	PUNCT
ejpam-5460	206	17	and	and	CCONJ
ejpam-5460	206	18	n	n	CCONJ
ejpam-5460	206	19	,	,	PUNCT
ejpam-5460	206	20	the	the	DET
ejpam-5460	206	21	above	above	ADJ
ejpam-5460	206	22	notions	notion	NOUN
ejpam-5460	206	23	generalize	generalize	VERB
ejpam-5460	206	24	several	several	ADJ
ejpam-5460	206	25	kinds	kind	NOUN
ejpam-5460	206	26	of	of	ADP
ejpam-5460	206	27	fuzzy	fuzzy	ADJ
ejpam-5460	206	28	ideals	ideal	NOUN
ejpam-5460	206	29	in	in	ADP
ejpam-5460	206	30	ordered	order	VERB
ejpam-5460	206	31	semigroups	semigroup	NOUN
ejpam-5460	206	32	investigated	investigate	VERB
ejpam-5460	206	33	in	in	ADP
ejpam-5460	206	34	some	some	DET
ejpam-5460	206	35	literature	literature	NOUN
ejpam-5460	206	36	described	describe	VERB
ejpam-5460	206	37	as	as	ADP
ejpam-5460	206	38	follows	follow	VERB
ejpam-5460	206	39	.	.	PUNCT
ejpam-5460	207	1	any	any	DET
ejpam-5460	207	2	fuzzy	fuzzy	ADJ
ejpam-5460	207	3	left	left	NOUN
ejpam-5460	207	4	(	(	PUNCT
ejpam-5460	207	5	resp	resp	NOUN
ejpam-5460	207	6	.	.	PUNCT
ejpam-5460	207	7	,	,	PUNCT
ejpam-5460	207	8	right	right	INTJ
ejpam-5460	207	9	,	,	PUNCT
ejpam-5460	207	10	bi-	bi-	NUM
ejpam-5460	207	11	,	,	PUNCT
ejpam-5460	207	12	interior	interior	ADJ
ejpam-5460	207	13	)	)	PUNCT
ejpam-5460	207	14	ideal	ideal	NOUN
ejpam-5460	207	15	is	be	AUX
ejpam-5460	207	16	a	a	DET
ejpam-5460	207	17	(	(	PUNCT
ejpam-5460	207	18	0	0	NUM
ejpam-5460	207	19	,	,	PUNCT
ejpam-5460	207	20	1)-fuzzy	1)-fuzzy	NUM
ejpam-5460	207	21	(	(	PUNCT
ejpam-5460	207	22	0	0	NUM
ejpam-5460	207	23	,	,	PUNCT
ejpam-5460	207	24	1)(resp	1)(resp	NUM
ejpam-5460	207	25	.	.	PUNCT
ejpam-5460	207	26	,	,	PUNCT
ejpam-5460	207	27	(	(	PUNCT
ejpam-5460	207	28	1	1	NUM
ejpam-5460	207	29	,	,	PUNCT
ejpam-5460	207	30	0)-	0)-	NUM
ejpam-5460	207	31	,	,	PUNCT
ejpam-5460	207	32	(	(	PUNCT
ejpam-5460	207	33	1	1	NUM
ejpam-5460	207	34	,	,	PUNCT
ejpam-5460	207	35	1)-	1)-	NUM
ejpam-5460	207	36	,	,	PUNCT
ejpam-5460	207	37	1	1	NUM
ejpam-5460	207	38	-	-	PUNCT
ejpam-5460	207	39	interior	interior	ADJ
ejpam-5460	207	40	)	)	PUNCT
ejpam-5460	207	41	ideal	ideal	NOUN
ejpam-5460	207	42	(	(	PUNCT
ejpam-5460	207	43	see	see	VERB
ejpam-5460	207	44	[	[	X
ejpam-5460	207	45	19	19	NUM
ejpam-5460	207	46	,	,	PUNCT
ejpam-5460	207	47	21	21	NUM
ejpam-5460	207	48	,	,	PUNCT
ejpam-5460	207	49	22	22	NUM
ejpam-5460	207	50	]	]	PUNCT
ejpam-5460	207	51	)	)	PUNCT
ejpam-5460	207	52	.	.	PUNCT
ejpam-5460	208	1	any	any	DET
ejpam-5460	208	2	(	(	PUNCT
ejpam-5460	208	3	∈,∈	∈,∈	X
ejpam-5460	208	4	∨q)-fuzzy	∨q)-fuzzy	ADJ
ejpam-5460	208	5	left	left	ADJ
ejpam-5460	208	6	(	(	PUNCT
ejpam-5460	208	7	resp	resp	NOUN
ejpam-5460	208	8	.	.	PUNCT
ejpam-5460	208	9	,	,	PUNCT
ejpam-5460	208	10	right	right	INTJ
ejpam-5460	208	11	,	,	PUNCT
ejpam-5460	208	12	bi-	bi-	NUM
ejpam-5460	208	13	,	,	PUNCT
ejpam-5460	208	14	interior	interior	ADJ
ejpam-5460	208	15	)	)	PUNCT
ejpam-5460	208	16	ideal	ideal	NOUN
ejpam-5460	208	17	is	be	AUX
ejpam-5460	208	18	a	a	DET
ejpam-5460	208	19	(	(	PUNCT
ejpam-5460	208	20	0	0	NUM
ejpam-5460	208	21	,	,	PUNCT
ejpam-5460	208	22	0.5)-fuzzy	0.5)-fuzzy	NUM
ejpam-5460	208	23	(	(	PUNCT
ejpam-5460	208	24	0	0	NUM
ejpam-5460	208	25	,	,	PUNCT
ejpam-5460	208	26	1)(resp	1)(resp	NUM
ejpam-5460	208	27	.	.	PUNCT
ejpam-5460	208	28	,	,	PUNCT
ejpam-5460	208	29	(	(	PUNCT
ejpam-5460	208	30	1	1	NUM
ejpam-5460	208	31	,	,	PUNCT
ejpam-5460	208	32	0)-	0)-	NUM
ejpam-5460	208	33	,	,	PUNCT
ejpam-5460	208	34	(	(	PUNCT
ejpam-5460	208	35	1	1	NUM
ejpam-5460	208	36	,	,	PUNCT
ejpam-5460	208	37	1)-	1)-	NUM
ejpam-5460	208	38	,	,	PUNCT
ejpam-5460	208	39	1	1	NUM
ejpam-5460	208	40	-	-	PUNCT
ejpam-5460	208	41	interior	interior	ADJ
ejpam-5460	208	42	)	)	PUNCT
ejpam-5460	208	43	ideal	ideal	NOUN
ejpam-5460	208	44	(	(	PUNCT
ejpam-5460	208	45	see	see	VERB
ejpam-5460	208	46	[	[	X
ejpam-5460	208	47	11	11	NUM
ejpam-5460	208	48	,	,	PUNCT
ejpam-5460	208	49	31	31	NUM
ejpam-5460	208	50	]	]	PUNCT
ejpam-5460	208	51	)	)	PUNCT
ejpam-5460	208	52	.	.	PUNCT
ejpam-5460	209	1	any	any	DET
ejpam-5460	209	2	(	(	PUNCT
ejpam-5460	209	3	∈,∈	∈,∈	X
ejpam-5460	209	4	∨qk)-fuzzy	∨qk)-fuzzy	ADJ
ejpam-5460	209	5	left	left	ADJ
ejpam-5460	209	6	(	(	PUNCT
ejpam-5460	209	7	resp	resp	NOUN
ejpam-5460	209	8	.	.	PUNCT
ejpam-5460	209	9	,	,	PUNCT
ejpam-5460	209	10	right	right	INTJ
ejpam-5460	209	11	,	,	PUNCT
ejpam-5460	209	12	bi-	bi-	NUM
ejpam-5460	209	13	,	,	PUNCT
ejpam-5460	209	14	interior	interior	ADJ
ejpam-5460	209	15	)	)	PUNCT
ejpam-5460	209	16	ideal	ideal	NOUN
ejpam-5460	209	17	is	be	AUX
ejpam-5460	209	18	a	a	DET
ejpam-5460	209	19	(	(	PUNCT
ejpam-5460	209	20	0	0	NUM
ejpam-5460	209	21	,	,	PUNCT
ejpam-5460	209	22	1−k	1−k	NUM
ejpam-5460	209	23	2	2	NUM
ejpam-5460	209	24	)	)	PUNCT
ejpam-5460	209	25	-fuzzy	-fuzzy	NOUN
ejpam-5460	209	26	(	(	PUNCT
ejpam-5460	209	27	0	0	NUM
ejpam-5460	209	28	,	,	PUNCT
ejpam-5460	209	29	1)(resp	1)(resp	NUM
ejpam-5460	209	30	.	.	PUNCT
ejpam-5460	209	31	,	,	PUNCT
ejpam-5460	209	32	(	(	PUNCT
ejpam-5460	209	33	1	1	NUM
ejpam-5460	209	34	,	,	PUNCT
ejpam-5460	209	35	0)-	0)-	NUM
ejpam-5460	209	36	,	,	PUNCT
ejpam-5460	209	37	(	(	PUNCT
ejpam-5460	209	38	1	1	NUM
ejpam-5460	209	39	,	,	PUNCT
ejpam-5460	209	40	1)-	1)-	NUM
ejpam-5460	209	41	,	,	PUNCT
ejpam-5460	209	42	1	1	NUM
ejpam-5460	209	43	-	-	PUNCT
ejpam-5460	209	44	interior	interior	ADJ
ejpam-5460	209	45	)	)	PUNCT
ejpam-5460	209	46	ideal	ideal	NOUN
ejpam-5460	209	47	,	,	PUNCT
ejpam-5460	209	48	where	where	SCONJ
ejpam-5460	209	49	0	0	NUM
ejpam-5460	209	50	≤	≤	X
ejpam-5460	209	51	k	k	X
ejpam-5460	209	52	<	<	X
ejpam-5460	209	53	1	1	NUM
ejpam-5460	209	54	(	(	PUNCT
ejpam-5460	209	55	see	see	VERB
ejpam-5460	209	56	[	[	X
ejpam-5460	209	57	29	29	NUM
ejpam-5460	209	58	,	,	PUNCT
ejpam-5460	209	59	47	47	NUM
ejpam-5460	209	60	]	]	PUNCT
ejpam-5460	209	61	)	)	PUNCT
ejpam-5460	209	62	.	.	PUNCT
ejpam-5460	210	1	any	any	DET
ejpam-5460	210	2	(	(	PUNCT
ejpam-5460	210	3	∈,∈	∈,∈	X
ejpam-5460	210	4	∨(k∗	∨(k∗	ADJ
ejpam-5460	210	5	,	,	PUNCT
ejpam-5460	210	6	qk))-fuzzy	qk))-fuzzy	ADV
ejpam-5460	210	7	left	left	ADJ
ejpam-5460	210	8	(	(	PUNCT
ejpam-5460	210	9	resp	resp	NOUN
ejpam-5460	210	10	.	.	PUNCT
ejpam-5460	210	11	,	,	PUNCT
ejpam-5460	210	12	right	right	INTJ
ejpam-5460	210	13	,	,	PUNCT
ejpam-5460	210	14	bi-	bi-	NUM
ejpam-5460	210	15	,	,	PUNCT
ejpam-5460	210	16	interior	interior	ADJ
ejpam-5460	210	17	)	)	PUNCT
ejpam-5460	210	18	ideal	ideal	NOUN
ejpam-5460	210	19	is	be	AUX
ejpam-5460	210	20	a	a	DET
ejpam-5460	210	21	(	(	PUNCT
ejpam-5460	210	22	0	0	NUM
ejpam-5460	210	23	,	,	PUNCT
ejpam-5460	210	24	k	k	PROPN
ejpam-5460	210	25	∗−k	∗−k	PROPN
ejpam-5460	210	26	2	2	NUM
ejpam-5460	210	27	)	)	PUNCT
ejpam-5460	210	28	-fuzzy	-fuzzy	NOUN
ejpam-5460	210	29	(	(	PUNCT
ejpam-5460	210	30	0	0	NUM
ejpam-5460	210	31	,	,	PUNCT
ejpam-5460	210	32	1)(resp	1)(resp	NUM
ejpam-5460	210	33	.	.	PUNCT
ejpam-5460	210	34	,	,	PUNCT
ejpam-5460	210	35	(	(	PUNCT
ejpam-5460	210	36	1	1	NUM
ejpam-5460	210	37	,	,	PUNCT
ejpam-5460	210	38	0)-	0)-	NUM
ejpam-5460	210	39	,	,	PUNCT
ejpam-5460	210	40	(	(	PUNCT
ejpam-5460	210	41	1	1	NUM
ejpam-5460	210	42	,	,	PUNCT
ejpam-5460	210	43	1)-	1)-	NUM
ejpam-5460	210	44	,	,	PUNCT
ejpam-5460	210	45	1	1	NUM
ejpam-5460	210	46	-	-	PUNCT
ejpam-5460	210	47	interior	interior	ADJ
ejpam-5460	210	48	)	)	PUNCT
ejpam-5460	210	49	ideal	ideal	NOUN
ejpam-5460	210	50	,	,	PUNCT
ejpam-5460	210	51	where	where	SCONJ
ejpam-5460	210	52	0	0	X
ejpam-5460	210	53	<	<	X
ejpam-5460	210	54	k∗	k∗	VERB
ejpam-5460	210	55	≤	≤	NUM
ejpam-5460	210	56	1	1	NUM
ejpam-5460	210	57	and	and	CCONJ
ejpam-5460	210	58	0	0	NUM
ejpam-5460	210	59	≤	≤	NUM
ejpam-5460	211	1	k	k	X
ejpam-5460	211	2	<	<	X
ejpam-5460	211	3	1	1	NUM
ejpam-5460	211	4	(	(	PUNCT
ejpam-5460	211	5	see	see	VERB
ejpam-5460	211	6	[	[	X
ejpam-5460	211	7	33	33	NUM
ejpam-5460	211	8	,	,	PUNCT
ejpam-5460	211	9	34	34	NUM
ejpam-5460	211	10	]	]	PUNCT
ejpam-5460	211	11	)	)	PUNCT
ejpam-5460	211	12	.	.	PUNCT
ejpam-5460	212	1	any	any	DET
ejpam-5460	212	2	(	(	PUNCT
ejpam-5460	212	3	α	α	NOUN
ejpam-5460	212	4	,	,	PUNCT
ejpam-5460	212	5	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	212	6	left	leave	VERB
ejpam-5460	212	7	(	(	PUNCT
ejpam-5460	212	8	resp	resp	NOUN
ejpam-5460	212	9	.	.	PUNCT
ejpam-5460	212	10	,	,	PUNCT
ejpam-5460	212	11	right	right	INTJ
ejpam-5460	212	12	,	,	PUNCT
ejpam-5460	212	13	bi-	bi-	NUM
ejpam-5460	212	14	,	,	PUNCT
ejpam-5460	212	15	interior	interior	ADJ
ejpam-5460	212	16	)	)	PUNCT
ejpam-5460	212	17	ideal	ideal	NOUN
ejpam-5460	212	18	is	be	AUX
ejpam-5460	212	19	a	a	DET
ejpam-5460	212	20	(	(	PUNCT
ejpam-5460	212	21	α	α	NOUN
ejpam-5460	212	22	,	,	PUNCT
ejpam-5460	212	23	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	212	24	(	(	PUNCT
ejpam-5460	212	25	0	0	NUM
ejpam-5460	212	26	,	,	PUNCT
ejpam-5460	212	27	1)(resp	1)(resp	NUM
ejpam-5460	212	28	.	.	PUNCT
ejpam-5460	212	29	,	,	PUNCT
ejpam-5460	212	30	(	(	PUNCT
ejpam-5460	212	31	1	1	NUM
ejpam-5460	212	32	,	,	PUNCT
ejpam-5460	212	33	0)-	0)-	NUM
ejpam-5460	212	34	,	,	PUNCT
ejpam-5460	212	35	(	(	PUNCT
ejpam-5460	212	36	1	1	NUM
ejpam-5460	212	37	,	,	PUNCT
ejpam-5460	212	38	1)-	1)-	NUM
ejpam-5460	212	39	,	,	PUNCT
ejpam-5460	212	40	1	1	NUM
ejpam-5460	212	41	-	-	PUNCT
ejpam-5460	212	42	interior	interior	ADJ
ejpam-5460	212	43	)	)	PUNCT
ejpam-5460	212	44	ideal	ideal	NOUN
ejpam-5460	212	45	(	(	PUNCT
ejpam-5460	212	46	see	see	VERB
ejpam-5460	212	47	[	[	X
ejpam-5460	212	48	6	6	NUM
ejpam-5460	212	49	,	,	PUNCT
ejpam-5460	212	50	7	7	NUM
ejpam-5460	212	51	,	,	PUNCT
ejpam-5460	212	52	30	30	NUM
ejpam-5460	212	53	]	]	PUNCT
ejpam-5460	212	54	)	)	PUNCT
ejpam-5460	212	55	.	.	PUNCT
ejpam-5460	213	1	example	example	NOUN
ejpam-5460	213	2	2	2	NUM
ejpam-5460	213	3	(	(	PUNCT
ejpam-5460	213	4	[	[	X
ejpam-5460	213	5	4	4	NUM
ejpam-5460	213	6	]	]	NUM
ejpam-5460	213	7	)	)	PUNCT
ejpam-5460	213	8	.	.	PUNCT
ejpam-5460	214	1	let	let	VERB
ejpam-5460	214	2	s	s	VERB
ejpam-5460	214	3	=	=	X
ejpam-5460	214	4	{	{	PUNCT
ejpam-5460	214	5	0	0	NUM
ejpam-5460	214	6	,	,	PUNCT
ejpam-5460	214	7	1	1	NUM
ejpam-5460	214	8	,	,	PUNCT
ejpam-5460	214	9	2	2	NUM
ejpam-5460	214	10	,	,	PUNCT
ejpam-5460	214	11	3	3	NUM
ejpam-5460	214	12	,	,	PUNCT
ejpam-5460	214	13	4	4	NUM
ejpam-5460	214	14	}	}	PUNCT
ejpam-5460	214	15	.	.	PUNCT
ejpam-5460	215	1	we	we	PRON
ejpam-5460	215	2	define	define	VERB
ejpam-5460	215	3	a	a	DET
ejpam-5460	215	4	binary	binary	ADJ
ejpam-5460	215	5	operation	operation	NOUN
ejpam-5460	215	6	·	·	PUNCT
ejpam-5460	215	7	and	and	CCONJ
ejpam-5460	215	8	a	a	DET
ejpam-5460	215	9	partial	partial	ADJ
ejpam-5460	215	10	order	order	NOUN
ejpam-5460	215	11	≤	≤	X
ejpam-5460	215	12	on	on	ADP
ejpam-5460	215	13	s	s	PRON
ejpam-5460	215	14	as	as	SCONJ
ejpam-5460	215	15	follows	follow	VERB
ejpam-5460	215	16	.	.	PUNCT
ejpam-5460	216	1	·	·	PUNCT
ejpam-5460	216	2	0	0	NUM
ejpam-5460	217	1	1	1	NUM
ejpam-5460	217	2	2	2	NUM
ejpam-5460	217	3	3	3	NUM
ejpam-5460	217	4	4	4	NUM
ejpam-5460	217	5	0	0	NUM
ejpam-5460	217	6	0	0	NUM
ejpam-5460	217	7	0	0	NUM
ejpam-5460	217	8	0	0	NUM
ejpam-5460	217	9	0	0	NUM
ejpam-5460	217	10	0	0	NUM
ejpam-5460	217	11	1	1	NUM
ejpam-5460	217	12	0	0	NUM
ejpam-5460	217	13	0	0	NUM
ejpam-5460	217	14	0	0	NUM
ejpam-5460	217	15	0	0	NUM
ejpam-5460	217	16	3	3	NUM
ejpam-5460	217	17	2	2	NUM
ejpam-5460	217	18	0	0	NUM
ejpam-5460	217	19	0	0	NUM
ejpam-5460	217	20	0	0	NUM
ejpam-5460	217	21	0	0	NUM
ejpam-5460	217	22	3	3	NUM
ejpam-5460	217	23	3	3	NUM
ejpam-5460	217	24	3	3	NUM
ejpam-5460	217	25	3	3	NUM
ejpam-5460	217	26	3	3	NUM
ejpam-5460	217	27	3	3	NUM
ejpam-5460	217	28	3	3	NUM
ejpam-5460	217	29	4	4	NUM
ejpam-5460	217	30	4	4	NUM
ejpam-5460	217	31	4	4	NUM
ejpam-5460	217	32	4	4	NUM
ejpam-5460	217	33	4	4	NUM
ejpam-5460	217	34	4	4	NUM
ejpam-5460	217	35	and	and	CCONJ
ejpam-5460	217	36	≤	≤	NUM
ejpam-5460	217	37	:	:	PUNCT
ejpam-5460	217	38	=	=	SYM
ejpam-5460	217	39	{	{	PUNCT
ejpam-5460	217	40	(	(	PUNCT
ejpam-5460	217	41	0	0	NUM
ejpam-5460	217	42	,	,	PUNCT
ejpam-5460	217	43	3	3	NUM
ejpam-5460	217	44	)	)	PUNCT
ejpam-5460	217	45	}	}	PUNCT
ejpam-5460	217	46	∪∆s	∪∆	NOUN
ejpam-5460	217	47	.	.	PUNCT
ejpam-5460	218	1	then	then	ADV
ejpam-5460	218	2	,	,	PUNCT
ejpam-5460	218	3	s	s	X
ejpam-5460	218	4	:	:	PUNCT
ejpam-5460	218	5	=	=	SYM
ejpam-5460	218	6	⟨s	⟨s	NOUN
ejpam-5460	218	7	;	;	PUNCT
ejpam-5460	218	8	·	·	PUNCT
ejpam-5460	218	9	,	,	PUNCT
ejpam-5460	218	10	≤⟩	≤⟩	VERB
ejpam-5460	218	11	is	be	AUX
ejpam-5460	218	12	an	an	DET
ejpam-5460	218	13	ordered	order	VERB
ejpam-5460	218	14	semigroup	semigroup	NOUN
ejpam-5460	218	15	.	.	PUNCT
ejpam-5460	219	1	define	define	VERB
ejpam-5460	219	2	a	a	DET
ejpam-5460	219	3	fuzzy	fuzzy	ADJ
ejpam-5460	219	4	set	set	NOUN
ejpam-5460	219	5	f	f	PROPN
ejpam-5460	219	6	in	in	ADP
ejpam-5460	219	7	s	s	PRON
ejpam-5460	219	8	by	by	ADP
ejpam-5460	219	9	f(0	f(0	NOUN
ejpam-5460	219	10	)	)	PUNCT
ejpam-5460	219	11	=	=	SYM
ejpam-5460	219	12	0.9	0.9	NUM
ejpam-5460	219	13	,	,	PUNCT
ejpam-5460	219	14	f(1	f(1	PROPN
ejpam-5460	219	15	)	)	PUNCT
ejpam-5460	219	16	=	=	SYM
ejpam-5460	219	17	1	1	NUM
ejpam-5460	219	18	,	,	PUNCT
ejpam-5460	219	19	f(2	f(2	PROPN
ejpam-5460	219	20	)	)	PUNCT
ejpam-5460	219	21	=	=	NUM
ejpam-5460	219	22	0.2	0.2	NUM
ejpam-5460	219	23	,	,	PUNCT
ejpam-5460	219	24	f(3	f(3	PROPN
ejpam-5460	219	25	)	)	PUNCT
ejpam-5460	219	26	=	=	SYM
ejpam-5460	219	27	0	0	NUM
ejpam-5460	219	28	and	and	CCONJ
ejpam-5460	219	29	f(4	f(4	PROPN
ejpam-5460	219	30	)	)	PUNCT
ejpam-5460	219	31	=	=	NOUN
ejpam-5460	219	32	0.2	0.2	NUM
ejpam-5460	219	33	.	.	PUNCT
ejpam-5460	220	1	we	we	PRON
ejpam-5460	220	2	can	can	AUX
ejpam-5460	220	3	calculate	calculate	VERB
ejpam-5460	220	4	that	that	SCONJ
ejpam-5460	220	5	f	f	PROPN
ejpam-5460	220	6	is	be	AUX
ejpam-5460	220	7	a	a	DET
ejpam-5460	220	8	(	(	PUNCT
ejpam-5460	220	9	0.3	0.3	NUM
ejpam-5460	220	10	,	,	PUNCT
ejpam-5460	220	11	0.6)-fuzzy	0.6)-fuzzy	NUM
ejpam-5460	220	12	(	(	PUNCT
ejpam-5460	220	13	2	2	NUM
ejpam-5460	220	14	,	,	PUNCT
ejpam-5460	220	15	2)-ideal	2)-ideal	NUM
ejpam-5460	220	16	of	of	ADP
ejpam-5460	220	17	s	s	PROPN
ejpam-5460	220	18	,	,	PUNCT
ejpam-5460	220	19	but	but	CCONJ
ejpam-5460	220	20	f	f	PROPN
ejpam-5460	220	21	is	be	AUX
ejpam-5460	220	22	not	not	PART
ejpam-5460	220	23	a	a	DET
ejpam-5460	220	24	fuzzy	fuzzy	ADJ
ejpam-5460	220	25	(	(	PUNCT
ejpam-5460	220	26	2	2	NUM
ejpam-5460	220	27	,	,	PUNCT
ejpam-5460	220	28	2)-ideal	2)-ideal	NUM
ejpam-5460	220	29	of	of	ADP
ejpam-5460	220	30	s.	s.	PROPN
ejpam-5460	220	31	example	example	PROPN
ejpam-5460	220	32	3	3	NUM
ejpam-5460	220	33	(	(	PUNCT
ejpam-5460	220	34	[	[	X
ejpam-5460	220	35	4	4	NUM
ejpam-5460	220	36	]	]	NUM
ejpam-5460	220	37	)	)	PUNCT
ejpam-5460	220	38	.	.	PUNCT
ejpam-5460	221	1	let	let	VERB
ejpam-5460	221	2	s	s	VERB
ejpam-5460	221	3	=	=	X
ejpam-5460	221	4	{	{	PUNCT
ejpam-5460	221	5	0	0	NUM
ejpam-5460	221	6	,	,	PUNCT
ejpam-5460	221	7	1	1	NUM
ejpam-5460	221	8	,	,	PUNCT
ejpam-5460	221	9	2	2	NUM
ejpam-5460	221	10	,	,	PUNCT
ejpam-5460	221	11	3	3	NUM
ejpam-5460	221	12	,	,	PUNCT
ejpam-5460	221	13	4	4	NUM
ejpam-5460	221	14	,	,	PUNCT
ejpam-5460	221	15	5	5	NUM
ejpam-5460	221	16	}	}	PUNCT
ejpam-5460	221	17	.	.	PUNCT
ejpam-5460	222	1	we	we	PRON
ejpam-5460	222	2	define	define	VERB
ejpam-5460	222	3	a	a	DET
ejpam-5460	222	4	binary	binary	ADJ
ejpam-5460	222	5	operation	operation	NOUN
ejpam-5460	222	6	·	·	PUNCT
ejpam-5460	222	7	and	and	CCONJ
ejpam-5460	222	8	a	a	DET
ejpam-5460	222	9	partial	partial	ADJ
ejpam-5460	222	10	order	order	NOUN
ejpam-5460	222	11	≤	≤	X
ejpam-5460	222	12	on	on	ADP
ejpam-5460	222	13	s	s	PRON
ejpam-5460	222	14	as	as	SCONJ
ejpam-5460	222	15	follows	follow	VERB
ejpam-5460	222	16	.	.	PUNCT
ejpam-5460	223	1	·	·	PUNCT
ejpam-5460	223	2	0	0	NUM
ejpam-5460	224	1	1	1	NUM
ejpam-5460	224	2	2	2	NUM
ejpam-5460	224	3	3	3	NUM
ejpam-5460	224	4	4	4	NUM
ejpam-5460	224	5	5	5	NUM
ejpam-5460	224	6	0	0	NUM
ejpam-5460	224	7	0	0	NUM
ejpam-5460	224	8	0	0	NUM
ejpam-5460	224	9	0	0	NUM
ejpam-5460	224	10	0	0	NUM
ejpam-5460	224	11	0	0	NUM
ejpam-5460	224	12	0	0	NUM
ejpam-5460	224	13	1	1	NUM
ejpam-5460	224	14	0	0	NUM
ejpam-5460	224	15	0	0	NUM
ejpam-5460	224	16	0	0	NUM
ejpam-5460	224	17	0	0	NUM
ejpam-5460	224	18	3	3	NUM
ejpam-5460	224	19	1	1	NUM
ejpam-5460	224	20	2	2	NUM
ejpam-5460	224	21	0	0	NUM
ejpam-5460	224	22	0	0	NUM
ejpam-5460	224	23	0	0	NUM
ejpam-5460	224	24	0	0	NUM
ejpam-5460	224	25	3	3	NUM
ejpam-5460	224	26	1	1	NUM
ejpam-5460	224	27	3	3	NUM
ejpam-5460	224	28	0	0	NUM
ejpam-5460	224	29	0	0	NUM
ejpam-5460	224	30	0	0	NUM
ejpam-5460	224	31	0	0	NUM
ejpam-5460	224	32	0	0	NUM
ejpam-5460	224	33	3	3	NUM
ejpam-5460	224	34	4	4	NUM
ejpam-5460	224	35	0	0	NUM
ejpam-5460	224	36	3	3	NUM
ejpam-5460	224	37	3	3	NUM
ejpam-5460	224	38	0	0	NUM
ejpam-5460	224	39	0	0	NUM
ejpam-5460	224	40	3	3	NUM
ejpam-5460	224	41	5	5	NUM
ejpam-5460	224	42	0	0	NUM
ejpam-5460	224	43	3	3	NUM
ejpam-5460	224	44	3	3	NUM
ejpam-5460	224	45	3	3	NUM
ejpam-5460	224	46	4	4	NUM
ejpam-5460	224	47	5	5	NUM
ejpam-5460	224	48	and	and	CCONJ
ejpam-5460	224	49	≤	≤	NUM
ejpam-5460	224	50	:	:	PUNCT
ejpam-5460	225	1	=	=	SYM
ejpam-5460	225	2	{	{	PUNCT
ejpam-5460	225	3	(	(	PUNCT
ejpam-5460	225	4	0	0	NUM
ejpam-5460	225	5	,	,	PUNCT
ejpam-5460	225	6	3	3	NUM
ejpam-5460	225	7	)	)	PUNCT
ejpam-5460	225	8	}	}	PUNCT
ejpam-5460	225	9	∪∆s	∪∆	NOUN
ejpam-5460	225	10	.	.	PUNCT
ejpam-5460	226	1	then	then	ADV
ejpam-5460	226	2	,	,	PUNCT
ejpam-5460	226	3	s	s	X
ejpam-5460	226	4	:	:	PUNCT
ejpam-5460	226	5	=	=	SYM
ejpam-5460	226	6	⟨s	⟨s	NOUN
ejpam-5460	226	7	;	;	PUNCT
ejpam-5460	226	8	·	·	PUNCT
ejpam-5460	226	9	,	,	PUNCT
ejpam-5460	226	10	≤⟩	≤⟩	VERB
ejpam-5460	226	11	is	be	AUX
ejpam-5460	226	12	an	an	DET
ejpam-5460	226	13	ordered	order	VERB
ejpam-5460	226	14	semigroup	semigroup	NOUN
ejpam-5460	226	15	.	.	PUNCT
ejpam-5460	227	1	define	define	VERB
ejpam-5460	227	2	a	a	DET
ejpam-5460	227	3	fuzzy	fuzzy	ADJ
ejpam-5460	227	4	set	set	NOUN
ejpam-5460	227	5	f	f	PROPN
ejpam-5460	227	6	in	in	ADP
ejpam-5460	227	7	s	s	PRON
ejpam-5460	227	8	by	by	ADP
ejpam-5460	227	9	f(0	f(0	NOUN
ejpam-5460	227	10	)	)	PUNCT
ejpam-5460	227	11	=	=	SYM
ejpam-5460	227	12	0.9	0.9	NUM
ejpam-5460	227	13	,	,	PUNCT
ejpam-5460	227	14	f(1	f(1	PROPN
ejpam-5460	227	15	)	)	PUNCT
ejpam-5460	227	16	=	=	SYM
ejpam-5460	227	17	0	0	NUM
ejpam-5460	227	18	,	,	PUNCT
ejpam-5460	227	19	f(2	f(2	PROPN
ejpam-5460	227	20	)	)	PUNCT
ejpam-5460	227	21	=	=	SYM
ejpam-5460	227	22	0.4	0.4	NUM
ejpam-5460	227	23	,	,	PUNCT
ejpam-5460	227	24	f(3	f(3	PROPN
ejpam-5460	227	25	)	)	PUNCT
ejpam-5460	227	26	=	=	SYM
ejpam-5460	227	27	0	0	NUM
ejpam-5460	227	28	,	,	PUNCT
ejpam-5460	227	29	f(4	f(4	PROPN
ejpam-5460	227	30	)	)	PUNCT
ejpam-5460	227	31	=	=	PUNCT
ejpam-5460	227	32	0.8	0.8	NUM
ejpam-5460	227	33	and	and	CCONJ
ejpam-5460	227	34	f(5	f(5	PROPN
ejpam-5460	227	35	)	)	PUNCT
ejpam-5460	228	1	=	=	NOUN
ejpam-5460	228	2	0.2	0.2	NUM
ejpam-5460	228	3	.	.	PUNCT
ejpam-5460	229	1	we	we	PRON
ejpam-5460	229	2	can	can	AUX
ejpam-5460	229	3	calculate	calculate	VERB
ejpam-5460	229	4	that	that	SCONJ
ejpam-5460	229	5	f	f	PROPN
ejpam-5460	229	6	is	be	AUX
ejpam-5460	229	7	a	a	DET
ejpam-5460	229	8	(	(	PUNCT
ejpam-5460	229	9	0.4	0.4	NUM
ejpam-5460	229	10	,	,	PUNCT
ejpam-5460	229	11	0.6)-fuzzy	0.6)-fuzzy	NUM
ejpam-5460	229	12	2	2	NUM
ejpam-5460	229	13	-	-	PUNCT
ejpam-5460	229	14	interior	interior	ADJ
ejpam-5460	229	15	ideal	ideal	NOUN
ejpam-5460	229	16	of	of	ADP
ejpam-5460	229	17	s	s	PROPN
ejpam-5460	229	18	,	,	PUNCT
ejpam-5460	229	19	but	but	CCONJ
ejpam-5460	229	20	f	f	PROPN
ejpam-5460	229	21	is	be	AUX
ejpam-5460	229	22	not	not	PART
ejpam-5460	229	23	a	a	DET
ejpam-5460	229	24	fuzzy	fuzzy	ADJ
ejpam-5460	229	25	2	2	NUM
ejpam-5460	229	26	-	-	PUNCT
ejpam-5460	229	27	interior	interior	ADJ
ejpam-5460	229	28	ideal	ideal	NOUN
ejpam-5460	229	29	of	of	ADP
ejpam-5460	229	30	s.	s.	PROPN
ejpam-5460	229	31	s.	s.	PROPN
ejpam-5460	229	32	lekkoksung	lekkoksung	PROPN
ejpam-5460	229	33	,	,	PUNCT
ejpam-5460	229	34	b.	b.	PROPN
ejpam-5460	229	35	davvaz	davvaz	PROPN
ejpam-5460	229	36	,	,	PUNCT
ejpam-5460	229	37	n.	n.	PROPN
ejpam-5460	229	38	lekkoksung	lekkoksung	PROPN
ejpam-5460	229	39	/	/	SYM
ejpam-5460	229	40	eur	eur	PROPN
ejpam-5460	229	41	.	.	PUNCT
ejpam-5460	230	1	j.	j.	PROPN
ejpam-5460	230	2	pure	pure	PROPN
ejpam-5460	230	3	appl	appl	PROPN
ejpam-5460	230	4	.	.	PROPN
ejpam-5460	230	5	math	math	PROPN
ejpam-5460	230	6	,	,	PUNCT
ejpam-5460	230	7	17	17	NUM
ejpam-5460	230	8	(	(	PUNCT
ejpam-5460	230	9	4	4	NUM
ejpam-5460	230	10	)	)	PUNCT
ejpam-5460	230	11	(	(	PUNCT
ejpam-5460	230	12	2024	2024	NUM
ejpam-5460	230	13	)	)	PUNCT
ejpam-5460	230	14	,	,	PUNCT
ejpam-5460	230	15	2962	2962	NUM
ejpam-5460	230	16	-	-	SYM
ejpam-5460	230	17	2984	2984	NUM
ejpam-5460	230	18	2970	2970	NUM
ejpam-5460	230	19	the	the	DET
ejpam-5460	230	20	above	above	ADJ
ejpam-5460	230	21	examples	example	NOUN
ejpam-5460	230	22	demonstrate	demonstrate	VERB
ejpam-5460	230	23	the	the	DET
ejpam-5460	230	24	distinct	distinct	ADJ
ejpam-5460	230	25	concepts	concept	NOUN
ejpam-5460	230	26	of	of	ADP
ejpam-5460	230	27	(	(	PUNCT
ejpam-5460	230	28	α	α	NOUN
ejpam-5460	230	29	,	,	PUNCT
ejpam-5460	230	30	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	230	31	(	(	PUNCT
ejpam-5460	230	32	m	m	NOUN
ejpam-5460	230	33	,	,	PUNCT
ejpam-5460	230	34	n)-ideals	n)-ideal	NOUN
ejpam-5460	230	35	and	and	CCONJ
ejpam-5460	230	36	(	(	PUNCT
ejpam-5460	230	37	α	α	NOUN
ejpam-5460	230	38	,	,	PUNCT
ejpam-5460	230	39	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	230	40	n	n	CCONJ
ejpam-5460	230	41	-	-	PUNCT
ejpam-5460	230	42	interior	interior	ADJ
ejpam-5460	230	43	ideals	ideal	NOUN
ejpam-5460	230	44	whenever	whenever	SCONJ
ejpam-5460	230	45	α	α	X
ejpam-5460	230	46	,	,	PUNCT
ejpam-5460	230	47	β	β	X
ejpam-5460	230	48	,	,	PUNCT
ejpam-5460	230	49	m	m	PROPN
ejpam-5460	230	50	,	,	PUNCT
ejpam-5460	230	51	and	and	CCONJ
ejpam-5460	230	52	n	n	PRON
ejpam-5460	230	53	differ	differ	VERB
ejpam-5460	230	54	in	in	ADP
ejpam-5460	230	55	ordered	order	VERB
ejpam-5460	230	56	semigroups	semigroup	NOUN
ejpam-5460	230	57	.	.	PUNCT
ejpam-5460	231	1	remark	remark	PROPN
ejpam-5460	231	2	1	1	NUM
ejpam-5460	231	3	.	.	PUNCT
ejpam-5460	232	1	by	by	ADP
ejpam-5460	232	2	the	the	DET
ejpam-5460	232	3	property	property	NOUN
ejpam-5460	232	4	of	of	ADP
ejpam-5460	232	5	fuzzy	fuzzy	ADJ
ejpam-5460	232	6	sets	set	NOUN
ejpam-5460	232	7	with	with	ADP
ejpam-5460	232	8	restricted	restricted	ADJ
ejpam-5460	232	9	range	range	NOUN
ejpam-5460	232	10	i	i	PRON
ejpam-5460	232	11	,	,	PUNCT
ejpam-5460	232	12	we	we	PRON
ejpam-5460	232	13	observe	observe	VERB
ejpam-5460	232	14	that	that	SCONJ
ejpam-5460	232	15	:	:	PUNCT
ejpam-5460	232	16	(	(	PUNCT
ejpam-5460	232	17	i	i	NOUN
ejpam-5460	232	18	)	)	PUNCT
ejpam-5460	232	19	if	if	SCONJ
ejpam-5460	232	20	f	f	PROPN
ejpam-5460	232	21	is	be	AUX
ejpam-5460	232	22	an	an	DET
ejpam-5460	232	23	(	(	PUNCT
ejpam-5460	232	24	α	α	NOUN
ejpam-5460	232	25	,	,	PUNCT
ejpam-5460	232	26	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	232	27	(	(	PUNCT
ejpam-5460	232	28	m	m	PROPN
ejpam-5460	232	29	,	,	PUNCT
ejpam-5460	232	30	n)-ideal	n)-ideal	NOUN
ejpam-5460	232	31	of	of	ADP
ejpam-5460	232	32	s	s	PROPN
ejpam-5460	232	33	,	,	PUNCT
ejpam-5460	232	34	then	then	ADV
ejpam-5460	232	35	fi(x	fi(x	NUM
ejpam-5460	232	36	m	m	PROPN
ejpam-5460	232	37	1	1	NUM
ejpam-5460	232	38	yzn1	yzn1	NOUN
ejpam-5460	232	39	)	)	PUNCT
ejpam-5460	232	40	≥	≥	PROPN
ejpam-5460	232	41	(	(	PUNCT
ejpam-5460	232	42	m∧	m∧	PROPN
ejpam-5460	232	43	i=1	i=1	PROPN
ejpam-5460	232	44	fi(xi	fi(xi	PROPN
ejpam-5460	232	45	)	)	PUNCT
ejpam-5460	232	46	)	)	PUNCT
ejpam-5460	233	1	∧	∧	NOUN
ejpam-5460	233	2	(	(	PUNCT
ejpam-5460	233	3	n∧	n∧	NUM
ejpam-5460	233	4	i=1	i=1	PROPN
ejpam-5460	233	5	fi(zi	fi(zi	PROPN
ejpam-5460	233	6	)	)	PUNCT
ejpam-5460	233	7	)	)	PUNCT
ejpam-5460	233	8	for	for	ADP
ejpam-5460	233	9	any	any	DET
ejpam-5460	233	10	x1	x1	PROPN
ejpam-5460	233	11	,	,	PUNCT
ejpam-5460	233	12	.	.	PUNCT
ejpam-5460	233	13	.	.	PUNCT
ejpam-5460	233	14	.	.	PUNCT
ejpam-5460	234	1	,	,	PUNCT
ejpam-5460	234	2	xm	xm	PROPN
ejpam-5460	234	3	,	,	PUNCT
ejpam-5460	234	4	y	y	PROPN
ejpam-5460	234	5	,	,	PUNCT
ejpam-5460	234	6	z1	z1	PROPN
ejpam-5460	234	7	,	,	PUNCT
ejpam-5460	234	8	.	.	PUNCT
ejpam-5460	234	9	.	.	PUNCT
ejpam-5460	234	10	.	.	PUNCT
ejpam-5460	235	1	,	,	PUNCT
ejpam-5460	235	2	zn	zn	PROPN
ejpam-5460	235	3	∈	∈	PROPN
ejpam-5460	235	4	s	s	PART
ejpam-5460	235	5	;	;	PUNCT
ejpam-5460	235	6	(	(	PUNCT
ejpam-5460	235	7	ii	ii	NOUN
ejpam-5460	235	8	)	)	PUNCT
ejpam-5460	235	9	if	if	SCONJ
ejpam-5460	235	10	f	f	PROPN
ejpam-5460	235	11	is	be	AUX
ejpam-5460	235	12	an	an	DET
ejpam-5460	235	13	(	(	PUNCT
ejpam-5460	235	14	α	α	NOUN
ejpam-5460	235	15	,	,	PUNCT
ejpam-5460	235	16	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	235	17	n	n	CCONJ
ejpam-5460	235	18	-	-	PUNCT
ejpam-5460	235	19	interior	interior	ADJ
ejpam-5460	235	20	ideal	ideal	NOUN
ejpam-5460	235	21	of	of	ADP
ejpam-5460	235	22	s	s	PROPN
ejpam-5460	235	23	,	,	PUNCT
ejpam-5460	235	24	then	then	ADV
ejpam-5460	235	25	fi(xy	fi(xy	PROPN
ejpam-5460	235	26	n	n	PROPN
ejpam-5460	235	27	1	1	NUM
ejpam-5460	235	28	z	z	NOUN
ejpam-5460	235	29	)	)	PUNCT
ejpam-5460	235	30	≥	≥	PROPN
ejpam-5460	235	31	n∧	n∧	NUM
ejpam-5460	235	32	i=1	i=1	PROPN
ejpam-5460	235	33	fi(yi	fi(yi	PROPN
ejpam-5460	235	34	)	)	PUNCT
ejpam-5460	235	35	for	for	ADP
ejpam-5460	235	36	any	any	DET
ejpam-5460	235	37	x	x	NOUN
ejpam-5460	235	38	,	,	PUNCT
ejpam-5460	235	39	y1	y1	NOUN
ejpam-5460	235	40	,	,	PUNCT
ejpam-5460	235	41	.	.	PUNCT
ejpam-5460	235	42	.	.	PUNCT
ejpam-5460	235	43	.	.	PUNCT
ejpam-5460	236	1	,	,	PUNCT
ejpam-5460	236	2	yn	yn	PROPN
ejpam-5460	236	3	,	,	PUNCT
ejpam-5460	236	4	z	z	PROPN
ejpam-5460	236	5	∈	∈	PROPN
ejpam-5460	236	6	s.	s.	PROPN
ejpam-5460	236	7	the	the	DET
ejpam-5460	236	8	subsequent	subsequent	ADJ
ejpam-5460	236	9	result	result	NOUN
ejpam-5460	236	10	is	be	AUX
ejpam-5460	236	11	essential	essential	ADJ
ejpam-5460	236	12	for	for	ADP
ejpam-5460	236	13	categorizing	categorize	VERB
ejpam-5460	236	14	ordered	order	VERB
ejpam-5460	236	15	semigroups	semigroup	NOUN
ejpam-5460	236	16	into	into	ADP
ejpam-5460	236	17	classes	class	NOUN
ejpam-5460	236	18	.	.	PUNCT
ejpam-5460	237	1	proposition	proposition	NOUN
ejpam-5460	237	2	2	2	NUM
ejpam-5460	237	3	.	.	PUNCT
ejpam-5460	238	1	let	let	VERB
ejpam-5460	238	2	s	s	PRON
ejpam-5460	238	3	be	be	AUX
ejpam-5460	238	4	an	an	DET
ejpam-5460	238	5	ordered	order	VERB
ejpam-5460	238	6	semigroup	semigroup	NOUN
ejpam-5460	238	7	,	,	PUNCT
ejpam-5460	238	8	a	a	PRON
ejpam-5460	238	9	and	and	CCONJ
ejpam-5460	238	10	b	b	NOUN
ejpam-5460	238	11	subsets	subset	NOUN
ejpam-5460	238	12	of	of	ADP
ejpam-5460	238	13	s.	s.	PROPN
ejpam-5460	238	14	then	then	ADV
ejpam-5460	238	15	,	,	PUNCT
ejpam-5460	238	16	the	the	DET
ejpam-5460	238	17	following	follow	VERB
ejpam-5460	238	18	statements	statement	NOUN
ejpam-5460	238	19	hold	hold	VERB
ejpam-5460	238	20	.	.	PUNCT
ejpam-5460	239	1	(	(	PUNCT
ejpam-5460	239	2	i	i	NOUN
ejpam-5460	239	3	)	)	PUNCT
ejpam-5460	239	4	(	(	PUNCT
ejpam-5460	239	5	χa)i	χa)i	PROPN
ejpam-5460	239	6	⊆	⊆	NUM
ejpam-5460	239	7	(	(	PUNCT
ejpam-5460	239	8	χb)i	χb)i	PROPN
ejpam-5460	239	9	if	if	SCONJ
ejpam-5460	239	10	and	and	CCONJ
ejpam-5460	239	11	only	only	ADV
ejpam-5460	239	12	if	if	SCONJ
ejpam-5460	239	13	a	a	DET
ejpam-5460	239	14	⊆	⊆	NUM
ejpam-5460	239	15	b.	b.	PROPN
ejpam-5460	239	16	(	(	PUNCT
ejpam-5460	239	17	ii	ii	PROPN
ejpam-5460	239	18	)	)	PUNCT
ejpam-5460	239	19	χa	χa	PROPN
ejpam-5460	239	20	◦	◦	NOUN
ejpam-5460	239	21	i	i	PRON
ejpam-5460	239	22	χb	χb	VERB
ejpam-5460	239	23	=	=	PUNCT
ejpam-5460	239	24	(	(	PUNCT
ejpam-5460	239	25	χ(ab])i	χ(ab])i	ADV
ejpam-5460	239	26	.	.	PUNCT
ejpam-5460	240	1	(	(	PUNCT
ejpam-5460	240	2	iii	iii	X
ejpam-5460	240	3	)	)	PUNCT
ejpam-5460	240	4	χa	χa	NOUN
ejpam-5460	240	5	∩i	∩i	ADV
ejpam-5460	240	6	χb	χb	PROPN
ejpam-5460	241	1	=	=	SYM
ejpam-5460	241	2	(	(	PUNCT
ejpam-5460	241	3	χa∩b)i	χa∩b)i	NOUN
ejpam-5460	241	4	.	.	PUNCT
ejpam-5460	242	1	(	(	PUNCT
ejpam-5460	242	2	iv	iv	X
ejpam-5460	242	3	)	)	PUNCT
ejpam-5460	242	4	(	(	PUNCT
ejpam-5460	242	5	χa)i	χa)i	PROPN
ejpam-5460	242	6	is	be	AUX
ejpam-5460	242	7	an	an	DET
ejpam-5460	242	8	(	(	PUNCT
ejpam-5460	242	9	α	α	NOUN
ejpam-5460	242	10	,	,	PUNCT
ejpam-5460	242	11	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	242	12	(	(	PUNCT
ejpam-5460	242	13	m	m	PROPN
ejpam-5460	242	14	,	,	PUNCT
ejpam-5460	242	15	n)(resp	n)(resp	PROPN
ejpam-5460	242	16	.	.	PROPN
ejpam-5460	242	17	,	,	PUNCT
ejpam-5460	242	18	n	n	CCONJ
ejpam-5460	242	19	-	-	ADJ
ejpam-5460	242	20	interior	interior	ADJ
ejpam-5460	242	21	ideal	ideal	NOUN
ejpam-5460	242	22	)	)	PUNCT
ejpam-5460	242	23	ideal	ideal	NOUN
ejpam-5460	242	24	of	of	ADP
ejpam-5460	242	25	s	s	PRON
ejpam-5460	242	26	if	if	SCONJ
ejpam-5460	242	27	and	and	CCONJ
ejpam-5460	242	28	only	only	ADV
ejpam-5460	242	29	if	if	SCONJ
ejpam-5460	242	30	a	a	PRON
ejpam-5460	242	31	is	be	AUX
ejpam-5460	242	32	an	an	DET
ejpam-5460	242	33	(	(	PUNCT
ejpam-5460	242	34	m	m	PROPN
ejpam-5460	242	35	,	,	PUNCT
ejpam-5460	242	36	n)(resp	n)(resp	PROPN
ejpam-5460	242	37	.	.	PROPN
ejpam-5460	242	38	,	,	PUNCT
ejpam-5460	242	39	n	n	CCONJ
ejpam-5460	242	40	-	-	ADJ
ejpam-5460	242	41	interior	interior	ADJ
ejpam-5460	242	42	ideal	ideal	NOUN
ejpam-5460	242	43	)	)	PUNCT
ejpam-5460	242	44	ideal	ideal	NOUN
ejpam-5460	242	45	of	of	ADP
ejpam-5460	242	46	s.	s.	PROPN
ejpam-5460	242	47	proof	proof	PROPN
ejpam-5460	242	48	.	.	PUNCT
ejpam-5460	243	1	(	(	PUNCT
ejpam-5460	243	2	1	1	NUM
ejpam-5460	243	3	)	)	PUNCT
ejpam-5460	243	4	.	.	PUNCT
ejpam-5460	244	1	let	let	VERB
ejpam-5460	244	2	x	x	PUNCT
ejpam-5460	244	3	∈	∈	VERB
ejpam-5460	244	4	a.	a.	NOUN
ejpam-5460	244	5	then	then	ADV
ejpam-5460	244	6	,	,	PUNCT
ejpam-5460	244	7	β	β	X
ejpam-5460	244	8	=	=	SYM
ejpam-5460	244	9	(	(	PUNCT
ejpam-5460	244	10	χa)i(x	χa)i(x	X
ejpam-5460	244	11	)	)	PUNCT
ejpam-5460	244	12	≤	≤	NOUN
ejpam-5460	244	13	(	(	PUNCT
ejpam-5460	244	14	χb)i(x	χb)i(x	NOUN
ejpam-5460	244	15	)	)	PUNCT
ejpam-5460	244	16	≤	≤	NOUN
ejpam-5460	245	1	β	β	X
ejpam-5460	245	2	.	.	PUNCT
ejpam-5460	246	1	this	this	PRON
ejpam-5460	246	2	means	mean	VERB
ejpam-5460	246	3	that	that	SCONJ
ejpam-5460	246	4	x	x	PROPN
ejpam-5460	246	5	∈	∈	PROPN
ejpam-5460	246	6	b.	b.	PROPN
ejpam-5460	246	7	on	on	ADP
ejpam-5460	246	8	the	the	DET
ejpam-5460	246	9	other	other	ADJ
ejpam-5460	246	10	hand	hand	NOUN
ejpam-5460	246	11	,	,	PUNCT
ejpam-5460	246	12	let	let	VERB
ejpam-5460	246	13	x	x	PROPN
ejpam-5460	246	14	∈	∈	PROPN
ejpam-5460	246	15	s.	s.	PROPN
ejpam-5460	246	16	if	if	SCONJ
ejpam-5460	246	17	x	x	PROPN
ejpam-5460	246	18	̸∈	̸∈	PROPN
ejpam-5460	246	19	a	a	PROPN
ejpam-5460	246	20	,	,	PUNCT
ejpam-5460	246	21	then	then	ADV
ejpam-5460	246	22	(	(	PUNCT
ejpam-5460	246	23	χa)i(x	χa)i(x	X
ejpam-5460	246	24	)	)	PUNCT
ejpam-5460	246	25	=	=	PUNCT
ejpam-5460	246	26	α	α	X
ejpam-5460	246	27	≤	≤	NOUN
ejpam-5460	246	28	(	(	PUNCT
ejpam-5460	246	29	χb)i(x	χb)i(x	NOUN
ejpam-5460	246	30	)	)	PUNCT
ejpam-5460	246	31	.	.	PUNCT
ejpam-5460	247	1	if	if	SCONJ
ejpam-5460	247	2	x	x	SYM
ejpam-5460	247	3	∈	∈	PROPN
ejpam-5460	247	4	a	a	DET
ejpam-5460	247	5	,	,	PUNCT
ejpam-5460	247	6	then	then	ADV
ejpam-5460	247	7	x	x	PART
ejpam-5460	247	8	∈	∈	PROPN
ejpam-5460	247	9	b.	b.	NOUN
ejpam-5460	247	10	this	this	PRON
ejpam-5460	247	11	implies	imply	VERB
ejpam-5460	247	12	(	(	PUNCT
ejpam-5460	247	13	χa)i(x	χa)i(x	X
ejpam-5460	247	14	)	)	PUNCT
ejpam-5460	247	15	=	=	NOUN
ejpam-5460	248	1	β	β	X
ejpam-5460	248	2	=	=	SYM
ejpam-5460	248	3	(	(	PUNCT
ejpam-5460	248	4	χb)i(x	χb)i(x	NOUN
ejpam-5460	248	5	)	)	PUNCT
ejpam-5460	248	6	.	.	PUNCT
ejpam-5460	249	1	(	(	PUNCT
ejpam-5460	249	2	2	2	NUM
ejpam-5460	249	3	)	)	PUNCT
ejpam-5460	249	4	.	.	PUNCT
ejpam-5460	250	1	it	it	PRON
ejpam-5460	250	2	was	be	AUX
ejpam-5460	250	3	shown	show	VERB
ejpam-5460	250	4	in	in	ADP
ejpam-5460	250	5	[	[	X
ejpam-5460	250	6	26	26	NUM
ejpam-5460	250	7	,	,	PUNCT
ejpam-5460	250	8	lemma	lemma	PROPN
ejpam-5460	250	9	2.5	2.5	NUM
ejpam-5460	250	10	]	]	PUNCT
ejpam-5460	250	11	that	that	PRON
ejpam-5460	250	12	χa	χa	AUX
ejpam-5460	250	13	◦	◦	VERB
ejpam-5460	250	14	χb	χb	ADV
ejpam-5460	250	15	=	=	SYM
ejpam-5460	250	16	χ(ab	χ(ab	PROPN
ejpam-5460	250	17	]	]	PUNCT
ejpam-5460	250	18	.	.	PUNCT
ejpam-5460	251	1	thus	thus	ADV
ejpam-5460	251	2	,	,	PUNCT
ejpam-5460	251	3	we	we	PRON
ejpam-5460	251	4	have	have	VERB
ejpam-5460	251	5	χa	χa	ADP
ejpam-5460	251	6	◦	◦	NOUN
ejpam-5460	252	1	i	i	PRON
ejpam-5460	252	2	χb	χb	VERB
ejpam-5460	252	3	=	=	PUNCT
ejpam-5460	252	4	(	(	PUNCT
ejpam-5460	252	5	χa	χa	NOUN
ejpam-5460	252	6	◦	◦	VERB
ejpam-5460	252	7	χb)i	χb)i	PROPN
ejpam-5460	252	8	=	=	SYM
ejpam-5460	252	9	(	(	PUNCT
ejpam-5460	252	10	χ(ab])i	χ(ab])i	ADV
ejpam-5460	252	11	.	.	PUNCT
ejpam-5460	253	1	(	(	PUNCT
ejpam-5460	253	2	3	3	NUM
ejpam-5460	253	3	)	)	PUNCT
ejpam-5460	253	4	.	.	PUNCT
ejpam-5460	254	1	in	in	ADP
ejpam-5460	254	2	[	[	X
ejpam-5460	254	3	24	24	NUM
ejpam-5460	254	4	,	,	PUNCT
ejpam-5460	254	5	proposition	proposition	NOUN
ejpam-5460	254	6	9	9	NUM
ejpam-5460	254	7	]	]	PUNCT
ejpam-5460	254	8	,	,	PUNCT
ejpam-5460	254	9	the	the	DET
ejpam-5460	254	10	authors	author	NOUN
ejpam-5460	254	11	illustrated	illustrate	VERB
ejpam-5460	254	12	that	that	SCONJ
ejpam-5460	254	13	χa	χa	ADP
ejpam-5460	254	14	∩	∩	NOUN
ejpam-5460	254	15	χb	χb	ADP
ejpam-5460	254	16	=	=	SYM
ejpam-5460	254	17	χa∩b	χa∩b	PROPN
ejpam-5460	254	18	.	.	PUNCT
ejpam-5460	255	1	then	then	ADV
ejpam-5460	255	2	,	,	PUNCT
ejpam-5460	255	3	we	we	PRON
ejpam-5460	255	4	obtain	obtain	VERB
ejpam-5460	255	5	χa	χa	ADP
ejpam-5460	255	6	∩i	∩i	NOUN
ejpam-5460	255	7	χb	χb	PROPN
ejpam-5460	256	1	=	=	PUNCT
ejpam-5460	256	2	(	(	PUNCT
ejpam-5460	256	3	χa	χa	PROPN
ejpam-5460	256	4	∩	∩	ADJ
ejpam-5460	256	5	χb)i	χb)i	PROPN
ejpam-5460	256	6	=	=	SYM
ejpam-5460	256	7	(	(	PUNCT
ejpam-5460	256	8	χa∩b)i	χa∩b)i	NOUN
ejpam-5460	256	9	as	as	SCONJ
ejpam-5460	256	10	required	require	VERB
ejpam-5460	256	11	.	.	PUNCT
ejpam-5460	257	1	(	(	PUNCT
ejpam-5460	257	2	4	4	NUM
ejpam-5460	257	3	)	)	PUNCT
ejpam-5460	257	4	.	.	PUNCT
ejpam-5460	258	1	the	the	DET
ejpam-5460	258	2	proof	proof	NOUN
ejpam-5460	258	3	can	can	AUX
ejpam-5460	258	4	be	be	AUX
ejpam-5460	258	5	found	find	VERB
ejpam-5460	258	6	in	in	ADP
ejpam-5460	258	7	[	[	X
ejpam-5460	258	8	4	4	NUM
ejpam-5460	258	9	]	]	PUNCT
ejpam-5460	258	10	.	.	PUNCT
ejpam-5460	259	1	by	by	ADP
ejpam-5460	259	2	the	the	DET
ejpam-5460	259	3	associativity	associativity	NOUN
ejpam-5460	259	4	of	of	ADP
ejpam-5460	259	5	◦	◦	NOUN
ejpam-5460	259	6	i	i	PROPN
ejpam-5460	259	7	and	and	CCONJ
ejpam-5460	259	8	lemma	lemma	PROPN
ejpam-5460	259	9	1	1	NUM
ejpam-5460	259	10	,	,	PUNCT
ejpam-5460	259	11	proposition	proposition	NOUN
ejpam-5460	259	12	2(ii	2(ii	NUM
ejpam-5460	259	13	)	)	PUNCT
ejpam-5460	259	14	and	and	CCONJ
ejpam-5460	259	15	2(iii	2(iii	NUM
ejpam-5460	259	16	)	)	PUNCT
ejpam-5460	259	17	can	can	AUX
ejpam-5460	259	18	be	be	AUX
ejpam-5460	259	19	extended	extend	VERB
ejpam-5460	259	20	as	as	SCONJ
ejpam-5460	259	21	follows	follow	VERB
ejpam-5460	259	22	.	.	PUNCT
ejpam-5460	260	1	corollary	corollary	ADJ
ejpam-5460	260	2	2	2	NUM
ejpam-5460	260	3	.	.	PUNCT
ejpam-5460	261	1	let	let	VERB
ejpam-5460	261	2	s	s	PRON
ejpam-5460	261	3	be	be	AUX
ejpam-5460	261	4	an	an	DET
ejpam-5460	261	5	ordered	order	VERB
ejpam-5460	261	6	semigroup	semigroup	NOUN
ejpam-5460	261	7	,	,	PUNCT
ejpam-5460	261	8	and	and	CCONJ
ejpam-5460	261	9	a1	a1	NOUN
ejpam-5460	261	10	,	,	PUNCT
ejpam-5460	261	11	.	.	PUNCT
ejpam-5460	261	12	.	.	PUNCT
ejpam-5460	262	1	.	.	PUNCT
ejpam-5460	263	1	,	,	PUNCT
ejpam-5460	263	2	an	an	DET
ejpam-5460	263	3	subsets	subset	NOUN
ejpam-5460	263	4	of	of	ADP
ejpam-5460	263	5	s.	s.	PROPN
ejpam-5460	263	6	then	then	ADV
ejpam-5460	263	7	,	,	PUNCT
ejpam-5460	263	8	we	we	PRON
ejpam-5460	263	9	have	have	VERB
ejpam-5460	263	10	χa1	χa1	NOUN
ejpam-5460	263	11	◦	◦	VERB
ejpam-5460	263	12	i	i	NOUN
ejpam-5460	263	13	·	·	PUNCT
ejpam-5460	263	14	·	·	PUNCT
ejpam-5460	263	15	·	·	PUNCT
ejpam-5460	264	1	◦	◦	NOUN
ejpam-5460	264	2	i	i	PRON
ejpam-5460	264	3	χan	χan	NOUN
ejpam-5460	264	4	=	=	SYM
ejpam-5460	264	5	(	(	PUNCT
ejpam-5460	264	6	χ(a1···an])i	χ(a1···an])i	PROPN
ejpam-5460	264	7	and	and	CCONJ
ejpam-5460	264	8	χa1	χa1	NOUN
ejpam-5460	264	9	∩i	∩i	PROPN
ejpam-5460	264	10	·	·	PUNCT
ejpam-5460	264	11	·	·	PUNCT
ejpam-5460	264	12	·	·	PUNCT
ejpam-5460	264	13	∩i	∩i	NOUN
ejpam-5460	264	14	χan	χan	NOUN
ejpam-5460	264	15	=	=	SYM
ejpam-5460	264	16	(	(	PUNCT
ejpam-5460	264	17	χa1∩···∩an)i	χa1∩···∩an)i	PROPN
ejpam-5460	264	18	.	.	PUNCT
ejpam-5460	265	1	s.	s.	PROPN
ejpam-5460	265	2	lekkoksung	lekkoksung	PROPN
ejpam-5460	265	3	,	,	PUNCT
ejpam-5460	265	4	b.	b.	PROPN
ejpam-5460	265	5	davvaz	davvaz	PROPN
ejpam-5460	265	6	,	,	PUNCT
ejpam-5460	265	7	n.	n.	PROPN
ejpam-5460	265	8	lekkoksung	lekkoksung	PROPN
ejpam-5460	265	9	/	/	SYM
ejpam-5460	265	10	eur	eur	PROPN
ejpam-5460	265	11	.	.	PUNCT
ejpam-5460	266	1	j.	j.	PROPN
ejpam-5460	266	2	pure	pure	PROPN
ejpam-5460	266	3	appl	appl	PROPN
ejpam-5460	266	4	.	.	PROPN
ejpam-5460	266	5	math	math	PROPN
ejpam-5460	266	6	,	,	PUNCT
ejpam-5460	266	7	17	17	NUM
ejpam-5460	266	8	(	(	PUNCT
ejpam-5460	266	9	4	4	NUM
ejpam-5460	266	10	)	)	PUNCT
ejpam-5460	266	11	(	(	PUNCT
ejpam-5460	266	12	2024	2024	NUM
ejpam-5460	266	13	)	)	PUNCT
ejpam-5460	266	14	,	,	PUNCT
ejpam-5460	266	15	2962	2962	NUM
ejpam-5460	266	16	-	-	SYM
ejpam-5460	266	17	2984	2984	NUM
ejpam-5460	266	18	2971	2971	NUM
ejpam-5460	266	19	4	4	NUM
ejpam-5460	266	20	.	.	PUNCT
ejpam-5460	266	21	main	main	ADJ
ejpam-5460	266	22	results	result	NOUN
ejpam-5460	266	23	in	in	ADP
ejpam-5460	266	24	this	this	DET
ejpam-5460	266	25	section	section	NOUN
ejpam-5460	267	1	,	,	PUNCT
ejpam-5460	267	2	we	we	PRON
ejpam-5460	267	3	delineate	delineate	VERB
ejpam-5460	267	4	the	the	DET
ejpam-5460	267	5	classification	classification	NOUN
ejpam-5460	267	6	of	of	ADP
ejpam-5460	267	7	ordered	order	VERB
ejpam-5460	267	8	semigroups	semigroup	NOUN
ejpam-5460	267	9	based	base	VERB
ejpam-5460	267	10	on	on	ADP
ejpam-5460	267	11	their	their	PRON
ejpam-5460	267	12	regularities	regularity	NOUN
ejpam-5460	267	13	,	,	PUNCT
ejpam-5460	267	14	employing	employ	VERB
ejpam-5460	267	15	the	the	DET
ejpam-5460	267	16	concepts	concept	NOUN
ejpam-5460	267	17	of	of	ADP
ejpam-5460	267	18	(	(	PUNCT
ejpam-5460	267	19	α	α	NOUN
ejpam-5460	267	20	,	,	PUNCT
ejpam-5460	267	21	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	267	22	(	(	PUNCT
ejpam-5460	267	23	m	m	NOUN
ejpam-5460	267	24	,	,	PUNCT
ejpam-5460	267	25	n)-ideals	n)-ideal	NOUN
ejpam-5460	267	26	and	and	CCONJ
ejpam-5460	267	27	(	(	PUNCT
ejpam-5460	267	28	α	α	NOUN
ejpam-5460	267	29	,	,	PUNCT
ejpam-5460	267	30	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	267	31	n	n	CCONJ
ejpam-5460	267	32	-	-	PUNCT
ejpam-5460	267	33	interior	interior	ADJ
ejpam-5460	267	34	ideals	ideal	NOUN
ejpam-5460	267	35	.	.	PUNCT
ejpam-5460	268	1	according	accord	VERB
ejpam-5460	268	2	to	to	ADP
ejpam-5460	268	3	their	their	PRON
ejpam-5460	268	4	regularities	regularity	NOUN
ejpam-5460	268	5	,	,	PUNCT
ejpam-5460	268	6	ordered	order	VERB
ejpam-5460	268	7	semigroups	semigroup	NOUN
ejpam-5460	268	8	can	can	AUX
ejpam-5460	268	9	be	be	AUX
ejpam-5460	268	10	categorized	categorize	VERB
ejpam-5460	268	11	into	into	ADP
ejpam-5460	268	12	16	16	NUM
ejpam-5460	268	13	classes	class	NOUN
ejpam-5460	268	14	,	,	PUNCT
ejpam-5460	268	15	as	as	SCONJ
ejpam-5460	268	16	outlined	outline	VERB
ejpam-5460	268	17	below	below	ADV
ejpam-5460	268	18	.	.	PUNCT
ejpam-5460	269	1	an	an	DET
ejpam-5460	269	2	ordered	order	VERB
ejpam-5460	269	3	semigroup	semigroup	NOUN
ejpam-5460	269	4	s	s	PART
ejpam-5460	269	5	satisfies	satisfie	NOUN
ejpam-5460	269	6	:	:	PUNCT
ejpam-5460	269	7	(	(	PUNCT
ejpam-5460	269	8	c1	c1	NOUN
ejpam-5460	269	9	)	)	PUNCT
ejpam-5460	269	10	if	if	SCONJ
ejpam-5460	269	11	a	a	DET
ejpam-5460	269	12	⊆	⊆	NUM
ejpam-5460	269	13	(	(	PUNCT
ejpam-5460	269	14	sas	sas	PROPN
ejpam-5460	269	15	]	]	PUNCT
ejpam-5460	269	16	for	for	ADP
ejpam-5460	269	17	all	all	DET
ejpam-5460	269	18	∅	∅	NOUN
ejpam-5460	269	19	=	=	NOUN
ejpam-5460	269	20	̸	̸	NUM
ejpam-5460	269	21	a	a	DET
ejpam-5460	269	22	⊆	⊆	NUM
ejpam-5460	269	23	s	s	NOUN
ejpam-5460	269	24	;	;	PUNCT
ejpam-5460	269	25	(	(	PUNCT
ejpam-5460	269	26	c2	c2	PROPN
ejpam-5460	269	27	)	)	PUNCT
ejpam-5460	269	28	if	if	SCONJ
ejpam-5460	269	29	a	a	DET
ejpam-5460	269	30	⊆	⊆	NUM
ejpam-5460	269	31	(	(	PUNCT
ejpam-5460	269	32	sa	sa	NOUN
ejpam-5460	269	33	]	]	PUNCT
ejpam-5460	269	34	for	for	ADP
ejpam-5460	269	35	all	all	DET
ejpam-5460	269	36	∅	∅	NOUN
ejpam-5460	269	37	=	=	NOUN
ejpam-5460	269	38	̸	̸	NUM
ejpam-5460	269	39	a	a	DET
ejpam-5460	269	40	⊆	⊆	NUM
ejpam-5460	269	41	s	s	NOUN
ejpam-5460	269	42	;	;	PUNCT
ejpam-5460	269	43	(	(	PUNCT
ejpam-5460	269	44	c3	c3	NOUN
ejpam-5460	269	45	)	)	PUNCT
ejpam-5460	269	46	if	if	SCONJ
ejpam-5460	269	47	a	a	DET
ejpam-5460	269	48	⊆	⊆	NUM
ejpam-5460	269	49	(	(	PUNCT
ejpam-5460	269	50	as	as	ADP
ejpam-5460	269	51	]	]	PUNCT
ejpam-5460	269	52	for	for	ADP
ejpam-5460	269	53	all	all	DET
ejpam-5460	269	54	∅	∅	NOUN
ejpam-5460	269	55	=	=	NOUN
ejpam-5460	269	56	̸	̸	NUM
ejpam-5460	269	57	a	a	DET
ejpam-5460	269	58	⊆	⊆	NUM
ejpam-5460	269	59	s	s	NOUN
ejpam-5460	269	60	;	;	PUNCT
ejpam-5460	269	61	(	(	PUNCT
ejpam-5460	269	62	c4	c4	NOUN
ejpam-5460	269	63	)	)	PUNCT
ejpam-5460	269	64	if	if	SCONJ
ejpam-5460	269	65	a	a	DET
ejpam-5460	269	66	⊆	⊆	NUM
ejpam-5460	269	67	(	(	PUNCT
ejpam-5460	269	68	sasas	sasa	NOUN
ejpam-5460	269	69	]	]	PUNCT
ejpam-5460	269	70	for	for	ADP
ejpam-5460	269	71	all	all	DET
ejpam-5460	269	72	∅	∅	NOUN
ejpam-5460	269	73	=	=	NOUN
ejpam-5460	269	74	̸	̸	NUM
ejpam-5460	269	75	a	a	DET
ejpam-5460	269	76	⊆	⊆	NUM
ejpam-5460	269	77	s	s	NOUN
ejpam-5460	269	78	;	;	PUNCT
ejpam-5460	269	79	(	(	PUNCT
ejpam-5460	269	80	c5	c5	PROPN
ejpam-5460	269	81	)	)	PUNCT
ejpam-5460	269	82	if	if	SCONJ
ejpam-5460	269	83	a	a	DET
ejpam-5460	269	84	⊆	⊆	NUM
ejpam-5460	269	85	(	(	PUNCT
ejpam-5460	269	86	sasa	sasa	PROPN
ejpam-5460	269	87	]	]	PUNCT
ejpam-5460	269	88	for	for	ADP
ejpam-5460	269	89	all	all	DET
ejpam-5460	269	90	∅	∅	NOUN
ejpam-5460	269	91	=	=	NOUN
ejpam-5460	269	92	̸	̸	NUM
ejpam-5460	269	93	a	a	DET
ejpam-5460	269	94	⊆	⊆	NUM
ejpam-5460	269	95	s	s	NOUN
ejpam-5460	269	96	;	;	PUNCT
ejpam-5460	269	97	(	(	PUNCT
ejpam-5460	269	98	c6	c6	PROPN
ejpam-5460	269	99	)	)	PUNCT
ejpam-5460	269	100	if	if	SCONJ
ejpam-5460	269	101	a	a	DET
ejpam-5460	269	102	⊆	⊆	NUM
ejpam-5460	269	103	(	(	PUNCT
ejpam-5460	269	104	asas	asa	NOUN
ejpam-5460	269	105	]	]	PUNCT
ejpam-5460	269	106	for	for	ADP
ejpam-5460	269	107	all	all	DET
ejpam-5460	269	108	∅	∅	NOUN
ejpam-5460	269	109	=	=	NOUN
ejpam-5460	269	110	̸	̸	NUM
ejpam-5460	269	111	a	a	DET
ejpam-5460	269	112	⊆	⊆	NUM
ejpam-5460	269	113	s	s	NOUN
ejpam-5460	269	114	;	;	PUNCT
ejpam-5460	269	115	(	(	PUNCT
ejpam-5460	269	116	c7	c7	PROPN
ejpam-5460	269	117	)	)	PUNCT
ejpam-5460	269	118	if	if	SCONJ
ejpam-5460	269	119	a	a	DET
ejpam-5460	269	120	⊆	⊆	NUM
ejpam-5460	269	121	(	(	PUNCT
ejpam-5460	269	122	asa	asa	PROPN
ejpam-5460	269	123	]	]	PUNCT
ejpam-5460	269	124	for	for	ADP
ejpam-5460	269	125	all	all	DET
ejpam-5460	269	126	∅	∅	NOUN
ejpam-5460	269	127	=	=	NOUN
ejpam-5460	269	128	̸	̸	NUM
ejpam-5460	269	129	a	a	DET
ejpam-5460	269	130	⊆	⊆	NUM
ejpam-5460	269	131	s	s	NOUN
ejpam-5460	269	132	;	;	PUNCT
ejpam-5460	269	133	(	(	PUNCT
ejpam-5460	269	134	c8	c8	PROPN
ejpam-5460	269	135	)	)	PUNCT
ejpam-5460	269	136	of	of	ADP
ejpam-5460	269	137	degree	degree	NOUN
ejpam-5460	269	138	n	n	NOUN
ejpam-5460	269	139	if	if	SCONJ
ejpam-5460	269	140	a	a	DET
ejpam-5460	269	141	⊆	⊆	NUM
ejpam-5460	269	142	(	(	PUNCT
ejpam-5460	269	143	sans	san	NOUN
ejpam-5460	269	144	]	]	PUNCT
ejpam-5460	269	145	for	for	ADP
ejpam-5460	269	146	all	all	DET
ejpam-5460	269	147	∅	∅	NOUN
ejpam-5460	269	148	=	=	NOUN
ejpam-5460	269	149	̸	̸	NUM
ejpam-5460	269	150	a	a	DET
ejpam-5460	269	151	⊆	⊆	NUM
ejpam-5460	269	152	s	s	NOUN
ejpam-5460	269	153	and	and	CCONJ
ejpam-5460	269	154	n	n	PRON
ejpam-5460	269	155	∈	∈	PROPN
ejpam-5460	269	156	n∖	n∖	X
ejpam-5460	269	157	{	{	PUNCT
ejpam-5460	269	158	1	1	NUM
ejpam-5460	269	159	}	}	PUNCT
ejpam-5460	269	160	;	;	PUNCT
ejpam-5460	269	161	(	(	PUNCT
ejpam-5460	269	162	c9	c9	NOUN
ejpam-5460	269	163	)	)	PUNCT
ejpam-5460	269	164	of	of	ADP
ejpam-5460	269	165	degree	degree	NOUN
ejpam-5460	269	166	n	n	NOUN
ejpam-5460	269	167	if	if	SCONJ
ejpam-5460	269	168	a	a	DET
ejpam-5460	269	169	⊆	⊆	NUM
ejpam-5460	269	170	(	(	PUNCT
ejpam-5460	269	171	sansa	sansa	PROPN
ejpam-5460	269	172	]	]	PUNCT
ejpam-5460	269	173	for	for	ADP
ejpam-5460	269	174	all	all	DET
ejpam-5460	269	175	∅	∅	NOUN
ejpam-5460	269	176	=	=	NOUN
ejpam-5460	269	177	̸	̸	NUM
ejpam-5460	269	178	a	a	DET
ejpam-5460	269	179	⊆	⊆	NUM
ejpam-5460	269	180	s	s	NOUN
ejpam-5460	269	181	and	and	CCONJ
ejpam-5460	269	182	n	n	PRON
ejpam-5460	269	183	∈	∈	PROPN
ejpam-5460	269	184	n∖	n∖	X
ejpam-5460	269	185	{	{	PUNCT
ejpam-5460	269	186	1	1	NUM
ejpam-5460	269	187	}	}	PUNCT
ejpam-5460	269	188	;	;	PUNCT
ejpam-5460	269	189	(	(	PUNCT
ejpam-5460	269	190	c10	c10	PROPN
ejpam-5460	269	191	)	)	PUNCT
ejpam-5460	269	192	of	of	ADP
ejpam-5460	269	193	degree	degree	NOUN
ejpam-5460	269	194	n	n	NOUN
ejpam-5460	269	195	if	if	SCONJ
ejpam-5460	269	196	a	a	DET
ejpam-5460	269	197	⊆	⊆	NUM
ejpam-5460	269	198	(	(	PUNCT
ejpam-5460	269	199	asans	asan	NOUN
ejpam-5460	269	200	]	]	PUNCT
ejpam-5460	269	201	for	for	ADP
ejpam-5460	269	202	all	all	DET
ejpam-5460	269	203	∅	∅	NOUN
ejpam-5460	269	204	=	=	NOUN
ejpam-5460	269	205	̸	̸	NUM
ejpam-5460	269	206	a	a	DET
ejpam-5460	269	207	⊆	⊆	NUM
ejpam-5460	269	208	s	s	NOUN
ejpam-5460	269	209	and	and	CCONJ
ejpam-5460	269	210	n	n	PRON
ejpam-5460	269	211	∈	∈	PROPN
ejpam-5460	269	212	n∖	n∖	X
ejpam-5460	269	213	{	{	PUNCT
ejpam-5460	269	214	1	1	NUM
ejpam-5460	269	215	}	}	PUNCT
ejpam-5460	269	216	;	;	PUNCT
ejpam-5460	269	217	(	(	PUNCT
ejpam-5460	269	218	c11	c11	NOUN
ejpam-5460	269	219	)	)	PUNCT
ejpam-5460	269	220	of	of	ADP
ejpam-5460	269	221	degree	degree	NOUN
ejpam-5460	269	222	n	n	NOUN
ejpam-5460	269	223	if	if	SCONJ
ejpam-5460	269	224	a	a	DET
ejpam-5460	269	225	⊆	⊆	NUM
ejpam-5460	269	226	(	(	PUNCT
ejpam-5460	269	227	asansa	asansa	PROPN
ejpam-5460	269	228	]	]	PUNCT
ejpam-5460	269	229	for	for	ADP
ejpam-5460	269	230	all	all	DET
ejpam-5460	269	231	∅	∅	NOUN
ejpam-5460	269	232	=	=	NOUN
ejpam-5460	269	233	̸	̸	NUM
ejpam-5460	269	234	a	a	DET
ejpam-5460	269	235	⊆	⊆	NUM
ejpam-5460	269	236	s	s	NOUN
ejpam-5460	269	237	and	and	CCONJ
ejpam-5460	269	238	n	n	PRON
ejpam-5460	269	239	∈	∈	PROPN
ejpam-5460	269	240	n∖	n∖	X
ejpam-5460	269	241	{	{	PUNCT
ejpam-5460	269	242	1	1	NUM
ejpam-5460	269	243	}	}	PUNCT
ejpam-5460	269	244	;	;	PUNCT
ejpam-5460	269	245	(	(	PUNCT
ejpam-5460	269	246	c12	c12	PROPN
ejpam-5460	269	247	)	)	PUNCT
ejpam-5460	269	248	if	if	SCONJ
ejpam-5460	269	249	a	a	DET
ejpam-5460	269	250	⊆	⊆	NUM
ejpam-5460	269	251	(	(	PUNCT
ejpam-5460	269	252	sa2	sa2	PROPN
ejpam-5460	269	253	]	]	PUNCT
ejpam-5460	269	254	for	for	ADP
ejpam-5460	269	255	all	all	DET
ejpam-5460	269	256	∅	∅	NOUN
ejpam-5460	269	257	=	=	NOUN
ejpam-5460	269	258	̸	̸	NUM
ejpam-5460	269	259	a	a	DET
ejpam-5460	269	260	⊆	⊆	NUM
ejpam-5460	269	261	s	s	NOUN
ejpam-5460	269	262	;	;	PUNCT
ejpam-5460	269	263	(	(	PUNCT
ejpam-5460	269	264	c13	c13	X
ejpam-5460	269	265	)	)	PUNCT
ejpam-5460	269	266	if	if	SCONJ
ejpam-5460	269	267	a	a	DET
ejpam-5460	269	268	⊆	⊆	NUM
ejpam-5460	269	269	(	(	PUNCT
ejpam-5460	269	270	a2s	a2s	NOUN
ejpam-5460	269	271	]	]	PUNCT
ejpam-5460	269	272	for	for	ADP
ejpam-5460	269	273	all	all	DET
ejpam-5460	269	274	∅	∅	NOUN
ejpam-5460	269	275	=	=	NOUN
ejpam-5460	269	276	̸	̸	NUM
ejpam-5460	269	277	a	a	DET
ejpam-5460	269	278	⊆	⊆	NUM
ejpam-5460	269	279	s	s	NOUN
ejpam-5460	269	280	;	;	PUNCT
ejpam-5460	269	281	(	(	PUNCT
ejpam-5460	269	282	c14	c14	NOUN
ejpam-5460	269	283	)	)	PUNCT
ejpam-5460	269	284	if	if	SCONJ
ejpam-5460	269	285	a	a	DET
ejpam-5460	269	286	⊆	⊆	NUM
ejpam-5460	269	287	(	(	PUNCT
ejpam-5460	269	288	a2sa2	a2sa2	NOUN
ejpam-5460	269	289	]	]	PUNCT
ejpam-5460	269	290	for	for	ADP
ejpam-5460	269	291	all	all	DET
ejpam-5460	269	292	∅	∅	NOUN
ejpam-5460	269	293	=	=	NOUN
ejpam-5460	269	294	̸	̸	NUM
ejpam-5460	269	295	a	a	DET
ejpam-5460	269	296	⊆	⊆	NUM
ejpam-5460	269	297	s	s	NOUN
ejpam-5460	269	298	;	;	PUNCT
ejpam-5460	269	299	(	(	PUNCT
ejpam-5460	269	300	c15	c15	NOUN
ejpam-5460	269	301	)	)	PUNCT
ejpam-5460	269	302	if	if	SCONJ
ejpam-5460	269	303	a	a	DET
ejpam-5460	269	304	⊆	⊆	NUM
ejpam-5460	269	305	(	(	PUNCT
ejpam-5460	269	306	asa2	asa2	PROPN
ejpam-5460	269	307	]	]	PUNCT
ejpam-5460	269	308	for	for	ADP
ejpam-5460	269	309	all	all	DET
ejpam-5460	269	310	∅	∅	NOUN
ejpam-5460	269	311	=	=	NOUN
ejpam-5460	269	312	̸	̸	NUM
ejpam-5460	269	313	a	a	DET
ejpam-5460	269	314	⊆	⊆	NUM
ejpam-5460	269	315	s	s	NOUN
ejpam-5460	269	316	;	;	PUNCT
ejpam-5460	269	317	(	(	PUNCT
ejpam-5460	269	318	c16	c16	NOUN
ejpam-5460	269	319	)	)	PUNCT
ejpam-5460	269	320	if	if	SCONJ
ejpam-5460	269	321	a	a	DET
ejpam-5460	269	322	⊆	⊆	NUM
ejpam-5460	269	323	(	(	PUNCT
ejpam-5460	269	324	a2sa	a2sa	PROPN
ejpam-5460	269	325	]	]	X
ejpam-5460	269	326	for	for	ADP
ejpam-5460	269	327	all	all	DET
ejpam-5460	269	328	∅	∅	NOUN
ejpam-5460	269	329	=	=	NOUN
ejpam-5460	269	330	̸	̸	NUM
ejpam-5460	269	331	a	a	DET
ejpam-5460	269	332	⊆	⊆	NUM
ejpam-5460	269	333	s.	s.	NOUN
ejpam-5460	269	334	the	the	DET
ejpam-5460	269	335	relation	relation	NOUN
ejpam-5460	269	336	of	of	ADP
ejpam-5460	269	337	such	such	ADJ
ejpam-5460	269	338	classes	class	NOUN
ejpam-5460	269	339	of	of	ADP
ejpam-5460	269	340	ordered	order	VERB
ejpam-5460	269	341	semigroups	semigroup	NOUN
ejpam-5460	269	342	can	can	AUX
ejpam-5460	269	343	be	be	AUX
ejpam-5460	269	344	represented	represent	VERB
ejpam-5460	269	345	by	by	ADP
ejpam-5460	269	346	figure	figure	NOUN
ejpam-5460	269	347	1	1	NUM
ejpam-5460	269	348	.	.	PUNCT
ejpam-5460	269	349	example	example	NOUN
ejpam-5460	269	350	4	4	NUM
ejpam-5460	269	351	.	.	PUNCT
ejpam-5460	269	352	by	by	ADP
ejpam-5460	269	353	example	example	NOUN
ejpam-5460	269	354	1	1	NUM
ejpam-5460	269	355	,	,	PUNCT
ejpam-5460	269	356	2	2	NUM
ejpam-5460	269	357	and	and	CCONJ
ejpam-5460	269	358	3	3	NUM
ejpam-5460	269	359	,	,	PUNCT
ejpam-5460	269	360	we	we	PRON
ejpam-5460	269	361	can	can	AUX
ejpam-5460	269	362	see	see	VERB
ejpam-5460	269	363	that	that	PRON
ejpam-5460	269	364	s	s	VERB
ejpam-5460	269	365	does	do	AUX
ejpam-5460	269	366	not	not	PART
ejpam-5460	269	367	satisfy	satisfy	VERB
ejpam-5460	269	368	(	(	PUNCT
ejpam-5460	269	369	c1)–(c16	c1)–(c16	NOUN
ejpam-5460	269	370	)	)	PUNCT
ejpam-5460	269	371	since	since	SCONJ
ejpam-5460	269	372	{	{	PUNCT
ejpam-5460	269	373	1	1	X
ejpam-5460	269	374	}	}	PUNCT
ejpam-5460	269	375	does	do	AUX
ejpam-5460	269	376	not	not	PART
ejpam-5460	269	377	meet	meet	VERB
ejpam-5460	269	378	any	any	DET
ejpam-5460	269	379	regularity	regularity	NOUN
ejpam-5460	269	380	condition	condition	NOUN
ejpam-5460	269	381	.	.	PUNCT
ejpam-5460	270	1	example	example	NOUN
ejpam-5460	271	1	5	5	NUM
ejpam-5460	271	2	.	.	PUNCT
ejpam-5460	272	1	let	let	VERB
ejpam-5460	272	2	s	s	VERB
ejpam-5460	272	3	=	=	X
ejpam-5460	272	4	{	{	PUNCT
ejpam-5460	272	5	0	0	NUM
ejpam-5460	272	6	,	,	PUNCT
ejpam-5460	272	7	1	1	NUM
ejpam-5460	272	8	,	,	PUNCT
ejpam-5460	272	9	2	2	NUM
ejpam-5460	272	10	,	,	PUNCT
ejpam-5460	272	11	3	3	NUM
ejpam-5460	272	12	,	,	PUNCT
ejpam-5460	272	13	4	4	NUM
ejpam-5460	272	14	}	}	PUNCT
ejpam-5460	272	15	.	.	PUNCT
ejpam-5460	273	1	we	we	PRON
ejpam-5460	273	2	define	define	VERB
ejpam-5460	273	3	a	a	DET
ejpam-5460	273	4	binary	binary	ADJ
ejpam-5460	273	5	operation	operation	NOUN
ejpam-5460	273	6	·	·	PUNCT
ejpam-5460	273	7	and	and	CCONJ
ejpam-5460	273	8	a	a	DET
ejpam-5460	273	9	partial	partial	ADJ
ejpam-5460	273	10	order	order	NOUN
ejpam-5460	273	11	≤	≤	X
ejpam-5460	273	12	on	on	ADP
ejpam-5460	273	13	s	s	PRON
ejpam-5460	273	14	as	as	SCONJ
ejpam-5460	273	15	follows	follow	VERB
ejpam-5460	273	16	.	.	PUNCT
ejpam-5460	274	1	·	·	PUNCT
ejpam-5460	274	2	0	0	NUM
ejpam-5460	274	3	1	1	NUM
ejpam-5460	274	4	2	2	NUM
ejpam-5460	274	5	3	3	NUM
ejpam-5460	274	6	4	4	NUM
ejpam-5460	274	7	0	0	NUM
ejpam-5460	274	8	0	0	NUM
ejpam-5460	274	9	0	0	NUM
ejpam-5460	274	10	0	0	NUM
ejpam-5460	274	11	0	0	NUM
ejpam-5460	274	12	0	0	NUM
ejpam-5460	274	13	1	1	NUM
ejpam-5460	274	14	0	0	NUM
ejpam-5460	274	15	0	0	NUM
ejpam-5460	274	16	3	3	NUM
ejpam-5460	274	17	0	0	NUM
ejpam-5460	274	18	1	1	NUM
ejpam-5460	274	19	2	2	NUM
ejpam-5460	274	20	0	0	NUM
ejpam-5460	274	21	4	4	NUM
ejpam-5460	274	22	0	0	NUM
ejpam-5460	274	23	2	2	NUM
ejpam-5460	274	24	0	0	NUM
ejpam-5460	274	25	3	3	NUM
ejpam-5460	274	26	0	0	NUM
ejpam-5460	274	27	1	1	NUM
ejpam-5460	274	28	0	0	NUM
ejpam-5460	274	29	3	3	NUM
ejpam-5460	274	30	0	0	NUM
ejpam-5460	274	31	4	4	NUM
ejpam-5460	274	32	0	0	NUM
ejpam-5460	274	33	0	0	NUM
ejpam-5460	274	34	2	2	NUM
ejpam-5460	274	35	0	0	NUM
ejpam-5460	274	36	4	4	NUM
ejpam-5460	274	37	s.	s.	PROPN
ejpam-5460	274	38	lekkoksung	lekkoksung	PROPN
ejpam-5460	274	39	,	,	PUNCT
ejpam-5460	274	40	b.	b.	PROPN
ejpam-5460	274	41	davvaz	davvaz	PROPN
ejpam-5460	274	42	,	,	PUNCT
ejpam-5460	274	43	n.	n.	PROPN
ejpam-5460	274	44	lekkoksung	lekkoksung	PROPN
ejpam-5460	274	45	/	/	SYM
ejpam-5460	274	46	eur	eur	PROPN
ejpam-5460	274	47	.	.	PUNCT
ejpam-5460	275	1	j.	j.	PROPN
ejpam-5460	275	2	pure	pure	PROPN
ejpam-5460	275	3	appl	appl	PROPN
ejpam-5460	275	4	.	.	PROPN
ejpam-5460	275	5	math	math	PROPN
ejpam-5460	275	6	,	,	PUNCT
ejpam-5460	275	7	17	17	NUM
ejpam-5460	275	8	(	(	PUNCT
ejpam-5460	275	9	4	4	NUM
ejpam-5460	275	10	)	)	PUNCT
ejpam-5460	275	11	(	(	PUNCT
ejpam-5460	275	12	2024	2024	NUM
ejpam-5460	275	13	)	)	PUNCT
ejpam-5460	275	14	,	,	PUNCT
ejpam-5460	275	15	2962	2962	NUM
ejpam-5460	275	16	-	-	SYM
ejpam-5460	275	17	2984	2984	NUM
ejpam-5460	275	18	2972	2972	NUM
ejpam-5460	275	19	and	and	CCONJ
ejpam-5460	275	20	≤	≤	NUM
ejpam-5460	275	21	:	:	PUNCT
ejpam-5460	275	22	=	=	SYM
ejpam-5460	275	23	{	{	PUNCT
ejpam-5460	275	24	(	(	PUNCT
ejpam-5460	275	25	0	0	NUM
ejpam-5460	275	26	,	,	PUNCT
ejpam-5460	275	27	1	1	NUM
ejpam-5460	275	28	)	)	PUNCT
ejpam-5460	275	29	,	,	PUNCT
ejpam-5460	275	30	(	(	PUNCT
ejpam-5460	275	31	0	0	NUM
ejpam-5460	275	32	,	,	PUNCT
ejpam-5460	275	33	2	2	NUM
ejpam-5460	275	34	)	)	PUNCT
ejpam-5460	275	35	,	,	PUNCT
ejpam-5460	275	36	(	(	PUNCT
ejpam-5460	275	37	0	0	NUM
ejpam-5460	275	38	,	,	PUNCT
ejpam-5460	275	39	3	3	NUM
ejpam-5460	275	40	)	)	PUNCT
ejpam-5460	275	41	,	,	PUNCT
ejpam-5460	275	42	(	(	PUNCT
ejpam-5460	275	43	0	0	NUM
ejpam-5460	275	44	,	,	PUNCT
ejpam-5460	275	45	4	4	NUM
ejpam-5460	275	46	)	)	PUNCT
ejpam-5460	275	47	}	}	PUNCT
ejpam-5460	275	48	∪∆s	∪∆	NOUN
ejpam-5460	275	49	.	.	PUNCT
ejpam-5460	276	1	then	then	ADV
ejpam-5460	276	2	,	,	PUNCT
ejpam-5460	276	3	s	s	X
ejpam-5460	276	4	:	:	PUNCT
ejpam-5460	276	5	=	=	SYM
ejpam-5460	276	6	⟨s	⟨s	NOUN
ejpam-5460	276	7	;	;	PUNCT
ejpam-5460	276	8	·	·	PUNCT
ejpam-5460	276	9	,	,	PUNCT
ejpam-5460	276	10	≤⟩	≤⟩	VERB
ejpam-5460	276	11	is	be	AUX
ejpam-5460	276	12	an	an	DET
ejpam-5460	276	13	ordered	order	VERB
ejpam-5460	276	14	semigroup	semigroup	NOUN
ejpam-5460	276	15	.	.	PUNCT
ejpam-5460	277	1	we	we	PRON
ejpam-5460	277	2	can	can	AUX
ejpam-5460	277	3	see	see	VERB
ejpam-5460	277	4	that	that	PRON
ejpam-5460	277	5	:	:	PUNCT
ejpam-5460	277	6	•	•	X
ejpam-5460	277	7	0	0	NUM
ejpam-5460	277	8	≤	≤	NUM
ejpam-5460	277	9	000	000	NUM
ejpam-5460	277	10	;	;	PUNCT
ejpam-5460	277	11	•	•	NUM
ejpam-5460	277	12	1	1	NUM
ejpam-5460	277	13	≤	≤	NUM
ejpam-5460	277	14	314	314	NUM
ejpam-5460	277	15	;	;	PUNCT
ejpam-5460	277	16	•	•	NUM
ejpam-5460	277	17	2	2	NUM
ejpam-5460	277	18	≤	≤	NUM
ejpam-5460	277	19	423	423	NUM
ejpam-5460	277	20	;	;	PUNCT
ejpam-5460	277	21	•	•	NUM
ejpam-5460	277	22	3	3	NUM
ejpam-5460	277	23	≤	≤	NUM
ejpam-5460	277	24	333	333	NUM
ejpam-5460	277	25	;	;	PUNCT
ejpam-5460	277	26	•	•	NUM
ejpam-5460	277	27	4	4	NUM
ejpam-5460	277	28	≤	≤	NUM
ejpam-5460	277	29	444	444	NUM
ejpam-5460	277	30	.	.	PUNCT
ejpam-5460	278	1	this	this	PRON
ejpam-5460	278	2	means	mean	VERB
ejpam-5460	278	3	that	that	SCONJ
ejpam-5460	278	4	s	s	VERB
ejpam-5460	278	5	satisfies	satisfie	NOUN
ejpam-5460	278	6	(	(	PUNCT
ejpam-5460	278	7	c7	c7	PROPN
ejpam-5460	278	8	)	)	PUNCT
ejpam-5460	278	9	.	.	PUNCT
ejpam-5460	279	1	for	for	ADP
ejpam-5460	279	2	more	more	ADJ
ejpam-5460	279	3	details	detail	NOUN
ejpam-5460	279	4	about	about	ADP
ejpam-5460	279	5	the	the	DET
ejpam-5460	279	6	regularities	regularity	NOUN
ejpam-5460	279	7	of	of	ADP
ejpam-5460	279	8	ordered	order	VERB
ejpam-5460	279	9	semigroups	semigroup	NOUN
ejpam-5460	279	10	,	,	PUNCT
ejpam-5460	279	11	the	the	DET
ejpam-5460	279	12	readers	reader	NOUN
ejpam-5460	279	13	can	can	AUX
ejpam-5460	279	14	refer	refer	VERB
ejpam-5460	279	15	to	to	ADP
ejpam-5460	279	16	[	[	X
ejpam-5460	279	17	43	43	NUM
ejpam-5460	279	18	]	]	PUNCT
ejpam-5460	279	19	.	.	PUNCT
ejpam-5460	280	1	(	(	PUNCT
ejpam-5460	280	2	c14	c14	NOUN
ejpam-5460	280	3	)	)	PUNCT
ejpam-5460	280	4	(	(	PUNCT
ejpam-5460	280	5	c15	c15	PROPN
ejpam-5460	280	6	)	)	PUNCT
ejpam-5460	280	7	(	(	PUNCT
ejpam-5460	280	8	c16	c16	PROPN
ejpam-5460	280	9	)	)	PUNCT
ejpam-5460	280	10	(	(	PUNCT
ejpam-5460	280	11	c11)(c12	c11)(c12	NOUN
ejpam-5460	280	12	)	)	PUNCT
ejpam-5460	280	13	(	(	PUNCT
ejpam-5460	280	14	c13	c13	PROPN
ejpam-5460	280	15	)	)	PUNCT
ejpam-5460	280	16	(	(	PUNCT
ejpam-5460	280	17	c7)(c9	c7)(c9	PROPN
ejpam-5460	280	18	)	)	PUNCT
ejpam-5460	280	19	(	(	PUNCT
ejpam-5460	280	20	c10	c10	PROPN
ejpam-5460	280	21	)	)	PUNCT
ejpam-5460	280	22	(	(	PUNCT
ejpam-5460	280	23	c8)(c5	c8)(c5	NOUN
ejpam-5460	280	24	)	)	PUNCT
ejpam-5460	280	25	(	(	PUNCT
ejpam-5460	280	26	c6	c6	PROPN
ejpam-5460	280	27	)	)	PUNCT
ejpam-5460	280	28	(	(	PUNCT
ejpam-5460	280	29	c4)(c2	c4)(c2	PROPN
ejpam-5460	280	30	)	)	PUNCT
ejpam-5460	280	31	(	(	PUNCT
ejpam-5460	280	32	c3	c3	PROPN
ejpam-5460	280	33	)	)	PUNCT
ejpam-5460	280	34	(	(	PUNCT
ejpam-5460	280	35	c1	c1	NOUN
ejpam-5460	280	36	)	)	PUNCT
ejpam-5460	280	37	figure	figure	NOUN
ejpam-5460	280	38	1	1	NUM
ejpam-5460	280	39	:	:	PUNCT
ejpam-5460	280	40	hesse	hesse	PROPN
ejpam-5460	280	41	diagram	diagram	PROPN
ejpam-5460	280	42	of	of	ADP
ejpam-5460	280	43	ordered	order	VERB
ejpam-5460	280	44	semigroups	semigroup	NOUN
ejpam-5460	280	45	classified	classify	VERB
ejpam-5460	280	46	by	by	ADP
ejpam-5460	280	47	regularities	regularity	NOUN
ejpam-5460	280	48	under	under	ADP
ejpam-5460	280	49	the	the	DET
ejpam-5460	280	50	inclusion	inclusion	NOUN
ejpam-5460	280	51	.	.	PUNCT
ejpam-5460	281	1	in	in	ADP
ejpam-5460	281	2	2022	2022	NUM
ejpam-5460	281	3	,	,	PUNCT
ejpam-5460	281	4	lekkoksung	lekkoksung	PROPN
ejpam-5460	281	5	et	et	PROPN
ejpam-5460	281	6	al	al	PROPN
ejpam-5460	281	7	.	.	PUNCT
ejpam-5460	282	1	[	[	X
ejpam-5460	282	2	36	36	NUM
ejpam-5460	282	3	]	]	PUNCT
ejpam-5460	282	4	considered	consider	VERB
ejpam-5460	282	5	ordered	order	VERB
ejpam-5460	282	6	semigroups	semigroup	NOUN
ejpam-5460	282	7	with	with	ADP
ejpam-5460	282	8	the	the	DET
ejpam-5460	282	9	greatest	great	ADJ
ejpam-5460	282	10	element	element	NOUN
ejpam-5460	282	11	,	,	PUNCT
ejpam-5460	282	12	commonly	commonly	ADV
ejpam-5460	282	13	known	know	VERB
ejpam-5460	282	14	as	as	ADP
ejpam-5460	282	15	poe	poe	PROPN
ejpam-5460	282	16	-	-	PUNCT
ejpam-5460	282	17	semigroups	semigroup	NOUN
ejpam-5460	282	18	.	.	PUNCT
ejpam-5460	283	1	notably	notably	ADV
ejpam-5460	283	2	,	,	PUNCT
ejpam-5460	283	3	any	any	DET
ejpam-5460	283	4	ordered	order	VERB
ejpam-5460	283	5	semigroup	semigroup	NOUN
ejpam-5460	283	6	can	can	AUX
ejpam-5460	283	7	be	be	AUX
ejpam-5460	283	8	embedded	embed	VERB
ejpam-5460	283	9	into	into	ADP
ejpam-5460	283	10	a	a	DET
ejpam-5460	283	11	poe	poe	PROPN
ejpam-5460	283	12	-	-	PUNCT
ejpam-5460	283	13	semigroup	semigroup	PROPN
ejpam-5460	283	14	.	.	PUNCT
ejpam-5460	284	1	consequently	consequently	ADV
ejpam-5460	284	2	,	,	PUNCT
ejpam-5460	284	3	the	the	DET
ejpam-5460	284	4	authors	author	NOUN
ejpam-5460	284	5	comprehensively	comprehensively	ADV
ejpam-5460	284	6	investigated	investigate	VERB
ejpam-5460	284	7	and	and	CCONJ
ejpam-5460	284	8	characterized	characterize	VERB
ejpam-5460	284	9	poe	poe	PROPN
ejpam-5460	284	10	-	-	PUNCT
ejpam-5460	284	11	semigroups	semigroup	NOUN
ejpam-5460	284	12	based	base	VERB
ejpam-5460	284	13	on	on	ADP
ejpam-5460	284	14	their	their	PRON
ejpam-5460	284	15	regularities	regularity	NOUN
ejpam-5460	284	16	.	.	PUNCT
ejpam-5460	285	1	furthermore	furthermore	ADV
ejpam-5460	285	2	,	,	PUNCT
ejpam-5460	285	3	they	they	PRON
ejpam-5460	285	4	extended	extend	VERB
ejpam-5460	285	5	the	the	DET
ejpam-5460	285	6	implications	implication	NOUN
ejpam-5460	285	7	of	of	ADP
ejpam-5460	285	8	their	their	PRON
ejpam-5460	285	9	findings	finding	NOUN
ejpam-5460	285	10	to	to	PART
ejpam-5460	285	11	semigroups	semigroup	NOUN
ejpam-5460	285	12	and	and	CCONJ
ejpam-5460	285	13	hypersemigroups	hypersemigroup	NOUN
ejpam-5460	285	14	.	.	PUNCT
ejpam-5460	286	1	in	in	ADP
ejpam-5460	286	2	this	this	DET
ejpam-5460	286	3	section	section	NOUN
ejpam-5460	286	4	,	,	PUNCT
ejpam-5460	286	5	we	we	PRON
ejpam-5460	286	6	use	use	VERB
ejpam-5460	286	7	the	the	DET
ejpam-5460	286	8	results	result	NOUN
ejpam-5460	286	9	outlined	outline	VERB
ejpam-5460	286	10	in	in	ADP
ejpam-5460	286	11	[	[	X
ejpam-5460	286	12	36	36	NUM
ejpam-5460	286	13	]	]	PUNCT
ejpam-5460	286	14	in	in	ADP
ejpam-5460	286	15	conjunction	conjunction	NOUN
ejpam-5460	286	16	with	with	ADP
ejpam-5460	286	17	the	the	DET
ejpam-5460	286	18	methodology	methodology	NOUN
ejpam-5460	286	19	given	give	VERB
ejpam-5460	286	20	by	by	ADP
ejpam-5460	286	21	kehayopulu	kehayopulu	PROPN
ejpam-5460	286	22	,	,	PUNCT
ejpam-5460	286	23	which	which	PRON
ejpam-5460	286	24	involves	involve	VERB
ejpam-5460	286	25	transposing	transpose	VERB
ejpam-5460	286	26	results	result	NOUN
ejpam-5460	286	27	from	from	ADP
ejpam-5460	286	28	poe	poe	PROPN
ejpam-5460	286	29	-	-	PUNCT
ejpam-5460	286	30	semigroups	semigroup	NOUN
ejpam-5460	286	31	to	to	PART
ejpam-5460	286	32	ordered	order	VERB
ejpam-5460	286	33	semigroups	semigroup	NOUN
ejpam-5460	286	34	and	and	CCONJ
ejpam-5460	286	35	ordered	order	VERB
ejpam-5460	286	36	hypersemigroups	hypersemigroup	NOUN
ejpam-5460	286	37	(	(	PUNCT
ejpam-5460	286	38	see	see	VERB
ejpam-5460	286	39	[	[	X
ejpam-5460	286	40	17	17	NUM
ejpam-5460	286	41	,	,	PUNCT
ejpam-5460	286	42	18	18	NUM
ejpam-5460	286	43	]	]	PUNCT
ejpam-5460	286	44	)	)	PUNCT
ejpam-5460	286	45	.	.	PUNCT
ejpam-5460	287	1	this	this	DET
ejpam-5460	287	2	strategic	strategic	ADJ
ejpam-5460	287	3	approach	approach	NOUN
ejpam-5460	287	4	enables	enable	VERB
ejpam-5460	287	5	us	we	PRON
ejpam-5460	287	6	to	to	PART
ejpam-5460	287	7	characterize	characterize	VERB
ejpam-5460	287	8	ordered	order	VERB
ejpam-5460	287	9	semigroups	semigroup	NOUN
ejpam-5460	287	10	by	by	ADP
ejpam-5460	287	11	(	(	PUNCT
ejpam-5460	287	12	α	α	X
ejpam-5460	287	13	,	,	PUNCT
ejpam-5460	287	14	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	287	15	(	(	PUNCT
ejpam-5460	287	16	m	m	NOUN
ejpam-5460	287	17	,	,	PUNCT
ejpam-5460	287	18	n)-ideals	n)-ideal	NOUN
ejpam-5460	287	19	and	and	CCONJ
ejpam-5460	287	20	(	(	PUNCT
ejpam-5460	287	21	α	α	NOUN
ejpam-5460	287	22	,	,	PUNCT
ejpam-5460	287	23	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	287	24	n	n	CCONJ
ejpam-5460	287	25	-	-	PUNCT
ejpam-5460	287	26	interior	interior	ADJ
ejpam-5460	287	27	ideals	ideal	NOUN
ejpam-5460	287	28	effectively	effectively	ADV
ejpam-5460	287	29	.	.	PUNCT
ejpam-5460	288	1	consequently	consequently	ADV
ejpam-5460	288	2	,	,	PUNCT
ejpam-5460	288	3	we	we	PRON
ejpam-5460	288	4	will	will	AUX
ejpam-5460	288	5	present	present	VERB
ejpam-5460	288	6	these	these	DET
ejpam-5460	288	7	facts	fact	NOUN
ejpam-5460	288	8	of	of	ADP
ejpam-5460	288	9	poe	poe	PROPN
ejpam-5460	288	10	-	-	PUNCT
ejpam-5460	288	11	semigroups	semigroup	NOUN
ejpam-5460	288	12	in	in	ADP
ejpam-5460	288	13	terms	term	NOUN
ejpam-5460	288	14	of	of	ADP
ejpam-5460	288	15	ordered	order	VERB
ejpam-5460	288	16	semigroups	semigroup	NOUN
ejpam-5460	288	17	.	.	PUNCT
ejpam-5460	289	1	we	we	PRON
ejpam-5460	289	2	begin	begin	VERB
ejpam-5460	289	3	categorizing	categorize	VERB
ejpam-5460	289	4	ordered	order	VERB
ejpam-5460	289	5	semigroups	semigroup	NOUN
ejpam-5460	289	6	into	into	ADP
ejpam-5460	289	7	distinct	distinct	ADJ
ejpam-5460	289	8	classes	class	NOUN
ejpam-5460	289	9	based	base	VERB
ejpam-5460	289	10	on	on	ADP
ejpam-5460	289	11	their	their	PRON
ejpam-5460	289	12	regularities	regularity	NOUN
ejpam-5460	289	13	through	through	ADP
ejpam-5460	289	14	(	(	PUNCT
ejpam-5460	289	15	α	α	NOUN
ejpam-5460	289	16	,	,	PUNCT
ejpam-5460	289	17	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	289	18	(	(	PUNCT
ejpam-5460	289	19	m	m	NOUN
ejpam-5460	289	20	,	,	PUNCT
ejpam-5460	289	21	n)-ideals	n)-ideal	NOUN
ejpam-5460	289	22	and	and	CCONJ
ejpam-5460	289	23	(	(	PUNCT
ejpam-5460	289	24	α	α	NOUN
ejpam-5460	289	25	,	,	PUNCT
ejpam-5460	289	26	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	289	27	n	n	CCONJ
ejpam-5460	289	28	-	-	PUNCT
ejpam-5460	289	29	interior	interior	ADJ
ejpam-5460	289	30	ideals	ideal	NOUN
ejpam-5460	289	31	as	as	SCONJ
ejpam-5460	289	32	follows	follow	VERB
ejpam-5460	289	33	.	.	PUNCT
ejpam-5460	290	1	lemma	lemma	PROPN
ejpam-5460	290	2	2	2	NUM
ejpam-5460	290	3	(	(	PUNCT
ejpam-5460	290	4	[	[	X
ejpam-5460	290	5	36	36	NUM
ejpam-5460	290	6	]	]	PUNCT
ejpam-5460	290	7	)	)	PUNCT
ejpam-5460	290	8	.	.	PUNCT
ejpam-5460	291	1	let	let	VERB
ejpam-5460	291	2	s	s	PRON
ejpam-5460	291	3	be	be	AUX
ejpam-5460	291	4	an	an	DET
ejpam-5460	291	5	ordered	order	VERB
ejpam-5460	291	6	semigroup	semigroup	NOUN
ejpam-5460	291	7	.	.	PUNCT
ejpam-5460	292	1	then	then	ADV
ejpam-5460	292	2	,	,	PUNCT
ejpam-5460	292	3	s	s	NOUN
ejpam-5460	292	4	satisfies	satisfie	NOUN
ejpam-5460	292	5	(	(	PUNCT
ejpam-5460	292	6	c1	c1	NOUN
ejpam-5460	292	7	)	)	PUNCT
ejpam-5460	292	8	if	if	SCONJ
ejpam-5460	292	9	and	and	CCONJ
ejpam-5460	292	10	only	only	ADV
ejpam-5460	292	11	if	if	SCONJ
ejpam-5460	292	12	a	a	DET
ejpam-5460	292	13	⊆	⊆	NUM
ejpam-5460	292	14	(	(	PUNCT
ejpam-5460	292	15	sas	sas	PROPN
ejpam-5460	292	16	]	]	X
ejpam-5460	292	17	for	for	ADP
ejpam-5460	292	18	any	any	DET
ejpam-5460	292	19	1	1	NUM
ejpam-5460	292	20	-	-	PUNCT
ejpam-5460	292	21	interior	interior	ADJ
ejpam-5460	292	22	ideal	ideal	NOUN
ejpam-5460	292	23	a	a	PRON
ejpam-5460	292	24	of	of	ADP
ejpam-5460	292	25	s.	s.	PROPN
ejpam-5460	292	26	theorem	theorem	VERB
ejpam-5460	292	27	2	2	X
ejpam-5460	292	28	.	.	PUNCT
ejpam-5460	292	29	let	let	VERB
ejpam-5460	292	30	s	s	PRON
ejpam-5460	292	31	be	be	AUX
ejpam-5460	292	32	an	an	DET
ejpam-5460	292	33	ordered	order	VERB
ejpam-5460	292	34	semigroup	semigroup	NOUN
ejpam-5460	292	35	.	.	PUNCT
ejpam-5460	293	1	then	then	ADV
ejpam-5460	293	2	,	,	PUNCT
ejpam-5460	293	3	the	the	DET
ejpam-5460	293	4	following	follow	VERB
ejpam-5460	293	5	statements	statement	NOUN
ejpam-5460	293	6	are	be	AUX
ejpam-5460	293	7	equivalent	equivalent	ADJ
ejpam-5460	293	8	.	.	PUNCT
ejpam-5460	294	1	s.	s.	PROPN
ejpam-5460	294	2	lekkoksung	lekkoksung	PROPN
ejpam-5460	294	3	,	,	PUNCT
ejpam-5460	294	4	b.	b.	PROPN
ejpam-5460	294	5	davvaz	davvaz	PROPN
ejpam-5460	294	6	,	,	PUNCT
ejpam-5460	294	7	n.	n.	PROPN
ejpam-5460	294	8	lekkoksung	lekkoksung	PROPN
ejpam-5460	294	9	/	/	SYM
ejpam-5460	294	10	eur	eur	PROPN
ejpam-5460	294	11	.	.	PUNCT
ejpam-5460	295	1	j.	j.	PROPN
ejpam-5460	295	2	pure	pure	PROPN
ejpam-5460	295	3	appl	appl	PROPN
ejpam-5460	295	4	.	.	PROPN
ejpam-5460	295	5	math	math	PROPN
ejpam-5460	295	6	,	,	PUNCT
ejpam-5460	295	7	17	17	NUM
ejpam-5460	295	8	(	(	PUNCT
ejpam-5460	295	9	4	4	NUM
ejpam-5460	295	10	)	)	PUNCT
ejpam-5460	295	11	(	(	PUNCT
ejpam-5460	295	12	2024	2024	NUM
ejpam-5460	295	13	)	)	PUNCT
ejpam-5460	295	14	,	,	PUNCT
ejpam-5460	295	15	2962	2962	NUM
ejpam-5460	295	16	-	-	SYM
ejpam-5460	295	17	2984	2984	NUM
ejpam-5460	295	18	2973	2973	NUM
ejpam-5460	295	19	(	(	PUNCT
ejpam-5460	295	20	i	i	NOUN
ejpam-5460	295	21	)	)	PUNCT
ejpam-5460	295	22	s	s	PART
ejpam-5460	295	23	satisfies	satisfie	NOUN
ejpam-5460	295	24	(	(	PUNCT
ejpam-5460	295	25	c1	c1	PROPN
ejpam-5460	295	26	)	)	PUNCT
ejpam-5460	295	27	.	.	PUNCT
ejpam-5460	296	1	(	(	PUNCT
ejpam-5460	296	2	ii	ii	NOUN
ejpam-5460	296	3	)	)	PUNCT
ejpam-5460	296	4	fi	fi	NOUN
ejpam-5460	297	1	⊆	⊆	NUM
ejpam-5460	297	2	β	β	X
ejpam-5460	297	3	◦	◦	NOUN
ejpam-5460	297	4	i	i	X
ejpam-5460	297	5	f	f	NOUN
ejpam-5460	297	6	◦	◦	NOUN
ejpam-5460	297	7	i	i	PRON
ejpam-5460	297	8	β	β	VERB
ejpam-5460	297	9	for	for	ADP
ejpam-5460	297	10	any	any	DET
ejpam-5460	297	11	(	(	PUNCT
ejpam-5460	297	12	α	α	NOUN
ejpam-5460	297	13	,	,	PUNCT
ejpam-5460	297	14	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	297	15	1	1	NUM
ejpam-5460	297	16	-	-	PUNCT
ejpam-5460	297	17	interior	interior	ADJ
ejpam-5460	297	18	ideal	ideal	NOUN
ejpam-5460	297	19	f	f	PROPN
ejpam-5460	297	20	of	of	ADP
ejpam-5460	297	21	s.	s.	PROPN
ejpam-5460	297	22	proof	proof	PROPN
ejpam-5460	297	23	.	.	PUNCT
ejpam-5460	298	1	(	(	PUNCT
ejpam-5460	298	2	1)⇒(2	1)⇒(2	NUM
ejpam-5460	298	3	)	)	PUNCT
ejpam-5460	298	4	.	.	PUNCT
ejpam-5460	299	1	let	let	VERB
ejpam-5460	299	2	f	f	PRON
ejpam-5460	299	3	be	be	AUX
ejpam-5460	299	4	an	an	DET
ejpam-5460	299	5	(	(	PUNCT
ejpam-5460	299	6	α	α	NOUN
ejpam-5460	299	7	,	,	PUNCT
ejpam-5460	299	8	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	299	9	1	1	NUM
ejpam-5460	299	10	-	-	PUNCT
ejpam-5460	299	11	interior	interior	ADJ
ejpam-5460	299	12	ideal	ideal	NOUN
ejpam-5460	299	13	of	of	ADP
ejpam-5460	299	14	s.	s.	PROPN
ejpam-5460	299	15	let	let	VERB
ejpam-5460	299	16	a	a	DET
ejpam-5460	299	17	∈	∈	NOUN
ejpam-5460	299	18	s.	s.	PROPN
ejpam-5460	299	19	since	since	SCONJ
ejpam-5460	299	20	s	s	PART
ejpam-5460	299	21	satisfies	satisfie	NOUN
ejpam-5460	299	22	(	(	PUNCT
ejpam-5460	299	23	c1	c1	PROPN
ejpam-5460	299	24	)	)	PUNCT
ejpam-5460	299	25	,	,	PUNCT
ejpam-5460	299	26	there	there	PRON
ejpam-5460	299	27	exist	exist	VERB
ejpam-5460	299	28	x1	x1	PROPN
ejpam-5460	299	29	,	,	PUNCT
ejpam-5460	299	30	x2	x2	PROPN
ejpam-5460	299	31	∈	∈	PROPN
ejpam-5460	299	32	s	s	VERB
ejpam-5460	299	33	such	such	ADJ
ejpam-5460	299	34	that	that	SCONJ
ejpam-5460	299	35	a	a	DET
ejpam-5460	299	36	≤	≤	NUM
ejpam-5460	299	37	x1ax2	x1ax2	NOUN
ejpam-5460	299	38	≤	≤	NUM
ejpam-5460	299	39	x1x1ax2x2	x1x1ax2x2	NUM
ejpam-5460	299	40	.	.	PUNCT
ejpam-5460	300	1	that	that	PRON
ejpam-5460	300	2	is	be	AUX
ejpam-5460	300	3	,	,	PUNCT
ejpam-5460	300	4	(	(	PUNCT
ejpam-5460	300	5	x1x1ax2	x1x1ax2	PROPN
ejpam-5460	300	6	,	,	PUNCT
ejpam-5460	300	7	x2	x2	PROPN
ejpam-5460	300	8	)	)	PUNCT
ejpam-5460	300	9	∈	∈	PROPN
ejpam-5460	300	10	sa	sa	PROPN
ejpam-5460	300	11	.	.	PUNCT
ejpam-5460	301	1	then	then	ADV
ejpam-5460	301	2	,	,	PUNCT
ejpam-5460	301	3	(	(	PUNCT
ejpam-5460	301	4	β	β	X
ejpam-5460	301	5	◦	◦	NOUN
ejpam-5460	301	6	i	i	NOUN
ejpam-5460	301	7	f	f	NOUN
ejpam-5460	301	8	◦	◦	NOUN
ejpam-5460	301	9	i	i	PRON
ejpam-5460	301	10	β)(a	β)(a	NOUN
ejpam-5460	301	11	)	)	PUNCT
ejpam-5460	301	12	≥	≥	NUM
ejpam-5460	301	13	(	(	PUNCT
ejpam-5460	301	14	β	β	X
ejpam-5460	301	15	◦	◦	NOUN
ejpam-5460	301	16	i	i	PRON
ejpam-5460	301	17	f)i(x1x1ax2	f)i(x1x1ax2	VERB
ejpam-5460	301	18	)	)	PUNCT
ejpam-5460	301	19	=	=	PUNCT
ejpam-5460	301	20	(	(	PUNCT
ejpam-5460	301	21	β	β	X
ejpam-5460	301	22	◦	◦	NOUN
ejpam-5460	301	23	f)i(x1x1ax2	f)i(x1x1ax2	NOUN
ejpam-5460	301	24	)	)	PUNCT
ejpam-5460	301	25	=	=	SYM
ejpam-5460	301	26	fi(x1ax2	fi(x1ax2	NOUN
ejpam-5460	301	27	)	)	PUNCT
ejpam-5460	301	28	≥	≥	NOUN
ejpam-5460	301	29	fi(a	fi(a	NOUN
ejpam-5460	301	30	)	)	PUNCT
ejpam-5460	301	31	.	.	PUNCT
ejpam-5460	302	1	(	(	PUNCT
ejpam-5460	302	2	2)⇒(1	2)⇒(1	NOUN
ejpam-5460	302	3	)	)	PUNCT
ejpam-5460	302	4	.	.	PUNCT
ejpam-5460	303	1	let	let	VERB
ejpam-5460	303	2	a	a	PRON
ejpam-5460	303	3	be	be	AUX
ejpam-5460	303	4	a	a	DET
ejpam-5460	303	5	1	1	NUM
ejpam-5460	303	6	-	-	PUNCT
ejpam-5460	303	7	interior	interior	ADJ
ejpam-5460	303	8	ideal	ideal	NOUN
ejpam-5460	303	9	of	of	ADP
ejpam-5460	303	10	s.	s.	PROPN
ejpam-5460	303	11	then	then	ADV
ejpam-5460	303	12	,	,	PUNCT
ejpam-5460	303	13	by	by	ADP
ejpam-5460	303	14	proposition	proposition	NOUN
ejpam-5460	303	15	2	2	NUM
ejpam-5460	303	16	,	,	PUNCT
ejpam-5460	303	17	(	(	PUNCT
ejpam-5460	303	18	χa)i	χa)i	PROPN
ejpam-5460	303	19	is	be	AUX
ejpam-5460	303	20	an	an	DET
ejpam-5460	303	21	(	(	PUNCT
ejpam-5460	303	22	α	α	NOUN
ejpam-5460	303	23	,	,	PUNCT
ejpam-5460	303	24	β)fuzzy	β)fuzzy	ADJ
ejpam-5460	303	25	1	1	NUM
ejpam-5460	303	26	-	-	ADJ
ejpam-5460	303	27	interior	interior	ADJ
ejpam-5460	303	28	ideal	ideal	NOUN
ejpam-5460	303	29	of	of	ADP
ejpam-5460	303	30	s.	s.	PROPN
ejpam-5460	303	31	by	by	ADP
ejpam-5460	303	32	our	our	PRON
ejpam-5460	303	33	presumption	presumption	NOUN
ejpam-5460	303	34	,	,	PUNCT
ejpam-5460	303	35	we	we	PRON
ejpam-5460	303	36	have	have	VERB
ejpam-5460	303	37	(	(	PUNCT
ejpam-5460	303	38	(	(	PUNCT
ejpam-5460	303	39	χa)i)i	χa)i)i	NOUN
ejpam-5460	303	40	⊆	⊆	NUM
ejpam-5460	303	41	β	β	X
ejpam-5460	303	42	◦	◦	NOUN
ejpam-5460	303	43	i	i	PRON
ejpam-5460	303	44	(	(	PUNCT
ejpam-5460	303	45	χa)i	χa)i	PROPN
ejpam-5460	303	46	◦	◦	NOUN
ejpam-5460	303	47	i	i	PRON
ejpam-5460	303	48	β	β	NOUN
ejpam-5460	303	49	.	.	PUNCT
ejpam-5460	304	1	this	this	PRON
ejpam-5460	304	2	implies	imply	VERB
ejpam-5460	304	3	that	that	SCONJ
ejpam-5460	304	4	(	(	PUNCT
ejpam-5460	304	5	χa)i	χa)i	PROPN
ejpam-5460	304	6	=	=	SYM
ejpam-5460	304	7	(	(	PUNCT
ejpam-5460	304	8	(	(	PUNCT
ejpam-5460	304	9	χa)i)i	χa)i)i	NOUN
ejpam-5460	304	10	⊆	⊆	NUM
ejpam-5460	304	11	β	β	X
ejpam-5460	304	12	◦	◦	NOUN
ejpam-5460	304	13	i	i	PRON
ejpam-5460	304	14	(	(	PUNCT
ejpam-5460	304	15	χa)i	χa)i	PROPN
ejpam-5460	304	16	◦	◦	NOUN
ejpam-5460	304	17	i	i	PRON
ejpam-5460	304	18	β	β	X
ejpam-5460	304	19	=	=	SYM
ejpam-5460	304	20	(	(	PUNCT
ejpam-5460	304	21	χs)i	χs)i	PROPN
ejpam-5460	304	22	◦	◦	NOUN
ejpam-5460	304	23	i	i	PROPN
ejpam-5460	304	24	(	(	PUNCT
ejpam-5460	304	25	χa)i	χa)i	PROPN
ejpam-5460	304	26	◦	◦	NOUN
ejpam-5460	304	27	i	i	PRON
ejpam-5460	304	28	(	(	PUNCT
ejpam-5460	304	29	χs)i	χs)i	PROPN
ejpam-5460	304	30	=	=	SYM
ejpam-5460	304	31	(	(	PUNCT
ejpam-5460	304	32	χs	χs	PART
ejpam-5460	304	33	◦	◦	VERB
ejpam-5460	304	34	χa	χa	PROPN
ejpam-5460	304	35	◦	◦	NOUN
ejpam-5460	304	36	χs)i	χs)i	PROPN
ejpam-5460	305	1	=	=	PRON
ejpam-5460	306	1	(	(	PUNCT
ejpam-5460	306	2	χ(sas])i	χ(sas])i	PROPN
ejpam-5460	306	3	.	.	PUNCT
ejpam-5460	307	1	by	by	ADP
ejpam-5460	307	2	proposition	proposition	NOUN
ejpam-5460	307	3	2	2	NUM
ejpam-5460	307	4	,	,	PUNCT
ejpam-5460	307	5	we	we	PRON
ejpam-5460	307	6	have	have	VERB
ejpam-5460	307	7	a	a	DET
ejpam-5460	307	8	⊆	⊆	NUM
ejpam-5460	307	9	(	(	PUNCT
ejpam-5460	307	10	sas	sas	PROPN
ejpam-5460	307	11	]	]	PUNCT
ejpam-5460	307	12	.	.	PUNCT
ejpam-5460	308	1	hence	hence	ADV
ejpam-5460	308	2	,	,	PUNCT
ejpam-5460	308	3	by	by	ADP
ejpam-5460	308	4	lemma	lemma	PROPN
ejpam-5460	308	5	2	2	NUM
ejpam-5460	308	6	,	,	PUNCT
ejpam-5460	308	7	s	s	PART
ejpam-5460	308	8	satisfies	satisfie	NOUN
ejpam-5460	308	9	(	(	PUNCT
ejpam-5460	308	10	c1	c1	PROPN
ejpam-5460	308	11	)	)	PUNCT
ejpam-5460	308	12	.	.	PUNCT
ejpam-5460	309	1	lemma	lemma	PROPN
ejpam-5460	309	2	3	3	NUM
ejpam-5460	309	3	(	(	PUNCT
ejpam-5460	309	4	[	[	X
ejpam-5460	309	5	36	36	NUM
ejpam-5460	309	6	]	]	PUNCT
ejpam-5460	309	7	)	)	PUNCT
ejpam-5460	309	8	.	.	PUNCT
ejpam-5460	310	1	let	let	VERB
ejpam-5460	310	2	s	s	PRON
ejpam-5460	310	3	be	be	AUX
ejpam-5460	310	4	an	an	DET
ejpam-5460	310	5	ordered	order	VERB
ejpam-5460	310	6	semigroup	semigroup	NOUN
ejpam-5460	310	7	.	.	PUNCT
ejpam-5460	311	1	then	then	ADV
ejpam-5460	311	2	,	,	PUNCT
ejpam-5460	311	3	s	s	NOUN
ejpam-5460	311	4	satisfies	satisfie	NOUN
ejpam-5460	311	5	(	(	PUNCT
ejpam-5460	311	6	c2	c2	PROPN
ejpam-5460	311	7	)	)	PUNCT
ejpam-5460	311	8	if	if	SCONJ
ejpam-5460	311	9	and	and	CCONJ
ejpam-5460	311	10	only	only	ADV
ejpam-5460	311	11	if	if	SCONJ
ejpam-5460	311	12	a	a	DET
ejpam-5460	311	13	⊆	⊆	NUM
ejpam-5460	311	14	(	(	PUNCT
ejpam-5460	311	15	sa	sa	NOUN
ejpam-5460	311	16	]	]	X
ejpam-5460	311	17	for	for	ADP
ejpam-5460	311	18	any	any	DET
ejpam-5460	311	19	(	(	PUNCT
ejpam-5460	311	20	0	0	NUM
ejpam-5460	311	21	,	,	PUNCT
ejpam-5460	311	22	1)-ideal	1)-ideal	NUM
ejpam-5460	311	23	a	a	PRON
ejpam-5460	311	24	of	of	ADP
ejpam-5460	311	25	s.	s.	PROPN
ejpam-5460	311	26	theorem	theorem	VERB
ejpam-5460	311	27	3	3	X
ejpam-5460	311	28	.	.	PUNCT
ejpam-5460	312	1	let	let	VERB
ejpam-5460	312	2	s	s	PRON
ejpam-5460	312	3	be	be	AUX
ejpam-5460	312	4	an	an	DET
ejpam-5460	312	5	ordered	order	VERB
ejpam-5460	312	6	semigroup	semigroup	NOUN
ejpam-5460	312	7	.	.	PUNCT
ejpam-5460	313	1	then	then	ADV
ejpam-5460	313	2	,	,	PUNCT
ejpam-5460	313	3	the	the	DET
ejpam-5460	313	4	following	follow	VERB
ejpam-5460	313	5	statements	statement	NOUN
ejpam-5460	313	6	are	be	AUX
ejpam-5460	313	7	equivalent	equivalent	ADJ
ejpam-5460	313	8	.	.	PUNCT
ejpam-5460	314	1	(	(	PUNCT
ejpam-5460	314	2	i	i	NOUN
ejpam-5460	314	3	)	)	PUNCT
ejpam-5460	314	4	s	s	PART
ejpam-5460	314	5	satisfies	satisfie	NOUN
ejpam-5460	314	6	(	(	PUNCT
ejpam-5460	314	7	c2	c2	PROPN
ejpam-5460	314	8	)	)	PUNCT
ejpam-5460	314	9	.	.	PUNCT
ejpam-5460	315	1	(	(	PUNCT
ejpam-5460	315	2	ii	ii	NOUN
ejpam-5460	315	3	)	)	PUNCT
ejpam-5460	315	4	fi	fi	NOUN
ejpam-5460	316	1	⊆	⊆	NUM
ejpam-5460	316	2	β	β	X
ejpam-5460	316	3	◦	◦	NOUN
ejpam-5460	316	4	i	i	PRON
ejpam-5460	316	5	f	f	NOUN
ejpam-5460	316	6	for	for	ADP
ejpam-5460	316	7	any	any	DET
ejpam-5460	316	8	(	(	PUNCT
ejpam-5460	316	9	α	α	NOUN
ejpam-5460	316	10	,	,	PUNCT
ejpam-5460	316	11	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	316	12	(	(	PUNCT
ejpam-5460	316	13	0	0	NUM
ejpam-5460	316	14	,	,	PUNCT
ejpam-5460	316	15	1)-ideal	1)-ideal	NUM
ejpam-5460	316	16	f	f	NOUN
ejpam-5460	316	17	of	of	ADP
ejpam-5460	316	18	s.	s.	PROPN
ejpam-5460	316	19	proof	proof	PROPN
ejpam-5460	316	20	.	.	PUNCT
ejpam-5460	317	1	(	(	PUNCT
ejpam-5460	317	2	1)⇒(2	1)⇒(2	NUM
ejpam-5460	317	3	)	)	PUNCT
ejpam-5460	317	4	.	.	PUNCT
ejpam-5460	318	1	let	let	VERB
ejpam-5460	318	2	f	f	PRON
ejpam-5460	318	3	be	be	AUX
ejpam-5460	318	4	an	an	DET
ejpam-5460	318	5	(	(	PUNCT
ejpam-5460	318	6	α	α	NOUN
ejpam-5460	318	7	,	,	PUNCT
ejpam-5460	318	8	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	318	9	(	(	PUNCT
ejpam-5460	318	10	0	0	NUM
ejpam-5460	318	11	,	,	PUNCT
ejpam-5460	318	12	1)-ideal	1)-ideal	NUM
ejpam-5460	318	13	of	of	ADP
ejpam-5460	318	14	s.	s.	PROPN
ejpam-5460	318	15	let	let	VERB
ejpam-5460	318	16	a	a	DET
ejpam-5460	318	17	∈	∈	NOUN
ejpam-5460	318	18	s.	s.	PROPN
ejpam-5460	318	19	since	since	SCONJ
ejpam-5460	318	20	s	s	PART
ejpam-5460	318	21	satisfies	satisfie	NOUN
ejpam-5460	318	22	(	(	PUNCT
ejpam-5460	318	23	c2	c2	PROPN
ejpam-5460	318	24	)	)	PUNCT
ejpam-5460	318	25	,	,	PUNCT
ejpam-5460	318	26	there	there	PRON
ejpam-5460	318	27	exist	exist	VERB
ejpam-5460	318	28	x	x	PUNCT
ejpam-5460	318	29	∈	∈	NOUN
ejpam-5460	318	30	s	s	VERB
ejpam-5460	318	31	such	such	ADJ
ejpam-5460	318	32	that	that	SCONJ
ejpam-5460	318	33	a	a	DET
ejpam-5460	318	34	≤	≤	ADJ
ejpam-5460	318	35	xa	xa	PROPN
ejpam-5460	318	36	.	.	PUNCT
ejpam-5460	319	1	that	that	PRON
ejpam-5460	319	2	is	be	AUX
ejpam-5460	319	3	,	,	PUNCT
ejpam-5460	319	4	(	(	PUNCT
ejpam-5460	319	5	x	x	NOUN
ejpam-5460	319	6	,	,	PUNCT
ejpam-5460	319	7	a	a	DET
ejpam-5460	319	8	)	)	PUNCT
ejpam-5460	319	9	∈	∈	PROPN
ejpam-5460	319	10	sa	sa	PROPN
ejpam-5460	319	11	.	.	PUNCT
ejpam-5460	320	1	then	then	ADV
ejpam-5460	320	2	,	,	PUNCT
ejpam-5460	320	3	(	(	PUNCT
ejpam-5460	320	4	β	β	X
ejpam-5460	320	5	◦	◦	NOUN
ejpam-5460	320	6	i	i	PRON
ejpam-5460	320	7	f)(a	f)(a	NOUN
ejpam-5460	320	8	)	)	PUNCT
ejpam-5460	320	9	≥	≥	NOUN
ejpam-5460	320	10	fi(a	fi(a	NOUN
ejpam-5460	320	11	)	)	PUNCT
ejpam-5460	320	12	.	.	PUNCT
ejpam-5460	321	1	(	(	PUNCT
ejpam-5460	321	2	2)⇒(1	2)⇒(1	NOUN
ejpam-5460	321	3	)	)	PUNCT
ejpam-5460	321	4	.	.	PUNCT
ejpam-5460	322	1	let	let	VERB
ejpam-5460	322	2	a	a	DET
ejpam-5460	322	3	be	be	AUX
ejpam-5460	322	4	a	a	DET
ejpam-5460	322	5	(	(	PUNCT
ejpam-5460	322	6	0	0	NUM
ejpam-5460	322	7	,	,	PUNCT
ejpam-5460	322	8	1)-ideal	1)-ideal	NUM
ejpam-5460	322	9	of	of	ADP
ejpam-5460	322	10	s.	s.	PROPN
ejpam-5460	322	11	then	then	ADV
ejpam-5460	322	12	,	,	PUNCT
ejpam-5460	322	13	by	by	ADP
ejpam-5460	322	14	proposition	proposition	NOUN
ejpam-5460	322	15	2	2	NUM
ejpam-5460	322	16	,	,	PUNCT
ejpam-5460	322	17	(	(	PUNCT
ejpam-5460	322	18	χa)i	χa)i	PROPN
ejpam-5460	322	19	is	be	AUX
ejpam-5460	322	20	an	an	DET
ejpam-5460	322	21	(	(	PUNCT
ejpam-5460	322	22	α	α	NOUN
ejpam-5460	322	23	,	,	PUNCT
ejpam-5460	322	24	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	322	25	(	(	PUNCT
ejpam-5460	322	26	0	0	NUM
ejpam-5460	322	27	,	,	PUNCT
ejpam-5460	322	28	1)-ideal	1)-ideal	NUM
ejpam-5460	322	29	of	of	ADP
ejpam-5460	322	30	s.	s.	PROPN
ejpam-5460	322	31	by	by	ADP
ejpam-5460	322	32	our	our	PRON
ejpam-5460	322	33	presumption	presumption	NOUN
ejpam-5460	322	34	,	,	PUNCT
ejpam-5460	322	35	we	we	PRON
ejpam-5460	322	36	have	have	VERB
ejpam-5460	322	37	(	(	PUNCT
ejpam-5460	322	38	(	(	PUNCT
ejpam-5460	322	39	χa)i)i	χa)i)i	NOUN
ejpam-5460	322	40	⊆	⊆	NUM
ejpam-5460	322	41	β	β	X
ejpam-5460	322	42	◦	◦	NOUN
ejpam-5460	322	43	i	i	PROPN
ejpam-5460	322	44	(	(	PUNCT
ejpam-5460	322	45	χa)i	χa)i	PROPN
ejpam-5460	322	46	.	.	PUNCT
ejpam-5460	323	1	this	this	PRON
ejpam-5460	323	2	implies	imply	VERB
ejpam-5460	323	3	that	that	SCONJ
ejpam-5460	323	4	(	(	PUNCT
ejpam-5460	323	5	χa)i	χa)i	PROPN
ejpam-5460	323	6	=	=	SYM
ejpam-5460	323	7	(	(	PUNCT
ejpam-5460	323	8	(	(	PUNCT
ejpam-5460	323	9	χa)i)i	χa)i)i	NOUN
ejpam-5460	323	10	⊆	⊆	NUM
ejpam-5460	323	11	β	β	X
ejpam-5460	323	12	◦	◦	NOUN
ejpam-5460	323	13	i	i	PROPN
ejpam-5460	323	14	(	(	PUNCT
ejpam-5460	323	15	χa)i	χa)i	PROPN
ejpam-5460	323	16	=	=	SYM
ejpam-5460	323	17	(	(	PUNCT
ejpam-5460	323	18	χs)i	χs)i	PROPN
ejpam-5460	323	19	◦	◦	NOUN
ejpam-5460	323	20	i	i	PROPN
ejpam-5460	323	21	(	(	PUNCT
ejpam-5460	323	22	χa)i	χa)i	PROPN
ejpam-5460	323	23	=	=	PUNCT
ejpam-5460	323	24	χs	χs	PART
ejpam-5460	323	25	◦	◦	NOUN
ejpam-5460	323	26	i	i	NOUN
ejpam-5460	323	27	χa	χa	VERB
ejpam-5460	323	28	=	=	SYM
ejpam-5460	323	29	(	(	PUNCT
ejpam-5460	323	30	χ(sa])i	χ(sa])i	PROPN
ejpam-5460	323	31	.	.	PUNCT
ejpam-5460	324	1	by	by	ADP
ejpam-5460	324	2	proposition	proposition	NOUN
ejpam-5460	324	3	2	2	NUM
ejpam-5460	324	4	,	,	PUNCT
ejpam-5460	324	5	we	we	PRON
ejpam-5460	324	6	have	have	VERB
ejpam-5460	324	7	a	a	DET
ejpam-5460	324	8	⊆	⊆	NUM
ejpam-5460	324	9	(	(	PUNCT
ejpam-5460	324	10	sa	sa	NOUN
ejpam-5460	324	11	]	]	PUNCT
ejpam-5460	324	12	.	.	PUNCT
ejpam-5460	325	1	hence	hence	ADV
ejpam-5460	325	2	,	,	PUNCT
ejpam-5460	325	3	by	by	ADP
ejpam-5460	325	4	lemma	lemma	PROPN
ejpam-5460	325	5	3	3	NUM
ejpam-5460	325	6	,	,	PUNCT
ejpam-5460	325	7	s	s	PART
ejpam-5460	325	8	satisfies	satisfie	NOUN
ejpam-5460	325	9	(	(	PUNCT
ejpam-5460	325	10	c2	c2	PROPN
ejpam-5460	325	11	)	)	PUNCT
ejpam-5460	325	12	.	.	PUNCT
ejpam-5460	326	1	similarly	similarly	ADV
ejpam-5460	326	2	,	,	PUNCT
ejpam-5460	326	3	we	we	PRON
ejpam-5460	326	4	obtain	obtain	VERB
ejpam-5460	326	5	a	a	DET
ejpam-5460	326	6	characterization	characterization	NOUN
ejpam-5460	326	7	of	of	ADP
ejpam-5460	326	8	ordered	order	VERB
ejpam-5460	326	9	semigroups	semigroup	NOUN
ejpam-5460	326	10	satisfying	satisfy	VERB
ejpam-5460	326	11	(	(	PUNCT
ejpam-5460	326	12	c3	c3	PROPN
ejpam-5460	326	13	)	)	PUNCT
ejpam-5460	326	14	as	as	SCONJ
ejpam-5460	326	15	follows	follow	VERB
ejpam-5460	326	16	.	.	PUNCT
ejpam-5460	327	1	theorem	theorem	ADJ
ejpam-5460	327	2	4	4	NUM
ejpam-5460	327	3	.	.	PUNCT
ejpam-5460	328	1	let	let	VERB
ejpam-5460	328	2	s	s	PRON
ejpam-5460	328	3	be	be	AUX
ejpam-5460	328	4	an	an	DET
ejpam-5460	328	5	ordered	order	VERB
ejpam-5460	328	6	semigroup	semigroup	NOUN
ejpam-5460	328	7	.	.	PUNCT
ejpam-5460	329	1	then	then	ADV
ejpam-5460	329	2	,	,	PUNCT
ejpam-5460	329	3	the	the	DET
ejpam-5460	329	4	following	follow	VERB
ejpam-5460	329	5	statements	statement	NOUN
ejpam-5460	329	6	are	be	AUX
ejpam-5460	329	7	equivalent	equivalent	ADJ
ejpam-5460	329	8	.	.	PUNCT
ejpam-5460	330	1	(	(	PUNCT
ejpam-5460	330	2	i	i	NOUN
ejpam-5460	330	3	)	)	PUNCT
ejpam-5460	330	4	s	s	PART
ejpam-5460	330	5	satisfies	satisfie	NOUN
ejpam-5460	330	6	(	(	PUNCT
ejpam-5460	330	7	c3	c3	PROPN
ejpam-5460	330	8	)	)	PUNCT
ejpam-5460	330	9	.	.	PUNCT
ejpam-5460	331	1	s.	s.	PROPN
ejpam-5460	331	2	lekkoksung	lekkoksung	PROPN
ejpam-5460	331	3	,	,	PUNCT
ejpam-5460	331	4	b.	b.	PROPN
ejpam-5460	331	5	davvaz	davvaz	PROPN
ejpam-5460	331	6	,	,	PUNCT
ejpam-5460	331	7	n.	n.	PROPN
ejpam-5460	331	8	lekkoksung	lekkoksung	PROPN
ejpam-5460	331	9	/	/	SYM
ejpam-5460	331	10	eur	eur	PROPN
ejpam-5460	331	11	.	.	PUNCT
ejpam-5460	332	1	j.	j.	PROPN
ejpam-5460	332	2	pure	pure	PROPN
ejpam-5460	332	3	appl	appl	PROPN
ejpam-5460	332	4	.	.	PROPN
ejpam-5460	332	5	math	math	PROPN
ejpam-5460	332	6	,	,	PUNCT
ejpam-5460	332	7	17	17	NUM
ejpam-5460	332	8	(	(	PUNCT
ejpam-5460	332	9	4	4	NUM
ejpam-5460	332	10	)	)	PUNCT
ejpam-5460	332	11	(	(	PUNCT
ejpam-5460	332	12	2024	2024	NUM
ejpam-5460	332	13	)	)	PUNCT
ejpam-5460	332	14	,	,	PUNCT
ejpam-5460	332	15	2962	2962	NUM
ejpam-5460	332	16	-	-	SYM
ejpam-5460	332	17	2984	2984	NUM
ejpam-5460	332	18	2974	2974	NUM
ejpam-5460	332	19	(	(	PUNCT
ejpam-5460	332	20	ii	ii	NOUN
ejpam-5460	332	21	)	)	PUNCT
ejpam-5460	332	22	fi	fi	NOUN
ejpam-5460	333	1	⊆	⊆	NUM
ejpam-5460	333	2	f	f	PROPN
ejpam-5460	333	3	◦	◦	NOUN
ejpam-5460	333	4	i	i	PRON
ejpam-5460	333	5	β	β	VERB
ejpam-5460	333	6	for	for	ADP
ejpam-5460	333	7	any	any	DET
ejpam-5460	333	8	(	(	PUNCT
ejpam-5460	333	9	α	α	NOUN
ejpam-5460	333	10	,	,	PUNCT
ejpam-5460	333	11	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	333	12	(	(	PUNCT
ejpam-5460	333	13	1	1	NUM
ejpam-5460	333	14	,	,	PUNCT
ejpam-5460	333	15	0)-ideal	0)-ideal	PROPN
ejpam-5460	333	16	f	f	PROPN
ejpam-5460	333	17	of	of	ADP
ejpam-5460	333	18	s.	s.	PROPN
ejpam-5460	333	19	lemma	lemma	PROPN
ejpam-5460	333	20	4	4	NUM
ejpam-5460	333	21	(	(	PUNCT
ejpam-5460	333	22	[	[	X
ejpam-5460	333	23	49	49	NUM
ejpam-5460	333	24	]	]	PUNCT
ejpam-5460	333	25	)	)	PUNCT
ejpam-5460	333	26	.	.	PUNCT
ejpam-5460	334	1	let	let	VERB
ejpam-5460	334	2	s	s	PRON
ejpam-5460	334	3	be	be	AUX
ejpam-5460	334	4	an	an	DET
ejpam-5460	334	5	ordered	order	VERB
ejpam-5460	334	6	semigroup	semigroup	NOUN
ejpam-5460	334	7	.	.	PUNCT
ejpam-5460	335	1	then	then	ADV
ejpam-5460	335	2	,	,	PUNCT
ejpam-5460	335	3	s	s	NOUN
ejpam-5460	335	4	satisfies	satisfie	NOUN
ejpam-5460	335	5	(	(	PUNCT
ejpam-5460	335	6	c4	c4	NOUN
ejpam-5460	335	7	)	)	PUNCT
ejpam-5460	335	8	if	if	SCONJ
ejpam-5460	335	9	and	and	CCONJ
ejpam-5460	335	10	only	only	ADV
ejpam-5460	335	11	if	if	SCONJ
ejpam-5460	335	12	a	a	DET
ejpam-5460	335	13	∩b	∩b	NOUN
ejpam-5460	335	14	⊆	⊆	NUM
ejpam-5460	335	15	(	(	PUNCT
ejpam-5460	335	16	aba	aba	X
ejpam-5460	335	17	]	]	PUNCT
ejpam-5460	335	18	for	for	ADP
ejpam-5460	335	19	any	any	DET
ejpam-5460	335	20	1	1	NUM
ejpam-5460	335	21	-	-	PUNCT
ejpam-5460	335	22	interior	interior	ADJ
ejpam-5460	335	23	ideal	ideal	NOUN
ejpam-5460	335	24	a	a	PRON
ejpam-5460	335	25	and	and	CCONJ
ejpam-5460	335	26	(	(	PUNCT
ejpam-5460	335	27	1	1	NUM
ejpam-5460	335	28	,	,	PUNCT
ejpam-5460	335	29	1)-ideal	1)-ideal	NUM
ejpam-5460	335	30	b	b	PROPN
ejpam-5460	335	31	of	of	ADP
ejpam-5460	335	32	s.	s.	PROPN
ejpam-5460	335	33	theorem	theorem	VERB
ejpam-5460	335	34	5	5	NUM
ejpam-5460	335	35	.	.	PUNCT
ejpam-5460	336	1	let	let	VERB
ejpam-5460	336	2	s	s	PRON
ejpam-5460	336	3	be	be	AUX
ejpam-5460	336	4	an	an	DET
ejpam-5460	336	5	ordered	order	VERB
ejpam-5460	336	6	semigroup	semigroup	NOUN
ejpam-5460	336	7	.	.	PUNCT
ejpam-5460	337	1	then	then	ADV
ejpam-5460	337	2	,	,	PUNCT
ejpam-5460	337	3	the	the	DET
ejpam-5460	337	4	following	follow	VERB
ejpam-5460	337	5	statements	statement	NOUN
ejpam-5460	337	6	are	be	AUX
ejpam-5460	337	7	equivalent	equivalent	ADJ
ejpam-5460	337	8	.	.	PUNCT
ejpam-5460	338	1	(	(	PUNCT
ejpam-5460	338	2	i	i	NOUN
ejpam-5460	338	3	)	)	PUNCT
ejpam-5460	338	4	s	s	PART
ejpam-5460	338	5	satisfies	satisfie	NOUN
ejpam-5460	338	6	(	(	PUNCT
ejpam-5460	338	7	c4	c4	NOUN
ejpam-5460	338	8	)	)	PUNCT
ejpam-5460	338	9	.	.	PUNCT
ejpam-5460	339	1	(	(	PUNCT
ejpam-5460	339	2	ii	ii	X
ejpam-5460	339	3	)	)	PUNCT
ejpam-5460	339	4	f	f	PROPN
ejpam-5460	339	5	∩i	∩i	PROPN
ejpam-5460	339	6	g	g	PROPN
ejpam-5460	339	7	⊆	⊆	NUM
ejpam-5460	339	8	f	f	PROPN
ejpam-5460	339	9	◦	◦	NOUN
ejpam-5460	339	10	i	i	PROPN
ejpam-5460	339	11	g	g	PROPN
ejpam-5460	339	12	◦	◦	NOUN
ejpam-5460	339	13	i	i	PRON
ejpam-5460	339	14	f	f	NOUN
ejpam-5460	339	15	for	for	ADP
ejpam-5460	339	16	any	any	DET
ejpam-5460	339	17	(	(	PUNCT
ejpam-5460	339	18	α	α	NOUN
ejpam-5460	339	19	,	,	PUNCT
ejpam-5460	339	20	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	339	21	1	1	NUM
ejpam-5460	339	22	-	-	PUNCT
ejpam-5460	339	23	interior	interior	ADJ
ejpam-5460	339	24	ideal	ideal	NOUN
ejpam-5460	339	25	f	f	PROPN
ejpam-5460	339	26	and	and	CCONJ
ejpam-5460	339	27	(	(	PUNCT
ejpam-5460	339	28	α	α	NOUN
ejpam-5460	339	29	,	,	PUNCT
ejpam-5460	339	30	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	339	31	(	(	PUNCT
ejpam-5460	339	32	1	1	NUM
ejpam-5460	339	33	,	,	PUNCT
ejpam-5460	339	34	1)-ideal	1)-ideal	NUM
ejpam-5460	339	35	g	g	NOUN
ejpam-5460	339	36	of	of	ADP
ejpam-5460	339	37	s.	s.	PROPN
ejpam-5460	339	38	proof	proof	PROPN
ejpam-5460	339	39	.	.	PUNCT
ejpam-5460	340	1	(	(	PUNCT
ejpam-5460	340	2	1)⇒(2	1)⇒(2	NUM
ejpam-5460	340	3	)	)	PUNCT
ejpam-5460	340	4	.	.	PUNCT
ejpam-5460	341	1	let	let	VERB
ejpam-5460	341	2	f	f	PROPN
ejpam-5460	341	3	and	and	CCONJ
ejpam-5460	341	4	g	g	PROPN
ejpam-5460	341	5	be	be	AUX
ejpam-5460	341	6	an	an	DET
ejpam-5460	341	7	(	(	PUNCT
ejpam-5460	341	8	α	α	NOUN
ejpam-5460	341	9	,	,	PUNCT
ejpam-5460	341	10	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	341	11	1	1	NUM
ejpam-5460	341	12	-	-	PUNCT
ejpam-5460	341	13	interior	interior	ADJ
ejpam-5460	341	14	ideal	ideal	NOUN
ejpam-5460	341	15	and	and	CCONJ
ejpam-5460	341	16	an	an	DET
ejpam-5460	341	17	(	(	PUNCT
ejpam-5460	341	18	α	α	NOUN
ejpam-5460	341	19	,	,	PUNCT
ejpam-5460	341	20	β)fuzzy	β)fuzzy	PUNCT
ejpam-5460	342	1	(	(	PUNCT
ejpam-5460	342	2	1	1	NUM
ejpam-5460	342	3	,	,	PUNCT
ejpam-5460	342	4	1)-ideal	1)-ideal	NUM
ejpam-5460	342	5	of	of	ADP
ejpam-5460	342	6	s	s	PROPN
ejpam-5460	342	7	,	,	PUNCT
ejpam-5460	342	8	respectively	respectively	ADV
ejpam-5460	342	9	.	.	PUNCT
ejpam-5460	343	1	let	let	VERB
ejpam-5460	343	2	a	a	DET
ejpam-5460	343	3	∈	∈	NOUN
ejpam-5460	343	4	s.	s.	PROPN
ejpam-5460	343	5	since	since	SCONJ
ejpam-5460	343	6	s	s	PART
ejpam-5460	343	7	satisfies	satisfie	NOUN
ejpam-5460	343	8	(	(	PUNCT
ejpam-5460	343	9	c4	c4	NOUN
ejpam-5460	343	10	)	)	PUNCT
ejpam-5460	343	11	,	,	PUNCT
ejpam-5460	343	12	there	there	PRON
ejpam-5460	343	13	exist	exist	VERB
ejpam-5460	343	14	y1	y1	NOUN
ejpam-5460	343	15	,	,	PUNCT
ejpam-5460	343	16	y2	y2	INTJ
ejpam-5460	343	17	,	,	PUNCT
ejpam-5460	343	18	y3	y3	PROPN
ejpam-5460	343	19	∈	∈	PROPN
ejpam-5460	343	20	s	s	VERB
ejpam-5460	343	21	such	such	ADJ
ejpam-5460	343	22	that	that	SCONJ
ejpam-5460	343	23	a	a	DET
ejpam-5460	343	24	≤	≤	NUM
ejpam-5460	343	25	y1ay2ay3	y1ay2ay3	NOUN
ejpam-5460	343	26	≤	≤	NUM
ejpam-5460	343	27	y1ay2y1ay2ay3y3	y1ay2y1ay2ay3y3	NUM
ejpam-5460	343	28	≤	≤	NOUN
ejpam-5460	343	29	y1ay2y1ay2y1ay2ay3y3y3	y1ay2y1ay2y1ay2ay3y3y3	NOUN
ejpam-5460	343	30	.	.	PUNCT
ejpam-5460	344	1	that	that	PRON
ejpam-5460	344	2	is	be	AUX
ejpam-5460	344	3	,	,	PUNCT
ejpam-5460	344	4	(	(	PUNCT
ejpam-5460	344	5	x1ax2ax3a	x1ax2ax3a	PROPN
ejpam-5460	344	6	,	,	PUNCT
ejpam-5460	344	7	x4ax5	x4ax5	PROPN
ejpam-5460	344	8	)	)	PUNCT
ejpam-5460	344	9	∈	∈	PROPN
ejpam-5460	344	10	sa	sa	NOUN
ejpam-5460	344	11	for	for	ADP
ejpam-5460	344	12	some	some	DET
ejpam-5460	344	13	x1	x1	PROPN
ejpam-5460	344	14	,	,	PUNCT
ejpam-5460	344	15	x2	x2	PROPN
ejpam-5460	344	16	,	,	PUNCT
ejpam-5460	344	17	x3	x3	PROPN
ejpam-5460	344	18	,	,	PUNCT
ejpam-5460	344	19	x4	x4	PROPN
ejpam-5460	344	20	,	,	PUNCT
ejpam-5460	344	21	x5	x5	PROPN
ejpam-5460	344	22	∈	∈	PROPN
ejpam-5460	344	23	s.	s.	PROPN
ejpam-5460	344	24	then	then	ADV
ejpam-5460	344	25	,	,	PUNCT
ejpam-5460	344	26	(	(	PUNCT
ejpam-5460	344	27	f	f	X
ejpam-5460	344	28	◦	◦	NOUN
ejpam-5460	344	29	i	i	PROPN
ejpam-5460	344	30	g	g	PROPN
ejpam-5460	344	31	◦	◦	NOUN
ejpam-5460	344	32	i	i	PRON
ejpam-5460	344	33	f)(a	f)(a	NOUN
ejpam-5460	344	34	)	)	PUNCT
ejpam-5460	344	35	=	=	PUNCT
ejpam-5460	345	1	[	[	X
ejpam-5460	345	2	(	(	PUNCT
ejpam-5460	345	3	f	f	NOUN
ejpam-5460	345	4	◦	◦	NOUN
ejpam-5460	345	5	i	i	NOUN
ejpam-5460	345	6	g	g	NOUN
ejpam-5460	345	7	)	)	PUNCT
ejpam-5460	345	8	◦	◦	NOUN
ejpam-5460	345	9	f	f	X
ejpam-5460	346	1	]	]	X
ejpam-5460	346	2	i	i	PRON
ejpam-5460	346	3	(	(	PUNCT
ejpam-5460	346	4	a	a	X
ejpam-5460	346	5	)	)	PUNCT
ejpam-5460	346	6	≥	≥	NOUN
ejpam-5460	346	7	(	(	PUNCT
ejpam-5460	346	8	f	f	PROPN
ejpam-5460	346	9	◦	◦	NOUN
ejpam-5460	346	10	i	i	NOUN
ejpam-5460	346	11	g)i(x1ax2ax3a	g)i(x1ax2ax3a	NOUN
ejpam-5460	346	12	)	)	PUNCT
ejpam-5460	346	13	∧	∧	PROPN
ejpam-5460	346	14	fi(x4ax5	fi(x4ax5	NUM
ejpam-5460	346	15	)	)	PUNCT
ejpam-5460	346	16	≥	≥	NUM
ejpam-5460	346	17	(	(	PUNCT
ejpam-5460	346	18	f	f	PROPN
ejpam-5460	346	19	◦	◦	NOUN
ejpam-5460	346	20	i	i	NOUN
ejpam-5460	346	21	g)i(x1ax2ax3a	g)i(x1ax2ax3a	NOUN
ejpam-5460	346	22	)	)	PUNCT
ejpam-5460	346	23	∧	∧	NOUN
ejpam-5460	346	24	fi(a	fi(a	NOUN
ejpam-5460	346	25	)	)	PUNCT
ejpam-5460	346	26	=	=	PUNCT
ejpam-5460	347	1	(	(	PUNCT
ejpam-5460	347	2	f	f	X
ejpam-5460	347	3	◦	◦	NOUN
ejpam-5460	347	4	g)i(x1ax2ax3a	g)i(x1ax2ax3a	NOUN
ejpam-5460	347	5	)	)	PUNCT
ejpam-5460	347	6	∧	∧	NOUN
ejpam-5460	347	7	fi(a	fi(a	NOUN
ejpam-5460	347	8	)	)	PUNCT
ejpam-5460	347	9	=	=	SYM
ejpam-5460	347	10	fi(x1ax2	fi(x1ax2	NOUN
ejpam-5460	347	11	)	)	PUNCT
ejpam-5460	347	12	∧	∧	PROPN
ejpam-5460	347	13	gi(ax3a	gi(ax3a	NOUN
ejpam-5460	347	14	)	)	PUNCT
ejpam-5460	347	15	∧	∧	PROPN
ejpam-5460	347	16	fi(a	fi(a	PROPN
ejpam-5460	347	17	)	)	PUNCT
ejpam-5460	347	18	≥	≥	NOUN
ejpam-5460	347	19	fi(a	fi(a	CCONJ
ejpam-5460	347	20	)	)	PUNCT
ejpam-5460	347	21	∧	∧	NOUN
ejpam-5460	347	22	gi(a	gi(a	X
ejpam-5460	347	23	)	)	PUNCT
ejpam-5460	347	24	∧	∧	NOUN
ejpam-5460	347	25	fi(a	fi(a	NOUN
ejpam-5460	347	26	)	)	PUNCT
ejpam-5460	347	27	=	=	PUNCT
ejpam-5460	348	1	(	(	PUNCT
ejpam-5460	348	2	f	f	PROPN
ejpam-5460	348	3	∩i	∩i	PROPN
ejpam-5460	348	4	g)(a	g)(a	PROPN
ejpam-5460	348	5	)	)	PUNCT
ejpam-5460	348	6	.	.	PUNCT
ejpam-5460	349	1	(	(	PUNCT
ejpam-5460	349	2	2)⇒(1	2)⇒(1	NOUN
ejpam-5460	349	3	)	)	PUNCT
ejpam-5460	349	4	.	.	PUNCT
ejpam-5460	350	1	let	let	VERB
ejpam-5460	350	2	a	a	PRON
ejpam-5460	350	3	and	and	CCONJ
ejpam-5460	350	4	b	b	NOUN
ejpam-5460	350	5	be	be	AUX
ejpam-5460	350	6	a	a	DET
ejpam-5460	350	7	1	1	NUM
ejpam-5460	350	8	-	-	PUNCT
ejpam-5460	350	9	interior	interior	ADJ
ejpam-5460	350	10	ideal	ideal	NOUN
ejpam-5460	350	11	and	and	CCONJ
ejpam-5460	350	12	a	a	DET
ejpam-5460	350	13	(	(	PUNCT
ejpam-5460	350	14	1	1	NUM
ejpam-5460	350	15	,	,	PUNCT
ejpam-5460	350	16	1)-ideal	1)-ideal	NUM
ejpam-5460	350	17	of	of	ADP
ejpam-5460	350	18	s	s	PROPN
ejpam-5460	350	19	,	,	PUNCT
ejpam-5460	350	20	respectively	respectively	ADV
ejpam-5460	350	21	.	.	PUNCT
ejpam-5460	351	1	then	then	ADV
ejpam-5460	351	2	,	,	PUNCT
ejpam-5460	351	3	by	by	ADP
ejpam-5460	351	4	proposition	proposition	NOUN
ejpam-5460	351	5	2	2	NUM
ejpam-5460	351	6	,	,	PUNCT
ejpam-5460	351	7	(	(	PUNCT
ejpam-5460	351	8	χa)i	χa)i	NUM
ejpam-5460	351	9	and	and	CCONJ
ejpam-5460	351	10	(	(	PUNCT
ejpam-5460	351	11	χb)i	χb)i	PROPN
ejpam-5460	351	12	is	be	AUX
ejpam-5460	351	13	an	an	DET
ejpam-5460	351	14	(	(	PUNCT
ejpam-5460	351	15	α	α	NOUN
ejpam-5460	351	16	,	,	PUNCT
ejpam-5460	351	17	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	351	18	1	1	NUM
ejpam-5460	351	19	-	-	PUNCT
ejpam-5460	351	20	interior	interior	ADJ
ejpam-5460	351	21	ideal	ideal	NOUN
ejpam-5460	351	22	and	and	CCONJ
ejpam-5460	351	23	an	an	DET
ejpam-5460	351	24	(	(	PUNCT
ejpam-5460	351	25	α	α	NOUN
ejpam-5460	351	26	,	,	PUNCT
ejpam-5460	351	27	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	351	28	(	(	PUNCT
ejpam-5460	351	29	1	1	NUM
ejpam-5460	351	30	,	,	PUNCT
ejpam-5460	351	31	1)-ideal	1)-ideal	NUM
ejpam-5460	351	32	of	of	ADP
ejpam-5460	351	33	s	s	PROPN
ejpam-5460	351	34	,	,	PUNCT
ejpam-5460	351	35	respectively	respectively	ADV
ejpam-5460	351	36	.	.	PUNCT
ejpam-5460	352	1	by	by	ADP
ejpam-5460	352	2	our	our	PRON
ejpam-5460	352	3	presumption	presumption	NOUN
ejpam-5460	352	4	,	,	PUNCT
ejpam-5460	352	5	we	we	PRON
ejpam-5460	352	6	have	have	VERB
ejpam-5460	352	7	(	(	PUNCT
ejpam-5460	352	8	χa)i	χa)i	PROPN
ejpam-5460	352	9	∩i	∩i	NOUN
ejpam-5460	352	10	(	(	PUNCT
ejpam-5460	352	11	χb)i	χb)i	PROPN
ejpam-5460	352	12	⊆	⊆	NUM
ejpam-5460	352	13	(	(	PUNCT
ejpam-5460	352	14	χa)i	χa)i	PROPN
ejpam-5460	352	15	◦	◦	NOUN
ejpam-5460	352	16	i	i	PROPN
ejpam-5460	352	17	(	(	PUNCT
ejpam-5460	352	18	χb)i	χb)i	PROPN
ejpam-5460	352	19	◦	◦	NOUN
ejpam-5460	352	20	i	i	NOUN
ejpam-5460	352	21	(	(	PUNCT
ejpam-5460	352	22	χa)i	χa)i	PROPN
ejpam-5460	352	23	.	.	PUNCT
ejpam-5460	353	1	this	this	PRON
ejpam-5460	353	2	implies	imply	VERB
ejpam-5460	353	3	that	that	SCONJ
ejpam-5460	353	4	(	(	PUNCT
ejpam-5460	353	5	χa∩b)i	χa∩b)i	NOUN
ejpam-5460	353	6	=	=	NOUN
ejpam-5460	353	7	χa	χa	NOUN
ejpam-5460	353	8	∩i	∩i	ADV
ejpam-5460	353	9	χb	χb	PROPN
ejpam-5460	353	10	=	=	PUNCT
ejpam-5460	353	11	(	(	PUNCT
ejpam-5460	353	12	χa)i	χa)i	PROPN
ejpam-5460	353	13	∩i	∩i	NOUN
ejpam-5460	353	14	(	(	PUNCT
ejpam-5460	353	15	χb)i	χb)i	PROPN
ejpam-5460	353	16	⊆	⊆	NUM
ejpam-5460	353	17	(	(	PUNCT
ejpam-5460	353	18	χa)i	χa)i	PROPN
ejpam-5460	353	19	◦	◦	NOUN
ejpam-5460	353	20	i	i	PROPN
ejpam-5460	353	21	(	(	PUNCT
ejpam-5460	353	22	χb)i	χb)i	PROPN
ejpam-5460	353	23	◦	◦	NOUN
ejpam-5460	353	24	i	i	NOUN
ejpam-5460	353	25	(	(	PUNCT
ejpam-5460	353	26	χa)i	χa)i	PROPN
ejpam-5460	353	27	=	=	SYM
ejpam-5460	353	28	χa	χa	NOUN
ejpam-5460	353	29	◦	◦	NOUN
ejpam-5460	353	30	i	i	PRON
ejpam-5460	353	31	χb	χb	VERB
ejpam-5460	353	32	◦	◦	VERB
ejpam-5460	353	33	i	i	PRON
ejpam-5460	353	34	χa	χa	VERB
ejpam-5460	353	35	=	=	SYM
ejpam-5460	353	36	(	(	PUNCT
ejpam-5460	353	37	χ(aba])i	χ(aba])i	PROPN
ejpam-5460	353	38	.	.	PUNCT
ejpam-5460	354	1	by	by	ADP
ejpam-5460	354	2	proposition	proposition	NOUN
ejpam-5460	354	3	2	2	NUM
ejpam-5460	354	4	,	,	PUNCT
ejpam-5460	354	5	we	we	PRON
ejpam-5460	354	6	have	have	VERB
ejpam-5460	354	7	a	a	DET
ejpam-5460	354	8	∩b	∩b	NOUN
ejpam-5460	354	9	⊆	⊆	NUM
ejpam-5460	354	10	(	(	PUNCT
ejpam-5460	354	11	aba	aba	PROPN
ejpam-5460	354	12	]	]	PUNCT
ejpam-5460	354	13	.	.	PUNCT
ejpam-5460	355	1	hence	hence	ADV
ejpam-5460	355	2	,	,	PUNCT
ejpam-5460	355	3	by	by	ADP
ejpam-5460	355	4	lemma	lemma	PROPN
ejpam-5460	355	5	4	4	NUM
ejpam-5460	355	6	,	,	PUNCT
ejpam-5460	355	7	s	s	PART
ejpam-5460	355	8	satisfies	satisfie	NOUN
ejpam-5460	355	9	(	(	PUNCT
ejpam-5460	355	10	c4	c4	NOUN
ejpam-5460	355	11	)	)	PUNCT
ejpam-5460	355	12	.	.	PUNCT
ejpam-5460	356	1	lemma	lemma	PROPN
ejpam-5460	356	2	5	5	NUM
ejpam-5460	356	3	(	(	PUNCT
ejpam-5460	356	4	[	[	X
ejpam-5460	356	5	49	49	NUM
ejpam-5460	356	6	]	]	PUNCT
ejpam-5460	356	7	)	)	PUNCT
ejpam-5460	356	8	.	.	PUNCT
ejpam-5460	357	1	let	let	VERB
ejpam-5460	357	2	s	s	PRON
ejpam-5460	357	3	be	be	AUX
ejpam-5460	357	4	an	an	DET
ejpam-5460	357	5	ordered	order	VERB
ejpam-5460	357	6	semigroup	semigroup	NOUN
ejpam-5460	357	7	.	.	PUNCT
ejpam-5460	358	1	then	then	ADV
ejpam-5460	358	2	,	,	PUNCT
ejpam-5460	358	3	s	s	NOUN
ejpam-5460	358	4	satisfies	satisfie	NOUN
ejpam-5460	358	5	(	(	PUNCT
ejpam-5460	358	6	c5	c5	PROPN
ejpam-5460	358	7	)	)	PUNCT
ejpam-5460	358	8	if	if	SCONJ
ejpam-5460	358	9	and	and	CCONJ
ejpam-5460	358	10	only	only	ADV
ejpam-5460	358	11	if	if	SCONJ
ejpam-5460	358	12	a	a	DET
ejpam-5460	358	13	∩b	∩b	NOUN
ejpam-5460	358	14	⊆	⊆	NUM
ejpam-5460	358	15	(	(	PUNCT
ejpam-5460	358	16	ab	ab	X
ejpam-5460	358	17	]	]	X
ejpam-5460	358	18	for	for	ADP
ejpam-5460	358	19	any	any	DET
ejpam-5460	358	20	1	1	NUM
ejpam-5460	358	21	-	-	PUNCT
ejpam-5460	358	22	interior	interior	ADJ
ejpam-5460	358	23	ideal	ideal	NOUN
ejpam-5460	358	24	a	a	PRON
ejpam-5460	358	25	and	and	CCONJ
ejpam-5460	358	26	(	(	PUNCT
ejpam-5460	358	27	1	1	NUM
ejpam-5460	358	28	,	,	PUNCT
ejpam-5460	358	29	1)-ideal	1)-ideal	NUM
ejpam-5460	358	30	b	b	PROPN
ejpam-5460	358	31	of	of	ADP
ejpam-5460	358	32	s.	s.	PROPN
ejpam-5460	358	33	theorem	theorem	VERB
ejpam-5460	358	34	6	6	NUM
ejpam-5460	358	35	.	.	PUNCT
ejpam-5460	359	1	let	let	VERB
ejpam-5460	359	2	s	s	PRON
ejpam-5460	359	3	be	be	AUX
ejpam-5460	359	4	an	an	DET
ejpam-5460	359	5	ordered	order	VERB
ejpam-5460	359	6	semigroup	semigroup	NOUN
ejpam-5460	359	7	.	.	PUNCT
ejpam-5460	360	1	then	then	ADV
ejpam-5460	360	2	,	,	PUNCT
ejpam-5460	360	3	the	the	DET
ejpam-5460	360	4	following	follow	VERB
ejpam-5460	360	5	statements	statement	NOUN
ejpam-5460	360	6	are	be	AUX
ejpam-5460	360	7	equivalent	equivalent	ADJ
ejpam-5460	360	8	.	.	PUNCT
ejpam-5460	361	1	s.	s.	PROPN
ejpam-5460	361	2	lekkoksung	lekkoksung	PROPN
ejpam-5460	361	3	,	,	PUNCT
ejpam-5460	361	4	b.	b.	PROPN
ejpam-5460	361	5	davvaz	davvaz	PROPN
ejpam-5460	361	6	,	,	PUNCT
ejpam-5460	361	7	n.	n.	PROPN
ejpam-5460	361	8	lekkoksung	lekkoksung	PROPN
ejpam-5460	361	9	/	/	SYM
ejpam-5460	361	10	eur	eur	PROPN
ejpam-5460	361	11	.	.	PUNCT
ejpam-5460	362	1	j.	j.	PROPN
ejpam-5460	362	2	pure	pure	PROPN
ejpam-5460	362	3	appl	appl	PROPN
ejpam-5460	362	4	.	.	PROPN
ejpam-5460	362	5	math	math	PROPN
ejpam-5460	362	6	,	,	PUNCT
ejpam-5460	362	7	17	17	NUM
ejpam-5460	362	8	(	(	PUNCT
ejpam-5460	362	9	4	4	NUM
ejpam-5460	362	10	)	)	PUNCT
ejpam-5460	362	11	(	(	PUNCT
ejpam-5460	362	12	2024	2024	NUM
ejpam-5460	362	13	)	)	PUNCT
ejpam-5460	362	14	,	,	PUNCT
ejpam-5460	362	15	2962	2962	NUM
ejpam-5460	362	16	-	-	SYM
ejpam-5460	362	17	2984	2984	NUM
ejpam-5460	362	18	2975	2975	NUM
ejpam-5460	362	19	(	(	PUNCT
ejpam-5460	362	20	i	i	NOUN
ejpam-5460	362	21	)	)	PUNCT
ejpam-5460	362	22	s	s	PART
ejpam-5460	362	23	satisfies	satisfie	NOUN
ejpam-5460	362	24	(	(	PUNCT
ejpam-5460	362	25	c5	c5	PROPN
ejpam-5460	362	26	)	)	PUNCT
ejpam-5460	362	27	.	.	PUNCT
ejpam-5460	363	1	(	(	PUNCT
ejpam-5460	363	2	ii	ii	X
ejpam-5460	363	3	)	)	PUNCT
ejpam-5460	363	4	f	f	PROPN
ejpam-5460	363	5	∩i	∩i	PROPN
ejpam-5460	363	6	g	g	PROPN
ejpam-5460	363	7	⊆	⊆	NUM
ejpam-5460	363	8	f	f	PROPN
ejpam-5460	363	9	◦	◦	NOUN
ejpam-5460	363	10	i	i	PRON
ejpam-5460	363	11	g	g	NOUN
ejpam-5460	363	12	for	for	ADP
ejpam-5460	363	13	any	any	DET
ejpam-5460	363	14	(	(	PUNCT
ejpam-5460	363	15	α	α	NOUN
ejpam-5460	363	16	,	,	PUNCT
ejpam-5460	363	17	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	363	18	1	1	NUM
ejpam-5460	363	19	-	-	PUNCT
ejpam-5460	363	20	interior	interior	ADJ
ejpam-5460	363	21	ideal	ideal	NOUN
ejpam-5460	363	22	f	f	PROPN
ejpam-5460	363	23	and	and	CCONJ
ejpam-5460	363	24	(	(	PUNCT
ejpam-5460	363	25	α	α	NOUN
ejpam-5460	363	26	,	,	PUNCT
ejpam-5460	363	27	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	363	28	(	(	PUNCT
ejpam-5460	363	29	1	1	NUM
ejpam-5460	363	30	,	,	PUNCT
ejpam-5460	363	31	1)-ideal	1)-ideal	NUM
ejpam-5460	363	32	g	g	NOUN
ejpam-5460	363	33	of	of	ADP
ejpam-5460	363	34	s.	s.	PROPN
ejpam-5460	363	35	proof	proof	PROPN
ejpam-5460	363	36	.	.	PUNCT
ejpam-5460	364	1	(	(	PUNCT
ejpam-5460	364	2	1)⇒(2	1)⇒(2	NUM
ejpam-5460	364	3	)	)	PUNCT
ejpam-5460	364	4	.	.	PUNCT
ejpam-5460	365	1	let	let	VERB
ejpam-5460	365	2	f	f	PROPN
ejpam-5460	365	3	and	and	CCONJ
ejpam-5460	365	4	g	g	PROPN
ejpam-5460	365	5	be	be	AUX
ejpam-5460	365	6	an	an	DET
ejpam-5460	365	7	(	(	PUNCT
ejpam-5460	365	8	α	α	NOUN
ejpam-5460	365	9	,	,	PUNCT
ejpam-5460	365	10	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	365	11	1	1	NUM
ejpam-5460	365	12	-	-	PUNCT
ejpam-5460	365	13	interior	interior	ADJ
ejpam-5460	365	14	ideal	ideal	NOUN
ejpam-5460	365	15	and	and	CCONJ
ejpam-5460	365	16	an	an	DET
ejpam-5460	365	17	(	(	PUNCT
ejpam-5460	365	18	α	α	NOUN
ejpam-5460	365	19	,	,	PUNCT
ejpam-5460	365	20	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	365	21	(	(	PUNCT
ejpam-5460	365	22	1	1	NUM
ejpam-5460	365	23	,	,	PUNCT
ejpam-5460	365	24	1)-ideal	1)-ideal	NUM
ejpam-5460	365	25	of	of	ADP
ejpam-5460	365	26	s	s	PROPN
ejpam-5460	365	27	,	,	PUNCT
ejpam-5460	365	28	respectively	respectively	ADV
ejpam-5460	365	29	.	.	PUNCT
ejpam-5460	366	1	let	let	VERB
ejpam-5460	366	2	a	a	DET
ejpam-5460	366	3	∈	∈	NOUN
ejpam-5460	366	4	s.	s.	PROPN
ejpam-5460	366	5	since	since	SCONJ
ejpam-5460	366	6	s	s	PART
ejpam-5460	366	7	satisfies	satisfie	NOUN
ejpam-5460	366	8	(	(	PUNCT
ejpam-5460	366	9	c5	c5	PROPN
ejpam-5460	366	10	)	)	PUNCT
ejpam-5460	366	11	,	,	PUNCT
ejpam-5460	366	12	there	there	PRON
ejpam-5460	366	13	exist	exist	VERB
ejpam-5460	366	14	y1	y1	NOUN
ejpam-5460	366	15	,	,	PUNCT
ejpam-5460	366	16	y2	y2	PROPN
ejpam-5460	366	17	∈	∈	PROPN
ejpam-5460	366	18	s	s	VERB
ejpam-5460	366	19	such	such	ADJ
ejpam-5460	366	20	that	that	SCONJ
ejpam-5460	366	21	a	a	DET
ejpam-5460	366	22	≤	≤	NOUN
ejpam-5460	366	23	y1ay2a	y1ay2a	NOUN
ejpam-5460	366	24	≤	≤	NOUN
ejpam-5460	366	25	y1ay2y1ay2a	y1ay2y1ay2a	AUX
ejpam-5460	366	26	.	.	PUNCT
ejpam-5460	367	1	that	that	PRON
ejpam-5460	367	2	is	be	AUX
ejpam-5460	367	3	,	,	PUNCT
ejpam-5460	367	4	(	(	PUNCT
ejpam-5460	367	5	x1ax2	x1ax2	NOUN
ejpam-5460	367	6	,	,	PUNCT
ejpam-5460	367	7	ax3a	ax3a	PROPN
ejpam-5460	367	8	)	)	PUNCT
ejpam-5460	367	9	∈	∈	PROPN
ejpam-5460	367	10	sa	sa	NOUN
ejpam-5460	367	11	for	for	ADP
ejpam-5460	367	12	some	some	DET
ejpam-5460	367	13	x1	x1	PROPN
ejpam-5460	367	14	,	,	PUNCT
ejpam-5460	367	15	x2	x2	PROPN
ejpam-5460	367	16	,	,	PUNCT
ejpam-5460	367	17	x3	x3	PROPN
ejpam-5460	367	18	∈	∈	PROPN
ejpam-5460	367	19	s.	s.	PROPN
ejpam-5460	367	20	then	then	ADV
ejpam-5460	367	21	,	,	PUNCT
ejpam-5460	367	22	(	(	PUNCT
ejpam-5460	367	23	f	f	X
ejpam-5460	367	24	◦	◦	NOUN
ejpam-5460	367	25	i	i	PRON
ejpam-5460	367	26	g)(a	g)(a	VERB
ejpam-5460	367	27	)	)	PUNCT
ejpam-5460	367	28	≥	≥	NOUN
ejpam-5460	367	29	fi(x1ax2	fi(x1ax2	NOUN
ejpam-5460	367	30	)	)	PUNCT
ejpam-5460	367	31	∧	∧	PROPN
ejpam-5460	367	32	gi(ax3a	gi(ax3a	NOUN
ejpam-5460	367	33	)	)	PUNCT
ejpam-5460	367	34	≥	≥	NOUN
ejpam-5460	367	35	fi(a	fi(a	NOUN
ejpam-5460	367	36	)	)	PUNCT
ejpam-5460	367	37	∧	∧	NOUN
ejpam-5460	367	38	gi(a	gi(a	X
ejpam-5460	367	39	)	)	PUNCT
ejpam-5460	367	40	=	=	SYM
ejpam-5460	368	1	(	(	PUNCT
ejpam-5460	368	2	f	f	PROPN
ejpam-5460	368	3	∩i	∩i	PROPN
ejpam-5460	368	4	g)(a	g)(a	PROPN
ejpam-5460	368	5	)	)	PUNCT
ejpam-5460	368	6	.	.	PUNCT
ejpam-5460	369	1	(	(	PUNCT
ejpam-5460	369	2	2)⇒(1	2)⇒(1	NOUN
ejpam-5460	369	3	)	)	PUNCT
ejpam-5460	369	4	.	.	PUNCT
ejpam-5460	370	1	leta	leta	PROPN
ejpam-5460	370	2	andb	andb	PROPN
ejpam-5460	370	3	be	be	VERB
ejpam-5460	370	4	a	a	DET
ejpam-5460	370	5	1	1	NUM
ejpam-5460	370	6	-	-	PUNCT
ejpam-5460	370	7	interior	interior	ADJ
ejpam-5460	370	8	ideal	ideal	NOUN
ejpam-5460	370	9	and	and	CCONJ
ejpam-5460	370	10	a	a	DET
ejpam-5460	370	11	(	(	PUNCT
ejpam-5460	370	12	1	1	NUM
ejpam-5460	370	13	,	,	PUNCT
ejpam-5460	370	14	1)-ideal	1)-ideal	NUM
ejpam-5460	370	15	of	of	ADP
ejpam-5460	370	16	s	s	PROPN
ejpam-5460	370	17	,	,	PUNCT
ejpam-5460	370	18	respectively	respectively	ADV
ejpam-5460	370	19	.	.	PUNCT
ejpam-5460	371	1	then	then	ADV
ejpam-5460	371	2	,	,	PUNCT
ejpam-5460	371	3	by	by	ADP
ejpam-5460	371	4	proposition	proposition	NOUN
ejpam-5460	371	5	2	2	NUM
ejpam-5460	371	6	,	,	PUNCT
ejpam-5460	371	7	(	(	PUNCT
ejpam-5460	371	8	χa)i	χa)i	NUM
ejpam-5460	371	9	and	and	CCONJ
ejpam-5460	371	10	(	(	PUNCT
ejpam-5460	371	11	χb)i	χb)i	PROPN
ejpam-5460	371	12	is	be	AUX
ejpam-5460	371	13	an	an	DET
ejpam-5460	371	14	(	(	PUNCT
ejpam-5460	371	15	α	α	NOUN
ejpam-5460	371	16	,	,	PUNCT
ejpam-5460	371	17	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	371	18	1	1	NUM
ejpam-5460	371	19	-	-	PUNCT
ejpam-5460	371	20	interior	interior	ADJ
ejpam-5460	371	21	ideal	ideal	NOUN
ejpam-5460	371	22	and	and	CCONJ
ejpam-5460	371	23	an	an	DET
ejpam-5460	371	24	(	(	PUNCT
ejpam-5460	371	25	α	α	NOUN
ejpam-5460	371	26	,	,	PUNCT
ejpam-5460	371	27	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	371	28	(	(	PUNCT
ejpam-5460	371	29	1	1	NUM
ejpam-5460	371	30	,	,	PUNCT
ejpam-5460	371	31	1)ideal	1)ideal	NUM
ejpam-5460	371	32	of	of	ADP
ejpam-5460	371	33	s	s	NOUN
ejpam-5460	371	34	,	,	PUNCT
ejpam-5460	371	35	respectively	respectively	ADV
ejpam-5460	371	36	.	.	PUNCT
ejpam-5460	372	1	by	by	ADP
ejpam-5460	372	2	our	our	PRON
ejpam-5460	372	3	presumption	presumption	NOUN
ejpam-5460	372	4	,	,	PUNCT
ejpam-5460	372	5	we	we	PRON
ejpam-5460	372	6	have	have	VERB
ejpam-5460	372	7	(	(	PUNCT
ejpam-5460	372	8	χa)i	χa)i	PROPN
ejpam-5460	372	9	∩i	∩i	NOUN
ejpam-5460	372	10	(	(	PUNCT
ejpam-5460	372	11	χb)i	χb)i	PROPN
ejpam-5460	372	12	⊆	⊆	NUM
ejpam-5460	372	13	(	(	PUNCT
ejpam-5460	372	14	χa)i	χa)i	PROPN
ejpam-5460	372	15	◦	◦	NOUN
ejpam-5460	372	16	i	i	PROPN
ejpam-5460	372	17	(	(	PUNCT
ejpam-5460	372	18	χb)i	χb)i	PROPN
ejpam-5460	372	19	.	.	PUNCT
ejpam-5460	373	1	this	this	PRON
ejpam-5460	373	2	implies	imply	VERB
ejpam-5460	373	3	that	that	SCONJ
ejpam-5460	373	4	(	(	PUNCT
ejpam-5460	373	5	χa∩b)i	χa∩b)i	NOUN
ejpam-5460	373	6	=	=	NOUN
ejpam-5460	373	7	χa	χa	NOUN
ejpam-5460	373	8	∩i	∩i	ADV
ejpam-5460	373	9	χb	χb	PROPN
ejpam-5460	373	10	=	=	PUNCT
ejpam-5460	373	11	(	(	PUNCT
ejpam-5460	373	12	χa)i	χa)i	PROPN
ejpam-5460	373	13	∩i	∩i	NOUN
ejpam-5460	373	14	(	(	PUNCT
ejpam-5460	373	15	χb)i	χb)i	PROPN
ejpam-5460	373	16	⊆	⊆	NUM
ejpam-5460	373	17	(	(	PUNCT
ejpam-5460	373	18	χa)i	χa)i	PROPN
ejpam-5460	373	19	◦	◦	NOUN
ejpam-5460	373	20	i	i	PROPN
ejpam-5460	373	21	(	(	PUNCT
ejpam-5460	373	22	χb)i	χb)i	PROPN
ejpam-5460	373	23	=	=	PUNCT
ejpam-5460	373	24	χa	χa	NOUN
ejpam-5460	373	25	◦	◦	NOUN
ejpam-5460	373	26	i	i	PRON
ejpam-5460	373	27	χb	χb	VERB
ejpam-5460	373	28	=	=	PUNCT
ejpam-5460	373	29	(	(	PUNCT
ejpam-5460	373	30	χ(ab])i	χ(ab])i	ADV
ejpam-5460	373	31	.	.	PUNCT
ejpam-5460	374	1	by	by	ADP
ejpam-5460	374	2	proposition	proposition	NOUN
ejpam-5460	374	3	2	2	NUM
ejpam-5460	374	4	,	,	PUNCT
ejpam-5460	374	5	we	we	PRON
ejpam-5460	374	6	have	have	VERB
ejpam-5460	374	7	a	a	DET
ejpam-5460	374	8	∩b	∩b	NOUN
ejpam-5460	374	9	⊆	⊆	NUM
ejpam-5460	374	10	(	(	PUNCT
ejpam-5460	374	11	ab	ab	X
ejpam-5460	374	12	]	]	X
ejpam-5460	374	13	.	.	PUNCT
ejpam-5460	375	1	hence	hence	ADV
ejpam-5460	375	2	,	,	PUNCT
ejpam-5460	375	3	by	by	ADP
ejpam-5460	375	4	lemma	lemma	PROPN
ejpam-5460	375	5	5	5	NUM
ejpam-5460	375	6	,	,	PUNCT
ejpam-5460	375	7	s	s	PART
ejpam-5460	375	8	satisfies	satisfie	NOUN
ejpam-5460	375	9	(	(	PUNCT
ejpam-5460	375	10	c5	c5	PROPN
ejpam-5460	375	11	)	)	PUNCT
ejpam-5460	375	12	.	.	PUNCT
ejpam-5460	376	1	similarly	similarly	ADV
ejpam-5460	376	2	to	to	ADP
ejpam-5460	376	3	the	the	DET
ejpam-5460	376	4	above	above	ADJ
ejpam-5460	376	5	result	result	NOUN
ejpam-5460	376	6	,	,	PUNCT
ejpam-5460	376	7	we	we	PRON
ejpam-5460	376	8	obtain	obtain	VERB
ejpam-5460	376	9	a	a	DET
ejpam-5460	376	10	characterization	characterization	NOUN
ejpam-5460	376	11	of	of	ADP
ejpam-5460	376	12	ordered	order	VERB
ejpam-5460	376	13	semigroups	semigroup	NOUN
ejpam-5460	376	14	satisfying	satisfy	VERB
ejpam-5460	376	15	(	(	PUNCT
ejpam-5460	376	16	c6	c6	PROPN
ejpam-5460	376	17	)	)	PUNCT
ejpam-5460	376	18	as	as	SCONJ
ejpam-5460	376	19	follows	follow	VERB
ejpam-5460	376	20	.	.	PUNCT
ejpam-5460	377	1	theorem	theorem	ADJ
ejpam-5460	377	2	7	7	NUM
ejpam-5460	377	3	.	.	PUNCT
ejpam-5460	378	1	let	let	VERB
ejpam-5460	378	2	s	s	PRON
ejpam-5460	378	3	be	be	AUX
ejpam-5460	378	4	an	an	DET
ejpam-5460	378	5	ordered	order	VERB
ejpam-5460	378	6	semigroup	semigroup	NOUN
ejpam-5460	378	7	.	.	PUNCT
ejpam-5460	379	1	then	then	ADV
ejpam-5460	379	2	,	,	PUNCT
ejpam-5460	379	3	the	the	DET
ejpam-5460	379	4	following	follow	VERB
ejpam-5460	379	5	statements	statement	NOUN
ejpam-5460	379	6	are	be	AUX
ejpam-5460	379	7	equivalent	equivalent	ADJ
ejpam-5460	379	8	.	.	PUNCT
ejpam-5460	380	1	(	(	PUNCT
ejpam-5460	380	2	i	i	NOUN
ejpam-5460	380	3	)	)	PUNCT
ejpam-5460	380	4	s	s	PART
ejpam-5460	380	5	satisfies	satisfie	NOUN
ejpam-5460	380	6	(	(	PUNCT
ejpam-5460	380	7	c6	c6	PROPN
ejpam-5460	380	8	)	)	PUNCT
ejpam-5460	380	9	.	.	PUNCT
ejpam-5460	381	1	(	(	PUNCT
ejpam-5460	381	2	ii	ii	X
ejpam-5460	381	3	)	)	PUNCT
ejpam-5460	381	4	f	f	PROPN
ejpam-5460	381	5	∩i	∩i	PROPN
ejpam-5460	381	6	g	g	PROPN
ejpam-5460	381	7	⊆	⊆	NUM
ejpam-5460	381	8	f	f	PROPN
ejpam-5460	381	9	◦	◦	NOUN
ejpam-5460	381	10	i	i	PRON
ejpam-5460	381	11	g	g	NOUN
ejpam-5460	381	12	for	for	ADP
ejpam-5460	381	13	any	any	DET
ejpam-5460	381	14	(	(	PUNCT
ejpam-5460	381	15	α	α	NOUN
ejpam-5460	381	16	,	,	PUNCT
ejpam-5460	381	17	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	381	18	(	(	PUNCT
ejpam-5460	381	19	1	1	NUM
ejpam-5460	381	20	,	,	PUNCT
ejpam-5460	381	21	1)-ideal	1)-ideal	NUM
ejpam-5460	381	22	f	f	NOUN
ejpam-5460	381	23	and	and	CCONJ
ejpam-5460	381	24	(	(	PUNCT
ejpam-5460	381	25	α	α	NOUN
ejpam-5460	381	26	,	,	PUNCT
ejpam-5460	381	27	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	381	28	1	1	NUM
ejpam-5460	381	29	-	-	PUNCT
ejpam-5460	381	30	interior	interior	ADJ
ejpam-5460	381	31	ideal	ideal	NOUN
ejpam-5460	381	32	g	g	PROPN
ejpam-5460	381	33	of	of	ADP
ejpam-5460	381	34	s.	s.	PROPN
ejpam-5460	381	35	lemma	lemma	PROPN
ejpam-5460	381	36	6	6	NUM
ejpam-5460	381	37	(	(	PUNCT
ejpam-5460	381	38	[	[	X
ejpam-5460	381	39	36	36	NUM
ejpam-5460	381	40	]	]	PUNCT
ejpam-5460	381	41	)	)	PUNCT
ejpam-5460	381	42	.	.	PUNCT
ejpam-5460	382	1	let	let	VERB
ejpam-5460	382	2	s	s	PRON
ejpam-5460	382	3	be	be	AUX
ejpam-5460	382	4	an	an	DET
ejpam-5460	382	5	ordered	order	VERB
ejpam-5460	382	6	semigroup	semigroup	NOUN
ejpam-5460	382	7	.	.	PUNCT
ejpam-5460	383	1	then	then	ADV
ejpam-5460	383	2	,	,	PUNCT
ejpam-5460	383	3	s	s	NOUN
ejpam-5460	383	4	satisfies	satisfie	NOUN
ejpam-5460	383	5	(	(	PUNCT
ejpam-5460	383	6	c7	c7	PROPN
ejpam-5460	383	7	)	)	PUNCT
ejpam-5460	383	8	if	if	SCONJ
ejpam-5460	383	9	and	and	CCONJ
ejpam-5460	383	10	only	only	ADV
ejpam-5460	383	11	if	if	SCONJ
ejpam-5460	383	12	a	a	DET
ejpam-5460	383	13	∩b	∩b	NOUN
ejpam-5460	383	14	∩	∩	NOUN
ejpam-5460	383	15	c	c	NOUN
ejpam-5460	383	16	⊆	⊆	NUM
ejpam-5460	383	17	(	(	PUNCT
ejpam-5460	383	18	abc	abc	PROPN
ejpam-5460	383	19	]	]	X
ejpam-5460	383	20	for	for	ADP
ejpam-5460	383	21	any	any	DET
ejpam-5460	383	22	(	(	PUNCT
ejpam-5460	383	23	1	1	NUM
ejpam-5460	383	24	,	,	PUNCT
ejpam-5460	383	25	0)-ideal	0)-ideal	NOUN
ejpam-5460	383	26	a	a	DET
ejpam-5460	383	27	,	,	PUNCT
ejpam-5460	383	28	1	1	NUM
ejpam-5460	383	29	-	-	ADJ
ejpam-5460	383	30	interior	interior	ADJ
ejpam-5460	383	31	ideal	ideal	NOUN
ejpam-5460	383	32	b	b	NOUN
ejpam-5460	383	33	,	,	PUNCT
ejpam-5460	383	34	and	and	CCONJ
ejpam-5460	383	35	(	(	PUNCT
ejpam-5460	383	36	0	0	NUM
ejpam-5460	383	37	,	,	PUNCT
ejpam-5460	383	38	1)-ideal	1)-ideal	NUM
ejpam-5460	383	39	c	c	NOUN
ejpam-5460	383	40	of	of	ADP
ejpam-5460	383	41	s.	s.	PROPN
ejpam-5460	383	42	theorem	theorem	VERB
ejpam-5460	383	43	8	8	NUM
ejpam-5460	383	44	.	.	PUNCT
ejpam-5460	384	1	let	let	VERB
ejpam-5460	384	2	s	s	PRON
ejpam-5460	384	3	be	be	AUX
ejpam-5460	384	4	an	an	DET
ejpam-5460	384	5	ordered	order	VERB
ejpam-5460	384	6	semigroup	semigroup	NOUN
ejpam-5460	384	7	.	.	PUNCT
ejpam-5460	385	1	then	then	ADV
ejpam-5460	385	2	,	,	PUNCT
ejpam-5460	385	3	the	the	DET
ejpam-5460	385	4	following	follow	VERB
ejpam-5460	385	5	statements	statement	NOUN
ejpam-5460	385	6	are	be	AUX
ejpam-5460	385	7	equivalent	equivalent	ADJ
ejpam-5460	385	8	.	.	PUNCT
ejpam-5460	386	1	(	(	PUNCT
ejpam-5460	386	2	i	i	NOUN
ejpam-5460	386	3	)	)	PUNCT
ejpam-5460	386	4	s	s	PART
ejpam-5460	386	5	satisfies	satisfie	NOUN
ejpam-5460	386	6	(	(	PUNCT
ejpam-5460	386	7	c7	c7	PROPN
ejpam-5460	386	8	)	)	PUNCT
ejpam-5460	386	9	.	.	PUNCT
ejpam-5460	387	1	(	(	PUNCT
ejpam-5460	387	2	ii	ii	X
ejpam-5460	387	3	)	)	PUNCT
ejpam-5460	387	4	f	f	PROPN
ejpam-5460	387	5	∩i	∩i	PROPN
ejpam-5460	387	6	g	g	PROPN
ejpam-5460	387	7	∩i	∩i	PROPN
ejpam-5460	387	8	h	h	NOUN
ejpam-5460	388	1	⊆	⊆	NUM
ejpam-5460	388	2	f	f	X
ejpam-5460	388	3	◦	◦	NOUN
ejpam-5460	388	4	i	i	PROPN
ejpam-5460	388	5	g	g	PROPN
ejpam-5460	388	6	◦	◦	NOUN
ejpam-5460	388	7	i	i	PRON
ejpam-5460	388	8	h	h	NOUN
ejpam-5460	388	9	for	for	ADP
ejpam-5460	388	10	any	any	DET
ejpam-5460	388	11	(	(	PUNCT
ejpam-5460	388	12	α	α	NOUN
ejpam-5460	388	13	,	,	PUNCT
ejpam-5460	388	14	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	388	15	(	(	PUNCT
ejpam-5460	388	16	1	1	NUM
ejpam-5460	388	17	,	,	PUNCT
ejpam-5460	388	18	0)-ideal	0)-ideal	PROPN
ejpam-5460	388	19	f	f	PROPN
ejpam-5460	388	20	,	,	PUNCT
ejpam-5460	388	21	(	(	PUNCT
ejpam-5460	388	22	α	α	NOUN
ejpam-5460	388	23	,	,	PUNCT
ejpam-5460	388	24	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	388	25	1	1	NUM
ejpam-5460	388	26	-	-	PUNCT
ejpam-5460	388	27	interior	interior	ADJ
ejpam-5460	388	28	ideal	ideal	NOUN
ejpam-5460	388	29	g	g	NOUN
ejpam-5460	388	30	,	,	PUNCT
ejpam-5460	388	31	and	and	CCONJ
ejpam-5460	388	32	(	(	PUNCT
ejpam-5460	388	33	α	α	NOUN
ejpam-5460	388	34	,	,	PUNCT
ejpam-5460	388	35	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	388	36	(	(	PUNCT
ejpam-5460	388	37	0	0	NUM
ejpam-5460	388	38	,	,	PUNCT
ejpam-5460	388	39	1)-ideal	1)-ideal	NUM
ejpam-5460	388	40	h	h	NOUN
ejpam-5460	388	41	of	of	ADP
ejpam-5460	388	42	s.	s.	PROPN
ejpam-5460	388	43	proof	proof	PROPN
ejpam-5460	388	44	.	.	PUNCT
ejpam-5460	389	1	(	(	PUNCT
ejpam-5460	389	2	1)⇒(2	1)⇒(2	NUM
ejpam-5460	389	3	)	)	PUNCT
ejpam-5460	389	4	.	.	PUNCT
ejpam-5460	390	1	let	let	VERB
ejpam-5460	390	2	f	f	X
ejpam-5460	390	3	,	,	PUNCT
ejpam-5460	390	4	g	g	PROPN
ejpam-5460	390	5	and	and	CCONJ
ejpam-5460	390	6	h	h	NOUN
ejpam-5460	390	7	be	be	VERB
ejpam-5460	390	8	an	an	DET
ejpam-5460	390	9	(	(	PUNCT
ejpam-5460	390	10	α	α	NOUN
ejpam-5460	390	11	,	,	PUNCT
ejpam-5460	390	12	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	390	13	(	(	PUNCT
ejpam-5460	390	14	1	1	NUM
ejpam-5460	390	15	,	,	PUNCT
ejpam-5460	390	16	0)-ideal	0)-ideal	PROPN
ejpam-5460	390	17	,	,	PUNCT
ejpam-5460	390	18	an	an	DET
ejpam-5460	390	19	(	(	PUNCT
ejpam-5460	390	20	α	α	NOUN
ejpam-5460	390	21	,	,	PUNCT
ejpam-5460	390	22	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	390	23	1	1	NUM
ejpam-5460	390	24	-	-	PUNCT
ejpam-5460	390	25	interior	interior	ADJ
ejpam-5460	390	26	ideal	ideal	NOUN
ejpam-5460	390	27	,	,	PUNCT
ejpam-5460	390	28	and	and	CCONJ
ejpam-5460	390	29	an	an	DET
ejpam-5460	390	30	(	(	PUNCT
ejpam-5460	390	31	α	α	NOUN
ejpam-5460	390	32	,	,	PUNCT
ejpam-5460	390	33	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	390	34	(	(	PUNCT
ejpam-5460	390	35	0	0	NUM
ejpam-5460	390	36	,	,	PUNCT
ejpam-5460	390	37	1)-ideal	1)-ideal	NUM
ejpam-5460	390	38	of	of	ADP
ejpam-5460	390	39	s	s	PROPN
ejpam-5460	390	40	,	,	PUNCT
ejpam-5460	390	41	respectively	respectively	ADV
ejpam-5460	390	42	.	.	PUNCT
ejpam-5460	391	1	let	let	VERB
ejpam-5460	391	2	a	a	DET
ejpam-5460	391	3	∈	∈	NOUN
ejpam-5460	391	4	s.	s.	PROPN
ejpam-5460	391	5	since	since	SCONJ
ejpam-5460	391	6	s	s	PART
ejpam-5460	391	7	satisfies	satisfie	NOUN
ejpam-5460	391	8	(	(	PUNCT
ejpam-5460	391	9	c7	c7	PROPN
ejpam-5460	391	10	)	)	PUNCT
ejpam-5460	391	11	,	,	PUNCT
ejpam-5460	391	12	there	there	PRON
ejpam-5460	391	13	exist	exist	VERB
ejpam-5460	391	14	y1	y1	PROPN
ejpam-5460	391	15	∈	∈	PROPN
ejpam-5460	391	16	s	s	VERB
ejpam-5460	391	17	such	such	ADJ
ejpam-5460	391	18	that	that	SCONJ
ejpam-5460	391	19	a	a	DET
ejpam-5460	391	20	≤	≤	ADJ
ejpam-5460	391	21	ay1a	ay1a	PROPN
ejpam-5460	391	22	≤	≤	NUM
ejpam-5460	391	23	ay1ay1a	ay1ay1a	PROPN
ejpam-5460	391	24	≤	≤	PROPN
ejpam-5460	391	25	ay1ay1ay1a	ay1ay1ay1a	PROPN
ejpam-5460	391	26	≤	≤	NOUN
ejpam-5460	391	27	ay1ay1ay1ay1a	ay1ay1ay1ay1a	NOUN
ejpam-5460	391	28	.	.	PUNCT
ejpam-5460	392	1	that	that	PRON
ejpam-5460	392	2	is	be	AUX
ejpam-5460	392	3	,	,	PUNCT
ejpam-5460	392	4	(	(	PUNCT
ejpam-5460	392	5	ax1x2ax3	ax1x2ax3	INTJ
ejpam-5460	392	6	,	,	PUNCT
ejpam-5460	392	7	x4a	x4a	NUM
ejpam-5460	392	8	)	)	PUNCT
ejpam-5460	392	9	∈	∈	PROPN
ejpam-5460	392	10	sa	sa	NOUN
ejpam-5460	392	11	for	for	ADP
ejpam-5460	392	12	some	some	PRON
ejpam-5460	392	13	x1	x1	PROPN
ejpam-5460	392	14	,	,	PUNCT
ejpam-5460	392	15	x2	x2	PROPN
ejpam-5460	392	16	,	,	PUNCT
ejpam-5460	392	17	x3	x3	ADJ
ejpam-5460	392	18	,	,	PUNCT
ejpam-5460	392	19	x4	x4	PROPN
ejpam-5460	392	20	∈	∈	PROPN
ejpam-5460	392	21	s.	s.	PROPN
ejpam-5460	392	22	then	then	ADV
ejpam-5460	392	23	,	,	PUNCT
ejpam-5460	392	24	(	(	PUNCT
ejpam-5460	392	25	f	f	X
ejpam-5460	392	26	◦	◦	NOUN
ejpam-5460	392	27	i	i	PROPN
ejpam-5460	392	28	g	g	PROPN
ejpam-5460	392	29	◦	◦	NOUN
ejpam-5460	392	30	i	i	PRON
ejpam-5460	392	31	h)(a	h)(a	NOUN
ejpam-5460	392	32	)	)	PUNCT
ejpam-5460	393	1	=	=	PUNCT
ejpam-5460	394	1	[	[	X
ejpam-5460	394	2	(	(	PUNCT
ejpam-5460	394	3	f	f	NOUN
ejpam-5460	394	4	◦	◦	NOUN
ejpam-5460	394	5	i	i	PRON
ejpam-5460	394	6	g	g	NOUN
ejpam-5460	394	7	)	)	PUNCT
ejpam-5460	394	8	◦	◦	NOUN
ejpam-5460	394	9	h]i	h]i	NOUN
ejpam-5460	394	10	(	(	PUNCT
ejpam-5460	394	11	a	a	X
ejpam-5460	394	12	)	)	PUNCT
ejpam-5460	394	13	s.	s.	PROPN
ejpam-5460	394	14	lekkoksung	lekkoksung	PROPN
ejpam-5460	394	15	,	,	PUNCT
ejpam-5460	394	16	b.	b.	PROPN
ejpam-5460	394	17	davvaz	davvaz	PROPN
ejpam-5460	394	18	,	,	PUNCT
ejpam-5460	394	19	n.	n.	PROPN
ejpam-5460	394	20	lekkoksung	lekkoksung	PROPN
ejpam-5460	394	21	/	/	SYM
ejpam-5460	394	22	eur	eur	PROPN
ejpam-5460	394	23	.	.	PUNCT
ejpam-5460	395	1	j.	j.	PROPN
ejpam-5460	395	2	pure	pure	PROPN
ejpam-5460	395	3	appl	appl	PROPN
ejpam-5460	395	4	.	.	PROPN
ejpam-5460	395	5	math	math	PROPN
ejpam-5460	395	6	,	,	PUNCT
ejpam-5460	395	7	17	17	NUM
ejpam-5460	395	8	(	(	PUNCT
ejpam-5460	395	9	4	4	NUM
ejpam-5460	395	10	)	)	PUNCT
ejpam-5460	395	11	(	(	PUNCT
ejpam-5460	395	12	2024	2024	NUM
ejpam-5460	395	13	)	)	PUNCT
ejpam-5460	395	14	,	,	PUNCT
ejpam-5460	395	15	2962	2962	NUM
ejpam-5460	395	16	-	-	SYM
ejpam-5460	395	17	2984	2984	NUM
ejpam-5460	395	18	2976	2976	NUM
ejpam-5460	395	19	≥	≥	NOUN
ejpam-5460	395	20	(	(	PUNCT
ejpam-5460	395	21	f	f	PROPN
ejpam-5460	395	22	◦	◦	NOUN
ejpam-5460	395	23	i	i	PRON
ejpam-5460	395	24	g)i(ax1x2ax3	g)i(ax1x2ax3	VERB
ejpam-5460	395	25	)	)	PUNCT
ejpam-5460	395	26	∧	∧	PROPN
ejpam-5460	395	27	hi(x4a	hi(x4a	PROPN
ejpam-5460	395	28	)	)	PUNCT
ejpam-5460	395	29	≥	≥	NOUN
ejpam-5460	395	30	(	(	PUNCT
ejpam-5460	395	31	f	f	X
ejpam-5460	395	32	◦	◦	NOUN
ejpam-5460	395	33	i	i	PRON
ejpam-5460	395	34	g)i(ax1x2ax3	g)i(ax1x2ax3	VERB
ejpam-5460	395	35	)	)	PUNCT
ejpam-5460	395	36	∧	∧	NOUN
ejpam-5460	395	37	hi(a	hi(a	ADJ
ejpam-5460	395	38	)	)	PUNCT
ejpam-5460	395	39	=	=	SYM
ejpam-5460	396	1	(	(	PUNCT
ejpam-5460	396	2	f	f	X
ejpam-5460	396	3	◦	◦	NOUN
ejpam-5460	396	4	g)i(ax1x2ax3	g)i(ax1x2ax3	NOUN
ejpam-5460	396	5	)	)	PUNCT
ejpam-5460	396	6	∧	∧	NOUN
ejpam-5460	396	7	hi(a	hi(a	ADJ
ejpam-5460	396	8	)	)	PUNCT
ejpam-5460	396	9	=	=	SYM
ejpam-5460	396	10	fi(ax1	fi(ax1	NUM
ejpam-5460	396	11	)	)	PUNCT
ejpam-5460	396	12	∧	∧	PROPN
ejpam-5460	396	13	gi(x2ax3	gi(x2ax3	PROPN
ejpam-5460	396	14	)	)	PUNCT
ejpam-5460	396	15	∧	∧	PROPN
ejpam-5460	396	16	h(a	h(a	PROPN
ejpam-5460	396	17	)	)	PUNCT
ejpam-5460	396	18	≥	≥	NOUN
ejpam-5460	396	19	fi(a	fi(a	NOUN
ejpam-5460	396	20	)	)	PUNCT
ejpam-5460	396	21	∧	∧	NOUN
ejpam-5460	396	22	gi(a	gi(a	X
ejpam-5460	396	23	)	)	PUNCT
ejpam-5460	396	24	∧	∧	NOUN
ejpam-5460	396	25	hi(a	hi(a	ADJ
ejpam-5460	396	26	)	)	PUNCT
ejpam-5460	396	27	=	=	SYM
ejpam-5460	397	1	(	(	PUNCT
ejpam-5460	397	2	f	f	PROPN
ejpam-5460	397	3	∩i	∩i	PROPN
ejpam-5460	397	4	g	g	PROPN
ejpam-5460	397	5	∩i	∩i	PROPN
ejpam-5460	397	6	h)(a	h)(a	PROPN
ejpam-5460	397	7	)	)	PUNCT
ejpam-5460	397	8	.	.	PUNCT
ejpam-5460	398	1	(	(	PUNCT
ejpam-5460	398	2	2)⇒(1	2)⇒(1	NOUN
ejpam-5460	398	3	)	)	PUNCT
ejpam-5460	398	4	.	.	PUNCT
ejpam-5460	399	1	let	let	VERB
ejpam-5460	399	2	a	a	DET
ejpam-5460	399	3	,	,	PUNCT
ejpam-5460	399	4	b	b	NOUN
ejpam-5460	399	5	and	and	CCONJ
ejpam-5460	399	6	c	c	PROPN
ejpam-5460	399	7	be	be	AUX
ejpam-5460	399	8	a	a	DET
ejpam-5460	399	9	(	(	PUNCT
ejpam-5460	399	10	1	1	NUM
ejpam-5460	399	11	,	,	PUNCT
ejpam-5460	399	12	0)-ideal	0)-ideal	PROPN
ejpam-5460	399	13	,	,	PUNCT
ejpam-5460	399	14	a	a	DET
ejpam-5460	399	15	1	1	NUM
ejpam-5460	399	16	-	-	PUNCT
ejpam-5460	399	17	interior	interior	ADJ
ejpam-5460	399	18	ideal	ideal	NOUN
ejpam-5460	399	19	and	and	CCONJ
ejpam-5460	399	20	a	a	DET
ejpam-5460	399	21	(	(	PUNCT
ejpam-5460	399	22	0	0	NUM
ejpam-5460	399	23	,	,	PUNCT
ejpam-5460	399	24	1)-ideal	1)-ideal	NUM
ejpam-5460	399	25	of	of	ADP
ejpam-5460	399	26	s	s	PROPN
ejpam-5460	399	27	,	,	PUNCT
ejpam-5460	399	28	respectively	respectively	ADV
ejpam-5460	399	29	.	.	PUNCT
ejpam-5460	400	1	then	then	ADV
ejpam-5460	400	2	,	,	PUNCT
ejpam-5460	400	3	by	by	ADP
ejpam-5460	400	4	proposition	proposition	NOUN
ejpam-5460	400	5	2	2	NUM
ejpam-5460	400	6	,	,	PUNCT
ejpam-5460	400	7	(	(	PUNCT
ejpam-5460	400	8	χa)i	χa)i	NUM
ejpam-5460	400	9	,	,	PUNCT
ejpam-5460	400	10	(	(	PUNCT
ejpam-5460	400	11	χb)i	χb)i	PROPN
ejpam-5460	400	12	and	and	CCONJ
ejpam-5460	400	13	(	(	PUNCT
ejpam-5460	400	14	χc)i	χc)i	PROPN
ejpam-5460	400	15	is	be	AUX
ejpam-5460	400	16	an	an	DET
ejpam-5460	400	17	(	(	PUNCT
ejpam-5460	400	18	α	α	NOUN
ejpam-5460	400	19	,	,	PUNCT
ejpam-5460	400	20	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	400	21	(	(	PUNCT
ejpam-5460	400	22	1	1	NUM
ejpam-5460	400	23	,	,	PUNCT
ejpam-5460	400	24	0)-ideal	0)-ideal	PROPN
ejpam-5460	400	25	,	,	PUNCT
ejpam-5460	400	26	an	an	DET
ejpam-5460	400	27	(	(	PUNCT
ejpam-5460	400	28	α	α	NOUN
ejpam-5460	400	29	,	,	PUNCT
ejpam-5460	400	30	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	400	31	1	1	NUM
ejpam-5460	400	32	-	-	PUNCT
ejpam-5460	400	33	interior	interior	ADJ
ejpam-5460	400	34	ideal	ideal	NOUN
ejpam-5460	400	35	and	and	CCONJ
ejpam-5460	400	36	an	an	DET
ejpam-5460	400	37	(	(	PUNCT
ejpam-5460	400	38	α	α	NOUN
ejpam-5460	400	39	,	,	PUNCT
ejpam-5460	400	40	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	400	41	(	(	PUNCT
ejpam-5460	400	42	0	0	NUM
ejpam-5460	400	43	,	,	PUNCT
ejpam-5460	400	44	1)-ideal	1)-ideal	NUM
ejpam-5460	400	45	of	of	ADP
ejpam-5460	400	46	s	s	PROPN
ejpam-5460	400	47	,	,	PUNCT
ejpam-5460	400	48	respectively	respectively	ADV
ejpam-5460	400	49	.	.	PUNCT
ejpam-5460	401	1	by	by	ADP
ejpam-5460	401	2	our	our	PRON
ejpam-5460	401	3	presumption	presumption	NOUN
ejpam-5460	401	4	,	,	PUNCT
ejpam-5460	401	5	we	we	PRON
ejpam-5460	401	6	have	have	VERB
ejpam-5460	401	7	(	(	PUNCT
ejpam-5460	401	8	χa)i	χa)i	PROPN
ejpam-5460	401	9	∩i	∩i	PROPN
ejpam-5460	401	10	(	(	PUNCT
ejpam-5460	401	11	χb)i	χb)i	PROPN
ejpam-5460	401	12	∩i	∩i	NOUN
ejpam-5460	401	13	(	(	PUNCT
ejpam-5460	401	14	χc)i	χc)i	PROPN
ejpam-5460	401	15	⊆	⊆	NUM
ejpam-5460	401	16	(	(	PUNCT
ejpam-5460	401	17	χa)i	χa)i	NUM
ejpam-5460	401	18	◦	◦	NOUN
ejpam-5460	401	19	i	i	PROPN
ejpam-5460	401	20	(	(	PUNCT
ejpam-5460	401	21	χb)i	χb)i	PROPN
ejpam-5460	401	22	◦	◦	NOUN
ejpam-5460	401	23	i	i	NOUN
ejpam-5460	401	24	(	(	PUNCT
ejpam-5460	401	25	χc)i	χc)i	PROPN
ejpam-5460	401	26	.	.	PUNCT
ejpam-5460	402	1	this	this	PRON
ejpam-5460	402	2	implies	imply	VERB
ejpam-5460	402	3	that	that	SCONJ
ejpam-5460	402	4	(	(	PUNCT
ejpam-5460	402	5	χa∩b∩c)i	χa∩b∩c)i	NUM
ejpam-5460	402	6	=	=	SYM
ejpam-5460	402	7	χa	χa	NOUN
ejpam-5460	402	8	∩i	∩i	ADV
ejpam-5460	402	9	χb	χb	INTJ
ejpam-5460	402	10	∩i	∩i	VERB
ejpam-5460	402	11	χc	χc	PROPN
ejpam-5460	403	1	=	=	PUNCT
ejpam-5460	404	1	(	(	PUNCT
ejpam-5460	404	2	χa)i	χa)i	PROPN
ejpam-5460	404	3	∩i	∩i	PROPN
ejpam-5460	404	4	(	(	PUNCT
ejpam-5460	404	5	χb)i	χb)i	PROPN
ejpam-5460	404	6	∩i	∩i	NOUN
ejpam-5460	404	7	(	(	PUNCT
ejpam-5460	404	8	χc)i	χc)i	PROPN
ejpam-5460	404	9	⊆	⊆	NUM
ejpam-5460	404	10	(	(	PUNCT
ejpam-5460	404	11	χa)i	χa)i	NUM
ejpam-5460	404	12	◦	◦	NOUN
ejpam-5460	404	13	i	i	PROPN
ejpam-5460	404	14	(	(	PUNCT
ejpam-5460	404	15	χb)i	χb)i	PROPN
ejpam-5460	404	16	◦	◦	NOUN
ejpam-5460	404	17	i	i	NOUN
ejpam-5460	404	18	(	(	PUNCT
ejpam-5460	404	19	χc)i	χc)i	PROPN
ejpam-5460	404	20	=	=	SYM
ejpam-5460	404	21	χa	χa	NOUN
ejpam-5460	404	22	◦	◦	NOUN
ejpam-5460	404	23	i	i	PRON
ejpam-5460	404	24	χb	χb	VERB
ejpam-5460	404	25	◦	◦	VERB
ejpam-5460	404	26	i	i	PRON
ejpam-5460	404	27	χc	χc	NOUN
ejpam-5460	404	28	=	=	SYM
ejpam-5460	404	29	(	(	PUNCT
ejpam-5460	404	30	χ(abc])i	χ(abc])i	PROPN
ejpam-5460	404	31	.	.	PUNCT
ejpam-5460	405	1	by	by	ADP
ejpam-5460	405	2	proposition	proposition	NOUN
ejpam-5460	405	3	2	2	NUM
ejpam-5460	405	4	,	,	PUNCT
ejpam-5460	405	5	we	we	PRON
ejpam-5460	405	6	have	have	VERB
ejpam-5460	405	7	a	a	DET
ejpam-5460	405	8	∩b	∩b	NOUN
ejpam-5460	405	9	∩	∩	NOUN
ejpam-5460	405	10	c	c	NOUN
ejpam-5460	405	11	⊆	⊆	NUM
ejpam-5460	405	12	(	(	PUNCT
ejpam-5460	405	13	abc	abc	PROPN
ejpam-5460	405	14	]	]	X
ejpam-5460	405	15	.	.	PUNCT
ejpam-5460	406	1	hence	hence	ADV
ejpam-5460	406	2	,	,	PUNCT
ejpam-5460	406	3	by	by	ADP
ejpam-5460	406	4	lemma	lemma	PROPN
ejpam-5460	406	5	6	6	NUM
ejpam-5460	406	6	,	,	PUNCT
ejpam-5460	406	7	s	s	PART
ejpam-5460	406	8	satisfies	satisfie	NOUN
ejpam-5460	406	9	(	(	PUNCT
ejpam-5460	406	10	c7	c7	PROPN
ejpam-5460	406	11	)	)	PUNCT
ejpam-5460	406	12	.	.	PUNCT
ejpam-5460	407	1	lemma	lemma	PROPN
ejpam-5460	407	2	7	7	NUM
ejpam-5460	407	3	(	(	PUNCT
ejpam-5460	407	4	[	[	X
ejpam-5460	407	5	49	49	NUM
ejpam-5460	407	6	]	]	PUNCT
ejpam-5460	407	7	)	)	PUNCT
ejpam-5460	407	8	.	.	PUNCT
ejpam-5460	408	1	let	let	VERB
ejpam-5460	408	2	s	s	PRON
ejpam-5460	408	3	be	be	AUX
ejpam-5460	408	4	an	an	DET
ejpam-5460	408	5	ordered	order	VERB
ejpam-5460	408	6	semigroup	semigroup	NOUN
ejpam-5460	408	7	.	.	PUNCT
ejpam-5460	409	1	then	then	ADV
ejpam-5460	409	2	,	,	PUNCT
ejpam-5460	409	3	s	s	NOUN
ejpam-5460	409	4	satisfies	satisfie	NOUN
ejpam-5460	409	5	(	(	PUNCT
ejpam-5460	409	6	c8	c8	PROPN
ejpam-5460	409	7	)	)	PUNCT
ejpam-5460	409	8	of	of	ADP
ejpam-5460	409	9	degree	degree	NOUN
ejpam-5460	409	10	n	n	NOUN
ejpam-5460	409	11	if	if	SCONJ
ejpam-5460	409	12	and	and	CCONJ
ejpam-5460	409	13	only	only	ADV
ejpam-5460	409	14	if	if	SCONJ
ejpam-5460	409	15	a	a	DET
ejpam-5460	409	16	⊆	⊆	NUM
ejpam-5460	409	17	(	(	PUNCT
ejpam-5460	409	18	a2	a2	PROPN
ejpam-5460	409	19	]	]	PUNCT
ejpam-5460	409	20	for	for	ADP
ejpam-5460	409	21	any	any	DET
ejpam-5460	409	22	n	n	CCONJ
ejpam-5460	409	23	-	-	PUNCT
ejpam-5460	409	24	interior	interior	ADJ
ejpam-5460	409	25	ideal	ideal	NOUN
ejpam-5460	409	26	a	a	PRON
ejpam-5460	409	27	of	of	ADP
ejpam-5460	409	28	s.	s.	PROPN
ejpam-5460	409	29	theorem	theorem	VERB
ejpam-5460	409	30	9	9	NUM
ejpam-5460	409	31	.	.	PUNCT
ejpam-5460	410	1	let	let	VERB
ejpam-5460	410	2	s	s	PRON
ejpam-5460	410	3	be	be	AUX
ejpam-5460	410	4	an	an	DET
ejpam-5460	410	5	ordered	order	VERB
ejpam-5460	410	6	semigroup	semigroup	NOUN
ejpam-5460	410	7	.	.	PUNCT
ejpam-5460	411	1	then	then	ADV
ejpam-5460	411	2	,	,	PUNCT
ejpam-5460	411	3	the	the	DET
ejpam-5460	411	4	following	follow	VERB
ejpam-5460	411	5	statements	statement	NOUN
ejpam-5460	411	6	are	be	AUX
ejpam-5460	411	7	equivalent	equivalent	ADJ
ejpam-5460	411	8	.	.	PUNCT
ejpam-5460	412	1	(	(	PUNCT
ejpam-5460	412	2	i	i	NOUN
ejpam-5460	412	3	)	)	PUNCT
ejpam-5460	412	4	s	s	PART
ejpam-5460	412	5	satisfies	satisfie	NOUN
ejpam-5460	412	6	(	(	PUNCT
ejpam-5460	412	7	c8	c8	PROPN
ejpam-5460	412	8	)	)	PUNCT
ejpam-5460	412	9	of	of	ADP
ejpam-5460	412	10	degree	degree	NOUN
ejpam-5460	412	11	n.	n.	PROPN
ejpam-5460	412	12	(	(	PUNCT
ejpam-5460	412	13	ii	ii	NOUN
ejpam-5460	412	14	)	)	PUNCT
ejpam-5460	412	15	fi	fi	NOUN
ejpam-5460	413	1	⊆	⊆	NUM
ejpam-5460	413	2	f	f	PROPN
ejpam-5460	413	3	◦	◦	NOUN
ejpam-5460	413	4	i	i	PRON
ejpam-5460	413	5	f	f	NOUN
ejpam-5460	413	6	for	for	ADP
ejpam-5460	413	7	any	any	DET
ejpam-5460	413	8	(	(	PUNCT
ejpam-5460	413	9	α	α	NOUN
ejpam-5460	413	10	,	,	PUNCT
ejpam-5460	413	11	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	413	12	n	n	CCONJ
ejpam-5460	413	13	-	-	PUNCT
ejpam-5460	413	14	interior	interior	ADJ
ejpam-5460	413	15	ideal	ideal	NOUN
ejpam-5460	413	16	f	f	PROPN
ejpam-5460	413	17	of	of	ADP
ejpam-5460	413	18	s.	s.	PROPN
ejpam-5460	413	19	proof	proof	PROPN
ejpam-5460	413	20	.	.	PUNCT
ejpam-5460	414	1	(	(	PUNCT
ejpam-5460	414	2	1)⇒(2	1)⇒(2	NUM
ejpam-5460	414	3	)	)	PUNCT
ejpam-5460	414	4	.	.	PUNCT
ejpam-5460	415	1	let	let	VERB
ejpam-5460	415	2	f	f	PRON
ejpam-5460	415	3	be	be	AUX
ejpam-5460	415	4	an	an	DET
ejpam-5460	415	5	(	(	PUNCT
ejpam-5460	415	6	α	α	NOUN
ejpam-5460	415	7	,	,	PUNCT
ejpam-5460	415	8	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	415	9	n	n	CCONJ
ejpam-5460	415	10	-	-	PUNCT
ejpam-5460	415	11	interior	interior	ADJ
ejpam-5460	415	12	ideal	ideal	NOUN
ejpam-5460	415	13	of	of	ADP
ejpam-5460	415	14	s.	s.	PROPN
ejpam-5460	415	15	let	let	VERB
ejpam-5460	415	16	a	a	DET
ejpam-5460	415	17	∈	∈	NOUN
ejpam-5460	415	18	s.	s.	PROPN
ejpam-5460	415	19	since	since	SCONJ
ejpam-5460	415	20	s	s	PART
ejpam-5460	415	21	satisfies	satisfie	NOUN
ejpam-5460	415	22	(	(	PUNCT
ejpam-5460	415	23	c8	c8	PROPN
ejpam-5460	415	24	)	)	PUNCT
ejpam-5460	415	25	of	of	ADP
ejpam-5460	415	26	degree	degree	NOUN
ejpam-5460	415	27	n	n	CCONJ
ejpam-5460	415	28	,	,	PUNCT
ejpam-5460	415	29	there	there	PRON
ejpam-5460	415	30	exist	exist	VERB
ejpam-5460	415	31	y1	y1	NOUN
ejpam-5460	415	32	,	,	PUNCT
ejpam-5460	415	33	y2	y2	PROPN
ejpam-5460	415	34	∈	∈	PROPN
ejpam-5460	415	35	s	s	VERB
ejpam-5460	415	36	such	such	ADJ
ejpam-5460	415	37	that	that	SCONJ
ejpam-5460	415	38	a	a	DET
ejpam-5460	415	39	≤	≤	NUM
ejpam-5460	415	40	y1a	y1a	ADJ
ejpam-5460	415	41	ny2	ny2	PROPN
ejpam-5460	415	42	=	=	SYM
ejpam-5460	415	43	y1a	y1a	PROPN
ejpam-5460	415	44	n−1aay2	n−1aay2	PROPN
ejpam-5460	415	45	≤	≤	PROPN
ejpam-5460	415	46	y1a	y1a	PROPN
ejpam-5460	415	47	n−2y1a	n−2y1a	PROPN
ejpam-5460	415	48	ny2y1a	ny2y1a	PROPN
ejpam-5460	415	49	ny2y2	ny2y2	PROPN
ejpam-5460	415	50	.	.	PUNCT
ejpam-5460	416	1	that	that	PRON
ejpam-5460	416	2	is	be	AUX
ejpam-5460	416	3	,	,	PUNCT
ejpam-5460	416	4	(	(	PUNCT
ejpam-5460	416	5	x1a	x1a	PROPN
ejpam-5460	416	6	nx2	nx2	PROPN
ejpam-5460	416	7	,	,	PUNCT
ejpam-5460	416	8	x3a	x3a	PROPN
ejpam-5460	416	9	nx4	nx4	PROPN
ejpam-5460	416	10	)	)	PUNCT
ejpam-5460	416	11	∈	∈	PROPN
ejpam-5460	416	12	sa	sa	NOUN
ejpam-5460	416	13	for	for	ADP
ejpam-5460	416	14	some	some	PRON
ejpam-5460	416	15	x1	x1	PROPN
ejpam-5460	416	16	,	,	PUNCT
ejpam-5460	416	17	x2	x2	PROPN
ejpam-5460	416	18	,	,	PUNCT
ejpam-5460	416	19	x3	x3	ADJ
ejpam-5460	416	20	,	,	PUNCT
ejpam-5460	416	21	x4	x4	PROPN
ejpam-5460	416	22	∈	∈	PROPN
ejpam-5460	416	23	s.	s.	PROPN
ejpam-5460	416	24	then	then	ADV
ejpam-5460	416	25	,	,	PUNCT
ejpam-5460	416	26	(	(	PUNCT
ejpam-5460	416	27	f	f	X
ejpam-5460	416	28	◦	◦	NOUN
ejpam-5460	416	29	i	i	PRON
ejpam-5460	416	30	f)(a	f)(a	NOUN
ejpam-5460	416	31	)	)	PUNCT
ejpam-5460	416	32	≥	≥	NOUN
ejpam-5460	416	33	fi(x1a	fi(x1a	PROPN
ejpam-5460	416	34	nx2	nx2	PROPN
ejpam-5460	416	35	)	)	PUNCT
ejpam-5460	416	36	∧	∧	PROPN
ejpam-5460	416	37	fi(x3a	fi(x3a	PROPN
ejpam-5460	416	38	nx4	nx4	PROPN
ejpam-5460	416	39	)	)	PUNCT
ejpam-5460	416	40	≥	≥	NOUN
ejpam-5460	416	41	fi(a	fi(a	NOUN
ejpam-5460	416	42	)	)	PUNCT
ejpam-5460	416	43	∧	∧	NOUN
ejpam-5460	416	44	fi(a	fi(a	NOUN
ejpam-5460	416	45	)	)	PUNCT
ejpam-5460	416	46	=	=	SYM
ejpam-5460	416	47	fi(a	fi(a	PROPN
ejpam-5460	416	48	)	)	PUNCT
ejpam-5460	416	49	.	.	PUNCT
ejpam-5460	417	1	(	(	PUNCT
ejpam-5460	417	2	2)⇒(1	2)⇒(1	NOUN
ejpam-5460	417	3	)	)	PUNCT
ejpam-5460	417	4	.	.	PUNCT
ejpam-5460	418	1	let	let	VERB
ejpam-5460	418	2	a	a	DET
ejpam-5460	418	3	be	be	AUX
ejpam-5460	418	4	an	an	DET
ejpam-5460	418	5	n	n	CCONJ
ejpam-5460	418	6	-	-	PUNCT
ejpam-5460	418	7	interior	interior	ADJ
ejpam-5460	418	8	ideal	ideal	NOUN
ejpam-5460	418	9	of	of	ADP
ejpam-5460	418	10	s.	s.	PROPN
ejpam-5460	418	11	then	then	ADV
ejpam-5460	418	12	,	,	PUNCT
ejpam-5460	418	13	by	by	ADP
ejpam-5460	418	14	proposition	proposition	NOUN
ejpam-5460	418	15	2	2	NUM
ejpam-5460	418	16	,	,	PUNCT
ejpam-5460	418	17	(	(	PUNCT
ejpam-5460	418	18	χa)i	χa)i	PROPN
ejpam-5460	418	19	is	be	AUX
ejpam-5460	418	20	an	an	DET
ejpam-5460	418	21	(	(	PUNCT
ejpam-5460	418	22	α	α	NOUN
ejpam-5460	418	23	,	,	PUNCT
ejpam-5460	418	24	β)fuzzy	β)fuzzy	ADJ
ejpam-5460	418	25	n	n	CCONJ
ejpam-5460	418	26	-	-	PUNCT
ejpam-5460	418	27	interior	interior	ADJ
ejpam-5460	418	28	ideal	ideal	NOUN
ejpam-5460	418	29	of	of	ADP
ejpam-5460	418	30	s.	s.	PROPN
ejpam-5460	418	31	by	by	ADP
ejpam-5460	418	32	our	our	PRON
ejpam-5460	418	33	presumption	presumption	NOUN
ejpam-5460	418	34	,	,	PUNCT
ejpam-5460	418	35	we	we	PRON
ejpam-5460	418	36	have	have	VERB
ejpam-5460	418	37	(	(	PUNCT
ejpam-5460	418	38	(	(	PUNCT
ejpam-5460	418	39	χa)i)i	χa)i)i	NOUN
ejpam-5460	418	40	⊆	⊆	NUM
ejpam-5460	418	41	(	(	PUNCT
ejpam-5460	418	42	χa)i	χa)i	NUM
ejpam-5460	418	43	◦	◦	NOUN
ejpam-5460	418	44	i	i	PROPN
ejpam-5460	418	45	(	(	PUNCT
ejpam-5460	418	46	χa)i	χa)i	PROPN
ejpam-5460	418	47	.	.	PUNCT
ejpam-5460	419	1	this	this	PRON
ejpam-5460	419	2	implies	imply	VERB
ejpam-5460	419	3	that	that	SCONJ
ejpam-5460	419	4	(	(	PUNCT
ejpam-5460	419	5	χa)i	χa)i	PROPN
ejpam-5460	419	6	=	=	SYM
ejpam-5460	419	7	(	(	PUNCT
ejpam-5460	419	8	(	(	PUNCT
ejpam-5460	419	9	χa)i)i	χa)i)i	NOUN
ejpam-5460	419	10	⊆	⊆	NUM
ejpam-5460	419	11	(	(	PUNCT
ejpam-5460	419	12	χa)i	χa)i	NUM
ejpam-5460	419	13	◦	◦	NOUN
ejpam-5460	419	14	i	i	PROPN
ejpam-5460	419	15	(	(	PUNCT
ejpam-5460	419	16	χa)i	χa)i	PROPN
ejpam-5460	419	17	=	=	SYM
ejpam-5460	419	18	χa	χa	NOUN
ejpam-5460	419	19	◦	◦	NOUN
ejpam-5460	419	20	i	i	PRON
ejpam-5460	419	21	χa	χa	VERB
ejpam-5460	419	22	=	=	SYM
ejpam-5460	419	23	(	(	PUNCT
ejpam-5460	419	24	χ(a2])i	χ(a2])i	PROPN
ejpam-5460	419	25	.	.	PUNCT
ejpam-5460	420	1	by	by	ADP
ejpam-5460	420	2	proposition	proposition	NOUN
ejpam-5460	420	3	2	2	NUM
ejpam-5460	420	4	,	,	PUNCT
ejpam-5460	420	5	we	we	PRON
ejpam-5460	420	6	have	have	VERB
ejpam-5460	420	7	a	a	DET
ejpam-5460	420	8	⊆	⊆	NUM
ejpam-5460	420	9	(	(	PUNCT
ejpam-5460	420	10	a2	a2	PROPN
ejpam-5460	420	11	]	]	PUNCT
ejpam-5460	420	12	.	.	PUNCT
ejpam-5460	421	1	hence	hence	ADV
ejpam-5460	421	2	,	,	PUNCT
ejpam-5460	421	3	by	by	ADP
ejpam-5460	421	4	lemma	lemma	PROPN
ejpam-5460	421	5	7	7	NUM
ejpam-5460	421	6	,	,	PUNCT
ejpam-5460	421	7	s	s	PART
ejpam-5460	421	8	satisfies	satisfie	NOUN
ejpam-5460	421	9	(	(	PUNCT
ejpam-5460	421	10	c8	c8	PROPN
ejpam-5460	421	11	)	)	PUNCT
ejpam-5460	421	12	of	of	ADP
ejpam-5460	421	13	degree	degree	NOUN
ejpam-5460	421	14	n.	n.	PROPN
ejpam-5460	421	15	s.	s.	PROPN
ejpam-5460	421	16	lekkoksung	lekkoksung	PROPN
ejpam-5460	421	17	,	,	PUNCT
ejpam-5460	421	18	b.	b.	PROPN
ejpam-5460	421	19	davvaz	davvaz	PROPN
ejpam-5460	421	20	,	,	PUNCT
ejpam-5460	421	21	n.	n.	PROPN
ejpam-5460	421	22	lekkoksung	lekkoksung	PROPN
ejpam-5460	421	23	/	/	SYM
ejpam-5460	421	24	eur	eur	PROPN
ejpam-5460	421	25	.	.	PUNCT
ejpam-5460	422	1	j.	j.	PROPN
ejpam-5460	422	2	pure	pure	PROPN
ejpam-5460	422	3	appl	appl	PROPN
ejpam-5460	422	4	.	.	PROPN
ejpam-5460	422	5	math	math	PROPN
ejpam-5460	422	6	,	,	PUNCT
ejpam-5460	422	7	17	17	NUM
ejpam-5460	422	8	(	(	PUNCT
ejpam-5460	422	9	4	4	NUM
ejpam-5460	422	10	)	)	PUNCT
ejpam-5460	422	11	(	(	PUNCT
ejpam-5460	422	12	2024	2024	NUM
ejpam-5460	422	13	)	)	PUNCT
ejpam-5460	422	14	,	,	PUNCT
ejpam-5460	422	15	2962	2962	NUM
ejpam-5460	422	16	-	-	SYM
ejpam-5460	422	17	2984	2984	NUM
ejpam-5460	422	18	2977	2977	NUM
ejpam-5460	422	19	lemma	lemma	PROPN
ejpam-5460	422	20	8	8	NUM
ejpam-5460	422	21	(	(	PUNCT
ejpam-5460	422	22	[	[	X
ejpam-5460	422	23	49	49	NUM
ejpam-5460	422	24	]	]	PUNCT
ejpam-5460	422	25	)	)	PUNCT
ejpam-5460	422	26	.	.	PUNCT
ejpam-5460	423	1	let	let	VERB
ejpam-5460	423	2	s	s	PRON
ejpam-5460	423	3	be	be	AUX
ejpam-5460	423	4	an	an	DET
ejpam-5460	423	5	ordered	order	VERB
ejpam-5460	423	6	semigroup	semigroup	NOUN
ejpam-5460	423	7	.	.	PUNCT
ejpam-5460	424	1	then	then	ADV
ejpam-5460	424	2	,	,	PUNCT
ejpam-5460	424	3	s	s	NOUN
ejpam-5460	424	4	satisfies	satisfie	NOUN
ejpam-5460	424	5	(	(	PUNCT
ejpam-5460	424	6	c9	c9	NOUN
ejpam-5460	424	7	)	)	PUNCT
ejpam-5460	424	8	of	of	ADP
ejpam-5460	424	9	degree	degree	NOUN
ejpam-5460	424	10	n	n	NOUN
ejpam-5460	424	11	if	if	SCONJ
ejpam-5460	424	12	and	and	CCONJ
ejpam-5460	424	13	only	only	ADV
ejpam-5460	424	14	if	if	SCONJ
ejpam-5460	424	15	a	a	DET
ejpam-5460	424	16	∩b	∩b	NOUN
ejpam-5460	424	17	⊆	⊆	NUM
ejpam-5460	424	18	(	(	PUNCT
ejpam-5460	424	19	ab	ab	X
ejpam-5460	424	20	]	]	X
ejpam-5460	424	21	for	for	ADP
ejpam-5460	424	22	any	any	DET
ejpam-5460	424	23	n	n	CCONJ
ejpam-5460	424	24	-	-	PUNCT
ejpam-5460	424	25	interior	interior	ADJ
ejpam-5460	424	26	ideal	ideal	NOUN
ejpam-5460	424	27	a	a	PRON
ejpam-5460	424	28	and	and	CCONJ
ejpam-5460	424	29	(	(	PUNCT
ejpam-5460	424	30	n	n	CCONJ
ejpam-5460	424	31	,	,	PUNCT
ejpam-5460	424	32	1)-ideal	1)-ideal	PROPN
ejpam-5460	424	33	b	b	PROPN
ejpam-5460	424	34	of	of	ADP
ejpam-5460	424	35	s.	s.	PROPN
ejpam-5460	424	36	theorem	theorem	VERB
ejpam-5460	424	37	10	10	NUM
ejpam-5460	424	38	.	.	PUNCT
ejpam-5460	425	1	let	let	VERB
ejpam-5460	425	2	s	s	PRON
ejpam-5460	425	3	be	be	AUX
ejpam-5460	425	4	an	an	DET
ejpam-5460	425	5	ordered	order	VERB
ejpam-5460	425	6	semigroup	semigroup	NOUN
ejpam-5460	425	7	.	.	PUNCT
ejpam-5460	426	1	then	then	ADV
ejpam-5460	426	2	,	,	PUNCT
ejpam-5460	426	3	the	the	DET
ejpam-5460	426	4	following	follow	VERB
ejpam-5460	426	5	statements	statement	NOUN
ejpam-5460	426	6	are	be	AUX
ejpam-5460	426	7	equivalent	equivalent	ADJ
ejpam-5460	426	8	.	.	PUNCT
ejpam-5460	427	1	(	(	PUNCT
ejpam-5460	427	2	i	i	NOUN
ejpam-5460	427	3	)	)	PUNCT
ejpam-5460	427	4	s	s	PART
ejpam-5460	427	5	satisfies	satisfie	NOUN
ejpam-5460	427	6	(	(	PUNCT
ejpam-5460	427	7	c9	c9	NOUN
ejpam-5460	427	8	)	)	PUNCT
ejpam-5460	427	9	of	of	ADP
ejpam-5460	427	10	degree	degree	NOUN
ejpam-5460	427	11	n.	n.	PROPN
ejpam-5460	427	12	(	(	PUNCT
ejpam-5460	427	13	ii	ii	PROPN
ejpam-5460	427	14	)	)	PUNCT
ejpam-5460	428	1	f	f	PROPN
ejpam-5460	428	2	∩i	∩i	PROPN
ejpam-5460	428	3	g	g	PROPN
ejpam-5460	428	4	⊆	⊆	NUM
ejpam-5460	428	5	f	f	PROPN
ejpam-5460	428	6	◦	◦	NOUN
ejpam-5460	428	7	i	i	PRON
ejpam-5460	428	8	g	g	NOUN
ejpam-5460	428	9	for	for	ADP
ejpam-5460	428	10	any	any	DET
ejpam-5460	428	11	(	(	PUNCT
ejpam-5460	428	12	α	α	NOUN
ejpam-5460	428	13	,	,	PUNCT
ejpam-5460	428	14	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	428	15	n	n	CCONJ
ejpam-5460	428	16	-	-	PUNCT
ejpam-5460	428	17	interior	interior	ADJ
ejpam-5460	428	18	ideal	ideal	NOUN
ejpam-5460	428	19	f	f	PROPN
ejpam-5460	428	20	and	and	CCONJ
ejpam-5460	428	21	(	(	PUNCT
ejpam-5460	428	22	α	α	NOUN
ejpam-5460	428	23	,	,	PUNCT
ejpam-5460	428	24	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	428	25	(	(	PUNCT
ejpam-5460	428	26	n	n	CCONJ
ejpam-5460	428	27	,	,	PUNCT
ejpam-5460	428	28	1)-ideal	1)-ideal	NUM
ejpam-5460	428	29	g	g	NOUN
ejpam-5460	428	30	of	of	ADP
ejpam-5460	428	31	s.	s.	PROPN
ejpam-5460	428	32	proof	proof	PROPN
ejpam-5460	428	33	.	.	PUNCT
ejpam-5460	429	1	(	(	PUNCT
ejpam-5460	429	2	1)⇒(2	1)⇒(2	NUM
ejpam-5460	429	3	)	)	PUNCT
ejpam-5460	429	4	.	.	PUNCT
ejpam-5460	430	1	let	let	VERB
ejpam-5460	430	2	f	f	PROPN
ejpam-5460	430	3	and	and	CCONJ
ejpam-5460	430	4	g	g	PROPN
ejpam-5460	430	5	be	be	AUX
ejpam-5460	430	6	an	an	DET
ejpam-5460	430	7	(	(	PUNCT
ejpam-5460	430	8	α	α	NOUN
ejpam-5460	430	9	,	,	PUNCT
ejpam-5460	430	10	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	430	11	n	n	CCONJ
ejpam-5460	430	12	-	-	PUNCT
ejpam-5460	430	13	interior	interior	ADJ
ejpam-5460	430	14	ideal	ideal	NOUN
ejpam-5460	430	15	and	and	CCONJ
ejpam-5460	430	16	an	an	DET
ejpam-5460	430	17	(	(	PUNCT
ejpam-5460	430	18	α	α	NOUN
ejpam-5460	430	19	,	,	PUNCT
ejpam-5460	430	20	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	430	21	(	(	PUNCT
ejpam-5460	430	22	n	n	CCONJ
ejpam-5460	430	23	,	,	PUNCT
ejpam-5460	430	24	1)-ideal	1)-ideal	NUM
ejpam-5460	430	25	of	of	ADP
ejpam-5460	430	26	s	s	PROPN
ejpam-5460	430	27	,	,	PUNCT
ejpam-5460	430	28	respectively	respectively	ADV
ejpam-5460	430	29	.	.	PUNCT
ejpam-5460	431	1	let	let	VERB
ejpam-5460	431	2	a	a	DET
ejpam-5460	431	3	∈	∈	NOUN
ejpam-5460	431	4	s.	s.	PROPN
ejpam-5460	431	5	since	since	SCONJ
ejpam-5460	431	6	s	s	PART
ejpam-5460	431	7	satisfies	satisfie	NOUN
ejpam-5460	431	8	(	(	PUNCT
ejpam-5460	431	9	c9	c9	NOUN
ejpam-5460	431	10	)	)	PUNCT
ejpam-5460	431	11	of	of	ADP
ejpam-5460	431	12	degree	degree	NOUN
ejpam-5460	431	13	n	n	CCONJ
ejpam-5460	431	14	,	,	PUNCT
ejpam-5460	431	15	there	there	PRON
ejpam-5460	431	16	exist	exist	VERB
ejpam-5460	431	17	y1	y1	NOUN
ejpam-5460	431	18	,	,	PUNCT
ejpam-5460	431	19	y2	y2	PROPN
ejpam-5460	431	20	∈	∈	PROPN
ejpam-5460	431	21	s	s	VERB
ejpam-5460	431	22	such	such	ADJ
ejpam-5460	431	23	that	that	SCONJ
ejpam-5460	431	24	a	a	DET
ejpam-5460	431	25	≤	≤	PROPN
ejpam-5460	431	26	y1a	y1a	PROPN
ejpam-5460	431	27	ny2a	ny2a	PROPN
ejpam-5460	432	1	=	=	SYM
ejpam-5460	432	2	y1a	y1a	PROPN
ejpam-5460	432	3	n−2aay2a	n−2aay2a	VERB
ejpam-5460	432	4	≤	≤	PROPN
ejpam-5460	432	5	y1a	y1a	PROPN
ejpam-5460	432	6	n−2y1a	n−2y1a	NUM
ejpam-5460	432	7	ny2ay1a	ny2ay1a	PROPN
ejpam-5460	432	8	ny2ay2a	ny2ay2a	PROPN
ejpam-5460	432	9	.	.	PUNCT
ejpam-5460	433	1	that	that	PRON
ejpam-5460	433	2	is	be	AUX
ejpam-5460	433	3	,	,	PUNCT
ejpam-5460	433	4	(	(	PUNCT
ejpam-5460	433	5	x1a	x1a	PROPN
ejpam-5460	433	6	nx2	nx2	PROPN
ejpam-5460	433	7	,	,	PUNCT
ejpam-5460	433	8	a	a	DET
ejpam-5460	433	9	nx3a	nx3a	PROPN
ejpam-5460	433	10	)	)	PUNCT
ejpam-5460	433	11	∈	∈	PROPN
ejpam-5460	433	12	sa	sa	NOUN
ejpam-5460	433	13	for	for	ADP
ejpam-5460	433	14	some	some	DET
ejpam-5460	433	15	x1	x1	PROPN
ejpam-5460	433	16	,	,	PUNCT
ejpam-5460	433	17	x2	x2	PROPN
ejpam-5460	433	18	,	,	PUNCT
ejpam-5460	433	19	x3	x3	PROPN
ejpam-5460	433	20	∈	∈	PROPN
ejpam-5460	433	21	s.	s.	PROPN
ejpam-5460	433	22	then	then	ADV
ejpam-5460	433	23	,	,	PUNCT
ejpam-5460	433	24	(	(	PUNCT
ejpam-5460	433	25	f	f	X
ejpam-5460	433	26	◦	◦	NOUN
ejpam-5460	433	27	i	i	PRON
ejpam-5460	433	28	g)(a	g)(a	VERB
ejpam-5460	433	29	)	)	PUNCT
ejpam-5460	433	30	≥	≥	NOUN
ejpam-5460	433	31	fi(x1a	fi(x1a	PROPN
ejpam-5460	433	32	nx2	nx2	PROPN
ejpam-5460	433	33	)	)	PUNCT
ejpam-5460	433	34	∧	∧	PROPN
ejpam-5460	433	35	gi(a	gi(a	X
ejpam-5460	433	36	nx3a	nx3a	PROPN
ejpam-5460	433	37	)	)	PUNCT
ejpam-5460	433	38	≥	≥	NOUN
ejpam-5460	433	39	fi(a	fi(a	NOUN
ejpam-5460	433	40	)	)	PUNCT
ejpam-5460	433	41	∧	∧	NOUN
ejpam-5460	433	42	gi(a	gi(a	X
ejpam-5460	433	43	)	)	PUNCT
ejpam-5460	433	44	=	=	SYM
ejpam-5460	434	1	(	(	PUNCT
ejpam-5460	434	2	f	f	PROPN
ejpam-5460	434	3	∩i	∩i	PROPN
ejpam-5460	434	4	g)(a	g)(a	PROPN
ejpam-5460	434	5	)	)	PUNCT
ejpam-5460	434	6	.	.	PUNCT
ejpam-5460	435	1	(	(	PUNCT
ejpam-5460	435	2	2)⇒(1	2)⇒(1	NOUN
ejpam-5460	435	3	)	)	PUNCT
ejpam-5460	435	4	.	.	PUNCT
ejpam-5460	436	1	let	let	VERB
ejpam-5460	436	2	a	a	PRON
ejpam-5460	436	3	and	and	CCONJ
ejpam-5460	436	4	b	b	NOUN
ejpam-5460	436	5	be	be	AUX
ejpam-5460	436	6	an	an	DET
ejpam-5460	436	7	n	n	CCONJ
ejpam-5460	436	8	-	-	PUNCT
ejpam-5460	436	9	interior	interior	ADJ
ejpam-5460	436	10	ideal	ideal	NOUN
ejpam-5460	436	11	and	and	CCONJ
ejpam-5460	436	12	an	an	DET
ejpam-5460	436	13	(	(	PUNCT
ejpam-5460	436	14	n	n	CCONJ
ejpam-5460	436	15	,	,	PUNCT
ejpam-5460	436	16	1)-ideal	1)-ideal	NUM
ejpam-5460	436	17	of	of	ADP
ejpam-5460	436	18	s	s	PROPN
ejpam-5460	436	19	,	,	PUNCT
ejpam-5460	436	20	respectively	respectively	ADV
ejpam-5460	436	21	.	.	PUNCT
ejpam-5460	437	1	then	then	ADV
ejpam-5460	437	2	,	,	PUNCT
ejpam-5460	437	3	by	by	ADP
ejpam-5460	437	4	proposition	proposition	NOUN
ejpam-5460	437	5	2	2	NUM
ejpam-5460	437	6	,	,	PUNCT
ejpam-5460	437	7	(	(	PUNCT
ejpam-5460	437	8	χa)i	χa)i	NUM
ejpam-5460	437	9	and	and	CCONJ
ejpam-5460	437	10	(	(	PUNCT
ejpam-5460	437	11	χb)i	χb)i	PROPN
ejpam-5460	437	12	is	be	AUX
ejpam-5460	437	13	an	an	DET
ejpam-5460	437	14	(	(	PUNCT
ejpam-5460	437	15	α	α	NOUN
ejpam-5460	437	16	,	,	PUNCT
ejpam-5460	437	17	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	437	18	n	n	CCONJ
ejpam-5460	437	19	-	-	PUNCT
ejpam-5460	437	20	interior	interior	ADJ
ejpam-5460	437	21	ideal	ideal	NOUN
ejpam-5460	437	22	and	and	CCONJ
ejpam-5460	437	23	(	(	PUNCT
ejpam-5460	437	24	α	α	NOUN
ejpam-5460	437	25	,	,	PUNCT
ejpam-5460	437	26	β)fuzzy	β)fuzzy	PUNCT
ejpam-5460	437	27	(	(	PUNCT
ejpam-5460	437	28	n	n	CCONJ
ejpam-5460	437	29	,	,	PUNCT
ejpam-5460	437	30	1)-ideal	1)-ideal	NUM
ejpam-5460	437	31	of	of	ADP
ejpam-5460	437	32	s	s	PROPN
ejpam-5460	437	33	,	,	PUNCT
ejpam-5460	437	34	respectively	respectively	ADV
ejpam-5460	437	35	.	.	PUNCT
ejpam-5460	438	1	by	by	ADP
ejpam-5460	438	2	our	our	PRON
ejpam-5460	438	3	presumption	presumption	NOUN
ejpam-5460	438	4	,	,	PUNCT
ejpam-5460	438	5	we	we	PRON
ejpam-5460	438	6	have	have	VERB
ejpam-5460	438	7	(	(	PUNCT
ejpam-5460	438	8	χa)i	χa)i	PROPN
ejpam-5460	438	9	∩i	∩i	NOUN
ejpam-5460	438	10	(	(	PUNCT
ejpam-5460	438	11	χb)i	χb)i	PROPN
ejpam-5460	438	12	⊆	⊆	NUM
ejpam-5460	438	13	(	(	PUNCT
ejpam-5460	438	14	χa)i	χa)i	PROPN
ejpam-5460	438	15	◦	◦	NOUN
ejpam-5460	438	16	i	i	PROPN
ejpam-5460	438	17	(	(	PUNCT
ejpam-5460	438	18	χb)i	χb)i	PROPN
ejpam-5460	438	19	.	.	PUNCT
ejpam-5460	439	1	this	this	PRON
ejpam-5460	439	2	implies	imply	VERB
ejpam-5460	439	3	that	that	SCONJ
ejpam-5460	439	4	(	(	PUNCT
ejpam-5460	439	5	χa∩b)i	χa∩b)i	NOUN
ejpam-5460	439	6	=	=	NOUN
ejpam-5460	439	7	χa	χa	NOUN
ejpam-5460	439	8	∩i	∩i	ADV
ejpam-5460	439	9	χb	χb	PROPN
ejpam-5460	439	10	=	=	PUNCT
ejpam-5460	439	11	(	(	PUNCT
ejpam-5460	439	12	χa)i	χa)i	PROPN
ejpam-5460	439	13	∩i	∩i	NOUN
ejpam-5460	439	14	(	(	PUNCT
ejpam-5460	439	15	χb)i	χb)i	PROPN
ejpam-5460	439	16	⊆	⊆	NUM
ejpam-5460	439	17	(	(	PUNCT
ejpam-5460	439	18	χa)i	χa)i	PROPN
ejpam-5460	439	19	◦	◦	NOUN
ejpam-5460	439	20	i	i	PROPN
ejpam-5460	439	21	(	(	PUNCT
ejpam-5460	439	22	χb)i	χb)i	PROPN
ejpam-5460	439	23	=	=	PUNCT
ejpam-5460	439	24	χa	χa	NOUN
ejpam-5460	439	25	◦	◦	NOUN
ejpam-5460	439	26	i	i	PRON
ejpam-5460	439	27	χb	χb	VERB
ejpam-5460	439	28	=	=	PUNCT
ejpam-5460	439	29	(	(	PUNCT
ejpam-5460	439	30	χ(ab])i	χ(ab])i	ADV
ejpam-5460	439	31	.	.	PUNCT
ejpam-5460	440	1	by	by	ADP
ejpam-5460	440	2	proposition	proposition	NOUN
ejpam-5460	440	3	2	2	NUM
ejpam-5460	440	4	,	,	PUNCT
ejpam-5460	440	5	we	we	PRON
ejpam-5460	440	6	have	have	VERB
ejpam-5460	440	7	a	a	DET
ejpam-5460	440	8	∩b	∩b	NOUN
ejpam-5460	440	9	⊆	⊆	NUM
ejpam-5460	440	10	(	(	PUNCT
ejpam-5460	440	11	ab	ab	X
ejpam-5460	440	12	]	]	X
ejpam-5460	440	13	.	.	PUNCT
ejpam-5460	441	1	hence	hence	ADV
ejpam-5460	441	2	,	,	PUNCT
ejpam-5460	441	3	by	by	ADP
ejpam-5460	441	4	lemma	lemma	PROPN
ejpam-5460	441	5	8	8	NUM
ejpam-5460	441	6	,	,	PUNCT
ejpam-5460	441	7	s	s	PART
ejpam-5460	441	8	satisfies	satisfie	NOUN
ejpam-5460	441	9	(	(	PUNCT
ejpam-5460	441	10	c9	c9	NOUN
ejpam-5460	441	11	)	)	PUNCT
ejpam-5460	441	12	of	of	ADP
ejpam-5460	441	13	degree	degree	NOUN
ejpam-5460	441	14	n.	n.	NOUN
ejpam-5460	441	15	we	we	PRON
ejpam-5460	441	16	can	can	AUX
ejpam-5460	441	17	characterize	characterize	VERB
ejpam-5460	441	18	ordered	order	VERB
ejpam-5460	441	19	semigroups	semigroup	NOUN
ejpam-5460	441	20	satisfying	satisfy	VERB
ejpam-5460	441	21	(	(	PUNCT
ejpam-5460	441	22	c10	c10	PROPN
ejpam-5460	441	23	)	)	PUNCT
ejpam-5460	441	24	in	in	ADP
ejpam-5460	441	25	the	the	DET
ejpam-5460	441	26	same	same	ADJ
ejpam-5460	441	27	way	way	NOUN
ejpam-5460	441	28	of	of	ADP
ejpam-5460	441	29	the	the	DET
ejpam-5460	441	30	above	above	ADJ
ejpam-5460	441	31	result	result	NOUN
ejpam-5460	441	32	using	use	VERB
ejpam-5460	441	33	the	the	DET
ejpam-5460	441	34	concepts	concept	NOUN
ejpam-5460	441	35	of	of	ADP
ejpam-5460	441	36	(	(	PUNCT
ejpam-5460	441	37	α	α	NOUN
ejpam-5460	441	38	,	,	PUNCT
ejpam-5460	441	39	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	441	40	(	(	PUNCT
ejpam-5460	441	41	1	1	NUM
ejpam-5460	441	42	,	,	PUNCT
ejpam-5460	441	43	n)-ideals	n)-ideal	NOUN
ejpam-5460	441	44	and	and	CCONJ
ejpam-5460	441	45	(	(	PUNCT
ejpam-5460	441	46	α	α	NOUN
ejpam-5460	441	47	,	,	PUNCT
ejpam-5460	441	48	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	441	49	n	n	CCONJ
ejpam-5460	441	50	-	-	PUNCT
ejpam-5460	441	51	interior	interior	ADJ
ejpam-5460	441	52	ideals	ideal	NOUN
ejpam-5460	441	53	as	as	SCONJ
ejpam-5460	441	54	follows	follow	VERB
ejpam-5460	441	55	.	.	PUNCT
ejpam-5460	442	1	theorem	theorem	ADJ
ejpam-5460	442	2	11	11	NUM
ejpam-5460	442	3	.	.	PUNCT
ejpam-5460	443	1	let	let	VERB
ejpam-5460	443	2	s	s	PRON
ejpam-5460	443	3	be	be	AUX
ejpam-5460	443	4	an	an	DET
ejpam-5460	443	5	ordered	order	VERB
ejpam-5460	443	6	semigroup	semigroup	NOUN
ejpam-5460	443	7	.	.	PUNCT
ejpam-5460	444	1	then	then	ADV
ejpam-5460	444	2	,	,	PUNCT
ejpam-5460	444	3	the	the	DET
ejpam-5460	444	4	following	follow	VERB
ejpam-5460	444	5	statements	statement	NOUN
ejpam-5460	444	6	are	be	AUX
ejpam-5460	444	7	equivalent	equivalent	ADJ
ejpam-5460	444	8	.	.	PUNCT
ejpam-5460	445	1	(	(	PUNCT
ejpam-5460	445	2	i	i	NOUN
ejpam-5460	445	3	)	)	PUNCT
ejpam-5460	445	4	s	s	PART
ejpam-5460	445	5	satisfies	satisfie	NOUN
ejpam-5460	445	6	(	(	PUNCT
ejpam-5460	445	7	c10	c10	PROPN
ejpam-5460	445	8	)	)	PUNCT
ejpam-5460	445	9	of	of	ADP
ejpam-5460	445	10	degree	degree	NOUN
ejpam-5460	445	11	n.	n.	PROPN
ejpam-5460	445	12	(	(	PUNCT
ejpam-5460	445	13	ii	ii	PROPN
ejpam-5460	445	14	)	)	PUNCT
ejpam-5460	446	1	f	f	PROPN
ejpam-5460	446	2	∩i	∩i	PROPN
ejpam-5460	446	3	g	g	PROPN
ejpam-5460	446	4	⊆	⊆	NUM
ejpam-5460	446	5	f	f	PROPN
ejpam-5460	446	6	◦	◦	NOUN
ejpam-5460	446	7	i	i	PRON
ejpam-5460	446	8	g	g	NOUN
ejpam-5460	446	9	for	for	ADP
ejpam-5460	446	10	any	any	DET
ejpam-5460	446	11	(	(	PUNCT
ejpam-5460	446	12	α	α	NOUN
ejpam-5460	446	13	,	,	PUNCT
ejpam-5460	446	14	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	446	15	(	(	PUNCT
ejpam-5460	446	16	1	1	NUM
ejpam-5460	446	17	,	,	PUNCT
ejpam-5460	446	18	n)-ideal	n)-ideal	PROPN
ejpam-5460	446	19	f	f	PROPN
ejpam-5460	446	20	and	and	CCONJ
ejpam-5460	446	21	(	(	PUNCT
ejpam-5460	446	22	α	α	NOUN
ejpam-5460	446	23	,	,	PUNCT
ejpam-5460	446	24	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	446	25	n	n	CCONJ
ejpam-5460	446	26	-	-	PUNCT
ejpam-5460	446	27	interior	interior	ADJ
ejpam-5460	446	28	ideal	ideal	NOUN
ejpam-5460	446	29	g	g	PROPN
ejpam-5460	446	30	of	of	ADP
ejpam-5460	446	31	s.	s.	PROPN
ejpam-5460	446	32	lemma	lemma	PROPN
ejpam-5460	446	33	9	9	NUM
ejpam-5460	446	34	(	(	PUNCT
ejpam-5460	446	35	[	[	X
ejpam-5460	446	36	36	36	NUM
ejpam-5460	446	37	]	]	PUNCT
ejpam-5460	446	38	)	)	PUNCT
ejpam-5460	446	39	.	.	PUNCT
ejpam-5460	447	1	let	let	VERB
ejpam-5460	447	2	s	s	PRON
ejpam-5460	447	3	be	be	AUX
ejpam-5460	447	4	an	an	DET
ejpam-5460	447	5	ordered	order	VERB
ejpam-5460	447	6	semigroup	semigroup	NOUN
ejpam-5460	447	7	.	.	PUNCT
ejpam-5460	448	1	then	then	ADV
ejpam-5460	448	2	,	,	PUNCT
ejpam-5460	448	3	s	s	NOUN
ejpam-5460	448	4	satisfies	satisfie	NOUN
ejpam-5460	448	5	(	(	PUNCT
ejpam-5460	448	6	c11	c11	NOUN
ejpam-5460	448	7	)	)	PUNCT
ejpam-5460	448	8	of	of	ADP
ejpam-5460	448	9	degree	degree	NOUN
ejpam-5460	448	10	n	n	NOUN
ejpam-5460	448	11	if	if	SCONJ
ejpam-5460	448	12	and	and	CCONJ
ejpam-5460	448	13	only	only	ADV
ejpam-5460	448	14	if	if	SCONJ
ejpam-5460	448	15	a	a	DET
ejpam-5460	448	16	∩b	∩b	NOUN
ejpam-5460	448	17	∩	∩	NOUN
ejpam-5460	448	18	c	c	NOUN
ejpam-5460	448	19	⊆	⊆	NUM
ejpam-5460	448	20	(	(	PUNCT
ejpam-5460	448	21	abc	abc	PROPN
ejpam-5460	448	22	]	]	X
ejpam-5460	448	23	for	for	ADP
ejpam-5460	448	24	any	any	DET
ejpam-5460	448	25	(	(	PUNCT
ejpam-5460	448	26	1	1	NUM
ejpam-5460	448	27	,	,	PUNCT
ejpam-5460	448	28	0)-ideal	0)-ideal	NOUN
ejpam-5460	448	29	a	a	PRON
ejpam-5460	448	30	,	,	PUNCT
ejpam-5460	448	31	n	n	CCONJ
ejpam-5460	448	32	-	-	ADJ
ejpam-5460	448	33	interior	interior	ADJ
ejpam-5460	448	34	ideal	ideal	NOUN
ejpam-5460	448	35	b	b	PROPN
ejpam-5460	448	36	and	and	CCONJ
ejpam-5460	448	37	(	(	PUNCT
ejpam-5460	448	38	0	0	NUM
ejpam-5460	448	39	,	,	PUNCT
ejpam-5460	448	40	1)-ideal	1)-ideal	NUM
ejpam-5460	448	41	c	c	NOUN
ejpam-5460	448	42	of	of	ADP
ejpam-5460	448	43	s.	s.	PROPN
ejpam-5460	448	44	theorem	theorem	VERB
ejpam-5460	448	45	12	12	NUM
ejpam-5460	448	46	.	.	PUNCT
ejpam-5460	449	1	let	let	VERB
ejpam-5460	449	2	s	s	PRON
ejpam-5460	449	3	be	be	AUX
ejpam-5460	449	4	an	an	DET
ejpam-5460	449	5	ordered	order	VERB
ejpam-5460	449	6	semigroup	semigroup	NOUN
ejpam-5460	449	7	.	.	PUNCT
ejpam-5460	450	1	then	then	ADV
ejpam-5460	450	2	,	,	PUNCT
ejpam-5460	450	3	the	the	DET
ejpam-5460	450	4	following	follow	VERB
ejpam-5460	450	5	statements	statement	NOUN
ejpam-5460	450	6	are	be	AUX
ejpam-5460	450	7	equivalent	equivalent	ADJ
ejpam-5460	450	8	.	.	PUNCT
ejpam-5460	451	1	(	(	PUNCT
ejpam-5460	451	2	i	i	NOUN
ejpam-5460	451	3	)	)	PUNCT
ejpam-5460	451	4	s	s	PART
ejpam-5460	451	5	satisfies	satisfie	NOUN
ejpam-5460	451	6	(	(	PUNCT
ejpam-5460	451	7	c11	c11	NOUN
ejpam-5460	451	8	)	)	PUNCT
ejpam-5460	451	9	of	of	ADP
ejpam-5460	451	10	degree	degree	NOUN
ejpam-5460	451	11	n.	n.	PROPN
ejpam-5460	451	12	s.	s.	PROPN
ejpam-5460	451	13	lekkoksung	lekkoksung	PROPN
ejpam-5460	451	14	,	,	PUNCT
ejpam-5460	451	15	b.	b.	PROPN
ejpam-5460	451	16	davvaz	davvaz	PROPN
ejpam-5460	451	17	,	,	PUNCT
ejpam-5460	451	18	n.	n.	PROPN
ejpam-5460	451	19	lekkoksung	lekkoksung	PROPN
ejpam-5460	451	20	/	/	SYM
ejpam-5460	451	21	eur	eur	PROPN
ejpam-5460	451	22	.	.	PUNCT
ejpam-5460	452	1	j.	j.	PROPN
ejpam-5460	452	2	pure	pure	PROPN
ejpam-5460	452	3	appl	appl	PROPN
ejpam-5460	452	4	.	.	PROPN
ejpam-5460	452	5	math	math	PROPN
ejpam-5460	452	6	,	,	PUNCT
ejpam-5460	452	7	17	17	NUM
ejpam-5460	452	8	(	(	PUNCT
ejpam-5460	452	9	4	4	NUM
ejpam-5460	452	10	)	)	PUNCT
ejpam-5460	452	11	(	(	PUNCT
ejpam-5460	452	12	2024	2024	NUM
ejpam-5460	452	13	)	)	PUNCT
ejpam-5460	452	14	,	,	PUNCT
ejpam-5460	452	15	2962	2962	NUM
ejpam-5460	452	16	-	-	SYM
ejpam-5460	452	17	2984	2984	NUM
ejpam-5460	452	18	2978	2978	NUM
ejpam-5460	452	19	(	(	PUNCT
ejpam-5460	452	20	ii	ii	NOUN
ejpam-5460	452	21	)	)	PUNCT
ejpam-5460	452	22	f	f	PROPN
ejpam-5460	452	23	∩i	∩i	PROPN
ejpam-5460	453	1	g	g	PROPN
ejpam-5460	453	2	∩i	∩i	PROPN
ejpam-5460	453	3	h	h	NOUN
ejpam-5460	454	1	⊆	⊆	NUM
ejpam-5460	454	2	f	f	X
ejpam-5460	454	3	◦	◦	NOUN
ejpam-5460	454	4	i	i	PROPN
ejpam-5460	454	5	g	g	PROPN
ejpam-5460	454	6	◦	◦	NOUN
ejpam-5460	454	7	i	i	PRON
ejpam-5460	454	8	h	h	NOUN
ejpam-5460	454	9	for	for	ADP
ejpam-5460	454	10	any	any	DET
ejpam-5460	454	11	(	(	PUNCT
ejpam-5460	454	12	α	α	NOUN
ejpam-5460	454	13	,	,	PUNCT
ejpam-5460	454	14	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	454	15	(	(	PUNCT
ejpam-5460	454	16	1	1	NUM
ejpam-5460	454	17	,	,	PUNCT
ejpam-5460	454	18	0)-ideal	0)-ideal	PROPN
ejpam-5460	454	19	f	f	PROPN
ejpam-5460	454	20	,	,	PUNCT
ejpam-5460	454	21	(	(	PUNCT
ejpam-5460	454	22	α	α	NOUN
ejpam-5460	454	23	,	,	PUNCT
ejpam-5460	454	24	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	454	25	n	n	CCONJ
ejpam-5460	454	26	-	-	PUNCT
ejpam-5460	454	27	interior	interior	ADJ
ejpam-5460	454	28	ideal	ideal	NOUN
ejpam-5460	454	29	g	g	PROPN
ejpam-5460	454	30	and	and	CCONJ
ejpam-5460	454	31	(	(	PUNCT
ejpam-5460	454	32	α	α	NOUN
ejpam-5460	454	33	,	,	PUNCT
ejpam-5460	454	34	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	454	35	(	(	PUNCT
ejpam-5460	454	36	0	0	NUM
ejpam-5460	454	37	,	,	PUNCT
ejpam-5460	454	38	1)-ideal	1)-ideal	NUM
ejpam-5460	454	39	h	h	NOUN
ejpam-5460	454	40	of	of	ADP
ejpam-5460	454	41	s.	s.	PROPN
ejpam-5460	454	42	proof	proof	PROPN
ejpam-5460	454	43	.	.	PUNCT
ejpam-5460	455	1	(	(	PUNCT
ejpam-5460	455	2	1)⇒(2	1)⇒(2	NUM
ejpam-5460	455	3	)	)	PUNCT
ejpam-5460	455	4	.	.	PUNCT
ejpam-5460	456	1	let	let	VERB
ejpam-5460	456	2	f	f	X
ejpam-5460	456	3	,	,	PUNCT
ejpam-5460	456	4	g	g	PROPN
ejpam-5460	456	5	and	and	CCONJ
ejpam-5460	456	6	h	h	NOUN
ejpam-5460	456	7	be	be	VERB
ejpam-5460	456	8	an	an	DET
ejpam-5460	456	9	(	(	PUNCT
ejpam-5460	456	10	α	α	NOUN
ejpam-5460	456	11	,	,	PUNCT
ejpam-5460	456	12	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	456	13	(	(	PUNCT
ejpam-5460	456	14	1	1	NUM
ejpam-5460	456	15	,	,	PUNCT
ejpam-5460	456	16	0)-ideal	0)-ideal	PROPN
ejpam-5460	456	17	,	,	PUNCT
ejpam-5460	456	18	an	an	DET
ejpam-5460	456	19	(	(	PUNCT
ejpam-5460	456	20	α	α	NOUN
ejpam-5460	456	21	,	,	PUNCT
ejpam-5460	456	22	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	456	23	n	n	CCONJ
ejpam-5460	456	24	-	-	PUNCT
ejpam-5460	456	25	interior	interior	ADJ
ejpam-5460	456	26	ideal	ideal	NOUN
ejpam-5460	456	27	and	and	CCONJ
ejpam-5460	456	28	an	an	DET
ejpam-5460	456	29	(	(	PUNCT
ejpam-5460	456	30	α	α	NOUN
ejpam-5460	456	31	,	,	PUNCT
ejpam-5460	456	32	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	456	33	(	(	PUNCT
ejpam-5460	456	34	0	0	NUM
ejpam-5460	456	35	,	,	PUNCT
ejpam-5460	456	36	1)-ideal	1)-ideal	NUM
ejpam-5460	456	37	of	of	ADP
ejpam-5460	456	38	s	s	PROPN
ejpam-5460	456	39	,	,	PUNCT
ejpam-5460	456	40	respectively	respectively	ADV
ejpam-5460	456	41	.	.	PUNCT
ejpam-5460	457	1	let	let	VERB
ejpam-5460	457	2	a	a	DET
ejpam-5460	457	3	∈	∈	NOUN
ejpam-5460	457	4	s.	s.	PROPN
ejpam-5460	457	5	since	since	SCONJ
ejpam-5460	457	6	s	s	PART
ejpam-5460	457	7	satisfies	satisfie	NOUN
ejpam-5460	457	8	(	(	PUNCT
ejpam-5460	457	9	c11	c11	NOUN
ejpam-5460	457	10	)	)	PUNCT
ejpam-5460	457	11	of	of	ADP
ejpam-5460	457	12	degree	degree	NOUN
ejpam-5460	457	13	n	n	CCONJ
ejpam-5460	457	14	,	,	PUNCT
ejpam-5460	457	15	there	there	PRON
ejpam-5460	457	16	exist	exist	VERB
ejpam-5460	457	17	y1	y1	NOUN
ejpam-5460	457	18	,	,	PUNCT
ejpam-5460	457	19	y2	y2	PROPN
ejpam-5460	457	20	∈	∈	PROPN
ejpam-5460	457	21	s	s	VERB
ejpam-5460	457	22	such	such	ADJ
ejpam-5460	457	23	that	that	SCONJ
ejpam-5460	457	24	a	a	DET
ejpam-5460	457	25	≤	≤	ADJ
ejpam-5460	457	26	ay1a	ay1a	PROPN
ejpam-5460	457	27	ny2a	ny2a	PROPN
ejpam-5460	457	28	≤	≤	PROPN
ejpam-5460	457	29	ay1a	ay1a	PROPN
ejpam-5460	457	30	ny2ay1a	ny2ay1a	PROPN
ejpam-5460	457	31	ny2a	ny2a	PROPN
ejpam-5460	457	32	.	.	PUNCT
ejpam-5460	458	1	that	that	PRON
ejpam-5460	458	2	is	be	AUX
ejpam-5460	458	3	,	,	PUNCT
ejpam-5460	458	4	(	(	PUNCT
ejpam-5460	458	5	ax1x2a	ax1x2a	PROPN
ejpam-5460	458	6	nx3	nx3	SYM
ejpam-5460	458	7	,	,	PUNCT
ejpam-5460	458	8	x4a	x4a	NUM
ejpam-5460	458	9	)	)	PUNCT
ejpam-5460	458	10	∈	∈	PROPN
ejpam-5460	458	11	sa	sa	NOUN
ejpam-5460	458	12	for	for	ADP
ejpam-5460	458	13	some	some	PRON
ejpam-5460	458	14	x1	x1	PROPN
ejpam-5460	458	15	,	,	PUNCT
ejpam-5460	458	16	x2	x2	PROPN
ejpam-5460	458	17	,	,	PUNCT
ejpam-5460	458	18	x3	x3	ADJ
ejpam-5460	458	19	,	,	PUNCT
ejpam-5460	458	20	x4	x4	PROPN
ejpam-5460	458	21	∈	∈	PROPN
ejpam-5460	458	22	s.	s.	PROPN
ejpam-5460	458	23	then	then	ADV
ejpam-5460	458	24	,	,	PUNCT
ejpam-5460	458	25	(	(	PUNCT
ejpam-5460	458	26	f	f	X
ejpam-5460	458	27	◦	◦	NOUN
ejpam-5460	458	28	i	i	PROPN
ejpam-5460	458	29	g	g	PROPN
ejpam-5460	458	30	◦	◦	NOUN
ejpam-5460	458	31	i	i	PRON
ejpam-5460	458	32	h)(a	h)(a	NOUN
ejpam-5460	458	33	)	)	PUNCT
ejpam-5460	459	1	=	=	PUNCT
ejpam-5460	460	1	[	[	X
ejpam-5460	460	2	(	(	PUNCT
ejpam-5460	460	3	f	f	NOUN
ejpam-5460	460	4	◦	◦	NOUN
ejpam-5460	460	5	i	i	PRON
ejpam-5460	460	6	g	g	NOUN
ejpam-5460	460	7	)	)	PUNCT
ejpam-5460	460	8	◦	◦	NOUN
ejpam-5460	460	9	h]i	h]i	NOUN
ejpam-5460	460	10	(	(	PUNCT
ejpam-5460	460	11	a	a	X
ejpam-5460	460	12	)	)	PUNCT
ejpam-5460	460	13	≥	≥	NOUN
ejpam-5460	460	14	(	(	PUNCT
ejpam-5460	460	15	f	f	PROPN
ejpam-5460	460	16	◦	◦	NOUN
ejpam-5460	460	17	i	i	PRON
ejpam-5460	460	18	g)i(ax1x2anx3	g)i(ax1x2anx3	ADJ
ejpam-5460	460	19	)	)	PUNCT
ejpam-5460	460	20	∧	∧	PROPN
ejpam-5460	460	21	hi(x4a	hi(x4a	PROPN
ejpam-5460	460	22	)	)	PUNCT
ejpam-5460	460	23	≥	≥	NOUN
ejpam-5460	460	24	(	(	PUNCT
ejpam-5460	460	25	f	f	PROPN
ejpam-5460	460	26	◦	◦	NOUN
ejpam-5460	460	27	i	i	PRON
ejpam-5460	460	28	g)i(ax1x2anx3	g)i(ax1x2anx3	VERB
ejpam-5460	460	29	)	)	PUNCT
ejpam-5460	460	30	∧	∧	NOUN
ejpam-5460	460	31	hi(a	hi(a	ADJ
ejpam-5460	460	32	)	)	PUNCT
ejpam-5460	460	33	=	=	SYM
ejpam-5460	461	1	(	(	PUNCT
ejpam-5460	461	2	f	f	X
ejpam-5460	461	3	◦	◦	NOUN
ejpam-5460	461	4	g)i(ax1x2anx3	g)i(ax1x2anx3	PROPN
ejpam-5460	461	5	)	)	PUNCT
ejpam-5460	461	6	∧	∧	NOUN
ejpam-5460	461	7	hi(a	hi(a	ADJ
ejpam-5460	461	8	)	)	PUNCT
ejpam-5460	461	9	=	=	SYM
ejpam-5460	461	10	fi(ax1	fi(ax1	NUM
ejpam-5460	461	11	)	)	PUNCT
ejpam-5460	461	12	∧	∧	PROPN
ejpam-5460	461	13	gi(x2a	gi(x2a	PROPN
ejpam-5460	461	14	nx3	nx3	PROPN
ejpam-5460	461	15	)	)	PUNCT
ejpam-5460	461	16	∧	∧	NOUN
ejpam-5460	461	17	hi(a	hi(a	ADV
ejpam-5460	461	18	)	)	PUNCT
ejpam-5460	461	19	≥	≥	NOUN
ejpam-5460	461	20	fi(a	fi(a	NOUN
ejpam-5460	461	21	)	)	PUNCT
ejpam-5460	461	22	∧	∧	NOUN
ejpam-5460	461	23	gi(a	gi(a	X
ejpam-5460	461	24	)	)	PUNCT
ejpam-5460	461	25	∧	∧	NOUN
ejpam-5460	461	26	hi(a	hi(a	ADJ
ejpam-5460	461	27	)	)	PUNCT
ejpam-5460	461	28	=	=	SYM
ejpam-5460	462	1	(	(	PUNCT
ejpam-5460	462	2	f	f	PROPN
ejpam-5460	462	3	∩i	∩i	PROPN
ejpam-5460	462	4	g	g	PROPN
ejpam-5460	462	5	∩i	∩i	PROPN
ejpam-5460	462	6	h)(a	h)(a	PROPN
ejpam-5460	462	7	)	)	PUNCT
ejpam-5460	462	8	.	.	PUNCT
ejpam-5460	463	1	(	(	PUNCT
ejpam-5460	463	2	2)⇒(1	2)⇒(1	NOUN
ejpam-5460	463	3	)	)	PUNCT
ejpam-5460	463	4	.	.	PUNCT
ejpam-5460	464	1	let	let	VERB
ejpam-5460	464	2	a	a	DET
ejpam-5460	464	3	,	,	PUNCT
ejpam-5460	464	4	b	b	NOUN
ejpam-5460	464	5	and	and	CCONJ
ejpam-5460	464	6	c	c	PROPN
ejpam-5460	464	7	be	be	AUX
ejpam-5460	464	8	a	a	DET
ejpam-5460	464	9	(	(	PUNCT
ejpam-5460	464	10	1	1	NUM
ejpam-5460	464	11	,	,	PUNCT
ejpam-5460	464	12	0)-ideal	0)-ideal	PROPN
ejpam-5460	464	13	,	,	PUNCT
ejpam-5460	464	14	an	an	DET
ejpam-5460	464	15	n	n	CCONJ
ejpam-5460	464	16	-	-	ADJ
ejpam-5460	464	17	interior	interior	ADJ
ejpam-5460	464	18	ideal	ideal	NOUN
ejpam-5460	464	19	and	and	CCONJ
ejpam-5460	464	20	a	a	DET
ejpam-5460	464	21	(	(	PUNCT
ejpam-5460	464	22	0	0	NUM
ejpam-5460	464	23	,	,	PUNCT
ejpam-5460	464	24	1)-ideal	1)-ideal	NUM
ejpam-5460	464	25	of	of	ADP
ejpam-5460	464	26	s	s	PROPN
ejpam-5460	464	27	,	,	PUNCT
ejpam-5460	464	28	respectively	respectively	ADV
ejpam-5460	464	29	.	.	PUNCT
ejpam-5460	465	1	then	then	ADV
ejpam-5460	465	2	,	,	PUNCT
ejpam-5460	465	3	by	by	ADP
ejpam-5460	465	4	proposition	proposition	NOUN
ejpam-5460	465	5	2	2	NUM
ejpam-5460	465	6	,	,	PUNCT
ejpam-5460	465	7	(	(	PUNCT
ejpam-5460	465	8	χa)i	χa)i	NUM
ejpam-5460	465	9	,	,	PUNCT
ejpam-5460	465	10	(	(	PUNCT
ejpam-5460	465	11	χb)i	χb)i	PROPN
ejpam-5460	465	12	and	and	CCONJ
ejpam-5460	465	13	(	(	PUNCT
ejpam-5460	465	14	χc)i	χc)i	PROPN
ejpam-5460	465	15	is	be	AUX
ejpam-5460	465	16	an	an	DET
ejpam-5460	465	17	(	(	PUNCT
ejpam-5460	465	18	α	α	NOUN
ejpam-5460	465	19	,	,	PUNCT
ejpam-5460	465	20	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	465	21	(	(	PUNCT
ejpam-5460	465	22	1	1	NUM
ejpam-5460	465	23	,	,	PUNCT
ejpam-5460	465	24	0)ideal	0)ideal	ADJ
ejpam-5460	465	25	,	,	PUNCT
ejpam-5460	465	26	(	(	PUNCT
ejpam-5460	465	27	α	α	NOUN
ejpam-5460	465	28	,	,	PUNCT
ejpam-5460	465	29	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	465	30	n	n	CCONJ
ejpam-5460	465	31	-	-	PUNCT
ejpam-5460	465	32	interior	interior	ADJ
ejpam-5460	465	33	ideal	ideal	NOUN
ejpam-5460	465	34	and	and	CCONJ
ejpam-5460	465	35	(	(	PUNCT
ejpam-5460	465	36	α	α	NOUN
ejpam-5460	465	37	,	,	PUNCT
ejpam-5460	465	38	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	465	39	(	(	PUNCT
ejpam-5460	465	40	0	0	NUM
ejpam-5460	465	41	,	,	PUNCT
ejpam-5460	465	42	1)-ideal	1)-ideal	NUM
ejpam-5460	465	43	of	of	ADP
ejpam-5460	465	44	s	s	PROPN
ejpam-5460	465	45	,	,	PUNCT
ejpam-5460	465	46	respectively	respectively	ADV
ejpam-5460	465	47	.	.	PUNCT
ejpam-5460	466	1	by	by	ADP
ejpam-5460	466	2	our	our	PRON
ejpam-5460	466	3	presumption	presumption	NOUN
ejpam-5460	466	4	,	,	PUNCT
ejpam-5460	466	5	we	we	PRON
ejpam-5460	466	6	have	have	VERB
ejpam-5460	466	7	(	(	PUNCT
ejpam-5460	466	8	χa)i	χa)i	PROPN
ejpam-5460	466	9	∩i	∩i	PROPN
ejpam-5460	466	10	(	(	PUNCT
ejpam-5460	466	11	χb)i	χb)i	PROPN
ejpam-5460	466	12	∩i	∩i	NOUN
ejpam-5460	466	13	(	(	PUNCT
ejpam-5460	466	14	χc)i	χc)i	PROPN
ejpam-5460	466	15	⊆	⊆	NUM
ejpam-5460	466	16	(	(	PUNCT
ejpam-5460	466	17	χa)i	χa)i	NUM
ejpam-5460	466	18	◦	◦	NOUN
ejpam-5460	466	19	i	i	PROPN
ejpam-5460	466	20	(	(	PUNCT
ejpam-5460	466	21	χb)i	χb)i	PROPN
ejpam-5460	466	22	◦	◦	NOUN
ejpam-5460	466	23	i	i	NOUN
ejpam-5460	466	24	(	(	PUNCT
ejpam-5460	466	25	χc)i	χc)i	PROPN
ejpam-5460	466	26	.	.	PUNCT
ejpam-5460	467	1	this	this	PRON
ejpam-5460	467	2	implies	imply	VERB
ejpam-5460	467	3	that	that	SCONJ
ejpam-5460	467	4	(	(	PUNCT
ejpam-5460	467	5	χa∩b∩c)i	χa∩b∩c)i	NUM
ejpam-5460	467	6	=	=	SYM
ejpam-5460	467	7	χa	χa	NOUN
ejpam-5460	467	8	∩i	∩i	ADV
ejpam-5460	467	9	χb	χb	INTJ
ejpam-5460	467	10	∩i	∩i	VERB
ejpam-5460	467	11	χc	χc	PROPN
ejpam-5460	468	1	=	=	PUNCT
ejpam-5460	469	1	(	(	PUNCT
ejpam-5460	469	2	χa)i	χa)i	PROPN
ejpam-5460	469	3	∩i	∩i	PROPN
ejpam-5460	469	4	(	(	PUNCT
ejpam-5460	469	5	χb)i	χb)i	PROPN
ejpam-5460	469	6	∩i	∩i	NOUN
ejpam-5460	469	7	(	(	PUNCT
ejpam-5460	469	8	χc)i	χc)i	PROPN
ejpam-5460	469	9	⊆	⊆	NUM
ejpam-5460	469	10	(	(	PUNCT
ejpam-5460	469	11	χa)i	χa)i	NUM
ejpam-5460	469	12	◦	◦	NOUN
ejpam-5460	469	13	i	i	PROPN
ejpam-5460	469	14	(	(	PUNCT
ejpam-5460	469	15	χb)i	χb)i	PROPN
ejpam-5460	469	16	◦	◦	NOUN
ejpam-5460	469	17	i	i	NOUN
ejpam-5460	469	18	(	(	PUNCT
ejpam-5460	469	19	χc)i	χc)i	PROPN
ejpam-5460	469	20	=	=	SYM
ejpam-5460	469	21	χa	χa	NOUN
ejpam-5460	469	22	◦	◦	NOUN
ejpam-5460	469	23	i	i	PRON
ejpam-5460	469	24	χb	χb	VERB
ejpam-5460	469	25	◦	◦	VERB
ejpam-5460	469	26	i	i	PRON
ejpam-5460	469	27	χc	χc	NOUN
ejpam-5460	469	28	=	=	SYM
ejpam-5460	469	29	(	(	PUNCT
ejpam-5460	469	30	χ(abc])i	χ(abc])i	PROPN
ejpam-5460	469	31	.	.	PUNCT
ejpam-5460	470	1	by	by	ADP
ejpam-5460	470	2	proposition	proposition	NOUN
ejpam-5460	470	3	2	2	NUM
ejpam-5460	470	4	,	,	PUNCT
ejpam-5460	470	5	we	we	PRON
ejpam-5460	470	6	have	have	VERB
ejpam-5460	470	7	a	a	DET
ejpam-5460	470	8	∩b	∩b	NOUN
ejpam-5460	470	9	∩c	∩c	NOUN
ejpam-5460	470	10	⊆	⊆	NUM
ejpam-5460	470	11	(	(	PUNCT
ejpam-5460	470	12	abc	abc	PROPN
ejpam-5460	470	13	]	]	X
ejpam-5460	470	14	.	.	PUNCT
ejpam-5460	471	1	hence	hence	ADV
ejpam-5460	471	2	,	,	PUNCT
ejpam-5460	471	3	by	by	ADP
ejpam-5460	471	4	lemma	lemma	PROPN
ejpam-5460	471	5	9	9	NUM
ejpam-5460	471	6	,	,	PUNCT
ejpam-5460	471	7	s	s	PART
ejpam-5460	471	8	satisfies	satisfie	NOUN
ejpam-5460	471	9	(	(	PUNCT
ejpam-5460	471	10	c11	c11	NOUN
ejpam-5460	471	11	)	)	PUNCT
ejpam-5460	471	12	of	of	ADP
ejpam-5460	471	13	degree	degree	NOUN
ejpam-5460	471	14	n.	n.	PROPN
ejpam-5460	471	15	lemma	lemma	PROPN
ejpam-5460	471	16	10	10	NUM
ejpam-5460	471	17	(	(	PUNCT
ejpam-5460	471	18	[	[	X
ejpam-5460	471	19	36	36	NUM
ejpam-5460	471	20	]	]	PUNCT
ejpam-5460	471	21	)	)	PUNCT
ejpam-5460	471	22	.	.	PUNCT
ejpam-5460	472	1	let	let	VERB
ejpam-5460	472	2	s	s	PRON
ejpam-5460	472	3	be	be	AUX
ejpam-5460	472	4	an	an	DET
ejpam-5460	472	5	ordered	order	VERB
ejpam-5460	472	6	semigroup	semigroup	NOUN
ejpam-5460	472	7	.	.	PUNCT
ejpam-5460	473	1	then	then	ADV
ejpam-5460	473	2	,	,	PUNCT
ejpam-5460	473	3	s	s	NOUN
ejpam-5460	473	4	satisfies	satisfie	NOUN
ejpam-5460	473	5	(	(	PUNCT
ejpam-5460	473	6	c12	c12	PROPN
ejpam-5460	473	7	)	)	PUNCT
ejpam-5460	473	8	if	if	SCONJ
ejpam-5460	473	9	and	and	CCONJ
ejpam-5460	473	10	only	only	ADV
ejpam-5460	473	11	if	if	SCONJ
ejpam-5460	473	12	a	a	DET
ejpam-5460	473	13	⊆	⊆	NUM
ejpam-5460	473	14	(	(	PUNCT
ejpam-5460	473	15	sa2	sa2	PROPN
ejpam-5460	473	16	]	]	PUNCT
ejpam-5460	473	17	for	for	ADP
ejpam-5460	473	18	any	any	DET
ejpam-5460	473	19	(	(	PUNCT
ejpam-5460	473	20	0	0	NUM
ejpam-5460	473	21	,	,	PUNCT
ejpam-5460	473	22	2)-ideal	2)-ideal	NUM
ejpam-5460	473	23	a	a	PRON
ejpam-5460	473	24	of	of	ADP
ejpam-5460	473	25	s.	s.	PROPN
ejpam-5460	473	26	theorem	theorem	VERB
ejpam-5460	473	27	13	13	NUM
ejpam-5460	473	28	.	.	PUNCT
ejpam-5460	474	1	let	let	VERB
ejpam-5460	474	2	s	s	PRON
ejpam-5460	474	3	be	be	AUX
ejpam-5460	474	4	an	an	DET
ejpam-5460	474	5	ordered	order	VERB
ejpam-5460	474	6	semigroup	semigroup	NOUN
ejpam-5460	474	7	.	.	PUNCT
ejpam-5460	475	1	then	then	ADV
ejpam-5460	475	2	,	,	PUNCT
ejpam-5460	475	3	the	the	DET
ejpam-5460	475	4	following	follow	VERB
ejpam-5460	475	5	statements	statement	NOUN
ejpam-5460	475	6	are	be	AUX
ejpam-5460	475	7	equivalent	equivalent	ADJ
ejpam-5460	475	8	.	.	PUNCT
ejpam-5460	476	1	(	(	PUNCT
ejpam-5460	476	2	i	i	NOUN
ejpam-5460	476	3	)	)	PUNCT
ejpam-5460	476	4	s	s	PART
ejpam-5460	476	5	satisfies	satisfie	NOUN
ejpam-5460	476	6	(	(	PUNCT
ejpam-5460	476	7	c12	c12	PROPN
ejpam-5460	476	8	)	)	PUNCT
ejpam-5460	476	9	.	.	PUNCT
ejpam-5460	477	1	(	(	PUNCT
ejpam-5460	477	2	ii	ii	NOUN
ejpam-5460	477	3	)	)	PUNCT
ejpam-5460	477	4	fi	fi	NOUN
ejpam-5460	478	1	⊆	⊆	NUM
ejpam-5460	478	2	β	β	X
ejpam-5460	478	3	◦	◦	NOUN
ejpam-5460	478	4	i	i	X
ejpam-5460	478	5	f	f	NOUN
ejpam-5460	478	6	◦	◦	VERB
ejpam-5460	478	7	i	i	PRON
ejpam-5460	478	8	f	f	NOUN
ejpam-5460	478	9	for	for	ADP
ejpam-5460	478	10	any	any	DET
ejpam-5460	478	11	(	(	PUNCT
ejpam-5460	478	12	α	α	NOUN
ejpam-5460	478	13	,	,	PUNCT
ejpam-5460	478	14	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	478	15	(	(	PUNCT
ejpam-5460	478	16	0	0	NUM
ejpam-5460	478	17	,	,	PUNCT
ejpam-5460	478	18	2)-ideal	2)-ideal	NUM
ejpam-5460	478	19	f	f	NOUN
ejpam-5460	478	20	of	of	ADP
ejpam-5460	478	21	s.	s.	PROPN
ejpam-5460	478	22	proof	proof	PROPN
ejpam-5460	478	23	.	.	PUNCT
ejpam-5460	479	1	(	(	PUNCT
ejpam-5460	479	2	1)⇒(2	1)⇒(2	NUM
ejpam-5460	479	3	)	)	PUNCT
ejpam-5460	479	4	.	.	PUNCT
ejpam-5460	480	1	let	let	VERB
ejpam-5460	480	2	f	f	PRON
ejpam-5460	480	3	be	be	AUX
ejpam-5460	480	4	an	an	DET
ejpam-5460	480	5	(	(	PUNCT
ejpam-5460	480	6	α	α	NOUN
ejpam-5460	480	7	,	,	PUNCT
ejpam-5460	480	8	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	480	9	(	(	PUNCT
ejpam-5460	480	10	0	0	NUM
ejpam-5460	480	11	,	,	PUNCT
ejpam-5460	480	12	2)-ideal	2)-ideal	NUM
ejpam-5460	480	13	of	of	ADP
ejpam-5460	480	14	s.	s.	PROPN
ejpam-5460	480	15	let	let	VERB
ejpam-5460	480	16	a	a	DET
ejpam-5460	480	17	∈	∈	NOUN
ejpam-5460	480	18	s.	s.	PROPN
ejpam-5460	480	19	since	since	SCONJ
ejpam-5460	480	20	s	s	PART
ejpam-5460	480	21	satisfies	satisfie	NOUN
ejpam-5460	480	22	(	(	PUNCT
ejpam-5460	480	23	c12	c12	PROPN
ejpam-5460	480	24	)	)	PUNCT
ejpam-5460	480	25	,	,	PUNCT
ejpam-5460	480	26	there	there	PRON
ejpam-5460	480	27	exist	exist	VERB
ejpam-5460	480	28	y1	y1	PROPN
ejpam-5460	480	29	∈	∈	PROPN
ejpam-5460	480	30	s	s	VERB
ejpam-5460	480	31	such	such	ADJ
ejpam-5460	480	32	that	that	SCONJ
ejpam-5460	480	33	a	a	DET
ejpam-5460	480	34	≤	≤	NUM
ejpam-5460	480	35	y1a	y1a	PROPN
ejpam-5460	480	36	2	2	NUM
ejpam-5460	480	37	≤	≤	NUM
ejpam-5460	480	38	y21a	y21a	NOUN
ejpam-5460	480	39	3	3	NUM
ejpam-5460	480	40	≤	≤	NUM
ejpam-5460	480	41	y31a	y31a	NUM
ejpam-5460	480	42	4	4	NUM
ejpam-5460	480	43	≤	≤	NUM
ejpam-5460	480	44	y41a	y41a	NOUN
ejpam-5460	480	45	5	5	NUM
ejpam-5460	480	46	.	.	PUNCT
ejpam-5460	481	1	that	that	ADV
ejpam-5460	481	2	is	is	ADV
ejpam-5460	481	3	,	,	PUNCT
ejpam-5460	481	4	(	(	PUNCT
ejpam-5460	481	5	x1x2a	x1x2a	X
ejpam-5460	481	6	2	2	NUM
ejpam-5460	481	7	,	,	PUNCT
ejpam-5460	481	8	x3a	x3a	PROPN
ejpam-5460	481	9	2	2	X
ejpam-5460	481	10	)	)	PUNCT
ejpam-5460	481	11	∈	∈	PROPN
ejpam-5460	481	12	sa	sa	NOUN
ejpam-5460	481	13	for	for	ADP
ejpam-5460	481	14	some	some	DET
ejpam-5460	481	15	x1	x1	PROPN
ejpam-5460	481	16	,	,	PUNCT
ejpam-5460	481	17	x2	x2	PROPN
ejpam-5460	481	18	,	,	PUNCT
ejpam-5460	481	19	x3	x3	PROPN
ejpam-5460	481	20	∈	∈	PROPN
ejpam-5460	481	21	s.	s.	PROPN
ejpam-5460	481	22	then	then	ADV
ejpam-5460	481	23	,	,	PUNCT
ejpam-5460	481	24	(	(	PUNCT
ejpam-5460	481	25	β	β	X
ejpam-5460	481	26	◦	◦	NOUN
ejpam-5460	481	27	i	i	NOUN
ejpam-5460	481	28	f	f	NOUN
ejpam-5460	481	29	◦	◦	NOUN
ejpam-5460	481	30	i	i	PRON
ejpam-5460	481	31	f)(a	f)(a	NOUN
ejpam-5460	481	32	)	)	PUNCT
ejpam-5460	481	33	=	=	PUNCT
ejpam-5460	482	1	[	[	X
ejpam-5460	482	2	(	(	PUNCT
ejpam-5460	482	3	β	β	X
ejpam-5460	482	4	◦	◦	NOUN
ejpam-5460	482	5	i	i	NOUN
ejpam-5460	482	6	f	f	X
ejpam-5460	482	7	)	)	PUNCT
ejpam-5460	482	8	◦	◦	NOUN
ejpam-5460	482	9	f	f	X
ejpam-5460	483	1	]	]	X
ejpam-5460	483	2	i	i	PRON
ejpam-5460	483	3	(	(	PUNCT
ejpam-5460	483	4	a	a	X
ejpam-5460	483	5	)	)	PUNCT
ejpam-5460	483	6	s.	s.	PROPN
ejpam-5460	483	7	lekkoksung	lekkoksung	PROPN
ejpam-5460	483	8	,	,	PUNCT
ejpam-5460	483	9	b.	b.	PROPN
ejpam-5460	483	10	davvaz	davvaz	PROPN
ejpam-5460	483	11	,	,	PUNCT
ejpam-5460	483	12	n.	n.	PROPN
ejpam-5460	483	13	lekkoksung	lekkoksung	PROPN
ejpam-5460	483	14	/	/	SYM
ejpam-5460	483	15	eur	eur	PROPN
ejpam-5460	483	16	.	.	PUNCT
ejpam-5460	484	1	j.	j.	PROPN
ejpam-5460	484	2	pure	pure	PROPN
ejpam-5460	484	3	appl	appl	PROPN
ejpam-5460	484	4	.	.	PROPN
ejpam-5460	484	5	math	math	PROPN
ejpam-5460	484	6	,	,	PUNCT
ejpam-5460	484	7	17	17	NUM
ejpam-5460	484	8	(	(	PUNCT
ejpam-5460	484	9	4	4	NUM
ejpam-5460	484	10	)	)	PUNCT
ejpam-5460	484	11	(	(	PUNCT
ejpam-5460	484	12	2024	2024	NUM
ejpam-5460	484	13	)	)	PUNCT
ejpam-5460	484	14	,	,	PUNCT
ejpam-5460	484	15	2962	2962	NUM
ejpam-5460	484	16	-	-	SYM
ejpam-5460	484	17	2984	2984	NUM
ejpam-5460	484	18	2979	2979	NUM
ejpam-5460	484	19	≥	≥	NOUN
ejpam-5460	484	20	(	(	PUNCT
ejpam-5460	484	21	β	β	X
ejpam-5460	484	22	◦	◦	NOUN
ejpam-5460	484	23	i	i	PRON
ejpam-5460	484	24	f)i(x1x2a2	f)i(x1x2a2	NOUN
ejpam-5460	484	25	)	)	PUNCT
ejpam-5460	484	26	∧	∧	PROPN
ejpam-5460	484	27	fi(x3a	fi(x3a	PROPN
ejpam-5460	484	28	2	2	NUM
ejpam-5460	484	29	)	)	PUNCT
ejpam-5460	484	30	≥	≥	NOUN
ejpam-5460	484	31	(	(	PUNCT
ejpam-5460	484	32	β	β	X
ejpam-5460	484	33	◦	◦	NOUN
ejpam-5460	484	34	i	i	PRON
ejpam-5460	484	35	f)i(x1x2a2	f)i(x1x2a2	NOUN
ejpam-5460	484	36	)	)	PUNCT
ejpam-5460	484	37	∧	∧	NOUN
ejpam-5460	484	38	fi(a	fi(a	NOUN
ejpam-5460	484	39	)	)	PUNCT
ejpam-5460	484	40	=	=	SYM
ejpam-5460	484	41	(	(	PUNCT
ejpam-5460	484	42	β	β	X
ejpam-5460	484	43	◦	◦	NOUN
ejpam-5460	484	44	f)i(x1x2a2	f)i(x1x2a2	NOUN
ejpam-5460	484	45	)	)	PUNCT
ejpam-5460	484	46	∧	∧	NOUN
ejpam-5460	484	47	fi(a	fi(a	NOUN
ejpam-5460	484	48	)	)	PUNCT
ejpam-5460	484	49	=	=	PUNCT
ejpam-5460	485	1	fi(x2a	fi(x2a	NOUN
ejpam-5460	485	2	2	2	X
ejpam-5460	485	3	)	)	PUNCT
ejpam-5460	485	4	∧	∧	NOUN
ejpam-5460	485	5	fi(a	fi(a	NOUN
ejpam-5460	485	6	)	)	PUNCT
ejpam-5460	485	7	=	=	SYM
ejpam-5460	485	8	fi(a	fi(a	X
ejpam-5460	485	9	)	)	PUNCT
ejpam-5460	485	10	∧	∧	NOUN
ejpam-5460	485	11	fi(a	fi(a	NOUN
ejpam-5460	485	12	)	)	PUNCT
ejpam-5460	485	13	=	=	SYM
ejpam-5460	485	14	fi(a	fi(a	PROPN
ejpam-5460	485	15	)	)	PUNCT
ejpam-5460	485	16	.	.	PUNCT
ejpam-5460	486	1	(	(	PUNCT
ejpam-5460	486	2	2)⇒(1	2)⇒(1	NOUN
ejpam-5460	486	3	)	)	PUNCT
ejpam-5460	486	4	.	.	PUNCT
ejpam-5460	487	1	let	let	VERB
ejpam-5460	487	2	a	a	DET
ejpam-5460	487	3	be	be	AUX
ejpam-5460	487	4	a	a	DET
ejpam-5460	487	5	(	(	PUNCT
ejpam-5460	487	6	0	0	NUM
ejpam-5460	487	7	,	,	PUNCT
ejpam-5460	487	8	2)-ideal	2)-ideal	NUM
ejpam-5460	487	9	of	of	ADP
ejpam-5460	487	10	s.	s.	PROPN
ejpam-5460	487	11	then	then	ADV
ejpam-5460	487	12	,	,	PUNCT
ejpam-5460	487	13	by	by	ADP
ejpam-5460	487	14	proposition	proposition	NOUN
ejpam-5460	487	15	2	2	NUM
ejpam-5460	487	16	,	,	PUNCT
ejpam-5460	487	17	(	(	PUNCT
ejpam-5460	487	18	χa)i	χa)i	PROPN
ejpam-5460	487	19	is	be	AUX
ejpam-5460	487	20	an	an	DET
ejpam-5460	487	21	(	(	PUNCT
ejpam-5460	487	22	α	α	NOUN
ejpam-5460	487	23	,	,	PUNCT
ejpam-5460	487	24	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	487	25	(	(	PUNCT
ejpam-5460	487	26	0	0	NUM
ejpam-5460	487	27	,	,	PUNCT
ejpam-5460	487	28	2)-ideal	2)-ideal	NUM
ejpam-5460	487	29	of	of	ADP
ejpam-5460	487	30	s.	s.	PROPN
ejpam-5460	487	31	by	by	ADP
ejpam-5460	487	32	our	our	PRON
ejpam-5460	487	33	presumption	presumption	NOUN
ejpam-5460	487	34	,	,	PUNCT
ejpam-5460	487	35	we	we	PRON
ejpam-5460	487	36	have	have	VERB
ejpam-5460	487	37	(	(	PUNCT
ejpam-5460	487	38	χa)i	χa)i	PROPN
ejpam-5460	487	39	=	=	SYM
ejpam-5460	487	40	(	(	PUNCT
ejpam-5460	487	41	(	(	PUNCT
ejpam-5460	487	42	χa)i)i	χa)i)i	NOUN
ejpam-5460	487	43	⊆	⊆	NUM
ejpam-5460	487	44	β	β	X
ejpam-5460	487	45	◦	◦	NOUN
ejpam-5460	487	46	i	i	PRON
ejpam-5460	487	47	(	(	PUNCT
ejpam-5460	487	48	χa)i	χa)i	PROPN
ejpam-5460	487	49	◦	◦	NOUN
ejpam-5460	487	50	i	i	PROPN
ejpam-5460	487	51	(	(	PUNCT
ejpam-5460	487	52	χa)i	χa)i	PROPN
ejpam-5460	487	53	=	=	SYM
ejpam-5460	487	54	(	(	PUNCT
ejpam-5460	487	55	χs)i	χs)i	PROPN
ejpam-5460	487	56	◦	◦	NOUN
ejpam-5460	487	57	i	i	PROPN
ejpam-5460	487	58	(	(	PUNCT
ejpam-5460	487	59	χa)i	χa)i	PROPN
ejpam-5460	487	60	◦	◦	NOUN
ejpam-5460	487	61	i	i	PROPN
ejpam-5460	487	62	(	(	PUNCT
ejpam-5460	487	63	χa)i	χa)i	PROPN
ejpam-5460	487	64	.	.	PUNCT
ejpam-5460	488	1	this	this	PRON
ejpam-5460	488	2	implies	imply	VERB
ejpam-5460	488	3	that	that	SCONJ
ejpam-5460	488	4	(	(	PUNCT
ejpam-5460	488	5	χa)i	χa)i	PROPN
ejpam-5460	488	6	=	=	SYM
ejpam-5460	488	7	(	(	PUNCT
ejpam-5460	488	8	χs∩a∩a)i	χs∩a∩a)i	X
ejpam-5460	488	9	=	=	SYM
ejpam-5460	488	10	χs	χs	PROPN
ejpam-5460	488	11	∩i	∩i	PROPN
ejpam-5460	488	12	χa	χa	PROPN
ejpam-5460	488	13	∩i	∩i	VERB
ejpam-5460	488	14	χa	χa	PROPN
ejpam-5460	489	1	=	=	PUNCT
ejpam-5460	490	1	(	(	PUNCT
ejpam-5460	490	2	χs)i	χs)i	PROPN
ejpam-5460	490	3	∩i	∩i	PROPN
ejpam-5460	490	4	(	(	PUNCT
ejpam-5460	490	5	χa)i	χa)i	PROPN
ejpam-5460	490	6	∩i	∩i	NOUN
ejpam-5460	490	7	(	(	PUNCT
ejpam-5460	490	8	χa)i	χa)i	PROPN
ejpam-5460	490	9	⊆	⊆	NUM
ejpam-5460	490	10	(	(	PUNCT
ejpam-5460	490	11	χs)i	χs)i	PROPN
ejpam-5460	490	12	◦	◦	NOUN
ejpam-5460	490	13	i	i	PROPN
ejpam-5460	490	14	(	(	PUNCT
ejpam-5460	490	15	χa)i	χa)i	PROPN
ejpam-5460	490	16	◦	◦	NOUN
ejpam-5460	490	17	i	i	PROPN
ejpam-5460	490	18	(	(	PUNCT
ejpam-5460	490	19	χa)i	χa)i	PROPN
ejpam-5460	490	20	=	=	PUNCT
ejpam-5460	490	21	χs	χs	PART
ejpam-5460	490	22	◦	◦	NOUN
ejpam-5460	490	23	i	i	PRON
ejpam-5460	490	24	χa	χa	VERB
ejpam-5460	490	25	◦	◦	NOUN
ejpam-5460	490	26	i	i	PRON
ejpam-5460	490	27	χa	χa	VERB
ejpam-5460	490	28	=	=	PUNCT
ejpam-5460	490	29	(	(	PUNCT
ejpam-5460	490	30	χ(sa2])i	χ(sa2])i	VERB
ejpam-5460	490	31	.	.	PUNCT
ejpam-5460	491	1	by	by	ADP
ejpam-5460	491	2	proposition	proposition	NOUN
ejpam-5460	491	3	2	2	NUM
ejpam-5460	491	4	,	,	PUNCT
ejpam-5460	491	5	we	we	PRON
ejpam-5460	491	6	have	have	VERB
ejpam-5460	491	7	a	a	DET
ejpam-5460	491	8	⊆	⊆	NUM
ejpam-5460	491	9	(	(	PUNCT
ejpam-5460	491	10	sa2	sa2	PROPN
ejpam-5460	491	11	]	]	PUNCT
ejpam-5460	491	12	.	.	PUNCT
ejpam-5460	492	1	hence	hence	ADV
ejpam-5460	492	2	,	,	PUNCT
ejpam-5460	492	3	by	by	ADP
ejpam-5460	492	4	lemma	lemma	PROPN
ejpam-5460	492	5	9	9	NUM
ejpam-5460	492	6	,	,	PUNCT
ejpam-5460	492	7	s	s	PART
ejpam-5460	492	8	satisfies	satisfie	NOUN
ejpam-5460	492	9	(	(	PUNCT
ejpam-5460	492	10	c12	c12	PROPN
ejpam-5460	492	11	)	)	PUNCT
ejpam-5460	492	12	.	.	PUNCT
ejpam-5460	493	1	the	the	DET
ejpam-5460	493	2	following	follow	VERB
ejpam-5460	493	3	theorem	theorem	VERB
ejpam-5460	493	4	,	,	PUNCT
ejpam-5460	493	5	we	we	PRON
ejpam-5460	493	6	provide	provide	VERB
ejpam-5460	493	7	a	a	DET
ejpam-5460	493	8	characterization	characterization	NOUN
ejpam-5460	493	9	of	of	ADP
ejpam-5460	493	10	ordered	order	VERB
ejpam-5460	493	11	semigroups	semigroup	NOUN
ejpam-5460	493	12	satisfying	satisfy	VERB
ejpam-5460	493	13	(	(	PUNCT
ejpam-5460	493	14	c13	c13	X
ejpam-5460	493	15	)	)	PUNCT
ejpam-5460	493	16	by	by	ADP
ejpam-5460	493	17	(	(	PUNCT
ejpam-5460	493	18	α	α	X
ejpam-5460	493	19	,	,	PUNCT
ejpam-5460	493	20	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	493	21	(	(	PUNCT
ejpam-5460	493	22	2	2	NUM
ejpam-5460	493	23	,	,	PUNCT
ejpam-5460	493	24	0)-ideals	0)-ideal	NOUN
ejpam-5460	493	25	.	.	PUNCT
ejpam-5460	494	1	the	the	DET
ejpam-5460	494	2	proof	proof	NOUN
ejpam-5460	494	3	is	be	AUX
ejpam-5460	494	4	similar	similar	ADJ
ejpam-5460	494	5	to	to	ADP
ejpam-5460	494	6	the	the	DET
ejpam-5460	494	7	above	above	ADJ
ejpam-5460	494	8	theorem	theorem	NOUN
ejpam-5460	494	9	,	,	PUNCT
ejpam-5460	494	10	so	so	SCONJ
ejpam-5460	494	11	we	we	PRON
ejpam-5460	494	12	skip	skip	VERB
ejpam-5460	494	13	the	the	DET
ejpam-5460	494	14	proof	proof	NOUN
ejpam-5460	494	15	.	.	PUNCT
ejpam-5460	495	1	theorem	theorem	VERB
ejpam-5460	495	2	14	14	NUM
ejpam-5460	495	3	.	.	PUNCT
ejpam-5460	496	1	let	let	VERB
ejpam-5460	496	2	s	s	PRON
ejpam-5460	496	3	be	be	AUX
ejpam-5460	496	4	an	an	DET
ejpam-5460	496	5	ordered	order	VERB
ejpam-5460	496	6	semigroup	semigroup	NOUN
ejpam-5460	496	7	.	.	PUNCT
ejpam-5460	497	1	then	then	ADV
ejpam-5460	497	2	,	,	PUNCT
ejpam-5460	497	3	the	the	DET
ejpam-5460	497	4	following	follow	VERB
ejpam-5460	497	5	statements	statement	NOUN
ejpam-5460	497	6	are	be	AUX
ejpam-5460	497	7	equivalent	equivalent	ADJ
ejpam-5460	497	8	.	.	PUNCT
ejpam-5460	498	1	(	(	PUNCT
ejpam-5460	498	2	i	i	NOUN
ejpam-5460	498	3	)	)	PUNCT
ejpam-5460	498	4	s	s	PART
ejpam-5460	498	5	satisfies	satisfie	NOUN
ejpam-5460	498	6	(	(	PUNCT
ejpam-5460	498	7	c13	c13	PROPN
ejpam-5460	498	8	)	)	PUNCT
ejpam-5460	498	9	.	.	PUNCT
ejpam-5460	499	1	(	(	PUNCT
ejpam-5460	499	2	ii	ii	NOUN
ejpam-5460	499	3	)	)	PUNCT
ejpam-5460	499	4	fi	fi	NOUN
ejpam-5460	500	1	⊆	⊆	NUM
ejpam-5460	500	2	f	f	PROPN
ejpam-5460	500	3	◦	◦	NOUN
ejpam-5460	500	4	i	i	NOUN
ejpam-5460	500	5	f	f	NOUN
ejpam-5460	500	6	◦	◦	NOUN
ejpam-5460	500	7	i	i	PRON
ejpam-5460	500	8	β	β	VERB
ejpam-5460	500	9	for	for	ADP
ejpam-5460	500	10	any	any	DET
ejpam-5460	500	11	(	(	PUNCT
ejpam-5460	500	12	α	α	NOUN
ejpam-5460	500	13	,	,	PUNCT
ejpam-5460	500	14	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	500	15	(	(	PUNCT
ejpam-5460	500	16	2	2	NUM
ejpam-5460	500	17	,	,	PUNCT
ejpam-5460	500	18	0)-ideal	0)-ideal	PROPN
ejpam-5460	500	19	f	f	PROPN
ejpam-5460	500	20	of	of	ADP
ejpam-5460	500	21	s.	s.	PROPN
ejpam-5460	500	22	lemma	lemma	PROPN
ejpam-5460	500	23	11	11	NUM
ejpam-5460	500	24	(	(	PUNCT
ejpam-5460	500	25	[	[	X
ejpam-5460	500	26	36	36	NUM
ejpam-5460	500	27	]	]	PUNCT
ejpam-5460	500	28	)	)	PUNCT
ejpam-5460	500	29	.	.	PUNCT
ejpam-5460	501	1	let	let	VERB
ejpam-5460	501	2	s	s	PRON
ejpam-5460	501	3	be	be	AUX
ejpam-5460	501	4	an	an	DET
ejpam-5460	501	5	ordered	order	VERB
ejpam-5460	501	6	semigroup	semigroup	NOUN
ejpam-5460	501	7	.	.	PUNCT
ejpam-5460	502	1	then	then	ADV
ejpam-5460	502	2	,	,	PUNCT
ejpam-5460	502	3	s	s	NOUN
ejpam-5460	502	4	satisfies	satisfie	NOUN
ejpam-5460	502	5	(	(	PUNCT
ejpam-5460	502	6	c14	c14	NOUN
ejpam-5460	502	7	)	)	PUNCT
ejpam-5460	502	8	if	if	SCONJ
ejpam-5460	502	9	and	and	CCONJ
ejpam-5460	502	10	only	only	ADV
ejpam-5460	502	11	if	if	SCONJ
ejpam-5460	502	12	a	a	DET
ejpam-5460	502	13	∩b	∩b	NOUN
ejpam-5460	502	14	⊆	⊆	NUM
ejpam-5460	502	15	(	(	PUNCT
ejpam-5460	502	16	ab	ab	X
ejpam-5460	502	17	]	]	X
ejpam-5460	502	18	for	for	ADP
ejpam-5460	502	19	any	any	DET
ejpam-5460	502	20	(	(	PUNCT
ejpam-5460	502	21	2	2	NUM
ejpam-5460	502	22	,	,	PUNCT
ejpam-5460	502	23	0)-ideal	0)-ideal	NOUN
ejpam-5460	502	24	a	a	NOUN
ejpam-5460	502	25	and	and	CCONJ
ejpam-5460	502	26	(	(	PUNCT
ejpam-5460	502	27	0	0	NUM
ejpam-5460	502	28	,	,	PUNCT
ejpam-5460	502	29	2)-ideal	2)-ideal	NUM
ejpam-5460	502	30	b	b	PROPN
ejpam-5460	502	31	of	of	ADP
ejpam-5460	502	32	s.	s.	PROPN
ejpam-5460	502	33	theorem	theorem	VERB
ejpam-5460	502	34	15	15	NUM
ejpam-5460	502	35	.	.	PUNCT
ejpam-5460	503	1	let	let	VERB
ejpam-5460	503	2	s	s	PRON
ejpam-5460	503	3	be	be	AUX
ejpam-5460	503	4	an	an	DET
ejpam-5460	503	5	ordered	order	VERB
ejpam-5460	503	6	semigroup	semigroup	NOUN
ejpam-5460	503	7	.	.	PUNCT
ejpam-5460	504	1	then	then	ADV
ejpam-5460	504	2	,	,	PUNCT
ejpam-5460	504	3	the	the	DET
ejpam-5460	504	4	following	follow	VERB
ejpam-5460	504	5	statements	statement	NOUN
ejpam-5460	504	6	are	be	AUX
ejpam-5460	504	7	equivalent	equivalent	ADJ
ejpam-5460	504	8	.	.	PUNCT
ejpam-5460	505	1	(	(	PUNCT
ejpam-5460	505	2	i	i	NOUN
ejpam-5460	505	3	)	)	PUNCT
ejpam-5460	505	4	s	s	PART
ejpam-5460	505	5	satisfies	satisfie	NOUN
ejpam-5460	505	6	(	(	PUNCT
ejpam-5460	505	7	c14	c14	NOUN
ejpam-5460	505	8	)	)	PUNCT
ejpam-5460	505	9	.	.	PUNCT
ejpam-5460	506	1	(	(	PUNCT
ejpam-5460	506	2	ii	ii	X
ejpam-5460	506	3	)	)	PUNCT
ejpam-5460	506	4	f	f	PROPN
ejpam-5460	506	5	∩i	∩i	PROPN
ejpam-5460	506	6	g	g	PROPN
ejpam-5460	506	7	⊆	⊆	NUM
ejpam-5460	506	8	f	f	PROPN
ejpam-5460	506	9	◦	◦	NOUN
ejpam-5460	506	10	i	i	PRON
ejpam-5460	506	11	g	g	NOUN
ejpam-5460	506	12	for	for	ADP
ejpam-5460	506	13	any	any	DET
ejpam-5460	506	14	(	(	PUNCT
ejpam-5460	506	15	α	α	NOUN
ejpam-5460	506	16	,	,	PUNCT
ejpam-5460	506	17	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	506	18	(	(	PUNCT
ejpam-5460	506	19	2	2	NUM
ejpam-5460	506	20	,	,	PUNCT
ejpam-5460	506	21	0)-ideal	0)-ideal	PROPN
ejpam-5460	506	22	f	f	PROPN
ejpam-5460	506	23	and	and	CCONJ
ejpam-5460	506	24	(	(	PUNCT
ejpam-5460	506	25	α	α	NOUN
ejpam-5460	506	26	,	,	PUNCT
ejpam-5460	506	27	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	506	28	(	(	PUNCT
ejpam-5460	506	29	0	0	NUM
ejpam-5460	506	30	,	,	PUNCT
ejpam-5460	506	31	2)-ideal	2)-ideal	NUM
ejpam-5460	506	32	g	g	NOUN
ejpam-5460	506	33	of	of	ADP
ejpam-5460	506	34	s.	s.	PROPN
ejpam-5460	506	35	proof	proof	PROPN
ejpam-5460	506	36	.	.	PUNCT
ejpam-5460	507	1	(	(	PUNCT
ejpam-5460	507	2	1)⇒(2	1)⇒(2	NUM
ejpam-5460	507	3	)	)	PUNCT
ejpam-5460	507	4	.	.	PUNCT
ejpam-5460	508	1	let	let	VERB
ejpam-5460	508	2	f	f	PROPN
ejpam-5460	508	3	and	and	CCONJ
ejpam-5460	508	4	g	g	PROPN
ejpam-5460	508	5	be	be	AUX
ejpam-5460	508	6	an	an	DET
ejpam-5460	508	7	(	(	PUNCT
ejpam-5460	508	8	α	α	NOUN
ejpam-5460	508	9	,	,	PUNCT
ejpam-5460	508	10	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	508	11	(	(	PUNCT
ejpam-5460	508	12	2	2	NUM
ejpam-5460	508	13	,	,	PUNCT
ejpam-5460	508	14	0)-ideal	0)-ideal	NUM
ejpam-5460	508	15	and	and	CCONJ
ejpam-5460	508	16	an	an	DET
ejpam-5460	508	17	(	(	PUNCT
ejpam-5460	508	18	α	α	NOUN
ejpam-5460	508	19	,	,	PUNCT
ejpam-5460	508	20	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	508	21	(	(	PUNCT
ejpam-5460	508	22	0	0	NUM
ejpam-5460	508	23	,	,	PUNCT
ejpam-5460	508	24	2)ideal	2)ideal	NUM
ejpam-5460	508	25	of	of	ADP
ejpam-5460	508	26	s	s	PROPN
ejpam-5460	508	27	,	,	PUNCT
ejpam-5460	508	28	respectively	respectively	ADV
ejpam-5460	508	29	.	.	PUNCT
ejpam-5460	509	1	let	let	VERB
ejpam-5460	509	2	a	a	DET
ejpam-5460	509	3	∈	∈	NOUN
ejpam-5460	509	4	s.	s.	PROPN
ejpam-5460	509	5	since	since	SCONJ
ejpam-5460	509	6	s	s	PART
ejpam-5460	509	7	satisfies	satisfie	NOUN
ejpam-5460	509	8	(	(	PUNCT
ejpam-5460	509	9	c14	c14	NOUN
ejpam-5460	509	10	)	)	PUNCT
ejpam-5460	509	11	,	,	PUNCT
ejpam-5460	509	12	there	there	PRON
ejpam-5460	509	13	exist	exist	VERB
ejpam-5460	509	14	y1	y1	PROPN
ejpam-5460	509	15	∈	∈	PROPN
ejpam-5460	509	16	s	s	VERB
ejpam-5460	509	17	such	such	ADJ
ejpam-5460	509	18	that	that	SCONJ
ejpam-5460	509	19	a	a	DET
ejpam-5460	509	20	≤	≤	NUM
ejpam-5460	509	21	a2y1a	a2y1a	SYM
ejpam-5460	509	22	2	2	NUM
ejpam-5460	509	23	≤	≤	NOUN
ejpam-5460	509	24	a2y1a	a2y1a	PUNCT
ejpam-5460	509	25	3y1a	3y1a	PROPN
ejpam-5460	509	26	2	2	NUM
ejpam-5460	509	27	.	.	X
ejpam-5460	509	28	that	that	PRON
ejpam-5460	509	29	is	is	ADV
ejpam-5460	509	30	,	,	PUNCT
ejpam-5460	509	31	(	(	PUNCT
ejpam-5460	509	32	a2x1	a2x1	X
ejpam-5460	509	33	,	,	PUNCT
ejpam-5460	509	34	x2a	x2a	PROPN
ejpam-5460	509	35	2	2	X
ejpam-5460	509	36	)	)	PUNCT
ejpam-5460	509	37	∈	∈	PROPN
ejpam-5460	509	38	sa	sa	NOUN
ejpam-5460	509	39	for	for	ADP
ejpam-5460	509	40	some	some	DET
ejpam-5460	509	41	x1	x1	PROPN
ejpam-5460	509	42	,	,	PUNCT
ejpam-5460	509	43	x2	x2	PROPN
ejpam-5460	509	44	∈	∈	PROPN
ejpam-5460	509	45	s.	s.	PROPN
ejpam-5460	509	46	then	then	ADV
ejpam-5460	509	47	,	,	PUNCT
ejpam-5460	509	48	(	(	PUNCT
ejpam-5460	509	49	f	f	X
ejpam-5460	509	50	◦	◦	NOUN
ejpam-5460	509	51	i	i	PRON
ejpam-5460	509	52	g)(a	g)(a	VERB
ejpam-5460	509	53	)	)	PUNCT
ejpam-5460	509	54	≥	≥	NOUN
ejpam-5460	509	55	fi(a	fi(a	NUM
ejpam-5460	509	56	2x1	2x1	NUM
ejpam-5460	509	57	)	)	PUNCT
ejpam-5460	509	58	∧	∧	PROPN
ejpam-5460	509	59	gi(x2a	gi(x2a	PROPN
ejpam-5460	509	60	2	2	NUM
ejpam-5460	509	61	)	)	PUNCT
ejpam-5460	509	62	≥	≥	NOUN
ejpam-5460	509	63	fi(a	fi(a	NOUN
ejpam-5460	509	64	)	)	PUNCT
ejpam-5460	509	65	∧	∧	NOUN
ejpam-5460	509	66	gi(a	gi(a	X
ejpam-5460	509	67	)	)	PUNCT
ejpam-5460	509	68	=	=	SYM
ejpam-5460	510	1	(	(	PUNCT
ejpam-5460	510	2	f	f	PROPN
ejpam-5460	510	3	∩i	∩i	PROPN
ejpam-5460	510	4	g)(a	g)(a	PROPN
ejpam-5460	510	5	)	)	PUNCT
ejpam-5460	510	6	.	.	PUNCT
ejpam-5460	511	1	s.	s.	PROPN
ejpam-5460	511	2	lekkoksung	lekkoksung	PROPN
ejpam-5460	511	3	,	,	PUNCT
ejpam-5460	511	4	b.	b.	PROPN
ejpam-5460	511	5	davvaz	davvaz	PROPN
ejpam-5460	511	6	,	,	PUNCT
ejpam-5460	511	7	n.	n.	PROPN
ejpam-5460	511	8	lekkoksung	lekkoksung	PROPN
ejpam-5460	511	9	/	/	SYM
ejpam-5460	511	10	eur	eur	PROPN
ejpam-5460	511	11	.	.	PUNCT
ejpam-5460	512	1	j.	j.	PROPN
ejpam-5460	512	2	pure	pure	PROPN
ejpam-5460	512	3	appl	appl	PROPN
ejpam-5460	512	4	.	.	PROPN
ejpam-5460	512	5	math	math	PROPN
ejpam-5460	512	6	,	,	PUNCT
ejpam-5460	512	7	17	17	NUM
ejpam-5460	512	8	(	(	PUNCT
ejpam-5460	512	9	4	4	NUM
ejpam-5460	512	10	)	)	PUNCT
ejpam-5460	512	11	(	(	PUNCT
ejpam-5460	512	12	2024	2024	NUM
ejpam-5460	512	13	)	)	PUNCT
ejpam-5460	512	14	,	,	PUNCT
ejpam-5460	512	15	2962	2962	NUM
ejpam-5460	512	16	-	-	SYM
ejpam-5460	512	17	2984	2984	NUM
ejpam-5460	512	18	2980	2980	NUM
ejpam-5460	512	19	(	(	PUNCT
ejpam-5460	512	20	2)⇒(1	2)⇒(1	NOUN
ejpam-5460	512	21	)	)	PUNCT
ejpam-5460	512	22	.	.	PUNCT
ejpam-5460	513	1	let	let	VERB
ejpam-5460	513	2	a	a	PRON
ejpam-5460	513	3	and	and	CCONJ
ejpam-5460	513	4	b	b	NOUN
ejpam-5460	513	5	be	be	AUX
ejpam-5460	513	6	a	a	DET
ejpam-5460	513	7	(	(	PUNCT
ejpam-5460	513	8	2	2	NUM
ejpam-5460	513	9	,	,	PUNCT
ejpam-5460	513	10	0)-ideal	0)-ideal	NUM
ejpam-5460	513	11	and	and	CCONJ
ejpam-5460	513	12	a	a	DET
ejpam-5460	513	13	(	(	PUNCT
ejpam-5460	513	14	0	0	NUM
ejpam-5460	513	15	,	,	PUNCT
ejpam-5460	513	16	2)-ideal	2)-ideal	NUM
ejpam-5460	513	17	of	of	ADP
ejpam-5460	513	18	s	s	PRON
ejpam-5460	513	19	,	,	PUNCT
ejpam-5460	513	20	respectively	respectively	ADV
ejpam-5460	513	21	.	.	PUNCT
ejpam-5460	514	1	then	then	ADV
ejpam-5460	514	2	,	,	PUNCT
ejpam-5460	514	3	by	by	ADP
ejpam-5460	514	4	proposition	proposition	NOUN
ejpam-5460	514	5	2	2	NUM
ejpam-5460	514	6	,	,	PUNCT
ejpam-5460	514	7	(	(	PUNCT
ejpam-5460	514	8	χa)i	χa)i	NUM
ejpam-5460	514	9	and	and	CCONJ
ejpam-5460	514	10	(	(	PUNCT
ejpam-5460	514	11	χb)i	χb)i	PROPN
ejpam-5460	514	12	is	be	AUX
ejpam-5460	514	13	an	an	DET
ejpam-5460	514	14	(	(	PUNCT
ejpam-5460	514	15	α	α	NOUN
ejpam-5460	514	16	,	,	PUNCT
ejpam-5460	514	17	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	514	18	(	(	PUNCT
ejpam-5460	514	19	2	2	NUM
ejpam-5460	514	20	,	,	PUNCT
ejpam-5460	514	21	0)-ideal	0)-ideal	NUM
ejpam-5460	514	22	and	and	CCONJ
ejpam-5460	514	23	an	an	DET
ejpam-5460	514	24	(	(	PUNCT
ejpam-5460	514	25	α	α	NOUN
ejpam-5460	514	26	,	,	PUNCT
ejpam-5460	514	27	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	514	28	(	(	PUNCT
ejpam-5460	514	29	0	0	NUM
ejpam-5460	514	30	,	,	PUNCT
ejpam-5460	514	31	2)ideal	2)ideal	NUM
ejpam-5460	514	32	of	of	ADP
ejpam-5460	514	33	s	s	PROPN
ejpam-5460	514	34	,	,	PUNCT
ejpam-5460	514	35	respectively	respectively	ADV
ejpam-5460	514	36	.	.	PUNCT
ejpam-5460	515	1	by	by	ADP
ejpam-5460	515	2	our	our	PRON
ejpam-5460	515	3	presumption	presumption	NOUN
ejpam-5460	515	4	,	,	PUNCT
ejpam-5460	515	5	we	we	PRON
ejpam-5460	515	6	have	have	VERB
ejpam-5460	515	7	(	(	PUNCT
ejpam-5460	515	8	χa)i	χa)i	PROPN
ejpam-5460	515	9	∩i	∩i	NOUN
ejpam-5460	515	10	(	(	PUNCT
ejpam-5460	515	11	χb)i	χb)i	PROPN
ejpam-5460	515	12	⊆	⊆	NUM
ejpam-5460	515	13	(	(	PUNCT
ejpam-5460	515	14	χa)i	χa)i	PROPN
ejpam-5460	515	15	◦	◦	NOUN
ejpam-5460	515	16	i	i	PROPN
ejpam-5460	515	17	(	(	PUNCT
ejpam-5460	515	18	χb)i	χb)i	PROPN
ejpam-5460	515	19	.	.	PUNCT
ejpam-5460	516	1	this	this	PRON
ejpam-5460	516	2	implies	imply	VERB
ejpam-5460	516	3	that	that	SCONJ
ejpam-5460	516	4	(	(	PUNCT
ejpam-5460	516	5	χa∩b)i	χa∩b)i	NOUN
ejpam-5460	516	6	=	=	NOUN
ejpam-5460	516	7	χa	χa	NOUN
ejpam-5460	516	8	∩i	∩i	ADV
ejpam-5460	516	9	χb	χb	PROPN
ejpam-5460	516	10	=	=	PUNCT
ejpam-5460	516	11	(	(	PUNCT
ejpam-5460	516	12	χa)i	χa)i	PROPN
ejpam-5460	516	13	∩i	∩i	NOUN
ejpam-5460	516	14	(	(	PUNCT
ejpam-5460	516	15	χb)i	χb)i	PROPN
ejpam-5460	516	16	⊆	⊆	NUM
ejpam-5460	516	17	(	(	PUNCT
ejpam-5460	516	18	χa)i	χa)i	PROPN
ejpam-5460	516	19	◦	◦	NOUN
ejpam-5460	516	20	i	i	PROPN
ejpam-5460	516	21	(	(	PUNCT
ejpam-5460	516	22	χb)i	χb)i	PROPN
ejpam-5460	516	23	=	=	PUNCT
ejpam-5460	516	24	χa	χa	NOUN
ejpam-5460	516	25	◦	◦	NOUN
ejpam-5460	516	26	i	i	PRON
ejpam-5460	516	27	χb	χb	VERB
ejpam-5460	516	28	=	=	PUNCT
ejpam-5460	516	29	(	(	PUNCT
ejpam-5460	516	30	χ(ab])i	χ(ab])i	ADV
ejpam-5460	516	31	.	.	PUNCT
ejpam-5460	517	1	by	by	ADP
ejpam-5460	517	2	proposition	proposition	NOUN
ejpam-5460	517	3	2	2	NUM
ejpam-5460	517	4	,	,	PUNCT
ejpam-5460	517	5	we	we	PRON
ejpam-5460	517	6	have	have	VERB
ejpam-5460	517	7	a	a	DET
ejpam-5460	517	8	⊆	⊆	NUM
ejpam-5460	517	9	(	(	PUNCT
ejpam-5460	517	10	ab	ab	NOUN
ejpam-5460	517	11	]	]	X
ejpam-5460	517	12	.	.	PUNCT
ejpam-5460	518	1	hence	hence	ADV
ejpam-5460	518	2	,	,	PUNCT
ejpam-5460	518	3	by	by	ADP
ejpam-5460	518	4	lemma	lemma	PROPN
ejpam-5460	518	5	11	11	NUM
ejpam-5460	518	6	,	,	PUNCT
ejpam-5460	518	7	s	s	PART
ejpam-5460	518	8	satisfies	satisfie	NOUN
ejpam-5460	518	9	(	(	PUNCT
ejpam-5460	518	10	c14	c14	NOUN
ejpam-5460	518	11	)	)	PUNCT
ejpam-5460	518	12	.	.	PUNCT
ejpam-5460	519	1	lemma	lemma	PROPN
ejpam-5460	519	2	12	12	NUM
ejpam-5460	519	3	(	(	PUNCT
ejpam-5460	519	4	[	[	X
ejpam-5460	519	5	36	36	NUM
ejpam-5460	519	6	]	]	PUNCT
ejpam-5460	519	7	)	)	PUNCT
ejpam-5460	519	8	.	.	PUNCT
ejpam-5460	520	1	let	let	VERB
ejpam-5460	520	2	s	s	PRON
ejpam-5460	520	3	be	be	AUX
ejpam-5460	520	4	an	an	DET
ejpam-5460	520	5	ordered	order	VERB
ejpam-5460	520	6	semigroup	semigroup	NOUN
ejpam-5460	520	7	.	.	PUNCT
ejpam-5460	521	1	then	then	ADV
ejpam-5460	521	2	,	,	PUNCT
ejpam-5460	521	3	s	s	NOUN
ejpam-5460	521	4	satisfies	satisfie	NOUN
ejpam-5460	521	5	(	(	PUNCT
ejpam-5460	521	6	c15	c15	PROPN
ejpam-5460	521	7	)	)	PUNCT
ejpam-5460	521	8	if	if	SCONJ
ejpam-5460	521	9	and	and	CCONJ
ejpam-5460	521	10	only	only	ADV
ejpam-5460	521	11	if	if	SCONJ
ejpam-5460	521	12	a	a	DET
ejpam-5460	521	13	∩b	∩b	NOUN
ejpam-5460	521	14	⊆	⊆	NUM
ejpam-5460	521	15	(	(	PUNCT
ejpam-5460	521	16	ab	ab	X
ejpam-5460	521	17	]	]	X
ejpam-5460	521	18	for	for	ADP
ejpam-5460	521	19	any	any	DET
ejpam-5460	521	20	(	(	PUNCT
ejpam-5460	521	21	1	1	NUM
ejpam-5460	521	22	,	,	PUNCT
ejpam-5460	521	23	0)-ideal	0)-ideal	NOUN
ejpam-5460	521	24	a	a	NOUN
ejpam-5460	521	25	and	and	CCONJ
ejpam-5460	521	26	(	(	PUNCT
ejpam-5460	521	27	0	0	NUM
ejpam-5460	521	28	,	,	PUNCT
ejpam-5460	521	29	2)-ideal	2)-ideal	NUM
ejpam-5460	521	30	b	b	PROPN
ejpam-5460	521	31	of	of	ADP
ejpam-5460	521	32	s.	s.	PROPN
ejpam-5460	521	33	theorem	theorem	VERB
ejpam-5460	521	34	16	16	NUM
ejpam-5460	521	35	.	.	PUNCT
ejpam-5460	522	1	let	let	VERB
ejpam-5460	522	2	s	s	PRON
ejpam-5460	522	3	be	be	AUX
ejpam-5460	522	4	an	an	DET
ejpam-5460	522	5	ordered	order	VERB
ejpam-5460	522	6	semigroup	semigroup	NOUN
ejpam-5460	522	7	.	.	PUNCT
ejpam-5460	523	1	then	then	ADV
ejpam-5460	523	2	,	,	PUNCT
ejpam-5460	523	3	the	the	DET
ejpam-5460	523	4	following	follow	VERB
ejpam-5460	523	5	statements	statement	NOUN
ejpam-5460	523	6	are	be	AUX
ejpam-5460	523	7	equivalent	equivalent	ADJ
ejpam-5460	523	8	.	.	PUNCT
ejpam-5460	524	1	(	(	PUNCT
ejpam-5460	524	2	i	i	NOUN
ejpam-5460	524	3	)	)	PUNCT
ejpam-5460	524	4	s	s	PART
ejpam-5460	524	5	satisfies	satisfie	NOUN
ejpam-5460	524	6	(	(	PUNCT
ejpam-5460	524	7	c15	c15	PROPN
ejpam-5460	524	8	)	)	PUNCT
ejpam-5460	524	9	.	.	PUNCT
ejpam-5460	525	1	(	(	PUNCT
ejpam-5460	525	2	ii	ii	X
ejpam-5460	525	3	)	)	PUNCT
ejpam-5460	525	4	f	f	PROPN
ejpam-5460	525	5	∩i	∩i	PROPN
ejpam-5460	525	6	g	g	PROPN
ejpam-5460	525	7	⊆	⊆	NUM
ejpam-5460	525	8	f	f	PROPN
ejpam-5460	525	9	◦	◦	NOUN
ejpam-5460	525	10	i	i	PRON
ejpam-5460	525	11	g	g	NOUN
ejpam-5460	525	12	for	for	ADP
ejpam-5460	525	13	any	any	DET
ejpam-5460	525	14	(	(	PUNCT
ejpam-5460	525	15	α	α	NOUN
ejpam-5460	525	16	,	,	PUNCT
ejpam-5460	525	17	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	525	18	(	(	PUNCT
ejpam-5460	525	19	1	1	NUM
ejpam-5460	525	20	,	,	PUNCT
ejpam-5460	525	21	0)-ideal	0)-ideal	PROPN
ejpam-5460	525	22	f	f	PROPN
ejpam-5460	525	23	and	and	CCONJ
ejpam-5460	525	24	(	(	PUNCT
ejpam-5460	525	25	α	α	NOUN
ejpam-5460	525	26	,	,	PUNCT
ejpam-5460	525	27	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	525	28	(	(	PUNCT
ejpam-5460	525	29	0	0	NUM
ejpam-5460	525	30	,	,	PUNCT
ejpam-5460	525	31	2)-ideal	2)-ideal	NUM
ejpam-5460	525	32	g	g	NOUN
ejpam-5460	525	33	of	of	ADP
ejpam-5460	525	34	s.	s.	PROPN
ejpam-5460	525	35	proof	proof	PROPN
ejpam-5460	525	36	.	.	PUNCT
ejpam-5460	526	1	(	(	PUNCT
ejpam-5460	526	2	1)⇒(2	1)⇒(2	NUM
ejpam-5460	526	3	)	)	PUNCT
ejpam-5460	526	4	.	.	PUNCT
ejpam-5460	527	1	let	let	VERB
ejpam-5460	527	2	f	f	PROPN
ejpam-5460	527	3	and	and	CCONJ
ejpam-5460	527	4	g	g	PROPN
ejpam-5460	527	5	be	be	AUX
ejpam-5460	527	6	an	an	DET
ejpam-5460	527	7	(	(	PUNCT
ejpam-5460	527	8	α	α	NOUN
ejpam-5460	527	9	,	,	PUNCT
ejpam-5460	527	10	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	527	11	(	(	PUNCT
ejpam-5460	527	12	1	1	NUM
ejpam-5460	527	13	,	,	PUNCT
ejpam-5460	527	14	0)-ideal	0)-ideal	PROPN
ejpam-5460	527	15	and	and	CCONJ
ejpam-5460	527	16	an	an	DET
ejpam-5460	527	17	(	(	PUNCT
ejpam-5460	527	18	α	α	NOUN
ejpam-5460	527	19	,	,	PUNCT
ejpam-5460	527	20	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	527	21	(	(	PUNCT
ejpam-5460	527	22	0	0	NUM
ejpam-5460	527	23	,	,	PUNCT
ejpam-5460	527	24	2)ideal	2)ideal	NUM
ejpam-5460	527	25	of	of	ADP
ejpam-5460	527	26	s	s	PROPN
ejpam-5460	527	27	,	,	PUNCT
ejpam-5460	527	28	respectively	respectively	ADV
ejpam-5460	527	29	.	.	PUNCT
ejpam-5460	528	1	let	let	VERB
ejpam-5460	528	2	a	a	DET
ejpam-5460	528	3	∈	∈	NOUN
ejpam-5460	528	4	s.	s.	PROPN
ejpam-5460	528	5	since	since	SCONJ
ejpam-5460	528	6	s	s	PART
ejpam-5460	528	7	satisfies	satisfie	NOUN
ejpam-5460	528	8	(	(	PUNCT
ejpam-5460	528	9	c15	c15	PROPN
ejpam-5460	528	10	)	)	PUNCT
ejpam-5460	528	11	,	,	PUNCT
ejpam-5460	528	12	there	there	PRON
ejpam-5460	528	13	exist	exist	VERB
ejpam-5460	528	14	y1	y1	PROPN
ejpam-5460	528	15	∈	∈	PROPN
ejpam-5460	528	16	s	s	VERB
ejpam-5460	528	17	such	such	ADJ
ejpam-5460	528	18	that	that	SCONJ
ejpam-5460	528	19	a	a	DET
ejpam-5460	528	20	≤	≤	ADJ
ejpam-5460	528	21	ay1a	ay1a	PROPN
ejpam-5460	528	22	2	2	NUM
ejpam-5460	528	23	≤	≤	NOUN
ejpam-5460	528	24	ay1a	ay1a	PROPN
ejpam-5460	528	25	2y1a	2y1a	NUM
ejpam-5460	528	26	2	2	NUM
ejpam-5460	528	27	.	.	PUNCT
ejpam-5460	529	1	that	that	PRON
ejpam-5460	529	2	is	is	ADV
ejpam-5460	529	3	,	,	PUNCT
ejpam-5460	529	4	(	(	PUNCT
ejpam-5460	529	5	ax1	ax1	PROPN
ejpam-5460	529	6	,	,	PUNCT
ejpam-5460	529	7	x2a	x2a	PROPN
ejpam-5460	529	8	2	2	X
ejpam-5460	529	9	)	)	PUNCT
ejpam-5460	529	10	∈	∈	PROPN
ejpam-5460	529	11	sa	sa	NOUN
ejpam-5460	529	12	for	for	ADP
ejpam-5460	529	13	some	some	DET
ejpam-5460	529	14	x1	x1	PROPN
ejpam-5460	529	15	,	,	PUNCT
ejpam-5460	529	16	x2	x2	PROPN
ejpam-5460	529	17	∈	∈	PROPN
ejpam-5460	529	18	s.	s.	PROPN
ejpam-5460	529	19	then	then	ADV
ejpam-5460	529	20	,	,	PUNCT
ejpam-5460	529	21	(	(	PUNCT
ejpam-5460	529	22	f	f	X
ejpam-5460	529	23	◦	◦	NOUN
ejpam-5460	529	24	i	i	PRON
ejpam-5460	529	25	g)(a	g)(a	VERB
ejpam-5460	529	26	)	)	PUNCT
ejpam-5460	529	27	≥	≥	NOUN
ejpam-5460	529	28	fi(ax1	fi(ax1	NUM
ejpam-5460	529	29	)	)	PUNCT
ejpam-5460	529	30	∧	∧	PROPN
ejpam-5460	529	31	gi(x2a	gi(x2a	PROPN
ejpam-5460	529	32	2	2	NUM
ejpam-5460	529	33	)	)	PUNCT
ejpam-5460	529	34	≥	≥	NOUN
ejpam-5460	529	35	fi(a	fi(a	NOUN
ejpam-5460	529	36	)	)	PUNCT
ejpam-5460	529	37	∧	∧	NOUN
ejpam-5460	529	38	gi(a	gi(a	X
ejpam-5460	529	39	)	)	PUNCT
ejpam-5460	529	40	=	=	SYM
ejpam-5460	530	1	(	(	PUNCT
ejpam-5460	530	2	f	f	PROPN
ejpam-5460	530	3	∩i	∩i	PROPN
ejpam-5460	530	4	g)(a	g)(a	PROPN
ejpam-5460	530	5	)	)	PUNCT
ejpam-5460	530	6	.	.	PUNCT
ejpam-5460	531	1	(	(	PUNCT
ejpam-5460	531	2	2)⇒(1	2)⇒(1	NOUN
ejpam-5460	531	3	)	)	PUNCT
ejpam-5460	531	4	.	.	PUNCT
ejpam-5460	532	1	let	let	VERB
ejpam-5460	532	2	a	a	PRON
ejpam-5460	532	3	and	and	CCONJ
ejpam-5460	532	4	b	b	NOUN
ejpam-5460	532	5	be	be	AUX
ejpam-5460	532	6	a	a	DET
ejpam-5460	532	7	(	(	PUNCT
ejpam-5460	532	8	1	1	NUM
ejpam-5460	532	9	,	,	PUNCT
ejpam-5460	532	10	0)-ideal	0)-ideal	NUM
ejpam-5460	532	11	and	and	CCONJ
ejpam-5460	532	12	a	a	DET
ejpam-5460	532	13	(	(	PUNCT
ejpam-5460	532	14	0	0	NUM
ejpam-5460	532	15	,	,	PUNCT
ejpam-5460	532	16	2)-ideal	2)-ideal	NUM
ejpam-5460	532	17	of	of	ADP
ejpam-5460	532	18	s	s	PRON
ejpam-5460	532	19	,	,	PUNCT
ejpam-5460	532	20	respectively	respectively	ADV
ejpam-5460	532	21	.	.	PUNCT
ejpam-5460	533	1	then	then	ADV
ejpam-5460	533	2	,	,	PUNCT
ejpam-5460	533	3	by	by	ADP
ejpam-5460	533	4	proposition	proposition	NOUN
ejpam-5460	533	5	2	2	NUM
ejpam-5460	533	6	,	,	PUNCT
ejpam-5460	533	7	(	(	PUNCT
ejpam-5460	533	8	χa)i	χa)i	NUM
ejpam-5460	533	9	and	and	CCONJ
ejpam-5460	533	10	(	(	PUNCT
ejpam-5460	533	11	χb)i	χb)i	PROPN
ejpam-5460	533	12	is	be	AUX
ejpam-5460	533	13	an	an	DET
ejpam-5460	533	14	(	(	PUNCT
ejpam-5460	533	15	α	α	NOUN
ejpam-5460	533	16	,	,	PUNCT
ejpam-5460	533	17	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	533	18	(	(	PUNCT
ejpam-5460	533	19	1	1	NUM
ejpam-5460	533	20	,	,	PUNCT
ejpam-5460	533	21	0)-ideal	0)-ideal	PROPN
ejpam-5460	533	22	and	and	CCONJ
ejpam-5460	533	23	an	an	DET
ejpam-5460	533	24	(	(	PUNCT
ejpam-5460	533	25	α	α	NOUN
ejpam-5460	533	26	,	,	PUNCT
ejpam-5460	533	27	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	533	28	(	(	PUNCT
ejpam-5460	533	29	0	0	NUM
ejpam-5460	533	30	,	,	PUNCT
ejpam-5460	533	31	2)ideal	2)ideal	NUM
ejpam-5460	533	32	of	of	ADP
ejpam-5460	533	33	s	s	PROPN
ejpam-5460	533	34	,	,	PUNCT
ejpam-5460	533	35	respectively	respectively	ADV
ejpam-5460	533	36	.	.	PUNCT
ejpam-5460	534	1	by	by	ADP
ejpam-5460	534	2	our	our	PRON
ejpam-5460	534	3	presumption	presumption	NOUN
ejpam-5460	534	4	,	,	PUNCT
ejpam-5460	534	5	we	we	PRON
ejpam-5460	534	6	have	have	VERB
ejpam-5460	534	7	(	(	PUNCT
ejpam-5460	534	8	χa)i	χa)i	PROPN
ejpam-5460	534	9	∩i	∩i	NOUN
ejpam-5460	534	10	(	(	PUNCT
ejpam-5460	534	11	χb)i	χb)i	PROPN
ejpam-5460	534	12	⊆	⊆	NUM
ejpam-5460	534	13	(	(	PUNCT
ejpam-5460	534	14	χa)i	χa)i	PROPN
ejpam-5460	534	15	◦	◦	NOUN
ejpam-5460	534	16	i	i	PROPN
ejpam-5460	534	17	(	(	PUNCT
ejpam-5460	534	18	χb)i	χb)i	PROPN
ejpam-5460	534	19	.	.	PUNCT
ejpam-5460	535	1	this	this	PRON
ejpam-5460	535	2	implies	imply	VERB
ejpam-5460	535	3	that	that	SCONJ
ejpam-5460	535	4	(	(	PUNCT
ejpam-5460	535	5	χa∩b)i	χa∩b)i	NOUN
ejpam-5460	535	6	=	=	NOUN
ejpam-5460	535	7	χa	χa	NOUN
ejpam-5460	535	8	∩i	∩i	ADV
ejpam-5460	535	9	χb	χb	PROPN
ejpam-5460	535	10	=	=	PUNCT
ejpam-5460	535	11	(	(	PUNCT
ejpam-5460	535	12	χa)i	χa)i	PROPN
ejpam-5460	535	13	∩i	∩i	NOUN
ejpam-5460	535	14	(	(	PUNCT
ejpam-5460	535	15	χb)i	χb)i	PROPN
ejpam-5460	535	16	⊆	⊆	NUM
ejpam-5460	535	17	(	(	PUNCT
ejpam-5460	535	18	χa)i	χa)i	PROPN
ejpam-5460	535	19	◦	◦	NOUN
ejpam-5460	535	20	i	i	PROPN
ejpam-5460	535	21	(	(	PUNCT
ejpam-5460	535	22	χb)i	χb)i	PROPN
ejpam-5460	535	23	=	=	PUNCT
ejpam-5460	535	24	χa	χa	NOUN
ejpam-5460	535	25	◦	◦	NOUN
ejpam-5460	535	26	i	i	PRON
ejpam-5460	535	27	χb	χb	VERB
ejpam-5460	535	28	=	=	PUNCT
ejpam-5460	535	29	(	(	PUNCT
ejpam-5460	535	30	χ(ab])i	χ(ab])i	ADV
ejpam-5460	535	31	.	.	PUNCT
ejpam-5460	536	1	by	by	ADP
ejpam-5460	536	2	proposition	proposition	NOUN
ejpam-5460	536	3	2	2	NUM
ejpam-5460	536	4	,	,	PUNCT
ejpam-5460	536	5	we	we	PRON
ejpam-5460	536	6	have	have	VERB
ejpam-5460	536	7	a	a	DET
ejpam-5460	536	8	⊆	⊆	NUM
ejpam-5460	536	9	(	(	PUNCT
ejpam-5460	536	10	ab	ab	NOUN
ejpam-5460	536	11	]	]	X
ejpam-5460	536	12	.	.	PUNCT
ejpam-5460	537	1	hence	hence	ADV
ejpam-5460	537	2	,	,	PUNCT
ejpam-5460	537	3	by	by	ADP
ejpam-5460	537	4	lemma	lemma	PROPN
ejpam-5460	537	5	12	12	NUM
ejpam-5460	537	6	,	,	PUNCT
ejpam-5460	537	7	s	s	PART
ejpam-5460	537	8	satisfies	satisfie	NOUN
ejpam-5460	537	9	(	(	PUNCT
ejpam-5460	537	10	c15	c15	PROPN
ejpam-5460	537	11	)	)	PUNCT
ejpam-5460	537	12	.	.	PUNCT
ejpam-5460	538	1	similarly	similarly	ADV
ejpam-5460	538	2	,	,	PUNCT
ejpam-5460	538	3	we	we	PRON
ejpam-5460	538	4	obtain	obtain	VERB
ejpam-5460	538	5	the	the	DET
ejpam-5460	538	6	following	follow	VERB
ejpam-5460	538	7	theorem	theorem	VERB
ejpam-5460	538	8	.	.	PUNCT
ejpam-5460	538	9	theorem	theorem	PROPN
ejpam-5460	538	10	17	17	NUM
ejpam-5460	538	11	.	.	PUNCT
ejpam-5460	539	1	let	let	VERB
ejpam-5460	539	2	s	s	PRON
ejpam-5460	539	3	be	be	AUX
ejpam-5460	539	4	an	an	DET
ejpam-5460	539	5	ordered	order	VERB
ejpam-5460	539	6	semigroup	semigroup	NOUN
ejpam-5460	539	7	.	.	PUNCT
ejpam-5460	540	1	then	then	ADV
ejpam-5460	540	2	,	,	PUNCT
ejpam-5460	540	3	the	the	DET
ejpam-5460	540	4	following	follow	VERB
ejpam-5460	540	5	statements	statement	NOUN
ejpam-5460	540	6	are	be	AUX
ejpam-5460	540	7	equivalent	equivalent	ADJ
ejpam-5460	540	8	.	.	PUNCT
ejpam-5460	541	1	(	(	PUNCT
ejpam-5460	541	2	i	i	NOUN
ejpam-5460	541	3	)	)	PUNCT
ejpam-5460	541	4	s	s	PART
ejpam-5460	541	5	satisfies	satisfie	NOUN
ejpam-5460	541	6	(	(	PUNCT
ejpam-5460	541	7	c16	c16	PROPN
ejpam-5460	541	8	)	)	PUNCT
ejpam-5460	541	9	.	.	PUNCT
ejpam-5460	542	1	(	(	PUNCT
ejpam-5460	542	2	ii	ii	X
ejpam-5460	542	3	)	)	PUNCT
ejpam-5460	542	4	f	f	PROPN
ejpam-5460	542	5	∩i	∩i	PROPN
ejpam-5460	542	6	g	g	PROPN
ejpam-5460	542	7	⊆	⊆	NUM
ejpam-5460	542	8	f	f	PROPN
ejpam-5460	542	9	◦	◦	NOUN
ejpam-5460	542	10	i	i	PRON
ejpam-5460	542	11	g	g	NOUN
ejpam-5460	542	12	for	for	ADP
ejpam-5460	542	13	any	any	DET
ejpam-5460	542	14	(	(	PUNCT
ejpam-5460	542	15	α	α	NOUN
ejpam-5460	542	16	,	,	PUNCT
ejpam-5460	542	17	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	542	18	(	(	PUNCT
ejpam-5460	542	19	2	2	NUM
ejpam-5460	542	20	,	,	PUNCT
ejpam-5460	542	21	0)-ideal	0)-ideal	PROPN
ejpam-5460	542	22	f	f	PROPN
ejpam-5460	542	23	and	and	CCONJ
ejpam-5460	542	24	(	(	PUNCT
ejpam-5460	542	25	α	α	NOUN
ejpam-5460	542	26	,	,	PUNCT
ejpam-5460	542	27	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	542	28	(	(	PUNCT
ejpam-5460	542	29	0	0	NUM
ejpam-5460	542	30	,	,	PUNCT
ejpam-5460	542	31	1)-ideal	1)-ideal	NUM
ejpam-5460	542	32	g	g	NOUN
ejpam-5460	542	33	of	of	ADP
ejpam-5460	542	34	s.	s.	PROPN
ejpam-5460	542	35	references	reference	NOUN
ejpam-5460	542	36	2981	2981	NUM
ejpam-5460	542	37	5	5	NUM
ejpam-5460	542	38	.	.	PUNCT
ejpam-5460	543	1	conclusion	conclusion	NOUN
ejpam-5460	543	2	this	this	DET
ejpam-5460	543	3	paper	paper	NOUN
ejpam-5460	543	4	uses	use	VERB
ejpam-5460	543	5	the	the	DET
ejpam-5460	543	6	concepts	concept	NOUN
ejpam-5460	543	7	of	of	ADP
ejpam-5460	543	8	(	(	PUNCT
ejpam-5460	543	9	α	α	NOUN
ejpam-5460	543	10	,	,	PUNCT
ejpam-5460	543	11	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	543	12	(	(	PUNCT
ejpam-5460	543	13	m	m	NOUN
ejpam-5460	543	14	,	,	PUNCT
ejpam-5460	543	15	n)-ideals	n)-ideal	NOUN
ejpam-5460	543	16	and	and	CCONJ
ejpam-5460	543	17	(	(	PUNCT
ejpam-5460	543	18	α	α	NOUN
ejpam-5460	543	19	,	,	PUNCT
ejpam-5460	543	20	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	543	21	n	n	CCONJ
ejpam-5460	543	22	-	-	PUNCT
ejpam-5460	543	23	interior	interior	ADJ
ejpam-5460	543	24	ideals	ideal	NOUN
ejpam-5460	543	25	to	to	PART
ejpam-5460	543	26	categorize	categorize	VERB
ejpam-5460	543	27	ordered	order	VERB
ejpam-5460	543	28	semigroups	semigroup	NOUN
ejpam-5460	543	29	into	into	ADP
ejpam-5460	543	30	sixteen	sixteen	NUM
ejpam-5460	543	31	distinct	distinct	ADJ
ejpam-5460	543	32	classes	class	NOUN
ejpam-5460	543	33	based	base	VERB
ejpam-5460	543	34	on	on	ADP
ejpam-5460	543	35	the	the	DET
ejpam-5460	543	36	regularities	regularity	NOUN
ejpam-5460	543	37	of	of	ADP
ejpam-5460	543	38	ordered	order	VERB
ejpam-5460	543	39	semigroups	semigroup	NOUN
ejpam-5460	543	40	.	.	PUNCT
ejpam-5460	544	1	an	an	DET
ejpam-5460	544	2	advantage	advantage	NOUN
ejpam-5460	544	3	of	of	ADP
ejpam-5460	544	4	the	the	DET
ejpam-5460	544	5	results	result	NOUN
ejpam-5460	544	6	in	in	ADP
ejpam-5460	544	7	the	the	DET
ejpam-5460	544	8	current	current	ADJ
ejpam-5460	544	9	study	study	NOUN
ejpam-5460	544	10	is	be	AUX
ejpam-5460	544	11	that	that	SCONJ
ejpam-5460	544	12	,	,	PUNCT
ejpam-5460	544	13	by	by	ADP
ejpam-5460	544	14	carefully	carefully	ADV
ejpam-5460	544	15	selecting	select	VERB
ejpam-5460	544	16	appropriate	appropriate	ADJ
ejpam-5460	544	17	parameters	parameter	NOUN
ejpam-5460	544	18	,	,	PUNCT
ejpam-5460	544	19	we	we	PRON
ejpam-5460	544	20	derive	derive	VERB
ejpam-5460	544	21	several	several	ADJ
ejpam-5460	544	22	characterizations	characterization	NOUN
ejpam-5460	544	23	of	of	ADP
ejpam-5460	544	24	ordered	order	VERB
ejpam-5460	544	25	semigroups	semigroup	NOUN
ejpam-5460	544	26	utilizing	utilize	VERB
ejpam-5460	544	27	fuzzy	fuzzy	ADJ
ejpam-5460	544	28	ideals	ideal	NOUN
ejpam-5460	544	29	that	that	SCONJ
ejpam-5460	544	30	several	several	ADJ
ejpam-5460	544	31	authors	author	NOUN
ejpam-5460	544	32	studied	study	VERB
ejpam-5460	544	33	.	.	PUNCT
ejpam-5460	545	1	however	however	ADV
ejpam-5460	545	2	,	,	PUNCT
ejpam-5460	545	3	in	in	ADP
ejpam-5460	545	4	the	the	DET
ejpam-5460	545	5	theorems	theorem	NOUN
ejpam-5460	545	6	we	we	PRON
ejpam-5460	545	7	obtained	obtain	VERB
ejpam-5460	545	8	,	,	PUNCT
ejpam-5460	545	9	in	in	ADP
ejpam-5460	545	10	practice	practice	NOUN
ejpam-5460	545	11	,	,	PUNCT
ejpam-5460	545	12	there	there	PRON
ejpam-5460	545	13	is	be	VERB
ejpam-5460	545	14	a	a	DET
ejpam-5460	545	15	challenging	challenging	ADJ
ejpam-5460	545	16	problem	problem	NOUN
ejpam-5460	545	17	:	:	PUNCT
ejpam-5460	545	18	are	be	AUX
ejpam-5460	545	19	there	there	PRON
ejpam-5460	545	20	any	any	DET
ejpam-5460	545	21	easier	easy	ADJ
ejpam-5460	545	22	conditions	condition	NOUN
ejpam-5460	545	23	to	to	PART
ejpam-5460	545	24	obtain	obtain	VERB
ejpam-5460	545	25	the	the	DET
ejpam-5460	545	26	set	set	NOUN
ejpam-5460	545	27	of	of	ADP
ejpam-5460	545	28	all	all	PRON
ejpam-5460	545	29	(	(	PUNCT
ejpam-5460	545	30	α	α	NOUN
ejpam-5460	545	31	,	,	PUNCT
ejpam-5460	545	32	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	545	33	(	(	PUNCT
ejpam-5460	545	34	m	m	NOUN
ejpam-5460	545	35	,	,	PUNCT
ejpam-5460	545	36	n)-ideals	n)-ideal	NOUN
ejpam-5460	545	37	and	and	CCONJ
ejpam-5460	545	38	(	(	PUNCT
ejpam-5460	545	39	α	α	NOUN
ejpam-5460	545	40	,	,	PUNCT
ejpam-5460	545	41	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	545	42	n	n	CCONJ
ejpam-5460	545	43	-	-	PUNCT
ejpam-5460	545	44	interior	interior	ADJ
ejpam-5460	545	45	ideals	ideal	NOUN
ejpam-5460	545	46	?	?	PUNCT
ejpam-5460	546	1	this	this	DET
ejpam-5460	546	2	question	question	NOUN
ejpam-5460	546	3	can	can	AUX
ejpam-5460	546	4	be	be	AUX
ejpam-5460	546	5	stated	state	VERB
ejpam-5460	546	6	as	as	ADP
ejpam-5460	546	7	an	an	DET
ejpam-5460	546	8	open	open	ADJ
ejpam-5460	546	9	problem	problem	NOUN
ejpam-5460	546	10	.	.	PUNCT
ejpam-5460	547	1	furthermore	furthermore	ADV
ejpam-5460	547	2	,	,	PUNCT
ejpam-5460	547	3	in	in	ADP
ejpam-5460	547	4	future	future	ADJ
ejpam-5460	547	5	research	research	NOUN
ejpam-5460	547	6	,	,	PUNCT
ejpam-5460	547	7	we	we	PRON
ejpam-5460	547	8	aim	aim	VERB
ejpam-5460	547	9	to	to	PART
ejpam-5460	547	10	delve	delve	VERB
ejpam-5460	547	11	deeper	deeply	ADV
ejpam-5460	547	12	into	into	ADP
ejpam-5460	547	13	the	the	DET
ejpam-5460	547	14	properties	property	NOUN
ejpam-5460	547	15	of	of	ADP
ejpam-5460	547	16	(	(	PUNCT
ejpam-5460	547	17	α	α	NOUN
ejpam-5460	547	18	,	,	PUNCT
ejpam-5460	547	19	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	547	20	(	(	PUNCT
ejpam-5460	547	21	m	m	NOUN
ejpam-5460	547	22	,	,	PUNCT
ejpam-5460	547	23	n)-ideals	n)-ideal	NOUN
ejpam-5460	547	24	and	and	CCONJ
ejpam-5460	547	25	(	(	PUNCT
ejpam-5460	547	26	α	α	NOUN
ejpam-5460	547	27	,	,	PUNCT
ejpam-5460	547	28	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	547	29	n	n	CCONJ
ejpam-5460	547	30	-	-	PUNCT
ejpam-5460	547	31	interior	interior	ADJ
ejpam-5460	547	32	ideals	ideal	NOUN
ejpam-5460	547	33	,	,	PUNCT
ejpam-5460	547	34	exploring	explore	VERB
ejpam-5460	547	35	their	their	PRON
ejpam-5460	547	36	pure	pure	ADJ
ejpam-5460	547	37	and	and	CCONJ
ejpam-5460	547	38	prime	prime	ADJ
ejpam-5460	547	39	properties	property	NOUN
ejpam-5460	547	40	.	.	PUNCT
ejpam-5460	548	1	acknowledgements	acknowledgement	NOUN
ejpam-5460	548	2	this	this	DET
ejpam-5460	548	3	work	work	NOUN
ejpam-5460	548	4	(	(	PUNCT
ejpam-5460	548	5	grant	grant	VERB
ejpam-5460	548	6	no	no	INTJ
ejpam-5460	548	7	.	.	PUNCT
ejpam-5460	549	1	rgns	rgn	VERB
ejpam-5460	549	2	65	65	NUM
ejpam-5460	549	3	-	-	SYM
ejpam-5460	549	4	097	097	NUM
ejpam-5460	549	5	)	)	PUNCT
ejpam-5460	549	6	was	be	AUX
ejpam-5460	549	7	supported	support	VERB
ejpam-5460	549	8	by	by	ADP
ejpam-5460	549	9	office	office	NOUN
ejpam-5460	549	10	of	of	ADP
ejpam-5460	549	11	the	the	DET
ejpam-5460	549	12	permanent	permanent	ADJ
ejpam-5460	549	13	secretary	secretary	NOUN
ejpam-5460	549	14	,	,	PUNCT
ejpam-5460	549	15	ministry	ministry	PROPN
ejpam-5460	549	16	of	of	ADP
ejpam-5460	549	17	higher	high	ADJ
ejpam-5460	549	18	education	education	NOUN
ejpam-5460	549	19	,	,	PUNCT
ejpam-5460	549	20	science	science	NOUN
ejpam-5460	549	21	,	,	PUNCT
ejpam-5460	549	22	research	research	NOUN
ejpam-5460	549	23	and	and	CCONJ
ejpam-5460	549	24	innovation	innovation	NOUN
ejpam-5460	549	25	(	(	PUNCT
ejpam-5460	549	26	ops	op	NOUN
ejpam-5460	549	27	mhesi	mhesi	PROPN
ejpam-5460	549	28	)	)	PUNCT
ejpam-5460	549	29	,	,	PUNCT
ejpam-5460	549	30	thailand	thailand	PROPN
ejpam-5460	549	31	science	science	PROPN
ejpam-5460	549	32	research	research	PROPN
ejpam-5460	549	33	and	and	CCONJ
ejpam-5460	549	34	innovation	innovation	NOUN
ejpam-5460	549	35	(	(	PUNCT
ejpam-5460	549	36	tsri	tsri	ADJ
ejpam-5460	549	37	)	)	PUNCT
ejpam-5460	549	38	and	and	CCONJ
ejpam-5460	549	39	rajamangala	rajamangala	PROPN
ejpam-5460	549	40	university	university	PROPN
ejpam-5460	549	41	of	of	ADP
ejpam-5460	549	42	technology	technology	PROPN
ejpam-5460	549	43	isan	isan	PROPN
ejpam-5460	549	44	,	,	PUNCT
ejpam-5460	549	45	khon	khon	PROPN
ejpam-5460	549	46	kaen	kaen	PROPN
ejpam-5460	549	47	campus	campus	PROPN
ejpam-5460	549	48	.	.	PUNCT
ejpam-5460	550	1	the	the	DET
ejpam-5460	550	2	authors	author	NOUN
ejpam-5460	550	3	are	be	AUX
ejpam-5460	550	4	immensely	immensely	ADV
ejpam-5460	550	5	grateful	grateful	ADJ
ejpam-5460	550	6	to	to	ADP
ejpam-5460	550	7	the	the	DET
ejpam-5460	550	8	reviewers	reviewer	NOUN
ejpam-5460	550	9	for	for	ADP
ejpam-5460	550	10	their	their	PRON
ejpam-5460	550	11	valuable	valuable	ADJ
ejpam-5460	550	12	suggestions	suggestion	NOUN
ejpam-5460	550	13	,	,	PUNCT
ejpam-5460	550	14	which	which	PRON
ejpam-5460	550	15	have	have	AUX
ejpam-5460	550	16	greatly	greatly	ADV
ejpam-5460	550	17	helped	help	VERB
ejpam-5460	550	18	to	to	PART
ejpam-5460	550	19	improve	improve	VERB
ejpam-5460	550	20	this	this	DET
ejpam-5460	550	21	work	work	NOUN
ejpam-5460	550	22	.	.	PUNCT
ejpam-5460	551	1	references	reference	NOUN
ejpam-5460	551	2	[	[	X
ejpam-5460	551	3	1	1	NUM
ejpam-5460	551	4	]	]	X
ejpam-5460	551	5	j	j	PROPN
ejpam-5460	551	6	ahsan	ahsan	PROPN
ejpam-5460	551	7	,	,	PUNCT
ejpam-5460	551	8	k	k	PROPN
ejpam-5460	551	9	saifullah	saifullah	PROPN
ejpam-5460	551	10	,	,	PUNCT
ejpam-5460	551	11	and	and	CCONJ
ejpam-5460	551	12	mf	mf	VERB
ejpam-5460	551	13	khan	khan	PROPN
ejpam-5460	551	14	.	.	PUNCT
ejpam-5460	552	1	fuzzy	fuzzy	ADJ
ejpam-5460	552	2	semirings	semiring	NOUN
ejpam-5460	552	3	.	.	PUNCT
ejpam-5460	553	1	fuzzy	fuzzy	ADJ
ejpam-5460	553	2	sets	set	NOUN
ejpam-5460	553	3	and	and	CCONJ
ejpam-5460	553	4	systems	system	NOUN
ejpam-5460	553	5	,	,	PUNCT
ejpam-5460	553	6	60:309–320	60:309–320	PROPN
ejpam-5460	553	7	,	,	PUNCT
ejpam-5460	553	8	1993	1993	NUM
ejpam-5460	553	9	.	.	PUNCT
ejpam-5460	554	1	[	[	X
ejpam-5460	554	2	2	2	NUM
ejpam-5460	554	3	]	]	SYM
ejpam-5460	554	4	g	g	NOUN
ejpam-5460	554	5	birkhoff	birkhoff	NOUN
ejpam-5460	554	6	.	.	PUNCT
ejpam-5460	555	1	lattice	lattice	PROPN
ejpam-5460	555	2	theory	theory	NOUN
ejpam-5460	555	3	.	.	PUNCT
ejpam-5460	556	1	american	american	PROPN
ejpam-5460	556	2	mathematical	mathematical	PROPN
ejpam-5460	556	3	society	society	NOUN
ejpam-5460	556	4	,	,	PUNCT
ejpam-5460	556	5	providence	providence	NOUN
ejpam-5460	556	6	,	,	PUNCT
ejpam-5460	556	7	1967	1967	NUM
ejpam-5460	556	8	.	.	PUNCT
ejpam-5460	557	1	[	[	X
ejpam-5460	557	2	3	3	NUM
ejpam-5460	557	3	]	]	X
ejpam-5460	557	4	l	l	NOUN
ejpam-5460	557	5	bussaban	bussaban	NOUN
ejpam-5460	557	6	and	and	CCONJ
ejpam-5460	557	7	t	t	PROPN
ejpam-5460	557	8	changphas	changpha	NOUN
ejpam-5460	557	9	.	.	PUNCT
ejpam-5460	558	1	a	a	DET
ejpam-5460	558	2	note	note	NOUN
ejpam-5460	558	3	on	on	ADP
ejpam-5460	558	4	(	(	PUNCT
ejpam-5460	558	5	m	m	PROPN
ejpam-5460	558	6	,	,	PUNCT
ejpam-5460	558	7	n)-ideals	n)-ideal	NOUN
ejpam-5460	558	8	in	in	ADP
ejpam-5460	558	9	regular	regular	ADJ
ejpam-5460	558	10	duo	duo	NOUN
ejpam-5460	558	11	ordered	order	VERB
ejpam-5460	558	12	semigroups	semigroup	NOUN
ejpam-5460	558	13	.	.	PUNCT
ejpam-5460	559	1	quasigroups	quasigroup	NOUN
ejpam-5460	559	2	and	and	CCONJ
ejpam-5460	559	3	related	related	ADJ
ejpam-5460	559	4	systems	system	NOUN
ejpam-5460	559	5	,	,	PUNCT
ejpam-5460	559	6	23:211–216	23:211–216	NUM
ejpam-5460	559	7	,	,	PUNCT
ejpam-5460	559	8	2015	2015	NUM
ejpam-5460	559	9	.	.	PUNCT
ejpam-5460	560	1	[	[	X
ejpam-5460	560	2	4	4	NUM
ejpam-5460	560	3	]	]	SYM
ejpam-5460	560	4	b	b	X
ejpam-5460	560	5	davvaz	davvaz	NOUN
ejpam-5460	560	6	,	,	PUNCT
ejpam-5460	560	7	r	r	NOUN
ejpam-5460	560	8	chinram	chinram	NOUN
ejpam-5460	560	9	,	,	PUNCT
ejpam-5460	560	10	s	s	NOUN
ejpam-5460	560	11	lekkoksung	lekkoksung	NOUN
ejpam-5460	560	12	,	,	PUNCT
ejpam-5460	560	13	and	and	CCONJ
ejpam-5460	561	1	n	n	DET
ejpam-5460	561	2	lekkoksung	lekkoksung	NOUN
ejpam-5460	561	3	.	.	PUNCT
ejpam-5460	562	1	characterizations	characterization	NOUN
ejpam-5460	562	2	of	of	ADP
ejpam-5460	562	3	generalized	generalized	ADJ
ejpam-5460	562	4	fuzzy	fuzzy	ADJ
ejpam-5460	562	5	ideals	ideal	NOUN
ejpam-5460	562	6	in	in	ADP
ejpam-5460	562	7	ordered	order	VERB
ejpam-5460	562	8	semigroups	semigroup	NOUN
ejpam-5460	562	9	.	.	PUNCT
ejpam-5460	563	1	journal	journal	NOUN
ejpam-5460	563	2	of	of	ADP
ejpam-5460	563	3	intelligent	intelligent	ADJ
ejpam-5460	563	4	and	and	CCONJ
ejpam-5460	563	5	fuzzy	fuzzy	ADJ
ejpam-5460	563	6	systems	system	NOUN
ejpam-5460	563	7	,	,	PUNCT
ejpam-5460	563	8	45:2367–2380	45:2367–2380	PROPN
ejpam-5460	563	9	,	,	PUNCT
ejpam-5460	563	10	2023	2023	NUM
ejpam-5460	563	11	.	.	PUNCT
ejpam-5460	564	1	[	[	X
ejpam-5460	564	2	5	5	NUM
ejpam-5460	564	3	]	]	PUNCT
ejpam-5460	564	4	vn	vn	PROPN
ejpam-5460	564	5	dixit	dixit	PROPN
ejpam-5460	564	6	,	,	PUNCT
ejpam-5460	564	7	r	r	PROPN
ejpam-5460	564	8	kumar	kumar	PROPN
ejpam-5460	564	9	,	,	PUNCT
ejpam-5460	564	10	and	and	CCONJ
ejpam-5460	564	11	n	n	ADV
ejpam-5460	564	12	ajmal	ajmal	ADJ
ejpam-5460	564	13	.	.	PUNCT
ejpam-5460	565	1	on	on	ADP
ejpam-5460	565	2	fuzzy	fuzzy	ADJ
ejpam-5460	565	3	rings	ring	NOUN
ejpam-5460	565	4	.	.	PUNCT
ejpam-5460	566	1	fuzzy	fuzzy	ADJ
ejpam-5460	566	2	sets	set	NOUN
ejpam-5460	566	3	and	and	CCONJ
ejpam-5460	566	4	systems	system	NOUN
ejpam-5460	566	5	,	,	PUNCT
ejpam-5460	566	6	49:205	49:205	NUM
ejpam-5460	566	7	–	–	PUNCT
ejpam-5460	566	8	213	213	NUM
ejpam-5460	566	9	,	,	PUNCT
ejpam-5460	566	10	1992	1992	NUM
ejpam-5460	566	11	.	.	PUNCT
ejpam-5460	567	1	[	[	X
ejpam-5460	567	2	6	6	NUM
ejpam-5460	567	3	]	]	X
ejpam-5460	567	4	y	y	PROPN
ejpam-5460	567	5	feng	feng	PROPN
ejpam-5460	567	6	and	and	CCONJ
ejpam-5460	567	7	p	p	NOUN
ejpam-5460	567	8	corsini	corsini	PROPN
ejpam-5460	567	9	.	.	PUNCT
ejpam-5460	568	1	(	(	PUNCT
ejpam-5460	568	2	λ	λ	X
ejpam-5460	568	3	,	,	PUNCT
ejpam-5460	568	4	µ)-fuzzy	µ)-fuzzy	ADJ
ejpam-5460	568	5	ideals	ideal	NOUN
ejpam-5460	568	6	of	of	ADP
ejpam-5460	568	7	ordered	order	VERB
ejpam-5460	568	8	semigroups	semigroup	NOUN
ejpam-5460	568	9	.	.	PUNCT
ejpam-5460	569	1	annals	annal	NOUN
ejpam-5460	569	2	of	of	ADP
ejpam-5460	569	3	fuzzy	fuzzy	ADJ
ejpam-5460	569	4	mathematics	mathematic	NOUN
ejpam-5460	569	5	and	and	CCONJ
ejpam-5460	569	6	informatics	informatic	NOUN
ejpam-5460	569	7	,	,	PUNCT
ejpam-5460	569	8	4:123–129	4:123–129	PROPN
ejpam-5460	569	9	,	,	PUNCT
ejpam-5460	569	10	2012	2012	NUM
ejpam-5460	569	11	.	.	PUNCT
ejpam-5460	570	1	[	[	X
ejpam-5460	570	2	7	7	X
ejpam-5460	570	3	]	]	X
ejpam-5460	570	4	y	y	PROPN
ejpam-5460	570	5	feng	feng	PROPN
ejpam-5460	570	6	and	and	CCONJ
ejpam-5460	570	7	p	p	NOUN
ejpam-5460	570	8	corsini	corsini	PROPN
ejpam-5460	570	9	.	.	PUNCT
ejpam-5460	571	1	(	(	PUNCT
ejpam-5460	571	2	λ	λ	PROPN
ejpam-5460	571	3	,	,	PUNCT
ejpam-5460	571	4	µ)-fuzzy	µ)-fuzzy	PUNCT
ejpam-5460	571	5	version	version	NOUN
ejpam-5460	571	6	of	of	ADP
ejpam-5460	571	7	ideals	ideal	NOUN
ejpam-5460	571	8	,	,	PUNCT
ejpam-5460	571	9	interior	interior	ADJ
ejpam-5460	571	10	ideals	ideal	NOUN
ejpam-5460	571	11	,	,	PUNCT
ejpam-5460	571	12	quasi	quasi	NOUN
ejpam-5460	571	13	-	-	NOUN
ejpam-5460	571	14	ideals	ideal	NOUN
ejpam-5460	571	15	,	,	PUNCT
ejpam-5460	571	16	and	and	CCONJ
ejpam-5460	571	17	bi	bi	NOUN
ejpam-5460	571	18	-	-	NOUN
ejpam-5460	571	19	ideals	ideal	NOUN
ejpam-5460	571	20	.	.	PUNCT
ejpam-5460	572	1	journal	journal	NOUN
ejpam-5460	572	2	of	of	ADP
ejpam-5460	572	3	applied	apply	VERB
ejpam-5460	572	4	mathematics	mathematic	NOUN
ejpam-5460	572	5	,	,	PUNCT
ejpam-5460	572	6	2012:425890	2012:425890	NUM
ejpam-5460	572	7	,	,	PUNCT
ejpam-5460	572	8	2012	2012	NUM
ejpam-5460	572	9	.	.	PUNCT
ejpam-5460	573	1	[	[	X
ejpam-5460	573	2	8	8	NUM
ejpam-5460	573	3	]	]	X
ejpam-5460	573	4	l	l	NOUN
ejpam-5460	573	5	fuch	fuch	NOUN
ejpam-5460	573	6	.	.	PUNCT
ejpam-5460	573	7	partially	partially	ADV
ejpam-5460	573	8	ordered	order	VERB
ejpam-5460	573	9	algebraic	algebraic	ADJ
ejpam-5460	573	10	systems	system	NOUN
ejpam-5460	573	11	.	.	PUNCT
ejpam-5460	574	1	dover	dover	PROPN
ejpam-5460	574	2	publications	publication	NOUN
ejpam-5460	574	3	,	,	PUNCT
ejpam-5460	574	4	2011	2011	NUM
ejpam-5460	574	5	.	.	PUNCT
ejpam-5460	575	1	references	reference	NOUN
ejpam-5460	575	2	2982	2982	NUM
ejpam-5460	575	3	[	[	X
ejpam-5460	575	4	9	9	NUM
ejpam-5460	575	5	]	]	X
ejpam-5460	575	6	z	z	NOUN
ejpam-5460	575	7	gu	gu	NOUN
ejpam-5460	575	8	.	.	PUNCT
ejpam-5460	576	1	on	on	ADP
ejpam-5460	576	2	bi	bi	NOUN
ejpam-5460	576	3	-	-	NOUN
ejpam-5460	576	4	ideals	ideal	NOUN
ejpam-5460	576	5	of	of	ADP
ejpam-5460	576	6	ordered	order	VERB
ejpam-5460	576	7	semigroups	semigroup	NOUN
ejpam-5460	576	8	.	.	PUNCT
ejpam-5460	576	9	quasigroups	quasigroup	NOUN
ejpam-5460	576	10	and	and	CCONJ
ejpam-5460	576	11	related	related	ADJ
ejpam-5460	576	12	systems	system	NOUN
ejpam-5460	576	13	,	,	PUNCT
ejpam-5460	576	14	26:149	26:149	NUM
ejpam-5460	576	15	–	–	PUNCT
ejpam-5460	576	16	154	154	NUM
ejpam-5460	576	17	,	,	PUNCT
ejpam-5460	576	18	2018	2018	NUM
ejpam-5460	576	19	.	.	PUNCT
ejpam-5460	577	1	[	[	X
ejpam-5460	577	2	10	10	NUM
ejpam-5460	577	3	]	]	X
ejpam-5460	577	4	k	k	PROPN
ejpam-5460	577	5	hansda	hansda	NOUN
ejpam-5460	577	6	.	.	PUNCT
ejpam-5460	578	1	minimal	minimal	ADJ
ejpam-5460	578	2	bi	bi	NOUN
ejpam-5460	578	3	-	-	NOUN
ejpam-5460	578	4	ideals	ideal	NOUN
ejpam-5460	578	5	in	in	ADP
ejpam-5460	578	6	regular	regular	ADJ
ejpam-5460	578	7	and	and	CCONJ
ejpam-5460	578	8	completely	completely	ADV
ejpam-5460	578	9	regular	regular	ADJ
ejpam-5460	578	10	ordered	order	VERB
ejpam-5460	578	11	semigroups	semigroup	NOUN
ejpam-5460	578	12	.	.	PUNCT
ejpam-5460	579	1	quasigroups	quasigroup	NOUN
ejpam-5460	579	2	and	and	CCONJ
ejpam-5460	579	3	related	related	ADJ
ejpam-5460	579	4	systems	system	NOUN
ejpam-5460	579	5	,	,	PUNCT
ejpam-5460	579	6	27:63–72	27:63–72	NUM
ejpam-5460	579	7	,	,	PUNCT
ejpam-5460	579	8	2019	2019	NUM
ejpam-5460	579	9	.	.	PUNCT
ejpam-5460	580	1	[	[	X
ejpam-5460	580	2	11	11	NUM
ejpam-5460	580	3	]	]	X
ejpam-5460	580	4	yb	yb	PROPN
ejpam-5460	580	5	jun	jun	PROPN
ejpam-5460	580	6	,	,	PUNCT
ejpam-5460	580	7	a	a	DET
ejpam-5460	580	8	khan	khan	PROPN
ejpam-5460	580	9	,	,	PUNCT
ejpam-5460	580	10	and	and	CCONJ
ejpam-5460	580	11	m	m	PROPN
ejpam-5460	580	12	shabir	shabir	PROPN
ejpam-5460	580	13	.	.	PUNCT
ejpam-5460	581	1	ordered	order	VERB
ejpam-5460	581	2	semigroups	semigroup	NOUN
ejpam-5460	581	3	characterized	characterize	VERB
ejpam-5460	581	4	by	by	ADP
ejpam-5460	581	5	their	their	PRON
ejpam-5460	581	6	fuzzy	fuzzy	ADJ
ejpam-5460	581	7	bi	bi	NOUN
ejpam-5460	581	8	-	-	NOUN
ejpam-5460	581	9	ideals	ideal	NOUN
ejpam-5460	581	10	.	.	PUNCT
ejpam-5460	582	1	bulletin	bulletin	NOUN
ejpam-5460	582	2	of	of	ADP
ejpam-5460	582	3	the	the	DET
ejpam-5460	582	4	malaysian	malaysian	PROPN
ejpam-5460	582	5	mathematical	mathematical	PROPN
ejpam-5460	582	6	sciences	sciences	PROPN
ejpam-5460	582	7	society	society	NOUN
ejpam-5460	582	8	,	,	PUNCT
ejpam-5460	582	9	32:391–408	32:391–408	NUM
ejpam-5460	582	10	,	,	PUNCT
ejpam-5460	582	11	2009	2009	NUM
ejpam-5460	582	12	.	.	PUNCT
ejpam-5460	583	1	[	[	X
ejpam-5460	583	2	12	12	NUM
ejpam-5460	583	3	]	]	X
ejpam-5460	583	4	yb	yb	PROPN
ejpam-5460	583	5	jun	jun	PROPN
ejpam-5460	583	6	,	,	PUNCT
ejpam-5460	583	7	k	k	PROPN
ejpam-5460	583	8	tinpun	tinpun	VERB
ejpam-5460	583	9	,	,	PUNCT
ejpam-5460	583	10	and	and	CCONJ
ejpam-5460	583	11	n	n	DET
ejpam-5460	583	12	lekkoksung	lekkoksung	NOUN
ejpam-5460	583	13	.	.	PUNCT
ejpam-5460	584	1	exploring	explore	VERB
ejpam-5460	584	2	regularities	regularity	NOUN
ejpam-5460	584	3	of	of	ADP
ejpam-5460	584	4	ordered	order	VERB
ejpam-5460	584	5	semigroups	semigroup	NOUN
ejpam-5460	584	6	through	through	ADP
ejpam-5460	584	7	generalized	generalized	ADJ
ejpam-5460	584	8	fuzzy	fuzzy	ADJ
ejpam-5460	584	9	ideals	ideal	NOUN
ejpam-5460	584	10	.	.	PUNCT
ejpam-5460	585	1	international	international	ADJ
ejpam-5460	585	2	journal	journal	NOUN
ejpam-5460	585	3	of	of	ADP
ejpam-5460	585	4	fuzzy	fuzzy	ADJ
ejpam-5460	585	5	logic	logic	NOUN
ejpam-5460	585	6	and	and	CCONJ
ejpam-5460	585	7	intelligent	intelligent	ADJ
ejpam-5460	585	8	systems	system	NOUN
ejpam-5460	585	9	,	,	PUNCT
ejpam-5460	585	10	24:141–152	24:141–152	NUM
ejpam-5460	585	11	,	,	PUNCT
ejpam-5460	585	12	2024	2024	NUM
ejpam-5460	585	13	.	.	PUNCT
ejpam-5460	586	1	[	[	X
ejpam-5460	586	2	13	13	NUM
ejpam-5460	586	3	]	]	PUNCT
ejpam-5460	586	4	n	n	CCONJ
ejpam-5460	586	5	kehayopulu	kehayopulu	VERB
ejpam-5460	586	6	.	.	PUNCT
ejpam-5460	587	1	on	on	ADP
ejpam-5460	587	2	weakly	weakly	ADJ
ejpam-5460	587	3	prime	prime	ADJ
ejpam-5460	587	4	ideals	ideal	NOUN
ejpam-5460	587	5	of	of	ADP
ejpam-5460	587	6	ordered	order	VERB
ejpam-5460	587	7	semigroups	semigroup	NOUN
ejpam-5460	587	8	.	.	PUNCT
ejpam-5460	588	1	mathematica	mathematica	PROPN
ejpam-5460	588	2	japonica	japonica	PROPN
ejpam-5460	588	3	,	,	PUNCT
ejpam-5460	588	4	35:1051–1056	35:1051–1056	NUM
ejpam-5460	588	5	,	,	PUNCT
ejpam-5460	588	6	1990	1990	NUM
ejpam-5460	588	7	.	.	PUNCT
ejpam-5460	589	1	[	[	X
ejpam-5460	589	2	14	14	NUM
ejpam-5460	589	3	]	]	PUNCT
ejpam-5460	589	4	n	n	CCONJ
ejpam-5460	589	5	kehayopulu	kehayopulu	VERB
ejpam-5460	589	6	.	.	PUNCT
ejpam-5460	590	1	on	on	ADP
ejpam-5460	590	2	completely	completely	ADV
ejpam-5460	590	3	regular	regular	ADJ
ejpam-5460	590	4	poe	poe	PROPN
ejpam-5460	590	5	-	-	PUNCT
ejpam-5460	590	6	semigroups	semigroups	PROPN
ejpam-5460	590	7	.	.	PUNCT
ejpam-5460	591	1	mathematica	mathematica	PROPN
ejpam-5460	591	2	japonica	japonica	PROPN
ejpam-5460	591	3	,	,	PUNCT
ejpam-5460	591	4	37:123–130	37:123–130	PROPN
ejpam-5460	591	5	,	,	PUNCT
ejpam-5460	591	6	1992	1992	NUM
ejpam-5460	591	7	.	.	PUNCT
ejpam-5460	592	1	[	[	X
ejpam-5460	592	2	15	15	NUM
ejpam-5460	592	3	]	]	X
ejpam-5460	592	4	n	n	PRON
ejpam-5460	592	5	kehayopulu	kehayopulu	VERB
ejpam-5460	592	6	.	.	PUNCT
ejpam-5460	593	1	interior	interior	ADJ
ejpam-5460	593	2	ideals	ideal	NOUN
ejpam-5460	593	3	and	and	CCONJ
ejpam-5460	593	4	interior	interior	ADJ
ejpam-5460	593	5	ideal	ideal	ADJ
ejpam-5460	593	6	elements	element	NOUN
ejpam-5460	593	7	in	in	ADP
ejpam-5460	593	8	ordered	order	VERB
ejpam-5460	593	9	semigroups	semigroup	NOUN
ejpam-5460	593	10	.	.	PUNCT
ejpam-5460	594	1	pure	pure	ADJ
ejpam-5460	594	2	mathematics	mathematic	NOUN
ejpam-5460	594	3	and	and	CCONJ
ejpam-5460	594	4	applications	application	NOUN
ejpam-5460	594	5	,	,	PUNCT
ejpam-5460	594	6	10:323–329	10:323–329	PROPN
ejpam-5460	594	7	,	,	PUNCT
ejpam-5460	594	8	1999	1999	NUM
ejpam-5460	594	9	.	.	PUNCT
ejpam-5460	595	1	[	[	X
ejpam-5460	595	2	16	16	NUM
ejpam-5460	595	3	]	]	PUNCT
ejpam-5460	595	4	n	n	CCONJ
ejpam-5460	595	5	kehayopulu	kehayopulu	ADJ
ejpam-5460	595	6	.	.	PUNCT
ejpam-5460	596	1	characterization	characterization	NOUN
ejpam-5460	596	2	of	of	ADP
ejpam-5460	596	3	left	left	ADJ
ejpam-5460	596	4	quasi	quasi	ADJ
ejpam-5460	596	5	-	-	ADJ
ejpam-5460	596	6	regular	regular	ADJ
ejpam-5460	596	7	and	and	CCONJ
ejpam-5460	596	8	semisimple	semisimple	NOUN
ejpam-5460	596	9	ordered	order	VERB
ejpam-5460	596	10	semigroups	semigroup	NOUN
ejpam-5460	596	11	in	in	ADP
ejpam-5460	596	12	terms	term	NOUN
ejpam-5460	596	13	of	of	ADP
ejpam-5460	596	14	fuzzy	fuzzy	ADJ
ejpam-5460	596	15	sets	set	NOUN
ejpam-5460	596	16	.	.	PUNCT
ejpam-5460	597	1	international	international	ADJ
ejpam-5460	597	2	journal	journal	NOUN
ejpam-5460	597	3	of	of	ADP
ejpam-5460	597	4	algebra	algebra	PROPN
ejpam-5460	597	5	,	,	PUNCT
ejpam-5460	597	6	6:747–755	6:747–755	NUM
ejpam-5460	597	7	,	,	PUNCT
ejpam-5460	597	8	2012	2012	NUM
ejpam-5460	597	9	.	.	PUNCT
ejpam-5460	598	1	[	[	X
ejpam-5460	598	2	17	17	NUM
ejpam-5460	598	3	]	]	PUNCT
ejpam-5460	598	4	n	n	CCONJ
ejpam-5460	598	5	kehayopulu	kehayopulu	VERB
ejpam-5460	598	6	.	.	PUNCT
ejpam-5460	599	1	adjunction	adjunction	VERB
ejpam-5460	599	2	greatest	great	ADJ
ejpam-5460	599	3	element	element	NOUN
ejpam-5460	599	4	to	to	AUX
ejpam-5460	599	5	ordered	order	VERB
ejpam-5460	599	6	hypersemigroups	hypersemigroup	NOUN
ejpam-5460	599	7	.	.	PUNCT
ejpam-5460	600	1	turkish	turkish	ADJ
ejpam-5460	600	2	journal	journal	NOUN
ejpam-5460	600	3	of	of	ADP
ejpam-5460	600	4	mathematics	mathematic	NOUN
ejpam-5460	600	5	,	,	PUNCT
ejpam-5460	600	6	47:1595–1615	47:1595–1615	NUM
ejpam-5460	600	7	,	,	PUNCT
ejpam-5460	600	8	2023	2023	NUM
ejpam-5460	600	9	.	.	PUNCT
ejpam-5460	601	1	[	[	X
ejpam-5460	601	2	18	18	NUM
ejpam-5460	601	3	]	]	PUNCT
ejpam-5460	601	4	n	n	PRON
ejpam-5460	601	5	kehayopulu	kehayopulu	VERB
ejpam-5460	601	6	.	.	PUNCT
ejpam-5460	602	1	what	what	PRON
ejpam-5460	602	2	can	can	AUX
ejpam-5460	602	3	lattices	lattice	NOUN
ejpam-5460	602	4	do	do	VERB
ejpam-5460	602	5	for	for	ADP
ejpam-5460	602	6	hypersemigroups	hypersemigroup	NOUN
ejpam-5460	602	7	?	?	PUNCT
ejpam-5460	602	8	.	.	PUNCT
ejpam-5460	603	1	turkish	turkish	ADJ
ejpam-5460	603	2	journal	journal	NOUN
ejpam-5460	603	3	of	of	ADP
ejpam-5460	603	4	mathematics	mathematic	NOUN
ejpam-5460	603	5	,	,	PUNCT
ejpam-5460	603	6	47:1558–1572	47:1558–1572	PROPN
ejpam-5460	603	7	,	,	PUNCT
ejpam-5460	603	8	2023	2023	NUM
ejpam-5460	603	9	.	.	PUNCT
ejpam-5460	604	1	[	[	X
ejpam-5460	604	2	19	19	NUM
ejpam-5460	604	3	]	]	PUNCT
ejpam-5460	604	4	n	n	CCONJ
ejpam-5460	604	5	kehayopulu	kehayopulu	ADJ
ejpam-5460	604	6	and	and	CCONJ
ejpam-5460	604	7	m	m	PROPN
ejpam-5460	604	8	tsingelis	tsingelis	PROPN
ejpam-5460	604	9	.	.	PUNCT
ejpam-5460	605	1	fuzzy	fuzzy	ADJ
ejpam-5460	605	2	sets	set	NOUN
ejpam-5460	605	3	in	in	ADP
ejpam-5460	605	4	ordered	order	VERB
ejpam-5460	605	5	groupoids	groupoid	NOUN
ejpam-5460	605	6	.	.	PUNCT
ejpam-5460	606	1	semigroup	semigroup	PROPN
ejpam-5460	606	2	forum	forum	PROPN
ejpam-5460	606	3	,	,	PUNCT
ejpam-5460	606	4	65:128–132	65:128–132	PROPN
ejpam-5460	606	5	,	,	PUNCT
ejpam-5460	606	6	2002	2002	NUM
ejpam-5460	606	7	.	.	PUNCT
ejpam-5460	607	1	[	[	X
ejpam-5460	607	2	20	20	NUM
ejpam-5460	607	3	]	]	PUNCT
ejpam-5460	607	4	n	n	CCONJ
ejpam-5460	607	5	kehayopulu	kehayopulu	ADJ
ejpam-5460	607	6	and	and	CCONJ
ejpam-5460	607	7	m	m	PROPN
ejpam-5460	607	8	tsingelis	tsingeli	NOUN
ejpam-5460	607	9	.	.	PUNCT
ejpam-5460	608	1	the	the	DET
ejpam-5460	608	2	embedding	embedding	NOUN
ejpam-5460	608	3	of	of	ADP
ejpam-5460	608	4	an	an	DET
ejpam-5460	608	5	ordered	order	VERB
ejpam-5460	608	6	groupoid	groupoid	NOUN
ejpam-5460	608	7	into	into	ADP
ejpam-5460	608	8	a	a	DET
ejpam-5460	608	9	poegroupoid	poegroupoid	NOUN
ejpam-5460	608	10	in	in	ADP
ejpam-5460	608	11	terms	term	NOUN
ejpam-5460	608	12	of	of	ADP
ejpam-5460	608	13	fuzzy	fuzzy	ADJ
ejpam-5460	608	14	sets	set	NOUN
ejpam-5460	608	15	.	.	PUNCT
ejpam-5460	609	1	information	information	NOUN
ejpam-5460	609	2	sciences	sciences	PROPN
ejpam-5460	609	3	,	,	PUNCT
ejpam-5460	609	4	152:231–236	152:231–236	NUM
ejpam-5460	609	5	,	,	PUNCT
ejpam-5460	609	6	2003	2003	NUM
ejpam-5460	609	7	.	.	PUNCT
ejpam-5460	610	1	[	[	X
ejpam-5460	610	2	21	21	NUM
ejpam-5460	610	3	]	]	PUNCT
ejpam-5460	610	4	n	n	CCONJ
ejpam-5460	610	5	kehayopulu	kehayopulu	ADJ
ejpam-5460	610	6	and	and	CCONJ
ejpam-5460	610	7	m	m	PROPN
ejpam-5460	610	8	tsingelis	tsingelis	PROPN
ejpam-5460	610	9	.	.	PUNCT
ejpam-5460	611	1	fuzzy	fuzzy	ADJ
ejpam-5460	611	2	bi	bi	NOUN
ejpam-5460	611	3	-	-	NOUN
ejpam-5460	611	4	ideals	ideal	NOUN
ejpam-5460	611	5	in	in	ADP
ejpam-5460	611	6	ordered	order	VERB
ejpam-5460	611	7	semigroups	semigroup	NOUN
ejpam-5460	611	8	.	.	PUNCT
ejpam-5460	612	1	information	information	NOUN
ejpam-5460	612	2	sciences	sciences	PROPN
ejpam-5460	612	3	,	,	PUNCT
ejpam-5460	612	4	171:13–28	171:13–28	NUM
ejpam-5460	612	5	,	,	PUNCT
ejpam-5460	612	6	2005	2005	NUM
ejpam-5460	612	7	.	.	PUNCT
ejpam-5460	613	1	[	[	X
ejpam-5460	613	2	22	22	NUM
ejpam-5460	613	3	]	]	PUNCT
ejpam-5460	613	4	n	n	CCONJ
ejpam-5460	613	5	kehayopulu	kehayopulu	ADJ
ejpam-5460	613	6	and	and	CCONJ
ejpam-5460	613	7	m	m	PROPN
ejpam-5460	613	8	tsingelis	tsingelis	PROPN
ejpam-5460	613	9	.	.	PUNCT
ejpam-5460	614	1	fuzzy	fuzzy	ADJ
ejpam-5460	614	2	interior	interior	ADJ
ejpam-5460	614	3	ideals	ideal	NOUN
ejpam-5460	614	4	in	in	ADP
ejpam-5460	614	5	ordered	order	VERB
ejpam-5460	614	6	semigroups	semigroup	NOUN
ejpam-5460	614	7	.	.	PUNCT
ejpam-5460	615	1	lobachevskii	lobachevskii	PROPN
ejpam-5460	615	2	journal	journal	PROPN
ejpam-5460	615	3	of	of	ADP
ejpam-5460	615	4	mathematics	mathematics	PROPN
ejpam-5460	615	5	,	,	PUNCT
ejpam-5460	615	6	21:65–71	21:65–71	NUM
ejpam-5460	615	7	,	,	PUNCT
ejpam-5460	615	8	2006	2006	NUM
ejpam-5460	615	9	.	.	PUNCT
ejpam-5460	616	1	[	[	X
ejpam-5460	616	2	23	23	NUM
ejpam-5460	616	3	]	]	PUNCT
ejpam-5460	616	4	n	n	CCONJ
ejpam-5460	616	5	kehayopulu	kehayopulu	ADJ
ejpam-5460	616	6	and	and	CCONJ
ejpam-5460	616	7	m	m	PROPN
ejpam-5460	616	8	tsingelis	tsingelis	PROPN
ejpam-5460	616	9	.	.	PUNCT
ejpam-5460	617	1	left	leave	VERB
ejpam-5460	617	2	regular	regular	ADJ
ejpam-5460	617	3	and	and	CCONJ
ejpam-5460	617	4	intra	intra	ADJ
ejpam-5460	617	5	-	-	ADJ
ejpam-5460	617	6	regular	regular	ADJ
ejpam-5460	617	7	ordered	order	VERB
ejpam-5460	617	8	semigroups	semigroup	NOUN
ejpam-5460	617	9	in	in	ADP
ejpam-5460	617	10	terms	term	NOUN
ejpam-5460	617	11	of	of	ADP
ejpam-5460	617	12	fuzzy	fuzzy	ADJ
ejpam-5460	617	13	subsets	subset	NOUN
ejpam-5460	617	14	.	.	PUNCT
ejpam-5460	618	1	quasigroups	quasigroup	NOUN
ejpam-5460	618	2	and	and	CCONJ
ejpam-5460	618	3	related	related	ADJ
ejpam-5460	618	4	systems	system	NOUN
ejpam-5460	618	5	,	,	PUNCT
ejpam-5460	618	6	14:169–178	14:169–178	NUM
ejpam-5460	618	7	,	,	PUNCT
ejpam-5460	618	8	2006	2006	NUM
ejpam-5460	618	9	.	.	PUNCT
ejpam-5460	619	1	[	[	X
ejpam-5460	619	2	24	24	NUM
ejpam-5460	619	3	]	]	PUNCT
ejpam-5460	619	4	n	n	CCONJ
ejpam-5460	619	5	kehayopulu	kehayopulu	ADJ
ejpam-5460	619	6	and	and	CCONJ
ejpam-5460	619	7	m	m	PROPN
ejpam-5460	619	8	tsingelis	tsingelis	PROPN
ejpam-5460	619	9	.	.	PUNCT
ejpam-5460	620	1	regular	regular	ADJ
ejpam-5460	620	2	ordered	order	VERB
ejpam-5460	620	3	semigroups	semigroup	NOUN
ejpam-5460	620	4	in	in	ADP
ejpam-5460	620	5	terms	term	NOUN
ejpam-5460	620	6	of	of	ADP
ejpam-5460	620	7	fuzzy	fuzzy	ADJ
ejpam-5460	620	8	subsets	subset	NOUN
ejpam-5460	620	9	.	.	PUNCT
ejpam-5460	621	1	information	information	NOUN
ejpam-5460	621	2	sciences	sciences	PROPN
ejpam-5460	621	3	,	,	PUNCT
ejpam-5460	621	4	176:3675–3693	176:3675–3693	NUM
ejpam-5460	621	5	,	,	PUNCT
ejpam-5460	621	6	2006	2006	NUM
ejpam-5460	621	7	.	.	PUNCT
ejpam-5460	622	1	references	reference	NOUN
ejpam-5460	622	2	2983	2983	NUM
ejpam-5460	622	3	[	[	X
ejpam-5460	622	4	25	25	NUM
ejpam-5460	622	5	]	]	PUNCT
ejpam-5460	622	6	n	n	CCONJ
ejpam-5460	622	7	kehayopulu	kehayopulu	ADJ
ejpam-5460	622	8	and	and	CCONJ
ejpam-5460	622	9	m	m	PROPN
ejpam-5460	622	10	tsingelis	tsingelis	PROPN
ejpam-5460	622	11	.	.	PUNCT
ejpam-5460	623	1	fuzzy	fuzzy	ADJ
ejpam-5460	623	2	ideals	ideal	NOUN
ejpam-5460	623	3	in	in	ADP
ejpam-5460	623	4	ordered	order	VERB
ejpam-5460	623	5	semigroups	semigroup	NOUN
ejpam-5460	623	6	.	.	PUNCT
ejpam-5460	624	1	quasigroups	quasigroup	NOUN
ejpam-5460	624	2	and	and	CCONJ
ejpam-5460	624	3	related	related	ADJ
ejpam-5460	624	4	systems	system	NOUN
ejpam-5460	624	5	,	,	PUNCT
ejpam-5460	624	6	15:279–289	15:279–289	NUM
ejpam-5460	624	7	,	,	PUNCT
ejpam-5460	624	8	2007	2007	NUM
ejpam-5460	624	9	.	.	PUNCT
ejpam-5460	625	1	[	[	X
ejpam-5460	625	2	26	26	NUM
ejpam-5460	625	3	]	]	PUNCT
ejpam-5460	625	4	n	n	CCONJ
ejpam-5460	625	5	kehayopulu	kehayopulu	ADJ
ejpam-5460	625	6	and	and	CCONJ
ejpam-5460	625	7	m	m	PROPN
ejpam-5460	625	8	tsingelis	tsingeli	NOUN
ejpam-5460	625	9	.	.	PUNCT
ejpam-5460	626	1	on	on	ADP
ejpam-5460	626	2	fuzzy	fuzzy	ADJ
ejpam-5460	626	3	ordered	order	VERB
ejpam-5460	626	4	semigroups	semigroup	NOUN
ejpam-5460	626	5	.	.	PUNCT
ejpam-5460	626	6	quasigroups	quasigroup	NOUN
ejpam-5460	626	7	and	and	CCONJ
ejpam-5460	626	8	related	related	ADJ
ejpam-5460	626	9	systems	system	NOUN
ejpam-5460	626	10	,	,	PUNCT
ejpam-5460	626	11	20:61–70	20:61–70	NUM
ejpam-5460	626	12	,	,	PUNCT
ejpam-5460	626	13	2012	2012	NUM
ejpam-5460	626	14	.	.	PUNCT
ejpam-5460	627	1	[	[	X
ejpam-5460	627	2	27	27	NUM
ejpam-5460	627	3	]	]	X
ejpam-5460	627	4	a	a	DET
ejpam-5460	627	5	khan	khan	PROPN
ejpam-5460	627	6	,	,	PUNCT
ejpam-5460	627	7	yb	yb	PROPN
ejpam-5460	627	8	jun	jun	PROPN
ejpam-5460	627	9	,	,	PUNCT
ejpam-5460	627	10	nh	nh	PROPN
ejpam-5460	627	11	sarmin	sarmin	NOUN
ejpam-5460	627	12	,	,	PUNCT
ejpam-5460	627	13	and	and	CCONJ
ejpam-5460	627	14	fm	fm	PROPN
ejpam-5460	627	15	khan	khan	PROPN
ejpam-5460	627	16	.	.	PUNCT
ejpam-5460	628	1	ordered	order	VERB
ejpam-5460	628	2	semigroups	semigroup	NOUN
ejpam-5460	628	3	characterized	characterize	VERB
ejpam-5460	628	4	by	by	ADP
ejpam-5460	628	5	(	(	PUNCT
ejpam-5460	628	6	∈,∈	∈,∈	X
ejpam-5460	628	7	∨qk)-fuzzy	∨qk)-fuzzy	ADJ
ejpam-5460	628	8	generalized	generalized	ADJ
ejpam-5460	628	9	bi	bi	NOUN
ejpam-5460	628	10	-	-	NOUN
ejpam-5460	628	11	ideals	ideal	NOUN
ejpam-5460	628	12	.	.	PUNCT
ejpam-5460	629	1	neural	neural	ADJ
ejpam-5460	629	2	computing	computing	NOUN
ejpam-5460	629	3	and	and	CCONJ
ejpam-5460	629	4	applications	application	NOUN
ejpam-5460	629	5	,	,	PUNCT
ejpam-5460	629	6	21:121	21:121	NUM
ejpam-5460	629	7	–	–	PUNCT
ejpam-5460	629	8	132	132	NUM
ejpam-5460	629	9	,	,	PUNCT
ejpam-5460	629	10	2012	2012	NUM
ejpam-5460	629	11	.	.	PUNCT
ejpam-5460	630	1	[	[	X
ejpam-5460	630	2	28	28	NUM
ejpam-5460	630	3	]	]	X
ejpam-5460	630	4	a	a	DET
ejpam-5460	630	5	khan	khan	PROPN
ejpam-5460	630	6	,	,	PUNCT
ejpam-5460	630	7	yb	yb	PROPN
ejpam-5460	630	8	jun	jun	PROPN
ejpam-5460	630	9	,	,	PUNCT
ejpam-5460	630	10	and	and	CCONJ
ejpam-5460	630	11	m	m	PROPN
ejpam-5460	630	12	shabir	shabir	PROPN
ejpam-5460	630	13	.	.	PUNCT
ejpam-5460	631	1	a	a	DET
ejpam-5460	631	2	study	study	NOUN
ejpam-5460	631	3	of	of	ADP
ejpam-5460	631	4	generalized	generalized	ADJ
ejpam-5460	631	5	fuzzy	fuzzy	ADJ
ejpam-5460	631	6	ideals	ideal	NOUN
ejpam-5460	631	7	in	in	ADP
ejpam-5460	631	8	ordered	order	VERB
ejpam-5460	631	9	semigroups	semigroup	NOUN
ejpam-5460	631	10	.	.	PUNCT
ejpam-5460	631	11	neural	neural	ADJ
ejpam-5460	631	12	computing	computing	NOUN
ejpam-5460	631	13	and	and	CCONJ
ejpam-5460	631	14	applications	application	NOUN
ejpam-5460	631	15	,	,	PUNCT
ejpam-5460	631	16	21:69–78	21:69–78	NUM
ejpam-5460	631	17	,	,	PUNCT
ejpam-5460	631	18	2012	2012	NUM
ejpam-5460	631	19	.	.	PUNCT
ejpam-5460	632	1	[	[	X
ejpam-5460	632	2	29	29	NUM
ejpam-5460	632	3	]	]	X
ejpam-5460	632	4	a	a	DET
ejpam-5460	632	5	khan	khan	PROPN
ejpam-5460	632	6	,	,	PUNCT
ejpam-5460	632	7	nh	nh	PROPN
ejpam-5460	632	8	sarmin	sarmin	NOUN
ejpam-5460	632	9	,	,	PUNCT
ejpam-5460	632	10	b	b	NOUN
ejpam-5460	632	11	davvaz	davvaz	NOUN
ejpam-5460	632	12	,	,	PUNCT
ejpam-5460	632	13	and	and	CCONJ
ejpam-5460	632	14	fm	fm	PROPN
ejpam-5460	632	15	khan	khan	PROPN
ejpam-5460	632	16	.	.	PUNCT
ejpam-5460	633	1	new	new	ADJ
ejpam-5460	633	2	types	type	NOUN
ejpam-5460	633	3	of	of	ADP
ejpam-5460	633	4	fuzzy	fuzzy	ADJ
ejpam-5460	633	5	bi	bi	NOUN
ejpam-5460	633	6	-	-	NOUN
ejpam-5460	633	7	ideals	ideal	NOUN
ejpam-5460	633	8	in	in	ADP
ejpam-5460	633	9	ordered	order	VERB
ejpam-5460	633	10	semigroups	semigroup	NOUN
ejpam-5460	633	11	.	.	PUNCT
ejpam-5460	634	1	neural	neural	ADJ
ejpam-5460	634	2	computing	computing	NOUN
ejpam-5460	634	3	and	and	CCONJ
ejpam-5460	634	4	applications	application	NOUN
ejpam-5460	634	5	,	,	PUNCT
ejpam-5460	634	6	21:295–305	21:295–305	NUM
ejpam-5460	634	7	,	,	PUNCT
ejpam-5460	634	8	2012	2012	NUM
ejpam-5460	634	9	.	.	PUNCT
ejpam-5460	635	1	[	[	X
ejpam-5460	635	2	30	30	NUM
ejpam-5460	635	3	]	]	X
ejpam-5460	635	4	a	a	DET
ejpam-5460	635	5	khan	khan	PROPN
ejpam-5460	635	6	,	,	PUNCT
ejpam-5460	635	7	nh	nh	PROPN
ejpam-5460	635	8	sarmin	sarmin	NOUN
ejpam-5460	635	9	,	,	PUNCT
ejpam-5460	635	10	b	b	NOUN
ejpam-5460	635	11	davvaz	davvaz	NOUN
ejpam-5460	635	12	,	,	PUNCT
ejpam-5460	635	13	and	and	CCONJ
ejpam-5460	635	14	fm	fm	PROPN
ejpam-5460	635	15	khan	khan	PROPN
ejpam-5460	635	16	.	.	PUNCT
ejpam-5460	636	1	some	some	DET
ejpam-5460	636	2	new	new	ADJ
ejpam-5460	636	3	characterization	characterization	NOUN
ejpam-5460	636	4	of	of	ADP
ejpam-5460	636	5	ordered	order	VERB
ejpam-5460	636	6	semigroups	semigroup	NOUN
ejpam-5460	636	7	in	in	ADP
ejpam-5460	636	8	terms	term	NOUN
ejpam-5460	636	9	of	of	ADP
ejpam-5460	636	10	(	(	PUNCT
ejpam-5460	636	11	λ	λ	PROPN
ejpam-5460	636	12	,	,	PUNCT
ejpam-5460	636	13	θ)-fuzzy	θ)-fuzzy	PUNCT
ejpam-5460	636	14	bi	bi	NOUN
ejpam-5460	636	15	-	-	NOUN
ejpam-5460	636	16	ideals	ideal	NOUN
ejpam-5460	636	17	.	.	PUNCT
ejpam-5460	637	1	international	international	ADJ
ejpam-5460	637	2	journal	journal	NOUN
ejpam-5460	637	3	of	of	ADP
ejpam-5460	637	4	algebra	algebra	PROPN
ejpam-5460	637	5	and	and	CCONJ
ejpam-5460	637	6	statistics	statistic	NOUN
ejpam-5460	637	7	,	,	PUNCT
ejpam-5460	637	8	1:22–32	1:22–32	NUM
ejpam-5460	637	9	,	,	PUNCT
ejpam-5460	637	10	2012	2012	NUM
ejpam-5460	637	11	.	.	PUNCT
ejpam-5460	638	1	[	[	X
ejpam-5460	638	2	31	31	NUM
ejpam-5460	638	3	]	]	PUNCT
ejpam-5460	638	4	a	a	DET
ejpam-5460	638	5	khan	khan	PROPN
ejpam-5460	638	6	and	and	CCONJ
ejpam-5460	638	7	m	m	PROPN
ejpam-5460	638	8	shabir	shabir	PROPN
ejpam-5460	638	9	.	.	PUNCT
ejpam-5460	639	1	(	(	PUNCT
ejpam-5460	639	2	α	α	NOUN
ejpam-5460	639	3	,	,	PUNCT
ejpam-5460	639	4	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	639	5	interior	interior	ADJ
ejpam-5460	639	6	ideals	ideal	NOUN
ejpam-5460	639	7	in	in	ADP
ejpam-5460	639	8	ordered	order	VERB
ejpam-5460	639	9	semigroups	semigroup	NOUN
ejpam-5460	639	10	.	.	PUNCT
ejpam-5460	640	1	lobachevskii	lobachevskii	PROPN
ejpam-5460	640	2	journal	journal	PROPN
ejpam-5460	640	3	of	of	ADP
ejpam-5460	640	4	mathematics	mathematic	NOUN
ejpam-5460	640	5	,	,	PUNCT
ejpam-5460	640	6	30:30–39	30:30–39	NUM
ejpam-5460	640	7	,	,	PUNCT
ejpam-5460	640	8	2009	2009	NUM
ejpam-5460	640	9	.	.	PUNCT
ejpam-5460	641	1	[	[	X
ejpam-5460	641	2	32	32	NUM
ejpam-5460	641	3	]	]	X
ejpam-5460	641	4	fm	fm	PROPN
ejpam-5460	641	5	khan	khan	PROPN
ejpam-5460	641	6	,	,	PUNCT
ejpam-5460	641	7	nh	nh	PROPN
ejpam-5460	641	8	sarmin	sarmin	NOUN
ejpam-5460	641	9	,	,	PUNCT
ejpam-5460	641	10	and	and	CCONJ
ejpam-5460	641	11	a	a	DET
ejpam-5460	641	12	khan	khan	PROPN
ejpam-5460	641	13	.	.	PUNCT
ejpam-5460	642	1	some	some	DET
ejpam-5460	642	2	study	study	NOUN
ejpam-5460	642	3	of	of	ADP
ejpam-5460	642	4	(	(	PUNCT
ejpam-5460	642	5	α	α	NOUN
ejpam-5460	642	6	,	,	PUNCT
ejpam-5460	642	7	β)-fuzzy	β)-fuzzy	PUNCT
ejpam-5460	642	8	ideals	ideal	NOUN
ejpam-5460	642	9	in	in	ADP
ejpam-5460	642	10	ordered	order	VERB
ejpam-5460	642	11	semigroups	semigroup	NOUN
ejpam-5460	642	12	.	.	PUNCT
ejpam-5460	643	1	annals	annal	NOUN
ejpam-5460	643	2	of	of	ADP
ejpam-5460	643	3	fuzzy	fuzzy	ADJ
ejpam-5460	643	4	mathematics	mathematic	NOUN
ejpam-5460	643	5	and	and	CCONJ
ejpam-5460	643	6	informatics	informatic	NOUN
ejpam-5460	643	7	,	,	PUNCT
ejpam-5460	643	8	3:213–227	3:213–227	PROPN
ejpam-5460	643	9	,	,	PUNCT
ejpam-5460	643	10	2012	2012	NUM
ejpam-5460	643	11	.	.	PUNCT
ejpam-5460	644	1	[	[	X
ejpam-5460	644	2	33	33	NUM
ejpam-5460	644	3	]	]	X
ejpam-5460	644	4	ms	ms	PROPN
ejpam-5460	644	5	ali	ali	PROPN
ejpam-5460	644	6	khan	khan	PROPN
ejpam-5460	644	7	,	,	PUNCT
ejpam-5460	644	8	s	s	PROPN
ejpam-5460	644	9	abdullah	abdullah	PROPN
ejpam-5460	644	10	,	,	PUNCT
ejpam-5460	644	11	a	a	DET
ejpam-5460	644	12	ali	ali	PROPN
ejpam-5460	644	13	,	,	PUNCT
ejpam-5460	644	14	f	f	PROPN
ejpam-5460	644	15	amin	amin	PROPN
ejpam-5460	644	16	,	,	PUNCT
ejpam-5460	644	17	and	and	CCONJ
ejpam-5460	644	18	k	k	PROPN
ejpam-5460	644	19	rahman	rahman	PROPN
ejpam-5460	644	20	.	.	PUNCT
ejpam-5460	645	1	on	on	ADP
ejpam-5460	645	2	generalized	generalized	ADJ
ejpam-5460	645	3	(	(	PUNCT
ejpam-5460	645	4	∈,∈	∈,∈	X
ejpam-5460	645	5	∨qk)fuzzy	∨qk)fuzzy	X
ejpam-5460	645	6	quasi	quasi	NOUN
ejpam-5460	645	7	-	-	NOUN
ejpam-5460	645	8	ideals	ideal	NOUN
ejpam-5460	645	9	in	in	ADP
ejpam-5460	645	10	ordered	order	VERB
ejpam-5460	645	11	semigroups	semigroup	NOUN
ejpam-5460	645	12	.	.	PUNCT
ejpam-5460	646	1	turkish	turkish	ADJ
ejpam-5460	646	2	journal	journal	NOUN
ejpam-5460	646	3	of	of	ADP
ejpam-5460	646	4	fuzzy	fuzzy	ADJ
ejpam-5460	646	5	systems	system	NOUN
ejpam-5460	646	6	,	,	PUNCT
ejpam-5460	646	7	8:33–51	8:33–51	NUM
ejpam-5460	646	8	,	,	PUNCT
ejpam-5460	646	9	2017	2017	NUM
ejpam-5460	646	10	.	.	PUNCT
ejpam-5460	647	1	[	[	X
ejpam-5460	647	2	34	34	NUM
ejpam-5460	647	3	]	]	X
ejpam-5460	647	4	nh	nh	PROPN
ejpam-5460	647	5	khan	khan	PROPN
ejpam-5460	647	6	,	,	PUNCT
ejpam-5460	647	7	b	b	PROPN
ejpam-5460	647	8	davvaz	davvaz	NOUN
ejpam-5460	647	9	,	,	PUNCT
ejpam-5460	647	10	and	and	CCONJ
ejpam-5460	647	11	ma	ma	PROPN
ejpam-5460	647	12	khan	khan	PROPN
ejpam-5460	647	13	.	.	PUNCT
ejpam-5460	648	1	ordered	order	VERB
ejpam-5460	648	2	semigroups	semigroup	NOUN
ejpam-5460	648	3	characterized	characterize	VERB
ejpam-5460	648	4	in	in	ADP
ejpam-5460	648	5	terms	term	NOUN
ejpam-5460	648	6	of	of	ADP
ejpam-5460	648	7	generalized	generalized	ADJ
ejpam-5460	648	8	fuzzy	fuzzy	ADJ
ejpam-5460	648	9	ideals	ideal	NOUN
ejpam-5460	648	10	.	.	PUNCT
ejpam-5460	649	1	journal	journal	NOUN
ejpam-5460	649	2	of	of	ADP
ejpam-5460	649	3	intelligent	intelligent	ADJ
ejpam-5460	649	4	and	and	CCONJ
ejpam-5460	649	5	fuzzy	fuzzy	ADJ
ejpam-5460	649	6	systems	system	NOUN
ejpam-5460	649	7	,	,	PUNCT
ejpam-5460	649	8	32:1045–1057	32:1045–1057	NUM
ejpam-5460	649	9	,	,	PUNCT
ejpam-5460	649	10	2017	2017	NUM
ejpam-5460	649	11	.	.	PUNCT
ejpam-5460	650	1	[	[	X
ejpam-5460	650	2	35	35	NUM
ejpam-5460	650	3	]	]	SYM
ejpam-5460	650	4	s	s	PART
ejpam-5460	650	5	lekkoksung	lekkoksung	NOUN
ejpam-5460	650	6	,	,	PUNCT
ejpam-5460	650	7	a	a	DET
ejpam-5460	650	8	iampan	iampan	NOUN
ejpam-5460	650	9	,	,	PUNCT
ejpam-5460	650	10	p	p	ADJ
ejpam-5460	650	11	julatha	julatha	NOUN
ejpam-5460	650	12	,	,	PUNCT
ejpam-5460	650	13	and	and	CCONJ
ejpam-5460	650	14	n	n	DET
ejpam-5460	650	15	lekkoksung	lekkoksung	NOUN
ejpam-5460	650	16	.	.	PUNCT
ejpam-5460	651	1	representations	representation	NOUN
ejpam-5460	651	2	of	of	ADP
ejpam-5460	651	3	ordered	order	VERB
ejpam-5460	651	4	semigroups	semigroup	NOUN
ejpam-5460	651	5	and	and	CCONJ
ejpam-5460	651	6	their	their	PRON
ejpam-5460	651	7	interconnection	interconnection	NOUN
ejpam-5460	651	8	.	.	PUNCT
ejpam-5460	652	1	journal	journal	NOUN
ejpam-5460	652	2	of	of	ADP
ejpam-5460	652	3	intelligent	intelligent	ADJ
ejpam-5460	652	4	and	and	CCONJ
ejpam-5460	652	5	fuzzy	fuzzy	ADJ
ejpam-5460	652	6	systems	system	NOUN
ejpam-5460	652	7	,	,	PUNCT
ejpam-5460	652	8	44:6877–6884	44:6877–6884	NUM
ejpam-5460	652	9	,	,	PUNCT
ejpam-5460	652	10	2023	2023	NUM
ejpam-5460	652	11	.	.	PUNCT
ejpam-5460	653	1	[	[	X
ejpam-5460	653	2	36	36	NUM
ejpam-5460	653	3	]	]	X
ejpam-5460	653	4	s	s	VERB
ejpam-5460	653	5	lekkoksung	lekkoksung	NOUN
ejpam-5460	653	6	,	,	PUNCT
ejpam-5460	653	7	a	a	DET
ejpam-5460	653	8	iampan	iampan	NOUN
ejpam-5460	653	9	,	,	PUNCT
ejpam-5460	653	10	and	and	CCONJ
ejpam-5460	653	11	n	n	DET
ejpam-5460	653	12	lekkoksung	lekkoksung	NOUN
ejpam-5460	653	13	.	.	PUNCT
ejpam-5460	654	1	on	on	ADP
ejpam-5460	654	2	ideal	ideal	ADJ
ejpam-5460	654	3	elements	element	NOUN
ejpam-5460	654	4	of	of	ADP
ejpam-5460	654	5	partially	partially	ADV
ejpam-5460	654	6	ordered	order	VERB
ejpam-5460	654	7	semigroups	semigroup	NOUN
ejpam-5460	654	8	with	with	ADP
ejpam-5460	654	9	the	the	DET
ejpam-5460	654	10	greatest	great	ADJ
ejpam-5460	654	11	element	element	NOUN
ejpam-5460	654	12	.	.	PUNCT
ejpam-5460	655	1	international	international	ADJ
ejpam-5460	655	2	journal	journal	NOUN
ejpam-5460	655	3	of	of	ADP
ejpam-5460	655	4	innovative	innovative	ADJ
ejpam-5460	655	5	computing	computing	NOUN
ejpam-5460	655	6	,	,	PUNCT
ejpam-5460	655	7	information	information	NOUN
ejpam-5460	655	8	and	and	CCONJ
ejpam-5460	655	9	control	control	NOUN
ejpam-5460	655	10	,	,	PUNCT
ejpam-5460	655	11	18:1941–1955	18:1941–1955	NUM
ejpam-5460	655	12	,	,	PUNCT
ejpam-5460	655	13	2022	2022	NUM
ejpam-5460	655	14	.	.	PUNCT
ejpam-5460	656	1	[	[	X
ejpam-5460	656	2	37	37	NUM
ejpam-5460	656	3	]	]	X
ejpam-5460	656	4	p	p	X
ejpam-5460	656	5	luangchaisri	luangchaisri	VERB
ejpam-5460	656	6	and	and	CCONJ
ejpam-5460	656	7	t	t	PROPN
ejpam-5460	656	8	changphas	changpha	NOUN
ejpam-5460	656	9	.	.	PUNCT
ejpam-5460	657	1	on	on	ADP
ejpam-5460	657	2	(	(	PUNCT
ejpam-5460	657	3	m	m	NOUN
ejpam-5460	657	4	,	,	PUNCT
ejpam-5460	657	5	n)-regular	n)-regular	ADJ
ejpam-5460	657	6	and	and	CCONJ
ejpam-5460	657	7	intra	intra	ADJ
ejpam-5460	657	8	-	-	ADJ
ejpam-5460	657	9	regular	regular	ADJ
ejpam-5460	657	10	ordere	ordere	ADJ
ejpam-5460	657	11	semigroups	semigroup	NOUN
ejpam-5460	657	12	.	.	PUNCT
ejpam-5460	658	1	quasigroups	quasigroup	NOUN
ejpam-5460	658	2	and	and	CCONJ
ejpam-5460	658	3	related	related	ADJ
ejpam-5460	658	4	systems	system	NOUN
ejpam-5460	658	5	,	,	PUNCT
ejpam-5460	658	6	27:267–272	27:267–272	NUM
ejpam-5460	658	7	,	,	PUNCT
ejpam-5460	658	8	2019	2019	NUM
ejpam-5460	658	9	.	.	PUNCT
ejpam-5460	659	1	[	[	X
ejpam-5460	659	2	38	38	NUM
ejpam-5460	659	3	]	]	PUNCT
ejpam-5460	659	4	a	a	DET
ejpam-5460	659	5	mahboob	mahboob	NOUN
ejpam-5460	659	6	,	,	PUNCT
ejpam-5460	659	7	b	b	NOUN
ejpam-5460	659	8	davvaz	davvaz	NOUN
ejpam-5460	659	9	,	,	PUNCT
ejpam-5460	659	10	and	and	CCONJ
ejpam-5460	659	11	nm	nm	ADJ
ejpam-5460	659	12	khan	khan	PROPN
ejpam-5460	659	13	.	.	PUNCT
ejpam-5460	660	1	fuzzy	fuzzy	ADJ
ejpam-5460	660	2	(	(	PUNCT
ejpam-5460	660	3	m	m	PROPN
ejpam-5460	660	4	,	,	PUNCT
ejpam-5460	660	5	n)-ideals	n)-ideal	NOUN
ejpam-5460	660	6	in	in	ADP
ejpam-5460	660	7	semigroups	semigroup	NOUN
ejpam-5460	660	8	.	.	PUNCT
ejpam-5460	661	1	computational	computational	ADJ
ejpam-5460	661	2	and	and	CCONJ
ejpam-5460	661	3	applied	applied	ADJ
ejpam-5460	661	4	mathematics	mathematic	NOUN
ejpam-5460	661	5	,	,	PUNCT
ejpam-5460	661	6	38:189	38:189	NUM
ejpam-5460	661	7	,	,	PUNCT
ejpam-5460	661	8	2019	2019	NUM
ejpam-5460	661	9	.	.	PUNCT
ejpam-5460	662	1	references	reference	NOUN
ejpam-5460	662	2	2984	2984	NUM
ejpam-5460	662	3	[	[	X
ejpam-5460	662	4	39	39	NUM
ejpam-5460	662	5	]	]	X
ejpam-5460	662	6	d	d	X
ejpam-5460	662	7	mandal	mandal	PROPN
ejpam-5460	662	8	.	.	PUNCT
ejpam-5460	663	1	fuzzy	fuzzy	ADJ
ejpam-5460	663	2	ideals	ideal	NOUN
ejpam-5460	663	3	and	and	CCONJ
ejpam-5460	663	4	fuzzy	fuzzy	ADJ
ejpam-5460	663	5	interior	interior	ADJ
ejpam-5460	663	6	ideals	ideal	NOUN
ejpam-5460	663	7	in	in	ADP
ejpam-5460	663	8	ordered	order	VERB
ejpam-5460	663	9	semirings	semiring	NOUN
ejpam-5460	663	10	.	.	PUNCT
ejpam-5460	664	1	fuzzy	fuzzy	ADJ
ejpam-5460	664	2	information	information	NOUN
ejpam-5460	664	3	and	and	CCONJ
ejpam-5460	664	4	engineering	engineering	NOUN
ejpam-5460	664	5	,	,	PUNCT
ejpam-5460	664	6	6:101–114	6:101–114	NUM
ejpam-5460	664	7	,	,	PUNCT
ejpam-5460	664	8	2014	2014	NUM
ejpam-5460	664	9	.	.	PUNCT
ejpam-5460	665	1	[	[	X
ejpam-5460	665	2	40	40	NUM
ejpam-5460	665	3	]	]	X
ejpam-5460	665	4	jn	jn	PROPN
ejpam-5460	665	5	mordeson	mordeson	PROPN
ejpam-5460	665	6	,	,	PUNCT
ejpam-5460	665	7	k	k	PROPN
ejpam-5460	665	8	bhutani	bhutani	NOUN
ejpam-5460	665	9	,	,	PUNCT
ejpam-5460	665	10	and	and	CCONJ
ejpam-5460	665	11	a	a	DET
ejpam-5460	665	12	rosenfeld	rosenfeld	NOUN
ejpam-5460	665	13	.	.	PUNCT
ejpam-5460	666	1	fuzzy	fuzzy	ADJ
ejpam-5460	666	2	group	group	PROPN
ejpam-5460	666	3	theory	theory	NOUN
ejpam-5460	666	4	.	.	PUNCT
ejpam-5460	667	1	springer	springer	NOUN
ejpam-5460	667	2	,	,	PUNCT
ejpam-5460	667	3	new	new	PROPN
ejpam-5460	667	4	york	york	PROPN
ejpam-5460	667	5	,	,	PUNCT
ejpam-5460	667	6	2005	2005	NUM
ejpam-5460	667	7	.	.	PUNCT
ejpam-5460	668	1	[	[	X
ejpam-5460	668	2	41	41	NUM
ejpam-5460	668	3	]	]	X
ejpam-5460	668	4	jn	jn	PROPN
ejpam-5460	668	5	mordeson	mordeson	PROPN
ejpam-5460	668	6	,	,	PUNCT
ejpam-5460	668	7	ds	ds	ADJ
ejpam-5460	668	8	malik	malik	NOUN
ejpam-5460	668	9	,	,	PUNCT
ejpam-5460	668	10	and	and	CCONJ
ejpam-5460	668	11	n	n	PRON
ejpam-5460	668	12	kuroki	kuroki	NOUN
ejpam-5460	668	13	.	.	PUNCT
ejpam-5460	669	1	fuzzy	fuzzy	ADJ
ejpam-5460	669	2	semigroups	semigroup	NOUN
ejpam-5460	669	3	.	.	PUNCT
ejpam-5460	670	1	springer	springer	NOUN
ejpam-5460	670	2	,	,	PUNCT
ejpam-5460	670	3	new	new	PROPN
ejpam-5460	670	4	york	york	PROPN
ejpam-5460	670	5	,	,	PUNCT
ejpam-5460	670	6	2012	2012	NUM
ejpam-5460	670	7	.	.	PUNCT
ejpam-5460	671	1	[	[	X
ejpam-5460	671	2	42	42	NUM
ejpam-5460	671	3	]	]	X
ejpam-5460	671	4	g	g	PROPN
ejpam-5460	671	5	muhiuddin	muhiuddin	PROPN
ejpam-5460	671	6	,	,	PUNCT
ejpam-5460	671	7	a	a	DET
ejpam-5460	671	8	mahboob	mahboob	NOUN
ejpam-5460	671	9	,	,	PUNCT
ejpam-5460	671	10	nm	nm	ADJ
ejpam-5460	671	11	khan	khan	PROPN
ejpam-5460	671	12	,	,	PUNCT
ejpam-5460	671	13	and	and	CCONJ
ejpam-5460	671	14	d	d	PROPN
ejpam-5460	671	15	al	al	PROPN
ejpam-5460	671	16	-	-	PUNCT
ejpam-5460	671	17	kadi	kadi	PROPN
ejpam-5460	671	18	.	.	PUNCT
ejpam-5460	672	1	new	new	ADJ
ejpam-5460	672	2	types	type	NOUN
ejpam-5460	672	3	of	of	ADP
ejpam-5460	672	4	fuzzy	fuzzy	ADJ
ejpam-5460	672	5	(	(	PUNCT
ejpam-5460	672	6	m	m	X
ejpam-5460	672	7	,	,	PUNCT
ejpam-5460	672	8	n)ideals	n)ideal	NOUN
ejpam-5460	672	9	in	in	ADP
ejpam-5460	672	10	ordered	order	VERB
ejpam-5460	672	11	semigroups	semigroup	NOUN
ejpam-5460	672	12	.	.	PUNCT
ejpam-5460	673	1	journal	journal	NOUN
ejpam-5460	673	2	of	of	ADP
ejpam-5460	673	3	intelligent	intelligent	ADJ
ejpam-5460	673	4	and	and	CCONJ
ejpam-5460	673	5	fuzzy	fuzzy	ADJ
ejpam-5460	673	6	systems	system	NOUN
ejpam-5460	673	7	,	,	PUNCT
ejpam-5460	673	8	41:6561–6574	41:6561–6574	NUM
ejpam-5460	673	9	,	,	PUNCT
ejpam-5460	673	10	2021	2021	NUM
ejpam-5460	673	11	.	.	PUNCT
ejpam-5460	674	1	[	[	X
ejpam-5460	674	2	43	43	NUM
ejpam-5460	674	3	]	]	X
ejpam-5460	674	4	t	t	X
ejpam-5460	674	5	phochai	phochai	NOUN
ejpam-5460	674	6	and	and	CCONJ
ejpam-5460	674	7	t	t	PROPN
ejpam-5460	674	8	changphas	changpha	NOUN
ejpam-5460	674	9	.	.	PUNCT
ejpam-5460	675	1	linear	linear	ADJ
ejpam-5460	675	2	inequations	inequation	NOUN
ejpam-5460	675	3	and	and	CCONJ
ejpam-5460	675	4	regularity	regularity	NOUN
ejpam-5460	675	5	conditions	condition	NOUN
ejpam-5460	675	6	on	on	ADP
ejpam-5460	675	7	ordered	order	VERB
ejpam-5460	675	8	semigroups	semigroup	NOUN
ejpam-5460	675	9	.	.	PUNCT
ejpam-5460	676	1	analele	analele	ADP
ejpam-5460	676	2	ştiinţifice	ştiinţifice	NOUN
ejpam-5460	676	3	ale	ale	NOUN
ejpam-5460	676	4	universităţii	universităţii	AUX
ejpam-5460	676	5	“	"	PUNCT
ejpam-5460	676	6	alexandru	alexandru	PROPN
ejpam-5460	676	7	ioan	ioan	PROPN
ejpam-5460	676	8	cuza	cuza	PROPN
ejpam-5460	676	9	”	"	PUNCT
ejpam-5460	676	10	din	din	VERB
ejpam-5460	676	11	iaşi	iaşi	PROPN
ejpam-5460	676	12	.	.	PUNCT
ejpam-5460	677	1	matematică	matematică	NOUN
ejpam-5460	677	2	(	(	PUNCT
ejpam-5460	677	3	serie	serie	VERB
ejpam-5460	677	4	nouă	nouă	PROPN
ejpam-5460	677	5	)	)	PUNCT
ejpam-5460	677	6	,	,	PUNCT
ejpam-5460	677	7	63:627–636	63:627–636	PROPN
ejpam-5460	677	8	,	,	PUNCT
ejpam-5460	677	9	2017	2017	NUM
ejpam-5460	677	10	.	.	PUNCT
ejpam-5460	678	1	[	[	X
ejpam-5460	678	2	44	44	NUM
ejpam-5460	678	3	]	]	X
ejpam-5460	678	4	j	j	PROPN
ejpam-5460	678	5	sanborisoot	sanborisoot	PROPN
ejpam-5460	678	6	and	and	CCONJ
ejpam-5460	678	7	t	t	PROPN
ejpam-5460	678	8	changphas	changpha	NOUN
ejpam-5460	678	9	.	.	PUNCT
ejpam-5460	679	1	on	on	ADP
ejpam-5460	679	2	characterizations	characterization	NOUN
ejpam-5460	679	3	of	of	ADP
ejpam-5460	679	4	(	(	PUNCT
ejpam-5460	679	5	m	m	PROPN
ejpam-5460	679	6	,	,	PUNCT
ejpam-5460	679	7	n)-regular	n)-regular	PRON
ejpam-5460	679	8	ordered	order	VERB
ejpam-5460	679	9	semigroup	semigroup	NOUN
ejpam-5460	679	10	.	.	PUNCT
ejpam-5460	680	1	far	far	PROPN
ejpam-5460	680	2	east	east	PROPN
ejpam-5460	680	3	journal	journal	PROPN
ejpam-5460	680	4	of	of	ADP
ejpam-5460	680	5	mathematical	mathematical	ADJ
ejpam-5460	680	6	sciences	science	NOUN
ejpam-5460	680	7	,	,	PUNCT
ejpam-5460	680	8	65:75–86	65:75–86	NUM
ejpam-5460	680	9	,	,	PUNCT
ejpam-5460	680	10	2012	2012	NUM
ejpam-5460	680	11	.	.	PUNCT
ejpam-5460	681	1	[	[	X
ejpam-5460	681	2	45	45	NUM
ejpam-5460	681	3	]	]	X
ejpam-5460	681	4	r	r	NOUN
ejpam-5460	681	5	sarita	sarita	PROPN
ejpam-5460	681	6	.	.	PUNCT
ejpam-5460	682	1	prime	prime	ADJ
ejpam-5460	682	2	and	and	CCONJ
ejpam-5460	682	3	semiprime	semiprime	NOUN
ejpam-5460	682	4	bi	bi	NOUN
ejpam-5460	682	5	-	-	NOUN
ejpam-5460	682	6	ideals	ideal	NOUN
ejpam-5460	682	7	in	in	ADP
ejpam-5460	682	8	ordered	order	VERB
ejpam-5460	682	9	semigroups	semigroup	NOUN
ejpam-5460	682	10	.	.	PUNCT
ejpam-5460	683	1	international	international	ADJ
ejpam-5460	683	2	journal	journal	NOUN
ejpam-5460	683	3	of	of	ADP
ejpam-5460	683	4	algebra	algebra	PROPN
ejpam-5460	683	5	,	,	PUNCT
ejpam-5460	683	6	7:839–845	7:839–845	NOUN
ejpam-5460	683	7	,	,	PUNCT
ejpam-5460	683	8	2013	2013	NUM
ejpam-5460	683	9	.	.	PUNCT
ejpam-5460	684	1	[	[	X
ejpam-5460	684	2	46	46	NUM
ejpam-5460	684	3	]	]	X
ejpam-5460	684	4	m	m	VERB
ejpam-5460	684	5	shabir	shabir	NOUN
ejpam-5460	684	6	and	and	CCONJ
ejpam-5460	684	7	a	a	DET
ejpam-5460	684	8	khan	khan	PROPN
ejpam-5460	684	9	.	.	PUNCT
ejpam-5460	685	1	characterizations	characterization	NOUN
ejpam-5460	685	2	of	of	ADP
ejpam-5460	685	3	ordered	order	VERB
ejpam-5460	685	4	semigroups	semigroup	NOUN
ejpam-5460	685	5	by	by	ADP
ejpam-5460	685	6	the	the	DET
ejpam-5460	685	7	properties	property	NOUN
ejpam-5460	685	8	of	of	ADP
ejpam-5460	685	9	their	their	PRON
ejpam-5460	685	10	fuzzy	fuzzy	ADJ
ejpam-5460	685	11	ideals	ideal	NOUN
ejpam-5460	685	12	.	.	PUNCT
ejpam-5460	686	1	computational	computational	ADJ
ejpam-5460	686	2	and	and	CCONJ
ejpam-5460	686	3	applied	applied	ADJ
ejpam-5460	686	4	mathematics	mathematic	NOUN
ejpam-5460	686	5	,	,	PUNCT
ejpam-5460	686	6	59:539–549	59:539–549	PROPN
ejpam-5460	686	7	,	,	PUNCT
ejpam-5460	686	8	2010	2010	NUM
ejpam-5460	686	9	.	.	PUNCT
ejpam-5460	687	1	[	[	X
ejpam-5460	687	2	47	47	NUM
ejpam-5460	687	3	]	]	X
ejpam-5460	687	4	j	j	PROPN
ejpam-5460	687	5	tang	tang	PROPN
ejpam-5460	687	6	and	and	CCONJ
ejpam-5460	687	7	xy	xy	PROPN
ejpam-5460	687	8	xie	xie	PROPN
ejpam-5460	687	9	.	.	PUNCT
ejpam-5460	688	1	on	on	ADP
ejpam-5460	688	2	(	(	PUNCT
ejpam-5460	688	3	∈,∈	∈,∈	X
ejpam-5460	688	4	∨qk)-fuzzy	∨qk)-fuzzy	ADJ
ejpam-5460	688	5	ideals	ideal	NOUN
ejpam-5460	688	6	of	of	ADP
ejpam-5460	688	7	ordered	order	VERB
ejpam-5460	688	8	semigroups	semigroup	NOUN
ejpam-5460	688	9	.	.	PUNCT
ejpam-5460	688	10	fuzzy	fuzzy	ADJ
ejpam-5460	688	11	information	information	NOUN
ejpam-5460	688	12	and	and	CCONJ
ejpam-5460	688	13	engineering	engineering	NOUN
ejpam-5460	688	14	,	,	PUNCT
ejpam-5460	688	15	5:57–67	5:57–67	NUM
ejpam-5460	688	16	,	,	PUNCT
ejpam-5460	688	17	2013	2013	NUM
ejpam-5460	688	18	.	.	PUNCT
ejpam-5460	689	1	[	[	X
ejpam-5460	689	2	48	48	NUM
ejpam-5460	689	3	]	]	PUNCT
ejpam-5460	689	4	n	n	DET
ejpam-5460	689	5	tiprachot	tiprachot	NOUN
ejpam-5460	689	6	,	,	PUNCT
ejpam-5460	689	7	n	n	DET
ejpam-5460	689	8	lekkoksung	lekkoksung	NOUN
ejpam-5460	689	9	,	,	PUNCT
ejpam-5460	689	10	and	and	CCONJ
ejpam-5460	689	11	b	b	X
ejpam-5460	689	12	pibaljommee	pibaljommee	NOUN
ejpam-5460	689	13	.	.	PUNCT
ejpam-5460	690	1	on	on	ADP
ejpam-5460	690	2	regularities	regularity	NOUN
ejpam-5460	690	3	of	of	ADP
ejpam-5460	690	4	ordered	order	VERB
ejpam-5460	690	5	semigroups	semigroup	NOUN
ejpam-5460	690	6	.	.	PUNCT
ejpam-5460	691	1	asian	asian	ADJ
ejpam-5460	691	2	-	-	PUNCT
ejpam-5460	691	3	european	european	ADJ
ejpam-5460	691	4	journal	journal	NOUN
ejpam-5460	691	5	of	of	ADP
ejpam-5460	691	6	mathematics	mathematic	NOUN
ejpam-5460	691	7	,	,	PUNCT
ejpam-5460	691	8	15:2250207	15:2250207	NOUN
ejpam-5460	691	9	,	,	PUNCT
ejpam-5460	691	10	2022	2022	NUM
ejpam-5460	691	11	.	.	PUNCT
ejpam-5460	692	1	[	[	X
ejpam-5460	692	2	49	49	NUM
ejpam-5460	692	3	]	]	PUNCT
ejpam-5460	692	4	n	n	PRON
ejpam-5460	692	5	tiprachot	tiprachot	NOUN
ejpam-5460	692	6	,	,	PUNCT
ejpam-5460	692	7	n	n	DET
ejpam-5460	692	8	lekkoksung	lekkoksung	NOUN
ejpam-5460	692	9	,	,	PUNCT
ejpam-5460	692	10	and	and	CCONJ
ejpam-5460	692	11	b	b	X
ejpam-5460	692	12	pibaljommee	pibaljommee	NOUN
ejpam-5460	692	13	.	.	PUNCT
ejpam-5460	693	1	regularities	regularity	NOUN
ejpam-5460	693	2	of	of	ADP
ejpam-5460	693	3	ordered	order	VERB
ejpam-5460	693	4	semigroups	semigroup	NOUN
ejpam-5460	693	5	in	in	ADP
ejpam-5460	693	6	terms	term	NOUN
ejpam-5460	693	7	of	of	ADP
ejpam-5460	693	8	(	(	PUNCT
ejpam-5460	693	9	m	m	PROPN
ejpam-5460	693	10	,	,	PUNCT
ejpam-5460	693	11	n)-ideals	n)-ideal	NOUN
ejpam-5460	693	12	and	and	CCONJ
ejpam-5460	693	13	n	n	CCONJ
ejpam-5460	693	14	-	-	ADJ
ejpam-5460	693	15	interior	interior	ADJ
ejpam-5460	693	16	ideals	ideal	NOUN
ejpam-5460	693	17	.	.	PUNCT
ejpam-5460	694	1	international	international	ADJ
ejpam-5460	694	2	journal	journal	NOUN
ejpam-5460	694	3	of	of	ADP
ejpam-5460	694	4	mathematics	mathematic	NOUN
ejpam-5460	694	5	and	and	CCONJ
ejpam-5460	694	6	computer	computer	NOUN
ejpam-5460	694	7	science	science	NOUN
ejpam-5460	694	8	,	,	PUNCT
ejpam-5460	694	9	17:723–730	17:723–730	NUM
ejpam-5460	694	10	,	,	PUNCT
ejpam-5460	694	11	2022	2022	NUM
ejpam-5460	694	12	.	.	PUNCT
ejpam-5460	695	1	[	[	X
ejpam-5460	695	2	50	50	NUM
ejpam-5460	695	3	]	]	PUNCT
ejpam-5460	695	4	xy	xy	PROPN
ejpam-5460	695	5	xie	xie	PROPN
ejpam-5460	695	6	and	and	CCONJ
ejpam-5460	695	7	j	j	PROPN
ejpam-5460	695	8	tang	tang	PROPN
ejpam-5460	695	9	.	.	PUNCT
ejpam-5460	696	1	regular	regular	ADJ
ejpam-5460	696	2	ordered	order	VERB
ejpam-5460	696	3	semigroups	semigroup	NOUN
ejpam-5460	696	4	and	and	CCONJ
ejpam-5460	696	5	intra	intra	ADJ
ejpam-5460	696	6	-	-	ADJ
ejpam-5460	696	7	regular	regular	ADJ
ejpam-5460	696	8	ordered	order	VERB
ejpam-5460	696	9	semigroups	semigroup	NOUN
ejpam-5460	696	10	in	in	ADP
ejpam-5460	696	11	terms	term	NOUN
ejpam-5460	696	12	of	of	ADP
ejpam-5460	696	13	fuzzy	fuzzy	ADJ
ejpam-5460	696	14	subsets	subset	NOUN
ejpam-5460	696	15	.	.	PUNCT
ejpam-5460	697	1	iranian	iranian	ADJ
ejpam-5460	697	2	journal	journal	PROPN
ejpam-5460	697	3	of	of	ADP
ejpam-5460	697	4	fuzzy	fuzzy	ADJ
ejpam-5460	697	5	systems	system	NOUN
ejpam-5460	697	6	,	,	PUNCT
ejpam-5460	697	7	7:121–140	7:121–140	NUM
ejpam-5460	697	8	,	,	PUNCT
ejpam-5460	697	9	2010	2010	NUM
ejpam-5460	697	10	.	.	PUNCT
ejpam-5460	698	1	[	[	X
ejpam-5460	698	2	51	51	NUM
ejpam-5460	698	3	]	]	X
ejpam-5460	698	4	la	la	PROPN
ejpam-5460	698	5	zadeh	zadeh	PROPN
ejpam-5460	698	6	.	.	PUNCT
ejpam-5460	698	7	fuzzy	fuzzy	ADJ
ejpam-5460	698	8	sets	set	NOUN
ejpam-5460	698	9	.	.	PUNCT
ejpam-5460	699	1	information	information	NOUN
ejpam-5460	699	2	and	and	CCONJ
ejpam-5460	699	3	control	control	NOUN
ejpam-5460	699	4	,	,	PUNCT
ejpam-5460	699	5	8:338–353	8:338–353	NUM
ejpam-5460	699	6	,	,	PUNCT
ejpam-5460	699	7	1965	1965	NUM
ejpam-5460	699	8	.	.	PUNCT
ejpam-5460	700	1	[	[	X
ejpam-5460	700	2	52	52	NUM
ejpam-5460	700	3	]	]	X
ejpam-5460	700	4	q	q	PROPN
ejpam-5460	700	5	zhu	zhu	PROPN
ejpam-5460	700	6	.	.	PUNCT
ejpam-5460	701	1	a	a	DET
ejpam-5460	701	2	note	note	NOUN
ejpam-5460	701	3	on	on	ADP
ejpam-5460	701	4	left	left	ADJ
ejpam-5460	701	5	(	(	PUNCT
ejpam-5460	701	6	right	right	ADJ
ejpam-5460	701	7	)	)	PUNCT
ejpam-5460	701	8	semiregular	semiregular	PROPN
ejpam-5460	701	9	po	po	PROPN
ejpam-5460	701	10	-	-	PUNCT
ejpam-5460	701	11	semigroups	semigroup	NOUN
ejpam-5460	701	12	.	.	PUNCT
ejpam-5460	702	1	far	far	PROPN
ejpam-5460	702	2	east	east	PROPN
ejpam-5460	702	3	journal	journal	PROPN
ejpam-5460	702	4	of	of	ADP
ejpam-5460	702	5	mathematical	mathematical	ADJ
ejpam-5460	702	6	sciences	science	NOUN
ejpam-5460	702	7	,	,	PUNCT
ejpam-5460	702	8	26:697–704	26:697–704	NUM
ejpam-5460	702	9	,	,	PUNCT
ejpam-5460	702	10	2007	2007	NUM
ejpam-5460	702	11	.	.	PUNCT