id	sid	tid	token	lemma	pos
ejpam-3504	1	1	european	european	PROPN
ejpam-3504	1	2	journal	journal	PROPN
ejpam-3504	1	3	of	of	ADP
ejpam-3504	1	4	pure	pure	ADJ
ejpam-3504	1	5	and	and	CCONJ
ejpam-3504	1	6	applied	apply	VERB
ejpam-3504	1	7	mathematics	mathematic	NOUN
ejpam-3504	1	8	vol	vol	NOUN
ejpam-3504	1	9	.	.	PROPN
ejpam-3504	2	1	12	12	NUM
ejpam-3504	2	2	,	,	PUNCT
ejpam-3504	2	3	no	no	INTJ
ejpam-3504	2	4	.	.	NOUN
ejpam-3504	2	5	3	3	NUM
ejpam-3504	2	6	,	,	PUNCT
ejpam-3504	2	7	2019	2019	NUM
ejpam-3504	2	8	,	,	PUNCT
ejpam-3504	3	1	709	709	NUM
ejpam-3504	3	2	-	-	SYM
ejpam-3504	3	3	721	721	NUM
ejpam-3504	3	4	issn	issn	PROPN
ejpam-3504	3	5	1307	1307	NUM
ejpam-3504	3	6	-	-	SYM
ejpam-3504	3	7	5543	5543	NUM
ejpam-3504	3	8	–	–	PUNCT
ejpam-3504	3	9	www.ejpam.com	www.ejpam.com	X
ejpam-3504	3	10	published	publish	VERB
ejpam-3504	3	11	by	by	ADP
ejpam-3504	3	12	new	new	PROPN
ejpam-3504	3	13	york	york	PROPN
ejpam-3504	3	14	business	business	PROPN
ejpam-3504	3	15	global	global	ADJ
ejpam-3504	3	16	on	on	ADP
ejpam-3504	3	17	fuzzy	fuzzy	ADJ
ejpam-3504	3	18	sets	set	NOUN
ejpam-3504	3	19	in	in	ADP
ejpam-3504	3	20	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	3	21	niovi	niovi	PROPN
ejpam-3504	3	22	kehayopulu	kehayopulu	ADJ
ejpam-3504	3	23	abstract	abstract	NOUN
ejpam-3504	3	24	.	.	PUNCT
ejpam-3504	4	1	the	the	DET
ejpam-3504	4	2	concept	concept	NOUN
ejpam-3504	4	3	of	of	ADP
ejpam-3504	4	4	a	a	DET
ejpam-3504	4	5	fuzzy	fuzzy	ADJ
ejpam-3504	4	6	set	set	NOUN
ejpam-3504	4	7	,	,	PUNCT
ejpam-3504	4	8	introduced	introduce	VERB
ejpam-3504	4	9	by	by	ADP
ejpam-3504	4	10	zadeh	zadeh	PROPN
ejpam-3504	4	11	,	,	PUNCT
ejpam-3504	4	12	was	be	AUX
ejpam-3504	4	13	first	first	ADV
ejpam-3504	4	14	applied	apply	VERB
ejpam-3504	4	15	by	by	ADP
ejpam-3504	4	16	rosenfeld	rosenfeld	PROPN
ejpam-3504	4	17	to	to	ADP
ejpam-3504	4	18	groups	group	NOUN
ejpam-3504	4	19	,	,	PUNCT
ejpam-3504	4	20	then	then	ADV
ejpam-3504	4	21	by	by	ADP
ejpam-3504	4	22	kuroki	kuroki	PROPN
ejpam-3504	4	23	to	to	ADP
ejpam-3504	4	24	semigroups	semigroup	NOUN
ejpam-3504	4	25	.	.	PUNCT
ejpam-3504	5	1	in	in	ADP
ejpam-3504	5	2	the	the	DET
ejpam-3504	5	3	present	present	ADJ
ejpam-3504	5	4	paper	paper	NOUN
ejpam-3504	5	5	it	it	PRON
ejpam-3504	5	6	is	be	AUX
ejpam-3504	5	7	applied	apply	VERB
ejpam-3504	5	8	to	to	ADP
ejpam-3504	5	9	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	5	10	,	,	PUNCT
ejpam-3504	5	11	and	and	CCONJ
ejpam-3504	5	12	some	some	DET
ejpam-3504	5	13	properties	property	NOUN
ejpam-3504	5	14	of	of	ADP
ejpam-3504	5	15	fuzzy	fuzzy	ADJ
ejpam-3504	5	16	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	5	17	are	be	AUX
ejpam-3504	5	18	discussed	discuss	VERB
ejpam-3504	5	19	.	.	PUNCT
ejpam-3504	6	1	2010	2010	NUM
ejpam-3504	6	2	mathematics	mathematic	NOUN
ejpam-3504	6	3	subject	subject	NOUN
ejpam-3504	6	4	classifications	classification	NOUN
ejpam-3504	6	5	:	:	PUNCT
ejpam-3504	6	6	20m99	20m99	NUM
ejpam-3504	6	7	,	,	PUNCT
ejpam-3504	6	8	08a72	08a72	NOUN
ejpam-3504	6	9	key	key	ADJ
ejpam-3504	6	10	words	word	NOUN
ejpam-3504	6	11	and	and	CCONJ
ejpam-3504	6	12	phrases	phrase	NOUN
ejpam-3504	6	13	:	:	PUNCT
ejpam-3504	6	14	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	6	15	,	,	PUNCT
ejpam-3504	6	16	fuzzy	fuzzy	ADJ
ejpam-3504	6	17	left	leave	VERB
ejpam-3504	6	18	duo	duo	NOUN
ejpam-3504	6	19	,	,	PUNCT
ejpam-3504	6	20	left	leave	VERB
ejpam-3504	6	21	ideal	ideal	ADJ
ejpam-3504	6	22	,	,	PUNCT
ejpam-3504	6	23	bi	bi	NOUN
ejpam-3504	6	24	-	-	ADJ
ejpam-3504	6	25	ideal	ideal	ADJ
ejpam-3504	6	26	,	,	PUNCT
ejpam-3504	6	27	fuzzy	fuzzy	ADJ
ejpam-3504	6	28	left	leave	VERB
ejpam-3504	6	29	ideal	ideal	ADJ
ejpam-3504	6	30	,	,	PUNCT
ejpam-3504	6	31	fuzzy	fuzzy	ADJ
ejpam-3504	6	32	bi	bi	NOUN
ejpam-3504	6	33	-	-	ADJ
ejpam-3504	6	34	ideal	ideal	ADJ
ejpam-3504	6	35	,	,	PUNCT
ejpam-3504	6	36	intra	intra	ADJ
ejpam-3504	6	37	-	-	ADJ
ejpam-3504	6	38	regular	regular	ADJ
ejpam-3504	6	39	,	,	PUNCT
ejpam-3504	6	40	left	leave	VERB
ejpam-3504	6	41	regular	regular	ADV
ejpam-3504	6	42	,	,	PUNCT
ejpam-3504	6	43	simple	simple	ADJ
ejpam-3504	6	44	1	1	NUM
ejpam-3504	6	45	.	.	PUNCT
ejpam-3504	7	1	introduction	introduction	NOUN
ejpam-3504	7	2	this	this	DET
ejpam-3504	7	3	note	note	NOUN
ejpam-3504	7	4	is	be	AUX
ejpam-3504	7	5	based	base	VERB
ejpam-3504	7	6	on	on	ADP
ejpam-3504	7	7	the	the	DET
ejpam-3504	7	8	papers	paper	NOUN
ejpam-3504	7	9	by	by	ADP
ejpam-3504	7	10	kuroki	kuroki	NOUN
ejpam-3504	7	11	[	[	X
ejpam-3504	7	12	7–8	7–8	X
ejpam-3504	7	13	]	]	X
ejpam-3504	7	14	and	and	CCONJ
ejpam-3504	7	15	its	its	PRON
ejpam-3504	7	16	aim	aim	NOUN
ejpam-3504	7	17	is	be	AUX
ejpam-3504	7	18	to	to	PART
ejpam-3504	7	19	show	show	VERB
ejpam-3504	7	20	the	the	DET
ejpam-3504	7	21	way	way	NOUN
ejpam-3504	7	22	we	we	PRON
ejpam-3504	7	23	pass	pass	VERB
ejpam-3504	7	24	from	from	ADP
ejpam-3504	7	25	fuzzy	fuzzy	ADJ
ejpam-3504	7	26	semigroups	semigroup	NOUN
ejpam-3504	7	27	to	to	ADP
ejpam-3504	7	28	fuzzy	fuzzy	ADJ
ejpam-3504	7	29	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	7	30	.	.	PUNCT
ejpam-3504	8	1	we	we	PRON
ejpam-3504	8	2	show	show	VERB
ejpam-3504	8	3	,	,	PUNCT
ejpam-3504	8	4	among	among	ADP
ejpam-3504	8	5	others	other	NOUN
ejpam-3504	8	6	,	,	PUNCT
ejpam-3504	8	7	that	that	SCONJ
ejpam-3504	8	8	a	a	DET
ejpam-3504	8	9	regular	regular	ADJ
ejpam-3504	8	10	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	8	11	is	be	AUX
ejpam-3504	8	12	left	leave	VERB
ejpam-3504	8	13	(	(	PUNCT
ejpam-3504	8	14	resp	resp	NOUN
ejpam-3504	8	15	.	.	PUNCT
ejpam-3504	9	1	right	right	ADJ
ejpam-3504	9	2	)	)	PUNCT
ejpam-3504	9	3	duo	duo	NOUN
ejpam-3504	9	4	if	if	SCONJ
ejpam-3504	9	5	and	and	CCONJ
ejpam-3504	9	6	only	only	ADV
ejpam-3504	9	7	if	if	SCONJ
ejpam-3504	9	8	it	it	PRON
ejpam-3504	9	9	is	be	AUX
ejpam-3504	9	10	fuzzy	fuzzy	ADJ
ejpam-3504	9	11	left	left	ADJ
ejpam-3504	9	12	(	(	PUNCT
ejpam-3504	9	13	resp	resp	NOUN
ejpam-3504	9	14	.	.	PUNCT
ejpam-3504	10	1	fuzzy	fuzzy	ADJ
ejpam-3504	10	2	right	right	ADJ
ejpam-3504	10	3	)	)	PUNCT
ejpam-3504	10	4	duo	duo	NOUN
ejpam-3504	10	5	.	.	PUNCT
ejpam-3504	11	1	in	in	ADP
ejpam-3504	11	2	a	a	DET
ejpam-3504	11	3	regular	regular	ADJ
ejpam-3504	11	4	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	11	5	,	,	PUNCT
ejpam-3504	11	6	every	every	DET
ejpam-3504	11	7	bi	bi	NOUN
ejpam-3504	11	8	-	-	ADJ
ejpam-3504	11	9	ideal	ideal	ADJ
ejpam-3504	11	10	is	be	AUX
ejpam-3504	11	11	a	a	DET
ejpam-3504	11	12	right	right	ADJ
ejpam-3504	11	13	(	(	PUNCT
ejpam-3504	11	14	resp	resp	NOUN
ejpam-3504	11	15	.	.	PUNCT
ejpam-3504	12	1	left	left	ADJ
ejpam-3504	12	2	)	)	PUNCT
ejpam-3504	12	3	ideal	ideal	NOUN
ejpam-3504	12	4	if	if	SCONJ
ejpam-3504	12	5	and	and	CCONJ
ejpam-3504	12	6	only	only	ADV
ejpam-3504	12	7	if	if	SCONJ
ejpam-3504	12	8	every	every	DET
ejpam-3504	12	9	fuzzy	fuzzy	ADJ
ejpam-3504	12	10	bi	bi	NOUN
ejpam-3504	12	11	-	-	ADJ
ejpam-3504	12	12	ideal	ideal	ADJ
ejpam-3504	12	13	is	be	AUX
ejpam-3504	12	14	a	a	DET
ejpam-3504	12	15	fuzzy	fuzzy	ADJ
ejpam-3504	12	16	right	right	NOUN
ejpam-3504	12	17	(	(	PUNCT
ejpam-3504	12	18	resp	resp	NOUN
ejpam-3504	12	19	.	.	PUNCT
ejpam-3504	13	1	fuzzy	fuzzy	ADJ
ejpam-3504	13	2	left	left	ADJ
ejpam-3504	13	3	)	)	PUNCT
ejpam-3504	13	4	ideal	ideal	ADJ
ejpam-3504	13	5	.	.	PUNCT
ejpam-3504	14	1	in	in	ADP
ejpam-3504	14	2	an	an	DET
ejpam-3504	14	3	intra	intra	ADJ
ejpam-3504	14	4	-	-	ADJ
ejpam-3504	14	5	regular	regular	ADJ
ejpam-3504	14	6	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	14	7	s	s	NOUN
ejpam-3504	14	8	for	for	ADP
ejpam-3504	14	9	every	every	DET
ejpam-3504	14	10	fuzzy	fuzzy	ADJ
ejpam-3504	14	11	ideal	ideal	NOUN
ejpam-3504	14	12	f	f	PROPN
ejpam-3504	14	13	of	of	ADP
ejpam-3504	14	14	s	s	PRON
ejpam-3504	14	15	and	and	CCONJ
ejpam-3504	14	16	any	any	DET
ejpam-3504	14	17	a	a	DET
ejpam-3504	14	18	∈	∈	NOUN
ejpam-3504	14	19	s	s	VERB
ejpam-3504	14	20	there	there	PRON
ejpam-3504	14	21	exists	exist	VERB
ejpam-3504	14	22	u	u	PROPN
ejpam-3504	14	23	∈	∈	PROPN
ejpam-3504	14	24	a	a	DET
ejpam-3504	14	25	◦	◦	NOUN
ejpam-3504	14	26	a	a	DET
ejpam-3504	14	27	such	such	ADJ
ejpam-3504	14	28	that	that	DET
ejpam-3504	14	29	f(a	f(a	NOUN
ejpam-3504	14	30	)	)	PUNCT
ejpam-3504	14	31	=	=	SYM
ejpam-3504	14	32	f(u	f(u	PROPN
ejpam-3504	14	33	)	)	PUNCT
ejpam-3504	14	34	.	.	PUNCT
ejpam-3504	15	1	“	"	PUNCT
ejpam-3504	15	2	conversely	conversely	ADV
ejpam-3504	15	3	”	"	PUNCT
ejpam-3504	15	4	if	if	SCONJ
ejpam-3504	15	5	s	s	PROPN
ejpam-3504	15	6	is	be	AUX
ejpam-3504	15	7	an	an	DET
ejpam-3504	15	8	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	15	9	such	such	ADJ
ejpam-3504	15	10	that	that	PRON
ejpam-3504	15	11	for	for	SCONJ
ejpam-3504	15	12	every	every	DET
ejpam-3504	15	13	fuzzy	fuzzy	ADJ
ejpam-3504	15	14	ideal	ideal	NOUN
ejpam-3504	15	15	f	f	PROPN
ejpam-3504	15	16	of	of	ADP
ejpam-3504	15	17	s	s	PROPN
ejpam-3504	15	18	,	,	PUNCT
ejpam-3504	15	19	any	any	DET
ejpam-3504	15	20	a	a	DET
ejpam-3504	15	21	∈	∈	NOUN
ejpam-3504	15	22	s	s	PART
ejpam-3504	15	23	and	and	CCONJ
ejpam-3504	15	24	any	any	DET
ejpam-3504	15	25	u	u	NOUN
ejpam-3504	15	26	∈	∈	PROPN
ejpam-3504	15	27	a	a	DET
ejpam-3504	15	28	◦	◦	NOUN
ejpam-3504	15	29	a	a	X
ejpam-3504	15	30	,	,	PUNCT
ejpam-3504	15	31	we	we	PRON
ejpam-3504	15	32	have	have	VERB
ejpam-3504	15	33	f(a	f(a	NOUN
ejpam-3504	15	34	)	)	PUNCT
ejpam-3504	15	35	=	=	SYM
ejpam-3504	15	36	f(u	f(u	PROPN
ejpam-3504	15	37	)	)	PUNCT
ejpam-3504	15	38	,	,	PUNCT
ejpam-3504	15	39	then	then	ADV
ejpam-3504	15	40	s	s	VERB
ejpam-3504	15	41	is	be	AUX
ejpam-3504	15	42	intra	intra	ADJ
ejpam-3504	15	43	-	-	ADJ
ejpam-3504	15	44	regular	regular	ADJ
ejpam-3504	15	45	.	.	PUNCT
ejpam-3504	16	1	if	if	SCONJ
ejpam-3504	16	2	s	s	NOUN
ejpam-3504	16	3	is	be	AUX
ejpam-3504	16	4	a	a	DET
ejpam-3504	16	5	left	left	ADJ
ejpam-3504	16	6	(	(	PUNCT
ejpam-3504	16	7	resp	resp	NOUN
ejpam-3504	16	8	.	.	PUNCT
ejpam-3504	17	1	right	right	ADJ
ejpam-3504	17	2	)	)	PUNCT
ejpam-3504	17	3	regular	regular	ADJ
ejpam-3504	17	4	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	17	5	,	,	PUNCT
ejpam-3504	17	6	then	then	ADV
ejpam-3504	17	7	for	for	ADP
ejpam-3504	17	8	every	every	DET
ejpam-3504	17	9	fuzzy	fuzzy	ADJ
ejpam-3504	17	10	left	left	NOUN
ejpam-3504	17	11	(	(	PUNCT
ejpam-3504	17	12	resp	resp	NOUN
ejpam-3504	17	13	.	.	PUNCT
ejpam-3504	18	1	fuzzy	fuzzy	ADJ
ejpam-3504	18	2	right	right	ADJ
ejpam-3504	18	3	)	)	PUNCT
ejpam-3504	18	4	ideal	ideal	PROPN
ejpam-3504	18	5	f	f	PROPN
ejpam-3504	18	6	of	of	ADP
ejpam-3504	18	7	s	s	PRON
ejpam-3504	18	8	and	and	CCONJ
ejpam-3504	18	9	any	any	DET
ejpam-3504	18	10	a	a	DET
ejpam-3504	18	11	∈	∈	NOUN
ejpam-3504	18	12	s	s	VERB
ejpam-3504	18	13	there	there	PRON
ejpam-3504	18	14	exists	exist	VERB
ejpam-3504	18	15	u	u	PROPN
ejpam-3504	18	16	∈	∈	PROPN
ejpam-3504	18	17	a	a	DET
ejpam-3504	18	18	◦	◦	NOUN
ejpam-3504	18	19	a	a	DET
ejpam-3504	18	20	such	such	ADJ
ejpam-3504	18	21	that	that	DET
ejpam-3504	18	22	f(a	f(a	NOUN
ejpam-3504	18	23	)	)	PUNCT
ejpam-3504	18	24	=	=	SYM
ejpam-3504	18	25	f(u	f(u	PROPN
ejpam-3504	18	26	)	)	PUNCT
ejpam-3504	18	27	.	.	PUNCT
ejpam-3504	19	1	“	"	PUNCT
ejpam-3504	19	2	conversely	conversely	ADV
ejpam-3504	19	3	”	"	PUNCT
ejpam-3504	19	4	if	if	SCONJ
ejpam-3504	19	5	s	s	PROPN
ejpam-3504	19	6	is	be	AUX
ejpam-3504	19	7	an	an	DET
ejpam-3504	19	8	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	19	9	such	such	ADJ
ejpam-3504	19	10	that	that	PRON
ejpam-3504	19	11	for	for	ADP
ejpam-3504	19	12	any	any	DET
ejpam-3504	19	13	fuzzy	fuzzy	ADJ
ejpam-3504	19	14	left	left	NOUN
ejpam-3504	19	15	(	(	PUNCT
ejpam-3504	19	16	resp	resp	NOUN
ejpam-3504	19	17	.	.	PUNCT
ejpam-3504	19	18	fuzzy	fuzzy	ADJ
ejpam-3504	19	19	right	right	ADJ
ejpam-3504	19	20	)	)	PUNCT
ejpam-3504	19	21	ideal	ideal	PROPN
ejpam-3504	19	22	f	f	PROPN
ejpam-3504	19	23	of	of	ADP
ejpam-3504	19	24	s	s	PROPN
ejpam-3504	19	25	,	,	PUNCT
ejpam-3504	19	26	any	any	DET
ejpam-3504	19	27	a	a	DET
ejpam-3504	19	28	∈	∈	NOUN
ejpam-3504	19	29	s	s	PART
ejpam-3504	19	30	and	and	CCONJ
ejpam-3504	19	31	any	any	DET
ejpam-3504	19	32	u	u	NOUN
ejpam-3504	19	33	∈	∈	PROPN
ejpam-3504	19	34	a	a	DET
ejpam-3504	19	35	◦	◦	NOUN
ejpam-3504	19	36	a	a	X
ejpam-3504	19	37	,	,	PUNCT
ejpam-3504	19	38	we	we	PRON
ejpam-3504	19	39	have	have	VERB
ejpam-3504	19	40	f(a	f(a	NOUN
ejpam-3504	19	41	)	)	PUNCT
ejpam-3504	20	1	=	=	SYM
ejpam-3504	20	2	f(u	f(u	PROPN
ejpam-3504	20	3	)	)	PUNCT
ejpam-3504	20	4	,	,	PUNCT
ejpam-3504	20	5	then	then	ADV
ejpam-3504	20	6	s	s	VERB
ejpam-3504	20	7	is	be	AUX
ejpam-3504	20	8	left	leave	VERB
ejpam-3504	20	9	(	(	PUNCT
ejpam-3504	20	10	resp	resp	NOUN
ejpam-3504	20	11	.	.	PUNCT
ejpam-3504	21	1	right	right	ADJ
ejpam-3504	21	2	)	)	PUNCT
ejpam-3504	21	3	regular	regular	ADV
ejpam-3504	21	4	.	.	PUNCT
ejpam-3504	22	1	an	an	DET
ejpam-3504	22	2	hypergroupoid	hypergroupoid	NOUN
ejpam-3504	22	3	is	be	AUX
ejpam-3504	22	4	left	leave	VERB
ejpam-3504	22	5	(	(	PUNCT
ejpam-3504	22	6	resp	resp	NOUN
ejpam-3504	22	7	.	.	PUNCT
ejpam-3504	23	1	right	right	ADJ
ejpam-3504	23	2	)	)	PUNCT
ejpam-3504	23	3	simple	simple	ADJ
ejpam-3504	23	4	if	if	SCONJ
ejpam-3504	24	1	and	and	CCONJ
ejpam-3504	24	2	only	only	ADV
ejpam-3504	24	3	if	if	SCONJ
ejpam-3504	24	4	it	it	PRON
ejpam-3504	24	5	is	be	AUX
ejpam-3504	24	6	fuzzy	fuzzy	ADJ
ejpam-3504	24	7	left	left	ADJ
ejpam-3504	24	8	(	(	PUNCT
ejpam-3504	24	9	resp	resp	NOUN
ejpam-3504	24	10	.	.	PUNCT
ejpam-3504	25	1	fuzzy	fuzzy	ADJ
ejpam-3504	25	2	right	right	ADJ
ejpam-3504	25	3	)	)	PUNCT
ejpam-3504	25	4	simple	simple	ADJ
ejpam-3504	26	1	and	and	CCONJ
ejpam-3504	26	2	so	so	ADV
ejpam-3504	26	3	it	it	PRON
ejpam-3504	26	4	is	be	AUX
ejpam-3504	26	5	simple	simple	ADJ
ejpam-3504	26	6	if	if	SCONJ
ejpam-3504	27	1	and	and	CCONJ
ejpam-3504	27	2	only	only	ADV
ejpam-3504	27	3	if	if	SCONJ
ejpam-3504	27	4	it	it	PRON
ejpam-3504	27	5	is	be	AUX
ejpam-3504	27	6	fuzzy	fuzzy	ADJ
ejpam-3504	27	7	simple	simple	ADJ
ejpam-3504	27	8	.	.	PUNCT
ejpam-3504	28	1	the	the	DET
ejpam-3504	28	2	simple	simple	ADJ
ejpam-3504	28	3	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	28	4	are	be	AUX
ejpam-3504	28	5	regular	regular	ADJ
ejpam-3504	28	6	and	and	CCONJ
ejpam-3504	28	7	intra	intra	ADJ
ejpam-3504	28	8	-	-	ADJ
ejpam-3504	28	9	regular	regular	ADJ
ejpam-3504	28	10	.	.	PUNCT
ejpam-3504	29	1	the	the	DET
ejpam-3504	29	2	left	left	ADJ
ejpam-3504	29	3	(	(	PUNCT
ejpam-3504	29	4	resp	resp	NOUN
ejpam-3504	29	5	.	.	PUNCT
ejpam-3504	30	1	right	right	ADJ
ejpam-3504	30	2	)	)	PUNCT
ejpam-3504	30	3	simple	simple	ADJ
ejpam-3504	30	4	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	30	5	are	be	AUX
ejpam-3504	30	6	left	leave	VERB
ejpam-3504	30	7	(	(	PUNCT
ejpam-3504	30	8	resp	resp	NOUN
ejpam-3504	30	9	.	.	PUNCT
ejpam-3504	31	1	right	right	ADJ
ejpam-3504	31	2	)	)	PUNCT
ejpam-3504	31	3	regular	regular	ADJ
ejpam-3504	31	4	.	.	PUNCT
ejpam-3504	32	1	finally	finally	ADV
ejpam-3504	32	2	,	,	PUNCT
ejpam-3504	32	3	in	in	ADP
ejpam-3504	32	4	left	left	ADJ
ejpam-3504	32	5	(	(	PUNCT
ejpam-3504	32	6	resp	resp	NOUN
ejpam-3504	32	7	.	.	PUNCT
ejpam-3504	33	1	right	right	ADJ
ejpam-3504	33	2	)	)	PUNCT
ejpam-3504	33	3	simple	simple	ADJ
ejpam-3504	33	4	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	33	5	,	,	PUNCT
ejpam-3504	33	6	every	every	DET
ejpam-3504	33	7	fuzzy	fuzzy	ADJ
ejpam-3504	33	8	bi	bi	NOUN
ejpam-3504	33	9	-	-	ADJ
ejpam-3504	33	10	ideal	ideal	ADJ
ejpam-3504	33	11	is	be	AUX
ejpam-3504	33	12	a	a	DET
ejpam-3504	33	13	fuzzy	fuzzy	ADJ
ejpam-3504	33	14	right	right	NOUN
ejpam-3504	33	15	(	(	PUNCT
ejpam-3504	33	16	resp	resp	NOUN
ejpam-3504	33	17	.	.	PUNCT
ejpam-3504	34	1	fuzzy	fuzzy	ADJ
ejpam-3504	34	2	left	left	ADJ
ejpam-3504	34	3	)	)	PUNCT
ejpam-3504	34	4	ideal	ideal	ADJ
ejpam-3504	34	5	.	.	PUNCT
ejpam-3504	35	1	as	as	ADP
ejpam-3504	35	2	a	a	DET
ejpam-3504	35	3	consequence	consequence	NOUN
ejpam-3504	35	4	,	,	PUNCT
ejpam-3504	35	5	in	in	ADP
ejpam-3504	35	6	a	a	DET
ejpam-3504	35	7	left	left	ADJ
ejpam-3504	35	8	(	(	PUNCT
ejpam-3504	35	9	resp	resp	NOUN
ejpam-3504	35	10	.	.	PUNCT
ejpam-3504	36	1	right	right	ADJ
ejpam-3504	36	2	)	)	PUNCT
ejpam-3504	36	3	simple	simple	ADJ
ejpam-3504	36	4	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	36	5	,	,	PUNCT
ejpam-3504	36	6	every	every	DET
ejpam-3504	36	7	bi	bi	NOUN
ejpam-3504	36	8	-	-	ADJ
ejpam-3504	36	9	ideal	ideal	ADJ
ejpam-3504	36	10	is	be	AUX
ejpam-3504	36	11	a	a	DET
ejpam-3504	36	12	right	right	ADJ
ejpam-3504	36	13	(	(	PUNCT
ejpam-3504	36	14	resp	resp	NOUN
ejpam-3504	36	15	.	.	PUNCT
ejpam-3504	37	1	left	left	ADJ
ejpam-3504	37	2	)	)	PUNCT
ejpam-3504	37	3	ideal	ideal	ADJ
ejpam-3504	37	4	.	.	PUNCT
ejpam-3504	38	1	doi	doi	NOUN
ejpam-3504	38	2	:	:	PUNCT
ejpam-3504	38	3	https://doi.org/10.29020/nybg.ejpam.v12i3.3504	https://doi.org/10.29020/nybg.ejpam.v12i3.3504	VERB
ejpam-3504	38	4	email	email	NOUN
ejpam-3504	38	5	address	address	NOUN
ejpam-3504	38	6	:	:	PUNCT
ejpam-3504	38	7	nkehayop@math.uoa.gr	nkehayop@math.uoa.gr	ADV
ejpam-3504	38	8	(	(	PUNCT
ejpam-3504	38	9	n.	n.	PROPN
ejpam-3504	38	10	kehayopulu	kehayopulu	PROPN
ejpam-3504	38	11	)	)	PUNCT
ejpam-3504	38	12	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3504	39	1	709	709	NUM
ejpam-3504	39	2	c	c	X
ejpam-3504	39	3	©	©	PROPN
ejpam-3504	39	4	2019	2019	NUM
ejpam-3504	39	5	ejpam	ejpam	NOUN
ejpam-3504	39	6	all	all	DET
ejpam-3504	39	7	rights	right	NOUN
ejpam-3504	39	8	reserved	reserve	VERB
ejpam-3504	39	9	.	.	PUNCT
ejpam-3504	40	1	n.	n.	PROPN
ejpam-3504	40	2	kehayopulu	kehayopulu	PROPN
ejpam-3504	40	3	/	/	SYM
ejpam-3504	40	4	eur	eur	PROPN
ejpam-3504	40	5	.	.	PUNCT
ejpam-3504	41	1	j.	j.	PROPN
ejpam-3504	41	2	pure	pure	PROPN
ejpam-3504	41	3	appl	appl	PROPN
ejpam-3504	41	4	.	.	PROPN
ejpam-3504	41	5	math	math	PROPN
ejpam-3504	41	6	,	,	PUNCT
ejpam-3504	41	7	12	12	NUM
ejpam-3504	41	8	(	(	PUNCT
ejpam-3504	41	9	3	3	NUM
ejpam-3504	41	10	)	)	PUNCT
ejpam-3504	41	11	(	(	PUNCT
ejpam-3504	41	12	2019	2019	NUM
ejpam-3504	41	13	)	)	PUNCT
ejpam-3504	41	14	,	,	PUNCT
ejpam-3504	41	15	709	709	NUM
ejpam-3504	41	16	-	-	SYM
ejpam-3504	41	17	721	721	NUM
ejpam-3504	41	18	710	710	NUM
ejpam-3504	41	19	2	2	NUM
ejpam-3504	41	20	.	.	PUNCT
ejpam-3504	41	21	necessary	necessary	ADJ
ejpam-3504	41	22	definitions	definition	NOUN
ejpam-3504	41	23	for	for	ADP
ejpam-3504	41	24	the	the	DET
ejpam-3504	41	25	sake	sake	NOUN
ejpam-3504	41	26	of	of	ADP
ejpam-3504	41	27	completeness	completeness	NOUN
ejpam-3504	41	28	,	,	PUNCT
ejpam-3504	41	29	we	we	PRON
ejpam-3504	41	30	give	give	VERB
ejpam-3504	41	31	the	the	DET
ejpam-3504	41	32	following	follow	VERB
ejpam-3504	41	33	definitions	definition	NOUN
ejpam-3504	41	34	:	:	PUNCT
ejpam-3504	41	35	p∗(s	p∗(s	X
ejpam-3504	41	36	)	)	PUNCT
ejpam-3504	41	37	denotes	denote	VERB
ejpam-3504	41	38	the	the	DET
ejpam-3504	41	39	set	set	NOUN
ejpam-3504	41	40	of	of	ADP
ejpam-3504	41	41	(	(	PUNCT
ejpam-3504	41	42	all	all	ADV
ejpam-3504	41	43	)	)	PUNCT
ejpam-3504	41	44	nonempty	nonempty	VERB
ejpam-3504	41	45	subsets	subset	NOUN
ejpam-3504	41	46	of	of	ADP
ejpam-3504	41	47	s.	s.	PROPN
ejpam-3504	41	48	an	an	DET
ejpam-3504	41	49	hypergroupoid	hypergroupoid	PROPN
ejpam-3504	41	50	is	be	AUX
ejpam-3504	41	51	a	a	DET
ejpam-3504	41	52	nonempty	nonempty	ADJ
ejpam-3504	41	53	set	set	VERB
ejpam-3504	41	54	s	s	NOUN
ejpam-3504	41	55	with	with	ADP
ejpam-3504	41	56	an	an	DET
ejpam-3504	41	57	“	"	PUNCT
ejpam-3504	41	58	operation	operation	NOUN
ejpam-3504	41	59	”	"	PUNCT
ejpam-3504	41	60	◦	◦	NOUN
ejpam-3504	41	61	:	:	PUNCT
ejpam-3504	41	62	s	s	VERB
ejpam-3504	41	63	×	×	PROPN
ejpam-3504	41	64	s	s	X
ejpam-3504	41	65	→	→	SYM
ejpam-3504	41	66	p∗(s	p∗(s	NOUN
ejpam-3504	41	67	)	)	PUNCT
ejpam-3504	41	68	|	|	NOUN
ejpam-3504	41	69	(	(	PUNCT
ejpam-3504	41	70	a	a	PRON
ejpam-3504	41	71	,	,	PUNCT
ejpam-3504	41	72	b)→	b)→	VERB
ejpam-3504	41	73	a	a	DET
ejpam-3504	41	74	◦	◦	NOUN
ejpam-3504	41	75	b	b	NOUN
ejpam-3504	41	76	on	on	ADP
ejpam-3504	41	77	s	s	NOUN
ejpam-3504	41	78	called	call	VERB
ejpam-3504	41	79	hyperoperation	hyperoperation	NOUN
ejpam-3504	41	80	(	(	PUNCT
ejpam-3504	41	81	as	as	SCONJ
ejpam-3504	41	82	it	it	PRON
ejpam-3504	41	83	assigns	assign	VERB
ejpam-3504	41	84	to	to	ADP
ejpam-3504	41	85	each	each	DET
ejpam-3504	41	86	couple	couple	NOUN
ejpam-3504	41	87	(	(	PUNCT
ejpam-3504	41	88	a	a	DET
ejpam-3504	41	89	,	,	PUNCT
ejpam-3504	41	90	b	b	NOUN
ejpam-3504	41	91	)	)	PUNCT
ejpam-3504	41	92	a	a	DET
ejpam-3504	41	93	nonempty	nonempty	NOUN
ejpam-3504	41	94	subset	subset	NOUN
ejpam-3504	41	95	of	of	ADP
ejpam-3504	41	96	s	s	NOUN
ejpam-3504	41	97	)	)	PUNCT
ejpam-3504	41	98	.	.	PUNCT
ejpam-3504	42	1	the	the	DET
ejpam-3504	42	2	hyperoperation	hyperoperation	NOUN
ejpam-3504	42	3	“	"	PUNCT
ejpam-3504	42	4	◦	◦	NOUN
ejpam-3504	42	5	”	"	PUNCT
ejpam-3504	42	6	induces	induce	VERB
ejpam-3504	42	7	an	an	DET
ejpam-3504	42	8	operation	operation	NOUN
ejpam-3504	42	9	on	on	ADP
ejpam-3504	42	10	p∗(s	p∗(s	NOUN
ejpam-3504	42	11	)	)	PUNCT
ejpam-3504	42	12	defined	define	VERB
ejpam-3504	42	13	by	by	ADP
ejpam-3504	42	14	a	a	DET
ejpam-3504	42	15	∗b	∗b	NOUN
ejpam-3504	42	16	=	=	SYM
ejpam-3504	42	17	⋃	⋃	NOUN
ejpam-3504	42	18	a∈a	a∈a	ADJ
ejpam-3504	42	19	,	,	PUNCT
ejpam-3504	42	20	b∈b	b∈b	VERB
ejpam-3504	42	21	a	a	DET
ejpam-3504	42	22	◦	◦	NOUN
ejpam-3504	42	23	b.	b.	NOUN
ejpam-3504	42	24	clearly	clearly	ADV
ejpam-3504	42	25	a	a	DET
ejpam-3504	42	26	∈	∈	PROPN
ejpam-3504	42	27	a	a	PRON
ejpam-3504	42	28	,	,	PUNCT
ejpam-3504	42	29	b	b	X
ejpam-3504	42	30	∈	∈	PROPN
ejpam-3504	42	31	b	b	PROPN
ejpam-3504	42	32	implies	imply	VERB
ejpam-3504	42	33	a	a	DET
ejpam-3504	42	34	◦	◦	NOUN
ejpam-3504	42	35	b	b	NOUN
ejpam-3504	42	36	⊆	⊆	NUM
ejpam-3504	42	37	a	a	DET
ejpam-3504	42	38	∗b	∗b	NOUN
ejpam-3504	42	39	;	;	PUNCT
ejpam-3504	42	40	and	and	CCONJ
ejpam-3504	42	41	if	if	SCONJ
ejpam-3504	42	42	x	x	SYM
ejpam-3504	42	43	∈	∈	PROPN
ejpam-3504	42	44	a	a	DET
ejpam-3504	42	45	∗b	∗b	NOUN
ejpam-3504	42	46	,	,	PUNCT
ejpam-3504	42	47	then	then	ADV
ejpam-3504	42	48	x	x	PART
ejpam-3504	42	49	∈	∈	PROPN
ejpam-3504	42	50	a	a	DET
ejpam-3504	42	51	◦	◦	NOUN
ejpam-3504	42	52	b	b	NOUN
ejpam-3504	42	53	for	for	ADP
ejpam-3504	42	54	some	some	DET
ejpam-3504	42	55	a	a	DET
ejpam-3504	42	56	∈	∈	PROPN
ejpam-3504	42	57	a	a	PRON
ejpam-3504	42	58	,	,	PUNCT
ejpam-3504	42	59	b	b	PROPN
ejpam-3504	42	60	∈	∈	PROPN
ejpam-3504	42	61	b.	b.	PROPN
ejpam-3504	42	62	also	also	ADV
ejpam-3504	42	63	,	,	PUNCT
ejpam-3504	42	64	for	for	ADP
ejpam-3504	42	65	any	any	DET
ejpam-3504	42	66	nonempty	nonempty	ADJ
ejpam-3504	42	67	subsets	subset	NOUN
ejpam-3504	42	68	a	a	DET
ejpam-3504	42	69	,	,	PUNCT
ejpam-3504	42	70	b	b	NOUN
ejpam-3504	42	71	,	,	PUNCT
ejpam-3504	42	72	c	c	PROPN
ejpam-3504	42	73	of	of	ADP
ejpam-3504	42	74	an	an	DET
ejpam-3504	42	75	hypergroupoid	hypergroupoid	PROPN
ejpam-3504	42	76	s	s	PROPN
ejpam-3504	42	77	,	,	PUNCT
ejpam-3504	42	78	a	a	DET
ejpam-3504	42	79	⊆	⊆	NUM
ejpam-3504	42	80	b	b	NOUN
ejpam-3504	42	81	implies	imply	VERB
ejpam-3504	42	82	a	a	DET
ejpam-3504	42	83	∗	∗	NOUN
ejpam-3504	42	84	c	c	NOUN
ejpam-3504	42	85	⊆	⊆	NUM
ejpam-3504	42	86	b	b	NOUN
ejpam-3504	42	87	∗	∗	NOUN
ejpam-3504	42	88	c	c	NOUN
ejpam-3504	42	89	and	and	CCONJ
ejpam-3504	42	90	c	c	PROPN
ejpam-3504	42	91	∗a	∗a	PROPN
ejpam-3504	42	92	⊆	⊆	PROPN
ejpam-3504	42	93	c	c	PROPN
ejpam-3504	42	94	∗b	∗b	PROPN
ejpam-3504	42	95	.	.	PUNCT
ejpam-3504	43	1	recall	recall	VERB
ejpam-3504	43	2	that	that	SCONJ
ejpam-3504	43	3	{	{	PUNCT
ejpam-3504	43	4	x	x	NOUN
ejpam-3504	43	5	}	}	PUNCT
ejpam-3504	43	6	∗	∗	NOUN
ejpam-3504	43	7	{	{	PUNCT
ejpam-3504	43	8	y	y	NOUN
ejpam-3504	43	9	}	}	PUNCT
ejpam-3504	43	10	=	=	SYM
ejpam-3504	43	11	x	x	PUNCT
ejpam-3504	43	12	◦	◦	NOUN
ejpam-3504	43	13	y	y	NOUN
ejpam-3504	43	14	for	for	ADP
ejpam-3504	43	15	every	every	DET
ejpam-3504	43	16	x	x	PROPN
ejpam-3504	43	17	,	,	PUNCT
ejpam-3504	43	18	y	y	PROPN
ejpam-3504	43	19	∈	∈	PROPN
ejpam-3504	43	20	s.	s.	PROPN
ejpam-3504	43	21	a	a	DET
ejpam-3504	43	22	nonempty	nonempty	NOUN
ejpam-3504	43	23	subset	subset	VERB
ejpam-3504	43	24	a	a	PRON
ejpam-3504	43	25	of	of	ADP
ejpam-3504	43	26	an	an	DET
ejpam-3504	43	27	hypergroupoid	hypergroupoid	NOUN
ejpam-3504	43	28	(	(	PUNCT
ejpam-3504	43	29	s	s	NOUN
ejpam-3504	43	30	,	,	PUNCT
ejpam-3504	43	31	◦	◦	NOUN
ejpam-3504	43	32	)	)	PUNCT
ejpam-3504	43	33	is	be	AUX
ejpam-3504	43	34	called	call	VERB
ejpam-3504	43	35	a	a	DET
ejpam-3504	43	36	subgroupoid	subgroupoid	NOUN
ejpam-3504	43	37	of	of	ADP
ejpam-3504	43	38	s	s	PROPN
ejpam-3504	43	39	is	be	AUX
ejpam-3504	43	40	a∗a	a∗a	NUM
ejpam-3504	43	41	⊆	⊆	NUM
ejpam-3504	43	42	a	a	PRON
ejpam-3504	43	43	;	;	PUNCT
ejpam-3504	43	44	that	that	PRON
ejpam-3504	43	45	is	be	AUX
ejpam-3504	43	46	if	if	SCONJ
ejpam-3504	43	47	a	a	PRON
ejpam-3504	43	48	,	,	PUNCT
ejpam-3504	43	49	b	b	X
ejpam-3504	43	50	∈	∈	PROPN
ejpam-3504	43	51	a	a	PRON
ejpam-3504	43	52	,	,	PUNCT
ejpam-3504	43	53	then	then	ADV
ejpam-3504	43	54	a	a	DET
ejpam-3504	43	55	◦	◦	NOUN
ejpam-3504	43	56	b	b	NOUN
ejpam-3504	43	57	⊆	⊆	NUM
ejpam-3504	43	58	a.	a.	NOUN
ejpam-3504	43	59	a	a	DET
ejpam-3504	43	60	nonempty	nonempty	NOUN
ejpam-3504	43	61	subset	subset	VERB
ejpam-3504	43	62	a	a	PRON
ejpam-3504	43	63	of	of	ADP
ejpam-3504	43	64	an	an	DET
ejpam-3504	43	65	hypergroupoid	hypergroupoid	NOUN
ejpam-3504	43	66	(	(	PUNCT
ejpam-3504	43	67	s	s	NOUN
ejpam-3504	43	68	,	,	PUNCT
ejpam-3504	43	69	◦	◦	NOUN
ejpam-3504	43	70	)	)	PUNCT
ejpam-3504	43	71	is	be	AUX
ejpam-3504	43	72	called	call	VERB
ejpam-3504	43	73	a	a	DET
ejpam-3504	43	74	left	left	ADJ
ejpam-3504	43	75	ideal	ideal	NOUN
ejpam-3504	43	76	of	of	ADP
ejpam-3504	43	77	s	s	PRON
ejpam-3504	43	78	if	if	SCONJ
ejpam-3504	43	79	s	s	PART
ejpam-3504	43	80	∗	∗	NOUN
ejpam-3504	43	81	a	a	DET
ejpam-3504	43	82	⊆	⊆	NUM
ejpam-3504	43	83	a	a	NOUN
ejpam-3504	43	84	;	;	PUNCT
ejpam-3504	43	85	that	that	PRON
ejpam-3504	43	86	is	be	AUX
ejpam-3504	43	87	if	if	SCONJ
ejpam-3504	43	88	u	u	PROPN
ejpam-3504	43	89	∈	∈	PROPN
ejpam-3504	43	90	s	s	PART
ejpam-3504	43	91	◦	◦	NOUN
ejpam-3504	43	92	a	a	PRON
ejpam-3504	43	93	for	for	ADP
ejpam-3504	43	94	some	some	DET
ejpam-3504	43	95	s	s	PART
ejpam-3504	43	96	∈	∈	PROPN
ejpam-3504	43	97	s	s	NOUN
ejpam-3504	43	98	,	,	PUNCT
ejpam-3504	43	99	a	a	DET
ejpam-3504	43	100	∈	∈	PROPN
ejpam-3504	43	101	a	a	PRON
ejpam-3504	43	102	,	,	PUNCT
ejpam-3504	43	103	then	then	ADV
ejpam-3504	43	104	u	u	PROPN
ejpam-3504	43	105	∈	∈	PROPN
ejpam-3504	43	106	a.	a.	NOUN
ejpam-3504	43	107	it	it	PRON
ejpam-3504	43	108	is	be	AUX
ejpam-3504	43	109	called	call	VERB
ejpam-3504	43	110	a	a	DET
ejpam-3504	43	111	right	right	ADJ
ejpam-3504	43	112	ideal	ideal	NOUN
ejpam-3504	43	113	of	of	ADP
ejpam-3504	43	114	s	s	PRON
ejpam-3504	43	115	if	if	SCONJ
ejpam-3504	43	116	a	a	DET
ejpam-3504	43	117	∗s	∗s	NOUN
ejpam-3504	43	118	⊆	⊆	NUM
ejpam-3504	43	119	a.	a.	NOUN
ejpam-3504	43	120	a	a	DET
ejpam-3504	43	121	subset	subset	NOUN
ejpam-3504	43	122	of	of	ADP
ejpam-3504	43	123	s	s	PRON
ejpam-3504	43	124	that	that	PRON
ejpam-3504	43	125	is	be	AUX
ejpam-3504	43	126	a	a	DET
ejpam-3504	43	127	both	both	CCONJ
ejpam-3504	43	128	a	a	DET
ejpam-3504	43	129	right	right	NOUN
ejpam-3504	43	130	and	and	CCONJ
ejpam-3504	43	131	a	a	DET
ejpam-3504	43	132	left	left	ADJ
ejpam-3504	43	133	ideal	ideal	NOUN
ejpam-3504	43	134	of	of	ADP
ejpam-3504	43	135	s	s	PRON
ejpam-3504	43	136	is	be	AUX
ejpam-3504	43	137	called	call	VERB
ejpam-3504	43	138	an	an	DET
ejpam-3504	43	139	ideal	ideal	NOUN
ejpam-3504	43	140	of	of	ADP
ejpam-3504	43	141	s.	s.	PROPN
ejpam-3504	43	142	an	an	DET
ejpam-3504	43	143	hypergroupoid	hypergroupoid	PROPN
ejpam-3504	43	144	s	s	PROPN
ejpam-3504	43	145	is	be	AUX
ejpam-3504	43	146	called	call	VERB
ejpam-3504	43	147	left	left	ADJ
ejpam-3504	43	148	(	(	PUNCT
ejpam-3504	43	149	resp	resp	NOUN
ejpam-3504	43	150	.	.	PUNCT
ejpam-3504	44	1	right	right	ADJ
ejpam-3504	44	2	)	)	PUNCT
ejpam-3504	44	3	duo	duo	NOUN
ejpam-3504	44	4	if	if	SCONJ
ejpam-3504	44	5	every	every	DET
ejpam-3504	44	6	left	left	NOUN
ejpam-3504	44	7	(	(	PUNCT
ejpam-3504	44	8	resp	resp	NOUN
ejpam-3504	44	9	.	.	PUNCT
ejpam-3504	45	1	right	right	ADJ
ejpam-3504	45	2	)	)	PUNCT
ejpam-3504	45	3	ideal	ideal	NOUN
ejpam-3504	45	4	of	of	ADP
ejpam-3504	45	5	s	s	PROPN
ejpam-3504	45	6	is	be	AUX
ejpam-3504	45	7	an	an	DET
ejpam-3504	45	8	ideal	ideal	NOUN
ejpam-3504	45	9	of	of	ADP
ejpam-3504	45	10	s.	s.	PROPN
ejpam-3504	45	11	it	it	PRON
ejpam-3504	45	12	is	be	AUX
ejpam-3504	45	13	called	call	VERB
ejpam-3504	45	14	duo	duo	NOUN
ejpam-3504	45	15	if	if	SCONJ
ejpam-3504	45	16	it	it	PRON
ejpam-3504	45	17	is	be	AUX
ejpam-3504	45	18	both	both	PRON
ejpam-3504	45	19	left	leave	VERB
ejpam-3504	45	20	and	and	CCONJ
ejpam-3504	45	21	right	right	ADJ
ejpam-3504	45	22	duo	duo	NOUN
ejpam-3504	45	23	.	.	PUNCT
ejpam-3504	46	1	an	an	DET
ejpam-3504	46	2	hypergroupoid	hypergroupoid	PROPN
ejpam-3504	46	3	s	s	PART
ejpam-3504	46	4	is	be	AUX
ejpam-3504	46	5	called	call	VERB
ejpam-3504	46	6	hypersemigroup	hypersemigroup	ADV
ejpam-3504	46	7	if	if	SCONJ
ejpam-3504	46	8	{	{	PUNCT
ejpam-3504	46	9	x	x	NOUN
ejpam-3504	46	10	}	}	PUNCT
ejpam-3504	46	11	∗	∗	NOUN
ejpam-3504	46	12	(	(	PUNCT
ejpam-3504	46	13	y	y	PROPN
ejpam-3504	46	14	◦	◦	PROPN
ejpam-3504	46	15	z	z	PROPN
ejpam-3504	46	16	)	)	PUNCT
ejpam-3504	46	17	=	=	SYM
ejpam-3504	47	1	(	(	PUNCT
ejpam-3504	47	2	x	x	SYM
ejpam-3504	47	3	◦	◦	VERB
ejpam-3504	47	4	y	y	NOUN
ejpam-3504	47	5	)	)	PUNCT
ejpam-3504	47	6	∗	∗	NOUN
ejpam-3504	47	7	{	{	PUNCT
ejpam-3504	47	8	z	z	NOUN
ejpam-3504	47	9	}	}	PUNCT
ejpam-3504	47	10	for	for	ADP
ejpam-3504	47	11	all	all	DET
ejpam-3504	47	12	x	x	NOUN
ejpam-3504	47	13	,	,	PUNCT
ejpam-3504	47	14	y	y	PROPN
ejpam-3504	47	15	,	,	PUNCT
ejpam-3504	47	16	z	z	PROPN
ejpam-3504	47	17	∈	∈	PROPN
ejpam-3504	47	18	s.	s.	PROPN
ejpam-3504	47	19	if	if	SCONJ
ejpam-3504	47	20	is	be	AUX
ejpam-3504	47	21	convenient	convenient	ADJ
ejpam-3504	47	22	and	and	CCONJ
ejpam-3504	47	23	no	no	DET
ejpam-3504	47	24	confusion	confusion	NOUN
ejpam-3504	47	25	is	be	AUX
ejpam-3504	47	26	possible	possible	ADJ
ejpam-3504	47	27	,	,	PUNCT
ejpam-3504	47	28	the	the	DET
ejpam-3504	47	29	singleton	singleton	NOUN
ejpam-3504	47	30	{	{	PUNCT
ejpam-3504	47	31	x	x	NOUN
ejpam-3504	47	32	}	}	PUNCT
ejpam-3504	47	33	can	can	AUX
ejpam-3504	47	34	be	be	AUX
ejpam-3504	47	35	identified	identify	VERB
ejpam-3504	47	36	by	by	ADP
ejpam-3504	47	37	the	the	DET
ejpam-3504	47	38	element	element	NOUN
ejpam-3504	47	39	x	x	PUNCT
ejpam-3504	47	40	and	and	CCONJ
ejpam-3504	47	41	write	write	VERB
ejpam-3504	47	42	,	,	PUNCT
ejpam-3504	47	43	for	for	ADP
ejpam-3504	47	44	short	short	ADJ
ejpam-3504	47	45	,	,	PUNCT
ejpam-3504	47	46	x∗	x∗	PROPN
ejpam-3504	47	47	(	(	PUNCT
ejpam-3504	47	48	y	y	PROPN
ejpam-3504	47	49	◦	◦	PROPN
ejpam-3504	47	50	z	z	NOUN
ejpam-3504	47	51	)	)	PUNCT
ejpam-3504	47	52	=	=	SYM
ejpam-3504	47	53	(	(	PUNCT
ejpam-3504	47	54	x	x	X
ejpam-3504	47	55	◦	◦	NOUN
ejpam-3504	47	56	y)∗z	y)∗z	NOUN
ejpam-3504	47	57	for	for	ADP
ejpam-3504	47	58	all	all	DET
ejpam-3504	47	59	x	x	NOUN
ejpam-3504	47	60	,	,	PUNCT
ejpam-3504	47	61	y	y	PROPN
ejpam-3504	47	62	,	,	PUNCT
ejpam-3504	47	63	z	z	PROPN
ejpam-3504	47	64	∈	∈	PROPN
ejpam-3504	47	65	s	s	VERB
ejpam-3504	47	66	for	for	ADP
ejpam-3504	47	67	the	the	DET
ejpam-3504	47	68	associativity	associativity	NOUN
ejpam-3504	47	69	relation	relation	NOUN
ejpam-3504	47	70	,	,	PUNCT
ejpam-3504	47	71	also	also	ADV
ejpam-3504	47	72	expressions	expression	NOUN
ejpam-3504	47	73	of	of	ADP
ejpam-3504	47	74	the	the	DET
ejpam-3504	47	75	form	form	NOUN
ejpam-3504	47	76	s	s	VERB
ejpam-3504	47	77	∗x	∗x	NOUN
ejpam-3504	47	78	,	,	PUNCT
ejpam-3504	47	79	x	x	X
ejpam-3504	47	80	∗s	∗s	ADP
ejpam-3504	47	81	∗x	∗x	NOUN
ejpam-3504	47	82	,	,	PUNCT
ejpam-3504	47	83	a	a	DET
ejpam-3504	47	84	∗	∗	NOUN
ejpam-3504	47	85	(	(	PUNCT
ejpam-3504	47	86	x	x	SYM
ejpam-3504	47	87	◦	◦	NOUN
ejpam-3504	47	88	a	a	X
ejpam-3504	47	89	)	)	PUNCT
ejpam-3504	47	90	etc	etc	X
ejpam-3504	47	91	.	.	X
ejpam-3504	47	92	;	;	PUNCT
ejpam-3504	47	93	we	we	PRON
ejpam-3504	47	94	will	will	AUX
ejpam-3504	47	95	freely	freely	ADV
ejpam-3504	47	96	use	use	VERB
ejpam-3504	47	97	both	both	PRON
ejpam-3504	47	98	of	of	ADP
ejpam-3504	47	99	them	they	PRON
ejpam-3504	47	100	in	in	ADP
ejpam-3504	47	101	the	the	DET
ejpam-3504	47	102	present	present	ADJ
ejpam-3504	47	103	paper	paper	NOUN
ejpam-3504	47	104	.	.	PUNCT
ejpam-3504	48	1	for	for	ADP
ejpam-3504	48	2	any	any	DET
ejpam-3504	48	3	nonempty	nonempty	ADJ
ejpam-3504	48	4	subsets	subset	NOUN
ejpam-3504	48	5	a	a	DET
ejpam-3504	48	6	,	,	PUNCT
ejpam-3504	48	7	b	b	NOUN
ejpam-3504	48	8	,	,	PUNCT
ejpam-3504	48	9	c	c	NOUN
ejpam-3504	48	10	of	of	ADP
ejpam-3504	48	11	an	an	DET
ejpam-3504	48	12	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	48	13	s	s	NOUN
ejpam-3504	48	14	,	,	PUNCT
ejpam-3504	48	15	we	we	PRON
ejpam-3504	48	16	have	have	VERB
ejpam-3504	48	17	a∗	a∗	NOUN
ejpam-3504	48	18	(	(	PUNCT
ejpam-3504	48	19	b	b	NOUN
ejpam-3504	48	20	∗c	∗c	PROPN
ejpam-3504	48	21	)	)	PUNCT
ejpam-3504	49	1	=	=	PUNCT
ejpam-3504	49	2	(	(	PUNCT
ejpam-3504	49	3	a∗	a∗	PROPN
ejpam-3504	49	4	b	b	NOUN
ejpam-3504	49	5	)	)	PUNCT
ejpam-3504	49	6	∗c	∗c	PROPN
ejpam-3504	49	7	,	,	PUNCT
ejpam-3504	49	8	that	that	PRON
ejpam-3504	49	9	is	be	AUX
ejpam-3504	49	10	the	the	DET
ejpam-3504	49	11	operation	operation	NOUN
ejpam-3504	49	12	“	"	PUNCT
ejpam-3504	49	13	∗	∗	NOUN
ejpam-3504	49	14	”	"	PUNCT
ejpam-3504	49	15	on	on	ADP
ejpam-3504	49	16	p∗(s	p∗(s	NOUN
ejpam-3504	49	17	)	)	PUNCT
ejpam-3504	49	18	is	be	AUX
ejpam-3504	49	19	associative	associative	ADJ
ejpam-3504	49	20	and	and	CCONJ
ejpam-3504	49	21	so	so	ADV
ejpam-3504	49	22	(	(	PUNCT
ejpam-3504	49	23	p∗(s	p∗(s	NOUN
ejpam-3504	49	24	)	)	PUNCT
ejpam-3504	49	25	,	,	PUNCT
ejpam-3504	49	26	∗	∗	NOUN
ejpam-3504	49	27	)	)	PUNCT
ejpam-3504	49	28	is	be	AUX
ejpam-3504	49	29	a	a	DET
ejpam-3504	49	30	semigroup	semigroup	NOUN
ejpam-3504	49	31	.	.	PUNCT
ejpam-3504	50	1	this	this	DET
ejpam-3504	50	2	important	important	ADJ
ejpam-3504	50	3	result	result	NOUN
ejpam-3504	50	4	that	that	PRON
ejpam-3504	50	5	allow	allow	VERB
ejpam-3504	50	6	us	we	PRON
ejpam-3504	50	7	to	to	PART
ejpam-3504	50	8	write	write	VERB
ejpam-3504	50	9	expressions	expression	NOUN
ejpam-3504	50	10	of	of	ADP
ejpam-3504	50	11	the	the	DET
ejpam-3504	50	12	form	form	NOUN
ejpam-3504	50	13	a1	a1	NOUN
ejpam-3504	50	14	∗	∗	NOUN
ejpam-3504	50	15	a2	a2	PROPN
ejpam-3504	50	16	∗	∗	NOUN
ejpam-3504	50	17	.....	.....	PUNCT
ejpam-3504	50	18	an	an	DET
ejpam-3504	50	19	(	(	PUNCT
ejpam-3504	50	20	n	n	CCONJ
ejpam-3504	50	21	natural	natural	ADJ
ejpam-3504	50	22	number	number	NOUN
ejpam-3504	50	23	)	)	PUNCT
ejpam-3504	50	24	without	without	ADP
ejpam-3504	50	25	using	use	VERB
ejpam-3504	50	26	parentheses	parenthesis	NOUN
ejpam-3504	50	27	,	,	PUNCT
ejpam-3504	50	28	has	have	AUX
ejpam-3504	50	29	been	be	AUX
ejpam-3504	50	30	first	first	ADV
ejpam-3504	50	31	proved	prove	VERB
ejpam-3504	50	32	in	in	ADP
ejpam-3504	50	33	[	[	X
ejpam-3504	50	34	5	5	NUM
ejpam-3504	50	35	]	]	PUNCT
ejpam-3504	50	36	.	.	PUNCT
ejpam-3504	51	1	let	let	VERB
ejpam-3504	51	2	us	we	PRON
ejpam-3504	51	3	give	give	VERB
ejpam-3504	51	4	here	here	ADV
ejpam-3504	51	5	a	a	DET
ejpam-3504	51	6	very	very	ADV
ejpam-3504	51	7	easy	easy	ADJ
ejpam-3504	51	8	proof	proof	NOUN
ejpam-3504	51	9	of	of	ADP
ejpam-3504	51	10	this	this	DET
ejpam-3504	51	11	statement	statement	NOUN
ejpam-3504	51	12	:	:	PUNCT
ejpam-3504	51	13	suppose	suppose	VERB
ejpam-3504	51	14	x	x	X
ejpam-3504	51	15	∈	∈	PROPN
ejpam-3504	51	16	a	a	DET
ejpam-3504	51	17	∗	∗	NOUN
ejpam-3504	51	18	(	(	PUNCT
ejpam-3504	51	19	b	b	NOUN
ejpam-3504	51	20	∗	∗	NOUN
ejpam-3504	51	21	c	c	NOUN
ejpam-3504	51	22	)	)	PUNCT
ejpam-3504	51	23	.	.	PUNCT
ejpam-3504	52	1	then	then	ADV
ejpam-3504	52	2	x	x	X
ejpam-3504	52	3	∈	∈	PROPN
ejpam-3504	52	4	a	a	DET
ejpam-3504	52	5	◦	◦	NOUN
ejpam-3504	52	6	v	v	NOUN
ejpam-3504	52	7	for	for	ADP
ejpam-3504	52	8	some	some	DET
ejpam-3504	52	9	a	a	DET
ejpam-3504	52	10	∈	∈	PROPN
ejpam-3504	52	11	a	a	PRON
ejpam-3504	52	12	,	,	PUNCT
ejpam-3504	52	13	v	v	PROPN
ejpam-3504	52	14	∈	∈	PROPN
ejpam-3504	52	15	b	b	NOUN
ejpam-3504	52	16	∗	∗	X
ejpam-3504	52	17	c	c	NOUN
ejpam-3504	52	18	and	and	CCONJ
ejpam-3504	52	19	v	v	ADP
ejpam-3504	52	20	∈	∈	PROPN
ejpam-3504	52	21	b	b	PROPN
ejpam-3504	52	22	◦	◦	NOUN
ejpam-3504	52	23	c	c	NOUN
ejpam-3504	52	24	for	for	ADP
ejpam-3504	52	25	some	some	DET
ejpam-3504	52	26	b	b	PROPN
ejpam-3504	52	27	∈	∈	PROPN
ejpam-3504	52	28	b	b	PROPN
ejpam-3504	52	29	,	,	PUNCT
ejpam-3504	52	30	c	c	PROPN
ejpam-3504	52	31	∈	∈	PROPN
ejpam-3504	52	32	s.	s.	PROPN
ejpam-3504	53	1	then	then	ADV
ejpam-3504	53	2	we	we	PRON
ejpam-3504	53	3	have	have	VERB
ejpam-3504	53	4	x	x	PROPN
ejpam-3504	53	5	∈	∈	PROPN
ejpam-3504	53	6	a	a	DET
ejpam-3504	53	7	◦	◦	NOUN
ejpam-3504	53	8	v	v	NUM
ejpam-3504	53	9	⊆	⊆	NUM
ejpam-3504	53	10	{	{	PUNCT
ejpam-3504	53	11	a	a	DET
ejpam-3504	53	12	}	}	PUNCT
ejpam-3504	53	13	∗	∗	NOUN
ejpam-3504	53	14	(	(	PUNCT
ejpam-3504	53	15	b	b	X
ejpam-3504	53	16	◦	◦	NOUN
ejpam-3504	53	17	c	c	NOUN
ejpam-3504	53	18	)	)	PUNCT
ejpam-3504	53	19	=	=	SYM
ejpam-3504	53	20	(	(	PUNCT
ejpam-3504	53	21	a	a	DET
ejpam-3504	53	22	◦	◦	NOUN
ejpam-3504	53	23	b	b	NOUN
ejpam-3504	53	24	)	)	PUNCT
ejpam-3504	53	25	∗	∗	NOUN
ejpam-3504	53	26	{	{	PUNCT
ejpam-3504	53	27	c	c	NOUN
ejpam-3504	53	28	}	}	PUNCT
ejpam-3504	53	29	⊆	⊆	NUM
ejpam-3504	53	30	(	(	PUNCT
ejpam-3504	53	31	a	a	DET
ejpam-3504	53	32	∗b	∗b	NOUN
ejpam-3504	53	33	)	)	PUNCT
ejpam-3504	53	34	∗	∗	NOUN
ejpam-3504	53	35	c	c	NOUN
ejpam-3504	54	1	and	and	CCONJ
ejpam-3504	54	2	so	so	ADV
ejpam-3504	54	3	a	a	DET
ejpam-3504	54	4	∗	∗	NOUN
ejpam-3504	54	5	(	(	PUNCT
ejpam-3504	54	6	b	b	NOUN
ejpam-3504	54	7	∗	∗	NOUN
ejpam-3504	54	8	c	c	NOUN
ejpam-3504	54	9	)	)	PUNCT
ejpam-3504	54	10	⊆	⊆	NUM
ejpam-3504	54	11	(	(	PUNCT
ejpam-3504	54	12	a	a	DET
ejpam-3504	54	13	∗b	∗b	NOUN
ejpam-3504	54	14	)	)	PUNCT
ejpam-3504	54	15	∗	∗	NOUN
ejpam-3504	54	16	c.	c.	PROPN
ejpam-3504	55	1	similarly	similarly	ADV
ejpam-3504	55	2	we	we	PRON
ejpam-3504	55	3	get	get	VERB
ejpam-3504	55	4	(	(	PUNCT
ejpam-3504	55	5	a	a	DET
ejpam-3504	55	6	∗b	∗b	NOUN
ejpam-3504	55	7	)	)	PUNCT
ejpam-3504	55	8	∗	∗	NOUN
ejpam-3504	55	9	c	c	NOUN
ejpam-3504	55	10	⊆	⊆	NUM
ejpam-3504	55	11	a	a	DET
ejpam-3504	55	12	∗	∗	NOUN
ejpam-3504	55	13	(	(	PUNCT
ejpam-3504	55	14	b	b	NOUN
ejpam-3504	55	15	∗	∗	NOUN
ejpam-3504	55	16	c	c	NOUN
ejpam-3504	55	17	)	)	PUNCT
ejpam-3504	55	18	.	.	PUNCT
ejpam-3504	56	1	a	a	DET
ejpam-3504	56	2	nonempty	nonempty	NOUN
ejpam-3504	56	3	subset	subset	VERB
ejpam-3504	56	4	a	a	PRON
ejpam-3504	56	5	of	of	ADP
ejpam-3504	56	6	an	an	DET
ejpam-3504	56	7	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	56	8	s	s	PART
ejpam-3504	56	9	is	be	AUX
ejpam-3504	56	10	called	call	VERB
ejpam-3504	56	11	a	a	DET
ejpam-3504	56	12	bi	bi	NOUN
ejpam-3504	56	13	-	-	NOUN
ejpam-3504	56	14	ideal	ideal	NOUN
ejpam-3504	56	15	of	of	ADP
ejpam-3504	56	16	s	s	PRON
ejpam-3504	56	17	if	if	SCONJ
ejpam-3504	56	18	a∗s	a∗	NOUN
ejpam-3504	56	19	∗a	∗a	PROPN
ejpam-3504	56	20	⊆	⊆	NUM
ejpam-3504	56	21	a	a	PRON
ejpam-3504	56	22	;	;	PUNCT
ejpam-3504	56	23	that	that	ADV
ejpam-3504	56	24	is	is	ADV
ejpam-3504	56	25	,	,	PUNCT
ejpam-3504	56	26	if	if	SCONJ
ejpam-3504	56	27	u	u	PROPN
ejpam-3504	56	28	∈	∈	PROPN
ejpam-3504	56	29	v	v	ADP
ejpam-3504	56	30	◦	◦	NOUN
ejpam-3504	56	31	a	a	DET
ejpam-3504	56	32	and	and	CCONJ
ejpam-3504	56	33	v	v	ADP
ejpam-3504	56	34	∈	∈	PROPN
ejpam-3504	56	35	w	w	PROPN
ejpam-3504	56	36	◦	◦	NOUN
ejpam-3504	56	37	s	s	NUM
ejpam-3504	56	38	for	for	ADP
ejpam-3504	56	39	some	some	DET
ejpam-3504	56	40	a	a	DET
ejpam-3504	56	41	,	,	PUNCT
ejpam-3504	56	42	w	w	PROPN
ejpam-3504	56	43	∈	∈	PROPN
ejpam-3504	56	44	a	a	PRON
ejpam-3504	56	45	,	,	PUNCT
ejpam-3504	56	46	s	s	NOUN
ejpam-3504	56	47	∈	∈	PROPN
ejpam-3504	56	48	s	s	NOUN
ejpam-3504	56	49	,	,	PUNCT
ejpam-3504	56	50	then	then	ADV
ejpam-3504	56	51	u	u	X
ejpam-3504	56	52	∈	∈	PROPN
ejpam-3504	56	53	a.	a.	NOUN
ejpam-3504	56	54	because	because	SCONJ
ejpam-3504	56	55	of	of	ADP
ejpam-3504	56	56	the	the	DET
ejpam-3504	56	57	associativity	associativity	NOUN
ejpam-3504	56	58	relation	relation	NOUN
ejpam-3504	56	59	of	of	ADP
ejpam-3504	56	60	the	the	DET
ejpam-3504	56	61	operation	operation	NOUN
ejpam-3504	56	62	“	"	PUNCT
ejpam-3504	56	63	∗	∗	NOUN
ejpam-3504	56	64	”	"	PUNCT
ejpam-3504	56	65	,	,	PUNCT
ejpam-3504	56	66	this	this	PRON
ejpam-3504	56	67	can	can	AUX
ejpam-3504	56	68	be	be	AUX
ejpam-3504	56	69	also	also	ADV
ejpam-3504	56	70	expressed	express	VERB
ejpam-3504	56	71	as	as	SCONJ
ejpam-3504	56	72	follows	follow	VERB
ejpam-3504	56	73	:	:	PUNCT
ejpam-3504	56	74	if	if	SCONJ
ejpam-3504	56	75	u	u	PROPN
ejpam-3504	56	76	∈	∈	PROPN
ejpam-3504	56	77	a	a	DET
ejpam-3504	56	78	◦	◦	NOUN
ejpam-3504	56	79	v	v	NOUN
ejpam-3504	56	80	and	and	CCONJ
ejpam-3504	56	81	v	v	ADP
ejpam-3504	56	82	∈	∈	NOUN
ejpam-3504	56	83	s	s	PART
ejpam-3504	56	84	◦	◦	NOUN
ejpam-3504	56	85	w	w	NOUN
ejpam-3504	56	86	for	for	ADP
ejpam-3504	56	87	some	some	DET
ejpam-3504	56	88	a	a	DET
ejpam-3504	56	89	,	,	PUNCT
ejpam-3504	56	90	w	w	PROPN
ejpam-3504	56	91	∈	∈	PROPN
ejpam-3504	56	92	a	a	PRON
ejpam-3504	56	93	,	,	PUNCT
ejpam-3504	56	94	s	s	NOUN
ejpam-3504	56	95	∈	∈	PROPN
ejpam-3504	56	96	s	s	NOUN
ejpam-3504	56	97	,	,	PUNCT
ejpam-3504	56	98	then	then	ADV
ejpam-3504	56	99	u	u	PROPN
ejpam-3504	56	100	∈	∈	PROPN
ejpam-3504	56	101	a.	a.	NOUN
ejpam-3504	56	102	n.	n.	PROPN
ejpam-3504	56	103	kehayopulu	kehayopulu	PROPN
ejpam-3504	56	104	/	/	SYM
ejpam-3504	56	105	eur	eur	PROPN
ejpam-3504	56	106	.	.	PUNCT
ejpam-3504	57	1	j.	j.	PROPN
ejpam-3504	57	2	pure	pure	PROPN
ejpam-3504	57	3	appl	appl	PROPN
ejpam-3504	57	4	.	.	PROPN
ejpam-3504	57	5	math	math	PROPN
ejpam-3504	57	6	,	,	PUNCT
ejpam-3504	57	7	12	12	NUM
ejpam-3504	57	8	(	(	PUNCT
ejpam-3504	57	9	3	3	NUM
ejpam-3504	57	10	)	)	PUNCT
ejpam-3504	57	11	(	(	PUNCT
ejpam-3504	57	12	2019	2019	NUM
ejpam-3504	57	13	)	)	PUNCT
ejpam-3504	57	14	,	,	PUNCT
ejpam-3504	57	15	709	709	NUM
ejpam-3504	57	16	-	-	SYM
ejpam-3504	57	17	721	721	NUM
ejpam-3504	57	18	711	711	NUM
ejpam-3504	57	19	an	an	DET
ejpam-3504	57	20	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	57	21	s	s	PART
ejpam-3504	57	22	is	be	AUX
ejpam-3504	57	23	called	call	VERB
ejpam-3504	57	24	regular	regular	ADV
ejpam-3504	57	25	if	if	SCONJ
ejpam-3504	57	26	for	for	SCONJ
ejpam-3504	57	27	every	every	DET
ejpam-3504	57	28	a	a	DET
ejpam-3504	57	29	∈	∈	NOUN
ejpam-3504	57	30	s	s	VERB
ejpam-3504	57	31	there	there	PRON
ejpam-3504	57	32	exists	exist	VERB
ejpam-3504	57	33	x	x	X
ejpam-3504	57	34	∈	∈	NOUN
ejpam-3504	57	35	s	s	VERB
ejpam-3504	57	36	such	such	ADJ
ejpam-3504	57	37	that	that	SCONJ
ejpam-3504	57	38	a	a	DET
ejpam-3504	57	39	∈	∈	NOUN
ejpam-3504	57	40	(	(	PUNCT
ejpam-3504	57	41	a	a	DET
ejpam-3504	57	42	◦	◦	NOUN
ejpam-3504	57	43	x	x	SYM
ejpam-3504	57	44	)	)	PUNCT
ejpam-3504	57	45	∗	∗	NOUN
ejpam-3504	57	46	{	{	PUNCT
ejpam-3504	57	47	a	a	PRON
ejpam-3504	57	48	}	}	PUNCT
ejpam-3504	57	49	.	.	PUNCT
ejpam-3504	58	1	that	that	PRON
ejpam-3504	58	2	is	be	AUX
ejpam-3504	58	3	,	,	PUNCT
ejpam-3504	58	4	for	for	ADP
ejpam-3504	58	5	every	every	DET
ejpam-3504	58	6	a	a	DET
ejpam-3504	58	7	∈	∈	NOUN
ejpam-3504	58	8	s	s	VERB
ejpam-3504	58	9	there	there	PRON
ejpam-3504	58	10	exist	exist	VERB
ejpam-3504	58	11	x	x	PUNCT
ejpam-3504	58	12	∈	∈	PROPN
ejpam-3504	58	13	s	s	PART
ejpam-3504	58	14	and	and	CCONJ
ejpam-3504	58	15	u	u	NOUN
ejpam-3504	58	16	∈	∈	PROPN
ejpam-3504	58	17	a	a	DET
ejpam-3504	58	18	◦	◦	NOUN
ejpam-3504	58	19	x	x	SYM
ejpam-3504	58	20	such	such	ADJ
ejpam-3504	58	21	that	that	SCONJ
ejpam-3504	58	22	a	a	DET
ejpam-3504	58	23	∈	∈	PROPN
ejpam-3504	58	24	u	u	NOUN
ejpam-3504	58	25	◦	◦	NOUN
ejpam-3504	58	26	a.	a.	NOUN
ejpam-3504	58	27	this	this	PRON
ejpam-3504	58	28	is	be	AUX
ejpam-3504	58	29	equivalent	equivalent	ADJ
ejpam-3504	58	30	to	to	ADP
ejpam-3504	58	31	saying	say	VERB
ejpam-3504	58	32	that	that	SCONJ
ejpam-3504	58	33	a	a	DET
ejpam-3504	58	34	∈	∈	PROPN
ejpam-3504	59	1	a	a	DET
ejpam-3504	59	2	∗	∗	NOUN
ejpam-3504	59	3	s	s	PART
ejpam-3504	59	4	∗	∗	NOUN
ejpam-3504	59	5	a	a	PRON
ejpam-3504	59	6	for	for	ADP
ejpam-3504	59	7	every	every	DET
ejpam-3504	59	8	a	a	DET
ejpam-3504	59	9	∈	∈	PROPN
ejpam-3504	59	10	s	s	NOUN
ejpam-3504	59	11	or	or	CCONJ
ejpam-3504	59	12	a	a	DET
ejpam-3504	59	13	⊆	⊆	NUM
ejpam-3504	59	14	a	a	DET
ejpam-3504	59	15	∗	∗	NOUN
ejpam-3504	59	16	s	s	NOUN
ejpam-3504	59	17	∗a	∗a	ADJ
ejpam-3504	59	18	for	for	ADP
ejpam-3504	59	19	every	every	DET
ejpam-3504	59	20	nonempty	nonempty	NOUN
ejpam-3504	59	21	subset	subset	VERB
ejpam-3504	59	22	a	a	PRON
ejpam-3504	59	23	of	of	ADP
ejpam-3504	59	24	s.	s.	PROPN
ejpam-3504	59	25	it	it	PRON
ejpam-3504	59	26	is	be	AUX
ejpam-3504	59	27	called	call	VERB
ejpam-3504	59	28	intra	intra	ADJ
ejpam-3504	59	29	-	-	ADJ
ejpam-3504	59	30	regular	regular	ADJ
ejpam-3504	59	31	if	if	SCONJ
ejpam-3504	59	32	for	for	SCONJ
ejpam-3504	59	33	every	every	DET
ejpam-3504	59	34	a	a	DET
ejpam-3504	59	35	∈	∈	NOUN
ejpam-3504	59	36	s	s	VERB
ejpam-3504	59	37	there	there	PRON
ejpam-3504	59	38	exist	exist	VERB
ejpam-3504	59	39	x	x	NOUN
ejpam-3504	59	40	,	,	PUNCT
ejpam-3504	59	41	y	y	PROPN
ejpam-3504	59	42	∈	∈	PROPN
ejpam-3504	59	43	s	s	VERB
ejpam-3504	59	44	such	such	ADJ
ejpam-3504	59	45	that	that	SCONJ
ejpam-3504	59	46	a	a	DET
ejpam-3504	59	47	∈	∈	NOUN
ejpam-3504	59	48	(	(	PUNCT
ejpam-3504	59	49	x	x	SYM
ejpam-3504	59	50	◦	◦	VERB
ejpam-3504	59	51	a	a	X
ejpam-3504	59	52	)	)	PUNCT
ejpam-3504	59	53	∗	∗	NOUN
ejpam-3504	59	54	(	(	PUNCT
ejpam-3504	59	55	a	a	DET
ejpam-3504	59	56	◦	◦	NOUN
ejpam-3504	59	57	y	y	NOUN
ejpam-3504	59	58	)	)	PUNCT
ejpam-3504	59	59	(=	(=	NOUN
ejpam-3504	59	60	{	{	PUNCT
ejpam-3504	59	61	x	x	SYM
ejpam-3504	59	62	}	}	PUNCT
ejpam-3504	59	63	∗	∗	NOUN
ejpam-3504	59	64	(	(	PUNCT
ejpam-3504	59	65	a	a	DET
ejpam-3504	59	66	◦	◦	NOUN
ejpam-3504	59	67	a	a	X
ejpam-3504	59	68	)	)	PUNCT
ejpam-3504	59	69	∗	∗	NOUN
ejpam-3504	59	70	{	{	PUNCT
ejpam-3504	59	71	y	y	NOUN
ejpam-3504	59	72	}	}	PUNCT
ejpam-3504	59	73	)	)	PUNCT
ejpam-3504	59	74	.	.	PUNCT
ejpam-3504	60	1	that	that	PRON
ejpam-3504	60	2	is	be	AUX
ejpam-3504	60	3	,	,	PUNCT
ejpam-3504	60	4	for	for	ADP
ejpam-3504	60	5	any	any	DET
ejpam-3504	60	6	a	a	DET
ejpam-3504	60	7	∈	∈	NOUN
ejpam-3504	60	8	s	s	VERB
ejpam-3504	60	9	there	there	PRON
ejpam-3504	60	10	exist	exist	VERB
ejpam-3504	60	11	x	x	NOUN
ejpam-3504	60	12	,	,	PUNCT
ejpam-3504	60	13	y	y	PROPN
ejpam-3504	60	14	∈	∈	PROPN
ejpam-3504	60	15	s	s	PROPN
ejpam-3504	60	16	,	,	PUNCT
ejpam-3504	60	17	u	u	PROPN
ejpam-3504	60	18	∈	∈	PROPN
ejpam-3504	60	19	x	x	PUNCT
ejpam-3504	60	20	◦	◦	VERB
ejpam-3504	60	21	a	a	PRON
ejpam-3504	60	22	and	and	CCONJ
ejpam-3504	60	23	v	v	ADP
ejpam-3504	60	24	∈	∈	PROPN
ejpam-3504	60	25	a	a	DET
ejpam-3504	60	26	◦	◦	NOUN
ejpam-3504	60	27	y	y	PRON
ejpam-3504	60	28	such	such	ADJ
ejpam-3504	60	29	that	that	SCONJ
ejpam-3504	60	30	a	a	DET
ejpam-3504	60	31	∈	∈	PROPN
ejpam-3504	60	32	u	u	NOUN
ejpam-3504	60	33	◦	◦	NOUN
ejpam-3504	60	34	v.	v.	ADP
ejpam-3504	60	35	this	this	PRON
ejpam-3504	60	36	is	be	AUX
ejpam-3504	60	37	equivalent	equivalent	ADJ
ejpam-3504	60	38	to	to	ADP
ejpam-3504	60	39	saying	say	VERB
ejpam-3504	60	40	that	that	SCONJ
ejpam-3504	60	41	a	a	DET
ejpam-3504	60	42	∈	∈	PROPN
ejpam-3504	60	43	s	s	PART
ejpam-3504	60	44	∗	∗	NOUN
ejpam-3504	60	45	(	(	PUNCT
ejpam-3504	60	46	a	a	DET
ejpam-3504	60	47	◦	◦	NOUN
ejpam-3504	60	48	a	a	X
ejpam-3504	60	49	)	)	PUNCT
ejpam-3504	60	50	∗	∗	NOUN
ejpam-3504	60	51	s	s	NOUN
ejpam-3504	60	52	for	for	ADP
ejpam-3504	60	53	every	every	DET
ejpam-3504	60	54	a	a	DET
ejpam-3504	60	55	∈	∈	PROPN
ejpam-3504	60	56	s	s	NOUN
ejpam-3504	60	57	or	or	CCONJ
ejpam-3504	60	58	a	a	DET
ejpam-3504	60	59	⊆	⊆	NUM
ejpam-3504	60	60	s	s	NOUN
ejpam-3504	60	61	∗	∗	NOUN
ejpam-3504	60	62	a	a	DET
ejpam-3504	60	63	∗	∗	NOUN
ejpam-3504	60	64	a	a	DET
ejpam-3504	60	65	∗	∗	NOUN
ejpam-3504	60	66	s	s	NOUN
ejpam-3504	60	67	for	for	ADP
ejpam-3504	60	68	every	every	DET
ejpam-3504	60	69	nonempty	nonempty	NOUN
ejpam-3504	60	70	subset	subset	VERB
ejpam-3504	60	71	a	a	PRON
ejpam-3504	60	72	of	of	ADP
ejpam-3504	60	73	s.	s.	PROPN
ejpam-3504	60	74	an	an	DET
ejpam-3504	60	75	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	60	76	s	s	PART
ejpam-3504	60	77	is	be	AUX
ejpam-3504	60	78	called	call	VERB
ejpam-3504	60	79	left	leave	VERB
ejpam-3504	60	80	regular	regular	ADV
ejpam-3504	60	81	if	if	SCONJ
ejpam-3504	60	82	for	for	SCONJ
ejpam-3504	60	83	every	every	DET
ejpam-3504	60	84	a	a	DET
ejpam-3504	60	85	∈	∈	NOUN
ejpam-3504	60	86	s	s	VERB
ejpam-3504	60	87	there	there	PRON
ejpam-3504	60	88	exists	exist	VERB
ejpam-3504	60	89	x	x	X
ejpam-3504	60	90	∈	∈	NOUN
ejpam-3504	60	91	s	s	VERB
ejpam-3504	60	92	such	such	ADJ
ejpam-3504	60	93	that	that	SCONJ
ejpam-3504	60	94	a	a	DET
ejpam-3504	60	95	∈	∈	PROPN
ejpam-3504	60	96	{	{	PUNCT
ejpam-3504	60	97	x	x	NOUN
ejpam-3504	60	98	}	}	PUNCT
ejpam-3504	60	99	∗	∗	NOUN
ejpam-3504	60	100	(	(	PUNCT
ejpam-3504	60	101	a	a	DET
ejpam-3504	60	102	◦	◦	NOUN
ejpam-3504	60	103	a	a	X
ejpam-3504	60	104	)	)	PUNCT
ejpam-3504	60	105	.	.	PUNCT
ejpam-3504	61	1	that	that	PRON
ejpam-3504	61	2	is	be	AUX
ejpam-3504	61	3	,	,	PUNCT
ejpam-3504	61	4	for	for	ADP
ejpam-3504	61	5	every	every	DET
ejpam-3504	61	6	a	a	DET
ejpam-3504	61	7	∈	∈	NOUN
ejpam-3504	61	8	s	s	VERB
ejpam-3504	61	9	there	there	PRON
ejpam-3504	61	10	exist	exist	VERB
ejpam-3504	61	11	x	x	PUNCT
ejpam-3504	61	12	∈	∈	PROPN
ejpam-3504	61	13	s	s	PART
ejpam-3504	61	14	and	and	CCONJ
ejpam-3504	61	15	u	u	PROPN
ejpam-3504	61	16	∈	∈	PROPN
ejpam-3504	61	17	a	a	DET
ejpam-3504	61	18	◦	◦	NOUN
ejpam-3504	61	19	a	a	DET
ejpam-3504	61	20	such	such	ADJ
ejpam-3504	61	21	that	that	SCONJ
ejpam-3504	61	22	a	a	DET
ejpam-3504	61	23	∈	∈	PROPN
ejpam-3504	61	24	x	x	PUNCT
ejpam-3504	61	25	◦	◦	NOUN
ejpam-3504	61	26	u.	u.	ADV
ejpam-3504	61	27	equivalently	equivalently	ADV
ejpam-3504	61	28	,	,	PUNCT
ejpam-3504	61	29	if	if	SCONJ
ejpam-3504	61	30	a	a	DET
ejpam-3504	61	31	∈	∈	PROPN
ejpam-3504	61	32	s	s	PART
ejpam-3504	61	33	∗	∗	NOUN
ejpam-3504	61	34	(	(	PUNCT
ejpam-3504	61	35	a	a	DET
ejpam-3504	61	36	◦	◦	NOUN
ejpam-3504	61	37	a	a	X
ejpam-3504	61	38	)	)	PUNCT
ejpam-3504	61	39	for	for	ADP
ejpam-3504	61	40	every	every	DET
ejpam-3504	61	41	a	a	DET
ejpam-3504	61	42	∈	∈	PROPN
ejpam-3504	61	43	s	s	NOUN
ejpam-3504	61	44	or	or	CCONJ
ejpam-3504	61	45	a	a	DET
ejpam-3504	61	46	⊆	⊆	NUM
ejpam-3504	61	47	s	s	NOUN
ejpam-3504	61	48	∗	∗	NOUN
ejpam-3504	61	49	a	a	DET
ejpam-3504	61	50	∗	∗	NOUN
ejpam-3504	61	51	a	a	PRON
ejpam-3504	61	52	for	for	ADP
ejpam-3504	61	53	any	any	DET
ejpam-3504	61	54	nonempty	nonempty	NOUN
ejpam-3504	61	55	subset	subset	VERB
ejpam-3504	61	56	a	a	PRON
ejpam-3504	61	57	of	of	ADP
ejpam-3504	61	58	s.	s.	PROPN
ejpam-3504	61	59	it	it	PRON
ejpam-3504	61	60	is	be	AUX
ejpam-3504	61	61	called	call	VERB
ejpam-3504	61	62	right	right	ADV
ejpam-3504	61	63	regular	regular	ADV
ejpam-3504	61	64	if	if	SCONJ
ejpam-3504	61	65	for	for	SCONJ
ejpam-3504	61	66	every	every	DET
ejpam-3504	61	67	a	a	DET
ejpam-3504	61	68	∈	∈	NOUN
ejpam-3504	61	69	s	s	VERB
ejpam-3504	61	70	there	there	PRON
ejpam-3504	61	71	exists	exist	VERB
ejpam-3504	61	72	x	x	X
ejpam-3504	61	73	∈	∈	NOUN
ejpam-3504	61	74	s	s	VERB
ejpam-3504	61	75	such	such	ADJ
ejpam-3504	61	76	that	that	SCONJ
ejpam-3504	61	77	a	a	DET
ejpam-3504	61	78	∈	∈	NOUN
ejpam-3504	61	79	(	(	PUNCT
ejpam-3504	61	80	a	a	DET
ejpam-3504	61	81	◦	◦	NOUN
ejpam-3504	61	82	a	a	X
ejpam-3504	61	83	)	)	PUNCT
ejpam-3504	61	84	∗	∗	NOUN
ejpam-3504	61	85	{	{	PUNCT
ejpam-3504	61	86	x	x	NOUN
ejpam-3504	61	87	}	}	PUNCT
ejpam-3504	61	88	.	.	PUNCT
ejpam-3504	62	1	that	that	PRON
ejpam-3504	62	2	is	be	AUX
ejpam-3504	62	3	,	,	PUNCT
ejpam-3504	62	4	for	for	ADP
ejpam-3504	62	5	every	every	DET
ejpam-3504	62	6	a	a	DET
ejpam-3504	62	7	∈	∈	NOUN
ejpam-3504	62	8	s	s	VERB
ejpam-3504	62	9	there	there	PRON
ejpam-3504	62	10	exist	exist	VERB
ejpam-3504	62	11	x	x	PUNCT
ejpam-3504	62	12	∈	∈	PROPN
ejpam-3504	62	13	s	s	PART
ejpam-3504	62	14	and	and	CCONJ
ejpam-3504	62	15	u	u	PROPN
ejpam-3504	62	16	∈	∈	PROPN
ejpam-3504	62	17	a	a	DET
ejpam-3504	62	18	◦	◦	NOUN
ejpam-3504	62	19	a	a	DET
ejpam-3504	62	20	such	such	ADJ
ejpam-3504	62	21	that	that	SCONJ
ejpam-3504	62	22	a	a	DET
ejpam-3504	62	23	∈	∈	PROPN
ejpam-3504	62	24	u	u	NOUN
ejpam-3504	62	25	◦	◦	NOUN
ejpam-3504	62	26	x.	x.	NOUN
ejpam-3504	62	27	equivalently	equivalently	ADV
ejpam-3504	62	28	,	,	PUNCT
ejpam-3504	62	29	if	if	SCONJ
ejpam-3504	62	30	a	a	DET
ejpam-3504	62	31	∈	∈	NOUN
ejpam-3504	62	32	(	(	PUNCT
ejpam-3504	62	33	a	a	DET
ejpam-3504	62	34	◦	◦	NOUN
ejpam-3504	62	35	a	a	X
ejpam-3504	62	36	)	)	PUNCT
ejpam-3504	62	37	∗	∗	NOUN
ejpam-3504	62	38	s	s	NOUN
ejpam-3504	62	39	for	for	ADP
ejpam-3504	62	40	any	any	DET
ejpam-3504	62	41	a	a	DET
ejpam-3504	62	42	∈	∈	NOUN
ejpam-3504	62	43	s	s	NOUN
ejpam-3504	62	44	or	or	CCONJ
ejpam-3504	62	45	a	a	DET
ejpam-3504	62	46	⊆	⊆	NUM
ejpam-3504	62	47	a	a	DET
ejpam-3504	62	48	∗	∗	NOUN
ejpam-3504	62	49	a	a	DET
ejpam-3504	62	50	∗	∗	NOUN
ejpam-3504	62	51	s	s	NOUN
ejpam-3504	62	52	for	for	ADP
ejpam-3504	62	53	any	any	DET
ejpam-3504	62	54	nonempty	nonempty	NOUN
ejpam-3504	62	55	subset	subset	VERB
ejpam-3504	62	56	a	a	PRON
ejpam-3504	62	57	of	of	ADP
ejpam-3504	62	58	s.	s.	PROPN
ejpam-3504	62	59	as	as	ADP
ejpam-3504	62	60	(	(	PUNCT
ejpam-3504	62	61	a	a	DET
ejpam-3504	62	62	◦	◦	NOUN
ejpam-3504	62	63	x	x	NOUN
ejpam-3504	62	64	)	)	PUNCT
ejpam-3504	62	65	∗{a	∗{a	NOUN
ejpam-3504	62	66	}	}	PUNCT
ejpam-3504	62	67	=	=	SYM
ejpam-3504	62	68	{	{	PUNCT
ejpam-3504	62	69	a}∗	a}∗	PROPN
ejpam-3504	62	70	(	(	PUNCT
ejpam-3504	62	71	x	x	INTJ
ejpam-3504	62	72	◦	◦	NOUN
ejpam-3504	62	73	a	a	NOUN
ejpam-3504	62	74	)	)	PUNCT
ejpam-3504	62	75	,	,	PUNCT
ejpam-3504	62	76	the	the	DET
ejpam-3504	62	77	regularity	regularity	NOUN
ejpam-3504	62	78	can	can	AUX
ejpam-3504	62	79	be	be	AUX
ejpam-3504	62	80	also	also	ADV
ejpam-3504	62	81	defined	define	VERB
ejpam-3504	62	82	as	as	SCONJ
ejpam-3504	62	83	follows	follow	VERB
ejpam-3504	62	84	:	:	PUNCT
ejpam-3504	62	85	for	for	SCONJ
ejpam-3504	62	86	every	every	DET
ejpam-3504	62	87	a	a	DET
ejpam-3504	62	88	∈	∈	NOUN
ejpam-3504	62	89	s	s	VERB
ejpam-3504	62	90	there	there	PRON
ejpam-3504	62	91	exist	exist	VERB
ejpam-3504	62	92	x	x	PUNCT
ejpam-3504	62	93	∈	∈	PROPN
ejpam-3504	62	94	s	s	PART
ejpam-3504	62	95	and	and	CCONJ
ejpam-3504	62	96	u	u	NOUN
ejpam-3504	62	97	∈	∈	PROPN
ejpam-3504	62	98	x	x	PUNCT
ejpam-3504	62	99	◦	◦	VERB
ejpam-3504	62	100	a	a	DET
ejpam-3504	62	101	such	such	ADJ
ejpam-3504	62	102	that	that	SCONJ
ejpam-3504	62	103	a	a	DET
ejpam-3504	62	104	∈	∈	PROPN
ejpam-3504	62	105	a	a	DET
ejpam-3504	62	106	◦	◦	NOUN
ejpam-3504	62	107	u.	u.	NOUN
ejpam-3504	62	108	of	of	ADP
ejpam-3504	62	109	course	course	NOUN
ejpam-3504	62	110	,	,	PUNCT
ejpam-3504	62	111	similar	similar	ADJ
ejpam-3504	62	112	arguments	argument	NOUN
ejpam-3504	62	113	for	for	ADP
ejpam-3504	62	114	intra	intra	ADJ
ejpam-3504	62	115	-	-	ADJ
ejpam-3504	62	116	regularity	regularity	NOUN
ejpam-3504	62	117	and	and	CCONJ
ejpam-3504	62	118	for	for	ADP
ejpam-3504	62	119	left	left	ADJ
ejpam-3504	62	120	and	and	CCONJ
ejpam-3504	62	121	right	right	ADJ
ejpam-3504	62	122	regularity	regularity	NOUN
ejpam-3504	62	123	also	also	ADV
ejpam-3504	62	124	hold	hold	VERB
ejpam-3504	62	125	.	.	PUNCT
ejpam-3504	63	1	following	follow	VERB
ejpam-3504	63	2	zadeh	zadeh	PROPN
ejpam-3504	63	3	,	,	PUNCT
ejpam-3504	63	4	if	if	SCONJ
ejpam-3504	63	5	(	(	PUNCT
ejpam-3504	63	6	s	s	NOUN
ejpam-3504	63	7	,	,	PUNCT
ejpam-3504	63	8	◦	◦	NOUN
ejpam-3504	63	9	)	)	PUNCT
ejpam-3504	63	10	is	be	AUX
ejpam-3504	63	11	an	an	DET
ejpam-3504	63	12	hypergroupoid	hypergroupoid	NOUN
ejpam-3504	63	13	,	,	PUNCT
ejpam-3504	63	14	then	then	ADV
ejpam-3504	63	15	every	every	DET
ejpam-3504	63	16	mapping	mapping	NOUN
ejpam-3504	63	17	f	f	X
ejpam-3504	63	18	of	of	ADP
ejpam-3504	63	19	s	s	PRON
ejpam-3504	63	20	into	into	ADP
ejpam-3504	63	21	the	the	DET
ejpam-3504	63	22	real	real	ADJ
ejpam-3504	63	23	closed	closed	ADJ
ejpam-3504	63	24	interval	interval	NOUN
ejpam-3504	63	25	[	[	X
ejpam-3504	63	26	0	0	NUM
ejpam-3504	63	27	,	,	PUNCT
ejpam-3504	63	28	1	1	NUM
ejpam-3504	63	29	]	]	PUNCT
ejpam-3504	63	30	is	be	AUX
ejpam-3504	63	31	called	call	VERB
ejpam-3504	63	32	a	a	DET
ejpam-3504	63	33	fuzzy	fuzzy	ADJ
ejpam-3504	63	34	subset	subset	NOUN
ejpam-3504	63	35	of	of	ADP
ejpam-3504	63	36	s	s	PRON
ejpam-3504	63	37	or	or	CCONJ
ejpam-3504	63	38	a	a	DET
ejpam-3504	63	39	fuzzy	fuzzy	ADJ
ejpam-3504	63	40	set	set	NOUN
ejpam-3504	63	41	in	in	ADP
ejpam-3504	63	42	s.	s.	PROPN
ejpam-3504	63	43	for	for	ADP
ejpam-3504	63	44	any	any	DET
ejpam-3504	63	45	nonempty	nonempty	NOUN
ejpam-3504	63	46	subset	subset	VERB
ejpam-3504	63	47	a	a	PRON
ejpam-3504	63	48	of	of	ADP
ejpam-3504	63	49	s	s	PROPN
ejpam-3504	63	50	,	,	PUNCT
ejpam-3504	63	51	the	the	DET
ejpam-3504	63	52	characteristic	characteristic	ADJ
ejpam-3504	63	53	function	function	NOUN
ejpam-3504	63	54	fa	fa	PROPN
ejpam-3504	63	55	is	be	AUX
ejpam-3504	63	56	the	the	DET
ejpam-3504	63	57	fuzzy	fuzzy	ADJ
ejpam-3504	63	58	subset	subset	NOUN
ejpam-3504	63	59	of	of	ADP
ejpam-3504	63	60	s	s	PRON
ejpam-3504	63	61	defined	define	VERB
ejpam-3504	63	62	by	by	ADP
ejpam-3504	63	63	fa	fa	PROPN
ejpam-3504	63	64	:	:	PUNCT
ejpam-3504	63	65	s	s	X
ejpam-3504	63	66	→	→	PUNCT
ejpam-3504	63	67	{	{	PUNCT
ejpam-3504	63	68	0	0	NUM
ejpam-3504	63	69	,	,	PUNCT
ejpam-3504	63	70	1	1	NUM
ejpam-3504	63	71	}	}	PUNCT
ejpam-3504	63	72	|	|	NOUN
ejpam-3504	63	73	x→	x→	PUNCT
ejpam-3504	63	74	fa(x	fa(x	PROPN
ejpam-3504	63	75	)	)	PUNCT
ejpam-3504	63	76	=	=	PRON
ejpam-3504	63	77	{	{	PUNCT
ejpam-3504	63	78	1	1	NUM
ejpam-3504	63	79	if	if	SCONJ
ejpam-3504	63	80	x	x	PROPN
ejpam-3504	63	81	∈	∈	PROPN
ejpam-3504	63	82	a	a	DET
ejpam-3504	63	83	0	0	NOUN
ejpam-3504	63	84	if	if	SCONJ
ejpam-3504	63	85	x	x	X
ejpam-3504	63	86	/∈	/∈	PUNCT
ejpam-3504	63	87	a.	a.	NOUN
ejpam-3504	63	88	a	a	DET
ejpam-3504	63	89	fuzzy	fuzzy	ADJ
ejpam-3504	63	90	subset	subset	NOUN
ejpam-3504	63	91	f	f	PROPN
ejpam-3504	63	92	of	of	ADP
ejpam-3504	63	93	s	s	PROPN
ejpam-3504	63	94	is	be	AUX
ejpam-3504	63	95	called	call	VERB
ejpam-3504	63	96	a	a	DET
ejpam-3504	63	97	fuzzy	fuzzy	ADJ
ejpam-3504	63	98	subgroupoid	subgroupoid	NOUN
ejpam-3504	63	99	of	of	ADP
ejpam-3504	63	100	s	s	PRON
ejpam-3504	63	101	if	if	SCONJ
ejpam-3504	63	102	f(x	f(x	PROPN
ejpam-3504	63	103	◦	◦	VERB
ejpam-3504	63	104	y	y	PROPN
ejpam-3504	63	105	)	)	PUNCT
ejpam-3504	63	106	≥	≥	NOUN
ejpam-3504	63	107	min{f(x	min{f(x	NOUN
ejpam-3504	63	108	)	)	PUNCT
ejpam-3504	63	109	,	,	PUNCT
ejpam-3504	63	110	f(y	f(y	NOUN
ejpam-3504	63	111	)	)	PUNCT
ejpam-3504	63	112	}	}	PUNCT
ejpam-3504	64	1	for	for	ADP
ejpam-3504	64	2	all	all	DET
ejpam-3504	64	3	x	x	NOUN
ejpam-3504	64	4	,	,	PUNCT
ejpam-3504	64	5	y	y	PROPN
ejpam-3504	64	6	∈	∈	PROPN
ejpam-3504	64	7	s	s	PART
ejpam-3504	64	8	,	,	PUNCT
ejpam-3504	64	9	in	in	ADP
ejpam-3504	64	10	the	the	DET
ejpam-3504	64	11	sense	sense	NOUN
ejpam-3504	64	12	that	that	SCONJ
ejpam-3504	64	13	if	if	SCONJ
ejpam-3504	64	14	u	u	PROPN
ejpam-3504	64	15	∈	∈	PROPN
ejpam-3504	64	16	x	x	PROPN
ejpam-3504	64	17	◦	◦	NOUN
ejpam-3504	64	18	y	y	PROPN
ejpam-3504	64	19	,	,	PUNCT
ejpam-3504	64	20	then	then	ADV
ejpam-3504	64	21	f(u	f(u	PROPN
ejpam-3504	64	22	)	)	PUNCT
ejpam-3504	64	23	≥	≥	NOUN
ejpam-3504	64	24	min{f(x	min{f(x	NOUN
ejpam-3504	64	25	)	)	PUNCT
ejpam-3504	64	26	,	,	PUNCT
ejpam-3504	64	27	f(y	f(y	NOUN
ejpam-3504	64	28	)	)	PUNCT
ejpam-3504	64	29	}	}	PUNCT
ejpam-3504	64	30	.	.	PUNCT
ejpam-3504	65	1	a	a	DET
ejpam-3504	65	2	fuzzy	fuzzy	ADJ
ejpam-3504	65	3	subset	subset	NOUN
ejpam-3504	65	4	f	f	PROPN
ejpam-3504	65	5	of	of	ADP
ejpam-3504	65	6	s	s	PROPN
ejpam-3504	65	7	is	be	AUX
ejpam-3504	65	8	called	call	VERB
ejpam-3504	65	9	a	a	DET
ejpam-3504	65	10	fuzzy	fuzzy	ADJ
ejpam-3504	65	11	left	leave	VERB
ejpam-3504	65	12	ideal	ideal	NOUN
ejpam-3504	65	13	of	of	ADP
ejpam-3504	65	14	s	s	PRON
ejpam-3504	65	15	if	if	SCONJ
ejpam-3504	65	16	f(x	f(x	PROPN
ejpam-3504	65	17	◦	◦	VERB
ejpam-3504	65	18	y	y	PROPN
ejpam-3504	65	19	)	)	PUNCT
ejpam-3504	65	20	≥	≥	NOUN
ejpam-3504	65	21	f(y	f(y	NOUN
ejpam-3504	65	22	)	)	PUNCT
ejpam-3504	65	23	for	for	ADP
ejpam-3504	65	24	every	every	DET
ejpam-3504	65	25	x	x	PROPN
ejpam-3504	65	26	,	,	PUNCT
ejpam-3504	65	27	y	y	PROPN
ejpam-3504	65	28	∈	∈	PROPN
ejpam-3504	65	29	s	s	PART
ejpam-3504	65	30	,	,	PUNCT
ejpam-3504	65	31	in	in	ADP
ejpam-3504	65	32	the	the	DET
ejpam-3504	65	33	sense	sense	NOUN
ejpam-3504	65	34	that	that	SCONJ
ejpam-3504	65	35	if	if	SCONJ
ejpam-3504	65	36	u	u	PROPN
ejpam-3504	65	37	∈	∈	X
ejpam-3504	65	38	x	x	VERB
ejpam-3504	65	39	◦	◦	NOUN
ejpam-3504	65	40	y	y	PROPN
ejpam-3504	65	41	,	,	PUNCT
ejpam-3504	65	42	then	then	ADV
ejpam-3504	65	43	f(u	f(u	PROPN
ejpam-3504	65	44	)	)	PUNCT
ejpam-3504	65	45	≥	≥	NOUN
ejpam-3504	65	46	f(y	f(y	NOUN
ejpam-3504	65	47	)	)	PUNCT
ejpam-3504	65	48	;	;	PUNCT
ejpam-3504	65	49	it	it	PRON
ejpam-3504	65	50	is	be	AUX
ejpam-3504	65	51	called	call	VERB
ejpam-3504	65	52	a	a	DET
ejpam-3504	65	53	fuzzy	fuzzy	ADJ
ejpam-3504	65	54	right	right	ADJ
ejpam-3504	65	55	ideal	ideal	NOUN
ejpam-3504	65	56	of	of	ADP
ejpam-3504	65	57	s	s	PRON
ejpam-3504	65	58	if	if	SCONJ
ejpam-3504	65	59	f(x	f(x	PROPN
ejpam-3504	65	60	◦	◦	VERB
ejpam-3504	65	61	y	y	PROPN
ejpam-3504	65	62	)	)	PUNCT
ejpam-3504	65	63	≥	≥	NOUN
ejpam-3504	65	64	f(x	f(x	PROPN
ejpam-3504	65	65	)	)	PUNCT
ejpam-3504	65	66	for	for	ADP
ejpam-3504	65	67	every	every	DET
ejpam-3504	65	68	x	x	PROPN
ejpam-3504	65	69	,	,	PUNCT
ejpam-3504	65	70	y	y	PROPN
ejpam-3504	65	71	∈	∈	PROPN
ejpam-3504	65	72	s	s	PART
ejpam-3504	65	73	,	,	PUNCT
ejpam-3504	65	74	in	in	ADP
ejpam-3504	65	75	the	the	DET
ejpam-3504	65	76	sense	sense	NOUN
ejpam-3504	65	77	that	that	SCONJ
ejpam-3504	65	78	if	if	SCONJ
ejpam-3504	65	79	u	u	PROPN
ejpam-3504	65	80	∈	∈	X
ejpam-3504	65	81	x	x	VERB
ejpam-3504	65	82	◦	◦	NOUN
ejpam-3504	65	83	y	y	PROPN
ejpam-3504	65	84	,	,	PUNCT
ejpam-3504	65	85	then	then	ADV
ejpam-3504	65	86	f(u	f(u	PROPN
ejpam-3504	65	87	)	)	PUNCT
ejpam-3504	65	88	≥	≥	NOUN
ejpam-3504	65	89	f(x	f(x	PROPN
ejpam-3504	65	90	)	)	PUNCT
ejpam-3504	66	1	[	[	X
ejpam-3504	66	2	2	2	NUM
ejpam-3504	66	3	]	]	PUNCT
ejpam-3504	66	4	.	.	PUNCT
ejpam-3504	67	1	by	by	ADP
ejpam-3504	67	2	a	a	DET
ejpam-3504	67	3	fuzzy	fuzzy	ADJ
ejpam-3504	67	4	ideal	ideal	NOUN
ejpam-3504	67	5	of	of	ADP
ejpam-3504	67	6	s	s	PRON
ejpam-3504	67	7	we	we	PRON
ejpam-3504	67	8	mean	mean	VERB
ejpam-3504	67	9	an	an	DET
ejpam-3504	67	10	ideal	ideal	NOUN
ejpam-3504	67	11	of	of	ADP
ejpam-3504	67	12	s	s	PRON
ejpam-3504	67	13	which	which	PRON
ejpam-3504	67	14	is	be	AUX
ejpam-3504	67	15	both	both	CCONJ
ejpam-3504	67	16	a	a	DET
ejpam-3504	67	17	fuzzy	fuzzy	ADJ
ejpam-3504	67	18	left	leave	VERB
ejpam-3504	67	19	ideal	ideal	NOUN
ejpam-3504	67	20	and	and	CCONJ
ejpam-3504	67	21	a	a	DET
ejpam-3504	67	22	fuzzy	fuzzy	ADJ
ejpam-3504	67	23	right	right	ADJ
ejpam-3504	67	24	ideal	ideal	NOUN
ejpam-3504	67	25	of	of	ADP
ejpam-3504	67	26	s.	s.	PROPN
ejpam-3504	67	27	an	an	DET
ejpam-3504	67	28	hypergroupoid	hypergroupoid	PROPN
ejpam-3504	67	29	s	s	PART
ejpam-3504	67	30	is	be	AUX
ejpam-3504	67	31	called	call	VERB
ejpam-3504	67	32	fuzzy	fuzzy	ADJ
ejpam-3504	67	33	left	left	ADJ
ejpam-3504	67	34	(	(	PUNCT
ejpam-3504	67	35	resp	resp	NOUN
ejpam-3504	67	36	.	.	PUNCT
ejpam-3504	68	1	fuzzy	fuzzy	ADJ
ejpam-3504	68	2	right	right	NOUN
ejpam-3504	68	3	)	)	PUNCT
ejpam-3504	69	1	duo	duo	NOUN
ejpam-3504	69	2	if	if	SCONJ
ejpam-3504	69	3	the	the	DET
ejpam-3504	69	4	fuzzy	fuzzy	ADJ
ejpam-3504	69	5	left	leave	VERB
ejpam-3504	69	6	(	(	PUNCT
ejpam-3504	69	7	resp	resp	NOUN
ejpam-3504	69	8	.	.	PUNCT
ejpam-3504	70	1	right	right	ADJ
ejpam-3504	70	2	)	)	PUNCT
ejpam-3504	70	3	ideals	ideal	NOUN
ejpam-3504	70	4	of	of	ADP
ejpam-3504	70	5	s	s	NOUN
ejpam-3504	70	6	are	be	AUX
ejpam-3504	70	7	at	at	ADP
ejpam-3504	70	8	the	the	DET
ejpam-3504	70	9	same	same	ADJ
ejpam-3504	70	10	time	time	NOUN
ejpam-3504	70	11	fuzzy	fuzzy	ADJ
ejpam-3504	70	12	right	right	INTJ
ejpam-3504	70	13	(	(	PUNCT
ejpam-3504	70	14	resp	resp	NOUN
ejpam-3504	70	15	.	.	PUNCT
ejpam-3504	71	1	left	left	ADJ
ejpam-3504	71	2	)	)	PUNCT
ejpam-3504	71	3	ideals	ideal	NOUN
ejpam-3504	71	4	of	of	ADP
ejpam-3504	71	5	s	s	NOUN
ejpam-3504	71	6	(	(	PUNCT
ejpam-3504	71	7	that	that	PRON
ejpam-3504	71	8	is	be	AUX
ejpam-3504	71	9	,	,	PUNCT
ejpam-3504	71	10	ideals	ideal	NOUN
ejpam-3504	71	11	of	of	ADP
ejpam-3504	71	12	s	s	NOUN
ejpam-3504	71	13	)	)	PUNCT
ejpam-3504	71	14	.	.	PUNCT
ejpam-3504	72	1	a	a	DET
ejpam-3504	72	2	fuzzy	fuzzy	ADJ
ejpam-3504	72	3	subset	subset	NOUN
ejpam-3504	72	4	f	f	PROPN
ejpam-3504	72	5	of	of	ADP
ejpam-3504	72	6	an	an	DET
ejpam-3504	72	7	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	72	8	s	s	PART
ejpam-3504	72	9	is	be	AUX
ejpam-3504	72	10	called	call	VERB
ejpam-3504	72	11	a	a	DET
ejpam-3504	72	12	fuzzy	fuzzy	ADJ
ejpam-3504	72	13	bi	bi	NOUN
ejpam-3504	72	14	-	-	NOUN
ejpam-3504	72	15	ideal	ideal	NOUN
ejpam-3504	72	16	of	of	ADP
ejpam-3504	72	17	s	s	PRON
ejpam-3504	72	18	if	if	SCONJ
ejpam-3504	72	19	f	f	PROPN
ejpam-3504	72	20	(	(	PUNCT
ejpam-3504	72	21	(	(	PUNCT
ejpam-3504	72	22	x	x	NOUN
ejpam-3504	72	23	◦	◦	NOUN
ejpam-3504	72	24	y)∗{z	y)∗{z	NOUN
ejpam-3504	72	25	}	}	PUNCT
ejpam-3504	72	26	)	)	PUNCT
ejpam-3504	72	27	≥	≥	NOUN
ejpam-3504	72	28	min{f(x	min{f(x	NOUN
ejpam-3504	72	29	)	)	PUNCT
ejpam-3504	72	30	,	,	PUNCT
ejpam-3504	72	31	f(z	f(z	PROPN
ejpam-3504	72	32	)	)	PUNCT
ejpam-3504	72	33	}	}	PUNCT
ejpam-3504	72	34	for	for	ADP
ejpam-3504	72	35	every	every	DET
ejpam-3504	72	36	x	x	PROPN
ejpam-3504	72	37	,	,	PUNCT
ejpam-3504	72	38	y	y	PROPN
ejpam-3504	72	39	,	,	PUNCT
ejpam-3504	72	40	z	z	PROPN
ejpam-3504	72	41	∈	∈	PROPN
ejpam-3504	72	42	s	s	PART
ejpam-3504	72	43	,	,	PUNCT
ejpam-3504	72	44	in	in	ADP
ejpam-3504	72	45	the	the	DET
ejpam-3504	72	46	sense	sense	NOUN
ejpam-3504	72	47	that	that	SCONJ
ejpam-3504	72	48	if	if	SCONJ
ejpam-3504	72	49	u	u	PROPN
ejpam-3504	72	50	∈	∈	PROPN
ejpam-3504	72	51	(	(	PUNCT
ejpam-3504	72	52	x	x	NOUN
ejpam-3504	72	53	◦	◦	NOUN
ejpam-3504	72	54	y)∗{z	y)∗{z	NOUN
ejpam-3504	72	55	}	}	PUNCT
ejpam-3504	72	56	,	,	PUNCT
ejpam-3504	72	57	then	then	ADV
ejpam-3504	72	58	f(u	f(u	PROPN
ejpam-3504	72	59	)	)	PUNCT
ejpam-3504	72	60	≥	≥	NOUN
ejpam-3504	72	61	min{f(x	min{f(x	NOUN
ejpam-3504	72	62	)	)	PUNCT
ejpam-3504	72	63	,	,	PUNCT
ejpam-3504	72	64	f(z	f(z	PROPN
ejpam-3504	72	65	)	)	PUNCT
ejpam-3504	72	66	}	}	PUNCT
ejpam-3504	72	67	.	.	PUNCT
ejpam-3504	73	1	as	as	ADP
ejpam-3504	73	2	(	(	PUNCT
ejpam-3504	73	3	x	x	PART
ejpam-3504	73	4	◦	◦	NOUN
ejpam-3504	73	5	y	y	NOUN
ejpam-3504	73	6	)	)	PUNCT
ejpam-3504	73	7	∗	∗	NOUN
ejpam-3504	73	8	{	{	PUNCT
ejpam-3504	73	9	z	z	NOUN
ejpam-3504	73	10	}	}	PUNCT
ejpam-3504	73	11	=	=	SYM
ejpam-3504	73	12	{	{	PUNCT
ejpam-3504	73	13	x	x	NOUN
ejpam-3504	73	14	}	}	PUNCT
ejpam-3504	73	15	∗	∗	NOUN
ejpam-3504	73	16	(	(	PUNCT
ejpam-3504	73	17	y	y	PROPN
ejpam-3504	73	18	◦	◦	PROPN
ejpam-3504	73	19	z	z	PROPN
ejpam-3504	73	20	)	)	PUNCT
ejpam-3504	73	21	,	,	PUNCT
ejpam-3504	73	22	the	the	DET
ejpam-3504	73	23	fuzzy	fuzzy	ADJ
ejpam-3504	73	24	bi	bi	NOUN
ejpam-3504	73	25	-	-	ADJ
ejpam-3504	73	26	ideal	ideal	ADJ
ejpam-3504	73	27	can	can	AUX
ejpam-3504	73	28	be	be	AUX
ejpam-3504	73	29	also	also	ADV
ejpam-3504	73	30	defined	define	VERB
ejpam-3504	73	31	by	by	ADP
ejpam-3504	73	32	f	f	PROPN
ejpam-3504	73	33	(	(	PUNCT
ejpam-3504	73	34	{	{	PUNCT
ejpam-3504	73	35	x	x	NOUN
ejpam-3504	73	36	}	}	PUNCT
ejpam-3504	73	37	∗	∗	NOUN
ejpam-3504	73	38	(	(	PUNCT
ejpam-3504	73	39	y	y	PROPN
ejpam-3504	73	40	◦	◦	PROPN
ejpam-3504	73	41	z	z	PROPN
ejpam-3504	73	42	)	)	PUNCT
ejpam-3504	73	43	)	)	PUNCT
ejpam-3504	73	44	≥	≥	PROPN
ejpam-3504	73	45	min{f(x	min{f(x	NOUN
ejpam-3504	73	46	)	)	PUNCT
ejpam-3504	73	47	,	,	PUNCT
ejpam-3504	73	48	f(z	f(z	PROPN
ejpam-3504	73	49	)	)	PUNCT
ejpam-3504	73	50	}	}	PUNCT
ejpam-3504	73	51	for	for	SCONJ
ejpam-3504	73	52	every	every	DET
ejpam-3504	73	53	x	x	PROPN
ejpam-3504	73	54	,	,	PUNCT
ejpam-3504	73	55	y	y	PROPN
ejpam-3504	73	56	,	,	PUNCT
ejpam-3504	73	57	z	z	PROPN
ejpam-3504	73	58	∈	∈	PROPN
ejpam-3504	73	59	s	s	NOUN
ejpam-3504	73	60	,	,	PUNCT
ejpam-3504	73	61	meaning	mean	VERB
ejpam-3504	73	62	that	that	SCONJ
ejpam-3504	73	63	if	if	SCONJ
ejpam-3504	73	64	u	u	PROPN
ejpam-3504	73	65	∈	∈	PROPN
ejpam-3504	73	66	{	{	PUNCT
ejpam-3504	73	67	x	x	NOUN
ejpam-3504	73	68	}	}	PUNCT
ejpam-3504	73	69	∗	∗	NOUN
ejpam-3504	73	70	(	(	PUNCT
ejpam-3504	73	71	y	y	PROPN
ejpam-3504	73	72	◦	◦	PROPN
ejpam-3504	73	73	z	z	PROPN
ejpam-3504	73	74	)	)	PUNCT
ejpam-3504	73	75	,	,	PUNCT
ejpam-3504	73	76	then	then	ADV
ejpam-3504	73	77	f(u	f(u	PROPN
ejpam-3504	73	78	)	)	PUNCT
ejpam-3504	73	79	≥	≥	NOUN
ejpam-3504	73	80	min{f(x	min{f(x	NOUN
ejpam-3504	73	81	)	)	PUNCT
ejpam-3504	73	82	,	,	PUNCT
ejpam-3504	73	83	f(z	f(z	PROPN
ejpam-3504	73	84	)	)	PUNCT
ejpam-3504	73	85	}	}	PUNCT
ejpam-3504	73	86	.	.	PUNCT
ejpam-3504	74	1	one	one	PRON
ejpam-3504	74	2	can	can	AUX
ejpam-3504	74	3	find	find	VERB
ejpam-3504	74	4	some	some	DET
ejpam-3504	74	5	further	further	ADJ
ejpam-3504	74	6	results	result	NOUN
ejpam-3504	74	7	related	relate	VERB
ejpam-3504	74	8	to	to	ADP
ejpam-3504	74	9	this	this	DET
ejpam-3504	74	10	subject	subject	NOUN
ejpam-3504	74	11	in	in	ADP
ejpam-3504	74	12	[	[	X
ejpam-3504	74	13	1–6	1–6	NUM
ejpam-3504	74	14	]	]	X
ejpam-3504	74	15	.	.	PUNCT
ejpam-3504	75	1	3	3	X
ejpam-3504	75	2	.	.	X
ejpam-3504	75	3	fuzzy	fuzzy	ADJ
ejpam-3504	75	4	ideals	ideal	NOUN
ejpam-3504	75	5	and	and	CCONJ
ejpam-3504	75	6	regular	regular	ADJ
ejpam-3504	75	7	fuzzy	fuzzy	ADJ
ejpam-3504	75	8	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	75	9	lemma	lemma	PROPN
ejpam-3504	75	10	3.1	3.1	NUM
ejpam-3504	75	11	.	.	PUNCT
ejpam-3504	76	1	let	let	AUX
ejpam-3504	76	2	(	(	PUNCT
ejpam-3504	76	3	s	s	NOUN
ejpam-3504	76	4	,	,	PUNCT
ejpam-3504	76	5	◦	◦	NOUN
ejpam-3504	76	6	)	)	PUNCT
ejpam-3504	76	7	be	be	AUX
ejpam-3504	76	8	an	an	DET
ejpam-3504	76	9	hypergroupoid	hypergroupoid	NOUN
ejpam-3504	76	10	.	.	PUNCT
ejpam-3504	77	1	if	if	SCONJ
ejpam-3504	77	2	a	a	PRON
ejpam-3504	77	3	is	be	AUX
ejpam-3504	77	4	a	a	DET
ejpam-3504	77	5	left	left	ADJ
ejpam-3504	77	6	(	(	PUNCT
ejpam-3504	77	7	resp	resp	NOUN
ejpam-3504	77	8	.	.	PUNCT
ejpam-3504	78	1	right	right	ADJ
ejpam-3504	78	2	)	)	PUNCT
ejpam-3504	78	3	ideal	ideal	NOUN
ejpam-3504	78	4	of	of	ADP
ejpam-3504	78	5	s	s	PROPN
ejpam-3504	78	6	,	,	PUNCT
ejpam-3504	78	7	then	then	ADV
ejpam-3504	78	8	the	the	DET
ejpam-3504	78	9	characteristic	characteristic	ADJ
ejpam-3504	78	10	function	function	NOUN
ejpam-3504	78	11	fa	fa	PROPN
ejpam-3504	78	12	is	be	AUX
ejpam-3504	78	13	a	a	DET
ejpam-3504	78	14	fuzzy	fuzzy	ADJ
ejpam-3504	78	15	left	left	NOUN
ejpam-3504	78	16	(	(	PUNCT
ejpam-3504	78	17	resp	resp	NOUN
ejpam-3504	78	18	.	.	PUNCT
ejpam-3504	79	1	fuzzy	fuzzy	ADJ
ejpam-3504	79	2	right	right	ADJ
ejpam-3504	79	3	)	)	PUNCT
ejpam-3504	79	4	ideal	ideal	NOUN
ejpam-3504	79	5	of	of	ADP
ejpam-3504	79	6	s.	s.	PROPN
ejpam-3504	79	7	“	"	PUNCT
ejpam-3504	79	8	conversely	conversely	ADV
ejpam-3504	79	9	”	"	PUNCT
ejpam-3504	79	10	,	,	PUNCT
ejpam-3504	79	11	n.	n.	NOUN
ejpam-3504	79	12	kehayopulu	kehayopulu	PROPN
ejpam-3504	79	13	/	/	SYM
ejpam-3504	79	14	eur	eur	PROPN
ejpam-3504	79	15	.	.	PUNCT
ejpam-3504	80	1	j.	j.	PROPN
ejpam-3504	80	2	pure	pure	PROPN
ejpam-3504	80	3	appl	appl	PROPN
ejpam-3504	80	4	.	.	PROPN
ejpam-3504	80	5	math	math	PROPN
ejpam-3504	80	6	,	,	PUNCT
ejpam-3504	80	7	12	12	NUM
ejpam-3504	80	8	(	(	PUNCT
ejpam-3504	80	9	3	3	NUM
ejpam-3504	80	10	)	)	PUNCT
ejpam-3504	80	11	(	(	PUNCT
ejpam-3504	80	12	2019	2019	NUM
ejpam-3504	80	13	)	)	PUNCT
ejpam-3504	80	14	,	,	PUNCT
ejpam-3504	80	15	709	709	NUM
ejpam-3504	80	16	-	-	SYM
ejpam-3504	80	17	721	721	NUM
ejpam-3504	80	18	712	712	NUM
ejpam-3504	80	19	if	if	SCONJ
ejpam-3504	80	20	a	a	PRON
ejpam-3504	80	21	is	be	AUX
ejpam-3504	80	22	a	a	DET
ejpam-3504	80	23	nonempty	nonempty	ADJ
ejpam-3504	80	24	subset	subset	NOUN
ejpam-3504	80	25	of	of	ADP
ejpam-3504	80	26	s	s	PRON
ejpam-3504	80	27	and	and	CCONJ
ejpam-3504	80	28	fa	fa	PROPN
ejpam-3504	80	29	is	be	AUX
ejpam-3504	80	30	a	a	DET
ejpam-3504	80	31	fuzzy	fuzzy	ADJ
ejpam-3504	80	32	left	left	NOUN
ejpam-3504	80	33	(	(	PUNCT
ejpam-3504	80	34	resp	resp	NOUN
ejpam-3504	80	35	.	.	PUNCT
ejpam-3504	81	1	fuzzy	fuzzy	ADJ
ejpam-3504	81	2	right	right	ADJ
ejpam-3504	81	3	)	)	PUNCT
ejpam-3504	81	4	ideal	ideal	NOUN
ejpam-3504	81	5	of	of	ADP
ejpam-3504	81	6	s	s	PROPN
ejpam-3504	81	7	,	,	PUNCT
ejpam-3504	81	8	then	then	ADV
ejpam-3504	81	9	a	a	PRON
ejpam-3504	81	10	is	be	AUX
ejpam-3504	81	11	a	a	DET
ejpam-3504	81	12	left	left	ADJ
ejpam-3504	81	13	(	(	PUNCT
ejpam-3504	81	14	resp	resp	NOUN
ejpam-3504	81	15	.	.	PUNCT
ejpam-3504	82	1	right	right	ADJ
ejpam-3504	82	2	)	)	PUNCT
ejpam-3504	82	3	ideal	ideal	NOUN
ejpam-3504	82	4	of	of	ADP
ejpam-3504	82	5	s.	s.	PROPN
ejpam-3504	82	6	proof	proof	PROPN
ejpam-3504	82	7	.	.	PUNCT
ejpam-3504	83	1	=	=	NOUN
ejpam-3504	83	2	⇒.	⇒.	NOUN
ejpam-3504	83	3	let	let	VERB
ejpam-3504	83	4	a	a	PRON
ejpam-3504	83	5	be	be	AUX
ejpam-3504	83	6	a	a	DET
ejpam-3504	83	7	left	left	ADJ
ejpam-3504	83	8	ideal	ideal	NOUN
ejpam-3504	83	9	of	of	ADP
ejpam-3504	83	10	s	s	PRON
ejpam-3504	83	11	and	and	CCONJ
ejpam-3504	83	12	x	x	NOUN
ejpam-3504	83	13	,	,	PUNCT
ejpam-3504	83	14	y	y	PROPN
ejpam-3504	83	15	∈	∈	PROPN
ejpam-3504	83	16	s.	s.	PROPN
ejpam-3504	83	17	then	then	ADV
ejpam-3504	83	18	fa(x	fa(x	VERB
ejpam-3504	83	19	◦	◦	NOUN
ejpam-3504	83	20	y	y	NOUN
ejpam-3504	83	21	)	)	PUNCT
ejpam-3504	83	22	≥	≥	NOUN
ejpam-3504	83	23	fa(y	fa(y	PROPN
ejpam-3504	83	24	)	)	PUNCT
ejpam-3504	83	25	.	.	PUNCT
ejpam-3504	84	1	indeed	indeed	ADV
ejpam-3504	84	2	:	:	PUNCT
ejpam-3504	84	3	let	let	VERB
ejpam-3504	84	4	u	u	PRON
ejpam-3504	84	5	∈	∈	PROPN
ejpam-3504	84	6	x	x	INTJ
ejpam-3504	84	7	◦	◦	NOUN
ejpam-3504	84	8	y.	y.	NOUN
ejpam-3504	84	9	if	if	SCONJ
ejpam-3504	84	10	y	y	PROPN
ejpam-3504	84	11	∈	∈	PROPN
ejpam-3504	84	12	a	a	PRON
ejpam-3504	84	13	,	,	PUNCT
ejpam-3504	84	14	then	then	ADV
ejpam-3504	84	15	fa(y	fa(y	NUM
ejpam-3504	84	16	)	)	PUNCT
ejpam-3504	84	17	=	=	SYM
ejpam-3504	84	18	1	1	NUM
ejpam-3504	84	19	,	,	PUNCT
ejpam-3504	84	20	x	x	PUNCT
ejpam-3504	84	21	◦	◦	VERB
ejpam-3504	84	22	y	y	NUM
ejpam-3504	85	1	⊆	⊆	NUM
ejpam-3504	85	2	s	s	PROPN
ejpam-3504	85	3	∗	∗	NOUN
ejpam-3504	85	4	a	a	DET
ejpam-3504	85	5	⊆	⊆	NUM
ejpam-3504	85	6	a	a	PRON
ejpam-3504	85	7	,	,	PUNCT
ejpam-3504	85	8	fa(u	fa(u	X
ejpam-3504	85	9	)	)	PUNCT
ejpam-3504	86	1	=	=	SYM
ejpam-3504	86	2	1	1	NUM
ejpam-3504	86	3	and	and	CCONJ
ejpam-3504	86	4	so	so	ADV
ejpam-3504	86	5	fa(u	fa(u	NOUN
ejpam-3504	86	6	)	)	PUNCT
ejpam-3504	86	7	≥	≥	X
ejpam-3504	86	8	fa(y	fa(y	PROPN
ejpam-3504	86	9	)	)	PUNCT
ejpam-3504	86	10	.	.	PUNCT
ejpam-3504	87	1	if	if	SCONJ
ejpam-3504	87	2	y	y	PROPN
ejpam-3504	87	3	/∈	/∈	VERB
ejpam-3504	88	1	a	a	PRON
ejpam-3504	88	2	,	,	PUNCT
ejpam-3504	88	3	then	then	ADV
ejpam-3504	88	4	fa(y	fa(y	NUM
ejpam-3504	88	5	)	)	PUNCT
ejpam-3504	88	6	=	=	SYM
ejpam-3504	89	1	0	0	NUM
ejpam-3504	89	2	≤	≤	NUM
ejpam-3504	89	3	fa(u	fa(u	NOUN
ejpam-3504	89	4	)	)	PUNCT
ejpam-3504	89	5	.	.	PUNCT
ejpam-3504	90	1	thus	thus	ADV
ejpam-3504	90	2	fa	fa	PROPN
ejpam-3504	90	3	is	be	AUX
ejpam-3504	90	4	a	a	DET
ejpam-3504	90	5	fuzzy	fuzzy	ADJ
ejpam-3504	90	6	left	leave	VERB
ejpam-3504	90	7	ideal	ideal	NOUN
ejpam-3504	90	8	of	of	ADP
ejpam-3504	90	9	s.	s.	PROPN
ejpam-3504	90	10	⇐	⇐	PROPN
ejpam-3504	90	11	=	=	PROPN
ejpam-3504	90	12	.	.	PUNCT
ejpam-3504	90	13	let	let	VERB
ejpam-3504	90	14	a	a	PRON
ejpam-3504	90	15	be	be	AUX
ejpam-3504	90	16	a	a	DET
ejpam-3504	90	17	nonempty	nonempty	ADJ
ejpam-3504	90	18	subset	subset	NOUN
ejpam-3504	90	19	of	of	ADP
ejpam-3504	90	20	s	s	PRON
ejpam-3504	90	21	such	such	ADJ
ejpam-3504	90	22	that	that	SCONJ
ejpam-3504	90	23	fa	fa	PROPN
ejpam-3504	90	24	is	be	AUX
ejpam-3504	90	25	a	a	DET
ejpam-3504	90	26	fuzzy	fuzzy	ADJ
ejpam-3504	90	27	left	leave	VERB
ejpam-3504	90	28	ideal	ideal	NOUN
ejpam-3504	90	29	of	of	ADP
ejpam-3504	90	30	s.	s.	PROPN
ejpam-3504	90	31	then	then	ADV
ejpam-3504	90	32	s	s	VERB
ejpam-3504	90	33	∗a	∗a	PROPN
ejpam-3504	90	34	⊆	⊆	NUM
ejpam-3504	90	35	a.	a.	NOUN
ejpam-3504	90	36	indeed	indeed	ADV
ejpam-3504	90	37	:	:	PUNCT
ejpam-3504	90	38	let	let	VERB
ejpam-3504	90	39	u	u	PRON
ejpam-3504	90	40	∈	∈	PROPN
ejpam-3504	90	41	s	s	PART
ejpam-3504	90	42	∗	∗	NOUN
ejpam-3504	90	43	a.	a.	NOUN
ejpam-3504	90	44	then	then	ADV
ejpam-3504	90	45	u	u	PROPN
ejpam-3504	90	46	∈	∈	PROPN
ejpam-3504	90	47	s	s	PART
ejpam-3504	90	48	◦	◦	NOUN
ejpam-3504	90	49	a	a	PRON
ejpam-3504	90	50	for	for	ADP
ejpam-3504	90	51	some	some	DET
ejpam-3504	90	52	s	s	PART
ejpam-3504	90	53	∈	∈	PROPN
ejpam-3504	90	54	s	s	NOUN
ejpam-3504	90	55	,	,	PUNCT
ejpam-3504	90	56	a	a	DET
ejpam-3504	90	57	∈	∈	PROPN
ejpam-3504	90	58	a	a	PRON
ejpam-3504	90	59	,	,	PUNCT
ejpam-3504	90	60	fa(s	fa(s	PUNCT
ejpam-3504	90	61	◦	◦	NOUN
ejpam-3504	90	62	a	a	X
ejpam-3504	90	63	)	)	PUNCT
ejpam-3504	90	64	≥	≥	NOUN
ejpam-3504	90	65	fa(a	fa(a	NUM
ejpam-3504	90	66	)	)	PUNCT
ejpam-3504	91	1	=	=	SYM
ejpam-3504	91	2	1	1	NUM
ejpam-3504	91	3	,	,	PUNCT
ejpam-3504	91	4	fa(u	fa(u	X
ejpam-3504	91	5	)	)	PUNCT
ejpam-3504	91	6	=	=	SYM
ejpam-3504	92	1	1	1	NUM
ejpam-3504	92	2	and	and	CCONJ
ejpam-3504	92	3	so	so	ADV
ejpam-3504	92	4	u	u	PROPN
ejpam-3504	92	5	∈	∈	PROPN
ejpam-3504	92	6	a.	a.	NOUN
ejpam-3504	92	7	thus	thus	ADV
ejpam-3504	92	8	a	a	PRON
ejpam-3504	92	9	is	be	AUX
ejpam-3504	92	10	a	a	DET
ejpam-3504	92	11	left	left	ADJ
ejpam-3504	92	12	ideal	ideal	NOUN
ejpam-3504	92	13	of	of	ADP
ejpam-3504	92	14	s.	s.	PROPN
ejpam-3504	92	15	�	�	PROPN
ejpam-3504	92	16	by	by	ADP
ejpam-3504	92	17	lemma	lemma	PROPN
ejpam-3504	92	18	3.1	3.1	NUM
ejpam-3504	92	19	,	,	PUNCT
ejpam-3504	92	20	we	we	PRON
ejpam-3504	92	21	clearly	clearly	ADV
ejpam-3504	92	22	have	have	VERB
ejpam-3504	92	23	the	the	DET
ejpam-3504	92	24	following	follow	VERB
ejpam-3504	92	25	lemma	lemma	PROPN
ejpam-3504	92	26	.	.	PUNCT
ejpam-3504	93	1	lemma	lemma	PROPN
ejpam-3504	93	2	3.2	3.2	NUM
ejpam-3504	93	3	.	.	PUNCT
ejpam-3504	94	1	let	let	AUX
ejpam-3504	94	2	(	(	PUNCT
ejpam-3504	94	3	s	s	NOUN
ejpam-3504	94	4	,	,	PUNCT
ejpam-3504	94	5	◦	◦	NOUN
ejpam-3504	94	6	)	)	PUNCT
ejpam-3504	94	7	be	be	AUX
ejpam-3504	94	8	an	an	DET
ejpam-3504	94	9	hypergroupoid	hypergroupoid	NOUN
ejpam-3504	94	10	.	.	PUNCT
ejpam-3504	95	1	if	if	SCONJ
ejpam-3504	95	2	a	a	PRON
ejpam-3504	95	3	is	be	AUX
ejpam-3504	95	4	an	an	DET
ejpam-3504	95	5	ideal	ideal	NOUN
ejpam-3504	95	6	of	of	ADP
ejpam-3504	95	7	s	s	PROPN
ejpam-3504	95	8	,	,	PUNCT
ejpam-3504	95	9	then	then	ADV
ejpam-3504	95	10	the	the	DET
ejpam-3504	95	11	characteristic	characteristic	ADJ
ejpam-3504	95	12	function	function	NOUN
ejpam-3504	95	13	fa	fa	PROPN
ejpam-3504	95	14	is	be	AUX
ejpam-3504	95	15	a	a	DET
ejpam-3504	95	16	fuzzy	fuzzy	ADJ
ejpam-3504	95	17	ideal	ideal	NOUN
ejpam-3504	95	18	of	of	ADP
ejpam-3504	95	19	s.	s.	PROPN
ejpam-3504	95	20	“	"	PUNCT
ejpam-3504	95	21	conversely	conversely	ADV
ejpam-3504	95	22	”	"	PUNCT
ejpam-3504	95	23	,	,	PUNCT
ejpam-3504	95	24	if	if	SCONJ
ejpam-3504	95	25	a	a	PRON
ejpam-3504	95	26	is	be	AUX
ejpam-3504	95	27	a	a	DET
ejpam-3504	95	28	nonempty	nonempty	ADJ
ejpam-3504	95	29	subset	subset	NOUN
ejpam-3504	95	30	of	of	ADP
ejpam-3504	95	31	s	s	PRON
ejpam-3504	95	32	such	such	ADJ
ejpam-3504	95	33	that	that	SCONJ
ejpam-3504	95	34	fa	fa	PROPN
ejpam-3504	95	35	is	be	AUX
ejpam-3504	95	36	a	a	DET
ejpam-3504	95	37	fuzzy	fuzzy	ADJ
ejpam-3504	95	38	ideal	ideal	NOUN
ejpam-3504	95	39	of	of	ADP
ejpam-3504	95	40	s	s	PROPN
ejpam-3504	95	41	,	,	PUNCT
ejpam-3504	95	42	then	then	ADV
ejpam-3504	95	43	a	a	PRON
ejpam-3504	95	44	is	be	AUX
ejpam-3504	95	45	an	an	DET
ejpam-3504	95	46	ideal	ideal	NOUN
ejpam-3504	95	47	of	of	ADP
ejpam-3504	95	48	s.	s.	PROPN
ejpam-3504	96	1	thus	thus	ADV
ejpam-3504	96	2	we	we	PRON
ejpam-3504	96	3	have	have	VERB
ejpam-3504	96	4	the	the	DET
ejpam-3504	96	5	following	follow	VERB
ejpam-3504	96	6	corollary	corollary	NOUN
ejpam-3504	96	7	.	.	PUNCT
ejpam-3504	97	1	corollary	corollary	ADJ
ejpam-3504	97	2	3.3	3.3	NUM
ejpam-3504	97	3	.	.	PUNCT
ejpam-3504	98	1	a	a	DET
ejpam-3504	98	2	nonempty	nonempty	NOUN
ejpam-3504	98	3	subset	subset	VERB
ejpam-3504	98	4	a	a	PRON
ejpam-3504	98	5	of	of	ADP
ejpam-3504	98	6	an	an	DET
ejpam-3504	98	7	hypergroupoid	hypergroupoid	PROPN
ejpam-3504	98	8	s	s	PART
ejpam-3504	98	9	is	be	AUX
ejpam-3504	98	10	an	an	DET
ejpam-3504	98	11	ideal	ideal	NOUN
ejpam-3504	98	12	of	of	ADP
ejpam-3504	98	13	s	s	PRON
ejpam-3504	98	14	if	if	SCONJ
ejpam-3504	98	15	and	and	CCONJ
ejpam-3504	98	16	only	only	ADV
ejpam-3504	98	17	if	if	SCONJ
ejpam-3504	98	18	the	the	DET
ejpam-3504	98	19	characteristic	characteristic	ADJ
ejpam-3504	98	20	function	function	NOUN
ejpam-3504	98	21	fa	fa	PROPN
ejpam-3504	98	22	is	be	AUX
ejpam-3504	98	23	a	a	DET
ejpam-3504	98	24	fuzzy	fuzzy	ADJ
ejpam-3504	98	25	ideal	ideal	NOUN
ejpam-3504	98	26	of	of	ADP
ejpam-3504	98	27	s.	s.	PROPN
ejpam-3504	98	28	corollary	corollary	PROPN
ejpam-3504	98	29	3.3	3.3	NUM
ejpam-3504	98	30	also	also	ADV
ejpam-3504	98	31	holds	hold	VERB
ejpam-3504	98	32	if	if	SCONJ
ejpam-3504	98	33	we	we	PRON
ejpam-3504	98	34	replace	replace	VERB
ejpam-3504	98	35	the	the	DET
ejpam-3504	98	36	word	word	NOUN
ejpam-3504	98	37	“	"	PUNCT
ejpam-3504	98	38	ideal	ideal	ADJ
ejpam-3504	98	39	”	"	PUNCT
ejpam-3504	98	40	by	by	ADP
ejpam-3504	98	41	“	"	PUNCT
ejpam-3504	98	42	subgroupoid	subgroupoid	NOUN
ejpam-3504	98	43	”	"	PUNCT
ejpam-3504	98	44	and	and	CCONJ
ejpam-3504	98	45	we	we	PRON
ejpam-3504	98	46	have	have	VERB
ejpam-3504	98	47	the	the	DET
ejpam-3504	98	48	following	follow	VERB
ejpam-3504	98	49	proposition	proposition	NOUN
ejpam-3504	98	50	.	.	PUNCT
ejpam-3504	99	1	proposition	proposition	NOUN
ejpam-3504	99	2	3.4	3.4	NUM
ejpam-3504	99	3	.	.	PUNCT
ejpam-3504	100	1	let	let	AUX
ejpam-3504	100	2	(	(	PUNCT
ejpam-3504	100	3	s	s	NOUN
ejpam-3504	100	4	,	,	PUNCT
ejpam-3504	100	5	◦	◦	NOUN
ejpam-3504	100	6	)	)	PUNCT
ejpam-3504	100	7	be	be	AUX
ejpam-3504	100	8	an	an	DET
ejpam-3504	100	9	hypergroupoid	hypergroupoid	NOUN
ejpam-3504	100	10	.	.	PUNCT
ejpam-3504	101	1	if	if	SCONJ
ejpam-3504	101	2	a	a	PRON
ejpam-3504	101	3	is	be	AUX
ejpam-3504	101	4	a	a	DET
ejpam-3504	101	5	subgroupoid	subgroupoid	NOUN
ejpam-3504	101	6	of	of	ADP
ejpam-3504	101	7	s	s	PROPN
ejpam-3504	101	8	,	,	PUNCT
ejpam-3504	101	9	then	then	ADV
ejpam-3504	101	10	the	the	DET
ejpam-3504	101	11	characteristic	characteristic	ADJ
ejpam-3504	101	12	function	function	NOUN
ejpam-3504	101	13	fa	fa	PROPN
ejpam-3504	101	14	is	be	AUX
ejpam-3504	101	15	a	a	DET
ejpam-3504	101	16	fuzzy	fuzzy	ADJ
ejpam-3504	101	17	subgroupoid	subgroupoid	NOUN
ejpam-3504	101	18	of	of	ADP
ejpam-3504	101	19	s.	s.	PROPN
ejpam-3504	101	20	“	"	PUNCT
ejpam-3504	101	21	conversely	conversely	ADV
ejpam-3504	101	22	”	"	PUNCT
ejpam-3504	101	23	,	,	PUNCT
ejpam-3504	101	24	if	if	SCONJ
ejpam-3504	101	25	a	a	PRON
ejpam-3504	101	26	is	be	AUX
ejpam-3504	101	27	a	a	DET
ejpam-3504	101	28	nonempty	nonempty	ADJ
ejpam-3504	101	29	subset	subset	NOUN
ejpam-3504	101	30	of	of	ADP
ejpam-3504	101	31	s	s	PRON
ejpam-3504	101	32	and	and	CCONJ
ejpam-3504	101	33	fa	fa	PROPN
ejpam-3504	101	34	is	be	AUX
ejpam-3504	101	35	a	a	DET
ejpam-3504	101	36	fuzzy	fuzzy	ADJ
ejpam-3504	101	37	subgroupoid	subgroupoid	NOUN
ejpam-3504	101	38	of	of	ADP
ejpam-3504	101	39	s	s	PROPN
ejpam-3504	101	40	,	,	PUNCT
ejpam-3504	101	41	then	then	ADV
ejpam-3504	101	42	a	a	PRON
ejpam-3504	101	43	is	be	AUX
ejpam-3504	101	44	a	a	DET
ejpam-3504	101	45	subgroupoid	subgroupoid	NOUN
ejpam-3504	101	46	of	of	ADP
ejpam-3504	101	47	s.	s.	PROPN
ejpam-3504	101	48	proof	proof	PROPN
ejpam-3504	101	49	.	.	PUNCT
ejpam-3504	102	1	let	let	VERB
ejpam-3504	102	2	a	a	PRON
ejpam-3504	102	3	be	be	AUX
ejpam-3504	102	4	a	a	DET
ejpam-3504	102	5	subgroupoid	subgroupoid	NOUN
ejpam-3504	102	6	of	of	ADP
ejpam-3504	102	7	s	s	PRON
ejpam-3504	102	8	and	and	CCONJ
ejpam-3504	102	9	x	x	X
ejpam-3504	102	10	,	,	PUNCT
ejpam-3504	102	11	y	y	PROPN
ejpam-3504	102	12	∈	∈	PROPN
ejpam-3504	102	13	a.	a.	NOUN
ejpam-3504	102	14	then	then	ADV
ejpam-3504	102	15	fa(x	fa(x	VERB
ejpam-3504	102	16	◦	◦	NOUN
ejpam-3504	102	17	y	y	NOUN
ejpam-3504	102	18	)	)	PUNCT
ejpam-3504	102	19	≥	≥	NOUN
ejpam-3504	102	20	min{fa(x	min{fa(x	NOUN
ejpam-3504	102	21	)	)	PUNCT
ejpam-3504	102	22	,	,	PUNCT
ejpam-3504	102	23	fa(y	fa(y	PROPN
ejpam-3504	102	24	)	)	PUNCT
ejpam-3504	102	25	}	}	PUNCT
ejpam-3504	102	26	.	.	PUNCT
ejpam-3504	103	1	indeed	indeed	ADV
ejpam-3504	103	2	:	:	PUNCT
ejpam-3504	103	3	let	let	VERB
ejpam-3504	103	4	u	u	PRON
ejpam-3504	103	5	∈	∈	PROPN
ejpam-3504	103	6	x	x	INTJ
ejpam-3504	103	7	◦	◦	NOUN
ejpam-3504	103	8	y.	y.	NOUN
ejpam-3504	103	9	since	since	SCONJ
ejpam-3504	103	10	x	x	SYM
ejpam-3504	103	11	◦	◦	VERB
ejpam-3504	103	12	y	y	PROPN
ejpam-3504	103	13	⊆	⊆	NUM
ejpam-3504	103	14	a	a	DET
ejpam-3504	103	15	∗	∗	NOUN
ejpam-3504	103	16	a	a	DET
ejpam-3504	103	17	⊆	⊆	NUM
ejpam-3504	103	18	a	a	NOUN
ejpam-3504	103	19	,	,	PUNCT
ejpam-3504	103	20	we	we	PRON
ejpam-3504	103	21	have	have	VERB
ejpam-3504	103	22	u	u	NOUN
ejpam-3504	103	23	∈	∈	PROPN
ejpam-3504	103	24	a	a	PRON
ejpam-3504	103	25	,	,	PUNCT
ejpam-3504	103	26	then	then	ADV
ejpam-3504	103	27	fa(u	fa(u	X
ejpam-3504	103	28	)	)	PUNCT
ejpam-3504	103	29	=	=	PUNCT
ejpam-3504	104	1	1	1	X
ejpam-3504	104	2	.	.	PUNCT
ejpam-3504	105	1	since	since	SCONJ
ejpam-3504	105	2	x	x	X
ejpam-3504	105	3	,	,	PUNCT
ejpam-3504	105	4	y	y	PROPN
ejpam-3504	105	5	∈	∈	PROPN
ejpam-3504	106	1	a	a	PRON
ejpam-3504	106	2	,	,	PUNCT
ejpam-3504	106	3	we	we	PRON
ejpam-3504	106	4	have	have	VERB
ejpam-3504	106	5	fa(x	fa(x	NOUN
ejpam-3504	106	6	)	)	PUNCT
ejpam-3504	106	7	=	=	SYM
ejpam-3504	106	8	fa(y	fa(y	NOUN
ejpam-3504	106	9	)	)	PUNCT
ejpam-3504	106	10	=	=	SYM
ejpam-3504	107	1	1	1	X
ejpam-3504	107	2	.	.	PUNCT
ejpam-3504	107	3	thus	thus	ADV
ejpam-3504	107	4	we	we	PRON
ejpam-3504	107	5	have	have	VERB
ejpam-3504	107	6	fa(u	fa(u	X
ejpam-3504	107	7	)	)	PUNCT
ejpam-3504	108	1	=	=	SYM
ejpam-3504	108	2	1	1	NUM
ejpam-3504	108	3	≥	≥	NOUN
ejpam-3504	108	4	min{fa(x	min{fa(x	NOUN
ejpam-3504	108	5	)	)	PUNCT
ejpam-3504	108	6	,	,	PUNCT
ejpam-3504	108	7	fa(y	fa(y	PROPN
ejpam-3504	108	8	)	)	PUNCT
ejpam-3504	108	9	}	}	PUNCT
ejpam-3504	108	10	.	.	PUNCT
ejpam-3504	109	1	for	for	ADP
ejpam-3504	109	2	the	the	DET
ejpam-3504	109	3	converse	converse	NOUN
ejpam-3504	109	4	statement	statement	NOUN
ejpam-3504	109	5	,	,	PUNCT
ejpam-3504	109	6	let	let	VERB
ejpam-3504	109	7	a	a	PRON
ejpam-3504	109	8	be	be	AUX
ejpam-3504	109	9	a	a	DET
ejpam-3504	109	10	nonempty	nonempty	ADJ
ejpam-3504	109	11	subset	subset	NOUN
ejpam-3504	109	12	of	of	ADP
ejpam-3504	109	13	s	s	PRON
ejpam-3504	109	14	such	such	ADJ
ejpam-3504	109	15	that	that	SCONJ
ejpam-3504	109	16	fa	fa	PROPN
ejpam-3504	109	17	is	be	AUX
ejpam-3504	109	18	a	a	DET
ejpam-3504	109	19	fuzzy	fuzzy	ADJ
ejpam-3504	109	20	subgroupoid	subgroupoid	NOUN
ejpam-3504	109	21	of	of	ADP
ejpam-3504	109	22	s.	s.	PROPN
ejpam-3504	109	23	then	then	ADV
ejpam-3504	109	24	a	a	DET
ejpam-3504	109	25	∗	∗	NOUN
ejpam-3504	109	26	a	a	DET
ejpam-3504	109	27	⊆	⊆	NUM
ejpam-3504	109	28	a.	a.	NOUN
ejpam-3504	109	29	indeed	indeed	ADV
ejpam-3504	109	30	:	:	PUNCT
ejpam-3504	109	31	let	let	VERB
ejpam-3504	109	32	u	u	PRON
ejpam-3504	109	33	∈	∈	PROPN
ejpam-3504	109	34	a	a	DET
ejpam-3504	109	35	∗	∗	NOUN
ejpam-3504	109	36	a.	a.	NOUN
ejpam-3504	109	37	then	then	ADV
ejpam-3504	109	38	u	u	NOUN
ejpam-3504	109	39	∈	∈	PROPN
ejpam-3504	109	40	a	a	DET
ejpam-3504	109	41	◦	◦	NOUN
ejpam-3504	109	42	b	b	NOUN
ejpam-3504	109	43	for	for	ADP
ejpam-3504	109	44	some	some	DET
ejpam-3504	109	45	a	a	PRON
ejpam-3504	109	46	,	,	PUNCT
ejpam-3504	109	47	b	b	X
ejpam-3504	109	48	∈	∈	PROPN
ejpam-3504	109	49	a.	a.	NOUN
ejpam-3504	109	50	since	since	SCONJ
ejpam-3504	109	51	fa	fa	PROPN
ejpam-3504	109	52	is	be	AUX
ejpam-3504	109	53	a	a	DET
ejpam-3504	109	54	fuzzy	fuzzy	ADJ
ejpam-3504	109	55	subgroupoid	subgroupoid	NOUN
ejpam-3504	109	56	of	of	ADP
ejpam-3504	109	57	s	s	PROPN
ejpam-3504	109	58	,	,	PUNCT
ejpam-3504	109	59	we	we	PRON
ejpam-3504	109	60	have	have	VERB
ejpam-3504	109	61	fa(a	fa(a	VERB
ejpam-3504	109	62	◦	◦	NOUN
ejpam-3504	109	63	b	b	NOUN
ejpam-3504	109	64	)	)	PUNCT
ejpam-3504	109	65	≥	≥	NOUN
ejpam-3504	109	66	min{fa(a	min{fa(a	PROPN
ejpam-3504	109	67	)	)	PUNCT
ejpam-3504	109	68	,	,	PUNCT
ejpam-3504	109	69	fa(b	fa(b	NOUN
ejpam-3504	109	70	)	)	PUNCT
ejpam-3504	109	71	}	}	PUNCT
ejpam-3504	109	72	and	and	CCONJ
ejpam-3504	109	73	,	,	PUNCT
ejpam-3504	109	74	since	since	SCONJ
ejpam-3504	109	75	u	u	PROPN
ejpam-3504	109	76	∈	∈	PROPN
ejpam-3504	109	77	a	a	DET
ejpam-3504	109	78	◦	◦	NOUN
ejpam-3504	109	79	b	b	NUM
ejpam-3504	109	80	,	,	PUNCT
ejpam-3504	109	81	we	we	PRON
ejpam-3504	109	82	have	have	VERB
ejpam-3504	109	83	fa(u	fa(u	NOUN
ejpam-3504	109	84	)	)	PUNCT
ejpam-3504	109	85	≥	≥	X
ejpam-3504	109	86	min{fa(a	min{fa(a	PROPN
ejpam-3504	109	87	)	)	PUNCT
ejpam-3504	109	88	,	,	PUNCT
ejpam-3504	109	89	fa(b	fa(b	NOUN
ejpam-3504	109	90	)	)	PUNCT
ejpam-3504	109	91	}	}	PUNCT
ejpam-3504	109	92	.	.	PUNCT
ejpam-3504	110	1	on	on	ADP
ejpam-3504	110	2	the	the	DET
ejpam-3504	110	3	other	other	ADJ
ejpam-3504	110	4	hand	hand	NOUN
ejpam-3504	110	5	,	,	PUNCT
ejpam-3504	110	6	since	since	SCONJ
ejpam-3504	110	7	a	a	DET
ejpam-3504	110	8	,	,	PUNCT
ejpam-3504	110	9	b	b	PROPN
ejpam-3504	110	10	∈	∈	PROPN
ejpam-3504	110	11	a	a	X
ejpam-3504	110	12	,	,	PUNCT
ejpam-3504	110	13	we	we	PRON
ejpam-3504	110	14	have	have	VERB
ejpam-3504	110	15	fa(a	fa(a	VERB
ejpam-3504	110	16	)	)	PUNCT
ejpam-3504	110	17	=	=	SYM
ejpam-3504	110	18	fa(b	fa(b	X
ejpam-3504	110	19	)	)	PUNCT
ejpam-3504	110	20	=	=	SYM
ejpam-3504	110	21	1	1	NUM
ejpam-3504	110	22	,	,	PUNCT
ejpam-3504	110	23	and	and	CCONJ
ejpam-3504	110	24	then	then	ADV
ejpam-3504	110	25	fa(u	fa(u	NOUN
ejpam-3504	110	26	)	)	PUNCT
ejpam-3504	110	27	≥	≥	NOUN
ejpam-3504	111	1	1	1	NUM
ejpam-3504	111	2	.	.	PUNCT
ejpam-3504	112	1	since	since	SCONJ
ejpam-3504	112	2	u	u	PROPN
ejpam-3504	112	3	∈	∈	PROPN
ejpam-3504	112	4	s	s	PART
ejpam-3504	112	5	,	,	PUNCT
ejpam-3504	112	6	we	we	PRON
ejpam-3504	112	7	have	have	VERB
ejpam-3504	112	8	fa(u	fa(u	NOUN
ejpam-3504	112	9	)	)	PUNCT
ejpam-3504	112	10	≤	≤	NUM
ejpam-3504	112	11	1	1	NUM
ejpam-3504	112	12	.	.	PUNCT
ejpam-3504	113	1	thus	thus	ADV
ejpam-3504	113	2	we	we	PRON
ejpam-3504	113	3	have	have	VERB
ejpam-3504	113	4	fa(u	fa(u	X
ejpam-3504	113	5	)	)	PUNCT
ejpam-3504	114	1	=	=	SYM
ejpam-3504	114	2	1	1	NUM
ejpam-3504	114	3	,	,	PUNCT
ejpam-3504	114	4	and	and	CCONJ
ejpam-3504	114	5	u	u	PROPN
ejpam-3504	114	6	∈	∈	PROPN
ejpam-3504	114	7	a.	a.	NOUN
ejpam-3504	114	8	therefore	therefore	ADV
ejpam-3504	114	9	a	a	DET
ejpam-3504	114	10	∗a	∗a	PROPN
ejpam-3504	114	11	⊆	⊆	NUM
ejpam-3504	114	12	a	a	PRON
ejpam-3504	114	13	and	and	CCONJ
ejpam-3504	114	14	the	the	DET
ejpam-3504	114	15	proof	proof	NOUN
ejpam-3504	114	16	is	be	AUX
ejpam-3504	114	17	complete	complete	ADJ
ejpam-3504	114	18	.	.	PUNCT
ejpam-3504	115	1	�	�	PROPN
ejpam-3504	115	2	corollary	corollary	ADJ
ejpam-3504	115	3	3.5	3.5	NUM
ejpam-3504	115	4	.	.	PUNCT
ejpam-3504	116	1	a	a	DET
ejpam-3504	116	2	nonempty	nonempty	NOUN
ejpam-3504	116	3	subset	subset	VERB
ejpam-3504	116	4	a	a	PRON
ejpam-3504	116	5	of	of	ADP
ejpam-3504	116	6	an	an	DET
ejpam-3504	116	7	hypergroupoid	hypergroupoid	PROPN
ejpam-3504	116	8	s	s	PART
ejpam-3504	116	9	is	be	AUX
ejpam-3504	116	10	a	a	DET
ejpam-3504	116	11	subgroupoid	subgroupoid	NOUN
ejpam-3504	116	12	of	of	ADP
ejpam-3504	116	13	s	s	PRON
ejpam-3504	116	14	if	if	SCONJ
ejpam-3504	116	15	and	and	CCONJ
ejpam-3504	116	16	only	only	ADV
ejpam-3504	116	17	if	if	SCONJ
ejpam-3504	116	18	it	it	PRON
ejpam-3504	116	19	is	be	AUX
ejpam-3504	116	20	a	a	DET
ejpam-3504	116	21	fuzzy	fuzzy	ADJ
ejpam-3504	116	22	subgroupoid	subgroupoid	NOUN
ejpam-3504	116	23	of	of	ADP
ejpam-3504	116	24	s.	s.	PROPN
ejpam-3504	116	25	definition	definition	NOUN
ejpam-3504	116	26	3.6	3.6	NUM
ejpam-3504	116	27	.	.	PUNCT
ejpam-3504	117	1	an	an	DET
ejpam-3504	117	2	hypergroupoid	hypergroupoid	PROPN
ejpam-3504	117	3	s	s	PART
ejpam-3504	117	4	is	be	AUX
ejpam-3504	117	5	called	call	VERB
ejpam-3504	117	6	left	left	ADJ
ejpam-3504	117	7	(	(	PUNCT
ejpam-3504	117	8	resp	resp	NOUN
ejpam-3504	117	9	.	.	PUNCT
ejpam-3504	118	1	right	right	ADJ
ejpam-3504	118	2	)	)	PUNCT
ejpam-3504	118	3	zero	zero	NUM
ejpam-3504	118	4	if	if	SCONJ
ejpam-3504	118	5	,	,	PUNCT
ejpam-3504	118	6	for	for	ADP
ejpam-3504	118	7	every	every	DET
ejpam-3504	118	8	x	x	NOUN
ejpam-3504	118	9	,	,	PUNCT
ejpam-3504	118	10	y	y	PROPN
ejpam-3504	118	11	∈	∈	PROPN
ejpam-3504	118	12	s	s	X
ejpam-3504	118	13	,	,	PUNCT
ejpam-3504	118	14	we	we	PRON
ejpam-3504	118	15	have	have	VERB
ejpam-3504	119	1	x	x	X
ejpam-3504	119	2	∈	∈	NOUN
ejpam-3504	119	3	x	x	PUNCT
ejpam-3504	119	4	◦	◦	NOUN
ejpam-3504	119	5	y	y	PROPN
ejpam-3504	119	6	(	(	PUNCT
ejpam-3504	119	7	resp	resp	NOUN
ejpam-3504	119	8	.	.	PUNCT
ejpam-3504	120	1	y	y	PROPN
ejpam-3504	120	2	∈	∈	PROPN
ejpam-3504	120	3	x	x	PUNCT
ejpam-3504	120	4	◦	◦	NOUN
ejpam-3504	120	5	y	y	PROPN
ejpam-3504	120	6	)	)	PUNCT
ejpam-3504	120	7	.	.	PUNCT
ejpam-3504	121	1	following	follow	VERB
ejpam-3504	121	2	kuroki	kuroki	PROPN
ejpam-3504	121	3	,	,	PUNCT
ejpam-3504	121	4	a	a	DET
ejpam-3504	121	5	fuzzy	fuzzy	ADJ
ejpam-3504	121	6	subset	subset	NOUN
ejpam-3504	121	7	f	f	PROPN
ejpam-3504	121	8	of	of	ADP
ejpam-3504	121	9	an	an	DET
ejpam-3504	121	10	hypergroupoid	hypergroupoid	PROPN
ejpam-3504	121	11	s	s	NOUN
ejpam-3504	121	12	is	be	AUX
ejpam-3504	121	13	said	say	VERB
ejpam-3504	121	14	to	to	PART
ejpam-3504	121	15	be	be	AUX
ejpam-3504	121	16	a	a	DET
ejpam-3504	121	17	constant	constant	ADJ
ejpam-3504	121	18	function	function	NOUN
ejpam-3504	121	19	if	if	SCONJ
ejpam-3504	121	20	,	,	PUNCT
ejpam-3504	121	21	for	for	ADP
ejpam-3504	121	22	any	any	DET
ejpam-3504	121	23	a	a	NOUN
ejpam-3504	121	24	,	,	PUNCT
ejpam-3504	121	25	b	b	PROPN
ejpam-3504	121	26	∈	∈	PROPN
ejpam-3504	121	27	s	s	X
ejpam-3504	121	28	,	,	PUNCT
ejpam-3504	121	29	we	we	PRON
ejpam-3504	121	30	have	have	VERB
ejpam-3504	121	31	f(a	f(a	NOUN
ejpam-3504	121	32	)	)	PUNCT
ejpam-3504	121	33	=	=	SYM
ejpam-3504	121	34	f(b	f(b	PROPN
ejpam-3504	121	35	)	)	PUNCT
ejpam-3504	121	36	.	.	PUNCT
ejpam-3504	122	1	n.	n.	PROPN
ejpam-3504	122	2	kehayopulu	kehayopulu	PROPN
ejpam-3504	122	3	/	/	SYM
ejpam-3504	122	4	eur	eur	PROPN
ejpam-3504	122	5	.	.	PUNCT
ejpam-3504	123	1	j.	j.	PROPN
ejpam-3504	123	2	pure	pure	PROPN
ejpam-3504	123	3	appl	appl	PROPN
ejpam-3504	123	4	.	.	PROPN
ejpam-3504	123	5	math	math	PROPN
ejpam-3504	123	6	,	,	PUNCT
ejpam-3504	123	7	12	12	NUM
ejpam-3504	123	8	(	(	PUNCT
ejpam-3504	123	9	3	3	NUM
ejpam-3504	123	10	)	)	PUNCT
ejpam-3504	123	11	(	(	PUNCT
ejpam-3504	123	12	2019	2019	NUM
ejpam-3504	123	13	)	)	PUNCT
ejpam-3504	123	14	,	,	PUNCT
ejpam-3504	123	15	709	709	NUM
ejpam-3504	123	16	-	-	SYM
ejpam-3504	123	17	721	721	NUM
ejpam-3504	123	18	713	713	NUM
ejpam-3504	123	19	proposition	proposition	NOUN
ejpam-3504	123	20	3.7	3.7	NUM
ejpam-3504	123	21	.	.	PUNCT
ejpam-3504	124	1	if	if	SCONJ
ejpam-3504	124	2	an	an	DET
ejpam-3504	124	3	hypergroupoid	hypergroupoid	PROPN
ejpam-3504	124	4	s	s	NOUN
ejpam-3504	124	5	is	be	AUX
ejpam-3504	124	6	left	leave	VERB
ejpam-3504	124	7	(	(	PUNCT
ejpam-3504	124	8	resp	resp	NOUN
ejpam-3504	124	9	.	.	PUNCT
ejpam-3504	125	1	right	right	ADJ
ejpam-3504	125	2	)	)	PUNCT
ejpam-3504	125	3	zero	zero	NUM
ejpam-3504	125	4	,	,	PUNCT
ejpam-3504	125	5	then	then	ADV
ejpam-3504	125	6	every	every	DET
ejpam-3504	125	7	fuzzy	fuzzy	ADJ
ejpam-3504	125	8	left	left	NOUN
ejpam-3504	125	9	(	(	PUNCT
ejpam-3504	125	10	resp	resp	NOUN
ejpam-3504	125	11	.	.	PUNCT
ejpam-3504	126	1	fuzzy	fuzzy	ADJ
ejpam-3504	126	2	right	right	ADJ
ejpam-3504	126	3	)	)	PUNCT
ejpam-3504	126	4	ideal	ideal	PROPN
ejpam-3504	126	5	f	f	PROPN
ejpam-3504	126	6	of	of	ADP
ejpam-3504	126	7	s	s	PROPN
ejpam-3504	126	8	is	be	AUX
ejpam-3504	126	9	a	a	DET
ejpam-3504	126	10	constant	constant	ADJ
ejpam-3504	126	11	function	function	NOUN
ejpam-3504	126	12	.	.	PUNCT
ejpam-3504	127	1	proof	proof	NOUN
ejpam-3504	127	2	.	.	PUNCT
ejpam-3504	128	1	let	let	VERB
ejpam-3504	128	2	s	s	PRON
ejpam-3504	128	3	be	be	AUX
ejpam-3504	128	4	left	leave	VERB
ejpam-3504	128	5	zero	zero	NUM
ejpam-3504	128	6	,	,	PUNCT
ejpam-3504	128	7	f	f	PROPN
ejpam-3504	128	8	a	a	DET
ejpam-3504	128	9	fuzzy	fuzzy	ADJ
ejpam-3504	128	10	left	leave	VERB
ejpam-3504	128	11	ideal	ideal	NOUN
ejpam-3504	128	12	of	of	ADP
ejpam-3504	128	13	s	s	PRON
ejpam-3504	128	14	and	and	CCONJ
ejpam-3504	128	15	a	a	DET
ejpam-3504	128	16	,	,	PUNCT
ejpam-3504	128	17	b	b	PROPN
ejpam-3504	128	18	∈	∈	PROPN
ejpam-3504	128	19	s.	s.	PROPN
ejpam-3504	128	20	since	since	SCONJ
ejpam-3504	128	21	s	s	PROPN
ejpam-3504	128	22	is	be	AUX
ejpam-3504	128	23	left	leave	VERB
ejpam-3504	128	24	zero	zero	NUM
ejpam-3504	128	25	,	,	PUNCT
ejpam-3504	128	26	for	for	ADP
ejpam-3504	128	27	every	every	DET
ejpam-3504	128	28	x	x	NOUN
ejpam-3504	128	29	,	,	PUNCT
ejpam-3504	128	30	y	y	PROPN
ejpam-3504	128	31	∈	∈	PROPN
ejpam-3504	128	32	s	s	X
ejpam-3504	128	33	,	,	PUNCT
ejpam-3504	128	34	we	we	PRON
ejpam-3504	128	35	have	have	VERB
ejpam-3504	128	36	x	x	X
ejpam-3504	128	37	∈	∈	NOUN
ejpam-3504	128	38	x	x	PUNCT
ejpam-3504	128	39	◦	◦	NOUN
ejpam-3504	128	40	y.	y.	NOUN
ejpam-3504	129	1	so	so	ADV
ejpam-3504	129	2	,	,	PUNCT
ejpam-3504	129	3	for	for	ADP
ejpam-3504	129	4	the	the	DET
ejpam-3504	129	5	elements	element	NOUN
ejpam-3504	129	6	a	a	DET
ejpam-3504	129	7	,	,	PUNCT
ejpam-3504	129	8	b	b	PROPN
ejpam-3504	129	9	of	of	ADP
ejpam-3504	129	10	s	s	PROPN
ejpam-3504	129	11	,	,	PUNCT
ejpam-3504	129	12	we	we	PRON
ejpam-3504	129	13	have	have	VERB
ejpam-3504	129	14	a	a	DET
ejpam-3504	129	15	∈	∈	PROPN
ejpam-3504	129	16	a	a	DET
ejpam-3504	129	17	◦	◦	NOUN
ejpam-3504	129	18	b	b	NOUN
ejpam-3504	129	19	and	and	CCONJ
ejpam-3504	129	20	b	b	PROPN
ejpam-3504	129	21	∈	∈	PROPN
ejpam-3504	129	22	b	b	PROPN
ejpam-3504	129	23	◦	◦	NOUN
ejpam-3504	129	24	a.	a.	NOUN
ejpam-3504	129	25	since	since	SCONJ
ejpam-3504	129	26	f	f	PROPN
ejpam-3504	129	27	is	be	AUX
ejpam-3504	129	28	a	a	DET
ejpam-3504	129	29	fuzzy	fuzzy	ADJ
ejpam-3504	129	30	left	leave	VERB
ejpam-3504	129	31	ideal	ideal	NOUN
ejpam-3504	129	32	of	of	ADP
ejpam-3504	129	33	s	s	PROPN
ejpam-3504	129	34	,	,	PUNCT
ejpam-3504	129	35	we	we	PRON
ejpam-3504	129	36	have	have	VERB
ejpam-3504	129	37	f(a	f(a	NOUN
ejpam-3504	129	38	◦	◦	NOUN
ejpam-3504	129	39	b	b	NUM
ejpam-3504	129	40	)	)	PUNCT
ejpam-3504	129	41	≥	≥	NOUN
ejpam-3504	129	42	f(b	f(b	PROPN
ejpam-3504	129	43	)	)	PUNCT
ejpam-3504	129	44	and	and	CCONJ
ejpam-3504	129	45	f(b	f(b	PROPN
ejpam-3504	129	46	◦	◦	VERB
ejpam-3504	129	47	a	a	DET
ejpam-3504	129	48	)	)	PUNCT
ejpam-3504	129	49	≥	≥	NOUN
ejpam-3504	129	50	f(a	f(a	NOUN
ejpam-3504	129	51	)	)	PUNCT
ejpam-3504	129	52	.	.	PUNCT
ejpam-3504	130	1	since	since	SCONJ
ejpam-3504	130	2	a	a	DET
ejpam-3504	130	3	∈	∈	PROPN
ejpam-3504	130	4	a	a	DET
ejpam-3504	130	5	◦	◦	NOUN
ejpam-3504	130	6	b	b	NUM
ejpam-3504	130	7	,	,	PUNCT
ejpam-3504	130	8	we	we	PRON
ejpam-3504	130	9	have	have	VERB
ejpam-3504	130	10	f(a	f(a	PROPN
ejpam-3504	130	11	)	)	PUNCT
ejpam-3504	130	12	≥	≥	NOUN
ejpam-3504	130	13	f(b	f(b	PROPN
ejpam-3504	130	14	)	)	PUNCT
ejpam-3504	130	15	and	and	CCONJ
ejpam-3504	130	16	since	since	SCONJ
ejpam-3504	130	17	b	b	PROPN
ejpam-3504	130	18	∈	∈	PROPN
ejpam-3504	130	19	b	b	PROPN
ejpam-3504	130	20	◦	◦	NOUN
ejpam-3504	130	21	a	a	X
ejpam-3504	130	22	,	,	PUNCT
ejpam-3504	130	23	we	we	PRON
ejpam-3504	130	24	have	have	VERB
ejpam-3504	130	25	f(b	f(b	PROPN
ejpam-3504	130	26	)	)	PUNCT
ejpam-3504	130	27	≥	≥	NOUN
ejpam-3504	130	28	f(a	f(a	NOUN
ejpam-3504	130	29	)	)	PUNCT
ejpam-3504	130	30	;	;	PUNCT
ejpam-3504	130	31	thus	thus	ADV
ejpam-3504	130	32	we	we	PRON
ejpam-3504	130	33	have	have	VERB
ejpam-3504	130	34	f(a	f(a	NOUN
ejpam-3504	130	35	)	)	PUNCT
ejpam-3504	130	36	=	=	SYM
ejpam-3504	130	37	f(b	f(b	PROPN
ejpam-3504	130	38	)	)	PUNCT
ejpam-3504	130	39	.	.	PUNCT
ejpam-3504	131	1	let	let	VERB
ejpam-3504	131	2	now	now	ADV
ejpam-3504	131	3	s	s	AUX
ejpam-3504	131	4	be	be	AUX
ejpam-3504	131	5	right	right	ADJ
ejpam-3504	131	6	zero	zero	NUM
ejpam-3504	131	7	,	,	PUNCT
ejpam-3504	131	8	f	f	PROPN
ejpam-3504	131	9	a	a	DET
ejpam-3504	131	10	fuzzy	fuzzy	ADJ
ejpam-3504	131	11	right	right	ADJ
ejpam-3504	131	12	ideal	ideal	NOUN
ejpam-3504	131	13	of	of	ADP
ejpam-3504	131	14	s	s	PRON
ejpam-3504	131	15	and	and	CCONJ
ejpam-3504	131	16	a	a	PRON
ejpam-3504	131	17	,	,	PUNCT
ejpam-3504	131	18	b	b	X
ejpam-3504	131	19	∈	∈	PROPN
ejpam-3504	131	20	s.	s.	PROPN
ejpam-3504	131	21	then	then	ADV
ejpam-3504	131	22	a	a	DET
ejpam-3504	131	23	∈	∈	PROPN
ejpam-3504	131	24	b	b	PROPN
ejpam-3504	131	25	◦	◦	NOUN
ejpam-3504	131	26	a	a	PRON
ejpam-3504	131	27	,	,	PUNCT
ejpam-3504	131	28	b	b	X
ejpam-3504	131	29	∈	∈	PROPN
ejpam-3504	131	30	a	a	DET
ejpam-3504	131	31	◦	◦	NOUN
ejpam-3504	131	32	b	b	NOUN
ejpam-3504	131	33	,	,	PUNCT
ejpam-3504	131	34	f(b	f(b	PROPN
ejpam-3504	131	35	◦	◦	NOUN
ejpam-3504	131	36	a	a	X
ejpam-3504	131	37	)	)	PUNCT
ejpam-3504	131	38	≥	≥	NOUN
ejpam-3504	131	39	f(b	f(b	PROPN
ejpam-3504	131	40	)	)	PUNCT
ejpam-3504	131	41	and	and	CCONJ
ejpam-3504	131	42	f(a	f(a	PROPN
ejpam-3504	131	43	◦	◦	PROPN
ejpam-3504	131	44	b	b	NUM
ejpam-3504	131	45	)	)	PUNCT
ejpam-3504	131	46	≥	≥	NOUN
ejpam-3504	131	47	f(a	f(a	NOUN
ejpam-3504	131	48	)	)	PUNCT
ejpam-3504	131	49	.	.	PUNCT
ejpam-3504	132	1	thus	thus	ADV
ejpam-3504	132	2	we	we	PRON
ejpam-3504	132	3	get	get	VERB
ejpam-3504	132	4	f(a	f(a	NOUN
ejpam-3504	132	5	)	)	PUNCT
ejpam-3504	132	6	≥	≥	NOUN
ejpam-3504	132	7	f(b	f(b	PROPN
ejpam-3504	132	8	)	)	PUNCT
ejpam-3504	132	9	and	and	CCONJ
ejpam-3504	132	10	f(b	f(b	PROPN
ejpam-3504	132	11	)	)	PUNCT
ejpam-3504	132	12	≥	≥	NOUN
ejpam-3504	132	13	f(a	f(a	NOUN
ejpam-3504	132	14	)	)	PUNCT
ejpam-3504	132	15	and	and	CCONJ
ejpam-3504	132	16	so	so	ADV
ejpam-3504	132	17	f(a	f(a	NOUN
ejpam-3504	132	18	)	)	PUNCT
ejpam-3504	132	19	=	=	SYM
ejpam-3504	132	20	f(b	f(b	PROPN
ejpam-3504	132	21	)	)	PUNCT
ejpam-3504	132	22	.	.	PUNCT
ejpam-3504	133	1	�	�	PROPN
ejpam-3504	133	2	proposition	proposition	NOUN
ejpam-3504	133	3	3.8	3.8	NUM
ejpam-3504	133	4	.	.	PUNCT
ejpam-3504	134	1	if	if	SCONJ
ejpam-3504	134	2	(	(	PUNCT
ejpam-3504	134	3	s	s	NOUN
ejpam-3504	134	4	,	,	PUNCT
ejpam-3504	134	5	◦	◦	NOUN
ejpam-3504	134	6	)	)	PUNCT
ejpam-3504	134	7	is	be	AUX
ejpam-3504	134	8	a	a	DET
ejpam-3504	134	9	fuzzy	fuzzy	ADJ
ejpam-3504	134	10	left	left	NOUN
ejpam-3504	134	11	(	(	PUNCT
ejpam-3504	134	12	resp	resp	NOUN
ejpam-3504	134	13	.	.	PUNCT
ejpam-3504	135	1	fuzzy	fuzzy	ADJ
ejpam-3504	135	2	right	right	NOUN
ejpam-3504	135	3	)	)	PUNCT
ejpam-3504	136	1	duo	duo	NOUN
ejpam-3504	136	2	hypergroupoid	hypergroupoid	NOUN
ejpam-3504	136	3	,	,	PUNCT
ejpam-3504	136	4	then	then	ADV
ejpam-3504	136	5	it	it	PRON
ejpam-3504	136	6	is	be	AUX
ejpam-3504	136	7	left	leave	VERB
ejpam-3504	136	8	(	(	PUNCT
ejpam-3504	136	9	resp	resp	NOUN
ejpam-3504	136	10	.	.	PUNCT
ejpam-3504	137	1	right	right	ADJ
ejpam-3504	137	2	)	)	PUNCT
ejpam-3504	137	3	duo	duo	NOUN
ejpam-3504	137	4	.	.	PUNCT
ejpam-3504	138	1	proof	proof	NOUN
ejpam-3504	138	2	.	.	PUNCT
ejpam-3504	139	1	let	let	VERB
ejpam-3504	139	2	a	a	DET
ejpam-3504	139	3	be	be	AUX
ejpam-3504	139	4	a	a	DET
ejpam-3504	139	5	left	left	ADJ
ejpam-3504	139	6	ideal	ideal	NOUN
ejpam-3504	139	7	of	of	ADP
ejpam-3504	139	8	s.	s.	PROPN
ejpam-3504	139	9	then	then	ADV
ejpam-3504	139	10	,	,	PUNCT
ejpam-3504	139	11	by	by	ADP
ejpam-3504	139	12	lemma	lemma	PROPN
ejpam-3504	139	13	3.1	3.1	NUM
ejpam-3504	139	14	,	,	PUNCT
ejpam-3504	139	15	the	the	DET
ejpam-3504	139	16	characteristic	characteristic	ADJ
ejpam-3504	139	17	function	function	NOUN
ejpam-3504	139	18	fa	fa	PROPN
ejpam-3504	139	19	is	be	AUX
ejpam-3504	139	20	a	a	DET
ejpam-3504	139	21	fuzzy	fuzzy	ADJ
ejpam-3504	139	22	left	leave	VERB
ejpam-3504	139	23	ideal	ideal	NOUN
ejpam-3504	139	24	of	of	ADP
ejpam-3504	139	25	s.	s.	PROPN
ejpam-3504	139	26	by	by	ADP
ejpam-3504	139	27	assumption	assumption	NOUN
ejpam-3504	139	28	,	,	PUNCT
ejpam-3504	139	29	fa	fa	PROPN
ejpam-3504	139	30	is	be	AUX
ejpam-3504	139	31	a	a	DET
ejpam-3504	139	32	fuzzy	fuzzy	ADJ
ejpam-3504	139	33	right	right	ADJ
ejpam-3504	139	34	ideal	ideal	NOUN
ejpam-3504	139	35	of	of	ADP
ejpam-3504	139	36	s	s	PRON
ejpam-3504	139	37	as	as	ADV
ejpam-3504	139	38	well	well	ADV
ejpam-3504	139	39	.	.	PUNCT
ejpam-3504	140	1	since	since	SCONJ
ejpam-3504	140	2	a	a	PRON
ejpam-3504	140	3	is	be	AUX
ejpam-3504	140	4	a	a	DET
ejpam-3504	140	5	nonempty	nonempty	ADV
ejpam-3504	140	6	set	set	VERB
ejpam-3504	140	7	and	and	CCONJ
ejpam-3504	140	8	fa	fa	PROPN
ejpam-3504	140	9	is	be	AUX
ejpam-3504	140	10	a	a	DET
ejpam-3504	140	11	fuzzy	fuzzy	ADJ
ejpam-3504	140	12	right	right	ADJ
ejpam-3504	140	13	ideal	ideal	NOUN
ejpam-3504	140	14	of	of	ADP
ejpam-3504	140	15	s	s	NOUN
ejpam-3504	140	16	,	,	PUNCT
ejpam-3504	140	17	again	again	ADV
ejpam-3504	140	18	by	by	ADP
ejpam-3504	140	19	lemma	lemma	PROPN
ejpam-3504	140	20	3.1	3.1	NUM
ejpam-3504	140	21	,	,	PUNCT
ejpam-3504	140	22	a	a	PRON
ejpam-3504	140	23	is	be	AUX
ejpam-3504	140	24	a	a	DET
ejpam-3504	140	25	right	right	ADJ
ejpam-3504	140	26	ideal	ideal	NOUN
ejpam-3504	140	27	of	of	ADP
ejpam-3504	140	28	s	s	PRON
ejpam-3504	140	29	and	and	CCONJ
ejpam-3504	140	30	so	so	ADV
ejpam-3504	140	31	s	s	NOUN
ejpam-3504	140	32	is	be	AUX
ejpam-3504	140	33	left	leave	VERB
ejpam-3504	140	34	duo	duo	NOUN
ejpam-3504	140	35	.	.	PUNCT
ejpam-3504	141	1	for	for	ADP
ejpam-3504	141	2	fuzzy	fuzzy	ADJ
ejpam-3504	141	3	right	right	ADJ
ejpam-3504	141	4	duo	duo	NOUN
ejpam-3504	141	5	hypergroupoids	hypergroupoid	NOUN
ejpam-3504	141	6	the	the	DET
ejpam-3504	141	7	proof	proof	NOUN
ejpam-3504	141	8	is	be	AUX
ejpam-3504	141	9	analogous	analogous	ADJ
ejpam-3504	141	10	.	.	PUNCT
ejpam-3504	142	1	�	�	PROPN
ejpam-3504	142	2	proposition	proposition	NOUN
ejpam-3504	142	3	3.9	3.9	NUM
ejpam-3504	142	4	.	.	PUNCT
ejpam-3504	143	1	let	let	AUX
ejpam-3504	143	2	(	(	PUNCT
ejpam-3504	143	3	s	s	NOUN
ejpam-3504	143	4	,	,	PUNCT
ejpam-3504	143	5	◦	◦	NOUN
ejpam-3504	143	6	)	)	PUNCT
ejpam-3504	143	7	be	be	AUX
ejpam-3504	143	8	a	a	DET
ejpam-3504	143	9	regular	regular	ADJ
ejpam-3504	143	10	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	143	11	.	.	PUNCT
ejpam-3504	144	1	if	if	SCONJ
ejpam-3504	144	2	s	s	NOUN
ejpam-3504	144	3	is	be	AUX
ejpam-3504	144	4	left	leave	VERB
ejpam-3504	144	5	(	(	PUNCT
ejpam-3504	144	6	resp	resp	NOUN
ejpam-3504	144	7	.	.	PUNCT
ejpam-3504	145	1	right	right	ADJ
ejpam-3504	145	2	)	)	PUNCT
ejpam-3504	145	3	duo	duo	NOUN
ejpam-3504	145	4	,	,	PUNCT
ejpam-3504	145	5	then	then	ADV
ejpam-3504	145	6	it	it	PRON
ejpam-3504	145	7	is	be	AUX
ejpam-3504	145	8	fuzzy	fuzzy	ADJ
ejpam-3504	145	9	left	left	ADJ
ejpam-3504	145	10	(	(	PUNCT
ejpam-3504	145	11	resp	resp	NOUN
ejpam-3504	145	12	.	.	PUNCT
ejpam-3504	146	1	fuzzy	fuzzy	ADJ
ejpam-3504	146	2	right	right	ADJ
ejpam-3504	146	3	)	)	PUNCT
ejpam-3504	146	4	duo	duo	NOUN
ejpam-3504	146	5	.	.	PUNCT
ejpam-3504	147	1	proof	proof	NOUN
ejpam-3504	147	2	.	.	PUNCT
ejpam-3504	148	1	let	let	VERB
ejpam-3504	148	2	s	s	PRON
ejpam-3504	148	3	be	be	AUX
ejpam-3504	148	4	left	leave	VERB
ejpam-3504	148	5	duo	duo	NOUN
ejpam-3504	149	1	and	and	CCONJ
ejpam-3504	149	2	f	f	PROPN
ejpam-3504	149	3	be	be	AUX
ejpam-3504	149	4	a	a	DET
ejpam-3504	149	5	fuzzy	fuzzy	ADJ
ejpam-3504	149	6	left	leave	VERB
ejpam-3504	149	7	ideal	ideal	NOUN
ejpam-3504	149	8	of	of	ADP
ejpam-3504	149	9	s.	s.	PROPN
ejpam-3504	149	10	then	then	ADV
ejpam-3504	149	11	f	f	PROPN
ejpam-3504	149	12	is	be	AUX
ejpam-3504	149	13	a	a	DET
ejpam-3504	149	14	fuzzy	fuzzy	ADJ
ejpam-3504	149	15	right	right	ADJ
ejpam-3504	149	16	ideal	ideal	NOUN
ejpam-3504	149	17	of	of	ADP
ejpam-3504	149	18	s	s	PROPN
ejpam-3504	149	19	,	,	PUNCT
ejpam-3504	149	20	that	that	PRON
ejpam-3504	149	21	is	be	AUX
ejpam-3504	149	22	f(x	f(x	PROPN
ejpam-3504	149	23	◦	◦	PROPN
ejpam-3504	149	24	y	y	PROPN
ejpam-3504	149	25	)	)	PUNCT
ejpam-3504	149	26	≥	≥	NOUN
ejpam-3504	149	27	f(x	f(x	PROPN
ejpam-3504	149	28	)	)	PUNCT
ejpam-3504	149	29	for	for	ADP
ejpam-3504	149	30	every	every	DET
ejpam-3504	149	31	x	x	PROPN
ejpam-3504	149	32	,	,	PUNCT
ejpam-3504	149	33	y	y	PROPN
ejpam-3504	149	34	∈	∈	PROPN
ejpam-3504	149	35	s.	s.	PROPN
ejpam-3504	149	36	indeed	indeed	ADV
ejpam-3504	149	37	:	:	PUNCT
ejpam-3504	149	38	let	let	VERB
ejpam-3504	149	39	x	x	PRON
ejpam-3504	149	40	,	,	PUNCT
ejpam-3504	149	41	y	y	PROPN
ejpam-3504	149	42	∈	∈	PROPN
ejpam-3504	149	43	s	s	PART
ejpam-3504	149	44	and	and	CCONJ
ejpam-3504	149	45	u	u	PROPN
ejpam-3504	149	46	∈	∈	PROPN
ejpam-3504	149	47	x	x	PUNCT
ejpam-3504	149	48	◦	◦	NOUN
ejpam-3504	149	49	y.	y.	NOUN
ejpam-3504	149	50	since	since	SCONJ
ejpam-3504	149	51	s	s	PROPN
ejpam-3504	149	52	is	be	AUX
ejpam-3504	149	53	regular	regular	ADJ
ejpam-3504	149	54	,	,	PUNCT
ejpam-3504	149	55	we	we	PRON
ejpam-3504	149	56	have	have	VERB
ejpam-3504	149	57	x	x	PART
ejpam-3504	149	58	◦	◦	VERB
ejpam-3504	149	59	y	y	PROPN
ejpam-3504	149	60	⊆	⊆	NUM
ejpam-3504	149	61	(	(	PUNCT
ejpam-3504	149	62	x	x	X
ejpam-3504	149	63	∗s	∗s	ADP
ejpam-3504	149	64	∗x	∗x	NOUN
ejpam-3504	149	65	)	)	PUNCT
ejpam-3504	149	66	∗	∗	NOUN
ejpam-3504	149	67	{	{	PUNCT
ejpam-3504	149	68	y	y	NOUN
ejpam-3504	149	69	}	}	PUNCT
ejpam-3504	149	70	⊆	⊆	NUM
ejpam-3504	149	71	(	(	PUNCT
ejpam-3504	149	72	s	s	NOUN
ejpam-3504	149	73	∗x	∗x	NOUN
ejpam-3504	149	74	)	)	PUNCT
ejpam-3504	149	75	∗s	∗s	NOUN
ejpam-3504	149	76	.	.	PUNCT
ejpam-3504	150	1	since	since	SCONJ
ejpam-3504	150	2	s	s	PRON
ejpam-3504	150	3	∗x	∗x	PRON
ejpam-3504	150	4	is	be	AUX
ejpam-3504	150	5	a	a	DET
ejpam-3504	150	6	left	left	ADJ
ejpam-3504	150	7	ideal	ideal	NOUN
ejpam-3504	150	8	of	of	ADP
ejpam-3504	150	9	s	s	PROPN
ejpam-3504	150	10	,	,	PUNCT
ejpam-3504	150	11	by	by	ADP
ejpam-3504	150	12	hypothesis	hypothesis	NOUN
ejpam-3504	150	13	,	,	PUNCT
ejpam-3504	150	14	it	it	PRON
ejpam-3504	150	15	is	be	AUX
ejpam-3504	150	16	a	a	DET
ejpam-3504	150	17	right	right	ADJ
ejpam-3504	150	18	ideal	ideal	NOUN
ejpam-3504	150	19	of	of	ADP
ejpam-3504	150	20	s	s	PRON
ejpam-3504	150	21	as	as	ADV
ejpam-3504	150	22	well	well	ADV
ejpam-3504	150	23	and	and	CCONJ
ejpam-3504	150	24	so	so	ADV
ejpam-3504	150	25	(	(	PUNCT
ejpam-3504	150	26	s	s	NOUN
ejpam-3504	150	27	∗	∗	X
ejpam-3504	150	28	x	x	NOUN
ejpam-3504	150	29	)	)	PUNCT
ejpam-3504	150	30	∗	∗	NOUN
ejpam-3504	150	31	s	s	NOUN
ejpam-3504	150	32	⊆	⊆	NUM
ejpam-3504	150	33	s	s	NOUN
ejpam-3504	150	34	∗	∗	NOUN
ejpam-3504	150	35	x	x	NOUN
ejpam-3504	150	36	,	,	PUNCT
ejpam-3504	150	37	hence	hence	ADV
ejpam-3504	150	38	x	x	PUNCT
ejpam-3504	150	39	◦	◦	NOUN
ejpam-3504	150	40	y	y	NUM
ejpam-3504	151	1	⊆	⊆	NUM
ejpam-3504	151	2	s	s	PART
ejpam-3504	151	3	∗	∗	NOUN
ejpam-3504	151	4	x	x	PUNCT
ejpam-3504	151	5	and	and	CCONJ
ejpam-3504	151	6	u	u	PROPN
ejpam-3504	151	7	∈	∈	PROPN
ejpam-3504	151	8	s	s	PART
ejpam-3504	151	9	∗	∗	NOUN
ejpam-3504	151	10	x.	x.	NOUN
ejpam-3504	151	11	then	then	ADV
ejpam-3504	151	12	there	there	PRON
ejpam-3504	151	13	exists	exist	VERB
ejpam-3504	151	14	v	v	ADP
ejpam-3504	151	15	∈	∈	PROPN
ejpam-3504	151	16	s	s	VERB
ejpam-3504	151	17	such	such	ADJ
ejpam-3504	151	18	that	that	SCONJ
ejpam-3504	151	19	u	u	PROPN
ejpam-3504	151	20	∈	∈	PROPN
ejpam-3504	151	21	v	v	ADP
ejpam-3504	151	22	◦	◦	NOUN
ejpam-3504	151	23	x.	x.	NOUN
ejpam-3504	151	24	since	since	SCONJ
ejpam-3504	151	25	f	f	PROPN
ejpam-3504	151	26	is	be	AUX
ejpam-3504	151	27	a	a	DET
ejpam-3504	151	28	fuzzy	fuzzy	ADJ
ejpam-3504	151	29	left	leave	VERB
ejpam-3504	151	30	ideal	ideal	NOUN
ejpam-3504	151	31	of	of	ADP
ejpam-3504	151	32	s	s	PROPN
ejpam-3504	151	33	,	,	PUNCT
ejpam-3504	151	34	we	we	PRON
ejpam-3504	151	35	have	have	VERB
ejpam-3504	151	36	f(v	f(v	VERB
ejpam-3504	151	37	◦	◦	NOUN
ejpam-3504	151	38	x	x	SYM
ejpam-3504	151	39	)	)	PUNCT
ejpam-3504	151	40	≥	≥	PROPN
ejpam-3504	151	41	f(x	f(x	PROPN
ejpam-3504	151	42	)	)	PUNCT
ejpam-3504	151	43	and	and	CCONJ
ejpam-3504	151	44	,	,	PUNCT
ejpam-3504	151	45	since	since	SCONJ
ejpam-3504	151	46	u	u	PROPN
ejpam-3504	151	47	∈	∈	PROPN
ejpam-3504	151	48	v	v	ADP
ejpam-3504	151	49	◦	◦	NOUN
ejpam-3504	151	50	x	x	SYM
ejpam-3504	151	51	,	,	PUNCT
ejpam-3504	151	52	we	we	PRON
ejpam-3504	151	53	have	have	VERB
ejpam-3504	151	54	f(u	f(u	PROPN
ejpam-3504	151	55	)	)	PUNCT
ejpam-3504	151	56	≥	≥	NOUN
ejpam-3504	151	57	f(x	f(x	PROPN
ejpam-3504	151	58	)	)	PUNCT
ejpam-3504	151	59	.	.	PUNCT
ejpam-3504	152	1	thus	thus	ADV
ejpam-3504	152	2	f	f	PROPN
ejpam-3504	152	3	is	be	AUX
ejpam-3504	152	4	a	a	DET
ejpam-3504	152	5	fuzzy	fuzzy	ADJ
ejpam-3504	152	6	right	right	ADJ
ejpam-3504	152	7	ideal	ideal	NOUN
ejpam-3504	152	8	of	of	ADP
ejpam-3504	152	9	s	s	PRON
ejpam-3504	152	10	and	and	CCONJ
ejpam-3504	152	11	f	f	PROPN
ejpam-3504	152	12	is	be	AUX
ejpam-3504	152	13	fuzzy	fuzzy	ADJ
ejpam-3504	152	14	left	leave	VERB
ejpam-3504	152	15	duo	duo	NOUN
ejpam-3504	152	16	.	.	PUNCT
ejpam-3504	153	1	let	let	VERB
ejpam-3504	153	2	now	now	ADV
ejpam-3504	153	3	s	s	AUX
ejpam-3504	153	4	be	be	AUX
ejpam-3504	153	5	right	right	ADJ
ejpam-3504	153	6	duo	duo	NOUN
ejpam-3504	153	7	,	,	PUNCT
ejpam-3504	153	8	f	f	PROPN
ejpam-3504	153	9	a	a	DET
ejpam-3504	153	10	fuzzy	fuzzy	ADJ
ejpam-3504	153	11	right	right	ADJ
ejpam-3504	153	12	ideal	ideal	NOUN
ejpam-3504	153	13	of	of	ADP
ejpam-3504	153	14	s	s	PROPN
ejpam-3504	153	15	,	,	PUNCT
ejpam-3504	153	16	x	x	PRON
ejpam-3504	153	17	,	,	PUNCT
ejpam-3504	153	18	y	y	PROPN
ejpam-3504	153	19	∈	∈	PROPN
ejpam-3504	153	20	s	s	PART
ejpam-3504	153	21	and	and	CCONJ
ejpam-3504	153	22	u	u	PROPN
ejpam-3504	153	23	∈	∈	PROPN
ejpam-3504	153	24	x	x	PUNCT
ejpam-3504	153	25	◦	◦	NOUN
ejpam-3504	153	26	y.	y.	NOUN
ejpam-3504	153	27	since	since	SCONJ
ejpam-3504	153	28	s	s	PROPN
ejpam-3504	153	29	is	be	AUX
ejpam-3504	153	30	regular	regular	ADJ
ejpam-3504	153	31	,	,	PUNCT
ejpam-3504	153	32	we	we	PRON
ejpam-3504	153	33	have	have	VERB
ejpam-3504	153	34	x	x	PART
ejpam-3504	153	35	◦	◦	VERB
ejpam-3504	153	36	y	y	PROPN
ejpam-3504	153	37	⊆	⊆	NUM
ejpam-3504	153	38	x	x	SYM
ejpam-3504	153	39	∗	∗	NOUN
ejpam-3504	153	40	(	(	PUNCT
ejpam-3504	153	41	y	y	PROPN
ejpam-3504	153	42	∗s	∗s	PROPN
ejpam-3504	153	43	∗	∗	PROPN
ejpam-3504	153	44	y	y	NOUN
ejpam-3504	153	45	)	)	PUNCT
ejpam-3504	153	46	⊆	⊆	NUM
ejpam-3504	153	47	s	s	NOUN
ejpam-3504	153	48	∗	∗	NOUN
ejpam-3504	153	49	(	(	PUNCT
ejpam-3504	153	50	y	y	PROPN
ejpam-3504	153	51	∗s	∗s	PROPN
ejpam-3504	153	52	)	)	PUNCT
ejpam-3504	153	53	⊆	⊆	NUM
ejpam-3504	153	54	y	y	PROPN
ejpam-3504	153	55	∗s	∗s	PROPN
ejpam-3504	153	56	.	.	PUNCT
ejpam-3504	154	1	then	then	ADV
ejpam-3504	154	2	u	u	PROPN
ejpam-3504	154	3	∈	∈	PROPN
ejpam-3504	154	4	y	y	PROPN
ejpam-3504	154	5	◦	◦	NOUN
ejpam-3504	154	6	s	s	VERB
ejpam-3504	154	7	for	for	ADP
ejpam-3504	154	8	some	some	DET
ejpam-3504	154	9	s	s	PART
ejpam-3504	154	10	∈	∈	PROPN
ejpam-3504	154	11	s	s	NOUN
ejpam-3504	154	12	,	,	PUNCT
ejpam-3504	154	13	f(y	f(y	PROPN
ejpam-3504	154	14	◦	◦	PROPN
ejpam-3504	154	15	s	s	PART
ejpam-3504	154	16	)	)	PUNCT
ejpam-3504	154	17	≥	≥	NOUN
ejpam-3504	154	18	f(y	f(y	NOUN
ejpam-3504	154	19	)	)	PUNCT
ejpam-3504	154	20	and	and	CCONJ
ejpam-3504	154	21	f(u	f(u	PROPN
ejpam-3504	154	22	)	)	PUNCT
ejpam-3504	154	23	≥	≥	NOUN
ejpam-3504	154	24	f(y	f(y	NOUN
ejpam-3504	154	25	)	)	PUNCT
ejpam-3504	154	26	.	.	PUNCT
ejpam-3504	155	1	thus	thus	ADV
ejpam-3504	155	2	f	f	PROPN
ejpam-3504	155	3	is	be	AUX
ejpam-3504	155	4	a	a	DET
ejpam-3504	155	5	fuzzy	fuzzy	ADJ
ejpam-3504	155	6	left	leave	VERB
ejpam-3504	155	7	ideal	ideal	NOUN
ejpam-3504	155	8	of	of	ADP
ejpam-3504	155	9	s.	s.	PROPN
ejpam-3504	155	10	�	�	PROPN
ejpam-3504	155	11	by	by	ADP
ejpam-3504	155	12	propositions	proposition	NOUN
ejpam-3504	155	13	3.8	3.8	NUM
ejpam-3504	155	14	and	and	CCONJ
ejpam-3504	155	15	3.9	3.9	NUM
ejpam-3504	155	16	,	,	PUNCT
ejpam-3504	155	17	we	we	PRON
ejpam-3504	155	18	have	have	VERB
ejpam-3504	155	19	the	the	DET
ejpam-3504	155	20	following	follow	VERB
ejpam-3504	155	21	theorem	theorem	VERB
ejpam-3504	155	22	.	.	PUNCT
ejpam-3504	155	23	theorem	theorem	VERB
ejpam-3504	155	24	3.10	3.10	NUM
ejpam-3504	155	25	.	.	PUNCT
ejpam-3504	156	1	let	let	VERB
ejpam-3504	156	2	s	s	PRON
ejpam-3504	156	3	be	be	AUX
ejpam-3504	156	4	a	a	DET
ejpam-3504	156	5	regular	regular	ADJ
ejpam-3504	156	6	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	156	7	.	.	PUNCT
ejpam-3504	157	1	then	then	ADV
ejpam-3504	157	2	we	we	PRON
ejpam-3504	157	3	have	have	VERB
ejpam-3504	157	4	the	the	DET
ejpam-3504	157	5	following	following	NOUN
ejpam-3504	157	6	:	:	PUNCT
ejpam-3504	157	7	(	(	PUNCT
ejpam-3504	157	8	1	1	X
ejpam-3504	157	9	)	)	PUNCT
ejpam-3504	157	10	s	s	VERB
ejpam-3504	157	11	is	be	AUX
ejpam-3504	157	12	left	leave	VERB
ejpam-3504	157	13	duo	duo	NOUN
ejpam-3504	157	14	if	if	SCONJ
ejpam-3504	158	1	and	and	CCONJ
ejpam-3504	158	2	only	only	ADV
ejpam-3504	158	3	if	if	SCONJ
ejpam-3504	158	4	it	it	PRON
ejpam-3504	158	5	is	be	AUX
ejpam-3504	158	6	fuzzy	fuzzy	ADJ
ejpam-3504	158	7	left	leave	VERB
ejpam-3504	158	8	duo	duo	NOUN
ejpam-3504	158	9	.	.	PUNCT
ejpam-3504	159	1	(	(	PUNCT
ejpam-3504	159	2	2	2	X
ejpam-3504	159	3	)	)	PUNCT
ejpam-3504	159	4	s	s	VERB
ejpam-3504	159	5	is	be	AUX
ejpam-3504	159	6	right	right	ADJ
ejpam-3504	159	7	duo	duo	NOUN
ejpam-3504	159	8	if	if	SCONJ
ejpam-3504	160	1	and	and	CCONJ
ejpam-3504	160	2	only	only	ADV
ejpam-3504	160	3	if	if	SCONJ
ejpam-3504	160	4	it	it	PRON
ejpam-3504	160	5	is	be	AUX
ejpam-3504	160	6	fuzzy	fuzzy	ADJ
ejpam-3504	160	7	right	right	ADJ
ejpam-3504	160	8	duo	duo	NOUN
ejpam-3504	160	9	.	.	PUNCT
ejpam-3504	161	1	as	as	ADP
ejpam-3504	161	2	a	a	DET
ejpam-3504	161	3	consequence	consequence	NOUN
ejpam-3504	161	4	,	,	PUNCT
ejpam-3504	161	5	a	a	DET
ejpam-3504	161	6	regular	regular	ADJ
ejpam-3504	161	7	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	161	8	is	be	AUX
ejpam-3504	161	9	duo	duo	ADJ
ejpam-3504	161	10	if	if	SCONJ
ejpam-3504	161	11	and	and	CCONJ
ejpam-3504	161	12	only	only	ADV
ejpam-3504	161	13	if	if	SCONJ
ejpam-3504	161	14	it	it	PRON
ejpam-3504	161	15	is	be	AUX
ejpam-3504	161	16	fuzzy	fuzzy	ADJ
ejpam-3504	161	17	duo	duo	NOUN
ejpam-3504	161	18	.	.	PUNCT
ejpam-3504	162	1	lemma	lemma	PROPN
ejpam-3504	162	2	3.11	3.11	NUM
ejpam-3504	162	3	.	.	PUNCT
ejpam-3504	163	1	let	let	AUX
ejpam-3504	163	2	(	(	PUNCT
ejpam-3504	163	3	s	s	NOUN
ejpam-3504	163	4	,	,	PUNCT
ejpam-3504	163	5	◦	◦	NOUN
ejpam-3504	163	6	)	)	PUNCT
ejpam-3504	163	7	be	be	VERB
ejpam-3504	163	8	an	an	DET
ejpam-3504	163	9	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	163	10	.	.	PUNCT
ejpam-3504	164	1	if	if	SCONJ
ejpam-3504	164	2	a	a	PRON
ejpam-3504	164	3	is	be	AUX
ejpam-3504	164	4	a	a	DET
ejpam-3504	164	5	bi	bi	NOUN
ejpam-3504	164	6	-	-	NOUN
ejpam-3504	164	7	ideal	ideal	NOUN
ejpam-3504	164	8	of	of	ADP
ejpam-3504	164	9	s	s	PROPN
ejpam-3504	164	10	,	,	PUNCT
ejpam-3504	164	11	then	then	ADV
ejpam-3504	164	12	the	the	DET
ejpam-3504	164	13	characteristic	characteristic	ADJ
ejpam-3504	164	14	function	function	NOUN
ejpam-3504	164	15	fa	fa	PROPN
ejpam-3504	164	16	is	be	AUX
ejpam-3504	164	17	a	a	DET
ejpam-3504	164	18	fuzzy	fuzzy	ADJ
ejpam-3504	164	19	bi	bi	NOUN
ejpam-3504	164	20	-	-	NOUN
ejpam-3504	164	21	ideal	ideal	NOUN
ejpam-3504	164	22	of	of	ADP
ejpam-3504	164	23	s.	s.	PROPN
ejpam-3504	164	24	“	"	PUNCT
ejpam-3504	164	25	conversely	conversely	ADV
ejpam-3504	164	26	”	"	PUNCT
ejpam-3504	164	27	,	,	PUNCT
ejpam-3504	164	28	if	if	SCONJ
ejpam-3504	164	29	a	a	PRON
ejpam-3504	164	30	is	be	AUX
ejpam-3504	164	31	a	a	DET
ejpam-3504	164	32	nonempty	nonempty	ADJ
ejpam-3504	164	33	subset	subset	NOUN
ejpam-3504	164	34	of	of	ADP
ejpam-3504	164	35	s	s	PRON
ejpam-3504	164	36	such	such	ADJ
ejpam-3504	164	37	that	that	SCONJ
ejpam-3504	164	38	fa	fa	PROPN
ejpam-3504	164	39	is	be	AUX
ejpam-3504	164	40	a	a	DET
ejpam-3504	164	41	fuzzy	fuzzy	ADJ
ejpam-3504	164	42	bi	bi	NOUN
ejpam-3504	164	43	-	-	NOUN
ejpam-3504	164	44	ideal	ideal	NOUN
ejpam-3504	164	45	of	of	ADP
ejpam-3504	164	46	s	s	PROPN
ejpam-3504	164	47	,	,	PUNCT
ejpam-3504	164	48	then	then	ADV
ejpam-3504	164	49	a	a	PRON
ejpam-3504	164	50	is	be	AUX
ejpam-3504	164	51	a	a	DET
ejpam-3504	164	52	bi	bi	NOUN
ejpam-3504	164	53	-	-	NOUN
ejpam-3504	164	54	ideal	ideal	NOUN
ejpam-3504	164	55	of	of	ADP
ejpam-3504	164	56	s.	s.	PROPN
ejpam-3504	164	57	proof	proof	PROPN
ejpam-3504	164	58	.	.	PUNCT
ejpam-3504	165	1	=	=	NOUN
ejpam-3504	165	2	⇒.	⇒.	NOUN
ejpam-3504	165	3	let	let	VERB
ejpam-3504	165	4	a	a	PRON
ejpam-3504	165	5	be	be	AUX
ejpam-3504	165	6	a	a	DET
ejpam-3504	165	7	bi	bi	NOUN
ejpam-3504	165	8	-	-	NOUN
ejpam-3504	165	9	ideal	ideal	NOUN
ejpam-3504	165	10	of	of	ADP
ejpam-3504	165	11	s	s	PRON
ejpam-3504	165	12	and	and	CCONJ
ejpam-3504	165	13	x	x	PROPN
ejpam-3504	165	14	,	,	PUNCT
ejpam-3504	165	15	y	y	PROPN
ejpam-3504	165	16	,	,	PUNCT
ejpam-3504	165	17	z	z	PROPN
ejpam-3504	165	18	∈	∈	PROPN
ejpam-3504	165	19	s.	s.	PROPN
ejpam-3504	165	20	then	then	ADV
ejpam-3504	165	21	fa	fa	INTJ
ejpam-3504	165	22	(	(	PUNCT
ejpam-3504	165	23	(	(	PUNCT
ejpam-3504	165	24	x	x	SYM
ejpam-3504	165	25	◦	◦	VERB
ejpam-3504	165	26	y	y	NOUN
ejpam-3504	165	27	)	)	PUNCT
ejpam-3504	165	28	∗	∗	NOUN
ejpam-3504	165	29	z	z	NOUN
ejpam-3504	165	30	)	)	PUNCT
ejpam-3504	165	31	≥	≥	X
ejpam-3504	165	32	min{fa(x	min{fa(x	NOUN
ejpam-3504	165	33	)	)	PUNCT
ejpam-3504	165	34	,	,	PUNCT
ejpam-3504	165	35	fa(z	fa(z	PUNCT
ejpam-3504	165	36	)	)	PUNCT
ejpam-3504	165	37	}	}	PUNCT
ejpam-3504	165	38	(	(	PUNCT
ejpam-3504	165	39	1	1	X
ejpam-3504	165	40	)	)	PUNCT
ejpam-3504	165	41	n.	n.	NOUN
ejpam-3504	165	42	kehayopulu	kehayopulu	PROPN
ejpam-3504	165	43	/	/	SYM
ejpam-3504	165	44	eur	eur	PROPN
ejpam-3504	165	45	.	.	PUNCT
ejpam-3504	166	1	j.	j.	PROPN
ejpam-3504	166	2	pure	pure	PROPN
ejpam-3504	166	3	appl	appl	PROPN
ejpam-3504	166	4	.	.	PROPN
ejpam-3504	166	5	math	math	PROPN
ejpam-3504	166	6	,	,	PUNCT
ejpam-3504	166	7	12	12	NUM
ejpam-3504	166	8	(	(	PUNCT
ejpam-3504	166	9	3	3	NUM
ejpam-3504	166	10	)	)	PUNCT
ejpam-3504	166	11	(	(	PUNCT
ejpam-3504	166	12	2019	2019	NUM
ejpam-3504	166	13	)	)	PUNCT
ejpam-3504	166	14	,	,	PUNCT
ejpam-3504	166	15	709	709	NUM
ejpam-3504	166	16	-	-	SYM
ejpam-3504	166	17	721	721	NUM
ejpam-3504	166	18	714	714	NUM
ejpam-3504	166	19	in	in	ADP
ejpam-3504	166	20	fact	fact	NOUN
ejpam-3504	166	21	:	:	PUNCT
ejpam-3504	166	22	let	let	VERB
ejpam-3504	166	23	u	u	PRON
ejpam-3504	166	24	∈	∈	PROPN
ejpam-3504	166	25	(	(	PUNCT
ejpam-3504	166	26	x	x	NOUN
ejpam-3504	166	27	◦	◦	NOUN
ejpam-3504	166	28	y)∗z	y)∗z	NOUN
ejpam-3504	166	29	.	.	PUNCT
ejpam-3504	167	1	if	if	SCONJ
ejpam-3504	167	2	x	x	SYM
ejpam-3504	167	3	∈	∈	PROPN
ejpam-3504	167	4	a	a	PRON
ejpam-3504	167	5	and	and	CCONJ
ejpam-3504	167	6	z	z	NOUN
ejpam-3504	167	7	∈	∈	PROPN
ejpam-3504	167	8	a	a	PRON
ejpam-3504	167	9	,	,	PUNCT
ejpam-3504	167	10	then	then	ADV
ejpam-3504	167	11	fa(x	fa(x	NOUN
ejpam-3504	167	12	)	)	PUNCT
ejpam-3504	167	13	=	=	PUNCT
ejpam-3504	167	14	fa(z	fa(z	PUNCT
ejpam-3504	167	15	)	)	PUNCT
ejpam-3504	167	16	=	=	SYM
ejpam-3504	167	17	1	1	NUM
ejpam-3504	167	18	,	,	PUNCT
ejpam-3504	167	19	min{fa(x	min{fa(x	NOUN
ejpam-3504	167	20	)	)	PUNCT
ejpam-3504	167	21	,	,	PUNCT
ejpam-3504	167	22	fa(z	fa(z	PUNCT
ejpam-3504	167	23	)	)	PUNCT
ejpam-3504	167	24	}	}	PUNCT
ejpam-3504	167	25	=	=	SYM
ejpam-3504	167	26	1	1	NUM
ejpam-3504	167	27	and	and	CCONJ
ejpam-3504	167	28	u	u	PROPN
ejpam-3504	167	29	∈	∈	PROPN
ejpam-3504	167	30	a	a	DET
ejpam-3504	167	31	∗	∗	NOUN
ejpam-3504	167	32	s	s	PART
ejpam-3504	167	33	∗	∗	NOUN
ejpam-3504	167	34	a	a	DET
ejpam-3504	167	35	⊆	⊆	NUM
ejpam-3504	167	36	a	a	NOUN
ejpam-3504	167	37	,	,	PUNCT
ejpam-3504	167	38	so	so	CCONJ
ejpam-3504	167	39	u	u	PROPN
ejpam-3504	167	40	∈	∈	PROPN
ejpam-3504	167	41	a	a	PRON
ejpam-3504	167	42	and	and	CCONJ
ejpam-3504	167	43	fa(u	fa(u	NOUN
ejpam-3504	167	44	)	)	PUNCT
ejpam-3504	167	45	=	=	PUNCT
ejpam-3504	168	1	1	1	X
ejpam-3504	168	2	.	.	PUNCT
ejpam-3504	168	3	thus	thus	ADV
ejpam-3504	168	4	we	we	PRON
ejpam-3504	168	5	have	have	VERB
ejpam-3504	168	6	fa(u	fa(u	X
ejpam-3504	168	7	)	)	PUNCT
ejpam-3504	169	1	=	=	SYM
ejpam-3504	169	2	1	1	NUM
ejpam-3504	169	3	≥	≥	NOUN
ejpam-3504	169	4	min{fa(x	min{fa(x	NOUN
ejpam-3504	169	5	)	)	PUNCT
ejpam-3504	169	6	,	,	PUNCT
ejpam-3504	169	7	fa(z	fa(z	PUNCT
ejpam-3504	169	8	)	)	PUNCT
ejpam-3504	169	9	}	}	PUNCT
ejpam-3504	169	10	and	and	CCONJ
ejpam-3504	169	11	property	property	NOUN
ejpam-3504	169	12	(	(	PUNCT
ejpam-3504	169	13	1	1	NUM
ejpam-3504	169	14	)	)	PUNCT
ejpam-3504	169	15	is	be	AUX
ejpam-3504	169	16	satisfied	satisfied	ADJ
ejpam-3504	169	17	.	.	PUNCT
ejpam-3504	170	1	if	if	SCONJ
ejpam-3504	170	2	x	x	X
ejpam-3504	170	3	/∈	/∈	NOUN
ejpam-3504	171	1	a	a	PRON
ejpam-3504	171	2	or	or	CCONJ
ejpam-3504	171	3	z	z	NOUN
ejpam-3504	171	4	/∈	/∈	PUNCT
ejpam-3504	172	1	a	a	DET
ejpam-3504	172	2	,	,	PUNCT
ejpam-3504	172	3	then	then	ADV
ejpam-3504	172	4	fa(x	fa(x	NOUN
ejpam-3504	172	5	)	)	PUNCT
ejpam-3504	172	6	=	=	SYM
ejpam-3504	172	7	0	0	NUM
ejpam-3504	172	8	or	or	CCONJ
ejpam-3504	172	9	fa(z	fa(z	PUNCT
ejpam-3504	172	10	)	)	PUNCT
ejpam-3504	172	11	=	=	SYM
ejpam-3504	172	12	0	0	NUM
ejpam-3504	172	13	,	,	PUNCT
ejpam-3504	172	14	so	so	ADV
ejpam-3504	172	15	min{fa(x	min{fa(x	NOUN
ejpam-3504	172	16	)	)	PUNCT
ejpam-3504	172	17	,	,	PUNCT
ejpam-3504	172	18	fa(z	fa(z	PUNCT
ejpam-3504	172	19	)	)	PUNCT
ejpam-3504	172	20	}	}	PUNCT
ejpam-3504	173	1	=	=	SYM
ejpam-3504	173	2	0	0	X
ejpam-3504	173	3	.	.	PUNCT
ejpam-3504	174	1	since	since	SCONJ
ejpam-3504	174	2	u	u	PROPN
ejpam-3504	174	3	∈	∈	PROPN
ejpam-3504	174	4	s	s	PART
ejpam-3504	174	5	,	,	PUNCT
ejpam-3504	174	6	we	we	PRON
ejpam-3504	174	7	have	have	VERB
ejpam-3504	174	8	fa(u	fa(u	NOUN
ejpam-3504	174	9	)	)	PUNCT
ejpam-3504	174	10	≥	≥	NOUN
ejpam-3504	174	11	0	0	NUM
ejpam-3504	174	12	,	,	PUNCT
ejpam-3504	174	13	then	then	ADV
ejpam-3504	174	14	fa(u	fa(u	X
ejpam-3504	174	15	)	)	PUNCT
ejpam-3504	175	1	=	=	SYM
ejpam-3504	175	2	0	0	NUM
ejpam-3504	175	3	≥	≥	NOUN
ejpam-3504	175	4	min{fa(x	min{fa(x	NOUN
ejpam-3504	175	5	)	)	PUNCT
ejpam-3504	175	6	,	,	PUNCT
ejpam-3504	175	7	fa(z	fa(z	PUNCT
ejpam-3504	175	8	)	)	PUNCT
ejpam-3504	175	9	}	}	PUNCT
ejpam-3504	175	10	and	and	CCONJ
ejpam-3504	175	11	again	again	ADV
ejpam-3504	175	12	property	property	NOUN
ejpam-3504	175	13	(	(	PUNCT
ejpam-3504	175	14	1	1	NUM
ejpam-3504	175	15	)	)	PUNCT
ejpam-3504	175	16	holds	hold	NOUN
ejpam-3504	175	17	.	.	PUNCT
ejpam-3504	176	1	⇐	⇐	PROPN
ejpam-3504	176	2	=	=	PRON
ejpam-3504	176	3	.	.	PUNCT
ejpam-3504	177	1	let	let	VERB
ejpam-3504	177	2	a	a	DET
ejpam-3504	177	3	be	be	AUX
ejpam-3504	177	4	a	a	DET
ejpam-3504	177	5	nonempty	nonempty	ADJ
ejpam-3504	177	6	subset	subset	NOUN
ejpam-3504	177	7	of	of	ADP
ejpam-3504	177	8	s	s	PRON
ejpam-3504	177	9	and	and	CCONJ
ejpam-3504	177	10	fa	fa	PROPN
ejpam-3504	177	11	be	be	AUX
ejpam-3504	177	12	a	a	DET
ejpam-3504	177	13	fuzzy	fuzzy	ADJ
ejpam-3504	177	14	bi	bi	NOUN
ejpam-3504	177	15	-	-	NOUN
ejpam-3504	177	16	ideal	ideal	NOUN
ejpam-3504	177	17	of	of	ADP
ejpam-3504	177	18	s.	s.	PROPN
ejpam-3504	177	19	then	then	ADV
ejpam-3504	177	20	a	a	PRON
ejpam-3504	177	21	is	be	AUX
ejpam-3504	177	22	a	a	DET
ejpam-3504	177	23	bi	bi	NOUN
ejpam-3504	177	24	-	-	NOUN
ejpam-3504	177	25	ideal	ideal	NOUN
ejpam-3504	177	26	of	of	ADP
ejpam-3504	177	27	s	s	PROPN
ejpam-3504	177	28	,	,	PUNCT
ejpam-3504	177	29	that	that	PRON
ejpam-3504	177	30	is	be	AUX
ejpam-3504	177	31	a	a	DET
ejpam-3504	177	32	∗	∗	NOUN
ejpam-3504	177	33	s	s	PART
ejpam-3504	177	34	∗	∗	NOUN
ejpam-3504	177	35	a	a	DET
ejpam-3504	177	36	⊆	⊆	NUM
ejpam-3504	177	37	a.	a.	NOUN
ejpam-3504	177	38	indeed	indeed	ADV
ejpam-3504	177	39	:	:	PUNCT
ejpam-3504	177	40	let	let	VERB
ejpam-3504	177	41	x	x	PART
ejpam-3504	177	42	∈	∈	VERB
ejpam-3504	177	43	a	a	DET
ejpam-3504	177	44	∗	∗	NOUN
ejpam-3504	177	45	s	s	PART
ejpam-3504	177	46	∗	∗	NOUN
ejpam-3504	177	47	a.	a.	NOUN
ejpam-3504	177	48	then	then	ADV
ejpam-3504	177	49	x	x	SYM
ejpam-3504	177	50	∈	∈	PROPN
ejpam-3504	177	51	u	u	NOUN
ejpam-3504	177	52	◦	◦	NOUN
ejpam-3504	177	53	a	a	PRON
ejpam-3504	177	54	for	for	ADP
ejpam-3504	177	55	some	some	DET
ejpam-3504	177	56	u	u	NOUN
ejpam-3504	177	57	∈	∈	PROPN
ejpam-3504	177	58	a	a	DET
ejpam-3504	177	59	∗s	∗s	NOUN
ejpam-3504	177	60	,	,	PUNCT
ejpam-3504	177	61	a	a	DET
ejpam-3504	177	62	∈	∈	PROPN
ejpam-3504	177	63	a	a	PRON
ejpam-3504	177	64	and	and	CCONJ
ejpam-3504	177	65	u	u	NOUN
ejpam-3504	177	66	∈	∈	PROPN
ejpam-3504	177	67	v	v	ADP
ejpam-3504	177	68	◦	◦	NOUN
ejpam-3504	177	69	s	s	NUM
ejpam-3504	177	70	for	for	ADP
ejpam-3504	177	71	some	some	PRON
ejpam-3504	177	72	v	v	ADP
ejpam-3504	177	73	∈	∈	PROPN
ejpam-3504	177	74	a	a	PRON
ejpam-3504	177	75	,	,	PUNCT
ejpam-3504	177	76	s	s	PROPN
ejpam-3504	177	77	∈	∈	PROPN
ejpam-3504	177	78	s.	s.	PROPN
ejpam-3504	177	79	then	then	ADV
ejpam-3504	177	80	we	we	PRON
ejpam-3504	177	81	have	have	VERB
ejpam-3504	177	82	x	x	X
ejpam-3504	177	83	∈	∈	PROPN
ejpam-3504	177	84	u	u	NOUN
ejpam-3504	177	85	◦	◦	NOUN
ejpam-3504	177	86	a	a	DET
ejpam-3504	177	87	⊆	⊆	NUM
ejpam-3504	177	88	(	(	PUNCT
ejpam-3504	177	89	v	v	NUM
ejpam-3504	177	90	◦	◦	NOUN
ejpam-3504	177	91	s	s	PART
ejpam-3504	177	92	)	)	PUNCT
ejpam-3504	177	93	∗	∗	NOUN
ejpam-3504	177	94	a.	a.	NOUN
ejpam-3504	177	95	since	since	SCONJ
ejpam-3504	177	96	fa	fa	PROPN
ejpam-3504	177	97	is	be	AUX
ejpam-3504	177	98	a	a	DET
ejpam-3504	177	99	fuzzy	fuzzy	ADJ
ejpam-3504	177	100	bi	bi	NOUN
ejpam-3504	177	101	-	-	NOUN
ejpam-3504	177	102	ideal	ideal	NOUN
ejpam-3504	177	103	of	of	ADP
ejpam-3504	177	104	s	s	PROPN
ejpam-3504	177	105	,	,	PUNCT
ejpam-3504	177	106	we	we	PRON
ejpam-3504	177	107	have	have	VERB
ejpam-3504	177	108	fa	fa	INTJ
ejpam-3504	177	109	(	(	PUNCT
ejpam-3504	177	110	(	(	PUNCT
ejpam-3504	177	111	v	v	NUM
ejpam-3504	177	112	◦	◦	NOUN
ejpam-3504	177	113	s	s	PART
ejpam-3504	177	114	)	)	PUNCT
ejpam-3504	177	115	∗	∗	NOUN
ejpam-3504	177	116	a	a	DET
ejpam-3504	177	117	)	)	PUNCT
ejpam-3504	177	118	≥	≥	NOUN
ejpam-3504	177	119	min{fa(v	min{fa(v	PROPN
ejpam-3504	177	120	)	)	PUNCT
ejpam-3504	177	121	,	,	PUNCT
ejpam-3504	177	122	fa(a	fa(a	NOUN
ejpam-3504	177	123	)	)	PUNCT
ejpam-3504	177	124	}	}	PUNCT
ejpam-3504	177	125	and	and	CCONJ
ejpam-3504	177	126	,	,	PUNCT
ejpam-3504	177	127	since	since	SCONJ
ejpam-3504	177	128	x	x	PROPN
ejpam-3504	177	129	∈	∈	PROPN
ejpam-3504	177	130	(	(	PUNCT
ejpam-3504	177	131	v	v	NOUN
ejpam-3504	177	132	∗	∗	PRON
ejpam-3504	177	133	s	s	NOUN
ejpam-3504	177	134	)	)	PUNCT
ejpam-3504	177	135	∗	∗	NOUN
ejpam-3504	177	136	a	a	X
ejpam-3504	177	137	,	,	PUNCT
ejpam-3504	177	138	we	we	PRON
ejpam-3504	177	139	have	have	VERB
ejpam-3504	177	140	fa(x	fa(x	NOUN
ejpam-3504	177	141	)	)	PUNCT
ejpam-3504	177	142	≥	≥	PROPN
ejpam-3504	177	143	min{fa(v	min{fa(v	PROPN
ejpam-3504	177	144	)	)	PUNCT
ejpam-3504	177	145	,	,	PUNCT
ejpam-3504	177	146	fa(a	fa(a	NOUN
ejpam-3504	177	147	)	)	PUNCT
ejpam-3504	177	148	}	}	PUNCT
ejpam-3504	177	149	.	.	PUNCT
ejpam-3504	178	1	since	since	SCONJ
ejpam-3504	178	2	v	v	NOUN
ejpam-3504	178	3	,	,	PUNCT
ejpam-3504	178	4	a	a	DET
ejpam-3504	178	5	∈	∈	PROPN
ejpam-3504	178	6	a	a	X
ejpam-3504	178	7	,	,	PUNCT
ejpam-3504	178	8	we	we	PRON
ejpam-3504	178	9	have	have	VERB
ejpam-3504	178	10	fa(v	fa(v	VERB
ejpam-3504	178	11	)	)	PUNCT
ejpam-3504	178	12	=	=	SYM
ejpam-3504	178	13	fa(a	fa(a	NOUN
ejpam-3504	178	14	)	)	PUNCT
ejpam-3504	178	15	=	=	SYM
ejpam-3504	179	1	1	1	X
ejpam-3504	179	2	.	.	PUNCT
ejpam-3504	179	3	then	then	ADV
ejpam-3504	179	4	fa(x	fa(x	NOUN
ejpam-3504	179	5	)	)	PUNCT
ejpam-3504	179	6	≥	≥	NOUN
ejpam-3504	179	7	1	1	NUM
ejpam-3504	179	8	,	,	PUNCT
ejpam-3504	179	9	hence	hence	ADV
ejpam-3504	179	10	fa(x	fa(x	NOUN
ejpam-3504	179	11	)	)	PUNCT
ejpam-3504	179	12	=	=	SYM
ejpam-3504	179	13	1	1	NUM
ejpam-3504	179	14	and	and	CCONJ
ejpam-3504	179	15	so	so	ADV
ejpam-3504	179	16	x	x	SYM
ejpam-3504	179	17	∈	∈	NOUN
ejpam-3504	179	18	a.	a.	NOUN
ejpam-3504	179	19	hence	hence	ADV
ejpam-3504	179	20	we	we	PRON
ejpam-3504	179	21	obtain	obtain	VERB
ejpam-3504	179	22	a	a	DET
ejpam-3504	179	23	∗	∗	NOUN
ejpam-3504	179	24	s	s	NOUN
ejpam-3504	179	25	∗a	∗a	PROPN
ejpam-3504	179	26	⊆	⊆	PROPN
ejpam-3504	179	27	a	a	PRON
ejpam-3504	179	28	and	and	CCONJ
ejpam-3504	179	29	the	the	DET
ejpam-3504	179	30	proof	proof	NOUN
ejpam-3504	179	31	is	be	AUX
ejpam-3504	179	32	complete	complete	ADJ
ejpam-3504	179	33	.	.	PUNCT
ejpam-3504	180	1	�	�	PROPN
ejpam-3504	180	2	proposition	proposition	PROPN
ejpam-3504	180	3	3.12	3.12	NUM
ejpam-3504	180	4	.	.	PUNCT
ejpam-3504	181	1	let	let	VERB
ejpam-3504	181	2	s	s	PRON
ejpam-3504	181	3	be	be	AUX
ejpam-3504	181	4	an	an	DET
ejpam-3504	181	5	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	181	6	in	in	ADP
ejpam-3504	181	7	which	which	PRON
ejpam-3504	181	8	the	the	DET
ejpam-3504	181	9	fuzzy	fuzzy	ADJ
ejpam-3504	181	10	bi	bi	NOUN
ejpam-3504	181	11	-	-	NOUN
ejpam-3504	181	12	ideals	ideal	NOUN
ejpam-3504	181	13	are	be	AUX
ejpam-3504	181	14	fuzzy	fuzzy	ADJ
ejpam-3504	182	1	right	right	ADJ
ejpam-3504	182	2	(	(	PUNCT
ejpam-3504	182	3	resp	resp	NOUN
ejpam-3504	182	4	.	.	PUNCT
ejpam-3504	183	1	fuzzy	fuzzy	ADJ
ejpam-3504	183	2	left	left	ADJ
ejpam-3504	183	3	)	)	PUNCT
ejpam-3504	183	4	ideals	ideal	NOUN
ejpam-3504	183	5	.	.	PUNCT
ejpam-3504	184	1	then	then	ADV
ejpam-3504	184	2	every	every	DET
ejpam-3504	184	3	bi	bi	NOUN
ejpam-3504	184	4	-	-	NOUN
ejpam-3504	184	5	ideal	ideal	NOUN
ejpam-3504	184	6	of	of	ADP
ejpam-3504	184	7	s	s	PROPN
ejpam-3504	184	8	is	be	AUX
ejpam-3504	184	9	a	a	DET
ejpam-3504	184	10	right	right	ADJ
ejpam-3504	184	11	(	(	PUNCT
ejpam-3504	184	12	resp	resp	NOUN
ejpam-3504	184	13	.	.	PUNCT
ejpam-3504	185	1	left	left	ADJ
ejpam-3504	185	2	)	)	PUNCT
ejpam-3504	185	3	ideal	ideal	NOUN
ejpam-3504	185	4	of	of	ADP
ejpam-3504	185	5	s.	s.	PROPN
ejpam-3504	185	6	proof	proof	PROPN
ejpam-3504	185	7	.	.	PUNCT
ejpam-3504	186	1	assuming	assume	VERB
ejpam-3504	186	2	the	the	DET
ejpam-3504	186	3	fuzzy	fuzzy	ADJ
ejpam-3504	186	4	bi	bi	NOUN
ejpam-3504	186	5	-	-	NOUN
ejpam-3504	186	6	ideals	ideal	NOUN
ejpam-3504	186	7	of	of	ADP
ejpam-3504	186	8	s	s	NOUN
ejpam-3504	186	9	are	be	AUX
ejpam-3504	186	10	fuzzy	fuzzy	ADJ
ejpam-3504	186	11	right	right	ADJ
ejpam-3504	186	12	ideals	ideal	NOUN
ejpam-3504	186	13	,	,	PUNCT
ejpam-3504	186	14	let	let	VERB
ejpam-3504	186	15	a	a	PRON
ejpam-3504	186	16	be	be	AUX
ejpam-3504	186	17	a	a	DET
ejpam-3504	186	18	bi	bi	NOUN
ejpam-3504	186	19	-	-	NOUN
ejpam-3504	186	20	ideal	ideal	NOUN
ejpam-3504	186	21	of	of	ADP
ejpam-3504	186	22	s.	s.	PROPN
ejpam-3504	186	23	by	by	ADP
ejpam-3504	186	24	lemma	lemma	PROPN
ejpam-3504	186	25	3.11	3.11	NUM
ejpam-3504	186	26	,	,	PUNCT
ejpam-3504	186	27	the	the	DET
ejpam-3504	186	28	characteristic	characteristic	ADJ
ejpam-3504	186	29	function	function	NOUN
ejpam-3504	186	30	fa	fa	PROPN
ejpam-3504	186	31	is	be	AUX
ejpam-3504	186	32	a	a	DET
ejpam-3504	186	33	fuzzy	fuzzy	ADJ
ejpam-3504	186	34	bi	bi	NOUN
ejpam-3504	186	35	-	-	NOUN
ejpam-3504	186	36	ideal	ideal	NOUN
ejpam-3504	186	37	of	of	ADP
ejpam-3504	186	38	s.	s.	PROPN
ejpam-3504	186	39	by	by	ADP
ejpam-3504	186	40	the	the	DET
ejpam-3504	186	41	assumption	assumption	NOUN
ejpam-3504	186	42	,	,	PUNCT
ejpam-3504	186	43	fa	fa	PROPN
ejpam-3504	186	44	is	be	AUX
ejpam-3504	186	45	a	a	DET
ejpam-3504	186	46	fuzzy	fuzzy	ADJ
ejpam-3504	186	47	right	right	ADJ
ejpam-3504	186	48	ideal	ideal	NOUN
ejpam-3504	186	49	of	of	ADP
ejpam-3504	186	50	s.	s.	PROPN
ejpam-3504	186	51	since	since	SCONJ
ejpam-3504	186	52	a	a	PRON
ejpam-3504	186	53	is	be	AUX
ejpam-3504	186	54	a	a	DET
ejpam-3504	186	55	nonempty	nonempty	ADV
ejpam-3504	186	56	set	set	VERB
ejpam-3504	186	57	and	and	CCONJ
ejpam-3504	186	58	fa	fa	PROPN
ejpam-3504	186	59	is	be	AUX
ejpam-3504	186	60	a	a	DET
ejpam-3504	186	61	fuzzy	fuzzy	ADJ
ejpam-3504	186	62	right	right	ADJ
ejpam-3504	186	63	ideal	ideal	NOUN
ejpam-3504	186	64	of	of	ADP
ejpam-3504	186	65	s	s	PROPN
ejpam-3504	186	66	,	,	PUNCT
ejpam-3504	186	67	by	by	ADP
ejpam-3504	186	68	lemma	lemma	PROPN
ejpam-3504	186	69	3.1	3.1	NUM
ejpam-3504	186	70	,	,	PUNCT
ejpam-3504	186	71	a	a	PRON
ejpam-3504	186	72	is	be	AUX
ejpam-3504	186	73	a	a	DET
ejpam-3504	186	74	right	right	ADJ
ejpam-3504	186	75	ideal	ideal	NOUN
ejpam-3504	186	76	of	of	ADP
ejpam-3504	186	77	s.	s.	PROPN
ejpam-3504	186	78	the	the	DET
ejpam-3504	186	79	other	other	ADJ
ejpam-3504	186	80	case	case	NOUN
ejpam-3504	186	81	can	can	AUX
ejpam-3504	186	82	be	be	AUX
ejpam-3504	186	83	proved	prove	VERB
ejpam-3504	186	84	similarly	similarly	ADV
ejpam-3504	186	85	.	.	PUNCT
ejpam-3504	187	1	�	�	PROPN
ejpam-3504	187	2	proposition	proposition	NOUN
ejpam-3504	187	3	3.13	3.13	NUM
ejpam-3504	187	4	.	.	PUNCT
ejpam-3504	188	1	let	let	AUX
ejpam-3504	188	2	(	(	PUNCT
ejpam-3504	188	3	s	s	NOUN
ejpam-3504	188	4	,	,	PUNCT
ejpam-3504	188	5	◦	◦	NOUN
ejpam-3504	188	6	)	)	PUNCT
ejpam-3504	188	7	be	be	VERB
ejpam-3504	188	8	a	a	DET
ejpam-3504	188	9	regular	regular	ADJ
ejpam-3504	188	10	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	188	11	in	in	ADP
ejpam-3504	188	12	which	which	PRON
ejpam-3504	188	13	every	every	DET
ejpam-3504	188	14	bi	bi	NOUN
ejpam-3504	188	15	-	-	ADJ
ejpam-3504	188	16	ideal	ideal	ADJ
ejpam-3504	188	17	is	be	AUX
ejpam-3504	188	18	a	a	DET
ejpam-3504	188	19	right	right	ADJ
ejpam-3504	188	20	(	(	PUNCT
ejpam-3504	188	21	resp	resp	NOUN
ejpam-3504	188	22	.	.	PUNCT
ejpam-3504	189	1	left	left	ADJ
ejpam-3504	189	2	)	)	PUNCT
ejpam-3504	189	3	ideal	ideal	NOUN
ejpam-3504	189	4	.	.	PUNCT
ejpam-3504	190	1	then	then	ADV
ejpam-3504	190	2	every	every	DET
ejpam-3504	190	3	fuzzy	fuzzy	ADJ
ejpam-3504	190	4	bi	bi	NOUN
ejpam-3504	190	5	-	-	NOUN
ejpam-3504	190	6	ideal	ideal	NOUN
ejpam-3504	190	7	of	of	ADP
ejpam-3504	190	8	s	s	PROPN
ejpam-3504	190	9	is	be	AUX
ejpam-3504	190	10	a	a	DET
ejpam-3504	190	11	fuzzy	fuzzy	ADJ
ejpam-3504	190	12	right	right	NOUN
ejpam-3504	190	13	(	(	PUNCT
ejpam-3504	190	14	resp	resp	NOUN
ejpam-3504	190	15	.	.	PUNCT
ejpam-3504	191	1	fuzzy	fuzzy	ADJ
ejpam-3504	191	2	left	left	ADJ
ejpam-3504	191	3	)	)	PUNCT
ejpam-3504	191	4	ideal	ideal	NOUN
ejpam-3504	191	5	of	of	ADP
ejpam-3504	191	6	s.	s.	PROPN
ejpam-3504	191	7	proof	proof	PROPN
ejpam-3504	191	8	.	.	PUNCT
ejpam-3504	192	1	assuming	assume	VERB
ejpam-3504	192	2	the	the	DET
ejpam-3504	192	3	bi	bi	NOUN
ejpam-3504	192	4	-	-	NOUN
ejpam-3504	192	5	ideals	ideal	NOUN
ejpam-3504	192	6	are	be	AUX
ejpam-3504	192	7	right	right	ADJ
ejpam-3504	192	8	ideals	ideal	NOUN
ejpam-3504	192	9	,	,	PUNCT
ejpam-3504	192	10	let	let	VERB
ejpam-3504	192	11	f	f	PRON
ejpam-3504	192	12	be	be	AUX
ejpam-3504	192	13	a	a	DET
ejpam-3504	192	14	fuzzy	fuzzy	ADJ
ejpam-3504	192	15	bi	bi	NOUN
ejpam-3504	192	16	-	-	NOUN
ejpam-3504	192	17	ideal	ideal	NOUN
ejpam-3504	192	18	of	of	ADP
ejpam-3504	192	19	s	s	PRON
ejpam-3504	192	20	and	and	CCONJ
ejpam-3504	192	21	x	x	NOUN
ejpam-3504	192	22	,	,	PUNCT
ejpam-3504	192	23	y	y	PROPN
ejpam-3504	192	24	∈	∈	PROPN
ejpam-3504	192	25	s.	s.	PROPN
ejpam-3504	192	26	then	then	ADV
ejpam-3504	192	27	f(x	f(x	PROPN
ejpam-3504	192	28	◦	◦	PROPN
ejpam-3504	192	29	y	y	PROPN
ejpam-3504	192	30	)	)	PUNCT
ejpam-3504	192	31	≥	≥	NOUN
ejpam-3504	192	32	f(x	f(x	PROPN
ejpam-3504	192	33	)	)	PUNCT
ejpam-3504	192	34	.	.	PUNCT
ejpam-3504	193	1	in	in	ADP
ejpam-3504	193	2	fact	fact	NOUN
ejpam-3504	193	3	:	:	PUNCT
ejpam-3504	193	4	let	let	VERB
ejpam-3504	193	5	u	u	PRON
ejpam-3504	193	6	∈	∈	PROPN
ejpam-3504	193	7	x	x	INTJ
ejpam-3504	193	8	◦	◦	NOUN
ejpam-3504	193	9	y.	y.	NOUN
ejpam-3504	193	10	since	since	SCONJ
ejpam-3504	193	11	s	s	PROPN
ejpam-3504	193	12	is	be	AUX
ejpam-3504	193	13	regular	regular	ADJ
ejpam-3504	193	14	,	,	PUNCT
ejpam-3504	193	15	we	we	PRON
ejpam-3504	193	16	have	have	VERB
ejpam-3504	193	17	x	x	PART
ejpam-3504	193	18	◦	◦	VERB
ejpam-3504	193	19	y	y	PROPN
ejpam-3504	193	20	⊆	⊆	NUM
ejpam-3504	193	21	(	(	PUNCT
ejpam-3504	193	22	x	x	NOUN
ejpam-3504	193	23	∗	∗	NOUN
ejpam-3504	193	24	s	s	NOUN
ejpam-3504	193	25	∗	∗	NOUN
ejpam-3504	193	26	x	x	NOUN
ejpam-3504	193	27	)	)	PUNCT
ejpam-3504	193	28	∗	∗	NOUN
ejpam-3504	193	29	{	{	PUNCT
ejpam-3504	193	30	y	y	NOUN
ejpam-3504	193	31	}	}	PUNCT
ejpam-3504	193	32	⊆	⊆	NUM
ejpam-3504	193	33	(	(	PUNCT
ejpam-3504	193	34	x	x	NOUN
ejpam-3504	193	35	∗	∗	NOUN
ejpam-3504	193	36	s	s	NOUN
ejpam-3504	193	37	∗	∗	NOUN
ejpam-3504	193	38	x	x	NOUN
ejpam-3504	193	39	)	)	PUNCT
ejpam-3504	193	40	∗	∗	NOUN
ejpam-3504	193	41	s.	s.	PROPN
ejpam-3504	193	42	the	the	DET
ejpam-3504	193	43	set	set	PROPN
ejpam-3504	193	44	x	x	PUNCT
ejpam-3504	193	45	∗	∗	X
ejpam-3504	193	46	s	s	PART
ejpam-3504	193	47	∗	∗	NOUN
ejpam-3504	193	48	x	x	PUNCT
ejpam-3504	193	49	is	be	AUX
ejpam-3504	193	50	a	a	DET
ejpam-3504	193	51	bi	bi	NOUN
ejpam-3504	193	52	-	-	NOUN
ejpam-3504	193	53	ideal	ideal	NOUN
ejpam-3504	193	54	of	of	ADP
ejpam-3504	193	55	s.	s.	PROPN
ejpam-3504	193	56	this	this	PRON
ejpam-3504	193	57	is	be	AUX
ejpam-3504	193	58	because	because	SCONJ
ejpam-3504	193	59	(	(	PUNCT
ejpam-3504	193	60	x	x	SYM
ejpam-3504	193	61	∗	∗	X
ejpam-3504	193	62	s	s	NOUN
ejpam-3504	193	63	∗	∗	NOUN
ejpam-3504	193	64	x	x	NOUN
ejpam-3504	193	65	)	)	PUNCT
ejpam-3504	193	66	∗	∗	NOUN
ejpam-3504	193	67	s	s	PART
ejpam-3504	193	68	∗	∗	NOUN
ejpam-3504	193	69	(	(	PUNCT
ejpam-3504	193	70	x	x	SYM
ejpam-3504	193	71	∗	∗	X
ejpam-3504	193	72	s	s	NOUN
ejpam-3504	193	73	∗	∗	NOUN
ejpam-3504	193	74	x	x	X
ejpam-3504	193	75	)	)	PUNCT
ejpam-3504	193	76	⊆	⊆	NUM
ejpam-3504	193	77	x	x	SYM
ejpam-3504	193	78	∗	∗	NOUN
ejpam-3504	193	79	s	s	PART
ejpam-3504	193	80	∗	∗	NOUN
ejpam-3504	193	81	x.	x.	NOUN
ejpam-3504	193	82	by	by	ADP
ejpam-3504	193	83	the	the	DET
ejpam-3504	193	84	assumption	assumption	NOUN
ejpam-3504	193	85	,	,	PUNCT
ejpam-3504	193	86	x	x	PUNCT
ejpam-3504	193	87	∗	∗	NOUN
ejpam-3504	193	88	s	s	PART
ejpam-3504	193	89	∗	∗	NOUN
ejpam-3504	193	90	x	x	PUNCT
ejpam-3504	193	91	is	be	AUX
ejpam-3504	193	92	a	a	DET
ejpam-3504	193	93	right	right	ADJ
ejpam-3504	193	94	ideal	ideal	NOUN
ejpam-3504	193	95	of	of	ADP
ejpam-3504	193	96	s	s	PROPN
ejpam-3504	193	97	,	,	PUNCT
ejpam-3504	193	98	that	that	ADV
ejpam-3504	193	99	is	is	ADV
ejpam-3504	193	100	(	(	PUNCT
ejpam-3504	193	101	x	x	NOUN
ejpam-3504	193	102	∗	∗	NOUN
ejpam-3504	193	103	s	s	NOUN
ejpam-3504	193	104	∗	∗	NOUN
ejpam-3504	193	105	x	x	NOUN
ejpam-3504	193	106	)	)	PUNCT
ejpam-3504	193	107	∗	∗	NOUN
ejpam-3504	193	108	s	s	NOUN
ejpam-3504	193	109	⊆	⊆	NUM
ejpam-3504	193	110	x	x	SYM
ejpam-3504	193	111	∗	∗	NOUN
ejpam-3504	193	112	s	s	PART
ejpam-3504	193	113	∗	∗	NOUN
ejpam-3504	193	114	x.	x.	NOUN
ejpam-3504	194	1	then	then	ADV
ejpam-3504	194	2	we	we	PRON
ejpam-3504	194	3	have	have	VERB
ejpam-3504	194	4	u	u	NOUN
ejpam-3504	194	5	∈	∈	NOUN
ejpam-3504	194	6	(	(	PUNCT
ejpam-3504	194	7	x	x	NOUN
ejpam-3504	194	8	∗	∗	NUM
ejpam-3504	194	9	s	s	PART
ejpam-3504	194	10	)	)	PUNCT
ejpam-3504	194	11	∗	∗	NOUN
ejpam-3504	194	12	x	x	NOUN
ejpam-3504	194	13	,	,	PUNCT
ejpam-3504	194	14	so	so	SCONJ
ejpam-3504	194	15	u	u	NOUN
ejpam-3504	194	16	∈	∈	PROPN
ejpam-3504	194	17	v	v	ADP
ejpam-3504	194	18	◦	◦	NOUN
ejpam-3504	194	19	x	x	PUNCT
ejpam-3504	194	20	for	for	ADP
ejpam-3504	194	21	some	some	DET
ejpam-3504	194	22	v	v	NOUN
ejpam-3504	194	23	∈	∈	NOUN
ejpam-3504	194	24	x	x	PUNCT
ejpam-3504	194	25	∗	∗	NOUN
ejpam-3504	194	26	s	s	NOUN
ejpam-3504	194	27	and	and	CCONJ
ejpam-3504	194	28	v	v	ADP
ejpam-3504	194	29	∈	∈	NOUN
ejpam-3504	194	30	x	x	PUNCT
ejpam-3504	194	31	◦	◦	NOUN
ejpam-3504	194	32	w	w	NOUN
ejpam-3504	194	33	for	for	ADP
ejpam-3504	194	34	some	some	DET
ejpam-3504	194	35	w	w	PROPN
ejpam-3504	194	36	∈	∈	PROPN
ejpam-3504	194	37	s.	s.	PROPN
ejpam-3504	194	38	since	since	SCONJ
ejpam-3504	194	39	v	v	NUM
ejpam-3504	194	40	◦	◦	NOUN
ejpam-3504	194	41	x	x	SYM
ejpam-3504	194	42	⊆	⊆	X
ejpam-3504	194	43	(	(	PUNCT
ejpam-3504	194	44	x	x	SYM
ejpam-3504	194	45	◦	◦	NOUN
ejpam-3504	194	46	w	w	NOUN
ejpam-3504	194	47	)	)	PUNCT
ejpam-3504	194	48	∗	∗	NOUN
ejpam-3504	194	49	x	x	SYM
ejpam-3504	194	50	,	,	PUNCT
ejpam-3504	194	51	we	we	PRON
ejpam-3504	194	52	have	have	VERB
ejpam-3504	194	53	u	u	NOUN
ejpam-3504	194	54	∈	∈	NOUN
ejpam-3504	194	55	(	(	PUNCT
ejpam-3504	194	56	x	x	SYM
ejpam-3504	194	57	◦	◦	NOUN
ejpam-3504	194	58	w	w	NOUN
ejpam-3504	194	59	)	)	PUNCT
ejpam-3504	194	60	∗	∗	NOUN
ejpam-3504	194	61	x.	x.	NOUN
ejpam-3504	195	1	since	since	SCONJ
ejpam-3504	195	2	f	f	PROPN
ejpam-3504	195	3	is	be	AUX
ejpam-3504	195	4	a	a	DET
ejpam-3504	195	5	fuzzy	fuzzy	ADJ
ejpam-3504	195	6	bi	bi	NOUN
ejpam-3504	195	7	-	-	NOUN
ejpam-3504	195	8	ideal	ideal	NOUN
ejpam-3504	195	9	of	of	ADP
ejpam-3504	195	10	s	s	PROPN
ejpam-3504	195	11	,	,	PUNCT
ejpam-3504	195	12	we	we	PRON
ejpam-3504	195	13	have	have	VERB
ejpam-3504	195	14	f	f	X
ejpam-3504	195	15	(	(	PUNCT
ejpam-3504	195	16	(	(	PUNCT
ejpam-3504	195	17	x	x	SYM
ejpam-3504	195	18	◦	◦	NOUN
ejpam-3504	195	19	w	w	NOUN
ejpam-3504	195	20	)	)	PUNCT
ejpam-3504	195	21	∗	∗	NOUN
ejpam-3504	195	22	x	x	SYM
ejpam-3504	195	23	)	)	PUNCT
ejpam-3504	195	24	≥	≥	NOUN
ejpam-3504	195	25	min{f(x	min{f(x	NOUN
ejpam-3504	195	26	)	)	PUNCT
ejpam-3504	195	27	,	,	PUNCT
ejpam-3504	195	28	f(x	f(x	PROPN
ejpam-3504	195	29	)	)	PUNCT
ejpam-3504	195	30	}	}	PUNCT
ejpam-3504	195	31	=	=	SYM
ejpam-3504	195	32	f(x	f(x	PROPN
ejpam-3504	195	33	)	)	PUNCT
ejpam-3504	195	34	and	and	CCONJ
ejpam-3504	195	35	,	,	PUNCT
ejpam-3504	195	36	since	since	SCONJ
ejpam-3504	195	37	u	u	PROPN
ejpam-3504	195	38	∈	∈	PROPN
ejpam-3504	195	39	(	(	PUNCT
ejpam-3504	195	40	x	x	SYM
ejpam-3504	195	41	◦	◦	NOUN
ejpam-3504	195	42	w	w	NOUN
ejpam-3504	195	43	)	)	PUNCT
ejpam-3504	195	44	∗	∗	NOUN
ejpam-3504	195	45	x	x	SYM
ejpam-3504	195	46	,	,	PUNCT
ejpam-3504	195	47	we	we	PRON
ejpam-3504	195	48	have	have	VERB
ejpam-3504	195	49	f(u	f(u	PROPN
ejpam-3504	195	50	)	)	PUNCT
ejpam-3504	195	51	≥	≥	NOUN
ejpam-3504	195	52	f(x	f(x	PROPN
ejpam-3504	195	53	)	)	PUNCT
ejpam-3504	195	54	.	.	PUNCT
ejpam-3504	196	1	thus	thus	ADV
ejpam-3504	196	2	f	f	PROPN
ejpam-3504	196	3	is	be	AUX
ejpam-3504	196	4	a	a	DET
ejpam-3504	196	5	fuzzy	fuzzy	ADJ
ejpam-3504	196	6	right	right	ADJ
ejpam-3504	196	7	ideal	ideal	NOUN
ejpam-3504	196	8	of	of	ADP
ejpam-3504	196	9	s.	s.	PROPN
ejpam-3504	196	10	suppose	suppose	VERB
ejpam-3504	196	11	now	now	ADV
ejpam-3504	196	12	that	that	SCONJ
ejpam-3504	196	13	every	every	DET
ejpam-3504	196	14	bi	bi	NOUN
ejpam-3504	196	15	-	-	NOUN
ejpam-3504	196	16	ideal	ideal	NOUN
ejpam-3504	196	17	of	of	ADP
ejpam-3504	196	18	s	s	PROPN
ejpam-3504	196	19	is	be	AUX
ejpam-3504	196	20	a	a	DET
ejpam-3504	196	21	left	left	ADJ
ejpam-3504	196	22	ideal	ideal	NOUN
ejpam-3504	196	23	of	of	ADP
ejpam-3504	196	24	s	s	PRON
ejpam-3504	196	25	and	and	CCONJ
ejpam-3504	196	26	let	let	VERB
ejpam-3504	196	27	f	f	PRON
ejpam-3504	196	28	be	be	AUX
ejpam-3504	196	29	a	a	DET
ejpam-3504	196	30	fuzzy	fuzzy	ADJ
ejpam-3504	196	31	bi	bi	NOUN
ejpam-3504	196	32	-	-	NOUN
ejpam-3504	196	33	ideal	ideal	NOUN
ejpam-3504	196	34	of	of	ADP
ejpam-3504	196	35	s	s	PROPN
ejpam-3504	196	36	,	,	PUNCT
ejpam-3504	196	37	x	x	PRON
ejpam-3504	196	38	,	,	PUNCT
ejpam-3504	196	39	y	y	PROPN
ejpam-3504	196	40	∈	∈	PROPN
ejpam-3504	196	41	s	s	PART
ejpam-3504	196	42	and	and	CCONJ
ejpam-3504	196	43	u	u	PROPN
ejpam-3504	196	44	∈	∈	PROPN
ejpam-3504	196	45	x	x	PUNCT
ejpam-3504	196	46	◦	◦	VERB
ejpam-3504	196	47	y.	y.	NOUN
ejpam-3504	196	48	then	then	ADV
ejpam-3504	196	49	x	x	VERB
ejpam-3504	196	50	◦	◦	VERB
ejpam-3504	196	51	y	y	NOUN
ejpam-3504	196	52	⊆	⊆	NUM
ejpam-3504	196	53	{	{	PUNCT
ejpam-3504	196	54	x}∗(y∗s∗y	x}∗(y∗s∗y	NUM
ejpam-3504	196	55	)	)	PUNCT
ejpam-3504	196	56	⊆	⊆	NUM
ejpam-3504	196	57	s∗(y∗s∗y	s∗(y∗s∗y	NOUN
ejpam-3504	196	58	)	)	PUNCT
ejpam-3504	196	59	⊆	⊆	NUM
ejpam-3504	196	60	y∗s∗y	y∗s∗y	NOUN
ejpam-3504	196	61	and	and	CCONJ
ejpam-3504	196	62	so	so	ADV
ejpam-3504	196	63	u	u	PROPN
ejpam-3504	196	64	∈	∈	PROPN
ejpam-3504	196	65	y	y	PROPN
ejpam-3504	196	66	◦	◦	NOUN
ejpam-3504	196	67	v	v	NOUN
ejpam-3504	196	68	for	for	ADP
ejpam-3504	196	69	some	some	DET
ejpam-3504	196	70	v	v	NOUN
ejpam-3504	196	71	∈	∈	NOUN
ejpam-3504	196	72	s	s	PART
ejpam-3504	196	73	∗y	∗y	PROPN
ejpam-3504	196	74	and	and	CCONJ
ejpam-3504	196	75	v	v	ADP
ejpam-3504	196	76	∈	∈	PROPN
ejpam-3504	196	77	w	w	PROPN
ejpam-3504	196	78	◦	◦	NOUN
ejpam-3504	196	79	y	y	NOUN
ejpam-3504	196	80	for	for	ADP
ejpam-3504	196	81	some	some	DET
ejpam-3504	196	82	w	w	PROPN
ejpam-3504	196	83	∈	∈	PROPN
ejpam-3504	196	84	s.	s.	PROPN
ejpam-3504	196	85	thus	thus	ADV
ejpam-3504	196	86	we	we	PRON
ejpam-3504	196	87	have	have	VERB
ejpam-3504	196	88	u	u	PROPN
ejpam-3504	196	89	∈	∈	PROPN
ejpam-3504	196	90	y	y	PROPN
ejpam-3504	196	91	∗	∗	NOUN
ejpam-3504	196	92	(	(	PUNCT
ejpam-3504	196	93	w	w	PROPN
ejpam-3504	196	94	◦	◦	NOUN
ejpam-3504	196	95	y	y	NOUN
ejpam-3504	196	96	)	)	PUNCT
ejpam-3504	196	97	=	=	PUNCT
ejpam-3504	197	1	(	(	PUNCT
ejpam-3504	197	2	y	y	NOUN
ejpam-3504	197	3	◦	◦	NOUN
ejpam-3504	197	4	w)∗y	w)∗y	PROPN
ejpam-3504	197	5	.	.	PUNCT
ejpam-3504	198	1	since	since	SCONJ
ejpam-3504	198	2	f	f	PROPN
ejpam-3504	198	3	is	be	AUX
ejpam-3504	198	4	a	a	DET
ejpam-3504	198	5	fuzzy	fuzzy	ADJ
ejpam-3504	198	6	bi	bi	NOUN
ejpam-3504	198	7	-	-	NOUN
ejpam-3504	198	8	ideal	ideal	NOUN
ejpam-3504	198	9	of	of	ADP
ejpam-3504	198	10	s	s	PROPN
ejpam-3504	198	11	,	,	PUNCT
ejpam-3504	198	12	we	we	PRON
ejpam-3504	198	13	have	have	VERB
ejpam-3504	198	14	f	f	X
ejpam-3504	198	15	(	(	PUNCT
ejpam-3504	198	16	(	(	PUNCT
ejpam-3504	198	17	y	y	PROPN
ejpam-3504	198	18	◦	◦	NOUN
ejpam-3504	198	19	w	w	PROPN
ejpam-3504	198	20	)	)	PUNCT
ejpam-3504	198	21	∗	∗	NOUN
ejpam-3504	198	22	y	y	PROPN
ejpam-3504	198	23	)	)	PUNCT
ejpam-3504	198	24	≥	≥	NOUN
ejpam-3504	198	25	min{f(y	min{f(y	NOUN
ejpam-3504	198	26	)	)	PUNCT
ejpam-3504	198	27	,	,	PUNCT
ejpam-3504	198	28	f(y	f(y	NOUN
ejpam-3504	198	29	)	)	PUNCT
ejpam-3504	198	30	}	}	PUNCT
ejpam-3504	198	31	=	=	SYM
ejpam-3504	198	32	f(y	f(y	NOUN
ejpam-3504	198	33	)	)	PUNCT
ejpam-3504	198	34	and	and	CCONJ
ejpam-3504	198	35	,	,	PUNCT
ejpam-3504	198	36	since	since	SCONJ
ejpam-3504	198	37	u	u	PROPN
ejpam-3504	198	38	∈	∈	PROPN
ejpam-3504	198	39	(	(	PUNCT
ejpam-3504	198	40	y	y	PROPN
ejpam-3504	198	41	◦	◦	PROPN
ejpam-3504	198	42	w	w	NOUN
ejpam-3504	198	43	)	)	PUNCT
ejpam-3504	198	44	∗	∗	NOUN
ejpam-3504	198	45	y	y	PROPN
ejpam-3504	198	46	,	,	PUNCT
ejpam-3504	198	47	we	we	PRON
ejpam-3504	198	48	have	have	VERB
ejpam-3504	198	49	f(u	f(u	PROPN
ejpam-3504	198	50	)	)	PUNCT
ejpam-3504	198	51	≥	≥	NOUN
ejpam-3504	198	52	f(y	f(y	NOUN
ejpam-3504	198	53	)	)	PUNCT
ejpam-3504	198	54	.	.	PUNCT
ejpam-3504	199	1	thus	thus	ADV
ejpam-3504	199	2	f	f	PROPN
ejpam-3504	199	3	is	be	AUX
ejpam-3504	199	4	a	a	DET
ejpam-3504	199	5	fuzzy	fuzzy	ADJ
ejpam-3504	199	6	left	leave	VERB
ejpam-3504	199	7	ideal	ideal	NOUN
ejpam-3504	199	8	of	of	ADP
ejpam-3504	199	9	s	s	PRON
ejpam-3504	199	10	and	and	CCONJ
ejpam-3504	199	11	the	the	DET
ejpam-3504	199	12	proof	proof	NOUN
ejpam-3504	199	13	is	be	AUX
ejpam-3504	199	14	complete	complete	ADJ
ejpam-3504	199	15	.	.	PUNCT
ejpam-3504	200	1	�	�	PROPN
ejpam-3504	200	2	n.	n.	PROPN
ejpam-3504	200	3	kehayopulu	kehayopulu	PROPN
ejpam-3504	200	4	/	/	SYM
ejpam-3504	200	5	eur	eur	PROPN
ejpam-3504	200	6	.	.	PUNCT
ejpam-3504	201	1	j.	j.	PROPN
ejpam-3504	201	2	pure	pure	PROPN
ejpam-3504	201	3	appl	appl	PROPN
ejpam-3504	201	4	.	.	PROPN
ejpam-3504	201	5	math	math	PROPN
ejpam-3504	201	6	,	,	PUNCT
ejpam-3504	201	7	12	12	NUM
ejpam-3504	201	8	(	(	PUNCT
ejpam-3504	201	9	3	3	NUM
ejpam-3504	201	10	)	)	PUNCT
ejpam-3504	201	11	(	(	PUNCT
ejpam-3504	201	12	2019	2019	NUM
ejpam-3504	201	13	)	)	PUNCT
ejpam-3504	201	14	,	,	PUNCT
ejpam-3504	201	15	709	709	NUM
ejpam-3504	201	16	-	-	SYM
ejpam-3504	201	17	721	721	NUM
ejpam-3504	201	18	715	715	NUM
ejpam-3504	201	19	combining	combine	VERB
ejpam-3504	201	20	propositions	proposition	NOUN
ejpam-3504	201	21	3.12	3.12	NUM
ejpam-3504	201	22	and	and	CCONJ
ejpam-3504	201	23	3.13	3.13	NUM
ejpam-3504	201	24	we	we	PRON
ejpam-3504	201	25	have	have	VERB
ejpam-3504	201	26	the	the	DET
ejpam-3504	201	27	following	follow	VERB
ejpam-3504	201	28	theorem	theorem	VERB
ejpam-3504	201	29	.	.	PUNCT
ejpam-3504	202	1	theorem	theorem	PROPN
ejpam-3504	202	2	3.14	3.14	NUM
ejpam-3504	202	3	.	.	PUNCT
ejpam-3504	203	1	let	let	VERB
ejpam-3504	203	2	s	s	PRON
ejpam-3504	203	3	be	be	AUX
ejpam-3504	203	4	a	a	DET
ejpam-3504	203	5	regular	regular	ADJ
ejpam-3504	203	6	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	203	7	.	.	PUNCT
ejpam-3504	204	1	then	then	ADV
ejpam-3504	204	2	we	we	PRON
ejpam-3504	204	3	have	have	VERB
ejpam-3504	204	4	the	the	DET
ejpam-3504	204	5	following	following	NOUN
ejpam-3504	204	6	:	:	PUNCT
ejpam-3504	204	7	(	(	PUNCT
ejpam-3504	204	8	1	1	X
ejpam-3504	204	9	)	)	PUNCT
ejpam-3504	204	10	every	every	DET
ejpam-3504	204	11	bi	bi	NOUN
ejpam-3504	204	12	-	-	NOUN
ejpam-3504	204	13	ideal	ideal	NOUN
ejpam-3504	204	14	of	of	ADP
ejpam-3504	204	15	s	s	PROPN
ejpam-3504	204	16	is	be	AUX
ejpam-3504	204	17	a	a	DET
ejpam-3504	204	18	right	right	ADJ
ejpam-3504	204	19	ideal	ideal	NOUN
ejpam-3504	204	20	of	of	ADP
ejpam-3504	204	21	s	s	PRON
ejpam-3504	204	22	if	if	SCONJ
ejpam-3504	205	1	and	and	CCONJ
ejpam-3504	205	2	only	only	ADV
ejpam-3504	205	3	if	if	SCONJ
ejpam-3504	205	4	every	every	DET
ejpam-3504	205	5	fuzzy	fuzzy	ADJ
ejpam-3504	205	6	bi	bi	NOUN
ejpam-3504	205	7	-	-	NOUN
ejpam-3504	205	8	ideal	ideal	NOUN
ejpam-3504	205	9	of	of	ADP
ejpam-3504	205	10	s	s	PROPN
ejpam-3504	205	11	is	be	AUX
ejpam-3504	205	12	a	a	DET
ejpam-3504	205	13	fuzzy	fuzzy	ADJ
ejpam-3504	205	14	right	right	ADJ
ejpam-3504	205	15	ideal	ideal	NOUN
ejpam-3504	205	16	of	of	ADP
ejpam-3504	205	17	s.	s.	PROPN
ejpam-3504	205	18	(	(	PUNCT
ejpam-3504	205	19	2	2	X
ejpam-3504	205	20	)	)	PUNCT
ejpam-3504	205	21	every	every	DET
ejpam-3504	205	22	bi	bi	NOUN
ejpam-3504	205	23	-	-	NOUN
ejpam-3504	205	24	ideal	ideal	NOUN
ejpam-3504	205	25	of	of	ADP
ejpam-3504	205	26	s	s	PROPN
ejpam-3504	205	27	is	be	AUX
ejpam-3504	205	28	a	a	DET
ejpam-3504	205	29	left	left	ADJ
ejpam-3504	205	30	ideal	ideal	NOUN
ejpam-3504	205	31	of	of	ADP
ejpam-3504	205	32	s	s	PRON
ejpam-3504	205	33	if	if	SCONJ
ejpam-3504	205	34	and	and	CCONJ
ejpam-3504	205	35	only	only	ADV
ejpam-3504	205	36	if	if	SCONJ
ejpam-3504	205	37	every	every	DET
ejpam-3504	205	38	fuzzy	fuzzy	ADJ
ejpam-3504	205	39	bi	bi	NOUN
ejpam-3504	205	40	-	-	NOUN
ejpam-3504	205	41	ideal	ideal	NOUN
ejpam-3504	205	42	of	of	ADP
ejpam-3504	205	43	s	s	PROPN
ejpam-3504	205	44	is	be	AUX
ejpam-3504	205	45	a	a	DET
ejpam-3504	205	46	fuzzy	fuzzy	ADJ
ejpam-3504	205	47	left	leave	VERB
ejpam-3504	205	48	ideal	ideal	NOUN
ejpam-3504	205	49	of	of	ADP
ejpam-3504	205	50	s.	s.	PROPN
ejpam-3504	205	51	as	as	ADP
ejpam-3504	205	52	a	a	DET
ejpam-3504	205	53	consequence	consequence	NOUN
ejpam-3504	205	54	,	,	PUNCT
ejpam-3504	205	55	in	in	ADP
ejpam-3504	205	56	a	a	DET
ejpam-3504	205	57	regular	regular	ADJ
ejpam-3504	205	58	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	205	59	every	every	DET
ejpam-3504	205	60	bi	bi	NOUN
ejpam-3504	205	61	-	-	ADJ
ejpam-3504	205	62	ideal	ideal	ADJ
ejpam-3504	205	63	is	be	AUX
ejpam-3504	205	64	an	an	DET
ejpam-3504	205	65	ideal	ideal	NOUN
ejpam-3504	205	66	if	if	SCONJ
ejpam-3504	206	1	and	and	CCONJ
ejpam-3504	206	2	only	only	ADV
ejpam-3504	206	3	if	if	SCONJ
ejpam-3504	206	4	every	every	DET
ejpam-3504	206	5	fuzzy	fuzzy	ADJ
ejpam-3504	206	6	bi	bi	NOUN
ejpam-3504	206	7	-	-	ADJ
ejpam-3504	206	8	ideal	ideal	ADJ
ejpam-3504	206	9	is	be	AUX
ejpam-3504	206	10	a	a	DET
ejpam-3504	206	11	fuzzy	fuzzy	ADJ
ejpam-3504	206	12	ideal	ideal	NOUN
ejpam-3504	206	13	.	.	PUNCT
ejpam-3504	207	1	4	4	X
ejpam-3504	207	2	.	.	X
ejpam-3504	207	3	on	on	ADP
ejpam-3504	207	4	intra	intra	ADJ
ejpam-3504	207	5	-	-	ADJ
ejpam-3504	207	6	regular	regular	ADJ
ejpam-3504	207	7	and	and	CCONJ
ejpam-3504	207	8	left	leave	VERB
ejpam-3504	207	9	regular	regular	ADJ
ejpam-3504	207	10	fuzzy	fuzzy	ADJ
ejpam-3504	207	11	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	207	12	kuroki	kuroki	PROPN
ejpam-3504	207	13	characterized	characterize	VERB
ejpam-3504	207	14	the	the	DET
ejpam-3504	207	15	intra	intra	ADJ
ejpam-3504	207	16	-	-	ADJ
ejpam-3504	207	17	regular	regular	ADJ
ejpam-3504	207	18	semigroup	semigroup	NOUN
ejpam-3504	207	19	as	as	ADP
ejpam-3504	207	20	a	a	DET
ejpam-3504	207	21	semigroup	semigroup	NOUN
ejpam-3504	207	22	s	s	VERB
ejpam-3504	207	23	such	such	ADJ
ejpam-3504	207	24	that	that	DET
ejpam-3504	207	25	f(a	f(a	NOUN
ejpam-3504	207	26	)	)	PUNCT
ejpam-3504	207	27	=	=	SYM
ejpam-3504	207	28	f(a2	f(a2	NOUN
ejpam-3504	207	29	)	)	PUNCT
ejpam-3504	207	30	for	for	ADP
ejpam-3504	207	31	every	every	DET
ejpam-3504	207	32	fuzzy	fuzzy	ADJ
ejpam-3504	207	33	ideal	ideal	NOUN
ejpam-3504	207	34	f	f	PROPN
ejpam-3504	207	35	of	of	ADP
ejpam-3504	207	36	s	s	PRON
ejpam-3504	207	37	and	and	CCONJ
ejpam-3504	207	38	every	every	DET
ejpam-3504	207	39	a	a	DET
ejpam-3504	207	40	∈	∈	NOUN
ejpam-3504	207	41	s	s	PART
ejpam-3504	207	42	[	[	X
ejpam-3504	207	43	8	8	NUM
ejpam-3504	207	44	;	;	PUNCT
ejpam-3504	207	45	theorem	theorem	VERB
ejpam-3504	207	46	4.1	4.1	NUM
ejpam-3504	207	47	]	]	PUNCT
ejpam-3504	207	48	.	.	PUNCT
ejpam-3504	208	1	he	he	PRON
ejpam-3504	208	2	also	also	ADV
ejpam-3504	208	3	characterized	characterize	VERB
ejpam-3504	208	4	the	the	DET
ejpam-3504	208	5	left	left	ADJ
ejpam-3504	208	6	(	(	PUNCT
ejpam-3504	208	7	right	right	ADJ
ejpam-3504	208	8	)	)	PUNCT
ejpam-3504	208	9	regular	regular	ADJ
ejpam-3504	208	10	semigroup	semigroup	NOUN
ejpam-3504	208	11	as	as	ADP
ejpam-3504	208	12	a	a	DET
ejpam-3504	208	13	semigroup	semigroup	NOUN
ejpam-3504	208	14	s	s	VERB
ejpam-3504	208	15	such	such	ADJ
ejpam-3504	208	16	that	that	DET
ejpam-3504	208	17	f(a	f(a	NOUN
ejpam-3504	208	18	)	)	PUNCT
ejpam-3504	208	19	=	=	SYM
ejpam-3504	208	20	f(a2	f(a2	NOUN
ejpam-3504	208	21	)	)	PUNCT
ejpam-3504	208	22	for	for	ADP
ejpam-3504	208	23	every	every	DET
ejpam-3504	208	24	fuzzy	fuzzy	ADJ
ejpam-3504	208	25	left	leave	VERB
ejpam-3504	208	26	(	(	PUNCT
ejpam-3504	208	27	right	right	ADJ
ejpam-3504	208	28	)	)	PUNCT
ejpam-3504	208	29	ideal	ideal	PROPN
ejpam-3504	208	30	f	f	PROPN
ejpam-3504	208	31	and	and	CCONJ
ejpam-3504	208	32	any	any	DET
ejpam-3504	208	33	a	a	DET
ejpam-3504	208	34	∈	∈	NOUN
ejpam-3504	208	35	s	s	PART
ejpam-3504	208	36	[	[	X
ejpam-3504	208	37	8	8	NUM
ejpam-3504	208	38	;	;	PUNCT
ejpam-3504	208	39	theorems	theorem	NOUN
ejpam-3504	208	40	5.1	5.1	NUM
ejpam-3504	208	41	and	and	CCONJ
ejpam-3504	208	42	5.2	5.2	NUM
ejpam-3504	208	43	]	]	PUNCT
ejpam-3504	208	44	.	.	PUNCT
ejpam-3504	209	1	in	in	ADP
ejpam-3504	209	2	this	this	DET
ejpam-3504	209	3	section	section	NOUN
ejpam-3504	209	4	we	we	PRON
ejpam-3504	209	5	examine	examine	VERB
ejpam-3504	209	6	these	these	DET
ejpam-3504	209	7	results	result	NOUN
ejpam-3504	209	8	in	in	ADP
ejpam-3504	209	9	case	case	NOUN
ejpam-3504	209	10	of	of	ADP
ejpam-3504	209	11	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	209	12	.	.	PUNCT
ejpam-3504	210	1	denote	denote	VERB
ejpam-3504	210	2	by	by	ADP
ejpam-3504	210	3	i(u	i(u	PROPN
ejpam-3504	210	4	)	)	PUNCT
ejpam-3504	210	5	the	the	DET
ejpam-3504	210	6	ideal	ideal	NOUN
ejpam-3504	210	7	of	of	ADP
ejpam-3504	210	8	s	s	AUX
ejpam-3504	210	9	generated	generate	VERB
ejpam-3504	210	10	by	by	ADP
ejpam-3504	210	11	the	the	DET
ejpam-3504	210	12	element	element	ADJ
ejpam-3504	210	13	u	u	NOUN
ejpam-3504	210	14	of	of	ADP
ejpam-3504	210	15	s	s	PRON
ejpam-3504	210	16	and	and	CCONJ
ejpam-3504	210	17	by	by	ADP
ejpam-3504	210	18	l(u	l(u	PROPN
ejpam-3504	210	19	)	)	PUNCT
ejpam-3504	210	20	the	the	DET
ejpam-3504	210	21	left	left	ADJ
ejpam-3504	210	22	ideal	ideal	NOUN
ejpam-3504	210	23	of	of	ADP
ejpam-3504	210	24	s	s	AUX
ejpam-3504	210	25	generated	generate	VERB
ejpam-3504	210	26	by	by	ADP
ejpam-3504	210	27	u.	u.	PROPN
ejpam-3504	210	28	for	for	ADP
ejpam-3504	210	29	an	an	DET
ejpam-3504	210	30	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	210	31	s	s	NOUN
ejpam-3504	210	32	and	and	CCONJ
ejpam-3504	210	33	any	any	DET
ejpam-3504	210	34	u	u	NOUN
ejpam-3504	210	35	∈	∈	PROPN
ejpam-3504	210	36	s	s	PART
ejpam-3504	210	37	,	,	PUNCT
ejpam-3504	210	38	we	we	PRON
ejpam-3504	210	39	have	have	VERB
ejpam-3504	210	40	i(u	i(u	PROPN
ejpam-3504	210	41	)	)	PUNCT
ejpam-3504	211	1	=	=	SYM
ejpam-3504	211	2	u	u	NOUN
ejpam-3504	211	3	∪	∪	X
ejpam-3504	211	4	(	(	PUNCT
ejpam-3504	211	5	s	s	NOUN
ejpam-3504	211	6	∗	∗	NOUN
ejpam-3504	211	7	u	u	NOUN
ejpam-3504	211	8	)	)	PUNCT
ejpam-3504	211	9	∪	∪	ADV
ejpam-3504	211	10	(	(	PUNCT
ejpam-3504	211	11	u	u	NOUN
ejpam-3504	211	12	∗	∗	NOUN
ejpam-3504	211	13	s	s	PART
ejpam-3504	211	14	)	)	PUNCT
ejpam-3504	211	15	∪	∪	ADP
ejpam-3504	211	16	s	s	PART
ejpam-3504	211	17	∗	∗	NOUN
ejpam-3504	211	18	u	u	NOUN
ejpam-3504	211	19	∗	∗	NOUN
ejpam-3504	211	20	s	s	NOUN
ejpam-3504	211	21	and	and	CCONJ
ejpam-3504	211	22	l(u	l(u	PROPN
ejpam-3504	211	23	)	)	PUNCT
ejpam-3504	212	1	=	=	SYM
ejpam-3504	212	2	u	u	NOUN
ejpam-3504	212	3	∪	∪	X
ejpam-3504	212	4	(	(	PUNCT
ejpam-3504	212	5	s	s	NOUN
ejpam-3504	212	6	∗	∗	NOUN
ejpam-3504	212	7	u	u	NOUN
ejpam-3504	212	8	)	)	PUNCT
ejpam-3504	212	9	.	.	PUNCT
ejpam-3504	213	1	every	every	PRON
ejpam-3504	213	2	left	leave	VERB
ejpam-3504	213	3	regular	regular	ADV
ejpam-3504	213	4	and	and	CCONJ
ejpam-3504	213	5	every	every	DET
ejpam-3504	213	6	right	right	ADJ
ejpam-3504	213	7	regular	regular	ADJ
ejpam-3504	213	8	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	213	9	is	be	AUX
ejpam-3504	213	10	intra	intra	ADJ
ejpam-3504	213	11	-	-	ADJ
ejpam-3504	213	12	regular	regular	ADJ
ejpam-3504	213	13	.	.	PUNCT
ejpam-3504	214	1	in	in	ADP
ejpam-3504	214	2	fact	fact	NOUN
ejpam-3504	214	3	,	,	PUNCT
ejpam-3504	214	4	if	if	SCONJ
ejpam-3504	214	5	s	s	NOUN
ejpam-3504	214	6	is	be	AUX
ejpam-3504	214	7	left	leave	VERB
ejpam-3504	214	8	regular	regular	ADV
ejpam-3504	214	9	then	then	ADV
ejpam-3504	214	10	,	,	PUNCT
ejpam-3504	214	11	for	for	ADP
ejpam-3504	214	12	any	any	DET
ejpam-3504	214	13	nonempty	nonempty	NOUN
ejpam-3504	214	14	subset	subset	VERB
ejpam-3504	214	15	a	a	PRON
ejpam-3504	214	16	of	of	ADP
ejpam-3504	214	17	s	s	PROPN
ejpam-3504	214	18	,	,	PUNCT
ejpam-3504	214	19	we	we	PRON
ejpam-3504	214	20	have	have	VERB
ejpam-3504	214	21	a	a	DET
ejpam-3504	214	22	⊆	⊆	NUM
ejpam-3504	214	23	s	s	NOUN
ejpam-3504	214	24	∗	∗	NOUN
ejpam-3504	214	25	a	a	DET
ejpam-3504	214	26	∗	∗	NOUN
ejpam-3504	215	1	a	a	PRON
ejpam-3504	216	1	,	,	PUNCT
ejpam-3504	216	2	then	then	ADV
ejpam-3504	216	3	we	we	PRON
ejpam-3504	216	4	have	have	VERB
ejpam-3504	216	5	a	a	DET
ejpam-3504	216	6	⊆	⊆	NUM
ejpam-3504	216	7	s	s	NOUN
ejpam-3504	216	8	∗	∗	NOUN
ejpam-3504	216	9	(	(	PUNCT
ejpam-3504	216	10	s	s	NOUN
ejpam-3504	216	11	∗	∗	NOUN
ejpam-3504	216	12	a	a	DET
ejpam-3504	216	13	∗	∗	NOUN
ejpam-3504	216	14	a	a	PRON
ejpam-3504	216	15	)	)	PUNCT
ejpam-3504	216	16	∗	∗	NOUN
ejpam-3504	216	17	a	a	DET
ejpam-3504	216	18	⊆	⊆	NUM
ejpam-3504	216	19	s	s	NOUN
ejpam-3504	216	20	∗	∗	NOUN
ejpam-3504	216	21	a	a	DET
ejpam-3504	216	22	∗	∗	NOUN
ejpam-3504	216	23	a	a	DET
ejpam-3504	216	24	∗	∗	NOUN
ejpam-3504	216	25	s	s	NOUN
ejpam-3504	216	26	,	,	PUNCT
ejpam-3504	216	27	and	and	CCONJ
ejpam-3504	216	28	so	so	ADV
ejpam-3504	216	29	s	s	VERB
ejpam-3504	216	30	is	be	AUX
ejpam-3504	216	31	intra	intra	ADJ
ejpam-3504	216	32	-	-	ADJ
ejpam-3504	216	33	regular	regular	ADJ
ejpam-3504	216	34	.	.	PUNCT
ejpam-3504	217	1	in	in	ADP
ejpam-3504	217	2	a	a	DET
ejpam-3504	217	3	similar	similar	ADJ
ejpam-3504	217	4	way	way	NOUN
ejpam-3504	217	5	the	the	DET
ejpam-3504	217	6	right	right	ADJ
ejpam-3504	217	7	regular	regular	ADJ
ejpam-3504	217	8	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	217	9	are	be	AUX
ejpam-3504	217	10	intra	intra	ADJ
ejpam-3504	217	11	-	-	ADJ
ejpam-3504	217	12	regular	regular	ADJ
ejpam-3504	217	13	.	.	PUNCT
ejpam-3504	218	1	theorem	theorem	VERB
ejpam-3504	218	2	4.1	4.1	NUM
ejpam-3504	218	3	.	.	PUNCT
ejpam-3504	219	1	let	let	AUX
ejpam-3504	219	2	(	(	PUNCT
ejpam-3504	219	3	s	s	NOUN
ejpam-3504	219	4	,	,	PUNCT
ejpam-3504	219	5	◦	◦	NOUN
ejpam-3504	219	6	)	)	PUNCT
ejpam-3504	219	7	be	be	VERB
ejpam-3504	219	8	an	an	DET
ejpam-3504	219	9	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	219	10	.	.	PUNCT
ejpam-3504	220	1	if	if	SCONJ
ejpam-3504	220	2	s	s	PROPN
ejpam-3504	220	3	is	be	AUX
ejpam-3504	220	4	intra	intra	ADJ
ejpam-3504	220	5	-	-	ADJ
ejpam-3504	220	6	regular	regular	ADJ
ejpam-3504	220	7	then	then	ADV
ejpam-3504	220	8	,	,	PUNCT
ejpam-3504	220	9	for	for	ADP
ejpam-3504	220	10	every	every	DET
ejpam-3504	220	11	fuzzy	fuzzy	ADJ
ejpam-3504	220	12	ideal	ideal	NOUN
ejpam-3504	220	13	f	f	PROPN
ejpam-3504	220	14	of	of	ADP
ejpam-3504	220	15	s	s	PRON
ejpam-3504	220	16	and	and	CCONJ
ejpam-3504	220	17	every	every	PRON
ejpam-3504	220	18	a	a	DET
ejpam-3504	220	19	∈	∈	PROPN
ejpam-3504	220	20	s	s	NOUN
ejpam-3504	220	21	,	,	PUNCT
ejpam-3504	220	22	we	we	PRON
ejpam-3504	220	23	have	have	VERB
ejpam-3504	220	24	f(a	f(a	NOUN
ejpam-3504	220	25	)	)	PUNCT
ejpam-3504	221	1	=	=	SYM
ejpam-3504	222	1	f(a	f(a	NOUN
ejpam-3504	222	2	◦	◦	NOUN
ejpam-3504	222	3	a	a	NOUN
ejpam-3504	222	4	)	)	PUNCT
ejpam-3504	222	5	in	in	ADP
ejpam-3504	222	6	the	the	DET
ejpam-3504	222	7	sense	sense	NOUN
ejpam-3504	222	8	that	that	SCONJ
ejpam-3504	222	9	there	there	PRON
ejpam-3504	222	10	exists	exist	VERB
ejpam-3504	222	11	u	u	PROPN
ejpam-3504	222	12	∈	∈	PROPN
ejpam-3504	222	13	a	a	DET
ejpam-3504	222	14	◦	◦	NOUN
ejpam-3504	222	15	a	a	DET
ejpam-3504	222	16	such	such	ADJ
ejpam-3504	222	17	that	that	DET
ejpam-3504	222	18	f(a	f(a	NOUN
ejpam-3504	222	19	)	)	PUNCT
ejpam-3504	222	20	=	=	SYM
ejpam-3504	222	21	f(u	f(u	PROPN
ejpam-3504	222	22	)	)	PUNCT
ejpam-3504	222	23	.	.	PUNCT
ejpam-3504	223	1	“	"	PUNCT
ejpam-3504	223	2	conversely	conversely	ADV
ejpam-3504	223	3	”	"	PUNCT
ejpam-3504	223	4	,	,	PUNCT
ejpam-3504	223	5	if	if	SCONJ
ejpam-3504	223	6	for	for	SCONJ
ejpam-3504	223	7	any	any	DET
ejpam-3504	223	8	fuzzy	fuzzy	ADJ
ejpam-3504	223	9	ideal	ideal	NOUN
ejpam-3504	223	10	f	f	PROPN
ejpam-3504	223	11	of	of	ADP
ejpam-3504	223	12	s	s	PRON
ejpam-3504	223	13	and	and	CCONJ
ejpam-3504	223	14	any	any	DET
ejpam-3504	223	15	a	a	DET
ejpam-3504	223	16	∈	∈	NOUN
ejpam-3504	223	17	s	s	VERB
ejpam-3504	223	18	we	we	PRON
ejpam-3504	223	19	have	have	VERB
ejpam-3504	223	20	f(a	f(a	NOUN
ejpam-3504	223	21	)	)	PUNCT
ejpam-3504	223	22	=	=	SYM
ejpam-3504	223	23	f̂(a	f̂(a	PUNCT
ejpam-3504	223	24	◦	◦	NOUN
ejpam-3504	223	25	a	a	X
ejpam-3504	223	26	)	)	PUNCT
ejpam-3504	223	27	in	in	ADP
ejpam-3504	223	28	the	the	DET
ejpam-3504	223	29	sense	sense	NOUN
ejpam-3504	223	30	that	that	SCONJ
ejpam-3504	223	31	for	for	ADP
ejpam-3504	223	32	each	each	DET
ejpam-3504	223	33	u	u	NOUN
ejpam-3504	223	34	∈	∈	PROPN
ejpam-3504	223	35	a	a	DET
ejpam-3504	223	36	◦	◦	NOUN
ejpam-3504	223	37	a	a	PRON
ejpam-3504	223	38	,	,	PUNCT
ejpam-3504	223	39	f(a	f(a	NOUN
ejpam-3504	223	40	)	)	PUNCT
ejpam-3504	223	41	=	=	SYM
ejpam-3504	223	42	f(u	f(u	PROPN
ejpam-3504	223	43	)	)	PUNCT
ejpam-3504	223	44	,	,	PUNCT
ejpam-3504	223	45	then	then	ADV
ejpam-3504	223	46	s	s	VERB
ejpam-3504	223	47	is	be	AUX
ejpam-3504	223	48	intra	intra	ADJ
ejpam-3504	223	49	-	-	ADJ
ejpam-3504	223	50	regular	regular	ADJ
ejpam-3504	223	51	.	.	PUNCT
ejpam-3504	224	1	proof	proof	NOUN
ejpam-3504	224	2	.	.	PUNCT
ejpam-3504	225	1	let	let	VERB
ejpam-3504	225	2	s	s	PRON
ejpam-3504	225	3	be	be	AUX
ejpam-3504	225	4	intra	intra	ADJ
ejpam-3504	225	5	-	-	ADJ
ejpam-3504	225	6	regular	regular	ADJ
ejpam-3504	225	7	,	,	PUNCT
ejpam-3504	225	8	f	f	PROPN
ejpam-3504	225	9	a	a	DET
ejpam-3504	225	10	fuzzy	fuzzy	ADJ
ejpam-3504	225	11	ideal	ideal	NOUN
ejpam-3504	225	12	of	of	ADP
ejpam-3504	225	13	s	s	PRON
ejpam-3504	225	14	and	and	CCONJ
ejpam-3504	225	15	a	a	DET
ejpam-3504	225	16	∈	∈	NOUN
ejpam-3504	225	17	s.	s.	PROPN
ejpam-3504	225	18	since	since	SCONJ
ejpam-3504	225	19	s	s	PROPN
ejpam-3504	225	20	is	be	AUX
ejpam-3504	225	21	intra	intra	ADJ
ejpam-3504	225	22	-	-	ADJ
ejpam-3504	225	23	regular	regular	ADJ
ejpam-3504	225	24	,	,	PUNCT
ejpam-3504	225	25	there	there	PRON
ejpam-3504	225	26	exist	exist	VERB
ejpam-3504	225	27	x	x	NOUN
ejpam-3504	225	28	,	,	PUNCT
ejpam-3504	225	29	y	y	PROPN
ejpam-3504	225	30	∈	∈	PROPN
ejpam-3504	225	31	s	s	VERB
ejpam-3504	225	32	such	such	ADJ
ejpam-3504	225	33	that	that	SCONJ
ejpam-3504	225	34	a	a	DET
ejpam-3504	225	35	∈	∈	PROPN
ejpam-3504	225	36	(	(	PUNCT
ejpam-3504	225	37	x∗	x∗	X
ejpam-3504	225	38	(	(	PUNCT
ejpam-3504	225	39	a	a	DET
ejpam-3504	225	40	◦	◦	NOUN
ejpam-3504	225	41	a	a	NOUN
ejpam-3504	225	42	)	)	PUNCT
ejpam-3504	225	43	)	)	PUNCT
ejpam-3504	226	1	∗{y	∗{y	VERB
ejpam-3504	226	2	}	}	PUNCT
ejpam-3504	226	3	.	.	PUNCT
ejpam-3504	227	1	then	then	ADV
ejpam-3504	227	2	a	a	DET
ejpam-3504	227	3	∈	∈	PROPN
ejpam-3504	227	4	v	v	ADP
ejpam-3504	227	5	◦	◦	NOUN
ejpam-3504	227	6	y	y	NOUN
ejpam-3504	227	7	for	for	ADP
ejpam-3504	227	8	some	some	DET
ejpam-3504	227	9	v	v	NOUN
ejpam-3504	227	10	∈	∈	NOUN
ejpam-3504	227	11	x∗	x∗	X
ejpam-3504	227	12	(	(	PUNCT
ejpam-3504	227	13	a	a	DET
ejpam-3504	227	14	◦	◦	NOUN
ejpam-3504	227	15	a	a	NOUN
ejpam-3504	227	16	)	)	PUNCT
ejpam-3504	227	17	and	and	CCONJ
ejpam-3504	227	18	v	v	ADP
ejpam-3504	227	19	∈	∈	NOUN
ejpam-3504	227	20	x	x	X
ejpam-3504	227	21	◦	◦	NOUN
ejpam-3504	227	22	u	u	NOUN
ejpam-3504	227	23	for	for	ADP
ejpam-3504	227	24	some	some	DET
ejpam-3504	227	25	u	u	NOUN
ejpam-3504	227	26	∈	∈	PROPN
ejpam-3504	227	27	a	a	DET
ejpam-3504	227	28	◦	◦	NOUN
ejpam-3504	227	29	a.	a.	NOUN
ejpam-3504	227	30	since	since	SCONJ
ejpam-3504	227	31	f	f	PROPN
ejpam-3504	227	32	is	be	AUX
ejpam-3504	227	33	a	a	DET
ejpam-3504	227	34	fuzzy	fuzzy	ADJ
ejpam-3504	227	35	right	right	ADJ
ejpam-3504	227	36	ideal	ideal	NOUN
ejpam-3504	227	37	of	of	ADP
ejpam-3504	227	38	s	s	PROPN
ejpam-3504	227	39	,	,	PUNCT
ejpam-3504	227	40	we	we	PRON
ejpam-3504	227	41	have	have	VERB
ejpam-3504	227	42	f(v	f(v	NOUN
ejpam-3504	227	43	◦	◦	NOUN
ejpam-3504	227	44	y	y	NOUN
ejpam-3504	227	45	)	)	PUNCT
ejpam-3504	227	46	≥	≥	NOUN
ejpam-3504	227	47	f(v	f(v	NOUN
ejpam-3504	227	48	)	)	PUNCT
ejpam-3504	227	49	and	and	CCONJ
ejpam-3504	227	50	,	,	PUNCT
ejpam-3504	227	51	since	since	SCONJ
ejpam-3504	227	52	a	a	DET
ejpam-3504	227	53	∈	∈	NOUN
ejpam-3504	227	54	v	v	ADP
ejpam-3504	227	55	◦	◦	NOUN
ejpam-3504	227	56	y	y	NOUN
ejpam-3504	227	57	,	,	PUNCT
ejpam-3504	227	58	we	we	PRON
ejpam-3504	227	59	have	have	VERB
ejpam-3504	227	60	f(a	f(a	PROPN
ejpam-3504	227	61	)	)	PUNCT
ejpam-3504	227	62	≥	≥	NOUN
ejpam-3504	227	63	f(v	f(v	NOUN
ejpam-3504	227	64	)	)	PUNCT
ejpam-3504	227	65	.	.	PUNCT
ejpam-3504	228	1	since	since	SCONJ
ejpam-3504	228	2	f	f	PROPN
ejpam-3504	228	3	is	be	AUX
ejpam-3504	228	4	a	a	DET
ejpam-3504	228	5	fuzzy	fuzzy	ADJ
ejpam-3504	228	6	left	leave	VERB
ejpam-3504	228	7	ideal	ideal	NOUN
ejpam-3504	228	8	of	of	ADP
ejpam-3504	228	9	s	s	PROPN
ejpam-3504	228	10	,	,	PUNCT
ejpam-3504	228	11	we	we	PRON
ejpam-3504	228	12	have	have	VERB
ejpam-3504	228	13	f(x	f(x	PROPN
ejpam-3504	228	14	◦	◦	PROPN
ejpam-3504	228	15	u	u	NOUN
ejpam-3504	228	16	)	)	PUNCT
ejpam-3504	228	17	≥	≥	NOUN
ejpam-3504	228	18	f(u	f(u	PROPN
ejpam-3504	228	19	)	)	PUNCT
ejpam-3504	228	20	and	and	CCONJ
ejpam-3504	228	21	,	,	PUNCT
ejpam-3504	228	22	since	since	SCONJ
ejpam-3504	228	23	v	v	NUM
ejpam-3504	228	24	∈	∈	NOUN
ejpam-3504	228	25	x	x	PUNCT
ejpam-3504	228	26	◦	◦	NOUN
ejpam-3504	228	27	u	u	NOUN
ejpam-3504	228	28	,	,	PUNCT
ejpam-3504	228	29	we	we	PRON
ejpam-3504	228	30	get	get	VERB
ejpam-3504	228	31	f(v	f(v	NOUN
ejpam-3504	228	32	)	)	PUNCT
ejpam-3504	228	33	≥	≥	NOUN
ejpam-3504	228	34	f(u	f(u	PROPN
ejpam-3504	228	35	)	)	PUNCT
ejpam-3504	228	36	.	.	PUNCT
ejpam-3504	229	1	thus	thus	ADV
ejpam-3504	229	2	we	we	PRON
ejpam-3504	229	3	have	have	VERB
ejpam-3504	229	4	f(a	f(a	PROPN
ejpam-3504	229	5	)	)	PUNCT
ejpam-3504	229	6	≥	≥	NOUN
ejpam-3504	229	7	f(u	f(u	PROPN
ejpam-3504	229	8	)	)	PUNCT
ejpam-3504	229	9	.	.	PUNCT
ejpam-3504	230	1	besides	besides	SCONJ
ejpam-3504	230	2	,	,	PUNCT
ejpam-3504	230	3	from	from	ADP
ejpam-3504	230	4	the	the	DET
ejpam-3504	230	5	fact	fact	NOUN
ejpam-3504	230	6	that	that	SCONJ
ejpam-3504	230	7	f	f	PROPN
ejpam-3504	230	8	is	be	AUX
ejpam-3504	230	9	a	a	DET
ejpam-3504	230	10	left	left	ADJ
ejpam-3504	230	11	(	(	PUNCT
ejpam-3504	230	12	or	or	CCONJ
ejpam-3504	230	13	right	right	ADJ
ejpam-3504	230	14	)	)	PUNCT
ejpam-3504	230	15	ideal	ideal	NOUN
ejpam-3504	230	16	of	of	ADP
ejpam-3504	230	17	s	s	PROPN
ejpam-3504	230	18	,	,	PUNCT
ejpam-3504	230	19	we	we	PRON
ejpam-3504	230	20	have	have	VERB
ejpam-3504	230	21	f(a	f(a	NOUN
ejpam-3504	230	22	◦	◦	NOUN
ejpam-3504	230	23	a	a	DET
ejpam-3504	230	24	)	)	PUNCT
ejpam-3504	230	25	≥	≥	NOUN
ejpam-3504	230	26	f(a	f(a	NOUN
ejpam-3504	230	27	)	)	PUNCT
ejpam-3504	230	28	and	and	CCONJ
ejpam-3504	230	29	since	since	SCONJ
ejpam-3504	230	30	u	u	PROPN
ejpam-3504	230	31	∈	∈	PROPN
ejpam-3504	230	32	a	a	DET
ejpam-3504	230	33	◦	◦	NOUN
ejpam-3504	230	34	a	a	X
ejpam-3504	230	35	,	,	PUNCT
ejpam-3504	230	36	we	we	PRON
ejpam-3504	230	37	have	have	VERB
ejpam-3504	230	38	f(u	f(u	PROPN
ejpam-3504	230	39	)	)	PUNCT
ejpam-3504	230	40	≥	≥	NOUN
ejpam-3504	230	41	f(a	f(a	NOUN
ejpam-3504	230	42	)	)	PUNCT
ejpam-3504	230	43	.	.	PUNCT
ejpam-3504	231	1	thus	thus	ADV
ejpam-3504	231	2	we	we	PRON
ejpam-3504	231	3	have	have	VERB
ejpam-3504	231	4	f(a	f(a	NOUN
ejpam-3504	231	5	)	)	PUNCT
ejpam-3504	231	6	=	=	SYM
ejpam-3504	231	7	f(u	f(u	PROPN
ejpam-3504	231	8	)	)	PUNCT
ejpam-3504	231	9	.	.	PUNCT
ejpam-3504	232	1	for	for	ADP
ejpam-3504	232	2	the	the	DET
ejpam-3504	232	3	“	"	PUNCT
ejpam-3504	232	4	converse	converse	NOUN
ejpam-3504	232	5	”	"	PUNCT
ejpam-3504	232	6	statement	statement	NOUN
ejpam-3504	232	7	,	,	PUNCT
ejpam-3504	232	8	let	let	VERB
ejpam-3504	232	9	a	a	DET
ejpam-3504	232	10	∈	∈	PROPN
ejpam-3504	232	11	s.	s.	PROPN
ejpam-3504	232	12	take	take	VERB
ejpam-3504	232	13	an	an	DET
ejpam-3504	232	14	element	element	NOUN
ejpam-3504	232	15	u	u	NOUN
ejpam-3504	232	16	∈	∈	PROPN
ejpam-3504	232	17	a	a	DET
ejpam-3504	232	18	◦	◦	NOUN
ejpam-3504	232	19	a	a	DET
ejpam-3504	232	20	(	(	PUNCT
ejpam-3504	232	21	a	a	DET
ejpam-3504	232	22	◦	◦	NOUN
ejpam-3504	232	23	a	a	DET
ejpam-3504	232	24	6=	6=	NOUN
ejpam-3504	232	25	∅	∅	NOUN
ejpam-3504	232	26	)	)	PUNCT
ejpam-3504	232	27	.	.	PUNCT
ejpam-3504	233	1	since	since	SCONJ
ejpam-3504	233	2	i(u	i(u	PROPN
ejpam-3504	233	3	)	)	PUNCT
ejpam-3504	233	4	is	be	AUX
ejpam-3504	233	5	an	an	DET
ejpam-3504	233	6	ideal	ideal	NOUN
ejpam-3504	233	7	of	of	ADP
ejpam-3504	233	8	s	s	PROPN
ejpam-3504	233	9	,	,	PUNCT
ejpam-3504	233	10	by	by	ADP
ejpam-3504	233	11	lemma	lemma	PROPN
ejpam-3504	233	12	3.2	3.2	NUM
ejpam-3504	233	13	,	,	PUNCT
ejpam-3504	233	14	the	the	DET
ejpam-3504	233	15	characteristic	characteristic	ADJ
ejpam-3504	233	16	function	function	NOUN
ejpam-3504	233	17	fi(u	fi(u	NOUN
ejpam-3504	233	18	)	)	PUNCT
ejpam-3504	233	19	is	be	AUX
ejpam-3504	233	20	a	a	DET
ejpam-3504	233	21	fuzzy	fuzzy	ADJ
ejpam-3504	233	22	ideal	ideal	NOUN
ejpam-3504	233	23	of	of	ADP
ejpam-3504	233	24	s.	s.	PROPN
ejpam-3504	233	25	by	by	ADP
ejpam-3504	233	26	hypothesis	hypothesis	NOUN
ejpam-3504	233	27	,	,	PUNCT
ejpam-3504	233	28	we	we	PRON
ejpam-3504	233	29	have	have	VERB
ejpam-3504	233	30	fi(u)(a	fi(u)(a	NUM
ejpam-3504	233	31	)	)	PUNCT
ejpam-3504	234	1	=	=	SYM
ejpam-3504	234	2	f̂i(u)(a	f̂i(u)(a	NOUN
ejpam-3504	234	3	◦	◦	VERB
ejpam-3504	234	4	a	a	X
ejpam-3504	234	5	)	)	PUNCT
ejpam-3504	234	6	and	and	CCONJ
ejpam-3504	234	7	,	,	PUNCT
ejpam-3504	234	8	since	since	SCONJ
ejpam-3504	234	9	u	u	PROPN
ejpam-3504	234	10	∈	∈	PROPN
ejpam-3504	234	11	a	a	DET
ejpam-3504	234	12	◦	◦	NOUN
ejpam-3504	234	13	a	a	X
ejpam-3504	234	14	,	,	PUNCT
ejpam-3504	234	15	we	we	PRON
ejpam-3504	234	16	have	have	VERB
ejpam-3504	234	17	n.	n.	NOUN
ejpam-3504	234	18	kehayopulu	kehayopulu	PROPN
ejpam-3504	234	19	/	/	SYM
ejpam-3504	234	20	eur	eur	PROPN
ejpam-3504	234	21	.	.	PUNCT
ejpam-3504	235	1	j.	j.	PROPN
ejpam-3504	235	2	pure	pure	PROPN
ejpam-3504	235	3	appl	appl	PROPN
ejpam-3504	235	4	.	.	PROPN
ejpam-3504	235	5	math	math	PROPN
ejpam-3504	235	6	,	,	PUNCT
ejpam-3504	235	7	12	12	NUM
ejpam-3504	235	8	(	(	PUNCT
ejpam-3504	235	9	3	3	NUM
ejpam-3504	235	10	)	)	PUNCT
ejpam-3504	235	11	(	(	PUNCT
ejpam-3504	235	12	2019	2019	NUM
ejpam-3504	235	13	)	)	PUNCT
ejpam-3504	235	14	,	,	PUNCT
ejpam-3504	235	15	709	709	NUM
ejpam-3504	235	16	-	-	SYM
ejpam-3504	235	17	721	721	NUM
ejpam-3504	235	18	716	716	NUM
ejpam-3504	235	19	fi(u)(a	fi(u)(a	NUM
ejpam-3504	235	20	)	)	PUNCT
ejpam-3504	235	21	=	=	SYM
ejpam-3504	235	22	fi(u)(u	fi(u)(u	NOUN
ejpam-3504	235	23	)	)	PUNCT
ejpam-3504	235	24	.	.	PUNCT
ejpam-3504	236	1	since	since	SCONJ
ejpam-3504	236	2	u	u	PROPN
ejpam-3504	236	3	∈	∈	PROPN
ejpam-3504	236	4	i(u	i(u	PROPN
ejpam-3504	236	5	)	)	PUNCT
ejpam-3504	236	6	,	,	PUNCT
ejpam-3504	236	7	we	we	PRON
ejpam-3504	236	8	have	have	VERB
ejpam-3504	236	9	fi(u)(u	fi(u)(u	NUM
ejpam-3504	236	10	)	)	PUNCT
ejpam-3504	236	11	=	=	SYM
ejpam-3504	237	1	1	1	X
ejpam-3504	237	2	.	.	PUNCT
ejpam-3504	237	3	then	then	ADV
ejpam-3504	237	4	we	we	PRON
ejpam-3504	237	5	fi(u)(a	fi(u)(a	VERB
ejpam-3504	237	6	)	)	PUNCT
ejpam-3504	238	1	=	=	SYM
ejpam-3504	238	2	1	1	NUM
ejpam-3504	238	3	,	,	PUNCT
ejpam-3504	238	4	and	and	CCONJ
ejpam-3504	238	5	then	then	ADV
ejpam-3504	238	6	a	a	DET
ejpam-3504	238	7	∈	∈	PROPN
ejpam-3504	238	8	i(u	i(u	PROPN
ejpam-3504	238	9	)	)	PUNCT
ejpam-3504	239	1	=	=	SYM
ejpam-3504	239	2	u	u	NOUN
ejpam-3504	239	3	∪	∪	VERB
ejpam-3504	239	4	s	s	PROPN
ejpam-3504	239	5	∗	∗	NOUN
ejpam-3504	239	6	u	u	NOUN
ejpam-3504	239	7	∪	∪	ADP
ejpam-3504	239	8	u	u	NOUN
ejpam-3504	239	9	∗	∗	NOUN
ejpam-3504	239	10	s	s	NOUN
ejpam-3504	239	11	∪	∪	NOUN
ejpam-3504	239	12	s	s	PART
ejpam-3504	239	13	∗	∗	NOUN
ejpam-3504	239	14	u	u	NOUN
ejpam-3504	239	15	∗	∗	NOUN
ejpam-3504	239	16	s.	s.	PROPN
ejpam-3504	239	17	if	if	SCONJ
ejpam-3504	239	18	a	a	DET
ejpam-3504	239	19	=	=	SYM
ejpam-3504	239	20	u	u	NOUN
ejpam-3504	239	21	,	,	PUNCT
ejpam-3504	239	22	then	then	ADV
ejpam-3504	239	23	a	a	DET
ejpam-3504	239	24	∈	∈	NOUN
ejpam-3504	239	25	{	{	PUNCT
ejpam-3504	239	26	a	a	NOUN
ejpam-3504	239	27	}	}	PUNCT
ejpam-3504	239	28	∗	∗	NOUN
ejpam-3504	239	29	{	{	PUNCT
ejpam-3504	239	30	a	a	PRON
ejpam-3504	239	31	}	}	PUNCT
ejpam-3504	239	32	⊆	⊆	NUM
ejpam-3504	239	33	{	{	PUNCT
ejpam-3504	239	34	a	a	PRON
ejpam-3504	239	35	}	}	PUNCT
ejpam-3504	239	36	∗	∗	NOUN
ejpam-3504	239	37	{	{	PUNCT
ejpam-3504	239	38	a	a	DET
ejpam-3504	239	39	}	}	PUNCT
ejpam-3504	239	40	∗	∗	NOUN
ejpam-3504	239	41	{	{	PUNCT
ejpam-3504	239	42	a	a	DET
ejpam-3504	239	43	}	}	PUNCT
ejpam-3504	239	44	∗	∗	NOUN
ejpam-3504	239	45	{	{	PUNCT
ejpam-3504	239	46	a	a	NOUN
ejpam-3504	239	47	}	}	PUNCT
ejpam-3504	239	48	⊆	⊆	NUM
ejpam-3504	239	49	s	s	NOUN
ejpam-3504	239	50	∗	∗	NOUN
ejpam-3504	239	51	(	(	PUNCT
ejpam-3504	239	52	a	a	DET
ejpam-3504	239	53	◦	◦	NOUN
ejpam-3504	239	54	a	a	X
ejpam-3504	239	55	)	)	PUNCT
ejpam-3504	239	56	∗	∗	NOUN
ejpam-3504	239	57	s.	s.	PROPN
ejpam-3504	239	58	if	if	SCONJ
ejpam-3504	239	59	a	a	DET
ejpam-3504	239	60	∈	∈	PROPN
ejpam-3504	239	61	s	s	PART
ejpam-3504	239	62	∗	∗	NOUN
ejpam-3504	239	63	u	u	PROPN
ejpam-3504	239	64	,	,	PUNCT
ejpam-3504	239	65	then	then	ADV
ejpam-3504	239	66	a	a	DET
ejpam-3504	239	67	∈	∈	PROPN
ejpam-3504	239	68	s	s	PART
ejpam-3504	239	69	∗	∗	NOUN
ejpam-3504	239	70	{	{	PUNCT
ejpam-3504	239	71	a	a	DET
ejpam-3504	239	72	}	}	PUNCT
ejpam-3504	239	73	∗	∗	NOUN
ejpam-3504	239	74	{	{	PUNCT
ejpam-3504	239	75	a	a	NOUN
ejpam-3504	239	76	}	}	PUNCT
ejpam-3504	239	77	⊆	⊆	NUM
ejpam-3504	239	78	s	s	NOUN
ejpam-3504	239	79	∗	∗	NOUN
ejpam-3504	239	80	(	(	PUNCT
ejpam-3504	239	81	s	s	NOUN
ejpam-3504	239	82	∗	∗	NOUN
ejpam-3504	239	83	{	{	PUNCT
ejpam-3504	239	84	a	a	DET
ejpam-3504	239	85	}	}	PUNCT
ejpam-3504	239	86	∗	∗	NOUN
ejpam-3504	239	87	{	{	PUNCT
ejpam-3504	239	88	a	a	NOUN
ejpam-3504	239	89	}	}	PUNCT
ejpam-3504	239	90	)	)	PUNCT
ejpam-3504	239	91	∗	∗	NOUN
ejpam-3504	239	92	{	{	PUNCT
ejpam-3504	239	93	a	a	NOUN
ejpam-3504	239	94	}	}	PUNCT
ejpam-3504	239	95	⊆	⊆	NUM
ejpam-3504	239	96	s	s	NOUN
ejpam-3504	239	97	∗	∗	NOUN
ejpam-3504	239	98	(	(	PUNCT
ejpam-3504	239	99	{	{	PUNCT
ejpam-3504	239	100	a	a	PRON
ejpam-3504	239	101	}	}	PUNCT
ejpam-3504	239	102	∗	∗	NOUN
ejpam-3504	239	103	{	{	PUNCT
ejpam-3504	239	104	a	a	NOUN
ejpam-3504	239	105	}	}	PUNCT
ejpam-3504	239	106	)	)	PUNCT
ejpam-3504	239	107	∗	∗	NOUN
ejpam-3504	239	108	s	s	PART
ejpam-3504	239	109	=	=	SYM
ejpam-3504	239	110	s	s	PART
ejpam-3504	239	111	∗	∗	NOUN
ejpam-3504	239	112	(	(	PUNCT
ejpam-3504	239	113	a	a	DET
ejpam-3504	239	114	◦	◦	NOUN
ejpam-3504	239	115	a	a	X
ejpam-3504	239	116	)	)	PUNCT
ejpam-3504	239	117	∗	∗	NOUN
ejpam-3504	239	118	s.	s.	PROPN
ejpam-3504	240	1	the	the	DET
ejpam-3504	240	2	case	case	NOUN
ejpam-3504	240	3	a	a	DET
ejpam-3504	240	4	∈	∈	NOUN
ejpam-3504	240	5	u	u	NOUN
ejpam-3504	240	6	∗	∗	NOUN
ejpam-3504	240	7	s	s	NOUN
ejpam-3504	240	8	is	be	AUX
ejpam-3504	240	9	similar	similar	ADJ
ejpam-3504	240	10	.	.	PUNCT
ejpam-3504	241	1	if	if	SCONJ
ejpam-3504	241	2	a	a	DET
ejpam-3504	241	3	∈	∈	PROPN
ejpam-3504	241	4	s	s	PART
ejpam-3504	241	5	∗	∗	NOUN
ejpam-3504	241	6	u	u	NOUN
ejpam-3504	241	7	∗	∗	NOUN
ejpam-3504	241	8	s	s	PROPN
ejpam-3504	241	9	,	,	PUNCT
ejpam-3504	241	10	then	then	ADV
ejpam-3504	241	11	a	a	DET
ejpam-3504	241	12	∈	∈	PROPN
ejpam-3504	241	13	s	s	PART
ejpam-3504	241	14	∗	∗	NOUN
ejpam-3504	241	15	(	(	PUNCT
ejpam-3504	241	16	a	a	DET
ejpam-3504	241	17	◦	◦	NOUN
ejpam-3504	241	18	a	a	X
ejpam-3504	241	19	)	)	PUNCT
ejpam-3504	241	20	∗	∗	NOUN
ejpam-3504	241	21	s.	s.	PROPN
ejpam-3504	241	22	in	in	ADP
ejpam-3504	241	23	any	any	DET
ejpam-3504	241	24	case	case	NOUN
ejpam-3504	241	25	,	,	PUNCT
ejpam-3504	241	26	we	we	PRON
ejpam-3504	241	27	have	have	VERB
ejpam-3504	241	28	a	a	DET
ejpam-3504	241	29	∈	∈	PROPN
ejpam-3504	241	30	s	s	PART
ejpam-3504	241	31	∗	∗	NOUN
ejpam-3504	241	32	(	(	PUNCT
ejpam-3504	241	33	a	a	DET
ejpam-3504	241	34	◦	◦	NOUN
ejpam-3504	241	35	a	a	X
ejpam-3504	241	36	)	)	PUNCT
ejpam-3504	241	37	∗	∗	NOUN
ejpam-3504	241	38	s	s	NOUN
ejpam-3504	241	39	and	and	CCONJ
ejpam-3504	241	40	so	so	ADV
ejpam-3504	241	41	s	s	NOUN
ejpam-3504	241	42	is	be	AUX
ejpam-3504	241	43	intra	intra	ADJ
ejpam-3504	241	44	-	-	ADJ
ejpam-3504	241	45	regular	regular	ADJ
ejpam-3504	241	46	.	.	PUNCT
ejpam-3504	242	1	�	�	PROPN
ejpam-3504	242	2	theorem	theorem	VERB
ejpam-3504	242	3	4.2	4.2	NUM
ejpam-3504	242	4	.	.	PUNCT
ejpam-3504	243	1	let	let	AUX
ejpam-3504	243	2	(	(	PUNCT
ejpam-3504	243	3	s	s	NOUN
ejpam-3504	243	4	,	,	PUNCT
ejpam-3504	243	5	◦	◦	NOUN
ejpam-3504	243	6	)	)	PUNCT
ejpam-3504	243	7	be	be	VERB
ejpam-3504	243	8	an	an	DET
ejpam-3504	243	9	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	243	10	.	.	PUNCT
ejpam-3504	244	1	if	if	SCONJ
ejpam-3504	244	2	s	s	NOUN
ejpam-3504	244	3	is	be	AUX
ejpam-3504	244	4	left	leave	VERB
ejpam-3504	244	5	regular	regular	ADV
ejpam-3504	244	6	,	,	PUNCT
ejpam-3504	244	7	then	then	ADV
ejpam-3504	244	8	for	for	SCONJ
ejpam-3504	244	9	every	every	DET
ejpam-3504	244	10	fuzzy	fuzzy	ADJ
ejpam-3504	244	11	left	leave	VERB
ejpam-3504	244	12	ideal	ideal	NOUN
ejpam-3504	244	13	f	f	PROPN
ejpam-3504	244	14	of	of	ADP
ejpam-3504	244	15	s	s	PRON
ejpam-3504	244	16	and	and	CCONJ
ejpam-3504	244	17	every	every	PRON
ejpam-3504	244	18	a	a	DET
ejpam-3504	244	19	∈	∈	PROPN
ejpam-3504	244	20	s	s	NOUN
ejpam-3504	244	21	,	,	PUNCT
ejpam-3504	244	22	we	we	PRON
ejpam-3504	244	23	have	have	VERB
ejpam-3504	244	24	f(a	f(a	NOUN
ejpam-3504	244	25	)	)	PUNCT
ejpam-3504	245	1	=	=	SYM
ejpam-3504	245	2	f(a	f(a	PROPN
ejpam-3504	245	3	◦	◦	NOUN
ejpam-3504	245	4	a	a	X
ejpam-3504	245	5	)	)	PUNCT
ejpam-3504	245	6	in	in	ADP
ejpam-3504	245	7	the	the	DET
ejpam-3504	245	8	sense	sense	NOUN
ejpam-3504	245	9	that	that	SCONJ
ejpam-3504	245	10	there	there	PRON
ejpam-3504	245	11	exists	exist	VERB
ejpam-3504	245	12	u	u	PROPN
ejpam-3504	245	13	∈	∈	PROPN
ejpam-3504	245	14	a	a	DET
ejpam-3504	245	15	◦	◦	NOUN
ejpam-3504	245	16	a	a	DET
ejpam-3504	245	17	such	such	ADJ
ejpam-3504	245	18	that	that	DET
ejpam-3504	245	19	f(a	f(a	NOUN
ejpam-3504	245	20	)	)	PUNCT
ejpam-3504	245	21	=	=	SYM
ejpam-3504	245	22	f(u	f(u	PROPN
ejpam-3504	245	23	)	)	PUNCT
ejpam-3504	245	24	.	.	PUNCT
ejpam-3504	246	1	“	"	PUNCT
ejpam-3504	246	2	conversely	conversely	ADV
ejpam-3504	246	3	”	"	PUNCT
ejpam-3504	246	4	if	if	SCONJ
ejpam-3504	246	5	,	,	PUNCT
ejpam-3504	246	6	for	for	ADP
ejpam-3504	246	7	any	any	DET
ejpam-3504	246	8	fuzzy	fuzzy	ADJ
ejpam-3504	246	9	left	leave	VERB
ejpam-3504	246	10	ideal	ideal	NOUN
ejpam-3504	246	11	f	f	PROPN
ejpam-3504	246	12	of	of	ADP
ejpam-3504	246	13	s	s	PRON
ejpam-3504	246	14	and	and	CCONJ
ejpam-3504	246	15	any	any	DET
ejpam-3504	246	16	a	a	DET
ejpam-3504	246	17	∈	∈	NOUN
ejpam-3504	246	18	s	s	VERB
ejpam-3504	246	19	we	we	PRON
ejpam-3504	246	20	have	have	VERB
ejpam-3504	246	21	f(a	f(a	NOUN
ejpam-3504	246	22	)	)	PUNCT
ejpam-3504	246	23	=	=	SYM
ejpam-3504	246	24	f̂(a	f̂(a	PUNCT
ejpam-3504	246	25	◦	◦	NOUN
ejpam-3504	246	26	a	a	X
ejpam-3504	246	27	)	)	PUNCT
ejpam-3504	246	28	in	in	ADP
ejpam-3504	246	29	the	the	DET
ejpam-3504	246	30	sense	sense	NOUN
ejpam-3504	246	31	that	that	SCONJ
ejpam-3504	246	32	if	if	SCONJ
ejpam-3504	246	33	u	u	PROPN
ejpam-3504	246	34	∈	∈	VERB
ejpam-3504	246	35	a	a	DET
ejpam-3504	246	36	◦	◦	NOUN
ejpam-3504	246	37	a	a	PRON
ejpam-3504	246	38	,	,	PUNCT
ejpam-3504	246	39	then	then	ADV
ejpam-3504	246	40	f(a	f(a	PROPN
ejpam-3504	246	41	)	)	PUNCT
ejpam-3504	246	42	=	=	SYM
ejpam-3504	246	43	f(u	f(u	PROPN
ejpam-3504	246	44	)	)	PUNCT
ejpam-3504	246	45	,	,	PUNCT
ejpam-3504	246	46	then	then	ADV
ejpam-3504	246	47	s	s	VERB
ejpam-3504	246	48	is	be	AUX
ejpam-3504	246	49	left	leave	VERB
ejpam-3504	246	50	regular	regular	ADV
ejpam-3504	246	51	.	.	PUNCT
ejpam-3504	247	1	proof	proof	NOUN
ejpam-3504	247	2	.	.	PUNCT
ejpam-3504	248	1	let	let	VERB
ejpam-3504	248	2	s	s	PRON
ejpam-3504	248	3	be	be	AUX
ejpam-3504	248	4	left	leave	VERB
ejpam-3504	248	5	regular	regular	ADV
ejpam-3504	248	6	,	,	PUNCT
ejpam-3504	248	7	f	f	PROPN
ejpam-3504	248	8	a	a	DET
ejpam-3504	248	9	fuzzy	fuzzy	ADJ
ejpam-3504	248	10	left	leave	VERB
ejpam-3504	248	11	ideal	ideal	NOUN
ejpam-3504	248	12	of	of	ADP
ejpam-3504	248	13	s	s	PRON
ejpam-3504	248	14	and	and	CCONJ
ejpam-3504	248	15	a	a	DET
ejpam-3504	248	16	∈	∈	NOUN
ejpam-3504	248	17	s.	s.	PROPN
ejpam-3504	248	18	since	since	SCONJ
ejpam-3504	248	19	s	s	PROPN
ejpam-3504	248	20	is	be	AUX
ejpam-3504	248	21	left	leave	VERB
ejpam-3504	248	22	regular	regular	ADV
ejpam-3504	248	23	,	,	PUNCT
ejpam-3504	248	24	there	there	PRON
ejpam-3504	248	25	exists	exist	VERB
ejpam-3504	248	26	x	x	X
ejpam-3504	248	27	∈	∈	PROPN
ejpam-3504	248	28	s	s	VERB
ejpam-3504	248	29	such	such	ADJ
ejpam-3504	248	30	that	that	SCONJ
ejpam-3504	248	31	a	a	DET
ejpam-3504	248	32	∈	∈	NOUN
ejpam-3504	248	33	x	x	PUNCT
ejpam-3504	248	34	∗	∗	NOUN
ejpam-3504	248	35	(	(	PUNCT
ejpam-3504	248	36	a	a	DET
ejpam-3504	248	37	◦	◦	NOUN
ejpam-3504	248	38	a	a	NOUN
ejpam-3504	248	39	)	)	PUNCT
ejpam-3504	248	40	.	.	PUNCT
ejpam-3504	249	1	then	then	ADV
ejpam-3504	249	2	there	there	PRON
ejpam-3504	249	3	exists	exist	VERB
ejpam-3504	249	4	u	u	PROPN
ejpam-3504	249	5	∈	∈	PROPN
ejpam-3504	249	6	a	a	DET
ejpam-3504	249	7	◦	◦	NOUN
ejpam-3504	249	8	a	a	PRON
ejpam-3504	249	9	such	such	ADJ
ejpam-3504	249	10	that	that	SCONJ
ejpam-3504	249	11	a	a	DET
ejpam-3504	249	12	∈	∈	PROPN
ejpam-3504	249	13	x	x	PUNCT
ejpam-3504	249	14	◦	◦	NOUN
ejpam-3504	249	15	u.	u.	NOUN
ejpam-3504	249	16	since	since	SCONJ
ejpam-3504	249	17	f	f	PROPN
ejpam-3504	249	18	is	be	AUX
ejpam-3504	249	19	a	a	DET
ejpam-3504	249	20	fuzzy	fuzzy	ADJ
ejpam-3504	249	21	left	leave	VERB
ejpam-3504	249	22	ideal	ideal	NOUN
ejpam-3504	249	23	of	of	ADP
ejpam-3504	249	24	s	s	PROPN
ejpam-3504	249	25	,	,	PUNCT
ejpam-3504	249	26	we	we	PRON
ejpam-3504	249	27	have	have	VERB
ejpam-3504	249	28	f(x	f(x	PROPN
ejpam-3504	249	29	◦	◦	PROPN
ejpam-3504	249	30	u	u	NOUN
ejpam-3504	249	31	)	)	PUNCT
ejpam-3504	249	32	≥	≥	NOUN
ejpam-3504	249	33	f(u	f(u	PROPN
ejpam-3504	249	34	)	)	PUNCT
ejpam-3504	249	35	and	and	CCONJ
ejpam-3504	249	36	,	,	PUNCT
ejpam-3504	249	37	since	since	SCONJ
ejpam-3504	249	38	a	a	DET
ejpam-3504	249	39	∈	∈	NOUN
ejpam-3504	249	40	x	x	PUNCT
ejpam-3504	249	41	◦	◦	NOUN
ejpam-3504	249	42	u	u	NOUN
ejpam-3504	249	43	,	,	PUNCT
ejpam-3504	249	44	we	we	PRON
ejpam-3504	249	45	have	have	VERB
ejpam-3504	249	46	f(a	f(a	PROPN
ejpam-3504	249	47	)	)	PUNCT
ejpam-3504	249	48	≥	≥	NOUN
ejpam-3504	249	49	f(u	f(u	PROPN
ejpam-3504	249	50	)	)	PUNCT
ejpam-3504	249	51	.	.	PUNCT
ejpam-3504	250	1	again	again	ADV
ejpam-3504	250	2	since	since	SCONJ
ejpam-3504	250	3	f	f	PROPN
ejpam-3504	250	4	is	be	AUX
ejpam-3504	250	5	a	a	DET
ejpam-3504	250	6	fuzzy	fuzzy	ADJ
ejpam-3504	250	7	left	leave	VERB
ejpam-3504	250	8	ideal	ideal	NOUN
ejpam-3504	250	9	of	of	ADP
ejpam-3504	250	10	s	s	PROPN
ejpam-3504	250	11	,	,	PUNCT
ejpam-3504	250	12	we	we	PRON
ejpam-3504	250	13	have	have	VERB
ejpam-3504	250	14	f(a	f(a	NOUN
ejpam-3504	250	15	◦	◦	NOUN
ejpam-3504	250	16	a	a	DET
ejpam-3504	250	17	)	)	PUNCT
ejpam-3504	250	18	≥	≥	NOUN
ejpam-3504	250	19	f(a	f(a	NOUN
ejpam-3504	250	20	)	)	PUNCT
ejpam-3504	250	21	and	and	CCONJ
ejpam-3504	250	22	,	,	PUNCT
ejpam-3504	250	23	since	since	SCONJ
ejpam-3504	250	24	u	u	PROPN
ejpam-3504	250	25	∈	∈	PROPN
ejpam-3504	250	26	a	a	DET
ejpam-3504	250	27	◦	◦	NOUN
ejpam-3504	250	28	a	a	X
ejpam-3504	250	29	,	,	PUNCT
ejpam-3504	250	30	we	we	PRON
ejpam-3504	250	31	have	have	VERB
ejpam-3504	250	32	f(u	f(u	PROPN
ejpam-3504	250	33	)	)	PUNCT
ejpam-3504	250	34	≥	≥	NOUN
ejpam-3504	250	35	f(a	f(a	NOUN
ejpam-3504	250	36	)	)	PUNCT
ejpam-3504	250	37	.	.	PUNCT
ejpam-3504	251	1	thus	thus	ADV
ejpam-3504	251	2	we	we	PRON
ejpam-3504	251	3	have	have	VERB
ejpam-3504	251	4	f(a	f(a	NOUN
ejpam-3504	251	5	)	)	PUNCT
ejpam-3504	251	6	=	=	SYM
ejpam-3504	251	7	f(u	f(u	PROPN
ejpam-3504	251	8	)	)	PUNCT
ejpam-3504	251	9	.	.	PUNCT
ejpam-3504	252	1	for	for	ADP
ejpam-3504	252	2	the	the	DET
ejpam-3504	252	3	“	"	PUNCT
ejpam-3504	252	4	converse	converse	NOUN
ejpam-3504	252	5	”	"	PUNCT
ejpam-3504	252	6	statement	statement	NOUN
ejpam-3504	252	7	,	,	PUNCT
ejpam-3504	252	8	let	let	VERB
ejpam-3504	252	9	a	a	DET
ejpam-3504	252	10	∈	∈	PROPN
ejpam-3504	252	11	s.	s.	PROPN
ejpam-3504	252	12	take	take	VERB
ejpam-3504	252	13	an	an	DET
ejpam-3504	252	14	element	element	NOUN
ejpam-3504	252	15	u	u	NOUN
ejpam-3504	252	16	∈	∈	PROPN
ejpam-3504	252	17	a	a	DET
ejpam-3504	252	18	◦	◦	NOUN
ejpam-3504	252	19	a	a	DET
ejpam-3504	252	20	(	(	PUNCT
ejpam-3504	252	21	a	a	DET
ejpam-3504	252	22	◦	◦	NOUN
ejpam-3504	252	23	a	a	DET
ejpam-3504	252	24	6=	6=	NOUN
ejpam-3504	252	25	∅	∅	NOUN
ejpam-3504	252	26	)	)	PUNCT
ejpam-3504	252	27	.	.	PUNCT
ejpam-3504	253	1	since	since	SCONJ
ejpam-3504	253	2	l(u	l(u	PROPN
ejpam-3504	253	3	)	)	PUNCT
ejpam-3504	253	4	is	be	AUX
ejpam-3504	253	5	a	a	DET
ejpam-3504	253	6	left	left	ADJ
ejpam-3504	253	7	ideal	ideal	NOUN
ejpam-3504	253	8	of	of	ADP
ejpam-3504	253	9	s	s	PROPN
ejpam-3504	253	10	,	,	PUNCT
ejpam-3504	253	11	by	by	ADP
ejpam-3504	253	12	lemma	lemma	PROPN
ejpam-3504	253	13	3.1	3.1	NUM
ejpam-3504	253	14	,	,	PUNCT
ejpam-3504	253	15	fl(u	fl(u	NUM
ejpam-3504	253	16	)	)	PUNCT
ejpam-3504	253	17	is	be	AUX
ejpam-3504	253	18	a	a	DET
ejpam-3504	253	19	fuzzy	fuzzy	ADJ
ejpam-3504	253	20	left	leave	VERB
ejpam-3504	253	21	ideal	ideal	NOUN
ejpam-3504	253	22	of	of	ADP
ejpam-3504	253	23	s.	s.	PROPN
ejpam-3504	253	24	by	by	ADP
ejpam-3504	253	25	hypothesis	hypothesis	NOUN
ejpam-3504	253	26	,	,	PUNCT
ejpam-3504	253	27	we	we	PRON
ejpam-3504	253	28	have	have	VERB
ejpam-3504	253	29	fl(u)(a	fl(u)(a	NUM
ejpam-3504	253	30	)	)	PUNCT
ejpam-3504	254	1	=	=	SYM
ejpam-3504	254	2	fl(u)(a	fl(u)(a	PROPN
ejpam-3504	254	3	◦	◦	NOUN
ejpam-3504	254	4	a	a	X
ejpam-3504	254	5	)	)	PUNCT
ejpam-3504	254	6	and	and	CCONJ
ejpam-3504	254	7	,	,	PUNCT
ejpam-3504	254	8	since	since	SCONJ
ejpam-3504	254	9	u	u	PROPN
ejpam-3504	254	10	∈	∈	PROPN
ejpam-3504	254	11	a	a	DET
ejpam-3504	254	12	◦	◦	NOUN
ejpam-3504	254	13	a	a	X
ejpam-3504	254	14	,	,	PUNCT
ejpam-3504	254	15	we	we	PRON
ejpam-3504	254	16	have	have	VERB
ejpam-3504	254	17	fl(u)(a	fl(u)(a	NUM
ejpam-3504	254	18	)	)	PUNCT
ejpam-3504	255	1	=	=	SYM
ejpam-3504	255	2	fl(u)(u	fl(u)(u	PROPN
ejpam-3504	255	3	)	)	PUNCT
ejpam-3504	255	4	.	.	PUNCT
ejpam-3504	256	1	since	since	SCONJ
ejpam-3504	256	2	u	u	PROPN
ejpam-3504	256	3	∈	∈	PROPN
ejpam-3504	256	4	l(u	l(u	PROPN
ejpam-3504	256	5	)	)	PUNCT
ejpam-3504	256	6	,	,	PUNCT
ejpam-3504	256	7	we	we	PRON
ejpam-3504	256	8	have	have	VERB
ejpam-3504	256	9	fl(u)(u	fl(u)(u	NOUN
ejpam-3504	256	10	)	)	PUNCT
ejpam-3504	257	1	=	=	SYM
ejpam-3504	258	1	1	1	X
ejpam-3504	258	2	.	.	PUNCT
ejpam-3504	258	3	then	then	ADV
ejpam-3504	258	4	fl(u)(a	fl(u)(a	NUM
ejpam-3504	258	5	)	)	PUNCT
ejpam-3504	259	1	=	=	SYM
ejpam-3504	259	2	1	1	NUM
ejpam-3504	259	3	,	,	PUNCT
ejpam-3504	259	4	and	and	CCONJ
ejpam-3504	259	5	a	a	DET
ejpam-3504	259	6	∈	∈	PROPN
ejpam-3504	259	7	l(u	l(u	PROPN
ejpam-3504	259	8	)	)	PUNCT
ejpam-3504	260	1	=	=	SYM
ejpam-3504	260	2	u	u	NOUN
ejpam-3504	260	3	∪	∪	VERB
ejpam-3504	260	4	s	s	PART
ejpam-3504	260	5	∗	∗	NOUN
ejpam-3504	260	6	u.	u.	NOUN
ejpam-3504	261	1	if	if	SCONJ
ejpam-3504	261	2	a	a	DET
ejpam-3504	261	3	=	=	SYM
ejpam-3504	261	4	u	u	NOUN
ejpam-3504	261	5	,	,	PUNCT
ejpam-3504	261	6	then	then	ADV
ejpam-3504	261	7	a	a	DET
ejpam-3504	261	8	∈	∈	NOUN
ejpam-3504	261	9	{	{	PUNCT
ejpam-3504	261	10	a	a	NOUN
ejpam-3504	261	11	}	}	PUNCT
ejpam-3504	261	12	∗	∗	NOUN
ejpam-3504	261	13	{	{	PUNCT
ejpam-3504	261	14	a	a	PRON
ejpam-3504	261	15	}	}	PUNCT
ejpam-3504	261	16	⊆	⊆	NUM
ejpam-3504	261	17	{	{	PUNCT
ejpam-3504	261	18	a	a	PRON
ejpam-3504	261	19	}	}	PUNCT
ejpam-3504	261	20	∗	∗	NOUN
ejpam-3504	261	21	{	{	PUNCT
ejpam-3504	261	22	a	a	DET
ejpam-3504	261	23	}	}	PUNCT
ejpam-3504	261	24	∗	∗	NOUN
ejpam-3504	261	25	{	{	PUNCT
ejpam-3504	261	26	a	a	NOUN
ejpam-3504	261	27	}	}	PUNCT
ejpam-3504	261	28	⊆	⊆	NUM
ejpam-3504	261	29	s	s	NOUN
ejpam-3504	261	30	∗	∗	NOUN
ejpam-3504	261	31	(	(	PUNCT
ejpam-3504	261	32	a	a	DET
ejpam-3504	261	33	◦	◦	NOUN
ejpam-3504	261	34	a	a	X
ejpam-3504	261	35	)	)	PUNCT
ejpam-3504	261	36	,	,	PUNCT
ejpam-3504	261	37	so	so	CCONJ
ejpam-3504	261	38	a	a	DET
ejpam-3504	261	39	∈	∈	PROPN
ejpam-3504	261	40	s	s	PART
ejpam-3504	261	41	∗	∗	NOUN
ejpam-3504	261	42	(	(	PUNCT
ejpam-3504	261	43	a	a	DET
ejpam-3504	261	44	◦	◦	NOUN
ejpam-3504	261	45	a	a	X
ejpam-3504	261	46	)	)	PUNCT
ejpam-3504	261	47	.	.	PUNCT
ejpam-3504	262	1	if	if	SCONJ
ejpam-3504	262	2	a	a	DET
ejpam-3504	262	3	∈	∈	PROPN
ejpam-3504	262	4	s	s	PART
ejpam-3504	262	5	∗	∗	NOUN
ejpam-3504	262	6	u	u	PROPN
ejpam-3504	262	7	,	,	PUNCT
ejpam-3504	262	8	then	then	ADV
ejpam-3504	262	9	again	again	ADV
ejpam-3504	262	10	a	a	DET
ejpam-3504	262	11	∈	∈	NOUN
ejpam-3504	262	12	s	s	PART
ejpam-3504	262	13	∗	∗	NOUN
ejpam-3504	262	14	(	(	PUNCT
ejpam-3504	262	15	a	a	DET
ejpam-3504	262	16	◦	◦	NOUN
ejpam-3504	262	17	a	a	X
ejpam-3504	262	18	)	)	PUNCT
ejpam-3504	262	19	.	.	PUNCT
ejpam-3504	263	1	in	in	ADP
ejpam-3504	263	2	any	any	DET
ejpam-3504	263	3	case	case	NOUN
ejpam-3504	263	4	,	,	PUNCT
ejpam-3504	263	5	a	a	DET
ejpam-3504	263	6	∈	∈	PROPN
ejpam-3504	263	7	s	s	PART
ejpam-3504	263	8	∗	∗	NOUN
ejpam-3504	263	9	(	(	PUNCT
ejpam-3504	263	10	a	a	DET
ejpam-3504	263	11	◦	◦	NOUN
ejpam-3504	263	12	a	a	X
ejpam-3504	263	13	)	)	PUNCT
ejpam-3504	263	14	holds	hold	VERB
ejpam-3504	263	15	and	and	CCONJ
ejpam-3504	263	16	so	so	ADV
ejpam-3504	263	17	s	s	NOUN
ejpam-3504	263	18	is	be	AUX
ejpam-3504	263	19	left	leave	VERB
ejpam-3504	263	20	regular	regular	ADV
ejpam-3504	263	21	.	.	PUNCT
ejpam-3504	264	1	�	�	PROPN
ejpam-3504	264	2	in	in	ADP
ejpam-3504	264	3	a	a	DET
ejpam-3504	264	4	similar	similar	ADJ
ejpam-3504	264	5	way	way	NOUN
ejpam-3504	264	6	we	we	PRON
ejpam-3504	264	7	can	can	AUX
ejpam-3504	264	8	prove	prove	VERB
ejpam-3504	264	9	the	the	DET
ejpam-3504	264	10	following	follow	VERB
ejpam-3504	264	11	theorem	theorem	ADJ
ejpam-3504	264	12	.	.	PUNCT
ejpam-3504	264	13	theorem	theorem	VERB
ejpam-3504	264	14	4.3	4.3	NUM
ejpam-3504	264	15	.	.	PUNCT
ejpam-3504	265	1	let	let	VERB
ejpam-3504	265	2	s	s	PRON
ejpam-3504	265	3	be	be	AUX
ejpam-3504	265	4	an	an	DET
ejpam-3504	265	5	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	265	6	.	.	PUNCT
ejpam-3504	266	1	if	if	SCONJ
ejpam-3504	266	2	s	s	VERB
ejpam-3504	266	3	is	be	AUX
ejpam-3504	266	4	right	right	ADV
ejpam-3504	266	5	regular	regular	ADJ
ejpam-3504	266	6	,	,	PUNCT
ejpam-3504	266	7	then	then	ADV
ejpam-3504	266	8	for	for	ADP
ejpam-3504	266	9	every	every	DET
ejpam-3504	266	10	fuzzy	fuzzy	ADJ
ejpam-3504	266	11	right	right	ADJ
ejpam-3504	266	12	ideal	ideal	NOUN
ejpam-3504	266	13	f	f	PROPN
ejpam-3504	266	14	of	of	ADP
ejpam-3504	266	15	s	s	PRON
ejpam-3504	266	16	and	and	CCONJ
ejpam-3504	266	17	every	every	PRON
ejpam-3504	266	18	a	a	DET
ejpam-3504	266	19	∈	∈	PROPN
ejpam-3504	266	20	s	s	NOUN
ejpam-3504	266	21	,	,	PUNCT
ejpam-3504	266	22	we	we	PRON
ejpam-3504	266	23	have	have	VERB
ejpam-3504	266	24	f(a	f(a	NOUN
ejpam-3504	266	25	)	)	PUNCT
ejpam-3504	267	1	=	=	SYM
ejpam-3504	267	2	f(a	f(a	PROPN
ejpam-3504	267	3	◦	◦	NOUN
ejpam-3504	267	4	a	a	X
ejpam-3504	267	5	)	)	PUNCT
ejpam-3504	267	6	in	in	ADP
ejpam-3504	267	7	the	the	DET
ejpam-3504	267	8	sense	sense	NOUN
ejpam-3504	267	9	that	that	SCONJ
ejpam-3504	267	10	there	there	PRON
ejpam-3504	267	11	exists	exist	VERB
ejpam-3504	267	12	u	u	PROPN
ejpam-3504	267	13	∈	∈	PROPN
ejpam-3504	267	14	a	a	DET
ejpam-3504	267	15	◦	◦	NOUN
ejpam-3504	267	16	a	a	DET
ejpam-3504	267	17	such	such	ADJ
ejpam-3504	267	18	that	that	DET
ejpam-3504	267	19	f(a	f(a	NOUN
ejpam-3504	267	20	)	)	PUNCT
ejpam-3504	267	21	=	=	SYM
ejpam-3504	267	22	f(u	f(u	PROPN
ejpam-3504	267	23	)	)	PUNCT
ejpam-3504	267	24	.	.	PUNCT
ejpam-3504	268	1	“	"	PUNCT
ejpam-3504	268	2	conversely	conversely	ADV
ejpam-3504	268	3	”	"	PUNCT
ejpam-3504	268	4	if	if	SCONJ
ejpam-3504	268	5	,	,	PUNCT
ejpam-3504	268	6	for	for	SCONJ
ejpam-3504	268	7	any	any	DET
ejpam-3504	268	8	fuzzy	fuzzy	ADJ
ejpam-3504	268	9	right	right	ADJ
ejpam-3504	268	10	ideal	ideal	NOUN
ejpam-3504	268	11	f	f	PROPN
ejpam-3504	268	12	of	of	ADP
ejpam-3504	268	13	s	s	PRON
ejpam-3504	268	14	and	and	CCONJ
ejpam-3504	268	15	any	any	DET
ejpam-3504	268	16	a	a	DET
ejpam-3504	268	17	∈	∈	NOUN
ejpam-3504	268	18	s	s	VERB
ejpam-3504	268	19	we	we	PRON
ejpam-3504	268	20	have	have	VERB
ejpam-3504	268	21	f(a	f(a	NOUN
ejpam-3504	268	22	)	)	PUNCT
ejpam-3504	268	23	=	=	SYM
ejpam-3504	268	24	f̂(a	f̂(a	PUNCT
ejpam-3504	268	25	◦	◦	NOUN
ejpam-3504	268	26	a	a	X
ejpam-3504	268	27	)	)	PUNCT
ejpam-3504	268	28	in	in	ADP
ejpam-3504	268	29	the	the	DET
ejpam-3504	268	30	sense	sense	NOUN
ejpam-3504	268	31	that	that	SCONJ
ejpam-3504	268	32	if	if	SCONJ
ejpam-3504	268	33	u	u	PROPN
ejpam-3504	268	34	∈	∈	VERB
ejpam-3504	268	35	a	a	DET
ejpam-3504	268	36	◦	◦	NOUN
ejpam-3504	268	37	a	a	PRON
ejpam-3504	268	38	,	,	PUNCT
ejpam-3504	268	39	then	then	ADV
ejpam-3504	268	40	f(a	f(a	PROPN
ejpam-3504	268	41	)	)	PUNCT
ejpam-3504	268	42	=	=	SYM
ejpam-3504	268	43	f(u	f(u	PROPN
ejpam-3504	268	44	)	)	PUNCT
ejpam-3504	268	45	,	,	PUNCT
ejpam-3504	268	46	then	then	ADV
ejpam-3504	268	47	s	s	VERB
ejpam-3504	268	48	is	be	AUX
ejpam-3504	268	49	right	right	ADV
ejpam-3504	268	50	regular	regular	ADJ
ejpam-3504	268	51	.	.	PUNCT
ejpam-3504	269	1	5	5	X
ejpam-3504	269	2	.	.	X
ejpam-3504	269	3	on	on	ADP
ejpam-3504	269	4	left	left	ADJ
ejpam-3504	269	5	simple	simple	ADJ
ejpam-3504	269	6	hypergroupoids	hypergroupoid	NOUN
ejpam-3504	269	7	an	an	PRON
ejpam-3504	269	8	hypergroupoid	hypergroupoid	NOUN
ejpam-3504	269	9	(	(	PUNCT
ejpam-3504	269	10	s	s	NOUN
ejpam-3504	269	11	,	,	PUNCT
ejpam-3504	269	12	◦	◦	NOUN
ejpam-3504	269	13	)	)	PUNCT
ejpam-3504	269	14	is	be	AUX
ejpam-3504	269	15	called	call	VERB
ejpam-3504	269	16	left	left	ADJ
ejpam-3504	269	17	(	(	PUNCT
ejpam-3504	269	18	resp	resp	NOUN
ejpam-3504	269	19	.	.	PUNCT
ejpam-3504	270	1	right	right	ADJ
ejpam-3504	270	2	)	)	PUNCT
ejpam-3504	270	3	simple	simple	ADJ
ejpam-3504	271	1	if	if	SCONJ
ejpam-3504	271	2	s	s	VERB
ejpam-3504	271	3	is	be	AUX
ejpam-3504	271	4	the	the	DET
ejpam-3504	271	5	only	only	ADJ
ejpam-3504	271	6	left	leave	VERB
ejpam-3504	271	7	(	(	PUNCT
ejpam-3504	271	8	resp	resp	NOUN
ejpam-3504	271	9	.	.	PUNCT
ejpam-3504	272	1	right	right	ADJ
ejpam-3504	272	2	)	)	PUNCT
ejpam-3504	272	3	ideal	ideal	NOUN
ejpam-3504	272	4	of	of	ADP
ejpam-3504	272	5	s	s	PROPN
ejpam-3504	272	6	,	,	PUNCT
ejpam-3504	272	7	that	that	PRON
ejpam-3504	272	8	is	be	AUX
ejpam-3504	272	9	if	if	SCONJ
ejpam-3504	272	10	m	m	NOUN
ejpam-3504	272	11	is	be	AUX
ejpam-3504	272	12	a	a	DET
ejpam-3504	272	13	left	left	ADJ
ejpam-3504	272	14	(	(	PUNCT
ejpam-3504	272	15	resp	resp	NOUN
ejpam-3504	272	16	.	.	PUNCT
ejpam-3504	273	1	right	right	ADJ
ejpam-3504	273	2	)	)	PUNCT
ejpam-3504	273	3	ideal	ideal	NOUN
ejpam-3504	273	4	of	of	ADP
ejpam-3504	273	5	s	s	PROPN
ejpam-3504	273	6	,	,	PUNCT
ejpam-3504	273	7	then	then	ADV
ejpam-3504	273	8	m	m	VERB
ejpam-3504	273	9	=	=	ADJ
ejpam-3504	273	10	s.	s.	PROPN
ejpam-3504	273	11	an	an	DET
ejpam-3504	273	12	hypergroupoid	hypergroupoid	PROPN
ejpam-3504	273	13	is	be	AUX
ejpam-3504	273	14	called	call	VERB
ejpam-3504	273	15	simple	simple	ADJ
ejpam-3504	273	16	if	if	SCONJ
ejpam-3504	273	17	it	it	PRON
ejpam-3504	273	18	is	be	AUX
ejpam-3504	273	19	both	both	PRON
ejpam-3504	273	20	left	left	ADJ
ejpam-3504	273	21	and	and	CCONJ
ejpam-3504	273	22	right	right	ADJ
ejpam-3504	273	23	simple	simple	NOUN
ejpam-3504	273	24	.	.	PUNCT
ejpam-3504	274	1	n.	n.	PROPN
ejpam-3504	274	2	kehayopulu	kehayopulu	PROPN
ejpam-3504	274	3	/	/	SYM
ejpam-3504	274	4	eur	eur	PROPN
ejpam-3504	274	5	.	.	PUNCT
ejpam-3504	275	1	j.	j.	PROPN
ejpam-3504	275	2	pure	pure	PROPN
ejpam-3504	275	3	appl	appl	PROPN
ejpam-3504	275	4	.	.	PROPN
ejpam-3504	275	5	math	math	PROPN
ejpam-3504	275	6	,	,	PUNCT
ejpam-3504	275	7	12	12	NUM
ejpam-3504	275	8	(	(	PUNCT
ejpam-3504	275	9	3	3	NUM
ejpam-3504	275	10	)	)	PUNCT
ejpam-3504	275	11	(	(	PUNCT
ejpam-3504	275	12	2019	2019	NUM
ejpam-3504	275	13	)	)	PUNCT
ejpam-3504	275	14	,	,	PUNCT
ejpam-3504	275	15	709	709	NUM
ejpam-3504	275	16	-	-	SYM
ejpam-3504	275	17	721	721	NUM
ejpam-3504	275	18	717	717	NUM
ejpam-3504	276	1	lemma	lemma	PROPN
ejpam-3504	276	2	5.1	5.1	NUM
ejpam-3504	276	3	.	.	PUNCT
ejpam-3504	277	1	let	let	VERB
ejpam-3504	277	2	s	s	PRON
ejpam-3504	277	3	be	be	AUX
ejpam-3504	277	4	an	an	DET
ejpam-3504	277	5	hypergroupoid	hypergroupoid	NOUN
ejpam-3504	277	6	.	.	PUNCT
ejpam-3504	278	1	if	if	SCONJ
ejpam-3504	278	2	s	s	X
ejpam-3504	278	3	∗	∗	NOUN
ejpam-3504	278	4	a	a	DET
ejpam-3504	278	5	=	=	X
ejpam-3504	278	6	s	s	NOUN
ejpam-3504	278	7	for	for	ADP
ejpam-3504	278	8	every	every	DET
ejpam-3504	278	9	a	a	DET
ejpam-3504	278	10	∈	∈	ADJ
ejpam-3504	278	11	s	s	NOUN
ejpam-3504	278	12	,	,	PUNCT
ejpam-3504	278	13	then	then	ADV
ejpam-3504	278	14	s	s	VERB
ejpam-3504	278	15	is	be	AUX
ejpam-3504	278	16	left	leave	VERB
ejpam-3504	278	17	simple	simple	ADJ
ejpam-3504	278	18	.	.	PUNCT
ejpam-3504	279	1	“	"	PUNCT
ejpam-3504	279	2	conversely	conversely	ADV
ejpam-3504	279	3	”	"	PUNCT
ejpam-3504	279	4	if	if	SCONJ
ejpam-3504	279	5	s	s	PROPN
ejpam-3504	279	6	is	be	AUX
ejpam-3504	279	7	a	a	DET
ejpam-3504	279	8	left	left	ADJ
ejpam-3504	279	9	simple	simple	ADJ
ejpam-3504	279	10	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	279	11	,	,	PUNCT
ejpam-3504	279	12	then	then	ADV
ejpam-3504	279	13	s	s	VERB
ejpam-3504	279	14	∗	∗	NOUN
ejpam-3504	279	15	a	a	DET
ejpam-3504	279	16	=	=	X
ejpam-3504	279	17	s	s	NOUN
ejpam-3504	279	18	for	for	ADP
ejpam-3504	279	19	every	every	DET
ejpam-3504	279	20	a	a	DET
ejpam-3504	279	21	∈	∈	PROPN
ejpam-3504	279	22	s.	s.	PROPN
ejpam-3504	279	23	proof	proof	NOUN
ejpam-3504	279	24	.	.	PUNCT
ejpam-3504	280	1	=	=	NOUN
ejpam-3504	280	2	⇒.	⇒.	NOUN
ejpam-3504	280	3	let	let	VERB
ejpam-3504	280	4	a	a	PRON
ejpam-3504	280	5	be	be	AUX
ejpam-3504	280	6	a	a	DET
ejpam-3504	280	7	left	left	ADJ
ejpam-3504	280	8	ideal	ideal	NOUN
ejpam-3504	280	9	of	of	ADP
ejpam-3504	280	10	s	s	PRON
ejpam-3504	280	11	and	and	CCONJ
ejpam-3504	280	12	a	a	DET
ejpam-3504	280	13	∈	∈	NOUN
ejpam-3504	280	14	s.	s.	PROPN
ejpam-3504	280	15	take	take	VERB
ejpam-3504	280	16	an	an	DET
ejpam-3504	280	17	element	element	NOUN
ejpam-3504	280	18	b	b	PROPN
ejpam-3504	280	19	∈	∈	PROPN
ejpam-3504	280	20	a	a	DET
ejpam-3504	280	21	(	(	PUNCT
ejpam-3504	280	22	a	a	PRON
ejpam-3504	280	23	6=	6=	NUM
ejpam-3504	280	24	∅	∅	NOUN
ejpam-3504	280	25	)	)	PUNCT
ejpam-3504	280	26	.	.	PUNCT
ejpam-3504	281	1	by	by	ADP
ejpam-3504	281	2	hypothesis	hypothesis	NOUN
ejpam-3504	281	3	,	,	PUNCT
ejpam-3504	281	4	we	we	PRON
ejpam-3504	281	5	have	have	VERB
ejpam-3504	281	6	s	s	NOUN
ejpam-3504	281	7	=	=	SYM
ejpam-3504	281	8	s	s	NOUN
ejpam-3504	281	9	∗	∗	NOUN
ejpam-3504	281	10	b	b	NOUN
ejpam-3504	281	11	⊆	⊆	NUM
ejpam-3504	281	12	s	s	NOUN
ejpam-3504	281	13	∗a	∗a	PROPN
ejpam-3504	281	14	⊆	⊆	PROPN
ejpam-3504	281	15	a	a	DET
ejpam-3504	281	16	so	so	ADV
ejpam-3504	281	17	a	a	DET
ejpam-3504	281	18	=	=	SYM
ejpam-3504	281	19	s	s	NOUN
ejpam-3504	281	20	,	,	PUNCT
ejpam-3504	281	21	and	and	CCONJ
ejpam-3504	281	22	s	s	VERB
ejpam-3504	281	23	is	be	AUX
ejpam-3504	281	24	left	leave	VERB
ejpam-3504	281	25	simple	simple	ADJ
ejpam-3504	281	26	.	.	PUNCT
ejpam-3504	282	1	⇐	⇐	PROPN
ejpam-3504	282	2	=	=	PRON
ejpam-3504	282	3	.	.	PUNCT
ejpam-3504	283	1	let	let	VERB
ejpam-3504	283	2	a	a	DET
ejpam-3504	283	3	∈	∈	NOUN
ejpam-3504	283	4	s.	s.	PROPN
ejpam-3504	283	5	the	the	DET
ejpam-3504	283	6	set	set	PROPN
ejpam-3504	283	7	s	s	PROPN
ejpam-3504	283	8	∗	∗	NOUN
ejpam-3504	283	9	a	a	PRON
ejpam-3504	283	10	is	be	AUX
ejpam-3504	283	11	a	a	DET
ejpam-3504	283	12	left	left	ADJ
ejpam-3504	283	13	ideal	ideal	NOUN
ejpam-3504	283	14	of	of	ADP
ejpam-3504	283	15	s	s	PRON
ejpam-3504	283	16	since	since	SCONJ
ejpam-3504	283	17	s	s	PART
ejpam-3504	283	18	∗	∗	NOUN
ejpam-3504	283	19	(	(	PUNCT
ejpam-3504	284	1	s	s	NOUN
ejpam-3504	284	2	∗	∗	NOUN
ejpam-3504	284	3	a	a	NOUN
ejpam-3504	284	4	)	)	PUNCT
ejpam-3504	285	1	=	=	PUNCT
ejpam-3504	285	2	(	(	PUNCT
ejpam-3504	285	3	s	s	NOUN
ejpam-3504	285	4	∗	∗	PRON
ejpam-3504	285	5	s	s	PART
ejpam-3504	285	6	)	)	PUNCT
ejpam-3504	285	7	∗	∗	NOUN
ejpam-3504	285	8	a	a	DET
ejpam-3504	285	9	⊆	⊆	NUM
ejpam-3504	285	10	s	s	NOUN
ejpam-3504	285	11	∗	∗	NOUN
ejpam-3504	285	12	a.	a.	NOUN
ejpam-3504	285	13	since	since	SCONJ
ejpam-3504	285	14	s	s	PROPN
ejpam-3504	285	15	is	be	AUX
ejpam-3504	285	16	left	leave	VERB
ejpam-3504	285	17	simple	simple	ADJ
ejpam-3504	285	18	,	,	PUNCT
ejpam-3504	285	19	we	we	PRON
ejpam-3504	285	20	have	have	VERB
ejpam-3504	285	21	s	s	PROPN
ejpam-3504	285	22	∗	∗	NOUN
ejpam-3504	285	23	a	a	DET
ejpam-3504	285	24	=	=	X
ejpam-3504	285	25	s.	s.	PROPN
ejpam-3504	285	26	�	�	PROPN
ejpam-3504	285	27	similarly	similarly	ADV
ejpam-3504	285	28	the	the	DET
ejpam-3504	285	29	hypergroupoids	hypergroupoid	NOUN
ejpam-3504	285	30	in	in	ADP
ejpam-3504	285	31	which	which	PRON
ejpam-3504	285	32	a	a	DET
ejpam-3504	285	33	∗	∗	NOUN
ejpam-3504	285	34	s	s	PART
ejpam-3504	285	35	=	=	NOUN
ejpam-3504	285	36	s	s	NOUN
ejpam-3504	285	37	for	for	ADP
ejpam-3504	285	38	every	every	DET
ejpam-3504	285	39	a	a	DET
ejpam-3504	285	40	∈	∈	NOUN
ejpam-3504	285	41	s	s	VERB
ejpam-3504	285	42	are	be	AUX
ejpam-3504	285	43	right	right	ADJ
ejpam-3504	285	44	simple	simple	ADJ
ejpam-3504	285	45	and	and	CCONJ
ejpam-3504	285	46	in	in	ADP
ejpam-3504	285	47	right	right	ADJ
ejpam-3504	285	48	simple	simple	ADJ
ejpam-3504	285	49	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	285	50	,	,	PUNCT
ejpam-3504	285	51	for	for	ADP
ejpam-3504	285	52	any	any	DET
ejpam-3504	285	53	a	a	DET
ejpam-3504	285	54	∈	∈	ADJ
ejpam-3504	285	55	s	s	NOUN
ejpam-3504	285	56	,	,	PUNCT
ejpam-3504	285	57	we	we	PRON
ejpam-3504	285	58	have	have	VERB
ejpam-3504	285	59	a	a	DET
ejpam-3504	285	60	∗	∗	NOUN
ejpam-3504	285	61	s	s	PART
ejpam-3504	285	62	=	=	PUNCT
ejpam-3504	285	63	s.	s.	PROPN
ejpam-3504	285	64	condition	condition	NOUN
ejpam-3504	285	65	s	s	PART
ejpam-3504	285	66	∗	∗	NOUN
ejpam-3504	285	67	a	a	DET
ejpam-3504	285	68	=	=	X
ejpam-3504	285	69	s	s	NOUN
ejpam-3504	285	70	for	for	ADP
ejpam-3504	285	71	every	every	DET
ejpam-3504	285	72	a	a	DET
ejpam-3504	285	73	∈	∈	NOUN
ejpam-3504	285	74	s	s	NOUN
ejpam-3504	285	75	is	be	AUX
ejpam-3504	285	76	equivalent	equivalent	ADJ
ejpam-3504	285	77	to	to	ADP
ejpam-3504	285	78	s	s	PROPN
ejpam-3504	285	79	∗	∗	NOUN
ejpam-3504	285	80	a	a	DET
ejpam-3504	285	81	=	=	SYM
ejpam-3504	285	82	s	s	NOUN
ejpam-3504	285	83	for	for	ADP
ejpam-3504	285	84	every	every	DET
ejpam-3504	285	85	nonempty	nonempty	NOUN
ejpam-3504	285	86	subset	subset	VERB
ejpam-3504	285	87	a	a	PRON
ejpam-3504	285	88	of	of	ADP
ejpam-3504	285	89	s.	s.	PROPN
ejpam-3504	285	90	indeed	indeed	ADV
ejpam-3504	285	91	,	,	PUNCT
ejpam-3504	285	92	if	if	SCONJ
ejpam-3504	285	93	s	s	VERB
ejpam-3504	285	94	∗	∗	NOUN
ejpam-3504	285	95	a	a	DET
ejpam-3504	285	96	=	=	X
ejpam-3504	285	97	s	s	NOUN
ejpam-3504	285	98	for	for	ADP
ejpam-3504	285	99	every	every	DET
ejpam-3504	285	100	a	a	DET
ejpam-3504	285	101	∈	∈	PROPN
ejpam-3504	285	102	s	s	NOUN
ejpam-3504	285	103	and	and	CCONJ
ejpam-3504	285	104	a	a	PRON
ejpam-3504	285	105	is	be	AUX
ejpam-3504	285	106	a	a	DET
ejpam-3504	285	107	nonempty	nonempty	ADJ
ejpam-3504	285	108	subset	subset	NOUN
ejpam-3504	285	109	of	of	ADP
ejpam-3504	285	110	s	s	PRON
ejpam-3504	285	111	then	then	ADV
ejpam-3504	285	112	,	,	PUNCT
ejpam-3504	285	113	for	for	ADP
ejpam-3504	285	114	an	an	DET
ejpam-3504	285	115	element	element	NOUN
ejpam-3504	285	116	a	a	DET
ejpam-3504	285	117	∈	∈	PROPN
ejpam-3504	285	118	a	a	PRON
ejpam-3504	285	119	,	,	PUNCT
ejpam-3504	285	120	we	we	PRON
ejpam-3504	285	121	have	have	VERB
ejpam-3504	285	122	s	s	NOUN
ejpam-3504	285	123	∗a	∗a	PROPN
ejpam-3504	285	124	⊇	⊇	NOUN
ejpam-3504	285	125	s	s	PART
ejpam-3504	285	126	∗a	∗a	PROPN
ejpam-3504	285	127	=	=	SYM
ejpam-3504	285	128	s	s	PROPN
ejpam-3504	286	1	and	and	CCONJ
ejpam-3504	286	2	so	so	ADV
ejpam-3504	286	3	s	s	ADJ
ejpam-3504	286	4	∗a	∗a	PROPN
ejpam-3504	286	5	=	=	PUNCT
ejpam-3504	286	6	s.	s.	PROPN
ejpam-3504	286	7	the	the	DET
ejpam-3504	286	8	⇐	⇐	PROPN
ejpam-3504	286	9	-part	-part	PROPN
ejpam-3504	286	10	is	be	AUX
ejpam-3504	286	11	obvious	obvious	ADJ
ejpam-3504	286	12	.	.	PUNCT
ejpam-3504	287	1	similarly	similarly	ADV
ejpam-3504	287	2	,	,	PUNCT
ejpam-3504	287	3	a	a	DET
ejpam-3504	287	4	∗	∗	NOUN
ejpam-3504	287	5	s	s	PART
ejpam-3504	287	6	=	=	SYM
ejpam-3504	287	7	s	s	PART
ejpam-3504	287	8	∀	∀	NOUN
ejpam-3504	287	9	a	a	DET
ejpam-3504	287	10	∈	∈	NOUN
ejpam-3504	287	11	s	s	PART
ejpam-3504	287	12	is	be	AUX
ejpam-3504	287	13	equivalent	equivalent	ADJ
ejpam-3504	287	14	to	to	ADP
ejpam-3504	287	15	a	a	DET
ejpam-3504	287	16	∗	∗	NOUN
ejpam-3504	287	17	s	s	PART
ejpam-3504	287	18	=	=	SYM
ejpam-3504	287	19	s	s	PART
ejpam-3504	287	20	∀	∀	NOUN
ejpam-3504	287	21	∅	∅	NOUN
ejpam-3504	287	22	6=	6=	ADP
ejpam-3504	287	23	a	a	DET
ejpam-3504	287	24	⊆	⊆	NUM
ejpam-3504	287	25	s.	s.	NOUN
ejpam-3504	287	26	as	as	ADP
ejpam-3504	287	27	a	a	DET
ejpam-3504	287	28	consequence	consequence	NOUN
ejpam-3504	287	29	,	,	PUNCT
ejpam-3504	287	30	we	we	PRON
ejpam-3504	287	31	immediately	immediately	ADV
ejpam-3504	287	32	have	have	VERB
ejpam-3504	287	33	the	the	DET
ejpam-3504	287	34	following	follow	VERB
ejpam-3504	287	35	corollary	corollary	NOUN
ejpam-3504	287	36	.	.	PUNCT
ejpam-3504	288	1	corollary	corollary	ADJ
ejpam-3504	288	2	5.2	5.2	NUM
ejpam-3504	288	3	.	.	PUNCT
ejpam-3504	289	1	let	let	VERB
ejpam-3504	289	2	s	s	PRON
ejpam-3504	289	3	be	be	AUX
ejpam-3504	289	4	an	an	DET
ejpam-3504	289	5	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	289	6	.	.	PUNCT
ejpam-3504	290	1	the	the	DET
ejpam-3504	290	2	following	follow	VERB
ejpam-3504	290	3	are	be	AUX
ejpam-3504	290	4	equivalent	equivalent	ADJ
ejpam-3504	290	5	:	:	PUNCT
ejpam-3504	290	6	(	(	PUNCT
ejpam-3504	290	7	1	1	X
ejpam-3504	290	8	)	)	PUNCT
ejpam-3504	290	9	s	s	VERB
ejpam-3504	290	10	is	be	AUX
ejpam-3504	290	11	left	leave	VERB
ejpam-3504	290	12	(	(	PUNCT
ejpam-3504	290	13	resp	resp	NOUN
ejpam-3504	290	14	.	.	PUNCT
ejpam-3504	291	1	right	right	ADJ
ejpam-3504	291	2	)	)	PUNCT
ejpam-3504	291	3	simple	simple	NOUN
ejpam-3504	291	4	.	.	PUNCT
ejpam-3504	292	1	(	(	PUNCT
ejpam-3504	292	2	2	2	X
ejpam-3504	292	3	)	)	PUNCT
ejpam-3504	292	4	s	s	PART
ejpam-3504	292	5	∗	∗	NOUN
ejpam-3504	292	6	a	a	DET
ejpam-3504	292	7	=	=	SYM
ejpam-3504	292	8	s	s	X
ejpam-3504	292	9	(	(	PUNCT
ejpam-3504	292	10	resp	resp	NOUN
ejpam-3504	292	11	.	.	PUNCT
ejpam-3504	293	1	a	a	DET
ejpam-3504	293	2	∗	∗	NOUN
ejpam-3504	293	3	s	s	PART
ejpam-3504	293	4	=	=	SYM
ejpam-3504	293	5	s	s	NOUN
ejpam-3504	293	6	)	)	PUNCT
ejpam-3504	293	7	for	for	ADP
ejpam-3504	293	8	every	every	DET
ejpam-3504	293	9	a	a	DET
ejpam-3504	293	10	∈	∈	PROPN
ejpam-3504	293	11	s.	s.	PROPN
ejpam-3504	293	12	(	(	PUNCT
ejpam-3504	293	13	3	3	NUM
ejpam-3504	293	14	)	)	PUNCT
ejpam-3504	293	15	s	s	NOUN
ejpam-3504	293	16	∗a	∗a	PROPN
ejpam-3504	293	17	=	=	SYM
ejpam-3504	293	18	s	s	X
ejpam-3504	293	19	(	(	PUNCT
ejpam-3504	293	20	resp	resp	NOUN
ejpam-3504	293	21	.	.	PUNCT
ejpam-3504	294	1	a	a	DET
ejpam-3504	294	2	∗	∗	NOUN
ejpam-3504	294	3	s	s	PART
ejpam-3504	294	4	=	=	SYM
ejpam-3504	294	5	s	s	NOUN
ejpam-3504	294	6	)	)	PUNCT
ejpam-3504	294	7	for	for	ADP
ejpam-3504	294	8	every	every	DET
ejpam-3504	294	9	nonempty	nonempty	NOUN
ejpam-3504	294	10	subset	subset	VERB
ejpam-3504	294	11	a	a	PRON
ejpam-3504	294	12	of	of	ADP
ejpam-3504	294	13	s.	s.	PROPN
ejpam-3504	294	14	remark	remark	PROPN
ejpam-3504	294	15	5.3	5.3	NUM
ejpam-3504	294	16	.	.	PUNCT
ejpam-3504	295	1	the	the	DET
ejpam-3504	295	2	left	left	ADJ
ejpam-3504	295	3	(	(	PUNCT
ejpam-3504	295	4	resp	resp	NOUN
ejpam-3504	295	5	.	.	PUNCT
ejpam-3504	296	1	right	right	ADJ
ejpam-3504	296	2	)	)	PUNCT
ejpam-3504	296	3	simple	simple	ADJ
ejpam-3504	296	4	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	296	5	are	be	AUX
ejpam-3504	296	6	left	leave	VERB
ejpam-3504	296	7	(	(	PUNCT
ejpam-3504	296	8	resp	resp	NOUN
ejpam-3504	296	9	.	.	PUNCT
ejpam-3504	297	1	right	right	ADJ
ejpam-3504	297	2	)	)	PUNCT
ejpam-3504	298	1	regular	regular	ADJ
ejpam-3504	298	2	and	and	CCONJ
ejpam-3504	298	3	intra	intra	ADJ
ejpam-3504	298	4	-	-	ADJ
ejpam-3504	298	5	regular	regular	ADJ
ejpam-3504	298	6	.	.	PUNCT
ejpam-3504	299	1	the	the	DET
ejpam-3504	299	2	simple	simple	ADJ
ejpam-3504	299	3	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	299	4	are	be	AUX
ejpam-3504	299	5	regular	regular	ADJ
ejpam-3504	299	6	.	.	PUNCT
ejpam-3504	300	1	proof	proof	NOUN
ejpam-3504	300	2	.	.	PUNCT
ejpam-3504	301	1	let	let	VERB
ejpam-3504	301	2	a	a	DET
ejpam-3504	301	3	be	be	AUX
ejpam-3504	301	4	a	a	DET
ejpam-3504	301	5	nonempty	nonempty	ADJ
ejpam-3504	301	6	subset	subset	NOUN
ejpam-3504	301	7	of	of	ADP
ejpam-3504	301	8	s.	s.	PROPN
ejpam-3504	301	9	if	if	SCONJ
ejpam-3504	301	10	s	s	NOUN
ejpam-3504	301	11	is	be	AUX
ejpam-3504	301	12	left	leave	VERB
ejpam-3504	301	13	simple	simple	ADJ
ejpam-3504	301	14	then	then	ADV
ejpam-3504	301	15	,	,	PUNCT
ejpam-3504	301	16	by	by	ADP
ejpam-3504	301	17	corollary	corollary	ADJ
ejpam-3504	301	18	5.2	5.2	NUM
ejpam-3504	301	19	,	,	PUNCT
ejpam-3504	301	20	we	we	PRON
ejpam-3504	301	21	have	have	VERB
ejpam-3504	301	22	s	s	PROPN
ejpam-3504	301	23	∗	∗	NOUN
ejpam-3504	301	24	a	a	DET
ejpam-3504	301	25	=	=	X
ejpam-3504	301	26	s.	s.	PROPN
ejpam-3504	301	27	then	then	ADV
ejpam-3504	301	28	a	a	DET
ejpam-3504	301	29	⊆	⊆	NUM
ejpam-3504	301	30	s	s	NOUN
ejpam-3504	301	31	=	=	X
ejpam-3504	301	32	s	s	NOUN
ejpam-3504	301	33	∗	∗	NOUN
ejpam-3504	301	34	a	a	DET
ejpam-3504	301	35	=	=	X
ejpam-3504	301	36	(	(	PUNCT
ejpam-3504	301	37	s	s	NOUN
ejpam-3504	301	38	∗	∗	X
ejpam-3504	301	39	a	a	NOUN
ejpam-3504	301	40	)	)	PUNCT
ejpam-3504	301	41	∗	∗	NOUN
ejpam-3504	301	42	a	a	DET
ejpam-3504	301	43	=	=	SYM
ejpam-3504	301	44	s	s	PROPN
ejpam-3504	301	45	∗	∗	NOUN
ejpam-3504	301	46	a	a	DET
ejpam-3504	301	47	∗	∗	NOUN
ejpam-3504	301	48	a	a	PRON
ejpam-3504	302	1	and	and	CCONJ
ejpam-3504	302	2	so	so	ADV
ejpam-3504	302	3	s	s	NOUN
ejpam-3504	302	4	is	be	AUX
ejpam-3504	302	5	left	leave	VERB
ejpam-3504	302	6	regular	regular	ADV
ejpam-3504	302	7	.	.	PUNCT
ejpam-3504	303	1	since	since	SCONJ
ejpam-3504	303	2	s	s	NOUN
ejpam-3504	303	3	is	be	AUX
ejpam-3504	303	4	left	leave	VERB
ejpam-3504	303	5	regular	regular	ADV
ejpam-3504	303	6	,	,	PUNCT
ejpam-3504	303	7	it	it	PRON
ejpam-3504	303	8	is	be	AUX
ejpam-3504	303	9	intra	intra	ADJ
ejpam-3504	303	10	-	-	ADJ
ejpam-3504	303	11	regular	regular	ADJ
ejpam-3504	303	12	as	as	ADV
ejpam-3504	303	13	well	well	ADV
ejpam-3504	303	14	.	.	PUNCT
ejpam-3504	304	1	if	if	SCONJ
ejpam-3504	304	2	s	s	NOUN
ejpam-3504	304	3	is	be	AUX
ejpam-3504	304	4	simple	simple	ADJ
ejpam-3504	304	5	then	then	ADV
ejpam-3504	304	6	,	,	PUNCT
ejpam-3504	304	7	by	by	ADP
ejpam-3504	304	8	corollary	corollary	ADJ
ejpam-3504	304	9	5.2	5.2	NUM
ejpam-3504	304	10	,	,	PUNCT
ejpam-3504	304	11	we	we	PRON
ejpam-3504	304	12	have	have	VERB
ejpam-3504	304	13	s	s	PROPN
ejpam-3504	304	14	∗	∗	NOUN
ejpam-3504	304	15	a	a	DET
ejpam-3504	304	16	=	=	NOUN
ejpam-3504	304	17	a	a	DET
ejpam-3504	304	18	∗	∗	NOUN
ejpam-3504	304	19	s	s	PART
ejpam-3504	304	20	=	=	PUNCT
ejpam-3504	304	21	s.	s.	PROPN
ejpam-3504	304	22	then	then	ADV
ejpam-3504	304	23	we	we	PRON
ejpam-3504	304	24	have	have	VERB
ejpam-3504	304	25	a	a	DET
ejpam-3504	304	26	⊆	⊆	NUM
ejpam-3504	304	27	s	s	NOUN
ejpam-3504	304	28	=	=	X
ejpam-3504	304	29	a	a	DET
ejpam-3504	304	30	∗	∗	NOUN
ejpam-3504	304	31	s	s	PART
ejpam-3504	304	32	=	=	NOUN
ejpam-3504	304	33	a	a	DET
ejpam-3504	304	34	∗	∗	NOUN
ejpam-3504	304	35	(	(	PUNCT
ejpam-3504	304	36	s	s	NOUN
ejpam-3504	304	37	∗	∗	NOUN
ejpam-3504	304	38	a	a	NOUN
ejpam-3504	304	39	)	)	PUNCT
ejpam-3504	304	40	=	=	PUNCT
ejpam-3504	304	41	a	a	DET
ejpam-3504	304	42	∗	∗	NOUN
ejpam-3504	304	43	s	s	PART
ejpam-3504	304	44	∗	∗	NOUN
ejpam-3504	304	45	a	a	PRON
ejpam-3504	305	1	and	and	CCONJ
ejpam-3504	305	2	so	so	ADV
ejpam-3504	305	3	s	s	VERB
ejpam-3504	305	4	is	be	AUX
ejpam-3504	305	5	regular	regular	ADJ
ejpam-3504	305	6	.	.	PUNCT
ejpam-3504	306	1	�	�	PROPN
ejpam-3504	306	2	definition	definition	NOUN
ejpam-3504	306	3	5.4	5.4	NUM
ejpam-3504	306	4	.	.	PUNCT
ejpam-3504	307	1	an	an	PRON
ejpam-3504	307	2	hypergroupoid	hypergroupoid	PROPN
ejpam-3504	307	3	(	(	PUNCT
ejpam-3504	307	4	s	s	NOUN
ejpam-3504	307	5	,	,	PUNCT
ejpam-3504	307	6	◦	◦	NOUN
ejpam-3504	307	7	)	)	PUNCT
ejpam-3504	307	8	is	be	AUX
ejpam-3504	307	9	called	call	VERB
ejpam-3504	307	10	fuzzy	fuzzy	ADJ
ejpam-3504	307	11	left	left	ADJ
ejpam-3504	307	12	(	(	PUNCT
ejpam-3504	307	13	resp	resp	NOUN
ejpam-3504	307	14	.	.	PUNCT
ejpam-3504	308	1	fuzzy	fuzzy	ADJ
ejpam-3504	308	2	right	right	ADJ
ejpam-3504	308	3	)	)	PUNCT
ejpam-3504	309	1	simple	simple	ADJ
ejpam-3504	310	1	if	if	SCONJ
ejpam-3504	310	2	every	every	DET
ejpam-3504	310	3	fuzzy	fuzzy	ADJ
ejpam-3504	310	4	left	left	NOUN
ejpam-3504	310	5	(	(	PUNCT
ejpam-3504	310	6	resp	resp	NOUN
ejpam-3504	310	7	.	.	PUNCT
ejpam-3504	311	1	fuzzy	fuzzy	ADJ
ejpam-3504	311	2	right	right	ADJ
ejpam-3504	311	3	)	)	PUNCT
ejpam-3504	311	4	ideal	ideal	NOUN
ejpam-3504	311	5	of	of	ADP
ejpam-3504	311	6	s	s	PROPN
ejpam-3504	311	7	is	be	AUX
ejpam-3504	311	8	a	a	DET
ejpam-3504	311	9	constant	constant	ADJ
ejpam-3504	311	10	function	function	NOUN
ejpam-3504	311	11	.	.	PUNCT
ejpam-3504	312	1	an	an	DET
ejpam-3504	312	2	hypergroupoid	hypergroupoid	NOUN
ejpam-3504	312	3	that	that	PRON
ejpam-3504	312	4	is	be	AUX
ejpam-3504	312	5	both	both	PRON
ejpam-3504	312	6	left	leave	VERB
ejpam-3504	312	7	simple	simple	ADJ
ejpam-3504	312	8	and	and	CCONJ
ejpam-3504	312	9	right	right	ADJ
ejpam-3504	312	10	simple	simple	ADJ
ejpam-3504	312	11	is	be	AUX
ejpam-3504	312	12	called	call	VERB
ejpam-3504	312	13	simple	simple	ADJ
ejpam-3504	312	14	.	.	PUNCT
ejpam-3504	313	1	theorem	theorem	VERB
ejpam-3504	313	2	5.5	5.5	NUM
ejpam-3504	313	3	.	.	PUNCT
ejpam-3504	314	1	an	an	DET
ejpam-3504	314	2	hypergroupoid	hypergroupoid	PROPN
ejpam-3504	314	3	(	(	PUNCT
ejpam-3504	314	4	s	s	NOUN
ejpam-3504	314	5	,	,	PUNCT
ejpam-3504	314	6	◦	◦	NOUN
ejpam-3504	314	7	)	)	PUNCT
ejpam-3504	314	8	is	be	AUX
ejpam-3504	314	9	left	leave	VERB
ejpam-3504	314	10	simple	simple	ADJ
ejpam-3504	314	11	if	if	SCONJ
ejpam-3504	315	1	and	and	CCONJ
ejpam-3504	315	2	only	only	ADV
ejpam-3504	315	3	if	if	SCONJ
ejpam-3504	315	4	it	it	PRON
ejpam-3504	315	5	is	be	AUX
ejpam-3504	315	6	fuzzy	fuzzy	ADJ
ejpam-3504	315	7	left	leave	VERB
ejpam-3504	315	8	simple	simple	ADJ
ejpam-3504	315	9	.	.	PUNCT
ejpam-3504	316	1	proof	proof	NOUN
ejpam-3504	316	2	.	.	PUNCT
ejpam-3504	317	1	=	=	NOUN
ejpam-3504	317	2	⇒.	⇒.	NOUN
ejpam-3504	317	3	let	let	VERB
ejpam-3504	317	4	f	f	PRON
ejpam-3504	317	5	be	be	AUX
ejpam-3504	317	6	a	a	DET
ejpam-3504	317	7	fuzzy	fuzzy	ADJ
ejpam-3504	317	8	left	leave	VERB
ejpam-3504	317	9	ideal	ideal	NOUN
ejpam-3504	317	10	of	of	ADP
ejpam-3504	317	11	s	s	PRON
ejpam-3504	317	12	and	and	CCONJ
ejpam-3504	317	13	a	a	PRON
ejpam-3504	317	14	,	,	PUNCT
ejpam-3504	317	15	b	b	X
ejpam-3504	317	16	∈	∈	PROPN
ejpam-3504	317	17	s.	s.	PROPN
ejpam-3504	317	18	then	then	ADV
ejpam-3504	317	19	f(a	f(a	PROPN
ejpam-3504	317	20	)	)	PUNCT
ejpam-3504	317	21	=	=	SYM
ejpam-3504	317	22	f(b	f(b	PROPN
ejpam-3504	317	23	)	)	PUNCT
ejpam-3504	317	24	.	.	PUNCT
ejpam-3504	318	1	indeed	indeed	ADV
ejpam-3504	318	2	:	:	PUNCT
ejpam-3504	318	3	since	since	SCONJ
ejpam-3504	318	4	s	s	NOUN
ejpam-3504	318	5	is	be	AUX
ejpam-3504	318	6	left	leave	VERB
ejpam-3504	318	7	simple	simple	ADJ
ejpam-3504	318	8	,	,	PUNCT
ejpam-3504	318	9	we	we	PRON
ejpam-3504	318	10	have	have	VERB
ejpam-3504	318	11	s	s	PROPN
ejpam-3504	318	12	∗	∗	NOUN
ejpam-3504	319	1	a	a	PRON
ejpam-3504	319	2	=	=	SYM
ejpam-3504	319	3	s	s	X
ejpam-3504	319	4	and	and	CCONJ
ejpam-3504	319	5	s	s	NOUN
ejpam-3504	319	6	∗	∗	NOUN
ejpam-3504	319	7	b	b	PROPN
ejpam-3504	319	8	=	=	PUNCT
ejpam-3504	319	9	s.	s.	PROPN
ejpam-3504	319	10	then	then	ADV
ejpam-3504	319	11	we	we	PRON
ejpam-3504	319	12	have	have	VERB
ejpam-3504	319	13	b	b	NUM
ejpam-3504	319	14	∈	∈	PROPN
ejpam-3504	319	15	x	x	PUNCT
ejpam-3504	319	16	◦	◦	NOUN
ejpam-3504	319	17	a	a	PRON
ejpam-3504	319	18	and	and	CCONJ
ejpam-3504	319	19	a	a	DET
ejpam-3504	319	20	∈	∈	PROPN
ejpam-3504	319	21	y	y	PROPN
ejpam-3504	319	22	◦	◦	NOUN
ejpam-3504	319	23	b	b	NOUN
ejpam-3504	319	24	for	for	ADP
ejpam-3504	319	25	some	some	DET
ejpam-3504	319	26	x	x	NOUN
ejpam-3504	319	27	,	,	PUNCT
ejpam-3504	319	28	y	y	PROPN
ejpam-3504	319	29	∈	∈	PROPN
ejpam-3504	319	30	s.	s.	PROPN
ejpam-3504	319	31	since	since	SCONJ
ejpam-3504	319	32	f	f	PROPN
ejpam-3504	319	33	is	be	AUX
ejpam-3504	319	34	a	a	DET
ejpam-3504	319	35	fuzzy	fuzzy	ADJ
ejpam-3504	319	36	left	leave	VERB
ejpam-3504	319	37	ideal	ideal	NOUN
ejpam-3504	319	38	of	of	ADP
ejpam-3504	319	39	s	s	PROPN
ejpam-3504	319	40	,	,	PUNCT
ejpam-3504	319	41	we	we	PRON
ejpam-3504	319	42	have	have	VERB
ejpam-3504	319	43	f(x	f(x	PROPN
ejpam-3504	319	44	◦	◦	VERB
ejpam-3504	319	45	a	a	DET
ejpam-3504	319	46	)	)	PUNCT
ejpam-3504	319	47	≥	≥	NOUN
ejpam-3504	319	48	f(a	f(a	NOUN
ejpam-3504	319	49	)	)	PUNCT
ejpam-3504	319	50	and	and	CCONJ
ejpam-3504	319	51	f(y	f(y	PROPN
ejpam-3504	319	52	◦	◦	PROPN
ejpam-3504	319	53	b	b	NOUN
ejpam-3504	319	54	)	)	PUNCT
ejpam-3504	319	55	≥	≥	NOUN
ejpam-3504	319	56	f(b	f(b	PROPN
ejpam-3504	319	57	)	)	PUNCT
ejpam-3504	319	58	.	.	PUNCT
ejpam-3504	320	1	since	since	SCONJ
ejpam-3504	320	2	b	b	PROPN
ejpam-3504	320	3	∈	∈	PROPN
ejpam-3504	320	4	x	x	PUNCT
ejpam-3504	320	5	◦	◦	NOUN
ejpam-3504	320	6	a	a	PRON
ejpam-3504	320	7	and	and	CCONJ
ejpam-3504	320	8	a	a	DET
ejpam-3504	320	9	∈	∈	PROPN
ejpam-3504	320	10	y	y	PROPN
ejpam-3504	320	11	◦	◦	NOUN
ejpam-3504	320	12	b	b	NUM
ejpam-3504	320	13	,	,	PUNCT
ejpam-3504	320	14	we	we	PRON
ejpam-3504	320	15	have	have	VERB
ejpam-3504	320	16	f(b	f(b	PROPN
ejpam-3504	320	17	)	)	PUNCT
ejpam-3504	320	18	≥	≥	NOUN
ejpam-3504	320	19	f(a	f(a	NOUN
ejpam-3504	320	20	)	)	PUNCT
ejpam-3504	320	21	and	and	CCONJ
ejpam-3504	320	22	f(a	f(a	PROPN
ejpam-3504	320	23	)	)	PUNCT
ejpam-3504	320	24	≥	≥	NOUN
ejpam-3504	320	25	f(b	f(b	PROPN
ejpam-3504	320	26	)	)	PUNCT
ejpam-3504	320	27	and	and	CCONJ
ejpam-3504	320	28	so	so	ADV
ejpam-3504	320	29	f(a	f(a	NOUN
ejpam-3504	320	30	)	)	PUNCT
ejpam-3504	320	31	=	=	SYM
ejpam-3504	320	32	f(b	f(b	PROPN
ejpam-3504	320	33	)	)	PUNCT
ejpam-3504	320	34	.	.	PUNCT
ejpam-3504	321	1	⇐	⇐	PROPN
ejpam-3504	321	2	=	=	PRON
ejpam-3504	321	3	.	.	PUNCT
ejpam-3504	321	4	let	let	VERB
ejpam-3504	321	5	a	a	DET
ejpam-3504	321	6	be	be	AUX
ejpam-3504	321	7	a	a	DET
ejpam-3504	321	8	left	left	ADJ
ejpam-3504	321	9	ideal	ideal	NOUN
ejpam-3504	321	10	of	of	ADP
ejpam-3504	321	11	s	s	PRON
ejpam-3504	321	12	and	and	CCONJ
ejpam-3504	321	13	b	b	PROPN
ejpam-3504	321	14	∈	∈	PROPN
ejpam-3504	321	15	s.	s.	PROPN
ejpam-3504	321	16	then	then	ADV
ejpam-3504	321	17	b	b	PROPN
ejpam-3504	321	18	∈	∈	PROPN
ejpam-3504	321	19	a.	a.	NOUN
ejpam-3504	321	20	indeed	indeed	ADV
ejpam-3504	321	21	:	:	PUNCT
ejpam-3504	321	22	by	by	ADP
ejpam-3504	321	23	lemma	lemma	PROPN
ejpam-3504	321	24	3.1	3.1	NUM
ejpam-3504	321	25	,	,	PUNCT
ejpam-3504	321	26	fa	fa	PROPN
ejpam-3504	321	27	is	be	AUX
ejpam-3504	321	28	a	a	DET
ejpam-3504	321	29	fuzzy	fuzzy	ADJ
ejpam-3504	321	30	left	leave	VERB
ejpam-3504	321	31	ideal	ideal	NOUN
ejpam-3504	321	32	of	of	ADP
ejpam-3504	321	33	s.	s.	PROPN
ejpam-3504	321	34	since	since	SCONJ
ejpam-3504	321	35	s	s	PROPN
ejpam-3504	321	36	is	be	AUX
ejpam-3504	321	37	fuzzy	fuzzy	ADJ
ejpam-3504	321	38	left	leave	VERB
ejpam-3504	321	39	simple	simple	NOUN
ejpam-3504	321	40	,	,	PUNCT
ejpam-3504	321	41	fa	fa	PROPN
ejpam-3504	321	42	is	be	AUX
ejpam-3504	321	43	a	a	DET
ejpam-3504	321	44	constant	constant	ADJ
ejpam-3504	321	45	function	function	NOUN
ejpam-3504	321	46	,	,	PUNCT
ejpam-3504	321	47	that	that	ADV
ejpam-3504	321	48	is	be	AUX
ejpam-3504	321	49	fa(x	fa(x	NOUN
ejpam-3504	321	50	)	)	PUNCT
ejpam-3504	321	51	=	=	SYM
ejpam-3504	321	52	fa(y	fa(y	NOUN
ejpam-3504	321	53	)	)	PUNCT
ejpam-3504	321	54	for	for	ADP
ejpam-3504	321	55	every	every	DET
ejpam-3504	321	56	x	x	PROPN
ejpam-3504	321	57	,	,	PUNCT
ejpam-3504	321	58	y	y	PROPN
ejpam-3504	321	59	∈	∈	PROPN
ejpam-3504	321	60	s.	s.	PROPN
ejpam-3504	321	61	take	take	VERB
ejpam-3504	321	62	an	an	DET
ejpam-3504	321	63	element	element	NOUN
ejpam-3504	321	64	a	a	DET
ejpam-3504	321	65	∈	∈	NOUN
ejpam-3504	321	66	a.	a.	NOUN
ejpam-3504	321	67	we	we	PRON
ejpam-3504	321	68	have	have	VERB
ejpam-3504	321	69	fa(a	fa(a	VERB
ejpam-3504	321	70	)	)	PUNCT
ejpam-3504	321	71	=	=	SYM
ejpam-3504	321	72	fa(b	fa(b	X
ejpam-3504	321	73	)	)	PUNCT
ejpam-3504	321	74	and	and	CCONJ
ejpam-3504	321	75	fa(a	fa(a	NUM
ejpam-3504	321	76	)	)	PUNCT
ejpam-3504	322	1	=	=	SYM
ejpam-3504	322	2	1	1	NUM
ejpam-3504	322	3	,	,	PUNCT
ejpam-3504	322	4	so	so	ADV
ejpam-3504	322	5	fa(b	fa(b	NUM
ejpam-3504	322	6	)	)	PUNCT
ejpam-3504	322	7	=	=	SYM
ejpam-3504	322	8	1	1	NUM
ejpam-3504	322	9	and	and	CCONJ
ejpam-3504	322	10	b	b	PROPN
ejpam-3504	322	11	∈	∈	NOUN
ejpam-3504	322	12	a.	a.	NOUN
ejpam-3504	322	13	thus	thus	ADV
ejpam-3504	322	14	we	we	PRON
ejpam-3504	322	15	have	have	VERB
ejpam-3504	322	16	a	a	DET
ejpam-3504	322	17	=	=	SYM
ejpam-3504	322	18	s	s	NOUN
ejpam-3504	322	19	,	,	PUNCT
ejpam-3504	322	20	and	and	CCONJ
ejpam-3504	322	21	s	s	VERB
ejpam-3504	322	22	is	be	AUX
ejpam-3504	322	23	left	leave	VERB
ejpam-3504	322	24	simple	simple	ADJ
ejpam-3504	322	25	.	.	PUNCT
ejpam-3504	323	1	�	�	PROPN
ejpam-3504	323	2	theorem	theorem	VERB
ejpam-3504	323	3	5.5	5.5	NUM
ejpam-3504	323	4	is	be	AUX
ejpam-3504	323	5	also	also	ADV
ejpam-3504	323	6	true	true	ADJ
ejpam-3504	323	7	if	if	SCONJ
ejpam-3504	323	8	we	we	PRON
ejpam-3504	323	9	replace	replace	VERB
ejpam-3504	323	10	the	the	DET
ejpam-3504	323	11	word	word	NOUN
ejpam-3504	323	12	“	"	PUNCT
ejpam-3504	323	13	left	leave	VERB
ejpam-3504	323	14	”	"	PUNCT
ejpam-3504	323	15	by	by	ADP
ejpam-3504	323	16	“	"	PUNCT
ejpam-3504	323	17	right	right	ADJ
ejpam-3504	323	18	”	"	PUNCT
ejpam-3504	323	19	.	.	PUNCT
ejpam-3504	324	1	as	as	ADP
ejpam-3504	324	2	a	a	DET
ejpam-3504	324	3	consequence	consequence	NOUN
ejpam-3504	324	4	,	,	PUNCT
ejpam-3504	324	5	the	the	DET
ejpam-3504	324	6	following	follow	VERB
ejpam-3504	324	7	theorem	theorem	NOUN
ejpam-3504	324	8	also	also	ADV
ejpam-3504	324	9	holds	hold	VERB
ejpam-3504	324	10	.	.	PUNCT
ejpam-3504	325	1	n.	n.	PROPN
ejpam-3504	325	2	kehayopulu	kehayopulu	PROPN
ejpam-3504	325	3	/	/	SYM
ejpam-3504	325	4	eur	eur	PROPN
ejpam-3504	325	5	.	.	PUNCT
ejpam-3504	326	1	j.	j.	PROPN
ejpam-3504	326	2	pure	pure	PROPN
ejpam-3504	326	3	appl	appl	PROPN
ejpam-3504	326	4	.	.	PROPN
ejpam-3504	326	5	math	math	PROPN
ejpam-3504	326	6	,	,	PUNCT
ejpam-3504	326	7	12	12	NUM
ejpam-3504	326	8	(	(	PUNCT
ejpam-3504	326	9	3	3	NUM
ejpam-3504	326	10	)	)	PUNCT
ejpam-3504	326	11	(	(	PUNCT
ejpam-3504	326	12	2019	2019	NUM
ejpam-3504	326	13	)	)	PUNCT
ejpam-3504	326	14	,	,	PUNCT
ejpam-3504	326	15	709	709	NUM
ejpam-3504	326	16	-	-	SYM
ejpam-3504	326	17	721	721	NUM
ejpam-3504	326	18	718	718	NUM
ejpam-3504	326	19	theorem	theorem	VERB
ejpam-3504	326	20	5.6	5.6	NUM
ejpam-3504	326	21	.	.	PUNCT
ejpam-3504	327	1	an	an	DET
ejpam-3504	327	2	hypergroupoid	hypergroupoid	PROPN
ejpam-3504	327	3	(	(	PUNCT
ejpam-3504	327	4	s	s	NOUN
ejpam-3504	327	5	,	,	PUNCT
ejpam-3504	327	6	◦	◦	NOUN
ejpam-3504	327	7	)	)	PUNCT
ejpam-3504	327	8	is	be	AUX
ejpam-3504	327	9	simple	simple	ADJ
ejpam-3504	327	10	if	if	SCONJ
ejpam-3504	328	1	and	and	CCONJ
ejpam-3504	328	2	only	only	ADV
ejpam-3504	328	3	if	if	SCONJ
ejpam-3504	328	4	it	it	PRON
ejpam-3504	328	5	is	be	AUX
ejpam-3504	328	6	fuzzy	fuzzy	ADJ
ejpam-3504	328	7	simple	simple	ADJ
ejpam-3504	328	8	.	.	PUNCT
ejpam-3504	329	1	proposition	proposition	NOUN
ejpam-3504	329	2	5.7	5.7	NUM
ejpam-3504	329	3	.	.	PUNCT
ejpam-3504	330	1	let	let	VERB
ejpam-3504	330	2	s	s	PRON
ejpam-3504	330	3	be	be	AUX
ejpam-3504	330	4	an	an	DET
ejpam-3504	330	5	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	330	6	.	.	PUNCT
ejpam-3504	331	1	then	then	ADV
ejpam-3504	331	2	every	every	DET
ejpam-3504	331	3	left	leave	VERB
ejpam-3504	331	4	ideal	ideal	NOUN
ejpam-3504	331	5	of	of	ADP
ejpam-3504	331	6	s	s	PRON
ejpam-3504	331	7	and	and	CCONJ
ejpam-3504	331	8	every	every	DET
ejpam-3504	331	9	right	right	ADJ
ejpam-3504	331	10	ideal	ideal	NOUN
ejpam-3504	331	11	of	of	ADP
ejpam-3504	331	12	s	s	PROPN
ejpam-3504	331	13	is	be	AUX
ejpam-3504	331	14	a	a	DET
ejpam-3504	331	15	bi	bi	NOUN
ejpam-3504	331	16	-	-	NOUN
ejpam-3504	331	17	ideal	ideal	NOUN
ejpam-3504	331	18	of	of	ADP
ejpam-3504	331	19	s.	s.	PROPN
ejpam-3504	331	20	in	in	ADP
ejpam-3504	331	21	particular	particular	ADJ
ejpam-3504	331	22	,	,	PUNCT
ejpam-3504	331	23	if	if	SCONJ
ejpam-3504	331	24	s	s	NOUN
ejpam-3504	331	25	is	be	AUX
ejpam-3504	331	26	left	leave	VERB
ejpam-3504	331	27	(	(	PUNCT
ejpam-3504	331	28	resp	resp	NOUN
ejpam-3504	331	29	.	.	PUNCT
ejpam-3504	332	1	right	right	ADJ
ejpam-3504	332	2	)	)	PUNCT
ejpam-3504	332	3	simple	simple	NOUN
ejpam-3504	332	4	,	,	PUNCT
ejpam-3504	332	5	then	then	ADV
ejpam-3504	332	6	every	every	DET
ejpam-3504	332	7	bi	bi	NOUN
ejpam-3504	332	8	-	-	NOUN
ejpam-3504	332	9	ideal	ideal	NOUN
ejpam-3504	332	10	of	of	ADP
ejpam-3504	332	11	s	s	PROPN
ejpam-3504	332	12	is	be	AUX
ejpam-3504	332	13	a	a	DET
ejpam-3504	332	14	right	right	ADJ
ejpam-3504	332	15	(	(	PUNCT
ejpam-3504	332	16	resp	resp	NOUN
ejpam-3504	332	17	.	.	PUNCT
ejpam-3504	333	1	left	left	ADJ
ejpam-3504	333	2	)	)	PUNCT
ejpam-3504	333	3	ideal	ideal	NOUN
ejpam-3504	333	4	of	of	ADP
ejpam-3504	333	5	s.	s.	PROPN
ejpam-3504	333	6	proof	proof	PROPN
ejpam-3504	333	7	.	.	PUNCT
ejpam-3504	334	1	let	let	VERB
ejpam-3504	334	2	a	a	DET
ejpam-3504	334	3	be	be	AUX
ejpam-3504	334	4	a	a	DET
ejpam-3504	334	5	left	left	ADJ
ejpam-3504	334	6	ideal	ideal	NOUN
ejpam-3504	334	7	of	of	ADP
ejpam-3504	334	8	s.	s.	PROPN
ejpam-3504	334	9	then	then	ADV
ejpam-3504	334	10	a∗	a∗	PROPN
ejpam-3504	334	11	(	(	PUNCT
ejpam-3504	334	12	s	s	NOUN
ejpam-3504	334	13	∗a	∗a	ADJ
ejpam-3504	334	14	)	)	PUNCT
ejpam-3504	334	15	⊆	⊆	NUM
ejpam-3504	334	16	a∗a	a∗a	ADP
ejpam-3504	334	17	⊆	⊆	NUM
ejpam-3504	334	18	s	s	NOUN
ejpam-3504	334	19	∗a	∗a	PROPN
ejpam-3504	334	20	⊆	⊆	NUM
ejpam-3504	334	21	a	a	PRON
ejpam-3504	334	22	,	,	PUNCT
ejpam-3504	334	23	so	so	SCONJ
ejpam-3504	334	24	a∗s	a∗s	PROPN
ejpam-3504	334	25	∗a	∗a	PROPN
ejpam-3504	334	26	⊆	⊆	NUM
ejpam-3504	334	27	a	a	PRON
ejpam-3504	334	28	and	and	CCONJ
ejpam-3504	334	29	a	a	PRON
ejpam-3504	334	30	is	be	AUX
ejpam-3504	334	31	a	a	DET
ejpam-3504	334	32	bi	bi	NOUN
ejpam-3504	334	33	-	-	NOUN
ejpam-3504	334	34	ideal	ideal	NOUN
ejpam-3504	334	35	of	of	ADP
ejpam-3504	334	36	s.	s.	PROPN
ejpam-3504	334	37	if	if	SCONJ
ejpam-3504	334	38	a	a	PRON
ejpam-3504	334	39	is	be	AUX
ejpam-3504	334	40	a	a	DET
ejpam-3504	334	41	right	right	ADJ
ejpam-3504	334	42	ideal	ideal	NOUN
ejpam-3504	334	43	of	of	ADP
ejpam-3504	334	44	s	s	PROPN
ejpam-3504	334	45	,	,	PUNCT
ejpam-3504	334	46	then	then	ADV
ejpam-3504	334	47	(	(	PUNCT
ejpam-3504	334	48	a	a	DET
ejpam-3504	334	49	∗	∗	NOUN
ejpam-3504	334	50	s	s	NOUN
ejpam-3504	334	51	)	)	PUNCT
ejpam-3504	334	52	∗	∗	NOUN
ejpam-3504	334	53	a	a	DET
ejpam-3504	334	54	⊆	⊆	NUM
ejpam-3504	334	55	a	a	DET
ejpam-3504	334	56	∗	∗	NOUN
ejpam-3504	334	57	a	a	DET
ejpam-3504	334	58	⊆	⊆	NUM
ejpam-3504	334	59	a	a	DET
ejpam-3504	334	60	∗	∗	NOUN
ejpam-3504	334	61	s	s	NOUN
ejpam-3504	334	62	⊆	⊆	NUM
ejpam-3504	334	63	a	a	PRON
ejpam-3504	334	64	and	and	CCONJ
ejpam-3504	334	65	again	again	ADV
ejpam-3504	334	66	a	a	PRON
ejpam-3504	334	67	is	be	AUX
ejpam-3504	334	68	a	a	DET
ejpam-3504	334	69	bi	bi	NOUN
ejpam-3504	334	70	-	-	NOUN
ejpam-3504	334	71	ideal	ideal	NOUN
ejpam-3504	334	72	of	of	ADP
ejpam-3504	334	73	s.	s.	PROPN
ejpam-3504	334	74	let	let	VERB
ejpam-3504	334	75	now	now	ADV
ejpam-3504	334	76	s	s	AUX
ejpam-3504	334	77	be	be	AUX
ejpam-3504	334	78	left	leave	VERB
ejpam-3504	334	79	simple	simple	ADJ
ejpam-3504	334	80	and	and	CCONJ
ejpam-3504	334	81	a	a	DET
ejpam-3504	334	82	be	be	AUX
ejpam-3504	334	83	a	a	DET
ejpam-3504	334	84	bi	bi	NOUN
ejpam-3504	334	85	-	-	NOUN
ejpam-3504	334	86	ideal	ideal	NOUN
ejpam-3504	334	87	of	of	ADP
ejpam-3504	334	88	s.	s.	PROPN
ejpam-3504	334	89	since	since	SCONJ
ejpam-3504	334	90	a	a	PRON
ejpam-3504	334	91	is	be	AUX
ejpam-3504	334	92	a	a	DET
ejpam-3504	334	93	bi	bi	NOUN
ejpam-3504	334	94	-	-	NOUN
ejpam-3504	334	95	ideal	ideal	NOUN
ejpam-3504	334	96	of	of	ADP
ejpam-3504	334	97	s	s	PROPN
ejpam-3504	334	98	,	,	PUNCT
ejpam-3504	334	99	we	we	PRON
ejpam-3504	334	100	have	have	VERB
ejpam-3504	334	101	a	a	DET
ejpam-3504	334	102	∗	∗	NOUN
ejpam-3504	334	103	(	(	PUNCT
ejpam-3504	334	104	s	s	NOUN
ejpam-3504	334	105	∗	∗	NOUN
ejpam-3504	334	106	a	a	NOUN
ejpam-3504	334	107	)	)	PUNCT
ejpam-3504	334	108	⊆	⊆	NUM
ejpam-3504	334	109	a.	a.	NOUN
ejpam-3504	334	110	since	since	SCONJ
ejpam-3504	334	111	s	s	NOUN
ejpam-3504	334	112	is	be	AUX
ejpam-3504	334	113	left	leave	VERB
ejpam-3504	334	114	simple	simple	ADJ
ejpam-3504	334	115	,	,	PUNCT
ejpam-3504	334	116	we	we	PRON
ejpam-3504	334	117	have	have	VERB
ejpam-3504	334	118	s	s	PROPN
ejpam-3504	334	119	∗	∗	NOUN
ejpam-3504	334	120	a	a	DET
ejpam-3504	334	121	=	=	SYM
ejpam-3504	334	122	s	s	NOUN
ejpam-3504	334	123	,	,	PUNCT
ejpam-3504	334	124	then	then	ADV
ejpam-3504	334	125	a	a	DET
ejpam-3504	334	126	∗	∗	NOUN
ejpam-3504	334	127	s	s	VERB
ejpam-3504	334	128	⊆	⊆	NUM
ejpam-3504	334	129	a	a	PRON
ejpam-3504	334	130	and	and	CCONJ
ejpam-3504	334	131	so	so	ADV
ejpam-3504	335	1	a	a	PRON
ejpam-3504	335	2	is	be	AUX
ejpam-3504	335	3	a	a	DET
ejpam-3504	335	4	right	right	ADJ
ejpam-3504	335	5	ideal	ideal	NOUN
ejpam-3504	335	6	of	of	ADP
ejpam-3504	335	7	s.	s.	PROPN
ejpam-3504	335	8	if	if	SCONJ
ejpam-3504	335	9	s	s	VERB
ejpam-3504	335	10	is	be	AUX
ejpam-3504	335	11	right	right	ADJ
ejpam-3504	335	12	simple	simple	ADJ
ejpam-3504	335	13	and	and	CCONJ
ejpam-3504	335	14	a	a	PRON
ejpam-3504	335	15	is	be	AUX
ejpam-3504	335	16	a	a	DET
ejpam-3504	335	17	bi	bi	NOUN
ejpam-3504	335	18	-	-	NOUN
ejpam-3504	335	19	ideal	ideal	NOUN
ejpam-3504	335	20	of	of	ADP
ejpam-3504	335	21	s	s	PROPN
ejpam-3504	335	22	,	,	PUNCT
ejpam-3504	335	23	then	then	ADV
ejpam-3504	335	24	(	(	PUNCT
ejpam-3504	335	25	a	a	DET
ejpam-3504	335	26	∗	∗	NOUN
ejpam-3504	335	27	s	s	NOUN
ejpam-3504	335	28	)	)	PUNCT
ejpam-3504	335	29	∗	∗	NOUN
ejpam-3504	335	30	a	a	DET
ejpam-3504	335	31	⊆	⊆	NUM
ejpam-3504	335	32	a	a	PRON
ejpam-3504	335	33	and	and	CCONJ
ejpam-3504	335	34	a	a	DET
ejpam-3504	335	35	∗	∗	NOUN
ejpam-3504	335	36	s	s	PART
ejpam-3504	335	37	=	=	SYM
ejpam-3504	335	38	s	s	NOUN
ejpam-3504	335	39	,	,	PUNCT
ejpam-3504	335	40	thus	thus	ADV
ejpam-3504	335	41	s	s	VERB
ejpam-3504	335	42	∗a	∗a	PROPN
ejpam-3504	335	43	⊆	⊆	NUM
ejpam-3504	335	44	a	a	PRON
ejpam-3504	335	45	and	and	CCONJ
ejpam-3504	335	46	so	so	ADV
ejpam-3504	335	47	a	a	PRON
ejpam-3504	335	48	is	be	AUX
ejpam-3504	335	49	a	a	DET
ejpam-3504	335	50	left	left	ADJ
ejpam-3504	335	51	ideal	ideal	NOUN
ejpam-3504	335	52	of	of	ADP
ejpam-3504	335	53	s.	s.	PROPN
ejpam-3504	335	54	�	�	PROPN
ejpam-3504	335	55	as	as	ADP
ejpam-3504	335	56	a	a	DET
ejpam-3504	335	57	consequence	consequence	NOUN
ejpam-3504	335	58	,	,	PUNCT
ejpam-3504	335	59	in	in	ADP
ejpam-3504	335	60	an	an	DET
ejpam-3504	335	61	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	335	62	,	,	PUNCT
ejpam-3504	335	63	every	every	DET
ejpam-3504	335	64	ideal	ideal	NOUN
ejpam-3504	335	65	is	be	AUX
ejpam-3504	335	66	a	a	DET
ejpam-3504	335	67	bi	bi	NOUN
ejpam-3504	335	68	-	-	NOUN
ejpam-3504	335	69	ideal	ideal	ADJ
ejpam-3504	335	70	and	and	CCONJ
ejpam-3504	335	71	in	in	ADP
ejpam-3504	335	72	simple	simple	ADJ
ejpam-3504	335	73	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	335	74	the	the	DET
ejpam-3504	335	75	ideals	ideal	NOUN
ejpam-3504	335	76	and	and	CCONJ
ejpam-3504	335	77	the	the	DET
ejpam-3504	335	78	bi	bi	ADJ
ejpam-3504	335	79	-	-	ADJ
ejpam-3504	335	80	ideals	ideal	NOUN
ejpam-3504	335	81	coincide	coincide	NOUN
ejpam-3504	335	82	.	.	PUNCT
ejpam-3504	336	1	it	it	PRON
ejpam-3504	336	2	is	be	AUX
ejpam-3504	336	3	natural	natural	ADJ
ejpam-3504	336	4	to	to	PART
ejpam-3504	336	5	ask	ask	VERB
ejpam-3504	336	6	if	if	SCONJ
ejpam-3504	336	7	proposition	proposition	NOUN
ejpam-3504	336	8	5.7	5.7	NUM
ejpam-3504	336	9	remains	remain	VERB
ejpam-3504	336	10	true	true	ADJ
ejpam-3504	336	11	if	if	SCONJ
ejpam-3504	336	12	we	we	PRON
ejpam-3504	336	13	replace	replace	VERB
ejpam-3504	336	14	the	the	DET
ejpam-3504	336	15	word	word	NOUN
ejpam-3504	336	16	“	"	PUNCT
ejpam-3504	336	17	left	left	ADJ
ejpam-3504	336	18	(	(	PUNCT
ejpam-3504	336	19	right	right	ADJ
ejpam-3504	336	20	)	)	PUNCT
ejpam-3504	336	21	”	"	PUNCT
ejpam-3504	336	22	by	by	ADP
ejpam-3504	336	23	“	"	PUNCT
ejpam-3504	336	24	fuzzy	fuzzy	ADJ
ejpam-3504	336	25	left	left	ADJ
ejpam-3504	336	26	(	(	PUNCT
ejpam-3504	336	27	fuzzy	fuzzy	ADJ
ejpam-3504	336	28	right	right	NOUN
ejpam-3504	336	29	)	)	PUNCT
ejpam-3504	336	30	”	"	PUNCT
ejpam-3504	336	31	and	and	CCONJ
ejpam-3504	336	32	the	the	DET
ejpam-3504	336	33	word	word	NOUN
ejpam-3504	336	34	“	"	PUNCT
ejpam-3504	336	35	bi	bi	NOUN
ejpam-3504	336	36	-	-	ADJ
ejpam-3504	336	37	ideal	ideal	NOUN
ejpam-3504	336	38	”	"	PUNCT
ejpam-3504	336	39	by	by	ADP
ejpam-3504	336	40	“	"	PUNCT
ejpam-3504	336	41	fuzzy	fuzzy	ADJ
ejpam-3504	336	42	bi	bi	NOUN
ejpam-3504	336	43	-	-	ADJ
ejpam-3504	336	44	ideal	ideal	NOUN
ejpam-3504	336	45	”	"	PUNCT
ejpam-3504	336	46	.	.	PUNCT
ejpam-3504	337	1	the	the	DET
ejpam-3504	337	2	answer	answer	NOUN
ejpam-3504	337	3	is	be	AUX
ejpam-3504	337	4	given	give	VERB
ejpam-3504	337	5	in	in	ADP
ejpam-3504	337	6	proposition	proposition	NOUN
ejpam-3504	337	7	5.10	5.10	NUM
ejpam-3504	337	8	below	below	ADV
ejpam-3504	337	9	and	and	CCONJ
ejpam-3504	337	10	proposition	proposition	NOUN
ejpam-3504	337	11	5.8	5.8	NUM
ejpam-3504	337	12	has	have	AUX
ejpam-3504	337	13	been	be	AUX
ejpam-3504	337	14	used	use	VERB
ejpam-3504	337	15	in	in	ADP
ejpam-3504	337	16	it	it	PRON
ejpam-3504	337	17	.	.	PUNCT
ejpam-3504	338	1	proposition	proposition	NOUN
ejpam-3504	338	2	5.8	5.8	NUM
ejpam-3504	338	3	.	.	PUNCT
ejpam-3504	339	1	let	let	AUX
ejpam-3504	339	2	(	(	PUNCT
ejpam-3504	339	3	s	s	NOUN
ejpam-3504	339	4	,	,	PUNCT
ejpam-3504	339	5	◦	◦	NOUN
ejpam-3504	339	6	)	)	PUNCT
ejpam-3504	339	7	be	be	VERB
ejpam-3504	339	8	an	an	DET
ejpam-3504	339	9	hypergroupoid	hypergroupoid	NOUN
ejpam-3504	339	10	and	and	CCONJ
ejpam-3504	339	11	a	a	DET
ejpam-3504	339	12	a	a	DET
ejpam-3504	339	13	nonempty	nonempty	ADJ
ejpam-3504	339	14	subset	subset	NOUN
ejpam-3504	339	15	of	of	ADP
ejpam-3504	339	16	s.	s.	PROPN
ejpam-3504	339	17	then	then	ADV
ejpam-3504	339	18	we	we	PRON
ejpam-3504	339	19	have	have	VERB
ejpam-3504	339	20	the	the	DET
ejpam-3504	339	21	following	following	NOUN
ejpam-3504	339	22	:	:	PUNCT
ejpam-3504	339	23	(	(	PUNCT
ejpam-3504	339	24	1	1	X
ejpam-3504	339	25	)	)	PUNCT
ejpam-3504	339	26	if	if	SCONJ
ejpam-3504	339	27	f	f	PROPN
ejpam-3504	339	28	is	be	AUX
ejpam-3504	339	29	a	a	DET
ejpam-3504	339	30	fuzzy	fuzzy	ADJ
ejpam-3504	339	31	left	leave	VERB
ejpam-3504	339	32	ideal	ideal	NOUN
ejpam-3504	339	33	of	of	ADP
ejpam-3504	339	34	s	s	PROPN
ejpam-3504	339	35	,	,	PUNCT
ejpam-3504	339	36	then	then	ADV
ejpam-3504	339	37	f(a	f(a	PROPN
ejpam-3504	339	38	∗	∗	PROPN
ejpam-3504	339	39	y	y	PROPN
ejpam-3504	339	40	)	)	PUNCT
ejpam-3504	339	41	≥	≥	NOUN
ejpam-3504	339	42	f(y	f(y	NOUN
ejpam-3504	339	43	)	)	PUNCT
ejpam-3504	339	44	for	for	ADP
ejpam-3504	339	45	any	any	DET
ejpam-3504	339	46	y	y	PROPN
ejpam-3504	339	47	∈	∈	PROPN
ejpam-3504	339	48	s	s	PART
ejpam-3504	339	49	;	;	PUNCT
ejpam-3504	339	50	in	in	ADP
ejpam-3504	339	51	the	the	DET
ejpam-3504	339	52	sense	sense	NOUN
ejpam-3504	339	53	that	that	SCONJ
ejpam-3504	339	54	if	if	SCONJ
ejpam-3504	339	55	y	y	PROPN
ejpam-3504	339	56	∈	∈	PROPN
ejpam-3504	339	57	s	s	PART
ejpam-3504	339	58	and	and	CCONJ
ejpam-3504	339	59	u	u	PROPN
ejpam-3504	339	60	∈	∈	PROPN
ejpam-3504	339	61	a	a	DET
ejpam-3504	339	62	∗	∗	NOUN
ejpam-3504	339	63	y	y	PROPN
ejpam-3504	339	64	,	,	PUNCT
ejpam-3504	339	65	then	then	ADV
ejpam-3504	339	66	f(u	f(u	PROPN
ejpam-3504	339	67	)	)	PUNCT
ejpam-3504	339	68	≥	≥	NOUN
ejpam-3504	339	69	f(y	f(y	NOUN
ejpam-3504	339	70	)	)	PUNCT
ejpam-3504	339	71	.	.	PUNCT
ejpam-3504	340	1	(	(	PUNCT
ejpam-3504	340	2	2	2	X
ejpam-3504	340	3	)	)	PUNCT
ejpam-3504	340	4	if	if	SCONJ
ejpam-3504	340	5	f	f	PROPN
ejpam-3504	340	6	is	be	AUX
ejpam-3504	340	7	a	a	DET
ejpam-3504	340	8	fuzzy	fuzzy	ADJ
ejpam-3504	340	9	right	right	ADJ
ejpam-3504	340	10	ideal	ideal	NOUN
ejpam-3504	340	11	of	of	ADP
ejpam-3504	340	12	s	s	PROPN
ejpam-3504	340	13	,	,	PUNCT
ejpam-3504	340	14	then	then	ADV
ejpam-3504	340	15	f(x	f(x	PROPN
ejpam-3504	340	16	∗a	∗a	PROPN
ejpam-3504	340	17	)	)	PUNCT
ejpam-3504	340	18	≥	≥	NOUN
ejpam-3504	340	19	f(x	f(x	PROPN
ejpam-3504	340	20	)	)	PUNCT
ejpam-3504	340	21	for	for	ADP
ejpam-3504	340	22	any	any	DET
ejpam-3504	340	23	x	x	SYM
ejpam-3504	340	24	∈	∈	PROPN
ejpam-3504	340	25	s	s	NOUN
ejpam-3504	340	26	;	;	PUNCT
ejpam-3504	340	27	in	in	ADP
ejpam-3504	340	28	the	the	DET
ejpam-3504	340	29	sense	sense	NOUN
ejpam-3504	340	30	that	that	SCONJ
ejpam-3504	340	31	if	if	SCONJ
ejpam-3504	340	32	x	x	PUNCT
ejpam-3504	340	33	∈	∈	PROPN
ejpam-3504	340	34	s	s	X
ejpam-3504	340	35	and	and	CCONJ
ejpam-3504	340	36	u	u	NOUN
ejpam-3504	340	37	∈	∈	PROPN
ejpam-3504	340	38	x	x	PUNCT
ejpam-3504	340	39	∗a	∗a	PROPN
ejpam-3504	340	40	,	,	PUNCT
ejpam-3504	340	41	then	then	ADV
ejpam-3504	340	42	f(u	f(u	PROPN
ejpam-3504	340	43	)	)	PUNCT
ejpam-3504	340	44	≥	≥	NOUN
ejpam-3504	340	45	f(x	f(x	PROPN
ejpam-3504	340	46	)	)	PUNCT
ejpam-3504	340	47	.	.	PUNCT
ejpam-3504	341	1	proof	proof	NOUN
ejpam-3504	341	2	.	.	PUNCT
ejpam-3504	342	1	(	(	PUNCT
ejpam-3504	342	2	1	1	X
ejpam-3504	342	3	)	)	PUNCT
ejpam-3504	342	4	let	let	VERB
ejpam-3504	342	5	y	y	PROPN
ejpam-3504	342	6	∈	∈	PROPN
ejpam-3504	342	7	s	s	PART
ejpam-3504	342	8	and	and	CCONJ
ejpam-3504	342	9	u	u	PROPN
ejpam-3504	342	10	∈	∈	PROPN
ejpam-3504	342	11	a	a	DET
ejpam-3504	342	12	∗	∗	X
ejpam-3504	342	13	y.	y.	NOUN
ejpam-3504	342	14	then	then	ADV
ejpam-3504	342	15	u	u	PROPN
ejpam-3504	342	16	∈	∈	PROPN
ejpam-3504	342	17	a	a	DET
ejpam-3504	342	18	◦	◦	NOUN
ejpam-3504	342	19	y	y	NOUN
ejpam-3504	342	20	for	for	ADP
ejpam-3504	342	21	some	some	PRON
ejpam-3504	342	22	a	a	DET
ejpam-3504	342	23	∈	∈	PROPN
ejpam-3504	342	24	a.	a.	NOUN
ejpam-3504	342	25	since	since	SCONJ
ejpam-3504	342	26	f	f	PROPN
ejpam-3504	342	27	is	be	AUX
ejpam-3504	342	28	a	a	DET
ejpam-3504	342	29	fuzzy	fuzzy	ADJ
ejpam-3504	342	30	left	leave	VERB
ejpam-3504	342	31	ideal	ideal	NOUN
ejpam-3504	342	32	of	of	ADP
ejpam-3504	342	33	s	s	PROPN
ejpam-3504	342	34	,	,	PUNCT
ejpam-3504	342	35	we	we	PRON
ejpam-3504	342	36	have	have	VERB
ejpam-3504	342	37	f(a	f(a	NOUN
ejpam-3504	342	38	◦	◦	NOUN
ejpam-3504	342	39	y	y	PROPN
ejpam-3504	342	40	)	)	PUNCT
ejpam-3504	342	41	≥	≥	NOUN
ejpam-3504	342	42	f(y	f(y	NOUN
ejpam-3504	342	43	)	)	PUNCT
ejpam-3504	342	44	and	and	CCONJ
ejpam-3504	342	45	,	,	PUNCT
ejpam-3504	342	46	since	since	SCONJ
ejpam-3504	342	47	u	u	PROPN
ejpam-3504	342	48	∈	∈	PROPN
ejpam-3504	342	49	a	a	DET
ejpam-3504	342	50	◦	◦	NOUN
ejpam-3504	342	51	y	y	PROPN
ejpam-3504	342	52	,	,	PUNCT
ejpam-3504	342	53	we	we	PRON
ejpam-3504	342	54	have	have	VERB
ejpam-3504	342	55	f(u	f(u	PROPN
ejpam-3504	342	56	)	)	PUNCT
ejpam-3504	342	57	≥	≥	NOUN
ejpam-3504	342	58	f(y	f(y	NOUN
ejpam-3504	342	59	)	)	PUNCT
ejpam-3504	342	60	.	.	PUNCT
ejpam-3504	343	1	(	(	PUNCT
ejpam-3504	343	2	2	2	X
ejpam-3504	343	3	)	)	PUNCT
ejpam-3504	343	4	let	let	VERB
ejpam-3504	343	5	x	x	PUNCT
ejpam-3504	343	6	∈	∈	PROPN
ejpam-3504	343	7	s	s	PART
ejpam-3504	343	8	and	and	CCONJ
ejpam-3504	343	9	u	u	NOUN
ejpam-3504	343	10	∈	∈	PROPN
ejpam-3504	343	11	x	x	PUNCT
ejpam-3504	343	12	∗a	∗a	PROPN
ejpam-3504	343	13	.	.	PUNCT
ejpam-3504	344	1	then	then	ADV
ejpam-3504	344	2	u	u	PROPN
ejpam-3504	344	3	∈	∈	PROPN
ejpam-3504	344	4	x	x	PUNCT
ejpam-3504	344	5	◦	◦	VERB
ejpam-3504	344	6	a	a	PRON
ejpam-3504	344	7	for	for	ADP
ejpam-3504	344	8	some	some	PRON
ejpam-3504	344	9	a	a	DET
ejpam-3504	344	10	∈	∈	PROPN
ejpam-3504	344	11	a.	a.	NOUN
ejpam-3504	344	12	since	since	SCONJ
ejpam-3504	344	13	f	f	PROPN
ejpam-3504	344	14	is	be	AUX
ejpam-3504	344	15	a	a	DET
ejpam-3504	344	16	fuzzy	fuzzy	ADJ
ejpam-3504	344	17	right	right	ADJ
ejpam-3504	344	18	ideal	ideal	NOUN
ejpam-3504	344	19	of	of	ADP
ejpam-3504	344	20	s	s	PROPN
ejpam-3504	344	21	,	,	PUNCT
ejpam-3504	344	22	we	we	PRON
ejpam-3504	344	23	have	have	VERB
ejpam-3504	344	24	f(x	f(x	PROPN
ejpam-3504	344	25	◦	◦	VERB
ejpam-3504	344	26	a	a	DET
ejpam-3504	344	27	)	)	PUNCT
ejpam-3504	344	28	≥	≥	NOUN
ejpam-3504	344	29	f(x	f(x	PROPN
ejpam-3504	344	30	)	)	PUNCT
ejpam-3504	344	31	and	and	CCONJ
ejpam-3504	344	32	,	,	PUNCT
ejpam-3504	344	33	since	since	SCONJ
ejpam-3504	344	34	u	u	NOUN
ejpam-3504	344	35	∈	∈	PROPN
ejpam-3504	344	36	x	x	PUNCT
ejpam-3504	344	37	◦	◦	NOUN
ejpam-3504	344	38	a	a	X
ejpam-3504	344	39	,	,	PUNCT
ejpam-3504	344	40	we	we	PRON
ejpam-3504	344	41	have	have	VERB
ejpam-3504	344	42	f(u	f(u	PROPN
ejpam-3504	344	43	)	)	PUNCT
ejpam-3504	344	44	≥	≥	NOUN
ejpam-3504	344	45	f(x	f(x	PROPN
ejpam-3504	344	46	)	)	PUNCT
ejpam-3504	344	47	.	.	PUNCT
ejpam-3504	345	1	�	�	PROPN
ejpam-3504	345	2	regarding	regard	VERB
ejpam-3504	345	3	the	the	DET
ejpam-3504	345	4	fuzzy	fuzzy	ADJ
ejpam-3504	345	5	bi	bi	NOUN
ejpam-3504	345	6	-	-	NOUN
ejpam-3504	345	7	ideals	ideal	NOUN
ejpam-3504	345	8	,	,	PUNCT
ejpam-3504	345	9	we	we	PRON
ejpam-3504	345	10	have	have	VERB
ejpam-3504	345	11	the	the	DET
ejpam-3504	345	12	following	follow	VERB
ejpam-3504	345	13	proposition	proposition	NOUN
ejpam-3504	345	14	.	.	PUNCT
ejpam-3504	346	1	proposition	proposition	NOUN
ejpam-3504	346	2	5.9	5.9	NUM
ejpam-3504	346	3	.	.	PUNCT
ejpam-3504	347	1	let	let	VERB
ejpam-3504	347	2	s	s	PRON
ejpam-3504	347	3	be	be	AUX
ejpam-3504	347	4	an	an	DET
ejpam-3504	347	5	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	347	6	.	.	PUNCT
ejpam-3504	348	1	if	if	SCONJ
ejpam-3504	348	2	f	f	PROPN
ejpam-3504	348	3	is	be	AUX
ejpam-3504	348	4	a	a	DET
ejpam-3504	348	5	fuzzy	fuzzy	ADJ
ejpam-3504	348	6	bi	bi	NOUN
ejpam-3504	348	7	-	-	NOUN
ejpam-3504	348	8	ideal	ideal	NOUN
ejpam-3504	348	9	of	of	ADP
ejpam-3504	348	10	s	s	PRON
ejpam-3504	348	11	then	then	ADV
ejpam-3504	348	12	,	,	PUNCT
ejpam-3504	348	13	for	for	ADP
ejpam-3504	348	14	any	any	DET
ejpam-3504	348	15	nonempty	nonempty	NOUN
ejpam-3504	348	16	subset	subset	VERB
ejpam-3504	348	17	b	b	NOUN
ejpam-3504	348	18	of	of	ADP
ejpam-3504	348	19	s	s	PROPN
ejpam-3504	348	20	,	,	PUNCT
ejpam-3504	348	21	we	we	PRON
ejpam-3504	348	22	have	have	VERB
ejpam-3504	348	23	f(a	f(a	PROPN
ejpam-3504	348	24	∗b	∗b	PROPN
ejpam-3504	348	25	∗	∗	NOUN
ejpam-3504	348	26	a	a	PRON
ejpam-3504	348	27	)	)	PUNCT
ejpam-3504	348	28	≥	≥	NOUN
ejpam-3504	348	29	f(a	f(a	NOUN
ejpam-3504	348	30	)	)	PUNCT
ejpam-3504	348	31	in	in	ADP
ejpam-3504	348	32	the	the	DET
ejpam-3504	348	33	sense	sense	NOUN
ejpam-3504	348	34	that	that	SCONJ
ejpam-3504	348	35	if	if	SCONJ
ejpam-3504	348	36	u	u	PROPN
ejpam-3504	348	37	∈	∈	VERB
ejpam-3504	348	38	a	a	DET
ejpam-3504	348	39	∗b	∗b	PROPN
ejpam-3504	348	40	∗	∗	NOUN
ejpam-3504	348	41	a	a	PRON
ejpam-3504	348	42	,	,	PUNCT
ejpam-3504	348	43	then	then	ADV
ejpam-3504	348	44	f(u	f(u	PROPN
ejpam-3504	348	45	)	)	PUNCT
ejpam-3504	348	46	≥	≥	NOUN
ejpam-3504	348	47	f(a	f(a	NOUN
ejpam-3504	348	48	)	)	PUNCT
ejpam-3504	348	49	.	.	PUNCT
ejpam-3504	349	1	proof	proof	NOUN
ejpam-3504	349	2	.	.	PUNCT
ejpam-3504	350	1	let	let	VERB
ejpam-3504	350	2	u	u	PRON
ejpam-3504	350	3	∈	∈	PROPN
ejpam-3504	350	4	a	a	DET
ejpam-3504	350	5	∗	∗	NOUN
ejpam-3504	350	6	b	b	NOUN
ejpam-3504	350	7	∗	∗	NOUN
ejpam-3504	350	8	a.	a.	NOUN
ejpam-3504	351	1	then	then	ADV
ejpam-3504	351	2	u	u	PROPN
ejpam-3504	351	3	∈	∈	PROPN
ejpam-3504	351	4	v	v	ADP
ejpam-3504	351	5	◦	◦	NOUN
ejpam-3504	351	6	a	a	PRON
ejpam-3504	351	7	for	for	ADP
ejpam-3504	351	8	some	some	PRON
ejpam-3504	351	9	v	v	ADP
ejpam-3504	351	10	∈	∈	PROPN
ejpam-3504	351	11	a	a	DET
ejpam-3504	351	12	∗	∗	NOUN
ejpam-3504	351	13	b	b	NOUN
ejpam-3504	351	14	and	and	CCONJ
ejpam-3504	351	15	v	v	ADP
ejpam-3504	351	16	∈	∈	PROPN
ejpam-3504	351	17	a	a	DET
ejpam-3504	351	18	◦	◦	NOUN
ejpam-3504	351	19	b	b	NOUN
ejpam-3504	351	20	for	for	ADP
ejpam-3504	351	21	some	some	DET
ejpam-3504	351	22	b	b	PROPN
ejpam-3504	351	23	∈	∈	PROPN
ejpam-3504	351	24	b.	b.	PROPN
ejpam-3504	352	1	then	then	ADV
ejpam-3504	352	2	we	we	PRON
ejpam-3504	352	3	have	have	VERB
ejpam-3504	352	4	u	u	NOUN
ejpam-3504	352	5	∈	∈	PROPN
ejpam-3504	352	6	v	v	ADP
ejpam-3504	352	7	◦	◦	NOUN
ejpam-3504	352	8	a	a	DET
ejpam-3504	352	9	⊆	⊆	NUM
ejpam-3504	352	10	(	(	PUNCT
ejpam-3504	352	11	a	a	DET
ejpam-3504	352	12	◦	◦	NOUN
ejpam-3504	352	13	b	b	NOUN
ejpam-3504	352	14	)	)	PUNCT
ejpam-3504	352	15	∗	∗	NOUN
ejpam-3504	352	16	a.	a.	NOUN
ejpam-3504	352	17	since	since	SCONJ
ejpam-3504	352	18	f	f	PROPN
ejpam-3504	352	19	is	be	AUX
ejpam-3504	352	20	a	a	DET
ejpam-3504	352	21	fuzzy	fuzzy	ADJ
ejpam-3504	352	22	bi	bi	NOUN
ejpam-3504	352	23	-	-	NOUN
ejpam-3504	352	24	ideal	ideal	NOUN
ejpam-3504	352	25	of	of	ADP
ejpam-3504	352	26	s	s	PROPN
ejpam-3504	352	27	,	,	PUNCT
ejpam-3504	352	28	we	we	PRON
ejpam-3504	352	29	have	have	VERB
ejpam-3504	352	30	f	f	X
ejpam-3504	352	31	(	(	PUNCT
ejpam-3504	352	32	(	(	PUNCT
ejpam-3504	352	33	a	a	DET
ejpam-3504	352	34	◦	◦	NOUN
ejpam-3504	352	35	b	b	NOUN
ejpam-3504	352	36	)	)	PUNCT
ejpam-3504	352	37	∗	∗	NOUN
ejpam-3504	352	38	a	a	PRON
ejpam-3504	352	39	)	)	PUNCT
ejpam-3504	352	40	≥	≥	NOUN
ejpam-3504	352	41	min{f(a	min{f(a	PROPN
ejpam-3504	352	42	)	)	PUNCT
ejpam-3504	352	43	,	,	PUNCT
ejpam-3504	352	44	f(a	f(a	NOUN
ejpam-3504	352	45	)	)	PUNCT
ejpam-3504	352	46	}	}	PUNCT
ejpam-3504	352	47	=	=	SYM
ejpam-3504	352	48	f(a	f(a	PROPN
ejpam-3504	352	49	)	)	PUNCT
ejpam-3504	352	50	;	;	PUNCT
ejpam-3504	352	51	and	and	CCONJ
ejpam-3504	352	52	since	since	SCONJ
ejpam-3504	352	53	u	u	PROPN
ejpam-3504	352	54	∈	∈	PROPN
ejpam-3504	352	55	(	(	PUNCT
ejpam-3504	352	56	a	a	DET
ejpam-3504	352	57	◦	◦	NOUN
ejpam-3504	352	58	b	b	NOUN
ejpam-3504	352	59	)	)	PUNCT
ejpam-3504	352	60	∗	∗	NOUN
ejpam-3504	352	61	a	a	X
ejpam-3504	352	62	,	,	PUNCT
ejpam-3504	352	63	we	we	PRON
ejpam-3504	352	64	have	have	VERB
ejpam-3504	352	65	f(u	f(u	PROPN
ejpam-3504	352	66	)	)	PUNCT
ejpam-3504	352	67	≥	≥	NOUN
ejpam-3504	352	68	f(a	f(a	NOUN
ejpam-3504	352	69	)	)	PUNCT
ejpam-3504	352	70	.	.	PUNCT
ejpam-3504	353	1	�	�	PROPN
ejpam-3504	353	2	proposition	proposition	PROPN
ejpam-3504	353	3	5.10	5.10	NUM
ejpam-3504	353	4	.	.	PUNCT
ejpam-3504	354	1	let	let	AUX
ejpam-3504	354	2	(	(	PUNCT
ejpam-3504	354	3	s	s	NOUN
ejpam-3504	354	4	,	,	PUNCT
ejpam-3504	354	5	◦	◦	NOUN
ejpam-3504	354	6	)	)	PUNCT
ejpam-3504	354	7	be	be	VERB
ejpam-3504	354	8	an	an	DET
ejpam-3504	354	9	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	354	10	.	.	PUNCT
ejpam-3504	355	1	then	then	ADV
ejpam-3504	355	2	every	every	DET
ejpam-3504	355	3	fuzzy	fuzzy	ADJ
ejpam-3504	355	4	left	leave	VERB
ejpam-3504	355	5	ideal	ideal	NOUN
ejpam-3504	355	6	and	and	CCONJ
ejpam-3504	355	7	every	every	DET
ejpam-3504	355	8	fuzzy	fuzzy	ADJ
ejpam-3504	355	9	right	right	ADJ
ejpam-3504	355	10	ideal	ideal	NOUN
ejpam-3504	355	11	of	of	ADP
ejpam-3504	355	12	s	s	PROPN
ejpam-3504	355	13	is	be	AUX
ejpam-3504	355	14	a	a	DET
ejpam-3504	355	15	fuzzy	fuzzy	ADJ
ejpam-3504	355	16	bi	bi	NOUN
ejpam-3504	355	17	-	-	NOUN
ejpam-3504	355	18	ideal	ideal	NOUN
ejpam-3504	355	19	of	of	ADP
ejpam-3504	355	20	s.	s.	PROPN
ejpam-3504	355	21	in	in	ADP
ejpam-3504	355	22	particular	particular	ADJ
ejpam-3504	355	23	,	,	PUNCT
ejpam-3504	355	24	if	if	SCONJ
ejpam-3504	355	25	s	s	NOUN
ejpam-3504	355	26	is	be	AUX
ejpam-3504	355	27	left	leave	VERB
ejpam-3504	355	28	(	(	PUNCT
ejpam-3504	355	29	resp	resp	NOUN
ejpam-3504	355	30	.	.	PUNCT
ejpam-3504	356	1	right	right	ADJ
ejpam-3504	356	2	)	)	PUNCT
ejpam-3504	356	3	simple	simple	NOUN
ejpam-3504	356	4	,	,	PUNCT
ejpam-3504	356	5	then	then	ADV
ejpam-3504	356	6	every	every	DET
ejpam-3504	356	7	fuzzy	fuzzy	ADJ
ejpam-3504	356	8	bi	bi	NOUN
ejpam-3504	356	9	-	-	NOUN
ejpam-3504	356	10	ideal	ideal	NOUN
ejpam-3504	356	11	of	of	ADP
ejpam-3504	356	12	s	s	PROPN
ejpam-3504	356	13	is	be	AUX
ejpam-3504	356	14	a	a	DET
ejpam-3504	356	15	fuzzy	fuzzy	ADJ
ejpam-3504	356	16	right	right	NOUN
ejpam-3504	356	17	(	(	PUNCT
ejpam-3504	356	18	resp	resp	NOUN
ejpam-3504	356	19	.	.	PUNCT
ejpam-3504	357	1	fuzzy	fuzzy	ADJ
ejpam-3504	357	2	left	left	ADJ
ejpam-3504	357	3	)	)	PUNCT
ejpam-3504	357	4	ideal	ideal	NOUN
ejpam-3504	357	5	of	of	ADP
ejpam-3504	357	6	s.	s.	PROPN
ejpam-3504	357	7	n.	n.	PROPN
ejpam-3504	357	8	kehayopulu	kehayopulu	PROPN
ejpam-3504	357	9	/	/	SYM
ejpam-3504	357	10	eur	eur	PROPN
ejpam-3504	357	11	.	.	PUNCT
ejpam-3504	358	1	j.	j.	PROPN
ejpam-3504	358	2	pure	pure	PROPN
ejpam-3504	358	3	appl	appl	PROPN
ejpam-3504	358	4	.	.	PROPN
ejpam-3504	358	5	math	math	PROPN
ejpam-3504	358	6	,	,	PUNCT
ejpam-3504	358	7	12	12	NUM
ejpam-3504	358	8	(	(	PUNCT
ejpam-3504	358	9	3	3	NUM
ejpam-3504	358	10	)	)	PUNCT
ejpam-3504	358	11	(	(	PUNCT
ejpam-3504	358	12	2019	2019	NUM
ejpam-3504	358	13	)	)	PUNCT
ejpam-3504	358	14	,	,	PUNCT
ejpam-3504	358	15	709	709	NUM
ejpam-3504	358	16	-	-	SYM
ejpam-3504	358	17	721	721	NUM
ejpam-3504	358	18	719	719	NUM
ejpam-3504	358	19	proof	proof	NOUN
ejpam-3504	358	20	.	.	PUNCT
ejpam-3504	359	1	let	let	VERB
ejpam-3504	359	2	f	f	PRON
ejpam-3504	359	3	be	be	AUX
ejpam-3504	359	4	a	a	DET
ejpam-3504	359	5	fuzzy	fuzzy	ADJ
ejpam-3504	359	6	left	leave	VERB
ejpam-3504	359	7	ideal	ideal	NOUN
ejpam-3504	359	8	of	of	ADP
ejpam-3504	359	9	s	s	PRON
ejpam-3504	359	10	and	and	CCONJ
ejpam-3504	359	11	x	x	PROPN
ejpam-3504	359	12	,	,	PUNCT
ejpam-3504	359	13	y	y	PROPN
ejpam-3504	359	14	,	,	PUNCT
ejpam-3504	359	15	z	z	PROPN
ejpam-3504	359	16	∈	∈	PROPN
ejpam-3504	359	17	s.	s.	PROPN
ejpam-3504	359	18	since	since	SCONJ
ejpam-3504	359	19	x	x	SYM
ejpam-3504	359	20	◦	◦	NOUN
ejpam-3504	359	21	y	y	PROPN
ejpam-3504	359	22	is	be	AUX
ejpam-3504	359	23	a	a	DET
ejpam-3504	359	24	nonempty	nonempty	ADJ
ejpam-3504	359	25	subset	subset	NOUN
ejpam-3504	359	26	of	of	ADP
ejpam-3504	359	27	s	s	PROPN
ejpam-3504	359	28	,	,	PUNCT
ejpam-3504	359	29	by	by	ADP
ejpam-3504	359	30	proposition	proposition	NOUN
ejpam-3504	359	31	5.8(1	5.8(1	NUM
ejpam-3504	359	32	)	)	PUNCT
ejpam-3504	359	33	,	,	PUNCT
ejpam-3504	359	34	we	we	PRON
ejpam-3504	359	35	have	have	VERB
ejpam-3504	359	36	f	f	X
ejpam-3504	359	37	(	(	PUNCT
ejpam-3504	359	38	(	(	PUNCT
ejpam-3504	359	39	x	x	SYM
ejpam-3504	359	40	◦	◦	VERB
ejpam-3504	359	41	y	y	NOUN
ejpam-3504	359	42	)	)	PUNCT
ejpam-3504	359	43	∗	∗	NOUN
ejpam-3504	359	44	z	z	PROPN
ejpam-3504	359	45	)	)	PUNCT
ejpam-3504	359	46	≥	≥	NOUN
ejpam-3504	359	47	f(z	f(z	PROPN
ejpam-3504	359	48	)	)	PUNCT
ejpam-3504	359	49	≥	≥	NOUN
ejpam-3504	359	50	min{f(x	min{f(x	NOUN
ejpam-3504	359	51	)	)	PUNCT
ejpam-3504	359	52	,	,	PUNCT
ejpam-3504	359	53	f(z	f(z	PROPN
ejpam-3504	359	54	)	)	PUNCT
ejpam-3504	359	55	}	}	PUNCT
ejpam-3504	359	56	.	.	PUNCT
ejpam-3504	360	1	thus	thus	ADV
ejpam-3504	360	2	f	f	PROPN
ejpam-3504	360	3	is	be	AUX
ejpam-3504	360	4	a	a	DET
ejpam-3504	360	5	fuzzy	fuzzy	ADJ
ejpam-3504	360	6	bi	bi	NOUN
ejpam-3504	360	7	-	-	NOUN
ejpam-3504	360	8	ideal	ideal	NOUN
ejpam-3504	360	9	of	of	ADP
ejpam-3504	360	10	s.	s.	PROPN
ejpam-3504	360	11	if	if	SCONJ
ejpam-3504	360	12	f	f	PROPN
ejpam-3504	360	13	is	be	AUX
ejpam-3504	360	14	a	a	DET
ejpam-3504	360	15	fuzzy	fuzzy	ADJ
ejpam-3504	360	16	right	right	ADJ
ejpam-3504	360	17	ideal	ideal	NOUN
ejpam-3504	360	18	of	of	ADP
ejpam-3504	360	19	s	s	PRON
ejpam-3504	360	20	and	and	CCONJ
ejpam-3504	360	21	x	x	PROPN
ejpam-3504	360	22	,	,	PUNCT
ejpam-3504	360	23	y	y	PROPN
ejpam-3504	360	24	,	,	PUNCT
ejpam-3504	360	25	z	z	PROPN
ejpam-3504	360	26	∈	∈	PROPN
ejpam-3504	360	27	s	s	VERB
ejpam-3504	360	28	then	then	ADV
ejpam-3504	360	29	,	,	PUNCT
ejpam-3504	360	30	by	by	ADP
ejpam-3504	360	31	proposition	proposition	NOUN
ejpam-3504	360	32	5.8(2	5.8(2	NUM
ejpam-3504	360	33	)	)	PUNCT
ejpam-3504	360	34	,	,	PUNCT
ejpam-3504	360	35	we	we	PRON
ejpam-3504	360	36	have	have	VERB
ejpam-3504	360	37	f	f	X
ejpam-3504	360	38	(	(	PUNCT
ejpam-3504	360	39	(	(	PUNCT
ejpam-3504	360	40	x	x	SYM
ejpam-3504	360	41	◦	◦	VERB
ejpam-3504	360	42	y	y	NOUN
ejpam-3504	360	43	)	)	PUNCT
ejpam-3504	360	44	∗	∗	NOUN
ejpam-3504	360	45	z	z	NOUN
ejpam-3504	360	46	)	)	PUNCT
ejpam-3504	361	1	=	=	PUNCT
ejpam-3504	361	2	f	f	X
ejpam-3504	361	3	(	(	PUNCT
ejpam-3504	361	4	x	x	SYM
ejpam-3504	361	5	∗	∗	NOUN
ejpam-3504	361	6	(	(	PUNCT
ejpam-3504	361	7	y	y	PROPN
ejpam-3504	361	8	◦	◦	PROPN
ejpam-3504	361	9	z	z	PROPN
ejpam-3504	361	10	)	)	PUNCT
ejpam-3504	361	11	)	)	PUNCT
ejpam-3504	361	12	≥	≥	PROPN
ejpam-3504	361	13	f(x	f(x	PROPN
ejpam-3504	361	14	)	)	PUNCT
ejpam-3504	361	15	≥	≥	NOUN
ejpam-3504	361	16	min{f(x	min{f(x	NOUN
ejpam-3504	361	17	)	)	PUNCT
ejpam-3504	361	18	,	,	PUNCT
ejpam-3504	361	19	f(z	f(z	NUM
ejpam-3504	361	20	)	)	PUNCT
ejpam-3504	361	21	}	}	PUNCT
ejpam-3504	361	22	and	and	CCONJ
ejpam-3504	361	23	so	so	ADV
ejpam-3504	361	24	f	f	PROPN
ejpam-3504	361	25	is	be	AUX
ejpam-3504	361	26	a	a	DET
ejpam-3504	361	27	fuzzy	fuzzy	ADJ
ejpam-3504	361	28	bi	bi	NOUN
ejpam-3504	361	29	-	-	NOUN
ejpam-3504	361	30	ideal	ideal	NOUN
ejpam-3504	361	31	of	of	ADP
ejpam-3504	361	32	s.	s.	PROPN
ejpam-3504	361	33	let	let	VERB
ejpam-3504	361	34	now	now	ADV
ejpam-3504	361	35	s	s	AUX
ejpam-3504	361	36	be	be	AUX
ejpam-3504	361	37	left	leave	VERB
ejpam-3504	361	38	simple	simple	ADJ
ejpam-3504	361	39	,	,	PUNCT
ejpam-3504	361	40	f	f	PROPN
ejpam-3504	361	41	be	be	AUX
ejpam-3504	361	42	a	a	DET
ejpam-3504	361	43	fuzzy	fuzzy	ADJ
ejpam-3504	361	44	bi	bi	NOUN
ejpam-3504	361	45	-	-	NOUN
ejpam-3504	361	46	ideal	ideal	NOUN
ejpam-3504	361	47	of	of	ADP
ejpam-3504	361	48	s	s	PRON
ejpam-3504	361	49	and	and	CCONJ
ejpam-3504	361	50	a	a	PRON
ejpam-3504	361	51	,	,	PUNCT
ejpam-3504	361	52	b	b	X
ejpam-3504	361	53	∈	∈	PROPN
ejpam-3504	361	54	s.	s.	PROPN
ejpam-3504	361	55	then	then	ADV
ejpam-3504	361	56	f(a	f(a	PROPN
ejpam-3504	361	57	◦	◦	PROPN
ejpam-3504	361	58	b	b	NOUN
ejpam-3504	361	59	)	)	PUNCT
ejpam-3504	361	60	≥	≥	NOUN
ejpam-3504	361	61	f(a	f(a	NOUN
ejpam-3504	361	62	)	)	PUNCT
ejpam-3504	361	63	.	.	PUNCT
ejpam-3504	362	1	indeed	indeed	ADV
ejpam-3504	362	2	:	:	PUNCT
ejpam-3504	362	3	let	let	VERB
ejpam-3504	362	4	u	u	PRON
ejpam-3504	362	5	∈	∈	PROPN
ejpam-3504	362	6	a	a	DET
ejpam-3504	362	7	◦	◦	NOUN
ejpam-3504	362	8	b.	b.	NOUN
ejpam-3504	362	9	since	since	SCONJ
ejpam-3504	362	10	s	s	PROPN
ejpam-3504	362	11	is	be	AUX
ejpam-3504	362	12	left	leave	VERB
ejpam-3504	362	13	simple	simple	ADJ
ejpam-3504	362	14	,	,	PUNCT
ejpam-3504	362	15	we	we	PRON
ejpam-3504	362	16	have	have	VERB
ejpam-3504	362	17	s	s	PROPN
ejpam-3504	362	18	∗	∗	NOUN
ejpam-3504	362	19	a	a	DET
ejpam-3504	362	20	=	=	SYM
ejpam-3504	362	21	s	s	NOUN
ejpam-3504	362	22	,	,	PUNCT
ejpam-3504	362	23	then	then	ADV
ejpam-3504	362	24	b	b	PROPN
ejpam-3504	362	25	∈	∈	PROPN
ejpam-3504	362	26	x	x	PUNCT
ejpam-3504	362	27	◦	◦	VERB
ejpam-3504	362	28	a	a	PRON
ejpam-3504	362	29	for	for	ADP
ejpam-3504	362	30	some	some	DET
ejpam-3504	362	31	x	x	SYM
ejpam-3504	362	32	∈	∈	PROPN
ejpam-3504	362	33	s	s	PART
ejpam-3504	362	34	and	and	CCONJ
ejpam-3504	362	35	u	u	PROPN
ejpam-3504	362	36	∈	∈	PROPN
ejpam-3504	362	37	a	a	DET
ejpam-3504	362	38	∗	∗	NOUN
ejpam-3504	362	39	(	(	PUNCT
ejpam-3504	362	40	x	x	SYM
ejpam-3504	362	41	◦	◦	VERB
ejpam-3504	362	42	a	a	X
ejpam-3504	362	43	)	)	PUNCT
ejpam-3504	363	1	=	=	SYM
ejpam-3504	363	2	(	(	PUNCT
ejpam-3504	363	3	a	a	DET
ejpam-3504	363	4	◦	◦	NOUN
ejpam-3504	363	5	x	x	SYM
ejpam-3504	363	6	)	)	PUNCT
ejpam-3504	363	7	∗	∗	NOUN
ejpam-3504	363	8	a.	a.	NOUN
ejpam-3504	363	9	since	since	SCONJ
ejpam-3504	363	10	f	f	PROPN
ejpam-3504	363	11	is	be	AUX
ejpam-3504	363	12	a	a	DET
ejpam-3504	363	13	fuzzy	fuzzy	ADJ
ejpam-3504	363	14	bi	bi	NOUN
ejpam-3504	363	15	-	-	NOUN
ejpam-3504	363	16	ideal	ideal	NOUN
ejpam-3504	363	17	of	of	ADP
ejpam-3504	363	18	s	s	PROPN
ejpam-3504	363	19	,	,	PUNCT
ejpam-3504	363	20	we	we	PRON
ejpam-3504	363	21	have	have	VERB
ejpam-3504	363	22	f	f	X
ejpam-3504	363	23	(	(	PUNCT
ejpam-3504	363	24	(	(	PUNCT
ejpam-3504	363	25	a	a	DET
ejpam-3504	363	26	◦	◦	NOUN
ejpam-3504	363	27	x	x	SYM
ejpam-3504	363	28	)	)	PUNCT
ejpam-3504	363	29	∗	∗	NOUN
ejpam-3504	363	30	a	a	PRON
ejpam-3504	363	31	)	)	PUNCT
ejpam-3504	363	32	≥	≥	NOUN
ejpam-3504	363	33	min{f(a	min{f(a	PROPN
ejpam-3504	363	34	)	)	PUNCT
ejpam-3504	363	35	,	,	PUNCT
ejpam-3504	363	36	f(a	f(a	NOUN
ejpam-3504	363	37	)	)	PUNCT
ejpam-3504	363	38	}	}	PUNCT
ejpam-3504	363	39	=	=	SYM
ejpam-3504	363	40	f(a	f(a	NOUN
ejpam-3504	363	41	)	)	PUNCT
ejpam-3504	363	42	and	and	CCONJ
ejpam-3504	363	43	,	,	PUNCT
ejpam-3504	363	44	since	since	SCONJ
ejpam-3504	363	45	u	u	PROPN
ejpam-3504	363	46	∈	∈	PROPN
ejpam-3504	363	47	(	(	PUNCT
ejpam-3504	363	48	a	a	DET
ejpam-3504	363	49	◦	◦	NOUN
ejpam-3504	363	50	x	x	SYM
ejpam-3504	363	51	)	)	PUNCT
ejpam-3504	363	52	∗	∗	NOUN
ejpam-3504	363	53	a	a	X
ejpam-3504	363	54	,	,	PUNCT
ejpam-3504	363	55	we	we	PRON
ejpam-3504	363	56	have	have	VERB
ejpam-3504	363	57	f(u	f(u	PROPN
ejpam-3504	363	58	)	)	PUNCT
ejpam-3504	363	59	≥	≥	NOUN
ejpam-3504	363	60	f(a	f(a	NOUN
ejpam-3504	363	61	)	)	PUNCT
ejpam-3504	363	62	;	;	PUNCT
ejpam-3504	363	63	thus	thus	ADV
ejpam-3504	363	64	f	f	PROPN
ejpam-3504	363	65	is	be	AUX
ejpam-3504	363	66	a	a	DET
ejpam-3504	363	67	fuzzy	fuzzy	ADJ
ejpam-3504	363	68	right	right	ADJ
ejpam-3504	363	69	ideal	ideal	NOUN
ejpam-3504	363	70	of	of	ADP
ejpam-3504	363	71	s.	s.	PROPN
ejpam-3504	363	72	finally	finally	ADV
ejpam-3504	363	73	,	,	PUNCT
ejpam-3504	363	74	let	let	VERB
ejpam-3504	363	75	s	s	PRON
ejpam-3504	363	76	be	be	AUX
ejpam-3504	363	77	right	right	ADJ
ejpam-3504	363	78	simple	simple	ADJ
ejpam-3504	363	79	,	,	PUNCT
ejpam-3504	363	80	f	f	PROPN
ejpam-3504	363	81	be	be	AUX
ejpam-3504	363	82	a	a	DET
ejpam-3504	363	83	fuzzy	fuzzy	ADJ
ejpam-3504	363	84	bi	bi	NOUN
ejpam-3504	363	85	-	-	NOUN
ejpam-3504	363	86	ideal	ideal	NOUN
ejpam-3504	363	87	of	of	ADP
ejpam-3504	363	88	s	s	PRON
ejpam-3504	363	89	and	and	CCONJ
ejpam-3504	363	90	a	a	PRON
ejpam-3504	363	91	,	,	PUNCT
ejpam-3504	363	92	b	b	X
ejpam-3504	363	93	∈	∈	PROPN
ejpam-3504	363	94	s.	s.	PROPN
ejpam-3504	363	95	then	then	ADV
ejpam-3504	363	96	f(a	f(a	PROPN
ejpam-3504	363	97	◦	◦	PROPN
ejpam-3504	363	98	b	b	NUM
ejpam-3504	363	99	)	)	PUNCT
ejpam-3504	363	100	≥	≥	NOUN
ejpam-3504	363	101	f(b	f(b	PROPN
ejpam-3504	363	102	)	)	PUNCT
ejpam-3504	363	103	.	.	PUNCT
ejpam-3504	364	1	indeed	indeed	ADV
ejpam-3504	364	2	:	:	PUNCT
ejpam-3504	364	3	let	let	VERB
ejpam-3504	364	4	u	u	PRON
ejpam-3504	364	5	∈	∈	PROPN
ejpam-3504	364	6	a	a	DET
ejpam-3504	364	7	◦	◦	NOUN
ejpam-3504	364	8	b.	b.	NOUN
ejpam-3504	364	9	since	since	SCONJ
ejpam-3504	364	10	s	s	NOUN
ejpam-3504	364	11	is	be	AUX
ejpam-3504	364	12	right	right	ADJ
ejpam-3504	364	13	simple	simple	ADJ
ejpam-3504	364	14	,	,	PUNCT
ejpam-3504	364	15	we	we	PRON
ejpam-3504	364	16	have	have	VERB
ejpam-3504	364	17	b	b	NUM
ejpam-3504	364	18	∗	∗	NOUN
ejpam-3504	364	19	s	s	PART
ejpam-3504	364	20	=	=	SYM
ejpam-3504	364	21	s	s	PROPN
ejpam-3504	364	22	,	,	PUNCT
ejpam-3504	364	23	then	then	ADV
ejpam-3504	364	24	a	a	DET
ejpam-3504	364	25	∈	∈	PROPN
ejpam-3504	364	26	b	b	NOUN
ejpam-3504	364	27	◦	◦	NOUN
ejpam-3504	364	28	x	x	SYM
ejpam-3504	364	29	for	for	ADP
ejpam-3504	364	30	some	some	DET
ejpam-3504	364	31	x	x	SYM
ejpam-3504	364	32	∈	∈	PROPN
ejpam-3504	364	33	s	s	PART
ejpam-3504	364	34	and	and	CCONJ
ejpam-3504	364	35	u	u	PROPN
ejpam-3504	364	36	∈	∈	PROPN
ejpam-3504	364	37	(	(	PUNCT
ejpam-3504	364	38	b	b	X
ejpam-3504	364	39	◦	◦	NOUN
ejpam-3504	364	40	x	x	NOUN
ejpam-3504	364	41	)	)	PUNCT
ejpam-3504	364	42	∗	∗	PROPN
ejpam-3504	364	43	b.	b.	PROPN
ejpam-3504	364	44	since	since	SCONJ
ejpam-3504	364	45	f	f	PROPN
ejpam-3504	364	46	is	be	AUX
ejpam-3504	364	47	a	a	DET
ejpam-3504	364	48	fuzzy	fuzzy	ADJ
ejpam-3504	364	49	bi	bi	NOUN
ejpam-3504	364	50	-	-	NOUN
ejpam-3504	364	51	ideal	ideal	NOUN
ejpam-3504	364	52	of	of	ADP
ejpam-3504	364	53	s	s	PROPN
ejpam-3504	364	54	,	,	PUNCT
ejpam-3504	364	55	we	we	PRON
ejpam-3504	364	56	have	have	VERB
ejpam-3504	364	57	f	f	X
ejpam-3504	364	58	(	(	PUNCT
ejpam-3504	364	59	(	(	PUNCT
ejpam-3504	364	60	b	b	X
ejpam-3504	364	61	◦	◦	NOUN
ejpam-3504	364	62	x	x	NOUN
ejpam-3504	364	63	)	)	PUNCT
ejpam-3504	364	64	∗	∗	PROPN
ejpam-3504	364	65	b	b	NOUN
ejpam-3504	364	66	)	)	PUNCT
ejpam-3504	364	67	≥	≥	NOUN
ejpam-3504	364	68	min{f(b	min{f(b	NUM
ejpam-3504	364	69	)	)	PUNCT
ejpam-3504	364	70	,	,	PUNCT
ejpam-3504	364	71	f(b	f(b	PROPN
ejpam-3504	364	72	)	)	PUNCT
ejpam-3504	364	73	}	}	PUNCT
ejpam-3504	364	74	=	=	SYM
ejpam-3504	364	75	f(b	f(b	X
ejpam-3504	364	76	)	)	PUNCT
ejpam-3504	364	77	and	and	CCONJ
ejpam-3504	364	78	,	,	PUNCT
ejpam-3504	364	79	since	since	SCONJ
ejpam-3504	364	80	u	u	PROPN
ejpam-3504	364	81	∈	∈	PROPN
ejpam-3504	364	82	(	(	PUNCT
ejpam-3504	364	83	b	b	X
ejpam-3504	364	84	◦	◦	NOUN
ejpam-3504	364	85	x	x	NOUN
ejpam-3504	364	86	)	)	PUNCT
ejpam-3504	364	87	∗	∗	PROPN
ejpam-3504	364	88	b	b	NOUN
ejpam-3504	364	89	,	,	PUNCT
ejpam-3504	364	90	we	we	PRON
ejpam-3504	364	91	have	have	VERB
ejpam-3504	364	92	f(u	f(u	PROPN
ejpam-3504	364	93	)	)	PUNCT
ejpam-3504	364	94	≥	≥	NOUN
ejpam-3504	364	95	f(b	f(b	PROPN
ejpam-3504	364	96	)	)	PUNCT
ejpam-3504	364	97	;	;	PUNCT
ejpam-3504	364	98	thus	thus	ADV
ejpam-3504	364	99	f	f	PROPN
ejpam-3504	364	100	is	be	AUX
ejpam-3504	364	101	a	a	DET
ejpam-3504	364	102	fuzzy	fuzzy	ADJ
ejpam-3504	364	103	left	leave	VERB
ejpam-3504	364	104	ideal	ideal	NOUN
ejpam-3504	364	105	of	of	ADP
ejpam-3504	364	106	s.	s.	PROPN
ejpam-3504	364	107	�	�	PROPN
ejpam-3504	364	108	remark	remark	VERB
ejpam-3504	364	109	5.11	5.11	NUM
ejpam-3504	364	110	.	.	PUNCT
ejpam-3504	365	1	proposition	proposition	NOUN
ejpam-3504	365	2	5.7	5.7	NUM
ejpam-3504	365	3	can	can	AUX
ejpam-3504	365	4	be	be	AUX
ejpam-3504	365	5	also	also	ADV
ejpam-3504	365	6	obtained	obtain	VERB
ejpam-3504	365	7	as	as	ADP
ejpam-3504	365	8	a	a	DET
ejpam-3504	365	9	corollary	corollary	NOUN
ejpam-3504	365	10	to	to	PART
ejpam-3504	365	11	proposition	proposition	NOUN
ejpam-3504	365	12	5.10	5.10	NUM
ejpam-3504	365	13	.	.	PUNCT
ejpam-3504	366	1	in	in	ADP
ejpam-3504	366	2	fact	fact	NOUN
ejpam-3504	366	3	:	:	PUNCT
ejpam-3504	366	4	if	if	SCONJ
ejpam-3504	366	5	a	a	PRON
ejpam-3504	366	6	is	be	AUX
ejpam-3504	366	7	a	a	DET
ejpam-3504	366	8	left	left	ADJ
ejpam-3504	366	9	ideal	ideal	NOUN
ejpam-3504	366	10	of	of	ADP
ejpam-3504	366	11	s	s	PRON
ejpam-3504	366	12	then	then	ADV
ejpam-3504	366	13	,	,	PUNCT
ejpam-3504	366	14	by	by	ADP
ejpam-3504	366	15	lemma	lemma	PROPN
ejpam-3504	366	16	3.1	3.1	NUM
ejpam-3504	366	17	,	,	PUNCT
ejpam-3504	366	18	fa	fa	PROPN
ejpam-3504	366	19	is	be	AUX
ejpam-3504	366	20	a	a	DET
ejpam-3504	366	21	fuzzy	fuzzy	ADJ
ejpam-3504	366	22	left	leave	VERB
ejpam-3504	366	23	ideal	ideal	NOUN
ejpam-3504	366	24	of	of	ADP
ejpam-3504	366	25	s	s	PROPN
ejpam-3504	366	26	,	,	PUNCT
ejpam-3504	366	27	by	by	ADP
ejpam-3504	366	28	proposition	proposition	NOUN
ejpam-3504	366	29	5.10	5.10	NUM
ejpam-3504	366	30	,	,	PUNCT
ejpam-3504	366	31	fa	fa	PROPN
ejpam-3504	366	32	is	be	AUX
ejpam-3504	366	33	a	a	DET
ejpam-3504	366	34	fuzzy	fuzzy	ADJ
ejpam-3504	366	35	bi	bi	NOUN
ejpam-3504	366	36	-	-	NOUN
ejpam-3504	366	37	ideal	ideal	NOUN
ejpam-3504	366	38	of	of	ADP
ejpam-3504	366	39	s	s	NOUN
ejpam-3504	366	40	and	and	CCONJ
ejpam-3504	366	41	,	,	PUNCT
ejpam-3504	366	42	by	by	ADP
ejpam-3504	366	43	lemma	lemma	PROPN
ejpam-3504	366	44	3.11	3.11	NUM
ejpam-3504	366	45	,	,	PUNCT
ejpam-3504	366	46	a	a	PRON
ejpam-3504	366	47	is	be	AUX
ejpam-3504	366	48	a	a	DET
ejpam-3504	366	49	bi	bi	NOUN
ejpam-3504	366	50	-	-	NOUN
ejpam-3504	366	51	ideal	ideal	NOUN
ejpam-3504	366	52	of	of	ADP
ejpam-3504	366	53	s.	s.	PROPN
ejpam-3504	366	54	if	if	SCONJ
ejpam-3504	366	55	s	s	NOUN
ejpam-3504	366	56	is	be	AUX
ejpam-3504	366	57	left	leave	VERB
ejpam-3504	366	58	simple	simple	ADJ
ejpam-3504	366	59	and	and	CCONJ
ejpam-3504	366	60	a	a	DET
ejpam-3504	366	61	a	a	DET
ejpam-3504	366	62	bi	bi	NOUN
ejpam-3504	366	63	-	-	NOUN
ejpam-3504	366	64	ideal	ideal	NOUN
ejpam-3504	366	65	of	of	ADP
ejpam-3504	366	66	s	s	PROPN
ejpam-3504	366	67	,	,	PUNCT
ejpam-3504	366	68	then	then	ADV
ejpam-3504	366	69	fa	fa	PROPN
ejpam-3504	366	70	is	be	AUX
ejpam-3504	366	71	a	a	DET
ejpam-3504	366	72	fuzzy	fuzzy	ADJ
ejpam-3504	366	73	bi	bi	NOUN
ejpam-3504	366	74	-	-	NOUN
ejpam-3504	366	75	ideal	ideal	NOUN
ejpam-3504	366	76	of	of	ADP
ejpam-3504	366	77	s	s	PRON
ejpam-3504	366	78	then	then	ADV
ejpam-3504	366	79	,	,	PUNCT
ejpam-3504	366	80	by	by	ADP
ejpam-3504	366	81	proposition	proposition	NOUN
ejpam-3504	366	82	5.10	5.10	NUM
ejpam-3504	366	83	,	,	PUNCT
ejpam-3504	366	84	fa	fa	PROPN
ejpam-3504	366	85	is	be	AUX
ejpam-3504	366	86	a	a	DET
ejpam-3504	366	87	fuzzy	fuzzy	ADJ
ejpam-3504	366	88	right	right	ADJ
ejpam-3504	366	89	ideal	ideal	NOUN
ejpam-3504	366	90	of	of	ADP
ejpam-3504	366	91	s	s	PRON
ejpam-3504	366	92	and	and	CCONJ
ejpam-3504	366	93	so	so	ADV
ejpam-3504	366	94	a	a	PRON
ejpam-3504	366	95	is	be	AUX
ejpam-3504	366	96	a	a	DET
ejpam-3504	366	97	right	right	ADJ
ejpam-3504	366	98	ideal	ideal	NOUN
ejpam-3504	366	99	of	of	ADP
ejpam-3504	366	100	s.	s.	PROPN
ejpam-3504	366	101	by	by	ADP
ejpam-3504	366	102	proposition	proposition	NOUN
ejpam-3504	366	103	5.10	5.10	NUM
ejpam-3504	366	104	,	,	PUNCT
ejpam-3504	366	105	we	we	PRON
ejpam-3504	366	106	have	have	VERB
ejpam-3504	366	107	the	the	DET
ejpam-3504	366	108	following	follow	VERB
ejpam-3504	366	109	theorem	theorem	PROPN
ejpam-3504	366	110	.	.	PUNCT
ejpam-3504	366	111	theorem	theorem	VERB
ejpam-3504	366	112	5.12	5.12	NUM
ejpam-3504	366	113	.	.	PUNCT
ejpam-3504	367	1	in	in	ADP
ejpam-3504	367	2	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	367	3	,	,	PUNCT
ejpam-3504	367	4	the	the	DET
ejpam-3504	367	5	fuzzy	fuzzy	ADJ
ejpam-3504	367	6	ideals	ideal	NOUN
ejpam-3504	367	7	are	be	AUX
ejpam-3504	367	8	fuzzy	fuzzy	ADJ
ejpam-3504	367	9	bi	bi	NOUN
ejpam-3504	367	10	-	-	NOUN
ejpam-3504	367	11	ideals	ideal	NOUN
ejpam-3504	367	12	and	and	CCONJ
ejpam-3504	367	13	in	in	ADP
ejpam-3504	367	14	simple	simple	ADJ
ejpam-3504	367	15	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	367	16	,	,	PUNCT
ejpam-3504	367	17	the	the	DET
ejpam-3504	367	18	fuzzy	fuzzy	ADJ
ejpam-3504	367	19	ideals	ideal	NOUN
ejpam-3504	367	20	and	and	CCONJ
ejpam-3504	367	21	the	the	DET
ejpam-3504	367	22	fuzzy	fuzzy	ADJ
ejpam-3504	367	23	bi	bi	ADJ
ejpam-3504	367	24	-	-	ADJ
ejpam-3504	367	25	ideals	ideal	NOUN
ejpam-3504	367	26	coincide	coincide	NOUN
ejpam-3504	367	27	.	.	PUNCT
ejpam-3504	368	1	theorem	theorem	VERB
ejpam-3504	368	2	5.13	5.13	NUM
ejpam-3504	368	3	.	.	PUNCT
ejpam-3504	369	1	let	let	VERB
ejpam-3504	369	2	s	s	PRON
ejpam-3504	369	3	be	be	AUX
ejpam-3504	369	4	an	an	DET
ejpam-3504	369	5	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	369	6	that	that	PRON
ejpam-3504	369	7	is	be	AUX
ejpam-3504	369	8	both	both	ADV
ejpam-3504	369	9	right	right	ADJ
ejpam-3504	369	10	(	(	PUNCT
ejpam-3504	369	11	resp	resp	NOUN
ejpam-3504	369	12	.	.	PUNCT
ejpam-3504	370	1	left	leave	VERB
ejpam-3504	370	2	)	)	PUNCT
ejpam-3504	370	3	regular	regular	ADV
ejpam-3504	370	4	and	and	CCONJ
ejpam-3504	370	5	left	left	ADJ
ejpam-3504	370	6	(	(	PUNCT
ejpam-3504	370	7	resp	resp	NOUN
ejpam-3504	370	8	.	.	PUNCT
ejpam-3504	371	1	right	right	ADJ
ejpam-3504	371	2	)	)	PUNCT
ejpam-3504	371	3	simple	simple	NOUN
ejpam-3504	371	4	.	.	PUNCT
ejpam-3504	372	1	then	then	ADV
ejpam-3504	372	2	,	,	PUNCT
ejpam-3504	372	3	for	for	ADP
ejpam-3504	372	4	any	any	DET
ejpam-3504	372	5	fuzzy	fuzzy	ADJ
ejpam-3504	372	6	bi	bi	ADJ
ejpam-3504	372	7	-	-	ADJ
ejpam-3504	372	8	ideal	ideal	ADJ
ejpam-3504	372	9	f	f	PROPN
ejpam-3504	372	10	of	of	ADP
ejpam-3504	372	11	s	s	PRON
ejpam-3504	372	12	and	and	CCONJ
ejpam-3504	372	13	any	any	DET
ejpam-3504	372	14	a	a	DET
ejpam-3504	372	15	∈	∈	ADJ
ejpam-3504	372	16	s	s	NOUN
ejpam-3504	372	17	,	,	PUNCT
ejpam-3504	372	18	we	we	PRON
ejpam-3504	372	19	have	have	VERB
ejpam-3504	372	20	f(a	f(a	NOUN
ejpam-3504	372	21	)	)	PUNCT
ejpam-3504	373	1	=	=	SYM
ejpam-3504	373	2	f(a	f(a	PROPN
ejpam-3504	373	3	◦	◦	NOUN
ejpam-3504	373	4	a	a	X
ejpam-3504	373	5	)	)	PUNCT
ejpam-3504	373	6	in	in	ADP
ejpam-3504	373	7	the	the	DET
ejpam-3504	373	8	sense	sense	NOUN
ejpam-3504	373	9	that	that	SCONJ
ejpam-3504	373	10	there	there	PRON
ejpam-3504	373	11	exists	exist	VERB
ejpam-3504	373	12	u	u	PROPN
ejpam-3504	373	13	∈	∈	PROPN
ejpam-3504	373	14	a	a	DET
ejpam-3504	373	15	◦	◦	NOUN
ejpam-3504	373	16	a	a	DET
ejpam-3504	373	17	such	such	ADJ
ejpam-3504	373	18	that	that	DET
ejpam-3504	373	19	f(a	f(a	NOUN
ejpam-3504	373	20	)	)	PUNCT
ejpam-3504	373	21	=	=	SYM
ejpam-3504	373	22	f(u	f(u	PROPN
ejpam-3504	373	23	)	)	PUNCT
ejpam-3504	373	24	.	.	PUNCT
ejpam-3504	374	1	proof	proof	NOUN
ejpam-3504	374	2	.	.	PUNCT
ejpam-3504	375	1	let	let	VERB
ejpam-3504	375	2	s	s	PRON
ejpam-3504	375	3	be	be	AUX
ejpam-3504	375	4	right	right	ADV
ejpam-3504	375	5	regular	regular	ADJ
ejpam-3504	375	6	and	and	CCONJ
ejpam-3504	375	7	left	leave	VERB
ejpam-3504	375	8	simple	simple	NOUN
ejpam-3504	375	9	,	,	PUNCT
ejpam-3504	375	10	f	f	PROPN
ejpam-3504	375	11	be	be	AUX
ejpam-3504	375	12	a	a	DET
ejpam-3504	375	13	fuzzy	fuzzy	ADJ
ejpam-3504	375	14	bi	bi	NOUN
ejpam-3504	375	15	-	-	NOUN
ejpam-3504	375	16	ideal	ideal	NOUN
ejpam-3504	375	17	of	of	ADP
ejpam-3504	375	18	s	s	PRON
ejpam-3504	375	19	and	and	CCONJ
ejpam-3504	375	20	a	a	DET
ejpam-3504	375	21	∈	∈	NOUN
ejpam-3504	375	22	s.	s.	PROPN
ejpam-3504	375	23	since	since	SCONJ
ejpam-3504	375	24	s	s	PROPN
ejpam-3504	375	25	is	be	AUX
ejpam-3504	375	26	left	leave	VERB
ejpam-3504	375	27	simple	simple	ADJ
ejpam-3504	375	28	,	,	PUNCT
ejpam-3504	375	29	by	by	ADP
ejpam-3504	375	30	proposition	proposition	NOUN
ejpam-3504	375	31	5.10	5.10	NUM
ejpam-3504	375	32	,	,	PUNCT
ejpam-3504	375	33	f	f	PROPN
ejpam-3504	375	34	is	be	AUX
ejpam-3504	375	35	a	a	DET
ejpam-3504	375	36	fuzzy	fuzzy	ADJ
ejpam-3504	375	37	right	right	ADJ
ejpam-3504	375	38	ideal	ideal	NOUN
ejpam-3504	375	39	of	of	ADP
ejpam-3504	375	40	s.	s.	PROPN
ejpam-3504	375	41	since	since	SCONJ
ejpam-3504	375	42	s	s	PROPN
ejpam-3504	375	43	is	be	AUX
ejpam-3504	375	44	right	right	ADV
ejpam-3504	375	45	regular	regular	ADJ
ejpam-3504	375	46	and	and	CCONJ
ejpam-3504	375	47	f	f	PROPN
ejpam-3504	375	48	is	be	AUX
ejpam-3504	375	49	a	a	DET
ejpam-3504	375	50	fuzzy	fuzzy	ADJ
ejpam-3504	375	51	right	right	ADJ
ejpam-3504	375	52	ideal	ideal	NOUN
ejpam-3504	375	53	of	of	ADP
ejpam-3504	375	54	s	s	PROPN
ejpam-3504	375	55	,	,	PUNCT
ejpam-3504	375	56	by	by	ADP
ejpam-3504	375	57	theorem	theorem	NOUN
ejpam-3504	375	58	4.3	4.3	NUM
ejpam-3504	375	59	,	,	PUNCT
ejpam-3504	375	60	we	we	PRON
ejpam-3504	375	61	have	have	VERB
ejpam-3504	375	62	f(a	f(a	NOUN
ejpam-3504	375	63	)	)	PUNCT
ejpam-3504	376	1	=	=	SYM
ejpam-3504	376	2	f(a	f(a	NOUN
ejpam-3504	376	3	◦	◦	NOUN
ejpam-3504	376	4	a	a	NOUN
ejpam-3504	376	5	)	)	PUNCT
ejpam-3504	376	6	.	.	PUNCT
ejpam-3504	377	1	for	for	ADP
ejpam-3504	377	2	left	left	ADJ
ejpam-3504	377	3	regular	regular	ADJ
ejpam-3504	377	4	and	and	CCONJ
ejpam-3504	377	5	right	right	ADJ
ejpam-3504	377	6	simple	simple	ADJ
ejpam-3504	377	7	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	377	8	the	the	DET
ejpam-3504	377	9	proof	proof	NOUN
ejpam-3504	377	10	is	be	AUX
ejpam-3504	377	11	analogous	analogous	ADJ
ejpam-3504	377	12	.	.	PUNCT
ejpam-3504	378	1	�	�	PROPN
ejpam-3504	378	2	problem	problem	NOUN
ejpam-3504	378	3	1	1	NUM
ejpam-3504	378	4	.	.	PUNCT
ejpam-3504	379	1	a	a	DET
ejpam-3504	379	2	semigroup	semigroup	NOUN
ejpam-3504	379	3	s	s	VERB
ejpam-3504	379	4	is	be	AUX
ejpam-3504	379	5	called	call	VERB
ejpam-3504	379	6	completely	completely	ADV
ejpam-3504	379	7	regular	regular	ADJ
ejpam-3504	379	8	if	if	SCONJ
ejpam-3504	379	9	for	for	SCONJ
ejpam-3504	379	10	every	every	DET
ejpam-3504	379	11	a	a	DET
ejpam-3504	379	12	∈	∈	NOUN
ejpam-3504	379	13	s	s	VERB
ejpam-3504	379	14	there	there	PRON
ejpam-3504	379	15	exists	exist	VERB
ejpam-3504	379	16	x	x	X
ejpam-3504	379	17	∈	∈	NOUN
ejpam-3504	379	18	s	s	VERB
ejpam-3504	379	19	such	such	ADJ
ejpam-3504	380	1	that	that	SCONJ
ejpam-3504	380	2	a	a	DET
ejpam-3504	380	3	=	=	X
ejpam-3504	380	4	axa	axa	NOUN
ejpam-3504	380	5	and	and	CCONJ
ejpam-3504	380	6	ax	ax	NOUN
ejpam-3504	380	7	=	=	X
ejpam-3504	380	8	xa	xa	PROPN
ejpam-3504	381	1	[	[	X
ejpam-3504	381	2	9	9	NUM
ejpam-3504	381	3	]	]	PUNCT
ejpam-3504	381	4	.	.	PUNCT
ejpam-3504	382	1	the	the	DET
ejpam-3504	382	2	completely	completely	ADV
ejpam-3504	382	3	regular	regular	ADJ
ejpam-3504	382	4	semigroups	semigroup	NOUN
ejpam-3504	382	5	are	be	AUX
ejpam-3504	382	6	regular	regular	ADJ
ejpam-3504	382	7	,	,	PUNCT
ejpam-3504	382	8	left	leave	VERB
ejpam-3504	382	9	regular	regular	ADV
ejpam-3504	382	10	and	and	CCONJ
ejpam-3504	382	11	right	right	ADV
ejpam-3504	382	12	regular	regular	ADJ
ejpam-3504	382	13	.	.	PUNCT
ejpam-3504	383	1	in	in	ADP
ejpam-3504	383	2	addition	addition	NOUN
ejpam-3504	383	3	,	,	PUNCT
ejpam-3504	383	4	a	a	DET
ejpam-3504	383	5	semigroup	semigroup	NOUN
ejpam-3504	383	6	s	s	VERB
ejpam-3504	383	7	is	be	AUX
ejpam-3504	383	8	completely	completely	ADV
ejpam-3504	383	9	regular	regular	ADJ
ejpam-3504	383	10	if	if	SCONJ
ejpam-3504	383	11	and	and	CCONJ
ejpam-3504	383	12	only	only	ADV
ejpam-3504	383	13	if	if	SCONJ
ejpam-3504	383	14	,	,	PUNCT
ejpam-3504	383	15	for	for	ADP
ejpam-3504	383	16	every	every	DET
ejpam-3504	383	17	a	a	DET
ejpam-3504	383	18	∈	∈	ADJ
ejpam-3504	383	19	s	s	NOUN
ejpam-3504	383	20	,	,	PUNCT
ejpam-3504	383	21	there	there	PRON
ejpam-3504	383	22	exists	exist	VERB
ejpam-3504	383	23	x	x	X
ejpam-3504	383	24	∈	∈	PROPN
ejpam-3504	383	25	s	s	VERB
ejpam-3504	383	26	such	such	ADJ
ejpam-3504	383	27	that	that	SCONJ
ejpam-3504	383	28	a	a	DET
ejpam-3504	383	29	∈	∈	PROPN
ejpam-3504	383	30	a2xa2	a2xa2	VERB
ejpam-3504	384	1	[	[	PUNCT
ejpam-3504	384	2	9	9	NUM
ejpam-3504	384	3	;	;	PUNCT
ejpam-3504	384	4	iv.1.2	iv.1.2	ADJ
ejpam-3504	384	5	proposition	proposition	NOUN
ejpam-3504	384	6	]	]	PUNCT
ejpam-3504	384	7	.	.	PUNCT
ejpam-3504	385	1	the	the	DET
ejpam-3504	385	2	concept	concept	NOUN
ejpam-3504	385	3	of	of	ADP
ejpam-3504	385	4	completely	completely	ADV
ejpam-3504	385	5	regular	regular	ADJ
ejpam-3504	385	6	semigroups	semigroup	NOUN
ejpam-3504	385	7	can	can	AUX
ejpam-3504	385	8	be	be	AUX
ejpam-3504	385	9	naturally	naturally	ADV
ejpam-3504	385	10	transferred	transfer	VERB
ejpam-3504	385	11	to	to	ADP
ejpam-3504	385	12	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	385	13	as	as	SCONJ
ejpam-3504	385	14	follows	follow	VERB
ejpam-3504	385	15	:	:	PUNCT
ejpam-3504	385	16	an	an	DET
ejpam-3504	385	17	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	385	18	s	s	X
ejpam-3504	385	19	is	be	AUX
ejpam-3504	385	20	called	call	VERB
ejpam-3504	385	21	completely	completely	ADV
ejpam-3504	385	22	regular	regular	ADJ
ejpam-3504	385	23	if	if	SCONJ
ejpam-3504	385	24	for	for	ADP
ejpam-3504	385	25	any	any	DET
ejpam-3504	385	26	a	a	DET
ejpam-3504	385	27	∈	∈	NOUN
ejpam-3504	385	28	s	s	VERB
ejpam-3504	385	29	there	there	PRON
ejpam-3504	385	30	exists	exist	VERB
ejpam-3504	385	31	x	x	X
ejpam-3504	385	32	∈	∈	NOUN
ejpam-3504	385	33	s	s	VERB
ejpam-3504	385	34	such	such	ADJ
ejpam-3504	385	35	that	that	SCONJ
ejpam-3504	385	36	a	a	DET
ejpam-3504	385	37	∈	∈	NOUN
ejpam-3504	385	38	(	(	PUNCT
ejpam-3504	385	39	a	a	DET
ejpam-3504	385	40	◦	◦	NOUN
ejpam-3504	385	41	x	x	SYM
ejpam-3504	385	42	)	)	PUNCT
ejpam-3504	385	43	∗	∗	NOUN
ejpam-3504	385	44	a	a	PRON
ejpam-3504	385	45	and	and	CCONJ
ejpam-3504	385	46	a	a	DET
ejpam-3504	385	47	◦	◦	NOUN
ejpam-3504	385	48	x	x	X
ejpam-3504	385	49	=	=	SYM
ejpam-3504	385	50	x	x	PUNCT
ejpam-3504	385	51	◦	◦	NOUN
ejpam-3504	385	52	a.	a.	NOUN
ejpam-3504	385	53	kuroki	kuroki	PROPN
ejpam-3504	385	54	has	have	AUX
ejpam-3504	385	55	shown	show	VERB
ejpam-3504	385	56	that	that	SCONJ
ejpam-3504	385	57	a	a	DET
ejpam-3504	385	58	semigroup	semigroup	NOUN
ejpam-3504	385	59	s	s	VERB
ejpam-3504	385	60	is	be	AUX
ejpam-3504	385	61	references	reference	NOUN
ejpam-3504	385	62	720	720	NUM
ejpam-3504	385	63	completely	completely	ADV
ejpam-3504	385	64	regular	regular	ADJ
ejpam-3504	385	65	if	if	SCONJ
ejpam-3504	385	66	and	and	CCONJ
ejpam-3504	385	67	only	only	ADV
ejpam-3504	385	68	if	if	SCONJ
ejpam-3504	385	69	for	for	ADP
ejpam-3504	385	70	every	every	DET
ejpam-3504	385	71	fuzzy	fuzzy	ADJ
ejpam-3504	385	72	bi	bi	ADJ
ejpam-3504	385	73	-	-	ADJ
ejpam-3504	385	74	ideal	ideal	ADJ
ejpam-3504	385	75	f	f	PROPN
ejpam-3504	385	76	of	of	ADP
ejpam-3504	385	77	s	s	PRON
ejpam-3504	385	78	and	and	CCONJ
ejpam-3504	385	79	every	every	PRON
ejpam-3504	385	80	a	a	DET
ejpam-3504	385	81	∈	∈	PROPN
ejpam-3504	385	82	s	s	NOUN
ejpam-3504	385	83	,	,	PUNCT
ejpam-3504	385	84	we	we	PRON
ejpam-3504	385	85	have	have	VERB
ejpam-3504	385	86	f(a	f(a	NOUN
ejpam-3504	385	87	)	)	PUNCT
ejpam-3504	385	88	=	=	SYM
ejpam-3504	385	89	f(a2	f(a2	NOUN
ejpam-3504	385	90	)	)	PUNCT
ejpam-3504	386	1	[	[	X
ejpam-3504	386	2	7	7	NUM
ejpam-3504	386	3	;	;	PUNCT
ejpam-3504	386	4	theorem	theorem	VERB
ejpam-3504	386	5	4	4	NUM
ejpam-3504	386	6	]	]	PUNCT
ejpam-3504	386	7	.	.	PUNCT
ejpam-3504	387	1	is	be	AUX
ejpam-3504	387	2	there	there	PRON
ejpam-3504	387	3	an	an	DET
ejpam-3504	387	4	analogous	analogous	ADJ
ejpam-3504	387	5	result	result	NOUN
ejpam-3504	387	6	in	in	ADP
ejpam-3504	387	7	case	case	NOUN
ejpam-3504	387	8	of	of	ADP
ejpam-3504	387	9	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	387	10	?	?	PUNCT
ejpam-3504	388	1	problem	problem	NOUN
ejpam-3504	388	2	2	2	NUM
ejpam-3504	388	3	.	.	PUNCT
ejpam-3504	389	1	according	accord	VERB
ejpam-3504	389	2	to	to	ADP
ejpam-3504	389	3	kuroki	kuroki	PROPN
ejpam-3504	389	4	[	[	X
ejpam-3504	389	5	8	8	NUM
ejpam-3504	389	6	;	;	PUNCT
ejpam-3504	389	7	theorem	theorem	VERB
ejpam-3504	389	8	3.9	3.9	NUM
ejpam-3504	389	9	]	]	PUNCT
ejpam-3504	389	10	,	,	PUNCT
ejpam-3504	389	11	for	for	ADP
ejpam-3504	389	12	a	a	DET
ejpam-3504	389	13	regular	regular	ADJ
ejpam-3504	389	14	semigroup	semigroup	NOUN
ejpam-3504	389	15	s	s	VERB
ejpam-3504	389	16	the	the	DET
ejpam-3504	389	17	following	follow	VERB
ejpam-3504	389	18	conditions	condition	NOUN
ejpam-3504	389	19	are	be	AUX
ejpam-3504	389	20	equivalent	equivalent	ADJ
ejpam-3504	389	21	:	:	PUNCT
ejpam-3504	389	22	(	(	PUNCT
ejpam-3504	389	23	1	1	X
ejpam-3504	389	24	)	)	PUNCT
ejpam-3504	389	25	the	the	DET
ejpam-3504	389	26	set	set	NOUN
ejpam-3504	389	27	of	of	ADP
ejpam-3504	389	28	all	all	DET
ejpam-3504	389	29	idempotent	idempotent	ADJ
ejpam-3504	389	30	elements	element	NOUN
ejpam-3504	389	31	of	of	ADP
ejpam-3504	389	32	s	s	NOUN
ejpam-3504	389	33	forms	form	NOUN
ejpam-3504	389	34	a	a	DET
ejpam-3504	389	35	left	left	ADJ
ejpam-3504	389	36	zero	zero	NUM
ejpam-3504	389	37	subsemigroup	subsemigroup	NOUN
ejpam-3504	389	38	of	of	ADP
ejpam-3504	389	39	s.	s.	PROPN
ejpam-3504	389	40	(	(	PUNCT
ejpam-3504	389	41	2	2	NUM
ejpam-3504	389	42	)	)	PUNCT
ejpam-3504	389	43	for	for	ADP
ejpam-3504	389	44	every	every	DET
ejpam-3504	389	45	fuzzy	fuzzy	ADJ
ejpam-3504	389	46	left	leave	VERB
ejpam-3504	389	47	ideal	ideal	NOUN
ejpam-3504	389	48	f	f	PROPN
ejpam-3504	389	49	of	of	ADP
ejpam-3504	389	50	s	s	PRON
ejpam-3504	389	51	and	and	CCONJ
ejpam-3504	389	52	any	any	DET
ejpam-3504	389	53	idempotent	idempotent	ADJ
ejpam-3504	389	54	elements	element	NOUN
ejpam-3504	389	55	a	a	PRON
ejpam-3504	389	56	,	,	PUNCT
ejpam-3504	389	57	b	b	X
ejpam-3504	389	58	∈	∈	PROPN
ejpam-3504	389	59	s	s	X
ejpam-3504	389	60	,	,	PUNCT
ejpam-3504	389	61	we	we	PRON
ejpam-3504	389	62	have	have	VERB
ejpam-3504	389	63	f(a	f(a	NOUN
ejpam-3504	389	64	)	)	PUNCT
ejpam-3504	389	65	=	=	SYM
ejpam-3504	389	66	f(b	f(b	X
ejpam-3504	389	67	)	)	PUNCT
ejpam-3504	389	68	(	(	PUNCT
ejpam-3504	389	69	in	in	ADP
ejpam-3504	389	70	fact	fact	NOUN
ejpam-3504	389	71	,	,	PUNCT
ejpam-3504	389	72	the	the	DET
ejpam-3504	389	73	implication	implication	NOUN
ejpam-3504	389	74	(	(	PUNCT
ejpam-3504	389	75	1	1	X
ejpam-3504	389	76	)	)	PUNCT
ejpam-3504	389	77	⇒	⇒	NOUN
ejpam-3504	389	78	(	(	PUNCT
ejpam-3504	389	79	2	2	X
ejpam-3504	389	80	)	)	PUNCT
ejpam-3504	389	81	holds	hold	VERB
ejpam-3504	389	82	in	in	ADP
ejpam-3504	389	83	groupoids	groupoid	NOUN
ejpam-3504	389	84	in	in	ADP
ejpam-3504	389	85	general	general	ADJ
ejpam-3504	389	86	–	–	PUNCT
ejpam-3504	389	87	this	this	PRON
ejpam-3504	389	88	being	be	AUX
ejpam-3504	389	89	so	so	ADV
ejpam-3504	389	90	,	,	PUNCT
ejpam-3504	389	91	the	the	DET
ejpam-3504	389	92	regularity	regularity	NOUN
ejpam-3504	389	93	does	do	AUX
ejpam-3504	389	94	not	not	PART
ejpam-3504	389	95	play	play	VERB
ejpam-3504	389	96	any	any	DET
ejpam-3504	389	97	role	role	NOUN
ejpam-3504	389	98	in	in	ADP
ejpam-3504	389	99	it	it	PRON
ejpam-3504	389	100	)	)	PUNCT
ejpam-3504	389	101	.	.	PUNCT
ejpam-3504	390	1	is	be	AUX
ejpam-3504	390	2	there	there	PRON
ejpam-3504	390	3	something	something	PRON
ejpam-3504	390	4	analogous	analogous	ADJ
ejpam-3504	390	5	if	if	SCONJ
ejpam-3504	390	6	we	we	PRON
ejpam-3504	390	7	replace	replace	VERB
ejpam-3504	390	8	the	the	DET
ejpam-3504	390	9	words	word	NOUN
ejpam-3504	390	10	“	"	PUNCT
ejpam-3504	390	11	groupoid	groupoid	PROPN
ejpam-3504	390	12	”	"	PUNCT
ejpam-3504	390	13	,	,	PUNCT
ejpam-3504	390	14	“	"	PUNCT
ejpam-3504	390	15	semigroup	semigroup	NOUN
ejpam-3504	390	16	”	"	PUNCT
ejpam-3504	390	17	by	by	ADP
ejpam-3504	390	18	“	"	PUNCT
ejpam-3504	390	19	hypergroupoid	hypergroupoid	PROPN
ejpam-3504	390	20	”	"	PUNCT
ejpam-3504	390	21	,	,	PUNCT
ejpam-3504	390	22	“	"	PUNCT
ejpam-3504	390	23	hypersemigroup	hypersemigroup	NOUN
ejpam-3504	390	24	”	"	PUNCT
ejpam-3504	390	25	?	?	PUNCT
ejpam-3504	391	1	problem	problem	NOUN
ejpam-3504	391	2	3	3	NUM
ejpam-3504	391	3	.	.	PUNCT
ejpam-3504	392	1	according	accord	VERB
ejpam-3504	392	2	to	to	ADP
ejpam-3504	392	3	[	[	X
ejpam-3504	392	4	8	8	NUM
ejpam-3504	392	5	;	;	PUNCT
ejpam-3504	392	6	theorem	theorem	VERB
ejpam-3504	392	7	6.3	6.3	NUM
ejpam-3504	392	8	]	]	PUNCT
ejpam-3504	392	9	,	,	PUNCT
ejpam-3504	392	10	a	a	DET
ejpam-3504	392	11	semigroup	semigroup	NOUN
ejpam-3504	392	12	s	s	VERB
ejpam-3504	392	13	is	be	AUX
ejpam-3504	392	14	a	a	DET
ejpam-3504	392	15	semilattice	semilattice	NOUN
ejpam-3504	392	16	of	of	ADP
ejpam-3504	392	17	left	left	ADJ
ejpam-3504	392	18	simple	simple	ADJ
ejpam-3504	392	19	semigroups	semigroup	NOUN
ejpam-3504	392	20	if	if	SCONJ
ejpam-3504	392	21	and	and	CCONJ
ejpam-3504	392	22	only	only	ADV
ejpam-3504	392	23	if	if	SCONJ
ejpam-3504	392	24	for	for	ADP
ejpam-3504	392	25	any	any	DET
ejpam-3504	392	26	left	left	ADJ
ejpam-3504	392	27	ideal	ideal	NOUN
ejpam-3504	392	28	f	f	PROPN
ejpam-3504	392	29	of	of	ADP
ejpam-3504	392	30	s	s	PRON
ejpam-3504	392	31	and	and	CCONJ
ejpam-3504	392	32	any	any	DET
ejpam-3504	392	33	a	a	PRON
ejpam-3504	392	34	,	,	PUNCT
ejpam-3504	392	35	b	b	PROPN
ejpam-3504	392	36	∈	∈	PROPN
ejpam-3504	392	37	s	s	X
ejpam-3504	392	38	,	,	PUNCT
ejpam-3504	392	39	we	we	PRON
ejpam-3504	392	40	have	have	AUX
ejpam-3504	392	41	f(a	f(a	NOUN
ejpam-3504	392	42	)	)	PUNCT
ejpam-3504	392	43	=	=	SYM
ejpam-3504	392	44	f(a2	f(a2	NOUN
ejpam-3504	392	45	)	)	PUNCT
ejpam-3504	392	46	and	and	CCONJ
ejpam-3504	392	47	f(ab	f(ab	NOUN
ejpam-3504	392	48	)	)	PUNCT
ejpam-3504	392	49	=	=	SYM
ejpam-3504	392	50	f(ba	f(ba	X
ejpam-3504	392	51	)	)	PUNCT
ejpam-3504	392	52	.	.	PUNCT
ejpam-3504	393	1	examine	examine	VERB
ejpam-3504	393	2	it	it	PRON
ejpam-3504	393	3	in	in	ADP
ejpam-3504	393	4	case	case	NOUN
ejpam-3504	393	5	of	of	ADP
ejpam-3504	393	6	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	393	7	.	.	PUNCT
ejpam-3504	394	1	note	note	VERB
ejpam-3504	394	2	.	.	PUNCT
ejpam-3504	395	1	the	the	DET
ejpam-3504	395	2	results	result	NOUN
ejpam-3504	395	3	of	of	ADP
ejpam-3504	395	4	the	the	DET
ejpam-3504	395	5	present	present	ADJ
ejpam-3504	395	6	paper	paper	NOUN
ejpam-3504	395	7	in	in	ADP
ejpam-3504	395	8	which	which	PRON
ejpam-3504	395	9	the	the	DET
ejpam-3504	395	10	word	word	NOUN
ejpam-3504	395	11	“	"	PUNCT
ejpam-3504	395	12	fuzzy	fuzzy	ADJ
ejpam-3504	395	13	”	"	PUNCT
ejpam-3504	395	14	is	be	AUX
ejpam-3504	395	15	not	not	PART
ejpam-3504	395	16	included	include	VERB
ejpam-3504	395	17	,	,	PUNCT
ejpam-3504	395	18	that	that	PRON
ejpam-3504	395	19	is	be	AUX
ejpam-3504	395	20	lemma	lemma	PROPN
ejpam-3504	395	21	5.1	5.1	NUM
ejpam-3504	395	22	,	,	PUNCT
ejpam-3504	395	23	corollary	corollary	ADJ
ejpam-3504	395	24	5.2	5.2	NUM
ejpam-3504	395	25	,	,	PUNCT
ejpam-3504	395	26	remark	remark	VERB
ejpam-3504	395	27	5.3	5.3	NUM
ejpam-3504	395	28	and	and	CCONJ
ejpam-3504	395	29	proposition	proposition	NOUN
ejpam-3504	395	30	5.7	5.7	NUM
ejpam-3504	395	31	come	come	VERB
ejpam-3504	395	32	directly	directly	ADV
ejpam-3504	395	33	from	from	ADP
ejpam-3504	395	34	the	the	DET
ejpam-3504	395	35	poe	poe	PROPN
ejpam-3504	395	36	-	-	PUNCT
ejpam-3504	395	37	semigroups	semigroups	X
ejpam-3504	395	38	(	(	PUNCT
ejpam-3504	395	39	that	that	PRON
ejpam-3504	395	40	is	is	ADV
ejpam-3504	395	41	,	,	PUNCT
ejpam-3504	395	42	from	from	ADP
ejpam-3504	395	43	ordered	order	VERB
ejpam-3504	395	44	semigroups	semigroup	NOUN
ejpam-3504	395	45	having	have	VERB
ejpam-3504	395	46	a	a	DET
ejpam-3504	395	47	greatest	great	ADJ
ejpam-3504	395	48	element	element	NOUN
ejpam-3504	395	49	)	)	PUNCT
ejpam-3504	395	50	.	.	PUNCT
ejpam-3504	396	1	analogous	analogous	ADJ
ejpam-3504	396	2	results	result	NOUN
ejpam-3504	396	3	to	to	ADP
ejpam-3504	396	4	the	the	DET
ejpam-3504	396	5	results	result	NOUN
ejpam-3504	396	6	of	of	ADP
ejpam-3504	396	7	the	the	DET
ejpam-3504	396	8	present	present	ADJ
ejpam-3504	396	9	paper	paper	NOUN
ejpam-3504	396	10	hold	hold	NOUN
ejpam-3504	396	11	for	for	ADP
ejpam-3504	396	12	γ	γ	NOUN
ejpam-3504	396	13	-	-	PUNCT
ejpam-3504	396	14	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	396	15	and	and	CCONJ
ejpam-3504	396	16	can	can	AUX
ejpam-3504	396	17	be	be	AUX
ejpam-3504	396	18	obtained	obtain	VERB
ejpam-3504	396	19	by	by	ADP
ejpam-3504	396	20	easy	easy	ADJ
ejpam-3504	396	21	modification	modification	NOUN
ejpam-3504	396	22	.	.	PUNCT
ejpam-3504	397	1	analogous	analogous	ADJ
ejpam-3504	397	2	results	result	NOUN
ejpam-3504	397	3	hold	hold	VERB
ejpam-3504	397	4	for	for	ADP
ejpam-3504	397	5	ordered	order	VERB
ejpam-3504	397	6	γ	γ	NOUN
ejpam-3504	397	7	-	-	PUNCT
ejpam-3504	397	8	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	397	9	as	as	ADV
ejpam-3504	397	10	well	well	ADV
ejpam-3504	397	11	.	.	PUNCT
ejpam-3504	398	1	references	reference	NOUN
ejpam-3504	398	2	[	[	X
ejpam-3504	398	3	1	1	NUM
ejpam-3504	398	4	]	]	PUNCT
ejpam-3504	398	5	n.	n.	NOUN
ejpam-3504	398	6	kehayopulu	kehayopulu	PROPN
ejpam-3504	398	7	.	.	PUNCT
ejpam-3504	399	1	a	a	DET
ejpam-3504	399	2	characterization	characterization	NOUN
ejpam-3504	399	3	of	of	ADP
ejpam-3504	399	4	regular	regular	ADJ
ejpam-3504	399	5	,	,	PUNCT
ejpam-3504	399	6	intra	intra	ADJ
ejpam-3504	399	7	-	-	ADJ
ejpam-3504	399	8	regular	regular	ADJ
ejpam-3504	399	9	,	,	PUNCT
ejpam-3504	399	10	left	leave	VERB
ejpam-3504	399	11	quasi	quasi	ADJ
ejpam-3504	399	12	-	-	ADJ
ejpam-3504	399	13	regular	regular	ADJ
ejpam-3504	399	14	and	and	CCONJ
ejpam-3504	399	15	semisimple	semisimple	ADJ
ejpam-3504	399	16	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	399	17	in	in	ADP
ejpam-3504	399	18	terms	term	NOUN
ejpam-3504	399	19	of	of	ADP
ejpam-3504	399	20	fuzzy	fuzzy	ADJ
ejpam-3504	399	21	sets	set	NOUN
ejpam-3504	399	22	.	.	PUNCT
ejpam-3504	400	1	pure	pure	ADJ
ejpam-3504	400	2	math	math	NOUN
ejpam-3504	400	3	.	.	PUNCT
ejpam-3504	401	1	appl	appl	PROPN
ejpam-3504	401	2	.	.	PUNCT
ejpam-3504	402	1	(	(	PUNCT
ejpam-3504	402	2	pu.m.a	pu.m.a	PROPN
ejpam-3504	402	3	.	.	PUNCT
ejpam-3504	402	4	)	)	PUNCT
ejpam-3504	403	1	26(1):46–56	26(1):46–56	NUM
ejpam-3504	403	2	,	,	PUNCT
ejpam-3504	403	3	2017	2017	NUM
ejpam-3504	403	4	.	.	PUNCT
ejpam-3504	404	1	[	[	X
ejpam-3504	404	2	2	2	NUM
ejpam-3504	404	3	]	]	PUNCT
ejpam-3504	404	4	n.	n.	NOUN
ejpam-3504	404	5	kehayopulu	kehayopulu	PROPN
ejpam-3504	404	6	.	.	PUNCT
ejpam-3504	405	1	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	405	2	and	and	CCONJ
ejpam-3504	405	3	fuzzy	fuzzy	ADJ
ejpam-3504	405	4	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	405	5	.	.	PUNCT
ejpam-3504	406	1	eur	eur	PROPN
ejpam-3504	406	2	.	.	PUNCT
ejpam-3504	407	1	j.	j.	PROPN
ejpam-3504	407	2	pure	pure	PROPN
ejpam-3504	407	3	appl	appl	PROPN
ejpam-3504	407	4	.	.	PUNCT
ejpam-3504	407	5	math	math	NOUN
ejpam-3504	407	6	.	.	PUNCT
ejpam-3504	408	1	10(5):929–945	10(5):929–945	NUM
ejpam-3504	408	2	,	,	PUNCT
ejpam-3504	408	3	2017	2017	NUM
ejpam-3504	408	4	.	.	PUNCT
ejpam-3504	409	1	[	[	X
ejpam-3504	409	2	3	3	X
ejpam-3504	409	3	]	]	X
ejpam-3504	409	4	n.	n.	NOUN
ejpam-3504	409	5	kehayopulu	kehayopulu	PROPN
ejpam-3504	409	6	.	.	PUNCT
ejpam-3504	410	1	left	leave	VERB
ejpam-3504	410	2	regular	regular	ADJ
ejpam-3504	410	3	and	and	CCONJ
ejpam-3504	410	4	intra	intra	ADJ
ejpam-3504	410	5	-	-	ADJ
ejpam-3504	410	6	regular	regular	ADJ
ejpam-3504	410	7	ordered	order	VERB
ejpam-3504	410	8	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	410	9	in	in	ADP
ejpam-3504	410	10	terms	term	NOUN
ejpam-3504	410	11	of	of	ADP
ejpam-3504	410	12	semiprime	semiprime	NOUN
ejpam-3504	410	13	and	and	CCONJ
ejpam-3504	410	14	fuzzy	fuzzy	ADJ
ejpam-3504	410	15	semiprime	semiprime	NOUN
ejpam-3504	410	16	subsets	subset	NOUN
ejpam-3504	410	17	.	.	PUNCT
ejpam-3504	411	1	sci	sci	PROPN
ejpam-3504	411	2	.	.	PROPN
ejpam-3504	411	3	math	math	PROPN
ejpam-3504	411	4	.	.	PUNCT
ejpam-3504	412	1	jpn	jpn	PROPN
ejpam-3504	412	2	.	.	PUNCT
ejpam-3504	413	1	80(3):295–305	80(3):295–305	PROPN
ejpam-3504	413	2	,	,	PUNCT
ejpam-3504	413	3	2017	2017	NUM
ejpam-3504	413	4	.	.	PUNCT
ejpam-3504	414	1	[	[	X
ejpam-3504	414	2	4	4	X
ejpam-3504	414	3	]	]	X
ejpam-3504	414	4	n.	n.	NOUN
ejpam-3504	414	5	kehayopulu	kehayopulu	PROPN
ejpam-3504	414	6	.	.	PUNCT
ejpam-3504	415	1	fuzzy	fuzzy	ADJ
ejpam-3504	415	2	sets	set	NOUN
ejpam-3504	415	3	in	in	ADP
ejpam-3504	415	4	≤-hypergroupoids	≤-hypergroupoids	PROPN
ejpam-3504	415	5	.	.	PUNCT
ejpam-3504	416	1	sci	sci	PROPN
ejpam-3504	416	2	.	.	PUNCT
ejpam-3504	416	3	math	math	PROPN
ejpam-3504	416	4	.	.	PUNCT
ejpam-3504	417	1	jpn	jpn	PROPN
ejpam-3504	417	2	.	.	PUNCT
ejpam-3504	418	1	80(3):307–314	80(3):307–314	PROPN
ejpam-3504	418	2	,	,	PUNCT
ejpam-3504	418	3	2017	2017	NUM
ejpam-3504	418	4	.	.	PUNCT
ejpam-3504	419	1	[	[	X
ejpam-3504	419	2	5	5	X
ejpam-3504	419	3	]	]	PUNCT
ejpam-3504	419	4	n.	n.	NOUN
ejpam-3504	419	5	kehayopulu	kehayopulu	PROPN
ejpam-3504	419	6	.	.	PUNCT
ejpam-3504	420	1	how	how	SCONJ
ejpam-3504	420	2	we	we	PRON
ejpam-3504	420	3	pass	pass	VERB
ejpam-3504	420	4	from	from	ADP
ejpam-3504	420	5	semigroups	semigroup	NOUN
ejpam-3504	420	6	to	to	ADP
ejpam-3504	420	7	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	420	8	.	.	PUNCT
ejpam-3504	421	1	lobachevskii	lobachevskii	PROPN
ejpam-3504	421	2	j.	j.	PROPN
ejpam-3504	421	3	math	math	PROPN
ejpam-3504	421	4	.	.	PUNCT
ejpam-3504	422	1	39(1):121–128	39(1):121–128	NUM
ejpam-3504	422	2	,	,	PUNCT
ejpam-3504	422	3	2018	2018	NUM
ejpam-3504	422	4	.	.	PUNCT
ejpam-3504	423	1	[	[	X
ejpam-3504	423	2	6	6	NUM
ejpam-3504	423	3	]	]	PUNCT
ejpam-3504	423	4	n.	n.	NOUN
ejpam-3504	423	5	kehayopulu	kehayopulu	PROPN
ejpam-3504	423	6	.	.	PUNCT
ejpam-3504	424	1	fuzzy	fuzzy	ADJ
ejpam-3504	424	2	right	right	INTJ
ejpam-3504	424	3	(	(	PUNCT
ejpam-3504	424	4	left	left	ADJ
ejpam-3504	424	5	)	)	PUNCT
ejpam-3504	424	6	ideals	ideal	NOUN
ejpam-3504	424	7	in	in	ADP
ejpam-3504	424	8	hypergroupoids	hypergroupoid	NOUN
ejpam-3504	424	9	and	and	CCONJ
ejpam-3504	424	10	fuzzy	fuzzy	ADJ
ejpam-3504	424	11	bi	bi	NOUN
ejpam-3504	424	12	-	-	NOUN
ejpam-3504	424	13	ideals	ideal	NOUN
ejpam-3504	424	14	in	in	ADP
ejpam-3504	424	15	hypersemigroups	hypersemigroup	NOUN
ejpam-3504	424	16	.	.	PUNCT
ejpam-3504	425	1	armen	armen	PROPN
ejpam-3504	425	2	.	.	PUNCT
ejpam-3504	426	1	j.	j.	PROPN
ejpam-3504	426	2	math	math	PROPN
ejpam-3504	426	3	.	.	PUNCT
ejpam-3504	427	1	10	10	NUM
ejpam-3504	427	2	,	,	PUNCT
ejpam-3504	427	3	paper	paper	NOUN
ejpam-3504	427	4	no	no	NOUN
ejpam-3504	427	5	.	.	NOUN
ejpam-3504	427	6	3	3	NUM
ejpam-3504	427	7	(	(	PUNCT
ejpam-3504	427	8	2018	2018	NUM
ejpam-3504	427	9	)	)	PUNCT
ejpam-3504	427	10	,	,	PUNCT
ejpam-3504	427	11	10	10	NUM
ejpam-3504	427	12	pp	pp	NOUN
ejpam-3504	427	13	.	.	PUNCT
ejpam-3504	428	1	[	[	X
ejpam-3504	428	2	7	7	X
ejpam-3504	428	3	]	]	X
ejpam-3504	428	4	n.	n.	PROPN
ejpam-3504	428	5	kuroki	kuroki	PROPN
ejpam-3504	428	6	.	.	PUNCT
ejpam-3504	429	1	fuzzy	fuzzy	ADJ
ejpam-3504	429	2	bi	bi	NOUN
ejpam-3504	429	3	-	-	NOUN
ejpam-3504	429	4	ideals	ideal	NOUN
ejpam-3504	429	5	in	in	ADP
ejpam-3504	429	6	semigroups	semigroup	NOUN
ejpam-3504	429	7	.	.	PUNCT
ejpam-3504	430	1	comment	comment	NOUN
ejpam-3504	430	2	.	.	PUNCT
ejpam-3504	431	1	math	math	NOUN
ejpam-3504	431	2	.	.	PUNCT
ejpam-3504	432	1	univ	univ	PROPN
ejpam-3504	432	2	.	.	PUNCT
ejpam-3504	432	3	st	st	PROPN
ejpam-3504	432	4	.	.	PROPN
ejpam-3504	432	5	pauli	pauli	PROPN
ejpam-3504	432	6	xxviii	xxviii	PROPN
ejpam-3504	432	7	–	–	PUNCT
ejpam-3504	432	8	1:17–21	1:17–21	NUM
ejpam-3504	432	9	,	,	PUNCT
ejpam-3504	432	10	1979	1979	NUM
ejpam-3504	432	11	.	.	PUNCT
ejpam-3504	433	1	[	[	X
ejpam-3504	433	2	8	8	NUM
ejpam-3504	433	3	]	]	X
ejpam-3504	433	4	n.	n.	PROPN
ejpam-3504	433	5	kuroki	kuroki	PROPN
ejpam-3504	433	6	.	.	PUNCT
ejpam-3504	434	1	on	on	ADP
ejpam-3504	434	2	fuzzy	fuzzy	ADJ
ejpam-3504	434	3	ideals	ideal	NOUN
ejpam-3504	434	4	and	and	CCONJ
ejpam-3504	434	5	fuzzy	fuzzy	ADJ
ejpam-3504	434	6	bi	bi	NOUN
ejpam-3504	434	7	-	-	NOUN
ejpam-3504	434	8	ideals	ideal	NOUN
ejpam-3504	434	9	in	in	ADP
ejpam-3504	434	10	semigroups	semigroup	NOUN
ejpam-3504	434	11	.	.	PUNCT
ejpam-3504	435	1	fuzzy	fuzzy	ADJ
ejpam-3504	435	2	sets	set	NOUN
ejpam-3504	435	3	systems	system	NOUN
ejpam-3504	435	4	5:203–215	5:203–215	NUM
ejpam-3504	435	5	,	,	PUNCT
ejpam-3504	435	6	1981	1981	NUM
ejpam-3504	435	7	.	.	PUNCT
ejpam-3504	436	1	references	reference	NOUN
ejpam-3504	436	2	721	721	NUM
ejpam-3504	437	1	[	[	X
ejpam-3504	437	2	9	9	NUM
ejpam-3504	437	3	]	]	PUNCT
ejpam-3504	437	4	m.	m.	NOUN
ejpam-3504	437	5	petrich	petrich	PROPN
ejpam-3504	437	6	.	.	PUNCT
ejpam-3504	438	1	introduction	introduction	NOUN
ejpam-3504	438	2	to	to	ADP
ejpam-3504	438	3	semigroups	semigroup	NOUN
ejpam-3504	438	4	.	.	PUNCT
ejpam-3504	439	1	merrill	merrill	NOUN
ejpam-3504	439	2	research	research	NOUN
ejpam-3504	439	3	and	and	CCONJ
ejpam-3504	439	4	lecture	lecture	NOUN
ejpam-3504	439	5	series	series	NOUN
ejpam-3504	439	6	.	.	PUNCT
ejpam-3504	440	1	charles	charles	PROPN
ejpam-3504	440	2	e.	e.	PROPN
ejpam-3504	440	3	merrill	merrill	PROPN
ejpam-3504	440	4	publishing	publishing	PROPN
ejpam-3504	440	5	co.	co.	PROPN
ejpam-3504	440	6	,	,	PUNCT
ejpam-3504	440	7	columbus	columbus	PROPN
ejpam-3504	440	8	,	,	PUNCT
ejpam-3504	440	9	ohio	ohio	PROPN
ejpam-3504	440	10	,	,	PUNCT
ejpam-3504	440	11	1973	1973	NUM
ejpam-3504	440	12	.	.	PUNCT
ejpam-3504	441	1	viii+198	viii+198	NOUN
ejpam-3504	441	2	pp	pp	ADP
ejpam-3504	441	3	.	.	PUNCT
ejpam-3504	442	1	[	[	X
ejpam-3504	442	2	10	10	NUM
ejpam-3504	442	3	]	]	PUNCT
ejpam-3504	442	4	a.	a.	NOUN
ejpam-3504	442	5	rosenfeld	rosenfeld	PROPN
ejpam-3504	442	6	.	.	PUNCT
ejpam-3504	443	1	fuzzy	fuzzy	ADJ
ejpam-3504	443	2	groups	group	NOUN
ejpam-3504	443	3	.	.	PUNCT
ejpam-3504	444	1	j.	j.	PROPN
ejpam-3504	444	2	math	math	PROPN
ejpam-3504	444	3	.	.	PUNCT
ejpam-3504	445	1	anal	anal	PROPN
ejpam-3504	445	2	.	.	PUNCT
ejpam-3504	445	3	appl	appl	PROPN
ejpam-3504	445	4	.	.	PUNCT
ejpam-3504	446	1	35:512–517	35:512–517	PROPN
ejpam-3504	446	2	,	,	PUNCT
ejpam-3504	446	3	1971	1971	NUM
ejpam-3504	446	4	.	.	PUNCT