id	sid	tid	token	lemma	pos
ejpam-3383	1	1	many	many	ADJ
ejpam-3383	1	2	things	thing	NOUN
ejpam-3383	1	3	come	come	VERB
ejpam-3383	1	4	from	from	ADP
ejpam-3383	1	5	$	$	SYM
ejpam-3383	1	6	le$-semigroups	le$-semigroup	NOUN
ejpam-3383	1	7	european	european	ADJ
ejpam-3383	1	8	journal	journal	NOUN
ejpam-3383	1	9	of	of	ADP
ejpam-3383	1	10	pure	pure	ADJ
ejpam-3383	1	11	and	and	CCONJ
ejpam-3383	1	12	applied	apply	VERB
ejpam-3383	1	13	mathematics	mathematic	NOUN
ejpam-3383	1	14	vol	vol	NOUN
ejpam-3383	1	15	.	.	PROPN
ejpam-3383	2	1	12	12	NUM
ejpam-3383	2	2	,	,	PUNCT
ejpam-3383	2	3	no	no	INTJ
ejpam-3383	2	4	.	.	NOUN
ejpam-3383	2	5	1	1	NUM
ejpam-3383	2	6	,	,	PUNCT
ejpam-3383	2	7	2019	2019	NUM
ejpam-3383	2	8	,	,	PUNCT
ejpam-3383	2	9	208	208	NUM
ejpam-3383	2	10	-	-	SYM
ejpam-3383	2	11	225	225	NUM
ejpam-3383	2	12	issn	issn	PROPN
ejpam-3383	2	13	1307	1307	NUM
ejpam-3383	2	14	-	-	SYM
ejpam-3383	2	15	5543	5543	NUM
ejpam-3383	2	16	–	–	PUNCT
ejpam-3383	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-3383	2	18	published	publish	VERB
ejpam-3383	2	19	by	by	ADP
ejpam-3383	2	20	new	new	PROPN
ejpam-3383	2	21	york	york	PROPN
ejpam-3383	2	22	business	business	PROPN
ejpam-3383	2	23	global	global	PROPN
ejpam-3383	2	24	an	an	DET
ejpam-3383	2	25	application	application	NOUN
ejpam-3383	2	26	of	of	ADP
ejpam-3383	2	27	le	le	PROPN
ejpam-3383	2	28	-	-	NOUN
ejpam-3383	2	29	semigroup	semigroup	PROPN
ejpam-3383	2	30	techniques	technique	NOUN
ejpam-3383	2	31	to	to	ADP
ejpam-3383	2	32	semigroups	semigroup	NOUN
ejpam-3383	2	33	,	,	PUNCT
ejpam-3383	2	34	γ	γ	NOUN
ejpam-3383	2	35	-	-	PUNCT
ejpam-3383	2	36	semigroups	semigroup	NOUN
ejpam-3383	2	37	and	and	CCONJ
ejpam-3383	2	38	to	to	ADP
ejpam-3383	2	39	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	2	40	niovi	niovi	PROPN
ejpam-3383	2	41	kehayopulu	kehayopulu	ADJ
ejpam-3383	2	42	abstract	abstract	NOUN
ejpam-3383	2	43	.	.	PUNCT
ejpam-3383	3	1	an	an	DET
ejpam-3383	3	2	le	le	PROPN
ejpam-3383	3	3	-	-	PUNCT
ejpam-3383	3	4	semigroup	semigroup	PROPN
ejpam-3383	3	5	,	,	PUNCT
ejpam-3383	3	6	is	be	AUX
ejpam-3383	3	7	a	a	DET
ejpam-3383	3	8	semigroup	semigroup	NOUN
ejpam-3383	3	9	s	s	NOUN
ejpam-3383	3	10	at	at	ADP
ejpam-3383	3	11	the	the	DET
ejpam-3383	3	12	same	same	ADJ
ejpam-3383	3	13	time	time	NOUN
ejpam-3383	3	14	a	a	DET
ejpam-3383	3	15	lattice	lattice	NOUN
ejpam-3383	3	16	with	with	ADP
ejpam-3383	3	17	a	a	DET
ejpam-3383	3	18	greatest	great	ADJ
ejpam-3383	3	19	element	element	NOUN
ejpam-3383	3	20	e	e	NOUN
ejpam-3383	3	21	(	(	PUNCT
ejpam-3383	3	22	e	e	X
ejpam-3383	3	23	≥	≥	NOUN
ejpam-3383	3	24	a	a	PRON
ejpam-3383	3	25	for	for	ADP
ejpam-3383	3	26	every	every	DET
ejpam-3383	3	27	a	a	DET
ejpam-3383	3	28	∈	∈	PROPN
ejpam-3383	3	29	s	s	NOUN
ejpam-3383	3	30	)	)	PUNCT
ejpam-3383	3	31	such	such	ADJ
ejpam-3383	3	32	that	that	SCONJ
ejpam-3383	3	33	a(b	a(b	PROPN
ejpam-3383	3	34	∨	∨	NUM
ejpam-3383	3	35	c	c	NOUN
ejpam-3383	3	36	)	)	PUNCT
ejpam-3383	3	37	=	=	SYM
ejpam-3383	3	38	ab	ab	PROPN
ejpam-3383	3	39	∨	∨	NUM
ejpam-3383	3	40	ac	ac	PROPN
ejpam-3383	3	41	and	and	CCONJ
ejpam-3383	3	42	(	(	PUNCT
ejpam-3383	3	43	a	a	DET
ejpam-3383	3	44	∨	∨	NOUN
ejpam-3383	3	45	b)c	b)c	X
ejpam-3383	3	46	=	=	PRON
ejpam-3383	3	47	ac	ac	PROPN
ejpam-3383	3	48	∨	∨	PROPN
ejpam-3383	3	49	bc	bc	PROPN
ejpam-3383	3	50	for	for	ADP
ejpam-3383	3	51	all	all	DET
ejpam-3383	3	52	a	a	DET
ejpam-3383	3	53	,	,	PUNCT
ejpam-3383	3	54	b	b	NOUN
ejpam-3383	3	55	,	,	PUNCT
ejpam-3383	3	56	c	c	PROPN
ejpam-3383	3	57	∈	∈	PROPN
ejpam-3383	3	58	s.	s.	PROPN
ejpam-3383	3	59	if	if	SCONJ
ejpam-3383	3	60	s	s	X
ejpam-3383	3	61	is	be	AUX
ejpam-3383	3	62	not	not	PART
ejpam-3383	3	63	a	a	DET
ejpam-3383	3	64	lattice	lattice	NOUN
ejpam-3383	3	65	but	but	CCONJ
ejpam-3383	3	66	only	only	ADV
ejpam-3383	3	67	an	an	DET
ejpam-3383	3	68	upper	upper	ADJ
ejpam-3383	3	69	semilattice	semilattice	NOUN
ejpam-3383	3	70	(	(	PUNCT
ejpam-3383	3	71	∨-semilattice	∨-semilattice	PROPN
ejpam-3383	3	72	)	)	PUNCT
ejpam-3383	3	73	,	,	PUNCT
ejpam-3383	3	74	then	then	ADV
ejpam-3383	3	75	is	be	AUX
ejpam-3383	3	76	called	call	VERB
ejpam-3383	3	77	∨e	∨e	NOUN
ejpam-3383	3	78	-	-	PUNCT
ejpam-3383	3	79	semigroup	semigroup	NOUN
ejpam-3383	3	80	.	.	PUNCT
ejpam-3383	4	1	a	a	DET
ejpam-3383	4	2	poe	poe	PROPN
ejpam-3383	4	3	-	-	PUNCT
ejpam-3383	4	4	semigroup	semigroup	PROPN
ejpam-3383	4	5	is	be	AUX
ejpam-3383	4	6	a	a	DET
ejpam-3383	4	7	semigroup	semigroup	NOUN
ejpam-3383	4	8	s	s	NOUN
ejpam-3383	4	9	at	at	ADP
ejpam-3383	4	10	the	the	DET
ejpam-3383	4	11	same	same	ADJ
ejpam-3383	4	12	time	time	NOUN
ejpam-3383	4	13	an	an	DET
ejpam-3383	4	14	ordered	order	VERB
ejpam-3383	4	15	set	set	NOUN
ejpam-3383	4	16	with	with	ADP
ejpam-3383	4	17	a	a	DET
ejpam-3383	4	18	greatest	great	ADJ
ejpam-3383	4	19	element	element	NOUN
ejpam-3383	4	20	e	e	NOUN
ejpam-3383	4	21	such	such	ADJ
ejpam-3383	4	22	that	that	SCONJ
ejpam-3383	4	23	a	a	DET
ejpam-3383	4	24	≤	≤	PROPN
ejpam-3383	4	25	b	b	NOUN
ejpam-3383	4	26	implies	imply	VERB
ejpam-3383	4	27	ac	ac	PROPN
ejpam-3383	4	28	≤	≤	PUNCT
ejpam-3383	4	29	bc	bc	PROPN
ejpam-3383	4	30	and	and	CCONJ
ejpam-3383	4	31	ca	can	AUX
ejpam-3383	4	32	≤	≤	NUM
ejpam-3383	4	33	cb	cb	NOUN
ejpam-3383	4	34	for	for	ADP
ejpam-3383	4	35	all	all	DET
ejpam-3383	4	36	c	c	PROPN
ejpam-3383	4	37	∈	∈	PROPN
ejpam-3383	4	38	s.	s.	PROPN
ejpam-3383	4	39	every	every	DET
ejpam-3383	4	40	∨e	∨e	PROPN
ejpam-3383	4	41	-	-	PUNCT
ejpam-3383	4	42	semigroup	semigroup	PROPN
ejpam-3383	4	43	is	be	AUX
ejpam-3383	4	44	a	a	DET
ejpam-3383	4	45	poe	poe	PROPN
ejpam-3383	4	46	-	-	PUNCT
ejpam-3383	4	47	semigroup	semigroup	NOUN
ejpam-3383	4	48	.	.	PUNCT
ejpam-3383	5	1	if	if	SCONJ
ejpam-3383	5	2	s	s	PROPN
ejpam-3383	5	3	is	be	AUX
ejpam-3383	5	4	a	a	DET
ejpam-3383	5	5	semigroup	semigroup	NOUN
ejpam-3383	5	6	or	or	CCONJ
ejpam-3383	5	7	a	a	DET
ejpam-3383	5	8	γ	γ	NOUN
ejpam-3383	5	9	-	-	PUNCT
ejpam-3383	5	10	semigroup	semigroup	NOUN
ejpam-3383	5	11	,	,	PUNCT
ejpam-3383	5	12	then	then	ADV
ejpam-3383	5	13	the	the	DET
ejpam-3383	5	14	set	set	ADJ
ejpam-3383	5	15	p(s	p(s	NOUN
ejpam-3383	5	16	)	)	PUNCT
ejpam-3383	5	17	of	of	ADP
ejpam-3383	5	18	all	all	DET
ejpam-3383	5	19	subsets	subset	NOUN
ejpam-3383	5	20	of	of	ADP
ejpam-3383	5	21	s	s	PROPN
ejpam-3383	5	22	is	be	AUX
ejpam-3383	5	23	an	an	DET
ejpam-3383	5	24	le	le	NOUN
ejpam-3383	5	25	-	-	NOUN
ejpam-3383	5	26	semigroup	semigroup	NOUN
ejpam-3383	5	27	.	.	PUNCT
ejpam-3383	6	1	if	if	SCONJ
ejpam-3383	6	2	s	s	PROPN
ejpam-3383	6	3	is	be	AUX
ejpam-3383	6	4	an	an	DET
ejpam-3383	6	5	hypersemigroup	hypersemigroup	NOUN
ejpam-3383	6	6	,	,	PUNCT
ejpam-3383	6	7	then	then	ADV
ejpam-3383	6	8	the	the	DET
ejpam-3383	6	9	set	set	NOUN
ejpam-3383	6	10	p∗(s	p∗(s	NOUN
ejpam-3383	6	11	)	)	PUNCT
ejpam-3383	6	12	of	of	ADP
ejpam-3383	6	13	all	all	DET
ejpam-3383	6	14	nonempty	nonempty	ADJ
ejpam-3383	6	15	subsets	subset	NOUN
ejpam-3383	6	16	of	of	ADP
ejpam-3383	6	17	s	s	PROPN
ejpam-3383	6	18	is	be	AUX
ejpam-3383	6	19	an	an	DET
ejpam-3383	6	20	le	le	NOUN
ejpam-3383	6	21	-	-	NOUN
ejpam-3383	6	22	semigroup	semigroup	NOUN
ejpam-3383	6	23	.	.	PUNCT
ejpam-3383	7	1	so	so	ADV
ejpam-3383	7	2	all	all	DET
ejpam-3383	7	3	the	the	DET
ejpam-3383	7	4	results	result	NOUN
ejpam-3383	7	5	of	of	ADP
ejpam-3383	7	6	le	le	X
ejpam-3383	7	7	-	-	PUNCT
ejpam-3383	7	8	semigroups	semigroup	NOUN
ejpam-3383	7	9	,	,	PUNCT
ejpam-3383	7	10	∨e	∨e	NOUN
ejpam-3383	7	11	-	-	PUNCT
ejpam-3383	7	12	semigroups	semigroup	NOUN
ejpam-3383	7	13	and	and	CCONJ
ejpam-3383	7	14	poe	poe	PROPN
ejpam-3383	7	15	-	-	PUNCT
ejpam-3383	7	16	semigroups	semigroup	NOUN
ejpam-3383	7	17	based	base	VERB
ejpam-3383	7	18	on	on	ADP
ejpam-3383	7	19	ideal	ideal	ADJ
ejpam-3383	7	20	elements	element	NOUN
ejpam-3383	7	21	,	,	PUNCT
ejpam-3383	7	22	automatically	automatically	ADV
ejpam-3383	7	23	hold	hold	VERB
ejpam-3383	7	24	for	for	ADP
ejpam-3383	7	25	semigroups	semigroup	NOUN
ejpam-3383	7	26	,	,	PUNCT
ejpam-3383	7	27	γ	γ	X
ejpam-3383	7	28	-	-	PUNCT
ejpam-3383	7	29	semigroups	semigroup	NOUN
ejpam-3383	7	30	and	and	CCONJ
ejpam-3383	7	31	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	7	32	.	.	PUNCT
ejpam-3383	8	1	this	this	PRON
ejpam-3383	8	2	is	be	AUX
ejpam-3383	8	3	not	not	PART
ejpam-3383	8	4	the	the	DET
ejpam-3383	8	5	case	case	NOUN
ejpam-3383	8	6	for	for	ADP
ejpam-3383	8	7	ordered	order	VERB
ejpam-3383	8	8	γ	γ	NOUN
ejpam-3383	8	9	-	-	PUNCT
ejpam-3383	8	10	semigroups	semigroup	NOUN
ejpam-3383	8	11	or	or	CCONJ
ejpam-3383	8	12	ordered	order	VERB
ejpam-3383	8	13	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	8	14	;	;	PUNCT
ejpam-3383	8	15	however	however	ADV
ejpam-3383	8	16	the	the	DET
ejpam-3383	8	17	main	main	ADJ
ejpam-3383	8	18	idea	idea	NOUN
ejpam-3383	8	19	,	,	PUNCT
ejpam-3383	8	20	even	even	ADV
ejpam-3383	8	21	in	in	ADP
ejpam-3383	8	22	these	these	DET
ejpam-3383	8	23	cases	case	NOUN
ejpam-3383	8	24	,	,	PUNCT
ejpam-3383	8	25	comes	come	VERB
ejpam-3383	8	26	from	from	ADP
ejpam-3383	8	27	the	the	DET
ejpam-3383	8	28	le	le	X
ejpam-3383	8	29	(	(	PUNCT
ejpam-3383	8	30	∨e)-semigroups	∨e)-semigroup	NOUN
ejpam-3383	8	31	.	.	PUNCT
ejpam-3383	9	1	as	as	ADP
ejpam-3383	9	2	an	an	DET
ejpam-3383	9	3	example	example	NOUN
ejpam-3383	9	4	,	,	PUNCT
ejpam-3383	9	5	we	we	PRON
ejpam-3383	9	6	study	study	VERB
ejpam-3383	9	7	the	the	DET
ejpam-3383	9	8	weakly	weakly	ADJ
ejpam-3383	9	9	prime	prime	ADJ
ejpam-3383	9	10	ideal	ideal	ADJ
ejpam-3383	9	11	elements	element	NOUN
ejpam-3383	9	12	of	of	ADP
ejpam-3383	9	13	a	a	DET
ejpam-3383	9	14	∨e	∨e	NOUN
ejpam-3383	9	15	-	-	PUNCT
ejpam-3383	9	16	semigroup	semigroup	NOUN
ejpam-3383	9	17	and	and	CCONJ
ejpam-3383	9	18	their	their	PRON
ejpam-3383	9	19	role	role	NOUN
ejpam-3383	9	20	to	to	ADP
ejpam-3383	9	21	the	the	DET
ejpam-3383	9	22	different	different	ADJ
ejpam-3383	9	23	type	type	NOUN
ejpam-3383	9	24	of	of	ADP
ejpam-3383	9	25	semigroups	semigroup	NOUN
ejpam-3383	9	26	mentioned	mention	VERB
ejpam-3383	9	27	above	above	ADV
ejpam-3383	9	28	.	.	PUNCT
ejpam-3383	10	1	2010	2010	NUM
ejpam-3383	10	2	mathematics	mathematic	NOUN
ejpam-3383	10	3	subject	subject	NOUN
ejpam-3383	10	4	classifications	classification	NOUN
ejpam-3383	10	5	:	:	PUNCT
ejpam-3383	10	6	06f05	06f05	NUM
ejpam-3383	10	7	,	,	PUNCT
ejpam-3383	10	8	06f99	06f99	NUM
ejpam-3383	10	9	,	,	PUNCT
ejpam-3383	10	10	20m99	20m99	NUM
ejpam-3383	10	11	,	,	PUNCT
ejpam-3383	10	12	20m10	20m10	NUM
ejpam-3383	10	13	key	key	ADJ
ejpam-3383	10	14	words	word	NOUN
ejpam-3383	10	15	and	and	CCONJ
ejpam-3383	10	16	phrases	phrase	NOUN
ejpam-3383	10	17	:	:	PUNCT
ejpam-3383	10	18	lattice	lattice	PROPN
ejpam-3383	10	19	ordered	order	VERB
ejpam-3383	10	20	semigroup	semigroup	PROPN
ejpam-3383	10	21	,	,	PUNCT
ejpam-3383	10	22	le	le	PROPN
ejpam-3383	10	23	-	-	PUNCT
ejpam-3383	10	24	semigroup	semigroup	PROPN
ejpam-3383	10	25	,	,	PUNCT
ejpam-3383	10	26	(	(	PUNCT
ejpam-3383	10	27	ordered	order	VERB
ejpam-3383	10	28	)	)	PUNCT
ejpam-3383	10	29	γ	γ	PROPN
ejpam-3383	10	30	-	-	PUNCT
ejpam-3383	10	31	semigroup	semigroup	NOUN
ejpam-3383	10	32	,	,	PUNCT
ejpam-3383	10	33	(	(	PUNCT
ejpam-3383	10	34	ordered	order	VERB
ejpam-3383	10	35	)	)	PUNCT
ejpam-3383	10	36	hypersemigroup	hypersemigroup	NOUN
ejpam-3383	10	37	,	,	PUNCT
ejpam-3383	10	38	weakly	weakly	ADJ
ejpam-3383	10	39	prime	prime	ADJ
ejpam-3383	10	40	,	,	PUNCT
ejpam-3383	10	41	weakly	weakly	ADJ
ejpam-3383	10	42	semiprime	semiprime	NOUN
ejpam-3383	10	43	,	,	PUNCT
ejpam-3383	10	44	prime	prime	ADJ
ejpam-3383	10	45	subset	subset	NOUN
ejpam-3383	10	46	,	,	PUNCT
ejpam-3383	10	47	right	right	INTJ
ejpam-3383	10	48	(	(	PUNCT
ejpam-3383	10	49	left	left	ADJ
ejpam-3383	10	50	)	)	PUNCT
ejpam-3383	10	51	ideal	ideal	ADJ
ejpam-3383	10	52	element	element	NOUN
ejpam-3383	10	53	,	,	PUNCT
ejpam-3383	10	54	right	right	INTJ
ejpam-3383	10	55	(	(	PUNCT
ejpam-3383	10	56	left	left	ADJ
ejpam-3383	10	57	)	)	PUNCT
ejpam-3383	10	58	ideal	ideal	NOUN
ejpam-3383	10	59	1	1	NUM
ejpam-3383	10	60	.	.	PUNCT
ejpam-3383	11	1	introduction	introduction	NOUN
ejpam-3383	11	2	if	if	SCONJ
ejpam-3383	11	3	s	s	VERB
ejpam-3383	11	4	is	be	AUX
ejpam-3383	11	5	a	a	DET
ejpam-3383	11	6	semigroup	semigroup	NOUN
ejpam-3383	11	7	or	or	CCONJ
ejpam-3383	11	8	a	a	DET
ejpam-3383	11	9	γ	γ	NOUN
ejpam-3383	11	10	-	-	PUNCT
ejpam-3383	11	11	semigroup	semigroup	NOUN
ejpam-3383	11	12	,	,	PUNCT
ejpam-3383	11	13	then	then	ADV
ejpam-3383	11	14	the	the	DET
ejpam-3383	11	15	set	set	NOUN
ejpam-3383	11	16	of	of	ADP
ejpam-3383	11	17	(	(	PUNCT
ejpam-3383	11	18	all	all	DET
ejpam-3383	11	19	)	)	PUNCT
ejpam-3383	11	20	subsets	subset	NOUN
ejpam-3383	11	21	of	of	ADP
ejpam-3383	11	22	s	s	PROPN
ejpam-3383	11	23	is	be	AUX
ejpam-3383	11	24	a	a	DET
ejpam-3383	11	25	poesemigroup	poesemigroup	NOUN
ejpam-3383	11	26	,	,	PUNCT
ejpam-3383	11	27	a	a	DET
ejpam-3383	11	28	∨e	∨e	NOUN
ejpam-3383	11	29	-	-	PUNCT
ejpam-3383	11	30	semigroup	semigroup	NOUN
ejpam-3383	11	31	and	and	CCONJ
ejpam-3383	11	32	an	an	DET
ejpam-3383	11	33	le	le	X
ejpam-3383	11	34	-	-	NOUN
ejpam-3383	11	35	semigroup	semigroup	NOUN
ejpam-3383	11	36	.	.	PUNCT
ejpam-3383	12	1	if	if	SCONJ
ejpam-3383	12	2	s	s	PROPN
ejpam-3383	12	3	is	be	AUX
ejpam-3383	12	4	an	an	DET
ejpam-3383	12	5	hypersemigroup	hypersemigroup	NOUN
ejpam-3383	12	6	,	,	PUNCT
ejpam-3383	12	7	then	then	ADV
ejpam-3383	12	8	the	the	DET
ejpam-3383	12	9	set	set	NOUN
ejpam-3383	12	10	of	of	ADP
ejpam-3383	12	11	(	(	PUNCT
ejpam-3383	12	12	all	all	ADV
ejpam-3383	12	13	)	)	PUNCT
ejpam-3383	12	14	nonempty	nonempty	ADJ
ejpam-3383	12	15	subsets	subset	NOUN
ejpam-3383	12	16	of	of	ADP
ejpam-3383	12	17	s	s	PROPN
ejpam-3383	12	18	is	be	AUX
ejpam-3383	12	19	a	a	DET
ejpam-3383	12	20	poe	poe	PROPN
ejpam-3383	12	21	-	-	PUNCT
ejpam-3383	12	22	semigroup	semigroup	PROPN
ejpam-3383	12	23	,	,	PUNCT
ejpam-3383	12	24	a	a	DET
ejpam-3383	12	25	∨e	∨e	NOUN
ejpam-3383	12	26	-	-	PUNCT
ejpam-3383	12	27	semigroup	semigroup	NOUN
ejpam-3383	12	28	and	and	CCONJ
ejpam-3383	12	29	an	an	DET
ejpam-3383	12	30	le	le	X
ejpam-3383	12	31	-	-	NOUN
ejpam-3383	12	32	semigroup	semigroup	NOUN
ejpam-3383	12	33	.	.	PUNCT
ejpam-3383	13	1	every	every	DET
ejpam-3383	13	2	le	le	PROPN
ejpam-3383	13	3	-	-	PUNCT
ejpam-3383	13	4	semigroup	semigroup	PROPN
ejpam-3383	13	5	is	be	AUX
ejpam-3383	13	6	a	a	DET
ejpam-3383	13	7	∨e	∨e	NOUN
ejpam-3383	13	8	-	-	PUNCT
ejpam-3383	13	9	semigroup	semigroup	NOUN
ejpam-3383	13	10	and	and	CCONJ
ejpam-3383	13	11	every	every	DET
ejpam-3383	13	12	∨e	∨e	NOUN
ejpam-3383	13	13	-	-	PUNCT
ejpam-3383	13	14	semigroup	semigroup	PROPN
ejpam-3383	13	15	is	be	AUX
ejpam-3383	13	16	a	a	DET
ejpam-3383	13	17	poe	poe	PROPN
ejpam-3383	13	18	-	-	PUNCT
ejpam-3383	13	19	semigroup	semigroup	PROPN
ejpam-3383	13	20	.	.	PUNCT
ejpam-3383	14	1	so	so	ADV
ejpam-3383	14	2	many	many	ADJ
ejpam-3383	14	3	results	result	NOUN
ejpam-3383	14	4	on	on	ADP
ejpam-3383	14	5	γ	γ	NOUN
ejpam-3383	14	6	-	-	PUNCT
ejpam-3383	14	7	semigroups	semigroup	NOUN
ejpam-3383	14	8	or	or	CCONJ
ejpam-3383	14	9	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	14	10	do	do	AUX
ejpam-3383	14	11	not	not	PART
ejpam-3383	14	12	need	need	VERB
ejpam-3383	14	13	any	any	DET
ejpam-3383	14	14	proof	proof	NOUN
ejpam-3383	14	15	,	,	PUNCT
ejpam-3383	14	16	unless	unless	SCONJ
ejpam-3383	14	17	we	we	PRON
ejpam-3383	14	18	would	would	AUX
ejpam-3383	14	19	like	like	VERB
ejpam-3383	14	20	to	to	PART
ejpam-3383	14	21	know	know	VERB
ejpam-3383	14	22	how	how	SCONJ
ejpam-3383	14	23	an	an	DET
ejpam-3383	14	24	independent	independent	ADJ
ejpam-3383	14	25	proof	proof	NOUN
ejpam-3383	14	26	on	on	ADP
ejpam-3383	14	27	these	these	DET
ejpam-3383	14	28	structures	structure	NOUN
ejpam-3383	14	29	works	work	VERB
ejpam-3383	14	30	.	.	PUNCT
ejpam-3383	15	1	this	this	PRON
ejpam-3383	15	2	is	be	AUX
ejpam-3383	15	3	not	not	PART
ejpam-3383	15	4	the	the	DET
ejpam-3383	15	5	case	case	NOUN
ejpam-3383	15	6	for	for	ADP
ejpam-3383	15	7	ordered	order	VERB
ejpam-3383	15	8	semigroups	semigroup	NOUN
ejpam-3383	15	9	,	,	PUNCT
ejpam-3383	15	10	ordered	order	VERB
ejpam-3383	15	11	γ	γ	NOUN
ejpam-3383	15	12	-	-	PUNCT
ejpam-3383	15	13	semigroups	semigroup	NOUN
ejpam-3383	15	14	or	or	CCONJ
ejpam-3383	15	15	ordered	order	VERB
ejpam-3383	15	16	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	15	17	;	;	PUNCT
ejpam-3383	15	18	however	however	ADV
ejpam-3383	15	19	the	the	DET
ejpam-3383	15	20	main	main	ADJ
ejpam-3383	15	21	idea	idea	NOUN
ejpam-3383	15	22	,	,	PUNCT
ejpam-3383	15	23	even	even	ADV
ejpam-3383	15	24	in	in	ADP
ejpam-3383	15	25	these	these	DET
ejpam-3383	15	26	cases	case	NOUN
ejpam-3383	15	27	,	,	PUNCT
ejpam-3383	15	28	comes	come	VERB
ejpam-3383	15	29	from	from	ADP
ejpam-3383	15	30	the	the	DET
ejpam-3383	15	31	le	le	X
ejpam-3383	15	32	(	(	PUNCT
ejpam-3383	15	33	∨e	∨e	PROPN
ejpam-3383	15	34	)	)	PUNCT
ejpam-3383	15	35	or	or	CCONJ
ejpam-3383	15	36	poe	poe	PROPN
ejpam-3383	15	37	-	-	PUNCT
ejpam-3383	15	38	semigroups	semigroup	NOUN
ejpam-3383	15	39	.	.	PUNCT
ejpam-3383	16	1	in	in	ADP
ejpam-3383	16	2	the	the	DET
ejpam-3383	16	3	present	present	ADJ
ejpam-3383	16	4	paper	paper	NOUN
ejpam-3383	16	5	we	we	PRON
ejpam-3383	16	6	first	first	ADV
ejpam-3383	16	7	characterize	characterize	VERB
ejpam-3383	16	8	the	the	DET
ejpam-3383	16	9	weakly	weakly	ADJ
ejpam-3383	16	10	prime	prime	ADJ
ejpam-3383	16	11	and	and	CCONJ
ejpam-3383	16	12	weakly	weakly	ADJ
ejpam-3383	16	13	semiprime	semiprime	NOUN
ejpam-3383	16	14	ideal	ideal	NOUN
ejpam-3383	16	15	elements	element	NOUN
ejpam-3383	16	16	of	of	ADP
ejpam-3383	16	17	a	a	DET
ejpam-3383	16	18	∨e	∨e	NOUN
ejpam-3383	16	19	-	-	PUNCT
ejpam-3383	16	20	semigroup	semigroup	NOUN
ejpam-3383	16	21	s	s	X
ejpam-3383	16	22	in	in	ADP
ejpam-3383	16	23	terms	term	NOUN
ejpam-3383	16	24	of	of	ADP
ejpam-3383	16	25	right	right	NOUN
ejpam-3383	16	26	and	and	CCONJ
ejpam-3383	16	27	left	leave	VERB
ejpam-3383	16	28	ideal	ideal	ADJ
ejpam-3383	16	29	elements	element	NOUN
ejpam-3383	16	30	of	of	ADP
ejpam-3383	16	31	s	s	PRON
ejpam-3383	16	32	and	and	CCONJ
ejpam-3383	16	33	show	show	VERB
ejpam-3383	16	34	that	that	SCONJ
ejpam-3383	16	35	doi	doi	NOUN
ejpam-3383	16	36	:	:	PUNCT
ejpam-3383	16	37	https://doi.org/10.29020/nybg.ejpam.v12i1.3383	https://doi.org/10.29020/nybg.ejpam.v12i1.3383	NOUN
ejpam-3383	16	38	email	email	NOUN
ejpam-3383	16	39	address	address	NOUN
ejpam-3383	16	40	:	:	PUNCT
ejpam-3383	16	41	nkehayop@math.uoa.gr	nkehayop@math.uoa.gr	ADV
ejpam-3383	16	42	(	(	PUNCT
ejpam-3383	16	43	n.	n.	PROPN
ejpam-3383	16	44	kehayopulu	kehayopulu	PROPN
ejpam-3383	16	45	)	)	PUNCT
ejpam-3383	16	46	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-3383	17	1	208	208	NUM
ejpam-3383	17	2	c	c	X
ejpam-3383	17	3	©	©	PROPN
ejpam-3383	17	4	2019	2019	NUM
ejpam-3383	17	5	ejpam	ejpam	NOUN
ejpam-3383	17	6	all	all	DET
ejpam-3383	17	7	rights	right	NOUN
ejpam-3383	17	8	reserved	reserve	VERB
ejpam-3383	17	9	.	.	PUNCT
ejpam-3383	18	1	n.	n.	PROPN
ejpam-3383	18	2	kehayopulu	kehayopulu	PROPN
ejpam-3383	18	3	/	/	SYM
ejpam-3383	18	4	eur	eur	PROPN
ejpam-3383	18	5	.	.	PUNCT
ejpam-3383	19	1	j.	j.	PROPN
ejpam-3383	19	2	pure	pure	PROPN
ejpam-3383	19	3	appl	appl	PROPN
ejpam-3383	19	4	.	.	PROPN
ejpam-3383	19	5	math	math	PROPN
ejpam-3383	19	6	,	,	PUNCT
ejpam-3383	19	7	12	12	NUM
ejpam-3383	19	8	(	(	PUNCT
ejpam-3383	19	9	1	1	NUM
ejpam-3383	19	10	)	)	PUNCT
ejpam-3383	19	11	(	(	PUNCT
ejpam-3383	19	12	2019	2019	NUM
ejpam-3383	19	13	)	)	PUNCT
ejpam-3383	19	14	,	,	PUNCT
ejpam-3383	19	15	208	208	NUM
ejpam-3383	19	16	-	-	SYM
ejpam-3383	19	17	225	225	NUM
ejpam-3383	19	18	209	209	NUM
ejpam-3383	19	19	one	one	NOUN
ejpam-3383	19	20	can	can	AUX
ejpam-3383	19	21	get	get	VERB
ejpam-3383	19	22	the	the	DET
ejpam-3383	19	23	corresponding	corresponding	ADJ
ejpam-3383	19	24	characterizations	characterization	NOUN
ejpam-3383	19	25	of	of	ADP
ejpam-3383	19	26	weakly	weakly	ADJ
ejpam-3383	19	27	prime	prime	ADJ
ejpam-3383	19	28	and	and	CCONJ
ejpam-3383	19	29	weakly	weakly	ADJ
ejpam-3383	19	30	semiprime	semiprime	NOUN
ejpam-3383	19	31	ideals	ideal	NOUN
ejpam-3383	19	32	of	of	ADP
ejpam-3383	19	33	semigroups	semigroup	NOUN
ejpam-3383	19	34	,	,	PUNCT
ejpam-3383	19	35	γ	γ	NOUN
ejpam-3383	19	36	-	-	PUNCT
ejpam-3383	19	37	semigroups	semigroup	NOUN
ejpam-3383	19	38	or	or	CCONJ
ejpam-3383	19	39	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	19	40	(	(	PUNCT
ejpam-3383	19	41	in	in	ADP
ejpam-3383	19	42	terms	term	NOUN
ejpam-3383	19	43	of	of	ADP
ejpam-3383	19	44	right	right	ADJ
ejpam-3383	19	45	and	and	CCONJ
ejpam-3383	19	46	left	leave	VERB
ejpam-3383	19	47	ideals	ideal	NOUN
ejpam-3383	19	48	)	)	PUNCT
ejpam-3383	19	49	,	,	PUNCT
ejpam-3383	19	50	as	as	ADP
ejpam-3383	19	51	corollaries	corollary	NOUN
ejpam-3383	19	52	.	.	PUNCT
ejpam-3383	20	1	then	then	ADV
ejpam-3383	20	2	we	we	PRON
ejpam-3383	20	3	examine	examine	VERB
ejpam-3383	20	4	the	the	DET
ejpam-3383	20	5	same	same	ADJ
ejpam-3383	20	6	results	result	NOUN
ejpam-3383	20	7	in	in	ADP
ejpam-3383	20	8	case	case	NOUN
ejpam-3383	20	9	of	of	ADP
ejpam-3383	20	10	an	an	DET
ejpam-3383	20	11	ordered	order	VERB
ejpam-3383	20	12	semigroup	semigroup	NOUN
ejpam-3383	20	13	,	,	PUNCT
ejpam-3383	20	14	an	an	DET
ejpam-3383	20	15	ordered	order	VERB
ejpam-3383	20	16	γ	γ	NOUN
ejpam-3383	20	17	-	-	PUNCT
ejpam-3383	20	18	semigroup	semigroup	NOUN
ejpam-3383	20	19	and	and	CCONJ
ejpam-3383	20	20	an	an	DET
ejpam-3383	20	21	ordered	order	VERB
ejpam-3383	20	22	hypersemigroup	hypersemigroup	NOUN
ejpam-3383	20	23	.	.	PUNCT
ejpam-3383	21	1	for	for	ADP
ejpam-3383	21	2	weakly	weakly	ADJ
ejpam-3383	21	3	prime	prime	ADJ
ejpam-3383	21	4	ideals	ideal	NOUN
ejpam-3383	21	5	of	of	ADP
ejpam-3383	21	6	rings	ring	NOUN
ejpam-3383	21	7	see	see	VERB
ejpam-3383	21	8	,	,	PUNCT
ejpam-3383	21	9	for	for	ADP
ejpam-3383	21	10	example	example	NOUN
ejpam-3383	21	11	,	,	PUNCT
ejpam-3383	21	12	[	[	X
ejpam-3383	21	13	5	5	NUM
ejpam-3383	21	14	]	]	PUNCT
ejpam-3383	21	15	.	.	PUNCT
ejpam-3383	22	1	this	this	PRON
ejpam-3383	22	2	is	be	AUX
ejpam-3383	22	3	from	from	ADP
ejpam-3383	22	4	the	the	DET
ejpam-3383	22	5	first	first	ADJ
ejpam-3383	22	6	chapter	chapter	NOUN
ejpam-3383	22	7	“	"	PUNCT
ejpam-3383	22	8	what	what	PRON
ejpam-3383	22	9	can	can	AUX
ejpam-3383	22	10	lattices	lattice	NOUN
ejpam-3383	22	11	do	do	VERB
ejpam-3383	22	12	for	for	ADP
ejpam-3383	22	13	you	you	PRON
ejpam-3383	22	14	?	?	PUNCT
ejpam-3383	22	15	”	"	PUNCT
ejpam-3383	22	16	by	by	ADP
ejpam-3383	22	17	garrett	garrett	PROPN
ejpam-3383	22	18	birkhoff	birkhoff	NOUN
ejpam-3383	22	19	in	in	ADP
ejpam-3383	22	20	[	[	X
ejpam-3383	22	21	trends	trend	NOUN
ejpam-3383	22	22	in	in	ADP
ejpam-3383	22	23	lattice	lattice	NOUN
ejpam-3383	22	24	theory	theory	NOUN
ejpam-3383	22	25	.	.	PUNCT
ejpam-3383	23	1	contributors	contributor	NOUN
ejpam-3383	23	2	:	:	PUNCT
ejpam-3383	23	3	garrett	garrett	PROPN
ejpam-3383	23	4	birkhoff	birkhoff	PROPN
ejpam-3383	23	5	,	,	PUNCT
ejpam-3383	23	6	samuel	samuel	PROPN
ejpam-3383	23	7	s.	s.	PROPN
ejpam-3383	23	8	holland	holland	PROPN
ejpam-3383	23	9	,	,	PUNCT
ejpam-3383	23	10	jr	jr	PROPN
ejpam-3383	23	11	.	.	PROPN
ejpam-3383	23	12	,	,	PUNCT
ejpam-3383	23	13	henry	henry	PROPN
ejpam-3383	23	14	crapo	crapo	PROPN
ejpam-3383	23	15	and	and	CCONJ
ejpam-3383	23	16	gian	gian	PROPN
ejpam-3383	23	17	-	-	PUNCT
ejpam-3383	23	18	carlo	carlo	PROPN
ejpam-3383	23	19	rota	rota	PROPN
ejpam-3383	23	20	,	,	PUNCT
ejpam-3383	23	21	george	george	PROPN
ejpam-3383	23	22	grätzer	grätzer	PROPN
ejpam-3383	23	23	.	.	PUNCT
ejpam-3383	24	1	van	van	PROPN
ejpam-3383	24	2	nostrand	nostrand	PROPN
ejpam-3383	24	3	reinhold	reinhold	PROPN
ejpam-3383	24	4	comp	comp	PROPN
ejpam-3383	24	5	.	.	PUNCT
ejpam-3383	25	1	1970	1970	NUM
ejpam-3383	25	2	]	]	SYM
ejpam-3383	25	3	:	:	PUNCT
ejpam-3383	25	4	“	"	PUNCT
ejpam-3383	25	5	in	in	ADP
ejpam-3383	25	6	general	general	ADJ
ejpam-3383	25	7	,	,	PUNCT
ejpam-3383	25	8	lattice	lattice	PROPN
ejpam-3383	25	9	theory	theory	NOUN
ejpam-3383	25	10	has	have	AUX
ejpam-3383	25	11	helped	help	VERB
ejpam-3383	25	12	to	to	PART
ejpam-3383	25	13	simplify	simplify	VERB
ejpam-3383	25	14	,	,	PUNCT
ejpam-3383	25	15	unify	unify	VERB
ejpam-3383	25	16	and	and	CCONJ
ejpam-3383	25	17	generalize	generalize	VERB
ejpam-3383	25	18	many	many	ADJ
ejpam-3383	25	19	aspects	aspect	NOUN
ejpam-3383	25	20	of	of	ADP
ejpam-3383	25	21	mathematics	mathematic	NOUN
ejpam-3383	25	22	...	...	PUNCT
ejpam-3383	25	23	”	"	PUNCT
ejpam-3383	26	1	(	(	PUNCT
ejpam-3383	26	2	p.	p.	NOUN
ejpam-3383	26	3	1	1	NUM
ejpam-3383	26	4	)	)	PUNCT
ejpam-3383	26	5	;	;	PUNCT
ejpam-3383	26	6	“	"	PUNCT
ejpam-3383	26	7	because	because	SCONJ
ejpam-3383	26	8	of	of	ADP
ejpam-3383	26	9	its	its	PRON
ejpam-3383	26	10	central	central	ADJ
ejpam-3383	26	11	concept	concept	NOUN
ejpam-3383	26	12	,	,	PUNCT
ejpam-3383	26	13	that	that	PRON
ejpam-3383	26	14	of	of	ADP
ejpam-3383	26	15	order	order	NOUN
ejpam-3383	26	16	,	,	PUNCT
ejpam-3383	26	17	intertwines	intertwine	VERB
ejpam-3383	26	18	through	through	ADP
ejpam-3383	26	19	almost	almost	ADV
ejpam-3383	26	20	all	all	PRON
ejpam-3383	26	21	of	of	ADP
ejpam-3383	26	22	mathematics	mathematic	NOUN
ejpam-3383	26	23	...	...	PUNCT
ejpam-3383	26	24	”	"	PUNCT
ejpam-3383	27	1	(	(	PUNCT
ejpam-3383	27	2	p.	p.	NOUN
ejpam-3383	27	3	1	1	NUM
ejpam-3383	27	4	)	)	PUNCT
ejpam-3383	27	5	;	;	PUNCT
ejpam-3383	27	6	“	"	PUNCT
ejpam-3383	27	7	...	...	PUNCT
ejpam-3383	27	8	lattices	lattice	NOUN
ejpam-3383	27	9	can	can	AUX
ejpam-3383	27	10	do	do	VERB
ejpam-3383	27	11	things	thing	NOUN
ejpam-3383	27	12	for	for	ADP
ejpam-3383	27	13	you	you	PRON
ejpam-3383	27	14	no	no	ADV
ejpam-3383	27	15	matter	matter	ADV
ejpam-3383	27	16	what	what	DET
ejpam-3383	27	17	kind	kind	NOUN
ejpam-3383	27	18	of	of	ADP
ejpam-3383	27	19	mathematician	mathematician	NOUN
ejpam-3383	27	20	you	you	PRON
ejpam-3383	27	21	are	be	AUX
ejpam-3383	27	22	!	!	PUNCT
ejpam-3383	27	23	”	"	PUNCT
ejpam-3383	28	1	(	(	PUNCT
ejpam-3383	28	2	p.	p.	NOUN
ejpam-3383	28	3	38	38	NUM
ejpam-3383	28	4	)	)	PUNCT
ejpam-3383	28	5	.	.	PUNCT
ejpam-3383	29	1	2	2	X
ejpam-3383	29	2	.	.	NUM
ejpam-3383	29	3	preliminaries	preliminary	NOUN
ejpam-3383	29	4	an	an	DET
ejpam-3383	29	5	ordered	order	VERB
ejpam-3383	29	6	groupoid	groupoid	NOUN
ejpam-3383	29	7	(	(	PUNCT
ejpam-3383	29	8	po	po	NOUN
ejpam-3383	29	9	-	-	NOUN
ejpam-3383	29	10	groupoid	groupoid	NOUN
ejpam-3383	29	11	)	)	PUNCT
ejpam-3383	29	12	is	be	AUX
ejpam-3383	29	13	a	a	DET
ejpam-3383	29	14	groupoid	groupoid	PROPN
ejpam-3383	29	15	s	s	NOUN
ejpam-3383	29	16	at	at	ADP
ejpam-3383	29	17	the	the	DET
ejpam-3383	29	18	same	same	ADJ
ejpam-3383	29	19	time	time	NOUN
ejpam-3383	29	20	an	an	DET
ejpam-3383	29	21	ordered	order	VERB
ejpam-3383	29	22	set	set	NOUN
ejpam-3383	29	23	such	such	ADJ
ejpam-3383	29	24	that	that	SCONJ
ejpam-3383	29	25	a	a	DET
ejpam-3383	29	26	≤	≤	PROPN
ejpam-3383	29	27	b	b	NOUN
ejpam-3383	29	28	implies	imply	VERB
ejpam-3383	29	29	ac	ac	PROPN
ejpam-3383	29	30	≤	≤	PUNCT
ejpam-3383	29	31	bc	bc	PROPN
ejpam-3383	29	32	and	and	CCONJ
ejpam-3383	29	33	ca	can	AUX
ejpam-3383	29	34	≤	≤	NUM
ejpam-3383	29	35	cb	cb	NOUN
ejpam-3383	29	36	for	for	ADP
ejpam-3383	29	37	all	all	PRON
ejpam-3383	29	38	c	c	PROPN
ejpam-3383	29	39	∈	∈	PROPN
ejpam-3383	29	40	s.	s.	PROPN
ejpam-3383	29	41	a	a	PRON
ejpam-3383	29	42	∨-groupoid	∨-groupoid	PROPN
ejpam-3383	29	43	is	be	AUX
ejpam-3383	29	44	a	a	DET
ejpam-3383	29	45	groupoid	groupoid	NOUN
ejpam-3383	29	46	s	s	NOUN
ejpam-3383	29	47	at	at	ADP
ejpam-3383	29	48	the	the	DET
ejpam-3383	29	49	same	same	ADJ
ejpam-3383	29	50	time	time	NOUN
ejpam-3383	29	51	an	an	DET
ejpam-3383	29	52	upper	upper	ADJ
ejpam-3383	29	53	semilattice	semilattice	NOUN
ejpam-3383	30	1	such	such	ADJ
ejpam-3383	30	2	that	that	SCONJ
ejpam-3383	30	3	(	(	PUNCT
ejpam-3383	30	4	a	a	DET
ejpam-3383	30	5	∨	∨	NOUN
ejpam-3383	30	6	b)c	b)c	X
ejpam-3383	30	7	=	=	PRON
ejpam-3383	30	8	ac	ac	PROPN
ejpam-3383	30	9	∨	∨	NUM
ejpam-3383	30	10	bc	bc	PROPN
ejpam-3383	30	11	and	and	CCONJ
ejpam-3383	30	12	a(b	a(b	PROPN
ejpam-3383	30	13	∨	∨	NUM
ejpam-3383	30	14	c	c	NOUN
ejpam-3383	30	15	)	)	PUNCT
ejpam-3383	30	16	=	=	SYM
ejpam-3383	30	17	ab	ab	PROPN
ejpam-3383	30	18	∨	∨	NUM
ejpam-3383	30	19	ac	ac	PROPN
ejpam-3383	30	20	for	for	ADP
ejpam-3383	30	21	all	all	DET
ejpam-3383	30	22	a	a	DET
ejpam-3383	30	23	,	,	PUNCT
ejpam-3383	30	24	b	b	NOUN
ejpam-3383	30	25	,	,	PUNCT
ejpam-3383	30	26	c	c	PROPN
ejpam-3383	30	27	∈	∈	PROPN
ejpam-3383	30	28	s.	s.	PROPN
ejpam-3383	30	29	if	if	SCONJ
ejpam-3383	30	30	s	s	X
ejpam-3383	30	31	is	be	AUX
ejpam-3383	30	32	not	not	PART
ejpam-3383	30	33	only	only	ADV
ejpam-3383	30	34	an	an	DET
ejpam-3383	30	35	upper	upper	ADJ
ejpam-3383	30	36	semilattice	semilattice	NOUN
ejpam-3383	30	37	,	,	PUNCT
ejpam-3383	30	38	but	but	CCONJ
ejpam-3383	30	39	a	a	DET
ejpam-3383	30	40	lattice	lattice	NOUN
ejpam-3383	30	41	,	,	PUNCT
ejpam-3383	30	42	then	then	ADV
ejpam-3383	30	43	it	it	PRON
ejpam-3383	30	44	is	be	AUX
ejpam-3383	30	45	called	call	VERB
ejpam-3383	30	46	a	a	DET
ejpam-3383	30	47	lattice	lattice	NOUN
ejpam-3383	30	48	ordered	order	VERB
ejpam-3383	30	49	semigroup	semigroup	NOUN
ejpam-3383	30	50	(	(	PUNCT
ejpam-3383	30	51	or	or	CCONJ
ejpam-3383	30	52	an	an	DET
ejpam-3383	30	53	l	l	NOUN
ejpam-3383	30	54	-	-	NOUN
ejpam-3383	30	55	semigroup	semigroup	NOUN
ejpam-3383	30	56	)	)	PUNCT
ejpam-3383	31	1	[	[	X
ejpam-3383	31	2	2	2	NUM
ejpam-3383	31	3	,	,	PUNCT
ejpam-3383	31	4	4	4	NUM
ejpam-3383	31	5	]	]	PUNCT
ejpam-3383	31	6	(	(	PUNCT
ejpam-3383	31	7	see	see	VERB
ejpam-3383	31	8	also	also	ADV
ejpam-3383	31	9	[	[	X
ejpam-3383	31	10	1	1	NUM
ejpam-3383	31	11	]	]	PUNCT
ejpam-3383	31	12	)	)	PUNCT
ejpam-3383	31	13	.	.	PUNCT
ejpam-3383	32	1	by	by	ADP
ejpam-3383	32	2	a	a	DET
ejpam-3383	32	3	poe	poe	PROPN
ejpam-3383	32	4	-	-	PUNCT
ejpam-3383	32	5	groupoid	groupoid	PROPN
ejpam-3383	32	6	,	,	PUNCT
ejpam-3383	32	7	∨e	∨e	NOUN
ejpam-3383	32	8	-	-	NOUN
ejpam-3383	32	9	groupoid	groupoid	PROPN
ejpam-3383	32	10	or	or	CCONJ
ejpam-3383	32	11	le	le	X
ejpam-3383	32	12	-	-	NOUN
ejpam-3383	32	13	groupoid	groupoid	NOUN
ejpam-3383	32	14	we	we	PRON
ejpam-3383	32	15	mean	mean	VERB
ejpam-3383	32	16	a	a	DET
ejpam-3383	32	17	po	po	NOUN
ejpam-3383	32	18	-	-	NOUN
ejpam-3383	32	19	groupoid	groupoid	NOUN
ejpam-3383	32	20	,	,	PUNCT
ejpam-3383	32	21	∨-groupoid	∨-groupoid	NUM
ejpam-3383	32	22	or	or	CCONJ
ejpam-3383	32	23	l	l	NOUN
ejpam-3383	32	24	-	-	NOUN
ejpam-3383	32	25	groupoid	groupoid	ADJ
ejpam-3383	32	26	s	s	NOUN
ejpam-3383	32	27	,	,	PUNCT
ejpam-3383	32	28	respectively	respectively	ADV
ejpam-3383	32	29	,	,	PUNCT
ejpam-3383	32	30	having	have	VERB
ejpam-3383	32	31	a	a	DET
ejpam-3383	32	32	greatest	great	ADJ
ejpam-3383	32	33	element	element	NOUN
ejpam-3383	32	34	usually	usually	ADV
ejpam-3383	32	35	denoted	denote	VERB
ejpam-3383	32	36	by	by	ADP
ejpam-3383	32	37	“	"	PUNCT
ejpam-3383	32	38	e	e	NOUN
ejpam-3383	32	39	”	"	PUNCT
ejpam-3383	32	40	(	(	PUNCT
ejpam-3383	32	41	i.e.	i.e.	X
ejpam-3383	32	42	e	e	X
ejpam-3383	32	43	≥	≥	X
ejpam-3383	32	44	a	a	PRON
ejpam-3383	32	45	for	for	ADP
ejpam-3383	32	46	every	every	DET
ejpam-3383	32	47	a	a	DET
ejpam-3383	32	48	∈	∈	PROPN
ejpam-3383	32	49	s	s	NOUN
ejpam-3383	32	50	)	)	PUNCT
ejpam-3383	32	51	.	.	PUNCT
ejpam-3383	33	1	if	if	SCONJ
ejpam-3383	33	2	the	the	DET
ejpam-3383	33	3	multiplication	multiplication	NOUN
ejpam-3383	33	4	on	on	ADP
ejpam-3383	33	5	a	a	DET
ejpam-3383	33	6	po	po	NOUN
ejpam-3383	33	7	-	-	NOUN
ejpam-3383	33	8	groupoid	groupoid	NOUN
ejpam-3383	33	9	s	s	PART
ejpam-3383	33	10	is	be	AUX
ejpam-3383	33	11	associative	associative	ADJ
ejpam-3383	33	12	,	,	PUNCT
ejpam-3383	33	13	then	then	ADV
ejpam-3383	33	14	s	s	VERB
ejpam-3383	33	15	is	be	AUX
ejpam-3383	33	16	called	call	VERB
ejpam-3383	33	17	a	a	DET
ejpam-3383	33	18	po	po	NOUN
ejpam-3383	33	19	-	-	PUNCT
ejpam-3383	33	20	semigroup	semigroup	NOUN
ejpam-3383	33	21	.	.	PUNCT
ejpam-3383	34	1	in	in	ADP
ejpam-3383	34	2	a	a	DET
ejpam-3383	34	3	similar	similar	ADJ
ejpam-3383	34	4	way	way	NOUN
ejpam-3383	34	5	we	we	PRON
ejpam-3383	34	6	have	have	VERB
ejpam-3383	34	7	the	the	DET
ejpam-3383	34	8	∨e	∨e	NOUN
ejpam-3383	34	9	-	-	PUNCT
ejpam-3383	34	10	semigroups	semigroup	NOUN
ejpam-3383	34	11	and	and	CCONJ
ejpam-3383	34	12	the	the	DET
ejpam-3383	34	13	le	le	NOUN
ejpam-3383	34	14	-	-	PUNCT
ejpam-3383	34	15	semigroups	semigroup	NOUN
ejpam-3383	34	16	.	.	PUNCT
ejpam-3383	35	1	in	in	ADP
ejpam-3383	35	2	a	a	DET
ejpam-3383	35	3	poe	poe	PROPN
ejpam-3383	35	4	-	-	PROPN
ejpam-3383	35	5	groupoid	groupoid	PROPN
ejpam-3383	35	6	s	s	PROPN
ejpam-3383	35	7	,	,	PUNCT
ejpam-3383	35	8	an	an	DET
ejpam-3383	35	9	element	element	NOUN
ejpam-3383	35	10	a	a	PRON
ejpam-3383	35	11	is	be	AUX
ejpam-3383	35	12	called	call	VERB
ejpam-3383	35	13	a	a	DET
ejpam-3383	35	14	right	right	ADJ
ejpam-3383	35	15	ideal	ideal	ADJ
ejpam-3383	35	16	element	element	NOUN
ejpam-3383	35	17	if	if	SCONJ
ejpam-3383	35	18	ae	ae	PROPN
ejpam-3383	35	19	≤	≤	VERB
ejpam-3383	35	20	a	a	X
ejpam-3383	35	21	,	,	PUNCT
ejpam-3383	35	22	it	it	PRON
ejpam-3383	35	23	is	be	AUX
ejpam-3383	35	24	called	call	VERB
ejpam-3383	35	25	a	a	DET
ejpam-3383	35	26	left	left	ADJ
ejpam-3383	35	27	ideal	ideal	ADJ
ejpam-3383	35	28	element	element	NOUN
ejpam-3383	35	29	if	if	SCONJ
ejpam-3383	35	30	ea	ea	NOUN
ejpam-3383	35	31	≤	≤	NUM
ejpam-3383	35	32	a.	a.	NOUN
ejpam-3383	35	33	an	an	DET
ejpam-3383	35	34	element	element	NOUN
ejpam-3383	35	35	which	which	PRON
ejpam-3383	35	36	is	be	AUX
ejpam-3383	35	37	both	both	CCONJ
ejpam-3383	35	38	a	a	DET
ejpam-3383	35	39	right	right	NOUN
ejpam-3383	35	40	and	and	CCONJ
ejpam-3383	35	41	a	a	DET
ejpam-3383	35	42	left	left	ADJ
ejpam-3383	35	43	ideal	ideal	ADJ
ejpam-3383	35	44	element	element	NOUN
ejpam-3383	35	45	is	be	AUX
ejpam-3383	35	46	called	call	VERB
ejpam-3383	35	47	an	an	DET
ejpam-3383	35	48	ideal	ideal	ADJ
ejpam-3383	35	49	element	element	NOUN
ejpam-3383	35	50	[	[	X
ejpam-3383	35	51	6	6	NUM
ejpam-3383	35	52	]	]	PUNCT
ejpam-3383	35	53	(	(	PUNCT
ejpam-3383	35	54	see	see	VERB
ejpam-3383	35	55	also	also	ADV
ejpam-3383	35	56	[	[	X
ejpam-3383	35	57	2	2	NUM
ejpam-3383	35	58	;	;	PUNCT
ejpam-3383	35	59	p.	p.	NOUN
ejpam-3383	35	60	328	328	NUM
ejpam-3383	35	61	]	]	PUNCT
ejpam-3383	35	62	)	)	PUNCT
ejpam-3383	35	63	.	.	PUNCT
ejpam-3383	36	1	we	we	PRON
ejpam-3383	36	2	denote	denote	VERB
ejpam-3383	36	3	by	by	ADP
ejpam-3383	36	4	fr	fr	PROPN
ejpam-3383	36	5	(	(	PUNCT
ejpam-3383	36	6	resp	resp	PROPN
ejpam-3383	36	7	.	.	PUNCT
ejpam-3383	37	1	fl	fl	X
ejpam-3383	37	2	)	)	PUNCT
ejpam-3383	37	3	the	the	DET
ejpam-3383	37	4	set	set	NOUN
ejpam-3383	37	5	of	of	ADP
ejpam-3383	37	6	right	right	NOUN
ejpam-3383	37	7	(	(	PUNCT
ejpam-3383	37	8	resp	resp	NOUN
ejpam-3383	37	9	.	.	PUNCT
ejpam-3383	38	1	left	left	ADJ
ejpam-3383	38	2	)	)	PUNCT
ejpam-3383	38	3	ideal	ideal	ADJ
ejpam-3383	38	4	elements	element	NOUN
ejpam-3383	38	5	of	of	ADP
ejpam-3383	38	6	s.	s.	PROPN
ejpam-3383	38	7	a	a	DET
ejpam-3383	38	8	nonempty	nonempty	ADV
ejpam-3383	38	9	set	set	VERB
ejpam-3383	38	10	a	a	PRON
ejpam-3383	38	11	of	of	ADP
ejpam-3383	38	12	a	a	DET
ejpam-3383	38	13	groupoid	groupoid	NOUN
ejpam-3383	38	14	(	(	PUNCT
ejpam-3383	38	15	s	s	PROPN
ejpam-3383	38	16	,	,	PUNCT
ejpam-3383	38	17	·	·	PUNCT
ejpam-3383	38	18	)	)	PUNCT
ejpam-3383	38	19	is	be	AUX
ejpam-3383	38	20	called	call	VERB
ejpam-3383	38	21	a	a	DET
ejpam-3383	38	22	right	right	NOUN
ejpam-3383	38	23	(	(	PUNCT
ejpam-3383	38	24	resp	resp	NOUN
ejpam-3383	38	25	.	.	PUNCT
ejpam-3383	39	1	left	left	ADJ
ejpam-3383	39	2	)	)	PUNCT
ejpam-3383	39	3	ideal	ideal	NOUN
ejpam-3383	39	4	of	of	ADP
ejpam-3383	39	5	s	s	PRON
ejpam-3383	39	6	if	if	SCONJ
ejpam-3383	39	7	as	as	ADP
ejpam-3383	39	8	⊆	⊆	NUM
ejpam-3383	39	9	a	a	DET
ejpam-3383	39	10	(	(	PUNCT
ejpam-3383	39	11	resp	resp	NOUN
ejpam-3383	39	12	.	.	PUNCT
ejpam-3383	40	1	sa	sa	PROPN
ejpam-3383	40	2	⊆	⊆	NUM
ejpam-3383	40	3	a	a	PRON
ejpam-3383	40	4	)	)	PUNCT
ejpam-3383	40	5	.	.	PUNCT
ejpam-3383	41	1	it	it	PRON
ejpam-3383	41	2	is	be	AUX
ejpam-3383	41	3	called	call	VERB
ejpam-3383	41	4	an	an	DET
ejpam-3383	41	5	ideal	ideal	NOUN
ejpam-3383	41	6	of	of	ADP
ejpam-3383	41	7	s	s	PRON
ejpam-3383	41	8	if	if	SCONJ
ejpam-3383	41	9	it	it	PRON
ejpam-3383	41	10	is	be	AUX
ejpam-3383	41	11	both	both	CCONJ
ejpam-3383	41	12	a	a	DET
ejpam-3383	41	13	right	right	NOUN
ejpam-3383	41	14	and	and	CCONJ
ejpam-3383	41	15	left	leave	VERB
ejpam-3383	41	16	ideal	ideal	NOUN
ejpam-3383	41	17	of	of	ADP
ejpam-3383	41	18	s	s	X
ejpam-3383	41	19	[	[	X
ejpam-3383	41	20	3	3	NUM
ejpam-3383	41	21	,	,	PUNCT
ejpam-3383	41	22	17	17	NUM
ejpam-3383	41	23	]	]	PUNCT
ejpam-3383	41	24	.	.	PUNCT
ejpam-3383	42	1	if	if	SCONJ
ejpam-3383	42	2	s	s	NOUN
ejpam-3383	42	3	is	be	AUX
ejpam-3383	42	4	an	an	DET
ejpam-3383	42	5	ordered	ordered	ADJ
ejpam-3383	42	6	groupoid	groupoid	NOUN
ejpam-3383	42	7	,	,	PUNCT
ejpam-3383	42	8	then	then	ADV
ejpam-3383	42	9	a	a	DET
ejpam-3383	42	10	nonempty	nonempty	NOUN
ejpam-3383	42	11	subset	subset	VERB
ejpam-3383	42	12	a	a	PRON
ejpam-3383	42	13	of	of	ADP
ejpam-3383	42	14	s	s	PRON
ejpam-3383	42	15	is	be	AUX
ejpam-3383	42	16	called	call	VERB
ejpam-3383	42	17	a	a	DET
ejpam-3383	42	18	right	right	NOUN
ejpam-3383	42	19	(	(	PUNCT
ejpam-3383	42	20	resp	resp	NOUN
ejpam-3383	42	21	.	.	PUNCT
ejpam-3383	43	1	left	left	ADJ
ejpam-3383	43	2	)	)	PUNCT
ejpam-3383	43	3	ideal	ideal	NOUN
ejpam-3383	43	4	of	of	ADP
ejpam-3383	43	5	s	s	PROPN
ejpam-3383	43	6	,	,	PUNCT
ejpam-3383	43	7	if	if	SCONJ
ejpam-3383	43	8	it	it	PRON
ejpam-3383	43	9	is	be	AUX
ejpam-3383	43	10	an	an	DET
ejpam-3383	43	11	ideal	ideal	NOUN
ejpam-3383	43	12	of	of	ADP
ejpam-3383	43	13	the	the	DET
ejpam-3383	43	14	semigroup	semigroup	NOUN
ejpam-3383	43	15	(	(	PUNCT
ejpam-3383	43	16	s	s	PROPN
ejpam-3383	43	17	,	,	PUNCT
ejpam-3383	43	18	·	·	PUNCT
ejpam-3383	43	19	)	)	PUNCT
ejpam-3383	43	20	and	and	CCONJ
ejpam-3383	43	21	,	,	PUNCT
ejpam-3383	43	22	in	in	ADP
ejpam-3383	43	23	addition	addition	NOUN
ejpam-3383	43	24	,	,	PUNCT
ejpam-3383	43	25	if	if	SCONJ
ejpam-3383	43	26	a	a	DET
ejpam-3383	43	27	∈	∈	PROPN
ejpam-3383	43	28	a	a	PRON
ejpam-3383	43	29	and	and	CCONJ
ejpam-3383	43	30	s	s	PROPN
ejpam-3383	43	31	3	3	NUM
ejpam-3383	43	32	b	b	NOUN
ejpam-3383	43	33	≤	≤	NUM
ejpam-3383	43	34	a	a	PRON
ejpam-3383	43	35	,	,	PUNCT
ejpam-3383	43	36	then	then	ADV
ejpam-3383	43	37	b	b	X
ejpam-3383	43	38	∈	∈	PROPN
ejpam-3383	43	39	a	a	PRON
ejpam-3383	44	1	[	[	X
ejpam-3383	44	2	7	7	NUM
ejpam-3383	44	3	]	]	PUNCT
ejpam-3383	44	4	.	.	PUNCT
ejpam-3383	45	1	either	either	CCONJ
ejpam-3383	45	2	for	for	ADP
ejpam-3383	45	3	a	a	DET
ejpam-3383	45	4	groupoid	groupoid	NOUN
ejpam-3383	45	5	or	or	CCONJ
ejpam-3383	45	6	for	for	ADP
ejpam-3383	45	7	an	an	DET
ejpam-3383	45	8	ordered	order	VERB
ejpam-3383	45	9	groupoid	groupoid	NOUN
ejpam-3383	45	10	,	,	PUNCT
ejpam-3383	45	11	sets	set	VERB
ejpam-3383	45	12	that	that	PRON
ejpam-3383	45	13	are	be	AUX
ejpam-3383	45	14	both	both	ADV
ejpam-3383	45	15	right	right	ADJ
ejpam-3383	45	16	and	and	CCONJ
ejpam-3383	45	17	left	leave	VERB
ejpam-3383	45	18	ideals	ideal	NOUN
ejpam-3383	45	19	are	be	AUX
ejpam-3383	45	20	called	call	VERB
ejpam-3383	45	21	ideals	ideal	NOUN
ejpam-3383	45	22	.	.	PUNCT
ejpam-3383	46	1	for	for	ADP
ejpam-3383	46	2	sets	set	NOUN
ejpam-3383	46	3	a	a	DET
ejpam-3383	46	4	,	,	PUNCT
ejpam-3383	46	5	b	b	NOUN
ejpam-3383	46	6	and	and	CCONJ
ejpam-3383	46	7	γ	γ	PROPN
ejpam-3383	46	8	,	,	PUNCT
ejpam-3383	46	9	the	the	DET
ejpam-3383	46	10	symbol	symbol	NOUN
ejpam-3383	46	11	aγb	aγb	PART
ejpam-3383	46	12	denotes	denote	VERB
ejpam-3383	46	13	the	the	DET
ejpam-3383	46	14	set	set	NOUN
ejpam-3383	46	15	of	of	ADP
ejpam-3383	46	16	all	all	DET
ejpam-3383	46	17	aγb	aγb	NOUN
ejpam-3383	46	18	with	with	ADP
ejpam-3383	46	19	a	a	DET
ejpam-3383	46	20	∈	∈	PROPN
ejpam-3383	46	21	a	a	PRON
ejpam-3383	46	22	,	,	PUNCT
ejpam-3383	46	23	b	b	PROPN
ejpam-3383	46	24	∈	∈	PROPN
ejpam-3383	46	25	b	b	PROPN
ejpam-3383	46	26	,	,	PUNCT
ejpam-3383	46	27	γ	γ	PROPN
ejpam-3383	46	28	∈	∈	PROPN
ejpam-3383	46	29	γ	γ	X
ejpam-3383	46	30	.	.	PUNCT
ejpam-3383	47	1	if	if	SCONJ
ejpam-3383	47	2	a	a	DET
ejpam-3383	47	3	=	=	NOUN
ejpam-3383	47	4	∅	∅	NOUN
ejpam-3383	47	5	or	or	CCONJ
ejpam-3383	47	6	b	b	NOUN
ejpam-3383	47	7	=	=	NOUN
ejpam-3383	47	8	∅	∅	NOUN
ejpam-3383	47	9	,	,	PUNCT
ejpam-3383	47	10	we	we	PRON
ejpam-3383	47	11	define	define	VERB
ejpam-3383	47	12	aγb	aγb	NOUN
ejpam-3383	47	13	=	=	SYM
ejpam-3383	47	14	∅.	∅.	PRON
ejpam-3383	47	15	definition	definition	NOUN
ejpam-3383	47	16	2.1	2.1	NUM
ejpam-3383	47	17	.	.	PUNCT
ejpam-3383	48	1	[	[	X
ejpam-3383	48	2	10	10	NUM
ejpam-3383	48	3	]	]	PUNCT
ejpam-3383	48	4	let	let	VERB
ejpam-3383	48	5	s	s	PRON
ejpam-3383	48	6	and	and	CCONJ
ejpam-3383	48	7	γ	γ	NOUN
ejpam-3383	48	8	be	be	AUX
ejpam-3383	48	9	two	two	NUM
ejpam-3383	48	10	nonempty	nonempty	ADJ
ejpam-3383	48	11	sets	set	NOUN
ejpam-3383	48	12	.	.	PUNCT
ejpam-3383	49	1	the	the	DET
ejpam-3383	49	2	set	set	NOUN
ejpam-3383	49	3	s	s	PART
ejpam-3383	49	4	is	be	AUX
ejpam-3383	49	5	called	call	VERB
ejpam-3383	49	6	a	a	DET
ejpam-3383	49	7	γ	γ	NOUN
ejpam-3383	49	8	-	-	NOUN
ejpam-3383	49	9	groupoid	groupoid	NOUN
ejpam-3383	49	10	if	if	SCONJ
ejpam-3383	49	11	the	the	DET
ejpam-3383	49	12	following	follow	VERB
ejpam-3383	49	13	two	two	NUM
ejpam-3383	49	14	assertions	assertion	NOUN
ejpam-3383	49	15	are	be	AUX
ejpam-3383	49	16	satisfied	satisfied	ADJ
ejpam-3383	49	17	:	:	PUNCT
ejpam-3383	49	18	(	(	PUNCT
ejpam-3383	49	19	1	1	X
ejpam-3383	49	20	)	)	PUNCT
ejpam-3383	49	21	sγs	sγs	NOUN
ejpam-3383	49	22	⊆	⊆	NUM
ejpam-3383	49	23	s.	s.	PROPN
ejpam-3383	49	24	(	(	PUNCT
ejpam-3383	49	25	2	2	NUM
ejpam-3383	49	26	)	)	PUNCT
ejpam-3383	49	27	if	if	SCONJ
ejpam-3383	49	28	a	a	DET
ejpam-3383	49	29	,	,	PUNCT
ejpam-3383	49	30	b	b	NOUN
ejpam-3383	49	31	,	,	PUNCT
ejpam-3383	49	32	c	c	NOUN
ejpam-3383	49	33	,	,	PUNCT
ejpam-3383	49	34	d	d	PROPN
ejpam-3383	49	35	∈	∈	PROPN
ejpam-3383	49	36	s	s	X
ejpam-3383	49	37	and	and	CCONJ
ejpam-3383	49	38	γ	γ	PROPN
ejpam-3383	49	39	,	,	PUNCT
ejpam-3383	49	40	µ	µ	PRON
ejpam-3383	49	41	∈	∈	NOUN
ejpam-3383	49	42	γ	γ	NOUN
ejpam-3383	49	43	such	such	ADJ
ejpam-3383	49	44	that	that	SCONJ
ejpam-3383	49	45	a	a	DET
ejpam-3383	49	46	=	=	SYM
ejpam-3383	49	47	c	c	NOUN
ejpam-3383	49	48	,	,	PUNCT
ejpam-3383	49	49	b	b	X
ejpam-3383	49	50	=	=	SYM
ejpam-3383	49	51	d	d	PROPN
ejpam-3383	49	52	and	and	CCONJ
ejpam-3383	49	53	γ	γ	X
ejpam-3383	49	54	=	=	SYM
ejpam-3383	49	55	µ	µ	PROPN
ejpam-3383	49	56	,	,	PUNCT
ejpam-3383	49	57	then	then	ADV
ejpam-3383	49	58	aγb	aγb	ADV
ejpam-3383	49	59	=	=	NOUN
ejpam-3383	49	60	cµd	cµd	PROPN
ejpam-3383	49	61	.	.	PUNCT
ejpam-3383	50	1	if	if	SCONJ
ejpam-3383	50	2	,	,	PUNCT
ejpam-3383	50	3	in	in	ADP
ejpam-3383	50	4	addition	addition	NOUN
ejpam-3383	50	5	,	,	PUNCT
ejpam-3383	50	6	for	for	ADP
ejpam-3383	50	7	all	all	DET
ejpam-3383	50	8	a	a	DET
ejpam-3383	50	9	,	,	PUNCT
ejpam-3383	50	10	b	b	NOUN
ejpam-3383	50	11	,	,	PUNCT
ejpam-3383	50	12	c	c	PROPN
ejpam-3383	50	13	∈	∈	PROPN
ejpam-3383	50	14	s	s	X
ejpam-3383	50	15	and	and	CCONJ
ejpam-3383	50	16	all	all	DET
ejpam-3383	50	17	γ	γ	PROPN
ejpam-3383	50	18	,	,	PUNCT
ejpam-3383	50	19	µ	µ	PRON
ejpam-3383	50	20	∈	∈	PROPN
ejpam-3383	50	21	γ	γ	X
ejpam-3383	50	22	,	,	PUNCT
ejpam-3383	50	23	we	we	PRON
ejpam-3383	50	24	have	have	VERB
ejpam-3383	50	25	the	the	DET
ejpam-3383	50	26	property	property	NOUN
ejpam-3383	50	27	n.	n.	NOUN
ejpam-3383	50	28	kehayopulu	kehayopulu	PROPN
ejpam-3383	50	29	/	/	SYM
ejpam-3383	50	30	eur	eur	PROPN
ejpam-3383	50	31	.	.	PUNCT
ejpam-3383	51	1	j.	j.	PROPN
ejpam-3383	51	2	pure	pure	PROPN
ejpam-3383	51	3	appl	appl	PROPN
ejpam-3383	51	4	.	.	PROPN
ejpam-3383	51	5	math	math	PROPN
ejpam-3383	51	6	,	,	PUNCT
ejpam-3383	51	7	12	12	NUM
ejpam-3383	51	8	(	(	PUNCT
ejpam-3383	51	9	1	1	NUM
ejpam-3383	51	10	)	)	PUNCT
ejpam-3383	51	11	(	(	PUNCT
ejpam-3383	51	12	2019	2019	NUM
ejpam-3383	51	13	)	)	PUNCT
ejpam-3383	51	14	,	,	PUNCT
ejpam-3383	51	15	208	208	NUM
ejpam-3383	51	16	-	-	SYM
ejpam-3383	51	17	225	225	NUM
ejpam-3383	51	18	210	210	NUM
ejpam-3383	51	19	(	(	PUNCT
ejpam-3383	51	20	3	3	NUM
ejpam-3383	51	21	)	)	PUNCT
ejpam-3383	51	22	(	(	PUNCT
ejpam-3383	51	23	aγb)µc	aγb)µc	NOUN
ejpam-3383	51	24	=	=	SYM
ejpam-3383	51	25	aγ(bµc	aγ(bµc	NOUN
ejpam-3383	51	26	)	)	PUNCT
ejpam-3383	51	27	then	then	ADV
ejpam-3383	51	28	s	s	VERB
ejpam-3383	51	29	is	be	AUX
ejpam-3383	51	30	called	call	VERB
ejpam-3383	51	31	a	a	DET
ejpam-3383	51	32	γ	γ	NOUN
ejpam-3383	51	33	-	-	PUNCT
ejpam-3383	51	34	semigroup	semigroup	NOUN
ejpam-3383	51	35	.	.	PUNCT
ejpam-3383	52	1	definition	definition	NOUN
ejpam-3383	52	2	2.1	2.1	NUM
ejpam-3383	52	3	is	be	AUX
ejpam-3383	52	4	the	the	DET
ejpam-3383	52	5	definition	definition	NOUN
ejpam-3383	52	6	of	of	ADP
ejpam-3383	52	7	a	a	DET
ejpam-3383	52	8	γ	γ	PROPN
ejpam-3383	52	9	-	-	PUNCT
ejpam-3383	52	10	semigroup	semigroup	NOUN
ejpam-3383	52	11	introduced	introduce	VERB
ejpam-3383	52	12	by	by	ADP
ejpam-3383	52	13	sen	sen	PROPN
ejpam-3383	52	14	and	and	CCONJ
ejpam-3383	52	15	saha	saha	PROPN
ejpam-3383	52	16	in	in	ADP
ejpam-3383	52	17	[	[	X
ejpam-3383	52	18	19	19	NUM
ejpam-3383	52	19	]	]	PUNCT
ejpam-3383	52	20	in	in	ADP
ejpam-3383	52	21	which	which	PRON
ejpam-3383	52	22	the	the	DET
ejpam-3383	52	23	missing	miss	VERB
ejpam-3383	52	24	uniqueness	uniqueness	NOUN
ejpam-3383	52	25	condition	condition	NOUN
ejpam-3383	52	26	(	(	PUNCT
ejpam-3383	52	27	2	2	X
ejpam-3383	52	28	)	)	PUNCT
ejpam-3383	52	29	has	have	AUX
ejpam-3383	52	30	been	be	AUX
ejpam-3383	52	31	added	add	VERB
ejpam-3383	52	32	(	(	PUNCT
ejpam-3383	52	33	for	for	ADP
ejpam-3383	52	34	details	detail	NOUN
ejpam-3383	52	35	,	,	PUNCT
ejpam-3383	52	36	see	see	VERB
ejpam-3383	52	37	[	[	X
ejpam-3383	52	38	10	10	NUM
ejpam-3383	52	39	]	]	NUM
ejpam-3383	52	40	)	)	PUNCT
ejpam-3383	52	41	.	.	PUNCT
ejpam-3383	53	1	in	in	ADP
ejpam-3383	53	2	other	other	ADJ
ejpam-3383	53	3	words	word	NOUN
ejpam-3383	53	4	,	,	PUNCT
ejpam-3383	53	5	a	a	DET
ejpam-3383	53	6	γ	γ	NOUN
ejpam-3383	53	7	-	-	PUNCT
ejpam-3383	53	8	semigroup	semigroup	NOUN
ejpam-3383	53	9	is	be	AUX
ejpam-3383	53	10	a	a	DET
ejpam-3383	53	11	set	set	NOUN
ejpam-3383	53	12	of	of	ADP
ejpam-3383	53	13	binary	binary	ADJ
ejpam-3383	53	14	operations	operation	NOUN
ejpam-3383	53	15	on	on	ADP
ejpam-3383	53	16	s	s	NOUN
ejpam-3383	53	17	and	and	CCONJ
ejpam-3383	53	18	condition	condition	NOUN
ejpam-3383	53	19	(	(	PUNCT
ejpam-3383	53	20	3	3	NUM
ejpam-3383	53	21	)	)	PUNCT
ejpam-3383	53	22	of	of	ADP
ejpam-3383	53	23	definition	definition	NOUN
ejpam-3383	53	24	2.1	2.1	NUM
ejpam-3383	53	25	is	be	AUX
ejpam-3383	53	26	satisfied	satisfied	ADJ
ejpam-3383	53	27	.	.	PUNCT
ejpam-3383	54	1	for	for	ADP
ejpam-3383	54	2	an	an	DET
ejpam-3383	54	3	application	application	NOUN
ejpam-3383	54	4	of	of	ADP
ejpam-3383	54	5	γ	γ	PROPN
ejpam-3383	54	6	-	-	PUNCT
ejpam-3383	54	7	semigroup	semigroup	NOUN
ejpam-3383	54	8	techniques	technique	NOUN
ejpam-3383	54	9	to	to	ADP
ejpam-3383	54	10	the	the	DET
ejpam-3383	54	11	green	green	PROPN
ejpam-3383	54	12	’s	’s	PART
ejpam-3383	54	13	theorem	theorem	NOUN
ejpam-3383	54	14	we	we	PRON
ejpam-3383	54	15	refer	refer	VERB
ejpam-3383	54	16	to	to	ADP
ejpam-3383	54	17	[	[	X
ejpam-3383	54	18	14	14	NUM
ejpam-3383	54	19	]	]	PUNCT
ejpam-3383	54	20	.	.	PUNCT
ejpam-3383	55	1	let	let	VERB
ejpam-3383	55	2	us	we	PRON
ejpam-3383	55	3	give	give	VERB
ejpam-3383	55	4	an	an	DET
ejpam-3383	55	5	example	example	NOUN
ejpam-3383	55	6	of	of	ADP
ejpam-3383	55	7	a	a	DET
ejpam-3383	55	8	γ	γ	NOUN
ejpam-3383	55	9	-	-	PUNCT
ejpam-3383	55	10	semigroup	semigroup	NOUN
ejpam-3383	55	11	(	(	PUNCT
ejpam-3383	55	12	see	see	VERB
ejpam-3383	55	13	[	[	X
ejpam-3383	55	14	10	10	NUM
ejpam-3383	55	15	;	;	PUNCT
ejpam-3383	55	16	example	example	NOUN
ejpam-3383	55	17	2(b	2(b	NUM
ejpam-3383	55	18	)	)	PUNCT
ejpam-3383	55	19	]	]	PUNCT
ejpam-3383	55	20	):	):	PUNCT
ejpam-3383	55	21	consider	consider	VERB
ejpam-3383	55	22	the	the	DET
ejpam-3383	55	23	set	set	NOUN
ejpam-3383	55	24	s	s	PART
ejpam-3383	55	25	=	=	PUNCT
ejpam-3383	55	26	{	{	PUNCT
ejpam-3383	55	27	a	a	PRON
ejpam-3383	55	28	,	,	PUNCT
ejpam-3383	55	29	b	b	NOUN
ejpam-3383	55	30	,	,	PUNCT
ejpam-3383	55	31	c	c	NOUN
ejpam-3383	55	32	}	}	PUNCT
ejpam-3383	55	33	and	and	CCONJ
ejpam-3383	55	34	let	let	VERB
ejpam-3383	55	35	γ	γ	X
ejpam-3383	55	36	=	=	SYM
ejpam-3383	55	37	{	{	PUNCT
ejpam-3383	55	38	γ	γ	PROPN
ejpam-3383	55	39	,	,	PUNCT
ejpam-3383	55	40	µ	µ	NOUN
ejpam-3383	55	41	}	}	PUNCT
ejpam-3383	55	42	be	be	AUX
ejpam-3383	55	43	the	the	DET
ejpam-3383	55	44	set	set	NOUN
ejpam-3383	55	45	of	of	ADP
ejpam-3383	55	46	two	two	NUM
ejpam-3383	55	47	binary	binary	ADJ
ejpam-3383	55	48	operations	operation	NOUN
ejpam-3383	55	49	on	on	ADP
ejpam-3383	55	50	s	s	PRON
ejpam-3383	55	51	defined	define	VERB
ejpam-3383	55	52	by	by	ADP
ejpam-3383	55	53	the	the	DET
ejpam-3383	55	54	tables	table	NOUN
ejpam-3383	55	55	below	below	ADV
ejpam-3383	55	56	:	:	PUNCT
ejpam-3383	55	57	γ	γ	PROPN
ejpam-3383	55	58	a	a	PROPN
ejpam-3383	55	59	b	b	X
ejpam-3383	55	60	c	c	ADP
ejpam-3383	55	61	a	a	DET
ejpam-3383	55	62	a	a	DET
ejpam-3383	55	63	b	b	NOUN
ejpam-3383	55	64	c	c	NOUN
ejpam-3383	55	65	b	b	PROPN
ejpam-3383	55	66	b	b	PROPN
ejpam-3383	55	67	c	c	PROPN
ejpam-3383	55	68	a	a	DET
ejpam-3383	55	69	c	c	NOUN
ejpam-3383	55	70	c	c	PROPN
ejpam-3383	55	71	a	a	DET
ejpam-3383	55	72	b	b	PROPN
ejpam-3383	55	73	µ	µ	NOUN
ejpam-3383	55	74	a	a	DET
ejpam-3383	55	75	b	b	NOUN
ejpam-3383	55	76	c	c	NOUN
ejpam-3383	55	77	a	a	DET
ejpam-3383	55	78	b	b	NOUN
ejpam-3383	55	79	c	c	NOUN
ejpam-3383	55	80	a	a	DET
ejpam-3383	55	81	b	b	NOUN
ejpam-3383	55	82	c	c	NOUN
ejpam-3383	55	83	a	a	DET
ejpam-3383	55	84	b	b	NOUN
ejpam-3383	55	85	c	c	NOUN
ejpam-3383	55	86	a	a	DET
ejpam-3383	55	87	b	b	NOUN
ejpam-3383	55	88	c	c	AUX
ejpam-3383	55	89	we	we	PRON
ejpam-3383	55	90	have	have	VERB
ejpam-3383	55	91	(	(	PUNCT
ejpam-3383	55	92	xρy)ωz	xρy)ωz	NOUN
ejpam-3383	55	93	=	=	PUNCT
ejpam-3383	55	94	xρ(yωz	xρ(yωz	PROPN
ejpam-3383	55	95	)	)	PUNCT
ejpam-3383	55	96	for	for	ADP
ejpam-3383	55	97	all	all	DET
ejpam-3383	55	98	x	x	NOUN
ejpam-3383	55	99	,	,	PUNCT
ejpam-3383	55	100	y	y	PROPN
ejpam-3383	55	101	,	,	PUNCT
ejpam-3383	55	102	z	z	PROPN
ejpam-3383	55	103	∈	∈	PROPN
ejpam-3383	55	104	s	s	PART
ejpam-3383	55	105	and	and	CCONJ
ejpam-3383	55	106	all	all	DET
ejpam-3383	55	107	ρ	ρ	PROPN
ejpam-3383	55	108	,	,	PUNCT
ejpam-3383	55	109	ω	ω	PROPN
ejpam-3383	55	110	∈	∈	PROPN
ejpam-3383	55	111	γ	γ	X
ejpam-3383	55	112	,	,	PUNCT
ejpam-3383	56	1	so	so	SCONJ
ejpam-3383	56	2	s	s	NOUN
ejpam-3383	56	3	is	be	AUX
ejpam-3383	56	4	a	a	DET
ejpam-3383	56	5	γ	γ	NOUN
ejpam-3383	56	6	-	-	PUNCT
ejpam-3383	56	7	semigroup	semigroup	NOUN
ejpam-3383	56	8	.	.	PUNCT
ejpam-3383	57	1	a	a	DET
ejpam-3383	57	2	γ	γ	X
ejpam-3383	57	3	-	-	ADJ
ejpam-3383	57	4	groupoid	groupoid	ADJ
ejpam-3383	57	5	(	(	PUNCT
ejpam-3383	57	6	s	s	PROPN
ejpam-3383	57	7	,	,	PUNCT
ejpam-3383	57	8	γ	γ	NOUN
ejpam-3383	57	9	)	)	PUNCT
ejpam-3383	57	10	endowed	endow	VERB
ejpam-3383	57	11	with	with	ADP
ejpam-3383	57	12	an	an	DET
ejpam-3383	57	13	order	order	NOUN
ejpam-3383	57	14	relation	relation	NOUN
ejpam-3383	57	15	“	"	PUNCT
ejpam-3383	57	16	≤	≤	NUM
ejpam-3383	57	17	”	"	PUNCT
ejpam-3383	57	18	such	such	ADJ
ejpam-3383	57	19	that	that	SCONJ
ejpam-3383	57	20	a	a	DET
ejpam-3383	57	21	≤	≤	PROPN
ejpam-3383	57	22	b	b	NOUN
ejpam-3383	57	23	implies	imply	VERB
ejpam-3383	57	24	aγc	aγc	PROPN
ejpam-3383	57	25	≤	≤	PROPN
ejpam-3383	57	26	bγc	bγc	NOUN
ejpam-3383	57	27	and	and	CCONJ
ejpam-3383	57	28	cγa	cγa	PROPN
ejpam-3383	57	29	≤	≤	NUM
ejpam-3383	57	30	cγb	cγb	NOUN
ejpam-3383	57	31	for	for	ADP
ejpam-3383	57	32	all	all	DET
ejpam-3383	57	33	c	c	NOUN
ejpam-3383	57	34	∈	∈	NOUN
ejpam-3383	57	35	s	s	VERB
ejpam-3383	57	36	is	be	AUX
ejpam-3383	57	37	called	call	VERB
ejpam-3383	57	38	an	an	DET
ejpam-3383	57	39	ordered	order	VERB
ejpam-3383	57	40	γ	γ	X
ejpam-3383	57	41	-	-	ADJ
ejpam-3383	57	42	groupoid	groupoid	ADJ
ejpam-3383	57	43	(	(	PUNCT
ejpam-3383	57	44	po	po	NOUN
ejpam-3383	57	45	-	-	PUNCT
ejpam-3383	57	46	γ	γ	NOUN
ejpam-3383	57	47	-	-	NOUN
ejpam-3383	57	48	groupoid	groupoid	NOUN
ejpam-3383	57	49	)	)	PUNCT
ejpam-3383	58	1	[	[	X
ejpam-3383	58	2	20	20	NUM
ejpam-3383	58	3	]	]	PUNCT
ejpam-3383	58	4	.	.	PUNCT
ejpam-3383	59	1	a	a	DET
ejpam-3383	59	2	nonempty	nonempty	NOUN
ejpam-3383	59	3	subset	subset	VERB
ejpam-3383	59	4	a	a	PRON
ejpam-3383	59	5	of	of	ADP
ejpam-3383	59	6	a	a	DET
ejpam-3383	59	7	γ	γ	NOUN
ejpam-3383	59	8	-	-	ADJ
ejpam-3383	59	9	groupoid	groupoid	ADJ
ejpam-3383	59	10	s	s	PART
ejpam-3383	59	11	is	be	AUX
ejpam-3383	59	12	called	call	VERB
ejpam-3383	59	13	a	a	DET
ejpam-3383	59	14	right	right	NOUN
ejpam-3383	59	15	(	(	PUNCT
ejpam-3383	59	16	resp	resp	NOUN
ejpam-3383	59	17	.	.	PUNCT
ejpam-3383	60	1	left	left	ADJ
ejpam-3383	60	2	)	)	PUNCT
ejpam-3383	60	3	ideal	ideal	NOUN
ejpam-3383	60	4	of	of	ADP
ejpam-3383	60	5	s	s	PRON
ejpam-3383	60	6	if	if	SCONJ
ejpam-3383	60	7	aγs	aγs	ADJ
ejpam-3383	60	8	⊆	⊆	NUM
ejpam-3383	60	9	a	a	DET
ejpam-3383	60	10	(	(	PUNCT
ejpam-3383	60	11	resp	resp	NOUN
ejpam-3383	60	12	.	.	PUNCT
ejpam-3383	61	1	sγa	sγa	PROPN
ejpam-3383	61	2	⊆	⊆	NUM
ejpam-3383	61	3	a	a	PRON
ejpam-3383	61	4	)	)	PUNCT
ejpam-3383	61	5	.	.	PUNCT
ejpam-3383	62	1	a	a	DET
ejpam-3383	62	2	nonempty	nonempty	NOUN
ejpam-3383	62	3	subset	subset	VERB
ejpam-3383	62	4	a	a	PRON
ejpam-3383	62	5	of	of	ADP
ejpam-3383	62	6	a	a	DET
ejpam-3383	62	7	po	po	NOUN
ejpam-3383	62	8	-	-	PUNCT
ejpam-3383	62	9	γ	γ	NOUN
ejpam-3383	62	10	-	-	NOUN
ejpam-3383	62	11	groupoid	groupoid	ADJ
ejpam-3383	62	12	s	s	PART
ejpam-3383	62	13	is	be	AUX
ejpam-3383	62	14	called	call	VERB
ejpam-3383	62	15	a	a	DET
ejpam-3383	62	16	right	right	NOUN
ejpam-3383	62	17	(	(	PUNCT
ejpam-3383	62	18	resp	resp	NOUN
ejpam-3383	62	19	.	.	PUNCT
ejpam-3383	63	1	left	left	ADJ
ejpam-3383	63	2	)	)	PUNCT
ejpam-3383	63	3	ideal	ideal	NOUN
ejpam-3383	63	4	of	of	ADP
ejpam-3383	63	5	s	s	PRON
ejpam-3383	63	6	if	if	SCONJ
ejpam-3383	63	7	it	it	PRON
ejpam-3383	63	8	is	be	AUX
ejpam-3383	63	9	a	a	DET
ejpam-3383	63	10	right	right	ADJ
ejpam-3383	63	11	(	(	PUNCT
ejpam-3383	63	12	resp	resp	NOUN
ejpam-3383	63	13	.	.	PUNCT
ejpam-3383	64	1	left	left	ADJ
ejpam-3383	64	2	)	)	PUNCT
ejpam-3383	64	3	ideal	ideal	NOUN
ejpam-3383	64	4	of	of	ADP
ejpam-3383	64	5	the	the	DET
ejpam-3383	64	6	γ	γ	PROPN
ejpam-3383	64	7	-	-	PUNCT
ejpam-3383	64	8	semigroup	semigroup	NOUN
ejpam-3383	64	9	s	s	PART
ejpam-3383	64	10	and	and	CCONJ
ejpam-3383	64	11	,	,	PUNCT
ejpam-3383	64	12	in	in	ADP
ejpam-3383	64	13	addition	addition	NOUN
ejpam-3383	64	14	,	,	PUNCT
ejpam-3383	64	15	if	if	SCONJ
ejpam-3383	64	16	a	a	DET
ejpam-3383	64	17	∈	∈	PROPN
ejpam-3383	64	18	a	a	PRON
ejpam-3383	64	19	and	and	CCONJ
ejpam-3383	64	20	s	s	PROPN
ejpam-3383	64	21	3	3	NUM
ejpam-3383	64	22	b	b	NOUN
ejpam-3383	64	23	≤	≤	NUM
ejpam-3383	64	24	a	a	PRON
ejpam-3383	64	25	,	,	PUNCT
ejpam-3383	64	26	then	then	ADV
ejpam-3383	64	27	b	b	X
ejpam-3383	64	28	∈	∈	PROPN
ejpam-3383	65	1	a	a	PRON
ejpam-3383	66	1	[	[	X
ejpam-3383	66	2	10	10	NUM
ejpam-3383	66	3	]	]	PUNCT
ejpam-3383	66	4	.	.	PUNCT
ejpam-3383	67	1	in	in	ADP
ejpam-3383	67	2	both	both	DET
ejpam-3383	67	3	cases	case	NOUN
ejpam-3383	67	4	,	,	PUNCT
ejpam-3383	67	5	a	a	PRON
ejpam-3383	67	6	is	be	AUX
ejpam-3383	67	7	called	call	VERB
ejpam-3383	67	8	an	an	DET
ejpam-3383	67	9	ideal	ideal	NOUN
ejpam-3383	67	10	of	of	ADP
ejpam-3383	67	11	s	s	PRON
ejpam-3383	67	12	if	if	SCONJ
ejpam-3383	67	13	it	it	PRON
ejpam-3383	67	14	is	be	AUX
ejpam-3383	67	15	both	both	CCONJ
ejpam-3383	67	16	a	a	DET
ejpam-3383	67	17	right	right	NOUN
ejpam-3383	67	18	and	and	CCONJ
ejpam-3383	67	19	a	a	DET
ejpam-3383	67	20	left	left	ADJ
ejpam-3383	67	21	ideal	ideal	NOUN
ejpam-3383	67	22	of	of	ADP
ejpam-3383	67	23	s.	s.	PROPN
ejpam-3383	67	24	remark	remark	PROPN
ejpam-3383	67	25	2.2	2.2	NUM
ejpam-3383	67	26	.	.	PUNCT
ejpam-3383	68	1	if	if	SCONJ
ejpam-3383	68	2	(	(	PUNCT
ejpam-3383	68	3	s	s	X
ejpam-3383	68	4	,	,	PUNCT
ejpam-3383	68	5	γ	γ	NOUN
ejpam-3383	68	6	)	)	PUNCT
ejpam-3383	68	7	is	be	AUX
ejpam-3383	68	8	a	a	DET
ejpam-3383	68	9	γ	γ	NOUN
ejpam-3383	68	10	-	-	PUNCT
ejpam-3383	68	11	semigroup	semigroup	NOUN
ejpam-3383	68	12	then	then	ADV
ejpam-3383	68	13	,	,	PUNCT
ejpam-3383	68	14	for	for	ADP
ejpam-3383	68	15	any	any	DET
ejpam-3383	68	16	subsets	subset	NOUN
ejpam-3383	68	17	a	a	DET
ejpam-3383	68	18	,	,	PUNCT
ejpam-3383	68	19	b	b	NOUN
ejpam-3383	68	20	,	,	PUNCT
ejpam-3383	68	21	c	c	NOUN
ejpam-3383	68	22	of	of	ADP
ejpam-3383	68	23	s	s	PROPN
ejpam-3383	68	24	,	,	PUNCT
ejpam-3383	68	25	we	we	PRON
ejpam-3383	68	26	have	have	VERB
ejpam-3383	68	27	(	(	PUNCT
ejpam-3383	68	28	aγb)γc	aγb)γc	VERB
ejpam-3383	68	29	=	=	PUNCT
ejpam-3383	68	30	aγ(bγc	aγ(bγc	PROPN
ejpam-3383	68	31	)	)	PUNCT
ejpam-3383	68	32	.	.	PUNCT
ejpam-3383	69	1	indeed	indeed	ADV
ejpam-3383	69	2	,	,	PUNCT
ejpam-3383	69	3	if	if	SCONJ
ejpam-3383	69	4	x	x	SYM
ejpam-3383	69	5	∈	∈	PROPN
ejpam-3383	69	6	(	(	PUNCT
ejpam-3383	69	7	aγb)γc	aγb)γc	VERB
ejpam-3383	69	8	,	,	PUNCT
ejpam-3383	69	9	then	then	ADV
ejpam-3383	69	10	x	x	X
ejpam-3383	69	11	=	=	PRON
ejpam-3383	69	12	(	(	PUNCT
ejpam-3383	69	13	aγb)µc	aγb)µc	NOUN
ejpam-3383	69	14	=	=	NOUN
ejpam-3383	69	15	aγ(bµc	aγ(bµc	NOUN
ejpam-3383	69	16	)	)	PUNCT
ejpam-3383	69	17	for	for	ADP
ejpam-3383	69	18	some	some	PRON
ejpam-3383	69	19	a	a	DET
ejpam-3383	69	20	,	,	PUNCT
ejpam-3383	69	21	b	b	NOUN
ejpam-3383	69	22	,	,	PUNCT
ejpam-3383	69	23	c	c	PROPN
ejpam-3383	69	24	∈	∈	PROPN
ejpam-3383	69	25	s	s	X
ejpam-3383	69	26	and	and	CCONJ
ejpam-3383	69	27	γ	γ	PROPN
ejpam-3383	69	28	,	,	PUNCT
ejpam-3383	69	29	µ	µ	X
ejpam-3383	69	30	∈	∈	NOUN
ejpam-3383	69	31	γ	γ	NOUN
ejpam-3383	69	32	and	and	CCONJ
ejpam-3383	69	33	so	so	ADV
ejpam-3383	69	34	x	x	SYM
ejpam-3383	69	35	∈	∈	PROPN
ejpam-3383	69	36	aγ(bγc	aγ(bγc	PROPN
ejpam-3383	69	37	)	)	PUNCT
ejpam-3383	69	38	.	.	PUNCT
ejpam-3383	70	1	similarly	similarly	ADV
ejpam-3383	70	2	we	we	PRON
ejpam-3383	70	3	have	have	AUX
ejpam-3383	70	4	aγ(bγc	aγ(bγc	VERB
ejpam-3383	70	5	)	)	PUNCT
ejpam-3383	71	1	⊆	⊆	NUM
ejpam-3383	71	2	(	(	PUNCT
ejpam-3383	71	3	aγb)γc	aγb)γc	NOUN
ejpam-3383	71	4	.	.	PUNCT
ejpam-3383	72	1	lemma	lemma	PROPN
ejpam-3383	72	2	2.3	2.3	NUM
ejpam-3383	72	3	.	.	PUNCT
ejpam-3383	73	1	if	if	SCONJ
ejpam-3383	73	2	(	(	PUNCT
ejpam-3383	73	3	s	s	X
ejpam-3383	73	4	,	,	PUNCT
ejpam-3383	73	5	γ	γ	NOUN
ejpam-3383	73	6	)	)	PUNCT
ejpam-3383	73	7	is	be	AUX
ejpam-3383	73	8	a	a	DET
ejpam-3383	73	9	γ	γ	NOUN
ejpam-3383	73	10	-	-	NOUN
ejpam-3383	73	11	groupoid	groupoid	NOUN
ejpam-3383	73	12	then	then	ADV
ejpam-3383	73	13	,	,	PUNCT
ejpam-3383	73	14	for	for	ADP
ejpam-3383	73	15	any	any	DET
ejpam-3383	73	16	subsets	subset	NOUN
ejpam-3383	73	17	a	a	DET
ejpam-3383	73	18	,	,	PUNCT
ejpam-3383	73	19	b	b	NOUN
ejpam-3383	73	20	,	,	PUNCT
ejpam-3383	73	21	c	c	NOUN
ejpam-3383	73	22	of	of	ADP
ejpam-3383	73	23	s	s	PROPN
ejpam-3383	73	24	,	,	PUNCT
ejpam-3383	73	25	we	we	PRON
ejpam-3383	73	26	have	have	VERB
ejpam-3383	73	27	(	(	PUNCT
ejpam-3383	73	28	1	1	NUM
ejpam-3383	73	29	)	)	PUNCT
ejpam-3383	73	30	(	(	PUNCT
ejpam-3383	73	31	a	a	DET
ejpam-3383	73	32	∪b)γc	∪b)γc	PROPN
ejpam-3383	73	33	=	=	SYM
ejpam-3383	73	34	aγc	aγc	PROPN
ejpam-3383	73	35	∪bγc	∪bγc	PROPN
ejpam-3383	74	1	and	and	CCONJ
ejpam-3383	74	2	(	(	PUNCT
ejpam-3383	74	3	2	2	NUM
ejpam-3383	74	4	)	)	PUNCT
ejpam-3383	74	5	aγ(b	aγ(b	X
ejpam-3383	74	6	∪	∪	ADP
ejpam-3383	74	7	c	c	NOUN
ejpam-3383	74	8	)	)	PUNCT
ejpam-3383	75	1	=	=	PRON
ejpam-3383	75	2	aγb	aγb	NOUN
ejpam-3383	75	3	∪aγc	∪aγc	NOUN
ejpam-3383	75	4	.	.	PUNCT
ejpam-3383	76	1	proof	proof	NOUN
ejpam-3383	76	2	.	.	PUNCT
ejpam-3383	77	1	(	(	PUNCT
ejpam-3383	77	2	1	1	X
ejpam-3383	77	3	)	)	PUNCT
ejpam-3383	77	4	since	since	SCONJ
ejpam-3383	77	5	a∪b	a∪b	NOUN
ejpam-3383	77	6	⊇	⊇	PROPN
ejpam-3383	77	7	a	a	PRON
ejpam-3383	77	8	and	and	CCONJ
ejpam-3383	77	9	a∪b	a∪b	ADJ
ejpam-3383	77	10	⊇	⊇	PROPN
ejpam-3383	77	11	b	b	PROPN
ejpam-3383	77	12	,	,	PUNCT
ejpam-3383	77	13	we	we	PRON
ejpam-3383	77	14	have	have	VERB
ejpam-3383	77	15	(	(	PUNCT
ejpam-3383	77	16	a∪b)γc	a∪b)γc	PROPN
ejpam-3383	77	17	⊇	⊇	NOUN
ejpam-3383	77	18	aγc	aγc	PROPN
ejpam-3383	77	19	and	and	CCONJ
ejpam-3383	77	20	(	(	PUNCT
ejpam-3383	77	21	a∪b)γc	a∪b)γc	PROPN
ejpam-3383	77	22	⊇	⊇	PROPN
ejpam-3383	77	23	bγc	bγc	NOUN
ejpam-3383	77	24	and	and	CCONJ
ejpam-3383	77	25	so	so	ADV
ejpam-3383	77	26	(	(	PUNCT
ejpam-3383	77	27	a	a	DET
ejpam-3383	77	28	∪	∪	ADJ
ejpam-3383	77	29	b)γc	b)γc	PROPN
ejpam-3383	77	30	⊇	⊇	PROPN
ejpam-3383	77	31	aγc	aγc	PROPN
ejpam-3383	77	32	∪	∪	PROPN
ejpam-3383	77	33	bγc	bγc	PROPN
ejpam-3383	77	34	.	.	PUNCT
ejpam-3383	78	1	let	let	VERB
ejpam-3383	78	2	now	now	ADV
ejpam-3383	78	3	x	x	X
ejpam-3383	78	4	∈	∈	PROPN
ejpam-3383	78	5	(	(	PUNCT
ejpam-3383	78	6	a	a	DET
ejpam-3383	78	7	∪	∪	ADJ
ejpam-3383	78	8	b)γc	b)γc	PROPN
ejpam-3383	78	9	.	.	PUNCT
ejpam-3383	79	1	then	then	ADV
ejpam-3383	79	2	x	x	X
ejpam-3383	79	3	=	=	PUNCT
ejpam-3383	79	4	tγc	tγc	NOUN
ejpam-3383	79	5	for	for	ADP
ejpam-3383	79	6	some	some	DET
ejpam-3383	79	7	t	t	NOUN
ejpam-3383	79	8	∈	∈	PROPN
ejpam-3383	79	9	a	a	DET
ejpam-3383	79	10	∪	∪	ADJ
ejpam-3383	79	11	b	b	NOUN
ejpam-3383	79	12	,	,	PUNCT
ejpam-3383	79	13	γ	γ	PROPN
ejpam-3383	79	14	∈	∈	PROPN
ejpam-3383	79	15	γ	γ	X
ejpam-3383	79	16	,	,	PUNCT
ejpam-3383	79	17	c	c	PROPN
ejpam-3383	79	18	∈	∈	PROPN
ejpam-3383	79	19	c.	c.	NOUN
ejpam-3383	79	20	if	if	SCONJ
ejpam-3383	79	21	t	t	PROPN
ejpam-3383	79	22	∈	∈	PROPN
ejpam-3383	79	23	a	a	PRON
ejpam-3383	79	24	,	,	PUNCT
ejpam-3383	79	25	then	then	ADV
ejpam-3383	79	26	x	x	PART
ejpam-3383	79	27	∈	∈	PROPN
ejpam-3383	79	28	aγc	aγc	NOUN
ejpam-3383	79	29	;	;	PUNCT
ejpam-3383	79	30	if	if	SCONJ
ejpam-3383	79	31	t	t	PROPN
ejpam-3383	79	32	∈	∈	PROPN
ejpam-3383	79	33	b	b	PROPN
ejpam-3383	79	34	,	,	PUNCT
ejpam-3383	79	35	then	then	ADV
ejpam-3383	79	36	x	x	PART
ejpam-3383	79	37	∈	∈	PROPN
ejpam-3383	79	38	bγc	bγc	NOUN
ejpam-3383	79	39	and	and	CCONJ
ejpam-3383	79	40	so	so	ADV
ejpam-3383	79	41	x	x	SYM
ejpam-3383	79	42	∈	∈	PROPN
ejpam-3383	79	43	aγc	aγc	NOUN
ejpam-3383	79	44	∪bγc	∪bγc	PROPN
ejpam-3383	79	45	.	.	PUNCT
ejpam-3383	80	1	the	the	DET
ejpam-3383	80	2	proof	proof	NOUN
ejpam-3383	80	3	of	of	ADP
ejpam-3383	80	4	property	property	NOUN
ejpam-3383	80	5	(	(	PUNCT
ejpam-3383	80	6	2	2	NUM
ejpam-3383	80	7	)	)	PUNCT
ejpam-3383	80	8	is	be	AUX
ejpam-3383	80	9	similar	similar	ADJ
ejpam-3383	80	10	.	.	PUNCT
ejpam-3383	81	1	�	�	PROPN
ejpam-3383	81	2	an	an	DET
ejpam-3383	81	3	hypergroupoid	hypergroupoid	PROPN
ejpam-3383	81	4	is	be	AUX
ejpam-3383	81	5	a	a	DET
ejpam-3383	81	6	nonempty	nonempty	ADJ
ejpam-3383	81	7	set	set	VERB
ejpam-3383	81	8	s	s	NOUN
ejpam-3383	81	9	with	with	ADP
ejpam-3383	81	10	an	an	DET
ejpam-3383	81	11	hyperoperation	hyperoperation	NOUN
ejpam-3383	81	12	◦	◦	NOUN
ejpam-3383	81	13	:	:	PUNCT
ejpam-3383	81	14	s	s	VERB
ejpam-3383	81	15	×	×	PROPN
ejpam-3383	81	16	s	s	X
ejpam-3383	81	17	→	→	SYM
ejpam-3383	81	18	p∗(s	p∗(s	NOUN
ejpam-3383	81	19	)	)	PUNCT
ejpam-3383	81	20	|	|	NOUN
ejpam-3383	81	21	(	(	PUNCT
ejpam-3383	81	22	a	a	PRON
ejpam-3383	81	23	,	,	PUNCT
ejpam-3383	81	24	b)→	b)→	VERB
ejpam-3383	81	25	a	a	DET
ejpam-3383	81	26	◦	◦	NOUN
ejpam-3383	81	27	b	b	NOUN
ejpam-3383	81	28	on	on	ADP
ejpam-3383	81	29	s	s	PRON
ejpam-3383	81	30	and	and	CCONJ
ejpam-3383	81	31	an	an	DET
ejpam-3383	81	32	operation	operation	NOUN
ejpam-3383	81	33	∗	∗	NOUN
ejpam-3383	81	34	:	:	PUNCT
ejpam-3383	81	35	p∗(s	p∗(s	X
ejpam-3383	81	36	)	)	PUNCT
ejpam-3383	81	37	×	×	NOUN
ejpam-3383	81	38	p∗(s	p∗(s	NOUN
ejpam-3383	81	39	)	)	PUNCT
ejpam-3383	81	40	→	→	SYM
ejpam-3383	81	41	p∗(s	p∗(s	NOUN
ejpam-3383	81	42	)	)	PUNCT
ejpam-3383	81	43	|	|	NOUN
ejpam-3383	81	44	(	(	PUNCT
ejpam-3383	81	45	a	a	DET
ejpam-3383	81	46	,	,	PUNCT
ejpam-3383	81	47	b	b	NOUN
ejpam-3383	81	48	)	)	PUNCT
ejpam-3383	81	49	→	→	ADP
ejpam-3383	81	50	a	a	DET
ejpam-3383	81	51	∗	∗	NOUN
ejpam-3383	81	52	b	b	NOUN
ejpam-3383	81	53	on	on	ADP
ejpam-3383	81	54	p∗(s	p∗(s	ADP
ejpam-3383	81	55	)	)	PUNCT
ejpam-3383	81	56	(	(	PUNCT
ejpam-3383	81	57	induced	induce	VERB
ejpam-3383	81	58	by	by	ADP
ejpam-3383	81	59	the	the	DET
ejpam-3383	81	60	operation	operation	NOUN
ejpam-3383	81	61	of	of	ADP
ejpam-3383	81	62	s	s	NOUN
ejpam-3383	81	63	)	)	PUNCT
ejpam-3383	81	64	such	such	ADJ
ejpam-3383	81	65	that	that	SCONJ
ejpam-3383	81	66	a	a	DET
ejpam-3383	81	67	∗	∗	NOUN
ejpam-3383	81	68	b	b	NOUN
ejpam-3383	81	69	=	=	X
ejpam-3383	81	70	⋃	⋃	PROPN
ejpam-3383	81	71	(	(	PUNCT
ejpam-3383	81	72	a	a	PRON
ejpam-3383	81	73	,	,	PUNCT
ejpam-3383	81	74	b)∈a×b	b)∈a×b	NUM
ejpam-3383	81	75	(	(	PUNCT
ejpam-3383	81	76	a	a	DET
ejpam-3383	81	77	◦	◦	NOUN
ejpam-3383	81	78	b	b	NOUN
ejpam-3383	81	79	)	)	PUNCT
ejpam-3383	81	80	for	for	ADP
ejpam-3383	81	81	every	every	DET
ejpam-3383	81	82	a	a	PROPN
ejpam-3383	81	83	,	,	PUNCT
ejpam-3383	81	84	b	b	PROPN
ejpam-3383	81	85	∈	∈	PROPN
ejpam-3383	81	86	p∗(s	p∗(s	NOUN
ejpam-3383	81	87	)	)	PUNCT
ejpam-3383	81	88	.	.	PUNCT
ejpam-3383	82	1	p∗(s	p∗(s	NOUN
ejpam-3383	82	2	)	)	PUNCT
ejpam-3383	82	3	denotes	denote	VERB
ejpam-3383	82	4	the	the	DET
ejpam-3383	82	5	set	set	NOUN
ejpam-3383	82	6	of	of	ADP
ejpam-3383	82	7	all	all	DET
ejpam-3383	82	8	nonempty	nonempty	ADJ
ejpam-3383	82	9	subsets	subset	NOUN
ejpam-3383	82	10	of	of	ADP
ejpam-3383	82	11	s.	s.	PROPN
ejpam-3383	82	12	clearly	clearly	ADV
ejpam-3383	82	13	,	,	PUNCT
ejpam-3383	82	14	x	x	PROPN
ejpam-3383	82	15	∈	∈	PROPN
ejpam-3383	82	16	a	a	DET
ejpam-3383	82	17	∗b	∗b	NOUN
ejpam-3383	82	18	if	if	SCONJ
ejpam-3383	83	1	and	and	CCONJ
ejpam-3383	83	2	only	only	ADV
ejpam-3383	83	3	if	if	SCONJ
ejpam-3383	83	4	there	there	PRON
ejpam-3383	83	5	exist	exist	VERB
ejpam-3383	83	6	n.	n.	NOUN
ejpam-3383	83	7	kehayopulu	kehayopulu	PROPN
ejpam-3383	83	8	/	/	SYM
ejpam-3383	83	9	eur	eur	PROPN
ejpam-3383	83	10	.	.	PUNCT
ejpam-3383	84	1	j.	j.	PROPN
ejpam-3383	84	2	pure	pure	PROPN
ejpam-3383	84	3	appl	appl	PROPN
ejpam-3383	84	4	.	.	PROPN
ejpam-3383	84	5	math	math	PROPN
ejpam-3383	84	6	,	,	PUNCT
ejpam-3383	84	7	12	12	NUM
ejpam-3383	84	8	(	(	PUNCT
ejpam-3383	84	9	1	1	NUM
ejpam-3383	84	10	)	)	PUNCT
ejpam-3383	84	11	(	(	PUNCT
ejpam-3383	84	12	2019	2019	NUM
ejpam-3383	84	13	)	)	PUNCT
ejpam-3383	84	14	,	,	PUNCT
ejpam-3383	84	15	208	208	NUM
ejpam-3383	84	16	-	-	SYM
ejpam-3383	84	17	225	225	NUM
ejpam-3383	84	18	211	211	NUM
ejpam-3383	84	19	a	a	DET
ejpam-3383	84	20	∈	∈	PROPN
ejpam-3383	84	21	a	a	PRON
ejpam-3383	84	22	and	and	CCONJ
ejpam-3383	84	23	b	b	NOUN
ejpam-3383	84	24	∈	∈	PROPN
ejpam-3383	84	25	b	b	NOUN
ejpam-3383	84	26	such	such	ADJ
ejpam-3383	84	27	that	that	SCONJ
ejpam-3383	84	28	x	x	SYM
ejpam-3383	84	29	∈	∈	PROPN
ejpam-3383	84	30	a	a	DET
ejpam-3383	84	31	◦	◦	NOUN
ejpam-3383	84	32	b.	b.	NOUN
ejpam-3383	85	1	as	as	SCONJ
ejpam-3383	85	2	one	one	PRON
ejpam-3383	85	3	can	can	AUX
ejpam-3383	85	4	easily	easily	ADV
ejpam-3383	85	5	see	see	VERB
ejpam-3383	85	6	,	,	PUNCT
ejpam-3383	85	7	for	for	ADP
ejpam-3383	85	8	any	any	DET
ejpam-3383	85	9	x	x	NOUN
ejpam-3383	85	10	,	,	PUNCT
ejpam-3383	85	11	y	y	PROPN
ejpam-3383	85	12	∈	∈	PROPN
ejpam-3383	85	13	s	s	PART
ejpam-3383	85	14	,	,	PUNCT
ejpam-3383	85	15	we	we	PRON
ejpam-3383	85	16	have	have	VERB
ejpam-3383	85	17	{	{	PUNCT
ejpam-3383	85	18	x}∗{y	x}∗{y	NOUN
ejpam-3383	85	19	}	}	PUNCT
ejpam-3383	85	20	=	=	SYM
ejpam-3383	85	21	x	x	SYM
ejpam-3383	85	22	◦	◦	VERB
ejpam-3383	85	23	y.	y.	NOUN
ejpam-3383	85	24	an	an	DET
ejpam-3383	85	25	hypergroupoid	hypergroupoid	PROPN
ejpam-3383	85	26	s	s	PART
ejpam-3383	85	27	is	be	AUX
ejpam-3383	85	28	called	call	VERB
ejpam-3383	85	29	hypersemigroup	hypersemigroup	ADV
ejpam-3383	85	30	if	if	SCONJ
ejpam-3383	85	31	{	{	PUNCT
ejpam-3383	85	32	x}∗(y	x}∗(y	PROPN
ejpam-3383	85	33	◦	◦	PROPN
ejpam-3383	85	34	z	z	PROPN
ejpam-3383	85	35	)	)	PUNCT
ejpam-3383	85	36	=	=	SYM
ejpam-3383	85	37	(	(	PUNCT
ejpam-3383	85	38	x	x	X
ejpam-3383	85	39	◦	◦	NOUN
ejpam-3383	85	40	y)∗{z	y)∗{z	NOUN
ejpam-3383	85	41	}	}	PUNCT
ejpam-3383	85	42	for	for	ADP
ejpam-3383	85	43	every	every	DET
ejpam-3383	85	44	x	x	PROPN
ejpam-3383	85	45	,	,	PUNCT
ejpam-3383	85	46	y	y	PROPN
ejpam-3383	85	47	,	,	PUNCT
ejpam-3383	85	48	z	z	PROPN
ejpam-3383	85	49	∈	∈	PROPN
ejpam-3383	85	50	s	s	PART
ejpam-3383	85	51	[	[	X
ejpam-3383	85	52	11	11	NUM
ejpam-3383	85	53	,	,	PUNCT
ejpam-3383	85	54	13	13	NUM
ejpam-3383	85	55	]	]	PUNCT
ejpam-3383	85	56	.	.	PUNCT
ejpam-3383	86	1	the	the	DET
ejpam-3383	86	2	concept	concept	NOUN
ejpam-3383	86	3	of	of	ADP
ejpam-3383	86	4	an	an	DET
ejpam-3383	86	5	ordered	order	VERB
ejpam-3383	86	6	groupoid	groupoid	NOUN
ejpam-3383	86	7	can	can	AUX
ejpam-3383	86	8	be	be	AUX
ejpam-3383	86	9	naturally	naturally	ADV
ejpam-3383	86	10	transferred	transfer	VERB
ejpam-3383	86	11	to	to	ADP
ejpam-3383	86	12	ordered	order	VERB
ejpam-3383	86	13	hypergroupoids	hypergroupoid	NOUN
ejpam-3383	86	14	as	as	SCONJ
ejpam-3383	86	15	follows	follow	VERB
ejpam-3383	86	16	:	:	PUNCT
ejpam-3383	86	17	an	an	DET
ejpam-3383	86	18	hypergroupoid	hypergroupoid	PROPN
ejpam-3383	86	19	(	(	PUNCT
ejpam-3383	86	20	s	s	NOUN
ejpam-3383	86	21	,	,	PUNCT
ejpam-3383	86	22	◦	◦	NOUN
ejpam-3383	86	23	)	)	PUNCT
ejpam-3383	86	24	is	be	AUX
ejpam-3383	86	25	called	call	VERB
ejpam-3383	86	26	ordered	order	VERB
ejpam-3383	86	27	hypergroupoid	hypergroupoid	PROPN
ejpam-3383	86	28	if	if	SCONJ
ejpam-3383	86	29	there	there	PRON
ejpam-3383	86	30	exists	exist	VERB
ejpam-3383	86	31	an	an	DET
ejpam-3383	86	32	order	order	NOUN
ejpam-3383	86	33	relation	relation	NOUN
ejpam-3383	86	34	“	"	PUNCT
ejpam-3383	86	35	≤	≤	NUM
ejpam-3383	86	36	”	"	PUNCT
ejpam-3383	86	37	on	on	ADP
ejpam-3383	86	38	s	s	PRON
ejpam-3383	86	39	such	such	ADJ
ejpam-3383	86	40	that	that	SCONJ
ejpam-3383	86	41	a	a	DET
ejpam-3383	86	42	≤	≤	PROPN
ejpam-3383	86	43	b	b	NOUN
ejpam-3383	86	44	implies	imply	VERB
ejpam-3383	86	45	a	a	DET
ejpam-3383	86	46	◦	◦	NOUN
ejpam-3383	86	47	c	c	X
ejpam-3383	86	48	�	�	PROPN
ejpam-3383	86	49	b	b	PROPN
ejpam-3383	86	50	◦	◦	NOUN
ejpam-3383	86	51	c	c	PROPN
ejpam-3383	86	52	and	and	CCONJ
ejpam-3383	86	53	c	c	AUX
ejpam-3383	86	54	◦	◦	VERB
ejpam-3383	86	55	a	a	DET
ejpam-3383	86	56	�	�	PROPN
ejpam-3383	86	57	c	c	PROPN
ejpam-3383	86	58	◦	◦	NOUN
ejpam-3383	86	59	b	b	NOUN
ejpam-3383	86	60	for	for	ADP
ejpam-3383	86	61	any	any	DET
ejpam-3383	86	62	c	c	PROPN
ejpam-3383	86	63	∈	∈	PROPN
ejpam-3383	86	64	s	s	X
ejpam-3383	86	65	in	in	ADP
ejpam-3383	86	66	the	the	DET
ejpam-3383	86	67	sense	sense	NOUN
ejpam-3383	86	68	that	that	SCONJ
ejpam-3383	86	69	for	for	ADP
ejpam-3383	86	70	every	every	DET
ejpam-3383	86	71	u	u	PROPN
ejpam-3383	86	72	∈	∈	PROPN
ejpam-3383	86	73	a	a	DET
ejpam-3383	86	74	◦	◦	NOUN
ejpam-3383	86	75	c	c	NOUN
ejpam-3383	86	76	there	there	PRON
ejpam-3383	86	77	exists	exist	VERB
ejpam-3383	86	78	v	v	ADP
ejpam-3383	86	79	∈	∈	PROPN
ejpam-3383	86	80	b	b	PROPN
ejpam-3383	86	81	◦	◦	NOUN
ejpam-3383	86	82	c	c	ADP
ejpam-3383	86	83	such	such	ADJ
ejpam-3383	86	84	that	that	SCONJ
ejpam-3383	86	85	u	u	PROPN
ejpam-3383	86	86	≤	≤	X
ejpam-3383	86	87	v	v	NOUN
ejpam-3383	86	88	and	and	CCONJ
ejpam-3383	86	89	for	for	ADP
ejpam-3383	86	90	every	every	DET
ejpam-3383	86	91	u	u	PROPN
ejpam-3383	86	92	∈	∈	PROPN
ejpam-3383	86	93	c	c	X
ejpam-3383	86	94	◦	◦	NOUN
ejpam-3383	86	95	a	a	DET
ejpam-3383	86	96	there	there	NOUN
ejpam-3383	86	97	exists	exist	VERB
ejpam-3383	87	1	v	v	ADP
ejpam-3383	87	2	∈	∈	PROPN
ejpam-3383	87	3	c	c	NOUN
ejpam-3383	87	4	◦	◦	NOUN
ejpam-3383	87	5	b	b	X
ejpam-3383	87	6	such	such	ADJ
ejpam-3383	87	7	that	that	DET
ejpam-3383	87	8	u	u	PROPN
ejpam-3383	87	9	≤	≤	X
ejpam-3383	87	10	v	v	ADP
ejpam-3383	87	11	[	[	X
ejpam-3383	87	12	18	18	NUM
ejpam-3383	87	13	]	]	PUNCT
ejpam-3383	87	14	.	.	PUNCT
ejpam-3383	88	1	a	a	DET
ejpam-3383	88	2	nonempty	nonempty	NOUN
ejpam-3383	88	3	subset	subset	VERB
ejpam-3383	88	4	a	a	PRON
ejpam-3383	88	5	of	of	ADP
ejpam-3383	88	6	an	an	DET
ejpam-3383	88	7	hypergroupoid	hypergroupoid	NOUN
ejpam-3383	88	8	(	(	PUNCT
ejpam-3383	88	9	s	s	NOUN
ejpam-3383	88	10	,	,	PUNCT
ejpam-3383	88	11	◦	◦	NOUN
ejpam-3383	88	12	)	)	PUNCT
ejpam-3383	88	13	is	be	AUX
ejpam-3383	88	14	called	call	VERB
ejpam-3383	88	15	a	a	DET
ejpam-3383	88	16	right	right	NOUN
ejpam-3383	88	17	(	(	PUNCT
ejpam-3383	88	18	resp	resp	NOUN
ejpam-3383	88	19	.	.	PUNCT
ejpam-3383	89	1	left	left	ADJ
ejpam-3383	89	2	)	)	PUNCT
ejpam-3383	89	3	ideal	ideal	NOUN
ejpam-3383	89	4	of	of	ADP
ejpam-3383	89	5	s	s	PRON
ejpam-3383	89	6	if	if	SCONJ
ejpam-3383	89	7	a	a	DET
ejpam-3383	89	8	∗	∗	NOUN
ejpam-3383	89	9	s	s	NOUN
ejpam-3383	89	10	⊆	⊆	NUM
ejpam-3383	89	11	a	a	DET
ejpam-3383	89	12	(	(	PUNCT
ejpam-3383	89	13	resp	resp	NOUN
ejpam-3383	89	14	.	.	PUNCT
ejpam-3383	90	1	s	s	PART
ejpam-3383	90	2	∗	∗	NOUN
ejpam-3383	90	3	a	a	DET
ejpam-3383	90	4	⊆	⊆	NUM
ejpam-3383	90	5	a	a	NOUN
ejpam-3383	90	6	)	)	PUNCT
ejpam-3383	90	7	.	.	PUNCT
ejpam-3383	91	1	if	if	SCONJ
ejpam-3383	91	2	,	,	PUNCT
ejpam-3383	91	3	in	in	ADP
ejpam-3383	91	4	particular	particular	ADJ
ejpam-3383	91	5	,	,	PUNCT
ejpam-3383	91	6	the	the	DET
ejpam-3383	91	7	hypergroupoid	hypergroupoid	PROPN
ejpam-3383	91	8	s	s	PROPN
ejpam-3383	91	9	is	be	AUX
ejpam-3383	91	10	an	an	DET
ejpam-3383	91	11	ordered	ordered	ADJ
ejpam-3383	91	12	hypergroupoid	hypergroupoid	NOUN
ejpam-3383	91	13	,	,	PUNCT
ejpam-3383	91	14	then	then	ADV
ejpam-3383	91	15	a	a	DET
ejpam-3383	91	16	set	set	NOUN
ejpam-3383	91	17	a	a	PRON
ejpam-3383	91	18	is	be	AUX
ejpam-3383	91	19	called	call	VERB
ejpam-3383	91	20	a	a	DET
ejpam-3383	91	21	right	right	NOUN
ejpam-3383	91	22	(	(	PUNCT
ejpam-3383	91	23	resp	resp	NOUN
ejpam-3383	91	24	.	.	PUNCT
ejpam-3383	92	1	left	left	ADJ
ejpam-3383	92	2	)	)	PUNCT
ejpam-3383	92	3	ideal	ideal	NOUN
ejpam-3383	92	4	of	of	ADP
ejpam-3383	92	5	s	s	PRON
ejpam-3383	92	6	if	if	SCONJ
ejpam-3383	92	7	a	a	PRON
ejpam-3383	92	8	is	be	AUX
ejpam-3383	92	9	a	a	DET
ejpam-3383	92	10	right	right	ADJ
ejpam-3383	92	11	(	(	PUNCT
ejpam-3383	92	12	resp	resp	NOUN
ejpam-3383	92	13	.	.	PUNCT
ejpam-3383	93	1	left	left	ADJ
ejpam-3383	93	2	)	)	PUNCT
ejpam-3383	93	3	ideal	ideal	NOUN
ejpam-3383	93	4	of	of	ADP
ejpam-3383	93	5	the	the	DET
ejpam-3383	93	6	hypergroupoid	hypergroupoid	PROPN
ejpam-3383	93	7	(	(	PUNCT
ejpam-3383	93	8	s	s	NOUN
ejpam-3383	93	9	,	,	PUNCT
ejpam-3383	93	10	◦	◦	NOUN
ejpam-3383	93	11	)	)	PUNCT
ejpam-3383	93	12	and	and	CCONJ
ejpam-3383	93	13	,	,	PUNCT
ejpam-3383	93	14	in	in	ADP
ejpam-3383	93	15	addition	addition	NOUN
ejpam-3383	93	16	if	if	SCONJ
ejpam-3383	93	17	a	a	DET
ejpam-3383	93	18	∈	∈	PROPN
ejpam-3383	93	19	a	a	PRON
ejpam-3383	93	20	and	and	CCONJ
ejpam-3383	93	21	s	s	PROPN
ejpam-3383	93	22	3	3	NUM
ejpam-3383	93	23	b	b	NOUN
ejpam-3383	93	24	≤	≤	NOUN
ejpam-3383	93	25	a	a	DET
ejpam-3383	93	26	implies	implie	NOUN
ejpam-3383	93	27	b	b	X
ejpam-3383	93	28	∈	∈	ADJ
ejpam-3383	93	29	a.	a.	NOUN
ejpam-3383	93	30	both	both	PRON
ejpam-3383	93	31	for	for	ADP
ejpam-3383	93	32	hypergroupoids	hypergroupoid	NOUN
ejpam-3383	93	33	or	or	CCONJ
ejpam-3383	93	34	ordered	order	VERB
ejpam-3383	93	35	hypergroupoids	hypergroupoid	NOUN
ejpam-3383	93	36	the	the	DET
ejpam-3383	93	37	sets	set	NOUN
ejpam-3383	93	38	that	that	PRON
ejpam-3383	93	39	are	be	AUX
ejpam-3383	93	40	at	at	ADP
ejpam-3383	93	41	the	the	DET
ejpam-3383	93	42	same	same	ADJ
ejpam-3383	93	43	time	time	NOUN
ejpam-3383	93	44	right	right	ADV
ejpam-3383	93	45	and	and	CCONJ
ejpam-3383	93	46	left	leave	VERB
ejpam-3383	93	47	ideals	ideal	NOUN
ejpam-3383	93	48	are	be	AUX
ejpam-3383	93	49	called	call	VERB
ejpam-3383	93	50	ideals	ideal	NOUN
ejpam-3383	93	51	.	.	PUNCT
ejpam-3383	94	1	lemma	lemma	PROPN
ejpam-3383	94	2	2.4	2.4	NUM
ejpam-3383	94	3	.	.	PUNCT
ejpam-3383	95	1	[	[	X
ejpam-3383	95	2	13	13	NUM
ejpam-3383	95	3	;	;	PUNCT
ejpam-3383	95	4	proposition	proposition	NOUN
ejpam-3383	95	5	9	9	NUM
ejpam-3383	95	6	]	]	PUNCT
ejpam-3383	95	7	for	for	ADP
ejpam-3383	95	8	an	an	DET
ejpam-3383	95	9	hypersemigroup	hypersemigroup	NOUN
ejpam-3383	95	10	(	(	PUNCT
ejpam-3383	95	11	s	s	NOUN
ejpam-3383	95	12	,	,	PUNCT
ejpam-3383	95	13	◦	◦	NOUN
ejpam-3383	95	14	)	)	PUNCT
ejpam-3383	95	15	and	and	CCONJ
ejpam-3383	95	16	nonempty	nonempty	ADJ
ejpam-3383	95	17	subsets	subset	NOUN
ejpam-3383	95	18	a	a	DET
ejpam-3383	95	19	,	,	PUNCT
ejpam-3383	95	20	b	b	NOUN
ejpam-3383	95	21	,	,	PUNCT
ejpam-3383	95	22	c	c	NOUN
ejpam-3383	95	23	of	of	ADP
ejpam-3383	95	24	s	s	PROPN
ejpam-3383	95	25	,	,	PUNCT
ejpam-3383	95	26	we	we	PRON
ejpam-3383	95	27	have	have	VERB
ejpam-3383	95	28	a	a	DET
ejpam-3383	95	29	∗	∗	NOUN
ejpam-3383	95	30	(	(	PUNCT
ejpam-3383	95	31	b	b	NOUN
ejpam-3383	95	32	∗	∗	NOUN
ejpam-3383	95	33	c	c	NOUN
ejpam-3383	95	34	)	)	PUNCT
ejpam-3383	95	35	=	=	SYM
ejpam-3383	95	36	(	(	PUNCT
ejpam-3383	95	37	a	a	DET
ejpam-3383	95	38	∗b	∗b	NOUN
ejpam-3383	95	39	)	)	PUNCT
ejpam-3383	95	40	∗	∗	NOUN
ejpam-3383	95	41	c.	c.	NOUN
ejpam-3383	96	1	the	the	DET
ejpam-3383	96	2	following	follow	VERB
ejpam-3383	96	3	lemma	lemma	PROPN
ejpam-3383	96	4	has	have	AUX
ejpam-3383	96	5	been	be	AUX
ejpam-3383	96	6	proved	prove	VERB
ejpam-3383	96	7	in	in	ADP
ejpam-3383	96	8	a	a	DET
ejpam-3383	96	9	more	more	ADV
ejpam-3383	96	10	general	general	ADJ
ejpam-3383	96	11	case	case	NOUN
ejpam-3383	96	12	in	in	ADP
ejpam-3383	96	13	[	[	NOUN
ejpam-3383	96	14	13	13	NUM
ejpam-3383	96	15	;	;	PUNCT
ejpam-3383	96	16	proposition	proposition	NOUN
ejpam-3383	96	17	7	7	NUM
ejpam-3383	96	18	]	]	PUNCT
ejpam-3383	96	19	.	.	PUNCT
ejpam-3383	97	1	since	since	SCONJ
ejpam-3383	97	2	this	this	DET
ejpam-3383	97	3	lemma	lemma	PROPN
ejpam-3383	97	4	plays	play	VERB
ejpam-3383	97	5	an	an	DET
ejpam-3383	97	6	essential	essential	ADJ
ejpam-3383	97	7	role	role	NOUN
ejpam-3383	97	8	in	in	ADP
ejpam-3383	97	9	the	the	DET
ejpam-3383	97	10	present	present	ADJ
ejpam-3383	97	11	paper	paper	NOUN
ejpam-3383	97	12	,	,	PUNCT
ejpam-3383	97	13	for	for	ADP
ejpam-3383	97	14	the	the	DET
ejpam-3383	97	15	sake	sake	NOUN
ejpam-3383	97	16	of	of	ADP
ejpam-3383	97	17	completeness	completeness	NOUN
ejpam-3383	97	18	,	,	PUNCT
ejpam-3383	97	19	we	we	PRON
ejpam-3383	97	20	will	will	AUX
ejpam-3383	97	21	give	give	VERB
ejpam-3383	97	22	its	its	PRON
ejpam-3383	97	23	proof	proof	NOUN
ejpam-3383	97	24	.	.	PUNCT
ejpam-3383	98	1	lemma	lemma	PROPN
ejpam-3383	98	2	2.5	2.5	NUM
ejpam-3383	98	3	.	.	PUNCT
ejpam-3383	99	1	if	if	SCONJ
ejpam-3383	99	2	(	(	PUNCT
ejpam-3383	99	3	s	s	NOUN
ejpam-3383	99	4	,	,	PUNCT
ejpam-3383	99	5	◦	◦	NOUN
ejpam-3383	99	6	)	)	PUNCT
ejpam-3383	99	7	is	be	AUX
ejpam-3383	99	8	an	an	DET
ejpam-3383	99	9	hypergroupoid	hypergroupoid	NOUN
ejpam-3383	99	10	then	then	ADV
ejpam-3383	99	11	,	,	PUNCT
ejpam-3383	99	12	for	for	ADP
ejpam-3383	99	13	any	any	DET
ejpam-3383	99	14	nonempty	nonempty	ADJ
ejpam-3383	99	15	subsets	subset	NOUN
ejpam-3383	99	16	a	a	DET
ejpam-3383	99	17	,	,	PUNCT
ejpam-3383	99	18	b	b	NOUN
ejpam-3383	99	19	,	,	PUNCT
ejpam-3383	99	20	c	c	NOUN
ejpam-3383	99	21	of	of	ADP
ejpam-3383	99	22	s	s	PROPN
ejpam-3383	99	23	,	,	PUNCT
ejpam-3383	99	24	we	we	PRON
ejpam-3383	99	25	have	have	VERB
ejpam-3383	99	26	(	(	PUNCT
ejpam-3383	99	27	1	1	NUM
ejpam-3383	99	28	)	)	PUNCT
ejpam-3383	99	29	(	(	PUNCT
ejpam-3383	99	30	a	a	DET
ejpam-3383	99	31	∪b	∪b	NOUN
ejpam-3383	99	32	)	)	PUNCT
ejpam-3383	99	33	∗	∗	NOUN
ejpam-3383	99	34	c	c	NOUN
ejpam-3383	100	1	=	=	SYM
ejpam-3383	100	2	(	(	PUNCT
ejpam-3383	100	3	a	a	DET
ejpam-3383	100	4	∗	∗	NOUN
ejpam-3383	100	5	c	c	NOUN
ejpam-3383	100	6	)	)	PUNCT
ejpam-3383	100	7	∪	∪	NOUN
ejpam-3383	100	8	(	(	PUNCT
ejpam-3383	100	9	b	b	NOUN
ejpam-3383	100	10	∗	∗	NOUN
ejpam-3383	100	11	c	c	NOUN
ejpam-3383	100	12	)	)	PUNCT
ejpam-3383	100	13	and	and	CCONJ
ejpam-3383	100	14	(	(	PUNCT
ejpam-3383	100	15	2	2	X
ejpam-3383	100	16	)	)	PUNCT
ejpam-3383	100	17	a	a	DET
ejpam-3383	100	18	∗	∗	NOUN
ejpam-3383	100	19	(	(	PUNCT
ejpam-3383	100	20	b	b	NOUN
ejpam-3383	100	21	∪	∪	X
ejpam-3383	100	22	c	c	NOUN
ejpam-3383	100	23	)	)	PUNCT
ejpam-3383	100	24	=	=	SYM
ejpam-3383	100	25	(	(	PUNCT
ejpam-3383	100	26	a	a	DET
ejpam-3383	100	27	∗b	∗b	NOUN
ejpam-3383	100	28	)	)	PUNCT
ejpam-3383	100	29	∪	∪	NOUN
ejpam-3383	100	30	(	(	PUNCT
ejpam-3383	100	31	a	a	DET
ejpam-3383	100	32	∗	∗	NOUN
ejpam-3383	100	33	c	c	NOUN
ejpam-3383	100	34	)	)	PUNCT
ejpam-3383	100	35	.	.	PUNCT
ejpam-3383	101	1	proof	proof	NOUN
ejpam-3383	101	2	.	.	PUNCT
ejpam-3383	102	1	since	since	SCONJ
ejpam-3383	102	2	a∪b	a∪b	PROPN
ejpam-3383	102	3	⊇	⊇	PROPN
ejpam-3383	102	4	a	a	DET
ejpam-3383	102	5	,	,	PUNCT
ejpam-3383	102	6	b	b	NOUN
ejpam-3383	102	7	,	,	PUNCT
ejpam-3383	102	8	we	we	PRON
ejpam-3383	102	9	have	have	VERB
ejpam-3383	102	10	(	(	PUNCT
ejpam-3383	102	11	a∪b	a∪b	ADJ
ejpam-3383	102	12	)	)	PUNCT
ejpam-3383	103	1	∗c	∗c	PROPN
ejpam-3383	103	2	⊇	⊇	NOUN
ejpam-3383	103	3	(	(	PUNCT
ejpam-3383	103	4	a	a	DET
ejpam-3383	103	5	∗c)∪	∗c)∪	PROPN
ejpam-3383	103	6	(	(	PUNCT
ejpam-3383	103	7	b	b	NOUN
ejpam-3383	103	8	∗c	∗c	NUM
ejpam-3383	103	9	)	)	PUNCT
ejpam-3383	103	10	.	.	PUNCT
ejpam-3383	104	1	if	if	SCONJ
ejpam-3383	104	2	x	x	SYM
ejpam-3383	104	3	∈	∈	PROPN
ejpam-3383	104	4	(	(	PUNCT
ejpam-3383	104	5	a∪b	a∪b	ADJ
ejpam-3383	104	6	)	)	PUNCT
ejpam-3383	104	7	∗c	∗c	PROPN
ejpam-3383	104	8	,	,	PUNCT
ejpam-3383	104	9	then	then	ADV
ejpam-3383	104	10	x	x	PART
ejpam-3383	104	11	∈	∈	PROPN
ejpam-3383	104	12	u	u	NOUN
ejpam-3383	104	13	◦	◦	NOUN
ejpam-3383	104	14	c	c	NOUN
ejpam-3383	104	15	for	for	ADP
ejpam-3383	104	16	some	some	DET
ejpam-3383	104	17	u	u	NOUN
ejpam-3383	104	18	∈	∈	PROPN
ejpam-3383	104	19	a	a	DET
ejpam-3383	104	20	∪	∪	ADJ
ejpam-3383	104	21	b	b	NOUN
ejpam-3383	104	22	,	,	PUNCT
ejpam-3383	104	23	c	c	PROPN
ejpam-3383	104	24	∈	∈	PROPN
ejpam-3383	104	25	c.	c.	NOUN
ejpam-3383	104	26	if	if	SCONJ
ejpam-3383	104	27	u	u	PROPN
ejpam-3383	104	28	∈	∈	PROPN
ejpam-3383	104	29	a	a	PRON
ejpam-3383	104	30	,	,	PUNCT
ejpam-3383	104	31	then	then	ADV
ejpam-3383	104	32	x	x	SYM
ejpam-3383	104	33	∈	∈	PROPN
ejpam-3383	104	34	a	a	DET
ejpam-3383	104	35	∗	∗	NOUN
ejpam-3383	104	36	c	c	NOUN
ejpam-3383	104	37	;	;	PUNCT
ejpam-3383	104	38	if	if	SCONJ
ejpam-3383	104	39	u	u	PROPN
ejpam-3383	104	40	∈	∈	PROPN
ejpam-3383	104	41	b	b	PROPN
ejpam-3383	104	42	,	,	PUNCT
ejpam-3383	104	43	then	then	ADV
ejpam-3383	104	44	x	x	PART
ejpam-3383	104	45	∈	∈	PROPN
ejpam-3383	104	46	b	b	PROPN
ejpam-3383	104	47	∗	∗	X
ejpam-3383	104	48	c	c	NOUN
ejpam-3383	104	49	,	,	PUNCT
ejpam-3383	104	50	so	so	ADV
ejpam-3383	104	51	x	x	SYM
ejpam-3383	104	52	∈	∈	PROPN
ejpam-3383	104	53	(	(	PUNCT
ejpam-3383	104	54	a	a	DET
ejpam-3383	104	55	∗	∗	NOUN
ejpam-3383	104	56	c	c	NOUN
ejpam-3383	104	57	)	)	PUNCT
ejpam-3383	104	58	∪	∪	NOUN
ejpam-3383	104	59	(	(	PUNCT
ejpam-3383	104	60	b	b	NOUN
ejpam-3383	104	61	∗	∗	NOUN
ejpam-3383	104	62	c	c	NOUN
ejpam-3383	104	63	)	)	PUNCT
ejpam-3383	104	64	.	.	PUNCT
ejpam-3383	105	1	�	�	PROPN
ejpam-3383	105	2	definition	definition	NOUN
ejpam-3383	105	3	2.6	2.6	NUM
ejpam-3383	105	4	.	.	PUNCT
ejpam-3383	106	1	[	[	X
ejpam-3383	106	2	7	7	NUM
ejpam-3383	106	3	,	,	PUNCT
ejpam-3383	106	4	8	8	NUM
ejpam-3383	106	5	]	]	PUNCT
ejpam-3383	106	6	let	let	VERB
ejpam-3383	106	7	s	s	PRON
ejpam-3383	106	8	be	be	AUX
ejpam-3383	106	9	a	a	DET
ejpam-3383	106	10	groupoid	groupoid	NOUN
ejpam-3383	106	11	or	or	CCONJ
ejpam-3383	106	12	an	an	DET
ejpam-3383	106	13	ordered	ordered	ADJ
ejpam-3383	106	14	groupoid	groupoid	NOUN
ejpam-3383	106	15	.	.	PUNCT
ejpam-3383	107	1	a	a	DET
ejpam-3383	107	2	subset	subset	NOUN
ejpam-3383	107	3	m	m	VERB
ejpam-3383	107	4	of	of	ADP
ejpam-3383	107	5	s	s	PRON
ejpam-3383	107	6	is	be	AUX
ejpam-3383	107	7	called	call	VERB
ejpam-3383	107	8	prime	prime	ADJ
ejpam-3383	107	9	if	if	SCONJ
ejpam-3383	107	10	for	for	ADP
ejpam-3383	107	11	any	any	DET
ejpam-3383	107	12	subsets	subset	NOUN
ejpam-3383	107	13	a	a	DET
ejpam-3383	107	14	,	,	PUNCT
ejpam-3383	107	15	b	b	NOUN
ejpam-3383	107	16	of	of	ADP
ejpam-3383	107	17	s	s	PRON
ejpam-3383	107	18	such	such	ADJ
ejpam-3383	107	19	that	that	SCONJ
ejpam-3383	107	20	ab	ab	PROPN
ejpam-3383	107	21	⊆m	⊆m	NOUN
ejpam-3383	107	22	,	,	PUNCT
ejpam-3383	107	23	we	we	PRON
ejpam-3383	107	24	have	have	VERB
ejpam-3383	107	25	a	a	DET
ejpam-3383	107	26	⊆m	⊆m	NOUN
ejpam-3383	107	27	or	or	CCONJ
ejpam-3383	107	28	b	b	NOUN
ejpam-3383	107	29	⊆m	⊆m	NOUN
ejpam-3383	107	30	.	.	PUNCT
ejpam-3383	108	1	it	it	PRON
ejpam-3383	108	2	is	be	AUX
ejpam-3383	108	3	called	call	VERB
ejpam-3383	108	4	semiprime	semiprime	NOUN
ejpam-3383	108	5	if	if	SCONJ
ejpam-3383	108	6	for	for	ADP
ejpam-3383	108	7	any	any	DET
ejpam-3383	108	8	subset	subset	NOUN
ejpam-3383	108	9	a	a	PRON
ejpam-3383	108	10	of	of	ADP
ejpam-3383	108	11	s	s	PRON
ejpam-3383	108	12	such	such	ADJ
ejpam-3383	108	13	that	that	DET
ejpam-3383	108	14	a2	a2	PROPN
ejpam-3383	108	15	⊆m	⊆m	NOUN
ejpam-3383	108	16	,	,	PUNCT
ejpam-3383	108	17	we	we	PRON
ejpam-3383	108	18	have	have	VERB
ejpam-3383	108	19	a	a	DET
ejpam-3383	108	20	⊆m	⊆m	NOUN
ejpam-3383	108	21	.	.	PUNCT
ejpam-3383	109	1	definition	definition	NOUN
ejpam-3383	109	2	2.7	2.7	NUM
ejpam-3383	109	3	.	.	PUNCT
ejpam-3383	110	1	[	[	X
ejpam-3383	110	2	7	7	NUM
ejpam-3383	110	3	,	,	PUNCT
ejpam-3383	110	4	8	8	NUM
ejpam-3383	110	5	]	]	PUNCT
ejpam-3383	110	6	let	let	VERB
ejpam-3383	110	7	s	s	PRON
ejpam-3383	110	8	be	be	AUX
ejpam-3383	110	9	a	a	DET
ejpam-3383	110	10	groupoid	groupoid	NOUN
ejpam-3383	110	11	or	or	CCONJ
ejpam-3383	110	12	an	an	DET
ejpam-3383	110	13	ordered	ordered	ADJ
ejpam-3383	110	14	groupoid	groupoid	NOUN
ejpam-3383	110	15	.	.	PUNCT
ejpam-3383	111	1	a	a	DET
ejpam-3383	111	2	subset	subset	NOUN
ejpam-3383	111	3	m	m	VERB
ejpam-3383	111	4	of	of	ADP
ejpam-3383	111	5	s	s	PRON
ejpam-3383	111	6	is	be	AUX
ejpam-3383	111	7	called	call	VERB
ejpam-3383	111	8	weakly	weakly	ADJ
ejpam-3383	111	9	prime	prime	NOUN
ejpam-3383	111	10	if	if	SCONJ
ejpam-3383	111	11	for	for	ADP
ejpam-3383	111	12	any	any	DET
ejpam-3383	111	13	ideals	ideal	NOUN
ejpam-3383	111	14	a	a	DET
ejpam-3383	111	15	,	,	PUNCT
ejpam-3383	111	16	b	b	NOUN
ejpam-3383	111	17	of	of	ADP
ejpam-3383	111	18	s	s	PRON
ejpam-3383	111	19	such	such	ADJ
ejpam-3383	111	20	that	that	SCONJ
ejpam-3383	111	21	ab	ab	PROPN
ejpam-3383	111	22	⊆	⊆	NUM
ejpam-3383	111	23	m	m	NOUN
ejpam-3383	111	24	,	,	PUNCT
ejpam-3383	111	25	we	we	PRON
ejpam-3383	111	26	have	have	VERB
ejpam-3383	111	27	a	a	DET
ejpam-3383	111	28	⊆	⊆	NUM
ejpam-3383	111	29	m	m	NOUN
ejpam-3383	111	30	or	or	CCONJ
ejpam-3383	111	31	b	b	NOUN
ejpam-3383	111	32	⊆m	⊆m	NOUN
ejpam-3383	111	33	.	.	PUNCT
ejpam-3383	112	1	it	it	PRON
ejpam-3383	112	2	is	be	AUX
ejpam-3383	112	3	called	call	VERB
ejpam-3383	112	4	weakly	weakly	ADJ
ejpam-3383	112	5	semiprime	semiprime	NOUN
ejpam-3383	112	6	if	if	SCONJ
ejpam-3383	112	7	for	for	ADP
ejpam-3383	112	8	any	any	DET
ejpam-3383	112	9	ideal	ideal	NOUN
ejpam-3383	112	10	a	a	PRON
ejpam-3383	112	11	of	of	ADP
ejpam-3383	112	12	s	s	PRON
ejpam-3383	112	13	such	such	ADJ
ejpam-3383	112	14	that	that	DET
ejpam-3383	112	15	a2	a2	PROPN
ejpam-3383	112	16	⊆m	⊆m	NOUN
ejpam-3383	112	17	,	,	PUNCT
ejpam-3383	112	18	we	we	PRON
ejpam-3383	112	19	have	have	VERB
ejpam-3383	112	20	a	a	DET
ejpam-3383	112	21	⊆m	⊆m	NOUN
ejpam-3383	112	22	.	.	PUNCT
ejpam-3383	113	1	definition	definition	NOUN
ejpam-3383	113	2	2.8	2.8	NUM
ejpam-3383	113	3	.	.	PUNCT
ejpam-3383	114	1	[	[	X
ejpam-3383	114	2	9	9	NUM
ejpam-3383	114	3	]	]	PUNCT
ejpam-3383	114	4	let	let	VERB
ejpam-3383	114	5	s	s	PRON
ejpam-3383	114	6	be	be	AUX
ejpam-3383	114	7	a	a	DET
ejpam-3383	114	8	γ	γ	NOUN
ejpam-3383	114	9	-	-	ADJ
ejpam-3383	114	10	groupoid	groupoid	NOUN
ejpam-3383	114	11	or	or	CCONJ
ejpam-3383	114	12	an	an	DET
ejpam-3383	114	13	ordered	order	VERB
ejpam-3383	114	14	γ	γ	NOUN
ejpam-3383	114	15	-	-	NOUN
ejpam-3383	114	16	groupoid	groupoid	NOUN
ejpam-3383	114	17	.	.	PUNCT
ejpam-3383	115	1	a	a	DET
ejpam-3383	115	2	subset	subset	NOUN
ejpam-3383	115	3	m	m	VERB
ejpam-3383	115	4	of	of	ADP
ejpam-3383	115	5	s	s	PRON
ejpam-3383	115	6	is	be	AUX
ejpam-3383	115	7	called	call	VERB
ejpam-3383	115	8	prime	prime	ADJ
ejpam-3383	115	9	if	if	SCONJ
ejpam-3383	115	10	for	for	ADP
ejpam-3383	115	11	any	any	DET
ejpam-3383	115	12	subsets	subset	NOUN
ejpam-3383	115	13	a	a	DET
ejpam-3383	115	14	,	,	PUNCT
ejpam-3383	115	15	b	b	NOUN
ejpam-3383	115	16	of	of	ADP
ejpam-3383	115	17	s	s	PRON
ejpam-3383	115	18	such	such	ADJ
ejpam-3383	115	19	that	that	SCONJ
ejpam-3383	115	20	aγb	aγb	NOUN
ejpam-3383	115	21	⊆m	⊆m	NOUN
ejpam-3383	115	22	,	,	PUNCT
ejpam-3383	115	23	we	we	PRON
ejpam-3383	115	24	have	have	VERB
ejpam-3383	115	25	a	a	DET
ejpam-3383	115	26	⊆m	⊆m	NOUN
ejpam-3383	115	27	or	or	CCONJ
ejpam-3383	115	28	b	b	NOUN
ejpam-3383	115	29	⊆m	⊆m	NOUN
ejpam-3383	115	30	.	.	PUNCT
ejpam-3383	116	1	it	it	PRON
ejpam-3383	116	2	is	be	AUX
ejpam-3383	116	3	called	call	VERB
ejpam-3383	116	4	semiprime	semiprime	NOUN
ejpam-3383	116	5	if	if	SCONJ
ejpam-3383	116	6	for	for	ADP
ejpam-3383	116	7	any	any	DET
ejpam-3383	116	8	subset	subset	NOUN
ejpam-3383	116	9	a	a	PRON
ejpam-3383	116	10	of	of	ADP
ejpam-3383	116	11	s	s	PRON
ejpam-3383	116	12	such	such	ADJ
ejpam-3383	116	13	that	that	DET
ejpam-3383	116	14	aγa	aγa	NOUN
ejpam-3383	116	15	⊆m	⊆m	NOUN
ejpam-3383	116	16	,	,	PUNCT
ejpam-3383	116	17	we	we	PRON
ejpam-3383	116	18	have	have	VERB
ejpam-3383	116	19	a	a	DET
ejpam-3383	116	20	⊆m	⊆m	NOUN
ejpam-3383	116	21	.	.	PUNCT
ejpam-3383	117	1	n.	n.	PROPN
ejpam-3383	117	2	kehayopulu	kehayopulu	PROPN
ejpam-3383	117	3	/	/	SYM
ejpam-3383	117	4	eur	eur	PROPN
ejpam-3383	117	5	.	.	PUNCT
ejpam-3383	118	1	j.	j.	PROPN
ejpam-3383	118	2	pure	pure	PROPN
ejpam-3383	118	3	appl	appl	PROPN
ejpam-3383	118	4	.	.	PROPN
ejpam-3383	118	5	math	math	PROPN
ejpam-3383	118	6	,	,	PUNCT
ejpam-3383	118	7	12	12	NUM
ejpam-3383	118	8	(	(	PUNCT
ejpam-3383	118	9	1	1	NUM
ejpam-3383	118	10	)	)	PUNCT
ejpam-3383	118	11	(	(	PUNCT
ejpam-3383	118	12	2019	2019	NUM
ejpam-3383	118	13	)	)	PUNCT
ejpam-3383	118	14	,	,	PUNCT
ejpam-3383	118	15	208	208	NUM
ejpam-3383	118	16	-	-	SYM
ejpam-3383	118	17	225	225	NUM
ejpam-3383	118	18	212	212	NUM
ejpam-3383	118	19	definition	definition	NOUN
ejpam-3383	118	20	2.9	2.9	NUM
ejpam-3383	118	21	.	.	PUNCT
ejpam-3383	119	1	[	[	X
ejpam-3383	119	2	9	9	NUM
ejpam-3383	119	3	]	]	PUNCT
ejpam-3383	119	4	let	let	VERB
ejpam-3383	119	5	s	s	PRON
ejpam-3383	119	6	be	be	AUX
ejpam-3383	119	7	a	a	DET
ejpam-3383	119	8	γ	γ	NOUN
ejpam-3383	119	9	-	-	ADJ
ejpam-3383	119	10	groupoid	groupoid	NOUN
ejpam-3383	119	11	or	or	CCONJ
ejpam-3383	119	12	an	an	DET
ejpam-3383	119	13	ordered	order	VERB
ejpam-3383	119	14	γ	γ	NOUN
ejpam-3383	119	15	-	-	NOUN
ejpam-3383	119	16	groupoid	groupoid	NOUN
ejpam-3383	119	17	.	.	PUNCT
ejpam-3383	120	1	a	a	DET
ejpam-3383	120	2	subset	subset	NOUN
ejpam-3383	120	3	m	m	VERB
ejpam-3383	120	4	of	of	ADP
ejpam-3383	120	5	s	s	PRON
ejpam-3383	120	6	is	be	AUX
ejpam-3383	120	7	called	call	VERB
ejpam-3383	120	8	weakly	weakly	ADJ
ejpam-3383	120	9	prime	prime	NOUN
ejpam-3383	120	10	if	if	SCONJ
ejpam-3383	120	11	for	for	ADP
ejpam-3383	120	12	any	any	DET
ejpam-3383	120	13	ideals	ideal	NOUN
ejpam-3383	120	14	a	a	DET
ejpam-3383	120	15	,	,	PUNCT
ejpam-3383	120	16	b	b	NOUN
ejpam-3383	120	17	of	of	ADP
ejpam-3383	120	18	s	s	PRON
ejpam-3383	120	19	such	such	ADJ
ejpam-3383	120	20	that	that	SCONJ
ejpam-3383	120	21	aγb	aγb	ADV
ejpam-3383	120	22	⊆	⊆	NUM
ejpam-3383	120	23	m	m	NOUN
ejpam-3383	120	24	,	,	PUNCT
ejpam-3383	120	25	we	we	PRON
ejpam-3383	120	26	have	have	VERB
ejpam-3383	120	27	a	a	DET
ejpam-3383	120	28	⊆	⊆	NUM
ejpam-3383	120	29	m	m	NOUN
ejpam-3383	120	30	or	or	CCONJ
ejpam-3383	120	31	b	b	NOUN
ejpam-3383	120	32	⊆	⊆	NUM
ejpam-3383	120	33	m	m	NOUN
ejpam-3383	120	34	.	.	PUNCT
ejpam-3383	121	1	it	it	PRON
ejpam-3383	121	2	is	be	AUX
ejpam-3383	121	3	called	call	VERB
ejpam-3383	121	4	weakly	weakly	ADJ
ejpam-3383	121	5	semiprime	semiprime	NOUN
ejpam-3383	121	6	if	if	SCONJ
ejpam-3383	121	7	for	for	ADP
ejpam-3383	121	8	any	any	DET
ejpam-3383	121	9	ideal	ideal	NOUN
ejpam-3383	121	10	a	a	PRON
ejpam-3383	121	11	of	of	ADP
ejpam-3383	121	12	s	s	PRON
ejpam-3383	121	13	such	such	ADJ
ejpam-3383	121	14	that	that	PRON
ejpam-3383	121	15	aγa	aγa	VERB
ejpam-3383	121	16	⊆	⊆	NUM
ejpam-3383	121	17	m	m	NOUN
ejpam-3383	121	18	,	,	PUNCT
ejpam-3383	121	19	we	we	PRON
ejpam-3383	121	20	have	have	VERB
ejpam-3383	121	21	a	a	DET
ejpam-3383	121	22	⊆m	⊆m	NOUN
ejpam-3383	121	23	.	.	PUNCT
ejpam-3383	122	1	definition	definition	NOUN
ejpam-3383	122	2	2.10	2.10	NUM
ejpam-3383	122	3	.	.	PUNCT
ejpam-3383	123	1	[	[	X
ejpam-3383	123	2	15	15	NUM
ejpam-3383	123	3	]	]	X
ejpam-3383	123	4	let	let	VERB
ejpam-3383	123	5	(	(	PUNCT
ejpam-3383	123	6	s	s	NOUN
ejpam-3383	123	7	,	,	PUNCT
ejpam-3383	123	8	◦	◦	NOUN
ejpam-3383	123	9	)	)	PUNCT
ejpam-3383	123	10	be	be	VERB
ejpam-3383	123	11	an	an	DET
ejpam-3383	123	12	hypergroupoid	hypergroupoid	NOUN
ejpam-3383	123	13	or	or	CCONJ
ejpam-3383	123	14	an	an	DET
ejpam-3383	123	15	ordered	order	VERB
ejpam-3383	123	16	hypergroupoid	hypergroupoid	PROPN
ejpam-3383	123	17	.	.	PUNCT
ejpam-3383	124	1	a	a	DET
ejpam-3383	124	2	subset	subset	NOUN
ejpam-3383	124	3	m	m	VERB
ejpam-3383	124	4	of	of	ADP
ejpam-3383	124	5	s	s	PRON
ejpam-3383	124	6	is	be	AUX
ejpam-3383	124	7	called	call	VERB
ejpam-3383	124	8	prime	prime	ADJ
ejpam-3383	124	9	if	if	SCONJ
ejpam-3383	124	10	for	for	ADP
ejpam-3383	124	11	any	any	DET
ejpam-3383	124	12	nonempty	nonempty	ADJ
ejpam-3383	124	13	subsets	subset	NOUN
ejpam-3383	124	14	a	a	DET
ejpam-3383	124	15	,	,	PUNCT
ejpam-3383	124	16	b	b	PROPN
ejpam-3383	124	17	of	of	ADP
ejpam-3383	124	18	s	s	PRON
ejpam-3383	124	19	such	such	ADJ
ejpam-3383	124	20	that	that	DET
ejpam-3383	124	21	a∗b	a∗b	NUM
ejpam-3383	124	22	⊆m	⊆m	NOUN
ejpam-3383	124	23	,	,	PUNCT
ejpam-3383	124	24	we	we	PRON
ejpam-3383	124	25	have	have	VERB
ejpam-3383	124	26	a	a	DET
ejpam-3383	124	27	⊆m	⊆m	NOUN
ejpam-3383	124	28	or	or	CCONJ
ejpam-3383	124	29	b	b	NOUN
ejpam-3383	124	30	⊆m	⊆m	NOUN
ejpam-3383	124	31	.	.	PUNCT
ejpam-3383	125	1	it	it	PRON
ejpam-3383	125	2	is	be	AUX
ejpam-3383	125	3	called	call	VERB
ejpam-3383	125	4	semiprime	semiprime	NOUN
ejpam-3383	125	5	if	if	SCONJ
ejpam-3383	125	6	for	for	ADP
ejpam-3383	125	7	any	any	DET
ejpam-3383	125	8	nonempty	nonempty	NOUN
ejpam-3383	125	9	subset	subset	VERB
ejpam-3383	125	10	a	a	PRON
ejpam-3383	125	11	of	of	ADP
ejpam-3383	125	12	s	s	PRON
ejpam-3383	125	13	such	such	ADJ
ejpam-3383	125	14	that	that	SCONJ
ejpam-3383	125	15	a	a	DET
ejpam-3383	125	16	∗a	∗a	ADJ
ejpam-3383	125	17	⊆m	⊆m	NOUN
ejpam-3383	125	18	,	,	PUNCT
ejpam-3383	125	19	we	we	PRON
ejpam-3383	125	20	have	have	VERB
ejpam-3383	125	21	a	a	DET
ejpam-3383	125	22	⊆m	⊆m	NOUN
ejpam-3383	125	23	.	.	PUNCT
ejpam-3383	126	1	definition	definition	NOUN
ejpam-3383	126	2	2.11	2.11	NUM
ejpam-3383	126	3	.	.	PUNCT
ejpam-3383	127	1	[	[	X
ejpam-3383	127	2	15	15	NUM
ejpam-3383	127	3	]	]	X
ejpam-3383	127	4	let	let	VERB
ejpam-3383	127	5	(	(	PUNCT
ejpam-3383	127	6	s	s	NOUN
ejpam-3383	127	7	,	,	PUNCT
ejpam-3383	127	8	◦	◦	NOUN
ejpam-3383	127	9	)	)	PUNCT
ejpam-3383	127	10	be	be	VERB
ejpam-3383	127	11	an	an	DET
ejpam-3383	127	12	hypergroupoid	hypergroupoid	NOUN
ejpam-3383	127	13	or	or	CCONJ
ejpam-3383	127	14	an	an	DET
ejpam-3383	127	15	ordered	order	VERB
ejpam-3383	127	16	hypergroupoid	hypergroupoid	PROPN
ejpam-3383	127	17	.	.	PUNCT
ejpam-3383	128	1	a	a	DET
ejpam-3383	128	2	subset	subset	NOUN
ejpam-3383	128	3	m	m	VERB
ejpam-3383	128	4	of	of	ADP
ejpam-3383	128	5	s	s	PRON
ejpam-3383	128	6	is	be	AUX
ejpam-3383	128	7	called	call	VERB
ejpam-3383	128	8	weakly	weakly	ADJ
ejpam-3383	128	9	prime	prime	NOUN
ejpam-3383	128	10	if	if	SCONJ
ejpam-3383	128	11	for	for	ADP
ejpam-3383	128	12	any	any	DET
ejpam-3383	128	13	ideals	ideal	NOUN
ejpam-3383	128	14	a	a	DET
ejpam-3383	128	15	,	,	PUNCT
ejpam-3383	128	16	b	b	NOUN
ejpam-3383	128	17	of	of	ADP
ejpam-3383	128	18	s	s	PRON
ejpam-3383	128	19	such	such	ADJ
ejpam-3383	128	20	that	that	SCONJ
ejpam-3383	128	21	a	a	DET
ejpam-3383	128	22	∗b	∗b	PROPN
ejpam-3383	128	23	⊆m	⊆m	NOUN
ejpam-3383	128	24	,	,	PUNCT
ejpam-3383	128	25	we	we	PRON
ejpam-3383	128	26	have	have	VERB
ejpam-3383	128	27	a	a	DET
ejpam-3383	128	28	⊆	⊆	NUM
ejpam-3383	128	29	m	m	NOUN
ejpam-3383	128	30	or	or	CCONJ
ejpam-3383	128	31	b	b	NOUN
ejpam-3383	128	32	⊆	⊆	NUM
ejpam-3383	128	33	m	m	NOUN
ejpam-3383	128	34	.	.	PUNCT
ejpam-3383	129	1	it	it	PRON
ejpam-3383	129	2	is	be	AUX
ejpam-3383	129	3	called	call	VERB
ejpam-3383	129	4	weakly	weakly	ADJ
ejpam-3383	129	5	semiprime	semiprime	NOUN
ejpam-3383	129	6	if	if	SCONJ
ejpam-3383	129	7	for	for	ADP
ejpam-3383	129	8	any	any	DET
ejpam-3383	129	9	ideal	ideal	NOUN
ejpam-3383	129	10	a	a	PRON
ejpam-3383	129	11	of	of	ADP
ejpam-3383	129	12	s	s	PRON
ejpam-3383	129	13	such	such	ADJ
ejpam-3383	129	14	that	that	SCONJ
ejpam-3383	129	15	a	a	DET
ejpam-3383	129	16	∗a	∗a	ADJ
ejpam-3383	129	17	⊆m	⊆m	NOUN
ejpam-3383	129	18	,	,	PUNCT
ejpam-3383	129	19	we	we	PRON
ejpam-3383	129	20	have	have	VERB
ejpam-3383	129	21	a	a	DET
ejpam-3383	129	22	⊆m	⊆m	NOUN
ejpam-3383	129	23	.	.	PUNCT
ejpam-3383	130	1	for	for	ADP
ejpam-3383	130	2	a	a	DET
ejpam-3383	130	3	nonempty	nonempty	NOUN
ejpam-3383	130	4	subset	subset	VERB
ejpam-3383	130	5	a	a	PRON
ejpam-3383	130	6	of	of	ADP
ejpam-3383	130	7	a	a	DET
ejpam-3383	130	8	semigroup	semigroup	NOUN
ejpam-3383	130	9	,	,	PUNCT
ejpam-3383	130	10	ordered	order	VERB
ejpam-3383	130	11	semigroup	semigroup	PROPN
ejpam-3383	130	12	,	,	PUNCT
ejpam-3383	130	13	γ	γ	PROPN
ejpam-3383	130	14	-	-	PUNCT
ejpam-3383	130	15	semigroup	semigroup	NOUN
ejpam-3383	130	16	,	,	PUNCT
ejpam-3383	130	17	ordered	order	VERB
ejpam-3383	130	18	γsemigroup	γsemigroup	NOUN
ejpam-3383	130	19	,	,	PUNCT
ejpam-3383	130	20	hypersemigroup	hypersemigroup	NOUN
ejpam-3383	130	21	or	or	CCONJ
ejpam-3383	130	22	ordered	order	VERB
ejpam-3383	130	23	hypersemigroup	hypersemigroup	PROPN
ejpam-3383	130	24	s	s	PROPN
ejpam-3383	130	25	,	,	PUNCT
ejpam-3383	130	26	we	we	PRON
ejpam-3383	130	27	denote	denote	VERB
ejpam-3383	130	28	by	by	ADP
ejpam-3383	130	29	i(a	i(a	PROPN
ejpam-3383	130	30	)	)	PUNCT
ejpam-3383	130	31	the	the	DET
ejpam-3383	130	32	ideal	ideal	NOUN
ejpam-3383	130	33	of	of	ADP
ejpam-3383	130	34	s	s	AUX
ejpam-3383	130	35	generated	generate	VERB
ejpam-3383	130	36	by	by	ADP
ejpam-3383	130	37	a	a	PRON
ejpam-3383	130	38	,	,	PUNCT
ejpam-3383	130	39	and	and	CCONJ
ejpam-3383	130	40	we	we	PRON
ejpam-3383	130	41	have	have	VERB
ejpam-3383	130	42	i(a	i(a	PROPN
ejpam-3383	130	43	)	)	PUNCT
ejpam-3383	130	44	=	=	SYM
ejpam-3383	131	1	a∪sa∪as∪sas	a∪sa∪as∪sa	NOUN
ejpam-3383	131	2	,	,	PUNCT
ejpam-3383	131	3	i(a	i(a	NUM
ejpam-3383	131	4	)	)	PUNCT
ejpam-3383	131	5	=	=	PUNCT
ejpam-3383	131	6	(	(	PUNCT
ejpam-3383	131	7	a∪sa∪as∪sas	a∪sa∪as∪sas	ADP
ejpam-3383	131	8	]	]	X
ejpam-3383	131	9	,	,	PUNCT
ejpam-3383	131	10	i(a	i(a	PROPN
ejpam-3383	131	11	)	)	PUNCT
ejpam-3383	131	12	=	=	SYM
ejpam-3383	131	13	a∪sγa∪aγs∪sγaγs	a∪sγa∪aγs∪sγaγs	PROPN
ejpam-3383	131	14	,	,	PUNCT
ejpam-3383	131	15	i(a	i(a	PROPN
ejpam-3383	131	16	)	)	PUNCT
ejpam-3383	131	17	=	=	PUNCT
ejpam-3383	131	18	(	(	PUNCT
ejpam-3383	131	19	a∪sγa∪aγs∪sγaγs	a∪sγa∪aγs∪sγaγs	PROPN
ejpam-3383	131	20	]	]	X
ejpam-3383	131	21	,	,	PUNCT
ejpam-3383	131	22	i(a	i(a	PROPN
ejpam-3383	131	23	)	)	PUNCT
ejpam-3383	131	24	=	=	SYM
ejpam-3383	132	1	a∪s	a∪s	ADV
ejpam-3383	132	2	∗a∪	∗a∪	ADV
ejpam-3383	132	3	a	a	DET
ejpam-3383	132	4	∗	∗	NOUN
ejpam-3383	132	5	s	s	NOUN
ejpam-3383	132	6	∪	∪	NOUN
ejpam-3383	132	7	s	s	PART
ejpam-3383	132	8	∗a	∗a	ADJ
ejpam-3383	132	9	∗	∗	NOUN
ejpam-3383	132	10	s	s	NOUN
ejpam-3383	132	11	,	,	PUNCT
ejpam-3383	132	12	i(a	i(a	PROPN
ejpam-3383	132	13	)	)	PUNCT
ejpam-3383	132	14	=	=	PUNCT
ejpam-3383	133	1	(	(	PUNCT
ejpam-3383	133	2	a	a	DET
ejpam-3383	133	3	∪	∪	NOUN
ejpam-3383	133	4	s	s	X
ejpam-3383	133	5	∗a	∗a	ADJ
ejpam-3383	133	6	∪a	∪a	X
ejpam-3383	133	7	∗	∗	NOUN
ejpam-3383	133	8	s	s	PART
ejpam-3383	133	9	∪	∪	NOUN
ejpam-3383	133	10	s	s	PART
ejpam-3383	133	11	∗a	∗a	ADJ
ejpam-3383	133	12	∗	∗	NOUN
ejpam-3383	133	13	s	s	NOUN
ejpam-3383	133	14	]	]	X
ejpam-3383	133	15	,	,	PUNCT
ejpam-3383	133	16	respectively	respectively	ADV
ejpam-3383	133	17	;	;	PUNCT
ejpam-3383	133	18	where	where	SCONJ
ejpam-3383	133	19	for	for	ADP
ejpam-3383	133	20	a	a	DET
ejpam-3383	133	21	subset	subset	NOUN
ejpam-3383	133	22	b	b	NOUN
ejpam-3383	133	23	of	of	ADP
ejpam-3383	133	24	s	s	PROPN
ejpam-3383	133	25	,	,	PUNCT
ejpam-3383	133	26	the	the	DET
ejpam-3383	133	27	symbol	symbol	NOUN
ejpam-3383	133	28	(	(	PUNCT
ejpam-3383	133	29	b	b	X
ejpam-3383	133	30	]	]	PUNCT
ejpam-3383	133	31	denotes	denote	VERB
ejpam-3383	133	32	the	the	DET
ejpam-3383	133	33	subset	subset	NOUN
ejpam-3383	133	34	of	of	ADP
ejpam-3383	133	35	s	s	PRON
ejpam-3383	133	36	defined	define	VERB
ejpam-3383	133	37	by	by	ADP
ejpam-3383	133	38	{	{	PUNCT
ejpam-3383	133	39	t	t	PROPN
ejpam-3383	133	40	∈	∈	PROPN
ejpam-3383	133	41	s	s	AUX
ejpam-3383	133	42	|	|	NOUN
ejpam-3383	133	43	t	t	X
ejpam-3383	133	44	≤	≤	NUM
ejpam-3383	133	45	b	b	NOUN
ejpam-3383	133	46	for	for	ADP
ejpam-3383	133	47	some	some	DET
ejpam-3383	133	48	b	b	NOUN
ejpam-3383	133	49	∈	∈	PROPN
ejpam-3383	133	50	b	b	NOUN
ejpam-3383	133	51	}	}	PUNCT
ejpam-3383	133	52	.	.	PUNCT
ejpam-3383	134	1	when	when	SCONJ
ejpam-3383	134	2	is	be	AUX
ejpam-3383	134	3	convenient	convenient	ADJ
ejpam-3383	134	4	and	and	CCONJ
ejpam-3383	134	5	no	no	DET
ejpam-3383	134	6	confusion	confusion	NOUN
ejpam-3383	134	7	is	be	AUX
ejpam-3383	134	8	possible	possible	ADJ
ejpam-3383	134	9	,	,	PUNCT
ejpam-3383	134	10	we	we	PRON
ejpam-3383	134	11	identify	identify	VERB
ejpam-3383	134	12	the	the	DET
ejpam-3383	134	13	singleton	singleton	NOUN
ejpam-3383	134	14	{	{	PUNCT
ejpam-3383	134	15	a	a	PROPN
ejpam-3383	134	16	}	}	PUNCT
ejpam-3383	134	17	by	by	ADP
ejpam-3383	134	18	the	the	DET
ejpam-3383	134	19	element	element	NOUN
ejpam-3383	134	20	a.	a.	NOUN
ejpam-3383	134	21	3	3	NUM
ejpam-3383	134	22	.	.	PUNCT
ejpam-3383	135	1	on	on	ADP
ejpam-3383	135	2	∨e	∨e	NOUN
ejpam-3383	135	3	-	-	PUNCT
ejpam-3383	135	4	semigroups	semigroup	NOUN
ejpam-3383	135	5	let	let	VERB
ejpam-3383	135	6	(	(	PUNCT
ejpam-3383	135	7	s	s	X
ejpam-3383	135	8	,	,	PUNCT
ejpam-3383	135	9	·	·	PUNCT
ejpam-3383	135	10	,	,	PUNCT
ejpam-3383	135	11	≤	≤	NUM
ejpam-3383	135	12	)	)	PUNCT
ejpam-3383	135	13	be	be	VERB
ejpam-3383	135	14	a	a	DET
ejpam-3383	135	15	poe	poe	PROPN
ejpam-3383	135	16	-	-	NOUN
ejpam-3383	135	17	groupoid	groupoid	PROPN
ejpam-3383	135	18	.	.	PUNCT
ejpam-3383	136	1	an	an	DET
ejpam-3383	136	2	element	element	NOUN
ejpam-3383	136	3	m	m	NOUN
ejpam-3383	136	4	of	of	ADP
ejpam-3383	136	5	s	s	PRON
ejpam-3383	136	6	is	be	AUX
ejpam-3383	136	7	called	call	VERB
ejpam-3383	136	8	weakly	weakly	ADJ
ejpam-3383	136	9	prime	prime	NOUN
ejpam-3383	136	10	if	if	SCONJ
ejpam-3383	136	11	for	for	ADP
ejpam-3383	136	12	any	any	DET
ejpam-3383	136	13	ideal	ideal	ADJ
ejpam-3383	136	14	elements	element	NOUN
ejpam-3383	136	15	a	a	DET
ejpam-3383	136	16	,	,	PUNCT
ejpam-3383	136	17	b	b	PROPN
ejpam-3383	136	18	of	of	ADP
ejpam-3383	136	19	s	s	PRON
ejpam-3383	136	20	such	such	ADJ
ejpam-3383	136	21	that	that	SCONJ
ejpam-3383	136	22	ab	ab	PROPN
ejpam-3383	136	23	≤	≤	PROPN
ejpam-3383	136	24	m	m	VERB
ejpam-3383	136	25	we	we	PRON
ejpam-3383	136	26	have	have	VERB
ejpam-3383	136	27	a	a	DET
ejpam-3383	136	28	≤	≤	NUM
ejpam-3383	136	29	m	m	NOUN
ejpam-3383	136	30	or	or	CCONJ
ejpam-3383	136	31	b	b	PROPN
ejpam-3383	136	32	≤	≤	NUM
ejpam-3383	136	33	m.	m.	NOUN
ejpam-3383	136	34	it	it	PRON
ejpam-3383	136	35	is	be	AUX
ejpam-3383	136	36	called	call	VERB
ejpam-3383	136	37	weakly	weakly	ADJ
ejpam-3383	136	38	semiprime	semiprime	NOUN
ejpam-3383	136	39	if	if	SCONJ
ejpam-3383	136	40	for	for	ADP
ejpam-3383	136	41	any	any	DET
ejpam-3383	136	42	ideal	ideal	ADJ
ejpam-3383	136	43	element	element	NOUN
ejpam-3383	136	44	a	a	PRON
ejpam-3383	136	45	of	of	ADP
ejpam-3383	136	46	s	s	PRON
ejpam-3383	136	47	such	such	ADJ
ejpam-3383	136	48	that	that	DET
ejpam-3383	136	49	a2	a2	PROPN
ejpam-3383	136	50	≤	≤	PROPN
ejpam-3383	137	1	m	m	VERB
ejpam-3383	138	1	we	we	PRON
ejpam-3383	138	2	have	have	VERB
ejpam-3383	138	3	a	a	DET
ejpam-3383	138	4	≤	≤	NUM
ejpam-3383	138	5	m.	m.	NOUN
ejpam-3383	138	6	an	an	DET
ejpam-3383	138	7	element	element	NOUN
ejpam-3383	138	8	m	m	NOUN
ejpam-3383	138	9	of	of	ADP
ejpam-3383	138	10	a	a	DET
ejpam-3383	138	11	poe	poe	PROPN
ejpam-3383	138	12	-	-	PROPN
ejpam-3383	138	13	groupoid	groupoid	PROPN
ejpam-3383	138	14	s	s	PART
ejpam-3383	138	15	is	be	AUX
ejpam-3383	138	16	called	call	VERB
ejpam-3383	138	17	prime	prime	ADJ
ejpam-3383	138	18	if	if	SCONJ
ejpam-3383	138	19	for	for	SCONJ
ejpam-3383	138	20	any	any	DET
ejpam-3383	138	21	elements	element	NOUN
ejpam-3383	138	22	a	a	DET
ejpam-3383	138	23	,	,	PUNCT
ejpam-3383	138	24	b	b	PROPN
ejpam-3383	138	25	of	of	ADP
ejpam-3383	138	26	s	s	PRON
ejpam-3383	138	27	such	such	ADJ
ejpam-3383	138	28	that	that	SCONJ
ejpam-3383	138	29	ab	ab	PROPN
ejpam-3383	138	30	≤	≤	PROPN
ejpam-3383	138	31	m	m	VERB
ejpam-3383	138	32	we	we	PRON
ejpam-3383	138	33	have	have	VERB
ejpam-3383	138	34	a	a	DET
ejpam-3383	138	35	≤	≤	NUM
ejpam-3383	138	36	m	m	NOUN
ejpam-3383	138	37	or	or	CCONJ
ejpam-3383	138	38	b	b	PROPN
ejpam-3383	138	39	≤	≤	NUM
ejpam-3383	138	40	m.	m.	NOUN
ejpam-3383	138	41	it	it	PRON
ejpam-3383	138	42	is	be	AUX
ejpam-3383	138	43	called	call	VERB
ejpam-3383	138	44	semiprime	semiprime	NOUN
ejpam-3383	138	45	if	if	SCONJ
ejpam-3383	138	46	for	for	ADP
ejpam-3383	138	47	any	any	DET
ejpam-3383	138	48	element	element	NOUN
ejpam-3383	138	49	a	a	PRON
ejpam-3383	138	50	of	of	ADP
ejpam-3383	138	51	s	s	PRON
ejpam-3383	139	1	such	such	ADJ
ejpam-3383	139	2	that	that	DET
ejpam-3383	139	3	a2	a2	PROPN
ejpam-3383	139	4	≤	≤	PROPN
ejpam-3383	140	1	m	m	VERB
ejpam-3383	141	1	we	we	PRON
ejpam-3383	141	2	have	have	VERB
ejpam-3383	141	3	a	a	DET
ejpam-3383	141	4	≤	≤	ADJ
ejpam-3383	141	5	m.	m.	NOUN
ejpam-3383	141	6	for	for	ADP
ejpam-3383	141	7	a	a	DET
ejpam-3383	141	8	∨e	∨e	NOUN
ejpam-3383	141	9	-	-	PUNCT
ejpam-3383	141	10	semigroup	semigroup	NOUN
ejpam-3383	141	11	s	s	PROPN
ejpam-3383	141	12	and	and	CCONJ
ejpam-3383	141	13	an	an	DET
ejpam-3383	141	14	element	element	NOUN
ejpam-3383	141	15	a	a	PRON
ejpam-3383	141	16	of	of	ADP
ejpam-3383	141	17	s	s	PROPN
ejpam-3383	141	18	,	,	PUNCT
ejpam-3383	141	19	we	we	PRON
ejpam-3383	141	20	denote	denote	VERB
ejpam-3383	141	21	by	by	ADP
ejpam-3383	141	22	i(a	i(a	PROPN
ejpam-3383	141	23	)	)	PUNCT
ejpam-3383	141	24	the	the	DET
ejpam-3383	141	25	ideal	ideal	ADJ
ejpam-3383	141	26	element	element	NOUN
ejpam-3383	141	27	of	of	ADP
ejpam-3383	141	28	s	s	PRON
ejpam-3383	141	29	generated	generate	VERB
ejpam-3383	141	30	by	by	ADP
ejpam-3383	141	31	a	a	PRON
ejpam-3383	141	32	,	,	PUNCT
ejpam-3383	141	33	and	and	CCONJ
ejpam-3383	141	34	we	we	PRON
ejpam-3383	141	35	have	have	VERB
ejpam-3383	141	36	i(a	i(a	PROPN
ejpam-3383	141	37	)	)	PUNCT
ejpam-3383	141	38	=	=	PUNCT
ejpam-3383	141	39	a	a	DET
ejpam-3383	141	40	∨	∨	NUM
ejpam-3383	141	41	ea	ea	NOUN
ejpam-3383	141	42	∨	∨	PROPN
ejpam-3383	141	43	ae	ae	PROPN
ejpam-3383	141	44	∨	∨	PROPN
ejpam-3383	141	45	eae	eae	PROPN
ejpam-3383	141	46	.	.	PUNCT
ejpam-3383	142	1	proposition	proposition	NOUN
ejpam-3383	142	2	3.1	3.1	NUM
ejpam-3383	142	3	.	.	PUNCT
ejpam-3383	143	1	let	let	VERB
ejpam-3383	143	2	s	s	PRON
ejpam-3383	143	3	be	be	AUX
ejpam-3383	143	4	a	a	DET
ejpam-3383	143	5	∨e	∨e	NOUN
ejpam-3383	143	6	-	-	PUNCT
ejpam-3383	143	7	semigroup	semigroup	NOUN
ejpam-3383	143	8	and	and	CCONJ
ejpam-3383	143	9	m	m	AUX
ejpam-3383	143	10	be	be	VERB
ejpam-3383	143	11	an	an	DET
ejpam-3383	143	12	ideal	ideal	ADJ
ejpam-3383	143	13	element	element	NOUN
ejpam-3383	143	14	of	of	ADP
ejpam-3383	143	15	s.	s.	PROPN
ejpam-3383	143	16	the	the	DET
ejpam-3383	143	17	following	follow	VERB
ejpam-3383	143	18	are	be	AUX
ejpam-3383	143	19	equivalent	equivalent	ADJ
ejpam-3383	143	20	:	:	PUNCT
ejpam-3383	143	21	(	(	PUNCT
ejpam-3383	143	22	1	1	X
ejpam-3383	143	23	)	)	PUNCT
ejpam-3383	143	24	m	m	VERB
ejpam-3383	143	25	is	be	AUX
ejpam-3383	143	26	weakly	weakly	ADV
ejpam-3383	143	27	prime	prime	ADJ
ejpam-3383	143	28	.	.	PUNCT
ejpam-3383	144	1	(	(	PUNCT
ejpam-3383	144	2	2	2	X
ejpam-3383	144	3	)	)	PUNCT
ejpam-3383	144	4	if	if	SCONJ
ejpam-3383	144	5	a	a	PRON
ejpam-3383	144	6	,	,	PUNCT
ejpam-3383	144	7	b	b	X
ejpam-3383	144	8	∈	∈	NOUN
ejpam-3383	144	9	fr	fr	NOUN
ejpam-3383	144	10	such	such	ADJ
ejpam-3383	144	11	that	that	SCONJ
ejpam-3383	144	12	ab	ab	PROPN
ejpam-3383	144	13	≤	≤	PROPN
ejpam-3383	144	14	m	m	PROPN
ejpam-3383	144	15	,	,	PUNCT
ejpam-3383	144	16	then	then	ADV
ejpam-3383	144	17	a	a	DET
ejpam-3383	144	18	≤	≤	NUM
ejpam-3383	144	19	m	m	VERB
ejpam-3383	144	20	or	or	CCONJ
ejpam-3383	144	21	b	b	NOUN
ejpam-3383	144	22	≤	≤	NUM
ejpam-3383	144	23	m.	m.	NOUN
ejpam-3383	144	24	(	(	PUNCT
ejpam-3383	144	25	3	3	X
ejpam-3383	144	26	)	)	PUNCT
ejpam-3383	144	27	if	if	SCONJ
ejpam-3383	144	28	a	a	PRON
ejpam-3383	144	29	,	,	PUNCT
ejpam-3383	144	30	b	b	X
ejpam-3383	144	31	∈	∈	NOUN
ejpam-3383	144	32	fl	fl	ADP
ejpam-3383	144	33	such	such	ADJ
ejpam-3383	144	34	that	that	SCONJ
ejpam-3383	144	35	ab	ab	PROPN
ejpam-3383	144	36	≤	≤	PROPN
ejpam-3383	144	37	m	m	PROPN
ejpam-3383	144	38	,	,	PUNCT
ejpam-3383	144	39	then	then	ADV
ejpam-3383	144	40	a	a	DET
ejpam-3383	144	41	≤	≤	NUM
ejpam-3383	144	42	m	m	VERB
ejpam-3383	144	43	or	or	CCONJ
ejpam-3383	144	44	b	b	NOUN
ejpam-3383	144	45	≤	≤	NUM
ejpam-3383	144	46	m.	m.	NOUN
ejpam-3383	144	47	(	(	PUNCT
ejpam-3383	144	48	4	4	NUM
ejpam-3383	144	49	)	)	PUNCT
ejpam-3383	144	50	if	if	SCONJ
ejpam-3383	144	51	a	a	DET
ejpam-3383	144	52	∈	∈	PROPN
ejpam-3383	144	53	fr	fr	NOUN
ejpam-3383	144	54	and	and	CCONJ
ejpam-3383	144	55	b	b	X
ejpam-3383	144	56	∈	∈	NOUN
ejpam-3383	144	57	fl	fl	ADP
ejpam-3383	144	58	such	such	ADJ
ejpam-3383	144	59	that	that	SCONJ
ejpam-3383	144	60	ab	ab	PROPN
ejpam-3383	144	61	≤	≤	PROPN
ejpam-3383	144	62	m	m	PROPN
ejpam-3383	144	63	,	,	PUNCT
ejpam-3383	144	64	then	then	ADV
ejpam-3383	144	65	a	a	DET
ejpam-3383	144	66	≤	≤	NUM
ejpam-3383	144	67	m	m	VERB
ejpam-3383	144	68	or	or	CCONJ
ejpam-3383	144	69	b	b	NOUN
ejpam-3383	144	70	≤	≤	NUM
ejpam-3383	144	71	m.	m.	NOUN
ejpam-3383	144	72	proof	proof	NOUN
ejpam-3383	144	73	.	.	PUNCT
ejpam-3383	145	1	(	(	PUNCT
ejpam-3383	145	2	1	1	X
ejpam-3383	145	3	)	)	PUNCT
ejpam-3383	145	4	=	=	NOUN
ejpam-3383	145	5	⇒	⇒	NOUN
ejpam-3383	145	6	(	(	PUNCT
ejpam-3383	145	7	2	2	NUM
ejpam-3383	145	8	)	)	PUNCT
ejpam-3383	145	9	.	.	PUNCT
ejpam-3383	146	1	let	let	VERB
ejpam-3383	146	2	a	a	DET
ejpam-3383	146	3	,	,	PUNCT
ejpam-3383	146	4	b	b	X
ejpam-3383	146	5	∈	∈	NOUN
ejpam-3383	146	6	fr	fr	NOUN
ejpam-3383	146	7	such	such	ADJ
ejpam-3383	146	8	that	that	SCONJ
ejpam-3383	146	9	ab	ab	PROPN
ejpam-3383	146	10	≤	≤	PROPN
ejpam-3383	146	11	m.	m.	NOUN
ejpam-3383	146	12	we	we	PRON
ejpam-3383	146	13	consider	consider	VERB
ejpam-3383	146	14	the	the	DET
ejpam-3383	146	15	ideal	ideal	ADJ
ejpam-3383	146	16	elements	element	NOUN
ejpam-3383	146	17	i(a	i(a	PROPN
ejpam-3383	146	18	)	)	PUNCT
ejpam-3383	146	19	and	and	CCONJ
ejpam-3383	146	20	i(b	i(b	NOUN
ejpam-3383	146	21	)	)	PUNCT
ejpam-3383	146	22	of	of	ADP
ejpam-3383	146	23	s	s	AUX
ejpam-3383	146	24	generated	generate	VERB
ejpam-3383	146	25	by	by	ADP
ejpam-3383	146	26	a	a	PRON
ejpam-3383	146	27	and	and	CCONJ
ejpam-3383	146	28	b	b	NOUN
ejpam-3383	146	29	,	,	PUNCT
ejpam-3383	146	30	respectively	respectively	ADV
ejpam-3383	146	31	,	,	PUNCT
ejpam-3383	146	32	and	and	CCONJ
ejpam-3383	146	33	we	we	PRON
ejpam-3383	146	34	have	have	VERB
ejpam-3383	146	35	i(a)i(b	i(a)i(b	NOUN
ejpam-3383	146	36	)	)	PUNCT
ejpam-3383	147	1	=	=	SYM
ejpam-3383	148	1	(	(	PUNCT
ejpam-3383	148	2	a	a	DET
ejpam-3383	148	3	∨	∨	NUM
ejpam-3383	148	4	ea	ea	NOUN
ejpam-3383	148	5	∨	∨	PROPN
ejpam-3383	148	6	ae	ae	PROPN
ejpam-3383	148	7	∨	∨	NUM
ejpam-3383	148	8	eae)(b	eae)(b	PROPN
ejpam-3383	148	9	∨	∨	PROPN
ejpam-3383	148	10	eb	eb	PROPN
ejpam-3383	148	11	∨	∨	NUM
ejpam-3383	148	12	be	be	PROPN
ejpam-3383	148	13	∨	∨	NUM
ejpam-3383	148	14	ebe	ebe	NOUN
ejpam-3383	148	15	)	)	PUNCT
ejpam-3383	148	16	n.	n.	NOUN
ejpam-3383	148	17	kehayopulu	kehayopulu	PROPN
ejpam-3383	148	18	/	/	SYM
ejpam-3383	148	19	eur	eur	PROPN
ejpam-3383	148	20	.	.	PUNCT
ejpam-3383	149	1	j.	j.	PROPN
ejpam-3383	149	2	pure	pure	PROPN
ejpam-3383	149	3	appl	appl	PROPN
ejpam-3383	149	4	.	.	PROPN
ejpam-3383	149	5	math	math	PROPN
ejpam-3383	149	6	,	,	PUNCT
ejpam-3383	149	7	12	12	NUM
ejpam-3383	149	8	(	(	PUNCT
ejpam-3383	149	9	1	1	NUM
ejpam-3383	149	10	)	)	PUNCT
ejpam-3383	149	11	(	(	PUNCT
ejpam-3383	149	12	2019	2019	NUM
ejpam-3383	149	13	)	)	PUNCT
ejpam-3383	149	14	,	,	PUNCT
ejpam-3383	149	15	208	208	NUM
ejpam-3383	149	16	-	-	SYM
ejpam-3383	149	17	225	225	NUM
ejpam-3383	149	18	213	213	NUM
ejpam-3383	149	19	=	=	SYM
ejpam-3383	149	20	ab	ab	PROPN
ejpam-3383	149	21	∨	∨	NUM
ejpam-3383	149	22	eab	eab	PROPN
ejpam-3383	149	23	∨	∨	PROPN
ejpam-3383	149	24	aeb	aeb	PROPN
ejpam-3383	149	25	∨	∨	NUM
ejpam-3383	149	26	eaeb	eaeb	PROPN
ejpam-3383	149	27	∨	∨	PROPN
ejpam-3383	149	28	abe	abe	PROPN
ejpam-3383	149	29	∨	∨	PROPN
ejpam-3383	149	30	eabe	eabe	PROPN
ejpam-3383	149	31	∨	∨	NUM
ejpam-3383	149	32	aebe	aebe	PROPN
ejpam-3383	149	33	∨	∨	NUM
ejpam-3383	149	34	eaebe	eaebe	NOUN
ejpam-3383	149	35	.	.	PUNCT
ejpam-3383	150	1	we	we	PRON
ejpam-3383	150	2	have	have	VERB
ejpam-3383	150	3	ab	ab	PROPN
ejpam-3383	150	4	≤	≤	PROPN
ejpam-3383	150	5	m	m	PROPN
ejpam-3383	150	6	,	,	PUNCT
ejpam-3383	150	7	e(ab	e(ab	PROPN
ejpam-3383	150	8	)	)	PUNCT
ejpam-3383	150	9	≤	≤	NOUN
ejpam-3383	150	10	em	em	PRON
ejpam-3383	150	11	≤	≤	NOUN
ejpam-3383	150	12	m	m	VERB
ejpam-3383	150	13	,	,	PUNCT
ejpam-3383	150	14	(	(	PUNCT
ejpam-3383	150	15	ae)b	ae)b	PROPN
ejpam-3383	150	16	≤	≤	PROPN
ejpam-3383	150	17	ab	ab	PROPN
ejpam-3383	150	18	≤	≤	PROPN
ejpam-3383	150	19	m	m	PROPN
ejpam-3383	150	20	,	,	PUNCT
ejpam-3383	150	21	e(ae)b	e(ae)b	PRON
ejpam-3383	150	22	≤	≤	ADJ
ejpam-3383	150	23	eab	eab	NOUN
ejpam-3383	150	24	≤	≤	PUNCT
ejpam-3383	150	25	m	m	PROPN
ejpam-3383	150	26	,	,	PUNCT
ejpam-3383	150	27	(	(	PUNCT
ejpam-3383	150	28	ab)e	ab)e	PROPN
ejpam-3383	150	29	≤	≤	NUM
ejpam-3383	150	30	me	i	PRON
ejpam-3383	150	31	≤	≤	NUM
ejpam-3383	150	32	m	m	PROPN
ejpam-3383	150	33	,	,	PUNCT
ejpam-3383	150	34	e(abe	e(abe	PROPN
ejpam-3383	150	35	)	)	PUNCT
ejpam-3383	150	36	≤	≤	PUNCT
ejpam-3383	151	1	em	em	PRON
ejpam-3383	151	2	≤	≤	NOUN
ejpam-3383	151	3	m	m	VERB
ejpam-3383	151	4	,	,	PUNCT
ejpam-3383	151	5	(	(	PUNCT
ejpam-3383	151	6	ae)(be	ae)(be	NOUN
ejpam-3383	151	7	)	)	PUNCT
ejpam-3383	151	8	≤	≤	PUNCT
ejpam-3383	151	9	ab	ab	PROPN
ejpam-3383	151	10	≤	≤	PROPN
ejpam-3383	151	11	m	m	PROPN
ejpam-3383	151	12	,	,	PUNCT
ejpam-3383	151	13	e(aebe	e(aebe	PROPN
ejpam-3383	151	14	)	)	PUNCT
ejpam-3383	151	15	≤	≤	PUNCT
ejpam-3383	151	16	em	em	PRON
ejpam-3383	151	17	≤	≤	NUM
ejpam-3383	151	18	m.	m.	NOUN
ejpam-3383	151	19	thus	thus	ADV
ejpam-3383	151	20	we	we	PRON
ejpam-3383	151	21	have	have	VERB
ejpam-3383	151	22	i(a)i(b	i(a)i(b	NOUN
ejpam-3383	151	23	)	)	PUNCT
ejpam-3383	151	24	≤	≤	NUM
ejpam-3383	151	25	m.	m.	NOUN
ejpam-3383	151	26	since	since	SCONJ
ejpam-3383	151	27	m	m	PROPN
ejpam-3383	151	28	is	be	AUX
ejpam-3383	151	29	weakly	weakly	ADV
ejpam-3383	151	30	prime	prime	ADJ
ejpam-3383	151	31	and	and	CCONJ
ejpam-3383	151	32	i(a	i(a	NOUN
ejpam-3383	151	33	)	)	PUNCT
ejpam-3383	151	34	,	,	PUNCT
ejpam-3383	151	35	i(b	i(b	PROPN
ejpam-3383	151	36	)	)	PUNCT
ejpam-3383	151	37	are	be	AUX
ejpam-3383	151	38	ideal	ideal	ADJ
ejpam-3383	151	39	elements	element	NOUN
ejpam-3383	151	40	of	of	ADP
ejpam-3383	151	41	s	s	PROPN
ejpam-3383	151	42	,	,	PUNCT
ejpam-3383	151	43	we	we	PRON
ejpam-3383	151	44	have	have	VERB
ejpam-3383	151	45	i(a	i(a	PROPN
ejpam-3383	151	46	)	)	PUNCT
ejpam-3383	151	47	≤	≤	NUM
ejpam-3383	151	48	m	m	ADP
ejpam-3383	151	49	or	or	CCONJ
ejpam-3383	151	50	i(b	i(b	NOUN
ejpam-3383	151	51	)	)	PUNCT
ejpam-3383	151	52	≤	≤	NUM
ejpam-3383	151	53	m	m	ADP
ejpam-3383	151	54	,	,	PUNCT
ejpam-3383	151	55	thus	thus	ADV
ejpam-3383	151	56	we	we	PRON
ejpam-3383	151	57	have	have	VERB
ejpam-3383	151	58	a	a	DET
ejpam-3383	151	59	≤	≤	NUM
ejpam-3383	151	60	m	m	NOUN
ejpam-3383	151	61	or	or	CCONJ
ejpam-3383	151	62	b	b	PROPN
ejpam-3383	151	63	≤	≤	NUM
ejpam-3383	151	64	m	m	NOUN
ejpam-3383	151	65	and	and	CCONJ
ejpam-3383	151	66	property	property	NOUN
ejpam-3383	151	67	(	(	PUNCT
ejpam-3383	151	68	2	2	NUM
ejpam-3383	151	69	)	)	PUNCT
ejpam-3383	151	70	holds	hold	VERB
ejpam-3383	151	71	.	.	PUNCT
ejpam-3383	152	1	(	(	PUNCT
ejpam-3383	152	2	1	1	X
ejpam-3383	152	3	)	)	PUNCT
ejpam-3383	152	4	=	=	NOUN
ejpam-3383	152	5	⇒	⇒	NOUN
ejpam-3383	152	6	(	(	PUNCT
ejpam-3383	152	7	3	3	NUM
ejpam-3383	152	8	)	)	PUNCT
ejpam-3383	152	9	.	.	PUNCT
ejpam-3383	153	1	let	let	VERB
ejpam-3383	153	2	a	a	DET
ejpam-3383	153	3	,	,	PUNCT
ejpam-3383	153	4	b	b	PROPN
ejpam-3383	153	5	∈	∈	PROPN
ejpam-3383	153	6	fl	fl	PROPN
ejpam-3383	153	7	and	and	CCONJ
ejpam-3383	153	8	ab	ab	PROPN
ejpam-3383	153	9	≤	≤	PROPN
ejpam-3383	153	10	m.	m.	NOUN
ejpam-3383	153	11	we	we	PRON
ejpam-3383	153	12	have	have	VERB
ejpam-3383	153	13	i(a)i(b	i(a)i(b	NOUN
ejpam-3383	153	14	)	)	PUNCT
ejpam-3383	154	1	=	=	SYM
ejpam-3383	154	2	ab	ab	PROPN
ejpam-3383	154	3	∨	∨	NUM
ejpam-3383	154	4	eab	eab	PROPN
ejpam-3383	154	5	∨	∨	PROPN
ejpam-3383	154	6	aeb	aeb	PROPN
ejpam-3383	154	7	∨	∨	NUM
ejpam-3383	154	8	eaeb	eaeb	PROPN
ejpam-3383	154	9	∨	∨	PROPN
ejpam-3383	154	10	abe	abe	PROPN
ejpam-3383	154	11	∨	∨	PROPN
ejpam-3383	154	12	eabe	eabe	PROPN
ejpam-3383	154	13	∨	∨	NUM
ejpam-3383	154	14	aebe	aebe	PROPN
ejpam-3383	154	15	∨	∨	NUM
ejpam-3383	154	16	eaebe	eaebe	NOUN
ejpam-3383	154	17	.	.	PUNCT
ejpam-3383	155	1	we	we	PRON
ejpam-3383	155	2	also	also	ADV
ejpam-3383	155	3	have	have	VERB
ejpam-3383	155	4	ab	ab	PROPN
ejpam-3383	155	5	≤	≤	PROPN
ejpam-3383	155	6	m	m	PROPN
ejpam-3383	155	7	,	,	PUNCT
ejpam-3383	155	8	e(ab	e(ab	PROPN
ejpam-3383	155	9	)	)	PUNCT
ejpam-3383	155	10	≤	≤	NOUN
ejpam-3383	155	11	em	em	PRON
ejpam-3383	155	12	≤	≤	PROPN
ejpam-3383	155	13	m	m	PROPN
ejpam-3383	155	14	,	,	PUNCT
ejpam-3383	155	15	a(eb	a(eb	PROPN
ejpam-3383	155	16	)	)	PUNCT
ejpam-3383	155	17	≤	≤	PUNCT
ejpam-3383	155	18	ab	ab	PROPN
ejpam-3383	155	19	≤	≤	PROPN
ejpam-3383	155	20	m	m	PROPN
ejpam-3383	155	21	,	,	PUNCT
ejpam-3383	155	22	(	(	PUNCT
ejpam-3383	155	23	ea)(eb	ea)(eb	NOUN
ejpam-3383	155	24	)	)	PUNCT
ejpam-3383	155	25	≤	≤	PUNCT
ejpam-3383	155	26	ab	ab	PROPN
ejpam-3383	155	27	≤	≤	PROPN
ejpam-3383	155	28	m	m	PROPN
ejpam-3383	155	29	,	,	PUNCT
ejpam-3383	155	30	(	(	PUNCT
ejpam-3383	155	31	ab)e	ab)e	PROPN
ejpam-3383	155	32	≤	≤	NUM
ejpam-3383	155	33	me	i	PRON
ejpam-3383	155	34	≤	≤	NUM
ejpam-3383	155	35	m	m	PROPN
ejpam-3383	155	36	,	,	PUNCT
ejpam-3383	155	37	e(abe	e(abe	PROPN
ejpam-3383	155	38	)	)	PUNCT
ejpam-3383	155	39	≤	≤	PUNCT
ejpam-3383	156	1	em	em	PRON
ejpam-3383	156	2	≤	≤	NOUN
ejpam-3383	156	3	m	m	VERB
ejpam-3383	156	4	,	,	PUNCT
ejpam-3383	156	5	(	(	PUNCT
ejpam-3383	156	6	aeb)e	aeb)e	ADV
ejpam-3383	156	7	≤	≤	PUNCT
ejpam-3383	156	8	me	i	PRON
ejpam-3383	156	9	≤	≤	NUM
ejpam-3383	156	10	m	m	PROPN
ejpam-3383	156	11	,	,	PUNCT
ejpam-3383	156	12	e(aebe	e(aebe	PROPN
ejpam-3383	156	13	)	)	PUNCT
ejpam-3383	156	14	≤	≤	PUNCT
ejpam-3383	157	1	em	em	PRON
ejpam-3383	157	2	≤	≤	NUM
ejpam-3383	157	3	m.	m.	NOUN
ejpam-3383	157	4	then	then	ADV
ejpam-3383	157	5	i(a)i(b	i(a)i(b	NOUN
ejpam-3383	157	6	)	)	PUNCT
ejpam-3383	157	7	≤	≤	NUM
ejpam-3383	157	8	m	m	ADP
ejpam-3383	157	9	and	and	CCONJ
ejpam-3383	157	10	,	,	PUNCT
ejpam-3383	157	11	by	by	ADP
ejpam-3383	157	12	(	(	PUNCT
ejpam-3383	157	13	1	1	NUM
ejpam-3383	157	14	)	)	PUNCT
ejpam-3383	157	15	,	,	PUNCT
ejpam-3383	157	16	i(a	i(a	PROPN
ejpam-3383	157	17	)	)	PUNCT
ejpam-3383	157	18	≤	≤	NUM
ejpam-3383	157	19	m	m	ADP
ejpam-3383	157	20	or	or	CCONJ
ejpam-3383	157	21	i(b	i(b	NOUN
ejpam-3383	157	22	)	)	PUNCT
ejpam-3383	157	23	≤	≤	NUM
ejpam-3383	157	24	m	m	PROPN
ejpam-3383	157	25	and	and	CCONJ
ejpam-3383	157	26	so	so	ADV
ejpam-3383	157	27	a	a	DET
ejpam-3383	157	28	≤	≤	NUM
ejpam-3383	157	29	m	m	NOUN
ejpam-3383	157	30	or	or	CCONJ
ejpam-3383	157	31	b	b	NOUN
ejpam-3383	157	32	≤	≤	NUM
ejpam-3383	157	33	m.	m.	NOUN
ejpam-3383	157	34	(	(	PUNCT
ejpam-3383	157	35	1	1	X
ejpam-3383	157	36	)	)	PUNCT
ejpam-3383	158	1	=	=	NOUN
ejpam-3383	158	2	⇒	⇒	NOUN
ejpam-3383	158	3	(	(	PUNCT
ejpam-3383	158	4	4	4	NUM
ejpam-3383	158	5	)	)	PUNCT
ejpam-3383	158	6	.	.	PUNCT
ejpam-3383	159	1	let	let	VERB
ejpam-3383	159	2	a	a	DET
ejpam-3383	159	3	∈	∈	ADJ
ejpam-3383	159	4	fr	fr	NOUN
ejpam-3383	159	5	,	,	PUNCT
ejpam-3383	159	6	b	b	X
ejpam-3383	159	7	∈	∈	PROPN
ejpam-3383	159	8	fl	fl	PROPN
ejpam-3383	159	9	and	and	CCONJ
ejpam-3383	159	10	ab	ab	PROPN
ejpam-3383	159	11	≤	≤	PROPN
ejpam-3383	159	12	m.	m.	NOUN
ejpam-3383	159	13	we	we	PRON
ejpam-3383	159	14	have	have	VERB
ejpam-3383	159	15	i(a)i(b	i(a)i(b	NOUN
ejpam-3383	159	16	)	)	PUNCT
ejpam-3383	160	1	=	=	SYM
ejpam-3383	160	2	ab	ab	PROPN
ejpam-3383	160	3	∨	∨	NUM
ejpam-3383	160	4	eab	eab	PROPN
ejpam-3383	160	5	∨	∨	PROPN
ejpam-3383	160	6	aeb	aeb	PROPN
ejpam-3383	160	7	∨	∨	NUM
ejpam-3383	160	8	eaeb	eaeb	PROPN
ejpam-3383	160	9	∨	∨	PROPN
ejpam-3383	160	10	abe	abe	PROPN
ejpam-3383	160	11	∨	∨	PROPN
ejpam-3383	160	12	eabe	eabe	PROPN
ejpam-3383	160	13	∨	∨	NUM
ejpam-3383	160	14	aebe	aebe	PROPN
ejpam-3383	160	15	∨	∨	NUM
ejpam-3383	160	16	eaebe	eaebe	NOUN
ejpam-3383	160	17	,	,	PUNCT
ejpam-3383	160	18	ab	ab	PROPN
ejpam-3383	160	19	≤	≤	PROPN
ejpam-3383	160	20	m	m	PROPN
ejpam-3383	160	21	,	,	PUNCT
ejpam-3383	160	22	e(ab	e(ab	PROPN
ejpam-3383	160	23	)	)	PUNCT
ejpam-3383	160	24	≤	≤	NOUN
ejpam-3383	160	25	em	em	PRON
ejpam-3383	160	26	≤	≤	NOUN
ejpam-3383	160	27	m	m	VERB
ejpam-3383	160	28	,	,	PUNCT
ejpam-3383	160	29	(	(	PUNCT
ejpam-3383	160	30	ae)b	ae)b	PROPN
ejpam-3383	160	31	≤	≤	PROPN
ejpam-3383	160	32	ab	ab	PROPN
ejpam-3383	160	33	≤	≤	PROPN
ejpam-3383	160	34	m	m	PROPN
ejpam-3383	160	35	,	,	PUNCT
ejpam-3383	160	36	e(aeb	e(aeb	PROPN
ejpam-3383	160	37	)	)	PUNCT
ejpam-3383	160	38	≤	≤	PUNCT
ejpam-3383	161	1	em	em	PRON
ejpam-3383	161	2	≤	≤	NOUN
ejpam-3383	161	3	m	m	VERB
ejpam-3383	161	4	,	,	PUNCT
ejpam-3383	161	5	(	(	PUNCT
ejpam-3383	161	6	ab)e	ab)e	PROPN
ejpam-3383	161	7	≤	≤	NUM
ejpam-3383	161	8	me	i	PRON
ejpam-3383	161	9	≤	≤	NUM
ejpam-3383	161	10	m	m	PROPN
ejpam-3383	161	11	,	,	PUNCT
ejpam-3383	161	12	e(abe	e(abe	PROPN
ejpam-3383	161	13	)	)	PUNCT
ejpam-3383	161	14	≤	≤	PUNCT
ejpam-3383	161	15	em	em	PRON
ejpam-3383	161	16	≤	≤	NOUN
ejpam-3383	161	17	m	m	PROPN
ejpam-3383	161	18	,	,	PUNCT
ejpam-3383	161	19	a(eb)e	a(eb)e	PROPN
ejpam-3383	161	20	≤	≤	PROPN
ejpam-3383	161	21	abe	abe	PROPN
ejpam-3383	161	22	≤	≤	PROPN
ejpam-3383	161	23	m	m	PROPN
ejpam-3383	161	24	,	,	PUNCT
ejpam-3383	161	25	e(aebe	e(aebe	PROPN
ejpam-3383	161	26	)	)	PUNCT
ejpam-3383	161	27	≤	≤	PUNCT
ejpam-3383	161	28	em	em	PRON
ejpam-3383	161	29	≤	≤	ADJ
ejpam-3383	161	30	m.	m.	NOUN
ejpam-3383	161	31	by	by	ADP
ejpam-3383	161	32	hypothesis	hypothesis	NOUN
ejpam-3383	161	33	,	,	PUNCT
ejpam-3383	161	34	we	we	PRON
ejpam-3383	161	35	have	have	VERB
ejpam-3383	161	36	i(a	i(a	PROPN
ejpam-3383	161	37	)	)	PUNCT
ejpam-3383	161	38	≤	≤	NUM
ejpam-3383	161	39	m	m	ADP
ejpam-3383	161	40	or	or	CCONJ
ejpam-3383	161	41	i(b	i(b	NOUN
ejpam-3383	161	42	)	)	PUNCT
ejpam-3383	161	43	≤	≤	NUM
ejpam-3383	162	1	m	m	PROPN
ejpam-3383	163	1	and	and	CCONJ
ejpam-3383	163	2	so	so	ADV
ejpam-3383	163	3	a	a	DET
ejpam-3383	163	4	≤	≤	NUM
ejpam-3383	163	5	m	m	NOUN
ejpam-3383	163	6	or	or	CCONJ
ejpam-3383	163	7	b	b	PROPN
ejpam-3383	163	8	≤	≤	NUM
ejpam-3383	163	9	m.	m.	NOUN
ejpam-3383	163	10	the	the	DET
ejpam-3383	163	11	implications	implication	NOUN
ejpam-3383	163	12	(	(	PUNCT
ejpam-3383	163	13	2	2	NUM
ejpam-3383	163	14	)	)	PUNCT
ejpam-3383	163	15	⇒	⇒	NOUN
ejpam-3383	163	16	(	(	PUNCT
ejpam-3383	163	17	1	1	NUM
ejpam-3383	163	18	)	)	PUNCT
ejpam-3383	163	19	,	,	PUNCT
ejpam-3383	163	20	(	(	PUNCT
ejpam-3383	163	21	3	3	X
ejpam-3383	163	22	)	)	PUNCT
ejpam-3383	163	23	⇒	⇒	NOUN
ejpam-3383	163	24	(	(	PUNCT
ejpam-3383	163	25	1	1	NUM
ejpam-3383	163	26	)	)	PUNCT
ejpam-3383	163	27	and	and	CCONJ
ejpam-3383	163	28	(	(	PUNCT
ejpam-3383	163	29	4	4	X
ejpam-3383	163	30	)	)	PUNCT
ejpam-3383	163	31	⇒	⇒	NOUN
ejpam-3383	163	32	(	(	PUNCT
ejpam-3383	163	33	1	1	X
ejpam-3383	163	34	)	)	PUNCT
ejpam-3383	163	35	are	be	AUX
ejpam-3383	163	36	obvious	obvious	ADJ
ejpam-3383	163	37	and	and	CCONJ
ejpam-3383	163	38	they	they	PRON
ejpam-3383	163	39	hold	hold	VERB
ejpam-3383	163	40	in	in	ADP
ejpam-3383	163	41	po	po	NOUN
ejpam-3383	163	42	-	-	NOUN
ejpam-3383	163	43	groupoids	groupoid	NOUN
ejpam-3383	163	44	in	in	ADP
ejpam-3383	163	45	general	general	ADJ
ejpam-3383	163	46	.	.	PUNCT
ejpam-3383	164	1	�	�	PROPN
ejpam-3383	164	2	proposition	proposition	PROPN
ejpam-3383	164	3	3.2	3.2	NUM
ejpam-3383	164	4	.	.	PUNCT
ejpam-3383	165	1	let	let	VERB
ejpam-3383	165	2	s	s	PRON
ejpam-3383	165	3	be	be	AUX
ejpam-3383	165	4	a	a	DET
ejpam-3383	165	5	∨e	∨e	NOUN
ejpam-3383	165	6	-	-	PUNCT
ejpam-3383	165	7	semigroup	semigroup	NOUN
ejpam-3383	165	8	and	and	CCONJ
ejpam-3383	165	9	m	m	VERB
ejpam-3383	165	10	an	an	DET
ejpam-3383	165	11	ideal	ideal	ADJ
ejpam-3383	165	12	element	element	NOUN
ejpam-3383	165	13	of	of	ADP
ejpam-3383	165	14	s.	s.	PROPN
ejpam-3383	165	15	the	the	DET
ejpam-3383	165	16	following	follow	VERB
ejpam-3383	165	17	are	be	AUX
ejpam-3383	165	18	equivalent	equivalent	ADJ
ejpam-3383	165	19	:	:	PUNCT
ejpam-3383	165	20	(	(	PUNCT
ejpam-3383	165	21	1	1	X
ejpam-3383	165	22	)	)	PUNCT
ejpam-3383	165	23	m	m	VERB
ejpam-3383	165	24	is	be	AUX
ejpam-3383	165	25	weakly	weakly	ADJ
ejpam-3383	165	26	semiprime	semiprime	NOUN
ejpam-3383	165	27	.	.	PUNCT
ejpam-3383	166	1	(	(	PUNCT
ejpam-3383	166	2	2	2	X
ejpam-3383	166	3	)	)	PUNCT
ejpam-3383	166	4	if	if	SCONJ
ejpam-3383	166	5	a	a	DET
ejpam-3383	166	6	∈	∈	NOUN
ejpam-3383	166	7	fr	fr	NOUN
ejpam-3383	166	8	such	such	ADJ
ejpam-3383	166	9	that	that	DET
ejpam-3383	166	10	a2	a2	PROPN
ejpam-3383	166	11	≤	≤	PROPN
ejpam-3383	166	12	m	m	PROPN
ejpam-3383	166	13	,	,	PUNCT
ejpam-3383	166	14	then	then	ADV
ejpam-3383	166	15	a	a	DET
ejpam-3383	166	16	≤	≤	ADJ
ejpam-3383	166	17	m.	m.	NOUN
ejpam-3383	166	18	(	(	PUNCT
ejpam-3383	166	19	3	3	X
ejpam-3383	166	20	)	)	PUNCT
ejpam-3383	166	21	if	if	SCONJ
ejpam-3383	166	22	b	b	PROPN
ejpam-3383	166	23	∈	∈	PROPN
ejpam-3383	166	24	fl	fl	ADP
ejpam-3383	166	25	such	such	ADJ
ejpam-3383	166	26	that	that	DET
ejpam-3383	166	27	b2	b2	NOUN
ejpam-3383	166	28	≤	≤	NUM
ejpam-3383	166	29	m	m	NOUN
ejpam-3383	166	30	,	,	PUNCT
ejpam-3383	166	31	then	then	ADV
ejpam-3383	166	32	b	b	PROPN
ejpam-3383	166	33	≤	≤	ADJ
ejpam-3383	166	34	m.	m.	NOUN
ejpam-3383	166	35	proof	proof	NOUN
ejpam-3383	166	36	.	.	PUNCT
ejpam-3383	167	1	(	(	PUNCT
ejpam-3383	167	2	1	1	X
ejpam-3383	167	3	)	)	PUNCT
ejpam-3383	167	4	=	=	NOUN
ejpam-3383	167	5	⇒	⇒	NOUN
ejpam-3383	167	6	(	(	PUNCT
ejpam-3383	167	7	2	2	NUM
ejpam-3383	167	8	)	)	PUNCT
ejpam-3383	167	9	.	.	PUNCT
ejpam-3383	168	1	let	let	VERB
ejpam-3383	168	2	a	a	DET
ejpam-3383	168	3	∈	∈	ADJ
ejpam-3383	168	4	fr	fr	NOUN
ejpam-3383	168	5	and	and	CCONJ
ejpam-3383	168	6	a2	a2	PROPN
ejpam-3383	168	7	≤	≤	PROPN
ejpam-3383	168	8	m.	m.	NOUN
ejpam-3383	168	9	then	then	ADV
ejpam-3383	168	10	i(a)2	i(a)2	ADP
ejpam-3383	168	11	=	=	PRON
ejpam-3383	168	12	(	(	PUNCT
ejpam-3383	168	13	a	a	DET
ejpam-3383	168	14	∨	∨	NUM
ejpam-3383	168	15	ea	ea	NOUN
ejpam-3383	168	16	∨	∨	PROPN
ejpam-3383	168	17	ae	ae	PROPN
ejpam-3383	168	18	∨	∨	PROPN
ejpam-3383	168	19	eae)(a	eae)(a	NOUN
ejpam-3383	168	20	∨	∨	PROPN
ejpam-3383	168	21	ea	ea	PROPN
ejpam-3383	168	22	∨	∨	NUM
ejpam-3383	168	23	ae	ae	PROPN
ejpam-3383	168	24	∨	∨	PROPN
ejpam-3383	168	25	eae	eae	PROPN
ejpam-3383	168	26	)	)	PUNCT
ejpam-3383	168	27	=	=	PROPN
ejpam-3383	168	28	a2	a2	PROPN
ejpam-3383	168	29	∨	∨	PROPN
ejpam-3383	168	30	ea2	ea2	PROPN
ejpam-3383	168	31	∨	∨	PROPN
ejpam-3383	168	32	aea	aea	PROPN
ejpam-3383	168	33	∨	∨	PROPN
ejpam-3383	168	34	eaea	eaea	PROPN
ejpam-3383	168	35	∨	∨	PROPN
ejpam-3383	168	36	a2e	a2e	X
ejpam-3383	168	37	∨	∨	NUM
ejpam-3383	168	38	ea2e	ea2e	PROPN
ejpam-3383	168	39	∨	∨	NUM
ejpam-3383	168	40	aeae	aeae	PROPN
ejpam-3383	168	41	∨	∨	NUM
ejpam-3383	168	42	eaeae	eaeae	PROPN
ejpam-3383	168	43	.	.	PUNCT
ejpam-3383	169	1	we	we	PRON
ejpam-3383	169	2	have	have	VERB
ejpam-3383	169	3	a2	a2	PROPN
ejpam-3383	169	4	≤	≤	PROPN
ejpam-3383	169	5	m	m	PROPN
ejpam-3383	169	6	,	,	PUNCT
ejpam-3383	169	7	ea2	ea2	VERB
ejpam-3383	169	8	≤	≤	NUM
ejpam-3383	169	9	em	em	PRON
ejpam-3383	169	10	≤	≤	NOUN
ejpam-3383	169	11	m	m	VERB
ejpam-3383	169	12	,	,	PUNCT
ejpam-3383	169	13	(	(	PUNCT
ejpam-3383	169	14	ae)a	ae)a	PROPN
ejpam-3383	169	15	≤	≤	NUM
ejpam-3383	169	16	a2	a2	PROPN
ejpam-3383	169	17	≤	≤	PROPN
ejpam-3383	169	18	m	m	PROPN
ejpam-3383	169	19	,	,	PUNCT
ejpam-3383	169	20	e(aea	e(aea	PROPN
ejpam-3383	169	21	)	)	PUNCT
ejpam-3383	169	22	≤	≤	NOUN
ejpam-3383	170	1	em	em	PRON
ejpam-3383	170	2	≤	≤	NOUN
ejpam-3383	170	3	m	m	PROPN
ejpam-3383	170	4	,	,	PUNCT
ejpam-3383	170	5	a2e	a2e	PROPN
ejpam-3383	170	6	≤	≤	VERB
ejpam-3383	170	7	me	i	PRON
ejpam-3383	170	8	≤	≤	NUM
ejpam-3383	170	9	m	m	PROPN
ejpam-3383	170	10	,	,	PUNCT
ejpam-3383	170	11	e(a2e	e(a2e	NOUN
ejpam-3383	170	12	)	)	PUNCT
ejpam-3383	170	13	≤	≤	NOUN
ejpam-3383	170	14	em	em	PRON
ejpam-3383	170	15	≤	≤	NOUN
ejpam-3383	170	16	m	m	VERB
ejpam-3383	170	17	,	,	PUNCT
ejpam-3383	170	18	(	(	PUNCT
ejpam-3383	170	19	ae)(ae	ae)(ae	NOUN
ejpam-3383	170	20	)	)	PUNCT
ejpam-3383	170	21	≤	≤	NUM
ejpam-3383	170	22	a2	a2	PROPN
ejpam-3383	170	23	≤	≤	PROPN
ejpam-3383	170	24	m	m	PROPN
ejpam-3383	170	25	,	,	PUNCT
ejpam-3383	170	26	e(aeae	e(aeae	PROPN
ejpam-3383	170	27	)	)	PUNCT
ejpam-3383	170	28	≤	≤	PUNCT
ejpam-3383	170	29	em	em	PRON
ejpam-3383	170	30	≤	≤	NUM
ejpam-3383	170	31	m.	m.	NOUN
ejpam-3383	170	32	since	since	SCONJ
ejpam-3383	170	33	i(a	i(a	PROPN
ejpam-3383	170	34	)	)	PUNCT
ejpam-3383	170	35	is	be	AUX
ejpam-3383	170	36	an	an	DET
ejpam-3383	170	37	ideal	ideal	ADJ
ejpam-3383	170	38	element	element	NOUN
ejpam-3383	170	39	of	of	ADP
ejpam-3383	170	40	s	s	PRON
ejpam-3383	170	41	and	and	CCONJ
ejpam-3383	170	42	i(a)2	i(a)2	ADP
ejpam-3383	170	43	≤	≤	NOUN
ejpam-3383	170	44	m	m	VERB
ejpam-3383	170	45	,	,	PUNCT
ejpam-3383	170	46	by	by	ADP
ejpam-3383	170	47	(	(	PUNCT
ejpam-3383	170	48	1	1	NUM
ejpam-3383	170	49	)	)	PUNCT
ejpam-3383	170	50	,	,	PUNCT
ejpam-3383	170	51	we	we	PRON
ejpam-3383	170	52	have	have	VERB
ejpam-3383	170	53	i(a	i(a	PROPN
ejpam-3383	170	54	)	)	PUNCT
ejpam-3383	170	55	≤	≤	NOUN
ejpam-3383	171	1	m	m	PROPN
ejpam-3383	172	1	and	and	CCONJ
ejpam-3383	172	2	so	so	ADV
ejpam-3383	172	3	a	a	DET
ejpam-3383	172	4	≤	≤	ADJ
ejpam-3383	172	5	m.	m.	NOUN
ejpam-3383	172	6	(	(	PUNCT
ejpam-3383	172	7	1	1	X
ejpam-3383	172	8	)	)	PUNCT
ejpam-3383	173	1	=	=	NOUN
ejpam-3383	173	2	⇒	⇒	NOUN
ejpam-3383	173	3	(	(	PUNCT
ejpam-3383	173	4	3	3	NUM
ejpam-3383	173	5	)	)	PUNCT
ejpam-3383	173	6	.	.	PUNCT
ejpam-3383	174	1	let	let	VERB
ejpam-3383	174	2	b	b	X
ejpam-3383	174	3	∈	∈	PROPN
ejpam-3383	174	4	fl	fl	NOUN
ejpam-3383	174	5	and	and	CCONJ
ejpam-3383	174	6	b2	b2	NOUN
ejpam-3383	174	7	≤	≤	ADJ
ejpam-3383	174	8	m.	m.	NOUN
ejpam-3383	174	9	then	then	ADV
ejpam-3383	174	10	i(b)2	i(b)2	PROPN
ejpam-3383	174	11	=	=	PROPN
ejpam-3383	174	12	b2	b2	PROPN
ejpam-3383	174	13	∨	∨	NUM
ejpam-3383	174	14	eb2	eb2	PROPN
ejpam-3383	174	15	∨	∨	PROPN
ejpam-3383	174	16	beb	beb	PROPN
ejpam-3383	174	17	∨	∨	PROPN
ejpam-3383	174	18	ebeb	ebeb	NOUN
ejpam-3383	174	19	∨	∨	NUM
ejpam-3383	174	20	b2e	b2e	X
ejpam-3383	174	21	∨	∨	NUM
ejpam-3383	174	22	eb2e	eb2e	VERB
ejpam-3383	174	23	∨	∨	NUM
ejpam-3383	174	24	bebe	bebe	PROPN
ejpam-3383	174	25	∨	∨	PROPN
ejpam-3383	174	26	ebebe	ebebe	PROPN
ejpam-3383	174	27	.	.	PUNCT
ejpam-3383	175	1	we	we	PRON
ejpam-3383	175	2	have	have	AUX
ejpam-3383	175	3	b2	b2	VERB
ejpam-3383	175	4	≤	≤	NUM
ejpam-3383	175	5	m	m	NOUN
ejpam-3383	175	6	,	,	PUNCT
ejpam-3383	175	7	eb2	eb2	PROPN
ejpam-3383	175	8	≤	≤	NUM
ejpam-3383	175	9	em	em	PRON
ejpam-3383	175	10	≤	≤	NOUN
ejpam-3383	175	11	m	m	PROPN
ejpam-3383	175	12	,	,	PUNCT
ejpam-3383	175	13	b(eb	b(eb	NOUN
ejpam-3383	175	14	)	)	PUNCT
ejpam-3383	175	15	≤	≤	NUM
ejpam-3383	175	16	b2	b2	NOUN
ejpam-3383	175	17	≤	≤	NUM
ejpam-3383	175	18	m	m	ADP
ejpam-3383	175	19	,	,	PUNCT
ejpam-3383	175	20	(	(	PUNCT
ejpam-3383	175	21	eb)(eb	eb)(eb	NOUN
ejpam-3383	175	22	)	)	PUNCT
ejpam-3383	175	23	≤	≤	NUM
ejpam-3383	175	24	b2	b2	NOUN
ejpam-3383	175	25	≤	≤	NUM
ejpam-3383	175	26	m	m	PROPN
ejpam-3383	175	27	,	,	PUNCT
ejpam-3383	175	28	b2e	b2e	X
ejpam-3383	175	29	≤	≤	VERB
ejpam-3383	175	30	me	i	PRON
ejpam-3383	175	31	≤	≤	NUM
ejpam-3383	175	32	m	m	PROPN
ejpam-3383	175	33	,	,	PUNCT
ejpam-3383	175	34	e(b2e	e(b2e	NOUN
ejpam-3383	175	35	)	)	PUNCT
ejpam-3383	175	36	≤	≤	NOUN
ejpam-3383	175	37	em	em	PRON
ejpam-3383	175	38	≤	≤	NOUN
ejpam-3383	175	39	m	m	VERB
ejpam-3383	175	40	,	,	PUNCT
ejpam-3383	175	41	(	(	PUNCT
ejpam-3383	175	42	beb)e	beb)e	PROPN
ejpam-3383	175	43	≤	≤	PROPN
ejpam-3383	175	44	me	i	PRON
ejpam-3383	175	45	≤	≤	NUM
ejpam-3383	175	46	m	m	PROPN
ejpam-3383	175	47	,	,	PUNCT
ejpam-3383	175	48	e(bebe	e(bebe	NOUN
ejpam-3383	175	49	)	)	PUNCT
ejpam-3383	175	50	≤	≤	NOUN
ejpam-3383	175	51	em	em	PRON
ejpam-3383	175	52	≤	≤	NUM
ejpam-3383	175	53	m.	m.	NOUN
ejpam-3383	175	54	since	since	SCONJ
ejpam-3383	175	55	i(b	i(b	PROPN
ejpam-3383	175	56	)	)	PUNCT
ejpam-3383	175	57	is	be	AUX
ejpam-3383	175	58	an	an	DET
ejpam-3383	175	59	ideal	ideal	ADJ
ejpam-3383	175	60	element	element	NOUN
ejpam-3383	175	61	of	of	ADP
ejpam-3383	175	62	s	s	NOUN
ejpam-3383	175	63	and	and	CCONJ
ejpam-3383	175	64	i(b)2	i(b)2	PROPN
ejpam-3383	175	65	≤	≤	PROPN
ejpam-3383	175	66	m	m	PROPN
ejpam-3383	175	67	,	,	PUNCT
ejpam-3383	175	68	by	by	ADP
ejpam-3383	175	69	(	(	PUNCT
ejpam-3383	175	70	1	1	NUM
ejpam-3383	175	71	)	)	PUNCT
ejpam-3383	175	72	,	,	PUNCT
ejpam-3383	175	73	we	we	PRON
ejpam-3383	175	74	have	have	VERB
ejpam-3383	175	75	i(b	i(b	NOUN
ejpam-3383	175	76	)	)	PUNCT
ejpam-3383	175	77	≤	≤	NUM
ejpam-3383	175	78	m	m	PROPN
ejpam-3383	175	79	and	and	CCONJ
ejpam-3383	175	80	b	b	PROPN
ejpam-3383	175	81	≤	≤	NUM
ejpam-3383	175	82	m.	m.	NOUN
ejpam-3383	175	83	the	the	DET
ejpam-3383	175	84	implications	implication	NOUN
ejpam-3383	175	85	(	(	PUNCT
ejpam-3383	175	86	2	2	NUM
ejpam-3383	175	87	)	)	PUNCT
ejpam-3383	175	88	⇒	⇒	NOUN
ejpam-3383	175	89	(	(	PUNCT
ejpam-3383	175	90	1	1	NUM
ejpam-3383	175	91	)	)	PUNCT
ejpam-3383	175	92	and	and	CCONJ
ejpam-3383	175	93	(	(	PUNCT
ejpam-3383	175	94	3	3	X
ejpam-3383	175	95	)	)	PUNCT
ejpam-3383	175	96	⇒	⇒	NOUN
ejpam-3383	175	97	(	(	PUNCT
ejpam-3383	175	98	1	1	X
ejpam-3383	175	99	)	)	PUNCT
ejpam-3383	175	100	are	be	AUX
ejpam-3383	175	101	obvious	obvious	ADJ
ejpam-3383	175	102	and	and	CCONJ
ejpam-3383	175	103	they	they	PRON
ejpam-3383	175	104	hold	hold	VERB
ejpam-3383	175	105	in	in	ADP
ejpam-3383	175	106	po	po	NOUN
ejpam-3383	175	107	-	-	NOUN
ejpam-3383	175	108	groupoids	groupoid	NOUN
ejpam-3383	175	109	in	in	ADP
ejpam-3383	175	110	general	general	ADJ
ejpam-3383	175	111	.	.	PUNCT
ejpam-3383	176	1	�	�	PROPN
ejpam-3383	176	2	n.	n.	PROPN
ejpam-3383	176	3	kehayopulu	kehayopulu	PROPN
ejpam-3383	176	4	/	/	SYM
ejpam-3383	176	5	eur	eur	PROPN
ejpam-3383	176	6	.	.	PUNCT
ejpam-3383	177	1	j.	j.	PROPN
ejpam-3383	177	2	pure	pure	PROPN
ejpam-3383	177	3	appl	appl	PROPN
ejpam-3383	177	4	.	.	PROPN
ejpam-3383	177	5	math	math	PROPN
ejpam-3383	177	6	,	,	PUNCT
ejpam-3383	177	7	12	12	NUM
ejpam-3383	177	8	(	(	PUNCT
ejpam-3383	177	9	1	1	NUM
ejpam-3383	177	10	)	)	PUNCT
ejpam-3383	177	11	(	(	PUNCT
ejpam-3383	177	12	2019	2019	NUM
ejpam-3383	177	13	)	)	PUNCT
ejpam-3383	177	14	,	,	PUNCT
ejpam-3383	177	15	208	208	NUM
ejpam-3383	177	16	-	-	SYM
ejpam-3383	177	17	225	225	NUM
ejpam-3383	177	18	214	214	NUM
ejpam-3383	177	19	4	4	NUM
ejpam-3383	177	20	.	.	PUNCT
ejpam-3383	177	21	applications	application	NOUN
ejpam-3383	177	22	to	to	ADP
ejpam-3383	177	23	semigroups	semigroup	NOUN
ejpam-3383	177	24	,	,	PUNCT
ejpam-3383	177	25	γ	γ	NOUN
ejpam-3383	177	26	-	-	PUNCT
ejpam-3383	177	27	semigroups	semigroup	NOUN
ejpam-3383	177	28	and	and	CCONJ
ejpam-3383	177	29	to	to	ADP
ejpam-3383	177	30	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	177	31	in	in	ADP
ejpam-3383	177	32	this	this	DET
ejpam-3383	177	33	section	section	NOUN
ejpam-3383	177	34	,	,	PUNCT
ejpam-3383	177	35	we	we	PRON
ejpam-3383	177	36	apply	apply	VERB
ejpam-3383	177	37	the	the	DET
ejpam-3383	177	38	above	above	ADJ
ejpam-3383	177	39	results	result	NOUN
ejpam-3383	177	40	to	to	ADP
ejpam-3383	177	41	semigroups	semigroup	NOUN
ejpam-3383	177	42	,	,	PUNCT
ejpam-3383	177	43	γ	γ	NOUN
ejpam-3383	177	44	-	-	PUNCT
ejpam-3383	177	45	semigroups	semigroup	NOUN
ejpam-3383	177	46	,	,	PUNCT
ejpam-3383	177	47	and	and	CCONJ
ejpam-3383	177	48	to	to	ADP
ejpam-3383	177	49	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	177	50	.	.	PUNCT
ejpam-3383	178	1	let	let	VERB
ejpam-3383	178	2	us	we	PRON
ejpam-3383	178	3	begin	begin	VERB
ejpam-3383	178	4	with	with	ADP
ejpam-3383	178	5	a	a	DET
ejpam-3383	178	6	semigroup	semigroup	NOUN
ejpam-3383	178	7	.	.	PUNCT
ejpam-3383	179	1	as	as	ADP
ejpam-3383	179	2	an	an	DET
ejpam-3383	179	3	application	application	NOUN
ejpam-3383	179	4	of	of	ADP
ejpam-3383	179	5	proposition	proposition	NOUN
ejpam-3383	179	6	3.1	3.1	NUM
ejpam-3383	179	7	to	to	ADP
ejpam-3383	179	8	semigroups	semigroup	NOUN
ejpam-3383	179	9	,	,	PUNCT
ejpam-3383	179	10	we	we	PRON
ejpam-3383	179	11	have	have	VERB
ejpam-3383	179	12	the	the	DET
ejpam-3383	179	13	following	follow	VERB
ejpam-3383	179	14	corollary	corollary	NOUN
ejpam-3383	179	15	.	.	PUNCT
ejpam-3383	180	1	corollary	corollary	ADJ
ejpam-3383	180	2	4.1	4.1	NUM
ejpam-3383	180	3	.	.	PUNCT
ejpam-3383	181	1	let	let	AUX
ejpam-3383	181	2	(	(	PUNCT
ejpam-3383	181	3	s	s	X
ejpam-3383	181	4	,	,	PUNCT
ejpam-3383	181	5	·	·	PUNCT
ejpam-3383	181	6	)	)	PUNCT
ejpam-3383	181	7	be	be	AUX
ejpam-3383	181	8	a	a	DET
ejpam-3383	181	9	semigroup	semigroup	NOUN
ejpam-3383	181	10	and	and	CCONJ
ejpam-3383	181	11	m	m	VERB
ejpam-3383	181	12	an	an	DET
ejpam-3383	181	13	ideal	ideal	NOUN
ejpam-3383	181	14	of	of	ADP
ejpam-3383	181	15	s.	s.	PROPN
ejpam-3383	181	16	the	the	DET
ejpam-3383	181	17	following	follow	VERB
ejpam-3383	181	18	are	be	AUX
ejpam-3383	181	19	equivalent	equivalent	ADJ
ejpam-3383	181	20	:	:	PUNCT
ejpam-3383	181	21	(	(	PUNCT
ejpam-3383	181	22	1	1	X
ejpam-3383	181	23	)	)	PUNCT
ejpam-3383	182	1	m	m	VERB
ejpam-3383	182	2	is	be	AUX
ejpam-3383	182	3	weakly	weakly	ADV
ejpam-3383	182	4	prime	prime	ADJ
ejpam-3383	182	5	.	.	PUNCT
ejpam-3383	183	1	(	(	PUNCT
ejpam-3383	183	2	2	2	X
ejpam-3383	183	3	)	)	PUNCT
ejpam-3383	183	4	if	if	SCONJ
ejpam-3383	183	5	a	a	PRON
ejpam-3383	183	6	,	,	PUNCT
ejpam-3383	183	7	b	b	NOUN
ejpam-3383	183	8	are	be	AUX
ejpam-3383	183	9	right	right	ADJ
ejpam-3383	183	10	ideals	ideal	NOUN
ejpam-3383	183	11	of	of	ADP
ejpam-3383	183	12	s	s	PRON
ejpam-3383	183	13	such	such	ADJ
ejpam-3383	183	14	that	that	SCONJ
ejpam-3383	183	15	ab	ab	PROPN
ejpam-3383	183	16	⊆m	⊆m	NOUN
ejpam-3383	183	17	,	,	PUNCT
ejpam-3383	183	18	then	then	ADV
ejpam-3383	183	19	a	a	DET
ejpam-3383	183	20	⊆m	⊆m	NOUN
ejpam-3383	183	21	or	or	CCONJ
ejpam-3383	183	22	b	b	NOUN
ejpam-3383	183	23	⊆m	⊆m	NOUN
ejpam-3383	183	24	.	.	PUNCT
ejpam-3383	184	1	(	(	PUNCT
ejpam-3383	184	2	3	3	X
ejpam-3383	184	3	)	)	PUNCT
ejpam-3383	184	4	if	if	SCONJ
ejpam-3383	184	5	a	a	DET
ejpam-3383	184	6	,	,	PUNCT
ejpam-3383	184	7	b	b	NOUN
ejpam-3383	184	8	are	be	AUX
ejpam-3383	184	9	left	leave	VERB
ejpam-3383	184	10	ideals	ideal	NOUN
ejpam-3383	184	11	of	of	ADP
ejpam-3383	184	12	s	s	PRON
ejpam-3383	184	13	such	such	ADJ
ejpam-3383	184	14	that	that	SCONJ
ejpam-3383	184	15	ab	ab	PROPN
ejpam-3383	184	16	⊆m	⊆m	NOUN
ejpam-3383	184	17	,	,	PUNCT
ejpam-3383	184	18	then	then	ADV
ejpam-3383	184	19	a	a	DET
ejpam-3383	184	20	⊆m	⊆m	NOUN
ejpam-3383	184	21	or	or	CCONJ
ejpam-3383	184	22	b	b	NOUN
ejpam-3383	184	23	⊆m	⊆m	NOUN
ejpam-3383	184	24	.	.	PUNCT
ejpam-3383	185	1	(	(	PUNCT
ejpam-3383	185	2	4	4	X
ejpam-3383	185	3	)	)	PUNCT
ejpam-3383	185	4	if	if	SCONJ
ejpam-3383	185	5	a	a	PRON
ejpam-3383	185	6	is	be	AUX
ejpam-3383	185	7	a	a	DET
ejpam-3383	185	8	right	right	ADJ
ejpam-3383	185	9	ideal	ideal	NOUN
ejpam-3383	185	10	and	and	CCONJ
ejpam-3383	185	11	b	b	NOUN
ejpam-3383	185	12	is	be	AUX
ejpam-3383	185	13	a	a	DET
ejpam-3383	185	14	left	left	ADJ
ejpam-3383	185	15	ideal	ideal	NOUN
ejpam-3383	185	16	of	of	ADP
ejpam-3383	185	17	s	s	PRON
ejpam-3383	185	18	such	such	ADJ
ejpam-3383	185	19	that	that	SCONJ
ejpam-3383	185	20	ab	ab	PROPN
ejpam-3383	185	21	⊆	⊆	NUM
ejpam-3383	185	22	m	m	NOUN
ejpam-3383	185	23	,	,	PUNCT
ejpam-3383	185	24	then	then	ADV
ejpam-3383	185	25	a	a	DET
ejpam-3383	185	26	⊆	⊆	NUM
ejpam-3383	185	27	m	m	NOUN
ejpam-3383	185	28	or	or	CCONJ
ejpam-3383	185	29	b	b	NOUN
ejpam-3383	185	30	⊆m	⊆m	NOUN
ejpam-3383	185	31	.	.	PUNCT
ejpam-3383	186	1	proof	proof	NOUN
ejpam-3383	186	2	.	.	PUNCT
ejpam-3383	187	1	(	(	PUNCT
ejpam-3383	187	2	1	1	X
ejpam-3383	187	3	)	)	PUNCT
ejpam-3383	187	4	=	=	NOUN
ejpam-3383	187	5	⇒	⇒	NOUN
ejpam-3383	187	6	(	(	PUNCT
ejpam-3383	187	7	2	2	NUM
ejpam-3383	187	8	)	)	PUNCT
ejpam-3383	187	9	.	.	PUNCT
ejpam-3383	188	1	let	let	VERB
ejpam-3383	188	2	a	a	DET
ejpam-3383	188	3	,	,	PUNCT
ejpam-3383	188	4	b	b	NOUN
ejpam-3383	188	5	be	be	AUX
ejpam-3383	188	6	right	right	ADJ
ejpam-3383	188	7	ideals	ideal	NOUN
ejpam-3383	188	8	of	of	ADP
ejpam-3383	188	9	s	s	PRON
ejpam-3383	188	10	such	such	ADJ
ejpam-3383	188	11	that	that	SCONJ
ejpam-3383	188	12	ab	ab	PROPN
ejpam-3383	188	13	⊆	⊆	NUM
ejpam-3383	188	14	m	m	NOUN
ejpam-3383	188	15	.	.	PUNCT
ejpam-3383	189	1	the	the	DET
ejpam-3383	189	2	set	set	VERB
ejpam-3383	189	3	p(s	p(s	NOUN
ejpam-3383	189	4	)	)	PUNCT
ejpam-3383	189	5	of	of	ADP
ejpam-3383	189	6	all	all	DET
ejpam-3383	189	7	subsets	subset	NOUN
ejpam-3383	189	8	of	of	ADP
ejpam-3383	189	9	s	s	PRON
ejpam-3383	189	10	with	with	ADP
ejpam-3383	189	11	the	the	DET
ejpam-3383	189	12	multiplication	multiplication	NOUN
ejpam-3383	189	13	a	a	DET
ejpam-3383	189	14	◦	◦	NOUN
ejpam-3383	189	15	b	b	X
ejpam-3383	189	16	:	:	PUNCT
ejpam-3383	189	17	=	=	SYM
ejpam-3383	189	18	ab	ab	PROPN
ejpam-3383	189	19	and	and	CCONJ
ejpam-3383	189	20	the	the	DET
ejpam-3383	189	21	inclusion	inclusion	NOUN
ejpam-3383	189	22	relation	relation	NOUN
ejpam-3383	189	23	“	"	PUNCT
ejpam-3383	189	24	⊆	⊆	X
ejpam-3383	189	25	”	"	PUNCT
ejpam-3383	189	26	is	be	AUX
ejpam-3383	189	27	a	a	DET
ejpam-3383	189	28	∨e	∨e	NOUN
ejpam-3383	189	29	-	-	PUNCT
ejpam-3383	189	30	semigroup	semigroup	PROPN
ejpam-3383	189	31	,	,	PUNCT
ejpam-3383	189	32	m	m	VERB
ejpam-3383	189	33	is	be	AUX
ejpam-3383	189	34	a	a	DET
ejpam-3383	189	35	weakly	weakly	ADJ
ejpam-3383	189	36	prime	prime	ADJ
ejpam-3383	189	37	ideal	ideal	ADJ
ejpam-3383	189	38	element	element	NOUN
ejpam-3383	189	39	of	of	ADP
ejpam-3383	189	40	(	(	PUNCT
ejpam-3383	189	41	p(s	p(s	PROPN
ejpam-3383	189	42	)	)	PUNCT
ejpam-3383	189	43	,	,	PUNCT
ejpam-3383	189	44	◦	◦	NOUN
ejpam-3383	189	45	,	,	PUNCT
ejpam-3383	189	46	⊆	⊆	NUM
ejpam-3383	189	47	)	)	PUNCT
ejpam-3383	189	48	and	and	CCONJ
ejpam-3383	189	49	a	a	DET
ejpam-3383	189	50	,	,	PUNCT
ejpam-3383	189	51	b	b	NOUN
ejpam-3383	189	52	are	be	AUX
ejpam-3383	189	53	right	right	ADJ
ejpam-3383	189	54	ideal	ideal	ADJ
ejpam-3383	189	55	elements	element	NOUN
ejpam-3383	189	56	of	of	ADP
ejpam-3383	189	57	(	(	PUNCT
ejpam-3383	189	58	p(s	p(s	PROPN
ejpam-3383	189	59	)	)	PUNCT
ejpam-3383	189	60	,	,	PUNCT
ejpam-3383	189	61	◦	◦	NOUN
ejpam-3383	189	62	,	,	PUNCT
ejpam-3383	189	63	⊆	⊆	NOUN
ejpam-3383	189	64	)	)	PUNCT
ejpam-3383	189	65	such	such	ADJ
ejpam-3383	189	66	that	that	SCONJ
ejpam-3383	189	67	a	a	DET
ejpam-3383	189	68	◦	◦	NOUN
ejpam-3383	189	69	b	b	NOUN
ejpam-3383	189	70	⊆	⊆	NUM
ejpam-3383	189	71	m	m	NOUN
ejpam-3383	189	72	.	.	PUNCT
ejpam-3383	190	1	by	by	ADP
ejpam-3383	190	2	proposition	proposition	NOUN
ejpam-3383	190	3	3.1(1	3.1(1	NUM
ejpam-3383	190	4	)	)	PUNCT
ejpam-3383	190	5	⇒	⇒	NOUN
ejpam-3383	190	6	(	(	PUNCT
ejpam-3383	190	7	2	2	NUM
ejpam-3383	190	8	)	)	PUNCT
ejpam-3383	190	9	,	,	PUNCT
ejpam-3383	190	10	we	we	PRON
ejpam-3383	190	11	have	have	VERB
ejpam-3383	190	12	a	a	DET
ejpam-3383	190	13	⊆m	⊆m	NOUN
ejpam-3383	190	14	or	or	CCONJ
ejpam-3383	190	15	b	b	NOUN
ejpam-3383	190	16	⊆m	⊆m	NOUN
ejpam-3383	190	17	,	,	PUNCT
ejpam-3383	190	18	so	so	CCONJ
ejpam-3383	190	19	property	property	NOUN
ejpam-3383	190	20	(	(	PUNCT
ejpam-3383	190	21	2	2	NUM
ejpam-3383	190	22	)	)	PUNCT
ejpam-3383	190	23	is	be	AUX
ejpam-3383	190	24	satisfied	satisfied	ADJ
ejpam-3383	190	25	.	.	PUNCT
ejpam-3383	191	1	the	the	DET
ejpam-3383	191	2	implications	implication	NOUN
ejpam-3383	191	3	(	(	PUNCT
ejpam-3383	191	4	1)⇒	1)⇒	NUM
ejpam-3383	191	5	(	(	PUNCT
ejpam-3383	191	6	3	3	NUM
ejpam-3383	191	7	)	)	PUNCT
ejpam-3383	191	8	and	and	CCONJ
ejpam-3383	191	9	(	(	PUNCT
ejpam-3383	191	10	1)⇒	1)⇒	NUM
ejpam-3383	191	11	(	(	PUNCT
ejpam-3383	191	12	4	4	NUM
ejpam-3383	191	13	)	)	PUNCT
ejpam-3383	191	14	can	can	AUX
ejpam-3383	191	15	be	be	AUX
ejpam-3383	191	16	proved	prove	VERB
ejpam-3383	191	17	in	in	ADP
ejpam-3383	191	18	a	a	DET
ejpam-3383	191	19	similar	similar	ADJ
ejpam-3383	191	20	way	way	NOUN
ejpam-3383	191	21	.	.	PUNCT
ejpam-3383	192	1	the	the	DET
ejpam-3383	192	2	implications	implication	NOUN
ejpam-3383	192	3	(	(	PUNCT
ejpam-3383	192	4	2	2	NUM
ejpam-3383	192	5	)	)	PUNCT
ejpam-3383	192	6	⇒	⇒	NOUN
ejpam-3383	192	7	(	(	PUNCT
ejpam-3383	192	8	1	1	NUM
ejpam-3383	192	9	)	)	PUNCT
ejpam-3383	192	10	,	,	PUNCT
ejpam-3383	192	11	(	(	PUNCT
ejpam-3383	192	12	3	3	X
ejpam-3383	192	13	)	)	PUNCT
ejpam-3383	192	14	⇒	⇒	NOUN
ejpam-3383	192	15	(	(	PUNCT
ejpam-3383	192	16	1	1	NUM
ejpam-3383	192	17	)	)	PUNCT
ejpam-3383	192	18	and	and	CCONJ
ejpam-3383	192	19	(	(	PUNCT
ejpam-3383	192	20	4	4	X
ejpam-3383	192	21	)	)	PUNCT
ejpam-3383	192	22	⇒	⇒	NOUN
ejpam-3383	192	23	(	(	PUNCT
ejpam-3383	192	24	1	1	X
ejpam-3383	192	25	)	)	PUNCT
ejpam-3383	192	26	are	be	AUX
ejpam-3383	192	27	obvious	obvious	ADJ
ejpam-3383	192	28	and	and	CCONJ
ejpam-3383	192	29	they	they	PRON
ejpam-3383	192	30	hold	hold	VERB
ejpam-3383	192	31	in	in	ADP
ejpam-3383	192	32	groupoids	groupoid	NOUN
ejpam-3383	192	33	in	in	ADP
ejpam-3383	192	34	general	general	ADJ
ejpam-3383	192	35	.	.	PUNCT
ejpam-3383	193	1	�	�	PROPN
ejpam-3383	193	2	in	in	ADP
ejpam-3383	193	3	a	a	DET
ejpam-3383	193	4	similar	similar	ADJ
ejpam-3383	193	5	way	way	NOUN
ejpam-3383	193	6	,	,	PUNCT
ejpam-3383	193	7	applying	apply	VERB
ejpam-3383	193	8	proposition	proposition	NOUN
ejpam-3383	193	9	3.2	3.2	NUM
ejpam-3383	193	10	to	to	ADP
ejpam-3383	193	11	semigroups	semigroups	X
ejpam-3383	193	12	we	we	PRON
ejpam-3383	193	13	get	get	VERB
ejpam-3383	193	14	the	the	DET
ejpam-3383	193	15	following	follow	VERB
ejpam-3383	193	16	corollary	corollary	NOUN
ejpam-3383	193	17	.	.	PUNCT
ejpam-3383	194	1	corollary	corollary	ADJ
ejpam-3383	194	2	4.2	4.2	NUM
ejpam-3383	194	3	.	.	PUNCT
ejpam-3383	195	1	let	let	VERB
ejpam-3383	195	2	s	s	PRON
ejpam-3383	195	3	be	be	AUX
ejpam-3383	195	4	a	a	DET
ejpam-3383	195	5	semigroup	semigroup	NOUN
ejpam-3383	195	6	and	and	CCONJ
ejpam-3383	195	7	m	m	VERB
ejpam-3383	195	8	an	an	DET
ejpam-3383	195	9	ideal	ideal	NOUN
ejpam-3383	195	10	of	of	ADP
ejpam-3383	195	11	s.	s.	PROPN
ejpam-3383	195	12	the	the	DET
ejpam-3383	195	13	following	follow	VERB
ejpam-3383	195	14	are	be	AUX
ejpam-3383	195	15	equivalent	equivalent	ADJ
ejpam-3383	195	16	:	:	PUNCT
ejpam-3383	195	17	(	(	PUNCT
ejpam-3383	195	18	1	1	X
ejpam-3383	195	19	)	)	PUNCT
ejpam-3383	196	1	m	m	VERB
ejpam-3383	196	2	is	be	AUX
ejpam-3383	196	3	weakly	weakly	ADJ
ejpam-3383	196	4	semiprime	semiprime	NOUN
ejpam-3383	196	5	.	.	PUNCT
ejpam-3383	197	1	(	(	PUNCT
ejpam-3383	197	2	2	2	X
ejpam-3383	197	3	)	)	PUNCT
ejpam-3383	197	4	if	if	SCONJ
ejpam-3383	197	5	a	a	PRON
ejpam-3383	197	6	is	be	AUX
ejpam-3383	197	7	a	a	DET
ejpam-3383	197	8	right	right	ADJ
ejpam-3383	197	9	ideal	ideal	NOUN
ejpam-3383	197	10	of	of	ADP
ejpam-3383	197	11	s	s	PRON
ejpam-3383	197	12	such	such	ADJ
ejpam-3383	197	13	that	that	DET
ejpam-3383	197	14	a2	a2	PROPN
ejpam-3383	197	15	⊆m	⊆m	NOUN
ejpam-3383	197	16	,	,	PUNCT
ejpam-3383	197	17	then	then	ADV
ejpam-3383	197	18	a	a	DET
ejpam-3383	197	19	⊆m	⊆m	NOUN
ejpam-3383	197	20	.	.	PUNCT
ejpam-3383	198	1	(	(	PUNCT
ejpam-3383	198	2	3	3	X
ejpam-3383	198	3	)	)	PUNCT
ejpam-3383	198	4	if	if	SCONJ
ejpam-3383	198	5	b	b	NOUN
ejpam-3383	198	6	is	be	AUX
ejpam-3383	198	7	a	a	DET
ejpam-3383	198	8	left	left	ADJ
ejpam-3383	198	9	ideal	ideal	NOUN
ejpam-3383	198	10	of	of	ADP
ejpam-3383	198	11	s	s	PRON
ejpam-3383	198	12	such	such	ADJ
ejpam-3383	198	13	that	that	DET
ejpam-3383	198	14	b2	b2	NOUN
ejpam-3383	198	15	⊆m	⊆m	NOUN
ejpam-3383	198	16	,	,	PUNCT
ejpam-3383	198	17	then	then	ADV
ejpam-3383	198	18	b	b	X
ejpam-3383	198	19	⊆m	⊆m	NOUN
ejpam-3383	198	20	.	.	PUNCT
ejpam-3383	199	1	we	we	PRON
ejpam-3383	199	2	apply	apply	VERB
ejpam-3383	199	3	now	now	ADV
ejpam-3383	199	4	proposition	proposition	NOUN
ejpam-3383	199	5	3.1	3.1	NUM
ejpam-3383	199	6	to	to	ADP
ejpam-3383	199	7	γ	γ	NOUN
ejpam-3383	199	8	-	-	PUNCT
ejpam-3383	199	9	semigroups	semigroup	NOUN
ejpam-3383	199	10	.	.	PUNCT
ejpam-3383	200	1	corollary	corollary	NOUN
ejpam-3383	200	2	4.3	4.3	NUM
ejpam-3383	200	3	.	.	PUNCT
ejpam-3383	201	1	let	let	AUX
ejpam-3383	201	2	(	(	PUNCT
ejpam-3383	201	3	s	s	X
ejpam-3383	201	4	,	,	PUNCT
ejpam-3383	201	5	γ	γ	NOUN
ejpam-3383	201	6	)	)	PUNCT
ejpam-3383	201	7	be	be	VERB
ejpam-3383	201	8	a	a	DET
ejpam-3383	201	9	γ	γ	NOUN
ejpam-3383	201	10	-	-	PUNCT
ejpam-3383	201	11	semigroup	semigroup	NOUN
ejpam-3383	201	12	and	and	CCONJ
ejpam-3383	201	13	m	m	AUX
ejpam-3383	201	14	be	be	VERB
ejpam-3383	201	15	an	an	DET
ejpam-3383	201	16	ideal	ideal	NOUN
ejpam-3383	201	17	of	of	ADP
ejpam-3383	201	18	s.	s.	PROPN
ejpam-3383	201	19	the	the	DET
ejpam-3383	201	20	following	follow	VERB
ejpam-3383	201	21	are	be	AUX
ejpam-3383	201	22	equivalent	equivalent	ADJ
ejpam-3383	201	23	:	:	PUNCT
ejpam-3383	201	24	(	(	PUNCT
ejpam-3383	201	25	1	1	X
ejpam-3383	201	26	)	)	PUNCT
ejpam-3383	202	1	m	m	VERB
ejpam-3383	202	2	is	be	AUX
ejpam-3383	202	3	weakly	weakly	ADV
ejpam-3383	202	4	prime	prime	ADJ
ejpam-3383	202	5	.	.	PUNCT
ejpam-3383	203	1	(	(	PUNCT
ejpam-3383	203	2	2	2	X
ejpam-3383	203	3	)	)	PUNCT
ejpam-3383	203	4	if	if	SCONJ
ejpam-3383	203	5	a	a	PRON
ejpam-3383	203	6	,	,	PUNCT
ejpam-3383	203	7	b	b	NOUN
ejpam-3383	203	8	are	be	AUX
ejpam-3383	203	9	right	right	ADJ
ejpam-3383	203	10	ideals	ideal	NOUN
ejpam-3383	203	11	of	of	ADP
ejpam-3383	203	12	s	s	PRON
ejpam-3383	203	13	such	such	ADJ
ejpam-3383	203	14	that	that	SCONJ
ejpam-3383	203	15	aγb	aγb	NOUN
ejpam-3383	203	16	⊆m	⊆m	NOUN
ejpam-3383	203	17	,	,	PUNCT
ejpam-3383	203	18	then	then	ADV
ejpam-3383	203	19	a	a	DET
ejpam-3383	203	20	⊆m	⊆m	NOUN
ejpam-3383	203	21	or	or	CCONJ
ejpam-3383	203	22	b	b	NOUN
ejpam-3383	203	23	⊆m	⊆m	NOUN
ejpam-3383	203	24	.	.	PUNCT
ejpam-3383	204	1	(	(	PUNCT
ejpam-3383	204	2	3	3	X
ejpam-3383	204	3	)	)	PUNCT
ejpam-3383	204	4	if	if	SCONJ
ejpam-3383	204	5	a	a	DET
ejpam-3383	204	6	,	,	PUNCT
ejpam-3383	204	7	b	b	NOUN
ejpam-3383	204	8	are	be	AUX
ejpam-3383	204	9	left	leave	VERB
ejpam-3383	204	10	ideals	ideal	NOUN
ejpam-3383	204	11	of	of	ADP
ejpam-3383	204	12	s	s	PRON
ejpam-3383	204	13	such	such	ADJ
ejpam-3383	204	14	that	that	SCONJ
ejpam-3383	204	15	aγb	aγb	NOUN
ejpam-3383	204	16	⊆m	⊆m	NOUN
ejpam-3383	204	17	,	,	PUNCT
ejpam-3383	204	18	then	then	ADV
ejpam-3383	204	19	a	a	DET
ejpam-3383	204	20	⊆m	⊆m	NOUN
ejpam-3383	204	21	or	or	CCONJ
ejpam-3383	204	22	b	b	NOUN
ejpam-3383	204	23	⊆m	⊆m	NOUN
ejpam-3383	204	24	.	.	PUNCT
ejpam-3383	205	1	(	(	PUNCT
ejpam-3383	205	2	4	4	X
ejpam-3383	205	3	)	)	PUNCT
ejpam-3383	205	4	if	if	SCONJ
ejpam-3383	205	5	a	a	PRON
ejpam-3383	205	6	is	be	AUX
ejpam-3383	205	7	a	a	DET
ejpam-3383	205	8	right	right	ADJ
ejpam-3383	205	9	ideal	ideal	NOUN
ejpam-3383	205	10	and	and	CCONJ
ejpam-3383	205	11	b	b	NOUN
ejpam-3383	205	12	is	be	AUX
ejpam-3383	205	13	a	a	DET
ejpam-3383	205	14	left	left	ADJ
ejpam-3383	205	15	ideal	ideal	NOUN
ejpam-3383	205	16	of	of	ADP
ejpam-3383	205	17	s	s	PRON
ejpam-3383	205	18	such	such	ADJ
ejpam-3383	205	19	that	that	SCONJ
ejpam-3383	205	20	aγb	aγb	NOUN
ejpam-3383	205	21	⊆m	⊆m	NOUN
ejpam-3383	205	22	,	,	PUNCT
ejpam-3383	205	23	then	then	ADV
ejpam-3383	205	24	a	a	DET
ejpam-3383	205	25	⊆m	⊆m	NOUN
ejpam-3383	205	26	or	or	CCONJ
ejpam-3383	205	27	b	b	NOUN
ejpam-3383	205	28	⊆m	⊆m	NOUN
ejpam-3383	205	29	.	.	PUNCT
ejpam-3383	206	1	n.	n.	PROPN
ejpam-3383	206	2	kehayopulu	kehayopulu	PROPN
ejpam-3383	206	3	/	/	SYM
ejpam-3383	206	4	eur	eur	PROPN
ejpam-3383	206	5	.	.	PUNCT
ejpam-3383	207	1	j.	j.	PROPN
ejpam-3383	207	2	pure	pure	PROPN
ejpam-3383	207	3	appl	appl	PROPN
ejpam-3383	207	4	.	.	PROPN
ejpam-3383	207	5	math	math	PROPN
ejpam-3383	207	6	,	,	PUNCT
ejpam-3383	207	7	12	12	NUM
ejpam-3383	207	8	(	(	PUNCT
ejpam-3383	207	9	1	1	NUM
ejpam-3383	207	10	)	)	PUNCT
ejpam-3383	207	11	(	(	PUNCT
ejpam-3383	207	12	2019	2019	NUM
ejpam-3383	207	13	)	)	PUNCT
ejpam-3383	207	14	,	,	PUNCT
ejpam-3383	207	15	208	208	NUM
ejpam-3383	207	16	-	-	SYM
ejpam-3383	207	17	225	225	NUM
ejpam-3383	207	18	215	215	NUM
ejpam-3383	207	19	proof	proof	NOUN
ejpam-3383	207	20	.	.	PUNCT
ejpam-3383	208	1	(	(	PUNCT
ejpam-3383	208	2	1	1	X
ejpam-3383	208	3	)	)	PUNCT
ejpam-3383	208	4	=	=	NOUN
ejpam-3383	208	5	⇒	⇒	NOUN
ejpam-3383	208	6	(	(	PUNCT
ejpam-3383	208	7	2	2	NUM
ejpam-3383	208	8	)	)	PUNCT
ejpam-3383	208	9	.	.	PUNCT
ejpam-3383	209	1	let	let	VERB
ejpam-3383	209	2	a	a	DET
ejpam-3383	209	3	,	,	PUNCT
ejpam-3383	209	4	b	b	NOUN
ejpam-3383	209	5	be	be	AUX
ejpam-3383	209	6	right	right	ADJ
ejpam-3383	209	7	ideals	ideal	NOUN
ejpam-3383	209	8	of	of	ADP
ejpam-3383	209	9	s	s	PRON
ejpam-3383	209	10	such	such	ADJ
ejpam-3383	209	11	that	that	SCONJ
ejpam-3383	209	12	aγb	aγb	ADV
ejpam-3383	209	13	⊆	⊆	NUM
ejpam-3383	209	14	m	m	NOUN
ejpam-3383	209	15	.	.	PUNCT
ejpam-3383	210	1	the	the	DET
ejpam-3383	210	2	set	set	VERB
ejpam-3383	210	3	p(s	p(s	NOUN
ejpam-3383	210	4	)	)	PUNCT
ejpam-3383	210	5	of	of	ADP
ejpam-3383	210	6	all	all	DET
ejpam-3383	210	7	subsets	subset	NOUN
ejpam-3383	210	8	of	of	ADP
ejpam-3383	210	9	s	s	PRON
ejpam-3383	210	10	with	with	ADP
ejpam-3383	210	11	the	the	DET
ejpam-3383	210	12	multiplication	multiplication	NOUN
ejpam-3383	210	13	a	a	DET
ejpam-3383	210	14	◦	◦	NOUN
ejpam-3383	210	15	b	b	NOUN
ejpam-3383	210	16	=	=	NOUN
ejpam-3383	210	17	aγb	aγb	NOUN
ejpam-3383	210	18	and	and	CCONJ
ejpam-3383	210	19	the	the	DET
ejpam-3383	210	20	inclusion	inclusion	NOUN
ejpam-3383	210	21	relation	relation	NOUN
ejpam-3383	210	22	“	"	PUNCT
ejpam-3383	210	23	⊆	⊆	X
ejpam-3383	210	24	”	"	PUNCT
ejpam-3383	210	25	is	be	AUX
ejpam-3383	210	26	a	a	DET
ejpam-3383	210	27	∨e	∨e	NOUN
ejpam-3383	210	28	-	-	PUNCT
ejpam-3383	210	29	semigroup	semigroup	PROPN
ejpam-3383	210	30	,	,	PUNCT
ejpam-3383	210	31	the	the	DET
ejpam-3383	210	32	set	set	NOUN
ejpam-3383	210	33	m	m	VERB
ejpam-3383	210	34	is	be	AUX
ejpam-3383	210	35	a	a	DET
ejpam-3383	210	36	weakly	weakly	ADJ
ejpam-3383	210	37	prime	prime	ADJ
ejpam-3383	210	38	ideal	ideal	ADJ
ejpam-3383	210	39	element	element	NOUN
ejpam-3383	210	40	of	of	ADP
ejpam-3383	210	41	(	(	PUNCT
ejpam-3383	210	42	p(s	p(s	PROPN
ejpam-3383	210	43	)	)	PUNCT
ejpam-3383	210	44	,	,	PUNCT
ejpam-3383	210	45	◦	◦	NOUN
ejpam-3383	210	46	,	,	PUNCT
ejpam-3383	210	47	⊆	⊆	NUM
ejpam-3383	210	48	)	)	PUNCT
ejpam-3383	210	49	and	and	CCONJ
ejpam-3383	210	50	,	,	PUNCT
ejpam-3383	210	51	a	a	DET
ejpam-3383	210	52	,	,	PUNCT
ejpam-3383	210	53	b	b	NOUN
ejpam-3383	210	54	are	be	AUX
ejpam-3383	210	55	ideal	ideal	ADJ
ejpam-3383	210	56	elements	element	NOUN
ejpam-3383	210	57	of	of	ADP
ejpam-3383	210	58	(	(	PUNCT
ejpam-3383	210	59	p(s	p(s	PROPN
ejpam-3383	210	60	)	)	PUNCT
ejpam-3383	210	61	,	,	PUNCT
ejpam-3383	210	62	◦	◦	NOUN
ejpam-3383	210	63	,	,	PUNCT
ejpam-3383	210	64	⊆	⊆	NOUN
ejpam-3383	210	65	)	)	PUNCT
ejpam-3383	210	66	such	such	ADJ
ejpam-3383	210	67	that	that	SCONJ
ejpam-3383	210	68	a	a	DET
ejpam-3383	210	69	◦	◦	NOUN
ejpam-3383	210	70	b	b	NOUN
ejpam-3383	210	71	⊆m	⊆m	NOUN
ejpam-3383	210	72	.	.	PUNCT
ejpam-3383	211	1	by	by	ADP
ejpam-3383	211	2	proposition	proposition	NOUN
ejpam-3383	211	3	3.1	3.1	NUM
ejpam-3383	211	4	,	,	PUNCT
ejpam-3383	211	5	we	we	PRON
ejpam-3383	211	6	have	have	VERB
ejpam-3383	211	7	a	a	DET
ejpam-3383	211	8	⊆m	⊆m	NOUN
ejpam-3383	211	9	or	or	CCONJ
ejpam-3383	211	10	b	b	NOUN
ejpam-3383	211	11	⊆m	⊆m	NOUN
ejpam-3383	211	12	.	.	PUNCT
ejpam-3383	212	1	�	�	PROPN
ejpam-3383	212	2	in	in	ADP
ejpam-3383	212	3	a	a	DET
ejpam-3383	212	4	similar	similar	ADJ
ejpam-3383	212	5	way	way	NOUN
ejpam-3383	212	6	,	,	PUNCT
ejpam-3383	212	7	proposition	proposition	NOUN
ejpam-3383	212	8	3.2	3.2	NUM
ejpam-3383	212	9	can	can	AUX
ejpam-3383	212	10	be	be	AUX
ejpam-3383	212	11	applied	apply	VERB
ejpam-3383	212	12	to	to	ADP
ejpam-3383	212	13	the	the	DET
ejpam-3383	212	14	following	follow	VERB
ejpam-3383	212	15	corollary	corollary	NOUN
ejpam-3383	212	16	.	.	PUNCT
ejpam-3383	213	1	corollary	corollary	NOUN
ejpam-3383	213	2	4.4	4.4	NUM
ejpam-3383	213	3	.	.	PUNCT
ejpam-3383	214	1	let	let	AUX
ejpam-3383	214	2	(	(	PUNCT
ejpam-3383	214	3	s	s	X
ejpam-3383	214	4	,	,	PUNCT
ejpam-3383	214	5	γ	γ	NOUN
ejpam-3383	214	6	)	)	PUNCT
ejpam-3383	214	7	be	be	VERB
ejpam-3383	214	8	a	a	DET
ejpam-3383	214	9	γ	γ	NOUN
ejpam-3383	214	10	-	-	PUNCT
ejpam-3383	214	11	semigroup	semigroup	NOUN
ejpam-3383	214	12	and	and	CCONJ
ejpam-3383	214	13	m	m	VERB
ejpam-3383	214	14	an	an	DET
ejpam-3383	214	15	ideal	ideal	NOUN
ejpam-3383	214	16	of	of	ADP
ejpam-3383	214	17	s.	s.	PROPN
ejpam-3383	214	18	the	the	DET
ejpam-3383	214	19	following	follow	VERB
ejpam-3383	214	20	are	be	AUX
ejpam-3383	214	21	equivalent	equivalent	ADJ
ejpam-3383	214	22	:	:	PUNCT
ejpam-3383	214	23	(	(	PUNCT
ejpam-3383	214	24	1	1	X
ejpam-3383	214	25	)	)	PUNCT
ejpam-3383	214	26	m	m	VERB
ejpam-3383	214	27	is	be	AUX
ejpam-3383	214	28	weakly	weakly	ADJ
ejpam-3383	214	29	semiprime	semiprime	NOUN
ejpam-3383	214	30	.	.	PUNCT
ejpam-3383	215	1	(	(	PUNCT
ejpam-3383	215	2	2	2	X
ejpam-3383	215	3	)	)	PUNCT
ejpam-3383	215	4	if	if	SCONJ
ejpam-3383	215	5	a	a	PRON
ejpam-3383	215	6	is	be	AUX
ejpam-3383	215	7	a	a	DET
ejpam-3383	215	8	right	right	ADJ
ejpam-3383	215	9	ideal	ideal	NOUN
ejpam-3383	215	10	of	of	ADP
ejpam-3383	215	11	s	s	PRON
ejpam-3383	215	12	such	such	ADJ
ejpam-3383	215	13	that	that	DET
ejpam-3383	215	14	aγa	aγa	NOUN
ejpam-3383	215	15	⊆m	⊆m	NOUN
ejpam-3383	215	16	,	,	PUNCT
ejpam-3383	215	17	then	then	ADV
ejpam-3383	215	18	a	a	DET
ejpam-3383	215	19	⊆m	⊆m	NOUN
ejpam-3383	215	20	.	.	PUNCT
ejpam-3383	216	1	(	(	PUNCT
ejpam-3383	216	2	3	3	X
ejpam-3383	216	3	)	)	PUNCT
ejpam-3383	216	4	if	if	SCONJ
ejpam-3383	216	5	b	b	NOUN
ejpam-3383	216	6	is	be	AUX
ejpam-3383	216	7	a	a	DET
ejpam-3383	216	8	left	left	ADJ
ejpam-3383	216	9	ideal	ideal	NOUN
ejpam-3383	216	10	of	of	ADP
ejpam-3383	216	11	s	s	PRON
ejpam-3383	216	12	such	such	ADJ
ejpam-3383	216	13	that	that	DET
ejpam-3383	216	14	bγb	bγb	NOUN
ejpam-3383	216	15	⊆m	⊆m	NOUN
ejpam-3383	216	16	,	,	PUNCT
ejpam-3383	216	17	then	then	ADV
ejpam-3383	216	18	b	b	X
ejpam-3383	216	19	⊆m	⊆m	NOUN
ejpam-3383	216	20	.	.	PUNCT
ejpam-3383	217	1	finally	finally	ADV
ejpam-3383	217	2	,	,	PUNCT
ejpam-3383	217	3	we	we	PRON
ejpam-3383	217	4	apply	apply	VERB
ejpam-3383	217	5	proposition	proposition	NOUN
ejpam-3383	217	6	3.1	3.1	NUM
ejpam-3383	217	7	to	to	ADP
ejpam-3383	217	8	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	217	9	.	.	PUNCT
ejpam-3383	218	1	corollary	corollary	ADJ
ejpam-3383	218	2	4.5	4.5	NUM
ejpam-3383	218	3	.	.	PUNCT
ejpam-3383	219	1	let	let	AUX
ejpam-3383	219	2	(	(	PUNCT
ejpam-3383	219	3	s	s	NOUN
ejpam-3383	219	4	,	,	PUNCT
ejpam-3383	219	5	◦	◦	NOUN
ejpam-3383	219	6	)	)	PUNCT
ejpam-3383	219	7	be	be	VERB
ejpam-3383	219	8	an	an	DET
ejpam-3383	219	9	hypersemigroup	hypersemigroup	NOUN
ejpam-3383	219	10	and	and	CCONJ
ejpam-3383	219	11	m	m	AUX
ejpam-3383	219	12	be	be	AUX
ejpam-3383	219	13	an	an	DET
ejpam-3383	219	14	ideal	ideal	NOUN
ejpam-3383	219	15	of	of	ADP
ejpam-3383	219	16	s.	s.	PROPN
ejpam-3383	219	17	the	the	DET
ejpam-3383	219	18	following	follow	VERB
ejpam-3383	219	19	are	be	AUX
ejpam-3383	219	20	equivalent	equivalent	ADJ
ejpam-3383	219	21	:	:	PUNCT
ejpam-3383	219	22	(	(	PUNCT
ejpam-3383	219	23	1	1	X
ejpam-3383	219	24	)	)	PUNCT
ejpam-3383	220	1	m	m	VERB
ejpam-3383	220	2	is	be	AUX
ejpam-3383	220	3	weakly	weakly	ADV
ejpam-3383	220	4	prime	prime	ADJ
ejpam-3383	220	5	.	.	PUNCT
ejpam-3383	221	1	(	(	PUNCT
ejpam-3383	221	2	2	2	X
ejpam-3383	221	3	)	)	PUNCT
ejpam-3383	221	4	if	if	SCONJ
ejpam-3383	221	5	a	a	PRON
ejpam-3383	221	6	,	,	PUNCT
ejpam-3383	221	7	b	b	NOUN
ejpam-3383	221	8	are	be	AUX
ejpam-3383	221	9	right	right	ADJ
ejpam-3383	221	10	ideals	ideal	NOUN
ejpam-3383	221	11	of	of	ADP
ejpam-3383	221	12	s	s	PRON
ejpam-3383	221	13	such	such	ADJ
ejpam-3383	221	14	that	that	SCONJ
ejpam-3383	221	15	a	a	DET
ejpam-3383	221	16	∗b	∗b	PROPN
ejpam-3383	221	17	⊆m	⊆m	NOUN
ejpam-3383	221	18	,	,	PUNCT
ejpam-3383	221	19	then	then	ADV
ejpam-3383	221	20	a	a	DET
ejpam-3383	221	21	⊆m	⊆m	NOUN
ejpam-3383	221	22	or	or	CCONJ
ejpam-3383	221	23	b	b	NOUN
ejpam-3383	221	24	⊆m	⊆m	NOUN
ejpam-3383	221	25	.	.	PUNCT
ejpam-3383	222	1	(	(	PUNCT
ejpam-3383	222	2	3	3	X
ejpam-3383	222	3	)	)	PUNCT
ejpam-3383	222	4	if	if	SCONJ
ejpam-3383	222	5	a	a	DET
ejpam-3383	222	6	,	,	PUNCT
ejpam-3383	222	7	b	b	NOUN
ejpam-3383	222	8	are	be	AUX
ejpam-3383	222	9	left	leave	VERB
ejpam-3383	222	10	ideals	ideal	NOUN
ejpam-3383	222	11	of	of	ADP
ejpam-3383	222	12	s	s	PRON
ejpam-3383	222	13	such	such	ADJ
ejpam-3383	222	14	that	that	SCONJ
ejpam-3383	222	15	a	a	DET
ejpam-3383	222	16	∗b	∗b	PROPN
ejpam-3383	222	17	⊆m	⊆m	NOUN
ejpam-3383	222	18	,	,	PUNCT
ejpam-3383	222	19	then	then	ADV
ejpam-3383	222	20	a	a	DET
ejpam-3383	222	21	⊆m	⊆m	NOUN
ejpam-3383	222	22	or	or	CCONJ
ejpam-3383	222	23	b	b	NOUN
ejpam-3383	222	24	⊆m	⊆m	NOUN
ejpam-3383	222	25	.	.	PUNCT
ejpam-3383	223	1	(	(	PUNCT
ejpam-3383	223	2	4	4	X
ejpam-3383	223	3	)	)	PUNCT
ejpam-3383	223	4	if	if	SCONJ
ejpam-3383	223	5	a	a	PRON
ejpam-3383	223	6	is	be	AUX
ejpam-3383	223	7	a	a	DET
ejpam-3383	223	8	right	right	ADJ
ejpam-3383	223	9	ideal	ideal	NOUN
ejpam-3383	223	10	and	and	CCONJ
ejpam-3383	223	11	b	b	DET
ejpam-3383	223	12	a	a	DET
ejpam-3383	223	13	left	left	ADJ
ejpam-3383	223	14	ideal	ideal	NOUN
ejpam-3383	223	15	of	of	ADP
ejpam-3383	223	16	s	s	PRON
ejpam-3383	223	17	such	such	ADJ
ejpam-3383	223	18	that	that	SCONJ
ejpam-3383	223	19	a	a	DET
ejpam-3383	223	20	∗	∗	NOUN
ejpam-3383	223	21	b	b	NOUN
ejpam-3383	223	22	⊆	⊆	NUM
ejpam-3383	223	23	m	m	NOUN
ejpam-3383	223	24	,	,	PUNCT
ejpam-3383	223	25	then	then	ADV
ejpam-3383	223	26	a	a	DET
ejpam-3383	223	27	⊆	⊆	NUM
ejpam-3383	223	28	m	m	NOUN
ejpam-3383	223	29	or	or	CCONJ
ejpam-3383	223	30	b	b	NOUN
ejpam-3383	223	31	⊆m	⊆m	NOUN
ejpam-3383	223	32	.	.	PUNCT
ejpam-3383	224	1	proof	proof	NOUN
ejpam-3383	224	2	.	.	PUNCT
ejpam-3383	225	1	(	(	PUNCT
ejpam-3383	225	2	1	1	X
ejpam-3383	225	3	)	)	PUNCT
ejpam-3383	225	4	=	=	NOUN
ejpam-3383	225	5	⇒	⇒	NOUN
ejpam-3383	225	6	(	(	PUNCT
ejpam-3383	225	7	2	2	NUM
ejpam-3383	225	8	)	)	PUNCT
ejpam-3383	225	9	.	.	PUNCT
ejpam-3383	226	1	let	let	VERB
ejpam-3383	226	2	a	a	PRON
ejpam-3383	226	3	,	,	PUNCT
ejpam-3383	226	4	b	b	NOUN
ejpam-3383	226	5	be	be	AUX
ejpam-3383	226	6	right	right	ADJ
ejpam-3383	226	7	ideals	ideal	NOUN
ejpam-3383	226	8	of	of	ADP
ejpam-3383	226	9	(	(	PUNCT
ejpam-3383	226	10	s	s	NOUN
ejpam-3383	226	11	,	,	PUNCT
ejpam-3383	226	12	◦	◦	NOUN
ejpam-3383	226	13	)	)	PUNCT
ejpam-3383	226	14	such	such	ADJ
ejpam-3383	226	15	that	that	SCONJ
ejpam-3383	226	16	a	a	DET
ejpam-3383	226	17	∗	∗	NOUN
ejpam-3383	226	18	b	b	NOUN
ejpam-3383	226	19	⊆	⊆	NUM
ejpam-3383	226	20	m	m	NOUN
ejpam-3383	226	21	.	.	PUNCT
ejpam-3383	227	1	by	by	ADP
ejpam-3383	227	2	lemma	lemma	PROPN
ejpam-3383	227	3	2.5	2.5	NUM
ejpam-3383	227	4	,	,	PUNCT
ejpam-3383	227	5	the	the	DET
ejpam-3383	227	6	set	set	NOUN
ejpam-3383	227	7	p∗(s	p∗(s	NOUN
ejpam-3383	227	8	)	)	PUNCT
ejpam-3383	227	9	of	of	ADP
ejpam-3383	227	10	all	all	DET
ejpam-3383	227	11	nonempty	nonempty	ADJ
ejpam-3383	227	12	subsets	subset	NOUN
ejpam-3383	227	13	of	of	ADP
ejpam-3383	227	14	s	s	PRON
ejpam-3383	227	15	with	with	ADP
ejpam-3383	227	16	the	the	DET
ejpam-3383	227	17	multiplication	multiplication	NOUN
ejpam-3383	227	18	a•b	a•b	NOUN
ejpam-3383	227	19	:	:	PUNCT
ejpam-3383	227	20	=	=	PUNCT
ejpam-3383	227	21	a∗b	a∗b	PROPN
ejpam-3383	227	22	and	and	CCONJ
ejpam-3383	227	23	the	the	DET
ejpam-3383	227	24	inclusion	inclusion	NOUN
ejpam-3383	227	25	relation	relation	NOUN
ejpam-3383	227	26	“	"	PUNCT
ejpam-3383	227	27	⊆	⊆	X
ejpam-3383	227	28	”	"	PUNCT
ejpam-3383	227	29	is	be	AUX
ejpam-3383	227	30	a	a	DET
ejpam-3383	227	31	∨e	∨e	NOUN
ejpam-3383	227	32	-	-	PUNCT
ejpam-3383	227	33	semigroup	semigroup	PROPN
ejpam-3383	227	34	;	;	PUNCT
ejpam-3383	227	35	the	the	DET
ejpam-3383	227	36	set	set	NOUN
ejpam-3383	227	37	m	m	VERB
ejpam-3383	227	38	is	be	AUX
ejpam-3383	227	39	a	a	DET
ejpam-3383	227	40	weakly	weakly	ADJ
ejpam-3383	227	41	prime	prime	ADJ
ejpam-3383	227	42	ideal	ideal	ADJ
ejpam-3383	227	43	element	element	NOUN
ejpam-3383	227	44	of	of	ADP
ejpam-3383	227	45	(	(	PUNCT
ejpam-3383	227	46	p∗(s	p∗(s	NOUN
ejpam-3383	227	47	)	)	PUNCT
ejpam-3383	227	48	,	,	PUNCT
ejpam-3383	227	49	•,⊆	•,⊆	PROPN
ejpam-3383	227	50	)	)	PUNCT
ejpam-3383	227	51	and	and	CCONJ
ejpam-3383	227	52	a	a	PRON
ejpam-3383	227	53	,	,	PUNCT
ejpam-3383	227	54	b	b	NOUN
ejpam-3383	227	55	are	be	AUX
ejpam-3383	227	56	right	right	ADJ
ejpam-3383	227	57	ideal	ideal	ADJ
ejpam-3383	227	58	element	element	NOUN
ejpam-3383	227	59	of	of	ADP
ejpam-3383	227	60	(	(	PUNCT
ejpam-3383	227	61	p∗(s	p∗(s	NOUN
ejpam-3383	227	62	)	)	PUNCT
ejpam-3383	227	63	,	,	PUNCT
ejpam-3383	227	64	•,⊆	•,⊆	PROPN
ejpam-3383	227	65	)	)	PUNCT
ejpam-3383	228	1	such	such	ADJ
ejpam-3383	228	2	that	that	SCONJ
ejpam-3383	228	3	a	a	DET
ejpam-3383	228	4	•	•	NOUN
ejpam-3383	228	5	b	b	NOUN
ejpam-3383	228	6	⊆	⊆	NUM
ejpam-3383	228	7	m	m	NOUN
ejpam-3383	228	8	.	.	PUNCT
ejpam-3383	229	1	by	by	ADP
ejpam-3383	229	2	proposition	proposition	NOUN
ejpam-3383	229	3	3.1(1)⇒	3.1(1)⇒	NUM
ejpam-3383	229	4	(	(	PUNCT
ejpam-3383	229	5	2	2	NUM
ejpam-3383	229	6	)	)	PUNCT
ejpam-3383	229	7	,	,	PUNCT
ejpam-3383	229	8	we	we	PRON
ejpam-3383	229	9	have	have	VERB
ejpam-3383	229	10	a	a	DET
ejpam-3383	229	11	⊆m	⊆m	NOUN
ejpam-3383	229	12	or	or	CCONJ
ejpam-3383	229	13	b	b	NOUN
ejpam-3383	229	14	⊆m	⊆m	NOUN
ejpam-3383	229	15	and	and	CCONJ
ejpam-3383	229	16	property	property	NOUN
ejpam-3383	229	17	(	(	PUNCT
ejpam-3383	229	18	2	2	NUM
ejpam-3383	229	19	)	)	PUNCT
ejpam-3383	229	20	holds	hold	VERB
ejpam-3383	229	21	.	.	PUNCT
ejpam-3383	230	1	the	the	DET
ejpam-3383	230	2	rest	rest	NOUN
ejpam-3383	230	3	of	of	ADP
ejpam-3383	230	4	the	the	DET
ejpam-3383	230	5	corollary	corollary	NOUN
ejpam-3383	230	6	can	can	AUX
ejpam-3383	230	7	be	be	AUX
ejpam-3383	230	8	proved	prove	VERB
ejpam-3383	230	9	at	at	ADP
ejpam-3383	230	10	a	a	DET
ejpam-3383	230	11	similar	similar	ADJ
ejpam-3383	230	12	way	way	NOUN
ejpam-3383	230	13	.	.	PUNCT
ejpam-3383	231	1	�	�	PROPN
ejpam-3383	231	2	similarly	similarly	ADV
ejpam-3383	231	3	,	,	PUNCT
ejpam-3383	231	4	applying	apply	VERB
ejpam-3383	231	5	proposition	proposition	NOUN
ejpam-3383	231	6	3.2	3.2	NUM
ejpam-3383	231	7	to	to	ADP
ejpam-3383	231	8	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	231	9	,	,	PUNCT
ejpam-3383	231	10	we	we	PRON
ejpam-3383	231	11	get	get	VERB
ejpam-3383	231	12	the	the	DET
ejpam-3383	231	13	following	follow	VERB
ejpam-3383	231	14	corollary	corollary	NOUN
ejpam-3383	231	15	.	.	PUNCT
ejpam-3383	232	1	corollary	corollary	ADJ
ejpam-3383	232	2	4.6	4.6	NUM
ejpam-3383	232	3	.	.	PUNCT
ejpam-3383	233	1	let	let	AUX
ejpam-3383	233	2	(	(	PUNCT
ejpam-3383	233	3	s	s	NOUN
ejpam-3383	233	4	,	,	PUNCT
ejpam-3383	233	5	◦	◦	NOUN
ejpam-3383	233	6	)	)	PUNCT
ejpam-3383	233	7	be	be	VERB
ejpam-3383	233	8	an	an	DET
ejpam-3383	233	9	hypersemigroup	hypersemigroup	NOUN
ejpam-3383	233	10	and	and	CCONJ
ejpam-3383	233	11	m	m	VERB
ejpam-3383	233	12	an	an	DET
ejpam-3383	233	13	ideal	ideal	NOUN
ejpam-3383	233	14	of	of	ADP
ejpam-3383	233	15	s.	s.	PROPN
ejpam-3383	233	16	the	the	DET
ejpam-3383	233	17	following	follow	VERB
ejpam-3383	233	18	are	be	AUX
ejpam-3383	233	19	equivalent	equivalent	ADJ
ejpam-3383	233	20	:	:	PUNCT
ejpam-3383	233	21	(	(	PUNCT
ejpam-3383	233	22	1	1	X
ejpam-3383	233	23	)	)	PUNCT
ejpam-3383	233	24	m	m	VERB
ejpam-3383	233	25	is	be	AUX
ejpam-3383	233	26	weakly	weakly	ADJ
ejpam-3383	233	27	semiprime	semiprime	NOUN
ejpam-3383	233	28	.	.	PUNCT
ejpam-3383	234	1	(	(	PUNCT
ejpam-3383	234	2	2	2	X
ejpam-3383	234	3	)	)	PUNCT
ejpam-3383	234	4	if	if	SCONJ
ejpam-3383	234	5	a	a	PRON
ejpam-3383	234	6	is	be	AUX
ejpam-3383	234	7	a	a	DET
ejpam-3383	234	8	right	right	ADJ
ejpam-3383	234	9	ideal	ideal	NOUN
ejpam-3383	234	10	of	of	ADP
ejpam-3383	234	11	s	s	PRON
ejpam-3383	234	12	such	such	ADJ
ejpam-3383	234	13	that	that	SCONJ
ejpam-3383	234	14	a	a	DET
ejpam-3383	234	15	∗a	∗a	ADJ
ejpam-3383	234	16	⊆m	⊆m	NOUN
ejpam-3383	234	17	,	,	PUNCT
ejpam-3383	234	18	then	then	ADV
ejpam-3383	234	19	a	a	DET
ejpam-3383	234	20	⊆m	⊆m	NOUN
ejpam-3383	234	21	.	.	PUNCT
ejpam-3383	235	1	(	(	PUNCT
ejpam-3383	235	2	3	3	X
ejpam-3383	235	3	)	)	PUNCT
ejpam-3383	235	4	if	if	SCONJ
ejpam-3383	235	5	b	b	NOUN
ejpam-3383	235	6	is	be	AUX
ejpam-3383	235	7	a	a	DET
ejpam-3383	235	8	left	left	ADJ
ejpam-3383	235	9	ideal	ideal	NOUN
ejpam-3383	235	10	of	of	ADP
ejpam-3383	235	11	s	s	PRON
ejpam-3383	235	12	such	such	ADJ
ejpam-3383	235	13	that	that	DET
ejpam-3383	235	14	b	b	X
ejpam-3383	235	15	∗b	∗b	PROPN
ejpam-3383	235	16	⊆m	⊆m	NOUN
ejpam-3383	235	17	,	,	PUNCT
ejpam-3383	235	18	then	then	ADV
ejpam-3383	235	19	b	b	X
ejpam-3383	235	20	⊆m	⊆m	NOUN
ejpam-3383	235	21	.	.	PUNCT
ejpam-3383	236	1	if	if	SCONJ
ejpam-3383	236	2	we	we	PRON
ejpam-3383	236	3	want	want	VERB
ejpam-3383	236	4	to	to	PART
ejpam-3383	236	5	obtain	obtain	VERB
ejpam-3383	236	6	the	the	DET
ejpam-3383	236	7	results	result	NOUN
ejpam-3383	236	8	of	of	ADP
ejpam-3383	236	9	this	this	DET
ejpam-3383	236	10	section	section	NOUN
ejpam-3383	236	11	independently	independently	ADV
ejpam-3383	236	12	,	,	PUNCT
ejpam-3383	236	13	then	then	ADV
ejpam-3383	236	14	their	their	PRON
ejpam-3383	236	15	proof	proof	NOUN
ejpam-3383	236	16	is	be	AUX
ejpam-3383	236	17	on	on	ADP
ejpam-3383	236	18	the	the	DET
ejpam-3383	236	19	line	line	NOUN
ejpam-3383	236	20	of	of	ADP
ejpam-3383	236	21	the	the	DET
ejpam-3383	236	22	corresponding	corresponding	ADJ
ejpam-3383	236	23	proofs	proof	NOUN
ejpam-3383	236	24	of	of	ADP
ejpam-3383	236	25	the	the	DET
ejpam-3383	236	26	∨e	∨e	NOUN
ejpam-3383	236	27	-	-	PUNCT
ejpam-3383	236	28	semigroups	semigroup	NOUN
ejpam-3383	236	29	given	give	VERB
ejpam-3383	236	30	in	in	ADP
ejpam-3383	236	31	the	the	DET
ejpam-3383	236	32	previous	previous	ADJ
ejpam-3383	236	33	section	section	NOUN
ejpam-3383	236	34	.	.	PUNCT
ejpam-3383	237	1	n.	n.	PROPN
ejpam-3383	237	2	kehayopulu	kehayopulu	PROPN
ejpam-3383	237	3	/	/	SYM
ejpam-3383	237	4	eur	eur	PROPN
ejpam-3383	237	5	.	.	PUNCT
ejpam-3383	238	1	j.	j.	PROPN
ejpam-3383	238	2	pure	pure	PROPN
ejpam-3383	238	3	appl	appl	PROPN
ejpam-3383	238	4	.	.	PROPN
ejpam-3383	238	5	math	math	PROPN
ejpam-3383	238	6	,	,	PUNCT
ejpam-3383	238	7	12	12	NUM
ejpam-3383	238	8	(	(	PUNCT
ejpam-3383	238	9	1	1	NUM
ejpam-3383	238	10	)	)	PUNCT
ejpam-3383	238	11	(	(	PUNCT
ejpam-3383	238	12	2019	2019	NUM
ejpam-3383	238	13	)	)	PUNCT
ejpam-3383	238	14	,	,	PUNCT
ejpam-3383	238	15	208	208	NUM
ejpam-3383	238	16	-	-	SYM
ejpam-3383	238	17	225	225	NUM
ejpam-3383	238	18	216	216	NUM
ejpam-3383	238	19	5	5	NUM
ejpam-3383	238	20	.	.	PUNCT
ejpam-3383	239	1	on	on	ADP
ejpam-3383	239	2	ordered	order	VERB
ejpam-3383	239	3	semigroups	semigroup	NOUN
ejpam-3383	239	4	if	if	SCONJ
ejpam-3383	239	5	s	s	NOUN
ejpam-3383	239	6	is	be	AUX
ejpam-3383	239	7	an	an	DET
ejpam-3383	239	8	ordered	order	VERB
ejpam-3383	239	9	semigroup	semigroup	NOUN
ejpam-3383	239	10	then	then	ADV
ejpam-3383	239	11	,	,	PUNCT
ejpam-3383	239	12	for	for	ADP
ejpam-3383	239	13	any	any	DET
ejpam-3383	239	14	nonempty	nonempty	ADJ
ejpam-3383	239	15	subsets	subset	NOUN
ejpam-3383	239	16	a	a	PRON
ejpam-3383	239	17	and	and	CCONJ
ejpam-3383	239	18	b	b	NOUN
ejpam-3383	239	19	of	of	ADP
ejpam-3383	239	20	s	s	PROPN
ejpam-3383	239	21	,	,	PUNCT
ejpam-3383	239	22	we	we	PRON
ejpam-3383	239	23	have	have	VERB
ejpam-3383	239	24	(	(	PUNCT
ejpam-3383	239	25	a](b	a](b	NOUN
ejpam-3383	239	26	]	]	PUNCT
ejpam-3383	239	27	⊆	⊆	NUM
ejpam-3383	239	28	(	(	PUNCT
ejpam-3383	239	29	ab	ab	X
ejpam-3383	239	30	]	]	X
ejpam-3383	239	31	;	;	PUNCT
ejpam-3383	239	32	a	a	DET
ejpam-3383	239	33	⊆	⊆	NUM
ejpam-3383	239	34	b	b	NOUN
ejpam-3383	239	35	⇒	⇒	NOUN
ejpam-3383	239	36	(	(	PUNCT
ejpam-3383	239	37	a	a	X
ejpam-3383	239	38	]	]	X
ejpam-3383	239	39	⊆	⊆	NUM
ejpam-3383	239	40	(	(	PUNCT
ejpam-3383	239	41	b	b	NOUN
ejpam-3383	239	42	]	]	X
ejpam-3383	239	43	;	;	PUNCT
ejpam-3383	239	44	and	and	CCONJ
ejpam-3383	239	45	if	if	SCONJ
ejpam-3383	239	46	m	m	NOUN
ejpam-3383	239	47	is	be	AUX
ejpam-3383	239	48	an	an	DET
ejpam-3383	239	49	ideal	ideal	NOUN
ejpam-3383	239	50	of	of	ADP
ejpam-3383	239	51	s	s	PROPN
ejpam-3383	239	52	,	,	PUNCT
ejpam-3383	239	53	then	then	ADV
ejpam-3383	239	54	(	(	PUNCT
ejpam-3383	239	55	m	m	NOUN
ejpam-3383	239	56	]	]	X
ejpam-3383	239	57	=	=	PUNCT
ejpam-3383	239	58	m	m	VERB
ejpam-3383	239	59	.	.	PUNCT
ejpam-3383	240	1	using	use	VERB
ejpam-3383	240	2	these	these	DET
ejpam-3383	240	3	properties	property	NOUN
ejpam-3383	240	4	,	,	PUNCT
ejpam-3383	240	5	we	we	PRON
ejpam-3383	240	6	prove	prove	VERB
ejpam-3383	240	7	the	the	DET
ejpam-3383	240	8	following	follow	VERB
ejpam-3383	240	9	proposition	proposition	NOUN
ejpam-3383	240	10	.	.	PUNCT
ejpam-3383	241	1	for	for	ADP
ejpam-3383	241	2	the	the	DET
ejpam-3383	241	3	sake	sake	NOUN
ejpam-3383	241	4	of	of	ADP
ejpam-3383	241	5	completeness	completeness	NOUN
ejpam-3383	241	6	we	we	PRON
ejpam-3383	241	7	will	will	AUX
ejpam-3383	241	8	give	give	VERB
ejpam-3383	241	9	its	its	PRON
ejpam-3383	241	10	proof	proof	NOUN
ejpam-3383	241	11	.	.	PUNCT
ejpam-3383	242	1	proposition	proposition	NOUN
ejpam-3383	242	2	5.1	5.1	NUM
ejpam-3383	242	3	.	.	PUNCT
ejpam-3383	243	1	(	(	PUNCT
ejpam-3383	243	2	cf	cf	NOUN
ejpam-3383	243	3	.	.	PUNCT
ejpam-3383	244	1	also	also	ADV
ejpam-3383	244	2	[	[	X
ejpam-3383	244	3	7	7	NUM
ejpam-3383	244	4	;	;	PUNCT
ejpam-3383	244	5	the	the	DET
ejpam-3383	244	6	theorem	theorem	NOUN
ejpam-3383	244	7	]	]	PUNCT
ejpam-3383	244	8	)	)	PUNCT
ejpam-3383	244	9	let	let	VERB
ejpam-3383	244	10	(	(	PUNCT
ejpam-3383	244	11	s	s	X
ejpam-3383	244	12	,	,	PUNCT
ejpam-3383	244	13	·	·	PUNCT
ejpam-3383	244	14	,	,	PUNCT
ejpam-3383	244	15	≤	≤	NUM
ejpam-3383	244	16	)	)	PUNCT
ejpam-3383	244	17	be	be	VERB
ejpam-3383	244	18	an	an	DET
ejpam-3383	244	19	ordered	order	VERB
ejpam-3383	244	20	semigroup	semigroup	NOUN
ejpam-3383	244	21	and	and	CCONJ
ejpam-3383	244	22	m	m	AUX
ejpam-3383	244	23	be	be	VERB
ejpam-3383	244	24	an	an	DET
ejpam-3383	244	25	ideal	ideal	NOUN
ejpam-3383	244	26	of	of	ADP
ejpam-3383	244	27	s.	s.	PROPN
ejpam-3383	244	28	the	the	DET
ejpam-3383	244	29	following	follow	VERB
ejpam-3383	244	30	are	be	AUX
ejpam-3383	244	31	equivalent	equivalent	ADJ
ejpam-3383	244	32	:	:	PUNCT
ejpam-3383	244	33	(	(	PUNCT
ejpam-3383	244	34	1	1	X
ejpam-3383	244	35	)	)	PUNCT
ejpam-3383	245	1	m	m	VERB
ejpam-3383	245	2	is	be	AUX
ejpam-3383	245	3	weakly	weakly	ADV
ejpam-3383	245	4	prime	prime	ADJ
ejpam-3383	245	5	.	.	PUNCT
ejpam-3383	246	1	(	(	PUNCT
ejpam-3383	246	2	2	2	X
ejpam-3383	246	3	)	)	PUNCT
ejpam-3383	246	4	if	if	SCONJ
ejpam-3383	246	5	a	a	PRON
ejpam-3383	246	6	,	,	PUNCT
ejpam-3383	246	7	b	b	NOUN
ejpam-3383	246	8	are	be	AUX
ejpam-3383	246	9	right	right	ADJ
ejpam-3383	246	10	ideals	ideal	NOUN
ejpam-3383	246	11	of	of	ADP
ejpam-3383	246	12	s	s	PRON
ejpam-3383	246	13	such	such	ADJ
ejpam-3383	246	14	that	that	SCONJ
ejpam-3383	246	15	ab	ab	PROPN
ejpam-3383	246	16	⊆m	⊆m	NOUN
ejpam-3383	246	17	,	,	PUNCT
ejpam-3383	246	18	then	then	ADV
ejpam-3383	246	19	a	a	DET
ejpam-3383	246	20	⊆m	⊆m	NOUN
ejpam-3383	246	21	or	or	CCONJ
ejpam-3383	246	22	b	b	NOUN
ejpam-3383	246	23	⊆m	⊆m	NOUN
ejpam-3383	246	24	.	.	PUNCT
ejpam-3383	247	1	(	(	PUNCT
ejpam-3383	247	2	3	3	X
ejpam-3383	247	3	)	)	PUNCT
ejpam-3383	247	4	if	if	SCONJ
ejpam-3383	247	5	a	a	DET
ejpam-3383	247	6	,	,	PUNCT
ejpam-3383	247	7	b	b	NOUN
ejpam-3383	247	8	are	be	AUX
ejpam-3383	247	9	left	leave	VERB
ejpam-3383	247	10	ideals	ideal	NOUN
ejpam-3383	247	11	of	of	ADP
ejpam-3383	247	12	s	s	PRON
ejpam-3383	247	13	such	such	ADJ
ejpam-3383	247	14	that	that	SCONJ
ejpam-3383	247	15	ab	ab	PROPN
ejpam-3383	247	16	⊆m	⊆m	NOUN
ejpam-3383	247	17	,	,	PUNCT
ejpam-3383	247	18	then	then	ADV
ejpam-3383	247	19	a	a	DET
ejpam-3383	247	20	⊆m	⊆m	NOUN
ejpam-3383	247	21	or	or	CCONJ
ejpam-3383	247	22	b	b	NOUN
ejpam-3383	247	23	⊆m	⊆m	NOUN
ejpam-3383	247	24	.	.	PUNCT
ejpam-3383	248	1	(	(	PUNCT
ejpam-3383	248	2	4	4	X
ejpam-3383	248	3	)	)	PUNCT
ejpam-3383	248	4	if	if	SCONJ
ejpam-3383	248	5	a	a	PRON
ejpam-3383	248	6	is	be	AUX
ejpam-3383	248	7	a	a	DET
ejpam-3383	248	8	right	right	ADJ
ejpam-3383	248	9	ideal	ideal	NOUN
ejpam-3383	248	10	and	and	CCONJ
ejpam-3383	248	11	b	b	DET
ejpam-3383	248	12	a	a	DET
ejpam-3383	248	13	left	left	ADJ
ejpam-3383	248	14	ideal	ideal	NOUN
ejpam-3383	248	15	of	of	ADP
ejpam-3383	248	16	s	s	PRON
ejpam-3383	248	17	such	such	ADJ
ejpam-3383	248	18	that	that	SCONJ
ejpam-3383	248	19	ab	ab	PROPN
ejpam-3383	248	20	⊆	⊆	NUM
ejpam-3383	248	21	m	m	NOUN
ejpam-3383	248	22	,	,	PUNCT
ejpam-3383	248	23	then	then	ADV
ejpam-3383	248	24	a	a	DET
ejpam-3383	248	25	⊆	⊆	NUM
ejpam-3383	248	26	m	m	NOUN
ejpam-3383	248	27	or	or	CCONJ
ejpam-3383	248	28	b	b	NOUN
ejpam-3383	248	29	⊆m	⊆m	NOUN
ejpam-3383	248	30	.	.	PUNCT
ejpam-3383	249	1	proof	proof	NOUN
ejpam-3383	249	2	.	.	PUNCT
ejpam-3383	250	1	(	(	PUNCT
ejpam-3383	250	2	1	1	X
ejpam-3383	250	3	)	)	PUNCT
ejpam-3383	250	4	=	=	NOUN
ejpam-3383	250	5	⇒	⇒	NOUN
ejpam-3383	250	6	(	(	PUNCT
ejpam-3383	250	7	2	2	NUM
ejpam-3383	250	8	)	)	PUNCT
ejpam-3383	250	9	.	.	PUNCT
ejpam-3383	251	1	let	let	VERB
ejpam-3383	251	2	a	a	DET
ejpam-3383	251	3	,	,	PUNCT
ejpam-3383	251	4	b	b	NOUN
ejpam-3383	251	5	be	be	AUX
ejpam-3383	251	6	right	right	ADJ
ejpam-3383	251	7	ideals	ideal	NOUN
ejpam-3383	251	8	of	of	ADP
ejpam-3383	251	9	s	s	PRON
ejpam-3383	251	10	such	such	ADJ
ejpam-3383	251	11	that	that	SCONJ
ejpam-3383	251	12	ab	ab	PROPN
ejpam-3383	251	13	⊆	⊆	NUM
ejpam-3383	251	14	m	m	NOUN
ejpam-3383	251	15	.	.	PUNCT
ejpam-3383	252	1	we	we	PRON
ejpam-3383	252	2	consider	consider	VERB
ejpam-3383	252	3	the	the	DET
ejpam-3383	252	4	ideals	ideal	NOUN
ejpam-3383	252	5	i(a	i(a	PROPN
ejpam-3383	252	6	)	)	PUNCT
ejpam-3383	252	7	and	and	CCONJ
ejpam-3383	252	8	i(b	i(b	NOUN
ejpam-3383	252	9	)	)	PUNCT
ejpam-3383	252	10	of	of	ADP
ejpam-3383	252	11	s	s	AUX
ejpam-3383	252	12	generated	generate	VERB
ejpam-3383	252	13	by	by	ADP
ejpam-3383	252	14	a	a	PRON
ejpam-3383	252	15	and	and	CCONJ
ejpam-3383	252	16	b	b	NOUN
ejpam-3383	252	17	respectively	respectively	ADV
ejpam-3383	252	18	.	.	PUNCT
ejpam-3383	253	1	we	we	PRON
ejpam-3383	253	2	have	have	VERB
ejpam-3383	253	3	i(a)i(b	i(a)i(b	NOUN
ejpam-3383	253	4	)	)	PUNCT
ejpam-3383	253	5	=	=	PUNCT
ejpam-3383	254	1	(	(	PUNCT
ejpam-3383	254	2	a	a	DET
ejpam-3383	254	3	∪	∪	NOUN
ejpam-3383	254	4	sa	sa	NOUN
ejpam-3383	254	5	∪as	∪as	NOUN
ejpam-3383	254	6	∪	∪	ADJ
ejpam-3383	254	7	sas](b	sas](b	PROPN
ejpam-3383	254	8	∪	∪	X
ejpam-3383	254	9	sb	sb	PROPN
ejpam-3383	254	10	∪bs	∪bs	ADV
ejpam-3383	254	11	∪	∪	ADJ
ejpam-3383	254	12	sbs	sbs	NOUN
ejpam-3383	254	13	]	]	PUNCT
ejpam-3383	254	14	⊆	⊆	NUM
ejpam-3383	254	15	(	(	PUNCT
ejpam-3383	254	16	(	(	PUNCT
ejpam-3383	254	17	a	a	DET
ejpam-3383	254	18	∪	∪	NOUN
ejpam-3383	254	19	sa	sa	NOUN
ejpam-3383	254	20	∪as	∪as	NOUN
ejpam-3383	254	21	∪	∪	ADJ
ejpam-3383	254	22	sas)(b	sas)(b	ADJ
ejpam-3383	254	23	∪	∪	SYM
ejpam-3383	254	24	sb	sb	PROPN
ejpam-3383	254	25	∪bs	∪bs	ADV
ejpam-3383	254	26	∪	∪	X
ejpam-3383	254	27	sbs	sbs	NOUN
ejpam-3383	254	28	)	)	PUNCT
ejpam-3383	254	29	]	]	PUNCT
ejpam-3383	255	1	=	=	PUNCT
ejpam-3383	255	2	(	(	PUNCT
ejpam-3383	255	3	ab	ab	PROPN
ejpam-3383	255	4	∪	∪	X
ejpam-3383	255	5	sab	sab	PROPN
ejpam-3383	255	6	∪asb	∪asb	NOUN
ejpam-3383	255	7	∪	∪	ADJ
ejpam-3383	255	8	sasb	sasb	NOUN
ejpam-3383	255	9	∪abs	∪ab	NOUN
ejpam-3383	255	10	∪	∪	PROPN
ejpam-3383	255	11	sabs	sabs	PROPN
ejpam-3383	255	12	∪asbs	∪asbs	PROPN
ejpam-3383	255	13	∪	∪	X
ejpam-3383	255	14	sasbs	sasbs	NOUN
ejpam-3383	255	15	]	]	PUNCT
ejpam-3383	255	16	⊆	⊆	NUM
ejpam-3383	255	17	(	(	PUNCT
ejpam-3383	255	18	m	m	NOUN
ejpam-3383	255	19	]	]	X
ejpam-3383	255	20	=	=	PUNCT
ejpam-3383	255	21	m.	m.	NOUN
ejpam-3383	255	22	since	since	SCONJ
ejpam-3383	255	23	i(a)i(b	i(a)i(b	NOUN
ejpam-3383	255	24	)	)	PUNCT
ejpam-3383	255	25	⊆m	⊆m	NOUN
ejpam-3383	255	26	,	,	PUNCT
ejpam-3383	255	27	by	by	ADP
ejpam-3383	255	28	(	(	PUNCT
ejpam-3383	255	29	1	1	NUM
ejpam-3383	255	30	)	)	PUNCT
ejpam-3383	255	31	,	,	PUNCT
ejpam-3383	255	32	we	we	PRON
ejpam-3383	255	33	have	have	VERB
ejpam-3383	255	34	i(a	i(a	PROPN
ejpam-3383	255	35	)	)	PUNCT
ejpam-3383	255	36	⊆m	⊆m	NOUN
ejpam-3383	255	37	or	or	CCONJ
ejpam-3383	255	38	i(b	i(b	NOUN
ejpam-3383	255	39	)	)	PUNCT
ejpam-3383	255	40	⊆m	⊆m	NOUN
ejpam-3383	256	1	and	and	CCONJ
ejpam-3383	256	2	so	so	ADV
ejpam-3383	256	3	a	a	DET
ejpam-3383	256	4	⊆m	⊆m	NOUN
ejpam-3383	256	5	or	or	CCONJ
ejpam-3383	256	6	b	b	NOUN
ejpam-3383	256	7	⊆m	⊆m	NOUN
ejpam-3383	256	8	.	.	PUNCT
ejpam-3383	257	1	(	(	PUNCT
ejpam-3383	257	2	1	1	X
ejpam-3383	257	3	)	)	PUNCT
ejpam-3383	257	4	=	=	NOUN
ejpam-3383	257	5	⇒	⇒	NOUN
ejpam-3383	257	6	(	(	PUNCT
ejpam-3383	257	7	3	3	NUM
ejpam-3383	257	8	)	)	PUNCT
ejpam-3383	257	9	.	.	PUNCT
ejpam-3383	258	1	let	let	VERB
ejpam-3383	258	2	a	a	DET
ejpam-3383	258	3	,	,	PUNCT
ejpam-3383	258	4	b	b	PROPN
ejpam-3383	258	5	be	be	AUX
ejpam-3383	258	6	left	leave	VERB
ejpam-3383	258	7	ideals	ideal	NOUN
ejpam-3383	258	8	of	of	ADP
ejpam-3383	258	9	s	s	PRON
ejpam-3383	258	10	such	such	ADJ
ejpam-3383	258	11	that	that	SCONJ
ejpam-3383	258	12	ab	ab	PROPN
ejpam-3383	258	13	⊆	⊆	NUM
ejpam-3383	258	14	m	m	NOUN
ejpam-3383	258	15	.	.	PUNCT
ejpam-3383	259	1	in	in	ADP
ejpam-3383	259	2	a	a	DET
ejpam-3383	259	3	similar	similar	ADJ
ejpam-3383	259	4	way	way	NOUN
ejpam-3383	259	5	as	as	ADP
ejpam-3383	259	6	in	in	ADP
ejpam-3383	259	7	the	the	DET
ejpam-3383	259	8	previous	previous	ADJ
ejpam-3383	259	9	case	case	NOUN
ejpam-3383	259	10	,	,	PUNCT
ejpam-3383	259	11	we	we	PRON
ejpam-3383	259	12	have	have	VERB
ejpam-3383	259	13	i(a)i(b	i(a)i(b	NOUN
ejpam-3383	259	14	)	)	PUNCT
ejpam-3383	259	15	⊆	⊆	NUM
ejpam-3383	259	16	m	m	NOUN
ejpam-3383	259	17	,	,	PUNCT
ejpam-3383	259	18	then	then	ADV
ejpam-3383	259	19	i(a	i(a	PROPN
ejpam-3383	259	20	)	)	PUNCT
ejpam-3383	259	21	⊆	⊆	NUM
ejpam-3383	259	22	m	m	NOUN
ejpam-3383	259	23	or	or	CCONJ
ejpam-3383	259	24	i(b	i(b	NOUN
ejpam-3383	259	25	)	)	PUNCT
ejpam-3383	259	26	⊆	⊆	NUM
ejpam-3383	259	27	m	m	NOUN
ejpam-3383	259	28	and	and	CCONJ
ejpam-3383	259	29	so	so	ADV
ejpam-3383	259	30	a	a	DET
ejpam-3383	259	31	⊆	⊆	NUM
ejpam-3383	259	32	m	m	NOUN
ejpam-3383	259	33	or	or	CCONJ
ejpam-3383	259	34	b	b	NOUN
ejpam-3383	259	35	⊆m	⊆m	NOUN
ejpam-3383	259	36	.	.	PUNCT
ejpam-3383	260	1	(	(	PUNCT
ejpam-3383	260	2	1	1	X
ejpam-3383	260	3	)	)	PUNCT
ejpam-3383	260	4	=	=	NOUN
ejpam-3383	260	5	⇒	⇒	NOUN
ejpam-3383	260	6	(	(	PUNCT
ejpam-3383	260	7	4	4	NUM
ejpam-3383	260	8	)	)	PUNCT
ejpam-3383	260	9	.	.	PUNCT
ejpam-3383	261	1	let	let	VERB
ejpam-3383	261	2	a	a	PRON
ejpam-3383	261	3	be	be	AUX
ejpam-3383	261	4	a	a	DET
ejpam-3383	261	5	right	right	ADJ
ejpam-3383	261	6	ideal	ideal	NOUN
ejpam-3383	261	7	and	and	CCONJ
ejpam-3383	261	8	b	b	DET
ejpam-3383	261	9	a	a	DET
ejpam-3383	261	10	left	left	ADJ
ejpam-3383	261	11	ideal	ideal	NOUN
ejpam-3383	261	12	of	of	ADP
ejpam-3383	261	13	s	s	PRON
ejpam-3383	261	14	such	such	ADJ
ejpam-3383	261	15	that	that	SCONJ
ejpam-3383	261	16	ab	ab	PROPN
ejpam-3383	261	17	⊆m	⊆m	PROPN
ejpam-3383	261	18	.	.	PUNCT
ejpam-3383	262	1	then	then	ADV
ejpam-3383	262	2	again	again	ADV
ejpam-3383	262	3	i(a)i(b	i(a)i(b	NOUN
ejpam-3383	262	4	)	)	PUNCT
ejpam-3383	262	5	⊆m	⊆m	NOUN
ejpam-3383	262	6	and	and	CCONJ
ejpam-3383	262	7	so	so	ADV
ejpam-3383	262	8	a	a	DET
ejpam-3383	262	9	⊆m	⊆m	NOUN
ejpam-3383	262	10	or	or	CCONJ
ejpam-3383	262	11	b	b	NOUN
ejpam-3383	262	12	⊆m	⊆m	NOUN
ejpam-3383	262	13	.	.	PUNCT
ejpam-3383	263	1	the	the	DET
ejpam-3383	263	2	implications	implication	NOUN
ejpam-3383	263	3	(	(	PUNCT
ejpam-3383	263	4	2)⇒	2)⇒	NUM
ejpam-3383	263	5	(	(	PUNCT
ejpam-3383	263	6	1	1	NUM
ejpam-3383	263	7	)	)	PUNCT
ejpam-3383	263	8	,	,	PUNCT
ejpam-3383	263	9	(	(	PUNCT
ejpam-3383	263	10	3)⇒	3)⇒	NUM
ejpam-3383	263	11	(	(	PUNCT
ejpam-3383	263	12	1	1	NUM
ejpam-3383	263	13	)	)	PUNCT
ejpam-3383	263	14	and	and	CCONJ
ejpam-3383	263	15	(	(	PUNCT
ejpam-3383	263	16	4)⇒	4)⇒	X
ejpam-3383	263	17	(	(	PUNCT
ejpam-3383	263	18	1	1	NUM
ejpam-3383	263	19	)	)	PUNCT
ejpam-3383	263	20	are	be	AUX
ejpam-3383	263	21	obvious	obvious	ADJ
ejpam-3383	263	22	.	.	PUNCT
ejpam-3383	264	1	�	�	PROPN
ejpam-3383	264	2	although	although	SCONJ
ejpam-3383	264	3	for	for	ADP
ejpam-3383	264	4	an	an	DET
ejpam-3383	264	5	ordered	order	VERB
ejpam-3383	264	6	semigroup	semigroup	NOUN
ejpam-3383	264	7	(	(	PUNCT
ejpam-3383	264	8	s	s	PROPN
ejpam-3383	264	9	,	,	PUNCT
ejpam-3383	264	10	·	·	PUNCT
ejpam-3383	264	11	,	,	PUNCT
ejpam-3383	264	12	≤	≤	NUM
ejpam-3383	264	13	)	)	PUNCT
ejpam-3383	264	14	,	,	PUNCT
ejpam-3383	264	15	the	the	DET
ejpam-3383	264	16	set	set	ADJ
ejpam-3383	264	17	p(s	p(s	NOUN
ejpam-3383	264	18	)	)	PUNCT
ejpam-3383	264	19	is	be	AUX
ejpam-3383	264	20	again	again	ADV
ejpam-3383	264	21	a	a	DET
ejpam-3383	264	22	∨e	∨e	NOUN
ejpam-3383	264	23	-	-	PUNCT
ejpam-3383	264	24	semigroup	semigroup	NOUN
ejpam-3383	264	25	,	,	PUNCT
ejpam-3383	264	26	one	one	PRON
ejpam-3383	264	27	can	can	AUX
ejpam-3383	264	28	easily	easily	ADV
ejpam-3383	264	29	check	check	VERB
ejpam-3383	264	30	that	that	DET
ejpam-3383	264	31	proposition	proposition	NOUN
ejpam-3383	264	32	3.1	3.1	NUM
ejpam-3383	264	33	can	can	AUX
ejpam-3383	264	34	not	not	PART
ejpam-3383	264	35	be	be	AUX
ejpam-3383	264	36	applied	apply	VERB
ejpam-3383	264	37	to	to	ADP
ejpam-3383	264	38	ordered	order	VERB
ejpam-3383	264	39	semigroups	semigroup	NOUN
ejpam-3383	264	40	and	and	CCONJ
ejpam-3383	264	41	an	an	DET
ejpam-3383	264	42	independent	independent	ADJ
ejpam-3383	264	43	proof	proof	NOUN
ejpam-3383	264	44	of	of	ADP
ejpam-3383	264	45	proposition	proposition	NOUN
ejpam-3383	264	46	5.1	5.1	NUM
ejpam-3383	264	47	is	be	AUX
ejpam-3383	264	48	needed	need	VERB
ejpam-3383	264	49	;	;	PUNCT
ejpam-3383	264	50	though	though	SCONJ
ejpam-3383	264	51	its	its	PRON
ejpam-3383	264	52	proof	proof	NOUN
ejpam-3383	264	53	is	be	AUX
ejpam-3383	264	54	again	again	ADV
ejpam-3383	264	55	on	on	ADP
ejpam-3383	264	56	the	the	DET
ejpam-3383	264	57	line	line	NOUN
ejpam-3383	264	58	of	of	ADP
ejpam-3383	264	59	proposition	proposition	NOUN
ejpam-3383	264	60	3.1	3.1	NUM
ejpam-3383	264	61	.	.	PUNCT
ejpam-3383	265	1	proposition	proposition	NOUN
ejpam-3383	265	2	5.2	5.2	NUM
ejpam-3383	265	3	.	.	PUNCT
ejpam-3383	266	1	(	(	PUNCT
ejpam-3383	266	2	cf	cf	NOUN
ejpam-3383	266	3	.	.	PUNCT
ejpam-3383	267	1	also	also	ADV
ejpam-3383	267	2	[	[	X
ejpam-3383	267	3	7	7	NUM
ejpam-3383	267	4	;	;	PUNCT
ejpam-3383	267	5	remark	remark	NOUN
ejpam-3383	267	6	4	4	NUM
ejpam-3383	267	7	]	]	PUNCT
ejpam-3383	267	8	)	)	PUNCT
ejpam-3383	267	9	let	let	VERB
ejpam-3383	267	10	s	s	PRON
ejpam-3383	267	11	be	be	AUX
ejpam-3383	267	12	an	an	DET
ejpam-3383	267	13	ordered	order	VERB
ejpam-3383	267	14	semigroup	semigroup	NOUN
ejpam-3383	267	15	and	and	CCONJ
ejpam-3383	267	16	m	m	VERB
ejpam-3383	267	17	an	an	DET
ejpam-3383	267	18	ideal	ideal	NOUN
ejpam-3383	267	19	of	of	ADP
ejpam-3383	267	20	s.	s.	PROPN
ejpam-3383	267	21	the	the	DET
ejpam-3383	267	22	following	follow	VERB
ejpam-3383	267	23	are	be	AUX
ejpam-3383	267	24	equivalent	equivalent	ADJ
ejpam-3383	267	25	:	:	PUNCT
ejpam-3383	267	26	(	(	PUNCT
ejpam-3383	267	27	1	1	X
ejpam-3383	267	28	)	)	PUNCT
ejpam-3383	267	29	m	m	VERB
ejpam-3383	267	30	is	be	AUX
ejpam-3383	267	31	weakly	weakly	ADJ
ejpam-3383	267	32	semiprime	semiprime	NOUN
ejpam-3383	267	33	.	.	PUNCT
ejpam-3383	268	1	(	(	PUNCT
ejpam-3383	268	2	2	2	X
ejpam-3383	268	3	)	)	PUNCT
ejpam-3383	268	4	if	if	SCONJ
ejpam-3383	268	5	a	a	PRON
ejpam-3383	268	6	is	be	AUX
ejpam-3383	268	7	a	a	DET
ejpam-3383	268	8	right	right	ADJ
ejpam-3383	268	9	ideal	ideal	NOUN
ejpam-3383	268	10	of	of	ADP
ejpam-3383	268	11	s	s	PRON
ejpam-3383	268	12	such	such	ADJ
ejpam-3383	268	13	that	that	DET
ejpam-3383	268	14	a2	a2	PROPN
ejpam-3383	268	15	⊆m	⊆m	NOUN
ejpam-3383	268	16	,	,	PUNCT
ejpam-3383	268	17	then	then	ADV
ejpam-3383	268	18	a	a	DET
ejpam-3383	268	19	⊆m	⊆m	NOUN
ejpam-3383	268	20	.	.	PUNCT
ejpam-3383	269	1	n.	n.	PROPN
ejpam-3383	269	2	kehayopulu	kehayopulu	PROPN
ejpam-3383	269	3	/	/	SYM
ejpam-3383	269	4	eur	eur	PROPN
ejpam-3383	269	5	.	.	PUNCT
ejpam-3383	270	1	j.	j.	PROPN
ejpam-3383	270	2	pure	pure	PROPN
ejpam-3383	270	3	appl	appl	PROPN
ejpam-3383	270	4	.	.	PROPN
ejpam-3383	270	5	math	math	PROPN
ejpam-3383	270	6	,	,	PUNCT
ejpam-3383	270	7	12	12	NUM
ejpam-3383	270	8	(	(	PUNCT
ejpam-3383	270	9	1	1	NUM
ejpam-3383	270	10	)	)	PUNCT
ejpam-3383	270	11	(	(	PUNCT
ejpam-3383	270	12	2019	2019	NUM
ejpam-3383	270	13	)	)	PUNCT
ejpam-3383	270	14	,	,	PUNCT
ejpam-3383	270	15	208	208	NUM
ejpam-3383	270	16	-	-	SYM
ejpam-3383	270	17	225	225	NUM
ejpam-3383	270	18	217	217	NUM
ejpam-3383	270	19	(	(	PUNCT
ejpam-3383	270	20	3	3	NUM
ejpam-3383	270	21	)	)	PUNCT
ejpam-3383	270	22	if	if	SCONJ
ejpam-3383	270	23	b	b	NOUN
ejpam-3383	270	24	is	be	AUX
ejpam-3383	270	25	a	a	DET
ejpam-3383	270	26	left	left	ADJ
ejpam-3383	270	27	ideal	ideal	NOUN
ejpam-3383	270	28	of	of	ADP
ejpam-3383	270	29	s	s	PRON
ejpam-3383	270	30	such	such	ADJ
ejpam-3383	270	31	that	that	DET
ejpam-3383	270	32	b2	b2	NOUN
ejpam-3383	270	33	⊆m	⊆m	NOUN
ejpam-3383	270	34	,	,	PUNCT
ejpam-3383	270	35	then	then	ADV
ejpam-3383	270	36	b	b	X
ejpam-3383	270	37	⊆m	⊆m	NOUN
ejpam-3383	270	38	.	.	PUNCT
ejpam-3383	271	1	note	note	VERB
ejpam-3383	271	2	.	.	PUNCT
ejpam-3383	272	1	the	the	DET
ejpam-3383	272	2	results	result	NOUN
ejpam-3383	272	3	on	on	ADP
ejpam-3383	272	4	semigroups	semigroup	NOUN
ejpam-3383	272	5	,	,	PUNCT
ejpam-3383	272	6	that	that	PRON
ejpam-3383	272	7	is	be	AUX
ejpam-3383	272	8	corollaries	corollary	NOUN
ejpam-3383	272	9	4.1	4.1	NUM
ejpam-3383	272	10	and	and	CCONJ
ejpam-3383	272	11	4.2	4.2	NUM
ejpam-3383	272	12	can	can	AUX
ejpam-3383	272	13	be	be	AUX
ejpam-3383	272	14	also	also	ADV
ejpam-3383	272	15	obtained	obtain	VERB
ejpam-3383	272	16	as	as	ADP
ejpam-3383	272	17	application	application	NOUN
ejpam-3383	272	18	of	of	ADP
ejpam-3383	272	19	the	the	DET
ejpam-3383	272	20	corresponding	corresponding	ADJ
ejpam-3383	272	21	results	result	NOUN
ejpam-3383	272	22	of	of	ADP
ejpam-3383	272	23	this	this	DET
ejpam-3383	272	24	section	section	NOUN
ejpam-3383	272	25	.	.	PUNCT
ejpam-3383	273	1	as	as	ADP
ejpam-3383	273	2	an	an	DET
ejpam-3383	273	3	example	example	NOUN
ejpam-3383	273	4	,	,	PUNCT
ejpam-3383	273	5	suppose	suppose	VERB
ejpam-3383	273	6	(	(	PUNCT
ejpam-3383	273	7	s	s	X
ejpam-3383	273	8	,	,	PUNCT
ejpam-3383	273	9	·	·	PUNCT
ejpam-3383	273	10	)	)	PUNCT
ejpam-3383	273	11	is	be	AUX
ejpam-3383	273	12	a	a	DET
ejpam-3383	273	13	semigroup	semigroup	NOUN
ejpam-3383	273	14	,	,	PUNCT
ejpam-3383	273	15	m	m	VERB
ejpam-3383	273	16	a	a	DET
ejpam-3383	273	17	weakly	weakly	ADJ
ejpam-3383	273	18	prime	prime	ADJ
ejpam-3383	273	19	ideal	ideal	NOUN
ejpam-3383	273	20	of	of	ADP
ejpam-3383	273	21	(	(	PUNCT
ejpam-3383	273	22	s	s	PROPN
ejpam-3383	273	23	,	,	PUNCT
ejpam-3383	273	24	·	·	PUNCT
ejpam-3383	273	25	)	)	PUNCT
ejpam-3383	273	26	and	and	CCONJ
ejpam-3383	273	27	a	a	PRON
ejpam-3383	273	28	,	,	PUNCT
ejpam-3383	273	29	b	b	NOUN
ejpam-3383	273	30	be	be	AUX
ejpam-3383	273	31	right	right	ADJ
ejpam-3383	273	32	ideals	ideal	NOUN
ejpam-3383	273	33	of	of	ADP
ejpam-3383	273	34	(	(	PUNCT
ejpam-3383	273	35	s	s	X
ejpam-3383	273	36	,	,	PUNCT
ejpam-3383	273	37	·	·	PUNCT
ejpam-3383	273	38	)	)	PUNCT
ejpam-3383	273	39	such	such	ADJ
ejpam-3383	273	40	that	that	SCONJ
ejpam-3383	273	41	ab	ab	PROPN
ejpam-3383	273	42	⊆	⊆	NUM
ejpam-3383	273	43	m	m	NOUN
ejpam-3383	273	44	.	.	PUNCT
ejpam-3383	274	1	we	we	PRON
ejpam-3383	274	2	endow	endow	VERB
ejpam-3383	274	3	(	(	PUNCT
ejpam-3383	274	4	s	s	PROPN
ejpam-3383	274	5	,	,	PUNCT
ejpam-3383	274	6	·	·	PUNCT
ejpam-3383	274	7	)	)	PUNCT
ejpam-3383	274	8	with	with	ADP
ejpam-3383	274	9	the	the	DET
ejpam-3383	274	10	order	order	NOUN
ejpam-3383	274	11	≤=	≤=	PUNCT
ejpam-3383	274	12	{	{	PUNCT
ejpam-3383	274	13	(	(	PUNCT
ejpam-3383	274	14	x	x	NOUN
ejpam-3383	274	15	,	,	PUNCT
ejpam-3383	274	16	y	y	NOUN
ejpam-3383	274	17	)	)	PUNCT
ejpam-3383	275	1	|	|	ADV
ejpam-3383	275	2	x	x	X
ejpam-3383	275	3	=	=	SYM
ejpam-3383	275	4	y	y	PROPN
ejpam-3383	275	5	}	}	PUNCT
ejpam-3383	275	6	.	.	PUNCT
ejpam-3383	276	1	then	then	ADV
ejpam-3383	276	2	(	(	PUNCT
ejpam-3383	276	3	s	s	X
ejpam-3383	276	4	,	,	PUNCT
ejpam-3383	276	5	·	·	PUNCT
ejpam-3383	276	6	,	,	PUNCT
ejpam-3383	276	7	≤	≤	NUM
ejpam-3383	276	8	)	)	PUNCT
ejpam-3383	276	9	is	be	AUX
ejpam-3383	276	10	an	an	DET
ejpam-3383	276	11	ordered	order	VERB
ejpam-3383	276	12	semigroup	semigroup	NOUN
ejpam-3383	276	13	,	,	PUNCT
ejpam-3383	276	14	the	the	DET
ejpam-3383	276	15	set	set	NOUN
ejpam-3383	276	16	m	m	VERB
ejpam-3383	276	17	is	be	AUX
ejpam-3383	276	18	a	a	DET
ejpam-3383	276	19	weakly	weakly	ADJ
ejpam-3383	276	20	prime	prime	ADJ
ejpam-3383	276	21	ideal	ideal	NOUN
ejpam-3383	276	22	of	of	ADP
ejpam-3383	276	23	(	(	PUNCT
ejpam-3383	276	24	s	s	PROPN
ejpam-3383	276	25	,	,	PUNCT
ejpam-3383	276	26	·	·	PUNCT
ejpam-3383	276	27	,	,	PUNCT
ejpam-3383	276	28	≤	≤	NUM
ejpam-3383	276	29	)	)	PUNCT
ejpam-3383	276	30	and	and	CCONJ
ejpam-3383	276	31	a	a	DET
ejpam-3383	276	32	,	,	PUNCT
ejpam-3383	276	33	b	b	NOUN
ejpam-3383	276	34	are	be	AUX
ejpam-3383	276	35	ideals	ideal	NOUN
ejpam-3383	276	36	of	of	ADP
ejpam-3383	276	37	(	(	PUNCT
ejpam-3383	276	38	s	s	X
ejpam-3383	276	39	,	,	PUNCT
ejpam-3383	276	40	·	·	PUNCT
ejpam-3383	276	41	,	,	PUNCT
ejpam-3383	276	42	≤	≤	NUM
ejpam-3383	276	43	)	)	PUNCT
ejpam-3383	276	44	such	such	ADJ
ejpam-3383	276	45	that	that	SCONJ
ejpam-3383	276	46	ab	ab	PROPN
ejpam-3383	276	47	⊆m	⊆m	NOUN
ejpam-3383	276	48	.	.	PUNCT
ejpam-3383	277	1	by	by	ADP
ejpam-3383	277	2	proposition	proposition	NOUN
ejpam-3383	277	3	5.1(1)⇒	5.1(1)⇒	NUM
ejpam-3383	277	4	(	(	PUNCT
ejpam-3383	277	5	2	2	NUM
ejpam-3383	277	6	)	)	PUNCT
ejpam-3383	277	7	,	,	PUNCT
ejpam-3383	277	8	we	we	PRON
ejpam-3383	277	9	have	have	VERB
ejpam-3383	277	10	a	a	DET
ejpam-3383	277	11	⊆m	⊆m	NOUN
ejpam-3383	277	12	or	or	CCONJ
ejpam-3383	277	13	b	b	NOUN
ejpam-3383	277	14	⊆m	⊆m	NOUN
ejpam-3383	277	15	and	and	CCONJ
ejpam-3383	277	16	property	property	NOUN
ejpam-3383	277	17	(	(	PUNCT
ejpam-3383	277	18	2	2	NUM
ejpam-3383	277	19	)	)	PUNCT
ejpam-3383	277	20	of	of	ADP
ejpam-3383	277	21	corollary	corollary	ADJ
ejpam-3383	277	22	4.1	4.1	NUM
ejpam-3383	277	23	is	be	AUX
ejpam-3383	277	24	satisfied	satisfied	ADJ
ejpam-3383	277	25	(	(	PUNCT
ejpam-3383	277	26	see	see	VERB
ejpam-3383	277	27	also	also	ADV
ejpam-3383	277	28	[	[	X
ejpam-3383	277	29	7	7	NUM
ejpam-3383	277	30	,	,	PUNCT
ejpam-3383	277	31	8	8	NUM
ejpam-3383	277	32	]	]	PUNCT
ejpam-3383	277	33	;	;	PUNCT
ejpam-3383	277	34	the	the	DET
ejpam-3383	277	35	introduction	introduction	NOUN
ejpam-3383	277	36	)	)	PUNCT
ejpam-3383	277	37	.	.	PUNCT
ejpam-3383	278	1	6	6	X
ejpam-3383	278	2	.	.	X
ejpam-3383	278	3	on	on	ADP
ejpam-3383	278	4	ordered	order	VERB
ejpam-3383	278	5	γ	γ	NOUN
ejpam-3383	278	6	-	-	PUNCT
ejpam-3383	278	7	semigroups	semigroup	NOUN
ejpam-3383	278	8	for	for	ADP
ejpam-3383	278	9	an	an	DET
ejpam-3383	278	10	ordered	order	VERB
ejpam-3383	278	11	γ	γ	NOUN
ejpam-3383	278	12	-	-	PUNCT
ejpam-3383	278	13	semigroup	semigroup	NOUN
ejpam-3383	278	14	s	s	PART
ejpam-3383	278	15	and	and	CCONJ
ejpam-3383	278	16	nonempty	nonempty	ADJ
ejpam-3383	278	17	subsets	subset	NOUN
ejpam-3383	278	18	a	a	PRON
ejpam-3383	278	19	and	and	CCONJ
ejpam-3383	278	20	b	b	NOUN
ejpam-3383	278	21	of	of	ADP
ejpam-3383	278	22	s	s	PROPN
ejpam-3383	278	23	,	,	PUNCT
ejpam-3383	278	24	we	we	PRON
ejpam-3383	278	25	have	have	VERB
ejpam-3383	278	26	(	(	PUNCT
ejpam-3383	278	27	a]γ(b	a]γ(b	ADP
ejpam-3383	278	28	]	]	PUNCT
ejpam-3383	278	29	⊆	⊆	NUM
ejpam-3383	278	30	(	(	PUNCT
ejpam-3383	278	31	aγb	aγb	NOUN
ejpam-3383	278	32	]	]	X
ejpam-3383	278	33	;	;	PUNCT
ejpam-3383	278	34	a	a	DET
ejpam-3383	278	35	⊆	⊆	NUM
ejpam-3383	278	36	b	b	NOUN
ejpam-3383	278	37	⇒	⇒	NOUN
ejpam-3383	278	38	(	(	PUNCT
ejpam-3383	278	39	a	a	X
ejpam-3383	278	40	]	]	X
ejpam-3383	278	41	⊆	⊆	NUM
ejpam-3383	278	42	(	(	PUNCT
ejpam-3383	278	43	b	b	NOUN
ejpam-3383	278	44	]	]	X
ejpam-3383	278	45	;	;	PUNCT
ejpam-3383	278	46	and	and	CCONJ
ejpam-3383	278	47	if	if	SCONJ
ejpam-3383	278	48	m	m	NOUN
ejpam-3383	278	49	is	be	AUX
ejpam-3383	278	50	an	an	DET
ejpam-3383	278	51	ideal	ideal	NOUN
ejpam-3383	278	52	of	of	ADP
ejpam-3383	278	53	s	s	PROPN
ejpam-3383	278	54	,	,	PUNCT
ejpam-3383	278	55	then	then	ADV
ejpam-3383	278	56	(	(	PUNCT
ejpam-3383	278	57	m	m	NOUN
ejpam-3383	278	58	]	]	X
ejpam-3383	279	1	=	=	PUNCT
ejpam-3383	279	2	m	m	VERB
ejpam-3383	279	3	.	.	PUNCT
ejpam-3383	280	1	using	use	VERB
ejpam-3383	280	2	these	these	DET
ejpam-3383	280	3	properties	property	NOUN
ejpam-3383	280	4	,	,	PUNCT
ejpam-3383	280	5	we	we	PRON
ejpam-3383	280	6	prove	prove	VERB
ejpam-3383	280	7	the	the	DET
ejpam-3383	280	8	following	follow	VERB
ejpam-3383	280	9	proposition	proposition	NOUN
ejpam-3383	280	10	.	.	PUNCT
ejpam-3383	281	1	proposition	proposition	NOUN
ejpam-3383	281	2	6.1	6.1	NUM
ejpam-3383	281	3	.	.	PUNCT
ejpam-3383	282	1	let	let	VERB
ejpam-3383	282	2	s	s	PRON
ejpam-3383	282	3	be	be	AUX
ejpam-3383	282	4	an	an	DET
ejpam-3383	282	5	ordered	order	VERB
ejpam-3383	282	6	γ	γ	NOUN
ejpam-3383	282	7	-	-	PUNCT
ejpam-3383	282	8	semigroup	semigroup	NOUN
ejpam-3383	282	9	and	and	CCONJ
ejpam-3383	282	10	m	m	AUX
ejpam-3383	282	11	be	be	VERB
ejpam-3383	282	12	an	an	DET
ejpam-3383	282	13	ideal	ideal	NOUN
ejpam-3383	282	14	of	of	ADP
ejpam-3383	282	15	s.	s.	PROPN
ejpam-3383	282	16	the	the	DET
ejpam-3383	282	17	following	follow	VERB
ejpam-3383	282	18	are	be	AUX
ejpam-3383	282	19	equivalent	equivalent	ADJ
ejpam-3383	282	20	:	:	PUNCT
ejpam-3383	282	21	(	(	PUNCT
ejpam-3383	282	22	1	1	X
ejpam-3383	282	23	)	)	PUNCT
ejpam-3383	282	24	m	m	VERB
ejpam-3383	282	25	is	be	AUX
ejpam-3383	282	26	weakly	weakly	ADV
ejpam-3383	282	27	prime	prime	ADJ
ejpam-3383	282	28	.	.	PUNCT
ejpam-3383	283	1	(	(	PUNCT
ejpam-3383	283	2	2	2	X
ejpam-3383	283	3	)	)	PUNCT
ejpam-3383	283	4	if	if	SCONJ
ejpam-3383	283	5	a	a	PRON
ejpam-3383	283	6	,	,	PUNCT
ejpam-3383	283	7	b	b	NOUN
ejpam-3383	283	8	are	be	AUX
ejpam-3383	283	9	right	right	ADJ
ejpam-3383	283	10	ideals	ideal	NOUN
ejpam-3383	283	11	of	of	ADP
ejpam-3383	283	12	s	s	PRON
ejpam-3383	283	13	such	such	ADJ
ejpam-3383	283	14	that	that	SCONJ
ejpam-3383	283	15	aγb	aγb	NOUN
ejpam-3383	283	16	⊆m	⊆m	NOUN
ejpam-3383	283	17	,	,	PUNCT
ejpam-3383	283	18	then	then	ADV
ejpam-3383	283	19	a	a	DET
ejpam-3383	283	20	⊆m	⊆m	NOUN
ejpam-3383	283	21	or	or	CCONJ
ejpam-3383	283	22	b	b	NOUN
ejpam-3383	283	23	⊆m	⊆m	NOUN
ejpam-3383	283	24	.	.	PUNCT
ejpam-3383	284	1	(	(	PUNCT
ejpam-3383	284	2	3	3	X
ejpam-3383	284	3	)	)	PUNCT
ejpam-3383	284	4	if	if	SCONJ
ejpam-3383	284	5	a	a	DET
ejpam-3383	284	6	,	,	PUNCT
ejpam-3383	284	7	b	b	NOUN
ejpam-3383	284	8	are	be	AUX
ejpam-3383	284	9	left	leave	VERB
ejpam-3383	284	10	ideals	ideal	NOUN
ejpam-3383	284	11	of	of	ADP
ejpam-3383	284	12	s	s	PRON
ejpam-3383	284	13	such	such	ADJ
ejpam-3383	284	14	that	that	SCONJ
ejpam-3383	284	15	aγb	aγb	NOUN
ejpam-3383	284	16	⊆m	⊆m	NOUN
ejpam-3383	284	17	,	,	PUNCT
ejpam-3383	284	18	then	then	ADV
ejpam-3383	284	19	a	a	DET
ejpam-3383	284	20	⊆m	⊆m	NOUN
ejpam-3383	284	21	or	or	CCONJ
ejpam-3383	284	22	b	b	NOUN
ejpam-3383	284	23	⊆m	⊆m	NOUN
ejpam-3383	284	24	.	.	PUNCT
ejpam-3383	285	1	(	(	PUNCT
ejpam-3383	285	2	4	4	X
ejpam-3383	285	3	)	)	PUNCT
ejpam-3383	285	4	if	if	SCONJ
ejpam-3383	285	5	a	a	PRON
ejpam-3383	285	6	is	be	AUX
ejpam-3383	285	7	a	a	DET
ejpam-3383	285	8	right	right	ADJ
ejpam-3383	285	9	ideal	ideal	NOUN
ejpam-3383	285	10	and	and	CCONJ
ejpam-3383	285	11	b	b	DET
ejpam-3383	285	12	a	a	DET
ejpam-3383	285	13	left	left	ADJ
ejpam-3383	285	14	ideal	ideal	NOUN
ejpam-3383	285	15	of	of	ADP
ejpam-3383	285	16	s	s	PRON
ejpam-3383	285	17	such	such	ADJ
ejpam-3383	285	18	that	that	SCONJ
ejpam-3383	285	19	aγb	aγb	ADV
ejpam-3383	285	20	⊆	⊆	NUM
ejpam-3383	285	21	m	m	NOUN
ejpam-3383	285	22	,	,	PUNCT
ejpam-3383	285	23	then	then	ADV
ejpam-3383	285	24	a	a	DET
ejpam-3383	285	25	⊆	⊆	NUM
ejpam-3383	285	26	m	m	NOUN
ejpam-3383	285	27	or	or	CCONJ
ejpam-3383	285	28	b	b	NOUN
ejpam-3383	285	29	⊆m	⊆m	NOUN
ejpam-3383	285	30	.	.	PUNCT
ejpam-3383	286	1	proof	proof	NOUN
ejpam-3383	286	2	.	.	PUNCT
ejpam-3383	287	1	let	let	VERB
ejpam-3383	287	2	us	we	PRON
ejpam-3383	287	3	prove	prove	VERB
ejpam-3383	287	4	the	the	DET
ejpam-3383	287	5	implication	implication	NOUN
ejpam-3383	287	6	(	(	PUNCT
ejpam-3383	287	7	1	1	X
ejpam-3383	287	8	)	)	PUNCT
ejpam-3383	287	9	⇒	⇒	NOUN
ejpam-3383	287	10	(	(	PUNCT
ejpam-3383	287	11	3	3	NUM
ejpam-3383	287	12	)	)	PUNCT
ejpam-3383	287	13	.	.	PUNCT
ejpam-3383	288	1	let	let	VERB
ejpam-3383	288	2	a	a	DET
ejpam-3383	288	3	,	,	PUNCT
ejpam-3383	288	4	b	b	PROPN
ejpam-3383	288	5	be	be	AUX
ejpam-3383	288	6	left	leave	VERB
ejpam-3383	288	7	ideals	ideal	NOUN
ejpam-3383	288	8	of	of	ADP
ejpam-3383	288	9	s	s	PRON
ejpam-3383	288	10	such	such	ADJ
ejpam-3383	288	11	that	that	SCONJ
ejpam-3383	288	12	aγb	aγb	NOUN
ejpam-3383	288	13	⊆m	⊆m	NOUN
ejpam-3383	288	14	.	.	PUNCT
ejpam-3383	289	1	then	then	ADV
ejpam-3383	289	2	we	we	PRON
ejpam-3383	289	3	have	have	VERB
ejpam-3383	289	4	i(a)γi(b	i(a)γi(b	NOUN
ejpam-3383	289	5	)	)	PUNCT
ejpam-3383	289	6	=	=	PUNCT
ejpam-3383	290	1	(	(	PUNCT
ejpam-3383	290	2	a	a	DET
ejpam-3383	290	3	∪	∪	ADJ
ejpam-3383	290	4	sγa	sγa	PROPN
ejpam-3383	290	5	∪aγs	∪aγs	PROPN
ejpam-3383	290	6	∪	∪	ADP
ejpam-3383	290	7	sγaγs](b	sγaγs](b	NOUN
ejpam-3383	290	8	∪	∪	SYM
ejpam-3383	290	9	sγb	sγb	NOUN
ejpam-3383	290	10	∪bγs	∪bγs	NOUN
ejpam-3383	290	11	∪	∪	ADP
ejpam-3383	290	12	sγbγs	sγbγs	NOUN
ejpam-3383	290	13	]	]	PUNCT
ejpam-3383	290	14	⊆	⊆	NUM
ejpam-3383	290	15	(	(	PUNCT
ejpam-3383	290	16	(	(	PUNCT
ejpam-3383	290	17	a	a	PRON
ejpam-3383	290	18	∪	∪	ADJ
ejpam-3383	290	19	sγa	sγa	PROPN
ejpam-3383	290	20	∪aγs	∪aγs	PROPN
ejpam-3383	290	21	∪	∪	PROPN
ejpam-3383	290	22	sγaγs)(b	sγaγs)(b	PROPN
ejpam-3383	290	23	∪	∪	ADP
ejpam-3383	290	24	sγb	sγb	PROPN
ejpam-3383	290	25	∪bγs	∪bγs	NOUN
ejpam-3383	290	26	∪	∪	ADP
ejpam-3383	290	27	sγbγs	sγbγs	NOUN
ejpam-3383	290	28	)	)	PUNCT
ejpam-3383	290	29	]	]	PUNCT
ejpam-3383	291	1	=	=	PUNCT
ejpam-3383	291	2	(	(	PUNCT
ejpam-3383	291	3	aγb	aγb	NOUN
ejpam-3383	291	4	∪	∪	VERB
ejpam-3383	291	5	sγaγb	sγaγb	ADJ
ejpam-3383	291	6	∪aγsγb	∪aγsγb	NOUN
ejpam-3383	291	7	∪	∪	ADP
ejpam-3383	291	8	sγaγsγb	sγaγsγb	NOUN
ejpam-3383	291	9	∪aγbγs	∪aγbγs	PROPN
ejpam-3383	291	10	∪sγaγbγs	∪sγaγbγs	NOUN
ejpam-3383	291	11	∪aγsγbγs	∪aγsγbγs	PROPN
ejpam-3383	291	12	∪	∪	ADJ
ejpam-3383	291	13	sγaγsγbγs	sγaγsγbγs	NOUN
ejpam-3383	291	14	]	]	PUNCT
ejpam-3383	291	15	⊆	⊆	NUM
ejpam-3383	291	16	(	(	PUNCT
ejpam-3383	291	17	m	m	NOUN
ejpam-3383	291	18	]	]	X
ejpam-3383	291	19	=	=	PUNCT
ejpam-3383	291	20	m.	m.	NOUN
ejpam-3383	291	21	since	since	SCONJ
ejpam-3383	291	22	i(a)γi(b	i(a)γi(b	PROPN
ejpam-3383	291	23	)	)	PUNCT
ejpam-3383	291	24	⊆m	⊆m	NOUN
ejpam-3383	291	25	,	,	PUNCT
ejpam-3383	291	26	by	by	ADP
ejpam-3383	291	27	hypothesis	hypothesis	NOUN
ejpam-3383	291	28	,	,	PUNCT
ejpam-3383	291	29	we	we	PRON
ejpam-3383	291	30	have	have	VERB
ejpam-3383	291	31	i(a	i(a	PROPN
ejpam-3383	291	32	)	)	PUNCT
ejpam-3383	291	33	⊆m	⊆m	NOUN
ejpam-3383	291	34	or	or	CCONJ
ejpam-3383	291	35	i(b	i(b	NOUN
ejpam-3383	291	36	)	)	PUNCT
ejpam-3383	291	37	⊆m	⊆m	NOUN
ejpam-3383	291	38	and	and	CCONJ
ejpam-3383	291	39	so	so	ADV
ejpam-3383	291	40	a	a	DET
ejpam-3383	291	41	⊆m	⊆m	NOUN
ejpam-3383	291	42	or	or	CCONJ
ejpam-3383	291	43	b	b	NOUN
ejpam-3383	291	44	⊆m	⊆m	NOUN
ejpam-3383	291	45	.	.	PUNCT
ejpam-3383	292	1	the	the	DET
ejpam-3383	292	2	implications	implication	NOUN
ejpam-3383	292	3	(	(	PUNCT
ejpam-3383	292	4	1	1	X
ejpam-3383	292	5	)	)	PUNCT
ejpam-3383	292	6	⇒	⇒	NOUN
ejpam-3383	292	7	(	(	PUNCT
ejpam-3383	292	8	2	2	NUM
ejpam-3383	292	9	)	)	PUNCT
ejpam-3383	292	10	and	and	CCONJ
ejpam-3383	292	11	(	(	PUNCT
ejpam-3383	292	12	1	1	X
ejpam-3383	292	13	)	)	PUNCT
ejpam-3383	292	14	⇒	⇒	NOUN
ejpam-3383	292	15	(	(	PUNCT
ejpam-3383	292	16	4	4	X
ejpam-3383	292	17	)	)	PUNCT
ejpam-3383	292	18	can	can	AUX
ejpam-3383	292	19	be	be	AUX
ejpam-3383	292	20	proved	prove	VERB
ejpam-3383	292	21	at	at	ADP
ejpam-3383	292	22	a	a	DET
ejpam-3383	292	23	similar	similar	ADJ
ejpam-3383	292	24	way	way	NOUN
ejpam-3383	292	25	;	;	PUNCT
ejpam-3383	292	26	and	and	CCONJ
ejpam-3383	292	27	the	the	DET
ejpam-3383	292	28	implications	implication	NOUN
ejpam-3383	292	29	(	(	PUNCT
ejpam-3383	292	30	2)⇒	2)⇒	NUM
ejpam-3383	292	31	(	(	PUNCT
ejpam-3383	292	32	1	1	NUM
ejpam-3383	292	33	)	)	PUNCT
ejpam-3383	292	34	,	,	PUNCT
ejpam-3383	292	35	(	(	PUNCT
ejpam-3383	292	36	3)⇒	3)⇒	NUM
ejpam-3383	292	37	(	(	PUNCT
ejpam-3383	292	38	1	1	NUM
ejpam-3383	292	39	)	)	PUNCT
ejpam-3383	292	40	and	and	CCONJ
ejpam-3383	292	41	(	(	PUNCT
ejpam-3383	292	42	4)⇒	4)⇒	X
ejpam-3383	292	43	(	(	PUNCT
ejpam-3383	292	44	1	1	NUM
ejpam-3383	292	45	)	)	PUNCT
ejpam-3383	292	46	are	be	AUX
ejpam-3383	292	47	obvious	obvious	ADJ
ejpam-3383	292	48	.	.	PUNCT
ejpam-3383	293	1	�	�	PROPN
ejpam-3383	293	2	in	in	ADP
ejpam-3383	293	3	a	a	DET
ejpam-3383	293	4	similar	similar	ADJ
ejpam-3383	293	5	way	way	NOUN
ejpam-3383	293	6	we	we	PRON
ejpam-3383	293	7	prove	prove	VERB
ejpam-3383	293	8	the	the	DET
ejpam-3383	293	9	following	follow	VERB
ejpam-3383	293	10	proposition	proposition	NOUN
ejpam-3383	293	11	.	.	PUNCT
ejpam-3383	294	1	proposition	proposition	NOUN
ejpam-3383	294	2	6.2	6.2	NUM
ejpam-3383	294	3	.	.	PUNCT
ejpam-3383	295	1	let	let	VERB
ejpam-3383	295	2	s	s	PRON
ejpam-3383	295	3	be	be	AUX
ejpam-3383	295	4	an	an	DET
ejpam-3383	295	5	ordered	order	VERB
ejpam-3383	295	6	γ	γ	NOUN
ejpam-3383	295	7	-	-	PUNCT
ejpam-3383	295	8	semigroup	semigroup	NOUN
ejpam-3383	295	9	and	and	CCONJ
ejpam-3383	295	10	m	m	VERB
ejpam-3383	295	11	an	an	DET
ejpam-3383	295	12	ideal	ideal	NOUN
ejpam-3383	295	13	of	of	ADP
ejpam-3383	295	14	s.	s.	PROPN
ejpam-3383	295	15	the	the	DET
ejpam-3383	295	16	following	follow	VERB
ejpam-3383	295	17	are	be	AUX
ejpam-3383	295	18	equivalent	equivalent	ADJ
ejpam-3383	295	19	:	:	PUNCT
ejpam-3383	295	20	n.	n.	PROPN
ejpam-3383	295	21	kehayopulu	kehayopulu	PROPN
ejpam-3383	295	22	/	/	SYM
ejpam-3383	295	23	eur	eur	PROPN
ejpam-3383	295	24	.	.	PUNCT
ejpam-3383	296	1	j.	j.	PROPN
ejpam-3383	296	2	pure	pure	PROPN
ejpam-3383	296	3	appl	appl	PROPN
ejpam-3383	296	4	.	.	PROPN
ejpam-3383	296	5	math	math	PROPN
ejpam-3383	296	6	,	,	PUNCT
ejpam-3383	296	7	12	12	NUM
ejpam-3383	296	8	(	(	PUNCT
ejpam-3383	296	9	1	1	NUM
ejpam-3383	296	10	)	)	PUNCT
ejpam-3383	296	11	(	(	PUNCT
ejpam-3383	296	12	2019	2019	NUM
ejpam-3383	296	13	)	)	PUNCT
ejpam-3383	296	14	,	,	PUNCT
ejpam-3383	296	15	208	208	NUM
ejpam-3383	296	16	-	-	SYM
ejpam-3383	296	17	225	225	NUM
ejpam-3383	296	18	218	218	NUM
ejpam-3383	296	19	(	(	PUNCT
ejpam-3383	296	20	1	1	NUM
ejpam-3383	296	21	)	)	PUNCT
ejpam-3383	296	22	m	m	VERB
ejpam-3383	296	23	is	be	AUX
ejpam-3383	296	24	weakly	weakly	ADJ
ejpam-3383	296	25	semiprime	semiprime	NOUN
ejpam-3383	296	26	.	.	PUNCT
ejpam-3383	297	1	(	(	PUNCT
ejpam-3383	297	2	2	2	X
ejpam-3383	297	3	)	)	PUNCT
ejpam-3383	297	4	if	if	SCONJ
ejpam-3383	297	5	a	a	PRON
ejpam-3383	297	6	is	be	AUX
ejpam-3383	297	7	a	a	DET
ejpam-3383	297	8	right	right	ADJ
ejpam-3383	297	9	ideal	ideal	NOUN
ejpam-3383	297	10	of	of	ADP
ejpam-3383	297	11	s	s	PRON
ejpam-3383	297	12	such	such	ADJ
ejpam-3383	297	13	that	that	DET
ejpam-3383	297	14	aγa	aγa	NOUN
ejpam-3383	297	15	⊆m	⊆m	NOUN
ejpam-3383	297	16	,	,	PUNCT
ejpam-3383	297	17	then	then	ADV
ejpam-3383	297	18	a	a	DET
ejpam-3383	297	19	⊆m	⊆m	NOUN
ejpam-3383	297	20	.	.	PUNCT
ejpam-3383	298	1	(	(	PUNCT
ejpam-3383	298	2	3	3	X
ejpam-3383	298	3	)	)	PUNCT
ejpam-3383	298	4	if	if	SCONJ
ejpam-3383	298	5	b	b	NOUN
ejpam-3383	298	6	is	be	AUX
ejpam-3383	298	7	a	a	DET
ejpam-3383	298	8	left	left	ADJ
ejpam-3383	298	9	ideal	ideal	NOUN
ejpam-3383	298	10	of	of	ADP
ejpam-3383	298	11	s	s	PRON
ejpam-3383	298	12	such	such	ADJ
ejpam-3383	298	13	that	that	DET
ejpam-3383	298	14	bγb	bγb	NOUN
ejpam-3383	298	15	⊆m	⊆m	NOUN
ejpam-3383	298	16	,	,	PUNCT
ejpam-3383	298	17	then	then	ADV
ejpam-3383	298	18	b	b	X
ejpam-3383	298	19	⊆m	⊆m	NOUN
ejpam-3383	298	20	.	.	PUNCT
ejpam-3383	299	1	the	the	DET
ejpam-3383	299	2	results	result	NOUN
ejpam-3383	299	3	on	on	ADP
ejpam-3383	299	4	γ	γ	NOUN
ejpam-3383	299	5	-	-	PUNCT
ejpam-3383	299	6	semigroups	semigroup	NOUN
ejpam-3383	299	7	,	,	PUNCT
ejpam-3383	299	8	that	that	PRON
ejpam-3383	299	9	is	be	AUX
ejpam-3383	299	10	corollaries	corollary	NOUN
ejpam-3383	299	11	4.3	4.3	NUM
ejpam-3383	299	12	and	and	CCONJ
ejpam-3383	299	13	4.4	4.4	NUM
ejpam-3383	299	14	can	can	AUX
ejpam-3383	299	15	be	be	AUX
ejpam-3383	299	16	also	also	ADV
ejpam-3383	299	17	obtained	obtain	VERB
ejpam-3383	299	18	as	as	ADP
ejpam-3383	299	19	application	application	NOUN
ejpam-3383	299	20	of	of	ADP
ejpam-3383	299	21	the	the	DET
ejpam-3383	299	22	corresponding	corresponding	ADJ
ejpam-3383	299	23	results	result	NOUN
ejpam-3383	299	24	of	of	ADP
ejpam-3383	299	25	this	this	DET
ejpam-3383	299	26	section	section	NOUN
ejpam-3383	299	27	in	in	ADP
ejpam-3383	299	28	the	the	DET
ejpam-3383	299	29	way	way	NOUN
ejpam-3383	299	30	indicated	indicate	VERB
ejpam-3383	299	31	in	in	ADP
ejpam-3383	299	32	the	the	DET
ejpam-3383	299	33	note	note	NOUN
ejpam-3383	299	34	of	of	ADP
ejpam-3383	299	35	section	section	NOUN
ejpam-3383	299	36	5	5	NUM
ejpam-3383	299	37	.	.	NOUN
ejpam-3383	300	1	7	7	NUM
ejpam-3383	300	2	.	.	X
ejpam-3383	300	3	on	on	ADP
ejpam-3383	300	4	ordered	order	VERB
ejpam-3383	300	5	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	300	6	for	for	ADP
ejpam-3383	300	7	an	an	DET
ejpam-3383	300	8	ordered	order	VERB
ejpam-3383	300	9	hypersemigroup	hypersemigroup	NOUN
ejpam-3383	300	10	(	(	PUNCT
ejpam-3383	300	11	s	s	NOUN
ejpam-3383	300	12	,	,	PUNCT
ejpam-3383	300	13	◦	◦	NOUN
ejpam-3383	300	14	,	,	PUNCT
ejpam-3383	300	15	≤	≤	NUM
ejpam-3383	300	16	)	)	PUNCT
ejpam-3383	300	17	and	and	CCONJ
ejpam-3383	300	18	nonempty	nonempty	ADJ
ejpam-3383	300	19	subsets	subset	NOUN
ejpam-3383	300	20	a	a	PRON
ejpam-3383	300	21	and	and	CCONJ
ejpam-3383	300	22	b	b	NOUN
ejpam-3383	300	23	of	of	ADP
ejpam-3383	300	24	s	s	PROPN
ejpam-3383	300	25	,	,	PUNCT
ejpam-3383	300	26	we	we	PRON
ejpam-3383	300	27	have	have	VERB
ejpam-3383	300	28	(	(	PUNCT
ejpam-3383	300	29	a	a	PRON
ejpam-3383	300	30	]	]	X
ejpam-3383	300	31	∗	∗	NOUN
ejpam-3383	300	32	(	(	PUNCT
ejpam-3383	300	33	b	b	X
ejpam-3383	300	34	]	]	X
ejpam-3383	300	35	⊆	⊆	NUM
ejpam-3383	300	36	(	(	PUNCT
ejpam-3383	300	37	a	a	DET
ejpam-3383	300	38	∗	∗	NOUN
ejpam-3383	300	39	b	b	NOUN
ejpam-3383	300	40	]	]	X
ejpam-3383	301	1	[	[	X
ejpam-3383	301	2	11	11	NUM
ejpam-3383	301	3	]	]	X
ejpam-3383	301	4	;	;	PUNCT
ejpam-3383	301	5	a	a	DET
ejpam-3383	301	6	⊆	⊆	NUM
ejpam-3383	301	7	b	b	NOUN
ejpam-3383	301	8	⇒	⇒	NOUN
ejpam-3383	301	9	(	(	PUNCT
ejpam-3383	301	10	a	a	X
ejpam-3383	301	11	]	]	X
ejpam-3383	301	12	⊆	⊆	NUM
ejpam-3383	301	13	(	(	PUNCT
ejpam-3383	301	14	b	b	NOUN
ejpam-3383	301	15	]	]	X
ejpam-3383	301	16	;	;	PUNCT
ejpam-3383	301	17	and	and	CCONJ
ejpam-3383	301	18	if	if	SCONJ
ejpam-3383	301	19	m	m	NOUN
ejpam-3383	301	20	is	be	AUX
ejpam-3383	301	21	an	an	DET
ejpam-3383	301	22	ideal	ideal	NOUN
ejpam-3383	301	23	of	of	ADP
ejpam-3383	301	24	s	s	PROPN
ejpam-3383	301	25	,	,	PUNCT
ejpam-3383	301	26	then	then	ADV
ejpam-3383	301	27	(	(	PUNCT
ejpam-3383	301	28	m	m	NOUN
ejpam-3383	301	29	]	]	X
ejpam-3383	301	30	=	=	PUNCT
ejpam-3383	301	31	m	m	VERB
ejpam-3383	301	32	.	.	PUNCT
ejpam-3383	302	1	using	use	VERB
ejpam-3383	302	2	these	these	DET
ejpam-3383	302	3	properties	property	NOUN
ejpam-3383	302	4	,	,	PUNCT
ejpam-3383	302	5	we	we	PRON
ejpam-3383	302	6	prove	prove	VERB
ejpam-3383	302	7	the	the	DET
ejpam-3383	302	8	following	follow	VERB
ejpam-3383	302	9	proposition	proposition	NOUN
ejpam-3383	302	10	.	.	PUNCT
ejpam-3383	303	1	proposition	proposition	NOUN
ejpam-3383	303	2	7.1	7.1	NUM
ejpam-3383	303	3	.	.	PUNCT
ejpam-3383	304	1	let	let	VERB
ejpam-3383	304	2	(	(	PUNCT
ejpam-3383	304	3	s	s	X
ejpam-3383	304	4	,	,	PUNCT
ejpam-3383	304	5	◦	◦	NOUN
ejpam-3383	304	6	,	,	PUNCT
ejpam-3383	304	7	≤	≤	NUM
ejpam-3383	304	8	)	)	PUNCT
ejpam-3383	304	9	be	be	AUX
ejpam-3383	304	10	an	an	DET
ejpam-3383	304	11	ordered	order	VERB
ejpam-3383	304	12	hypersemigroup	hypersemigroup	NOUN
ejpam-3383	304	13	and	and	CCONJ
ejpam-3383	304	14	m	m	VERB
ejpam-3383	304	15	an	an	DET
ejpam-3383	304	16	ideal	ideal	NOUN
ejpam-3383	304	17	of	of	ADP
ejpam-3383	304	18	s.	s.	PROPN
ejpam-3383	304	19	the	the	DET
ejpam-3383	304	20	following	follow	VERB
ejpam-3383	304	21	are	be	AUX
ejpam-3383	304	22	equivalent	equivalent	ADJ
ejpam-3383	304	23	:	:	PUNCT
ejpam-3383	304	24	(	(	PUNCT
ejpam-3383	304	25	1	1	X
ejpam-3383	304	26	)	)	PUNCT
ejpam-3383	304	27	m	m	VERB
ejpam-3383	304	28	is	be	AUX
ejpam-3383	304	29	weakly	weakly	ADV
ejpam-3383	304	30	prime	prime	ADJ
ejpam-3383	304	31	.	.	PUNCT
ejpam-3383	305	1	(	(	PUNCT
ejpam-3383	305	2	2	2	X
ejpam-3383	305	3	)	)	PUNCT
ejpam-3383	305	4	if	if	SCONJ
ejpam-3383	305	5	a	a	PRON
ejpam-3383	305	6	,	,	PUNCT
ejpam-3383	305	7	b	b	NOUN
ejpam-3383	305	8	are	be	AUX
ejpam-3383	305	9	right	right	ADJ
ejpam-3383	305	10	ideals	ideal	NOUN
ejpam-3383	305	11	of	of	ADP
ejpam-3383	305	12	s	s	PRON
ejpam-3383	305	13	such	such	ADJ
ejpam-3383	305	14	that	that	SCONJ
ejpam-3383	305	15	a	a	DET
ejpam-3383	305	16	∗b	∗b	PROPN
ejpam-3383	305	17	⊆m	⊆m	NOUN
ejpam-3383	305	18	,	,	PUNCT
ejpam-3383	305	19	then	then	ADV
ejpam-3383	305	20	a	a	DET
ejpam-3383	305	21	⊆m	⊆m	NOUN
ejpam-3383	305	22	or	or	CCONJ
ejpam-3383	305	23	b	b	NOUN
ejpam-3383	305	24	⊆m	⊆m	NOUN
ejpam-3383	305	25	.	.	PUNCT
ejpam-3383	306	1	(	(	PUNCT
ejpam-3383	306	2	3	3	X
ejpam-3383	306	3	)	)	PUNCT
ejpam-3383	306	4	if	if	SCONJ
ejpam-3383	306	5	a	a	DET
ejpam-3383	306	6	,	,	PUNCT
ejpam-3383	306	7	b	b	NOUN
ejpam-3383	306	8	are	be	AUX
ejpam-3383	306	9	left	leave	VERB
ejpam-3383	306	10	ideals	ideal	NOUN
ejpam-3383	306	11	of	of	ADP
ejpam-3383	306	12	s	s	PRON
ejpam-3383	306	13	such	such	ADJ
ejpam-3383	306	14	that	that	SCONJ
ejpam-3383	306	15	a	a	DET
ejpam-3383	306	16	∗b	∗b	PROPN
ejpam-3383	306	17	⊆m	⊆m	NOUN
ejpam-3383	306	18	,	,	PUNCT
ejpam-3383	306	19	then	then	ADV
ejpam-3383	306	20	a	a	DET
ejpam-3383	306	21	⊆m	⊆m	NOUN
ejpam-3383	306	22	or	or	CCONJ
ejpam-3383	306	23	b	b	NOUN
ejpam-3383	306	24	⊆m	⊆m	NOUN
ejpam-3383	306	25	.	.	PUNCT
ejpam-3383	307	1	(	(	PUNCT
ejpam-3383	307	2	4	4	X
ejpam-3383	307	3	)	)	PUNCT
ejpam-3383	307	4	if	if	SCONJ
ejpam-3383	307	5	a	a	PRON
ejpam-3383	307	6	is	be	AUX
ejpam-3383	307	7	a	a	DET
ejpam-3383	307	8	right	right	ADJ
ejpam-3383	307	9	ideal	ideal	NOUN
ejpam-3383	307	10	and	and	CCONJ
ejpam-3383	307	11	b	b	DET
ejpam-3383	307	12	a	a	DET
ejpam-3383	307	13	left	left	ADJ
ejpam-3383	307	14	ideal	ideal	NOUN
ejpam-3383	307	15	of	of	ADP
ejpam-3383	307	16	s	s	PRON
ejpam-3383	307	17	such	such	ADJ
ejpam-3383	307	18	that	that	SCONJ
ejpam-3383	307	19	a	a	DET
ejpam-3383	307	20	∗	∗	NOUN
ejpam-3383	307	21	b	b	NOUN
ejpam-3383	307	22	⊆	⊆	NUM
ejpam-3383	307	23	m	m	NOUN
ejpam-3383	307	24	,	,	PUNCT
ejpam-3383	307	25	then	then	ADV
ejpam-3383	307	26	a	a	DET
ejpam-3383	307	27	⊆	⊆	NUM
ejpam-3383	307	28	m	m	NOUN
ejpam-3383	307	29	or	or	CCONJ
ejpam-3383	307	30	b	b	NOUN
ejpam-3383	307	31	⊆m	⊆m	NOUN
ejpam-3383	307	32	.	.	PUNCT
ejpam-3383	308	1	proof	proof	NOUN
ejpam-3383	308	2	.	.	PUNCT
ejpam-3383	309	1	let	let	VERB
ejpam-3383	309	2	us	we	PRON
ejpam-3383	309	3	prove	prove	VERB
ejpam-3383	309	4	the	the	DET
ejpam-3383	309	5	implication	implication	NOUN
ejpam-3383	309	6	(	(	PUNCT
ejpam-3383	309	7	1)⇒	1)⇒	NUM
ejpam-3383	309	8	(	(	PUNCT
ejpam-3383	309	9	4	4	NUM
ejpam-3383	309	10	)	)	PUNCT
ejpam-3383	309	11	.	.	PUNCT
ejpam-3383	310	1	let	let	VERB
ejpam-3383	310	2	a	a	PRON
ejpam-3383	310	3	be	be	AUX
ejpam-3383	310	4	a	a	DET
ejpam-3383	310	5	right	right	ADJ
ejpam-3383	310	6	ideal	ideal	NOUN
ejpam-3383	310	7	and	and	CCONJ
ejpam-3383	310	8	b	b	DET
ejpam-3383	310	9	a	a	DET
ejpam-3383	310	10	left	left	ADJ
ejpam-3383	310	11	ideal	ideal	NOUN
ejpam-3383	310	12	of	of	ADP
ejpam-3383	310	13	s	s	PRON
ejpam-3383	310	14	such	such	ADJ
ejpam-3383	310	15	that	that	SCONJ
ejpam-3383	310	16	a	a	DET
ejpam-3383	310	17	∗b	∗b	PROPN
ejpam-3383	310	18	⊆m	⊆m	NOUN
ejpam-3383	310	19	.	.	PUNCT
ejpam-3383	311	1	we	we	PRON
ejpam-3383	311	2	have	have	VERB
ejpam-3383	311	3	i(a	i(a	PROPN
ejpam-3383	311	4	)	)	PUNCT
ejpam-3383	311	5	∗	∗	NOUN
ejpam-3383	311	6	i(b	i(b	NOUN
ejpam-3383	311	7	)	)	PUNCT
ejpam-3383	311	8	=	=	PUNCT
ejpam-3383	312	1	(	(	PUNCT
ejpam-3383	312	2	a	a	DET
ejpam-3383	312	3	∪	∪	NOUN
ejpam-3383	312	4	s	s	X
ejpam-3383	312	5	∗a	∗a	ADJ
ejpam-3383	312	6	∪a	∪a	X
ejpam-3383	312	7	∗	∗	NOUN
ejpam-3383	312	8	s	s	PART
ejpam-3383	312	9	∪	∪	NOUN
ejpam-3383	312	10	s	s	PART
ejpam-3383	312	11	∗a	∗a	ADJ
ejpam-3383	312	12	∗	∗	NOUN
ejpam-3383	312	13	s	s	PART
ejpam-3383	312	14	]	]	X
ejpam-3383	312	15	∗	∗	NOUN
ejpam-3383	312	16	(	(	PUNCT
ejpam-3383	312	17	b	b	NOUN
ejpam-3383	312	18	∪	∪	ADP
ejpam-3383	312	19	s	s	X
ejpam-3383	312	20	∗b	∗b	NOUN
ejpam-3383	312	21	∪b	∪b	X
ejpam-3383	313	1	∗	∗	NOUN
ejpam-3383	313	2	s	s	PART
ejpam-3383	313	3	∪	∪	NOUN
ejpam-3383	313	4	s	s	PART
ejpam-3383	313	5	∗b	∗b	NOUN
ejpam-3383	313	6	∗	∗	NOUN
ejpam-3383	313	7	s	s	PART
ejpam-3383	313	8	]	]	X
ejpam-3383	313	9	⊆	⊆	NUM
ejpam-3383	313	10	(	(	PUNCT
ejpam-3383	313	11	(	(	PUNCT
ejpam-3383	313	12	a	a	DET
ejpam-3383	313	13	∪	∪	NOUN
ejpam-3383	313	14	s	s	X
ejpam-3383	313	15	∗a	∗a	ADJ
ejpam-3383	313	16	∪a	∪a	X
ejpam-3383	313	17	∗	∗	NOUN
ejpam-3383	313	18	s	s	PART
ejpam-3383	313	19	∪	∪	NOUN
ejpam-3383	313	20	s	s	PART
ejpam-3383	313	21	∗a	∗a	ADJ
ejpam-3383	313	22	∗	∗	NOUN
ejpam-3383	313	23	s)(b	s)(b	PUNCT
ejpam-3383	313	24	∪	∪	ADP
ejpam-3383	313	25	s	s	PART
ejpam-3383	313	26	∗b	∗b	NOUN
ejpam-3383	313	27	∪b	∪b	X
ejpam-3383	314	1	∗	∗	NOUN
ejpam-3383	314	2	s	s	PART
ejpam-3383	314	3	∪	∪	NOUN
ejpam-3383	314	4	s	s	PART
ejpam-3383	314	5	∗b	∗b	PROPN
ejpam-3383	314	6	∗	∗	NOUN
ejpam-3383	314	7	s	s	NOUN
ejpam-3383	314	8	)	)	PUNCT
ejpam-3383	314	9	]	]	PUNCT
ejpam-3383	315	1	=	=	PUNCT
ejpam-3383	315	2	(	(	PUNCT
ejpam-3383	315	3	a	a	DET
ejpam-3383	315	4	∗b	∗b	PROPN
ejpam-3383	315	5	∪	∪	NOUN
ejpam-3383	315	6	s	s	PRON
ejpam-3383	315	7	∗a	∗a	ADJ
ejpam-3383	315	8	∗b	∗b	PROPN
ejpam-3383	315	9	∪a	∪a	X
ejpam-3383	315	10	∗	∗	NOUN
ejpam-3383	315	11	s	s	PART
ejpam-3383	315	12	∗b	∗b	NOUN
ejpam-3383	315	13	∪	∪	NOUN
ejpam-3383	315	14	s	s	PART
ejpam-3383	315	15	∗a	∗a	ADJ
ejpam-3383	315	16	∗	∗	NOUN
ejpam-3383	315	17	s	s	PART
ejpam-3383	315	18	∗b	∗b	PROPN
ejpam-3383	315	19	∪a	∪a	X
ejpam-3383	315	20	∗b	∗b	PROPN
ejpam-3383	315	21	∗	∗	NOUN
ejpam-3383	315	22	s	s	PART
ejpam-3383	315	23	∪s	∪s	NUM
ejpam-3383	315	24	∗a	∗a	PROPN
ejpam-3383	315	25	∗b	∗b	PROPN
ejpam-3383	315	26	∗	∗	NOUN
ejpam-3383	315	27	s	s	PART
ejpam-3383	315	28	∪a	∪a	X
ejpam-3383	315	29	∗	∗	NOUN
ejpam-3383	315	30	s	s	PART
ejpam-3383	315	31	∗b	∗b	NOUN
ejpam-3383	315	32	∗	∗	NOUN
ejpam-3383	315	33	s	s	NOUN
ejpam-3383	315	34	∪	∪	NOUN
ejpam-3383	315	35	s	s	PART
ejpam-3383	315	36	∗a	∗a	ADJ
ejpam-3383	315	37	∗	∗	NOUN
ejpam-3383	315	38	s	s	PART
ejpam-3383	315	39	∗b	∗b	NOUN
ejpam-3383	315	40	∗	∗	NOUN
ejpam-3383	315	41	s	s	PART
ejpam-3383	315	42	]	]	X
ejpam-3383	315	43	⊆	⊆	NUM
ejpam-3383	315	44	(	(	PUNCT
ejpam-3383	315	45	m	m	NOUN
ejpam-3383	315	46	]	]	X
ejpam-3383	315	47	=	=	PUNCT
ejpam-3383	315	48	m.	m.	NOUN
ejpam-3383	315	49	since	since	SCONJ
ejpam-3383	315	50	i(a)∗i(b	i(a)∗i(b	PROPN
ejpam-3383	315	51	)	)	PUNCT
ejpam-3383	315	52	⊆m	⊆m	NOUN
ejpam-3383	315	53	,	,	PUNCT
ejpam-3383	315	54	by	by	ADP
ejpam-3383	315	55	(	(	PUNCT
ejpam-3383	315	56	1	1	NUM
ejpam-3383	315	57	)	)	PUNCT
ejpam-3383	315	58	,	,	PUNCT
ejpam-3383	315	59	we	we	PRON
ejpam-3383	315	60	have	have	VERB
ejpam-3383	315	61	i(a	i(a	PROPN
ejpam-3383	315	62	)	)	PUNCT
ejpam-3383	315	63	⊆m	⊆m	NOUN
ejpam-3383	315	64	or	or	CCONJ
ejpam-3383	315	65	i(b	i(b	NOUN
ejpam-3383	315	66	)	)	PUNCT
ejpam-3383	315	67	⊆m	⊆m	NOUN
ejpam-3383	315	68	and	and	CCONJ
ejpam-3383	315	69	so	so	ADV
ejpam-3383	315	70	a	a	DET
ejpam-3383	315	71	⊆m	⊆m	NOUN
ejpam-3383	315	72	or	or	CCONJ
ejpam-3383	315	73	b	b	NOUN
ejpam-3383	315	74	⊆m	⊆m	NOUN
ejpam-3383	315	75	.	.	PUNCT
ejpam-3383	316	1	the	the	DET
ejpam-3383	316	2	implications	implication	NOUN
ejpam-3383	316	3	(	(	PUNCT
ejpam-3383	316	4	1	1	X
ejpam-3383	316	5	)	)	PUNCT
ejpam-3383	316	6	⇒	⇒	NOUN
ejpam-3383	316	7	(	(	PUNCT
ejpam-3383	316	8	2	2	NUM
ejpam-3383	316	9	)	)	PUNCT
ejpam-3383	316	10	and	and	CCONJ
ejpam-3383	316	11	(	(	PUNCT
ejpam-3383	316	12	1	1	X
ejpam-3383	316	13	)	)	PUNCT
ejpam-3383	316	14	⇒	⇒	NOUN
ejpam-3383	316	15	(	(	PUNCT
ejpam-3383	316	16	3	3	X
ejpam-3383	316	17	)	)	PUNCT
ejpam-3383	316	18	can	can	AUX
ejpam-3383	316	19	be	be	AUX
ejpam-3383	316	20	proved	prove	VERB
ejpam-3383	316	21	at	at	ADP
ejpam-3383	316	22	a	a	DET
ejpam-3383	316	23	similar	similar	ADJ
ejpam-3383	316	24	way	way	NOUN
ejpam-3383	316	25	and	and	CCONJ
ejpam-3383	316	26	the	the	DET
ejpam-3383	316	27	implications	implication	NOUN
ejpam-3383	316	28	(	(	PUNCT
ejpam-3383	316	29	2)⇒	2)⇒	NUM
ejpam-3383	316	30	(	(	PUNCT
ejpam-3383	316	31	1	1	NUM
ejpam-3383	316	32	)	)	PUNCT
ejpam-3383	316	33	,	,	PUNCT
ejpam-3383	316	34	(	(	PUNCT
ejpam-3383	316	35	3)⇒	3)⇒	NUM
ejpam-3383	316	36	(	(	PUNCT
ejpam-3383	316	37	1	1	NUM
ejpam-3383	316	38	)	)	PUNCT
ejpam-3383	316	39	and	and	CCONJ
ejpam-3383	316	40	(	(	PUNCT
ejpam-3383	316	41	4)⇒	4)⇒	X
ejpam-3383	316	42	(	(	PUNCT
ejpam-3383	316	43	1	1	NUM
ejpam-3383	316	44	)	)	PUNCT
ejpam-3383	316	45	are	be	AUX
ejpam-3383	316	46	obvious	obvious	ADJ
ejpam-3383	316	47	.	.	PUNCT
ejpam-3383	317	1	�	�	NOUN
ejpam-3383	317	2	the	the	DET
ejpam-3383	317	3	proof	proof	NOUN
ejpam-3383	317	4	of	of	ADP
ejpam-3383	317	5	the	the	DET
ejpam-3383	317	6	following	follow	VERB
ejpam-3383	317	7	proposition	proposition	NOUN
ejpam-3383	317	8	is	be	AUX
ejpam-3383	317	9	analogous	analogous	ADJ
ejpam-3383	317	10	.	.	PUNCT
ejpam-3383	318	1	proposition	proposition	NOUN
ejpam-3383	318	2	7.2	7.2	NUM
ejpam-3383	318	3	.	.	PUNCT
ejpam-3383	319	1	let	let	VERB
ejpam-3383	319	2	(	(	PUNCT
ejpam-3383	319	3	s	s	NOUN
ejpam-3383	319	4	,	,	PUNCT
ejpam-3383	319	5	◦	◦	NOUN
ejpam-3383	319	6	,	,	PUNCT
ejpam-3383	319	7	≤	≤	NUM
ejpam-3383	319	8	)	)	PUNCT
ejpam-3383	319	9	be	be	AUX
ejpam-3383	319	10	an	an	DET
ejpam-3383	319	11	ordered	order	VERB
ejpam-3383	319	12	hypersemigroup	hypersemigroup	NOUN
ejpam-3383	319	13	and	and	CCONJ
ejpam-3383	319	14	m	m	VERB
ejpam-3383	319	15	an	an	DET
ejpam-3383	319	16	ideal	ideal	NOUN
ejpam-3383	319	17	of	of	ADP
ejpam-3383	319	18	s.	s.	PROPN
ejpam-3383	319	19	the	the	DET
ejpam-3383	319	20	following	follow	VERB
ejpam-3383	319	21	are	be	AUX
ejpam-3383	319	22	equivalent	equivalent	ADJ
ejpam-3383	319	23	:	:	PUNCT
ejpam-3383	319	24	(	(	PUNCT
ejpam-3383	319	25	1	1	X
ejpam-3383	319	26	)	)	PUNCT
ejpam-3383	319	27	m	m	VERB
ejpam-3383	319	28	is	be	AUX
ejpam-3383	319	29	weakly	weakly	ADJ
ejpam-3383	319	30	semiprime	semiprime	NOUN
ejpam-3383	319	31	.	.	PUNCT
ejpam-3383	320	1	n.	n.	PROPN
ejpam-3383	320	2	kehayopulu	kehayopulu	PROPN
ejpam-3383	320	3	/	/	SYM
ejpam-3383	320	4	eur	eur	PROPN
ejpam-3383	320	5	.	.	PUNCT
ejpam-3383	321	1	j.	j.	PROPN
ejpam-3383	321	2	pure	pure	PROPN
ejpam-3383	321	3	appl	appl	PROPN
ejpam-3383	321	4	.	.	PROPN
ejpam-3383	321	5	math	math	PROPN
ejpam-3383	321	6	,	,	PUNCT
ejpam-3383	321	7	12	12	NUM
ejpam-3383	321	8	(	(	PUNCT
ejpam-3383	321	9	1	1	NUM
ejpam-3383	321	10	)	)	PUNCT
ejpam-3383	321	11	(	(	PUNCT
ejpam-3383	321	12	2019	2019	NUM
ejpam-3383	321	13	)	)	PUNCT
ejpam-3383	321	14	,	,	PUNCT
ejpam-3383	321	15	208	208	NUM
ejpam-3383	321	16	-	-	SYM
ejpam-3383	321	17	225	225	NUM
ejpam-3383	321	18	219	219	NUM
ejpam-3383	321	19	(	(	PUNCT
ejpam-3383	321	20	2	2	NUM
ejpam-3383	321	21	)	)	PUNCT
ejpam-3383	321	22	if	if	SCONJ
ejpam-3383	321	23	a	a	PRON
ejpam-3383	321	24	is	be	AUX
ejpam-3383	321	25	a	a	DET
ejpam-3383	321	26	right	right	ADJ
ejpam-3383	321	27	ideal	ideal	NOUN
ejpam-3383	321	28	of	of	ADP
ejpam-3383	321	29	s	s	PRON
ejpam-3383	321	30	such	such	ADJ
ejpam-3383	321	31	that	that	SCONJ
ejpam-3383	321	32	a	a	DET
ejpam-3383	321	33	∗a	∗a	ADJ
ejpam-3383	321	34	⊆m	⊆m	NOUN
ejpam-3383	321	35	,	,	PUNCT
ejpam-3383	321	36	then	then	ADV
ejpam-3383	321	37	a	a	DET
ejpam-3383	321	38	⊆m	⊆m	NOUN
ejpam-3383	321	39	.	.	PUNCT
ejpam-3383	322	1	(	(	PUNCT
ejpam-3383	322	2	3	3	X
ejpam-3383	322	3	)	)	PUNCT
ejpam-3383	322	4	if	if	SCONJ
ejpam-3383	322	5	b	b	NOUN
ejpam-3383	322	6	is	be	AUX
ejpam-3383	322	7	a	a	DET
ejpam-3383	322	8	left	left	ADJ
ejpam-3383	322	9	ideal	ideal	NOUN
ejpam-3383	322	10	of	of	ADP
ejpam-3383	322	11	s	s	PRON
ejpam-3383	322	12	such	such	ADJ
ejpam-3383	322	13	that	that	DET
ejpam-3383	322	14	b	b	X
ejpam-3383	322	15	∗b	∗b	PROPN
ejpam-3383	322	16	⊆m	⊆m	NOUN
ejpam-3383	322	17	,	,	PUNCT
ejpam-3383	322	18	then	then	ADV
ejpam-3383	322	19	b	b	X
ejpam-3383	322	20	⊆m	⊆m	NOUN
ejpam-3383	322	21	.	.	PUNCT
ejpam-3383	323	1	the	the	DET
ejpam-3383	323	2	following	follow	VERB
ejpam-3383	323	3	note	note	NOUN
ejpam-3383	323	4	is	be	AUX
ejpam-3383	323	5	referred	refer	VERB
ejpam-3383	323	6	to	to	ADP
ejpam-3383	323	7	sections	section	NOUN
ejpam-3383	323	8	5–7	5–7	NUM
ejpam-3383	323	9	.	.	PUNCT
ejpam-3383	324	1	note	note	NOUN
ejpam-3383	324	2	.	.	PUNCT
ejpam-3383	325	1	to	to	PART
ejpam-3383	325	2	pass	pass	VERB
ejpam-3383	325	3	from	from	ADP
ejpam-3383	325	4	ordered	order	VERB
ejpam-3383	325	5	semigroups	semigroup	NOUN
ejpam-3383	325	6	to	to	PART
ejpam-3383	325	7	ordered	ordered	VERB
ejpam-3383	325	8	γ	γ	NOUN
ejpam-3383	325	9	-	-	PUNCT
ejpam-3383	325	10	semigroups	semigroup	NOUN
ejpam-3383	325	11	or	or	CCONJ
ejpam-3383	325	12	to	to	PART
ejpam-3383	325	13	ordered	order	VERB
ejpam-3383	325	14	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	325	15	,	,	PUNCT
ejpam-3383	325	16	it	it	PRON
ejpam-3383	325	17	is	be	AUX
ejpam-3383	325	18	enough	enough	ADJ
ejpam-3383	325	19	to	to	PART
ejpam-3383	325	20	put	put	VERB
ejpam-3383	325	21	a	a	DET
ejpam-3383	325	22	“	"	PUNCT
ejpam-3383	325	23	γ	γ	X
ejpam-3383	325	24	”	"	PUNCT
ejpam-3383	325	25	or	or	CCONJ
ejpam-3383	325	26	“	"	PUNCT
ejpam-3383	325	27	∗	∗	NOUN
ejpam-3383	325	28	”	"	PUNCT
ejpam-3383	325	29	in	in	ADP
ejpam-3383	325	30	the	the	DET
ejpam-3383	325	31	appropriate	appropriate	ADJ
ejpam-3383	325	32	place	place	NOUN
ejpam-3383	325	33	.	.	PUNCT
ejpam-3383	326	1	8	8	X
ejpam-3383	326	2	.	.	X
ejpam-3383	327	1	another	another	DET
ejpam-3383	327	2	characterization	characterization	NOUN
ejpam-3383	327	3	of	of	ADP
ejpam-3383	327	4	weakly	weakly	ADJ
ejpam-3383	327	5	prime	prime	ADJ
ejpam-3383	327	6	ideal	ideal	ADJ
ejpam-3383	327	7	elements	element	NOUN
ejpam-3383	327	8	-	-	PUNCT
ejpam-3383	327	9	ideals	ideal	NOUN
ejpam-3383	327	10	proposition	proposition	NOUN
ejpam-3383	327	11	8.1	8.1	NUM
ejpam-3383	327	12	.	.	PUNCT
ejpam-3383	328	1	let	let	VERB
ejpam-3383	328	2	s	s	PRON
ejpam-3383	328	3	be	be	AUX
ejpam-3383	328	4	a	a	DET
ejpam-3383	328	5	poe	poe	PROPN
ejpam-3383	328	6	-	-	PUNCT
ejpam-3383	328	7	semigroup	semigroup	PROPN
ejpam-3383	328	8	and	and	CCONJ
ejpam-3383	328	9	m	m	PROPN
ejpam-3383	328	10	∈	∈	PROPN
ejpam-3383	328	11	s.	s.	PROPN
ejpam-3383	328	12	suppose	suppose	VERB
ejpam-3383	328	13	that	that	SCONJ
ejpam-3383	328	14	for	for	ADP
ejpam-3383	328	15	every	every	DET
ejpam-3383	328	16	right	right	ADJ
ejpam-3383	328	17	ideal	ideal	ADJ
ejpam-3383	328	18	element	element	NOUN
ejpam-3383	328	19	a	a	PRON
ejpam-3383	328	20	of	of	ADP
ejpam-3383	328	21	s	s	PRON
ejpam-3383	328	22	and	and	CCONJ
ejpam-3383	328	23	any	any	DET
ejpam-3383	328	24	b	b	PROPN
ejpam-3383	328	25	∈	∈	PROPN
ejpam-3383	328	26	s	s	PROPN
ejpam-3383	328	27	,	,	PUNCT
ejpam-3383	328	28	aeb	aeb	NOUN
ejpam-3383	328	29	≤	≤	NUM
ejpam-3383	328	30	m	m	VERB
ejpam-3383	328	31	implies	imply	VERB
ejpam-3383	328	32	a	a	DET
ejpam-3383	328	33	≤	≤	NUM
ejpam-3383	328	34	m	m	NOUN
ejpam-3383	328	35	or	or	CCONJ
ejpam-3383	328	36	b	b	PROPN
ejpam-3383	328	37	≤	≤	NUM
ejpam-3383	328	38	m.	m.	NOUN
ejpam-3383	328	39	then	then	ADV
ejpam-3383	328	40	m	m	VERB
ejpam-3383	328	41	is	be	AUX
ejpam-3383	328	42	weakly	weakly	ADV
ejpam-3383	328	43	prime	prime	ADJ
ejpam-3383	328	44	.	.	PUNCT
ejpam-3383	329	1	proof	proof	NOUN
ejpam-3383	329	2	.	.	PUNCT
ejpam-3383	330	1	let	let	VERB
ejpam-3383	330	2	a	a	DET
ejpam-3383	330	3	,	,	PUNCT
ejpam-3383	330	4	b	b	NOUN
ejpam-3383	330	5	be	be	AUX
ejpam-3383	330	6	ideal	ideal	ADJ
ejpam-3383	330	7	elements	element	NOUN
ejpam-3383	330	8	of	of	ADP
ejpam-3383	330	9	s	s	PRON
ejpam-3383	330	10	such	such	ADJ
ejpam-3383	330	11	that	that	SCONJ
ejpam-3383	330	12	ab	ab	PROPN
ejpam-3383	330	13	≤	≤	PROPN
ejpam-3383	330	14	m.	m.	NOUN
ejpam-3383	330	15	since	since	SCONJ
ejpam-3383	330	16	a	a	PRON
ejpam-3383	330	17	is	be	AUX
ejpam-3383	330	18	a	a	DET
ejpam-3383	330	19	right	right	ADJ
ejpam-3383	330	20	ideal	ideal	ADJ
ejpam-3383	330	21	element	element	NOUN
ejpam-3383	330	22	of	of	ADP
ejpam-3383	330	23	s	s	PROPN
ejpam-3383	330	24	,	,	PUNCT
ejpam-3383	330	25	b	b	PROPN
ejpam-3383	330	26	∈	∈	PROPN
ejpam-3383	330	27	s	s	X
ejpam-3383	330	28	and	and	CCONJ
ejpam-3383	330	29	aeb	aeb	NOUN
ejpam-3383	330	30	=	=	PUNCT
ejpam-3383	330	31	(	(	PUNCT
ejpam-3383	330	32	ae)b	ae)b	PROPN
ejpam-3383	330	33	≤	≤	PROPN
ejpam-3383	330	34	ab	ab	PROPN
ejpam-3383	330	35	≤	≤	PROPN
ejpam-3383	330	36	m	m	PROPN
ejpam-3383	330	37	,	,	PUNCT
ejpam-3383	330	38	by	by	ADP
ejpam-3383	330	39	hypothesis	hypothesis	NOUN
ejpam-3383	330	40	,	,	PUNCT
ejpam-3383	330	41	we	we	PRON
ejpam-3383	330	42	have	have	VERB
ejpam-3383	330	43	a	a	DET
ejpam-3383	330	44	≤	≤	NUM
ejpam-3383	330	45	m	m	NOUN
ejpam-3383	330	46	or	or	CCONJ
ejpam-3383	330	47	b	b	PROPN
ejpam-3383	330	48	≤	≤	NUM
ejpam-3383	330	49	m	m	NOUN
ejpam-3383	330	50	,	,	PUNCT
ejpam-3383	330	51	thus	thus	ADV
ejpam-3383	330	52	m	m	NOUN
ejpam-3383	330	53	is	be	AUX
ejpam-3383	330	54	weakly	weakly	ADV
ejpam-3383	330	55	prime	prime	ADJ
ejpam-3383	330	56	.	.	PUNCT
ejpam-3383	331	1	�	�	PROPN
ejpam-3383	331	2	proposition	proposition	PROPN
ejpam-3383	331	3	8.2	8.2	NUM
ejpam-3383	331	4	.	.	PUNCT
ejpam-3383	332	1	let	let	VERB
ejpam-3383	332	2	s	s	PRON
ejpam-3383	332	3	be	be	AUX
ejpam-3383	332	4	a	a	DET
ejpam-3383	332	5	poe	poe	PROPN
ejpam-3383	332	6	-	-	PUNCT
ejpam-3383	332	7	semigroup	semigroup	PROPN
ejpam-3383	332	8	and	and	CCONJ
ejpam-3383	332	9	m	m	PROPN
ejpam-3383	332	10	∈	∈	PROPN
ejpam-3383	332	11	s.	s.	PROPN
ejpam-3383	332	12	suppose	suppose	VERB
ejpam-3383	332	13	that	that	SCONJ
ejpam-3383	332	14	for	for	ADP
ejpam-3383	332	15	every	every	DET
ejpam-3383	332	16	left	leave	VERB
ejpam-3383	332	17	ideal	ideal	ADJ
ejpam-3383	332	18	element	element	PROPN
ejpam-3383	332	19	b	b	PROPN
ejpam-3383	332	20	of	of	ADP
ejpam-3383	332	21	s	s	PRON
ejpam-3383	332	22	and	and	CCONJ
ejpam-3383	332	23	any	any	DET
ejpam-3383	332	24	a	a	DET
ejpam-3383	332	25	∈	∈	PROPN
ejpam-3383	332	26	s	s	NOUN
ejpam-3383	332	27	,	,	PUNCT
ejpam-3383	332	28	aeb	aeb	NOUN
ejpam-3383	332	29	≤	≤	NUM
ejpam-3383	332	30	m	m	VERB
ejpam-3383	332	31	implies	imply	VERB
ejpam-3383	332	32	a	a	DET
ejpam-3383	332	33	≤	≤	NUM
ejpam-3383	332	34	m	m	NOUN
ejpam-3383	332	35	or	or	CCONJ
ejpam-3383	332	36	b	b	PROPN
ejpam-3383	332	37	≤	≤	NUM
ejpam-3383	332	38	m.	m.	NOUN
ejpam-3383	332	39	then	then	ADV
ejpam-3383	332	40	m	m	VERB
ejpam-3383	332	41	is	be	AUX
ejpam-3383	332	42	weakly	weakly	ADV
ejpam-3383	332	43	prime	prime	ADJ
ejpam-3383	332	44	.	.	PUNCT
ejpam-3383	333	1	proof	proof	NOUN
ejpam-3383	333	2	.	.	PUNCT
ejpam-3383	334	1	let	let	VERB
ejpam-3383	334	2	a	a	DET
ejpam-3383	334	3	,	,	PUNCT
ejpam-3383	334	4	b	b	NOUN
ejpam-3383	334	5	be	be	AUX
ejpam-3383	334	6	ideal	ideal	ADJ
ejpam-3383	334	7	elements	element	NOUN
ejpam-3383	334	8	of	of	ADP
ejpam-3383	334	9	s	s	PRON
ejpam-3383	334	10	such	such	ADJ
ejpam-3383	334	11	that	that	SCONJ
ejpam-3383	334	12	ab	ab	PROPN
ejpam-3383	334	13	≤	≤	PROPN
ejpam-3383	334	14	m.	m.	NOUN
ejpam-3383	334	15	since	since	SCONJ
ejpam-3383	334	16	b	b	PROPN
ejpam-3383	334	17	is	be	AUX
ejpam-3383	334	18	a	a	DET
ejpam-3383	334	19	left	left	ADJ
ejpam-3383	334	20	ideal	ideal	ADJ
ejpam-3383	334	21	element	element	NOUN
ejpam-3383	334	22	of	of	ADP
ejpam-3383	334	23	s	s	PROPN
ejpam-3383	334	24	,	,	PUNCT
ejpam-3383	334	25	a	a	DET
ejpam-3383	334	26	∈	∈	PROPN
ejpam-3383	334	27	s	s	NOUN
ejpam-3383	334	28	and	and	CCONJ
ejpam-3383	334	29	aeb	aeb	NOUN
ejpam-3383	334	30	=	=	SYM
ejpam-3383	334	31	a(eb	a(eb	PROPN
ejpam-3383	334	32	)	)	PUNCT
ejpam-3383	334	33	≤	≤	PUNCT
ejpam-3383	334	34	ab	ab	PROPN
ejpam-3383	334	35	≤	≤	PROPN
ejpam-3383	334	36	m	m	PROPN
ejpam-3383	334	37	,	,	PUNCT
ejpam-3383	334	38	by	by	ADP
ejpam-3383	334	39	hypothesis	hypothesis	NOUN
ejpam-3383	334	40	,	,	PUNCT
ejpam-3383	334	41	we	we	PRON
ejpam-3383	334	42	have	have	VERB
ejpam-3383	334	43	a	a	DET
ejpam-3383	334	44	≤	≤	NUM
ejpam-3383	334	45	m	m	NOUN
ejpam-3383	334	46	or	or	CCONJ
ejpam-3383	334	47	b	b	PROPN
ejpam-3383	334	48	≤	≤	NUM
ejpam-3383	334	49	m	m	NOUN
ejpam-3383	334	50	,	,	PUNCT
ejpam-3383	334	51	thus	thus	ADV
ejpam-3383	334	52	m	m	NOUN
ejpam-3383	334	53	is	be	AUX
ejpam-3383	334	54	weakly	weakly	ADV
ejpam-3383	334	55	prime	prime	ADJ
ejpam-3383	334	56	.	.	PUNCT
ejpam-3383	335	1	�	�	PROPN
ejpam-3383	335	2	corollary	corollary	NOUN
ejpam-3383	335	3	8.3	8.3	NUM
ejpam-3383	335	4	.	.	PUNCT
ejpam-3383	336	1	let	let	AUX
ejpam-3383	336	2	(	(	PUNCT
ejpam-3383	336	3	s	s	X
ejpam-3383	336	4	,	,	PUNCT
ejpam-3383	336	5	·	·	PUNCT
ejpam-3383	336	6	)	)	PUNCT
ejpam-3383	336	7	be	be	AUX
ejpam-3383	336	8	a	a	DET
ejpam-3383	336	9	semigroup	semigroup	NOUN
ejpam-3383	336	10	and	and	CCONJ
ejpam-3383	336	11	m	m	PROPN
ejpam-3383	336	12	a	a	DET
ejpam-3383	336	13	subset	subset	NOUN
ejpam-3383	336	14	of	of	ADP
ejpam-3383	336	15	s.	s.	PROPN
ejpam-3383	336	16	suppose	suppose	VERB
ejpam-3383	336	17	that	that	SCONJ
ejpam-3383	336	18	for	for	ADP
ejpam-3383	336	19	any	any	DET
ejpam-3383	336	20	right	right	ADJ
ejpam-3383	336	21	ideal	ideal	NOUN
ejpam-3383	336	22	a	a	PRON
ejpam-3383	336	23	of	of	ADP
ejpam-3383	336	24	s	s	PRON
ejpam-3383	336	25	and	and	CCONJ
ejpam-3383	336	26	any	any	DET
ejpam-3383	336	27	b	b	NOUN
ejpam-3383	336	28	⊆	⊆	NUM
ejpam-3383	336	29	s	s	X
ejpam-3383	336	30	(	(	PUNCT
ejpam-3383	336	31	or	or	CCONJ
ejpam-3383	336	32	for	for	ADP
ejpam-3383	336	33	any	any	DET
ejpam-3383	336	34	left	left	ADJ
ejpam-3383	336	35	ideal	ideal	NOUN
ejpam-3383	336	36	b	b	PROPN
ejpam-3383	336	37	of	of	ADP
ejpam-3383	336	38	s	s	PRON
ejpam-3383	336	39	and	and	CCONJ
ejpam-3383	336	40	any	any	DET
ejpam-3383	336	41	a	a	DET
ejpam-3383	336	42	⊆	⊆	NUM
ejpam-3383	336	43	s	s	NOUN
ejpam-3383	336	44	)	)	PUNCT
ejpam-3383	336	45	,	,	PUNCT
ejpam-3383	336	46	asb	asb	PROPN
ejpam-3383	336	47	⊆	⊆	NUM
ejpam-3383	336	48	m	m	NOUN
ejpam-3383	336	49	,	,	PUNCT
ejpam-3383	336	50	implies	imply	VERB
ejpam-3383	336	51	a	a	DET
ejpam-3383	336	52	⊆m	⊆m	NOUN
ejpam-3383	336	53	or	or	CCONJ
ejpam-3383	336	54	b	b	NOUN
ejpam-3383	336	55	⊆m	⊆m	NOUN
ejpam-3383	336	56	.	.	PUNCT
ejpam-3383	337	1	then	then	ADV
ejpam-3383	337	2	m	m	PROPN
ejpam-3383	337	3	is	be	AUX
ejpam-3383	337	4	weakly	weakly	ADV
ejpam-3383	337	5	prime	prime	ADJ
ejpam-3383	337	6	.	.	PUNCT
ejpam-3383	338	1	corollary	corollary	ADJ
ejpam-3383	338	2	8.4	8.4	NUM
ejpam-3383	338	3	.	.	PUNCT
ejpam-3383	339	1	let	let	VERB
ejpam-3383	339	2	(	(	PUNCT
ejpam-3383	339	3	s	s	X
ejpam-3383	339	4	,	,	PUNCT
ejpam-3383	339	5	γ	γ	NOUN
ejpam-3383	339	6	)	)	PUNCT
ejpam-3383	339	7	be	be	AUX
ejpam-3383	339	8	a	a	DET
ejpam-3383	339	9	γ	γ	NOUN
ejpam-3383	339	10	-	-	PUNCT
ejpam-3383	339	11	semigroup	semigroup	NOUN
ejpam-3383	339	12	and	and	CCONJ
ejpam-3383	339	13	m	m	VERB
ejpam-3383	339	14	a	a	DET
ejpam-3383	339	15	subset	subset	NOUN
ejpam-3383	339	16	of	of	ADP
ejpam-3383	339	17	s.	s.	PROPN
ejpam-3383	339	18	suppose	suppose	VERB
ejpam-3383	339	19	that	that	SCONJ
ejpam-3383	339	20	for	for	ADP
ejpam-3383	339	21	any	any	DET
ejpam-3383	339	22	right	right	ADJ
ejpam-3383	339	23	ideal	ideal	NOUN
ejpam-3383	339	24	a	a	PRON
ejpam-3383	339	25	of	of	ADP
ejpam-3383	339	26	s	s	PRON
ejpam-3383	339	27	and	and	CCONJ
ejpam-3383	339	28	any	any	DET
ejpam-3383	339	29	b	b	NOUN
ejpam-3383	339	30	⊆	⊆	NUM
ejpam-3383	339	31	s	s	X
ejpam-3383	339	32	(	(	PUNCT
ejpam-3383	339	33	or	or	CCONJ
ejpam-3383	339	34	for	for	ADP
ejpam-3383	339	35	any	any	DET
ejpam-3383	339	36	left	left	ADJ
ejpam-3383	339	37	ideal	ideal	NOUN
ejpam-3383	339	38	b	b	PROPN
ejpam-3383	339	39	of	of	ADP
ejpam-3383	339	40	s	s	PRON
ejpam-3383	339	41	and	and	CCONJ
ejpam-3383	339	42	any	any	DET
ejpam-3383	339	43	a	a	DET
ejpam-3383	339	44	⊆	⊆	NUM
ejpam-3383	339	45	s	s	NOUN
ejpam-3383	339	46	)	)	PUNCT
ejpam-3383	339	47	,	,	PUNCT
ejpam-3383	339	48	aγsγb	aγsγb	ADJ
ejpam-3383	339	49	⊆m	⊆m	PROPN
ejpam-3383	339	50	,	,	PUNCT
ejpam-3383	339	51	implies	imply	VERB
ejpam-3383	339	52	a	a	DET
ejpam-3383	339	53	⊆m	⊆m	NOUN
ejpam-3383	339	54	or	or	CCONJ
ejpam-3383	339	55	b	b	NOUN
ejpam-3383	339	56	⊆m	⊆m	NOUN
ejpam-3383	339	57	.	.	PUNCT
ejpam-3383	340	1	then	then	ADV
ejpam-3383	340	2	m	m	PROPN
ejpam-3383	340	3	is	be	AUX
ejpam-3383	340	4	weakly	weakly	ADV
ejpam-3383	340	5	prime	prime	ADJ
ejpam-3383	340	6	.	.	PUNCT
ejpam-3383	341	1	for	for	ADP
ejpam-3383	341	2	its	its	PRON
ejpam-3383	341	3	proof	proof	NOUN
ejpam-3383	341	4	,	,	PUNCT
ejpam-3383	341	5	it	it	PRON
ejpam-3383	341	6	is	be	AUX
ejpam-3383	341	7	enough	enough	ADJ
ejpam-3383	341	8	to	to	PART
ejpam-3383	341	9	remark	remark	VERB
ejpam-3383	341	10	that	that	SCONJ
ejpam-3383	341	11	(	(	PUNCT
ejpam-3383	341	12	p(s	p(s	NOUN
ejpam-3383	341	13	)	)	PUNCT
ejpam-3383	341	14	,	,	PUNCT
ejpam-3383	341	15	·	·	PUNCT
ejpam-3383	341	16	,	,	PUNCT
ejpam-3383	341	17	⊆	⊆	NUM
ejpam-3383	341	18	)	)	PUNCT
ejpam-3383	341	19	is	be	AUX
ejpam-3383	341	20	a	a	DET
ejpam-3383	341	21	poe	poe	PROPN
ejpam-3383	341	22	-	-	PUNCT
ejpam-3383	341	23	semigroup	semigroup	PROPN
ejpam-3383	341	24	(	(	PUNCT
ejpam-3383	341	25	p(s	p(s	PROPN
ejpam-3383	341	26	)	)	PUNCT
ejpam-3383	341	27	is	be	AUX
ejpam-3383	341	28	the	the	DET
ejpam-3383	341	29	set	set	NOUN
ejpam-3383	341	30	of	of	ADP
ejpam-3383	341	31	subsets	subset	NOUN
ejpam-3383	341	32	of	of	ADP
ejpam-3383	341	33	s	s	NOUN
ejpam-3383	341	34	)	)	PUNCT
ejpam-3383	341	35	.	.	PUNCT
ejpam-3383	342	1	if	if	SCONJ
ejpam-3383	342	2	we	we	PRON
ejpam-3383	342	3	put	put	VERB
ejpam-3383	342	4	γ	γ	X
ejpam-3383	342	5	=	=	SYM
ejpam-3383	342	6	{	{	PUNCT
ejpam-3383	342	7	·	·	PUNCT
ejpam-3383	342	8	}	}	PUNCT
ejpam-3383	342	9	,	,	PUNCT
ejpam-3383	342	10	then	then	ADV
ejpam-3383	342	11	corollary	corollary	NOUN
ejpam-3383	342	12	8.3	8.3	NUM
ejpam-3383	342	13	can	can	AUX
ejpam-3383	342	14	be	be	AUX
ejpam-3383	342	15	obtained	obtain	VERB
ejpam-3383	342	16	.	.	PUNCT
ejpam-3383	343	1	corollary	corollary	ADJ
ejpam-3383	343	2	8.5	8.5	NUM
ejpam-3383	343	3	.	.	PUNCT
ejpam-3383	344	1	let	let	AUX
ejpam-3383	344	2	(	(	PUNCT
ejpam-3383	344	3	s	s	NOUN
ejpam-3383	344	4	,	,	PUNCT
ejpam-3383	344	5	◦	◦	NOUN
ejpam-3383	344	6	)	)	PUNCT
ejpam-3383	344	7	be	be	VERB
ejpam-3383	344	8	an	an	DET
ejpam-3383	344	9	hypersemigroup	hypersemigroup	NOUN
ejpam-3383	344	10	and	and	CCONJ
ejpam-3383	344	11	m	m	PROPN
ejpam-3383	344	12	a	a	DET
ejpam-3383	344	13	subset	subset	NOUN
ejpam-3383	344	14	of	of	ADP
ejpam-3383	344	15	s.	s.	PROPN
ejpam-3383	344	16	suppose	suppose	VERB
ejpam-3383	344	17	that	that	SCONJ
ejpam-3383	344	18	for	for	ADP
ejpam-3383	344	19	any	any	DET
ejpam-3383	344	20	right	right	ADJ
ejpam-3383	344	21	ideal	ideal	NOUN
ejpam-3383	344	22	a	a	PRON
ejpam-3383	344	23	of	of	ADP
ejpam-3383	344	24	s	s	PRON
ejpam-3383	344	25	and	and	CCONJ
ejpam-3383	344	26	any	any	DET
ejpam-3383	344	27	b	b	NOUN
ejpam-3383	344	28	⊆	⊆	NUM
ejpam-3383	344	29	s	s	X
ejpam-3383	344	30	(	(	PUNCT
ejpam-3383	344	31	or	or	CCONJ
ejpam-3383	344	32	for	for	ADP
ejpam-3383	344	33	any	any	DET
ejpam-3383	344	34	left	left	ADJ
ejpam-3383	344	35	ideal	ideal	NOUN
ejpam-3383	344	36	b	b	PROPN
ejpam-3383	344	37	of	of	ADP
ejpam-3383	344	38	s	s	PRON
ejpam-3383	344	39	and	and	CCONJ
ejpam-3383	344	40	any	any	DET
ejpam-3383	344	41	a	a	DET
ejpam-3383	344	42	⊆	⊆	NUM
ejpam-3383	344	43	s	s	NOUN
ejpam-3383	344	44	)	)	PUNCT
ejpam-3383	344	45	,	,	PUNCT
ejpam-3383	344	46	a	a	DET
ejpam-3383	344	47	∗	∗	NOUN
ejpam-3383	344	48	s	s	NOUN
ejpam-3383	344	49	∗b	∗b	NOUN
ejpam-3383	344	50	⊆m	⊆m	NOUN
ejpam-3383	344	51	,	,	PUNCT
ejpam-3383	344	52	implies	imply	VERB
ejpam-3383	344	53	a	a	DET
ejpam-3383	344	54	⊆m	⊆m	NOUN
ejpam-3383	344	55	or	or	CCONJ
ejpam-3383	344	56	b	b	NOUN
ejpam-3383	344	57	⊆m	⊆m	NOUN
ejpam-3383	344	58	.	.	PUNCT
ejpam-3383	345	1	then	then	ADV
ejpam-3383	345	2	m	m	PROPN
ejpam-3383	345	3	is	be	AUX
ejpam-3383	345	4	weakly	weakly	ADV
ejpam-3383	345	5	prime	prime	ADJ
ejpam-3383	345	6	.	.	PUNCT
ejpam-3383	346	1	proposition	proposition	NOUN
ejpam-3383	346	2	8.6	8.6	NUM
ejpam-3383	346	3	.	.	PUNCT
ejpam-3383	347	1	let	let	VERB
ejpam-3383	347	2	s	s	PRON
ejpam-3383	347	3	be	be	AUX
ejpam-3383	347	4	a	a	DET
ejpam-3383	347	5	∨e	∨e	NOUN
ejpam-3383	347	6	-	-	PUNCT
ejpam-3383	347	7	semigroup	semigroup	NOUN
ejpam-3383	347	8	and	and	CCONJ
ejpam-3383	347	9	m	m	VERB
ejpam-3383	347	10	an	an	DET
ejpam-3383	347	11	ideal	ideal	ADJ
ejpam-3383	347	12	element	element	NOUN
ejpam-3383	347	13	of	of	ADP
ejpam-3383	347	14	s.	s.	PROPN
ejpam-3383	347	15	if	if	SCONJ
ejpam-3383	347	16	m	m	NOUN
ejpam-3383	347	17	is	be	AUX
ejpam-3383	347	18	weakly	weakly	ADV
ejpam-3383	347	19	prime	prime	ADJ
ejpam-3383	347	20	then	then	ADV
ejpam-3383	347	21	,	,	PUNCT
ejpam-3383	347	22	for	for	ADP
ejpam-3383	347	23	every	every	DET
ejpam-3383	347	24	a	a	PROPN
ejpam-3383	347	25	,	,	PUNCT
ejpam-3383	347	26	b	b	X
ejpam-3383	347	27	∈	∈	NOUN
ejpam-3383	347	28	s	s	VERB
ejpam-3383	347	29	such	such	ADJ
ejpam-3383	347	30	that	that	DET
ejpam-3383	347	31	aeb	aeb	NOUN
ejpam-3383	347	32	≤	≤	NUM
ejpam-3383	347	33	m	m	ADP
ejpam-3383	347	34	,	,	PUNCT
ejpam-3383	347	35	we	we	PRON
ejpam-3383	347	36	have	have	VERB
ejpam-3383	347	37	a	a	DET
ejpam-3383	347	38	≤	≤	NUM
ejpam-3383	347	39	m	m	NOUN
ejpam-3383	347	40	or	or	CCONJ
ejpam-3383	347	41	b	b	NOUN
ejpam-3383	347	42	≤	≤	NUM
ejpam-3383	347	43	m.	m.	NOUN
ejpam-3383	347	44	proof	proof	NOUN
ejpam-3383	347	45	.	.	PUNCT
ejpam-3383	348	1	let	let	VERB
ejpam-3383	348	2	a	a	DET
ejpam-3383	348	3	,	,	PUNCT
ejpam-3383	348	4	b	b	X
ejpam-3383	348	5	∈	∈	NOUN
ejpam-3383	348	6	s	s	VERB
ejpam-3383	348	7	such	such	ADJ
ejpam-3383	348	8	that	that	DET
ejpam-3383	348	9	aeb	aeb	NOUN
ejpam-3383	348	10	≤	≤	ADJ
ejpam-3383	348	11	m.	m.	NOUN
ejpam-3383	349	1	then	then	ADV
ejpam-3383	349	2	we	we	PRON
ejpam-3383	349	3	have	have	VERB
ejpam-3383	349	4	(	(	PUNCT
ejpam-3383	349	5	eae)(ebe	eae)(ebe	NOUN
ejpam-3383	349	6	)	)	PUNCT
ejpam-3383	349	7	≤	≤	NOUN
ejpam-3383	349	8	e(aeb)e	e(aeb)e	ADJ
ejpam-3383	349	9	≤	≤	NUM
ejpam-3383	349	10	eme	eme	NOUN
ejpam-3383	349	11	≤	≤	PROPN
ejpam-3383	349	12	m.	m.	NOUN
ejpam-3383	349	13	since	since	SCONJ
ejpam-3383	349	14	eae	eae	PROPN
ejpam-3383	349	15	,	,	PUNCT
ejpam-3383	349	16	ebe	ebe	PROPN
ejpam-3383	349	17	are	be	AUX
ejpam-3383	349	18	ideal	ideal	ADJ
ejpam-3383	349	19	elements	element	NOUN
ejpam-3383	349	20	of	of	ADP
ejpam-3383	349	21	s	s	PRON
ejpam-3383	349	22	and	and	CCONJ
ejpam-3383	349	23	m	m	PROPN
ejpam-3383	349	24	is	be	AUX
ejpam-3383	349	25	weakly	weakly	ADV
ejpam-3383	349	26	prime	prime	ADJ
ejpam-3383	349	27	,	,	PUNCT
ejpam-3383	349	28	we	we	PRON
ejpam-3383	349	29	have	have	VERB
ejpam-3383	349	30	eae	eae	PROPN
ejpam-3383	349	31	≤	≤	PROPN
ejpam-3383	349	32	m	m	PROPN
ejpam-3383	349	33	or	or	CCONJ
ejpam-3383	349	34	ebe	ebe	PROPN
ejpam-3383	349	35	≤	≤	PROPN
ejpam-3383	349	36	m.	m.	NOUN
ejpam-3383	349	37	let	let	VERB
ejpam-3383	349	38	eae	eae	PROPN
ejpam-3383	349	39	≤	≤	PROPN
ejpam-3383	349	40	m.	m.	NOUN
ejpam-3383	349	41	then	then	ADV
ejpam-3383	349	42	i(a)2i(a	i(a)2i(a	PROPN
ejpam-3383	349	43	)	)	PUNCT
ejpam-3383	350	1	=	=	SYM
ejpam-3383	350	2	(	(	PUNCT
ejpam-3383	350	3	a	a	DET
ejpam-3383	350	4	∨	∨	NUM
ejpam-3383	350	5	ea	ea	NOUN
ejpam-3383	350	6	∨	∨	NUM
ejpam-3383	350	7	ae	ae	PROPN
ejpam-3383	350	8	∨	∨	NUM
ejpam-3383	350	9	eae)2(a	eae)2(a	PROPN
ejpam-3383	350	10	∨	∨	PROPN
ejpam-3383	350	11	ea	ea	PROPN
ejpam-3383	350	12	∨	∨	NUM
ejpam-3383	350	13	ae	ae	PROPN
ejpam-3383	350	14	∨	∨	PROPN
ejpam-3383	350	15	eae	eae	PROPN
ejpam-3383	350	16	)	)	PUNCT
ejpam-3383	350	17	≤	≤	NUM
ejpam-3383	350	18	(	(	PUNCT
ejpam-3383	350	19	ea	ea	NOUN
ejpam-3383	350	20	∨	∨	NUM
ejpam-3383	350	21	eae)(a	eae)(a	NOUN
ejpam-3383	350	22	∨	∨	NUM
ejpam-3383	350	23	ea	ea	PROPN
ejpam-3383	350	24	∨	∨	NUM
ejpam-3383	350	25	ae	ae	PROPN
ejpam-3383	350	26	∨	∨	PROPN
ejpam-3383	350	27	eae	eae	PROPN
ejpam-3383	350	28	)	)	PUNCT
ejpam-3383	350	29	≤	≤	PUNCT
ejpam-3383	350	30	eae	eae	PROPN
ejpam-3383	350	31	≤	≤	PROPN
ejpam-3383	350	32	m.	m.	NOUN
ejpam-3383	350	33	n.	n.	PROPN
ejpam-3383	350	34	kehayopulu	kehayopulu	PROPN
ejpam-3383	350	35	/	/	SYM
ejpam-3383	350	36	eur	eur	PROPN
ejpam-3383	350	37	.	.	PUNCT
ejpam-3383	351	1	j.	j.	PROPN
ejpam-3383	351	2	pure	pure	PROPN
ejpam-3383	351	3	appl	appl	PROPN
ejpam-3383	351	4	.	.	PROPN
ejpam-3383	351	5	math	math	PROPN
ejpam-3383	351	6	,	,	PUNCT
ejpam-3383	351	7	12	12	NUM
ejpam-3383	351	8	(	(	PUNCT
ejpam-3383	351	9	1	1	NUM
ejpam-3383	351	10	)	)	PUNCT
ejpam-3383	351	11	(	(	PUNCT
ejpam-3383	351	12	2019	2019	NUM
ejpam-3383	351	13	)	)	PUNCT
ejpam-3383	351	14	,	,	PUNCT
ejpam-3383	351	15	208	208	NUM
ejpam-3383	351	16	-	-	SYM
ejpam-3383	351	17	225	225	NUM
ejpam-3383	351	18	220	220	NUM
ejpam-3383	351	19	since	since	SCONJ
ejpam-3383	351	20	the	the	DET
ejpam-3383	351	21	elements	element	NOUN
ejpam-3383	351	22	i(a)2	i(a)2	PROPN
ejpam-3383	351	23	and	and	CCONJ
ejpam-3383	351	24	i(a	i(a	NOUN
ejpam-3383	351	25	)	)	PUNCT
ejpam-3383	351	26	are	be	AUX
ejpam-3383	351	27	ideal	ideal	ADJ
ejpam-3383	351	28	elements	element	NOUN
ejpam-3383	351	29	of	of	ADP
ejpam-3383	351	30	s	s	PRON
ejpam-3383	351	31	and	and	CCONJ
ejpam-3383	351	32	m	m	PROPN
ejpam-3383	351	33	is	be	AUX
ejpam-3383	351	34	weakly	weakly	ADV
ejpam-3383	351	35	prime	prime	ADJ
ejpam-3383	351	36	,	,	PUNCT
ejpam-3383	351	37	we	we	PRON
ejpam-3383	351	38	have	have	VERB
ejpam-3383	351	39	i(a)2	i(a)2	NOUN
ejpam-3383	351	40	≤	≤	NOUN
ejpam-3383	351	41	m	m	NOUN
ejpam-3383	351	42	or	or	CCONJ
ejpam-3383	351	43	i(a	i(a	NUM
ejpam-3383	351	44	)	)	PUNCT
ejpam-3383	351	45	≤	≤	NOUN
ejpam-3383	351	46	m.	m.	NOUN
ejpam-3383	351	47	if	if	SCONJ
ejpam-3383	351	48	i(a	i(a	PROPN
ejpam-3383	351	49	)	)	PUNCT
ejpam-3383	351	50	≤	≤	NOUN
ejpam-3383	351	51	m	m	PROPN
ejpam-3383	351	52	,	,	PUNCT
ejpam-3383	351	53	then	then	ADV
ejpam-3383	351	54	we	we	PRON
ejpam-3383	351	55	have	have	VERB
ejpam-3383	351	56	a	a	DET
ejpam-3383	351	57	≤	≤	NUM
ejpam-3383	351	58	i(a	i(a	PROPN
ejpam-3383	351	59	)	)	PUNCT
ejpam-3383	351	60	≤	≤	NOUN
ejpam-3383	351	61	m.	m.	NOUN
ejpam-3383	351	62	if	if	SCONJ
ejpam-3383	351	63	i(a)2	i(a)2	PROPN
ejpam-3383	351	64	≤	≤	NOUN
ejpam-3383	351	65	m	m	VERB
ejpam-3383	351	66	then	then	ADV
ejpam-3383	351	67	,	,	PUNCT
ejpam-3383	351	68	since	since	SCONJ
ejpam-3383	351	69	i(a	i(a	PROPN
ejpam-3383	351	70	)	)	PUNCT
ejpam-3383	351	71	is	be	AUX
ejpam-3383	351	72	an	an	DET
ejpam-3383	351	73	ideal	ideal	ADJ
ejpam-3383	351	74	element	element	NOUN
ejpam-3383	351	75	of	of	ADP
ejpam-3383	351	76	s	s	PRON
ejpam-3383	351	77	and	and	CCONJ
ejpam-3383	351	78	m	m	PROPN
ejpam-3383	351	79	is	be	AUX
ejpam-3383	351	80	weakly	weakly	ADV
ejpam-3383	351	81	prime	prime	ADJ
ejpam-3383	351	82	,	,	PUNCT
ejpam-3383	351	83	we	we	PRON
ejpam-3383	351	84	have	have	VERB
ejpam-3383	351	85	i(a	i(a	PROPN
ejpam-3383	351	86	)	)	PUNCT
ejpam-3383	352	1	≤	≤	NOUN
ejpam-3383	353	1	m	m	PROPN
ejpam-3383	354	1	and	and	CCONJ
ejpam-3383	354	2	so	so	ADV
ejpam-3383	354	3	a	a	DET
ejpam-3383	354	4	≤	≤	ADJ
ejpam-3383	354	5	m.	m.	NOUN
ejpam-3383	354	6	from	from	ADP
ejpam-3383	354	7	ebe	ebe	PROPN
ejpam-3383	354	8	≤	≤	PROPN
ejpam-3383	354	9	m	m	PROPN
ejpam-3383	354	10	,	,	PUNCT
ejpam-3383	354	11	by	by	ADP
ejpam-3383	354	12	symmetry	symmetry	NOUN
ejpam-3383	354	13	,	,	PUNCT
ejpam-3383	354	14	we	we	PRON
ejpam-3383	354	15	have	have	VERB
ejpam-3383	354	16	b	b	NOUN
ejpam-3383	354	17	≤	≤	NUM
ejpam-3383	354	18	m.	m.	NOUN
ejpam-3383	354	19	�	�	PROPN
ejpam-3383	354	20	corollary	corollary	PROPN
ejpam-3383	354	21	8.7	8.7	NUM
ejpam-3383	354	22	.	.	PUNCT
ejpam-3383	355	1	let	let	VERB
ejpam-3383	355	2	(	(	PUNCT
ejpam-3383	355	3	s	s	X
ejpam-3383	355	4	,	,	PUNCT
ejpam-3383	355	5	·	·	PUNCT
ejpam-3383	355	6	)	)	PUNCT
ejpam-3383	355	7	be	be	AUX
ejpam-3383	355	8	a	a	DET
ejpam-3383	355	9	semigroup	semigroup	NOUN
ejpam-3383	355	10	and	and	CCONJ
ejpam-3383	355	11	m	m	VERB
ejpam-3383	355	12	an	an	DET
ejpam-3383	355	13	ideal	ideal	NOUN
ejpam-3383	355	14	of	of	ADP
ejpam-3383	355	15	s.	s.	PROPN
ejpam-3383	355	16	if	if	SCONJ
ejpam-3383	355	17	m	m	NOUN
ejpam-3383	355	18	is	be	AUX
ejpam-3383	355	19	weakly	weakly	ADV
ejpam-3383	355	20	prime	prime	ADJ
ejpam-3383	355	21	then	then	ADV
ejpam-3383	355	22	,	,	PUNCT
ejpam-3383	355	23	for	for	ADP
ejpam-3383	355	24	every	every	DET
ejpam-3383	355	25	subsets	subset	NOUN
ejpam-3383	355	26	a	a	PRON
ejpam-3383	355	27	,	,	PUNCT
ejpam-3383	355	28	b	b	NOUN
ejpam-3383	355	29	of	of	ADP
ejpam-3383	355	30	s	s	PRON
ejpam-3383	355	31	such	such	ADJ
ejpam-3383	355	32	that	that	SCONJ
ejpam-3383	355	33	asb	asb	PROPN
ejpam-3383	355	34	⊆m	⊆m	NOUN
ejpam-3383	355	35	,	,	PUNCT
ejpam-3383	355	36	we	we	PRON
ejpam-3383	355	37	have	have	VERB
ejpam-3383	355	38	a	a	DET
ejpam-3383	355	39	⊆m	⊆m	NOUN
ejpam-3383	355	40	or	or	CCONJ
ejpam-3383	355	41	b	b	NOUN
ejpam-3383	355	42	⊆m	⊆m	NOUN
ejpam-3383	355	43	.	.	PUNCT
ejpam-3383	356	1	corollary	corollary	ADJ
ejpam-3383	356	2	8.8	8.8	NUM
ejpam-3383	356	3	.	.	PUNCT
ejpam-3383	357	1	let	let	AUX
ejpam-3383	357	2	(	(	PUNCT
ejpam-3383	357	3	s	s	X
ejpam-3383	357	4	,	,	PUNCT
ejpam-3383	357	5	γ	γ	NOUN
ejpam-3383	357	6	)	)	PUNCT
ejpam-3383	357	7	be	be	VERB
ejpam-3383	357	8	a	a	DET
ejpam-3383	357	9	γ	γ	NOUN
ejpam-3383	357	10	-	-	PUNCT
ejpam-3383	357	11	semigroup	semigroup	NOUN
ejpam-3383	357	12	and	and	CCONJ
ejpam-3383	357	13	m	m	VERB
ejpam-3383	357	14	an	an	DET
ejpam-3383	357	15	ideal	ideal	NOUN
ejpam-3383	357	16	of	of	ADP
ejpam-3383	357	17	s.	s.	PROPN
ejpam-3383	357	18	if	if	SCONJ
ejpam-3383	357	19	m	m	NOUN
ejpam-3383	357	20	is	be	AUX
ejpam-3383	357	21	weakly	weakly	ADV
ejpam-3383	357	22	prime	prime	ADJ
ejpam-3383	357	23	then	then	ADV
ejpam-3383	357	24	,	,	PUNCT
ejpam-3383	357	25	for	for	ADP
ejpam-3383	357	26	every	every	DET
ejpam-3383	357	27	subsets	subset	NOUN
ejpam-3383	357	28	a	a	PRON
ejpam-3383	357	29	,	,	PUNCT
ejpam-3383	357	30	b	b	NOUN
ejpam-3383	357	31	of	of	ADP
ejpam-3383	357	32	s	s	PRON
ejpam-3383	357	33	such	such	ADJ
ejpam-3383	357	34	that	that	SCONJ
ejpam-3383	357	35	aγsγb	aγsγb	ADJ
ejpam-3383	357	36	⊆m	⊆m	NOUN
ejpam-3383	357	37	,	,	PUNCT
ejpam-3383	357	38	we	we	PRON
ejpam-3383	357	39	have	have	VERB
ejpam-3383	357	40	a	a	DET
ejpam-3383	357	41	⊆m	⊆m	NOUN
ejpam-3383	357	42	or	or	CCONJ
ejpam-3383	357	43	b	b	NOUN
ejpam-3383	357	44	⊆m	⊆m	NOUN
ejpam-3383	357	45	.	.	PUNCT
ejpam-3383	358	1	corollary	corollary	ADJ
ejpam-3383	358	2	8.9	8.9	NUM
ejpam-3383	358	3	.	.	PUNCT
ejpam-3383	359	1	let	let	AUX
ejpam-3383	359	2	(	(	PUNCT
ejpam-3383	359	3	s	s	NOUN
ejpam-3383	359	4	,	,	PUNCT
ejpam-3383	359	5	◦	◦	NOUN
ejpam-3383	359	6	)	)	PUNCT
ejpam-3383	359	7	be	be	VERB
ejpam-3383	359	8	an	an	DET
ejpam-3383	359	9	hypersemigroup	hypersemigroup	NOUN
ejpam-3383	359	10	and	and	CCONJ
ejpam-3383	359	11	m	m	VERB
ejpam-3383	359	12	an	an	DET
ejpam-3383	359	13	ideal	ideal	NOUN
ejpam-3383	359	14	of	of	ADP
ejpam-3383	359	15	s.	s.	PROPN
ejpam-3383	359	16	if	if	SCONJ
ejpam-3383	359	17	m	m	NOUN
ejpam-3383	359	18	is	be	AUX
ejpam-3383	359	19	weakly	weakly	ADV
ejpam-3383	359	20	prime	prime	ADJ
ejpam-3383	359	21	then	then	ADV
ejpam-3383	359	22	,	,	PUNCT
ejpam-3383	359	23	for	for	ADP
ejpam-3383	359	24	every	every	DET
ejpam-3383	359	25	nonempty	nonempty	NOUN
ejpam-3383	359	26	subsets	subset	NOUN
ejpam-3383	359	27	a	a	DET
ejpam-3383	359	28	,	,	PUNCT
ejpam-3383	359	29	b	b	PROPN
ejpam-3383	359	30	of	of	ADP
ejpam-3383	359	31	s	s	PRON
ejpam-3383	359	32	such	such	ADJ
ejpam-3383	359	33	that	that	SCONJ
ejpam-3383	359	34	a	a	DET
ejpam-3383	359	35	∗	∗	NOUN
ejpam-3383	359	36	s	s	NOUN
ejpam-3383	359	37	∗b	∗b	NOUN
ejpam-3383	359	38	⊆m	⊆m	NOUN
ejpam-3383	359	39	,	,	PUNCT
ejpam-3383	359	40	we	we	PRON
ejpam-3383	359	41	have	have	VERB
ejpam-3383	359	42	a	a	DET
ejpam-3383	359	43	⊆m	⊆m	NOUN
ejpam-3383	359	44	or	or	CCONJ
ejpam-3383	359	45	b	b	NOUN
ejpam-3383	359	46	⊆m	⊆m	NOUN
ejpam-3383	359	47	.	.	PUNCT
ejpam-3383	360	1	in	in	ADP
ejpam-3383	360	2	addition	addition	NOUN
ejpam-3383	360	3	,	,	PUNCT
ejpam-3383	360	4	by	by	ADP
ejpam-3383	360	5	proposition	proposition	NOUN
ejpam-3383	360	6	8.1	8.1	NUM
ejpam-3383	360	7	or	or	CCONJ
ejpam-3383	360	8	proposition	proposition	VERB
ejpam-3383	360	9	8.2	8.2	NUM
ejpam-3383	360	10	and	and	CCONJ
ejpam-3383	360	11	proposition	proposition	NOUN
ejpam-3383	360	12	8.6	8.6	NUM
ejpam-3383	360	13	,	,	PUNCT
ejpam-3383	360	14	the	the	DET
ejpam-3383	360	15	following	follow	VERB
ejpam-3383	360	16	proposition	proposition	NOUN
ejpam-3383	360	17	holds	hold	VERB
ejpam-3383	360	18	.	.	PUNCT
ejpam-3383	361	1	proposition	proposition	NOUN
ejpam-3383	361	2	8.10	8.10	NUM
ejpam-3383	361	3	.	.	PUNCT
ejpam-3383	362	1	let	let	VERB
ejpam-3383	362	2	s	s	PRON
ejpam-3383	362	3	be	be	AUX
ejpam-3383	362	4	a	a	DET
ejpam-3383	362	5	∨e	∨e	NOUN
ejpam-3383	362	6	-	-	PUNCT
ejpam-3383	362	7	semigroup	semigroup	NOUN
ejpam-3383	362	8	and	and	CCONJ
ejpam-3383	362	9	m	m	VERB
ejpam-3383	362	10	an	an	DET
ejpam-3383	362	11	ideal	ideal	ADJ
ejpam-3383	362	12	element	element	NOUN
ejpam-3383	362	13	of	of	ADP
ejpam-3383	362	14	s.	s.	PROPN
ejpam-3383	362	15	the	the	DET
ejpam-3383	362	16	following	follow	VERB
ejpam-3383	362	17	are	be	AUX
ejpam-3383	362	18	equivalent	equivalent	ADJ
ejpam-3383	362	19	:	:	PUNCT
ejpam-3383	362	20	(	(	PUNCT
ejpam-3383	362	21	1	1	X
ejpam-3383	362	22	)	)	PUNCT
ejpam-3383	362	23	m	m	VERB
ejpam-3383	362	24	is	be	AUX
ejpam-3383	362	25	weakly	weakly	ADV
ejpam-3383	362	26	prime	prime	ADJ
ejpam-3383	362	27	.	.	PUNCT
ejpam-3383	363	1	(	(	PUNCT
ejpam-3383	363	2	2	2	X
ejpam-3383	363	3	)	)	PUNCT
ejpam-3383	363	4	for	for	ADP
ejpam-3383	363	5	every	every	DET
ejpam-3383	363	6	a	a	PROPN
ejpam-3383	363	7	,	,	PUNCT
ejpam-3383	363	8	b	b	X
ejpam-3383	363	9	∈	∈	NOUN
ejpam-3383	363	10	s	s	VERB
ejpam-3383	363	11	such	such	ADJ
ejpam-3383	363	12	that	that	DET
ejpam-3383	363	13	aeb	aeb	NOUN
ejpam-3383	363	14	≤	≤	NUM
ejpam-3383	363	15	m	m	ADP
ejpam-3383	363	16	,	,	PUNCT
ejpam-3383	363	17	we	we	PRON
ejpam-3383	363	18	have	have	VERB
ejpam-3383	363	19	a	a	DET
ejpam-3383	363	20	≤	≤	NUM
ejpam-3383	363	21	m	m	NOUN
ejpam-3383	363	22	or	or	CCONJ
ejpam-3383	363	23	b	b	PROPN
ejpam-3383	363	24	≤	≤	NUM
ejpam-3383	363	25	m.	m.	NOUN
ejpam-3383	363	26	clearly	clearly	ADV
ejpam-3383	363	27	,	,	PUNCT
ejpam-3383	363	28	the	the	DET
ejpam-3383	363	29	analogous	analogous	ADJ
ejpam-3383	363	30	for	for	ADP
ejpam-3383	363	31	semigroups	semigroup	NOUN
ejpam-3383	363	32	,	,	PUNCT
ejpam-3383	363	33	γ	γ	X
ejpam-3383	363	34	-	-	PUNCT
ejpam-3383	363	35	semigroups	semigroup	NOUN
ejpam-3383	363	36	and	and	CCONJ
ejpam-3383	363	37	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	363	38	also	also	ADV
ejpam-3383	363	39	holds	hold	VERB
ejpam-3383	363	40	.	.	PUNCT
ejpam-3383	364	1	an	an	DET
ejpam-3383	364	2	ideal	ideal	ADJ
ejpam-3383	364	3	m	m	NOUN
ejpam-3383	364	4	of	of	ADP
ejpam-3383	364	5	an	an	DET
ejpam-3383	364	6	hypersemigroup	hypersemigroup	NOUN
ejpam-3383	364	7	(	(	PUNCT
ejpam-3383	364	8	s	s	NOUN
ejpam-3383	364	9	,	,	PUNCT
ejpam-3383	364	10	◦	◦	NOUN
ejpam-3383	364	11	)	)	PUNCT
ejpam-3383	364	12	,	,	PUNCT
ejpam-3383	364	13	for	for	ADP
ejpam-3383	364	14	example	example	NOUN
ejpam-3383	364	15	is	be	AUX
ejpam-3383	364	16	weakly	weakly	ADV
ejpam-3383	364	17	prime	prime	ADJ
ejpam-3383	364	18	if	if	SCONJ
ejpam-3383	365	1	and	and	CCONJ
ejpam-3383	365	2	only	only	ADV
ejpam-3383	365	3	if	if	SCONJ
ejpam-3383	365	4	for	for	ADP
ejpam-3383	365	5	any	any	DET
ejpam-3383	365	6	nonempty	nonempty	ADJ
ejpam-3383	365	7	subsets	subset	NOUN
ejpam-3383	365	8	a	a	DET
ejpam-3383	365	9	,	,	PUNCT
ejpam-3383	365	10	b	b	PROPN
ejpam-3383	365	11	of	of	ADP
ejpam-3383	365	12	s	s	PRON
ejpam-3383	365	13	such	such	ADJ
ejpam-3383	365	14	that	that	SCONJ
ejpam-3383	365	15	a	a	DET
ejpam-3383	365	16	∗	∗	NOUN
ejpam-3383	365	17	s	s	NOUN
ejpam-3383	365	18	∗	∗	NOUN
ejpam-3383	365	19	b	b	NOUN
ejpam-3383	365	20	⊆	⊆	NUM
ejpam-3383	365	21	m	m	NOUN
ejpam-3383	365	22	,	,	PUNCT
ejpam-3383	365	23	we	we	PRON
ejpam-3383	365	24	have	have	VERB
ejpam-3383	365	25	a	a	DET
ejpam-3383	365	26	⊆	⊆	NUM
ejpam-3383	365	27	m	m	NOUN
ejpam-3383	365	28	or	or	CCONJ
ejpam-3383	365	29	b	b	NOUN
ejpam-3383	365	30	⊆	⊆	NUM
ejpam-3383	365	31	m	m	NOUN
ejpam-3383	365	32	.	.	PUNCT
ejpam-3383	366	1	this	this	PRON
ejpam-3383	366	2	is	be	AUX
ejpam-3383	366	3	equivalent	equivalent	ADJ
ejpam-3383	366	4	to	to	ADP
ejpam-3383	366	5	saying	say	VERB
ejpam-3383	366	6	that	that	SCONJ
ejpam-3383	366	7	for	for	ADP
ejpam-3383	366	8	any	any	DET
ejpam-3383	366	9	a	a	NOUN
ejpam-3383	366	10	,	,	PUNCT
ejpam-3383	366	11	b	b	X
ejpam-3383	366	12	∈	∈	NOUN
ejpam-3383	366	13	s	s	VERB
ejpam-3383	366	14	such	such	ADJ
ejpam-3383	366	15	that	that	SCONJ
ejpam-3383	366	16	a	a	DET
ejpam-3383	366	17	∗	∗	NOUN
ejpam-3383	366	18	s	s	NOUN
ejpam-3383	366	19	∗	∗	NOUN
ejpam-3383	366	20	b	b	NOUN
ejpam-3383	366	21	⊆m	⊆m	NOUN
ejpam-3383	366	22	we	we	PRON
ejpam-3383	366	23	have	have	VERB
ejpam-3383	366	24	a	a	DET
ejpam-3383	366	25	∈m	∈m	NOUN
ejpam-3383	366	26	or	or	CCONJ
ejpam-3383	366	27	b	b	NOUN
ejpam-3383	366	28	∈m	∈m	NOUN
ejpam-3383	366	29	.	.	PUNCT
ejpam-3383	367	1	in	in	ADP
ejpam-3383	367	2	the	the	DET
ejpam-3383	367	3	last	last	ADJ
ejpam-3383	367	4	characterization	characterization	NOUN
ejpam-3383	367	5	,	,	PUNCT
ejpam-3383	367	6	if	if	SCONJ
ejpam-3383	367	7	we	we	PRON
ejpam-3383	367	8	delete	delete	VERB
ejpam-3383	367	9	the	the	DET
ejpam-3383	367	10	operation	operation	NOUN
ejpam-3383	367	11	“	"	PUNCT
ejpam-3383	367	12	∗	∗	NOUN
ejpam-3383	367	13	”	"	PUNCT
ejpam-3383	367	14	and	and	CCONJ
ejpam-3383	367	15	put	put	VERB
ejpam-3383	367	16	“	"	PUNCT
ejpam-3383	367	17	·	·	PUNCT
ejpam-3383	367	18	”	"	PUNCT
ejpam-3383	367	19	instead	instead	ADV
ejpam-3383	367	20	,	,	PUNCT
ejpam-3383	367	21	then	then	ADV
ejpam-3383	367	22	this	this	PRON
ejpam-3383	367	23	is	be	AUX
ejpam-3383	367	24	the	the	DET
ejpam-3383	367	25	definition	definition	NOUN
ejpam-3383	367	26	of	of	ADP
ejpam-3383	367	27	a	a	DET
ejpam-3383	367	28	weakly	weakly	ADJ
ejpam-3383	367	29	prime	prime	ADJ
ejpam-3383	367	30	ideal	ideal	NOUN
ejpam-3383	367	31	of	of	ADP
ejpam-3383	367	32	a	a	DET
ejpam-3383	367	33	semigroup	semigroup	NOUN
ejpam-3383	367	34	given	give	VERB
ejpam-3383	367	35	by	by	ADP
ejpam-3383	367	36	petrich	petrich	NOUN
ejpam-3383	367	37	in	in	ADP
ejpam-3383	367	38	[	[	X
ejpam-3383	367	39	17	17	NUM
ejpam-3383	367	40	;	;	PUNCT
ejpam-3383	367	41	ii.3.1	ii.3.1	ADJ
ejpam-3383	367	42	definition	definition	NOUN
ejpam-3383	367	43	]	]	PUNCT
ejpam-3383	367	44	.	.	PUNCT
ejpam-3383	368	1	petrich	petrich	PROPN
ejpam-3383	368	2	uses	use	VERB
ejpam-3383	368	3	the	the	DET
ejpam-3383	368	4	terms	term	NOUN
ejpam-3383	368	5	“	"	PUNCT
ejpam-3383	368	6	prime	prime	ADJ
ejpam-3383	368	7	”	"	PUNCT
ejpam-3383	368	8	,	,	PUNCT
ejpam-3383	368	9	“	"	PUNCT
ejpam-3383	368	10	completely	completely	ADV
ejpam-3383	368	11	prime	prime	ADJ
ejpam-3383	368	12	”	"	PUNCT
ejpam-3383	368	13	while	while	SCONJ
ejpam-3383	368	14	we	we	PRON
ejpam-3383	368	15	use	use	VERB
ejpam-3383	368	16	the	the	DET
ejpam-3383	368	17	terms	term	NOUN
ejpam-3383	368	18	“	"	PUNCT
ejpam-3383	368	19	weakly	weakly	ADJ
ejpam-3383	368	20	prime	prime	NOUN
ejpam-3383	368	21	”	"	PUNCT
ejpam-3383	368	22	,	,	PUNCT
ejpam-3383	368	23	“	"	PUNCT
ejpam-3383	368	24	prime	prime	ADJ
ejpam-3383	368	25	”	"	PUNCT
ejpam-3383	368	26	.	.	PUNCT
ejpam-3383	369	1	in	in	ADP
ejpam-3383	369	2	the	the	DET
ejpam-3383	369	3	rest	rest	NOUN
ejpam-3383	369	4	of	of	ADP
ejpam-3383	369	5	this	this	DET
ejpam-3383	369	6	section	section	NOUN
ejpam-3383	369	7	,	,	PUNCT
ejpam-3383	369	8	we	we	PRON
ejpam-3383	369	9	consider	consider	VERB
ejpam-3383	369	10	the	the	DET
ejpam-3383	369	11	case	case	NOUN
ejpam-3383	369	12	of	of	ADP
ejpam-3383	369	13	ordered	order	VERB
ejpam-3383	369	14	semigroups	semigroup	NOUN
ejpam-3383	369	15	,	,	PUNCT
ejpam-3383	369	16	ordered	order	VERB
ejpam-3383	369	17	γsemigroups	γsemigroup	NOUN
ejpam-3383	369	18	and	and	CCONJ
ejpam-3383	369	19	ordered	order	VERB
ejpam-3383	369	20	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	369	21	.	.	PUNCT
ejpam-3383	370	1	proposition	proposition	NOUN
ejpam-3383	370	2	8.11	8.11	NUM
ejpam-3383	370	3	.	.	PUNCT
ejpam-3383	371	1	let	let	VERB
ejpam-3383	371	2	s	s	PRON
ejpam-3383	371	3	be	be	AUX
ejpam-3383	371	4	an	an	DET
ejpam-3383	371	5	ordered	order	VERB
ejpam-3383	371	6	semigroup	semigroup	NOUN
ejpam-3383	371	7	and	and	CCONJ
ejpam-3383	371	8	m	m	AUX
ejpam-3383	371	9	be	be	AUX
ejpam-3383	371	10	a	a	DET
ejpam-3383	371	11	subset	subset	NOUN
ejpam-3383	371	12	of	of	ADP
ejpam-3383	371	13	s	s	PRON
ejpam-3383	371	14	such	such	ADJ
ejpam-3383	371	15	that	that	SCONJ
ejpam-3383	371	16	m	m	VERB
ejpam-3383	371	17	=	=	SYM
ejpam-3383	371	18	(	(	PUNCT
ejpam-3383	371	19	m	m	PROPN
ejpam-3383	371	20	]	]	X
ejpam-3383	371	21	.	.	PUNCT
ejpam-3383	372	1	suppose	suppose	VERB
ejpam-3383	372	2	that	that	SCONJ
ejpam-3383	372	3	for	for	ADP
ejpam-3383	372	4	any	any	DET
ejpam-3383	372	5	right	right	ADJ
ejpam-3383	372	6	ideal	ideal	NOUN
ejpam-3383	372	7	a	a	PRON
ejpam-3383	372	8	of	of	ADP
ejpam-3383	372	9	s	s	PRON
ejpam-3383	372	10	and	and	CCONJ
ejpam-3383	372	11	any	any	DET
ejpam-3383	372	12	b	b	NOUN
ejpam-3383	372	13	⊆	⊆	NUM
ejpam-3383	372	14	s	s	NOUN
ejpam-3383	372	15	,	,	PUNCT
ejpam-3383	372	16	(	(	PUNCT
ejpam-3383	372	17	asb	asb	PROPN
ejpam-3383	372	18	]	]	X
ejpam-3383	372	19	⊆	⊆	NUM
ejpam-3383	372	20	m	m	NOUN
ejpam-3383	372	21	implies	imply	VERB
ejpam-3383	372	22	a	a	DET
ejpam-3383	372	23	⊆m	⊆m	NOUN
ejpam-3383	372	24	or	or	CCONJ
ejpam-3383	372	25	b	b	NOUN
ejpam-3383	372	26	⊆m	⊆m	NOUN
ejpam-3383	372	27	.	.	PUNCT
ejpam-3383	373	1	then	then	ADV
ejpam-3383	373	2	m	m	PROPN
ejpam-3383	373	3	is	be	AUX
ejpam-3383	373	4	weakly	weakly	ADV
ejpam-3383	373	5	prime	prime	ADJ
ejpam-3383	373	6	.	.	PUNCT
ejpam-3383	374	1	proof	proof	NOUN
ejpam-3383	374	2	.	.	PUNCT
ejpam-3383	375	1	let	let	VERB
ejpam-3383	375	2	a	a	DET
ejpam-3383	375	3	,	,	PUNCT
ejpam-3383	375	4	b	b	NOUN
ejpam-3383	375	5	be	be	AUX
ejpam-3383	375	6	ideals	ideal	NOUN
ejpam-3383	375	7	of	of	ADP
ejpam-3383	375	8	s	s	PRON
ejpam-3383	375	9	such	such	ADJ
ejpam-3383	375	10	that	that	SCONJ
ejpam-3383	375	11	ab	ab	PROPN
ejpam-3383	375	12	⊆	⊆	NUM
ejpam-3383	375	13	m	m	NOUN
ejpam-3383	375	14	.	.	PUNCT
ejpam-3383	376	1	then	then	ADV
ejpam-3383	376	2	(	(	PUNCT
ejpam-3383	376	3	asb	asb	PROPN
ejpam-3383	376	4	]	]	X
ejpam-3383	376	5	⊆	⊆	NUM
ejpam-3383	376	6	(	(	PUNCT
ejpam-3383	376	7	ab	ab	X
ejpam-3383	376	8	]	]	X
ejpam-3383	376	9	⊆	⊆	NUM
ejpam-3383	376	10	(	(	PUNCT
ejpam-3383	376	11	m	m	NOUN
ejpam-3383	376	12	]	]	X
ejpam-3383	377	1	=	=	PUNCT
ejpam-3383	377	2	m	m	VERB
ejpam-3383	377	3	.	.	PUNCT
ejpam-3383	378	1	since	since	SCONJ
ejpam-3383	378	2	a	a	PRON
ejpam-3383	378	3	is	be	AUX
ejpam-3383	378	4	a	a	DET
ejpam-3383	378	5	right	right	ADJ
ejpam-3383	378	6	ideal	ideal	NOUN
ejpam-3383	378	7	of	of	ADP
ejpam-3383	378	8	s	s	PROPN
ejpam-3383	378	9	,	,	PUNCT
ejpam-3383	378	10	b	b	PROPN
ejpam-3383	378	11	⊆	⊆	NUM
ejpam-3383	378	12	s	s	ADP
ejpam-3383	378	13	and	and	CCONJ
ejpam-3383	378	14	(	(	PUNCT
ejpam-3383	378	15	asb	asb	PROPN
ejpam-3383	378	16	]	]	X
ejpam-3383	378	17	⊆	⊆	NUM
ejpam-3383	378	18	m	m	NOUN
ejpam-3383	378	19	,	,	PUNCT
ejpam-3383	378	20	by	by	ADP
ejpam-3383	378	21	hypothesis	hypothesis	NOUN
ejpam-3383	378	22	,	,	PUNCT
ejpam-3383	378	23	we	we	PRON
ejpam-3383	378	24	have	have	VERB
ejpam-3383	378	25	a	a	DET
ejpam-3383	378	26	⊆	⊆	NUM
ejpam-3383	378	27	m	m	NOUN
ejpam-3383	378	28	or	or	CCONJ
ejpam-3383	378	29	b	b	NOUN
ejpam-3383	378	30	⊆m	⊆m	NOUN
ejpam-3383	378	31	,	,	PUNCT
ejpam-3383	378	32	thus	thus	ADV
ejpam-3383	378	33	m	m	NOUN
ejpam-3383	378	34	is	be	AUX
ejpam-3383	378	35	weakly	weakly	ADV
ejpam-3383	378	36	prime	prime	ADJ
ejpam-3383	378	37	.	.	PUNCT
ejpam-3383	379	1	�	�	PROPN
ejpam-3383	379	2	in	in	ADP
ejpam-3383	379	3	a	a	DET
ejpam-3383	379	4	similar	similar	ADJ
ejpam-3383	379	5	way	way	NOUN
ejpam-3383	379	6	the	the	DET
ejpam-3383	379	7	following	follow	VERB
ejpam-3383	379	8	proposition	proposition	NOUN
ejpam-3383	379	9	holds	hold	VERB
ejpam-3383	379	10	.	.	PUNCT
ejpam-3383	380	1	proposition	proposition	NOUN
ejpam-3383	380	2	8.12	8.12	NUM
ejpam-3383	380	3	.	.	PUNCT
ejpam-3383	381	1	let	let	VERB
ejpam-3383	381	2	s	s	PRON
ejpam-3383	381	3	be	be	AUX
ejpam-3383	381	4	an	an	DET
ejpam-3383	381	5	ordered	order	VERB
ejpam-3383	381	6	semigroup	semigroup	NOUN
ejpam-3383	381	7	and	and	CCONJ
ejpam-3383	381	8	m	m	AUX
ejpam-3383	381	9	be	be	AUX
ejpam-3383	381	10	a	a	DET
ejpam-3383	381	11	subset	subset	NOUN
ejpam-3383	381	12	of	of	ADP
ejpam-3383	381	13	s	s	PRON
ejpam-3383	381	14	such	such	ADJ
ejpam-3383	381	15	that	that	SCONJ
ejpam-3383	381	16	m	m	VERB
ejpam-3383	381	17	=	=	SYM
ejpam-3383	381	18	(	(	PUNCT
ejpam-3383	381	19	m	m	PROPN
ejpam-3383	381	20	]	]	X
ejpam-3383	381	21	.	.	PUNCT
ejpam-3383	382	1	suppose	suppose	VERB
ejpam-3383	382	2	that	that	SCONJ
ejpam-3383	382	3	for	for	ADP
ejpam-3383	382	4	any	any	DET
ejpam-3383	382	5	left	left	ADJ
ejpam-3383	382	6	ideal	ideal	NOUN
ejpam-3383	382	7	b	b	PROPN
ejpam-3383	382	8	of	of	ADP
ejpam-3383	382	9	s	s	PRON
ejpam-3383	382	10	and	and	CCONJ
ejpam-3383	382	11	any	any	DET
ejpam-3383	382	12	a	a	DET
ejpam-3383	382	13	⊆	⊆	NUM
ejpam-3383	382	14	s	s	NOUN
ejpam-3383	382	15	,	,	PUNCT
ejpam-3383	382	16	(	(	PUNCT
ejpam-3383	382	17	asb	asb	PROPN
ejpam-3383	382	18	]	]	X
ejpam-3383	382	19	⊆	⊆	NUM
ejpam-3383	382	20	m	m	NOUN
ejpam-3383	382	21	implies	imply	VERB
ejpam-3383	382	22	a	a	DET
ejpam-3383	382	23	⊆m	⊆m	NOUN
ejpam-3383	382	24	or	or	CCONJ
ejpam-3383	382	25	b	b	NOUN
ejpam-3383	382	26	⊆m	⊆m	NOUN
ejpam-3383	382	27	.	.	PUNCT
ejpam-3383	383	1	then	then	ADV
ejpam-3383	383	2	m	m	PROPN
ejpam-3383	383	3	is	be	AUX
ejpam-3383	383	4	weakly	weakly	ADV
ejpam-3383	383	5	prime	prime	ADJ
ejpam-3383	383	6	.	.	PUNCT
ejpam-3383	384	1	n.	n.	PROPN
ejpam-3383	384	2	kehayopulu	kehayopulu	PROPN
ejpam-3383	384	3	/	/	SYM
ejpam-3383	384	4	eur	eur	PROPN
ejpam-3383	384	5	.	.	PUNCT
ejpam-3383	385	1	j.	j.	PROPN
ejpam-3383	385	2	pure	pure	PROPN
ejpam-3383	385	3	appl	appl	PROPN
ejpam-3383	385	4	.	.	PROPN
ejpam-3383	385	5	math	math	PROPN
ejpam-3383	385	6	,	,	PUNCT
ejpam-3383	385	7	12	12	NUM
ejpam-3383	385	8	(	(	PUNCT
ejpam-3383	385	9	1	1	NUM
ejpam-3383	385	10	)	)	PUNCT
ejpam-3383	385	11	(	(	PUNCT
ejpam-3383	385	12	2019	2019	NUM
ejpam-3383	385	13	)	)	PUNCT
ejpam-3383	385	14	,	,	PUNCT
ejpam-3383	385	15	208	208	NUM
ejpam-3383	385	16	-	-	SYM
ejpam-3383	385	17	225	225	NUM
ejpam-3383	385	18	221	221	NUM
ejpam-3383	385	19	proposition	proposition	NOUN
ejpam-3383	385	20	8.13	8.13	NUM
ejpam-3383	385	21	.	.	PUNCT
ejpam-3383	386	1	let	let	VERB
ejpam-3383	386	2	s	s	PRON
ejpam-3383	386	3	be	be	AUX
ejpam-3383	386	4	an	an	DET
ejpam-3383	386	5	ordered	order	VERB
ejpam-3383	386	6	semigroup	semigroup	NOUN
ejpam-3383	386	7	and	and	CCONJ
ejpam-3383	386	8	m	m	AUX
ejpam-3383	386	9	be	be	VERB
ejpam-3383	386	10	an	an	DET
ejpam-3383	386	11	ideal	ideal	NOUN
ejpam-3383	386	12	of	of	ADP
ejpam-3383	386	13	s.	s.	PROPN
ejpam-3383	386	14	suppose	suppose	VERB
ejpam-3383	386	15	that	that	SCONJ
ejpam-3383	386	16	for	for	ADP
ejpam-3383	386	17	any	any	DET
ejpam-3383	386	18	right	right	ADJ
ejpam-3383	386	19	ideal	ideal	NOUN
ejpam-3383	386	20	a	a	PRON
ejpam-3383	386	21	of	of	ADP
ejpam-3383	386	22	s	s	PRON
ejpam-3383	386	23	and	and	CCONJ
ejpam-3383	386	24	any	any	DET
ejpam-3383	386	25	b	b	NOUN
ejpam-3383	386	26	⊆	⊆	NUM
ejpam-3383	386	27	s	s	NOUN
ejpam-3383	386	28	,	,	PUNCT
ejpam-3383	386	29	asb	asb	PROPN
ejpam-3383	386	30	⊆	⊆	NUM
ejpam-3383	386	31	m	m	NOUN
ejpam-3383	386	32	implies	imply	VERB
ejpam-3383	386	33	a	a	DET
ejpam-3383	386	34	⊆	⊆	NUM
ejpam-3383	386	35	m	m	NOUN
ejpam-3383	386	36	or	or	CCONJ
ejpam-3383	386	37	b	b	NOUN
ejpam-3383	386	38	⊆	⊆	NUM
ejpam-3383	386	39	m	m	NOUN
ejpam-3383	386	40	.	.	PUNCT
ejpam-3383	387	1	then	then	ADV
ejpam-3383	387	2	m	m	VERB
ejpam-3383	387	3	is	be	AUX
ejpam-3383	387	4	weakly	weakly	ADV
ejpam-3383	387	5	prime	prime	ADJ
ejpam-3383	387	6	.	.	PUNCT
ejpam-3383	388	1	proposition	proposition	NOUN
ejpam-3383	388	2	8.14	8.14	NUM
ejpam-3383	388	3	.	.	PUNCT
ejpam-3383	389	1	let	let	VERB
ejpam-3383	389	2	s	s	PRON
ejpam-3383	389	3	be	be	AUX
ejpam-3383	389	4	an	an	DET
ejpam-3383	389	5	ordered	order	VERB
ejpam-3383	389	6	semigroup	semigroup	NOUN
ejpam-3383	389	7	and	and	CCONJ
ejpam-3383	389	8	m	m	AUX
ejpam-3383	389	9	be	be	VERB
ejpam-3383	389	10	an	an	DET
ejpam-3383	389	11	ideal	ideal	NOUN
ejpam-3383	389	12	of	of	ADP
ejpam-3383	389	13	s.	s.	PROPN
ejpam-3383	389	14	suppose	suppose	VERB
ejpam-3383	389	15	that	that	SCONJ
ejpam-3383	389	16	for	for	ADP
ejpam-3383	389	17	any	any	DET
ejpam-3383	389	18	left	left	ADJ
ejpam-3383	389	19	ideal	ideal	NOUN
ejpam-3383	389	20	b	b	PROPN
ejpam-3383	389	21	of	of	ADP
ejpam-3383	389	22	s	s	PRON
ejpam-3383	389	23	and	and	CCONJ
ejpam-3383	389	24	any	any	DET
ejpam-3383	389	25	a	a	DET
ejpam-3383	389	26	⊆	⊆	NUM
ejpam-3383	389	27	s	s	NOUN
ejpam-3383	389	28	,	,	PUNCT
ejpam-3383	389	29	asb	asb	PROPN
ejpam-3383	389	30	⊆m	⊆m	PROPN
ejpam-3383	389	31	implies	imply	VERB
ejpam-3383	389	32	a	a	DET
ejpam-3383	389	33	⊆m	⊆m	NOUN
ejpam-3383	389	34	or	or	CCONJ
ejpam-3383	389	35	b	b	NOUN
ejpam-3383	389	36	⊆m	⊆m	NOUN
ejpam-3383	389	37	.	.	PUNCT
ejpam-3383	390	1	then	then	ADV
ejpam-3383	390	2	m	m	PROPN
ejpam-3383	390	3	is	be	AUX
ejpam-3383	390	4	weakly	weakly	ADV
ejpam-3383	390	5	prime	prime	ADJ
ejpam-3383	390	6	.	.	PUNCT
ejpam-3383	391	1	the	the	DET
ejpam-3383	391	2	proof	proof	NOUN
ejpam-3383	391	3	of	of	ADP
ejpam-3383	391	4	the	the	DET
ejpam-3383	391	5	last	last	ADJ
ejpam-3383	391	6	two	two	NUM
ejpam-3383	391	7	propositions	proposition	NOUN
ejpam-3383	391	8	mimics	mimic	VERB
ejpam-3383	391	9	the	the	DET
ejpam-3383	391	10	proof	proof	NOUN
ejpam-3383	391	11	of	of	ADP
ejpam-3383	391	12	propositions	proposition	NOUN
ejpam-3383	391	13	8.11	8.11	NUM
ejpam-3383	391	14	and	and	CCONJ
ejpam-3383	391	15	8.12	8.12	NUM
ejpam-3383	391	16	;	;	PUNCT
ejpam-3383	391	17	however	however	ADV
ejpam-3383	391	18	they	they	PRON
ejpam-3383	391	19	can	can	AUX
ejpam-3383	391	20	be	be	AUX
ejpam-3383	391	21	also	also	ADV
ejpam-3383	391	22	obtained	obtain	VERB
ejpam-3383	391	23	as	as	ADP
ejpam-3383	391	24	corollaries	corollary	NOUN
ejpam-3383	391	25	to	to	ADP
ejpam-3383	391	26	propositions	proposition	NOUN
ejpam-3383	391	27	8.11	8.11	NUM
ejpam-3383	391	28	and	and	CCONJ
ejpam-3383	391	29	8.12	8.12	NUM
ejpam-3383	391	30	.	.	PUNCT
ejpam-3383	392	1	proposition	proposition	NOUN
ejpam-3383	392	2	8.15	8.15	NUM
ejpam-3383	392	3	.	.	PUNCT
ejpam-3383	393	1	let	let	VERB
ejpam-3383	393	2	s	s	PRON
ejpam-3383	393	3	be	be	AUX
ejpam-3383	393	4	an	an	DET
ejpam-3383	393	5	ordered	order	VERB
ejpam-3383	393	6	semigroup	semigroup	NOUN
ejpam-3383	393	7	and	and	CCONJ
ejpam-3383	393	8	m	m	VERB
ejpam-3383	393	9	an	an	DET
ejpam-3383	393	10	ideal	ideal	NOUN
ejpam-3383	393	11	of	of	ADP
ejpam-3383	393	12	s.	s.	PROPN
ejpam-3383	393	13	if	if	SCONJ
ejpam-3383	393	14	m	m	NOUN
ejpam-3383	393	15	is	be	AUX
ejpam-3383	393	16	weakly	weakly	ADV
ejpam-3383	393	17	prime	prime	ADJ
ejpam-3383	393	18	then	then	ADV
ejpam-3383	393	19	,	,	PUNCT
ejpam-3383	393	20	for	for	ADP
ejpam-3383	393	21	any	any	DET
ejpam-3383	393	22	a	a	PRON
ejpam-3383	393	23	,	,	PUNCT
ejpam-3383	393	24	b	b	NOUN
ejpam-3383	394	1	⊆	⊆	NUM
ejpam-3383	394	2	s	s	ADP
ejpam-3383	394	3	such	such	ADJ
ejpam-3383	394	4	that	that	SCONJ
ejpam-3383	394	5	(	(	PUNCT
ejpam-3383	394	6	asb	asb	PROPN
ejpam-3383	394	7	]	]	X
ejpam-3383	394	8	⊆m	⊆m	NOUN
ejpam-3383	394	9	,	,	PUNCT
ejpam-3383	394	10	we	we	PRON
ejpam-3383	394	11	have	have	VERB
ejpam-3383	394	12	a	a	DET
ejpam-3383	394	13	⊆m	⊆m	NOUN
ejpam-3383	394	14	or	or	CCONJ
ejpam-3383	394	15	b	b	NOUN
ejpam-3383	394	16	⊆m	⊆m	NOUN
ejpam-3383	394	17	.	.	PUNCT
ejpam-3383	395	1	proof	proof	NOUN
ejpam-3383	395	2	.	.	PUNCT
ejpam-3383	396	1	for	for	ADP
ejpam-3383	396	2	a	a	DET
ejpam-3383	396	3	=	=	SYM
ejpam-3383	396	4	∅	∅	NOUN
ejpam-3383	396	5	or	or	CCONJ
ejpam-3383	396	6	b	b	NOUN
ejpam-3383	396	7	=	=	SYM
ejpam-3383	396	8	∅	∅	NOUN
ejpam-3383	396	9	,	,	PUNCT
ejpam-3383	396	10	the	the	DET
ejpam-3383	396	11	proposition	proposition	NOUN
ejpam-3383	396	12	holds	hold	VERB
ejpam-3383	396	13	.	.	PUNCT
ejpam-3383	397	1	let	let	VERB
ejpam-3383	397	2	a	a	DET
ejpam-3383	397	3	,	,	PUNCT
ejpam-3383	397	4	b	b	NOUN
ejpam-3383	397	5	nonempty	nonempty	ADJ
ejpam-3383	397	6	subsets	subset	NOUN
ejpam-3383	397	7	of	of	ADP
ejpam-3383	397	8	s	s	PRON
ejpam-3383	397	9	such	such	ADJ
ejpam-3383	397	10	that	that	SCONJ
ejpam-3383	397	11	(	(	PUNCT
ejpam-3383	397	12	asb	asb	PROPN
ejpam-3383	397	13	]	]	X
ejpam-3383	397	14	⊆m	⊆m	NOUN
ejpam-3383	397	15	.	.	PUNCT
ejpam-3383	398	1	then	then	ADV
ejpam-3383	398	2	we	we	PRON
ejpam-3383	398	3	have	have	VERB
ejpam-3383	398	4	(	(	PUNCT
ejpam-3383	398	5	sas](sbs	sas](sbs	VERB
ejpam-3383	398	6	]	]	X
ejpam-3383	398	7	⊆	⊆	NUM
ejpam-3383	398	8	(	(	PUNCT
ejpam-3383	398	9	sas2bs	sas2bs	X
ejpam-3383	398	10	]	]	X
ejpam-3383	398	11	⊆	⊆	NUM
ejpam-3383	398	12	(	(	PUNCT
ejpam-3383	398	13	s(asb)s	s(asb)s	NOUN
ejpam-3383	398	14	]	]	X
ejpam-3383	398	15	=	=	SYM
ejpam-3383	398	16	(	(	PUNCT
ejpam-3383	398	17	s(asb]s	s(asb]s	PROPN
ejpam-3383	398	18	]	]	X
ejpam-3383	398	19	⊆	⊆	NUM
ejpam-3383	398	20	(	(	PUNCT
ejpam-3383	398	21	sms	sms	PROPN
ejpam-3383	398	22	]	]	PUNCT
ejpam-3383	398	23	⊆	⊆	NUM
ejpam-3383	398	24	(	(	PUNCT
ejpam-3383	398	25	m	m	NOUN
ejpam-3383	398	26	]	]	X
ejpam-3383	398	27	=	=	PUNCT
ejpam-3383	398	28	m.	m.	NOUN
ejpam-3383	398	29	since	since	SCONJ
ejpam-3383	398	30	(	(	PUNCT
ejpam-3383	398	31	sas	sas	PROPN
ejpam-3383	398	32	]	]	PUNCT
ejpam-3383	398	33	,	,	PUNCT
ejpam-3383	398	34	(	(	PUNCT
ejpam-3383	398	35	sbs	sbs	PROPN
ejpam-3383	398	36	]	]	PUNCT
ejpam-3383	398	37	are	be	AUX
ejpam-3383	398	38	ideals	ideal	NOUN
ejpam-3383	398	39	of	of	ADP
ejpam-3383	398	40	s	s	PRON
ejpam-3383	398	41	and	and	CCONJ
ejpam-3383	398	42	m	m	PROPN
ejpam-3383	398	43	is	be	AUX
ejpam-3383	398	44	weakly	weakly	ADV
ejpam-3383	398	45	prime	prime	ADJ
ejpam-3383	398	46	,	,	PUNCT
ejpam-3383	398	47	we	we	PRON
ejpam-3383	398	48	have	have	AUX
ejpam-3383	398	49	(	(	PUNCT
ejpam-3383	398	50	sas	sas	PROPN
ejpam-3383	398	51	]	]	X
ejpam-3383	398	52	⊆	⊆	NUM
ejpam-3383	398	53	m	m	NOUN
ejpam-3383	398	54	or	or	CCONJ
ejpam-3383	398	55	(	(	PUNCT
ejpam-3383	398	56	sbs	sbs	NOUN
ejpam-3383	398	57	]	]	PUNCT
ejpam-3383	398	58	⊆m	⊆m	NOUN
ejpam-3383	398	59	.	.	PUNCT
ejpam-3383	399	1	let	let	VERB
ejpam-3383	399	2	(	(	PUNCT
ejpam-3383	399	3	sas	sas	VERB
ejpam-3383	399	4	]	]	PUNCT
ejpam-3383	399	5	⊆m	⊆m	NOUN
ejpam-3383	399	6	.	.	PUNCT
ejpam-3383	400	1	then	then	ADV
ejpam-3383	400	2	i(a)3	i(a)3	ADJ
ejpam-3383	400	3	=	=	PUNCT
ejpam-3383	400	4	(	(	PUNCT
ejpam-3383	400	5	a	a	DET
ejpam-3383	400	6	∪	∪	NOUN
ejpam-3383	400	7	sa	sa	NOUN
ejpam-3383	400	8	∪as	∪as	NOUN
ejpam-3383	400	9	∪	∪	ADJ
ejpam-3383	400	10	sas]2(a	sas]2(a	NOUN
ejpam-3383	400	11	∪	∪	NOUN
ejpam-3383	400	12	sa	sa	NOUN
ejpam-3383	400	13	∪as	∪as	NOUN
ejpam-3383	400	14	∪	∪	ADJ
ejpam-3383	400	15	sas	sas	X
ejpam-3383	400	16	]	]	X
ejpam-3383	400	17	⊆	⊆	NUM
ejpam-3383	400	18	(	(	PUNCT
ejpam-3383	400	19	sa	sa	NOUN
ejpam-3383	400	20	∪	∪	ADP
ejpam-3383	400	21	sas](a	sas](a	PROPN
ejpam-3383	400	22	∪	∪	NOUN
ejpam-3383	400	23	sa	sa	NOUN
ejpam-3383	400	24	∪as	∪as	NOUN
ejpam-3383	400	25	∪	∪	ADJ
ejpam-3383	400	26	sas	sas	X
ejpam-3383	400	27	]	]	X
ejpam-3383	400	28	⊆	⊆	NUM
ejpam-3383	400	29	(	(	PUNCT
ejpam-3383	400	30	(	(	PUNCT
ejpam-3383	400	31	sa	sa	PROPN
ejpam-3383	400	32	∪	∪	PROPN
ejpam-3383	400	33	sas)(a	sas)(a	PROPN
ejpam-3383	400	34	∪	∪	PROPN
ejpam-3383	400	35	sa	sa	PROPN
ejpam-3383	400	36	∪as	∪as	NOUN
ejpam-3383	400	37	∪	∪	ADJ
ejpam-3383	400	38	sas	sas	PROPN
ejpam-3383	400	39	)	)	PUNCT
ejpam-3383	400	40	]	]	PUNCT
ejpam-3383	401	1	⊆	⊆	NUM
ejpam-3383	401	2	(	(	PUNCT
ejpam-3383	401	3	sas	sas	PROPN
ejpam-3383	401	4	]	]	PUNCT
ejpam-3383	401	5	⊆m	⊆m	NOUN
ejpam-3383	401	6	.	.	PUNCT
ejpam-3383	402	1	then	then	ADV
ejpam-3383	402	2	we	we	PRON
ejpam-3383	402	3	have	have	VERB
ejpam-3383	402	4	(	(	PUNCT
ejpam-3383	402	5	i(a)2]i(a	i(a)2]i(a	ADJ
ejpam-3383	402	6	)	)	PUNCT
ejpam-3383	402	7	=	=	SYM
ejpam-3383	402	8	(	(	PUNCT
ejpam-3383	402	9	i(a)2](i(a	i(a)2](i(a	ADJ
ejpam-3383	402	10	)	)	PUNCT
ejpam-3383	402	11	]	]	PUNCT
ejpam-3383	403	1	⊆	⊆	X
ejpam-3383	403	2	(	(	PUNCT
ejpam-3383	403	3	i(a)3	i(a)3	PROPN
ejpam-3383	403	4	]	]	X
ejpam-3383	403	5	⊆	⊆	NUM
ejpam-3383	403	6	(	(	PUNCT
ejpam-3383	403	7	m	m	NOUN
ejpam-3383	403	8	]	]	X
ejpam-3383	403	9	=	=	PUNCT
ejpam-3383	403	10	m.	m.	NOUN
ejpam-3383	403	11	since	since	SCONJ
ejpam-3383	403	12	i(a	i(a	PROPN
ejpam-3383	403	13	)	)	PUNCT
ejpam-3383	403	14	is	be	AUX
ejpam-3383	403	15	an	an	DET
ejpam-3383	403	16	ideal	ideal	NOUN
ejpam-3383	403	17	of	of	ADP
ejpam-3383	403	18	s	s	PROPN
ejpam-3383	403	19	,	,	PUNCT
ejpam-3383	403	20	(	(	PUNCT
ejpam-3383	403	21	i(a)2	i(a)2	PROPN
ejpam-3383	403	22	]	]	PUNCT
ejpam-3383	403	23	is	be	AUX
ejpam-3383	403	24	an	an	DET
ejpam-3383	403	25	ideal	ideal	NOUN
ejpam-3383	403	26	of	of	ADP
ejpam-3383	403	27	s	s	PRON
ejpam-3383	403	28	as	as	ADV
ejpam-3383	403	29	well	well	ADV
ejpam-3383	403	30	(	(	PUNCT
ejpam-3383	403	31	in	in	ADP
ejpam-3383	403	32	general	general	ADJ
ejpam-3383	403	33	,	,	PUNCT
ejpam-3383	403	34	a	a	PRON
ejpam-3383	403	35	,	,	PUNCT
ejpam-3383	403	36	b	b	PROPN
ejpam-3383	403	37	ideals⇒	ideals⇒	PROPN
ejpam-3383	403	38	(	(	PUNCT
ejpam-3383	403	39	ab	ab	PROPN
ejpam-3383	403	40	]	]	X
ejpam-3383	403	41	ideal	ideal	NOUN
ejpam-3383	403	42	)	)	PUNCT
ejpam-3383	403	43	.	.	PUNCT
ejpam-3383	404	1	since	since	SCONJ
ejpam-3383	404	2	(	(	PUNCT
ejpam-3383	404	3	i(a)2	i(a)2	PROPN
ejpam-3383	404	4	]	]	PUNCT
ejpam-3383	404	5	,	,	PUNCT
ejpam-3383	404	6	i(a	i(a	PROPN
ejpam-3383	404	7	)	)	PUNCT
ejpam-3383	404	8	are	be	AUX
ejpam-3383	404	9	ideals	ideal	NOUN
ejpam-3383	404	10	of	of	ADP
ejpam-3383	404	11	s	s	PRON
ejpam-3383	404	12	and	and	CCONJ
ejpam-3383	404	13	m	m	PROPN
ejpam-3383	404	14	is	be	AUX
ejpam-3383	404	15	weakly	weakly	ADV
ejpam-3383	404	16	prime	prime	ADJ
ejpam-3383	404	17	,	,	PUNCT
ejpam-3383	404	18	we	we	PRON
ejpam-3383	404	19	have	have	VERB
ejpam-3383	404	20	(	(	PUNCT
ejpam-3383	404	21	i(a)2	i(a)2	PROPN
ejpam-3383	404	22	]	]	X
ejpam-3383	404	23	⊆	⊆	NUM
ejpam-3383	404	24	m	m	NOUN
ejpam-3383	404	25	or	or	CCONJ
ejpam-3383	404	26	i(a	i(a	NOUN
ejpam-3383	404	27	)	)	PUNCT
ejpam-3383	404	28	⊆	⊆	NUM
ejpam-3383	404	29	m	m	NOUN
ejpam-3383	404	30	.	.	PUNCT
ejpam-3383	405	1	if	if	SCONJ
ejpam-3383	405	2	i(a	i(a	NOUN
ejpam-3383	405	3	)	)	PUNCT
ejpam-3383	405	4	⊆	⊆	NUM
ejpam-3383	405	5	m	m	NOUN
ejpam-3383	405	6	,	,	PUNCT
ejpam-3383	405	7	then	then	ADV
ejpam-3383	405	8	a	a	DET
ejpam-3383	405	9	⊆	⊆	NUM
ejpam-3383	405	10	m	m	NOUN
ejpam-3383	405	11	.	.	PUNCT
ejpam-3383	406	1	if	if	SCONJ
ejpam-3383	406	2	(	(	PUNCT
ejpam-3383	406	3	i(a)2	i(a)2	PROPN
ejpam-3383	406	4	]	]	X
ejpam-3383	406	5	⊆	⊆	NUM
ejpam-3383	406	6	m	m	NOUN
ejpam-3383	406	7	,	,	PUNCT
ejpam-3383	406	8	then	then	ADV
ejpam-3383	406	9	i(a)2	i(a)2	ADP
ejpam-3383	406	10	⊆	⊆	NUM
ejpam-3383	406	11	(	(	PUNCT
ejpam-3383	406	12	i(a)2	i(a)2	PROPN
ejpam-3383	406	13	]	]	PUNCT
ejpam-3383	406	14	⊆	⊆	NUM
ejpam-3383	406	15	m	m	NOUN
ejpam-3383	406	16	.	.	PUNCT
ejpam-3383	407	1	since	since	SCONJ
ejpam-3383	407	2	i(a	i(a	PROPN
ejpam-3383	407	3	)	)	PUNCT
ejpam-3383	407	4	is	be	AUX
ejpam-3383	407	5	an	an	DET
ejpam-3383	407	6	ideal	ideal	NOUN
ejpam-3383	407	7	of	of	ADP
ejpam-3383	407	8	s	s	PRON
ejpam-3383	407	9	and	and	CCONJ
ejpam-3383	407	10	m	m	PROPN
ejpam-3383	407	11	is	be	AUX
ejpam-3383	407	12	weakly	weakly	ADV
ejpam-3383	407	13	prime	prime	ADJ
ejpam-3383	407	14	,	,	PUNCT
ejpam-3383	407	15	we	we	PRON
ejpam-3383	407	16	have	have	VERB
ejpam-3383	407	17	i(a	i(a	PROPN
ejpam-3383	407	18	)	)	PUNCT
ejpam-3383	407	19	⊆m	⊆m	NOUN
ejpam-3383	407	20	and	and	CCONJ
ejpam-3383	407	21	again	again	ADV
ejpam-3383	407	22	a	a	DET
ejpam-3383	407	23	⊆m	⊆m	NOUN
ejpam-3383	407	24	.	.	PUNCT
ejpam-3383	408	1	�	�	PROPN
ejpam-3383	408	2	remark	remark	VERB
ejpam-3383	408	3	8.16	8.16	NUM
ejpam-3383	408	4	.	.	PUNCT
ejpam-3383	409	1	if	if	SCONJ
ejpam-3383	409	2	s	s	NOUN
ejpam-3383	409	3	is	be	AUX
ejpam-3383	409	4	an	an	DET
ejpam-3383	409	5	ordered	order	VERB
ejpam-3383	409	6	semigroup	semigroup	NOUN
ejpam-3383	409	7	and	and	CCONJ
ejpam-3383	409	8	m	m	PROPN
ejpam-3383	409	9	is	be	AUX
ejpam-3383	409	10	an	an	DET
ejpam-3383	409	11	ideal	ideal	NOUN
ejpam-3383	409	12	of	of	ADP
ejpam-3383	409	13	s	s	PROPN
ejpam-3383	409	14	,	,	PUNCT
ejpam-3383	409	15	then	then	ADV
ejpam-3383	409	16	the	the	DET
ejpam-3383	409	17	following	following	NOUN
ejpam-3383	409	18	are	be	AUX
ejpam-3383	409	19	equivalent	equivalent	ADJ
ejpam-3383	409	20	:	:	PUNCT
ejpam-3383	409	21	(	(	PUNCT
ejpam-3383	409	22	1	1	X
ejpam-3383	409	23	)	)	PUNCT
ejpam-3383	409	24	for	for	ADP
ejpam-3383	409	25	any	any	PRON
ejpam-3383	409	26	a	a	PRON
ejpam-3383	409	27	,	,	PUNCT
ejpam-3383	409	28	b	b	PROPN
ejpam-3383	409	29	⊆	⊆	NUM
ejpam-3383	409	30	s	s	NOUN
ejpam-3383	409	31	,	,	PUNCT
ejpam-3383	409	32	(	(	PUNCT
ejpam-3383	409	33	asb	asb	PROPN
ejpam-3383	409	34	]	]	PUNCT
ejpam-3383	409	35	⊆m	⊆m	NOUN
ejpam-3383	409	36	implies	imply	VERB
ejpam-3383	409	37	a	a	DET
ejpam-3383	409	38	⊆m	⊆m	NOUN
ejpam-3383	409	39	or	or	CCONJ
ejpam-3383	409	40	b	b	NOUN
ejpam-3383	409	41	⊆m	⊆m	NOUN
ejpam-3383	409	42	.	.	PUNCT
ejpam-3383	410	1	(	(	PUNCT
ejpam-3383	410	2	2	2	X
ejpam-3383	410	3	)	)	PUNCT
ejpam-3383	410	4	for	for	ADP
ejpam-3383	410	5	any	any	DET
ejpam-3383	410	6	a	a	PRON
ejpam-3383	410	7	,	,	PUNCT
ejpam-3383	410	8	b	b	PROPN
ejpam-3383	410	9	⊆	⊆	NUM
ejpam-3383	410	10	s	s	NOUN
ejpam-3383	410	11	,	,	PUNCT
ejpam-3383	410	12	asb	asb	PROPN
ejpam-3383	410	13	⊆m	⊆m	PROPN
ejpam-3383	410	14	implies	imply	VERB
ejpam-3383	410	15	a	a	DET
ejpam-3383	410	16	⊆m	⊆m	NOUN
ejpam-3383	410	17	or	or	CCONJ
ejpam-3383	410	18	b	b	NOUN
ejpam-3383	410	19	⊆m	⊆m	NOUN
ejpam-3383	410	20	.	.	PUNCT
ejpam-3383	411	1	by	by	ADP
ejpam-3383	411	2	proposition	proposition	NOUN
ejpam-3383	411	3	8.11	8.11	NUM
ejpam-3383	411	4	(	(	PUNCT
ejpam-3383	411	5	or	or	CCONJ
ejpam-3383	411	6	8.12	8.12	NUM
ejpam-3383	411	7	)	)	PUNCT
ejpam-3383	411	8	,	,	PUNCT
ejpam-3383	411	9	proposition	proposition	NOUN
ejpam-3383	411	10	8.15	8.15	NUM
ejpam-3383	411	11	and	and	CCONJ
ejpam-3383	411	12	remark	remark	NOUN
ejpam-3383	411	13	8.16	8.16	NUM
ejpam-3383	411	14	,	,	PUNCT
ejpam-3383	411	15	the	the	DET
ejpam-3383	411	16	following	follow	VERB
ejpam-3383	411	17	proposition	proposition	NOUN
ejpam-3383	411	18	holds	hold	VERB
ejpam-3383	411	19	.	.	PUNCT
ejpam-3383	412	1	proposition	proposition	NOUN
ejpam-3383	412	2	8.17	8.17	NUM
ejpam-3383	412	3	.	.	PUNCT
ejpam-3383	413	1	(	(	PUNCT
ejpam-3383	413	2	see	see	VERB
ejpam-3383	413	3	also	also	ADV
ejpam-3383	413	4	[	[	X
ejpam-3383	413	5	7	7	NUM
ejpam-3383	413	6	;	;	PUNCT
ejpam-3383	413	7	the	the	DET
ejpam-3383	413	8	theorem	theorem	NOUN
ejpam-3383	413	9	]	]	PUNCT
ejpam-3383	413	10	)	)	PUNCT
ejpam-3383	413	11	let	let	VERB
ejpam-3383	413	12	(	(	PUNCT
ejpam-3383	413	13	s	s	X
ejpam-3383	413	14	,	,	PUNCT
ejpam-3383	413	15	·	·	PUNCT
ejpam-3383	413	16	)	)	PUNCT
ejpam-3383	413	17	be	be	AUX
ejpam-3383	413	18	an	an	DET
ejpam-3383	413	19	ordered	order	VERB
ejpam-3383	413	20	semigroup	semigroup	NOUN
ejpam-3383	413	21	and	and	CCONJ
ejpam-3383	413	22	m	m	VERB
ejpam-3383	413	23	an	an	DET
ejpam-3383	413	24	ideal	ideal	NOUN
ejpam-3383	413	25	of	of	ADP
ejpam-3383	413	26	s.	s.	PROPN
ejpam-3383	413	27	the	the	DET
ejpam-3383	413	28	following	follow	VERB
ejpam-3383	413	29	are	be	AUX
ejpam-3383	413	30	equivalent	equivalent	ADJ
ejpam-3383	413	31	:	:	PUNCT
ejpam-3383	413	32	n.	n.	PROPN
ejpam-3383	413	33	kehayopulu	kehayopulu	PROPN
ejpam-3383	413	34	/	/	SYM
ejpam-3383	413	35	eur	eur	PROPN
ejpam-3383	413	36	.	.	PUNCT
ejpam-3383	414	1	j.	j.	PROPN
ejpam-3383	414	2	pure	pure	PROPN
ejpam-3383	414	3	appl	appl	PROPN
ejpam-3383	414	4	.	.	PROPN
ejpam-3383	414	5	math	math	PROPN
ejpam-3383	414	6	,	,	PUNCT
ejpam-3383	414	7	12	12	NUM
ejpam-3383	414	8	(	(	PUNCT
ejpam-3383	414	9	1	1	NUM
ejpam-3383	414	10	)	)	PUNCT
ejpam-3383	414	11	(	(	PUNCT
ejpam-3383	414	12	2019	2019	NUM
ejpam-3383	414	13	)	)	PUNCT
ejpam-3383	414	14	,	,	PUNCT
ejpam-3383	414	15	208	208	NUM
ejpam-3383	414	16	-	-	SYM
ejpam-3383	414	17	225	225	NUM
ejpam-3383	414	18	222	222	NUM
ejpam-3383	414	19	(	(	PUNCT
ejpam-3383	414	20	1	1	NUM
ejpam-3383	414	21	)	)	PUNCT
ejpam-3383	414	22	m	m	VERB
ejpam-3383	414	23	is	be	AUX
ejpam-3383	414	24	weakly	weakly	ADV
ejpam-3383	414	25	prime	prime	ADJ
ejpam-3383	414	26	.	.	PUNCT
ejpam-3383	415	1	(	(	PUNCT
ejpam-3383	415	2	2	2	X
ejpam-3383	415	3	)	)	PUNCT
ejpam-3383	415	4	for	for	ADP
ejpam-3383	415	5	any	any	DET
ejpam-3383	415	6	subsets	subset	NOUN
ejpam-3383	415	7	a	a	PRON
ejpam-3383	415	8	and	and	CCONJ
ejpam-3383	415	9	b	b	NOUN
ejpam-3383	415	10	of	of	ADP
ejpam-3383	415	11	s	s	PRON
ejpam-3383	415	12	such	such	ADJ
ejpam-3383	415	13	that	that	SCONJ
ejpam-3383	415	14	(	(	PUNCT
ejpam-3383	415	15	asb	asb	PROPN
ejpam-3383	415	16	]	]	X
ejpam-3383	415	17	⊆m	⊆m	NOUN
ejpam-3383	415	18	,	,	PUNCT
ejpam-3383	415	19	we	we	PRON
ejpam-3383	415	20	have	have	VERB
ejpam-3383	415	21	a	a	DET
ejpam-3383	415	22	⊆m	⊆m	NOUN
ejpam-3383	415	23	or	or	CCONJ
ejpam-3383	415	24	b	b	NOUN
ejpam-3383	415	25	⊆m	⊆m	NOUN
ejpam-3383	415	26	.	.	PUNCT
ejpam-3383	416	1	(	(	PUNCT
ejpam-3383	416	2	3	3	X
ejpam-3383	416	3	)	)	PUNCT
ejpam-3383	416	4	for	for	ADP
ejpam-3383	416	5	any	any	DET
ejpam-3383	416	6	a	a	DET
ejpam-3383	416	7	,	,	PUNCT
ejpam-3383	416	8	b	b	X
ejpam-3383	416	9	∈	∈	NOUN
ejpam-3383	416	10	s	s	VERB
ejpam-3383	416	11	such	such	ADJ
ejpam-3383	416	12	that	that	SCONJ
ejpam-3383	416	13	(	(	PUNCT
ejpam-3383	416	14	asb	asb	PROPN
ejpam-3383	416	15	]	]	X
ejpam-3383	416	16	⊆m	⊆m	NOUN
ejpam-3383	416	17	,	,	PUNCT
ejpam-3383	416	18	we	we	PRON
ejpam-3383	416	19	have	have	VERB
ejpam-3383	416	20	a	a	DET
ejpam-3383	416	21	∈m	∈m	NOUN
ejpam-3383	416	22	or	or	CCONJ
ejpam-3383	416	23	b	b	NOUN
ejpam-3383	416	24	∈m	∈m	NOUN
ejpam-3383	416	25	.	.	PUNCT
ejpam-3383	417	1	(	(	PUNCT
ejpam-3383	417	2	4	4	NUM
ejpam-3383	417	3	)	)	PUNCT
ejpam-3383	417	4	for	for	ADP
ejpam-3383	417	5	any	any	DET
ejpam-3383	417	6	subsets	subset	NOUN
ejpam-3383	417	7	a	a	DET
ejpam-3383	417	8	and	and	CCONJ
ejpam-3383	417	9	b	b	NOUN
ejpam-3383	417	10	of	of	ADP
ejpam-3383	417	11	s	s	PRON
ejpam-3383	417	12	such	such	ADJ
ejpam-3383	417	13	that	that	SCONJ
ejpam-3383	417	14	asb	asb	PROPN
ejpam-3383	417	15	⊆m	⊆m	NOUN
ejpam-3383	417	16	,	,	PUNCT
ejpam-3383	417	17	we	we	PRON
ejpam-3383	417	18	have	have	VERB
ejpam-3383	417	19	a	a	DET
ejpam-3383	417	20	⊆m	⊆m	NOUN
ejpam-3383	417	21	or	or	CCONJ
ejpam-3383	417	22	b	b	NOUN
ejpam-3383	417	23	⊆m	⊆m	NOUN
ejpam-3383	417	24	.	.	PUNCT
ejpam-3383	418	1	(	(	PUNCT
ejpam-3383	418	2	5	5	NUM
ejpam-3383	418	3	)	)	PUNCT
ejpam-3383	418	4	for	for	ADP
ejpam-3383	418	5	any	any	DET
ejpam-3383	418	6	a	a	DET
ejpam-3383	418	7	,	,	PUNCT
ejpam-3383	418	8	b	b	X
ejpam-3383	418	9	∈	∈	NOUN
ejpam-3383	418	10	s	s	VERB
ejpam-3383	418	11	such	such	ADJ
ejpam-3383	418	12	that	that	SCONJ
ejpam-3383	418	13	asb	asb	PROPN
ejpam-3383	418	14	⊆m	⊆m	NOUN
ejpam-3383	418	15	,	,	PUNCT
ejpam-3383	418	16	we	we	PRON
ejpam-3383	418	17	have	have	VERB
ejpam-3383	418	18	a	a	DET
ejpam-3383	418	19	∈m	∈m	NOUN
ejpam-3383	418	20	or	or	CCONJ
ejpam-3383	418	21	b	b	NOUN
ejpam-3383	418	22	∈m	∈m	NOUN
ejpam-3383	418	23	.	.	PUNCT
ejpam-3383	419	1	the	the	DET
ejpam-3383	419	2	above	above	ADJ
ejpam-3383	419	3	results	result	NOUN
ejpam-3383	419	4	on	on	ADP
ejpam-3383	419	5	ordered	order	VERB
ejpam-3383	419	6	semigroups	semigroup	NOUN
ejpam-3383	419	7	can	can	AUX
ejpam-3383	419	8	be	be	AUX
ejpam-3383	419	9	transferred	transfer	VERB
ejpam-3383	419	10	to	to	PART
ejpam-3383	419	11	ordered	order	VERB
ejpam-3383	419	12	γ	γ	NOUN
ejpam-3383	419	13	-	-	PUNCT
ejpam-3383	419	14	semigroups	semigroup	NOUN
ejpam-3383	419	15	(	(	PUNCT
ejpam-3383	419	16	resp	resp	NOUN
ejpam-3383	419	17	.	.	PUNCT
ejpam-3383	420	1	ordered	order	VERB
ejpam-3383	420	2	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	420	3	)	)	PUNCT
ejpam-3383	420	4	by	by	ADP
ejpam-3383	420	5	putting	put	VERB
ejpam-3383	420	6	a	a	DET
ejpam-3383	420	7	“	"	PUNCT
ejpam-3383	420	8	γ	γ	X
ejpam-3383	420	9	”	"	PUNCT
ejpam-3383	420	10	(	(	PUNCT
ejpam-3383	420	11	resp	resp	NOUN
ejpam-3383	420	12	.	.	PUNCT
ejpam-3383	421	1	“	"	PUNCT
ejpam-3383	421	2	∗	∗	NOUN
ejpam-3383	421	3	”	"	PUNCT
ejpam-3383	421	4	)	)	PUNCT
ejpam-3383	421	5	in	in	ADP
ejpam-3383	421	6	the	the	DET
ejpam-3383	421	7	appropriate	appropriate	ADJ
ejpam-3383	421	8	place	place	NOUN
ejpam-3383	421	9	and	and	CCONJ
ejpam-3383	421	10	we	we	PRON
ejpam-3383	421	11	have	have	VERB
ejpam-3383	421	12	the	the	DET
ejpam-3383	421	13	following	follow	VERB
ejpam-3383	421	14	results	result	NOUN
ejpam-3383	421	15	.	.	PUNCT
ejpam-3383	422	1	proposition	proposition	NOUN
ejpam-3383	422	2	8.18	8.18	NUM
ejpam-3383	422	3	.	.	PUNCT
ejpam-3383	423	1	let	let	VERB
ejpam-3383	423	2	(	(	PUNCT
ejpam-3383	423	3	s	s	X
ejpam-3383	423	4	,	,	PUNCT
ejpam-3383	423	5	·	·	PUNCT
ejpam-3383	423	6	)	)	PUNCT
ejpam-3383	423	7	be	be	AUX
ejpam-3383	423	8	an	an	DET
ejpam-3383	423	9	ordered	order	VERB
ejpam-3383	423	10	γ	γ	NOUN
ejpam-3383	423	11	-	-	PUNCT
ejpam-3383	423	12	semigroup	semigroup	NOUN
ejpam-3383	423	13	and	and	CCONJ
ejpam-3383	423	14	m	m	VERB
ejpam-3383	423	15	an	an	DET
ejpam-3383	423	16	ideal	ideal	NOUN
ejpam-3383	423	17	of	of	ADP
ejpam-3383	423	18	s.	s.	PROPN
ejpam-3383	423	19	the	the	DET
ejpam-3383	423	20	following	follow	VERB
ejpam-3383	423	21	are	be	AUX
ejpam-3383	423	22	equivalent	equivalent	ADJ
ejpam-3383	423	23	:	:	PUNCT
ejpam-3383	423	24	(	(	PUNCT
ejpam-3383	423	25	1	1	X
ejpam-3383	423	26	)	)	PUNCT
ejpam-3383	423	27	m	m	VERB
ejpam-3383	423	28	is	be	AUX
ejpam-3383	423	29	weakly	weakly	ADV
ejpam-3383	423	30	prime	prime	ADJ
ejpam-3383	423	31	.	.	PUNCT
ejpam-3383	424	1	(	(	PUNCT
ejpam-3383	424	2	2	2	X
ejpam-3383	424	3	)	)	PUNCT
ejpam-3383	424	4	for	for	ADP
ejpam-3383	424	5	any	any	DET
ejpam-3383	424	6	subsets	subset	NOUN
ejpam-3383	424	7	a	a	PRON
ejpam-3383	424	8	and	and	CCONJ
ejpam-3383	424	9	b	b	NOUN
ejpam-3383	424	10	of	of	ADP
ejpam-3383	424	11	s	s	PRON
ejpam-3383	424	12	such	such	ADJ
ejpam-3383	424	13	that	that	SCONJ
ejpam-3383	424	14	(	(	PUNCT
ejpam-3383	424	15	aγsγb	aγsγb	ADJ
ejpam-3383	424	16	]	]	PUNCT
ejpam-3383	424	17	⊆m	⊆m	NOUN
ejpam-3383	424	18	,	,	PUNCT
ejpam-3383	424	19	we	we	PRON
ejpam-3383	424	20	have	have	VERB
ejpam-3383	424	21	a	a	DET
ejpam-3383	424	22	⊆m	⊆m	NOUN
ejpam-3383	424	23	or	or	CCONJ
ejpam-3383	424	24	b	b	NOUN
ejpam-3383	424	25	⊆m	⊆m	NOUN
ejpam-3383	424	26	.	.	PUNCT
ejpam-3383	425	1	(	(	PUNCT
ejpam-3383	425	2	3	3	X
ejpam-3383	425	3	)	)	PUNCT
ejpam-3383	425	4	for	for	ADP
ejpam-3383	425	5	any	any	DET
ejpam-3383	425	6	a	a	DET
ejpam-3383	425	7	,	,	PUNCT
ejpam-3383	425	8	b	b	X
ejpam-3383	425	9	∈	∈	NOUN
ejpam-3383	425	10	s	s	VERB
ejpam-3383	425	11	such	such	ADJ
ejpam-3383	425	12	that	that	SCONJ
ejpam-3383	425	13	(	(	PUNCT
ejpam-3383	425	14	aγsγb	aγsγb	ADJ
ejpam-3383	425	15	]	]	PUNCT
ejpam-3383	425	16	⊆m	⊆m	NOUN
ejpam-3383	425	17	,	,	PUNCT
ejpam-3383	425	18	we	we	PRON
ejpam-3383	425	19	have	have	VERB
ejpam-3383	425	20	a	a	DET
ejpam-3383	425	21	∈m	∈m	NOUN
ejpam-3383	425	22	or	or	CCONJ
ejpam-3383	425	23	b	b	NOUN
ejpam-3383	425	24	∈m	∈m	NOUN
ejpam-3383	425	25	.	.	PUNCT
ejpam-3383	426	1	(	(	PUNCT
ejpam-3383	426	2	4	4	NUM
ejpam-3383	426	3	)	)	PUNCT
ejpam-3383	426	4	for	for	ADP
ejpam-3383	426	5	any	any	DET
ejpam-3383	426	6	subsets	subset	NOUN
ejpam-3383	426	7	a	a	DET
ejpam-3383	426	8	and	and	CCONJ
ejpam-3383	426	9	b	b	NOUN
ejpam-3383	426	10	of	of	ADP
ejpam-3383	426	11	s	s	PRON
ejpam-3383	426	12	such	such	ADJ
ejpam-3383	426	13	that	that	SCONJ
ejpam-3383	426	14	aγsγb	aγsγb	ADJ
ejpam-3383	426	15	⊆m	⊆m	NOUN
ejpam-3383	426	16	,	,	PUNCT
ejpam-3383	426	17	we	we	PRON
ejpam-3383	426	18	have	have	VERB
ejpam-3383	426	19	a	a	DET
ejpam-3383	426	20	⊆m	⊆m	NOUN
ejpam-3383	426	21	or	or	CCONJ
ejpam-3383	426	22	b	b	NOUN
ejpam-3383	426	23	⊆m	⊆m	NOUN
ejpam-3383	426	24	.	.	PUNCT
ejpam-3383	427	1	(	(	PUNCT
ejpam-3383	427	2	5	5	NUM
ejpam-3383	427	3	)	)	PUNCT
ejpam-3383	427	4	for	for	ADP
ejpam-3383	427	5	any	any	DET
ejpam-3383	427	6	a	a	DET
ejpam-3383	427	7	,	,	PUNCT
ejpam-3383	427	8	b	b	X
ejpam-3383	427	9	∈	∈	NOUN
ejpam-3383	427	10	s	s	VERB
ejpam-3383	427	11	such	such	ADJ
ejpam-3383	427	12	that	that	SCONJ
ejpam-3383	427	13	aγsγb	aγsγb	ADJ
ejpam-3383	427	14	⊆m	⊆m	NOUN
ejpam-3383	427	15	,	,	PUNCT
ejpam-3383	427	16	we	we	PRON
ejpam-3383	427	17	have	have	VERB
ejpam-3383	427	18	a	a	DET
ejpam-3383	427	19	∈m	∈m	NOUN
ejpam-3383	427	20	or	or	CCONJ
ejpam-3383	427	21	b	b	NOUN
ejpam-3383	427	22	∈m	∈m	NOUN
ejpam-3383	427	23	.	.	PUNCT
ejpam-3383	428	1	proposition	proposition	NOUN
ejpam-3383	428	2	8.19	8.19	NUM
ejpam-3383	428	3	.	.	PUNCT
ejpam-3383	429	1	let	let	AUX
ejpam-3383	429	2	(	(	PUNCT
ejpam-3383	429	3	s	s	NOUN
ejpam-3383	429	4	,	,	PUNCT
ejpam-3383	429	5	◦	◦	NOUN
ejpam-3383	429	6	)	)	PUNCT
ejpam-3383	429	7	be	be	VERB
ejpam-3383	429	8	an	an	DET
ejpam-3383	429	9	ordered	order	VERB
ejpam-3383	429	10	hypersemigroup	hypersemigroup	NOUN
ejpam-3383	429	11	and	and	CCONJ
ejpam-3383	429	12	m	m	VERB
ejpam-3383	429	13	an	an	DET
ejpam-3383	429	14	ideal	ideal	NOUN
ejpam-3383	429	15	of	of	ADP
ejpam-3383	429	16	s.	s.	PROPN
ejpam-3383	429	17	the	the	DET
ejpam-3383	429	18	following	follow	VERB
ejpam-3383	429	19	are	be	AUX
ejpam-3383	429	20	equivalent	equivalent	ADJ
ejpam-3383	429	21	:	:	PUNCT
ejpam-3383	429	22	(	(	PUNCT
ejpam-3383	429	23	1	1	X
ejpam-3383	429	24	)	)	PUNCT
ejpam-3383	429	25	m	m	VERB
ejpam-3383	429	26	is	be	AUX
ejpam-3383	429	27	weakly	weakly	ADV
ejpam-3383	429	28	prime	prime	ADJ
ejpam-3383	429	29	.	.	PUNCT
ejpam-3383	430	1	(	(	PUNCT
ejpam-3383	430	2	2	2	X
ejpam-3383	430	3	)	)	PUNCT
ejpam-3383	430	4	for	for	ADP
ejpam-3383	430	5	every	every	DET
ejpam-3383	430	6	nonempty	nonempty	NOUN
ejpam-3383	430	7	subsets	subset	NOUN
ejpam-3383	430	8	a	a	DET
ejpam-3383	430	9	and	and	CCONJ
ejpam-3383	430	10	b	b	NOUN
ejpam-3383	430	11	of	of	ADP
ejpam-3383	430	12	s	s	PRON
ejpam-3383	430	13	such	such	ADJ
ejpam-3383	430	14	that	that	PRON
ejpam-3383	430	15	(	(	PUNCT
ejpam-3383	430	16	a∗s∗b	a∗s∗b	PROPN
ejpam-3383	430	17	]	]	PUNCT
ejpam-3383	430	18	⊆m	⊆m	NOUN
ejpam-3383	430	19	,	,	PUNCT
ejpam-3383	430	20	we	we	PRON
ejpam-3383	430	21	have	have	VERB
ejpam-3383	430	22	a	a	DET
ejpam-3383	430	23	⊆m	⊆m	NOUN
ejpam-3383	430	24	or	or	CCONJ
ejpam-3383	430	25	b	b	NOUN
ejpam-3383	430	26	⊆m	⊆m	NOUN
ejpam-3383	430	27	.	.	PUNCT
ejpam-3383	431	1	(	(	PUNCT
ejpam-3383	431	2	3	3	X
ejpam-3383	431	3	)	)	PUNCT
ejpam-3383	431	4	for	for	ADP
ejpam-3383	431	5	every	every	DET
ejpam-3383	431	6	a	a	PROPN
ejpam-3383	431	7	,	,	PUNCT
ejpam-3383	431	8	b	b	X
ejpam-3383	431	9	∈	∈	NOUN
ejpam-3383	431	10	s	s	VERB
ejpam-3383	431	11	such	such	ADJ
ejpam-3383	431	12	that	that	SCONJ
ejpam-3383	431	13	(	(	PUNCT
ejpam-3383	431	14	a	a	DET
ejpam-3383	431	15	∗	∗	NOUN
ejpam-3383	431	16	s	s	NOUN
ejpam-3383	431	17	∗	∗	NOUN
ejpam-3383	431	18	b	b	NOUN
ejpam-3383	431	19	]	]	X
ejpam-3383	431	20	⊆m	⊆m	NOUN
ejpam-3383	431	21	,	,	PUNCT
ejpam-3383	431	22	we	we	PRON
ejpam-3383	431	23	have	have	VERB
ejpam-3383	431	24	a	a	DET
ejpam-3383	431	25	∈m	∈m	NOUN
ejpam-3383	431	26	or	or	CCONJ
ejpam-3383	431	27	b	b	NOUN
ejpam-3383	431	28	∈m	∈m	NOUN
ejpam-3383	431	29	.	.	PUNCT
ejpam-3383	432	1	(	(	PUNCT
ejpam-3383	432	2	4	4	NUM
ejpam-3383	432	3	)	)	PUNCT
ejpam-3383	432	4	for	for	ADP
ejpam-3383	432	5	every	every	DET
ejpam-3383	432	6	nonempty	nonempty	NOUN
ejpam-3383	432	7	subsets	subset	NOUN
ejpam-3383	432	8	a	a	PRON
ejpam-3383	432	9	and	and	CCONJ
ejpam-3383	432	10	b	b	NOUN
ejpam-3383	432	11	of	of	ADP
ejpam-3383	432	12	s	s	PRON
ejpam-3383	432	13	such	such	ADJ
ejpam-3383	432	14	that	that	SCONJ
ejpam-3383	432	15	a∗s	a∗s	NUM
ejpam-3383	432	16	∗b	∗b	PROPN
ejpam-3383	432	17	⊆m	⊆m	NOUN
ejpam-3383	432	18	,	,	PUNCT
ejpam-3383	432	19	we	we	PRON
ejpam-3383	432	20	have	have	VERB
ejpam-3383	432	21	a	a	DET
ejpam-3383	432	22	⊆m	⊆m	NOUN
ejpam-3383	432	23	or	or	CCONJ
ejpam-3383	432	24	b	b	NOUN
ejpam-3383	432	25	⊆m	⊆m	NOUN
ejpam-3383	432	26	.	.	PUNCT
ejpam-3383	433	1	(	(	PUNCT
ejpam-3383	433	2	5	5	NUM
ejpam-3383	433	3	)	)	PUNCT
ejpam-3383	433	4	for	for	ADP
ejpam-3383	433	5	every	every	DET
ejpam-3383	433	6	a	a	PROPN
ejpam-3383	433	7	,	,	PUNCT
ejpam-3383	433	8	b	b	X
ejpam-3383	433	9	∈	∈	NOUN
ejpam-3383	433	10	s	s	VERB
ejpam-3383	433	11	such	such	ADJ
ejpam-3383	433	12	that	that	SCONJ
ejpam-3383	433	13	a	a	DET
ejpam-3383	433	14	∗	∗	NOUN
ejpam-3383	433	15	s	s	NOUN
ejpam-3383	433	16	∗	∗	NOUN
ejpam-3383	433	17	b	b	NOUN
ejpam-3383	433	18	⊆m	⊆m	NOUN
ejpam-3383	433	19	,	,	PUNCT
ejpam-3383	433	20	we	we	PRON
ejpam-3383	433	21	have	have	VERB
ejpam-3383	433	22	a	a	DET
ejpam-3383	433	23	∈m	∈m	NOUN
ejpam-3383	433	24	or	or	CCONJ
ejpam-3383	433	25	b	b	NOUN
ejpam-3383	433	26	∈m	∈m	NOUN
ejpam-3383	433	27	.	.	PUNCT
ejpam-3383	434	1	by	by	ADP
ejpam-3383	434	2	easy	easy	ADJ
ejpam-3383	434	3	modification	modification	NOUN
ejpam-3383	434	4	of	of	ADP
ejpam-3383	434	5	the	the	DET
ejpam-3383	434	6	proof	proof	NOUN
ejpam-3383	434	7	of	of	ADP
ejpam-3383	434	8	proposition	proposition	NOUN
ejpam-3383	434	9	8.17	8.17	NUM
ejpam-3383	434	10	,	,	PUNCT
ejpam-3383	434	11	the	the	DET
ejpam-3383	434	12	following	follow	VERB
ejpam-3383	434	13	proposition	proposition	NOUN
ejpam-3383	434	14	holds	hold	VERB
ejpam-3383	434	15	.	.	PUNCT
ejpam-3383	435	1	proposition	proposition	NOUN
ejpam-3383	435	2	8.20	8.20	NUM
ejpam-3383	435	3	.	.	PUNCT
ejpam-3383	436	1	(	(	PUNCT
ejpam-3383	436	2	see	see	VERB
ejpam-3383	436	3	also	also	ADV
ejpam-3383	436	4	[	[	X
ejpam-3383	436	5	7	7	NUM
ejpam-3383	436	6	;	;	PUNCT
ejpam-3383	436	7	remark	remark	NOUN
ejpam-3383	436	8	4	4	NUM
ejpam-3383	436	9	]	]	PUNCT
ejpam-3383	436	10	)	)	PUNCT
ejpam-3383	436	11	for	for	ADP
ejpam-3383	436	12	an	an	DET
ejpam-3383	436	13	ideal	ideal	ADJ
ejpam-3383	436	14	m	m	NOUN
ejpam-3383	436	15	of	of	ADP
ejpam-3383	436	16	an	an	DET
ejpam-3383	436	17	ordered	order	VERB
ejpam-3383	436	18	semigroup	semigroup	PROPN
ejpam-3383	436	19	s	s	PROPN
ejpam-3383	436	20	,	,	PUNCT
ejpam-3383	436	21	the	the	DET
ejpam-3383	436	22	following	follow	VERB
ejpam-3383	436	23	are	be	AUX
ejpam-3383	436	24	equivalent	equivalent	ADJ
ejpam-3383	436	25	:	:	PUNCT
ejpam-3383	436	26	(	(	PUNCT
ejpam-3383	436	27	1	1	X
ejpam-3383	436	28	)	)	PUNCT
ejpam-3383	436	29	m	m	VERB
ejpam-3383	436	30	is	be	AUX
ejpam-3383	436	31	weakly	weakly	ADJ
ejpam-3383	436	32	semiprime	semiprime	NOUN
ejpam-3383	436	33	.	.	PUNCT
ejpam-3383	437	1	(	(	PUNCT
ejpam-3383	437	2	2	2	X
ejpam-3383	437	3	)	)	PUNCT
ejpam-3383	437	4	for	for	ADP
ejpam-3383	437	5	any	any	DET
ejpam-3383	437	6	subset	subset	NOUN
ejpam-3383	437	7	a	a	PRON
ejpam-3383	437	8	of	of	ADP
ejpam-3383	437	9	s	s	PRON
ejpam-3383	437	10	such	such	ADJ
ejpam-3383	437	11	that	that	SCONJ
ejpam-3383	437	12	(	(	PUNCT
ejpam-3383	437	13	asa	asa	PROPN
ejpam-3383	437	14	]	]	PUNCT
ejpam-3383	437	15	⊆m	⊆m	NOUN
ejpam-3383	437	16	,	,	PUNCT
ejpam-3383	437	17	we	we	PRON
ejpam-3383	437	18	have	have	VERB
ejpam-3383	437	19	a	a	DET
ejpam-3383	437	20	⊆m	⊆m	NOUN
ejpam-3383	437	21	.	.	PUNCT
ejpam-3383	438	1	n.	n.	PROPN
ejpam-3383	438	2	kehayopulu	kehayopulu	PROPN
ejpam-3383	438	3	/	/	SYM
ejpam-3383	438	4	eur	eur	PROPN
ejpam-3383	438	5	.	.	PUNCT
ejpam-3383	439	1	j.	j.	PROPN
ejpam-3383	439	2	pure	pure	PROPN
ejpam-3383	439	3	appl	appl	PROPN
ejpam-3383	439	4	.	.	PROPN
ejpam-3383	439	5	math	math	PROPN
ejpam-3383	439	6	,	,	PUNCT
ejpam-3383	439	7	12	12	NUM
ejpam-3383	439	8	(	(	PUNCT
ejpam-3383	439	9	1	1	NUM
ejpam-3383	439	10	)	)	PUNCT
ejpam-3383	439	11	(	(	PUNCT
ejpam-3383	439	12	2019	2019	NUM
ejpam-3383	439	13	)	)	PUNCT
ejpam-3383	439	14	,	,	PUNCT
ejpam-3383	439	15	208	208	NUM
ejpam-3383	439	16	-	-	SYM
ejpam-3383	439	17	225	225	NUM
ejpam-3383	439	18	223	223	NUM
ejpam-3383	439	19	(	(	PUNCT
ejpam-3383	439	20	3	3	NUM
ejpam-3383	439	21	)	)	PUNCT
ejpam-3383	439	22	for	for	ADP
ejpam-3383	439	23	any	any	DET
ejpam-3383	439	24	a	a	DET
ejpam-3383	439	25	∈	∈	NOUN
ejpam-3383	439	26	s	s	VERB
ejpam-3383	439	27	such	such	ADJ
ejpam-3383	439	28	that	that	SCONJ
ejpam-3383	439	29	(	(	PUNCT
ejpam-3383	439	30	asa	asa	PROPN
ejpam-3383	439	31	]	]	PUNCT
ejpam-3383	439	32	⊆m	⊆m	NOUN
ejpam-3383	439	33	,	,	PUNCT
ejpam-3383	439	34	we	we	PRON
ejpam-3383	439	35	have	have	VERB
ejpam-3383	439	36	a	a	DET
ejpam-3383	439	37	∈m	∈m	NOUN
ejpam-3383	439	38	.	.	PUNCT
ejpam-3383	440	1	(	(	PUNCT
ejpam-3383	440	2	4	4	NUM
ejpam-3383	440	3	)	)	PUNCT
ejpam-3383	440	4	for	for	ADP
ejpam-3383	440	5	any	any	DET
ejpam-3383	440	6	subset	subset	NOUN
ejpam-3383	440	7	a	a	PRON
ejpam-3383	440	8	of	of	ADP
ejpam-3383	440	9	s	s	PRON
ejpam-3383	440	10	such	such	ADJ
ejpam-3383	440	11	that	that	SCONJ
ejpam-3383	440	12	asa	asa	PROPN
ejpam-3383	440	13	⊆m	⊆m	NOUN
ejpam-3383	440	14	,	,	PUNCT
ejpam-3383	440	15	we	we	PRON
ejpam-3383	440	16	have	have	VERB
ejpam-3383	440	17	a	a	DET
ejpam-3383	440	18	⊆m	⊆m	NOUN
ejpam-3383	440	19	.	.	PUNCT
ejpam-3383	441	1	(	(	PUNCT
ejpam-3383	441	2	5	5	NUM
ejpam-3383	441	3	)	)	PUNCT
ejpam-3383	441	4	for	for	ADP
ejpam-3383	441	5	any	any	DET
ejpam-3383	441	6	a	a	DET
ejpam-3383	441	7	∈	∈	NOUN
ejpam-3383	441	8	s	s	VERB
ejpam-3383	441	9	such	such	ADJ
ejpam-3383	441	10	that	that	SCONJ
ejpam-3383	441	11	asa	asa	PROPN
ejpam-3383	441	12	⊆m	⊆m	NOUN
ejpam-3383	441	13	,	,	PUNCT
ejpam-3383	441	14	we	we	PRON
ejpam-3383	441	15	have	have	VERB
ejpam-3383	441	16	a	a	DET
ejpam-3383	441	17	∈m	∈m	NOUN
ejpam-3383	441	18	.	.	PUNCT
ejpam-3383	442	1	the	the	DET
ejpam-3383	442	2	analogous	analogous	ADJ
ejpam-3383	442	3	result	result	NOUN
ejpam-3383	442	4	for	for	ADP
ejpam-3383	442	5	ordered	order	VERB
ejpam-3383	442	6	γ	γ	NOUN
ejpam-3383	442	7	-	-	PUNCT
ejpam-3383	442	8	semigroups	semigroup	NOUN
ejpam-3383	442	9	and	and	CCONJ
ejpam-3383	442	10	ordered	order	VERB
ejpam-3383	442	11	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	442	12	also	also	ADV
ejpam-3383	442	13	holds	hold	VERB
ejpam-3383	442	14	.	.	PUNCT
ejpam-3383	443	1	in	in	ADP
ejpam-3383	443	2	case	case	NOUN
ejpam-3383	443	3	of	of	ADP
ejpam-3383	443	4	an	an	DET
ejpam-3383	443	5	ordered	order	VERB
ejpam-3383	443	6	hypersemigroup	hypersemigroup	NOUN
ejpam-3383	443	7	s	s	PROPN
ejpam-3383	443	8	,	,	PUNCT
ejpam-3383	443	9	the	the	DET
ejpam-3383	443	10	subset	subset	NOUN
ejpam-3383	443	11	a	a	PRON
ejpam-3383	443	12	of	of	ADP
ejpam-3383	443	13	s	s	PRON
ejpam-3383	443	14	in	in	ADP
ejpam-3383	443	15	properties	property	NOUN
ejpam-3383	443	16	(	(	PUNCT
ejpam-3383	443	17	2	2	NUM
ejpam-3383	443	18	)	)	PUNCT
ejpam-3383	443	19	and	and	CCONJ
ejpam-3383	443	20	(	(	PUNCT
ejpam-3383	443	21	4	4	NUM
ejpam-3383	443	22	)	)	PUNCT
ejpam-3383	443	23	of	of	ADP
ejpam-3383	443	24	the	the	DET
ejpam-3383	443	25	above	above	ADJ
ejpam-3383	443	26	proposition	proposition	NOUN
ejpam-3383	443	27	should	should	AUX
ejpam-3383	443	28	be	be	AUX
ejpam-3383	443	29	a	a	DET
ejpam-3383	443	30	nonempty	nonempty	ADV
ejpam-3383	443	31	set	set	VERB
ejpam-3383	443	32	.	.	PUNCT
ejpam-3383	444	1	9	9	X
ejpam-3383	444	2	.	.	X
ejpam-3383	445	1	some	some	DET
ejpam-3383	445	2	further	further	ADJ
ejpam-3383	445	3	results	result	NOUN
ejpam-3383	445	4	related	relate	VERB
ejpam-3383	445	5	to	to	ADP
ejpam-3383	445	6	ordered	order	VERB
ejpam-3383	445	7	hypergroupoids	hypergroupoid	NOUN
ejpam-3383	445	8	let	let	VERB
ejpam-3383	445	9	(	(	PUNCT
ejpam-3383	445	10	s	s	X
ejpam-3383	445	11	,	,	PUNCT
ejpam-3383	445	12	·	·	PUNCT
ejpam-3383	445	13	,	,	PUNCT
ejpam-3383	445	14	≤	≤	NUM
ejpam-3383	445	15	)	)	PUNCT
ejpam-3383	445	16	be	be	VERB
ejpam-3383	445	17	an	an	DET
ejpam-3383	445	18	ordered	order	VERB
ejpam-3383	445	19	groupoid	groupoid	NOUN
ejpam-3383	445	20	and	and	CCONJ
ejpam-3383	445	21	“	"	PUNCT
ejpam-3383	445	22	◦	◦	NOUN
ejpam-3383	445	23	”	"	PUNCT
ejpam-3383	445	24	the	the	DET
ejpam-3383	445	25	hyperoperation	hyperoperation	NOUN
ejpam-3383	445	26	on	on	ADP
ejpam-3383	445	27	s	s	PRON
ejpam-3383	445	28	defined	define	VERB
ejpam-3383	445	29	by	by	ADP
ejpam-3383	445	30	a	a	DET
ejpam-3383	445	31	◦	◦	NOUN
ejpam-3383	445	32	b	b	X
ejpam-3383	445	33	:	:	PUNCT
ejpam-3383	445	34	=	=	SYM
ejpam-3383	445	35	{	{	PUNCT
ejpam-3383	445	36	x	x	PUNCT
ejpam-3383	445	37	∈	∈	PROPN
ejpam-3383	445	38	s	s	VERB
ejpam-3383	445	39	|	|	ADV
ejpam-3383	445	40	x	x	SYM
ejpam-3383	445	41	≤	≤	PROPN
ejpam-3383	445	42	ab	ab	NOUN
ejpam-3383	445	43	}	}	PUNCT
ejpam-3383	445	44	and	and	CCONJ
ejpam-3383	445	45	the	the	DET
ejpam-3383	445	46	same	same	ADJ
ejpam-3383	445	47	order	order	NOUN
ejpam-3383	445	48	“	"	PUNCT
ejpam-3383	445	49	≤	≤	NUM
ejpam-3383	445	50	”	"	PUNCT
ejpam-3383	445	51	.	.	PUNCT
ejpam-3383	446	1	then	then	ADV
ejpam-3383	446	2	(	(	PUNCT
ejpam-3383	446	3	s	s	X
ejpam-3383	446	4	,	,	PUNCT
ejpam-3383	446	5	◦	◦	NOUN
ejpam-3383	446	6	,	,	PUNCT
ejpam-3383	446	7	≤	≤	NUM
ejpam-3383	446	8	)	)	PUNCT
ejpam-3383	446	9	is	be	AUX
ejpam-3383	446	10	an	an	DET
ejpam-3383	446	11	ordered	ordered	ADJ
ejpam-3383	446	12	hypergroupoid	hypergroupoid	NOUN
ejpam-3383	446	13	.	.	PUNCT
ejpam-3383	447	1	in	in	ADP
ejpam-3383	447	2	particular	particular	ADJ
ejpam-3383	447	3	,	,	PUNCT
ejpam-3383	447	4	if	if	SCONJ
ejpam-3383	447	5	(	(	PUNCT
ejpam-3383	447	6	s	s	X
ejpam-3383	447	7	,	,	PUNCT
ejpam-3383	447	8	·	·	PUNCT
ejpam-3383	447	9	,	,	PUNCT
ejpam-3383	447	10	≤	≤	NUM
ejpam-3383	447	11	)	)	PUNCT
ejpam-3383	447	12	is	be	AUX
ejpam-3383	447	13	an	an	DET
ejpam-3383	447	14	ordered	order	VERB
ejpam-3383	447	15	semigroup	semigroup	NOUN
ejpam-3383	447	16	,	,	PUNCT
ejpam-3383	447	17	then	then	ADV
ejpam-3383	447	18	(	(	PUNCT
ejpam-3383	447	19	s	s	X
ejpam-3383	447	20	,	,	PUNCT
ejpam-3383	447	21	◦	◦	NOUN
ejpam-3383	447	22	,	,	PUNCT
ejpam-3383	447	23	≤	≤	NUM
ejpam-3383	447	24	)	)	PUNCT
ejpam-3383	447	25	is	be	AUX
ejpam-3383	447	26	an	an	DET
ejpam-3383	447	27	ordered	order	VERB
ejpam-3383	447	28	hypersemigroup	hypersemigroup	NOUN
ejpam-3383	447	29	[	[	X
ejpam-3383	447	30	16	16	NUM
ejpam-3383	447	31	;	;	PUNCT
ejpam-3383	447	32	lemma	lemma	PROPN
ejpam-3383	447	33	1	1	NUM
ejpam-3383	447	34	]	]	PUNCT
ejpam-3383	447	35	.	.	PUNCT
ejpam-3383	448	1	proposition	proposition	NOUN
ejpam-3383	448	2	9.1	9.1	NUM
ejpam-3383	448	3	.	.	PUNCT
ejpam-3383	449	1	we	we	PRON
ejpam-3383	449	2	have	have	VERB
ejpam-3383	449	3	a	a	DET
ejpam-3383	449	4	∗b	∗b	NOUN
ejpam-3383	449	5	=	=	SYM
ejpam-3383	449	6	(	(	PUNCT
ejpam-3383	449	7	ab	ab	X
ejpam-3383	449	8	]	]	PUNCT
ejpam-3383	449	9	.	.	PUNCT
ejpam-3383	450	1	proof	proof	NOUN
ejpam-3383	450	2	.	.	PUNCT
ejpam-3383	451	1	indeed	indeed	ADV
ejpam-3383	451	2	,	,	PUNCT
ejpam-3383	451	3	we	we	PRON
ejpam-3383	451	4	have	have	VERB
ejpam-3383	451	5	x	x	PROPN
ejpam-3383	451	6	∈	∈	PROPN
ejpam-3383	451	7	a	a	DET
ejpam-3383	451	8	∗b	∗b	ADJ
ejpam-3383	451	9	⇐	⇐	ADJ
ejpam-3383	451	10	⇒	⇒	NOUN
ejpam-3383	451	11	x	x	SYM
ejpam-3383	451	12	∈	∈	PROPN
ejpam-3383	451	13	a	a	DET
ejpam-3383	451	14	◦	◦	NOUN
ejpam-3383	451	15	b	b	NOUN
ejpam-3383	451	16	for	for	ADP
ejpam-3383	451	17	some	some	DET
ejpam-3383	451	18	a	a	DET
ejpam-3383	451	19	∈	∈	PROPN
ejpam-3383	451	20	a	a	DET
ejpam-3383	451	21	,	,	PUNCT
ejpam-3383	451	22	b	b	PROPN
ejpam-3383	451	23	∈	∈	PROPN
ejpam-3383	451	24	b	b	X
ejpam-3383	451	25	⇐	⇐	ADJ
ejpam-3383	451	26	⇒	⇒	NOUN
ejpam-3383	451	27	x	x	PUNCT
ejpam-3383	452	1	≤	≤	ADJ
ejpam-3383	452	2	ab	ab	PROPN
ejpam-3383	452	3	∈	∈	PROPN
ejpam-3383	452	4	ab	ab	PROPN
ejpam-3383	452	5	⇐	⇐	PROPN
ejpam-3383	452	6	⇒	⇒	PROPN
ejpam-3383	452	7	x	x	X
ejpam-3383	452	8	∈	∈	PROPN
ejpam-3383	452	9	(	(	PUNCT
ejpam-3383	452	10	ab	ab	PROPN
ejpam-3383	452	11	]	]	PUNCT
ejpam-3383	452	12	.	.	PUNCT
ejpam-3383	453	1	�	�	PROPN
ejpam-3383	453	2	proposition	proposition	NOUN
ejpam-3383	453	3	9.2	9.2	NUM
ejpam-3383	453	4	.	.	PUNCT
ejpam-3383	454	1	if	if	SCONJ
ejpam-3383	454	2	m	m	NOUN
ejpam-3383	454	3	is	be	AUX
ejpam-3383	454	4	a	a	DET
ejpam-3383	454	5	prime	prime	ADJ
ejpam-3383	454	6	(	(	PUNCT
ejpam-3383	454	7	resp	resp	NOUN
ejpam-3383	454	8	.	.	PUNCT
ejpam-3383	454	9	semiprime	semiprime	NOUN
ejpam-3383	454	10	)	)	PUNCT
ejpam-3383	454	11	subset	subset	NOUN
ejpam-3383	454	12	of	of	ADP
ejpam-3383	454	13	an	an	DET
ejpam-3383	454	14	ordered	order	VERB
ejpam-3383	454	15	groupoid	groupoid	NOUN
ejpam-3383	454	16	(	(	PUNCT
ejpam-3383	454	17	s	s	PROPN
ejpam-3383	454	18	,	,	PUNCT
ejpam-3383	454	19	·	·	PUNCT
ejpam-3383	454	20	,	,	PUNCT
ejpam-3383	454	21	≤	≤	NUM
ejpam-3383	454	22	)	)	PUNCT
ejpam-3383	454	23	,	,	PUNCT
ejpam-3383	454	24	then	then	ADV
ejpam-3383	454	25	it	it	PRON
ejpam-3383	454	26	is	be	AUX
ejpam-3383	454	27	a	a	DET
ejpam-3383	454	28	prime	prime	ADJ
ejpam-3383	454	29	(	(	PUNCT
ejpam-3383	454	30	resp	resp	NOUN
ejpam-3383	454	31	.	.	PUNCT
ejpam-3383	454	32	semiprime	semiprime	NOUN
ejpam-3383	454	33	)	)	PUNCT
ejpam-3383	454	34	subset	subset	NOUN
ejpam-3383	454	35	of	of	ADP
ejpam-3383	454	36	the	the	DET
ejpam-3383	454	37	ordered	order	VERB
ejpam-3383	454	38	hypergroupoid	hypergroupoid	PROPN
ejpam-3383	454	39	(	(	PUNCT
ejpam-3383	454	40	s	s	NOUN
ejpam-3383	454	41	,	,	PUNCT
ejpam-3383	454	42	◦	◦	NOUN
ejpam-3383	454	43	,	,	PUNCT
ejpam-3383	454	44	≤	≤	NUM
ejpam-3383	454	45	)	)	PUNCT
ejpam-3383	454	46	.	.	PUNCT
ejpam-3383	455	1	proof	proof	NOUN
ejpam-3383	455	2	.	.	PUNCT
ejpam-3383	456	1	let	let	VERB
ejpam-3383	456	2	m	m	PRON
ejpam-3383	456	3	be	be	AUX
ejpam-3383	456	4	a	a	DET
ejpam-3383	456	5	prime	prime	NOUN
ejpam-3383	456	6	subset	subset	NOUN
ejpam-3383	456	7	of	of	ADP
ejpam-3383	456	8	(	(	PUNCT
ejpam-3383	456	9	s	s	PROPN
ejpam-3383	456	10	,	,	PUNCT
ejpam-3383	456	11	·	·	PUNCT
ejpam-3383	456	12	,	,	PUNCT
ejpam-3383	456	13	≤	≤	NUM
ejpam-3383	456	14	)	)	PUNCT
ejpam-3383	456	15	and	and	CCONJ
ejpam-3383	456	16	a	a	PRON
ejpam-3383	456	17	,	,	PUNCT
ejpam-3383	456	18	b	b	PROPN
ejpam-3383	456	19	be	be	AUX
ejpam-3383	456	20	nonempty	nonempty	X
ejpam-3383	456	21	subsets	subset	NOUN
ejpam-3383	456	22	of	of	ADP
ejpam-3383	456	23	(	(	PUNCT
ejpam-3383	456	24	s	s	X
ejpam-3383	456	25	,	,	PUNCT
ejpam-3383	456	26	◦	◦	NOUN
ejpam-3383	456	27	,	,	PUNCT
ejpam-3383	456	28	≤	≤	NUM
ejpam-3383	456	29	)	)	PUNCT
ejpam-3383	456	30	such	such	ADJ
ejpam-3383	456	31	that	that	SCONJ
ejpam-3383	456	32	a	a	DET
ejpam-3383	456	33	∗	∗	NOUN
ejpam-3383	456	34	b	b	NOUN
ejpam-3383	456	35	⊆	⊆	NUM
ejpam-3383	456	36	m	m	NOUN
ejpam-3383	456	37	.	.	PUNCT
ejpam-3383	457	1	by	by	ADP
ejpam-3383	457	2	proposition	proposition	NOUN
ejpam-3383	457	3	9.1	9.1	NUM
ejpam-3383	457	4	,	,	PUNCT
ejpam-3383	457	5	we	we	PRON
ejpam-3383	457	6	have	have	VERB
ejpam-3383	457	7	ab	ab	PROPN
ejpam-3383	457	8	⊆	⊆	NUM
ejpam-3383	457	9	(	(	PUNCT
ejpam-3383	457	10	ab	ab	X
ejpam-3383	457	11	]	]	X
ejpam-3383	457	12	=	=	PUNCT
ejpam-3383	457	13	a	a	DET
ejpam-3383	457	14	∗	∗	NOUN
ejpam-3383	457	15	b	b	NOUN
ejpam-3383	457	16	⊆	⊆	NUM
ejpam-3383	457	17	m	m	NOUN
ejpam-3383	457	18	.	.	PUNCT
ejpam-3383	458	1	by	by	ADP
ejpam-3383	458	2	hypothesis	hypothesis	NOUN
ejpam-3383	458	3	,	,	PUNCT
ejpam-3383	458	4	we	we	PRON
ejpam-3383	458	5	have	have	VERB
ejpam-3383	458	6	a	a	DET
ejpam-3383	458	7	⊆m	⊆m	NOUN
ejpam-3383	458	8	or	or	CCONJ
ejpam-3383	458	9	b	b	NOUN
ejpam-3383	458	10	⊆m	⊆m	NOUN
ejpam-3383	458	11	,	,	PUNCT
ejpam-3383	458	12	thus	thus	ADV
ejpam-3383	458	13	m	m	NOUN
ejpam-3383	458	14	is	be	AUX
ejpam-3383	458	15	a	a	DET
ejpam-3383	458	16	prime	prime	ADJ
ejpam-3383	458	17	subset	subset	NOUN
ejpam-3383	458	18	of	of	ADP
ejpam-3383	458	19	(	(	PUNCT
ejpam-3383	458	20	s	s	PROPN
ejpam-3383	458	21	,	,	PUNCT
ejpam-3383	458	22	◦	◦	NOUN
ejpam-3383	458	23	,	,	PUNCT
ejpam-3383	458	24	≤	≤	NUM
ejpam-3383	458	25	)	)	PUNCT
ejpam-3383	458	26	.	.	PUNCT
ejpam-3383	459	1	similarly	similarly	ADV
ejpam-3383	459	2	,	,	PUNCT
ejpam-3383	459	3	if	if	SCONJ
ejpam-3383	459	4	m	m	NOUN
ejpam-3383	459	5	is	be	AUX
ejpam-3383	459	6	a	a	DET
ejpam-3383	459	7	semiprime	semiprime	NOUN
ejpam-3383	459	8	subset	subset	NOUN
ejpam-3383	459	9	of	of	ADP
ejpam-3383	459	10	(	(	PUNCT
ejpam-3383	459	11	s	s	PROPN
ejpam-3383	459	12	,	,	PUNCT
ejpam-3383	459	13	·	·	PUNCT
ejpam-3383	459	14	,	,	PUNCT
ejpam-3383	459	15	≤	≤	NUM
ejpam-3383	459	16	)	)	PUNCT
ejpam-3383	459	17	and	and	CCONJ
ejpam-3383	459	18	a	a	PRON
ejpam-3383	459	19	is	be	AUX
ejpam-3383	459	20	a	a	DET
ejpam-3383	459	21	nonempty	nonempty	ADJ
ejpam-3383	459	22	subset	subset	NOUN
ejpam-3383	459	23	of	of	ADP
ejpam-3383	459	24	s	s	PRON
ejpam-3383	459	25	such	such	ADJ
ejpam-3383	459	26	that	that	DET
ejpam-3383	459	27	a∗a	a∗a	NUM
ejpam-3383	459	28	⊆m	⊆m	NOUN
ejpam-3383	459	29	then	then	ADV
ejpam-3383	459	30	,	,	PUNCT
ejpam-3383	459	31	by	by	ADP
ejpam-3383	459	32	proposition	proposition	NOUN
ejpam-3383	459	33	9.1	9.1	NUM
ejpam-3383	459	34	,	,	PUNCT
ejpam-3383	459	35	we	we	PRON
ejpam-3383	459	36	have	have	VERB
ejpam-3383	459	37	a2	a2	PROPN
ejpam-3383	459	38	⊆	⊆	NUM
ejpam-3383	459	39	(	(	PUNCT
ejpam-3383	459	40	a2	a2	PROPN
ejpam-3383	459	41	]	]	PUNCT
ejpam-3383	459	42	⊆	⊆	NUM
ejpam-3383	459	43	a	a	DET
ejpam-3383	459	44	∗	∗	NOUN
ejpam-3383	459	45	a	a	DET
ejpam-3383	459	46	⊆	⊆	NUM
ejpam-3383	459	47	m	m	NOUN
ejpam-3383	459	48	and	and	CCONJ
ejpam-3383	459	49	so	so	ADV
ejpam-3383	459	50	a	a	DET
ejpam-3383	459	51	⊆	⊆	NUM
ejpam-3383	459	52	m	m	NOUN
ejpam-3383	459	53	,	,	PUNCT
ejpam-3383	459	54	thus	thus	ADV
ejpam-3383	459	55	m	m	VERB
ejpam-3383	459	56	is	be	AUX
ejpam-3383	459	57	a	a	DET
ejpam-3383	459	58	semiprime	semiprime	NOUN
ejpam-3383	459	59	subset	subset	NOUN
ejpam-3383	459	60	of	of	ADP
ejpam-3383	459	61	(	(	PUNCT
ejpam-3383	459	62	s	s	PROPN
ejpam-3383	459	63	,	,	PUNCT
ejpam-3383	459	64	◦	◦	NOUN
ejpam-3383	459	65	,	,	PUNCT
ejpam-3383	459	66	≤	≤	NUM
ejpam-3383	459	67	)	)	PUNCT
ejpam-3383	459	68	.	.	PUNCT
ejpam-3383	460	1	�	�	PROPN
ejpam-3383	460	2	proposition	proposition	NOUN
ejpam-3383	460	3	9.3	9.3	NUM
ejpam-3383	460	4	.	.	PUNCT
ejpam-3383	461	1	[	[	X
ejpam-3383	461	2	16	16	NUM
ejpam-3383	461	3	;	;	PUNCT
ejpam-3383	461	4	theorem	theorem	VERB
ejpam-3383	461	5	3	3	NUM
ejpam-3383	461	6	]	]	PUNCT
ejpam-3383	461	7	a	a	DET
ejpam-3383	461	8	set	set	NOUN
ejpam-3383	461	9	m	m	VERB
ejpam-3383	461	10	is	be	AUX
ejpam-3383	461	11	an	an	DET
ejpam-3383	461	12	ideal	ideal	NOUN
ejpam-3383	461	13	of	of	ADP
ejpam-3383	461	14	an	an	DET
ejpam-3383	461	15	ordered	order	VERB
ejpam-3383	461	16	groupoid	groupoid	NOUN
ejpam-3383	461	17	(	(	PUNCT
ejpam-3383	461	18	s	s	PROPN
ejpam-3383	461	19	,	,	PUNCT
ejpam-3383	461	20	·	·	PUNCT
ejpam-3383	461	21	,	,	PUNCT
ejpam-3383	461	22	≤	≤	NUM
ejpam-3383	461	23	)	)	PUNCT
ejpam-3383	461	24	if	if	SCONJ
ejpam-3383	461	25	and	and	CCONJ
ejpam-3383	461	26	only	only	ADV
ejpam-3383	461	27	if	if	SCONJ
ejpam-3383	461	28	m	m	NOUN
ejpam-3383	461	29	is	be	AUX
ejpam-3383	461	30	an	an	DET
ejpam-3383	461	31	ideal	ideal	NOUN
ejpam-3383	461	32	of	of	ADP
ejpam-3383	461	33	the	the	DET
ejpam-3383	461	34	ordered	order	VERB
ejpam-3383	461	35	hypergroupoid	hypergroupoid	PROPN
ejpam-3383	461	36	(	(	PUNCT
ejpam-3383	461	37	s	s	NOUN
ejpam-3383	461	38	,	,	PUNCT
ejpam-3383	461	39	◦	◦	NOUN
ejpam-3383	461	40	,	,	PUNCT
ejpam-3383	461	41	≤	≤	NUM
ejpam-3383	461	42	)	)	PUNCT
ejpam-3383	461	43	.	.	PUNCT
ejpam-3383	462	1	proposition	proposition	NOUN
ejpam-3383	462	2	9.4	9.4	NUM
ejpam-3383	462	3	.	.	PUNCT
ejpam-3383	463	1	a	a	DET
ejpam-3383	463	2	set	set	NOUN
ejpam-3383	463	3	m	m	VERB
ejpam-3383	463	4	is	be	AUX
ejpam-3383	463	5	a	a	DET
ejpam-3383	463	6	prime	prime	ADJ
ejpam-3383	463	7	(	(	PUNCT
ejpam-3383	463	8	resp	resp	NOUN
ejpam-3383	463	9	.	.	PUNCT
ejpam-3383	463	10	semiprime	semiprime	NOUN
ejpam-3383	463	11	)	)	PUNCT
ejpam-3383	463	12	ideal	ideal	NOUN
ejpam-3383	463	13	of	of	ADP
ejpam-3383	463	14	an	an	DET
ejpam-3383	463	15	ordered	order	VERB
ejpam-3383	463	16	groupoid	groupoid	NOUN
ejpam-3383	463	17	(	(	PUNCT
ejpam-3383	463	18	s	s	PROPN
ejpam-3383	463	19	,	,	PUNCT
ejpam-3383	463	20	·	·	PUNCT
ejpam-3383	463	21	,	,	PUNCT
ejpam-3383	463	22	≤	≤	NUM
ejpam-3383	463	23	)	)	PUNCT
ejpam-3383	464	1	if	if	SCONJ
ejpam-3383	464	2	and	and	CCONJ
ejpam-3383	464	3	only	only	ADV
ejpam-3383	464	4	if	if	SCONJ
ejpam-3383	464	5	m	m	NOUN
ejpam-3383	464	6	is	be	AUX
ejpam-3383	464	7	a	a	DET
ejpam-3383	464	8	prime	prime	ADJ
ejpam-3383	464	9	ideal	ideal	NOUN
ejpam-3383	464	10	of	of	ADP
ejpam-3383	464	11	the	the	DET
ejpam-3383	464	12	ordered	order	VERB
ejpam-3383	464	13	hypergroupoid	hypergroupoid	PROPN
ejpam-3383	464	14	(	(	PUNCT
ejpam-3383	464	15	s	s	NOUN
ejpam-3383	464	16	,	,	PUNCT
ejpam-3383	464	17	◦	◦	NOUN
ejpam-3383	464	18	,	,	PUNCT
ejpam-3383	464	19	≤	≤	NUM
ejpam-3383	464	20	)	)	PUNCT
ejpam-3383	464	21	.	.	PUNCT
ejpam-3383	465	1	proof	proof	NOUN
ejpam-3383	465	2	.	.	PUNCT
ejpam-3383	466	1	=	=	NOUN
ejpam-3383	466	2	⇒.	⇒.	NOUN
ejpam-3383	466	3	this	this	PRON
ejpam-3383	466	4	follows	follow	VERB
ejpam-3383	466	5	from	from	ADP
ejpam-3383	466	6	propositions	proposition	NOUN
ejpam-3383	466	7	9.2	9.2	NUM
ejpam-3383	466	8	and	and	CCONJ
ejpam-3383	466	9	9.3	9.3	NUM
ejpam-3383	466	10	.	.	PUNCT
ejpam-3383	467	1	⇐	⇐	PROPN
ejpam-3383	467	2	=	=	PROPN
ejpam-3383	467	3	.	.	PUNCT
ejpam-3383	468	1	let	let	VERB
ejpam-3383	468	2	m	m	PRON
ejpam-3383	468	3	be	be	AUX
ejpam-3383	468	4	a	a	DET
ejpam-3383	468	5	prime	prime	ADJ
ejpam-3383	468	6	ideal	ideal	NOUN
ejpam-3383	468	7	of	of	ADP
ejpam-3383	468	8	(	(	PUNCT
ejpam-3383	468	9	s	s	PROPN
ejpam-3383	468	10	,	,	PUNCT
ejpam-3383	468	11	◦	◦	NOUN
ejpam-3383	468	12	,	,	PUNCT
ejpam-3383	468	13	≤	≤	NUM
ejpam-3383	468	14	)	)	PUNCT
ejpam-3383	468	15	and	and	CCONJ
ejpam-3383	468	16	a	a	DET
ejpam-3383	468	17	,	,	PUNCT
ejpam-3383	468	18	b	b	PROPN
ejpam-3383	468	19	be	be	AUX
ejpam-3383	468	20	subsets	subset	NOUN
ejpam-3383	468	21	of	of	ADP
ejpam-3383	468	22	(	(	PUNCT
ejpam-3383	468	23	s	s	X
ejpam-3383	468	24	,	,	PUNCT
ejpam-3383	468	25	·	·	PUNCT
ejpam-3383	468	26	,	,	PUNCT
ejpam-3383	468	27	≤	≤	NUM
ejpam-3383	468	28	)	)	PUNCT
ejpam-3383	468	29	such	such	ADJ
ejpam-3383	468	30	that	that	SCONJ
ejpam-3383	468	31	ab	ab	PROPN
ejpam-3383	468	32	⊆m	⊆m	NOUN
ejpam-3383	468	33	.	.	PUNCT
ejpam-3383	469	1	by	by	ADP
ejpam-3383	469	2	proposition	proposition	NOUN
ejpam-3383	469	3	9.1	9.1	NUM
ejpam-3383	469	4	,	,	PUNCT
ejpam-3383	469	5	we	we	PRON
ejpam-3383	469	6	have	have	VERB
ejpam-3383	469	7	a	a	DET
ejpam-3383	469	8	∗	∗	NOUN
ejpam-3383	469	9	b	b	NOUN
ejpam-3383	469	10	=	=	SYM
ejpam-3383	469	11	(	(	PUNCT
ejpam-3383	469	12	ab	ab	X
ejpam-3383	469	13	]	]	X
ejpam-3383	469	14	⊆	⊆	NUM
ejpam-3383	469	15	(	(	PUNCT
ejpam-3383	469	16	m	m	NOUN
ejpam-3383	469	17	]	]	X
ejpam-3383	470	1	=	=	PUNCT
ejpam-3383	470	2	m	m	NOUN
ejpam-3383	470	3	.	.	PUNCT
ejpam-3383	471	1	by	by	ADP
ejpam-3383	471	2	hypothesis	hypothesis	NOUN
ejpam-3383	471	3	,	,	PUNCT
ejpam-3383	471	4	we	we	PRON
ejpam-3383	471	5	have	have	VERB
ejpam-3383	471	6	a	a	DET
ejpam-3383	471	7	⊆	⊆	NUM
ejpam-3383	471	8	m	m	NOUN
ejpam-3383	471	9	references	reference	NOUN
ejpam-3383	471	10	224	224	NUM
ejpam-3383	471	11	or	or	CCONJ
ejpam-3383	471	12	b	b	NOUN
ejpam-3383	471	13	⊆m	⊆m	NOUN
ejpam-3383	471	14	,	,	PUNCT
ejpam-3383	471	15	thus	thus	ADV
ejpam-3383	471	16	m	m	VERB
ejpam-3383	471	17	is	be	AUX
ejpam-3383	471	18	a	a	DET
ejpam-3383	471	19	prime	prime	ADJ
ejpam-3383	471	20	ideal	ideal	NOUN
ejpam-3383	471	21	of	of	ADP
ejpam-3383	471	22	(	(	PUNCT
ejpam-3383	471	23	s	s	PROPN
ejpam-3383	471	24	,	,	PUNCT
ejpam-3383	471	25	·	·	PUNCT
ejpam-3383	471	26	,	,	PUNCT
ejpam-3383	471	27	≤	≤	NUM
ejpam-3383	471	28	)	)	PUNCT
ejpam-3383	471	29	.	.	PUNCT
ejpam-3383	472	1	the	the	DET
ejpam-3383	472	2	proof	proof	NOUN
ejpam-3383	472	3	for	for	ADP
ejpam-3383	472	4	semiprime	semiprime	NOUN
ejpam-3383	472	5	ideals	ideal	NOUN
ejpam-3383	472	6	is	be	AUX
ejpam-3383	472	7	similar	similar	ADJ
ejpam-3383	472	8	.	.	PUNCT
ejpam-3383	473	1	�	�	PROPN
ejpam-3383	473	2	proposition	proposition	NOUN
ejpam-3383	473	3	9.4	9.4	NUM
ejpam-3383	473	4	remains	remain	VERB
ejpam-3383	473	5	true	true	ADJ
ejpam-3383	473	6	if	if	SCONJ
ejpam-3383	473	7	we	we	PRON
ejpam-3383	473	8	replace	replace	VERB
ejpam-3383	473	9	the	the	DET
ejpam-3383	473	10	word	word	NOUN
ejpam-3383	473	11	“	"	PUNCT
ejpam-3383	473	12	prime	prime	ADJ
ejpam-3383	473	13	”	"	PUNCT
ejpam-3383	473	14	by	by	ADP
ejpam-3383	473	15	“	"	PUNCT
ejpam-3383	473	16	weakly	weakly	ADJ
ejpam-3383	473	17	prime	prime	NOUN
ejpam-3383	473	18	”	"	PUNCT
ejpam-3383	473	19	and	and	CCONJ
ejpam-3383	473	20	the	the	DET
ejpam-3383	473	21	following	follow	VERB
ejpam-3383	473	22	proposition	proposition	NOUN
ejpam-3383	473	23	holds	hold	VERB
ejpam-3383	473	24	.	.	PUNCT
ejpam-3383	474	1	proposition	proposition	NOUN
ejpam-3383	474	2	9.5	9.5	NUM
ejpam-3383	474	3	.	.	PUNCT
ejpam-3383	475	1	a	a	DET
ejpam-3383	475	2	set	set	NOUN
ejpam-3383	475	3	m	m	VERB
ejpam-3383	475	4	is	be	AUX
ejpam-3383	475	5	a	a	DET
ejpam-3383	475	6	weakly	weakly	ADJ
ejpam-3383	475	7	prime	prime	ADJ
ejpam-3383	475	8	ideal	ideal	NOUN
ejpam-3383	475	9	of	of	ADP
ejpam-3383	475	10	an	an	DET
ejpam-3383	475	11	ordered	order	VERB
ejpam-3383	475	12	groupoid	groupoid	NOUN
ejpam-3383	475	13	(	(	PUNCT
ejpam-3383	475	14	s	s	PROPN
ejpam-3383	475	15	,	,	PUNCT
ejpam-3383	475	16	·	·	PUNCT
ejpam-3383	475	17	,	,	PUNCT
ejpam-3383	475	18	≤	≤	NUM
ejpam-3383	475	19	)	)	PUNCT
ejpam-3383	475	20	if	if	SCONJ
ejpam-3383	475	21	and	and	CCONJ
ejpam-3383	475	22	only	only	ADV
ejpam-3383	475	23	if	if	SCONJ
ejpam-3383	475	24	m	m	NOUN
ejpam-3383	475	25	is	be	AUX
ejpam-3383	475	26	a	a	DET
ejpam-3383	475	27	weakly	weakly	ADJ
ejpam-3383	475	28	prime	prime	ADJ
ejpam-3383	475	29	ideal	ideal	NOUN
ejpam-3383	475	30	of	of	ADP
ejpam-3383	475	31	the	the	DET
ejpam-3383	475	32	ordered	order	VERB
ejpam-3383	475	33	hypergroupoid	hypergroupoid	PROPN
ejpam-3383	475	34	(	(	PUNCT
ejpam-3383	475	35	s	s	NOUN
ejpam-3383	475	36	,	,	PUNCT
ejpam-3383	475	37	◦	◦	NOUN
ejpam-3383	475	38	,	,	PUNCT
ejpam-3383	475	39	≤	≤	NUM
ejpam-3383	475	40	)	)	PUNCT
ejpam-3383	475	41	.	.	PUNCT
ejpam-3383	476	1	i	i	PRON
ejpam-3383	476	2	would	would	AUX
ejpam-3383	476	3	like	like	VERB
ejpam-3383	476	4	to	to	PART
ejpam-3383	476	5	thank	thank	VERB
ejpam-3383	476	6	the	the	DET
ejpam-3383	476	7	anonymous	anonymous	ADJ
ejpam-3383	476	8	referee	referee	NOUN
ejpam-3383	476	9	for	for	ADP
ejpam-3383	476	10	his	his	PRON
ejpam-3383	476	11	/	/	SYM
ejpam-3383	476	12	her	her	PRON
ejpam-3383	476	13	time	time	NOUN
ejpam-3383	476	14	to	to	PART
ejpam-3383	476	15	read	read	VERB
ejpam-3383	476	16	the	the	DET
ejpam-3383	476	17	paper	paper	NOUN
ejpam-3383	476	18	carefully	carefully	ADV
ejpam-3383	476	19	and	and	CCONJ
ejpam-3383	476	20	his	his	PRON
ejpam-3383	476	21	/	/	SYM
ejpam-3383	476	22	her	her	PRON
ejpam-3383	476	23	prompt	prompt	ADJ
ejpam-3383	476	24	reply	reply	NOUN
ejpam-3383	476	25	.	.	PUNCT
ejpam-3383	477	1	references	reference	NOUN
ejpam-3383	477	2	[	[	X
ejpam-3383	477	3	1	1	NUM
ejpam-3383	477	4	]	]	PUNCT
ejpam-3383	477	5	g.	g.	NOUN
ejpam-3383	477	6	birkhoff	birkhoff	PROPN
ejpam-3383	477	7	.	.	PUNCT
ejpam-3383	478	1	lattice	lattice	PROPN
ejpam-3383	478	2	theory	theory	PROPN
ejpam-3383	478	3	.	.	PUNCT
ejpam-3383	479	1	revised	revise	VERB
ejpam-3383	479	2	reprint	reprint	NOUN
ejpam-3383	479	3	of	of	ADP
ejpam-3383	479	4	the	the	DET
ejpam-3383	479	5	1948	1948	NUM
ejpam-3383	479	6	edition	edition	NOUN
ejpam-3383	479	7	.	.	PUNCT
ejpam-3383	480	1	american	american	PROPN
ejpam-3383	480	2	mathematical	mathematical	PROPN
ejpam-3383	480	3	society	society	NOUN
ejpam-3383	480	4	colloquium	colloquium	NOUN
ejpam-3383	480	5	publications	publication	NOUN
ejpam-3383	480	6	,	,	PUNCT
ejpam-3383	480	7	25	25	NUM
ejpam-3383	480	8	.	.	PUNCT
ejpam-3383	481	1	american	american	PROPN
ejpam-3383	481	2	mathematical	mathematical	PROPN
ejpam-3383	481	3	society	society	NOUN
ejpam-3383	481	4	,	,	PUNCT
ejpam-3383	481	5	providence	providence	NOUN
ejpam-3383	481	6	,	,	PUNCT
ejpam-3383	481	7	r.i	r.i	PROPN
ejpam-3383	481	8	.	.	PROPN
ejpam-3383	481	9	1960	1960	NUM
ejpam-3383	481	10	xiii+283	xiii+283	PROPN
ejpam-3383	481	11	pp	pp	ADV
ejpam-3383	481	12	.	.	PUNCT
ejpam-3383	482	1	[	[	X
ejpam-3383	482	2	2	2	X
ejpam-3383	482	3	]	]	X
ejpam-3383	482	4	g.	g.	NOUN
ejpam-3383	482	5	birkhoff	birkhoff	PROPN
ejpam-3383	482	6	.	.	PUNCT
ejpam-3383	483	1	lattice	lattice	PROPN
ejpam-3383	483	2	theory	theory	PROPN
ejpam-3383	483	3	.	.	PUNCT
ejpam-3383	484	1	corrected	correct	VERB
ejpam-3383	484	2	reprint	reprint	NOUN
ejpam-3383	484	3	of	of	ADP
ejpam-3383	484	4	the	the	DET
ejpam-3383	484	5	1967	1967	NUM
ejpam-3383	484	6	third	third	PROPN
ejpam-3383	484	7	edition	edition	NOUN
ejpam-3383	484	8	.	.	PUNCT
ejpam-3383	485	1	american	american	PROPN
ejpam-3383	485	2	mathematical	mathematical	PROPN
ejpam-3383	485	3	society	society	NOUN
ejpam-3383	485	4	colloquium	colloquium	NOUN
ejpam-3383	485	5	publications	publication	NOUN
ejpam-3383	485	6	,	,	PUNCT
ejpam-3383	485	7	25	25	NUM
ejpam-3383	485	8	.	.	PUNCT
ejpam-3383	486	1	american	american	PROPN
ejpam-3383	486	2	mathematical	mathematical	PROPN
ejpam-3383	486	3	society	society	NOUN
ejpam-3383	486	4	,	,	PUNCT
ejpam-3383	486	5	providence	providence	NOUN
ejpam-3383	486	6	,	,	PUNCT
ejpam-3383	486	7	r.i	r.i	PROPN
ejpam-3383	486	8	.	.	PROPN
ejpam-3383	486	9	1979	1979	NUM
ejpam-3383	486	10	vi+418	vi+418	NOUN
ejpam-3383	486	11	pp	pp	ADV
ejpam-3383	486	12	.	.	PUNCT
ejpam-3383	487	1	[	[	X
ejpam-3383	487	2	3	3	X
ejpam-3383	487	3	]	]	X
ejpam-3383	487	4	a.h	a.h	PROPN
ejpam-3383	487	5	.	.	PROPN
ejpam-3383	487	6	clifford	clifford	PROPN
ejpam-3383	487	7	,	,	PUNCT
ejpam-3383	487	8	g.b	g.b	PROPN
ejpam-3383	487	9	.	.	PROPN
ejpam-3383	487	10	preston	preston	PROPN
ejpam-3383	487	11	.	.	PUNCT
ejpam-3383	488	1	the	the	DET
ejpam-3383	488	2	algebraic	algebraic	PROPN
ejpam-3383	488	3	theory	theory	NOUN
ejpam-3383	488	4	of	of	ADP
ejpam-3383	488	5	semigroups	semigroup	NOUN
ejpam-3383	488	6	.	.	PUNCT
ejpam-3383	489	1	vol	vol	NOUN
ejpam-3383	489	2	.	.	PUNCT
ejpam-3383	489	3	i.	i.	PROPN
ejpam-3383	489	4	mathematical	mathematical	PROPN
ejpam-3383	489	5	surveys	survey	NOUN
ejpam-3383	489	6	,	,	PUNCT
ejpam-3383	489	7	no	no	INTJ
ejpam-3383	489	8	.	.	NOUN
ejpam-3383	489	9	7	7	NUM
ejpam-3383	489	10	american	american	PROPN
ejpam-3383	489	11	mathematical	mathematical	ADJ
ejpam-3383	489	12	society	society	NOUN
ejpam-3383	489	13	,	,	PUNCT
ejpam-3383	489	14	providence	providence	NOUN
ejpam-3383	489	15	,	,	PUNCT
ejpam-3383	489	16	r.i	r.i	PROPN
ejpam-3383	489	17	.	.	PROPN
ejpam-3383	489	18	1961	1961	NUM
ejpam-3383	489	19	xv+224	xv+224	SYM
ejpam-3383	490	1	pp	pp	ADV
ejpam-3383	490	2	.	.	PUNCT
ejpam-3383	491	1	[	[	X
ejpam-3383	491	2	4	4	X
ejpam-3383	491	3	]	]	PUNCT
ejpam-3383	491	4	l.	l.	PROPN
ejpam-3383	491	5	fuchs	fuchs	PROPN
ejpam-3383	491	6	.	.	PUNCT
ejpam-3383	492	1	partially	partially	ADV
ejpam-3383	492	2	ordered	order	VERB
ejpam-3383	492	3	algebraic	algebraic	ADJ
ejpam-3383	492	4	systems	system	NOUN
ejpam-3383	492	5	.	.	PUNCT
ejpam-3383	493	1	pergamon	pergamon	PROPN
ejpam-3383	493	2	press	press	PROPN
ejpam-3383	493	3	,	,	PUNCT
ejpam-3383	493	4	oxford	oxford	PROPN
ejpam-3383	493	5	-	-	PUNCT
ejpam-3383	493	6	london	london	PROPN
ejpam-3383	493	7	-	-	PUNCT
ejpam-3383	493	8	new	new	PROPN
ejpam-3383	493	9	york	york	PROPN
ejpam-3383	493	10	-	-	PUNCT
ejpam-3383	493	11	paris	paris	PROPN
ejpam-3383	493	12	;	;	PUNCT
ejpam-3383	493	13	addison	addison	PROPN
ejpam-3383	493	14	-	-	PUNCT
ejpam-3383	493	15	wesley	wesley	PROPN
ejpam-3383	493	16	publishing	publishing	PROPN
ejpam-3383	493	17	co.	co.	PROPN
ejpam-3383	493	18	,	,	PUNCT
ejpam-3383	493	19	inc	inc	PROPN
ejpam-3383	493	20	.	.	PROPN
ejpam-3383	493	21	,	,	PUNCT
ejpam-3383	493	22	reading	reading	NOUN
ejpam-3383	493	23	,	,	PUNCT
ejpam-3383	493	24	mass.-palo	mass.-palo	NOUN
ejpam-3383	493	25	alto	alto	NOUN
ejpam-3383	493	26	,	,	PUNCT
ejpam-3383	493	27	calif.london	calif.london	PROPN
ejpam-3383	493	28	1963	1963	NUM
ejpam-3383	493	29	ix+229	ix+229	NUM
ejpam-3383	493	30	pp	pp	ADV
ejpam-3383	493	31	.	.	PUNCT
ejpam-3383	494	1	[	[	X
ejpam-3383	494	2	5	5	X
ejpam-3383	494	3	]	]	X
ejpam-3383	494	4	y.	y.	PROPN
ejpam-3383	494	5	hirano	hirano	PROPN
ejpam-3383	494	6	,	,	PUNCT
ejpam-3383	494	7	e.	e.	PROPN
ejpam-3383	494	8	poon	poon	PROPN
ejpam-3383	494	9	,	,	PUNCT
ejpam-3383	494	10	h.	h.	PROPN
ejpam-3383	494	11	tsutsui	tsutsui	PROPN
ejpam-3383	494	12	.	.	PUNCT
ejpam-3383	495	1	on	on	ADP
ejpam-3383	495	2	rings	ring	NOUN
ejpam-3383	495	3	in	in	ADP
ejpam-3383	495	4	which	which	PRON
ejpam-3383	495	5	every	every	DET
ejpam-3383	495	6	ideal	ideal	NOUN
ejpam-3383	495	7	is	be	AUX
ejpam-3383	495	8	weakly	weakly	ADV
ejpam-3383	495	9	prime	prime	ADJ
ejpam-3383	495	10	.	.	PUNCT
ejpam-3383	496	1	bull	bull	NOUN
ejpam-3383	496	2	.	.	PUNCT
ejpam-3383	497	1	korean	korean	ADJ
ejpam-3383	497	2	math	math	PROPN
ejpam-3383	497	3	.	.	PUNCT
ejpam-3383	498	1	soc	soc	PROPN
ejpam-3383	498	2	.	.	PUNCT
ejpam-3383	499	1	47(5):1077–1087	47(5):1077–1087	NUM
ejpam-3383	499	2	,	,	PUNCT
ejpam-3383	499	3	2010	2010	NUM
ejpam-3383	499	4	.	.	PUNCT
ejpam-3383	500	1	[	[	X
ejpam-3383	500	2	6	6	NUM
ejpam-3383	500	3	]	]	X
ejpam-3383	500	4	n.	n.	NOUN
ejpam-3383	500	5	kehayopulu	kehayopulu	PROPN
ejpam-3383	500	6	.	.	PUNCT
ejpam-3383	501	1	on	on	ADP
ejpam-3383	501	2	intra	intra	ADJ
ejpam-3383	501	3	-	-	ADJ
ejpam-3383	501	4	regular	regular	ADJ
ejpam-3383	501	5	∨e	∨e	NOUN
ejpam-3383	501	6	-	-	PUNCT
ejpam-3383	501	7	semigroups	semigroup	NOUN
ejpam-3383	501	8	.	.	PUNCT
ejpam-3383	502	1	semigroup	semigroup	PROPN
ejpam-3383	502	2	forum	forum	PROPN
ejpam-3383	502	3	19(2):111–121	19(2):111–121	PROPN
ejpam-3383	502	4	,	,	PUNCT
ejpam-3383	502	5	1980	1980	NUM
ejpam-3383	502	6	.	.	PUNCT
ejpam-3383	503	1	[	[	X
ejpam-3383	503	2	7	7	X
ejpam-3383	503	3	]	]	X
ejpam-3383	503	4	n.	n.	NOUN
ejpam-3383	503	5	kehayopulu	kehayopulu	PROPN
ejpam-3383	503	6	.	.	PUNCT
ejpam-3383	504	1	on	on	ADP
ejpam-3383	504	2	weakly	weakly	ADJ
ejpam-3383	504	3	prime	prime	ADJ
ejpam-3383	504	4	ideals	ideal	NOUN
ejpam-3383	504	5	of	of	ADP
ejpam-3383	504	6	ordered	order	VERB
ejpam-3383	504	7	semigroups	semigroup	NOUN
ejpam-3383	504	8	.	.	PUNCT
ejpam-3383	504	9	math	math	NOUN
ejpam-3383	504	10	.	.	PUNCT
ejpam-3383	505	1	japon	japon	PROPN
ejpam-3383	505	2	.	.	PUNCT
ejpam-3383	506	1	35(6):1051–1056	35(6):1051–1056	NUM
ejpam-3383	506	2	,	,	PUNCT
ejpam-3383	506	3	1990	1990	NUM
ejpam-3383	506	4	.	.	PUNCT
ejpam-3383	507	1	[	[	X
ejpam-3383	507	2	8	8	NUM
ejpam-3383	507	3	]	]	X
ejpam-3383	507	4	n.	n.	NOUN
ejpam-3383	507	5	kehayopulu	kehayopulu	PROPN
ejpam-3383	507	6	.	.	PUNCT
ejpam-3383	508	1	on	on	ADP
ejpam-3383	508	2	prime	prime	ADJ
ejpam-3383	508	3	,	,	PUNCT
ejpam-3383	508	4	weakly	weakly	ADJ
ejpam-3383	508	5	prime	prime	ADJ
ejpam-3383	508	6	ideals	ideal	NOUN
ejpam-3383	508	7	in	in	ADP
ejpam-3383	508	8	ordered	order	VERB
ejpam-3383	508	9	semigroups	semigroup	NOUN
ejpam-3383	508	10	.	.	PUNCT
ejpam-3383	509	1	semigroup	semigroup	PROPN
ejpam-3383	509	2	forum	forum	PROPN
ejpam-3383	509	3	44(3):341–346	44(3):341–346	PROPN
ejpam-3383	509	4	,	,	PUNCT
ejpam-3383	509	5	1992	1992	NUM
ejpam-3383	509	6	.	.	PUNCT
ejpam-3383	510	1	[	[	X
ejpam-3383	510	2	9	9	NUM
ejpam-3383	510	3	]	]	X
ejpam-3383	510	4	n.	n.	NOUN
ejpam-3383	510	5	kehayopulu	kehayopulu	PROPN
ejpam-3383	510	6	.	.	PUNCT
ejpam-3383	511	1	on	on	ADP
ejpam-3383	511	2	prime	prime	ADJ
ejpam-3383	511	3	,	,	PUNCT
ejpam-3383	511	4	weakly	weakly	ADJ
ejpam-3383	511	5	prime	prime	ADJ
ejpam-3383	511	6	ideals	ideal	NOUN
ejpam-3383	511	7	in	in	ADP
ejpam-3383	511	8	po	po	NOUN
ejpam-3383	511	9	-	-	PUNCT
ejpam-3383	511	10	γ	γ	NOUN
ejpam-3383	511	11	-	-	PUNCT
ejpam-3383	511	12	semigroups	semigroup	NOUN
ejpam-3383	511	13	.	.	PUNCT
ejpam-3383	512	1	lobachevskii	lobachevskii	PROPN
ejpam-3383	512	2	j.	j.	PROPN
ejpam-3383	512	3	math	math	PROPN
ejpam-3383	512	4	.	.	PUNCT
ejpam-3383	513	1	30(4):257–262	30(4):257–262	NUM
ejpam-3383	513	2	,	,	PUNCT
ejpam-3383	513	3	2009	2009	NUM
ejpam-3383	513	4	.	.	PUNCT
ejpam-3383	514	1	[	[	X
ejpam-3383	514	2	10	10	NUM
ejpam-3383	514	3	]	]	X
ejpam-3383	514	4	n.	n.	NOUN
ejpam-3383	514	5	kehayopulu	kehayopulu	PROPN
ejpam-3383	514	6	.	.	PUNCT
ejpam-3383	515	1	on	on	ADP
ejpam-3383	515	2	ordered	order	VERB
ejpam-3383	515	3	γ	γ	NOUN
ejpam-3383	515	4	-	-	PUNCT
ejpam-3383	515	5	semigroups	semigroup	NOUN
ejpam-3383	515	6	.	.	PUNCT
ejpam-3383	516	1	sci	sci	PROPN
ejpam-3383	516	2	.	.	PROPN
ejpam-3383	516	3	math	math	PROPN
ejpam-3383	516	4	.	.	PUNCT
ejpam-3383	517	1	jpn	jpn	PROPN
ejpam-3383	517	2	.	.	PUNCT
ejpam-3383	518	1	71(2):179–185	71(2):179–185	NUM
ejpam-3383	518	2	,	,	PUNCT
ejpam-3383	518	3	2010	2010	NUM
ejpam-3383	518	4	.	.	PUNCT
ejpam-3383	519	1	[	[	X
ejpam-3383	519	2	11	11	NUM
ejpam-3383	519	3	]	]	X
ejpam-3383	519	4	n.	n.	NOUN
ejpam-3383	519	5	kehayopulu	kehayopulu	PROPN
ejpam-3383	519	6	.	.	PUNCT
ejpam-3383	520	1	on	on	ADP
ejpam-3383	520	2	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	520	3	.	.	PUNCT
ejpam-3383	521	1	pure	pure	ADJ
ejpam-3383	521	2	math	math	NOUN
ejpam-3383	521	3	.	.	PUNCT
ejpam-3383	522	1	appl	appl	PROPN
ejpam-3383	522	2	.	.	PUNCT
ejpam-3383	523	1	(	(	PUNCT
ejpam-3383	523	2	pu.m.a	pu.m.a	PROPN
ejpam-3383	523	3	.	.	PUNCT
ejpam-3383	523	4	)	)	PUNCT
ejpam-3383	524	1	25(2):151–156	25(2):151–156	PROPN
ejpam-3383	524	2	,	,	PUNCT
ejpam-3383	524	3	2015	2015	NUM
ejpam-3383	524	4	.	.	PUNCT
ejpam-3383	525	1	references	reference	NOUN
ejpam-3383	525	2	225	225	NUM
ejpam-3383	525	3	[	[	X
ejpam-3383	525	4	12	12	NUM
ejpam-3383	525	5	]	]	X
ejpam-3383	525	6	n.	n.	PROPN
ejpam-3383	525	7	kehayopulu	kehayopulu	PROPN
ejpam-3383	525	8	.	.	PUNCT
ejpam-3383	526	1	left	leave	VERB
ejpam-3383	526	2	regular	regular	ADJ
ejpam-3383	526	3	and	and	CCONJ
ejpam-3383	526	4	intra	intra	ADJ
ejpam-3383	526	5	-	-	ADJ
ejpam-3383	526	6	regular	regular	ADJ
ejpam-3383	526	7	ordered	order	VERB
ejpam-3383	526	8	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	526	9	in	in	ADP
ejpam-3383	526	10	terms	term	NOUN
ejpam-3383	526	11	of	of	ADP
ejpam-3383	526	12	semiprime	semiprime	NOUN
ejpam-3383	526	13	and	and	CCONJ
ejpam-3383	526	14	fuzzy	fuzzy	ADJ
ejpam-3383	526	15	semiprime	semiprime	NOUN
ejpam-3383	526	16	subsets	subset	NOUN
ejpam-3383	526	17	.	.	PUNCT
ejpam-3383	527	1	sci	sci	PROPN
ejpam-3383	527	2	.	.	PROPN
ejpam-3383	527	3	math	math	PROPN
ejpam-3383	527	4	.	.	PUNCT
ejpam-3383	528	1	jpn	jpn	PROPN
ejpam-3383	528	2	.	.	PUNCT
ejpam-3383	529	1	80(3):295–305	80(3):295–305	PROPN
ejpam-3383	529	2	,	,	PUNCT
ejpam-3383	529	3	2017	2017	NUM
ejpam-3383	529	4	.	.	PUNCT
ejpam-3383	530	1	[	[	X
ejpam-3383	530	2	13	13	NUM
ejpam-3383	530	3	]	]	X
ejpam-3383	530	4	n.	n.	NOUN
ejpam-3383	530	5	kehayopulu	kehayopulu	PROPN
ejpam-3383	530	6	.	.	PUNCT
ejpam-3383	531	1	how	how	SCONJ
ejpam-3383	531	2	we	we	PRON
ejpam-3383	531	3	pass	pass	VERB
ejpam-3383	531	4	from	from	ADP
ejpam-3383	531	5	semigroups	semigroup	NOUN
ejpam-3383	531	6	to	to	ADP
ejpam-3383	531	7	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	531	8	.	.	PUNCT
ejpam-3383	532	1	lobachevskii	lobachevskii	PROPN
ejpam-3383	532	2	j.	j.	PROPN
ejpam-3383	532	3	math	math	PROPN
ejpam-3383	532	4	.	.	PUNCT
ejpam-3383	533	1	39(1):121–128	39(1):121–128	NUM
ejpam-3383	533	2	,	,	PUNCT
ejpam-3383	533	3	2018	2018	NUM
ejpam-3383	533	4	.	.	PUNCT
ejpam-3383	534	1	[	[	X
ejpam-3383	534	2	14	14	NUM
ejpam-3383	534	3	]	]	X
ejpam-3383	534	4	n.	n.	PROPN
ejpam-3383	534	5	kehayopulu	kehayopulu	PROPN
ejpam-3383	534	6	.	.	PUNCT
ejpam-3383	535	1	an	an	DET
ejpam-3383	535	2	application	application	NOUN
ejpam-3383	535	3	of	of	ADP
ejpam-3383	535	4	γ	γ	NOUN
ejpam-3383	535	5	-	-	PUNCT
ejpam-3383	535	6	semigroups	semigroup	NOUN
ejpam-3383	535	7	techniques	technique	NOUN
ejpam-3383	535	8	to	to	ADP
ejpam-3383	535	9	the	the	DET
ejpam-3383	535	10	green	green	PROPN
ejpam-3383	535	11	’s	’s	PART
ejpam-3383	535	12	theorem	theorem	PROPN
ejpam-3383	535	13	.	.	PUNCT
ejpam-3383	536	1	afr	afr	PROPN
ejpam-3383	536	2	.	.	PUNCT
ejpam-3383	537	1	mat	mat	PROPN
ejpam-3383	537	2	.	.	PUNCT
ejpam-3383	537	3	29(1	29(1	PROPN
ejpam-3383	537	4	-	-	PUNCT
ejpam-3383	537	5	2):65–71	2):65–71	NUM
ejpam-3383	537	6	,	,	PUNCT
ejpam-3383	537	7	2018	2018	NUM
ejpam-3383	537	8	.	.	PUNCT
ejpam-3383	538	1	[	[	X
ejpam-3383	538	2	15	15	NUM
ejpam-3383	538	3	]	]	X
ejpam-3383	538	4	n.	n.	NOUN
ejpam-3383	538	5	kehayopulu	kehayopulu	PROPN
ejpam-3383	538	6	.	.	PUNCT
ejpam-3383	539	1	on	on	ADP
ejpam-3383	539	2	ordered	order	VERB
ejpam-3383	539	3	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	539	4	with	with	ADP
ejpam-3383	539	5	idempotent	idempotent	ADJ
ejpam-3383	539	6	ideals	ideal	NOUN
ejpam-3383	539	7	,	,	PUNCT
ejpam-3383	539	8	prime	prime	ADJ
ejpam-3383	539	9	or	or	CCONJ
ejpam-3383	539	10	weakly	weakly	ADJ
ejpam-3383	539	11	prime	prime	ADJ
ejpam-3383	539	12	ideals	ideal	NOUN
ejpam-3383	539	13	.	.	PUNCT
ejpam-3383	540	1	eur	eur	PROPN
ejpam-3383	540	2	.	.	PUNCT
ejpam-3383	541	1	j.	j.	PROPN
ejpam-3383	541	2	pure	pure	PROPN
ejpam-3383	541	3	appl	appl	PROPN
ejpam-3383	541	4	.	.	PUNCT
ejpam-3383	541	5	math	math	NOUN
ejpam-3383	541	6	.	.	PUNCT
ejpam-3383	542	1	11(1):10–22	11(1):10–22	NUM
ejpam-3383	542	2	,	,	PUNCT
ejpam-3383	542	3	2018	2018	NUM
ejpam-3383	542	4	.	.	PUNCT
ejpam-3383	543	1	[	[	X
ejpam-3383	543	2	16	16	NUM
ejpam-3383	543	3	]	]	X
ejpam-3383	543	4	n.	n.	PROPN
ejpam-3383	543	5	kehayopulu	kehayopulu	PROPN
ejpam-3383	543	6	.	.	PUNCT
ejpam-3383	544	1	on	on	ADP
ejpam-3383	544	2	ordered	order	VERB
ejpam-3383	544	3	hypersemigroups	hypersemigroup	NOUN
ejpam-3383	544	4	given	give	VERB
ejpam-3383	544	5	by	by	ADP
ejpam-3383	544	6	a	a	DET
ejpam-3383	544	7	table	table	NOUN
ejpam-3383	544	8	of	of	ADP
ejpam-3383	544	9	multiplication	multiplication	NOUN
ejpam-3383	544	10	and	and	CCONJ
ejpam-3383	544	11	a	a	DET
ejpam-3383	544	12	figure	figure	NOUN
ejpam-3383	544	13	.	.	PUNCT
ejpam-3383	545	1	turkish	turkish	ADJ
ejpam-3383	545	2	j.	j.	PROPN
ejpam-3383	545	3	math	math	PROPN
ejpam-3383	545	4	.	.	PUNCT
ejpam-3383	546	1	42(4):2045–2060	42(4):2045–2060	NOUN
ejpam-3383	546	2	,	,	PUNCT
ejpam-3383	546	3	2018	2018	NUM
ejpam-3383	546	4	.	.	PUNCT
ejpam-3383	547	1	[	[	X
ejpam-3383	547	2	17	17	NUM
ejpam-3383	547	3	]	]	PUNCT
ejpam-3383	547	4	m.	m.	NOUN
ejpam-3383	547	5	petrich	petrich	PROPN
ejpam-3383	547	6	.	.	PUNCT
ejpam-3383	548	1	introduction	introduction	NOUN
ejpam-3383	548	2	to	to	ADP
ejpam-3383	548	3	semigroups	semigroup	NOUN
ejpam-3383	548	4	.	.	PUNCT
ejpam-3383	549	1	merrill	merrill	NOUN
ejpam-3383	549	2	research	research	NOUN
ejpam-3383	549	3	and	and	CCONJ
ejpam-3383	549	4	lecture	lecture	NOUN
ejpam-3383	549	5	series	series	NOUN
ejpam-3383	549	6	.	.	PUNCT
ejpam-3383	550	1	charles	charles	PROPN
ejpam-3383	550	2	e.	e.	PROPN
ejpam-3383	550	3	merrill	merrill	PROPN
ejpam-3383	550	4	publishing	publishing	PROPN
ejpam-3383	550	5	co.	co.	PROPN
ejpam-3383	550	6	,	,	PUNCT
ejpam-3383	550	7	columbus	columbus	PROPN
ejpam-3383	550	8	,	,	PUNCT
ejpam-3383	550	9	ohio	ohio	PROPN
ejpam-3383	550	10	1973	1973	NUM
ejpam-3383	550	11	viii+198	viii+198	PROPN
ejpam-3383	550	12	pp	pp	NOUN
ejpam-3383	550	13	.	.	PUNCT
ejpam-3383	551	1	[	[	X
ejpam-3383	551	2	18	18	NUM
ejpam-3383	551	3	]	]	X
ejpam-3383	551	4	b.	b.	PROPN
ejpam-3383	551	5	pibaljommee	pibaljommee	PROPN
ejpam-3383	551	6	,	,	PUNCT
ejpam-3383	551	7	k.	k.	PROPN
ejpam-3383	551	8	wannatong	wannatong	PROPN
ejpam-3383	551	9	,	,	PUNCT
ejpam-3383	551	10	b.	b.	PROPN
ejpam-3383	551	11	davvaz	davvaz	PROPN
ejpam-3383	551	12	.	.	PUNCT
ejpam-3383	552	1	an	an	DET
ejpam-3383	552	2	investigation	investigation	NOUN
ejpam-3383	552	3	on	on	ADP
ejpam-3383	552	4	fuzzy	fuzzy	ADJ
ejpam-3383	552	5	hyperideals	hyperideal	NOUN
ejpam-3383	552	6	of	of	ADP
ejpam-3383	552	7	ordered	order	VERB
ejpam-3383	552	8	semihypergroups	semihypergroup	NOUN
ejpam-3383	552	9	.	.	PUNCT
ejpam-3383	553	1	quasigroups	quasigroups	PROPN
ejpam-3383	553	2	related	related	ADJ
ejpam-3383	553	3	systems	system	NOUN
ejpam-3383	553	4	23(2):297–308	23(2):297–308	NUM
ejpam-3383	553	5	,	,	PUNCT
ejpam-3383	553	6	2015	2015	NUM
ejpam-3383	553	7	.	.	PUNCT
ejpam-3383	554	1	[	[	X
ejpam-3383	554	2	19	19	NUM
ejpam-3383	554	3	]	]	X
ejpam-3383	554	4	m.k	m.k	PROPN
ejpam-3383	554	5	.	.	PUNCT
ejpam-3383	554	6	sen	sen	PROPN
ejpam-3383	554	7	,	,	PUNCT
ejpam-3383	554	8	n.k	n.k	PROPN
ejpam-3383	554	9	.	.	PROPN
ejpam-3383	554	10	saha	saha	PROPN
ejpam-3383	554	11	.	.	PUNCT
ejpam-3383	555	1	on	on	ADP
ejpam-3383	555	2	γ	γ	PROPN
ejpam-3383	555	3	-	-	PUNCT
ejpam-3383	555	4	semigroup	semigroup	NOUN
ejpam-3383	555	5	.	.	PUNCT
ejpam-3383	555	6	i.	i.	PROPN
ejpam-3383	555	7	bull	bull	PROPN
ejpam-3383	555	8	.	.	PUNCT
ejpam-3383	556	1	calcutta	calcutta	PROPN
ejpam-3383	556	2	math	math	PROPN
ejpam-3383	556	3	.	.	PUNCT
ejpam-3383	557	1	soc	soc	PROPN
ejpam-3383	557	2	.	.	PUNCT
ejpam-3383	558	1	78(3):180–186	78(3):180–186	PROPN
ejpam-3383	558	2	,	,	PUNCT
ejpam-3383	558	3	1986	1986	NUM
ejpam-3383	558	4	.	.	PUNCT
ejpam-3383	559	1	[	[	X
ejpam-3383	559	2	20	20	NUM
ejpam-3383	559	3	]	]	X
ejpam-3383	559	4	m.k	m.k	PROPN
ejpam-3383	559	5	.	.	PUNCT
ejpam-3383	559	6	sen	sen	PROPN
ejpam-3383	559	7	,	,	PUNCT
ejpam-3383	559	8	a.	a.	PROPN
ejpam-3383	559	9	seth	seth	PROPN
ejpam-3383	559	10	.	.	PUNCT
ejpam-3383	560	1	on	on	ADP
ejpam-3383	560	2	po	po	NOUN
ejpam-3383	560	3	-	-	PUNCT
ejpam-3383	560	4	γ	γ	NOUN
ejpam-3383	560	5	-	-	PUNCT
ejpam-3383	560	6	semigroups	semigroup	NOUN
ejpam-3383	560	7	.	.	PUNCT
ejpam-3383	561	1	bull	bull	NOUN
ejpam-3383	561	2	.	.	PUNCT
ejpam-3383	562	1	calcutta	calcutta	PROPN
ejpam-3383	562	2	math	math	PROPN
ejpam-3383	562	3	.	.	PUNCT
ejpam-3383	563	1	soc	soc	PROPN
ejpam-3383	563	2	.	.	PUNCT
ejpam-3383	564	1	85(5):445–450	85(5):445–450	NUM
ejpam-3383	564	2	,	,	PUNCT
ejpam-3383	564	3	1993	1993	NUM
ejpam-3383	564	4	.	.	PUNCT