id	sid	tid	token	lemma	pos
ejpam-2562	1	1	compile	compile	NOUN
ejpam-2562	1	2	/	/	SYM
ejpam-2562	1	3	output.dvi	output.dvi	NOUN
ejpam-2562	1	4	european	european	ADJ
ejpam-2562	1	5	journal	journal	NOUN
ejpam-2562	1	6	of	of	ADP
ejpam-2562	1	7	pure	pure	ADJ
ejpam-2562	1	8	and	and	CCONJ
ejpam-2562	1	9	applied	apply	VERB
ejpam-2562	1	10	mathematics	mathematic	NOUN
ejpam-2562	1	11	vol	vol	NOUN
ejpam-2562	1	12	.	.	PROPN
ejpam-2562	2	1	9	9	NUM
ejpam-2562	2	2	,	,	PUNCT
ejpam-2562	2	3	no	no	INTJ
ejpam-2562	2	4	.	.	NOUN
ejpam-2562	2	5	3	3	NUM
ejpam-2562	2	6	,	,	PUNCT
ejpam-2562	2	7	2016	2016	NUM
ejpam-2562	2	8	,	,	PUNCT
ejpam-2562	2	9	250	250	NUM
ejpam-2562	2	10	-	-	SYM
ejpam-2562	2	11	265	265	NUM
ejpam-2562	2	12	issn	issn	PROPN
ejpam-2562	2	13	1307	1307	NUM
ejpam-2562	2	14	-	-	SYM
ejpam-2562	2	15	5543	5543	NUM
ejpam-2562	2	16	–	–	PUNCT
ejpam-2562	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-2562	2	18	on	on	ADP
ejpam-2562	2	19	"	"	PUNCT
ejpam-2562	2	20	essential	essential	ADJ
ejpam-2562	2	21	"	"	PUNCT
ejpam-2562	2	22	subsemimodules	subsemimodule	NOUN
ejpam-2562	2	23	and	and	CCONJ
ejpam-2562	2	24	weakly	weakly	ADJ
ejpam-2562	2	25	co	co	ADJ
ejpam-2562	2	26	-	-	ADJ
ejpam-2562	2	27	hopfian	hopfian	ADJ
ejpam-2562	2	28	semimodules	semimodule	NOUN
ejpam-2562	2	29	el	el	PROPN
ejpam-2562	2	30	hadji	hadji	PROPN
ejpam-2562	2	31	demba	demba	PROPN
ejpam-2562	2	32	wade	wade	PROPN
ejpam-2562	2	33	diop∗	diop∗	PROPN
ejpam-2562	2	34	,	,	PUNCT
ejpam-2562	2	35	djiby	djiby	PROPN
ejpam-2562	2	36	sow	sow	PROPN
ejpam-2562	2	37	department	department	PROPN
ejpam-2562	2	38	de	de	ADP
ejpam-2562	2	39	mathematiques	mathematiques	PROPN
ejpam-2562	2	40	et	et	PROPN
ejpam-2562	2	41	informatique	informatique	PROPN
ejpam-2562	2	42	,	,	PUNCT
ejpam-2562	2	43	faculté	faculté	PROPN
ejpam-2562	2	44	des	des	PROPN
ejpam-2562	2	45	sciences	sciences	PROPN
ejpam-2562	2	46	et	et	PROPN
ejpam-2562	2	47	techniques	technique	NOUN
ejpam-2562	2	48	,	,	PUNCT
ejpam-2562	2	49	université	université	PROPN
ejpam-2562	2	50	cheikh	cheikh	PROPN
ejpam-2562	2	51	anta	anta	PROPN
ejpam-2562	2	52	diop	diop	PROPN
ejpam-2562	2	53	,	,	PUNCT
ejpam-2562	2	54	bp	bp	PROPN
ejpam-2562	2	55	5005	5005	NUM
ejpam-2562	2	56	dakar	dakar	PROPN
ejpam-2562	2	57	fann	fann	PROPN
ejpam-2562	2	58	,	,	PUNCT
ejpam-2562	2	59	sénégal	sénégal	ADJ
ejpam-2562	2	60	abstract	abstract	NOUN
ejpam-2562	2	61	.	.	PUNCT
ejpam-2562	3	1	two	two	NUM
ejpam-2562	3	2	different	different	ADJ
ejpam-2562	3	3	notions	notion	NOUN
ejpam-2562	3	4	of	of	ADP
ejpam-2562	3	5	"	"	PUNCT
ejpam-2562	3	6	essential	essential	ADJ
ejpam-2562	3	7	"	"	PUNCT
ejpam-2562	3	8	subsemimodules	subsemimodule	NOUN
ejpam-2562	3	9	were	be	AUX
ejpam-2562	3	10	introduced	introduce	VERB
ejpam-2562	3	11	in	in	ADP
ejpam-2562	3	12	the	the	DET
ejpam-2562	3	13	theory	theory	NOUN
ejpam-2562	3	14	of	of	ADP
ejpam-2562	3	15	semimodules	semimodule	NOUN
ejpam-2562	3	16	over	over	ADP
ejpam-2562	3	17	a	a	DET
ejpam-2562	3	18	semiring	semiring	NOUN
ejpam-2562	3	19	with	with	ADP
ejpam-2562	3	20	identity	identity	NOUN
ejpam-2562	3	21	,	,	PUNCT
ejpam-2562	3	22	in	in	ADP
ejpam-2562	3	23	order	order	NOUN
ejpam-2562	3	24	to	to	PART
ejpam-2562	3	25	generalize	generalize	VERB
ejpam-2562	3	26	the	the	DET
ejpam-2562	3	27	same	same	ADJ
ejpam-2562	3	28	notion	notion	NOUN
ejpam-2562	3	29	of	of	ADP
ejpam-2562	3	30	"	"	PUNCT
ejpam-2562	3	31	essential	essential	ADJ
ejpam-2562	3	32	"	"	PUNCT
ejpam-2562	3	33	submodules	submodule	NOUN
ejpam-2562	3	34	in	in	ADP
ejpam-2562	3	35	the	the	DET
ejpam-2562	3	36	theory	theory	NOUN
ejpam-2562	3	37	of	of	ADP
ejpam-2562	3	38	modules	module	NOUN
ejpam-2562	3	39	over	over	ADP
ejpam-2562	3	40	a	a	DET
ejpam-2562	3	41	ring	ring	NOUN
ejpam-2562	3	42	with	with	ADP
ejpam-2562	3	43	identity	identity	NOUN
ejpam-2562	3	44	.	.	PUNCT
ejpam-2562	4	1	in	in	ADP
ejpam-2562	4	2	this	this	DET
ejpam-2562	4	3	paper	paper	NOUN
ejpam-2562	4	4	,	,	PUNCT
ejpam-2562	4	5	we	we	PRON
ejpam-2562	4	6	introduce	introduce	VERB
ejpam-2562	4	7	a	a	DET
ejpam-2562	4	8	new	new	ADJ
ejpam-2562	4	9	class	class	NOUN
ejpam-2562	4	10	of	of	ADP
ejpam-2562	4	11	essential	essential	ADJ
ejpam-2562	4	12	subsemimodules	subsemimodule	NOUN
ejpam-2562	4	13	called	call	VERB
ejpam-2562	4	14	weakly	weakly	ADJ
ejpam-2562	4	15	essential	essential	ADJ
ejpam-2562	4	16	subsemimodules	subsemimodule	NOUN
ejpam-2562	4	17	.	.	PUNCT
ejpam-2562	5	1	we	we	PRON
ejpam-2562	5	2	prove	prove	VERB
ejpam-2562	5	3	that	that	SCONJ
ejpam-2562	5	4	this	this	DET
ejpam-2562	5	5	new	new	ADJ
ejpam-2562	5	6	class	class	NOUN
ejpam-2562	5	7	contains	contain	VERB
ejpam-2562	5	8	the	the	DET
ejpam-2562	5	9	others	other	NOUN
ejpam-2562	5	10	two	two	NUM
ejpam-2562	5	11	kind	kind	NOUN
ejpam-2562	5	12	of	of	ADP
ejpam-2562	5	13	classes	class	NOUN
ejpam-2562	5	14	of	of	ADP
ejpam-2562	5	15	"	"	PUNCT
ejpam-2562	5	16	essential	essential	ADJ
ejpam-2562	5	17	"	"	PUNCT
ejpam-2562	5	18	subsemimodules	subsemimodule	NOUN
ejpam-2562	5	19	.	.	PUNCT
ejpam-2562	6	1	futhermore	futhermore	NOUN
ejpam-2562	6	2	,	,	PUNCT
ejpam-2562	6	3	we	we	PRON
ejpam-2562	6	4	studie	studie	VERB
ejpam-2562	6	5	the	the	DET
ejpam-2562	6	6	properties	property	NOUN
ejpam-2562	6	7	of	of	ADP
ejpam-2562	6	8	weakly	weakly	ADJ
ejpam-2562	6	9	essential	essential	ADJ
ejpam-2562	6	10	subsemimodules	subsemimodule	NOUN
ejpam-2562	6	11	.	.	PUNCT
ejpam-2562	7	1	for	for	ADP
ejpam-2562	7	2	applications	application	NOUN
ejpam-2562	7	3	we	we	PRON
ejpam-2562	7	4	introduce	introduce	VERB
ejpam-2562	7	5	and	and	CCONJ
ejpam-2562	7	6	investigate	investigate	VERB
ejpam-2562	7	7	the	the	DET
ejpam-2562	7	8	co	co	ADJ
ejpam-2562	7	9	-	-	ADJ
ejpam-2562	7	10	hopfian	hopfian	ADJ
ejpam-2562	7	11	semimodules	semimodule	NOUN
ejpam-2562	7	12	with	with	ADP
ejpam-2562	7	13	this	this	DET
ejpam-2562	7	14	new	new	ADJ
ejpam-2562	7	15	definition	definition	NOUN
ejpam-2562	7	16	of	of	ADP
ejpam-2562	7	17	semi	semi	ADJ
ejpam-2562	7	18	-	-	ADJ
ejpam-2562	7	19	essential	essential	ADJ
ejpam-2562	7	20	.	.	PUNCT
ejpam-2562	8	1	2010	2010	NUM
ejpam-2562	8	2	mathematics	mathematic	NOUN
ejpam-2562	8	3	subject	subject	NOUN
ejpam-2562	8	4	classifications	classification	NOUN
ejpam-2562	8	5	:	:	PUNCT
ejpam-2562	8	6	16y60	16y60	NUM
ejpam-2562	8	7	key	key	ADJ
ejpam-2562	8	8	words	word	NOUN
ejpam-2562	8	9	and	and	CCONJ
ejpam-2562	8	10	phrases	phrase	NOUN
ejpam-2562	8	11	:	:	PUNCT
ejpam-2562	8	12	semiring	semiring	NOUN
ejpam-2562	8	13	,	,	PUNCT
ejpam-2562	8	14	subsemimodule	subsemimodule	NOUN
ejpam-2562	8	15	,	,	PUNCT
ejpam-2562	8	16	r	r	NOUN
ejpam-2562	8	17	-	-	PUNCT
ejpam-2562	8	18	congruence	congruence	NOUN
ejpam-2562	8	19	relation	relation	NOUN
ejpam-2562	8	20	,	,	PUNCT
ejpam-2562	8	21	essential	essential	ADJ
ejpam-2562	8	22	,	,	PUNCT
ejpam-2562	8	23	cancellative	cancellative	ADJ
ejpam-2562	8	24	,	,	PUNCT
ejpam-2562	8	25	subtractive	subtractive	NOUN
ejpam-2562	8	26	,	,	PUNCT
ejpam-2562	8	27	direct	direct	ADJ
ejpam-2562	8	28	sum	sum	NOUN
ejpam-2562	8	29	,	,	PUNCT
ejpam-2562	8	30	cohopfian	cohopfian	ADJ
ejpam-2562	8	31	semimodules	semimodule	NOUN
ejpam-2562	8	32	1	1	NUM
ejpam-2562	8	33	.	.	X
ejpam-2562	8	34	introduction	introduction	NOUN
ejpam-2562	8	35	the	the	DET
ejpam-2562	8	36	theory	theory	NOUN
ejpam-2562	8	37	of	of	ADP
ejpam-2562	8	38	semimodules	semimodule	NOUN
ejpam-2562	8	39	over	over	ADP
ejpam-2562	8	40	semirings	semiring	NOUN
ejpam-2562	8	41	with	with	ADP
ejpam-2562	8	42	identity	identity	NOUN
ejpam-2562	8	43	(	(	PUNCT
ejpam-2562	8	44	see	see	VERB
ejpam-2562	8	45	golan	golan	PROPN
ejpam-2562	8	46	[	[	X
ejpam-2562	8	47	6	6	NUM
ejpam-2562	8	48	]	]	PUNCT
ejpam-2562	8	49	,	,	PUNCT
ejpam-2562	8	50	abuhlail	abuhlail	NOUN
ejpam-2562	8	51	[	[	X
ejpam-2562	8	52	1	1	NUM
ejpam-2562	8	53	]	]	PUNCT
ejpam-2562	8	54	,	,	PUNCT
ejpam-2562	8	55	takahashi	takahashi	PROPN
ejpam-2562	9	1	[	[	X
ejpam-2562	9	2	9–11	9–11	PROPN
ejpam-2562	9	3	]	]	PUNCT
ejpam-2562	9	4	)	)	PUNCT
ejpam-2562	9	5	can	can	AUX
ejpam-2562	9	6	be	be	AUX
ejpam-2562	9	7	regarded	regard	VERB
ejpam-2562	9	8	as	as	ADP
ejpam-2562	9	9	a	a	DET
ejpam-2562	9	10	generalization	generalization	NOUN
ejpam-2562	9	11	of	of	ADP
ejpam-2562	9	12	the	the	DET
ejpam-2562	9	13	theory	theory	NOUN
ejpam-2562	9	14	of	of	ADP
ejpam-2562	9	15	modules	module	NOUN
ejpam-2562	9	16	over	over	ADP
ejpam-2562	9	17	rings	ring	NOUN
ejpam-2562	9	18	with	with	ADP
ejpam-2562	9	19	identity	identity	NOUN
ejpam-2562	9	20	(	(	PUNCT
ejpam-2562	9	21	see	see	VERB
ejpam-2562	9	22	anderson	anderson	PROPN
ejpam-2562	9	23	-	-	PUNCT
ejpam-2562	9	24	fuller[2	fuller[2	PROPN
ejpam-2562	9	25	]	]	X
ejpam-2562	9	26	,	,	PUNCT
ejpam-2562	9	27	lam	lam	PROPN
ejpam-2562	10	1	[	[	X
ejpam-2562	10	2	8	8	NUM
ejpam-2562	10	3	]	]	PUNCT
ejpam-2562	10	4	,	,	PUNCT
ejpam-2562	10	5	wisbauer	wisbauer	NOUN
ejpam-2562	10	6	[	[	X
ejpam-2562	10	7	15	15	NUM
ejpam-2562	10	8	]	]	PUNCT
ejpam-2562	10	9	)	)	PUNCT
ejpam-2562	10	10	.	.	PUNCT
ejpam-2562	11	1	many	many	ADJ
ejpam-2562	11	2	results	result	NOUN
ejpam-2562	11	3	for	for	ADP
ejpam-2562	11	4	semirings	semiring	NOUN
ejpam-2562	11	5	and	and	CCONJ
ejpam-2562	11	6	semimodules	semimodule	NOUN
ejpam-2562	11	7	also	also	ADV
ejpam-2562	11	8	hold	hold	VERB
ejpam-2562	11	9	for	for	ADP
ejpam-2562	11	10	rings	ring	NOUN
ejpam-2562	11	11	and	and	CCONJ
ejpam-2562	11	12	modules	module	NOUN
ejpam-2562	11	13	,	,	PUNCT
ejpam-2562	11	14	but	but	CCONJ
ejpam-2562	11	15	not	not	PART
ejpam-2562	11	16	conversely	conversely	ADV
ejpam-2562	11	17	[	[	X
ejpam-2562	11	18	5	5	NUM
ejpam-2562	11	19	,	,	PUNCT
ejpam-2562	11	20	12–14	12–14	NUM
ejpam-2562	11	21	]	]	PUNCT
ejpam-2562	11	22	.	.	PUNCT
ejpam-2562	12	1	the	the	DET
ejpam-2562	12	2	concept	concept	NOUN
ejpam-2562	12	3	of	of	ADP
ejpam-2562	12	4	"	"	PUNCT
ejpam-2562	12	5	essential	essential	ADJ
ejpam-2562	12	6	submodule	submodule	NOUN
ejpam-2562	12	7	"	"	PUNCT
ejpam-2562	12	8	in	in	ADP
ejpam-2562	12	9	an	an	DET
ejpam-2562	12	10	r	r	NOUN
ejpam-2562	12	11	-	-	PUNCT
ejpam-2562	12	12	module	module	NOUN
ejpam-2562	12	13	m	m	NOUN
ejpam-2562	12	14	[	[	X
ejpam-2562	12	15	2	2	NUM
ejpam-2562	12	16	]	]	PUNCT
ejpam-2562	12	17	,	,	PUNCT
ejpam-2562	12	18	introduced	introduce	VERB
ejpam-2562	12	19	by	by	ADP
ejpam-2562	12	20	johnson	johnson	PROPN
ejpam-2562	12	21	,	,	PUNCT
ejpam-2562	12	22	eckman	eckman	NOUN
ejpam-2562	12	23	and	and	CCONJ
ejpam-2562	12	24	schpof	schpof	VERB
ejpam-2562	12	25	[	[	X
ejpam-2562	12	26	4	4	NUM
ejpam-2562	12	27	,	,	PUNCT
ejpam-2562	12	28	7	7	NUM
ejpam-2562	12	29	]	]	PUNCT
ejpam-2562	12	30	,	,	PUNCT
ejpam-2562	12	31	plays	play	VERB
ejpam-2562	12	32	an	an	DET
ejpam-2562	12	33	important	important	ADJ
ejpam-2562	12	34	role	role	NOUN
ejpam-2562	12	35	in	in	ADP
ejpam-2562	12	36	the	the	DET
ejpam-2562	12	37	context	context	NOUN
ejpam-2562	12	38	of	of	ADP
ejpam-2562	12	39	commutative	commutative	ADJ
ejpam-2562	12	40	and	and	CCONJ
ejpam-2562	12	41	noncommutative	noncommutative	ADJ
ejpam-2562	12	42	algebras	algebra	NOUN
ejpam-2562	12	43	.	.	PUNCT
ejpam-2562	13	1	in	in	ADP
ejpam-2562	13	2	module	module	NOUN
ejpam-2562	13	3	theory	theory	NOUN
ejpam-2562	13	4	,	,	PUNCT
ejpam-2562	13	5	for	for	ADP
ejpam-2562	13	6	a	a	DET
ejpam-2562	13	7	ring	ring	NOUN
ejpam-2562	13	8	r	r	NOUN
ejpam-2562	13	9	,	,	PUNCT
ejpam-2562	13	10	a	a	DET
ejpam-2562	13	11	submodule	submodule	NOUN
ejpam-2562	13	12	n	n	PROPN
ejpam-2562	13	13	of	of	ADP
ejpam-2562	13	14	a	a	DET
ejpam-2562	13	15	module	module	NOUN
ejpam-2562	13	16	m	m	NOUN
ejpam-2562	13	17	is	be	AUX
ejpam-2562	13	18	said	say	VERB
ejpam-2562	13	19	to	to	PART
ejpam-2562	13	20	be	be	AUX
ejpam-2562	13	21	essential	essential	ADJ
ejpam-2562	13	22	(	(	PUNCT
ejpam-2562	13	23	denoted	denote	VERB
ejpam-2562	13	24	by	by	ADP
ejpam-2562	13	25	n	n	PRON
ejpam-2562	13	26	ã	ã	PROPN
ejpam-2562	13	27	m	m	NOUN
ejpam-2562	13	28	)	)	PUNCT
ejpam-2562	13	29	if	if	SCONJ
ejpam-2562	13	30	k	k	PROPN
ejpam-2562	13	31	∩n	∩n	PROPN
ejpam-2562	13	32	=	=	PUNCT
ejpam-2562	14	1	0=⇒	0=⇒	NUM
ejpam-2562	15	1	k	k	X
ejpam-2562	15	2	=	=	PUNCT
ejpam-2562	15	3	0	0	NUM
ejpam-2562	15	4	for	for	ADP
ejpam-2562	15	5	all	all	DET
ejpam-2562	15	6	submodule	submodule	NOUN
ejpam-2562	15	7	k	k	PROPN
ejpam-2562	15	8	of	of	ADP
ejpam-2562	15	9	m	m	PROPN
ejpam-2562	15	10	.	.	PUNCT
ejpam-2562	16	1	a	a	DET
ejpam-2562	16	2	monomorphism	monomorphism	NOUN
ejpam-2562	16	3	f	f	X
ejpam-2562	16	4	:	:	PUNCT
ejpam-2562	16	5	m	m	VERB
ejpam-2562	16	6	−→	−→	ADJ
ejpam-2562	16	7	m	m	VERB
ejpam-2562	16	8	′	′	ADJ
ejpam-2562	16	9	is	be	AUX
ejpam-2562	16	10	said	say	VERB
ejpam-2562	16	11	to	to	PART
ejpam-2562	16	12	be	be	AUX
ejpam-2562	16	13	essential	essential	ADJ
ejpam-2562	16	14	if	if	SCONJ
ejpam-2562	16	15	f	f	PROPN
ejpam-2562	16	16	(	(	PUNCT
ejpam-2562	16	17	m)ã	m)ã	NOUN
ejpam-2562	16	18	m	m	NOUN
ejpam-2562	16	19	′.	′.	NOUN
ejpam-2562	16	20	it	it	PRON
ejpam-2562	16	21	is	be	AUX
ejpam-2562	16	22	known	know	VERB
ejpam-2562	16	23	that	that	SCONJ
ejpam-2562	16	24	an	an	DET
ejpam-2562	16	25	r	r	NOUN
ejpam-2562	16	26	-	-	PUNCT
ejpam-2562	16	27	monomorphism	monomorphism	NOUN
ejpam-2562	16	28	f	f	NOUN
ejpam-2562	16	29	:	:	PUNCT
ejpam-2562	16	30	m	m	VERB
ejpam-2562	16	31	−→	−→	ADJ
ejpam-2562	16	32	m	m	VERB
ejpam-2562	16	33	′	′	NOUN
ejpam-2562	16	34	of	of	ADP
ejpam-2562	16	35	left	left	ADJ
ejpam-2562	16	36	r	r	NOUN
ejpam-2562	16	37	-	-	PUNCT
ejpam-2562	16	38	modules	module	NOUN
ejpam-2562	16	39	is	be	AUX
ejpam-2562	16	40	essential	essential	ADJ
ejpam-2562	16	41	iff	iff	NOUN
ejpam-2562	16	42	for	for	ADP
ejpam-2562	16	43	any	any	DET
ejpam-2562	16	44	r	r	NOUN
ejpam-2562	16	45	-	-	PUNCT
ejpam-2562	16	46	homomorphism	homomorphism	NOUN
ejpam-2562	16	47	g	g	NOUN
ejpam-2562	16	48	:	:	PUNCT
ejpam-2562	16	49	m	m	VERB
ejpam-2562	16	50	′	′	VERB
ejpam-2562	16	51	−→	−→	ADJ
ejpam-2562	16	52	m	m	PUNCT
ejpam-2562	16	53	”	"	PUNCT
ejpam-2562	16	54	,	,	PUNCT
ejpam-2562	16	55	g	g	PROPN
ejpam-2562	16	56	◦	◦	NOUN
ejpam-2562	16	57	f	f	PROPN
ejpam-2562	16	58	is	be	AUX
ejpam-2562	16	59	a	a	DET
ejpam-2562	16	60	monomorphism	monomorphism	NOUN
ejpam-2562	16	61	implies	imply	VERB
ejpam-2562	16	62	that	that	SCONJ
ejpam-2562	16	63	g	g	PROPN
ejpam-2562	16	64	is	be	AUX
ejpam-2562	16	65	a	a	DET
ejpam-2562	16	66	monomorphism	monomorphism	NOUN
ejpam-2562	16	67	.	.	PUNCT
ejpam-2562	17	1	a	a	DET
ejpam-2562	17	2	submodule	submodule	PROPN
ejpam-2562	17	3	n	n	PROPN
ejpam-2562	17	4	of	of	ADP
ejpam-2562	17	5	a	a	DET
ejpam-2562	17	6	module	module	NOUN
ejpam-2562	17	7	m	m	NOUN
ejpam-2562	17	8	is	be	AUX
ejpam-2562	17	9	essential	essential	ADJ
ejpam-2562	17	10	iff	iff	PROPN
ejpam-2562	17	11	the	the	DET
ejpam-2562	17	12	injective	injective	ADJ
ejpam-2562	17	13	map	map	NOUN
ejpam-2562	18	1	i	i	PRON
ejpam-2562	18	2	:	:	PUNCT
ejpam-2562	18	3	n	n	CCONJ
ejpam-2562	18	4	−→	−→	NOUN
ejpam-2562	18	5	m	m	VERB
ejpam-2562	18	6	:	:	PUNCT
ejpam-2562	18	7	x	x	SYM
ejpam-2562	18	8	7→	7→	NUM
ejpam-2562	18	9	x	x	PUNCT
ejpam-2562	18	10	is	be	AUX
ejpam-2562	18	11	essential	essential	ADJ
ejpam-2562	18	12	.	.	PUNCT
ejpam-2562	19	1	∗corresponding	∗corresponde	VERB
ejpam-2562	19	2	author	author	NOUN
ejpam-2562	19	3	.	.	PUNCT
ejpam-2562	20	1	email	email	NOUN
ejpam-2562	20	2	addresses	address	NOUN
ejpam-2562	20	3	:	:	PUNCT
ejpam-2562	20	4	elhjoop@gmail.com	elhjoop@gmail.com	X
ejpam-2562	20	5	(	(	PUNCT
ejpam-2562	20	6	e.	e.	PROPN
ejpam-2562	20	7	diop	diop	PROPN
ejpam-2562	20	8	)	)	PUNCT
ejpam-2562	20	9	,	,	PUNCT
ejpam-2562	20	10	sowdjibab@yahoo.fr	sowdjibab@yahoo.fr	NOUN
ejpam-2562	20	11	(	(	PUNCT
ejpam-2562	20	12	d.	d.	PROPN
ejpam-2562	20	13	sow	sow	PROPN
ejpam-2562	20	14	)	)	PUNCT
ejpam-2562	20	15	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-2562	21	1	250	250	NUM
ejpam-2562	21	2	c	c	X
ejpam-2562	21	3	©	©	PROPN
ejpam-2562	21	4	2016	2016	NUM
ejpam-2562	21	5	ejpam	ejpam	VERB
ejpam-2562	21	6	all	all	DET
ejpam-2562	21	7	rights	right	NOUN
ejpam-2562	21	8	reserved	reserve	VERB
ejpam-2562	21	9	.	.	PUNCT
ejpam-2562	22	1	e.	e.	PROPN
ejpam-2562	22	2	diop	diop	PROPN
ejpam-2562	22	3	,	,	PUNCT
ejpam-2562	22	4	d.	d.	PROPN
ejpam-2562	22	5	sow	sow	PROPN
ejpam-2562	22	6	/	/	SYM
ejpam-2562	22	7	eur	eur	PROPN
ejpam-2562	22	8	.	.	PUNCT
ejpam-2562	23	1	j.	j.	PROPN
ejpam-2562	23	2	pure	pure	PROPN
ejpam-2562	23	3	appl	appl	PROPN
ejpam-2562	23	4	.	.	PROPN
ejpam-2562	23	5	math	math	PROPN
ejpam-2562	23	6	,	,	PUNCT
ejpam-2562	23	7	9	9	NUM
ejpam-2562	23	8	(	(	PUNCT
ejpam-2562	23	9	2016	2016	NUM
ejpam-2562	23	10	)	)	PUNCT
ejpam-2562	23	11	,	,	PUNCT
ejpam-2562	23	12	250	250	NUM
ejpam-2562	23	13	-	-	SYM
ejpam-2562	23	14	265	265	NUM
ejpam-2562	23	15	251	251	NUM
ejpam-2562	23	16	the	the	DET
ejpam-2562	23	17	previous	previous	ADJ
ejpam-2562	23	18	characterisations	characterisation	NOUN
ejpam-2562	23	19	of	of	ADP
ejpam-2562	23	20	"	"	PUNCT
ejpam-2562	23	21	essential	essential	ADJ
ejpam-2562	23	22	"	"	PUNCT
ejpam-2562	23	23	are	be	AUX
ejpam-2562	23	24	not	not	PART
ejpam-2562	23	25	equivalents	equivalent	NOUN
ejpam-2562	23	26	in	in	ADP
ejpam-2562	23	27	semimodules	semimodule	NOUN
ejpam-2562	23	28	theory	theory	NOUN
ejpam-2562	23	29	.	.	PUNCT
ejpam-2562	24	1	hence	hence	ADV
ejpam-2562	24	2	this	this	DET
ejpam-2562	24	3	notion	notion	NOUN
ejpam-2562	24	4	of	of	ADP
ejpam-2562	24	5	"	"	PUNCT
ejpam-2562	24	6	essential	essential	ADJ
ejpam-2562	24	7	"	"	PUNCT
ejpam-2562	24	8	was	be	AUX
ejpam-2562	24	9	generalized	generalize	VERB
ejpam-2562	24	10	in	in	ADP
ejpam-2562	24	11	semimodules	semimodule	NOUN
ejpam-2562	24	12	theory	theory	NOUN
ejpam-2562	24	13	in	in	ADP
ejpam-2562	24	14	two	two	NUM
ejpam-2562	24	15	different	different	ADJ
ejpam-2562	24	16	ways	way	NOUN
ejpam-2562	24	17	.	.	PUNCT
ejpam-2562	25	1	in	in	ADP
ejpam-2562	25	2	golan	golan	PROPN
ejpam-2562	25	3	book	book	PROPN
ejpam-2562	25	4	’s	’s	PART
ejpam-2562	25	5	[	[	X
ejpam-2562	25	6	6	6	NUM
ejpam-2562	25	7	]	]	PUNCT
ejpam-2562	25	8	,	,	PUNCT
ejpam-2562	25	9	it	it	PRON
ejpam-2562	25	10	was	be	AUX
ejpam-2562	25	11	proposed	propose	VERB
ejpam-2562	25	12	the	the	DET
ejpam-2562	25	13	following	follow	VERB
ejpam-2562	25	14	definitions	definition	NOUN
ejpam-2562	25	15	.	.	PUNCT
ejpam-2562	26	1	an	an	DET
ejpam-2562	26	2	r	r	NOUN
ejpam-2562	26	3	-	-	PUNCT
ejpam-2562	26	4	monomorphism	monomorphism	NOUN
ejpam-2562	26	5	f	f	NOUN
ejpam-2562	26	6	:	:	PUNCT
ejpam-2562	26	7	m	m	VERB
ejpam-2562	26	8	−→	−→	ADJ
ejpam-2562	26	9	m	m	VERB
ejpam-2562	26	10	′	′	NOUN
ejpam-2562	26	11	of	of	ADP
ejpam-2562	26	12	left	left	ADJ
ejpam-2562	26	13	r	r	NOUN
ejpam-2562	26	14	-	-	PUNCT
ejpam-2562	26	15	semimodules	semimodule	NOUN
ejpam-2562	26	16	is	be	AUX
ejpam-2562	26	17	essential	essential	ADJ
ejpam-2562	26	18	if	if	SCONJ
ejpam-2562	26	19	for	for	ADP
ejpam-2562	26	20	any	any	DET
ejpam-2562	26	21	r	r	NOUN
ejpam-2562	26	22	-	-	PUNCT
ejpam-2562	26	23	homomorphism	homomorphism	NOUN
ejpam-2562	26	24	g	g	NOUN
ejpam-2562	26	25	:	:	PUNCT
ejpam-2562	26	26	m	m	VERB
ejpam-2562	26	27	′	′	VERB
ejpam-2562	27	1	−→	−→	ADJ
ejpam-2562	27	2	m	m	VERB
ejpam-2562	27	3	′′	′′	PROPN
ejpam-2562	27	4	,	,	PUNCT
ejpam-2562	27	5	g	g	PROPN
ejpam-2562	27	6	◦	◦	NOUN
ejpam-2562	27	7	f	f	PROPN
ejpam-2562	27	8	is	be	AUX
ejpam-2562	27	9	a	a	DET
ejpam-2562	27	10	monomorphism	monomorphism	NOUN
ejpam-2562	27	11	implies	imply	VERB
ejpam-2562	27	12	that	that	SCONJ
ejpam-2562	27	13	g	g	PROPN
ejpam-2562	27	14	is	be	AUX
ejpam-2562	27	15	a	a	DET
ejpam-2562	27	16	monomorphism	monomorphism	NOUN
ejpam-2562	27	17	.	.	PUNCT
ejpam-2562	28	1	a	a	DET
ejpam-2562	28	2	subsemimodule	subsemimodule	NOUN
ejpam-2562	28	3	n	n	NOUN
ejpam-2562	28	4	of	of	ADP
ejpam-2562	28	5	a	a	DET
ejpam-2562	28	6	left	left	ADJ
ejpam-2562	28	7	r	r	NOUN
ejpam-2562	28	8	-	-	PUNCT
ejpam-2562	28	9	semimodule	semimodule	NOUN
ejpam-2562	28	10	m	m	NOUN
ejpam-2562	28	11	is	be	AUX
ejpam-2562	28	12	essential	essential	ADJ
ejpam-2562	28	13	(	(	PUNCT
ejpam-2562	28	14	or	or	CCONJ
ejpam-2562	28	15	large	large	ADJ
ejpam-2562	28	16	)	)	PUNCT
ejpam-2562	28	17	in	in	ADP
ejpam-2562	28	18	m	m	PROPN
ejpam-2562	28	19	if	if	SCONJ
ejpam-2562	28	20	the	the	DET
ejpam-2562	28	21	inclusion	inclusion	NOUN
ejpam-2562	28	22	map	map	NOUN
ejpam-2562	28	23	in	in	ADP
ejpam-2562	28	24	:	:	PUNCT
ejpam-2562	28	25	n	n	CCONJ
ejpam-2562	29	1	−→	−→	NOUN
ejpam-2562	29	2	m	m	VERB
ejpam-2562	29	3	is	be	AUX
ejpam-2562	29	4	an	an	DET
ejpam-2562	29	5	essential	essential	ADJ
ejpam-2562	29	6	r	r	NOUN
ejpam-2562	29	7	-	-	PUNCT
ejpam-2562	29	8	monomorphism	monomorphism	NOUN
ejpam-2562	29	9	.	.	PUNCT
ejpam-2562	30	1	note	note	VERB
ejpam-2562	30	2	that	that	SCONJ
ejpam-2562	31	1	f	f	X
ejpam-2562	31	2	:	:	PUNCT
ejpam-2562	31	3	m	m	VERB
ejpam-2562	31	4	−→	−→	ADJ
ejpam-2562	32	1	m	m	VERB
ejpam-2562	32	2	′	′	ADJ
ejpam-2562	32	3	is	be	AUX
ejpam-2562	32	4	an	an	DET
ejpam-2562	32	5	essential	essential	ADJ
ejpam-2562	32	6	r	r	NOUN
ejpam-2562	32	7	-	-	PUNCT
ejpam-2562	32	8	homomorphism	homomorphism	NOUN
ejpam-2562	32	9	if	if	SCONJ
ejpam-2562	32	10	and	and	CCONJ
ejpam-2562	32	11	only	only	ADV
ejpam-2562	32	12	if	if	SCONJ
ejpam-2562	32	13	f	f	PROPN
ejpam-2562	32	14	(	(	PUNCT
ejpam-2562	32	15	m	m	PROPN
ejpam-2562	32	16	)	)	PUNCT
ejpam-2562	32	17	is	be	AUX
ejpam-2562	32	18	a	a	DET
ejpam-2562	32	19	large	large	ADJ
ejpam-2562	32	20	subsemimodule	subsemimodule	NOUN
ejpam-2562	32	21	of	of	ADP
ejpam-2562	32	22	m	m	PROPN
ejpam-2562	32	23	′.	′.	NOUN
ejpam-2562	32	24	another	another	DET
ejpam-2562	32	25	way	way	NOUN
ejpam-2562	32	26	for	for	ADP
ejpam-2562	32	27	defining	define	VERB
ejpam-2562	32	28	the	the	DET
ejpam-2562	32	29	notion	notion	NOUN
ejpam-2562	32	30	of	of	ADP
ejpam-2562	32	31	"	"	PUNCT
ejpam-2562	32	32	essential	essential	ADJ
ejpam-2562	32	33	"	"	PUNCT
ejpam-2562	32	34	is	be	AUX
ejpam-2562	32	35	proposed	propose	VERB
ejpam-2562	32	36	in	in	ADP
ejpam-2562	32	37	[	[	X
ejpam-2562	32	38	5	5	NUM
ejpam-2562	32	39	]	]	PUNCT
ejpam-2562	32	40	as	as	SCONJ
ejpam-2562	32	41	follows	follow	VERB
ejpam-2562	32	42	.	.	PUNCT
ejpam-2562	33	1	a	a	DET
ejpam-2562	33	2	left	left	ADJ
ejpam-2562	33	3	r	r	NOUN
ejpam-2562	33	4	-	-	PUNCT
ejpam-2562	33	5	subsemimodule	subsemimodule	NOUN
ejpam-2562	33	6	n	n	PROPN
ejpam-2562	33	7	of	of	ADP
ejpam-2562	33	8	m	m	PROPN
ejpam-2562	33	9	is	be	AUX
ejpam-2562	33	10	said	say	VERB
ejpam-2562	33	11	to	to	PART
ejpam-2562	33	12	be	be	AUX
ejpam-2562	33	13	semi	semi	ADJ
ejpam-2562	33	14	-	-	ADJ
ejpam-2562	33	15	essential	essential	ADJ
ejpam-2562	33	16	in	in	ADP
ejpam-2562	33	17	m	m	PRON
ejpam-2562	33	18	,	,	PUNCT
ejpam-2562	33	19	written	write	VERB
ejpam-2562	33	20	as	as	ADP
ejpam-2562	33	21	n	n	ADP
ejpam-2562	33	22	ãs	ãs	NOUN
ejpam-2562	33	23	m	m	ADJ
ejpam-2562	33	24	,	,	PUNCT
ejpam-2562	33	25	if	if	SCONJ
ejpam-2562	33	26	for	for	ADP
ejpam-2562	33	27	every	every	DET
ejpam-2562	33	28	rsubsemimodule	rsubsemimodule	NOUN
ejpam-2562	33	29	k	k	PROPN
ejpam-2562	33	30	of	of	ADP
ejpam-2562	33	31	m	m	PROPN
ejpam-2562	33	32	:	:	PUNCT
ejpam-2562	33	33	n	n	PRON
ejpam-2562	33	34	∩k	∩k	NOUN
ejpam-2562	33	35	=	=	SYM
ejpam-2562	34	1	0⇒	0⇒	NUM
ejpam-2562	35	1	k	k	X
ejpam-2562	35	2	=	=	PUNCT
ejpam-2562	35	3	0	0	PROPN
ejpam-2562	35	4	.	.	PUNCT
ejpam-2562	36	1	a	a	DET
ejpam-2562	36	2	monomorphism	monomorphism	NOUN
ejpam-2562	36	3	(	(	PUNCT
ejpam-2562	36	4	respectively	respectively	ADV
ejpam-2562	36	5	:	:	PUNCT
ejpam-2562	36	6	semimonomorphism	semimonomorphism	NOUN
ejpam-2562	36	7	)	)	PUNCT
ejpam-2562	36	8	f	f	NOUN
ejpam-2562	36	9	:	:	PUNCT
ejpam-2562	36	10	m	m	VERB
ejpam-2562	36	11	−→	−→	ADJ
ejpam-2562	36	12	m	m	VERB
ejpam-2562	36	13	′	′	NOUN
ejpam-2562	36	14	of	of	ADP
ejpam-2562	36	15	left	left	ADJ
ejpam-2562	36	16	r	r	NOUN
ejpam-2562	36	17	-	-	PUNCT
ejpam-2562	36	18	semimodules	semimodule	NOUN
ejpam-2562	36	19	is	be	AUX
ejpam-2562	36	20	said	say	VERB
ejpam-2562	36	21	to	to	PART
ejpam-2562	36	22	be	be	AUX
ejpam-2562	36	23	semi	semi	ADJ
ejpam-2562	36	24	-	-	ADJ
ejpam-2562	36	25	essential	essential	ADJ
ejpam-2562	36	26	if	if	SCONJ
ejpam-2562	36	27	:	:	PUNCT
ejpam-2562	36	28	f	f	PROPN
ejpam-2562	36	29	(	(	PUNCT
ejpam-2562	36	30	m)ãs	m)ãs	PROPN
ejpam-2562	36	31	m	m	NOUN
ejpam-2562	36	32	′.	′.	NOUN
ejpam-2562	36	33	these	these	DET
ejpam-2562	36	34	two	two	NUM
ejpam-2562	36	35	differents	different	NOUN
ejpam-2562	36	36	notions	notion	NOUN
ejpam-2562	36	37	of	of	ADP
ejpam-2562	36	38	"	"	PUNCT
ejpam-2562	36	39	essential	essential	ADJ
ejpam-2562	36	40	subsemimodules	subsemimodule	NOUN
ejpam-2562	36	41	"	"	PUNCT
ejpam-2562	36	42	in	in	ADP
ejpam-2562	36	43	the	the	DET
ejpam-2562	36	44	theory	theory	NOUN
ejpam-2562	36	45	of	of	ADP
ejpam-2562	36	46	semimodules	semimodule	NOUN
ejpam-2562	36	47	(	(	PUNCT
ejpam-2562	36	48	see	see	VERB
ejpam-2562	36	49	[	[	X
ejpam-2562	36	50	5	5	NUM
ejpam-2562	36	51	]	]	PUNCT
ejpam-2562	36	52	)	)	PUNCT
ejpam-2562	36	53	are	be	AUX
ejpam-2562	36	54	the	the	DET
ejpam-2562	36	55	same	same	ADJ
ejpam-2562	36	56	in	in	ADP
ejpam-2562	36	57	the	the	DET
ejpam-2562	36	58	theory	theory	NOUN
ejpam-2562	36	59	of	of	ADP
ejpam-2562	36	60	modules	module	NOUN
ejpam-2562	36	61	.	.	PUNCT
ejpam-2562	37	1	also	also	ADV
ejpam-2562	37	2	,	,	PUNCT
ejpam-2562	37	3	it	it	PRON
ejpam-2562	37	4	was	be	AUX
ejpam-2562	37	5	proved	prove	VERB
ejpam-2562	37	6	[	[	X
ejpam-2562	37	7	5	5	X
ejpam-2562	37	8	]	]	PUNCT
ejpam-2562	37	9	that	that	SCONJ
ejpam-2562	37	10	the	the	DET
ejpam-2562	37	11	class	class	NOUN
ejpam-2562	37	12	of	of	ADP
ejpam-2562	37	13	essential	essential	ADJ
ejpam-2562	37	14	subsemimodules	subsemimodule	NOUN
ejpam-2562	37	15	is	be	AUX
ejpam-2562	37	16	not	not	PART
ejpam-2562	37	17	contained	contain	VERB
ejpam-2562	37	18	and	and	CCONJ
ejpam-2562	37	19	does	do	AUX
ejpam-2562	37	20	n’t	not	PART
ejpam-2562	37	21	contain	contain	VERB
ejpam-2562	37	22	the	the	DET
ejpam-2562	37	23	class	class	NOUN
ejpam-2562	37	24	of	of	ADP
ejpam-2562	37	25	semi	semi	ADJ
ejpam-2562	37	26	-	-	ADJ
ejpam-2562	37	27	essential	essential	ADJ
ejpam-2562	37	28	subsemimodules	subsemimodule	NOUN
ejpam-2562	37	29	.	.	PUNCT
ejpam-2562	38	1	furthermore	furthermore	ADV
ejpam-2562	38	2	,	,	PUNCT
ejpam-2562	38	3	the	the	DET
ejpam-2562	38	4	intersection	intersection	NOUN
ejpam-2562	38	5	of	of	ADP
ejpam-2562	38	6	these	these	DET
ejpam-2562	38	7	two	two	NUM
ejpam-2562	38	8	classes	class	NOUN
ejpam-2562	38	9	is	be	AUX
ejpam-2562	38	10	not	not	PART
ejpam-2562	38	11	empty	empty	ADJ
ejpam-2562	38	12	.	.	PUNCT
ejpam-2562	39	1	in	in	ADP
ejpam-2562	39	2	this	this	DET
ejpam-2562	39	3	paper	paper	NOUN
ejpam-2562	39	4	,	,	PUNCT
ejpam-2562	39	5	we	we	PRON
ejpam-2562	39	6	investigate	investigate	VERB
ejpam-2562	39	7	a	a	DET
ejpam-2562	39	8	new	new	ADJ
ejpam-2562	39	9	class	class	NOUN
ejpam-2562	39	10	of	of	ADP
ejpam-2562	39	11	"	"	PUNCT
ejpam-2562	39	12	essential	essential	ADJ
ejpam-2562	39	13	"	"	PUNCT
ejpam-2562	39	14	subsemimodules	subsemimodule	NOUN
ejpam-2562	39	15	(	(	PUNCT
ejpam-2562	39	16	called	call	VERB
ejpam-2562	39	17	here	here	ADV
ejpam-2562	39	18	:	:	PUNCT
ejpam-2562	39	19	semiweakly	semiweakly	ADJ
ejpam-2562	39	20	-	-	PUNCT
ejpam-2562	39	21	essential	essential	ADJ
ejpam-2562	39	22	subsemimodules	subsemimodule	NOUN
ejpam-2562	39	23	)	)	PUNCT
ejpam-2562	39	24	.	.	PUNCT
ejpam-2562	40	1	in	in	ADP
ejpam-2562	40	2	modules	module	NOUN
ejpam-2562	40	3	theory	theory	NOUN
ejpam-2562	40	4	,	,	PUNCT
ejpam-2562	40	5	it	it	PRON
ejpam-2562	40	6	is	be	AUX
ejpam-2562	40	7	well	well	ADV
ejpam-2562	40	8	known	know	VERB
ejpam-2562	40	9	that	that	SCONJ
ejpam-2562	40	10	the	the	DET
ejpam-2562	40	11	congruence	congruence	PROPN
ejpam-2562	40	12	relations	relation	NOUN
ejpam-2562	40	13	are	be	AUX
ejpam-2562	40	14	defined	define	VERB
ejpam-2562	40	15	by	by	ADP
ejpam-2562	40	16	the	the	DET
ejpam-2562	40	17	submodules	submodule	NOUN
ejpam-2562	40	18	but	but	CCONJ
ejpam-2562	40	19	not	not	PART
ejpam-2562	40	20	in	in	ADP
ejpam-2562	40	21	theory	theory	NOUN
ejpam-2562	40	22	of	of	ADP
ejpam-2562	40	23	semimodules	semimodule	NOUN
ejpam-2562	40	24	.	.	PUNCT
ejpam-2562	41	1	so	so	ADV
ejpam-2562	41	2	in	in	ADP
ejpam-2562	41	3	the	the	DET
ejpam-2562	41	4	new	new	ADJ
ejpam-2562	41	5	class	class	NOUN
ejpam-2562	41	6	we	we	PRON
ejpam-2562	41	7	consider	consider	VERB
ejpam-2562	41	8	the	the	DET
ejpam-2562	41	9	congruence	congruence	NOUN
ejpam-2562	41	10	relations	relation	NOUN
ejpam-2562	41	11	defined	define	VERB
ejpam-2562	41	12	only	only	ADV
ejpam-2562	41	13	by	by	ADP
ejpam-2562	41	14	the	the	DET
ejpam-2562	41	15	subsemimodules	subsemimodule	NOUN
ejpam-2562	41	16	.	.	PUNCT
ejpam-2562	42	1	we	we	PRON
ejpam-2562	42	2	show	show	VERB
ejpam-2562	42	3	that	that	SCONJ
ejpam-2562	42	4	this	this	DET
ejpam-2562	42	5	new	new	ADJ
ejpam-2562	42	6	class	class	NOUN
ejpam-2562	42	7	contains	contain	VERB
ejpam-2562	42	8	the	the	DET
ejpam-2562	42	9	two	two	NUM
ejpam-2562	42	10	known	know	VERB
ejpam-2562	42	11	classes	class	NOUN
ejpam-2562	42	12	of	of	ADP
ejpam-2562	42	13	"	"	PUNCT
ejpam-2562	42	14	essential	essential	ADJ
ejpam-2562	42	15	"	"	PUNCT
ejpam-2562	42	16	subsemimodules	subsemimodule	NOUN
ejpam-2562	42	17	.	.	PUNCT
ejpam-2562	43	1	futhermore	futhermore	NOUN
ejpam-2562	43	2	,	,	PUNCT
ejpam-2562	43	3	we	we	PRON
ejpam-2562	43	4	study	study	VERB
ejpam-2562	43	5	some	some	DET
ejpam-2562	43	6	interesting	interesting	ADJ
ejpam-2562	43	7	properties	property	NOUN
ejpam-2562	43	8	of	of	ADP
ejpam-2562	43	9	this	this	DET
ejpam-2562	43	10	new	new	ADJ
ejpam-2562	43	11	class	class	NOUN
ejpam-2562	43	12	.	.	PUNCT
ejpam-2562	44	1	as	as	ADP
ejpam-2562	44	2	applications	application	NOUN
ejpam-2562	44	3	we	we	PRON
ejpam-2562	44	4	introduce	introduce	VERB
ejpam-2562	44	5	three	three	NUM
ejpam-2562	44	6	notions	notion	NOUN
ejpam-2562	44	7	of	of	ADP
ejpam-2562	44	8	semi	semi	ADJ
ejpam-2562	44	9	-	-	ADJ
ejpam-2562	44	10	weakly	weakly	ADJ
ejpam-2562	44	11	-	-	PUNCT
ejpam-2562	44	12	co	co	ADJ
ejpam-2562	44	13	-	-	ADJ
ejpam-2562	44	14	hopfian	hopfian	ADJ
ejpam-2562	44	15	semimodules	semimodule	NOUN
ejpam-2562	44	16	.	.	PUNCT
ejpam-2562	45	1	all	all	DET
ejpam-2562	45	2	semirings	semiring	NOUN
ejpam-2562	45	3	are	be	AUX
ejpam-2562	45	4	associative	associative	ADJ
ejpam-2562	45	5	with	with	ADP
ejpam-2562	45	6	identity	identity	NOUN
ejpam-2562	45	7	1	1	NUM
ejpam-2562	45	8	(	(	PUNCT
ejpam-2562	45	9	if	if	SCONJ
ejpam-2562	45	10	r	r	NOUN
ejpam-2562	45	11	is	be	AUX
ejpam-2562	45	12	a	a	DET
ejpam-2562	45	13	semiring	semiring	NOUN
ejpam-2562	45	14	,	,	PUNCT
ejpam-2562	45	15	we	we	PRON
ejpam-2562	45	16	assume	assume	VERB
ejpam-2562	45	17	that	that	SCONJ
ejpam-2562	45	18	1	1	NUM
ejpam-2562	45	19	6=	6=	NUM
ejpam-2562	45	20	0	0	NUM
ejpam-2562	45	21	)	)	PUNCT
ejpam-2562	45	22	,	,	PUNCT
ejpam-2562	45	23	all	all	DET
ejpam-2562	45	24	semimodules	semimodule	NOUN
ejpam-2562	45	25	are	be	AUX
ejpam-2562	45	26	unital	unital	ADJ
ejpam-2562	45	27	and	and	CCONJ
ejpam-2562	45	28	all	all	DET
ejpam-2562	45	29	semiring	semire	VERB
ejpam-2562	45	30	extensions	extension	NOUN
ejpam-2562	45	31	contain	contain	VERB
ejpam-2562	45	32	the	the	DET
ejpam-2562	45	33	common	common	ADJ
ejpam-2562	45	34	identity	identity	NOUN
ejpam-2562	45	35	.	.	PUNCT
ejpam-2562	46	1	throughout	throughout	ADP
ejpam-2562	46	2	this	this	DET
ejpam-2562	46	3	paper	paper	NOUN
ejpam-2562	46	4	,	,	PUNCT
ejpam-2562	46	5	for	for	ADP
ejpam-2562	46	6	semimodule	semimodule	NOUN
ejpam-2562	46	7	theoretic	theoretic	ADJ
ejpam-2562	46	8	notions	notion	NOUN
ejpam-2562	46	9	and	and	CCONJ
ejpam-2562	46	10	notations	notation	NOUN
ejpam-2562	46	11	we	we	PRON
ejpam-2562	46	12	will	will	AUX
ejpam-2562	46	13	follow	follow	VERB
ejpam-2562	46	14	[	[	X
ejpam-2562	46	15	1	1	NUM
ejpam-2562	46	16	]	]	PUNCT
ejpam-2562	46	17	and	and	CCONJ
ejpam-2562	46	18	[	[	X
ejpam-2562	46	19	6	6	NUM
ejpam-2562	46	20	]	]	PUNCT
ejpam-2562	46	21	.	.	PUNCT
ejpam-2562	47	1	in	in	ADP
ejpam-2562	47	2	the	the	DET
ejpam-2562	47	3	following	following	NOUN
ejpam-2562	47	4	,	,	PUNCT
ejpam-2562	47	5	we	we	PRON
ejpam-2562	47	6	recall	recall	VERB
ejpam-2562	47	7	some	some	DET
ejpam-2562	47	8	definitions	definition	NOUN
ejpam-2562	47	9	and	and	CCONJ
ejpam-2562	47	10	notations	notation	NOUN
ejpam-2562	47	11	that	that	PRON
ejpam-2562	47	12	will	will	AUX
ejpam-2562	47	13	be	be	AUX
ejpam-2562	47	14	used	use	VERB
ejpam-2562	47	15	in	in	ADP
ejpam-2562	47	16	this	this	DET
ejpam-2562	47	17	paper	paper	NOUN
ejpam-2562	47	18	.	.	PUNCT
ejpam-2562	48	1	this	this	DET
ejpam-2562	48	2	work	work	NOUN
ejpam-2562	48	3	is	be	AUX
ejpam-2562	48	4	organized	organize	VERB
ejpam-2562	48	5	as	as	SCONJ
ejpam-2562	48	6	follows	follow	VERB
ejpam-2562	48	7	:	:	PUNCT
ejpam-2562	48	8	•	•	NOUN
ejpam-2562	48	9	in	in	ADP
ejpam-2562	48	10	section	section	NOUN
ejpam-2562	48	11	1	1	NUM
ejpam-2562	48	12	:	:	PUNCT
ejpam-2562	48	13	preliminaries	preliminary	NOUN
ejpam-2562	48	14	:	:	PUNCT
ejpam-2562	48	15	we	we	PRON
ejpam-2562	48	16	give	give	VERB
ejpam-2562	48	17	some	some	DET
ejpam-2562	48	18	results	result	NOUN
ejpam-2562	48	19	which	which	PRON
ejpam-2562	48	20	we	we	PRON
ejpam-2562	48	21	will	will	AUX
ejpam-2562	48	22	use	use	VERB
ejpam-2562	48	23	in	in	ADP
ejpam-2562	48	24	the	the	DET
ejpam-2562	48	25	sequel	sequel	NOUN
ejpam-2562	48	26	.	.	PUNCT
ejpam-2562	49	1	•	•	NUM
ejpam-2562	49	2	in	in	ADP
ejpam-2562	49	3	section	section	NOUN
ejpam-2562	49	4	2	2	NUM
ejpam-2562	49	5	:	:	PUNCT
ejpam-2562	49	6	new	new	ADJ
ejpam-2562	49	7	notions	notion	NOUN
ejpam-2562	49	8	of	of	ADP
ejpam-2562	49	9	essential	essential	ADJ
ejpam-2562	49	10	:	:	PUNCT
ejpam-2562	49	11	some	some	DET
ejpam-2562	49	12	properties	property	NOUN
ejpam-2562	49	13	of	of	ADP
ejpam-2562	49	14	semi	semi	ADJ
ejpam-2562	49	15	-	-	ADJ
ejpam-2562	49	16	weakly	weakly	ADJ
ejpam-2562	49	17	-	-	PUNCT
ejpam-2562	49	18	essential	essential	ADJ
ejpam-2562	49	19	subsemimodules	subsemimodule	NOUN
ejpam-2562	49	20	are	be	AUX
ejpam-2562	49	21	investigated	investigate	VERB
ejpam-2562	49	22	.	.	PUNCT
ejpam-2562	50	1	•	•	NUM
ejpam-2562	50	2	in	in	ADP
ejpam-2562	50	3	section	section	NOUN
ejpam-2562	50	4	3	3	NUM
ejpam-2562	50	5	:	:	PUNCT
ejpam-2562	50	6	applications	application	NOUN
ejpam-2562	50	7	:	:	PUNCT
ejpam-2562	50	8	three	three	NUM
ejpam-2562	50	9	types	type	NOUN
ejpam-2562	50	10	of	of	ADP
ejpam-2562	50	11	semi	semi	ADJ
ejpam-2562	50	12	-	-	ADJ
ejpam-2562	50	13	weakly	weakly	ADJ
ejpam-2562	50	14	-	-	PUNCT
ejpam-2562	50	15	co	co	NOUN
ejpam-2562	50	16	-	-	ADJ
ejpam-2562	50	17	hopfian	hopfian	ADJ
ejpam-2562	50	18	semimodules	semimodule	NOUN
ejpam-2562	50	19	are	be	AUX
ejpam-2562	50	20	introduced	introduce	VERB
ejpam-2562	50	21	.	.	PUNCT
ejpam-2562	51	1	2	2	X
ejpam-2562	51	2	.	.	X
ejpam-2562	51	3	preliminaries	preliminary	NOUN
ejpam-2562	51	4	we	we	PRON
ejpam-2562	51	5	recall	recall	VERB
ejpam-2562	51	6	briefly	briefly	ADV
ejpam-2562	51	7	some	some	DET
ejpam-2562	51	8	basic	basic	ADJ
ejpam-2562	51	9	notions	notion	NOUN
ejpam-2562	51	10	about	about	ADP
ejpam-2562	51	11	semimodules	semimodule	NOUN
ejpam-2562	51	12	.	.	PUNCT
ejpam-2562	52	1	e.	e.	PROPN
ejpam-2562	52	2	diop	diop	PROPN
ejpam-2562	52	3	,	,	PUNCT
ejpam-2562	52	4	d.	d.	PROPN
ejpam-2562	52	5	sow	sow	PROPN
ejpam-2562	52	6	/	/	SYM
ejpam-2562	52	7	eur	eur	PROPN
ejpam-2562	52	8	.	.	PUNCT
ejpam-2562	53	1	j.	j.	PROPN
ejpam-2562	53	2	pure	pure	PROPN
ejpam-2562	53	3	appl	appl	PROPN
ejpam-2562	53	4	.	.	PROPN
ejpam-2562	53	5	math	math	PROPN
ejpam-2562	53	6	,	,	PUNCT
ejpam-2562	53	7	9	9	NUM
ejpam-2562	53	8	(	(	PUNCT
ejpam-2562	53	9	2016	2016	NUM
ejpam-2562	53	10	)	)	PUNCT
ejpam-2562	53	11	,	,	PUNCT
ejpam-2562	53	12	250	250	NUM
ejpam-2562	53	13	-	-	SYM
ejpam-2562	53	14	265	265	NUM
ejpam-2562	53	15	252	252	NUM
ejpam-2562	53	16	we	we	PRON
ejpam-2562	53	17	denote	denote	VERB
ejpam-2562	53	18	by	by	ADP
ejpam-2562	53	19	n	n	DET
ejpam-2562	53	20	≤	≤	NOUN
ejpam-2562	53	21	m	m	VERB
ejpam-2562	53	22	,	,	PUNCT
ejpam-2562	53	23	if	if	SCONJ
ejpam-2562	53	24	n	n	PRON
ejpam-2562	53	25	is	be	AUX
ejpam-2562	53	26	a	a	DET
ejpam-2562	53	27	subsemimodule	subsemimodule	NOUN
ejpam-2562	53	28	of	of	ADP
ejpam-2562	53	29	a	a	DET
ejpam-2562	53	30	semimodule	semimodule	NOUN
ejpam-2562	53	31	m	m	PROPN
ejpam-2562	53	32	and	and	CCONJ
ejpam-2562	53	33	by	by	ADP
ejpam-2562	53	34	homomorphism	homomorphism	NOUN
ejpam-2562	53	35	,	,	PUNCT
ejpam-2562	53	36	we	we	PRON
ejpam-2562	53	37	mean	mean	VERB
ejpam-2562	53	38	a	a	DET
ejpam-2562	53	39	homomorphism	homomorphism	NOUN
ejpam-2562	53	40	of	of	ADP
ejpam-2562	53	41	left	left	ADJ
ejpam-2562	53	42	r	r	NOUN
ejpam-2562	53	43	-	-	PUNCT
ejpam-2562	53	44	semimodules	semimodule	NOUN
ejpam-2562	53	45	.	.	PUNCT
ejpam-2562	54	1	throughout	throughout	ADP
ejpam-2562	54	2	this	this	DET
ejpam-2562	54	3	paper	paper	NOUN
ejpam-2562	54	4	,	,	PUNCT
ejpam-2562	54	5	we	we	PRON
ejpam-2562	54	6	consider	consider	VERB
ejpam-2562	54	7	the	the	DET
ejpam-2562	54	8	left	left	ADJ
ejpam-2562	54	9	r	r	NOUN
ejpam-2562	54	10	-	-	PUNCT
ejpam-2562	54	11	semimodules	semimodule	NOUN
ejpam-2562	54	12	,	,	PUNCT
ejpam-2562	54	13	but	but	CCONJ
ejpam-2562	54	14	the	the	DET
ejpam-2562	54	15	results	result	NOUN
ejpam-2562	54	16	are	be	AUX
ejpam-2562	54	17	also	also	ADV
ejpam-2562	54	18	true	true	ADJ
ejpam-2562	54	19	for	for	ADP
ejpam-2562	54	20	right	right	ADJ
ejpam-2562	54	21	r	r	NOUN
ejpam-2562	54	22	-	-	PUNCT
ejpam-2562	54	23	semimodules	semimodule	NOUN
ejpam-2562	54	24	and	and	CCONJ
ejpam-2562	54	25	the	the	DET
ejpam-2562	54	26	proofs	proof	NOUN
ejpam-2562	54	27	are	be	AUX
ejpam-2562	54	28	similar	similar	ADJ
ejpam-2562	54	29	.	.	PUNCT
ejpam-2562	55	1	definition	definition	NOUN
ejpam-2562	55	2	1	1	NUM
ejpam-2562	55	3	.	.	PUNCT
ejpam-2562	56	1	let	let	VERB
ejpam-2562	56	2	m	m	PRON
ejpam-2562	56	3	be	be	AUX
ejpam-2562	56	4	a	a	DET
ejpam-2562	56	5	left	left	ADJ
ejpam-2562	56	6	r	r	NOUN
ejpam-2562	56	7	-	-	PUNCT
ejpam-2562	56	8	semimodule	semimodule	NOUN
ejpam-2562	56	9	.	.	PUNCT
ejpam-2562	57	1	•	•	NUM
ejpam-2562	57	2	an	an	DET
ejpam-2562	57	3	r	r	NOUN
ejpam-2562	57	4	-	-	PUNCT
ejpam-2562	57	5	congruence	congruence	NOUN
ejpam-2562	57	6	relation	relation	NOUN
ejpam-2562	57	7	on	on	ADP
ejpam-2562	57	8	a	a	DET
ejpam-2562	57	9	semimodule	semimodule	NOUN
ejpam-2562	57	10	m	m	VERB
ejpam-2562	57	11	is	be	AUX
ejpam-2562	57	12	an	an	DET
ejpam-2562	57	13	equivalence	equivalence	NOUN
ejpam-2562	57	14	relation	relation	NOUN
ejpam-2562	57	15	ρ	ρ	PROPN
ejpam-2562	57	16	on	on	ADP
ejpam-2562	57	17	m	m	PRON
ejpam-2562	57	18	such	such	ADJ
ejpam-2562	57	19	that	that	SCONJ
ejpam-2562	57	20	mρm′	mρm′	NOUN
ejpam-2562	57	21	and	and	CCONJ
ejpam-2562	57	22	nρn′	nρn′	NOUN
ejpam-2562	57	23	=	=	NOUN
ejpam-2562	57	24	⇒	⇒	NOUN
ejpam-2562	57	25	(	(	PUNCT
ejpam-2562	57	26	m+	m+	NUM
ejpam-2562	57	27	n)ρ(m′	n)ρ(m′	NOUN
ejpam-2562	57	28	+	+	NOUN
ejpam-2562	57	29	n′	n′	NUM
ejpam-2562	57	30	)	)	PUNCT
ejpam-2562	57	31	and	and	CCONJ
ejpam-2562	57	32	(	(	PUNCT
ejpam-2562	57	33	rm)ρ(rm′	rm)ρ(rm′	PROPN
ejpam-2562	57	34	)	)	PUNCT
ejpam-2562	57	35	,	,	PUNCT
ejpam-2562	57	36	∀m	∀m	PROPN
ejpam-2562	57	37	,	,	PUNCT
ejpam-2562	57	38	m′	m′	NUM
ejpam-2562	57	39	,	,	PUNCT
ejpam-2562	57	40	n	n	CCONJ
ejpam-2562	57	41	,	,	PUNCT
ejpam-2562	57	42	n′	n′	PROPN
ejpam-2562	57	43	∈	∈	PROPN
ejpam-2562	57	44	m	m	NOUN
ejpam-2562	57	45	and	and	CCONJ
ejpam-2562	57	46	r	r	PROPN
ejpam-2562	57	47	∈	∈	PROPN
ejpam-2562	57	48	r.	r.	NOUN
ejpam-2562	57	49	•	•	NOUN
ejpam-2562	58	1	the	the	DET
ejpam-2562	58	2	congruence	congruence	PROPN
ejpam-2562	58	3	relation	relation	NOUN
ejpam-2562	58	4	ρ	ρ	PROPN
ejpam-2562	58	5	defined	define	VERB
ejpam-2562	58	6	on	on	ADP
ejpam-2562	58	7	m	m	NOUN
ejpam-2562	58	8	by	by	ADP
ejpam-2562	58	9	mρm′	mρm′	NOUN
ejpam-2562	58	10	⇐	⇐	ADJ
ejpam-2562	58	11	⇒	⇒	NOUN
ejpam-2562	58	12	m=	m=	X
ejpam-2562	58	13	m′	m′	NOUN
ejpam-2562	58	14	is	be	AUX
ejpam-2562	58	15	called	call	VERB
ejpam-2562	58	16	a	a	DET
ejpam-2562	58	17	trivial	trivial	ADJ
ejpam-2562	58	18	congruence	congruence	NOUN
ejpam-2562	58	19	relation	relation	NOUN
ejpam-2562	58	20	on	on	ADP
ejpam-2562	58	21	m.	m.	NOUN
ejpam-2562	58	22	•	•	ADP
ejpam-2562	58	23	the	the	DET
ejpam-2562	58	24	congruence	congruence	PROPN
ejpam-2562	58	25	relation	relation	NOUN
ejpam-2562	58	26	ρ	ρ	PROPN
ejpam-2562	58	27	defined	define	VERB
ejpam-2562	58	28	on	on	ADP
ejpam-2562	58	29	m	m	NOUN
ejpam-2562	58	30	by	by	ADP
ejpam-2562	58	31	mρm′	mρm′	PROPN
ejpam-2562	58	32	∀m	∀m	NUM
ejpam-2562	58	33	,	,	PUNCT
ejpam-2562	58	34	m′	m′	NOUN
ejpam-2562	58	35	∈	∈	NOUN
ejpam-2562	58	36	m	m	VERB
ejpam-2562	58	37	is	be	AUX
ejpam-2562	58	38	called	call	VERB
ejpam-2562	58	39	universal	universal	ADJ
ejpam-2562	58	40	congruence	congruence	NOUN
ejpam-2562	58	41	relation	relation	NOUN
ejpam-2562	58	42	on	on	ADP
ejpam-2562	58	43	m.	m.	NOUN
ejpam-2562	58	44	•	•	ADP
ejpam-2562	58	45	m	m	VERB
ejpam-2562	58	46	is	be	AUX
ejpam-2562	58	47	a	a	DET
ejpam-2562	58	48	r	r	NOUN
ejpam-2562	58	49	-	-	PUNCT
ejpam-2562	58	50	simple	simple	ADJ
ejpam-2562	58	51	semimodule	semimodule	NOUN
ejpam-2562	58	52	if	if	SCONJ
ejpam-2562	58	53	any	any	DET
ejpam-2562	58	54	congruence	congruence	NOUN
ejpam-2562	58	55	relation	relation	NOUN
ejpam-2562	58	56	defined	define	VERB
ejpam-2562	58	57	over	over	ADP
ejpam-2562	58	58	m	m	PROPN
ejpam-2562	58	59	is	be	AUX
ejpam-2562	58	60	trivial	trivial	ADJ
ejpam-2562	58	61	or	or	CCONJ
ejpam-2562	58	62	universal	universal	ADJ
ejpam-2562	58	63	.	.	PUNCT
ejpam-2562	59	1	remark	remark	PROPN
ejpam-2562	59	2	1	1	NUM
ejpam-2562	59	3	.	.	PUNCT
ejpam-2562	60	1	the	the	DET
ejpam-2562	60	2	set	set	NOUN
ejpam-2562	60	3	of	of	ADP
ejpam-2562	60	4	all	all	DET
ejpam-2562	60	5	r	r	NOUN
ejpam-2562	60	6	-	-	PUNCT
ejpam-2562	60	7	congruence	congruence	NOUN
ejpam-2562	60	8	relation	relation	NOUN
ejpam-2562	60	9	on	on	ADP
ejpam-2562	60	10	m	m	PROPN
ejpam-2562	60	11	,	,	PUNCT
ejpam-2562	60	12	r−	r−	PROPN
ejpam-2562	60	13	cong(m	cong(m	PROPN
ejpam-2562	60	14	)	)	PUNCT
ejpam-2562	60	15	,	,	PUNCT
ejpam-2562	60	16	is	be	AUX
ejpam-2562	60	17	partially	partially	ADV
ejpam-2562	60	18	-	-	PUNCT
ejpam-2562	60	19	ordered	order	VERB
ejpam-2562	60	20	by	by	ADP
ejpam-2562	60	21	the	the	DET
ejpam-2562	60	22	relation	relation	NOUN
ejpam-2562	60	23	≤	≤	NUM
ejpam-2562	60	24	defined	define	VERB
ejpam-2562	60	25	by	by	ADP
ejpam-2562	60	26	ρ	ρ	PROPN
ejpam-2562	60	27	≤	≤	NOUN
ejpam-2562	60	28	ρ′	ρ′	PUNCT
ejpam-2562	60	29	if	if	SCONJ
ejpam-2562	60	30	and	and	CCONJ
ejpam-2562	60	31	only	only	ADV
ejpam-2562	60	32	if	if	SCONJ
ejpam-2562	60	33	mρm′	mρm′	PROPN
ejpam-2562	60	34	=	=	PRON
ejpam-2562	60	35	⇒	⇒	NOUN
ejpam-2562	60	36	mρ′m′	mρ′m′	ADV
ejpam-2562	60	37	∀m	∀m	PROPN
ejpam-2562	60	38	,	,	PUNCT
ejpam-2562	60	39	m′	m′	NOUN
ejpam-2562	60	40	∈	∈	NOUN
ejpam-2562	60	41	m.	m.	NOUN
ejpam-2562	60	42	for	for	ADP
ejpam-2562	60	43	m	m	PROPN
ejpam-2562	60	44	,	,	PUNCT
ejpam-2562	60	45	m′	m′	NOUN
ejpam-2562	60	46	∈	∈	PROPN
ejpam-2562	60	47	m	m	PROPN
ejpam-2562	60	48	,	,	PUNCT
ejpam-2562	60	49	ρ(m	ρ(m	NUM
ejpam-2562	60	50	,	,	PUNCT
ejpam-2562	60	51	m′	m′	NUM
ejpam-2562	60	52	)	)	PUNCT
ejpam-2562	60	53	is	be	AUX
ejpam-2562	60	54	the	the	DET
ejpam-2562	60	55	unique	unique	ADJ
ejpam-2562	60	56	smallest	small	ADJ
ejpam-2562	60	57	element	element	NOUN
ejpam-2562	60	58	ρ	ρ	NOUN
ejpam-2562	60	59	of	of	ADP
ejpam-2562	60	60	r−	r−	PROPN
ejpam-2562	60	61	cong(m	cong(m	VERB
ejpam-2562	60	62	)	)	PUNCT
ejpam-2562	60	63	satisfying	satisfy	VERB
ejpam-2562	60	64	mρm′.	mρm′.	PROPN
ejpam-2562	60	65	definition	definition	NOUN
ejpam-2562	60	66	2	2	NUM
ejpam-2562	60	67	.	.	NOUN
ejpam-2562	60	68	•	•	NUM
ejpam-2562	60	69	a	a	DET
ejpam-2562	60	70	subsemimodule	subsemimodule	NOUN
ejpam-2562	60	71	n	n	NOUN
ejpam-2562	60	72	of	of	ADP
ejpam-2562	60	73	a	a	DET
ejpam-2562	60	74	semimodule	semimodule	NOUN
ejpam-2562	60	75	m	m	VERB
ejpam-2562	60	76	is	be	AUX
ejpam-2562	60	77	called	call	VERB
ejpam-2562	60	78	subtractive	subtractive	NOUN
ejpam-2562	60	79	if	if	SCONJ
ejpam-2562	60	80	for	for	ADP
ejpam-2562	60	81	all	all	DET
ejpam-2562	60	82	m	m	PROPN
ejpam-2562	60	83	,	,	PUNCT
ejpam-2562	60	84	m′	m′	NOUN
ejpam-2562	60	85	∈	∈	PROPN
ejpam-2562	60	86	m	m	PROPN
ejpam-2562	60	87	,	,	PUNCT
ejpam-2562	60	88	m+m′	m+m′	PROPN
ejpam-2562	60	89	∈	∈	PROPN
ejpam-2562	60	90	n	n	PRON
ejpam-2562	60	91	and	and	CCONJ
ejpam-2562	60	92	m	m	PROPN
ejpam-2562	60	93	∈	∈	NOUN
ejpam-2562	60	94	n	n	NOUN
ejpam-2562	60	95	implies	imply	VERB
ejpam-2562	60	96	m′	m′	NUM
ejpam-2562	60	97	∈	∈	PROPN
ejpam-2562	60	98	n.	n.	NOUN
ejpam-2562	60	99	•	•	ADP
ejpam-2562	60	100	the	the	DET
ejpam-2562	60	101	subtractive	subtractive	NOUN
ejpam-2562	60	102	closure	closure	NOUN
ejpam-2562	60	103	of	of	ADP
ejpam-2562	60	104	a	a	DET
ejpam-2562	60	105	subsemimodule	subsemimodule	NOUN
ejpam-2562	60	106	n	n	NOUN
ejpam-2562	60	107	of	of	ADP
ejpam-2562	60	108	a	a	DET
ejpam-2562	60	109	semimodule	semimodule	NOUN
ejpam-2562	60	110	m	m	VERB
ejpam-2562	60	111	is	be	AUX
ejpam-2562	60	112	the	the	DET
ejpam-2562	60	113	smallest	small	ADJ
ejpam-2562	60	114	subtractive	subtractive	NOUN
ejpam-2562	60	115	subsemimodule	subsemimodule	NOUN
ejpam-2562	60	116	of	of	ADP
ejpam-2562	60	117	m	m	AUX
ejpam-2562	60	118	containing	contain	VERB
ejpam-2562	60	119	n.	n.	NOUN
ejpam-2562	60	120	•	•	ADP
ejpam-2562	60	121	a	a	DET
ejpam-2562	60	122	semimodule	semimodule	NOUN
ejpam-2562	60	123	m	m	VERB
ejpam-2562	60	124	is	be	AUX
ejpam-2562	60	125	said	say	VERB
ejpam-2562	60	126	to	to	PART
ejpam-2562	60	127	be	be	AUX
ejpam-2562	60	128	cancellative	cancellative	ADJ
ejpam-2562	60	129	(	(	PUNCT
ejpam-2562	60	130	additively	additively	ADV
ejpam-2562	60	131	cancellative	cancellative	ADJ
ejpam-2562	60	132	)	)	PUNCT
ejpam-2562	60	133	if	if	SCONJ
ejpam-2562	60	134	for	for	ADP
ejpam-2562	60	135	all	all	DET
ejpam-2562	60	136	m	m	PROPN
ejpam-2562	60	137	,	,	PUNCT
ejpam-2562	60	138	m′	m′	PROPN
ejpam-2562	60	139	,	,	PUNCT
ejpam-2562	60	140	m′′	m′′	PROPN
ejpam-2562	60	141	∈	∈	PROPN
ejpam-2562	60	142	m	m	PROPN
ejpam-2562	60	143	,	,	PUNCT
ejpam-2562	60	144	m+m′	m+m′	PROPN
ejpam-2562	60	145	=	=	PUNCT
ejpam-2562	60	146	m+m′′	m+m′′	PROPN
ejpam-2562	60	147	=	=	NOUN
ejpam-2562	60	148	⇒	⇒	VERB
ejpam-2562	60	149	m′	m′	NOUN
ejpam-2562	60	150	=	=	SYM
ejpam-2562	61	1	m′′.	m′′.	NOUN
ejpam-2562	61	2	definition	definition	NOUN
ejpam-2562	61	3	3	3	X
ejpam-2562	61	4	.	.	PUNCT
ejpam-2562	62	1	let	let	VERB
ejpam-2562	62	2	n	n	PRON
ejpam-2562	62	3	be	be	AUX
ejpam-2562	62	4	a	a	DET
ejpam-2562	62	5	subsemimodule	subsemimodule	NOUN
ejpam-2562	62	6	of	of	ADP
ejpam-2562	62	7	a	a	DET
ejpam-2562	62	8	left	left	ADJ
ejpam-2562	62	9	r	r	NOUN
ejpam-2562	62	10	-	-	PUNCT
ejpam-2562	62	11	semimodule	semimodule	NOUN
ejpam-2562	62	12	m.	m.	NOUN
ejpam-2562	62	13	n	n	X
ejpam-2562	62	14	induces	induce	VERB
ejpam-2562	62	15	on	on	ADP
ejpam-2562	62	16	m	m	PRON
ejpam-2562	62	17	an	an	DET
ejpam-2562	62	18	rcongruence	rcongruence	NOUN
ejpam-2562	62	19	relation	relation	NOUN
ejpam-2562	62	20	≡n	≡n	PROPN
ejpam-2562	62	21	,	,	PUNCT
ejpam-2562	62	22	know	know	VERB
ejpam-2562	62	23	as	as	ADP
ejpam-2562	62	24	the	the	DET
ejpam-2562	62	25	bourne	bourne	NOUN
ejpam-2562	62	26	relation	relation	NOUN
ejpam-2562	62	27	:	:	PUNCT
ejpam-2562	62	28	∀m	∀m	NUM
ejpam-2562	62	29	,	,	PUNCT
ejpam-2562	62	30	m′	m′	NOUN
ejpam-2562	62	31	∈	∈	PROPN
ejpam-2562	62	32	m	m	PROPN
ejpam-2562	62	33	;	;	PUNCT
ejpam-2562	62	34	m	m	VERB
ejpam-2562	62	35	≡n	≡n	PROPN
ejpam-2562	62	36	m′	m′	NUM
ejpam-2562	62	37	⇐	⇐	ADJ
ejpam-2562	62	38	⇒	⇒	PROPN
ejpam-2562	62	39	∃n	∃n	PROPN
ejpam-2562	62	40	,	,	PUNCT
ejpam-2562	62	41	n′	n′	PROPN
ejpam-2562	62	42	∈	∈	PROPN
ejpam-2562	62	43	n	n	PRON
ejpam-2562	62	44	such	such	ADJ
ejpam-2562	62	45	that	that	PRON
ejpam-2562	62	46	:	:	PUNCT
ejpam-2562	62	47	m+	m+	NUM
ejpam-2562	62	48	n=	n=	ADJ
ejpam-2562	62	49	m′	m′	NOUN
ejpam-2562	62	50	+	+	CCONJ
ejpam-2562	62	51	n′.	n′.	PROPN
ejpam-2562	62	52	•	•	PROPN
ejpam-2562	62	53	m	m	PROPN
ejpam-2562	62	54	/	/	SYM
ejpam-2562	62	55	n	n	PROPN
ejpam-2562	62	56	denotes	denote	VERB
ejpam-2562	62	57	the	the	DET
ejpam-2562	62	58	factor	factor	NOUN
ejpam-2562	62	59	r	r	NOUN
ejpam-2562	62	60	-	-	PUNCT
ejpam-2562	62	61	semimodule	semimodule	NOUN
ejpam-2562	62	62	m/	m/	NOUN
ejpam-2562	62	63	≡n	≡n	NOUN
ejpam-2562	62	64	,	,	PUNCT
ejpam-2562	62	65	and	and	CCONJ
ejpam-2562	62	66	m	m	PROPN
ejpam-2562	62	67	/	/	SYM
ejpam-2562	62	68	n	n	PROPN
ejpam-2562	62	69	denotes	denote	VERB
ejpam-2562	62	70	an	an	DET
ejpam-2562	62	71	element	element	NOUN
ejpam-2562	62	72	of	of	ADP
ejpam-2562	62	73	m	m	PROPN
ejpam-2562	62	74	/	/	SYM
ejpam-2562	62	75	n	n	PROPN
ejpam-2562	62	76	for	for	ADP
ejpam-2562	62	77	some	some	DET
ejpam-2562	62	78	m	m	NOUN
ejpam-2562	62	79	∈	∈	NOUN
ejpam-2562	62	80	m.	m.	NOUN
ejpam-2562	62	81	•	•	NOUN
ejpam-2562	62	82	0	0	NUM
ejpam-2562	62	83	/	/	SYM
ejpam-2562	62	84	n	n	NOUN
ejpam-2562	62	85	=	=	SYM
ejpam-2562	62	86	n	n	NOUN
ejpam-2562	62	87	=	=	PUNCT
ejpam-2562	62	88	{	{	PUNCT
ejpam-2562	62	89	m	m	PROPN
ejpam-2562	62	90	∈	∈	PROPN
ejpam-2562	62	91	m/∃n	m/∃n	PROPN
ejpam-2562	62	92	∈	∈	PROPN
ejpam-2562	62	93	n	n	CCONJ
ejpam-2562	62	94	/	/	SYM
ejpam-2562	62	95	m+	m+	NUM
ejpam-2562	62	96	n	n	CCONJ
ejpam-2562	62	97	∈	∈	PROPN
ejpam-2562	62	98	n	n	CCONJ
ejpam-2562	62	99	}	}	PUNCT
ejpam-2562	62	100	is	be	AUX
ejpam-2562	62	101	the	the	DET
ejpam-2562	62	102	subtractive	subtractive	NOUN
ejpam-2562	62	103	closure	closure	NOUN
ejpam-2562	62	104	of	of	ADP
ejpam-2562	62	105	n.	n.	PROPN
ejpam-2562	62	106	definition	definition	NOUN
ejpam-2562	62	107	4	4	NUM
ejpam-2562	62	108	.	.	PUNCT
ejpam-2562	63	1	let	let	VERB
ejpam-2562	63	2	m1	m1	PROPN
ejpam-2562	63	3	and	and	CCONJ
ejpam-2562	63	4	m2	m2	PROPN
ejpam-2562	63	5	be	be	VERB
ejpam-2562	63	6	subsemimodules	subsemimodule	NOUN
ejpam-2562	63	7	of	of	ADP
ejpam-2562	63	8	a	a	DET
ejpam-2562	63	9	left	left	ADJ
ejpam-2562	63	10	r	r	NOUN
ejpam-2562	63	11	-	-	PUNCT
ejpam-2562	63	12	semimodule	semimodule	NOUN
ejpam-2562	63	13	m.	m.	NOUN
ejpam-2562	63	14	if	if	SCONJ
ejpam-2562	63	15	m1	m1	PROPN
ejpam-2562	63	16	and	and	CCONJ
ejpam-2562	63	17	m2	m2	PROPN
ejpam-2562	63	18	span	span	PROPN
ejpam-2562	63	19	m	m	PROPN
ejpam-2562	63	20	(	(	PUNCT
ejpam-2562	63	21	i.e	i.e	NOUN
ejpam-2562	63	22	m	m	NOUN
ejpam-2562	63	23	=	=	ADJ
ejpam-2562	63	24	m1	m1	PROPN
ejpam-2562	63	25	+	+	CCONJ
ejpam-2562	63	26	m2	m2	PROPN
ejpam-2562	63	27	)	)	PUNCT
ejpam-2562	63	28	,	,	PUNCT
ejpam-2562	63	29	and	and	CCONJ
ejpam-2562	63	30	the	the	DET
ejpam-2562	63	31	restriction	restriction	NOUN
ejpam-2562	63	32	of	of	ADP
ejpam-2562	63	33	≡m2	≡m2	PRON
ejpam-2562	63	34	to	to	ADP
ejpam-2562	63	35	m1	m1	PROPN
ejpam-2562	63	36	and	and	CCONJ
ejpam-2562	63	37	the	the	DET
ejpam-2562	63	38	restriction	restriction	NOUN
ejpam-2562	63	39	of	of	ADP
ejpam-2562	63	40	≡m1	≡m1	NUM
ejpam-2562	63	41	to	to	ADP
ejpam-2562	63	42	m2	m2	PROPN
ejpam-2562	63	43	are	be	AUX
ejpam-2562	63	44	trivial	trivial	ADJ
ejpam-2562	63	45	,	,	PUNCT
ejpam-2562	63	46	then	then	ADV
ejpam-2562	63	47	m	m	VERB
ejpam-2562	63	48	is	be	AUX
ejpam-2562	63	49	the	the	DET
ejpam-2562	63	50	direct	direct	ADJ
ejpam-2562	63	51	sum	sum	NOUN
ejpam-2562	63	52	of	of	ADP
ejpam-2562	63	53	its	its	PRON
ejpam-2562	63	54	subsemimodules	subsemimodule	NOUN
ejpam-2562	63	55	m1	m1	PROPN
ejpam-2562	63	56	and	and	CCONJ
ejpam-2562	63	57	m2	m2	PROPN
ejpam-2562	63	58	.	.	PROPN
ejpam-2562	64	1	and	and	CCONJ
ejpam-2562	64	2	we	we	PRON
ejpam-2562	64	3	write	write	VERB
ejpam-2562	64	4	m	m	NOUN
ejpam-2562	64	5	=	=	SYM
ejpam-2562	64	6	m1	m1	PROPN
ejpam-2562	64	7	⊕m2	⊕m2	NUM
ejpam-2562	64	8	.	.	PUNCT
ejpam-2562	65	1	in	in	ADP
ejpam-2562	65	2	this	this	DET
ejpam-2562	65	3	case	case	NOUN
ejpam-2562	65	4	for	for	ADP
ejpam-2562	65	5	each	each	DET
ejpam-2562	65	6	m	m	NOUN
ejpam-2562	65	7	∈	∈	PROPN
ejpam-2562	65	8	m	m	PRON
ejpam-2562	65	9	,	,	PUNCT
ejpam-2562	65	10	there	there	PRON
ejpam-2562	65	11	exists	exist	VERB
ejpam-2562	65	12	unique	unique	ADJ
ejpam-2562	65	13	pair	pair	NOUN
ejpam-2562	65	14	(	(	PUNCT
ejpam-2562	65	15	m1	m1	NOUN
ejpam-2562	65	16	,	,	PUNCT
ejpam-2562	65	17	m2	m2	PROPN
ejpam-2562	65	18	)	)	PUNCT
ejpam-2562	65	19	∈	∈	PROPN
ejpam-2562	65	20	m1×m2	m1×m2	PROPN
ejpam-2562	65	21	such	such	ADJ
ejpam-2562	65	22	that	that	PRON
ejpam-2562	65	23	:	:	PUNCT
ejpam-2562	66	1	m=	m=	X
ejpam-2562	66	2	m1+m2	m1+m2	X
ejpam-2562	66	3	.	.	PUNCT
ejpam-2562	67	1	in	in	ADP
ejpam-2562	67	2	[	[	X
ejpam-2562	67	3	6	6	NUM
ejpam-2562	67	4	]	]	PUNCT
ejpam-2562	67	5	,	,	PUNCT
ejpam-2562	67	6	we	we	PRON
ejpam-2562	67	7	have	have	VERB
ejpam-2562	67	8	the	the	DET
ejpam-2562	67	9	following	follow	VERB
ejpam-2562	67	10	characterization	characterization	NOUN
ejpam-2562	67	11	of	of	ADP
ejpam-2562	67	12	essential	essential	ADJ
ejpam-2562	67	13	subsemimodules	subsemimodule	NOUN
ejpam-2562	67	14	.	.	PUNCT
ejpam-2562	68	1	notation	notation	NOUN
ejpam-2562	68	2	:	:	PUNCT
ejpam-2562	68	3	the	the	DET
ejpam-2562	68	4	class	class	NOUN
ejpam-2562	68	5	of	of	ADP
ejpam-2562	68	6	essential	essential	ADJ
ejpam-2562	68	7	subsemimodules	subsemimodule	NOUN
ejpam-2562	68	8	of	of	ADP
ejpam-2562	68	9	a	a	DET
ejpam-2562	68	10	left	left	ADJ
ejpam-2562	68	11	r	r	NOUN
ejpam-2562	68	12	-	-	PUNCT
ejpam-2562	68	13	semimodule	semimodule	NOUN
ejpam-2562	68	14	m	m	VERB
ejpam-2562	68	15	is	be	AUX
ejpam-2562	68	16	denoted	denote	VERB
ejpam-2562	68	17	by	by	ADP
ejpam-2562	68	18	c	c	PROPN
ejpam-2562	68	19	rm	rm	PROPN
ejpam-2562	68	20	.	.	PUNCT
ejpam-2562	69	1	e.	e.	PROPN
ejpam-2562	69	2	diop	diop	PROPN
ejpam-2562	69	3	,	,	PUNCT
ejpam-2562	69	4	d.	d.	PROPN
ejpam-2562	69	5	sow	sow	PROPN
ejpam-2562	69	6	/	/	SYM
ejpam-2562	69	7	eur	eur	PROPN
ejpam-2562	69	8	.	.	PUNCT
ejpam-2562	70	1	j.	j.	PROPN
ejpam-2562	70	2	pure	pure	PROPN
ejpam-2562	70	3	appl	appl	PROPN
ejpam-2562	70	4	.	.	PROPN
ejpam-2562	70	5	math	math	PROPN
ejpam-2562	70	6	,	,	PUNCT
ejpam-2562	70	7	9	9	NUM
ejpam-2562	70	8	(	(	PUNCT
ejpam-2562	70	9	2016	2016	NUM
ejpam-2562	70	10	)	)	PUNCT
ejpam-2562	70	11	,	,	PUNCT
ejpam-2562	70	12	250	250	NUM
ejpam-2562	70	13	-	-	SYM
ejpam-2562	70	14	265	265	NUM
ejpam-2562	70	15	253	253	NUM
ejpam-2562	70	16	lemma	lemma	PROPN
ejpam-2562	70	17	1	1	NUM
ejpam-2562	70	18	.	.	PUNCT
ejpam-2562	71	1	if	if	SCONJ
ejpam-2562	71	2	n	n	PRON
ejpam-2562	71	3	is	be	AUX
ejpam-2562	71	4	a	a	DET
ejpam-2562	71	5	subsemimodule	subsemimodule	NOUN
ejpam-2562	71	6	of	of	ADP
ejpam-2562	71	7	a	a	DET
ejpam-2562	71	8	left	left	ADJ
ejpam-2562	71	9	r	r	NOUN
ejpam-2562	71	10	-	-	PUNCT
ejpam-2562	71	11	semimodule	semimodule	NOUN
ejpam-2562	71	12	,	,	PUNCT
ejpam-2562	71	13	m	m	VERB
ejpam-2562	71	14	then	then	ADV
ejpam-2562	71	15	the	the	DET
ejpam-2562	71	16	following	follow	VERB
ejpam-2562	71	17	conditions	condition	NOUN
ejpam-2562	71	18	are	be	AUX
ejpam-2562	71	19	equivalent	equivalent	ADJ
ejpam-2562	71	20	:	:	PUNCT
ejpam-2562	71	21	(	(	PUNCT
ejpam-2562	71	22	i	i	NOUN
ejpam-2562	71	23	)	)	PUNCT
ejpam-2562	71	24	n	n	PRON
ejpam-2562	71	25	is	be	AUX
ejpam-2562	71	26	essential	essential	ADJ
ejpam-2562	71	27	(	(	PUNCT
ejpam-2562	71	28	or	or	CCONJ
ejpam-2562	71	29	large	large	ADJ
ejpam-2562	71	30	)	)	PUNCT
ejpam-2562	71	31	in	in	ADP
ejpam-2562	71	32	m	m	PROPN
ejpam-2562	71	33	;	;	PUNCT
ejpam-2562	71	34	(	(	PUNCT
ejpam-2562	71	35	ii	ii	NOUN
ejpam-2562	71	36	)	)	PUNCT
ejpam-2562	71	37	if	if	SCONJ
ejpam-2562	71	38	ρ	ρ	PROPN
ejpam-2562	71	39	is	be	AUX
ejpam-2562	71	40	a	a	DET
ejpam-2562	71	41	nontrivial	nontrivial	ADJ
ejpam-2562	71	42	r	r	VERB
ejpam-2562	71	43	-	-	PUNCT
ejpam-2562	71	44	congruence	congruence	NOUN
ejpam-2562	71	45	relation	relation	NOUN
ejpam-2562	71	46	on	on	ADP
ejpam-2562	71	47	m	m	PROPN
ejpam-2562	71	48	then	then	ADV
ejpam-2562	71	49	the	the	DET
ejpam-2562	71	50	restriction	restriction	NOUN
ejpam-2562	71	51	of	of	ADP
ejpam-2562	71	52	ρ	ρ	PROPN
ejpam-2562	71	53	to	to	ADP
ejpam-2562	71	54	n	n	PROPN
ejpam-2562	71	55	is	be	AUX
ejpam-2562	71	56	also	also	ADV
ejpam-2562	71	57	nontrivial	nontrivial	ADJ
ejpam-2562	71	58	;	;	PUNCT
ejpam-2562	71	59	(	(	PUNCT
ejpam-2562	71	60	iii	iii	X
ejpam-2562	71	61	)	)	PUNCT
ejpam-2562	71	62	if	if	SCONJ
ejpam-2562	71	63	m	m	NOUN
ejpam-2562	71	64	and	and	CCONJ
ejpam-2562	71	65	m′	m′	NOUN
ejpam-2562	71	66	are	be	AUX
ejpam-2562	71	67	distinct	distinct	ADJ
ejpam-2562	71	68	elements	element	NOUN
ejpam-2562	71	69	of	of	ADP
ejpam-2562	71	70	m	m	PRON
ejpam-2562	71	71	then	then	ADV
ejpam-2562	71	72	there	there	PRON
ejpam-2562	71	73	exist	exist	VERB
ejpam-2562	71	74	distinct	distinct	ADJ
ejpam-2562	71	75	elements	element	NOUN
ejpam-2562	71	76	n	n	PRON
ejpam-2562	71	77	and	and	CCONJ
ejpam-2562	71	78	n′	n′	PROPN
ejpam-2562	71	79	of	of	ADP
ejpam-2562	71	80	n	n	PRON
ejpam-2562	71	81	satisfying	satisfy	VERB
ejpam-2562	71	82	nρ(m	nρ(m	NOUN
ejpam-2562	71	83	,	,	PUNCT
ejpam-2562	71	84	m′)n	m′)n	X
ejpam-2562	71	85	′.	′.	NOUN
ejpam-2562	71	86	in	in	ADP
ejpam-2562	71	87	[	[	X
ejpam-2562	71	88	5	5	NUM
ejpam-2562	71	89	]	]	PUNCT
ejpam-2562	71	90	,	,	PUNCT
ejpam-2562	71	91	we	we	PRON
ejpam-2562	71	92	have	have	VERB
ejpam-2562	71	93	the	the	DET
ejpam-2562	71	94	following	following	ADJ
ejpam-2562	71	95	characterisation	characterisation	NOUN
ejpam-2562	71	96	of	of	ADP
ejpam-2562	71	97	semi	semi	ADJ
ejpam-2562	71	98	-	-	ADJ
ejpam-2562	71	99	essential	essential	ADJ
ejpam-2562	71	100	subsemimodules	subsemimodule	NOUN
ejpam-2562	71	101	.	.	PUNCT
ejpam-2562	72	1	notation	notation	NOUN
ejpam-2562	72	2	:	:	PUNCT
ejpam-2562	72	3	the	the	DET
ejpam-2562	72	4	class	class	NOUN
ejpam-2562	72	5	of	of	ADP
ejpam-2562	72	6	semi	semi	ADJ
ejpam-2562	72	7	-	-	ADJ
ejpam-2562	72	8	essential	essential	ADJ
ejpam-2562	72	9	subsemimodules	subsemimodule	NOUN
ejpam-2562	72	10	of	of	ADP
ejpam-2562	72	11	a	a	DET
ejpam-2562	72	12	left	left	ADJ
ejpam-2562	72	13	r	r	NOUN
ejpam-2562	72	14	-	-	PUNCT
ejpam-2562	72	15	semimodule	semimodule	NOUN
ejpam-2562	72	16	m	m	VERB
ejpam-2562	72	17	is	be	AUX
ejpam-2562	72	18	denoted	denote	VERB
ejpam-2562	72	19	by	by	ADP
ejpam-2562	72	20	c	c	PROPN
ejpam-2562	72	21	rm	rm	PROPN
ejpam-2562	72	22	.	.	PUNCT
ejpam-2562	73	1	lemma	lemma	PROPN
ejpam-2562	73	2	2	2	NUM
ejpam-2562	73	3	.	.	PUNCT
ejpam-2562	74	1	a	a	DET
ejpam-2562	74	2	subsemimodule	subsemimodule	NOUN
ejpam-2562	74	3	k	k	NOUN
ejpam-2562	74	4	of	of	ADP
ejpam-2562	74	5	a	a	DET
ejpam-2562	74	6	left	left	ADJ
ejpam-2562	74	7	r	r	NOUN
ejpam-2562	74	8	-	-	PUNCT
ejpam-2562	74	9	semimodule	semimodule	NOUN
ejpam-2562	74	10	m	m	NOUN
ejpam-2562	74	11	is	be	AUX
ejpam-2562	74	12	semi	semi	ADJ
ejpam-2562	74	13	-	-	ADJ
ejpam-2562	74	14	essential	essential	ADJ
ejpam-2562	74	15	if	if	SCONJ
ejpam-2562	74	16	,	,	PUNCT
ejpam-2562	74	17	and	and	CCONJ
ejpam-2562	74	18	only	only	ADV
ejpam-2562	74	19	if	if	SCONJ
ejpam-2562	74	20	,	,	PUNCT
ejpam-2562	74	21	for	for	ADP
ejpam-2562	74	22	all	all	DET
ejpam-2562	74	23	x	x	SYM
ejpam-2562	74	24	6=	6=	ADP
ejpam-2562	74	25	0	0	NUM
ejpam-2562	74	26	,	,	PUNCT
ejpam-2562	74	27	elements	element	NOUN
ejpam-2562	74	28	of	of	ADP
ejpam-2562	74	29	m	m	PRON
ejpam-2562	74	30	,	,	PUNCT
ejpam-2562	74	31	there	there	PRON
ejpam-2562	74	32	exists	exist	VERB
ejpam-2562	74	33	r	r	NOUN
ejpam-2562	74	34	∈	∈	PROPN
ejpam-2562	74	35	r	r	NOUN
ejpam-2562	74	36	such	such	ADJ
ejpam-2562	74	37	that	that	PRON
ejpam-2562	74	38	:	:	PUNCT
ejpam-2562	75	1	0	0	NUM
ejpam-2562	75	2	6=	6=	NUM
ejpam-2562	75	3	r	r	NOUN
ejpam-2562	75	4	x	x	SYM
ejpam-2562	75	5	∈	∈	PROPN
ejpam-2562	75	6	k.	k.	NOUN
ejpam-2562	76	1	3	3	X
ejpam-2562	76	2	.	.	PUNCT
ejpam-2562	76	3	new	new	ADJ
ejpam-2562	76	4	notions	notion	NOUN
ejpam-2562	76	5	of	of	ADP
ejpam-2562	76	6	essential	essential	ADJ
ejpam-2562	76	7	:	:	PUNCT
ejpam-2562	76	8	on	on	ADP
ejpam-2562	76	9	semi	semi	ADJ
ejpam-2562	76	10	-	-	ADJ
ejpam-2562	76	11	weakly	weakly	ADJ
ejpam-2562	76	12	-	-	PUNCT
ejpam-2562	76	13	essential	essential	ADJ
ejpam-2562	76	14	subsemimodules	subsemimodule	NOUN
ejpam-2562	76	15	in	in	ADP
ejpam-2562	76	16	[	[	X
ejpam-2562	76	17	5	5	NUM
ejpam-2562	76	18	]	]	PUNCT
ejpam-2562	76	19	the	the	DET
ejpam-2562	76	20	class	class	NOUN
ejpam-2562	76	21	c	c	PROPN
ejpam-2562	76	22	rm	rm	NOUN
ejpam-2562	76	23	of	of	ADP
ejpam-2562	76	24	essential	essential	ADJ
ejpam-2562	76	25	subsemimodules	subsemimodule	NOUN
ejpam-2562	76	26	and	and	CCONJ
ejpam-2562	76	27	the	the	DET
ejpam-2562	76	28	class	class	NOUN
ejpam-2562	76	29	c	c	PROPN
ejpam-2562	76	30	rm	rm	NOUN
ejpam-2562	76	31	of	of	ADP
ejpam-2562	76	32	semi	semi	ADJ
ejpam-2562	76	33	-	-	ADJ
ejpam-2562	76	34	essential	essential	ADJ
ejpam-2562	76	35	subsemimodules	subsemimodule	NOUN
ejpam-2562	76	36	were	be	AUX
ejpam-2562	76	37	studied	study	VERB
ejpam-2562	76	38	and	and	CCONJ
ejpam-2562	76	39	neither	neither	DET
ejpam-2562	76	40	one	one	NUM
ejpam-2562	76	41	of	of	ADP
ejpam-2562	76	42	these	these	DET
ejpam-2562	76	43	two	two	NUM
ejpam-2562	76	44	classes	class	NOUN
ejpam-2562	76	45	is	be	AUX
ejpam-2562	76	46	contained	contain	VERB
ejpam-2562	76	47	in	in	ADP
ejpam-2562	76	48	the	the	DET
ejpam-2562	76	49	other	other	ADJ
ejpam-2562	76	50	.	.	PUNCT
ejpam-2562	77	1	in	in	ADP
ejpam-2562	77	2	this	this	DET
ejpam-2562	77	3	section	section	NOUN
ejpam-2562	77	4	these	these	DET
ejpam-2562	77	5	two	two	NUM
ejpam-2562	77	6	classes	class	NOUN
ejpam-2562	77	7	are	be	AUX
ejpam-2562	77	8	embedded	embed	VERB
ejpam-2562	77	9	in	in	ADP
ejpam-2562	77	10	a	a	DET
ejpam-2562	77	11	new	new	ADJ
ejpam-2562	77	12	class	class	NOUN
ejpam-2562	77	13	namely	namely	ADV
ejpam-2562	77	14	the	the	DET
ejpam-2562	77	15	class	class	NOUN
ejpam-2562	77	16	wc	wc	PROPN
ejpam-2562	77	17	rm	rm	PROPN
ejpam-2562	77	18	of	of	ADP
ejpam-2562	77	19	semiweakly	semiweakly	ADJ
ejpam-2562	77	20	-	-	PUNCT
ejpam-2562	77	21	essential	essential	ADJ
ejpam-2562	77	22	subsemimodules	subsemimodule	NOUN
ejpam-2562	77	23	.	.	PUNCT
ejpam-2562	78	1	definition	definition	NOUN
ejpam-2562	78	2	5	5	NUM
ejpam-2562	78	3	.	.	PUNCT
ejpam-2562	79	1	a	a	DET
ejpam-2562	79	2	subsemimodule	subsemimodule	NOUN
ejpam-2562	79	3	n	n	NOUN
ejpam-2562	79	4	of	of	ADP
ejpam-2562	79	5	a	a	DET
ejpam-2562	79	6	semimodule	semimodule	NOUN
ejpam-2562	79	7	m	m	VERB
ejpam-2562	79	8	is	be	AUX
ejpam-2562	79	9	said	say	VERB
ejpam-2562	79	10	to	to	PART
ejpam-2562	79	11	be	be	AUX
ejpam-2562	79	12	semi	semi	ADJ
ejpam-2562	79	13	-	-	ADJ
ejpam-2562	79	14	weakly	weakly	ADV
ejpam-2562	79	15	-	-	PUNCT
ejpam-2562	79	16	essential	essential	ADJ
ejpam-2562	79	17	in	in	ADP
ejpam-2562	79	18	m	m	PROPN
ejpam-2562	79	19	(	(	PUNCT
ejpam-2562	79	20	denoted	denote	VERB
ejpam-2562	79	21	by	by	ADP
ejpam-2562	79	22	n	n	X
ejpam-2562	79	23	ãswe	ãswe	NOUN
ejpam-2562	79	24	m	m	PROPN
ejpam-2562	79	25	)	)	PUNCT
ejpam-2562	79	26	,	,	PUNCT
ejpam-2562	79	27	if	if	SCONJ
ejpam-2562	79	28	∀h	∀h	NOUN
ejpam-2562	79	29	:	:	PUNCT
ejpam-2562	79	30	m	m	VERB
ejpam-2562	79	31	−→	−→	ADJ
ejpam-2562	79	32	m	m	VERB
ejpam-2562	79	33	′	′	NUM
ejpam-2562	80	1	r	r	X
ejpam-2562	80	2	-	-	PUNCT
ejpam-2562	80	3	homomorphism	homomorphism	NOUN
ejpam-2562	80	4	,	,	PUNCT
ejpam-2562	80	5	the	the	DET
ejpam-2562	80	6	restriction	restriction	NOUN
ejpam-2562	80	7	of	of	ADP
ejpam-2562	80	8	≡kerh	≡kerh	PROPN
ejpam-2562	80	9	to	to	ADP
ejpam-2562	80	10	n	n	PROPN
ejpam-2562	80	11	is	be	AUX
ejpam-2562	80	12	trivial	trivial	ADJ
ejpam-2562	80	13	=	=	NOUN
ejpam-2562	80	14	⇒	⇒	NOUN
ejpam-2562	80	15	kerh=	kerh=	NUM
ejpam-2562	80	16	0	0	NUM
ejpam-2562	80	17	.	.	PUNCT
ejpam-2562	81	1	proposition	proposition	NOUN
ejpam-2562	81	2	1	1	NUM
ejpam-2562	81	3	.	.	PUNCT
ejpam-2562	82	1	if	if	SCONJ
ejpam-2562	82	2	n	n	PRON
ejpam-2562	82	3	is	be	AUX
ejpam-2562	82	4	a	a	DET
ejpam-2562	82	5	subsemimodule	subsemimodule	NOUN
ejpam-2562	82	6	of	of	ADP
ejpam-2562	82	7	a	a	DET
ejpam-2562	82	8	left	left	ADJ
ejpam-2562	82	9	r	r	NOUN
ejpam-2562	82	10	-	-	PUNCT
ejpam-2562	82	11	semimodule	semimodule	NOUN
ejpam-2562	82	12	m	m	VERB
ejpam-2562	82	13	then	then	ADV
ejpam-2562	82	14	the	the	DET
ejpam-2562	82	15	following	follow	VERB
ejpam-2562	82	16	conditions	condition	NOUN
ejpam-2562	82	17	are	be	AUX
ejpam-2562	82	18	equivalent	equivalent	ADJ
ejpam-2562	82	19	:	:	PUNCT
ejpam-2562	82	20	(	(	PUNCT
ejpam-2562	82	21	i	i	NOUN
ejpam-2562	82	22	)	)	PUNCT
ejpam-2562	82	23	n	n	PRON
ejpam-2562	82	24	is	be	AUX
ejpam-2562	82	25	semi	semi	ADJ
ejpam-2562	82	26	-	-	ADJ
ejpam-2562	82	27	weakly	weakly	ADV
ejpam-2562	82	28	-	-	PUNCT
ejpam-2562	82	29	essential	essential	ADJ
ejpam-2562	82	30	in	in	ADP
ejpam-2562	82	31	m	m	PROPN
ejpam-2562	82	32	;	;	PUNCT
ejpam-2562	82	33	(	(	PUNCT
ejpam-2562	82	34	ii	ii	NOUN
ejpam-2562	82	35	)	)	PUNCT
ejpam-2562	82	36	∀k	∀k	NOUN
ejpam-2562	82	37	≤	≤	NUM
ejpam-2562	82	38	m	m	PROPN
ejpam-2562	82	39	,	,	PUNCT
ejpam-2562	82	40	the	the	DET
ejpam-2562	82	41	restriction	restriction	NOUN
ejpam-2562	82	42	of	of	ADP
ejpam-2562	82	43	≡k	≡k	NOUN
ejpam-2562	82	44	to	to	ADP
ejpam-2562	82	45	n	n	PROPN
ejpam-2562	82	46	is	be	AUX
ejpam-2562	82	47	trivial⇒	trivial⇒	PROPN
ejpam-2562	82	48	k	k	PROPN
ejpam-2562	82	49	=	=	PUNCT
ejpam-2562	82	50	0	0	X
ejpam-2562	82	51	.	.	PUNCT
ejpam-2562	83	1	proof	proof	NOUN
ejpam-2562	83	2	.	.	PUNCT
ejpam-2562	84	1	(	(	PUNCT
ejpam-2562	84	2	i	i	NOUN
ejpam-2562	84	3	)	)	PUNCT
ejpam-2562	85	1	=	=	NOUN
ejpam-2562	85	2	⇒	⇒	NOUN
ejpam-2562	85	3	(	(	PUNCT
ejpam-2562	85	4	ii	ii	NOUN
ejpam-2562	85	5	)	)	PUNCT
ejpam-2562	85	6	.	.	PUNCT
ejpam-2562	86	1	let	let	VERB
ejpam-2562	86	2	k	k	PROPN
ejpam-2562	86	3	≤	≤	NUM
ejpam-2562	86	4	m	m	VERB
ejpam-2562	86	5	such	such	ADJ
ejpam-2562	86	6	that	that	SCONJ
ejpam-2562	86	7	the	the	DET
ejpam-2562	86	8	restriction	restriction	NOUN
ejpam-2562	86	9	of	of	ADP
ejpam-2562	86	10	≡k	≡k	NOUN
ejpam-2562	86	11	to	to	ADP
ejpam-2562	86	12	n	n	PROPN
ejpam-2562	86	13	is	be	AUX
ejpam-2562	86	14	trivial	trivial	ADJ
ejpam-2562	86	15	.	.	PUNCT
ejpam-2562	87	1	let	let	VERB
ejpam-2562	87	2	h	h	NOUN
ejpam-2562	87	3	:	:	PUNCT
ejpam-2562	87	4	m	m	VERB
ejpam-2562	87	5	−→	−→	ADJ
ejpam-2562	87	6	m	m	ADJ
ejpam-2562	87	7	/	/	SYM
ejpam-2562	87	8	k	k	PROPN
ejpam-2562	87	9	be	be	VERB
ejpam-2562	87	10	the	the	DET
ejpam-2562	87	11	surjection	surjection	NOUN
ejpam-2562	87	12	map	map	NOUN
ejpam-2562	87	13	.	.	PUNCT
ejpam-2562	88	1	so	so	ADV
ejpam-2562	88	2	kerh=	kerh=	PROPN
ejpam-2562	88	3	k	k	X
ejpam-2562	88	4	.	.	PUNCT
ejpam-2562	89	1	since	since	SCONJ
ejpam-2562	89	2	≡k	≡k	NOUN
ejpam-2562	89	3	and	and	CCONJ
ejpam-2562	89	4	≡k	≡k	NOUN
ejpam-2562	89	5	are	be	AUX
ejpam-2562	89	6	equivalent	equivalent	ADJ
ejpam-2562	89	7	and	and	CCONJ
ejpam-2562	89	8	the	the	DET
ejpam-2562	89	9	restriction	restriction	NOUN
ejpam-2562	89	10	of	of	ADP
ejpam-2562	89	11	≡k	≡k	NOUN
ejpam-2562	89	12	to	to	ADP
ejpam-2562	89	13	n	n	PROPN
ejpam-2562	89	14	is	be	AUX
ejpam-2562	89	15	trivial	trivial	ADJ
ejpam-2562	89	16	then	then	ADV
ejpam-2562	89	17	restriction	restriction	NOUN
ejpam-2562	89	18	of	of	ADP
ejpam-2562	89	19	≡kerh	≡kerh	PROPN
ejpam-2562	89	20	to	to	ADP
ejpam-2562	89	21	n	n	PROPN
ejpam-2562	89	22	is	be	AUX
ejpam-2562	89	23	trivial	trivial	ADJ
ejpam-2562	89	24	.	.	PUNCT
ejpam-2562	90	1	n	n	PRON
ejpam-2562	90	2	is	be	AUX
ejpam-2562	90	3	semi	semi	ADJ
ejpam-2562	90	4	-	-	ADJ
ejpam-2562	90	5	weakly	weakly	ADV
ejpam-2562	90	6	-	-	PUNCT
ejpam-2562	90	7	essential	essential	ADJ
ejpam-2562	90	8	in	in	ADP
ejpam-2562	90	9	m	m	NOUN
ejpam-2562	90	10	by	by	ADP
ejpam-2562	90	11	assumption	assumption	NOUN
ejpam-2562	90	12	,	,	PUNCT
ejpam-2562	90	13	so	so	CCONJ
ejpam-2562	90	14	the	the	DET
ejpam-2562	90	15	restriction	restriction	NOUN
ejpam-2562	90	16	of	of	ADP
ejpam-2562	90	17	≡kerh	≡kerh	PROPN
ejpam-2562	90	18	to	to	ADP
ejpam-2562	90	19	n	n	PROPN
ejpam-2562	90	20	is	be	AUX
ejpam-2562	90	21	trivial	trivial	ADJ
ejpam-2562	90	22	=	=	NOUN
ejpam-2562	90	23	⇒	⇒	NOUN
ejpam-2562	90	24	kerh=	kerh=	NUM
ejpam-2562	90	25	0	0	NUM
ejpam-2562	91	1	i.e.	i.e.	X
ejpam-2562	91	2	k	k	X
ejpam-2562	91	3	=	=	SYM
ejpam-2562	91	4	0	0	PUNCT
ejpam-2562	91	5	whence	whence	NOUN
ejpam-2562	91	6	k	k	PROPN
ejpam-2562	92	1	=	=	SYM
ejpam-2562	92	2	0	0	NUM
ejpam-2562	92	3	is	be	AUX
ejpam-2562	92	4	trivial	trivial	ADJ
ejpam-2562	92	5	.	.	PUNCT
ejpam-2562	93	1	(	(	PUNCT
ejpam-2562	93	2	ii	ii	NOUN
ejpam-2562	93	3	)	)	PUNCT
ejpam-2562	94	1	=	=	NOUN
ejpam-2562	94	2	⇒	⇒	NOUN
ejpam-2562	94	3	(	(	PUNCT
ejpam-2562	94	4	i	i	NOUN
ejpam-2562	94	5	)	)	PUNCT
ejpam-2562	94	6	.	.	PUNCT
ejpam-2562	95	1	indeed	indeed	ADV
ejpam-2562	95	2	let	let	VERB
ejpam-2562	95	3	h	h	NOUN
ejpam-2562	95	4	:	:	PUNCT
ejpam-2562	95	5	m	m	VERB
ejpam-2562	95	6	−→	−→	ADJ
ejpam-2562	95	7	m	m	VERB
ejpam-2562	95	8	′	′	NUM
ejpam-2562	95	9	an	an	DET
ejpam-2562	95	10	r	r	NOUN
ejpam-2562	95	11	-	-	PUNCT
ejpam-2562	95	12	homomorphism	homomorphism	NOUN
ejpam-2562	95	13	such	such	ADJ
ejpam-2562	95	14	that	that	SCONJ
ejpam-2562	95	15	the	the	DET
ejpam-2562	95	16	restriction	restriction	NOUN
ejpam-2562	95	17	of	of	ADP
ejpam-2562	95	18	≡kerh	≡kerh	PROPN
ejpam-2562	95	19	to	to	ADP
ejpam-2562	95	20	n	n	PROPN
ejpam-2562	95	21	is	be	AUX
ejpam-2562	95	22	trivial	trivial	ADJ
ejpam-2562	95	23	.	.	PUNCT
ejpam-2562	96	1	so	so	ADV
ejpam-2562	96	2	by	by	ADP
ejpam-2562	96	3	assumption	assumption	NOUN
ejpam-2562	96	4	we	we	PRON
ejpam-2562	96	5	have	have	VERB
ejpam-2562	96	6	the	the	DET
ejpam-2562	96	7	restriction	restriction	NOUN
ejpam-2562	96	8	of	of	ADP
ejpam-2562	96	9	≡kerh	≡kerh	PROPN
ejpam-2562	96	10	to	to	ADP
ejpam-2562	96	11	n	n	PROPN
ejpam-2562	96	12	is	be	AUX
ejpam-2562	96	13	trivial	trivial	ADJ
ejpam-2562	96	14	=	=	NOUN
ejpam-2562	96	15	⇒	⇒	NOUN
ejpam-2562	96	16	kerh=	kerh=	NUM
ejpam-2562	96	17	0	0	NUM
ejpam-2562	96	18	.	.	PUNCT
ejpam-2562	97	1	proposition	proposition	NOUN
ejpam-2562	97	2	2	2	NUM
ejpam-2562	97	3	.	.	PUNCT
ejpam-2562	98	1	if	if	SCONJ
ejpam-2562	98	2	n	n	PRON
ejpam-2562	98	3	is	be	AUX
ejpam-2562	98	4	a	a	DET
ejpam-2562	98	5	subsemimodule	subsemimodule	NOUN
ejpam-2562	98	6	of	of	ADP
ejpam-2562	98	7	a	a	DET
ejpam-2562	98	8	left	left	ADJ
ejpam-2562	98	9	r	r	NOUN
ejpam-2562	98	10	-	-	PUNCT
ejpam-2562	98	11	semimodule	semimodule	NOUN
ejpam-2562	98	12	m	m	NOUN
ejpam-2562	98	13	,	,	PUNCT
ejpam-2562	98	14	then	then	ADV
ejpam-2562	98	15	we	we	PRON
ejpam-2562	98	16	have	have	VERB
ejpam-2562	98	17	:	:	PUNCT
ejpam-2562	98	18	e.	e.	PROPN
ejpam-2562	98	19	diop	diop	PROPN
ejpam-2562	98	20	,	,	PUNCT
ejpam-2562	98	21	d.	d.	PROPN
ejpam-2562	98	22	sow	sow	PROPN
ejpam-2562	98	23	/	/	SYM
ejpam-2562	98	24	eur	eur	PROPN
ejpam-2562	98	25	.	.	PUNCT
ejpam-2562	99	1	j.	j.	PROPN
ejpam-2562	99	2	pure	pure	PROPN
ejpam-2562	99	3	appl	appl	PROPN
ejpam-2562	99	4	.	.	PROPN
ejpam-2562	99	5	math	math	PROPN
ejpam-2562	99	6	,	,	PUNCT
ejpam-2562	99	7	9	9	NUM
ejpam-2562	99	8	(	(	PUNCT
ejpam-2562	99	9	2016	2016	NUM
ejpam-2562	99	10	)	)	PUNCT
ejpam-2562	99	11	,	,	PUNCT
ejpam-2562	99	12	250	250	NUM
ejpam-2562	99	13	-	-	SYM
ejpam-2562	99	14	265	265	NUM
ejpam-2562	99	15	254	254	NUM
ejpam-2562	99	16	(	(	PUNCT
ejpam-2562	99	17	i	i	NOUN
ejpam-2562	99	18	)	)	PUNCT
ejpam-2562	99	19	n	n	CCONJ
ejpam-2562	100	1	ã	ã	NOUN
ejpam-2562	100	2	m	m	NOUN
ejpam-2562	100	3	=	=	VERB
ejpam-2562	100	4	⇒	⇒	NOUN
ejpam-2562	100	5	n	n	PRON
ejpam-2562	100	6	ãswe	ãswe	VERB
ejpam-2562	100	7	m.	m.	NOUN
ejpam-2562	100	8	(	(	PUNCT
ejpam-2562	100	9	ii	ii	NOUN
ejpam-2562	100	10	)	)	PUNCT
ejpam-2562	100	11	n	n	CCONJ
ejpam-2562	100	12	ãs	ãs	PRON
ejpam-2562	100	13	m	m	NOUN
ejpam-2562	100	14	=	=	VERB
ejpam-2562	100	15	⇒	⇒	NOUN
ejpam-2562	100	16	n	n	PRON
ejpam-2562	100	17	ãswe	ãswe	VERB
ejpam-2562	100	18	m.	m.	NOUN
ejpam-2562	100	19	proof	proof	NOUN
ejpam-2562	100	20	.	.	PUNCT
ejpam-2562	101	1	(	(	PUNCT
ejpam-2562	101	2	i	i	NOUN
ejpam-2562	101	3	)	)	PUNCT
ejpam-2562	101	4	n	n	CCONJ
ejpam-2562	101	5	ã	ã	NOUN
ejpam-2562	101	6	m	m	NOUN
ejpam-2562	101	7	=	=	VERB
ejpam-2562	101	8	⇒	⇒	NOUN
ejpam-2562	101	9	n	n	X
ejpam-2562	101	10	ãswe	ãswe	VERB
ejpam-2562	101	11	m	m	VERB
ejpam-2562	101	12	?	?	PUNCT
ejpam-2562	102	1	let	let	VERB
ejpam-2562	102	2	h	h	NOUN
ejpam-2562	102	3	:	:	PUNCT
ejpam-2562	102	4	m	m	VERB
ejpam-2562	102	5	−→	−→	ADJ
ejpam-2562	102	6	m	m	VERB
ejpam-2562	102	7	′	′	NUM
ejpam-2562	102	8	an	an	DET
ejpam-2562	102	9	r	r	NOUN
ejpam-2562	102	10	-	-	PUNCT
ejpam-2562	102	11	homomorphism	homomorphism	NOUN
ejpam-2562	102	12	such	such	ADJ
ejpam-2562	102	13	that	that	SCONJ
ejpam-2562	102	14	the	the	DET
ejpam-2562	102	15	restriction	restriction	NOUN
ejpam-2562	102	16	of	of	ADP
ejpam-2562	102	17	≡kerh	≡kerh	PROPN
ejpam-2562	102	18	to	to	ADP
ejpam-2562	102	19	n	n	PROPN
ejpam-2562	102	20	is	be	AUX
ejpam-2562	102	21	trivial	trivial	ADJ
ejpam-2562	102	22	.	.	PUNCT
ejpam-2562	103	1	suppose	suppose	VERB
ejpam-2562	103	2	that	that	SCONJ
ejpam-2562	103	3	≡kerh	≡kerh	PROPN
ejpam-2562	103	4	is	be	AUX
ejpam-2562	103	5	nontrivial	nontrivial	ADJ
ejpam-2562	103	6	on	on	ADP
ejpam-2562	103	7	m	m	PROPN
ejpam-2562	103	8	.	.	PUNCT
ejpam-2562	104	1	so	so	ADV
ejpam-2562	104	2	,	,	PUNCT
ejpam-2562	104	3	we	we	PRON
ejpam-2562	104	4	have	have	VERB
ejpam-2562	104	5	an	an	DET
ejpam-2562	104	6	r	r	NOUN
ejpam-2562	104	7	-	-	PUNCT
ejpam-2562	104	8	congruence	congruence	NOUN
ejpam-2562	104	9	relation	relation	NOUN
ejpam-2562	104	10	on	on	ADP
ejpam-2562	104	11	m	m	PRON
ejpam-2562	104	12	namely	namely	ADV
ejpam-2562	104	13	ρ	ρ	PROPN
ejpam-2562	105	1	=	=	PROPN
ejpam-2562	105	2	≡kerh	≡kerh	PROPN
ejpam-2562	105	3	which	which	PRON
ejpam-2562	105	4	is	be	AUX
ejpam-2562	105	5	nontrivial	nontrivial	ADJ
ejpam-2562	105	6	on	on	ADP
ejpam-2562	105	7	m	m	PRON
ejpam-2562	105	8	and	and	CCONJ
ejpam-2562	105	9	whose	whose	DET
ejpam-2562	105	10	restriction	restriction	NOUN
ejpam-2562	105	11	on	on	ADP
ejpam-2562	105	12	n	n	PRON
ejpam-2562	105	13	is	be	AUX
ejpam-2562	105	14	trivial	trivial	ADJ
ejpam-2562	105	15	;	;	PUNCT
ejpam-2562	105	16	contradiction	contradiction	NOUN
ejpam-2562	105	17	because	because	SCONJ
ejpam-2562	105	18	n	n	PROPN
ejpam-2562	105	19	ã	ã	NOUN
ejpam-2562	105	20	m	m	NOUN
ejpam-2562	105	21	.	.	PUNCT
ejpam-2562	106	1	so	so	ADV
ejpam-2562	106	2	≡kerh	≡kerh	PROPN
ejpam-2562	106	3	is	be	AUX
ejpam-2562	106	4	trivial	trivial	ADJ
ejpam-2562	106	5	on	on	ADP
ejpam-2562	106	6	m	m	PRON
ejpam-2562	106	7	and	and	CCONJ
ejpam-2562	106	8	we	we	PRON
ejpam-2562	106	9	deduce	deduce	VERB
ejpam-2562	106	10	that	that	DET
ejpam-2562	106	11	kerh	kerh	NOUN
ejpam-2562	107	1	=	=	PUNCT
ejpam-2562	107	2	0	0	NUM
ejpam-2562	107	3	,	,	PUNCT
ejpam-2562	107	4	then	then	ADV
ejpam-2562	107	5	n	n	PRON
ejpam-2562	107	6	ãswe	ãswe	NOUN
ejpam-2562	107	7	m	m	PROPN
ejpam-2562	107	8	.	.	PUNCT
ejpam-2562	108	1	(	(	PUNCT
ejpam-2562	108	2	ii	ii	NOUN
ejpam-2562	108	3	)	)	PUNCT
ejpam-2562	108	4	n	n	CCONJ
ejpam-2562	108	5	ãs	ãs	PRON
ejpam-2562	108	6	m	m	NOUN
ejpam-2562	108	7	=	=	VERB
ejpam-2562	108	8	⇒	⇒	NOUN
ejpam-2562	108	9	n	n	X
ejpam-2562	108	10	ãswe	ãswe	VERB
ejpam-2562	108	11	m	m	VERB
ejpam-2562	108	12	?	?	PUNCT
ejpam-2562	109	1	let	let	VERB
ejpam-2562	109	2	h	h	NOUN
ejpam-2562	109	3	:	:	PUNCT
ejpam-2562	109	4	m	m	VERB
ejpam-2562	109	5	−→	−→	ADJ
ejpam-2562	109	6	m	m	VERB
ejpam-2562	109	7	′	′	NUM
ejpam-2562	109	8	an	an	DET
ejpam-2562	109	9	r	r	NOUN
ejpam-2562	109	10	-	-	PUNCT
ejpam-2562	109	11	homomorphism	homomorphism	NOUN
ejpam-2562	109	12	such	such	ADJ
ejpam-2562	109	13	that	that	SCONJ
ejpam-2562	109	14	the	the	DET
ejpam-2562	109	15	restriction	restriction	NOUN
ejpam-2562	109	16	of	of	ADP
ejpam-2562	109	17	≡kerh	≡kerh	PROPN
ejpam-2562	109	18	to	to	ADP
ejpam-2562	109	19	n	n	PROPN
ejpam-2562	109	20	is	be	AUX
ejpam-2562	109	21	trivial	trivial	ADJ
ejpam-2562	109	22	.	.	PUNCT
ejpam-2562	110	1	so	so	ADV
ejpam-2562	110	2	kerh∩	kerh∩	PROPN
ejpam-2562	110	3	n	n	PROPN
ejpam-2562	110	4	=	=	SYM
ejpam-2562	110	5	0	0	NUM
ejpam-2562	110	6	.	.	PUNCT
ejpam-2562	111	1	since	since	SCONJ
ejpam-2562	111	2	n	n	PRON
ejpam-2562	111	3	ãs	ãs	PRON
ejpam-2562	111	4	m	m	PROPN
ejpam-2562	111	5	,	,	PUNCT
ejpam-2562	111	6	then	then	ADV
ejpam-2562	111	7	kerh∩	kerh∩	PROPN
ejpam-2562	111	8	n	n	PROPN
ejpam-2562	111	9	=	=	SYM
ejpam-2562	111	10	0=⇒	0=⇒	PROPN
ejpam-2562	111	11	kerh=	kerh=	NUM
ejpam-2562	111	12	0	0	NUM
ejpam-2562	111	13	,	,	PUNCT
ejpam-2562	111	14	then	then	ADV
ejpam-2562	111	15	n	n	PRON
ejpam-2562	111	16	ãswe	ãswe	NOUN
ejpam-2562	111	17	m	m	PROPN
ejpam-2562	111	18	.	.	PUNCT
ejpam-2562	112	1	the	the	DET
ejpam-2562	112	2	previous	previous	ADJ
ejpam-2562	112	3	proposition	proposition	NOUN
ejpam-2562	112	4	shows	show	VERB
ejpam-2562	112	5	that	that	SCONJ
ejpam-2562	112	6	the	the	DET
ejpam-2562	112	7	class	class	NOUN
ejpam-2562	112	8	of	of	ADP
ejpam-2562	112	9	semi	semi	ADJ
ejpam-2562	112	10	-	-	ADJ
ejpam-2562	112	11	weakly	weakly	ADJ
ejpam-2562	112	12	-	-	PUNCT
ejpam-2562	112	13	essential	essential	ADJ
ejpam-2562	112	14	subsemimodules	subsemimodule	NOUN
ejpam-2562	112	15	of	of	ADP
ejpam-2562	112	16	a	a	DET
ejpam-2562	112	17	semimodule	semimodule	NOUN
ejpam-2562	112	18	m	m	NOUN
ejpam-2562	112	19	contains	contain	VERB
ejpam-2562	112	20	the	the	DET
ejpam-2562	112	21	two	two	NUM
ejpam-2562	112	22	classes	class	NOUN
ejpam-2562	112	23	constituted	constitute	VERB
ejpam-2562	112	24	by	by	ADP
ejpam-2562	112	25	essential	essential	ADJ
ejpam-2562	112	26	subsemimodules	subsemimodule	NOUN
ejpam-2562	112	27	and	and	CCONJ
ejpam-2562	112	28	semi	semi	ADJ
ejpam-2562	112	29	-	-	ADJ
ejpam-2562	112	30	essential	essential	ADJ
ejpam-2562	112	31	subsemimodules	subsemimodule	NOUN
ejpam-2562	112	32	of	of	ADP
ejpam-2562	112	33	m	m	PROPN
ejpam-2562	112	34	.	.	PUNCT
ejpam-2562	113	1	since	since	SCONJ
ejpam-2562	113	2	r	r	NOUN
ejpam-2562	113	3	-	-	PUNCT
ejpam-2562	113	4	congruence	congruence	NOUN
ejpam-2562	113	5	relations	relation	NOUN
ejpam-2562	113	6	in	in	ADP
ejpam-2562	113	7	modules	module	NOUN
ejpam-2562	113	8	theory	theory	NOUN
ejpam-2562	113	9	are	be	AUX
ejpam-2562	113	10	characterized	characterize	VERB
ejpam-2562	113	11	by	by	ADP
ejpam-2562	113	12	submodules	submodule	NOUN
ejpam-2562	113	13	,	,	PUNCT
ejpam-2562	113	14	then	then	ADV
ejpam-2562	113	15	the	the	DET
ejpam-2562	113	16	three	three	NUM
ejpam-2562	113	17	notions	notion	NOUN
ejpam-2562	113	18	of	of	ADP
ejpam-2562	113	19	essential	essential	ADJ
ejpam-2562	113	20	in	in	ADP
ejpam-2562	113	21	semimodules	semimodule	NOUN
ejpam-2562	113	22	theory	theory	NOUN
ejpam-2562	113	23	,	,	PUNCT
ejpam-2562	113	24	defined	define	VERB
ejpam-2562	113	25	in	in	ADP
ejpam-2562	113	26	this	this	DET
ejpam-2562	113	27	paper	paper	NOUN
ejpam-2562	113	28	,	,	PUNCT
ejpam-2562	113	29	are	be	AUX
ejpam-2562	113	30	the	the	DET
ejpam-2562	113	31	same	same	ADJ
ejpam-2562	113	32	for	for	ADP
ejpam-2562	113	33	modules	module	NOUN
ejpam-2562	113	34	theory	theory	NOUN
ejpam-2562	113	35	.	.	PUNCT
ejpam-2562	114	1	recall	recall	VERB
ejpam-2562	114	2	that	that	PRON
ejpam-2562	114	3	:	:	PUNCT
ejpam-2562	114	4	essential	essential	ADJ
ejpam-2562	114	5	:	:	PUNCT
ejpam-2562	114	6	∀ρ	∀ρ	X
ejpam-2562	114	7	∈	∈	PROPN
ejpam-2562	114	8	r	r	NOUN
ejpam-2562	114	9	-	-	PUNCT
ejpam-2562	114	10	cong(m	cong(m	VERB
ejpam-2562	114	11	)	)	PUNCT
ejpam-2562	114	12	,	,	PUNCT
ejpam-2562	114	13	ρ	ρ	PROPN
ejpam-2562	114	14	trivial	trivial	ADJ
ejpam-2562	114	15	on	on	ADP
ejpam-2562	114	16	n	n	PRON
ejpam-2562	114	17	=	=	NOUN
ejpam-2562	114	18	⇒	⇒	NOUN
ejpam-2562	114	19	ρ	ρ	NOUN
ejpam-2562	114	20	trivial	trivial	ADJ
ejpam-2562	114	21	on	on	ADP
ejpam-2562	114	22	m	m	NOUN
ejpam-2562	114	23	semi	semi	ADJ
ejpam-2562	114	24	-	-	ADJ
ejpam-2562	114	25	essential	essential	ADJ
ejpam-2562	114	26	:	:	PUNCT
ejpam-2562	114	27	∀l	∀l	NOUN
ejpam-2562	114	28	≤	≤	NOUN
ejpam-2562	114	29	m	m	VERB
ejpam-2562	114	30	,	,	PUNCT
ejpam-2562	114	31	l	l	NOUN
ejpam-2562	114	32	∩	∩	X
ejpam-2562	114	33	n	n	NOUN
ejpam-2562	114	34	=	=	SYM
ejpam-2562	114	35	0=⇒	0=⇒	NUM
ejpam-2562	114	36	l	l	NOUN
ejpam-2562	114	37	=	=	SYM
ejpam-2562	114	38	0	0	NUM
ejpam-2562	114	39	semi	semi	ADJ
ejpam-2562	114	40	-	-	ADJ
ejpam-2562	114	41	weakly	weakly	ADJ
ejpam-2562	114	42	essential	essential	ADJ
ejpam-2562	114	43	:	:	PUNCT
ejpam-2562	114	44	∀l	∀l	NOUN
ejpam-2562	114	45	≤	≤	NOUN
ejpam-2562	114	46	m	m	VERB
ejpam-2562	114	47	,	,	PUNCT
ejpam-2562	114	48	≡l	≡l	NOUN
ejpam-2562	114	49	trivial	trivial	ADJ
ejpam-2562	114	50	on	on	ADP
ejpam-2562	114	51	n	n	PRON
ejpam-2562	114	52	=	=	NOUN
ejpam-2562	114	53	⇒	⇒	NOUN
ejpam-2562	114	54	l	l	NOUN
ejpam-2562	115	1	=	=	PUNCT
ejpam-2562	115	2	0	0	PUNCT
ejpam-2562	115	3	here	here	ADV
ejpam-2562	115	4	we	we	PRON
ejpam-2562	115	5	give	give	VERB
ejpam-2562	115	6	some	some	DET
ejpam-2562	115	7	examples	example	NOUN
ejpam-2562	115	8	for	for	ADP
ejpam-2562	115	9	the	the	DET
ejpam-2562	115	10	different	different	ADJ
ejpam-2562	115	11	notions	notion	NOUN
ejpam-2562	115	12	of	of	ADP
ejpam-2562	115	13	"	"	PUNCT
ejpam-2562	115	14	essential	essential	ADJ
ejpam-2562	115	15	"	"	PUNCT
ejpam-2562	115	16	in	in	ADP
ejpam-2562	115	17	semimodules	semimodule	NOUN
ejpam-2562	115	18	theory	theory	NOUN
ejpam-2562	115	19	.	.	PUNCT
ejpam-2562	116	1	example	example	NOUN
ejpam-2562	117	1	1	1	NUM
ejpam-2562	117	2	.	.	PUNCT
ejpam-2562	117	3	in	in	ADP
ejpam-2562	117	4	this	this	DET
ejpam-2562	117	5	example	example	NOUN
ejpam-2562	117	6	,	,	PUNCT
ejpam-2562	117	7	we	we	PRON
ejpam-2562	117	8	propose	propose	VERB
ejpam-2562	117	9	a	a	DET
ejpam-2562	117	10	finite	finite	NOUN
ejpam-2562	117	11	semi	semi	ADJ
ejpam-2562	117	12	-	-	ADJ
ejpam-2562	117	13	weakly	weakly	ADJ
ejpam-2562	117	14	-	-	PUNCT
ejpam-2562	117	15	essential	essential	ADJ
ejpam-2562	117	16	subsemimodule	subsemimodule	NOUN
ejpam-2562	117	17	which	which	PRON
ejpam-2562	117	18	is	be	AUX
ejpam-2562	117	19	neither	neither	CCONJ
ejpam-2562	117	20	essential	essential	ADJ
ejpam-2562	117	21	nor	nor	CCONJ
ejpam-2562	117	22	semi	semi	ADJ
ejpam-2562	117	23	-	-	ADJ
ejpam-2562	117	24	essential	essential	ADJ
ejpam-2562	117	25	.	.	PUNCT
ejpam-2562	118	1	set	set	VERB
ejpam-2562	118	2	r=	r=	PROPN
ejpam-2562	118	3	{	{	PUNCT
ejpam-2562	118	4	0,1	0,1	NOUN
ejpam-2562	118	5	,	,	PUNCT
ejpam-2562	118	6	.	.	PUNCT
ejpam-2562	118	7	.	.	PUNCT
ejpam-2562	119	1	.	.	PUNCT
ejpam-2562	120	1	,	,	PUNCT
ejpam-2562	120	2	n	n	CCONJ
ejpam-2562	120	3	}	}	PUNCT
ejpam-2562	120	4	(	(	PUNCT
ejpam-2562	120	5	with	with	ADP
ejpam-2562	120	6	n	n	PRON
ejpam-2562	120	7	∈	∈	PROPN
ejpam-2562	120	8	n	n	NOUN
ejpam-2562	120	9	and	and	CCONJ
ejpam-2562	120	10	n≥	n≥	NOUN
ejpam-2562	120	11	2	2	NUM
ejpam-2562	120	12	)	)	PUNCT
ejpam-2562	120	13	and	and	CCONJ
ejpam-2562	120	14	define	define	VERB
ejpam-2562	120	15	on	on	ADP
ejpam-2562	120	16	r	r	NOUN
ejpam-2562	120	17	the	the	DET
ejpam-2562	120	18	two	two	NUM
ejpam-2562	120	19	commutative	commutative	ADJ
ejpam-2562	120	20	operations	operation	NOUN
ejpam-2562	120	21	(	(	PUNCT
ejpam-2562	120	22	⊕,⊗	⊕,⊗	NOUN
ejpam-2562	120	23	)	)	PUNCT
ejpam-2562	120	24	as	as	SCONJ
ejpam-2562	120	25	follows	follow	VERB
ejpam-2562	120	26	:	:	PUNCT
ejpam-2562	120	27	(	(	PUNCT
ejpam-2562	120	28	i	i	NOUN
ejpam-2562	120	29	)	)	PUNCT
ejpam-2562	120	30	∀x	∀x	X
ejpam-2562	120	31	,	,	PUNCT
ejpam-2562	120	32	y	y	PROPN
ejpam-2562	120	33	∈	∈	PROPN
ejpam-2562	120	34	r	r	NOUN
ejpam-2562	120	35	:	:	PUNCT
ejpam-2562	120	36	x	x	PROPN
ejpam-2562	120	37	⊕	⊕	NOUN
ejpam-2562	120	38	y	y	NOUN
ejpam-2562	120	39	=	=	PRON
ejpam-2562	120	40	min(x	min(x	PROPN
ejpam-2562	120	41	,	,	PUNCT
ejpam-2562	120	42	y	y	PROPN
ejpam-2562	120	43	)	)	PUNCT
ejpam-2562	120	44	(	(	PUNCT
ejpam-2562	120	45	ii	ii	NOUN
ejpam-2562	120	46	)	)	PUNCT
ejpam-2562	120	47	∀x	∀x	X
ejpam-2562	120	48	,	,	PUNCT
ejpam-2562	120	49	y	y	PROPN
ejpam-2562	120	50	∈	∈	PROPN
ejpam-2562	120	51	r	r	NOUN
ejpam-2562	120	52	:	:	PUNCT
ejpam-2562	120	53	x	x	SYM
ejpam-2562	120	54	⊗	⊗	NUM
ejpam-2562	120	55	y	y	PROPN
ejpam-2562	120	56	=	=	PUNCT
ejpam-2562	120	57	max(x	max(x	PROPN
ejpam-2562	120	58	,	,	PUNCT
ejpam-2562	120	59	y	y	PROPN
ejpam-2562	120	60	)	)	PUNCT
ejpam-2562	120	61	.	.	PUNCT
ejpam-2562	121	1	(	(	PUNCT
ejpam-2562	121	2	r,⊕,⊗	r,⊕,⊗	NOUN
ejpam-2562	121	3	)	)	PUNCT
ejpam-2562	121	4	is	be	AUX
ejpam-2562	121	5	a	a	DET
ejpam-2562	121	6	semiring	semiring	NOUN
ejpam-2562	121	7	with	with	ADP
ejpam-2562	121	8	0r	0r	PROPN
ejpam-2562	121	9	=	=	SYM
ejpam-2562	121	10	n	n	CCONJ
ejpam-2562	121	11	,	,	PUNCT
ejpam-2562	121	12	1r	1r	NUM
ejpam-2562	121	13	=	=	SYM
ejpam-2562	121	14	0	0	X
ejpam-2562	121	15	.	.	PUNCT
ejpam-2562	122	1	let	let	VERB
ejpam-2562	122	2	m	m	VERB
ejpam-2562	122	3	=	=	VERB
ejpam-2562	122	4	{	{	PUNCT
ejpam-2562	122	5	0	0	NUM
ejpam-2562	122	6	,	,	PUNCT
ejpam-2562	122	7	1	1	NUM
ejpam-2562	122	8	n	n	NUM
ejpam-2562	122	9	,	,	PUNCT
ejpam-2562	122	10	1	1	NUM
ejpam-2562	122	11	n−	n−	NOUN
ejpam-2562	122	12	1	1	NUM
ejpam-2562	122	13	,	,	PUNCT
ejpam-2562	122	14	1	1	NUM
ejpam-2562	122	15	n−	n−	NOUN
ejpam-2562	122	16	2	2	NUM
ejpam-2562	122	17	,	,	PUNCT
ejpam-2562	122	18	.	.	PUNCT
ejpam-2562	122	19	.	.	PUNCT
ejpam-2562	123	1	.	.	PUNCT
ejpam-2562	124	1	1	1	NUM
ejpam-2562	124	2	2	2	NUM
ejpam-2562	124	3	,	,	PUNCT
ejpam-2562	124	4	1,1	1,1	NUM
ejpam-2562	124	5	+	+	SYM
ejpam-2562	124	6	1	1	NUM
ejpam-2562	124	7	n	n	NOUN
ejpam-2562	124	8	,	,	PUNCT
ejpam-2562	124	9	1	1	NUM
ejpam-2562	124	10	+	+	SYM
ejpam-2562	124	11	1	1	NUM
ejpam-2562	124	12	n−	n−	NOUN
ejpam-2562	124	13	1	1	NUM
ejpam-2562	124	14	,	,	PUNCT
ejpam-2562	124	15	e.	e.	PROPN
ejpam-2562	124	16	diop	diop	PROPN
ejpam-2562	124	17	,	,	PUNCT
ejpam-2562	124	18	d.	d.	PROPN
ejpam-2562	124	19	sow	sow	PROPN
ejpam-2562	124	20	/	/	SYM
ejpam-2562	124	21	eur	eur	PROPN
ejpam-2562	124	22	.	.	PUNCT
ejpam-2562	125	1	j.	j.	PROPN
ejpam-2562	125	2	pure	pure	PROPN
ejpam-2562	125	3	appl	appl	PROPN
ejpam-2562	125	4	.	.	PROPN
ejpam-2562	125	5	math	math	PROPN
ejpam-2562	125	6	,	,	PUNCT
ejpam-2562	125	7	9	9	NUM
ejpam-2562	125	8	(	(	PUNCT
ejpam-2562	125	9	2016	2016	NUM
ejpam-2562	125	10	)	)	PUNCT
ejpam-2562	125	11	,	,	PUNCT
ejpam-2562	125	12	250	250	NUM
ejpam-2562	125	13	-	-	SYM
ejpam-2562	125	14	265	265	NUM
ejpam-2562	125	15	255	255	NUM
ejpam-2562	125	16	.	.	PUNCT
ejpam-2562	125	17	.	.	PUNCT
ejpam-2562	126	1	.	.	PUNCT
ejpam-2562	127	1	1	1	NUM
ejpam-2562	127	2	+	+	NUM
ejpam-2562	127	3	1	1	NUM
ejpam-2562	127	4	2	2	NUM
ejpam-2562	127	5	,	,	PUNCT
ejpam-2562	127	6	2,2	2,2	NUM
ejpam-2562	127	7	+	+	SYM
ejpam-2562	127	8	1	1	NUM
ejpam-2562	127	9	n	n	NOUN
ejpam-2562	127	10	,	,	PUNCT
ejpam-2562	127	11	.	.	PUNCT
ejpam-2562	127	12	.	.	PUNCT
ejpam-2562	127	13	.	.	PUNCT
ejpam-2562	128	1	n−	n−	NOUN
ejpam-2562	128	2	1	1	NUM
ejpam-2562	128	3	,	,	PUNCT
ejpam-2562	128	4	(	(	PUNCT
ejpam-2562	128	5	n−	n−	NOUN
ejpam-2562	128	6	1	1	NUM
ejpam-2562	128	7	)	)	PUNCT
ejpam-2562	128	8	+	+	CCONJ
ejpam-2562	128	9	1	1	NUM
ejpam-2562	128	10	n	n	NOUN
ejpam-2562	128	11	,	,	PUNCT
ejpam-2562	128	12	.	.	PUNCT
ejpam-2562	128	13	.	.	PUNCT
ejpam-2562	128	14	.	.	PUNCT
ejpam-2562	129	1	(	(	PUNCT
ejpam-2562	129	2	n−	n−	NOUN
ejpam-2562	129	3	1	1	NUM
ejpam-2562	129	4	)	)	PUNCT
ejpam-2562	129	5	+	+	CCONJ
ejpam-2562	129	6	1	1	NUM
ejpam-2562	129	7	2	2	NUM
ejpam-2562	129	8	,	,	PUNCT
ejpam-2562	129	9	n}.(m	n}.(m	PROPN
ejpam-2562	129	10	,	,	PUNCT
ejpam-2562	129	11	⊕	⊕	PROPN
ejpam-2562	129	12	)	)	PUNCT
ejpam-2562	129	13	is	be	AUX
ejpam-2562	129	14	a	a	DET
ejpam-2562	129	15	commutative	commutative	ADJ
ejpam-2562	129	16	monoid	monoid	NOUN
ejpam-2562	129	17	with	with	ADP
ejpam-2562	129	18	0	0	NUM
ejpam-2562	129	19	m	m	NOUN
ejpam-2562	129	20	=	=	NOUN
ejpam-2562	129	21	n.	n.	NOUN
ejpam-2562	129	22	let	let	VERB
ejpam-2562	129	23	"	"	PUNCT
ejpam-2562	129	24	⋆	⋆	VERB
ejpam-2562	129	25	"	"	PUNCT
ejpam-2562	129	26	be	be	AUX
ejpam-2562	129	27	the	the	DET
ejpam-2562	129	28	external	external	ADJ
ejpam-2562	129	29	operation	operation	NOUN
ejpam-2562	129	30	defined	define	VERB
ejpam-2562	129	31	by	by	ADP
ejpam-2562	129	32	:	:	PUNCT
ejpam-2562	129	33	⋆	⋆	NOUN
ejpam-2562	129	34	:	:	PUNCT
ejpam-2562	129	35	r×m	r×m	ADP
ejpam-2562	129	36	−→	−→	ADJ
ejpam-2562	129	37	m	m	PROPN
ejpam-2562	129	38	(	(	PUNCT
ejpam-2562	129	39	r	r	NOUN
ejpam-2562	129	40	,	,	PUNCT
ejpam-2562	129	41	m	m	NOUN
ejpam-2562	129	42	)	)	PUNCT
ejpam-2562	129	43	7−→	7−→	NOUN
ejpam-2562	129	44	r	r	NOUN
ejpam-2562	129	45	⋆m	⋆m	PROPN
ejpam-2562	129	46	=	=	SYM
ejpam-2562	129	47	max(r	max(r	PROPN
ejpam-2562	129	48	,	,	PUNCT
ejpam-2562	129	49	m	m	VERB
ejpam-2562	129	50	)	)	PUNCT
ejpam-2562	129	51	it	it	PRON
ejpam-2562	129	52	is	be	AUX
ejpam-2562	129	53	clear	clear	ADJ
ejpam-2562	129	54	that	that	SCONJ
ejpam-2562	129	55	(	(	PUNCT
ejpam-2562	129	56	m	m	NOUN
ejpam-2562	129	57	,	,	PUNCT
ejpam-2562	129	58	⊕,⋆	⊕,⋆	PROPN
ejpam-2562	129	59	)	)	PUNCT
ejpam-2562	129	60	is	be	AUX
ejpam-2562	129	61	an	an	DET
ejpam-2562	129	62	r	r	NOUN
ejpam-2562	129	63	-	-	PUNCT
ejpam-2562	129	64	semimodule	semimodule	NOUN
ejpam-2562	129	65	.	.	PUNCT
ejpam-2562	130	1	let	let	VERB
ejpam-2562	130	2	a	a	DET
ejpam-2562	130	3	∈	∈	PROPN
ejpam-2562	130	4	{	{	PUNCT
ejpam-2562	130	5	0,1	0,1	NOUN
ejpam-2562	130	6	,	,	PUNCT
ejpam-2562	130	7	.	.	PUNCT
ejpam-2562	130	8	.	.	PUNCT
ejpam-2562	131	1	.	.	PUNCT
ejpam-2562	132	1	,	,	PUNCT
ejpam-2562	132	2	n−	n−	NOUN
ejpam-2562	132	3	1	1	NUM
ejpam-2562	132	4	}	}	PUNCT
ejpam-2562	132	5	.	.	PUNCT
ejpam-2562	133	1	set	set	VERB
ejpam-2562	133	2	na	na	NOUN
ejpam-2562	133	3	=	=	PUNCT
ejpam-2562	133	4	{	{	PUNCT
ejpam-2562	133	5	a	a	X
ejpam-2562	133	6	,	,	PUNCT
ejpam-2562	133	7	.	.	PUNCT
ejpam-2562	133	8	.	.	PUNCT
ejpam-2562	134	1	.	.	PUNCT
ejpam-2562	135	1	,	,	PUNCT
ejpam-2562	135	2	n−	n−	NOUN
ejpam-2562	135	3	1	1	NUM
ejpam-2562	135	4	2	2	NUM
ejpam-2562	135	5	,	,	PUNCT
ejpam-2562	135	6	n	n	CCONJ
ejpam-2562	135	7	}	}	PUNCT
ejpam-2562	135	8	;	;	PUNCT
ejpam-2562	135	9	then	then	ADV
ejpam-2562	135	10	na	na	INTJ
ejpam-2562	135	11	is	be	AUX
ejpam-2562	135	12	a	a	DET
ejpam-2562	135	13	subsemimodule	subsemimodule	NOUN
ejpam-2562	135	14	of	of	ADP
ejpam-2562	135	15	m.	m.	NOUN
ejpam-2562	135	16	now	now	ADV
ejpam-2562	135	17	let	let	VERB
ejpam-2562	135	18	us	we	PRON
ejpam-2562	135	19	show	show	VERB
ejpam-2562	135	20	that	that	SCONJ
ejpam-2562	135	21	na	na	NOUN
ejpam-2562	135	22	is	be	AUX
ejpam-2562	135	23	semi	semi	ADJ
ejpam-2562	135	24	-	-	ADJ
ejpam-2562	135	25	weakly	weakly	ADV
ejpam-2562	135	26	-	-	PUNCT
ejpam-2562	135	27	essential	essential	ADJ
ejpam-2562	135	28	but	but	CCONJ
ejpam-2562	135	29	is	be	AUX
ejpam-2562	135	30	neither	neither	CCONJ
ejpam-2562	135	31	essential	essential	ADJ
ejpam-2562	135	32	nor	nor	CCONJ
ejpam-2562	135	33	semi	semi	ADJ
ejpam-2562	135	34	-	-	ADJ
ejpam-2562	135	35	essential	essential	ADJ
ejpam-2562	135	36	.	.	PUNCT
ejpam-2562	136	1	•	•	NUM
ejpam-2562	136	2	let	let	VERB
ejpam-2562	136	3	us	we	PRON
ejpam-2562	136	4	prove	prove	VERB
ejpam-2562	136	5	that	that	SCONJ
ejpam-2562	136	6	na	na	NOUN
ejpam-2562	136	7	is	be	AUX
ejpam-2562	136	8	semi	semi	ADJ
ejpam-2562	136	9	-	-	ADJ
ejpam-2562	136	10	weakly	weakly	ADJ
ejpam-2562	136	11	-	-	PUNCT
ejpam-2562	136	12	essential	essential	ADJ
ejpam-2562	136	13	.	.	PUNCT
ejpam-2562	137	1	let	let	VERB
ejpam-2562	137	2	k	k	PROPN
ejpam-2562	137	3	≤	≤	PROPN
ejpam-2562	137	4	m.	m.	NOUN
ejpam-2562	137	5	let	let	VERB
ejpam-2562	137	6	us	we	PRON
ejpam-2562	137	7	prove	prove	VERB
ejpam-2562	137	8	that	that	SCONJ
ejpam-2562	137	9	the	the	DET
ejpam-2562	137	10	restriction	restriction	NOUN
ejpam-2562	137	11	of	of	ADP
ejpam-2562	137	12	≡k	≡k	NOUN
ejpam-2562	137	13	to	to	PART
ejpam-2562	137	14	na	na	VERB
ejpam-2562	137	15	is	be	AUX
ejpam-2562	137	16	trivial	trivial	ADJ
ejpam-2562	137	17	if	if	SCONJ
ejpam-2562	137	18	and	and	CCONJ
ejpam-2562	137	19	only	only	ADV
ejpam-2562	137	20	if	if	SCONJ
ejpam-2562	137	21	k	k	PROPN
ejpam-2562	137	22	=	=	PRON
ejpam-2562	137	23	{	{	PUNCT
ejpam-2562	137	24	0	0	NUM
ejpam-2562	137	25	m	m	NOUN
ejpam-2562	137	26	}	}	PUNCT
ejpam-2562	137	27	.	.	PUNCT
ejpam-2562	138	1	(=	(=	AUX
ejpam-2562	138	2	⇒	⇒	NOUN
ejpam-2562	138	3	)	)	PUNCT
ejpam-2562	138	4	if	if	SCONJ
ejpam-2562	138	5	k	k	PROPN
ejpam-2562	138	6	6=	6=	PROPN
ejpam-2562	138	7	{	{	PUNCT
ejpam-2562	138	8	0	0	NUM
ejpam-2562	138	9	m	m	VERB
ejpam-2562	138	10	}	}	PUNCT
ejpam-2562	138	11	then	then	ADV
ejpam-2562	138	12	let	let	VERB
ejpam-2562	138	13	k	k	PRON
ejpam-2562	138	14	be	be	AUX
ejpam-2562	138	15	an	an	DET
ejpam-2562	138	16	element	element	NOUN
ejpam-2562	138	17	of	of	ADP
ejpam-2562	138	18	k	k	PROPN
ejpam-2562	138	19	such	such	ADJ
ejpam-2562	138	20	that	that	SCONJ
ejpam-2562	138	21	k	k	PROPN
ejpam-2562	138	22	6=	6=	PROPN
ejpam-2562	138	23	0	0	NUM
ejpam-2562	138	24	m	m	NOUN
ejpam-2562	138	25	.	.	PUNCT
ejpam-2562	139	1	since	since	SCONJ
ejpam-2562	139	2	k	k	PROPN
ejpam-2562	139	3	6=	6=	PROPN
ejpam-2562	139	4	n	n	CCONJ
ejpam-2562	139	5	,	,	PUNCT
ejpam-2562	139	6	we	we	PRON
ejpam-2562	139	7	have	have	VERB
ejpam-2562	139	8	[	[	PUNCT
ejpam-2562	139	9	n−	n−	NOUN
ejpam-2562	139	10	1	1	NUM
ejpam-2562	139	11	2]⊕	2]⊕	NUM
ejpam-2562	139	12	k	k	NOUN
ejpam-2562	139	13	=	=	PUNCT
ejpam-2562	139	14	n⊕	n⊕	PROPN
ejpam-2562	139	15	k	k	PROPN
ejpam-2562	139	16	,	,	PUNCT
ejpam-2562	139	17	so	so	SCONJ
ejpam-2562	139	18	n	n	ADP
ejpam-2562	139	19	≡k	≡k	NOUN
ejpam-2562	139	20	n−	n−	NOUN
ejpam-2562	139	21	1	1	NUM
ejpam-2562	139	22	2	2	NUM
ejpam-2562	139	23	with	with	ADP
ejpam-2562	139	24	n	n	NUM
ejpam-2562	139	25	6=	6=	NUM
ejpam-2562	139	26	n−	n−	NOUN
ejpam-2562	139	27	1	1	NUM
ejpam-2562	139	28	2	2	NUM
ejpam-2562	139	29	,	,	PUNCT
ejpam-2562	139	30	contradiction	contradiction	NOUN
ejpam-2562	139	31	.	.	PUNCT
ejpam-2562	140	1	thus	thus	ADV
ejpam-2562	140	2	k	k	X
ejpam-2562	140	3	=	=	PUNCT
ejpam-2562	140	4	{	{	PUNCT
ejpam-2562	140	5	0	0	NUM
ejpam-2562	140	6	m	m	NOUN
ejpam-2562	140	7	}	}	PUNCT
ejpam-2562	140	8	.	.	PUNCT
ejpam-2562	141	1	(	(	PUNCT
ejpam-2562	141	2	⇐	⇐	ADJ
ejpam-2562	141	3	=)	=)	PROPN
ejpam-2562	141	4	trivial	trivial	ADJ
ejpam-2562	141	5	.	.	PUNCT
ejpam-2562	142	1	•	•	INTJ
ejpam-2562	142	2	let	let	VERB
ejpam-2562	142	3	us	we	PRON
ejpam-2562	142	4	prove	prove	VERB
ejpam-2562	142	5	that	that	SCONJ
ejpam-2562	142	6	na	na	NOUN
ejpam-2562	142	7	is	be	AUX
ejpam-2562	142	8	not	not	PART
ejpam-2562	142	9	semi	semi	ADJ
ejpam-2562	142	10	-	-	ADJ
ejpam-2562	142	11	essential	essential	ADJ
ejpam-2562	142	12	.	.	PUNCT
ejpam-2562	143	1	set	set	VERB
ejpam-2562	143	2	l	l	NOUN
ejpam-2562	143	3	=	=	SYM
ejpam-2562	143	4	{	{	PUNCT
ejpam-2562	143	5	(	(	PUNCT
ejpam-2562	143	6	n−	n−	NOUN
ejpam-2562	143	7	1	1	NUM
ejpam-2562	143	8	)	)	PUNCT
ejpam-2562	143	9	+	+	CCONJ
ejpam-2562	143	10	1	1	NUM
ejpam-2562	143	11	3	3	NUM
ejpam-2562	143	12	,	,	PUNCT
ejpam-2562	143	13	n	n	CCONJ
ejpam-2562	143	14	}	}	PUNCT
ejpam-2562	143	15	.	.	PUNCT
ejpam-2562	144	1	so	so	ADV
ejpam-2562	144	2	l	l	NOUN
ejpam-2562	144	3	is	be	AUX
ejpam-2562	144	4	a	a	DET
ejpam-2562	144	5	subsemimodule	subsemimodule	NOUN
ejpam-2562	144	6	of	of	ADP
ejpam-2562	144	7	m	m	NOUN
ejpam-2562	144	8	such	such	ADJ
ejpam-2562	144	9	that	that	PRON
ejpam-2562	144	10	l	l	NOUN
ejpam-2562	144	11	6=	6=	ADP
ejpam-2562	144	12	0	0	NUM
ejpam-2562	144	13	and	and	CCONJ
ejpam-2562	144	14	l	l	NOUN
ejpam-2562	144	15	∩	∩	NOUN
ejpam-2562	144	16	na	na	NOUN
ejpam-2562	144	17	=	=	SYM
ejpam-2562	144	18	0	0	NUM
ejpam-2562	144	19	,	,	PUNCT
ejpam-2562	144	20	then	then	ADV
ejpam-2562	144	21	na	na	INTJ
ejpam-2562	144	22	is	be	AUX
ejpam-2562	144	23	not	not	PART
ejpam-2562	144	24	semi	semi	ADJ
ejpam-2562	144	25	-	-	ADJ
ejpam-2562	144	26	essential	essential	ADJ
ejpam-2562	144	27	.	.	PUNCT
ejpam-2562	145	1	•	•	NUM
ejpam-2562	145	2	let	let	VERB
ejpam-2562	145	3	us	we	PRON
ejpam-2562	145	4	prove	prove	VERB
ejpam-2562	145	5	that	that	SCONJ
ejpam-2562	145	6	na	na	NOUN
ejpam-2562	145	7	is	be	AUX
ejpam-2562	145	8	not	not	PART
ejpam-2562	145	9	semi	semi	ADJ
ejpam-2562	145	10	-	-	ADJ
ejpam-2562	145	11	essential	essential	ADJ
ejpam-2562	145	12	.	.	PUNCT
ejpam-2562	146	1	let	let	VERB
ejpam-2562	146	2	ρ	ρ	NOUN
ejpam-2562	146	3	be	be	AUX
ejpam-2562	146	4	the	the	DET
ejpam-2562	146	5	relation	relation	NOUN
ejpam-2562	146	6	defined	define	VERB
ejpam-2562	146	7	on	on	ADP
ejpam-2562	146	8	m	m	PRON
ejpam-2562	146	9	by	by	ADP
ejpam-2562	146	10	:	:	PUNCT
ejpam-2562	146	11	∀(x	∀(x	NUM
ejpam-2562	146	12	,	,	PUNCT
ejpam-2562	146	13	y	y	PROPN
ejpam-2562	146	14	)	)	PUNCT
ejpam-2562	146	15	∈	∈	PROPN
ejpam-2562	146	16	m×m	m×m	PROPN
ejpam-2562	146	17	,	,	PUNCT
ejpam-2562	146	18	xρ	xρ	PROPN
ejpam-2562	146	19	y	y	PROPN
ejpam-2562	146	20	⇐	⇐	PROPN
ejpam-2562	146	21	⇒max(x	⇒max(x	NOUN
ejpam-2562	146	22	,	,	PUNCT
ejpam-2562	146	23	1	1	X
ejpam-2562	146	24	)	)	PUNCT
ejpam-2562	146	25	=	=	NOUN
ejpam-2562	146	26	max(y	max(y	X
ejpam-2562	146	27	,	,	PUNCT
ejpam-2562	146	28	1	1	NUM
ejpam-2562	146	29	)	)	PUNCT
ejpam-2562	146	30	.	.	PUNCT
ejpam-2562	147	1	ρ	ρ	PROPN
ejpam-2562	147	2	is	be	AUX
ejpam-2562	147	3	a	a	DET
ejpam-2562	147	4	congruence	congruence	NOUN
ejpam-2562	147	5	relation	relation	NOUN
ejpam-2562	147	6	on	on	ADP
ejpam-2562	147	7	m.	m.	NOUN
ejpam-2562	147	8	clearly	clearly	ADV
ejpam-2562	147	9	ρ	ρ	PROPN
ejpam-2562	147	10	is	be	AUX
ejpam-2562	147	11	not	not	PART
ejpam-2562	147	12	trivial	trivial	ADJ
ejpam-2562	147	13	on	on	ADP
ejpam-2562	147	14	m	m	PROPN
ejpam-2562	147	15	and	and	CCONJ
ejpam-2562	147	16	ρ	ρ	PROPN
ejpam-2562	147	17	is	be	AUX
ejpam-2562	147	18	trivial	trivial	ADJ
ejpam-2562	147	19	on	on	ADP
ejpam-2562	147	20	na	na	ADP
ejpam-2562	147	21	,	,	PUNCT
ejpam-2562	147	22	hence	hence	ADV
ejpam-2562	147	23	na	na	ADV
ejpam-2562	147	24	is	be	AUX
ejpam-2562	147	25	not	not	PART
ejpam-2562	147	26	essential	essential	ADJ
ejpam-2562	147	27	.	.	PUNCT
ejpam-2562	147	28	example	example	NOUN
ejpam-2562	148	1	2	2	NUM
ejpam-2562	148	2	.	.	X
ejpam-2562	148	3	in	in	ADP
ejpam-2562	148	4	this	this	DET
ejpam-2562	148	5	example	example	NOUN
ejpam-2562	148	6	,	,	PUNCT
ejpam-2562	148	7	we	we	PRON
ejpam-2562	148	8	give	give	VERB
ejpam-2562	148	9	an	an	DET
ejpam-2562	148	10	infinite	infinite	ADJ
ejpam-2562	148	11	semi	semi	ADJ
ejpam-2562	148	12	-	-	ADJ
ejpam-2562	148	13	weakly	weakly	ADJ
ejpam-2562	148	14	-	-	PUNCT
ejpam-2562	148	15	essential	essential	ADJ
ejpam-2562	148	16	subsemimodule	subsemimodule	NOUN
ejpam-2562	148	17	which	which	PRON
ejpam-2562	148	18	is	be	AUX
ejpam-2562	148	19	neither	neither	PRON
ejpam-2562	148	20	semi	semi	ADJ
ejpam-2562	148	21	-	-	ADJ
ejpam-2562	148	22	essential	essential	ADJ
ejpam-2562	148	23	nor	nor	CCONJ
ejpam-2562	148	24	essential	essential	ADJ
ejpam-2562	148	25	.	.	PUNCT
ejpam-2562	149	1	recall	recall	VERB
ejpam-2562	149	2	the	the	DET
ejpam-2562	149	3	semiring	semire	VERB
ejpam-2562	149	4	r	r	NOUN
ejpam-2562	149	5	of	of	ADP
ejpam-2562	149	6	the	the	DET
ejpam-2562	149	7	previous	previous	ADJ
ejpam-2562	149	8	example	example	NOUN
ejpam-2562	149	9	.	.	PUNCT
ejpam-2562	150	1	so	so	ADV
ejpam-2562	150	2	m	m	VERB
ejpam-2562	150	3	=	=	PUNCT
ejpam-2562	151	1	[	[	X
ejpam-2562	151	2	0	0	NUM
ejpam-2562	151	3	,	,	PUNCT
ejpam-2562	151	4	n	n	CCONJ
ejpam-2562	151	5	]	]	PUNCT
ejpam-2562	151	6	is	be	AUX
ejpam-2562	151	7	a	a	DET
ejpam-2562	151	8	r	r	NOUN
ejpam-2562	151	9	-	-	PUNCT
ejpam-2562	151	10	semimodule	semimodule	NOUN
ejpam-2562	151	11	.	.	PUNCT
ejpam-2562	152	1	set	set	VERB
ejpam-2562	152	2	na	na	NOUN
ejpam-2562	152	3	=	=	PUNCT
ejpam-2562	152	4	q	q	NOUN
ejpam-2562	152	5	∩	∩	ADJ
ejpam-2562	152	6	[	[	X
ejpam-2562	152	7	a	a	X
ejpam-2562	152	8	,	,	PUNCT
ejpam-2562	152	9	n	n	CCONJ
ejpam-2562	152	10	]	]	PUNCT
ejpam-2562	153	1	where	where	SCONJ
ejpam-2562	153	2	a	a	DET
ejpam-2562	153	3	∈	∈	PROPN
ejpam-2562	153	4	q∗+	q∗+	NOUN
ejpam-2562	153	5	.	.	PUNCT
ejpam-2562	153	6	by	by	ADP
ejpam-2562	153	7	a	a	DET
ejpam-2562	153	8	same	same	ADJ
ejpam-2562	153	9	way	way	NOUN
ejpam-2562	153	10	as	as	SCONJ
ejpam-2562	153	11	in	in	ADP
ejpam-2562	153	12	previous	previous	ADJ
ejpam-2562	153	13	example	example	NOUN
ejpam-2562	153	14	,	,	PUNCT
ejpam-2562	153	15	we	we	PRON
ejpam-2562	153	16	show	show	VERB
ejpam-2562	153	17	that	that	SCONJ
ejpam-2562	153	18	na	na	NOUN
ejpam-2562	153	19	is	be	AUX
ejpam-2562	153	20	semi	semi	ADJ
ejpam-2562	153	21	-	-	ADJ
ejpam-2562	153	22	weakly	weakly	ADV
ejpam-2562	153	23	-	-	PUNCT
ejpam-2562	153	24	essential	essential	ADJ
ejpam-2562	153	25	but	but	CCONJ
ejpam-2562	153	26	is	be	AUX
ejpam-2562	153	27	neither	neither	CCONJ
ejpam-2562	153	28	essential	essential	ADJ
ejpam-2562	153	29	nor	nor	CCONJ
ejpam-2562	153	30	semi	semi	ADJ
ejpam-2562	153	31	-	-	ADJ
ejpam-2562	153	32	essential	essential	ADJ
ejpam-2562	153	33	.	.	PUNCT
ejpam-2562	154	1	example	example	NOUN
ejpam-2562	154	2	3	3	NUM
ejpam-2562	154	3	(	(	PUNCT
ejpam-2562	154	4	[	[	X
ejpam-2562	154	5	5	5	NUM
ejpam-2562	154	6	]	]	PUNCT
ejpam-2562	154	7	)	)	PUNCT
ejpam-2562	154	8	.	.	PUNCT
ejpam-2562	155	1	in	in	ADP
ejpam-2562	155	2	this	this	DET
ejpam-2562	155	3	example	example	NOUN
ejpam-2562	155	4	,	,	PUNCT
ejpam-2562	155	5	we	we	PRON
ejpam-2562	155	6	propose	propose	VERB
ejpam-2562	155	7	a	a	DET
ejpam-2562	155	8	semi	semi	ADJ
ejpam-2562	155	9	-	-	ADJ
ejpam-2562	155	10	essential	essential	ADJ
ejpam-2562	155	11	subsemimodule	subsemimodule	NOUN
ejpam-2562	155	12	which	which	PRON
ejpam-2562	155	13	is	be	AUX
ejpam-2562	155	14	a	a	DET
ejpam-2562	155	15	semiweakly	semiweakly	ADJ
ejpam-2562	155	16	essential	essential	ADJ
ejpam-2562	155	17	subsemimodule	subsemimodule	NOUN
ejpam-2562	155	18	but	but	CCONJ
ejpam-2562	155	19	not	not	PART
ejpam-2562	155	20	an	an	DET
ejpam-2562	155	21	essential	essential	ADJ
ejpam-2562	155	22	subsemimodule	subsemimodule	NOUN
ejpam-2562	155	23	.	.	PUNCT
ejpam-2562	156	1	let	let	VERB
ejpam-2562	156	2	n	n	PRON
ejpam-2562	156	3	≥	≥	X
ejpam-2562	156	4	1	1	NUM
ejpam-2562	156	5	be	be	AUX
ejpam-2562	156	6	an	an	DET
ejpam-2562	156	7	integer	integer	NOUN
ejpam-2562	156	8	.	.	PUNCT
ejpam-2562	157	1	consider	consider	VERB
ejpam-2562	157	2	the	the	DET
ejpam-2562	157	3	set	set	NOUN
ejpam-2562	157	4	r	r	NOUN
ejpam-2562	157	5	=	=	PUNCT
ejpam-2562	157	6	{	{	PUNCT
ejpam-2562	157	7	r	r	NOUN
ejpam-2562	157	8	∈	∈	PROPN
ejpam-2562	157	9	q+/r	q+/r	PUNCT
ejpam-2562	157	10	≤	≤	NOUN
ejpam-2562	157	11	n	n	CCONJ
ejpam-2562	157	12	}	}	PUNCT
ejpam-2562	157	13	∪	∪	X
ejpam-2562	157	14	{	{	PUNCT
ejpam-2562	157	15	−∞	−∞	NOUN
ejpam-2562	157	16	}	}	PUNCT
ejpam-2562	157	17	in	in	ADP
ejpam-2562	157	18	which	which	PRON
ejpam-2562	157	19	q+	q+	ADV
ejpam-2562	157	20	is	be	AUX
ejpam-2562	157	21	the	the	DET
ejpam-2562	157	22	set	set	NOUN
ejpam-2562	157	23	of	of	ADP
ejpam-2562	157	24	all	all	DET
ejpam-2562	157	25	nonnegative	nonnegative	ADJ
ejpam-2562	157	26	rational	rational	ADJ
ejpam-2562	157	27	numbers	number	NOUN
ejpam-2562	157	28	,	,	PUNCT
ejpam-2562	157	29	−∞	−∞	PRON
ejpam-2562	157	30	is	be	AUX
ejpam-2562	157	31	assumed	assume	VERB
ejpam-2562	157	32	to	to	PART
ejpam-2562	157	33	satisfy	satisfy	VERB
ejpam-2562	157	34	the	the	DET
ejpam-2562	157	35	conditions	condition	NOUN
ejpam-2562	157	36	that	that	SCONJ
ejpam-2562	157	37	−∞	−∞	ADP
ejpam-2562	157	38	≤	≤	ADJ
ejpam-2562	157	39	i	i	PROPN
ejpam-2562	157	40	and	and	CCONJ
ejpam-2562	157	41	−∞	−∞	PROPN
ejpam-2562	157	42	+	+	CCONJ
ejpam-2562	157	43	i	i	PROPN
ejpam-2562	157	44	=	=	SYM
ejpam-2562	157	45	−∞	−∞	NOUN
ejpam-2562	157	46	,	,	PUNCT
ejpam-2562	157	47	∀i	∀i	X
ejpam-2562	157	48	∈	∈	PROPN
ejpam-2562	157	49	r.	r.	NOUN
ejpam-2562	157	50	define	define	VERB
ejpam-2562	157	51	on	on	ADP
ejpam-2562	157	52	r	r	NOUN
ejpam-2562	157	53	the	the	DET
ejpam-2562	157	54	operations	operation	NOUN
ejpam-2562	157	55	⊕	⊕	PROPN
ejpam-2562	157	56	and	and	CCONJ
ejpam-2562	157	57	⊗	⊗	PROPN
ejpam-2562	157	58	as	as	ADP
ejpam-2562	157	59	following	follow	VERB
ejpam-2562	157	60	:	:	PUNCT
ejpam-2562	157	61	∀i	∀i	NOUN
ejpam-2562	157	62	,	,	PUNCT
ejpam-2562	157	63	j	j	PROPN
ejpam-2562	157	64	∈	∈	PROPN
ejpam-2562	157	65	r	r	PROPN
ejpam-2562	157	66	;	;	PUNCT
ejpam-2562	157	67	i	i	PROPN
ejpam-2562	157	68	⊕	⊕	PROPN
ejpam-2562	157	69	j	j	PROPN
ejpam-2562	158	1	=	=	SYM
ejpam-2562	158	2	max(i	max(i	PROPN
ejpam-2562	158	3	;	;	PUNCT
ejpam-2562	158	4	j	j	NOUN
ejpam-2562	158	5	)	)	PUNCT
ejpam-2562	158	6	and	and	CCONJ
ejpam-2562	158	7	i	i	PRON
ejpam-2562	159	1	⊗	⊗	PROPN
ejpam-2562	159	2	j	j	PROPN
ejpam-2562	160	1	=	=	SYM
ejpam-2562	160	2	min(i	min(i	PROPN
ejpam-2562	160	3	+	+	CCONJ
ejpam-2562	160	4	j	j	PROPN
ejpam-2562	160	5	;	;	PUNCT
ejpam-2562	160	6	n	n	CCONJ
ejpam-2562	160	7	)	)	PUNCT
ejpam-2562	160	8	.	.	PUNCT
ejpam-2562	161	1	we	we	PRON
ejpam-2562	161	2	easily	easily	ADV
ejpam-2562	161	3	verify	verify	VERB
ejpam-2562	161	4	that	that	SCONJ
ejpam-2562	161	5	(	(	PUNCT
ejpam-2562	161	6	r;⊕;⊗	r;⊕;⊗	PROPN
ejpam-2562	161	7	)	)	PUNCT
ejpam-2562	161	8	is	be	AUX
ejpam-2562	161	9	a	a	DET
ejpam-2562	161	10	commutative	commutative	ADJ
ejpam-2562	161	11	semiring	semiring	NOUN
ejpam-2562	161	12	having	have	VERB
ejpam-2562	161	13	−∞	−∞	X
ejpam-2562	161	14	as	as	ADP
ejpam-2562	161	15	additive	additive	ADJ
ejpam-2562	161	16	identity	identity	NOUN
ejpam-2562	161	17	.	.	PUNCT
ejpam-2562	162	1	r	r	NOUN
ejpam-2562	162	2	is	be	AUX
ejpam-2562	162	3	also	also	ADV
ejpam-2562	162	4	a	a	DET
ejpam-2562	162	5	left	left	ADJ
ejpam-2562	162	6	r	r	NOUN
ejpam-2562	162	7	-	-	PUNCT
ejpam-2562	162	8	semimodule	semimodule	NOUN
ejpam-2562	162	9	.	.	PUNCT
ejpam-2562	163	1	put	put	VERB
ejpam-2562	163	2	r∗	r∗	PROPN
ejpam-2562	163	3	=	=	PUNCT
ejpam-2562	163	4	r\	r\	X
ejpam-2562	163	5	{	{	PUNCT
ejpam-2562	163	6	0	0	NUM
ejpam-2562	163	7	}	}	PUNCT
ejpam-2562	163	8	,	,	PUNCT
ejpam-2562	163	9	then	then	ADV
ejpam-2562	163	10	r∗	r∗	PROPN
ejpam-2562	163	11	is	be	AUX
ejpam-2562	163	12	an	an	DET
ejpam-2562	163	13	ideal	ideal	NOUN
ejpam-2562	163	14	of	of	ADP
ejpam-2562	163	15	r.	r.	PROPN
ejpam-2562	163	16	r∗	r∗	PROPN
ejpam-2562	163	17	is	be	AUX
ejpam-2562	163	18	a	a	DET
ejpam-2562	163	19	semi	semi	ADJ
ejpam-2562	163	20	-	-	ADJ
ejpam-2562	163	21	essential	essential	ADJ
ejpam-2562	163	22	subsemimodule	subsemimodule	NOUN
ejpam-2562	163	23	of	of	ADP
ejpam-2562	163	24	r	r	NOUN
ejpam-2562	163	25	,	,	PUNCT
ejpam-2562	163	26	hence	hence	ADV
ejpam-2562	163	27	r∗	r∗	NOUN
ejpam-2562	163	28	is	be	AUX
ejpam-2562	163	29	a	a	DET
ejpam-2562	163	30	semi	semi	ADJ
ejpam-2562	163	31	-	-	ADJ
ejpam-2562	163	32	weakly	weakly	ADJ
ejpam-2562	163	33	essential	essential	ADJ
ejpam-2562	163	34	subsemimodule	subsemimodule	NOUN
ejpam-2562	163	35	,	,	PUNCT
ejpam-2562	163	36	but	but	CCONJ
ejpam-2562	163	37	r∗	r∗	PROPN
ejpam-2562	163	38	is	be	AUX
ejpam-2562	163	39	not	not	PART
ejpam-2562	163	40	essential	essential	ADJ
ejpam-2562	163	41	.	.	PUNCT
ejpam-2562	164	1	e.	e.	PROPN
ejpam-2562	164	2	diop	diop	PROPN
ejpam-2562	164	3	,	,	PUNCT
ejpam-2562	164	4	d.	d.	PROPN
ejpam-2562	164	5	sow	sow	PROPN
ejpam-2562	164	6	/	/	SYM
ejpam-2562	164	7	eur	eur	PROPN
ejpam-2562	164	8	.	.	PUNCT
ejpam-2562	165	1	j.	j.	PROPN
ejpam-2562	165	2	pure	pure	PROPN
ejpam-2562	165	3	appl	appl	PROPN
ejpam-2562	165	4	.	.	PROPN
ejpam-2562	165	5	math	math	PROPN
ejpam-2562	165	6	,	,	PUNCT
ejpam-2562	165	7	9	9	NUM
ejpam-2562	165	8	(	(	PUNCT
ejpam-2562	165	9	2016	2016	NUM
ejpam-2562	165	10	)	)	PUNCT
ejpam-2562	165	11	,	,	PUNCT
ejpam-2562	165	12	250	250	NUM
ejpam-2562	165	13	-	-	SYM
ejpam-2562	165	14	265	265	NUM
ejpam-2562	165	15	256	256	NUM
ejpam-2562	165	16	example	example	NOUN
ejpam-2562	165	17	4	4	NUM
ejpam-2562	165	18	(	(	PUNCT
ejpam-2562	165	19	[	[	X
ejpam-2562	165	20	5	5	NUM
ejpam-2562	165	21	]	]	PUNCT
ejpam-2562	165	22	)	)	PUNCT
ejpam-2562	165	23	.	.	PUNCT
ejpam-2562	166	1	in	in	ADP
ejpam-2562	166	2	this	this	DET
ejpam-2562	166	3	example	example	NOUN
ejpam-2562	166	4	,	,	PUNCT
ejpam-2562	166	5	we	we	PRON
ejpam-2562	166	6	propose	propose	VERB
ejpam-2562	166	7	an	an	DET
ejpam-2562	166	8	essential	essential	ADJ
ejpam-2562	166	9	subsemimodule	subsemimodule	NOUN
ejpam-2562	166	10	which	which	PRON
ejpam-2562	166	11	is	be	AUX
ejpam-2562	166	12	a	a	DET
ejpam-2562	166	13	semi	semi	ADJ
ejpam-2562	166	14	-	-	ADJ
ejpam-2562	166	15	weakly	weakly	ADJ
ejpam-2562	166	16	essential	essential	ADJ
ejpam-2562	166	17	subsemimodule	subsemimodule	NOUN
ejpam-2562	166	18	but	but	CCONJ
ejpam-2562	166	19	not	not	PART
ejpam-2562	166	20	an	an	DET
ejpam-2562	166	21	semi	semi	ADJ
ejpam-2562	166	22	-	-	ADJ
ejpam-2562	166	23	essential	essential	ADJ
ejpam-2562	166	24	subsemimodule	subsemimodule	NOUN
ejpam-2562	166	25	.	.	PUNCT
ejpam-2562	167	1	set	set	VERB
ejpam-2562	167	2	r=	r=	PROPN
ejpam-2562	167	3	{	{	PUNCT
ejpam-2562	167	4	0,1	0,1	NUM
ejpam-2562	167	5	,	,	PUNCT
ejpam-2562	167	6	a	a	PRON
ejpam-2562	167	7	}	}	PUNCT
ejpam-2562	167	8	and	and	CCONJ
ejpam-2562	167	9	define	define	VERB
ejpam-2562	167	10	on	on	ADP
ejpam-2562	167	11	r	r	NOUN
ejpam-2562	167	12	the	the	DET
ejpam-2562	167	13	two	two	NUM
ejpam-2562	167	14	commutative	commutative	ADJ
ejpam-2562	167	15	operations	operation	NOUN
ejpam-2562	167	16	(	(	PUNCT
ejpam-2562	167	17	+	+	ADJ
ejpam-2562	167	18	,	,	PUNCT
ejpam-2562	167	19	×	×	NOUN
ejpam-2562	167	20	)	)	PUNCT
ejpam-2562	167	21	as	as	ADP
ejpam-2562	167	22	following	follow	VERB
ejpam-2562	167	23	:	:	PUNCT
ejpam-2562	167	24	(	(	PUNCT
ejpam-2562	167	25	i	i	NOUN
ejpam-2562	167	26	)	)	PUNCT
ejpam-2562	167	27	0r	0r	PART
ejpam-2562	168	1	=	=	SYM
ejpam-2562	168	2	0	0	NUM
ejpam-2562	168	3	;	;	PUNCT
ejpam-2562	168	4	1r	1r	NUM
ejpam-2562	168	5	=	=	SYM
ejpam-2562	168	6	1	1	NUM
ejpam-2562	168	7	(	(	PUNCT
ejpam-2562	168	8	ii	ii	NOUN
ejpam-2562	168	9	)	)	PUNCT
ejpam-2562	168	10	1	1	NUM
ejpam-2562	168	11	+	+	NUM
ejpam-2562	168	12	1=	1=	NUM
ejpam-2562	168	13	1	1	NUM
ejpam-2562	168	14	+	+	CCONJ
ejpam-2562	168	15	a	a	DET
ejpam-2562	168	16	=	=	ADJ
ejpam-2562	168	17	1	1	NUM
ejpam-2562	168	18	;	;	PUNCT
ejpam-2562	168	19	a+	a+	PUNCT
ejpam-2562	168	20	0=	0=	NUM
ejpam-2562	168	21	a+	a+	PUNCT
ejpam-2562	168	22	a	a	DET
ejpam-2562	168	23	=	=	X
ejpam-2562	168	24	a	a	DET
ejpam-2562	168	25	(	(	PUNCT
ejpam-2562	168	26	iii	iii	NOUN
ejpam-2562	168	27	)	)	PUNCT
ejpam-2562	168	28	0×	0×	NOUN
ejpam-2562	168	29	0=	0=	NUM
ejpam-2562	169	1	0×	0×	NUM
ejpam-2562	169	2	1=	1=	NUM
ejpam-2562	169	3	0×	0×	X
ejpam-2562	169	4	a	a	DET
ejpam-2562	169	5	=	=	NOUN
ejpam-2562	169	6	0	0	NUM
ejpam-2562	169	7	;	;	PUNCT
ejpam-2562	169	8	1×	1×	NUM
ejpam-2562	169	9	1=	1=	NUM
ejpam-2562	169	10	1	1	NUM
ejpam-2562	169	11	;	;	PUNCT
ejpam-2562	169	12	1×	1×	NUM
ejpam-2562	169	13	a	a	DET
ejpam-2562	169	14	=	=	SYM
ejpam-2562	169	15	a×	a×	PROPN
ejpam-2562	169	16	a	a	DET
ejpam-2562	169	17	=	=	NOUN
ejpam-2562	169	18	a.	a.	NOUN
ejpam-2562	169	19	then	then	ADV
ejpam-2562	169	20	(	(	PUNCT
ejpam-2562	169	21	r,+,×	r,+,×	PROPN
ejpam-2562	169	22	,	,	PUNCT
ejpam-2562	169	23	0	0	NUM
ejpam-2562	169	24	,	,	PUNCT
ejpam-2562	169	25	1	1	NUM
ejpam-2562	169	26	)	)	PUNCT
ejpam-2562	169	27	is	be	AUX
ejpam-2562	169	28	a	a	DET
ejpam-2562	169	29	commutative	commutative	ADJ
ejpam-2562	169	30	semiring	semiring	NOUN
ejpam-2562	169	31	.	.	PUNCT
ejpam-2562	170	1	let	let	VERB
ejpam-2562	170	2	m	m	VERB
ejpam-2562	170	3	=	=	PUNCT
ejpam-2562	170	4	{	{	PUNCT
ejpam-2562	170	5	0,1	0,1	NUM
ejpam-2562	170	6	,	,	PUNCT
ejpam-2562	170	7	a	a	DET
ejpam-2562	170	8	,	,	PUNCT
ejpam-2562	170	9	b	b	NOUN
ejpam-2562	170	10	}	}	PUNCT
ejpam-2562	170	11	with	with	ADP
ejpam-2562	170	12	the	the	DET
ejpam-2562	170	13	same	same	ADJ
ejpam-2562	170	14	operations	operation	NOUN
ejpam-2562	170	15	defined	define	VERB
ejpam-2562	170	16	in	in	ADP
ejpam-2562	170	17	r	r	NOUN
ejpam-2562	170	18	and	and	CCONJ
ejpam-2562	170	19	1	1	NUM
ejpam-2562	170	20	m	m	NOUN
ejpam-2562	170	21	=	=	NOUN
ejpam-2562	170	22	1r	1r	NUM
ejpam-2562	170	23	=	=	SYM
ejpam-2562	170	24	1	1	NUM
ejpam-2562	170	25	,	,	PUNCT
ejpam-2562	170	26	0	0	NUM
ejpam-2562	170	27	m	m	NOUN
ejpam-2562	170	28	=	=	SYM
ejpam-2562	170	29	0r	0r	X
ejpam-2562	171	1	=	=	SYM
ejpam-2562	171	2	0	0	NUM
ejpam-2562	171	3	,	,	PUNCT
ejpam-2562	171	4	b+	b+	X
ejpam-2562	171	5	0=	0=	NUM
ejpam-2562	171	6	b+	b+	X
ejpam-2562	172	1	b	b	X
ejpam-2562	172	2	=	=	SYM
ejpam-2562	172	3	b	b	PROPN
ejpam-2562	172	4	,	,	PUNCT
ejpam-2562	172	5	b+	b+	X
ejpam-2562	172	6	1=	1=	X
ejpam-2562	172	7	b+	b+	X
ejpam-2562	172	8	a	a	PRON
ejpam-2562	172	9	=	=	SYM
ejpam-2562	172	10	a	a	PROPN
ejpam-2562	172	11	,	,	PUNCT
ejpam-2562	172	12	0×	0×	PROPN
ejpam-2562	172	13	b	b	X
ejpam-2562	172	14	=	=	PUNCT
ejpam-2562	172	15	b×	b×	VERB
ejpam-2562	172	16	a	a	PRON
ejpam-2562	172	17	=	=	NOUN
ejpam-2562	172	18	0	0	NUM
ejpam-2562	172	19	,	,	PUNCT
ejpam-2562	172	20	b×	b×	VERB
ejpam-2562	172	21	1=	1=	X
ejpam-2562	172	22	b×	b×	NOUN
ejpam-2562	172	23	b	b	X
ejpam-2562	172	24	=	=	SYM
ejpam-2562	172	25	b.	b.	PROPN
ejpam-2562	173	1	it	it	PRON
ejpam-2562	173	2	’s	’	VERB
ejpam-2562	173	3	easy	easy	ADJ
ejpam-2562	173	4	to	to	PART
ejpam-2562	173	5	see	see	VERB
ejpam-2562	173	6	that	that	PRON
ejpam-2562	173	7	(	(	PUNCT
ejpam-2562	173	8	m	m	NOUN
ejpam-2562	173	9	,	,	PUNCT
ejpam-2562	173	10	+	+	ADJ
ejpam-2562	173	11	,	,	PUNCT
ejpam-2562	173	12	×	×	NOUN
ejpam-2562	173	13	,	,	PUNCT
ejpam-2562	173	14	0	0	NUM
ejpam-2562	173	15	,	,	PUNCT
ejpam-2562	173	16	1	1	NUM
ejpam-2562	173	17	)	)	PUNCT
ejpam-2562	173	18	is	be	AUX
ejpam-2562	173	19	a	a	DET
ejpam-2562	173	20	commutative	commutative	ADJ
ejpam-2562	173	21	r	r	NOUN
ejpam-2562	173	22	-	-	PUNCT
ejpam-2562	173	23	semimodule	semimodule	NOUN
ejpam-2562	173	24	.	.	PUNCT
ejpam-2562	174	1	now	now	ADV
ejpam-2562	174	2	,	,	PUNCT
ejpam-2562	174	3	put	put	VERB
ejpam-2562	174	4	n	n	PRON
ejpam-2562	174	5	=	=	PUNCT
ejpam-2562	174	6	r=	r=	X
ejpam-2562	174	7	{	{	PUNCT
ejpam-2562	174	8	0	0	NUM
ejpam-2562	174	9	;	;	PUNCT
ejpam-2562	174	10	1	1	NUM
ejpam-2562	174	11	;	;	PUNCT
ejpam-2562	174	12	a	a	PRON
ejpam-2562	174	13	}	}	PUNCT
ejpam-2562	174	14	,	,	PUNCT
ejpam-2562	174	15	then	then	ADV
ejpam-2562	174	16	n	n	PRON
ejpam-2562	174	17	is	be	AUX
ejpam-2562	174	18	a	a	DET
ejpam-2562	174	19	semi	semi	ADJ
ejpam-2562	174	20	-	-	ADJ
ejpam-2562	174	21	weakly	weakly	ADJ
ejpam-2562	174	22	essential	essential	ADJ
ejpam-2562	174	23	subsemimodule	subsemimodule	NOUN
ejpam-2562	174	24	of	of	ADP
ejpam-2562	174	25	m	m	PROPN
ejpam-2562	174	26	,	,	PUNCT
ejpam-2562	174	27	but	but	CCONJ
ejpam-2562	174	28	n	n	PRON
ejpam-2562	174	29	is	be	AUX
ejpam-2562	174	30	not	not	PART
ejpam-2562	174	31	semi	semi	ADJ
ejpam-2562	174	32	-	-	ADJ
ejpam-2562	174	33	essential	essential	ADJ
ejpam-2562	174	34	.	.	PUNCT
ejpam-2562	175	1	in	in	ADP
ejpam-2562	175	2	the	the	DET
ejpam-2562	175	3	class	class	NOUN
ejpam-2562	175	4	of	of	ADP
ejpam-2562	175	5	semi	semi	ADJ
ejpam-2562	175	6	-	-	ADJ
ejpam-2562	175	7	weakly	weakly	ADJ
ejpam-2562	175	8	-	-	PUNCT
ejpam-2562	175	9	essential	essential	ADJ
ejpam-2562	175	10	subsemimodules	subsemimodule	NOUN
ejpam-2562	175	11	,	,	PUNCT
ejpam-2562	175	12	we	we	PRON
ejpam-2562	175	13	have	have	VERB
ejpam-2562	175	14	the	the	DET
ejpam-2562	175	15	following	follow	VERB
ejpam-2562	175	16	result	result	NOUN
ejpam-2562	175	17	:	:	PUNCT
ejpam-2562	175	18	proposition	proposition	NOUN
ejpam-2562	175	19	3	3	X
ejpam-2562	175	20	.	.	PUNCT
ejpam-2562	176	1	let	let	VERB
ejpam-2562	176	2	m	m	PRON
ejpam-2562	176	3	be	be	AUX
ejpam-2562	176	4	a	a	DET
ejpam-2562	176	5	left	left	ADJ
ejpam-2562	176	6	r	r	NOUN
ejpam-2562	176	7	-	-	PUNCT
ejpam-2562	176	8	semimodule	semimodule	NOUN
ejpam-2562	176	9	,	,	PUNCT
ejpam-2562	176	10	k	k	PROPN
ejpam-2562	176	11	and	and	CCONJ
ejpam-2562	176	12	n	n	PROPN
ejpam-2562	176	13	be	be	VERB
ejpam-2562	176	14	subsemimodules	subsemimodule	NOUN
ejpam-2562	176	15	of	of	ADP
ejpam-2562	176	16	m	m	NOUN
ejpam-2562	176	17	such	such	ADJ
ejpam-2562	176	18	that	that	SCONJ
ejpam-2562	176	19	:	:	PUNCT
ejpam-2562	176	20	k	k	PROPN
ejpam-2562	176	21	≤	≤	PUNCT
ejpam-2562	176	22	n	n	CCONJ
ejpam-2562	176	23	≤	≤	NOUN
ejpam-2562	176	24	m	m	VERB
ejpam-2562	176	25	.	.	PUNCT
ejpam-2562	177	1	then	then	ADV
ejpam-2562	177	2	we	we	PRON
ejpam-2562	177	3	have	have	VERB
ejpam-2562	177	4	:	:	PUNCT
ejpam-2562	177	5	(	(	PUNCT
ejpam-2562	177	6	i	i	NOUN
ejpam-2562	177	7	)	)	PUNCT
ejpam-2562	178	1	k	k	NOUN
ejpam-2562	179	1	ãs	ãs	PRON
ejpam-2562	179	2	m	m	VERB
ejpam-2562	179	3	⇐	⇐	ADJ
ejpam-2562	179	4	⇒	⇒	NOUN
ejpam-2562	179	5	(	(	PUNCT
ejpam-2562	179	6	k	k	X
ejpam-2562	179	7	ãs	ãs	NOUN
ejpam-2562	179	8	n	n	PROPN
ejpam-2562	179	9	and	and	CCONJ
ejpam-2562	179	10	n	n	CCONJ
ejpam-2562	179	11	ãs	ãs	PRON
ejpam-2562	179	12	m	m	NOUN
ejpam-2562	179	13	)	)	PUNCT
ejpam-2562	179	14	.	.	PUNCT
ejpam-2562	180	1	(	(	PUNCT
ejpam-2562	180	2	ii	ii	X
ejpam-2562	180	3	)	)	PUNCT
ejpam-2562	180	4	k	k	NOUN
ejpam-2562	181	1	ã	ã	X
ejpam-2562	181	2	m	m	VERB
ejpam-2562	181	3	⇐	⇐	ADJ
ejpam-2562	181	4	⇒	⇒	NOUN
ejpam-2562	181	5	(	(	PUNCT
ejpam-2562	181	6	k	k	NOUN
ejpam-2562	181	7	ã	ã	X
ejpam-2562	181	8	n	n	PROPN
ejpam-2562	181	9	and	and	CCONJ
ejpam-2562	181	10	n	n	CCONJ
ejpam-2562	181	11	ã	ã	NOUN
ejpam-2562	181	12	m	m	NOUN
ejpam-2562	181	13	)	)	PUNCT
ejpam-2562	181	14	.	.	PUNCT
ejpam-2562	182	1	(	(	PUNCT
ejpam-2562	182	2	iii	iii	X
ejpam-2562	182	3	)	)	PUNCT
ejpam-2562	182	4	k	k	NOUN
ejpam-2562	182	5	ãswe	ãswe	NOUN
ejpam-2562	182	6	m	m	VERB
ejpam-2562	182	7	=	=	VERB
ejpam-2562	182	8	⇒	⇒	NOUN
ejpam-2562	182	9	(	(	PUNCT
ejpam-2562	182	10	k	k	X
ejpam-2562	182	11	ãswe	ãswe	PROPN
ejpam-2562	182	12	n	n	PROPN
ejpam-2562	182	13	and	and	CCONJ
ejpam-2562	182	14	n	n	ADV
ejpam-2562	182	15	ãswe	ãswe	NOUN
ejpam-2562	182	16	m	m	PROPN
ejpam-2562	182	17	)	)	PUNCT
ejpam-2562	182	18	.	.	PUNCT
ejpam-2562	183	1	proof	proof	NOUN
ejpam-2562	183	2	.	.	PUNCT
ejpam-2562	184	1	(	(	PUNCT
ejpam-2562	184	2	i	i	NOUN
ejpam-2562	184	3	)	)	PUNCT
ejpam-2562	184	4	similar	similar	ADJ
ejpam-2562	184	5	methods	method	NOUN
ejpam-2562	184	6	to	to	ADP
ejpam-2562	184	7	[	[	X
ejpam-2562	184	8	5	5	NUM
ejpam-2562	184	9	]	]	PUNCT
ejpam-2562	184	10	.	.	PUNCT
ejpam-2562	185	1	(	(	PUNCT
ejpam-2562	185	2	ii	ii	NOUN
ejpam-2562	185	3	)	)	PUNCT
ejpam-2562	185	4	=	=	NOUN
ejpam-2562	185	5	⇒	⇒	NOUN
ejpam-2562	185	6	suppose	suppose	VERB
ejpam-2562	185	7	that	that	SCONJ
ejpam-2562	185	8	k	k	PROPN
ejpam-2562	186	1	ã	ã	X
ejpam-2562	186	2	m	m	NOUN
ejpam-2562	186	3	.	.	PUNCT
ejpam-2562	187	1	•	•	INTJ
ejpam-2562	187	2	let	let	VERB
ejpam-2562	187	3	n	n	PRON
ejpam-2562	187	4	6=	6=	ADP
ejpam-2562	187	5	n′	n′	DET
ejpam-2562	187	6	∈	∈	PROPN
ejpam-2562	187	7	n	n	ADV
ejpam-2562	187	8	.	.	PUNCT
ejpam-2562	188	1	since	since	SCONJ
ejpam-2562	188	2	n	n	ADV
ejpam-2562	188	3	≤	≤	NOUN
ejpam-2562	188	4	m	m	PROPN
ejpam-2562	188	5	and	and	CCONJ
ejpam-2562	188	6	k	k	X
ejpam-2562	188	7	ã	ã	X
ejpam-2562	188	8	m	m	NOUN
ejpam-2562	188	9	,	,	PUNCT
ejpam-2562	188	10	there	there	PRON
ejpam-2562	188	11	exist	exist	VERB
ejpam-2562	188	12	k	k	PROPN
ejpam-2562	188	13	6=	6=	ADP
ejpam-2562	188	14	k′	k′	PROPN
ejpam-2562	188	15	∈	∈	PROPN
ejpam-2562	188	16	k	k	ADP
ejpam-2562	188	17	such	such	ADJ
ejpam-2562	188	18	that	that	DET
ejpam-2562	188	19	kρ(n	kρ(n	NOUN
ejpam-2562	188	20	,	,	PUNCT
ejpam-2562	188	21	n′)k	n′)k	X
ejpam-2562	188	22	′.	′.	NOUN
ejpam-2562	188	23	therefore	therefore	ADV
ejpam-2562	188	24	k	k	PROPN
ejpam-2562	188	25	ã	ã	X
ejpam-2562	189	1	n	n	PROPN
ejpam-2562	189	2	.	.	PUNCT
ejpam-2562	190	1	•	•	INTJ
ejpam-2562	190	2	let	let	VERB
ejpam-2562	190	3	m	m	PRON
ejpam-2562	190	4	6=	6=	NUM
ejpam-2562	190	5	m′	m′	NUM
ejpam-2562	190	6	∈	∈	PROPN
ejpam-2562	190	7	m	m	NOUN
ejpam-2562	190	8	.	.	PUNCT
ejpam-2562	191	1	since	since	SCONJ
ejpam-2562	191	2	k	k	PROPN
ejpam-2562	191	3	ã	ã	X
ejpam-2562	191	4	m	m	NOUN
ejpam-2562	191	5	,	,	PUNCT
ejpam-2562	191	6	there	there	PRON
ejpam-2562	191	7	exist	exist	VERB
ejpam-2562	191	8	k	k	PROPN
ejpam-2562	191	9	6=	6=	ADP
ejpam-2562	191	10	k′	k′	PROPN
ejpam-2562	191	11	∈	∈	PROPN
ejpam-2562	192	1	k	k	NOUN
ejpam-2562	192	2	such	such	ADJ
ejpam-2562	192	3	that	that	DET
ejpam-2562	192	4	kρ(m	kρ(m	NOUN
ejpam-2562	192	5	,	,	PUNCT
ejpam-2562	192	6	m′)k	m′)k	PUNCT
ejpam-2562	192	7	′.	′.	NOUN
ejpam-2562	192	8	but	but	CCONJ
ejpam-2562	192	9	k	k	PROPN
ejpam-2562	192	10	≤	≤	PROPN
ejpam-2562	192	11	n	n	CCONJ
ejpam-2562	192	12	,	,	PUNCT
ejpam-2562	192	13	so	so	ADV
ejpam-2562	192	14	k	k	PROPN
ejpam-2562	192	15	,	,	PUNCT
ejpam-2562	192	16	k′	k′	PROPN
ejpam-2562	192	17	∈	∈	PROPN
ejpam-2562	192	18	n	n	NOUN
ejpam-2562	192	19	,	,	PUNCT
ejpam-2562	192	20	therefore	therefore	ADV
ejpam-2562	192	21	n	n	CCONJ
ejpam-2562	192	22	ã	ã	NOUN
ejpam-2562	192	23	m	m	NOUN
ejpam-2562	192	24	.	.	PUNCT
ejpam-2562	193	1	⇐	⇐	PROPN
ejpam-2562	193	2	=	=	PRON
ejpam-2562	193	3	suppose	suppose	VERB
ejpam-2562	193	4	that	that	SCONJ
ejpam-2562	193	5	k	k	PROPN
ejpam-2562	193	6	ã	ã	X
ejpam-2562	193	7	n	n	PROPN
ejpam-2562	193	8	and	and	CCONJ
ejpam-2562	193	9	n	n	CCONJ
ejpam-2562	193	10	ã	ã	NOUN
ejpam-2562	193	11	m	m	NOUN
ejpam-2562	193	12	.	.	PUNCT
ejpam-2562	194	1	let	let	VERB
ejpam-2562	194	2	ρ	ρ	NOUN
ejpam-2562	194	3	be	be	AUX
ejpam-2562	194	4	an	an	DET
ejpam-2562	194	5	r	r	NOUN
ejpam-2562	194	6	-	-	PUNCT
ejpam-2562	194	7	congruence	congruence	NOUN
ejpam-2562	194	8	of	of	ADP
ejpam-2562	194	9	m	m	PRON
ejpam-2562	194	10	such	such	ADJ
ejpam-2562	194	11	that	that	SCONJ
ejpam-2562	194	12	ρ	ρ	PROPN
ejpam-2562	194	13	is	be	AUX
ejpam-2562	194	14	not	not	PART
ejpam-2562	194	15	trivial	trivial	ADJ
ejpam-2562	194	16	on	on	ADP
ejpam-2562	194	17	m	m	PROPN
ejpam-2562	194	18	.	.	PUNCT
ejpam-2562	195	1	since	since	SCONJ
ejpam-2562	195	2	n	n	PROPN
ejpam-2562	195	3	ã	ã	NOUN
ejpam-2562	195	4	m	m	NOUN
ejpam-2562	195	5	,	,	PUNCT
ejpam-2562	195	6	we	we	PRON
ejpam-2562	195	7	have	have	VERB
ejpam-2562	195	8	the	the	DET
ejpam-2562	195	9	restriction	restriction	NOUN
ejpam-2562	195	10	of	of	ADP
ejpam-2562	195	11	ρ	ρ	PROPN
ejpam-2562	195	12	to	to	ADP
ejpam-2562	195	13	n	n	PROPN
ejpam-2562	195	14	is	be	AUX
ejpam-2562	195	15	not	not	PART
ejpam-2562	195	16	trivial	trivial	ADJ
ejpam-2562	195	17	and	and	CCONJ
ejpam-2562	195	18	since	since	SCONJ
ejpam-2562	195	19	k	k	PROPN
ejpam-2562	195	20	ã	ã	X
ejpam-2562	195	21	n	n	X
ejpam-2562	195	22	,	,	PUNCT
ejpam-2562	195	23	we	we	PRON
ejpam-2562	195	24	have	have	VERB
ejpam-2562	195	25	the	the	DET
ejpam-2562	195	26	restriction	restriction	NOUN
ejpam-2562	195	27	of	of	ADP
ejpam-2562	195	28	ρ	ρ	PROPN
ejpam-2562	195	29	to	to	ADP
ejpam-2562	195	30	k	k	PROPN
ejpam-2562	195	31	is	be	AUX
ejpam-2562	195	32	not	not	PART
ejpam-2562	195	33	trivial	trivial	ADJ
ejpam-2562	195	34	,	,	PUNCT
ejpam-2562	195	35	hence	hence	ADV
ejpam-2562	195	36	k	k	PROPN
ejpam-2562	196	1	ã	ã	X
ejpam-2562	196	2	m	m	NOUN
ejpam-2562	196	3	.	.	PUNCT
ejpam-2562	197	1	(	(	PUNCT
ejpam-2562	197	2	iii	iii	NOUN
ejpam-2562	197	3	)	)	PUNCT
ejpam-2562	197	4	trivial	trivial	ADJ
ejpam-2562	197	5	.	.	PUNCT
ejpam-2562	198	1	e.	e.	PROPN
ejpam-2562	198	2	diop	diop	PROPN
ejpam-2562	198	3	,	,	PUNCT
ejpam-2562	198	4	d.	d.	PROPN
ejpam-2562	198	5	sow	sow	PROPN
ejpam-2562	198	6	/	/	SYM
ejpam-2562	198	7	eur	eur	PROPN
ejpam-2562	198	8	.	.	PUNCT
ejpam-2562	199	1	j.	j.	PROPN
ejpam-2562	199	2	pure	pure	PROPN
ejpam-2562	199	3	appl	appl	PROPN
ejpam-2562	199	4	.	.	PROPN
ejpam-2562	199	5	math	math	PROPN
ejpam-2562	199	6	,	,	PUNCT
ejpam-2562	199	7	9	9	NUM
ejpam-2562	199	8	(	(	PUNCT
ejpam-2562	199	9	2016	2016	NUM
ejpam-2562	199	10	)	)	PUNCT
ejpam-2562	199	11	,	,	PUNCT
ejpam-2562	199	12	250	250	NUM
ejpam-2562	199	13	-	-	SYM
ejpam-2562	199	14	265	265	NUM
ejpam-2562	199	15	257	257	NUM
ejpam-2562	199	16	remark	remark	NOUN
ejpam-2562	199	17	2	2	NUM
ejpam-2562	199	18	.	.	PUNCT
ejpam-2562	200	1	we	we	PRON
ejpam-2562	200	2	have	have	VERB
ejpam-2562	200	3	the	the	DET
ejpam-2562	200	4	following	follow	VERB
ejpam-2562	200	5	remark	remark	NOUN
ejpam-2562	200	6	.	.	PUNCT
ejpam-2562	201	1	let	let	VERB
ejpam-2562	201	2	n	n	PRON
ejpam-2562	201	3	≤	≤	X
ejpam-2562	202	1	k	k	PROPN
ejpam-2562	202	2	≤	≤	NUM
ejpam-2562	202	3	m	m	VERB
ejpam-2562	203	1	and	and	CCONJ
ejpam-2562	203	2	let	let	VERB
ejpam-2562	203	3	ρ	ρ	NOUN
ejpam-2562	203	4	be	be	AUX
ejpam-2562	203	5	and	and	CCONJ
ejpam-2562	203	6	congruence	congruence	NOUN
ejpam-2562	203	7	relation	relation	NOUN
ejpam-2562	203	8	which	which	PRON
ejpam-2562	203	9	is	be	AUX
ejpam-2562	203	10	trivial	trivial	ADJ
ejpam-2562	203	11	on	on	ADP
ejpam-2562	203	12	k.	k.	PROPN
ejpam-2562	204	1	so	so	PROPN
ejpam-2562	204	2	ρ	ρ	PROPN
ejpam-2562	204	3	is	be	AUX
ejpam-2562	204	4	trivial	trivial	ADJ
ejpam-2562	204	5	on	on	ADP
ejpam-2562	204	6	n	n	NOUN
ejpam-2562	204	7	but	but	CCONJ
ejpam-2562	204	8	is	be	AUX
ejpam-2562	204	9	not	not	PART
ejpam-2562	204	10	trivial	trivial	ADJ
ejpam-2562	204	11	in	in	ADP
ejpam-2562	204	12	general	general	ADJ
ejpam-2562	204	13	on	on	ADP
ejpam-2562	204	14	m.	m.	NOUN
ejpam-2562	204	15	in	in	ADP
ejpam-2562	204	16	the	the	DET
ejpam-2562	204	17	following	follow	VERB
ejpam-2562	204	18	lemma	lemma	PROPN
ejpam-2562	204	19	,	,	PUNCT
ejpam-2562	204	20	we	we	PRON
ejpam-2562	204	21	prove	prove	VERB
ejpam-2562	204	22	a	a	DET
ejpam-2562	204	23	particular	particular	ADJ
ejpam-2562	204	24	case	case	NOUN
ejpam-2562	204	25	where	where	SCONJ
ejpam-2562	204	26	we	we	PRON
ejpam-2562	204	27	can	can	AUX
ejpam-2562	204	28	enlarge	enlarge	VERB
ejpam-2562	204	29	a	a	DET
ejpam-2562	204	30	trivial	trivial	ADJ
ejpam-2562	204	31	congruence	congruence	NOUN
ejpam-2562	204	32	relation	relation	NOUN
ejpam-2562	204	33	.	.	PUNCT
ejpam-2562	205	1	lemma	lemma	PROPN
ejpam-2562	205	2	3	3	X
ejpam-2562	205	3	.	.	PUNCT
ejpam-2562	206	1	let	let	VERB
ejpam-2562	206	2	m	m	PRON
ejpam-2562	206	3	be	be	AUX
ejpam-2562	206	4	a	a	DET
ejpam-2562	206	5	left	left	ADJ
ejpam-2562	206	6	r	r	NOUN
ejpam-2562	206	7	-	-	PUNCT
ejpam-2562	206	8	semimodule	semimodule	NOUN
ejpam-2562	206	9	.	.	PUNCT
ejpam-2562	207	1	suppose	suppose	VERB
ejpam-2562	207	2	that	that	SCONJ
ejpam-2562	207	3	k1	k1	NOUN
ejpam-2562	207	4	≤	≤	PROPN
ejpam-2562	207	5	m1	m1	PROPN
ejpam-2562	207	6	≤	≤	NUM
ejpam-2562	207	7	m	m	PROPN
ejpam-2562	207	8	;	;	PUNCT
ejpam-2562	207	9	k2	k2	ADJ
ejpam-2562	207	10	≤	≤	PROPN
ejpam-2562	207	11	m2	m2	PROPN
ejpam-2562	207	12	≤	≤	PROPN
ejpam-2562	207	13	m	m	PROPN
ejpam-2562	207	14	;	;	PUNCT
ejpam-2562	207	15	l1	l1	PROPN
ejpam-2562	207	16	≤	≤	PROPN
ejpam-2562	207	17	m1	m1	NOUN
ejpam-2562	207	18	;	;	PUNCT
ejpam-2562	207	19	l2	l2	VERB
ejpam-2562	207	20	≤	≤	NUM
ejpam-2562	207	21	m2	m2	PROPN
ejpam-2562	207	22	and	and	CCONJ
ejpam-2562	207	23	m	m	PROPN
ejpam-2562	207	24	=	=	PROPN
ejpam-2562	207	25	m1	m1	PROPN
ejpam-2562	207	26	⊕m2	⊕m2	NUM
ejpam-2562	207	27	.	.	PUNCT
ejpam-2562	208	1	then	then	ADV
ejpam-2562	208	2	:	:	PUNCT
ejpam-2562	208	3	(	(	PUNCT
ejpam-2562	208	4	i	i	NOUN
ejpam-2562	208	5	)	)	PUNCT
ejpam-2562	208	6	≡l1	≡l1	VERB
ejpam-2562	208	7	trivial	trivial	ADJ
ejpam-2562	208	8	on	on	ADP
ejpam-2562	208	9	k1	k1	NOUN
ejpam-2562	208	10	=	=	NOUN
ejpam-2562	208	11	⇒≡l1	⇒≡l1	VERB
ejpam-2562	208	12	trivial	trivial	ADJ
ejpam-2562	208	13	on	on	ADP
ejpam-2562	208	14	k1	k1	PROPN
ejpam-2562	208	15	⊕	⊕	PROPN
ejpam-2562	208	16	k2	k2	PROPN
ejpam-2562	208	17	.	.	PUNCT
ejpam-2562	209	1	(	(	PUNCT
ejpam-2562	209	2	ii	ii	NOUN
ejpam-2562	209	3	)	)	PUNCT
ejpam-2562	209	4	≡l1⊕l2	≡l1⊕l2	NOUN
ejpam-2562	209	5	trivial	trivial	ADJ
ejpam-2562	209	6	on	on	ADP
ejpam-2562	209	7	k1	k1	PROPN
ejpam-2562	209	8	⊕	⊕	PROPN
ejpam-2562	209	9	k2	k2	PROPN
ejpam-2562	209	10	=	=	NOUN
ejpam-2562	209	11	⇒≡l1	⇒≡l1	VERB
ejpam-2562	209	12	trivial	trivial	ADJ
ejpam-2562	209	13	on	on	ADP
ejpam-2562	209	14	k1	k1	NOUN
ejpam-2562	209	15	and	and	CCONJ
ejpam-2562	209	16	≡l2	≡l2	NOUN
ejpam-2562	209	17	trivial	trivial	ADJ
ejpam-2562	209	18	on	on	ADP
ejpam-2562	209	19	k2	k2	PROPN
ejpam-2562	209	20	.	.	PUNCT
ejpam-2562	210	1	proof	proof	NOUN
ejpam-2562	210	2	.	.	PUNCT
ejpam-2562	211	1	(	(	PUNCT
ejpam-2562	211	2	i	i	NOUN
ejpam-2562	211	3	)	)	PUNCT
ejpam-2562	211	4	let	let	VERB
ejpam-2562	211	5	k1	k1	NOUN
ejpam-2562	211	6	+	+	CCONJ
ejpam-2562	211	7	k2	k2	ADJ
ejpam-2562	211	8	,	,	PUNCT
ejpam-2562	211	9	k′1	k′1	NOUN
ejpam-2562	211	10	+	+	CCONJ
ejpam-2562	211	11	k′2	k′2	NOUN
ejpam-2562	211	12	be	be	VERB
ejpam-2562	211	13	two	two	NUM
ejpam-2562	211	14	elements	element	NOUN
ejpam-2562	211	15	of	of	ADP
ejpam-2562	211	16	k1	k1	PROPN
ejpam-2562	211	17	⊕	⊕	PROPN
ejpam-2562	211	18	k2	k2	PROPN
ejpam-2562	211	19	.	.	PUNCT
ejpam-2562	212	1	we	we	PRON
ejpam-2562	212	2	have	have	VERB
ejpam-2562	212	3	k1	k1	NOUN
ejpam-2562	212	4	+	+	CCONJ
ejpam-2562	212	5	k2	k2	ADJ
ejpam-2562	212	6	≡l1	≡l1	NUM
ejpam-2562	212	7	k′1	k′1	X
ejpam-2562	212	8	+	+	CCONJ
ejpam-2562	212	9	k′2	k′2	X
ejpam-2562	212	10	=	=	NOUN
ejpam-2562	212	11	⇒	⇒	NOUN
ejpam-2562	212	12	k1	k1	NOUN
ejpam-2562	212	13	+	+	CCONJ
ejpam-2562	212	14	k2	k2	PROPN
ejpam-2562	212	15	+	+	CCONJ
ejpam-2562	212	16	l1	l1	PROPN
ejpam-2562	212	17	=	=	PROPN
ejpam-2562	212	18	k′1	k′1	NOUN
ejpam-2562	213	1	+	+	CCONJ
ejpam-2562	213	2	k′2	k′2	X
ejpam-2562	213	3	+	+	CCONJ
ejpam-2562	213	4	l	l	NOUN
ejpam-2562	213	5	′1	′1	NOUN
ejpam-2562	213	6	=	=	NOUN
ejpam-2562	213	7	⇒	⇒	NOUN
ejpam-2562	213	8	k1	k1	NOUN
ejpam-2562	213	9	+	+	CCONJ
ejpam-2562	213	10	l1	l1	PROPN
ejpam-2562	213	11	≡m2	≡m2	NOUN
ejpam-2562	213	12	k′1	k′1	NOUN
ejpam-2562	213	13	+	+	CCONJ
ejpam-2562	213	14	l	l	NOUN
ejpam-2562	213	15	′1	′1	X
ejpam-2562	213	16	and	and	CCONJ
ejpam-2562	213	17	k2	k2	PROPN
ejpam-2562	213	18	≡m1	≡m1	PROPN
ejpam-2562	213	19	k′2	k′2	PROPN
ejpam-2562	213	20	.	.	PUNCT
ejpam-2562	214	1	now	now	ADV
ejpam-2562	214	2	by	by	ADP
ejpam-2562	214	3	definition	definition	NOUN
ejpam-2562	214	4	of	of	ADP
ejpam-2562	214	5	m	m	PROPN
ejpam-2562	214	6	=	=	SYM
ejpam-2562	214	7	m1	m1	PROPN
ejpam-2562	214	8	⊕m2	⊕m2	NUM
ejpam-2562	214	9	and	and	CCONJ
ejpam-2562	214	10	by	by	ADP
ejpam-2562	214	11	hypothesis	hypothesis	NOUN
ejpam-2562	214	12	we	we	PRON
ejpam-2562	214	13	have	have	VERB
ejpam-2562	214	14	:	:	PUNCT
ejpam-2562	214	15	k1	k1	PROPN
ejpam-2562	214	16	+	+	CCONJ
ejpam-2562	214	17	l1	l1	PROPN
ejpam-2562	214	18	≡m2	≡m2	PROPN
ejpam-2562	214	19	k′1	k′1	NOUN
ejpam-2562	214	20	+	+	CCONJ
ejpam-2562	214	21	l	l	NOUN
ejpam-2562	215	1	′1	′1	NOUN
ejpam-2562	215	2	=	=	NOUN
ejpam-2562	215	3	⇒	⇒	NOUN
ejpam-2562	215	4	k1	k1	NOUN
ejpam-2562	215	5	=	=	SYM
ejpam-2562	215	6	k′1	k′1	NOUN
ejpam-2562	215	7	and	and	CCONJ
ejpam-2562	215	8	k2	k2	PROPN
ejpam-2562	215	9	≡m1	≡m1	ADJ
ejpam-2562	215	10	k′2	k′2	NOUN
ejpam-2562	215	11	=	=	AUX
ejpam-2562	215	12	⇒	⇒	PROPN
ejpam-2562	215	13	k2	k2	PROPN
ejpam-2562	215	14	=	=	SYM
ejpam-2562	215	15	k′2	k′2	PROPN
ejpam-2562	215	16	.	.	PUNCT
ejpam-2562	216	1	so	so	ADV
ejpam-2562	216	2	k1	k1	PROPN
ejpam-2562	216	3	+	+	CCONJ
ejpam-2562	216	4	k2	k2	NOUN
ejpam-2562	216	5	=	=	PROPN
ejpam-2562	216	6	k′1	k′1	NOUN
ejpam-2562	216	7	+	+	CCONJ
ejpam-2562	216	8	k′2	k′2	PROPN
ejpam-2562	216	9	and	and	CCONJ
ejpam-2562	216	10	therefore	therefore	ADV
ejpam-2562	216	11	≡l1	≡l1	ADV
ejpam-2562	216	12	is	be	AUX
ejpam-2562	216	13	trivial	trivial	ADJ
ejpam-2562	216	14	on	on	ADP
ejpam-2562	216	15	k1	k1	PROPN
ejpam-2562	216	16	⊕	⊕	PROPN
ejpam-2562	216	17	k2	k2	PROPN
ejpam-2562	216	18	.	.	PUNCT
ejpam-2562	217	1	(	(	PUNCT
ejpam-2562	217	2	ii	ii	NOUN
ejpam-2562	217	3	)	)	PUNCT
ejpam-2562	217	4	suppose	suppose	VERB
ejpam-2562	217	5	that	that	SCONJ
ejpam-2562	217	6	≡l1⊕l2	≡l1⊕l2	PROPN
ejpam-2562	217	7	trivial	trivial	ADJ
ejpam-2562	217	8	on	on	ADP
ejpam-2562	217	9	k1	k1	PROPN
ejpam-2562	217	10	⊕	⊕	PROPN
ejpam-2562	217	11	k2	k2	PROPN
ejpam-2562	217	12	.	.	PUNCT
ejpam-2562	218	1	let	let	VERB
ejpam-2562	218	2	k1	k1	PROPN
ejpam-2562	218	3	,	,	PUNCT
ejpam-2562	218	4	k′1	k′1	X
ejpam-2562	218	5	be	be	AUX
ejpam-2562	218	6	two	two	NUM
ejpam-2562	218	7	elements	element	NOUN
ejpam-2562	218	8	of	of	ADP
ejpam-2562	218	9	k1	k1	NOUN
ejpam-2562	218	10	such	such	ADJ
ejpam-2562	218	11	that	that	SCONJ
ejpam-2562	218	12	k1	k1	PROPN
ejpam-2562	218	13	≡l1	≡l1	X
ejpam-2562	218	14	k′1	k′1	NOUN
ejpam-2562	218	15	.	.	PUNCT
ejpam-2562	219	1	so	so	ADV
ejpam-2562	219	2	there	there	PRON
ejpam-2562	219	3	exist	exist	VERB
ejpam-2562	219	4	l1	l1	PROPN
ejpam-2562	219	5	,	,	PUNCT
ejpam-2562	219	6	l	l	PROPN
ejpam-2562	219	7	′1	′1	PROPN
ejpam-2562	219	8	∈	∈	PROPN
ejpam-2562	219	9	l1	l1	PROPN
ejpam-2562	219	10	such	such	ADJ
ejpam-2562	219	11	that	that	SCONJ
ejpam-2562	219	12	k1	k1	PROPN
ejpam-2562	219	13	+	+	CCONJ
ejpam-2562	219	14	l1	l1	PROPN
ejpam-2562	220	1	=	=	PROPN
ejpam-2562	220	2	k′1	k′1	NOUN
ejpam-2562	220	3	+	+	CCONJ
ejpam-2562	220	4	l	l	NOUN
ejpam-2562	220	5	′1	′1	NOUN
ejpam-2562	220	6	.	.	PUNCT
ejpam-2562	221	1	but	but	CCONJ
ejpam-2562	221	2	k1	k1	NOUN
ejpam-2562	221	3	=	=	SYM
ejpam-2562	221	4	k1	k1	PROPN
ejpam-2562	221	5	+	+	CCONJ
ejpam-2562	221	6	0	0	NUM
ejpam-2562	221	7	∈	∈	PROPN
ejpam-2562	221	8	k1	k1	PROPN
ejpam-2562	221	9	⊕	⊕	PROPN
ejpam-2562	221	10	k2	k2	PROPN
ejpam-2562	221	11	,	,	PUNCT
ejpam-2562	221	12	l1	l1	PROPN
ejpam-2562	221	13	=	=	PROPN
ejpam-2562	221	14	l1	l1	PROPN
ejpam-2562	221	15	+	+	CCONJ
ejpam-2562	221	16	0	0	NUM
ejpam-2562	221	17	∈	∈	PROPN
ejpam-2562	221	18	l1	l1	PROPN
ejpam-2562	221	19	⊕	⊕	PROPN
ejpam-2562	221	20	l2	l2	PROPN
ejpam-2562	221	21	,	,	PUNCT
ejpam-2562	221	22	idem	idem	PROPN
ejpam-2562	221	23	we	we	PRON
ejpam-2562	221	24	have	have	VERB
ejpam-2562	221	25	k′1	k′1	VERB
ejpam-2562	221	26	∈	∈	PROPN
ejpam-2562	221	27	k1	k1	PROPN
ejpam-2562	221	28	⊕	⊕	PROPN
ejpam-2562	221	29	k2	k2	PROPN
ejpam-2562	221	30	,	,	PUNCT
ejpam-2562	221	31	l1	l1	PROPN
ejpam-2562	221	32	∈	∈	PROPN
ejpam-2562	221	33	l1	l1	PROPN
ejpam-2562	221	34	⊕	⊕	PROPN
ejpam-2562	221	35	l2	l2	PROPN
ejpam-2562	221	36	.	.	PUNCT
ejpam-2562	222	1	since	since	SCONJ
ejpam-2562	222	2	≡l1⊕l2	≡l1⊕l2	PROPN
ejpam-2562	222	3	trivial	trivial	ADJ
ejpam-2562	222	4	on	on	ADP
ejpam-2562	222	5	k1	k1	PROPN
ejpam-2562	222	6	⊕	⊕	PROPN
ejpam-2562	222	7	k2	k2	PROPN
ejpam-2562	222	8	,	,	PUNCT
ejpam-2562	222	9	we	we	PRON
ejpam-2562	222	10	deduce	deduce	VERB
ejpam-2562	222	11	that	that	SCONJ
ejpam-2562	222	12	k1	k1	NOUN
ejpam-2562	222	13	=	=	SYM
ejpam-2562	222	14	k′1	k′1	NOUN
ejpam-2562	222	15	.	.	PUNCT
ejpam-2562	223	1	thus	thus	ADV
ejpam-2562	223	2	≡l1	≡l1	X
ejpam-2562	223	3	is	be	AUX
ejpam-2562	223	4	trivial	trivial	ADJ
ejpam-2562	223	5	on	on	ADP
ejpam-2562	223	6	k1	k1	NOUN
ejpam-2562	223	7	and	and	CCONJ
ejpam-2562	223	8	a	a	DET
ejpam-2562	223	9	same	same	ADJ
ejpam-2562	223	10	way	way	NOUN
ejpam-2562	223	11	shows	show	VERB
ejpam-2562	223	12	that	that	SCONJ
ejpam-2562	223	13	≡l2	≡l2	PROPN
ejpam-2562	223	14	is	be	AUX
ejpam-2562	223	15	trivial	trivial	ADJ
ejpam-2562	223	16	on	on	ADP
ejpam-2562	223	17	k2	k2	PROPN
ejpam-2562	223	18	.	.	PUNCT
ejpam-2562	224	1	proposition	proposition	NOUN
ejpam-2562	224	2	4	4	NUM
ejpam-2562	224	3	.	.	PUNCT
ejpam-2562	225	1	let	let	VERB
ejpam-2562	225	2	m	m	PRON
ejpam-2562	225	3	be	be	AUX
ejpam-2562	225	4	a	a	DET
ejpam-2562	225	5	left	left	ADJ
ejpam-2562	225	6	r	r	NOUN
ejpam-2562	225	7	-	-	PUNCT
ejpam-2562	225	8	semimodule	semimodule	NOUN
ejpam-2562	225	9	.	.	PUNCT
ejpam-2562	226	1	suppose	suppose	VERB
ejpam-2562	226	2	that	that	SCONJ
ejpam-2562	226	3	k1	k1	NOUN
ejpam-2562	226	4	≤	≤	PROPN
ejpam-2562	226	5	m1	m1	PROPN
ejpam-2562	226	6	≤	≤	NUM
ejpam-2562	226	7	m	m	PROPN
ejpam-2562	226	8	;	;	PUNCT
ejpam-2562	226	9	k2	k2	ADJ
ejpam-2562	226	10	≤	≤	PUNCT
ejpam-2562	226	11	m2	m2	PROPN
ejpam-2562	226	12	≤	≤	PROPN
ejpam-2562	226	13	m	m	PROPN
ejpam-2562	226	14	and	and	CCONJ
ejpam-2562	226	15	m	m	PROPN
ejpam-2562	226	16	=	=	PROPN
ejpam-2562	226	17	m1	m1	PROPN
ejpam-2562	226	18	⊕m2	⊕m2	NUM
ejpam-2562	226	19	.	.	PUNCT
ejpam-2562	227	1	then	then	ADV
ejpam-2562	227	2	(	(	PUNCT
ejpam-2562	227	3	k1	k1	PROPN
ejpam-2562	227	4	⊕	⊕	PROPN
ejpam-2562	227	5	k2)ãswe	k2)ãswe	PROPN
ejpam-2562	227	6	(	(	PUNCT
ejpam-2562	227	7	m1	m1	PROPN
ejpam-2562	227	8	⊕m2	⊕m2	NUM
ejpam-2562	227	9	)	)	PUNCT
ejpam-2562	228	1	=	=	NOUN
ejpam-2562	228	2	⇒	⇒	NOUN
ejpam-2562	228	3	(	(	PUNCT
ejpam-2562	228	4	k1	k1	X
ejpam-2562	228	5	ãswe	ãswe	NOUN
ejpam-2562	228	6	m1	m1	PROPN
ejpam-2562	228	7	and	and	CCONJ
ejpam-2562	228	8	k2	k2	PROPN
ejpam-2562	228	9	ãswe	ãswe	NOUN
ejpam-2562	228	10	m2	m2	PROPN
ejpam-2562	228	11	)	)	PUNCT
ejpam-2562	228	12	proof	proof	NOUN
ejpam-2562	228	13	.	.	PUNCT
ejpam-2562	229	1	let	let	VERB
ejpam-2562	229	2	us	we	PRON
ejpam-2562	229	3	show	show	VERB
ejpam-2562	229	4	that	that	SCONJ
ejpam-2562	229	5	k1	k1	NOUN
ejpam-2562	229	6	ãswe	ãswe	NOUN
ejpam-2562	229	7	m1	m1	PROPN
ejpam-2562	229	8	.	.	PUNCT
ejpam-2562	230	1	let	let	VERB
ejpam-2562	230	2	l1	l1	PROPN
ejpam-2562	230	3	≤	≤	PROPN
ejpam-2562	230	4	m1	m1	PROPN
ejpam-2562	230	5	such	such	ADJ
ejpam-2562	230	6	that	that	SCONJ
ejpam-2562	230	7	≡l1	≡l1	NOUN
ejpam-2562	230	8	is	be	AUX
ejpam-2562	230	9	trivial	trivial	ADJ
ejpam-2562	230	10	on	on	ADP
ejpam-2562	230	11	k1	k1	NOUN
ejpam-2562	230	12	.	.	PUNCT
ejpam-2562	231	1	by	by	ADP
ejpam-2562	231	2	lemma	lemma	PROPN
ejpam-2562	231	3	3(i	3(i	NUM
ejpam-2562	231	4	)	)	PUNCT
ejpam-2562	231	5	we	we	PRON
ejpam-2562	231	6	have	have	VERB
ejpam-2562	231	7	≡l1	≡l1	VERB
ejpam-2562	231	8	trivial	trivial	ADJ
ejpam-2562	231	9	on	on	ADP
ejpam-2562	231	10	k1	k1	PROPN
ejpam-2562	231	11	⊕	⊕	PROPN
ejpam-2562	231	12	k2	k2	PROPN
ejpam-2562	231	13	.	.	PUNCT
ejpam-2562	232	1	now	now	ADV
ejpam-2562	232	2	≡l1	≡l1	ADV
ejpam-2562	232	3	is	be	AUX
ejpam-2562	232	4	trivial	trivial	ADJ
ejpam-2562	232	5	on	on	ADP
ejpam-2562	232	6	k1	k1	PROPN
ejpam-2562	232	7	⊕	⊕	PROPN
ejpam-2562	232	8	k2	k2	PROPN
ejpam-2562	232	9	and	and	CCONJ
ejpam-2562	232	10	k1	k1	PROPN
ejpam-2562	232	11	⊕	⊕	PROPN
ejpam-2562	232	12	k2	k2	PROPN
ejpam-2562	232	13	ãswe	ãswe	PROPN
ejpam-2562	232	14	m1	m1	PROPN
ejpam-2562	232	15	⊕m2	⊕m2	PROPN
ejpam-2562	233	1	so	so	PROPN
ejpam-2562	233	2	l1	l1	PROPN
ejpam-2562	233	3	=	=	SYM
ejpam-2562	233	4	0	0	PROPN
ejpam-2562	233	5	.	.	PUNCT
ejpam-2562	234	1	proposition	proposition	NOUN
ejpam-2562	234	2	5	5	NUM
ejpam-2562	234	3	.	.	PUNCT
ejpam-2562	235	1	let	let	VERB
ejpam-2562	235	2	m	m	PRON
ejpam-2562	235	3	be	be	AUX
ejpam-2562	235	4	a	a	DET
ejpam-2562	235	5	left	left	ADJ
ejpam-2562	235	6	r	r	NOUN
ejpam-2562	235	7	-	-	PUNCT
ejpam-2562	235	8	semimodule	semimodule	NOUN
ejpam-2562	235	9	.	.	PUNCT
ejpam-2562	236	1	suppose	suppose	VERB
ejpam-2562	236	2	that	that	SCONJ
ejpam-2562	236	3	k1	k1	NOUN
ejpam-2562	236	4	≤	≤	PROPN
ejpam-2562	236	5	m1	m1	PROPN
ejpam-2562	236	6	≤	≤	NUM
ejpam-2562	236	7	m	m	PROPN
ejpam-2562	236	8	;	;	PUNCT
ejpam-2562	236	9	k2	k2	ADJ
ejpam-2562	236	10	≤	≤	PUNCT
ejpam-2562	236	11	m2	m2	PROPN
ejpam-2562	236	12	≤	≤	PROPN
ejpam-2562	236	13	m	m	PROPN
ejpam-2562	236	14	and	and	CCONJ
ejpam-2562	236	15	m	m	PROPN
ejpam-2562	236	16	=	=	PROPN
ejpam-2562	236	17	m1	m1	PROPN
ejpam-2562	236	18	⊕m2	⊕m2	NUM
ejpam-2562	236	19	.	.	PUNCT
ejpam-2562	237	1	then	then	ADV
ejpam-2562	237	2	(	(	PUNCT
ejpam-2562	237	3	k1	k1	PROPN
ejpam-2562	237	4	⊕	⊕	PROPN
ejpam-2562	237	5	k2)ãs	k2)ãs	PROPN
ejpam-2562	237	6	(	(	PUNCT
ejpam-2562	237	7	m1	m1	PROPN
ejpam-2562	237	8	⊕m2)	⊕m2)	ADV
ejpam-2562	237	9	⇐	⇐	ADJ
ejpam-2562	237	10	⇒	⇒	NOUN
ejpam-2562	237	11	(	(	PUNCT
ejpam-2562	237	12	k1	k1	X
ejpam-2562	237	13	ãs	ãs	DET
ejpam-2562	237	14	m1	m1	PROPN
ejpam-2562	237	15	and	and	CCONJ
ejpam-2562	237	16	k2	k2	PROPN
ejpam-2562	237	17	ãs	ãs	PRON
ejpam-2562	237	18	m2	m2	PROPN
ejpam-2562	237	19	)	)	PUNCT
ejpam-2562	237	20	.	.	PUNCT
ejpam-2562	238	1	e.	e.	PROPN
ejpam-2562	238	2	diop	diop	PROPN
ejpam-2562	238	3	,	,	PUNCT
ejpam-2562	238	4	d.	d.	PROPN
ejpam-2562	238	5	sow	sow	PROPN
ejpam-2562	238	6	/	/	SYM
ejpam-2562	238	7	eur	eur	PROPN
ejpam-2562	238	8	.	.	PUNCT
ejpam-2562	239	1	j.	j.	PROPN
ejpam-2562	239	2	pure	pure	PROPN
ejpam-2562	239	3	appl	appl	PROPN
ejpam-2562	239	4	.	.	PROPN
ejpam-2562	239	5	math	math	PROPN
ejpam-2562	239	6	,	,	PUNCT
ejpam-2562	239	7	9	9	NUM
ejpam-2562	239	8	(	(	PUNCT
ejpam-2562	239	9	2016	2016	NUM
ejpam-2562	239	10	)	)	PUNCT
ejpam-2562	239	11	,	,	PUNCT
ejpam-2562	239	12	250	250	NUM
ejpam-2562	239	13	-	-	SYM
ejpam-2562	239	14	265	265	NUM
ejpam-2562	239	15	258	258	NUM
ejpam-2562	239	16	proof	proof	NOUN
ejpam-2562	239	17	.	.	PUNCT
ejpam-2562	240	1	=	=	NOUN
ejpam-2562	240	2	⇒.	⇒.	NOUN
ejpam-2562	240	3	suppose	suppose	VERB
ejpam-2562	240	4	for	for	ADP
ejpam-2562	240	5	example	example	NOUN
ejpam-2562	240	6	k1	k1	PROPN
ejpam-2562	240	7	s	s	PROPN
ejpam-2562	240	8	m1	m1	PROPN
ejpam-2562	240	9	.	.	PUNCT
ejpam-2562	241	1	then	then	ADV
ejpam-2562	241	2	there	there	PRON
ejpam-2562	241	3	exists	exist	VERB
ejpam-2562	241	4	a	a	DET
ejpam-2562	241	5	subsemimodule	subsemimodule	NOUN
ejpam-2562	241	6	l1	l1	PROPN
ejpam-2562	241	7	6=	6=	ADP
ejpam-2562	241	8	0	0	NUM
ejpam-2562	241	9	of	of	ADP
ejpam-2562	241	10	m1	m1	PROPN
ejpam-2562	241	11	such	such	ADJ
ejpam-2562	241	12	that	that	PRON
ejpam-2562	241	13	:	:	PUNCT
ejpam-2562	241	14	l1	l1	PROPN
ejpam-2562	241	15	∩	∩	PROPN
ejpam-2562	241	16	k1	k1	NOUN
ejpam-2562	241	17	=	=	SYM
ejpam-2562	241	18	0	0	NUM
ejpam-2562	241	19	.	.	PUNCT
ejpam-2562	242	1	so	so	ADV
ejpam-2562	242	2	let	let	VERB
ejpam-2562	242	3	us	we	PRON
ejpam-2562	242	4	prove	prove	VERB
ejpam-2562	242	5	that	that	SCONJ
ejpam-2562	242	6	l1	l1	PROPN
ejpam-2562	242	7	∩	∩	PROPN
ejpam-2562	242	8	(	(	PUNCT
ejpam-2562	242	9	k1	k1	NOUN
ejpam-2562	242	10	+	+	CCONJ
ejpam-2562	242	11	k2	k2	ADJ
ejpam-2562	242	12	)	)	PUNCT
ejpam-2562	242	13	=	=	SYM
ejpam-2562	243	1	0	0	X
ejpam-2562	243	2	.	.	PUNCT
ejpam-2562	244	1	let	let	VERB
ejpam-2562	244	2	l1	l1	PROPN
ejpam-2562	244	3	∈	∈	PROPN
ejpam-2562	244	4	l1	l1	PROPN
ejpam-2562	244	5	∩	∩	PROPN
ejpam-2562	244	6	(	(	PUNCT
ejpam-2562	244	7	k1	k1	NOUN
ejpam-2562	244	8	+	+	CCONJ
ejpam-2562	244	9	k2	k2	ADJ
ejpam-2562	244	10	)	)	PUNCT
ejpam-2562	244	11	.	.	PUNCT
ejpam-2562	245	1	there	there	PRON
ejpam-2562	245	2	exists	exist	VERB
ejpam-2562	245	3	(	(	PUNCT
ejpam-2562	245	4	k1	k1	NOUN
ejpam-2562	245	5	;	;	PUNCT
ejpam-2562	245	6	k2	k2	ADJ
ejpam-2562	245	7	)	)	PUNCT
ejpam-2562	245	8	∈	∈	PROPN
ejpam-2562	245	9	k1	k1	PROPN
ejpam-2562	245	10	×	×	PROPN
ejpam-2562	245	11	k2	k2	PROPN
ejpam-2562	245	12	such	such	ADJ
ejpam-2562	245	13	that	that	PRON
ejpam-2562	245	14	:	:	PUNCT
ejpam-2562	245	15	l1	l1	PROPN
ejpam-2562	245	16	=	=	PROPN
ejpam-2562	245	17	k1	k1	PROPN
ejpam-2562	245	18	+	+	X
ejpam-2562	245	19	k2	k2	ADJ
ejpam-2562	245	20	.	.	PUNCT
ejpam-2562	246	1	we	we	PRON
ejpam-2562	246	2	have	have	VERB
ejpam-2562	246	3	:	:	PUNCT
ejpam-2562	246	4	l1	l1	PROPN
ejpam-2562	246	5	∈	∈	PROPN
ejpam-2562	246	6	l1	l1	PROPN
ejpam-2562	246	7	≤	≤	PROPN
ejpam-2562	246	8	m1	m1	NOUN
ejpam-2562	246	9	;	;	PUNCT
ejpam-2562	246	10	k1	k1	PROPN
ejpam-2562	246	11	∈	∈	PROPN
ejpam-2562	246	12	k1	k1	NOUN
ejpam-2562	246	13	≤	≤	PROPN
ejpam-2562	246	14	m1	m1	PROPN
ejpam-2562	246	15	and	and	CCONJ
ejpam-2562	246	16	k2	k2	PROPN
ejpam-2562	246	17	∈	∈	PROPN
ejpam-2562	246	18	k2	k2	PROPN
ejpam-2562	246	19	≤	≤	PROPN
ejpam-2562	246	20	m2	m2	PROPN
ejpam-2562	246	21	,	,	PUNCT
ejpam-2562	246	22	so	so	ADV
ejpam-2562	246	23	by	by	ADP
ejpam-2562	246	24	the	the	DET
ejpam-2562	246	25	direct	direct	ADJ
ejpam-2562	246	26	sum	sum	NOUN
ejpam-2562	246	27	m1	m1	PROPN
ejpam-2562	246	28	⊕m2	⊕m2	PROPN
ejpam-2562	247	1	we	we	PRON
ejpam-2562	247	2	deduce	deduce	VERB
ejpam-2562	247	3	that	that	DET
ejpam-2562	247	4	k2	k2	PROPN
ejpam-2562	247	5	=	=	SYM
ejpam-2562	247	6	0	0	PROPN
ejpam-2562	247	7	and	and	CCONJ
ejpam-2562	247	8	l1	l1	PROPN
ejpam-2562	247	9	=	=	PUNCT
ejpam-2562	247	10	k1	k1	PROPN
ejpam-2562	247	11	;	;	PUNCT
ejpam-2562	247	12	hence	hence	ADV
ejpam-2562	247	13	l1	l1	PROPN
ejpam-2562	247	14	∈	∈	PROPN
ejpam-2562	247	15	l1	l1	PROPN
ejpam-2562	247	16	∩	∩	PROPN
ejpam-2562	247	17	k1	k1	PROPN
ejpam-2562	247	18	=	=	SYM
ejpam-2562	247	19	0	0	NUM
ejpam-2562	247	20	,	,	PUNCT
ejpam-2562	247	21	whence	whence	NOUN
ejpam-2562	247	22	l1	l1	PROPN
ejpam-2562	247	23	=	=	PROPN
ejpam-2562	247	24	0	0	PROPN
ejpam-2562	247	25	.	.	PUNCT
ejpam-2562	248	1	consequently	consequently	ADV
ejpam-2562	248	2	l1	l1	PROPN
ejpam-2562	248	3	∩	∩	PROPN
ejpam-2562	248	4	(	(	PUNCT
ejpam-2562	248	5	k1	k1	PROPN
ejpam-2562	248	6	⊕	⊕	PROPN
ejpam-2562	248	7	k2	k2	PROPN
ejpam-2562	248	8	)	)	PUNCT
ejpam-2562	248	9	=	=	SYM
ejpam-2562	248	10	0	0	NUM
ejpam-2562	248	11	which	which	PRON
ejpam-2562	248	12	contradicts	contradict	VERB
ejpam-2562	248	13	the	the	DET
ejpam-2562	248	14	fact	fact	NOUN
ejpam-2562	248	15	that	that	SCONJ
ejpam-2562	248	16	(	(	PUNCT
ejpam-2562	248	17	k1⊕k2)ãs	k1⊕k2)ãs	X
ejpam-2562	248	18	(	(	PUNCT
ejpam-2562	248	19	m1⊕m2	m1⊕m2	NOUN
ejpam-2562	248	20	)	)	PUNCT
ejpam-2562	248	21	.	.	PUNCT
ejpam-2562	249	1	so	so	ADV
ejpam-2562	249	2	k1	k1	PROPN
ejpam-2562	249	3	ãs	ãs	DET
ejpam-2562	249	4	m1	m1	NOUN
ejpam-2562	249	5	.	.	PUNCT
ejpam-2562	250	1	a	a	DET
ejpam-2562	250	2	same	same	ADJ
ejpam-2562	250	3	argument	argument	NOUN
ejpam-2562	250	4	prove	prove	VERB
ejpam-2562	250	5	that	that	SCONJ
ejpam-2562	250	6	k2	k2	PROPN
ejpam-2562	250	7	ãs	ãs	PRON
ejpam-2562	250	8	m2	m2	PROPN
ejpam-2562	250	9	.	.	PUNCT
ejpam-2562	251	1	⇐	⇐	PROPN
ejpam-2562	251	2	=	=	PRON
ejpam-2562	251	3	.	.	PROPN
ejpam-2562	251	4	suppose	suppose	VERB
ejpam-2562	251	5	that	that	SCONJ
ejpam-2562	251	6	ki	ki	PROPN
ejpam-2562	251	7	ãs	ãs	INTJ
ejpam-2562	251	8	mi	mi	PROPN
ejpam-2562	251	9	for	for	ADP
ejpam-2562	251	10	all	all	PRON
ejpam-2562	251	11	i	i	PRON
ejpam-2562	251	12	∈	∈	PROPN
ejpam-2562	251	13	{	{	PUNCT
ejpam-2562	251	14	1,2	1,2	NUM
ejpam-2562	251	15	}	}	PUNCT
ejpam-2562	251	16	.	.	PUNCT
ejpam-2562	252	1	let	let	VERB
ejpam-2562	252	2	0	0	NUM
ejpam-2562	253	1	6=	6=	NUM
ejpam-2562	253	2	x	x	SYM
ejpam-2562	253	3	∈	∈	PROPN
ejpam-2562	253	4	m1	m1	PROPN
ejpam-2562	253	5	⊕m2	⊕m2	PROPN
ejpam-2562	253	6	.	.	PUNCT
ejpam-2562	254	1	then	then	ADV
ejpam-2562	254	2	,	,	PUNCT
ejpam-2562	254	3	there	there	PRON
ejpam-2562	254	4	exists	exist	VERB
ejpam-2562	254	5	(	(	PUNCT
ejpam-2562	254	6	0,0	0,0	NOUN
ejpam-2562	254	7	)	)	PUNCT
ejpam-2562	254	8	6=	6=	NUM
ejpam-2562	255	1	(	(	PUNCT
ejpam-2562	255	2	x1	x1	PROPN
ejpam-2562	255	3	,	,	PUNCT
ejpam-2562	255	4	x2	x2	ADJ
ejpam-2562	255	5	)	)	PUNCT
ejpam-2562	255	6	∈	∈	PROPN
ejpam-2562	255	7	m1	m1	NOUN
ejpam-2562	255	8	×m2	×m2	NOUN
ejpam-2562	255	9	such	such	ADJ
ejpam-2562	255	10	that	that	SCONJ
ejpam-2562	255	11	:	:	PUNCT
ejpam-2562	255	12	0	0	NUM
ejpam-2562	256	1	6=	6=	NUM
ejpam-2562	256	2	x	x	SYM
ejpam-2562	256	3	=	=	SYM
ejpam-2562	257	1	x1	x1	PROPN
ejpam-2562	258	1	+	+	NUM
ejpam-2562	258	2	x2	x2	PROPN
ejpam-2562	258	3	.	.	PUNCT
ejpam-2562	259	1	without	without	ADP
ejpam-2562	259	2	lost	lose	VERB
ejpam-2562	259	3	of	of	ADP
ejpam-2562	259	4	generality	generality	NOUN
ejpam-2562	259	5	we	we	PRON
ejpam-2562	259	6	can	can	AUX
ejpam-2562	259	7	suppose	suppose	VERB
ejpam-2562	259	8	that	that	SCONJ
ejpam-2562	259	9	:	:	PUNCT
ejpam-2562	259	10	0	0	NUM
ejpam-2562	259	11	6=	6=	NUM
ejpam-2562	259	12	x1	x1	PROPN
ejpam-2562	259	13	∈	∈	PROPN
ejpam-2562	259	14	m1	m1	NOUN
ejpam-2562	259	15	.	.	PUNCT
ejpam-2562	260	1	since	since	SCONJ
ejpam-2562	260	2	k1	k1	PROPN
ejpam-2562	260	3	ãs	ãs	DET
ejpam-2562	260	4	m1	m1	PROPN
ejpam-2562	260	5	then	then	ADV
ejpam-2562	260	6	from	from	ADP
ejpam-2562	260	7	lemma	lemma	PROPN
ejpam-2562	260	8	1	1	NUM
ejpam-2562	260	9	,	,	PUNCT
ejpam-2562	260	10	there	there	PRON
ejpam-2562	260	11	exists	exist	VERB
ejpam-2562	260	12	a	a	DET
ejpam-2562	260	13	r1	r1	NOUN
ejpam-2562	260	14	∈	∈	PROPN
ejpam-2562	260	15	r	r	NOUN
ejpam-2562	260	16	such	such	ADJ
ejpam-2562	260	17	that	that	PRON
ejpam-2562	260	18	:	:	PUNCT
ejpam-2562	260	19	r1	r1	PROPN
ejpam-2562	260	20	x1	x1	PROPN
ejpam-2562	260	21	∈	∈	PROPN
ejpam-2562	260	22	k1	k1	NOUN
ejpam-2562	260	23	and	and	CCONJ
ejpam-2562	260	24	r1	r1	PROPN
ejpam-2562	260	25	x1	x1	PROPN
ejpam-2562	261	1	6=	6=	ADP
ejpam-2562	261	2	0	0	NUM
ejpam-2562	261	3	.	.	NOUN
ejpam-2562	261	4	•	•	NOUN
ejpam-2562	261	5	if	if	SCONJ
ejpam-2562	261	6	r1	r1	PROPN
ejpam-2562	261	7	x2	x2	PROPN
ejpam-2562	261	8	∈	∈	PROPN
ejpam-2562	261	9	k2	k2	PROPN
ejpam-2562	261	10	then	then	ADV
ejpam-2562	261	11	r1	r1	PROPN
ejpam-2562	261	12	x1+r1	x1+r1	PROPN
ejpam-2562	261	13	x2	x2	PROPN
ejpam-2562	261	14	∈	∈	PROPN
ejpam-2562	261	15	k1+k2	k1+k2	PROPN
ejpam-2562	261	16	therefore	therefore	ADV
ejpam-2562	261	17	r1(x1+x2	r1(x1+x2	NOUN
ejpam-2562	261	18	)	)	PUNCT
ejpam-2562	262	1	∈	∈	PROPN
ejpam-2562	262	2	k1⊕k2	k1⊕k2	PROPN
ejpam-2562	262	3	with	with	ADP
ejpam-2562	262	4	r1(x1+x2	r1(x1+x2	NOUN
ejpam-2562	262	5	)	)	PUNCT
ejpam-2562	263	1	6=	6=	ADP
ejpam-2562	263	2	0	0	PUNCT
ejpam-2562	264	1	because	because	SCONJ
ejpam-2562	264	2	if	if	SCONJ
ejpam-2562	264	3	r1	r1	PROPN
ejpam-2562	264	4	x1	x1	PROPN
ejpam-2562	264	5	+	+	CCONJ
ejpam-2562	264	6	r1	r1	NOUN
ejpam-2562	264	7	x2	x2	PROPN
ejpam-2562	264	8	=	=	SYM
ejpam-2562	264	9	0	0	NUM
ejpam-2562	264	10	,	,	PUNCT
ejpam-2562	264	11	then	then	ADV
ejpam-2562	264	12	by	by	ADP
ejpam-2562	264	13	the	the	DET
ejpam-2562	264	14	sum	sum	NOUN
ejpam-2562	264	15	direct	direct	ADV
ejpam-2562	264	16	we	we	PRON
ejpam-2562	264	17	have	have	AUX
ejpam-2562	264	18	r1	r1	VERB
ejpam-2562	264	19	x1	x1	PROPN
ejpam-2562	265	1	=	=	SYM
ejpam-2562	265	2	0	0	PROPN
ejpam-2562	265	3	,	,	PUNCT
ejpam-2562	265	4	which	which	PRON
ejpam-2562	265	5	is	be	AUX
ejpam-2562	265	6	absurd	absurd	ADJ
ejpam-2562	265	7	.	.	PUNCT
ejpam-2562	266	1	consequently	consequently	ADV
ejpam-2562	266	2	(	(	PUNCT
ejpam-2562	266	3	k1	k1	PROPN
ejpam-2562	266	4	⊕	⊕	PROPN
ejpam-2562	266	5	k2)ãs	k2)ãs	PROPN
ejpam-2562	266	6	(	(	PUNCT
ejpam-2562	266	7	m1	m1	PROPN
ejpam-2562	266	8	⊕m2	⊕m2	NUM
ejpam-2562	266	9	)	)	PUNCT
ejpam-2562	266	10	.	.	PUNCT
ejpam-2562	267	1	•	•	INTJ
ejpam-2562	268	1	if	if	SCONJ
ejpam-2562	268	2	r1	r1	PROPN
ejpam-2562	268	3	x2	x2	PROPN
ejpam-2562	268	4	is	be	AUX
ejpam-2562	268	5	n’t	not	PART
ejpam-2562	268	6	in	in	ADP
ejpam-2562	268	7	k2	k2	PROPN
ejpam-2562	268	8	then	then	ADV
ejpam-2562	268	9	there	there	PRON
ejpam-2562	268	10	exists	exist	VERB
ejpam-2562	268	11	r2	r2	PROPN
ejpam-2562	268	12	∈	∈	PROPN
ejpam-2562	268	13	r	r	NOUN
ejpam-2562	268	14	such	such	ADJ
ejpam-2562	268	15	that	that	PRON
ejpam-2562	268	16	:	:	PUNCT
ejpam-2562	268	17	0	0	NUM
ejpam-2562	268	18	6=	6=	NUM
ejpam-2562	268	19	r2r1	r2r1	PROPN
ejpam-2562	268	20	x2	x2	PROPN
ejpam-2562	268	21	∈	∈	PROPN
ejpam-2562	268	22	k2	k2	PROPN
ejpam-2562	268	23	.	.	PUNCT
ejpam-2562	269	1	we	we	PRON
ejpam-2562	269	2	have	have	AUX
ejpam-2562	269	3	r2r1	r2r1	VERB
ejpam-2562	269	4	x1	x1	PROPN
ejpam-2562	269	5	∈	∈	PROPN
ejpam-2562	269	6	k1	k1	NOUN
ejpam-2562	269	7	then	then	ADV
ejpam-2562	269	8	r2r1(x1	r2r1(x1	ADJ
ejpam-2562	269	9	+	+	SYM
ejpam-2562	269	10	x2	x2	ADJ
ejpam-2562	269	11	)	)	PUNCT
ejpam-2562	269	12	∈	∈	PROPN
ejpam-2562	269	13	k1⊕k2	k1⊕k2	PROPN
ejpam-2562	269	14	.	.	PUNCT
ejpam-2562	270	1	if	if	SCONJ
ejpam-2562	270	2	we	we	PRON
ejpam-2562	270	3	put	put	VERB
ejpam-2562	270	4	r	r	NOUN
ejpam-2562	270	5	=	=	PUNCT
ejpam-2562	270	6	r2r1	r2r1	NOUN
ejpam-2562	270	7	then	then	ADV
ejpam-2562	270	8	there	there	PRON
ejpam-2562	270	9	exists	exist	VERB
ejpam-2562	270	10	r	r	NOUN
ejpam-2562	270	11	∈	∈	PROPN
ejpam-2562	270	12	r	r	NOUN
ejpam-2562	270	13	such	such	ADJ
ejpam-2562	270	14	that	that	PRON
ejpam-2562	270	15	:	:	PUNCT
ejpam-2562	270	16	r(x1	r(x1	PROPN
ejpam-2562	270	17	+	+	CCONJ
ejpam-2562	270	18	x2	x2	X
ejpam-2562	270	19	)	)	PUNCT
ejpam-2562	270	20	∈	∈	PROPN
ejpam-2562	270	21	k1	k1	PROPN
ejpam-2562	270	22	⊕	⊕	PROPN
ejpam-2562	270	23	k2	k2	PROPN
ejpam-2562	270	24	with	with	ADP
ejpam-2562	270	25	r(x1	r(x1	NOUN
ejpam-2562	270	26	+	+	CCONJ
ejpam-2562	270	27	x2	x2	PROPN
ejpam-2562	270	28	)	)	PUNCT
ejpam-2562	270	29	6=	6=	ADP
ejpam-2562	270	30	0	0	PUNCT
ejpam-2562	271	1	because	because	SCONJ
ejpam-2562	271	2	if	if	SCONJ
ejpam-2562	271	3	r	r	NOUN
ejpam-2562	271	4	x1	x1	NOUN
ejpam-2562	272	1	+	+	CCONJ
ejpam-2562	272	2	r	r	NOUN
ejpam-2562	272	3	x2	x2	NOUN
ejpam-2562	272	4	=	=	SYM
ejpam-2562	272	5	0	0	NUM
ejpam-2562	272	6	,	,	PUNCT
ejpam-2562	272	7	then	then	ADV
ejpam-2562	272	8	by	by	ADP
ejpam-2562	272	9	the	the	DET
ejpam-2562	272	10	direct	direct	ADJ
ejpam-2562	272	11	sum	sum	NOUN
ejpam-2562	272	12	we	we	PRON
ejpam-2562	272	13	have	have	VERB
ejpam-2562	272	14	r	r	NOUN
ejpam-2562	272	15	x1	x1	PROPN
ejpam-2562	272	16	=	=	SYM
ejpam-2562	272	17	0	0	PROPN
ejpam-2562	272	18	,	,	PUNCT
ejpam-2562	272	19	which	which	PRON
ejpam-2562	272	20	is	be	AUX
ejpam-2562	272	21	absurd	absurd	ADJ
ejpam-2562	272	22	.	.	PUNCT
ejpam-2562	273	1	therefore	therefore	ADV
ejpam-2562	273	2	(	(	PUNCT
ejpam-2562	273	3	k1	k1	PROPN
ejpam-2562	273	4	⊕	⊕	PROPN
ejpam-2562	273	5	k2	k2	PROPN
ejpam-2562	273	6	)	)	PUNCT
ejpam-2562	273	7	ãs	ãs	X
ejpam-2562	273	8	(	(	PUNCT
ejpam-2562	273	9	m1	m1	PROPN
ejpam-2562	273	10	⊕m2	⊕m2	NUM
ejpam-2562	273	11	)	)	PUNCT
ejpam-2562	273	12	.	.	PUNCT
ejpam-2562	274	1	thus	thus	ADV
ejpam-2562	274	2	ki	ki	PROPN
ejpam-2562	274	3	ãs	ãs	INTJ
ejpam-2562	274	4	mi	mi	PROPN
ejpam-2562	274	5	for	for	ADP
ejpam-2562	274	6	all	all	PRON
ejpam-2562	274	7	i	i	PRON
ejpam-2562	274	8	∈	∈	PROPN
ejpam-2562	274	9	{	{	PUNCT
ejpam-2562	274	10	1	1	NUM
ejpam-2562	274	11	;	;	PUNCT
ejpam-2562	274	12	2	2	NUM
ejpam-2562	274	13	}	}	PUNCT
ejpam-2562	274	14	=	=	NOUN
ejpam-2562	274	15	⇒	⇒	NOUN
ejpam-2562	274	16	(	(	PUNCT
ejpam-2562	274	17	k1	k1	PROPN
ejpam-2562	274	18	⊕	⊕	PROPN
ejpam-2562	274	19	k2)ãs	k2)ãs	PROPN
ejpam-2562	274	20	(	(	PUNCT
ejpam-2562	274	21	m1	m1	PROPN
ejpam-2562	274	22	⊕m2	⊕m2	NUM
ejpam-2562	274	23	)	)	PUNCT
ejpam-2562	274	24	.	.	PUNCT
ejpam-2562	275	1	definition	definition	NOUN
ejpam-2562	275	2	6	6	NUM
ejpam-2562	275	3	.	.	PUNCT
ejpam-2562	276	1	let	let	VERB
ejpam-2562	276	2	n	n	PRON
ejpam-2562	276	3	be	be	AUX
ejpam-2562	276	4	a	a	DET
ejpam-2562	276	5	subsemimodule	subsemimodule	NOUN
ejpam-2562	276	6	of	of	ADP
ejpam-2562	276	7	a	a	DET
ejpam-2562	276	8	left	left	ADJ
ejpam-2562	276	9	r	r	NOUN
ejpam-2562	276	10	-	-	PUNCT
ejpam-2562	276	11	semimodule	semimodule	NOUN
ejpam-2562	276	12	m.	m.	NOUN
ejpam-2562	276	13	a	a	DET
ejpam-2562	276	14	subsemimodule	subsemimodule	NOUN
ejpam-2562	276	15	n	n	NOUN
ejpam-2562	276	16	′	′	NOUN
ejpam-2562	276	17	of	of	ADP
ejpam-2562	276	18	m	m	PROPN
ejpam-2562	276	19	is	be	AUX
ejpam-2562	276	20	called	call	VERB
ejpam-2562	276	21	m	m	PROPN
ejpam-2562	276	22	-	-	PUNCT
ejpam-2562	276	23	w	w	NOUN
ejpam-2562	276	24	-	-	NOUN
ejpam-2562	276	25	complement	complement	NOUN
ejpam-2562	276	26	of	of	ADP
ejpam-2562	276	27	n	n	PRON
ejpam-2562	276	28	if	if	SCONJ
ejpam-2562	276	29	the	the	DET
ejpam-2562	276	30	restriction	restriction	NOUN
ejpam-2562	276	31	of	of	ADP
ejpam-2562	276	32	≡n	≡n	PROPN
ejpam-2562	276	33	′	′	NOUN
ejpam-2562	276	34	to	to	ADP
ejpam-2562	276	35	n	n	PROPN
ejpam-2562	276	36	is	be	AUX
ejpam-2562	276	37	trivial	trivial	ADJ
ejpam-2562	276	38	and	and	CCONJ
ejpam-2562	276	39	n	n	PRON
ejpam-2562	276	40	′	′	NOUN
ejpam-2562	276	41	is	be	AUX
ejpam-2562	276	42	maximal	maximal	ADJ
ejpam-2562	276	43	with	with	ADP
ejpam-2562	276	44	this	this	DET
ejpam-2562	276	45	property	property	NOUN
ejpam-2562	276	46	.	.	PUNCT
ejpam-2562	277	1	proposition	proposition	NOUN
ejpam-2562	277	2	6	6	NUM
ejpam-2562	277	3	.	.	PUNCT
ejpam-2562	278	1	(	(	PUNCT
ejpam-2562	278	2	i	i	NOUN
ejpam-2562	278	3	)	)	PUNCT
ejpam-2562	278	4	every	every	DET
ejpam-2562	278	5	subsemimodule	subsemimodule	NOUN
ejpam-2562	278	6	n	n	PROPN
ejpam-2562	278	7	of	of	ADP
ejpam-2562	278	8	a	a	DET
ejpam-2562	278	9	left	left	ADJ
ejpam-2562	278	10	r	r	NOUN
ejpam-2562	278	11	-	-	PUNCT
ejpam-2562	278	12	semimodule	semimodule	NOUN
ejpam-2562	278	13	m	m	VERB
ejpam-2562	278	14	has	have	VERB
ejpam-2562	278	15	a	a	DET
ejpam-2562	278	16	m	m	NOUN
ejpam-2562	278	17	-	-	PUNCT
ejpam-2562	278	18	w	w	NOUN
ejpam-2562	278	19	-	-	NOUN
ejpam-2562	278	20	complement	complement	NOUN
ejpam-2562	278	21	.	.	PUNCT
ejpam-2562	279	1	(	(	PUNCT
ejpam-2562	279	2	ii	ii	NOUN
ejpam-2562	279	3	)	)	PUNCT
ejpam-2562	279	4	if	if	SCONJ
ejpam-2562	279	5	n	n	NOUN
ejpam-2562	279	6	′	′	NOUN
ejpam-2562	279	7	is	be	AUX
ejpam-2562	279	8	a	a	DET
ejpam-2562	279	9	m	m	PROPN
ejpam-2562	279	10	-	-	PUNCT
ejpam-2562	279	11	w	w	NOUN
ejpam-2562	279	12	-	-	NOUN
ejpam-2562	279	13	complement	complement	NOUN
ejpam-2562	279	14	of	of	ADP
ejpam-2562	279	15	a	a	DET
ejpam-2562	279	16	subsemimodule	subsemimodule	NOUN
ejpam-2562	279	17	n	n	PROPN
ejpam-2562	279	18	of	of	ADP
ejpam-2562	279	19	m	m	PRON
ejpam-2562	279	20	and	and	CCONJ
ejpam-2562	279	21	if	if	SCONJ
ejpam-2562	279	22	n	n	PRON
ejpam-2562	279	23	⊕	⊕	PROPN
ejpam-2562	279	24	n	n	PRON
ejpam-2562	279	25	′	′	NUM
ejpam-2562	279	26	exists	exist	VERB
ejpam-2562	279	27	then	then	ADV
ejpam-2562	279	28	:	:	PUNCT
ejpam-2562	279	29	n	n	PROPN
ejpam-2562	279	30	⊕	⊕	PROPN
ejpam-2562	279	31	n	n	CCONJ
ejpam-2562	279	32	′	′	NUM
ejpam-2562	279	33	ãswe	ãswe	NOUN
ejpam-2562	279	34	m.	m.	NOUN
ejpam-2562	279	35	proof	proof	NOUN
ejpam-2562	279	36	.	.	PUNCT
ejpam-2562	280	1	(	(	PUNCT
ejpam-2562	280	2	i	i	NOUN
ejpam-2562	280	3	)	)	PUNCT
ejpam-2562	280	4	let	let	VERB
ejpam-2562	280	5	$	$	SYM
ejpam-2562	280	6	=	=	PRON
ejpam-2562	280	7	{	{	PUNCT
ejpam-2562	280	8	a≤	a≤	NOUN
ejpam-2562	280	9	m/	m/	VERB
ejpam-2562	280	10	the	the	DET
ejpam-2562	280	11	restriction	restriction	NOUN
ejpam-2562	280	12	of	of	ADP
ejpam-2562	280	13	≡a	≡a	NOUN
ejpam-2562	280	14	to	to	ADP
ejpam-2562	280	15	n	n	PROPN
ejpam-2562	280	16	,	,	PUNCT
ejpam-2562	280	17	is	be	AUX
ejpam-2562	280	18	trivial	trivial	ADJ
ejpam-2562	280	19	}	}	PUNCT
ejpam-2562	280	20	.	.	PUNCT
ejpam-2562	281	1	0	0	NUM
ejpam-2562	281	2	∈	∈	PROPN
ejpam-2562	281	3	$	$	ADP
ejpam-2562	281	4	,	,	PUNCT
ejpam-2562	281	5	then	then	ADV
ejpam-2562	281	6	$	$	SYM
ejpam-2562	281	7	6=	6=	ADP
ejpam-2562	281	8	∅.	∅.	NOUN
ejpam-2562	281	9	(	(	PUNCT
ejpam-2562	281	10	$	$	SYM
ejpam-2562	281	11	;	;	PUNCT
ejpam-2562	281	12	⊆	⊆	X
ejpam-2562	281	13	)	)	PUNCT
ejpam-2562	281	14	is	be	AUX
ejpam-2562	281	15	an	an	DET
ejpam-2562	281	16	ordered	order	VERB
ejpam-2562	281	17	poset	poset	NOUN
ejpam-2562	281	18	.	.	PUNCT
ejpam-2562	282	1	it	it	PRON
ejpam-2562	282	2	is	be	AUX
ejpam-2562	282	3	easy	easy	ADJ
ejpam-2562	282	4	to	to	PART
ejpam-2562	282	5	show	show	VERB
ejpam-2562	282	6	that	that	SCONJ
ejpam-2562	282	7	$	$	PRON
ejpam-2562	282	8	is	be	AUX
ejpam-2562	282	9	a	a	DET
ejpam-2562	282	10	non	non	ADJ
ejpam-2562	282	11	-	-	ADJ
ejpam-2562	282	12	empty	empty	ADJ
ejpam-2562	282	13	inductive	inductive	ADJ
ejpam-2562	282	14	poset	poset	NOUN
ejpam-2562	282	15	.	.	PUNCT
ejpam-2562	283	1	therefore	therefore	ADV
ejpam-2562	283	2	$	$	PRON
ejpam-2562	283	3	has	have	VERB
ejpam-2562	283	4	at	at	ADV
ejpam-2562	283	5	least	least	ADV
ejpam-2562	283	6	one	one	NUM
ejpam-2562	283	7	maximal	maximal	ADJ
ejpam-2562	283	8	element	element	NOUN
ejpam-2562	283	9	n	n	PRON
ejpam-2562	283	10	′.	′.	NOUN
ejpam-2562	283	11	and	and	CCONJ
ejpam-2562	283	12	n	n	PRON
ejpam-2562	283	13	′	′	NOUN
ejpam-2562	283	14	is	be	AUX
ejpam-2562	283	15	a	a	DET
ejpam-2562	283	16	m	m	NOUN
ejpam-2562	283	17	-ω	-ω	NOUN
ejpam-2562	283	18	-	-	NOUN
ejpam-2562	283	19	complement	complement	NOUN
ejpam-2562	283	20	of	of	ADP
ejpam-2562	283	21	n	n	PROPN
ejpam-2562	283	22	.	.	PUNCT
ejpam-2562	284	1	(	(	PUNCT
ejpam-2562	284	2	ii	ii	NOUN
ejpam-2562	284	3	)	)	PUNCT
ejpam-2562	284	4	•	•	NOUN
ejpam-2562	284	5	if	if	SCONJ
ejpam-2562	284	6	n	n	ADV
ejpam-2562	284	7	=	=	SYM
ejpam-2562	284	8	0	0	PUNCT
ejpam-2562	285	1	then	then	ADV
ejpam-2562	285	2	n	n	PRON
ejpam-2562	285	3	′	′	NOUN
ejpam-2562	286	1	=	=	VERB
ejpam-2562	286	2	m	m	NOUN
ejpam-2562	286	3	,	,	PUNCT
ejpam-2562	286	4	and	and	CCONJ
ejpam-2562	287	1	so	so	ADV
ejpam-2562	287	2	n	n	ADV
ejpam-2562	287	3	⊕	⊕	PROPN
ejpam-2562	287	4	n	n	CCONJ
ejpam-2562	287	5	′	′	NUM
ejpam-2562	287	6	ãswe	ãswe	NOUN
ejpam-2562	287	7	m	m	PROPN
ejpam-2562	287	8	.	.	PUNCT
ejpam-2562	288	1	•	•	INTJ
ejpam-2562	288	2	if	if	SCONJ
ejpam-2562	288	3	n	n	PROPN
ejpam-2562	288	4	6=	6=	NUM
ejpam-2562	288	5	0	0	NUM
ejpam-2562	288	6	,	,	PUNCT
ejpam-2562	288	7	then	then	ADV
ejpam-2562	288	8	let	let	VERB
ejpam-2562	288	9	0	0	NUM
ejpam-2562	288	10	6=	6=	ADP
ejpam-2562	288	11	l	l	NOUN
ejpam-2562	288	12	≤	≤	NUM
ejpam-2562	288	13	m	m	VERB
ejpam-2562	288	14	such	such	ADJ
ejpam-2562	288	15	that	that	SCONJ
ejpam-2562	288	16	the	the	DET
ejpam-2562	288	17	restriction	restriction	NOUN
ejpam-2562	288	18	of	of	ADP
ejpam-2562	288	19	≡l	≡l	NOUN
ejpam-2562	288	20	to	to	ADP
ejpam-2562	288	21	n	n	PROPN
ejpam-2562	288	22	⊕	⊕	PROPN
ejpam-2562	288	23	n	n	CCONJ
ejpam-2562	288	24	′	′	NOUN
ejpam-2562	288	25	is	be	AUX
ejpam-2562	288	26	trivial	trivial	ADJ
ejpam-2562	288	27	.	.	PUNCT
ejpam-2562	289	1	let	let	VERB
ejpam-2562	289	2	us	we	PRON
ejpam-2562	289	3	show	show	VERB
ejpam-2562	289	4	that	that	SCONJ
ejpam-2562	289	5	≡l	≡l	NOUN
ejpam-2562	289	6	is	be	AUX
ejpam-2562	289	7	trivial	trivial	ADJ
ejpam-2562	289	8	on	on	ADP
ejpam-2562	289	9	m	m	PROPN
ejpam-2562	289	10	.	.	PUNCT
ejpam-2562	290	1	first	first	ADV
ejpam-2562	290	2	of	of	ADP
ejpam-2562	290	3	all	all	PRON
ejpam-2562	290	4	we	we	PRON
ejpam-2562	290	5	show	show	VERB
ejpam-2562	290	6	that	that	SCONJ
ejpam-2562	290	7	the	the	DET
ejpam-2562	290	8	restriction	restriction	NOUN
ejpam-2562	290	9	of	of	ADP
ejpam-2562	290	10	≡n	≡n	PROPN
ejpam-2562	290	11	′+l	′+l	PROPN
ejpam-2562	290	12	to	to	ADP
ejpam-2562	290	13	n	n	PROPN
ejpam-2562	290	14	is	be	AUX
ejpam-2562	290	15	trivial	trivial	ADJ
ejpam-2562	290	16	.	.	PUNCT
ejpam-2562	291	1	let	let	VERB
ejpam-2562	291	2	n1	n1	NOUN
ejpam-2562	291	3	,	,	PUNCT
ejpam-2562	291	4	n2	n2	NOUN
ejpam-2562	291	5	∈	∈	PROPN
ejpam-2562	291	6	n	n	CCONJ
ejpam-2562	291	7	such	such	ADJ
ejpam-2562	291	8	that	that	SCONJ
ejpam-2562	291	9	n1	n1	PROPN
ejpam-2562	291	10	≡n	≡n	PROPN
ejpam-2562	291	11	′+l	′+l	PROPN
ejpam-2562	291	12	n2	n2	PROPN
ejpam-2562	291	13	.	.	PUNCT
ejpam-2562	292	1	e.	e.	PROPN
ejpam-2562	292	2	diop	diop	PROPN
ejpam-2562	292	3	,	,	PUNCT
ejpam-2562	292	4	d.	d.	PROPN
ejpam-2562	292	5	sow	sow	PROPN
ejpam-2562	292	6	/	/	SYM
ejpam-2562	292	7	eur	eur	PROPN
ejpam-2562	292	8	.	.	PUNCT
ejpam-2562	293	1	j.	j.	PROPN
ejpam-2562	293	2	pure	pure	PROPN
ejpam-2562	293	3	appl	appl	PROPN
ejpam-2562	293	4	.	.	PROPN
ejpam-2562	293	5	math	math	PROPN
ejpam-2562	293	6	,	,	PUNCT
ejpam-2562	293	7	9	9	NUM
ejpam-2562	293	8	(	(	PUNCT
ejpam-2562	293	9	2016	2016	NUM
ejpam-2562	293	10	)	)	PUNCT
ejpam-2562	293	11	,	,	PUNCT
ejpam-2562	293	12	250	250	NUM
ejpam-2562	293	13	-	-	SYM
ejpam-2562	293	14	265	265	NUM
ejpam-2562	293	15	259	259	NUM
ejpam-2562	293	16	n1	n1	PROPN
ejpam-2562	293	17	≡n	≡n	PROPN
ejpam-2562	293	18	′+l	′+l	PROPN
ejpam-2562	293	19	n2	n2	NOUN
ejpam-2562	293	20	=	=	NOUN
ejpam-2562	293	21	⇒	⇒	NOUN
ejpam-2562	294	1	∃n	∃n	INTJ
ejpam-2562	294	2	′	′	NUM
ejpam-2562	294	3	1	1	NUM
ejpam-2562	294	4	,	,	PUNCT
ejpam-2562	294	5	n′2	n′2	NOUN
ejpam-2562	294	6	∈	∈	PROPN
ejpam-2562	294	7	n	n	PROPN
ejpam-2562	294	8	,	,	PUNCT
ejpam-2562	294	9	l1	l1	PROPN
ejpam-2562	294	10	,	,	PUNCT
ejpam-2562	294	11	l2	l2	NOUN
ejpam-2562	294	12	∈	∈	PROPN
ejpam-2562	294	13	l	l	NOUN
ejpam-2562	294	14	such	such	ADJ
ejpam-2562	294	15	that	that	SCONJ
ejpam-2562	294	16	n1	n1	PROPN
ejpam-2562	294	17	+	+	CCONJ
ejpam-2562	294	18	n′1	n′1	NOUN
ejpam-2562	294	19	+	+	CCONJ
ejpam-2562	294	20	l1	l1	PROPN
ejpam-2562	294	21	=	=	PROPN
ejpam-2562	294	22	n2	n2	PROPN
ejpam-2562	294	23	+	+	CCONJ
ejpam-2562	294	24	n′2	n′2	ADJ
ejpam-2562	294	25	+	+	CCONJ
ejpam-2562	294	26	l2	l2	NOUN
ejpam-2562	294	27	.	.	PUNCT
ejpam-2562	295	1	so	so	ADV
ejpam-2562	295	2	by	by	ADP
ejpam-2562	295	3	hypothesis	hypothesis	NOUN
ejpam-2562	295	4	and	and	CCONJ
ejpam-2562	295	5	by	by	ADP
ejpam-2562	295	6	the	the	DET
ejpam-2562	295	7	direct	direct	ADJ
ejpam-2562	295	8	sum	sum	NOUN
ejpam-2562	295	9	n	n	PROPN
ejpam-2562	295	10	⊕	⊕	PROPN
ejpam-2562	295	11	n	n	ADP
ejpam-2562	295	12	′	′	NOUN
ejpam-2562	295	13	we	we	PRON
ejpam-2562	295	14	deduce	deduce	VERB
ejpam-2562	295	15	that	that	SCONJ
ejpam-2562	295	16	n1	n1	PROPN
ejpam-2562	295	17	=	=	SYM
ejpam-2562	295	18	n2	n2	PROPN
ejpam-2562	295	19	.	.	PUNCT
ejpam-2562	296	1	therefore	therefore	ADV
ejpam-2562	296	2	the	the	DET
ejpam-2562	296	3	restriction	restriction	NOUN
ejpam-2562	296	4	of	of	ADP
ejpam-2562	296	5	≡n	≡n	PROPN
ejpam-2562	296	6	′+l	′+l	PROPN
ejpam-2562	296	7	to	to	ADP
ejpam-2562	296	8	n	n	PROPN
ejpam-2562	296	9	is	be	AUX
ejpam-2562	296	10	trivial	trivial	ADJ
ejpam-2562	296	11	.	.	PUNCT
ejpam-2562	297	1	we	we	PRON
ejpam-2562	297	2	deduce	deduce	VERB
ejpam-2562	297	3	that	that	PRON
ejpam-2562	297	4	n	n	NUM
ejpam-2562	297	5	′	′	VERB
ejpam-2562	298	1	+	+	CCONJ
ejpam-2562	298	2	l	l	NOUN
ejpam-2562	298	3	∈	∈	ADJ
ejpam-2562	298	4	$	$	SYM
ejpam-2562	298	5	.	.	PUNCT
ejpam-2562	299	1	since	since	SCONJ
ejpam-2562	299	2	n	n	NUM
ejpam-2562	299	3	′	′	NUM
ejpam-2562	299	4	is	be	AUX
ejpam-2562	299	5	maximal	maximal	ADJ
ejpam-2562	299	6	,	,	PUNCT
ejpam-2562	299	7	then	then	ADV
ejpam-2562	299	8	n	n	PRON
ejpam-2562	299	9	′	′	NOUN
ejpam-2562	300	1	+	+	CCONJ
ejpam-2562	300	2	l	l	NOUN
ejpam-2562	300	3	=	=	SYM
ejpam-2562	300	4	n	n	CCONJ
ejpam-2562	300	5	′	′	NOUN
ejpam-2562	300	6	or	or	CCONJ
ejpam-2562	300	7	n	n	PRON
ejpam-2562	300	8	′	′	NUM
ejpam-2562	301	1	+	+	CCONJ
ejpam-2562	301	2	l	l	X
ejpam-2562	301	3	=	=	NOUN
ejpam-2562	301	4	m	m	VERB
ejpam-2562	301	5	.	.	PUNCT
ejpam-2562	302	1	we	we	PRON
ejpam-2562	302	2	have	have	VERB
ejpam-2562	302	3	n	n	NUM
ejpam-2562	302	4	′	′	NUM
ejpam-2562	303	1	+	+	CCONJ
ejpam-2562	303	2	l	l	X
ejpam-2562	303	3	6=	6=	NUM
ejpam-2562	303	4	m	m	PROPN
ejpam-2562	303	5	,	,	PUNCT
ejpam-2562	303	6	otherwise	otherwise	ADV
ejpam-2562	303	7	,	,	PUNCT
ejpam-2562	303	8	since	since	SCONJ
ejpam-2562	303	9	the	the	DET
ejpam-2562	303	10	restriction	restriction	NOUN
ejpam-2562	303	11	of	of	ADP
ejpam-2562	303	12	≡n	≡n	PROPN
ejpam-2562	303	13	′+l	′+l	PROPN
ejpam-2562	303	14	to	to	ADP
ejpam-2562	303	15	n	n	PROPN
ejpam-2562	303	16	is	be	AUX
ejpam-2562	303	17	trivial	trivial	ADJ
ejpam-2562	303	18	,	,	PUNCT
ejpam-2562	303	19	we	we	PRON
ejpam-2562	303	20	obtain	obtain	VERB
ejpam-2562	303	21	(	(	PUNCT
ejpam-2562	303	22	n	n	NOUN
ejpam-2562	303	23	′	′	NOUN
ejpam-2562	303	24	+	+	CCONJ
ejpam-2562	303	25	l	l	NOUN
ejpam-2562	303	26	)	)	PUNCT
ejpam-2562	303	27	∩	∩	NOUN
ejpam-2562	303	28	n	n	NOUN
ejpam-2562	303	29	=	=	SYM
ejpam-2562	303	30	m	m	NOUN
ejpam-2562	303	31	∩	∩	ADJ
ejpam-2562	303	32	n	n	NOUN
ejpam-2562	303	33	=	=	SYM
ejpam-2562	303	34	n	n	PROPN
ejpam-2562	303	35	=	=	SYM
ejpam-2562	303	36	0	0	NUM
ejpam-2562	303	37	which	which	PRON
ejpam-2562	303	38	is	be	AUX
ejpam-2562	303	39	a	a	DET
ejpam-2562	303	40	contradiction	contradiction	NOUN
ejpam-2562	303	41	.	.	PUNCT
ejpam-2562	304	1	so	so	ADV
ejpam-2562	304	2	n	n	ADV
ejpam-2562	304	3	′	′	VERB
ejpam-2562	305	1	+	+	CCONJ
ejpam-2562	305	2	l	l	NOUN
ejpam-2562	305	3	=	=	SYM
ejpam-2562	305	4	n	n	NOUN
ejpam-2562	305	5	′	′	NOUN
ejpam-2562	305	6	and	and	CCONJ
ejpam-2562	305	7	consequently	consequently	ADV
ejpam-2562	305	8	l	l	NOUN
ejpam-2562	305	9	⊆	⊆	NUM
ejpam-2562	305	10	n	n	PRON
ejpam-2562	305	11	′.	′.	NOUN
ejpam-2562	305	12	moreover	moreover	ADP
ejpam-2562	305	13	we	we	PRON
ejpam-2562	305	14	have	have	VERB
ejpam-2562	305	15	(	(	PUNCT
ejpam-2562	305	16	n⊕n	n⊕n	NOUN
ejpam-2562	305	17	′)∩l	′)∩l	NOUN
ejpam-2562	305	18	=	=	SYM
ejpam-2562	305	19	0	0	PUNCT
ejpam-2562	306	1	because	because	SCONJ
ejpam-2562	306	2	the	the	DET
ejpam-2562	306	3	restriction	restriction	NOUN
ejpam-2562	306	4	of≡l	of≡l	NOUN
ejpam-2562	306	5	to	to	ADP
ejpam-2562	306	6	n⊕n	n⊕n	NOUN
ejpam-2562	306	7	′	′	NUM
ejpam-2562	306	8	is	be	AUX
ejpam-2562	306	9	trivial	trivial	ADJ
ejpam-2562	306	10	;	;	PUNCT
ejpam-2562	306	11	so	so	ADV
ejpam-2562	306	12	n	n	PRON
ejpam-2562	306	13	′∩l	′∩l	PROPN
ejpam-2562	306	14	=	=	SYM
ejpam-2562	306	15	0	0	NUM
ejpam-2562	306	16	.	.	PUNCT
ejpam-2562	307	1	thus	thus	ADV
ejpam-2562	307	2	we	we	PRON
ejpam-2562	307	3	have	have	VERB
ejpam-2562	307	4	l	l	NOUN
ejpam-2562	307	5	⊆	⊆	NUM
ejpam-2562	307	6	n	n	SYM
ejpam-2562	307	7	′	′	NOUN
ejpam-2562	307	8	and	and	CCONJ
ejpam-2562	307	9	n	n	CCONJ
ejpam-2562	307	10	′	′	NUM
ejpam-2562	307	11	∩	∩	NOUN
ejpam-2562	307	12	l	l	NOUN
ejpam-2562	307	13	=	=	SYM
ejpam-2562	307	14	0	0	NUM
ejpam-2562	307	15	,	,	PUNCT
ejpam-2562	308	1	so	so	ADV
ejpam-2562	308	2	l	l	NOUN
ejpam-2562	309	1	=	=	NOUN
ejpam-2562	309	2	0	0	NUM
ejpam-2562	309	3	.	.	PUNCT
ejpam-2562	310	1	we	we	PRON
ejpam-2562	310	2	conclude	conclude	VERB
ejpam-2562	310	3	that	that	SCONJ
ejpam-2562	310	4	n	n	PROPN
ejpam-2562	310	5	⊕	⊕	PROPN
ejpam-2562	310	6	n	n	CCONJ
ejpam-2562	310	7	′	′	NUM
ejpam-2562	310	8	ãswe	ãswe	NOUN
ejpam-2562	310	9	m	m	VERB
ejpam-2562	310	10	definition	definition	NOUN
ejpam-2562	310	11	7	7	NUM
ejpam-2562	310	12	.	.	PUNCT
ejpam-2562	311	1	let	let	VERB
ejpam-2562	311	2	n	n	PRON
ejpam-2562	311	3	be	be	AUX
ejpam-2562	311	4	a	a	DET
ejpam-2562	311	5	subsemimodule	subsemimodule	NOUN
ejpam-2562	311	6	of	of	ADP
ejpam-2562	311	7	a	a	DET
ejpam-2562	311	8	left	left	ADJ
ejpam-2562	311	9	r	r	NOUN
ejpam-2562	311	10	-	-	PUNCT
ejpam-2562	311	11	semimodule	semimodule	NOUN
ejpam-2562	311	12	m.	m.	NOUN
ejpam-2562	311	13	a	a	DET
ejpam-2562	311	14	subsemimodule	subsemimodule	NOUN
ejpam-2562	311	15	n	n	NOUN
ejpam-2562	311	16	′	′	NOUN
ejpam-2562	311	17	of	of	ADP
ejpam-2562	311	18	m	m	PROPN
ejpam-2562	311	19	is	be	AUX
ejpam-2562	311	20	called	call	VERB
ejpam-2562	311	21	m	m	PROPN
ejpam-2562	311	22	-	-	PUNCT
ejpam-2562	311	23	s	s	NOUN
ejpam-2562	311	24	-	-	NOUN
ejpam-2562	311	25	complement	complement	NOUN
ejpam-2562	311	26	of	of	ADP
ejpam-2562	311	27	n	n	PRON
ejpam-2562	311	28	if	if	SCONJ
ejpam-2562	311	29	the	the	DET
ejpam-2562	311	30	restriction	restriction	NOUN
ejpam-2562	311	31	of	of	ADP
ejpam-2562	311	32	≡n	≡n	PROPN
ejpam-2562	311	33	to	to	ADP
ejpam-2562	311	34	n	n	PROPN
ejpam-2562	311	35	′	′	NUM
ejpam-2562	311	36	is	be	AUX
ejpam-2562	311	37	trivial	trivial	ADJ
ejpam-2562	311	38	,	,	PUNCT
ejpam-2562	311	39	the	the	DET
ejpam-2562	311	40	restriction	restriction	NOUN
ejpam-2562	311	41	of	of	ADP
ejpam-2562	311	42	≡n	≡n	NOUN
ejpam-2562	311	43	′	′	NOUN
ejpam-2562	311	44	to	to	ADP
ejpam-2562	311	45	n	n	PROPN
ejpam-2562	311	46	is	be	AUX
ejpam-2562	311	47	trivial	trivial	ADJ
ejpam-2562	311	48	and	and	CCONJ
ejpam-2562	311	49	n	n	PRON
ejpam-2562	311	50	′	′	NOUN
ejpam-2562	311	51	is	be	AUX
ejpam-2562	311	52	maximal	maximal	ADJ
ejpam-2562	311	53	with	with	ADP
ejpam-2562	311	54	this	this	DET
ejpam-2562	311	55	property	property	NOUN
ejpam-2562	311	56	.	.	PUNCT
ejpam-2562	312	1	proposition	proposition	NOUN
ejpam-2562	312	2	7	7	NUM
ejpam-2562	312	3	.	.	PUNCT
ejpam-2562	313	1	(	(	PUNCT
ejpam-2562	313	2	i	i	NOUN
ejpam-2562	313	3	)	)	PUNCT
ejpam-2562	313	4	every	every	DET
ejpam-2562	313	5	subsemimodule	subsemimodule	NOUN
ejpam-2562	313	6	n	n	PROPN
ejpam-2562	313	7	of	of	ADP
ejpam-2562	313	8	a	a	DET
ejpam-2562	313	9	left	left	ADJ
ejpam-2562	313	10	r	r	NOUN
ejpam-2562	313	11	-	-	PUNCT
ejpam-2562	313	12	semimodule	semimodule	NOUN
ejpam-2562	313	13	m	m	VERB
ejpam-2562	313	14	has	have	VERB
ejpam-2562	313	15	a	a	DET
ejpam-2562	313	16	m	m	NOUN
ejpam-2562	313	17	-	-	PUNCT
ejpam-2562	313	18	s	s	NOUN
ejpam-2562	313	19	-	-	NOUN
ejpam-2562	313	20	complement	complement	NOUN
ejpam-2562	313	21	.	.	PUNCT
ejpam-2562	314	1	(	(	PUNCT
ejpam-2562	314	2	ii	ii	NOUN
ejpam-2562	314	3	)	)	PUNCT
ejpam-2562	314	4	if	if	SCONJ
ejpam-2562	314	5	m	m	NOUN
ejpam-2562	314	6	is	be	AUX
ejpam-2562	314	7	a	a	DET
ejpam-2562	314	8	cancellative	cancellative	ADJ
ejpam-2562	314	9	left	leave	VERB
ejpam-2562	314	10	r	r	NOUN
ejpam-2562	314	11	-	-	PUNCT
ejpam-2562	314	12	semimodule	semimodule	NOUN
ejpam-2562	314	13	and	and	CCONJ
ejpam-2562	314	14	n	n	DET
ejpam-2562	314	15	a	a	DET
ejpam-2562	314	16	subsemimodule	subsemimodule	NOUN
ejpam-2562	314	17	of	of	ADP
ejpam-2562	314	18	m	m	PROPN
ejpam-2562	314	19	,	,	PUNCT
ejpam-2562	314	20	then	then	ADV
ejpam-2562	314	21	:	:	PUNCT
ejpam-2562	314	22	(	(	PUNCT
ejpam-2562	314	23	a	a	X
ejpam-2562	314	24	)	)	PUNCT
ejpam-2562	314	25	every	every	DET
ejpam-2562	314	26	m	m	PROPN
ejpam-2562	314	27	-	-	PUNCT
ejpam-2562	314	28	w	w	NOUN
ejpam-2562	314	29	-	-	NOUN
ejpam-2562	314	30	complement	complement	NOUN
ejpam-2562	314	31	of	of	ADP
ejpam-2562	314	32	n	n	PROPN
ejpam-2562	314	33	is	be	AUX
ejpam-2562	314	34	a	a	DET
ejpam-2562	314	35	m	m	PROPN
ejpam-2562	314	36	-	-	PUNCT
ejpam-2562	314	37	s	s	NOUN
ejpam-2562	314	38	-	-	NOUN
ejpam-2562	314	39	complement	complement	NOUN
ejpam-2562	314	40	of	of	ADP
ejpam-2562	314	41	n	n	PROPN
ejpam-2562	314	42	(	(	PUNCT
ejpam-2562	314	43	b	b	NOUN
ejpam-2562	314	44	)	)	PUNCT
ejpam-2562	314	45	if	if	SCONJ
ejpam-2562	314	46	n	n	NOUN
ejpam-2562	314	47	′	′	NOUN
ejpam-2562	314	48	is	be	AUX
ejpam-2562	314	49	a	a	DET
ejpam-2562	314	50	m	m	PROPN
ejpam-2562	314	51	-	-	PUNCT
ejpam-2562	314	52	s	s	NOUN
ejpam-2562	314	53	-	-	NOUN
ejpam-2562	314	54	complement	complement	NOUN
ejpam-2562	314	55	of	of	ADP
ejpam-2562	314	56	n	n	PRON
ejpam-2562	314	57	then	then	ADV
ejpam-2562	314	58	:	:	PUNCT
ejpam-2562	314	59	n	n	PROPN
ejpam-2562	314	60	⊕	⊕	PROPN
ejpam-2562	314	61	n	n	CCONJ
ejpam-2562	314	62	′	′	NUM
ejpam-2562	314	63	ãswe	ãswe	NOUN
ejpam-2562	314	64	m.	m.	NOUN
ejpam-2562	314	65	proof	proof	NOUN
ejpam-2562	314	66	.	.	PUNCT
ejpam-2562	315	1	(	(	PUNCT
ejpam-2562	315	2	i	i	NOUN
ejpam-2562	315	3	)	)	PUNCT
ejpam-2562	315	4	.	.	PUNCT
ejpam-2562	316	1	similar	similar	ADJ
ejpam-2562	316	2	to	to	ADP
ejpam-2562	316	3	the	the	DET
ejpam-2562	316	4	above	above	ADJ
ejpam-2562	316	5	proof	proof	NOUN
ejpam-2562	316	6	.	.	PUNCT
ejpam-2562	317	1	for	for	ADP
ejpam-2562	317	2	(	(	PUNCT
ejpam-2562	317	3	2)(a	2)(a	NUM
ejpam-2562	317	4	)	)	PUNCT
ejpam-2562	317	5	it	it	PRON
ejpam-2562	317	6	suffices	suffice	VERB
ejpam-2562	317	7	to	to	PART
ejpam-2562	317	8	see	see	VERB
ejpam-2562	317	9	that	that	SCONJ
ejpam-2562	317	10	,	,	PUNCT
ejpam-2562	317	11	if	if	SCONJ
ejpam-2562	317	12	m	m	NOUN
ejpam-2562	317	13	is	be	AUX
ejpam-2562	317	14	cancellative	cancellative	ADJ
ejpam-2562	317	15	and	and	CCONJ
ejpam-2562	317	16	if	if	SCONJ
ejpam-2562	317	17	the	the	DET
ejpam-2562	317	18	restriction	restriction	NOUN
ejpam-2562	317	19	of	of	ADP
ejpam-2562	317	20	≡n	≡n	PROPN
ejpam-2562	317	21	′	′	NOUN
ejpam-2562	317	22	to	to	ADP
ejpam-2562	317	23	n	n	PROPN
ejpam-2562	317	24	is	be	AUX
ejpam-2562	317	25	trivial	trivial	ADJ
ejpam-2562	317	26	,	,	PUNCT
ejpam-2562	317	27	then	then	ADV
ejpam-2562	317	28	the	the	DET
ejpam-2562	317	29	restriction	restriction	NOUN
ejpam-2562	317	30	of	of	ADP
ejpam-2562	317	31	≡n	≡n	NOUN
ejpam-2562	317	32	to	to	ADP
ejpam-2562	317	33	n	n	PROPN
ejpam-2562	317	34	′	′	NUM
ejpam-2562	317	35	is	be	AUX
ejpam-2562	317	36	trivial	trivial	ADJ
ejpam-2562	317	37	.	.	PUNCT
ejpam-2562	318	1	(	(	PUNCT
ejpam-2562	318	2	2)(b	2)(b	NUM
ejpam-2562	318	3	)	)	PUNCT
ejpam-2562	318	4	.	.	PUNCT
ejpam-2562	319	1	similar	similar	ADJ
ejpam-2562	319	2	to	to	ADP
ejpam-2562	319	3	the	the	DET
ejpam-2562	319	4	above	above	ADJ
ejpam-2562	319	5	proof	proof	NOUN
ejpam-2562	319	6	by	by	ADP
ejpam-2562	319	7	using	use	VERB
ejpam-2562	319	8	the	the	DET
ejpam-2562	319	9	fact	fact	NOUN
ejpam-2562	319	10	that	that	SCONJ
ejpam-2562	319	11	m	m	NOUN
ejpam-2562	319	12	is	be	AUX
ejpam-2562	319	13	cancellative	cancellative	ADJ
ejpam-2562	319	14	.	.	PUNCT
ejpam-2562	320	1	4	4	X
ejpam-2562	320	2	.	.	X
ejpam-2562	320	3	applications	application	NOUN
ejpam-2562	320	4	:	:	PUNCT
ejpam-2562	320	5	on	on	ADP
ejpam-2562	320	6	weakly	weakly	ADJ
ejpam-2562	320	7	co	co	ADJ
ejpam-2562	320	8	-	-	ADJ
ejpam-2562	320	9	hopfian	hopfian	ADJ
ejpam-2562	320	10	semimodules	semimodule	NOUN
ejpam-2562	320	11	we	we	PRON
ejpam-2562	320	12	have	have	VERB
ejpam-2562	320	13	three	three	NUM
ejpam-2562	320	14	differents	different	NOUN
ejpam-2562	320	15	notions	notion	NOUN
ejpam-2562	320	16	of	of	ADP
ejpam-2562	320	17	essential	essential	ADJ
ejpam-2562	320	18	subsemimodules	subsemimodule	NOUN
ejpam-2562	320	19	,	,	PUNCT
ejpam-2562	320	20	so	so	SCONJ
ejpam-2562	320	21	we	we	PRON
ejpam-2562	320	22	can	can	AUX
ejpam-2562	320	23	define	define	VERB
ejpam-2562	320	24	three	three	NUM
ejpam-2562	320	25	differents	different	NOUN
ejpam-2562	320	26	types	type	NOUN
ejpam-2562	320	27	of	of	ADP
ejpam-2562	320	28	weakly	weakly	ADJ
ejpam-2562	320	29	co	co	ADJ
ejpam-2562	320	30	-	-	ADJ
ejpam-2562	320	31	hopfian	hopfian	ADJ
ejpam-2562	320	32	semimodules	semimodule	VERB
ejpam-2562	320	33	4.1	4.1	NUM
ejpam-2562	320	34	.	.	PUNCT
ejpam-2562	321	1	on	on	ADP
ejpam-2562	321	2	weakly	weakly	ADJ
ejpam-2562	321	3	co	co	ADJ
ejpam-2562	321	4	-	-	ADJ
ejpam-2562	321	5	hopfian	hopfian	ADJ
ejpam-2562	321	6	semimodules	semimodule	NOUN
ejpam-2562	321	7	of	of	ADP
ejpam-2562	321	8	type	type	NOUN
ejpam-2562	321	9	1	1	NUM
ejpam-2562	321	10	definition	definition	NOUN
ejpam-2562	321	11	8	8	NUM
ejpam-2562	321	12	.	.	PUNCT
ejpam-2562	322	1	a	a	DET
ejpam-2562	322	2	nonzero	nonzero	NOUN
ejpam-2562	322	3	left	leave	VERB
ejpam-2562	322	4	r	r	NOUN
ejpam-2562	322	5	-	-	PUNCT
ejpam-2562	322	6	semimodule	semimodule	NOUN
ejpam-2562	322	7	rm	rm	NOUN
ejpam-2562	322	8	is	be	AUX
ejpam-2562	322	9	said	say	VERB
ejpam-2562	322	10	to	to	PART
ejpam-2562	322	11	be	be	AUX
ejpam-2562	322	12	weakly	weakly	ADJ
ejpam-2562	322	13	co	co	ADJ
ejpam-2562	322	14	-	-	VERB
ejpam-2562	322	15	hopfian-1	hopfian-1	NUM
ejpam-2562	322	16	(	(	PUNCT
ejpam-2562	322	17	denoted	denote	VERB
ejpam-2562	322	18	by	by	ADP
ejpam-2562	322	19	wch-1	wch-1	X
ejpam-2562	322	20	)	)	PUNCT
ejpam-2562	322	21	if	if	SCONJ
ejpam-2562	322	22	every	every	DET
ejpam-2562	322	23	monomorphism	monomorphism	NOUN
ejpam-2562	322	24	f	f	X
ejpam-2562	322	25	:	:	PUNCT
ejpam-2562	322	26	m	m	AUX
ejpam-2562	322	27	→	→	NOUN
ejpam-2562	322	28	m	m	NOUN
ejpam-2562	322	29	is	be	AUX
ejpam-2562	322	30	semi	semi	ADJ
ejpam-2562	322	31	-	-	ADJ
ejpam-2562	322	32	weakly	weakly	ADV
ejpam-2562	322	33	-	-	PUNCT
ejpam-2562	322	34	essential	essential	ADJ
ejpam-2562	322	35	i.e	i.e	PROPN
ejpam-2562	322	36	f	f	X
ejpam-2562	322	37	(	(	PUNCT
ejpam-2562	322	38	m)ãswe	m)ãswe	NOUN
ejpam-2562	322	39	m.	m.	NOUN
ejpam-2562	322	40	proposition	proposition	NOUN
ejpam-2562	322	41	8	8	NUM
ejpam-2562	322	42	.	.	PUNCT
ejpam-2562	323	1	the	the	DET
ejpam-2562	323	2	following	follow	VERB
ejpam-2562	323	3	are	be	AUX
ejpam-2562	323	4	equivalent	equivalent	ADJ
ejpam-2562	323	5	conditions	condition	NOUN
ejpam-2562	323	6	on	on	ADP
ejpam-2562	323	7	a	a	DET
ejpam-2562	323	8	left	left	ADJ
ejpam-2562	323	9	r	r	NOUN
ejpam-2562	323	10	-	-	PUNCT
ejpam-2562	323	11	semimodule	semimodule	NOUN
ejpam-2562	323	12	m.	m.	NOUN
ejpam-2562	323	13	(	(	PUNCT
ejpam-2562	323	14	i	i	NOUN
ejpam-2562	323	15	)	)	PUNCT
ejpam-2562	323	16	m	m	VERB
ejpam-2562	323	17	is	be	AUX
ejpam-2562	323	18	weakly	weakly	ADJ
ejpam-2562	323	19	co	co	NOUN
ejpam-2562	323	20	-	-	NOUN
ejpam-2562	323	21	hopfian-1	hopfian-1	NOUN
ejpam-2562	323	22	.	.	PUNCT
ejpam-2562	323	23	(	(	PUNCT
ejpam-2562	323	24	ii	ii	NOUN
ejpam-2562	323	25	)	)	PUNCT
ejpam-2562	323	26	∀{0	∀{0	PROPN
ejpam-2562	323	27	}	}	PUNCT
ejpam-2562	323	28	6=	6=	ADP
ejpam-2562	323	29	n	n	ADP
ejpam-2562	323	30	≤	≤	NOUN
ejpam-2562	323	31	m	m	ADP
ejpam-2562	323	32	,	,	PUNCT
ejpam-2562	323	33	if	if	SCONJ
ejpam-2562	323	34	g	g	PROPN
ejpam-2562	323	35	∈	∈	PROPN
ejpam-2562	323	36	end(m	end(m	PROPN
ejpam-2562	323	37	)	)	PUNCT
ejpam-2562	323	38	is	be	AUX
ejpam-2562	323	39	injective	injective	ADJ
ejpam-2562	323	40	,	,	PUNCT
ejpam-2562	323	41	then	then	ADV
ejpam-2562	323	42	the	the	DET
ejpam-2562	323	43	restriction	restriction	NOUN
ejpam-2562	323	44	of≡n	of≡n	NUM
ejpam-2562	323	45	to	to	ADP
ejpam-2562	323	46	g(m	g(m	NUM
ejpam-2562	323	47	)	)	PUNCT
ejpam-2562	323	48	is	be	AUX
ejpam-2562	323	49	not	not	PART
ejpam-2562	323	50	trivial	trivial	ADJ
ejpam-2562	323	51	.	.	PUNCT
ejpam-2562	324	1	e.	e.	PROPN
ejpam-2562	324	2	diop	diop	PROPN
ejpam-2562	324	3	,	,	PUNCT
ejpam-2562	324	4	d.	d.	PROPN
ejpam-2562	324	5	sow	sow	PROPN
ejpam-2562	324	6	/	/	SYM
ejpam-2562	324	7	eur	eur	PROPN
ejpam-2562	324	8	.	.	PUNCT
ejpam-2562	325	1	j.	j.	PROPN
ejpam-2562	325	2	pure	pure	PROPN
ejpam-2562	325	3	appl	appl	PROPN
ejpam-2562	325	4	.	.	PROPN
ejpam-2562	325	5	math	math	PROPN
ejpam-2562	325	6	,	,	PUNCT
ejpam-2562	325	7	9	9	NUM
ejpam-2562	325	8	(	(	PUNCT
ejpam-2562	325	9	2016	2016	NUM
ejpam-2562	325	10	)	)	PUNCT
ejpam-2562	325	11	,	,	PUNCT
ejpam-2562	325	12	250	250	NUM
ejpam-2562	325	13	-	-	SYM
ejpam-2562	325	14	265	265	NUM
ejpam-2562	325	15	260	260	NUM
ejpam-2562	325	16	proof	proof	NOUN
ejpam-2562	325	17	.	.	PUNCT
ejpam-2562	326	1	(	(	PUNCT
ejpam-2562	326	2	i)⇒(ii	i)⇒(ii	ADV
ejpam-2562	326	3	)	)	PUNCT
ejpam-2562	326	4	let	let	VERB
ejpam-2562	326	5	{	{	PUNCT
ejpam-2562	326	6	0	0	NUM
ejpam-2562	326	7	}	}	PUNCT
ejpam-2562	326	8	6=	6=	NUM
ejpam-2562	327	1	n	n	DET
ejpam-2562	327	2	≤	≤	NOUN
ejpam-2562	327	3	m	m	NOUN
ejpam-2562	328	1	and	and	CCONJ
ejpam-2562	328	2	let	let	VERB
ejpam-2562	328	3	g	g	PRON
ejpam-2562	328	4	be	be	AUX
ejpam-2562	328	5	an	an	DET
ejpam-2562	328	6	injective	injective	ADJ
ejpam-2562	328	7	endomorphism	endomorphism	NOUN
ejpam-2562	328	8	of	of	ADP
ejpam-2562	328	9	m	m	PRON
ejpam-2562	328	10	.	.	PUNCT
ejpam-2562	329	1	by	by	ADP
ejpam-2562	329	2	assumption	assumption	NOUN
ejpam-2562	329	3	g(m	g(m	PROPN
ejpam-2562	329	4	)	)	PUNCT
ejpam-2562	329	5	is	be	AUX
ejpam-2562	329	6	semi	semi	ADJ
ejpam-2562	329	7	-	-	ADJ
ejpam-2562	329	8	weakly	weakly	ADV
ejpam-2562	329	9	-	-	PUNCT
ejpam-2562	329	10	essential	essential	ADJ
ejpam-2562	329	11	in	in	ADP
ejpam-2562	329	12	m	m	PROPN
ejpam-2562	329	13	,	,	PUNCT
ejpam-2562	329	14	so	so	ADV
ejpam-2562	329	15	,	,	PUNCT
ejpam-2562	329	16	since	since	SCONJ
ejpam-2562	329	17	n	n	ADV
ejpam-2562	329	18	≤	≤	NOUN
ejpam-2562	329	19	m	m	PROPN
ejpam-2562	329	20	and	and	CCONJ
ejpam-2562	329	21	n	n	CCONJ
ejpam-2562	329	22	6=	6=	X
ejpam-2562	329	23	{	{	PUNCT
ejpam-2562	329	24	0	0	NUM
ejpam-2562	329	25	}	}	PUNCT
ejpam-2562	329	26	,	,	PUNCT
ejpam-2562	329	27	we	we	PRON
ejpam-2562	329	28	deduce	deduce	VERB
ejpam-2562	329	29	that	that	SCONJ
ejpam-2562	329	30	the	the	DET
ejpam-2562	329	31	restriction	restriction	NOUN
ejpam-2562	329	32	of	of	ADP
ejpam-2562	329	33	≡n	≡n	PROPN
ejpam-2562	329	34	to	to	ADP
ejpam-2562	329	35	g(m	g(m	NUM
ejpam-2562	329	36	)	)	PUNCT
ejpam-2562	329	37	is	be	AUX
ejpam-2562	329	38	not	not	PART
ejpam-2562	329	39	trivial	trivial	ADJ
ejpam-2562	329	40	.	.	PUNCT
ejpam-2562	330	1	(	(	PUNCT
ejpam-2562	330	2	ii)⇒(i	ii)⇒(i	NOUN
ejpam-2562	330	3	)	)	PUNCT
ejpam-2562	330	4	let	let	VERB
ejpam-2562	330	5	g	g	NOUN
ejpam-2562	330	6	:	:	PUNCT
ejpam-2562	330	7	m	m	VERB
ejpam-2562	330	8	→	→	NOUN
ejpam-2562	330	9	m	m	AUX
ejpam-2562	330	10	be	be	VERB
ejpam-2562	330	11	an	an	DET
ejpam-2562	330	12	injective	injective	ADJ
ejpam-2562	330	13	endomorphism	endomorphism	NOUN
ejpam-2562	330	14	of	of	ADP
ejpam-2562	330	15	m	m	PRON
ejpam-2562	330	16	and	and	CCONJ
ejpam-2562	330	17	let	let	VERB
ejpam-2562	330	18	n	n	PRON
ejpam-2562	330	19	be	be	AUX
ejpam-2562	330	20	a	a	DET
ejpam-2562	330	21	subsemimodule	subsemimodule	NOUN
ejpam-2562	330	22	of	of	ADP
ejpam-2562	330	23	m	m	NOUN
ejpam-2562	330	24	such	such	ADJ
ejpam-2562	330	25	that	that	SCONJ
ejpam-2562	330	26	the	the	DET
ejpam-2562	330	27	restriction	restriction	NOUN
ejpam-2562	330	28	of≡n	of≡n	NUM
ejpam-2562	330	29	to	to	ADP
ejpam-2562	330	30	g(m	g(m	NUM
ejpam-2562	330	31	)	)	PUNCT
ejpam-2562	330	32	is	be	AUX
ejpam-2562	330	33	trivial	trivial	ADJ
ejpam-2562	330	34	.	.	PUNCT
ejpam-2562	331	1	suppose	suppose	VERB
ejpam-2562	331	2	that	that	SCONJ
ejpam-2562	331	3	n	n	PROPN
ejpam-2562	331	4	6=	6=	NUM
ejpam-2562	331	5	{	{	PUNCT
ejpam-2562	331	6	0	0	NUM
ejpam-2562	331	7	}	}	PUNCT
ejpam-2562	331	8	.	.	PUNCT
ejpam-2562	332	1	so	so	ADV
ejpam-2562	332	2	by	by	ADP
ejpam-2562	332	3	hypothesis	hypothesis	NOUN
ejpam-2562	332	4	the	the	DET
ejpam-2562	332	5	restriction	restriction	NOUN
ejpam-2562	332	6	of	of	ADP
ejpam-2562	332	7	≡n	≡n	PROPN
ejpam-2562	332	8	to	to	ADP
ejpam-2562	332	9	g(m	g(m	NUM
ejpam-2562	332	10	)	)	PUNCT
ejpam-2562	332	11	is	be	AUX
ejpam-2562	332	12	not	not	PART
ejpam-2562	332	13	trivial	trivial	ADJ
ejpam-2562	332	14	which	which	PRON
ejpam-2562	332	15	contradicts	contradict	VERB
ejpam-2562	332	16	the	the	DET
ejpam-2562	332	17	assumption	assumption	NOUN
ejpam-2562	332	18	.	.	PUNCT
ejpam-2562	333	1	so	so	ADV
ejpam-2562	333	2	n	n	ADV
ejpam-2562	333	3	=	=	SYM
ejpam-2562	333	4	{	{	PUNCT
ejpam-2562	333	5	0	0	NUM
ejpam-2562	333	6	}	}	PUNCT
ejpam-2562	333	7	whence	whence	NOUN
ejpam-2562	333	8	g(m	g(m	VERB
ejpam-2562	333	9	)	)	PUNCT
ejpam-2562	333	10	is	be	AUX
ejpam-2562	333	11	semi	semi	ADJ
ejpam-2562	333	12	-	-	ADJ
ejpam-2562	333	13	weakly	weakly	ADJ
ejpam-2562	333	14	-	-	PUNCT
ejpam-2562	333	15	essential	essential	ADJ
ejpam-2562	333	16	and	and	CCONJ
ejpam-2562	333	17	so	so	ADV
ejpam-2562	333	18	m	m	VERB
ejpam-2562	333	19	is	be	AUX
ejpam-2562	333	20	weakly	weakly	ADJ
ejpam-2562	333	21	co	co	NOUN
ejpam-2562	333	22	-	-	ADJ
ejpam-2562	333	23	hopfian-1	hopfian-1	ADJ
ejpam-2562	333	24	.	.	PUNCT
ejpam-2562	334	1	proposition	proposition	NOUN
ejpam-2562	334	2	9	9	NUM
ejpam-2562	334	3	.	.	PUNCT
ejpam-2562	334	4	for	for	ADP
ejpam-2562	334	5	a	a	DET
ejpam-2562	334	6	left	left	ADJ
ejpam-2562	334	7	r	r	NOUN
ejpam-2562	334	8	-	-	PUNCT
ejpam-2562	334	9	semimodule	semimodule	NOUN
ejpam-2562	334	10	m	m	PROPN
ejpam-2562	334	11	,	,	PUNCT
ejpam-2562	334	12	consider	consider	VERB
ejpam-2562	334	13	the	the	DET
ejpam-2562	334	14	following	follow	VERB
ejpam-2562	334	15	statements	statement	NOUN
ejpam-2562	334	16	.	.	PUNCT
ejpam-2562	335	1	(	(	PUNCT
ejpam-2562	335	2	i	i	NOUN
ejpam-2562	335	3	)	)	PUNCT
ejpam-2562	335	4	m	m	VERB
ejpam-2562	335	5	is	be	AUX
ejpam-2562	335	6	weakly	weakly	ADJ
ejpam-2562	335	7	co	co	NOUN
ejpam-2562	335	8	-	-	NOUN
ejpam-2562	335	9	hopfian-1	hopfian-1	NOUN
ejpam-2562	335	10	.	.	PUNCT
ejpam-2562	336	1	(	(	PUNCT
ejpam-2562	336	2	ii	ii	NOUN
ejpam-2562	336	3	)	)	PUNCT
ejpam-2562	336	4	for	for	ADP
ejpam-2562	336	5	any	any	DET
ejpam-2562	336	6	left	left	ADJ
ejpam-2562	336	7	r	r	NOUN
ejpam-2562	336	8	-	-	PUNCT
ejpam-2562	336	9	semimodule	semimodule	NOUN
ejpam-2562	336	10	n	n	CCONJ
ejpam-2562	336	11	,	,	PUNCT
ejpam-2562	336	12	if	if	SCONJ
ejpam-2562	336	13	there	there	PRON
ejpam-2562	336	14	is	be	VERB
ejpam-2562	336	15	an	an	DET
ejpam-2562	336	16	r	r	NOUN
ejpam-2562	336	17	-	-	PUNCT
ejpam-2562	336	18	monomorphism	monomorphism	NOUN
ejpam-2562	336	19	˙m	˙m	VERB
ejpam-2562	336	20	⊕	⊕	PROPN
ejpam-2562	336	21	n	n	PROPN
ejpam-2562	336	22	→	→	SYM
ejpam-2562	336	23	m	m	VERB
ejpam-2562	336	24	then	then	ADV
ejpam-2562	336	25	n	n	NOUN
ejpam-2562	336	26	=	=	PUNCT
ejpam-2562	336	27	{	{	PUNCT
ejpam-2562	336	28	0	0	NUM
ejpam-2562	336	29	}	}	PUNCT
ejpam-2562	336	30	.	.	PUNCT
ejpam-2562	337	1	then	then	ADV
ejpam-2562	337	2	(	(	PUNCT
ejpam-2562	337	3	i	i	NOUN
ejpam-2562	337	4	)	)	PUNCT
ejpam-2562	337	5	=	=	NOUN
ejpam-2562	337	6	⇒	⇒	NOUN
ejpam-2562	337	7	(	(	PUNCT
ejpam-2562	337	8	ii	ii	NOUN
ejpam-2562	337	9	)	)	PUNCT
ejpam-2562	337	10	and	and	CCONJ
ejpam-2562	337	11	if	if	SCONJ
ejpam-2562	337	12	m	m	NOUN
ejpam-2562	337	13	is	be	AUX
ejpam-2562	337	14	cancellative	cancellative	ADJ
ejpam-2562	337	15	we	we	PRON
ejpam-2562	337	16	have	have	VERB
ejpam-2562	337	17	(	(	PUNCT
ejpam-2562	337	18	i)	i)	NOUN
ejpam-2562	337	19	⇐	⇐	ADJ
ejpam-2562	337	20	⇒	⇒	PROPN
ejpam-2562	337	21	(	(	PUNCT
ejpam-2562	337	22	ii	ii	NOUN
ejpam-2562	337	23	)	)	PUNCT
ejpam-2562	337	24	.	.	PUNCT
ejpam-2562	338	1	nb	nb	INTJ
ejpam-2562	338	2	:	:	PUNCT
ejpam-2562	338	3	a	a	DET
ejpam-2562	338	4	module	module	NOUN
ejpam-2562	338	5	m	m	VERB
ejpam-2562	338	6	which	which	PRON
ejpam-2562	338	7	verifies	verifie	NOUN
ejpam-2562	338	8	(	(	PUNCT
ejpam-2562	338	9	ii	ii	NOUN
ejpam-2562	338	10	)	)	PUNCT
ejpam-2562	338	11	is	be	AUX
ejpam-2562	338	12	said	say	VERB
ejpam-2562	338	13	dedekind	dedekind	PROPN
ejpam-2562	338	14	finite	finite	PROPN
ejpam-2562	338	15	.	.	PUNCT
ejpam-2562	339	1	a	a	DET
ejpam-2562	339	2	semimodule	semimodule	NOUN
ejpam-2562	339	3	which	which	PRON
ejpam-2562	339	4	is	be	AUX
ejpam-2562	339	5	finite	finite	ADJ
ejpam-2562	339	6	verifies	verifie	NOUN
ejpam-2562	339	7	(	(	PUNCT
ejpam-2562	339	8	ii	ii	NOUN
ejpam-2562	339	9	)	)	PUNCT
ejpam-2562	339	10	.	.	PUNCT
ejpam-2562	340	1	in	in	ADP
ejpam-2562	340	2	the	the	DET
ejpam-2562	340	3	sequel	sequel	NOUN
ejpam-2562	340	4	,	,	PUNCT
ejpam-2562	340	5	a	a	DET
ejpam-2562	340	6	semimodule	semimodule	NOUN
ejpam-2562	340	7	m	m	VERB
ejpam-2562	340	8	which	which	PRON
ejpam-2562	340	9	verifies	verifie	NOUN
ejpam-2562	340	10	(	(	PUNCT
ejpam-2562	340	11	ii	ii	NOUN
ejpam-2562	340	12	)	)	PUNCT
ejpam-2562	340	13	is	be	AUX
ejpam-2562	340	14	said	say	VERB
ejpam-2562	340	15	a	a	DET
ejpam-2562	340	16	f	f	PROPN
ejpam-2562	340	17	-semimodule	-semimodule	PROPN
ejpam-2562	340	18	.	.	PUNCT
ejpam-2562	341	1	proof	proof	NOUN
ejpam-2562	341	2	.	.	PUNCT
ejpam-2562	342	1	(	(	PUNCT
ejpam-2562	342	2	i)⇒(ii	i)⇒(ii	ADV
ejpam-2562	342	3	)	)	PUNCT
ejpam-2562	342	4	suppose	suppose	VERB
ejpam-2562	342	5	that	that	SCONJ
ejpam-2562	342	6	f	f	PROPN
ejpam-2562	342	7	:	:	PUNCT
ejpam-2562	342	8	˙m	˙m	PROPN
ejpam-2562	342	9	⊕	⊕	PROPN
ejpam-2562	342	10	n	n	PROPN
ejpam-2562	342	11	→	→	SYM
ejpam-2562	342	12	m	m	NOUN
ejpam-2562	342	13	is	be	AUX
ejpam-2562	342	14	a	a	DET
ejpam-2562	342	15	monomorphism	monomorphism	NOUN
ejpam-2562	342	16	where	where	SCONJ
ejpam-2562	342	17	n	n	PRON
ejpam-2562	342	18	is	be	AUX
ejpam-2562	342	19	a	a	DET
ejpam-2562	342	20	left	left	ADJ
ejpam-2562	342	21	rsemimodule	rsemimodule	NOUN
ejpam-2562	342	22	.	.	PUNCT
ejpam-2562	343	1	let	let	VERB
ejpam-2562	343	2	m	m	PRON
ejpam-2562	343	3	i	i	PRON
ejpam-2562	343	4	−→	−→	ADV
ejpam-2562	343	5	˙m	˙m	VERB
ejpam-2562	343	6	⊕	⊕	PROPN
ejpam-2562	343	7	n	n	CCONJ
ejpam-2562	343	8	f	f	PROPN
ejpam-2562	343	9	−→	−→	NOUN
ejpam-2562	343	10	m	m	AUX
ejpam-2562	343	11	be	be	AUX
ejpam-2562	343	12	an	an	DET
ejpam-2562	343	13	sequence	sequence	NOUN
ejpam-2562	343	14	where	where	SCONJ
ejpam-2562	343	15	i	i	PRON
ejpam-2562	343	16	is	be	AUX
ejpam-2562	343	17	the	the	DET
ejpam-2562	343	18	canonical	canonical	ADJ
ejpam-2562	343	19	injection	injection	NOUN
ejpam-2562	343	20	.	.	PUNCT
ejpam-2562	344	1	then	then	ADV
ejpam-2562	344	2	f	f	X
ejpam-2562	344	3	◦	◦	NOUN
ejpam-2562	344	4	i	i	PRON
ejpam-2562	344	5	is	be	AUX
ejpam-2562	344	6	a	a	DET
ejpam-2562	344	7	monomorphism	monomorphism	NOUN
ejpam-2562	344	8	,	,	PUNCT
ejpam-2562	344	9	hence	hence	ADV
ejpam-2562	344	10	(	(	PUNCT
ejpam-2562	344	11	f	f	X
ejpam-2562	344	12	◦	◦	NOUN
ejpam-2562	344	13	i)(m	i)(m	NOUN
ejpam-2562	344	14	)	)	PUNCT
ejpam-2562	344	15	is	be	AUX
ejpam-2562	344	16	semi	semi	ADJ
ejpam-2562	344	17	-	-	ADJ
ejpam-2562	344	18	weakly	weakly	ADV
ejpam-2562	344	19	-	-	PUNCT
ejpam-2562	344	20	essential	essential	ADJ
ejpam-2562	344	21	in	in	ADP
ejpam-2562	344	22	m	m	NOUN
ejpam-2562	344	23	by	by	ADP
ejpam-2562	344	24	assumption	assumption	NOUN
ejpam-2562	344	25	.	.	PUNCT
ejpam-2562	345	1	in	in	ADP
ejpam-2562	345	2	an	an	DET
ejpam-2562	345	3	easy	easy	ADJ
ejpam-2562	345	4	way	way	NOUN
ejpam-2562	345	5	one	one	PRON
ejpam-2562	345	6	can	can	AUX
ejpam-2562	345	7	show	show	VERB
ejpam-2562	345	8	that	that	SCONJ
ejpam-2562	345	9	f	f	PROPN
ejpam-2562	345	10	(	(	PUNCT
ejpam-2562	345	11	n	n	CCONJ
ejpam-2562	345	12	)	)	PUNCT
ejpam-2562	345	13	=	=	SYM
ejpam-2562	345	14	f	f	PROPN
ejpam-2562	345	15	(	(	PUNCT
ejpam-2562	345	16	˙0⊕	˙0⊕	NOUN
ejpam-2562	345	17	n	n	CCONJ
ejpam-2562	345	18	)	)	PUNCT
ejpam-2562	345	19	=	=	SYM
ejpam-2562	345	20	0	0	X
ejpam-2562	345	21	.	.	PUNCT
ejpam-2562	346	1	since	since	SCONJ
ejpam-2562	346	2	f	f	PROPN
ejpam-2562	346	3	is	be	AUX
ejpam-2562	346	4	monic	monic	ADJ
ejpam-2562	346	5	,	,	PUNCT
ejpam-2562	346	6	f	f	PROPN
ejpam-2562	346	7	(	(	PUNCT
ejpam-2562	346	8	n	n	CCONJ
ejpam-2562	346	9	)	)	PUNCT
ejpam-2562	347	1	=	=	SYM
ejpam-2562	347	2	0=⇒	0=⇒	NUM
ejpam-2562	348	1	n	n	NOUN
ejpam-2562	348	2	=	=	SYM
ejpam-2562	348	3	0	0	PROPN
ejpam-2562	348	4	.	.	PUNCT
ejpam-2562	348	5	(	(	PUNCT
ejpam-2562	348	6	ii)⇒(i	ii)⇒(i	NOUN
ejpam-2562	348	7	)	)	PUNCT
ejpam-2562	348	8	by	by	ADP
ejpam-2562	348	9	hypothesis	hypothesis	NOUN
ejpam-2562	348	10	we	we	PRON
ejpam-2562	348	11	deduce	deduce	VERB
ejpam-2562	348	12	this	this	DET
ejpam-2562	348	13	property	property	NOUN
ejpam-2562	348	14	(	(	PUNCT
ejpam-2562	348	15	p	p	NOUN
ejpam-2562	348	16	):	):	PUNCT
ejpam-2562	348	17	for	for	ADP
ejpam-2562	348	18	any	any	DET
ejpam-2562	348	19	left	left	ADJ
ejpam-2562	348	20	r	r	NOUN
ejpam-2562	348	21	-	-	PUNCT
ejpam-2562	348	22	semimodule	semimodule	NOUN
ejpam-2562	348	23	n	n	NOUN
ejpam-2562	348	24	,	,	PUNCT
ejpam-2562	348	25	if	if	SCONJ
ejpam-2562	348	26	˙m	˙m	VERB
ejpam-2562	348	27	⊕	⊕	PROPN
ejpam-2562	348	28	n	n	PROPN
ejpam-2562	348	29	→	→	PUNCT
ejpam-2562	348	30	m	m	NOUN
ejpam-2562	348	31	is	be	AUX
ejpam-2562	348	32	an	an	DET
ejpam-2562	348	33	semi	semi	ADJ
ejpam-2562	348	34	-	-	ADJ
ejpam-2562	348	35	weakly	weakly	ADJ
ejpam-2562	348	36	-	-	PUNCT
ejpam-2562	348	37	essential	essential	ADJ
ejpam-2562	348	38	monomorphism	monomorphism	NOUN
ejpam-2562	348	39	then	then	ADV
ejpam-2562	348	40	n	n	NOUN
ejpam-2562	348	41	=	=	PUNCT
ejpam-2562	348	42	{	{	PUNCT
ejpam-2562	348	43	0	0	NUM
ejpam-2562	348	44	}	}	PUNCT
ejpam-2562	348	45	.	.	PUNCT
ejpam-2562	349	1	now	now	ADV
ejpam-2562	349	2	let	let	VERB
ejpam-2562	349	3	g	g	NOUN
ejpam-2562	349	4	:	:	PUNCT
ejpam-2562	349	5	m	m	VERB
ejpam-2562	349	6	→	→	PUNCT
ejpam-2562	349	7	m	m	AUX
ejpam-2562	349	8	be	be	AUX
ejpam-2562	349	9	a	a	DET
ejpam-2562	349	10	monomorphism	monomorphism	NOUN
ejpam-2562	349	11	with	with	ADP
ejpam-2562	349	12	non	non	NOUN
ejpam-2562	349	13	semi	semi	ADJ
ejpam-2562	349	14	-	-	ADJ
ejpam-2562	349	15	weakly	weakly	ADJ
ejpam-2562	349	16	-	-	PUNCT
ejpam-2562	349	17	essential	essential	ADJ
ejpam-2562	349	18	image	image	NOUN
ejpam-2562	349	19	.	.	PUNCT
ejpam-2562	350	1	then	then	ADV
ejpam-2562	350	2	,	,	PUNCT
ejpam-2562	350	3	by	by	ADP
ejpam-2562	350	4	proposition	proposition	NOUN
ejpam-2562	350	5	7	7	NUM
ejpam-2562	350	6	there	there	ADV
ejpam-2562	350	7	exists	exist	VERB
ejpam-2562	350	8	a	a	DET
ejpam-2562	350	9	nonzero	nonzero	PROPN
ejpam-2562	350	10	subsemimodule	subsemimodule	NOUN
ejpam-2562	350	11	k	k	PROPN
ejpam-2562	350	12	with	with	ADP
ejpam-2562	350	13	g(m)⊕	g(m)⊕	PROPN
ejpam-2562	350	14	k	k	PROPN
ejpam-2562	350	15	ãswe	ãswe	NOUN
ejpam-2562	350	16	m	m	PROPN
ejpam-2562	350	17	.	.	PUNCT
ejpam-2562	351	1	define	define	VERB
ejpam-2562	351	2	f	f	PROPN
ejpam-2562	351	3	:	:	PUNCT
ejpam-2562	351	4	˙m	˙m	PROPN
ejpam-2562	352	1	⊕	⊕	PROPN
ejpam-2562	352	2	k	k	PROPN
ejpam-2562	353	1	→	→	PROPN
ejpam-2562	353	2	m	m	NOUN
ejpam-2562	353	3	:	:	PUNCT
ejpam-2562	353	4	(	(	PUNCT
ejpam-2562	353	5	m	m	PROPN
ejpam-2562	353	6	,	,	PUNCT
ejpam-2562	353	7	k	k	NOUN
ejpam-2562	353	8	)	)	PUNCT
ejpam-2562	353	9	7→	7→	NUM
ejpam-2562	353	10	g(m	g(m	ADJ
ejpam-2562	353	11	)	)	PUNCT
ejpam-2562	354	1	+	+	CCONJ
ejpam-2562	354	2	k.	k.	X
ejpam-2562	354	3	by	by	ADP
ejpam-2562	354	4	the	the	DET
ejpam-2562	354	5	direct	direct	ADJ
ejpam-2562	354	6	sum	sum	NOUN
ejpam-2562	354	7	g(m	g(m	PROPN
ejpam-2562	354	8	)	)	PUNCT
ejpam-2562	354	9	⊕	⊕	PROPN
ejpam-2562	354	10	k	k	PROPN
ejpam-2562	355	1	and	and	CCONJ
ejpam-2562	355	2	the	the	DET
ejpam-2562	355	3	fact	fact	NOUN
ejpam-2562	355	4	that	that	SCONJ
ejpam-2562	355	5	m	m	NOUN
ejpam-2562	355	6	is	be	AUX
ejpam-2562	355	7	cancellative	cancellative	ADJ
ejpam-2562	355	8	,	,	PUNCT
ejpam-2562	355	9	f	f	PROPN
ejpam-2562	355	10	is	be	AUX
ejpam-2562	355	11	monic	monic	ADJ
ejpam-2562	355	12	.	.	PUNCT
ejpam-2562	356	1	we	we	PRON
ejpam-2562	356	2	have	have	VERB
ejpam-2562	356	3	g(m	g(m	NOUN
ejpam-2562	356	4	)	)	PUNCT
ejpam-2562	356	5	⊕	⊕	PROPN
ejpam-2562	356	6	k	k	NOUN
ejpam-2562	357	1	⊆	⊆	NUM
ejpam-2562	357	2	f	f	X
ejpam-2562	357	3	(	(	PUNCT
ejpam-2562	357	4	˙m	˙m	VERB
ejpam-2562	357	5	⊕	⊕	PROPN
ejpam-2562	357	6	k	k	NOUN
ejpam-2562	357	7	)	)	PUNCT
ejpam-2562	357	8	⊆	⊆	NUM
ejpam-2562	357	9	m	m	NOUN
ejpam-2562	357	10	,	,	PUNCT
ejpam-2562	357	11	hence	hence	ADV
ejpam-2562	357	12	by	by	ADP
ejpam-2562	357	13	proposition	proposition	NOUN
ejpam-2562	357	14	3	3	NUM
ejpam-2562	357	15	f	f	PROPN
ejpam-2562	357	16	(	(	PUNCT
ejpam-2562	357	17	˙m	˙m	VERB
ejpam-2562	357	18	⊕	⊕	PROPN
ejpam-2562	357	19	k	k	NOUN
ejpam-2562	357	20	)	)	PUNCT
ejpam-2562	357	21	ãswe	ãswe	NOUN
ejpam-2562	357	22	m	m	PROPN
ejpam-2562	357	23	(	(	PUNCT
ejpam-2562	357	24	because	because	SCONJ
ejpam-2562	357	25	g(m	g(m	ADJ
ejpam-2562	357	26	)	)	PUNCT
ejpam-2562	357	27	⊕	⊕	PROPN
ejpam-2562	357	28	k	k	PROPN
ejpam-2562	357	29	ãswe	ãswe	PROPN
ejpam-2562	357	30	m	m	PROPN
ejpam-2562	357	31	)	)	PUNCT
ejpam-2562	357	32	.	.	PUNCT
ejpam-2562	358	1	so	so	ADV
ejpam-2562	358	2	f	f	PROPN
ejpam-2562	358	3	is	be	AUX
ejpam-2562	358	4	an	an	DET
ejpam-2562	358	5	semi	semi	ADJ
ejpam-2562	358	6	-	-	ADJ
ejpam-2562	358	7	weakly	weakly	ADJ
ejpam-2562	358	8	-	-	PUNCT
ejpam-2562	358	9	essential	essential	ADJ
ejpam-2562	358	10	monomorphism	monomorphism	NOUN
ejpam-2562	358	11	contradicting	contradict	VERB
ejpam-2562	358	12	the	the	DET
ejpam-2562	358	13	property	property	NOUN
ejpam-2562	358	14	(	(	PUNCT
ejpam-2562	358	15	p	p	NOUN
ejpam-2562	358	16	)	)	PUNCT
ejpam-2562	358	17	.	.	PUNCT
ejpam-2562	359	1	hence	hence	ADV
ejpam-2562	359	2	g(m)ãswe	g(m)ãswe	VERB
ejpam-2562	359	3	m	m	PROPN
ejpam-2562	359	4	as	as	SCONJ
ejpam-2562	359	5	desired	desire	VERB
ejpam-2562	359	6	.	.	PUNCT
ejpam-2562	360	1	proposition	proposition	NOUN
ejpam-2562	360	2	10	10	NUM
ejpam-2562	360	3	.	.	PUNCT
ejpam-2562	361	1	the	the	DET
ejpam-2562	361	2	following	follow	VERB
ejpam-2562	361	3	are	be	AUX
ejpam-2562	361	4	equivalent	equivalent	ADJ
ejpam-2562	361	5	conditions	condition	NOUN
ejpam-2562	361	6	on	on	ADP
ejpam-2562	361	7	a	a	DET
ejpam-2562	361	8	left	left	ADJ
ejpam-2562	361	9	r	r	NOUN
ejpam-2562	361	10	-	-	PUNCT
ejpam-2562	361	11	semimodule	semimodule	NOUN
ejpam-2562	361	12	m.	m.	NOUN
ejpam-2562	361	13	(	(	PUNCT
ejpam-2562	361	14	i	i	NOUN
ejpam-2562	361	15	)	)	PUNCT
ejpam-2562	361	16	m	m	VERB
ejpam-2562	361	17	is	be	AUX
ejpam-2562	361	18	weakly	weakly	ADJ
ejpam-2562	361	19	co	co	NOUN
ejpam-2562	361	20	-	-	NOUN
ejpam-2562	361	21	hopfian-1	hopfian-1	NOUN
ejpam-2562	361	22	.	.	PUNCT
ejpam-2562	362	1	(	(	PUNCT
ejpam-2562	362	2	ii	ii	NOUN
ejpam-2562	362	3	)	)	PUNCT
ejpam-2562	362	4	m	m	VERB
ejpam-2562	362	5	is	be	AUX
ejpam-2562	362	6	a	a	DET
ejpam-2562	362	7	f	f	NOUN
ejpam-2562	362	8	-	-	PUNCT
ejpam-2562	362	9	semimodule	semimodule	NOUN
ejpam-2562	362	10	and	and	CCONJ
ejpam-2562	362	11	the	the	DET
ejpam-2562	362	12	image	image	NOUN
ejpam-2562	362	13	of	of	ADP
ejpam-2562	362	14	any	any	DET
ejpam-2562	362	15	injective	injective	ADJ
ejpam-2562	362	16	endomorphism	endomorphism	NOUN
ejpam-2562	362	17	of	of	ADP
ejpam-2562	362	18	m	m	PROPN
ejpam-2562	362	19	is	be	AUX
ejpam-2562	362	20	either	either	CCONJ
ejpam-2562	362	21	semiweakly	semiweakly	ADV
ejpam-2562	362	22	-	-	PUNCT
ejpam-2562	362	23	essential	essential	ADJ
ejpam-2562	362	24	or	or	CCONJ
ejpam-2562	362	25	a	a	DET
ejpam-2562	362	26	proper	proper	ADJ
ejpam-2562	362	27	direct	direct	ADJ
ejpam-2562	362	28	summand	summand	NOUN
ejpam-2562	362	29	.	.	PUNCT
ejpam-2562	363	1	e.	e.	PROPN
ejpam-2562	363	2	diop	diop	PROPN
ejpam-2562	363	3	,	,	PUNCT
ejpam-2562	363	4	d.	d.	PROPN
ejpam-2562	363	5	sow	sow	PROPN
ejpam-2562	363	6	/	/	SYM
ejpam-2562	363	7	eur	eur	PROPN
ejpam-2562	363	8	.	.	PUNCT
ejpam-2562	364	1	j.	j.	PROPN
ejpam-2562	364	2	pure	pure	PROPN
ejpam-2562	364	3	appl	appl	PROPN
ejpam-2562	364	4	.	.	PROPN
ejpam-2562	364	5	math	math	PROPN
ejpam-2562	364	6	,	,	PUNCT
ejpam-2562	364	7	9	9	NUM
ejpam-2562	364	8	(	(	PUNCT
ejpam-2562	364	9	2016	2016	NUM
ejpam-2562	364	10	)	)	PUNCT
ejpam-2562	364	11	,	,	PUNCT
ejpam-2562	364	12	250	250	NUM
ejpam-2562	364	13	-	-	SYM
ejpam-2562	364	14	265	265	NUM
ejpam-2562	364	15	261	261	NUM
ejpam-2562	364	16	proof	proof	NOUN
ejpam-2562	364	17	.	.	PUNCT
ejpam-2562	365	1	(	(	PUNCT
ejpam-2562	365	2	i)⇒(ii	i)⇒(ii	ADV
ejpam-2562	365	3	)	)	PUNCT
ejpam-2562	365	4	by	by	ADP
ejpam-2562	365	5	hypothesis	hypothesis	NOUN
ejpam-2562	365	6	and	and	CCONJ
ejpam-2562	365	7	by	by	ADP
ejpam-2562	365	8	the	the	DET
ejpam-2562	365	9	previous	previous	ADJ
ejpam-2562	365	10	proposition	proposition	NOUN
ejpam-2562	365	11	,	,	PUNCT
ejpam-2562	365	12	we	we	PRON
ejpam-2562	365	13	have	have	VERB
ejpam-2562	365	14	:	:	PUNCT
ejpam-2562	365	15	for	for	ADP
ejpam-2562	365	16	any	any	DET
ejpam-2562	365	17	left	left	ADJ
ejpam-2562	365	18	rsemimodule	rsemimodule	NOUN
ejpam-2562	365	19	n	n	NOUN
ejpam-2562	365	20	,	,	PUNCT
ejpam-2562	365	21	if	if	SCONJ
ejpam-2562	365	22	there	there	PRON
ejpam-2562	365	23	is	be	VERB
ejpam-2562	365	24	an	an	DET
ejpam-2562	365	25	r	r	NOUN
ejpam-2562	365	26	-	-	PUNCT
ejpam-2562	365	27	monomorphism	monomorphism	NOUN
ejpam-2562	365	28	˙m	˙m	VERB
ejpam-2562	366	1	⊕	⊕	PROPN
ejpam-2562	366	2	n	n	PROPN
ejpam-2562	366	3	→	→	SYM
ejpam-2562	366	4	m	m	VERB
ejpam-2562	366	5	then	then	ADV
ejpam-2562	366	6	n	n	NOUN
ejpam-2562	366	7	=	=	PUNCT
ejpam-2562	366	8	{	{	PUNCT
ejpam-2562	366	9	0	0	NUM
ejpam-2562	366	10	}	}	PUNCT
ejpam-2562	366	11	.	.	PUNCT
ejpam-2562	367	1	now	now	ADV
ejpam-2562	367	2	suppose	suppose	VERB
ejpam-2562	367	3	that	that	SCONJ
ejpam-2562	367	4	˙m	˙m	VERB
ejpam-2562	367	5	⊕	⊕	PROPN
ejpam-2562	367	6	k	k	PROPN
ejpam-2562	367	7	∼=	∼=	PROPN
ejpam-2562	367	8	m	m	PROPN
ejpam-2562	367	9	,	,	PUNCT
ejpam-2562	367	10	then	then	ADV
ejpam-2562	367	11	there	there	PRON
ejpam-2562	367	12	is	be	VERB
ejpam-2562	367	13	a	a	DET
ejpam-2562	367	14	monomorphism	monomorphism	NOUN
ejpam-2562	367	15	f	f	X
ejpam-2562	367	16	:	:	PUNCT
ejpam-2562	367	17	˙m	˙m	PROPN
ejpam-2562	368	1	⊕	⊕	PROPN
ejpam-2562	368	2	k	k	PROPN
ejpam-2562	369	1	→	→	PUNCT
ejpam-2562	369	2	m	m	VERB
ejpam-2562	369	3	and	and	CCONJ
ejpam-2562	369	4	we	we	PRON
ejpam-2562	369	5	have	have	VERB
ejpam-2562	369	6	k	k	NOUN
ejpam-2562	369	7	=	=	PUNCT
ejpam-2562	369	8	{	{	PUNCT
ejpam-2562	369	9	0	0	NUM
ejpam-2562	369	10	}	}	PUNCT
ejpam-2562	369	11	,	,	PUNCT
ejpam-2562	369	12	hence	hence	ADV
ejpam-2562	369	13	m	m	VERB
ejpam-2562	369	14	is	be	AUX
ejpam-2562	369	15	a	a	DET
ejpam-2562	369	16	f	f	PROPN
ejpam-2562	369	17	-semimodule	-semimodule	PROPN
ejpam-2562	369	18	.	.	PUNCT
ejpam-2562	370	1	by	by	ADP
ejpam-2562	370	2	hypothesis	hypothesis	NOUN
ejpam-2562	370	3	the	the	DET
ejpam-2562	370	4	image	image	NOUN
ejpam-2562	370	5	of	of	ADP
ejpam-2562	370	6	any	any	DET
ejpam-2562	370	7	injective	injective	ADJ
ejpam-2562	370	8	endomorphism	endomorphism	NOUN
ejpam-2562	370	9	of	of	ADP
ejpam-2562	370	10	m	m	PROPN
ejpam-2562	370	11	is	be	AUX
ejpam-2562	370	12	in	in	ADP
ejpam-2562	370	13	fact	fact	NOUN
ejpam-2562	370	14	a	a	DET
ejpam-2562	370	15	semi	semi	ADJ
ejpam-2562	370	16	-	-	ADJ
ejpam-2562	370	17	weakly	weakly	ADJ
ejpam-2562	370	18	-	-	PUNCT
ejpam-2562	370	19	essential	essential	ADJ
ejpam-2562	370	20	subsemimodule	subsemimodule	NOUN
ejpam-2562	370	21	.	.	PUNCT
ejpam-2562	371	1	(	(	PUNCT
ejpam-2562	371	2	ii)⇒(i	ii)⇒(i	NOUN
ejpam-2562	371	3	)	)	PUNCT
ejpam-2562	371	4	let	let	VERB
ejpam-2562	371	5	g	g	NOUN
ejpam-2562	371	6	:	:	PUNCT
ejpam-2562	371	7	m	m	VERB
ejpam-2562	371	8	→	→	NOUN
ejpam-2562	371	9	m	m	AUX
ejpam-2562	371	10	be	be	VERB
ejpam-2562	371	11	an	an	DET
ejpam-2562	371	12	injective	injective	ADJ
ejpam-2562	371	13	endomorphism	endomorphism	NOUN
ejpam-2562	371	14	and	and	CCONJ
ejpam-2562	371	15	suppose	suppose	VERB
ejpam-2562	371	16	that	that	SCONJ
ejpam-2562	371	17	g(m	g(m	NOUN
ejpam-2562	371	18	)	)	PUNCT
ejpam-2562	371	19	is	be	AUX
ejpam-2562	371	20	not	not	PART
ejpam-2562	371	21	semiweakly	semiweakly	ADV
ejpam-2562	371	22	-	-	PUNCT
ejpam-2562	371	23	essential	essential	ADJ
ejpam-2562	371	24	in	in	ADP
ejpam-2562	371	25	m	m	PROPN
ejpam-2562	371	26	.	.	PUNCT
ejpam-2562	372	1	then	then	ADV
ejpam-2562	372	2	by	by	ADP
ejpam-2562	372	3	hypothesis	hypothesis	NOUN
ejpam-2562	372	4	g(m	g(m	VERB
ejpam-2562	372	5	)	)	PUNCT
ejpam-2562	372	6	is	be	AUX
ejpam-2562	372	7	a	a	DET
ejpam-2562	372	8	proper	proper	ADJ
ejpam-2562	372	9	direct	direct	ADJ
ejpam-2562	372	10	summand	summand	NOUN
ejpam-2562	372	11	of	of	ADP
ejpam-2562	372	12	m	m	PROPN
ejpam-2562	372	13	.	.	PUNCT
ejpam-2562	373	1	so	so	ADV
ejpam-2562	373	2	there	there	PRON
ejpam-2562	373	3	exits	exit	VERB
ejpam-2562	373	4	a	a	DET
ejpam-2562	373	5	nonzero	nonzero	ADJ
ejpam-2562	373	6	subsemimodule	subsemimodule	NOUN
ejpam-2562	373	7	k	k	PROPN
ejpam-2562	373	8	of	of	ADP
ejpam-2562	373	9	m	m	PROPN
ejpam-2562	373	10	such	such	ADJ
ejpam-2562	373	11	that	that	SCONJ
ejpam-2562	373	12	g(m)⊕	g(m)⊕	NOUN
ejpam-2562	373	13	k	k	X
ejpam-2562	374	1	=	=	PUNCT
ejpam-2562	374	2	m	m	VERB
ejpam-2562	374	3	.	.	PUNCT
ejpam-2562	375	1	define	define	VERB
ejpam-2562	375	2	f	f	PROPN
ejpam-2562	375	3	:	:	PUNCT
ejpam-2562	375	4	˙m	˙m	PROPN
ejpam-2562	376	1	⊕	⊕	PROPN
ejpam-2562	376	2	k	k	PROPN
ejpam-2562	377	1	→	→	PROPN
ejpam-2562	377	2	m	m	NOUN
ejpam-2562	377	3	:	:	PUNCT
ejpam-2562	377	4	(	(	PUNCT
ejpam-2562	377	5	m	m	PROPN
ejpam-2562	377	6	,	,	PUNCT
ejpam-2562	377	7	k	k	NOUN
ejpam-2562	377	8	)	)	PUNCT
ejpam-2562	377	9	7→	7→	NUM
ejpam-2562	377	10	g(m	g(m	ADJ
ejpam-2562	377	11	)	)	PUNCT
ejpam-2562	378	1	+	+	CCONJ
ejpam-2562	378	2	k.	k.	PROPN
ejpam-2562	378	3	then	then	ADV
ejpam-2562	378	4	f	f	PROPN
ejpam-2562	378	5	is	be	AUX
ejpam-2562	378	6	a	a	DET
ejpam-2562	378	7	monomorphism	monomorphism	NOUN
ejpam-2562	378	8	.	.	PUNCT
ejpam-2562	379	1	we	we	PRON
ejpam-2562	379	2	have	have	VERB
ejpam-2562	379	3	m	m	NOUN
ejpam-2562	379	4	=	=	PUNCT
ejpam-2562	380	1	g(m)⊕	g(m)⊕	X
ejpam-2562	380	2	k	k	PROPN
ejpam-2562	380	3	⊆	⊆	NUM
ejpam-2562	380	4	f	f	PROPN
ejpam-2562	380	5	(	(	PUNCT
ejpam-2562	380	6	˙m	˙m	VERB
ejpam-2562	380	7	⊕	⊕	PROPN
ejpam-2562	380	8	k	k	NOUN
ejpam-2562	380	9	)	)	PUNCT
ejpam-2562	380	10	⊆	⊆	NUM
ejpam-2562	380	11	m	m	NOUN
ejpam-2562	380	12	,	,	PUNCT
ejpam-2562	380	13	hence	hence	ADV
ejpam-2562	380	14	f	f	PROPN
ejpam-2562	380	15	(	(	PUNCT
ejpam-2562	380	16	˙m	˙m	VERB
ejpam-2562	380	17	⊕	⊕	PROPN
ejpam-2562	380	18	k	k	NOUN
ejpam-2562	380	19	)	)	PUNCT
ejpam-2562	381	1	=	=	NOUN
ejpam-2562	381	2	m	m	NOUN
ejpam-2562	381	3	,	,	PUNCT
ejpam-2562	381	4	whence	whence	ADP
ejpam-2562	381	5	f	f	PROPN
ejpam-2562	381	6	is	be	AUX
ejpam-2562	381	7	surjective	surjective	ADJ
ejpam-2562	381	8	.	.	PUNCT
ejpam-2562	382	1	therefore	therefore	ADV
ejpam-2562	382	2	there	there	PRON
ejpam-2562	382	3	is	be	VERB
ejpam-2562	382	4	an	an	DET
ejpam-2562	382	5	isomorphism	isomorphism	NOUN
ejpam-2562	382	6	˙m	˙m	VERB
ejpam-2562	383	1	⊕	⊕	PROPN
ejpam-2562	383	2	k	k	PROPN
ejpam-2562	383	3	∼=	∼=	PROPN
ejpam-2562	383	4	m	m	VERB
ejpam-2562	383	5	which	which	PRON
ejpam-2562	383	6	contradicts	contradict	VERB
ejpam-2562	383	7	the	the	DET
ejpam-2562	383	8	fact	fact	NOUN
ejpam-2562	383	9	that	that	SCONJ
ejpam-2562	383	10	m	m	NOUN
ejpam-2562	383	11	is	be	AUX
ejpam-2562	383	12	a	a	DET
ejpam-2562	383	13	f	f	PROPN
ejpam-2562	383	14	semimodule	semimodule	NOUN
ejpam-2562	383	15	.	.	PUNCT
ejpam-2562	384	1	thus	thus	ADV
ejpam-2562	384	2	g(m)ãswe	g(m)ãswe	PROPN
ejpam-2562	384	3	m	m	PROPN
ejpam-2562	384	4	.	.	PUNCT
ejpam-2562	385	1	proposition	proposition	NOUN
ejpam-2562	385	2	11	11	NUM
ejpam-2562	385	3	.	.	PUNCT
ejpam-2562	386	1	(	(	PUNCT
ejpam-2562	386	2	i	i	NOUN
ejpam-2562	386	3	)	)	PUNCT
ejpam-2562	386	4	the	the	DET
ejpam-2562	386	5	following	follow	VERB
ejpam-2562	386	6	are	be	AUX
ejpam-2562	386	7	equivalent	equivalent	ADJ
ejpam-2562	386	8	conditions	condition	NOUN
ejpam-2562	386	9	on	on	ADP
ejpam-2562	386	10	a	a	DET
ejpam-2562	386	11	left	left	ADJ
ejpam-2562	386	12	r	r	NOUN
ejpam-2562	386	13	-	-	PUNCT
ejpam-2562	386	14	semimodule	semimodule	NOUN
ejpam-2562	386	15	m.	m.	NOUN
ejpam-2562	386	16	(	(	PUNCT
ejpam-2562	386	17	a	a	X
ejpam-2562	386	18	)	)	PUNCT
ejpam-2562	386	19	m	m	VERB
ejpam-2562	386	20	is	be	AUX
ejpam-2562	386	21	weakly	weakly	ADJ
ejpam-2562	386	22	co	co	NOUN
ejpam-2562	386	23	-	-	NOUN
ejpam-2562	386	24	hopfian-1	hopfian-1	NOUN
ejpam-2562	386	25	.	.	PUNCT
ejpam-2562	387	1	(	(	PUNCT
ejpam-2562	387	2	b	b	X
ejpam-2562	387	3	)	)	PUNCT
ejpam-2562	387	4	there	there	PRON
ejpam-2562	387	5	exists	exist	VERB
ejpam-2562	387	6	a	a	DET
ejpam-2562	387	7	subsemimodule	subsemimodule	NOUN
ejpam-2562	387	8	k	k	NOUN
ejpam-2562	387	9	of	of	ADP
ejpam-2562	387	10	m	m	PROPN
ejpam-2562	387	11	such	such	ADJ
ejpam-2562	387	12	that	that	SCONJ
ejpam-2562	387	13	g(k)ãswe	g(k)ãswe	PROPN
ejpam-2562	387	14	m	m	VERB
ejpam-2562	387	15	for	for	ADP
ejpam-2562	387	16	all	all	DET
ejpam-2562	387	17	injective	injective	ADJ
ejpam-2562	387	18	g	g	PROPN
ejpam-2562	387	19	∈	∈	PROPN
ejpam-2562	387	20	end(m	end(m	PROPN
ejpam-2562	387	21	)	)	PUNCT
ejpam-2562	387	22	.	.	PUNCT
ejpam-2562	388	1	(	(	PUNCT
ejpam-2562	388	2	ii	ii	X
ejpam-2562	388	3	)	)	PUNCT
ejpam-2562	388	4	a	a	DET
ejpam-2562	388	5	direct	direct	ADJ
ejpam-2562	388	6	summand	summand	NOUN
ejpam-2562	388	7	of	of	ADP
ejpam-2562	388	8	a	a	DET
ejpam-2562	388	9	weakly	weakly	ADJ
ejpam-2562	388	10	co	co	NOUN
ejpam-2562	388	11	-	-	ADJ
ejpam-2562	388	12	hopfian-1	hopfian-1	NUM
ejpam-2562	388	13	semimodule	semimodule	NOUN
ejpam-2562	388	14	is	be	AUX
ejpam-2562	388	15	weakly	weakly	ADJ
ejpam-2562	388	16	co	co	NOUN
ejpam-2562	388	17	-	-	NOUN
ejpam-2562	388	18	hopfian-1	hopfian-1	ADJ
ejpam-2562	388	19	.	.	PUNCT
ejpam-2562	389	1	proof	proof	NOUN
ejpam-2562	389	2	.	.	PUNCT
ejpam-2562	390	1	(	(	PUNCT
ejpam-2562	390	2	i	i	NOUN
ejpam-2562	390	3	)	)	PUNCT
ejpam-2562	390	4	(	(	PUNCT
ejpam-2562	390	5	b)⇒(a	b)⇒(a	ADJ
ejpam-2562	390	6	)	)	PUNCT
ejpam-2562	390	7	trivial	trivial	ADJ
ejpam-2562	390	8	by	by	ADP
ejpam-2562	390	9	proposition	proposition	NOUN
ejpam-2562	390	10	3	3	NUM
ejpam-2562	390	11	.	.	PUNCT
ejpam-2562	390	12	(	(	PUNCT
ejpam-2562	390	13	a)⇒(b	a)⇒(b	NOUN
ejpam-2562	390	14	)	)	PUNCT
ejpam-2562	390	15	trivial	trivial	ADJ
ejpam-2562	390	16	.	.	PUNCT
ejpam-2562	391	1	(	(	PUNCT
ejpam-2562	391	2	ii	ii	NOUN
ejpam-2562	391	3	)	)	PUNCT
ejpam-2562	391	4	suppose	suppose	VERB
ejpam-2562	391	5	that	that	SCONJ
ejpam-2562	391	6	m	m	PROPN
ejpam-2562	391	7	is	be	AUX
ejpam-2562	391	8	a	a	DET
ejpam-2562	391	9	weakly	weakly	ADJ
ejpam-2562	391	10	co	co	NOUN
ejpam-2562	391	11	-	-	ADJ
ejpam-2562	391	12	hopfian-1	hopfian-1	NUM
ejpam-2562	391	13	semimodule	semimodule	NOUN
ejpam-2562	391	14	.	.	PUNCT
ejpam-2562	392	1	let	let	VERB
ejpam-2562	392	2	m	m	VERB
ejpam-2562	392	3	=	=	SYM
ejpam-2562	392	4	n	n	PROPN
ejpam-2562	392	5	⊕	⊕	PROPN
ejpam-2562	392	6	k	k	PROPN
ejpam-2562	392	7	and	and	CCONJ
ejpam-2562	392	8	let	let	VERB
ejpam-2562	392	9	f	f	X
ejpam-2562	392	10	:	:	PUNCT
ejpam-2562	392	11	n	n	CCONJ
ejpam-2562	392	12	−→	−→	NOUN
ejpam-2562	392	13	n	n	AUX
ejpam-2562	392	14	be	be	AUX
ejpam-2562	392	15	an	an	DET
ejpam-2562	392	16	injective	injective	ADJ
ejpam-2562	392	17	endomorphism	endomorphism	NOUN
ejpam-2562	392	18	of	of	ADP
ejpam-2562	392	19	n	n	PROPN
ejpam-2562	392	20	.	.	PUNCT
ejpam-2562	393	1	then	then	ADV
ejpam-2562	393	2	the	the	DET
ejpam-2562	393	3	map	map	NOUN
ejpam-2562	393	4	f	f	PROPN
ejpam-2562	393	5	⊕	⊕	PROPN
ejpam-2562	393	6	idk	idk	INTJ
ejpam-2562	393	7	:	:	PUNCT
ejpam-2562	393	8	m	m	VERB
ejpam-2562	393	9	=	=	SYM
ejpam-2562	393	10	n	n	PROPN
ejpam-2562	393	11	⊕	⊕	PROPN
ejpam-2562	393	12	k	k	PROPN
ejpam-2562	393	13	→	→	PUNCT
ejpam-2562	393	14	m	m	NOUN
ejpam-2562	393	15	=	=	SYM
ejpam-2562	393	16	n	n	PROPN
ejpam-2562	393	17	⊕	⊕	PROPN
ejpam-2562	393	18	k	k	PROPN
ejpam-2562	393	19	defined	define	VERB
ejpam-2562	393	20	by	by	ADP
ejpam-2562	393	21	n+	n+	PRON
ejpam-2562	393	22	k	k	PROPN
ejpam-2562	393	23	7→	7→	PROPN
ejpam-2562	393	24	(	(	PUNCT
ejpam-2562	393	25	f	f	PROPN
ejpam-2562	393	26	⊕	⊕	PROPN
ejpam-2562	393	27	idk)(n+	idk)(n+	X
ejpam-2562	394	1	k	k	X
ejpam-2562	394	2	)	)	PUNCT
ejpam-2562	394	3	=	=	SYM
ejpam-2562	394	4	f	f	PROPN
ejpam-2562	394	5	(	(	PUNCT
ejpam-2562	394	6	n	n	CCONJ
ejpam-2562	394	7	)	)	PUNCT
ejpam-2562	395	1	+	+	CCONJ
ejpam-2562	395	2	k	k	PROPN
ejpam-2562	395	3	is	be	AUX
ejpam-2562	395	4	an	an	DET
ejpam-2562	395	5	injective	injective	ADJ
ejpam-2562	395	6	endomorphism	endomorphism	NOUN
ejpam-2562	395	7	of	of	ADP
ejpam-2562	395	8	m	m	PRON
ejpam-2562	395	9	and	and	CCONJ
ejpam-2562	395	10	(	(	PUNCT
ejpam-2562	395	11	f	f	PROPN
ejpam-2562	395	12	⊕	⊕	PROPN
ejpam-2562	395	13	idk)(m	idk)(m	NOUN
ejpam-2562	395	14	)	)	PUNCT
ejpam-2562	395	15	ãswe	ãswe	NOUN
ejpam-2562	395	16	m	m	PROPN
ejpam-2562	395	17	.	.	PUNCT
ejpam-2562	396	1	then	then	ADV
ejpam-2562	396	2	(	(	PUNCT
ejpam-2562	396	3	f	f	PROPN
ejpam-2562	396	4	⊕	⊕	PROPN
ejpam-2562	396	5	idk)(m	idk)(m	NOUN
ejpam-2562	396	6	)	)	PUNCT
ejpam-2562	396	7	ãswe	ãswe	NOUN
ejpam-2562	396	8	m	m	VERB
ejpam-2562	396	9	⇒	⇒	PROPN
ejpam-2562	396	10	f	f	PROPN
ejpam-2562	396	11	(	(	PUNCT
ejpam-2562	396	12	n	n	CCONJ
ejpam-2562	396	13	)	)	PUNCT
ejpam-2562	396	14	⊕	⊕	PROPN
ejpam-2562	396	15	k	k	PROPN
ejpam-2562	397	1	ãswe	ãswe	PROPN
ejpam-2562	397	2	n	n	PROPN
ejpam-2562	397	3	⊕	⊕	PROPN
ejpam-2562	397	4	k	k	PROPN
ejpam-2562	397	5	and	and	CCONJ
ejpam-2562	397	6	by	by	ADP
ejpam-2562	397	7	proposition	proposition	NOUN
ejpam-2562	397	8	4	4	NUM
ejpam-2562	397	9	f	f	NOUN
ejpam-2562	397	10	(	(	PUNCT
ejpam-2562	397	11	n)ãswe	n)ãswe	NUM
ejpam-2562	397	12	n	n	NOUN
ejpam-2562	397	13	,	,	PUNCT
ejpam-2562	397	14	therefore	therefore	ADV
ejpam-2562	397	15	n	n	PRON
ejpam-2562	397	16	is	be	AUX
ejpam-2562	397	17	weakly	weakly	ADJ
ejpam-2562	397	18	co	co	NOUN
ejpam-2562	397	19	-	-	NOUN
ejpam-2562	397	20	hopfian-1	hopfian-1	NOUN
ejpam-2562	397	21	.	.	NOUN
ejpam-2562	398	1	4.2	4.2	NUM
ejpam-2562	398	2	.	.	PUNCT
ejpam-2562	399	1	on	on	ADP
ejpam-2562	399	2	weakly	weakly	ADJ
ejpam-2562	399	3	co	co	ADJ
ejpam-2562	399	4	-	-	ADJ
ejpam-2562	399	5	hopfian	hopfian	ADJ
ejpam-2562	399	6	semimodules	semimodule	NOUN
ejpam-2562	399	7	of	of	ADP
ejpam-2562	399	8	type	type	NOUN
ejpam-2562	399	9	2	2	NUM
ejpam-2562	399	10	definition	definition	NOUN
ejpam-2562	399	11	9	9	NUM
ejpam-2562	399	12	.	.	PUNCT
ejpam-2562	400	1	a	a	DET
ejpam-2562	400	2	nonzero	nonzero	NOUN
ejpam-2562	400	3	left	leave	VERB
ejpam-2562	400	4	r	r	NOUN
ejpam-2562	400	5	-	-	PUNCT
ejpam-2562	400	6	semimodule	semimodule	NOUN
ejpam-2562	400	7	rm	rm	NOUN
ejpam-2562	400	8	is	be	AUX
ejpam-2562	400	9	said	say	VERB
ejpam-2562	400	10	to	to	PART
ejpam-2562	400	11	be	be	AUX
ejpam-2562	400	12	weakly	weakly	ADJ
ejpam-2562	400	13	co	co	NOUN
ejpam-2562	400	14	-	-	NOUN
ejpam-2562	400	15	hopfian-2	hopfian-2	NUM
ejpam-2562	400	16	(	(	PUNCT
ejpam-2562	400	17	denoted	denote	VERB
ejpam-2562	400	18	by	by	ADP
ejpam-2562	400	19	wch-2	wch-2	PRON
ejpam-2562	400	20	)	)	PUNCT
ejpam-2562	400	21	if	if	SCONJ
ejpam-2562	400	22	every	every	DET
ejpam-2562	400	23	monomorphism	monomorphism	NOUN
ejpam-2562	400	24	f	f	X
ejpam-2562	400	25	:	:	PUNCT
ejpam-2562	400	26	m	m	VERB
ejpam-2562	400	27	→	→	NOUN
ejpam-2562	400	28	m	m	NOUN
ejpam-2562	400	29	is	be	AUX
ejpam-2562	400	30	semi	semi	ADJ
ejpam-2562	400	31	-	-	ADJ
ejpam-2562	400	32	essential	essential	ADJ
ejpam-2562	400	33	i.e	i.e	X
ejpam-2562	400	34	f	f	X
ejpam-2562	400	35	(	(	PUNCT
ejpam-2562	400	36	m)ãs	m)ãs	PROPN
ejpam-2562	400	37	m	m	PROPN
ejpam-2562	400	38	(	(	PUNCT
ejpam-2562	400	39	see	see	VERB
ejpam-2562	400	40	definition	definition	NOUN
ejpam-2562	400	41	in	in	ADP
ejpam-2562	400	42	the	the	DET
ejpam-2562	400	43	introduction	introduction	NOUN
ejpam-2562	400	44	)	)	PUNCT
ejpam-2562	400	45	.	.	PUNCT
ejpam-2562	401	1	e.	e.	PROPN
ejpam-2562	401	2	diop	diop	PROPN
ejpam-2562	401	3	,	,	PUNCT
ejpam-2562	401	4	d.	d.	PROPN
ejpam-2562	401	5	sow	sow	PROPN
ejpam-2562	401	6	/	/	SYM
ejpam-2562	401	7	eur	eur	PROPN
ejpam-2562	401	8	.	.	PUNCT
ejpam-2562	402	1	j.	j.	PROPN
ejpam-2562	402	2	pure	pure	PROPN
ejpam-2562	402	3	appl	appl	PROPN
ejpam-2562	402	4	.	.	PROPN
ejpam-2562	402	5	math	math	PROPN
ejpam-2562	402	6	,	,	PUNCT
ejpam-2562	402	7	9	9	NUM
ejpam-2562	402	8	(	(	PUNCT
ejpam-2562	402	9	2016	2016	NUM
ejpam-2562	402	10	)	)	PUNCT
ejpam-2562	402	11	,	,	PUNCT
ejpam-2562	402	12	250	250	NUM
ejpam-2562	402	13	-	-	SYM
ejpam-2562	402	14	265	265	NUM
ejpam-2562	402	15	262	262	NUM
ejpam-2562	402	16	proposition	proposition	NOUN
ejpam-2562	402	17	12	12	NUM
ejpam-2562	402	18	.	.	PUNCT
ejpam-2562	403	1	if	if	SCONJ
ejpam-2562	403	2	m	m	NOUN
ejpam-2562	403	3	is	be	AUX
ejpam-2562	403	4	wch-2	wch-2	ADP
ejpam-2562	403	5	,	,	PUNCT
ejpam-2562	403	6	then	then	ADV
ejpam-2562	403	7	m	m	PROPN
ejpam-2562	403	8	is	be	AUX
ejpam-2562	403	9	wch-1	wch-1	X
ejpam-2562	403	10	.	.	PUNCT
ejpam-2562	404	1	proof	proof	NOUN
ejpam-2562	404	2	.	.	PUNCT
ejpam-2562	405	1	by	by	ADP
ejpam-2562	405	2	the	the	DET
ejpam-2562	405	3	proposition	proposition	NOUN
ejpam-2562	405	4	2	2	NUM
ejpam-2562	405	5	.	.	X
ejpam-2562	405	6	proposition	proposition	NOUN
ejpam-2562	405	7	13	13	NUM
ejpam-2562	405	8	.	.	PUNCT
ejpam-2562	406	1	the	the	DET
ejpam-2562	406	2	following	follow	VERB
ejpam-2562	406	3	are	be	AUX
ejpam-2562	406	4	equivalent	equivalent	ADJ
ejpam-2562	406	5	conditions	condition	NOUN
ejpam-2562	406	6	on	on	ADP
ejpam-2562	406	7	a	a	DET
ejpam-2562	406	8	left	left	ADJ
ejpam-2562	406	9	r	r	NOUN
ejpam-2562	406	10	-	-	PUNCT
ejpam-2562	406	11	semimodule	semimodule	NOUN
ejpam-2562	406	12	m.	m.	NOUN
ejpam-2562	406	13	(	(	PUNCT
ejpam-2562	406	14	i	i	NOUN
ejpam-2562	406	15	)	)	PUNCT
ejpam-2562	406	16	m	m	VERB
ejpam-2562	406	17	is	be	AUX
ejpam-2562	406	18	weakly	weakly	ADJ
ejpam-2562	406	19	co	co	NOUN
ejpam-2562	406	20	-	-	NOUN
ejpam-2562	406	21	hopfian-2	hopfian-2	NOUN
ejpam-2562	406	22	.	.	PUNCT
ejpam-2562	407	1	(	(	PUNCT
ejpam-2562	407	2	ii	ii	NOUN
ejpam-2562	407	3	)	)	PUNCT
ejpam-2562	407	4	for	for	ADP
ejpam-2562	407	5	every	every	DET
ejpam-2562	407	6	monomorphism	monomorphism	NOUN
ejpam-2562	408	1	f	f	X
ejpam-2562	408	2	:	:	PUNCT
ejpam-2562	408	3	m	m	VERB
ejpam-2562	408	4	→	→	SYM
ejpam-2562	408	5	m	m	PROPN
ejpam-2562	408	6	,	,	PUNCT
ejpam-2562	408	7	if	if	SCONJ
ejpam-2562	408	8	x	x	PRON
ejpam-2562	408	9	is	be	AUX
ejpam-2562	408	10	an	an	DET
ejpam-2562	408	11	nonzero	nonzero	ADJ
ejpam-2562	408	12	element	element	NOUN
ejpam-2562	408	13	of	of	ADP
ejpam-2562	408	14	m	m	PROPN
ejpam-2562	408	15	,	,	PUNCT
ejpam-2562	408	16	then	then	ADV
ejpam-2562	408	17	there	there	PRON
ejpam-2562	408	18	exists	exist	VERB
ejpam-2562	408	19	r	r	NOUN
ejpam-2562	408	20	∈	∈	PROPN
ejpam-2562	408	21	r	r	NOUN
ejpam-2562	408	22	such	such	ADJ
ejpam-2562	408	23	that	that	DET
ejpam-2562	408	24	0	0	NUM
ejpam-2562	409	1	6=	6=	NUM
ejpam-2562	409	2	r	r	NOUN
ejpam-2562	409	3	x	x	SYM
ejpam-2562	409	4	∈	∈	PROPN
ejpam-2562	409	5	f	f	X
ejpam-2562	409	6	(	(	PUNCT
ejpam-2562	409	7	m	m	PROPN
ejpam-2562	409	8	)	)	PUNCT
ejpam-2562	409	9	.	.	PUNCT
ejpam-2562	410	1	(	(	PUNCT
ejpam-2562	410	2	iii	iii	X
ejpam-2562	410	3	)	)	PUNCT
ejpam-2562	410	4	injective	injective	ADJ
ejpam-2562	410	5	endomorphisms	endomorphism	NOUN
ejpam-2562	410	6	of	of	ADP
ejpam-2562	410	7	m	m	PROPN
ejpam-2562	410	8	map	map	VERB
ejpam-2562	410	9	semi	semi	ADJ
ejpam-2562	410	10	-	-	ADJ
ejpam-2562	410	11	essential	essential	ADJ
ejpam-2562	410	12	subsemimodules	subsemimodule	NOUN
ejpam-2562	410	13	to	to	ADP
ejpam-2562	410	14	semi	semi	ADJ
ejpam-2562	410	15	-	-	ADJ
ejpam-2562	410	16	essential	essential	ADJ
ejpam-2562	410	17	subsemimodules	subsemimodule	NOUN
ejpam-2562	410	18	.	.	PUNCT
ejpam-2562	411	1	proof	proof	NOUN
ejpam-2562	411	2	.	.	PUNCT
ejpam-2562	412	1	(	(	PUNCT
ejpam-2562	412	2	i)	i)	X
ejpam-2562	412	3	⇐	⇐	NOUN
ejpam-2562	412	4	⇒(ii	⇒(ii	ADV
ejpam-2562	412	5	)	)	PUNCT
ejpam-2562	412	6	by	by	ADP
ejpam-2562	412	7	definition	definition	NOUN
ejpam-2562	412	8	9	9	NUM
ejpam-2562	412	9	and	and	CCONJ
ejpam-2562	412	10	lemma	lemma	PROPN
ejpam-2562	412	11	2	2	NUM
ejpam-2562	412	12	.	.	PUNCT
ejpam-2562	412	13	(	(	PUNCT
ejpam-2562	412	14	iii)	iii)	NOUN
ejpam-2562	412	15	⇐	⇐	NOUN
ejpam-2562	412	16	⇒(i	⇒(i	NOUN
ejpam-2562	412	17	)	)	PUNCT
ejpam-2562	412	18	is	be	AUX
ejpam-2562	412	19	trivial	trivial	ADJ
ejpam-2562	412	20	.	.	PUNCT
ejpam-2562	413	1	(	(	PUNCT
ejpam-2562	413	2	i)	i)	X
ejpam-2562	413	3	⇐	⇐	NOUN
ejpam-2562	413	4	⇒(iii	⇒(iii	X
ejpam-2562	413	5	)	)	PUNCT
ejpam-2562	413	6	let	let	VERB
ejpam-2562	413	7	g	g	NOUN
ejpam-2562	413	8	:	:	PUNCT
ejpam-2562	413	9	m	m	VERB
ejpam-2562	413	10	→	→	NOUN
ejpam-2562	413	11	m	m	AUX
ejpam-2562	413	12	be	be	VERB
ejpam-2562	413	13	an	an	DET
ejpam-2562	413	14	injective	injective	ADJ
ejpam-2562	413	15	endomorphism	endomorphism	NOUN
ejpam-2562	413	16	of	of	ADP
ejpam-2562	413	17	m	m	PRON
ejpam-2562	413	18	,	,	PUNCT
ejpam-2562	413	19	and	and	CCONJ
ejpam-2562	413	20	let	let	VERB
ejpam-2562	413	21	k	k	PRON
ejpam-2562	413	22	be	be	AUX
ejpam-2562	413	23	a	a	DET
ejpam-2562	413	24	semi	semi	ADJ
ejpam-2562	413	25	-	-	ADJ
ejpam-2562	413	26	essential	essential	ADJ
ejpam-2562	413	27	subsemimodule	subsemimodule	NOUN
ejpam-2562	413	28	of	of	ADP
ejpam-2562	413	29	m.	m.	NOUN
ejpam-2562	413	30	let	let	VERB
ejpam-2562	413	31	us	we	PRON
ejpam-2562	413	32	prove	prove	VERB
ejpam-2562	413	33	that	that	PRON
ejpam-2562	413	34	g(k)ãs	g(k)ãs	VERB
ejpam-2562	413	35	m.	m.	NOUN
ejpam-2562	413	36	we	we	PRON
ejpam-2562	413	37	have	have	VERB
ejpam-2562	413	38	g(k)≤	g(k)≤	VERB
ejpam-2562	413	39	g(m)≤	g(m)≤	ADJ
ejpam-2562	413	40	m	m	PROPN
ejpam-2562	413	41	and	and	CCONJ
ejpam-2562	413	42	by	by	ADP
ejpam-2562	413	43	hypothesis	hypothesis	NOUN
ejpam-2562	413	44	g(m)ãs	g(m)ãs	NOUN
ejpam-2562	413	45	m	m	NOUN
ejpam-2562	413	46	,	,	PUNCT
ejpam-2562	413	47	so	so	ADV
ejpam-2562	413	48	,	,	PUNCT
ejpam-2562	413	49	according	accord	VERB
ejpam-2562	413	50	to	to	ADP
ejpam-2562	413	51	the	the	DET
ejpam-2562	413	52	proposition	proposition	NOUN
ejpam-2562	413	53	3	3	NUM
ejpam-2562	413	54	,	,	PUNCT
ejpam-2562	413	55	to	to	PART
ejpam-2562	413	56	obtain	obtain	VERB
ejpam-2562	413	57	g(k)ãs	g(k)ãs	NOUN
ejpam-2562	413	58	m	m	PRON
ejpam-2562	413	59	,	,	PUNCT
ejpam-2562	413	60	it	it	PRON
ejpam-2562	413	61	suffices	suffice	VERB
ejpam-2562	413	62	to	to	PART
ejpam-2562	413	63	prove	prove	VERB
ejpam-2562	413	64	that	that	PRON
ejpam-2562	413	65	g(k)ãs	g(k)ãs	ADV
ejpam-2562	413	66	g(m	g(m	NOUN
ejpam-2562	413	67	)	)	PUNCT
ejpam-2562	413	68	.	.	PUNCT
ejpam-2562	414	1	let	let	VERB
ejpam-2562	414	2	x	x	X
ejpam-2562	414	3	∈	∈	PROPN
ejpam-2562	414	4	g(m	g(m	VERB
ejpam-2562	414	5	)	)	PUNCT
ejpam-2562	414	6	.	.	PUNCT
ejpam-2562	415	1	so	so	ADV
ejpam-2562	415	2	there	there	PRON
ejpam-2562	415	3	exists	exist	VERB
ejpam-2562	415	4	m	m	VERB
ejpam-2562	415	5	∈	∈	PROPN
ejpam-2562	415	6	m	m	VERB
ejpam-2562	415	7	such	such	ADJ
ejpam-2562	415	8	that	that	SCONJ
ejpam-2562	415	9	x	x	SYM
ejpam-2562	415	10	=	=	PUNCT
ejpam-2562	415	11	g(m	g(m	PROPN
ejpam-2562	415	12	)	)	PUNCT
ejpam-2562	415	13	.	.	PUNCT
ejpam-2562	416	1	since	since	SCONJ
ejpam-2562	416	2	k	k	PROPN
ejpam-2562	416	3	ãs	ãs	INTJ
ejpam-2562	416	4	m	m	VERB
ejpam-2562	416	5	,	,	PUNCT
ejpam-2562	416	6	we	we	PRON
ejpam-2562	416	7	deduce	deduce	VERB
ejpam-2562	416	8	that	that	SCONJ
ejpam-2562	416	9	there	there	PRON
ejpam-2562	416	10	exists	exist	VERB
ejpam-2562	416	11	an	an	DET
ejpam-2562	416	12	r	r	NOUN
ejpam-2562	416	13	∈	∈	NOUN
ejpam-2562	416	14	r	r	NOUN
ejpam-2562	417	1	such	such	ADJ
ejpam-2562	417	2	that	that	PRON
ejpam-2562	417	3	rm	rm	PROPN
ejpam-2562	417	4	∈	∈	PROPN
ejpam-2562	418	1	k.	k.	PROPN
ejpam-2562	419	1	so	so	ADV
ejpam-2562	419	2	g(rm	g(rm	PROPN
ejpam-2562	419	3	)	)	PUNCT
ejpam-2562	419	4	=	=	SYM
ejpam-2562	419	5	r	r	NOUN
ejpam-2562	419	6	g(m	g(m	PROPN
ejpam-2562	419	7	)	)	PUNCT
ejpam-2562	419	8	=	=	SYM
ejpam-2562	419	9	r	r	NOUN
ejpam-2562	419	10	x	x	SYM
ejpam-2562	419	11	∈	∈	PROPN
ejpam-2562	419	12	g(k	g(k	PROPN
ejpam-2562	419	13	)	)	PUNCT
ejpam-2562	419	14	;	;	PUNCT
ejpam-2562	419	15	thus	thus	ADV
ejpam-2562	419	16	g(k)ãs	g(k)ãs	ADP
ejpam-2562	419	17	g(m	g(m	NOUN
ejpam-2562	419	18	)	)	PUNCT
ejpam-2562	419	19	.	.	PUNCT
ejpam-2562	420	1	proposition	proposition	NOUN
ejpam-2562	420	2	14	14	NUM
ejpam-2562	420	3	.	.	PUNCT
ejpam-2562	421	1	(	(	PUNCT
ejpam-2562	421	2	i	i	NOUN
ejpam-2562	421	3	)	)	PUNCT
ejpam-2562	421	4	a	a	DET
ejpam-2562	421	5	direct	direct	ADJ
ejpam-2562	421	6	summand	summand	NOUN
ejpam-2562	421	7	of	of	ADP
ejpam-2562	421	8	a	a	DET
ejpam-2562	421	9	weakly	weakly	ADJ
ejpam-2562	421	10	co	co	NOUN
ejpam-2562	421	11	-	-	NOUN
ejpam-2562	421	12	hopfian-2	hopfian-2	NUM
ejpam-2562	421	13	semimodule	semimodule	NOUN
ejpam-2562	421	14	is	be	AUX
ejpam-2562	421	15	weakly	weakly	ADJ
ejpam-2562	421	16	co	co	NOUN
ejpam-2562	421	17	-	-	NOUN
ejpam-2562	421	18	hopfian-2	hopfian-2	NOUN
ejpam-2562	421	19	.	.	PUNCT
ejpam-2562	422	1	(	(	PUNCT
ejpam-2562	422	2	ii	ii	NOUN
ejpam-2562	422	3	)	)	PUNCT
ejpam-2562	422	4	let	let	VERB
ejpam-2562	422	5	m	m	NOUN
ejpam-2562	422	6	=	=	SYM
ejpam-2562	422	7	m1	m1	PROPN
ejpam-2562	422	8	⊕	⊕	PROPN
ejpam-2562	422	9	m2	m2	PROPN
ejpam-2562	422	10	such	such	ADJ
ejpam-2562	422	11	that	that	SCONJ
ejpam-2562	422	12	each	each	DET
ejpam-2562	422	13	mi	mi	PROPN
ejpam-2562	422	14	is	be	AUX
ejpam-2562	422	15	fully	fully	ADV
ejpam-2562	422	16	invariant	invariant	ADJ
ejpam-2562	422	17	.	.	PUNCT
ejpam-2562	423	1	then	then	ADV
ejpam-2562	423	2	m	m	VERB
ejpam-2562	423	3	is	be	AUX
ejpam-2562	423	4	weakly	weakly	ADJ
ejpam-2562	423	5	co	co	NOUN
ejpam-2562	423	6	-	-	NOUN
ejpam-2562	423	7	hopfian-2	hopfian-2	NOUN
ejpam-2562	423	8	if	if	SCONJ
ejpam-2562	423	9	and	and	CCONJ
ejpam-2562	423	10	only	only	ADV
ejpam-2562	423	11	if	if	SCONJ
ejpam-2562	423	12	so	so	ADV
ejpam-2562	423	13	is	be	AUX
ejpam-2562	423	14	each	each	DET
ejpam-2562	423	15	mi	mi	PROPN
ejpam-2562	423	16	.	.	PUNCT
ejpam-2562	424	1	proof	proof	NOUN
ejpam-2562	424	2	.	.	PUNCT
ejpam-2562	425	1	(	(	PUNCT
ejpam-2562	425	2	i	i	NOUN
ejpam-2562	425	3	)	)	PUNCT
ejpam-2562	425	4	suppose	suppose	VERB
ejpam-2562	425	5	that	that	SCONJ
ejpam-2562	425	6	m	m	PROPN
ejpam-2562	425	7	is	be	AUX
ejpam-2562	425	8	a	a	DET
ejpam-2562	425	9	weakly	weakly	ADJ
ejpam-2562	425	10	co	co	NOUN
ejpam-2562	425	11	-	-	ADJ
ejpam-2562	425	12	hopfian-2	hopfian-2	NUM
ejpam-2562	425	13	semimodule	semimodule	NOUN
ejpam-2562	425	14	.	.	PUNCT
ejpam-2562	426	1	let	let	VERB
ejpam-2562	426	2	m	m	VERB
ejpam-2562	426	3	=	=	SYM
ejpam-2562	426	4	n	n	PROPN
ejpam-2562	426	5	⊕	⊕	PROPN
ejpam-2562	426	6	k	k	PROPN
ejpam-2562	426	7	and	and	CCONJ
ejpam-2562	426	8	let	let	VERB
ejpam-2562	426	9	f	f	X
ejpam-2562	426	10	:	:	PUNCT
ejpam-2562	426	11	n	n	CCONJ
ejpam-2562	426	12	−→	−→	NOUN
ejpam-2562	426	13	n	n	AUX
ejpam-2562	426	14	be	be	AUX
ejpam-2562	426	15	an	an	DET
ejpam-2562	426	16	injective	injective	ADJ
ejpam-2562	426	17	endomorphism	endomorphism	NOUN
ejpam-2562	426	18	of	of	ADP
ejpam-2562	426	19	n	n	PROPN
ejpam-2562	426	20	.	.	PUNCT
ejpam-2562	427	1	then	then	ADV
ejpam-2562	427	2	the	the	DET
ejpam-2562	427	3	map	map	NOUN
ejpam-2562	427	4	f	f	PROPN
ejpam-2562	427	5	⊕	⊕	PROPN
ejpam-2562	427	6	idk	idk	INTJ
ejpam-2562	427	7	:	:	PUNCT
ejpam-2562	427	8	m	m	VERB
ejpam-2562	427	9	=	=	SYM
ejpam-2562	427	10	n	n	PROPN
ejpam-2562	427	11	⊕	⊕	PROPN
ejpam-2562	427	12	k	k	PROPN
ejpam-2562	427	13	→	→	PUNCT
ejpam-2562	427	14	m	m	NOUN
ejpam-2562	427	15	=	=	SYM
ejpam-2562	427	16	n	n	PROPN
ejpam-2562	427	17	⊕	⊕	PROPN
ejpam-2562	427	18	k	k	PROPN
ejpam-2562	427	19	defined	define	VERB
ejpam-2562	427	20	by	by	ADP
ejpam-2562	427	21	n	n	PROPN
ejpam-2562	427	22	+	+	CCONJ
ejpam-2562	427	23	k	k	PROPN
ejpam-2562	427	24	7→	7→	PROPN
ejpam-2562	427	25	(	(	PUNCT
ejpam-2562	427	26	f	f	PROPN
ejpam-2562	427	27	⊕	⊕	PROPN
ejpam-2562	427	28	idk)(n	idk)(n	NOUN
ejpam-2562	428	1	+	+	X
ejpam-2562	428	2	k	k	X
ejpam-2562	428	3	)	)	PUNCT
ejpam-2562	428	4	=	=	SYM
ejpam-2562	428	5	f	f	PROPN
ejpam-2562	428	6	(	(	PUNCT
ejpam-2562	428	7	n	n	CCONJ
ejpam-2562	428	8	)	)	PUNCT
ejpam-2562	429	1	+	+	CCONJ
ejpam-2562	429	2	k	k	PROPN
ejpam-2562	429	3	is	be	AUX
ejpam-2562	429	4	an	an	DET
ejpam-2562	429	5	injective	injective	ADJ
ejpam-2562	429	6	endomorphism	endomorphism	NOUN
ejpam-2562	429	7	of	of	ADP
ejpam-2562	429	8	m	m	PRON
ejpam-2562	429	9	and	and	CCONJ
ejpam-2562	429	10	so	so	ADV
ejpam-2562	429	11	(	(	PUNCT
ejpam-2562	429	12	f	f	PROPN
ejpam-2562	429	13	⊕	⊕	PROPN
ejpam-2562	429	14	idk)(m	idk)(m	VERB
ejpam-2562	429	15	)	)	PUNCT
ejpam-2562	430	1	ãs	ãs	ADP
ejpam-2562	430	2	m	m	VERB
ejpam-2562	430	3	.	.	PUNCT
ejpam-2562	431	1	since	since	SCONJ
ejpam-2562	431	2	(	(	PUNCT
ejpam-2562	431	3	f	f	PROPN
ejpam-2562	431	4	⊕	⊕	PROPN
ejpam-2562	431	5	idk)(m	idk)(m	VERB
ejpam-2562	431	6	)	)	PUNCT
ejpam-2562	432	1	≤	≤	NUM
ejpam-2562	432	2	f	f	X
ejpam-2562	432	3	(	(	PUNCT
ejpam-2562	432	4	n)⊕	n)⊕	NOUN
ejpam-2562	432	5	k	k	PROPN
ejpam-2562	432	6	,	,	PUNCT
ejpam-2562	432	7	we	we	PRON
ejpam-2562	432	8	deduce	deduce	VERB
ejpam-2562	432	9	by	by	ADP
ejpam-2562	432	10	proposition	proposition	NOUN
ejpam-2562	432	11	3	3	NUM
ejpam-2562	432	12	that	that	SCONJ
ejpam-2562	432	13	f	f	PROPN
ejpam-2562	432	14	(	(	PUNCT
ejpam-2562	432	15	n	n	CCONJ
ejpam-2562	432	16	)	)	PUNCT
ejpam-2562	432	17	⊕	⊕	PROPN
ejpam-2562	433	1	k	k	PROPN
ejpam-2562	434	1	ãs	ãs	PRON
ejpam-2562	434	2	n	n	PROPN
ejpam-2562	434	3	⊕	⊕	PROPN
ejpam-2562	434	4	k	k	PROPN
ejpam-2562	434	5	and	and	CCONJ
ejpam-2562	434	6	by	by	ADP
ejpam-2562	434	7	proposition	proposition	NOUN
ejpam-2562	434	8	5	5	NUM
ejpam-2562	435	1	that	that	SCONJ
ejpam-2562	435	2	f	f	PROPN
ejpam-2562	435	3	(	(	PUNCT
ejpam-2562	435	4	n	n	CCONJ
ejpam-2562	435	5	)	)	PUNCT
ejpam-2562	435	6	ãs	ãs	ADP
ejpam-2562	435	7	n	n	PROPN
ejpam-2562	435	8	,	,	PUNCT
ejpam-2562	435	9	therefore	therefore	ADV
ejpam-2562	435	10	n	n	PRON
ejpam-2562	435	11	is	be	AUX
ejpam-2562	435	12	weakly	weakly	ADJ
ejpam-2562	435	13	co	co	NOUN
ejpam-2562	435	14	-	-	NOUN
ejpam-2562	435	15	hopfian-2	hopfian-2	NOUN
ejpam-2562	435	16	.	.	PUNCT
ejpam-2562	436	1	(	(	PUNCT
ejpam-2562	436	2	2)=⇒	2)=⇒	NUM
ejpam-2562	436	3	)	)	PUNCT
ejpam-2562	436	4	by	by	ADP
ejpam-2562	436	5	(	(	PUNCT
ejpam-2562	436	6	i	i	NOUN
ejpam-2562	436	7	)	)	PUNCT
ejpam-2562	436	8	.	.	PUNCT
ejpam-2562	437	1	⇐	⇐	PROPN
ejpam-2562	437	2	=)	=)	PROPN
ejpam-2562	437	3	let	let	VERB
ejpam-2562	437	4	f	f	NOUN
ejpam-2562	437	5	:	:	PUNCT
ejpam-2562	437	6	m	m	VERB
ejpam-2562	437	7	=	=	PUNCT
ejpam-2562	438	1	m1⊕m2	m1⊕m2	ADJ
ejpam-2562	438	2	−→	−→	NOUN
ejpam-2562	438	3	m	m	NOUN
ejpam-2562	438	4	=	=	NOUN
ejpam-2562	438	5	m1⊕m2	m1⊕m2	NOUN
ejpam-2562	438	6	be	be	AUX
ejpam-2562	438	7	an	an	DET
ejpam-2562	438	8	injective	injective	ADJ
ejpam-2562	438	9	endomorphism	endomorphism	NOUN
ejpam-2562	438	10	of	of	ADP
ejpam-2562	438	11	m	m	PROPN
ejpam-2562	438	12	=	=	SYM
ejpam-2562	438	13	m1⊕m2	m1⊕m2	PROPN
ejpam-2562	438	14	.	.	PUNCT
ejpam-2562	439	1	we	we	PRON
ejpam-2562	439	2	have	have	VERB
ejpam-2562	439	3	f	f	X
ejpam-2562	439	4	(	(	PUNCT
ejpam-2562	439	5	m1)⊕	m1)⊕	NOUN
ejpam-2562	439	6	f	f	X
ejpam-2562	439	7	(	(	PUNCT
ejpam-2562	439	8	m2	m2	PROPN
ejpam-2562	439	9	)	)	PUNCT
ejpam-2562	439	10	⊆	⊆	NUM
ejpam-2562	439	11	f	f	X
ejpam-2562	439	12	(	(	PUNCT
ejpam-2562	439	13	m1	m1	PROPN
ejpam-2562	439	14	⊕m2	⊕m2	NUM
ejpam-2562	439	15	)	)	PUNCT
ejpam-2562	439	16	⊆	⊆	NUM
ejpam-2562	439	17	m1	m1	PROPN
ejpam-2562	439	18	⊕m2	⊕m2	NOUN
ejpam-2562	439	19	.	.	PUNCT
ejpam-2562	440	1	by	by	ADP
ejpam-2562	440	2	assumption	assumption	NOUN
ejpam-2562	440	3	we	we	PRON
ejpam-2562	440	4	have	have	VERB
ejpam-2562	440	5	f	f	PROPN
ejpam-2562	440	6	(	(	PUNCT
ejpam-2562	440	7	m1	m1	PROPN
ejpam-2562	440	8	)	)	PUNCT
ejpam-2562	440	9	ãs	ãs	DET
ejpam-2562	440	10	m1	m1	PROPN
ejpam-2562	440	11	and	and	CCONJ
ejpam-2562	440	12	f	f	PROPN
ejpam-2562	440	13	(	(	PUNCT
ejpam-2562	440	14	m2	m2	PROPN
ejpam-2562	440	15	)	)	PUNCT
ejpam-2562	440	16	ãs	ãs	DET
ejpam-2562	440	17	m2	m2	PROPN
ejpam-2562	440	18	,	,	PUNCT
ejpam-2562	440	19	and	and	CCONJ
ejpam-2562	440	20	by	by	ADP
ejpam-2562	440	21	the	the	DET
ejpam-2562	440	22	proposition	proposition	NOUN
ejpam-2562	440	23	5	5	NUM
ejpam-2562	440	24	,	,	PUNCT
ejpam-2562	440	25	we	we	PRON
ejpam-2562	440	26	obtain	obtain	VERB
ejpam-2562	440	27	f	f	X
ejpam-2562	440	28	(	(	PUNCT
ejpam-2562	440	29	m1	m1	NOUN
ejpam-2562	440	30	)	)	PUNCT
ejpam-2562	440	31	⊕	⊕	PROPN
ejpam-2562	440	32	f	f	PROPN
ejpam-2562	441	1	(	(	PUNCT
ejpam-2562	441	2	m2	m2	PROPN
ejpam-2562	441	3	)	)	PUNCT
ejpam-2562	441	4	ãs	ãs	PRON
ejpam-2562	441	5	m1	m1	PROPN
ejpam-2562	441	6	⊕	⊕	PROPN
ejpam-2562	441	7	m2	m2	PROPN
ejpam-2562	441	8	and	and	CCONJ
ejpam-2562	441	9	consequently	consequently	ADV
ejpam-2562	441	10	f	f	PROPN
ejpam-2562	442	1	(	(	PUNCT
ejpam-2562	442	2	m1	m1	PROPN
ejpam-2562	442	3	⊕m2)ãs	⊕m2)ãs	PROPN
ejpam-2562	442	4	m1	m1	PROPN
ejpam-2562	442	5	⊕m2	⊕m2	PROPN
ejpam-2562	442	6	.	.	PUNCT
ejpam-2562	443	1	e.	e.	PROPN
ejpam-2562	443	2	diop	diop	PROPN
ejpam-2562	443	3	,	,	PUNCT
ejpam-2562	443	4	d.	d.	PROPN
ejpam-2562	443	5	sow	sow	PROPN
ejpam-2562	443	6	/	/	SYM
ejpam-2562	443	7	eur	eur	PROPN
ejpam-2562	443	8	.	.	PUNCT
ejpam-2562	444	1	j.	j.	PROPN
ejpam-2562	444	2	pure	pure	PROPN
ejpam-2562	444	3	appl	appl	PROPN
ejpam-2562	444	4	.	.	PROPN
ejpam-2562	444	5	math	math	PROPN
ejpam-2562	444	6	,	,	PUNCT
ejpam-2562	444	7	9	9	NUM
ejpam-2562	444	8	(	(	PUNCT
ejpam-2562	444	9	2016	2016	NUM
ejpam-2562	444	10	)	)	PUNCT
ejpam-2562	444	11	,	,	PUNCT
ejpam-2562	444	12	250	250	NUM
ejpam-2562	444	13	-	-	SYM
ejpam-2562	444	14	265	265	NUM
ejpam-2562	444	15	263	263	NUM
ejpam-2562	444	16	4.3	4.3	NUM
ejpam-2562	444	17	.	.	PUNCT
ejpam-2562	445	1	on	on	ADP
ejpam-2562	445	2	weakly	weakly	ADJ
ejpam-2562	445	3	co	co	ADJ
ejpam-2562	445	4	-	-	ADJ
ejpam-2562	445	5	hopfian	hopfian	ADJ
ejpam-2562	445	6	semimodules	semimodule	NOUN
ejpam-2562	445	7	of	of	ADP
ejpam-2562	445	8	type	type	NOUN
ejpam-2562	445	9	3	3	NUM
ejpam-2562	445	10	definition	definition	NOUN
ejpam-2562	445	11	10	10	NUM
ejpam-2562	445	12	.	.	PUNCT
ejpam-2562	446	1	a	a	DET
ejpam-2562	446	2	nonzero	nonzero	NOUN
ejpam-2562	446	3	left	leave	VERB
ejpam-2562	446	4	r	r	NOUN
ejpam-2562	446	5	-	-	PUNCT
ejpam-2562	446	6	semimodule	semimodule	NOUN
ejpam-2562	446	7	rm	rm	NOUN
ejpam-2562	446	8	is	be	AUX
ejpam-2562	446	9	said	say	VERB
ejpam-2562	446	10	to	to	PART
ejpam-2562	446	11	be	be	AUX
ejpam-2562	446	12	weakly	weakly	ADJ
ejpam-2562	446	13	co	co	NOUN
ejpam-2562	446	14	-	-	NOUN
ejpam-2562	446	15	hopfian-3	hopfian-3	NOUN
ejpam-2562	446	16	(	(	PUNCT
ejpam-2562	446	17	denoted	denote	VERB
ejpam-2562	446	18	by	by	ADP
ejpam-2562	446	19	wch-3	wch-3	NOUN
ejpam-2562	446	20	)	)	PUNCT
ejpam-2562	446	21	if	if	SCONJ
ejpam-2562	446	22	every	every	DET
ejpam-2562	446	23	monomorphism	monomorphism	NOUN
ejpam-2562	446	24	f	f	X
ejpam-2562	446	25	:	:	PUNCT
ejpam-2562	446	26	m	m	AUX
ejpam-2562	446	27	→	→	NOUN
ejpam-2562	446	28	m	m	VERB
ejpam-2562	446	29	is	be	AUX
ejpam-2562	446	30	essential	essential	ADJ
ejpam-2562	447	1	i.e	i.e	PRON
ejpam-2562	447	2	f	f	X
ejpam-2562	447	3	(	(	PUNCT
ejpam-2562	447	4	m)ã	m)ã	NOUN
ejpam-2562	447	5	m.	m.	NOUN
ejpam-2562	447	6	proposition	proposition	NOUN
ejpam-2562	447	7	15	15	NUM
ejpam-2562	447	8	.	.	PUNCT
ejpam-2562	448	1	the	the	DET
ejpam-2562	448	2	following	follow	VERB
ejpam-2562	448	3	are	be	AUX
ejpam-2562	448	4	equivalent	equivalent	ADJ
ejpam-2562	448	5	conditions	condition	NOUN
ejpam-2562	448	6	on	on	ADP
ejpam-2562	448	7	a	a	DET
ejpam-2562	448	8	left	left	ADJ
ejpam-2562	448	9	r	r	NOUN
ejpam-2562	448	10	-	-	PUNCT
ejpam-2562	448	11	semimodule	semimodule	NOUN
ejpam-2562	448	12	m.	m.	NOUN
ejpam-2562	448	13	(	(	PUNCT
ejpam-2562	448	14	i	i	NOUN
ejpam-2562	448	15	)	)	PUNCT
ejpam-2562	448	16	m	m	VERB
ejpam-2562	448	17	is	be	AUX
ejpam-2562	448	18	weakly	weakly	ADJ
ejpam-2562	448	19	co	co	NOUN
ejpam-2562	448	20	-	-	NOUN
ejpam-2562	448	21	hopfian-3	hopfian-3	NOUN
ejpam-2562	448	22	.	.	PUNCT
ejpam-2562	449	1	(	(	PUNCT
ejpam-2562	449	2	ii	ii	NOUN
ejpam-2562	449	3	)	)	PUNCT
ejpam-2562	449	4	if	if	SCONJ
ejpam-2562	449	5	m	m	ADJ
ejpam-2562	449	6	and	and	CCONJ
ejpam-2562	449	7	m′	m′	NOUN
ejpam-2562	449	8	are	be	AUX
ejpam-2562	449	9	distinct	distinct	ADJ
ejpam-2562	449	10	elements	element	NOUN
ejpam-2562	449	11	of	of	ADP
ejpam-2562	449	12	m	m	PRON
ejpam-2562	449	13	then	then	ADV
ejpam-2562	449	14	for	for	ADP
ejpam-2562	449	15	every	every	DET
ejpam-2562	449	16	monomorphism	monomorphism	NOUN
ejpam-2562	450	1	f	f	X
ejpam-2562	450	2	:	:	PUNCT
ejpam-2562	450	3	m	m	VERB
ejpam-2562	450	4	→	→	SYM
ejpam-2562	450	5	m	m	NOUN
ejpam-2562	450	6	,	,	PUNCT
ejpam-2562	450	7	there	there	PRON
ejpam-2562	450	8	exist	exist	VERB
ejpam-2562	450	9	distinct	distinct	ADJ
ejpam-2562	450	10	elements	element	NOUN
ejpam-2562	450	11	f	f	X
ejpam-2562	450	12	(	(	PUNCT
ejpam-2562	450	13	m1	m1	PROPN
ejpam-2562	450	14	)	)	PUNCT
ejpam-2562	450	15	and	and	CCONJ
ejpam-2562	450	16	f	f	PROPN
ejpam-2562	450	17	(	(	PUNCT
ejpam-2562	450	18	m2	m2	PROPN
ejpam-2562	450	19	)	)	PUNCT
ejpam-2562	450	20	of	of	ADP
ejpam-2562	450	21	f	f	PROPN
ejpam-2562	450	22	(	(	PUNCT
ejpam-2562	450	23	m	m	NOUN
ejpam-2562	450	24	)	)	PUNCT
ejpam-2562	450	25	satisfying	satisfy	VERB
ejpam-2562	450	26	f	f	X
ejpam-2562	450	27	(	(	PUNCT
ejpam-2562	450	28	m1)ρ(m	m1)ρ(m	PROPN
ejpam-2562	450	29	,	,	PUNCT
ejpam-2562	450	30	m′	m′	NUM
ejpam-2562	450	31	)	)	PUNCT
ejpam-2562	450	32	f	f	PROPN
ejpam-2562	450	33	(	(	PUNCT
ejpam-2562	450	34	m2	m2	PROPN
ejpam-2562	450	35	)	)	PUNCT
ejpam-2562	450	36	.	.	PUNCT
ejpam-2562	451	1	proof	proof	NOUN
ejpam-2562	451	2	.	.	PUNCT
ejpam-2562	452	1	by	by	ADP
ejpam-2562	452	2	definition	definition	NOUN
ejpam-2562	452	3	10	10	NUM
ejpam-2562	452	4	and	and	CCONJ
ejpam-2562	452	5	lemma	lemma	PROPN
ejpam-2562	452	6	1	1	NUM
ejpam-2562	452	7	.	.	PUNCT
ejpam-2562	452	8	proposition	proposition	NOUN
ejpam-2562	452	9	16	16	NUM
ejpam-2562	452	10	.	.	PUNCT
ejpam-2562	453	1	if	if	SCONJ
ejpam-2562	453	2	m	m	NOUN
ejpam-2562	453	3	is	be	AUX
ejpam-2562	453	4	wch-3	wch-3	VERB
ejpam-2562	453	5	,	,	PUNCT
ejpam-2562	453	6	then	then	ADV
ejpam-2562	453	7	m	m	PROPN
ejpam-2562	453	8	is	be	AUX
ejpam-2562	453	9	wch-1	wch-1	X
ejpam-2562	453	10	.	.	PUNCT
ejpam-2562	454	1	proof	proof	NOUN
ejpam-2562	454	2	.	.	PUNCT
ejpam-2562	455	1	by	by	ADP
ejpam-2562	455	2	proposition	proposition	NOUN
ejpam-2562	455	3	2	2	NUM
ejpam-2562	455	4	.	.	PUNCT
ejpam-2562	455	5	by	by	ADP
ejpam-2562	455	6	a	a	DET
ejpam-2562	455	7	same	same	ADJ
ejpam-2562	455	8	way	way	NOUN
ejpam-2562	455	9	as	as	ADP
ejpam-2562	455	10	above	above	ADV
ejpam-2562	455	11	,	,	PUNCT
ejpam-2562	455	12	we	we	PRON
ejpam-2562	455	13	show	show	VERB
ejpam-2562	455	14	the	the	DET
ejpam-2562	455	15	following	follow	VERB
ejpam-2562	455	16	results	result	NOUN
ejpam-2562	455	17	.	.	PUNCT
ejpam-2562	456	1	proposition	proposition	NOUN
ejpam-2562	456	2	17	17	NUM
ejpam-2562	456	3	.	.	PUNCT
ejpam-2562	457	1	(	(	PUNCT
ejpam-2562	457	2	i	i	NOUN
ejpam-2562	457	3	)	)	PUNCT
ejpam-2562	457	4	the	the	DET
ejpam-2562	457	5	following	follow	VERB
ejpam-2562	457	6	are	be	AUX
ejpam-2562	457	7	equivalent	equivalent	ADJ
ejpam-2562	457	8	conditions	condition	NOUN
ejpam-2562	457	9	on	on	ADP
ejpam-2562	457	10	a	a	DET
ejpam-2562	457	11	left	left	ADJ
ejpam-2562	457	12	r	r	NOUN
ejpam-2562	457	13	-	-	PUNCT
ejpam-2562	457	14	semimodule	semimodule	NOUN
ejpam-2562	457	15	m.	m.	NOUN
ejpam-2562	457	16	(	(	PUNCT
ejpam-2562	457	17	a	a	X
ejpam-2562	457	18	)	)	PUNCT
ejpam-2562	457	19	m	m	VERB
ejpam-2562	457	20	is	be	AUX
ejpam-2562	457	21	weakly	weakly	ADJ
ejpam-2562	457	22	co	co	NOUN
ejpam-2562	457	23	-	-	NOUN
ejpam-2562	457	24	hopfian-3	hopfian-3	NOUN
ejpam-2562	457	25	.	.	PUNCT
ejpam-2562	458	1	(	(	PUNCT
ejpam-2562	458	2	b	b	X
ejpam-2562	458	3	)	)	PUNCT
ejpam-2562	458	4	there	there	PRON
ejpam-2562	458	5	exists	exist	VERB
ejpam-2562	458	6	a	a	DET
ejpam-2562	458	7	subsemimodule	subsemimodule	NOUN
ejpam-2562	458	8	k	k	NOUN
ejpam-2562	458	9	of	of	ADP
ejpam-2562	458	10	m	m	PROPN
ejpam-2562	458	11	such	such	ADJ
ejpam-2562	458	12	that	that	SCONJ
ejpam-2562	458	13	g(k)ã	g(k)ã	PROPN
ejpam-2562	458	14	m	m	NOUN
ejpam-2562	458	15	for	for	ADP
ejpam-2562	458	16	all	all	DET
ejpam-2562	458	17	injective	injective	ADJ
ejpam-2562	458	18	g	g	PROPN
ejpam-2562	458	19	∈	∈	PROPN
ejpam-2562	458	20	end(m	end(m	PROPN
ejpam-2562	458	21	)	)	PUNCT
ejpam-2562	458	22	.	.	PUNCT
ejpam-2562	459	1	(	(	PUNCT
ejpam-2562	459	2	ii	ii	X
ejpam-2562	459	3	)	)	PUNCT
ejpam-2562	459	4	a	a	DET
ejpam-2562	459	5	direct	direct	ADJ
ejpam-2562	459	6	summand	summand	NOUN
ejpam-2562	459	7	of	of	ADP
ejpam-2562	459	8	a	a	DET
ejpam-2562	459	9	weakly	weakly	ADJ
ejpam-2562	459	10	co	co	NOUN
ejpam-2562	459	11	-	-	NOUN
ejpam-2562	459	12	hopfian-3	hopfian-3	NUM
ejpam-2562	459	13	semimodule	semimodule	NOUN
ejpam-2562	459	14	is	be	AUX
ejpam-2562	459	15	weakly	weakly	ADJ
ejpam-2562	459	16	co	co	NOUN
ejpam-2562	459	17	-	-	NOUN
ejpam-2562	459	18	hopfian-3	hopfian-3	NOUN
ejpam-2562	459	19	.	.	PUNCT
ejpam-2562	460	1	(	(	PUNCT
ejpam-2562	460	2	iii	iii	X
ejpam-2562	460	3	)	)	PUNCT
ejpam-2562	460	4	let	let	VERB
ejpam-2562	460	5	m	m	NOUN
ejpam-2562	460	6	=	=	SYM
ejpam-2562	460	7	m1	m1	PROPN
ejpam-2562	460	8	⊕	⊕	PROPN
ejpam-2562	460	9	m2	m2	PROPN
ejpam-2562	460	10	such	such	ADJ
ejpam-2562	460	11	that	that	SCONJ
ejpam-2562	460	12	each	each	DET
ejpam-2562	460	13	mi	mi	PROPN
ejpam-2562	460	14	is	be	AUX
ejpam-2562	460	15	fully	fully	ADV
ejpam-2562	460	16	invariant.then	invariant.then	ADP
ejpam-2562	460	17	m	m	NOUN
ejpam-2562	460	18	is	be	AUX
ejpam-2562	460	19	weakly	weakly	ADJ
ejpam-2562	460	20	co	co	NOUN
ejpam-2562	460	21	-	-	NOUN
ejpam-2562	460	22	hopfian-3	hopfian-3	NOUN
ejpam-2562	460	23	if	if	SCONJ
ejpam-2562	460	24	and	and	CCONJ
ejpam-2562	460	25	only	only	ADV
ejpam-2562	460	26	if	if	SCONJ
ejpam-2562	460	27	so	so	ADV
ejpam-2562	460	28	is	be	AUX
ejpam-2562	460	29	each	each	DET
ejpam-2562	460	30	mi	mi	PROPN
ejpam-2562	460	31	.	.	PUNCT
ejpam-2562	461	1	proposition	proposition	PROPN
ejpam-2562	461	2	18	18	NUM
ejpam-2562	461	3	.	.	PUNCT
ejpam-2562	462	1	any	any	PRON
ejpam-2562	462	2	r−simple	r−simple	ADP
ejpam-2562	462	3	semimodule	semimodule	NOUN
ejpam-2562	462	4	is	be	AUX
ejpam-2562	462	5	wch-3	wch-3	X
ejpam-2562	462	6	proof	proof	NOUN
ejpam-2562	462	7	.	.	PUNCT
ejpam-2562	463	1	obvious	obvious	ADJ
ejpam-2562	463	2	.	.	PUNCT
ejpam-2562	464	1	4.4	4.4	NUM
ejpam-2562	464	2	.	.	PUNCT
ejpam-2562	465	1	examples	example	NOUN
ejpam-2562	465	2	of	of	ADP
ejpam-2562	465	3	co	co	ADJ
ejpam-2562	465	4	-	-	ADJ
ejpam-2562	465	5	hopfian	hopfian	ADJ
ejpam-2562	465	6	semimodules	semimodule	NOUN
ejpam-2562	465	7	here	here	ADV
ejpam-2562	465	8	we	we	PRON
ejpam-2562	465	9	give	give	VERB
ejpam-2562	465	10	examples	example	NOUN
ejpam-2562	465	11	of	of	ADP
ejpam-2562	465	12	co	co	ADJ
ejpam-2562	465	13	-	-	ADJ
ejpam-2562	465	14	hopfian	hopfian	ADJ
ejpam-2562	465	15	semimodules	semimodule	NOUN
ejpam-2562	465	16	.	.	PUNCT
ejpam-2562	466	1	example	example	NOUN
ejpam-2562	467	1	5	5	NUM
ejpam-2562	467	2	.	.	PUNCT
ejpam-2562	467	3	recall	recall	VERB
ejpam-2562	467	4	the	the	DET
ejpam-2562	467	5	semimodule	semimodule	NOUN
ejpam-2562	467	6	(	(	PUNCT
ejpam-2562	467	7	m	m	PROPN
ejpam-2562	467	8	,	,	PUNCT
ejpam-2562	467	9	⊕,⋆	⊕,⋆	NOUN
ejpam-2562	467	10	)	)	PUNCT
ejpam-2562	467	11	of	of	ADP
ejpam-2562	467	12	example	example	NOUN
ejpam-2562	468	1	1	1	X
ejpam-2562	468	2	.	.	PUNCT
ejpam-2562	469	1	let	let	VERB
ejpam-2562	469	2	us	we	PRON
ejpam-2562	469	3	show	show	VERB
ejpam-2562	469	4	that	that	SCONJ
ejpam-2562	469	5	m	m	PROPN
ejpam-2562	469	6	is	be	AUX
ejpam-2562	469	7	wch-1	wch-1	ADV
ejpam-2562	469	8	,	,	PUNCT
ejpam-2562	469	9	wch-2	wch-2	ADP
ejpam-2562	469	10	and	and	CCONJ
ejpam-2562	469	11	wch-3	wch-3	X
ejpam-2562	469	12	.	.	PUNCT
ejpam-2562	470	1	let	let	VERB
ejpam-2562	470	2	g	g	NOUN
ejpam-2562	470	3	:	:	PUNCT
ejpam-2562	470	4	m	m	VERB
ejpam-2562	470	5	−→	−→	ADJ
ejpam-2562	470	6	m	m	AUX
ejpam-2562	470	7	be	be	VERB
ejpam-2562	470	8	an	an	DET
ejpam-2562	470	9	injective	injective	ADJ
ejpam-2562	470	10	endomorphism	endomorphism	NOUN
ejpam-2562	470	11	of	of	ADP
ejpam-2562	470	12	m.	m.	NOUN
ejpam-2562	470	13	so	so	SCONJ
ejpam-2562	470	14	g	g	PROPN
ejpam-2562	470	15	verifies	verifie	NOUN
ejpam-2562	470	16	the	the	DET
ejpam-2562	470	17	following	follow	VERB
ejpam-2562	470	18	property	property	NOUN
ejpam-2562	470	19	:	:	PUNCT
ejpam-2562	470	20	∀m	∀m	NUM
ejpam-2562	470	21	,	,	PUNCT
ejpam-2562	470	22	m′	m′	NOUN
ejpam-2562	470	23	∈	∈	NOUN
ejpam-2562	470	24	m	m	NOUN
ejpam-2562	470	25	:	:	PUNCT
ejpam-2562	470	26	m≤	m≤	VERB
ejpam-2562	470	27	m′	m′	ADJ
ejpam-2562	470	28	=	=	NOUN
ejpam-2562	470	29	⇒	⇒	PROPN
ejpam-2562	470	30	g(m)≤	g(m)≤	NOUN
ejpam-2562	470	31	g(m′	g(m′	PROPN
ejpam-2562	470	32	)	)	PUNCT
ejpam-2562	470	33	.	.	PUNCT
ejpam-2562	471	1	this	this	DET
ejpam-2562	471	2	property	property	NOUN
ejpam-2562	471	3	with	with	ADP
ejpam-2562	471	4	the	the	DET
ejpam-2562	471	5	injectivity	injectivity	NOUN
ejpam-2562	471	6	of	of	ADP
ejpam-2562	471	7	g	g	NOUN
ejpam-2562	471	8	make	make	VERB
ejpam-2562	471	9	that	that	SCONJ
ejpam-2562	471	10	there	there	PRON
ejpam-2562	471	11	exists	exist	VERB
ejpam-2562	471	12	an	an	DET
ejpam-2562	471	13	unique	unique	ADJ
ejpam-2562	471	14	injective	injective	ADJ
ejpam-2562	471	15	endomorphism	endomorphism	NOUN
ejpam-2562	471	16	of	of	ADP
ejpam-2562	471	17	m	m	PRON
ejpam-2562	471	18	,	,	PUNCT
ejpam-2562	471	19	namely	namely	ADV
ejpam-2562	471	20	the	the	DET
ejpam-2562	471	21	identity	identity	NOUN
ejpam-2562	471	22	map	map	NOUN
ejpam-2562	471	23	on	on	ADP
ejpam-2562	471	24	m.	m.	NOUN
ejpam-2562	471	25	we	we	PRON
ejpam-2562	471	26	deduce	deduce	VERB
ejpam-2562	471	27	that	that	SCONJ
ejpam-2562	471	28	the	the	DET
ejpam-2562	471	29	image	image	NOUN
ejpam-2562	471	30	of	of	ADP
ejpam-2562	471	31	any	any	DET
ejpam-2562	471	32	injective	injective	ADJ
ejpam-2562	471	33	endomorphism	endomorphism	NOUN
ejpam-2562	471	34	of	of	ADP
ejpam-2562	471	35	m	m	PROPN
ejpam-2562	471	36	is	be	AUX
ejpam-2562	471	37	both	both	PRON
ejpam-2562	471	38	semi	semi	ADJ
ejpam-2562	471	39	-	-	ADJ
ejpam-2562	471	40	weakly	weakly	ADJ
ejpam-2562	471	41	-	-	PUNCT
ejpam-2562	471	42	essential	essential	ADJ
ejpam-2562	471	43	,	,	PUNCT
ejpam-2562	471	44	semi	semi	ADJ
ejpam-2562	471	45	-	-	ADJ
ejpam-2562	471	46	essential	essential	ADJ
ejpam-2562	471	47	and	and	CCONJ
ejpam-2562	471	48	essential	essential	ADJ
ejpam-2562	471	49	and	and	CCONJ
ejpam-2562	471	50	consequently	consequently	ADV
ejpam-2562	471	51	that	that	SCONJ
ejpam-2562	471	52	m	m	PROPN
ejpam-2562	471	53	is	be	AUX
ejpam-2562	471	54	wch-1	wch-1	ADV
ejpam-2562	471	55	,	,	PUNCT
ejpam-2562	471	56	wch-2	wch-2	ADP
ejpam-2562	471	57	and	and	CCONJ
ejpam-2562	471	58	wch-3	wch-3	X
ejpam-2562	471	59	.	.	PUNCT
ejpam-2562	472	1	references	reference	NOUN
ejpam-2562	472	2	264	264	NUM
ejpam-2562	472	3	example	example	NOUN
ejpam-2562	472	4	6	6	NUM
ejpam-2562	472	5	.	.	PUNCT
ejpam-2562	473	1	let	let	VERB
ejpam-2562	473	2	n	n	PRON
ejpam-2562	473	3	be	be	AUX
ejpam-2562	473	4	the	the	DET
ejpam-2562	473	5	set	set	NOUN
ejpam-2562	473	6	of	of	ADP
ejpam-2562	473	7	natural	natural	ADJ
ejpam-2562	473	8	integers	integer	NOUN
ejpam-2562	473	9	and	and	CCONJ
ejpam-2562	473	10	put	put	NOUN
ejpam-2562	473	11	(	(	PUNCT
ejpam-2562	473	12	n,⊕,⊙	n,⊕,⊙	PROPN
ejpam-2562	473	13	)	)	PUNCT
ejpam-2562	473	14	the	the	DET
ejpam-2562	473	15	semiring	semiring	NOUN
ejpam-2562	473	16	where	where	SCONJ
ejpam-2562	473	17	∀a	∀a	NOUN
ejpam-2562	473	18	,	,	PUNCT
ejpam-2562	473	19	b	b	PROPN
ejpam-2562	473	20	∈	∈	PROPN
ejpam-2562	473	21	n	n	CCONJ
ejpam-2562	473	22	,	,	PUNCT
ejpam-2562	473	23	a⊕b	a⊕b	PROPN
ejpam-2562	473	24	=	=	SYM
ejpam-2562	473	25	gcd(a	gcd(a	PROPN
ejpam-2562	473	26	,	,	PUNCT
ejpam-2562	473	27	b	b	NOUN
ejpam-2562	473	28	)	)	PUNCT
ejpam-2562	473	29	is	be	AUX
ejpam-2562	473	30	the	the	DET
ejpam-2562	473	31	greatest	great	ADJ
ejpam-2562	473	32	common	common	ADJ
ejpam-2562	473	33	divisor	divisor	NOUN
ejpam-2562	473	34	of	of	ADP
ejpam-2562	473	35	a	a	DET
ejpam-2562	473	36	,	,	PUNCT
ejpam-2562	473	37	b	b	PROPN
ejpam-2562	473	38	and	and	CCONJ
ejpam-2562	473	39	a⊙b	a⊙b	NOUN
ejpam-2562	473	40	=	=	SYM
ejpam-2562	473	41	lcm(a	lcm(a	PROPN
ejpam-2562	473	42	,	,	PUNCT
ejpam-2562	473	43	b	b	NOUN
ejpam-2562	473	44	)	)	PUNCT
ejpam-2562	473	45	is	be	AUX
ejpam-2562	473	46	the	the	DET
ejpam-2562	473	47	least	least	ADJ
ejpam-2562	473	48	common	common	ADJ
ejpam-2562	473	49	divisor	divisor	NOUN
ejpam-2562	473	50	of	of	ADP
ejpam-2562	473	51	a	a	PRON
ejpam-2562	473	52	,	,	PUNCT
ejpam-2562	473	53	b.	b.	PROPN
ejpam-2562	474	1	so	so	ADV
ejpam-2562	474	2	n	n	PROPN
ejpam-2562	474	3	is	be	AUX
ejpam-2562	474	4	a	a	DET
ejpam-2562	474	5	left	left	ADJ
ejpam-2562	474	6	n	n	CCONJ
ejpam-2562	474	7	-	-	PUNCT
ejpam-2562	474	8	semimodule	semimodule	NOUN
ejpam-2562	474	9	.	.	PUNCT
ejpam-2562	475	1	every	every	DET
ejpam-2562	475	2	subsemimodule	subsemimodule	NOUN
ejpam-2562	475	3	of	of	ADP
ejpam-2562	475	4	n	n	NUM
ejpam-2562	475	5	is	be	AUX
ejpam-2562	475	6	in	in	ADP
ejpam-2562	475	7	form	form	NOUN
ejpam-2562	475	8	<	<	X
ejpam-2562	475	9	p	p	X
ejpam-2562	475	10	>	>	X
ejpam-2562	475	11	=	=	PROPN
ejpam-2562	475	12	pn	pn	PROPN
ejpam-2562	475	13	=	=	PUNCT
ejpam-2562	475	14	{	{	PUNCT
ejpam-2562	475	15	pn	pn	PROPN
ejpam-2562	475	16	/	/	SYM
ejpam-2562	475	17	n	n	CCONJ
ejpam-2562	475	18	∈	∈	PROPN
ejpam-2562	475	19	n	n	CCONJ
ejpam-2562	475	20	}	}	PUNCT
ejpam-2562	475	21	where	where	SCONJ
ejpam-2562	475	22	p	p	PROPN
ejpam-2562	475	23	∈	∈	PROPN
ejpam-2562	475	24	n	n	CCONJ
ejpam-2562	475	25	and	and	CCONJ
ejpam-2562	475	26	pn	pn	PROPN
ejpam-2562	475	27	is	be	AUX
ejpam-2562	475	28	the	the	DET
ejpam-2562	475	29	usual	usual	ADJ
ejpam-2562	475	30	product	product	NOUN
ejpam-2562	475	31	of	of	ADP
ejpam-2562	475	32	p	p	PROPN
ejpam-2562	475	33	and	and	CCONJ
ejpam-2562	475	34	n	n	CCONJ
ejpam-2562	475	35	(	(	PUNCT
ejpam-2562	475	36	see	see	VERB
ejpam-2562	475	37	[	[	X
ejpam-2562	475	38	3	3	NUM
ejpam-2562	475	39	]	]	NUM
ejpam-2562	475	40	)	)	PUNCT
ejpam-2562	475	41	.	.	PUNCT
ejpam-2562	476	1	every	every	DET
ejpam-2562	476	2	subsemimodule	subsemimodule	PROPN
ejpam-2562	476	3	pn	pn	PROPN
ejpam-2562	476	4	of	of	ADP
ejpam-2562	476	5	n	n	PROPN
ejpam-2562	476	6	is	be	AUX
ejpam-2562	476	7	semiessential	semiessential	ADJ
ejpam-2562	476	8	on	on	ADP
ejpam-2562	476	9	n	n	NOUN
ejpam-2562	476	10	because	because	SCONJ
ejpam-2562	476	11	∀n	∀n	NUM
ejpam-2562	476	12	∈	∈	NOUN
ejpam-2562	476	13	n	n	CCONJ
ejpam-2562	476	14	there	there	PRON
ejpam-2562	476	15	exists	exist	VERB
ejpam-2562	476	16	r	r	NOUN
ejpam-2562	476	17	∈	∈	PROPN
ejpam-2562	476	18	n	n	PRON
ejpam-2562	477	1	such	such	ADJ
ejpam-2562	477	2	that	that	SCONJ
ejpam-2562	477	3	rn	rn	PROPN
ejpam-2562	477	4	∈	∈	PROPN
ejpam-2562	477	5	pn	pn	PROPN
ejpam-2562	477	6	.	.	PUNCT
ejpam-2562	478	1	so	so	ADV
ejpam-2562	478	2	the	the	DET
ejpam-2562	478	3	image	image	NOUN
ejpam-2562	478	4	of	of	ADP
ejpam-2562	478	5	any	any	DET
ejpam-2562	478	6	injective	injective	ADJ
ejpam-2562	478	7	endomorphism	endomorphism	NOUN
ejpam-2562	478	8	of	of	ADP
ejpam-2562	478	9	n	n	PROPN
ejpam-2562	478	10	is	be	AUX
ejpam-2562	478	11	of	of	ADP
ejpam-2562	478	12	the	the	DET
ejpam-2562	478	13	form	form	NOUN
ejpam-2562	478	14	pn	pn	NOUN
ejpam-2562	478	15	,	,	PUNCT
ejpam-2562	478	16	and	and	CCONJ
ejpam-2562	478	17	therefore	therefore	ADV
ejpam-2562	478	18	we	we	PRON
ejpam-2562	478	19	deduce	deduce	VERB
ejpam-2562	478	20	that	that	SCONJ
ejpam-2562	478	21	n	n	NOUN
ejpam-2562	478	22	is	be	AUX
ejpam-2562	478	23	weakly	weakly	ADV
ejpam-2562	478	24	-	-	PUNCT
ejpam-2562	478	25	co	co	NOUN
ejpam-2562	478	26	-	-	NOUN
ejpam-2562	478	27	hopfian-2	hopfian-2	NOUN
ejpam-2562	478	28	.	.	NOUN
ejpam-2562	478	29	example	example	NOUN
ejpam-2562	479	1	7	7	NUM
ejpam-2562	479	2	.	.	PUNCT
ejpam-2562	480	1	let	let	VERB
ejpam-2562	480	2	r=	r=	ADJ
ejpam-2562	480	3	{	{	PUNCT
ejpam-2562	480	4	0	0	NUM
ejpam-2562	480	5	;	;	PUNCT
ejpam-2562	480	6	1	1	NUM
ejpam-2562	480	7	}	}	PUNCT
ejpam-2562	480	8	be	be	AUX
ejpam-2562	480	9	a	a	DET
ejpam-2562	480	10	set	set	NOUN
ejpam-2562	480	11	.	.	PUNCT
ejpam-2562	481	1	(	(	PUNCT
ejpam-2562	481	2	r;+;×	r;+;×	NOUN
ejpam-2562	481	3	)	)	PUNCT
ejpam-2562	481	4	and	and	CCONJ
ejpam-2562	481	5	(	(	PUNCT
ejpam-2562	481	6	r;+;⊗	r;+;⊗	PROPN
ejpam-2562	481	7	)	)	PUNCT
ejpam-2562	481	8	(	(	PUNCT
ejpam-2562	481	9	where	where	SCONJ
ejpam-2562	481	10	for	for	ADP
ejpam-2562	481	11	all	all	DET
ejpam-2562	481	12	i	i	PRON
ejpam-2562	481	13	,	,	PUNCT
ejpam-2562	481	14	j	j	PROPN
ejpam-2562	481	15	∈	∈	PROPN
ejpam-2562	481	16	r	r	PROPN
ejpam-2562	481	17	,	,	PUNCT
ejpam-2562	481	18	i	i	PROPN
ejpam-2562	481	19	+	+	NUM
ejpam-2562	481	20	j	j	PROPN
ejpam-2562	481	21	=	=	SYM
ejpam-2562	481	22	max(i	max(i	PROPN
ejpam-2562	481	23	,	,	PUNCT
ejpam-2562	481	24	j	j	PROPN
ejpam-2562	481	25	)	)	PUNCT
ejpam-2562	481	26	,	,	PUNCT
ejpam-2562	481	27	i	i	PRON
ejpam-2562	482	1	×	×	VERB
ejpam-2562	482	2	j	j	NOUN
ejpam-2562	482	3	=	=	SYM
ejpam-2562	482	4	0	0	PROPN
ejpam-2562	482	5	,	,	PUNCT
ejpam-2562	483	1	i	i	PRON
ejpam-2562	483	2	⊗	⊗	PROPN
ejpam-2562	483	3	j	j	PROPN
ejpam-2562	484	1	=	=	NOUN
ejpam-2562	484	2	0	0	NUM
ejpam-2562	484	3	except	except	SCONJ
ejpam-2562	484	4	1⊗	1⊗	NUM
ejpam-2562	484	5	1	1	NUM
ejpam-2562	484	6	=	=	SYM
ejpam-2562	484	7	1	1	NUM
ejpam-2562	484	8	)	)	PUNCT
ejpam-2562	484	9	are	be	AUX
ejpam-2562	484	10	semirings	semiring	NOUN
ejpam-2562	484	11	(	(	PUNCT
ejpam-2562	484	12	see	see	VERB
ejpam-2562	484	13	[	[	X
ejpam-2562	484	14	16	16	NUM
ejpam-2562	484	15	]	]	PUNCT
ejpam-2562	484	16	)	)	PUNCT
ejpam-2562	484	17	.	.	PUNCT
ejpam-2562	485	1	it	it	PRON
ejpam-2562	485	2	’s	’	VERB
ejpam-2562	485	3	easy	easy	ADJ
ejpam-2562	485	4	to	to	PART
ejpam-2562	485	5	see	see	VERB
ejpam-2562	485	6	that	that	DET
ejpam-2562	485	7	(	(	PUNCT
ejpam-2562	485	8	rr;+;×	rr;+;×	NOUN
ejpam-2562	485	9	)	)	PUNCT
ejpam-2562	485	10	and	and	CCONJ
ejpam-2562	485	11	(	(	PUNCT
ejpam-2562	485	12	rr;+;⊗	rr;+;⊗	PRON
ejpam-2562	485	13	)	)	PUNCT
ejpam-2562	485	14	are	be	AUX
ejpam-2562	485	15	two	two	NUM
ejpam-2562	485	16	r	r	NOUN
ejpam-2562	485	17	-	-	PUNCT
ejpam-2562	485	18	simples	simple	NOUN
ejpam-2562	485	19	semimodules	semimodule	NOUN
ejpam-2562	485	20	,	,	PUNCT
ejpam-2562	485	21	therefore	therefore	ADV
ejpam-2562	485	22	they	they	PRON
ejpam-2562	485	23	are	be	AUX
ejpam-2562	485	24	wch-3	wch-3	VERB
ejpam-2562	485	25	by	by	ADP
ejpam-2562	485	26	the	the	DET
ejpam-2562	485	27	proposition	proposition	NOUN
ejpam-2562	485	28	18	18	NUM
ejpam-2562	485	29	.	.	PUNCT
ejpam-2562	486	1	references	reference	NOUN
ejpam-2562	486	2	[	[	X
ejpam-2562	486	3	1	1	X
ejpam-2562	486	4	]	]	PUNCT
ejpam-2562	486	5	j.	j.	PROPN
ejpam-2562	486	6	abuhlail	abuhlail	PROPN
ejpam-2562	486	7	.	.	PUNCT
ejpam-2562	487	1	semicorings	semicoring	NOUN
ejpam-2562	487	2	and	and	CCONJ
ejpam-2562	487	3	semicomodules	semicomodule	NOUN
ejpam-2562	487	4	.	.	PUNCT
ejpam-2562	488	1	research	research	NOUN
ejpam-2562	488	2	report	report	NOUN
ejpam-2562	488	3	s-1	s-1	PROPN
ejpam-2562	488	4	,	,	PUNCT
ejpam-2562	488	5	department	department	NOUN
ejpam-2562	488	6	of	of	ADP
ejpam-2562	488	7	mathematics	mathematics	PROPN
ejpam-2562	488	8	,	,	PUNCT
ejpam-2562	488	9	suny	suny	PROPN
ejpam-2562	488	10	at	at	ADP
ejpam-2562	488	11	albany	albany	PROPN
ejpam-2562	488	12	,	,	PUNCT
ejpam-2562	488	13	2008	2008	NUM
ejpam-2562	488	14	.	.	PUNCT
ejpam-2562	489	1	[	[	X
ejpam-2562	489	2	2	2	X
ejpam-2562	489	3	]	]	PUNCT
ejpam-2562	489	4	f.	f.	PROPN
ejpam-2562	489	5	w.	w.	PROPN
ejpam-2562	489	6	anderson	anderson	PROPN
ejpam-2562	489	7	and	and	CCONJ
ejpam-2562	489	8	k.	k.	PROPN
ejpam-2562	489	9	r.	r.	PROPN
ejpam-2562	489	10	fuller	fuller	PROPN
ejpam-2562	489	11	.	.	PUNCT
ejpam-2562	490	1	rings	ring	NOUN
ejpam-2562	490	2	and	and	CCONJ
ejpam-2562	490	3	categories	category	NOUN
ejpam-2562	490	4	of	of	ADP
ejpam-2562	490	5	modules	module	NOUN
ejpam-2562	490	6	.	.	PUNCT
ejpam-2562	491	1	springer	springer	NOUN
ejpam-2562	491	2	-	-	PUNCT
ejpam-2562	491	3	verlag	verlag	PROPN
ejpam-2562	491	4	,	,	PUNCT
ejpam-2562	491	5	new	new	PROPN
ejpam-2562	491	6	york	york	PROPN
ejpam-2562	491	7	,	,	PUNCT
ejpam-2562	491	8	1975	1975	NUM
ejpam-2562	491	9	.	.	PUNCT
ejpam-2562	492	1	[	[	X
ejpam-2562	492	2	3	3	X
ejpam-2562	492	3	]	]	PUNCT
ejpam-2562	492	4	j.	j.	PROPN
ejpam-2562	492	5	n.	n.	PROPN
ejpam-2562	492	6	chaudhari	chaudhari	PROPN
ejpam-2562	492	7	and	and	CCONJ
ejpam-2562	492	8	k.	k.	PROPN
ejpam-2562	492	9	j.	j.	PROPN
ejpam-2562	492	10	ingale	ingale	PROPN
ejpam-2562	492	11	.	.	PUNCT
ejpam-2562	493	1	a	a	DET
ejpam-2562	493	2	note	note	NOUN
ejpam-2562	493	3	on	on	ADP
ejpam-2562	493	4	strongly	strongly	ADV
ejpam-2562	493	5	euclidean	euclidean	ADJ
ejpam-2562	493	6	semirings	semiring	NOUN
ejpam-2562	493	7	.	.	PUNCT
ejpam-2562	494	1	international	international	ADJ
ejpam-2562	494	2	journal	journal	PROPN
ejpam-2562	494	3	of	of	ADP
ejpam-2562	494	4	algebra	algebra	PROPN
ejpam-2562	494	5	,	,	PUNCT
ejpam-2562	494	6	6(6):271–275	6(6):271–275	NUM
ejpam-2562	494	7	,	,	PUNCT
ejpam-2562	494	8	2012	2012	NUM
ejpam-2562	494	9	.	.	PUNCT
ejpam-2562	495	1	[	[	X
ejpam-2562	495	2	4	4	X
ejpam-2562	495	3	]	]	X
ejpam-2562	495	4	b.	b.	PROPN
ejpam-2562	495	5	eckman	eckman	PROPN
ejpam-2562	495	6	and	and	CCONJ
ejpam-2562	495	7	a.	a.	NOUN
ejpam-2562	495	8	schpof	schpof	PROPN
ejpam-2562	495	9	.	.	PUNCT
ejpam-2562	496	1	uber	uber	ADJ
ejpam-2562	496	2	injective	injective	ADJ
ejpam-2562	496	3	moduln	moduln	NOUN
ejpam-2562	496	4	.	.	PUNCT
ejpam-2562	497	1	arch	arch	ADJ
ejpam-2562	497	2	der	der	PROPN
ejpam-2562	497	3	math	math	NOUN
ejpam-2562	497	4	,	,	PUNCT
ejpam-2562	497	5	4:75–78	4:75–78	PROPN
ejpam-2562	497	6	,	,	PUNCT
ejpam-2562	497	7	1953	1953	NUM
ejpam-2562	497	8	.	.	PUNCT
ejpam-2562	498	1	[	[	X
ejpam-2562	498	2	5	5	X
ejpam-2562	498	3	]	]	PUNCT
ejpam-2562	498	4	l.	l.	PROPN
ejpam-2562	498	5	fall	fall	PROPN
ejpam-2562	498	6	,	,	PUNCT
ejpam-2562	498	7	j.	j.	PROPN
ejpam-2562	498	8	r.	r.	PROPN
ejpam-2562	498	9	tsiba	tsiba	PROPN
ejpam-2562	498	10	,	,	PUNCT
ejpam-2562	498	11	and	and	CCONJ
ejpam-2562	498	12	d.	d.	PROPN
ejpam-2562	498	13	sow	sow	PROPN
ejpam-2562	498	14	.	.	PUNCT
ejpam-2562	499	1	on	on	ADP
ejpam-2562	499	2	semi	semi	ADJ
ejpam-2562	499	3	-	-	ADJ
ejpam-2562	499	4	essential	essential	ADJ
ejpam-2562	499	5	subsemimodules	subsemimodule	NOUN
ejpam-2562	499	6	.	.	PUNCT
ejpam-2562	500	1	jp	jp	PROPN
ejpam-2562	500	2	journal	journal	PROPN
ejpam-2562	500	3	of	of	ADP
ejpam-2562	500	4	algebra	algebra	PROPN
ejpam-2562	500	5	,	,	PUNCT
ejpam-2562	500	6	22(2):187–203	22(2):187–203	PROPN
ejpam-2562	500	7	,	,	PUNCT
ejpam-2562	500	8	2011	2011	NUM
ejpam-2562	500	9	.	.	PUNCT
ejpam-2562	501	1	[	[	X
ejpam-2562	501	2	6	6	NUM
ejpam-2562	501	3	]	]	PUNCT
ejpam-2562	501	4	j.	j.	PROPN
ejpam-2562	501	5	s.	s.	PROPN
ejpam-2562	501	6	golan	golan	PROPN
ejpam-2562	501	7	.	.	PUNCT
ejpam-2562	502	1	the	the	DET
ejpam-2562	502	2	theory	theory	NOUN
ejpam-2562	502	3	of	of	ADP
ejpam-2562	502	4	semirings	semiring	NOUN
ejpam-2562	502	5	with	with	ADP
ejpam-2562	502	6	applications	application	NOUN
ejpam-2562	502	7	to	to	ADP
ejpam-2562	502	8	mathematics	mathematic	NOUN
ejpam-2562	502	9	and	and	CCONJ
ejpam-2562	502	10	theoretical	theoretical	ADJ
ejpam-2562	502	11	computer	computer	NOUN
ejpam-2562	502	12	science	science	NOUN
ejpam-2562	502	13	.	.	PUNCT
ejpam-2562	503	1	kluwer	kluwer	NOUN
ejpam-2562	503	2	academic	academic	ADJ
ejpam-2562	503	3	publishers	publisher	NOUN
ejpam-2562	503	4	,	,	PUNCT
ejpam-2562	503	5	dordrecht	dordrecht	PROPN
ejpam-2562	503	6	,	,	PUNCT
ejpam-2562	503	7	1999	1999	NUM
ejpam-2562	503	8	.	.	PUNCT
ejpam-2562	504	1	[	[	X
ejpam-2562	504	2	7	7	X
ejpam-2562	504	3	]	]	PUNCT
ejpam-2562	504	4	r.	r.	PROPN
ejpam-2562	504	5	e.	e.	PROPN
ejpam-2562	504	6	johnson	johnson	PROPN
ejpam-2562	504	7	.	.	PUNCT
ejpam-2562	505	1	the	the	DET
ejpam-2562	505	2	extended	extended	ADJ
ejpam-2562	505	3	centralizer	centralizer	NOUN
ejpam-2562	505	4	of	of	ADP
ejpam-2562	505	5	module	module	NOUN
ejpam-2562	505	6	.	.	PUNCT
ejpam-2562	506	1	proc	proc	PROPN
ejpam-2562	506	2	.	.	PUNCT
ejpam-2562	507	1	amer	amer	PROPN
ejpam-2562	507	2	.	.	PUNCT
ejpam-2562	507	3	math	math	PROPN
ejpam-2562	507	4	.	.	PUNCT
ejpam-2562	508	1	soc	soc	PROPN
ejpam-2562	508	2	.	.	PUNCT
ejpam-2562	508	3	,	,	PUNCT
ejpam-2562	508	4	2:891–895	2:891–895	PROPN
ejpam-2562	508	5	,	,	PUNCT
ejpam-2562	508	6	1951	1951	NUM
ejpam-2562	508	7	.	.	PUNCT
ejpam-2562	509	1	[	[	X
ejpam-2562	509	2	8	8	X
ejpam-2562	509	3	]	]	PUNCT
ejpam-2562	509	4	t.	t.	PROPN
ejpam-2562	509	5	y.	y.	PROPN
ejpam-2562	509	6	lam	lam	PROPN
ejpam-2562	509	7	.	.	PUNCT
ejpam-2562	510	1	lectures	lecture	NOUN
ejpam-2562	510	2	on	on	ADP
ejpam-2562	510	3	modules	module	NOUN
ejpam-2562	510	4	and	and	CCONJ
ejpam-2562	510	5	rings	ring	NOUN
ejpam-2562	510	6	.	.	PUNCT
ejpam-2562	511	1	springer	springer	NOUN
ejpam-2562	511	2	-	-	PUNCT
ejpam-2562	511	3	verlag	verlag	PROPN
ejpam-2562	511	4	,	,	PUNCT
ejpam-2562	511	5	new	new	PROPN
ejpam-2562	511	6	york	york	PROPN
ejpam-2562	511	7	,	,	PUNCT
ejpam-2562	511	8	1998	1998	NUM
ejpam-2562	511	9	.	.	PUNCT
ejpam-2562	512	1	[	[	X
ejpam-2562	512	2	9	9	NUM
ejpam-2562	512	3	]	]	PUNCT
ejpam-2562	512	4	m.	m.	NOUN
ejpam-2562	512	5	takahashi	takahashi	PROPN
ejpam-2562	512	6	.	.	PUNCT
ejpam-2562	513	1	on	on	ADP
ejpam-2562	513	2	the	the	DET
ejpam-2562	513	3	bordism	bordism	NOUN
ejpam-2562	513	4	categories	category	NOUN
ejpam-2562	513	5	ii	ii	PROPN
ejpam-2562	513	6	(	(	PUNCT
ejpam-2562	513	7	elementary	elementary	ADJ
ejpam-2562	513	8	properties	property	NOUN
ejpam-2562	513	9	of	of	ADP
ejpam-2562	513	10	semimodules	semimodule	NOUN
ejpam-2562	513	11	)	)	PUNCT
ejpam-2562	513	12	.	.	PUNCT
ejpam-2562	514	1	mathematics	mathematic	NOUN
ejpam-2562	514	2	seminar	seminar	NOUN
ejpam-2562	514	3	notes	note	NOUN
ejpam-2562	514	4	,	,	PUNCT
ejpam-2562	514	5	9:445–530	9:445–530	NUM
ejpam-2562	514	6	,	,	PUNCT
ejpam-2562	514	7	1981	1981	NUM
ejpam-2562	514	8	.	.	PUNCT
ejpam-2562	515	1	[	[	X
ejpam-2562	515	2	10	10	NUM
ejpam-2562	515	3	]	]	PUNCT
ejpam-2562	515	4	m.	m.	NOUN
ejpam-2562	515	5	takahashi	takahashi	PROPN
ejpam-2562	515	6	.	.	PUNCT
ejpam-2562	516	1	on	on	ADP
ejpam-2562	516	2	the	the	DET
ejpam-2562	516	3	bordism	bordism	NOUN
ejpam-2562	516	4	categories	category	NOUN
ejpam-2562	516	5	iii	iii	X
ejpam-2562	516	6	(	(	PUNCT
ejpam-2562	516	7	fonctors	fonctor	NOUN
ejpam-2562	516	8	hom	hom	ADV
ejpam-2562	516	9	and	and	CCONJ
ejpam-2562	516	10	⊗	⊗	PROPN
ejpam-2562	516	11	for	for	ADP
ejpam-2562	516	12	semimodules	semimodule	NOUN
ejpam-2562	516	13	)	)	PUNCT
ejpam-2562	516	14	.	.	PUNCT
ejpam-2562	517	1	mathematics	mathematic	NOUN
ejpam-2562	517	2	seminar	seminar	NOUN
ejpam-2562	517	3	notes	note	NOUN
ejpam-2562	517	4	,	,	PUNCT
ejpam-2562	517	5	10:211–236	10:211–236	NUM
ejpam-2562	517	6	,	,	PUNCT
ejpam-2562	517	7	1982	1982	NUM
ejpam-2562	517	8	.	.	PUNCT
ejpam-2562	518	1	[	[	X
ejpam-2562	518	2	11	11	NUM
ejpam-2562	518	3	]	]	PUNCT
ejpam-2562	518	4	m.	m.	NOUN
ejpam-2562	518	5	takahashi	takahashi	PROPN
ejpam-2562	518	6	and	and	CCONJ
ejpam-2562	518	7	h.	h.	PROPN
ejpam-2562	518	8	x.	x.	PROPN
ejpam-2562	518	9	wang	wang	PROPN
ejpam-2562	518	10	.	.	PUNCT
ejpam-2562	519	1	on	on	ADP
ejpam-2562	519	2	epimorphisms	epimorphism	NOUN
ejpam-2562	519	3	of	of	ADP
ejpam-2562	519	4	semimodules	semimodule	NOUN
ejpam-2562	519	5	.	.	PUNCT
ejpam-2562	520	1	kobe	kobe	PROPN
ejpam-2562	520	2	j.	j.	PROPN
ejpam-2562	520	3	math	math	PROPN
ejpam-2562	520	4	.	.	PUNCT
ejpam-2562	520	5	,	,	PUNCT
ejpam-2562	520	6	6(2):297–298	6(2):297–298	NUM
ejpam-2562	520	7	,	,	PUNCT
ejpam-2562	520	8	1989	1989	NUM
ejpam-2562	520	9	.	.	PUNCT
ejpam-2562	521	1	references	reference	NOUN
ejpam-2562	521	2	265	265	NUM
ejpam-2562	522	1	[	[	X
ejpam-2562	522	2	12	12	NUM
ejpam-2562	522	3	]	]	PUNCT
ejpam-2562	522	4	j.	j.	PROPN
ejpam-2562	522	5	r.	r.	PROPN
ejpam-2562	522	6	tsiba	tsiba	PROPN
ejpam-2562	522	7	,	,	PUNCT
ejpam-2562	522	8	l.	l.	PROPN
ejpam-2562	522	9	fall	fall	PROPN
ejpam-2562	522	10	,	,	PUNCT
ejpam-2562	522	11	and	and	CCONJ
ejpam-2562	522	12	d.	d.	PROPN
ejpam-2562	522	13	sow	sow	PROPN
ejpam-2562	522	14	.	.	PUNCT
ejpam-2562	523	1	introduction	introduction	NOUN
ejpam-2562	523	2	to	to	ADP
ejpam-2562	523	3	s	s	NOUN
ejpam-2562	523	4	-	-	NOUN
ejpam-2562	523	5	hopfian	hopfian	ADJ
ejpam-2562	523	6	and	and	CCONJ
ejpam-2562	523	7	s	s	NOUN
ejpam-2562	523	8	-	-	ADJ
ejpam-2562	523	9	cohopfian	cohopfian	ADJ
ejpam-2562	523	10	semimodules	semimodule	NOUN
ejpam-2562	523	11	.	.	PUNCT
ejpam-2562	524	1	jp	jp	PROPN
ejpam-2562	524	2	journal	journal	PROPN
ejpam-2562	524	3	of	of	ADP
ejpam-2562	524	4	algebra	algebra	PROPN
ejpam-2562	524	5	,	,	PUNCT
ejpam-2562	524	6	number	number	NOUN
ejpam-2562	524	7	theory	theory	NOUN
ejpam-2562	524	8	and	and	CCONJ
ejpam-2562	524	9	applications	application	NOUN
ejpam-2562	524	10	,	,	PUNCT
ejpam-2562	524	11	22(1):81–109	22(1):81–109	NUM
ejpam-2562	524	12	,	,	PUNCT
ejpam-2562	524	13	2011	2011	NUM
ejpam-2562	524	14	.	.	PUNCT
ejpam-2562	525	1	[	[	X
ejpam-2562	525	2	13	13	NUM
ejpam-2562	525	3	]	]	PUNCT
ejpam-2562	525	4	j.	j.	PROPN
ejpam-2562	525	5	r.	r.	PROPN
ejpam-2562	525	6	tsiba	tsiba	PROPN
ejpam-2562	525	7	and	and	CCONJ
ejpam-2562	525	8	d.	d.	PROPN
ejpam-2562	525	9	sow	sow	PROPN
ejpam-2562	525	10	.	.	PUNCT
ejpam-2562	526	1	on	on	ADP
ejpam-2562	526	2	generator	generator	NOUN
ejpam-2562	526	3	and	and	CCONJ
ejpam-2562	526	4	projective	projective	ADJ
ejpam-2562	526	5	semimodules	semimodule	NOUN
ejpam-2562	526	6	.	.	PUNCT
ejpam-2562	527	1	international	international	ADJ
ejpam-2562	527	2	journal	journal	PROPN
ejpam-2562	527	3	of	of	ADP
ejpam-2562	527	4	algebra	algebra	PROPN
ejpam-2562	527	5	,	,	PUNCT
ejpam-2562	527	6	4(2):1153–1167	4(2):1153–1167	NOUN
ejpam-2562	527	7	,	,	PUNCT
ejpam-2562	527	8	2010	2010	NUM
ejpam-2562	527	9	.	.	PUNCT
ejpam-2562	528	1	[	[	X
ejpam-2562	528	2	14	14	NUM
ejpam-2562	528	3	]	]	X
ejpam-2562	528	4	n.	n.	PROPN
ejpam-2562	528	5	x.	x.	NOUN
ejpam-2562	528	6	tuyen	tuyen	PROPN
ejpam-2562	528	7	and	and	CCONJ
ejpam-2562	528	8	t.	t.	PROPN
ejpam-2562	528	9	g.	g.	PROPN
ejpam-2562	528	10	nam	nam	PROPN
ejpam-2562	528	11	.	.	PUNCT
ejpam-2562	529	1	on	on	ADP
ejpam-2562	529	2	projective	projective	ADJ
ejpam-2562	529	3	covers	cover	NOUN
ejpam-2562	529	4	of	of	ADP
ejpam-2562	529	5	semimodules	semimodule	NOUN
ejpam-2562	529	6	in	in	ADP
ejpam-2562	529	7	the	the	DET
ejpam-2562	529	8	category	category	NOUN
ejpam-2562	529	9	cssmod	cssmod	NOUN
ejpam-2562	529	10	and	and	CCONJ
ejpam-2562	529	11	their	their	PRON
ejpam-2562	529	12	applications	application	NOUN
ejpam-2562	529	13	.	.	PUNCT
ejpam-2562	530	1	southeast	southeast	ADJ
ejpam-2562	530	2	asian	asian	ADJ
ejpam-2562	530	3	bulletin	bulletin	NOUN
ejpam-2562	530	4	of	of	ADP
ejpam-2562	530	5	mathematics	mathematic	NOUN
ejpam-2562	530	6	,	,	PUNCT
ejpam-2562	530	7	31:363–377	31:363–377	NUM
ejpam-2562	530	8	,	,	PUNCT
ejpam-2562	530	9	2007	2007	NUM
ejpam-2562	530	10	.	.	PUNCT
ejpam-2562	531	1	[	[	X
ejpam-2562	531	2	15	15	NUM
ejpam-2562	531	3	]	]	X
ejpam-2562	531	4	r.	r.	PROPN
ejpam-2562	531	5	wisbauer	wisbauer	NOUN
ejpam-2562	531	6	.	.	PUNCT
ejpam-2562	532	1	foundations	foundation	NOUN
ejpam-2562	532	2	of	of	ADP
ejpam-2562	532	3	module	module	NOUN
ejpam-2562	532	4	and	and	CCONJ
ejpam-2562	532	5	ring	ring	NOUN
ejpam-2562	532	6	theory	theory	NOUN
ejpam-2562	532	7	.	.	PUNCT
ejpam-2562	533	1	gordon	gordon	PROPN
ejpam-2562	533	2	and	and	CCONJ
ejpam-2562	533	3	breach	breach	PROPN
ejpam-2562	533	4	,	,	PUNCT
ejpam-2562	533	5	philadelphia	philadelphia	PROPN
ejpam-2562	533	6	,	,	PUNCT
ejpam-2562	533	7	1991	1991	NUM
ejpam-2562	533	8	.	.	PUNCT
ejpam-2562	534	1	[	[	X
ejpam-2562	534	2	16	16	NUM
ejpam-2562	534	3	]	]	X
ejpam-2562	534	4	j.	j.	PROPN
ejpam-2562	534	5	zumbraegel	zumbraegel	PROPN
ejpam-2562	534	6	.	.	PUNCT
ejpam-2562	535	1	classification	classification	NOUN
ejpam-2562	535	2	of	of	ADP
ejpam-2562	535	3	finite	finite	ADJ
ejpam-2562	535	4	congruence	congruence	PROPN
ejpam-2562	535	5	-	-	PUNCT
ejpam-2562	535	6	simple	simple	ADJ
ejpam-2562	535	7	semiring	semiring	NOUN
ejpam-2562	535	8	with	with	ADP
ejpam-2562	535	9	zero	zero	NUM
ejpam-2562	535	10	.	.	PUNCT
ejpam-2562	536	1	journal	journal	NOUN
ejpam-2562	536	2	of	of	ADP
ejpam-2562	536	3	algebra	algebra	PROPN
ejpam-2562	536	4	and	and	CCONJ
ejpam-2562	536	5	its	its	PRON
ejpam-2562	536	6	applications	application	NOUN
ejpam-2562	536	7	,	,	PUNCT
ejpam-2562	536	8	7(3):363–377	7(3):363–377	NOUN
ejpam-2562	536	9	,	,	PUNCT
ejpam-2562	536	10	2008	2008	NUM
ejpam-2562	536	11	.	.	PUNCT