id	sid	tid	token	lemma	pos
ejpam-1373	1	1	6_xxx_sankar.dvi	6_xxx_sankar.dvi	NUM
ejpam-1373	1	2	european	european	ADJ
ejpam-1373	1	3	journal	journal	NOUN
ejpam-1373	1	4	of	of	ADP
ejpam-1373	1	5	pure	pure	ADJ
ejpam-1373	1	6	and	and	CCONJ
ejpam-1373	1	7	applied	apply	VERB
ejpam-1373	1	8	mathematics	mathematic	NOUN
ejpam-1373	1	9	vol	vol	NOUN
ejpam-1373	1	10	.	.	PROPN
ejpam-1373	2	1	4	4	NUM
ejpam-1373	2	2	,	,	PUNCT
ejpam-1373	2	3	no	no	INTJ
ejpam-1373	2	4	.	.	NOUN
ejpam-1373	2	5	4	4	NUM
ejpam-1373	2	6	,	,	PUNCT
ejpam-1373	2	7	2011	2011	NUM
ejpam-1373	2	8	,	,	PUNCT
ejpam-1373	2	9	424	424	NUM
ejpam-1373	2	10	-	-	SYM
ejpam-1373	2	11	434	434	NUM
ejpam-1373	2	12	issn	issn	PROPN
ejpam-1373	2	13	1307	1307	NUM
ejpam-1373	2	14	-	-	SYM
ejpam-1373	2	15	5543	5543	NUM
ejpam-1373	2	16	–	–	PUNCT
ejpam-1373	2	17	www.ejpam.com	www.ejpam.com	X
ejpam-1373	2	18	lattice	lattice	VERB
ejpam-1373	2	19	structures	structure	NOUN
ejpam-1373	2	20	on	on	ADP
ejpam-1373	2	21	z+	z+	NUM
ejpam-1373	2	22	induced	induce	VERB
ejpam-1373	2	23	by	by	ADP
ejpam-1373	2	24	convolutions	convolution	NOUN
ejpam-1373	2	25	u.m	u.m	PROPN
ejpam-1373	2	26	.	.	PUNCT
ejpam-1373	2	27	swamy1	swamy1	PROPN
ejpam-1373	3	1	and	and	CCONJ
ejpam-1373	3	2	sagi	sagi	PROPN
ejpam-1373	3	3	sankar2,∗	sankar2,∗	VERB
ejpam-1373	3	4	1	1	NUM
ejpam-1373	3	5	department	department	NOUN
ejpam-1373	3	6	of	of	ADP
ejpam-1373	3	7	mathematics	mathematic	NOUN
ejpam-1373	3	8	,	,	PUNCT
ejpam-1373	3	9	g.v.p.college	g.v.p.college	PROPN
ejpam-1373	3	10	of	of	ADP
ejpam-1373	3	11	engineering	engineering	NOUN
ejpam-1373	3	12	,	,	PUNCT
ejpam-1373	3	13	visakhapatnam	visakhapatnam	PROPN
ejpam-1373	3	14	,	,	PUNCT
ejpam-1373	3	15	india	india	PROPN
ejpam-1373	3	16	.	.	PROPN
ejpam-1373	4	1	2	2	NUM
ejpam-1373	4	2	department	department	NOUN
ejpam-1373	4	3	of	of	ADP
ejpam-1373	4	4	mathematics	mathematics	PROPN
ejpam-1373	4	5	,	,	PUNCT
ejpam-1373	4	6	m.v.g.r	m.v.g.r	PROPN
ejpam-1373	4	7	.	.	PUNCT
ejpam-1373	4	8	college	college	PROPN
ejpam-1373	4	9	of	of	ADP
ejpam-1373	4	10	engineering	engineering	PROPN
ejpam-1373	4	11	,	,	PUNCT
ejpam-1373	4	12	vizianagaram	vizianagaram	PROPN
ejpam-1373	4	13	,	,	PUNCT
ejpam-1373	4	14	india	india	PROPN
ejpam-1373	4	15	.	.	PUNCT
ejpam-1373	5	1	abstract	abstract	PROPN
ejpam-1373	5	2	.	.	PUNCT
ejpam-1373	6	1	a	a	DET
ejpam-1373	6	2	convolution	convolution	NOUN
ejpam-1373	6	3	c	c	NOUN
ejpam-1373	6	4	is	be	AUX
ejpam-1373	6	5	a	a	DET
ejpam-1373	6	6	mapping	mapping	NOUN
ejpam-1373	6	7	of	of	ADP
ejpam-1373	6	8	the	the	DET
ejpam-1373	6	9	set	set	NOUN
ejpam-1373	6	10	z+	z+	NUM
ejpam-1373	6	11	of	of	ADP
ejpam-1373	6	12	positive	positive	ADJ
ejpam-1373	6	13	integers	integer	NOUN
ejpam-1373	6	14	into	into	ADP
ejpam-1373	6	15	the	the	DET
ejpam-1373	6	16	power	power	NOUN
ejpam-1373	6	17	set	set	VERB
ejpam-1373	6	18	p(z+	p(z+	NOUN
ejpam-1373	6	19	)	)	PUNCT
ejpam-1373	6	20	such	such	ADJ
ejpam-1373	6	21	that	that	SCONJ
ejpam-1373	6	22	every	every	DET
ejpam-1373	6	23	member	member	NOUN
ejpam-1373	6	24	of	of	ADP
ejpam-1373	6	25	c(n	c(n	PROPN
ejpam-1373	6	26	)	)	PUNCT
ejpam-1373	6	27	is	be	AUX
ejpam-1373	6	28	a	a	DET
ejpam-1373	6	29	divisor	divisor	NOUN
ejpam-1373	6	30	of	of	ADP
ejpam-1373	6	31	n.	n.	NOUN
ejpam-1373	6	32	if	if	SCONJ
ejpam-1373	6	33	for	for	ADP
ejpam-1373	6	34	any	any	DET
ejpam-1373	6	35	n	n	CCONJ
ejpam-1373	6	36	,	,	PUNCT
ejpam-1373	6	37	d(n	d(n	NOUN
ejpam-1373	6	38	)	)	PUNCT
ejpam-1373	6	39	is	be	AUX
ejpam-1373	6	40	the	the	DET
ejpam-1373	6	41	set	set	NOUN
ejpam-1373	6	42	of	of	ADP
ejpam-1373	6	43	all	all	DET
ejpam-1373	6	44	positive	positive	ADJ
ejpam-1373	6	45	divisors	divisor	NOUN
ejpam-1373	6	46	of	of	ADP
ejpam-1373	6	47	n	n	NOUN
ejpam-1373	6	48	,	,	PUNCT
ejpam-1373	6	49	then	then	ADV
ejpam-1373	6	50	d	d	PROPN
ejpam-1373	6	51	is	be	AUX
ejpam-1373	6	52	called	call	VERB
ejpam-1373	6	53	the	the	DET
ejpam-1373	6	54	dirichlet	dirichlet	PROPN
ejpam-1373	6	55	’s	’s	PART
ejpam-1373	6	56	convolution	convolution	NOUN
ejpam-1373	6	57	.	.	PUNCT
ejpam-1373	7	1	it	it	PRON
ejpam-1373	7	2	is	be	AUX
ejpam-1373	7	3	well	well	ADV
ejpam-1373	7	4	known	know	VERB
ejpam-1373	7	5	that	that	SCONJ
ejpam-1373	7	6	z+	z+	NUM
ejpam-1373	7	7	has	have	VERB
ejpam-1373	7	8	the	the	DET
ejpam-1373	7	9	structure	structure	NOUN
ejpam-1373	7	10	of	of	ADP
ejpam-1373	7	11	a	a	DET
ejpam-1373	7	12	distributive	distributive	ADJ
ejpam-1373	7	13	lattice	lattice	NOUN
ejpam-1373	7	14	with	with	ADP
ejpam-1373	7	15	respect	respect	NOUN
ejpam-1373	7	16	to	to	ADP
ejpam-1373	7	17	the	the	DET
ejpam-1373	7	18	division	division	NOUN
ejpam-1373	7	19	order	order	NOUN
ejpam-1373	7	20	.	.	PUNCT
ejpam-1373	8	1	corresponding	correspond	VERB
ejpam-1373	8	2	to	to	ADP
ejpam-1373	8	3	any	any	DET
ejpam-1373	8	4	general	general	ADJ
ejpam-1373	8	5	convolution	convolution	NOUN
ejpam-1373	8	6	c	c	NOUN
ejpam-1373	8	7	,	,	PUNCT
ejpam-1373	8	8	one	one	PRON
ejpam-1373	8	9	can	can	AUX
ejpam-1373	8	10	define	define	VERB
ejpam-1373	8	11	a	a	DET
ejpam-1373	8	12	binary	binary	ADJ
ejpam-1373	8	13	relation	relation	NOUN
ejpam-1373	8	14	≤c	≤c	PROPN
ejpam-1373	8	15	on	on	ADP
ejpam-1373	8	16	z+	z+	NUM
ejpam-1373	8	17	by	by	ADP
ejpam-1373	8	18	‘	'	PUNCT
ejpam-1373	8	19	m	m	PROPN
ejpam-1373	8	20	≤c	≤c	PROPN
ejpam-1373	8	21	n	n	NOUN
ejpam-1373	8	22	if	if	SCONJ
ejpam-1373	8	23	and	and	CCONJ
ejpam-1373	8	24	only	only	ADV
ejpam-1373	8	25	if	if	SCONJ
ejpam-1373	8	26	m	m	VERB
ejpam-1373	8	27	∈	∈	PROPN
ejpam-1373	8	28	c(n	c(n	NOUN
ejpam-1373	8	29	)	)	PUNCT
ejpam-1373	8	30	’	'	PUNCT
ejpam-1373	8	31	.	.	PUNCT
ejpam-1373	9	1	in	in	ADP
ejpam-1373	9	2	this	this	DET
ejpam-1373	9	3	paper	paper	NOUN
ejpam-1373	9	4	we	we	PRON
ejpam-1373	9	5	characterize	characterize	VERB
ejpam-1373	9	6	convolutions	convolution	NOUN
ejpam-1373	9	7	c	c	X
ejpam-1373	9	8	which	which	PRON
ejpam-1373	9	9	induce	induce	VERB
ejpam-1373	9	10	partial	partial	ADJ
ejpam-1373	9	11	orders	order	NOUN
ejpam-1373	9	12	with	with	ADP
ejpam-1373	9	13	respect	respect	NOUN
ejpam-1373	9	14	to	to	ADP
ejpam-1373	9	15	which	which	PRON
ejpam-1373	9	16	z+	z+	NUM
ejpam-1373	9	17	has	have	VERB
ejpam-1373	9	18	the	the	DET
ejpam-1373	9	19	structure	structure	NOUN
ejpam-1373	9	20	of	of	ADP
ejpam-1373	9	21	a	a	DET
ejpam-1373	9	22	semi	semi	ADJ
ejpam-1373	9	23	lattice	lattice	NOUN
ejpam-1373	9	24	or	or	CCONJ
ejpam-1373	9	25	lattice	lattice	NOUN
ejpam-1373	9	26	and	and	CCONJ
ejpam-1373	9	27	various	various	ADJ
ejpam-1373	9	28	lattice	lattice	NOUN
ejpam-1373	9	29	theoretic	theoretic	NOUN
ejpam-1373	9	30	properties	property	NOUN
ejpam-1373	9	31	are	be	AUX
ejpam-1373	9	32	discussed	discuss	VERB
ejpam-1373	9	33	in	in	ADP
ejpam-1373	9	34	terms	term	NOUN
ejpam-1373	9	35	of	of	ADP
ejpam-1373	9	36	convolution	convolution	NOUN
ejpam-1373	9	37	.	.	PUNCT
ejpam-1373	10	1	2000	2000	NUM
ejpam-1373	10	2	mathematics	mathematic	NOUN
ejpam-1373	10	3	subject	subject	NOUN
ejpam-1373	10	4	classifications	classification	NOUN
ejpam-1373	10	5	:	:	PUNCT
ejpam-1373	10	6	06b99,11a99	06b99,11a99	X
ejpam-1373	10	7	.	.	PUNCT
ejpam-1373	11	1	key	key	ADJ
ejpam-1373	11	2	words	word	NOUN
ejpam-1373	11	3	and	and	CCONJ
ejpam-1373	11	4	phrases	phrase	NOUN
ejpam-1373	11	5	:	:	PUNCT
ejpam-1373	11	6	poset	poset	NOUN
ejpam-1373	11	7	,	,	PUNCT
ejpam-1373	11	8	lattice	lattice	NOUN
ejpam-1373	11	9	,	,	PUNCT
ejpam-1373	11	10	support	support	NOUN
ejpam-1373	11	11	,	,	PUNCT
ejpam-1373	11	12	convolution	convolution	NOUN
ejpam-1373	11	13	,	,	PUNCT
ejpam-1373	11	14	multiplicative	multiplicative	ADJ
ejpam-1373	11	15	,	,	PUNCT
ejpam-1373	11	16	relatively	relatively	ADV
ejpam-1373	11	17	prime	prime	ADJ
ejpam-1373	11	18	.	.	PUNCT
ejpam-1373	12	1	1	1	X
ejpam-1373	12	2	.	.	X
ejpam-1373	12	3	introduction	introduction	NOUN
ejpam-1373	12	4	a	a	DET
ejpam-1373	12	5	convolution	convolution	NOUN
ejpam-1373	12	6	is	be	AUX
ejpam-1373	12	7	a	a	DET
ejpam-1373	12	8	mapping	mapping	NOUN
ejpam-1373	12	9	c	c	NOUN
ejpam-1373	12	10	:	:	PUNCT
ejpam-1373	12	11	z+	z+	NUM
ejpam-1373	12	12	−→	−→	NOUN
ejpam-1373	12	13	p	p	X
ejpam-1373	12	14	(	(	PUNCT
ejpam-1373	12	15	z+	z+	NOUN
ejpam-1373	12	16	)	)	PUNCT
ejpam-1373	12	17	such	such	ADJ
ejpam-1373	12	18	that	that	SCONJ
ejpam-1373	12	19	c	c	NOUN
ejpam-1373	12	20	(	(	PUNCT
ejpam-1373	12	21	n	n	CCONJ
ejpam-1373	12	22	)	)	PUNCT
ejpam-1373	12	23	is	be	AUX
ejpam-1373	12	24	a	a	DET
ejpam-1373	12	25	set	set	NOUN
ejpam-1373	12	26	of	of	ADP
ejpam-1373	12	27	positive	positive	ADJ
ejpam-1373	12	28	divisors	divisor	NOUN
ejpam-1373	12	29	on	on	ADP
ejpam-1373	12	30	n	n	CCONJ
ejpam-1373	12	31	,	,	PUNCT
ejpam-1373	12	32	n	n	PROPN
ejpam-1373	12	33	∈	∈	PROPN
ejpam-1373	12	34	c	c	X
ejpam-1373	12	35	(	(	PUNCT
ejpam-1373	12	36	n	n	CCONJ
ejpam-1373	12	37	)	)	PUNCT
ejpam-1373	12	38	and	and	CCONJ
ejpam-1373	12	39	c	c	PROPN
ejpam-1373	12	40	(	(	PUNCT
ejpam-1373	12	41	n	n	CCONJ
ejpam-1373	12	42	)	)	PUNCT
ejpam-1373	12	43	=	=	SYM
ejpam-1373	13	1	⋃	⋃	NOUN
ejpam-1373	13	2	m∈c	m∈c	NOUN
ejpam-1373	13	3	(	(	PUNCT
ejpam-1373	13	4	n	n	CCONJ
ejpam-1373	13	5	)	)	PUNCT
ejpam-1373	13	6	c	c	NOUN
ejpam-1373	13	7	(	(	PUNCT
ejpam-1373	13	8	m	m	NOUN
ejpam-1373	13	9	)	)	PUNCT
ejpam-1373	13	10	,	,	PUNCT
ejpam-1373	13	11	for	for	ADP
ejpam-1373	13	12	any	any	DET
ejpam-1373	13	13	n	n	PRON
ejpam-1373	13	14	∈	∈	PROPN
ejpam-1373	13	15	z+	z+	X
ejpam-1373	13	16	.	.	PUNCT
ejpam-1373	14	1	popular	popular	ADJ
ejpam-1373	14	2	examples	example	NOUN
ejpam-1373	14	3	are	be	AUX
ejpam-1373	14	4	the	the	DET
ejpam-1373	14	5	dirichlet	dirichlet	PROPN
ejpam-1373	14	6	’s	’s	PART
ejpam-1373	14	7	convolution	convolution	NOUN
ejpam-1373	14	8	d	d	PROPN
ejpam-1373	14	9	and	and	CCONJ
ejpam-1373	14	10	the	the	DET
ejpam-1373	14	11	unitary	unitary	ADJ
ejpam-1373	14	12	convolution	convolution	NOUN
ejpam-1373	14	13	u	u	NOUN
ejpam-1373	14	14	defined	define	VERB
ejpam-1373	14	15	respectively	respectively	ADV
ejpam-1373	14	16	by	by	ADP
ejpam-1373	14	17	d(n	d(n	NOUN
ejpam-1373	14	18	)	)	PUNCT
ejpam-1373	14	19	=	=	SYM
ejpam-1373	14	20	the	the	DET
ejpam-1373	14	21	set	set	NOUN
ejpam-1373	14	22	of	of	ADP
ejpam-1373	14	23	all	all	DET
ejpam-1373	14	24	positive	positive	ADJ
ejpam-1373	14	25	divisors	divisor	NOUN
ejpam-1373	14	26	of	of	ADP
ejpam-1373	14	27	n	n	NOUN
ejpam-1373	14	28	and	and	CCONJ
ejpam-1373	14	29	u(n	u(n	PROPN
ejpam-1373	14	30	)	)	PUNCT
ejpam-1373	14	31	=	=	NOUN
ejpam-1373	15	1	the	the	DET
ejpam-1373	15	2	set	set	NOUN
ejpam-1373	15	3	of	of	ADP
ejpam-1373	15	4	unitary	unitary	ADJ
ejpam-1373	15	5	divisors	divisor	NOUN
ejpam-1373	15	6	of	of	ADP
ejpam-1373	15	7	n	n	PRON
ejpam-1373	15	8	for	for	ADP
ejpam-1373	15	9	any	any	DET
ejpam-1373	15	10	n	n	PRON
ejpam-1373	15	11	∈	∈	NOUN
ejpam-1373	15	12	z+	z+	NOUN
ejpam-1373	15	13	.	.	PUNCT
ejpam-1373	16	1	if	if	SCONJ
ejpam-1373	16	2	c	c	PROPN
ejpam-1373	16	3	is	be	AUX
ejpam-1373	16	4	a	a	DET
ejpam-1373	16	5	convolution	convolution	NOUN
ejpam-1373	16	6	,	,	PUNCT
ejpam-1373	16	7	then	then	ADV
ejpam-1373	16	8	the	the	DET
ejpam-1373	16	9	binary	binary	PROPN
ejpam-1373	16	10	relation	relation	PROPN
ejpam-1373	16	11	≤c	≤c	PROPN
ejpam-1373	16	12	on	on	ADP
ejpam-1373	16	13	z+	z+	NUM
ejpam-1373	16	14	,	,	PUNCT
ejpam-1373	16	15	defined	define	VERB
ejpam-1373	16	16	by	by	ADP
ejpam-1373	16	17	,	,	PUNCT
ejpam-1373	16	18	m	m	PROPN
ejpam-1373	16	19	≤c	≤c	PROPN
ejpam-1373	16	20	n	n	CCONJ
ejpam-1373	16	21	if	if	SCONJ
ejpam-1373	16	22	and	and	CCONJ
ejpam-1373	16	23	only	only	ADV
ejpam-1373	16	24	if	if	SCONJ
ejpam-1373	16	25	m	m	VERB
ejpam-1373	16	26	∈	∈	PROPN
ejpam-1373	16	27	c	c	X
ejpam-1373	16	28	(	(	PUNCT
ejpam-1373	16	29	n	n	CCONJ
ejpam-1373	16	30	)	)	PUNCT
ejpam-1373	16	31	,	,	PUNCT
ejpam-1373	16	32	is	be	AUX
ejpam-1373	16	33	a	a	DET
ejpam-1373	16	34	partial	partial	ADJ
ejpam-1373	16	35	order	order	NOUN
ejpam-1373	16	36	on	on	ADP
ejpam-1373	16	37	z+	z+	NUM
ejpam-1373	16	38	and	and	CCONJ
ejpam-1373	16	39	is	be	AUX
ejpam-1373	16	40	called	call	VERB
ejpam-1373	16	41	the	the	DET
ejpam-1373	16	42	partial	partial	ADJ
ejpam-1373	16	43	order	order	NOUN
ejpam-1373	16	44	induced	induce	VERB
ejpam-1373	16	45	by	by	ADP
ejpam-1373	16	46	c	c	PROPN
ejpam-1373	16	47	[	[	X
ejpam-1373	16	48	3	3	NUM
ejpam-1373	16	49	]	]	PUNCT
ejpam-1373	16	50	.	.	PUNCT
ejpam-1373	17	1	it	it	PRON
ejpam-1373	17	2	is	be	AUX
ejpam-1373	17	3	well	well	ADV
ejpam-1373	17	4	known	know	VERB
ejpam-1373	17	5	that	that	SCONJ
ejpam-1373	17	6	the	the	DET
ejpam-1373	17	7	dirichlet	dirichlet	PROPN
ejpam-1373	17	8	’s	’s	PART
ejpam-1373	17	9	convolution	convolution	NOUN
ejpam-1373	17	10	induces	induce	VERB
ejpam-1373	17	11	the	the	DET
ejpam-1373	17	12	division	division	NOUN
ejpam-1373	17	13	order	order	NOUN
ejpam-1373	17	14	on	on	ADP
ejpam-1373	17	15	z+	z+	NUM
ejpam-1373	17	16	with	with	ADP
ejpam-1373	17	17	respect	respect	NOUN
ejpam-1373	17	18	to	to	ADP
ejpam-1373	17	19	which	which	PRON
ejpam-1373	17	20	z+	z+	NUM
ejpam-1373	17	21	becomes	become	VERB
ejpam-1373	17	22	a	a	DET
ejpam-1373	17	23	distributive	distributive	ADJ
ejpam-1373	17	24	lattice	lattice	NOUN
ejpam-1373	17	25	,	,	PUNCT
ejpam-1373	17	26	where	where	SCONJ
ejpam-1373	17	27	,	,	PUNCT
ejpam-1373	17	28	for	for	ADP
ejpam-1373	17	29	any	any	DET
ejpam-1373	17	30	a	a	PRON
ejpam-1373	17	31	,	,	PUNCT
ejpam-1373	17	32	b	b	PROPN
ejpam-1373	17	33	∈	∈	PROPN
ejpam-1373	17	34	z+	z+	NUM
ejpam-1373	17	35	,	,	PUNCT
ejpam-1373	17	36	the	the	DET
ejpam-1373	17	37	greatest	great	ADJ
ejpam-1373	17	38	common	common	ADJ
ejpam-1373	17	39	divisor(gcd	divisor(gcd	NOUN
ejpam-1373	17	40	)	)	PUNCT
ejpam-1373	17	41	and	and	CCONJ
ejpam-1373	17	42	the	the	DET
ejpam-1373	17	43	least	least	ADJ
ejpam-1373	17	44	common	common	ADJ
ejpam-1373	17	45	multiple(lcm	multiple(lcm	NOUN
ejpam-1373	17	46	)	)	PUNCT
ejpam-1373	17	47	of	of	ADP
ejpam-1373	17	48	a	a	PRON
ejpam-1373	17	49	and	and	CCONJ
ejpam-1373	17	50	b	b	NOUN
ejpam-1373	17	51	are	be	AUX
ejpam-1373	17	52	respectively	respectively	ADV
ejpam-1373	17	53	the	the	DET
ejpam-1373	17	54	greatest	greatest	ADV
ejpam-1373	17	55	lower	low	ADJ
ejpam-1373	17	56	bound(glb	bound(glb	NOUN
ejpam-1373	17	57	)	)	PUNCT
ejpam-1373	17	58	∗corresponding	∗corresponde	VERB
ejpam-1373	17	59	author	author	NOUN
ejpam-1373	17	60	.	.	PUNCT
ejpam-1373	18	1	email	email	NOUN
ejpam-1373	18	2	addresses	address	NOUN
ejpam-1373	18	3	:	:	PUNCT
ejpam-1373	18	4	umswamy	umswamy	NOUN
ejpam-1373	18	5	�	�	PROPN
ejpam-1373	18	6	yahoo	yahoo	PROPN
ejpam-1373	18	7	.	.	PUNCT
ejpam-1373	19	1	om	om	PROPN
ejpam-1373	19	2	(	(	PUNCT
ejpam-1373	19	3	u.	u.	PROPN
ejpam-1373	19	4	swamy	swamy	PROPN
ejpam-1373	19	5	)	)	PUNCT
ejpam-1373	19	6	,	,	PUNCT
ejpam-1373	19	7	sagi_sankar	sagi_sankar	NOUN
ejpam-1373	19	8	�	�	PROPN
ejpam-1373	19	9	yahoo	yahoo	PROPN
ejpam-1373	19	10	.	.	PUNCT
ejpam-1373	20	1	o.in	o.in	PROPN
ejpam-1373	20	2	(	(	PUNCT
ejpam-1373	20	3	s.	s.	PROPN
ejpam-1373	20	4	sankar	sankar	PROPN
ejpam-1373	20	5	)	)	PUNCT
ejpam-1373	20	6	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-1373	21	1	424	424	NUM
ejpam-1373	21	2	c	c	NOUN
ejpam-1373	21	3	©	©	PROPN
ejpam-1373	21	4	2011	2011	NUM
ejpam-1373	21	5	ejpam	ejpam	VERB
ejpam-1373	21	6	all	all	DET
ejpam-1373	21	7	rights	right	NOUN
ejpam-1373	21	8	reserved	reserve	VERB
ejpam-1373	21	9	.	.	PUNCT
ejpam-1373	22	1	u.	u.	PROPN
ejpam-1373	22	2	swamy	swamy	PROPN
ejpam-1373	22	3	and	and	CCONJ
ejpam-1373	22	4	s.	s.	PROPN
ejpam-1373	22	5	sankar	sankar	PROPN
ejpam-1373	22	6	/	/	SYM
ejpam-1373	22	7	eur	eur	PROPN
ejpam-1373	22	8	.	.	PUNCT
ejpam-1373	23	1	j.	j.	PROPN
ejpam-1373	23	2	pure	pure	PROPN
ejpam-1373	23	3	appl	appl	PROPN
ejpam-1373	23	4	.	.	PROPN
ejpam-1373	23	5	math	math	PROPN
ejpam-1373	23	6	,	,	PUNCT
ejpam-1373	23	7	4	4	NUM
ejpam-1373	23	8	(	(	PUNCT
ejpam-1373	23	9	2011	2011	NUM
ejpam-1373	23	10	)	)	PUNCT
ejpam-1373	23	11	,	,	PUNCT
ejpam-1373	23	12	424	424	NUM
ejpam-1373	23	13	-	-	SYM
ejpam-1373	23	14	434	434	NUM
ejpam-1373	23	15	425	425	NUM
ejpam-1373	23	16	and	and	CCONJ
ejpam-1373	23	17	the	the	DET
ejpam-1373	23	18	least	least	ADJ
ejpam-1373	23	19	upper	upper	ADJ
ejpam-1373	23	20	bound(lub	bound(lub	NOUN
ejpam-1373	23	21	)	)	PUNCT
ejpam-1373	23	22	of	of	ADP
ejpam-1373	23	23	a	a	PRON
ejpam-1373	23	24	and	and	CCONJ
ejpam-1373	23	25	b	b	NOUN
ejpam-1373	23	26	.	.	PUNCT
ejpam-1373	24	1	in	in	ADP
ejpam-1373	24	2	fact	fact	NOUN
ejpam-1373	24	3	,	,	PUNCT
ejpam-1373	24	4	with	with	ADP
ejpam-1373	24	5	respect	respect	NOUN
ejpam-1373	24	6	to	to	ADP
ejpam-1373	24	7	the	the	DET
ejpam-1373	24	8	division	division	NOUN
ejpam-1373	24	9	order	order	NOUN
ejpam-1373	24	10	,	,	PUNCT
ejpam-1373	24	11	the	the	DET
ejpam-1373	24	12	lattice	lattice	NOUN
ejpam-1373	24	13	z+	z+	NUM
ejpam-1373	24	14	satisfies	satisfy	VERB
ejpam-1373	24	15	the	the	DET
ejpam-1373	24	16	infinite	infinite	ADJ
ejpam-1373	24	17	join	join	NOUN
ejpam-1373	24	18	distributive	distributive	ADJ
ejpam-1373	24	19	law	law	NOUN
ejpam-1373	24	20	given	give	VERB
ejpam-1373	24	21	by	by	ADP
ejpam-1373	24	22	(	(	PUNCT
ejpam-1373	24	23	a	a	DET
ejpam-1373	24	24	∨	∨	NOUN
ejpam-1373	24	25	(	(	PUNCT
ejpam-1373	24	26	∧	∧	PROPN
ejpam-1373	24	27	i∈i	i∈i	ADJ
ejpam-1373	24	28	bi	bi	NOUN
ejpam-1373	24	29	)	)	PUNCT
ejpam-1373	24	30	=	=	SYM
ejpam-1373	25	1	∧	∧	PROPN
ejpam-1373	25	2	i∈i	i∈i	ADJ
ejpam-1373	25	3	(	(	PUNCT
ejpam-1373	25	4	a	a	DET
ejpam-1373	25	5	∨	∨	NUM
ejpam-1373	25	6	bi	bi	NOUN
ejpam-1373	25	7	)	)	PUNCT
ejpam-1373	25	8	)	)	PUNCT
ejpam-1373	26	1	for	for	ADP
ejpam-1373	26	2	any	any	DET
ejpam-1373	26	3	a	a	DET
ejpam-1373	26	4	∈	∈	NOUN
ejpam-1373	26	5	z+	z+	NUM
ejpam-1373	26	6	and	and	CCONJ
ejpam-1373	26	7	{	{	PUNCT
ejpam-1373	26	8	bi}i∈i	bi}i∈i	NOUN
ejpam-1373	26	9	⊆	⊆	NUM
ejpam-1373	26	10	z	z	NOUN
ejpam-1373	26	11	+	+	NOUN
ejpam-1373	26	12	.	.	PUNCT
ejpam-1373	27	1	in	in	ADP
ejpam-1373	27	2	this	this	DET
ejpam-1373	27	3	paper	paper	NOUN
ejpam-1373	27	4	,	,	PUNCT
ejpam-1373	27	5	we	we	PRON
ejpam-1373	27	6	discuss	discuss	VERB
ejpam-1373	27	7	various	various	ADJ
ejpam-1373	27	8	aspects	aspect	NOUN
ejpam-1373	27	9	of	of	ADP
ejpam-1373	27	10	the	the	DET
ejpam-1373	27	11	lattice	lattice	NOUN
ejpam-1373	27	12	structures	structure	NOUN
ejpam-1373	27	13	on	on	ADP
ejpam-1373	27	14	z+	z+	NUM
ejpam-1373	27	15	induced	induce	VERB
ejpam-1373	27	16	by	by	ADP
ejpam-1373	27	17	general	general	ADJ
ejpam-1373	27	18	convolutions	convolution	NOUN
ejpam-1373	27	19	.	.	PUNCT
ejpam-1373	28	1	2	2	X
ejpam-1373	28	2	.	.	X
ejpam-1373	28	3	preliminaries	preliminary	NOUN
ejpam-1373	28	4	let	let	VERB
ejpam-1373	28	5	us	we	PRON
ejpam-1373	28	6	recall	recall	VERB
ejpam-1373	28	7	that	that	SCONJ
ejpam-1373	28	8	a	a	DET
ejpam-1373	28	9	partial	partial	ADJ
ejpam-1373	28	10	order	order	NOUN
ejpam-1373	28	11	on	on	ADP
ejpam-1373	28	12	a	a	DET
ejpam-1373	28	13	non	non	ADJ
ejpam-1373	28	14	-	-	ADJ
ejpam-1373	28	15	empty	empty	ADJ
ejpam-1373	28	16	set	set	NOUN
ejpam-1373	28	17	x	x	PUNCT
ejpam-1373	28	18	is	be	AUX
ejpam-1373	28	19	defined	define	VERB
ejpam-1373	28	20	as	as	ADP
ejpam-1373	28	21	a	a	DET
ejpam-1373	28	22	binary	binary	ADJ
ejpam-1373	28	23	relation	relation	NOUN
ejpam-1373	28	24	≤	≤	PUNCT
ejpam-1373	28	25	on	on	ADP
ejpam-1373	28	26	x	x	PUNCT
ejpam-1373	28	27	which	which	PRON
ejpam-1373	28	28	is	be	AUX
ejpam-1373	28	29	reflexive	reflexive	ADJ
ejpam-1373	28	30	(	(	PUNCT
ejpam-1373	28	31	a	a	DET
ejpam-1373	28	32	≤	≤	PROPN
ejpam-1373	28	33	a	a	X
ejpam-1373	28	34	)	)	PUNCT
ejpam-1373	28	35	,	,	PUNCT
ejpam-1373	28	36	transitive	transitive	ADJ
ejpam-1373	28	37	(	(	PUNCT
ejpam-1373	28	38	a	a	DET
ejpam-1373	28	39	≤	≤	NUM
ejpam-1373	28	40	b	b	NUM
ejpam-1373	28	41	,	,	PUNCT
ejpam-1373	28	42	b	b	PROPN
ejpam-1373	28	43	≤	≤	NUM
ejpam-1373	28	44	c	c	X
ejpam-1373	29	1	=	=	PRON
ejpam-1373	29	2	⇒	⇒	VERB
ejpam-1373	29	3	a	a	DET
ejpam-1373	29	4	≤	≤	NUM
ejpam-1373	29	5	c	c	NOUN
ejpam-1373	29	6	)	)	PUNCT
ejpam-1373	29	7	and	and	CCONJ
ejpam-1373	29	8	antisymmetric	antisymmetric	ADJ
ejpam-1373	29	9	(	(	PUNCT
ejpam-1373	29	10	a	a	DET
ejpam-1373	29	11	≤	≤	NUM
ejpam-1373	29	12	b	b	NUM
ejpam-1373	29	13	,	,	PUNCT
ejpam-1373	29	14	b	b	PROPN
ejpam-1373	29	15	≤	≤	NOUN
ejpam-1373	29	16	a	a	DET
ejpam-1373	29	17	=	=	NOUN
ejpam-1373	29	18	⇒	⇒	VERB
ejpam-1373	29	19	a	a	DET
ejpam-1373	29	20	=	=	SYM
ejpam-1373	29	21	b	b	NOUN
ejpam-1373	29	22	)	)	PUNCT
ejpam-1373	29	23	and	and	CCONJ
ejpam-1373	29	24	that	that	SCONJ
ejpam-1373	29	25	a	a	DET
ejpam-1373	29	26	pair	pair	NOUN
ejpam-1373	29	27	(	(	PUNCT
ejpam-1373	29	28	x	x	NOUN
ejpam-1373	29	29	,	,	PUNCT
ejpam-1373	29	30	≤	≤	NUM
ejpam-1373	29	31	)	)	PUNCT
ejpam-1373	29	32	is	be	AUX
ejpam-1373	29	33	called	call	VERB
ejpam-1373	29	34	a	a	DET
ejpam-1373	29	35	partially	partially	ADV
ejpam-1373	29	36	ordered	order	VERB
ejpam-1373	29	37	set(poset	set(poset	NOUN
ejpam-1373	29	38	)	)	PUNCT
ejpam-1373	29	39	if	if	SCONJ
ejpam-1373	29	40	x	x	PRON
ejpam-1373	29	41	is	be	AUX
ejpam-1373	29	42	a	a	DET
ejpam-1373	29	43	non	non	ADJ
ejpam-1373	29	44	-	-	ADJ
ejpam-1373	29	45	empty	empty	ADJ
ejpam-1373	29	46	set	set	NOUN
ejpam-1373	29	47	and	and	CCONJ
ejpam-1373	29	48	≤	≤	NOUN
ejpam-1373	29	49	is	be	AUX
ejpam-1373	29	50	a	a	DET
ejpam-1373	29	51	partial	partial	ADJ
ejpam-1373	29	52	order	order	NOUN
ejpam-1373	29	53	on	on	ADP
ejpam-1373	29	54	x	x	X
ejpam-1373	29	55	.	.	PUNCT
ejpam-1373	30	1	for	for	ADP
ejpam-1373	30	2	any	any	DET
ejpam-1373	30	3	a	a	DET
ejpam-1373	30	4	⊆	⊆	NUM
ejpam-1373	30	5	x	x	SYM
ejpam-1373	30	6	and	and	CCONJ
ejpam-1373	30	7	x	x	SYM
ejpam-1373	30	8	∈	∈	PROPN
ejpam-1373	30	9	x	x	X
ejpam-1373	30	10	,	,	PUNCT
ejpam-1373	30	11	x	x	X
ejpam-1373	30	12	is	be	AUX
ejpam-1373	30	13	called	call	VERB
ejpam-1373	30	14	a	a	DET
ejpam-1373	30	15	lower(upper	lower(upper	PROPN
ejpam-1373	30	16	)	)	PUNCT
ejpam-1373	30	17	bound	bind	VERB
ejpam-1373	30	18	of	of	ADP
ejpam-1373	30	19	a	a	DET
ejpam-1373	30	20	if	if	NOUN
ejpam-1373	30	21	x	x	SYM
ejpam-1373	30	22	≤	≤	NUM
ejpam-1373	30	23	a(respectively	a(respectively	ADV
ejpam-1373	30	24	a	a	DET
ejpam-1373	30	25	≤	≤	NUM
ejpam-1373	30	26	x	x	X
ejpam-1373	30	27	)	)	PUNCT
ejpam-1373	30	28	for	for	ADP
ejpam-1373	30	29	all	all	DET
ejpam-1373	30	30	a	a	DET
ejpam-1373	30	31	∈	∈	NOUN
ejpam-1373	30	32	a.	a.	NOUN
ejpam-1373	30	33	we	we	PRON
ejpam-1373	30	34	have	have	VERB
ejpam-1373	30	35	the	the	DET
ejpam-1373	30	36	usual	usual	ADJ
ejpam-1373	30	37	notations	notation	NOUN
ejpam-1373	30	38	of	of	ADP
ejpam-1373	30	39	the	the	DET
ejpam-1373	30	40	greatest	greatest	ADV
ejpam-1373	30	41	lower	low	ADJ
ejpam-1373	30	42	bound(glb	bound(glb	NOUN
ejpam-1373	30	43	)	)	PUNCT
ejpam-1373	30	44	and	and	CCONJ
ejpam-1373	30	45	least	least	ADJ
ejpam-1373	30	46	upper	upper	ADJ
ejpam-1373	30	47	bound(lub	bound(lub	NOUN
ejpam-1373	30	48	)	)	PUNCT
ejpam-1373	30	49	of	of	ADP
ejpam-1373	30	50	a	a	DET
ejpam-1373	30	51	in	in	ADP
ejpam-1373	30	52	x	x	X
ejpam-1373	30	53	.	.	PUNCT
ejpam-1373	31	1	if	if	SCONJ
ejpam-1373	31	2	a	a	PRON
ejpam-1373	31	3	is	be	AUX
ejpam-1373	31	4	a	a	DET
ejpam-1373	31	5	finite	finite	NOUN
ejpam-1373	31	6	subset	subset	NOUN
ejpam-1373	31	7	{	{	PUNCT
ejpam-1373	31	8	a1	a1	PROPN
ejpam-1373	31	9	,	,	PUNCT
ejpam-1373	31	10	a2	a2	PROPN
ejpam-1373	31	11	,	,	PUNCT
ejpam-1373	31	12	·	·	PUNCT
ejpam-1373	31	13	·	·	PUNCT
ejpam-1373	31	14	·	·	PUNCT
ejpam-1373	31	15	,	,	PUNCT
ejpam-1373	31	16	an	an	X
ejpam-1373	31	17	}	}	PUNCT
ejpam-1373	31	18	,	,	PUNCT
ejpam-1373	31	19	the	the	DET
ejpam-1373	31	20	glb	glb	NOUN
ejpam-1373	31	21	of	of	ADP
ejpam-1373	31	22	a(lub	a(lub	NOUN
ejpam-1373	31	23	of	of	ADP
ejpam-1373	31	24	a	a	PRON
ejpam-1373	31	25	)	)	PUNCT
ejpam-1373	31	26	is	be	AUX
ejpam-1373	31	27	denoted	denote	VERB
ejpam-1373	31	28	by	by	ADP
ejpam-1373	31	29	a1	a1	NOUN
ejpam-1373	31	30	∧	∧	PROPN
ejpam-1373	31	31	a2	a2	PROPN
ejpam-1373	31	32	∧	∧	PROPN
ejpam-1373	31	33	·	·	PUNCT
ejpam-1373	31	34	·	·	PUNCT
ejpam-1373	31	35	·	·	PUNCT
ejpam-1373	32	1	∧	∧	NOUN
ejpam-1373	32	2	an	an	PRON
ejpam-1373	32	3	or	or	CCONJ
ejpam-1373	32	4	n	n	PRON
ejpam-1373	32	5	∧	∧	NOUN
ejpam-1373	32	6	i=1	i=1	PROPN
ejpam-1373	32	7	ai	ai	VERB
ejpam-1373	32	8	(	(	PUNCT
ejpam-1373	32	9	respectively	respectively	ADV
ejpam-1373	32	10	by	by	ADP
ejpam-1373	32	11	a1∨a2∨	a1∨a2∨	PROPN
ejpam-1373	32	12	·	·	PUNCT
ejpam-1373	32	13	·	·	PUNCT
ejpam-1373	32	14	·	·	PUNCT
ejpam-1373	32	15	∨an	∨an	NOUN
ejpam-1373	32	16	or	or	CCONJ
ejpam-1373	32	17	n	n	PRON
ejpam-1373	32	18	∨	∨	NUM
ejpam-1373	32	19	i=1	i=1	PROPN
ejpam-1373	32	20	ai	ai	PROPN
ejpam-1373	32	21	)	)	PUNCT
ejpam-1373	32	22	.	.	PUNCT
ejpam-1373	33	1	a	a	DET
ejpam-1373	33	2	partially	partially	ADV
ejpam-1373	33	3	ordered	order	VERB
ejpam-1373	33	4	set	set	NOUN
ejpam-1373	33	5	(	(	PUNCT
ejpam-1373	33	6	x	x	INTJ
ejpam-1373	33	7	,	,	PUNCT
ejpam-1373	33	8	≤	≤	NUM
ejpam-1373	33	9	)	)	PUNCT
ejpam-1373	33	10	is	be	AUX
ejpam-1373	33	11	called	call	VERB
ejpam-1373	33	12	a	a	DET
ejpam-1373	33	13	meet	meet	NOUN
ejpam-1373	33	14	semi	semi	ADJ
ejpam-1373	33	15	lattice	lattice	NOUN
ejpam-1373	33	16	if	if	SCONJ
ejpam-1373	33	17	a∧	a∧	PROPN
ejpam-1373	33	18	b	b	PROPN
ejpam-1373	33	19	(=	(=	ADP
ejpam-1373	33	20	glb{a	glb{a	NOUN
ejpam-1373	33	21	,	,	PUNCT
ejpam-1373	33	22	b	b	NOUN
ejpam-1373	33	23	}	}	PUNCT
ejpam-1373	33	24	)	)	PUNCT
ejpam-1373	33	25	exists	exist	VERB
ejpam-1373	33	26	for	for	ADP
ejpam-1373	33	27	all	all	DET
ejpam-1373	33	28	a	a	PRON
ejpam-1373	33	29	and	and	CCONJ
ejpam-1373	33	30	b	b	NOUN
ejpam-1373	33	31	∈	∈	PROPN
ejpam-1373	33	32	x	x	X
ejpam-1373	33	33	.	.	PUNCT
ejpam-1373	34	1	(	(	PUNCT
ejpam-1373	34	2	x	x	X
ejpam-1373	34	3	,	,	PUNCT
ejpam-1373	34	4	≤	≤	NUM
ejpam-1373	34	5	)	)	PUNCT
ejpam-1373	34	6	is	be	AUX
ejpam-1373	34	7	called	call	VERB
ejpam-1373	34	8	a	a	DET
ejpam-1373	34	9	join	join	NOUN
ejpam-1373	34	10	semi	semi	ADV
ejpam-1373	34	11	lattice	lattice	NOUN
ejpam-1373	34	12	if	if	SCONJ
ejpam-1373	34	13	a∨	a∨	PROPN
ejpam-1373	34	14	b	b	PROPN
ejpam-1373	34	15	(=	(=	NOUN
ejpam-1373	34	16	lub{a	lub{a	ADV
ejpam-1373	34	17	,	,	PUNCT
ejpam-1373	34	18	b	b	NOUN
ejpam-1373	34	19	}	}	PUNCT
ejpam-1373	34	20	)	)	PUNCT
ejpam-1373	34	21	exists	exist	VERB
ejpam-1373	34	22	for	for	ADP
ejpam-1373	34	23	all	all	DET
ejpam-1373	34	24	a	a	PRON
ejpam-1373	34	25	and	and	CCONJ
ejpam-1373	34	26	b	b	NOUN
ejpam-1373	34	27	∈	∈	NOUN
ejpam-1373	34	28	x	x	X
ejpam-1373	34	29	.	.	PUNCT
ejpam-1373	35	1	a	a	DET
ejpam-1373	35	2	poset	poset	NOUN
ejpam-1373	35	3	(	(	PUNCT
ejpam-1373	35	4	x	x	INTJ
ejpam-1373	35	5	,	,	PUNCT
ejpam-1373	35	6	≤	≤	NUM
ejpam-1373	35	7	)	)	PUNCT
ejpam-1373	35	8	is	be	AUX
ejpam-1373	35	9	called	call	VERB
ejpam-1373	35	10	a	a	DET
ejpam-1373	35	11	lattice	lattice	NOUN
ejpam-1373	35	12	if	if	SCONJ
ejpam-1373	35	13	it	it	PRON
ejpam-1373	35	14	is	be	AUX
ejpam-1373	35	15	both	both	CCONJ
ejpam-1373	35	16	a	a	DET
ejpam-1373	35	17	meet	meet	NOUN
ejpam-1373	35	18	and	and	CCONJ
ejpam-1373	35	19	join	join	VERB
ejpam-1373	35	20	semi	semi	ADV
ejpam-1373	35	21	lattice	lattice	PROPN
ejpam-1373	35	22	.	.	PUNCT
ejpam-1373	36	1	equivalently	equivalently	ADV
ejpam-1373	36	2	,	,	PUNCT
ejpam-1373	36	3	lattice	lattice	PROPN
ejpam-1373	36	4	can	can	AUX
ejpam-1373	36	5	also	also	ADV
ejpam-1373	36	6	be	be	AUX
ejpam-1373	36	7	defined	define	VERB
ejpam-1373	36	8	as	as	ADP
ejpam-1373	36	9	an	an	DET
ejpam-1373	36	10	algebraic	algebraic	ADJ
ejpam-1373	36	11	system	system	NOUN
ejpam-1373	36	12	(	(	PUNCT
ejpam-1373	36	13	x	x	INTJ
ejpam-1373	36	14	,	,	PUNCT
ejpam-1373	36	15	∧,∨	∧,∨	ADJ
ejpam-1373	36	16	)	)	PUNCT
ejpam-1373	36	17	,	,	PUNCT
ejpam-1373	36	18	where	where	SCONJ
ejpam-1373	36	19	∧	∧	PROPN
ejpam-1373	36	20	and	and	CCONJ
ejpam-1373	36	21	∨	∨	NUM
ejpam-1373	36	22	are	be	AUX
ejpam-1373	36	23	binary	binary	ADJ
ejpam-1373	36	24	operations	operation	NOUN
ejpam-1373	36	25	which	which	PRON
ejpam-1373	36	26	are	be	AUX
ejpam-1373	36	27	associative	associative	ADJ
ejpam-1373	36	28	,	,	PUNCT
ejpam-1373	36	29	commutative	commutative	ADJ
ejpam-1373	36	30	and	and	CCONJ
ejpam-1373	36	31	idempotent	idempotent	NOUN
ejpam-1373	36	32	and	and	CCONJ
ejpam-1373	36	33	satisfying	satisfy	VERB
ejpam-1373	36	34	the	the	DET
ejpam-1373	36	35	absorption	absorption	NOUN
ejpam-1373	36	36	laws	law	NOUN
ejpam-1373	36	37	,	,	PUNCT
ejpam-1373	36	38	namely	namely	ADV
ejpam-1373	36	39	a	a	DET
ejpam-1373	36	40	∧	∧	NOUN
ejpam-1373	36	41	(	(	PUNCT
ejpam-1373	36	42	a	a	DET
ejpam-1373	36	43	∨	∨	NUM
ejpam-1373	36	44	b	b	NOUN
ejpam-1373	36	45	)	)	PUNCT
ejpam-1373	36	46	=	=	PUNCT
ejpam-1373	37	1	a	a	PRON
ejpam-1373	37	2	=	=	PUNCT
ejpam-1373	37	3	a	a	DET
ejpam-1373	37	4	∨	∨	NOUN
ejpam-1373	37	5	(	(	PUNCT
ejpam-1373	37	6	a	a	DET
ejpam-1373	37	7	∧	∧	PROPN
ejpam-1373	37	8	b	b	NOUN
ejpam-1373	37	9	)	)	PUNCT
ejpam-1373	37	10	for	for	ADP
ejpam-1373	37	11	all	all	DET
ejpam-1373	37	12	a	a	DET
ejpam-1373	37	13	,	,	PUNCT
ejpam-1373	37	14	b	b	X
ejpam-1373	37	15	∈	∈	PROPN
ejpam-1373	37	16	x	x	X
ejpam-1373	37	17	;	;	PUNCT
ejpam-1373	37	18	in	in	ADP
ejpam-1373	37	19	this	this	DET
ejpam-1373	37	20	case	case	NOUN
ejpam-1373	37	21	the	the	DET
ejpam-1373	37	22	partial	partial	ADJ
ejpam-1373	37	23	order	order	NOUN
ejpam-1373	37	24	≤	≤	X
ejpam-1373	37	25	on	on	ADP
ejpam-1373	37	26	x	x	SYM
ejpam-1373	37	27	is	be	AUX
ejpam-1373	37	28	such	such	ADJ
ejpam-1373	37	29	that	that	SCONJ
ejpam-1373	37	30	a	a	DET
ejpam-1373	37	31	∧	∧	PROPN
ejpam-1373	37	32	b	b	PROPN
ejpam-1373	37	33	and	and	CCONJ
ejpam-1373	37	34	a	a	DET
ejpam-1373	37	35	∨	∨	PROPN
ejpam-1373	37	36	b	b	NOUN
ejpam-1373	37	37	are	be	AUX
ejpam-1373	37	38	respectively	respectively	ADV
ejpam-1373	37	39	the	the	DET
ejpam-1373	37	40	glb	glb	NOUN
ejpam-1373	37	41	and	and	CCONJ
ejpam-1373	37	42	lub	lub	NOUN
ejpam-1373	37	43	of	of	ADP
ejpam-1373	37	44	{	{	PUNCT
ejpam-1373	37	45	a	a	PROPN
ejpam-1373	37	46	,	,	PUNCT
ejpam-1373	37	47	b	b	NOUN
ejpam-1373	37	48	}	}	PUNCT
ejpam-1373	37	49	.	.	PUNCT
ejpam-1373	38	1	the	the	DET
ejpam-1373	38	2	algebraic	algebraic	ADJ
ejpam-1373	38	3	operations	operation	NOUN
ejpam-1373	38	4	∧	∧	PROPN
ejpam-1373	38	5	and	and	CCONJ
ejpam-1373	38	6	∨	∨	NUM
ejpam-1373	38	7	and	and	CCONJ
ejpam-1373	38	8	the	the	DET
ejpam-1373	38	9	partial	partial	ADJ
ejpam-1373	38	10	order	order	NOUN
ejpam-1373	38	11	≤	≤	NOUN
ejpam-1373	38	12	are	be	AUX
ejpam-1373	38	13	related	relate	VERB
ejpam-1373	38	14	by	by	ADP
ejpam-1373	38	15	(	(	PUNCT
ejpam-1373	38	16	a	a	DET
ejpam-1373	38	17	=	=	X
ejpam-1373	38	18	a	a	DET
ejpam-1373	38	19	∧	∧	PROPN
ejpam-1373	38	20	b	b	PROPN
ejpam-1373	38	21	⇐	⇐	PROPN
ejpam-1373	38	22	⇒	⇒	NOUN
ejpam-1373	38	23	a	a	DET
ejpam-1373	38	24	≤	≤	PROPN
ejpam-1373	38	25	b	b	PUNCT
ejpam-1373	38	26	⇐	⇐	ADJ
ejpam-1373	38	27	⇒	⇒	NOUN
ejpam-1373	38	28	a	a	DET
ejpam-1373	38	29	∨	∨	NOUN
ejpam-1373	38	30	b	b	PROPN
ejpam-1373	38	31	=	=	SYM
ejpam-1373	38	32	b	b	PROPN
ejpam-1373	38	33	)	)	PUNCT
ejpam-1373	38	34	.	.	PUNCT
ejpam-1373	39	1	throughout	throughout	ADP
ejpam-1373	39	2	the	the	DET
ejpam-1373	39	3	paper	paper	NOUN
ejpam-1373	39	4	,	,	PUNCT
ejpam-1373	39	5	z+	z+	NUM
ejpam-1373	39	6	and	and	CCONJ
ejpam-1373	39	7	n	n	PRON
ejpam-1373	39	8	denote	denote	VERB
ejpam-1373	39	9	the	the	DET
ejpam-1373	39	10	set	set	NOUN
ejpam-1373	39	11	of	of	ADP
ejpam-1373	39	12	positive	positive	ADJ
ejpam-1373	39	13	integers	integer	NOUN
ejpam-1373	39	14	and	and	CCONJ
ejpam-1373	39	15	the	the	DET
ejpam-1373	39	16	set	set	NOUN
ejpam-1373	39	17	of	of	ADP
ejpam-1373	39	18	nonnegative	nonnegative	ADJ
ejpam-1373	39	19	integers	integer	NOUN
ejpam-1373	39	20	respectively	respectively	ADV
ejpam-1373	39	21	.	.	PUNCT
ejpam-1373	40	1	definition	definition	NOUN
ejpam-1373	40	2	1	1	NUM
ejpam-1373	40	3	.	.	PUNCT
ejpam-1373	41	1	a	a	DET
ejpam-1373	41	2	mappingc	mappingc	NOUN
ejpam-1373	41	3	:	:	PUNCT
ejpam-1373	41	4	z+	z+	NUM
ejpam-1373	41	5	−→p	−→p	NOUN
ejpam-1373	41	6	(	(	PUNCT
ejpam-1373	41	7	z+	z+	NUM
ejpam-1373	41	8	)	)	PUNCT
ejpam-1373	41	9	is	be	AUX
ejpam-1373	41	10	called	call	VERB
ejpam-1373	41	11	a	a	DET
ejpam-1373	41	12	convolution	convolution	NOUN
ejpam-1373	41	13	if	if	SCONJ
ejpam-1373	41	14	the	the	DET
ejpam-1373	41	15	following	following	NOUN
ejpam-1373	41	16	are	be	AUX
ejpam-1373	41	17	satisfied	satisfied	ADJ
ejpam-1373	41	18	for	for	ADP
ejpam-1373	41	19	any	any	DET
ejpam-1373	41	20	n	n	PRON
ejpam-1373	41	21	∈	∈	PROPN
ejpam-1373	41	22	z+	z+	NUM
ejpam-1373	41	23	.	.	PUNCT
ejpam-1373	42	1	(	(	PUNCT
ejpam-1373	42	2	1	1	NUM
ejpam-1373	42	3	)	)	PUNCT
ejpam-1373	42	4	.	.	PUNCT
ejpam-1373	43	1	c	c	NOUN
ejpam-1373	43	2	(	(	PUNCT
ejpam-1373	43	3	n	n	CCONJ
ejpam-1373	43	4	)	)	PUNCT
ejpam-1373	43	5	is	be	AUX
ejpam-1373	43	6	a	a	DET
ejpam-1373	43	7	set	set	NOUN
ejpam-1373	43	8	of	of	ADP
ejpam-1373	43	9	positive	positive	ADJ
ejpam-1373	43	10	divisors	divisor	NOUN
ejpam-1373	43	11	of	of	ADP
ejpam-1373	43	12	n	n	PROPN
ejpam-1373	43	13	(	(	PUNCT
ejpam-1373	43	14	2	2	NUM
ejpam-1373	43	15	)	)	PUNCT
ejpam-1373	43	16	.	.	PUNCT
ejpam-1373	44	1	n	n	PROPN
ejpam-1373	44	2	∈	∈	PROPN
ejpam-1373	44	3	c	c	X
ejpam-1373	44	4	(	(	PUNCT
ejpam-1373	44	5	n	n	CCONJ
ejpam-1373	44	6	)	)	PUNCT
ejpam-1373	44	7	(	(	PUNCT
ejpam-1373	44	8	3	3	NUM
ejpam-1373	44	9	)	)	PUNCT
ejpam-1373	44	10	.	.	PUNCT
ejpam-1373	45	1	c	c	NOUN
ejpam-1373	45	2	(	(	PUNCT
ejpam-1373	45	3	n	n	CCONJ
ejpam-1373	45	4	)	)	PUNCT
ejpam-1373	45	5	=	=	SYM
ejpam-1373	45	6	⋃	⋃	NOUN
ejpam-1373	45	7	m∈c	m∈c	NOUN
ejpam-1373	45	8	(	(	PUNCT
ejpam-1373	45	9	n	n	CCONJ
ejpam-1373	45	10	)	)	PUNCT
ejpam-1373	45	11	c	c	NOUN
ejpam-1373	45	12	(	(	PUNCT
ejpam-1373	45	13	m	m	NOUN
ejpam-1373	45	14	)	)	PUNCT
ejpam-1373	45	15	.	.	PUNCT
ejpam-1373	46	1	definition	definition	NOUN
ejpam-1373	46	2	2	2	NUM
ejpam-1373	46	3	.	.	X
ejpam-1373	47	1	for	for	ADP
ejpam-1373	47	2	any	any	DET
ejpam-1373	47	3	convolution	convolution	NOUN
ejpam-1373	47	4	c	c	PROPN
ejpam-1373	47	5	and	and	CCONJ
ejpam-1373	47	6	m	m	PROPN
ejpam-1373	47	7	and	and	CCONJ
ejpam-1373	47	8	n	n	PRON
ejpam-1373	47	9	∈	∈	PROPN
ejpam-1373	47	10	z+	z+	NUM
ejpam-1373	47	11	,	,	PUNCT
ejpam-1373	47	12	we	we	PRON
ejpam-1373	47	13	define	define	VERB
ejpam-1373	47	14	�	�	PROPN
ejpam-1373	47	15	m≤	m≤	PROPN
ejpam-1373	47	16	n	n	CCONJ
ejpam-1373	47	17	if	if	ADV
ejpam-1373	48	1	and	and	CCONJ
ejpam-1373	48	2	only	only	ADV
ejpam-1373	48	3	if	if	SCONJ
ejpam-1373	48	4	m	m	VERB
ejpam-1373	48	5	∈	∈	PROPN
ejpam-1373	48	6	c	c	X
ejpam-1373	48	7	(	(	PUNCT
ejpam-1373	48	8	n	n	CCONJ
ejpam-1373	48	9	)	)	PUNCT
ejpam-1373	48	10	�	�	PROPN
ejpam-1373	48	11	then	then	ADV
ejpam-1373	48	12	≤c	≤c	PROPN
ejpam-1373	48	13	is	be	AUX
ejpam-1373	48	14	a	a	DET
ejpam-1373	48	15	partial	partial	ADJ
ejpam-1373	48	16	order	order	NOUN
ejpam-1373	48	17	on	on	ADP
ejpam-1373	48	18	z+	z+	NUM
ejpam-1373	48	19	and	and	CCONJ
ejpam-1373	48	20	is	be	AUX
ejpam-1373	48	21	called	call	VERB
ejpam-1373	48	22	the	the	DET
ejpam-1373	48	23	partial	partial	ADJ
ejpam-1373	48	24	order	order	NOUN
ejpam-1373	48	25	induced	induce	VERB
ejpam-1373	48	26	by	by	ADP
ejpam-1373	48	27	c	c	PROPN
ejpam-1373	48	28	on	on	ADP
ejpam-1373	48	29	z+	z+	NUM
ejpam-1373	48	30	.	.	PUNCT
ejpam-1373	49	1	u.	u.	PROPN
ejpam-1373	49	2	swamy	swamy	PROPN
ejpam-1373	49	3	and	and	CCONJ
ejpam-1373	49	4	s.	s.	PROPN
ejpam-1373	49	5	sankar	sankar	PROPN
ejpam-1373	49	6	/	/	SYM
ejpam-1373	49	7	eur	eur	PROPN
ejpam-1373	49	8	.	.	PUNCT
ejpam-1373	50	1	j.	j.	PROPN
ejpam-1373	50	2	pure	pure	PROPN
ejpam-1373	50	3	appl	appl	PROPN
ejpam-1373	50	4	.	.	PROPN
ejpam-1373	50	5	math	math	PROPN
ejpam-1373	50	6	,	,	PUNCT
ejpam-1373	50	7	4	4	NUM
ejpam-1373	50	8	(	(	PUNCT
ejpam-1373	50	9	2011	2011	NUM
ejpam-1373	50	10	)	)	PUNCT
ejpam-1373	50	11	,	,	PUNCT
ejpam-1373	50	12	424	424	NUM
ejpam-1373	50	13	-	-	SYM
ejpam-1373	50	14	434	434	NUM
ejpam-1373	50	15	426	426	NUM
ejpam-1373	50	16	in	in	ADP
ejpam-1373	50	17	fact	fact	NOUN
ejpam-1373	50	18	,	,	PUNCT
ejpam-1373	50	19	for	for	ADP
ejpam-1373	50	20	any	any	DET
ejpam-1373	50	21	mapping	mapping	NOUN
ejpam-1373	50	22	c	c	NOUN
ejpam-1373	50	23	:	:	PUNCT
ejpam-1373	50	24	z+	z+	NUM
ejpam-1373	50	25	−→p	−→p	NOUN
ejpam-1373	50	26	(	(	PUNCT
ejpam-1373	50	27	z+	z+	NOUN
ejpam-1373	50	28	)	)	PUNCT
ejpam-1373	50	29	such	such	ADJ
ejpam-1373	50	30	that	that	SCONJ
ejpam-1373	50	31	each	each	DET
ejpam-1373	50	32	member	member	NOUN
ejpam-1373	50	33	of	of	ADP
ejpam-1373	50	34	c	c	PROPN
ejpam-1373	50	35	(	(	PUNCT
ejpam-1373	50	36	n	n	CCONJ
ejpam-1373	50	37	)	)	PUNCT
ejpam-1373	50	38	is	be	AUX
ejpam-1373	50	39	a	a	DET
ejpam-1373	50	40	divisor	divisor	NOUN
ejpam-1373	50	41	of	of	ADP
ejpam-1373	50	42	n	n	CCONJ
ejpam-1373	50	43	,	,	PUNCT
ejpam-1373	50	44	≤c	≤c	PROPN
ejpam-1373	50	45	is	be	AUX
ejpam-1373	50	46	a	a	DET
ejpam-1373	50	47	partial	partial	ADJ
ejpam-1373	50	48	order	order	NOUN
ejpam-1373	50	49	on	on	ADP
ejpam-1373	50	50	z+	z+	NUM
ejpam-1373	50	51	if	if	SCONJ
ejpam-1373	50	52	and	and	CCONJ
ejpam-1373	50	53	only	only	ADV
ejpam-1373	50	54	if	if	SCONJ
ejpam-1373	50	55	c	c	PROPN
ejpam-1373	50	56	is	be	AUX
ejpam-1373	50	57	a	a	DET
ejpam-1373	50	58	convolution	convolution	NOUN
ejpam-1373	50	59	,	,	PUNCT
ejpam-1373	50	60	as	as	SCONJ
ejpam-1373	50	61	defined	define	VERB
ejpam-1373	50	62	above[4	above[4	NOUN
ejpam-1373	50	63	]	]	PUNCT
ejpam-1373	50	64	.	.	PUNCT
ejpam-1373	51	1	it	it	PRON
ejpam-1373	51	2	is	be	AUX
ejpam-1373	51	3	known	know	VERB
ejpam-1373	51	4	that	that	SCONJ
ejpam-1373	51	5	,	,	PUNCT
ejpam-1373	51	6	for	for	ADP
ejpam-1373	51	7	any	any	DET
ejpam-1373	51	8	convolution	convolution	NOUN
ejpam-1373	51	9	c	c	NOUN
ejpam-1373	51	10	,	,	PUNCT
ejpam-1373	51	11	the	the	DET
ejpam-1373	51	12	poset	poset	NOUN
ejpam-1373	51	13	(	(	PUNCT
ejpam-1373	51	14	z+,≤c	z+,≤c	NUM
ejpam-1373	51	15	)	)	PUNCT
ejpam-1373	51	16	satisfies	satisfy	VERB
ejpam-1373	51	17	the	the	DET
ejpam-1373	51	18	descending	descend	VERB
ejpam-1373	51	19	chain	chain	NOUN
ejpam-1373	51	20	condition(dcc	condition(dcc	NOUN
ejpam-1373	51	21	)	)	PUNCT
ejpam-1373	51	22	in	in	ADP
ejpam-1373	51	23	the	the	DET
ejpam-1373	51	24	sense	sense	NOUN
ejpam-1373	51	25	that	that	SCONJ
ejpam-1373	51	26	any	any	DET
ejpam-1373	51	27	non	non	ADJ
ejpam-1373	51	28	-	-	ADJ
ejpam-1373	51	29	empty	empty	ADJ
ejpam-1373	51	30	subset	subset	NOUN
ejpam-1373	51	31	of	of	ADP
ejpam-1373	51	32	z+	z+	NUM
ejpam-1373	51	33	has	have	AUX
ejpam-1373	51	34	minimal	minimal	ADJ
ejpam-1373	51	35	member	member	NOUN
ejpam-1373	51	36	.	.	PUNCT
ejpam-1373	52	1	3	3	X
ejpam-1373	52	2	.	.	X
ejpam-1373	52	3	semilattice	semilattice	NOUN
ejpam-1373	52	4	structures	structure	NOUN
ejpam-1373	52	5	on	on	ADP
ejpam-1373	52	6	z+	z+	NUM
ejpam-1373	52	7	in	in	ADP
ejpam-1373	52	8	this	this	DET
ejpam-1373	52	9	section	section	NOUN
ejpam-1373	52	10	,	,	PUNCT
ejpam-1373	52	11	we	we	PRON
ejpam-1373	52	12	discuss	discuss	VERB
ejpam-1373	52	13	possible	possible	ADJ
ejpam-1373	52	14	semi	semi	ADJ
ejpam-1373	52	15	lattice	lattice	NOUN
ejpam-1373	52	16	structures	structure	NOUN
ejpam-1373	52	17	on	on	ADP
ejpam-1373	52	18	z+	z+	NUM
ejpam-1373	52	19	induced	induce	VERB
ejpam-1373	52	20	by	by	ADP
ejpam-1373	52	21	convolutions	convolution	NOUN
ejpam-1373	52	22	.	.	PUNCT
ejpam-1373	53	1	recall	recall	VERB
ejpam-1373	53	2	that	that	SCONJ
ejpam-1373	53	3	the	the	DET
ejpam-1373	53	4	dirichlet	dirichlet	PROPN
ejpam-1373	53	5	’s	’s	PART
ejpam-1373	53	6	convolution	convolution	NOUN
ejpam-1373	53	7	induces	induce	VERB
ejpam-1373	53	8	a	a	DET
ejpam-1373	53	9	lattice	lattice	NOUN
ejpam-1373	53	10	structure	structure	NOUN
ejpam-1373	53	11	on	on	ADP
ejpam-1373	53	12	z+	z+	NUM
ejpam-1373	53	13	,	,	PUNCT
ejpam-1373	53	14	while	while	SCONJ
ejpam-1373	53	15	the	the	DET
ejpam-1373	53	16	unitary	unitary	ADJ
ejpam-1373	53	17	convolution	convolution	NOUN
ejpam-1373	53	18	induces	induce	VERB
ejpam-1373	53	19	only	only	ADV
ejpam-1373	53	20	a	a	DET
ejpam-1373	53	21	meet	meet	NOUN
ejpam-1373	53	22	semi	semi	ADJ
ejpam-1373	53	23	lattice	lattice	NOUN
ejpam-1373	53	24	structure	structure	NOUN
ejpam-1373	53	25	on	on	ADP
ejpam-1373	53	26	z+	z+	NUM
ejpam-1373	53	27	.	.	PUNCT
ejpam-1373	54	1	definition	definition	NOUN
ejpam-1373	54	2	3	3	X
ejpam-1373	54	3	.	.	PUNCT
ejpam-1373	55	1	let	let	VERB
ejpam-1373	55	2	c	c	PRON
ejpam-1373	55	3	be	be	AUX
ejpam-1373	55	4	a	a	DET
ejpam-1373	55	5	convolution	convolution	NOUN
ejpam-1373	55	6	.	.	PUNCT
ejpam-1373	56	1	(	(	PUNCT
ejpam-1373	56	2	1	1	NUM
ejpam-1373	56	3	)	)	PUNCT
ejpam-1373	56	4	.	.	PUNCT
ejpam-1373	57	1	c	c	PROPN
ejpam-1373	57	2	is	be	AUX
ejpam-1373	57	3	said	say	VERB
ejpam-1373	57	4	to	to	PART
ejpam-1373	57	5	satisfy	satisfy	VERB
ejpam-1373	57	6	the	the	DET
ejpam-1373	57	7	finite	finite	PROPN
ejpam-1373	57	8	intersection	intersection	NOUN
ejpam-1373	57	9	property(fip	property(fip	PROPN
ejpam-1373	57	10	)	)	PUNCT
ejpam-1373	57	11	if	if	SCONJ
ejpam-1373	57	12	c	c	PROPN
ejpam-1373	57	13	(	(	PUNCT
ejpam-1373	57	14	n1)∩c	n1)∩c	PROPN
ejpam-1373	57	15	(	(	PUNCT
ejpam-1373	57	16	n2)∩	n2)∩	X
ejpam-1373	57	17	·	·	PUNCT
ejpam-1373	57	18	·	·	PUNCT
ejpam-1373	57	19	·	·	PUNCT
ejpam-1373	58	1	∩c	∩c	NOUN
ejpam-1373	58	2	(	(	PUNCT
ejpam-1373	58	3	nr	nr	X
ejpam-1373	58	4	)	)	PUNCT
ejpam-1373	58	5	6=	6=	NUM
ejpam-1373	58	6	;	;	PUNCT
ejpam-1373	58	7	for	for	ADP
ejpam-1373	58	8	any	any	DET
ejpam-1373	58	9	n1	n1	NOUN
ejpam-1373	58	10	,	,	PUNCT
ejpam-1373	58	11	n2	n2	NOUN
ejpam-1373	58	12	,	,	PUNCT
ejpam-1373	58	13	·	·	PUNCT
ejpam-1373	58	14	·	·	PUNCT
ejpam-1373	58	15	·	·	PUNCT
ejpam-1373	58	16	,	,	PUNCT
ejpam-1373	58	17	nr	nr	PROPN
ejpam-1373	58	18	∈	∈	PROPN
ejpam-1373	58	19	z	z	PROPN
ejpam-1373	58	20	+	+	NOUN
ejpam-1373	58	21	.	.	PUNCT
ejpam-1373	58	22	(	(	PUNCT
ejpam-1373	58	23	2	2	NUM
ejpam-1373	58	24	)	)	PUNCT
ejpam-1373	58	25	.	.	PUNCT
ejpam-1373	59	1	c	c	PROPN
ejpam-1373	59	2	is	be	AUX
ejpam-1373	59	3	said	say	VERB
ejpam-1373	59	4	to	to	PART
ejpam-1373	59	5	be	be	AUX
ejpam-1373	59	6	closed	close	VERB
ejpam-1373	59	7	under	under	ADP
ejpam-1373	59	8	finite	finite	ADJ
ejpam-1373	59	9	intersections(unions	intersections(union	NOUN
ejpam-1373	59	10	)	)	PUNCT
ejpam-1373	59	11	if	if	SCONJ
ejpam-1373	59	12	,	,	PUNCT
ejpam-1373	59	13	for	for	ADP
ejpam-1373	59	14	any	any	DET
ejpam-1373	59	15	n1	n1	NOUN
ejpam-1373	59	16	,	,	PUNCT
ejpam-1373	59	17	n2	n2	NOUN
ejpam-1373	59	18	,	,	PUNCT
ejpam-1373	59	19	·	·	PUNCT
ejpam-1373	59	20	·	·	PUNCT
ejpam-1373	59	21	·	·	PUNCT
ejpam-1373	59	22	,	,	PUNCT
ejpam-1373	59	23	nr	nr	PROPN
ejpam-1373	59	24	∈	∈	PROPN
ejpam-1373	59	25	z	z	PROPN
ejpam-1373	60	1	+	+	ADV
ejpam-1373	60	2	,	,	PUNCT
ejpam-1373	60	3	there	there	PRON
ejpam-1373	60	4	exists	exist	VERB
ejpam-1373	60	5	n	n	PRON
ejpam-1373	60	6	∈	∈	PROPN
ejpam-1373	60	7	z+	z+	NUM
ejpam-1373	60	8	such	such	ADJ
ejpam-1373	60	9	that	that	SCONJ
ejpam-1373	60	10	c	c	NOUN
ejpam-1373	60	11	(	(	PUNCT
ejpam-1373	60	12	n1)∩c	n1)∩c	PROPN
ejpam-1373	60	13	(	(	PUNCT
ejpam-1373	60	14	n2)∩	n2)∩	X
ejpam-1373	60	15	·	·	PUNCT
ejpam-1373	60	16	·	·	PUNCT
ejpam-1373	60	17	·	·	PUNCT
ejpam-1373	61	1	∩c	∩c	NOUN
ejpam-1373	61	2	(	(	PUNCT
ejpam-1373	61	3	nr	nr	NOUN
ejpam-1373	61	4	)	)	PUNCT
ejpam-1373	61	5	=	=	SYM
ejpam-1373	61	6	c	c	X
ejpam-1373	61	7	(	(	PUNCT
ejpam-1373	61	8	n	n	CCONJ
ejpam-1373	61	9	)	)	PUNCT
ejpam-1373	61	10	.	.	PUNCT
ejpam-1373	62	1	(	(	PUNCT
ejpam-1373	62	2	respectively	respectively	ADV
ejpam-1373	62	3	c	c	X
ejpam-1373	62	4	(	(	PUNCT
ejpam-1373	62	5	n1)∪c	n1)∪c	ADV
ejpam-1373	62	6	(	(	PUNCT
ejpam-1373	62	7	n2)∪···∪c	n2)∪···∪c	X
ejpam-1373	62	8	(	(	PUNCT
ejpam-1373	62	9	nr	nr	NOUN
ejpam-1373	62	10	)	)	PUNCT
ejpam-1373	63	1	=	=	SYM
ejpam-1373	63	2	c	c	X
ejpam-1373	63	3	(	(	PUNCT
ejpam-1373	63	4	n	n	CCONJ
ejpam-1373	63	5	)	)	PUNCT
ejpam-1373	63	6	)	)	PUNCT
ejpam-1373	63	7	(	(	PUNCT
ejpam-1373	63	8	3	3	NUM
ejpam-1373	63	9	)	)	PUNCT
ejpam-1373	63	10	.	.	PUNCT
ejpam-1373	64	1	c	c	PROPN
ejpam-1373	64	2	is	be	AUX
ejpam-1373	64	3	said	say	VERB
ejpam-1373	64	4	to	to	PART
ejpam-1373	64	5	be	be	AUX
ejpam-1373	64	6	closed	close	VERB
ejpam-1373	64	7	under	under	ADP
ejpam-1373	64	8	non	non	ADJ
ejpam-1373	64	9	-	-	ADJ
ejpam-1373	64	10	empty	empty	ADJ
ejpam-1373	64	11	intersections	intersection	NOUN
ejpam-1373	64	12	if	if	SCONJ
ejpam-1373	64	13	,	,	PUNCT
ejpam-1373	64	14	for	for	ADP
ejpam-1373	64	15	any	any	DET
ejpam-1373	64	16	non	non	ADJ
ejpam-1373	64	17	-	-	ADJ
ejpam-1373	64	18	empty	empty	ADJ
ejpam-1373	64	19	subset	subset	NOUN
ejpam-1373	64	20	a	a	PRON
ejpam-1373	64	21	of	of	ADP
ejpam-1373	64	22	z+	z+	NOUN
ejpam-1373	64	23	,	,	PUNCT
ejpam-1373	64	24	there	there	PRON
ejpam-1373	64	25	exists	exist	VERB
ejpam-1373	64	26	n	n	PRON
ejpam-1373	64	27	∈	∈	PROPN
ejpam-1373	64	28	z+	z+	NUM
ejpam-1373	64	29	such	such	ADJ
ejpam-1373	64	30	that	that	SCONJ
ejpam-1373	64	31	⋂	⋂	PROPN
ejpam-1373	64	32	a∈a	a∈a	ADJ
ejpam-1373	64	33	c	c	NOUN
ejpam-1373	64	34	(	(	PUNCT
ejpam-1373	64	35	a	a	X
ejpam-1373	64	36	)	)	PUNCT
ejpam-1373	64	37	=	=	SYM
ejpam-1373	64	38	c	c	X
ejpam-1373	64	39	(	(	PUNCT
ejpam-1373	64	40	n	n	CCONJ
ejpam-1373	64	41	)	)	PUNCT
ejpam-1373	64	42	.	.	PUNCT
ejpam-1373	65	1	theorem	theorem	NOUN
ejpam-1373	65	2	1	1	NUM
ejpam-1373	65	3	.	.	PUNCT
ejpam-1373	66	1	let	let	VERB
ejpam-1373	66	2	c	c	PRON
ejpam-1373	66	3	be	be	AUX
ejpam-1373	66	4	a	a	DET
ejpam-1373	66	5	convolution	convolution	NOUN
ejpam-1373	66	6	and	and	CCONJ
ejpam-1373	66	7	≤c	≤c	NOUN
ejpam-1373	66	8	the	the	DET
ejpam-1373	66	9	partial	partial	ADJ
ejpam-1373	66	10	order	order	NOUN
ejpam-1373	66	11	induced	induce	VERB
ejpam-1373	66	12	by	by	ADP
ejpam-1373	66	13	c	c	PROPN
ejpam-1373	66	14	on	on	ADP
ejpam-1373	66	15	z+	z+	NUM
ejpam-1373	66	16	.	.	PUNCT
ejpam-1373	67	1	then	then	ADV
ejpam-1373	67	2	the	the	DET
ejpam-1373	67	3	following	follow	VERB
ejpam-1373	67	4	are	be	AUX
ejpam-1373	67	5	equivalent	equivalent	ADJ
ejpam-1373	67	6	to	to	ADP
ejpam-1373	67	7	each	each	DET
ejpam-1373	67	8	other	other	ADJ
ejpam-1373	67	9	.	.	PUNCT
ejpam-1373	68	1	(	(	PUNCT
ejpam-1373	68	2	1	1	NUM
ejpam-1373	68	3	)	)	PUNCT
ejpam-1373	68	4	.	.	PUNCT
ejpam-1373	69	1	(	(	PUNCT
ejpam-1373	69	2	z+,≤c	z+,≤c	PUNCT
ejpam-1373	69	3	)	)	PUNCT
ejpam-1373	69	4	is	be	AUX
ejpam-1373	69	5	a	a	DET
ejpam-1373	69	6	semilattice	semilattice	NOUN
ejpam-1373	69	7	(	(	PUNCT
ejpam-1373	69	8	2	2	NUM
ejpam-1373	69	9	)	)	PUNCT
ejpam-1373	69	10	.	.	PUNCT
ejpam-1373	70	1	c	c	PROPN
ejpam-1373	70	2	is	be	AUX
ejpam-1373	70	3	closed	close	VERB
ejpam-1373	70	4	under	under	ADP
ejpam-1373	70	5	finite	finite	ADJ
ejpam-1373	70	6	intersections	intersection	NOUN
ejpam-1373	70	7	(	(	PUNCT
ejpam-1373	70	8	3	3	NUM
ejpam-1373	70	9	)	)	PUNCT
ejpam-1373	70	10	.	.	PUNCT
ejpam-1373	71	1	c	c	PROPN
ejpam-1373	71	2	is	be	AUX
ejpam-1373	71	3	closed	close	VERB
ejpam-1373	71	4	under	under	ADP
ejpam-1373	71	5	non	non	ADJ
ejpam-1373	71	6	-	-	ADJ
ejpam-1373	71	7	empty	empty	ADJ
ejpam-1373	71	8	intersections	intersection	NOUN
ejpam-1373	71	9	(	(	PUNCT
ejpam-1373	71	10	4	4	NUM
ejpam-1373	71	11	)	)	PUNCT
ejpam-1373	71	12	.	.	PUNCT
ejpam-1373	72	1	every	every	DET
ejpam-1373	72	2	non	non	ADJ
ejpam-1373	72	3	-	-	ADJ
ejpam-1373	72	4	empty	empty	ADJ
ejpam-1373	72	5	subset	subset	NOUN
ejpam-1373	72	6	of	of	ADP
ejpam-1373	72	7	z+	z+	NUM
ejpam-1373	72	8	has	have	VERB
ejpam-1373	72	9	glb	glb	NOUN
ejpam-1373	72	10	in	in	ADP
ejpam-1373	72	11	(	(	PUNCT
ejpam-1373	72	12	z+,≤c	z+,≤c	NUM
ejpam-1373	72	13	)	)	PUNCT
ejpam-1373	72	14	.	.	PUNCT
ejpam-1373	73	1	proof	proof	NOUN
ejpam-1373	73	2	.	.	PUNCT
ejpam-1373	74	1	(	(	PUNCT
ejpam-1373	74	2	1	1	X
ejpam-1373	74	3	)	)	PUNCT
ejpam-1373	74	4	=	=	NOUN
ejpam-1373	74	5	⇒	⇒	NOUN
ejpam-1373	74	6	(	(	PUNCT
ejpam-1373	74	7	2	2	NUM
ejpam-1373	74	8	)	)	PUNCT
ejpam-1373	74	9	:	:	PUNCT
ejpam-1373	74	10	suppose	suppose	VERB
ejpam-1373	74	11	that	that	SCONJ
ejpam-1373	74	12	(	(	PUNCT
ejpam-1373	74	13	z+,≤c	z+,≤c	NUM
ejpam-1373	74	14	)	)	PUNCT
ejpam-1373	74	15	is	be	AUX
ejpam-1373	74	16	a	a	DET
ejpam-1373	74	17	meet	meet	ADJ
ejpam-1373	74	18	semilattice	semilattice	NOUN
ejpam-1373	74	19	.	.	PUNCT
ejpam-1373	75	1	then	then	ADV
ejpam-1373	75	2	every	every	DET
ejpam-1373	75	3	non	non	ADJ
ejpam-1373	75	4	-	-	ADJ
ejpam-1373	75	5	empty	empty	ADJ
ejpam-1373	75	6	finite	finite	NOUN
ejpam-1373	75	7	subset	subset	NOUN
ejpam-1373	75	8	of	of	ADP
ejpam-1373	75	9	(	(	PUNCT
ejpam-1373	75	10	z+,≤c	z+,≤c	PROPN
ejpam-1373	75	11	)	)	PUNCT
ejpam-1373	75	12	is	be	AUX
ejpam-1373	75	13	a	a	DET
ejpam-1373	75	14	semilattice	semilattice	NOUN
ejpam-1373	75	15	.	.	PUNCT
ejpam-1373	76	1	let	let	VERB
ejpam-1373	76	2	n1	n1	NOUN
ejpam-1373	76	3	,	,	PUNCT
ejpam-1373	76	4	n2	n2	ADJ
ejpam-1373	76	5	,	,	PUNCT
ejpam-1373	76	6	·	·	PUNCT
ejpam-1373	76	7	·	·	PUNCT
ejpam-1373	76	8	·	·	PUNCT
ejpam-1373	76	9	,	,	PUNCT
ejpam-1373	76	10	nr	nr	PROPN
ejpam-1373	76	11	∈	∈	PROPN
ejpam-1373	76	12	z	z	NOUN
ejpam-1373	76	13	+	+	CCONJ
ejpam-1373	76	14	and	and	CCONJ
ejpam-1373	76	15	�	�	PROPN
ejpam-1373	76	16	n=	n=	PROPN
ejpam-1373	76	17	gl	gl	PROPN
ejpam-1373	76	18	b	b	PROPN
ejpam-1373	76	19	{	{	PUNCT
ejpam-1373	76	20	n1	n1	NOUN
ejpam-1373	76	21	,	,	PUNCT
ejpam-1373	76	22	n2	n2	NOUN
ejpam-1373	76	23	,	,	PUNCT
ejpam-1373	76	24	·	·	PUNCT
ejpam-1373	76	25	·	·	PUNCT
ejpam-1373	76	26	·	·	PUNCT
ejpam-1373	76	27	,	,	PUNCT
ejpam-1373	76	28	nr	nr	CCONJ
ejpam-1373	76	29	}	}	PUNCT
ejpam-1373	76	30	�	�	PROPN
ejpam-1373	76	31	.	.	PUNCT
ejpam-1373	77	1	then	then	ADV
ejpam-1373	77	2	,	,	PUNCT
ejpam-1373	77	3	for	for	ADP
ejpam-1373	77	4	any	any	DET
ejpam-1373	77	5	a	a	DET
ejpam-1373	77	6	∈	∈	PROPN
ejpam-1373	77	7	z+	z+	NUM
ejpam-1373	77	8	,	,	PUNCT
ejpam-1373	77	9	a	a	DET
ejpam-1373	77	10	∈	∈	NOUN
ejpam-1373	77	11	r	r	NOUN
ejpam-1373	77	12	⋂	⋂	PROPN
ejpam-1373	77	13	i=1	i=1	PROPN
ejpam-1373	77	14	c	c	PROPN
ejpam-1373	77	15	(	(	PUNCT
ejpam-1373	77	16	ni	ni	PROPN
ejpam-1373	77	17	)	)	PUNCT
ejpam-1373	77	18	⇐	⇐	ADJ
ejpam-1373	77	19	⇒	⇒	PROPN
ejpam-1373	77	20	a	a	DET
ejpam-1373	77	21	≤c	≤c	PROPN
ejpam-1373	77	22	ni	ni	PROPN
ejpam-1373	77	23	for	for	ADP
ejpam-1373	77	24	all	all	DET
ejpam-1373	77	25	1≤	1≤	NUM
ejpam-1373	78	1	i	i	PRON
ejpam-1373	78	2	≤	≤	ADJ
ejpam-1373	78	3	r	r	VERB
ejpam-1373	78	4	⇐	⇐	ADJ
ejpam-1373	78	5	⇒	⇒	PROPN
ejpam-1373	78	6	a	a	DET
ejpam-1373	78	7	≤c	≤c	PROPN
ejpam-1373	78	8	n	n	CCONJ
ejpam-1373	78	9	,	,	PUNCT
ejpam-1373	78	10	since	since	SCONJ
ejpam-1373	78	11	n=	n=	ADJ
ejpam-1373	78	12	gl	gl	PROPN
ejpam-1373	78	13	b{n1	b{n1	PROPN
ejpam-1373	78	14	,	,	PUNCT
ejpam-1373	78	15	n2	n2	NOUN
ejpam-1373	78	16	,	,	PUNCT
ejpam-1373	78	17	·	·	PUNCT
ejpam-1373	78	18	·	·	PUNCT
ejpam-1373	78	19	·	·	PUNCT
ejpam-1373	78	20	,	,	PUNCT
ejpam-1373	78	21	nr	nr	CCONJ
ejpam-1373	78	22	}	}	PUNCT
ejpam-1373	78	23	⇐	⇐	ADJ
ejpam-1373	78	24	⇒	⇒	NOUN
ejpam-1373	78	25	a	a	DET
ejpam-1373	78	26	∈	∈	PROPN
ejpam-1373	78	27	c	c	X
ejpam-1373	78	28	(	(	PUNCT
ejpam-1373	78	29	n	n	CCONJ
ejpam-1373	78	30	)	)	PUNCT
ejpam-1373	78	31	u.	u.	NOUN
ejpam-1373	78	32	swamy	swamy	PROPN
ejpam-1373	78	33	and	and	CCONJ
ejpam-1373	78	34	s.	s.	PROPN
ejpam-1373	78	35	sankar	sankar	PROPN
ejpam-1373	78	36	/	/	SYM
ejpam-1373	78	37	eur	eur	PROPN
ejpam-1373	78	38	.	.	PUNCT
ejpam-1373	79	1	j.	j.	PROPN
ejpam-1373	79	2	pure	pure	PROPN
ejpam-1373	79	3	appl	appl	PROPN
ejpam-1373	79	4	.	.	PROPN
ejpam-1373	79	5	math	math	PROPN
ejpam-1373	79	6	,	,	PUNCT
ejpam-1373	79	7	4	4	NUM
ejpam-1373	79	8	(	(	PUNCT
ejpam-1373	79	9	2011	2011	NUM
ejpam-1373	79	10	)	)	PUNCT
ejpam-1373	79	11	,	,	PUNCT
ejpam-1373	79	12	424	424	NUM
ejpam-1373	79	13	-	-	SYM
ejpam-1373	79	14	434	434	NUM
ejpam-1373	79	15	427	427	NUM
ejpam-1373	79	16	and	and	CCONJ
ejpam-1373	79	17	hence	hence	ADV
ejpam-1373	79	18	�	�	PROPN
ejpam-1373	80	1	r	r	NOUN
ejpam-1373	80	2	⋂	⋂	PROPN
ejpam-1373	80	3	i=1	i=1	PROPN
ejpam-1373	80	4	c	c	PROPN
ejpam-1373	80	5	(	(	PUNCT
ejpam-1373	80	6	ni	ni	PROPN
ejpam-1373	80	7	)	)	PUNCT
ejpam-1373	80	8	=	=	SYM
ejpam-1373	80	9	c	c	X
ejpam-1373	80	10	(	(	PUNCT
ejpam-1373	80	11	n	n	CCONJ
ejpam-1373	80	12	)	)	PUNCT
ejpam-1373	80	13	�	�	PROPN
ejpam-1373	80	14	.	.	PUNCT
ejpam-1373	81	1	thus	thus	ADV
ejpam-1373	81	2	c	c	PROPN
ejpam-1373	81	3	is	be	AUX
ejpam-1373	81	4	closed	close	VERB
ejpam-1373	81	5	under	under	ADP
ejpam-1373	81	6	finite	finite	ADJ
ejpam-1373	81	7	intersections	intersection	NOUN
ejpam-1373	81	8	.	.	PUNCT
ejpam-1373	82	1	(	(	PUNCT
ejpam-1373	82	2	2	2	X
ejpam-1373	82	3	)	)	PUNCT
ejpam-1373	82	4	=	=	NOUN
ejpam-1373	82	5	⇒	⇒	NOUN
ejpam-1373	82	6	(	(	PUNCT
ejpam-1373	82	7	3	3	NUM
ejpam-1373	82	8	)	)	PUNCT
ejpam-1373	82	9	:	:	PUNCT
ejpam-1373	82	10	suppose	suppose	VERB
ejpam-1373	82	11	that	that	SCONJ
ejpam-1373	82	12	c	c	PROPN
ejpam-1373	82	13	is	be	AUX
ejpam-1373	82	14	closed	close	VERB
ejpam-1373	82	15	under	under	ADP
ejpam-1373	82	16	finite	finite	ADJ
ejpam-1373	82	17	intersections	intersection	NOUN
ejpam-1373	82	18	.	.	PUNCT
ejpam-1373	83	1	let	let	VERB
ejpam-1373	83	2	a	a	PRON
ejpam-1373	83	3	be	be	AUX
ejpam-1373	83	4	a	a	DET
ejpam-1373	83	5	non	non	ADJ
ejpam-1373	83	6	-	-	ADJ
ejpam-1373	83	7	empty	empty	ADJ
ejpam-1373	83	8	subset	subset	NOUN
ejpam-1373	83	9	of	of	ADP
ejpam-1373	83	10	z+	z+	PROPN
ejpam-1373	83	11	.	.	PUNCT
ejpam-1373	84	1	we	we	PRON
ejpam-1373	84	2	have	have	VERB
ejpam-1373	84	3	to	to	PART
ejpam-1373	84	4	prove	prove	VERB
ejpam-1373	84	5	that	that	SCONJ
ejpam-1373	84	6	⋂	⋂	PROPN
ejpam-1373	84	7	a∈a	a∈a	VERB
ejpam-1373	84	8	c	c	NOUN
ejpam-1373	84	9	(	(	PUNCT
ejpam-1373	84	10	a	a	X
ejpam-1373	84	11	)	)	PUNCT
ejpam-1373	84	12	=	=	SYM
ejpam-1373	84	13	c	c	X
ejpam-1373	84	14	(	(	PUNCT
ejpam-1373	84	15	n	n	CCONJ
ejpam-1373	84	16	)	)	PUNCT
ejpam-1373	84	17	for	for	ADP
ejpam-1373	84	18	some	some	DET
ejpam-1373	84	19	n	n	PRON
ejpam-1373	84	20	∈	∈	NOUN
ejpam-1373	84	21	z+	z+	PUNCT
ejpam-1373	84	22	.	.	PUNCT
ejpam-1373	85	1	if	if	SCONJ
ejpam-1373	85	2	a	a	PRON
ejpam-1373	85	3	is	be	AUX
ejpam-1373	85	4	finite	finite	ADJ
ejpam-1373	85	5	,	,	PUNCT
ejpam-1373	85	6	we	we	PRON
ejpam-1373	85	7	are	be	AUX
ejpam-1373	85	8	through	through	ADP
ejpam-1373	85	9	.	.	PUNCT
ejpam-1373	86	1	suppose	suppose	VERB
ejpam-1373	86	2	that	that	SCONJ
ejpam-1373	86	3	a	a	PRON
ejpam-1373	86	4	is	be	AUX
ejpam-1373	86	5	an	an	DET
ejpam-1373	86	6	infinite	infinite	ADJ
ejpam-1373	86	7	subset	subset	NOUN
ejpam-1373	86	8	of	of	ADP
ejpam-1373	86	9	z+	z+	PROPN
ejpam-1373	86	10	.	.	PUNCT
ejpam-1373	87	1	then	then	ADV
ejpam-1373	87	2	a	a	PRON
ejpam-1373	87	3	is	be	AUX
ejpam-1373	87	4	countably	countably	ADV
ejpam-1373	87	5	infinite	infinite	ADJ
ejpam-1373	87	6	and	and	CCONJ
ejpam-1373	87	7	hence	hence	ADV
ejpam-1373	87	8	we	we	PRON
ejpam-1373	87	9	can	can	AUX
ejpam-1373	87	10	write	write	VERB
ejpam-1373	87	11	a=	a=	ADJ
ejpam-1373	87	12	{	{	PUNCT
ejpam-1373	87	13	a1	a1	NOUN
ejpam-1373	87	14	,	,	PUNCT
ejpam-1373	87	15	a2	a2	PROPN
ejpam-1373	87	16	,	,	PUNCT
ejpam-1373	87	17	a3	a3	NOUN
ejpam-1373	87	18	,	,	PUNCT
ejpam-1373	87	19	·	·	PUNCT
ejpam-1373	87	20	·	·	PUNCT
ejpam-1373	87	21	·	·	PUNCT
ejpam-1373	87	22	}	}	PUNCT
ejpam-1373	87	23	for	for	ADP
ejpam-1373	87	24	each	each	DET
ejpam-1373	87	25	r	r	NOUN
ejpam-1373	87	26	∈	∈	NOUN
ejpam-1373	87	27	z+	z+	NOUN
ejpam-1373	87	28	,	,	PUNCT
ejpam-1373	87	29	there	there	PRON
ejpam-1373	87	30	exists	exist	VERB
ejpam-1373	87	31	br	br	PROPN
ejpam-1373	87	32	∈	∈	PROPN
ejpam-1373	87	33	z	z	PROPN
ejpam-1373	88	1	+	+	CCONJ
ejpam-1373	89	1	such	such	ADJ
ejpam-1373	89	2	that	that	SCONJ
ejpam-1373	89	3	r	r	NOUN
ejpam-1373	89	4	⋂	⋂	PROPN
ejpam-1373	89	5	i=1	i=1	PROPN
ejpam-1373	89	6	c	c	PROPN
ejpam-1373	89	7	(	(	PUNCT
ejpam-1373	89	8	ai	ai	NOUN
ejpam-1373	89	9	)	)	PUNCT
ejpam-1373	89	10	=	=	SYM
ejpam-1373	89	11	c	c	X
ejpam-1373	89	12	(	(	PUNCT
ejpam-1373	89	13	br	br	NOUN
ejpam-1373	89	14	)	)	PUNCT
ejpam-1373	89	15	.	.	PUNCT
ejpam-1373	90	1	then	then	ADV
ejpam-1373	90	2	,	,	PUNCT
ejpam-1373	90	3	we	we	PRON
ejpam-1373	90	4	have	have	VERB
ejpam-1373	90	5	,	,	PUNCT
ejpam-1373	90	6	for	for	ADP
ejpam-1373	90	7	any	any	DET
ejpam-1373	90	8	r	r	NOUN
ejpam-1373	90	9	∈	∈	NOUN
ejpam-1373	90	10	z+	z+	NUM
ejpam-1373	90	11	,	,	PUNCT
ejpam-1373	90	12	br+1	br+1	X
ejpam-1373	90	13	∈	∈	PROPN
ejpam-1373	90	14	c	c	X
ejpam-1373	90	15	(	(	PUNCT
ejpam-1373	90	16	br+1	br+1	X
ejpam-1373	90	17	)	)	PUNCT
ejpam-1373	90	18	=	=	SYM
ejpam-1373	91	1	r+1	r+1	PROPN
ejpam-1373	91	2	⋂	⋂	PROPN
ejpam-1373	91	3	i=1	i=1	PROPN
ejpam-1373	91	4	c	c	PROPN
ejpam-1373	91	5	(	(	PUNCT
ejpam-1373	91	6	ai)⊆	ai)⊆	PROPN
ejpam-1373	91	7	r	r	NOUN
ejpam-1373	91	8	⋂	⋂	PROPN
ejpam-1373	91	9	i=1	i=1	PROPN
ejpam-1373	91	10	c	c	PROPN
ejpam-1373	91	11	(	(	PUNCT
ejpam-1373	91	12	ai	ai	NOUN
ejpam-1373	91	13	)	)	PUNCT
ejpam-1373	91	14	=	=	SYM
ejpam-1373	91	15	c	c	X
ejpam-1373	91	16	(	(	PUNCT
ejpam-1373	91	17	br	br	NOUN
ejpam-1373	91	18	)	)	PUNCT
ejpam-1373	91	19	so	so	SCONJ
ejpam-1373	91	20	that	that	SCONJ
ejpam-1373	91	21	br+1	br+1	NUM
ejpam-1373	91	22	≤c	≤c	PROPN
ejpam-1373	91	23	br	br	NOUN
ejpam-1373	91	24	for	for	ADP
ejpam-1373	91	25	all	all	DET
ejpam-1373	91	26	r	r	NOUN
ejpam-1373	91	27	∈	∈	NOUN
ejpam-1373	91	28	z+	z+	PUNCT
ejpam-1373	91	29	.	.	PUNCT
ejpam-1373	92	1	let	let	VERB
ejpam-1373	92	2	b	b	NOUN
ejpam-1373	92	3	=	=	PRON
ejpam-1373	92	4	{	{	PUNCT
ejpam-1373	92	5	b1	b1	PROPN
ejpam-1373	92	6	,	,	PUNCT
ejpam-1373	92	7	b2	b2	NOUN
ejpam-1373	92	8	,	,	PUNCT
ejpam-1373	92	9	b3	b3	PROPN
ejpam-1373	92	10	,	,	PUNCT
ejpam-1373	92	11	·	·	PUNCT
ejpam-1373	92	12	·	·	PUNCT
ejpam-1373	92	13	·	·	PUNCT
ejpam-1373	92	14	}	}	PUNCT
ejpam-1373	92	15	.	.	PUNCT
ejpam-1373	93	1	since	since	SCONJ
ejpam-1373	93	2	(	(	PUNCT
ejpam-1373	93	3	z+,≤c	z+,≤c	PRON
ejpam-1373	93	4	)	)	PUNCT
ejpam-1373	93	5	satisfies	satisfy	VERB
ejpam-1373	93	6	the	the	DET
ejpam-1373	93	7	descending	descend	VERB
ejpam-1373	93	8	chain	chain	NOUN
ejpam-1373	93	9	condition	condition	NOUN
ejpam-1373	93	10	,	,	PUNCT
ejpam-1373	93	11	b	b	PROPN
ejpam-1373	93	12	has	have	VERB
ejpam-1373	93	13	a	a	DET
ejpam-1373	93	14	minimal	minimal	ADJ
ejpam-1373	93	15	element	element	NOUN
ejpam-1373	93	16	,	,	PUNCT
ejpam-1373	93	17	say	say	VERB
ejpam-1373	93	18	n.	n.	NOUN
ejpam-1373	93	19	then	then	ADV
ejpam-1373	93	20	n	n	NOUN
ejpam-1373	93	21	=	=	SYM
ejpam-1373	93	22	br	br	NOUN
ejpam-1373	93	23	for	for	ADP
ejpam-1373	93	24	some	some	DET
ejpam-1373	93	25	r	r	NOUN
ejpam-1373	93	26	and	and	CCONJ
ejpam-1373	93	27	,	,	PUNCT
ejpam-1373	93	28	since	since	SCONJ
ejpam-1373	93	29	br+k	br+k	PROPN
ejpam-1373	93	30	≤	≤	NUM
ejpam-1373	93	31	br	br	NOUN
ejpam-1373	93	32	=	=	SYM
ejpam-1373	93	33	n	n	NOUN
ejpam-1373	93	34	and	and	CCONJ
ejpam-1373	93	35	since	since	SCONJ
ejpam-1373	93	36	n	n	PRON
ejpam-1373	93	37	is	be	AUX
ejpam-1373	93	38	minimal	minimal	ADJ
ejpam-1373	93	39	in	in	ADP
ejpam-1373	93	40	b	b	NUM
ejpam-1373	93	41	,	,	PUNCT
ejpam-1373	93	42	it	it	PRON
ejpam-1373	93	43	follows	follow	VERB
ejpam-1373	93	44	that	that	SCONJ
ejpam-1373	93	45	br+k	br+k	PROPN
ejpam-1373	93	46	=	=	SYM
ejpam-1373	93	47	n=	n=	ADJ
ejpam-1373	93	48	br	br	NOUN
ejpam-1373	93	49	for	for	ADP
ejpam-1373	93	50	all	all	DET
ejpam-1373	93	51	k	k	PROPN
ejpam-1373	93	52	∈	∈	PROPN
ejpam-1373	93	53	z+	z+	PUNCT
ejpam-1373	93	54	.	.	PUNCT
ejpam-1373	94	1	now	now	ADV
ejpam-1373	94	2	,	,	PUNCT
ejpam-1373	94	3	∞	∞	PROPN
ejpam-1373	94	4	⋂	⋂	PROPN
ejpam-1373	94	5	i=1	i=1	PROPN
ejpam-1373	94	6	c	c	PROPN
ejpam-1373	94	7	(	(	PUNCT
ejpam-1373	94	8	ai	ai	NOUN
ejpam-1373	94	9	)	)	PUNCT
ejpam-1373	94	10	=	=	SYM
ejpam-1373	95	1	∞	∞	NUM
ejpam-1373	95	2	⋂	⋂	PROPN
ejpam-1373	95	3	i=1	i=1	PROPN
ejpam-1373	95	4	c	c	PROPN
ejpam-1373	95	5	(	(	PUNCT
ejpam-1373	95	6	bi	bi	NOUN
ejpam-1373	95	7	)	)	PUNCT
ejpam-1373	95	8	=	=	SYM
ejpam-1373	95	9	c	c	X
ejpam-1373	95	10	(	(	PUNCT
ejpam-1373	95	11	b1)∩	b1)∩	X
ejpam-1373	95	12	·	·	PUNCT
ejpam-1373	95	13	·	·	PUNCT
ejpam-1373	95	14	·	·	PUNCT
ejpam-1373	96	1	∩	∩	NOUN
ejpam-1373	96	2	c	c	X
ejpam-1373	96	3	(	(	PUNCT
ejpam-1373	96	4	br	br	NOUN
ejpam-1373	96	5	)	)	PUNCT
ejpam-1373	96	6	=	=	SYM
ejpam-1373	97	1	c	c	X
ejpam-1373	97	2	(	(	PUNCT
ejpam-1373	97	3	br	br	NOUN
ejpam-1373	97	4	)	)	PUNCT
ejpam-1373	97	5	=	=	SYM
ejpam-1373	97	6	c	c	X
ejpam-1373	97	7	(	(	PUNCT
ejpam-1373	97	8	n	n	CCONJ
ejpam-1373	97	9	)	)	PUNCT
ejpam-1373	97	10	.	.	PUNCT
ejpam-1373	98	1	thus	thus	ADV
ejpam-1373	98	2	c	c	PROPN
ejpam-1373	98	3	is	be	AUX
ejpam-1373	98	4	closed	close	VERB
ejpam-1373	98	5	under	under	ADP
ejpam-1373	98	6	non	non	ADJ
ejpam-1373	98	7	-	-	ADJ
ejpam-1373	98	8	empty	empty	ADJ
ejpam-1373	98	9	intersections	intersection	NOUN
ejpam-1373	98	10	.	.	PUNCT
ejpam-1373	99	1	(	(	PUNCT
ejpam-1373	99	2	3	3	X
ejpam-1373	99	3	)	)	PUNCT
ejpam-1373	99	4	=	=	NOUN
ejpam-1373	99	5	⇒	⇒	NOUN
ejpam-1373	99	6	(	(	PUNCT
ejpam-1373	99	7	4	4	X
ejpam-1373	99	8	)	)	PUNCT
ejpam-1373	99	9	is	be	AUX
ejpam-1373	99	10	similar	similar	ADJ
ejpam-1373	99	11	to	to	ADP
ejpam-1373	99	12	that	that	PRON
ejpam-1373	99	13	of	of	ADP
ejpam-1373	99	14	(	(	PUNCT
ejpam-1373	99	15	1	1	X
ejpam-1373	99	16	)	)	PUNCT
ejpam-1373	99	17	=	=	NOUN
ejpam-1373	99	18	⇒	⇒	NOUN
ejpam-1373	99	19	(	(	PUNCT
ejpam-1373	99	20	2	2	NUM
ejpam-1373	99	21	)	)	PUNCT
ejpam-1373	99	22	.	.	PUNCT
ejpam-1373	100	1	(	(	PUNCT
ejpam-1373	100	2	4	4	X
ejpam-1373	100	3	)	)	PUNCT
ejpam-1373	100	4	=	=	NOUN
ejpam-1373	100	5	⇒	⇒	NOUN
ejpam-1373	100	6	(	(	PUNCT
ejpam-1373	100	7	1	1	X
ejpam-1373	100	8	)	)	PUNCT
ejpam-1373	100	9	is	be	AUX
ejpam-1373	100	10	trivial	trivial	ADJ
ejpam-1373	100	11	.	.	PUNCT
ejpam-1373	101	1	the	the	DET
ejpam-1373	101	2	dual	dual	ADJ
ejpam-1373	101	3	of	of	ADP
ejpam-1373	101	4	the	the	DET
ejpam-1373	101	5	above	above	ADJ
ejpam-1373	101	6	theorem	theorem	NOUN
ejpam-1373	101	7	is	be	AUX
ejpam-1373	101	8	not	not	PART
ejpam-1373	101	9	true	true	ADJ
ejpam-1373	101	10	;	;	PUNCT
ejpam-1373	101	11	that	that	PRON
ejpam-1373	101	12	is	is	ADV
ejpam-1373	101	13	,	,	PUNCT
ejpam-1373	101	14	even	even	ADV
ejpam-1373	101	15	if	if	SCONJ
ejpam-1373	101	16	(	(	PUNCT
ejpam-1373	101	17	z+,≤c	z+,≤c	NUM
ejpam-1373	101	18	)	)	PUNCT
ejpam-1373	101	19	is	be	AUX
ejpam-1373	101	20	a	a	DET
ejpam-1373	101	21	join	join	NOUN
ejpam-1373	101	22	semilattice	semilattice	NOUN
ejpam-1373	101	23	,	,	PUNCT
ejpam-1373	101	24	c	c	PROPN
ejpam-1373	101	25	may	may	AUX
ejpam-1373	101	26	not	not	PART
ejpam-1373	101	27	be	be	AUX
ejpam-1373	101	28	closed	close	VERB
ejpam-1373	101	29	under	under	ADP
ejpam-1373	101	30	finite	finite	ADJ
ejpam-1373	101	31	unions	union	NOUN
ejpam-1373	101	32	.	.	PUNCT
ejpam-1373	102	1	however	however	ADV
ejpam-1373	102	2	,	,	PUNCT
ejpam-1373	102	3	we	we	PRON
ejpam-1373	102	4	have	have	VERB
ejpam-1373	102	5	the	the	DET
ejpam-1373	102	6	following	follow	VERB
ejpam-1373	102	7	other	other	ADJ
ejpam-1373	102	8	extreme	extreme	NOUN
ejpam-1373	102	9	.	.	PUNCT
ejpam-1373	103	1	theorem	theorem	NOUN
ejpam-1373	103	2	2	2	NUM
ejpam-1373	103	3	.	.	PUNCT
ejpam-1373	104	1	let	let	VERB
ejpam-1373	104	2	c	c	PRON
ejpam-1373	104	3	be	be	AUX
ejpam-1373	104	4	a	a	DET
ejpam-1373	104	5	convolution	convolution	NOUN
ejpam-1373	104	6	and	and	CCONJ
ejpam-1373	104	7	m	m	PROPN
ejpam-1373	104	8	and	and	CCONJ
ejpam-1373	104	9	n	n	PRON
ejpam-1373	104	10	∈	∈	PROPN
ejpam-1373	104	11	z+	z+	NUM
ejpam-1373	104	12	.	.	PUNCT
ejpam-1373	105	1	then	then	ADV
ejpam-1373	105	2	c	c	X
ejpam-1373	105	3	(	(	PUNCT
ejpam-1373	105	4	m)∪c	m)∪c	X
ejpam-1373	105	5	(	(	PUNCT
ejpam-1373	105	6	n	n	CCONJ
ejpam-1373	105	7	)	)	PUNCT
ejpam-1373	105	8	=	=	SYM
ejpam-1373	105	9	c	c	X
ejpam-1373	105	10	(	(	PUNCT
ejpam-1373	105	11	a	a	NOUN
ejpam-1373	105	12	)	)	PUNCT
ejpam-1373	105	13	for	for	ADP
ejpam-1373	105	14	some	some	DET
ejpam-1373	105	15	a	a	DET
ejpam-1373	105	16	∈	∈	NOUN
ejpam-1373	105	17	z+	z+	NUM
ejpam-1373	105	18	if	if	SCONJ
ejpam-1373	105	19	and	and	CCONJ
ejpam-1373	105	20	only	only	ADV
ejpam-1373	105	21	if	if	SCONJ
ejpam-1373	105	22	c	c	PROPN
ejpam-1373	105	23	(	(	PUNCT
ejpam-1373	105	24	m)⊆	m)⊆	PROPN
ejpam-1373	105	25	c	c	PROPN
ejpam-1373	105	26	(	(	PUNCT
ejpam-1373	105	27	n	n	CCONJ
ejpam-1373	105	28	)	)	PUNCT
ejpam-1373	105	29	or	or	CCONJ
ejpam-1373	105	30	c	c	X
ejpam-1373	105	31	(	(	PUNCT
ejpam-1373	105	32	n)⊆	n)⊆	PROPN
ejpam-1373	105	33	c	c	PROPN
ejpam-1373	105	34	(	(	PUNCT
ejpam-1373	105	35	m	m	PROPN
ejpam-1373	105	36	)	)	PUNCT
ejpam-1373	105	37	.	.	PUNCT
ejpam-1373	106	1	proof	proof	NOUN
ejpam-1373	106	2	.	.	PUNCT
ejpam-1373	107	1	if	if	SCONJ
ejpam-1373	107	2	c	c	PROPN
ejpam-1373	107	3	(	(	PUNCT
ejpam-1373	107	4	m	m	NOUN
ejpam-1373	107	5	)	)	PUNCT
ejpam-1373	107	6	∪c	∪c	NOUN
ejpam-1373	107	7	(	(	PUNCT
ejpam-1373	107	8	n	n	CCONJ
ejpam-1373	107	9	)	)	PUNCT
ejpam-1373	107	10	=	=	SYM
ejpam-1373	107	11	c	c	X
ejpam-1373	107	12	(	(	PUNCT
ejpam-1373	107	13	a	a	NOUN
ejpam-1373	107	14	)	)	PUNCT
ejpam-1373	107	15	,	,	PUNCT
ejpam-1373	107	16	then	then	ADV
ejpam-1373	107	17	a	a	DET
ejpam-1373	107	18	∈	∈	PROPN
ejpam-1373	107	19	c	c	X
ejpam-1373	107	20	(	(	PUNCT
ejpam-1373	107	21	a	a	X
ejpam-1373	107	22	)	)	PUNCT
ejpam-1373	107	23	=	=	SYM
ejpam-1373	107	24	c	c	X
ejpam-1373	107	25	(	(	PUNCT
ejpam-1373	107	26	m	m	NOUN
ejpam-1373	107	27	)	)	PUNCT
ejpam-1373	107	28	∪c	∪c	NOUN
ejpam-1373	107	29	(	(	PUNCT
ejpam-1373	107	30	n	n	CCONJ
ejpam-1373	107	31	)	)	PUNCT
ejpam-1373	107	32	and	and	CCONJ
ejpam-1373	107	33	hence	hence	ADV
ejpam-1373	107	34	a	a	DET
ejpam-1373	107	35	∈	∈	NOUN
ejpam-1373	107	36	c	c	X
ejpam-1373	107	37	(	(	PUNCT
ejpam-1373	107	38	m	m	NOUN
ejpam-1373	107	39	)	)	PUNCT
ejpam-1373	107	40	or	or	CCONJ
ejpam-1373	107	41	a	a	DET
ejpam-1373	107	42	∈	∈	PROPN
ejpam-1373	107	43	c	c	X
ejpam-1373	107	44	(	(	PUNCT
ejpam-1373	107	45	n	n	CCONJ
ejpam-1373	107	46	)	)	PUNCT
ejpam-1373	107	47	so	so	SCONJ
ejpam-1373	107	48	that	that	SCONJ
ejpam-1373	107	49	c	c	NOUN
ejpam-1373	107	50	(	(	PUNCT
ejpam-1373	107	51	n)⊆	n)⊆	PROPN
ejpam-1373	107	52	c	c	PROPN
ejpam-1373	107	53	(	(	PUNCT
ejpam-1373	107	54	a)⊆	a)⊆	PROPN
ejpam-1373	107	55	c	c	PROPN
ejpam-1373	107	56	(	(	PUNCT
ejpam-1373	107	57	m	m	NOUN
ejpam-1373	107	58	)	)	PUNCT
ejpam-1373	107	59	or	or	CCONJ
ejpam-1373	107	60	c	c	X
ejpam-1373	107	61	(	(	PUNCT
ejpam-1373	107	62	m)⊆	m)⊆	PROPN
ejpam-1373	107	63	c	c	PROPN
ejpam-1373	107	64	(	(	PUNCT
ejpam-1373	107	65	a)⊆	a)⊆	PROPN
ejpam-1373	107	66	c	c	PROPN
ejpam-1373	107	67	(	(	PUNCT
ejpam-1373	107	68	n	n	CCONJ
ejpam-1373	107	69	)	)	PUNCT
ejpam-1373	107	70	.	.	PUNCT
ejpam-1373	108	1	the	the	DET
ejpam-1373	108	2	converse	converse	NOUN
ejpam-1373	108	3	is	be	AUX
ejpam-1373	108	4	trivial	trivial	ADJ
ejpam-1373	108	5	.	.	PUNCT
ejpam-1373	109	1	in	in	ADP
ejpam-1373	109	2	fact	fact	NOUN
ejpam-1373	109	3	,	,	PUNCT
ejpam-1373	109	4	any	any	DET
ejpam-1373	109	5	convolution	convolution	NOUN
ejpam-1373	109	6	c	c	NOUN
ejpam-1373	109	7	is	be	AUX
ejpam-1373	109	8	never	never	ADV
ejpam-1373	109	9	closed	close	VERB
ejpam-1373	109	10	under	under	ADP
ejpam-1373	109	11	finite	finite	NOUN
ejpam-1373	109	12	(	(	PUNCT
ejpam-1373	109	13	or	or	CCONJ
ejpam-1373	109	14	infinite	infinite	ADJ
ejpam-1373	109	15	)	)	PUNCT
ejpam-1373	109	16	unions	union	NOUN
ejpam-1373	109	17	.	.	PUNCT
ejpam-1373	110	1	for	for	ADP
ejpam-1373	110	2	,	,	PUNCT
ejpam-1373	110	3	consider	consider	VERB
ejpam-1373	110	4	two	two	NUM
ejpam-1373	110	5	distinct	distinct	ADJ
ejpam-1373	110	6	primes	prime	NOUN
ejpam-1373	110	7	p	p	NOUN
ejpam-1373	110	8	and	and	CCONJ
ejpam-1373	110	9	q.	q.	NOUN
ejpam-1373	110	10	then	then	ADV
ejpam-1373	110	11	neither	neither	CCONJ
ejpam-1373	110	12	c	c	PROPN
ejpam-1373	110	13	(	(	PUNCT
ejpam-1373	110	14	p	p	NOUN
ejpam-1373	110	15	)	)	PUNCT
ejpam-1373	110	16	⊆	⊆	NUM
ejpam-1373	110	17	c	c	X
ejpam-1373	110	18	(	(	PUNCT
ejpam-1373	110	19	q	q	NOUN
ejpam-1373	110	20	)	)	PUNCT
ejpam-1373	110	21	nor	nor	CCONJ
ejpam-1373	110	22	c	c	NOUN
ejpam-1373	110	23	(	(	PUNCT
ejpam-1373	110	24	q	q	X
ejpam-1373	110	25	)	)	PUNCT
ejpam-1373	110	26	⊆	⊆	NUM
ejpam-1373	110	27	c	c	X
ejpam-1373	110	28	(	(	PUNCT
ejpam-1373	110	29	p	p	NOUN
ejpam-1373	110	30	)	)	PUNCT
ejpam-1373	110	31	and	and	CCONJ
ejpam-1373	110	32	hence	hence	ADV
ejpam-1373	110	33	,	,	PUNCT
ejpam-1373	110	34	by	by	ADP
ejpam-1373	110	35	the	the	DET
ejpam-1373	110	36	above	above	ADJ
ejpam-1373	110	37	theorem	theorem	ADJ
ejpam-1373	110	38	c	c	PROPN
ejpam-1373	110	39	(	(	PUNCT
ejpam-1373	110	40	p)∪c	p)∪c	NOUN
ejpam-1373	110	41	(	(	PUNCT
ejpam-1373	110	42	q	q	NOUN
ejpam-1373	110	43	)	)	PUNCT
ejpam-1373	110	44	6=	6=	ADP
ejpam-1373	110	45	c	c	X
ejpam-1373	110	46	(	(	PUNCT
ejpam-1373	110	47	a	a	NOUN
ejpam-1373	110	48	)	)	PUNCT
ejpam-1373	110	49	for	for	ADP
ejpam-1373	110	50	any	any	DET
ejpam-1373	110	51	a	a	DET
ejpam-1373	110	52	∈	∈	PROPN
ejpam-1373	110	53	z+	z+	PUNCT
ejpam-1373	110	54	.	.	PUNCT
ejpam-1373	111	1	though	though	SCONJ
ejpam-1373	111	2	c	c	PROPN
ejpam-1373	111	3	is	be	AUX
ejpam-1373	111	4	never	never	ADV
ejpam-1373	111	5	closed	close	VERB
ejpam-1373	111	6	under	under	ADP
ejpam-1373	111	7	finite	finite	ADJ
ejpam-1373	111	8	unions	union	NOUN
ejpam-1373	111	9	,	,	PUNCT
ejpam-1373	111	10	it	it	PRON
ejpam-1373	111	11	is	be	AUX
ejpam-1373	111	12	quite	quite	ADV
ejpam-1373	111	13	possible	possible	ADJ
ejpam-1373	111	14	that	that	SCONJ
ejpam-1373	111	15	(	(	PUNCT
ejpam-1373	111	16	z+,≤c	z+,≤c	NUM
ejpam-1373	111	17	)	)	PUNCT
ejpam-1373	111	18	is	be	AUX
ejpam-1373	111	19	a	a	DET
ejpam-1373	111	20	join	join	NOUN
ejpam-1373	111	21	semi	semi	NOUN
ejpam-1373	111	22	lattice	lattice	PROPN
ejpam-1373	111	23	.	.	PUNCT
ejpam-1373	112	1	for	for	ADP
ejpam-1373	112	2	,	,	PUNCT
ejpam-1373	112	3	consider	consider	VERB
ejpam-1373	112	4	the	the	DET
ejpam-1373	112	5	dirichlet	dirichlet	PROPN
ejpam-1373	112	6	’s	’s	PART
ejpam-1373	112	7	convolution	convolution	PROPN
ejpam-1373	112	8	d.	d.	PROPN
ejpam-1373	112	9	then	then	ADV
ejpam-1373	112	10	(	(	PUNCT
ejpam-1373	112	11	z+,≤d	z+,≤d	X
ejpam-1373	112	12	)	)	PUNCT
ejpam-1373	112	13	is	be	AUX
ejpam-1373	112	14	a	a	DET
ejpam-1373	112	15	lattice	lattice	NOUN
ejpam-1373	112	16	.	.	PUNCT
ejpam-1373	113	1	recall	recall	VERB
ejpam-1373	113	2	that	that	SCONJ
ejpam-1373	113	3	a	a	DET
ejpam-1373	113	4	partially	partially	ADV
ejpam-1373	113	5	ordered	order	VERB
ejpam-1373	113	6	set	set	NOUN
ejpam-1373	113	7	(	(	PUNCT
ejpam-1373	113	8	x	x	INTJ
ejpam-1373	113	9	,	,	PUNCT
ejpam-1373	113	10	≤	≤	NUM
ejpam-1373	113	11	)	)	PUNCT
ejpam-1373	113	12	is	be	AUX
ejpam-1373	113	13	called	call	VERB
ejpam-1373	113	14	directed	direct	VERB
ejpam-1373	113	15	below(above	below(above	NOUN
ejpam-1373	113	16	)	)	PUNCT
ejpam-1373	113	17	if	if	SCONJ
ejpam-1373	113	18	,	,	PUNCT
ejpam-1373	113	19	for	for	ADP
ejpam-1373	113	20	any	any	DET
ejpam-1373	113	21	a	a	PRON
ejpam-1373	113	22	and	and	CCONJ
ejpam-1373	113	23	b	b	NOUN
ejpam-1373	113	24	∈	∈	PROPN
ejpam-1373	113	25	x	x	INTJ
ejpam-1373	113	26	,	,	PUNCT
ejpam-1373	113	27	there	there	PRON
ejpam-1373	113	28	exists	exist	VERB
ejpam-1373	113	29	x	x	X
ejpam-1373	113	30	∈	∈	PROPN
ejpam-1373	113	31	x	x	X
ejpam-1373	113	32	such	such	ADJ
ejpam-1373	113	33	that	that	SCONJ
ejpam-1373	113	34	x	x	SYM
ejpam-1373	113	35	≤	≤	NOUN
ejpam-1373	113	36	a	a	PRON
ejpam-1373	113	37	and	and	CCONJ
ejpam-1373	113	38	x	x	SYM
ejpam-1373	113	39	≤	≤	NUM
ejpam-1373	113	40	b	b	X
ejpam-1373	113	41	(	(	PUNCT
ejpam-1373	113	42	respectively	respectively	ADV
ejpam-1373	113	43	a	a	DET
ejpam-1373	113	44	≤	≤	NOUN
ejpam-1373	113	45	x	x	PUNCT
ejpam-1373	113	46	and	and	CCONJ
ejpam-1373	113	47	b	b	NOUN
ejpam-1373	113	48	≤	≤	NUM
ejpam-1373	113	49	x	x	X
ejpam-1373	113	50	)	)	PUNCT
ejpam-1373	113	51	.	.	PUNCT
ejpam-1373	114	1	u.	u.	PROPN
ejpam-1373	114	2	swamy	swamy	PROPN
ejpam-1373	114	3	and	and	CCONJ
ejpam-1373	114	4	s.	s.	PROPN
ejpam-1373	114	5	sankar	sankar	PROPN
ejpam-1373	114	6	/	/	SYM
ejpam-1373	114	7	eur	eur	PROPN
ejpam-1373	114	8	.	.	PUNCT
ejpam-1373	115	1	j.	j.	PROPN
ejpam-1373	115	2	pure	pure	PROPN
ejpam-1373	115	3	appl	appl	PROPN
ejpam-1373	115	4	.	.	PROPN
ejpam-1373	115	5	math	math	PROPN
ejpam-1373	115	6	,	,	PUNCT
ejpam-1373	115	7	4	4	NUM
ejpam-1373	115	8	(	(	PUNCT
ejpam-1373	115	9	2011	2011	NUM
ejpam-1373	115	10	)	)	PUNCT
ejpam-1373	115	11	,	,	PUNCT
ejpam-1373	115	12	424	424	NUM
ejpam-1373	115	13	-	-	SYM
ejpam-1373	115	14	434	434	NUM
ejpam-1373	115	15	428	428	NUM
ejpam-1373	115	16	theorem	theorem	NOUN
ejpam-1373	115	17	3	3	X
ejpam-1373	115	18	.	.	PUNCT
ejpam-1373	116	1	let	let	VERB
ejpam-1373	116	2	c	c	PRON
ejpam-1373	116	3	be	be	AUX
ejpam-1373	116	4	a	a	DET
ejpam-1373	116	5	convolution	convolution	NOUN
ejpam-1373	116	6	which	which	PRON
ejpam-1373	116	7	is	be	AUX
ejpam-1373	116	8	closed	close	VERB
ejpam-1373	116	9	under	under	ADP
ejpam-1373	116	10	finite	finite	ADJ
ejpam-1373	116	11	intersections	intersection	NOUN
ejpam-1373	116	12	and	and	CCONJ
ejpam-1373	116	13	≤c	≤c	PROPN
ejpam-1373	116	14	be	be	VERB
ejpam-1373	116	15	the	the	DET
ejpam-1373	116	16	partial	partial	ADJ
ejpam-1373	116	17	order	order	NOUN
ejpam-1373	116	18	on	on	ADP
ejpam-1373	116	19	z+	z+	NUM
ejpam-1373	116	20	induced	induce	VERB
ejpam-1373	116	21	by	by	ADP
ejpam-1373	116	22	c	c	PROPN
ejpam-1373	116	23	.	.	PUNCT
ejpam-1373	117	1	then	then	ADV
ejpam-1373	117	2	(	(	PUNCT
ejpam-1373	117	3	z+,≤c	z+,≤c	X
ejpam-1373	117	4	)	)	PUNCT
ejpam-1373	117	5	is	be	AUX
ejpam-1373	117	6	a	a	DET
ejpam-1373	117	7	lattice	lattice	NOUN
ejpam-1373	117	8	if	if	SCONJ
ejpam-1373	118	1	and	and	CCONJ
ejpam-1373	118	2	only	only	ADV
ejpam-1373	118	3	if	if	SCONJ
ejpam-1373	118	4	it	it	PRON
ejpam-1373	118	5	is	be	AUX
ejpam-1373	118	6	directed	direct	VERB
ejpam-1373	118	7	above	above	ADV
ejpam-1373	118	8	.	.	PUNCT
ejpam-1373	119	1	proof	proof	NOUN
ejpam-1373	119	2	.	.	PUNCT
ejpam-1373	120	1	from	from	ADP
ejpam-1373	120	2	the	the	DET
ejpam-1373	120	3	hypothesis	hypothesis	NOUN
ejpam-1373	120	4	and	and	CCONJ
ejpam-1373	120	5	theorem	theorem	NOUN
ejpam-1373	120	6	1	1	NUM
ejpam-1373	120	7	,	,	PUNCT
ejpam-1373	120	8	it	it	PRON
ejpam-1373	120	9	follows	follow	VERB
ejpam-1373	120	10	that	that	SCONJ
ejpam-1373	120	11	(	(	PUNCT
ejpam-1373	120	12	z+,≤c	z+,≤c	NUM
ejpam-1373	120	13	)	)	PUNCT
ejpam-1373	120	14	is	be	AUX
ejpam-1373	120	15	a	a	DET
ejpam-1373	120	16	meet	meet	ADJ
ejpam-1373	120	17	semilattice	semilattice	NOUN
ejpam-1373	120	18	.	.	PUNCT
ejpam-1373	121	1	also	also	ADV
ejpam-1373	121	2	,	,	PUNCT
ejpam-1373	121	3	every	every	DET
ejpam-1373	121	4	non	non	ADJ
ejpam-1373	121	5	-	-	ADJ
ejpam-1373	121	6	empty	empty	ADJ
ejpam-1373	121	7	subset	subset	NOUN
ejpam-1373	121	8	of	of	ADP
ejpam-1373	121	9	z+	z+	NUM
ejpam-1373	121	10	has	have	VERB
ejpam-1373	121	11	glb	glb	NOUN
ejpam-1373	121	12	in	in	ADP
ejpam-1373	121	13	(	(	PUNCT
ejpam-1373	121	14	z+,≤c	z+,≤c	PROPN
ejpam-1373	121	15	)	)	PUNCT
ejpam-1373	121	16	.	.	PUNCT
ejpam-1373	122	1	now	now	ADV
ejpam-1373	122	2	,	,	PUNCT
ejpam-1373	122	3	suppose	suppose	VERB
ejpam-1373	122	4	that	that	SCONJ
ejpam-1373	122	5	(	(	PUNCT
ejpam-1373	122	6	z+,≤c	z+,≤c	NUM
ejpam-1373	122	7	)	)	PUNCT
ejpam-1373	122	8	is	be	AUX
ejpam-1373	122	9	directed	direct	VERB
ejpam-1373	122	10	above	above	ADV
ejpam-1373	122	11	.	.	PUNCT
ejpam-1373	123	1	let	let	VERB
ejpam-1373	123	2	a	a	DET
ejpam-1373	123	3	and	and	CCONJ
ejpam-1373	123	4	b	b	NOUN
ejpam-1373	123	5	∈	∈	PROPN
ejpam-1373	123	6	z+	z+	NUM
ejpam-1373	123	7	and	and	CCONJ
ejpam-1373	123	8	�	�	PROPN
ejpam-1373	123	9	a=	a=	PROPN
ejpam-1373	123	10	{	{	PUNCT
ejpam-1373	123	11	n	n	CCONJ
ejpam-1373	123	12	∈	∈	PROPN
ejpam-1373	123	13	z+|a	z+|a	PROPN
ejpam-1373	123	14	≤c	≤c	PROPN
ejpam-1373	123	15	n	n	PROPN
ejpam-1373	123	16	and	and	CCONJ
ejpam-1373	123	17	b	b	PROPN
ejpam-1373	123	18	≤c	≤c	PROPN
ejpam-1373	123	19	n	n	CCONJ
ejpam-1373	123	20	}	}	PUNCT
ejpam-1373	123	21	�	�	PROPN
ejpam-1373	123	22	then	then	ADV
ejpam-1373	123	23	,	,	PUNCT
ejpam-1373	123	24	since	since	SCONJ
ejpam-1373	123	25	the	the	DET
ejpam-1373	123	26	poset	poset	NOUN
ejpam-1373	123	27	is	be	AUX
ejpam-1373	123	28	directed	direct	VERB
ejpam-1373	123	29	above	above	ADV
ejpam-1373	123	30	,	,	PUNCT
ejpam-1373	123	31	a	a	PRON
ejpam-1373	123	32	is	be	AUX
ejpam-1373	123	33	a	a	DET
ejpam-1373	123	34	non	non	ADJ
ejpam-1373	123	35	-	-	ADJ
ejpam-1373	123	36	empty	empty	ADJ
ejpam-1373	123	37	subset	subset	NOUN
ejpam-1373	123	38	of	of	ADP
ejpam-1373	123	39	z+	z+	NUM
ejpam-1373	123	40	and	and	CCONJ
ejpam-1373	123	41	hence	hence	ADV
ejpam-1373	123	42	a	a	PRON
ejpam-1373	123	43	has	have	VERB
ejpam-1373	123	44	glb	glb	NOUN
ejpam-1373	123	45	.	.	PUNCT
ejpam-1373	124	1	then	then	ADV
ejpam-1373	124	2	it	it	PRON
ejpam-1373	124	3	can	can	AUX
ejpam-1373	124	4	be	be	AUX
ejpam-1373	124	5	easily	easily	ADV
ejpam-1373	124	6	verified	verify	VERB
ejpam-1373	124	7	that	that	SCONJ
ejpam-1373	124	8	the	the	DET
ejpam-1373	124	9	glb	glb	NOUN
ejpam-1373	124	10	of	of	ADP
ejpam-1373	124	11	a	a	PRON
ejpam-1373	124	12	is	be	AUX
ejpam-1373	124	13	the	the	DET
ejpam-1373	124	14	lub	lub	NOUN
ejpam-1373	124	15	of	of	ADP
ejpam-1373	124	16	{	{	PUNCT
ejpam-1373	124	17	a	a	PROPN
ejpam-1373	124	18	,	,	PUNCT
ejpam-1373	124	19	b	b	NOUN
ejpam-1373	124	20	}	}	PUNCT
ejpam-1373	124	21	.	.	PUNCT
ejpam-1373	125	1	thus	thus	ADV
ejpam-1373	125	2	(	(	PUNCT
ejpam-1373	125	3	z+,≤c	z+,≤c	X
ejpam-1373	125	4	)	)	PUNCT
ejpam-1373	125	5	is	be	AUX
ejpam-1373	125	6	a	a	DET
ejpam-1373	125	7	join	join	NOUN
ejpam-1373	125	8	semilattice	semilattice	NOUN
ejpam-1373	125	9	also	also	ADV
ejpam-1373	125	10	and	and	CCONJ
ejpam-1373	125	11	hence	hence	ADV
ejpam-1373	125	12	a	a	DET
ejpam-1373	125	13	lattice	lattice	NOUN
ejpam-1373	125	14	.	.	PUNCT
ejpam-1373	126	1	the	the	DET
ejpam-1373	126	2	converse	converse	NOUN
ejpam-1373	126	3	is	be	AUX
ejpam-1373	126	4	trivial	trivial	ADJ
ejpam-1373	126	5	.	.	PUNCT
ejpam-1373	127	1	unlike	unlike	ADP
ejpam-1373	127	2	in	in	ADP
ejpam-1373	127	3	theorem	theorem	NOUN
ejpam-1373	127	4	1	1	NUM
ejpam-1373	127	5	,	,	PUNCT
ejpam-1373	127	6	(	(	PUNCT
ejpam-1373	127	7	z+,≤c	z+,≤c	PUNCT
ejpam-1373	127	8	)	)	PUNCT
ejpam-1373	127	9	may	may	AUX
ejpam-1373	127	10	be	be	AUX
ejpam-1373	127	11	a	a	DET
ejpam-1373	127	12	join	join	NOUN
ejpam-1373	127	13	semi	semi	ADV
ejpam-1373	127	14	lattice	lattice	NOUN
ejpam-1373	127	15	and	and	CCONJ
ejpam-1373	127	16	not	not	PART
ejpam-1373	127	17	every	every	DET
ejpam-1373	127	18	non	non	ADJ
ejpam-1373	127	19	-	-	ADJ
ejpam-1373	127	20	empty	empty	ADJ
ejpam-1373	127	21	subset	subset	NOUN
ejpam-1373	127	22	has	have	AUX
ejpam-1373	127	23	lub	lub	VERB
ejpam-1373	127	24	in	in	ADP
ejpam-1373	127	25	(	(	PUNCT
ejpam-1373	127	26	z+,≤c	z+,≤c	PROPN
ejpam-1373	127	27	)	)	PUNCT
ejpam-1373	127	28	.	.	PUNCT
ejpam-1373	128	1	in	in	ADP
ejpam-1373	128	2	this	this	DET
ejpam-1373	128	3	context	context	NOUN
ejpam-1373	128	4	,	,	PUNCT
ejpam-1373	128	5	note	note	VERB
ejpam-1373	128	6	that	that	SCONJ
ejpam-1373	128	7	(	(	PUNCT
ejpam-1373	128	8	z+,≤c	z+,≤c	NUM
ejpam-1373	128	9	)	)	PUNCT
ejpam-1373	128	10	can	can	AUX
ejpam-1373	128	11	never	never	ADV
ejpam-1373	128	12	possess	possess	VERB
ejpam-1373	128	13	the	the	DET
ejpam-1373	128	14	largest	large	ADJ
ejpam-1373	128	15	element	element	NOUN
ejpam-1373	128	16	;	;	PUNCT
ejpam-1373	128	17	for	for	SCONJ
ejpam-1373	128	18	,	,	PUNCT
ejpam-1373	128	19	c	c	PROPN
ejpam-1373	128	20	(	(	PUNCT
ejpam-1373	128	21	n	n	CCONJ
ejpam-1373	128	22	)	)	PUNCT
ejpam-1373	128	23	is	be	AUX
ejpam-1373	128	24	a	a	DET
ejpam-1373	128	25	finite	finite	NOUN
ejpam-1373	128	26	set	set	VERB
ejpam-1373	128	27	for	for	ADP
ejpam-1373	128	28	all	all	DET
ejpam-1373	128	29	n	n	PRON
ejpam-1373	128	30	∈	∈	PROPN
ejpam-1373	128	31	z+	z+	NUM
ejpam-1373	128	32	.	.	PUNCT
ejpam-1373	129	1	theorem	theorem	NOUN
ejpam-1373	129	2	3	3	NUM
ejpam-1373	129	3	can	can	AUX
ejpam-1373	129	4	be	be	AUX
ejpam-1373	129	5	dualised	dualise	VERB
ejpam-1373	129	6	as	as	SCONJ
ejpam-1373	129	7	given	give	VERB
ejpam-1373	129	8	in	in	ADP
ejpam-1373	129	9	the	the	DET
ejpam-1373	129	10	following	following	NOUN
ejpam-1373	129	11	theorem	theorem	VERB
ejpam-1373	129	12	,	,	PUNCT
ejpam-1373	129	13	whose	whose	DET
ejpam-1373	129	14	proof	proof	NOUN
ejpam-1373	129	15	is	be	AUX
ejpam-1373	129	16	a	a	DET
ejpam-1373	129	17	consequence	consequence	NOUN
ejpam-1373	129	18	of	of	ADP
ejpam-1373	129	19	the	the	DET
ejpam-1373	129	20	fact	fact	NOUN
ejpam-1373	129	21	that	that	SCONJ
ejpam-1373	129	22	the	the	DET
ejpam-1373	129	23	set	set	NOUN
ejpam-1373	129	24	of	of	ADP
ejpam-1373	129	25	lower	low	ADJ
ejpam-1373	129	26	bounds	bound	NOUN
ejpam-1373	129	27	of	of	ADP
ejpam-1373	129	28	any	any	DET
ejpam-1373	129	29	non	non	ADJ
ejpam-1373	129	30	-	-	ADJ
ejpam-1373	129	31	empty	empty	ADJ
ejpam-1373	129	32	subset	subset	NOUN
ejpam-1373	129	33	of	of	ADP
ejpam-1373	129	34	z+	z+	NUM
ejpam-1373	129	35	is	be	AUX
ejpam-1373	129	36	finite	finite	ADJ
ejpam-1373	129	37	.	.	PUNCT
ejpam-1373	130	1	theorem	theorem	ADJ
ejpam-1373	130	2	4	4	NUM
ejpam-1373	130	3	.	.	PUNCT
ejpam-1373	131	1	let	let	VERB
ejpam-1373	131	2	c	c	PRON
ejpam-1373	131	3	be	be	AUX
ejpam-1373	131	4	a	a	DET
ejpam-1373	131	5	convolution	convolution	NOUN
ejpam-1373	131	6	such	such	ADJ
ejpam-1373	131	7	that	that	SCONJ
ejpam-1373	131	8	(	(	PUNCT
ejpam-1373	131	9	z+,≤c	z+,≤c	NUM
ejpam-1373	131	10	)	)	PUNCT
ejpam-1373	131	11	is	be	AUX
ejpam-1373	131	12	a	a	DET
ejpam-1373	131	13	join	join	NOUN
ejpam-1373	131	14	semilattice	semilattice	NOUN
ejpam-1373	131	15	.	.	PUNCT
ejpam-1373	132	1	then	then	ADV
ejpam-1373	132	2	(	(	PUNCT
ejpam-1373	132	3	z+,≤c	z+,≤c	X
ejpam-1373	132	4	)	)	PUNCT
ejpam-1373	132	5	is	be	AUX
ejpam-1373	132	6	a	a	DET
ejpam-1373	132	7	lattice	lattice	NOUN
ejpam-1373	132	8	if	if	SCONJ
ejpam-1373	133	1	and	and	CCONJ
ejpam-1373	133	2	if	if	SCONJ
ejpam-1373	133	3	it	it	PRON
ejpam-1373	133	4	is	be	AUX
ejpam-1373	133	5	directed	direct	VERB
ejpam-1373	133	6	below	below	ADV
ejpam-1373	133	7	.	.	PUNCT
ejpam-1373	134	1	definition	definition	NOUN
ejpam-1373	134	2	4	4	NUM
ejpam-1373	134	3	.	.	PUNCT
ejpam-1373	135	1	let	let	VERB
ejpam-1373	135	2	c	c	PRON
ejpam-1373	135	3	be	be	AUX
ejpam-1373	135	4	a	a	DET
ejpam-1373	135	5	convolution	convolution	NOUN
ejpam-1373	135	6	and	and	CCONJ
ejpam-1373	135	7	p	p	X
ejpam-1373	135	8	a	a	DET
ejpam-1373	135	9	prime	prime	ADJ
ejpam-1373	135	10	number	number	NOUN
ejpam-1373	135	11	.	.	PUNCT
ejpam-1373	136	1	define	define	VERB
ejpam-1373	136	2	a	a	DET
ejpam-1373	136	3	relation	relation	NOUN
ejpam-1373	136	4	≤p	≤p	NOUN
ejpam-1373	136	5	c	c	PROPN
ejpam-1373	136	6	on	on	ADP
ejpam-1373	136	7	the	the	DET
ejpam-1373	136	8	set	set	NOUN
ejpam-1373	136	9	n	n	PROPN
ejpam-1373	136	10	of	of	ADP
ejpam-1373	136	11	non	non	ADJ
ejpam-1373	136	12	-	-	ADJ
ejpam-1373	136	13	negative	negative	ADJ
ejpam-1373	136	14	integers	integer	NOUN
ejpam-1373	136	15	by	by	ADP
ejpam-1373	136	16	�	�	PROPN
ejpam-1373	136	17	a	a	DET
ejpam-1373	136	18	≤p	≤p	NOUN
ejpam-1373	136	19	c	c	PROPN
ejpam-1373	136	20	b	b	NOUN
ejpam-1373	137	1	if	if	SCONJ
ejpam-1373	137	2	and	and	CCONJ
ejpam-1373	137	3	only	only	ADV
ejpam-1373	137	4	if	if	SCONJ
ejpam-1373	137	5	pa	pa	PROPN
ejpam-1373	137	6	∈	∈	PROPN
ejpam-1373	137	7	c	c	X
ejpam-1373	137	8	(	(	PUNCT
ejpam-1373	137	9	pb	pb	NOUN
ejpam-1373	137	10	)	)	PUNCT
ejpam-1373	137	11	�	�	PROPN
ejpam-1373	137	12	for	for	ADP
ejpam-1373	137	13	any	any	DET
ejpam-1373	137	14	a	a	DET
ejpam-1373	137	15	and	and	CCONJ
ejpam-1373	137	16	b	b	NOUN
ejpam-1373	137	17	∈	∈	PROPN
ejpam-1373	137	18	n	n	NOUN
ejpam-1373	137	19	.	.	PUNCT
ejpam-1373	138	1	it	it	PRON
ejpam-1373	138	2	can	can	AUX
ejpam-1373	138	3	be	be	AUX
ejpam-1373	138	4	easily	easily	ADV
ejpam-1373	138	5	verified	verify	VERB
ejpam-1373	138	6	that	that	SCONJ
ejpam-1373	138	7	≤p	≤p	PROPN
ejpam-1373	138	8	c	c	PROPN
ejpam-1373	138	9	is	be	AUX
ejpam-1373	138	10	a	a	DET
ejpam-1373	138	11	partial	partial	ADJ
ejpam-1373	138	12	order	order	NOUN
ejpam-1373	138	13	on	on	ADP
ejpam-1373	138	14	n	n	PROPN
ejpam-1373	138	15	,	,	PUNCT
ejpam-1373	138	16	for	for	ADP
ejpam-1373	138	17	each	each	DET
ejpam-1373	138	18	prime	prime	NOUN
ejpam-1373	138	19	p.	p.	NOUN
ejpam-1373	138	20	the	the	DET
ejpam-1373	138	21	following	follow	VERB
ejpam-1373	138	22	is	be	AUX
ejpam-1373	138	23	a	a	DET
ejpam-1373	138	24	direct	direct	ADJ
ejpam-1373	138	25	verification	verification	NOUN
ejpam-1373	138	26	.	.	PUNCT
ejpam-1373	139	1	theorem	theorem	NOUN
ejpam-1373	139	2	5	5	NUM
ejpam-1373	139	3	.	.	PUNCT
ejpam-1373	140	1	let	let	VERB
ejpam-1373	140	2	c	c	PRON
ejpam-1373	140	3	be	be	AUX
ejpam-1373	140	4	a	a	DET
ejpam-1373	140	5	convolution	convolution	NOUN
ejpam-1373	140	6	.	.	PUNCT
ejpam-1373	141	1	(	(	PUNCT
ejpam-1373	141	2	1	1	NUM
ejpam-1373	141	3	)	)	PUNCT
ejpam-1373	141	4	.	.	PUNCT
ejpam-1373	142	1	if	if	SCONJ
ejpam-1373	142	2	(	(	PUNCT
ejpam-1373	142	3	z+,≤c	z+,≤c	NUM
ejpam-1373	142	4	)	)	PUNCT
ejpam-1373	142	5	is	be	AUX
ejpam-1373	142	6	a	a	DET
ejpam-1373	142	7	meet(join	meet(join	NOUN
ejpam-1373	142	8	)	)	PUNCT
ejpam-1373	142	9	semilattice	semilattice	NOUN
ejpam-1373	142	10	,	,	PUNCT
ejpam-1373	142	11	then	then	ADV
ejpam-1373	142	12	so	so	ADV
ejpam-1373	142	13	is	be	AUX
ejpam-1373	142	14	(	(	PUNCT
ejpam-1373	142	15	n	n	X
ejpam-1373	142	16	,	,	PUNCT
ejpam-1373	142	17	≤p	≤p	PROPN
ejpam-1373	142	18	c	c	PROPN
ejpam-1373	142	19	)	)	PUNCT
ejpam-1373	142	20	for	for	ADP
ejpam-1373	142	21	each	each	DET
ejpam-1373	142	22	prime	prime	ADJ
ejpam-1373	142	23	p.	p.	NOUN
ejpam-1373	142	24	(	(	PUNCT
ejpam-1373	142	25	2	2	NUM
ejpam-1373	142	26	)	)	PUNCT
ejpam-1373	142	27	.	.	PUNCT
ejpam-1373	143	1	if	if	SCONJ
ejpam-1373	143	2	(	(	PUNCT
ejpam-1373	143	3	z+,≤c	z+,≤c	NUM
ejpam-1373	143	4	)	)	PUNCT
ejpam-1373	143	5	is	be	AUX
ejpam-1373	143	6	a	a	DET
ejpam-1373	143	7	lattice	lattice	NOUN
ejpam-1373	143	8	,	,	PUNCT
ejpam-1373	143	9	then	then	ADV
ejpam-1373	143	10	so	so	ADV
ejpam-1373	143	11	is	be	AUX
ejpam-1373	143	12	(	(	PUNCT
ejpam-1373	143	13	n	n	X
ejpam-1373	143	14	,	,	PUNCT
ejpam-1373	143	15	≤p	≤p	PROPN
ejpam-1373	143	16	c	c	PROPN
ejpam-1373	143	17	)	)	PUNCT
ejpam-1373	143	18	for	for	ADP
ejpam-1373	143	19	each	each	DET
ejpam-1373	143	20	prime	prime	NOUN
ejpam-1373	143	21	p.	p.	NOUN
ejpam-1373	143	22	the	the	DET
ejpam-1373	143	23	converse	converse	NOUN
ejpam-1373	143	24	of	of	ADP
ejpam-1373	143	25	the	the	DET
ejpam-1373	143	26	assertions	assertion	NOUN
ejpam-1373	143	27	made	make	VERB
ejpam-1373	143	28	in	in	ADP
ejpam-1373	143	29	the	the	DET
ejpam-1373	143	30	above	above	ADJ
ejpam-1373	143	31	theorem	theorem	NOUN
ejpam-1373	143	32	are	be	AUX
ejpam-1373	143	33	not	not	PART
ejpam-1373	143	34	true	true	ADJ
ejpam-1373	143	35	in	in	ADP
ejpam-1373	143	36	general	general	ADJ
ejpam-1373	143	37	.	.	PUNCT
ejpam-1373	144	1	for	for	ADP
ejpam-1373	144	2	,	,	PUNCT
ejpam-1373	144	3	consider	consider	VERB
ejpam-1373	144	4	the	the	DET
ejpam-1373	144	5	following	following	NOUN
ejpam-1373	144	6	.	.	PUNCT
ejpam-1373	145	1	example	example	NOUN
ejpam-1373	146	1	1	1	NUM
ejpam-1373	146	2	.	.	X
ejpam-1373	146	3	define	define	VERB
ejpam-1373	146	4	c	c	NOUN
ejpam-1373	146	5	:	:	PUNCT
ejpam-1373	146	6	z+	z+	NUM
ejpam-1373	146	7	−→p	−→p	NOUN
ejpam-1373	146	8	(	(	PUNCT
ejpam-1373	146	9	z+	z+	NUM
ejpam-1373	146	10	)	)	PUNCT
ejpam-1373	146	11	by	by	ADP
ejpam-1373	146	12	c	c	PROPN
ejpam-1373	146	13	(	(	PUNCT
ejpam-1373	146	14	n	n	CCONJ
ejpam-1373	146	15	)	)	PUNCT
ejpam-1373	146	16	=	=	PUNCT
ejpam-1373	146	17			PROPN
ejpam-1373	146	18			PROPN
ejpam-1373	146	19			PROPN
ejpam-1373	146	20	{	{	PUNCT
ejpam-1373	146	21	1,2,5,10	1,2,5,10	PROPN
ejpam-1373	146	22	}	}	PUNCT
ejpam-1373	146	23	if	if	SCONJ
ejpam-1373	146	24	n=	n=	ADJ
ejpam-1373	146	25	10	10	NUM
ejpam-1373	146	26	{	{	PUNCT
ejpam-1373	146	27	1,2,5,20	1,2,5,20	NOUN
ejpam-1373	146	28	}	}	PUNCT
ejpam-1373	146	29	if	if	SCONJ
ejpam-1373	146	30	n=	n=	ADJ
ejpam-1373	146	31	20	20	NUM
ejpam-1373	146	32	{	{	SYM
ejpam-1373	146	33	1	1	NUM
ejpam-1373	146	34	,	,	PUNCT
ejpam-1373	146	35	n	n	CCONJ
ejpam-1373	146	36	}	}	PUNCT
ejpam-1373	146	37	otherwise	otherwise	ADV
ejpam-1373	146	38	then	then	ADV
ejpam-1373	146	39	c	c	PROPN
ejpam-1373	146	40	is	be	AUX
ejpam-1373	146	41	a	a	DET
ejpam-1373	146	42	convolution	convolution	NOUN
ejpam-1373	146	43	.	.	PUNCT
ejpam-1373	147	1	in	in	ADP
ejpam-1373	147	2	this	this	DET
ejpam-1373	147	3	case	case	NOUN
ejpam-1373	147	4	,	,	PUNCT
ejpam-1373	147	5	for	for	ADP
ejpam-1373	147	6	any	any	DET
ejpam-1373	147	7	prime	prime	ADJ
ejpam-1373	147	8	p	p	NOUN
ejpam-1373	147	9	and	and	CCONJ
ejpam-1373	147	10	a	a	DET
ejpam-1373	147	11	∈	∈	PROPN
ejpam-1373	147	12	n	n	X
ejpam-1373	147	13	,	,	PUNCT
ejpam-1373	147	14	we	we	PRON
ejpam-1373	147	15	have	have	VERB
ejpam-1373	147	16	c	c	NOUN
ejpam-1373	147	17	(	(	PUNCT
ejpam-1373	147	18	pa	pa	PROPN
ejpam-1373	147	19	)	)	PUNCT
ejpam-1373	147	20	=	=	PUNCT
ejpam-1373	147	21	{	{	PUNCT
ejpam-1373	147	22	1	1	NUM
ejpam-1373	147	23	,	,	PUNCT
ejpam-1373	147	24	pa	pa	NOUN
ejpam-1373	147	25	}	}	PUNCT
ejpam-1373	147	26	and	and	CCONJ
ejpam-1373	147	27	hence	hence	ADV
ejpam-1373	147	28	0	0	NUM
ejpam-1373	147	29	is	be	AUX
ejpam-1373	147	30	the	the	DET
ejpam-1373	147	31	only	only	ADV
ejpam-1373	147	32	lower	lower	ADV
ejpam-1373	147	33	bound	bind	VERB
ejpam-1373	147	34	for	for	ADP
ejpam-1373	147	35	any	any	DET
ejpam-1373	147	36	two	two	NUM
ejpam-1373	147	37	distinct	distinct	ADJ
ejpam-1373	147	38	a	a	PRON
ejpam-1373	147	39	and	and	CCONJ
ejpam-1373	147	40	b	b	NOUN
ejpam-1373	147	41	in	in	ADP
ejpam-1373	147	42	(	(	PUNCT
ejpam-1373	147	43	n	n	NOUN
ejpam-1373	147	44	,	,	PUNCT
ejpam-1373	147	45	≤p	≤p	PROPN
ejpam-1373	147	46	c	c	PROPN
ejpam-1373	147	47	)	)	PUNCT
ejpam-1373	147	48	.	.	PUNCT
ejpam-1373	148	1	this	this	PRON
ejpam-1373	148	2	implies	imply	VERB
ejpam-1373	148	3	that	that	SCONJ
ejpam-1373	148	4	,	,	PUNCT
ejpam-1373	148	5	for	for	ADP
ejpam-1373	148	6	each	each	DET
ejpam-1373	148	7	prime	prime	NOUN
ejpam-1373	148	8	p	p	X
ejpam-1373	148	9	,	,	PUNCT
ejpam-1373	148	10	(	(	PUNCT
ejpam-1373	148	11	n	n	CCONJ
ejpam-1373	148	12	,	,	PUNCT
ejpam-1373	148	13	≤p	≤p	PROPN
ejpam-1373	148	14	c	c	PROPN
ejpam-1373	148	15	)	)	PUNCT
ejpam-1373	148	16	is	be	AUX
ejpam-1373	148	17	a	a	DET
ejpam-1373	148	18	meet	meet	ADJ
ejpam-1373	148	19	semilattice	semilattice	NOUN
ejpam-1373	148	20	.	.	PUNCT
ejpam-1373	149	1	however	however	ADV
ejpam-1373	149	2	,	,	PUNCT
ejpam-1373	149	3	(	(	PUNCT
ejpam-1373	149	4	z+,≤c	z+,≤c	X
ejpam-1373	149	5	)	)	PUNCT
ejpam-1373	149	6	is	be	AUX
ejpam-1373	149	7	not	not	PART
ejpam-1373	149	8	a	a	DET
ejpam-1373	149	9	meet	meet	NOUN
ejpam-1373	149	10	semilattice	semilattice	NOUN
ejpam-1373	149	11	,	,	PUNCT
ejpam-1373	149	12	since	since	SCONJ
ejpam-1373	149	13	the	the	DET
ejpam-1373	149	14	set	set	NOUN
ejpam-1373	149	15	{	{	PUNCT
ejpam-1373	149	16	10,20	10,20	NUM
ejpam-1373	149	17	}	}	PUNCT
ejpam-1373	149	18	has	have	VERB
ejpam-1373	149	19	no	no	DET
ejpam-1373	149	20	glb	glb	NOUN
ejpam-1373	149	21	in	in	ADP
ejpam-1373	149	22	(	(	PUNCT
ejpam-1373	149	23	z+,≤c	z+,≤c	PROPN
ejpam-1373	149	24	)	)	PUNCT
ejpam-1373	149	25	.	.	PUNCT
ejpam-1373	150	1	u.	u.	PROPN
ejpam-1373	150	2	swamy	swamy	PROPN
ejpam-1373	150	3	and	and	CCONJ
ejpam-1373	150	4	s.	s.	PROPN
ejpam-1373	150	5	sankar	sankar	PROPN
ejpam-1373	150	6	/	/	SYM
ejpam-1373	150	7	eur	eur	PROPN
ejpam-1373	150	8	.	.	PUNCT
ejpam-1373	151	1	j.	j.	PROPN
ejpam-1373	151	2	pure	pure	PROPN
ejpam-1373	151	3	appl	appl	PROPN
ejpam-1373	151	4	.	.	PROPN
ejpam-1373	151	5	math	math	PROPN
ejpam-1373	151	6	,	,	PUNCT
ejpam-1373	151	7	4	4	NUM
ejpam-1373	151	8	(	(	PUNCT
ejpam-1373	151	9	2011	2011	NUM
ejpam-1373	151	10	)	)	PUNCT
ejpam-1373	151	11	,	,	PUNCT
ejpam-1373	151	12	424	424	NUM
ejpam-1373	151	13	-	-	SYM
ejpam-1373	151	14	434	434	NUM
ejpam-1373	151	15	429	429	NUM
ejpam-1373	151	16	however	however	ADV
ejpam-1373	151	17	,	,	PUNCT
ejpam-1373	151	18	the	the	DET
ejpam-1373	151	19	converse	converse	NOUN
ejpam-1373	151	20	of	of	ADP
ejpam-1373	151	21	theorem	theorem	NOUN
ejpam-1373	151	22	5	5	NUM
ejpam-1373	151	23	are	be	AUX
ejpam-1373	151	24	true	true	ADJ
ejpam-1373	151	25	if	if	SCONJ
ejpam-1373	151	26	the	the	DET
ejpam-1373	151	27	convolution	convolution	NOUN
ejpam-1373	151	28	satisfies	satisfy	VERB
ejpam-1373	151	29	certain	certain	ADJ
ejpam-1373	151	30	additional	additional	ADJ
ejpam-1373	151	31	conditions	condition	NOUN
ejpam-1373	151	32	.	.	PUNCT
ejpam-1373	152	1	4	4	X
ejpam-1373	152	2	.	.	X
ejpam-1373	152	3	multiplicative	multiplicative	ADJ
ejpam-1373	152	4	convolutions	convolution	NOUN
ejpam-1373	152	5	multiplicative	multiplicative	ADJ
ejpam-1373	152	6	convolutions	convolution	NOUN
ejpam-1373	152	7	are	be	AUX
ejpam-1373	152	8	of	of	ADP
ejpam-1373	152	9	special	special	ADJ
ejpam-1373	152	10	importance	importance	NOUN
ejpam-1373	152	11	,	,	PUNCT
ejpam-1373	152	12	for	for	ADP
ejpam-1373	152	13	the	the	DET
ejpam-1373	152	14	single	single	ADJ
ejpam-1373	152	15	reason	reason	NOUN
ejpam-1373	152	16	that	that	PRON
ejpam-1373	152	17	the	the	DET
ejpam-1373	152	18	partial	partial	ADJ
ejpam-1373	152	19	orders	order	NOUN
ejpam-1373	152	20	induced	induce	VERB
ejpam-1373	152	21	by	by	ADP
ejpam-1373	152	22	them	they	PRON
ejpam-1373	152	23	onz+	onz+	VERB
ejpam-1373	152	24	can	can	AUX
ejpam-1373	152	25	be	be	AUX
ejpam-1373	152	26	characterized	characterize	VERB
ejpam-1373	152	27	by	by	ADP
ejpam-1373	152	28	those	those	DET
ejpam-1373	152	29	onn	onn	NOUN
ejpam-1373	152	30	.	.	PUNCT
ejpam-1373	153	1	in	in	ADP
ejpam-1373	153	2	this	this	DET
ejpam-1373	153	3	section	section	NOUN
ejpam-1373	153	4	,	,	PUNCT
ejpam-1373	153	5	we	we	PRON
ejpam-1373	153	6	discuss	discuss	VERB
ejpam-1373	153	7	the	the	DET
ejpam-1373	153	8	order	order	NOUN
ejpam-1373	153	9	structures	structure	NOUN
ejpam-1373	153	10	on	on	ADP
ejpam-1373	153	11	z+	z+	NUM
ejpam-1373	153	12	induced	induce	VERB
ejpam-1373	153	13	by	by	ADP
ejpam-1373	153	14	multiplicative	multiplicative	ADJ
ejpam-1373	153	15	convolutions	convolution	NOUN
ejpam-1373	153	16	.	.	PUNCT
ejpam-1373	154	1	definition	definition	NOUN
ejpam-1373	154	2	5	5	NUM
ejpam-1373	154	3	.	.	PUNCT
ejpam-1373	155	1	a	a	DET
ejpam-1373	155	2	convolution	convolution	NOUN
ejpam-1373	155	3	c	c	NOUN
ejpam-1373	155	4	is	be	AUX
ejpam-1373	155	5	said	say	VERB
ejpam-1373	155	6	to	to	PART
ejpam-1373	155	7	be	be	AUX
ejpam-1373	155	8	multiplicative	multiplicative	ADJ
ejpam-1373	155	9	if	if	SCONJ
ejpam-1373	155	10	,	,	PUNCT
ejpam-1373	155	11	for	for	ADP
ejpam-1373	155	12	any	any	DET
ejpam-1373	155	13	relatively	relatively	ADV
ejpam-1373	155	14	prime	prime	ADJ
ejpam-1373	155	15	integers	integer	NOUN
ejpam-1373	155	16	m	m	VERB
ejpam-1373	155	17	and	and	CCONJ
ejpam-1373	155	18	n	n	CCONJ
ejpam-1373	155	19	,	,	PUNCT
ejpam-1373	155	20	c	c	PROPN
ejpam-1373	155	21	(	(	PUNCT
ejpam-1373	155	22	mn	mn	PROPN
ejpam-1373	155	23	)	)	PUNCT
ejpam-1373	156	1	=	=	SYM
ejpam-1373	156	2	c	c	X
ejpam-1373	156	3	(	(	PUNCT
ejpam-1373	156	4	m)c	m)c	X
ejpam-1373	156	5	(	(	PUNCT
ejpam-1373	156	6	n	n	CCONJ
ejpam-1373	156	7	)	)	PUNCT
ejpam-1373	156	8	:	:	PUNCT
ejpam-1373	156	9	=	=	SYM
ejpam-1373	156	10	{	{	PUNCT
ejpam-1373	156	11	ab|a	ab|a	NOUN
ejpam-1373	156	12	∈	∈	PROPN
ejpam-1373	156	13	c	c	PROPN
ejpam-1373	156	14	(	(	PUNCT
ejpam-1373	156	15	m	m	NOUN
ejpam-1373	156	16	)	)	PUNCT
ejpam-1373	156	17	and	and	CCONJ
ejpam-1373	156	18	b	b	X
ejpam-1373	156	19	∈	∈	PROPN
ejpam-1373	156	20	c	c	X
ejpam-1373	156	21	(	(	PUNCT
ejpam-1373	156	22	n	n	CCONJ
ejpam-1373	156	23	)	)	PUNCT
ejpam-1373	156	24	}	}	PUNCT
ejpam-1373	156	25	.	.	PUNCT
ejpam-1373	157	1	it	it	PRON
ejpam-1373	157	2	can	can	AUX
ejpam-1373	157	3	be	be	AUX
ejpam-1373	157	4	verified	verify	VERB
ejpam-1373	157	5	that	that	SCONJ
ejpam-1373	157	6	a	a	DET
ejpam-1373	157	7	convolution	convolution	NOUN
ejpam-1373	157	8	c	c	NOUN
ejpam-1373	157	9	is	be	AUX
ejpam-1373	157	10	multiplicative	multiplicative	ADJ
ejpam-1373	157	11	if	if	SCONJ
ejpam-1373	157	12	and	and	CCONJ
ejpam-1373	157	13	only	only	ADV
ejpam-1373	157	14	if	if	SCONJ
ejpam-1373	157	15	,	,	PUNCT
ejpam-1373	157	16	for	for	ADP
ejpam-1373	157	17	any	any	DET
ejpam-1373	157	18	distinct	distinct	ADJ
ejpam-1373	157	19	primes	prime	NOUN
ejpam-1373	157	20	p1	p1	NOUN
ejpam-1373	157	21	,	,	PUNCT
ejpam-1373	157	22	p2	p2	NOUN
ejpam-1373	157	23	,	,	PUNCT
ejpam-1373	157	24	·	·	PUNCT
ejpam-1373	157	25	·	·	PUNCT
ejpam-1373	157	26	·	·	PUNCT
ejpam-1373	157	27	,	,	PUNCT
ejpam-1373	157	28	pr	pr	NOUN
ejpam-1373	157	29	and	and	CCONJ
ejpam-1373	157	30	non	non	ADJ
ejpam-1373	157	31	-	-	ADJ
ejpam-1373	157	32	negative	negative	ADJ
ejpam-1373	157	33	integers	integer	NOUN
ejpam-1373	157	34	a1	a1	NOUN
ejpam-1373	157	35	,	,	PUNCT
ejpam-1373	157	36	a2	a2	PROPN
ejpam-1373	157	37	,	,	PUNCT
ejpam-1373	157	38	·	·	PUNCT
ejpam-1373	157	39	·	·	PUNCT
ejpam-1373	157	40	·	·	PUNCT
ejpam-1373	157	41	,	,	PUNCT
ejpam-1373	157	42	ar	ar	PROPN
ejpam-1373	157	43	,	,	PUNCT
ejpam-1373	157	44	c	c	PROPN
ejpam-1373	158	1	(	(	PUNCT
ejpam-1373	158	2	r	r	NOUN
ejpam-1373	158	3	∏	∏	PROPN
ejpam-1373	158	4	i=1	i=1	X
ejpam-1373	159	1	p	p	X
ejpam-1373	159	2	ai	ai	VERB
ejpam-1373	159	3	i	i	PRON
ejpam-1373	159	4	)	)	PUNCT
ejpam-1373	160	1	=	=	PUNCT
ejpam-1373	160	2	r	r	NOUN
ejpam-1373	160	3	∏	∏	NUM
ejpam-1373	160	4	i=1	i=1	X
ejpam-1373	160	5	c	c	PROPN
ejpam-1373	160	6	(	(	PUNCT
ejpam-1373	160	7	pai	pai	PROPN
ejpam-1373	160	8	i	i	PROPN
ejpam-1373	160	9	)	)	PUNCT
ejpam-1373	160	10	:	:	PUNCT
ejpam-1373	160	11	=	=	SYM
ejpam-1373	160	12	{	{	PUNCT
ejpam-1373	160	13	m1m2	m1m2	X
ejpam-1373	160	14	·	·	PUNCT
ejpam-1373	160	15	·	·	PUNCT
ejpam-1373	160	16	·	·	PUNCT
ejpam-1373	160	17	mr	mr	PROPN
ejpam-1373	160	18	|mi	|mi	PROPN
ejpam-1373	160	19	∈	∈	PROPN
ejpam-1373	160	20	c	c	X
ejpam-1373	160	21	(	(	PUNCT
ejpam-1373	160	22	p	p	X
ejpam-1373	160	23	ai	ai	VERB
ejpam-1373	160	24	i	i	PRON
ejpam-1373	160	25	)	)	PUNCT
ejpam-1373	160	26	}	}	PUNCT
ejpam-1373	160	27	.	.	PUNCT
ejpam-1373	161	1	multiplicative	multiplicative	ADJ
ejpam-1373	161	2	convolutions	convolution	NOUN
ejpam-1373	161	3	can	can	AUX
ejpam-1373	161	4	be	be	AUX
ejpam-1373	161	5	characterized	characterize	VERB
ejpam-1373	161	6	in	in	ADP
ejpam-1373	161	7	terms	term	NOUN
ejpam-1373	161	8	of	of	ADP
ejpam-1373	161	9	the	the	DET
ejpam-1373	161	10	orders	order	NOUN
ejpam-1373	161	11	induced	induce	VERB
ejpam-1373	161	12	by	by	ADP
ejpam-1373	161	13	them	they	PRON
ejpam-1373	161	14	on	on	ADP
ejpam-1373	161	15	z+	z+	NUM
ejpam-1373	161	16	and	and	CCONJ
ejpam-1373	161	17	n	n	CCONJ
ejpam-1373	161	18	,	,	PUNCT
ejpam-1373	161	19	as	as	SCONJ
ejpam-1373	161	20	given	give	VERB
ejpam-1373	161	21	in	in	ADP
ejpam-1373	161	22	the	the	DET
ejpam-1373	161	23	following	following	NOUN
ejpam-1373	161	24	whose	whose	DET
ejpam-1373	161	25	proof	proof	NOUN
ejpam-1373	161	26	is	be	AUX
ejpam-1373	161	27	a	a	DET
ejpam-1373	161	28	straight	straight	ADJ
ejpam-1373	161	29	forward	forward	ADJ
ejpam-1373	161	30	verification	verification	NOUN
ejpam-1373	161	31	.	.	PUNCT
ejpam-1373	162	1	theorem	theorem	VERB
ejpam-1373	162	2	6	6	NUM
ejpam-1373	162	3	.	.	PUNCT
ejpam-1373	163	1	let	let	VERB
ejpam-1373	163	2	c	c	PRON
ejpam-1373	163	3	be	be	AUX
ejpam-1373	163	4	a	a	DET
ejpam-1373	163	5	convolution	convolution	NOUN
ejpam-1373	163	6	and	and	CCONJ
ejpam-1373	163	7	≤c	≤c	PROPN
ejpam-1373	163	8	and	and	CCONJ
ejpam-1373	163	9	≤p	≤p	PROPN
ejpam-1373	163	10	c	c	PROPN
ejpam-1373	163	11	be	be	AUX
ejpam-1373	163	12	the	the	DET
ejpam-1373	163	13	partial	partial	ADJ
ejpam-1373	163	14	orders	order	NOUN
ejpam-1373	163	15	induced	induce	VERB
ejpam-1373	163	16	by	by	ADP
ejpam-1373	163	17	c	c	PROPN
ejpam-1373	163	18	on	on	ADP
ejpam-1373	163	19	z+	z+	NUM
ejpam-1373	163	20	and	and	CCONJ
ejpam-1373	163	21	n	n	PRON
ejpam-1373	163	22	respectively	respectively	ADV
ejpam-1373	163	23	,	,	PUNCT
ejpam-1373	163	24	for	for	SCONJ
ejpam-1373	163	25	each	each	DET
ejpam-1373	163	26	prime	prime	ADJ
ejpam-1373	163	27	p.	p.	NOUN
ejpam-1373	163	28	let	let	VERB
ejpam-1373	163	29	∑	∑	PROPN
ejpam-1373	163	30	p	p	NOUN
ejpam-1373	163	31	n	n	NOUN
ejpam-1373	163	32	=	=	PRON
ejpam-1373	163	33	{	{	PUNCT
ejpam-1373	163	34	f	f	NOUN
ejpam-1373	163	35	:	:	PUNCT
ejpam-1373	164	1	p	p	X
ejpam-1373	164	2	−→n	−→n	VERB
ejpam-1373	164	3	|	|	NOUN
ejpam-1373	164	4	f	f	X
ejpam-1373	164	5	(	(	PUNCT
ejpam-1373	164	6	p	p	NOUN
ejpam-1373	164	7	)	)	PUNCT
ejpam-1373	164	8	=	=	SYM
ejpam-1373	164	9	0	0	NUM
ejpam-1373	164	10	for	for	ADP
ejpam-1373	164	11	all	all	DET
ejpam-1373	164	12	but	but	CCONJ
ejpam-1373	164	13	finite	finite	ADJ
ejpam-1373	164	14	number	number	NOUN
ejpam-1373	164	15	of	of	ADP
ejpam-1373	164	16	p′s	p′	NOUN
ejpam-1373	164	17	}	}	PUNCT
ejpam-1373	164	18	where	where	SCONJ
ejpam-1373	164	19	p	p	NOUN
ejpam-1373	164	20	is	be	AUX
ejpam-1373	164	21	the	the	DET
ejpam-1373	164	22	set	set	NOUN
ejpam-1373	164	23	of	of	ADP
ejpam-1373	164	24	all	all	DET
ejpam-1373	164	25	primes	prime	NOUN
ejpam-1373	164	26	.	.	PUNCT
ejpam-1373	165	1	define	define	VERB
ejpam-1373	165	2	θ	θ	PROPN
ejpam-1373	165	3	:	:	PUNCT
ejpam-1373	165	4	z+	z+	PUNCT
ejpam-1373	165	5	−→	−→	NOUN
ejpam-1373	165	6	∑	∑	ADP
ejpam-1373	165	7	p	p	NOUN
ejpam-1373	165	8	n	n	X
ejpam-1373	165	9	by	by	ADP
ejpam-1373	165	10	θ(n)(p	θ(n)(p	NOUN
ejpam-1373	165	11	)	)	PUNCT
ejpam-1373	165	12	=	=	SYM
ejpam-1373	165	13	a	a	NOUN
ejpam-1373	165	14	,	,	PUNCT
ejpam-1373	165	15	where	where	SCONJ
ejpam-1373	165	16	a	a	PRON
ejpam-1373	165	17	is	be	AUX
ejpam-1373	165	18	the	the	DET
ejpam-1373	165	19	largest	large	ADJ
ejpam-1373	165	20	inn	inn	NOUN
ejpam-1373	165	21	such	such	ADJ
ejpam-1373	165	22	that	that	SCONJ
ejpam-1373	165	23	pa	pa	PROPN
ejpam-1373	165	24	divides	divide	VERB
ejpam-1373	165	25	n.	n.	PROPN
ejpam-1373	165	26	then	then	ADV
ejpam-1373	165	27	θ	θ	PROPN
ejpam-1373	165	28	is	be	AUX
ejpam-1373	165	29	a	a	DET
ejpam-1373	165	30	bijection	bijection	NOUN
ejpam-1373	165	31	.	.	PUNCT
ejpam-1373	166	1	further	far	ADV
ejpam-1373	166	2	the	the	DET
ejpam-1373	166	3	convolution	convolution	NOUN
ejpam-1373	166	4	c	c	NOUN
ejpam-1373	166	5	is	be	AUX
ejpam-1373	166	6	multiplicative	multiplicative	ADJ
ejpam-1373	166	7	if	if	SCONJ
ejpam-1373	166	8	and	and	CCONJ
ejpam-1373	166	9	only	only	ADV
ejpam-1373	166	10	if	if	SCONJ
ejpam-1373	166	11	α	α	PRON
ejpam-1373	166	12	is	be	AUX
ejpam-1373	166	13	an	an	DET
ejpam-1373	166	14	order	order	NOUN
ejpam-1373	166	15	isomorphism	isomorphism	NOUN
ejpam-1373	166	16	of	of	ADP
ejpam-1373	166	17	(	(	PUNCT
ejpam-1373	166	18	z+,≤c	z+,≤c	PROPN
ejpam-1373	166	19	)	)	PUNCT
ejpam-1373	166	20	onto	onto	ADP
ejpam-1373	166	21	(	(	PUNCT
ejpam-1373	166	22	∑	∑	PROPN
ejpam-1373	166	23	p	p	NOUN
ejpam-1373	166	24	n	n	NOUN
ejpam-1373	166	25	,	,	PUNCT
ejpam-1373	166	26	≤c	≤c	PROPN
ejpam-1373	166	27	)	)	PUNCT
ejpam-1373	166	28	,	,	PUNCT
ejpam-1373	166	29	where	where	SCONJ
ejpam-1373	166	30	≤c	≤c	PROPN
ejpam-1373	166	31	also	also	ADV
ejpam-1373	166	32	denotes	denote	VERB
ejpam-1373	166	33	the	the	DET
ejpam-1373	166	34	point	point	ADV
ejpam-1373	166	35	-	-	PUNCT
ejpam-1373	166	36	wise	wise	ADJ
ejpam-1373	166	37	order	order	NOUN
ejpam-1373	166	38	on	on	ADP
ejpam-1373	166	39	∑	∑	PROPN
ejpam-1373	166	40	p	p	NOUN
ejpam-1373	166	41	n	n	PRON
ejpam-1373	166	42	defined	define	VERB
ejpam-1373	166	43	by	by	ADP
ejpam-1373	166	44	f	f	PROPN
ejpam-1373	166	45	≤c	≤c	PROPN
ejpam-1373	166	46	g	g	PROPN
ejpam-1373	166	47	⇐	⇐	ADJ
ejpam-1373	166	48	⇒	⇒	PROPN
ejpam-1373	166	49	f	f	PROPN
ejpam-1373	166	50	(	(	PUNCT
ejpam-1373	166	51	p)≤p	p)≤p	PROPN
ejpam-1373	166	52	c	c	PROPN
ejpam-1373	166	53	g(p	g(p	PROPN
ejpam-1373	166	54	)	)	PUNCT
ejpam-1373	166	55	for	for	ADP
ejpam-1373	166	56	all	all	DET
ejpam-1373	166	57	p	p	PROPN
ejpam-1373	166	58	∈	∈	PROPN
ejpam-1373	166	59	p.	p.	NOUN
ejpam-1373	166	60	corollary	corollary	NOUN
ejpam-1373	167	1	1	1	NUM
ejpam-1373	167	2	.	.	PUNCT
ejpam-1373	168	1	let	let	VERB
ejpam-1373	168	2	c	c	PRON
ejpam-1373	168	3	be	be	AUX
ejpam-1373	168	4	a	a	DET
ejpam-1373	168	5	multiplicative	multiplicative	ADJ
ejpam-1373	168	6	convolution	convolution	NOUN
ejpam-1373	168	7	.	.	PUNCT
ejpam-1373	169	1	then	then	ADV
ejpam-1373	169	2	(	(	PUNCT
ejpam-1373	169	3	n	n	X
ejpam-1373	169	4	,	,	PUNCT
ejpam-1373	169	5	≤p	≤p	PROPN
ejpam-1373	169	6	c	c	PROPN
ejpam-1373	169	7	)	)	PUNCT
ejpam-1373	169	8	is	be	AUX
ejpam-1373	169	9	a	a	DET
ejpam-1373	169	10	meet(join	meet(join	NOUN
ejpam-1373	169	11	)	)	PUNCT
ejpam-1373	169	12	semilattice	semilattice	NOUN
ejpam-1373	169	13	for	for	ADP
ejpam-1373	169	14	each	each	DET
ejpam-1373	169	15	prime	prime	NOUN
ejpam-1373	169	16	p	p	NOUN
ejpam-1373	169	17	if	if	SCONJ
ejpam-1373	170	1	and	and	CCONJ
ejpam-1373	170	2	only	only	ADV
ejpam-1373	170	3	if	if	SCONJ
ejpam-1373	170	4	(	(	PUNCT
ejpam-1373	170	5	z+,≤c	z+,≤c	NUM
ejpam-1373	170	6	)	)	PUNCT
ejpam-1373	170	7	is	be	AUX
ejpam-1373	170	8	a	a	DET
ejpam-1373	170	9	meet	meet	NOUN
ejpam-1373	170	10	(	(	PUNCT
ejpam-1373	170	11	respectively	respectively	ADV
ejpam-1373	170	12	join	join	NOUN
ejpam-1373	170	13	)	)	PUNCT
ejpam-1373	170	14	semilattice	semilattice	NOUN
ejpam-1373	170	15	.	.	PUNCT
ejpam-1373	171	1	corollary	corollary	ADJ
ejpam-1373	171	2	2	2	NUM
ejpam-1373	171	3	.	.	PUNCT
ejpam-1373	172	1	for	for	ADP
ejpam-1373	172	2	any	any	DET
ejpam-1373	172	3	multiplicative	multiplicative	ADJ
ejpam-1373	172	4	convolution	convolution	NOUN
ejpam-1373	172	5	c	c	PROPN
ejpam-1373	172	6	,	,	PUNCT
ejpam-1373	172	7	(	(	PUNCT
ejpam-1373	172	8	z+,≤c	z+,≤c	X
ejpam-1373	172	9	)	)	PUNCT
ejpam-1373	172	10	is	be	AUX
ejpam-1373	172	11	a	a	DET
ejpam-1373	172	12	lattice	lattice	NOUN
ejpam-1373	172	13	if	if	SCONJ
ejpam-1373	173	1	and	and	CCONJ
ejpam-1373	173	2	only	only	ADV
ejpam-1373	173	3	if	if	SCONJ
ejpam-1373	173	4	(	(	PUNCT
ejpam-1373	173	5	n	n	X
ejpam-1373	173	6	,	,	PUNCT
ejpam-1373	173	7	≤p	≤p	PROPN
ejpam-1373	173	8	c	c	PROPN
ejpam-1373	173	9	)	)	PUNCT
ejpam-1373	173	10	is	be	AUX
ejpam-1373	173	11	a	a	DET
ejpam-1373	173	12	lattice	lattice	NOUN
ejpam-1373	173	13	for	for	ADP
ejpam-1373	173	14	each	each	DET
ejpam-1373	173	15	prime	prime	ADJ
ejpam-1373	173	16	p.	p.	PROPN
ejpam-1373	173	17	u.	u.	PROPN
ejpam-1373	173	18	swamy	swamy	PROPN
ejpam-1373	173	19	and	and	CCONJ
ejpam-1373	173	20	s.	s.	PROPN
ejpam-1373	173	21	sankar	sankar	PROPN
ejpam-1373	173	22	/	/	SYM
ejpam-1373	173	23	eur	eur	PROPN
ejpam-1373	173	24	.	.	PUNCT
ejpam-1373	174	1	j.	j.	PROPN
ejpam-1373	174	2	pure	pure	PROPN
ejpam-1373	174	3	appl	appl	PROPN
ejpam-1373	174	4	.	.	PROPN
ejpam-1373	174	5	math	math	PROPN
ejpam-1373	174	6	,	,	PUNCT
ejpam-1373	174	7	4	4	NUM
ejpam-1373	174	8	(	(	PUNCT
ejpam-1373	174	9	2011	2011	NUM
ejpam-1373	174	10	)	)	PUNCT
ejpam-1373	174	11	,	,	PUNCT
ejpam-1373	174	12	424	424	NUM
ejpam-1373	174	13	-	-	SYM
ejpam-1373	174	14	434	434	NUM
ejpam-1373	174	15	430	430	NUM
ejpam-1373	174	16	example	example	NOUN
ejpam-1373	174	17	2	2	NUM
ejpam-1373	174	18	.	.	PUNCT
ejpam-1373	174	19	(	(	PUNCT
ejpam-1373	174	20	1	1	NUM
ejpam-1373	174	21	)	)	PUNCT
ejpam-1373	174	22	.	.	PUNCT
ejpam-1373	175	1	let	let	VERB
ejpam-1373	175	2	d	d	PRON
ejpam-1373	175	3	be	be	AUX
ejpam-1373	175	4	the	the	DET
ejpam-1373	175	5	dirichlet	dirichlet	PROPN
ejpam-1373	175	6	’s	’s	PART
ejpam-1373	175	7	convolution	convolution	NOUN
ejpam-1373	175	8	defined	define	VERB
ejpam-1373	175	9	by	by	ADP
ejpam-1373	175	10	d(n	d(n	NOUN
ejpam-1373	175	11	)	)	PUNCT
ejpam-1373	175	12	=	=	SYM
ejpam-1373	176	1	the	the	DET
ejpam-1373	176	2	set	set	NOUN
ejpam-1373	176	3	of	of	ADP
ejpam-1373	176	4	all	all	DET
ejpam-1373	176	5	positive	positive	ADJ
ejpam-1373	176	6	divisors	divisor	NOUN
ejpam-1373	176	7	of	of	ADP
ejpam-1373	176	8	n.	n.	NOUN
ejpam-1373	176	9	then	then	ADV
ejpam-1373	176	10	d	d	PROPN
ejpam-1373	176	11	is	be	AUX
ejpam-1373	176	12	multiplicative	multiplicative	ADJ
ejpam-1373	176	13	.	.	PUNCT
ejpam-1373	177	1	≤d	≤d	NOUN
ejpam-1373	177	2	is	be	AUX
ejpam-1373	177	3	precisely	precisely	ADV
ejpam-1373	177	4	the	the	DET
ejpam-1373	177	5	division	division	NOUN
ejpam-1373	177	6	order	order	NOUN
ejpam-1373	177	7	on	on	ADP
ejpam-1373	177	8	z+	z+	NUM
ejpam-1373	177	9	and	and	CCONJ
ejpam-1373	177	10	,	,	PUNCT
ejpam-1373	177	11	for	for	ADP
ejpam-1373	177	12	each	each	DET
ejpam-1373	177	13	prime	prime	NOUN
ejpam-1373	177	14	p	p	NOUN
ejpam-1373	177	15	,	,	PUNCT
ejpam-1373	177	16	≤p	≤p	PROPN
ejpam-1373	177	17	d	d	PROPN
ejpam-1373	177	18	is	be	AUX
ejpam-1373	177	19	the	the	DET
ejpam-1373	177	20	usual	usual	ADJ
ejpam-1373	177	21	order	order	NOUN
ejpam-1373	177	22	on	on	ADP
ejpam-1373	177	23	n	n	PROPN
ejpam-1373	177	24	.	.	PUNCT
ejpam-1373	178	1	(	(	PUNCT
ejpam-1373	178	2	z+,≤d	z+,≤d	NUM
ejpam-1373	178	3	)	)	PUNCT
ejpam-1373	178	4	is	be	AUX
ejpam-1373	178	5	known	know	VERB
ejpam-1373	178	6	to	to	PART
ejpam-1373	178	7	be	be	AUX
ejpam-1373	178	8	distributive	distributive	ADJ
ejpam-1373	178	9	lattice	lattice	NOUN
ejpam-1373	178	10	.	.	PUNCT
ejpam-1373	179	1	(	(	PUNCT
ejpam-1373	179	2	2	2	NUM
ejpam-1373	179	3	)	)	PUNCT
ejpam-1373	179	4	.	.	PUNCT
ejpam-1373	180	1	let	let	VERB
ejpam-1373	180	2	u(n	u(n	PROPN
ejpam-1373	180	3	)	)	PUNCT
ejpam-1373	180	4	be	be	AUX
ejpam-1373	180	5	the	the	DET
ejpam-1373	180	6	unitary	unitary	ADJ
ejpam-1373	180	7	convolution	convolution	NOUN
ejpam-1373	180	8	defined	define	VERB
ejpam-1373	180	9	by	by	ADP
ejpam-1373	180	10	u(n	u(n	NOUN
ejpam-1373	180	11	)	)	PUNCT
ejpam-1373	180	12	=	=	PRON
ejpam-1373	181	1	{	{	PUNCT
ejpam-1373	181	2	d	d	PUNCT
ejpam-1373	181	3	∈	∈	PROPN
ejpam-1373	181	4	d(n)|d	d(n)|d	NOUN
ejpam-1373	181	5	and	and	CCONJ
ejpam-1373	181	6	n	n	CCONJ
ejpam-1373	181	7	d	d	NOUN
ejpam-1373	181	8	are	be	AUX
ejpam-1373	181	9	relatively	relatively	ADV
ejpam-1373	181	10	prime	prime	ADJ
ejpam-1373	181	11	}	}	PUNCT
ejpam-1373	181	12	.	.	PUNCT
ejpam-1373	182	1	then	then	ADV
ejpam-1373	182	2	u	u	PRON
ejpam-1373	182	3	is	be	AUX
ejpam-1373	182	4	multiplicative	multiplicative	ADJ
ejpam-1373	182	5	and	and	CCONJ
ejpam-1373	182	6	(	(	PUNCT
ejpam-1373	182	7	z+,≤u	z+,≤u	NUM
ejpam-1373	182	8	)	)	PUNCT
ejpam-1373	182	9	is	be	AUX
ejpam-1373	182	10	a	a	DET
ejpam-1373	182	11	meet	meet	ADJ
ejpam-1373	182	12	semilattice	semilattice	NOUN
ejpam-1373	182	13	,	,	PUNCT
ejpam-1373	182	14	but	but	CCONJ
ejpam-1373	182	15	not	not	PART
ejpam-1373	182	16	a	a	DET
ejpam-1373	182	17	join	join	NOUN
ejpam-1373	182	18	semilattice	semilattice	NOUN
ejpam-1373	182	19	.	.	PUNCT
ejpam-1373	183	1	(	(	PUNCT
ejpam-1373	183	2	3	3	NUM
ejpam-1373	183	3	)	)	PUNCT
ejpam-1373	183	4	.	.	PUNCT
ejpam-1373	184	1	let	let	VERB
ejpam-1373	184	2	f2	f2	PROPN
ejpam-1373	184	3	be	be	AUX
ejpam-1373	184	4	the	the	DET
ejpam-1373	184	5	square	square	ADJ
ejpam-1373	184	6	-	-	PUNCT
ejpam-1373	184	7	free	free	ADJ
ejpam-1373	184	8	convolution	convolution	NOUN
ejpam-1373	184	9	defined	define	VERB
ejpam-1373	184	10	by	by	ADP
ejpam-1373	184	11	f2(n	f2(n	NOUN
ejpam-1373	184	12	)	)	PUNCT
ejpam-1373	184	13	=	=	SYM
ejpam-1373	184	14	{	{	PUNCT
ejpam-1373	184	15	n	n	CCONJ
ejpam-1373	184	16	}	}	PUNCT
ejpam-1373	184	17	∪	∪	X
ejpam-1373	184	18	{	{	PUNCT
ejpam-1373	184	19	d	d	PROPN
ejpam-1373	184	20	∈	∈	PROPN
ejpam-1373	184	21	d(n)|p2	d(n)|p2	NOUN
ejpam-1373	184	22	does	do	AUX
ejpam-1373	184	23	not	not	PART
ejpam-1373	184	24	divide	divide	VERB
ejpam-1373	184	25	n	n	ADP
ejpam-1373	184	26	for	for	ADP
ejpam-1373	184	27	any	any	DET
ejpam-1373	184	28	prime	prime	NOUN
ejpam-1373	184	29	p	p	NOUN
ejpam-1373	184	30	}	}	PUNCT
ejpam-1373	184	31	.	.	PUNCT
ejpam-1373	185	1	then	then	ADV
ejpam-1373	185	2	f2	f2	PROPN
ejpam-1373	185	3	is	be	AUX
ejpam-1373	185	4	a	a	DET
ejpam-1373	185	5	multiplicative	multiplicative	ADJ
ejpam-1373	185	6	convolution	convolution	NOUN
ejpam-1373	185	7	and	and	CCONJ
ejpam-1373	185	8	(	(	PUNCT
ejpam-1373	185	9	z+,≤f2	z+,≤f2	PROPN
ejpam-1373	185	10	)	)	PUNCT
ejpam-1373	185	11	is	be	AUX
ejpam-1373	185	12	a	a	DET
ejpam-1373	185	13	meet	meet	ADJ
ejpam-1373	185	14	semilattice	semilattice	NOUN
ejpam-1373	185	15	but	but	CCONJ
ejpam-1373	185	16	not	not	PART
ejpam-1373	185	17	a	a	DET
ejpam-1373	185	18	join	join	NOUN
ejpam-1373	185	19	semilattice	semilattice	NOUN
ejpam-1373	185	20	.	.	PUNCT
ejpam-1373	186	1	(	(	PUNCT
ejpam-1373	186	2	4	4	NUM
ejpam-1373	186	3	)	)	PUNCT
ejpam-1373	186	4	.	.	PUNCT
ejpam-1373	187	1	for	for	ADP
ejpam-1373	187	2	any	any	DET
ejpam-1373	187	3	k	k	PROPN
ejpam-1373	187	4	∈	∈	PROPN
ejpam-1373	187	5	z+	z+	PRON
ejpam-1373	187	6	,	,	PUNCT
ejpam-1373	187	7	a	a	DET
ejpam-1373	187	8	positive	positive	ADJ
ejpam-1373	187	9	integer	integer	NOUN
ejpam-1373	187	10	d	d	NOUN
ejpam-1373	187	11	is	be	AUX
ejpam-1373	187	12	said	say	VERB
ejpam-1373	187	13	to	to	PART
ejpam-1373	187	14	be	be	AUX
ejpam-1373	187	15	k	k	NOUN
ejpam-1373	187	16	-	-	ADJ
ejpam-1373	187	17	free	free	ADJ
ejpam-1373	187	18	if	if	SCONJ
ejpam-1373	187	19	pk	pk	NOUN
ejpam-1373	187	20	does	do	AUX
ejpam-1373	187	21	not	not	PART
ejpam-1373	187	22	divide	divide	VERB
ejpam-1373	187	23	d	d	NOUN
ejpam-1373	187	24	for	for	ADP
ejpam-1373	187	25	any	any	DET
ejpam-1373	187	26	prime	prime	ADJ
ejpam-1373	187	27	p.	p.	NOUN
ejpam-1373	187	28	let	let	VERB
ejpam-1373	187	29	fk(n	fk(n	NOUN
ejpam-1373	187	30	)	)	PUNCT
ejpam-1373	187	31	be	be	AUX
ejpam-1373	187	32	the	the	DET
ejpam-1373	187	33	set	set	NOUN
ejpam-1373	187	34	of	of	ADP
ejpam-1373	187	35	all	all	DET
ejpam-1373	187	36	k	k	ADJ
ejpam-1373	187	37	-	-	ADJ
ejpam-1373	187	38	free	free	ADJ
ejpam-1373	187	39	divisors	divisor	NOUN
ejpam-1373	187	40	of	of	ADP
ejpam-1373	187	41	n	n	CCONJ
ejpam-1373	187	42	together	together	ADV
ejpam-1373	187	43	with	with	ADP
ejpam-1373	187	44	n.	n.	PROPN
ejpam-1373	187	45	then	then	ADV
ejpam-1373	187	46	fk	fk	INTJ
ejpam-1373	187	47	is	be	AUX
ejpam-1373	187	48	a	a	DET
ejpam-1373	187	49	multiplicative	multiplicative	ADJ
ejpam-1373	187	50	convolution	convolution	NOUN
ejpam-1373	187	51	and	and	CCONJ
ejpam-1373	187	52	(	(	PUNCT
ejpam-1373	187	53	z+,≤fk	z+,≤fk	PROPN
ejpam-1373	187	54	)	)	PUNCT
ejpam-1373	187	55	is	be	AUX
ejpam-1373	187	56	a	a	DET
ejpam-1373	187	57	meet	meet	ADJ
ejpam-1373	187	58	semilattice	semilattice	NOUN
ejpam-1373	187	59	but	but	CCONJ
ejpam-1373	187	60	not	not	PART
ejpam-1373	187	61	a	a	DET
ejpam-1373	187	62	join	join	NOUN
ejpam-1373	187	63	semi	semi	NOUN
ejpam-1373	187	64	lattice	lattice	PROPN
ejpam-1373	187	65	.	.	PUNCT
ejpam-1373	188	1	in	in	ADP
ejpam-1373	188	2	our	our	PRON
ejpam-1373	188	3	further	further	ADJ
ejpam-1373	188	4	discussions	discussion	NOUN
ejpam-1373	188	5	,	,	PUNCT
ejpam-1373	188	6	we	we	PRON
ejpam-1373	188	7	assume	assume	VERB
ejpam-1373	188	8	that	that	SCONJ
ejpam-1373	188	9	a	a	DET
ejpam-1373	188	10	convolution	convolution	NOUN
ejpam-1373	188	11	c	c	NOUN
ejpam-1373	188	12	satisfies	satisfy	VERB
ejpam-1373	188	13	the	the	DET
ejpam-1373	188	14	additional	additional	ADJ
ejpam-1373	188	15	property	property	NOUN
ejpam-1373	188	16	that	that	SCONJ
ejpam-1373	188	17	1	1	NUM
ejpam-1373	188	18	∈	∈	PROPN
ejpam-1373	188	19	c	c	X
ejpam-1373	188	20	(	(	PUNCT
ejpam-1373	188	21	n	n	CCONJ
ejpam-1373	188	22	)	)	PUNCT
ejpam-1373	188	23	for	for	ADP
ejpam-1373	188	24	all	all	DET
ejpam-1373	188	25	n	n	PRON
ejpam-1373	188	26	∈	∈	NOUN
ejpam-1373	188	27	z+	z+	PUNCT
ejpam-1373	188	28	.	.	PUNCT
ejpam-1373	189	1	note	note	VERB
ejpam-1373	189	2	that	that	SCONJ
ejpam-1373	189	3	this	this	PRON
ejpam-1373	189	4	is	be	AUX
ejpam-1373	189	5	equivalent	equivalent	ADJ
ejpam-1373	189	6	to	to	ADP
ejpam-1373	189	7	saying	say	VERB
ejpam-1373	189	8	that	that	SCONJ
ejpam-1373	189	9	(	(	PUNCT
ejpam-1373	189	10	z+,≤c	z+,≤c	NUM
ejpam-1373	189	11	)	)	PUNCT
ejpam-1373	189	12	has	have	AUX
ejpam-1373	189	13	least	least	ADJ
ejpam-1373	189	14	element	element	ADJ
ejpam-1373	189	15	and	and	CCONJ
ejpam-1373	189	16	that	that	SCONJ
ejpam-1373	189	17	this	this	PRON
ejpam-1373	189	18	is	be	AUX
ejpam-1373	189	19	further	far	ADV
ejpam-1373	189	20	equivalent	equivalent	ADJ
ejpam-1373	189	21	to	to	ADP
ejpam-1373	189	22	saying	say	VERB
ejpam-1373	189	23	(	(	PUNCT
ejpam-1373	189	24	z+,≤c	z+,≤c	NUM
ejpam-1373	189	25	)	)	PUNCT
ejpam-1373	189	26	is	be	AUX
ejpam-1373	189	27	directed	direct	VERB
ejpam-1373	189	28	below	below	ADV
ejpam-1373	189	29	.	.	PUNCT
ejpam-1373	190	1	by	by	ADP
ejpam-1373	190	2	assuming	assume	VERB
ejpam-1373	190	3	that	that	SCONJ
ejpam-1373	190	4	1	1	NUM
ejpam-1373	190	5	∈	∈	PROPN
ejpam-1373	190	6	c	c	X
ejpam-1373	190	7	(	(	PUNCT
ejpam-1373	190	8	n	n	CCONJ
ejpam-1373	190	9	)	)	PUNCT
ejpam-1373	190	10	,	,	PUNCT
ejpam-1373	190	11	we	we	PRON
ejpam-1373	190	12	are	be	AUX
ejpam-1373	190	13	not	not	PART
ejpam-1373	190	14	losing	lose	VERB
ejpam-1373	190	15	any	any	DET
ejpam-1373	190	16	generality	generality	NOUN
ejpam-1373	190	17	,	,	PUNCT
ejpam-1373	190	18	since	since	SCONJ
ejpam-1373	190	19	we	we	PRON
ejpam-1373	190	20	are	be	AUX
ejpam-1373	190	21	interested	interested	ADJ
ejpam-1373	190	22	in	in	ADP
ejpam-1373	190	23	convolutions	convolution	NOUN
ejpam-1373	190	24	c	c	NOUN
ejpam-1373	190	25	with	with	ADP
ejpam-1373	190	26	respect	respect	NOUN
ejpam-1373	190	27	to	to	ADP
ejpam-1373	190	28	which	which	PRON
ejpam-1373	190	29	(	(	PUNCT
ejpam-1373	190	30	z+,≤c	z+,≤c	NUM
ejpam-1373	190	31	)	)	PUNCT
ejpam-1373	190	32	is	be	AUX
ejpam-1373	190	33	a	a	DET
ejpam-1373	190	34	meet	meet	ADJ
ejpam-1373	190	35	semilattice	semilattice	NOUN
ejpam-1373	190	36	.	.	PUNCT
ejpam-1373	191	1	theorem	theorem	VERB
ejpam-1373	191	2	7	7	NUM
ejpam-1373	191	3	.	.	PUNCT
ejpam-1373	192	1	a	a	DET
ejpam-1373	192	2	convolution	convolution	NOUN
ejpam-1373	192	3	c	c	NOUN
ejpam-1373	192	4	is	be	AUX
ejpam-1373	192	5	multiplicative	multiplicative	ADJ
ejpam-1373	192	6	if	if	SCONJ
ejpam-1373	192	7	and	and	CCONJ
ejpam-1373	192	8	only	only	ADV
ejpam-1373	192	9	if	if	SCONJ
ejpam-1373	192	10	the	the	DET
ejpam-1373	192	11	following	follow	VERB
ejpam-1373	192	12	conditions	condition	NOUN
ejpam-1373	192	13	are	be	AUX
ejpam-1373	192	14	satisfied	satisfied	ADJ
ejpam-1373	192	15	for	for	ADP
ejpam-1373	192	16	any	any	DET
ejpam-1373	192	17	relatively	relatively	ADV
ejpam-1373	192	18	prime	prime	ADJ
ejpam-1373	192	19	integers	integer	NOUN
ejpam-1373	192	20	m	m	VERB
ejpam-1373	192	21	and	and	CCONJ
ejpam-1373	192	22	n.	n.	NOUN
ejpam-1373	192	23	(	(	PUNCT
ejpam-1373	192	24	1	1	NUM
ejpam-1373	192	25	)	)	PUNCT
ejpam-1373	192	26	.	.	PUNCT
ejpam-1373	193	1	m∨	m∨	PROPN
ejpam-1373	193	2	n	n	PROPN
ejpam-1373	193	3	exists	exist	VERB
ejpam-1373	193	4	in	in	ADP
ejpam-1373	193	5	(	(	PUNCT
ejpam-1373	193	6	z+,≤c	z+,≤c	PUNCT
ejpam-1373	193	7	)	)	PUNCT
ejpam-1373	193	8	and	and	CCONJ
ejpam-1373	193	9	is	be	AUX
ejpam-1373	193	10	equal	equal	ADJ
ejpam-1373	193	11	to	to	ADP
ejpam-1373	193	12	mn	mn	PROPN
ejpam-1373	193	13	.	.	PUNCT
ejpam-1373	194	1	(	(	PUNCT
ejpam-1373	194	2	2	2	NUM
ejpam-1373	194	3	)	)	PUNCT
ejpam-1373	194	4	.	.	PUNCT
ejpam-1373	195	1	x	x	X
ejpam-1373	195	2	∧	∧	PROPN
ejpam-1373	195	3	(	(	PUNCT
ejpam-1373	195	4	m∨n	m∨n	PROPN
ejpam-1373	195	5	)	)	PUNCT
ejpam-1373	195	6	=	=	PUNCT
ejpam-1373	196	1	(	(	PUNCT
ejpam-1373	196	2	x	x	SYM
ejpam-1373	196	3	∧m)∨	∧m)∨	PROPN
ejpam-1373	196	4	(	(	PUNCT
ejpam-1373	196	5	x	x	NOUN
ejpam-1373	196	6	∧n	∧n	PROPN
ejpam-1373	196	7	)	)	PUNCT
ejpam-1373	196	8	for	for	ADP
ejpam-1373	196	9	all	all	DET
ejpam-1373	196	10	x	x	SYM
ejpam-1373	196	11	∈	∈	PROPN
ejpam-1373	196	12	z+	z+	NUM
ejpam-1373	196	13	,	,	PUNCT
ejpam-1373	196	14	in	in	ADP
ejpam-1373	196	15	the	the	DET
ejpam-1373	196	16	sense	sense	NOUN
ejpam-1373	196	17	that	that	SCONJ
ejpam-1373	196	18	,	,	PUNCT
ejpam-1373	196	19	if	if	SCONJ
ejpam-1373	196	20	one	one	NUM
ejpam-1373	196	21	side	side	NOUN
ejpam-1373	196	22	is	be	AUX
ejpam-1373	196	23	defined	define	VERB
ejpam-1373	196	24	then	then	ADV
ejpam-1373	196	25	the	the	DET
ejpam-1373	196	26	other	other	ADJ
ejpam-1373	196	27	side	side	NOUN
ejpam-1373	196	28	is	be	AUX
ejpam-1373	196	29	also	also	ADV
ejpam-1373	196	30	defined	define	VERB
ejpam-1373	196	31	and	and	CCONJ
ejpam-1373	196	32	they	they	PRON
ejpam-1373	196	33	are	be	AUX
ejpam-1373	196	34	equal	equal	ADJ
ejpam-1373	196	35	.	.	PUNCT
ejpam-1373	197	1	proof	proof	NOUN
ejpam-1373	197	2	.	.	PUNCT
ejpam-1373	198	1	let	let	VERB
ejpam-1373	198	2	c	c	PRON
ejpam-1373	198	3	be	be	AUX
ejpam-1373	198	4	a	a	DET
ejpam-1373	198	5	convolution	convolution	NOUN
ejpam-1373	198	6	.	.	PUNCT
ejpam-1373	199	1	suppose	suppose	VERB
ejpam-1373	199	2	that	that	SCONJ
ejpam-1373	199	3	c	c	PROPN
ejpam-1373	199	4	is	be	AUX
ejpam-1373	199	5	multiplicative	multiplicative	ADJ
ejpam-1373	199	6	.	.	PUNCT
ejpam-1373	200	1	then	then	ADV
ejpam-1373	200	2	the	the	DET
ejpam-1373	200	3	mapping	mapping	NOUN
ejpam-1373	200	4	θ	θ	NOUN
ejpam-1373	200	5	:	:	PUNCT
ejpam-1373	200	6	z+	z+	PUNCT
ejpam-1373	200	7	−→	−→	NOUN
ejpam-1373	200	8	∑	∑	PUNCT
ejpam-1373	200	9	p	p	NOUN
ejpam-1373	200	10	n	n	NOUN
ejpam-1373	200	11	,	,	PUNCT
ejpam-1373	200	12	defined	define	VERB
ejpam-1373	200	13	in	in	ADP
ejpam-1373	200	14	theorem	theorem	NOUN
ejpam-1373	200	15	6	6	NUM
ejpam-1373	200	16	,	,	PUNCT
ejpam-1373	200	17	is	be	AUX
ejpam-1373	200	18	an	an	DET
ejpam-1373	200	19	order	order	NOUN
ejpam-1373	200	20	isomorphism	isomorphism	NOUN
ejpam-1373	200	21	of	of	ADP
ejpam-1373	200	22	(	(	PUNCT
ejpam-1373	200	23	z+,≤c	z+,≤c	PROPN
ejpam-1373	200	24	)	)	PUNCT
ejpam-1373	200	25	onto	onto	ADP
ejpam-1373	200	26	(	(	PUNCT
ejpam-1373	200	27	∑	∑	PROPN
ejpam-1373	200	28	p	p	NOUN
ejpam-1373	200	29	n	n	NOUN
ejpam-1373	200	30	,	,	PUNCT
ejpam-1373	200	31	≤c	≤c	PROPN
ejpam-1373	200	32	)	)	PUNCT
ejpam-1373	200	33	.	.	PUNCT
ejpam-1373	201	1	let	let	VERB
ejpam-1373	201	2	m	m	PRON
ejpam-1373	201	3	and	and	CCONJ
ejpam-1373	201	4	n	n	PRON
ejpam-1373	201	5	∈	∈	PROPN
ejpam-1373	201	6	z+	z+	NUM
ejpam-1373	201	7	such	such	ADJ
ejpam-1373	201	8	that	that	SCONJ
ejpam-1373	201	9	(	(	PUNCT
ejpam-1373	201	10	m	m	NOUN
ejpam-1373	201	11	,	,	PUNCT
ejpam-1373	201	12	n	n	CCONJ
ejpam-1373	201	13	)	)	PUNCT
ejpam-1373	201	14	=	=	SYM
ejpam-1373	202	1	1	1	X
ejpam-1373	202	2	.	.	X
ejpam-1373	203	1	we	we	PRON
ejpam-1373	203	2	shall	shall	AUX
ejpam-1373	203	3	prove	prove	VERB
ejpam-1373	203	4	that	that	SCONJ
ejpam-1373	203	5	mn	mn	PROPN
ejpam-1373	203	6	=	=	SYM
ejpam-1373	203	7	lub{m	lub{m	PROPN
ejpam-1373	203	8	,	,	PUNCT
ejpam-1373	203	9	n	n	CCONJ
ejpam-1373	203	10	}	}	PUNCT
ejpam-1373	203	11	in	in	ADP
ejpam-1373	203	12	(	(	PUNCT
ejpam-1373	203	13	z+,≤c	z+,≤c	NUM
ejpam-1373	203	14	)	)	PUNCT
ejpam-1373	203	15	.	.	PUNCT
ejpam-1373	204	1	since	since	SCONJ
ejpam-1373	204	2	m	m	PROPN
ejpam-1373	204	3	∈	∈	PROPN
ejpam-1373	204	4	c	c	X
ejpam-1373	204	5	(	(	PUNCT
ejpam-1373	204	6	m	m	NOUN
ejpam-1373	204	7	)	)	PUNCT
ejpam-1373	204	8	and	and	CCONJ
ejpam-1373	204	9	1	1	NUM
ejpam-1373	204	10	∈	∈	NOUN
ejpam-1373	204	11	c	c	X
ejpam-1373	204	12	(	(	PUNCT
ejpam-1373	204	13	n	n	CCONJ
ejpam-1373	204	14	)	)	PUNCT
ejpam-1373	204	15	,	,	PUNCT
ejpam-1373	204	16	we	we	PRON
ejpam-1373	204	17	have	have	VERB
ejpam-1373	204	18	,	,	PUNCT
ejpam-1373	204	19	m	m	VERB
ejpam-1373	204	20	=	=	PUNCT
ejpam-1373	204	21	m.1	m.1	PROPN
ejpam-1373	204	22	∈	∈	X
ejpam-1373	204	23	c	c	NOUN
ejpam-1373	204	24	(	(	PUNCT
ejpam-1373	204	25	m).c	m).c	PROPN
ejpam-1373	204	26	(	(	PUNCT
ejpam-1373	204	27	n	n	CCONJ
ejpam-1373	204	28	)	)	PUNCT
ejpam-1373	204	29	=	=	SYM
ejpam-1373	204	30	c	c	X
ejpam-1373	204	31	(	(	PUNCT
ejpam-1373	204	32	mn	mn	PROPN
ejpam-1373	204	33	)	)	PUNCT
ejpam-1373	204	34	.	.	PUNCT
ejpam-1373	205	1	and	and	CCONJ
ejpam-1373	205	2	hence	hence	ADV
ejpam-1373	205	3	m	m	PROPN
ejpam-1373	205	4	≤c	≤c	PROPN
ejpam-1373	205	5	mn	mn	PROPN
ejpam-1373	205	6	and	and	CCONJ
ejpam-1373	205	7	similarly	similarly	ADV
ejpam-1373	205	8	n≤c	n≤c	PROPN
ejpam-1373	205	9	mn	mn	PROPN
ejpam-1373	205	10	.	.	PUNCT
ejpam-1373	206	1	if	if	SCONJ
ejpam-1373	206	2	r	r	NOUN
ejpam-1373	206	3	is	be	AUX
ejpam-1373	206	4	any	any	DET
ejpam-1373	206	5	upper	upper	ADJ
ejpam-1373	206	6	bound	bind	VERB
ejpam-1373	206	7	of	of	ADP
ejpam-1373	206	8	{	{	PUNCT
ejpam-1373	206	9	m	m	PROPN
ejpam-1373	206	10	,	,	PUNCT
ejpam-1373	206	11	n	n	CCONJ
ejpam-1373	206	12	}	}	PUNCT
ejpam-1373	206	13	in	in	ADP
ejpam-1373	206	14	(	(	PUNCT
ejpam-1373	206	15	z+,≤c	z+,≤c	PROPN
ejpam-1373	206	16	)	)	PUNCT
ejpam-1373	206	17	,	,	PUNCT
ejpam-1373	206	18	then	then	ADV
ejpam-1373	206	19	θ(m	θ(m	PROPN
ejpam-1373	206	20	)	)	PUNCT
ejpam-1373	206	21	≤c	≤c	PROPN
ejpam-1373	206	22	θ(r	θ(r	X
ejpam-1373	206	23	)	)	PUNCT
ejpam-1373	206	24	and	and	CCONJ
ejpam-1373	206	25	θ(n)≤c	θ(n)≤c	VERB
ejpam-1373	206	26	θ(r	θ(r	NOUN
ejpam-1373	206	27	)	)	PUNCT
ejpam-1373	206	28	in	in	ADP
ejpam-1373	206	29	(	(	PUNCT
ejpam-1373	206	30	∑	∑	PROPN
ejpam-1373	206	31	p	p	NOUN
ejpam-1373	206	32	n	n	PROPN
ejpam-1373	206	33	,	,	PUNCT
ejpam-1373	206	34	≤c	≤c	PROPN
ejpam-1373	206	35	)	)	PUNCT
ejpam-1373	206	36	u.	u.	PROPN
ejpam-1373	206	37	swamy	swamy	PROPN
ejpam-1373	206	38	and	and	CCONJ
ejpam-1373	206	39	s.	s.	PROPN
ejpam-1373	206	40	sankar	sankar	PROPN
ejpam-1373	206	41	/	/	SYM
ejpam-1373	206	42	eur	eur	PROPN
ejpam-1373	206	43	.	.	PUNCT
ejpam-1373	207	1	j.	j.	PROPN
ejpam-1373	207	2	pure	pure	PROPN
ejpam-1373	207	3	appl	appl	PROPN
ejpam-1373	207	4	.	.	PROPN
ejpam-1373	207	5	math	math	PROPN
ejpam-1373	207	6	,	,	PUNCT
ejpam-1373	207	7	4	4	NUM
ejpam-1373	207	8	(	(	PUNCT
ejpam-1373	207	9	2011	2011	NUM
ejpam-1373	207	10	)	)	PUNCT
ejpam-1373	207	11	,	,	PUNCT
ejpam-1373	207	12	424	424	NUM
ejpam-1373	207	13	-	-	SYM
ejpam-1373	207	14	434	434	NUM
ejpam-1373	207	15	431	431	NUM
ejpam-1373	207	16	and	and	CCONJ
ejpam-1373	207	17	hence	hence	ADV
ejpam-1373	207	18	,	,	PUNCT
ejpam-1373	207	19	for	for	ADP
ejpam-1373	207	20	any	any	DET
ejpam-1373	207	21	prime	prime	ADJ
ejpam-1373	207	22	p	p	X
ejpam-1373	207	23	,	,	PUNCT
ejpam-1373	207	24	θ(m)(p	θ(m)(p	ADJ
ejpam-1373	207	25	)	)	PUNCT
ejpam-1373	207	26	≤p	≤p	NOUN
ejpam-1373	207	27	c	c	NOUN
ejpam-1373	207	28	θ(r)(p	θ(r)(p	X
ejpam-1373	207	29	)	)	PUNCT
ejpam-1373	207	30	and	and	CCONJ
ejpam-1373	207	31	θ(n)(p)≤p	θ(n)(p)≤p	NOUN
ejpam-1373	207	32	c	c	X
ejpam-1373	207	33	θ(r)(p	θ(r)(p	NOUN
ejpam-1373	207	34	)	)	PUNCT
ejpam-1373	207	35	.	.	PUNCT
ejpam-1373	208	1	also	also	ADV
ejpam-1373	208	2	,	,	PUNCT
ejpam-1373	208	3	since	since	SCONJ
ejpam-1373	208	4	m	m	PROPN
ejpam-1373	208	5	and	and	CCONJ
ejpam-1373	208	6	n	n	CCONJ
ejpam-1373	208	7	are	be	AUX
ejpam-1373	208	8	relatively	relatively	ADV
ejpam-1373	208	9	prime	prime	ADJ
ejpam-1373	208	10	,	,	PUNCT
ejpam-1373	208	11	we	we	PRON
ejpam-1373	208	12	have	have	VERB
ejpam-1373	208	13	for	for	ADP
ejpam-1373	208	14	any	any	DET
ejpam-1373	208	15	prime	prime	ADJ
ejpam-1373	208	16	p	p	X
ejpam-1373	208	17	,	,	PUNCT
ejpam-1373	208	18	θ(m)(p	θ(m)(p	NOUN
ejpam-1373	208	19	)	)	PUNCT
ejpam-1373	208	20	=	=	SYM
ejpam-1373	208	21	0	0	NUM
ejpam-1373	208	22	or	or	CCONJ
ejpam-1373	208	23	θ(n)(p	θ(n)(p	NUM
ejpam-1373	208	24	)	)	PUNCT
ejpam-1373	208	25	=	=	SYM
ejpam-1373	208	26	0	0	X
ejpam-1373	208	27	.	.	PUNCT
ejpam-1373	209	1	now	now	ADV
ejpam-1373	209	2	,	,	PUNCT
ejpam-1373	209	3	θ(mn)(p	θ(mn)(p	ADJ
ejpam-1373	209	4	)	)	PUNCT
ejpam-1373	209	5	=	=	SYM
ejpam-1373	210	1	θ(m)(p	θ(m)(p	X
ejpam-1373	210	2	)	)	PUNCT
ejpam-1373	210	3	+	+	CCONJ
ejpam-1373	210	4	θ(n)(p	θ(n)(p	X
ejpam-1373	210	5	)	)	PUNCT
ejpam-1373	210	6	=	=	SYM
ejpam-1373	210	7	θ(m)(p	θ(m)(p	X
ejpam-1373	210	8	)	)	PUNCT
ejpam-1373	210	9	or	or	CCONJ
ejpam-1373	210	10	θ(n)(p	θ(n)(p	NUM
ejpam-1373	210	11	)	)	PUNCT
ejpam-1373	210	12	.	.	PUNCT
ejpam-1373	211	1	and	and	CCONJ
ejpam-1373	211	2	hence	hence	ADV
ejpam-1373	211	3	θ((mn))(p)≤p	θ((mn))(p)≤p	VERB
ejpam-1373	211	4	c	c	NOUN
ejpam-1373	211	5	θ(r)(p	θ(r)(p	NOUN
ejpam-1373	211	6	)	)	PUNCT
ejpam-1373	211	7	.	.	PUNCT
ejpam-1373	212	1	therefore	therefore	ADV
ejpam-1373	212	2	θ(mn)≤c	θ(mn)≤c	NOUN
ejpam-1373	212	3	θ(r	θ(r	NOUN
ejpam-1373	212	4	)	)	PUNCT
ejpam-1373	212	5	and	and	CCONJ
ejpam-1373	212	6	mn≤c	mn≤c	PROPN
ejpam-1373	212	7	r.	r.	PROPN
ejpam-1373	212	8	thus	thus	ADV
ejpam-1373	212	9	mn	mn	PROPN
ejpam-1373	212	10	is	be	AUX
ejpam-1373	212	11	the	the	DET
ejpam-1373	212	12	least	least	ADJ
ejpam-1373	212	13	upper	upper	ADJ
ejpam-1373	212	14	bound	bind	VERB
ejpam-1373	212	15	of	of	ADP
ejpam-1373	212	16	m	m	PROPN
ejpam-1373	212	17	and	and	CCONJ
ejpam-1373	212	18	n.	n.	VERB
ejpam-1373	212	19	this	this	PRON
ejpam-1373	212	20	proves	prove	VERB
ejpam-1373	212	21	(	(	PUNCT
ejpam-1373	212	22	1	1	NUM
ejpam-1373	212	23	)	)	PUNCT
ejpam-1373	212	24	.	.	PUNCT
ejpam-1373	213	1	to	to	PART
ejpam-1373	213	2	prove	prove	VERB
ejpam-1373	213	3	(	(	PUNCT
ejpam-1373	213	4	2	2	NUM
ejpam-1373	213	5	)	)	PUNCT
ejpam-1373	213	6	,	,	PUNCT
ejpam-1373	213	7	let	let	VERB
ejpam-1373	213	8	x	x	PUNCT
ejpam-1373	213	9	∈	∈	PROPN
ejpam-1373	213	10	z+	z+	PUNCT
ejpam-1373	213	11	.	.	PUNCT
ejpam-1373	213	12	suppose	suppose	VERB
ejpam-1373	213	13	that	that	SCONJ
ejpam-1373	213	14	x	x	SYM
ejpam-1373	213	15	∧	∧	PROPN
ejpam-1373	213	16	(	(	PUNCT
ejpam-1373	213	17	m∨	m∨	NOUN
ejpam-1373	213	18	n	n	CCONJ
ejpam-1373	213	19	)	)	PUNCT
ejpam-1373	213	20	exists	exist	VERB
ejpam-1373	213	21	in	in	ADP
ejpam-1373	213	22	(	(	PUNCT
ejpam-1373	213	23	z+,≤c	z+,≤c	PROPN
ejpam-1373	213	24	)	)	PUNCT
ejpam-1373	213	25	.	.	PUNCT
ejpam-1373	214	1	suppose	suppose	VERB
ejpam-1373	214	2	m∨	m∨	NOUN
ejpam-1373	214	3	n	n	PROPN
ejpam-1373	214	4	exists	exist	VERB
ejpam-1373	214	5	and	and	CCONJ
ejpam-1373	214	6	is	be	AUX
ejpam-1373	214	7	equal	equal	ADJ
ejpam-1373	214	8	to	to	ADP
ejpam-1373	214	9	mn	mn	PROPN
ejpam-1373	214	10	,	,	PUNCT
ejpam-1373	214	11	we	we	PRON
ejpam-1373	214	12	are	be	AUX
ejpam-1373	214	13	given	give	VERB
ejpam-1373	214	14	that	that	SCONJ
ejpam-1373	214	15	x	x	SYM
ejpam-1373	214	16	∧	∧	PROPN
ejpam-1373	214	17	(	(	PUNCT
ejpam-1373	214	18	mn	mn	NOUN
ejpam-1373	214	19	)	)	PUNCT
ejpam-1373	214	20	exists	exist	VERB
ejpam-1373	214	21	.	.	PUNCT
ejpam-1373	215	1	we	we	PRON
ejpam-1373	215	2	shall	shall	AUX
ejpam-1373	215	3	prove	prove	VERB
ejpam-1373	215	4	that	that	SCONJ
ejpam-1373	215	5	x	x	PROPN
ejpam-1373	215	6	∧m	∧m	ADJ
ejpam-1373	215	7	and	and	CCONJ
ejpam-1373	215	8	x	x	SYM
ejpam-1373	215	9	∧	∧	NOUN
ejpam-1373	215	10	n	n	PRON
ejpam-1373	215	11	exists	exist	VERB
ejpam-1373	215	12	in	in	ADP
ejpam-1373	215	13	(	(	PUNCT
ejpam-1373	215	14	z+,≤c	z+,≤c	PROPN
ejpam-1373	215	15	)	)	PUNCT
ejpam-1373	215	16	.	.	PUNCT
ejpam-1373	216	1	to	to	PART
ejpam-1373	216	2	prove	prove	VERB
ejpam-1373	216	3	this	this	PRON
ejpam-1373	216	4	,	,	PUNCT
ejpam-1373	216	5	it	it	PRON
ejpam-1373	216	6	is	be	AUX
ejpam-1373	216	7	enough	enough	ADJ
ejpam-1373	216	8	if	if	SCONJ
ejpam-1373	216	9	we	we	PRON
ejpam-1373	216	10	prove	prove	VERB
ejpam-1373	216	11	that	that	SCONJ
ejpam-1373	216	12	θ(x)(p)∧	θ(x)(p)∧	PROPN
ejpam-1373	216	13	θ(m)(p	θ(m)(p	ADJ
ejpam-1373	216	14	)	)	PUNCT
ejpam-1373	216	15	and	and	CCONJ
ejpam-1373	216	16	θ(x)(p)∧	θ(x)(p)∧	PROPN
ejpam-1373	216	17	θ(n)(p	θ(n)(p	X
ejpam-1373	216	18	)	)	PUNCT
ejpam-1373	216	19	exist	exist	VERB
ejpam-1373	216	20	in	in	ADP
ejpam-1373	216	21	(	(	PUNCT
ejpam-1373	216	22	n	n	X
ejpam-1373	216	23	,	,	PUNCT
ejpam-1373	216	24	≤p	≤p	PROPN
ejpam-1373	216	25	c	c	PROPN
ejpam-1373	216	26	)	)	PUNCT
ejpam-1373	216	27	for	for	ADP
ejpam-1373	216	28	any	any	DET
ejpam-1373	216	29	prime	prime	ADJ
ejpam-1373	216	30	p.	p.	NOUN
ejpam-1373	216	31	since	since	SCONJ
ejpam-1373	216	32	θ(mn)(p	θ(mn)(p	PROPN
ejpam-1373	216	33	)	)	PUNCT
ejpam-1373	216	34	=	=	SYM
ejpam-1373	216	35	θ(m)(p	θ(m)(p	X
ejpam-1373	216	36	)	)	PUNCT
ejpam-1373	216	37	+	+	CCONJ
ejpam-1373	216	38	θ(n)(p	θ(n)(p	X
ejpam-1373	216	39	)	)	PUNCT
ejpam-1373	216	40	=	=	SYM
ejpam-1373	216	41	θ(m)(p	θ(m)(p	X
ejpam-1373	216	42	)	)	PUNCT
ejpam-1373	216	43	or	or	CCONJ
ejpam-1373	216	44	θ(n)(p	θ(n)(p	X
ejpam-1373	216	45	)	)	PUNCT
ejpam-1373	216	46	and	and	CCONJ
ejpam-1373	216	47	since	since	SCONJ
ejpam-1373	216	48	θ(x)(p	θ(x)(p	NOUN
ejpam-1373	216	49	)	)	PUNCT
ejpam-1373	216	50	∧	∧	PROPN
ejpam-1373	216	51	θ(m)(p	θ(m)(p	NUM
ejpam-1373	216	52	)	)	PUNCT
ejpam-1373	216	53	exists	exist	VERB
ejpam-1373	216	54	in	in	ADP
ejpam-1373	216	55	(	(	PUNCT
ejpam-1373	216	56	n	n	X
ejpam-1373	216	57	,	,	PUNCT
ejpam-1373	216	58	≤p	≤p	PROPN
ejpam-1373	216	59	c	c	PROPN
ejpam-1373	216	60	)	)	PUNCT
ejpam-1373	217	1	it	it	PRON
ejpam-1373	217	2	follows	follow	VERB
ejpam-1373	217	3	that	that	SCONJ
ejpam-1373	217	4	θ(x)(p	θ(x)(p	VERB
ejpam-1373	217	5	)	)	PUNCT
ejpam-1373	217	6	∧	∧	PROPN
ejpam-1373	217	7	θ(m)(p	θ(m)(p	NUM
ejpam-1373	217	8	)	)	PUNCT
ejpam-1373	217	9	and	and	CCONJ
ejpam-1373	217	10	θ(x)(p	θ(x)(p	NOUN
ejpam-1373	217	11	)	)	PUNCT
ejpam-1373	217	12	∧	∧	NOUN
ejpam-1373	217	13	θ(n)(p	θ(n)(p	NOUN
ejpam-1373	217	14	)	)	PUNCT
ejpam-1373	217	15	exist	exist	VERB
ejpam-1373	217	16	in	in	ADP
ejpam-1373	217	17	(	(	PUNCT
ejpam-1373	217	18	n	n	X
ejpam-1373	217	19	,	,	PUNCT
ejpam-1373	217	20	≤p	≤p	PROPN
ejpam-1373	217	21	c	c	PROPN
ejpam-1373	217	22	)	)	PUNCT
ejpam-1373	217	23	for	for	ADP
ejpam-1373	217	24	any	any	DET
ejpam-1373	217	25	prime	prime	ADJ
ejpam-1373	217	26	p.	p.	NOUN
ejpam-1373	217	27	therefore	therefore	ADV
ejpam-1373	218	1	x	x	X
ejpam-1373	218	2	∧	∧	NOUN
ejpam-1373	218	3	m	m	ADJ
ejpam-1373	218	4	and	and	CCONJ
ejpam-1373	218	5	x	x	SYM
ejpam-1373	218	6	∧	∧	NOUN
ejpam-1373	218	7	n	n	PRON
ejpam-1373	218	8	exist	exist	VERB
ejpam-1373	218	9	in	in	ADP
ejpam-1373	218	10	(	(	PUNCT
ejpam-1373	218	11	z+,≤c	z+,≤c	PROPN
ejpam-1373	218	12	)	)	PUNCT
ejpam-1373	218	13	.	.	PUNCT
ejpam-1373	219	1	now	now	ADV
ejpam-1373	219	2	,	,	PUNCT
ejpam-1373	219	3	since	since	SCONJ
ejpam-1373	219	4	x	x	PROPN
ejpam-1373	219	5	∧	∧	PROPN
ejpam-1373	219	6	m	m	PROPN
ejpam-1373	219	7	≤c	≤c	PROPN
ejpam-1373	219	8	m	m	PROPN
ejpam-1373	219	9	,	,	PUNCT
ejpam-1373	219	10	x	x	PUNCT
ejpam-1373	219	11	∧	∧	NOUN
ejpam-1373	219	12	m	m	VERB
ejpam-1373	219	13	∈	∈	PROPN
ejpam-1373	219	14	c	c	X
ejpam-1373	219	15	(	(	PUNCT
ejpam-1373	219	16	m	m	PROPN
ejpam-1373	219	17	)	)	PUNCT
ejpam-1373	219	18	⊆	⊆	NUM
ejpam-1373	219	19	d(m	d(m	NOUN
ejpam-1373	219	20	)	)	PUNCT
ejpam-1373	219	21	and	and	CCONJ
ejpam-1373	219	22	hence	hence	ADV
ejpam-1373	219	23	x	x	PART
ejpam-1373	219	24	∧	∧	NOUN
ejpam-1373	219	25	m	m	VERB
ejpam-1373	219	26	is	be	AUX
ejpam-1373	219	27	a	a	DET
ejpam-1373	219	28	divisor	divisor	NOUN
ejpam-1373	219	29	of	of	ADP
ejpam-1373	219	30	m.	m.	NOUN
ejpam-1373	219	31	similarly	similarly	ADV
ejpam-1373	219	32	x	x	PUNCT
ejpam-1373	220	1	∧	∧	NOUN
ejpam-1373	220	2	n	n	PRON
ejpam-1373	220	3	is	be	AUX
ejpam-1373	220	4	a	a	DET
ejpam-1373	220	5	divisor	divisor	NOUN
ejpam-1373	220	6	of	of	ADP
ejpam-1373	220	7	n.	n.	NOUN
ejpam-1373	220	8	since	since	SCONJ
ejpam-1373	220	9	(	(	PUNCT
ejpam-1373	220	10	m	m	PROPN
ejpam-1373	220	11	,	,	PUNCT
ejpam-1373	220	12	n	n	CCONJ
ejpam-1373	220	13	)	)	PUNCT
ejpam-1373	220	14	=	=	SYM
ejpam-1373	220	15	1	1	NUM
ejpam-1373	220	16	,	,	PUNCT
ejpam-1373	220	17	we	we	PRON
ejpam-1373	220	18	get	get	VERB
ejpam-1373	220	19	that	that	PRON
ejpam-1373	220	20	(	(	PUNCT
ejpam-1373	220	21	x	x	X
ejpam-1373	220	22	∧m	∧m	PROPN
ejpam-1373	220	23	,	,	PUNCT
ejpam-1373	220	24	x	x	PART
ejpam-1373	220	25	∧	∧	PROPN
ejpam-1373	220	26	n	n	CCONJ
ejpam-1373	220	27	)	)	PUNCT
ejpam-1373	220	28	=	=	SYM
ejpam-1373	220	29	1	1	NUM
ejpam-1373	220	30	and	and	CCONJ
ejpam-1373	220	31	hence	hence	ADV
ejpam-1373	220	32	by	by	ADP
ejpam-1373	220	33	(	(	PUNCT
ejpam-1373	220	34	1	1	NUM
ejpam-1373	220	35	)	)	PUNCT
ejpam-1373	220	36	,	,	PUNCT
ejpam-1373	220	37	(	(	PUNCT
ejpam-1373	220	38	x	x	X
ejpam-1373	220	39	∧m)∨	∧m)∨	PROPN
ejpam-1373	220	40	(	(	PUNCT
ejpam-1373	220	41	x	x	PART
ejpam-1373	220	42	∧	∧	PROPN
ejpam-1373	220	43	n	n	CCONJ
ejpam-1373	220	44	)	)	PUNCT
ejpam-1373	220	45	exists	exist	VERB
ejpam-1373	220	46	in	in	ADP
ejpam-1373	220	47	(	(	PUNCT
ejpam-1373	220	48	z+,≤c	z+,≤c	PUNCT
ejpam-1373	220	49	)	)	PUNCT
ejpam-1373	220	50	and	and	CCONJ
ejpam-1373	220	51	is	be	AUX
ejpam-1373	220	52	equal	equal	ADJ
ejpam-1373	220	53	to	to	ADP
ejpam-1373	220	54	the	the	DET
ejpam-1373	220	55	product	product	NOUN
ejpam-1373	220	56	(	(	PUNCT
ejpam-1373	220	57	x	x	SYM
ejpam-1373	220	58	∧m)(x	∧m)(x	SYM
ejpam-1373	220	59	∧	∧	PROPN
ejpam-1373	220	60	n	n	CCONJ
ejpam-1373	220	61	)	)	PUNCT
ejpam-1373	220	62	.	.	PUNCT
ejpam-1373	221	1	now	now	ADV
ejpam-1373	221	2	,	,	PUNCT
ejpam-1373	221	3	for	for	ADP
ejpam-1373	221	4	any	any	DET
ejpam-1373	221	5	p	p	NOUN
ejpam-1373	221	6	∈	∈	PROPN
ejpam-1373	221	7	p	p	NOUN
ejpam-1373	221	8	,	,	PUNCT
ejpam-1373	221	9	we	we	PRON
ejpam-1373	221	10	have	have	VERB
ejpam-1373	221	11	θ(x	θ(x	PROPN
ejpam-1373	221	12	∧	∧	PROPN
ejpam-1373	221	13	(	(	PUNCT
ejpam-1373	221	14	m∨	m∨	NOUN
ejpam-1373	221	15	n))(p	n))(p	NOUN
ejpam-1373	221	16	)	)	PUNCT
ejpam-1373	221	17	=	=	SYM
ejpam-1373	221	18	θ(x	θ(x	PROPN
ejpam-1373	221	19	∧	∧	PROPN
ejpam-1373	221	20	(	(	PUNCT
ejpam-1373	221	21	mn))(p	mn))(p	NOUN
ejpam-1373	221	22	)	)	PUNCT
ejpam-1373	221	23	=	=	SYM
ejpam-1373	221	24	θ(x)(p)∧	θ(x)(p)∧	NOUN
ejpam-1373	221	25	θ(mn)(p	θ(mn)(p	PROPN
ejpam-1373	221	26	)	)	PUNCT
ejpam-1373	221	27	=	=	SYM
ejpam-1373	221	28	θ(x)(p)∧	θ(x)(p)∧	PROPN
ejpam-1373	221	29	θ(m)(p	θ(m)(p	PROPN
ejpam-1373	221	30	)	)	PUNCT
ejpam-1373	221	31	or	or	CCONJ
ejpam-1373	221	32	θ(x)(p)∧	θ(x)(p)∧	NOUN
ejpam-1373	221	33	θ(n)(p	θ(n)(p	ADJ
ejpam-1373	221	34	)	)	PUNCT
ejpam-1373	221	35	=	=	SYM
ejpam-1373	221	36	θ(x	θ(x	PROPN
ejpam-1373	221	37	∧m)(p	∧m)(p	PUNCT
ejpam-1373	221	38	)	)	PUNCT
ejpam-1373	221	39	or	or	CCONJ
ejpam-1373	221	40	θ(x	θ(x	PROPN
ejpam-1373	221	41	∧	∧	PROPN
ejpam-1373	221	42	n)(p	n)(p	NOUN
ejpam-1373	221	43	)	)	PUNCT
ejpam-1373	221	44	=	=	SYM
ejpam-1373	222	1	θ((x	θ((x	NOUN
ejpam-1373	222	2	∧m)(x	∧m)(x	NUM
ejpam-1373	222	3	∧	∧	PROPN
ejpam-1373	222	4	n))(p	n))(p	NOUN
ejpam-1373	222	5	)	)	PUNCT
ejpam-1373	222	6	=	=	SYM
ejpam-1373	223	1	θ((x	θ((x	NOUN
ejpam-1373	223	2	∧m)∨	∧m)∨	NOUN
ejpam-1373	223	3	(	(	PUNCT
ejpam-1373	223	4	x	x	PART
ejpam-1373	223	5	∧	∧	NOUN
ejpam-1373	223	6	n))(p	n))(p	NOUN
ejpam-1373	223	7	)	)	PUNCT
ejpam-1373	223	8	and	and	CCONJ
ejpam-1373	223	9	hence	hence	ADV
ejpam-1373	223	10	θ(x	θ(x	PROPN
ejpam-1373	223	11	∧	∧	PROPN
ejpam-1373	223	12	(	(	PUNCT
ejpam-1373	223	13	m∨	m∨	NOUN
ejpam-1373	223	14	n	n	NUM
ejpam-1373	223	15	)	)	PUNCT
ejpam-1373	223	16	)	)	PUNCT
ejpam-1373	224	1	=	=	SYM
ejpam-1373	224	2	θ((x	θ((x	NOUN
ejpam-1373	224	3	∧m)∨	∧m)∨	NOUN
ejpam-1373	224	4	(	(	PUNCT
ejpam-1373	224	5	x	x	PART
ejpam-1373	224	6	∧	∧	PROPN
ejpam-1373	224	7	n	n	NUM
ejpam-1373	224	8	)	)	PUNCT
ejpam-1373	224	9	)	)	PUNCT
ejpam-1373	224	10	,	,	PUNCT
ejpam-1373	224	11	so	so	SCONJ
ejpam-1373	224	12	that	that	SCONJ
ejpam-1373	224	13	x	x	SYM
ejpam-1373	224	14	∧	∧	NOUN
ejpam-1373	224	15	(	(	PUNCT
ejpam-1373	224	16	m∨	m∨	NOUN
ejpam-1373	224	17	n	n	CCONJ
ejpam-1373	224	18	)	)	PUNCT
ejpam-1373	224	19	=	=	SYM
ejpam-1373	225	1	(	(	PUNCT
ejpam-1373	225	2	x	x	PUNCT
ejpam-1373	225	3	∧m)∨	∧m)∨	PROPN
ejpam-1373	225	4	(	(	PUNCT
ejpam-1373	225	5	x	x	PART
ejpam-1373	225	6	∧	∧	PROPN
ejpam-1373	225	7	n	n	NUM
ejpam-1373	225	8	)	)	PUNCT
ejpam-1373	225	9	.	.	PUNCT
ejpam-1373	226	1	similarly	similarly	ADV
ejpam-1373	226	2	,	,	PUNCT
ejpam-1373	226	3	we	we	PRON
ejpam-1373	226	4	can	can	AUX
ejpam-1373	226	5	prove	prove	VERB
ejpam-1373	226	6	that	that	SCONJ
ejpam-1373	226	7	the	the	DET
ejpam-1373	226	8	left	left	ADJ
ejpam-1373	226	9	hand	hand	NOUN
ejpam-1373	226	10	side	side	NOUN
ejpam-1373	226	11	of	of	ADP
ejpam-1373	226	12	the	the	DET
ejpam-1373	226	13	equation	equation	NOUN
ejpam-1373	226	14	exists	exist	VERB
ejpam-1373	226	15	if	if	SCONJ
ejpam-1373	226	16	the	the	DET
ejpam-1373	226	17	right	right	ADJ
ejpam-1373	226	18	hand	hand	NOUN
ejpam-1373	226	19	side	side	NOUN
ejpam-1373	226	20	exists	exist	VERB
ejpam-1373	226	21	and	and	CCONJ
ejpam-1373	226	22	that	that	SCONJ
ejpam-1373	226	23	they	they	PRON
ejpam-1373	226	24	are	be	AUX
ejpam-1373	226	25	equal	equal	ADJ
ejpam-1373	226	26	.	.	PUNCT
ejpam-1373	227	1	this	this	PRON
ejpam-1373	227	2	proves	prove	VERB
ejpam-1373	227	3	(	(	PUNCT
ejpam-1373	227	4	2	2	NUM
ejpam-1373	227	5	)	)	PUNCT
ejpam-1373	227	6	.	.	PUNCT
ejpam-1373	228	1	conversely	conversely	ADV
ejpam-1373	228	2	suppose	suppose	VERB
ejpam-1373	228	3	that	that	SCONJ
ejpam-1373	228	4	the	the	DET
ejpam-1373	228	5	conditions	condition	NOUN
ejpam-1373	228	6	(	(	PUNCT
ejpam-1373	228	7	1	1	NUM
ejpam-1373	228	8	)	)	PUNCT
ejpam-1373	228	9	and	and	CCONJ
ejpam-1373	228	10	(	(	PUNCT
ejpam-1373	228	11	2	2	X
ejpam-1373	228	12	)	)	PUNCT
ejpam-1373	228	13	are	be	AUX
ejpam-1373	228	14	satisfied	satisfied	ADJ
ejpam-1373	228	15	for	for	ADP
ejpam-1373	228	16	any	any	DET
ejpam-1373	228	17	relatively	relatively	ADV
ejpam-1373	228	18	prime	prime	ADJ
ejpam-1373	228	19	positive	positive	ADJ
ejpam-1373	228	20	integers	integer	NOUN
ejpam-1373	228	21	m	m	VERB
ejpam-1373	228	22	and	and	CCONJ
ejpam-1373	228	23	n.	n.	VERB
ejpam-1373	228	24	to	to	PART
ejpam-1373	228	25	prove	prove	VERB
ejpam-1373	228	26	that	that	SCONJ
ejpam-1373	228	27	c	c	PROPN
ejpam-1373	228	28	is	be	AUX
ejpam-1373	228	29	multiplicative	multiplicative	ADJ
ejpam-1373	228	30	,	,	PUNCT
ejpam-1373	228	31	let	let	VERB
ejpam-1373	228	32	us	we	PRON
ejpam-1373	228	33	consider	consider	VERB
ejpam-1373	228	34	,	,	PUNCT
ejpam-1373	228	35	m	m	VERB
ejpam-1373	228	36	and	and	CCONJ
ejpam-1373	228	37	n	n	PRON
ejpam-1373	228	38	∈	∈	PROPN
ejpam-1373	228	39	z+	z+	NUM
ejpam-1373	228	40	such	such	ADJ
ejpam-1373	228	41	that	that	SCONJ
ejpam-1373	228	42	(	(	PUNCT
ejpam-1373	228	43	m	m	NOUN
ejpam-1373	228	44	,	,	PUNCT
ejpam-1373	228	45	n	n	CCONJ
ejpam-1373	228	46	)	)	PUNCT
ejpam-1373	228	47	=	=	SYM
ejpam-1373	229	1	1	1	X
ejpam-1373	229	2	.	.	PUNCT
ejpam-1373	229	3	then	then	ADV
ejpam-1373	229	4	(	(	PUNCT
ejpam-1373	229	5	y	y	PROPN
ejpam-1373	229	6	,	,	PUNCT
ejpam-1373	229	7	z	z	NOUN
ejpam-1373	229	8	)	)	PUNCT
ejpam-1373	229	9	=	=	SYM
ejpam-1373	229	10	1	1	NUM
ejpam-1373	229	11	for	for	ADP
ejpam-1373	229	12	all	all	DET
ejpam-1373	229	13	y	y	PROPN
ejpam-1373	229	14	∈	∈	PROPN
ejpam-1373	229	15	c	c	X
ejpam-1373	229	16	(	(	PUNCT
ejpam-1373	229	17	m	m	NOUN
ejpam-1373	229	18	)	)	PUNCT
ejpam-1373	229	19	and	and	CCONJ
ejpam-1373	229	20	z	z	NOUN
ejpam-1373	229	21	∈	∈	PROPN
ejpam-1373	229	22	(	(	PUNCT
ejpam-1373	229	23	c)(n	c)(n	PROPN
ejpam-1373	229	24	)	)	PUNCT
ejpam-1373	229	25	and	and	CCONJ
ejpam-1373	229	26	hence	hence	ADV
ejpam-1373	229	27	,	,	PUNCT
ejpam-1373	229	28	by	by	ADP
ejpam-1373	229	29	(	(	PUNCT
ejpam-1373	229	30	1	1	NUM
ejpam-1373	229	31	)	)	PUNCT
ejpam-1373	229	32	,	,	PUNCT
ejpam-1373	229	33	y	y	PROPN
ejpam-1373	229	34	∨	∨	PROPN
ejpam-1373	229	35	z	z	PROPN
ejpam-1373	229	36	exists	exist	VERB
ejpam-1373	229	37	and	and	CCONJ
ejpam-1373	229	38	is	be	AUX
ejpam-1373	229	39	equal	equal	ADJ
ejpam-1373	229	40	to	to	ADP
ejpam-1373	229	41	yz	yz	PROPN
ejpam-1373	229	42	in	in	ADP
ejpam-1373	229	43	(	(	PUNCT
ejpam-1373	229	44	z+,≤c	z+,≤c	PROPN
ejpam-1373	229	45	)	)	PUNCT
ejpam-1373	230	1	whenever	whenever	SCONJ
ejpam-1373	230	2	y	y	PROPN
ejpam-1373	230	3	≤c	≤c	PROPN
ejpam-1373	230	4	m	m	PROPN
ejpam-1373	230	5	and	and	CCONJ
ejpam-1373	230	6	z	z	PROPN
ejpam-1373	230	7	≤c	≤c	PROPN
ejpam-1373	230	8	n	n	PROPN
ejpam-1373	230	9	and	and	CCONJ
ejpam-1373	230	10	,	,	PUNCT
ejpam-1373	230	11	by	by	ADP
ejpam-1373	230	12	(	(	PUNCT
ejpam-1373	230	13	2	2	NUM
ejpam-1373	230	14	)	)	PUNCT
ejpam-1373	230	15	,	,	PUNCT
ejpam-1373	230	16	x	x	PUNCT
ejpam-1373	230	17	∧	∧	PROPN
ejpam-1373	230	18	(	(	PUNCT
ejpam-1373	230	19	y	y	PROPN
ejpam-1373	230	20	∨	∨	PROPN
ejpam-1373	230	21	z	z	PROPN
ejpam-1373	230	22	)	)	PUNCT
ejpam-1373	230	23	=	=	SYM
ejpam-1373	230	24	(	(	PUNCT
ejpam-1373	230	25	x	x	PUNCT
ejpam-1373	230	26	∧	∧	PROPN
ejpam-1373	230	27	y)∨	y)∨	PROPN
ejpam-1373	230	28	(	(	PUNCT
ejpam-1373	230	29	x	x	PROPN
ejpam-1373	230	30	∧	∧	PROPN
ejpam-1373	230	31	z	z	PROPN
ejpam-1373	230	32	)	)	PUNCT
ejpam-1373	230	33	for	for	ADP
ejpam-1373	230	34	all	all	DET
ejpam-1373	230	35	x	x	SYM
ejpam-1373	230	36	∈	∈	PROPN
ejpam-1373	230	37	z+	z+	PUNCT
ejpam-1373	230	38	.	.	PUNCT
ejpam-1373	230	39	now	now	ADV
ejpam-1373	230	40	consider	consider	VERB
ejpam-1373	230	41	x	x	X
ejpam-1373	230	42	∈	∈	PROPN
ejpam-1373	230	43	c	c	X
ejpam-1373	230	44	(	(	PUNCT
ejpam-1373	230	45	mn	mn	NOUN
ejpam-1373	230	46	)	)	PUNCT
ejpam-1373	230	47	=	=	NOUN
ejpam-1373	230	48	⇒	⇒	NOUN
ejpam-1373	230	49	x	x	PUNCT
ejpam-1373	230	50	≤c	≤c	PROPN
ejpam-1373	230	51	mn	mn	PROPN
ejpam-1373	230	52	=	=	PROPN
ejpam-1373	230	53	m∨	m∨	PROPN
ejpam-1373	230	54	n	n	PRON
ejpam-1373	230	55	=	=	NOUN
ejpam-1373	230	56	⇒	⇒	NOUN
ejpam-1373	230	57	x	x	PUNCT
ejpam-1373	230	58	=	=	PUNCT
ejpam-1373	230	59	x	x	SYM
ejpam-1373	230	60	∧	∧	PROPN
ejpam-1373	230	61	(	(	PUNCT
ejpam-1373	230	62	m∨	m∨	NOUN
ejpam-1373	230	63	n	n	CCONJ
ejpam-1373	230	64	)	)	PUNCT
ejpam-1373	230	65	=	=	SYM
ejpam-1373	231	1	(	(	PUNCT
ejpam-1373	231	2	x	x	PUNCT
ejpam-1373	231	3	∧m)∨	∧m)∨	PROPN
ejpam-1373	231	4	(	(	PUNCT
ejpam-1373	231	5	x	x	PART
ejpam-1373	231	6	∧	∧	PROPN
ejpam-1373	231	7	n	n	CCONJ
ejpam-1373	231	8	)	)	PUNCT
ejpam-1373	231	9	=	=	NOUN
ejpam-1373	231	10	⇒	⇒	NOUN
ejpam-1373	231	11	x	x	PUNCT
ejpam-1373	231	12	=	=	SYM
ejpam-1373	231	13	(	(	PUNCT
ejpam-1373	231	14	x	x	SYM
ejpam-1373	231	15	∧m)(x	∧m)(x	SYM
ejpam-1373	231	16	∧	∧	PROPN
ejpam-1373	231	17	n	n	CCONJ
ejpam-1373	231	18	)	)	PUNCT
ejpam-1373	231	19	,	,	PUNCT
ejpam-1373	231	20	x	x	X
ejpam-1373	231	21	∧m	∧m	PROPN
ejpam-1373	231	22	∈	∈	PROPN
ejpam-1373	231	23	c	c	X
ejpam-1373	231	24	(	(	PUNCT
ejpam-1373	231	25	m	m	PROPN
ejpam-1373	231	26	)	)	PUNCT
ejpam-1373	231	27	,	,	PUNCT
ejpam-1373	231	28	x	x	PUNCT
ejpam-1373	231	29	∧	∧	NOUN
ejpam-1373	231	30	n	n	CCONJ
ejpam-1373	231	31	∈	∈	PROPN
ejpam-1373	231	32	c	c	X
ejpam-1373	231	33	(	(	PUNCT
ejpam-1373	231	34	n	n	CCONJ
ejpam-1373	231	35	)	)	PUNCT
ejpam-1373	232	1	=	=	NOUN
ejpam-1373	232	2	⇒	⇒	NOUN
ejpam-1373	232	3	x	x	X
ejpam-1373	232	4	∈	∈	PROPN
ejpam-1373	232	5	c	c	X
ejpam-1373	232	6	(	(	PUNCT
ejpam-1373	232	7	m)c	m)c	X
ejpam-1373	232	8	(	(	PUNCT
ejpam-1373	232	9	n	n	CCONJ
ejpam-1373	232	10	)	)	PUNCT
ejpam-1373	232	11	u.	u.	NOUN
ejpam-1373	232	12	swamy	swamy	PROPN
ejpam-1373	232	13	and	and	CCONJ
ejpam-1373	232	14	s.	s.	PROPN
ejpam-1373	232	15	sankar	sankar	PROPN
ejpam-1373	232	16	/	/	SYM
ejpam-1373	232	17	eur	eur	PROPN
ejpam-1373	232	18	.	.	PUNCT
ejpam-1373	233	1	j.	j.	PROPN
ejpam-1373	233	2	pure	pure	PROPN
ejpam-1373	233	3	appl	appl	PROPN
ejpam-1373	233	4	.	.	PROPN
ejpam-1373	233	5	math	math	PROPN
ejpam-1373	233	6	,	,	PUNCT
ejpam-1373	233	7	4	4	NUM
ejpam-1373	233	8	(	(	PUNCT
ejpam-1373	233	9	2011	2011	NUM
ejpam-1373	233	10	)	)	PUNCT
ejpam-1373	233	11	,	,	PUNCT
ejpam-1373	233	12	424	424	NUM
ejpam-1373	233	13	-	-	SYM
ejpam-1373	233	14	434	434	NUM
ejpam-1373	233	15	432	432	NUM
ejpam-1373	234	1	therefore	therefore	ADV
ejpam-1373	234	2	c	c	X
ejpam-1373	234	3	(	(	PUNCT
ejpam-1373	234	4	mn)⊆	mn)⊆	PROPN
ejpam-1373	234	5	c	c	PROPN
ejpam-1373	234	6	(	(	PUNCT
ejpam-1373	234	7	m)c	m)c	X
ejpam-1373	234	8	(	(	PUNCT
ejpam-1373	234	9	n	n	CCONJ
ejpam-1373	234	10	)	)	PUNCT
ejpam-1373	234	11	.	.	PUNCT
ejpam-1373	235	1	on	on	ADP
ejpam-1373	235	2	the	the	DET
ejpam-1373	235	3	other	other	ADJ
ejpam-1373	235	4	hand	hand	NOUN
ejpam-1373	235	5	x	x	PUNCT
ejpam-1373	235	6	∈	∈	PROPN
ejpam-1373	235	7	c	c	X
ejpam-1373	235	8	(	(	PUNCT
ejpam-1373	235	9	m)c	m)c	X
ejpam-1373	235	10	(	(	PUNCT
ejpam-1373	235	11	n	n	CCONJ
ejpam-1373	235	12	)	)	PUNCT
ejpam-1373	236	1	=	=	NOUN
ejpam-1373	236	2	⇒	⇒	NOUN
ejpam-1373	236	3	x	x	PUNCT
ejpam-1373	236	4	=	=	SYM
ejpam-1373	236	5	yz	yz	PROPN
ejpam-1373	236	6	,	,	PUNCT
ejpam-1373	236	7	y	y	PROPN
ejpam-1373	236	8	∈	∈	PROPN
ejpam-1373	236	9	c	c	X
ejpam-1373	236	10	(	(	PUNCT
ejpam-1373	236	11	m	m	NOUN
ejpam-1373	236	12	)	)	PUNCT
ejpam-1373	236	13	and	and	CCONJ
ejpam-1373	236	14	z	z	NOUN
ejpam-1373	236	15	∈	∈	PROPN
ejpam-1373	236	16	c	c	X
ejpam-1373	236	17	(	(	PUNCT
ejpam-1373	236	18	n	n	CCONJ
ejpam-1373	236	19	)	)	PUNCT
ejpam-1373	237	1	=	=	NOUN
ejpam-1373	237	2	⇒	⇒	NOUN
ejpam-1373	237	3	x	x	PUNCT
ejpam-1373	237	4	=	=	SYM
ejpam-1373	237	5	y	y	PROPN
ejpam-1373	237	6	∨	∨	PROPN
ejpam-1373	237	7	z	z	PROPN
ejpam-1373	237	8	,	,	PUNCT
ejpam-1373	237	9	y	y	PROPN
ejpam-1373	237	10	≤c	≤c	PROPN
ejpam-1373	237	11	m	m	PROPN
ejpam-1373	237	12	,	,	PUNCT
ejpam-1373	237	13	and	and	CCONJ
ejpam-1373	237	14	z	z	PROPN
ejpam-1373	237	15	≤c	≤c	PROPN
ejpam-1373	237	16	n	n	CCONJ
ejpam-1373	237	17	=	=	NOUN
ejpam-1373	237	18	⇒	⇒	NOUN
ejpam-1373	237	19	x	x	PUNCT
ejpam-1373	237	20	=	=	SYM
ejpam-1373	237	21	y	y	PROPN
ejpam-1373	237	22	∨	∨	PROPN
ejpam-1373	237	23	z	z	PROPN
ejpam-1373	237	24	≤c	≤c	PROPN
ejpam-1373	237	25	m∨	m∨	PROPN
ejpam-1373	237	26	n=	n=	PROPN
ejpam-1373	237	27	mn	mn	PROPN
ejpam-1373	238	1	=	=	NOUN
ejpam-1373	238	2	⇒	⇒	VERB
ejpam-1373	238	3	x	x	PUNCT
ejpam-1373	238	4	∈	∈	PROPN
ejpam-1373	238	5	c	c	X
ejpam-1373	238	6	(	(	PUNCT
ejpam-1373	238	7	mn	mn	PROPN
ejpam-1373	238	8	)	)	PUNCT
ejpam-1373	238	9	therefore	therefore	ADV
ejpam-1373	238	10	c	c	PROPN
ejpam-1373	238	11	(	(	PUNCT
ejpam-1373	238	12	m)c	m)c	X
ejpam-1373	238	13	(	(	PUNCT
ejpam-1373	238	14	n)⊆	n)⊆	PROPN
ejpam-1373	238	15	c	c	PROPN
ejpam-1373	238	16	(	(	PUNCT
ejpam-1373	238	17	mn	mn	PROPN
ejpam-1373	238	18	)	)	PUNCT
ejpam-1373	238	19	.	.	PUNCT
ejpam-1373	239	1	thus	thus	ADV
ejpam-1373	239	2	c	c	PROPN
ejpam-1373	239	3	(	(	PUNCT
ejpam-1373	239	4	mn	mn	PROPN
ejpam-1373	239	5	)	)	PUNCT
ejpam-1373	239	6	=	=	SYM
ejpam-1373	239	7	c	c	X
ejpam-1373	239	8	(	(	PUNCT
ejpam-1373	239	9	m)c	m)c	X
ejpam-1373	239	10	(	(	PUNCT
ejpam-1373	239	11	n	n	CCONJ
ejpam-1373	239	12	)	)	PUNCT
ejpam-1373	239	13	and	and	CCONJ
ejpam-1373	239	14	hence	hence	ADV
ejpam-1373	239	15	c	c	PROPN
ejpam-1373	239	16	is	be	AUX
ejpam-1373	239	17	multiplicative	multiplicative	ADJ
ejpam-1373	239	18	.	.	PUNCT
ejpam-1373	240	1	actually	actually	ADV
ejpam-1373	240	2	,	,	PUNCT
ejpam-1373	240	3	the	the	DET
ejpam-1373	240	4	existence	existence	NOUN
ejpam-1373	240	5	of	of	ADP
ejpam-1373	240	6	m∨n	m∨n	PROPN
ejpam-1373	240	7	in	in	ADP
ejpam-1373	240	8	condition	condition	NOUN
ejpam-1373	240	9	(	(	PUNCT
ejpam-1373	240	10	1	1	NUM
ejpam-1373	240	11	)	)	PUNCT
ejpam-1373	240	12	above	above	ADV
ejpam-1373	240	13	is	be	AUX
ejpam-1373	240	14	a	a	DET
ejpam-1373	240	15	consequence	consequence	NOUN
ejpam-1373	240	16	of	of	ADP
ejpam-1373	240	17	(	(	PUNCT
ejpam-1373	240	18	2	2	NUM
ejpam-1373	240	19	)	)	PUNCT
ejpam-1373	240	20	.	.	PUNCT
ejpam-1373	241	1	for	for	ADP
ejpam-1373	241	2	,	,	PUNCT
ejpam-1373	241	3	choose	choose	VERB
ejpam-1373	241	4	a	a	DET
ejpam-1373	241	5	prime	prime	NOUN
ejpam-1373	241	6	p	p	NOUN
ejpam-1373	241	7	which	which	PRON
ejpam-1373	241	8	divides	divide	VERB
ejpam-1373	241	9	neither	neither	DET
ejpam-1373	241	10	m	m	PROPN
ejpam-1373	241	11	nor	nor	CCONJ
ejpam-1373	241	12	n.	n.	VERB
ejpam-1373	241	13	then	then	ADV
ejpam-1373	241	14	p	p	NOUN
ejpam-1373	241	15	∧m	∧m	PROPN
ejpam-1373	241	16	=	=	SYM
ejpam-1373	241	17	1=	1=	NUM
ejpam-1373	241	18	p∧	p∧	NOUN
ejpam-1373	241	19	n	n	CCONJ
ejpam-1373	241	20	and	and	CCONJ
ejpam-1373	241	21	hence	hence	ADV
ejpam-1373	241	22	(	(	PUNCT
ejpam-1373	241	23	p∧m)∨	p∧m)∨	PROPN
ejpam-1373	241	24	(	(	PUNCT
ejpam-1373	241	25	p∧	p∧	NOUN
ejpam-1373	241	26	n	n	CCONJ
ejpam-1373	241	27	)	)	PUNCT
ejpam-1373	241	28	exists	exist	VERB
ejpam-1373	241	29	which	which	PRON
ejpam-1373	241	30	,	,	PUNCT
ejpam-1373	241	31	by(2	by(2	NOUN
ejpam-1373	241	32	)	)	PUNCT
ejpam-1373	241	33	,	,	PUNCT
ejpam-1373	241	34	implies	imply	VERB
ejpam-1373	241	35	that	that	SCONJ
ejpam-1373	241	36	p	p	PROPN
ejpam-1373	241	37	∧	∧	PROPN
ejpam-1373	241	38	(	(	PUNCT
ejpam-1373	241	39	m∨	m∨	NOUN
ejpam-1373	241	40	n	n	CCONJ
ejpam-1373	241	41	)	)	PUNCT
ejpam-1373	241	42	exists	exist	VERB
ejpam-1373	241	43	and	and	CCONJ
ejpam-1373	241	44	,	,	PUNCT
ejpam-1373	241	45	in	in	ADP
ejpam-1373	241	46	particular	particular	ADJ
ejpam-1373	241	47	m∨	m∨	NOUN
ejpam-1373	241	48	n	n	PRON
ejpam-1373	241	49	exists	exist	VERB
ejpam-1373	241	50	.	.	PUNCT
ejpam-1373	242	1	theorem	theorem	VERB
ejpam-1373	242	2	8	8	NUM
ejpam-1373	242	3	.	.	PUNCT
ejpam-1373	243	1	a	a	DET
ejpam-1373	243	2	convolution	convolution	NOUN
ejpam-1373	243	3	c	c	NOUN
ejpam-1373	243	4	is	be	AUX
ejpam-1373	243	5	multiplicative	multiplicative	ADJ
ejpam-1373	243	6	if	if	SCONJ
ejpam-1373	243	7	and	and	CCONJ
ejpam-1373	243	8	only	only	ADV
ejpam-1373	243	9	if	if	SCONJ
ejpam-1373	243	10	the	the	DET
ejpam-1373	243	11	following	following	NOUN
ejpam-1373	243	12	are	be	AUX
ejpam-1373	243	13	satisfied	satisfied	ADJ
ejpam-1373	243	14	in	in	ADP
ejpam-1373	243	15	the	the	DET
ejpam-1373	243	16	poset	poset	NOUN
ejpam-1373	243	17	(	(	PUNCT
ejpam-1373	243	18	z+,≤c	z+,≤c	NUM
ejpam-1373	243	19	)	)	PUNCT
ejpam-1373	243	20	.	.	PUNCT
ejpam-1373	244	1	(	(	PUNCT
ejpam-1373	244	2	1	1	NUM
ejpam-1373	244	3	)	)	PUNCT
ejpam-1373	244	4	.	.	PUNCT
ejpam-1373	245	1	for	for	ADP
ejpam-1373	245	2	any	any	DET
ejpam-1373	245	3	m	m	NOUN
ejpam-1373	245	4	,	,	PUNCT
ejpam-1373	245	5	n	n	PRON
ejpam-1373	245	6	∈	∈	NOUN
ejpam-1373	245	7	z+	z+	NUM
ejpam-1373	245	8	with	with	ADP
ejpam-1373	245	9	(	(	PUNCT
ejpam-1373	245	10	m	m	PROPN
ejpam-1373	245	11	,	,	PUNCT
ejpam-1373	245	12	n	n	CCONJ
ejpam-1373	245	13	)	)	PUNCT
ejpam-1373	245	14	=	=	SYM
ejpam-1373	245	15	1	1	NUM
ejpam-1373	245	16	,	,	PUNCT
ejpam-1373	245	17	m∨	m∨	NOUN
ejpam-1373	245	18	n	n	PRON
ejpam-1373	245	19	exists	exist	VERB
ejpam-1373	245	20	and	and	CCONJ
ejpam-1373	245	21	is	be	AUX
ejpam-1373	245	22	equal	equal	ADJ
ejpam-1373	245	23	to	to	ADP
ejpam-1373	245	24	the	the	DET
ejpam-1373	245	25	product	product	NOUN
ejpam-1373	245	26	mn	mn	PROPN
ejpam-1373	245	27	(	(	PUNCT
ejpam-1373	245	28	2	2	NUM
ejpam-1373	245	29	)	)	PUNCT
ejpam-1373	245	30	.	.	PUNCT
ejpam-1373	246	1	for	for	ADP
ejpam-1373	246	2	any	any	DET
ejpam-1373	246	3	x	x	SYM
ejpam-1373	246	4	,	,	PUNCT
ejpam-1373	246	5	m	m	PROPN
ejpam-1373	246	6	and	and	CCONJ
ejpam-1373	246	7	n	n	PRON
ejpam-1373	246	8	∈	∈	NOUN
ejpam-1373	246	9	z+	z+	NUM
ejpam-1373	246	10	with	with	ADP
ejpam-1373	246	11	(	(	PUNCT
ejpam-1373	246	12	x	x	INTJ
ejpam-1373	246	13	,	,	PUNCT
ejpam-1373	246	14	m	m	NOUN
ejpam-1373	246	15	)	)	PUNCT
ejpam-1373	246	16	=	=	PUNCT
ejpam-1373	246	17	1=	1=	X
ejpam-1373	246	18	(	(	PUNCT
ejpam-1373	246	19	x	x	NOUN
ejpam-1373	246	20	,	,	PUNCT
ejpam-1373	246	21	n	n	CCONJ
ejpam-1373	246	22	)	)	PUNCT
ejpam-1373	246	23	,	,	PUNCT
ejpam-1373	246	24	x	x	X
ejpam-1373	246	25	∨	∨	X
ejpam-1373	246	26	(	(	PUNCT
ejpam-1373	246	27	m∧	m∧	NOUN
ejpam-1373	246	28	n	n	CCONJ
ejpam-1373	246	29	)	)	PUNCT
ejpam-1373	246	30	=	=	SYM
ejpam-1373	246	31	(	(	PUNCT
ejpam-1373	246	32	x	x	X
ejpam-1373	246	33	∨m)∧	∨m)∧	X
ejpam-1373	246	34	(	(	PUNCT
ejpam-1373	246	35	x	x	PROPN
ejpam-1373	246	36	∨	∨	NUM
ejpam-1373	246	37	n	n	CCONJ
ejpam-1373	246	38	)	)	PUNCT
ejpam-1373	246	39	.	.	PUNCT
ejpam-1373	247	1	proof	proof	NOUN
ejpam-1373	247	2	.	.	PUNCT
ejpam-1373	248	1	let	let	VERB
ejpam-1373	248	2	c	c	PRON
ejpam-1373	248	3	be	be	AUX
ejpam-1373	248	4	a	a	DET
ejpam-1373	248	5	convolution	convolution	NOUN
ejpam-1373	248	6	and	and	CCONJ
ejpam-1373	248	7	≤c	≤c	PROPN
ejpam-1373	248	8	be	be	VERB
ejpam-1373	248	9	the	the	DET
ejpam-1373	248	10	corresponding	corresponding	ADJ
ejpam-1373	248	11	partial	partial	ADJ
ejpam-1373	248	12	order	order	NOUN
ejpam-1373	248	13	on	on	ADP
ejpam-1373	248	14	z+	z+	NUM
ejpam-1373	248	15	.	.	PUNCT
ejpam-1373	248	16	suppose	suppose	VERB
ejpam-1373	248	17	that	that	SCONJ
ejpam-1373	248	18	c	c	PROPN
ejpam-1373	248	19	is	be	AUX
ejpam-1373	248	20	multiplicative	multiplicative	ADJ
ejpam-1373	248	21	.	.	PUNCT
ejpam-1373	249	1	then	then	ADV
ejpam-1373	249	2	by	by	ADP
ejpam-1373	249	3	theorem	theorem	NOUN
ejpam-1373	249	4	7	7	NUM
ejpam-1373	249	5	,	,	PUNCT
ejpam-1373	249	6	(	(	PUNCT
ejpam-1373	249	7	1	1	X
ejpam-1373	249	8	)	)	PUNCT
ejpam-1373	249	9	holds	hold	VERB
ejpam-1373	249	10	good	good	ADJ
ejpam-1373	249	11	.	.	PUNCT
ejpam-1373	250	1	to	to	PART
ejpam-1373	250	2	prove	prove	VERB
ejpam-1373	250	3	(	(	PUNCT
ejpam-1373	250	4	2	2	NUM
ejpam-1373	250	5	)	)	PUNCT
ejpam-1373	250	6	,	,	PUNCT
ejpam-1373	250	7	let	let	VERB
ejpam-1373	250	8	x	x	PRON
ejpam-1373	250	9	,	,	PUNCT
ejpam-1373	250	10	m	m	VERB
ejpam-1373	250	11	and	and	CCONJ
ejpam-1373	250	12	n	n	PRON
ejpam-1373	250	13	∈	∈	PROPN
ejpam-1373	250	14	z+	z+	NUM
ejpam-1373	250	15	such	such	ADJ
ejpam-1373	250	16	that	that	SCONJ
ejpam-1373	250	17	(	(	PUNCT
ejpam-1373	250	18	x	x	X
ejpam-1373	250	19	,	,	PUNCT
ejpam-1373	250	20	m	m	NOUN
ejpam-1373	250	21	)	)	PUNCT
ejpam-1373	251	1	=	=	PUNCT
ejpam-1373	251	2	1=	1=	X
ejpam-1373	251	3	(	(	PUNCT
ejpam-1373	251	4	x	x	NOUN
ejpam-1373	251	5	,	,	PUNCT
ejpam-1373	251	6	n	n	CCONJ
ejpam-1373	251	7	)	)	PUNCT
ejpam-1373	251	8	.	.	PUNCT
ejpam-1373	252	1	by	by	ADP
ejpam-1373	252	2	(	(	PUNCT
ejpam-1373	252	3	1	1	NUM
ejpam-1373	252	4	)	)	PUNCT
ejpam-1373	252	5	,	,	PUNCT
ejpam-1373	252	6	x	x	X
ejpam-1373	252	7	∨m	∨m	NOUN
ejpam-1373	252	8	and	and	CCONJ
ejpam-1373	252	9	x	x	SYM
ejpam-1373	252	10	∨	∨	NOUN
ejpam-1373	252	11	n	n	CCONJ
ejpam-1373	252	12	exist	exist	VERB
ejpam-1373	252	13	and	and	CCONJ
ejpam-1373	252	14	are	be	AUX
ejpam-1373	252	15	equal	equal	ADJ
ejpam-1373	252	16	to	to	ADP
ejpam-1373	252	17	xm	xm	PROPN
ejpam-1373	252	18	and	and	CCONJ
ejpam-1373	252	19	xn	xn	PROPN
ejpam-1373	252	20	respectively	respectively	ADV
ejpam-1373	252	21	in	in	ADP
ejpam-1373	252	22	(	(	PUNCT
ejpam-1373	252	23	z+,≤	z+,≤	PROPN
ejpam-1373	252	24	c	c	NOUN
ejpam-1373	252	25	)	)	PUNCT
ejpam-1373	252	26	.	.	PUNCT
ejpam-1373	253	1	suppose	suppose	VERB
ejpam-1373	253	2	that	that	SCONJ
ejpam-1373	253	3	(	(	PUNCT
ejpam-1373	253	4	x∨m)∧(x∨n	x∨m)∧(x∨n	NUM
ejpam-1373	253	5	)	)	PUNCT
ejpam-1373	253	6	exists	exist	VERB
ejpam-1373	253	7	.	.	PUNCT
ejpam-1373	254	1	then	then	ADV
ejpam-1373	254	2	θ(xm)(p)∧θ(xn)(p	θ(xm)(p)∧θ(xn)(p	NOUN
ejpam-1373	254	3	)	)	PUNCT
ejpam-1373	254	4	exists	exist	VERB
ejpam-1373	254	5	in	in	ADP
ejpam-1373	254	6	(	(	PUNCT
ejpam-1373	254	7	n	n	X
ejpam-1373	254	8	,	,	PUNCT
ejpam-1373	254	9	≤p	≤p	PROPN
ejpam-1373	254	10	c	c	PROPN
ejpam-1373	254	11	)	)	PUNCT
ejpam-1373	254	12	for	for	ADP
ejpam-1373	254	13	any	any	DET
ejpam-1373	254	14	prime	prime	ADJ
ejpam-1373	254	15	p.	p.	NOUN
ejpam-1373	254	16	we	we	PRON
ejpam-1373	254	17	have	have	VERB
ejpam-1373	254	18	θ(xm)(p	θ(xm)(p	NOUN
ejpam-1373	254	19	)	)	PUNCT
ejpam-1373	255	1	=	=	SYM
ejpam-1373	255	2	θ(x)(p)+	θ(x)(p)+	PROPN
ejpam-1373	255	3	θ(m)(p	θ(m)(p	NOUN
ejpam-1373	255	4	)	)	PUNCT
ejpam-1373	255	5	and	and	CCONJ
ejpam-1373	255	6	θ(xn)(p	θ(xn)(p	NOUN
ejpam-1373	255	7	)	)	PUNCT
ejpam-1373	255	8	=	=	PUNCT
ejpam-1373	255	9	θ(x)(p	θ(x)(p	X
ejpam-1373	255	10	)	)	PUNCT
ejpam-1373	255	11	+	+	CCONJ
ejpam-1373	255	12	θ(n)(p	θ(n)(p	NOUN
ejpam-1373	255	13	)	)	PUNCT
ejpam-1373	255	14	.	.	PUNCT
ejpam-1373	256	1	if	if	SCONJ
ejpam-1373	256	2	θ(x)(p	θ(x)(p	VERB
ejpam-1373	256	3	)	)	PUNCT
ejpam-1373	256	4	=	=	SYM
ejpam-1373	256	5	0	0	NUM
ejpam-1373	256	6	,	,	PUNCT
ejpam-1373	256	7	then	then	ADV
ejpam-1373	256	8	θ(m)(p	θ(m)(p	NUM
ejpam-1373	256	9	)	)	PUNCT
ejpam-1373	256	10	=	=	SYM
ejpam-1373	256	11	θ(xm)(p	θ(xm)(p	NOUN
ejpam-1373	256	12	)	)	PUNCT
ejpam-1373	256	13	and	and	CCONJ
ejpam-1373	256	14	θ(n)(p	θ(n)(p	NOUN
ejpam-1373	256	15	)	)	PUNCT
ejpam-1373	256	16	=	=	SYM
ejpam-1373	256	17	θ(xn)(p	θ(xn)(p	NOUN
ejpam-1373	256	18	)	)	PUNCT
ejpam-1373	256	19	which	which	PRON
ejpam-1373	256	20	implies	imply	VERB
ejpam-1373	256	21	that	that	SCONJ
ejpam-1373	256	22	θ(m)(p	θ(m)(p	ADJ
ejpam-1373	256	23	)	)	PUNCT
ejpam-1373	256	24	∧	∧	NOUN
ejpam-1373	256	25	θ(n)(p	θ(n)(p	NOUN
ejpam-1373	256	26	)	)	PUNCT
ejpam-1373	256	27	exists	exist	VERB
ejpam-1373	256	28	in	in	ADP
ejpam-1373	256	29	(	(	PUNCT
ejpam-1373	256	30	n	n	X
ejpam-1373	256	31	,	,	PUNCT
ejpam-1373	256	32	≤p	≤p	PROPN
ejpam-1373	256	33	c	c	PROPN
ejpam-1373	256	34	)	)	PUNCT
ejpam-1373	256	35	.	.	PUNCT
ejpam-1373	257	1	on	on	ADP
ejpam-1373	257	2	the	the	DET
ejpam-1373	257	3	other	other	ADJ
ejpam-1373	257	4	hand	hand	NOUN
ejpam-1373	257	5	,	,	PUNCT
ejpam-1373	257	6	if	if	SCONJ
ejpam-1373	257	7	θ(x)(p	θ(x)(p	VERB
ejpam-1373	257	8	)	)	PUNCT
ejpam-1373	257	9	6=	6=	ADP
ejpam-1373	257	10	0	0	NUM
ejpam-1373	257	11	,	,	PUNCT
ejpam-1373	257	12	then	then	ADV
ejpam-1373	257	13	θ(m)(p	θ(m)(p	NUM
ejpam-1373	257	14	)	)	PUNCT
ejpam-1373	257	15	=	=	SYM
ejpam-1373	257	16	0	0	PUNCT
ejpam-1373	257	17	=	=	SYM
ejpam-1373	257	18	θ(n)(p	θ(n)(p	X
ejpam-1373	257	19	)	)	PUNCT
ejpam-1373	257	20	and	and	CCONJ
ejpam-1373	257	21	trivially	trivially	ADV
ejpam-1373	257	22	θ(m)(p	θ(m)(p	ADJ
ejpam-1373	257	23	)	)	PUNCT
ejpam-1373	257	24	∧	∧	NOUN
ejpam-1373	257	25	θ(n)(p	θ(n)(p	NOUN
ejpam-1373	257	26	)	)	PUNCT
ejpam-1373	257	27	exists	exist	VERB
ejpam-1373	257	28	.	.	PUNCT
ejpam-1373	258	1	thus	thus	ADV
ejpam-1373	258	2	θ(m)(p)∧	θ(m)(p)∧	NOUN
ejpam-1373	258	3	θ(n)(p	θ(n)(p	X
ejpam-1373	258	4	)	)	PUNCT
ejpam-1373	258	5	exists	exist	VERB
ejpam-1373	258	6	in	in	ADP
ejpam-1373	258	7	(	(	PUNCT
ejpam-1373	258	8	n	n	X
ejpam-1373	258	9	,	,	PUNCT
ejpam-1373	258	10	≤p	≤p	PROPN
ejpam-1373	258	11	c	c	PROPN
ejpam-1373	258	12	)	)	PUNCT
ejpam-1373	258	13	for	for	ADP
ejpam-1373	258	14	all	all	DET
ejpam-1373	258	15	primes	prime	NOUN
ejpam-1373	259	1	p.	p.	NOUN
ejpam-1373	259	2	again	again	ADV
ejpam-1373	259	3	,	,	PUNCT
ejpam-1373	259	4	m∧	m∧	X
ejpam-1373	259	5	n	n	PRON
ejpam-1373	259	6	exists	exist	VERB
ejpam-1373	259	7	in	in	ADP
ejpam-1373	259	8	(	(	PUNCT
ejpam-1373	259	9	z+,≤c	z+,≤c	PROPN
ejpam-1373	259	10	)	)	PUNCT
ejpam-1373	259	11	.	.	PUNCT
ejpam-1373	260	1	also	also	ADV
ejpam-1373	260	2	,	,	PUNCT
ejpam-1373	260	3	since	since	SCONJ
ejpam-1373	260	4	m∧n≤c	m∧n≤c	ADJ
ejpam-1373	260	5	n	n	CCONJ
ejpam-1373	260	6	,	,	PUNCT
ejpam-1373	260	7	m∧n	m∧n	PROPN
ejpam-1373	260	8	is	be	AUX
ejpam-1373	260	9	a	a	DET
ejpam-1373	260	10	divisor	divisor	NOUN
ejpam-1373	260	11	of	of	ADP
ejpam-1373	260	12	m	m	PRON
ejpam-1373	260	13	and	and	CCONJ
ejpam-1373	260	14	hence	hence	ADV
ejpam-1373	260	15	(	(	PUNCT
ejpam-1373	260	16	x	x	INTJ
ejpam-1373	260	17	,	,	PUNCT
ejpam-1373	260	18	m∧n	m∧n	PROPN
ejpam-1373	260	19	)	)	PUNCT
ejpam-1373	260	20	=	=	SYM
ejpam-1373	261	1	1	1	X
ejpam-1373	261	2	.	.	X
ejpam-1373	261	3	therefore	therefore	ADV
ejpam-1373	261	4	x	x	X
ejpam-1373	261	5	∨	∨	X
ejpam-1373	261	6	(	(	PUNCT
ejpam-1373	261	7	m∧n	m∧n	PROPN
ejpam-1373	261	8	)	)	PUNCT
ejpam-1373	261	9	exists	exist	VERB
ejpam-1373	261	10	.	.	PUNCT
ejpam-1373	262	1	by	by	ADP
ejpam-1373	262	2	evaluating	evaluate	VERB
ejpam-1373	262	3	θ(x	θ(x	PROPN
ejpam-1373	262	4	∨(m∧n))(p	∨(m∧n))(p	PUNCT
ejpam-1373	262	5	)	)	PUNCT
ejpam-1373	262	6	and	and	CCONJ
ejpam-1373	262	7	θ((x	θ((x	VERB
ejpam-1373	262	8	∨m)∧(x	∨m)∧(x	PRON
ejpam-1373	262	9	∨n))(p	∨n))(p	NOUN
ejpam-1373	262	10	)	)	PUNCT
ejpam-1373	262	11	,	,	PUNCT
ejpam-1373	262	12	we	we	PRON
ejpam-1373	262	13	get	get	VERB
ejpam-1373	262	14	that	that	SCONJ
ejpam-1373	262	15	they	they	PRON
ejpam-1373	262	16	are	be	AUX
ejpam-1373	262	17	equal	equal	ADJ
ejpam-1373	262	18	for	for	ADP
ejpam-1373	262	19	all	all	DET
ejpam-1373	262	20	primes	prime	NOUN
ejpam-1373	262	21	p.	p.	NOUN
ejpam-1373	262	22	thus	thus	ADV
ejpam-1373	262	23	x∨(m∧n	x∨(m∧n	NOUN
ejpam-1373	262	24	)	)	PUNCT
ejpam-1373	262	25	exists	exist	VERB
ejpam-1373	262	26	and	and	CCONJ
ejpam-1373	262	27	is	be	AUX
ejpam-1373	262	28	equal	equal	ADJ
ejpam-1373	262	29	to	to	ADP
ejpam-1373	262	30	(	(	PUNCT
ejpam-1373	262	31	x∨m)∧(x∨n	x∨m)∧(x∨n	NUM
ejpam-1373	262	32	)	)	PUNCT
ejpam-1373	262	33	.	.	PUNCT
ejpam-1373	263	1	the	the	DET
ejpam-1373	263	2	other	other	ADJ
ejpam-1373	263	3	implication	implication	NOUN
ejpam-1373	263	4	can	can	AUX
ejpam-1373	263	5	also	also	ADV
ejpam-1373	263	6	be	be	AUX
ejpam-1373	263	7	proved	prove	VERB
ejpam-1373	263	8	similarly	similarly	ADV
ejpam-1373	263	9	.	.	PUNCT
ejpam-1373	264	1	thus	thus	ADV
ejpam-1373	264	2	the	the	DET
ejpam-1373	264	3	condition	condition	NOUN
ejpam-1373	264	4	(	(	PUNCT
ejpam-1373	264	5	2	2	X
ejpam-1373	264	6	)	)	PUNCT
ejpam-1373	264	7	holds	hold	VERB
ejpam-1373	264	8	good	good	ADJ
ejpam-1373	264	9	.	.	PUNCT
ejpam-1373	265	1	conversely	conversely	ADV
ejpam-1373	265	2	suppose	suppose	VERB
ejpam-1373	265	3	that	that	SCONJ
ejpam-1373	265	4	the	the	DET
ejpam-1373	265	5	conditions	condition	NOUN
ejpam-1373	265	6	(	(	PUNCT
ejpam-1373	265	7	1	1	NUM
ejpam-1373	265	8	)	)	PUNCT
ejpam-1373	265	9	and	and	CCONJ
ejpam-1373	265	10	(	(	PUNCT
ejpam-1373	265	11	2	2	X
ejpam-1373	265	12	)	)	PUNCT
ejpam-1373	265	13	are	be	AUX
ejpam-1373	265	14	satisfied	satisfied	ADJ
ejpam-1373	265	15	in	in	ADP
ejpam-1373	265	16	(	(	PUNCT
ejpam-1373	265	17	z+,≤c	z+,≤c	PROPN
ejpam-1373	265	18	)	)	PUNCT
ejpam-1373	265	19	.	.	PUNCT
ejpam-1373	266	1	to	to	PART
ejpam-1373	266	2	prove	prove	VERB
ejpam-1373	266	3	the	the	DET
ejpam-1373	266	4	multiplicativity	multiplicativity	NOUN
ejpam-1373	266	5	of	of	ADP
ejpam-1373	266	6	c	c	NOUN
ejpam-1373	266	7	,	,	PUNCT
ejpam-1373	266	8	let	let	VERB
ejpam-1373	266	9	m	m	PRON
ejpam-1373	266	10	and	and	CCONJ
ejpam-1373	266	11	n	n	PRON
ejpam-1373	266	12	∈	∈	PROPN
ejpam-1373	266	13	z+	z+	NUM
ejpam-1373	266	14	such	such	ADJ
ejpam-1373	266	15	that	that	SCONJ
ejpam-1373	266	16	(	(	PUNCT
ejpam-1373	266	17	m	m	NOUN
ejpam-1373	266	18	,	,	PUNCT
ejpam-1373	266	19	n	n	CCONJ
ejpam-1373	266	20	)	)	PUNCT
ejpam-1373	266	21	=	=	SYM
ejpam-1373	267	1	1	1	X
ejpam-1373	267	2	.	.	PUNCT
ejpam-1373	267	3	then	then	ADV
ejpam-1373	267	4	by	by	ADP
ejpam-1373	267	5	(	(	PUNCT
ejpam-1373	267	6	1	1	NUM
ejpam-1373	267	7	)	)	PUNCT
ejpam-1373	267	8	,	,	PUNCT
ejpam-1373	267	9	m	m	VERB
ejpam-1373	267	10	∨	∨	NOUN
ejpam-1373	267	11	n	n	PRON
ejpam-1373	267	12	exists	exist	VERB
ejpam-1373	267	13	and	and	CCONJ
ejpam-1373	267	14	is	be	AUX
ejpam-1373	267	15	equal	equal	ADJ
ejpam-1373	267	16	to	to	ADP
ejpam-1373	267	17	mn	mn	PROPN
ejpam-1373	267	18	.	.	PUNCT
ejpam-1373	268	1	in	in	ADP
ejpam-1373	268	2	particular	particular	ADJ
ejpam-1373	268	3	m≤c	m≤c	PROPN
ejpam-1373	268	4	mn	mn	PROPN
ejpam-1373	268	5	and	and	CCONJ
ejpam-1373	268	6	n≤c	n≤c	PROPN
ejpam-1373	268	7	mn	mn	PROPN
ejpam-1373	268	8	.	.	PUNCT
ejpam-1373	268	9	x	x	X
ejpam-1373	268	10	∈	∈	PROPN
ejpam-1373	268	11	c	c	X
ejpam-1373	268	12	(	(	PUNCT
ejpam-1373	268	13	m)c	m)c	X
ejpam-1373	268	14	(	(	PUNCT
ejpam-1373	268	15	n	n	CCONJ
ejpam-1373	268	16	)	)	PUNCT
ejpam-1373	269	1	=	=	NOUN
ejpam-1373	269	2	⇒	⇒	NOUN
ejpam-1373	269	3	x	x	X
ejpam-1373	269	4	=	=	SYM
ejpam-1373	269	5	ab	ab	PROPN
ejpam-1373	269	6	,	,	PUNCT
ejpam-1373	269	7	a	a	DET
ejpam-1373	269	8	∈	∈	PROPN
ejpam-1373	269	9	c	c	X
ejpam-1373	269	10	(	(	PUNCT
ejpam-1373	269	11	m	m	NOUN
ejpam-1373	269	12	)	)	PUNCT
ejpam-1373	269	13	and	and	CCONJ
ejpam-1373	269	14	b	b	X
ejpam-1373	269	15	∈	∈	PROPN
ejpam-1373	269	16	c	c	X
ejpam-1373	269	17	(	(	PUNCT
ejpam-1373	269	18	n	n	CCONJ
ejpam-1373	269	19	)	)	PUNCT
ejpam-1373	270	1	=	=	NOUN
ejpam-1373	270	2	⇒	⇒	NOUN
ejpam-1373	270	3	x	x	PUNCT
ejpam-1373	270	4	=	=	PUNCT
ejpam-1373	270	5	ab	ab	PROPN
ejpam-1373	270	6	=	=	PUNCT
ejpam-1373	270	7	a	a	DET
ejpam-1373	270	8	∨	∨	NUM
ejpam-1373	270	9	b	b	NOUN
ejpam-1373	270	10	,	,	PUNCT
ejpam-1373	270	11	(	(	PUNCT
ejpam-1373	270	12	since	since	SCONJ
ejpam-1373	270	13	(	(	PUNCT
ejpam-1373	270	14	a	a	DET
ejpam-1373	270	15	,	,	PUNCT
ejpam-1373	270	16	b	b	NOUN
ejpam-1373	270	17	)	)	PUNCT
ejpam-1373	270	18	=	=	SYM
ejpam-1373	270	19	1	1	X
ejpam-1373	270	20	)	)	PUNCT
ejpam-1373	271	1	=	=	NOUN
ejpam-1373	271	2	⇒	⇒	NOUN
ejpam-1373	271	3	x	x	PROPN
ejpam-1373	271	4	≤c	≤c	PROPN
ejpam-1373	271	5	m∨	m∨	PROPN
ejpam-1373	271	6	n=	n=	PROPN
ejpam-1373	271	7	mn	mn	PROPN
ejpam-1373	271	8	(	(	PUNCT
ejpam-1373	271	9	since	since	SCONJ
ejpam-1373	271	10	a	a	DET
ejpam-1373	271	11	≤c	≤c	PROPN
ejpam-1373	271	12	m	m	PROPN
ejpam-1373	271	13	,	,	PUNCT
ejpam-1373	271	14	b	b	PROPN
ejpam-1373	271	15	≤c	≤c	PROPN
ejpam-1373	271	16	n	n	CCONJ
ejpam-1373	271	17	)	)	PUNCT
ejpam-1373	271	18	=	=	NOUN
ejpam-1373	271	19	⇒	⇒	NOUN
ejpam-1373	271	20	x	x	X
ejpam-1373	271	21	∈	∈	PROPN
ejpam-1373	271	22	c	c	X
ejpam-1373	271	23	(	(	PUNCT
ejpam-1373	271	24	mn	mn	PROPN
ejpam-1373	271	25	)	)	PUNCT
ejpam-1373	271	26	therefore	therefore	ADV
ejpam-1373	271	27	c	c	PROPN
ejpam-1373	271	28	(	(	PUNCT
ejpam-1373	271	29	m)c	m)c	X
ejpam-1373	271	30	(	(	PUNCT
ejpam-1373	271	31	n)⊆	n)⊆	PROPN
ejpam-1373	271	32	c	c	PROPN
ejpam-1373	271	33	(	(	PUNCT
ejpam-1373	271	34	n	n	CCONJ
ejpam-1373	271	35	)	)	PUNCT
ejpam-1373	271	36	.	.	PUNCT
ejpam-1373	272	1	on	on	ADP
ejpam-1373	272	2	the	the	DET
ejpam-1373	272	3	other	other	ADJ
ejpam-1373	272	4	hand	hand	NOUN
ejpam-1373	272	5	,	,	PUNCT
ejpam-1373	272	6	let	let	VERB
ejpam-1373	272	7	x	x	SYM
ejpam-1373	272	8	∈	∈	PROPN
ejpam-1373	272	9	c	c	X
ejpam-1373	272	10	(	(	PUNCT
ejpam-1373	272	11	mn	mn	PROPN
ejpam-1373	272	12	)	)	PUNCT
ejpam-1373	272	13	.	.	PUNCT
ejpam-1373	273	1	put	put	VERB
ejpam-1373	273	2	y	y	NOUN
ejpam-1373	273	3	=	=	PUNCT
ejpam-1373	273	4	(	(	PUNCT
ejpam-1373	273	5	x	x	INTJ
ejpam-1373	273	6	,	,	PUNCT
ejpam-1373	273	7	m	m	PROPN
ejpam-1373	273	8	)	)	PUNCT
ejpam-1373	273	9	and	and	CCONJ
ejpam-1373	273	10	z	z	NOUN
ejpam-1373	273	11	=	=	SYM
ejpam-1373	273	12	(	(	PUNCT
ejpam-1373	273	13	x	x	NOUN
ejpam-1373	273	14	,	,	PUNCT
ejpam-1373	273	15	n	n	CCONJ
ejpam-1373	273	16	)	)	PUNCT
ejpam-1373	273	17	.	.	PUNCT
ejpam-1373	274	1	since	since	SCONJ
ejpam-1373	274	2	(	(	PUNCT
ejpam-1373	274	3	m	m	PROPN
ejpam-1373	274	4	,	,	PUNCT
ejpam-1373	274	5	n	n	CCONJ
ejpam-1373	274	6	)	)	PUNCT
ejpam-1373	274	7	=	=	SYM
ejpam-1373	274	8	1	1	NUM
ejpam-1373	274	9	,	,	PUNCT
ejpam-1373	274	10	it	it	PRON
ejpam-1373	274	11	follows	follow	VERB
ejpam-1373	274	12	that	that	SCONJ
ejpam-1373	274	13	x	x	X
ejpam-1373	274	14	=	=	PUNCT
ejpam-1373	274	15	yz	yz	PROPN
ejpam-1373	274	16	and	and	CCONJ
ejpam-1373	274	17	(	(	PUNCT
ejpam-1373	274	18	y	y	PROPN
ejpam-1373	274	19	,	,	PUNCT
ejpam-1373	274	20	z	z	NOUN
ejpam-1373	274	21	)	)	PUNCT
ejpam-1373	274	22	=	=	SYM
ejpam-1373	274	23	1	1	NUM
ejpam-1373	274	24	=	=	SYM
ejpam-1373	274	25	(	(	PUNCT
ejpam-1373	274	26	y	y	PROPN
ejpam-1373	274	27	,	,	PUNCT
ejpam-1373	274	28	n	n	CCONJ
ejpam-1373	274	29	)	)	PUNCT
ejpam-1373	274	30	=	=	SYM
ejpam-1373	274	31	(	(	PUNCT
ejpam-1373	274	32	z	z	NOUN
ejpam-1373	274	33	,	,	PUNCT
ejpam-1373	274	34	m	m	NOUN
ejpam-1373	274	35	)	)	PUNCT
ejpam-1373	275	1	=	=	SYM
ejpam-1373	275	2	(	(	PUNCT
ejpam-1373	275	3	m	m	PROPN
ejpam-1373	275	4	,	,	PUNCT
ejpam-1373	275	5	n	n	CCONJ
ejpam-1373	275	6	)	)	PUNCT
ejpam-1373	275	7	references	reference	NOUN
ejpam-1373	275	8	433	433	NUM
ejpam-1373	275	9	and	and	CCONJ
ejpam-1373	275	10	hence	hence	ADV
ejpam-1373	275	11	m	m	VERB
ejpam-1373	275	12	∨	∨	NUM
ejpam-1373	275	13	n	n	PROPN
ejpam-1373	275	14	=	=	PROPN
ejpam-1373	275	15	mn	mn	PROPN
ejpam-1373	275	16	and	and	CCONJ
ejpam-1373	275	17	m	m	PROPN
ejpam-1373	275	18	∨	∨	NOUN
ejpam-1373	275	19	z	z	PROPN
ejpam-1373	275	20	=	=	SYM
ejpam-1373	275	21	mz	mz	PROPN
ejpam-1373	275	22	.	.	PROPN
ejpam-1373	276	1	since	since	SCONJ
ejpam-1373	276	2	m	m	PROPN
ejpam-1373	276	3	≤c	≤c	PROPN
ejpam-1373	276	4	m	m	PROPN
ejpam-1373	276	5	∨	∨	NUM
ejpam-1373	276	6	n	n	PROPN
ejpam-1373	276	7	=	=	SYM
ejpam-1373	276	8	mn	mn	PROPN
ejpam-1373	276	9	and	and	CCONJ
ejpam-1373	276	10	z	z	PROPN
ejpam-1373	276	11	≤c	≤c	PROPN
ejpam-1373	276	12	y	y	PROPN
ejpam-1373	276	13	∨	∨	NUM
ejpam-1373	276	14	z	z	PROPN
ejpam-1373	276	15	=	=	PUNCT
ejpam-1373	276	16	yz	yz	PROPN
ejpam-1373	276	17	=	=	PUNCT
ejpam-1373	276	18	x	x	PROPN
ejpam-1373	276	19	≤c	≤c	PROPN
ejpam-1373	276	20	mn	mn	PROPN
ejpam-1373	276	21	,	,	PUNCT
ejpam-1373	276	22	we	we	PRON
ejpam-1373	276	23	get	get	VERB
ejpam-1373	276	24	that	that	DET
ejpam-1373	276	25	m∨	m∨	PROPN
ejpam-1373	276	26	z	z	PROPN
ejpam-1373	276	27	≤c	≤c	PROPN
ejpam-1373	276	28	mn=	mn=	PROPN
ejpam-1373	276	29	m∨n	m∨n	PROPN
ejpam-1373	276	30	.	.	PUNCT
ejpam-1373	277	1	therefore	therefore	ADV
ejpam-1373	277	2	(	(	PUNCT
ejpam-1373	277	3	m∨	m∨	PROPN
ejpam-1373	277	4	z)∧	z)∧	PROPN
ejpam-1373	277	5	(	(	PUNCT
ejpam-1373	277	6	m∨n	m∨n	PROPN
ejpam-1373	277	7	)	)	PUNCT
ejpam-1373	277	8	exists	exist	VERB
ejpam-1373	277	9	and	and	CCONJ
ejpam-1373	277	10	is	be	AUX
ejpam-1373	277	11	equal	equal	ADJ
ejpam-1373	277	12	to	to	ADP
ejpam-1373	277	13	m∨	m∨	PROPN
ejpam-1373	277	14	z.	z.	PROPN
ejpam-1373	277	15	by	by	ADP
ejpam-1373	277	16	(	(	PUNCT
ejpam-1373	277	17	2	2	NUM
ejpam-1373	277	18	)	)	PUNCT
ejpam-1373	277	19	,	,	PUNCT
ejpam-1373	277	20	m∨	m∨	PROPN
ejpam-1373	277	21	(	(	PUNCT
ejpam-1373	277	22	z	z	PROPN
ejpam-1373	277	23	∧	∧	PROPN
ejpam-1373	277	24	n	n	CCONJ
ejpam-1373	277	25	)	)	PUNCT
ejpam-1373	277	26	exists	exist	VERB
ejpam-1373	277	27	and	and	CCONJ
ejpam-1373	277	28	m∨	m∨	PROPN
ejpam-1373	277	29	z	z	PROPN
ejpam-1373	277	30	=	=	PUNCT
ejpam-1373	277	31	(	(	PUNCT
ejpam-1373	277	32	m∨	m∨	PROPN
ejpam-1373	277	33	z)∧	z)∧	PROPN
ejpam-1373	277	34	(	(	PUNCT
ejpam-1373	277	35	m∨	m∨	NOUN
ejpam-1373	277	36	n	n	CCONJ
ejpam-1373	277	37	)	)	PUNCT
ejpam-1373	278	1	=	=	SYM
ejpam-1373	278	2	m∨	m∨	NOUN
ejpam-1373	278	3	(	(	PUNCT
ejpam-1373	278	4	z	z	NOUN
ejpam-1373	278	5	∧	∧	PROPN
ejpam-1373	278	6	n	n	CCONJ
ejpam-1373	278	7	)	)	PUNCT
ejpam-1373	278	8	=	=	SYM
ejpam-1373	278	9	m(z	m(z	PROPN
ejpam-1373	278	10	∧	∧	PROPN
ejpam-1373	278	11	n	n	CCONJ
ejpam-1373	278	12	)	)	PUNCT
ejpam-1373	278	13	.	.	PUNCT
ejpam-1373	279	1	since	since	SCONJ
ejpam-1373	279	2	(	(	PUNCT
ejpam-1373	279	3	z	z	NOUN
ejpam-1373	279	4	,	,	PUNCT
ejpam-1373	279	5	m	m	NOUN
ejpam-1373	279	6	)	)	PUNCT
ejpam-1373	279	7	=	=	SYM
ejpam-1373	279	8	1	1	NUM
ejpam-1373	279	9	,	,	PUNCT
ejpam-1373	279	10	z	z	PROPN
ejpam-1373	279	11	should	should	AUX
ejpam-1373	279	12	divide	divide	VERB
ejpam-1373	279	13	z	z	NOUN
ejpam-1373	279	14	∧	∧	PROPN
ejpam-1373	279	15	n.	n.	NOUN
ejpam-1373	279	16	but	but	CCONJ
ejpam-1373	279	17	z	z	NOUN
ejpam-1373	279	18	∧	∧	PROPN
ejpam-1373	279	19	n≤c	n≤c	PROPN
ejpam-1373	279	20	z	z	PROPN
ejpam-1373	279	21	and	and	CCONJ
ejpam-1373	279	22	hence	hence	ADV
ejpam-1373	279	23	z	z	PROPN
ejpam-1373	279	24	∧	∧	PROPN
ejpam-1373	279	25	n	n	CCONJ
ejpam-1373	279	26	∈	∈	NOUN
ejpam-1373	279	27	c	c	X
ejpam-1373	279	28	(	(	PUNCT
ejpam-1373	279	29	z	z	NOUN
ejpam-1373	279	30	)	)	PUNCT
ejpam-1373	279	31	.	.	PUNCT
ejpam-1373	280	1	therefore	therefore	ADV
ejpam-1373	280	2	z	z	X
ejpam-1373	280	3	=	=	PUNCT
ejpam-1373	280	4	z	z	NOUN
ejpam-1373	280	5	∧	∧	PROPN
ejpam-1373	280	6	n≤c	n≤c	PROPN
ejpam-1373	280	7	n.	n.	PROPN
ejpam-1373	280	8	similarly	similarly	ADV
ejpam-1373	280	9	y	y	PROPN
ejpam-1373	280	10	≤c	≤c	PROPN
ejpam-1373	280	11	m	m	PROPN
ejpam-1373	280	12	and	and	CCONJ
ejpam-1373	280	13	hence	hence	ADV
ejpam-1373	280	14	y	y	PROPN
ejpam-1373	280	15	∈	∈	PROPN
ejpam-1373	280	16	c	c	X
ejpam-1373	280	17	(	(	PUNCT
ejpam-1373	280	18	m	m	NOUN
ejpam-1373	280	19	)	)	PUNCT
ejpam-1373	280	20	and	and	CCONJ
ejpam-1373	280	21	z	z	NOUN
ejpam-1373	280	22	∈	∈	PROPN
ejpam-1373	280	23	c	c	X
ejpam-1373	280	24	(	(	PUNCT
ejpam-1373	280	25	n	n	CCONJ
ejpam-1373	280	26	)	)	PUNCT
ejpam-1373	280	27	and	and	CCONJ
ejpam-1373	280	28	x	x	X
ejpam-1373	280	29	=	=	PUNCT
ejpam-1373	280	30	yz	yz	PROPN
ejpam-1373	280	31	∈	∈	PROPN
ejpam-1373	280	32	c	c	PROPN
ejpam-1373	280	33	(	(	PUNCT
ejpam-1373	280	34	m)c	m)c	X
ejpam-1373	280	35	(	(	PUNCT
ejpam-1373	280	36	n	n	CCONJ
ejpam-1373	280	37	)	)	PUNCT
ejpam-1373	280	38	.	.	PUNCT
ejpam-1373	281	1	therefore	therefore	ADV
ejpam-1373	281	2	c	c	X
ejpam-1373	281	3	(	(	PUNCT
ejpam-1373	281	4	mn)⊆	mn)⊆	PROPN
ejpam-1373	281	5	c	c	PROPN
ejpam-1373	281	6	(	(	PUNCT
ejpam-1373	281	7	m)c	m)c	X
ejpam-1373	281	8	(	(	PUNCT
ejpam-1373	281	9	n).thus	n).thus	PROPN
ejpam-1373	281	10	c	c	PROPN
ejpam-1373	281	11	(	(	PUNCT
ejpam-1373	281	12	mn	mn	PROPN
ejpam-1373	281	13	)	)	PUNCT
ejpam-1373	281	14	=	=	SYM
ejpam-1373	282	1	c	c	X
ejpam-1373	282	2	(	(	PUNCT
ejpam-1373	282	3	m)c	m)c	X
ejpam-1373	282	4	(	(	PUNCT
ejpam-1373	282	5	n	n	CCONJ
ejpam-1373	282	6	)	)	PUNCT
ejpam-1373	282	7	.	.	PUNCT
ejpam-1373	283	1	thus	thus	ADV
ejpam-1373	283	2	c	c	PROPN
ejpam-1373	283	3	is	be	AUX
ejpam-1373	283	4	multiplicative	multiplicative	ADJ
ejpam-1373	283	5	.	.	PUNCT
ejpam-1373	284	1	let	let	VERB
ejpam-1373	284	2	us	we	PRON
ejpam-1373	284	3	recall	recall	VERB
ejpam-1373	284	4	a	a	DET
ejpam-1373	284	5	subset	subset	NOUN
ejpam-1373	284	6	a	a	PRON
ejpam-1373	284	7	of	of	ADP
ejpam-1373	284	8	z+	z+	NUM
ejpam-1373	284	9	is	be	AUX
ejpam-1373	284	10	multiplicatively	multiplicatively	ADV
ejpam-1373	284	11	closed	close	VERB
ejpam-1373	284	12	if	if	SCONJ
ejpam-1373	284	13	the	the	DET
ejpam-1373	284	14	product	product	NOUN
ejpam-1373	284	15	of	of	ADP
ejpam-1373	284	16	any	any	DET
ejpam-1373	284	17	two	two	NUM
ejpam-1373	284	18	relatively	relatively	ADV
ejpam-1373	284	19	prime	prime	ADJ
ejpam-1373	284	20	elements	element	NOUN
ejpam-1373	284	21	of	of	ADP
ejpam-1373	284	22	a	a	PRON
ejpam-1373	284	23	is	be	AUX
ejpam-1373	284	24	again	again	ADV
ejpam-1373	284	25	an	an	DET
ejpam-1373	284	26	element	element	NOUN
ejpam-1373	284	27	of	of	ADP
ejpam-1373	284	28	a.	a.	NOUN
ejpam-1373	284	29	in	in	ADP
ejpam-1373	284	30	the	the	DET
ejpam-1373	284	31	following	following	NOUN
ejpam-1373	284	32	,	,	PUNCT
ejpam-1373	284	33	we	we	PRON
ejpam-1373	284	34	obtain	obtain	VERB
ejpam-1373	284	35	a	a	DET
ejpam-1373	284	36	condition	condition	NOUN
ejpam-1373	284	37	equivalent	equivalent	ADJ
ejpam-1373	284	38	to	to	ADP
ejpam-1373	284	39	(	(	PUNCT
ejpam-1373	284	40	1	1	NUM
ejpam-1373	284	41	)	)	PUNCT
ejpam-1373	284	42	above	above	ADV
ejpam-1373	284	43	.	.	PUNCT
ejpam-1373	285	1	theorem	theorem	VERB
ejpam-1373	285	2	9	9	NUM
ejpam-1373	285	3	.	.	PUNCT
ejpam-1373	286	1	the	the	DET
ejpam-1373	286	2	following	follow	VERB
ejpam-1373	286	3	are	be	AUX
ejpam-1373	286	4	equivalent	equivalent	ADJ
ejpam-1373	286	5	for	for	ADP
ejpam-1373	286	6	any	any	DET
ejpam-1373	286	7	convolution	convolution	NOUN
ejpam-1373	286	8	c	c	NOUN
ejpam-1373	286	9	.	.	PUNCT
ejpam-1373	287	1	(	(	PUNCT
ejpam-1373	287	2	1	1	NUM
ejpam-1373	287	3	)	)	PUNCT
ejpam-1373	287	4	.	.	PUNCT
ejpam-1373	288	1	m∨	m∨	PROPN
ejpam-1373	288	2	n	n	PROPN
ejpam-1373	288	3	exists	exist	VERB
ejpam-1373	288	4	in	in	ADP
ejpam-1373	288	5	(	(	PUNCT
ejpam-1373	288	6	z+,≤c	z+,≤c	PUNCT
ejpam-1373	288	7	)	)	PUNCT
ejpam-1373	288	8	and	and	CCONJ
ejpam-1373	288	9	is	be	AUX
ejpam-1373	288	10	equal	equal	ADJ
ejpam-1373	288	11	to	to	ADP
ejpam-1373	288	12	mn	mn	PROPN
ejpam-1373	288	13	,	,	PUNCT
ejpam-1373	288	14	for	for	ADP
ejpam-1373	288	15	all	all	DET
ejpam-1373	288	16	relatively	relatively	ADV
ejpam-1373	288	17	prime	prime	ADJ
ejpam-1373	288	18	m	m	PROPN
ejpam-1373	288	19	,	,	PUNCT
ejpam-1373	288	20	n	n	CCONJ
ejpam-1373	288	21	in	in	ADP
ejpam-1373	288	22	z+	z+	NUM
ejpam-1373	288	23	.	.	PUNCT
ejpam-1373	289	1	(	(	PUNCT
ejpam-1373	289	2	2	2	NUM
ejpam-1373	289	3	)	)	PUNCT
ejpam-1373	289	4	.	.	PUNCT
ejpam-1373	290	1	(	(	PUNCT
ejpam-1373	290	2	i	i	NOUN
ejpam-1373	290	3	)	)	PUNCT
ejpam-1373	290	4	c	c	PROPN
ejpam-1373	290	5	is	be	AUX
ejpam-1373	290	6	multiplicatively	multiplicatively	ADV
ejpam-1373	290	7	closed	close	VERB
ejpam-1373	290	8	for	for	ADP
ejpam-1373	290	9	all	all	DET
ejpam-1373	290	10	n	n	PRON
ejpam-1373	290	11	∈	∈	NOUN
ejpam-1373	290	12	z+	z+	NUM
ejpam-1373	290	13	and	and	CCONJ
ejpam-1373	290	14	(	(	PUNCT
ejpam-1373	290	15	ii	ii	NOUN
ejpam-1373	290	16	)	)	PUNCT
ejpam-1373	290	17	c	c	PROPN
ejpam-1373	290	18	(	(	PUNCT
ejpam-1373	290	19	m)c	m)c	X
ejpam-1373	290	20	(	(	PUNCT
ejpam-1373	290	21	n)⊆	n)⊆	PROPN
ejpam-1373	290	22	c	c	PROPN
ejpam-1373	290	23	(	(	PUNCT
ejpam-1373	290	24	mn	mn	PROPN
ejpam-1373	290	25	)	)	PUNCT
ejpam-1373	291	1	whenever	whenever	SCONJ
ejpam-1373	291	2	(	(	PUNCT
ejpam-1373	291	3	m	m	NOUN
ejpam-1373	291	4	,	,	PUNCT
ejpam-1373	291	5	n	n	CCONJ
ejpam-1373	291	6	)	)	PUNCT
ejpam-1373	291	7	=	=	SYM
ejpam-1373	291	8	1	1	X
ejpam-1373	291	9	.	.	PUNCT
ejpam-1373	291	10	proof	proof	NOUN
ejpam-1373	291	11	.	.	PUNCT
ejpam-1373	292	1	(	(	PUNCT
ejpam-1373	292	2	1)=⇒(2	1)=⇒(2	NUM
ejpam-1373	292	3	):	):	PUNCT
ejpam-1373	292	4	let	let	VERB
ejpam-1373	292	5	n	n	PRON
ejpam-1373	292	6	∈	∈	PROPN
ejpam-1373	292	7	z+	z+	NUM
ejpam-1373	292	8	and	and	CCONJ
ejpam-1373	292	9	x	x	X
ejpam-1373	292	10	,	,	PUNCT
ejpam-1373	292	11	y	y	PROPN
ejpam-1373	292	12	∈	∈	PROPN
ejpam-1373	292	13	c	c	X
ejpam-1373	292	14	(	(	PUNCT
ejpam-1373	292	15	n	n	CCONJ
ejpam-1373	292	16	)	)	PUNCT
ejpam-1373	292	17	such	such	ADJ
ejpam-1373	292	18	that	that	SCONJ
ejpam-1373	292	19	(	(	PUNCT
ejpam-1373	292	20	x	x	X
ejpam-1373	292	21	,	,	PUNCT
ejpam-1373	292	22	y	y	PROPN
ejpam-1373	292	23	)	)	PUNCT
ejpam-1373	292	24	=	=	SYM
ejpam-1373	293	1	1	1	X
ejpam-1373	293	2	.	.	PUNCT
ejpam-1373	293	3	then	then	ADV
ejpam-1373	293	4	x	x	X
ejpam-1373	293	5	∨	∨	PROPN
ejpam-1373	293	6	y	y	PROPN
ejpam-1373	293	7	exists	exist	VERB
ejpam-1373	293	8	and	and	CCONJ
ejpam-1373	293	9	is	be	AUX
ejpam-1373	293	10	equal	equal	ADJ
ejpam-1373	293	11	to	to	ADP
ejpam-1373	293	12	x	x	PROPN
ejpam-1373	293	13	y	y	PROPN
ejpam-1373	293	14	.	.	PUNCT
ejpam-1373	294	1	since	since	SCONJ
ejpam-1373	294	2	x	x	PROPN
ejpam-1373	294	3	≤c	≤c	PROPN
ejpam-1373	294	4	n	n	PROPN
ejpam-1373	294	5	and	and	CCONJ
ejpam-1373	294	6	y	y	PROPN
ejpam-1373	294	7	≤c	≤c	PROPN
ejpam-1373	294	8	n	n	CCONJ
ejpam-1373	294	9	,	,	PUNCT
ejpam-1373	294	10	we	we	PRON
ejpam-1373	294	11	get	get	VERB
ejpam-1373	294	12	that	that	PRON
ejpam-1373	294	13	x	x	PROPN
ejpam-1373	294	14	y	y	PROPN
ejpam-1373	294	15	=	=	SYM
ejpam-1373	294	16	x∨	x∨	PROPN
ejpam-1373	294	17	y	y	PROPN
ejpam-1373	294	18	≤c	≤c	PROPN
ejpam-1373	294	19	n	n	PROPN
ejpam-1373	294	20	and	and	CCONJ
ejpam-1373	294	21	hence	hence	ADV
ejpam-1373	294	22	x	x	VERB
ejpam-1373	294	23	y	y	PROPN
ejpam-1373	294	24	∈	∈	PROPN
ejpam-1373	294	25	c	c	X
ejpam-1373	294	26	(	(	PUNCT
ejpam-1373	294	27	n	n	CCONJ
ejpam-1373	294	28	)	)	PUNCT
ejpam-1373	294	29	.	.	PUNCT
ejpam-1373	295	1	therefore	therefore	ADV
ejpam-1373	295	2	c	c	PROPN
ejpam-1373	295	3	(	(	PUNCT
ejpam-1373	295	4	n	n	CCONJ
ejpam-1373	295	5	)	)	PUNCT
ejpam-1373	295	6	is	be	AUX
ejpam-1373	295	7	multiplicatively	multiplicatively	ADV
ejpam-1373	295	8	closed	closed	ADJ
ejpam-1373	295	9	.	.	PUNCT
ejpam-1373	296	1	next	next	ADV
ejpam-1373	296	2	,	,	PUNCT
ejpam-1373	296	3	let	let	VERB
ejpam-1373	296	4	m	m	PRON
ejpam-1373	296	5	,	,	PUNCT
ejpam-1373	296	6	n	n	PRON
ejpam-1373	296	7	∈	∈	PROPN
ejpam-1373	296	8	z+	z+	NUM
ejpam-1373	296	9	such	such	ADJ
ejpam-1373	296	10	that	that	SCONJ
ejpam-1373	296	11	(	(	PUNCT
ejpam-1373	296	12	m	m	NOUN
ejpam-1373	296	13	,	,	PUNCT
ejpam-1373	296	14	n	n	CCONJ
ejpam-1373	296	15	)	)	PUNCT
ejpam-1373	296	16	=	=	SYM
ejpam-1373	297	1	1	1	X
ejpam-1373	297	2	.	.	PUNCT
ejpam-1373	298	1	then	then	ADV
ejpam-1373	298	2	,	,	PUNCT
ejpam-1373	298	3	m∨	m∨	PROPN
ejpam-1373	298	4	n	n	PRON
ejpam-1373	298	5	exists	exist	VERB
ejpam-1373	298	6	and	and	CCONJ
ejpam-1373	298	7	is	be	AUX
ejpam-1373	298	8	equal	equal	ADJ
ejpam-1373	298	9	to	to	ADP
ejpam-1373	298	10	mn	mn	PROPN
ejpam-1373	298	11	and	and	CCONJ
ejpam-1373	298	12	x	x	AUX
ejpam-1373	298	13	∈	∈	PROPN
ejpam-1373	298	14	c	c	X
ejpam-1373	298	15	(	(	PUNCT
ejpam-1373	298	16	m	m	NOUN
ejpam-1373	298	17	)	)	PUNCT
ejpam-1373	298	18	and	and	CCONJ
ejpam-1373	298	19	y	y	PROPN
ejpam-1373	298	20	∈	∈	PROPN
ejpam-1373	298	21	c	c	X
ejpam-1373	298	22	(	(	PUNCT
ejpam-1373	298	23	n	n	CCONJ
ejpam-1373	298	24	)	)	PUNCT
ejpam-1373	298	25	=	=	NOUN
ejpam-1373	298	26	⇒	⇒	NOUN
ejpam-1373	298	27	x	x	PUNCT
ejpam-1373	298	28	≤c	≤c	PROPN
ejpam-1373	298	29	m	m	PROPN
ejpam-1373	298	30	and	and	CCONJ
ejpam-1373	298	31	y	y	PROPN
ejpam-1373	298	32	≤c	≤c	PROPN
ejpam-1373	298	33	n	n	PROPN
ejpam-1373	298	34	=	=	PROPN
ejpam-1373	298	35	⇒	⇒	NOUN
ejpam-1373	298	36	(	(	PUNCT
ejpam-1373	298	37	x	x	X
ejpam-1373	298	38	,	,	PUNCT
ejpam-1373	298	39	y	y	PROPN
ejpam-1373	298	40	)	)	PUNCT
ejpam-1373	298	41	=	=	SYM
ejpam-1373	298	42	1	1	NUM
ejpam-1373	298	43	and	and	CCONJ
ejpam-1373	298	44	x	x	PROPN
ejpam-1373	298	45	∨	∨	NUM
ejpam-1373	298	46	y	y	PROPN
ejpam-1373	298	47	≤c	≤c	PROPN
ejpam-1373	298	48	m∨	m∨	PROPN
ejpam-1373	298	49	n	n	PROPN
ejpam-1373	298	50	=	=	NOUN
ejpam-1373	298	51	⇒	⇒	NOUN
ejpam-1373	298	52	x	x	PUNCT
ejpam-1373	298	53	y	y	NOUN
ejpam-1373	298	54	=	=	PUNCT
ejpam-1373	298	55	x	x	PROPN
ejpam-1373	298	56	∨	∨	NUM
ejpam-1373	298	57	y	y	PROPN
ejpam-1373	298	58	∈	∈	PROPN
ejpam-1373	298	59	c	c	PROPN
ejpam-1373	298	60	(	(	PUNCT
ejpam-1373	298	61	m∨	m∨	NOUN
ejpam-1373	298	62	n	n	CCONJ
ejpam-1373	298	63	)	)	PUNCT
ejpam-1373	299	1	=	=	SYM
ejpam-1373	299	2	c	c	PROPN
ejpam-1373	299	3	(	(	PUNCT
ejpam-1373	299	4	mn	mn	PROPN
ejpam-1373	299	5	)	)	PUNCT
ejpam-1373	299	6	thus	thus	ADV
ejpam-1373	299	7	c	c	X
ejpam-1373	299	8	(	(	PUNCT
ejpam-1373	299	9	m).c	m).c	PROPN
ejpam-1373	299	10	(	(	PUNCT
ejpam-1373	299	11	n)⊆	n)⊆	PROPN
ejpam-1373	299	12	c	c	PROPN
ejpam-1373	299	13	(	(	PUNCT
ejpam-1373	299	14	mn	mn	PROPN
ejpam-1373	299	15	)	)	PUNCT
ejpam-1373	299	16	.	.	PUNCT
ejpam-1373	300	1	(	(	PUNCT
ejpam-1373	300	2	2)=⇒(1	2)=⇒(1	NUM
ejpam-1373	300	3	)	)	PUNCT
ejpam-1373	300	4	:	:	PUNCT
ejpam-1373	300	5	let	let	VERB
ejpam-1373	300	6	m	m	PRON
ejpam-1373	300	7	,	,	PUNCT
ejpam-1373	300	8	n	n	PRON
ejpam-1373	300	9	∈	∈	PROPN
ejpam-1373	300	10	z+	z+	NUM
ejpam-1373	300	11	such	such	ADJ
ejpam-1373	300	12	that	that	SCONJ
ejpam-1373	300	13	(	(	PUNCT
ejpam-1373	300	14	m	m	NOUN
ejpam-1373	300	15	,	,	PUNCT
ejpam-1373	300	16	n	n	CCONJ
ejpam-1373	300	17	)	)	PUNCT
ejpam-1373	300	18	=	=	SYM
ejpam-1373	301	1	1	1	X
ejpam-1373	301	2	.	.	PUNCT
ejpam-1373	301	3	then	then	ADV
ejpam-1373	301	4	by	by	ADP
ejpam-1373	301	5	(	(	PUNCT
ejpam-1373	301	6	2)(ii	2)(ii	NUM
ejpam-1373	301	7	)	)	PUNCT
ejpam-1373	301	8	,	,	PUNCT
ejpam-1373	301	9	c	c	X
ejpam-1373	301	10	(	(	PUNCT
ejpam-1373	301	11	m).c	m).c	PROPN
ejpam-1373	301	12	(	(	PUNCT
ejpam-1373	301	13	n	n	CCONJ
ejpam-1373	301	14	)	)	PUNCT
ejpam-1373	301	15	⊆	⊆	NUM
ejpam-1373	301	16	c	c	X
ejpam-1373	301	17	(	(	PUNCT
ejpam-1373	301	18	mn	mn	PROPN
ejpam-1373	301	19	)	)	PUNCT
ejpam-1373	301	20	and	and	CCONJ
ejpam-1373	301	21	therefore	therefore	ADV
ejpam-1373	301	22	m	m	VERB
ejpam-1373	301	23	=	=	SYM
ejpam-1373	301	24	m.1	m.1	PROPN
ejpam-1373	301	25	∈	∈	PROPN
ejpam-1373	301	26	c	c	NOUN
ejpam-1373	301	27	(	(	PUNCT
ejpam-1373	301	28	m).c	m).c	PROPN
ejpam-1373	301	29	(	(	PUNCT
ejpam-1373	301	30	n)⊆	n)⊆	PROPN
ejpam-1373	301	31	c	c	PROPN
ejpam-1373	301	32	(	(	PUNCT
ejpam-1373	301	33	mn	mn	PROPN
ejpam-1373	301	34	)	)	PUNCT
ejpam-1373	301	35	and	and	CCONJ
ejpam-1373	301	36	n=	n=	ADJ
ejpam-1373	301	37	n.1	n.1	PROPN
ejpam-1373	301	38	∈	∈	PROPN
ejpam-1373	301	39	c	c	X
ejpam-1373	301	40	(	(	PUNCT
ejpam-1373	301	41	m).c	m).c	PROPN
ejpam-1373	301	42	(	(	PUNCT
ejpam-1373	301	43	n)⊆	n)⊆	PROPN
ejpam-1373	301	44	c	c	PROPN
ejpam-1373	301	45	(	(	PUNCT
ejpam-1373	301	46	mn	mn	PROPN
ejpam-1373	301	47	)	)	PUNCT
ejpam-1373	301	48	and	and	CCONJ
ejpam-1373	301	49	hence	hence	ADV
ejpam-1373	301	50	m	m	PROPN
ejpam-1373	301	51	≤c	≤c	PROPN
ejpam-1373	301	52	mn	mn	PROPN
ejpam-1373	301	53	and	and	CCONJ
ejpam-1373	301	54	n≤c	n≤c	PROPN
ejpam-1373	301	55	mn	mn	PROPN
ejpam-1373	301	56	.	.	PUNCT
ejpam-1373	302	1	if	if	SCONJ
ejpam-1373	302	2	x	x	PRON
ejpam-1373	302	3	is	be	AUX
ejpam-1373	302	4	any	any	DET
ejpam-1373	302	5	upper	upper	ADJ
ejpam-1373	302	6	bound	bound	NOUN
ejpam-1373	302	7	of	of	ADP
ejpam-1373	302	8	m	m	PROPN
ejpam-1373	302	9	and	and	CCONJ
ejpam-1373	302	10	n	n	ADV
ejpam-1373	302	11	in	in	ADP
ejpam-1373	302	12	(	(	PUNCT
ejpam-1373	302	13	z+,≤c	z+,≤c	PROPN
ejpam-1373	302	14	)	)	PUNCT
ejpam-1373	302	15	,	,	PUNCT
ejpam-1373	302	16	then	then	ADV
ejpam-1373	302	17	m	m	PROPN
ejpam-1373	302	18	and	and	CCONJ
ejpam-1373	302	19	n	n	PRON
ejpam-1373	302	20	∈	∈	PROPN
ejpam-1373	302	21	c	c	X
ejpam-1373	302	22	(	(	PUNCT
ejpam-1373	302	23	x	x	NOUN
ejpam-1373	302	24	)	)	PUNCT
ejpam-1373	302	25	and	and	CCONJ
ejpam-1373	302	26	hence	hence	ADV
ejpam-1373	302	27	mn=	mn=	PROPN
ejpam-1373	302	28	m∨	m∨	PROPN
ejpam-1373	302	29	n≤c	n≤c	PROPN
ejpam-1373	302	30	x	x	X
ejpam-1373	302	31	.	.	PUNCT
ejpam-1373	303	1	thus	thus	ADV
ejpam-1373	303	2	,	,	PUNCT
ejpam-1373	303	3	mn	mn	PROPN
ejpam-1373	303	4	is	be	AUX
ejpam-1373	303	5	the	the	DET
ejpam-1373	303	6	least	least	ADJ
ejpam-1373	303	7	upper	upper	ADJ
ejpam-1373	303	8	bound	bind	VERB
ejpam-1373	303	9	of	of	ADP
ejpam-1373	303	10	m	m	PROPN
ejpam-1373	303	11	and	and	CCONJ
ejpam-1373	303	12	n.	n.	NOUN
ejpam-1373	303	13	references	reference	NOUN
ejpam-1373	303	14	[	[	X
ejpam-1373	303	15	1	1	X
ejpam-1373	303	16	]	]	PUNCT
ejpam-1373	303	17	e.	e.	PROPN
ejpam-1373	303	18	cohen	cohen	PROPN
ejpam-1373	303	19	,	,	PUNCT
ejpam-1373	303	20	arithmetical	arithmetical	ADJ
ejpam-1373	303	21	functions	function	NOUN
ejpam-1373	303	22	associated	associate	VERB
ejpam-1373	303	23	with	with	ADP
ejpam-1373	303	24	the	the	DET
ejpam-1373	303	25	unitary	unitary	ADJ
ejpam-1373	303	26	divisors	divisor	NOUN
ejpam-1373	303	27	of	of	ADP
ejpam-1373	303	28	an	an	DET
ejpam-1373	303	29	integer	integer	NOUN
ejpam-1373	303	30	,	,	PUNCT
ejpam-1373	303	31	math	math	NOUN
ejpam-1373	303	32	.	.	PUNCT
ejpam-1373	304	1	z.74	z.74	NOUN
ejpam-1373	304	2	,	,	PUNCT
ejpam-1373	304	3	66	66	NUM
ejpam-1373	304	4	-	-	SYM
ejpam-1373	304	5	80	80	NUM
ejpam-1373	304	6	.	.	PUNCT
ejpam-1373	305	1	1960	1960	NUM
ejpam-1373	305	2	.	.	PUNCT
ejpam-1373	306	1	[	[	X
ejpam-1373	306	2	2	2	X
ejpam-1373	306	3	]	]	PUNCT
ejpam-1373	306	4	w.	w.	PROPN
ejpam-1373	306	5	narkiewicz	narkiewicz	PROPN
ejpam-1373	306	6	,	,	PUNCT
ejpam-1373	306	7	on	on	ADP
ejpam-1373	306	8	a	a	DET
ejpam-1373	306	9	class	class	NOUN
ejpam-1373	306	10	of	of	ADP
ejpam-1373	306	11	arithmetical	arithmetical	ADJ
ejpam-1373	306	12	convolutions	convolution	NOUN
ejpam-1373	306	13	.	.	PUNCT
ejpam-1373	307	1	colloq	colloq	PROPN
ejpam-1373	307	2	.	.	PUNCT
ejpam-1373	308	1	math	math	PROPN
ejpam-1373	308	2	.	.	PUNCT
ejpam-1373	309	1	,10	,10	PUNCT
ejpam-1373	309	2	,	,	PUNCT
ejpam-1373	309	3	81	81	NUM
ejpam-1373	309	4	-	-	SYM
ejpam-1373	309	5	94	94	NUM
ejpam-1373	309	6	.	.	PUNCT
ejpam-1373	310	1	1963	1963	NUM
ejpam-1373	310	2	.	.	PUNCT
ejpam-1373	311	1	references	reference	NOUN
ejpam-1373	311	2	434	434	NUM
ejpam-1373	312	1	[	[	X
ejpam-1373	312	2	3	3	NUM
ejpam-1373	312	3	]	]	X
ejpam-1373	312	4	u.m	u.m	PROPN
ejpam-1373	312	5	.	.	PROPN
ejpam-1373	312	6	swamy	swamy	PROPN
ejpam-1373	312	7	,	,	PUNCT
ejpam-1373	312	8	g.c	g.c	PROPN
ejpam-1373	312	9	.	.	PROPN
ejpam-1373	312	10	rao	rao	PROPN
ejpam-1373	312	11	and	and	CCONJ
ejpam-1373	312	12	v.	v.	PROPN
ejpam-1373	312	13	sita	sita	PROPN
ejpam-1373	312	14	ramaiah	ramaiah	PROPN
ejpam-1373	312	15	,	,	PUNCT
ejpam-1373	312	16	on	on	ADP
ejpam-1373	312	17	a	a	DET
ejpam-1373	312	18	conjecture	conjecture	NOUN
ejpam-1373	312	19	in	in	ADP
ejpam-1373	312	20	a	a	DET
ejpam-1373	312	21	ring	ring	NOUN
ejpam-1373	312	22	of	of	ADP
ejpam-1373	312	23	arithmetic	arithmetic	ADJ
ejpam-1373	312	24	functions	function	NOUN
ejpam-1373	312	25	.	.	PUNCT
ejpam-1373	313	1	indian	indian	PROPN
ejpam-1373	313	2	j.	j.	PROPN
ejpam-1373	313	3	pure	pure	PROPN
ejpam-1373	313	4	appl	appl	PROPN
ejpam-1373	313	5	.	.	PUNCT
ejpam-1373	313	6	math	math	PROPN
ejpam-1373	313	7	.	.	PUNCT
ejpam-1373	313	8	,	,	PUNCT
ejpam-1373	313	9	14(12	14(12	NUM
ejpam-1373	313	10	)	)	PUNCT
ejpam-1373	313	11	,	,	PUNCT
ejpam-1373	313	12	1519	1519	NUM
ejpam-1373	313	13	-	-	SYM
ejpam-1373	313	14	1530	1530	NUM
ejpam-1373	313	15	.	.	PUNCT
ejpam-1373	314	1	1983	1983	NUM
ejpam-1373	314	2	.	.	PUNCT
ejpam-1373	315	1	[	[	X
ejpam-1373	315	2	4	4	NUM
ejpam-1373	315	3	]	]	X
ejpam-1373	315	4	u.m	u.m	PROPN
ejpam-1373	315	5	.	.	PROPN
ejpam-1373	315	6	swamy	swamy	PROPN
ejpam-1373	315	7	and	and	CCONJ
ejpam-1373	315	8	sagi	sagi	PROPN
ejpam-1373	315	9	sankar	sankar	PROPN
ejpam-1373	315	10	,	,	PUNCT
ejpam-1373	315	11	partial	partial	ADJ
ejpam-1373	315	12	orders	order	NOUN
ejpam-1373	315	13	induced	induce	VERB
ejpam-1373	315	14	by	by	ADP
ejpam-1373	315	15	convolutions	convolution	NOUN
ejpam-1373	315	16	.	.	PUNCT
ejpam-1373	315	17	(	(	PUNCT
ejpam-1373	315	18	accepted	accept	VERB
ejpam-1373	315	19	for	for	ADP
ejpam-1373	315	20	publication	publication	NOUN
ejpam-1373	315	21	)	)	PUNCT
ejpam-1373	315	22	int	int	NOUN
ejpam-1373	315	23	.	.	PUNCT
ejpam-1373	316	1	electron	electron	PROPN
ejpam-1373	316	2	.	.	PUNCT
ejpam-1373	317	1	j.	j.	PROPN
ejpam-1373	317	2	algebra	algebra	PROPN
ejpam-1373	317	3	.	.	PUNCT
ejpam-1373	318	1	2011	2011	NUM
ejpam-1373	318	2	.	.	PUNCT