id	sid	tid	token	lemma	pos
ejpam-1878	1	1	compile	compile	NOUN
ejpam-1878	1	2	/	/	SYM
ejpam-1878	1	3	output.dvi	output.dvi	NOUN
ejpam-1878	1	4	european	european	ADJ
ejpam-1878	1	5	journal	journal	NOUN
ejpam-1878	1	6	of	of	ADP
ejpam-1878	1	7	pure	pure	ADJ
ejpam-1878	1	8	and	and	CCONJ
ejpam-1878	1	9	applied	apply	VERB
ejpam-1878	1	10	mathematics	mathematic	NOUN
ejpam-1878	1	11	vol	vol	NOUN
ejpam-1878	1	12	.	.	PROPN
ejpam-1878	1	13	8	8	NUM
ejpam-1878	1	14	,	,	PUNCT
ejpam-1878	1	15	no	no	INTJ
ejpam-1878	1	16	.	.	NOUN
ejpam-1878	1	17	1	1	NUM
ejpam-1878	1	18	,	,	PUNCT
ejpam-1878	1	19	2015	2015	NUM
ejpam-1878	1	20	,	,	PUNCT
ejpam-1878	1	21	15	15	NUM
ejpam-1878	1	22	-	-	SYM
ejpam-1878	1	23	25	25	NUM
ejpam-1878	1	24	issn	issn	PROPN
ejpam-1878	1	25	1307	1307	NUM
ejpam-1878	1	26	-	-	SYM
ejpam-1878	1	27	5543	5543	NUM
ejpam-1878	1	28	–	–	PUNCT
ejpam-1878	1	29	www.ejpam.com	www.ejpam.com	X
ejpam-1878	1	30	characterization	characterization	NOUN
ejpam-1878	1	31	of	of	ADP
ejpam-1878	1	32	prime	prime	ADJ
ejpam-1878	1	33	ideals	ideal	NOUN
ejpam-1878	1	34	in	in	ADP
ejpam-1878	1	35	(	(	PUNCT
ejpam-1878	1	36	z+,≤d	z+,≤d	NUM
ejpam-1878	1	37	)	)	PUNCT
ejpam-1878	1	38	sankar	sankar	PROPN
ejpam-1878	1	39	sagi	sagi	PROPN
ejpam-1878	1	40	college	college	PROPN
ejpam-1878	1	41	of	of	ADP
ejpam-1878	1	42	applied	apply	VERB
ejpam-1878	1	43	sciences	science	NOUN
ejpam-1878	1	44	,	,	PUNCT
ejpam-1878	1	45	sohar	sohar	PROPN
ejpam-1878	1	46	,	,	PUNCT
ejpam-1878	1	47	ministry	ministry	PROPN
ejpam-1878	1	48	of	of	ADP
ejpam-1878	1	49	higher	high	ADJ
ejpam-1878	1	50	education	education	NOUN
ejpam-1878	1	51	,	,	PUNCT
ejpam-1878	1	52	sultanate	sultanate	NOUN
ejpam-1878	1	53	of	of	ADP
ejpam-1878	1	54	oman	oman	PROPN
ejpam-1878	1	55	abstract	abstract	PROPN
ejpam-1878	1	56	.	.	PUNCT
ejpam-1878	2	1	a	a	DET
ejpam-1878	2	2	convolution	convolution	NOUN
ejpam-1878	2	3	is	be	AUX
ejpam-1878	2	4	a	a	DET
ejpam-1878	2	5	mapping	mapping	NOUN
ejpam-1878	2	6	c	c	NOUN
ejpam-1878	2	7	of	of	ADP
ejpam-1878	2	8	the	the	DET
ejpam-1878	2	9	set	set	NOUN
ejpam-1878	2	10	z+	z+	NUM
ejpam-1878	2	11	of	of	ADP
ejpam-1878	2	12	positive	positive	ADJ
ejpam-1878	2	13	integers	integer	NOUN
ejpam-1878	2	14	into	into	ADP
ejpam-1878	2	15	the	the	DET
ejpam-1878	2	16	set	set	NOUN
ejpam-1878	2	17	p	p	NOUN
ejpam-1878	2	18	(	(	PUNCT
ejpam-1878	2	19	z+	z+	X
ejpam-1878	2	20	)	)	PUNCT
ejpam-1878	2	21	of	of	ADP
ejpam-1878	2	22	all	all	DET
ejpam-1878	2	23	subsets	subset	NOUN
ejpam-1878	2	24	of	of	ADP
ejpam-1878	2	25	z+	z+	NUM
ejpam-1878	2	26	such	such	ADJ
ejpam-1878	2	27	that	that	SCONJ
ejpam-1878	2	28	,	,	PUNCT
ejpam-1878	2	29	for	for	ADP
ejpam-1878	2	30	any	any	DET
ejpam-1878	2	31	n	n	PRON
ejpam-1878	2	32	∈	∈	PROPN
ejpam-1878	2	33	z+	z+	NUM
ejpam-1878	2	34	,	,	PUNCT
ejpam-1878	2	35	each	each	DET
ejpam-1878	2	36	member	member	NOUN
ejpam-1878	2	37	of	of	ADP
ejpam-1878	2	38	c	c	PROPN
ejpam-1878	2	39	(	(	PUNCT
ejpam-1878	2	40	n	n	CCONJ
ejpam-1878	2	41	)	)	PUNCT
ejpam-1878	2	42	is	be	AUX
ejpam-1878	2	43	a	a	DET
ejpam-1878	2	44	divisor	divisor	NOUN
ejpam-1878	2	45	of	of	ADP
ejpam-1878	2	46	n.	n.	NOUN
ejpam-1878	2	47	if	if	SCONJ
ejpam-1878	2	48	d(n	d(n	NOUN
ejpam-1878	2	49	)	)	PUNCT
ejpam-1878	2	50	is	be	AUX
ejpam-1878	2	51	the	the	DET
ejpam-1878	2	52	set	set	NOUN
ejpam-1878	2	53	of	of	ADP
ejpam-1878	2	54	all	all	DET
ejpam-1878	2	55	divisors	divisor	NOUN
ejpam-1878	2	56	of	of	ADP
ejpam-1878	2	57	n	n	CCONJ
ejpam-1878	2	58	,	,	PUNCT
ejpam-1878	2	59	for	for	ADP
ejpam-1878	2	60	any	any	DET
ejpam-1878	2	61	n	n	CCONJ
ejpam-1878	2	62	,	,	PUNCT
ejpam-1878	2	63	then	then	ADV
ejpam-1878	2	64	d	d	PROPN
ejpam-1878	2	65	is	be	AUX
ejpam-1878	2	66	called	call	VERB
ejpam-1878	2	67	the	the	DET
ejpam-1878	2	68	dirichlet	dirichlet	PROPN
ejpam-1878	2	69	’s	’s	PART
ejpam-1878	2	70	convolution	convolution	NOUN
ejpam-1878	2	71	.	.	PUNCT
ejpam-1878	3	1	corresponding	correspond	VERB
ejpam-1878	3	2	to	to	ADP
ejpam-1878	3	3	any	any	DET
ejpam-1878	3	4	general	general	ADJ
ejpam-1878	3	5	convolution	convolution	NOUN
ejpam-1878	3	6	c	c	NOUN
ejpam-1878	3	7	,	,	PUNCT
ejpam-1878	3	8	we	we	PRON
ejpam-1878	3	9	can	can	AUX
ejpam-1878	3	10	define	define	VERB
ejpam-1878	3	11	a	a	DET
ejpam-1878	3	12	binary	binary	ADJ
ejpam-1878	3	13	relation	relation	NOUN
ejpam-1878	3	14	≤c	≤c	PROPN
ejpam-1878	3	15	on	on	ADP
ejpam-1878	3	16	z+	z+	NUM
ejpam-1878	3	17	by	by	ADP
ejpam-1878	3	18	“	"	PUNCT
ejpam-1878	3	19	m	m	PROPN
ejpam-1878	3	20	≤c	≤c	PROPN
ejpam-1878	3	21	n	n	NOUN
ejpam-1878	3	22	if	if	SCONJ
ejpam-1878	4	1	and	and	CCONJ
ejpam-1878	4	2	only	only	ADV
ejpam-1878	4	3	if	if	SCONJ
ejpam-1878	4	4	m	m	VERB
ejpam-1878	4	5	∈	∈	PROPN
ejpam-1878	4	6	c	c	X
ejpam-1878	4	7	(	(	PUNCT
ejpam-1878	4	8	n	n	CCONJ
ejpam-1878	4	9	)	)	PUNCT
ejpam-1878	4	10	”	"	PUNCT
ejpam-1878	4	11	.	.	PUNCT
ejpam-1878	5	1	it	it	PRON
ejpam-1878	5	2	is	be	AUX
ejpam-1878	5	3	well	well	ADV
ejpam-1878	5	4	known	know	VERB
ejpam-1878	5	5	that	that	SCONJ
ejpam-1878	5	6	z+	z+	NUM
ejpam-1878	5	7	has	have	VERB
ejpam-1878	5	8	the	the	DET
ejpam-1878	5	9	structure	structure	NOUN
ejpam-1878	5	10	of	of	ADP
ejpam-1878	5	11	a	a	DET
ejpam-1878	5	12	distributive	distributive	ADJ
ejpam-1878	5	13	lattice	lattice	NOUN
ejpam-1878	5	14	with	with	ADP
ejpam-1878	5	15	respect	respect	NOUN
ejpam-1878	5	16	to	to	ADP
ejpam-1878	5	17	the	the	DET
ejpam-1878	5	18	division	division	NOUN
ejpam-1878	5	19	order	order	NOUN
ejpam-1878	5	20	.	.	PUNCT
ejpam-1878	6	1	the	the	DET
ejpam-1878	6	2	division	division	NOUN
ejpam-1878	6	3	ordering	ordering	NOUN
ejpam-1878	6	4	is	be	AUX
ejpam-1878	6	5	precisely	precisely	ADV
ejpam-1878	6	6	the	the	DET
ejpam-1878	6	7	partial	partial	ADJ
ejpam-1878	6	8	ordering	ordering	NOUN
ejpam-1878	6	9	≤d	≤d	NOUN
ejpam-1878	6	10	induced	induce	VERB
ejpam-1878	6	11	by	by	ADP
ejpam-1878	6	12	the	the	DET
ejpam-1878	6	13	dirichlet	dirichlet	PROPN
ejpam-1878	6	14	’s	’s	PART
ejpam-1878	6	15	convolution	convolution	PROPN
ejpam-1878	6	16	d.	d.	PROPN
ejpam-1878	6	17	in	in	ADP
ejpam-1878	6	18	this	this	DET
ejpam-1878	6	19	paper	paper	NOUN
ejpam-1878	6	20	,	,	PUNCT
ejpam-1878	6	21	we	we	PRON
ejpam-1878	6	22	present	present	VERB
ejpam-1878	6	23	a	a	DET
ejpam-1878	6	24	characterization	characterization	NOUN
ejpam-1878	6	25	for	for	ADP
ejpam-1878	6	26	the	the	DET
ejpam-1878	6	27	prime	prime	ADJ
ejpam-1878	6	28	ideals	ideal	NOUN
ejpam-1878	6	29	in	in	ADP
ejpam-1878	6	30	(	(	PUNCT
ejpam-1878	6	31	z+,≤d	z+,≤d	NUM
ejpam-1878	6	32	)	)	PUNCT
ejpam-1878	6	33	,	,	PUNCT
ejpam-1878	6	34	where	where	SCONJ
ejpam-1878	6	35	d	d	NOUN
ejpam-1878	6	36	is	be	AUX
ejpam-1878	6	37	the	the	DET
ejpam-1878	6	38	dirichlet	dirichlet	PROPN
ejpam-1878	6	39	’s	’s	PART
ejpam-1878	6	40	convolution	convolution	NOUN
ejpam-1878	6	41	.	.	PUNCT
ejpam-1878	7	1	2010	2010	NUM
ejpam-1878	7	2	mathematics	mathematic	NOUN
ejpam-1878	7	3	subject	subject	NOUN
ejpam-1878	7	4	classifications	classification	NOUN
ejpam-1878	7	5	:	:	PUNCT
ejpam-1878	7	6	06b10,11a99	06b10,11a99	PROPN
ejpam-1878	7	7	key	key	ADJ
ejpam-1878	7	8	words	word	NOUN
ejpam-1878	7	9	and	and	CCONJ
ejpam-1878	7	10	phrases	phrase	NOUN
ejpam-1878	7	11	:	:	PUNCT
ejpam-1878	7	12	poset	poset	NOUN
ejpam-1878	7	13	,	,	PUNCT
ejpam-1878	7	14	lattice	lattice	PROPN
ejpam-1878	7	15	,	,	PUNCT
ejpam-1878	7	16	semi	semi	ADJ
ejpam-1878	7	17	lattice	lattice	NOUN
ejpam-1878	7	18	,	,	PUNCT
ejpam-1878	7	19	convolution	convolution	NOUN
ejpam-1878	7	20	,	,	PUNCT
ejpam-1878	7	21	ideal	ideal	ADJ
ejpam-1878	7	22	1	1	NUM
ejpam-1878	7	23	.	.	PUNCT
ejpam-1878	8	1	introduction	introduction	NOUN
ejpam-1878	8	2	a	a	DET
ejpam-1878	8	3	convolution	convolution	NOUN
ejpam-1878	8	4	is	be	AUX
ejpam-1878	8	5	a	a	DET
ejpam-1878	8	6	mapping	mapping	NOUN
ejpam-1878	8	7	c	c	NOUN
ejpam-1878	8	8	:	:	PUNCT
ejpam-1878	8	9	z+	z+	NUM
ejpam-1878	8	10	−→p	−→p	NOUN
ejpam-1878	8	11	(	(	PUNCT
ejpam-1878	8	12	z+	z+	NOUN
ejpam-1878	8	13	)	)	PUNCT
ejpam-1878	8	14	such	such	ADJ
ejpam-1878	8	15	that	that	SCONJ
ejpam-1878	8	16	c	c	NOUN
ejpam-1878	8	17	(	(	PUNCT
ejpam-1878	8	18	n	n	CCONJ
ejpam-1878	8	19	)	)	PUNCT
ejpam-1878	8	20	is	be	AUX
ejpam-1878	8	21	a	a	DET
ejpam-1878	8	22	set	set	NOUN
ejpam-1878	8	23	of	of	ADP
ejpam-1878	8	24	positive	positive	ADJ
ejpam-1878	8	25	divisors	divisor	NOUN
ejpam-1878	8	26	on	on	ADP
ejpam-1878	8	27	n	n	CCONJ
ejpam-1878	8	28	,	,	PUNCT
ejpam-1878	8	29	n	n	PROPN
ejpam-1878	8	30	∈	∈	PROPN
ejpam-1878	8	31	c	c	X
ejpam-1878	8	32	(	(	PUNCT
ejpam-1878	8	33	n	n	CCONJ
ejpam-1878	8	34	)	)	PUNCT
ejpam-1878	8	35	andc	andc	NOUN
ejpam-1878	8	36	(	(	PUNCT
ejpam-1878	8	37	n	n	CCONJ
ejpam-1878	8	38	)	)	PUNCT
ejpam-1878	8	39	=	=	SYM
ejpam-1878	9	1	⋃	⋃	NOUN
ejpam-1878	9	2	m∈c	m∈c	NOUN
ejpam-1878	9	3	(	(	PUNCT
ejpam-1878	9	4	n	n	CCONJ
ejpam-1878	9	5	)	)	PUNCT
ejpam-1878	9	6	c	c	NOUN
ejpam-1878	9	7	(	(	PUNCT
ejpam-1878	9	8	m	m	NOUN
ejpam-1878	9	9	)	)	PUNCT
ejpam-1878	9	10	,	,	PUNCT
ejpam-1878	9	11	for	for	ADP
ejpam-1878	9	12	any	any	DET
ejpam-1878	9	13	n	n	PRON
ejpam-1878	9	14	∈	∈	PROPN
ejpam-1878	9	15	z+	z+	X
ejpam-1878	9	16	.	.	PUNCT
ejpam-1878	10	1	popular	popular	ADJ
ejpam-1878	10	2	examples	example	NOUN
ejpam-1878	10	3	are	be	AUX
ejpam-1878	10	4	the	the	DET
ejpam-1878	10	5	dirichlet	dirichlet	PROPN
ejpam-1878	10	6	’s	’s	PART
ejpam-1878	10	7	convolution	convolution	NOUN
ejpam-1878	10	8	d	d	PROPN
ejpam-1878	10	9	and	and	CCONJ
ejpam-1878	10	10	the	the	DET
ejpam-1878	10	11	unitary	unitary	ADJ
ejpam-1878	10	12	convolution	convolution	NOUN
ejpam-1878	10	13	u	u	NOUN
ejpam-1878	10	14	defined	define	VERB
ejpam-1878	10	15	respectively	respectively	ADV
ejpam-1878	10	16	by	by	ADP
ejpam-1878	10	17	d(n	d(n	NOUN
ejpam-1878	10	18	)	)	PUNCT
ejpam-1878	10	19	=	=	SYM
ejpam-1878	10	20	the	the	DET
ejpam-1878	10	21	set	set	NOUN
ejpam-1878	10	22	of	of	ADP
ejpam-1878	10	23	all	all	DET
ejpam-1878	10	24	positive	positive	ADJ
ejpam-1878	10	25	divisors	divisor	NOUN
ejpam-1878	10	26	of	of	ADP
ejpam-1878	10	27	n	n	NOUN
ejpam-1878	10	28	and	and	CCONJ
ejpam-1878	10	29	u	u	NOUN
ejpam-1878	10	30	(	(	PUNCT
ejpam-1878	10	31	n	n	CCONJ
ejpam-1878	10	32	)	)	PUNCT
ejpam-1878	10	33	=	=	NOUN
ejpam-1878	10	34	the	the	DET
ejpam-1878	10	35	set	set	NOUN
ejpam-1878	10	36	of	of	ADP
ejpam-1878	10	37	unitary	unitary	ADJ
ejpam-1878	10	38	divisors	divisor	NOUN
ejpam-1878	10	39	of	of	ADP
ejpam-1878	10	40	n	n	PRON
ejpam-1878	10	41	for	for	ADP
ejpam-1878	10	42	any	any	DET
ejpam-1878	10	43	n	n	PRON
ejpam-1878	10	44	∈	∈	NOUN
ejpam-1878	10	45	z+	z+	NOUN
ejpam-1878	10	46	.	.	PUNCT
ejpam-1878	11	1	if	if	SCONJ
ejpam-1878	11	2	c	c	PROPN
ejpam-1878	11	3	is	be	AUX
ejpam-1878	11	4	a	a	DET
ejpam-1878	11	5	convolution	convolution	NOUN
ejpam-1878	11	6	,	,	PUNCT
ejpam-1878	11	7	then	then	ADV
ejpam-1878	11	8	the	the	DET
ejpam-1878	11	9	binary	binary	PROPN
ejpam-1878	11	10	relation	relation	PROPN
ejpam-1878	11	11	≤c	≤c	PROPN
ejpam-1878	11	12	on	on	ADP
ejpam-1878	11	13	z+	z+	NUM
ejpam-1878	11	14	,	,	PUNCT
ejpam-1878	11	15	defined	define	VERB
ejpam-1878	11	16	by	by	ADP
ejpam-1878	11	17	,	,	PUNCT
ejpam-1878	11	18	m≤c	m≤c	PROPN
ejpam-1878	11	19	n	n	CCONJ
ejpam-1878	11	20	if	if	ADV
ejpam-1878	12	1	and	and	CCONJ
ejpam-1878	12	2	only	only	ADV
ejpam-1878	12	3	if	if	SCONJ
ejpam-1878	12	4	m	m	VERB
ejpam-1878	12	5	∈	∈	PROPN
ejpam-1878	12	6	c	c	X
ejpam-1878	12	7	(	(	PUNCT
ejpam-1878	12	8	n	n	CCONJ
ejpam-1878	12	9	)	)	PUNCT
ejpam-1878	12	10	,	,	PUNCT
ejpam-1878	12	11	is	be	AUX
ejpam-1878	12	12	a	a	DET
ejpam-1878	12	13	partial	partial	ADJ
ejpam-1878	12	14	order	order	NOUN
ejpam-1878	12	15	on	on	ADP
ejpam-1878	12	16	z+	z+	NUM
ejpam-1878	12	17	and	and	CCONJ
ejpam-1878	12	18	is	be	AUX
ejpam-1878	12	19	called	call	VERB
ejpam-1878	12	20	the	the	DET
ejpam-1878	12	21	partial	partial	ADJ
ejpam-1878	12	22	order	order	NOUN
ejpam-1878	12	23	induced	induce	VERB
ejpam-1878	12	24	by	by	ADP
ejpam-1878	12	25	c	c	PROPN
ejpam-1878	13	1	[	[	X
ejpam-1878	13	2	2	2	NUM
ejpam-1878	13	3	]	]	PUNCT
ejpam-1878	13	4	.	.	PUNCT
ejpam-1878	14	1	it	it	PRON
ejpam-1878	14	2	is	be	AUX
ejpam-1878	14	3	well	well	ADV
ejpam-1878	14	4	known	know	VERB
ejpam-1878	14	5	that	that	SCONJ
ejpam-1878	14	6	the	the	DET
ejpam-1878	14	7	dirichlet	dirichlet	PROPN
ejpam-1878	14	8	’s	’s	PART
ejpam-1878	14	9	convolution	convolution	NOUN
ejpam-1878	14	10	induces	induce	VERB
ejpam-1878	14	11	the	the	DET
ejpam-1878	14	12	division	division	NOUN
ejpam-1878	14	13	order	order	NOUN
ejpam-1878	14	14	onz+	onz+	VERB
ejpam-1878	14	15	with	with	ADP
ejpam-1878	14	16	respect	respect	NOUN
ejpam-1878	14	17	to	to	ADP
ejpam-1878	14	18	whichz+	whichz+	NOUN
ejpam-1878	14	19	becomes	become	VERB
ejpam-1878	14	20	a	a	DET
ejpam-1878	14	21	distributive	distributive	ADJ
ejpam-1878	14	22	lattice	lattice	NOUN
ejpam-1878	14	23	,	,	PUNCT
ejpam-1878	14	24	where	where	SCONJ
ejpam-1878	14	25	,	,	PUNCT
ejpam-1878	14	26	for	for	ADP
ejpam-1878	14	27	any	any	DET
ejpam-1878	14	28	a	a	PRON
ejpam-1878	14	29	,	,	PUNCT
ejpam-1878	14	30	b	b	PROPN
ejpam-1878	14	31	∈	∈	PROPN
ejpam-1878	14	32	z+	z+	NUM
ejpam-1878	14	33	,	,	PUNCT
ejpam-1878	14	34	the	the	DET
ejpam-1878	14	35	greatest	great	ADJ
ejpam-1878	14	36	common	common	ADJ
ejpam-1878	14	37	divisor(gcd	divisor(gcd	NOUN
ejpam-1878	14	38	)	)	PUNCT
ejpam-1878	14	39	and	and	CCONJ
ejpam-1878	14	40	the	the	DET
ejpam-1878	14	41	email	email	NOUN
ejpam-1878	14	42	address	address	NOUN
ejpam-1878	14	43	:	:	PUNCT
ejpam-1878	14	44	sagi_sankar@yahoo.co.in	sagi_sankar@yahoo.co.in	NUM
ejpam-1878	14	45	http://www.ejpam.com	http://www.ejpam.com	X
ejpam-1878	14	46	15	15	NUM
ejpam-1878	14	47	c	c	NOUN
ejpam-1878	14	48	©	©	PROPN
ejpam-1878	14	49	2015	2015	NUM
ejpam-1878	14	50	ejpam	ejpam	VERB
ejpam-1878	14	51	all	all	DET
ejpam-1878	14	52	rights	right	NOUN
ejpam-1878	14	53	reserved	reserve	VERB
ejpam-1878	14	54	.	.	PUNCT
ejpam-1878	15	1	s.	s.	PROPN
ejpam-1878	15	2	sagi	sagi	PROPN
ejpam-1878	15	3	/	/	SYM
ejpam-1878	15	4	eur	eur	PROPN
ejpam-1878	15	5	.	.	PUNCT
ejpam-1878	16	1	j.	j.	PROPN
ejpam-1878	16	2	pure	pure	PROPN
ejpam-1878	16	3	appl	appl	PROPN
ejpam-1878	16	4	.	.	PROPN
ejpam-1878	16	5	math	math	PROPN
ejpam-1878	16	6	,	,	PUNCT
ejpam-1878	16	7	8	8	NUM
ejpam-1878	16	8	(	(	PUNCT
ejpam-1878	16	9	2015	2015	NUM
ejpam-1878	16	10	)	)	PUNCT
ejpam-1878	16	11	,	,	PUNCT
ejpam-1878	16	12	15	15	NUM
ejpam-1878	16	13	-	-	SYM
ejpam-1878	16	14	25	25	NUM
ejpam-1878	16	15	16	16	NUM
ejpam-1878	16	16	least	least	ADJ
ejpam-1878	16	17	common	common	ADJ
ejpam-1878	16	18	multiple(lcm	multiple(lcm	NOUN
ejpam-1878	16	19	)	)	PUNCT
ejpam-1878	16	20	of	of	ADP
ejpam-1878	16	21	a	a	PRON
ejpam-1878	16	22	and	and	CCONJ
ejpam-1878	16	23	b	b	NOUN
ejpam-1878	16	24	are	be	AUX
ejpam-1878	16	25	respectively	respectively	ADV
ejpam-1878	16	26	the	the	DET
ejpam-1878	16	27	greatest	greatest	ADV
ejpam-1878	16	28	lower	low	ADJ
ejpam-1878	16	29	bound(glb	bound(glb	NOUN
ejpam-1878	16	30	)	)	PUNCT
ejpam-1878	16	31	and	and	CCONJ
ejpam-1878	16	32	the	the	DET
ejpam-1878	16	33	least	least	ADJ
ejpam-1878	16	34	upper	upper	ADJ
ejpam-1878	16	35	bound(lub	bound(lub	NOUN
ejpam-1878	16	36	)	)	PUNCT
ejpam-1878	16	37	of	of	ADP
ejpam-1878	16	38	a	a	PRON
ejpam-1878	16	39	and	and	CCONJ
ejpam-1878	16	40	b	b	NOUN
ejpam-1878	16	41	.	.	PUNCT
ejpam-1878	17	1	in	in	ADP
ejpam-1878	17	2	fact	fact	NOUN
ejpam-1878	17	3	,	,	PUNCT
ejpam-1878	17	4	with	with	ADP
ejpam-1878	17	5	respect	respect	NOUN
ejpam-1878	17	6	to	to	ADP
ejpam-1878	17	7	the	the	DET
ejpam-1878	17	8	division	division	NOUN
ejpam-1878	17	9	order	order	NOUN
ejpam-1878	17	10	,	,	PUNCT
ejpam-1878	17	11	the	the	DET
ejpam-1878	17	12	lattice	lattice	NOUN
ejpam-1878	17	13	z+	z+	NUM
ejpam-1878	17	14	satisfies	satisfy	VERB
ejpam-1878	17	15	the	the	DET
ejpam-1878	17	16	infinite	infinite	ADJ
ejpam-1878	17	17	join	join	NOUN
ejpam-1878	17	18	distributive	distributive	ADJ
ejpam-1878	17	19	law	law	NOUN
ejpam-1878	17	20	given	give	VERB
ejpam-1878	17	21	by	by	ADP
ejpam-1878	17	22	�	�	PROPN
ejpam-1878	17	23	a	a	DET
ejpam-1878	17	24	∨	∨	NOUN
ejpam-1878	17	25	(	(	PUNCT
ejpam-1878	17	26	∧	∧	PROPN
ejpam-1878	17	27	i∈i	i∈i	ADJ
ejpam-1878	17	28	bi	bi	NOUN
ejpam-1878	17	29	)	)	PUNCT
ejpam-1878	17	30	=	=	SYM
ejpam-1878	18	1	∧	∧	PROPN
ejpam-1878	18	2	i∈i	i∈i	ADJ
ejpam-1878	18	3	(	(	PUNCT
ejpam-1878	18	4	a	a	DET
ejpam-1878	18	5	∨	∨	NUM
ejpam-1878	18	6	bi	bi	NOUN
ejpam-1878	18	7	)	)	PUNCT
ejpam-1878	18	8	�	�	PROPN
ejpam-1878	18	9	for	for	ADP
ejpam-1878	18	10	any	any	DET
ejpam-1878	18	11	a	a	DET
ejpam-1878	18	12	∈	∈	NOUN
ejpam-1878	18	13	z+	z+	NUM
ejpam-1878	18	14	and	and	CCONJ
ejpam-1878	18	15	{	{	PUNCT
ejpam-1878	18	16	bi}i∈i	bi}i∈i	NOUN
ejpam-1878	18	17	⊆	⊆	NUM
ejpam-1878	18	18	z	z	NOUN
ejpam-1878	18	19	+	+	NOUN
ejpam-1878	18	20	.	.	PUNCT
ejpam-1878	19	1	in	in	ADP
ejpam-1878	19	2	this	this	DET
ejpam-1878	19	3	paper	paper	NOUN
ejpam-1878	19	4	,	,	PUNCT
ejpam-1878	19	5	we	we	PRON
ejpam-1878	19	6	discuss	discuss	VERB
ejpam-1878	19	7	various	various	ADJ
ejpam-1878	19	8	aspects	aspect	NOUN
ejpam-1878	19	9	of	of	ADP
ejpam-1878	19	10	ideals	ideal	NOUN
ejpam-1878	19	11	and	and	CCONJ
ejpam-1878	19	12	filters	filter	NOUN
ejpam-1878	19	13	in	in	ADP
ejpam-1878	19	14	(	(	PUNCT
ejpam-1878	19	15	z+,≤c	z+,≤c	NUM
ejpam-1878	19	16	)	)	PUNCT
ejpam-1878	19	17	and	and	CCONJ
ejpam-1878	19	18	eventually	eventually	ADV
ejpam-1878	19	19	present	present	VERB
ejpam-1878	19	20	a	a	DET
ejpam-1878	19	21	characterization	characterization	NOUN
ejpam-1878	19	22	of	of	ADP
ejpam-1878	19	23	prime	prime	ADJ
ejpam-1878	19	24	ideals	ideal	NOUN
ejpam-1878	19	25	in	in	ADP
ejpam-1878	19	26	(	(	PUNCT
ejpam-1878	19	27	z+,≤d	z+,≤d	NOUN
ejpam-1878	19	28	)	)	PUNCT
ejpam-1878	19	29	where	where	SCONJ
ejpam-1878	19	30	d	d	NOUN
ejpam-1878	19	31	is	be	AUX
ejpam-1878	19	32	the	the	DET
ejpam-1878	19	33	dirichlet	dirichlet	PROPN
ejpam-1878	19	34	’s	’s	PART
ejpam-1878	19	35	convolution	convolution	NOUN
ejpam-1878	19	36	actually	actually	ADV
ejpam-1878	19	37	a	a	DET
ejpam-1878	19	38	general	general	ADJ
ejpam-1878	19	39	convolution	convolution	NOUN
ejpam-1878	19	40	may	may	AUX
ejpam-1878	19	41	not	not	PART
ejpam-1878	19	42	induce	induce	VERB
ejpam-1878	19	43	a	a	DET
ejpam-1878	19	44	lattice	lattice	NOUN
ejpam-1878	19	45	structure	structure	NOUN
ejpam-1878	19	46	on	on	ADP
ejpam-1878	19	47	z+	z+	NUM
ejpam-1878	19	48	.	.	PUNCT
ejpam-1878	20	1	however	however	ADV
ejpam-1878	20	2	,	,	PUNCT
ejpam-1878	20	3	most	most	ADJ
ejpam-1878	20	4	of	of	ADP
ejpam-1878	20	5	the	the	DET
ejpam-1878	20	6	convolutions	convolution	NOUN
ejpam-1878	20	7	we	we	PRON
ejpam-1878	20	8	are	be	AUX
ejpam-1878	20	9	considering	consider	VERB
ejpam-1878	20	10	induce	induce	VERB
ejpam-1878	20	11	a	a	DET
ejpam-1878	20	12	meet	meet	NOUN
ejpam-1878	20	13	semi	semi	ADJ
ejpam-1878	20	14	lattice	lattice	NOUN
ejpam-1878	20	15	structure	structure	NOUN
ejpam-1878	20	16	on	on	ADP
ejpam-1878	20	17	z+	z+	NUM
ejpam-1878	20	18	.	.	PUNCT
ejpam-1878	21	1	for	for	ADP
ejpam-1878	21	2	this	this	DET
ejpam-1878	21	3	reason	reason	NOUN
ejpam-1878	21	4	,	,	PUNCT
ejpam-1878	21	5	we	we	PRON
ejpam-1878	21	6	first	first	ADV
ejpam-1878	21	7	consider	consider	VERB
ejpam-1878	21	8	a	a	DET
ejpam-1878	21	9	general	general	ADJ
ejpam-1878	21	10	semi	semi	ADJ
ejpam-1878	21	11	lattice	lattice	NOUN
ejpam-1878	21	12	and	and	CCONJ
ejpam-1878	21	13	study	study	VERB
ejpam-1878	21	14	it	it	PRON
ejpam-1878	21	15	’s	’	VERB
ejpam-1878	21	16	ideals	ideal	NOUN
ejpam-1878	21	17	and	and	CCONJ
ejpam-1878	21	18	later	later	ADV
ejpam-1878	21	19	extend	extend	VERB
ejpam-1878	21	20	these	these	PRON
ejpam-1878	21	21	to	to	ADP
ejpam-1878	21	22	(	(	PUNCT
ejpam-1878	21	23	z+,≤d	z+,≤d	NUM
ejpam-1878	21	24	)	)	PUNCT
ejpam-1878	21	25	.	.	PUNCT
ejpam-1878	22	1	2	2	X
ejpam-1878	22	2	.	.	X
ejpam-1878	22	3	preliminaries	preliminary	NOUN
ejpam-1878	22	4	let	let	VERB
ejpam-1878	22	5	us	we	PRON
ejpam-1878	22	6	recall	recall	VERB
ejpam-1878	22	7	that	that	SCONJ
ejpam-1878	22	8	a	a	DET
ejpam-1878	22	9	partial	partial	ADJ
ejpam-1878	22	10	order	order	NOUN
ejpam-1878	22	11	on	on	ADP
ejpam-1878	22	12	a	a	DET
ejpam-1878	22	13	non	non	ADJ
ejpam-1878	22	14	-	-	ADJ
ejpam-1878	22	15	empty	empty	ADJ
ejpam-1878	22	16	set	set	NOUN
ejpam-1878	22	17	x	x	PUNCT
ejpam-1878	22	18	is	be	AUX
ejpam-1878	22	19	defined	define	VERB
ejpam-1878	22	20	as	as	ADP
ejpam-1878	22	21	a	a	DET
ejpam-1878	22	22	binary	binary	ADJ
ejpam-1878	22	23	relation	relation	NOUN
ejpam-1878	22	24	≤	≤	PUNCT
ejpam-1878	22	25	on	on	ADP
ejpam-1878	22	26	x	x	PUNCT
ejpam-1878	22	27	which	which	PRON
ejpam-1878	22	28	is	be	AUX
ejpam-1878	22	29	reflexive	reflexive	ADJ
ejpam-1878	22	30	(	(	PUNCT
ejpam-1878	22	31	a	a	DET
ejpam-1878	22	32	≤	≤	PROPN
ejpam-1878	22	33	a	a	X
ejpam-1878	22	34	)	)	PUNCT
ejpam-1878	22	35	,	,	PUNCT
ejpam-1878	22	36	transitive	transitive	ADJ
ejpam-1878	22	37	(	(	PUNCT
ejpam-1878	22	38	a	a	DET
ejpam-1878	22	39	≤	≤	NUM
ejpam-1878	22	40	b	b	NUM
ejpam-1878	22	41	,	,	PUNCT
ejpam-1878	22	42	b	b	PROPN
ejpam-1878	22	43	≤	≤	NUM
ejpam-1878	22	44	c	c	X
ejpam-1878	23	1	=	=	PRON
ejpam-1878	23	2	⇒	⇒	VERB
ejpam-1878	23	3	a	a	DET
ejpam-1878	23	4	≤	≤	NUM
ejpam-1878	23	5	c	c	NOUN
ejpam-1878	23	6	)	)	PUNCT
ejpam-1878	23	7	and	and	CCONJ
ejpam-1878	23	8	antisymmetric	antisymmetric	ADJ
ejpam-1878	23	9	(	(	PUNCT
ejpam-1878	23	10	a	a	DET
ejpam-1878	23	11	≤	≤	NUM
ejpam-1878	23	12	b	b	NUM
ejpam-1878	23	13	,	,	PUNCT
ejpam-1878	23	14	b	b	PROPN
ejpam-1878	23	15	≤	≤	NOUN
ejpam-1878	23	16	a	a	DET
ejpam-1878	23	17	=	=	NOUN
ejpam-1878	23	18	⇒	⇒	VERB
ejpam-1878	23	19	a	a	DET
ejpam-1878	23	20	=	=	SYM
ejpam-1878	23	21	b	b	NOUN
ejpam-1878	23	22	)	)	PUNCT
ejpam-1878	23	23	and	and	CCONJ
ejpam-1878	23	24	that	that	SCONJ
ejpam-1878	23	25	a	a	DET
ejpam-1878	23	26	pair	pair	NOUN
ejpam-1878	23	27	(	(	PUNCT
ejpam-1878	23	28	x	x	NOUN
ejpam-1878	23	29	,	,	PUNCT
ejpam-1878	23	30	≤	≤	NUM
ejpam-1878	23	31	)	)	PUNCT
ejpam-1878	23	32	is	be	AUX
ejpam-1878	23	33	called	call	VERB
ejpam-1878	23	34	a	a	DET
ejpam-1878	23	35	partially	partially	ADV
ejpam-1878	23	36	ordered	order	VERB
ejpam-1878	23	37	set(poset	set(poset	NOUN
ejpam-1878	23	38	)	)	PUNCT
ejpam-1878	23	39	if	if	SCONJ
ejpam-1878	23	40	x	x	PRON
ejpam-1878	23	41	is	be	AUX
ejpam-1878	23	42	a	a	DET
ejpam-1878	23	43	non	non	ADJ
ejpam-1878	23	44	-	-	ADJ
ejpam-1878	23	45	empty	empty	ADJ
ejpam-1878	23	46	set	set	NOUN
ejpam-1878	23	47	and	and	CCONJ
ejpam-1878	23	48	≤	≤	NOUN
ejpam-1878	23	49	is	be	AUX
ejpam-1878	23	50	a	a	DET
ejpam-1878	23	51	partial	partial	ADJ
ejpam-1878	23	52	order	order	NOUN
ejpam-1878	23	53	on	on	ADP
ejpam-1878	23	54	x	x	X
ejpam-1878	23	55	.	.	PUNCT
ejpam-1878	24	1	for	for	ADP
ejpam-1878	24	2	any	any	DET
ejpam-1878	24	3	a	a	DET
ejpam-1878	24	4	⊆	⊆	NUM
ejpam-1878	24	5	x	x	SYM
ejpam-1878	24	6	and	and	CCONJ
ejpam-1878	24	7	x	x	SYM
ejpam-1878	24	8	∈	∈	PROPN
ejpam-1878	24	9	x	x	X
ejpam-1878	24	10	,	,	PUNCT
ejpam-1878	24	11	x	x	X
ejpam-1878	24	12	is	be	AUX
ejpam-1878	24	13	called	call	VERB
ejpam-1878	24	14	a	a	DET
ejpam-1878	24	15	lower(upper	lower(upper	PROPN
ejpam-1878	24	16	)	)	PUNCT
ejpam-1878	24	17	bound	bind	VERB
ejpam-1878	24	18	of	of	ADP
ejpam-1878	24	19	a	a	DET
ejpam-1878	24	20	if	if	NOUN
ejpam-1878	24	21	x	x	SYM
ejpam-1878	24	22	≤	≤	NUM
ejpam-1878	24	23	a(respectively	a(respectively	ADV
ejpam-1878	24	24	a	a	DET
ejpam-1878	24	25	≤	≤	NUM
ejpam-1878	24	26	x	x	X
ejpam-1878	24	27	)	)	PUNCT
ejpam-1878	24	28	for	for	ADP
ejpam-1878	24	29	all	all	DET
ejpam-1878	24	30	a	a	DET
ejpam-1878	24	31	∈	∈	NOUN
ejpam-1878	24	32	a.	a.	NOUN
ejpam-1878	24	33	we	we	PRON
ejpam-1878	24	34	have	have	VERB
ejpam-1878	24	35	the	the	DET
ejpam-1878	24	36	usual	usual	ADJ
ejpam-1878	24	37	notations	notation	NOUN
ejpam-1878	24	38	of	of	ADP
ejpam-1878	24	39	the	the	DET
ejpam-1878	24	40	greatest	greatest	ADV
ejpam-1878	24	41	lower	low	ADJ
ejpam-1878	24	42	bound(glb	bound(glb	NOUN
ejpam-1878	24	43	)	)	PUNCT
ejpam-1878	24	44	and	and	CCONJ
ejpam-1878	24	45	least	least	ADJ
ejpam-1878	24	46	upper	upper	ADJ
ejpam-1878	24	47	bound(lub	bound(lub	NOUN
ejpam-1878	24	48	)	)	PUNCT
ejpam-1878	24	49	of	of	ADP
ejpam-1878	24	50	a	a	DET
ejpam-1878	24	51	in	in	ADP
ejpam-1878	24	52	x	x	X
ejpam-1878	24	53	.	.	PUNCT
ejpam-1878	25	1	if	if	SCONJ
ejpam-1878	25	2	a	a	PRON
ejpam-1878	25	3	is	be	AUX
ejpam-1878	25	4	a	a	DET
ejpam-1878	25	5	finite	finite	NOUN
ejpam-1878	25	6	subset	subset	NOUN
ejpam-1878	25	7	{	{	PUNCT
ejpam-1878	25	8	a1	a1	PROPN
ejpam-1878	25	9	,	,	PUNCT
ejpam-1878	25	10	a2	a2	PROPN
ejpam-1878	25	11	,	,	PUNCT
ejpam-1878	25	12	·	·	PUNCT
ejpam-1878	25	13	·	·	PUNCT
ejpam-1878	25	14	·	·	PUNCT
ejpam-1878	25	15	,	,	PUNCT
ejpam-1878	25	16	an	an	X
ejpam-1878	25	17	}	}	PUNCT
ejpam-1878	25	18	,	,	PUNCT
ejpam-1878	25	19	the	the	DET
ejpam-1878	25	20	glb	glb	NOUN
ejpam-1878	25	21	of	of	ADP
ejpam-1878	25	22	a(lub	a(lub	NOUN
ejpam-1878	25	23	of	of	ADP
ejpam-1878	25	24	a	a	PRON
ejpam-1878	25	25	)	)	PUNCT
ejpam-1878	25	26	is	be	AUX
ejpam-1878	25	27	denoted	denote	VERB
ejpam-1878	25	28	by	by	ADP
ejpam-1878	25	29	a1	a1	NOUN
ejpam-1878	25	30	∧	∧	PROPN
ejpam-1878	25	31	a2	a2	PROPN
ejpam-1878	25	32	∧	∧	PROPN
ejpam-1878	25	33	·	·	PUNCT
ejpam-1878	25	34	·	·	PUNCT
ejpam-1878	25	35	·	·	PUNCT
ejpam-1878	26	1	∧	∧	NOUN
ejpam-1878	26	2	an	an	PRON
ejpam-1878	26	3	or	or	CCONJ
ejpam-1878	26	4	n	n	PRON
ejpam-1878	26	5	∧	∧	NOUN
ejpam-1878	26	6	i=1	i=1	PROPN
ejpam-1878	26	7	ai	ai	VERB
ejpam-1878	26	8	(	(	PUNCT
ejpam-1878	26	9	respectively	respectively	ADV
ejpam-1878	26	10	by	by	ADP
ejpam-1878	26	11	a1∨	a1∨	ADJ
ejpam-1878	26	12	a2∨	a2∨	NOUN
ejpam-1878	26	13	·	·	PUNCT
ejpam-1878	26	14	·	·	PUNCT
ejpam-1878	26	15	·	·	PUNCT
ejpam-1878	26	16	∨	∨	NUM
ejpam-1878	26	17	an	an	DET
ejpam-1878	26	18	or	or	CCONJ
ejpam-1878	26	19	n	n	PRON
ejpam-1878	26	20	∨	∨	NOUN
ejpam-1878	26	21	i=1	i=1	PROPN
ejpam-1878	26	22	ai	ai	PROPN
ejpam-1878	26	23	)	)	PUNCT
ejpam-1878	26	24	.	.	PUNCT
ejpam-1878	27	1	a	a	DET
ejpam-1878	27	2	partially	partially	ADV
ejpam-1878	27	3	ordered	order	VERB
ejpam-1878	27	4	set	set	NOUN
ejpam-1878	27	5	(	(	PUNCT
ejpam-1878	27	6	x	x	INTJ
ejpam-1878	27	7	,	,	PUNCT
ejpam-1878	27	8	≤	≤	NUM
ejpam-1878	27	9	)	)	PUNCT
ejpam-1878	27	10	is	be	AUX
ejpam-1878	27	11	called	call	VERB
ejpam-1878	27	12	a	a	DET
ejpam-1878	27	13	meet	meet	NOUN
ejpam-1878	27	14	semi	semi	ADJ
ejpam-1878	27	15	lattice	lattice	NOUN
ejpam-1878	27	16	if	if	SCONJ
ejpam-1878	27	17	a∧	a∧	PROPN
ejpam-1878	27	18	b	b	PROPN
ejpam-1878	27	19	(=	(=	ADP
ejpam-1878	27	20	glb{a	glb{a	NOUN
ejpam-1878	27	21	,	,	PUNCT
ejpam-1878	27	22	b	b	NOUN
ejpam-1878	27	23	}	}	PUNCT
ejpam-1878	27	24	)	)	PUNCT
ejpam-1878	27	25	exists	exist	VERB
ejpam-1878	27	26	for	for	ADP
ejpam-1878	27	27	all	all	DET
ejpam-1878	27	28	a	a	PRON
ejpam-1878	27	29	and	and	CCONJ
ejpam-1878	27	30	b	b	NOUN
ejpam-1878	27	31	∈	∈	PROPN
ejpam-1878	27	32	x	x	X
ejpam-1878	27	33	.	.	PUNCT
ejpam-1878	28	1	(	(	PUNCT
ejpam-1878	28	2	x	x	X
ejpam-1878	28	3	,	,	PUNCT
ejpam-1878	28	4	≤	≤	NUM
ejpam-1878	28	5	)	)	PUNCT
ejpam-1878	28	6	is	be	AUX
ejpam-1878	28	7	called	call	VERB
ejpam-1878	28	8	a	a	DET
ejpam-1878	28	9	join	join	NOUN
ejpam-1878	28	10	semi	semi	ADV
ejpam-1878	28	11	lattice	lattice	NOUN
ejpam-1878	28	12	if	if	SCONJ
ejpam-1878	28	13	a∨	a∨	PROPN
ejpam-1878	28	14	b	b	PROPN
ejpam-1878	28	15	(=	(=	NOUN
ejpam-1878	28	16	lub{a	lub{a	ADV
ejpam-1878	28	17	,	,	PUNCT
ejpam-1878	28	18	b	b	NOUN
ejpam-1878	28	19	}	}	PUNCT
ejpam-1878	28	20	)	)	PUNCT
ejpam-1878	28	21	exists	exist	VERB
ejpam-1878	28	22	for	for	ADP
ejpam-1878	28	23	all	all	DET
ejpam-1878	28	24	a	a	PRON
ejpam-1878	28	25	and	and	CCONJ
ejpam-1878	28	26	b	b	NOUN
ejpam-1878	28	27	∈	∈	NOUN
ejpam-1878	28	28	x	x	X
ejpam-1878	28	29	.	.	PUNCT
ejpam-1878	29	1	a	a	DET
ejpam-1878	29	2	poset	poset	NOUN
ejpam-1878	29	3	(	(	PUNCT
ejpam-1878	29	4	x	x	INTJ
ejpam-1878	29	5	,	,	PUNCT
ejpam-1878	29	6	≤	≤	NUM
ejpam-1878	29	7	)	)	PUNCT
ejpam-1878	29	8	is	be	AUX
ejpam-1878	29	9	called	call	VERB
ejpam-1878	29	10	a	a	DET
ejpam-1878	29	11	lattice	lattice	NOUN
ejpam-1878	29	12	if	if	SCONJ
ejpam-1878	29	13	it	it	PRON
ejpam-1878	29	14	is	be	AUX
ejpam-1878	29	15	both	both	CCONJ
ejpam-1878	29	16	a	a	DET
ejpam-1878	29	17	meet	meet	NOUN
ejpam-1878	29	18	and	and	CCONJ
ejpam-1878	29	19	join	join	VERB
ejpam-1878	29	20	semi	semi	ADV
ejpam-1878	29	21	lattice	lattice	PROPN
ejpam-1878	29	22	.	.	PUNCT
ejpam-1878	30	1	equivalently	equivalently	ADV
ejpam-1878	30	2	,	,	PUNCT
ejpam-1878	30	3	lattice	lattice	PROPN
ejpam-1878	30	4	can	can	AUX
ejpam-1878	30	5	also	also	ADV
ejpam-1878	30	6	be	be	AUX
ejpam-1878	30	7	defined	define	VERB
ejpam-1878	30	8	as	as	ADP
ejpam-1878	30	9	an	an	DET
ejpam-1878	30	10	algebraic	algebraic	ADJ
ejpam-1878	30	11	system	system	NOUN
ejpam-1878	30	12	(	(	PUNCT
ejpam-1878	30	13	x	x	INTJ
ejpam-1878	30	14	,	,	PUNCT
ejpam-1878	30	15	∧,∨	∧,∨	ADJ
ejpam-1878	30	16	)	)	PUNCT
ejpam-1878	30	17	,	,	PUNCT
ejpam-1878	30	18	where	where	SCONJ
ejpam-1878	30	19	∧	∧	PROPN
ejpam-1878	30	20	and	and	CCONJ
ejpam-1878	30	21	∨	∨	NUM
ejpam-1878	30	22	are	be	AUX
ejpam-1878	30	23	binary	binary	ADJ
ejpam-1878	30	24	operations	operation	NOUN
ejpam-1878	30	25	which	which	PRON
ejpam-1878	30	26	are	be	AUX
ejpam-1878	30	27	associative	associative	ADJ
ejpam-1878	30	28	,	,	PUNCT
ejpam-1878	30	29	commutative	commutative	ADJ
ejpam-1878	30	30	and	and	CCONJ
ejpam-1878	30	31	idempotent	idempotent	NOUN
ejpam-1878	30	32	and	and	CCONJ
ejpam-1878	30	33	satisfying	satisfy	VERB
ejpam-1878	30	34	the	the	DET
ejpam-1878	30	35	absorption	absorption	NOUN
ejpam-1878	30	36	laws	law	NOUN
ejpam-1878	30	37	,	,	PUNCT
ejpam-1878	30	38	namely	namely	ADV
ejpam-1878	30	39	a	a	DET
ejpam-1878	30	40	∧	∧	NOUN
ejpam-1878	30	41	(	(	PUNCT
ejpam-1878	30	42	a	a	DET
ejpam-1878	30	43	∨	∨	NUM
ejpam-1878	30	44	b	b	NOUN
ejpam-1878	30	45	)	)	PUNCT
ejpam-1878	30	46	=	=	PUNCT
ejpam-1878	31	1	a	a	PRON
ejpam-1878	31	2	=	=	PUNCT
ejpam-1878	31	3	a	a	DET
ejpam-1878	31	4	∨	∨	NOUN
ejpam-1878	31	5	(	(	PUNCT
ejpam-1878	31	6	a	a	DET
ejpam-1878	31	7	∧	∧	PROPN
ejpam-1878	31	8	b	b	NOUN
ejpam-1878	31	9	)	)	PUNCT
ejpam-1878	31	10	for	for	ADP
ejpam-1878	31	11	all	all	DET
ejpam-1878	31	12	a	a	DET
ejpam-1878	31	13	,	,	PUNCT
ejpam-1878	31	14	b	b	X
ejpam-1878	31	15	∈	∈	PROPN
ejpam-1878	31	16	x	x	X
ejpam-1878	31	17	;	;	PUNCT
ejpam-1878	31	18	in	in	ADP
ejpam-1878	31	19	this	this	DET
ejpam-1878	31	20	case	case	NOUN
ejpam-1878	31	21	the	the	DET
ejpam-1878	31	22	partial	partial	ADJ
ejpam-1878	31	23	order	order	NOUN
ejpam-1878	31	24	≤	≤	X
ejpam-1878	31	25	on	on	ADP
ejpam-1878	31	26	x	x	SYM
ejpam-1878	31	27	is	be	AUX
ejpam-1878	31	28	such	such	ADJ
ejpam-1878	31	29	that	that	SCONJ
ejpam-1878	31	30	a	a	DET
ejpam-1878	31	31	∧	∧	PROPN
ejpam-1878	31	32	b	b	PROPN
ejpam-1878	31	33	and	and	CCONJ
ejpam-1878	31	34	a	a	DET
ejpam-1878	31	35	∨	∨	PROPN
ejpam-1878	31	36	b	b	NOUN
ejpam-1878	31	37	are	be	AUX
ejpam-1878	31	38	respectively	respectively	ADV
ejpam-1878	31	39	the	the	DET
ejpam-1878	31	40	glb	glb	NOUN
ejpam-1878	31	41	and	and	CCONJ
ejpam-1878	31	42	lub	lub	NOUN
ejpam-1878	31	43	of	of	ADP
ejpam-1878	31	44	{	{	PUNCT
ejpam-1878	31	45	a	a	PROPN
ejpam-1878	31	46	,	,	PUNCT
ejpam-1878	31	47	b	b	NOUN
ejpam-1878	31	48	}	}	PUNCT
ejpam-1878	31	49	.	.	PUNCT
ejpam-1878	32	1	the	the	DET
ejpam-1878	32	2	algebraic	algebraic	ADJ
ejpam-1878	32	3	operations	operation	NOUN
ejpam-1878	32	4	∧	∧	PROPN
ejpam-1878	32	5	and	and	CCONJ
ejpam-1878	32	6	∨	∨	NUM
ejpam-1878	32	7	and	and	CCONJ
ejpam-1878	32	8	the	the	DET
ejpam-1878	32	9	partial	partial	ADJ
ejpam-1878	32	10	order	order	NOUN
ejpam-1878	32	11	≤	≤	NOUN
ejpam-1878	32	12	are	be	AUX
ejpam-1878	32	13	related	relate	VERB
ejpam-1878	32	14	by	by	ADP
ejpam-1878	32	15	(	(	PUNCT
ejpam-1878	32	16	a	a	DET
ejpam-1878	32	17	=	=	X
ejpam-1878	32	18	a	a	DET
ejpam-1878	32	19	∧	∧	PROPN
ejpam-1878	32	20	b	b	PROPN
ejpam-1878	32	21	⇐	⇐	PROPN
ejpam-1878	32	22	⇒	⇒	NOUN
ejpam-1878	32	23	a	a	DET
ejpam-1878	32	24	≤	≤	NUM
ejpam-1878	32	25	b	b	NUM
ejpam-1878	32	26	⇐	⇐	ADJ
ejpam-1878	32	27	⇒	⇒	NOUN
ejpam-1878	32	28	a	a	DET
ejpam-1878	32	29	∨	∨	NOUN
ejpam-1878	32	30	b	b	PROPN
ejpam-1878	32	31	=	=	SYM
ejpam-1878	32	32	b	b	PROPN
ejpam-1878	32	33	)	)	PUNCT
ejpam-1878	32	34	.	.	PUNCT
ejpam-1878	33	1	throughout	throughout	ADP
ejpam-1878	33	2	the	the	DET
ejpam-1878	33	3	paper	paper	NOUN
ejpam-1878	33	4	,	,	PUNCT
ejpam-1878	33	5	z+	z+	NUM
ejpam-1878	33	6	and	and	CCONJ
ejpam-1878	33	7	n	n	PRON
ejpam-1878	33	8	denote	denote	VERB
ejpam-1878	33	9	the	the	DET
ejpam-1878	33	10	set	set	NOUN
ejpam-1878	33	11	of	of	ADP
ejpam-1878	33	12	positive	positive	ADJ
ejpam-1878	33	13	integers	integer	NOUN
ejpam-1878	33	14	and	and	CCONJ
ejpam-1878	33	15	the	the	DET
ejpam-1878	33	16	set	set	NOUN
ejpam-1878	33	17	of	of	ADP
ejpam-1878	33	18	nonnegative	nonnegative	ADJ
ejpam-1878	33	19	integers	integer	NOUN
ejpam-1878	33	20	respectively	respectively	ADV
ejpam-1878	33	21	.	.	PUNCT
ejpam-1878	34	1	definition	definition	NOUN
ejpam-1878	34	2	1	1	NUM
ejpam-1878	34	3	.	.	PUNCT
ejpam-1878	35	1	a	a	DET
ejpam-1878	35	2	mappingc	mappingc	NOUN
ejpam-1878	35	3	:	:	PUNCT
ejpam-1878	35	4	z+	z+	NUM
ejpam-1878	35	5	−→p	−→p	NOUN
ejpam-1878	35	6	(	(	PUNCT
ejpam-1878	35	7	z+	z+	NUM
ejpam-1878	35	8	)	)	PUNCT
ejpam-1878	35	9	is	be	AUX
ejpam-1878	35	10	called	call	VERB
ejpam-1878	35	11	a	a	DET
ejpam-1878	35	12	convolution	convolution	NOUN
ejpam-1878	35	13	if	if	SCONJ
ejpam-1878	35	14	the	the	DET
ejpam-1878	35	15	following	following	NOUN
ejpam-1878	35	16	are	be	AUX
ejpam-1878	35	17	satisfied	satisfied	ADJ
ejpam-1878	35	18	for	for	ADP
ejpam-1878	35	19	any	any	DET
ejpam-1878	35	20	n	n	PRON
ejpam-1878	35	21	∈	∈	PROPN
ejpam-1878	35	22	z+	z+	NUM
ejpam-1878	35	23	.	.	PUNCT
ejpam-1878	36	1	(	(	PUNCT
ejpam-1878	36	2	1	1	NUM
ejpam-1878	36	3	)	)	PUNCT
ejpam-1878	36	4	.	.	PUNCT
ejpam-1878	37	1	c	c	NOUN
ejpam-1878	37	2	(	(	PUNCT
ejpam-1878	37	3	n	n	CCONJ
ejpam-1878	37	4	)	)	PUNCT
ejpam-1878	37	5	is	be	AUX
ejpam-1878	37	6	a	a	DET
ejpam-1878	37	7	set	set	NOUN
ejpam-1878	37	8	of	of	ADP
ejpam-1878	37	9	positive	positive	ADJ
ejpam-1878	37	10	divisors	divisor	NOUN
ejpam-1878	37	11	of	of	ADP
ejpam-1878	37	12	n	n	PROPN
ejpam-1878	37	13	(	(	PUNCT
ejpam-1878	37	14	2	2	NUM
ejpam-1878	37	15	)	)	PUNCT
ejpam-1878	37	16	.	.	PUNCT
ejpam-1878	38	1	n	n	PROPN
ejpam-1878	38	2	∈	∈	PROPN
ejpam-1878	38	3	c	c	X
ejpam-1878	38	4	(	(	PUNCT
ejpam-1878	38	5	n	n	CCONJ
ejpam-1878	38	6	)	)	PUNCT
ejpam-1878	38	7	s.	s.	PROPN
ejpam-1878	38	8	sagi	sagi	PROPN
ejpam-1878	38	9	/	/	SYM
ejpam-1878	38	10	eur	eur	PROPN
ejpam-1878	38	11	.	.	PUNCT
ejpam-1878	39	1	j.	j.	PROPN
ejpam-1878	39	2	pure	pure	PROPN
ejpam-1878	39	3	appl	appl	PROPN
ejpam-1878	39	4	.	.	PROPN
ejpam-1878	39	5	math	math	PROPN
ejpam-1878	39	6	,	,	PUNCT
ejpam-1878	39	7	8	8	NUM
ejpam-1878	39	8	(	(	PUNCT
ejpam-1878	39	9	2015	2015	NUM
ejpam-1878	39	10	)	)	PUNCT
ejpam-1878	39	11	,	,	PUNCT
ejpam-1878	39	12	15	15	NUM
ejpam-1878	39	13	-	-	SYM
ejpam-1878	39	14	25	25	NUM
ejpam-1878	39	15	17	17	NUM
ejpam-1878	39	16	(	(	PUNCT
ejpam-1878	39	17	3	3	NUM
ejpam-1878	39	18	)	)	PUNCT
ejpam-1878	39	19	.	.	PUNCT
ejpam-1878	40	1	c	c	NOUN
ejpam-1878	40	2	(	(	PUNCT
ejpam-1878	40	3	n	n	CCONJ
ejpam-1878	40	4	)	)	PUNCT
ejpam-1878	40	5	=	=	SYM
ejpam-1878	40	6	⋃	⋃	NOUN
ejpam-1878	40	7	m∈c	m∈c	NOUN
ejpam-1878	40	8	(	(	PUNCT
ejpam-1878	40	9	n	n	CCONJ
ejpam-1878	40	10	)	)	PUNCT
ejpam-1878	40	11	c	c	NOUN
ejpam-1878	40	12	(	(	PUNCT
ejpam-1878	40	13	m	m	NOUN
ejpam-1878	40	14	)	)	PUNCT
ejpam-1878	40	15	.	.	PUNCT
ejpam-1878	41	1	definition	definition	NOUN
ejpam-1878	41	2	2	2	NUM
ejpam-1878	41	3	.	.	X
ejpam-1878	42	1	for	for	ADP
ejpam-1878	42	2	any	any	DET
ejpam-1878	42	3	convolution	convolution	NOUN
ejpam-1878	42	4	c	c	PROPN
ejpam-1878	42	5	and	and	CCONJ
ejpam-1878	42	6	m	m	PROPN
ejpam-1878	42	7	and	and	CCONJ
ejpam-1878	42	8	n	n	PRON
ejpam-1878	42	9	∈	∈	PROPN
ejpam-1878	42	10	z+	z+	NUM
ejpam-1878	42	11	,	,	PUNCT
ejpam-1878	42	12	we	we	PRON
ejpam-1878	42	13	define	define	VERB
ejpam-1878	42	14	�	�	PROPN
ejpam-1878	42	15	m≤	m≤	PROPN
ejpam-1878	42	16	n	n	CCONJ
ejpam-1878	42	17	if	if	ADV
ejpam-1878	43	1	and	and	CCONJ
ejpam-1878	43	2	only	only	ADV
ejpam-1878	43	3	if	if	SCONJ
ejpam-1878	43	4	m	m	VERB
ejpam-1878	43	5	∈	∈	PROPN
ejpam-1878	43	6	c	c	X
ejpam-1878	43	7	(	(	PUNCT
ejpam-1878	43	8	n	n	CCONJ
ejpam-1878	43	9	)	)	PUNCT
ejpam-1878	43	10	�	�	PROPN
ejpam-1878	43	11	then	then	ADV
ejpam-1878	43	12	≤c	≤c	PROPN
ejpam-1878	43	13	is	be	AUX
ejpam-1878	43	14	a	a	DET
ejpam-1878	43	15	partial	partial	ADJ
ejpam-1878	43	16	order	order	NOUN
ejpam-1878	43	17	on	on	ADP
ejpam-1878	43	18	z+	z+	NUM
ejpam-1878	43	19	and	and	CCONJ
ejpam-1878	43	20	is	be	AUX
ejpam-1878	43	21	called	call	VERB
ejpam-1878	43	22	the	the	DET
ejpam-1878	43	23	partial	partial	ADJ
ejpam-1878	43	24	order	order	NOUN
ejpam-1878	43	25	induced	induce	VERB
ejpam-1878	43	26	by	by	ADP
ejpam-1878	43	27	c	c	PROPN
ejpam-1878	43	28	on	on	ADP
ejpam-1878	43	29	z+	z+	NUM
ejpam-1878	43	30	.	.	PUNCT
ejpam-1878	44	1	in	in	ADP
ejpam-1878	44	2	fact	fact	NOUN
ejpam-1878	44	3	,	,	PUNCT
ejpam-1878	44	4	for	for	ADP
ejpam-1878	44	5	any	any	DET
ejpam-1878	44	6	mapping	mapping	NOUN
ejpam-1878	44	7	c	c	NOUN
ejpam-1878	44	8	:	:	PUNCT
ejpam-1878	44	9	z+	z+	NUM
ejpam-1878	44	10	−→	−→	NOUN
ejpam-1878	44	11	p	p	X
ejpam-1878	44	12	(	(	PUNCT
ejpam-1878	44	13	z+	z+	NOUN
ejpam-1878	44	14	)	)	PUNCT
ejpam-1878	44	15	such	such	ADJ
ejpam-1878	44	16	that	that	SCONJ
ejpam-1878	44	17	each	each	DET
ejpam-1878	44	18	member	member	NOUN
ejpam-1878	44	19	of	of	ADP
ejpam-1878	44	20	c	c	PROPN
ejpam-1878	44	21	(	(	PUNCT
ejpam-1878	44	22	n	n	CCONJ
ejpam-1878	44	23	)	)	PUNCT
ejpam-1878	44	24	is	be	AUX
ejpam-1878	44	25	a	a	DET
ejpam-1878	44	26	divisor	divisor	NOUN
ejpam-1878	44	27	of	of	ADP
ejpam-1878	44	28	n	n	CCONJ
ejpam-1878	44	29	,	,	PUNCT
ejpam-1878	44	30	≤c	≤c	PROPN
ejpam-1878	44	31	is	be	AUX
ejpam-1878	44	32	a	a	DET
ejpam-1878	44	33	partial	partial	ADJ
ejpam-1878	44	34	order	order	NOUN
ejpam-1878	44	35	on	on	ADP
ejpam-1878	44	36	z+	z+	NUM
ejpam-1878	44	37	if	if	SCONJ
ejpam-1878	44	38	and	and	CCONJ
ejpam-1878	44	39	only	only	ADV
ejpam-1878	44	40	if	if	SCONJ
ejpam-1878	44	41	c	c	PROPN
ejpam-1878	44	42	is	be	AUX
ejpam-1878	44	43	a	a	DET
ejpam-1878	44	44	convolution	convolution	NOUN
ejpam-1878	44	45	,	,	PUNCT
ejpam-1878	44	46	as	as	SCONJ
ejpam-1878	44	47	defined	define	VERB
ejpam-1878	44	48	above	above	ADP
ejpam-1878	44	49	[	[	X
ejpam-1878	44	50	1	1	NUM
ejpam-1878	44	51	,	,	PUNCT
ejpam-1878	44	52	4	4	NUM
ejpam-1878	44	53	]	]	PUNCT
ejpam-1878	44	54	.	.	PUNCT
ejpam-1878	45	1	definition	definition	NOUN
ejpam-1878	45	2	3	3	X
ejpam-1878	45	3	.	.	PUNCT
ejpam-1878	46	1	let	let	VERB
ejpam-1878	46	2	c	c	PRON
ejpam-1878	46	3	be	be	AUX
ejpam-1878	46	4	a	a	DET
ejpam-1878	46	5	convolution	convolution	NOUN
ejpam-1878	46	6	and	and	CCONJ
ejpam-1878	46	7	p	p	X
ejpam-1878	46	8	a	a	DET
ejpam-1878	46	9	prime	prime	ADJ
ejpam-1878	46	10	number	number	NOUN
ejpam-1878	46	11	.	.	PUNCT
ejpam-1878	47	1	define	define	VERB
ejpam-1878	47	2	a	a	DET
ejpam-1878	47	3	relation	relation	NOUN
ejpam-1878	47	4	≤p	≤p	NOUN
ejpam-1878	47	5	c	c	PROPN
ejpam-1878	47	6	on	on	ADP
ejpam-1878	47	7	the	the	DET
ejpam-1878	47	8	set	set	NOUN
ejpam-1878	47	9	n	n	PROPN
ejpam-1878	47	10	of	of	ADP
ejpam-1878	47	11	non	non	ADJ
ejpam-1878	47	12	-	-	ADJ
ejpam-1878	47	13	negative	negative	ADJ
ejpam-1878	47	14	integers	integer	NOUN
ejpam-1878	47	15	by	by	ADP
ejpam-1878	47	16	�	�	PROPN
ejpam-1878	47	17	a	a	DET
ejpam-1878	47	18	≤p	≤p	NOUN
ejpam-1878	47	19	c	c	PROPN
ejpam-1878	47	20	b	b	NOUN
ejpam-1878	48	1	if	if	SCONJ
ejpam-1878	48	2	and	and	CCONJ
ejpam-1878	48	3	only	only	ADV
ejpam-1878	48	4	if	if	SCONJ
ejpam-1878	48	5	pa	pa	PROPN
ejpam-1878	48	6	∈	∈	PROPN
ejpam-1878	48	7	c	c	X
ejpam-1878	48	8	(	(	PUNCT
ejpam-1878	48	9	pb	pb	NOUN
ejpam-1878	48	10	)	)	PUNCT
ejpam-1878	48	11	�	�	PROPN
ejpam-1878	48	12	for	for	ADP
ejpam-1878	48	13	any	any	DET
ejpam-1878	48	14	a	a	DET
ejpam-1878	48	15	and	and	CCONJ
ejpam-1878	48	16	b	b	NOUN
ejpam-1878	48	17	∈	∈	PROPN
ejpam-1878	48	18	n	n	NOUN
ejpam-1878	48	19	.	.	PUNCT
ejpam-1878	49	1	it	it	PRON
ejpam-1878	49	2	can	can	AUX
ejpam-1878	49	3	be	be	AUX
ejpam-1878	49	4	easily	easily	ADV
ejpam-1878	49	5	verified	verify	VERB
ejpam-1878	49	6	that	that	SCONJ
ejpam-1878	49	7	≤p	≤p	PROPN
ejpam-1878	49	8	c	c	PROPN
ejpam-1878	49	9	is	be	AUX
ejpam-1878	49	10	a	a	DET
ejpam-1878	49	11	partial	partial	ADJ
ejpam-1878	49	12	order	order	NOUN
ejpam-1878	49	13	on	on	ADP
ejpam-1878	49	14	n	n	PROPN
ejpam-1878	49	15	,	,	PUNCT
ejpam-1878	49	16	for	for	ADP
ejpam-1878	49	17	each	each	DET
ejpam-1878	49	18	prime	prime	NOUN
ejpam-1878	49	19	p.	p.	NOUN
ejpam-1878	49	20	the	the	DET
ejpam-1878	49	21	following	follow	VERB
ejpam-1878	49	22	is	be	AUX
ejpam-1878	49	23	a	a	DET
ejpam-1878	49	24	direct	direct	ADJ
ejpam-1878	49	25	verification	verification	NOUN
ejpam-1878	49	26	.	.	PUNCT
ejpam-1878	50	1	theorem	theorem	NOUN
ejpam-1878	50	2	1	1	NUM
ejpam-1878	50	3	.	.	PUNCT
ejpam-1878	51	1	let	let	VERB
ejpam-1878	51	2	c	c	PRON
ejpam-1878	51	3	be	be	AUX
ejpam-1878	51	4	a	a	DET
ejpam-1878	51	5	convolution	convolution	NOUN
ejpam-1878	51	6	.	.	PUNCT
ejpam-1878	52	1	(	(	PUNCT
ejpam-1878	52	2	1	1	NUM
ejpam-1878	52	3	)	)	PUNCT
ejpam-1878	52	4	.	.	PUNCT
ejpam-1878	53	1	if	if	SCONJ
ejpam-1878	53	2	(	(	PUNCT
ejpam-1878	53	3	z+,≤c	z+,≤c	NUM
ejpam-1878	53	4	)	)	PUNCT
ejpam-1878	53	5	is	be	AUX
ejpam-1878	53	6	a	a	DET
ejpam-1878	53	7	meet(join	meet(join	NOUN
ejpam-1878	53	8	)	)	PUNCT
ejpam-1878	53	9	semilattice	semilattice	NOUN
ejpam-1878	53	10	,	,	PUNCT
ejpam-1878	53	11	then	then	ADV
ejpam-1878	53	12	so	so	ADV
ejpam-1878	53	13	is	be	AUX
ejpam-1878	53	14	(	(	PUNCT
ejpam-1878	53	15	n	n	X
ejpam-1878	53	16	,	,	PUNCT
ejpam-1878	53	17	≤p	≤p	PROPN
ejpam-1878	53	18	c	c	PROPN
ejpam-1878	53	19	)	)	PUNCT
ejpam-1878	53	20	for	for	ADP
ejpam-1878	53	21	each	each	DET
ejpam-1878	53	22	prime	prime	ADJ
ejpam-1878	53	23	p.	p.	NOUN
ejpam-1878	53	24	(	(	PUNCT
ejpam-1878	53	25	2	2	NUM
ejpam-1878	53	26	)	)	PUNCT
ejpam-1878	53	27	.	.	PUNCT
ejpam-1878	54	1	if	if	SCONJ
ejpam-1878	54	2	(	(	PUNCT
ejpam-1878	54	3	z+,≤c	z+,≤c	NUM
ejpam-1878	54	4	)	)	PUNCT
ejpam-1878	54	5	is	be	AUX
ejpam-1878	54	6	a	a	DET
ejpam-1878	54	7	lattice	lattice	NOUN
ejpam-1878	54	8	,	,	PUNCT
ejpam-1878	54	9	then	then	ADV
ejpam-1878	54	10	so	so	ADV
ejpam-1878	54	11	is	be	AUX
ejpam-1878	54	12	(	(	PUNCT
ejpam-1878	54	13	n	n	X
ejpam-1878	54	14	,	,	PUNCT
ejpam-1878	54	15	≤p	≤p	PROPN
ejpam-1878	54	16	c	c	PROPN
ejpam-1878	54	17	)	)	PUNCT
ejpam-1878	54	18	for	for	ADP
ejpam-1878	54	19	each	each	DET
ejpam-1878	54	20	prime	prime	NOUN
ejpam-1878	54	21	p.	p.	NOUN
ejpam-1878	54	22	3	3	NUM
ejpam-1878	54	23	.	.	PUNCT
ejpam-1878	54	24	ideals	ideal	NOUN
ejpam-1878	54	25	in	in	ADP
ejpam-1878	54	26	(	(	PUNCT
ejpam-1878	54	27	z	z	NOUN
ejpam-1878	54	28	+	+	ADJ
ejpam-1878	54	29	,	,	PUNCT
ejpam-1878	54	30	≤d	≤d	NOUN
ejpam-1878	54	31	)	)	PUNCT
ejpam-1878	54	32	recall	recall	VERB
ejpam-1878	54	33	that	that	SCONJ
ejpam-1878	54	34	most	most	ADJ
ejpam-1878	54	35	of	of	ADP
ejpam-1878	54	36	the	the	DET
ejpam-1878	54	37	convolutions	convolution	NOUN
ejpam-1878	54	38	like	like	ADP
ejpam-1878	54	39	dirichlet	dirichlet	PROPN
ejpam-1878	54	40	’s	’s	PART
ejpam-1878	54	41	convolution	convolution	NOUN
ejpam-1878	54	42	,	,	PUNCT
ejpam-1878	54	43	unitary	unitary	ADJ
ejpam-1878	54	44	convolution	convolution	NOUN
ejpam-1878	54	45	and	and	CCONJ
ejpam-1878	54	46	k	k	ADV
ejpam-1878	54	47	-	-	PUNCT
ejpam-1878	54	48	free	free	ADJ
ejpam-1878	54	49	convolution	convolution	NOUN
ejpam-1878	54	50	induce	induce	VERB
ejpam-1878	54	51	meet	meet	NOUN
ejpam-1878	54	52	semi	semi	ADJ
ejpam-1878	54	53	lattice	lattice	NOUN
ejpam-1878	54	54	structure	structure	NOUN
ejpam-1878	54	55	on	on	ADP
ejpam-1878	54	56	z+	z+	NUM
ejpam-1878	54	57	[	[	X
ejpam-1878	54	58	3	3	NUM
ejpam-1878	54	59	]	]	PUNCT
ejpam-1878	54	60	.	.	PUNCT
ejpam-1878	55	1	for	for	ADP
ejpam-1878	55	2	this	this	DET
ejpam-1878	55	3	reason	reason	NOUN
ejpam-1878	55	4	we	we	PRON
ejpam-1878	55	5	study	study	VERB
ejpam-1878	55	6	ideals	ideal	NOUN
ejpam-1878	55	7	in	in	ADP
ejpam-1878	55	8	a	a	DET
ejpam-1878	55	9	general	general	ADJ
ejpam-1878	55	10	meet	meet	VERB
ejpam-1878	55	11	semi	semi	ADV
ejpam-1878	55	12	lattice	lattice	NOUN
ejpam-1878	55	13	and	and	CCONJ
ejpam-1878	55	14	later	later	ADV
ejpam-1878	55	15	study	study	NOUN
ejpam-1878	55	16	ideals	ideal	NOUN
ejpam-1878	55	17	in	in	ADP
ejpam-1878	55	18	the	the	DET
ejpam-1878	55	19	lattice	lattice	NOUN
ejpam-1878	55	20	structure	structure	NOUN
ejpam-1878	55	21	z+	z+	NUM
ejpam-1878	55	22	induced	induce	VERB
ejpam-1878	55	23	by	by	ADP
ejpam-1878	55	24	the	the	DET
ejpam-1878	55	25	division	division	NOUN
ejpam-1878	55	26	ordering	ordering	NOUN
ejpam-1878	55	27	/.	/.	PUNCT
ejpam-1878	56	1	the	the	DET
ejpam-1878	56	2	division	division	NOUN
ejpam-1878	56	3	ordering	ordering	NOUN
ejpam-1878	56	4	/	/	PUNCT
ejpam-1878	56	5	is	be	AUX
ejpam-1878	56	6	precisely	precisely	ADV
ejpam-1878	56	7	the	the	DET
ejpam-1878	56	8	partial	partial	ADJ
ejpam-1878	56	9	ordering≤d	ordering≤d	NOUN
ejpam-1878	56	10	induced	induce	VERB
ejpam-1878	56	11	by	by	ADP
ejpam-1878	56	12	the	the	DET
ejpam-1878	56	13	dirichlet	dirichlet	PROPN
ejpam-1878	56	14	’s	’s	PART
ejpam-1878	56	15	convolution	convolution	PROPN
ejpam-1878	56	16	d.	d.	PROPN
ejpam-1878	56	17	throughout	throughout	ADP
ejpam-1878	56	18	this	this	DET
ejpam-1878	56	19	section	section	NOUN
ejpam-1878	56	20	,	,	PUNCT
ejpam-1878	56	21	unless	unless	SCONJ
ejpam-1878	56	22	otherwise	otherwise	ADV
ejpam-1878	56	23	stated	state	VERB
ejpam-1878	56	24	,	,	PUNCT
ejpam-1878	56	25	by	by	ADP
ejpam-1878	56	26	a	a	DET
ejpam-1878	56	27	semi	semi	ADJ
ejpam-1878	56	28	lattice	lattice	NOUN
ejpam-1878	56	29	we	we	PRON
ejpam-1878	56	30	mean	mean	VERB
ejpam-1878	56	31	a	a	DET
ejpam-1878	56	32	meet	meet	NOUN
ejpam-1878	56	33	semi	semi	ADJ
ejpam-1878	56	34	lattice	lattice	NOUN
ejpam-1878	56	35	only	only	ADV
ejpam-1878	56	36	.	.	PUNCT
ejpam-1878	57	1	definition	definition	NOUN
ejpam-1878	57	2	4	4	NUM
ejpam-1878	57	3	.	.	PUNCT
ejpam-1878	58	1	let	let	AUX
ejpam-1878	58	2	(	(	PUNCT
ejpam-1878	58	3	x	x	X
ejpam-1878	58	4	,	,	PUNCT
ejpam-1878	58	5	≤	≤	NUM
ejpam-1878	58	6	)	)	PUNCT
ejpam-1878	58	7	be	be	VERB
ejpam-1878	58	8	a	a	DET
ejpam-1878	58	9	poset	poset	NOUN
ejpam-1878	58	10	.	.	PUNCT
ejpam-1878	59	1	a	a	DET
ejpam-1878	59	2	non	non	ADJ
ejpam-1878	59	3	-	-	ADJ
ejpam-1878	59	4	empty	empty	ADJ
ejpam-1878	59	5	subset	subset	NOUN
ejpam-1878	59	6	i	i	PRON
ejpam-1878	59	7	of	of	ADP
ejpam-1878	59	8	x	x	PRON
ejpam-1878	59	9	is	be	AUX
ejpam-1878	59	10	called	call	VERB
ejpam-1878	59	11	an	an	DET
ejpam-1878	59	12	initial	initial	ADJ
ejpam-1878	59	13	segment	segment	NOUN
ejpam-1878	59	14	if	if	SCONJ
ejpam-1878	59	15	a	a	DET
ejpam-1878	59	16	∈	∈	NOUN
ejpam-1878	59	17	i	i	PRON
ejpam-1878	59	18	,	,	PUNCT
ejpam-1878	59	19	x	x	PUNCT
ejpam-1878	59	20	∈	∈	NOUN
ejpam-1878	59	21	x	x	X
ejpam-1878	59	22	and	and	CCONJ
ejpam-1878	59	23	x	x	SYM
ejpam-1878	59	24	≤	≤	NOUN
ejpam-1878	60	1	a	a	DET
ejpam-1878	60	2	=	=	NOUN
ejpam-1878	60	3	⇒	⇒	NOUN
ejpam-1878	60	4	x	x	PUNCT
ejpam-1878	60	5	∈	∈	PROPN
ejpam-1878	60	6	i	i	PRON
ejpam-1878	60	7	.	.	PUNCT
ejpam-1878	61	1	definition	definition	NOUN
ejpam-1878	61	2	5	5	NUM
ejpam-1878	61	3	.	.	PUNCT
ejpam-1878	62	1	let	let	VERB
ejpam-1878	62	2	(	(	PUNCT
ejpam-1878	62	3	s,∧	s,∧	NOUN
ejpam-1878	62	4	)	)	PUNCT
ejpam-1878	62	5	be	be	AUX
ejpam-1878	62	6	a	a	DET
ejpam-1878	62	7	semi	semi	ADJ
ejpam-1878	62	8	lattice	lattice	NOUN
ejpam-1878	62	9	.	.	PUNCT
ejpam-1878	63	1	a	a	DET
ejpam-1878	63	2	non	non	ADJ
ejpam-1878	63	3	-	-	ADJ
ejpam-1878	63	4	empty	empty	ADJ
ejpam-1878	63	5	subset	subset	NOUN
ejpam-1878	63	6	i	i	PRON
ejpam-1878	63	7	of	of	ADP
ejpam-1878	63	8	s	s	PROPN
ejpam-1878	63	9	is	be	AUX
ejpam-1878	63	10	called	call	VERB
ejpam-1878	63	11	an	an	DET
ejpam-1878	63	12	ideal	ideal	NOUN
ejpam-1878	63	13	of	of	ADP
ejpam-1878	63	14	s	s	PRON
ejpam-1878	63	15	if	if	SCONJ
ejpam-1878	63	16	the	the	DET
ejpam-1878	63	17	following	following	NOUN
ejpam-1878	63	18	are	be	AUX
ejpam-1878	63	19	satisfied	satisfied	ADJ
ejpam-1878	63	20	(	(	PUNCT
ejpam-1878	63	21	1	1	NUM
ejpam-1878	63	22	)	)	PUNCT
ejpam-1878	63	23	.	.	PUNCT
ejpam-1878	64	1	x	x	PUNCT
ejpam-1878	64	2	∈	∈	NOUN
ejpam-1878	64	3	s	s	X
ejpam-1878	64	4	and	and	CCONJ
ejpam-1878	64	5	x	x	SYM
ejpam-1878	64	6	≤	≤	NOUN
ejpam-1878	64	7	a	a	DET
ejpam-1878	64	8	∈	∈	NOUN
ejpam-1878	65	1	i	i	PRON
ejpam-1878	65	2	=	=	VERB
ejpam-1878	65	3	⇒	⇒	VERB
ejpam-1878	65	4	x	x	PUNCT
ejpam-1878	65	5	∈	∈	PROPN
ejpam-1878	66	1	i	i	PRON
ejpam-1878	66	2	(	(	PUNCT
ejpam-1878	66	3	2	2	NUM
ejpam-1878	66	4	)	)	PUNCT
ejpam-1878	66	5	.	.	PUNCT
ejpam-1878	67	1	for	for	ADP
ejpam-1878	67	2	any	any	DET
ejpam-1878	67	3	a	a	DET
ejpam-1878	67	4	and	and	CCONJ
ejpam-1878	67	5	b	b	NOUN
ejpam-1878	67	6	∈	∈	PROPN
ejpam-1878	67	7	i	i	PRON
ejpam-1878	67	8	,	,	PUNCT
ejpam-1878	67	9	there	there	PRON
ejpam-1878	67	10	exists	exist	VERB
ejpam-1878	67	11	c	c	NOUN
ejpam-1878	67	12	∈	∈	PROPN
ejpam-1878	67	13	i	i	PRON
ejpam-1878	67	14	such	such	ADJ
ejpam-1878	67	15	that	that	SCONJ
ejpam-1878	67	16	a	a	DET
ejpam-1878	67	17	≤	≤	PROPN
ejpam-1878	67	18	c	c	NOUN
ejpam-1878	67	19	and	and	CCONJ
ejpam-1878	67	20	b	b	NOUN
ejpam-1878	67	21	≤	≤	PROPN
ejpam-1878	67	22	c	c	PROPN
ejpam-1878	67	23	s.	s.	PROPN
ejpam-1878	67	24	sagi	sagi	PROPN
ejpam-1878	67	25	/	/	SYM
ejpam-1878	67	26	eur	eur	PROPN
ejpam-1878	67	27	.	.	PUNCT
ejpam-1878	68	1	j.	j.	PROPN
ejpam-1878	68	2	pure	pure	PROPN
ejpam-1878	68	3	appl	appl	PROPN
ejpam-1878	68	4	.	.	PROPN
ejpam-1878	68	5	math	math	PROPN
ejpam-1878	68	6	,	,	PUNCT
ejpam-1878	68	7	8	8	NUM
ejpam-1878	68	8	(	(	PUNCT
ejpam-1878	68	9	2015	2015	NUM
ejpam-1878	68	10	)	)	PUNCT
ejpam-1878	68	11	,	,	PUNCT
ejpam-1878	68	12	15	15	NUM
ejpam-1878	68	13	-	-	SYM
ejpam-1878	68	14	25	25	NUM
ejpam-1878	68	15	18	18	NUM
ejpam-1878	68	16	definition	definition	NOUN
ejpam-1878	68	17	6	6	NUM
ejpam-1878	68	18	.	.	PUNCT
ejpam-1878	69	1	let	let	VERB
ejpam-1878	69	2	(	(	PUNCT
ejpam-1878	69	3	s,∧	s,∧	NOUN
ejpam-1878	69	4	)	)	PUNCT
ejpam-1878	69	5	be	be	AUX
ejpam-1878	69	6	a	a	DET
ejpam-1878	69	7	semi	semi	ADJ
ejpam-1878	69	8	lattice	lattice	NOUN
ejpam-1878	69	9	and	and	CCONJ
ejpam-1878	69	10	a	a	DET
ejpam-1878	69	11	∈	∈	NOUN
ejpam-1878	69	12	s.	s.	PROPN
ejpam-1878	69	13	then	then	ADV
ejpam-1878	69	14	the	the	DET
ejpam-1878	69	15	set	set	NOUN
ejpam-1878	69	16	(	(	PUNCT
ejpam-1878	69	17	a	a	X
ejpam-1878	69	18	]	]	X
ejpam-1878	69	19	:	:	PUNCT
ejpam-1878	69	20	=	=	SYM
ejpam-1878	69	21	{	{	PUNCT
ejpam-1878	69	22	x	x	SYM
ejpam-1878	69	23	∈	∈	PROPN
ejpam-1878	69	24	s|x	s|x	NOUN
ejpam-1878	70	1	≤	≤	ADJ
ejpam-1878	70	2	a}=	a}=	PROPN
ejpam-1878	70	3	{	{	PUNCT
ejpam-1878	70	4	y	y	NOUN
ejpam-1878	70	5	∧	∧	PROPN
ejpam-1878	70	6	a|y	a|y	NOUN
ejpam-1878	70	7	∈	∈	PRON
ejpam-1878	70	8	s	s	AUX
ejpam-1878	70	9	}	}	PUNCT
ejpam-1878	70	10	is	be	AUX
ejpam-1878	70	11	an	an	DET
ejpam-1878	70	12	ideal	ideal	NOUN
ejpam-1878	70	13	of	of	ADP
ejpam-1878	70	14	s	s	PRON
ejpam-1878	70	15	and	and	CCONJ
ejpam-1878	70	16	is	be	AUX
ejpam-1878	70	17	called	call	VERB
ejpam-1878	70	18	the	the	DET
ejpam-1878	70	19	principal	principal	ADJ
ejpam-1878	70	20	ideal	ideal	NOUN
ejpam-1878	70	21	generated	generate	VERB
ejpam-1878	70	22	by	by	ADP
ejpam-1878	70	23	a	a	PRON
ejpam-1878	70	24	in	in	ADP
ejpam-1878	70	25	s.	s.	PROPN
ejpam-1878	70	26	note	note	VERB
ejpam-1878	70	27	that	that	SCONJ
ejpam-1878	70	28	(	(	PUNCT
ejpam-1878	70	29	a	a	X
ejpam-1878	70	30	]	]	X
ejpam-1878	70	31	is	be	AUX
ejpam-1878	70	32	the	the	DET
ejpam-1878	70	33	smallest	small	ADJ
ejpam-1878	70	34	ideal	ideal	NOUN
ejpam-1878	70	35	of	of	ADP
ejpam-1878	70	36	s	s	NUM
ejpam-1878	70	37	containing	contain	VERB
ejpam-1878	70	38	a.	a.	NOUN
ejpam-1878	70	39	now	now	ADV
ejpam-1878	70	40	,	,	PUNCT
ejpam-1878	70	41	we	we	PRON
ejpam-1878	70	42	present	present	VERB
ejpam-1878	70	43	the	the	DET
ejpam-1878	70	44	following	follow	VERB
ejpam-1878	70	45	theorem	theorem	NOUN
ejpam-1878	70	46	2	2	NUM
ejpam-1878	70	47	.	.	PUNCT
ejpam-1878	70	48	let	let	VERB
ejpam-1878	70	49	a	a	PRON
ejpam-1878	70	50	and	and	CCONJ
ejpam-1878	70	51	b	b	NOUN
ejpam-1878	70	52	be	be	AUX
ejpam-1878	70	53	elements	element	NOUN
ejpam-1878	70	54	of	of	ADP
ejpam-1878	70	55	a	a	DET
ejpam-1878	70	56	meet	meet	NOUN
ejpam-1878	70	57	semi	semi	ADJ
ejpam-1878	70	58	lattice	lattice	PROPN
ejpam-1878	70	59	(	(	PUNCT
ejpam-1878	70	60	s,∧	s,∧	NOUN
ejpam-1878	70	61	)	)	PUNCT
ejpam-1878	70	62	.	.	PUNCT
ejpam-1878	71	1	then	then	ADV
ejpam-1878	71	2	the	the	DET
ejpam-1878	71	3	following	follow	VERB
ejpam-1878	71	4	are	be	AUX
ejpam-1878	71	5	equivalent	equivalent	ADJ
ejpam-1878	71	6	to	to	ADP
ejpam-1878	71	7	each	each	DET
ejpam-1878	71	8	other	other	ADJ
ejpam-1878	71	9	.	.	PUNCT
ejpam-1878	72	1	(	(	PUNCT
ejpam-1878	72	2	1	1	NUM
ejpam-1878	72	3	)	)	PUNCT
ejpam-1878	72	4	.	.	PUNCT
ejpam-1878	73	1	there	there	PRON
ejpam-1878	73	2	exists	exist	VERB
ejpam-1878	73	3	smallest	small	ADJ
ejpam-1878	73	4	ideal	ideal	NOUN
ejpam-1878	73	5	of	of	ADP
ejpam-1878	73	6	s	s	AUX
ejpam-1878	73	7	containing	contain	VERB
ejpam-1878	73	8	a	a	PRON
ejpam-1878	73	9	and	and	CCONJ
ejpam-1878	73	10	b.	b.	PROPN
ejpam-1878	73	11	(	(	PUNCT
ejpam-1878	73	12	2	2	NUM
ejpam-1878	73	13	)	)	PUNCT
ejpam-1878	73	14	.	.	PUNCT
ejpam-1878	74	1	the	the	DET
ejpam-1878	74	2	intersection	intersection	NOUN
ejpam-1878	74	3	of	of	ADP
ejpam-1878	74	4	all	all	DET
ejpam-1878	74	5	ideals	ideal	NOUN
ejpam-1878	74	6	of	of	ADP
ejpam-1878	74	7	s	s	PRON
ejpam-1878	74	8	containing	contain	VERB
ejpam-1878	74	9	a	a	PRON
ejpam-1878	74	10	and	and	CCONJ
ejpam-1878	74	11	b	b	NOUN
ejpam-1878	74	12	is	be	AUX
ejpam-1878	74	13	again	again	ADV
ejpam-1878	74	14	an	an	DET
ejpam-1878	74	15	ideal	ideal	NOUN
ejpam-1878	74	16	of	of	ADP
ejpam-1878	74	17	s.	s.	PROPN
ejpam-1878	74	18	(	(	PUNCT
ejpam-1878	74	19	3	3	NUM
ejpam-1878	74	20	)	)	PUNCT
ejpam-1878	74	21	.	.	PUNCT
ejpam-1878	75	1	a	a	PRON
ejpam-1878	75	2	and	and	CCONJ
ejpam-1878	75	3	b	b	NOUN
ejpam-1878	75	4	have	have	AUX
ejpam-1878	75	5	least	least	ADJ
ejpam-1878	75	6	upper	upper	ADJ
ejpam-1878	75	7	bound	bind	VERB
ejpam-1878	75	8	in	in	ADP
ejpam-1878	75	9	s.	s.	PROPN
ejpam-1878	75	10	proof	proof	PROPN
ejpam-1878	75	11	.	.	PUNCT
ejpam-1878	76	1	(	(	PUNCT
ejpam-1878	76	2	1)	1)	NUM
ejpam-1878	76	3	⇐	⇐	ADJ
ejpam-1878	76	4	⇒	⇒	NOUN
ejpam-1878	76	5	(	(	PUNCT
ejpam-1878	76	6	2	2	NUM
ejpam-1878	76	7	)	)	PUNCT
ejpam-1878	76	8	:	:	PUNCT
ejpam-1878	76	9	is	be	AUX
ejpam-1878	76	10	trivial	trivial	ADJ
ejpam-1878	76	11	.	.	PUNCT
ejpam-1878	77	1	(	(	PUNCT
ejpam-1878	77	2	1	1	X
ejpam-1878	77	3	)	)	PUNCT
ejpam-1878	77	4	=	=	NOUN
ejpam-1878	77	5	⇒	⇒	NOUN
ejpam-1878	77	6	(	(	PUNCT
ejpam-1878	77	7	3	3	NUM
ejpam-1878	77	8	)	)	PUNCT
ejpam-1878	77	9	:	:	PUNCT
ejpam-1878	77	10	let	let	VERB
ejpam-1878	77	11	i	i	PRON
ejpam-1878	77	12	be	be	AUX
ejpam-1878	77	13	the	the	DET
ejpam-1878	77	14	smallest	small	ADJ
ejpam-1878	77	15	ideal	ideal	NOUN
ejpam-1878	77	16	of	of	ADP
ejpam-1878	77	17	s	s	AUX
ejpam-1878	77	18	containing	contain	VERB
ejpam-1878	77	19	a	a	PRON
ejpam-1878	77	20	and	and	CCONJ
ejpam-1878	77	21	b.	b.	PROPN
ejpam-1878	77	22	then	then	ADV
ejpam-1878	77	23	,	,	PUNCT
ejpam-1878	77	24	there	there	PRON
ejpam-1878	77	25	exists	exist	VERB
ejpam-1878	77	26	x	x	X
ejpam-1878	77	27	∈	∈	PROPN
ejpam-1878	77	28	i	i	PRON
ejpam-1878	77	29	such	such	ADJ
ejpam-1878	77	30	that	that	SCONJ
ejpam-1878	77	31	a	a	DET
ejpam-1878	77	32	≤	≤	PROPN
ejpam-1878	77	33	x	x	PUNCT
ejpam-1878	77	34	and	and	CCONJ
ejpam-1878	77	35	b	b	NOUN
ejpam-1878	77	36	≤	≤	NOUN
ejpam-1878	77	37	x	x	PUNCT
ejpam-1878	77	38	therefore	therefore	ADV
ejpam-1878	77	39	x	x	X
ejpam-1878	77	40	is	be	AUX
ejpam-1878	77	41	an	an	DET
ejpam-1878	77	42	upper	upper	ADJ
ejpam-1878	77	43	bound	bound	NOUN
ejpam-1878	77	44	of	of	ADP
ejpam-1878	77	45	a	a	PRON
ejpam-1878	77	46	and	and	CCONJ
ejpam-1878	77	47	b.	b.	NOUN
ejpam-1878	77	48	if	if	SCONJ
ejpam-1878	77	49	y	y	PROPN
ejpam-1878	77	50	is	be	AUX
ejpam-1878	77	51	any	any	DET
ejpam-1878	77	52	other	other	ADJ
ejpam-1878	77	53	upper	upper	ADJ
ejpam-1878	77	54	bound	bind	VERB
ejpam-1878	77	55	of	of	ADP
ejpam-1878	77	56	a	a	PRON
ejpam-1878	77	57	and	and	CCONJ
ejpam-1878	77	58	b	b	NOUN
ejpam-1878	77	59	,	,	PUNCT
ejpam-1878	77	60	then	then	ADV
ejpam-1878	77	61	(	(	PUNCT
ejpam-1878	77	62	y	y	NOUN
ejpam-1878	77	63	]	]	X
ejpam-1878	77	64	is	be	AUX
ejpam-1878	77	65	an	an	DET
ejpam-1878	77	66	ideal	ideal	NOUN
ejpam-1878	77	67	of	of	ADP
ejpam-1878	77	68	s	s	AUX
ejpam-1878	77	69	containing	contain	VERB
ejpam-1878	77	70	a	a	PRON
ejpam-1878	77	71	and	and	CCONJ
ejpam-1878	77	72	b	b	NOUN
ejpam-1878	77	73	and	and	CCONJ
ejpam-1878	77	74	hence	hence	ADV
ejpam-1878	77	75	i	i	PRON
ejpam-1878	77	76	⊆	⊆	NUM
ejpam-1878	77	77	(	(	PUNCT
ejpam-1878	77	78	y	y	NOUN
ejpam-1878	77	79	]	]	X
ejpam-1878	77	80	.	.	PUNCT
ejpam-1878	78	1	since	since	SCONJ
ejpam-1878	78	2	x	x	PROPN
ejpam-1878	78	3	∈	∈	PROPN
ejpam-1878	78	4	i	i	PRON
ejpam-1878	78	5	,	,	PUNCT
ejpam-1878	78	6	we	we	PRON
ejpam-1878	78	7	get	get	VERB
ejpam-1878	78	8	that	that	PRON
ejpam-1878	78	9	x	x	SYM
ejpam-1878	78	10	∈	∈	PROPN
ejpam-1878	78	11	(	(	PUNCT
ejpam-1878	78	12	y	y	NOUN
ejpam-1878	78	13	]	]	PUNCT
ejpam-1878	78	14	and	and	CCONJ
ejpam-1878	78	15	therefore	therefore	ADV
ejpam-1878	78	16	x	x	X
ejpam-1878	78	17	≤	≤	ADJ
ejpam-1878	78	18	y	y	NOUN
ejpam-1878	78	19	.	.	PUNCT
ejpam-1878	79	1	thus	thus	ADV
ejpam-1878	79	2	x	x	PRON
ejpam-1878	79	3	is	be	AUX
ejpam-1878	79	4	the	the	DET
ejpam-1878	79	5	least	least	ADJ
ejpam-1878	79	6	upper	upper	ADJ
ejpam-1878	79	7	bound	bind	VERB
ejpam-1878	79	8	of	of	ADP
ejpam-1878	79	9	a	a	PRON
ejpam-1878	79	10	and	and	CCONJ
ejpam-1878	79	11	b.	b.	PROPN
ejpam-1878	79	12	(	(	PUNCT
ejpam-1878	80	1	3	3	X
ejpam-1878	80	2	)	)	PUNCT
ejpam-1878	80	3	=	=	NOUN
ejpam-1878	80	4	⇒	⇒	NOUN
ejpam-1878	80	5	(	(	PUNCT
ejpam-1878	80	6	1	1	NUM
ejpam-1878	80	7	)	)	PUNCT
ejpam-1878	80	8	:	:	PUNCT
ejpam-1878	80	9	let	let	VERB
ejpam-1878	80	10	a∨	a∨	PROPN
ejpam-1878	80	11	b	b	PROPN
ejpam-1878	80	12	be	be	AUX
ejpam-1878	80	13	the	the	DET
ejpam-1878	80	14	least	least	ADJ
ejpam-1878	80	15	upper	upper	ADJ
ejpam-1878	80	16	bound	bind	VERB
ejpam-1878	80	17	of	of	ADP
ejpam-1878	80	18	a	a	PRON
ejpam-1878	80	19	and	and	CCONJ
ejpam-1878	80	20	b.	b.	PROPN
ejpam-1878	81	1	then	then	ADV
ejpam-1878	81	2	a	a	DET
ejpam-1878	81	3	≤	≤	ADJ
ejpam-1878	81	4	a∨	a∨	PROPN
ejpam-1878	81	5	b	b	PROPN
ejpam-1878	81	6	and	and	CCONJ
ejpam-1878	81	7	b	b	PROPN
ejpam-1878	81	8	≤	≤	PROPN
ejpam-1878	81	9	a∨	a∨	PROPN
ejpam-1878	81	10	b	b	PROPN
ejpam-1878	81	11	and	and	CCONJ
ejpam-1878	81	12	hence	hence	ADV
ejpam-1878	81	13	(	(	PUNCT
ejpam-1878	81	14	a	a	DET
ejpam-1878	81	15	∨	∨	PROPN
ejpam-1878	81	16	b	b	NOUN
ejpam-1878	81	17	]	]	X
ejpam-1878	81	18	is	be	AUX
ejpam-1878	81	19	an	an	DET
ejpam-1878	81	20	ideal	ideal	NOUN
ejpam-1878	81	21	containing	contain	VERB
ejpam-1878	81	22	a	a	PRON
ejpam-1878	81	23	and	and	CCONJ
ejpam-1878	81	24	b.	b.	NOUN
ejpam-1878	81	25	if	if	SCONJ
ejpam-1878	81	26	i	i	PRON
ejpam-1878	81	27	is	be	AUX
ejpam-1878	81	28	any	any	DET
ejpam-1878	81	29	ideal	ideal	NOUN
ejpam-1878	81	30	containing	contain	VERB
ejpam-1878	81	31	a	a	PRON
ejpam-1878	81	32	and	and	CCONJ
ejpam-1878	81	33	b	b	NOUN
ejpam-1878	81	34	,	,	PUNCT
ejpam-1878	81	35	then	then	ADV
ejpam-1878	81	36	there	there	PRON
ejpam-1878	81	37	exists	exist	VERB
ejpam-1878	81	38	x	x	X
ejpam-1878	81	39	∈	∈	PROPN
ejpam-1878	81	40	i	i	PRON
ejpam-1878	82	1	such	such	ADJ
ejpam-1878	82	2	that	that	SCONJ
ejpam-1878	82	3	a	a	DET
ejpam-1878	82	4	≤	≤	PROPN
ejpam-1878	82	5	x	x	PUNCT
ejpam-1878	82	6	and	and	CCONJ
ejpam-1878	82	7	b	b	NOUN
ejpam-1878	82	8	≤	≤	NOUN
ejpam-1878	82	9	x	x	PUNCT
ejpam-1878	82	10	and	and	CCONJ
ejpam-1878	82	11	hence	hence	ADV
ejpam-1878	82	12	a	a	DET
ejpam-1878	82	13	∨	∨	NUM
ejpam-1878	82	14	b	b	NOUN
ejpam-1878	82	15	≤	≤	NOUN
ejpam-1878	82	16	x	x	PUNCT
ejpam-1878	82	17	so	so	SCONJ
ejpam-1878	82	18	that	that	SCONJ
ejpam-1878	82	19	a∨	a∨	PROPN
ejpam-1878	82	20	b	b	PROPN
ejpam-1878	82	21	∈	∈	PROPN
ejpam-1878	83	1	i	i	PRON
ejpam-1878	83	2	and	and	CCONJ
ejpam-1878	83	3	(	(	PUNCT
ejpam-1878	83	4	a∨	a∨	PROPN
ejpam-1878	83	5	b	b	PROPN
ejpam-1878	83	6	]	]	PUNCT
ejpam-1878	83	7	⊆	⊆	NUM
ejpam-1878	83	8	i	i	PRON
ejpam-1878	83	9	.	.	PUNCT
ejpam-1878	84	1	thus	thus	ADV
ejpam-1878	84	2	(	(	PUNCT
ejpam-1878	84	3	a∨	a∨	PROPN
ejpam-1878	84	4	b	b	AUX
ejpam-1878	84	5	]	]	X
ejpam-1878	84	6	is	be	AUX
ejpam-1878	84	7	the	the	DET
ejpam-1878	84	8	smallest	small	ADJ
ejpam-1878	84	9	ideal	ideal	NOUN
ejpam-1878	84	10	of	of	ADP
ejpam-1878	84	11	s	s	AUX
ejpam-1878	84	12	containing	contain	VERB
ejpam-1878	84	13	a	a	PRON
ejpam-1878	84	14	and	and	CCONJ
ejpam-1878	84	15	b.	b.	PROPN
ejpam-1878	84	16	although	although	SCONJ
ejpam-1878	84	17	the	the	DET
ejpam-1878	84	18	intersection	intersection	NOUN
ejpam-1878	84	19	of	of	ADP
ejpam-1878	84	20	an	an	DET
ejpam-1878	84	21	arbitrary	arbitrary	ADJ
ejpam-1878	84	22	class	class	NOUN
ejpam-1878	84	23	of	of	ADP
ejpam-1878	84	24	ideals	ideal	NOUN
ejpam-1878	84	25	need	need	AUX
ejpam-1878	84	26	not	not	PART
ejpam-1878	84	27	be	be	AUX
ejpam-1878	84	28	an	an	DET
ejpam-1878	84	29	ideal	ideal	NOUN
ejpam-1878	84	30	,	,	PUNCT
ejpam-1878	84	31	a	a	DET
ejpam-1878	84	32	finite	finite	ADJ
ejpam-1878	84	33	intersection	intersection	NOUN
ejpam-1878	84	34	is	be	AUX
ejpam-1878	84	35	always	always	ADV
ejpam-1878	84	36	an	an	DET
ejpam-1878	84	37	ideal	ideal	NOUN
ejpam-1878	84	38	.	.	PUNCT
ejpam-1878	85	1	theorem	theorem	NOUN
ejpam-1878	85	2	3	3	X
ejpam-1878	85	3	.	.	PUNCT
ejpam-1878	86	1	let	let	ADJ
ejpam-1878	86	2	(	(	PUNCT
ejpam-1878	86	3	s,∧	s,∧	NOUN
ejpam-1878	86	4	)	)	PUNCT
ejpam-1878	86	5	be	be	AUX
ejpam-1878	86	6	a	a	DET
ejpam-1878	86	7	semi	semi	ADJ
ejpam-1878	86	8	lattice	lattice	NOUN
ejpam-1878	87	1	and	and	CCONJ
ejpam-1878	87	2	i	i	PRON
ejpam-1878	87	3	(	(	PUNCT
ejpam-1878	87	4	s	s	X
ejpam-1878	87	5	)	)	PUNCT
ejpam-1878	87	6	the	the	DET
ejpam-1878	87	7	set	set	NOUN
ejpam-1878	87	8	of	of	ADP
ejpam-1878	87	9	all	all	DET
ejpam-1878	87	10	ideals	ideal	NOUN
ejpam-1878	87	11	of	of	ADP
ejpam-1878	87	12	s.	s.	PROPN
ejpam-1878	87	13	then	then	ADV
ejpam-1878	87	14	(	(	PUNCT
ejpam-1878	87	15	i	i	PRON
ejpam-1878	87	16	(	(	PUNCT
ejpam-1878	87	17	s),∩	s),∩	PROPN
ejpam-1878	87	18	)	)	PUNCT
ejpam-1878	87	19	is	be	AUX
ejpam-1878	87	20	a	a	DET
ejpam-1878	87	21	semilattice	semilattice	NOUN
ejpam-1878	87	22	and	and	CCONJ
ejpam-1878	87	23	a	a	DET
ejpam-1878	87	24	7→	7→	NOUN
ejpam-1878	87	25	(	(	PUNCT
ejpam-1878	87	26	a	a	PRON
ejpam-1878	87	27	]	]	X
ejpam-1878	87	28	is	be	AUX
ejpam-1878	87	29	an	an	DET
ejpam-1878	87	30	embedding	embedding	NOUN
ejpam-1878	87	31	of	of	ADP
ejpam-1878	87	32	(	(	PUNCT
ejpam-1878	87	33	s,∧	s,∧	NOUN
ejpam-1878	87	34	)	)	PUNCT
ejpam-1878	87	35	onto	onto	ADP
ejpam-1878	87	36	(	(	PUNCT
ejpam-1878	87	37	i	i	PRON
ejpam-1878	87	38	(	(	PUNCT
ejpam-1878	87	39	s),∩	s),∩	PROPN
ejpam-1878	87	40	)	)	PUNCT
ejpam-1878	87	41	.	.	PUNCT
ejpam-1878	88	1	proof	proof	NOUN
ejpam-1878	88	2	.	.	PUNCT
ejpam-1878	89	1	by	by	ADP
ejpam-1878	89	2	the	the	DET
ejpam-1878	89	3	above	above	ADJ
ejpam-1878	89	4	theorem	theorem	NOUN
ejpam-1878	89	5	,	,	PUNCT
ejpam-1878	89	6	it	it	PRON
ejpam-1878	89	7	follows	follow	VERB
ejpam-1878	89	8	that	that	SCONJ
ejpam-1878	89	9	(	(	PUNCT
ejpam-1878	89	10	i	i	PRON
ejpam-1878	89	11	(	(	PUNCT
ejpam-1878	89	12	s),∩	s),∩	PROPN
ejpam-1878	89	13	)	)	PUNCT
ejpam-1878	89	14	is	be	AUX
ejpam-1878	89	15	a	a	DET
ejpam-1878	89	16	semi	semi	NOUN
ejpam-1878	89	17	lattice.also	lattice.also	ADV
ejpam-1878	89	18	,	,	PUNCT
ejpam-1878	89	19	for	for	ADP
ejpam-1878	89	20	any	any	DET
ejpam-1878	89	21	a	a	PRON
ejpam-1878	89	22	and	and	CCONJ
ejpam-1878	89	23	b	b	NOUN
ejpam-1878	89	24	in	in	ADP
ejpam-1878	89	25	s	s	PROPN
ejpam-1878	89	26	,	,	PUNCT
ejpam-1878	89	27	we	we	PRON
ejpam-1878	89	28	have	have	VERB
ejpam-1878	89	29	(	(	PUNCT
ejpam-1878	89	30	a]∩	a]∩	X
ejpam-1878	89	31	(	(	PUNCT
ejpam-1878	89	32	b	b	X
ejpam-1878	89	33	]	]	X
ejpam-1878	89	34	=	=	X
ejpam-1878	89	35	(	(	PUNCT
ejpam-1878	89	36	a	a	DET
ejpam-1878	89	37	∧	∧	PROPN
ejpam-1878	89	38	b	b	PROPN
ejpam-1878	89	39	]	]	PUNCT
ejpam-1878	89	40	and	and	CCONJ
ejpam-1878	89	41	(	(	PUNCT
ejpam-1878	89	42	a	a	X
ejpam-1878	89	43	]	]	X
ejpam-1878	89	44	⊆	⊆	NUM
ejpam-1878	89	45	(	(	PUNCT
ejpam-1878	89	46	b]	b]	ADJ
ejpam-1878	89	47	⇐	⇐	ADJ
ejpam-1878	89	48	⇒	⇒	NOUN
ejpam-1878	89	49	a	a	DET
ejpam-1878	89	50	∈	∈	PROPN
ejpam-1878	89	51	(	(	PUNCT
ejpam-1878	89	52	b]	b]	ADJ
ejpam-1878	89	53	⇐	⇐	ADJ
ejpam-1878	89	54	⇒	⇒	NOUN
ejpam-1878	89	55	a	a	DET
ejpam-1878	89	56	≤	≤	NUM
ejpam-1878	89	57	b	b	NOUN
ejpam-1878	89	58	therefore	therefore	ADV
ejpam-1878	89	59	a	a	DET
ejpam-1878	89	60	7→	7→	NOUN
ejpam-1878	89	61	(	(	PUNCT
ejpam-1878	89	62	a	a	PRON
ejpam-1878	89	63	]	]	X
ejpam-1878	89	64	is	be	AUX
ejpam-1878	89	65	an	an	DET
ejpam-1878	89	66	embedding	embedding	NOUN
ejpam-1878	89	67	of	of	ADP
ejpam-1878	89	68	s	s	NOUN
ejpam-1878	89	69	into	into	ADP
ejpam-1878	89	70	i	i	PRON
ejpam-1878	89	71	(	(	PUNCT
ejpam-1878	89	72	s	s	NOUN
ejpam-1878	89	73	)	)	PUNCT
ejpam-1878	89	74	.	.	PUNCT
ejpam-1878	90	1	theorem	theorem	ADJ
ejpam-1878	90	2	4	4	NUM
ejpam-1878	90	3	.	.	PUNCT
ejpam-1878	91	1	a	a	DET
ejpam-1878	91	2	semi	semi	ADJ
ejpam-1878	91	3	lattice	lattice	NOUN
ejpam-1878	91	4	(	(	PUNCT
ejpam-1878	91	5	s,∧	s,∧	PROPN
ejpam-1878	91	6	)	)	PUNCT
ejpam-1878	91	7	is	be	AUX
ejpam-1878	91	8	a	a	DET
ejpam-1878	91	9	lattice	lattice	NOUN
ejpam-1878	91	10	if	if	SCONJ
ejpam-1878	92	1	and	and	CCONJ
ejpam-1878	92	2	only	only	ADV
ejpam-1878	92	3	if	if	SCONJ
ejpam-1878	92	4	i	i	PRON
ejpam-1878	92	5	(	(	PUNCT
ejpam-1878	92	6	s	s	X
ejpam-1878	92	7	)	)	PUNCT
ejpam-1878	92	8	is	be	AUX
ejpam-1878	92	9	a	a	DET
ejpam-1878	92	10	lattice	lattice	NOUN
ejpam-1878	92	11	and	and	CCONJ
ejpam-1878	92	12	,	,	PUNCT
ejpam-1878	92	13	in	in	ADP
ejpam-1878	92	14	this	this	DET
ejpam-1878	92	15	case	case	NOUN
ejpam-1878	92	16	,	,	PUNCT
ejpam-1878	92	17	a	a	DET
ejpam-1878	92	18	7→	7→	NUM
ejpam-1878	92	19	(	(	PUNCT
ejpam-1878	92	20	a	a	PRON
ejpam-1878	92	21	]	]	X
ejpam-1878	92	22	is	be	AUX
ejpam-1878	92	23	an	an	DET
ejpam-1878	92	24	embedding	embedding	NOUN
ejpam-1878	92	25	of	of	ADP
ejpam-1878	92	26	the	the	DET
ejpam-1878	92	27	lattice	lattice	NOUN
ejpam-1878	92	28	s	s	NOUN
ejpam-1878	92	29	into	into	ADP
ejpam-1878	92	30	the	the	DET
ejpam-1878	92	31	lattice	lattice	NOUN
ejpam-1878	93	1	i	i	PRON
ejpam-1878	93	2	(	(	PUNCT
ejpam-1878	93	3	s	s	NOUN
ejpam-1878	93	4	)	)	PUNCT
ejpam-1878	93	5	.	.	PUNCT
ejpam-1878	94	1	s.	s.	PROPN
ejpam-1878	94	2	sagi	sagi	PROPN
ejpam-1878	94	3	/	/	SYM
ejpam-1878	94	4	eur	eur	PROPN
ejpam-1878	94	5	.	.	PUNCT
ejpam-1878	95	1	j.	j.	PROPN
ejpam-1878	95	2	pure	pure	PROPN
ejpam-1878	95	3	appl	appl	PROPN
ejpam-1878	95	4	.	.	PROPN
ejpam-1878	95	5	math	math	PROPN
ejpam-1878	95	6	,	,	PUNCT
ejpam-1878	95	7	8	8	NUM
ejpam-1878	95	8	(	(	PUNCT
ejpam-1878	95	9	2015	2015	NUM
ejpam-1878	95	10	)	)	PUNCT
ejpam-1878	95	11	,	,	PUNCT
ejpam-1878	95	12	15	15	NUM
ejpam-1878	95	13	-	-	SYM
ejpam-1878	95	14	25	25	NUM
ejpam-1878	95	15	19	19	NUM
ejpam-1878	95	16	proof	proof	NOUN
ejpam-1878	95	17	.	.	PUNCT
ejpam-1878	96	1	it	it	PRON
ejpam-1878	96	2	is	be	AUX
ejpam-1878	96	3	well	well	ADV
ejpam-1878	96	4	known	know	VERB
ejpam-1878	96	5	that	that	SCONJ
ejpam-1878	96	6	the	the	DET
ejpam-1878	96	7	set	set	NOUN
ejpam-1878	96	8	i	i	PRON
ejpam-1878	96	9	(	(	PUNCT
ejpam-1878	96	10	s	s	NOUN
ejpam-1878	96	11	)	)	PUNCT
ejpam-1878	96	12	of	of	ADP
ejpam-1878	96	13	ideals	ideal	NOUN
ejpam-1878	96	14	of	of	ADP
ejpam-1878	96	15	a	a	DET
ejpam-1878	96	16	lattice	lattice	NOUN
ejpam-1878	96	17	(	(	PUNCT
ejpam-1878	96	18	s,∧,∨	s,∧,∨	NOUN
ejpam-1878	96	19	)	)	PUNCT
ejpam-1878	96	20	is	be	AUX
ejpam-1878	96	21	again	again	ADV
ejpam-1878	96	22	a	a	DET
ejpam-1878	96	23	lattice	lattice	NOUN
ejpam-1878	96	24	in	in	ADP
ejpam-1878	96	25	which	which	PRON
ejpam-1878	96	26	,	,	PUNCT
ejpam-1878	96	27	i	i	PRON
ejpam-1878	97	1	∧	∧	PROPN
ejpam-1878	97	2	j	j	PROPN
ejpam-1878	98	1	=	=	SYM
ejpam-1878	98	2	i	i	PROPN
ejpam-1878	98	3	∩	∩	PROPN
ejpam-1878	98	4	j	j	PROPN
ejpam-1878	98	5	and	and	CCONJ
ejpam-1878	98	6	i	i	PROPN
ejpam-1878	98	7	∨	∨	PROPN
ejpam-1878	98	8	j	j	PROPN
ejpam-1878	99	1	=	=	PRON
ejpam-1878	99	2	{	{	PUNCT
ejpam-1878	99	3	x	x	SYM
ejpam-1878	99	4	∈	∈	PROPN
ejpam-1878	99	5	s|x	s|x	NOUN
ejpam-1878	99	6	≤	≤	VERB
ejpam-1878	99	7	a	a	DET
ejpam-1878	99	8	∧	∧	PROPN
ejpam-1878	99	9	b	b	PROPN
ejpam-1878	99	10	,	,	PUNCT
ejpam-1878	99	11	for	for	ADP
ejpam-1878	99	12	some	some	DET
ejpam-1878	99	13	a	a	DET
ejpam-1878	99	14	∈	∈	NOUN
ejpam-1878	100	1	i	i	PRON
ejpam-1878	100	2	and	and	CCONJ
ejpam-1878	100	3	b	b	PROPN
ejpam-1878	100	4	∈	∈	PROPN
ejpam-1878	100	5	j	j	PROPN
ejpam-1878	100	6	}	}	PUNCT
ejpam-1878	100	7	for	for	ADP
ejpam-1878	100	8	any	any	DET
ejpam-1878	100	9	ideals	ideal	NOUN
ejpam-1878	100	10	i	i	PRON
ejpam-1878	100	11	and	and	CCONJ
ejpam-1878	100	12	j	j	PROPN
ejpam-1878	100	13	,	,	PUNCT
ejpam-1878	100	14	in	in	ADP
ejpam-1878	100	15	this	this	DET
ejpam-1878	100	16	case	case	NOUN
ejpam-1878	100	17	,	,	PUNCT
ejpam-1878	100	18	(	(	PUNCT
ejpam-1878	100	19	a]∨	a]∨	PROPN
ejpam-1878	100	20	(	(	PUNCT
ejpam-1878	100	21	b	b	NOUN
ejpam-1878	100	22	]	]	X
ejpam-1878	100	23	=	=	X
ejpam-1878	100	24	(	(	PUNCT
ejpam-1878	100	25	a	a	DET
ejpam-1878	100	26	∨	∨	NUM
ejpam-1878	100	27	b	b	NOUN
ejpam-1878	100	28	]	]	X
ejpam-1878	100	29	for	for	ADP
ejpam-1878	100	30	any	any	DET
ejpam-1878	100	31	a	a	PRON
ejpam-1878	100	32	and	and	CCONJ
ejpam-1878	100	33	b	b	NOUN
ejpam-1878	100	34	in	in	ADP
ejpam-1878	100	35	s	s	NOUN
ejpam-1878	100	36	,	,	PUNCT
ejpam-1878	100	37	so	so	SCONJ
ejpam-1878	100	38	that	that	SCONJ
ejpam-1878	100	39	a	a	DET
ejpam-1878	100	40	7→	7→	NOUN
ejpam-1878	100	41	(	(	PUNCT
ejpam-1878	100	42	a	a	PRON
ejpam-1878	100	43	]	]	X
ejpam-1878	100	44	is	be	AUX
ejpam-1878	100	45	an	an	DET
ejpam-1878	100	46	embedding	embedding	NOUN
ejpam-1878	100	47	of	of	ADP
ejpam-1878	100	48	lattices	lattice	NOUN
ejpam-1878	100	49	.	.	PUNCT
ejpam-1878	101	1	conversely	conversely	ADV
ejpam-1878	101	2	,	,	PUNCT
ejpam-1878	101	3	suppose	suppose	VERB
ejpam-1878	101	4	that	that	SCONJ
ejpam-1878	101	5	i	i	PRON
ejpam-1878	101	6	(	(	PUNCT
ejpam-1878	101	7	s	s	X
ejpam-1878	101	8	)	)	PUNCT
ejpam-1878	101	9	is	be	AUX
ejpam-1878	101	10	a	a	DET
ejpam-1878	101	11	lattice	lattice	NOUN
ejpam-1878	101	12	.	.	PUNCT
ejpam-1878	102	1	let	let	VERB
ejpam-1878	102	2	a	a	DET
ejpam-1878	102	3	and	and	CCONJ
ejpam-1878	102	4	b	b	NOUN
ejpam-1878	102	5	∈	∈	NOUN
ejpam-1878	102	6	s	s	VERB
ejpam-1878	103	1	and	and	CCONJ
ejpam-1878	103	2	i	i	PRON
ejpam-1878	103	3	be	be	VERB
ejpam-1878	103	4	the	the	DET
ejpam-1878	103	5	least	least	ADJ
ejpam-1878	103	6	upper	upper	ADJ
ejpam-1878	103	7	bound	bind	VERB
ejpam-1878	103	8	of	of	ADP
ejpam-1878	103	9	(	(	PUNCT
ejpam-1878	103	10	a	a	PRON
ejpam-1878	103	11	]	]	X
ejpam-1878	103	12	and	and	CCONJ
ejpam-1878	103	13	(	(	PUNCT
ejpam-1878	103	14	b	b	X
ejpam-1878	103	15	]	]	X
ejpam-1878	103	16	in	in	ADP
ejpam-1878	103	17	i	i	PROPN
ejpam-1878	103	18	(	(	PUNCT
ejpam-1878	103	19	s	s	NOUN
ejpam-1878	103	20	)	)	PUNCT
ejpam-1878	103	21	.	.	PUNCT
ejpam-1878	104	1	then	then	ADV
ejpam-1878	104	2	i	i	PRON
ejpam-1878	104	3	is	be	AUX
ejpam-1878	104	4	the	the	DET
ejpam-1878	104	5	smallest	small	ADJ
ejpam-1878	104	6	ideal	ideal	NOUN
ejpam-1878	104	7	containing	contain	VERB
ejpam-1878	104	8	a	a	PRON
ejpam-1878	104	9	and	and	CCONJ
ejpam-1878	104	10	b	b	NOUN
ejpam-1878	104	11	and	and	CCONJ
ejpam-1878	104	12	hence	hence	ADV
ejpam-1878	104	13	by	by	ADP
ejpam-1878	104	14	theorem	theorem	NOUN
ejpam-1878	104	15	2	2	NUM
ejpam-1878	104	16	,	,	PUNCT
ejpam-1878	104	17	a	a	DET
ejpam-1878	104	18	∨	∨	NUM
ejpam-1878	104	19	b	b	NOUN
ejpam-1878	104	20	exists	exist	VERB
ejpam-1878	104	21	in	in	ADP
ejpam-1878	104	22	s.	s.	PROPN
ejpam-1878	104	23	therefore	therefore	ADV
ejpam-1878	104	24	s	s	VERB
ejpam-1878	104	25	is	be	AUX
ejpam-1878	104	26	a	a	DET
ejpam-1878	104	27	lattice	lattice	NOUN
ejpam-1878	104	28	.	.	PUNCT
ejpam-1878	105	1	for	for	ADP
ejpam-1878	105	2	a	a	DET
ejpam-1878	105	3	lattice	lattice	NOUN
ejpam-1878	105	4	(	(	PUNCT
ejpam-1878	105	5	l,∧,∨	l,∧,∨	ADJ
ejpam-1878	105	6	)	)	PUNCT
ejpam-1878	105	7	,	,	PUNCT
ejpam-1878	105	8	any	any	DET
ejpam-1878	105	9	ideal	ideal	NOUN
ejpam-1878	105	10	of	of	ADP
ejpam-1878	105	11	the	the	DET
ejpam-1878	105	12	semi	semi	ADJ
ejpam-1878	105	13	lattice	lattice	PROPN
ejpam-1878	105	14	(	(	PUNCT
ejpam-1878	105	15	l,∧	l,∧	NOUN
ejpam-1878	105	16	)	)	PUNCT
ejpam-1878	105	17	turns	turn	VERB
ejpam-1878	105	18	out	out	ADP
ejpam-1878	105	19	to	to	PART
ejpam-1878	105	20	be	be	AUX
ejpam-1878	105	21	the	the	DET
ejpam-1878	105	22	usual	usual	ADJ
ejpam-1878	105	23	ideal	ideal	NOUN
ejpam-1878	105	24	of	of	ADP
ejpam-1878	105	25	the	the	DET
ejpam-1878	105	26	lattice	lattice	NOUN
ejpam-1878	105	27	(	(	PUNCT
ejpam-1878	105	28	l,∧,∨	l,∧,∨	ADJ
ejpam-1878	105	29	)	)	PUNCT
ejpam-1878	105	30	.	.	PUNCT
ejpam-1878	106	1	definition	definition	NOUN
ejpam-1878	106	2	7	7	NUM
ejpam-1878	106	3	.	.	PUNCT
ejpam-1878	107	1	let	let	VERB
ejpam-1878	107	2	(	(	PUNCT
ejpam-1878	107	3	s,∧	s,∧	NOUN
ejpam-1878	107	4	)	)	PUNCT
ejpam-1878	107	5	be	be	AUX
ejpam-1878	107	6	a	a	DET
ejpam-1878	107	7	semi	semi	ADJ
ejpam-1878	107	8	lattice	lattice	NOUN
ejpam-1878	107	9	.	.	PUNCT
ejpam-1878	108	1	a	a	DET
ejpam-1878	108	2	non	non	ADJ
ejpam-1878	108	3	-	-	ADJ
ejpam-1878	108	4	empty	empty	ADJ
ejpam-1878	108	5	subset	subset	NOUN
ejpam-1878	108	6	f	f	NOUN
ejpam-1878	108	7	of	of	ADP
ejpam-1878	108	8	s	s	PROPN
ejpam-1878	108	9	is	be	AUX
ejpam-1878	108	10	called	call	VERB
ejpam-1878	108	11	filter	filter	NOUN
ejpam-1878	108	12	of	of	ADP
ejpam-1878	108	13	s	s	PRON
ejpam-1878	108	14	if	if	SCONJ
ejpam-1878	108	15	,	,	PUNCT
ejpam-1878	108	16	for	for	ADP
ejpam-1878	108	17	any	any	DET
ejpam-1878	108	18	a	a	NOUN
ejpam-1878	108	19	,	,	PUNCT
ejpam-1878	108	20	b	b	PROPN
ejpam-1878	108	21	∈	∈	PROPN
ejpam-1878	108	22	s	s	PROPN
ejpam-1878	108	23	,	,	PUNCT
ejpam-1878	108	24	a	a	DET
ejpam-1878	108	25	∧	∧	PROPN
ejpam-1878	108	26	b	b	PROPN
ejpam-1878	108	27	∈	∈	PROPN
ejpam-1878	108	28	f⇔	f⇔	NOUN
ejpam-1878	108	29	a	a	DET
ejpam-1878	108	30	∈	∈	PROPN
ejpam-1878	108	31	f	f	NOUN
ejpam-1878	108	32	and	and	CCONJ
ejpam-1878	108	33	b	b	PROPN
ejpam-1878	108	34	∈	∈	PROPN
ejpam-1878	108	35	f	f	PROPN
ejpam-1878	108	36	theorem	theorem	VERB
ejpam-1878	108	37	5	5	NUM
ejpam-1878	108	38	.	.	PUNCT
ejpam-1878	109	1	let	let	VERB
ejpam-1878	109	2	(	(	PUNCT
ejpam-1878	109	3	s,∧	s,∧	NOUN
ejpam-1878	109	4	)	)	PUNCT
ejpam-1878	109	5	be	be	AUX
ejpam-1878	109	6	a	a	DET
ejpam-1878	109	7	semi	semi	ADJ
ejpam-1878	109	8	lattice	lattice	NOUN
ejpam-1878	109	9	and	and	CCONJ
ejpam-1878	109	10	p	p	X
ejpam-1878	109	11	a	a	DET
ejpam-1878	109	12	proper	proper	ADJ
ejpam-1878	109	13	ideal	ideal	NOUN
ejpam-1878	109	14	of	of	ADP
ejpam-1878	109	15	s.	s.	PROPN
ejpam-1878	109	16	then	then	ADV
ejpam-1878	109	17	the	the	DET
ejpam-1878	109	18	following	follow	VERB
ejpam-1878	109	19	are	be	AUX
ejpam-1878	109	20	equivalent	equivalent	ADJ
ejpam-1878	109	21	to	to	ADP
ejpam-1878	109	22	each	each	DET
ejpam-1878	109	23	other	other	ADJ
ejpam-1878	109	24	(	(	PUNCT
ejpam-1878	109	25	1	1	NUM
ejpam-1878	109	26	)	)	PUNCT
ejpam-1878	109	27	.	.	PUNCT
ejpam-1878	110	1	for	for	ADP
ejpam-1878	110	2	any	any	DET
ejpam-1878	110	3	elements	element	NOUN
ejpam-1878	110	4	a	a	PRON
ejpam-1878	110	5	and	and	CCONJ
ejpam-1878	110	6	b	b	NOUN
ejpam-1878	110	7	in	in	ADP
ejpam-1878	110	8	s	s	PROPN
ejpam-1878	110	9	,	,	PUNCT
ejpam-1878	110	10	a	a	DET
ejpam-1878	110	11	∧	∧	PROPN
ejpam-1878	110	12	b	b	PROPN
ejpam-1878	110	13	∈	∈	PROPN
ejpam-1878	110	14	p	p	NOUN
ejpam-1878	110	15	=	=	NOUN
ejpam-1878	110	16	⇒	⇒	VERB
ejpam-1878	110	17	a	a	DET
ejpam-1878	110	18	∈	∈	PROPN
ejpam-1878	110	19	p	p	NOUN
ejpam-1878	110	20	or	or	CCONJ
ejpam-1878	110	21	b	b	NOUN
ejpam-1878	110	22	∈	∈	PROPN
ejpam-1878	110	23	p	p	X
ejpam-1878	110	24	(	(	PUNCT
ejpam-1878	110	25	2	2	NUM
ejpam-1878	110	26	)	)	PUNCT
ejpam-1878	110	27	.	.	PUNCT
ejpam-1878	111	1	for	for	ADP
ejpam-1878	111	2	any	any	DET
ejpam-1878	111	3	ideals	ideal	NOUN
ejpam-1878	111	4	i	i	PRON
ejpam-1878	111	5	and	and	CCONJ
ejpam-1878	111	6	j	j	PROPN
ejpam-1878	111	7	of	of	ADP
ejpam-1878	111	8	s	s	PROPN
ejpam-1878	111	9	,	,	PUNCT
ejpam-1878	111	10	i	i	PROPN
ejpam-1878	111	11	∩	∩	NOUN
ejpam-1878	111	12	j	j	PROPN
ejpam-1878	111	13	⊆	⊆	NUM
ejpam-1878	111	14	p	p	NOUN
ejpam-1878	111	15	=	=	NOUN
ejpam-1878	111	16	⇒	⇒	NOUN
ejpam-1878	111	17	i	i	PRON
ejpam-1878	111	18	⊆	⊆	NUM
ejpam-1878	111	19	p	p	NOUN
ejpam-1878	111	20	or	or	CCONJ
ejpam-1878	111	21	j	j	PROPN
ejpam-1878	111	22	⊆	⊆	NUM
ejpam-1878	111	23	p	p	NOUN
ejpam-1878	111	24	(	(	PUNCT
ejpam-1878	111	25	3	3	NUM
ejpam-1878	111	26	)	)	PUNCT
ejpam-1878	111	27	.	.	PUNCT
ejpam-1878	112	1	s	s	VERB
ejpam-1878	113	1	−	−	PROPN
ejpam-1878	113	2	p	p	NOUN
ejpam-1878	113	3	is	be	AUX
ejpam-1878	113	4	a	a	DET
ejpam-1878	113	5	filter	filter	NOUN
ejpam-1878	113	6	of	of	ADP
ejpam-1878	113	7	s.	s.	PROPN
ejpam-1878	113	8	proof	proof	PROPN
ejpam-1878	113	9	.	.	PUNCT
ejpam-1878	114	1	(	(	PUNCT
ejpam-1878	114	2	1	1	X
ejpam-1878	114	3	)	)	PUNCT
ejpam-1878	114	4	=	=	NOUN
ejpam-1878	114	5	⇒	⇒	NOUN
ejpam-1878	114	6	(	(	PUNCT
ejpam-1878	114	7	2	2	NUM
ejpam-1878	114	8	):	):	PUNCT
ejpam-1878	114	9	let	let	VERB
ejpam-1878	114	10	i	i	PRON
ejpam-1878	114	11	and	and	CCONJ
ejpam-1878	114	12	j	j	PROPN
ejpam-1878	114	13	be	be	VERB
ejpam-1878	114	14	ideals	ideal	NOUN
ejpam-1878	114	15	of	of	ADP
ejpam-1878	114	16	s.	s.	PROPN
ejpam-1878	114	17	suppose	suppose	VERB
ejpam-1878	114	18	that	that	SCONJ
ejpam-1878	114	19	i	i	PRON
ejpam-1878	114	20	6⊆	6⊆	VERB
ejpam-1878	114	21	p	p	NOUN
ejpam-1878	114	22	and	and	CCONJ
ejpam-1878	114	23	j	j	PROPN
ejpam-1878	114	24	6⊆	6⊆	PROPN
ejpam-1878	114	25	p.	p.	NOUN
ejpam-1878	114	26	then	then	ADV
ejpam-1878	114	27	there	there	PRON
ejpam-1878	114	28	exist	exist	VERB
ejpam-1878	114	29	a	a	DET
ejpam-1878	114	30	∈	∈	NOUN
ejpam-1878	114	31	i	i	PRON
ejpam-1878	114	32	and	and	CCONJ
ejpam-1878	114	33	b	b	PROPN
ejpam-1878	114	34	∈	∈	PROPN
ejpam-1878	114	35	j	j	NOUN
ejpam-1878	114	36	such	such	ADJ
ejpam-1878	114	37	that	that	SCONJ
ejpam-1878	114	38	a	a	DET
ejpam-1878	114	39	/∈	/∈	NOUN
ejpam-1878	114	40	p	p	NOUN
ejpam-1878	114	41	and	and	CCONJ
ejpam-1878	114	42	b	b	PROPN
ejpam-1878	114	43	/∈	/∈	PUNCT
ejpam-1878	115	1	p.	p.	NOUN
ejpam-1878	115	2	then	then	ADV
ejpam-1878	115	3	,	,	PUNCT
ejpam-1878	115	4	by	by	ADP
ejpam-1878	115	5	(	(	PUNCT
ejpam-1878	115	6	1	1	NUM
ejpam-1878	115	7	)	)	PUNCT
ejpam-1878	115	8	,	,	PUNCT
ejpam-1878	115	9	a∧	a∧	NOUN
ejpam-1878	115	10	b	b	PROPN
ejpam-1878	115	11	/∈	/∈	PUNCT
ejpam-1878	116	1	p.	p.	NOUN
ejpam-1878	117	1	but	but	CCONJ
ejpam-1878	117	2	a∧	a∧	PROPN
ejpam-1878	117	3	b	b	PROPN
ejpam-1878	117	4	≤	≤	ADV
ejpam-1878	117	5	a	a	DET
ejpam-1878	117	6	∈	∈	NOUN
ejpam-1878	118	1	i	i	PRON
ejpam-1878	118	2	and	and	CCONJ
ejpam-1878	118	3	a	a	DET
ejpam-1878	118	4	∧	∧	PROPN
ejpam-1878	118	5	b	b	PROPN
ejpam-1878	118	6	≤	≤	NUM
ejpam-1878	118	7	b	b	PROPN
ejpam-1878	118	8	∈	∈	PROPN
ejpam-1878	118	9	j	j	PROPN
ejpam-1878	118	10	and	and	CCONJ
ejpam-1878	118	11	hence	hence	ADV
ejpam-1878	118	12	a	a	DET
ejpam-1878	118	13	∧	∧	PROPN
ejpam-1878	118	14	b	b	PROPN
ejpam-1878	118	15	∈	∈	PROPN
ejpam-1878	118	16	i	i	PRON
ejpam-1878	118	17	∩	∩	PROPN
ejpam-1878	118	18	j	j	PROPN
ejpam-1878	118	19	.	.	PUNCT
ejpam-1878	119	1	therefore	therefore	ADV
ejpam-1878	119	2	i	i	PRON
ejpam-1878	119	3	∩	∩	PROPN
ejpam-1878	119	4	j	j	PROPN
ejpam-1878	119	5	6⊆	6⊆	NUM
ejpam-1878	119	6	p.	p.	NOUN
ejpam-1878	119	7	(	(	PUNCT
ejpam-1878	119	8	2	2	X
ejpam-1878	119	9	)	)	PUNCT
ejpam-1878	120	1	=	=	NOUN
ejpam-1878	120	2	⇒	⇒	NOUN
ejpam-1878	120	3	(	(	PUNCT
ejpam-1878	120	4	3	3	NUM
ejpam-1878	120	5	):	):	PUNCT
ejpam-1878	120	6	if	if	SCONJ
ejpam-1878	120	7	a	a	DET
ejpam-1878	120	8	≤	≤	NUM
ejpam-1878	120	9	b	b	NOUN
ejpam-1878	120	10	and	and	CCONJ
ejpam-1878	120	11	a	a	DET
ejpam-1878	120	12	∈	∈	NOUN
ejpam-1878	120	13	s	s	PART
ejpam-1878	120	14	−	−	PROPN
ejpam-1878	120	15	p	p	NOUN
ejpam-1878	120	16	,	,	PUNCT
ejpam-1878	120	17	then	then	ADV
ejpam-1878	120	18	clearly	clearly	ADV
ejpam-1878	120	19	b	b	PROPN
ejpam-1878	120	20	∈	∈	NOUN
ejpam-1878	120	21	s	s	PART
ejpam-1878	120	22	−	−	PROPN
ejpam-1878	120	23	p.	p.	NOUN
ejpam-1878	120	24	also	also	ADV
ejpam-1878	120	25	,	,	PUNCT
ejpam-1878	120	26	a	a	PRON
ejpam-1878	120	27	and	and	CCONJ
ejpam-1878	120	28	b	b	X
ejpam-1878	120	29	∈	∈	NOUN
ejpam-1878	120	30	s	s	VERB
ejpam-1878	120	31	−	−	X
ejpam-1878	120	32	p	p	X
ejpam-1878	120	33	=	=	PROPN
ejpam-1878	120	34	⇒a	⇒a	PROPN
ejpam-1878	120	35	/∈	/∈	PUNCT
ejpam-1878	121	1	p	p	NOUN
ejpam-1878	121	2	and	and	CCONJ
ejpam-1878	121	3	b	b	NOUN
ejpam-1878	121	4	/∈	/∈	PUNCT
ejpam-1878	122	1	p	p	X
ejpam-1878	122	2	=	=	ADJ
ejpam-1878	122	3	⇒(a	⇒(a	ADJ
ejpam-1878	122	4	]	]	PUNCT
ejpam-1878	122	5	6⊆	6⊆	NUM
ejpam-1878	122	6	p	p	NOUN
ejpam-1878	122	7	and	and	CCONJ
ejpam-1878	122	8	(	(	PUNCT
ejpam-1878	122	9	b	b	X
ejpam-1878	122	10	]	]	X
ejpam-1878	122	11	6⊆	6⊆	NUM
ejpam-1878	122	12	p	p	NOUN
ejpam-1878	122	13	=	=	PROPN
ejpam-1878	122	14	⇒(a	⇒(a	PROPN
ejpam-1878	122	15	∧	∧	PROPN
ejpam-1878	122	16	b	b	PROPN
ejpam-1878	122	17	]	]	X
ejpam-1878	122	18	=	=	SYM
ejpam-1878	122	19	(	(	PUNCT
ejpam-1878	122	20	a]∩	a]∩	PROPN
ejpam-1878	122	21	(	(	PUNCT
ejpam-1878	122	22	b	b	X
ejpam-1878	122	23	]	]	X
ejpam-1878	122	24	6⊆	6⊆	NUM
ejpam-1878	122	25	p	p	NOUN
ejpam-1878	122	26	=	=	NOUN
ejpam-1878	122	27	⇒x	⇒x	VERB
ejpam-1878	122	28	/∈	/∈	PUNCT
ejpam-1878	123	1	p	p	NOUN
ejpam-1878	123	2	for	for	ADP
ejpam-1878	123	3	some	some	DET
ejpam-1878	123	4	x	x	SYM
ejpam-1878	123	5	≤	≤	NOUN
ejpam-1878	123	6	a	a	DET
ejpam-1878	123	7	∧	∧	PROPN
ejpam-1878	123	8	b	b	NOUN
ejpam-1878	123	9	=	=	NOUN
ejpam-1878	123	10	⇒x	⇒x	VERB
ejpam-1878	123	11	≤	≤	NUM
ejpam-1878	123	12	a	a	DET
ejpam-1878	123	13	∧	∧	PROPN
ejpam-1878	123	14	b	b	PROPN
ejpam-1878	123	15	and	and	CCONJ
ejpam-1878	123	16	x	x	PROPN
ejpam-1878	123	17	∈	∈	PROPN
ejpam-1878	123	18	s	s	VERB
ejpam-1878	123	19	−	−	X
ejpam-1878	123	20	p	p	PROPN
ejpam-1878	123	21	=	=	PROPN
ejpam-1878	123	22	⇒a	⇒a	PROPN
ejpam-1878	123	23	∧	∧	PROPN
ejpam-1878	123	24	b	b	PROPN
ejpam-1878	123	25	∈	∈	PROPN
ejpam-1878	123	26	s	s	VERB
ejpam-1878	123	27	−	−	NOUN
ejpam-1878	123	28	p	p	NOUN
ejpam-1878	123	29	thus	thus	ADV
ejpam-1878	123	30	s	s	VERB
ejpam-1878	123	31	−	−	PROPN
ejpam-1878	123	32	p	p	NOUN
ejpam-1878	123	33	is	be	AUX
ejpam-1878	123	34	a	a	DET
ejpam-1878	123	35	filter	filter	NOUN
ejpam-1878	123	36	of	of	ADP
ejpam-1878	123	37	s.	s.	PROPN
ejpam-1878	123	38	(	(	PUNCT
ejpam-1878	123	39	3	3	X
ejpam-1878	123	40	)	)	PUNCT
ejpam-1878	123	41	=	=	NOUN
ejpam-1878	123	42	⇒	⇒	NOUN
ejpam-1878	123	43	(	(	PUNCT
ejpam-1878	123	44	1	1	NUM
ejpam-1878	123	45	):	):	PUNCT
ejpam-1878	123	46	for	for	ADP
ejpam-1878	123	47	any	any	DET
ejpam-1878	123	48	a	a	DET
ejpam-1878	123	49	and	and	CCONJ
ejpam-1878	123	50	b	b	NOUN
ejpam-1878	123	51	∈	∈	PROPN
ejpam-1878	123	52	s	s	PROPN
ejpam-1878	123	53	,	,	PUNCT
ejpam-1878	123	54	s.	s.	PROPN
ejpam-1878	123	55	sagi	sagi	PROPN
ejpam-1878	123	56	/	/	SYM
ejpam-1878	123	57	eur	eur	PROPN
ejpam-1878	123	58	.	.	PUNCT
ejpam-1878	124	1	j.	j.	PROPN
ejpam-1878	124	2	pure	pure	PROPN
ejpam-1878	124	3	appl	appl	PROPN
ejpam-1878	124	4	.	.	PROPN
ejpam-1878	124	5	math	math	PROPN
ejpam-1878	124	6	,	,	PUNCT
ejpam-1878	124	7	8	8	NUM
ejpam-1878	124	8	(	(	PUNCT
ejpam-1878	124	9	2015	2015	NUM
ejpam-1878	124	10	)	)	PUNCT
ejpam-1878	124	11	,	,	PUNCT
ejpam-1878	124	12	15	15	NUM
ejpam-1878	124	13	-	-	SYM
ejpam-1878	124	14	25	25	NUM
ejpam-1878	124	15	20	20	NUM
ejpam-1878	124	16	a	a	DET
ejpam-1878	124	17	/∈	/∈	NOUN
ejpam-1878	124	18	p	p	NOUN
ejpam-1878	124	19	and	and	CCONJ
ejpam-1878	124	20	b	b	NOUN
ejpam-1878	124	21	/∈	/∈	PUNCT
ejpam-1878	125	1	p	p	PROPN
ejpam-1878	126	1	=	=	ADJ
ejpam-1878	126	2	⇒a	⇒a	PROPN
ejpam-1878	126	3	and	and	CCONJ
ejpam-1878	126	4	b	b	PROPN
ejpam-1878	126	5	∈	∈	NOUN
ejpam-1878	126	6	s	s	VERB
ejpam-1878	126	7	−	−	X
ejpam-1878	126	8	p	p	X
ejpam-1878	126	9	=	=	PROPN
ejpam-1878	127	1	⇒a	⇒a	PROPN
ejpam-1878	127	2	∧	∧	PROPN
ejpam-1878	127	3	b	b	PROPN
ejpam-1878	127	4	∈	∈	PROPN
ejpam-1878	127	5	s	s	VERB
ejpam-1878	127	6	−	−	NOUN
ejpam-1878	127	7	p	p	X
ejpam-1878	127	8	=	=	PROPN
ejpam-1878	127	9	⇒a	⇒a	PROPN
ejpam-1878	127	10	∧	∧	PROPN
ejpam-1878	127	11	b	b	PROPN
ejpam-1878	127	12	/∈	/∈	PROPN
ejpam-1878	127	13	p	p	NOUN
ejpam-1878	127	14	definition	definition	NOUN
ejpam-1878	127	15	8	8	NUM
ejpam-1878	127	16	.	.	PUNCT
ejpam-1878	128	1	any	any	DET
ejpam-1878	128	2	proper	proper	ADJ
ejpam-1878	128	3	ideal	ideal	NOUN
ejpam-1878	128	4	p	p	NOUN
ejpam-1878	128	5	of	of	ADP
ejpam-1878	128	6	a	a	DET
ejpam-1878	128	7	semi	semi	ADJ
ejpam-1878	128	8	lattice	lattice	NOUN
ejpam-1878	128	9	(	(	PUNCT
ejpam-1878	128	10	s,∧	s,∧	PROPN
ejpam-1878	128	11	)	)	PUNCT
ejpam-1878	128	12	is	be	AUX
ejpam-1878	128	13	said	say	VERB
ejpam-1878	128	14	to	to	PART
ejpam-1878	128	15	be	be	AUX
ejpam-1878	128	16	a	a	DET
ejpam-1878	128	17	prime	prime	ADJ
ejpam-1878	128	18	ideal	ideal	NOUN
ejpam-1878	128	19	if	if	SCONJ
ejpam-1878	128	20	any	any	DET
ejpam-1878	128	21	one	one	NOUN
ejpam-1878	128	22	(	(	PUNCT
ejpam-1878	128	23	and	and	CCONJ
ejpam-1878	128	24	hence	hence	ADV
ejpam-1878	128	25	all	all	PRON
ejpam-1878	128	26	)	)	PUNCT
ejpam-1878	128	27	of	of	ADP
ejpam-1878	128	28	the	the	DET
ejpam-1878	128	29	conditions	condition	NOUN
ejpam-1878	128	30	in	in	ADP
ejpam-1878	128	31	theorem	theorem	NOUN
ejpam-1878	128	32	5	5	NUM
ejpam-1878	128	33	is	be	AUX
ejpam-1878	128	34	satisfied	satisfied	ADJ
ejpam-1878	128	35	.	.	PUNCT
ejpam-1878	129	1	4	4	X
ejpam-1878	129	2	.	.	X
ejpam-1878	129	3	prime	prime	ADJ
ejpam-1878	129	4	ideals	ideal	NOUN
ejpam-1878	129	5	in	in	ADP
ejpam-1878	129	6	(	(	PUNCT
ejpam-1878	129	7	z	z	NOUN
ejpam-1878	129	8	+	+	ADJ
ejpam-1878	129	9	,	,	PUNCT
ejpam-1878	129	10	≤d	≤d	NOUN
ejpam-1878	129	11	)	)	PUNCT
ejpam-1878	129	12	now	now	ADV
ejpam-1878	129	13	we	we	PRON
ejpam-1878	129	14	shall	shall	AUX
ejpam-1878	129	15	turn	turn	VERB
ejpam-1878	129	16	our	our	PRON
ejpam-1878	129	17	attention	attention	NOUN
ejpam-1878	129	18	to	to	ADP
ejpam-1878	129	19	the	the	DET
ejpam-1878	129	20	particular	particular	ADJ
ejpam-1878	129	21	case	case	NOUN
ejpam-1878	129	22	of	of	ADP
ejpam-1878	129	23	the	the	DET
ejpam-1878	129	24	lattice	lattice	NOUN
ejpam-1878	129	25	structure	structure	NOUN
ejpam-1878	129	26	onz+	onz+	AUX
ejpam-1878	129	27	induced	induce	VERB
ejpam-1878	129	28	by	by	ADP
ejpam-1878	129	29	the	the	DET
ejpam-1878	129	30	division	division	NOUN
ejpam-1878	129	31	ordering	ordering	NOUN
ejpam-1878	129	32	/	/	PUNCT
ejpam-1878	129	33	and	and	CCONJ
ejpam-1878	129	34	study	study	VERB
ejpam-1878	129	35	the	the	DET
ejpam-1878	129	36	ideals	ideal	NOUN
ejpam-1878	129	37	and	and	CCONJ
ejpam-1878	129	38	prime	prime	ADJ
ejpam-1878	129	39	ideals	ideal	NOUN
ejpam-1878	129	40	of	of	ADP
ejpam-1878	129	41	z+	z+	NUM
ejpam-1878	129	42	.	.	PUNCT
ejpam-1878	130	1	the	the	DET
ejpam-1878	130	2	division	division	NOUN
ejpam-1878	130	3	ordering	ordering	NOUN
ejpam-1878	130	4	is	be	AUX
ejpam-1878	130	5	precisely	precisely	ADV
ejpam-1878	130	6	the	the	DET
ejpam-1878	130	7	partial	partial	ADJ
ejpam-1878	130	8	ordering	ordering	NOUN
ejpam-1878	130	9	≤d	≤d	NOUN
ejpam-1878	130	10	induced	induce	VERB
ejpam-1878	130	11	by	by	ADP
ejpam-1878	130	12	the	the	DET
ejpam-1878	130	13	dirichlet	dirichlet	PROPN
ejpam-1878	130	14	’s	’s	PART
ejpam-1878	130	15	convolution	convolution	PROPN
ejpam-1878	130	16	d.	d.	PROPN
ejpam-1878	131	1	first	first	ADV
ejpam-1878	131	2	we	we	PRON
ejpam-1878	131	3	observe	observe	VERB
ejpam-1878	131	4	that	that	SCONJ
ejpam-1878	131	5	�	�	PROPN
ejpam-1878	131	6	θ	θ	PROPN
ejpam-1878	131	7	:	:	PUNCT
ejpam-1878	131	8	(	(	PUNCT
ejpam-1878	131	9	z+,/	z+,/	NOUN
ejpam-1878	131	10	)	)	PUNCT
ejpam-1878	131	11	−→	−→	NOUN
ejpam-1878	131	12	(	(	PUNCT
ejpam-1878	131	13	∑	∑	PROPN
ejpam-1878	131	14	p	p	NOUN
ejpam-1878	131	15	n	n	NOUN
ejpam-1878	131	16	,	,	PUNCT
ejpam-1878	131	17	≤	≤	NUM
ejpam-1878	131	18	)	)	PUNCT
ejpam-1878	131	19	�	�	PROPN
ejpam-1878	131	20	is	be	AUX
ejpam-1878	131	21	an	an	DET
ejpam-1878	131	22	order	order	NOUN
ejpam-1878	131	23	isomorphism	isomorphism	NOUN
ejpam-1878	131	24	where	where	SCONJ
ejpam-1878	131	25	θ	θ	PROPN
ejpam-1878	131	26	is	be	AUX
ejpam-1878	131	27	defined	define	VERB
ejpam-1878	131	28	by	by	ADP
ejpam-1878	131	29	�	�	PROPN
ejpam-1878	131	30	θ	θ	PROPN
ejpam-1878	131	31	(	(	PUNCT
ejpam-1878	131	32	a)(p	a)(p	VERB
ejpam-1878	131	33	)	)	PUNCT
ejpam-1878	131	34	=	=	PUNCT
ejpam-1878	131	35	the	the	DET
ejpam-1878	131	36	largest	large	ADJ
ejpam-1878	131	37	n	n	PRON
ejpam-1878	131	38	∈	∈	NOUN
ejpam-1878	131	39	n	n	PRON
ejpam-1878	131	40	such	such	ADJ
ejpam-1878	131	41	that	that	SCONJ
ejpam-1878	131	42	pn	pn	PROPN
ejpam-1878	131	43	divides	divide	VERB
ejpam-1878	131	44	a	a	PRON
ejpam-1878	131	45	,	,	PUNCT
ejpam-1878	131	46	for	for	ADP
ejpam-1878	131	47	any	any	DET
ejpam-1878	131	48	a	a	DET
ejpam-1878	131	49	∈	∈	NOUN
ejpam-1878	131	50	z+	z+	NUM
ejpam-1878	131	51	and	and	CCONJ
ejpam-1878	131	52	p	p	NOUN
ejpam-1878	131	53	∈	∈	PROPN
ejpam-1878	131	54	p	p	PROPN
ejpam-1878	131	55	�	�	PROPN
ejpam-1878	131	56	and	and	CCONJ
ejpam-1878	131	57	�	�	PROPN
ejpam-1878	131	58	∑	∑	PROPN
ejpam-1878	131	59	p	p	PROPN
ejpam-1878	131	60	n	n	PRON
ejpam-1878	131	61	�	�	PROPN
ejpam-1878	131	62	=	=	PUNCT
ejpam-1878	131	63	{	{	PUNCT
ejpam-1878	131	64	f	f	NOUN
ejpam-1878	131	65	:p	:p	INTJ
ejpam-1878	131	66	−→	−→	NOUN
ejpam-1878	132	1	n	n	CCONJ
ejpam-1878	133	1	|	|	ADV
ejpam-1878	133	2	f	f	X
ejpam-1878	133	3	(	(	PUNCT
ejpam-1878	133	4	p	p	NOUN
ejpam-1878	133	5	)	)	PUNCT
ejpam-1878	133	6	=	=	SYM
ejpam-1878	133	7	0	0	NUM
ejpam-1878	133	8	for	for	ADP
ejpam-1878	133	9	all	all	PRON
ejpam-1878	133	10	but	but	CCONJ
ejpam-1878	133	11	finite	finite	VERB
ejpam-1878	133	12	p	p	NOUN
ejpam-1878	133	13	}	}	PUNCT
ejpam-1878	133	14	.	.	PUNCT
ejpam-1878	134	1	here	here	ADV
ejpam-1878	134	2	p	p	NOUN
ejpam-1878	134	3	stands	stand	VERB
ejpam-1878	134	4	for	for	ADP
ejpam-1878	134	5	the	the	DET
ejpam-1878	134	6	set	set	NOUN
ejpam-1878	134	7	of	of	ADP
ejpam-1878	134	8	primes	prime	NOUN
ejpam-1878	134	9	and	and	CCONJ
ejpam-1878	134	10	n	n	PRON
ejpam-1878	134	11	stands	stand	VERB
ejpam-1878	134	12	for	for	ADP
ejpam-1878	134	13	the	the	DET
ejpam-1878	134	14	set	set	NOUN
ejpam-1878	134	15	of	of	ADP
ejpam-1878	134	16	non	non	ADJ
ejpam-1878	134	17	-	-	ADJ
ejpam-1878	134	18	negative	negative	ADJ
ejpam-1878	134	19	integers	integer	NOUN
ejpam-1878	134	20	.	.	PUNCT
ejpam-1878	135	1	definition	definition	NOUN
ejpam-1878	135	2	9	9	NUM
ejpam-1878	135	3	.	.	PUNCT
ejpam-1878	135	4	adjoin	adjoin	VERB
ejpam-1878	135	5	an	an	DET
ejpam-1878	135	6	external	external	ADJ
ejpam-1878	135	7	element∞	element∞	NOUN
ejpam-1878	135	8	to	to	ADP
ejpam-1878	135	9	n	n	NOUN
ejpam-1878	135	10	and	and	CCONJ
ejpam-1878	135	11	extend	extend	VERB
ejpam-1878	135	12	the	the	DET
ejpam-1878	135	13	usual	usual	ADJ
ejpam-1878	135	14	ordering	ordering	NOUN
ejpam-1878	135	15	≤	≤	NOUN
ejpam-1878	135	16	on	on	ADP
ejpam-1878	135	17	n	n	PART
ejpam-1878	135	18	to	to	ADP
ejpam-1878	135	19	n	n	PRON
ejpam-1878	135	20	∪	∪	VERB
ejpam-1878	135	21	{	{	PUNCT
ejpam-1878	135	22	∞	∞	NOUN
ejpam-1878	135	23	}	}	PUNCT
ejpam-1878	135	24	by	by	ADP
ejpam-1878	135	25	defining	define	VERB
ejpam-1878	135	26	a	a	DET
ejpam-1878	135	27	<	<	X
ejpam-1878	135	28	∞	∞	NOUN
ejpam-1878	135	29	for	for	ADP
ejpam-1878	135	30	all	all	DET
ejpam-1878	135	31	a	a	DET
ejpam-1878	135	32	∈	∈	NOUN
ejpam-1878	135	33	n	n	NOUN
ejpam-1878	135	34	.	.	PUNCT
ejpam-1878	136	1	we	we	PRON
ejpam-1878	136	2	shall	shall	AUX
ejpam-1878	136	3	denote	denote	VERB
ejpam-1878	136	4	n	n	CCONJ
ejpam-1878	136	5	∪	∪	X
ejpam-1878	136	6	{	{	PUNCT
ejpam-1878	136	7	∞	∞	NOUN
ejpam-1878	136	8	}	}	PUNCT
ejpam-1878	136	9	together	together	ADV
ejpam-1878	136	10	with	with	ADP
ejpam-1878	136	11	this	this	DET
ejpam-1878	136	12	extended	extend	VERB
ejpam-1878	136	13	usual	usual	ADJ
ejpam-1878	136	14	order	order	NOUN
ejpam-1878	136	15	by	by	ADP
ejpam-1878	136	16	n	n	PRON
ejpam-1878	136	17	∞	∞	PROPN
ejpam-1878	136	18	.	.	PUNCT
ejpam-1878	137	1	theorem	theorem	VERB
ejpam-1878	137	2	6	6	NUM
ejpam-1878	137	3	.	.	PUNCT
ejpam-1878	138	1	let	let	VERB
ejpam-1878	138	2	α	α	PRON
ejpam-1878	138	3	:p	:p	PUNCT
ejpam-1878	139	1	−→n	−→n	VERB
ejpam-1878	139	2	∞	∞	PROPN
ejpam-1878	139	3	be	be	VERB
ejpam-1878	139	4	a	a	DET
ejpam-1878	139	5	mapping	mapping	NOUN
ejpam-1878	139	6	and	and	CCONJ
ejpam-1878	139	7	define	define	VERB
ejpam-1878	139	8	iα	iα	INTJ
ejpam-1878	139	9	=	=	PUNCT
ejpam-1878	139	10	{	{	PUNCT
ejpam-1878	139	11	n	n	X
ejpam-1878	139	12	∈	∈	PROPN
ejpam-1878	139	13	z	z	PROPN
ejpam-1878	139	14	+	+	PROPN
ejpam-1878	139	15	|θ	|θ	PRON
ejpam-1878	139	16	(	(	PUNCT
ejpam-1878	139	17	n)(p)≤	n)(p)≤	PROPN
ejpam-1878	139	18	α(p	α(p	PROPN
ejpam-1878	139	19	)	)	PUNCT
ejpam-1878	139	20	for	for	ADP
ejpam-1878	139	21	all	all	DET
ejpam-1878	139	22	p	p	NOUN
ejpam-1878	139	23	∈	∈	PROPN
ejpam-1878	139	24	p	p	X
ejpam-1878	139	25	}	}	PUNCT
ejpam-1878	139	26	then	then	ADV
ejpam-1878	139	27	iα	iα	VERB
ejpam-1878	139	28	is	be	AUX
ejpam-1878	139	29	an	an	DET
ejpam-1878	139	30	ideal	ideal	NOUN
ejpam-1878	139	31	of	of	ADP
ejpam-1878	139	32	(	(	PUNCT
ejpam-1878	139	33	z+,/	z+,/	NOUN
ejpam-1878	139	34	)	)	PUNCT
ejpam-1878	139	35	and	and	CCONJ
ejpam-1878	139	36	every	every	DET
ejpam-1878	139	37	ideal	ideal	NOUN
ejpam-1878	139	38	of	of	ADP
ejpam-1878	139	39	(	(	PUNCT
ejpam-1878	139	40	z+,/	z+,/	NOUN
ejpam-1878	139	41	)	)	PUNCT
ejpam-1878	139	42	is	be	AUX
ejpam-1878	139	43	of	of	ADP
ejpam-1878	139	44	the	the	DET
ejpam-1878	139	45	form	form	NOUN
ejpam-1878	139	46	iα	iα	VERB
ejpam-1878	139	47	for	for	ADP
ejpam-1878	139	48	some	some	DET
ejpam-1878	139	49	mapping	mapping	NOUN
ejpam-1878	140	1	α	α	NOUN
ejpam-1878	140	2	:p	:p	PUNCT
ejpam-1878	141	1	−→n	−→n	VERB
ejpam-1878	141	2	∞	∞	NUM
ejpam-1878	141	3	proof	proof	NOUN
ejpam-1878	141	4	.	.	PUNCT
ejpam-1878	142	1	since	since	SCONJ
ejpam-1878	142	2	no	no	DET
ejpam-1878	142	3	prime	prime	NOUN
ejpam-1878	142	4	divides	divide	VERB
ejpam-1878	142	5	the	the	DET
ejpam-1878	142	6	integer	integer	NOUN
ejpam-1878	142	7	1	1	NUM
ejpam-1878	142	8	,	,	PUNCT
ejpam-1878	142	9	we	we	PRON
ejpam-1878	142	10	get	get	VERB
ejpam-1878	142	11	that	that	DET
ejpam-1878	142	12	θ	θ	PROPN
ejpam-1878	142	13	(	(	PUNCT
ejpam-1878	142	14	1)(p	1)(p	NUM
ejpam-1878	142	15	)	)	PUNCT
ejpam-1878	142	16	=	=	SYM
ejpam-1878	142	17	0	0	NUM
ejpam-1878	142	18	≤	≤	NUM
ejpam-1878	142	19	α(p	α(p	PROPN
ejpam-1878	142	20	)	)	PUNCT
ejpam-1878	142	21	for	for	ADP
ejpam-1878	142	22	all	all	DET
ejpam-1878	142	23	p	p	NOUN
ejpam-1878	142	24	∈	∈	PROPN
ejpam-1878	142	25	p	p	NOUN
ejpam-1878	142	26	and	and	CCONJ
ejpam-1878	142	27	hence	hence	ADV
ejpam-1878	142	28	1	1	NUM
ejpam-1878	142	29	∈	∈	NOUN
ejpam-1878	142	30	iα	iα	NOUN
ejpam-1878	142	31	.	.	PUNCT
ejpam-1878	142	32	therefore	therefore	ADV
ejpam-1878	142	33	iα	iα	VERB
ejpam-1878	142	34	is	be	AUX
ejpam-1878	142	35	a	a	DET
ejpam-1878	142	36	non	non	ADJ
ejpam-1878	142	37	-	-	ADJ
ejpam-1878	142	38	empty	empty	ADJ
ejpam-1878	142	39	subset	subset	NOUN
ejpam-1878	142	40	of	of	ADP
ejpam-1878	142	41	z+	z+	NUM
ejpam-1878	142	42	.	.	PUNCT
ejpam-1878	143	1	m	m	VERB
ejpam-1878	143	2	and	and	CCONJ
ejpam-1878	143	3	n	n	PRON
ejpam-1878	143	4	∈	∈	NOUN
ejpam-1878	143	5	iα	iα	NOUN
ejpam-1878	143	6	=	=	NOUN
ejpam-1878	143	7	⇒θ	⇒θ	PROPN
ejpam-1878	143	8	(	(	PUNCT
ejpam-1878	143	9	m)(p)≤	m)(p)≤	PROPN
ejpam-1878	143	10	α(p	α(p	PROPN
ejpam-1878	143	11	)	)	PUNCT
ejpam-1878	143	12	and	and	CCONJ
ejpam-1878	143	13	θ	θ	PROPN
ejpam-1878	143	14	(	(	PUNCT
ejpam-1878	143	15	n)(p)≤	n)(p)≤	PROPN
ejpam-1878	143	16	α(p	α(p	PROPN
ejpam-1878	143	17	)	)	PUNCT
ejpam-1878	143	18	for	for	ADP
ejpam-1878	143	19	all	all	DET
ejpam-1878	143	20	p	p	NOUN
ejpam-1878	143	21	∈	∈	PROPN
ejpam-1878	143	22	p	p	NOUN
ejpam-1878	143	23	=	=	NOUN
ejpam-1878	143	24	⇒θ	⇒θ	PROPN
ejpam-1878	143	25	(	(	PUNCT
ejpam-1878	143	26	m∨	m∨	NOUN
ejpam-1878	143	27	n)(p	n)(p	NOUN
ejpam-1878	143	28	)	)	PUNCT
ejpam-1878	143	29	=	=	PUNCT
ejpam-1878	143	30	max{θ	max{θ	X
ejpam-1878	143	31	(	(	PUNCT
ejpam-1878	143	32	m)(p),θ	m)(p),θ	PROPN
ejpam-1878	143	33	(	(	PUNCT
ejpam-1878	143	34	n)(p	n)(p	NOUN
ejpam-1878	143	35	)	)	PUNCT
ejpam-1878	143	36	}	}	PUNCT
ejpam-1878	143	37	≤	≤	NUM
ejpam-1878	143	38	α(p	α(p	PROPN
ejpam-1878	143	39	)	)	PUNCT
ejpam-1878	143	40	for	for	ADP
ejpam-1878	143	41	all	all	DET
ejpam-1878	143	42	p	p	NOUN
ejpam-1878	143	43	∈	∈	PROPN
ejpam-1878	143	44	p	p	PROPN
ejpam-1878	143	45	s.	s.	PROPN
ejpam-1878	143	46	sagi	sagi	PROPN
ejpam-1878	143	47	/	/	SYM
ejpam-1878	143	48	eur	eur	PROPN
ejpam-1878	143	49	.	.	PUNCT
ejpam-1878	144	1	j.	j.	PROPN
ejpam-1878	144	2	pure	pure	PROPN
ejpam-1878	144	3	appl	appl	PROPN
ejpam-1878	144	4	.	.	PROPN
ejpam-1878	144	5	math	math	PROPN
ejpam-1878	144	6	,	,	PUNCT
ejpam-1878	144	7	8	8	NUM
ejpam-1878	144	8	(	(	PUNCT
ejpam-1878	144	9	2015	2015	NUM
ejpam-1878	144	10	)	)	PUNCT
ejpam-1878	144	11	,	,	PUNCT
ejpam-1878	144	12	15	15	NUM
ejpam-1878	144	13	-	-	SYM
ejpam-1878	144	14	25	25	NUM
ejpam-1878	144	15	21	21	NUM
ejpam-1878	144	16	=	=	NUM
ejpam-1878	144	17	⇒m∨	⇒m∨	NOUN
ejpam-1878	144	18	n	n	PRON
ejpam-1878	144	19	∈	∈	PROPN
ejpam-1878	144	20	iα	iα	NOUN
ejpam-1878	144	21	and	and	CCONJ
ejpam-1878	144	22	m≤d	m≤d	PROPN
ejpam-1878	144	23	n	n	CCONJ
ejpam-1878	144	24	∈	∈	NOUN
ejpam-1878	144	25	iα	iα	NOUN
ejpam-1878	144	26	=	=	NOUN
ejpam-1878	144	27	⇒θ	⇒θ	PROPN
ejpam-1878	144	28	(	(	PUNCT
ejpam-1878	144	29	m)(p)≤	m)(p)≤	PROPN
ejpam-1878	144	30	θ	θ	PROPN
ejpam-1878	144	31	(	(	PUNCT
ejpam-1878	144	32	n)(p)≤	n)(p)≤	PROPN
ejpam-1878	144	33	α(p	α(p	PROPN
ejpam-1878	144	34	)	)	PUNCT
ejpam-1878	144	35	for	for	ADP
ejpam-1878	144	36	all	all	DET
ejpam-1878	144	37	p	p	NOUN
ejpam-1878	144	38	∈	∈	PROPN
ejpam-1878	144	39	p	p	NOUN
ejpam-1878	144	40	=	=	NOUN
ejpam-1878	144	41	⇒θ	⇒θ	NOUN
ejpam-1878	144	42	(	(	PUNCT
ejpam-1878	144	43	m)(p)≤	m)(p)≤	PROPN
ejpam-1878	144	44	α(p	α(p	PROPN
ejpam-1878	144	45	)	)	PUNCT
ejpam-1878	144	46	for	for	ADP
ejpam-1878	144	47	all	all	DET
ejpam-1878	144	48	p	p	NOUN
ejpam-1878	144	49	∈	∈	PROPN
ejpam-1878	144	50	p	p	NOUN
ejpam-1878	144	51	=	=	NOUN
ejpam-1878	144	52	⇒m	⇒m	NOUN
ejpam-1878	144	53	∈	∈	PROPN
ejpam-1878	144	54	iα	iα	NOUN
ejpam-1878	144	55	.	.	PUNCT
ejpam-1878	144	56	thus	thus	ADV
ejpam-1878	144	57	iα	iα	VERB
ejpam-1878	144	58	is	be	AUX
ejpam-1878	144	59	an	an	DET
ejpam-1878	144	60	ideal	ideal	NOUN
ejpam-1878	144	61	of	of	ADP
ejpam-1878	144	62	(	(	PUNCT
ejpam-1878	144	63	z+,/	z+,/	NOUN
ejpam-1878	144	64	)	)	PUNCT
ejpam-1878	144	65	.	.	PUNCT
ejpam-1878	145	1	conversely	conversely	ADV
ejpam-1878	145	2	suppose	suppose	VERB
ejpam-1878	145	3	that	that	SCONJ
ejpam-1878	145	4	i	i	PRON
ejpam-1878	145	5	is	be	AUX
ejpam-1878	145	6	any	any	DET
ejpam-1878	145	7	ideal	ideal	NOUN
ejpam-1878	145	8	of	of	ADP
ejpam-1878	145	9	(	(	PUNCT
ejpam-1878	145	10	z+,/	z+,/	NOUN
ejpam-1878	145	11	)	)	PUNCT
ejpam-1878	145	12	.	.	PUNCT
ejpam-1878	146	1	define	define	VERB
ejpam-1878	146	2	α	α	NOUN
ejpam-1878	146	3	:p	:p	PROPN
ejpam-1878	146	4	−→n	−→n	VERB
ejpam-1878	146	5	∞	∞	NUM
ejpam-1878	146	6	by	by	ADP
ejpam-1878	146	7	α(p	α(p	PROPN
ejpam-1878	146	8	)	)	PUNCT
ejpam-1878	146	9	=	=	SYM
ejpam-1878	146	10	sup{θ	sup{θ	X
ejpam-1878	146	11	(	(	PUNCT
ejpam-1878	146	12	n)(p)|n	n)(p)|n	NOUN
ejpam-1878	146	13	∈	∈	PROPN
ejpam-1878	146	14	i	i	X
ejpam-1878	146	15	}	}	PUNCT
ejpam-1878	146	16	for	for	ADP
ejpam-1878	146	17	any	any	DET
ejpam-1878	146	18	p	p	NOUN
ejpam-1878	146	19	∈	∈	PROPN
ejpam-1878	146	20	p	p	NOUN
ejpam-1878	146	21	note	note	NOUN
ejpam-1878	146	22	that	that	SCONJ
ejpam-1878	146	23	α(p	α(p	PROPN
ejpam-1878	146	24	)	)	PUNCT
ejpam-1878	146	25	is	be	AUX
ejpam-1878	146	26	either	either	CCONJ
ejpam-1878	146	27	a	a	DET
ejpam-1878	146	28	non	non	ADJ
ejpam-1878	146	29	-	-	ADJ
ejpam-1878	146	30	negative	negative	ADJ
ejpam-1878	146	31	integer	integer	NOUN
ejpam-1878	146	32	or∞	or∞	PROPN
ejpam-1878	146	33	,	,	PUNCT
ejpam-1878	146	34	for	for	ADP
ejpam-1878	146	35	any	any	DET
ejpam-1878	146	36	p	p	NOUN
ejpam-1878	146	37	∈	∈	PROPN
ejpam-1878	146	38	p	p	NOUN
ejpam-1878	146	39	.	.	PUNCT
ejpam-1878	147	1	therefore	therefore	ADV
ejpam-1878	147	2	α	α	PROPN
ejpam-1878	147	3	is	be	AUX
ejpam-1878	147	4	a	a	DET
ejpam-1878	147	5	mapping	mapping	NOUN
ejpam-1878	147	6	of	of	ADP
ejpam-1878	147	7	p	p	NOUN
ejpam-1878	147	8	into	into	ADP
ejpam-1878	147	9	n	n	DET
ejpam-1878	147	10	∞.	∞.	PROPN
ejpam-1878	147	11	n	n	CCONJ
ejpam-1878	147	12	∈	∈	PROPN
ejpam-1878	148	1	i	i	PRON
ejpam-1878	148	2	=	=	NOUN
ejpam-1878	148	3	⇒θ	⇒θ	PROPN
ejpam-1878	148	4	(	(	PUNCT
ejpam-1878	148	5	n)(p)≤	n)(p)≤	PROPN
ejpam-1878	148	6	α(p	α(p	PROPN
ejpam-1878	148	7	)	)	PUNCT
ejpam-1878	148	8	for	for	ADP
ejpam-1878	148	9	all	all	DET
ejpam-1878	148	10	p	p	NOUN
ejpam-1878	148	11	∈	∈	PROPN
ejpam-1878	148	12	p	p	X
ejpam-1878	148	13	=	=	NOUN
ejpam-1878	148	14	⇒n	⇒n	NOUN
ejpam-1878	148	15	∈	∈	NOUN
ejpam-1878	148	16	iα	iα	NOUN
ejpam-1878	148	17	therefore	therefore	ADV
ejpam-1878	148	18	i	i	PROPN
ejpam-1878	148	19	⊆	⊆	NUM
ejpam-1878	148	20	iα	iα	NOUN
ejpam-1878	148	21	.	.	PROPN
ejpam-1878	149	1	on	on	ADP
ejpam-1878	149	2	the	the	DET
ejpam-1878	149	3	other	other	ADJ
ejpam-1878	149	4	hand	hand	NOUN
ejpam-1878	149	5	,	,	PUNCT
ejpam-1878	149	6	suppose	suppose	VERB
ejpam-1878	149	7	n	n	PRON
ejpam-1878	149	8	∈	∈	PROPN
ejpam-1878	149	9	iα	iα	NOUN
ejpam-1878	149	10	.	.	PUNCT
ejpam-1878	150	1	then	then	ADV
ejpam-1878	150	2	θ	θ	X
ejpam-1878	150	3	(	(	PUNCT
ejpam-1878	150	4	n)(p	n)(p	NOUN
ejpam-1878	150	5	)	)	PUNCT
ejpam-1878	150	6	≤	≤	NOUN
ejpam-1878	150	7	α(p	α(p	PROPN
ejpam-1878	150	8	)	)	PUNCT
ejpam-1878	150	9	for	for	ADP
ejpam-1878	150	10	all	all	DET
ejpam-1878	150	11	p	p	NOUN
ejpam-1878	150	12	∈	∈	PROPN
ejpam-1878	150	13	p	p	NOUN
ejpam-1878	150	14	.	.	PUNCT
ejpam-1878	151	1	since	since	SCONJ
ejpam-1878	151	2	θ	θ	PROPN
ejpam-1878	151	3	(	(	PUNCT
ejpam-1878	151	4	n	n	CCONJ
ejpam-1878	151	5	)	)	PUNCT
ejpam-1878	151	6	∈	∈	PROPN
ejpam-1878	151	7	∑	∑	PUNCT
ejpam-1878	151	8	p	p	NOUN
ejpam-1878	151	9	n	n	PROPN
ejpam-1878	151	10	,	,	PUNCT
ejpam-1878	151	11	|θ	|θ	PRON
ejpam-1878	151	12	(	(	PUNCT
ejpam-1878	151	13	n)|	n)|	NOUN
ejpam-1878	151	14	is	be	AUX
ejpam-1878	151	15	finite	finite	ADJ
ejpam-1878	151	16	.	.	PUNCT
ejpam-1878	152	1	if	if	SCONJ
ejpam-1878	152	2	|θ	|θ	PRON
ejpam-1878	152	3	(	(	PUNCT
ejpam-1878	152	4	n)|=	n)|=	PROPN
ejpam-1878	152	5	φ	φ	NUM
ejpam-1878	152	6	,	,	PUNCT
ejpam-1878	152	7	then	then	ADV
ejpam-1878	152	8	n=	n=	ADJ
ejpam-1878	152	9	1	1	NUM
ejpam-1878	152	10	∈	∈	NOUN
ejpam-1878	152	11	i	i	PRON
ejpam-1878	152	12	.	.	PUNCT
ejpam-1878	152	13	suppose	suppose	VERB
ejpam-1878	152	14	|θ	|θ	PRON
ejpam-1878	152	15	(	(	PUNCT
ejpam-1878	152	16	n)|	n)|	NOUN
ejpam-1878	152	17	is	be	AUX
ejpam-1878	152	18	non	non	ADJ
ejpam-1878	152	19	-	-	ADJ
ejpam-1878	152	20	empty	empty	ADJ
ejpam-1878	152	21	.	.	PUNCT
ejpam-1878	153	1	let	let	VERB
ejpam-1878	153	2	|θ	|θ	PRON
ejpam-1878	153	3	(	(	PUNCT
ejpam-1878	153	4	n)|	n)|	NOUN
ejpam-1878	153	5	=	=	SYM
ejpam-1878	153	6	{	{	PUNCT
ejpam-1878	153	7	p1	p1	PROPN
ejpam-1878	153	8	,	,	PUNCT
ejpam-1878	153	9	p2	p2	X
ejpam-1878	153	10	·	·	PUNCT
ejpam-1878	153	11	·	·	PUNCT
ejpam-1878	153	12	·	·	PUNCT
ejpam-1878	153	13	,	,	PUNCT
ejpam-1878	153	14	pr	pr	NOUN
ejpam-1878	153	15	}	}	PUNCT
ejpam-1878	153	16	.	.	PUNCT
ejpam-1878	154	1	then	then	ADV
ejpam-1878	154	2	θ	θ	X
ejpam-1878	154	3	(	(	PUNCT
ejpam-1878	154	4	n)(p	n)(p	X
ejpam-1878	154	5	)	)	PUNCT
ejpam-1878	154	6	=	=	SYM
ejpam-1878	154	7	0	0	NUM
ejpam-1878	154	8	for	for	ADP
ejpam-1878	154	9	all	all	DET
ejpam-1878	154	10	p	p	NOUN
ejpam-1878	154	11	6=	6=	NUM
ejpam-1878	154	12	pi	pi	NOUN
ejpam-1878	154	13	,	,	PUNCT
ejpam-1878	154	14	1	1	NUM
ejpam-1878	154	15	≤	≤	NUM
ejpam-1878	154	16	i	i	NOUN
ejpam-1878	154	17	≤	≤	ADJ
ejpam-1878	154	18	r	r	NOUN
ejpam-1878	154	19	and	and	CCONJ
ejpam-1878	154	20	θ	θ	PROPN
ejpam-1878	154	21	(	(	PUNCT
ejpam-1878	154	22	n)(pi	n)(pi	NOUN
ejpam-1878	154	23	)	)	PUNCT
ejpam-1878	154	24	∈	∈	PROPN
ejpam-1878	154	25	n	n	NOUN
ejpam-1878	154	26	.	.	PUNCT
ejpam-1878	155	1	now	now	ADV
ejpam-1878	155	2	,	,	PUNCT
ejpam-1878	155	3	for	for	ADP
ejpam-1878	155	4	each	each	DET
ejpam-1878	155	5	1	1	NUM
ejpam-1878	155	6	≤	≤	NUM
ejpam-1878	155	7	i	i	PRON
ejpam-1878	155	8	≤	≤	ADJ
ejpam-1878	155	9	r	r	NOUN
ejpam-1878	155	10	,	,	PUNCT
ejpam-1878	155	11	θ	θ	PROPN
ejpam-1878	155	12	(	(	PUNCT
ejpam-1878	155	13	n)(pi	n)(pi	NOUN
ejpam-1878	155	14	)	)	PUNCT
ejpam-1878	155	15	≤	≤	NOUN
ejpam-1878	155	16	α(pi	α(pi	NUM
ejpam-1878	155	17	)	)	PUNCT
ejpam-1878	155	18	=	=	SYM
ejpam-1878	155	19	sup{θ	sup{θ	NOUN
ejpam-1878	155	20	(	(	PUNCT
ejpam-1878	155	21	m)(pi)|m	m)(pi)|m	PROPN
ejpam-1878	155	22	∈	∈	PROPN
ejpam-1878	155	23	i	i	PROPN
ejpam-1878	155	24	}	}	PUNCT
ejpam-1878	155	25	and	and	CCONJ
ejpam-1878	155	26	hence	hence	ADV
ejpam-1878	155	27	there	there	PRON
ejpam-1878	155	28	exists	exist	VERB
ejpam-1878	155	29	mi	mi	PROPN
ejpam-1878	155	30	∈	∈	PROPN
ejpam-1878	156	1	i	i	PRON
ejpam-1878	156	2	such	such	ADJ
ejpam-1878	156	3	that	that	SCONJ
ejpam-1878	156	4	θ	θ	PROPN
ejpam-1878	156	5	(	(	PUNCT
ejpam-1878	156	6	n)(pi)≤	n)(pi)≤	ADV
ejpam-1878	156	7	θ	θ	PROPN
ejpam-1878	156	8	(	(	PUNCT
ejpam-1878	156	9	m)(pi	m)(pi	NUM
ejpam-1878	156	10	)	)	PUNCT
ejpam-1878	156	11	.	.	PUNCT
ejpam-1878	157	1	now	now	ADV
ejpam-1878	157	2	,	,	PUNCT
ejpam-1878	157	3	put	put	VERB
ejpam-1878	157	4	m=	m=	X
ejpam-1878	157	5	m1	m1	PROPN
ejpam-1878	157	6	∨m2	∨m2	ADV
ejpam-1878	157	7	∨	∨	NUM
ejpam-1878	157	8	·	·	PUNCT
ejpam-1878	157	9	·	·	PUNCT
ejpam-1878	157	10	·	·	PUNCT
ejpam-1878	157	11	∨mr	∨mr	VERB
ejpam-1878	157	12	,	,	PUNCT
ejpam-1878	157	13	then	then	ADV
ejpam-1878	157	14	m	m	VERB
ejpam-1878	157	15	∈	∈	ADJ
ejpam-1878	157	16	i	i	PRON
ejpam-1878	157	17	and	and	CCONJ
ejpam-1878	157	18	θ	θ	PROPN
ejpam-1878	157	19	(	(	PUNCT
ejpam-1878	157	20	n)(pi)≤	n)(pi)≤	ADV
ejpam-1878	157	21	max{θ	max{θ	ADV
ejpam-1878	157	22	(	(	PUNCT
ejpam-1878	157	23	m1)(pi	m1)(pi	NOUN
ejpam-1878	157	24	)	)	PUNCT
ejpam-1878	157	25	,	,	PUNCT
ejpam-1878	157	26	.	.	PUNCT
ejpam-1878	157	27	.	.	PUNCT
ejpam-1878	158	1	.	.	PUNCT
ejpam-1878	159	1	,	,	PUNCT
ejpam-1878	159	2	θ	θ	PROPN
ejpam-1878	159	3	(	(	PUNCT
ejpam-1878	159	4	mi)(pi)}=	mi)(pi)}=	PROPN
ejpam-1878	159	5	θ	θ	PROPN
ejpam-1878	159	6	(	(	PUNCT
ejpam-1878	159	7	m)(pi	m)(pi	X
ejpam-1878	159	8	)	)	PUNCT
ejpam-1878	159	9	for	for	ADP
ejpam-1878	159	10	all	all	DET
ejpam-1878	159	11	1	1	NUM
ejpam-1878	159	12	≤	≤	NUM
ejpam-1878	159	13	i	i	PRON
ejpam-1878	159	14	≤	≤	PROPN
ejpam-1878	159	15	r.	r.	PROPN
ejpam-1878	159	16	also	also	ADV
ejpam-1878	159	17	,	,	PUNCT
ejpam-1878	159	18	since	since	SCONJ
ejpam-1878	159	19	θ	θ	PROPN
ejpam-1878	159	20	(	(	PUNCT
ejpam-1878	159	21	n)(p	n)(p	NOUN
ejpam-1878	159	22	)	)	PUNCT
ejpam-1878	159	23	=	=	SYM
ejpam-1878	159	24	0	0	NUM
ejpam-1878	159	25	for	for	ADP
ejpam-1878	159	26	all	all	DET
ejpam-1878	159	27	p	p	NOUN
ejpam-1878	159	28	6=	6=	NUM
ejpam-1878	159	29	pi	pi	NOUN
ejpam-1878	159	30	,	,	PUNCT
ejpam-1878	159	31	we	we	PRON
ejpam-1878	159	32	get	get	VERB
ejpam-1878	159	33	that	that	DET
ejpam-1878	159	34	θ	θ	PROPN
ejpam-1878	159	35	(	(	PUNCT
ejpam-1878	159	36	n)(p	n)(p	NOUN
ejpam-1878	159	37	)	)	PUNCT
ejpam-1878	159	38	≤	≤	NUM
ejpam-1878	159	39	θ	θ	PROPN
ejpam-1878	159	40	(	(	PUNCT
ejpam-1878	159	41	m)(p	m)(p	X
ejpam-1878	159	42	)	)	PUNCT
ejpam-1878	159	43	for	for	ADP
ejpam-1878	159	44	all	all	DET
ejpam-1878	159	45	p	p	NOUN
ejpam-1878	159	46	∈	∈	NOUN
ejpam-1878	159	47	p	p	NOUN
ejpam-1878	159	48	so	so	SCONJ
ejpam-1878	159	49	that	that	SCONJ
ejpam-1878	159	50	n≤d	n≤d	PROPN
ejpam-1878	159	51	m	m	VERB
ejpam-1878	159	52	∈	∈	NOUN
ejpam-1878	160	1	i	i	PRON
ejpam-1878	160	2	and	and	CCONJ
ejpam-1878	160	3	therefore	therefore	ADV
ejpam-1878	160	4	n	n	ADV
ejpam-1878	160	5	∈	∈	NOUN
ejpam-1878	161	1	i	i	PRON
ejpam-1878	161	2	.	.	PUNCT
ejpam-1878	162	1	therefore	therefore	ADV
ejpam-1878	162	2	iα	iα	VERB
ejpam-1878	162	3	⊆	⊆	NUM
ejpam-1878	162	4	i	i	PRON
ejpam-1878	162	5	.	.	PUNCT
ejpam-1878	163	1	thus	thus	ADV
ejpam-1878	163	2	i	i	PRON
ejpam-1878	163	3	=	=	SYM
ejpam-1878	163	4	iα	iα	PROPN
ejpam-1878	163	5	.	.	PUNCT
ejpam-1878	163	6	note	note	VERB
ejpam-1878	163	7	that	that	SCONJ
ejpam-1878	163	8	,	,	PUNCT
ejpam-1878	163	9	if	if	SCONJ
ejpam-1878	163	10	α	α	PRON
ejpam-1878	163	11	is	be	AUX
ejpam-1878	163	12	the	the	DET
ejpam-1878	163	13	constant	constant	ADJ
ejpam-1878	163	14	map	map	NOUN
ejpam-1878	163	15	0	0	NUM
ejpam-1878	163	16	defined	define	VERB
ejpam-1878	163	17	by	by	ADP
ejpam-1878	163	18	α(p	α(p	PROPN
ejpam-1878	163	19	)	)	PUNCT
ejpam-1878	164	1	=	=	SYM
ejpam-1878	164	2	0	0	NUM
ejpam-1878	165	1	for	for	ADP
ejpam-1878	165	2	all	all	PRON
ejpam-1878	165	3	p	p	NOUN
ejpam-1878	165	4	∈	∈	PROPN
ejpam-1878	165	5	p	p	NOUN
ejpam-1878	165	6	,	,	PUNCT
ejpam-1878	165	7	then	then	ADV
ejpam-1878	165	8	iα	iα	VERB
ejpam-1878	165	9	=	=	PUNCT
ejpam-1878	165	10	{	{	PUNCT
ejpam-1878	165	11	1	1	NUM
ejpam-1878	165	12	}	}	PUNCT
ejpam-1878	165	13	and	and	CCONJ
ejpam-1878	165	14	that	that	SCONJ
ejpam-1878	165	15	,	,	PUNCT
ejpam-1878	165	16	if	if	SCONJ
ejpam-1878	165	17	α	α	PRON
ejpam-1878	165	18	is	be	AUX
ejpam-1878	165	19	the	the	DET
ejpam-1878	165	20	constant	constant	ADJ
ejpam-1878	165	21	map∞	map∞	NOUN
ejpam-1878	165	22	,	,	PUNCT
ejpam-1878	165	23	then	then	ADV
ejpam-1878	165	24	iα	iα	VERB
ejpam-1878	165	25	=	=	PUNCT
ejpam-1878	165	26	z	z	PROPN
ejpam-1878	166	1	+	+	PROPN
ejpam-1878	166	2	.	.	PUNCT
ejpam-1878	166	3	definition	definition	NOUN
ejpam-1878	166	4	10	10	NUM
ejpam-1878	166	5	.	.	PUNCT
ejpam-1878	167	1	for	for	ADP
ejpam-1878	167	2	any	any	DET
ejpam-1878	167	3	mappings	mapping	NOUN
ejpam-1878	167	4	α	α	NOUN
ejpam-1878	167	5	and	and	CCONJ
ejpam-1878	167	6	β	β	X
ejpam-1878	167	7	from	from	ADP
ejpam-1878	167	8	p	p	NOUN
ejpam-1878	167	9	into	into	ADP
ejpam-1878	167	10	n	n	PRON
ejpam-1878	167	11	∞	∞	NOUN
ejpam-1878	167	12	,	,	PUNCT
ejpam-1878	167	13	define	define	VERB
ejpam-1878	167	14	α≤	α≤	PROPN
ejpam-1878	167	15	β	β	NOUN
ejpam-1878	167	16	if	if	SCONJ
ejpam-1878	168	1	and	and	CCONJ
ejpam-1878	168	2	only	only	ADV
ejpam-1878	168	3	if	if	SCONJ
ejpam-1878	168	4	α(p)≤	α(p)≤	PROPN
ejpam-1878	168	5	β(p	β(p	PROPN
ejpam-1878	168	6	)	)	PUNCT
ejpam-1878	168	7	for	for	ADP
ejpam-1878	168	8	all	all	PRON
ejpam-1878	168	9	p	p	NOUN
ejpam-1878	168	10	∈	∈	PROPN
ejpam-1878	168	11	p	p	NOUN
ejpam-1878	168	12	.	.	PUNCT
ejpam-1878	169	1	thus	thus	ADV
ejpam-1878	169	2	≤	≤	NUM
ejpam-1878	169	3	is	be	AUX
ejpam-1878	169	4	a	a	DET
ejpam-1878	169	5	partial	partial	ADJ
ejpam-1878	169	6	order	order	NOUN
ejpam-1878	169	7	on	on	ADP
ejpam-1878	169	8	(	(	PUNCT
ejpam-1878	169	9	n	n	X
ejpam-1878	169	10	∞)p	∞)p	X
ejpam-1878	169	11	.	.	PUNCT
ejpam-1878	170	1	theorem	theorem	ADJ
ejpam-1878	170	2	7	7	NUM
ejpam-1878	170	3	.	.	PUNCT
ejpam-1878	171	1	the	the	DET
ejpam-1878	171	2	map	map	NOUN
ejpam-1878	171	3	α	α	PROPN
ejpam-1878	171	4	7→	7→	NUM
ejpam-1878	171	5	iα	iα	NOUN
ejpam-1878	171	6	is	be	AUX
ejpam-1878	171	7	an	an	DET
ejpam-1878	171	8	order	order	NOUN
ejpam-1878	171	9	isomorphism	isomorphism	NOUN
ejpam-1878	171	10	of	of	ADP
ejpam-1878	171	11	the	the	DET
ejpam-1878	171	12	poset	poset	NOUN
ejpam-1878	171	13	(	(	PUNCT
ejpam-1878	171	14	(	(	PUNCT
ejpam-1878	171	15	n	n	X
ejpam-1878	171	16	∞)p	∞)p	PRON
ejpam-1878	171	17	,	,	PUNCT
ejpam-1878	171	18	≤	≤	NUM
ejpam-1878	171	19	)	)	PUNCT
ejpam-1878	171	20	,	,	PUNCT
ejpam-1878	171	21	onto	onto	ADP
ejpam-1878	171	22	the	the	DET
ejpam-1878	171	23	poset	poset	NOUN
ejpam-1878	171	24	(	(	PUNCT
ejpam-1878	171	25	i	i	PRON
ejpam-1878	171	26	(	(	PUNCT
ejpam-1878	171	27	z+),⊆	z+),⊆	PROPN
ejpam-1878	171	28	)	)	PUNCT
ejpam-1878	171	29	of	of	ADP
ejpam-1878	171	30	all	all	DET
ejpam-1878	171	31	ideals	ideal	NOUN
ejpam-1878	171	32	of	of	ADP
ejpam-1878	171	33	(	(	PUNCT
ejpam-1878	171	34	z+,/	z+,/	NOUN
ejpam-1878	171	35	)	)	PUNCT
ejpam-1878	171	36	.	.	PUNCT
ejpam-1878	172	1	proof	proof	NOUN
ejpam-1878	172	2	.	.	PUNCT
ejpam-1878	173	1	let	let	VERB
ejpam-1878	173	2	α	α	NOUN
ejpam-1878	173	3	and	and	CCONJ
ejpam-1878	173	4	β	β	X
ejpam-1878	173	5	:	:	PUNCT
ejpam-1878	173	6	p	p	PROPN
ejpam-1878	173	7	7→	7→	NUM
ejpam-1878	173	8	n	n	CCONJ
ejpam-1878	173	9	∞	∞	NUM
ejpam-1878	173	10	be	be	AUX
ejpam-1878	173	11	any	any	DET
ejpam-1878	173	12	mappings	mapping	NOUN
ejpam-1878	173	13	.	.	PUNCT
ejpam-1878	174	1	clearly	clearly	ADV
ejpam-1878	174	2	,	,	PUNCT
ejpam-1878	174	3	α	α	PROPN
ejpam-1878	174	4	≤	≤	NOUN
ejpam-1878	174	5	β	β	X
ejpam-1878	174	6	⇒	⇒	NOUN
ejpam-1878	174	7	iα	iα	VERB
ejpam-1878	174	8	⊆	⊆	NUM
ejpam-1878	174	9	iβ	iβ	ADP
ejpam-1878	174	10	.	.	PUNCT
ejpam-1878	175	1	on	on	ADP
ejpam-1878	175	2	the	the	DET
ejpam-1878	175	3	other	other	ADJ
ejpam-1878	175	4	hand	hand	NOUN
ejpam-1878	175	5	,	,	PUNCT
ejpam-1878	175	6	suppose	suppose	VERB
ejpam-1878	175	7	that	that	SCONJ
ejpam-1878	175	8	iα	iα	VERB
ejpam-1878	175	9	⊆	⊆	NUM
ejpam-1878	175	10	iβ	iβ	ADP
ejpam-1878	175	11	.	.	PUNCT
ejpam-1878	176	1	we	we	PRON
ejpam-1878	176	2	shall	shall	AUX
ejpam-1878	176	3	prove	prove	VERB
ejpam-1878	176	4	that	that	SCONJ
ejpam-1878	176	5	α(p	α(p	PROPN
ejpam-1878	176	6	)	)	PUNCT
ejpam-1878	176	7	≤	≤	PROPN
ejpam-1878	176	8	β(p	β(p	PROPN
ejpam-1878	176	9	)	)	PUNCT
ejpam-1878	176	10	for	for	ADP
ejpam-1878	176	11	all	all	PRON
ejpam-1878	176	12	p	p	NOUN
ejpam-1878	176	13	∈	∈	NOUN
ejpam-1878	176	14	p	p	NOUN
ejpam-1878	176	15	so	so	SCONJ
ejpam-1878	176	16	that	that	SCONJ
ejpam-1878	176	17	α	α	NOUN
ejpam-1878	176	18	≤	≤	NOUN
ejpam-1878	176	19	β	β	X
ejpam-1878	176	20	.	.	PUNCT
ejpam-1878	177	1	to	to	PART
ejpam-1878	177	2	prove	prove	VERB
ejpam-1878	177	3	this	this	PRON
ejpam-1878	177	4	,	,	PUNCT
ejpam-1878	177	5	let	let	VERB
ejpam-1878	177	6	us	we	PRON
ejpam-1878	177	7	fix	fix	VERB
ejpam-1878	177	8	p	p	NOUN
ejpam-1878	177	9	∈	∈	PROPN
ejpam-1878	177	10	p	p	NOUN
ejpam-1878	177	11	.	.	PUNCT
ejpam-1878	178	1	if	if	SCONJ
ejpam-1878	178	2	β(p	β(p	PROPN
ejpam-1878	178	3	)	)	PUNCT
ejpam-1878	179	1	=	=	NOUN
ejpam-1878	179	2	∞	∞	NUM
ejpam-1878	179	3	or	or	CCONJ
ejpam-1878	179	4	α(p	α(p	NUM
ejpam-1878	179	5	)	)	PUNCT
ejpam-1878	180	1	=	=	SYM
ejpam-1878	180	2	0	0	NUM
ejpam-1878	180	3	,	,	PUNCT
ejpam-1878	180	4	trivially	trivially	ADV
ejpam-1878	180	5	α(p	α(p	PROPN
ejpam-1878	180	6	)	)	PUNCT
ejpam-1878	180	7	≤	≤	NOUN
ejpam-1878	180	8	β(p	β(p	PROPN
ejpam-1878	180	9	)	)	PUNCT
ejpam-1878	180	10	.	.	PUNCT
ejpam-1878	181	1	therefore	therefore	ADV
ejpam-1878	181	2	,	,	PUNCT
ejpam-1878	181	3	we	we	PRON
ejpam-1878	181	4	can	can	AUX
ejpam-1878	181	5	assume	assume	VERB
ejpam-1878	181	6	that	that	SCONJ
ejpam-1878	181	7	β(p)<∞	β(p)<∞	PROPN
ejpam-1878	181	8	and	and	CCONJ
ejpam-1878	181	9	α(p	α(p	NUM
ejpam-1878	181	10	)	)	PUNCT
ejpam-1878	181	11	>	>	X
ejpam-1878	181	12	0	0	X
ejpam-1878	181	13	.	.	PUNCT
ejpam-1878	181	14	consider	consider	VERB
ejpam-1878	181	15	n=	n=	ADJ
ejpam-1878	181	16	pβ(p)+1	pβ(p)+1	PROPN
ejpam-1878	181	17	.	.	PUNCT
ejpam-1878	182	1	then	then	ADV
ejpam-1878	182	2	θ	θ	X
ejpam-1878	182	3	(	(	PUNCT
ejpam-1878	182	4	n)(p	n)(p	NOUN
ejpam-1878	182	5	)	)	PUNCT
ejpam-1878	182	6	=	=	SYM
ejpam-1878	182	7	β(p	β(p	NUM
ejpam-1878	182	8	)	)	PUNCT
ejpam-1878	183	1	+	+	CCONJ
ejpam-1878	183	2	1	1	NUM
ejpam-1878	183	3	6≤	6≤	NUM
ejpam-1878	183	4	β(p	β(p	NUM
ejpam-1878	183	5	)	)	PUNCT
ejpam-1878	183	6	.	.	PUNCT
ejpam-1878	184	1	s.	s.	PROPN
ejpam-1878	184	2	sagi	sagi	PROPN
ejpam-1878	184	3	/	/	SYM
ejpam-1878	184	4	eur	eur	PROPN
ejpam-1878	184	5	.	.	PUNCT
ejpam-1878	185	1	j.	j.	PROPN
ejpam-1878	185	2	pure	pure	PROPN
ejpam-1878	185	3	appl	appl	PROPN
ejpam-1878	185	4	.	.	PROPN
ejpam-1878	185	5	math	math	PROPN
ejpam-1878	185	6	,	,	PUNCT
ejpam-1878	185	7	8	8	NUM
ejpam-1878	185	8	(	(	PUNCT
ejpam-1878	185	9	2015	2015	NUM
ejpam-1878	185	10	)	)	PUNCT
ejpam-1878	185	11	,	,	PUNCT
ejpam-1878	185	12	15	15	NUM
ejpam-1878	185	13	-	-	SYM
ejpam-1878	185	14	25	25	NUM
ejpam-1878	185	15	22	22	NUM
ejpam-1878	185	16	and	and	CCONJ
ejpam-1878	185	17	hence	hence	ADV
ejpam-1878	185	18	n	n	NOUN
ejpam-1878	185	19	/∈	/∈	PUNCT
ejpam-1878	186	1	iβ	iβ	ADP
ejpam-1878	186	2	.	.	PUNCT
ejpam-1878	187	1	since	since	SCONJ
ejpam-1878	187	2	iα	iα	VERB
ejpam-1878	187	3	⊆	⊆	NUM
ejpam-1878	187	4	iβ	iβ	ADP
ejpam-1878	187	5	,	,	PUNCT
ejpam-1878	187	6	n	n	CCONJ
ejpam-1878	187	7	/∈	/∈	CCONJ
ejpam-1878	187	8	iα	iα	INTJ
ejpam-1878	187	9	and	and	CCONJ
ejpam-1878	187	10	therefore	therefore	ADV
ejpam-1878	187	11	θ	θ	X
ejpam-1878	187	12	(	(	PUNCT
ejpam-1878	187	13	n)(q	n)(q	PROPN
ejpam-1878	187	14	)	)	PUNCT
ejpam-1878	187	15	6≤	6≤	NUM
ejpam-1878	187	16	α(q	α(q	NUM
ejpam-1878	187	17	)	)	PUNCT
ejpam-1878	187	18	for	for	ADP
ejpam-1878	187	19	some	some	DET
ejpam-1878	187	20	q	q	NOUN
ejpam-1878	187	21	∈	∈	PROPN
ejpam-1878	187	22	p	p	NOUN
ejpam-1878	187	23	.	.	PUNCT
ejpam-1878	188	1	but	but	CCONJ
ejpam-1878	188	2	θ	θ	PROPN
ejpam-1878	188	3	(	(	PUNCT
ejpam-1878	188	4	n)(q	n)(q	PROPN
ejpam-1878	188	5	)	)	PUNCT
ejpam-1878	189	1	=	=	SYM
ejpam-1878	189	2	0	0	NUM
ejpam-1878	189	3	for	for	ADP
ejpam-1878	189	4	all	all	DET
ejpam-1878	189	5	q	q	NOUN
ejpam-1878	189	6	6=	6=	NUM
ejpam-1878	189	7	p.	p.	NOUN
ejpam-1878	189	8	thus	thus	ADV
ejpam-1878	189	9	β(p	β(p	NUM
ejpam-1878	189	10	)	)	PUNCT
ejpam-1878	190	1	+	+	CCONJ
ejpam-1878	190	2	1	1	NUM
ejpam-1878	190	3	=	=	SYM
ejpam-1878	190	4	θ	θ	X
ejpam-1878	190	5	(	(	PUNCT
ejpam-1878	190	6	n)(p	n)(p	NOUN
ejpam-1878	190	7	)	)	PUNCT
ejpam-1878	190	8	6≤	6≤	NUM
ejpam-1878	190	9	α(p	α(p	PROPN
ejpam-1878	190	10	)	)	PUNCT
ejpam-1878	190	11	α(p)<β(p	α(p)<β(p	PROPN
ejpam-1878	190	12	)	)	PUNCT
ejpam-1878	191	1	+	+	NUM
ejpam-1878	191	2	1	1	X
ejpam-1878	191	3	.	.	X
ejpam-1878	191	4	therefore	therefore	ADV
ejpam-1878	191	5	α(p	α(p	PROPN
ejpam-1878	191	6	)	)	PUNCT
ejpam-1878	191	7	≤	≤	PROPN
ejpam-1878	191	8	β(p	β(p	PROPN
ejpam-1878	191	9	)	)	PUNCT
ejpam-1878	191	10	.	.	PUNCT
ejpam-1878	192	1	this	this	PRON
ejpam-1878	192	2	is	be	AUX
ejpam-1878	192	3	true	true	ADJ
ejpam-1878	192	4	for	for	ADP
ejpam-1878	192	5	all	all	PRON
ejpam-1878	192	6	p	p	NOUN
ejpam-1878	192	7	∈	∈	PROPN
ejpam-1878	192	8	p	p	NOUN
ejpam-1878	192	9	.	.	PUNCT
ejpam-1878	193	1	thus	thus	ADV
ejpam-1878	193	2	α	α	X
ejpam-1878	193	3	≤	≤	ADJ
ejpam-1878	193	4	β	β	X
ejpam-1878	193	5	.	.	PUNCT
ejpam-1878	194	1	also	also	ADV
ejpam-1878	194	2	α	α	PROPN
ejpam-1878	194	3	7→	7→	NUM
ejpam-1878	194	4	iα	iα	NOUN
ejpam-1878	194	5	is	be	AUX
ejpam-1878	194	6	a	a	DET
ejpam-1878	194	7	surjection	surjection	NOUN
ejpam-1878	194	8	.	.	PUNCT
ejpam-1878	195	1	thus	thus	ADV
ejpam-1878	195	2	α	α	PRON
ejpam-1878	195	3	7→	7→	NUM
ejpam-1878	195	4	iα	iα	NOUN
ejpam-1878	195	5	is	be	AUX
ejpam-1878	195	6	an	an	DET
ejpam-1878	195	7	order	order	NOUN
ejpam-1878	195	8	isomorphism	isomorphism	NOUN
ejpam-1878	195	9	of	of	ADP
ejpam-1878	195	10	(	(	PUNCT
ejpam-1878	195	11	(	(	PUNCT
ejpam-1878	195	12	n	n	X
ejpam-1878	195	13	∞)p	∞)p	PRON
ejpam-1878	195	14	,	,	PUNCT
ejpam-1878	195	15	≤	≤	NUM
ejpam-1878	195	16	)	)	PUNCT
ejpam-1878	195	17	,	,	PUNCT
ejpam-1878	195	18	onto	onto	ADP
ejpam-1878	195	19	(	(	PUNCT
ejpam-1878	195	20	i	i	PRON
ejpam-1878	195	21	(	(	PUNCT
ejpam-1878	195	22	z+),⊆	z+),⊆	PROPN
ejpam-1878	195	23	)	)	PUNCT
ejpam-1878	195	24	.	.	PUNCT
ejpam-1878	196	1	corollary	corollary	ADJ
ejpam-1878	196	2	1	1	NUM
ejpam-1878	196	3	.	.	PUNCT
ejpam-1878	197	1	for	for	ADP
ejpam-1878	197	2	any	any	DET
ejpam-1878	197	3	α	α	NOUN
ejpam-1878	197	4	and	and	CCONJ
ejpam-1878	197	5	β	β	X
ejpam-1878	197	6	:p	:p	NOUN
ejpam-1878	197	7	→n	→n	NUM
ejpam-1878	197	8	∞	∞	NOUN
ejpam-1878	197	9	,	,	PUNCT
ejpam-1878	197	10	iα	iα	ADP
ejpam-1878	197	11	∩	∩	NOUN
ejpam-1878	197	12	iβ	iβ	ADP
ejpam-1878	197	13	=	=	NOUN
ejpam-1878	197	14	iα∧β	iα∧β	PROPN
ejpam-1878	197	15	.	.	PUNCT
ejpam-1878	198	1	and	and	CCONJ
ejpam-1878	198	2	iα	iα	VERB
ejpam-1878	198	3	∪	∪	ADV
ejpam-1878	198	4	iβ	iβ	ADP
ejpam-1878	198	5	=	=	PUNCT
ejpam-1878	198	6	iα∨β	iα∨β	PROPN
ejpam-1878	198	7	.	.	PUNCT
ejpam-1878	199	1	where	where	SCONJ
ejpam-1878	199	2	α∧	α∧	PROPN
ejpam-1878	199	3	β	β	X
ejpam-1878	199	4	and	and	CCONJ
ejpam-1878	199	5	α∨	α∨	PROPN
ejpam-1878	199	6	β	β	NOUN
ejpam-1878	199	7	are	be	AUX
ejpam-1878	199	8	point	point	ADV
ejpam-1878	199	9	-	-	PUNCT
ejpam-1878	199	10	wise	wise	ADJ
ejpam-1878	199	11	g.l.b	g.l.b	NOUN
ejpam-1878	199	12	and	and	CCONJ
ejpam-1878	199	13	l.u.b	l.u.b	NOUN
ejpam-1878	199	14	of	of	ADP
ejpam-1878	199	15	α	α	PROPN
ejpam-1878	199	16	and	and	CCONJ
ejpam-1878	199	17	β	β	PROPN
ejpam-1878	199	18	.	.	PUNCT
ejpam-1878	200	1	first	first	ADV
ejpam-1878	200	2	we	we	PRON
ejpam-1878	200	3	state	state	VERB
ejpam-1878	200	4	the	the	DET
ejpam-1878	200	5	following	follow	VERB
ejpam-1878	200	6	two	two	NUM
ejpam-1878	200	7	theorems	theorem	NOUN
ejpam-1878	200	8	from	from	ADP
ejpam-1878	200	9	“	"	PUNCT
ejpam-1878	200	10	lattice	lattice	NOUN
ejpam-1878	200	11	structures	structure	NOUN
ejpam-1878	200	12	on	on	ADP
ejpam-1878	200	13	z+	z+	NUM
ejpam-1878	200	14	induced	induce	VERB
ejpam-1878	200	15	by	by	ADP
ejpam-1878	200	16	convolutions	convolution	NOUN
ejpam-1878	200	17	”	"	PUNCT
ejpam-1878	201	1	[	[	X
ejpam-1878	201	2	3	3	NUM
ejpam-1878	201	3	]	]	PUNCT
ejpam-1878	201	4	.	.	PUNCT
ejpam-1878	202	1	theorem	theorem	ADJ
ejpam-1878	202	2	8	8	NUM
ejpam-1878	202	3	.	.	PUNCT
ejpam-1878	203	1	let	let	VERB
ejpam-1878	203	2	c	c	PRON
ejpam-1878	203	3	be	be	AUX
ejpam-1878	203	4	a	a	DET
ejpam-1878	203	5	convolution	convolution	NOUN
ejpam-1878	203	6	which	which	PRON
ejpam-1878	203	7	is	be	AUX
ejpam-1878	203	8	closed	close	VERB
ejpam-1878	203	9	under	under	ADP
ejpam-1878	203	10	finite	finite	ADJ
ejpam-1878	203	11	intersections	intersection	NOUN
ejpam-1878	203	12	and	and	CCONJ
ejpam-1878	203	13	≤c	≤c	PROPN
ejpam-1878	203	14	be	be	VERB
ejpam-1878	203	15	the	the	DET
ejpam-1878	203	16	partial	partial	ADJ
ejpam-1878	203	17	order	order	NOUN
ejpam-1878	203	18	on	on	ADP
ejpam-1878	203	19	z+	z+	NUM
ejpam-1878	203	20	induced	induce	VERB
ejpam-1878	203	21	by	by	ADP
ejpam-1878	203	22	c	c	PROPN
ejpam-1878	203	23	.	.	PUNCT
ejpam-1878	204	1	then	then	ADV
ejpam-1878	204	2	(	(	PUNCT
ejpam-1878	204	3	z+,≤c	z+,≤c	X
ejpam-1878	204	4	)	)	PUNCT
ejpam-1878	204	5	is	be	AUX
ejpam-1878	204	6	a	a	DET
ejpam-1878	204	7	lattice	lattice	NOUN
ejpam-1878	204	8	if	if	SCONJ
ejpam-1878	205	1	and	and	CCONJ
ejpam-1878	205	2	only	only	ADV
ejpam-1878	205	3	if	if	SCONJ
ejpam-1878	205	4	it	it	PRON
ejpam-1878	205	5	is	be	AUX
ejpam-1878	205	6	directed	direct	VERB
ejpam-1878	205	7	above	above	ADV
ejpam-1878	205	8	.	.	PUNCT
ejpam-1878	206	1	theorem	theorem	NOUN
ejpam-1878	206	2	9	9	NUM
ejpam-1878	206	3	.	.	PUNCT
ejpam-1878	207	1	let	let	VERB
ejpam-1878	207	2	c	c	PRON
ejpam-1878	207	3	be	be	AUX
ejpam-1878	207	4	a	a	DET
ejpam-1878	207	5	convolution	convolution	NOUN
ejpam-1878	207	6	.	.	PUNCT
ejpam-1878	208	1	(	(	PUNCT
ejpam-1878	208	2	1	1	NUM
ejpam-1878	208	3	)	)	PUNCT
ejpam-1878	208	4	.	.	PUNCT
ejpam-1878	209	1	if	if	SCONJ
ejpam-1878	209	2	(	(	PUNCT
ejpam-1878	209	3	z+,≤c	z+,≤c	NUM
ejpam-1878	209	4	)	)	PUNCT
ejpam-1878	209	5	is	be	AUX
ejpam-1878	209	6	a	a	DET
ejpam-1878	209	7	meet(join	meet(join	NOUN
ejpam-1878	209	8	)	)	PUNCT
ejpam-1878	209	9	semilattice	semilattice	NOUN
ejpam-1878	209	10	,	,	PUNCT
ejpam-1878	209	11	then	then	ADV
ejpam-1878	209	12	so	so	ADV
ejpam-1878	209	13	is	be	AUX
ejpam-1878	209	14	(	(	PUNCT
ejpam-1878	209	15	n	n	X
ejpam-1878	209	16	,	,	PUNCT
ejpam-1878	209	17	≤p	≤p	PROPN
ejpam-1878	209	18	c	c	PROPN
ejpam-1878	209	19	)	)	PUNCT
ejpam-1878	209	20	for	for	ADP
ejpam-1878	209	21	each	each	DET
ejpam-1878	209	22	prime	prime	NOUN
ejpam-1878	209	23	p	p	NOUN
ejpam-1878	209	24	(	(	PUNCT
ejpam-1878	209	25	2	2	NUM
ejpam-1878	209	26	)	)	PUNCT
ejpam-1878	209	27	.	.	PUNCT
ejpam-1878	210	1	if	if	SCONJ
ejpam-1878	210	2	(	(	PUNCT
ejpam-1878	210	3	z+,≤c	z+,≤c	NUM
ejpam-1878	210	4	)	)	PUNCT
ejpam-1878	210	5	is	be	AUX
ejpam-1878	210	6	a	a	DET
ejpam-1878	210	7	lattice	lattice	NOUN
ejpam-1878	210	8	,	,	PUNCT
ejpam-1878	210	9	then	then	ADV
ejpam-1878	210	10	so	so	ADV
ejpam-1878	210	11	is	be	AUX
ejpam-1878	210	12	(	(	PUNCT
ejpam-1878	210	13	n	n	X
ejpam-1878	210	14	,	,	PUNCT
ejpam-1878	210	15	≤p	≤p	PROPN
ejpam-1878	210	16	c	c	PROPN
ejpam-1878	210	17	)	)	PUNCT
ejpam-1878	210	18	for	for	ADP
ejpam-1878	210	19	each	each	DET
ejpam-1878	210	20	prime	prime	ADJ
ejpam-1878	210	21	p.	p.	NOUN
ejpam-1878	210	22	theorem	theorem	VERB
ejpam-1878	210	23	10	10	NUM
ejpam-1878	210	24	.	.	PUNCT
ejpam-1878	211	1	let	let	VERB
ejpam-1878	211	2	c	c	PRON
ejpam-1878	211	3	be	be	AUX
ejpam-1878	211	4	a	a	DET
ejpam-1878	211	5	multiplicative	multiplicative	ADJ
ejpam-1878	211	6	convolution	convolution	NOUN
ejpam-1878	211	7	such	such	ADJ
ejpam-1878	211	8	that	that	SCONJ
ejpam-1878	211	9	(	(	PUNCT
ejpam-1878	211	10	z+,/	z+,/	NOUN
ejpam-1878	211	11	)	)	PUNCT
ejpam-1878	211	12	is	be	AUX
ejpam-1878	211	13	a	a	DET
ejpam-1878	211	14	meet	meet	NOUN
ejpam-1878	211	15	semi	semi	ADJ
ejpam-1878	211	16	lattice	lattice	NOUN
ejpam-1878	211	17	.	.	PUNCT
ejpam-1878	212	1	for	for	ADP
ejpam-1878	212	2	any	any	DET
ejpam-1878	212	3	α	α	NOUN
ejpam-1878	212	4	:p	:p	NOUN
ejpam-1878	212	5	→n	→n	NUM
ejpam-1878	212	6	∞	∞	NOUN
ejpam-1878	212	7	,	,	PUNCT
ejpam-1878	212	8	let	let	VERB
ejpam-1878	212	9	iα	iα	INTJ
ejpam-1878	212	10	=	=	PUNCT
ejpam-1878	212	11	{	{	PUNCT
ejpam-1878	212	12	n	n	X
ejpam-1878	212	13	∈	∈	PROPN
ejpam-1878	212	14	z	z	PROPN
ejpam-1878	213	1	+	+	NOUN
ejpam-1878	213	2	|θ	|θ	PRON
ejpam-1878	213	3	(	(	PUNCT
ejpam-1878	213	4	n)(p)≤pc	n)(p)≤pc	NUM
ejpam-1878	213	5	α(p	α(p	NUM
ejpam-1878	213	6	)	)	PUNCT
ejpam-1878	213	7	for	for	ADP
ejpam-1878	213	8	all	all	DET
ejpam-1878	213	9	p	p	NOUN
ejpam-1878	213	10	∈	∈	PROPN
ejpam-1878	213	11	p	p	X
ejpam-1878	213	12	}	}	PUNCT
ejpam-1878	213	13	.	.	PUNCT
ejpam-1878	214	1	then	then	ADV
ejpam-1878	214	2	the	the	DET
ejpam-1878	214	3	following	follow	VERB
ejpam-1878	214	4	are	be	AUX
ejpam-1878	214	5	equivalent	equivalent	ADJ
ejpam-1878	214	6	to	to	ADP
ejpam-1878	214	7	each	each	DET
ejpam-1878	214	8	other	other	ADJ
ejpam-1878	214	9	.	.	PUNCT
ejpam-1878	215	1	(	(	PUNCT
ejpam-1878	215	2	1	1	NUM
ejpam-1878	215	3	)	)	PUNCT
ejpam-1878	215	4	.	.	PUNCT
ejpam-1878	216	1	iα	iα	PROPN
ejpam-1878	216	2	is	be	AUX
ejpam-1878	216	3	an	an	DET
ejpam-1878	216	4	ideal	ideal	NOUN
ejpam-1878	216	5	of	of	ADP
ejpam-1878	216	6	(	(	PUNCT
ejpam-1878	216	7	z+,≤c	z+,≤c	PROPN
ejpam-1878	216	8	)	)	PUNCT
ejpam-1878	216	9	for	for	ADP
ejpam-1878	216	10	any	any	DET
ejpam-1878	216	11	α	α	NOUN
ejpam-1878	216	12	:p	:p	PUNCT
ejpam-1878	217	1	→n	→n	PUNCT
ejpam-1878	217	2	∞.	∞.	PROPN
ejpam-1878	217	3	(	(	PUNCT
ejpam-1878	217	4	2	2	NUM
ejpam-1878	217	5	)	)	PUNCT
ejpam-1878	217	6	.	.	PUNCT
ejpam-1878	218	1	(	(	PUNCT
ejpam-1878	218	2	z+,≤c	z+,≤c	PUNCT
ejpam-1878	218	3	)	)	PUNCT
ejpam-1878	218	4	is	be	AUX
ejpam-1878	218	5	directed	direct	VERB
ejpam-1878	218	6	below	below	ADP
ejpam-1878	218	7	(	(	PUNCT
ejpam-1878	218	8	3	3	NUM
ejpam-1878	218	9	)	)	PUNCT
ejpam-1878	218	10	.	.	PUNCT
ejpam-1878	219	1	(	(	PUNCT
ejpam-1878	219	2	z+,≤c	z+,≤c	PUNCT
ejpam-1878	219	3	)	)	PUNCT
ejpam-1878	219	4	is	be	AUX
ejpam-1878	219	5	a	a	DET
ejpam-1878	219	6	lattice	lattice	NOUN
ejpam-1878	219	7	.	.	PUNCT
ejpam-1878	220	1	proof	proof	NOUN
ejpam-1878	220	2	.	.	PUNCT
ejpam-1878	221	1	(	(	PUNCT
ejpam-1878	221	2	2)⇔	2)⇔	NUM
ejpam-1878	221	3	(	(	PUNCT
ejpam-1878	221	4	3	3	NUM
ejpam-1878	221	5	)	)	PUNCT
ejpam-1878	221	6	follows	follow	VERB
ejpam-1878	221	7	from	from	ADP
ejpam-1878	221	8	theorem	theorem	ADJ
ejpam-1878	221	9	8	8	NUM
ejpam-1878	221	10	(	(	PUNCT
ejpam-1878	221	11	1)⇒	1)⇒	NUM
ejpam-1878	221	12	(	(	PUNCT
ejpam-1878	221	13	2	2	NUM
ejpam-1878	221	14	)	)	PUNCT
ejpam-1878	221	15	:	:	PUNCT
ejpam-1878	221	16	let	let	VERB
ejpam-1878	221	17	α	α	PRON
ejpam-1878	221	18	:p	:p	PUNCT
ejpam-1878	221	19	→n	→n	PUNCT
ejpam-1878	221	20	∞	∞	NUM
ejpam-1878	221	21	be	be	AUX
ejpam-1878	221	22	defined	define	VERB
ejpam-1878	221	23	by	by	ADP
ejpam-1878	221	24	α(p	α(p	PROPN
ejpam-1878	221	25	)	)	PUNCT
ejpam-1878	222	1	=	=	SYM
ejpam-1878	222	2	∞	∞	NOUN
ejpam-1878	222	3	for	for	ADP
ejpam-1878	222	4	all	all	PRON
ejpam-1878	222	5	p	p	NOUN
ejpam-1878	222	6	∈	∈	PROPN
ejpam-1878	222	7	p	p	NOUN
ejpam-1878	222	8	.	.	PUNCT
ejpam-1878	223	1	then	then	ADV
ejpam-1878	223	2	iα	iα	VERB
ejpam-1878	223	3	=	=	PUNCT
ejpam-1878	223	4	{	{	PUNCT
ejpam-1878	223	5	n	n	X
ejpam-1878	223	6	∈	∈	PROPN
ejpam-1878	223	7	z	z	PROPN
ejpam-1878	224	1	+	+	NOUN
ejpam-1878	224	2	|θ	|θ	PRON
ejpam-1878	224	3	(	(	PUNCT
ejpam-1878	224	4	n)(p)≤pc	n)(p)≤pc	NUM
ejpam-1878	224	5	α(p	α(p	NUM
ejpam-1878	224	6	)	)	PUNCT
ejpam-1878	225	1	=	=	SYM
ejpam-1878	225	2	∞	∞	NOUN
ejpam-1878	225	3	for	for	ADP
ejpam-1878	225	4	all	all	DET
ejpam-1878	225	5	p	p	NOUN
ejpam-1878	225	6	∈	∈	PROPN
ejpam-1878	225	7	p	p	X
ejpam-1878	225	8	}	}	PUNCT
ejpam-1878	225	9	and	and	CCONJ
ejpam-1878	225	10	hence	hence	ADV
ejpam-1878	225	11	,	,	PUNCT
ejpam-1878	225	12	by	by	ADP
ejpam-1878	225	13	(	(	PUNCT
ejpam-1878	225	14	1	1	NUM
ejpam-1878	225	15	)	)	PUNCT
ejpam-1878	225	16	,	,	PUNCT
ejpam-1878	225	17	z+	z+	NUM
ejpam-1878	225	18	is	be	AUX
ejpam-1878	225	19	an	an	DET
ejpam-1878	225	20	ideal	ideal	NOUN
ejpam-1878	225	21	of	of	ADP
ejpam-1878	225	22	(	(	PUNCT
ejpam-1878	225	23	z+,≤c	z+,≤c	PROPN
ejpam-1878	225	24	)	)	PUNCT
ejpam-1878	225	25	which	which	PRON
ejpam-1878	225	26	implies	imply	VERB
ejpam-1878	225	27	that	that	SCONJ
ejpam-1878	225	28	(	(	PUNCT
ejpam-1878	225	29	z+,≤c	z+,≤c	NUM
ejpam-1878	225	30	)	)	PUNCT
ejpam-1878	225	31	is	be	AUX
ejpam-1878	225	32	directed	direct	VERB
ejpam-1878	225	33	above	above	ADV
ejpam-1878	225	34	.	.	PUNCT
ejpam-1878	226	1	(	(	PUNCT
ejpam-1878	226	2	3)⇒	3)⇒	NUM
ejpam-1878	226	3	(	(	PUNCT
ejpam-1878	226	4	1	1	NUM
ejpam-1878	226	5	)	)	PUNCT
ejpam-1878	226	6	:	:	PUNCT
ejpam-1878	226	7	from	from	ADP
ejpam-1878	226	8	(	(	PUNCT
ejpam-1878	226	9	3	3	NUM
ejpam-1878	226	10	)	)	PUNCT
ejpam-1878	226	11	and	and	CCONJ
ejpam-1878	226	12	theorems	theorem	NOUN
ejpam-1878	226	13	8	8	NUM
ejpam-1878	226	14	and	and	CCONJ
ejpam-1878	226	15	9	9	NUM
ejpam-1878	226	16	,	,	PUNCT
ejpam-1878	226	17	it	it	PRON
ejpam-1878	226	18	follows	follow	VERB
ejpam-1878	226	19	that	that	SCONJ
ejpam-1878	226	20	(	(	PUNCT
ejpam-1878	226	21	n	n	X
ejpam-1878	226	22	,	,	PUNCT
ejpam-1878	226	23	≤p	≤p	PROPN
ejpam-1878	226	24	c	c	PROPN
ejpam-1878	226	25	)	)	PUNCT
ejpam-1878	226	26	is	be	AUX
ejpam-1878	226	27	a	a	DET
ejpam-1878	226	28	lattice	lattice	NOUN
ejpam-1878	226	29	for	for	ADP
ejpam-1878	226	30	each	each	DET
ejpam-1878	226	31	p	p	NOUN
ejpam-1878	226	32	∈	∈	PROPN
ejpam-1878	226	33	p	p	PROPN
ejpam-1878	226	34	s.	s.	PROPN
ejpam-1878	226	35	sagi	sagi	PROPN
ejpam-1878	226	36	/	/	SYM
ejpam-1878	226	37	eur	eur	PROPN
ejpam-1878	226	38	.	.	PUNCT
ejpam-1878	227	1	j.	j.	PROPN
ejpam-1878	227	2	pure	pure	PROPN
ejpam-1878	227	3	appl	appl	PROPN
ejpam-1878	227	4	.	.	PROPN
ejpam-1878	227	5	math	math	PROPN
ejpam-1878	227	6	,	,	PUNCT
ejpam-1878	227	7	8	8	NUM
ejpam-1878	227	8	(	(	PUNCT
ejpam-1878	227	9	2015	2015	NUM
ejpam-1878	227	10	)	)	PUNCT
ejpam-1878	227	11	,	,	PUNCT
ejpam-1878	227	12	15	15	NUM
ejpam-1878	227	13	-	-	SYM
ejpam-1878	227	14	25	25	NUM
ejpam-1878	227	15	23	23	NUM
ejpam-1878	227	16	and	and	CCONJ
ejpam-1878	227	17	θ	θ	PROPN
ejpam-1878	227	18	(	(	PUNCT
ejpam-1878	227	19	m	m	NOUN
ejpam-1878	227	20	∨	∨	NOUN
ejpam-1878	227	21	n)(p	n)(p	X
ejpam-1878	227	22	)	)	PUNCT
ejpam-1878	227	23	=	=	SYM
ejpam-1878	227	24	θ	θ	PROPN
ejpam-1878	227	25	(	(	PUNCT
ejpam-1878	227	26	m)(p	m)(p	X
ejpam-1878	227	27	)	)	PUNCT
ejpam-1878	227	28	∨	∨	NUM
ejpam-1878	227	29	θ	θ	PROPN
ejpam-1878	227	30	(	(	PUNCT
ejpam-1878	227	31	n)(p	n)(p	NOUN
ejpam-1878	227	32	)	)	PUNCT
ejpam-1878	227	33	in	in	ADP
ejpam-1878	227	34	(	(	PUNCT
ejpam-1878	227	35	n	n	X
ejpam-1878	227	36	,	,	PUNCT
ejpam-1878	227	37	≤p	≤p	PROPN
ejpam-1878	227	38	c	c	PROPN
ejpam-1878	227	39	)	)	PUNCT
ejpam-1878	227	40	for	for	ADP
ejpam-1878	227	41	any	any	DET
ejpam-1878	227	42	m	m	NOUN
ejpam-1878	227	43	and	and	CCONJ
ejpam-1878	227	44	n	n	PRON
ejpam-1878	227	45	∈	∈	PROPN
ejpam-1878	227	46	z+	z+	NUM
ejpam-1878	227	47	and	and	CCONJ
ejpam-1878	227	48	p	p	NOUN
ejpam-1878	227	49	∈	∈	PROPN
ejpam-1878	227	50	p	p	X
ejpam-1878	227	51	.	.	PUNCT
ejpam-1878	228	1	let	let	VERB
ejpam-1878	228	2	α	α	PRON
ejpam-1878	228	3	:p	:p	PUNCT
ejpam-1878	228	4	→n	→n	PUNCT
ejpam-1878	228	5	∞	∞	NUM
ejpam-1878	228	6	be	be	AUX
ejpam-1878	228	7	any	any	DET
ejpam-1878	228	8	mapping	mapping	NOUN
ejpam-1878	228	9	.	.	PUNCT
ejpam-1878	229	1	then	then	ADV
ejpam-1878	229	2	,	,	PUNCT
ejpam-1878	229	3	for	for	ADP
ejpam-1878	229	4	any	any	DET
ejpam-1878	229	5	m	m	NOUN
ejpam-1878	229	6	and	and	CCONJ
ejpam-1878	229	7	n	n	PRON
ejpam-1878	229	8	∈	∈	PROPN
ejpam-1878	229	9	z+	z+	NUM
ejpam-1878	229	10	,	,	PUNCT
ejpam-1878	229	11	m≤c	m≤c	PROPN
ejpam-1878	229	12	n	n	CCONJ
ejpam-1878	229	13	∈	∈	PROPN
ejpam-1878	229	14	iα	iα	NOUN
ejpam-1878	229	15	=	=	NOUN
ejpam-1878	229	16	⇒θ	⇒θ	PROPN
ejpam-1878	229	17	(	(	PUNCT
ejpam-1878	229	18	m)(p)≤	m)(p)≤	PROPN
ejpam-1878	229	19	p	p	NOUN
ejpam-1878	229	20	c	c	NOUN
ejpam-1878	229	21	θ	θ	NOUN
ejpam-1878	229	22	(	(	PUNCT
ejpam-1878	229	23	n)(p)≤	n)(p)≤	NOUN
ejpam-1878	229	24	p	p	X
ejpam-1878	229	25	c	c	PROPN
ejpam-1878	229	26	α(p	α(p	PROPN
ejpam-1878	229	27	)	)	PUNCT
ejpam-1878	229	28	for	for	ADP
ejpam-1878	229	29	all	all	PRON
ejpam-1878	229	30	p	p	NOUN
ejpam-1878	229	31	∈	∈	PROPN
ejpam-1878	229	32	p	p	NOUN
ejpam-1878	229	33	.	.	PUNCT
ejpam-1878	230	1	=	=	PRON
ejpam-1878	230	2	⇒θ	⇒θ	NOUN
ejpam-1878	230	3	(	(	PUNCT
ejpam-1878	230	4	m)(p)≤p	m)(p)≤p	PROPN
ejpam-1878	230	5	c	c	NOUN
ejpam-1878	230	6	for	for	ADP
ejpam-1878	230	7	all	all	DET
ejpam-1878	230	8	p	p	NOUN
ejpam-1878	230	9	∈	∈	PROPN
ejpam-1878	230	10	p	p	NOUN
ejpam-1878	230	11	.	.	PUNCT
ejpam-1878	231	1	=	=	NOUN
ejpam-1878	231	2	⇒m	⇒m	NOUN
ejpam-1878	231	3	∈	∈	PROPN
ejpam-1878	231	4	iα	iα	NOUN
ejpam-1878	231	5	.	.	PROPN
ejpam-1878	231	6	and	and	CCONJ
ejpam-1878	231	7	m	m	PROPN
ejpam-1878	231	8	and	and	CCONJ
ejpam-1878	231	9	n	n	PRON
ejpam-1878	231	10	∈	∈	NOUN
ejpam-1878	231	11	iα	iα	NOUN
ejpam-1878	231	12	=	=	NOUN
ejpam-1878	231	13	⇒θ	⇒θ	PROPN
ejpam-1878	231	14	(	(	PUNCT
ejpam-1878	231	15	m)(p)≤	m)(p)≤	PROPN
ejpam-1878	231	16	p	p	PROPN
ejpam-1878	231	17	c	c	NOUN
ejpam-1878	231	18	α(p	α(p	PROPN
ejpam-1878	231	19	)	)	PUNCT
ejpam-1878	231	20	and	and	CCONJ
ejpam-1878	231	21	θ	θ	PROPN
ejpam-1878	231	22	(	(	PUNCT
ejpam-1878	231	23	n)(p)≤p	n)(p)≤p	PROPN
ejpam-1878	231	24	c	c	PROPN
ejpam-1878	231	25	α(p	α(p	PROPN
ejpam-1878	231	26	)	)	PUNCT
ejpam-1878	231	27	for	for	ADP
ejpam-1878	231	28	all	all	DET
ejpam-1878	231	29	p	p	NOUN
ejpam-1878	231	30	∈	∈	PROPN
ejpam-1878	231	31	p	p	NOUN
ejpam-1878	231	32	.	.	PUNCT
ejpam-1878	232	1	=	=	PRON
ejpam-1878	232	2	⇒θ	⇒θ	NOUN
ejpam-1878	232	3	(	(	PUNCT
ejpam-1878	232	4	m)(p)∨	m)(p)∨	PROPN
ejpam-1878	232	5	θ	θ	PROPN
ejpam-1878	232	6	(	(	PUNCT
ejpam-1878	232	7	n)(p)≤p	n)(p)≤p	PROPN
ejpam-1878	232	8	c	c	PROPN
ejpam-1878	232	9	α(p	α(p	PROPN
ejpam-1878	232	10	)	)	PUNCT
ejpam-1878	232	11	for	for	ADP
ejpam-1878	232	12	all	all	DET
ejpam-1878	232	13	p	p	NOUN
ejpam-1878	232	14	∈	∈	PROPN
ejpam-1878	232	15	p	p	NOUN
ejpam-1878	232	16	.	.	PUNCT
ejpam-1878	233	1	=	=	AUX
ejpam-1878	233	2	⇒m∨	⇒m∨	VERB
ejpam-1878	233	3	n	n	PRON
ejpam-1878	233	4	∈	∈	PROPN
ejpam-1878	233	5	iα	iα	NOUN
ejpam-1878	233	6	.	.	PUNCT
ejpam-1878	233	7	therefore	therefore	ADV
ejpam-1878	233	8	iα	iα	VERB
ejpam-1878	233	9	is	be	AUX
ejpam-1878	233	10	an	an	DET
ejpam-1878	233	11	ideal	ideal	NOUN
ejpam-1878	233	12	of	of	ADP
ejpam-1878	233	13	(	(	PUNCT
ejpam-1878	233	14	z+,≤c	z+,≤c	PROPN
ejpam-1878	233	15	)	)	PUNCT
ejpam-1878	233	16	.	.	PUNCT
ejpam-1878	234	1	now	now	ADV
ejpam-1878	234	2	,	,	PUNCT
ejpam-1878	234	3	we	we	PRON
ejpam-1878	234	4	have	have	VERB
ejpam-1878	234	5	the	the	DET
ejpam-1878	234	6	following	follow	VERB
ejpam-1878	234	7	theorems	theorem	NOUN
ejpam-1878	234	8	which	which	PRON
ejpam-1878	234	9	characterize	characterize	VERB
ejpam-1878	234	10	the	the	DET
ejpam-1878	234	11	prime	prime	ADJ
ejpam-1878	234	12	ideals	ideal	NOUN
ejpam-1878	234	13	of	of	ADP
ejpam-1878	234	14	the	the	DET
ejpam-1878	234	15	lattice	lattice	NOUN
ejpam-1878	234	16	(	(	PUNCT
ejpam-1878	234	17	z+,≤d	z+,≤d	NUM
ejpam-1878	234	18	)	)	PUNCT
ejpam-1878	234	19	where	where	SCONJ
ejpam-1878	234	20	d	d	NOUN
ejpam-1878	234	21	is	be	AUX
ejpam-1878	234	22	the	the	DET
ejpam-1878	234	23	dirichlet	dirichlet	PROPN
ejpam-1878	234	24	’s	’s	PART
ejpam-1878	234	25	convolution	convolution	NOUN
ejpam-1878	234	26	.	.	PUNCT
ejpam-1878	235	1	theorem	theorem	VERB
ejpam-1878	235	2	11	11	NUM
ejpam-1878	235	3	.	.	PUNCT
ejpam-1878	236	1	let	let	VERB
ejpam-1878	236	2	α	α	PRON
ejpam-1878	236	3	:	:	PUNCT
ejpam-1878	236	4	p	p	X
ejpam-1878	236	5	→	→	SYM
ejpam-1878	236	6	n	n	CCONJ
ejpam-1878	236	7	∞	∞	PROPN
ejpam-1878	236	8	be	be	AUX
ejpam-1878	236	9	a	a	DET
ejpam-1878	236	10	mapping	mapping	NOUN
ejpam-1878	236	11	and	and	CCONJ
ejpam-1878	236	12	iα	iα	NOUN
ejpam-1878	236	13	is	be	AUX
ejpam-1878	236	14	an	an	DET
ejpam-1878	236	15	ideal	ideal	NOUN
ejpam-1878	236	16	of	of	ADP
ejpam-1878	236	17	(	(	PUNCT
ejpam-1878	236	18	z+,≤d	z+,≤d	NOUN
ejpam-1878	236	19	)	)	PUNCT
ejpam-1878	236	20	defined	define	VERB
ejpam-1878	236	21	by	by	ADP
ejpam-1878	236	22	iα	iα	NOUN
ejpam-1878	236	23	=	=	PUNCT
ejpam-1878	236	24	{	{	PUNCT
ejpam-1878	236	25	n	n	X
ejpam-1878	236	26	∈	∈	PROPN
ejpam-1878	236	27	z	z	PROPN
ejpam-1878	236	28	+	+	PROPN
ejpam-1878	236	29	|θ	|θ	PRON
ejpam-1878	236	30	(	(	PUNCT
ejpam-1878	236	31	n)(p)≤	n)(p)≤	PROPN
ejpam-1878	236	32	α(p	α(p	PROPN
ejpam-1878	236	33	)	)	PUNCT
ejpam-1878	236	34	for	for	ADP
ejpam-1878	236	35	all	all	DET
ejpam-1878	236	36	p	p	NOUN
ejpam-1878	236	37	∈	∈	PROPN
ejpam-1878	236	38	p	p	X
ejpam-1878	236	39	}	}	PUNCT
ejpam-1878	236	40	.	.	PUNCT
ejpam-1878	237	1	then	then	ADV
ejpam-1878	237	2	the	the	DET
ejpam-1878	237	3	following	follow	VERB
ejpam-1878	237	4	are	be	AUX
ejpam-1878	237	5	equivalent	equivalent	ADJ
ejpam-1878	237	6	to	to	ADP
ejpam-1878	237	7	each	each	DET
ejpam-1878	237	8	other	other	ADJ
ejpam-1878	237	9	.	.	PUNCT
ejpam-1878	238	1	(	(	PUNCT
ejpam-1878	238	2	1	1	NUM
ejpam-1878	238	3	)	)	PUNCT
ejpam-1878	238	4	.	.	PUNCT
ejpam-1878	239	1	iα	iα	PROPN
ejpam-1878	239	2	is	be	AUX
ejpam-1878	239	3	a	a	DET
ejpam-1878	239	4	prime	prime	ADJ
ejpam-1878	239	5	ideal	ideal	NOUN
ejpam-1878	239	6	of	of	ADP
ejpam-1878	239	7	(	(	PUNCT
ejpam-1878	239	8	z+,≤d	z+,≤d	NUM
ejpam-1878	239	9	)	)	PUNCT
ejpam-1878	239	10	.	.	PUNCT
ejpam-1878	240	1	(	(	PUNCT
ejpam-1878	240	2	2	2	NUM
ejpam-1878	240	3	)	)	PUNCT
ejpam-1878	240	4	.	.	PUNCT
ejpam-1878	241	1	α(p	α(p	X
ejpam-1878	241	2	)	)	PUNCT
ejpam-1878	242	1	6=∞	6=∞	NUM
ejpam-1878	242	2	for	for	ADP
ejpam-1878	242	3	some	some	DET
ejpam-1878	242	4	p	p	NOUN
ejpam-1878	242	5	∈	∈	PROPN
ejpam-1878	242	6	p	p	NOUN
ejpam-1878	242	7	and	and	CCONJ
ejpam-1878	242	8	for	for	ADP
ejpam-1878	242	9	any	any	DET
ejpam-1878	242	10	β	β	NOUN
ejpam-1878	242	11	and	and	CCONJ
ejpam-1878	242	12	γ	γ	X
ejpam-1878	242	13	:p	:p	PROPN
ejpam-1878	242	14	−→n	−→n	VERB
ejpam-1878	242	15	∞	∞	PROPN
ejpam-1878	242	16	,	,	PUNCT
ejpam-1878	242	17	β	β	X
ejpam-1878	242	18	∧	∧	NOUN
ejpam-1878	242	19	γ≤	γ≤	NUM
ejpam-1878	242	20	α=⇒	α=⇒	NOUN
ejpam-1878	242	21	β	β	NOUN
ejpam-1878	242	22	≤	≤	ADJ
ejpam-1878	242	23	α	α	NOUN
ejpam-1878	242	24	or	or	CCONJ
ejpam-1878	242	25	γ≤	γ≤	NUM
ejpam-1878	242	26	α	α	NOUN
ejpam-1878	242	27	.	.	PUNCT
ejpam-1878	243	1	(	(	PUNCT
ejpam-1878	243	2	3	3	NUM
ejpam-1878	243	3	)	)	PUNCT
ejpam-1878	243	4	.	.	PUNCT
ejpam-1878	244	1	there	there	PRON
ejpam-1878	244	2	exists	exist	VERB
ejpam-1878	244	3	unique	unique	ADJ
ejpam-1878	244	4	p	p	NOUN
ejpam-1878	244	5	∈	∈	PROPN
ejpam-1878	244	6	p	p	NOUN
ejpam-1878	244	7	such	such	ADJ
ejpam-1878	244	8	that	that	PRON
ejpam-1878	244	9	α(p	α(p	PROPN
ejpam-1878	244	10	)	)	PUNCT
ejpam-1878	244	11	6=∞	6=∞	NUM
ejpam-1878	244	12	and	and	CCONJ
ejpam-1878	244	13	α(q	α(q	NUM
ejpam-1878	244	14	)	)	PUNCT
ejpam-1878	244	15	=	=	SYM
ejpam-1878	244	16	∞	∞	NOUN
ejpam-1878	244	17	for	for	ADP
ejpam-1878	244	18	all	all	DET
ejpam-1878	244	19	q	q	NOUN
ejpam-1878	245	1	6=	6=	PUNCT
ejpam-1878	245	2	p	p	X
ejpam-1878	245	3	∈	∈	PROPN
ejpam-1878	245	4	p	p	NOUN
ejpam-1878	245	5	.	.	PUNCT
ejpam-1878	246	1	proof	proof	NOUN
ejpam-1878	246	2	.	.	PUNCT
ejpam-1878	247	1	(	(	PUNCT
ejpam-1878	247	2	1	1	X
ejpam-1878	247	3	)	)	PUNCT
ejpam-1878	247	4	=	=	NOUN
ejpam-1878	247	5	⇒	⇒	NOUN
ejpam-1878	247	6	(	(	PUNCT
ejpam-1878	247	7	2	2	X
ejpam-1878	247	8	)	)	PUNCT
ejpam-1878	247	9	follows	follow	VERB
ejpam-1878	247	10	from	from	ADP
ejpam-1878	247	11	theorem	theorem	ADJ
ejpam-1878	247	12	7	7	NUM
ejpam-1878	247	13	,	,	PUNCT
ejpam-1878	247	14	in	in	ADP
ejpam-1878	247	15	which	which	PRON
ejpam-1878	247	16	we	we	PRON
ejpam-1878	247	17	have	have	AUX
ejpam-1878	247	18	proved	prove	VERB
ejpam-1878	247	19	that	that	SCONJ
ejpam-1878	247	20	β	β	X
ejpam-1878	247	21	7→	7→	NUM
ejpam-1878	247	22	iβ	iβ	X
ejpam-1878	247	23	is	be	AUX
ejpam-1878	247	24	an	an	DET
ejpam-1878	247	25	isomorphism	isomorphism	NOUN
ejpam-1878	247	26	of	of	ADP
ejpam-1878	247	27	the	the	DET
ejpam-1878	247	28	lattice	lattice	NOUN
ejpam-1878	247	29	(	(	PUNCT
ejpam-1878	247	30	(	(	PUNCT
ejpam-1878	247	31	n	n	X
ejpam-1878	247	32	∞)p	∞)p	PRON
ejpam-1878	247	33	,	,	PUNCT
ejpam-1878	247	34	≤	≤	NUM
ejpam-1878	247	35	)	)	PUNCT
ejpam-1878	247	36	onto	onto	ADP
ejpam-1878	247	37	the	the	DET
ejpam-1878	247	38	lattice	lattice	NOUN
ejpam-1878	247	39	of	of	ADP
ejpam-1878	247	40	ideals	ideal	NOUN
ejpam-1878	247	41	of	of	ADP
ejpam-1878	247	42	(	(	PUNCT
ejpam-1878	247	43	z+,≤d	z+,≤d	NOUN
ejpam-1878	247	44	)	)	PUNCT
ejpam-1878	247	45	from	from	ADP
ejpam-1878	247	46	the	the	DET
ejpam-1878	247	47	fact	fact	NOUN
ejpam-1878	248	1	that	that	SCONJ
ejpam-1878	248	2	iβ	iβ	ADP
ejpam-1878	248	3	∩	∩	NOUN
ejpam-1878	248	4	iγ	iγ	NOUN
ejpam-1878	248	5	=	=	SYM
ejpam-1878	248	6	iβ∧γ	iβ∧γ	NOUN
ejpam-1878	248	7	for	for	ADP
ejpam-1878	248	8	any	any	DET
ejpam-1878	248	9	β	β	NOUN
ejpam-1878	248	10	and	and	CCONJ
ejpam-1878	248	11	γ	γ	X
ejpam-1878	248	12	:	:	PUNCT
ejpam-1878	248	13	p	p	X
ejpam-1878	248	14	−→	−→	NOUN
ejpam-1878	248	15	n	n	PRON
ejpam-1878	248	16	∞.	∞.	PROPN
ejpam-1878	248	17	if	if	SCONJ
ejpam-1878	248	18	α(p	α(p	PROPN
ejpam-1878	248	19	)	)	PUNCT
ejpam-1878	248	20	=	=	SYM
ejpam-1878	248	21	∞	∞	NOUN
ejpam-1878	248	22	for	for	ADP
ejpam-1878	248	23	all	all	PRON
ejpam-1878	248	24	p	p	NOUN
ejpam-1878	248	25	∈	∈	PROPN
ejpam-1878	248	26	p	p	NOUN
ejpam-1878	248	27	,	,	PUNCT
ejpam-1878	248	28	then	then	ADV
ejpam-1878	248	29	,	,	PUNCT
ejpam-1878	248	30	since	since	SCONJ
ejpam-1878	248	31	θ	θ	PROPN
ejpam-1878	248	32	(	(	PUNCT
ejpam-1878	248	33	n)(p	n)(p	X
ejpam-1878	248	34	)	)	PUNCT
ejpam-1878	248	35	∈	∈	PROPN
ejpam-1878	248	36	n	n	X
ejpam-1878	248	37	for	for	ADP
ejpam-1878	248	38	all	all	DET
ejpam-1878	248	39	n	n	PRON
ejpam-1878	248	40	∈	∈	NOUN
ejpam-1878	248	41	z+	z+	NUM
ejpam-1878	248	42	and	and	CCONJ
ejpam-1878	248	43	p	p	NOUN
ejpam-1878	248	44	∈	∈	PROPN
ejpam-1878	248	45	p	p	NOUN
ejpam-1878	248	46	,	,	PUNCT
ejpam-1878	248	47	iα	iα	INTJ
ejpam-1878	248	48	=	=	PUNCT
ejpam-1878	248	49	{	{	PUNCT
ejpam-1878	248	50	n	n	X
ejpam-1878	248	51	∈	∈	PROPN
ejpam-1878	248	52	z	z	PROPN
ejpam-1878	248	53	+	+	PROPN
ejpam-1878	248	54	|θ	|θ	NUM
ejpam-1878	248	55	(	(	PUNCT
ejpam-1878	248	56	n)(p)<∞}=	n)(p)<∞}=	X
ejpam-1878	248	57	z+	z+	NUM
ejpam-1878	248	58	which	which	PRON
ejpam-1878	248	59	is	be	AUX
ejpam-1878	248	60	a	a	DET
ejpam-1878	248	61	contradiction	contradiction	NOUN
ejpam-1878	248	62	to	to	ADP
ejpam-1878	248	63	the	the	DET
ejpam-1878	248	64	fact	fact	NOUN
ejpam-1878	248	65	that	that	SCONJ
ejpam-1878	248	66	every	every	DET
ejpam-1878	248	67	prime	prime	ADJ
ejpam-1878	248	68	ideal	ideal	NOUN
ejpam-1878	248	69	is	be	AUX
ejpam-1878	248	70	a	a	DET
ejpam-1878	248	71	proper	proper	ADJ
ejpam-1878	248	72	ideal	ideal	NOUN
ejpam-1878	248	73	.	.	PUNCT
ejpam-1878	249	1	thus	thus	ADV
ejpam-1878	249	2	α(p	α(p	X
ejpam-1878	249	3	)	)	PUNCT
ejpam-1878	250	1	6=∞	6=∞	NUM
ejpam-1878	250	2	for	for	ADP
ejpam-1878	250	3	some	some	DET
ejpam-1878	250	4	p	p	NOUN
ejpam-1878	250	5	∈	∈	PROPN
ejpam-1878	250	6	p	p	NOUN
ejpam-1878	250	7	.	.	PUNCT
ejpam-1878	251	1	(	(	PUNCT
ejpam-1878	251	2	2	2	X
ejpam-1878	251	3	)	)	PUNCT
ejpam-1878	251	4	=	=	NOUN
ejpam-1878	251	5	⇒	⇒	NOUN
ejpam-1878	251	6	(	(	PUNCT
ejpam-1878	251	7	3	3	NUM
ejpam-1878	251	8	):	):	PUNCT
ejpam-1878	251	9	suppose	suppose	VERB
ejpam-1878	251	10	that	that	SCONJ
ejpam-1878	251	11	α	α	PROPN
ejpam-1878	251	12	satisfies	satisfie	NOUN
ejpam-1878	251	13	(	(	PUNCT
ejpam-1878	251	14	2	2	NUM
ejpam-1878	251	15	)	)	PUNCT
ejpam-1878	251	16	.	.	PUNCT
ejpam-1878	252	1	fix	fix	VERB
ejpam-1878	252	2	p	p	NOUN
ejpam-1878	252	3	∈	∈	NOUN
ejpam-1878	252	4	p	p	NOUN
ejpam-1878	252	5	such	such	ADJ
ejpam-1878	252	6	that	that	PRON
ejpam-1878	252	7	α(p	α(p	PROPN
ejpam-1878	252	8	)	)	PUNCT
ejpam-1878	253	1	6=∞.	6=∞.	ADV
ejpam-1878	253	2	then	then	ADV
ejpam-1878	253	3	α(p	α(p	NUM
ejpam-1878	253	4	)	)	PUNCT
ejpam-1878	253	5	∈	∈	PROPN
ejpam-1878	253	6	n	n	NOUN
ejpam-1878	253	7	.	.	PUNCT
ejpam-1878	254	1	now	now	ADV
ejpam-1878	254	2	,	,	PUNCT
ejpam-1878	254	3	define	define	VERB
ejpam-1878	254	4	β	β	PROPN
ejpam-1878	254	5	and	and	CCONJ
ejpam-1878	254	6	γ	γ	X
ejpam-1878	254	7	:p	:p	PROPN
ejpam-1878	254	8	−→n	−→n	VERB
ejpam-1878	254	9	∞	∞	NUM
ejpam-1878	254	10	by	by	ADP
ejpam-1878	254	11	β(q	β(q	PROPN
ejpam-1878	254	12	)	)	PUNCT
ejpam-1878	255	1	=	=	PUNCT
ejpam-1878	256	1	¨	¨	NOUN
ejpam-1878	256	2	0	0	PUNCT
ejpam-1878	257	1	if	if	SCONJ
ejpam-1878	257	2	q	q	NOUN
ejpam-1878	257	3	=	=	PUNCT
ejpam-1878	257	4	p	p	NOUN
ejpam-1878	257	5	∞	∞	PROPN
ejpam-1878	257	6	if	if	SCONJ
ejpam-1878	257	7	q	q	PROPN
ejpam-1878	257	8	6=	6=	ADP
ejpam-1878	257	9	p	p	PROPN
ejpam-1878	257	10	s.	s.	PROPN
ejpam-1878	257	11	sagi	sagi	PROPN
ejpam-1878	257	12	/	/	SYM
ejpam-1878	257	13	eur	eur	PROPN
ejpam-1878	257	14	.	.	PUNCT
ejpam-1878	258	1	j.	j.	PROPN
ejpam-1878	258	2	pure	pure	PROPN
ejpam-1878	258	3	appl	appl	PROPN
ejpam-1878	258	4	.	.	PROPN
ejpam-1878	258	5	math	math	PROPN
ejpam-1878	258	6	,	,	PUNCT
ejpam-1878	258	7	8	8	NUM
ejpam-1878	258	8	(	(	PUNCT
ejpam-1878	258	9	2015	2015	NUM
ejpam-1878	258	10	)	)	PUNCT
ejpam-1878	258	11	,	,	PUNCT
ejpam-1878	258	12	15	15	NUM
ejpam-1878	258	13	-	-	SYM
ejpam-1878	258	14	25	25	NUM
ejpam-1878	258	15	24	24	NUM
ejpam-1878	258	16	and	and	CCONJ
ejpam-1878	258	17	γ(q	γ(q	PROPN
ejpam-1878	258	18	)	)	PUNCT
ejpam-1878	258	19	=	=	PUNCT
ejpam-1878	259	1	¨	¨	NOUN
ejpam-1878	259	2	∞	∞	NUM
ejpam-1878	259	3	if	if	SCONJ
ejpam-1878	259	4	q	q	PROPN
ejpam-1878	259	5	=	=	SYM
ejpam-1878	259	6	p	p	NOUN
ejpam-1878	259	7	0	0	PUNCT
ejpam-1878	259	8	if	if	SCONJ
ejpam-1878	259	9	q	q	PROPN
ejpam-1878	259	10	6=	6=	ADP
ejpam-1878	259	11	p	p	NOUN
ejpam-1878	259	12	for	for	ADP
ejpam-1878	259	13	any	any	DET
ejpam-1878	259	14	q	q	NOUN
ejpam-1878	259	15	∈	∈	PROPN
ejpam-1878	259	16	p	p	NOUN
ejpam-1878	259	17	.	.	PUNCT
ejpam-1878	260	1	then	then	ADV
ejpam-1878	260	2	,	,	PUNCT
ejpam-1878	260	3	(	(	PUNCT
ejpam-1878	260	4	β	β	X
ejpam-1878	260	5	∧	∧	NOUN
ejpam-1878	260	6	γ)(q	γ)(q	PROPN
ejpam-1878	260	7	)	)	PUNCT
ejpam-1878	261	1	=	=	SYM
ejpam-1878	261	2	β(q)∧	β(q)∧	PROPN
ejpam-1878	261	3	γ(q	γ(q	PROPN
ejpam-1878	261	4	)	)	PUNCT
ejpam-1878	261	5	=	=	SYM
ejpam-1878	261	6	0≤	0≤	NUM
ejpam-1878	261	7	α(q	α(q	NUM
ejpam-1878	261	8	)	)	PUNCT
ejpam-1878	261	9	for	for	ADP
ejpam-1878	261	10	all	all	DET
ejpam-1878	261	11	q	q	PROPN
ejpam-1878	261	12	∈	∈	PROPN
ejpam-1878	261	13	p	p	NOUN
ejpam-1878	261	14	and	and	CCONJ
ejpam-1878	261	15	hence	hence	ADV
ejpam-1878	261	16	β	β	X
ejpam-1878	261	17	∧γ≤	∧γ≤	ADP
ejpam-1878	261	18	α	α	X
ejpam-1878	261	19	.	.	PUNCT
ejpam-1878	262	1	since	since	SCONJ
ejpam-1878	262	2	α(p	α(p	PROPN
ejpam-1878	262	3	)	)	PUNCT
ejpam-1878	262	4	6=∞	6=∞	NUM
ejpam-1878	262	5	and	and	CCONJ
ejpam-1878	262	6	γ(p	γ(p	PROPN
ejpam-1878	262	7	)	)	PUNCT
ejpam-1878	262	8	=	=	SYM
ejpam-1878	262	9	∞	∞	PROPN
ejpam-1878	262	10	,	,	PUNCT
ejpam-1878	262	11	(	(	PUNCT
ejpam-1878	262	12	γ)(p	γ)(p	VERB
ejpam-1878	262	13	)	)	PUNCT
ejpam-1878	262	14	6≤	6≤	NUM
ejpam-1878	262	15	α(p	α(p	PROPN
ejpam-1878	262	16	)	)	PUNCT
ejpam-1878	262	17	and	and	CCONJ
ejpam-1878	262	18	hence	hence	ADV
ejpam-1878	262	19	γ	γ	X
ejpam-1878	262	20	6≤	6≤	NUM
ejpam-1878	262	21	α	α	NOUN
ejpam-1878	262	22	.	.	PUNCT
ejpam-1878	263	1	therefore	therefore	ADV
ejpam-1878	263	2	,	,	PUNCT
ejpam-1878	263	3	by	by	ADP
ejpam-1878	263	4	(	(	PUNCT
ejpam-1878	263	5	2	2	NUM
ejpam-1878	263	6	)	)	PUNCT
ejpam-1878	263	7	,	,	PUNCT
ejpam-1878	263	8	β	β	X
ejpam-1878	263	9	≤	≤	ADJ
ejpam-1878	263	10	α	α	NOUN
ejpam-1878	263	11	and	and	CCONJ
ejpam-1878	263	12	hence	hence	ADV
ejpam-1878	263	13	∞	∞	NUM
ejpam-1878	263	14	=	=	SYM
ejpam-1878	263	15	β(q)≤	β(q)≤	NOUN
ejpam-1878	263	16	α(q	α(q	NUM
ejpam-1878	263	17	)	)	PUNCT
ejpam-1878	263	18	for	for	ADP
ejpam-1878	263	19	all	all	DET
ejpam-1878	263	20	q	q	NOUN
ejpam-1878	263	21	6=	6=	NUM
ejpam-1878	264	1	p.	p.	NOUN
ejpam-1878	264	2	therefore	therefore	ADV
ejpam-1878	264	3	q(p	q(p	PROPN
ejpam-1878	264	4	)	)	PUNCT
ejpam-1878	265	1	=	=	SYM
ejpam-1878	265	2	∞	∞	NOUN
ejpam-1878	265	3	for	for	ADP
ejpam-1878	265	4	all	all	DET
ejpam-1878	265	5	q	q	NOUN
ejpam-1878	265	6	6=	6=	ADP
ejpam-1878	265	7	p	p	NOUN
ejpam-1878	265	8	in	in	ADP
ejpam-1878	265	9	p	p	PROPN
ejpam-1878	265	10	.	.	PUNCT
ejpam-1878	266	1	this	this	PRON
ejpam-1878	266	2	also	also	ADV
ejpam-1878	266	3	implies	imply	VERB
ejpam-1878	266	4	the	the	DET
ejpam-1878	266	5	uniqueness	uniqueness	NOUN
ejpam-1878	266	6	of	of	ADP
ejpam-1878	266	7	p.	p.	NOUN
ejpam-1878	266	8	(	(	PUNCT
ejpam-1878	266	9	3	3	X
ejpam-1878	266	10	)	)	PUNCT
ejpam-1878	267	1	=	=	NOUN
ejpam-1878	267	2	⇒	⇒	NOUN
ejpam-1878	267	3	(	(	PUNCT
ejpam-1878	267	4	1	1	NUM
ejpam-1878	267	5	):	):	PUNCT
ejpam-1878	267	6	let	let	VERB
ejpam-1878	267	7	p	p	PRON
ejpam-1878	267	8	∈	∈	PROPN
ejpam-1878	267	9	p	p	NOUN
ejpam-1878	267	10	such	such	ADJ
ejpam-1878	267	11	that	that	PRON
ejpam-1878	267	12	α(p	α(p	PROPN
ejpam-1878	267	13	)	)	PUNCT
ejpam-1878	267	14	6=∞	6=∞	NUM
ejpam-1878	267	15	and	and	CCONJ
ejpam-1878	267	16	α(q	α(q	NUM
ejpam-1878	267	17	)	)	PUNCT
ejpam-1878	268	1	=	=	SYM
ejpam-1878	268	2	∞	∞	NOUN
ejpam-1878	268	3	for	for	ADP
ejpam-1878	268	4	all	all	DET
ejpam-1878	268	5	q	q	NOUN
ejpam-1878	268	6	6=	6=	PUNCT
ejpam-1878	268	7	p	p	X
ejpam-1878	268	8	∈	∈	PROPN
ejpam-1878	268	9	p	p	NOUN
ejpam-1878	268	10	.	.	PUNCT
ejpam-1878	269	1	then	then	ADV
ejpam-1878	269	2	iα	iα	VERB
ejpam-1878	269	3	is	be	AUX
ejpam-1878	269	4	a	a	DET
ejpam-1878	269	5	proper	proper	ADJ
ejpam-1878	269	6	ideal	ideal	NOUN
ejpam-1878	269	7	of	of	ADP
ejpam-1878	269	8	(	(	PUNCT
ejpam-1878	269	9	z+,≤d	z+,≤d	NUM
ejpam-1878	269	10	)	)	PUNCT
ejpam-1878	269	11	.	.	PUNCT
ejpam-1878	270	1	let	let	VERB
ejpam-1878	270	2	j	j	PROPN
ejpam-1878	270	3	and	and	CCONJ
ejpam-1878	270	4	k	k	PROPN
ejpam-1878	270	5	be	be	VERB
ejpam-1878	270	6	any	any	DET
ejpam-1878	270	7	ideals	ideal	NOUN
ejpam-1878	270	8	of	of	ADP
ejpam-1878	270	9	(	(	PUNCT
ejpam-1878	270	10	z+,≤d	z+,≤d	NOUN
ejpam-1878	270	11	)	)	PUNCT
ejpam-1878	270	12	such	such	ADJ
ejpam-1878	270	13	that	that	SCONJ
ejpam-1878	270	14	j	j	PROPN
ejpam-1878	270	15	∩	∩	NOUN
ejpam-1878	270	16	k	k	PROPN
ejpam-1878	270	17	⊆	⊆	NUM
ejpam-1878	270	18	iα	iα	NOUN
ejpam-1878	270	19	.	.	PUNCT
ejpam-1878	271	1	then	then	ADV
ejpam-1878	271	2	there	there	PRON
ejpam-1878	271	3	exists	exist	VERB
ejpam-1878	271	4	β	β	PROPN
ejpam-1878	271	5	and	and	CCONJ
ejpam-1878	271	6	γ	γ	X
ejpam-1878	271	7	:	:	PUNCT
ejpam-1878	271	8	p	p	X
ejpam-1878	271	9	−→	−→	NOUN
ejpam-1878	271	10	n	n	CCONJ
ejpam-1878	271	11	∞	∞	NUM
ejpam-1878	271	12	such	such	ADJ
ejpam-1878	271	13	that	that	PRON
ejpam-1878	271	14	j	j	PROPN
ejpam-1878	271	15	=	=	PUNCT
ejpam-1878	271	16	iβ	iβ	NOUN
ejpam-1878	271	17	and	and	CCONJ
ejpam-1878	271	18	k	k	PROPN
ejpam-1878	271	19	=	=	NOUN
ejpam-1878	271	20	iγ	iγ	PROPN
ejpam-1878	271	21	.	.	PUNCT
ejpam-1878	272	1	now	now	ADV
ejpam-1878	272	2	,	,	PUNCT
ejpam-1878	272	3	iβ∧γ	iβ∧γ	ADV
ejpam-1878	272	4	=	=	SYM
ejpam-1878	272	5	iβ	iβ	ADP
ejpam-1878	272	6	∩	∩	ADJ
ejpam-1878	272	7	iγ	iγ	NOUN
ejpam-1878	272	8	=	=	SYM
ejpam-1878	272	9	j	j	PROPN
ejpam-1878	272	10	∩	∩	NOUN
ejpam-1878	272	11	k	k	PROPN
ejpam-1878	272	12	⊆	⊆	NUM
ejpam-1878	272	13	iα	iα	NOUN
ejpam-1878	272	14	and	and	CCONJ
ejpam-1878	272	15	hence	hence	ADV
ejpam-1878	272	16	β	β	X
ejpam-1878	272	17	∧	∧	NOUN
ejpam-1878	272	18	γ≤	γ≤	NUM
ejpam-1878	272	19	α	α	NOUN
ejpam-1878	272	20	so	so	SCONJ
ejpam-1878	272	21	that	that	SCONJ
ejpam-1878	272	22	min{β(p),γ(p	min{β(p),γ(p	NOUN
ejpam-1878	272	23	)	)	PUNCT
ejpam-1878	272	24	}	}	PUNCT
ejpam-1878	272	25	=	=	SYM
ejpam-1878	273	1	(	(	PUNCT
ejpam-1878	273	2	β	β	X
ejpam-1878	273	3	∧	∧	PROPN
ejpam-1878	273	4	γ)(p)≤	γ)(p)≤	PROPN
ejpam-1878	273	5	α(p	α(p	PROPN
ejpam-1878	273	6	)	)	PUNCT
ejpam-1878	273	7	.	.	PUNCT
ejpam-1878	274	1	therefore	therefore	ADV
ejpam-1878	274	2	β(p)≤	β(p)≤	X
ejpam-1878	274	3	α(p	α(p	PROPN
ejpam-1878	274	4	)	)	PUNCT
ejpam-1878	274	5	or	or	CCONJ
ejpam-1878	274	6	γ(p)≤	γ(p)≤	NOUN
ejpam-1878	274	7	α(p	α(p	PROPN
ejpam-1878	274	8	)	)	PUNCT
ejpam-1878	274	9	.	.	PUNCT
ejpam-1878	275	1	since	since	SCONJ
ejpam-1878	275	2	α(q	α(q	NUM
ejpam-1878	275	3	)	)	PUNCT
ejpam-1878	275	4	=	=	SYM
ejpam-1878	275	5	∞	∞	NOUN
ejpam-1878	275	6	for	for	ADP
ejpam-1878	275	7	all	all	DET
ejpam-1878	275	8	q	q	NOUN
ejpam-1878	276	1	6=	6=	ADP
ejpam-1878	276	2	p	p	X
ejpam-1878	276	3	,	,	PUNCT
ejpam-1878	276	4	it	it	PRON
ejpam-1878	276	5	follows	follow	VERB
ejpam-1878	276	6	that	that	SCONJ
ejpam-1878	276	7	β	β	VERB
ejpam-1878	276	8	≤	≤	ADJ
ejpam-1878	276	9	α	α	NOUN
ejpam-1878	276	10	or	or	CCONJ
ejpam-1878	276	11	γ	γ	X
ejpam-1878	276	12	≤	≤	NUM
ejpam-1878	276	13	α	α	NOUN
ejpam-1878	276	14	and	and	CCONJ
ejpam-1878	276	15	hence	hence	ADV
ejpam-1878	276	16	iβ	iβ	ADP
ejpam-1878	276	17	⊆	⊆	NUM
ejpam-1878	276	18	iα	iα	NOUN
ejpam-1878	276	19	or	or	CCONJ
ejpam-1878	276	20	iγ	iγ	VERB
ejpam-1878	276	21	⊆	⊆	NUM
ejpam-1878	276	22	iα	iα	NOUN
ejpam-1878	276	23	.	.	PUNCT
ejpam-1878	277	1	therefore	therefore	ADV
ejpam-1878	277	2	j	j	PROPN
ejpam-1878	277	3	⊆	⊆	NUM
ejpam-1878	277	4	iα	iα	NOUN
ejpam-1878	277	5	or	or	CCONJ
ejpam-1878	277	6	k	k	PROPN
ejpam-1878	277	7	⊆	⊆	NUM
ejpam-1878	277	8	iα	iα	NOUN
ejpam-1878	277	9	.	.	PUNCT
ejpam-1878	277	10	thus	thus	ADV
ejpam-1878	277	11	iα	iα	VERB
ejpam-1878	277	12	is	be	AUX
ejpam-1878	277	13	a	a	DET
ejpam-1878	277	14	prime	prime	ADJ
ejpam-1878	277	15	ideal	ideal	NOUN
ejpam-1878	277	16	of	of	ADP
ejpam-1878	277	17	(	(	PUNCT
ejpam-1878	277	18	z+,≤d	z+,≤d	NUM
ejpam-1878	277	19	)	)	PUNCT
ejpam-1878	277	20	.	.	PUNCT
ejpam-1878	278	1	definition	definition	NOUN
ejpam-1878	278	2	11	11	NUM
ejpam-1878	278	3	.	.	PUNCT
ejpam-1878	279	1	for	for	ADP
ejpam-1878	279	2	any	any	DET
ejpam-1878	279	3	prime	prime	ADJ
ejpam-1878	279	4	number	number	NOUN
ejpam-1878	279	5	p	p	NOUN
ejpam-1878	279	6	and	and	CCONJ
ejpam-1878	279	7	a	a	DET
ejpam-1878	279	8	∈	∈	PROPN
ejpam-1878	279	9	n	n	CCONJ
ejpam-1878	279	10	,	,	PUNCT
ejpam-1878	279	11	define	define	VERB
ejpam-1878	279	12	ip	ip	NOUN
ejpam-1878	279	13	,	,	PUNCT
ejpam-1878	279	14	a	a	PRON
ejpam-1878	279	15	=	=	X
ejpam-1878	279	16	{	{	PUNCT
ejpam-1878	279	17	n	n	NOUN
ejpam-1878	279	18	∈	∈	PROPN
ejpam-1878	279	19	z	z	PROPN
ejpam-1878	280	1	+	+	PROPN
ejpam-1878	280	2	|θ	|θ	PRON
ejpam-1878	280	3	(	(	PUNCT
ejpam-1878	280	4	n)(p)≤	n)(p)≤	VERB
ejpam-1878	280	5	a	a	X
ejpam-1878	280	6	}	}	PUNCT
ejpam-1878	280	7	.	.	PUNCT
ejpam-1878	281	1	then	then	ADV
ejpam-1878	281	2	ip	ip	X
ejpam-1878	281	3	,	,	PUNCT
ejpam-1878	281	4	a	a	PRON
ejpam-1878	281	5	is	be	AUX
ejpam-1878	281	6	an	an	DET
ejpam-1878	281	7	ideal	ideal	NOUN
ejpam-1878	281	8	of	of	ADP
ejpam-1878	281	9	(	(	PUNCT
ejpam-1878	281	10	z+,≤d	z+,≤d	NUM
ejpam-1878	281	11	)	)	PUNCT
ejpam-1878	281	12	.	.	PUNCT
ejpam-1878	282	1	in	in	ADP
ejpam-1878	282	2	fact	fact	NOUN
ejpam-1878	282	3	ip	ip	ADP
ejpam-1878	282	4	,	,	PUNCT
ejpam-1878	282	5	a	a	DET
ejpam-1878	282	6	=	=	X
ejpam-1878	282	7	iα	iα	NOUN
ejpam-1878	282	8	,	,	PUNCT
ejpam-1878	282	9	where	where	SCONJ
ejpam-1878	282	10	α	α	NOUN
ejpam-1878	282	11	:p	:p	PUNCT
ejpam-1878	282	12	−→n	−→n	VERB
ejpam-1878	282	13	∞	∞	PROPN
ejpam-1878	282	14	is	be	AUX
ejpam-1878	282	15	defined	define	VERB
ejpam-1878	282	16	by	by	ADP
ejpam-1878	282	17	α(q	α(q	ADJ
ejpam-1878	282	18	)	)	PUNCT
ejpam-1878	282	19	=	=	PUNCT
ejpam-1878	283	1	¨	¨	NOUN
ejpam-1878	283	2	a	a	PRON
ejpam-1878	283	3	if	if	NOUN
ejpam-1878	283	4	q	q	NOUN
ejpam-1878	283	5	=	=	PUNCT
ejpam-1878	283	6	p	p	NOUN
ejpam-1878	283	7	∞	∞	PROPN
ejpam-1878	283	8	if	if	SCONJ
ejpam-1878	283	9	q	q	PROPN
ejpam-1878	283	10	6=	6=	PROPN
ejpam-1878	283	11	p	p	DET
ejpam-1878	283	12	note	note	NOUN
ejpam-1878	283	13	that	that	SCONJ
ejpam-1878	283	14	ip	ip	ADP
ejpam-1878	283	15	,	,	PUNCT
ejpam-1878	283	16	a	a	PRON
ejpam-1878	283	17	=	=	X
ejpam-1878	283	18	{	{	PUNCT
ejpam-1878	283	19	n	n	NOUN
ejpam-1878	283	20	∈	∈	PROPN
ejpam-1878	283	21	z	z	NOUN
ejpam-1878	283	22	+	+	NOUN
ejpam-1878	283	23	|pa+1	|pa+1	NOUN
ejpam-1878	283	24	does	do	AUX
ejpam-1878	283	25	not	not	PART
ejpam-1878	283	26	divide	divide	VERB
ejpam-1878	283	27	n	n	CCONJ
ejpam-1878	283	28	}	}	PUNCT
ejpam-1878	283	29	.	.	PUNCT
ejpam-1878	284	1	theorem	theorem	NOUN
ejpam-1878	284	2	12	12	NUM
ejpam-1878	284	3	.	.	PUNCT
ejpam-1878	285	1	an	an	DET
ejpam-1878	285	2	ideal	ideal	NOUN
ejpam-1878	285	3	of	of	ADP
ejpam-1878	285	4	(	(	PUNCT
ejpam-1878	285	5	z+,≤d	z+,≤d	NUM
ejpam-1878	285	6	)	)	PUNCT
ejpam-1878	285	7	is	be	AUX
ejpam-1878	285	8	prime	prime	ADJ
ejpam-1878	285	9	if	if	SCONJ
ejpam-1878	286	1	and	and	CCONJ
ejpam-1878	286	2	only	only	ADV
ejpam-1878	286	3	if	if	SCONJ
ejpam-1878	286	4	it	it	PRON
ejpam-1878	286	5	is	be	AUX
ejpam-1878	286	6	of	of	ADP
ejpam-1878	286	7	the	the	DET
ejpam-1878	286	8	form	form	NOUN
ejpam-1878	286	9	ip	ip	NOUN
ejpam-1878	286	10	,	,	PUNCT
ejpam-1878	286	11	a	a	PRON
ejpam-1878	286	12	for	for	ADP
ejpam-1878	286	13	some	some	DET
ejpam-1878	286	14	p	p	NOUN
ejpam-1878	286	15	∈	∈	PROPN
ejpam-1878	286	16	p	p	NOUN
ejpam-1878	286	17	and	and	CCONJ
ejpam-1878	286	18	a	a	DET
ejpam-1878	286	19	∈	∈	PROPN
ejpam-1878	286	20	n	n	NOUN
ejpam-1878	286	21	.	.	PUNCT
ejpam-1878	287	1	proof	proof	NOUN
ejpam-1878	287	2	.	.	PUNCT
ejpam-1878	288	1	let	let	VERB
ejpam-1878	288	2	i	i	PRON
ejpam-1878	288	3	be	be	AUX
ejpam-1878	288	4	an	an	DET
ejpam-1878	288	5	ideal	ideal	NOUN
ejpam-1878	288	6	of	of	ADP
ejpam-1878	288	7	(	(	PUNCT
ejpam-1878	288	8	z+,≤d	z+,≤d	NUM
ejpam-1878	288	9	)	)	PUNCT
ejpam-1878	288	10	.	.	PUNCT
ejpam-1878	289	1	then	then	ADV
ejpam-1878	289	2	i	i	PRON
ejpam-1878	289	3	=	=	PUNCT
ejpam-1878	289	4	iα	iα	VERB
ejpam-1878	289	5	for	for	ADP
ejpam-1878	289	6	some	some	DET
ejpam-1878	289	7	mapping	mapping	NOUN
ejpam-1878	290	1	α	α	NOUN
ejpam-1878	290	2	:p	:p	PROPN
ejpam-1878	291	1	−→n	−→n	VERB
ejpam-1878	291	2	∞.	∞.	PROPN
ejpam-1878	291	3	now	now	ADV
ejpam-1878	291	4	,	,	PUNCT
ejpam-1878	291	5	by	by	ADP
ejpam-1878	291	6	theorem	theorem	NOUN
ejpam-1878	291	7	11	11	NUM
ejpam-1878	291	8	,	,	PUNCT
ejpam-1878	291	9	i	i	PRON
ejpam-1878	291	10	is	be	AUX
ejpam-1878	291	11	prime	prime	ADJ
ejpam-1878	291	12	⇐	⇐	ADJ
ejpam-1878	291	13	⇒	⇒	NOUN
ejpam-1878	291	14	there	there	PRON
ejpam-1878	291	15	exists	exist	VERB
ejpam-1878	291	16	p	p	PROPN
ejpam-1878	291	17	∈	∈	PROPN
ejpam-1878	291	18	p	p	NOUN
ejpam-1878	291	19	such	such	ADJ
ejpam-1878	291	20	that	that	PRON
ejpam-1878	291	21	α(p	α(p	PROPN
ejpam-1878	291	22	)	)	PUNCT
ejpam-1878	291	23	6=∞	6=∞	NUM
ejpam-1878	291	24	and	and	CCONJ
ejpam-1878	291	25	α(q	α(q	NUM
ejpam-1878	291	26	)	)	PUNCT
ejpam-1878	291	27	=	=	SYM
ejpam-1878	291	28	∞	∞	NOUN
ejpam-1878	291	29	for	for	ADP
ejpam-1878	291	30	all	all	DET
ejpam-1878	291	31	q	q	NOUN
ejpam-1878	291	32	6=	6=	ADP
ejpam-1878	291	33	p	p	NOUN
ejpam-1878	292	1	and	and	CCONJ
ejpam-1878	292	2	i	i	PRON
ejpam-1878	292	3	=	=	NOUN
ejpam-1878	292	4	iα	iα	VERB
ejpam-1878	292	5	⇐	⇐	PROPN
ejpam-1878	292	6	⇒	⇒	NOUN
ejpam-1878	293	1	i	i	NOUN
ejpam-1878	293	2	=	=	SYM
ejpam-1878	293	3	ip	ip	PROPN
ejpam-1878	293	4	,	,	PUNCT
ejpam-1878	293	5	a	a	PRON
ejpam-1878	293	6	,	,	PUNCT
ejpam-1878	293	7	where	where	SCONJ
ejpam-1878	293	8	a	a	DET
ejpam-1878	293	9	=	=	SYM
ejpam-1878	293	10	α(p	α(p	PROPN
ejpam-1878	293	11	)	)	PUNCT
ejpam-1878	293	12	.	.	PUNCT
ejpam-1878	294	1	theorem	theorem	VERB
ejpam-1878	294	2	13	13	NUM
ejpam-1878	294	3	.	.	PUNCT
ejpam-1878	295	1	for	for	ADP
ejpam-1878	295	2	any	any	DET
ejpam-1878	295	3	p	p	NOUN
ejpam-1878	295	4	and	and	CCONJ
ejpam-1878	295	5	q	q	NOUN
ejpam-1878	295	6	∈	∈	PROPN
ejpam-1878	295	7	p	p	NOUN
ejpam-1878	295	8	and	and	CCONJ
ejpam-1878	295	9	a	a	PRON
ejpam-1878	295	10	and	and	CCONJ
ejpam-1878	295	11	b	b	NOUN
ejpam-1878	295	12	∈	∈	PROPN
ejpam-1878	295	13	n	n	CCONJ
ejpam-1878	295	14	,	,	PUNCT
ejpam-1878	295	15	ip	ip	PROPN
ejpam-1878	295	16	,	,	PUNCT
ejpam-1878	295	17	a	a	DET
ejpam-1878	295	18	⊆	⊆	NUM
ejpam-1878	295	19	iq	iq	NOUN
ejpam-1878	295	20	,	,	PUNCT
ejpam-1878	295	21	b	b	X
ejpam-1878	295	22	⇐	⇐	ADJ
ejpam-1878	295	23	⇒	⇒	NOUN
ejpam-1878	295	24	p	p	X
ejpam-1878	295	25	=	=	X
ejpam-1878	295	26	q	q	X
ejpam-1878	295	27	and	and	CCONJ
ejpam-1878	295	28	a	a	DET
ejpam-1878	295	29	≤	≤	NUM
ejpam-1878	295	30	b	b	NUM
ejpam-1878	295	31	references	reference	NOUN
ejpam-1878	295	32	25	25	NUM
ejpam-1878	295	33	proof	proof	NOUN
ejpam-1878	295	34	.	.	PUNCT
ejpam-1878	296	1	if	if	SCONJ
ejpam-1878	296	2	p	p	NOUN
ejpam-1878	296	3	=	=	X
ejpam-1878	296	4	q	q	X
ejpam-1878	296	5	and	and	CCONJ
ejpam-1878	296	6	a	a	DET
ejpam-1878	296	7	≤	≤	NUM
ejpam-1878	296	8	b	b	NOUN
ejpam-1878	296	9	,	,	PUNCT
ejpam-1878	296	10	then	then	ADV
ejpam-1878	296	11	n	n	CCONJ
ejpam-1878	296	12	∈	∈	PROPN
ejpam-1878	296	13	ip	ip	NOUN
ejpam-1878	296	14	,	,	PUNCT
ejpam-1878	296	15	q	q	PROPN
ejpam-1878	296	16	=	=	NOUN
ejpam-1878	296	17	⇒θ	⇒θ	NOUN
ejpam-1878	296	18	(	(	PUNCT
ejpam-1878	296	19	n)(p)≤	n)(p)≤	VERB
ejpam-1878	296	20	a	a	DET
ejpam-1878	296	21	≤	≤	NUM
ejpam-1878	296	22	b	b	X
ejpam-1878	296	23	=	=	NOUN
ejpam-1878	296	24	⇒θ	⇒θ	NOUN
ejpam-1878	296	25	(	(	PUNCT
ejpam-1878	296	26	n)(q)≤	n)(q)≤	NOUN
ejpam-1878	296	27	b	b	SYM
ejpam-1878	296	28	=	=	PRON
ejpam-1878	296	29	⇒n	⇒n	PROPN
ejpam-1878	296	30	∈	∈	PROPN
ejpam-1878	296	31	iq	iq	PROPN
ejpam-1878	296	32	,	,	PUNCT
ejpam-1878	296	33	b	b	PROPN
ejpam-1878	296	34	and	and	CCONJ
ejpam-1878	296	35	hence	hence	ADV
ejpam-1878	296	36	ip	ip	NOUN
ejpam-1878	296	37	,	,	PUNCT
ejpam-1878	296	38	a	a	DET
ejpam-1878	296	39	⊆	⊆	NUM
ejpam-1878	296	40	iq	iq	NOUN
ejpam-1878	296	41	,	,	PUNCT
ejpam-1878	296	42	b.	b.	PROPN
ejpam-1878	296	43	conversely	conversely	ADV
ejpam-1878	296	44	suppose	suppose	VERB
ejpam-1878	296	45	that	that	SCONJ
ejpam-1878	296	46	ip	ip	NOUN
ejpam-1878	296	47	,	,	PUNCT
ejpam-1878	296	48	a	a	DET
ejpam-1878	296	49	⊆	⊆	NUM
ejpam-1878	296	50	iq	iq	NOUN
ejpam-1878	296	51	,	,	PUNCT
ejpam-1878	296	52	b.	b.	PROPN
ejpam-1878	296	53	if	if	SCONJ
ejpam-1878	296	54	p	p	PROPN
ejpam-1878	296	55	6=	6=	PROPN
ejpam-1878	296	56	q	q	PROPN
ejpam-1878	296	57	,	,	PUNCT
ejpam-1878	296	58	then	then	ADV
ejpam-1878	296	59	θ	θ	PROPN
ejpam-1878	296	60	(	(	PUNCT
ejpam-1878	296	61	qb+1)(p	qb+1)(p	NOUN
ejpam-1878	296	62	)	)	PUNCT
ejpam-1878	296	63	=	=	SYM
ejpam-1878	296	64	0≤	0≤	NUM
ejpam-1878	296	65	a	a	PRON
ejpam-1878	296	66	and	and	CCONJ
ejpam-1878	296	67	hence	hence	ADV
ejpam-1878	296	68	qb+1	qb+1	ADV
ejpam-1878	296	69	∈	∈	PROPN
ejpam-1878	296	70	ip	ip	NOUN
ejpam-1878	296	71	,	,	PUNCT
ejpam-1878	296	72	a	a	DET
ejpam-1878	296	73	⊆	⊆	NUM
ejpam-1878	296	74	iq	iq	NOUN
ejpam-1878	296	75	,	,	PUNCT
ejpam-1878	296	76	b	b	NOUN
ejpam-1878	296	77	so	so	ADV
ejpam-1878	296	78	that	that	SCONJ
ejpam-1878	296	79	θ	θ	PROPN
ejpam-1878	296	80	(	(	PUNCT
ejpam-1878	296	81	qb+1)(b)≤	qb+1)(b)≤	PROPN
ejpam-1878	296	82	b	b	PROPN
ejpam-1878	296	83	,	,	PUNCT
ejpam-1878	296	84	which	which	PRON
ejpam-1878	296	85	is	be	AUX
ejpam-1878	296	86	a	a	DET
ejpam-1878	296	87	contradiction	contradiction	NOUN
ejpam-1878	296	88	.	.	PUNCT
ejpam-1878	297	1	therefore	therefore	ADV
ejpam-1878	297	2	p	p	X
ejpam-1878	297	3	=	=	PUNCT
ejpam-1878	297	4	q.	q.	PROPN
ejpam-1878	297	5	now	now	ADV
ejpam-1878	297	6	,	,	PUNCT
ejpam-1878	297	7	since	since	SCONJ
ejpam-1878	297	8	θ	θ	PROPN
ejpam-1878	297	9	(	(	PUNCT
ejpam-1878	297	10	pa)(p	pa)(p	NOUN
ejpam-1878	297	11	)	)	PUNCT
ejpam-1878	297	12	=	=	SYM
ejpam-1878	298	1	a	a	PROPN
ejpam-1878	298	2	,	,	PUNCT
ejpam-1878	298	3	pa	pa	PROPN
ejpam-1878	298	4	∈	∈	PROPN
ejpam-1878	298	5	ip	ip	NOUN
ejpam-1878	298	6	,	,	PUNCT
ejpam-1878	298	7	a	a	DET
ejpam-1878	298	8	⊆	⊆	NUM
ejpam-1878	298	9	iq	iq	NOUN
ejpam-1878	298	10	,	,	PUNCT
ejpam-1878	298	11	b	b	NOUN
ejpam-1878	298	12	and	and	CCONJ
ejpam-1878	298	13	hence	hence	ADV
ejpam-1878	298	14	a	a	DET
ejpam-1878	298	15	=	=	SYM
ejpam-1878	298	16	θ	θ	PROPN
ejpam-1878	298	17	(	(	PUNCT
ejpam-1878	298	18	pa)(q	pa)(q	PROPN
ejpam-1878	298	19	)	)	PUNCT
ejpam-1878	298	20	≤	≤	NUM
ejpam-1878	298	21	b.	b.	X
ejpam-1878	299	1	thus	thus	ADV
ejpam-1878	299	2	p	p	X
ejpam-1878	299	3	=	=	X
ejpam-1878	299	4	q	q	X
ejpam-1878	299	5	and	and	CCONJ
ejpam-1878	299	6	a	a	DET
ejpam-1878	299	7	≤	≤	NUM
ejpam-1878	299	8	b.	b.	NOUN
ejpam-1878	300	1	the	the	DET
ejpam-1878	300	2	following	following	NOUN
ejpam-1878	300	3	are	be	AUX
ejpam-1878	300	4	immediate	immediate	ADJ
ejpam-1878	300	5	consequences	consequence	NOUN
ejpam-1878	300	6	of	of	ADP
ejpam-1878	300	7	theorems	theorem	NOUN
ejpam-1878	300	8	11,12	11,12	NUM
ejpam-1878	300	9	and	and	CCONJ
ejpam-1878	300	10	13	13	NUM
ejpam-1878	300	11	.	.	PUNCT
ejpam-1878	300	12	corollary	corollary	ADJ
ejpam-1878	300	13	2	2	NUM
ejpam-1878	300	14	.	.	PUNCT
ejpam-1878	300	15	for	for	ADP
ejpam-1878	300	16	each	each	DET
ejpam-1878	300	17	p	p	NOUN
ejpam-1878	300	18	∈	∈	PROPN
ejpam-1878	300	19	p	p	NOUN
ejpam-1878	300	20	,	,	PUNCT
ejpam-1878	300	21	let	let	VERB
ejpam-1878	300	22	pp	pp	ADV
ejpam-1878	300	23	=	=	PUNCT
ejpam-1878	300	24	{	{	PUNCT
ejpam-1878	300	25	ip	ip	NOUN
ejpam-1878	300	26	,	,	PUNCT
ejpam-1878	300	27	a|a	a|a	NOUN
ejpam-1878	300	28	∈	∈	PROPN
ejpam-1878	300	29	n	n	CCONJ
ejpam-1878	300	30	}	}	PUNCT
ejpam-1878	300	31	.	.	PUNCT
ejpam-1878	301	1	then	then	ADV
ejpam-1878	301	2	the	the	DET
ejpam-1878	301	3	following	follow	VERB
ejpam-1878	301	4	hold	hold	NOUN
ejpam-1878	301	5	.	.	PUNCT
ejpam-1878	302	1	(	(	PUNCT
ejpam-1878	302	2	1	1	NUM
ejpam-1878	302	3	)	)	PUNCT
ejpam-1878	302	4	.	.	PUNCT
ejpam-1878	303	1	pp	pp	ADV
ejpam-1878	303	2	is	be	AUX
ejpam-1878	303	3	a	a	DET
ejpam-1878	303	4	chain	chain	NOUN
ejpam-1878	303	5	of	of	ADP
ejpam-1878	303	6	prime	prime	ADJ
ejpam-1878	303	7	ideals	ideal	NOUN
ejpam-1878	303	8	of	of	ADP
ejpam-1878	303	9	(	(	PUNCT
ejpam-1878	303	10	z+,≤d	z+,≤d	NOUN
ejpam-1878	303	11	)	)	PUNCT
ejpam-1878	303	12	for	for	ADP
ejpam-1878	303	13	each	each	DET
ejpam-1878	303	14	p	p	NOUN
ejpam-1878	303	15	∈	∈	PROPN
ejpam-1878	303	16	p	p	X
ejpam-1878	303	17	.	.	PUNCT
ejpam-1878	304	1	(	(	PUNCT
ejpam-1878	304	2	2	2	NUM
ejpam-1878	304	3	)	)	PUNCT
ejpam-1878	304	4	.	.	PUNCT
ejpam-1878	305	1	pp	pp	ADV
ejpam-1878	305	2	∩pq	∩pq	NOUN
ejpam-1878	306	1	=	=	PROPN
ejpam-1878	306	2	φ	φ	PROPN
ejpam-1878	306	3	for	for	ADP
ejpam-1878	306	4	all	all	DET
ejpam-1878	306	5	p	p	NOUN
ejpam-1878	306	6	6=	6=	ADP
ejpam-1878	306	7	q	q	PROPN
ejpam-1878	306	8	∈	∈	PROPN
ejpam-1878	306	9	p	p	NOUN
ejpam-1878	306	10	.	.	PUNCT
ejpam-1878	307	1	(	(	PUNCT
ejpam-1878	307	2	3	3	NUM
ejpam-1878	307	3	)	)	PUNCT
ejpam-1878	307	4	.	.	PUNCT
ejpam-1878	308	1	⋃	⋃	VERB
ejpam-1878	308	2	p∈p	p∈p	PROPN
ejpam-1878	308	3	pp	pp	ADV
ejpam-1878	308	4	is	be	AUX
ejpam-1878	308	5	the	the	DET
ejpam-1878	308	6	set	set	NOUN
ejpam-1878	308	7	of	of	ADP
ejpam-1878	308	8	all	all	DET
ejpam-1878	308	9	prime	prime	ADJ
ejpam-1878	308	10	ideals	ideal	NOUN
ejpam-1878	308	11	of	of	ADP
ejpam-1878	308	12	(	(	PUNCT
ejpam-1878	308	13	z+,≤d	z+,≤d	NUM
ejpam-1878	308	14	)	)	PUNCT
ejpam-1878	308	15	.	.	PUNCT
ejpam-1878	309	1	corollary	corollary	ADJ
ejpam-1878	309	2	3	3	X
ejpam-1878	309	3	.	.	PUNCT
ejpam-1878	310	1	i	i	PRON
ejpam-1878	310	2	is	be	AUX
ejpam-1878	310	3	a	a	DET
ejpam-1878	310	4	minimal	minimal	ADJ
ejpam-1878	310	5	prime	prime	ADJ
ejpam-1878	310	6	ideal	ideal	NOUN
ejpam-1878	310	7	of	of	ADP
ejpam-1878	310	8	(	(	PUNCT
ejpam-1878	310	9	z+,≤d	z+,≤d	NOUN
ejpam-1878	310	10	)	)	PUNCT
ejpam-1878	310	11	if	if	SCONJ
ejpam-1878	311	1	and	and	CCONJ
ejpam-1878	311	2	only	only	ADV
ejpam-1878	311	3	if	if	SCONJ
ejpam-1878	311	4	i	i	PRON
ejpam-1878	311	5	=	=	PUNCT
ejpam-1878	311	6	ip,0	ip,0	NOUN
ejpam-1878	311	7	=	=	PUNCT
ejpam-1878	311	8	{	{	PUNCT
ejpam-1878	311	9	n	n	X
ejpam-1878	311	10	∈	∈	PROPN
ejpam-1878	311	11	z	z	X
ejpam-1878	311	12	+	+	NOUN
ejpam-1878	311	13	|p	|p	X
ejpam-1878	311	14	does	do	AUX
ejpam-1878	311	15	not	not	PART
ejpam-1878	311	16	divide	divide	VERB
ejpam-1878	311	17	n	n	CCONJ
ejpam-1878	311	18	}	}	PUNCT
ejpam-1878	311	19	for	for	ADP
ejpam-1878	311	20	some	some	DET
ejpam-1878	311	21	p	p	NOUN
ejpam-1878	311	22	∈	∈	PROPN
ejpam-1878	311	23	p	p	NOUN
ejpam-1878	311	24	.	.	PUNCT
ejpam-1878	312	1	references	reference	NOUN
ejpam-1878	312	2	[	[	X
ejpam-1878	312	3	1	1	X
ejpam-1878	312	4	]	]	PUNCT
ejpam-1878	312	5	s.	s.	PROPN
ejpam-1878	312	6	sagi	sagi	PROPN
ejpam-1878	312	7	.	.	PUNCT
ejpam-1878	313	1	lattice	lattice	PROPN
ejpam-1878	313	2	theory	theory	NOUN
ejpam-1878	313	3	of	of	ADP
ejpam-1878	313	4	convolutions	convolution	NOUN
ejpam-1878	313	5	,	,	PUNCT
ejpam-1878	313	6	ph.d	ph.d	PROPN
ejpam-1878	313	7	.	.	PUNCT
ejpam-1878	314	1	thesis	thesis	PROPN
ejpam-1878	314	2	,	,	PUNCT
ejpam-1878	314	3	andhra	andhra	PROPN
ejpam-1878	314	4	university	university	PROPN
ejpam-1878	314	5	,	,	PUNCT
ejpam-1878	314	6	waltair	waltair	NOUN
ejpam-1878	314	7	,	,	PUNCT
ejpam-1878	314	8	visakhapatnam	visakhapatnam	PROPN
ejpam-1878	314	9	,	,	PUNCT
ejpam-1878	314	10	india	india	PROPN
ejpam-1878	314	11	,	,	PUNCT
ejpam-1878	314	12	2010	2010	NUM
ejpam-1878	314	13	.	.	PUNCT
ejpam-1878	315	1	[	[	X
ejpam-1878	315	2	2	2	NUM
ejpam-1878	315	3	]	]	X
ejpam-1878	315	4	u.m	u.m	PROPN
ejpam-1878	315	5	.	.	PROPN
ejpam-1878	315	6	swamy	swamy	PROPN
ejpam-1878	315	7	,	,	PUNCT
ejpam-1878	315	8	g.c	g.c	PROPN
ejpam-1878	315	9	.	.	PROPN
ejpam-1878	315	10	rao	rao	PROPN
ejpam-1878	315	11	,	,	PUNCT
ejpam-1878	315	12	v.	v.	PROPN
ejpam-1878	315	13	s.	s.	PROPN
ejpam-1878	315	14	ramaiah	ramaiah	PROPN
ejpam-1878	315	15	.	.	PUNCT
ejpam-1878	316	1	on	on	ADP
ejpam-1878	316	2	a	a	DET
ejpam-1878	316	3	conjecture	conjecture	NOUN
ejpam-1878	316	4	in	in	ADP
ejpam-1878	316	5	a	a	DET
ejpam-1878	316	6	ring	ring	NOUN
ejpam-1878	316	7	of	of	ADP
ejpam-1878	316	8	arithmetic	arithmetic	ADJ
ejpam-1878	316	9	functions	function	NOUN
ejpam-1878	316	10	.	.	PUNCT
ejpam-1878	317	1	indian	indian	ADJ
ejpam-1878	317	2	journal	journal	PROPN
ejpam-1878	317	3	of	of	ADP
ejpam-1878	317	4	pure	pure	ADJ
ejpam-1878	317	5	and	and	CCONJ
ejpam-1878	317	6	applied	applied	ADJ
ejpam-1878	317	7	mathematics	mathematic	NOUN
ejpam-1878	317	8	,	,	PUNCT
ejpam-1878	317	9	14(12	14(12	NUM
ejpam-1878	317	10	)	)	PUNCT
ejpam-1878	317	11	,	,	PUNCT
ejpam-1878	317	12	1519	1519	NUM
ejpam-1878	317	13	-	-	SYM
ejpam-1878	317	14	1530	1530	NUM
ejpam-1878	317	15	.	.	PUNCT
ejpam-1878	318	1	1983	1983	NUM
ejpam-1878	318	2	.	.	PUNCT
ejpam-1878	319	1	[	[	X
ejpam-1878	319	2	3	3	X
ejpam-1878	319	3	]	]	X
ejpam-1878	319	4	u.m	u.m	PROPN
ejpam-1878	319	5	.	.	PROPN
ejpam-1878	319	6	swamy	swamy	PROPN
ejpam-1878	319	7	,	,	PUNCT
ejpam-1878	319	8	s.	s.	PROPN
ejpam-1878	319	9	sagi	sagi	PROPN
ejpam-1878	319	10	.	.	PUNCT
ejpam-1878	319	11	lattice	lattice	PROPN
ejpam-1878	319	12	structures	structure	NOUN
ejpam-1878	319	13	onz+	onz+	AUX
ejpam-1878	319	14	induced	induce	VERB
ejpam-1878	319	15	by	by	ADP
ejpam-1878	319	16	convolutions	convolution	NOUN
ejpam-1878	319	17	.	.	PUNCT
ejpam-1878	320	1	european	european	ADJ
ejpam-1878	320	2	jounal	jounal	NOUN
ejpam-1878	320	3	of	of	ADP
ejpam-1878	320	4	pure	pure	ADJ
ejpam-1878	320	5	and	and	CCONJ
ejpam-1878	320	6	applied	apply	VERB
ejpam-1878	320	7	mathematics,4(4	mathematics,4(4	NOUN
ejpam-1878	320	8	)	)	PUNCT
ejpam-1878	320	9	,	,	PUNCT
ejpam-1878	320	10	424	424	NUM
ejpam-1878	320	11	-	-	SYM
ejpam-1878	320	12	434	434	NUM
ejpam-1878	320	13	.	.	PUNCT
ejpam-1878	320	14	2011	2011	NUM
ejpam-1878	320	15	.	.	PUNCT
ejpam-1878	321	1	[	[	X
ejpam-1878	321	2	4	4	NUM
ejpam-1878	321	3	]	]	X
ejpam-1878	321	4	u.m	u.m	PROPN
ejpam-1878	321	5	.	.	PROPN
ejpam-1878	321	6	swamy	swamy	PROPN
ejpam-1878	321	7	,	,	PUNCT
ejpam-1878	321	8	s.	s.	PROPN
ejpam-1878	321	9	sagi	sagi	PROPN
ejpam-1878	321	10	.	.	PUNCT
ejpam-1878	322	1	partial	partial	ADJ
ejpam-1878	322	2	orders	order	NOUN
ejpam-1878	322	3	induced	induce	VERB
ejpam-1878	322	4	by	by	ADP
ejpam-1878	322	5	convolutions	convolution	NOUN
ejpam-1878	322	6	.	.	PUNCT
ejpam-1878	323	1	international	international	ADJ
ejpam-1878	323	2	journal	journal	NOUN
ejpam-1878	323	3	of	of	ADP
ejpam-1878	323	4	mathematics	mathematic	NOUN
ejpam-1878	323	5	and	and	CCONJ
ejpam-1878	323	6	soft	soft	ADJ
ejpam-1878	323	7	computing	computing	NOUN
ejpam-1878	323	8	,	,	PUNCT
ejpam-1878	323	9	2(1	2(1	NUM
ejpam-1878	323	10	)	)	PUNCT
ejpam-1878	323	11	,	,	PUNCT
ejpam-1878	323	12	2011	2011	NUM
ejpam-1878	323	13	,	,	PUNCT
ejpam-1878	323	14	25	25	NUM
ejpam-1878	323	15	-	-	SYM
ejpam-1878	323	16	33	33	NUM
ejpam-1878	323	17	.	.	PUNCT