id	sid	tid	token	lemma	pos
ejpam-5532	1	1	european	european	PROPN
ejpam-5532	1	2	journal	journal	PROPN
ejpam-5532	1	3	of	of	ADP
ejpam-5532	1	4	pure	pure	ADJ
ejpam-5532	1	5	and	and	CCONJ
ejpam-5532	1	6	applied	applied	ADJ
ejpam-5532	1	7	mathematics	mathematic	NOUN
ejpam-5532	1	8	2025	2025	NUM
ejpam-5532	1	9	,	,	PUNCT
ejpam-5532	1	10	vol	vol	NOUN
ejpam-5532	1	11	.	.	PROPN
ejpam-5532	1	12	18	18	NUM
ejpam-5532	1	13	,	,	PUNCT
ejpam-5532	1	14	issue	issue	NOUN
ejpam-5532	1	15	1	1	NUM
ejpam-5532	1	16	,	,	PUNCT
ejpam-5532	1	17	article	article	NOUN
ejpam-5532	1	18	number	number	NOUN
ejpam-5532	1	19	5532	5532	NUM
ejpam-5532	1	20	issn	issn	PROPN
ejpam-5532	1	21	1307	1307	NUM
ejpam-5532	1	22	-	-	SYM
ejpam-5532	1	23	5543	5543	NUM
ejpam-5532	1	24	–	–	PUNCT
ejpam-5532	1	25	ejpam.com	ejpam.com	X
ejpam-5532	1	26	published	publish	VERB
ejpam-5532	1	27	by	by	ADP
ejpam-5532	1	28	new	new	PROPN
ejpam-5532	1	29	york	york	PROPN
ejpam-5532	1	30	business	business	PROPN
ejpam-5532	1	31	global	global	PROPN
ejpam-5532	1	32	l	l	ADJ
ejpam-5532	1	33	-	-	ADJ
ejpam-5532	1	34	convex	convex	ADJ
ejpam-5532	1	35	sublattice	sublattice	NOUN
ejpam-5532	1	36	of	of	ADP
ejpam-5532	1	37	an	an	DET
ejpam-5532	1	38	l	l	NOUN
ejpam-5532	1	39	-	-	NOUN
ejpam-5532	1	40	lattice	lattice	NOUN
ejpam-5532	1	41	and	and	CCONJ
ejpam-5532	1	42	complement	complement	NOUN
ejpam-5532	1	43	of	of	ADP
ejpam-5532	1	44	an	an	DET
ejpam-5532	1	45	l	l	NOUN
ejpam-5532	1	46	-	-	ADJ
ejpam-5532	1	47	set	set	VERB
ejpam-5532	1	48	aparna	aparna	NOUN
ejpam-5532	1	49	jain1	jain1	NOUN
ejpam-5532	1	50	,	,	PUNCT
ejpam-5532	1	51	iffat	iffat	NOUN
ejpam-5532	1	52	jahan2	jahan2	PROPN
ejpam-5532	1	53	∗	∗	PROPN
ejpam-5532	1	54	1	1	NUM
ejpam-5532	1	55	department	department	NOUN
ejpam-5532	1	56	of	of	ADP
ejpam-5532	1	57	mathematics	mathematics	PROPN
ejpam-5532	1	58	,	,	PUNCT
ejpam-5532	1	59	shivaji	shivaji	PROPN
ejpam-5532	1	60	college	college	PROPN
ejpam-5532	1	61	,	,	PUNCT
ejpam-5532	1	62	university	university	PROPN
ejpam-5532	1	63	of	of	ADP
ejpam-5532	1	64	delhi	delhi	PROPN
ejpam-5532	1	65	,	,	PUNCT
ejpam-5532	1	66	delhi	delhi	PROPN
ejpam-5532	1	67	,	,	PUNCT
ejpam-5532	1	68	india	india	PROPN
ejpam-5532	1	69	2	2	NUM
ejpam-5532	1	70	department	department	NOUN
ejpam-5532	1	71	of	of	ADP
ejpam-5532	1	72	mathematics	mathematics	PROPN
ejpam-5532	1	73	,	,	PUNCT
ejpam-5532	1	74	ramjas	ramjas	PROPN
ejpam-5532	1	75	college	college	PROPN
ejpam-5532	1	76	,	,	PUNCT
ejpam-5532	1	77	university	university	NOUN
ejpam-5532	1	78	of	of	ADP
ejpam-5532	1	79	delhi	delhi	PROPN
ejpam-5532	1	80	,	,	PUNCT
ejpam-5532	1	81	delhi	delhi	PROPN
ejpam-5532	1	82	,	,	PUNCT
ejpam-5532	1	83	india	india	PROPN
ejpam-5532	1	84	abstract	abstract	NOUN
ejpam-5532	1	85	.	.	PUNCT
ejpam-5532	2	1	in	in	ADP
ejpam-5532	2	2	this	this	DET
ejpam-5532	2	3	paper	paper	NOUN
ejpam-5532	2	4	,	,	PUNCT
ejpam-5532	2	5	the	the	DET
ejpam-5532	2	6	authors	author	NOUN
ejpam-5532	2	7	define	define	VERB
ejpam-5532	2	8	and	and	CCONJ
ejpam-5532	2	9	explore	explore	VERB
ejpam-5532	2	10	the	the	DET
ejpam-5532	2	11	notion	notion	NOUN
ejpam-5532	2	12	of	of	ADP
ejpam-5532	2	13	an	an	DET
ejpam-5532	2	14	l	l	ADJ
ejpam-5532	2	15	-	-	ADJ
ejpam-5532	2	16	convex	convex	ADJ
ejpam-5532	2	17	sublattice	sublattice	NOUN
ejpam-5532	2	18	in	in	ADP
ejpam-5532	2	19	an	an	DET
ejpam-5532	2	20	l	l	NOUN
ejpam-5532	2	21	-	-	NOUN
ejpam-5532	2	22	lattice	lattice	NOUN
ejpam-5532	2	23	.	.	PUNCT
ejpam-5532	3	1	the	the	DET
ejpam-5532	3	2	investigations	investigation	NOUN
ejpam-5532	3	3	in	in	ADP
ejpam-5532	3	4	this	this	DET
ejpam-5532	3	5	paper	paper	NOUN
ejpam-5532	3	6	lead	lead	NOUN
ejpam-5532	3	7	to	to	ADP
ejpam-5532	3	8	’	'	PUNCT
ejpam-5532	3	9	the	the	DET
ejpam-5532	3	10	unique	unique	ADJ
ejpam-5532	3	11	representation	representation	NOUN
ejpam-5532	3	12	theorem	theorem	VERB
ejpam-5532	3	13	’	'	PUNCT
ejpam-5532	3	14	for	for	ADP
ejpam-5532	3	15	l	l	ADJ
ejpam-5532	3	16	-	-	ADJ
ejpam-5532	3	17	convex	convex	ADJ
ejpam-5532	3	18	sublattices	sublattice	NOUN
ejpam-5532	3	19	.	.	PUNCT
ejpam-5532	4	1	also	also	ADV
ejpam-5532	4	2	,	,	PUNCT
ejpam-5532	4	3	the	the	DET
ejpam-5532	4	4	authors	author	NOUN
ejpam-5532	4	5	effectively	effectively	ADV
ejpam-5532	4	6	use	use	VERB
ejpam-5532	4	7	the	the	DET
ejpam-5532	4	8	concept	concept	NOUN
ejpam-5532	4	9	of	of	ADP
ejpam-5532	4	10	order	order	NOUN
ejpam-5532	4	11	reversing	reverse	VERB
ejpam-5532	4	12	involution	involution	NOUN
ejpam-5532	4	13	on	on	ADP
ejpam-5532	4	14	the	the	DET
ejpam-5532	4	15	lattice	lattice	PROPN
ejpam-5532	4	16	l	l	PROPN
ejpam-5532	4	17	of	of	ADP
ejpam-5532	4	18	truth	truth	NOUN
ejpam-5532	4	19	values	value	NOUN
ejpam-5532	4	20	to	to	PART
ejpam-5532	4	21	define	define	VERB
ejpam-5532	4	22	complement	complement	NOUN
ejpam-5532	4	23	of	of	ADP
ejpam-5532	4	24	an	an	DET
ejpam-5532	4	25	l	l	NOUN
ejpam-5532	4	26	-	-	NOUN
ejpam-5532	4	27	set	set	NOUN
ejpam-5532	4	28	.	.	PUNCT
ejpam-5532	5	1	further	far	ADV
ejpam-5532	5	2	,	,	PUNCT
ejpam-5532	5	3	they	they	PRON
ejpam-5532	5	4	employ	employ	VERB
ejpam-5532	5	5	this	this	DET
ejpam-5532	5	6	notion	notion	NOUN
ejpam-5532	5	7	in	in	ADP
ejpam-5532	5	8	the	the	DET
ejpam-5532	5	9	studies	study	NOUN
ejpam-5532	5	10	of	of	ADP
ejpam-5532	5	11	lprime	lprime	ADJ
ejpam-5532	5	12	ideals	ideal	NOUN
ejpam-5532	5	13	and	and	CCONJ
ejpam-5532	5	14	l	l	NOUN
ejpam-5532	5	15	-	-	ADJ
ejpam-5532	5	16	maximal	maximal	ADJ
ejpam-5532	5	17	ideals	ideal	NOUN
ejpam-5532	5	18	.	.	PUNCT
ejpam-5532	6	1	2020	2020	NUM
ejpam-5532	6	2	mathematics	mathematic	NOUN
ejpam-5532	6	3	subject	subject	NOUN
ejpam-5532	6	4	classifications	classification	NOUN
ejpam-5532	6	5	:	:	PUNCT
ejpam-5532	6	6	06b10	06b10	NUM
ejpam-5532	6	7	,	,	PUNCT
ejpam-5532	6	8	06d72	06d72	NOUN
ejpam-5532	6	9	,	,	PUNCT
ejpam-5532	6	10	06d75	06d75	NUM
ejpam-5532	6	11	,	,	PUNCT
ejpam-5532	6	12	08a72	08a72	NOUN
ejpam-5532	6	13	key	key	ADJ
ejpam-5532	6	14	words	word	NOUN
ejpam-5532	6	15	and	and	CCONJ
ejpam-5532	6	16	phrases	phrase	NOUN
ejpam-5532	6	17	:	:	PUNCT
ejpam-5532	6	18	lattices	lattice	NOUN
ejpam-5532	6	19	,	,	PUNCT
ejpam-5532	6	20	generated	generate	VERB
ejpam-5532	6	21	l	l	NOUN
ejpam-5532	6	22	-	-	NOUN
ejpam-5532	6	23	sublattice	sublattice	NOUN
ejpam-5532	6	24	,	,	PUNCT
ejpam-5532	6	25	generated	generate	VERB
ejpam-5532	6	26	l	l	NOUN
ejpam-5532	6	27	-	-	NOUN
ejpam-5532	6	28	ideal	ideal	ADJ
ejpam-5532	6	29	,	,	PUNCT
ejpam-5532	6	30	generated	generate	VERB
ejpam-5532	6	31	l	l	ADJ
ejpam-5532	6	32	-	-	ADJ
ejpam-5532	6	33	dual	dual	ADJ
ejpam-5532	6	34	ideal	ideal	NOUN
ejpam-5532	6	35	,	,	PUNCT
ejpam-5532	6	36	l	l	ADJ
ejpam-5532	6	37	-	-	ADJ
ejpam-5532	6	38	convex	convex	ADJ
ejpam-5532	6	39	sublattice	sublattice	NOUN
ejpam-5532	6	40	,	,	PUNCT
ejpam-5532	6	41	complement	complement	NOUN
ejpam-5532	6	42	of	of	ADP
ejpam-5532	6	43	an	an	DET
ejpam-5532	6	44	l	l	NOUN
ejpam-5532	6	45	-	-	NOUN
ejpam-5532	6	46	set	set	ADJ
ejpam-5532	6	47	,	,	PUNCT
ejpam-5532	6	48	l	l	ADJ
ejpam-5532	6	49	-	-	ADJ
ejpam-5532	6	50	maximal	maximal	ADJ
ejpam-5532	6	51	ideal	ideal	NOUN
ejpam-5532	6	52	,	,	PUNCT
ejpam-5532	6	53	l	l	ADJ
ejpam-5532	6	54	-	-	ADJ
ejpam-5532	6	55	prime	prime	ADJ
ejpam-5532	6	56	ideal	ideal	NOUN
ejpam-5532	6	57	1	1	NUM
ejpam-5532	6	58	.	.	PUNCT
ejpam-5532	7	1	introduction	introduction	NOUN
ejpam-5532	7	2	the	the	DET
ejpam-5532	7	3	literature	literature	NOUN
ejpam-5532	7	4	on	on	ADP
ejpam-5532	7	5	fuzzy	fuzzy	ADJ
ejpam-5532	7	6	algebraic	algebraic	ADJ
ejpam-5532	7	7	structures	structure	NOUN
ejpam-5532	7	8	has	have	AUX
ejpam-5532	7	9	been	be	AUX
ejpam-5532	7	10	growing	grow	VERB
ejpam-5532	7	11	ever	ever	ADV
ejpam-5532	7	12	since	since	SCONJ
ejpam-5532	7	13	the	the	DET
ejpam-5532	7	14	introduction	introduction	NOUN
ejpam-5532	7	15	of	of	ADP
ejpam-5532	7	16	the	the	DET
ejpam-5532	7	17	concept	concept	NOUN
ejpam-5532	7	18	of	of	ADP
ejpam-5532	7	19	a	a	DET
ejpam-5532	7	20	fuzzy	fuzzy	ADJ
ejpam-5532	7	21	subgroup	subgroup	NOUN
ejpam-5532	7	22	by	by	ADP
ejpam-5532	7	23	a.	a.	PROPN
ejpam-5532	7	24	rosenfeld	rosenfeld	PROPN
ejpam-5532	8	1	[	[	X
ejpam-5532	8	2	14	14	NUM
ejpam-5532	8	3	]	]	PUNCT
ejpam-5532	8	4	in	in	ADP
ejpam-5532	8	5	the	the	DET
ejpam-5532	8	6	year	year	NOUN
ejpam-5532	8	7	1971	1971	NUM
ejpam-5532	8	8	.	.	PUNCT
ejpam-5532	9	1	ajmal	ajmal	PROPN
ejpam-5532	9	2	and	and	CCONJ
ejpam-5532	9	3	thomas	thomas	PROPN
ejpam-5532	10	1	[	[	X
ejpam-5532	10	2	4–6	4–6	X
ejpam-5532	10	3	]	]	X
ejpam-5532	10	4	systematically	systematically	ADV
ejpam-5532	10	5	developed	develop	VERB
ejpam-5532	10	6	the	the	DET
ejpam-5532	10	7	theory	theory	NOUN
ejpam-5532	10	8	of	of	ADP
ejpam-5532	10	9	fuzzy	fuzzy	ADJ
ejpam-5532	10	10	sublattices	sublattice	NOUN
ejpam-5532	10	11	in	in	ADP
ejpam-5532	10	12	a	a	DET
ejpam-5532	10	13	lattice	lattice	NOUN
ejpam-5532	10	14	.	.	PUNCT
ejpam-5532	11	1	they	they	PRON
ejpam-5532	11	2	introduced	introduce	VERB
ejpam-5532	11	3	the	the	DET
ejpam-5532	11	4	notions	notion	NOUN
ejpam-5532	11	5	of	of	ADP
ejpam-5532	11	6	a	a	DET
ejpam-5532	11	7	fuzzy	fuzzy	ADJ
ejpam-5532	11	8	sublattice	sublattice	NOUN
ejpam-5532	11	9	,	,	PUNCT
ejpam-5532	11	10	fuzzy	fuzzy	ADJ
ejpam-5532	11	11	ideal	ideal	NOUN
ejpam-5532	11	12	(	(	PUNCT
ejpam-5532	11	13	dual	dual	ADJ
ejpam-5532	11	14	ideal	ideal	NOUN
ejpam-5532	11	15	)	)	PUNCT
ejpam-5532	11	16	,	,	PUNCT
ejpam-5532	11	17	fuzzy	fuzzy	ADJ
ejpam-5532	11	18	prime	prime	ADJ
ejpam-5532	11	19	ideal	ideal	NOUN
ejpam-5532	11	20	(	(	PUNCT
ejpam-5532	11	21	dual	dual	ADJ
ejpam-5532	11	22	ideal	ideal	NOUN
ejpam-5532	11	23	)	)	PUNCT
ejpam-5532	11	24	,	,	PUNCT
ejpam-5532	11	25	fuzzy	fuzzy	ADJ
ejpam-5532	11	26	ideal	ideal	NOUN
ejpam-5532	11	27	(	(	PUNCT
ejpam-5532	11	28	dual	dual	ADJ
ejpam-5532	11	29	ideal	ideal	NOUN
ejpam-5532	11	30	)	)	PUNCT
ejpam-5532	11	31	generated	generate	VERB
ejpam-5532	11	32	by	by	ADP
ejpam-5532	11	33	a	a	DET
ejpam-5532	11	34	fuzzy	fuzzy	ADJ
ejpam-5532	11	35	set	set	NOUN
ejpam-5532	11	36	and	and	CCONJ
ejpam-5532	11	37	studied	study	VERB
ejpam-5532	11	38	their	their	PRON
ejpam-5532	11	39	properties	property	NOUN
ejpam-5532	11	40	.	.	PUNCT
ejpam-5532	12	1	the	the	DET
ejpam-5532	12	2	concept	concept	NOUN
ejpam-5532	12	3	of	of	ADP
ejpam-5532	12	4	a	a	DET
ejpam-5532	12	5	fuzzy	fuzzy	ADJ
ejpam-5532	12	6	convex	convex	NOUN
ejpam-5532	12	7	sublattice	sublattice	NOUN
ejpam-5532	12	8	was	be	AUX
ejpam-5532	12	9	also	also	ADV
ejpam-5532	12	10	introduced	introduce	VERB
ejpam-5532	12	11	by	by	ADP
ejpam-5532	12	12	ajmal	ajmal	PROPN
ejpam-5532	12	13	and	and	CCONJ
ejpam-5532	12	14	thomas	thomas	PROPN
ejpam-5532	12	15	in	in	ADP
ejpam-5532	12	16	[	[	X
ejpam-5532	12	17	4	4	NUM
ejpam-5532	12	18	,	,	PUNCT
ejpam-5532	12	19	5	5	NUM
ejpam-5532	12	20	]	]	PUNCT
ejpam-5532	12	21	,	,	PUNCT
ejpam-5532	12	22	wherein	wherein	SCONJ
ejpam-5532	12	23	the	the	DET
ejpam-5532	12	24	unique	unique	ADJ
ejpam-5532	12	25	representation	representation	NOUN
ejpam-5532	12	26	theorem	theorem	NOUN
ejpam-5532	12	27	for	for	ADP
ejpam-5532	12	28	convex	convex	ADJ
ejpam-5532	12	29	sublattices	sublattice	NOUN
ejpam-5532	12	30	was	be	AUX
ejpam-5532	12	31	extended	extend	VERB
ejpam-5532	12	32	to	to	ADP
ejpam-5532	12	33	fuzzy	fuzzy	ADJ
ejpam-5532	12	34	setting	setting	NOUN
ejpam-5532	12	35	.	.	PUNCT
ejpam-5532	13	1	the	the	DET
ejpam-5532	13	2	concept	concept	NOUN
ejpam-5532	13	3	of	of	ADP
ejpam-5532	13	4	an	an	DET
ejpam-5532	13	5	l	l	NOUN
ejpam-5532	13	6	-	-	ADJ
ejpam-5532	13	7	fuzzy	fuzzy	ADJ
ejpam-5532	13	8	set	set	NOUN
ejpam-5532	13	9	was	be	AUX
ejpam-5532	13	10	pioneered	pioneer	VERB
ejpam-5532	13	11	by	by	ADP
ejpam-5532	13	12	goguen	goguen	PROPN
ejpam-5532	14	1	[	[	X
ejpam-5532	14	2	7	7	X
ejpam-5532	14	3	]	]	PUNCT
ejpam-5532	14	4	in	in	ADP
ejpam-5532	14	5	the	the	DET
ejpam-5532	14	6	year	year	NOUN
ejpam-5532	14	7	1967	1967	NUM
ejpam-5532	14	8	.in	.in	PUNCT
ejpam-5532	15	1	[	[	X
ejpam-5532	15	2	10	10	NUM
ejpam-5532	15	3	]	]	PUNCT
ejpam-5532	15	4	,	,	PUNCT
ejpam-5532	15	5	the	the	DET
ejpam-5532	15	6	authors	author	NOUN
ejpam-5532	15	7	studied	study	VERB
ejpam-5532	15	8	the	the	DET
ejpam-5532	15	9	concept	concept	NOUN
ejpam-5532	15	10	of	of	ADP
ejpam-5532	15	11	an	an	DET
ejpam-5532	15	12	l	l	NOUN
ejpam-5532	15	13	-	-	NOUN
ejpam-5532	15	14	lattice	lattice	NOUN
ejpam-5532	15	15	.	.	PUNCT
ejpam-5532	16	1	this	this	DET
ejpam-5532	16	2	shifts	shift	NOUN
ejpam-5532	16	3	their	their	PRON
ejpam-5532	16	4	studies	study	NOUN
ejpam-5532	16	5	from	from	ADP
ejpam-5532	16	6	the	the	DET
ejpam-5532	16	7	evaluation	evaluation	NOUN
ejpam-5532	16	8	lattice	lattice	NOUN
ejpam-5532	17	1	[	[	X
ejpam-5532	17	2	0	0	NUM
ejpam-5532	17	3	,	,	PUNCT
ejpam-5532	17	4	1	1	NUM
ejpam-5532	17	5	]	]	PUNCT
ejpam-5532	17	6	to	to	ADP
ejpam-5532	17	7	a	a	DET
ejpam-5532	17	8	more	more	ADV
ejpam-5532	17	9	general	general	ADJ
ejpam-5532	17	10	lattice	lattice	PROPN
ejpam-5532	17	11	l.	l.	PROPN
ejpam-5532	17	12	moreover	moreover	ADV
ejpam-5532	17	13	in	in	ADP
ejpam-5532	17	14	[	[	X
ejpam-5532	17	15	10	10	NUM
ejpam-5532	17	16	]	]	PUNCT
ejpam-5532	17	17	,	,	PUNCT
ejpam-5532	17	18	authors	author	NOUN
ejpam-5532	17	19	made	make	VERB
ejpam-5532	17	20	one	one	NUM
ejpam-5532	17	21	more	more	ADJ
ejpam-5532	17	22	transition	transition	NOUN
ejpam-5532	17	23	by	by	ADP
ejpam-5532	17	24	studying	study	VERB
ejpam-5532	17	25	the	the	DET
ejpam-5532	17	26	notions	notion	NOUN
ejpam-5532	17	27	of	of	ADP
ejpam-5532	17	28	l	l	NOUN
ejpam-5532	17	29	-	-	NOUN
ejpam-5532	17	30	substructures	substructure	NOUN
ejpam-5532	17	31	in	in	ADP
ejpam-5532	17	32	an	an	DET
ejpam-5532	17	33	l	l	NOUN
ejpam-5532	17	34	-	-	NOUN
ejpam-5532	17	35	lattice	lattice	NOUN
ejpam-5532	17	36	instead	instead	ADV
ejpam-5532	17	37	of	of	ADP
ejpam-5532	17	38	fuzzy	fuzzy	ADJ
ejpam-5532	17	39	substructures	substructure	NOUN
ejpam-5532	17	40	of	of	ADP
ejpam-5532	17	41	an	an	DET
ejpam-5532	17	42	ordinary	ordinary	ADJ
ejpam-5532	17	43	lattice	lattice	NOUN
ejpam-5532	17	44	.	.	PUNCT
ejpam-5532	18	1	thus	thus	ADV
ejpam-5532	18	2	,	,	PUNCT
ejpam-5532	18	3	the	the	DET
ejpam-5532	18	4	parent	parent	NOUN
ejpam-5532	18	5	structure	structure	NOUN
ejpam-5532	18	6	also	also	ADV
ejpam-5532	18	7	shifts	shift	NOUN
ejpam-5532	18	8	from	from	ADP
ejpam-5532	18	9	a	a	DET
ejpam-5532	18	10	lattice	lattice	NOUN
ejpam-5532	18	11	to	to	ADP
ejpam-5532	18	12	an	an	DET
ejpam-5532	18	13	l	l	NOUN
ejpam-5532	18	14	-	-	NOUN
ejpam-5532	18	15	lattice	lattice	NOUN
ejpam-5532	18	16	.	.	PUNCT
ejpam-5532	19	1	it	it	PRON
ejpam-5532	19	2	is	be	AUX
ejpam-5532	19	3	worthwhile	worthwhile	ADJ
ejpam-5532	19	4	to	to	PART
ejpam-5532	19	5	mention	mention	VERB
ejpam-5532	19	6	here	here	ADV
ejpam-5532	19	7	that	that	SCONJ
ejpam-5532	19	8	under	under	ADP
ejpam-5532	19	9	this	this	DET
ejpam-5532	19	10	arrangement	arrangement	NOUN
ejpam-5532	19	11	,	,	PUNCT
ejpam-5532	19	12	some	some	DET
ejpam-5532	19	13	notions	notion	NOUN
ejpam-5532	19	14	∗corresponding	∗corresponde	VERB
ejpam-5532	19	15	author	author	NOUN
ejpam-5532	19	16	.	.	PUNCT
ejpam-5532	20	1	doi	doi	NOUN
ejpam-5532	20	2	:	:	PUNCT
ejpam-5532	20	3	https://doi.org/10.29020/nybg.ejpam.v18i1.5532	https://doi.org/10.29020/nybg.ejpam.v18i1.5532	NOUN
ejpam-5532	20	4	email	email	NOUN
ejpam-5532	20	5	addresses	address	VERB
ejpam-5532	20	6	:	:	PUNCT
ejpam-5532	21	1	jainaparna@yahoo.com	jainaparna@yahoo.com	X
ejpam-5532	21	2	(	(	PUNCT
ejpam-5532	21	3	a.	a.	NOUN
ejpam-5532	21	4	jain	jain	PROPN
ejpam-5532	21	5	)	)	PUNCT
ejpam-5532	21	6	,	,	PUNCT
ejpam-5532	21	7	ij.umar@yahoo.com	ij.umar@yahoo.com	X
ejpam-5532	21	8	(	(	PUNCT
ejpam-5532	21	9	i.	i.	PROPN
ejpam-5532	21	10	jahan	jahan	PROPN
ejpam-5532	21	11	)	)	PUNCT
ejpam-5532	21	12	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-5532	22	1	1	1	NUM
ejpam-5532	22	2	copyright	copyright	NOUN
ejpam-5532	22	3	:	:	PUNCT
ejpam-5532	22	4	©	©	PROPN
ejpam-5532	22	5	2025	2025	NUM
ejpam-5532	22	6	the	the	DET
ejpam-5532	22	7	author(s	author(s	NOUN
ejpam-5532	22	8	)	)	PUNCT
ejpam-5532	22	9	.	.	PUNCT
ejpam-5532	23	1	(	(	PUNCT
ejpam-5532	23	2	cc	cc	NOUN
ejpam-5532	23	3	by	by	ADP
ejpam-5532	23	4	-	-	PUNCT
ejpam-5532	23	5	nc	nc	PROPN
ejpam-5532	23	6	4.0	4.0	NUM
ejpam-5532	23	7	)	)	PUNCT
ejpam-5532	23	8	a.	a.	NOUN
ejpam-5532	23	9	jain	jain	PROPN
ejpam-5532	23	10	,	,	PUNCT
ejpam-5532	23	11	i.	i.	PROPN
ejpam-5532	23	12	jahan	jahan	PROPN
ejpam-5532	23	13	/	/	SYM
ejpam-5532	23	14	eur	eur	PROPN
ejpam-5532	23	15	.	.	PUNCT
ejpam-5532	24	1	j.	j.	PROPN
ejpam-5532	24	2	pure	pure	PROPN
ejpam-5532	24	3	appl	appl	PROPN
ejpam-5532	24	4	.	.	PROPN
ejpam-5532	24	5	math	math	PROPN
ejpam-5532	24	6	,	,	PUNCT
ejpam-5532	24	7	18	18	NUM
ejpam-5532	24	8	(	(	PUNCT
ejpam-5532	24	9	1	1	NUM
ejpam-5532	24	10	)	)	PUNCT
ejpam-5532	24	11	(	(	PUNCT
ejpam-5532	24	12	2025	2025	NUM
ejpam-5532	24	13	)	)	PUNCT
ejpam-5532	24	14	,	,	PUNCT
ejpam-5532	24	15	5532	5532	NUM
ejpam-5532	24	16	2	2	NUM
ejpam-5532	24	17	of	of	ADP
ejpam-5532	24	18	20	20	NUM
ejpam-5532	24	19	such	such	ADJ
ejpam-5532	24	20	as	as	ADP
ejpam-5532	24	21	l	l	ADJ
ejpam-5532	24	22	-	-	ADJ
ejpam-5532	24	23	maximal	maximal	ADJ
ejpam-5532	24	24	ideal	ideal	NOUN
ejpam-5532	24	25	can	can	AUX
ejpam-5532	24	26	be	be	AUX
ejpam-5532	24	27	defined	define	VERB
ejpam-5532	24	28	more	more	ADV
ejpam-5532	24	29	meaningfully	meaningfully	ADV
ejpam-5532	24	30	in	in	ADP
ejpam-5532	24	31	the	the	DET
ejpam-5532	24	32	l	l	NOUN
ejpam-5532	24	33	-	-	NOUN
ejpam-5532	24	34	setting	setting	NOUN
ejpam-5532	24	35	.	.	PUNCT
ejpam-5532	25	1	in	in	ADP
ejpam-5532	25	2	past	past	ADJ
ejpam-5532	25	3	few	few	ADJ
ejpam-5532	25	4	years	year	NOUN
ejpam-5532	25	5	,	,	PUNCT
ejpam-5532	25	6	ajmal	ajmal	PROPN
ejpam-5532	25	7	and	and	CCONJ
ejpam-5532	25	8	jahan	jahan	PROPN
ejpam-5532	25	9	have	have	AUX
ejpam-5532	25	10	successfully	successfully	ADV
ejpam-5532	25	11	developed	develop	VERB
ejpam-5532	25	12	the	the	DET
ejpam-5532	25	13	theory	theory	NOUN
ejpam-5532	25	14	of	of	ADP
ejpam-5532	25	15	l	l	NOUN
ejpam-5532	25	16	-	-	NOUN
ejpam-5532	25	17	subgroups	subgroup	NOUN
ejpam-5532	25	18	in	in	ADP
ejpam-5532	25	19	[	[	X
ejpam-5532	25	20	1–3	1–3	NOUN
ejpam-5532	25	21	,	,	PUNCT
ejpam-5532	25	22	8	8	NUM
ejpam-5532	25	23	,	,	PUNCT
ejpam-5532	25	24	9	9	NUM
ejpam-5532	25	25	,	,	PUNCT
ejpam-5532	25	26	12	12	NUM
ejpam-5532	25	27	]	]	PUNCT
ejpam-5532	25	28	]	]	PUNCT
ejpam-5532	25	29	.	.	PUNCT
ejpam-5532	26	1	they	they	PRON
ejpam-5532	26	2	have	have	AUX
ejpam-5532	26	3	taken	take	VERB
ejpam-5532	26	4	the	the	DET
ejpam-5532	26	5	theory	theory	NOUN
ejpam-5532	26	6	of	of	ADP
ejpam-5532	26	7	l	l	NOUN
ejpam-5532	26	8	-	-	NOUN
ejpam-5532	26	9	subgroups	subgroup	NOUN
ejpam-5532	26	10	towards	towards	ADP
ejpam-5532	26	11	completion	completion	NOUN
ejpam-5532	26	12	by	by	ADP
ejpam-5532	26	13	studying	study	VERB
ejpam-5532	26	14	the	the	DET
ejpam-5532	26	15	concepts	concept	NOUN
ejpam-5532	26	16	of	of	ADP
ejpam-5532	26	17	characteristic	characteristic	ADJ
ejpam-5532	26	18	subgroups	subgroup	NOUN
ejpam-5532	26	19	,	,	PUNCT
ejpam-5532	26	20	normalizer	normalizer	NOUN
ejpam-5532	26	21	of	of	ADP
ejpam-5532	26	22	a	a	DET
ejpam-5532	26	23	subgroup	subgroup	NOUN
ejpam-5532	26	24	,	,	PUNCT
ejpam-5532	26	25	nilpotent	nilpotent	ADJ
ejpam-5532	26	26	subgroups	subgroup	NOUN
ejpam-5532	26	27	,	,	PUNCT
ejpam-5532	26	28	solvable	solvable	ADJ
ejpam-5532	26	29	subgroups	subgroup	NOUN
ejpam-5532	26	30	,	,	PUNCT
ejpam-5532	26	31	normal	normal	ADJ
ejpam-5532	26	32	closure	closure	NOUN
ejpam-5532	26	33	of	of	ADP
ejpam-5532	26	34	a	a	DET
ejpam-5532	26	35	subgroup	subgroup	NOUN
ejpam-5532	26	36	etc	etc	X
ejpam-5532	26	37	.	.	X
ejpam-5532	26	38	,	,	PUNCT
ejpam-5532	26	39	within	within	ADP
ejpam-5532	26	40	the	the	DET
ejpam-5532	26	41	framework	framework	NOUN
ejpam-5532	26	42	of	of	ADP
ejpam-5532	26	43	l	l	NOUN
ejpam-5532	26	44	-	-	NOUN
ejpam-5532	26	45	groups	group	NOUN
ejpam-5532	26	46	.	.	PUNCT
ejpam-5532	27	1	in	in	ADP
ejpam-5532	27	2	[	[	X
ejpam-5532	27	3	9	9	NUM
ejpam-5532	27	4	]	]	PUNCT
ejpam-5532	27	5	,	,	PUNCT
ejpam-5532	27	6	jahan	jahan	PROPN
ejpam-5532	27	7	and	and	CCONJ
ejpam-5532	27	8	manas	mana	NOUN
ejpam-5532	27	9	studied	study	VERB
ejpam-5532	27	10	maximal	maximal	ADJ
ejpam-5532	27	11	and	and	CCONJ
ejpam-5532	27	12	frattini	frattini	ADJ
ejpam-5532	27	13	l	l	NOUN
ejpam-5532	27	14	-	-	NOUN
ejpam-5532	27	15	subgroups	subgroup	NOUN
ejpam-5532	27	16	of	of	ADP
ejpam-5532	27	17	an	an	DET
ejpam-5532	27	18	l	l	NOUN
ejpam-5532	27	19	-	-	NOUN
ejpam-5532	27	20	group	group	NOUN
ejpam-5532	27	21	.	.	PUNCT
ejpam-5532	28	1	in	in	ADP
ejpam-5532	28	2	[	[	X
ejpam-5532	28	3	10	10	NUM
ejpam-5532	28	4	]	]	PUNCT
ejpam-5532	28	5	,	,	PUNCT
ejpam-5532	28	6	the	the	DET
ejpam-5532	28	7	notions	notion	NOUN
ejpam-5532	28	8	of	of	ADP
ejpam-5532	28	9	an	an	DET
ejpam-5532	28	10	l	l	NOUN
ejpam-5532	28	11	-	-	ADJ
ejpam-5532	28	12	maximal	maximal	ADJ
ejpam-5532	28	13	ideal	ideal	NOUN
ejpam-5532	28	14	and	and	CCONJ
ejpam-5532	28	15	l	l	ADJ
ejpam-5532	28	16	-	-	ADJ
ejpam-5532	28	17	prime	prime	ADJ
ejpam-5532	28	18	ideal	ideal	NOUN
ejpam-5532	28	19	in	in	ADP
ejpam-5532	28	20	an	an	DET
ejpam-5532	28	21	l	l	NOUN
ejpam-5532	28	22	-	-	PUNCT
ejpam-5532	28	23	lattice	lattice	NOUN
ejpam-5532	28	24	are	be	AUX
ejpam-5532	28	25	defined	define	VERB
ejpam-5532	28	26	and	and	CCONJ
ejpam-5532	28	27	various	various	ADJ
ejpam-5532	28	28	related	related	ADJ
ejpam-5532	28	29	results	result	NOUN
ejpam-5532	28	30	are	be	AUX
ejpam-5532	28	31	studied	study	VERB
ejpam-5532	28	32	.	.	PUNCT
ejpam-5532	29	1	in	in	ADP
ejpam-5532	29	2	order	order	NOUN
ejpam-5532	29	3	to	to	PART
ejpam-5532	29	4	take	take	VERB
ejpam-5532	29	5	such	such	ADJ
ejpam-5532	29	6	studies	study	NOUN
ejpam-5532	29	7	further	far	ADV
ejpam-5532	29	8	,	,	PUNCT
ejpam-5532	29	9	in	in	ADP
ejpam-5532	29	10	the	the	DET
ejpam-5532	29	11	present	present	ADJ
ejpam-5532	29	12	paper	paper	NOUN
ejpam-5532	29	13	,	,	PUNCT
ejpam-5532	29	14	we	we	PRON
ejpam-5532	29	15	introduce	introduce	VERB
ejpam-5532	29	16	the	the	DET
ejpam-5532	29	17	notion	notion	NOUN
ejpam-5532	29	18	of	of	ADP
ejpam-5532	29	19	an	an	DET
ejpam-5532	29	20	l	l	ADJ
ejpam-5532	29	21	-	-	ADJ
ejpam-5532	29	22	convex	convex	ADJ
ejpam-5532	29	23	sublattice	sublattice	NOUN
ejpam-5532	29	24	in	in	ADP
ejpam-5532	29	25	an	an	DET
ejpam-5532	29	26	l-lattice.then	l-lattice.then	ADV
ejpam-5532	29	27	this	this	DET
ejpam-5532	29	28	notion	notion	NOUN
ejpam-5532	29	29	of	of	ADP
ejpam-5532	29	30	convex	convex	ADJ
ejpam-5532	29	31	l	l	NOUN
ejpam-5532	29	32	-	-	NOUN
ejpam-5532	29	33	sublattice	sublattice	NOUN
ejpam-5532	29	34	is	be	AUX
ejpam-5532	29	35	used	use	VERB
ejpam-5532	29	36	to	to	PART
ejpam-5532	29	37	demonstrate	demonstrate	VERB
ejpam-5532	29	38	that	that	SCONJ
ejpam-5532	29	39	the	the	DET
ejpam-5532	29	40	unique	unique	ADJ
ejpam-5532	29	41	representation	representation	NOUN
ejpam-5532	29	42	theorem	theorem	NOUN
ejpam-5532	29	43	of	of	ADP
ejpam-5532	29	44	classical	classical	ADJ
ejpam-5532	29	45	lattice	lattice	NOUN
ejpam-5532	29	46	theory	theory	NOUN
ejpam-5532	29	47	for	for	ADP
ejpam-5532	29	48	convex	convex	ADJ
ejpam-5532	29	49	sublattices	sublattice	NOUN
ejpam-5532	29	50	also	also	ADV
ejpam-5532	29	51	holds	hold	VERB
ejpam-5532	29	52	under	under	ADP
ejpam-5532	29	53	the	the	DET
ejpam-5532	29	54	l	l	NOUN
ejpam-5532	29	55	-	-	ADJ
ejpam-5532	29	56	setting	set	VERB
ejpam-5532	29	57	wherein	wherein	SCONJ
ejpam-5532	29	58	the	the	DET
ejpam-5532	29	59	parent	parent	NOUN
ejpam-5532	29	60	structure	structure	NOUN
ejpam-5532	29	61	is	be	AUX
ejpam-5532	29	62	an	an	DET
ejpam-5532	29	63	l	l	NOUN
ejpam-5532	29	64	-	-	NOUN
ejpam-5532	29	65	lattice	lattice	NOUN
ejpam-5532	29	66	.	.	PUNCT
ejpam-5532	30	1	in	in	ADP
ejpam-5532	30	2	the	the	DET
ejpam-5532	30	3	last	last	ADJ
ejpam-5532	30	4	section	section	NOUN
ejpam-5532	30	5	of	of	ADP
ejpam-5532	30	6	this	this	DET
ejpam-5532	30	7	paper	paper	NOUN
ejpam-5532	30	8	,	,	PUNCT
ejpam-5532	30	9	we	we	PRON
ejpam-5532	30	10	use	use	VERB
ejpam-5532	30	11	the	the	DET
ejpam-5532	30	12	notion	notion	NOUN
ejpam-5532	30	13	of	of	ADP
ejpam-5532	30	14	an	an	DET
ejpam-5532	30	15	order	order	NOUN
ejpam-5532	30	16	reversing	reverse	VERB
ejpam-5532	30	17	involution	involution	NOUN
ejpam-5532	30	18	on	on	ADP
ejpam-5532	30	19	a	a	DET
ejpam-5532	30	20	lattice	lattice	NOUN
ejpam-5532	30	21	to	to	PART
ejpam-5532	30	22	define	define	VERB
ejpam-5532	30	23	the	the	DET
ejpam-5532	30	24	concept	concept	NOUN
ejpam-5532	30	25	of	of	ADP
ejpam-5532	30	26	complement	complement	NOUN
ejpam-5532	30	27	of	of	ADP
ejpam-5532	30	28	an	an	DET
ejpam-5532	30	29	l	l	NOUN
ejpam-5532	30	30	-	-	NOUN
ejpam-5532	30	31	set	set	NOUN
ejpam-5532	30	32	.	.	PUNCT
ejpam-5532	31	1	the	the	DET
ejpam-5532	31	2	notion	notion	NOUN
ejpam-5532	31	3	of	of	ADP
ejpam-5532	31	4	order	order	NOUN
ejpam-5532	31	5	reversing	reverse	VERB
ejpam-5532	31	6	involution	involution	NOUN
ejpam-5532	31	7	occurs	occur	VERB
ejpam-5532	31	8	frequently	frequently	ADV
ejpam-5532	31	9	in	in	ADP
ejpam-5532	31	10	fuzzy	fuzzy	ADJ
ejpam-5532	31	11	topological	topological	ADJ
ejpam-5532	31	12	spaces	space	NOUN
ejpam-5532	31	13	and	and	CCONJ
ejpam-5532	31	14	fuzzy	fuzzy	ADJ
ejpam-5532	31	15	implication	implication	NOUN
ejpam-5532	31	16	algebras	algebra	NOUN
ejpam-5532	32	1	[	[	X
ejpam-5532	32	2	11	11	NUM
ejpam-5532	32	3	,	,	PUNCT
ejpam-5532	32	4	13	13	NUM
ejpam-5532	32	5	,	,	PUNCT
ejpam-5532	32	6	15–17	15–17	NUM
ejpam-5532	32	7	]	]	PUNCT
ejpam-5532	32	8	.	.	PUNCT
ejpam-5532	33	1	thereafter	thereafter	ADV
ejpam-5532	33	2	,	,	PUNCT
ejpam-5532	33	3	we	we	PRON
ejpam-5532	33	4	establish	establish	VERB
ejpam-5532	33	5	some	some	DET
ejpam-5532	33	6	significant	significant	ADJ
ejpam-5532	33	7	analogues	analogue	NOUN
ejpam-5532	33	8	of	of	ADP
ejpam-5532	33	9	results	result	NOUN
ejpam-5532	33	10	of	of	ADP
ejpam-5532	33	11	classical	classical	ADJ
ejpam-5532	33	12	lattice	lattice	NOUN
ejpam-5532	33	13	theory	theory	NOUN
ejpam-5532	33	14	to	to	ADP
ejpam-5532	33	15	l	l	NOUN
ejpam-5532	33	16	-	-	PUNCT
ejpam-5532	33	17	setting	set	VERB
ejpam-5532	33	18	using	use	VERB
ejpam-5532	33	19	complement	complement	NOUN
ejpam-5532	33	20	of	of	ADP
ejpam-5532	33	21	an	an	DET
ejpam-5532	33	22	l	l	NOUN
ejpam-5532	33	23	-	-	NOUN
ejpam-5532	33	24	set	set	NOUN
ejpam-5532	33	25	,	,	PUNCT
ejpam-5532	33	26	thereby	thereby	ADV
ejpam-5532	33	27	taking	take	VERB
ejpam-5532	33	28	the	the	DET
ejpam-5532	33	29	theory	theory	NOUN
ejpam-5532	33	30	of	of	ADP
ejpam-5532	33	31	l	l	NOUN
ejpam-5532	33	32	-	-	NOUN
ejpam-5532	33	33	lattices	lattice	NOUN
ejpam-5532	33	34	to	to	ADP
ejpam-5532	33	35	a	a	DET
ejpam-5532	33	36	more	more	ADV
ejpam-5532	33	37	developed	develop	VERB
ejpam-5532	33	38	stage	stage	NOUN
ejpam-5532	33	39	.	.	PUNCT
ejpam-5532	34	1	2	2	X
ejpam-5532	34	2	.	.	X
ejpam-5532	34	3	preliminaries	preliminary	NOUN
ejpam-5532	34	4	in	in	ADP
ejpam-5532	34	5	this	this	DET
ejpam-5532	34	6	work	work	NOUN
ejpam-5532	34	7	,	,	PUNCT
ejpam-5532	34	8	(	(	PUNCT
ejpam-5532	34	9	m,≤,∧,∨	m,≤,∧,∨	X
ejpam-5532	34	10	)	)	PUNCT
ejpam-5532	34	11	denotes	denote	VERB
ejpam-5532	34	12	a	a	DET
ejpam-5532	34	13	bounded	bounded	ADJ
ejpam-5532	34	14	lattice	lattice	NOUN
ejpam-5532	34	15	and	and	CCONJ
ejpam-5532	34	16	(	(	PUNCT
ejpam-5532	34	17	l,≤,∧,∨	l,≤,∧,∨	NOUN
ejpam-5532	34	18	)	)	PUNCT
ejpam-5532	34	19	a	a	DET
ejpam-5532	34	20	complete	complete	ADJ
ejpam-5532	34	21	lattice	lattice	NOUN
ejpam-5532	34	22	.	.	PUNCT
ejpam-5532	35	1	the	the	DET
ejpam-5532	35	2	maximal	maximal	ADJ
ejpam-5532	35	3	and	and	CCONJ
ejpam-5532	35	4	minimal	minimal	ADJ
ejpam-5532	35	5	elements	element	NOUN
ejpam-5532	35	6	of	of	ADP
ejpam-5532	35	7	both	both	CCONJ
ejpam-5532	35	8	the	the	DET
ejpam-5532	35	9	lattices	lattice	NOUN
ejpam-5532	35	10	l	l	NOUN
ejpam-5532	35	11	and	and	CCONJ
ejpam-5532	35	12	m	m	PROPN
ejpam-5532	35	13	are	be	AUX
ejpam-5532	35	14	denoted	denote	VERB
ejpam-5532	35	15	by	by	ADP
ejpam-5532	35	16	1	1	NUM
ejpam-5532	35	17	and	and	CCONJ
ejpam-5532	35	18	0	0	NUM
ejpam-5532	35	19	respectively	respectively	ADV
ejpam-5532	35	20	.	.	PUNCT
ejpam-5532	36	1	the	the	DET
ejpam-5532	36	2	notations	notation	NOUN
ejpam-5532	36	3	’	'	PUNCT
ejpam-5532	36	4	≤	≤	NUM
ejpam-5532	36	5	’	'	PUNCT
ejpam-5532	36	6	,	,	PUNCT
ejpam-5532	36	7	’	'	PUNCT
ejpam-5532	36	8	∧	∧	PROPN
ejpam-5532	36	9	’	'	PUNCT
ejpam-5532	36	10	and	and	CCONJ
ejpam-5532	36	11	’	'	PUNCT
ejpam-5532	36	12	∨	∨	NOUN
ejpam-5532	36	13	’	'	PUNCT
ejpam-5532	36	14	denote	denote	VERB
ejpam-5532	36	15	the	the	DET
ejpam-5532	36	16	partial	partial	ADJ
ejpam-5532	36	17	order	order	NOUN
ejpam-5532	36	18	,	,	PUNCT
ejpam-5532	36	19	meet	meet	VERB
ejpam-5532	36	20	and	and	CCONJ
ejpam-5532	36	21	join	join	VERB
ejpam-5532	36	22	operations	operation	NOUN
ejpam-5532	36	23	respectively	respectively	ADV
ejpam-5532	36	24	of	of	ADP
ejpam-5532	36	25	both	both	CCONJ
ejpam-5532	36	26	the	the	DET
ejpam-5532	36	27	lattices	lattice	NOUN
ejpam-5532	36	28	l	l	NOUN
ejpam-5532	36	29	and	and	CCONJ
ejpam-5532	36	30	m	m	PROPN
ejpam-5532	36	31	.	.	PUNCT
ejpam-5532	37	1	an	an	DET
ejpam-5532	37	2	l	l	NOUN
ejpam-5532	37	3	-	-	NOUN
ejpam-5532	37	4	subset	subset	NOUN
ejpam-5532	37	5	of	of	ADP
ejpam-5532	37	6	m	m	PROPN
ejpam-5532	37	7	is	be	AUX
ejpam-5532	37	8	defined	define	VERB
ejpam-5532	37	9	as	as	ADP
ejpam-5532	37	10	a	a	DET
ejpam-5532	37	11	mapping	mapping	NOUN
ejpam-5532	37	12	µ	µ	NOUN
ejpam-5532	37	13	:	:	PUNCT
ejpam-5532	37	14	m	m	AUX
ejpam-5532	37	15	→	→	SYM
ejpam-5532	37	16	l.	l.	X
ejpam-5532	37	17	the	the	DET
ejpam-5532	37	18	collection	collection	NOUN
ejpam-5532	37	19	of	of	ADP
ejpam-5532	37	20	all	all	DET
ejpam-5532	37	21	l	l	NOUN
ejpam-5532	37	22	-	-	NOUN
ejpam-5532	37	23	subsets	subset	NOUN
ejpam-5532	37	24	of	of	ADP
ejpam-5532	37	25	m	m	PROPN
ejpam-5532	37	26	is	be	AUX
ejpam-5532	37	27	denoted	denote	VERB
ejpam-5532	37	28	by	by	ADP
ejpam-5532	37	29	lm	lm	INTJ
ejpam-5532	37	30	and	and	CCONJ
ejpam-5532	37	31	is	be	AUX
ejpam-5532	37	32	called	call	VERB
ejpam-5532	37	33	the	the	DET
ejpam-5532	37	34	l	l	NOUN
ejpam-5532	37	35	-	-	NOUN
ejpam-5532	37	36	power	power	NOUN
ejpam-5532	37	37	set	set	NOUN
ejpam-5532	37	38	of	of	ADP
ejpam-5532	37	39	m	m	PROPN
ejpam-5532	37	40	.	.	PUNCT
ejpam-5532	38	1	if	if	SCONJ
ejpam-5532	38	2	µ	µ	NUM
ejpam-5532	38	3	,	,	PUNCT
ejpam-5532	38	4	η	η	PROPN
ejpam-5532	38	5	∈	∈	PROPN
ejpam-5532	38	6	lm	lm	INTJ
ejpam-5532	38	7	,	,	PUNCT
ejpam-5532	38	8	η	η	PROPN
ejpam-5532	38	9	is	be	AUX
ejpam-5532	38	10	said	say	VERB
ejpam-5532	38	11	to	to	PART
ejpam-5532	38	12	be	be	AUX
ejpam-5532	38	13	contained	contain	VERB
ejpam-5532	38	14	in	in	ADP
ejpam-5532	38	15	µ(denoted	µ(denote	VERB
ejpam-5532	38	16	by	by	ADP
ejpam-5532	38	17	η	η	PROPN
ejpam-5532	38	18	⊆	⊆	PROPN
ejpam-5532	38	19	µ	µ	NUM
ejpam-5532	38	20	)	)	PUNCT
ejpam-5532	38	21	,	,	PUNCT
ejpam-5532	38	22	if	if	SCONJ
ejpam-5532	38	23	η(x	η(x	NOUN
ejpam-5532	38	24	)	)	PUNCT
ejpam-5532	38	25	≤	≤	NOUN
ejpam-5532	38	26	µ(x	µ(x	NOUN
ejpam-5532	38	27	)	)	PUNCT
ejpam-5532	38	28	,	,	PUNCT
ejpam-5532	38	29	∀	∀	PUNCT
ejpam-5532	38	30	x	x	SYM
ejpam-5532	39	1	∈	∈	NOUN
ejpam-5532	39	2	m	m	VERB
ejpam-5532	39	3	.	.	PUNCT
ejpam-5532	40	1	moreover	moreover	ADV
ejpam-5532	40	2	,	,	PUNCT
ejpam-5532	40	3	η	η	PROPN
ejpam-5532	40	4	is	be	AUX
ejpam-5532	40	5	said	say	VERB
ejpam-5532	40	6	to	to	PART
ejpam-5532	40	7	be	be	AUX
ejpam-5532	40	8	properly	properly	ADV
ejpam-5532	40	9	contained	contain	VERB
ejpam-5532	40	10	in	in	ADP
ejpam-5532	40	11	µ	µ	NUM
ejpam-5532	40	12	,	,	PUNCT
ejpam-5532	40	13	if	if	SCONJ
ejpam-5532	40	14	η	η	PROPN
ejpam-5532	40	15	⊆	⊆	PROPN
ejpam-5532	40	16	µ	µ	X
ejpam-5532	40	17	and	and	CCONJ
ejpam-5532	40	18	there	there	PRON
ejpam-5532	40	19	exists	exist	VERB
ejpam-5532	40	20	x	x	X
ejpam-5532	40	21	∈	∈	NOUN
ejpam-5532	40	22	m	m	VERB
ejpam-5532	40	23	such	such	ADJ
ejpam-5532	40	24	that	that	SCONJ
ejpam-5532	40	25	η(x	η(x	NOUN
ejpam-5532	40	26	)	)	PUNCT
ejpam-5532	40	27	<	<	X
ejpam-5532	40	28	µ(x	µ(x	NOUN
ejpam-5532	40	29	)	)	PUNCT
ejpam-5532	40	30	.	.	PUNCT
ejpam-5532	41	1	if	if	SCONJ
ejpam-5532	41	2	η	η	PROPN
ejpam-5532	41	3	⊆	⊆	NUM
ejpam-5532	41	4	µ	µ	NUM
ejpam-5532	41	5	,	,	PUNCT
ejpam-5532	41	6	then	then	ADV
ejpam-5532	41	7	η	η	PROPN
ejpam-5532	41	8	is	be	AUX
ejpam-5532	41	9	said	say	VERB
ejpam-5532	41	10	to	to	PART
ejpam-5532	41	11	be	be	AUX
ejpam-5532	41	12	an	an	DET
ejpam-5532	41	13	l	l	NOUN
ejpam-5532	41	14	-	-	NOUN
ejpam-5532	41	15	subset	subset	NOUN
ejpam-5532	41	16	of	of	ADP
ejpam-5532	41	17	an	an	DET
ejpam-5532	41	18	l	l	NOUN
ejpam-5532	41	19	-	-	ADJ
ejpam-5532	41	20	set	set	VERB
ejpam-5532	41	21	µ.	µ.	NOUN
ejpam-5532	41	22	the	the	DET
ejpam-5532	41	23	set	set	NOUN
ejpam-5532	41	24	of	of	ADP
ejpam-5532	41	25	all	all	DET
ejpam-5532	41	26	l	l	NOUN
ejpam-5532	41	27	-	-	NOUN
ejpam-5532	41	28	subsets	subset	NOUN
ejpam-5532	41	29	of	of	ADP
ejpam-5532	41	30	µ	µ	NOUN
ejpam-5532	41	31	is	be	AUX
ejpam-5532	41	32	called	call	VERB
ejpam-5532	41	33	the	the	DET
ejpam-5532	41	34	l	l	NOUN
ejpam-5532	41	35	-	-	NOUN
ejpam-5532	41	36	power	power	NOUN
ejpam-5532	41	37	set	set	NOUN
ejpam-5532	41	38	of	of	ADP
ejpam-5532	41	39	µ	µ	NUM
ejpam-5532	41	40	and	and	CCONJ
ejpam-5532	41	41	is	be	AUX
ejpam-5532	41	42	denoted	denote	VERB
ejpam-5532	41	43	by	by	ADP
ejpam-5532	41	44	lµ.	lµ.	NOUN
ejpam-5532	41	45	if	if	SCONJ
ejpam-5532	41	46	µ	µ	X
ejpam-5532	41	47	∈	∈	NOUN
ejpam-5532	41	48	lm	lm	INTJ
ejpam-5532	41	49	and	and	CCONJ
ejpam-5532	41	50	α	α	PRON
ejpam-5532	41	51	∈	∈	PROPN
ejpam-5532	41	52	l	l	NOUN
ejpam-5532	41	53	,	,	PUNCT
ejpam-5532	41	54	the	the	DET
ejpam-5532	41	55	level	level	NOUN
ejpam-5532	41	56	subset	subset	VERB
ejpam-5532	41	57	µα	µα	ADP
ejpam-5532	41	58	and	and	CCONJ
ejpam-5532	41	59	the	the	DET
ejpam-5532	41	60	strong	strong	ADJ
ejpam-5532	41	61	level	level	NOUN
ejpam-5532	41	62	subset	subset	VERB
ejpam-5532	41	63	µ	µ	X
ejpam-5532	41	64	>	>	X
ejpam-5532	41	65	α	α	PROPN
ejpam-5532	41	66	are	be	AUX
ejpam-5532	41	67	defined	define	VERB
ejpam-5532	41	68	as	as	SCONJ
ejpam-5532	41	69	follows	follow	VERB
ejpam-5532	41	70	:	:	PUNCT
ejpam-5532	41	71	µα	µα	ADP
ejpam-5532	41	72	=	=	SYM
ejpam-5532	41	73	{	{	PUNCT
ejpam-5532	41	74	x	x	PROPN
ejpam-5532	41	75	∈	∈	PROPN
ejpam-5532	41	76	m/µ(x	m/µ(x	NOUN
ejpam-5532	41	77	)	)	PUNCT
ejpam-5532	41	78	≥	≥	NOUN
ejpam-5532	41	79	α	α	NOUN
ejpam-5532	41	80	}	}	PUNCT
ejpam-5532	41	81	and	and	CCONJ
ejpam-5532	41	82	µ	µ	X
ejpam-5532	41	83	>	>	X
ejpam-5532	41	84	α	α	NOUN
ejpam-5532	41	85	=	=	SYM
ejpam-5532	41	86	{	{	PUNCT
ejpam-5532	41	87	x	x	PUNCT
ejpam-5532	41	88	∈	∈	PROPN
ejpam-5532	41	89	m/µ(x	m/µ(x	NOUN
ejpam-5532	41	90	)	)	PUNCT
ejpam-5532	41	91	>	>	X
ejpam-5532	41	92	α	α	X
ejpam-5532	41	93	}	}	PUNCT
ejpam-5532	41	94	.	.	PUNCT
ejpam-5532	42	1	clearly	clearly	ADV
ejpam-5532	42	2	,	,	PUNCT
ejpam-5532	42	3	µ	µ	X
ejpam-5532	42	4	>	>	X
ejpam-5532	42	5	α	α	PROPN
ejpam-5532	42	6	⊆	⊆	PROPN
ejpam-5532	42	7	µα	µα	NOUN
ejpam-5532	42	8	,	,	PUNCT
ejpam-5532	42	9	∀	∀	X
ejpam-5532	42	10	α	α	NOUN
ejpam-5532	42	11	∈	∈	NOUN
ejpam-5532	42	12	l	l	NOUN
ejpam-5532	42	13	and	and	CCONJ
ejpam-5532	42	14	if	if	SCONJ
ejpam-5532	42	15	α	α	PRON
ejpam-5532	42	16	≤	≤	X
ejpam-5532	42	17	β	β	X
ejpam-5532	42	18	in	in	ADP
ejpam-5532	42	19	l	l	NOUN
ejpam-5532	42	20	,	,	PUNCT
ejpam-5532	42	21	then	then	ADV
ejpam-5532	42	22	µβ	µβ	VERB
ejpam-5532	42	23	⊆	⊆	NUM
ejpam-5532	42	24	µα	µα	ADP
ejpam-5532	42	25	and	and	CCONJ
ejpam-5532	42	26	µ	µ	X
ejpam-5532	42	27	>	>	X
ejpam-5532	42	28	β	β	X
ejpam-5532	42	29	⊆	⊆	NUM
ejpam-5532	42	30	µ	µ	X
ejpam-5532	42	31	>	>	X
ejpam-5532	42	32	α	α	PROPN
ejpam-5532	42	33	.	.	PUNCT
ejpam-5532	43	1	a.	a.	PROPN
ejpam-5532	43	2	jain	jain	PROPN
ejpam-5532	43	3	,	,	PUNCT
ejpam-5532	43	4	i.	i.	PROPN
ejpam-5532	43	5	jahan	jahan	PROPN
ejpam-5532	43	6	/	/	SYM
ejpam-5532	43	7	eur	eur	PROPN
ejpam-5532	43	8	.	.	PUNCT
ejpam-5532	44	1	j.	j.	PROPN
ejpam-5532	44	2	pure	pure	PROPN
ejpam-5532	44	3	appl	appl	PROPN
ejpam-5532	44	4	.	.	PROPN
ejpam-5532	44	5	math	math	PROPN
ejpam-5532	44	6	,	,	PUNCT
ejpam-5532	44	7	18	18	NUM
ejpam-5532	44	8	(	(	PUNCT
ejpam-5532	44	9	1	1	NUM
ejpam-5532	44	10	)	)	PUNCT
ejpam-5532	44	11	(	(	PUNCT
ejpam-5532	44	12	2025	2025	NUM
ejpam-5532	44	13	)	)	PUNCT
ejpam-5532	44	14	,	,	PUNCT
ejpam-5532	44	15	5532	5532	NUM
ejpam-5532	44	16	3	3	NUM
ejpam-5532	44	17	of	of	ADP
ejpam-5532	44	18	20	20	NUM
ejpam-5532	44	19	if	if	SCONJ
ejpam-5532	44	20	µ	µ	X
ejpam-5532	44	21	∈	∈	NOUN
ejpam-5532	44	22	lm	lm	INTJ
ejpam-5532	44	23	,	,	PUNCT
ejpam-5532	44	24	then	then	ADV
ejpam-5532	44	25	∨x∈mµ(x	∨x∈mµ(x	NOUN
ejpam-5532	44	26	)	)	PUNCT
ejpam-5532	44	27	and	and	CCONJ
ejpam-5532	44	28	∧x∈mµ(x	∧x∈mµ(x	PROPN
ejpam-5532	44	29	)	)	PUNCT
ejpam-5532	44	30	are	be	AUX
ejpam-5532	44	31	called	call	VERB
ejpam-5532	44	32	the	the	DET
ejpam-5532	44	33	tip	tip	NOUN
ejpam-5532	44	34	and	and	CCONJ
ejpam-5532	44	35	tail	tail	NOUN
ejpam-5532	44	36	of	of	ADP
ejpam-5532	44	37	µ	µ	NOUN
ejpam-5532	44	38	,	,	PUNCT
ejpam-5532	44	39	respectively	respectively	ADV
ejpam-5532	44	40	.	.	PUNCT
ejpam-5532	45	1	the	the	DET
ejpam-5532	45	2	arbitrary	arbitrary	ADJ
ejpam-5532	45	3	union	union	NOUN
ejpam-5532	45	4	∪i∈i(µi	∪i∈i(µi	X
ejpam-5532	45	5	)	)	PUNCT
ejpam-5532	45	6	and	and	CCONJ
ejpam-5532	45	7	intersection	intersection	NOUN
ejpam-5532	45	8	∩i∈i(µi	∩i∈i(µi	PROPN
ejpam-5532	45	9	)	)	PUNCT
ejpam-5532	45	10	of	of	ADP
ejpam-5532	45	11	a	a	DET
ejpam-5532	45	12	family	family	NOUN
ejpam-5532	45	13	{	{	PUNCT
ejpam-5532	45	14	µi}i∈i	µi}i∈i	ADV
ejpam-5532	45	15	of	of	ADP
ejpam-5532	45	16	l	l	NOUN
ejpam-5532	45	17	-	-	NOUN
ejpam-5532	45	18	subsets	subset	NOUN
ejpam-5532	45	19	of	of	ADP
ejpam-5532	45	20	m	m	NOUN
ejpam-5532	45	21	are	be	AUX
ejpam-5532	45	22	given	give	VERB
ejpam-5532	45	23	by	by	ADP
ejpam-5532	45	24	:	:	PUNCT
ejpam-5532	45	25	(	(	PUNCT
ejpam-5532	45	26	∪i∈iµi)(x	∪i∈iµi)(x	NOUN
ejpam-5532	45	27	)	)	PUNCT
ejpam-5532	45	28	=	=	SYM
ejpam-5532	45	29	∨i∈iµi(x	∨i∈iµi(x	X
ejpam-5532	45	30	)	)	PUNCT
ejpam-5532	45	31	and	and	CCONJ
ejpam-5532	45	32	(	(	PUNCT
ejpam-5532	45	33	∩i∈iµi)(x	∩i∈iµi)(x	NOUN
ejpam-5532	45	34	)	)	PUNCT
ejpam-5532	45	35	=	=	SYM
ejpam-5532	45	36	∧i∈iµi(x	∧i∈iµi(x	NOUN
ejpam-5532	45	37	)	)	PUNCT
ejpam-5532	45	38	.	.	PUNCT
ejpam-5532	46	1	definition	definition	NOUN
ejpam-5532	46	2	1	1	NUM
ejpam-5532	46	3	(	(	PUNCT
ejpam-5532	46	4	[	[	X
ejpam-5532	46	5	4	4	NUM
ejpam-5532	46	6	]	]	NUM
ejpam-5532	46	7	)	)	PUNCT
ejpam-5532	46	8	.	.	PUNCT
ejpam-5532	47	1	let	let	VERB
ejpam-5532	47	2	µ	µ	X
ejpam-5532	47	3	∈	∈	X
ejpam-5532	47	4	lm	lm	INTJ
ejpam-5532	47	5	.	.	PUNCT
ejpam-5532	48	1	then	then	ADV
ejpam-5532	48	2	,	,	PUNCT
ejpam-5532	48	3	µ	µ	X
ejpam-5532	48	4	is	be	AUX
ejpam-5532	48	5	said	say	VERB
ejpam-5532	48	6	to	to	PART
ejpam-5532	48	7	be	be	AUX
ejpam-5532	48	8	an	an	DET
ejpam-5532	48	9	l	l	NOUN
ejpam-5532	48	10	-	-	NOUN
ejpam-5532	48	11	sublattice	sublattice	NOUN
ejpam-5532	48	12	of	of	ADP
ejpam-5532	48	13	m	m	NOUN
ejpam-5532	48	14	if	if	SCONJ
ejpam-5532	48	15	∀	∀	NOUN
ejpam-5532	48	16	x	x	X
ejpam-5532	48	17	,	,	PUNCT
ejpam-5532	48	18	y	y	PROPN
ejpam-5532	48	19	∈	∈	PROPN
ejpam-5532	48	20	m	m	VERB
ejpam-5532	48	21	µ(x	µ(x	X
ejpam-5532	48	22	∨	∨	NUM
ejpam-5532	48	23	y	y	PROPN
ejpam-5532	48	24	)	)	PUNCT
ejpam-5532	48	25	≥	≥	NOUN
ejpam-5532	48	26	µ(x	µ(x	NOUN
ejpam-5532	48	27	)	)	PUNCT
ejpam-5532	48	28	∧	∧	PROPN
ejpam-5532	48	29	µ(y	µ(y	PROPN
ejpam-5532	48	30	)	)	PUNCT
ejpam-5532	48	31	and	and	CCONJ
ejpam-5532	48	32	µ(x	µ(x	ADJ
ejpam-5532	48	33	∧	∧	PROPN
ejpam-5532	48	34	y	y	PROPN
ejpam-5532	48	35	)	)	PUNCT
ejpam-5532	48	36	≥	≥	NOUN
ejpam-5532	48	37	µ(x	µ(x	NOUN
ejpam-5532	48	38	)	)	PUNCT
ejpam-5532	48	39	∧	∧	PROPN
ejpam-5532	48	40	µ(y	µ(y	PROPN
ejpam-5532	48	41	)	)	PUNCT
ejpam-5532	48	42	.	.	PUNCT
ejpam-5532	49	1	let	let	VERB
ejpam-5532	50	1	l(m	l(m	PROPN
ejpam-5532	50	2	)	)	PUNCT
ejpam-5532	50	3	denote	denote	VERB
ejpam-5532	50	4	the	the	DET
ejpam-5532	50	5	set	set	NOUN
ejpam-5532	50	6	of	of	ADP
ejpam-5532	50	7	all	all	DET
ejpam-5532	50	8	l	l	NOUN
ejpam-5532	50	9	-	-	NOUN
ejpam-5532	50	10	sublattices	sublattice	NOUN
ejpam-5532	50	11	of	of	ADP
ejpam-5532	50	12	m	m	PROPN
ejpam-5532	50	13	.	.	PUNCT
ejpam-5532	51	1	if	if	SCONJ
ejpam-5532	51	2	µ	µ	PRON
ejpam-5532	51	3	∈	∈	PROPN
ejpam-5532	51	4	l(m	l(m	PROPN
ejpam-5532	51	5	)	)	PUNCT
ejpam-5532	51	6	,	,	PUNCT
ejpam-5532	51	7	µ	µ	PROPN
ejpam-5532	51	8	is	be	AUX
ejpam-5532	51	9	called	call	VERB
ejpam-5532	51	10	an	an	DET
ejpam-5532	51	11	l	l	NOUN
ejpam-5532	51	12	-	-	NOUN
ejpam-5532	51	13	lattice	lattice	NOUN
ejpam-5532	51	14	and	and	CCONJ
ejpam-5532	51	15	is	be	AUX
ejpam-5532	51	16	denoted	denote	VERB
ejpam-5532	51	17	by	by	ADP
ejpam-5532	51	18	l(µ,m	l(µ,m	ADJ
ejpam-5532	51	19	)	)	PUNCT
ejpam-5532	51	20	.	.	PUNCT
ejpam-5532	52	1	if	if	SCONJ
ejpam-5532	52	2	µ	µ	NUM
ejpam-5532	52	3	,	,	PUNCT
ejpam-5532	52	4	η	η	PROPN
ejpam-5532	52	5	∈	∈	PROPN
ejpam-5532	52	6	l(m	l(m	PROPN
ejpam-5532	52	7	)	)	PUNCT
ejpam-5532	52	8	and	and	CCONJ
ejpam-5532	52	9	η	η	PROPN
ejpam-5532	52	10	⊆	⊆	NUM
ejpam-5532	52	11	µ	µ	NUM
ejpam-5532	52	12	,	,	PUNCT
ejpam-5532	52	13	then	then	ADV
ejpam-5532	52	14	η	η	PROPN
ejpam-5532	52	15	is	be	AUX
ejpam-5532	52	16	called	call	VERB
ejpam-5532	52	17	an	an	DET
ejpam-5532	52	18	l	l	NOUN
ejpam-5532	52	19	-	-	NOUN
ejpam-5532	52	20	sublattice	sublattice	NOUN
ejpam-5532	52	21	of	of	ADP
ejpam-5532	52	22	the	the	DET
ejpam-5532	52	23	l	l	NOUN
ejpam-5532	52	24	-	-	PUNCT
ejpam-5532	52	25	lattice	lattice	NOUN
ejpam-5532	52	26	µ.	µ.	NOUN
ejpam-5532	52	27	the	the	DET
ejpam-5532	52	28	collection	collection	NOUN
ejpam-5532	52	29	of	of	ADP
ejpam-5532	52	30	all	all	DET
ejpam-5532	52	31	l	l	NOUN
ejpam-5532	52	32	-	-	NOUN
ejpam-5532	52	33	sublattices	sublattice	NOUN
ejpam-5532	52	34	of	of	ADP
ejpam-5532	52	35	µ	µ	NOUN
ejpam-5532	52	36	is	be	AUX
ejpam-5532	52	37	denoted	denote	VERB
ejpam-5532	52	38	by	by	ADP
ejpam-5532	52	39	l(µ	l(µ	PROPN
ejpam-5532	52	40	)	)	PUNCT
ejpam-5532	52	41	.	.	PUNCT
ejpam-5532	53	1	in	in	ADP
ejpam-5532	53	2	this	this	DET
ejpam-5532	53	3	paper	paper	NOUN
ejpam-5532	53	4	,	,	PUNCT
ejpam-5532	53	5	we	we	PRON
ejpam-5532	53	6	shall	shall	AUX
ejpam-5532	53	7	study	study	VERB
ejpam-5532	53	8	the	the	DET
ejpam-5532	53	9	l	l	ADJ
ejpam-5532	53	10	-	-	ADJ
ejpam-5532	53	11	convex	convex	ADJ
ejpam-5532	53	12	sublattices	sublattice	NOUN
ejpam-5532	53	13	of	of	ADP
ejpam-5532	53	14	an	an	DET
ejpam-5532	53	15	l	l	NOUN
ejpam-5532	53	16	-	-	PUNCT
ejpam-5532	53	17	lattice	lattice	PROPN
ejpam-5532	53	18	µ	µ	PROPN
ejpam-5532	53	19	rather	rather	ADV
ejpam-5532	53	20	convex	convex	VERB
ejpam-5532	53	21	sublattices	sublattice	NOUN
ejpam-5532	53	22	of	of	ADP
ejpam-5532	53	23	an	an	DET
ejpam-5532	53	24	ordinary	ordinary	ADJ
ejpam-5532	53	25	lattice	lattice	NOUN
ejpam-5532	53	26	.	.	PUNCT
ejpam-5532	54	1	the	the	DET
ejpam-5532	54	2	following	follow	VERB
ejpam-5532	54	3	theorems	theorem	NOUN
ejpam-5532	54	4	provide	provide	VERB
ejpam-5532	54	5	the	the	DET
ejpam-5532	54	6	level	level	NOUN
ejpam-5532	54	7	subset	subset	NOUN
ejpam-5532	54	8	characterizations	characterization	NOUN
ejpam-5532	54	9	and	and	CCONJ
ejpam-5532	54	10	strong	strong	ADJ
ejpam-5532	54	11	level	level	NOUN
ejpam-5532	54	12	subset	subset	NOUN
ejpam-5532	54	13	characterizations	characterization	NOUN
ejpam-5532	54	14	of	of	ADP
ejpam-5532	54	15	an	an	DET
ejpam-5532	54	16	l	l	NOUN
ejpam-5532	54	17	-	-	NOUN
ejpam-5532	54	18	sublattice	sublattice	NOUN
ejpam-5532	54	19	of	of	ADP
ejpam-5532	54	20	µ.	µ.	NOUN
ejpam-5532	54	21	for	for	ADP
ejpam-5532	54	22	similar	similar	ADJ
ejpam-5532	54	23	characterizations	characterization	NOUN
ejpam-5532	54	24	of	of	ADP
ejpam-5532	54	25	l	l	NOUN
ejpam-5532	54	26	-	-	NOUN
ejpam-5532	54	27	sublattices	sublattice	NOUN
ejpam-5532	54	28	of	of	ADP
ejpam-5532	54	29	m	m	PROPN
ejpam-5532	54	30	,	,	PUNCT
ejpam-5532	54	31	we	we	PRON
ejpam-5532	54	32	refer	refer	VERB
ejpam-5532	54	33	to	to	ADP
ejpam-5532	54	34	[	[	X
ejpam-5532	54	35	10	10	NUM
ejpam-5532	54	36	]	]	PUNCT
ejpam-5532	54	37	.	.	PUNCT
ejpam-5532	55	1	theorem	theorem	ADJ
ejpam-5532	55	2	1	1	NUM
ejpam-5532	55	3	(	(	PUNCT
ejpam-5532	55	4	[	[	X
ejpam-5532	55	5	10	10	NUM
ejpam-5532	55	6	]	]	NUM
ejpam-5532	55	7	)	)	PUNCT
ejpam-5532	55	8	.	.	PUNCT
ejpam-5532	56	1	let	let	VERB
ejpam-5532	56	2	µ	µ	NUM
ejpam-5532	56	3	,	,	PUNCT
ejpam-5532	56	4	η	η	PROPN
ejpam-5532	56	5	∈	∈	PROPN
ejpam-5532	56	6	lm	lm	AUX
ejpam-5532	56	7	be	be	AUX
ejpam-5532	56	8	such	such	ADJ
ejpam-5532	56	9	that	that	SCONJ
ejpam-5532	56	10	η	η	PROPN
ejpam-5532	56	11	⊆	⊆	NUM
ejpam-5532	56	12	µ.	µ.	NOUN
ejpam-5532	56	13	also	also	ADV
ejpam-5532	56	14	,	,	PUNCT
ejpam-5532	56	15	let	let	VERB
ejpam-5532	56	16	l(µ,m	l(µ,m	ADV
ejpam-5532	56	17	)	)	PUNCT
ejpam-5532	56	18	be	be	AUX
ejpam-5532	56	19	an	an	DET
ejpam-5532	56	20	l	l	NOUN
ejpam-5532	56	21	-	-	NOUN
ejpam-5532	56	22	lattice	lattice	NOUN
ejpam-5532	56	23	and	and	CCONJ
ejpam-5532	56	24	ao	ao	NOUN
ejpam-5532	56	25	=	=	SYM
ejpam-5532	56	26	tip{η	tip{η	PROPN
ejpam-5532	56	27	}	}	PUNCT
ejpam-5532	56	28	.	.	PUNCT
ejpam-5532	57	1	then	then	ADV
ejpam-5532	57	2	,	,	PUNCT
ejpam-5532	57	3	η	η	PROPN
ejpam-5532	57	4	is	be	AUX
ejpam-5532	57	5	an	an	DET
ejpam-5532	57	6	l	l	NOUN
ejpam-5532	57	7	-	-	NOUN
ejpam-5532	57	8	sublattice	sublattice	NOUN
ejpam-5532	57	9	of	of	ADP
ejpam-5532	57	10	µ	µ	NOUN
ejpam-5532	57	11	if	if	NOUN
ejpam-5532	58	1	and	and	CCONJ
ejpam-5532	58	2	only	only	ADV
ejpam-5532	58	3	if	if	SCONJ
ejpam-5532	58	4	each	each	DET
ejpam-5532	58	5	level	level	NOUN
ejpam-5532	58	6	subset	subset	VERB
ejpam-5532	58	7	ηα	ηα	NOUN
ejpam-5532	58	8	is	be	AUX
ejpam-5532	58	9	a	a	DET
ejpam-5532	58	10	sublattice	sublattice	NOUN
ejpam-5532	58	11	of	of	ADP
ejpam-5532	58	12	µα	µα	ADP
ejpam-5532	58	13	,	,	PUNCT
ejpam-5532	58	14	∀α	∀α	VERB
ejpam-5532	58	15	≤	≤	NUM
ejpam-5532	58	16	ao	ao	NOUN
ejpam-5532	58	17	.	.	PUNCT
ejpam-5532	59	1	equivalently	equivalently	ADV
ejpam-5532	59	2	,	,	PUNCT
ejpam-5532	59	3	η	η	PROPN
ejpam-5532	59	4	is	be	AUX
ejpam-5532	59	5	an	an	DET
ejpam-5532	59	6	l	l	NOUN
ejpam-5532	59	7	-	-	NOUN
ejpam-5532	59	8	sublattice	sublattice	NOUN
ejpam-5532	59	9	of	of	ADP
ejpam-5532	59	10	µ	µ	NOUN
ejpam-5532	59	11	if	if	NOUN
ejpam-5532	59	12	and	and	CCONJ
ejpam-5532	59	13	only	only	ADV
ejpam-5532	59	14	if	if	SCONJ
ejpam-5532	59	15	each	each	DET
ejpam-5532	59	16	nonempty	nonempty	ADJ
ejpam-5532	59	17	level	level	NOUN
ejpam-5532	59	18	subset	subset	VERB
ejpam-5532	59	19	ηα	ηα	NOUN
ejpam-5532	59	20	is	be	AUX
ejpam-5532	59	21	a	a	DET
ejpam-5532	59	22	sublattice	sublattice	NOUN
ejpam-5532	59	23	of	of	ADP
ejpam-5532	59	24	µα	µα	ADP
ejpam-5532	59	25	.	.	PUNCT
ejpam-5532	60	1	theorem	theorem	NOUN
ejpam-5532	60	2	2	2	NUM
ejpam-5532	60	3	(	(	PUNCT
ejpam-5532	60	4	[	[	X
ejpam-5532	60	5	10	10	NUM
ejpam-5532	60	6	]	]	NUM
ejpam-5532	60	7	)	)	PUNCT
ejpam-5532	60	8	.	.	PUNCT
ejpam-5532	61	1	let	let	VERB
ejpam-5532	61	2	l	l	NOUN
ejpam-5532	61	3	be	be	AUX
ejpam-5532	61	4	a	a	DET
ejpam-5532	61	5	chain	chain	NOUN
ejpam-5532	61	6	.	.	PUNCT
ejpam-5532	62	1	let	let	VERB
ejpam-5532	62	2	µ	µ	NUM
ejpam-5532	62	3	,	,	PUNCT
ejpam-5532	62	4	η	η	PROPN
ejpam-5532	62	5	∈	∈	PROPN
ejpam-5532	62	6	lm	lm	AUX
ejpam-5532	62	7	be	be	AUX
ejpam-5532	62	8	such	such	ADJ
ejpam-5532	62	9	that	that	SCONJ
ejpam-5532	62	10	η	η	PROPN
ejpam-5532	62	11	⊆	⊆	NUM
ejpam-5532	62	12	µ.	µ.	NOUN
ejpam-5532	62	13	also	also	ADV
ejpam-5532	62	14	,	,	PUNCT
ejpam-5532	62	15	let	let	VERB
ejpam-5532	62	16	l(µ,m	l(µ,m	ADV
ejpam-5532	62	17	)	)	PUNCT
ejpam-5532	62	18	be	be	AUX
ejpam-5532	62	19	an	an	DET
ejpam-5532	62	20	l	l	NOUN
ejpam-5532	62	21	-	-	NOUN
ejpam-5532	62	22	lattice	lattice	NOUN
ejpam-5532	62	23	and	and	CCONJ
ejpam-5532	62	24	ao	ao	NOUN
ejpam-5532	62	25	=	=	SYM
ejpam-5532	62	26	tip{η	tip{η	PROPN
ejpam-5532	62	27	}	}	PUNCT
ejpam-5532	62	28	.	.	PUNCT
ejpam-5532	63	1	then	then	ADV
ejpam-5532	63	2	,	,	PUNCT
ejpam-5532	63	3	η	η	PROPN
ejpam-5532	63	4	is	be	AUX
ejpam-5532	63	5	an	an	DET
ejpam-5532	63	6	l	l	NOUN
ejpam-5532	63	7	-	-	NOUN
ejpam-5532	63	8	sublattice	sublattice	NOUN
ejpam-5532	63	9	of	of	ADP
ejpam-5532	63	10	µ	µ	NOUN
ejpam-5532	63	11	if	if	NOUN
ejpam-5532	64	1	and	and	CCONJ
ejpam-5532	64	2	only	only	ADV
ejpam-5532	64	3	if	if	SCONJ
ejpam-5532	64	4	each	each	DET
ejpam-5532	64	5	strong	strong	ADJ
ejpam-5532	64	6	level	level	NOUN
ejpam-5532	64	7	subset	subset	VERB
ejpam-5532	64	8	η	η	PROPN
ejpam-5532	64	9	>	>	PROPN
ejpam-5532	64	10	α	α	PROPN
ejpam-5532	64	11	is	be	AUX
ejpam-5532	64	12	a	a	DET
ejpam-5532	64	13	sublattice	sublattice	NOUN
ejpam-5532	64	14	of	of	ADP
ejpam-5532	64	15	µ	µ	NOUN
ejpam-5532	64	16	>	>	X
ejpam-5532	64	17	α	α	PROPN
ejpam-5532	64	18	,	,	PUNCT
ejpam-5532	64	19	∀	∀	X
ejpam-5532	64	20	α	α	X
ejpam-5532	64	21	<	<	X
ejpam-5532	64	22	ao	ao	PROPN
ejpam-5532	64	23	.	.	PUNCT
ejpam-5532	65	1	equivalently	equivalently	PROPN
ejpam-5532	65	2	,	,	PUNCT
ejpam-5532	65	3	η	η	PROPN
ejpam-5532	65	4	is	be	AUX
ejpam-5532	65	5	an	an	DET
ejpam-5532	65	6	l	l	NOUN
ejpam-5532	65	7	-	-	NOUN
ejpam-5532	65	8	sublattice	sublattice	NOUN
ejpam-5532	65	9	of	of	ADP
ejpam-5532	65	10	µ	µ	NOUN
ejpam-5532	65	11	if	if	NOUN
ejpam-5532	65	12	and	and	CCONJ
ejpam-5532	65	13	only	only	ADV
ejpam-5532	65	14	if	if	SCONJ
ejpam-5532	65	15	each	each	DET
ejpam-5532	65	16	nonempty	nonempty	ADV
ejpam-5532	65	17	strong	strong	ADJ
ejpam-5532	65	18	level	level	NOUN
ejpam-5532	65	19	subset	subset	VERB
ejpam-5532	65	20	η	η	PROPN
ejpam-5532	65	21	>	>	PROPN
ejpam-5532	65	22	α	α	PROPN
ejpam-5532	65	23	is	be	AUX
ejpam-5532	65	24	a	a	DET
ejpam-5532	65	25	sublattice	sublattice	NOUN
ejpam-5532	65	26	of	of	ADP
ejpam-5532	65	27	µ	µ	NOUN
ejpam-5532	65	28	>	>	X
ejpam-5532	65	29	α	α	NOUN
ejpam-5532	65	30	.	.	PUNCT
ejpam-5532	66	1	the	the	DET
ejpam-5532	66	2	notions	notion	NOUN
ejpam-5532	66	3	of	of	ADP
ejpam-5532	66	4	l	l	NOUN
ejpam-5532	66	5	-	-	NOUN
ejpam-5532	66	6	ideal	ideal	ADJ
ejpam-5532	66	7	,	,	PUNCT
ejpam-5532	66	8	l	l	ADJ
ejpam-5532	66	9	-	-	ADJ
ejpam-5532	66	10	dual	dual	ADJ
ejpam-5532	66	11	ideal	ideal	NOUN
ejpam-5532	66	12	in	in	ADP
ejpam-5532	66	13	lattice	lattice	PROPN
ejpam-5532	66	14	m	m	PROPN
ejpam-5532	66	15	and	and	CCONJ
ejpam-5532	66	16	l	l	NOUN
ejpam-5532	66	17	-	-	NOUN
ejpam-5532	66	18	ideal	ideal	ADJ
ejpam-5532	66	19	,	,	PUNCT
ejpam-5532	66	20	l	l	ADJ
ejpam-5532	66	21	-	-	ADJ
ejpam-5532	66	22	dual	dual	ADJ
ejpam-5532	66	23	ideal	ideal	NOUN
ejpam-5532	66	24	in	in	ADP
ejpam-5532	66	25	an	an	DET
ejpam-5532	66	26	l	l	NOUN
ejpam-5532	66	27	-	-	PUNCT
ejpam-5532	66	28	lattice	lattice	ADJ
ejpam-5532	66	29	µ	µ	X
ejpam-5532	66	30	are	be	AUX
ejpam-5532	66	31	defined	define	VERB
ejpam-5532	66	32	as	as	SCONJ
ejpam-5532	66	33	follows	follow	VERB
ejpam-5532	66	34	:	:	PUNCT
ejpam-5532	66	35	definition	definition	NOUN
ejpam-5532	66	36	2	2	NUM
ejpam-5532	66	37	(	(	PUNCT
ejpam-5532	66	38	[	[	X
ejpam-5532	66	39	10	10	NUM
ejpam-5532	66	40	]	]	NUM
ejpam-5532	66	41	)	)	PUNCT
ejpam-5532	66	42	.	.	PUNCT
ejpam-5532	67	1	let	let	VERB
ejpam-5532	67	2	µ	µ	X
ejpam-5532	67	3	∈	∈	X
ejpam-5532	67	4	lm	lm	INTJ
ejpam-5532	67	5	.	.	PUNCT
ejpam-5532	68	1	then	then	ADV
ejpam-5532	68	2	,	,	PUNCT
ejpam-5532	68	3	[	[	X
ejpam-5532	68	4	(	(	PUNCT
ejpam-5532	68	5	i)]µ	i)]µ	NOUN
ejpam-5532	68	6	is	be	AUX
ejpam-5532	68	7	called	call	VERB
ejpam-5532	68	8	an	an	DET
ejpam-5532	68	9	l	l	NOUN
ejpam-5532	68	10	-	-	PUNCT
ejpam-5532	68	11	ideal	ideal	ADJ
ejpam-5532	68	12	ofm	ofm	PROPN
ejpam-5532	68	13	if	if	SCONJ
ejpam-5532	68	14	µ	µ	PRON
ejpam-5532	68	15	∈	∈	PROPN
ejpam-5532	68	16	l(m	l(m	PROPN
ejpam-5532	68	17	)	)	PUNCT
ejpam-5532	68	18	and	and	CCONJ
ejpam-5532	68	19	x	x	SYM
ejpam-5532	68	20	≤	≤	PROPN
ejpam-5532	68	21	y	y	PROPN
ejpam-5532	68	22	inm	inm	PROPN
ejpam-5532	68	23	implies	imply	VERB
ejpam-5532	68	24	µ(x	µ(x	NOUN
ejpam-5532	68	25	)	)	PUNCT
ejpam-5532	68	26	≥	≥	NOUN
ejpam-5532	68	27	µ(y	µ(y	PROPN
ejpam-5532	68	28	)	)	PUNCT
ejpam-5532	68	29	in	in	ADP
ejpam-5532	68	30	l	l	NOUN
ejpam-5532	68	31	;	;	PUNCT
ejpam-5532	68	32	µ	µ	X
ejpam-5532	68	33	is	be	AUX
ejpam-5532	68	34	called	call	VERB
ejpam-5532	68	35	an	an	DET
ejpam-5532	68	36	l	l	ADJ
ejpam-5532	68	37	-	-	ADJ
ejpam-5532	68	38	dual	dual	ADJ
ejpam-5532	68	39	ideal	ideal	NOUN
ejpam-5532	68	40	of	of	ADP
ejpam-5532	68	41	m	m	PROPN
ejpam-5532	68	42	if	if	SCONJ
ejpam-5532	68	43	µ	µ	PRON
ejpam-5532	68	44	∈	∈	PROPN
ejpam-5532	68	45	l(m	l(m	PROPN
ejpam-5532	68	46	)	)	PUNCT
ejpam-5532	68	47	and	and	CCONJ
ejpam-5532	68	48	x	x	SYM
ejpam-5532	68	49	≤	≤	ADJ
ejpam-5532	68	50	y	y	NOUN
ejpam-5532	68	51	in	in	ADP
ejpam-5532	68	52	m	m	PROPN
ejpam-5532	68	53	implies	imply	VERB
ejpam-5532	68	54	µ(x	µ(x	NOUN
ejpam-5532	68	55	)	)	PUNCT
ejpam-5532	68	56	≤	≤	NOUN
ejpam-5532	68	57	µ(y	µ(y	PROPN
ejpam-5532	68	58	)	)	PUNCT
ejpam-5532	68	59	in	in	ADP
ejpam-5532	68	60	l.	l.	PROPN
ejpam-5532	68	61	definition	definition	NOUN
ejpam-5532	68	62	3	3	NUM
ejpam-5532	68	63	(	(	PUNCT
ejpam-5532	68	64	[	[	X
ejpam-5532	68	65	10	10	NUM
ejpam-5532	68	66	]	]	NUM
ejpam-5532	68	67	)	)	PUNCT
ejpam-5532	68	68	.	.	PUNCT
ejpam-5532	69	1	let	let	VERB
ejpam-5532	69	2	µ	µ	NUM
ejpam-5532	69	3	,	,	PUNCT
ejpam-5532	69	4	η	η	PROPN
ejpam-5532	69	5	∈	∈	PROPN
ejpam-5532	69	6	lm	lm	AUX
ejpam-5532	69	7	be	be	AUX
ejpam-5532	69	8	such	such	ADJ
ejpam-5532	69	9	that	that	SCONJ
ejpam-5532	69	10	η	η	PROPN
ejpam-5532	69	11	⊆	⊆	NUM
ejpam-5532	69	12	µ.	µ.	NOUN
ejpam-5532	69	13	also	also	ADV
ejpam-5532	69	14	,	,	PUNCT
ejpam-5532	69	15	let	let	VERB
ejpam-5532	69	16	l(µ,m	l(µ,m	ADV
ejpam-5532	69	17	)	)	PUNCT
ejpam-5532	69	18	be	be	AUX
ejpam-5532	69	19	an	an	DET
ejpam-5532	69	20	l	l	NOUN
ejpam-5532	69	21	-	-	NOUN
ejpam-5532	69	22	lattice	lattice	NOUN
ejpam-5532	69	23	.	.	PUNCT
ejpam-5532	70	1	then	then	ADV
ejpam-5532	70	2	,	,	PUNCT
ejpam-5532	70	3	[	[	X
ejpam-5532	70	4	(	(	PUNCT
ejpam-5532	70	5	i)]η	i)]η	NOUN
ejpam-5532	70	6	is	be	AUX
ejpam-5532	70	7	called	call	VERB
ejpam-5532	70	8	an	an	DET
ejpam-5532	70	9	l	l	NOUN
ejpam-5532	70	10	-	-	NOUN
ejpam-5532	70	11	ideal	ideal	NOUN
ejpam-5532	70	12	of	of	ADP
ejpam-5532	70	13	µ	µ	NOUN
ejpam-5532	70	14	if	if	SCONJ
ejpam-5532	70	15	η(x	η(x	PROPN
ejpam-5532	70	16	∨	∨	NUM
ejpam-5532	70	17	y	y	NOUN
ejpam-5532	70	18	)	)	PUNCT
ejpam-5532	70	19	≥	≥	NOUN
ejpam-5532	70	20	η(x	η(x	NOUN
ejpam-5532	70	21	)	)	PUNCT
ejpam-5532	70	22	∧	∧	PROPN
ejpam-5532	70	23	η(y	η(y	NOUN
ejpam-5532	70	24	)	)	PUNCT
ejpam-5532	70	25	and	and	CCONJ
ejpam-5532	70	26	η(x	η(x	PROPN
ejpam-5532	70	27	∧	∧	PROPN
ejpam-5532	70	28	y	y	PROPN
ejpam-5532	70	29	)	)	PUNCT
ejpam-5532	70	30	≥	≥	NOUN
ejpam-5532	70	31	µ(x	µ(x	NOUN
ejpam-5532	70	32	)	)	PUNCT
ejpam-5532	70	33	∧	∧	PROPN
ejpam-5532	70	34	η(y	η(y	NOUN
ejpam-5532	70	35	)	)	PUNCT
ejpam-5532	70	36	;	;	PUNCT
ejpam-5532	70	37	∀x	∀x	NUM
ejpam-5532	70	38	,	,	PUNCT
ejpam-5532	70	39	y	y	PROPN
ejpam-5532	70	40	∈	∈	PROPN
ejpam-5532	70	41	m.	m.	NOUN
ejpam-5532	70	42	η	η	PROPN
ejpam-5532	70	43	is	be	AUX
ejpam-5532	70	44	called	call	VERB
ejpam-5532	70	45	an	an	DET
ejpam-5532	70	46	l	l	ADJ
ejpam-5532	70	47	-	-	ADJ
ejpam-5532	70	48	dual	dual	ADJ
ejpam-5532	70	49	ideal	ideal	NOUN
ejpam-5532	70	50	of	of	ADP
ejpam-5532	70	51	µ	µ	NOUN
ejpam-5532	70	52	if	if	SCONJ
ejpam-5532	70	53	η(x	η(x	PROPN
ejpam-5532	70	54	∧	∧	PROPN
ejpam-5532	70	55	y	y	PROPN
ejpam-5532	70	56	)	)	PUNCT
ejpam-5532	70	57	≥	≥	NOUN
ejpam-5532	70	58	η(x	η(x	NOUN
ejpam-5532	70	59	)	)	PUNCT
ejpam-5532	70	60	∧	∧	PROPN
ejpam-5532	70	61	η(y	η(y	NOUN
ejpam-5532	70	62	)	)	PUNCT
ejpam-5532	70	63	and	and	CCONJ
ejpam-5532	70	64	η(x	η(x	PROPN
ejpam-5532	70	65	∨	∨	NUM
ejpam-5532	70	66	y	y	PROPN
ejpam-5532	70	67	)	)	PUNCT
ejpam-5532	70	68	≥	≥	NOUN
ejpam-5532	70	69	η(x	η(x	NOUN
ejpam-5532	70	70	)	)	PUNCT
ejpam-5532	70	71	∧	∧	PROPN
ejpam-5532	70	72	µ(y	µ(y	PROPN
ejpam-5532	70	73	)	)	PUNCT
ejpam-5532	70	74	;	;	PUNCT
ejpam-5532	70	75	∀x	∀x	X
ejpam-5532	70	76	,	,	PUNCT
ejpam-5532	70	77	y	y	PROPN
ejpam-5532	70	78	∈	∈	PROPN
ejpam-5532	70	79	m.	m.	NOUN
ejpam-5532	70	80	a.	a.	PROPN
ejpam-5532	70	81	jain	jain	PROPN
ejpam-5532	70	82	,	,	PUNCT
ejpam-5532	70	83	i.	i.	PROPN
ejpam-5532	70	84	jahan	jahan	PROPN
ejpam-5532	70	85	/	/	SYM
ejpam-5532	70	86	eur	eur	PROPN
ejpam-5532	70	87	.	.	PUNCT
ejpam-5532	71	1	j.	j.	PROPN
ejpam-5532	71	2	pure	pure	PROPN
ejpam-5532	71	3	appl	appl	PROPN
ejpam-5532	71	4	.	.	PROPN
ejpam-5532	71	5	math	math	PROPN
ejpam-5532	71	6	,	,	PUNCT
ejpam-5532	71	7	18	18	NUM
ejpam-5532	71	8	(	(	PUNCT
ejpam-5532	71	9	1	1	NUM
ejpam-5532	71	10	)	)	PUNCT
ejpam-5532	71	11	(	(	PUNCT
ejpam-5532	71	12	2025	2025	NUM
ejpam-5532	71	13	)	)	PUNCT
ejpam-5532	71	14	,	,	PUNCT
ejpam-5532	71	15	5532	5532	NUM
ejpam-5532	71	16	4	4	NUM
ejpam-5532	71	17	of	of	ADP
ejpam-5532	71	18	20	20	NUM
ejpam-5532	71	19	it	it	PRON
ejpam-5532	71	20	is	be	AUX
ejpam-5532	71	21	important	important	ADJ
ejpam-5532	71	22	to	to	PART
ejpam-5532	71	23	note	note	VERB
ejpam-5532	71	24	here	here	ADV
ejpam-5532	71	25	that	that	SCONJ
ejpam-5532	71	26	in	in	ADP
ejpam-5532	71	27	a	a	DET
ejpam-5532	71	28	bounded	bounded	ADJ
ejpam-5532	71	29	lattice	lattice	PROPN
ejpam-5532	71	30	m	m	PROPN
ejpam-5532	71	31	,	,	PUNCT
ejpam-5532	71	32	an	an	DET
ejpam-5532	71	33	l	l	ADJ
ejpam-5532	71	34	-	-	PUNCT
ejpam-5532	71	35	ideal	ideal	ADJ
ejpam-5532	71	36	attains	attain	VERB
ejpam-5532	71	37	its	its	PRON
ejpam-5532	71	38	supremum	supremum	NOUN
ejpam-5532	71	39	at	at	ADP
ejpam-5532	71	40	the	the	DET
ejpam-5532	71	41	least	least	ADJ
ejpam-5532	71	42	element	element	NOUN
ejpam-5532	71	43	of	of	ADP
ejpam-5532	71	44	m	m	PROPN
ejpam-5532	71	45	,	,	PUNCT
ejpam-5532	71	46	whereas	whereas	SCONJ
ejpam-5532	71	47	an	an	DET
ejpam-5532	71	48	l	l	ADJ
ejpam-5532	71	49	-	-	ADJ
ejpam-5532	71	50	dual	dual	ADJ
ejpam-5532	71	51	ideal	ideal	ADJ
ejpam-5532	71	52	attains	attain	VERB
ejpam-5532	71	53	its	its	PRON
ejpam-5532	71	54	supremum	supremum	NOUN
ejpam-5532	71	55	at	at	ADP
ejpam-5532	71	56	the	the	DET
ejpam-5532	71	57	greatest	great	ADJ
ejpam-5532	71	58	element	element	NOUN
ejpam-5532	71	59	of	of	ADP
ejpam-5532	71	60	m	m	PROPN
ejpam-5532	71	61	.	.	PUNCT
ejpam-5532	72	1	the	the	DET
ejpam-5532	72	2	following	follow	VERB
ejpam-5532	72	3	theorems	theorem	NOUN
ejpam-5532	72	4	provide	provide	VERB
ejpam-5532	72	5	the	the	DET
ejpam-5532	72	6	level	level	NOUN
ejpam-5532	72	7	subset	subset	NOUN
ejpam-5532	72	8	characterizations	characterization	NOUN
ejpam-5532	72	9	and	and	CCONJ
ejpam-5532	72	10	strong	strong	ADJ
ejpam-5532	72	11	level	level	NOUN
ejpam-5532	72	12	subset	subset	NOUN
ejpam-5532	72	13	characterizations	characterization	NOUN
ejpam-5532	72	14	of	of	ADP
ejpam-5532	72	15	an	an	DET
ejpam-5532	72	16	l	l	NOUN
ejpam-5532	72	17	-	-	NOUN
ejpam-5532	72	18	ideal	ideal	ADJ
ejpam-5532	72	19	(	(	PUNCT
ejpam-5532	72	20	l	l	ADJ
ejpam-5532	72	21	-	-	ADJ
ejpam-5532	72	22	dual	dual	ADJ
ejpam-5532	72	23	ideal	ideal	NOUN
ejpam-5532	72	24	)	)	PUNCT
ejpam-5532	72	25	of	of	ADP
ejpam-5532	72	26	µ.	µ.	NOUN
ejpam-5532	72	27	for	for	ADP
ejpam-5532	72	28	similar	similar	ADJ
ejpam-5532	72	29	characterizations	characterization	NOUN
ejpam-5532	72	30	of	of	ADP
ejpam-5532	72	31	l	l	NOUN
ejpam-5532	72	32	-	-	NOUN
ejpam-5532	72	33	ideal	ideal	ADJ
ejpam-5532	72	34	(	(	PUNCT
ejpam-5532	72	35	l	l	ADJ
ejpam-5532	72	36	-	-	ADJ
ejpam-5532	72	37	dual	dual	ADJ
ejpam-5532	72	38	ideal	ideal	NOUN
ejpam-5532	72	39	)	)	PUNCT
ejpam-5532	72	40	of	of	ADP
ejpam-5532	72	41	m	m	PRON
ejpam-5532	72	42	,	,	PUNCT
ejpam-5532	72	43	refer	refer	VERB
ejpam-5532	72	44	to	to	ADP
ejpam-5532	72	45	[	[	X
ejpam-5532	72	46	10	10	NUM
ejpam-5532	72	47	]	]	PUNCT
ejpam-5532	72	48	.	.	PUNCT
ejpam-5532	73	1	theorem	theorem	ADJ
ejpam-5532	73	2	3	3	NUM
ejpam-5532	73	3	(	(	PUNCT
ejpam-5532	73	4	[	[	X
ejpam-5532	73	5	10	10	NUM
ejpam-5532	73	6	]	]	NUM
ejpam-5532	73	7	)	)	PUNCT
ejpam-5532	73	8	.	.	PUNCT
ejpam-5532	74	1	let	let	VERB
ejpam-5532	74	2	µ	µ	NUM
ejpam-5532	74	3	,	,	PUNCT
ejpam-5532	74	4	η	η	PROPN
ejpam-5532	74	5	∈	∈	PROPN
ejpam-5532	74	6	lm	lm	AUX
ejpam-5532	74	7	be	be	AUX
ejpam-5532	74	8	such	such	ADJ
ejpam-5532	74	9	that	that	SCONJ
ejpam-5532	74	10	η	η	PROPN
ejpam-5532	74	11	⊆	⊆	NUM
ejpam-5532	74	12	µ.	µ.	NOUN
ejpam-5532	74	13	also	also	ADV
ejpam-5532	74	14	,	,	PUNCT
ejpam-5532	74	15	let	let	VERB
ejpam-5532	74	16	l(µ,m	l(µ,m	ADV
ejpam-5532	74	17	)	)	PUNCT
ejpam-5532	74	18	be	be	AUX
ejpam-5532	74	19	an	an	DET
ejpam-5532	74	20	l	l	NOUN
ejpam-5532	74	21	-	-	NOUN
ejpam-5532	74	22	lattice	lattice	NOUN
ejpam-5532	74	23	and	and	CCONJ
ejpam-5532	74	24	ao	ao	NOUN
ejpam-5532	74	25	=	=	SYM
ejpam-5532	74	26	tip{η	tip{η	PROPN
ejpam-5532	74	27	}	}	PUNCT
ejpam-5532	74	28	.	.	PUNCT
ejpam-5532	75	1	then	then	ADV
ejpam-5532	75	2	,	,	PUNCT
ejpam-5532	75	3	η	η	PROPN
ejpam-5532	75	4	is	be	AUX
ejpam-5532	75	5	an	an	DET
ejpam-5532	75	6	l	l	NOUN
ejpam-5532	75	7	-	-	PUNCT
ejpam-5532	75	8	ideal(l	ideal(l	ADJ
ejpam-5532	75	9	-	-	PUNCT
ejpam-5532	75	10	dual	dual	ADJ
ejpam-5532	75	11	ideal	ideal	NOUN
ejpam-5532	75	12	)	)	PUNCT
ejpam-5532	75	13	of	of	ADP
ejpam-5532	75	14	µ	µ	NOUN
ejpam-5532	75	15	if	if	NOUN
ejpam-5532	75	16	and	and	CCONJ
ejpam-5532	75	17	only	only	ADV
ejpam-5532	75	18	if	if	SCONJ
ejpam-5532	75	19	each	each	DET
ejpam-5532	75	20	level	level	NOUN
ejpam-5532	75	21	subset	subset	VERB
ejpam-5532	75	22	ηα	ηα	NOUN
ejpam-5532	75	23	is	be	AUX
ejpam-5532	75	24	an	an	DET
ejpam-5532	75	25	ideal(dual	ideal(dual	ADJ
ejpam-5532	75	26	ideal	ideal	NOUN
ejpam-5532	75	27	)	)	PUNCT
ejpam-5532	75	28	of	of	ADP
ejpam-5532	75	29	µα	µα	ADP
ejpam-5532	75	30	,	,	PUNCT
ejpam-5532	75	31	∀α	∀α	VERB
ejpam-5532	75	32	≤	≤	NUM
ejpam-5532	75	33	ao	ao	NOUN
ejpam-5532	75	34	.	.	PUNCT
ejpam-5532	76	1	equivalently	equivalently	ADV
ejpam-5532	76	2	,	,	PUNCT
ejpam-5532	76	3	η	η	PROPN
ejpam-5532	76	4	is	be	AUX
ejpam-5532	76	5	an	an	DET
ejpam-5532	76	6	l	l	NOUN
ejpam-5532	76	7	-	-	PUNCT
ejpam-5532	76	8	ideal(l	ideal(l	ADJ
ejpam-5532	76	9	-	-	PUNCT
ejpam-5532	76	10	dual	dual	ADJ
ejpam-5532	76	11	ideal	ideal	NOUN
ejpam-5532	76	12	)	)	PUNCT
ejpam-5532	76	13	of	of	ADP
ejpam-5532	76	14	µ	µ	NOUN
ejpam-5532	76	15	if	if	NOUN
ejpam-5532	76	16	and	and	CCONJ
ejpam-5532	76	17	only	only	ADV
ejpam-5532	76	18	if	if	SCONJ
ejpam-5532	76	19	each	each	DET
ejpam-5532	76	20	nonempty	nonempty	ADJ
ejpam-5532	76	21	level	level	NOUN
ejpam-5532	76	22	subset	subset	NOUN
ejpam-5532	76	23	ηα	ηα	NOUN
ejpam-5532	76	24	is	be	AUX
ejpam-5532	76	25	an	an	DET
ejpam-5532	76	26	ideal(dual	ideal(dual	ADJ
ejpam-5532	76	27	ideal	ideal	NOUN
ejpam-5532	76	28	)	)	PUNCT
ejpam-5532	76	29	of	of	ADP
ejpam-5532	76	30	µα	µα	ADP
ejpam-5532	76	31	.	.	PUNCT
ejpam-5532	76	32	theorem	theorem	NOUN
ejpam-5532	76	33	4	4	NUM
ejpam-5532	76	34	(	(	PUNCT
ejpam-5532	76	35	[	[	X
ejpam-5532	76	36	10	10	NUM
ejpam-5532	76	37	]	]	NUM
ejpam-5532	76	38	)	)	PUNCT
ejpam-5532	76	39	.	.	PUNCT
ejpam-5532	77	1	let	let	VERB
ejpam-5532	77	2	l	l	NOUN
ejpam-5532	77	3	is	be	AUX
ejpam-5532	77	4	a	a	DET
ejpam-5532	77	5	chain	chain	NOUN
ejpam-5532	77	6	.	.	PUNCT
ejpam-5532	78	1	let	let	VERB
ejpam-5532	78	2	µ	µ	NUM
ejpam-5532	78	3	,	,	PUNCT
ejpam-5532	78	4	η	η	PROPN
ejpam-5532	78	5	∈	∈	PROPN
ejpam-5532	78	6	lm	lm	AUX
ejpam-5532	78	7	be	be	AUX
ejpam-5532	78	8	such	such	ADJ
ejpam-5532	78	9	that	that	SCONJ
ejpam-5532	78	10	η	η	PROPN
ejpam-5532	78	11	⊆	⊆	NUM
ejpam-5532	78	12	µ.	µ.	NOUN
ejpam-5532	78	13	also	also	ADV
ejpam-5532	78	14	,	,	PUNCT
ejpam-5532	78	15	let	let	VERB
ejpam-5532	78	16	l(µ,m	l(µ,m	ADV
ejpam-5532	78	17	)	)	PUNCT
ejpam-5532	78	18	be	be	AUX
ejpam-5532	78	19	an	an	DET
ejpam-5532	78	20	l	l	NOUN
ejpam-5532	78	21	-	-	NOUN
ejpam-5532	78	22	lattice	lattice	NOUN
ejpam-5532	78	23	and	and	CCONJ
ejpam-5532	78	24	ao	ao	NOUN
ejpam-5532	78	25	=	=	SYM
ejpam-5532	78	26	tip{η	tip{η	PROPN
ejpam-5532	78	27	}	}	PUNCT
ejpam-5532	78	28	.	.	PUNCT
ejpam-5532	79	1	then	then	ADV
ejpam-5532	79	2	,	,	PUNCT
ejpam-5532	79	3	η	η	PROPN
ejpam-5532	79	4	is	be	AUX
ejpam-5532	79	5	an	an	DET
ejpam-5532	79	6	l	l	NOUN
ejpam-5532	79	7	-	-	PUNCT
ejpam-5532	79	8	ideal(l	ideal(l	ADJ
ejpam-5532	79	9	-	-	PUNCT
ejpam-5532	79	10	dual	dual	ADJ
ejpam-5532	79	11	ideal	ideal	NOUN
ejpam-5532	79	12	)	)	PUNCT
ejpam-5532	79	13	of	of	ADP
ejpam-5532	79	14	µ	µ	NOUN
ejpam-5532	79	15	if	if	NOUN
ejpam-5532	79	16	and	and	CCONJ
ejpam-5532	79	17	only	only	ADV
ejpam-5532	79	18	if	if	SCONJ
ejpam-5532	79	19	each	each	DET
ejpam-5532	79	20	strong	strong	ADJ
ejpam-5532	79	21	level	level	NOUN
ejpam-5532	79	22	subset	subset	VERB
ejpam-5532	79	23	η	η	PROPN
ejpam-5532	79	24	>	>	PROPN
ejpam-5532	79	25	α	α	PROPN
ejpam-5532	79	26	∀α	∀α	NOUN
ejpam-5532	79	27	<	<	X
ejpam-5532	79	28	ao	ao	PROPN
ejpam-5532	79	29	,	,	PUNCT
ejpam-5532	79	30	is	be	AUX
ejpam-5532	79	31	an	an	DET
ejpam-5532	79	32	ideal	ideal	ADJ
ejpam-5532	79	33	(	(	PUNCT
ejpam-5532	79	34	dual	dual	ADJ
ejpam-5532	79	35	ideal	ideal	NOUN
ejpam-5532	79	36	)	)	PUNCT
ejpam-5532	79	37	of	of	ADP
ejpam-5532	79	38	µ	µ	PROPN
ejpam-5532	79	39	>	>	X
ejpam-5532	79	40	α	α	PROPN
ejpam-5532	79	41	.	.	PUNCT
ejpam-5532	80	1	equivalently	equivalently	ADV
ejpam-5532	80	2	,	,	PUNCT
ejpam-5532	80	3	η	η	PROPN
ejpam-5532	80	4	is	be	AUX
ejpam-5532	80	5	an	an	DET
ejpam-5532	80	6	l	l	NOUN
ejpam-5532	80	7	-	-	PUNCT
ejpam-5532	80	8	ideal(l	ideal(l	ADJ
ejpam-5532	80	9	-	-	PUNCT
ejpam-5532	80	10	dual	dual	ADJ
ejpam-5532	80	11	ideal	ideal	NOUN
ejpam-5532	80	12	)	)	PUNCT
ejpam-5532	80	13	of	of	ADP
ejpam-5532	80	14	µ	µ	NOUN
ejpam-5532	80	15	if	if	NOUN
ejpam-5532	80	16	and	and	CCONJ
ejpam-5532	80	17	only	only	ADV
ejpam-5532	80	18	if	if	SCONJ
ejpam-5532	80	19	each	each	DET
ejpam-5532	80	20	nonempty	nonempty	ADV
ejpam-5532	80	21	strong	strong	ADJ
ejpam-5532	80	22	level	level	NOUN
ejpam-5532	80	23	subset	subset	VERB
ejpam-5532	80	24	η	η	PROPN
ejpam-5532	80	25	>	>	PROPN
ejpam-5532	80	26	α	α	PROPN
ejpam-5532	80	27	is	be	AUX
ejpam-5532	80	28	an	an	DET
ejpam-5532	80	29	ideal	ideal	ADJ
ejpam-5532	80	30	(	(	PUNCT
ejpam-5532	80	31	dual	dual	ADJ
ejpam-5532	80	32	ideal	ideal	NOUN
ejpam-5532	80	33	)	)	PUNCT
ejpam-5532	80	34	of	of	ADP
ejpam-5532	80	35	µ	µ	X
ejpam-5532	80	36	>	>	X
ejpam-5532	80	37	α	α	NOUN
ejpam-5532	80	38	.	.	PUNCT
ejpam-5532	81	1	it	it	PRON
ejpam-5532	81	2	can	can	AUX
ejpam-5532	81	3	be	be	AUX
ejpam-5532	81	4	easily	easily	ADV
ejpam-5532	81	5	verified	verify	VERB
ejpam-5532	81	6	that	that	SCONJ
ejpam-5532	81	7	the	the	DET
ejpam-5532	81	8	intersection	intersection	NOUN
ejpam-5532	81	9	of	of	ADP
ejpam-5532	81	10	an	an	DET
ejpam-5532	81	11	arbitrary	arbitrary	ADJ
ejpam-5532	81	12	family	family	NOUN
ejpam-5532	81	13	of	of	ADP
ejpam-5532	81	14	l	l	PROPN
ejpam-5532	81	15	-	-	NOUN
ejpam-5532	81	16	sublattices(lideals	sublattices(lideal	NOUN
ejpam-5532	81	17	,	,	PUNCT
ejpam-5532	81	18	l	l	ADJ
ejpam-5532	81	19	-	-	ADJ
ejpam-5532	81	20	dual	dual	ADJ
ejpam-5532	81	21	ideals	ideal	NOUN
ejpam-5532	81	22	)	)	PUNCT
ejpam-5532	81	23	of	of	ADP
ejpam-5532	81	24	µ	µ	PROPN
ejpam-5532	81	25	is	be	AUX
ejpam-5532	81	26	an	an	DET
ejpam-5532	81	27	l	l	NOUN
ejpam-5532	81	28	-	-	PUNCT
ejpam-5532	81	29	sublattice(l	sublattice(l	NOUN
ejpam-5532	81	30	-	-	PUNCT
ejpam-5532	81	31	ideal	ideal	NOUN
ejpam-5532	81	32	,	,	PUNCT
ejpam-5532	81	33	l	l	ADJ
ejpam-5532	81	34	-	-	ADJ
ejpam-5532	81	35	dual	dual	ADJ
ejpam-5532	81	36	ideal	ideal	NOUN
ejpam-5532	81	37	)	)	PUNCT
ejpam-5532	81	38	of	of	ADP
ejpam-5532	81	39	µ.	µ.	NOUN
ejpam-5532	81	40	this	this	PRON
ejpam-5532	81	41	leads	lead	VERB
ejpam-5532	81	42	to	to	ADP
ejpam-5532	81	43	the	the	DET
ejpam-5532	81	44	definition	definition	NOUN
ejpam-5532	81	45	of	of	ADP
ejpam-5532	81	46	an	an	DET
ejpam-5532	81	47	l	l	NOUN
ejpam-5532	81	48	-	-	PUNCT
ejpam-5532	81	49	sublattice(l	sublattice(l	NOUN
ejpam-5532	81	50	-	-	PUNCT
ejpam-5532	81	51	ideal	ideal	NOUN
ejpam-5532	81	52	,	,	PUNCT
ejpam-5532	81	53	l	l	ADJ
ejpam-5532	81	54	-	-	ADJ
ejpam-5532	81	55	dual	dual	ADJ
ejpam-5532	81	56	ideal	ideal	NOUN
ejpam-5532	81	57	)	)	PUNCT
ejpam-5532	81	58	generated	generate	VERB
ejpam-5532	81	59	by	by	ADP
ejpam-5532	81	60	an	an	DET
ejpam-5532	81	61	l	l	NOUN
ejpam-5532	81	62	-	-	PUNCT
ejpam-5532	81	63	subset	subset	VERB
ejpam-5532	81	64	η	η	PROPN
ejpam-5532	81	65	of	of	ADP
ejpam-5532	81	66	µ	µ	PROPN
ejpam-5532	81	67	as	as	ADP
ejpam-5532	81	68	the	the	DET
ejpam-5532	81	69	intersection	intersection	NOUN
ejpam-5532	81	70	of	of	ADP
ejpam-5532	81	71	all	all	DET
ejpam-5532	81	72	l	l	NOUN
ejpam-5532	81	73	-	-	ADJ
ejpam-5532	81	74	sublattices(resp	sublattices(resp	ADJ
ejpam-5532	81	75	.	.	PUNCT
ejpam-5532	82	1	l	l	NOUN
ejpam-5532	82	2	-	-	NOUN
ejpam-5532	82	3	ideals	ideal	NOUN
ejpam-5532	82	4	,	,	PUNCT
ejpam-5532	82	5	l	l	ADJ
ejpam-5532	82	6	-	-	ADJ
ejpam-5532	82	7	dual	dual	ADJ
ejpam-5532	82	8	ideals	ideal	NOUN
ejpam-5532	82	9	)	)	PUNCT
ejpam-5532	82	10	of	of	ADP
ejpam-5532	82	11	µ	µ	X
ejpam-5532	82	12	containing	contain	VERB
ejpam-5532	82	13	η	η	PROPN
ejpam-5532	82	14	.	.	PROPN
ejpam-5532	82	15	these	these	PRON
ejpam-5532	82	16	are	be	AUX
ejpam-5532	82	17	denoted	denote	VERB
ejpam-5532	82	18	by	by	ADP
ejpam-5532	82	19	[	[	X
ejpam-5532	82	20	η]µ	η]µ	ADJ
ejpam-5532	82	21	,	,	PUNCT
ejpam-5532	82	22	(	(	PUNCT
ejpam-5532	82	23	η]µ	η]µ	ADJ
ejpam-5532	82	24	and	and	CCONJ
ejpam-5532	82	25	[	[	X
ejpam-5532	82	26	η)µ	η)µ	NOUN
ejpam-5532	82	27	respectively	respectively	ADV
ejpam-5532	82	28	.	.	PUNCT
ejpam-5532	83	1	the	the	DET
ejpam-5532	83	2	following	follow	VERB
ejpam-5532	83	3	result	result	NOUN
ejpam-5532	83	4	from	from	ADP
ejpam-5532	83	5	[	[	X
ejpam-5532	83	6	3	3	NUM
ejpam-5532	83	7	]	]	PUNCT
ejpam-5532	83	8	gives	give	VERB
ejpam-5532	83	9	the	the	DET
ejpam-5532	83	10	structural	structural	ADJ
ejpam-5532	83	11	compositions	composition	NOUN
ejpam-5532	83	12	of	of	ADP
ejpam-5532	83	13	an	an	DET
ejpam-5532	83	14	l	l	NOUN
ejpam-5532	83	15	-	-	NOUN
ejpam-5532	83	16	sublattice	sublattice	NOUN
ejpam-5532	83	17	,	,	PUNCT
ejpam-5532	83	18	l	l	NOUN
ejpam-5532	83	19	-	-	NOUN
ejpam-5532	83	20	ideal	ideal	ADJ
ejpam-5532	83	21	,	,	PUNCT
ejpam-5532	83	22	l	l	ADJ
ejpam-5532	83	23	-	-	ADJ
ejpam-5532	83	24	dual	dual	ADJ
ejpam-5532	83	25	ideal	ideal	NOUN
ejpam-5532	83	26	of	of	ADP
ejpam-5532	83	27	an	an	DET
ejpam-5532	83	28	l	l	NOUN
ejpam-5532	83	29	-	-	PUNCT
ejpam-5532	83	30	lattice	lattice	NOUN
ejpam-5532	83	31	µ	µ	NOUN
ejpam-5532	83	32	generated	generate	VERB
ejpam-5532	83	33	by	by	ADP
ejpam-5532	83	34	an	an	DET
ejpam-5532	83	35	l	l	NOUN
ejpam-5532	83	36	-	-	PUNCT
ejpam-5532	83	37	subset	subset	VERB
ejpam-5532	83	38	η	η	PROPN
ejpam-5532	83	39	of	of	ADP
ejpam-5532	83	40	µ	µ	NUM
ejpam-5532	83	41	in	in	ADP
ejpam-5532	83	42	terms	term	NOUN
ejpam-5532	83	43	of	of	ADP
ejpam-5532	83	44	level	level	NOUN
ejpam-5532	83	45	subsets	subset	NOUN
ejpam-5532	83	46	.	.	PUNCT
ejpam-5532	84	1	theorem	theorem	NOUN
ejpam-5532	84	2	5	5	NUM
ejpam-5532	84	3	(	(	PUNCT
ejpam-5532	84	4	[	[	X
ejpam-5532	84	5	10	10	NUM
ejpam-5532	84	6	]	]	NUM
ejpam-5532	84	7	)	)	PUNCT
ejpam-5532	84	8	.	.	PUNCT
ejpam-5532	85	1	let	let	AUX
ejpam-5532	85	2	l(µ,m	l(µ,m	PUNCT
ejpam-5532	85	3	)	)	PUNCT
ejpam-5532	85	4	be	be	AUX
ejpam-5532	85	5	an	an	DET
ejpam-5532	85	6	l	l	NOUN
ejpam-5532	85	7	-	-	NOUN
ejpam-5532	85	8	lattice	lattice	NOUN
ejpam-5532	85	9	,	,	PUNCT
ejpam-5532	85	10	η	η	PROPN
ejpam-5532	85	11	∈	∈	PROPN
ejpam-5532	85	12	lm	lm	INTJ
ejpam-5532	85	13	,	,	PUNCT
ejpam-5532	85	14	η	η	PROPN
ejpam-5532	85	15	⊆	⊆	X
ejpam-5532	85	16	µ	µ	X
ejpam-5532	85	17	with	with	ADP
ejpam-5532	85	18	ao	ao	PROPN
ejpam-5532	85	19	=	=	SYM
ejpam-5532	85	20	tip{η	tip{η	NOUN
ejpam-5532	85	21	}	}	PUNCT
ejpam-5532	85	22	.	.	PUNCT
ejpam-5532	86	1	[	[	X
ejpam-5532	86	2	(	(	PUNCT
ejpam-5532	86	3	i)]define	i)]define	VERB
ejpam-5532	86	4	an	an	DET
ejpam-5532	86	5	l	l	NOUN
ejpam-5532	86	6	-	-	NOUN
ejpam-5532	86	7	subset	subset	ADJ
ejpam-5532	86	8	ηo	ηo	NOUN
ejpam-5532	86	9	of	of	ADP
ejpam-5532	86	10	m	m	PRON
ejpam-5532	86	11	as	as	ADP
ejpam-5532	86	12	:	:	PUNCT
ejpam-5532	86	13	ηo(x	ηo(x	X
ejpam-5532	86	14	)	)	PUNCT
ejpam-5532	86	15	=	=	PUNCT
ejpam-5532	86	16	∨	∨	NUM
ejpam-5532	86	17	t≤ao	t≤ao	PROPN
ejpam-5532	86	18	{	{	PUNCT
ejpam-5532	86	19	t	t	NOUN
ejpam-5532	86	20	:	:	PUNCT
ejpam-5532	86	21	x	x	X
ejpam-5532	86	22	∈	∈	PROPN
ejpam-5532	87	1	[	[	X
ejpam-5532	87	2	ηt	ηt	ADP
ejpam-5532	87	3	]	]	PUNCT
ejpam-5532	87	4	}	}	PUNCT
ejpam-5532	87	5	,	,	PUNCT
ejpam-5532	87	6	where	where	SCONJ
ejpam-5532	87	7	[	[	X
ejpam-5532	87	8	ηt	ηt	ADP
ejpam-5532	87	9	]	]	PUNCT
ejpam-5532	87	10	is	be	AUX
ejpam-5532	87	11	a	a	DET
ejpam-5532	87	12	sublattice	sublattice	NOUN
ejpam-5532	87	13	of	of	ADP
ejpam-5532	87	14	µt	µt	PRON
ejpam-5532	87	15	generated	generate	VERB
ejpam-5532	87	16	by	by	ADP
ejpam-5532	87	17	ηt	ηt	ADP
ejpam-5532	87	18	.	.	PUNCT
ejpam-5532	88	1	then	then	ADV
ejpam-5532	88	2	,	,	PUNCT
ejpam-5532	88	3	ηo	ηo	INTJ
ejpam-5532	88	4	is	be	AUX
ejpam-5532	88	5	an	an	DET
ejpam-5532	88	6	l	l	NOUN
ejpam-5532	88	7	-	-	NOUN
ejpam-5532	88	8	sublattice	sublattice	NOUN
ejpam-5532	88	9	of	of	ADP
ejpam-5532	88	10	µ	µ	NOUN
ejpam-5532	88	11	and	and	CCONJ
ejpam-5532	88	12	ηo	ηo	ADJ
ejpam-5532	88	13	=	=	SYM
ejpam-5532	89	1	[	[	X
ejpam-5532	89	2	η]µ.	η]µ.	PRON
ejpam-5532	89	3	define	define	VERB
ejpam-5532	89	4	an	an	DET
ejpam-5532	89	5	l	l	NOUN
ejpam-5532	89	6	-	-	PUNCT
ejpam-5532	89	7	subset	subset	ADJ
ejpam-5532	89	8	η1	η1	NOUN
ejpam-5532	89	9	of	of	ADP
ejpam-5532	89	10	m	m	PRON
ejpam-5532	89	11	as	as	ADP
ejpam-5532	89	12	:	:	PUNCT
ejpam-5532	89	13	η1(x	η1(x	NOUN
ejpam-5532	89	14	)	)	PUNCT
ejpam-5532	89	15	=	=	PUNCT
ejpam-5532	89	16	∨	∨	NUM
ejpam-5532	89	17	t≤ao	t≤ao	PROPN
ejpam-5532	89	18	{	{	PUNCT
ejpam-5532	89	19	t	t	NOUN
ejpam-5532	89	20	:	:	PUNCT
ejpam-5532	89	21	x	x	X
ejpam-5532	89	22	∈	∈	PROPN
ejpam-5532	89	23	(	(	PUNCT
ejpam-5532	89	24	ηt	ηt	ADP
ejpam-5532	89	25	]	]	PUNCT
ejpam-5532	89	26	}	}	PUNCT
ejpam-5532	89	27	,	,	PUNCT
ejpam-5532	89	28	where	where	SCONJ
ejpam-5532	89	29	(	(	PUNCT
ejpam-5532	89	30	ηt	ηt	ADP
ejpam-5532	89	31	]	]	PUNCT
ejpam-5532	89	32	is	be	AUX
ejpam-5532	89	33	an	an	DET
ejpam-5532	89	34	ideal	ideal	NOUN
ejpam-5532	89	35	of	of	ADP
ejpam-5532	89	36	µt	µt	PRON
ejpam-5532	89	37	generated	generate	VERB
ejpam-5532	89	38	by	by	ADP
ejpam-5532	89	39	ηt	ηt	ADP
ejpam-5532	89	40	.	.	PUNCT
ejpam-5532	90	1	then	then	ADV
ejpam-5532	90	2	,	,	PUNCT
ejpam-5532	90	3	η1	η1	NOUN
ejpam-5532	90	4	is	be	AUX
ejpam-5532	90	5	an	an	DET
ejpam-5532	90	6	l	l	NOUN
ejpam-5532	90	7	-	-	NOUN
ejpam-5532	90	8	ideal	ideal	NOUN
ejpam-5532	90	9	of	of	ADP
ejpam-5532	90	10	µ	µ	NOUN
ejpam-5532	90	11	and	and	CCONJ
ejpam-5532	90	12	η1	η1	NOUN
ejpam-5532	90	13	=	=	SYM
ejpam-5532	90	14	(	(	PUNCT
ejpam-5532	90	15	η]µ.	η]µ.	X
ejpam-5532	90	16	define	define	VERB
ejpam-5532	90	17	an	an	DET
ejpam-5532	90	18	l	l	NOUN
ejpam-5532	90	19	-	-	PUNCT
ejpam-5532	90	20	subset	subset	ADJ
ejpam-5532	90	21	η2	η2	NOUN
ejpam-5532	90	22	of	of	ADP
ejpam-5532	90	23	m	m	PRON
ejpam-5532	90	24	as	as	ADP
ejpam-5532	90	25	:	:	PUNCT
ejpam-5532	90	26	η2(x	η2(x	PROPN
ejpam-5532	90	27	)	)	PUNCT
ejpam-5532	90	28	=	=	SYM
ejpam-5532	91	1	∨	∨	NUM
ejpam-5532	91	2	t≤ao	t≤ao	PROPN
ejpam-5532	91	3	{	{	PUNCT
ejpam-5532	91	4	t	t	NOUN
ejpam-5532	91	5	:	:	PUNCT
ejpam-5532	91	6	x	x	X
ejpam-5532	91	7	∈	∈	PROPN
ejpam-5532	92	1	[	[	X
ejpam-5532	92	2	ηt	ηt	ADP
ejpam-5532	92	3	)	)	PUNCT
ejpam-5532	92	4	}	}	PUNCT
ejpam-5532	92	5	,	,	PUNCT
ejpam-5532	92	6	where	where	SCONJ
ejpam-5532	92	7	[	[	X
ejpam-5532	92	8	ηt	ηt	ADP
ejpam-5532	92	9	)	)	PUNCT
ejpam-5532	92	10	is	be	AUX
ejpam-5532	92	11	an	an	DET
ejpam-5532	92	12	dual	dual	ADJ
ejpam-5532	92	13	ideal	ideal	NOUN
ejpam-5532	92	14	of	of	ADP
ejpam-5532	92	15	µt	µt	PRON
ejpam-5532	92	16	generated	generate	VERB
ejpam-5532	92	17	by	by	ADP
ejpam-5532	92	18	ηt	ηt	ADP
ejpam-5532	92	19	.	.	PUNCT
ejpam-5532	93	1	then	then	ADV
ejpam-5532	93	2	,	,	PUNCT
ejpam-5532	93	3	η2	η2	PROPN
ejpam-5532	93	4	is	be	AUX
ejpam-5532	93	5	an	an	DET
ejpam-5532	93	6	l	l	ADJ
ejpam-5532	93	7	-	-	ADJ
ejpam-5532	93	8	dual	dual	ADJ
ejpam-5532	93	9	ideal	ideal	NOUN
ejpam-5532	93	10	of	of	ADP
ejpam-5532	93	11	µ	µ	NOUN
ejpam-5532	93	12	and	and	CCONJ
ejpam-5532	93	13	η2	η2	ADJ
ejpam-5532	93	14	=	=	PUNCT
ejpam-5532	94	1	[	[	X
ejpam-5532	94	2	η)µ.	η)µ.	PROPN
ejpam-5532	94	3	a.	a.	PROPN
ejpam-5532	94	4	jain	jain	PROPN
ejpam-5532	94	5	,	,	PUNCT
ejpam-5532	94	6	i.	i.	PROPN
ejpam-5532	94	7	jahan	jahan	PROPN
ejpam-5532	94	8	/	/	SYM
ejpam-5532	94	9	eur	eur	PROPN
ejpam-5532	94	10	.	.	PUNCT
ejpam-5532	95	1	j.	j.	PROPN
ejpam-5532	95	2	pure	pure	PROPN
ejpam-5532	95	3	appl	appl	PROPN
ejpam-5532	95	4	.	.	PROPN
ejpam-5532	95	5	math	math	PROPN
ejpam-5532	95	6	,	,	PUNCT
ejpam-5532	95	7	18	18	NUM
ejpam-5532	95	8	(	(	PUNCT
ejpam-5532	95	9	1	1	NUM
ejpam-5532	95	10	)	)	PUNCT
ejpam-5532	95	11	(	(	PUNCT
ejpam-5532	95	12	2025	2025	NUM
ejpam-5532	95	13	)	)	PUNCT
ejpam-5532	95	14	,	,	PUNCT
ejpam-5532	95	15	5532	5532	NUM
ejpam-5532	95	16	5	5	NUM
ejpam-5532	95	17	of	of	ADP
ejpam-5532	95	18	20	20	NUM
ejpam-5532	95	19	the	the	DET
ejpam-5532	95	20	concept	concept	NOUN
ejpam-5532	95	21	of	of	ADP
ejpam-5532	95	22	a	a	DET
ejpam-5532	95	23	maximal	maximal	ADJ
ejpam-5532	95	24	ideal	ideal	NOUN
ejpam-5532	95	25	could	could	AUX
ejpam-5532	95	26	be	be	AUX
ejpam-5532	95	27	meaningfully	meaningfully	ADV
ejpam-5532	95	28	extended	extend	VERB
ejpam-5532	95	29	from	from	ADP
ejpam-5532	95	30	classical	classical	ADJ
ejpam-5532	95	31	setting	setting	NOUN
ejpam-5532	95	32	to	to	ADP
ejpam-5532	95	33	fuzzy	fuzzy	ADJ
ejpam-5532	95	34	setting	setting	NOUN
ejpam-5532	95	35	by	by	ADP
ejpam-5532	95	36	the	the	DET
ejpam-5532	95	37	authors	author	NOUN
ejpam-5532	95	38	in	in	ADP
ejpam-5532	95	39	[	[	X
ejpam-5532	95	40	10	10	NUM
ejpam-5532	95	41	]	]	PUNCT
ejpam-5532	95	42	by	by	ADP
ejpam-5532	95	43	shifting	shift	VERB
ejpam-5532	95	44	the	the	DET
ejpam-5532	95	45	parent	parent	NOUN
ejpam-5532	95	46	structure	structure	NOUN
ejpam-5532	95	47	from	from	ADP
ejpam-5532	95	48	classical	classical	ADJ
ejpam-5532	95	49	lattice	lattice	NOUN
ejpam-5532	95	50	to	to	ADP
ejpam-5532	95	51	an	an	DET
ejpam-5532	95	52	l	l	NOUN
ejpam-5532	95	53	-	-	NOUN
ejpam-5532	95	54	structure	structure	NOUN
ejpam-5532	95	55	as	as	SCONJ
ejpam-5532	95	56	follows	follow	VERB
ejpam-5532	95	57	:	:	PUNCT
ejpam-5532	95	58	definition	definition	NOUN
ejpam-5532	95	59	4	4	NUM
ejpam-5532	95	60	(	(	PUNCT
ejpam-5532	95	61	[	[	X
ejpam-5532	95	62	10	10	NUM
ejpam-5532	95	63	]	]	NUM
ejpam-5532	95	64	)	)	PUNCT
ejpam-5532	95	65	.	.	PUNCT
ejpam-5532	96	1	let	let	AUX
ejpam-5532	96	2	l(µ,m	l(µ,m	PUNCT
ejpam-5532	96	3	)	)	PUNCT
ejpam-5532	96	4	be	be	AUX
ejpam-5532	96	5	an	an	DET
ejpam-5532	96	6	l	l	NOUN
ejpam-5532	96	7	-	-	NOUN
ejpam-5532	96	8	lattice	lattice	NOUN
ejpam-5532	96	9	.	.	PUNCT
ejpam-5532	97	1	[	[	X
ejpam-5532	97	2	(	(	PUNCT
ejpam-5532	97	3	i)]a	i)]a	PROPN
ejpam-5532	97	4	proper	proper	ADJ
ejpam-5532	97	5	l	l	NOUN
ejpam-5532	97	6	-	-	PUNCT
ejpam-5532	97	7	ideal	ideal	ADJ
ejpam-5532	97	8	η	η	PROPN
ejpam-5532	97	9	of	of	ADP
ejpam-5532	97	10	µ	µ	PROPN
ejpam-5532	97	11	is	be	AUX
ejpam-5532	97	12	called	call	VERB
ejpam-5532	97	13	an	an	DET
ejpam-5532	97	14	l	l	NOUN
ejpam-5532	97	15	-	-	ADJ
ejpam-5532	97	16	maximal	maximal	ADJ
ejpam-5532	97	17	ideal	ideal	NOUN
ejpam-5532	97	18	of	of	ADP
ejpam-5532	97	19	µ	µ	PRON
ejpam-5532	97	20	if	if	SCONJ
ejpam-5532	97	21	for	for	ADP
ejpam-5532	97	22	any	any	DET
ejpam-5532	97	23	l	l	NOUN
ejpam-5532	97	24	-	-	PUNCT
ejpam-5532	97	25	ideal	ideal	ADJ
ejpam-5532	97	26	θ	θ	PROPN
ejpam-5532	97	27	of	of	ADP
ejpam-5532	97	28	µ	µ	NOUN
ejpam-5532	97	29	,	,	PUNCT
ejpam-5532	97	30	whenever	whenever	SCONJ
ejpam-5532	97	31	η	η	PROPN
ejpam-5532	97	32	⊆	⊆	NUM
ejpam-5532	97	33	θ	θ	PROPN
ejpam-5532	97	34	⊆	⊆	NUM
ejpam-5532	97	35	µ	µ	NUM
ejpam-5532	97	36	,	,	PUNCT
ejpam-5532	97	37	then	then	ADV
ejpam-5532	97	38	θ	θ	PROPN
ejpam-5532	97	39	=	=	SYM
ejpam-5532	97	40	η	η	PROPN
ejpam-5532	97	41	or	or	CCONJ
ejpam-5532	97	42	θ	θ	PROPN
ejpam-5532	97	43	=	=	SYM
ejpam-5532	97	44	µ.	µ.	NOUN
ejpam-5532	97	45	a	a	DET
ejpam-5532	97	46	proper	proper	ADJ
ejpam-5532	97	47	l	l	ADJ
ejpam-5532	97	48	-	-	ADJ
ejpam-5532	97	49	dual	dual	ADJ
ejpam-5532	97	50	ideal	ideal	PROPN
ejpam-5532	97	51	η	η	PROPN
ejpam-5532	97	52	of	of	ADP
ejpam-5532	97	53	µ	µ	PROPN
ejpam-5532	97	54	is	be	AUX
ejpam-5532	97	55	called	call	VERB
ejpam-5532	97	56	an	an	DET
ejpam-5532	97	57	l	l	NOUN
ejpam-5532	97	58	-	-	ADJ
ejpam-5532	97	59	dual	dual	ADJ
ejpam-5532	97	60	maximal	maximal	ADJ
ejpam-5532	97	61	ideal	ideal	NOUN
ejpam-5532	97	62	of	of	ADP
ejpam-5532	97	63	µ	µ	PRON
ejpam-5532	97	64	if	if	SCONJ
ejpam-5532	97	65	for	for	ADP
ejpam-5532	97	66	any	any	DET
ejpam-5532	97	67	l	l	ADJ
ejpam-5532	97	68	-	-	ADJ
ejpam-5532	97	69	dual	dual	ADJ
ejpam-5532	97	70	ideal	ideal	ADJ
ejpam-5532	97	71	θ	θ	PROPN
ejpam-5532	97	72	of	of	ADP
ejpam-5532	97	73	µ	µ	NOUN
ejpam-5532	97	74	,	,	PUNCT
ejpam-5532	98	1	whenever	whenever	SCONJ
ejpam-5532	98	2	η	η	PROPN
ejpam-5532	98	3	⊆	⊆	NUM
ejpam-5532	98	4	θ	θ	PROPN
ejpam-5532	98	5	⊆	⊆	NUM
ejpam-5532	98	6	µ	µ	NUM
ejpam-5532	98	7	,	,	PUNCT
ejpam-5532	98	8	then	then	ADV
ejpam-5532	98	9	θ	θ	PROPN
ejpam-5532	98	10	=	=	SYM
ejpam-5532	98	11	η	η	PROPN
ejpam-5532	98	12	or	or	CCONJ
ejpam-5532	98	13	θ	θ	PROPN
ejpam-5532	98	14	=	=	SYM
ejpam-5532	98	15	µ.	µ.	NOUN
ejpam-5532	98	16	in	in	ADP
ejpam-5532	98	17	[	[	X
ejpam-5532	98	18	10	10	NUM
ejpam-5532	98	19	]	]	PUNCT
ejpam-5532	98	20	,	,	PUNCT
ejpam-5532	98	21	some	some	DET
ejpam-5532	98	22	characterizations	characterization	NOUN
ejpam-5532	98	23	of	of	ADP
ejpam-5532	98	24	an	an	DET
ejpam-5532	98	25	l	l	NOUN
ejpam-5532	98	26	-	-	ADJ
ejpam-5532	98	27	maximal	maximal	ADJ
ejpam-5532	98	28	ideal	ideal	NOUN
ejpam-5532	98	29	and	and	CCONJ
ejpam-5532	98	30	l	l	ADJ
ejpam-5532	98	31	-	-	ADJ
ejpam-5532	98	32	maximal	maximal	ADJ
ejpam-5532	98	33	dual	dual	ADJ
ejpam-5532	98	34	ideal	ideal	NOUN
ejpam-5532	98	35	of	of	ADP
ejpam-5532	98	36	µ	µ	PROPN
ejpam-5532	98	37	were	be	AUX
ejpam-5532	98	38	provided	provide	VERB
ejpam-5532	98	39	.	.	PUNCT
ejpam-5532	99	1	further	far	ADV
ejpam-5532	99	2	,	,	PUNCT
ejpam-5532	99	3	an	an	DET
ejpam-5532	99	4	l	l	NOUN
ejpam-5532	99	5	-	-	ADJ
ejpam-5532	99	6	prime	prime	ADJ
ejpam-5532	99	7	ideal(l	ideal(l	NOUN
ejpam-5532	99	8	-	-	PUNCT
ejpam-5532	99	9	prime	prime	ADJ
ejpam-5532	99	10	dual	dual	ADJ
ejpam-5532	99	11	ideal	ideal	NOUN
ejpam-5532	99	12	)	)	PUNCT
ejpam-5532	99	13	in	in	ADP
ejpam-5532	99	14	lattice	lattice	PROPN
ejpam-5532	99	15	m	m	PROPN
ejpam-5532	99	16	and	and	CCONJ
ejpam-5532	99	17	an	an	DET
ejpam-5532	99	18	l	l	NOUN
ejpam-5532	99	19	-	-	ADJ
ejpam-5532	99	20	prime	prime	ADJ
ejpam-5532	99	21	ideal(l	ideal(l	NOUN
ejpam-5532	99	22	-	-	PUNCT
ejpam-5532	99	23	prime	prime	ADJ
ejpam-5532	99	24	dual	dual	ADJ
ejpam-5532	99	25	ideal	ideal	NOUN
ejpam-5532	99	26	)	)	PUNCT
ejpam-5532	99	27	in	in	ADP
ejpam-5532	99	28	an	an	DET
ejpam-5532	99	29	l	l	NOUN
ejpam-5532	99	30	-	-	PUNCT
ejpam-5532	99	31	lattice	lattice	ADJ
ejpam-5532	99	32	µ	µ	X
ejpam-5532	99	33	are	be	AUX
ejpam-5532	99	34	defined	define	VERB
ejpam-5532	99	35	as	as	SCONJ
ejpam-5532	99	36	follows	follow	VERB
ejpam-5532	99	37	:	:	PUNCT
ejpam-5532	99	38	definition	definition	NOUN
ejpam-5532	99	39	5	5	NUM
ejpam-5532	99	40	(	(	PUNCT
ejpam-5532	99	41	[	[	X
ejpam-5532	99	42	4	4	NUM
ejpam-5532	99	43	]	]	NUM
ejpam-5532	99	44	)	)	PUNCT
ejpam-5532	99	45	.	.	PUNCT
ejpam-5532	100	1	[	[	X
ejpam-5532	100	2	(	(	PUNCT
ejpam-5532	100	3	i	i	NOUN
ejpam-5532	100	4	)	)	PUNCT
ejpam-5532	100	5	]	]	PUNCT
ejpam-5532	100	6	(	(	PUNCT
ejpam-5532	100	7	i	i	NOUN
ejpam-5532	100	8	)	)	PUNCT
ejpam-5532	100	9	an	an	DET
ejpam-5532	100	10	l	l	NOUN
ejpam-5532	100	11	-	-	PUNCT
ejpam-5532	100	12	ideal	ideal	ADJ
ejpam-5532	100	13	µ	µ	PROPN
ejpam-5532	100	14	of	of	ADP
ejpam-5532	100	15	m	m	PROPN
ejpam-5532	100	16	is	be	AUX
ejpam-5532	100	17	called	call	VERB
ejpam-5532	100	18	an	an	DET
ejpam-5532	100	19	l	l	ADJ
ejpam-5532	100	20	-	-	ADJ
ejpam-5532	100	21	prime	prime	ADJ
ejpam-5532	100	22	ideal	ideal	NOUN
ejpam-5532	100	23	of	of	ADP
ejpam-5532	100	24	m	m	PRON
ejpam-5532	100	25	if	if	SCONJ
ejpam-5532	100	26	µ(x	µ(x	ADJ
ejpam-5532	100	27	∧	∧	PROPN
ejpam-5532	100	28	y	y	NOUN
ejpam-5532	100	29	)	)	PUNCT
ejpam-5532	100	30	≤	≤	NOUN
ejpam-5532	100	31	µ(x	µ(x	X
ejpam-5532	100	32	)	)	PUNCT
ejpam-5532	100	33	∨	∨	NUM
ejpam-5532	100	34	µ(y	µ(y	PROPN
ejpam-5532	100	35	)	)	PUNCT
ejpam-5532	100	36	;	;	PUNCT
ejpam-5532	100	37	∀x	∀x	X
ejpam-5532	100	38	,	,	PUNCT
ejpam-5532	100	39	y	y	PROPN
ejpam-5532	100	40	∈	∈	PROPN
ejpam-5532	100	41	m.	m.	NOUN
ejpam-5532	100	42	(	(	PUNCT
ejpam-5532	100	43	ii	ii	NOUN
ejpam-5532	100	44	)	)	PUNCT
ejpam-5532	100	45	an	an	DET
ejpam-5532	100	46	l	l	ADJ
ejpam-5532	100	47	-	-	ADJ
ejpam-5532	100	48	dual	dual	ADJ
ejpam-5532	100	49	ideal	ideal	NOUN
ejpam-5532	100	50	µ	µ	PROPN
ejpam-5532	100	51	of	of	ADP
ejpam-5532	100	52	m	m	PROPN
ejpam-5532	100	53	is	be	AUX
ejpam-5532	100	54	called	call	VERB
ejpam-5532	100	55	an	an	DET
ejpam-5532	100	56	l	l	NOUN
ejpam-5532	100	57	-	-	ADJ
ejpam-5532	100	58	prime	prime	ADJ
ejpam-5532	100	59	dual	dual	ADJ
ejpam-5532	100	60	ideal	ideal	NOUN
ejpam-5532	100	61	of	of	ADP
ejpam-5532	100	62	m	m	PRON
ejpam-5532	100	63	if	if	SCONJ
ejpam-5532	100	64	µ(x	µ(x	PROPN
ejpam-5532	100	65	∨	∨	NUM
ejpam-5532	100	66	y	y	NOUN
ejpam-5532	100	67	)	)	PUNCT
ejpam-5532	100	68	≤	≤	NOUN
ejpam-5532	100	69	µ(x	µ(x	X
ejpam-5532	100	70	)	)	PUNCT
ejpam-5532	100	71	∨	∨	NUM
ejpam-5532	100	72	µ(y	µ(y	PROPN
ejpam-5532	100	73	)	)	PUNCT
ejpam-5532	100	74	;	;	PUNCT
ejpam-5532	100	75	∀x	∀x	X
ejpam-5532	100	76	,	,	PUNCT
ejpam-5532	100	77	y	y	PROPN
ejpam-5532	100	78	∈	∈	PROPN
ejpam-5532	100	79	m.	m.	NOUN
ejpam-5532	100	80	definition	definition	NOUN
ejpam-5532	100	81	6	6	NUM
ejpam-5532	100	82	(	(	PUNCT
ejpam-5532	100	83	[	[	X
ejpam-5532	100	84	10	10	NUM
ejpam-5532	100	85	]	]	NUM
ejpam-5532	100	86	)	)	PUNCT
ejpam-5532	100	87	.	.	PUNCT
ejpam-5532	101	1	let	let	AUX
ejpam-5532	101	2	l(µ,m	l(µ,m	PUNCT
ejpam-5532	101	3	)	)	PUNCT
ejpam-5532	101	4	be	be	AUX
ejpam-5532	101	5	an	an	DET
ejpam-5532	101	6	l	l	NOUN
ejpam-5532	101	7	-	-	NOUN
ejpam-5532	101	8	lattice	lattice	NOUN
ejpam-5532	101	9	.	.	PUNCT
ejpam-5532	102	1	[	[	X
ejpam-5532	102	2	(	(	PUNCT
ejpam-5532	102	3	i)]a	i)]a	PROPN
ejpam-5532	102	4	proper	proper	ADJ
ejpam-5532	102	5	l	l	NOUN
ejpam-5532	102	6	-	-	PUNCT
ejpam-5532	102	7	ideal	ideal	ADJ
ejpam-5532	102	8	η	η	PROPN
ejpam-5532	102	9	of	of	ADP
ejpam-5532	102	10	µ	µ	PROPN
ejpam-5532	102	11	is	be	AUX
ejpam-5532	102	12	called	call	VERB
ejpam-5532	102	13	an	an	DET
ejpam-5532	102	14	l	l	ADJ
ejpam-5532	102	15	-	-	ADJ
ejpam-5532	102	16	prime	prime	ADJ
ejpam-5532	102	17	ideal	ideal	NOUN
ejpam-5532	102	18	of	of	ADP
ejpam-5532	102	19	µ	µ	PRON
ejpam-5532	102	20	if	if	SCONJ
ejpam-5532	102	21	,	,	PUNCT
ejpam-5532	102	22	∀	∀	X
ejpam-5532	102	23	x	x	NOUN
ejpam-5532	102	24	,	,	PUNCT
ejpam-5532	102	25	y	y	PROPN
ejpam-5532	102	26	∈	∈	PROPN
ejpam-5532	102	27	m	m	VERB
ejpam-5532	102	28	η(x	η(x	PROPN
ejpam-5532	102	29	∧	∧	PROPN
ejpam-5532	102	30	y	y	PROPN
ejpam-5532	102	31	)	)	PUNCT
ejpam-5532	102	32	∧	∧	PROPN
ejpam-5532	102	33	µ(x	µ(x	VERB
ejpam-5532	102	34	)	)	PUNCT
ejpam-5532	102	35	∧	∧	PROPN
ejpam-5532	102	36	µ(y	µ(y	NOUN
ejpam-5532	102	37	)	)	PUNCT
ejpam-5532	102	38	≤	≤	NOUN
ejpam-5532	102	39	η(x	η(x	NOUN
ejpam-5532	102	40	)	)	PUNCT
ejpam-5532	102	41	or	or	CCONJ
ejpam-5532	102	42	η(x	η(x	PRON
ejpam-5532	102	43	∧	∧	PROPN
ejpam-5532	102	44	y	y	NOUN
ejpam-5532	102	45	)	)	PUNCT
ejpam-5532	102	46	∧	∧	PROPN
ejpam-5532	102	47	µ(x	µ(x	VERB
ejpam-5532	102	48	)	)	PUNCT
ejpam-5532	102	49	∧	∧	PROPN
ejpam-5532	102	50	µ(y	µ(y	NOUN
ejpam-5532	102	51	)	)	PUNCT
ejpam-5532	102	52	≤	≤	NUM
ejpam-5532	102	53	η(y	η(y	NOUN
ejpam-5532	102	54	)	)	PUNCT
ejpam-5532	102	55	.	.	PUNCT
ejpam-5532	103	1	a	a	DET
ejpam-5532	103	2	proper	proper	ADJ
ejpam-5532	103	3	l	l	ADJ
ejpam-5532	103	4	-	-	ADJ
ejpam-5532	103	5	dual	dual	ADJ
ejpam-5532	103	6	ideal	ideal	PROPN
ejpam-5532	103	7	η	η	PROPN
ejpam-5532	103	8	of	of	ADP
ejpam-5532	103	9	µ	µ	PROPN
ejpam-5532	103	10	is	be	AUX
ejpam-5532	103	11	called	call	VERB
ejpam-5532	103	12	an	an	DET
ejpam-5532	103	13	l	l	ADJ
ejpam-5532	103	14	-	-	ADJ
ejpam-5532	103	15	dual	dual	ADJ
ejpam-5532	103	16	prime	prime	ADJ
ejpam-5532	103	17	ideal	ideal	NOUN
ejpam-5532	103	18	of	of	ADP
ejpam-5532	103	19	µ	µ	PRON
ejpam-5532	103	20	if	if	SCONJ
ejpam-5532	103	21	,	,	PUNCT
ejpam-5532	103	22	∀	∀	X
ejpam-5532	103	23	x	x	NOUN
ejpam-5532	103	24	,	,	PUNCT
ejpam-5532	103	25	y	y	PROPN
ejpam-5532	103	26	∈	∈	PROPN
ejpam-5532	103	27	m	m	VERB
ejpam-5532	103	28	η(x	η(x	PROPN
ejpam-5532	103	29	∨	∨	NUM
ejpam-5532	103	30	y	y	NOUN
ejpam-5532	103	31	)	)	PUNCT
ejpam-5532	103	32	∧	∧	PROPN
ejpam-5532	103	33	µ(x	µ(x	VERB
ejpam-5532	103	34	)	)	PUNCT
ejpam-5532	103	35	∧	∧	PROPN
ejpam-5532	103	36	µ(y	µ(y	NOUN
ejpam-5532	103	37	)	)	PUNCT
ejpam-5532	103	38	≤	≤	NOUN
ejpam-5532	103	39	η(x	η(x	NOUN
ejpam-5532	103	40	)	)	PUNCT
ejpam-5532	103	41	or	or	CCONJ
ejpam-5532	103	42	η(x	η(x	PRON
ejpam-5532	103	43	∨	∨	NUM
ejpam-5532	103	44	y	y	NOUN
ejpam-5532	103	45	)	)	PUNCT
ejpam-5532	103	46	∧	∧	PROPN
ejpam-5532	103	47	µ(x	µ(x	VERB
ejpam-5532	103	48	)	)	PUNCT
ejpam-5532	103	49	∧	∧	PROPN
ejpam-5532	103	50	µ(y	µ(y	NOUN
ejpam-5532	103	51	)	)	PUNCT
ejpam-5532	103	52	≤	≤	NUM
ejpam-5532	103	53	η(y	η(y	NOUN
ejpam-5532	103	54	)	)	PUNCT
ejpam-5532	103	55	.	.	PUNCT
ejpam-5532	104	1	in	in	ADP
ejpam-5532	104	2	[	[	X
ejpam-5532	104	3	10	10	NUM
ejpam-5532	104	4	]	]	PUNCT
ejpam-5532	104	5	,	,	PUNCT
ejpam-5532	104	6	the	the	DET
ejpam-5532	104	7	authors	author	NOUN
ejpam-5532	104	8	defined	define	VERB
ejpam-5532	104	9	a	a	DET
ejpam-5532	104	10	fuzzy	fuzzy	ADJ
ejpam-5532	104	11	convex	convex	NOUN
ejpam-5532	104	12	sublattice	sublattice	NOUN
ejpam-5532	104	13	of	of	ADP
ejpam-5532	104	14	a	a	DET
ejpam-5532	104	15	lattice	lattice	NOUN
ejpam-5532	104	16	and	and	CCONJ
ejpam-5532	104	17	studied	study	VERB
ejpam-5532	104	18	the	the	DET
ejpam-5532	104	19	related	related	ADJ
ejpam-5532	104	20	properties	property	NOUN
ejpam-5532	104	21	.	.	PUNCT
ejpam-5532	105	1	on	on	ADP
ejpam-5532	105	2	similar	similar	ADJ
ejpam-5532	105	3	lines	line	NOUN
ejpam-5532	105	4	,	,	PUNCT
ejpam-5532	105	5	an	an	DET
ejpam-5532	105	6	l	l	ADJ
ejpam-5532	105	7	-	-	ADJ
ejpam-5532	105	8	convex	convex	ADJ
ejpam-5532	105	9	sublattice	sublattice	NOUN
ejpam-5532	105	10	of	of	ADP
ejpam-5532	105	11	a	a	DET
ejpam-5532	105	12	lattice	lattice	NOUN
ejpam-5532	105	13	m	m	VERB
ejpam-5532	105	14	can	can	AUX
ejpam-5532	105	15	be	be	AUX
ejpam-5532	105	16	defined	define	VERB
ejpam-5532	105	17	as	as	SCONJ
ejpam-5532	105	18	follows	follow	VERB
ejpam-5532	105	19	:	:	PUNCT
ejpam-5532	105	20	definition	definition	NOUN
ejpam-5532	105	21	7	7	NUM
ejpam-5532	105	22	.	.	PUNCT
ejpam-5532	106	1	if	if	SCONJ
ejpam-5532	106	2	µ	µ	PRON
ejpam-5532	106	3	∈	∈	PROPN
ejpam-5532	106	4	l(m	l(m	PROPN
ejpam-5532	106	5	)	)	PUNCT
ejpam-5532	106	6	,	,	PUNCT
ejpam-5532	106	7	then	then	ADV
ejpam-5532	106	8	µ	µ	NOUN
ejpam-5532	106	9	is	be	AUX
ejpam-5532	106	10	called	call	VERB
ejpam-5532	106	11	an	an	DET
ejpam-5532	106	12	l	l	NOUN
ejpam-5532	106	13	-	-	ADJ
ejpam-5532	106	14	convex	convex	ADJ
ejpam-5532	106	15	sublattice	sublattice	NOUN
ejpam-5532	106	16	of	of	ADP
ejpam-5532	106	17	m	m	PRON
ejpam-5532	106	18	if	if	SCONJ
ejpam-5532	106	19	for	for	ADP
ejpam-5532	106	20	each	each	DET
ejpam-5532	106	21	interval	interval	NOUN
ejpam-5532	106	22	[	[	X
ejpam-5532	106	23	a	a	X
ejpam-5532	106	24	,	,	PUNCT
ejpam-5532	106	25	b	b	NOUN
ejpam-5532	106	26	]	]	X
ejpam-5532	106	27	⊆	⊆	NUM
ejpam-5532	106	28	m	m	NOUN
ejpam-5532	106	29	,	,	PUNCT
ejpam-5532	106	30	µ(x	µ(x	ADJ
ejpam-5532	106	31	)	)	PUNCT
ejpam-5532	106	32	≥	≥	NOUN
ejpam-5532	106	33	µ(a	µ(a	NOUN
ejpam-5532	106	34	)	)	PUNCT
ejpam-5532	106	35	∧	∧	PROPN
ejpam-5532	106	36	µ(b	µ(b	PROPN
ejpam-5532	106	37	)	)	PUNCT
ejpam-5532	106	38	,	,	PUNCT
ejpam-5532	106	39	∀	∀	PUNCT
ejpam-5532	107	1	x	x	SYM
ejpam-5532	107	2	∈	∈	PROPN
ejpam-5532	107	3	[	[	X
ejpam-5532	107	4	a	a	X
ejpam-5532	107	5	,	,	PUNCT
ejpam-5532	107	6	b	b	NOUN
ejpam-5532	107	7	]	]	PUNCT
ejpam-5532	107	8	.	.	PUNCT
ejpam-5532	108	1	a.	a.	PROPN
ejpam-5532	108	2	jain	jain	PROPN
ejpam-5532	108	3	,	,	PUNCT
ejpam-5532	108	4	i.	i.	PROPN
ejpam-5532	108	5	jahan	jahan	PROPN
ejpam-5532	108	6	/	/	SYM
ejpam-5532	108	7	eur	eur	PROPN
ejpam-5532	108	8	.	.	PUNCT
ejpam-5532	109	1	j.	j.	PROPN
ejpam-5532	109	2	pure	pure	PROPN
ejpam-5532	109	3	appl	appl	PROPN
ejpam-5532	109	4	.	.	PROPN
ejpam-5532	109	5	math	math	PROPN
ejpam-5532	109	6	,	,	PUNCT
ejpam-5532	109	7	18	18	NUM
ejpam-5532	109	8	(	(	PUNCT
ejpam-5532	109	9	1	1	NUM
ejpam-5532	109	10	)	)	PUNCT
ejpam-5532	109	11	(	(	PUNCT
ejpam-5532	109	12	2025	2025	NUM
ejpam-5532	109	13	)	)	PUNCT
ejpam-5532	109	14	,	,	PUNCT
ejpam-5532	109	15	5532	5532	NUM
ejpam-5532	109	16	6	6	NUM
ejpam-5532	109	17	of	of	ADP
ejpam-5532	109	18	20	20	NUM
ejpam-5532	109	19	3	3	NUM
ejpam-5532	109	20	.	.	PUNCT
ejpam-5532	110	1	l	l	ADJ
ejpam-5532	110	2	-	-	ADJ
ejpam-5532	110	3	convex	convex	ADJ
ejpam-5532	110	4	sublattice	sublattice	NOUN
ejpam-5532	110	5	of	of	ADP
ejpam-5532	110	6	an	an	DET
ejpam-5532	110	7	l	l	NOUN
ejpam-5532	110	8	-	-	NOUN
ejpam-5532	110	9	lattice	lattice	NOUN
ejpam-5532	110	10	in	in	ADP
ejpam-5532	110	11	this	this	DET
ejpam-5532	110	12	section	section	NOUN
ejpam-5532	110	13	,	,	PUNCT
ejpam-5532	110	14	l	l	NOUN
ejpam-5532	110	15	is	be	AUX
ejpam-5532	110	16	taken	take	VERB
ejpam-5532	110	17	to	to	PART
ejpam-5532	110	18	be	be	AUX
ejpam-5532	110	19	a	a	DET
ejpam-5532	110	20	complete	complete	ADJ
ejpam-5532	110	21	and	and	CCONJ
ejpam-5532	110	22	completely	completely	ADV
ejpam-5532	110	23	distributive	distributive	ADJ
ejpam-5532	110	24	lattice	lattice	NOUN
ejpam-5532	110	25	in	in	ADP
ejpam-5532	110	26	some	some	DET
ejpam-5532	110	27	results	result	NOUN
ejpam-5532	110	28	.	.	PUNCT
ejpam-5532	111	1	the	the	DET
ejpam-5532	111	2	definition	definition	NOUN
ejpam-5532	111	3	of	of	ADP
ejpam-5532	111	4	a	a	DET
ejpam-5532	111	5	completely	completely	ADV
ejpam-5532	111	6	distributive	distributive	ADJ
ejpam-5532	111	7	lattice	lattice	NOUN
ejpam-5532	111	8	is	be	AUX
ejpam-5532	111	9	well	well	ADV
ejpam-5532	111	10	known	know	VERB
ejpam-5532	111	11	in	in	ADP
ejpam-5532	111	12	the	the	DET
ejpam-5532	111	13	literature	literature	NOUN
ejpam-5532	111	14	and	and	CCONJ
ejpam-5532	111	15	can	can	AUX
ejpam-5532	111	16	be	be	AUX
ejpam-5532	111	17	found	find	VERB
ejpam-5532	111	18	in	in	ADP
ejpam-5532	111	19	any	any	DET
ejpam-5532	111	20	standard	standard	ADJ
ejpam-5532	111	21	text	text	NOUN
ejpam-5532	111	22	on	on	ADP
ejpam-5532	111	23	the	the	DET
ejpam-5532	111	24	subject	subject	NOUN
ejpam-5532	111	25	.	.	PUNCT
ejpam-5532	112	1	let	let	VERB
ejpam-5532	112	2	{	{	PUNCT
ejpam-5532	112	3	ji	ji	NOUN
ejpam-5532	112	4	:	:	PUNCT
ejpam-5532	113	1	i	i	PRON
ejpam-5532	113	2	∈	∈	VERB
ejpam-5532	113	3	i	i	PRON
ejpam-5532	113	4	}	}	PUNCT
ejpam-5532	113	5	be	be	VERB
ejpam-5532	113	6	any	any	DET
ejpam-5532	113	7	family	family	NOUN
ejpam-5532	113	8	of	of	ADP
ejpam-5532	113	9	subsets	subset	NOUN
ejpam-5532	113	10	of	of	ADP
ejpam-5532	113	11	a	a	DET
ejpam-5532	113	12	complete	complete	ADJ
ejpam-5532	113	13	lattice	lattice	NOUN
ejpam-5532	113	14	l	l	NOUN
ejpam-5532	113	15	and	and	CCONJ
ejpam-5532	113	16	f	f	PROPN
ejpam-5532	113	17	denote	denote	VERB
ejpam-5532	113	18	the	the	DET
ejpam-5532	113	19	set	set	NOUN
ejpam-5532	113	20	of	of	ADP
ejpam-5532	113	21	choice	choice	NOUN
ejpam-5532	113	22	functions	function	NOUN
ejpam-5532	113	23	for	for	ADP
ejpam-5532	113	24	ji	ji	PROPN
ejpam-5532	113	25	,	,	PUNCT
ejpam-5532	113	26	i.e.	i.e.	X
ejpam-5532	113	27	functions	function	NOUN
ejpam-5532	114	1	f	f	NOUN
ejpam-5532	114	2	:	:	PUNCT
ejpam-5532	114	3	i	i	PROPN
ejpam-5532	114	4	→	→	SYM
ejpam-5532	114	5	∏	∏	PROPN
ejpam-5532	114	6	i∈i	i∈i	NOUN
ejpam-5532	114	7	ji	ji	PROPN
ejpam-5532	114	8	such	such	ADJ
ejpam-5532	114	9	that	that	SCONJ
ejpam-5532	114	10	f(i	f(i	PROPN
ejpam-5532	114	11	)	)	PUNCT
ejpam-5532	114	12	∈	∈	PROPN
ejpam-5532	114	13	ji	ji	PROPN
ejpam-5532	114	14	for	for	ADP
ejpam-5532	114	15	each	each	DET
ejpam-5532	114	16	i	i	PRON
ejpam-5532	114	17	∈	∈	PROPN
ejpam-5532	114	18	i.	i.	NOUN
ejpam-5532	114	19	then	then	ADV
ejpam-5532	114	20	,	,	PUNCT
ejpam-5532	114	21	we	we	PRON
ejpam-5532	114	22	say	say	VERB
ejpam-5532	114	23	that	that	SCONJ
ejpam-5532	114	24	l	l	NOUN
ejpam-5532	114	25	is	be	AUX
ejpam-5532	114	26	a	a	DET
ejpam-5532	114	27	completely	completely	ADV
ejpam-5532	114	28	distributive	distributive	ADJ
ejpam-5532	114	29	lattice	lattice	NOUN
ejpam-5532	114	30	,	,	PUNCT
ejpam-5532	114	31	if	if	SCONJ
ejpam-5532	114	32	∧{∨	∧{∨	PROPN
ejpam-5532	114	33	i∈i	i∈i	VERB
ejpam-5532	114	34	ji	ji	PROPN
ejpam-5532	114	35	}	}	PUNCT
ejpam-5532	114	36	=	=	PUNCT
ejpam-5532	114	37	∨	∨	NUM
ejpam-5532	114	38	f∈f	f∈f	NOUN
ejpam-5532	114	39	{	{	PUNCT
ejpam-5532	114	40	∧	∧	PROPN
ejpam-5532	114	41	i∈i	i∈i	ADJ
ejpam-5532	114	42	f(i	f(i	PROPN
ejpam-5532	114	43	)	)	PUNCT
ejpam-5532	114	44	}	}	PUNCT
ejpam-5532	114	45	.	.	PUNCT
ejpam-5532	115	1	the	the	DET
ejpam-5532	115	2	above	above	ADJ
ejpam-5532	115	3	law	law	NOUN
ejpam-5532	115	4	is	be	AUX
ejpam-5532	115	5	known	know	VERB
ejpam-5532	115	6	as	as	ADP
ejpam-5532	115	7	the	the	DET
ejpam-5532	115	8	complete	complete	ADJ
ejpam-5532	115	9	distributive	distributive	ADJ
ejpam-5532	115	10	law	law	NOUN
ejpam-5532	115	11	.	.	PUNCT
ejpam-5532	116	1	thus	thus	ADV
ejpam-5532	116	2	,	,	PUNCT
ejpam-5532	116	3	in	in	ADP
ejpam-5532	116	4	order	order	NOUN
ejpam-5532	116	5	theory	theory	NOUN
ejpam-5532	116	6	,	,	PUNCT
ejpam-5532	116	7	a	a	DET
ejpam-5532	116	8	complete	complete	ADJ
ejpam-5532	116	9	lattice	lattice	NOUN
ejpam-5532	116	10	is	be	AUX
ejpam-5532	116	11	completely	completely	ADV
ejpam-5532	116	12	distributive	distributive	ADJ
ejpam-5532	116	13	if	if	SCONJ
ejpam-5532	116	14	arbitrary	arbitrary	ADJ
ejpam-5532	116	15	joins	join	VERB
ejpam-5532	116	16	distribute	distribute	VERB
ejpam-5532	116	17	over	over	ADP
ejpam-5532	116	18	arbitrary	arbitrary	ADJ
ejpam-5532	116	19	meets	meet	NOUN
ejpam-5532	116	20	.	.	PUNCT
ejpam-5532	117	1	note	note	VERB
ejpam-5532	117	2	that	that	SCONJ
ejpam-5532	117	3	the	the	DET
ejpam-5532	117	4	dual	dual	ADJ
ejpam-5532	117	5	of	of	ADP
ejpam-5532	117	6	completely	completely	ADV
ejpam-5532	117	7	distributive	distributive	ADJ
ejpam-5532	117	8	law	law	NOUN
ejpam-5532	117	9	is	be	AUX
ejpam-5532	117	10	valid	valid	ADJ
ejpam-5532	117	11	in	in	ADP
ejpam-5532	117	12	a	a	DET
ejpam-5532	117	13	completely	completely	ADV
ejpam-5532	117	14	distributive	distributive	ADJ
ejpam-5532	117	15	lattice	lattice	NOUN
ejpam-5532	117	16	.	.	PUNCT
ejpam-5532	118	1	we	we	PRON
ejpam-5532	118	2	begin	begin	VERB
ejpam-5532	118	3	this	this	DET
ejpam-5532	118	4	section	section	NOUN
ejpam-5532	118	5	by	by	ADP
ejpam-5532	118	6	defining	define	VERB
ejpam-5532	118	7	an	an	DET
ejpam-5532	118	8	l	l	ADJ
ejpam-5532	118	9	-	-	ADJ
ejpam-5532	118	10	convex	convex	ADJ
ejpam-5532	118	11	sublattice	sublattice	NOUN
ejpam-5532	118	12	of	of	ADP
ejpam-5532	118	13	an	an	DET
ejpam-5532	118	14	l	l	NOUN
ejpam-5532	118	15	-	-	PUNCT
ejpam-5532	118	16	lattice	lattice	NOUN
ejpam-5532	118	17	µ	µ	NOUN
ejpam-5532	118	18	and	and	CCONJ
ejpam-5532	118	19	study	study	VERB
ejpam-5532	118	20	its	its	PRON
ejpam-5532	118	21	properties	property	NOUN
ejpam-5532	118	22	.	.	PUNCT
ejpam-5532	119	1	definition	definition	NOUN
ejpam-5532	119	2	8	8	NUM
ejpam-5532	119	3	.	.	PUNCT
ejpam-5532	120	1	let	let	AUX
ejpam-5532	120	2	l(µ,m	l(µ,m	ADV
ejpam-5532	120	3	)	)	PUNCT
ejpam-5532	120	4	be	be	AUX
ejpam-5532	120	5	an	an	DET
ejpam-5532	120	6	l	l	NOUN
ejpam-5532	120	7	-	-	NOUN
ejpam-5532	120	8	lattice	lattice	NOUN
ejpam-5532	120	9	.	.	PUNCT
ejpam-5532	121	1	an	an	DET
ejpam-5532	121	2	l	l	NOUN
ejpam-5532	121	3	-	-	PUNCT
ejpam-5532	121	4	sublattice	sublattice	NOUN
ejpam-5532	121	5	η	η	PROPN
ejpam-5532	121	6	of	of	ADP
ejpam-5532	121	7	µ	µ	PROPN
ejpam-5532	121	8	is	be	AUX
ejpam-5532	121	9	called	call	VERB
ejpam-5532	121	10	an	an	DET
ejpam-5532	121	11	l	l	NOUN
ejpam-5532	121	12	-	-	ADJ
ejpam-5532	121	13	convex	convex	ADJ
ejpam-5532	121	14	sublattice	sublattice	NOUN
ejpam-5532	121	15	of	of	ADP
ejpam-5532	121	16	µ	µ	NOUN
ejpam-5532	121	17	if	if	SCONJ
ejpam-5532	121	18	η(x	η(x	NOUN
ejpam-5532	121	19	)	)	PUNCT
ejpam-5532	121	20	≥	≥	NOUN
ejpam-5532	121	21	η(a	η(a	VERB
ejpam-5532	121	22	)	)	PUNCT
ejpam-5532	121	23	∧	∧	PROPN
ejpam-5532	121	24	η(b	η(b	NOUN
ejpam-5532	121	25	)	)	PUNCT
ejpam-5532	121	26	∧	∧	NOUN
ejpam-5532	121	27	µ(x	µ(x	VERB
ejpam-5532	121	28	)	)	PUNCT
ejpam-5532	121	29	where	where	SCONJ
ejpam-5532	121	30	a	a	DET
ejpam-5532	121	31	≤	≤	NOUN
ejpam-5532	121	32	x	x	SYM
ejpam-5532	121	33	≤	≤	NUM
ejpam-5532	121	34	b	b	NOUN
ejpam-5532	121	35	in	in	ADP
ejpam-5532	121	36	m.	m.	NOUN
ejpam-5532	121	37	the	the	DET
ejpam-5532	121	38	following	follow	VERB
ejpam-5532	121	39	characterisations	characterisation	NOUN
ejpam-5532	121	40	of	of	ADP
ejpam-5532	121	41	an	an	DET
ejpam-5532	121	42	l	l	NOUN
ejpam-5532	121	43	-	-	ADJ
ejpam-5532	121	44	convex	convex	ADJ
ejpam-5532	121	45	sublattice	sublattice	PROPN
ejpam-5532	121	46	η	η	PROPN
ejpam-5532	121	47	of	of	ADP
ejpam-5532	121	48	µ	µ	PROPN
ejpam-5532	121	49	with	with	ADP
ejpam-5532	121	50	the	the	DET
ejpam-5532	121	51	help	help	NOUN
ejpam-5532	121	52	of	of	ADP
ejpam-5532	121	53	level	level	NOUN
ejpam-5532	121	54	subsets	subset	NOUN
ejpam-5532	121	55	and	and	CCONJ
ejpam-5532	121	56	strong	strong	ADJ
ejpam-5532	121	57	level	level	NOUN
ejpam-5532	121	58	subsets	subset	NOUN
ejpam-5532	121	59	of	of	ADP
ejpam-5532	121	60	η	η	PROPN
ejpam-5532	121	61	can	can	AUX
ejpam-5532	121	62	be	be	AUX
ejpam-5532	121	63	verified	verify	VERB
ejpam-5532	121	64	easily	easily	ADV
ejpam-5532	121	65	.	.	PUNCT
ejpam-5532	122	1	theorem	theorem	ADJ
ejpam-5532	122	2	6	6	NUM
ejpam-5532	122	3	.	.	PUNCT
ejpam-5532	123	1	let	let	AUX
ejpam-5532	123	2	l(µ,m	l(µ,m	ADV
ejpam-5532	123	3	)	)	PUNCT
ejpam-5532	123	4	be	be	AUX
ejpam-5532	123	5	an	an	DET
ejpam-5532	123	6	l	l	NOUN
ejpam-5532	123	7	-	-	NOUN
ejpam-5532	123	8	lattice	lattice	NOUN
ejpam-5532	123	9	and	and	CCONJ
ejpam-5532	123	10	η	η	PROPN
ejpam-5532	123	11	∈	∈	PROPN
ejpam-5532	123	12	l(µ	l(µ	PROPN
ejpam-5532	123	13	)	)	PUNCT
ejpam-5532	123	14	with	with	ADP
ejpam-5532	123	15	ao	ao	PROPN
ejpam-5532	123	16	=	=	SYM
ejpam-5532	123	17	tip{η	tip{η	NOUN
ejpam-5532	123	18	}	}	PUNCT
ejpam-5532	123	19	.	.	PUNCT
ejpam-5532	124	1	then	then	ADV
ejpam-5532	124	2	,	,	PUNCT
ejpam-5532	124	3	η	η	PROPN
ejpam-5532	124	4	is	be	AUX
ejpam-5532	124	5	an	an	DET
ejpam-5532	124	6	l	l	ADJ
ejpam-5532	124	7	-	-	ADJ
ejpam-5532	124	8	convex	convex	ADJ
ejpam-5532	124	9	sublattice	sublattice	NOUN
ejpam-5532	124	10	of	of	ADP
ejpam-5532	124	11	µ	µ	NOUN
ejpam-5532	124	12	if	if	NOUN
ejpam-5532	125	1	and	and	CCONJ
ejpam-5532	125	2	only	only	ADV
ejpam-5532	125	3	if	if	SCONJ
ejpam-5532	125	4	each	each	DET
ejpam-5532	125	5	level	level	NOUN
ejpam-5532	125	6	subset	subset	VERB
ejpam-5532	125	7	ηt	ηt	ADP
ejpam-5532	125	8	,	,	PUNCT
ejpam-5532	125	9	∀	∀	X
ejpam-5532	125	10	t	t	NOUN
ejpam-5532	125	11	≤	≤	NUM
ejpam-5532	125	12	ao	ao	PROPN
ejpam-5532	125	13	,	,	PUNCT
ejpam-5532	125	14	is	be	AUX
ejpam-5532	125	15	a	a	DET
ejpam-5532	125	16	convex	convex	ADJ
ejpam-5532	125	17	sublattice	sublattice	NOUN
ejpam-5532	125	18	of	of	ADP
ejpam-5532	125	19	µt	µt	PROPN
ejpam-5532	125	20	.	.	PUNCT
ejpam-5532	125	21	equivalently	equivalently	PROPN
ejpam-5532	125	22	,	,	PUNCT
ejpam-5532	125	23	η	η	PROPN
ejpam-5532	125	24	is	be	AUX
ejpam-5532	125	25	an	an	DET
ejpam-5532	125	26	l	l	ADJ
ejpam-5532	125	27	-	-	ADJ
ejpam-5532	125	28	convex	convex	ADJ
ejpam-5532	125	29	sublattice	sublattice	NOUN
ejpam-5532	125	30	of	of	ADP
ejpam-5532	125	31	µ	µ	NOUN
ejpam-5532	125	32	if	if	NOUN
ejpam-5532	126	1	and	and	CCONJ
ejpam-5532	126	2	only	only	ADV
ejpam-5532	126	3	if	if	SCONJ
ejpam-5532	126	4	each	each	DET
ejpam-5532	126	5	nonempty	nonempty	ADJ
ejpam-5532	126	6	level	level	NOUN
ejpam-5532	126	7	subset	subset	VERB
ejpam-5532	126	8	ηt	ηt	ADP
ejpam-5532	126	9	is	be	AUX
ejpam-5532	126	10	a	a	DET
ejpam-5532	126	11	convex	convex	ADJ
ejpam-5532	126	12	sublattice	sublattice	NOUN
ejpam-5532	126	13	of	of	ADP
ejpam-5532	126	14	µt	µt	PROPN
ejpam-5532	126	15	.	.	PROPN
ejpam-5532	126	16	theorem	theorem	PROPN
ejpam-5532	126	17	7	7	NUM
ejpam-5532	126	18	.	.	PUNCT
ejpam-5532	127	1	let	let	VERB
ejpam-5532	127	2	l	l	NOUN
ejpam-5532	127	3	be	be	AUX
ejpam-5532	127	4	a	a	DET
ejpam-5532	127	5	chain	chain	NOUN
ejpam-5532	127	6	.	.	PUNCT
ejpam-5532	128	1	let	let	AUX
ejpam-5532	128	2	l(µ,m	l(µ,m	ADV
ejpam-5532	128	3	)	)	PUNCT
ejpam-5532	128	4	be	be	AUX
ejpam-5532	128	5	an	an	DET
ejpam-5532	128	6	l	l	NOUN
ejpam-5532	128	7	-	-	NOUN
ejpam-5532	128	8	lattice	lattice	NOUN
ejpam-5532	128	9	and	and	CCONJ
ejpam-5532	128	10	η	η	PROPN
ejpam-5532	128	11	∈	∈	PROPN
ejpam-5532	128	12	l(µ	l(µ	PROPN
ejpam-5532	128	13	)	)	PUNCT
ejpam-5532	128	14	with	with	ADP
ejpam-5532	128	15	ao	ao	PROPN
ejpam-5532	128	16	=	=	SYM
ejpam-5532	128	17	tip{η	tip{η	NOUN
ejpam-5532	128	18	}	}	PUNCT
ejpam-5532	128	19	.	.	PUNCT
ejpam-5532	129	1	then	then	ADV
ejpam-5532	129	2	,	,	PUNCT
ejpam-5532	129	3	η	η	PROPN
ejpam-5532	129	4	is	be	AUX
ejpam-5532	129	5	an	an	DET
ejpam-5532	129	6	l	l	ADJ
ejpam-5532	129	7	-	-	ADJ
ejpam-5532	129	8	convex	convex	ADJ
ejpam-5532	129	9	sublattice	sublattice	NOUN
ejpam-5532	129	10	of	of	ADP
ejpam-5532	129	11	µ	µ	NOUN
ejpam-5532	129	12	if	if	NOUN
ejpam-5532	130	1	and	and	CCONJ
ejpam-5532	130	2	only	only	ADV
ejpam-5532	130	3	if	if	SCONJ
ejpam-5532	130	4	each	each	DET
ejpam-5532	130	5	strong	strong	ADJ
ejpam-5532	130	6	level	level	NOUN
ejpam-5532	130	7	subset	subset	VERB
ejpam-5532	130	8	η	η	PROPN
ejpam-5532	130	9	>	>	PROPN
ejpam-5532	130	10	t	t	PROPN
ejpam-5532	130	11	is	be	AUX
ejpam-5532	130	12	a	a	DET
ejpam-5532	130	13	convex	convex	ADJ
ejpam-5532	130	14	sublattice	sublattice	NOUN
ejpam-5532	130	15	of	of	ADP
ejpam-5532	130	16	µ	µ	NOUN
ejpam-5532	130	17	>	>	X
ejpam-5532	130	18	t	t	PROPN
ejpam-5532	130	19	,	,	PUNCT
ejpam-5532	130	20	∀t	∀t	PROPN
ejpam-5532	130	21	<	<	X
ejpam-5532	130	22	ao	ao	PROPN
ejpam-5532	130	23	.	.	PUNCT
ejpam-5532	131	1	equivalently	equivalently	PROPN
ejpam-5532	131	2	,	,	PUNCT
ejpam-5532	131	3	η	η	PROPN
ejpam-5532	131	4	is	be	AUX
ejpam-5532	131	5	an	an	DET
ejpam-5532	131	6	l	l	ADJ
ejpam-5532	131	7	-	-	ADJ
ejpam-5532	131	8	convex	convex	ADJ
ejpam-5532	131	9	sublattice	sublattice	NOUN
ejpam-5532	131	10	of	of	ADP
ejpam-5532	131	11	µ	µ	NOUN
ejpam-5532	131	12	if	if	NOUN
ejpam-5532	131	13	and	and	CCONJ
ejpam-5532	131	14	only	only	ADV
ejpam-5532	131	15	if	if	SCONJ
ejpam-5532	131	16	each	each	DET
ejpam-5532	131	17	nonempty	nonempty	ADV
ejpam-5532	131	18	strong	strong	ADJ
ejpam-5532	131	19	level	level	NOUN
ejpam-5532	131	20	subset	subset	VERB
ejpam-5532	131	21	η	η	PROPN
ejpam-5532	131	22	>	>	PROPN
ejpam-5532	131	23	t	t	PROPN
ejpam-5532	131	24	is	be	AUX
ejpam-5532	131	25	a	a	DET
ejpam-5532	131	26	convex	convex	ADJ
ejpam-5532	131	27	sublattice	sublattice	NOUN
ejpam-5532	131	28	of	of	ADP
ejpam-5532	131	29	µ	µ	NOUN
ejpam-5532	131	30	>	>	X
ejpam-5532	131	31	t	t	PROPN
ejpam-5532	131	32	.	.	PUNCT
ejpam-5532	132	1	we	we	PRON
ejpam-5532	132	2	now	now	ADV
ejpam-5532	132	3	provide	provide	VERB
ejpam-5532	132	4	the	the	DET
ejpam-5532	132	5	following	follow	VERB
ejpam-5532	132	6	examples	example	NOUN
ejpam-5532	132	7	of	of	ADP
ejpam-5532	132	8	l	l	ADJ
ejpam-5532	132	9	-	-	ADJ
ejpam-5532	132	10	convex	convex	ADJ
ejpam-5532	132	11	sublattices	sublattice	NOUN
ejpam-5532	132	12	in	in	ADP
ejpam-5532	132	13	an	an	DET
ejpam-5532	132	14	l	l	NOUN
ejpam-5532	132	15	-	-	NOUN
ejpam-5532	132	16	lattice	lattice	NOUN
ejpam-5532	132	17	:	:	PUNCT
ejpam-5532	132	18	example	example	NOUN
ejpam-5532	133	1	1	1	X
ejpam-5532	133	2	.	.	PUNCT
ejpam-5532	134	1	let	let	VERB
ejpam-5532	134	2	m	m	NOUN
ejpam-5532	134	3	=	=	VERB
ejpam-5532	134	4	ℵ	ℵ	X
ejpam-5532	134	5	be	be	VERB
ejpam-5532	134	6	the	the	DET
ejpam-5532	134	7	chain	chain	NOUN
ejpam-5532	134	8	of	of	ADP
ejpam-5532	134	9	natural	natural	ADJ
ejpam-5532	134	10	numbers	number	NOUN
ejpam-5532	134	11	and	and	CCONJ
ejpam-5532	134	12	l	l	NOUN
ejpam-5532	134	13	=	=	SYM
ejpam-5532	134	14	p	p	X
ejpam-5532	134	15	(	(	PUNCT
ejpam-5532	134	16	ℵ	ℵ	NOUN
ejpam-5532	134	17	)	)	PUNCT
ejpam-5532	134	18	,	,	PUNCT
ejpam-5532	134	19	the	the	DET
ejpam-5532	134	20	power	power	NOUN
ejpam-5532	134	21	set	set	NOUN
ejpam-5532	134	22	of	of	ADP
ejpam-5532	134	23	ℵ	ℵ	NOUN
ejpam-5532	134	24	,	,	PUNCT
ejpam-5532	134	25	be	be	VERB
ejpam-5532	134	26	the	the	DET
ejpam-5532	134	27	boolean	boolean	ADJ
ejpam-5532	134	28	algebra	algebra	NOUN
ejpam-5532	134	29	.	.	PUNCT
ejpam-5532	135	1	define	define	VERB
ejpam-5532	135	2	the	the	DET
ejpam-5532	135	3	following	follow	VERB
ejpam-5532	135	4	l	l	NOUN
ejpam-5532	135	5	-	-	NOUN
ejpam-5532	135	6	subsets	subset	NOUN
ejpam-5532	135	7	of	of	ADP
ejpam-5532	135	8	ℵ	ℵ	NOUN
ejpam-5532	135	9	:	:	PUNCT
ejpam-5532	135	10	η(n	η(n	NUM
ejpam-5532	135	11	)	)	PUNCT
ejpam-5532	135	12	=	=	PRON
ejpam-5532	135	13	{	{	PUNCT
ejpam-5532	135	14	∅	∅	NOUN
ejpam-5532	135	15	if	if	SCONJ
ejpam-5532	135	16	n	n	X
ejpam-5532	135	17	=	=	SYM
ejpam-5532	135	18	1	1	NUM
ejpam-5532	135	19	,	,	PUNCT
ejpam-5532	135	20	{	{	PUNCT
ejpam-5532	135	21	1	1	NUM
ejpam-5532	135	22	,	,	PUNCT
ejpam-5532	135	23	2	2	NUM
ejpam-5532	135	24	,	,	PUNCT
ejpam-5532	135	25	...	...	PUNCT
ejpam-5532	135	26	,	,	PUNCT
ejpam-5532	135	27	n−	n−	NOUN
ejpam-5532	135	28	1	1	NUM
ejpam-5532	135	29	}	}	PUNCT
ejpam-5532	135	30	∀n	∀n	NUM
ejpam-5532	135	31	≥	≥	NOUN
ejpam-5532	135	32	2	2	NUM
ejpam-5532	135	33	;	;	PUNCT
ejpam-5532	135	34	a.	a.	NOUN
ejpam-5532	135	35	jain	jain	PROPN
ejpam-5532	135	36	,	,	PUNCT
ejpam-5532	135	37	i.	i.	PROPN
ejpam-5532	135	38	jahan	jahan	PROPN
ejpam-5532	135	39	/	/	SYM
ejpam-5532	135	40	eur	eur	PROPN
ejpam-5532	135	41	.	.	PUNCT
ejpam-5532	136	1	j.	j.	PROPN
ejpam-5532	136	2	pure	pure	PROPN
ejpam-5532	136	3	appl	appl	PROPN
ejpam-5532	136	4	.	.	PROPN
ejpam-5532	136	5	math	math	PROPN
ejpam-5532	136	6	,	,	PUNCT
ejpam-5532	136	7	18	18	NUM
ejpam-5532	136	8	(	(	PUNCT
ejpam-5532	136	9	1	1	NUM
ejpam-5532	136	10	)	)	PUNCT
ejpam-5532	136	11	(	(	PUNCT
ejpam-5532	136	12	2025	2025	NUM
ejpam-5532	136	13	)	)	PUNCT
ejpam-5532	136	14	,	,	PUNCT
ejpam-5532	136	15	5532	5532	NUM
ejpam-5532	136	16	7	7	NUM
ejpam-5532	136	17	of	of	ADP
ejpam-5532	136	18	20	20	NUM
ejpam-5532	136	19	and	and	CCONJ
ejpam-5532	136	20	µ(n	µ(n	ADJ
ejpam-5532	136	21	)	)	PUNCT
ejpam-5532	136	22	=	=	SYM
ejpam-5532	136	23	{	{	PUNCT
ejpam-5532	136	24	1	1	NUM
ejpam-5532	136	25	,	,	PUNCT
ejpam-5532	136	26	2	2	NUM
ejpam-5532	136	27	,	,	PUNCT
ejpam-5532	136	28	...	...	PUNCT
ejpam-5532	136	29	,	,	PUNCT
ejpam-5532	136	30	n	n	CCONJ
ejpam-5532	136	31	}	}	PUNCT
ejpam-5532	136	32	∀n	∀n	NUM
ejpam-5532	136	33	∈	∈	PROPN
ejpam-5532	137	1	ℵ.	ℵ.	PROPN
ejpam-5532	137	2	then	then	ADV
ejpam-5532	137	3	η	η	PROPN
ejpam-5532	137	4	⊆	⊆	NUM
ejpam-5532	137	5	µ.	µ.	NOUN
ejpam-5532	137	6	moreover	moreover	ADV
ejpam-5532	137	7	,	,	PUNCT
ejpam-5532	137	8	it	it	PRON
ejpam-5532	137	9	is	be	AUX
ejpam-5532	137	10	easy	easy	ADJ
ejpam-5532	137	11	to	to	PART
ejpam-5532	137	12	verify	verify	VERB
ejpam-5532	137	13	that	that	SCONJ
ejpam-5532	137	14	η	η	PROPN
ejpam-5532	137	15	and	and	CCONJ
ejpam-5532	137	16	µ	µ	PROPN
ejpam-5532	137	17	are	be	AUX
ejpam-5532	137	18	l	l	NOUN
ejpam-5532	137	19	-	-	NOUN
ejpam-5532	137	20	sublattices	sublattice	NOUN
ejpam-5532	137	21	of	of	ADP
ejpam-5532	137	22	ℵ.	ℵ.	PROPN
ejpam-5532	137	23	futhermore	futhermore	PROPN
ejpam-5532	137	24	,	,	PUNCT
ejpam-5532	137	25	η	η	PROPN
ejpam-5532	137	26	turns	turn	VERB
ejpam-5532	137	27	out	out	ADP
ejpam-5532	137	28	to	to	PART
ejpam-5532	137	29	be	be	AUX
ejpam-5532	137	30	an	an	DET
ejpam-5532	137	31	l	l	NOUN
ejpam-5532	137	32	-	-	ADJ
ejpam-5532	137	33	convex	convex	ADJ
ejpam-5532	137	34	sublattice	sublattice	NOUN
ejpam-5532	137	35	of	of	ADP
ejpam-5532	137	36	µ.	µ.	NOUN
ejpam-5532	137	37	it	it	PRON
ejpam-5532	137	38	is	be	AUX
ejpam-5532	137	39	wothwhile	wothwhile	ADJ
ejpam-5532	137	40	to	to	PART
ejpam-5532	137	41	note	note	VERB
ejpam-5532	137	42	here	here	ADV
ejpam-5532	137	43	that	that	SCONJ
ejpam-5532	137	44	the	the	DET
ejpam-5532	137	45	set	set	NOUN
ejpam-5532	137	46	of	of	ADP
ejpam-5532	137	47	all	all	DET
ejpam-5532	137	48	level	level	NOUN
ejpam-5532	137	49	subsets	subset	NOUN
ejpam-5532	137	50	of	of	ADP
ejpam-5532	137	51	η	η	PROPN
ejpam-5532	137	52	form	form	VERB
ejpam-5532	137	53	a	a	DET
ejpam-5532	137	54	chain	chain	NOUN
ejpam-5532	137	55	in	in	ADP
ejpam-5532	137	56	the	the	DET
ejpam-5532	137	57	lattice	lattice	NOUN
ejpam-5532	137	58	m	m	VERB
ejpam-5532	137	59	=	=	NOUN
ejpam-5532	137	60	ℵ.	ℵ.	PROPN
ejpam-5532	138	1	here	here	ADV
ejpam-5532	138	2	,	,	PUNCT
ejpam-5532	138	3	we	we	PRON
ejpam-5532	138	4	write	write	VERB
ejpam-5532	138	5	ai	ai	INTJ
ejpam-5532	138	6	=	=	PUNCT
ejpam-5532	138	7	{	{	PUNCT
ejpam-5532	138	8	1	1	NUM
ejpam-5532	138	9	,	,	PUNCT
ejpam-5532	138	10	2	2	NUM
ejpam-5532	138	11	,	,	PUNCT
ejpam-5532	138	12	·	·	PUNCT
ejpam-5532	138	13	·	·	PUNCT
ejpam-5532	138	14	·	·	PUNCT
ejpam-5532	138	15	,	,	PUNCT
ejpam-5532	138	16	i−	i−	PROPN
ejpam-5532	138	17	1	1	NUM
ejpam-5532	138	18	}	}	PUNCT
ejpam-5532	138	19	∀i	∀i	NOUN
ejpam-5532	138	20	≥	≥	NOUN
ejpam-5532	138	21	2	2	NUM
ejpam-5532	138	22	.	.	PUNCT
ejpam-5532	138	23	further	far	ADV
ejpam-5532	138	24	,	,	PUNCT
ejpam-5532	138	25	note	note	VERB
ejpam-5532	138	26	that	that	SCONJ
ejpam-5532	138	27	ηai	ηai	NOUN
ejpam-5532	138	28	=	=	SYM
ejpam-5532	138	29	ai	ai	VERB
ejpam-5532	138	30	and	and	CCONJ
ejpam-5532	138	31	η∅	η∅	VERB
ejpam-5532	138	32	=	=	PUNCT
ejpam-5532	138	33	∅.	∅.	ADP
ejpam-5532	138	34	thus	thus	ADV
ejpam-5532	138	35	,	,	PUNCT
ejpam-5532	138	36	we	we	PRON
ejpam-5532	138	37	have	have	VERB
ejpam-5532	138	38	a1	a1	NOUN
ejpam-5532	138	39	=	=	PUNCT
ejpam-5532	139	1	∅	∅	NOUN
ejpam-5532	139	2	⊆	⊆	NUM
ejpam-5532	139	3	a2	a2	PROPN
ejpam-5532	139	4	⊆	⊆	NUM
ejpam-5532	139	5	a3	a3	NOUN
ejpam-5532	139	6	⊆	⊆	NUM
ejpam-5532	139	7	·	·	PUNCT
ejpam-5532	139	8	·	·	PUNCT
ejpam-5532	139	9	·	·	PUNCT
ejpam-5532	140	1	⊆	⊆	NUM
ejpam-5532	140	2	an	an	DET
ejpam-5532	140	3	⊆	⊆	NUM
ejpam-5532	140	4	·	·	PUNCT
ejpam-5532	140	5	·	·	PUNCT
ejpam-5532	140	6	·	·	PUNCT
ejpam-5532	140	7	⊆	⊆	NUM
ejpam-5532	140	8	ℵ.	ℵ.	NOUN
ejpam-5532	140	9	example	example	NOUN
ejpam-5532	140	10	2	2	X
ejpam-5532	140	11	.	.	PUNCT
ejpam-5532	141	1	let	let	VERB
ejpam-5532	141	2	x	x	PUNCT
ejpam-5532	141	3	=	=	PRON
ejpam-5532	141	4	{	{	PUNCT
ejpam-5532	141	5	a	a	PRON
ejpam-5532	141	6	,	,	PUNCT
ejpam-5532	141	7	b	b	NOUN
ejpam-5532	141	8	,	,	PUNCT
ejpam-5532	141	9	c	c	NOUN
ejpam-5532	141	10	}	}	PUNCT
ejpam-5532	141	11	and	and	CCONJ
ejpam-5532	141	12	l	l	NOUN
ejpam-5532	141	13	=	=	SYM
ejpam-5532	141	14	p	p	X
ejpam-5532	141	15	(	(	PUNCT
ejpam-5532	141	16	x	x	NOUN
ejpam-5532	141	17	)	)	PUNCT
ejpam-5532	141	18	be	be	VERB
ejpam-5532	141	19	the	the	DET
ejpam-5532	141	20	power	power	NOUN
ejpam-5532	141	21	set	set	NOUN
ejpam-5532	141	22	of	of	ADP
ejpam-5532	141	23	x.	x.	NOUN
ejpam-5532	141	24	then	then	ADV
ejpam-5532	141	25	⟨l	⟨l	NOUN
ejpam-5532	141	26	,	,	PUNCT
ejpam-5532	141	27	∩	∩	NOUN
ejpam-5532	141	28	,	,	PUNCT
ejpam-5532	141	29	∪	∪	NOUN
ejpam-5532	141	30	,	,	PUNCT
ejpam-5532	141	31	′⟩	′⟩	NOUN
ejpam-5532	141	32	is	be	AUX
ejpam-5532	141	33	a	a	DET
ejpam-5532	141	34	boolean	boolean	ADJ
ejpam-5532	141	35	algebra	algebra	NOUN
ejpam-5532	141	36	where	where	SCONJ
ejpam-5532	141	37	′∪′	′∪′	ADV
ejpam-5532	141	38	,	,	PUNCT
ejpam-5532	141	39	′∩′	′∩′	PROPN
ejpam-5532	141	40	and	and	CCONJ
ejpam-5532	141	41	′′′	′′′	PROPN
ejpam-5532	141	42	denote	denote	VERB
ejpam-5532	141	43	the	the	DET
ejpam-5532	141	44	ordinary	ordinary	ADJ
ejpam-5532	141	45	intersection	intersection	NOUN
ejpam-5532	141	46	,	,	PUNCT
ejpam-5532	141	47	union	union	NOUN
ejpam-5532	141	48	and	and	CCONJ
ejpam-5532	141	49	complement	complement	NOUN
ejpam-5532	141	50	of	of	ADP
ejpam-5532	141	51	members	member	NOUN
ejpam-5532	141	52	of	of	ADP
ejpam-5532	141	53	l	l	NOUN
ejpam-5532	141	54	respectively	respectively	ADV
ejpam-5532	141	55	.	.	PUNCT
ejpam-5532	142	1	further	far	ADV
ejpam-5532	142	2	,	,	PUNCT
ejpam-5532	142	3	it	it	PRON
ejpam-5532	142	4	is	be	AUX
ejpam-5532	142	5	easy	easy	ADJ
ejpam-5532	142	6	to	to	PART
ejpam-5532	142	7	see	see	VERB
ejpam-5532	142	8	that	that	SCONJ
ejpam-5532	142	9	l	l	NOUN
ejpam-5532	142	10	is	be	AUX
ejpam-5532	142	11	boolean	boolean	ADJ
ejpam-5532	142	12	algebra	algebra	NOUN
ejpam-5532	142	13	with	with	ADP
ejpam-5532	142	14	order	order	NOUN
ejpam-5532	142	15	reversing	reverse	VERB
ejpam-5532	142	16	involution	involution	NOUN
ejpam-5532	142	17	given	give	VERB
ejpam-5532	142	18	by	by	ADP
ejpam-5532	142	19	:	:	PUNCT
ejpam-5532	142	20	τ	τ	X
ejpam-5532	142	21	:	:	PUNCT
ejpam-5532	142	22	l	l	NOUN
ejpam-5532	142	23	−→	−→	ADJ
ejpam-5532	142	24	l∗	l∗	NOUN
ejpam-5532	142	25	,	,	PUNCT
ejpam-5532	142	26	τ(a	τ(a	NOUN
ejpam-5532	142	27	)	)	PUNCT
ejpam-5532	142	28	=	=	PUNCT
ejpam-5532	142	29	a′.	a′.	NOUN
ejpam-5532	142	30	let	let	VERB
ejpam-5532	142	31	m	m	VERB
ejpam-5532	142	32	=	=	PUNCT
ejpam-5532	142	33	{	{	PUNCT
ejpam-5532	142	34	1	1	NUM
ejpam-5532	142	35	,	,	PUNCT
ejpam-5532	142	36	2	2	NUM
ejpam-5532	142	37	,	,	PUNCT
ejpam-5532	142	38	3	3	NUM
ejpam-5532	142	39	,	,	PUNCT
ejpam-5532	142	40	6	6	NUM
ejpam-5532	142	41	}	}	PUNCT
ejpam-5532	142	42	denote	denote	VERB
ejpam-5532	142	43	the	the	DET
ejpam-5532	142	44	set	set	NOUN
ejpam-5532	142	45	of	of	ADP
ejpam-5532	142	46	all	all	DET
ejpam-5532	142	47	factors	factor	NOUN
ejpam-5532	142	48	of	of	ADP
ejpam-5532	142	49	′6′.	′6′.	NOUN
ejpam-5532	142	50	then	then	ADV
ejpam-5532	142	51	⟨m,∨,∧	⟨m,∨,∧	PRON
ejpam-5532	142	52	,	,	PUNCT
ejpam-5532	142	53	′⟩	′⟩	NOUN
ejpam-5532	142	54	,	,	PUNCT
ejpam-5532	142	55	where	where	SCONJ
ejpam-5532	142	56	a	a	DET
ejpam-5532	142	57	∨	∨	NUM
ejpam-5532	142	58	b	b	NOUN
ejpam-5532	142	59	=	=	PUNCT
ejpam-5532	142	60	lcm{a	lcm{a	NOUN
ejpam-5532	142	61	,	,	PUNCT
ejpam-5532	142	62	b	b	NOUN
ejpam-5532	142	63	}	}	PUNCT
ejpam-5532	142	64	,	,	PUNCT
ejpam-5532	142	65	a	a	DET
ejpam-5532	142	66	∧	∧	PROPN
ejpam-5532	142	67	b	b	PROPN
ejpam-5532	142	68	=	=	SYM
ejpam-5532	142	69	gcd{a	gcd{a	PROPN
ejpam-5532	142	70	,	,	PUNCT
ejpam-5532	142	71	b	b	NOUN
ejpam-5532	142	72	}	}	PUNCT
ejpam-5532	142	73	and	and	CCONJ
ejpam-5532	142	74	a′	a′	PROPN
ejpam-5532	142	75	=	=	SYM
ejpam-5532	142	76	6	6	NUM
ejpam-5532	142	77	a	a	PRON
ejpam-5532	142	78	;	;	PUNCT
ejpam-5532	142	79	∀	∀	X
ejpam-5532	142	80	a	a	NOUN
ejpam-5532	142	81	,	,	PUNCT
ejpam-5532	142	82	b	b	X
ejpam-5532	142	83	∈	∈	ADV
ejpam-5532	142	84	m	m	VERB
ejpam-5532	142	85	,	,	PUNCT
ejpam-5532	142	86	is	be	AUX
ejpam-5532	142	87	also	also	ADV
ejpam-5532	142	88	a	a	DET
ejpam-5532	142	89	boolean	boolean	ADJ
ejpam-5532	142	90	algebra	algebra	NOUN
ejpam-5532	142	91	.	.	PUNCT
ejpam-5532	143	1	in	in	ADP
ejpam-5532	143	2	the	the	DET
ejpam-5532	143	3	following	following	ADJ
ejpam-5532	143	4	diagram	diagram	NOUN
ejpam-5532	143	5	,	,	PUNCT
ejpam-5532	143	6	(	(	PUNCT
ejpam-5532	143	7	i	i	NOUN
ejpam-5532	143	8	)	)	PUNCT
ejpam-5532	143	9	and	and	CCONJ
ejpam-5532	143	10	(	(	PUNCT
ejpam-5532	143	11	ii	ii	NOUN
ejpam-5532	143	12	)	)	PUNCT
ejpam-5532	143	13	represent	represent	VERB
ejpam-5532	143	14	boolean	boolean	ADJ
ejpam-5532	143	15	algebras	algebra	NOUN
ejpam-5532	143	16	m	m	VERB
ejpam-5532	143	17	and	and	CCONJ
ejpam-5532	143	18	l	l	NOUN
ejpam-5532	143	19	respectively	respectively	ADV
ejpam-5532	143	20	.	.	PUNCT
ejpam-5532	144	1	(	(	PUNCT
ejpam-5532	144	2	i	i	NOUN
ejpam-5532	144	3	)	)	PUNCT
ejpam-5532	144	4	(	(	PUNCT
ejpam-5532	144	5	ii	ii	NOUN
ejpam-5532	144	6	)	)	PUNCT
ejpam-5532	144	7	define	define	VERB
ejpam-5532	144	8	the	the	DET
ejpam-5532	144	9	following	follow	VERB
ejpam-5532	144	10	l	l	NOUN
ejpam-5532	144	11	-	-	PUNCT
ejpam-5532	144	12	subsets	subsets	PROPN
ejpam-5532	144	13	µ	µ	NOUN
ejpam-5532	144	14	and	and	CCONJ
ejpam-5532	144	15	η	η	PROPN
ejpam-5532	144	16	of	of	ADP
ejpam-5532	144	17	m	m	PROPN
ejpam-5532	144	18	:	:	PUNCT
ejpam-5532	144	19	µ(a	µ(a	PROPN
ejpam-5532	144	20	)	)	PUNCT
ejpam-5532	145	1	=	=	PRON
ejpam-5532	145	2	{	{	PUNCT
ejpam-5532	145	3	2	2	NUM
ejpam-5532	145	4	if	if	SCONJ
ejpam-5532	145	5	a	a	DET
ejpam-5532	145	6	∈	∈	PROPN
ejpam-5532	145	7	{	{	PUNCT
ejpam-5532	145	8	∅	∅	NOUN
ejpam-5532	145	9	,	,	PUNCT
ejpam-5532	145	10	{	{	PUNCT
ejpam-5532	145	11	a	a	X
ejpam-5532	145	12	}	}	PUNCT
ejpam-5532	145	13	,	,	PUNCT
ejpam-5532	145	14	{	{	PUNCT
ejpam-5532	145	15	b	b	NOUN
ejpam-5532	145	16	}	}	PUNCT
ejpam-5532	145	17	,	,	PUNCT
ejpam-5532	145	18	{	{	PUNCT
ejpam-5532	145	19	a	a	DET
ejpam-5532	145	20	,	,	PUNCT
ejpam-5532	145	21	b	b	NOUN
ejpam-5532	145	22	}	}	PUNCT
ejpam-5532	145	23	}	}	PUNCT
ejpam-5532	145	24	,	,	PUNCT
ejpam-5532	145	25	6	6	NUM
ejpam-5532	145	26	if	if	SCONJ
ejpam-5532	145	27	a	a	PRON
ejpam-5532	145	28	=	=	X
ejpam-5532	145	29	p	p	X
ejpam-5532	145	30	(	(	PUNCT
ejpam-5532	145	31	x	x	NOUN
ejpam-5532	145	32	)	)	PUNCT
ejpam-5532	145	33	\	\	NOUN
ejpam-5532	145	34	{	{	PUNCT
ejpam-5532	145	35	∅	∅	NOUN
ejpam-5532	145	36	,	,	PUNCT
ejpam-5532	145	37	{	{	PUNCT
ejpam-5532	145	38	a	a	X
ejpam-5532	145	39	}	}	PUNCT
ejpam-5532	145	40	,	,	PUNCT
ejpam-5532	145	41	{	{	PUNCT
ejpam-5532	145	42	b	b	NOUN
ejpam-5532	145	43	}	}	PUNCT
ejpam-5532	145	44	,	,	PUNCT
ejpam-5532	145	45	{	{	PUNCT
ejpam-5532	145	46	a	a	DET
ejpam-5532	145	47	,	,	PUNCT
ejpam-5532	145	48	b	b	NOUN
ejpam-5532	145	49	}	}	PUNCT
ejpam-5532	145	50	}	}	PUNCT
ejpam-5532	145	51	;	;	PUNCT
ejpam-5532	145	52	a.	a.	NOUN
ejpam-5532	145	53	jain	jain	PROPN
ejpam-5532	145	54	,	,	PUNCT
ejpam-5532	145	55	i.	i.	PROPN
ejpam-5532	145	56	jahan	jahan	PROPN
ejpam-5532	145	57	/	/	SYM
ejpam-5532	145	58	eur	eur	PROPN
ejpam-5532	145	59	.	.	PUNCT
ejpam-5532	146	1	j.	j.	PROPN
ejpam-5532	146	2	pure	pure	PROPN
ejpam-5532	146	3	appl	appl	PROPN
ejpam-5532	146	4	.	.	PROPN
ejpam-5532	146	5	math	math	PROPN
ejpam-5532	146	6	,	,	PUNCT
ejpam-5532	146	7	18	18	NUM
ejpam-5532	146	8	(	(	PUNCT
ejpam-5532	146	9	1	1	NUM
ejpam-5532	146	10	)	)	PUNCT
ejpam-5532	146	11	(	(	PUNCT
ejpam-5532	146	12	2025	2025	NUM
ejpam-5532	146	13	)	)	PUNCT
ejpam-5532	146	14	,	,	PUNCT
ejpam-5532	146	15	5532	5532	NUM
ejpam-5532	146	16	8	8	NUM
ejpam-5532	146	17	of	of	ADP
ejpam-5532	146	18	20	20	NUM
ejpam-5532	146	19	and	and	CCONJ
ejpam-5532	146	20	η(a	η(a	ADJ
ejpam-5532	146	21	)	)	PUNCT
ejpam-5532	146	22	=	=	PUNCT
ejpam-5532	147	1			NOUN
ejpam-5532	147	2	1	1	NUM
ejpam-5532	147	3	if	if	SCONJ
ejpam-5532	147	4	a	a	DET
ejpam-5532	147	5	∈	∈	PROPN
ejpam-5532	147	6	{	{	PUNCT
ejpam-5532	147	7	∅	∅	NOUN
ejpam-5532	147	8	,	,	PUNCT
ejpam-5532	147	9	{	{	PUNCT
ejpam-5532	147	10	b	b	NOUN
ejpam-5532	147	11	}	}	PUNCT
ejpam-5532	147	12	}	}	PUNCT
ejpam-5532	147	13	,	,	PUNCT
ejpam-5532	147	14	2	2	NUM
ejpam-5532	147	15	if	if	SCONJ
ejpam-5532	147	16	a	a	DET
ejpam-5532	147	17	∈	∈	PROPN
ejpam-5532	147	18	{	{	PUNCT
ejpam-5532	147	19	{	{	PUNCT
ejpam-5532	147	20	a	a	NOUN
ejpam-5532	147	21	}	}	PUNCT
ejpam-5532	147	22	,	,	PUNCT
ejpam-5532	147	23	{	{	PUNCT
ejpam-5532	147	24	a	a	PRON
ejpam-5532	147	25	,	,	PUNCT
ejpam-5532	147	26	b	b	NOUN
ejpam-5532	147	27	}	}	PUNCT
ejpam-5532	147	28	}	}	PUNCT
ejpam-5532	147	29	,	,	PUNCT
ejpam-5532	147	30	3	3	NUM
ejpam-5532	147	31	if	if	SCONJ
ejpam-5532	147	32	a	a	DET
ejpam-5532	147	33	∈	∈	PROPN
ejpam-5532	147	34	{	{	PUNCT
ejpam-5532	147	35	c	c	NOUN
ejpam-5532	147	36	,	,	PUNCT
ejpam-5532	147	37	{	{	PUNCT
ejpam-5532	147	38	{	{	PUNCT
ejpam-5532	147	39	b	b	NOUN
ejpam-5532	147	40	,	,	PUNCT
ejpam-5532	147	41	c	c	NOUN
ejpam-5532	147	42	}	}	PUNCT
ejpam-5532	147	43	}	}	PUNCT
ejpam-5532	147	44	,	,	PUNCT
ejpam-5532	147	45	6	6	NUM
ejpam-5532	147	46	if	if	SCONJ
ejpam-5532	147	47	a	a	DET
ejpam-5532	147	48	∈	∈	PROPN
ejpam-5532	147	49	{	{	PUNCT
ejpam-5532	147	50	{	{	PUNCT
ejpam-5532	147	51	a	a	X
ejpam-5532	147	52	,	,	PUNCT
ejpam-5532	147	53	c	c	NOUN
ejpam-5532	147	54	}	}	PUNCT
ejpam-5532	147	55	,	,	PUNCT
ejpam-5532	147	56	x	x	NOUN
ejpam-5532	147	57	}	}	PUNCT
ejpam-5532	147	58	.	.	PUNCT
ejpam-5532	148	1	now	now	ADV
ejpam-5532	148	2	,	,	PUNCT
ejpam-5532	148	3	note	note	VERB
ejpam-5532	148	4	that	that	SCONJ
ejpam-5532	148	5	η	η	PROPN
ejpam-5532	148	6	⊆	⊆	NUM
ejpam-5532	148	7	µ.	µ.	NOUN
ejpam-5532	148	8	the	the	DET
ejpam-5532	148	9	set	set	NOUN
ejpam-5532	148	10	{	{	PUNCT
ejpam-5532	148	11	ηa	ηa	INTJ
ejpam-5532	148	12	:	:	PUNCT
ejpam-5532	148	13	a	a	DET
ejpam-5532	148	14	∈	∈	PROPN
ejpam-5532	149	1	i	i	NOUN
ejpam-5532	149	2	m	m	NOUN
ejpam-5532	149	3	η	η	X
ejpam-5532	149	4	}	}	PUNCT
ejpam-5532	149	5	of	of	ADP
ejpam-5532	149	6	all	all	DET
ejpam-5532	149	7	level	level	NOUN
ejpam-5532	149	8	subset	subset	NOUN
ejpam-5532	149	9	of	of	ADP
ejpam-5532	149	10	η	η	PROPN
ejpam-5532	149	11	is	be	AUX
ejpam-5532	149	12	determined	determine	VERB
ejpam-5532	149	13	below	below	ADP
ejpam-5532	149	14	:	:	PUNCT
ejpam-5532	149	15	η1	η1	NOUN
ejpam-5532	149	16	=	=	SYM
ejpam-5532	149	17	m	m	PROPN
ejpam-5532	149	18	,	,	PUNCT
ejpam-5532	149	19	η2	η2	PROPN
ejpam-5532	149	20	=	=	PUNCT
ejpam-5532	149	21	{	{	PUNCT
ejpam-5532	149	22	{	{	PUNCT
ejpam-5532	149	23	a	a	NOUN
ejpam-5532	149	24	}	}	PUNCT
ejpam-5532	149	25	,	,	PUNCT
ejpam-5532	149	26	{	{	PUNCT
ejpam-5532	149	27	a	a	DET
ejpam-5532	149	28	,	,	PUNCT
ejpam-5532	149	29	b	b	NOUN
ejpam-5532	149	30	}	}	PUNCT
ejpam-5532	149	31	,	,	PUNCT
ejpam-5532	149	32	{	{	PUNCT
ejpam-5532	149	33	a	a	X
ejpam-5532	149	34	,	,	PUNCT
ejpam-5532	149	35	c	c	NOUN
ejpam-5532	149	36	}	}	PUNCT
ejpam-5532	149	37	,	,	PUNCT
ejpam-5532	149	38	x	x	X
ejpam-5532	149	39	}	}	PUNCT
ejpam-5532	149	40	,	,	PUNCT
ejpam-5532	149	41	η3	η3	PROPN
ejpam-5532	149	42	=	=	PRON
ejpam-5532	149	43	{	{	PUNCT
ejpam-5532	149	44	{	{	PUNCT
ejpam-5532	149	45	c	c	NOUN
ejpam-5532	149	46	}	}	PUNCT
ejpam-5532	149	47	,	,	PUNCT
ejpam-5532	149	48	{	{	PUNCT
ejpam-5532	149	49	b	b	X
ejpam-5532	149	50	,	,	PUNCT
ejpam-5532	149	51	c	c	NOUN
ejpam-5532	149	52	}	}	PUNCT
ejpam-5532	149	53	,	,	PUNCT
ejpam-5532	149	54	{	{	PUNCT
ejpam-5532	149	55	a	a	X
ejpam-5532	149	56	,	,	PUNCT
ejpam-5532	149	57	c	c	NOUN
ejpam-5532	149	58	}	}	PUNCT
ejpam-5532	149	59	,	,	PUNCT
ejpam-5532	149	60	x	x	NOUN
ejpam-5532	149	61	}	}	PUNCT
ejpam-5532	149	62	and	and	CCONJ
ejpam-5532	149	63	η6	η6	PROPN
ejpam-5532	149	64	=	=	SYM
ejpam-5532	149	65	{	{	PUNCT
ejpam-5532	149	66	{	{	PUNCT
ejpam-5532	149	67	a	a	X
ejpam-5532	149	68	,	,	PUNCT
ejpam-5532	149	69	c	c	NOUN
ejpam-5532	149	70	}	}	PUNCT
ejpam-5532	149	71	,	,	PUNCT
ejpam-5532	149	72	x	x	NOUN
ejpam-5532	149	73	}	}	PUNCT
ejpam-5532	149	74	.	.	PUNCT
ejpam-5532	150	1	further	far	ADV
ejpam-5532	150	2	,	,	PUNCT
ejpam-5532	150	3	the	the	DET
ejpam-5532	150	4	set	set	NOUN
ejpam-5532	150	5	{	{	PUNCT
ejpam-5532	150	6	µa	µa	NOUN
ejpam-5532	150	7	:	:	PUNCT
ejpam-5532	150	8	a	a	DET
ejpam-5532	150	9	∈	∈	PROPN
ejpam-5532	151	1	i	i	PRON
ejpam-5532	151	2	m	m	VERB
ejpam-5532	151	3	µ	µ	VERB
ejpam-5532	151	4	}	}	PUNCT
ejpam-5532	151	5	of	of	ADP
ejpam-5532	151	6	all	all	DET
ejpam-5532	151	7	level	level	NOUN
ejpam-5532	151	8	subset	subset	NOUN
ejpam-5532	151	9	of	of	ADP
ejpam-5532	151	10	µ	µ	NOUN
ejpam-5532	151	11	is	be	AUX
ejpam-5532	151	12	determined	determine	VERB
ejpam-5532	151	13	below	below	ADV
ejpam-5532	151	14	:	:	PUNCT
ejpam-5532	151	15	µ2	µ2	PROPN
ejpam-5532	151	16	=	=	PROPN
ejpam-5532	151	17	m	m	PROPN
ejpam-5532	151	18	,	,	PUNCT
ejpam-5532	151	19	and	and	CCONJ
ejpam-5532	151	20	µ6	µ6	PROPN
ejpam-5532	151	21	=	=	SYM
ejpam-5532	151	22	{	{	PUNCT
ejpam-5532	151	23	{	{	PUNCT
ejpam-5532	151	24	c	c	NOUN
ejpam-5532	151	25	}	}	PUNCT
ejpam-5532	151	26	,	,	PUNCT
ejpam-5532	151	27	{	{	PUNCT
ejpam-5532	151	28	b	b	NOUN
ejpam-5532	151	29	,	,	PUNCT
ejpam-5532	151	30	c}{a	c}{a	PROPN
ejpam-5532	151	31	,	,	PUNCT
ejpam-5532	151	32	c	c	NOUN
ejpam-5532	151	33	}	}	PUNCT
ejpam-5532	151	34	,	,	PUNCT
ejpam-5532	151	35	x	x	NOUN
ejpam-5532	151	36	}	}	PUNCT
ejpam-5532	151	37	.	.	PUNCT
ejpam-5532	152	1	now	now	ADV
ejpam-5532	152	2	,	,	PUNCT
ejpam-5532	152	3	it	it	PRON
ejpam-5532	152	4	is	be	AUX
ejpam-5532	152	5	easy	easy	ADJ
ejpam-5532	152	6	to	to	PART
ejpam-5532	152	7	see	see	VERB
ejpam-5532	152	8	that	that	SCONJ
ejpam-5532	152	9	η	η	PROPN
ejpam-5532	152	10	and	and	CCONJ
ejpam-5532	152	11	µ	µ	PROPN
ejpam-5532	152	12	are	be	AUX
ejpam-5532	152	13	l	l	NOUN
ejpam-5532	152	14	-	-	NOUN
ejpam-5532	152	15	sublattices	sublattice	NOUN
ejpam-5532	152	16	of	of	ADP
ejpam-5532	152	17	m	m	PROPN
ejpam-5532	152	18	.	.	PUNCT
ejpam-5532	153	1	futhermore	futhermore	NOUN
ejpam-5532	153	2	,	,	PUNCT
ejpam-5532	153	3	η	η	PROPN
ejpam-5532	153	4	forms	form	VERB
ejpam-5532	153	5	an	an	DET
ejpam-5532	153	6	lconvex	lconvex	NOUN
ejpam-5532	153	7	sublattice	sublattice	NOUN
ejpam-5532	153	8	of	of	ADP
ejpam-5532	153	9	µ.	µ.	NOUN
ejpam-5532	153	10	observe	observe	VERB
ejpam-5532	153	11	that	that	SCONJ
ejpam-5532	153	12	in	in	ADP
ejpam-5532	153	13	this	this	DET
ejpam-5532	153	14	example	example	NOUN
ejpam-5532	153	15	the	the	DET
ejpam-5532	153	16	set	set	NOUN
ejpam-5532	153	17	of	of	ADP
ejpam-5532	153	18	all	all	DET
ejpam-5532	153	19	level	level	NOUN
ejpam-5532	153	20	subsets	subset	NOUN
ejpam-5532	153	21	of	of	ADP
ejpam-5532	153	22	η	η	PROPN
ejpam-5532	153	23	does	do	AUX
ejpam-5532	153	24	not	not	PART
ejpam-5532	153	25	form	form	VERB
ejpam-5532	153	26	a	a	DET
ejpam-5532	153	27	chain	chain	NOUN
ejpam-5532	153	28	.	.	PUNCT
ejpam-5532	154	1	infact	infact	PROPN
ejpam-5532	154	2	,	,	PUNCT
ejpam-5532	154	3	the	the	DET
ejpam-5532	154	4	set	set	NOUN
ejpam-5532	154	5	of	of	ADP
ejpam-5532	154	6	all	all	DET
ejpam-5532	154	7	level	level	NOUN
ejpam-5532	154	8	subsets	subset	NOUN
ejpam-5532	154	9	{	{	PUNCT
ejpam-5532	154	10	ηa	ηa	INTJ
ejpam-5532	154	11	:	:	PUNCT
ejpam-5532	154	12	a	a	DET
ejpam-5532	154	13	∈	∈	PROPN
ejpam-5532	155	1	i	i	NOUN
ejpam-5532	155	2	m	m	PROPN
ejpam-5532	155	3	η	η	X
ejpam-5532	155	4	}	}	PUNCT
ejpam-5532	155	5	turns	turn	VERB
ejpam-5532	155	6	out	out	ADP
ejpam-5532	155	7	to	to	PART
ejpam-5532	155	8	be	be	AUX
ejpam-5532	155	9	only	only	ADV
ejpam-5532	155	10	a	a	DET
ejpam-5532	155	11	poset	poset	NOUN
ejpam-5532	155	12	under	under	ADP
ejpam-5532	155	13	the	the	DET
ejpam-5532	155	14	ordering	ordering	NOUN
ejpam-5532	155	15	of	of	ADP
ejpam-5532	155	16	usual	usual	ADJ
ejpam-5532	155	17	set	set	ADJ
ejpam-5532	155	18	theoretic	theoretic	ADJ
ejpam-5532	155	19	containment	containment	NOUN
ejpam-5532	155	20	.	.	PUNCT
ejpam-5532	156	1	example	example	NOUN
ejpam-5532	157	1	3	3	X
ejpam-5532	157	2	.	.	PUNCT
ejpam-5532	157	3	let	let	VERB
ejpam-5532	157	4	m	m	VERB
ejpam-5532	157	5	=	=	VERB
ejpam-5532	157	6	∅	∅	NOUN
ejpam-5532	157	7	∪	∪	ADP
ejpam-5532	157	8	z	z	NOUN
ejpam-5532	157	9	∪	∪	X
ejpam-5532	157	10	{	{	PUNCT
ejpam-5532	157	11	{	{	PUNCT
ejpam-5532	157	12	n	n	CCONJ
ejpam-5532	157	13	}	}	PUNCT
ejpam-5532	157	14	:	:	PUNCT
ejpam-5532	157	15	n	n	X
ejpam-5532	157	16	∈	∈	PROPN
ejpam-5532	157	17	z	z	NOUN
ejpam-5532	157	18	}	}	PUNCT
ejpam-5532	157	19	.	.	PUNCT
ejpam-5532	158	1	then	then	ADV
ejpam-5532	158	2	m	m	PROPN
ejpam-5532	158	3	is	be	AUX
ejpam-5532	158	4	a	a	DET
ejpam-5532	158	5	boolean	boolean	ADJ
ejpam-5532	158	6	algebra	algebra	NOUN
ejpam-5532	158	7	with	with	ADP
ejpam-5532	158	8	the	the	DET
ejpam-5532	158	9	following	follow	VERB
ejpam-5532	158	10	hasse	hasse	PROPN
ejpam-5532	158	11	diagram	diagram	NOUN
ejpam-5532	158	12	:	:	PUNCT
ejpam-5532	158	13	further	far	ADV
ejpam-5532	158	14	,	,	PUNCT
ejpam-5532	158	15	let	let	VERB
ejpam-5532	158	16	l	l	NOUN
ejpam-5532	158	17	=	=	PRON
ejpam-5532	158	18	{	{	PUNCT
ejpam-5532	158	19	a	a	DET
ejpam-5532	158	20	⊆	⊆	NUM
ejpam-5532	158	21	r	r	NOUN
ejpam-5532	158	22	:	:	PUNCT
ejpam-5532	158	23	either	either	CCONJ
ejpam-5532	158	24	a	a	PRON
ejpam-5532	158	25	or	or	CCONJ
ejpam-5532	158	26	a′	a′	NOUN
ejpam-5532	158	27	is	be	AUX
ejpam-5532	158	28	finite	finite	ADJ
ejpam-5532	158	29	}	}	PUNCT
ejpam-5532	158	30	.	.	PUNCT
ejpam-5532	159	1	here	here	ADV
ejpam-5532	159	2	a′	a′	PROPN
ejpam-5532	159	3	is	be	AUX
ejpam-5532	159	4	complement	complement	NOUN
ejpam-5532	159	5	of	of	ADP
ejpam-5532	159	6	a	a	PRON
ejpam-5532	159	7	in	in	ADP
ejpam-5532	159	8	r.	r.	NOUN
ejpam-5532	159	9	it	it	PRON
ejpam-5532	159	10	is	be	AUX
ejpam-5532	159	11	easy	easy	ADJ
ejpam-5532	159	12	to	to	PART
ejpam-5532	159	13	see	see	VERB
ejpam-5532	159	14	that	that	SCONJ
ejpam-5532	159	15	l	l	NOUN
ejpam-5532	159	16	is	be	AUX
ejpam-5532	159	17	boolean	boolean	ADJ
ejpam-5532	159	18	algebra	algebra	NOUN
ejpam-5532	159	19	with	with	ADP
ejpam-5532	159	20	order	order	NOUN
ejpam-5532	159	21	reversing	reverse	VERB
ejpam-5532	159	22	involution	involution	NOUN
ejpam-5532	159	23	given	give	VERB
ejpam-5532	159	24	by	by	ADP
ejpam-5532	159	25	:	:	PUNCT
ejpam-5532	159	26	τ	τ	X
ejpam-5532	159	27	:	:	PUNCT
ejpam-5532	159	28	l	l	NOUN
ejpam-5532	159	29	−→	−→	ADJ
ejpam-5532	159	30	l∗	l∗	NOUN
ejpam-5532	159	31	,	,	PUNCT
ejpam-5532	159	32	τ(a	τ(a	NOUN
ejpam-5532	159	33	)	)	PUNCT
ejpam-5532	159	34	=	=	PUNCT
ejpam-5532	159	35	a′.	a′.	NOUN
ejpam-5532	159	36	define	define	VERB
ejpam-5532	159	37	the	the	DET
ejpam-5532	159	38	following	follow	VERB
ejpam-5532	159	39	l	l	NOUN
ejpam-5532	159	40	-	-	NOUN
ejpam-5532	159	41	subsets	subset	NOUN
ejpam-5532	159	42	of	of	ADP
ejpam-5532	159	43	m	m	PRON
ejpam-5532	159	44	:	:	PUNCT
ejpam-5532	159	45	η(a	η(a	VERB
ejpam-5532	159	46	)	)	PUNCT
ejpam-5532	160	1	=	=	SYM
ejpam-5532	160	2			PRON
ejpam-5532	160	3	∅	∅	NOUN
ejpam-5532	160	4	if	if	SCONJ
ejpam-5532	160	5	a	a	DET
ejpam-5532	160	6	=	=	SYM
ejpam-5532	160	7	z	z	NOUN
ejpam-5532	160	8	,	,	PUNCT
ejpam-5532	160	9	r	r	NOUN
ejpam-5532	160	10	if	if	SCONJ
ejpam-5532	160	11	a	a	DET
ejpam-5532	160	12	=	=	NOUN
ejpam-5532	160	13	∅	∅	NOUN
ejpam-5532	160	14	,	,	PUNCT
ejpam-5532	160	15	{	{	PUNCT
ejpam-5532	160	16	n	n	CCONJ
ejpam-5532	160	17	}	}	PUNCT
ejpam-5532	160	18	if	if	SCONJ
ejpam-5532	160	19	n	n	PRON
ejpam-5532	160	20	∈	∈	PROPN
ejpam-5532	160	21	z	z	NOUN
ejpam-5532	160	22	;	;	PUNCT
ejpam-5532	160	23	a.	a.	NOUN
ejpam-5532	160	24	jain	jain	PROPN
ejpam-5532	160	25	,	,	PUNCT
ejpam-5532	160	26	i.	i.	PROPN
ejpam-5532	160	27	jahan	jahan	PROPN
ejpam-5532	160	28	/	/	SYM
ejpam-5532	160	29	eur	eur	PROPN
ejpam-5532	160	30	.	.	PUNCT
ejpam-5532	161	1	j.	j.	PROPN
ejpam-5532	161	2	pure	pure	PROPN
ejpam-5532	161	3	appl	appl	PROPN
ejpam-5532	161	4	.	.	PROPN
ejpam-5532	161	5	math	math	PROPN
ejpam-5532	161	6	,	,	PUNCT
ejpam-5532	161	7	18	18	NUM
ejpam-5532	161	8	(	(	PUNCT
ejpam-5532	161	9	1	1	NUM
ejpam-5532	161	10	)	)	PUNCT
ejpam-5532	161	11	(	(	PUNCT
ejpam-5532	161	12	2025	2025	NUM
ejpam-5532	161	13	)	)	PUNCT
ejpam-5532	161	14	,	,	PUNCT
ejpam-5532	161	15	5532	5532	NUM
ejpam-5532	161	16	9	9	NUM
ejpam-5532	161	17	of	of	ADP
ejpam-5532	161	18	20	20	NUM
ejpam-5532	161	19	and	and	CCONJ
ejpam-5532	161	20	µ(a	µ(a	PROPN
ejpam-5532	161	21	)	)	PUNCT
ejpam-5532	162	1	=	=	SYM
ejpam-5532	163	1			PRON
ejpam-5532	163	2	∅	∅	NOUN
ejpam-5532	163	3	if	if	SCONJ
ejpam-5532	163	4	a	a	DET
ejpam-5532	163	5	=	=	SYM
ejpam-5532	163	6	z	z	NOUN
ejpam-5532	163	7	,	,	PUNCT
ejpam-5532	163	8	r	r	NOUN
ejpam-5532	163	9	if	if	SCONJ
ejpam-5532	163	10	a	a	DET
ejpam-5532	163	11	=	=	NOUN
ejpam-5532	163	12	∅	∅	NOUN
ejpam-5532	163	13	,	,	PUNCT
ejpam-5532	163	14	{	{	PUNCT
ejpam-5532	163	15	n,−n	n,−n	NOUN
ejpam-5532	163	16	}	}	PUNCT
ejpam-5532	163	17	if	if	SCONJ
ejpam-5532	163	18	n	n	PRON
ejpam-5532	163	19	∈	∈	PROPN
ejpam-5532	163	20	z.	z.	PROPN
ejpam-5532	163	21	now	now	ADV
ejpam-5532	163	22	,	,	PUNCT
ejpam-5532	163	23	note	note	VERB
ejpam-5532	163	24	that	that	SCONJ
ejpam-5532	163	25	η	η	PROPN
ejpam-5532	163	26	⊆	⊆	NUM
ejpam-5532	163	27	µ.	µ.	NOUN
ejpam-5532	163	28	the	the	DET
ejpam-5532	163	29	set	set	NOUN
ejpam-5532	163	30	{	{	PUNCT
ejpam-5532	163	31	ηa	ηa	INTJ
ejpam-5532	163	32	:	:	PUNCT
ejpam-5532	163	33	a	a	DET
ejpam-5532	163	34	∈	∈	PROPN
ejpam-5532	164	1	i	i	NOUN
ejpam-5532	164	2	m	m	NOUN
ejpam-5532	164	3	η	η	X
ejpam-5532	164	4	}	}	PUNCT
ejpam-5532	164	5	of	of	ADP
ejpam-5532	164	6	all	all	DET
ejpam-5532	164	7	level	level	NOUN
ejpam-5532	164	8	subset	subset	NOUN
ejpam-5532	164	9	of	of	ADP
ejpam-5532	164	10	η	η	PROPN
ejpam-5532	164	11	is	be	AUX
ejpam-5532	164	12	determined	determine	VERB
ejpam-5532	164	13	below	below	ADV
ejpam-5532	164	14	:	:	PUNCT
ejpam-5532	164	15	ηr	ηr	PROPN
ejpam-5532	164	16	=	=	NOUN
ejpam-5532	164	17	∅	∅	NOUN
ejpam-5532	164	18	,	,	PUNCT
ejpam-5532	164	19	η{n	η{n	NOUN
ejpam-5532	164	20	}	}	PUNCT
ejpam-5532	164	21	=	=	SYM
ejpam-5532	164	22	{	{	PUNCT
ejpam-5532	164	23	z	z	NOUN
ejpam-5532	164	24	,	,	PUNCT
ejpam-5532	164	25	{	{	PUNCT
ejpam-5532	164	26	n	n	CCONJ
ejpam-5532	164	27	}	}	PUNCT
ejpam-5532	164	28	}	}	PUNCT
ejpam-5532	164	29	and	and	CCONJ
ejpam-5532	164	30	η∅	η∅	VERB
ejpam-5532	164	31	=	=	PUNCT
ejpam-5532	164	32	m.	m.	NOUN
ejpam-5532	164	33	further	far	ADV
ejpam-5532	164	34	,	,	PUNCT
ejpam-5532	164	35	the	the	DET
ejpam-5532	164	36	set	set	NOUN
ejpam-5532	164	37	{	{	PUNCT
ejpam-5532	164	38	µa	µa	NOUN
ejpam-5532	164	39	:	:	PUNCT
ejpam-5532	164	40	a	a	DET
ejpam-5532	164	41	∈	∈	PROPN
ejpam-5532	165	1	i	i	PRON
ejpam-5532	165	2	m	m	VERB
ejpam-5532	165	3	µ	µ	VERB
ejpam-5532	165	4	}	}	PUNCT
ejpam-5532	165	5	of	of	ADP
ejpam-5532	165	6	all	all	DET
ejpam-5532	165	7	level	level	NOUN
ejpam-5532	165	8	subset	subset	NOUN
ejpam-5532	165	9	of	of	ADP
ejpam-5532	165	10	µ	µ	NOUN
ejpam-5532	165	11	is	be	AUX
ejpam-5532	165	12	determined	determine	VERB
ejpam-5532	165	13	below	below	ADP
ejpam-5532	165	14	:	:	PUNCT
ejpam-5532	165	15	µr	µr	ADP
ejpam-5532	165	16	=	=	ADJ
ejpam-5532	165	17	∅	∅	NOUN
ejpam-5532	165	18	,	,	PUNCT
ejpam-5532	165	19	µ{±	µ{±	NOUN
ejpam-5532	165	20	n	n	CCONJ
ejpam-5532	165	21	}	}	PUNCT
ejpam-5532	165	22	=	=	SYM
ejpam-5532	165	23	{	{	PUNCT
ejpam-5532	165	24	z	z	NOUN
ejpam-5532	165	25	,	,	PUNCT
ejpam-5532	165	26	{	{	PUNCT
ejpam-5532	165	27	n	n	CCONJ
ejpam-5532	165	28	}	}	PUNCT
ejpam-5532	165	29	}	}	PUNCT
ejpam-5532	165	30	,	,	PUNCT
ejpam-5532	165	31	and	and	CCONJ
ejpam-5532	165	32	µ∅	µ∅	PUNCT
ejpam-5532	165	33	=	=	PUNCT
ejpam-5532	165	34	m.	m.	NOUN
ejpam-5532	165	35	now	now	ADV
ejpam-5532	165	36	,	,	PUNCT
ejpam-5532	165	37	it	it	PRON
ejpam-5532	165	38	is	be	AUX
ejpam-5532	165	39	easy	easy	ADJ
ejpam-5532	165	40	to	to	PART
ejpam-5532	165	41	see	see	VERB
ejpam-5532	165	42	that	that	SCONJ
ejpam-5532	165	43	η	η	PROPN
ejpam-5532	165	44	and	and	CCONJ
ejpam-5532	165	45	µ	µ	PROPN
ejpam-5532	165	46	are	be	AUX
ejpam-5532	165	47	l	l	NOUN
ejpam-5532	165	48	-	-	NOUN
ejpam-5532	165	49	sublattices	sublattice	NOUN
ejpam-5532	165	50	of	of	ADP
ejpam-5532	165	51	m	m	PROPN
ejpam-5532	165	52	.	.	PUNCT
ejpam-5532	166	1	futhermore	futhermore	NOUN
ejpam-5532	166	2	,	,	PUNCT
ejpam-5532	166	3	η	η	PROPN
ejpam-5532	166	4	forms	form	VERB
ejpam-5532	166	5	an	an	DET
ejpam-5532	166	6	lconvex	lconvex	NOUN
ejpam-5532	166	7	sublattice	sublattice	NOUN
ejpam-5532	166	8	of	of	ADP
ejpam-5532	166	9	µ.	µ.	NOUN
ejpam-5532	166	10	observe	observe	VERB
ejpam-5532	166	11	that	that	SCONJ
ejpam-5532	166	12	in	in	ADP
ejpam-5532	166	13	this	this	DET
ejpam-5532	166	14	example	example	NOUN
ejpam-5532	166	15	the	the	DET
ejpam-5532	166	16	hesse	hesse	PROPN
ejpam-5532	166	17	diagram	diagram	NOUN
ejpam-5532	166	18	of	of	ADP
ejpam-5532	166	19	set	set	NOUN
ejpam-5532	166	20	of	of	ADP
ejpam-5532	166	21	all	all	DET
ejpam-5532	166	22	level	level	NOUN
ejpam-5532	166	23	subsets	subset	NOUN
ejpam-5532	166	24	of	of	ADP
ejpam-5532	166	25	both	both	DET
ejpam-5532	166	26	η	η	PROPN
ejpam-5532	166	27	and	and	CCONJ
ejpam-5532	166	28	µ	µ	DET
ejpam-5532	166	29	coincide	coincide	NOUN
ejpam-5532	166	30	with	with	ADP
ejpam-5532	166	31	that	that	PRON
ejpam-5532	166	32	of	of	ADP
ejpam-5532	166	33	hasse	hasse	PROPN
ejpam-5532	166	34	diagram	diagram	NOUN
ejpam-5532	166	35	of	of	ADP
ejpam-5532	166	36	the	the	DET
ejpam-5532	166	37	lattice	lattice	NOUN
ejpam-5532	166	38	m	m	AUX
ejpam-5532	166	39	given	give	VERB
ejpam-5532	166	40	above	above	ADV
ejpam-5532	166	41	.	.	PUNCT
ejpam-5532	167	1	infact	infact	PROPN
ejpam-5532	167	2	,	,	PUNCT
ejpam-5532	167	3	the	the	DET
ejpam-5532	167	4	set	set	NOUN
ejpam-5532	167	5	of	of	ADP
ejpam-5532	167	6	all	all	DET
ejpam-5532	167	7	level	level	NOUN
ejpam-5532	167	8	subsets	subset	NOUN
ejpam-5532	167	9	{	{	PUNCT
ejpam-5532	167	10	ηa	ηa	INTJ
ejpam-5532	167	11	:	:	PUNCT
ejpam-5532	167	12	a	a	DET
ejpam-5532	167	13	∈	∈	PROPN
ejpam-5532	168	1	i	i	NOUN
ejpam-5532	168	2	m	m	PROPN
ejpam-5532	168	3	η	η	X
ejpam-5532	168	4	}	}	PUNCT
ejpam-5532	168	5	turns	turn	VERB
ejpam-5532	168	6	out	out	ADP
ejpam-5532	168	7	to	to	PART
ejpam-5532	168	8	be	be	AUX
ejpam-5532	168	9	lattice	lattice	ADJ
ejpam-5532	168	10	under	under	ADP
ejpam-5532	168	11	the	the	DET
ejpam-5532	168	12	usual	usual	ADJ
ejpam-5532	168	13	set	set	ADJ
ejpam-5532	168	14	theoretic	theoretic	ADJ
ejpam-5532	168	15	containment	containment	NOUN
ejpam-5532	168	16	.	.	PUNCT
ejpam-5532	169	1	the	the	DET
ejpam-5532	169	2	following	following	ADJ
ejpam-5532	169	3	result	result	NOUN
ejpam-5532	169	4	is	be	AUX
ejpam-5532	169	5	also	also	ADV
ejpam-5532	169	6	straightforward	straightforward	ADJ
ejpam-5532	169	7	.	.	PUNCT
ejpam-5532	170	1	theorem	theorem	ADJ
ejpam-5532	170	2	8	8	NUM
ejpam-5532	170	3	.	.	PUNCT
ejpam-5532	171	1	the	the	DET
ejpam-5532	171	2	intersection	intersection	NOUN
ejpam-5532	171	3	of	of	ADP
ejpam-5532	171	4	an	an	DET
ejpam-5532	171	5	arbitrary	arbitrary	ADJ
ejpam-5532	171	6	family	family	NOUN
ejpam-5532	171	7	of	of	ADP
ejpam-5532	171	8	l	l	ADJ
ejpam-5532	171	9	-	-	ADJ
ejpam-5532	171	10	convex	convex	ADJ
ejpam-5532	171	11	sublattices	sublattice	NOUN
ejpam-5532	171	12	of	of	ADP
ejpam-5532	171	13	l	l	PROPN
ejpam-5532	171	14	-	-	PUNCT
ejpam-5532	171	15	lattice	lattice	PROPN
ejpam-5532	171	16	µ	µ	PROPN
ejpam-5532	171	17	is	be	AUX
ejpam-5532	171	18	an	an	DET
ejpam-5532	171	19	l	l	ADJ
ejpam-5532	171	20	-	-	ADJ
ejpam-5532	171	21	convex	convex	ADJ
ejpam-5532	171	22	sublattice	sublattice	NOUN
ejpam-5532	171	23	of	of	ADP
ejpam-5532	171	24	µ.	µ.	NOUN
ejpam-5532	171	25	the	the	DET
ejpam-5532	171	26	above	above	ADJ
ejpam-5532	171	27	result	result	NOUN
ejpam-5532	171	28	is	be	AUX
ejpam-5532	171	29	instrumental	instrumental	ADJ
ejpam-5532	171	30	in	in	ADP
ejpam-5532	171	31	defining	define	VERB
ejpam-5532	171	32	an	an	DET
ejpam-5532	171	33	l	l	ADJ
ejpam-5532	171	34	-	-	ADJ
ejpam-5532	171	35	convex	convex	ADJ
ejpam-5532	171	36	sublattice	sublattice	NOUN
ejpam-5532	171	37	of	of	ADP
ejpam-5532	171	38	µ	µ	PRON
ejpam-5532	171	39	generated	generate	VERB
ejpam-5532	171	40	by	by	ADP
ejpam-5532	171	41	an	an	DET
ejpam-5532	171	42	l	l	NOUN
ejpam-5532	171	43	-	-	PUNCT
ejpam-5532	171	44	subset	subset	VERB
ejpam-5532	171	45	η	η	PROPN
ejpam-5532	171	46	of	of	ADP
ejpam-5532	171	47	µ.	µ.	PROPN
ejpam-5532	171	48	definition	definition	NOUN
ejpam-5532	171	49	9	9	NUM
ejpam-5532	171	50	.	.	PUNCT
ejpam-5532	172	1	an	an	DET
ejpam-5532	172	2	l	l	ADJ
ejpam-5532	172	3	-	-	ADJ
ejpam-5532	172	4	convex	convex	ADJ
ejpam-5532	172	5	sublattice	sublattice	NOUN
ejpam-5532	172	6	of	of	ADP
ejpam-5532	172	7	l	l	NOUN
ejpam-5532	172	8	-	-	PUNCT
ejpam-5532	172	9	lattice	lattice	NOUN
ejpam-5532	172	10	µ	µ	NOUN
ejpam-5532	172	11	generated	generate	VERB
ejpam-5532	172	12	by	by	ADP
ejpam-5532	172	13	an	an	DET
ejpam-5532	172	14	l	l	NOUN
ejpam-5532	172	15	-	-	PUNCT
ejpam-5532	172	16	subset	subset	VERB
ejpam-5532	172	17	η	η	PROPN
ejpam-5532	172	18	of	of	ADP
ejpam-5532	172	19	µ	µ	PROPN
ejpam-5532	172	20	is	be	AUX
ejpam-5532	172	21	defined	define	VERB
ejpam-5532	172	22	as	as	ADP
ejpam-5532	172	23	the	the	DET
ejpam-5532	172	24	intersection	intersection	NOUN
ejpam-5532	172	25	of	of	ADP
ejpam-5532	172	26	all	all	DET
ejpam-5532	172	27	l	l	ADJ
ejpam-5532	172	28	-	-	ADJ
ejpam-5532	172	29	convex	convex	ADJ
ejpam-5532	172	30	sublattices	sublattice	NOUN
ejpam-5532	172	31	of	of	ADP
ejpam-5532	172	32	µ	µ	X
ejpam-5532	172	33	containing	contain	VERB
ejpam-5532	172	34	η	η	PROPN
ejpam-5532	172	35	and	and	CCONJ
ejpam-5532	172	36	is	be	AUX
ejpam-5532	172	37	denoted	denote	VERB
ejpam-5532	172	38	by	by	ADP
ejpam-5532	172	39	[	[	X
ejpam-5532	172	40	η]cµ.	η]cµ.	NOUN
ejpam-5532	172	41	thus	thus	ADV
ejpam-5532	172	42	,	,	PUNCT
ejpam-5532	172	43	[	[	X
ejpam-5532	172	44	η]cµ	η]cµ	PROPN
ejpam-5532	172	45	=	=	SYM
ejpam-5532	172	46	⋂	⋂	PROPN
ejpam-5532	172	47	{	{	PUNCT
ejpam-5532	172	48	ηi	ηi	X
ejpam-5532	172	49	:	:	PUNCT
ejpam-5532	172	50	ηi	ηi	PROPN
ejpam-5532	172	51	is	be	AUX
ejpam-5532	172	52	an	an	DET
ejpam-5532	172	53	l	l	ADJ
ejpam-5532	172	54	-	-	ADJ
ejpam-5532	172	55	convex	convex	ADJ
ejpam-5532	172	56	sublattice	sublattice	NOUN
ejpam-5532	172	57	of	of	ADP
ejpam-5532	172	58	µ	µ	NUM
ejpam-5532	172	59	,	,	PUNCT
ejpam-5532	172	60	η	η	PROPN
ejpam-5532	172	61	⊆	⊆	NUM
ejpam-5532	172	62	ηi	ηi	NOUN
ejpam-5532	172	63	,	,	PUNCT
ejpam-5532	172	64	∀	∀	VERB
ejpam-5532	173	1	i	i	PRON
ejpam-5532	173	2	∈	∈	VERB
ejpam-5532	173	3	i	i	PRON
ejpam-5532	173	4	}	}	PUNCT
ejpam-5532	173	5	.	.	PUNCT
ejpam-5532	174	1	the	the	DET
ejpam-5532	174	2	next	next	ADJ
ejpam-5532	174	3	result	result	NOUN
ejpam-5532	174	4	provides	provide	VERB
ejpam-5532	174	5	a	a	DET
ejpam-5532	174	6	complete	complete	ADJ
ejpam-5532	174	7	structure	structure	NOUN
ejpam-5532	174	8	of	of	ADP
ejpam-5532	174	9	l	l	ADJ
ejpam-5532	174	10	-	-	ADJ
ejpam-5532	174	11	convex	convex	ADJ
ejpam-5532	174	12	sublattice	sublattice	NOUN
ejpam-5532	174	13	generated	generate	VERB
ejpam-5532	174	14	by	by	ADP
ejpam-5532	174	15	l	l	NOUN
ejpam-5532	174	16	-	-	PUNCT
ejpam-5532	174	17	subset	subset	VERB
ejpam-5532	174	18	η	η	PROPN
ejpam-5532	174	19	of	of	ADP
ejpam-5532	174	20	µ	µ	NUM
ejpam-5532	174	21	in	in	ADP
ejpam-5532	174	22	terms	term	NOUN
ejpam-5532	174	23	of	of	ADP
ejpam-5532	174	24	level	level	NOUN
ejpam-5532	174	25	subsets	subset	NOUN
ejpam-5532	174	26	.	.	PUNCT
ejpam-5532	175	1	theorem	theorem	NOUN
ejpam-5532	175	2	9	9	NUM
ejpam-5532	175	3	.	.	PUNCT
ejpam-5532	176	1	let	let	VERB
ejpam-5532	176	2	l	l	NOUN
ejpam-5532	176	3	be	be	AUX
ejpam-5532	176	4	a	a	DET
ejpam-5532	176	5	complete	complete	ADJ
ejpam-5532	176	6	and	and	CCONJ
ejpam-5532	176	7	completely	completely	ADV
ejpam-5532	176	8	distributive	distributive	ADJ
ejpam-5532	176	9	lattice	lattice	NOUN
ejpam-5532	176	10	and	and	CCONJ
ejpam-5532	176	11	l(µ,m	l(µ,m	ADV
ejpam-5532	176	12	)	)	PUNCT
ejpam-5532	176	13	be	be	AUX
ejpam-5532	176	14	an	an	DET
ejpam-5532	176	15	l	l	NOUN
ejpam-5532	176	16	-	-	NOUN
ejpam-5532	176	17	lattice	lattice	NOUN
ejpam-5532	176	18	.	.	PUNCT
ejpam-5532	177	1	let	let	VERB
ejpam-5532	177	2	η	η	PROPN
ejpam-5532	177	3	∈	∈	PROPN
ejpam-5532	177	4	lm	lm	X
ejpam-5532	177	5	with	with	ADP
ejpam-5532	177	6	η	η	PROPN
ejpam-5532	177	7	⊆	⊆	PROPN
ejpam-5532	177	8	µ	µ	X
ejpam-5532	177	9	and	and	CCONJ
ejpam-5532	177	10	a0	a0	PROPN
ejpam-5532	177	11	=	=	SYM
ejpam-5532	177	12	tip{η	tip{η	PROPN
ejpam-5532	177	13	}	}	PUNCT
ejpam-5532	177	14	.	.	PUNCT
ejpam-5532	178	1	define	define	VERB
ejpam-5532	178	2	an	an	DET
ejpam-5532	178	3	l	l	NOUN
ejpam-5532	178	4	-	-	NOUN
ejpam-5532	178	5	subset	subset	ADJ
ejpam-5532	178	6	η′	η′	NOUN
ejpam-5532	178	7	of	of	ADP
ejpam-5532	178	8	m	m	PRON
ejpam-5532	178	9	as	as	ADP
ejpam-5532	178	10	:	:	PUNCT
ejpam-5532	178	11	η′(x	η′(x	X
ejpam-5532	178	12	)	)	PUNCT
ejpam-5532	178	13	=	=	PUNCT
ejpam-5532	179	1	∨	∨	NUM
ejpam-5532	179	2	t≤ao	t≤ao	PROPN
ejpam-5532	179	3	{	{	PUNCT
ejpam-5532	179	4	t	t	NOUN
ejpam-5532	179	5	:	:	PUNCT
ejpam-5532	179	6	x	x	PUNCT
ejpam-5532	179	7	∈	∈	PROPN
ejpam-5532	179	8	[	[	X
ejpam-5532	179	9	ηt]c	ηt]c	PROPN
ejpam-5532	179	10	}	}	PUNCT
ejpam-5532	179	11	,	,	PUNCT
ejpam-5532	179	12	∀	∀	PUNCT
ejpam-5532	179	13	x	x	SYM
ejpam-5532	179	14	∈	∈	NOUN
ejpam-5532	179	15	m	m	NOUN
ejpam-5532	179	16	;	;	PUNCT
ejpam-5532	179	17	where	where	SCONJ
ejpam-5532	179	18	[	[	X
ejpam-5532	179	19	ηt]c	ηt]c	PROPN
ejpam-5532	179	20	is	be	AUX
ejpam-5532	179	21	the	the	DET
ejpam-5532	179	22	convex	convex	ADJ
ejpam-5532	179	23	sublattice	sublattice	NOUN
ejpam-5532	179	24	of	of	ADP
ejpam-5532	179	25	lattice	lattice	NOUN
ejpam-5532	179	26	µt	µt	AUX
ejpam-5532	179	27	generated	generate	VERB
ejpam-5532	179	28	by	by	ADP
ejpam-5532	179	29	ηt	ηt	ADP
ejpam-5532	179	30	.	.	PUNCT
ejpam-5532	180	1	then	then	ADV
ejpam-5532	180	2	,	,	PUNCT
ejpam-5532	180	3	η′	η′	PROPN
ejpam-5532	180	4	is	be	AUX
ejpam-5532	180	5	an	an	DET
ejpam-5532	180	6	l	l	ADJ
ejpam-5532	180	7	-	-	ADJ
ejpam-5532	180	8	convex	convex	ADJ
ejpam-5532	180	9	sublattice	sublattice	NOUN
ejpam-5532	180	10	of	of	ADP
ejpam-5532	180	11	µ	µ	NOUN
ejpam-5532	180	12	and	and	CCONJ
ejpam-5532	180	13	η′	η′	PRON
ejpam-5532	180	14	=	=	PUNCT
ejpam-5532	181	1	[	[	X
ejpam-5532	181	2	η]cµ.	η]cµ.	PROPN
ejpam-5532	181	3	a.	a.	NOUN
ejpam-5532	181	4	jain	jain	PROPN
ejpam-5532	181	5	,	,	PUNCT
ejpam-5532	181	6	i.	i.	PROPN
ejpam-5532	181	7	jahan	jahan	PROPN
ejpam-5532	181	8	/	/	SYM
ejpam-5532	181	9	eur	eur	PROPN
ejpam-5532	181	10	.	.	PUNCT
ejpam-5532	182	1	j.	j.	PROPN
ejpam-5532	182	2	pure	pure	PROPN
ejpam-5532	182	3	appl	appl	PROPN
ejpam-5532	182	4	.	.	PROPN
ejpam-5532	182	5	math	math	PROPN
ejpam-5532	182	6	,	,	PUNCT
ejpam-5532	182	7	18	18	NUM
ejpam-5532	182	8	(	(	PUNCT
ejpam-5532	182	9	1	1	NUM
ejpam-5532	182	10	)	)	PUNCT
ejpam-5532	182	11	(	(	PUNCT
ejpam-5532	182	12	2025	2025	NUM
ejpam-5532	182	13	)	)	PUNCT
ejpam-5532	182	14	,	,	PUNCT
ejpam-5532	182	15	5532	5532	NUM
ejpam-5532	182	16	10	10	NUM
ejpam-5532	182	17	of	of	ADP
ejpam-5532	182	18	20	20	NUM
ejpam-5532	182	19	proof	proof	NOUN
ejpam-5532	182	20	.	.	PUNCT
ejpam-5532	183	1	since	since	SCONJ
ejpam-5532	183	2	η	η	PROPN
ejpam-5532	183	3	⊆	⊆	NUM
ejpam-5532	183	4	µ	µ	NUM
ejpam-5532	183	5	,	,	PUNCT
ejpam-5532	183	6	ηt	ηt	ADP
ejpam-5532	183	7	⊆	⊆	NUM
ejpam-5532	183	8	µt	µt	NOUN
ejpam-5532	183	9	,	,	PUNCT
ejpam-5532	183	10	∀	∀	X
ejpam-5532	183	11	t	t	NOUN
ejpam-5532	183	12	∈	∈	PROPN
ejpam-5532	183	13	l.	l.	PROPN
ejpam-5532	183	14	as	as	SCONJ
ejpam-5532	183	15	µ	µ	PROPN
ejpam-5532	183	16	is	be	AUX
ejpam-5532	183	17	an	an	DET
ejpam-5532	183	18	l	l	NOUN
ejpam-5532	183	19	-	-	NOUN
ejpam-5532	183	20	lattice	lattice	NOUN
ejpam-5532	183	21	,	,	PUNCT
ejpam-5532	183	22	µt	µt	PRON
ejpam-5532	183	23	is	be	AUX
ejpam-5532	183	24	a	a	DET
ejpam-5532	183	25	sublattice	sublattice	NOUN
ejpam-5532	183	26	of	of	ADP
ejpam-5532	183	27	m	m	PRON
ejpam-5532	183	28	,	,	PUNCT
ejpam-5532	183	29	∀	∀	X
ejpam-5532	183	30	t	t	NOUN
ejpam-5532	183	31	≤	≤	NOUN
ejpam-5532	183	32	tip{µ	tip{µ	NOUN
ejpam-5532	183	33	}	}	PUNCT
ejpam-5532	183	34	.	.	PUNCT
ejpam-5532	184	1	moreover	moreover	ADV
ejpam-5532	184	2	,	,	PUNCT
ejpam-5532	184	3	[	[	X
ejpam-5532	184	4	ηt]c	ηt]c	NOUN
ejpam-5532	184	5	⊆	⊆	NUM
ejpam-5532	184	6	µt	µt	NOUN
ejpam-5532	184	7	,	,	PUNCT
ejpam-5532	184	8	as	as	SCONJ
ejpam-5532	184	9	[	[	X
ejpam-5532	184	10	ηt]c	ηt]c	NOUN
ejpam-5532	184	11	is	be	AUX
ejpam-5532	184	12	the	the	DET
ejpam-5532	184	13	convex	convex	ADJ
ejpam-5532	184	14	sublattice	sublattice	NOUN
ejpam-5532	184	15	of	of	ADP
ejpam-5532	184	16	µt	µt	PRON
ejpam-5532	184	17	generated	generate	VERB
ejpam-5532	184	18	by	by	ADP
ejpam-5532	184	19	ηt	ηt	PRON
ejpam-5532	184	20	.	.	PUNCT
ejpam-5532	185	1	thus	thus	ADV
ejpam-5532	185	2	,	,	PUNCT
ejpam-5532	185	3	η′(x	η′(x	PROPN
ejpam-5532	185	4	)	)	PUNCT
ejpam-5532	185	5	=	=	PUNCT
ejpam-5532	185	6	∨	∨	NUM
ejpam-5532	185	7	t≤ao	t≤ao	PROPN
ejpam-5532	185	8	{	{	PUNCT
ejpam-5532	185	9	t	t	NOUN
ejpam-5532	185	10	:	:	PUNCT
ejpam-5532	185	11	x	x	PUNCT
ejpam-5532	185	12	∈	∈	PROPN
ejpam-5532	185	13	[	[	X
ejpam-5532	185	14	ηt]c	ηt]c	PROPN
ejpam-5532	185	15	}	}	PUNCT
ejpam-5532	185	16	≤	≤	NOUN
ejpam-5532	185	17	∨	∨	NUM
ejpam-5532	185	18	t≤tip{µ	t≤tip{µ	PROPN
ejpam-5532	185	19	}	}	PUNCT
ejpam-5532	185	20	{	{	PUNCT
ejpam-5532	185	21	t	t	NOUN
ejpam-5532	185	22	:	:	PUNCT
ejpam-5532	185	23	x	x	PUNCT
ejpam-5532	185	24	∈	∈	PROPN
ejpam-5532	185	25	µt	µt	PRON
ejpam-5532	185	26	}	}	PUNCT
ejpam-5532	185	27	=	=	SYM
ejpam-5532	185	28	µ(x	µ(x	NUM
ejpam-5532	185	29	)	)	PUNCT
ejpam-5532	185	30	.	.	PUNCT
ejpam-5532	186	1	we	we	PRON
ejpam-5532	186	2	thus	thus	ADV
ejpam-5532	186	3	have	have	VERB
ejpam-5532	186	4	η′	η′	PROPN
ejpam-5532	186	5	⊆	⊆	NUM
ejpam-5532	186	6	µ.	µ.	NOUN
ejpam-5532	186	7	further	far	ADV
ejpam-5532	186	8	,	,	PUNCT
ejpam-5532	186	9	to	to	PART
ejpam-5532	186	10	prove	prove	VERB
ejpam-5532	186	11	that	that	SCONJ
ejpam-5532	186	12	η	η	PROPN
ejpam-5532	186	13	⊆	⊆	X
ejpam-5532	186	14	η′	η′	NUM
ejpam-5532	186	15	,	,	PUNCT
ejpam-5532	186	16	let	let	VERB
ejpam-5532	186	17	x	x	PUNCT
ejpam-5532	186	18	∈	∈	NOUN
ejpam-5532	186	19	m	m	NOUN
ejpam-5532	186	20	and	and	CCONJ
ejpam-5532	186	21	let	let	VERB
ejpam-5532	186	22	η(x	η(x	NOUN
ejpam-5532	186	23	)	)	PUNCT
ejpam-5532	186	24	=	=	PUNCT
ejpam-5532	186	25	α	α	PROPN
ejpam-5532	186	26	≤	≤	NUM
ejpam-5532	186	27	a0	a0	NOUN
ejpam-5532	186	28	.	.	PUNCT
ejpam-5532	187	1	then	then	ADV
ejpam-5532	187	2	,	,	PUNCT
ejpam-5532	187	3	x	x	PUNCT
ejpam-5532	187	4	∈	∈	NOUN
ejpam-5532	187	5	ηα	ηα	VERB
ejpam-5532	187	6	⊆	⊆	NUM
ejpam-5532	187	7	[	[	X
ejpam-5532	187	8	ηα]c	ηα]c	NOUN
ejpam-5532	187	9	.	.	PUNCT
ejpam-5532	188	1	therefore	therefore	ADV
ejpam-5532	188	2	,	,	PUNCT
ejpam-5532	188	3	by	by	ADP
ejpam-5532	188	4	definition	definition	NOUN
ejpam-5532	188	5	of	of	ADP
ejpam-5532	188	6	η′	η′	PROPN
ejpam-5532	188	7	,	,	PUNCT
ejpam-5532	188	8	α	α	NOUN
ejpam-5532	188	9	≤	≤	PUNCT
ejpam-5532	188	10	η′(x	η′(x	NOUN
ejpam-5532	188	11	)	)	PUNCT
ejpam-5532	188	12	.	.	PUNCT
ejpam-5532	189	1	that	that	PRON
ejpam-5532	189	2	is	be	AUX
ejpam-5532	189	3	,	,	PUNCT
ejpam-5532	189	4	η(x	η(x	NOUN
ejpam-5532	189	5	)	)	PUNCT
ejpam-5532	189	6	≤	≤	NUM
ejpam-5532	189	7	η′(x	η′(x	NOUN
ejpam-5532	189	8	)	)	PUNCT
ejpam-5532	189	9	.	.	PUNCT
ejpam-5532	190	1	thus	thus	ADV
ejpam-5532	190	2	,	,	PUNCT
ejpam-5532	190	3	η	η	PROPN
ejpam-5532	190	4	⊆	⊆	NUM
ejpam-5532	190	5	η′.	η′.	NOUN
ejpam-5532	190	6	we	we	PRON
ejpam-5532	190	7	now	now	ADV
ejpam-5532	190	8	prove	prove	VERB
ejpam-5532	190	9	that	that	SCONJ
ejpam-5532	190	10	η′	η′	PRON
ejpam-5532	190	11	is	be	AUX
ejpam-5532	190	12	an	an	DET
ejpam-5532	190	13	l	l	NOUN
ejpam-5532	190	14	-	-	NOUN
ejpam-5532	190	15	sublattice	sublattice	NOUN
ejpam-5532	190	16	of	of	ADP
ejpam-5532	190	17	µ.	µ.	NOUN
ejpam-5532	190	18	for	for	ADP
ejpam-5532	190	19	any	any	DET
ejpam-5532	190	20	z	z	NOUN
ejpam-5532	190	21	∈	∈	PROPN
ejpam-5532	190	22	m	m	VERB
ejpam-5532	190	23	,	,	PUNCT
ejpam-5532	190	24	define	define	VERB
ejpam-5532	190	25	a	a	DET
ejpam-5532	190	26	subset	subset	NOUN
ejpam-5532	190	27	lη(z	lη(z	NOUN
ejpam-5532	190	28	)	)	PUNCT
ejpam-5532	190	29	of	of	ADP
ejpam-5532	190	30	l	l	NOUN
ejpam-5532	190	31	as	as	SCONJ
ejpam-5532	190	32	follows	follow	VERB
ejpam-5532	190	33	:	:	PUNCT
ejpam-5532	190	34	lη(z	lη(z	X
ejpam-5532	190	35	)	)	PUNCT
ejpam-5532	190	36	=	=	PRON
ejpam-5532	190	37	{	{	PUNCT
ejpam-5532	190	38	t	t	NOUN
ejpam-5532	190	39	∈	∈	PROPN
ejpam-5532	190	40	l	l	PROPN
ejpam-5532	190	41	/	/	SYM
ejpam-5532	190	42	t	t	PROPN
ejpam-5532	190	43	≤	≤	NUM
ejpam-5532	190	44	a0	a0	NOUN
ejpam-5532	190	45	,	,	PUNCT
ejpam-5532	190	46	z	z	PROPN
ejpam-5532	190	47	∈	∈	PROPN
ejpam-5532	191	1	[	[	X
ejpam-5532	191	2	ηt]c	ηt]c	PROPN
ejpam-5532	191	3	}	}	PUNCT
ejpam-5532	191	4	.	.	PUNCT
ejpam-5532	192	1	clearly	clearly	ADV
ejpam-5532	192	2	,	,	PUNCT
ejpam-5532	192	3	η′(x	η′(x	PROPN
ejpam-5532	192	4	)	)	PUNCT
ejpam-5532	192	5	=	=	SYM
ejpam-5532	192	6	∨	∨	X
ejpam-5532	192	7	lη(x	lη(x	NUM
ejpam-5532	192	8	)	)	PUNCT
ejpam-5532	192	9	.	.	PUNCT
ejpam-5532	193	1	let	let	VERB
ejpam-5532	193	2	x	x	PRON
ejpam-5532	193	3	,	,	PUNCT
ejpam-5532	193	4	y	y	PROPN
ejpam-5532	193	5	∈	∈	PROPN
ejpam-5532	193	6	m	m	PROPN
ejpam-5532	193	7	,	,	PUNCT
ejpam-5532	193	8	a	a	DET
ejpam-5532	193	9	∈	∈	NOUN
ejpam-5532	193	10	lη(x	lη(x	X
ejpam-5532	193	11	)	)	PUNCT
ejpam-5532	193	12	and	and	CCONJ
ejpam-5532	193	13	b	b	X
ejpam-5532	193	14	∈	∈	PROPN
ejpam-5532	193	15	lη(y	lη(y	PUNCT
ejpam-5532	193	16	)	)	PUNCT
ejpam-5532	193	17	.	.	PUNCT
ejpam-5532	194	1	we	we	PRON
ejpam-5532	194	2	claim	claim	VERB
ejpam-5532	194	3	that	that	SCONJ
ejpam-5532	194	4	a	a	DET
ejpam-5532	194	5	∧	∧	PROPN
ejpam-5532	194	6	b	b	PROPN
ejpam-5532	194	7	∈	∈	PROPN
ejpam-5532	194	8	lη(x	lη(x	X
ejpam-5532	194	9	∨	∨	PROPN
ejpam-5532	194	10	y	y	NOUN
ejpam-5532	194	11	)	)	PUNCT
ejpam-5532	194	12	.	.	PUNCT
ejpam-5532	195	1	first	first	ADV
ejpam-5532	195	2	note	note	VERB
ejpam-5532	195	3	that	that	SCONJ
ejpam-5532	195	4	ηa	ηa	PROPN
ejpam-5532	195	5	∪	∪	VERB
ejpam-5532	195	6	ηb	ηb	ADP
ejpam-5532	195	7	⊆	⊆	NUM
ejpam-5532	195	8	ηa∧b	ηa∧b	NOUN
ejpam-5532	195	9	.	.	PUNCT
ejpam-5532	196	1	since	since	SCONJ
ejpam-5532	196	2	a	a	DET
ejpam-5532	196	3	∈	∈	PROPN
ejpam-5532	196	4	lη(x	lη(x	X
ejpam-5532	196	5	)	)	PUNCT
ejpam-5532	196	6	and	and	CCONJ
ejpam-5532	196	7	b	b	X
ejpam-5532	196	8	∈	∈	PROPN
ejpam-5532	196	9	lη(y	lη(y	PUNCT
ejpam-5532	196	10	)	)	PUNCT
ejpam-5532	196	11	,	,	PUNCT
ejpam-5532	196	12	we	we	PRON
ejpam-5532	196	13	have	have	VERB
ejpam-5532	196	14	a	a	DET
ejpam-5532	196	15	,	,	PUNCT
ejpam-5532	196	16	b	b	PROPN
ejpam-5532	196	17	≤	≤	NUM
ejpam-5532	196	18	a0	a0	NOUN
ejpam-5532	196	19	,	,	PUNCT
ejpam-5532	196	20	x	x	PROPN
ejpam-5532	196	21	∈	∈	PROPN
ejpam-5532	196	22	[	[	X
ejpam-5532	196	23	ηa]c	ηa]c	PROPN
ejpam-5532	196	24	,	,	PUNCT
ejpam-5532	196	25	y	y	PROPN
ejpam-5532	196	26	∈	∈	PROPN
ejpam-5532	197	1	[	[	X
ejpam-5532	197	2	ηb]c	ηb]c	PROPN
ejpam-5532	197	3	.	.	PUNCT
ejpam-5532	198	1	therefore	therefore	ADV
ejpam-5532	198	2	,	,	PUNCT
ejpam-5532	198	3	x	x	PROPN
ejpam-5532	198	4	=	=	SYM
ejpam-5532	198	5	p{xi	p{xi	PROPN
ejpam-5532	198	6	}	}	PUNCT
ejpam-5532	198	7	(	(	PUNCT
ejpam-5532	198	8	a	a	DET
ejpam-5532	198	9	lattice	lattice	NOUN
ejpam-5532	198	10	polynomial	polynomial	NOUN
ejpam-5532	198	11	in	in	ADP
ejpam-5532	198	12	variables	variable	NOUN
ejpam-5532	198	13	xi	xi	ADP
ejpam-5532	198	14	’s	’s	NOUN
ejpam-5532	198	15	.	.	PUNCT
ejpam-5532	199	1	where	where	SCONJ
ejpam-5532	199	2	xi	xi	PROPN
ejpam-5532	199	3	∈	∈	PROPN
ejpam-5532	199	4	ηa	ηa	PROPN
ejpam-5532	199	5	,	,	PUNCT
ejpam-5532	199	6	∀i	∀i	NOUN
ejpam-5532	199	7	)	)	PUNCT
ejpam-5532	199	8	.	.	PUNCT
ejpam-5532	200	1	similarly	similarly	ADV
ejpam-5532	200	2	,	,	PUNCT
ejpam-5532	200	3	y	y	PROPN
ejpam-5532	200	4	=	=	SYM
ejpam-5532	200	5	q{yj	q{yj	PROPN
ejpam-5532	200	6	}	}	PUNCT
ejpam-5532	200	7	(	(	PUNCT
ejpam-5532	200	8	a	a	DET
ejpam-5532	200	9	lattice	lattice	NOUN
ejpam-5532	200	10	polynomial	polynomial	NOUN
ejpam-5532	200	11	in	in	ADP
ejpam-5532	200	12	variables	variable	NOUN
ejpam-5532	200	13	yj	yj	PROPN
ejpam-5532	200	14	’	'	PUNCT
ejpam-5532	200	15	s.	s.	PROPN
ejpam-5532	200	16	where	where	SCONJ
ejpam-5532	200	17	yj	yj	PROPN
ejpam-5532	200	18	∈	∈	PROPN
ejpam-5532	200	19	ηb	ηb	PROPN
ejpam-5532	200	20	,	,	PUNCT
ejpam-5532	200	21	∀j	∀j	NOUN
ejpam-5532	200	22	)	)	PUNCT
ejpam-5532	200	23	.	.	PUNCT
ejpam-5532	201	1	thus	thus	ADV
ejpam-5532	201	2	,	,	PUNCT
ejpam-5532	201	3	x	x	PROPN
ejpam-5532	201	4	∨	∨	PROPN
ejpam-5532	201	5	y	y	PROPN
ejpam-5532	201	6	is	be	AUX
ejpam-5532	201	7	also	also	ADV
ejpam-5532	201	8	a	a	DET
ejpam-5532	201	9	lattice	lattice	ADJ
ejpam-5532	201	10	polynomial	polynomial	NOUN
ejpam-5532	201	11	in	in	ADP
ejpam-5532	201	12	variables	variable	NOUN
ejpam-5532	201	13	xi	xi	NOUN
ejpam-5532	201	14	’s	’s	PART
ejpam-5532	201	15	and	and	CCONJ
ejpam-5532	201	16	yj	yj	PROPN
ejpam-5532	201	17	’s	’s	X
ejpam-5532	201	18	,	,	PUNCT
ejpam-5532	201	19	where	where	SCONJ
ejpam-5532	201	20	xi	xi	PROPN
ejpam-5532	201	21	,	,	PUNCT
ejpam-5532	201	22	yj	yj	PROPN
ejpam-5532	201	23	∈	∈	PROPN
ejpam-5532	201	24	ηa	ηa	NOUN
ejpam-5532	201	25	∪	∪	VERB
ejpam-5532	201	26	ηb	ηb	ADP
ejpam-5532	201	27	⊆	⊆	NUM
ejpam-5532	201	28	ηa∧b	ηa∧b	NOUN
ejpam-5532	201	29	.	.	PUNCT
ejpam-5532	202	1	that	that	PRON
ejpam-5532	202	2	is	be	AUX
ejpam-5532	202	3	,	,	PUNCT
ejpam-5532	202	4	x	x	PROPN
ejpam-5532	202	5	∨	∨	NUM
ejpam-5532	202	6	y	y	PROPN
ejpam-5532	202	7	∈	∈	PROPN
ejpam-5532	203	1	[	[	X
ejpam-5532	203	2	ηa∧b]c	ηa∧b]c	ADJ
ejpam-5532	203	3	.	.	PUNCT
ejpam-5532	204	1	we	we	PRON
ejpam-5532	204	2	also	also	ADV
ejpam-5532	204	3	have	have	VERB
ejpam-5532	204	4	a	a	DET
ejpam-5532	204	5	∧	∧	PROPN
ejpam-5532	204	6	b	b	PROPN
ejpam-5532	204	7	≤	≤	PROPN
ejpam-5532	204	8	a0	a0	NOUN
ejpam-5532	204	9	.	.	PUNCT
ejpam-5532	205	1	thus	thus	ADV
ejpam-5532	205	2	,	,	PUNCT
ejpam-5532	205	3	a	a	DET
ejpam-5532	205	4	∧	∧	PROPN
ejpam-5532	205	5	b	b	PROPN
ejpam-5532	205	6	∈	∈	PROPN
ejpam-5532	205	7	lη(x	lη(x	X
ejpam-5532	205	8	∨	∨	PROPN
ejpam-5532	205	9	y	y	NOUN
ejpam-5532	205	10	)	)	PUNCT
ejpam-5532	205	11	.	.	PUNCT
ejpam-5532	206	1	this	this	PRON
ejpam-5532	206	2	implies	imply	VERB
ejpam-5532	206	3	that	that	SCONJ
ejpam-5532	206	4	η′(x	η′(x	PROPN
ejpam-5532	206	5	∨	∨	NUM
ejpam-5532	206	6	y	y	PROPN
ejpam-5532	206	7	)	)	PUNCT
ejpam-5532	206	8	≥	≥	NOUN
ejpam-5532	206	9	a	a	DET
ejpam-5532	206	10	∧	∧	PROPN
ejpam-5532	206	11	b	b	PROPN
ejpam-5532	206	12	;	;	PUNCT
ejpam-5532	206	13	∀	∀	X
ejpam-5532	206	14	a	a	DET
ejpam-5532	206	15	∈	∈	NOUN
ejpam-5532	206	16	lη(x	lη(x	X
ejpam-5532	206	17	)	)	PUNCT
ejpam-5532	206	18	and	and	CCONJ
ejpam-5532	206	19	b	b	X
ejpam-5532	206	20	∈	∈	PROPN
ejpam-5532	206	21	lη(y	lη(y	PUNCT
ejpam-5532	206	22	)	)	PUNCT
ejpam-5532	206	23	.	.	PUNCT
ejpam-5532	207	1	consequently	consequently	ADV
ejpam-5532	207	2	,	,	PUNCT
ejpam-5532	207	3	η′(x	η′(x	PROPN
ejpam-5532	207	4	∨	∨	NUM
ejpam-5532	207	5	y	y	PROPN
ejpam-5532	207	6	)	)	PUNCT
ejpam-5532	207	7	≥	≥	NOUN
ejpam-5532	207	8	∨{a	∨{a	PROPN
ejpam-5532	207	9	∧	∧	PROPN
ejpam-5532	207	10	b	b	PROPN
ejpam-5532	207	11	/	/	SYM
ejpam-5532	207	12	a	a	DET
ejpam-5532	207	13	∈	∈	NOUN
ejpam-5532	207	14	lη(x	lη(x	X
ejpam-5532	207	15	)	)	PUNCT
ejpam-5532	207	16	,	,	PUNCT
ejpam-5532	207	17	b	b	X
ejpam-5532	207	18	∈	∈	PROPN
ejpam-5532	207	19	lη(y	lη(y	PUNCT
ejpam-5532	207	20	)	)	PUNCT
ejpam-5532	207	21	}	}	PUNCT
ejpam-5532	208	1	=	=	SYM
ejpam-5532	208	2	{	{	PUNCT
ejpam-5532	208	3	∨{a	∨{a	PROPN
ejpam-5532	208	4	/	/	SYM
ejpam-5532	208	5	a	a	DET
ejpam-5532	208	6	∈	∈	NOUN
ejpam-5532	208	7	lη(x	lη(x	X
ejpam-5532	208	8	)	)	PUNCT
ejpam-5532	208	9	}	}	PUNCT
ejpam-5532	208	10	}	}	PUNCT
ejpam-5532	208	11	∧	∧	NOUN
ejpam-5532	208	12	{	{	PUNCT
ejpam-5532	208	13	∨{b	∨{b	NOUN
ejpam-5532	208	14	/	/	SYM
ejpam-5532	208	15	b	b	NOUN
ejpam-5532	208	16	∈	∈	NOUN
ejpam-5532	208	17	lη(y	lη(y	PUNCT
ejpam-5532	208	18	)	)	PUNCT
ejpam-5532	208	19	}	}	PUNCT
ejpam-5532	208	20	}	}	PUNCT
ejpam-5532	208	21	(	(	PUNCT
ejpam-5532	208	22	as	as	SCONJ
ejpam-5532	208	23	l	l	NOUN
ejpam-5532	208	24	is	be	AUX
ejpam-5532	208	25	a	a	DET
ejpam-5532	208	26	completely	completely	ADV
ejpam-5532	208	27	distributive	distributive	ADJ
ejpam-5532	208	28	lattice	lattice	NOUN
ejpam-5532	208	29	)	)	PUNCT
ejpam-5532	208	30	=	=	PUNCT
ejpam-5532	208	31	η′(x	η′(x	X
ejpam-5532	208	32	)	)	PUNCT
ejpam-5532	208	33	∧	∧	PROPN
ejpam-5532	208	34	η′(y	η′(y	NOUN
ejpam-5532	208	35	)	)	PUNCT
ejpam-5532	208	36	.	.	PUNCT
ejpam-5532	209	1	similarly	similarly	ADV
ejpam-5532	209	2	,	,	PUNCT
ejpam-5532	209	3	it	it	PRON
ejpam-5532	209	4	can	can	AUX
ejpam-5532	209	5	be	be	AUX
ejpam-5532	209	6	proved	prove	VERB
ejpam-5532	209	7	that	that	SCONJ
ejpam-5532	209	8	η′(x	η′(x	PROPN
ejpam-5532	209	9	∧	∧	PROPN
ejpam-5532	209	10	y	y	PROPN
ejpam-5532	209	11	)	)	PUNCT
ejpam-5532	209	12	≥	≥	NOUN
ejpam-5532	209	13	η′(x	η′(x	NOUN
ejpam-5532	209	14	)	)	PUNCT
ejpam-5532	209	15	∧	∧	PROPN
ejpam-5532	209	16	η′(y	η′(y	PROPN
ejpam-5532	209	17	)	)	PUNCT
ejpam-5532	209	18	;	;	PUNCT
ejpam-5532	209	19	∀x	∀x	NUM
ejpam-5532	209	20	,	,	PUNCT
ejpam-5532	209	21	y	y	PROPN
ejpam-5532	209	22	∈	∈	PROPN
ejpam-5532	209	23	m.	m.	NOUN
ejpam-5532	209	24	thus	thus	ADV
ejpam-5532	209	25	,	,	PUNCT
ejpam-5532	209	26	η′	η′	PROPN
ejpam-5532	209	27	is	be	AUX
ejpam-5532	209	28	an	an	DET
ejpam-5532	209	29	l	l	NOUN
ejpam-5532	209	30	-	-	NOUN
ejpam-5532	209	31	sublattice	sublattice	NOUN
ejpam-5532	209	32	of	of	ADP
ejpam-5532	209	33	µ.	µ.	NOUN
ejpam-5532	209	34	further	far	ADV
ejpam-5532	209	35	,	,	PUNCT
ejpam-5532	209	36	to	to	PART
ejpam-5532	209	37	establish	establish	VERB
ejpam-5532	209	38	that	that	SCONJ
ejpam-5532	209	39	η′	η′	PROPN
ejpam-5532	209	40	is	be	AUX
ejpam-5532	209	41	an	an	DET
ejpam-5532	209	42	l	l	ADJ
ejpam-5532	209	43	-	-	ADJ
ejpam-5532	209	44	convex	convex	ADJ
ejpam-5532	209	45	sublattice	sublattice	NOUN
ejpam-5532	209	46	of	of	ADP
ejpam-5532	209	47	µ	µ	NUM
ejpam-5532	209	48	,	,	PUNCT
ejpam-5532	209	49	we	we	PRON
ejpam-5532	209	50	shall	shall	AUX
ejpam-5532	209	51	prove	prove	VERB
ejpam-5532	209	52	that	that	SCONJ
ejpam-5532	209	53	,	,	PUNCT
ejpam-5532	209	54	(	(	PUNCT
ejpam-5532	209	55	i)(ii	i)(ii	X
ejpam-5532	209	56	)	)	PUNCT
ejpam-5532	209	57	η′(w	η′(w	NOUN
ejpam-5532	209	58	)	)	PUNCT
ejpam-5532	209	59	≥	≥	NOUN
ejpam-5532	209	60	η′(x	η′(x	NOUN
ejpam-5532	209	61	)	)	PUNCT
ejpam-5532	209	62	∧	∧	PROPN
ejpam-5532	209	63	η′(y	η′(y	NOUN
ejpam-5532	209	64	)	)	PUNCT
ejpam-5532	209	65	∧	∧	NOUN
ejpam-5532	209	66	µ(w	µ(w	PROPN
ejpam-5532	209	67	)	)	PUNCT
ejpam-5532	209	68	where	where	SCONJ
ejpam-5532	209	69	x	x	X
ejpam-5532	209	70	≤	≤	NUM
ejpam-5532	209	71	w	w	NOUN
ejpam-5532	209	72	≤	≤	NUM
ejpam-5532	209	73	y	y	NOUN
ejpam-5532	209	74	in	in	ADP
ejpam-5532	209	75	m	m	PROPN
ejpam-5532	209	76	.	.	PUNCT
ejpam-5532	210	1	a.	a.	PROPN
ejpam-5532	210	2	jain	jain	PROPN
ejpam-5532	210	3	,	,	PUNCT
ejpam-5532	210	4	i.	i.	PROPN
ejpam-5532	210	5	jahan	jahan	PROPN
ejpam-5532	210	6	/	/	SYM
ejpam-5532	210	7	eur	eur	PROPN
ejpam-5532	210	8	.	.	PUNCT
ejpam-5532	211	1	j.	j.	PROPN
ejpam-5532	211	2	pure	pure	PROPN
ejpam-5532	211	3	appl	appl	PROPN
ejpam-5532	211	4	.	.	PROPN
ejpam-5532	211	5	math	math	PROPN
ejpam-5532	211	6	,	,	PUNCT
ejpam-5532	211	7	18	18	NUM
ejpam-5532	211	8	(	(	PUNCT
ejpam-5532	211	9	1	1	NUM
ejpam-5532	211	10	)	)	PUNCT
ejpam-5532	211	11	(	(	PUNCT
ejpam-5532	211	12	2025	2025	NUM
ejpam-5532	211	13	)	)	PUNCT
ejpam-5532	211	14	,	,	PUNCT
ejpam-5532	211	15	5532	5532	NUM
ejpam-5532	211	16	11	11	NUM
ejpam-5532	211	17	of	of	ADP
ejpam-5532	211	18	20	20	NUM
ejpam-5532	211	19	for	for	ADP
ejpam-5532	211	20	any	any	DET
ejpam-5532	211	21	z	z	NOUN
ejpam-5532	211	22	∈	∈	NOUN
ejpam-5532	211	23	m	m	VERB
ejpam-5532	211	24	,	,	PUNCT
ejpam-5532	211	25	let	let	VERB
ejpam-5532	211	26	lη(z	lη(z	PUNCT
ejpam-5532	211	27	)	)	PUNCT
ejpam-5532	212	1	=	=	PRON
ejpam-5532	212	2	{	{	PUNCT
ejpam-5532	212	3	t	t	NOUN
ejpam-5532	212	4	∈	∈	PROPN
ejpam-5532	212	5	l	l	PROPN
ejpam-5532	212	6	/	/	SYM
ejpam-5532	212	7	t	t	PROPN
ejpam-5532	212	8	≤	≤	NUM
ejpam-5532	212	9	a0	a0	NOUN
ejpam-5532	212	10	,	,	PUNCT
ejpam-5532	212	11	z	z	PROPN
ejpam-5532	212	12	∈	∈	PROPN
ejpam-5532	213	1	[	[	X
ejpam-5532	213	2	ηt]c	ηt]c	PROPN
ejpam-5532	213	3	}	}	PUNCT
ejpam-5532	213	4	.	.	PUNCT
ejpam-5532	214	1	then	then	ADV
ejpam-5532	214	2	,	,	PUNCT
ejpam-5532	214	3	η′(x	η′(x	PROPN
ejpam-5532	214	4	)	)	PUNCT
ejpam-5532	214	5	=	=	SYM
ejpam-5532	214	6	∨	∨	X
ejpam-5532	214	7	lη(x	lη(x	NUM
ejpam-5532	214	8	)	)	PUNCT
ejpam-5532	214	9	.	.	PUNCT
ejpam-5532	215	1	also	also	ADV
ejpam-5532	215	2	let	let	VERB
ejpam-5532	215	3	lµ(z	lµ(z	NOUN
ejpam-5532	215	4	)	)	PUNCT
ejpam-5532	216	1	=	=	PRON
ejpam-5532	216	2	{	{	PUNCT
ejpam-5532	216	3	t	t	PROPN
ejpam-5532	216	4	∈	∈	PROPN
ejpam-5532	216	5	l/	l/	PROPN
ejpam-5532	216	6	t	t	PROPN
ejpam-5532	216	7	≤	≤	NOUN
ejpam-5532	216	8	tip	tip	NOUN
ejpam-5532	216	9	{	{	PUNCT
ejpam-5532	216	10	µ	µ	NOUN
ejpam-5532	216	11	}	}	PUNCT
ejpam-5532	216	12	,	,	PUNCT
ejpam-5532	216	13	z	z	NOUN
ejpam-5532	216	14	∈	∈	PROPN
ejpam-5532	216	15	µt	µt	PRON
ejpam-5532	217	1	=	=	PUNCT
ejpam-5532	218	1	[	[	X
ejpam-5532	218	2	µt	µt	X
ejpam-5532	218	3	]	]	X
ejpam-5532	218	4	}	}	PUNCT
ejpam-5532	218	5	.	.	PUNCT
ejpam-5532	219	1	let	let	VERB
ejpam-5532	219	2	r	r	NOUN
ejpam-5532	219	3	∈	∈	PROPN
ejpam-5532	219	4	lµ(w	lµ(w	X
ejpam-5532	219	5	)	)	PUNCT
ejpam-5532	219	6	,	,	PUNCT
ejpam-5532	219	7	s	s	PROPN
ejpam-5532	219	8	∈	∈	PROPN
ejpam-5532	219	9	lη(x	lη(x	X
ejpam-5532	219	10	)	)	PUNCT
ejpam-5532	219	11	and	and	CCONJ
ejpam-5532	219	12	t	t	PROPN
ejpam-5532	219	13	∈	∈	PROPN
ejpam-5532	219	14	lη(y	lη(y	PUNCT
ejpam-5532	219	15	)	)	PUNCT
ejpam-5532	219	16	.	.	PUNCT
ejpam-5532	220	1	then	then	ADV
ejpam-5532	220	2	,	,	PUNCT
ejpam-5532	220	3	r	r	NOUN
ejpam-5532	220	4	≤	≤	NUM
ejpam-5532	220	5	tip	tip	NOUN
ejpam-5532	220	6	{	{	PUNCT
ejpam-5532	220	7	µ	µ	NOUN
ejpam-5532	220	8	}	}	PUNCT
ejpam-5532	220	9	;	;	PUNCT
ejpam-5532	220	10	s	s	X
ejpam-5532	220	11	,	,	PUNCT
ejpam-5532	220	12	t	t	PROPN
ejpam-5532	220	13	≤	≤	PROPN
ejpam-5532	220	14	a0	a0	NOUN
ejpam-5532	220	15	,	,	PUNCT
ejpam-5532	220	16	w	w	PROPN
ejpam-5532	220	17	∈	∈	PROPN
ejpam-5532	221	1	[	[	X
ejpam-5532	221	2	µr	µr	ADP
ejpam-5532	221	3	]	]	X
ejpam-5532	221	4	=	=	PUNCT
ejpam-5532	221	5	µr	µr	ADP
ejpam-5532	221	6	,	,	PUNCT
ejpam-5532	221	7	x	x	SYM
ejpam-5532	221	8	∈	∈	PROPN
ejpam-5532	222	1	[	[	X
ejpam-5532	222	2	ηs]c	ηs]c	NOUN
ejpam-5532	222	3	and	and	CCONJ
ejpam-5532	222	4	y	y	PROPN
ejpam-5532	222	5	∈	∈	PROPN
ejpam-5532	223	1	[	[	X
ejpam-5532	223	2	ηt]c	ηt]c	PROPN
ejpam-5532	223	3	.	.	PUNCT
ejpam-5532	224	1	hence	hence	ADV
ejpam-5532	224	2	,	,	PUNCT
ejpam-5532	224	3	s	s	VERB
ejpam-5532	224	4	∧	∧	PROPN
ejpam-5532	224	5	t	t	NOUN
ejpam-5532	224	6	∧	∧	PROPN
ejpam-5532	224	7	r	r	NOUN
ejpam-5532	224	8	≤	≤	PUNCT
ejpam-5532	224	9	a0	a0	NOUN
ejpam-5532	225	1	and	and	CCONJ
ejpam-5532	225	2	we	we	PRON
ejpam-5532	225	3	have	have	VERB
ejpam-5532	225	4	x	x	PART
ejpam-5532	225	5	∈	∈	NOUN
ejpam-5532	225	6	[	[	X
ejpam-5532	225	7	ηs]c	ηs]c	NOUN
ejpam-5532	225	8	⊆	⊆	NUM
ejpam-5532	225	9	µs	µs	NOUN
ejpam-5532	225	10	⊆	⊆	NUM
ejpam-5532	225	11	µs∧t∧r	µs∧t∧r	NOUN
ejpam-5532	225	12	;	;	PUNCT
ejpam-5532	225	13	y	y	PROPN
ejpam-5532	225	14	∈	∈	PROPN
ejpam-5532	226	1	[	[	X
ejpam-5532	226	2	ηt]c	ηt]c	NOUN
ejpam-5532	226	3	⊆	⊆	NUM
ejpam-5532	226	4	µt	µt	ADP
ejpam-5532	226	5	⊆	⊆	NUM
ejpam-5532	226	6	µs∧t∧r	µs∧t∧r	NOUN
ejpam-5532	226	7	;	;	PUNCT
ejpam-5532	226	8	and	and	CCONJ
ejpam-5532	226	9	w	w	PROPN
ejpam-5532	226	10	∈	∈	PROPN
ejpam-5532	227	1	[	[	X
ejpam-5532	227	2	µr	µr	ADP
ejpam-5532	227	3	]	]	X
ejpam-5532	227	4	=	=	PUNCT
ejpam-5532	227	5	µr	µr	ADP
ejpam-5532	227	6	⊆	⊆	NUM
ejpam-5532	227	7	µs∧t∧r	µs∧t∧r	NOUN
ejpam-5532	227	8	.	.	PUNCT
ejpam-5532	228	1	thus	thus	ADV
ejpam-5532	228	2	,	,	PUNCT
ejpam-5532	228	3	x	x	X
ejpam-5532	228	4	,	,	PUNCT
ejpam-5532	228	5	y	y	PROPN
ejpam-5532	228	6	,	,	PUNCT
ejpam-5532	228	7	w	w	PROPN
ejpam-5532	228	8	∈	∈	PROPN
ejpam-5532	228	9	µs∧t∧r	µs∧t∧r	PROPN
ejpam-5532	228	10	.	.	PUNCT
ejpam-5532	229	1	moreover	moreover	ADV
ejpam-5532	229	2	,	,	PUNCT
ejpam-5532	229	3	x	x	SYM
ejpam-5532	229	4	∈	∈	NOUN
ejpam-5532	229	5	[	[	X
ejpam-5532	229	6	ηs]c	ηs]c	VERB
ejpam-5532	229	7	⊆	⊆	NUM
ejpam-5532	229	8	[	[	X
ejpam-5532	229	9	ηs∧t∧r]c	ηs∧t∧r]c	NOUN
ejpam-5532	229	10	and	and	CCONJ
ejpam-5532	229	11	y	y	PROPN
ejpam-5532	229	12	∈	∈	PROPN
ejpam-5532	230	1	[	[	X
ejpam-5532	230	2	ηt]c	ηt]c	NOUN
ejpam-5532	230	3	⊆	⊆	NUM
ejpam-5532	230	4	[	[	X
ejpam-5532	230	5	ηs∧t∧r]c	ηs∧t∧r]c	NOUN
ejpam-5532	230	6	;	;	PUNCT
ejpam-5532	230	7	and	and	CCONJ
ejpam-5532	230	8	[	[	X
ejpam-5532	230	9	ηs∧t∧r]c	ηs∧t∧r]c	NOUN
ejpam-5532	230	10	is	be	AUX
ejpam-5532	230	11	a	a	DET
ejpam-5532	230	12	convex	convex	ADJ
ejpam-5532	230	13	sublattice	sublattice	NOUN
ejpam-5532	230	14	of	of	ADP
ejpam-5532	230	15	µs∧t∧r	µs∧t∧r	PROPN
ejpam-5532	230	16	generated	generate	VERB
ejpam-5532	230	17	by	by	ADP
ejpam-5532	230	18	the	the	DET
ejpam-5532	230	19	l	l	NOUN
ejpam-5532	230	20	-	-	PUNCT
ejpam-5532	230	21	set	set	VERB
ejpam-5532	230	22	ηs∧t∧r	ηs∧t∧r	NOUN
ejpam-5532	230	23	.	.	PUNCT
ejpam-5532	231	1	therefore	therefore	ADV
ejpam-5532	231	2	,	,	PUNCT
ejpam-5532	231	3	we	we	PRON
ejpam-5532	231	4	have	have	VERB
ejpam-5532	231	5	w	w	NOUN
ejpam-5532	231	6	∈	∈	PROPN
ejpam-5532	232	1	[	[	X
ejpam-5532	232	2	ηs∧t∧r]c	ηs∧t∧r]c	NOUN
ejpam-5532	232	3	(	(	PUNCT
ejpam-5532	232	4	as	as	ADP
ejpam-5532	232	5	x	x	X
ejpam-5532	232	6	≤	≤	NUM
ejpam-5532	232	7	w	w	NOUN
ejpam-5532	232	8	≤	≤	NUM
ejpam-5532	232	9	y	y	NOUN
ejpam-5532	232	10	in	in	ADP
ejpam-5532	232	11	m	m	PROPN
ejpam-5532	232	12	)	)	PUNCT
ejpam-5532	232	13	.	.	PUNCT
ejpam-5532	233	1	this	this	PRON
ejpam-5532	233	2	implies	imply	VERB
ejpam-5532	233	3	η′(w	η′(w	NOUN
ejpam-5532	233	4	)	)	PUNCT
ejpam-5532	233	5	≥	≥	NOUN
ejpam-5532	233	6	s	s	PART
ejpam-5532	233	7	∧	∧	PROPN
ejpam-5532	233	8	t	t	PROPN
ejpam-5532	233	9	∧	∧	PROPN
ejpam-5532	233	10	r	r	NOUN
ejpam-5532	233	11	;	;	PUNCT
ejpam-5532	233	12	∀	∀	NUM
ejpam-5532	233	13	r	r	NOUN
ejpam-5532	233	14	∈	∈	NOUN
ejpam-5532	233	15	lµ(w	lµ(w	X
ejpam-5532	233	16	)	)	PUNCT
ejpam-5532	233	17	,	,	PUNCT
ejpam-5532	233	18	s	s	PROPN
ejpam-5532	233	19	∈	∈	PROPN
ejpam-5532	233	20	lη(x	lη(x	X
ejpam-5532	233	21	)	)	PUNCT
ejpam-5532	233	22	and	and	CCONJ
ejpam-5532	233	23	t	t	PROPN
ejpam-5532	233	24	∈	∈	PROPN
ejpam-5532	233	25	lη(y	lη(y	PUNCT
ejpam-5532	233	26	)	)	PUNCT
ejpam-5532	233	27	.	.	PUNCT
ejpam-5532	234	1	that	that	PRON
ejpam-5532	234	2	is	be	AUX
ejpam-5532	234	3	,	,	PUNCT
ejpam-5532	234	4	η′(w	η′(w	NOUN
ejpam-5532	234	5	)	)	PUNCT
ejpam-5532	234	6	≥	≥	NOUN
ejpam-5532	234	7	∨{s	∨{s	NOUN
ejpam-5532	234	8	∧	∧	PROPN
ejpam-5532	234	9	t	t	NOUN
ejpam-5532	234	10	∧	∧	PROPN
ejpam-5532	234	11	r	r	NOUN
ejpam-5532	234	12	/	/	SYM
ejpam-5532	234	13	r	r	NOUN
ejpam-5532	234	14	∈	∈	NOUN
ejpam-5532	234	15	lµ(w	lµ(w	X
ejpam-5532	234	16	)	)	PUNCT
ejpam-5532	234	17	,	,	PUNCT
ejpam-5532	234	18	s	s	PROPN
ejpam-5532	234	19	∈	∈	PROPN
ejpam-5532	234	20	lη(x	lη(x	X
ejpam-5532	234	21	)	)	PUNCT
ejpam-5532	234	22	and	and	CCONJ
ejpam-5532	234	23	t	t	PROPN
ejpam-5532	234	24	∈	∈	PROPN
ejpam-5532	234	25	lη(y	lη(y	PUNCT
ejpam-5532	234	26	)	)	PUNCT
ejpam-5532	234	27	}	}	PUNCT
ejpam-5532	234	28	=	=	PRON
ejpam-5532	234	29	{	{	PUNCT
ejpam-5532	234	30	∨{s	∨{s	NOUN
ejpam-5532	234	31	/	/	SYM
ejpam-5532	234	32	s	s	NOUN
ejpam-5532	234	33	∈	∈	NOUN
ejpam-5532	234	34	lη(x	lη(x	X
ejpam-5532	234	35	)	)	PUNCT
ejpam-5532	234	36	}	}	PUNCT
ejpam-5532	234	37	}	}	PUNCT
ejpam-5532	234	38	∧	∧	NOUN
ejpam-5532	234	39	{	{	PUNCT
ejpam-5532	234	40	∨{t	∨{t	PROPN
ejpam-5532	234	41	/	/	SYM
ejpam-5532	234	42	t	t	NOUN
ejpam-5532	234	43	∈	∈	PROPN
ejpam-5532	234	44	lη(y	lη(y	PUNCT
ejpam-5532	234	45	)	)	PUNCT
ejpam-5532	234	46	}	}	PUNCT
ejpam-5532	234	47	}	}	PUNCT
ejpam-5532	234	48	∧	∧	PROPN
ejpam-5532	234	49	{	{	PUNCT
ejpam-5532	234	50	∨{r	∨{r	NOUN
ejpam-5532	234	51	/	/	SYM
ejpam-5532	234	52	r	r	NOUN
ejpam-5532	234	53	∈	∈	NOUN
ejpam-5532	234	54	lµ(w	lµ(w	X
ejpam-5532	234	55	)	)	PUNCT
ejpam-5532	234	56	}	}	PUNCT
ejpam-5532	234	57	}	}	PUNCT
ejpam-5532	234	58	(	(	PUNCT
ejpam-5532	234	59	as	as	SCONJ
ejpam-5532	234	60	l	l	NOUN
ejpam-5532	234	61	is	be	AUX
ejpam-5532	234	62	a	a	DET
ejpam-5532	234	63	completely	completely	ADV
ejpam-5532	234	64	distributive	distributive	ADJ
ejpam-5532	234	65	lattice	lattice	NOUN
ejpam-5532	234	66	)	)	PUNCT
ejpam-5532	234	67	=	=	PUNCT
ejpam-5532	234	68	η′(x	η′(x	X
ejpam-5532	234	69	)	)	PUNCT
ejpam-5532	234	70	∧	∧	NOUN
ejpam-5532	234	71	η′(y	η′(y	NOUN
ejpam-5532	234	72	)	)	PUNCT
ejpam-5532	234	73	∧	∧	NOUN
ejpam-5532	234	74	µ(w	µ(w	PROPN
ejpam-5532	234	75	)	)	PUNCT
ejpam-5532	234	76	.	.	PUNCT
ejpam-5532	235	1	hence	hence	ADV
ejpam-5532	235	2	,	,	PUNCT
ejpam-5532	235	3	η′	η′	PROPN
ejpam-5532	235	4	is	be	AUX
ejpam-5532	235	5	an	an	DET
ejpam-5532	235	6	l	l	ADJ
ejpam-5532	235	7	-	-	ADJ
ejpam-5532	235	8	convex	convex	ADJ
ejpam-5532	235	9	sublattice	sublattice	NOUN
ejpam-5532	235	10	of	of	ADP
ejpam-5532	235	11	µ.	µ.	NOUN
ejpam-5532	235	12	now	now	ADV
ejpam-5532	235	13	it	it	PRON
ejpam-5532	235	14	is	be	AUX
ejpam-5532	235	15	left	leave	VERB
ejpam-5532	235	16	to	to	PART
ejpam-5532	235	17	prove	prove	VERB
ejpam-5532	235	18	that	that	SCONJ
ejpam-5532	235	19	η′	η′	PROPN
ejpam-5532	235	20	is	be	AUX
ejpam-5532	235	21	the	the	DET
ejpam-5532	235	22	smallest	small	ADJ
ejpam-5532	235	23	l	l	ADJ
ejpam-5532	235	24	-	-	ADJ
ejpam-5532	235	25	convex	convex	ADJ
ejpam-5532	235	26	sublattice	sublattice	NOUN
ejpam-5532	235	27	of	of	ADP
ejpam-5532	235	28	µ	µ	X
ejpam-5532	235	29	containing	contain	VERB
ejpam-5532	235	30	η	η	PROPN
ejpam-5532	235	31	.	.	PROPN
ejpam-5532	235	32	for	for	ADP
ejpam-5532	235	33	this	this	PRON
ejpam-5532	235	34	,	,	PUNCT
ejpam-5532	235	35	suppose	suppose	VERB
ejpam-5532	235	36	θ	θ	NOUN
ejpam-5532	235	37	is	be	AUX
ejpam-5532	235	38	an	an	DET
ejpam-5532	235	39	l	l	ADJ
ejpam-5532	235	40	-	-	ADJ
ejpam-5532	235	41	convex	convex	ADJ
ejpam-5532	235	42	sublattice	sublattice	NOUN
ejpam-5532	235	43	of	of	ADP
ejpam-5532	235	44	µ	µ	PRON
ejpam-5532	235	45	such	such	ADJ
ejpam-5532	235	46	that	that	SCONJ
ejpam-5532	235	47	η	η	PROPN
ejpam-5532	235	48	⊆	⊆	NUM
ejpam-5532	235	49	θ	θ	PROPN
ejpam-5532	235	50	.	.	PUNCT
ejpam-5532	236	1	then	then	ADV
ejpam-5532	236	2	,	,	PUNCT
ejpam-5532	236	3	ηt	ηt	ADP
ejpam-5532	236	4	⊆	⊆	NUM
ejpam-5532	236	5	θt	θt	ADJ
ejpam-5532	236	6	,	,	PUNCT
ejpam-5532	236	7	∀	∀	NOUN
ejpam-5532	236	8	t	t	NOUN
ejpam-5532	236	9	∈	∈	PROPN
ejpam-5532	236	10	l.	l.	PROPN
ejpam-5532	236	11	since	since	SCONJ
ejpam-5532	236	12	θ	θ	PROPN
ejpam-5532	236	13	is	be	AUX
ejpam-5532	236	14	an	an	DET
ejpam-5532	236	15	l	l	ADJ
ejpam-5532	236	16	-	-	ADJ
ejpam-5532	236	17	convex	convex	ADJ
ejpam-5532	236	18	sublattice	sublattice	NOUN
ejpam-5532	236	19	of	of	ADP
ejpam-5532	236	20	µ	µ	NOUN
ejpam-5532	236	21	,	,	PUNCT
ejpam-5532	236	22	therefore	therefore	ADV
ejpam-5532	236	23	by	by	ADP
ejpam-5532	236	24	theorem	theorem	NOUN
ejpam-5532	236	25	6	6	NUM
ejpam-5532	236	26	,	,	PUNCT
ejpam-5532	236	27	each	each	PRON
ejpam-5532	236	28	nonempty	nonempty	NOUN
ejpam-5532	236	29	θt	θt	PROPN
ejpam-5532	236	30	is	be	AUX
ejpam-5532	236	31	a	a	DET
ejpam-5532	236	32	convex	convex	ADJ
ejpam-5532	236	33	sublattice	sublattice	NOUN
ejpam-5532	236	34	of	of	ADP
ejpam-5532	236	35	µt	µt	PROPN
ejpam-5532	236	36	.	.	PUNCT
ejpam-5532	237	1	therefore	therefore	ADV
ejpam-5532	237	2	,	,	PUNCT
ejpam-5532	237	3	[	[	X
ejpam-5532	237	4	θt]c	θt]c	X
ejpam-5532	237	5	=	=	SYM
ejpam-5532	237	6	θt	θt	PROPN
ejpam-5532	237	7	.	.	PUNCT
ejpam-5532	238	1	this	this	PRON
ejpam-5532	238	2	implies	imply	VERB
ejpam-5532	238	3	that	that	SCONJ
ejpam-5532	239	1	[	[	X
ejpam-5532	239	2	ηt]c	ηt]c	PROPN
ejpam-5532	239	3	⊆	⊆	NUM
ejpam-5532	239	4	θt	θt	PROPN
ejpam-5532	239	5	,	,	PUNCT
ejpam-5532	239	6	∀	∀	X
ejpam-5532	239	7	t	t	PROPN
ejpam-5532	239	8	≤	≤	PROPN
ejpam-5532	239	9	a0	a0	NOUN
ejpam-5532	239	10	.	.	PUNCT
ejpam-5532	240	1	also	also	ADV
ejpam-5532	240	2	,	,	PUNCT
ejpam-5532	240	3	a0	a0	PROPN
ejpam-5532	240	4	=	=	SYM
ejpam-5532	240	5	tip{η	tip{η	PROPN
ejpam-5532	240	6	}	}	PUNCT
ejpam-5532	240	7	≤	≤	NOUN
ejpam-5532	240	8	tip{θ	tip{θ	PRON
ejpam-5532	240	9	}	}	PUNCT
ejpam-5532	240	10	.	.	PUNCT
ejpam-5532	241	1	thus	thus	ADV
ejpam-5532	241	2	,	,	PUNCT
ejpam-5532	241	3	∀	∀	X
ejpam-5532	241	4	x	x	SYM
ejpam-5532	241	5	∈	∈	NOUN
ejpam-5532	241	6	m	m	VERB
ejpam-5532	241	7	,	,	PUNCT
ejpam-5532	241	8	we	we	PRON
ejpam-5532	241	9	have	have	VERB
ejpam-5532	241	10	η′(x	η′(x	NOUN
ejpam-5532	241	11	)	)	PUNCT
ejpam-5532	241	12	=	=	PUNCT
ejpam-5532	242	1	∨	∨	NUM
ejpam-5532	242	2	t≤ao	t≤ao	PROPN
ejpam-5532	242	3	{	{	PUNCT
ejpam-5532	242	4	t	t	NOUN
ejpam-5532	242	5	:	:	PUNCT
ejpam-5532	242	6	x	x	PUNCT
ejpam-5532	242	7	∈	∈	PROPN
ejpam-5532	242	8	[	[	X
ejpam-5532	242	9	ηt]c	ηt]c	PROPN
ejpam-5532	242	10	}	}	PUNCT
ejpam-5532	242	11	≤	≤	NOUN
ejpam-5532	242	12	∨	∨	NUM
ejpam-5532	242	13	t≤tip{θ	t≤tip{θ	PROPN
ejpam-5532	242	14	}	}	PUNCT
ejpam-5532	242	15	{	{	PUNCT
ejpam-5532	242	16	t	t	NOUN
ejpam-5532	242	17	:	:	PUNCT
ejpam-5532	242	18	x	x	SYM
ejpam-5532	242	19	∈	∈	PROPN
ejpam-5532	242	20	θt	θt	NOUN
ejpam-5532	242	21	}	}	PUNCT
ejpam-5532	242	22	a.	a.	NOUN
ejpam-5532	242	23	jain	jain	PROPN
ejpam-5532	242	24	,	,	PUNCT
ejpam-5532	242	25	i.	i.	PROPN
ejpam-5532	242	26	jahan	jahan	PROPN
ejpam-5532	242	27	/	/	SYM
ejpam-5532	242	28	eur	eur	PROPN
ejpam-5532	242	29	.	.	PUNCT
ejpam-5532	243	1	j.	j.	PROPN
ejpam-5532	243	2	pure	pure	PROPN
ejpam-5532	243	3	appl	appl	PROPN
ejpam-5532	243	4	.	.	PROPN
ejpam-5532	243	5	math	math	PROPN
ejpam-5532	243	6	,	,	PUNCT
ejpam-5532	243	7	18	18	NUM
ejpam-5532	243	8	(	(	PUNCT
ejpam-5532	243	9	1	1	NUM
ejpam-5532	243	10	)	)	PUNCT
ejpam-5532	243	11	(	(	PUNCT
ejpam-5532	243	12	2025	2025	NUM
ejpam-5532	243	13	)	)	PUNCT
ejpam-5532	243	14	,	,	PUNCT
ejpam-5532	243	15	5532	5532	NUM
ejpam-5532	243	16	12	12	NUM
ejpam-5532	243	17	of	of	ADP
ejpam-5532	243	18	20	20	NUM
ejpam-5532	243	19	=	=	SYM
ejpam-5532	243	20	θ(x	θ(x	PROPN
ejpam-5532	243	21	)	)	PUNCT
ejpam-5532	243	22	.	.	PUNCT
ejpam-5532	244	1	that	that	PRON
ejpam-5532	244	2	is	be	AUX
ejpam-5532	244	3	,	,	PUNCT
ejpam-5532	244	4	η′	η′	PROPN
ejpam-5532	244	5	⊆	⊆	NUM
ejpam-5532	244	6	θ	θ	NOUN
ejpam-5532	244	7	.	.	PUNCT
ejpam-5532	245	1	hence	hence	ADV
ejpam-5532	245	2	,	,	PUNCT
ejpam-5532	245	3	η′	η′	X
ejpam-5532	245	4	=	=	PUNCT
ejpam-5532	246	1	[	[	X
ejpam-5532	246	2	η]cµ.	η]cµ.	VERB
ejpam-5532	246	3	the	the	DET
ejpam-5532	246	4	next	next	ADJ
ejpam-5532	246	5	result	result	NOUN
ejpam-5532	246	6	is	be	AUX
ejpam-5532	246	7	significant	significant	ADJ
ejpam-5532	246	8	for	for	ADP
ejpam-5532	246	9	establishing	establish	VERB
ejpam-5532	246	10	the	the	DET
ejpam-5532	246	11	unique	unique	ADJ
ejpam-5532	246	12	representation	representation	NOUN
ejpam-5532	246	13	theorem	theorem	NOUN
ejpam-5532	246	14	for	for	ADP
ejpam-5532	246	15	l	l	ADJ
ejpam-5532	246	16	-	-	ADJ
ejpam-5532	246	17	convex	convex	ADJ
ejpam-5532	246	18	sublattice	sublattice	NOUN
ejpam-5532	246	19	of	of	ADP
ejpam-5532	246	20	an	an	DET
ejpam-5532	246	21	l	l	NOUN
ejpam-5532	246	22	-	-	PUNCT
ejpam-5532	246	23	lattice	lattice	NOUN
ejpam-5532	246	24	µ.	µ.	NOUN
ejpam-5532	246	25	the	the	DET
ejpam-5532	246	26	proof	proof	NOUN
ejpam-5532	246	27	being	be	AUX
ejpam-5532	246	28	trivial	trivial	ADJ
ejpam-5532	246	29	is	be	AUX
ejpam-5532	246	30	omitted	omit	VERB
ejpam-5532	246	31	.	.	PUNCT
ejpam-5532	247	1	theorem	theorem	VERB
ejpam-5532	247	2	10	10	NUM
ejpam-5532	247	3	.	.	PUNCT
ejpam-5532	248	1	let	let	AUX
ejpam-5532	248	2	l(µ,m	l(µ,m	ADV
ejpam-5532	248	3	)	)	PUNCT
ejpam-5532	248	4	be	be	AUX
ejpam-5532	248	5	an	an	DET
ejpam-5532	248	6	l	l	NOUN
ejpam-5532	248	7	-	-	NOUN
ejpam-5532	248	8	lattice	lattice	NOUN
ejpam-5532	248	9	and	and	CCONJ
ejpam-5532	248	10	η	η	PROPN
ejpam-5532	248	11	⊆	⊆	PROPN
ejpam-5532	248	12	µ	µ	X
ejpam-5532	248	13	be	be	AUX
ejpam-5532	248	14	an	an	DET
ejpam-5532	248	15	l	l	NOUN
ejpam-5532	248	16	-	-	NOUN
ejpam-5532	248	17	ideal	ideal	ADJ
ejpam-5532	248	18	(	(	PUNCT
ejpam-5532	248	19	l	l	ADJ
ejpam-5532	248	20	-	-	ADJ
ejpam-5532	248	21	dual	dual	ADJ
ejpam-5532	248	22	ideal	ideal	NOUN
ejpam-5532	248	23	)	)	PUNCT
ejpam-5532	248	24	of	of	ADP
ejpam-5532	248	25	µ.	µ.	PROPN
ejpam-5532	248	26	then	then	ADV
ejpam-5532	248	27	,	,	PUNCT
ejpam-5532	248	28	η	η	PROPN
ejpam-5532	248	29	is	be	AUX
ejpam-5532	248	30	an	an	DET
ejpam-5532	248	31	l	l	ADJ
ejpam-5532	248	32	-	-	ADJ
ejpam-5532	248	33	convex	convex	ADJ
ejpam-5532	248	34	sublattice	sublattice	NOUN
ejpam-5532	248	35	of	of	ADP
ejpam-5532	248	36	µ.	µ.	NOUN
ejpam-5532	248	37	before	before	ADP
ejpam-5532	248	38	discussing	discuss	VERB
ejpam-5532	248	39	the	the	DET
ejpam-5532	248	40	unique	unique	ADJ
ejpam-5532	248	41	representation	representation	NOUN
ejpam-5532	248	42	theorem	theorem	NOUN
ejpam-5532	248	43	for	for	ADP
ejpam-5532	248	44	l	l	ADJ
ejpam-5532	248	45	-	-	ADJ
ejpam-5532	248	46	convex	convex	ADJ
ejpam-5532	248	47	sublattices	sublattice	NOUN
ejpam-5532	248	48	in	in	ADP
ejpam-5532	248	49	an	an	DET
ejpam-5532	248	50	llattice	llattice	NOUN
ejpam-5532	248	51	µ	µ	NOUN
ejpam-5532	248	52	,	,	PUNCT
ejpam-5532	248	53	we	we	PRON
ejpam-5532	248	54	provide	provide	VERB
ejpam-5532	248	55	the	the	DET
ejpam-5532	248	56	structural	structural	ADJ
ejpam-5532	248	57	composition	composition	NOUN
ejpam-5532	248	58	of	of	ADP
ejpam-5532	248	59	an	an	DET
ejpam-5532	248	60	l	l	NOUN
ejpam-5532	248	61	-	-	NOUN
ejpam-5532	248	62	ideal	ideal	NOUN
ejpam-5532	248	63	of	of	ADP
ejpam-5532	248	64	µ	µ	PRON
ejpam-5532	248	65	generated	generate	VERB
ejpam-5532	248	66	by	by	ADP
ejpam-5532	248	67	an	an	DET
ejpam-5532	248	68	l	l	NOUN
ejpam-5532	248	69	-	-	PUNCT
ejpam-5532	248	70	subset	subset	VERB
ejpam-5532	248	71	η	η	PROPN
ejpam-5532	248	72	of	of	ADP
ejpam-5532	248	73	µ	µ	NUM
ejpam-5532	248	74	in	in	ADP
ejpam-5532	248	75	terms	term	NOUN
ejpam-5532	248	76	of	of	ADP
ejpam-5532	248	77	strong	strong	ADJ
ejpam-5532	248	78	level	level	NOUN
ejpam-5532	248	79	subsets	subset	NOUN
ejpam-5532	248	80	of	of	ADP
ejpam-5532	248	81	η	η	PROPN
ejpam-5532	248	82	.	.	PUNCT
ejpam-5532	248	83	the	the	DET
ejpam-5532	248	84	following	following	ADJ
ejpam-5532	248	85	result	result	NOUN
ejpam-5532	248	86	is	be	AUX
ejpam-5532	248	87	proved	prove	VERB
ejpam-5532	248	88	by	by	ADP
ejpam-5532	248	89	taking	take	VERB
ejpam-5532	248	90	l	l	NOUN
ejpam-5532	248	91	to	to	PART
ejpam-5532	248	92	be	be	AUX
ejpam-5532	248	93	a	a	DET
ejpam-5532	248	94	dense	dense	ADJ
ejpam-5532	248	95	chain	chain	NOUN
ejpam-5532	248	96	.	.	PUNCT
ejpam-5532	249	1	note	note	VERB
ejpam-5532	249	2	that	that	SCONJ
ejpam-5532	249	3	a	a	DET
ejpam-5532	249	4	dense	dense	ADJ
ejpam-5532	249	5	chain	chain	NOUN
ejpam-5532	249	6	is	be	AUX
ejpam-5532	249	7	a	a	DET
ejpam-5532	249	8	completely	completely	ADV
ejpam-5532	249	9	distributive	distributive	ADJ
ejpam-5532	249	10	lattice	lattice	NOUN
ejpam-5532	249	11	.	.	PUNCT
ejpam-5532	250	1	theorem	theorem	NOUN
ejpam-5532	250	2	11	11	NUM
ejpam-5532	250	3	.	.	PUNCT
ejpam-5532	251	1	let	let	VERB
ejpam-5532	251	2	l	l	NOUN
ejpam-5532	251	3	be	be	AUX
ejpam-5532	251	4	a	a	DET
ejpam-5532	251	5	dense	dense	ADJ
ejpam-5532	251	6	chain	chain	NOUN
ejpam-5532	251	7	and	and	CCONJ
ejpam-5532	251	8	l(µ,m	l(µ,m	ADV
ejpam-5532	251	9	)	)	PUNCT
ejpam-5532	251	10	be	be	AUX
ejpam-5532	251	11	an	an	DET
ejpam-5532	251	12	l	l	NOUN
ejpam-5532	251	13	-	-	NOUN
ejpam-5532	251	14	lattice	lattice	NOUN
ejpam-5532	251	15	.	.	PUNCT
ejpam-5532	252	1	let	let	VERB
ejpam-5532	252	2	η	η	PROPN
ejpam-5532	252	3	∈	∈	PROPN
ejpam-5532	252	4	lm	lm	INTJ
ejpam-5532	252	5	,	,	PUNCT
ejpam-5532	252	6	η	η	PROPN
ejpam-5532	252	7	⊆	⊆	PROPN
ejpam-5532	252	8	µ	µ	X
ejpam-5532	252	9	and	and	CCONJ
ejpam-5532	252	10	a0	a0	PROPN
ejpam-5532	252	11	=	=	SYM
ejpam-5532	252	12	tip{η	tip{η	PROPN
ejpam-5532	252	13	}	}	PUNCT
ejpam-5532	252	14	.	.	PUNCT
ejpam-5532	253	1	define	define	VERB
ejpam-5532	253	2	an	an	DET
ejpam-5532	253	3	l	l	NOUN
ejpam-5532	253	4	-	-	ADJ
ejpam-5532	253	5	set	set	VERB
ejpam-5532	253	6	η̂	η̂	PROPN
ejpam-5532	253	7	of	of	ADP
ejpam-5532	253	8	m	m	PRON
ejpam-5532	253	9	as	as	SCONJ
ejpam-5532	253	10	follows	follow	VERB
ejpam-5532	253	11	:	:	PUNCT
ejpam-5532	253	12	η̂(x	η̂(x	NUM
ejpam-5532	253	13	)	)	PUNCT
ejpam-5532	253	14	=	=	SYM
ejpam-5532	253	15	∨t	∨t	NOUN
ejpam-5532	253	16	<	<	X
ejpam-5532	253	17	a0{t	a0{t	X
ejpam-5532	253	18	:	:	PUNCT
ejpam-5532	253	19	x	x	SYM
ejpam-5532	253	20	∈	∈	PROPN
ejpam-5532	253	21	(	(	PUNCT
ejpam-5532	253	22	η	η	X
ejpam-5532	253	23	>	>	PROPN
ejpam-5532	253	24	t	t	PROPN
ejpam-5532	253	25	]	]	PUNCT
ejpam-5532	253	26	}	}	PUNCT
ejpam-5532	253	27	,	,	PUNCT
ejpam-5532	253	28	where	where	SCONJ
ejpam-5532	253	29	(	(	PUNCT
ejpam-5532	253	30	η	η	X
ejpam-5532	253	31	>	>	X
ejpam-5532	253	32	t	t	PROPN
ejpam-5532	253	33	]	]	PUNCT
ejpam-5532	253	34	is	be	AUX
ejpam-5532	253	35	an	an	DET
ejpam-5532	253	36	ideal	ideal	NOUN
ejpam-5532	253	37	of	of	ADP
ejpam-5532	253	38	µ	µ	PROPN
ejpam-5532	253	39	>	>	X
ejpam-5532	253	40	t	t	PROPN
ejpam-5532	253	41	generated	generate	VERB
ejpam-5532	253	42	by	by	ADP
ejpam-5532	253	43	η	η	PROPN
ejpam-5532	253	44	>	>	PROPN
ejpam-5532	253	45	t	t	PROPN
ejpam-5532	253	46	.	.	PUNCT
ejpam-5532	254	1	then	then	ADV
ejpam-5532	254	2	,	,	PUNCT
ejpam-5532	254	3	η̂	η̂	PROPN
ejpam-5532	254	4	=	=	SYM
ejpam-5532	254	5	(	(	PUNCT
ejpam-5532	254	6	η]µ.	η]µ.	X
ejpam-5532	254	7	proof	proof	NOUN
ejpam-5532	254	8	.	.	PUNCT
ejpam-5532	255	1	since	since	SCONJ
ejpam-5532	255	2	η	η	PROPN
ejpam-5532	255	3	⊆	⊆	PROPN
ejpam-5532	255	4	µ	µ	NUM
ejpam-5532	255	5	,	,	PUNCT
ejpam-5532	255	6	η	η	PROPN
ejpam-5532	255	7	>	>	PROPN
ejpam-5532	255	8	t	t	PROPN
ejpam-5532	255	9	⊆	⊆	NUM
ejpam-5532	255	10	µ	µ	X
ejpam-5532	255	11	>	>	X
ejpam-5532	255	12	t	t	PROPN
ejpam-5532	255	13	,	,	PUNCT
ejpam-5532	255	14	∀	∀	X
ejpam-5532	255	15	t	t	NOUN
ejpam-5532	255	16	∈	∈	PROPN
ejpam-5532	255	17	l.	l.	PROPN
ejpam-5532	255	18	as	as	ADP
ejpam-5532	255	19	µ	µ	PROPN
ejpam-5532	255	20	is	be	AUX
ejpam-5532	255	21	an	an	DET
ejpam-5532	255	22	l	l	NOUN
ejpam-5532	255	23	-	-	NOUN
ejpam-5532	255	24	lattice	lattice	NOUN
ejpam-5532	255	25	,	,	PUNCT
ejpam-5532	255	26	µ	µ	X
ejpam-5532	255	27	>	>	X
ejpam-5532	255	28	t	t	PROPN
ejpam-5532	255	29	is	be	AUX
ejpam-5532	255	30	a	a	DET
ejpam-5532	255	31	sublattice	sublattice	NOUN
ejpam-5532	255	32	of	of	ADP
ejpam-5532	255	33	m	m	PRON
ejpam-5532	255	34	,	,	PUNCT
ejpam-5532	255	35	∀	∀	X
ejpam-5532	255	36	t	t	NOUN
ejpam-5532	255	37	<	<	X
ejpam-5532	255	38	tip{µ	tip{µ	NOUN
ejpam-5532	255	39	}	}	PUNCT
ejpam-5532	255	40	.	.	PUNCT
ejpam-5532	256	1	also	also	ADV
ejpam-5532	256	2	,	,	PUNCT
ejpam-5532	256	3	as	as	ADP
ejpam-5532	256	4	(	(	PUNCT
ejpam-5532	256	5	η	η	X
ejpam-5532	256	6	>	>	PROPN
ejpam-5532	256	7	t	t	PROPN
ejpam-5532	256	8	]	]	PUNCT
ejpam-5532	256	9	is	be	AUX
ejpam-5532	256	10	an	an	DET
ejpam-5532	256	11	ideal	ideal	NOUN
ejpam-5532	256	12	of	of	ADP
ejpam-5532	256	13	µ	µ	PROPN
ejpam-5532	256	14	>	>	X
ejpam-5532	256	15	t	t	PROPN
ejpam-5532	256	16	generated	generate	VERB
ejpam-5532	256	17	by	by	ADP
ejpam-5532	256	18	η	η	PROPN
ejpam-5532	256	19	>	>	PROPN
ejpam-5532	256	20	t	t	PROPN
ejpam-5532	256	21	,	,	PUNCT
ejpam-5532	256	22	we	we	PRON
ejpam-5532	256	23	have	have	VERB
ejpam-5532	256	24	(	(	PUNCT
ejpam-5532	256	25	η	η	X
ejpam-5532	256	26	>	>	X
ejpam-5532	256	27	t	t	X
ejpam-5532	256	28	]	]	PUNCT
ejpam-5532	256	29	⊆	⊆	NUM
ejpam-5532	256	30	µ	µ	X
ejpam-5532	256	31	>	>	X
ejpam-5532	256	32	t	t	PROPN
ejpam-5532	256	33	,	,	PUNCT
ejpam-5532	256	34	∀	∀	X
ejpam-5532	256	35	t	t	PROPN
ejpam-5532	256	36	<	<	X
ejpam-5532	256	37	a0	a0	PROPN
ejpam-5532	256	38	.	.	PUNCT
ejpam-5532	257	1	thus	thus	ADV
ejpam-5532	257	2	,	,	PUNCT
ejpam-5532	257	3	η̂(x	η̂(x	NUM
ejpam-5532	257	4	)	)	PUNCT
ejpam-5532	257	5	=	=	SYM
ejpam-5532	257	6	∨t	∨t	NOUN
ejpam-5532	257	7	<	<	X
ejpam-5532	257	8	a0{t	a0{t	X
ejpam-5532	257	9	:	:	PUNCT
ejpam-5532	257	10	x	x	SYM
ejpam-5532	257	11	∈	∈	PROPN
ejpam-5532	257	12	(	(	PUNCT
ejpam-5532	257	13	η	η	X
ejpam-5532	257	14	>	>	PROPN
ejpam-5532	257	15	t	t	PROPN
ejpam-5532	257	16	]	]	PUNCT
ejpam-5532	257	17	}	}	PUNCT
ejpam-5532	257	18	≤	≤	NUM
ejpam-5532	257	19	∨t	∨t	NOUN
ejpam-5532	257	20	<	<	X
ejpam-5532	257	21	tip{µ}{t	tip{µ}{t	NUM
ejpam-5532	257	22	:	:	PUNCT
ejpam-5532	257	23	x	x	SYM
ejpam-5532	257	24	∈	∈	PROPN
ejpam-5532	257	25	µ	µ	X
ejpam-5532	257	26	>	>	X
ejpam-5532	257	27	t	t	PROPN
ejpam-5532	257	28	}	}	PUNCT
ejpam-5532	257	29	≤	≤	NOUN
ejpam-5532	257	30	µ(x	µ(x	NOUN
ejpam-5532	257	31	)	)	PUNCT
ejpam-5532	257	32	.	.	PUNCT
ejpam-5532	258	1	we	we	PRON
ejpam-5532	258	2	thus	thus	ADV
ejpam-5532	258	3	have	have	VERB
ejpam-5532	258	4	η̂	η̂	NUM
ejpam-5532	258	5	⊆	⊆	NUM
ejpam-5532	258	6	µ.	µ.	NOUN
ejpam-5532	258	7	to	to	PART
ejpam-5532	258	8	establish	establish	VERB
ejpam-5532	258	9	that	that	SCONJ
ejpam-5532	258	10	η	η	PROPN
ejpam-5532	258	11	⊆	⊆	X
ejpam-5532	258	12	η̂	η̂	NUM
ejpam-5532	258	13	,	,	PUNCT
ejpam-5532	258	14	we	we	PRON
ejpam-5532	258	15	prove	prove	VERB
ejpam-5532	258	16	that	that	SCONJ
ejpam-5532	258	17	η	η	PROPN
ejpam-5532	258	18	>	>	X
ejpam-5532	258	19	α	α	PRON
ejpam-5532	258	20	⊆	⊆	NUM
ejpam-5532	258	21	(	(	PUNCT
ejpam-5532	258	22	η̂)>α	η̂)>α	NOUN
ejpam-5532	258	23	,	,	PUNCT
ejpam-5532	258	24	∀	∀	X
ejpam-5532	258	25	α	α	PRON
ejpam-5532	258	26	∈	∈	PROPN
ejpam-5532	258	27	l.	l.	NOUN
ejpam-5532	258	28	let	let	VERB
ejpam-5532	258	29	α	α	PRON
ejpam-5532	258	30	∈	∈	PROPN
ejpam-5532	258	31	l	l	NOUN
ejpam-5532	258	32	and	and	CCONJ
ejpam-5532	258	33	x	x	PROPN
ejpam-5532	258	34	∈	∈	PROPN
ejpam-5532	258	35	η	η	PROPN
ejpam-5532	258	36	>	>	PROPN
ejpam-5532	258	37	α	α	PROPN
ejpam-5532	258	38	.	.	PUNCT
ejpam-5532	259	1	then	then	ADV
ejpam-5532	259	2	,	,	PUNCT
ejpam-5532	259	3	η(x	η(x	X
ejpam-5532	259	4	)	)	PUNCT
ejpam-5532	259	5	>	>	X
ejpam-5532	259	6	α	α	X
ejpam-5532	259	7	.	.	PUNCT
ejpam-5532	260	1	since	since	SCONJ
ejpam-5532	260	2	l	l	NOUN
ejpam-5532	260	3	is	be	AUX
ejpam-5532	260	4	a	a	DET
ejpam-5532	260	5	dense	dense	ADJ
ejpam-5532	260	6	chain	chain	NOUN
ejpam-5532	260	7	,	,	PUNCT
ejpam-5532	260	8	∃	∃	PROPN
ejpam-5532	260	9	β	β	X
ejpam-5532	260	10	∈	∈	PROPN
ejpam-5532	260	11	l	l	NOUN
ejpam-5532	260	12	such	such	ADJ
ejpam-5532	260	13	that	that	PRON
ejpam-5532	260	14	η(x	η(x	NOUN
ejpam-5532	260	15	)	)	PUNCT
ejpam-5532	260	16	>	>	X
ejpam-5532	260	17	β	β	X
ejpam-5532	260	18	>	>	X
ejpam-5532	260	19	α	α	X
ejpam-5532	260	20	.	.	PUNCT
ejpam-5532	261	1	this	this	PRON
ejpam-5532	261	2	implies	imply	VERB
ejpam-5532	261	3	x	x	X
ejpam-5532	261	4	∈	∈	PROPN
ejpam-5532	261	5	η	η	PROPN
ejpam-5532	261	6	>	>	NOUN
ejpam-5532	261	7	β	β	X
ejpam-5532	261	8	and	and	CCONJ
ejpam-5532	261	9	hence	hence	ADV
ejpam-5532	261	10	x	x	X
ejpam-5532	261	11	∈	∈	PROPN
ejpam-5532	261	12	(	(	PUNCT
ejpam-5532	261	13	η	η	X
ejpam-5532	261	14	>	>	X
ejpam-5532	261	15	β	β	X
ejpam-5532	261	16	]	]	X
ejpam-5532	261	17	.	.	PUNCT
ejpam-5532	262	1	consequently	consequently	ADV
ejpam-5532	262	2	,	,	PUNCT
ejpam-5532	262	3	η̂(x	η̂(x	NUM
ejpam-5532	262	4	)	)	PUNCT
ejpam-5532	262	5	=	=	SYM
ejpam-5532	262	6	∨t	∨t	NOUN
ejpam-5532	262	7	<	<	X
ejpam-5532	262	8	a0{t	a0{t	X
ejpam-5532	262	9	:	:	PUNCT
ejpam-5532	262	10	x	x	SYM
ejpam-5532	262	11	∈	∈	PROPN
ejpam-5532	262	12	(	(	PUNCT
ejpam-5532	262	13	η	η	X
ejpam-5532	262	14	>	>	PROPN
ejpam-5532	262	15	t	t	PROPN
ejpam-5532	262	16	]	]	PUNCT
ejpam-5532	262	17	}	}	PUNCT
ejpam-5532	262	18	≥	≥	X
ejpam-5532	262	19	β	β	X
ejpam-5532	262	20	>	>	X
ejpam-5532	262	21	α	α	X
ejpam-5532	262	22	.	.	PUNCT
ejpam-5532	263	1	that	that	PRON
ejpam-5532	263	2	is	be	AUX
ejpam-5532	263	3	,	,	PUNCT
ejpam-5532	263	4	x	x	SYM
ejpam-5532	263	5	∈	∈	PROPN
ejpam-5532	263	6	(	(	PUNCT
ejpam-5532	263	7	η̂)>α	η̂)>α	NOUN
ejpam-5532	263	8	.	.	PUNCT
ejpam-5532	264	1	this	this	PRON
ejpam-5532	264	2	proves	prove	VERB
ejpam-5532	264	3	that	that	SCONJ
ejpam-5532	264	4	η	η	PROPN
ejpam-5532	264	5	⊆	⊆	NUM
ejpam-5532	264	6	η̂.	η̂.	NOUN
ejpam-5532	264	7	we	we	PRON
ejpam-5532	264	8	now	now	ADV
ejpam-5532	264	9	prove	prove	VERB
ejpam-5532	264	10	that	that	SCONJ
ejpam-5532	264	11	η̂	η̂	PUNCT
ejpam-5532	264	12	is	be	AUX
ejpam-5532	264	13	an	an	DET
ejpam-5532	264	14	l	l	NOUN
ejpam-5532	264	15	-	-	NOUN
ejpam-5532	264	16	ideal	ideal	NOUN
ejpam-5532	264	17	of	of	ADP
ejpam-5532	264	18	µ.	µ.	NOUN
ejpam-5532	264	19	for	for	ADP
ejpam-5532	264	20	any	any	DET
ejpam-5532	264	21	z	z	NOUN
ejpam-5532	264	22	∈	∈	PROPN
ejpam-5532	264	23	m	m	VERB
ejpam-5532	264	24	,	,	PUNCT
ejpam-5532	264	25	define	define	VERB
ejpam-5532	264	26	a	a	DET
ejpam-5532	264	27	set	set	NOUN
ejpam-5532	264	28	lη(z	lη(z	NOUN
ejpam-5532	264	29	)	)	PUNCT
ejpam-5532	264	30	=	=	PRON
ejpam-5532	264	31	{	{	PUNCT
ejpam-5532	264	32	t	t	NOUN
ejpam-5532	264	33	∈	∈	PROPN
ejpam-5532	264	34	l	l	PROPN
ejpam-5532	264	35	/	/	SYM
ejpam-5532	264	36	t	t	PROPN
ejpam-5532	264	37	<	<	X
ejpam-5532	264	38	a0	a0	PROPN
ejpam-5532	264	39	,	,	PUNCT
ejpam-5532	264	40	z	z	PROPN
ejpam-5532	264	41	∈	∈	PROPN
ejpam-5532	264	42	(	(	PUNCT
ejpam-5532	264	43	η	η	X
ejpam-5532	264	44	>	>	PROPN
ejpam-5532	264	45	t	t	PROPN
ejpam-5532	264	46	]	]	PUNCT
ejpam-5532	264	47	}	}	PUNCT
ejpam-5532	264	48	.	.	PUNCT
ejpam-5532	265	1	then	then	ADV
ejpam-5532	265	2	,	,	PUNCT
ejpam-5532	265	3	η̂(x	η̂(x	NUM
ejpam-5532	265	4	)	)	PUNCT
ejpam-5532	265	5	=	=	SYM
ejpam-5532	265	6	∨	∨	NUM
ejpam-5532	265	7	lη(x	lη(x	NUM
ejpam-5532	265	8	)	)	PUNCT
ejpam-5532	265	9	.	.	PUNCT
ejpam-5532	266	1	let	let	VERB
ejpam-5532	266	2	x	x	PRON
ejpam-5532	266	3	,	,	PUNCT
ejpam-5532	266	4	y	y	PROPN
ejpam-5532	266	5	∈	∈	PROPN
ejpam-5532	266	6	m	m	VERB
ejpam-5532	266	7	.	.	PUNCT
ejpam-5532	267	1	we	we	PRON
ejpam-5532	267	2	claim	claim	VERB
ejpam-5532	267	3	that	that	SCONJ
ejpam-5532	267	4	for	for	ADP
ejpam-5532	267	5	any	any	DET
ejpam-5532	267	6	a	a	DET
ejpam-5532	267	7	∈	∈	NOUN
ejpam-5532	267	8	lη(x	lη(x	X
ejpam-5532	267	9	)	)	PUNCT
ejpam-5532	267	10	and	and	CCONJ
ejpam-5532	267	11	b	b	X
ejpam-5532	267	12	∈	∈	PROPN
ejpam-5532	267	13	lη(y	lη(y	PUNCT
ejpam-5532	267	14	)	)	PUNCT
ejpam-5532	267	15	,	,	PUNCT
ejpam-5532	267	16	a	a	DET
ejpam-5532	267	17	∧	∧	PROPN
ejpam-5532	267	18	b	b	PROPN
ejpam-5532	267	19	∈	∈	PROPN
ejpam-5532	267	20	lη(x	lη(x	X
ejpam-5532	267	21	∨	∨	PROPN
ejpam-5532	267	22	y	y	NOUN
ejpam-5532	267	23	)	)	PUNCT
ejpam-5532	267	24	.	.	PUNCT
ejpam-5532	268	1	suppose	suppose	VERB
ejpam-5532	268	2	,	,	PUNCT
ejpam-5532	268	3	a	a	DET
ejpam-5532	268	4	∈	∈	NOUN
ejpam-5532	268	5	lη(x	lη(x	X
ejpam-5532	268	6	)	)	PUNCT
ejpam-5532	268	7	and	and	CCONJ
ejpam-5532	268	8	b	b	X
ejpam-5532	268	9	∈	∈	PROPN
ejpam-5532	268	10	lη(y	lη(y	PUNCT
ejpam-5532	268	11	)	)	PUNCT
ejpam-5532	268	12	.	.	PUNCT
ejpam-5532	269	1	then	then	ADV
ejpam-5532	269	2	,	,	PUNCT
ejpam-5532	269	3	a	a	DET
ejpam-5532	269	4	<	<	X
ejpam-5532	269	5	a0	a0	PROPN
ejpam-5532	269	6	,	,	PUNCT
ejpam-5532	269	7	b	b	X
ejpam-5532	269	8	<	<	X
ejpam-5532	269	9	b0	b0	NOUN
ejpam-5532	269	10	,	,	PUNCT
ejpam-5532	269	11	x	x	SYM
ejpam-5532	269	12	∈	∈	PROPN
ejpam-5532	269	13	(	(	PUNCT
ejpam-5532	269	14	η	η	X
ejpam-5532	269	15	>	>	X
ejpam-5532	269	16	a	a	PRON
ejpam-5532	269	17	]	]	X
ejpam-5532	269	18	,	,	PUNCT
ejpam-5532	269	19	y	y	PROPN
ejpam-5532	269	20	∈	∈	PROPN
ejpam-5532	269	21	(	(	PUNCT
ejpam-5532	269	22	η	η	X
ejpam-5532	269	23	>	>	X
ejpam-5532	269	24	b	b	X
ejpam-5532	269	25	]	]	PUNCT
ejpam-5532	269	26	.	.	PUNCT
ejpam-5532	270	1	a.	a.	PROPN
ejpam-5532	270	2	jain	jain	PROPN
ejpam-5532	270	3	,	,	PUNCT
ejpam-5532	270	4	i.	i.	PROPN
ejpam-5532	270	5	jahan	jahan	PROPN
ejpam-5532	270	6	/	/	SYM
ejpam-5532	270	7	eur	eur	PROPN
ejpam-5532	270	8	.	.	PUNCT
ejpam-5532	271	1	j.	j.	PROPN
ejpam-5532	271	2	pure	pure	PROPN
ejpam-5532	271	3	appl	appl	PROPN
ejpam-5532	271	4	.	.	PROPN
ejpam-5532	271	5	math	math	PROPN
ejpam-5532	271	6	,	,	PUNCT
ejpam-5532	271	7	18	18	NUM
ejpam-5532	271	8	(	(	PUNCT
ejpam-5532	271	9	1	1	NUM
ejpam-5532	271	10	)	)	PUNCT
ejpam-5532	271	11	(	(	PUNCT
ejpam-5532	271	12	2025	2025	NUM
ejpam-5532	271	13	)	)	PUNCT
ejpam-5532	271	14	,	,	PUNCT
ejpam-5532	271	15	5532	5532	NUM
ejpam-5532	271	16	13	13	NUM
ejpam-5532	271	17	of	of	ADP
ejpam-5532	271	18	20	20	NUM
ejpam-5532	271	19	since	since	SCONJ
ejpam-5532	271	20	l	l	NOUN
ejpam-5532	271	21	is	be	AUX
ejpam-5532	271	22	a	a	DET
ejpam-5532	271	23	chain	chain	NOUN
ejpam-5532	271	24	,	,	PUNCT
ejpam-5532	271	25	a	a	DET
ejpam-5532	271	26	∧	∧	PROPN
ejpam-5532	271	27	b	b	PROPN
ejpam-5532	271	28	<	<	X
ejpam-5532	271	29	a0	a0	PROPN
ejpam-5532	271	30	.	.	PUNCT
ejpam-5532	272	1	now	now	ADV
ejpam-5532	272	2	,	,	PUNCT
ejpam-5532	272	3	x	x	SYM
ejpam-5532	272	4	∈	∈	PROPN
ejpam-5532	272	5	(	(	PUNCT
ejpam-5532	272	6	η	η	X
ejpam-5532	272	7	>	>	X
ejpam-5532	272	8	a	a	PRON
ejpam-5532	272	9	]	]	X
ejpam-5532	272	10	,	,	PUNCT
ejpam-5532	272	11	y	y	PROPN
ejpam-5532	272	12	∈	∈	PROPN
ejpam-5532	272	13	(	(	PUNCT
ejpam-5532	272	14	η	η	X
ejpam-5532	272	15	>	>	X
ejpam-5532	272	16	b	b	X
ejpam-5532	272	17	]	]	PUNCT
ejpam-5532	272	18	implies	imply	VERB
ejpam-5532	272	19	that	that	SCONJ
ejpam-5532	272	20	x	x	SYM
ejpam-5532	272	21	≤	≤	NUM
ejpam-5532	272	22	x1	x1	PRON
ejpam-5532	272	23	∨	∨	NOUN
ejpam-5532	272	24	.	.	PUNCT
ejpam-5532	272	25	.	.	PUNCT
ejpam-5532	272	26	.	.	PUNCT
ejpam-5532	273	1	∨	∨	NUM
ejpam-5532	273	2	xn	xn	PROPN
ejpam-5532	273	3	,	,	PUNCT
ejpam-5532	273	4	xi	xi	PROPN
ejpam-5532	273	5	∈	∈	PROPN
ejpam-5532	273	6	η	η	PROPN
ejpam-5532	273	7	>	>	PROPN
ejpam-5532	273	8	a	a	PRON
ejpam-5532	273	9	,	,	PUNCT
ejpam-5532	273	10	∀	∀	PUNCT
ejpam-5532	273	11	i	i	NOUN
ejpam-5532	273	12	;	;	PUNCT
ejpam-5532	273	13	and	and	CCONJ
ejpam-5532	273	14	y	y	PROPN
ejpam-5532	273	15	≤	≤	PROPN
ejpam-5532	273	16	y1	y1	NOUN
ejpam-5532	273	17	∨	∨	NOUN
ejpam-5532	273	18	.	.	PUNCT
ejpam-5532	273	19	.	.	PUNCT
ejpam-5532	273	20	.	.	PUNCT
ejpam-5532	274	1	∨	∨	NUM
ejpam-5532	274	2	ym	ym	PROPN
ejpam-5532	274	3	,	,	PUNCT
ejpam-5532	274	4	yj	yj	PROPN
ejpam-5532	274	5	∈	∈	PROPN
ejpam-5532	274	6	η	η	PROPN
ejpam-5532	274	7	>	>	PROPN
ejpam-5532	274	8	b	b	PROPN
ejpam-5532	274	9	,	,	PUNCT
ejpam-5532	274	10	∀	∀	X
ejpam-5532	274	11	j.	j.	PROPN
ejpam-5532	274	12	then	then	ADV
ejpam-5532	274	13	,	,	PUNCT
ejpam-5532	274	14	we	we	PRON
ejpam-5532	274	15	have	have	VERB
ejpam-5532	274	16	x	x	NOUN
ejpam-5532	274	17	∨	∨	NUM
ejpam-5532	274	18	y	y	PROPN
ejpam-5532	274	19	≤	≤	NUM
ejpam-5532	274	20	(	(	PUNCT
ejpam-5532	274	21	∨xi	∨xi	NOUN
ejpam-5532	274	22	)	)	PUNCT
ejpam-5532	274	23	∨	∨	NOUN
ejpam-5532	274	24	(	(	PUNCT
ejpam-5532	274	25	∨yj	∨yj	X
ejpam-5532	274	26	)	)	PUNCT
ejpam-5532	274	27	,	,	PUNCT
ejpam-5532	274	28	which	which	PRON
ejpam-5532	274	29	is	be	AUX
ejpam-5532	274	30	a	a	DET
ejpam-5532	274	31	finite	finite	ADJ
ejpam-5532	274	32	join	join	NOUN
ejpam-5532	274	33	of	of	ADP
ejpam-5532	274	34	elements	element	NOUN
ejpam-5532	274	35	of	of	ADP
ejpam-5532	274	36	η	η	PROPN
ejpam-5532	274	37	>	>	X
ejpam-5532	274	38	a	a	PRON
ejpam-5532	274	39	∪	∪	X
ejpam-5532	274	40	η	η	PROPN
ejpam-5532	274	41	>	>	X
ejpam-5532	274	42	b	b	PROPN
ejpam-5532	274	43	and	and	CCONJ
ejpam-5532	274	44	η	η	PROPN
ejpam-5532	274	45	>	>	X
ejpam-5532	274	46	a	a	DET
ejpam-5532	274	47	∪	∪	X
ejpam-5532	274	48	η	η	PROPN
ejpam-5532	274	49	>	>	X
ejpam-5532	274	50	b	b	PROPN
ejpam-5532	274	51	⊆	⊆	NUM
ejpam-5532	274	52	η	η	X
ejpam-5532	274	53	>	>	PROPN
ejpam-5532	274	54	a∧b	a∧b	PROPN
ejpam-5532	274	55	.	.	PUNCT
ejpam-5532	275	1	therefore	therefore	ADV
ejpam-5532	275	2	,	,	PUNCT
ejpam-5532	275	3	x	x	PROPN
ejpam-5532	275	4	∨	∨	NUM
ejpam-5532	275	5	y	y	PROPN
ejpam-5532	275	6	∈	∈	PROPN
ejpam-5532	275	7	(	(	PUNCT
ejpam-5532	275	8	η	η	X
ejpam-5532	275	9	>	>	X
ejpam-5532	275	10	a∧b	a∧b	PROPN
ejpam-5532	275	11	]	]	X
ejpam-5532	275	12	and	and	CCONJ
ejpam-5532	275	13	a	a	DET
ejpam-5532	275	14	∧	∧	PROPN
ejpam-5532	275	15	b	b	PROPN
ejpam-5532	275	16	<	<	X
ejpam-5532	275	17	a0	a0	PROPN
ejpam-5532	275	18	.	.	PUNCT
ejpam-5532	276	1	thus	thus	ADV
ejpam-5532	276	2	,	,	PUNCT
ejpam-5532	276	3	a	a	DET
ejpam-5532	276	4	∧	∧	PROPN
ejpam-5532	276	5	b	b	PROPN
ejpam-5532	276	6	∈	∈	PROPN
ejpam-5532	276	7	lη(x	lη(x	X
ejpam-5532	276	8	∨	∨	PROPN
ejpam-5532	276	9	y	y	NOUN
ejpam-5532	276	10	)	)	PUNCT
ejpam-5532	276	11	.	.	PUNCT
ejpam-5532	277	1	consequently	consequently	ADV
ejpam-5532	277	2	,	,	PUNCT
ejpam-5532	277	3	η̂(x	η̂(x	NUM
ejpam-5532	277	4	∨	∨	NUM
ejpam-5532	277	5	y	y	PROPN
ejpam-5532	277	6	)	)	PUNCT
ejpam-5532	277	7	≥	≥	NOUN
ejpam-5532	277	8	a	a	DET
ejpam-5532	277	9	∧	∧	PROPN
ejpam-5532	277	10	b	b	PROPN
ejpam-5532	277	11	;	;	PUNCT
ejpam-5532	277	12	∀	∀	X
ejpam-5532	277	13	a	a	DET
ejpam-5532	277	14	∈	∈	NOUN
ejpam-5532	277	15	lη(x	lη(x	X
ejpam-5532	277	16	)	)	PUNCT
ejpam-5532	277	17	and	and	CCONJ
ejpam-5532	277	18	b	b	X
ejpam-5532	277	19	∈	∈	PROPN
ejpam-5532	277	20	lη(y	lη(y	PUNCT
ejpam-5532	277	21	)	)	PUNCT
ejpam-5532	277	22	.	.	PUNCT
ejpam-5532	278	1	hence	hence	ADV
ejpam-5532	278	2	,	,	PUNCT
ejpam-5532	278	3	η̂(x	η̂(x	NUM
ejpam-5532	278	4	∨	∨	NUM
ejpam-5532	278	5	y	y	PROPN
ejpam-5532	278	6	)	)	PUNCT
ejpam-5532	278	7	≥	≥	NOUN
ejpam-5532	278	8	∨{a	∨{a	PROPN
ejpam-5532	278	9	∧	∧	PROPN
ejpam-5532	278	10	b	b	PROPN
ejpam-5532	278	11	/	/	SYM
ejpam-5532	278	12	a	a	DET
ejpam-5532	278	13	∈	∈	NOUN
ejpam-5532	278	14	lη(x	lη(x	X
ejpam-5532	278	15	)	)	PUNCT
ejpam-5532	278	16	,	,	PUNCT
ejpam-5532	278	17	b	b	X
ejpam-5532	278	18	∈	∈	PROPN
ejpam-5532	278	19	lη(y	lη(y	PUNCT
ejpam-5532	278	20	)	)	PUNCT
ejpam-5532	278	21	}	}	PUNCT
ejpam-5532	279	1	=	=	SYM
ejpam-5532	279	2	{	{	PUNCT
ejpam-5532	279	3	∨{a	∨{a	PROPN
ejpam-5532	279	4	/	/	SYM
ejpam-5532	279	5	a	a	DET
ejpam-5532	279	6	∈	∈	NOUN
ejpam-5532	279	7	lη(x	lη(x	X
ejpam-5532	279	8	)	)	PUNCT
ejpam-5532	279	9	}	}	PUNCT
ejpam-5532	279	10	}	}	PUNCT
ejpam-5532	279	11	∧	∧	NOUN
ejpam-5532	279	12	{	{	PUNCT
ejpam-5532	279	13	∨{b	∨{b	NOUN
ejpam-5532	279	14	/	/	SYM
ejpam-5532	279	15	b	b	NOUN
ejpam-5532	279	16	∈	∈	NOUN
ejpam-5532	279	17	lη(y	lη(y	PUNCT
ejpam-5532	279	18	)	)	PUNCT
ejpam-5532	279	19	}	}	PUNCT
ejpam-5532	279	20	}	}	PUNCT
ejpam-5532	279	21	(	(	PUNCT
ejpam-5532	279	22	as	as	SCONJ
ejpam-5532	279	23	l	l	NOUN
ejpam-5532	279	24	is	be	AUX
ejpam-5532	279	25	a	a	DET
ejpam-5532	279	26	completely	completely	ADV
ejpam-5532	279	27	distributive	distributive	ADJ
ejpam-5532	279	28	lattice	lattice	NOUN
ejpam-5532	279	29	)	)	PUNCT
ejpam-5532	279	30	=	=	PUNCT
ejpam-5532	279	31	η′(x	η′(x	X
ejpam-5532	279	32	)	)	PUNCT
ejpam-5532	279	33	∧	∧	PROPN
ejpam-5532	279	34	η′(y	η′(y	NOUN
ejpam-5532	279	35	)	)	PUNCT
ejpam-5532	279	36	.	.	PUNCT
ejpam-5532	280	1	now	now	ADV
ejpam-5532	280	2	,	,	PUNCT
ejpam-5532	280	3	to	to	PART
ejpam-5532	280	4	verify	verify	VERB
ejpam-5532	280	5	that	that	SCONJ
ejpam-5532	280	6	η̂(x	η̂(x	NUM
ejpam-5532	280	7	∧	∧	PROPN
ejpam-5532	280	8	y	y	PROPN
ejpam-5532	280	9	)	)	PUNCT
ejpam-5532	280	10	≥	≥	NOUN
ejpam-5532	280	11	µ(x	µ(x	NOUN
ejpam-5532	280	12	)	)	PUNCT
ejpam-5532	280	13	∧	∧	NOUN
ejpam-5532	280	14	η̂(y	η̂(y	PROPN
ejpam-5532	280	15	)	)	PUNCT
ejpam-5532	280	16	,	,	PUNCT
ejpam-5532	280	17	we	we	PRON
ejpam-5532	280	18	again	again	ADV
ejpam-5532	280	19	define	define	VERB
ejpam-5532	280	20	the	the	DET
ejpam-5532	280	21	following	follow	VERB
ejpam-5532	280	22	subsets	subset	NOUN
ejpam-5532	280	23	of	of	ADP
ejpam-5532	280	24	l	l	NOUN
ejpam-5532	280	25	for	for	ADP
ejpam-5532	280	26	z	z	PROPN
ejpam-5532	280	27	∈	∈	PROPN
ejpam-5532	280	28	m	m	NOUN
ejpam-5532	280	29	:	:	PUNCT
ejpam-5532	280	30	lη(z	lη(z	X
ejpam-5532	280	31	)	)	PUNCT
ejpam-5532	281	1	=	=	PRON
ejpam-5532	281	2	{	{	PUNCT
ejpam-5532	281	3	t	t	NOUN
ejpam-5532	281	4	∈	∈	PROPN
ejpam-5532	281	5	l	l	PROPN
ejpam-5532	281	6	/	/	SYM
ejpam-5532	281	7	t	t	PROPN
ejpam-5532	281	8	<	<	X
ejpam-5532	281	9	a0	a0	PROPN
ejpam-5532	281	10	,	,	PUNCT
ejpam-5532	281	11	z	z	PROPN
ejpam-5532	281	12	∈	∈	PROPN
ejpam-5532	281	13	(	(	PUNCT
ejpam-5532	281	14	η	η	X
ejpam-5532	281	15	>	>	PROPN
ejpam-5532	281	16	t	t	PROPN
ejpam-5532	281	17	]	]	PUNCT
ejpam-5532	281	18	}	}	PUNCT
ejpam-5532	281	19	and	and	CCONJ
ejpam-5532	281	20	lµ(z	lµ(z	PROPN
ejpam-5532	281	21	)	)	PUNCT
ejpam-5532	281	22	=	=	PRON
ejpam-5532	281	23	{	{	PUNCT
ejpam-5532	281	24	t	t	NOUN
ejpam-5532	281	25	∈	∈	PROPN
ejpam-5532	281	26	l	l	PROPN
ejpam-5532	281	27	/	/	SYM
ejpam-5532	281	28	t	t	NOUN
ejpam-5532	281	29	≤	≤	NUM
ejpam-5532	281	30	tip{µ	tip{µ	NOUN
ejpam-5532	281	31	}	}	PUNCT
ejpam-5532	281	32	,	,	PUNCT
ejpam-5532	281	33	z	z	NOUN
ejpam-5532	281	34	∈	∈	PROPN
ejpam-5532	281	35	µt	µt	PRON
ejpam-5532	281	36	=	=	PUNCT
ejpam-5532	282	1	[	[	X
ejpam-5532	282	2	µt	µt	X
ejpam-5532	282	3	]	]	X
ejpam-5532	282	4	}	}	PUNCT
ejpam-5532	282	5	.	.	PUNCT
ejpam-5532	283	1	thus	thus	ADV
ejpam-5532	283	2	,	,	PUNCT
ejpam-5532	283	3	η̂(x	η̂(x	NUM
ejpam-5532	283	4	)	)	PUNCT
ejpam-5532	283	5	=	=	SYM
ejpam-5532	283	6	∨	∨	NOUN
ejpam-5532	283	7	lη(x	lη(x	NUM
ejpam-5532	283	8	)	)	PUNCT
ejpam-5532	283	9	and	and	CCONJ
ejpam-5532	283	10	µ(x	µ(x	NOUN
ejpam-5532	283	11	)	)	PUNCT
ejpam-5532	283	12	=	=	SYM
ejpam-5532	283	13	∨	∨	NUM
ejpam-5532	283	14	lµ(x	lµ(x	NUM
ejpam-5532	283	15	)	)	PUNCT
ejpam-5532	283	16	.	.	PUNCT
ejpam-5532	284	1	if	if	SCONJ
ejpam-5532	284	2	a	a	DET
ejpam-5532	284	3	∈	∈	PROPN
ejpam-5532	284	4	lµ(x	lµ(x	X
ejpam-5532	284	5	)	)	PUNCT
ejpam-5532	284	6	and	and	CCONJ
ejpam-5532	284	7	b	b	X
ejpam-5532	284	8	∈	∈	PROPN
ejpam-5532	284	9	lη(y	lη(y	PUNCT
ejpam-5532	284	10	)	)	PUNCT
ejpam-5532	284	11	,	,	PUNCT
ejpam-5532	284	12	then	then	ADV
ejpam-5532	284	13	a	a	DET
ejpam-5532	284	14	≤	≤	ADJ
ejpam-5532	284	15	tip	tip	NOUN
ejpam-5532	284	16	{	{	PUNCT
ejpam-5532	284	17	µ	µ	NOUN
ejpam-5532	284	18	}	}	PUNCT
ejpam-5532	284	19	,	,	PUNCT
ejpam-5532	284	20	b	b	PROPN
ejpam-5532	284	21	<	<	X
ejpam-5532	284	22	a0	a0	PROPN
ejpam-5532	284	23	,	,	PUNCT
ejpam-5532	284	24	x	x	SYM
ejpam-5532	284	25	∈	∈	NOUN
ejpam-5532	284	26	µa	µa	NOUN
ejpam-5532	285	1	=	=	PUNCT
ejpam-5532	286	1	[	[	X
ejpam-5532	286	2	µa	µa	X
ejpam-5532	286	3	]	]	X
ejpam-5532	286	4	and	and	CCONJ
ejpam-5532	286	5	y	y	PROPN
ejpam-5532	286	6	∈	∈	PROPN
ejpam-5532	286	7	(	(	PUNCT
ejpam-5532	286	8	η	η	X
ejpam-5532	286	9	>	>	X
ejpam-5532	286	10	b	b	X
ejpam-5532	286	11	]	]	PUNCT
ejpam-5532	286	12	.	.	PUNCT
ejpam-5532	286	13	therefore	therefore	ADV
ejpam-5532	286	14	,	,	PUNCT
ejpam-5532	286	15	a	a	DET
ejpam-5532	286	16	∧	∧	PROPN
ejpam-5532	286	17	b	b	PROPN
ejpam-5532	286	18	<	<	X
ejpam-5532	286	19	a0	a0	PROPN
ejpam-5532	286	20	and	and	CCONJ
ejpam-5532	286	21	y	y	PROPN
ejpam-5532	286	22	≤	≤	PROPN
ejpam-5532	286	23	y1	y1	NOUN
ejpam-5532	286	24	∨	∨	NOUN
ejpam-5532	286	25	.	.	PUNCT
ejpam-5532	286	26	.	.	PUNCT
ejpam-5532	286	27	.	.	PUNCT
ejpam-5532	287	1	∨	∨	NUM
ejpam-5532	287	2	ym	ym	PROPN
ejpam-5532	287	3	,	,	PUNCT
ejpam-5532	287	4	yj	yj	PROPN
ejpam-5532	287	5	∈	∈	PROPN
ejpam-5532	287	6	η	η	PROPN
ejpam-5532	287	7	>	>	PROPN
ejpam-5532	287	8	b	b	PROPN
ejpam-5532	287	9	,	,	PUNCT
ejpam-5532	287	10	∀	∀	PUNCT
ejpam-5532	287	11	j.	j.	PROPN
ejpam-5532	288	1	this	this	PRON
ejpam-5532	288	2	implies	imply	VERB
ejpam-5532	288	3	x	x	PUNCT
ejpam-5532	288	4	∧	∧	NOUN
ejpam-5532	288	5	y	y	PROPN
ejpam-5532	288	6	≤	≤	NUM
ejpam-5532	288	7	y1	y1	NOUN
ejpam-5532	288	8	∨	∨	NOUN
ejpam-5532	288	9	.	.	PUNCT
ejpam-5532	288	10	.	.	PUNCT
ejpam-5532	288	11	.	.	PUNCT
ejpam-5532	289	1	∨	∨	NUM
ejpam-5532	289	2	ym	ym	PROPN
ejpam-5532	289	3	,	,	PUNCT
ejpam-5532	289	4	yj	yj	PROPN
ejpam-5532	289	5	∈	∈	PROPN
ejpam-5532	289	6	η	η	PROPN
ejpam-5532	289	7	>	>	PROPN
ejpam-5532	289	8	b	b	PROPN
ejpam-5532	289	9	⊆	⊆	NUM
ejpam-5532	289	10	η	η	X
ejpam-5532	289	11	>	>	PROPN
ejpam-5532	289	12	a∧b	a∧b	PROPN
ejpam-5532	289	13	,	,	PUNCT
ejpam-5532	289	14	∀	∀	PUNCT
ejpam-5532	289	15	j.	j.	PROPN
ejpam-5532	290	1	we	we	PRON
ejpam-5532	290	2	thus	thus	ADV
ejpam-5532	290	3	have	have	VERB
ejpam-5532	290	4	x	x	ADJ
ejpam-5532	290	5	∧	∧	PROPN
ejpam-5532	290	6	y	y	PROPN
ejpam-5532	290	7	∈	∈	PROPN
ejpam-5532	290	8	(	(	PUNCT
ejpam-5532	290	9	η	η	X
ejpam-5532	290	10	>	>	X
ejpam-5532	290	11	b	b	X
ejpam-5532	290	12	]	]	X
ejpam-5532	290	13	⊆	⊆	NUM
ejpam-5532	290	14	(	(	PUNCT
ejpam-5532	290	15	η	η	X
ejpam-5532	290	16	>	>	X
ejpam-5532	290	17	a∧b	a∧b	PROPN
ejpam-5532	290	18	]	]	PUNCT
ejpam-5532	290	19	and	and	CCONJ
ejpam-5532	290	20	therefore	therefore	ADV
ejpam-5532	290	21	,	,	PUNCT
ejpam-5532	290	22	a	a	DET
ejpam-5532	290	23	∧	∧	PROPN
ejpam-5532	290	24	b	b	PROPN
ejpam-5532	290	25	∈	∈	PROPN
ejpam-5532	290	26	lη(x	lη(x	X
ejpam-5532	290	27	∧	∧	PROPN
ejpam-5532	290	28	y	y	PROPN
ejpam-5532	290	29	)	)	PUNCT
ejpam-5532	290	30	;	;	PUNCT
ejpam-5532	290	31	∀	∀	X
ejpam-5532	290	32	a	a	DET
ejpam-5532	290	33	∈	∈	PROPN
ejpam-5532	290	34	lµ(x	lµ(x	X
ejpam-5532	290	35	)	)	PUNCT
ejpam-5532	290	36	and	and	CCONJ
ejpam-5532	290	37	b	b	X
ejpam-5532	290	38	∈	∈	PROPN
ejpam-5532	290	39	lη(y	lη(y	PUNCT
ejpam-5532	290	40	)	)	PUNCT
ejpam-5532	290	41	.	.	PUNCT
ejpam-5532	291	1	that	that	PRON
ejpam-5532	291	2	is	be	AUX
ejpam-5532	291	3	,	,	PUNCT
ejpam-5532	291	4	η̂(x	η̂(x	NUM
ejpam-5532	291	5	∧	∧	PROPN
ejpam-5532	291	6	y	y	PROPN
ejpam-5532	291	7	)	)	PUNCT
ejpam-5532	291	8	≥	≥	NOUN
ejpam-5532	291	9	a	a	DET
ejpam-5532	291	10	∧	∧	PROPN
ejpam-5532	291	11	b	b	PROPN
ejpam-5532	291	12	;	;	PUNCT
ejpam-5532	291	13	∀	∀	X
ejpam-5532	291	14	a	a	DET
ejpam-5532	291	15	∈	∈	PROPN
ejpam-5532	291	16	lµ(x	lµ(x	X
ejpam-5532	291	17	)	)	PUNCT
ejpam-5532	291	18	and	and	CCONJ
ejpam-5532	291	19	b	b	X
ejpam-5532	291	20	∈	∈	PROPN
ejpam-5532	291	21	lη(y	lη(y	PUNCT
ejpam-5532	291	22	)	)	PUNCT
ejpam-5532	291	23	.	.	PUNCT
ejpam-5532	292	1	consequently	consequently	ADV
ejpam-5532	292	2	,	,	PUNCT
ejpam-5532	292	3	η̂(x	η̂(x	NUM
ejpam-5532	292	4	∧	∧	PROPN
ejpam-5532	292	5	y	y	PROPN
ejpam-5532	292	6	)	)	PUNCT
ejpam-5532	292	7	≥	≥	NOUN
ejpam-5532	292	8	∨{a	∨{a	PROPN
ejpam-5532	292	9	∧	∧	PROPN
ejpam-5532	292	10	b	b	PROPN
ejpam-5532	292	11	/	/	SYM
ejpam-5532	292	12	a	a	DET
ejpam-5532	292	13	∈	∈	NOUN
ejpam-5532	292	14	lµ(x	lµ(x	X
ejpam-5532	292	15	)	)	PUNCT
ejpam-5532	292	16	and	and	CCONJ
ejpam-5532	292	17	b	b	X
ejpam-5532	292	18	∈	∈	PROPN
ejpam-5532	292	19	lη(y	lη(y	PUNCT
ejpam-5532	292	20	)	)	PUNCT
ejpam-5532	292	21	}	}	PUNCT
ejpam-5532	293	1	=	=	SYM
ejpam-5532	293	2	{	{	PUNCT
ejpam-5532	293	3	∨{a	∨{a	PROPN
ejpam-5532	293	4	/	/	SYM
ejpam-5532	293	5	a	a	DET
ejpam-5532	293	6	∈	∈	PROPN
ejpam-5532	293	7	lµ(x	lµ(x	NOUN
ejpam-5532	293	8	)	)	PUNCT
ejpam-5532	293	9	}	}	PUNCT
ejpam-5532	293	10	}	}	PUNCT
ejpam-5532	293	11	∧	∧	NOUN
ejpam-5532	293	12	{	{	PUNCT
ejpam-5532	293	13	∨{b	∨{b	NOUN
ejpam-5532	293	14	/	/	SYM
ejpam-5532	293	15	b	b	NOUN
ejpam-5532	293	16	∈	∈	NOUN
ejpam-5532	293	17	lη(y	lη(y	PUNCT
ejpam-5532	293	18	)	)	PUNCT
ejpam-5532	293	19	}	}	PUNCT
ejpam-5532	293	20	}	}	PUNCT
ejpam-5532	293	21	(	(	PUNCT
ejpam-5532	293	22	as	as	SCONJ
ejpam-5532	293	23	l	l	NOUN
ejpam-5532	293	24	is	be	AUX
ejpam-5532	293	25	a	a	DET
ejpam-5532	293	26	completely	completely	ADV
ejpam-5532	293	27	distributive	distributive	ADJ
ejpam-5532	293	28	lattice	lattice	NOUN
ejpam-5532	293	29	)	)	PUNCT
ejpam-5532	293	30	=	=	SYM
ejpam-5532	293	31	µ(x	µ(x	X
ejpam-5532	293	32	)	)	PUNCT
ejpam-5532	293	33	∧	∧	NOUN
ejpam-5532	293	34	η̂(y	η̂(y	PROPN
ejpam-5532	293	35	)	)	PUNCT
ejpam-5532	293	36	.	.	PUNCT
ejpam-5532	294	1	a.	a.	PROPN
ejpam-5532	294	2	jain	jain	PROPN
ejpam-5532	294	3	,	,	PUNCT
ejpam-5532	294	4	i.	i.	PROPN
ejpam-5532	294	5	jahan	jahan	PROPN
ejpam-5532	294	6	/	/	SYM
ejpam-5532	294	7	eur	eur	PROPN
ejpam-5532	294	8	.	.	PUNCT
ejpam-5532	295	1	j.	j.	PROPN
ejpam-5532	295	2	pure	pure	PROPN
ejpam-5532	295	3	appl	appl	PROPN
ejpam-5532	295	4	.	.	PROPN
ejpam-5532	295	5	math	math	PROPN
ejpam-5532	295	6	,	,	PUNCT
ejpam-5532	295	7	18	18	NUM
ejpam-5532	295	8	(	(	PUNCT
ejpam-5532	295	9	1	1	NUM
ejpam-5532	295	10	)	)	PUNCT
ejpam-5532	295	11	(	(	PUNCT
ejpam-5532	295	12	2025	2025	NUM
ejpam-5532	295	13	)	)	PUNCT
ejpam-5532	295	14	,	,	PUNCT
ejpam-5532	295	15	5532	5532	NUM
ejpam-5532	295	16	14	14	NUM
ejpam-5532	295	17	of	of	ADP
ejpam-5532	295	18	20	20	NUM
ejpam-5532	295	19	we	we	PRON
ejpam-5532	295	20	thus	thus	ADV
ejpam-5532	295	21	get	get	VERB
ejpam-5532	295	22	that	that	PRON
ejpam-5532	295	23	η̂	η̂	PUNCT
ejpam-5532	295	24	is	be	AUX
ejpam-5532	295	25	an	an	DET
ejpam-5532	295	26	l	l	NOUN
ejpam-5532	295	27	-	-	NOUN
ejpam-5532	295	28	ideal	ideal	NOUN
ejpam-5532	295	29	of	of	ADP
ejpam-5532	295	30	µ.	µ.	NOUN
ejpam-5532	295	31	finally	finally	ADV
ejpam-5532	295	32	,	,	PUNCT
ejpam-5532	295	33	to	to	PART
ejpam-5532	295	34	prove	prove	VERB
ejpam-5532	295	35	that	that	SCONJ
ejpam-5532	295	36	η̂	η̂	PUNCT
ejpam-5532	295	37	is	be	AUX
ejpam-5532	295	38	the	the	DET
ejpam-5532	295	39	smallest	small	ADJ
ejpam-5532	295	40	l	l	NOUN
ejpam-5532	295	41	-	-	NOUN
ejpam-5532	295	42	ideal	ideal	NOUN
ejpam-5532	295	43	of	of	ADP
ejpam-5532	295	44	µ	µ	X
ejpam-5532	295	45	containing	contain	VERB
ejpam-5532	295	46	η	η	PROPN
ejpam-5532	295	47	,	,	PUNCT
ejpam-5532	295	48	suppose	suppose	VERB
ejpam-5532	295	49	θ	θ	NOUN
ejpam-5532	295	50	is	be	AUX
ejpam-5532	295	51	an	an	DET
ejpam-5532	295	52	l	l	NOUN
ejpam-5532	295	53	-	-	NOUN
ejpam-5532	295	54	ideal	ideal	NOUN
ejpam-5532	295	55	of	of	ADP
ejpam-5532	295	56	µ	µ	PRON
ejpam-5532	295	57	such	such	ADJ
ejpam-5532	295	58	that	that	SCONJ
ejpam-5532	295	59	η	η	PROPN
ejpam-5532	295	60	⊆	⊆	NUM
ejpam-5532	295	61	θ	θ	PROPN
ejpam-5532	295	62	.	.	PUNCT
ejpam-5532	296	1	then	then	ADV
ejpam-5532	296	2	,	,	PUNCT
ejpam-5532	296	3	η	η	PROPN
ejpam-5532	296	4	>	>	PROPN
ejpam-5532	296	5	t	t	PROPN
ejpam-5532	296	6	⊆	⊆	NUM
ejpam-5532	296	7	θ	θ	PROPN
ejpam-5532	296	8	>	>	X
ejpam-5532	296	9	t	t	PROPN
ejpam-5532	296	10	and	and	CCONJ
ejpam-5532	296	11	θ	θ	PROPN
ejpam-5532	296	12	>	>	PROPN
ejpam-5532	296	13	t	t	PROPN
ejpam-5532	296	14	is	be	AUX
ejpam-5532	296	15	an	an	DET
ejpam-5532	296	16	ideal	ideal	NOUN
ejpam-5532	296	17	of	of	ADP
ejpam-5532	296	18	µ	µ	NOUN
ejpam-5532	296	19	>	>	X
ejpam-5532	296	20	t	t	PROPN
ejpam-5532	296	21	.	.	PUNCT
ejpam-5532	297	1	therefore	therefore	ADV
ejpam-5532	297	2	,	,	PUNCT
ejpam-5532	297	3	(	(	PUNCT
ejpam-5532	297	4	θ	θ	X
ejpam-5532	297	5	>	>	X
ejpam-5532	297	6	t	t	X
ejpam-5532	297	7	]	]	PUNCT
ejpam-5532	297	8	=	=	PUNCT
ejpam-5532	297	9	θ	θ	PROPN
ejpam-5532	297	10	>	>	X
ejpam-5532	297	11	t	t	PROPN
ejpam-5532	297	12	.	.	PUNCT
ejpam-5532	298	1	we	we	PRON
ejpam-5532	298	2	thus	thus	ADV
ejpam-5532	298	3	have	have	VERB
ejpam-5532	298	4	:	:	PUNCT
ejpam-5532	298	5	η̂(x	η̂(x	NUM
ejpam-5532	298	6	)	)	PUNCT
ejpam-5532	298	7	=	=	SYM
ejpam-5532	298	8	∨t	∨t	NOUN
ejpam-5532	298	9	<	<	X
ejpam-5532	298	10	a0{t	a0{t	X
ejpam-5532	298	11	:	:	PUNCT
ejpam-5532	298	12	x	x	SYM
ejpam-5532	298	13	∈	∈	PROPN
ejpam-5532	298	14	(	(	PUNCT
ejpam-5532	298	15	η	η	X
ejpam-5532	298	16	>	>	PROPN
ejpam-5532	298	17	t	t	PROPN
ejpam-5532	298	18	]	]	PUNCT
ejpam-5532	298	19	}	}	PUNCT
ejpam-5532	298	20	≤	≤	NUM
ejpam-5532	298	21	∨	∨	NUM
ejpam-5532	298	22	t≤tip{θ	t≤tip{θ	NOUN
ejpam-5532	298	23	}	}	PUNCT
ejpam-5532	298	24	{	{	PUNCT
ejpam-5532	298	25	t	t	NOUN
ejpam-5532	298	26	:	:	PUNCT
ejpam-5532	298	27	x	x	SYM
ejpam-5532	298	28	∈	∈	PROPN
ejpam-5532	298	29	θ	θ	PROPN
ejpam-5532	298	30	>	>	X
ejpam-5532	298	31	t	t	PROPN
ejpam-5532	298	32	}	}	PUNCT
ejpam-5532	298	33	≤	≤	NUM
ejpam-5532	298	34	θ(x	θ(x	PROPN
ejpam-5532	298	35	)	)	PUNCT
ejpam-5532	298	36	.	.	PUNCT
ejpam-5532	299	1	that	that	PRON
ejpam-5532	299	2	is	be	AUX
ejpam-5532	299	3	,	,	PUNCT
ejpam-5532	299	4	η̂	η̂	PROPN
ejpam-5532	299	5	⊆	⊆	NUM
ejpam-5532	299	6	θ	θ	NOUN
ejpam-5532	299	7	.	.	PUNCT
ejpam-5532	299	8	hence	hence	ADV
ejpam-5532	299	9	,	,	PUNCT
ejpam-5532	299	10	η̂	η̂	PROPN
ejpam-5532	299	11	=	=	SYM
ejpam-5532	299	12	(	(	PUNCT
ejpam-5532	299	13	η]µ.	η]µ.	PROPN
ejpam-5532	299	14	a	a	DET
ejpam-5532	299	15	similar	similar	ADJ
ejpam-5532	299	16	result	result	NOUN
ejpam-5532	299	17	holds	hold	VERB
ejpam-5532	299	18	for	for	ADP
ejpam-5532	299	19	l	l	ADJ
ejpam-5532	299	20	-	-	ADJ
ejpam-5532	299	21	dual	dual	ADJ
ejpam-5532	299	22	ideal	ideal	NOUN
ejpam-5532	299	23	generated	generate	VERB
ejpam-5532	299	24	by	by	ADP
ejpam-5532	299	25	η	η	PROPN
ejpam-5532	299	26	.	.	PROPN
ejpam-5532	299	27	theorem	theorem	PROPN
ejpam-5532	299	28	12	12	NUM
ejpam-5532	299	29	.	.	PUNCT
ejpam-5532	300	1	let	let	AUX
ejpam-5532	300	2	l	l	NOUN
ejpam-5532	300	3	be	be	AUX
ejpam-5532	300	4	a	a	DET
ejpam-5532	300	5	dense	dense	ADJ
ejpam-5532	300	6	chain	chain	NOUN
ejpam-5532	300	7	and	and	CCONJ
ejpam-5532	300	8	l(µ,m	l(µ,m	ADV
ejpam-5532	300	9	)	)	PUNCT
ejpam-5532	300	10	be	be	AUX
ejpam-5532	300	11	an	an	DET
ejpam-5532	300	12	l	l	NOUN
ejpam-5532	300	13	-	-	NOUN
ejpam-5532	300	14	lattice	lattice	NOUN
ejpam-5532	300	15	.	.	PUNCT
ejpam-5532	301	1	let	let	VERB
ejpam-5532	301	2	η	η	PROPN
ejpam-5532	301	3	∈	∈	PROPN
ejpam-5532	301	4	lm	lm	INTJ
ejpam-5532	301	5	,	,	PUNCT
ejpam-5532	301	6	η	η	PROPN
ejpam-5532	301	7	⊆	⊆	PROPN
ejpam-5532	301	8	µ	µ	X
ejpam-5532	301	9	and	and	CCONJ
ejpam-5532	301	10	a0	a0	PROPN
ejpam-5532	301	11	=	=	SYM
ejpam-5532	301	12	tip{η	tip{η	PROPN
ejpam-5532	301	13	}	}	PUNCT
ejpam-5532	301	14	.	.	PUNCT
ejpam-5532	302	1	define	define	VERB
ejpam-5532	302	2	an	an	DET
ejpam-5532	302	3	l	l	NOUN
ejpam-5532	302	4	-	-	NOUN
ejpam-5532	302	5	subset	subset	NOUN
ejpam-5532	302	6	η̌	η̌	NOUN
ejpam-5532	302	7	of	of	ADP
ejpam-5532	302	8	m	m	PRON
ejpam-5532	302	9	as	as	SCONJ
ejpam-5532	302	10	follows	follow	VERB
ejpam-5532	302	11	:	:	PUNCT
ejpam-5532	302	12	η̌(x	η̌(x	NUM
ejpam-5532	302	13	)	)	PUNCT
ejpam-5532	302	14	=	=	SYM
ejpam-5532	302	15	∨t	∨t	NOUN
ejpam-5532	302	16	<	<	X
ejpam-5532	302	17	a0{t	a0{t	X
ejpam-5532	302	18	:	:	PUNCT
ejpam-5532	302	19	x	x	SYM
ejpam-5532	302	20	∈	∈	PROPN
ejpam-5532	302	21	[	[	X
ejpam-5532	302	22	η	η	X
ejpam-5532	302	23	>	>	PROPN
ejpam-5532	302	24	t	t	PROPN
ejpam-5532	302	25	)	)	PUNCT
ejpam-5532	302	26	}	}	PUNCT
ejpam-5532	302	27	,	,	PUNCT
ejpam-5532	302	28	where	where	SCONJ
ejpam-5532	302	29	[	[	X
ejpam-5532	302	30	η	η	X
ejpam-5532	302	31	>	>	PROPN
ejpam-5532	302	32	t	t	PROPN
ejpam-5532	302	33	)	)	PUNCT
ejpam-5532	302	34	is	be	AUX
ejpam-5532	302	35	a	a	DET
ejpam-5532	302	36	dual	dual	ADJ
ejpam-5532	302	37	ideal	ideal	NOUN
ejpam-5532	302	38	of	of	ADP
ejpam-5532	302	39	µ	µ	PROPN
ejpam-5532	302	40	>	>	X
ejpam-5532	302	41	t	t	PROPN
ejpam-5532	302	42	generated	generate	VERB
ejpam-5532	302	43	by	by	ADP
ejpam-5532	302	44	η	η	PROPN
ejpam-5532	302	45	>	>	PROPN
ejpam-5532	302	46	t	t	PROPN
ejpam-5532	302	47	.	.	PUNCT
ejpam-5532	303	1	then	then	ADV
ejpam-5532	303	2	,	,	PUNCT
ejpam-5532	303	3	η̌	η̌	X
ejpam-5532	303	4	=	=	PUNCT
ejpam-5532	304	1	[	[	X
ejpam-5532	304	2	η)µ.	η)µ.	X
ejpam-5532	304	3	we	we	PRON
ejpam-5532	304	4	now	now	ADV
ejpam-5532	304	5	establish	establish	VERB
ejpam-5532	304	6	the	the	DET
ejpam-5532	304	7	unique	unique	ADJ
ejpam-5532	304	8	representation	representation	NOUN
ejpam-5532	304	9	theorem	theorem	NOUN
ejpam-5532	304	10	for	for	ADP
ejpam-5532	304	11	l	l	ADJ
ejpam-5532	304	12	-	-	ADJ
ejpam-5532	304	13	convex	convex	ADJ
ejpam-5532	304	14	sublattices	sublattice	NOUN
ejpam-5532	304	15	in	in	ADP
ejpam-5532	304	16	an	an	DET
ejpam-5532	304	17	llattice	llattice	NOUN
ejpam-5532	304	18	.	.	PUNCT
ejpam-5532	305	1	in	in	ADP
ejpam-5532	305	2	the	the	DET
ejpam-5532	305	3	following	following	NOUN
ejpam-5532	305	4	theorem	theorem	NOUN
ejpam-5532	305	5	,	,	PUNCT
ejpam-5532	305	6	o	o	NOUN
ejpam-5532	305	7	represents	represent	VERB
ejpam-5532	305	8	the	the	DET
ejpam-5532	305	9	constant	constant	ADJ
ejpam-5532	305	10	l	l	NOUN
ejpam-5532	305	11	-	-	NOUN
ejpam-5532	305	12	subset	subset	NOUN
ejpam-5532	305	13	with	with	ADP
ejpam-5532	305	14	all	all	DET
ejpam-5532	305	15	truth	truth	NOUN
ejpam-5532	305	16	values	value	NOUN
ejpam-5532	305	17	equal	equal	ADJ
ejpam-5532	305	18	to	to	ADP
ejpam-5532	305	19	0	0	NUM
ejpam-5532	305	20	of	of	ADP
ejpam-5532	305	21	lattice	lattice	PROPN
ejpam-5532	305	22	l.	l.	PROPN
ejpam-5532	305	23	theorem	theorem	VERB
ejpam-5532	305	24	13	13	NUM
ejpam-5532	305	25	.	.	PUNCT
ejpam-5532	306	1	let	let	VERB
ejpam-5532	306	2	l	l	NOUN
ejpam-5532	306	3	be	be	AUX
ejpam-5532	306	4	a	a	DET
ejpam-5532	306	5	dense	dense	ADJ
ejpam-5532	306	6	chain	chain	NOUN
ejpam-5532	306	7	,	,	PUNCT
ejpam-5532	306	8	l(µ,m	l(µ,m	X
ejpam-5532	306	9	)	)	PUNCT
ejpam-5532	306	10	be	be	AUX
ejpam-5532	306	11	an	an	DET
ejpam-5532	306	12	l	l	NOUN
ejpam-5532	306	13	-	-	NOUN
ejpam-5532	306	14	lattice	lattice	ADJ
ejpam-5532	306	15	,	,	PUNCT
ejpam-5532	306	16	η	η	PROPN
ejpam-5532	306	17	,	,	PUNCT
ejpam-5532	306	18	θ	θ	PROPN
ejpam-5532	306	19	⊆	⊆	NUM
ejpam-5532	306	20	µ	µ	PRON
ejpam-5532	306	21	such	such	ADJ
ejpam-5532	306	22	that	that	SCONJ
ejpam-5532	306	23	η	η	PROPN
ejpam-5532	306	24	is	be	AUX
ejpam-5532	306	25	an	an	DET
ejpam-5532	306	26	l	l	NOUN
ejpam-5532	306	27	-	-	NOUN
ejpam-5532	306	28	ideal	ideal	NOUN
ejpam-5532	306	29	of	of	ADP
ejpam-5532	306	30	µ	µ	NUM
ejpam-5532	306	31	and	and	CCONJ
ejpam-5532	306	32	θ	θ	PROPN
ejpam-5532	306	33	is	be	AUX
ejpam-5532	306	34	an	an	DET
ejpam-5532	306	35	l	l	ADJ
ejpam-5532	306	36	-	-	ADJ
ejpam-5532	306	37	dual	dual	ADJ
ejpam-5532	306	38	ideal	ideal	NOUN
ejpam-5532	306	39	of	of	ADP
ejpam-5532	306	40	µ.	µ.	NOUN
ejpam-5532	306	41	then	then	ADV
ejpam-5532	306	42	,	,	PUNCT
ejpam-5532	306	43	η	η	PROPN
ejpam-5532	306	44	∩	∩	PROPN
ejpam-5532	306	45	θ	θ	PROPN
ejpam-5532	306	46	is	be	AUX
ejpam-5532	306	47	an	an	DET
ejpam-5532	306	48	l	l	ADJ
ejpam-5532	306	49	-	-	ADJ
ejpam-5532	306	50	convex	convex	ADJ
ejpam-5532	306	51	sublattice	sublattice	NOUN
ejpam-5532	306	52	of	of	ADP
ejpam-5532	306	53	µ	µ	NOUN
ejpam-5532	306	54	if	if	SCONJ
ejpam-5532	306	55	η	η	PROPN
ejpam-5532	306	56	∩	∩	PROPN
ejpam-5532	306	57	θ	θ	PROPN
ejpam-5532	306	58	̸=	̸=	PROPN
ejpam-5532	306	59	o.	o.	NOUN
ejpam-5532	306	60	further	far	ADV
ejpam-5532	306	61	,	,	PUNCT
ejpam-5532	306	62	every	every	DET
ejpam-5532	306	63	l	l	NOUN
ejpam-5532	306	64	-	-	ADJ
ejpam-5532	306	65	convex	convex	ADJ
ejpam-5532	306	66	sublattice	sublattice	NOUN
ejpam-5532	306	67	of	of	ADP
ejpam-5532	306	68	µ	µ	PRON
ejpam-5532	306	69	can	can	AUX
ejpam-5532	306	70	be	be	AUX
ejpam-5532	306	71	expressed	express	VERB
ejpam-5532	306	72	in	in	ADP
ejpam-5532	306	73	this	this	DET
ejpam-5532	306	74	form	form	NOUN
ejpam-5532	306	75	in	in	ADP
ejpam-5532	306	76	one	one	NUM
ejpam-5532	306	77	and	and	CCONJ
ejpam-5532	306	78	only	only	ADV
ejpam-5532	306	79	one	one	NUM
ejpam-5532	306	80	way	way	NOUN
ejpam-5532	306	81	.	.	PUNCT
ejpam-5532	307	1	proof	proof	NOUN
ejpam-5532	307	2	.	.	PUNCT
ejpam-5532	308	1	let	let	VERB
ejpam-5532	308	2	η	η	PROPN
ejpam-5532	308	3	be	be	AUX
ejpam-5532	308	4	an	an	DET
ejpam-5532	308	5	l	l	NOUN
ejpam-5532	308	6	-	-	NOUN
ejpam-5532	308	7	ideal	ideal	NOUN
ejpam-5532	308	8	of	of	ADP
ejpam-5532	308	9	µ	µ	NUM
ejpam-5532	308	10	and	and	CCONJ
ejpam-5532	308	11	θ	θ	PROPN
ejpam-5532	308	12	be	be	AUX
ejpam-5532	308	13	an	an	DET
ejpam-5532	308	14	l	l	ADJ
ejpam-5532	308	15	-	-	ADJ
ejpam-5532	308	16	dual	dual	ADJ
ejpam-5532	308	17	ideal	ideal	NOUN
ejpam-5532	308	18	of	of	ADP
ejpam-5532	308	19	µ.	µ.	NOUN
ejpam-5532	308	20	then	then	ADV
ejpam-5532	308	21	by	by	ADP
ejpam-5532	308	22	theorem	theorem	ADJ
ejpam-5532	308	23	10	10	NUM
ejpam-5532	308	24	,	,	PUNCT
ejpam-5532	308	25	η	η	PROPN
ejpam-5532	308	26	and	and	CCONJ
ejpam-5532	308	27	θ	θ	PROPN
ejpam-5532	308	28	are	be	AUX
ejpam-5532	308	29	l	l	ADJ
ejpam-5532	308	30	-	-	ADJ
ejpam-5532	308	31	convex	convex	ADJ
ejpam-5532	308	32	sublattices	sublattice	NOUN
ejpam-5532	308	33	of	of	ADP
ejpam-5532	308	34	µ.	µ.	NOUN
ejpam-5532	308	35	since	since	SCONJ
ejpam-5532	308	36	intersection	intersection	NOUN
ejpam-5532	308	37	of	of	ADP
ejpam-5532	308	38	l	l	ADJ
ejpam-5532	308	39	-	-	ADJ
ejpam-5532	308	40	convex	convex	ADJ
ejpam-5532	308	41	sublattices	sublattice	NOUN
ejpam-5532	308	42	of	of	ADP
ejpam-5532	308	43	µ	µ	PROPN
ejpam-5532	308	44	is	be	AUX
ejpam-5532	308	45	an	an	DET
ejpam-5532	308	46	l	l	ADJ
ejpam-5532	308	47	-	-	ADJ
ejpam-5532	308	48	convex	convex	ADJ
ejpam-5532	308	49	sublattice	sublattice	NOUN
ejpam-5532	308	50	of	of	ADP
ejpam-5532	308	51	µ	µ	NUM
ejpam-5532	308	52	,	,	PUNCT
ejpam-5532	308	53	therefore	therefore	ADV
ejpam-5532	308	54	η	η	PROPN
ejpam-5532	308	55	∩	∩	PROPN
ejpam-5532	308	56	θ	θ	PROPN
ejpam-5532	308	57	is	be	AUX
ejpam-5532	308	58	an	an	DET
ejpam-5532	308	59	l	l	ADJ
ejpam-5532	308	60	-	-	ADJ
ejpam-5532	308	61	convex	convex	ADJ
ejpam-5532	308	62	sublattice	sublattice	NOUN
ejpam-5532	308	63	of	of	ADP
ejpam-5532	308	64	µ	µ	PRON
ejpam-5532	308	65	provided	provide	VERB
ejpam-5532	308	66	η	η	PROPN
ejpam-5532	308	67	∩	∩	NOUN
ejpam-5532	308	68	θ	θ	PROPN
ejpam-5532	308	69	̸=	̸=	PROPN
ejpam-5532	308	70	o	o	NOUN
ejpam-5532	308	71	(	(	PUNCT
ejpam-5532	308	72	i.e.	i.e.	X
ejpam-5532	308	73	,	,	PUNCT
ejpam-5532	308	74	∃	∃	PROPN
ejpam-5532	308	75	x	x	SYM
ejpam-5532	308	76	∈	∈	PROPN
ejpam-5532	308	77	m	m	VERB
ejpam-5532	308	78	such	such	ADJ
ejpam-5532	308	79	that	that	SCONJ
ejpam-5532	308	80	(	(	PUNCT
ejpam-5532	308	81	η	η	PROPN
ejpam-5532	308	82	∩	∩	NOUN
ejpam-5532	308	83	θ)(x	θ)(x	NOUN
ejpam-5532	308	84	)	)	PUNCT
ejpam-5532	308	85	>	>	X
ejpam-5532	308	86	0	0	NUM
ejpam-5532	308	87	)	)	PUNCT
ejpam-5532	308	88	.	.	PUNCT
ejpam-5532	309	1	next	next	ADV
ejpam-5532	309	2	,	,	PUNCT
ejpam-5532	309	3	let	let	VERB
ejpam-5532	309	4	γ	γ	PRON
ejpam-5532	309	5	be	be	AUX
ejpam-5532	309	6	an	an	DET
ejpam-5532	309	7	l	l	NOUN
ejpam-5532	309	8	-	-	ADJ
ejpam-5532	309	9	convex	convex	ADJ
ejpam-5532	309	10	sublattice	sublattice	NOUN
ejpam-5532	309	11	of	of	ADP
ejpam-5532	309	12	µ.	µ.	NOUN
ejpam-5532	309	13	we	we	PRON
ejpam-5532	309	14	take	take	VERB
ejpam-5532	309	15	η	η	NOUN
ejpam-5532	309	16	=	=	PROPN
ejpam-5532	309	17	(	(	PUNCT
ejpam-5532	309	18	γ]µ	γ]µ	PROPN
ejpam-5532	309	19	and	and	CCONJ
ejpam-5532	309	20	θ	θ	NOUN
ejpam-5532	309	21	=	=	PUNCT
ejpam-5532	310	1	[	[	X
ejpam-5532	310	2	γ)µ.	γ)µ.	X
ejpam-5532	310	3	we	we	PRON
ejpam-5532	310	4	prove	prove	VERB
ejpam-5532	310	5	that	that	SCONJ
ejpam-5532	310	6	γ	γ	PROPN
ejpam-5532	310	7	=	=	SYM
ejpam-5532	310	8	η	η	PROPN
ejpam-5532	310	9	∩	∩	PROPN
ejpam-5532	310	10	θ	θ	PROPN
ejpam-5532	310	11	.	.	PUNCT
ejpam-5532	310	12	clearly	clearly	ADV
ejpam-5532	310	13	,	,	PUNCT
ejpam-5532	310	14	γ	γ	PROPN
ejpam-5532	310	15	⊆	⊆	NUM
ejpam-5532	310	16	η	η	NOUN
ejpam-5532	310	17	and	and	CCONJ
ejpam-5532	310	18	γ	γ	PROPN
ejpam-5532	310	19	⊆	⊆	NUM
ejpam-5532	310	20	θ	θ	PROPN
ejpam-5532	310	21	.	.	PUNCT
ejpam-5532	311	1	therefore	therefore	ADV
ejpam-5532	311	2	,	,	PUNCT
ejpam-5532	311	3	γ	γ	PROPN
ejpam-5532	311	4	⊆	⊆	NUM
ejpam-5532	311	5	η	η	PROPN
ejpam-5532	311	6	∩	∩	PROPN
ejpam-5532	311	7	θ	θ	PROPN
ejpam-5532	311	8	.	.	PROPN
ejpam-5532	311	9	suppose	suppose	VERB
ejpam-5532	311	10	,	,	PUNCT
ejpam-5532	311	11	γ	γ	PROPN
ejpam-5532	311	12	⊊	⊊	VERB
ejpam-5532	311	13	η	η	PROPN
ejpam-5532	311	14	∩	∩	ADJ
ejpam-5532	311	15	θ	θ	PROPN
ejpam-5532	311	16	.	.	PUNCT
ejpam-5532	312	1	then	then	ADV
ejpam-5532	312	2	,	,	PUNCT
ejpam-5532	312	3	∃	∃	PROPN
ejpam-5532	312	4	x	x	X
ejpam-5532	312	5	∈	∈	PROPN
ejpam-5532	312	6	m	m	VERB
ejpam-5532	312	7	such	such	ADJ
ejpam-5532	312	8	that	that	SCONJ
ejpam-5532	312	9	γ(x	γ(x	NOUN
ejpam-5532	312	10	)	)	PUNCT
ejpam-5532	312	11	<	<	X
ejpam-5532	312	12	η(x	η(x	ADJ
ejpam-5532	312	13	)	)	PUNCT
ejpam-5532	312	14	∧	∧	PROPN
ejpam-5532	312	15	θ(x	θ(x	PROPN
ejpam-5532	312	16	)	)	PUNCT
ejpam-5532	312	17	.	.	PUNCT
ejpam-5532	313	1	let	let	VERB
ejpam-5532	313	2	γ(x	γ(x	NOUN
ejpam-5532	313	3	)	)	PUNCT
ejpam-5532	314	1	=	=	SYM
ejpam-5532	315	1	t.	t.	NOUN
ejpam-5532	315	2	then	then	ADV
ejpam-5532	315	3	,	,	PUNCT
ejpam-5532	315	4	η(x	η(x	X
ejpam-5532	315	5	)	)	PUNCT
ejpam-5532	315	6	>	>	X
ejpam-5532	315	7	t	t	PROPN
ejpam-5532	315	8	and	and	CCONJ
ejpam-5532	315	9	θ(x	θ(x	PROPN
ejpam-5532	315	10	)	)	PUNCT
ejpam-5532	315	11	>	>	X
ejpam-5532	316	1	t.	t.	NOUN
ejpam-5532	316	2	that	that	PRON
ejpam-5532	316	3	is	be	AUX
ejpam-5532	316	4	,	,	PUNCT
ejpam-5532	316	5	x	x	PROPN
ejpam-5532	316	6	∈	∈	PROPN
ejpam-5532	316	7	η	η	PROPN
ejpam-5532	316	8	>	>	PROPN
ejpam-5532	316	9	t	t	PROPN
ejpam-5532	316	10	and	and	CCONJ
ejpam-5532	316	11	x	x	ADP
ejpam-5532	316	12	∈	∈	PROPN
ejpam-5532	316	13	θ	θ	PROPN
ejpam-5532	316	14	>	>	X
ejpam-5532	316	15	t	t	PROPN
ejpam-5532	316	16	.	.	PUNCT
ejpam-5532	317	1	by	by	ADP
ejpam-5532	317	2	theorem	theorem	ADJ
ejpam-5532	317	3	4	4	NUM
ejpam-5532	317	4	,	,	PUNCT
ejpam-5532	317	5	η	η	PROPN
ejpam-5532	317	6	>	>	PROPN
ejpam-5532	317	7	t	t	PROPN
ejpam-5532	317	8	is	be	AUX
ejpam-5532	317	9	an	an	DET
ejpam-5532	317	10	ideal	ideal	NOUN
ejpam-5532	317	11	of	of	ADP
ejpam-5532	317	12	µ	µ	NOUN
ejpam-5532	317	13	>	>	X
ejpam-5532	317	14	t	t	PROPN
ejpam-5532	317	15	and	and	CCONJ
ejpam-5532	317	16	θ	θ	PROPN
ejpam-5532	317	17	>	>	PROPN
ejpam-5532	317	18	t	t	PROPN
ejpam-5532	317	19	is	be	AUX
ejpam-5532	317	20	a	a	DET
ejpam-5532	317	21	dual	dual	ADJ
ejpam-5532	317	22	ideal	ideal	NOUN
ejpam-5532	317	23	of	of	ADP
ejpam-5532	317	24	µ	µ	NOUN
ejpam-5532	317	25	>	>	X
ejpam-5532	317	26	t	t	PROPN
ejpam-5532	317	27	.	.	PUNCT
ejpam-5532	318	1	this	this	PRON
ejpam-5532	318	2	implies	imply	VERB
ejpam-5532	318	3	η	η	PROPN
ejpam-5532	318	4	>	>	PROPN
ejpam-5532	318	5	t	t	PROPN
ejpam-5532	318	6	=	=	SYM
ejpam-5532	318	7	(	(	PUNCT
ejpam-5532	318	8	η	η	X
ejpam-5532	318	9	>	>	X
ejpam-5532	318	10	t	t	X
ejpam-5532	318	11	]	]	PUNCT
ejpam-5532	318	12	and	and	CCONJ
ejpam-5532	318	13	θ	θ	PROPN
ejpam-5532	318	14	>	>	X
ejpam-5532	318	15	t	t	NOUN
ejpam-5532	318	16	=	=	PUNCT
ejpam-5532	319	1	[	[	X
ejpam-5532	319	2	θ	θ	X
ejpam-5532	319	3	>	>	X
ejpam-5532	319	4	t	t	PROPN
ejpam-5532	319	5	)	)	PUNCT
ejpam-5532	319	6	.	.	PUNCT
ejpam-5532	320	1	since	since	SCONJ
ejpam-5532	320	2	η	η	PROPN
ejpam-5532	320	3	=	=	PROPN
ejpam-5532	320	4	(	(	PUNCT
ejpam-5532	320	5	γ]µ	γ]µ	PROPN
ejpam-5532	320	6	,	,	PUNCT
ejpam-5532	320	7	θ	θ	X
ejpam-5532	320	8	=	=	PUNCT
ejpam-5532	321	1	[	[	X
ejpam-5532	321	2	γ)µ	γ)µ	ADJ
ejpam-5532	321	3	and	and	CCONJ
ejpam-5532	321	4	η(x	η(x	NOUN
ejpam-5532	321	5	)	)	PUNCT
ejpam-5532	321	6	>	>	X
ejpam-5532	321	7	t	t	PROPN
ejpam-5532	321	8	,	,	PUNCT
ejpam-5532	321	9	θ(x	θ(x	PROPN
ejpam-5532	321	10	)	)	PUNCT
ejpam-5532	321	11	>	>	X
ejpam-5532	321	12	t	t	PROPN
ejpam-5532	321	13	,	,	PUNCT
ejpam-5532	321	14	therefore	therefore	ADV
ejpam-5532	321	15	by	by	ADP
ejpam-5532	321	16	theorem	theorem	NOUN
ejpam-5532	321	17	11	11	NUM
ejpam-5532	321	18	,	,	PUNCT
ejpam-5532	321	19	12	12	NUM
ejpam-5532	321	20	,	,	PUNCT
ejpam-5532	321	21	∃	∃	PROPN
ejpam-5532	321	22	r	r	PROPN
ejpam-5532	321	23	,	,	PUNCT
ejpam-5532	321	24	s	s	PART
ejpam-5532	321	25	<	<	X
ejpam-5532	321	26	a0	a0	NOUN
ejpam-5532	321	27	,	,	PUNCT
ejpam-5532	321	28	x	x	SYM
ejpam-5532	321	29	∈	∈	PROPN
ejpam-5532	321	30	(	(	PUNCT
ejpam-5532	321	31	γ	γ	X
ejpam-5532	321	32	>	>	X
ejpam-5532	321	33	r	r	NOUN
ejpam-5532	321	34	]	]	PUNCT
ejpam-5532	321	35	,	,	PUNCT
ejpam-5532	321	36	x	x	SYM
ejpam-5532	321	37	∈	∈	PROPN
ejpam-5532	322	1	[	[	X
ejpam-5532	322	2	γ	γ	X
ejpam-5532	322	3	>	>	X
ejpam-5532	322	4	s	s	PART
ejpam-5532	322	5	)	)	PUNCT
ejpam-5532	322	6	,	,	PUNCT
ejpam-5532	322	7	such	such	ADJ
ejpam-5532	322	8	that	that	SCONJ
ejpam-5532	322	9	t	t	NOUN
ejpam-5532	322	10	<	<	X
ejpam-5532	322	11	r	r	NOUN
ejpam-5532	322	12	and	and	CCONJ
ejpam-5532	322	13	t	t	NOUN
ejpam-5532	322	14	<	<	X
ejpam-5532	322	15	s.	s.	PROPN
ejpam-5532	323	1	thus	thus	ADV
ejpam-5532	323	2	we	we	PRON
ejpam-5532	323	3	have	have	VERB
ejpam-5532	323	4	,	,	PUNCT
ejpam-5532	323	5	x	x	SYM
ejpam-5532	323	6	∈	∈	PROPN
ejpam-5532	323	7	(	(	PUNCT
ejpam-5532	323	8	γ	γ	X
ejpam-5532	323	9	>	>	X
ejpam-5532	323	10	r	r	NOUN
ejpam-5532	323	11	]	]	PUNCT
ejpam-5532	323	12	⊆	⊆	NUM
ejpam-5532	323	13	(	(	PUNCT
ejpam-5532	323	14	γ	γ	X
ejpam-5532	323	15	>	>	X
ejpam-5532	323	16	r∧s	r∧s	X
ejpam-5532	323	17	]	]	PUNCT
ejpam-5532	323	18	,	,	PUNCT
ejpam-5532	323	19	t	t	X
ejpam-5532	323	20	<	<	X
ejpam-5532	323	21	r	r	NOUN
ejpam-5532	323	22	;	;	PUNCT
ejpam-5532	323	23	and	and	CCONJ
ejpam-5532	323	24	x	x	PUNCT
ejpam-5532	323	25	∈	∈	PROPN
ejpam-5532	323	26	[	[	X
ejpam-5532	323	27	γ	γ	X
ejpam-5532	323	28	>	>	X
ejpam-5532	323	29	s	s	PART
ejpam-5532	323	30	)	)	PUNCT
ejpam-5532	323	31	⊆	⊆	NUM
ejpam-5532	323	32	[	[	X
ejpam-5532	323	33	γ	γ	X
ejpam-5532	323	34	>	>	X
ejpam-5532	323	35	r∧s	r∧	NOUN
ejpam-5532	323	36	)	)	PUNCT
ejpam-5532	323	37	,	,	PUNCT
ejpam-5532	323	38	t	t	PROPN
ejpam-5532	323	39	<	<	X
ejpam-5532	323	40	s.	s.	PROPN
ejpam-5532	323	41	that	that	PRON
ejpam-5532	323	42	is	be	AUX
ejpam-5532	323	43	,	,	PUNCT
ejpam-5532	323	44	t	t	PROPN
ejpam-5532	323	45	≤	≤	NUM
ejpam-5532	323	46	r	r	NOUN
ejpam-5532	323	47	∧	∧	PROPN
ejpam-5532	323	48	s.	s.	PROPN
ejpam-5532	323	49	moreover	moreover	ADV
ejpam-5532	323	50	,	,	PUNCT
ejpam-5532	323	51	a.	a.	NOUN
ejpam-5532	323	52	jain	jain	PROPN
ejpam-5532	323	53	,	,	PUNCT
ejpam-5532	323	54	i.	i.	PROPN
ejpam-5532	323	55	jahan	jahan	PROPN
ejpam-5532	323	56	/	/	SYM
ejpam-5532	323	57	eur	eur	PROPN
ejpam-5532	323	58	.	.	PUNCT
ejpam-5532	324	1	j.	j.	PROPN
ejpam-5532	324	2	pure	pure	PROPN
ejpam-5532	324	3	appl	appl	PROPN
ejpam-5532	324	4	.	.	PROPN
ejpam-5532	324	5	math	math	PROPN
ejpam-5532	324	6	,	,	PUNCT
ejpam-5532	324	7	18	18	NUM
ejpam-5532	324	8	(	(	PUNCT
ejpam-5532	324	9	1	1	NUM
ejpam-5532	324	10	)	)	PUNCT
ejpam-5532	324	11	(	(	PUNCT
ejpam-5532	324	12	2025	2025	NUM
ejpam-5532	324	13	)	)	PUNCT
ejpam-5532	324	14	,	,	PUNCT
ejpam-5532	324	15	5532	5532	NUM
ejpam-5532	324	16	15	15	NUM
ejpam-5532	324	17	of	of	ADP
ejpam-5532	324	18	20	20	NUM
ejpam-5532	324	19	∃	∃	NUM
ejpam-5532	324	20	x1	x1	PROPN
ejpam-5532	324	21	,	,	PUNCT
ejpam-5532	324	22	.	.	PUNCT
ejpam-5532	324	23	.	.	PUNCT
ejpam-5532	325	1	.	.	PUNCT
ejpam-5532	326	1	,	,	PUNCT
ejpam-5532	326	2	xn	xn	PROPN
ejpam-5532	326	3	∈	∈	PROPN
ejpam-5532	326	4	γ	γ	X
ejpam-5532	326	5	>	>	PROPN
ejpam-5532	326	6	r∧s	r∧s	PROPN
ejpam-5532	326	7	⊆	⊆	NUM
ejpam-5532	326	8	µ	µ	SYM
ejpam-5532	326	9	>	>	X
ejpam-5532	326	10	r∧s	r∧s	PROPN
ejpam-5532	326	11	∃	∃	PROPN
ejpam-5532	326	12	y1	y1	PROPN
ejpam-5532	326	13	,	,	PUNCT
ejpam-5532	326	14	.	.	PUNCT
ejpam-5532	326	15	.	.	PUNCT
ejpam-5532	327	1	.	.	PUNCT
ejpam-5532	328	1	,	,	PUNCT
ejpam-5532	328	2	ym	ym	PROPN
ejpam-5532	328	3	∈	∈	PROPN
ejpam-5532	328	4	γ	γ	X
ejpam-5532	328	5	>	>	PROPN
ejpam-5532	328	6	r∧s	r∧s	PROPN
ejpam-5532	328	7	⊆	⊆	NUM
ejpam-5532	328	8	µ	µ	X
ejpam-5532	328	9	>	>	X
ejpam-5532	328	10	r∧s	r∧	NOUN
ejpam-5532	328	11	such	such	ADJ
ejpam-5532	328	12	that	that	SCONJ
ejpam-5532	328	13	a	a	DET
ejpam-5532	328	14	=	=	X
ejpam-5532	328	15	y1	y1	NOUN
ejpam-5532	328	16	∧	∧	PROPN
ejpam-5532	328	17	.	.	PUNCT
ejpam-5532	328	18	.	.	PUNCT
ejpam-5532	328	19	.	.	PUNCT
ejpam-5532	329	1	∧	∧	NOUN
ejpam-5532	329	2	ym	ym	NOUN
ejpam-5532	329	3	≤	≤	NUM
ejpam-5532	329	4	x	x	PUNCT
ejpam-5532	329	5	≤	≤	NUM
ejpam-5532	329	6	x1	x1	PRON
ejpam-5532	329	7	∨	∨	NOUN
ejpam-5532	329	8	.	.	PUNCT
ejpam-5532	329	9	.	.	PUNCT
ejpam-5532	329	10	.	.	PUNCT
ejpam-5532	330	1	∨	∨	NUM
ejpam-5532	330	2	xn	xn	X
ejpam-5532	331	1	=	=	PROPN
ejpam-5532	331	2	b.	b.	PROPN
ejpam-5532	331	3	since	since	SCONJ
ejpam-5532	331	4	µ	µ	PROPN
ejpam-5532	331	5	>	>	X
ejpam-5532	331	6	r∧s	r∧s	PROPN
ejpam-5532	331	7	is	be	AUX
ejpam-5532	331	8	a	a	DET
ejpam-5532	331	9	sublattice	sublattice	NOUN
ejpam-5532	331	10	of	of	ADP
ejpam-5532	331	11	m	m	PRON
ejpam-5532	331	12	and	and	CCONJ
ejpam-5532	331	13	(	(	PUNCT
ejpam-5532	331	14	γ	γ	X
ejpam-5532	331	15	>	>	X
ejpam-5532	331	16	r∧s	r∧s	X
ejpam-5532	331	17	]	]	PUNCT
ejpam-5532	331	18	is	be	AUX
ejpam-5532	331	19	an	an	DET
ejpam-5532	331	20	ideal	ideal	NOUN
ejpam-5532	331	21	of	of	ADP
ejpam-5532	331	22	µ	µ	X
ejpam-5532	331	23	>	>	X
ejpam-5532	331	24	r∧s	r∧	NOUN
ejpam-5532	331	25	,	,	PUNCT
ejpam-5532	331	26	we	we	PRON
ejpam-5532	331	27	have	have	VERB
ejpam-5532	331	28	a	a	DET
ejpam-5532	331	29	,	,	PUNCT
ejpam-5532	331	30	b	b	NOUN
ejpam-5532	331	31	,	,	PUNCT
ejpam-5532	331	32	x	x	PROPN
ejpam-5532	331	33	∈	∈	PROPN
ejpam-5532	331	34	µ	µ	X
ejpam-5532	331	35	>	>	X
ejpam-5532	331	36	r∧s	r∧	NOUN
ejpam-5532	331	37	.	.	PUNCT
ejpam-5532	332	1	we	we	PRON
ejpam-5532	332	2	also	also	ADV
ejpam-5532	332	3	have	have	VERB
ejpam-5532	332	4	a	a	DET
ejpam-5532	332	5	,	,	PUNCT
ejpam-5532	332	6	b	b	PROPN
ejpam-5532	332	7	∈	∈	PROPN
ejpam-5532	332	8	γ	γ	X
ejpam-5532	332	9	>	>	X
ejpam-5532	332	10	r∧s	r∧	NOUN
ejpam-5532	332	11	as	as	ADP
ejpam-5532	332	12	γ	γ	PROPN
ejpam-5532	332	13	>	>	X
ejpam-5532	332	14	r∧s	r∧s	PROPN
ejpam-5532	332	15	is	be	AUX
ejpam-5532	332	16	a	a	DET
ejpam-5532	332	17	sublattice	sublattice	NOUN
ejpam-5532	332	18	of	of	ADP
ejpam-5532	332	19	µ	µ	NOUN
ejpam-5532	332	20	>	>	X
ejpam-5532	332	21	r∧s	r∧	NOUN
ejpam-5532	332	22	.	.	PUNCT
ejpam-5532	333	1	now	now	ADV
ejpam-5532	333	2	,	,	PUNCT
ejpam-5532	333	3	γ	γ	X
ejpam-5532	333	4	being	be	AUX
ejpam-5532	333	5	an	an	DET
ejpam-5532	333	6	l	l	NOUN
ejpam-5532	333	7	-	-	ADJ
ejpam-5532	333	8	convex	convex	ADJ
ejpam-5532	333	9	sublattice	sublattice	NOUN
ejpam-5532	333	10	of	of	ADP
ejpam-5532	333	11	µ	µ	NUM
ejpam-5532	333	12	,	,	PUNCT
ejpam-5532	333	13	γ	γ	PROPN
ejpam-5532	333	14	>	>	X
ejpam-5532	333	15	r∧s	r∧s	PROPN
ejpam-5532	333	16	is	be	AUX
ejpam-5532	333	17	a	a	DET
ejpam-5532	333	18	convex	convex	ADJ
ejpam-5532	333	19	sublattice	sublattice	NOUN
ejpam-5532	333	20	of	of	ADP
ejpam-5532	333	21	µ	µ	NOUN
ejpam-5532	333	22	>	>	X
ejpam-5532	333	23	r∧s	r∧	NOUN
ejpam-5532	333	24	.	.	PUNCT
ejpam-5532	334	1	consequently	consequently	ADV
ejpam-5532	334	2	,	,	PUNCT
ejpam-5532	334	3	x	x	PROPN
ejpam-5532	334	4	∈	∈	PROPN
ejpam-5532	334	5	γ	γ	X
ejpam-5532	334	6	>	>	X
ejpam-5532	334	7	r∧s	r∧	NOUN
ejpam-5532	334	8	.	.	PUNCT
ejpam-5532	335	1	that	that	ADV
ejpam-5532	335	2	is	be	AUX
ejpam-5532	335	3	,	,	PUNCT
ejpam-5532	335	4	t	t	NOUN
ejpam-5532	335	5	=	=	PUNCT
ejpam-5532	335	6	γ(x	γ(x	PROPN
ejpam-5532	335	7	)	)	PUNCT
ejpam-5532	335	8	>	>	X
ejpam-5532	336	1	r∧s	r∧s	PROPN
ejpam-5532	336	2	,	,	PUNCT
ejpam-5532	336	3	which	which	PRON
ejpam-5532	336	4	contradicts	contradict	VERB
ejpam-5532	336	5	the	the	DET
ejpam-5532	336	6	fact	fact	NOUN
ejpam-5532	336	7	that	that	SCONJ
ejpam-5532	336	8	t	t	NOUN
ejpam-5532	336	9	<	<	X
ejpam-5532	336	10	r	r	NOUN
ejpam-5532	336	11	and	and	CCONJ
ejpam-5532	336	12	t	t	X
ejpam-5532	336	13	<	<	X
ejpam-5532	336	14	s	s	X
ejpam-5532	336	15	(	(	PUNCT
ejpam-5532	336	16	as	as	SCONJ
ejpam-5532	336	17	l	l	NOUN
ejpam-5532	336	18	is	be	AUX
ejpam-5532	336	19	a	a	DET
ejpam-5532	336	20	chain	chain	NOUN
ejpam-5532	336	21	)	)	PUNCT
ejpam-5532	336	22	.	.	PUNCT
ejpam-5532	337	1	hence	hence	ADV
ejpam-5532	337	2	,	,	PUNCT
ejpam-5532	337	3	γ(x	γ(x	ADV
ejpam-5532	337	4	)	)	PUNCT
ejpam-5532	337	5	=	=	SYM
ejpam-5532	337	6	(	(	PUNCT
ejpam-5532	337	7	η	η	PROPN
ejpam-5532	337	8	∩	∩	NOUN
ejpam-5532	337	9	θ)(x	θ)(x	PROPN
ejpam-5532	337	10	)	)	PUNCT
ejpam-5532	337	11	,	,	PUNCT
ejpam-5532	337	12	∀	∀	PUNCT
ejpam-5532	337	13	x	x	SYM
ejpam-5532	337	14	∈	∈	NOUN
ejpam-5532	337	15	m.	m.	NOUN
ejpam-5532	337	16	that	that	PRON
ejpam-5532	337	17	is	be	AUX
ejpam-5532	337	18	γ	γ	PROPN
ejpam-5532	337	19	=	=	SYM
ejpam-5532	337	20	η	η	PROPN
ejpam-5532	337	21	∩	∩	PROPN
ejpam-5532	337	22	θ	θ	PROPN
ejpam-5532	337	23	.	.	PUNCT
ejpam-5532	338	1	thus	thus	ADV
ejpam-5532	338	2	each	each	DET
ejpam-5532	338	3	convex	convex	NOUN
ejpam-5532	338	4	sublattice	sublattice	NOUN
ejpam-5532	338	5	γ	γ	NOUN
ejpam-5532	338	6	of	of	ADP
ejpam-5532	338	7	µ	µ	PROPN
ejpam-5532	338	8	can	can	AUX
ejpam-5532	338	9	be	be	AUX
ejpam-5532	338	10	expressed	express	VERB
ejpam-5532	338	11	in	in	ADP
ejpam-5532	338	12	this	this	DET
ejpam-5532	338	13	form	form	NOUN
ejpam-5532	338	14	.	.	PUNCT
ejpam-5532	339	1	to	to	PART
ejpam-5532	339	2	prove	prove	VERB
ejpam-5532	339	3	the	the	DET
ejpam-5532	339	4	uniqueness	uniqueness	NOUN
ejpam-5532	339	5	of	of	ADP
ejpam-5532	339	6	this	this	DET
ejpam-5532	339	7	representation	representation	NOUN
ejpam-5532	339	8	,	,	PUNCT
ejpam-5532	339	9	suppose	suppose	VERB
ejpam-5532	339	10	there	there	PRON
ejpam-5532	339	11	exists	exist	VERB
ejpam-5532	339	12	an	an	DET
ejpam-5532	339	13	l	l	NOUN
ejpam-5532	339	14	-	-	PUNCT
ejpam-5532	339	15	ideal	ideal	ADJ
ejpam-5532	339	16	η	η	PROPN
ejpam-5532	339	17	of	of	ADP
ejpam-5532	339	18	µ	µ	PROPN
ejpam-5532	339	19	and	and	CCONJ
ejpam-5532	339	20	an	an	DET
ejpam-5532	339	21	l	l	ADJ
ejpam-5532	339	22	-	-	ADJ
ejpam-5532	339	23	dual	dual	ADJ
ejpam-5532	339	24	ideal	ideal	ADJ
ejpam-5532	339	25	θ	θ	PROPN
ejpam-5532	339	26	of	of	ADP
ejpam-5532	339	27	µ	µ	PRON
ejpam-5532	339	28	such	such	ADJ
ejpam-5532	339	29	that	that	SCONJ
ejpam-5532	339	30	γ	γ	NOUN
ejpam-5532	339	31	=	=	SYM
ejpam-5532	339	32	η∩θ	η∩θ	PROPN
ejpam-5532	339	33	.	.	PUNCT
ejpam-5532	340	1	we	we	PRON
ejpam-5532	340	2	prove	prove	VERB
ejpam-5532	340	3	that	that	SCONJ
ejpam-5532	340	4	η	η	PROPN
ejpam-5532	340	5	=	=	PROPN
ejpam-5532	340	6	(	(	PUNCT
ejpam-5532	340	7	γ]µ	γ]µ	PROPN
ejpam-5532	340	8	and	and	CCONJ
ejpam-5532	340	9	θ	θ	NOUN
ejpam-5532	340	10	=	=	PUNCT
ejpam-5532	341	1	[	[	X
ejpam-5532	341	2	γ)µ.	γ)µ.	X
ejpam-5532	341	3	since	since	SCONJ
ejpam-5532	341	4	γ	γ	PROPN
ejpam-5532	341	5	⊆	⊆	NUM
ejpam-5532	341	6	η	η	NOUN
ejpam-5532	341	7	,	,	PUNCT
ejpam-5532	341	8	therefore	therefore	ADV
ejpam-5532	341	9	(	(	PUNCT
ejpam-5532	341	10	γ]µ	γ]µ	PROPN
ejpam-5532	341	11	⊆	⊆	NUM
ejpam-5532	341	12	η	η	PROPN
ejpam-5532	341	13	.	.	PROPN
ejpam-5532	341	14	for	for	ADP
ejpam-5532	341	15	reverse	reverse	ADJ
ejpam-5532	341	16	inclusion	inclusion	NOUN
ejpam-5532	341	17	,	,	PUNCT
ejpam-5532	341	18	let	let	VERB
ejpam-5532	341	19	x	x	X
ejpam-5532	341	20	∈	∈	NOUN
ejpam-5532	341	21	m	m	NOUN
ejpam-5532	341	22	and	and	CCONJ
ejpam-5532	341	23	η(x	η(x	NOUN
ejpam-5532	341	24	)	)	PUNCT
ejpam-5532	341	25	=	=	SYM
ejpam-5532	342	1	t.	t.	NOUN
ejpam-5532	342	2	then	then	ADV
ejpam-5532	342	3	clearly	clearly	ADV
ejpam-5532	342	4	,	,	PUNCT
ejpam-5532	342	5	t	t	PROPN
ejpam-5532	342	6	≤	≤	PROPN
ejpam-5532	342	7	a0	a0	PROPN
ejpam-5532	342	8	.	.	PUNCT
ejpam-5532	343	1	since	since	SCONJ
ejpam-5532	343	2	γ	γ	PROPN
ejpam-5532	343	3	⊆	⊆	NUM
ejpam-5532	343	4	η	η	NOUN
ejpam-5532	343	5	,	,	PUNCT
ejpam-5532	343	6	therefore	therefore	ADV
ejpam-5532	343	7	γt	γt	VERB
ejpam-5532	343	8	⊆	⊆	NUM
ejpam-5532	343	9	ηt	ηt	ADP
ejpam-5532	343	10	.	.	PUNCT
ejpam-5532	344	1	if	if	SCONJ
ejpam-5532	344	2	y	y	PROPN
ejpam-5532	344	3	∈	∈	PROPN
ejpam-5532	344	4	γt	γt	X
ejpam-5532	344	5	,	,	PUNCT
ejpam-5532	344	6	then	then	ADV
ejpam-5532	344	7	x	x	X
ejpam-5532	344	8	,	,	PUNCT
ejpam-5532	344	9	y	y	PROPN
ejpam-5532	344	10	∈	∈	PROPN
ejpam-5532	344	11	ηt	ηt	ADP
ejpam-5532	344	12	.	.	PUNCT
ejpam-5532	345	1	this	this	PRON
ejpam-5532	345	2	implies	imply	VERB
ejpam-5532	345	3	x	x	PUNCT
ejpam-5532	345	4	∨	∨	NUM
ejpam-5532	345	5	y	y	PROPN
ejpam-5532	345	6	∈	∈	PROPN
ejpam-5532	345	7	ηt	ηt	ADP
ejpam-5532	345	8	.	.	PUNCT
ejpam-5532	346	1	we	we	PRON
ejpam-5532	346	2	also	also	ADV
ejpam-5532	346	3	have	have	VERB
ejpam-5532	346	4	y	y	PROPN
ejpam-5532	346	5	∈	∈	PROPN
ejpam-5532	346	6	γt	γt	ADP
ejpam-5532	346	7	⊆	⊆	NUM
ejpam-5532	346	8	[	[	X
ejpam-5532	346	9	γt	γt	NOUN
ejpam-5532	346	10	)	)	PUNCT
ejpam-5532	346	11	,	,	PUNCT
ejpam-5532	346	12	where	where	SCONJ
ejpam-5532	346	13	[	[	X
ejpam-5532	346	14	γt	γt	NOUN
ejpam-5532	346	15	)	)	PUNCT
ejpam-5532	346	16	is	be	AUX
ejpam-5532	346	17	a	a	DET
ejpam-5532	346	18	dual	dual	ADJ
ejpam-5532	346	19	ideal	ideal	NOUN
ejpam-5532	346	20	of	of	ADP
ejpam-5532	346	21	µt	µt	PRON
ejpam-5532	346	22	and	and	CCONJ
ejpam-5532	346	23	y	y	PROPN
ejpam-5532	346	24	≤	≤	PROPN
ejpam-5532	346	25	x	x	PUNCT
ejpam-5532	346	26	∨	∨	NUM
ejpam-5532	346	27	y	y	PROPN
ejpam-5532	346	28	in	in	ADP
ejpam-5532	346	29	µt	µt	PROPN
ejpam-5532	346	30	.	.	PUNCT
ejpam-5532	347	1	therefore	therefore	ADV
ejpam-5532	347	2	,	,	PUNCT
ejpam-5532	347	3	x	x	PROPN
ejpam-5532	347	4	∨	∨	NUM
ejpam-5532	347	5	y	y	PROPN
ejpam-5532	347	6	∈	∈	PROPN
ejpam-5532	347	7	[	[	X
ejpam-5532	347	8	γt	γt	NOUN
ejpam-5532	347	9	)	)	PUNCT
ejpam-5532	347	10	and	and	CCONJ
ejpam-5532	347	11	t	t	PROPN
ejpam-5532	347	12	≤	≤	PROPN
ejpam-5532	347	13	a0	a0	PROPN
ejpam-5532	347	14	.	.	PUNCT
ejpam-5532	348	1	this	this	PRON
ejpam-5532	348	2	implies	imply	VERB
ejpam-5532	348	3	t	t	NOUN
ejpam-5532	348	4	≤	≤	X
ejpam-5532	349	1	[	[	PUNCT
ejpam-5532	349	2	γ)(x	γ)(x	PROPN
ejpam-5532	349	3	∨	∨	NUM
ejpam-5532	349	4	y	y	NOUN
ejpam-5532	349	5	)	)	PUNCT
ejpam-5532	349	6	=	=	SYM
ejpam-5532	349	7	θ(x	θ(x	PROPN
ejpam-5532	349	8	∨	∨	NUM
ejpam-5532	349	9	y	y	PROPN
ejpam-5532	349	10	)	)	PUNCT
ejpam-5532	349	11	.	.	PUNCT
ejpam-5532	350	1	we	we	PRON
ejpam-5532	350	2	also	also	ADV
ejpam-5532	350	3	have	have	AUX
ejpam-5532	350	4	η(x∨	η(x∨	VERB
ejpam-5532	350	5	y	y	NOUN
ejpam-5532	350	6	)	)	PUNCT
ejpam-5532	350	7	≥	≥	NOUN
ejpam-5532	350	8	t.	t.	PROPN
ejpam-5532	350	9	therefore	therefore	ADV
ejpam-5532	350	10	,	,	PUNCT
ejpam-5532	350	11	θ(x∨	θ(x∨	VERB
ejpam-5532	350	12	y)∧	y)∧	PROPN
ejpam-5532	350	13	η(x∨	η(x∨	VERB
ejpam-5532	350	14	y	y	NOUN
ejpam-5532	350	15	)	)	PUNCT
ejpam-5532	350	16	≥	≥	PROPN
ejpam-5532	350	17	t.	t.	NOUN
ejpam-5532	350	18	that	that	PRON
ejpam-5532	350	19	is	be	AUX
ejpam-5532	350	20	,	,	PUNCT
ejpam-5532	350	21	γ(x∨	γ(x∨	VERB
ejpam-5532	350	22	y	y	NOUN
ejpam-5532	350	23	)	)	PUNCT
ejpam-5532	350	24	≥	≥	NOUN
ejpam-5532	350	25	t.	t.	PROPN
ejpam-5532	350	26	thus	thus	ADV
ejpam-5532	350	27	,	,	PUNCT
ejpam-5532	350	28	x	x	PROPN
ejpam-5532	350	29	∨	∨	NUM
ejpam-5532	350	30	y	y	PROPN
ejpam-5532	350	31	∈	∈	PROPN
ejpam-5532	350	32	ηt	ηt	ADP
ejpam-5532	350	33	⊆	⊆	NUM
ejpam-5532	350	34	(	(	PUNCT
ejpam-5532	350	35	γt	γt	NOUN
ejpam-5532	350	36	]	]	PUNCT
ejpam-5532	350	37	.	.	PUNCT
ejpam-5532	351	1	note	note	VERB
ejpam-5532	351	2	that	that	SCONJ
ejpam-5532	351	3	(	(	PUNCT
ejpam-5532	351	4	γt	γt	NOUN
ejpam-5532	351	5	]	]	X
ejpam-5532	351	6	is	be	AUX
ejpam-5532	351	7	an	an	DET
ejpam-5532	351	8	ideal	ideal	NOUN
ejpam-5532	351	9	of	of	ADP
ejpam-5532	351	10	µt	µt	PRON
ejpam-5532	351	11	and	and	CCONJ
ejpam-5532	351	12	x	x	SYM
ejpam-5532	351	13	≤	≤	NUM
ejpam-5532	351	14	x	x	PUNCT
ejpam-5532	351	15	∨	∨	NUM
ejpam-5532	351	16	y.	y.	PROPN
ejpam-5532	351	17	hence	hence	ADV
ejpam-5532	351	18	,	,	PUNCT
ejpam-5532	351	19	x	x	SYM
ejpam-5532	351	20	∈	∈	PROPN
ejpam-5532	351	21	(	(	PUNCT
ejpam-5532	351	22	γt	γt	NOUN
ejpam-5532	351	23	]	]	PUNCT
ejpam-5532	351	24	and	and	CCONJ
ejpam-5532	351	25	t	t	PROPN
ejpam-5532	351	26	≤	≤	PROPN
ejpam-5532	351	27	a0	a0	PROPN
ejpam-5532	351	28	.	.	PUNCT
ejpam-5532	352	1	thus	thus	ADV
ejpam-5532	352	2	,	,	PUNCT
ejpam-5532	352	3	η(x	η(x	X
ejpam-5532	352	4	)	)	PUNCT
ejpam-5532	352	5	=	=	SYM
ejpam-5532	352	6	t	t	PROPN
ejpam-5532	352	7	≤	≤	NOUN
ejpam-5532	352	8	∨r≤a0{r	∨r≤a0{r	VERB
ejpam-5532	352	9	:	:	PUNCT
ejpam-5532	352	10	x	x	SYM
ejpam-5532	352	11	∈	∈	PROPN
ejpam-5532	352	12	(	(	PUNCT
ejpam-5532	352	13	γr	γr	PROPN
ejpam-5532	352	14	]	]	X
ejpam-5532	352	15	}	}	PUNCT
ejpam-5532	352	16	=	=	SYM
ejpam-5532	352	17	(	(	PUNCT
ejpam-5532	352	18	γ](x	γ](x	NOUN
ejpam-5532	352	19	)	)	PUNCT
ejpam-5532	352	20	.	.	PUNCT
ejpam-5532	353	1	hence	hence	ADV
ejpam-5532	353	2	,	,	PUNCT
ejpam-5532	353	3	η	η	PROPN
ejpam-5532	353	4	⊆	⊆	NUM
ejpam-5532	353	5	(	(	PUNCT
ejpam-5532	353	6	γ	γ	X
ejpam-5532	353	7	]	]	X
ejpam-5532	353	8	and	and	CCONJ
ejpam-5532	353	9	therefore	therefore	ADV
ejpam-5532	353	10	,	,	PUNCT
ejpam-5532	353	11	η	η	PROPN
ejpam-5532	353	12	=	=	X
ejpam-5532	353	13	(	(	PUNCT
ejpam-5532	353	14	γ	γ	X
ejpam-5532	353	15	]	]	X
ejpam-5532	353	16	.	.	PUNCT
ejpam-5532	354	1	similarly	similarly	ADV
ejpam-5532	354	2	,	,	PUNCT
ejpam-5532	354	3	θ	θ	X
ejpam-5532	354	4	=	=	PUNCT
ejpam-5532	355	1	[	[	X
ejpam-5532	355	2	γ	γ	X
ejpam-5532	355	3	)	)	PUNCT
ejpam-5532	355	4	.	.	PUNCT
ejpam-5532	356	1	consequently	consequently	ADV
ejpam-5532	356	2	,	,	PUNCT
ejpam-5532	356	3	we	we	PRON
ejpam-5532	356	4	get	get	VERB
ejpam-5532	356	5	the	the	DET
ejpam-5532	356	6	uniqueness	uniqueness	NOUN
ejpam-5532	356	7	of	of	ADP
ejpam-5532	356	8	the	the	DET
ejpam-5532	356	9	representation	representation	NOUN
ejpam-5532	356	10	of	of	ADP
ejpam-5532	356	11	γ	γ	PROPN
ejpam-5532	356	12	.	.	PROPN
ejpam-5532	356	13	4	4	NUM
ejpam-5532	356	14	.	.	X
ejpam-5532	356	15	complement	complement	NOUN
ejpam-5532	356	16	of	of	ADP
ejpam-5532	356	17	an	an	DET
ejpam-5532	356	18	l	l	NOUN
ejpam-5532	356	19	-	-	ADJ
ejpam-5532	356	20	set	set	VERB
ejpam-5532	356	21	and	and	CCONJ
ejpam-5532	356	22	l	l	ADJ
ejpam-5532	356	23	-	-	ADJ
ejpam-5532	356	24	prime	prime	ADJ
ejpam-5532	356	25	ideal	ideal	NOUN
ejpam-5532	356	26	and	and	CCONJ
ejpam-5532	356	27	l	l	ADJ
ejpam-5532	356	28	-	-	ADJ
ejpam-5532	356	29	maximal	maximal	ADJ
ejpam-5532	356	30	ideal	ideal	NOUN
ejpam-5532	356	31	of	of	ADP
ejpam-5532	356	32	an	an	DET
ejpam-5532	356	33	l	l	NOUN
ejpam-5532	356	34	-	-	NOUN
ejpam-5532	356	35	lattice	lattice	NOUN
ejpam-5532	356	36	in	in	ADP
ejpam-5532	356	37	this	this	DET
ejpam-5532	356	38	section	section	NOUN
ejpam-5532	356	39	,	,	PUNCT
ejpam-5532	356	40	the	the	DET
ejpam-5532	356	41	important	important	ADJ
ejpam-5532	356	42	concept	concept	NOUN
ejpam-5532	356	43	of	of	ADP
ejpam-5532	356	44	an	an	DET
ejpam-5532	356	45	order	order	NOUN
ejpam-5532	356	46	reversing	reverse	VERB
ejpam-5532	356	47	involution	involution	NOUN
ejpam-5532	356	48	on	on	ADP
ejpam-5532	356	49	a	a	DET
ejpam-5532	356	50	lattice	lattice	NOUN
ejpam-5532	356	51	is	be	AUX
ejpam-5532	356	52	discussed	discuss	VERB
ejpam-5532	356	53	.	.	PUNCT
ejpam-5532	357	1	based	base	VERB
ejpam-5532	357	2	on	on	ADP
ejpam-5532	357	3	this	this	DET
ejpam-5532	357	4	notion	notion	NOUN
ejpam-5532	357	5	,	,	PUNCT
ejpam-5532	357	6	the	the	DET
ejpam-5532	357	7	complement	complement	NOUN
ejpam-5532	357	8	of	of	ADP
ejpam-5532	357	9	an	an	DET
ejpam-5532	357	10	l	l	NOUN
ejpam-5532	357	11	-	-	PUNCT
ejpam-5532	357	12	lattice	lattice	NOUN
ejpam-5532	357	13	is	be	AUX
ejpam-5532	357	14	defined	define	VERB
ejpam-5532	357	15	.	.	PUNCT
ejpam-5532	358	1	these	these	DET
ejpam-5532	358	2	notions	notion	NOUN
ejpam-5532	358	3	occur	occur	VERB
ejpam-5532	358	4	frequently	frequently	ADV
ejpam-5532	358	5	in	in	ADP
ejpam-5532	358	6	lattice	lattice	PROPN
ejpam-5532	358	7	implication	implication	NOUN
ejpam-5532	358	8	algebras	algebra	NOUN
ejpam-5532	358	9	and	and	CCONJ
ejpam-5532	358	10	l	l	ADJ
ejpam-5532	358	11	-	-	ADJ
ejpam-5532	358	12	topological	topological	ADJ
ejpam-5532	358	13	spaces	space	NOUN
ejpam-5532	358	14	[	[	X
ejpam-5532	358	15	11	11	NUM
ejpam-5532	358	16	,	,	PUNCT
ejpam-5532	358	17	13	13	NUM
ejpam-5532	358	18	,	,	PUNCT
ejpam-5532	358	19	16	16	NUM
ejpam-5532	358	20	,	,	PUNCT
ejpam-5532	358	21	17	17	NUM
ejpam-5532	358	22	]	]	PUNCT
ejpam-5532	358	23	.	.	PUNCT
ejpam-5532	359	1	if	if	SCONJ
ejpam-5532	359	2	(	(	PUNCT
ejpam-5532	359	3	l,≤,∧,∨	l,≤,∧,∨	NOUN
ejpam-5532	359	4	)	)	PUNCT
ejpam-5532	359	5	is	be	AUX
ejpam-5532	359	6	a	a	DET
ejpam-5532	359	7	lattice	lattice	NOUN
ejpam-5532	359	8	,	,	PUNCT
ejpam-5532	359	9	then	then	ADV
ejpam-5532	359	10	l∗(=	l∗(=	PROPN
ejpam-5532	359	11	l	l	PROPN
ejpam-5532	359	12	)	)	PUNCT
ejpam-5532	359	13	is	be	AUX
ejpam-5532	359	14	also	also	ADV
ejpam-5532	359	15	a	a	DET
ejpam-5532	359	16	lattice	lattice	NOUN
ejpam-5532	359	17	with	with	ADP
ejpam-5532	359	18	respect	respect	NOUN
ejpam-5532	359	19	to	to	PART
ejpam-5532	359	20	reverse	reverse	VERB
ejpam-5532	359	21	order	order	NOUN
ejpam-5532	359	22	“	"	PUNCT
ejpam-5532	359	23	≥	≥	X
ejpam-5532	359	24	”	"	PUNCT
ejpam-5532	359	25	,	,	PUNCT
ejpam-5532	359	26	where	where	SCONJ
ejpam-5532	359	27	y	y	PROPN
ejpam-5532	359	28	≥	≥	VERB
ejpam-5532	359	29	x	x	X
ejpam-5532	359	30	in	in	ADP
ejpam-5532	359	31	l∗	l∗	PROPN
ejpam-5532	359	32	if	if	SCONJ
ejpam-5532	359	33	and	and	CCONJ
ejpam-5532	359	34	only	only	ADV
ejpam-5532	359	35	if	if	SCONJ
ejpam-5532	359	36	x	x	PROPN
ejpam-5532	359	37	≤	≤	X
ejpam-5532	359	38	y	y	NOUN
ejpam-5532	359	39	in	in	ADP
ejpam-5532	359	40	l.	l.	PROPN
ejpam-5532	359	41	an	an	DET
ejpam-5532	359	42	order	order	NOUN
ejpam-5532	359	43	reversing	reverse	VERB
ejpam-5532	359	44	involution	involution	NOUN
ejpam-5532	359	45	on	on	ADP
ejpam-5532	359	46	a	a	DET
ejpam-5532	359	47	lattice	lattice	NOUN
ejpam-5532	359	48	l	l	NOUN
ejpam-5532	359	49	is	be	AUX
ejpam-5532	359	50	defined	define	VERB
ejpam-5532	359	51	as	as	ADP
ejpam-5532	359	52	a	a	DET
ejpam-5532	359	53	bijection	bijection	NOUN
ejpam-5532	359	54	τ	τ	X
ejpam-5532	359	55	:	:	PUNCT
ejpam-5532	359	56	l	l	X
ejpam-5532	359	57	→	→	SYM
ejpam-5532	359	58	l∗	l∗	PROPN
ejpam-5532	359	59	satisfying	satisfying	NOUN
ejpam-5532	359	60	τ(τ(x	τ(τ(x	NOUN
ejpam-5532	359	61	)	)	PUNCT
ejpam-5532	359	62	)	)	PUNCT
ejpam-5532	360	1	=	=	SYM
ejpam-5532	360	2	x	x	NOUN
ejpam-5532	360	3	,	,	PUNCT
ejpam-5532	360	4	∀	∀	X
ejpam-5532	360	5	x	x	SYM
ejpam-5532	360	6	∈	∈	NOUN
ejpam-5532	360	7	l	l	NOUN
ejpam-5532	360	8	and	and	CCONJ
ejpam-5532	360	9	x	x	SYM
ejpam-5532	360	10	≤	≤	ADJ
ejpam-5532	360	11	y	y	NOUN
ejpam-5532	360	12	in	in	ADP
ejpam-5532	360	13	l	l	NOUN
ejpam-5532	360	14	if	if	SCONJ
ejpam-5532	360	15	and	and	CCONJ
ejpam-5532	360	16	only	only	ADV
ejpam-5532	360	17	if	if	SCONJ
ejpam-5532	360	18	τ(y	τ(y	NOUN
ejpam-5532	360	19	)	)	PUNCT
ejpam-5532	360	20	≤	≤	NOUN
ejpam-5532	360	21	τ(x	τ(x	NOUN
ejpam-5532	360	22	)	)	PUNCT
ejpam-5532	360	23	in	in	ADP
ejpam-5532	360	24	l	l	NOUN
ejpam-5532	360	25	=	=	PUNCT
ejpam-5532	361	1	l∗.	l∗.	NOUN
ejpam-5532	361	2	it	it	PRON
ejpam-5532	361	3	is	be	AUX
ejpam-5532	361	4	interesting	interesting	ADJ
ejpam-5532	361	5	to	to	PART
ejpam-5532	361	6	note	note	VERB
ejpam-5532	361	7	that	that	SCONJ
ejpam-5532	361	8	in	in	ADP
ejpam-5532	361	9	all	all	DET
ejpam-5532	361	10	the	the	DET
ejpam-5532	361	11	examples	example	NOUN
ejpam-5532	361	12	provided	provide	VERB
ejpam-5532	361	13	in	in	ADP
ejpam-5532	361	14	section	section	NOUN
ejpam-5532	361	15	3	3	NUM
ejpam-5532	361	16	,	,	PUNCT
ejpam-5532	361	17	there	there	PRON
ejpam-5532	361	18	is	be	VERB
ejpam-5532	361	19	an	an	DET
ejpam-5532	361	20	order	order	NOUN
ejpam-5532	361	21	reversing	reverse	VERB
ejpam-5532	361	22	involution	involution	NOUN
ejpam-5532	361	23	on	on	ADP
ejpam-5532	361	24	the	the	DET
ejpam-5532	361	25	lattice	lattice	PROPN
ejpam-5532	361	26	l	l	PROPN
ejpam-5532	361	27	of	of	ADP
ejpam-5532	361	28	truth	truth	NOUN
ejpam-5532	361	29	values	value	NOUN
ejpam-5532	361	30	.	.	PUNCT
ejpam-5532	362	1	the	the	DET
ejpam-5532	362	2	following	follow	VERB
ejpam-5532	362	3	result	result	NOUN
ejpam-5532	362	4	displays	display	VERB
ejpam-5532	362	5	an	an	DET
ejpam-5532	362	6	inherent	inherent	ADJ
ejpam-5532	362	7	property	property	NOUN
ejpam-5532	362	8	of	of	ADP
ejpam-5532	362	9	an	an	DET
ejpam-5532	362	10	order	order	NOUN
ejpam-5532	362	11	reversing	reverse	VERB
ejpam-5532	362	12	involution	involution	NOUN
ejpam-5532	362	13	.	.	PUNCT
ejpam-5532	363	1	a.	a.	PROPN
ejpam-5532	363	2	jain	jain	PROPN
ejpam-5532	363	3	,	,	PUNCT
ejpam-5532	363	4	i.	i.	PROPN
ejpam-5532	363	5	jahan	jahan	PROPN
ejpam-5532	363	6	/	/	SYM
ejpam-5532	363	7	eur	eur	PROPN
ejpam-5532	363	8	.	.	PUNCT
ejpam-5532	364	1	j.	j.	PROPN
ejpam-5532	364	2	pure	pure	PROPN
ejpam-5532	364	3	appl	appl	PROPN
ejpam-5532	364	4	.	.	PROPN
ejpam-5532	364	5	math	math	PROPN
ejpam-5532	364	6	,	,	PUNCT
ejpam-5532	364	7	18	18	NUM
ejpam-5532	364	8	(	(	PUNCT
ejpam-5532	364	9	1	1	NUM
ejpam-5532	364	10	)	)	PUNCT
ejpam-5532	364	11	(	(	PUNCT
ejpam-5532	364	12	2025	2025	NUM
ejpam-5532	364	13	)	)	PUNCT
ejpam-5532	364	14	,	,	PUNCT
ejpam-5532	364	15	5532	5532	NUM
ejpam-5532	364	16	16	16	NUM
ejpam-5532	364	17	of	of	ADP
ejpam-5532	364	18	20	20	NUM
ejpam-5532	364	19	lemma	lemma	PROPN
ejpam-5532	364	20	1	1	NUM
ejpam-5532	364	21	.	.	PUNCT
ejpam-5532	365	1	if	if	SCONJ
ejpam-5532	365	2	l	l	NOUN
ejpam-5532	365	3	and	and	CCONJ
ejpam-5532	365	4	l∗	l∗	PROPN
ejpam-5532	365	5	are	be	AUX
ejpam-5532	365	6	lattices	lattice	NOUN
ejpam-5532	365	7	and	and	CCONJ
ejpam-5532	365	8	τ	τ	PRON
ejpam-5532	365	9	:	:	PUNCT
ejpam-5532	365	10	l	l	X
ejpam-5532	365	11	→	→	PUNCT
ejpam-5532	365	12	l∗	l∗	PROPN
ejpam-5532	365	13	is	be	AUX
ejpam-5532	365	14	an	an	DET
ejpam-5532	365	15	order	order	NOUN
ejpam-5532	365	16	reversing	reverse	VERB
ejpam-5532	365	17	bijection	bijection	NOUN
ejpam-5532	365	18	,	,	PUNCT
ejpam-5532	365	19	then	then	ADV
ejpam-5532	365	20	τ(a	τ(a	NOUN
ejpam-5532	365	21	∨	∨	NUM
ejpam-5532	365	22	b	b	NOUN
ejpam-5532	365	23	)	)	PUNCT
ejpam-5532	365	24	=	=	SYM
ejpam-5532	365	25	τ(a	τ(a	NOUN
ejpam-5532	365	26	)	)	PUNCT
ejpam-5532	365	27	∧	∧	PROPN
ejpam-5532	365	28	τ(b	τ(b	PROPN
ejpam-5532	365	29	)	)	PUNCT
ejpam-5532	365	30	and	and	CCONJ
ejpam-5532	365	31	τ(a	τ(a	NOUN
ejpam-5532	365	32	∧	∧	PROPN
ejpam-5532	365	33	b	b	NOUN
ejpam-5532	365	34	)	)	PUNCT
ejpam-5532	365	35	=	=	SYM
ejpam-5532	365	36	τ(a	τ(a	NOUN
ejpam-5532	365	37	)	)	PUNCT
ejpam-5532	365	38	∨	∨	NUM
ejpam-5532	365	39	τ(b	τ(b	PROPN
ejpam-5532	365	40	)	)	PUNCT
ejpam-5532	365	41	;	;	PUNCT
ejpam-5532	365	42	∀	∀	X
ejpam-5532	365	43	a	a	PRON
ejpam-5532	365	44	,	,	PUNCT
ejpam-5532	365	45	b	b	X
ejpam-5532	365	46	∈	∈	PROPN
ejpam-5532	365	47	l.	l.	NOUN
ejpam-5532	365	48	an	an	DET
ejpam-5532	365	49	order	order	NOUN
ejpam-5532	365	50	reversing	reverse	VERB
ejpam-5532	365	51	involution	involution	NOUN
ejpam-5532	365	52	defined	define	VERB
ejpam-5532	365	53	on	on	ADP
ejpam-5532	365	54	a	a	DET
ejpam-5532	365	55	lattice	lattice	ADJ
ejpam-5532	365	56	l	l	NOUN
ejpam-5532	365	57	of	of	ADP
ejpam-5532	365	58	truth	truth	NOUN
ejpam-5532	365	59	values	value	NOUN
ejpam-5532	365	60	leads	lead	VERB
ejpam-5532	365	61	to	to	ADP
ejpam-5532	365	62	the	the	DET
ejpam-5532	365	63	definition	definition	NOUN
ejpam-5532	365	64	of	of	ADP
ejpam-5532	365	65	complement	complement	NOUN
ejpam-5532	365	66	of	of	ADP
ejpam-5532	365	67	an	an	DET
ejpam-5532	365	68	l	l	NOUN
ejpam-5532	365	69	-	-	NOUN
ejpam-5532	365	70	set	set	NOUN
ejpam-5532	365	71	as	as	SCONJ
ejpam-5532	365	72	follows	follow	VERB
ejpam-5532	365	73	:	:	PUNCT
ejpam-5532	365	74	definition	definition	NOUN
ejpam-5532	365	75	10	10	NUM
ejpam-5532	365	76	.	.	PUNCT
ejpam-5532	366	1	let	let	VERB
ejpam-5532	366	2	µ	µ	X
ejpam-5532	366	3	∈	∈	X
ejpam-5532	366	4	lm	lm	INTJ
ejpam-5532	366	5	and	and	CCONJ
ejpam-5532	366	6	τ	τ	PROPN
ejpam-5532	366	7	be	be	VERB
ejpam-5532	366	8	an	an	DET
ejpam-5532	366	9	order	order	NOUN
ejpam-5532	366	10	reversing	reverse	VERB
ejpam-5532	366	11	involution	involution	NOUN
ejpam-5532	366	12	on	on	ADP
ejpam-5532	366	13	l	l	NOUN
ejpam-5532	366	14	,	,	PUNCT
ejpam-5532	366	15	i.e.	i.e.	X
ejpam-5532	366	16	,	,	PUNCT
ejpam-5532	366	17	τ	τ	X
ejpam-5532	366	18	:	:	PUNCT
ejpam-5532	366	19	l	l	NOUN
ejpam-5532	366	20	→	→	PUNCT
ejpam-5532	366	21	l∗	l∗	PROPN
ejpam-5532	366	22	is	be	AUX
ejpam-5532	366	23	a	a	DET
ejpam-5532	366	24	bijection	bijection	ADJ
ejpam-5532	366	25	satisfying	satisfying	ADJ
ejpam-5532	366	26	τ(τ(x	τ(τ(x	NOUN
ejpam-5532	366	27	)	)	PUNCT
ejpam-5532	366	28	)	)	PUNCT
ejpam-5532	367	1	=	=	SYM
ejpam-5532	367	2	x	x	NOUN
ejpam-5532	367	3	,	,	PUNCT
ejpam-5532	367	4	∀	∀	X
ejpam-5532	367	5	x	x	SYM
ejpam-5532	367	6	∈	∈	NOUN
ejpam-5532	367	7	l	l	NOUN
ejpam-5532	367	8	and	and	CCONJ
ejpam-5532	367	9	x	x	SYM
ejpam-5532	367	10	≤	≤	ADJ
ejpam-5532	367	11	y	y	NOUN
ejpam-5532	367	12	in	in	ADP
ejpam-5532	367	13	l	l	NOUN
ejpam-5532	367	14	if	if	SCONJ
ejpam-5532	367	15	and	and	CCONJ
ejpam-5532	367	16	only	only	ADV
ejpam-5532	367	17	if	if	SCONJ
ejpam-5532	367	18	τ(y	τ(y	NOUN
ejpam-5532	367	19	)	)	PUNCT
ejpam-5532	367	20	≤	≤	NOUN
ejpam-5532	367	21	τ(x	τ(x	NOUN
ejpam-5532	367	22	)	)	PUNCT
ejpam-5532	367	23	in	in	ADP
ejpam-5532	367	24	l	l	NOUN
ejpam-5532	367	25	=	=	PUNCT
ejpam-5532	367	26	l∗.	l∗.	NOUN
ejpam-5532	367	27	define	define	VERB
ejpam-5532	367	28	an	an	DET
ejpam-5532	367	29	l	l	NOUN
ejpam-5532	367	30	-	-	ADJ
ejpam-5532	367	31	set	set	VERB
ejpam-5532	367	32	µ′	µ′	NOUN
ejpam-5532	367	33	:	:	PUNCT
ejpam-5532	367	34	m∗	m∗	PROPN
ejpam-5532	367	35	→	→	SYM
ejpam-5532	367	36	l∗	l∗	NOUN
ejpam-5532	367	37	as	as	ADP
ejpam-5532	367	38	µ′(x	µ′(x	NOUN
ejpam-5532	367	39	)	)	PUNCT
ejpam-5532	367	40	=	=	SYM
ejpam-5532	367	41	τ(µ(x	τ(µ(x	NOUN
ejpam-5532	367	42	)	)	PUNCT
ejpam-5532	367	43	)	)	PUNCT
ejpam-5532	367	44	,	,	PUNCT
ejpam-5532	367	45	∀	∀	PUNCT
ejpam-5532	367	46	x	x	SYM
ejpam-5532	367	47	∈	∈	PROPN
ejpam-5532	367	48	m∗(=	m∗(=	NOUN
ejpam-5532	367	49	m	m	NOUN
ejpam-5532	367	50	)	)	PUNCT
ejpam-5532	367	51	.	.	PUNCT
ejpam-5532	368	1	then	then	ADV
ejpam-5532	368	2	,	,	PUNCT
ejpam-5532	368	3	µ′	µ′	NOUN
ejpam-5532	368	4	∈	∈	PROPN
ejpam-5532	368	5	lm	lm	NOUN
ejpam-5532	368	6	and	and	CCONJ
ejpam-5532	368	7	µ′	µ′	NOUN
ejpam-5532	368	8	is	be	AUX
ejpam-5532	368	9	called	call	VERB
ejpam-5532	368	10	the	the	DET
ejpam-5532	368	11	complement	complement	NOUN
ejpam-5532	368	12	of	of	ADP
ejpam-5532	368	13	µ	µ	NOUN
ejpam-5532	368	14	in	in	ADP
ejpam-5532	368	15	lm	lm	INTJ
ejpam-5532	368	16	.	.	PUNCT
ejpam-5532	369	1	the	the	DET
ejpam-5532	369	2	following	follow	VERB
ejpam-5532	369	3	lemma	lemma	PROPN
ejpam-5532	369	4	establishes	establish	VERB
ejpam-5532	369	5	the	the	DET
ejpam-5532	369	6	de	de	PROPN
ejpam-5532	369	7	morgan	morgan	PROPN
ejpam-5532	369	8	’s	’s	PART
ejpam-5532	369	9	laws	law	NOUN
ejpam-5532	369	10	in	in	ADP
ejpam-5532	369	11	lm	lm	PROPN
ejpam-5532	369	12	:	:	PUNCT
ejpam-5532	369	13	lemma	lemma	PROPN
ejpam-5532	369	14	2	2	X
ejpam-5532	369	15	.	.	PUNCT
ejpam-5532	369	16	let	let	VERB
ejpam-5532	369	17	µ	µ	NUM
ejpam-5532	369	18	,	,	PUNCT
ejpam-5532	369	19	η	η	PROPN
ejpam-5532	369	20	∈	∈	PROPN
ejpam-5532	370	1	lm	lm	X
ejpam-5532	370	2	and	and	CCONJ
ejpam-5532	370	3	τ	τ	PROPN
ejpam-5532	370	4	be	be	VERB
ejpam-5532	370	5	an	an	DET
ejpam-5532	370	6	order	order	NOUN
ejpam-5532	370	7	reversing	reverse	VERB
ejpam-5532	370	8	involution	involution	NOUN
ejpam-5532	370	9	on	on	ADP
ejpam-5532	370	10	l.	l.	PROPN
ejpam-5532	370	11	then	then	ADV
ejpam-5532	370	12	,	,	PUNCT
ejpam-5532	370	13	(	(	PUNCT
ejpam-5532	370	14	µ	µ	X
ejpam-5532	370	15	∪	∪	X
ejpam-5532	370	16	η)′	η)′	PROPN
ejpam-5532	370	17	=	=	PUNCT
ejpam-5532	370	18	µ′	µ′	NOUN
ejpam-5532	371	1	∩	∩	ADJ
ejpam-5532	371	2	η′	η′	X
ejpam-5532	371	3	and	and	CCONJ
ejpam-5532	371	4	(	(	PUNCT
ejpam-5532	371	5	µ	µ	X
ejpam-5532	371	6	∩	∩	X
ejpam-5532	371	7	η)′	η)′	PROPN
ejpam-5532	371	8	=	=	PUNCT
ejpam-5532	371	9	µ′	µ′	NOUN
ejpam-5532	371	10	∪	∪	NOUN
ejpam-5532	371	11	η′.	η′.	NOUN
ejpam-5532	371	12	proof	proof	NOUN
ejpam-5532	371	13	.	.	PUNCT
ejpam-5532	372	1	let	let	VERB
ejpam-5532	372	2	x	x	PUNCT
ejpam-5532	372	3	∈	∈	NOUN
ejpam-5532	372	4	m	m	VERB
ejpam-5532	372	5	.	.	PUNCT
ejpam-5532	373	1	then	then	ADV
ejpam-5532	373	2	,	,	PUNCT
ejpam-5532	373	3	(	(	PUNCT
ejpam-5532	373	4	µ	µ	X
ejpam-5532	373	5	∪	∪	X
ejpam-5532	373	6	η)′(x	η)′(x	NOUN
ejpam-5532	373	7	)	)	PUNCT
ejpam-5532	373	8	=	=	PUNCT
ejpam-5532	374	1	τ	τ	X
ejpam-5532	375	1	[	[	X
ejpam-5532	375	2	(	(	PUNCT
ejpam-5532	375	3	µ	µ	X
ejpam-5532	375	4	∪	∪	X
ejpam-5532	375	5	η)(x	η)(x	PROPN
ejpam-5532	375	6	)	)	PUNCT
ejpam-5532	375	7	]	]	PUNCT
ejpam-5532	376	1	=	=	PUNCT
ejpam-5532	376	2	τ	τ	X
ejpam-5532	377	1	[	[	X
ejpam-5532	377	2	µ(x	µ(x	X
ejpam-5532	377	3	)	)	PUNCT
ejpam-5532	377	4	∨	∨	NUM
ejpam-5532	377	5	η(x	η(x	X
ejpam-5532	377	6	)	)	PUNCT
ejpam-5532	377	7	]	]	PUNCT
ejpam-5532	377	8	=	=	SYM
ejpam-5532	377	9	τ(µ(x	τ(µ(x	NOUN
ejpam-5532	377	10	)	)	PUNCT
ejpam-5532	377	11	)	)	PUNCT
ejpam-5532	378	1	∧	∧	PROPN
ejpam-5532	378	2	τ(η(x	τ(η(x	PROPN
ejpam-5532	378	3	)	)	PUNCT
ejpam-5532	378	4	)	)	PUNCT
ejpam-5532	379	1	=	=	PUNCT
ejpam-5532	379	2	µ′(x	µ′(x	X
ejpam-5532	379	3	)	)	PUNCT
ejpam-5532	379	4	∧	∧	PROPN
ejpam-5532	379	5	η′(x	η′(x	NOUN
ejpam-5532	379	6	)	)	PUNCT
ejpam-5532	379	7	=	=	PUNCT
ejpam-5532	379	8	(	(	PUNCT
ejpam-5532	379	9	µ′	µ′	NOUN
ejpam-5532	379	10	∩	∩	ADJ
ejpam-5532	379	11	η′)(x	η′)(x	NOUN
ejpam-5532	379	12	)	)	PUNCT
ejpam-5532	379	13	.	.	PUNCT
ejpam-5532	380	1	hence	hence	ADV
ejpam-5532	380	2	,	,	PUNCT
ejpam-5532	380	3	(	(	PUNCT
ejpam-5532	380	4	µ	µ	X
ejpam-5532	380	5	∪	∪	X
ejpam-5532	380	6	η)′	η)′	PROPN
ejpam-5532	380	7	=	=	PUNCT
ejpam-5532	380	8	µ′	µ′	NOUN
ejpam-5532	380	9	∩	∩	NOUN
ejpam-5532	380	10	η′.	η′.	VERB
ejpam-5532	380	11	the	the	DET
ejpam-5532	380	12	proof	proof	NOUN
ejpam-5532	380	13	of	of	ADP
ejpam-5532	380	14	the	the	DET
ejpam-5532	380	15	other	other	ADJ
ejpam-5532	380	16	part	part	NOUN
ejpam-5532	380	17	follows	follow	VERB
ejpam-5532	380	18	similarly	similarly	ADV
ejpam-5532	380	19	.	.	PUNCT
ejpam-5532	381	1	in	in	ADP
ejpam-5532	381	2	the	the	DET
ejpam-5532	381	3	next	next	ADJ
ejpam-5532	381	4	result	result	NOUN
ejpam-5532	381	5	,	,	PUNCT
ejpam-5532	381	6	it	it	PRON
ejpam-5532	381	7	is	be	AUX
ejpam-5532	381	8	proved	prove	VERB
ejpam-5532	381	9	that	that	SCONJ
ejpam-5532	381	10	the	the	DET
ejpam-5532	381	11	complement	complement	NOUN
ejpam-5532	381	12	of	of	ADP
ejpam-5532	381	13	an	an	DET
ejpam-5532	381	14	l	l	ADJ
ejpam-5532	381	15	-	-	ADJ
ejpam-5532	381	16	prime	prime	ADJ
ejpam-5532	381	17	ideal	ideal	NOUN
ejpam-5532	381	18	in	in	ADP
ejpam-5532	381	19	m	m	PROPN
ejpam-5532	381	20	is	be	AUX
ejpam-5532	381	21	an	an	DET
ejpam-5532	381	22	l	l	ADJ
ejpam-5532	381	23	-	-	ADJ
ejpam-5532	381	24	dual	dual	ADJ
ejpam-5532	381	25	prime	prime	ADJ
ejpam-5532	381	26	ideal	ideal	NOUN
ejpam-5532	381	27	in	in	ADP
ejpam-5532	381	28	m	m	PROPN
ejpam-5532	381	29	.	.	PUNCT
ejpam-5532	382	1	theorem	theorem	ADJ
ejpam-5532	382	2	14	14	NUM
ejpam-5532	382	3	.	.	PUNCT
ejpam-5532	383	1	let	let	VERB
ejpam-5532	383	2	τ	τ	PRON
ejpam-5532	383	3	be	be	AUX
ejpam-5532	383	4	an	an	DET
ejpam-5532	383	5	order	order	NOUN
ejpam-5532	383	6	reversing	reverse	VERB
ejpam-5532	383	7	involution	involution	NOUN
ejpam-5532	383	8	on	on	ADP
ejpam-5532	383	9	lattice	lattice	PROPN
ejpam-5532	383	10	l	l	PROPN
ejpam-5532	383	11	and	and	CCONJ
ejpam-5532	383	12	µ	µ	PRON
ejpam-5532	383	13	be	be	AUX
ejpam-5532	383	14	an	an	DET
ejpam-5532	383	15	l	l	ADJ
ejpam-5532	383	16	-	-	ADJ
ejpam-5532	383	17	prime	prime	ADJ
ejpam-5532	383	18	ideal	ideal	NOUN
ejpam-5532	383	19	of	of	ADP
ejpam-5532	383	20	m	m	PROPN
ejpam-5532	383	21	.	.	PUNCT
ejpam-5532	384	1	then	then	ADV
ejpam-5532	384	2	µ′	µ′	NUM
ejpam-5532	384	3	,	,	PUNCT
ejpam-5532	384	4	the	the	DET
ejpam-5532	384	5	complement	complement	NOUN
ejpam-5532	384	6	of	of	ADP
ejpam-5532	384	7	µ	µ	NOUN
ejpam-5532	384	8	in	in	ADP
ejpam-5532	384	9	lm	lm	INTJ
ejpam-5532	384	10	,	,	PUNCT
ejpam-5532	384	11	is	be	AUX
ejpam-5532	384	12	an	an	DET
ejpam-5532	384	13	l	l	ADJ
ejpam-5532	384	14	-	-	ADJ
ejpam-5532	384	15	dual	dual	ADJ
ejpam-5532	384	16	prime	prime	ADJ
ejpam-5532	384	17	ideal	ideal	NOUN
ejpam-5532	384	18	of	of	ADP
ejpam-5532	384	19	m	m	PROPN
ejpam-5532	384	20	.	.	PUNCT
ejpam-5532	385	1	proof	proof	NOUN
ejpam-5532	385	2	.	.	PUNCT
ejpam-5532	386	1	let	let	VERB
ejpam-5532	386	2	x	x	PRON
ejpam-5532	386	3	,	,	PUNCT
ejpam-5532	386	4	y	y	PROPN
ejpam-5532	386	5	∈	∈	PROPN
ejpam-5532	386	6	m	m	VERB
ejpam-5532	386	7	.	.	PUNCT
ejpam-5532	387	1	since	since	SCONJ
ejpam-5532	387	2	µ	µ	NOUN
ejpam-5532	387	3	is	be	AUX
ejpam-5532	387	4	an	an	DET
ejpam-5532	387	5	l	l	ADJ
ejpam-5532	387	6	-	-	ADJ
ejpam-5532	387	7	prime	prime	ADJ
ejpam-5532	387	8	ideal	ideal	NOUN
ejpam-5532	387	9	of	of	ADP
ejpam-5532	387	10	m	m	PROPN
ejpam-5532	387	11	,	,	PUNCT
ejpam-5532	387	12	we	we	PRON
ejpam-5532	387	13	have	have	VERB
ejpam-5532	387	14	µ(x	µ(x	VERB
ejpam-5532	387	15	∧	∧	PROPN
ejpam-5532	387	16	y	y	PROPN
ejpam-5532	387	17	)	)	PUNCT
ejpam-5532	387	18	≤	≤	NOUN
ejpam-5532	387	19	µ(x	µ(x	X
ejpam-5532	387	20	)	)	PUNCT
ejpam-5532	387	21	∨	∨	NUM
ejpam-5532	387	22	µ(y	µ(y	PROPN
ejpam-5532	387	23	)	)	PUNCT
ejpam-5532	387	24	.	.	PUNCT
ejpam-5532	388	1	this	this	PRON
ejpam-5532	388	2	implies	imply	VERB
ejpam-5532	388	3	,	,	PUNCT
ejpam-5532	388	4	τ(µ(x	τ(µ(x	VERB
ejpam-5532	388	5	∧	∧	PROPN
ejpam-5532	388	6	y	y	NOUN
ejpam-5532	388	7	)	)	PUNCT
ejpam-5532	388	8	)	)	PUNCT
ejpam-5532	389	1	≥	≥	X
ejpam-5532	389	2	τ	τ	X
ejpam-5532	390	1	[	[	X
ejpam-5532	390	2	µ(x	µ(x	X
ejpam-5532	390	3	)	)	PUNCT
ejpam-5532	390	4	∨	∨	NUM
ejpam-5532	390	5	µ(y	µ(y	PROPN
ejpam-5532	390	6	)	)	PUNCT
ejpam-5532	390	7	]	]	PUNCT
ejpam-5532	390	8	as	as	SCONJ
ejpam-5532	390	9	τ	τ	PROPN
ejpam-5532	390	10	is	be	AUX
ejpam-5532	390	11	an	an	DET
ejpam-5532	390	12	order	order	NOUN
ejpam-5532	390	13	reversing	reverse	VERB
ejpam-5532	390	14	involution	involution	NOUN
ejpam-5532	390	15	.	.	PUNCT
ejpam-5532	391	1	that	that	PRON
ejpam-5532	391	2	is	be	AUX
ejpam-5532	391	3	,	,	PUNCT
ejpam-5532	391	4	µ′(x	µ′(x	NUM
ejpam-5532	391	5	∧	∧	PROPN
ejpam-5532	391	6	y	y	PROPN
ejpam-5532	391	7	)	)	PUNCT
ejpam-5532	391	8	≥	≥	NOUN
ejpam-5532	391	9	τ	τ	X
ejpam-5532	392	1	[	[	X
ejpam-5532	392	2	µ(x	µ(x	X
ejpam-5532	392	3	)	)	PUNCT
ejpam-5532	392	4	∨	∨	NUM
ejpam-5532	392	5	µ(y	µ(y	PROPN
ejpam-5532	392	6	)	)	PUNCT
ejpam-5532	392	7	]	]	PUNCT
ejpam-5532	392	8	=	=	PUNCT
ejpam-5532	392	9	τ(µ(x	τ(µ(x	NOUN
ejpam-5532	392	10	)	)	PUNCT
ejpam-5532	392	11	)	)	PUNCT
ejpam-5532	392	12	∧	∧	PROPN
ejpam-5532	392	13	τ(µ(y	τ(µ(y	NUM
ejpam-5532	392	14	)	)	PUNCT
ejpam-5532	392	15	)	)	PUNCT
ejpam-5532	393	1	=	=	PUNCT
ejpam-5532	393	2	µ′(x	µ′(x	X
ejpam-5532	393	3	)	)	PUNCT
ejpam-5532	393	4	∧	∧	NOUN
ejpam-5532	393	5	µ′(y	µ′(y	PUNCT
ejpam-5532	393	6	)	)	PUNCT
ejpam-5532	393	7	.	.	PUNCT
ejpam-5532	394	1	(	(	PUNCT
ejpam-5532	394	2	1	1	X
ejpam-5532	394	3	)	)	PUNCT
ejpam-5532	394	4	a.	a.	NOUN
ejpam-5532	394	5	jain	jain	PROPN
ejpam-5532	394	6	,	,	PUNCT
ejpam-5532	394	7	i.	i.	PROPN
ejpam-5532	394	8	jahan	jahan	PROPN
ejpam-5532	394	9	/	/	SYM
ejpam-5532	394	10	eur	eur	PROPN
ejpam-5532	394	11	.	.	PUNCT
ejpam-5532	395	1	j.	j.	PROPN
ejpam-5532	395	2	pure	pure	PROPN
ejpam-5532	395	3	appl	appl	PROPN
ejpam-5532	395	4	.	.	PROPN
ejpam-5532	395	5	math	math	PROPN
ejpam-5532	395	6	,	,	PUNCT
ejpam-5532	395	7	18	18	NUM
ejpam-5532	395	8	(	(	PUNCT
ejpam-5532	395	9	1	1	NUM
ejpam-5532	395	10	)	)	PUNCT
ejpam-5532	395	11	(	(	PUNCT
ejpam-5532	395	12	2025	2025	NUM
ejpam-5532	395	13	)	)	PUNCT
ejpam-5532	395	14	,	,	PUNCT
ejpam-5532	395	15	5532	5532	NUM
ejpam-5532	395	16	17	17	NUM
ejpam-5532	395	17	of	of	ADP
ejpam-5532	395	18	20	20	NUM
ejpam-5532	395	19	further	far	ADV
ejpam-5532	395	20	,	,	PUNCT
ejpam-5532	395	21	if	if	SCONJ
ejpam-5532	395	22	x	x	ADP
ejpam-5532	395	23	≤	≤	NUM
ejpam-5532	395	24	y	y	NOUN
ejpam-5532	395	25	in	in	ADP
ejpam-5532	395	26	m	m	PROPN
ejpam-5532	395	27	,	,	PUNCT
ejpam-5532	395	28	then	then	ADV
ejpam-5532	395	29	µ(x	µ(x	NOUN
ejpam-5532	395	30	)	)	PUNCT
ejpam-5532	395	31	≥	≥	NOUN
ejpam-5532	395	32	µ(y	µ(y	PROPN
ejpam-5532	395	33	)	)	PUNCT
ejpam-5532	395	34	in	in	ADP
ejpam-5532	395	35	l	l	NOUN
ejpam-5532	395	36	(	(	PUNCT
ejpam-5532	395	37	as	as	SCONJ
ejpam-5532	395	38	µ	µ	NOUN
ejpam-5532	395	39	is	be	AUX
ejpam-5532	395	40	an	an	DET
ejpam-5532	395	41	l	l	NOUN
ejpam-5532	395	42	-	-	NOUN
ejpam-5532	395	43	ideal	ideal	NOUN
ejpam-5532	395	44	of	of	ADP
ejpam-5532	395	45	m	m	PROPN
ejpam-5532	395	46	)	)	PUNCT
ejpam-5532	395	47	.	.	PUNCT
ejpam-5532	396	1	this	this	PRON
ejpam-5532	396	2	implies	imply	VERB
ejpam-5532	396	3	τ(µ(x	τ(µ(x	NOUN
ejpam-5532	396	4	)	)	PUNCT
ejpam-5532	396	5	)	)	PUNCT
ejpam-5532	396	6	≤	≤	NOUN
ejpam-5532	397	1	τ(µ(y	τ(µ(y	NUM
ejpam-5532	397	2	)	)	PUNCT
ejpam-5532	397	3	)	)	PUNCT
ejpam-5532	398	1	in	in	ADP
ejpam-5532	398	2	l∗.	l∗.	NOUN
ejpam-5532	398	3	thus	thus	ADV
ejpam-5532	398	4	,	,	PUNCT
ejpam-5532	398	5	µ′(x	µ′(x	SYM
ejpam-5532	398	6	)	)	PUNCT
ejpam-5532	398	7	≤	≤	NOUN
ejpam-5532	398	8	µ′(y	µ′(y	PUNCT
ejpam-5532	398	9	)	)	PUNCT
ejpam-5532	398	10	.	.	PUNCT
ejpam-5532	399	1	(	(	PUNCT
ejpam-5532	399	2	2	2	X
ejpam-5532	399	3	)	)	PUNCT
ejpam-5532	399	4	moreover	moreover	ADV
ejpam-5532	399	5	,	,	PUNCT
ejpam-5532	399	6	as	as	SCONJ
ejpam-5532	399	7	µ	µ	NOUN
ejpam-5532	399	8	is	be	AUX
ejpam-5532	399	9	an	an	DET
ejpam-5532	399	10	l	l	NOUN
ejpam-5532	399	11	-	-	NOUN
ejpam-5532	399	12	ideal	ideal	NOUN
ejpam-5532	399	13	of	of	ADP
ejpam-5532	399	14	m	m	PRON
ejpam-5532	399	15	,	,	PUNCT
ejpam-5532	399	16	x	x	SYM
ejpam-5532	399	17	≤	≤	NUM
ejpam-5532	399	18	x	x	PUNCT
ejpam-5532	399	19	∨	∨	NUM
ejpam-5532	399	20	y	y	PROPN
ejpam-5532	399	21	and	and	CCONJ
ejpam-5532	399	22	y	y	PROPN
ejpam-5532	399	23	≤	≤	NUM
ejpam-5532	399	24	x	x	PUNCT
ejpam-5532	399	25	∨	∨	NUM
ejpam-5532	399	26	y	y	PROPN
ejpam-5532	399	27	in	in	ADP
ejpam-5532	399	28	m	m	PROPN
ejpam-5532	399	29	implies	imply	VERB
ejpam-5532	399	30	µ(x	µ(x	NOUN
ejpam-5532	399	31	)	)	PUNCT
ejpam-5532	399	32	≥	≥	NOUN
ejpam-5532	399	33	µ(x	µ(x	X
ejpam-5532	399	34	∨	∨	NUM
ejpam-5532	399	35	y	y	NOUN
ejpam-5532	399	36	)	)	PUNCT
ejpam-5532	399	37	and	and	CCONJ
ejpam-5532	399	38	µ(y	µ(y	PROPN
ejpam-5532	399	39	)	)	PUNCT
ejpam-5532	399	40	≥	≥	NOUN
ejpam-5532	399	41	µ(x	µ(x	X
ejpam-5532	399	42	∨	∨	NUM
ejpam-5532	399	43	y	y	NOUN
ejpam-5532	399	44	)	)	PUNCT
ejpam-5532	399	45	.	.	PUNCT
ejpam-5532	400	1	thus	thus	ADV
ejpam-5532	400	2	,	,	PUNCT
ejpam-5532	400	3	µ′(x	µ′(x	X
ejpam-5532	400	4	)	)	PUNCT
ejpam-5532	400	5	≤	≤	NOUN
ejpam-5532	400	6	µ′(x	µ′(x	PUNCT
ejpam-5532	400	7	∨	∨	NUM
ejpam-5532	400	8	y	y	NOUN
ejpam-5532	400	9	)	)	PUNCT
ejpam-5532	400	10	and	and	CCONJ
ejpam-5532	400	11	µ′(y	µ′(y	PROPN
ejpam-5532	400	12	)	)	PUNCT
ejpam-5532	400	13	≤	≤	NOUN
ejpam-5532	400	14	µ′(x	µ′(x	PUNCT
ejpam-5532	400	15	∨	∨	NUM
ejpam-5532	400	16	y	y	NOUN
ejpam-5532	400	17	)	)	PUNCT
ejpam-5532	400	18	and	and	CCONJ
ejpam-5532	400	19	hence	hence	ADV
ejpam-5532	400	20	µ′(x	µ′(x	X
ejpam-5532	400	21	)	)	PUNCT
ejpam-5532	400	22	∧	∧	NOUN
ejpam-5532	400	23	µ′(y	µ′(y	NOUN
ejpam-5532	400	24	)	)	PUNCT
ejpam-5532	400	25	≤	≤	NUM
ejpam-5532	400	26	µ′(x	µ′(x	X
ejpam-5532	400	27	)	)	PUNCT
ejpam-5532	400	28	≤	≤	NOUN
ejpam-5532	400	29	µ′(x	µ′(x	PUNCT
ejpam-5532	400	30	∨	∨	NUM
ejpam-5532	400	31	y	y	NOUN
ejpam-5532	400	32	)	)	PUNCT
ejpam-5532	400	33	.	.	PUNCT
ejpam-5532	401	1	(	(	PUNCT
ejpam-5532	401	2	3	3	X
ejpam-5532	401	3	)	)	PUNCT
ejpam-5532	401	4	by	by	ADP
ejpam-5532	401	5	(	(	PUNCT
ejpam-5532	401	6	1	1	NUM
ejpam-5532	401	7	)	)	PUNCT
ejpam-5532	401	8	,	,	PUNCT
ejpam-5532	401	9	(	(	PUNCT
ejpam-5532	401	10	2	2	X
ejpam-5532	401	11	)	)	PUNCT
ejpam-5532	401	12	and	and	CCONJ
ejpam-5532	401	13	(	(	PUNCT
ejpam-5532	401	14	3	3	NUM
ejpam-5532	401	15	)	)	PUNCT
ejpam-5532	401	16	,	,	PUNCT
ejpam-5532	401	17	we	we	PRON
ejpam-5532	401	18	get	get	VERB
ejpam-5532	401	19	that	that	PRON
ejpam-5532	401	20	µ′	µ′	NOUN
ejpam-5532	401	21	is	be	AUX
ejpam-5532	401	22	an	an	DET
ejpam-5532	401	23	l	l	ADJ
ejpam-5532	401	24	-	-	ADJ
ejpam-5532	401	25	dual	dual	ADJ
ejpam-5532	401	26	ideal	ideal	NOUN
ejpam-5532	401	27	of	of	ADP
ejpam-5532	401	28	m	m	PROPN
ejpam-5532	401	29	.	.	PUNCT
ejpam-5532	402	1	to	to	PART
ejpam-5532	402	2	establish	establish	VERB
ejpam-5532	402	3	that	that	SCONJ
ejpam-5532	402	4	µ′	µ′	NOUN
ejpam-5532	402	5	is	be	AUX
ejpam-5532	402	6	an	an	DET
ejpam-5532	402	7	l	l	ADJ
ejpam-5532	402	8	-	-	ADJ
ejpam-5532	402	9	dual	dual	ADJ
ejpam-5532	402	10	prime	prime	ADJ
ejpam-5532	402	11	ideal	ideal	NOUN
ejpam-5532	402	12	of	of	ADP
ejpam-5532	402	13	m	m	PRON
ejpam-5532	402	14	,	,	PUNCT
ejpam-5532	402	15	note	note	VERB
ejpam-5532	402	16	that	that	SCONJ
ejpam-5532	402	17	µ(x	µ(x	VERB
ejpam-5532	402	18	∨	∨	NUM
ejpam-5532	402	19	y	y	NOUN
ejpam-5532	402	20	)	)	PUNCT
ejpam-5532	402	21	≥	≥	NOUN
ejpam-5532	402	22	µ(x	µ(x	NOUN
ejpam-5532	402	23	)	)	PUNCT
ejpam-5532	402	24	∧	∧	PROPN
ejpam-5532	402	25	µ(y	µ(y	PROPN
ejpam-5532	402	26	)	)	PUNCT
ejpam-5532	402	27	.	.	PUNCT
ejpam-5532	403	1	this	this	PRON
ejpam-5532	403	2	implies	imply	VERB
ejpam-5532	403	3	µ′(x	µ′(x	ADJ
ejpam-5532	403	4	∨	∨	NUM
ejpam-5532	403	5	y	y	NOUN
ejpam-5532	403	6	)	)	PUNCT
ejpam-5532	403	7	=	=	PUNCT
ejpam-5532	404	1	τ(µ(x	τ(µ(x	VERB
ejpam-5532	404	2	∨	∨	NUM
ejpam-5532	404	3	y	y	NOUN
ejpam-5532	404	4	)	)	PUNCT
ejpam-5532	404	5	)	)	PUNCT
ejpam-5532	405	1	≤	≤	PUNCT
ejpam-5532	405	2	τ	τ	PUNCT
ejpam-5532	406	1	[	[	X
ejpam-5532	406	2	µ(x	µ(x	X
ejpam-5532	406	3	)	)	PUNCT
ejpam-5532	406	4	∧	∧	PROPN
ejpam-5532	406	5	µ(y	µ(y	PROPN
ejpam-5532	406	6	)	)	PUNCT
ejpam-5532	406	7	]	]	PUNCT
ejpam-5532	407	1	=	=	PUNCT
ejpam-5532	407	2	τ(µ(x	τ(µ(x	NOUN
ejpam-5532	407	3	)	)	PUNCT
ejpam-5532	407	4	)	)	PUNCT
ejpam-5532	407	5	∨	∨	NUM
ejpam-5532	407	6	τ(µ(y	τ(µ(y	NUM
ejpam-5532	407	7	)	)	PUNCT
ejpam-5532	407	8	)	)	PUNCT
ejpam-5532	408	1	=	=	PUNCT
ejpam-5532	408	2	µ′(x	µ′(x	X
ejpam-5532	408	3	)	)	PUNCT
ejpam-5532	408	4	∨	∨	NUM
ejpam-5532	408	5	µ′(y	µ′(y	PUNCT
ejpam-5532	408	6	)	)	PUNCT
ejpam-5532	408	7	.	.	PUNCT
ejpam-5532	409	1	consequently	consequently	ADV
ejpam-5532	409	2	,	,	PUNCT
ejpam-5532	409	3	µ′	µ′	PRON
ejpam-5532	409	4	is	be	AUX
ejpam-5532	409	5	an	an	DET
ejpam-5532	409	6	l	l	ADJ
ejpam-5532	409	7	-	-	ADJ
ejpam-5532	409	8	dual	dual	ADJ
ejpam-5532	409	9	prime	prime	ADJ
ejpam-5532	409	10	ideal	ideal	NOUN
ejpam-5532	409	11	of	of	ADP
ejpam-5532	409	12	m	m	PROPN
ejpam-5532	409	13	.	.	PUNCT
ejpam-5532	410	1	by	by	ADP
ejpam-5532	410	2	the	the	DET
ejpam-5532	410	3	above	above	ADJ
ejpam-5532	410	4	theorem	theorem	NOUN
ejpam-5532	410	5	,	,	PUNCT
ejpam-5532	410	6	it	it	PRON
ejpam-5532	410	7	can	can	AUX
ejpam-5532	410	8	be	be	AUX
ejpam-5532	410	9	concluded	conclude	VERB
ejpam-5532	410	10	that	that	SCONJ
ejpam-5532	410	11	if	if	SCONJ
ejpam-5532	410	12	l	l	NOUN
ejpam-5532	410	13	is	be	AUX
ejpam-5532	410	14	a	a	DET
ejpam-5532	410	15	lattice	lattice	NOUN
ejpam-5532	410	16	with	with	ADP
ejpam-5532	410	17	an	an	DET
ejpam-5532	410	18	order	order	NOUN
ejpam-5532	410	19	reversing	reverse	VERB
ejpam-5532	410	20	involution	involution	NOUN
ejpam-5532	410	21	τ	τ	PROPN
ejpam-5532	410	22	,	,	PUNCT
ejpam-5532	410	23	then	then	ADV
ejpam-5532	410	24	µ	µ	NOUN
ejpam-5532	410	25	is	be	AUX
ejpam-5532	410	26	an	an	DET
ejpam-5532	410	27	l	l	ADJ
ejpam-5532	410	28	-	-	ADJ
ejpam-5532	410	29	prime	prime	ADJ
ejpam-5532	410	30	ideal	ideal	NOUN
ejpam-5532	410	31	of	of	ADP
ejpam-5532	410	32	m	m	PRON
ejpam-5532	410	33	if	if	SCONJ
ejpam-5532	410	34	and	and	CCONJ
ejpam-5532	410	35	only	only	ADV
ejpam-5532	410	36	if	if	SCONJ
ejpam-5532	410	37	µ′	µ′	NOUN
ejpam-5532	410	38	is	be	AUX
ejpam-5532	410	39	an	an	DET
ejpam-5532	410	40	l	l	ADJ
ejpam-5532	410	41	-	-	ADJ
ejpam-5532	410	42	dual	dual	ADJ
ejpam-5532	410	43	prime	prime	ADJ
ejpam-5532	410	44	ideal	ideal	NOUN
ejpam-5532	410	45	of	of	ADP
ejpam-5532	410	46	m	m	PROPN
ejpam-5532	410	47	.	.	PUNCT
ejpam-5532	411	1	the	the	DET
ejpam-5532	411	2	next	next	ADJ
ejpam-5532	411	3	theorem	theorem	NOUN
ejpam-5532	411	4	,	,	PUNCT
ejpam-5532	411	5	combined	combine	VERB
ejpam-5532	411	6	with	with	ADP
ejpam-5532	411	7	theorem	theorem	ADJ
ejpam-5532	411	8	14	14	NUM
ejpam-5532	411	9	,	,	PUNCT
ejpam-5532	411	10	leads	lead	VERB
ejpam-5532	411	11	to	to	ADP
ejpam-5532	411	12	the	the	DET
ejpam-5532	411	13	following	follow	VERB
ejpam-5532	411	14	interesting	interesting	ADJ
ejpam-5532	411	15	analogue	analogue	NOUN
ejpam-5532	411	16	of	of	ADP
ejpam-5532	411	17	a	a	DET
ejpam-5532	411	18	result	result	NOUN
ejpam-5532	411	19	from	from	ADP
ejpam-5532	411	20	classical	classical	ADJ
ejpam-5532	411	21	lattice	lattice	NOUN
ejpam-5532	411	22	theory	theory	NOUN
ejpam-5532	411	23	:	:	PUNCT
ejpam-5532	411	24	an	an	DET
ejpam-5532	411	25	l	l	NOUN
ejpam-5532	411	26	-	-	PUNCT
ejpam-5532	411	27	ideal	ideal	ADJ
ejpam-5532	411	28	η	η	PROPN
ejpam-5532	411	29	of	of	ADP
ejpam-5532	411	30	m	m	PROPN
ejpam-5532	411	31	is	be	AUX
ejpam-5532	411	32	an	an	DET
ejpam-5532	411	33	l	l	ADJ
ejpam-5532	411	34	-	-	ADJ
ejpam-5532	411	35	prime	prime	ADJ
ejpam-5532	411	36	ideal	ideal	NOUN
ejpam-5532	411	37	of	of	ADP
ejpam-5532	411	38	m	m	PRON
ejpam-5532	411	39	if	if	SCONJ
ejpam-5532	411	40	and	and	CCONJ
ejpam-5532	411	41	only	only	ADV
ejpam-5532	411	42	if	if	SCONJ
ejpam-5532	411	43	η′	η′	PROPN
ejpam-5532	411	44	is	be	AUX
ejpam-5532	411	45	an	an	DET
ejpam-5532	411	46	l	l	ADJ
ejpam-5532	411	47	-	-	ADJ
ejpam-5532	411	48	dual	dual	ADJ
ejpam-5532	411	49	ideal	ideal	NOUN
ejpam-5532	411	50	of	of	ADP
ejpam-5532	411	51	m	m	PRON
ejpam-5532	411	52	.	.	PUNCT
ejpam-5532	412	1	in	in	ADP
ejpam-5532	412	2	fact	fact	NOUN
ejpam-5532	412	3	,	,	PUNCT
ejpam-5532	412	4	η′	η′	PROPN
ejpam-5532	412	5	is	be	AUX
ejpam-5532	412	6	an	an	DET
ejpam-5532	412	7	l	l	ADJ
ejpam-5532	412	8	-	-	ADJ
ejpam-5532	412	9	dual	dual	ADJ
ejpam-5532	412	10	prime	prime	ADJ
ejpam-5532	412	11	ideal	ideal	NOUN
ejpam-5532	412	12	of	of	ADP
ejpam-5532	412	13	m	m	PROPN
ejpam-5532	412	14	.	.	PUNCT
ejpam-5532	413	1	theorem	theorem	ADJ
ejpam-5532	413	2	15	15	NUM
ejpam-5532	413	3	.	.	PUNCT
ejpam-5532	414	1	let	let	VERB
ejpam-5532	414	2	τ	τ	PRON
ejpam-5532	414	3	be	be	AUX
ejpam-5532	414	4	an	an	DET
ejpam-5532	414	5	order	order	NOUN
ejpam-5532	414	6	reversing	reverse	VERB
ejpam-5532	414	7	involution	involution	NOUN
ejpam-5532	414	8	on	on	ADP
ejpam-5532	414	9	the	the	DET
ejpam-5532	414	10	lattice	lattice	PROPN
ejpam-5532	414	11	l	l	PROPN
ejpam-5532	414	12	and	and	CCONJ
ejpam-5532	414	13	η	η	PROPN
ejpam-5532	414	14	be	be	AUX
ejpam-5532	414	15	an	an	DET
ejpam-5532	414	16	l	l	NOUN
ejpam-5532	414	17	-	-	NOUN
ejpam-5532	414	18	ideal	ideal	NOUN
ejpam-5532	414	19	of	of	ADP
ejpam-5532	414	20	m	m	VERB
ejpam-5532	414	21	such	such	ADJ
ejpam-5532	414	22	that	that	SCONJ
ejpam-5532	414	23	η′	η′	PROPN
ejpam-5532	414	24	is	be	AUX
ejpam-5532	414	25	an	an	DET
ejpam-5532	414	26	l	l	ADJ
ejpam-5532	414	27	-	-	ADJ
ejpam-5532	414	28	dual	dual	ADJ
ejpam-5532	414	29	ideal	ideal	NOUN
ejpam-5532	414	30	of	of	ADP
ejpam-5532	414	31	m	m	PROPN
ejpam-5532	414	32	.	.	PUNCT
ejpam-5532	415	1	then	then	ADV
ejpam-5532	415	2	,	,	PUNCT
ejpam-5532	415	3	η	η	PROPN
ejpam-5532	415	4	and	and	CCONJ
ejpam-5532	415	5	η′	η′	PROPN
ejpam-5532	415	6	are	be	AUX
ejpam-5532	415	7	l	l	ADJ
ejpam-5532	415	8	-	-	ADJ
ejpam-5532	415	9	prime	prime	ADJ
ejpam-5532	415	10	ideals	ideal	NOUN
ejpam-5532	415	11	of	of	ADP
ejpam-5532	415	12	m	m	PROPN
ejpam-5532	415	13	.	.	PUNCT
ejpam-5532	416	1	proof	proof	NOUN
ejpam-5532	416	2	.	.	PUNCT
ejpam-5532	417	1	suppose	suppose	VERB
ejpam-5532	417	2	η	η	PROPN
ejpam-5532	417	3	is	be	AUX
ejpam-5532	417	4	an	an	DET
ejpam-5532	417	5	l	l	NOUN
ejpam-5532	417	6	-	-	NOUN
ejpam-5532	417	7	ideal	ideal	NOUN
ejpam-5532	417	8	of	of	ADP
ejpam-5532	417	9	m	m	VERB
ejpam-5532	417	10	such	such	ADJ
ejpam-5532	417	11	that	that	SCONJ
ejpam-5532	417	12	η′	η′	PROPN
ejpam-5532	417	13	is	be	AUX
ejpam-5532	417	14	an	an	DET
ejpam-5532	417	15	l	l	ADJ
ejpam-5532	417	16	-	-	ADJ
ejpam-5532	417	17	dual	dual	ADJ
ejpam-5532	417	18	ideal	ideal	NOUN
ejpam-5532	417	19	of	of	ADP
ejpam-5532	417	20	m	m	PROPN
ejpam-5532	417	21	.	.	PUNCT
ejpam-5532	418	1	we	we	PRON
ejpam-5532	418	2	have	have	VERB
ejpam-5532	418	3	η′(x	η′(x	NOUN
ejpam-5532	418	4	∧	∧	PROPN
ejpam-5532	418	5	y	y	PROPN
ejpam-5532	418	6	)	)	PUNCT
ejpam-5532	418	7	≥	≥	NOUN
ejpam-5532	418	8	η′(x	η′(x	NOUN
ejpam-5532	418	9	)	)	PUNCT
ejpam-5532	418	10	∧	∧	PROPN
ejpam-5532	418	11	η′(y	η′(y	PROPN
ejpam-5532	418	12	)	)	PUNCT
ejpam-5532	418	13	;	;	PUNCT
ejpam-5532	418	14	∀	∀	X
ejpam-5532	418	15	x	x	X
ejpam-5532	418	16	,	,	PUNCT
ejpam-5532	418	17	y	y	PROPN
ejpam-5532	418	18	∈	∈	PROPN
ejpam-5532	418	19	m.	m.	NOUN
ejpam-5532	418	20	this	this	PRON
ejpam-5532	418	21	implies	imply	VERB
ejpam-5532	418	22	η(x	η(x	PROPN
ejpam-5532	418	23	∧	∧	PROPN
ejpam-5532	418	24	y	y	NOUN
ejpam-5532	418	25	)	)	PUNCT
ejpam-5532	418	26	=	=	PUNCT
ejpam-5532	419	1	τ	τ	PROPN
ejpam-5532	420	1	[	[	X
ejpam-5532	420	2	η′(x	η′(x	X
ejpam-5532	420	3	∧	∧	PROPN
ejpam-5532	420	4	y	y	PROPN
ejpam-5532	420	5	)	)	PUNCT
ejpam-5532	420	6	]	]	PUNCT
ejpam-5532	421	1	≤	≤	NUM
ejpam-5532	421	2	τ	τ	X
ejpam-5532	422	1	[	[	X
ejpam-5532	422	2	η′(x	η′(x	X
ejpam-5532	422	3	)	)	PUNCT
ejpam-5532	422	4	∧	∧	PROPN
ejpam-5532	422	5	η′(y	η′(y	PROPN
ejpam-5532	422	6	)	)	PUNCT
ejpam-5532	422	7	]	]	PUNCT
ejpam-5532	423	1	=	=	PUNCT
ejpam-5532	423	2	τ(η′(x	τ(η′(x	NOUN
ejpam-5532	423	3	)	)	PUNCT
ejpam-5532	423	4	)	)	PUNCT
ejpam-5532	424	1	∨	∨	NUM
ejpam-5532	424	2	τ(η′(y	τ(η′(y	PROPN
ejpam-5532	424	3	)	)	PUNCT
ejpam-5532	424	4	)	)	PUNCT
ejpam-5532	425	1	=	=	SYM
ejpam-5532	425	2	η(x	η(x	X
ejpam-5532	425	3	)	)	PUNCT
ejpam-5532	425	4	∨	∨	NUM
ejpam-5532	425	5	η(y	η(y	NOUN
ejpam-5532	425	6	)	)	PUNCT
ejpam-5532	425	7	.	.	PUNCT
ejpam-5532	426	1	a.	a.	PROPN
ejpam-5532	426	2	jain	jain	PROPN
ejpam-5532	426	3	,	,	PUNCT
ejpam-5532	426	4	i.	i.	PROPN
ejpam-5532	426	5	jahan	jahan	PROPN
ejpam-5532	426	6	/	/	SYM
ejpam-5532	426	7	eur	eur	PROPN
ejpam-5532	426	8	.	.	PUNCT
ejpam-5532	427	1	j.	j.	PROPN
ejpam-5532	427	2	pure	pure	PROPN
ejpam-5532	427	3	appl	appl	PROPN
ejpam-5532	427	4	.	.	PROPN
ejpam-5532	427	5	math	math	PROPN
ejpam-5532	427	6	,	,	PUNCT
ejpam-5532	427	7	18	18	NUM
ejpam-5532	427	8	(	(	PUNCT
ejpam-5532	427	9	1	1	NUM
ejpam-5532	427	10	)	)	PUNCT
ejpam-5532	427	11	(	(	PUNCT
ejpam-5532	427	12	2025	2025	NUM
ejpam-5532	427	13	)	)	PUNCT
ejpam-5532	427	14	,	,	PUNCT
ejpam-5532	427	15	5532	5532	NUM
ejpam-5532	427	16	18	18	NUM
ejpam-5532	427	17	of	of	ADP
ejpam-5532	427	18	20	20	NUM
ejpam-5532	427	19	thus	thus	ADV
ejpam-5532	427	20	,	,	PUNCT
ejpam-5532	427	21	η	η	PROPN
ejpam-5532	427	22	is	be	AUX
ejpam-5532	427	23	an	an	DET
ejpam-5532	427	24	l	l	ADJ
ejpam-5532	427	25	-	-	ADJ
ejpam-5532	427	26	prime	prime	ADJ
ejpam-5532	427	27	ideal	ideal	NOUN
ejpam-5532	427	28	of	of	ADP
ejpam-5532	427	29	m	m	PROPN
ejpam-5532	427	30	.	.	PUNCT
ejpam-5532	428	1	similarly	similarly	ADV
ejpam-5532	428	2	,	,	PUNCT
ejpam-5532	428	3	we	we	PRON
ejpam-5532	428	4	have	have	VERB
ejpam-5532	428	5	η(x	η(x	ADJ
ejpam-5532	428	6	∧	∧	PROPN
ejpam-5532	428	7	y	y	PROPN
ejpam-5532	428	8	)	)	PUNCT
ejpam-5532	428	9	≥	≥	NOUN
ejpam-5532	428	10	η(x	η(x	NOUN
ejpam-5532	428	11	)	)	PUNCT
ejpam-5532	428	12	∧	∧	PROPN
ejpam-5532	428	13	η(y	η(y	NOUN
ejpam-5532	428	14	)	)	PUNCT
ejpam-5532	428	15	;	;	PUNCT
ejpam-5532	428	16	∀	∀	X
ejpam-5532	428	17	x	x	X
ejpam-5532	428	18	,	,	PUNCT
ejpam-5532	428	19	y	y	PROPN
ejpam-5532	428	20	∈	∈	PROPN
ejpam-5532	428	21	m.	m.	NOUN
ejpam-5532	428	22	this	this	PRON
ejpam-5532	428	23	implies	imply	VERB
ejpam-5532	428	24	η′(x	η′(x	X
ejpam-5532	428	25	∧	∧	PROPN
ejpam-5532	428	26	y	y	PROPN
ejpam-5532	428	27	)	)	PUNCT
ejpam-5532	428	28	=	=	PUNCT
ejpam-5532	429	1	τ	τ	PROPN
ejpam-5532	430	1	[	[	X
ejpam-5532	430	2	η(x	η(x	X
ejpam-5532	430	3	∧	∧	PROPN
ejpam-5532	430	4	y	y	PROPN
ejpam-5532	430	5	)	)	PUNCT
ejpam-5532	430	6	]	]	PUNCT
ejpam-5532	431	1	≤	≤	NUM
ejpam-5532	431	2	τ	τ	PUNCT
ejpam-5532	431	3	[	[	X
ejpam-5532	431	4	η(x	η(x	NOUN
ejpam-5532	431	5	)	)	PUNCT
ejpam-5532	431	6	∧	∧	PROPN
ejpam-5532	431	7	η(y	η(y	NOUN
ejpam-5532	431	8	)	)	PUNCT
ejpam-5532	431	9	]	]	PUNCT
ejpam-5532	431	10	=	=	PUNCT
ejpam-5532	431	11	τ(η(x	τ(η(x	PROPN
ejpam-5532	431	12	)	)	PUNCT
ejpam-5532	431	13	)	)	PUNCT
ejpam-5532	431	14	∨	∨	NUM
ejpam-5532	431	15	τ(η(y	τ(η(y	PROPN
ejpam-5532	431	16	)	)	PUNCT
ejpam-5532	431	17	)	)	PUNCT
ejpam-5532	432	1	=	=	PUNCT
ejpam-5532	432	2	η′(x	η′(x	X
ejpam-5532	432	3	)	)	PUNCT
ejpam-5532	432	4	∨	∨	NUM
ejpam-5532	432	5	η′(y	η′(y	NOUN
ejpam-5532	432	6	)	)	PUNCT
ejpam-5532	432	7	.	.	PUNCT
ejpam-5532	433	1	hence	hence	ADV
ejpam-5532	433	2	,	,	PUNCT
ejpam-5532	433	3	η	η	PROPN
ejpam-5532	433	4	and	and	CCONJ
ejpam-5532	433	5	η′	η′	PROPN
ejpam-5532	433	6	are	be	AUX
ejpam-5532	433	7	l	l	ADJ
ejpam-5532	433	8	-	-	ADJ
ejpam-5532	433	9	prime	prime	ADJ
ejpam-5532	433	10	ideals	ideal	NOUN
ejpam-5532	433	11	of	of	ADP
ejpam-5532	433	12	m	m	PROPN
ejpam-5532	433	13	.	.	PUNCT
ejpam-5532	434	1	in	in	ADP
ejpam-5532	434	2	fact	fact	NOUN
ejpam-5532	434	3	,	,	PUNCT
ejpam-5532	434	4	in	in	ADP
ejpam-5532	434	5	the	the	DET
ejpam-5532	434	6	above	above	ADJ
ejpam-5532	434	7	theorem	theorem	NOUN
ejpam-5532	434	8	,	,	PUNCT
ejpam-5532	434	9	it	it	PRON
ejpam-5532	434	10	can	can	AUX
ejpam-5532	434	11	be	be	AUX
ejpam-5532	434	12	proved	prove	VERB
ejpam-5532	434	13	that	that	SCONJ
ejpam-5532	434	14	η′	η′	PROPN
ejpam-5532	434	15	is	be	AUX
ejpam-5532	434	16	an	an	DET
ejpam-5532	434	17	l	l	ADJ
ejpam-5532	434	18	-	-	ADJ
ejpam-5532	434	19	dual	dual	ADJ
ejpam-5532	434	20	prime	prime	ADJ
ejpam-5532	434	21	ideal	ideal	NOUN
ejpam-5532	434	22	of	of	ADP
ejpam-5532	434	23	m	m	PROPN
ejpam-5532	434	24	.	.	PUNCT
ejpam-5532	435	1	we	we	PRON
ejpam-5532	435	2	conclude	conclude	VERB
ejpam-5532	435	3	this	this	DET
ejpam-5532	435	4	paper	paper	NOUN
ejpam-5532	435	5	by	by	ADP
ejpam-5532	435	6	discussing	discuss	VERB
ejpam-5532	435	7	an	an	DET
ejpam-5532	435	8	analogue	analogue	NOUN
ejpam-5532	435	9	of	of	ADP
ejpam-5532	435	10	another	another	DET
ejpam-5532	435	11	well	well	ADV
ejpam-5532	435	12	known	know	VERB
ejpam-5532	435	13	fact	fact	NOUN
ejpam-5532	435	14	in	in	ADP
ejpam-5532	435	15	classical	classical	ADJ
ejpam-5532	435	16	lattice	lattice	NOUN
ejpam-5532	435	17	theory	theory	NOUN
ejpam-5532	435	18	that	that	SCONJ
ejpam-5532	435	19	,	,	PUNCT
ejpam-5532	435	20	in	in	ADP
ejpam-5532	435	21	a	a	DET
ejpam-5532	435	22	distributive	distributive	ADJ
ejpam-5532	435	23	lattice	lattice	NOUN
ejpam-5532	435	24	with	with	ADP
ejpam-5532	435	25	the	the	DET
ejpam-5532	435	26	maximal	maximal	ADJ
ejpam-5532	435	27	element	element	NOUN
ejpam-5532	435	28	1	1	NUM
ejpam-5532	435	29	,	,	PUNCT
ejpam-5532	435	30	every	every	DET
ejpam-5532	435	31	proper	proper	ADJ
ejpam-5532	435	32	ideal	ideal	NOUN
ejpam-5532	435	33	is	be	AUX
ejpam-5532	435	34	contained	contain	VERB
ejpam-5532	435	35	in	in	ADP
ejpam-5532	435	36	a	a	DET
ejpam-5532	435	37	maximal	maximal	ADJ
ejpam-5532	435	38	ideal	ideal	NOUN
ejpam-5532	435	39	.	.	PUNCT
ejpam-5532	436	1	the	the	DET
ejpam-5532	436	2	next	next	ADJ
ejpam-5532	436	3	theorem	theorem	NOUN
ejpam-5532	436	4	proves	prove	VERB
ejpam-5532	436	5	a	a	DET
ejpam-5532	436	6	similar	similar	ADJ
ejpam-5532	436	7	result	result	NOUN
ejpam-5532	436	8	in	in	ADP
ejpam-5532	436	9	an	an	DET
ejpam-5532	436	10	l	l	ADJ
ejpam-5532	436	11	-	-	PUNCT
ejpam-5532	436	12	lattice	lattice	PROPN
ejpam-5532	436	13	µ.	µ.	PROPN
ejpam-5532	436	14	theorem	theorem	VERB
ejpam-5532	436	15	16	16	NUM
ejpam-5532	436	16	.	.	PUNCT
ejpam-5532	437	1	let	let	AUX
ejpam-5532	437	2	l(µ,m	l(µ,m	ADV
ejpam-5532	437	3	)	)	PUNCT
ejpam-5532	437	4	be	be	AUX
ejpam-5532	437	5	an	an	DET
ejpam-5532	437	6	l	l	NOUN
ejpam-5532	437	7	-	-	NOUN
ejpam-5532	437	8	lattice	lattice	NOUN
ejpam-5532	437	9	,	,	PUNCT
ejpam-5532	437	10	where	where	SCONJ
ejpam-5532	437	11	l	l	NOUN
ejpam-5532	437	12	is	be	AUX
ejpam-5532	437	13	a	a	DET
ejpam-5532	437	14	completely	completely	ADV
ejpam-5532	437	15	distributive	distributive	ADJ
ejpam-5532	437	16	lattice	lattice	NOUN
ejpam-5532	437	17	.	.	PUNCT
ejpam-5532	438	1	let	let	VERB
ejpam-5532	438	2	η	η	PROPN
ejpam-5532	438	3	⊆	⊆	PROPN
ejpam-5532	438	4	µ	µ	X
ejpam-5532	438	5	be	be	AUX
ejpam-5532	438	6	an	an	DET
ejpam-5532	438	7	l	l	NOUN
ejpam-5532	438	8	-	-	NOUN
ejpam-5532	438	9	ideal	ideal	NOUN
ejpam-5532	438	10	of	of	ADP
ejpam-5532	438	11	µ.	µ.	NOUN
ejpam-5532	438	12	then	then	ADV
ejpam-5532	438	13	,	,	PUNCT
ejpam-5532	438	14	there	there	PRON
ejpam-5532	438	15	exists	exist	VERB
ejpam-5532	438	16	an	an	DET
ejpam-5532	438	17	l	l	NOUN
ejpam-5532	438	18	-	-	ADJ
ejpam-5532	438	19	maximal	maximal	ADJ
ejpam-5532	438	20	ideal	ideal	ADJ
ejpam-5532	438	21	θ	θ	PROPN
ejpam-5532	438	22	of	of	ADP
ejpam-5532	438	23	µ	µ	PRON
ejpam-5532	439	1	such	such	ADJ
ejpam-5532	439	2	that	that	SCONJ
ejpam-5532	439	3	η	η	PROPN
ejpam-5532	439	4	⊆	⊆	NUM
ejpam-5532	439	5	θ	θ	PROPN
ejpam-5532	439	6	.	.	PUNCT
ejpam-5532	439	7	proof	proof	NOUN
ejpam-5532	439	8	.	.	PUNCT
ejpam-5532	440	1	let	let	VERB
ejpam-5532	440	2	i	i	PRON
ejpam-5532	440	3	=	=	PUNCT
ejpam-5532	440	4	{	{	PUNCT
ejpam-5532	440	5	γ	γ	X
ejpam-5532	440	6	/	/	SYM
ejpam-5532	440	7	γ	γ	X
ejpam-5532	440	8	is	be	AUX
ejpam-5532	440	9	an	an	DET
ejpam-5532	440	10	l	l	NOUN
ejpam-5532	440	11	-	-	NOUN
ejpam-5532	440	12	ideal	ideal	NOUN
ejpam-5532	440	13	of	of	ADP
ejpam-5532	440	14	µ	µ	PRON
ejpam-5532	440	15	such	such	ADJ
ejpam-5532	440	16	that	that	SCONJ
ejpam-5532	440	17	η	η	PROPN
ejpam-5532	440	18	⊆	⊆	X
ejpam-5532	440	19	γ	γ	X
ejpam-5532	440	20	}	}	PUNCT
ejpam-5532	440	21	.	.	PUNCT
ejpam-5532	441	1	let	let	VERB
ejpam-5532	441	2	φ	φ	PROPN
ejpam-5532	441	3	=	=	SYM
ejpam-5532	441	4	{	{	PUNCT
ejpam-5532	441	5	γi}i∈λ	γi}i∈λ	INTJ
ejpam-5532	441	6	be	be	AUX
ejpam-5532	441	7	a	a	DET
ejpam-5532	441	8	chain	chain	NOUN
ejpam-5532	441	9	in	in	ADP
ejpam-5532	441	10	i.	i.	PROPN
ejpam-5532	441	11	then	then	ADV
ejpam-5532	441	12	clearly	clearly	ADV
ejpam-5532	441	13	,	,	PUNCT
ejpam-5532	441	14	η	η	PROPN
ejpam-5532	441	15	⊆	⊆	NUM
ejpam-5532	441	16	∪{γi	∪{γi	NOUN
ejpam-5532	441	17	}	}	PUNCT
ejpam-5532	441	18	and	and	CCONJ
ejpam-5532	441	19	∪	∪	ADJ
ejpam-5532	441	20	{	{	PUNCT
ejpam-5532	441	21	γi	γi	INTJ
ejpam-5532	441	22	}	}	PUNCT
ejpam-5532	441	23	⊆	⊆	NUM
ejpam-5532	441	24	µ.	µ.	NOUN
ejpam-5532	441	25	if	if	SCONJ
ejpam-5532	441	26	x	x	NOUN
ejpam-5532	441	27	,	,	PUNCT
ejpam-5532	441	28	y	y	PROPN
ejpam-5532	441	29	∈	∈	PROPN
ejpam-5532	441	30	m	m	PROPN
ejpam-5532	441	31	,	,	PUNCT
ejpam-5532	441	32	{	{	PUNCT
ejpam-5532	441	33	∪γi}(x	∪γi}(x	PROPN
ejpam-5532	441	34	∨	∨	PROPN
ejpam-5532	441	35	y	y	PROPN
ejpam-5532	441	36	)	)	PUNCT
ejpam-5532	441	37	=	=	SYM
ejpam-5532	442	1	∨γi(x	∨γi(x	PROPN
ejpam-5532	442	2	∨	∨	NUM
ejpam-5532	442	3	y	y	PROPN
ejpam-5532	442	4	)	)	PUNCT
ejpam-5532	442	5	≥	≥	NOUN
ejpam-5532	442	6	∨[γi(x	∨[γi(x	NUM
ejpam-5532	442	7	)	)	PUNCT
ejpam-5532	442	8	∧	∧	NOUN
ejpam-5532	442	9	γi(y	γi(y	NOUN
ejpam-5532	442	10	)	)	PUNCT
ejpam-5532	442	11	]	]	PUNCT
ejpam-5532	443	1	(	(	PUNCT
ejpam-5532	443	2	as	as	SCONJ
ejpam-5532	443	3	each	each	DET
ejpam-5532	443	4	γi	γi	X
ejpam-5532	443	5	is	be	AUX
ejpam-5532	443	6	an	an	DET
ejpam-5532	443	7	l	l	NOUN
ejpam-5532	443	8	-	-	NOUN
ejpam-5532	443	9	ideal	ideal	NOUN
ejpam-5532	443	10	of	of	ADP
ejpam-5532	443	11	µ	µ	NOUN
ejpam-5532	443	12	)	)	PUNCT
ejpam-5532	443	13	=	=	PUNCT
ejpam-5532	444	1	[	[	X
ejpam-5532	444	2	∨γi(x	∨γi(x	PROPN
ejpam-5532	444	3	)	)	PUNCT
ejpam-5532	444	4	]	]	PUNCT
ejpam-5532	445	1	∧	∧	PROPN
ejpam-5532	445	2	[	[	X
ejpam-5532	445	3	∨γi(y	∨γi(y	PROPN
ejpam-5532	445	4	)	)	PUNCT
ejpam-5532	445	5	]	]	PUNCT
ejpam-5532	446	1	=	=	SYM
ejpam-5532	446	2	(	(	PUNCT
ejpam-5532	446	3	∪γi)(x	∪γi)(x	PROPN
ejpam-5532	446	4	)	)	PUNCT
ejpam-5532	446	5	∧	∧	PROPN
ejpam-5532	446	6	(	(	PUNCT
ejpam-5532	446	7	∪γi)(y	∪γi)(y	PROPN
ejpam-5532	446	8	)	)	PUNCT
ejpam-5532	446	9	.	.	PUNCT
ejpam-5532	447	1	moreover	moreover	ADV
ejpam-5532	447	2	,	,	PUNCT
ejpam-5532	447	3	{	{	PUNCT
ejpam-5532	447	4	∪γi}(x	∪γi}(x	PROPN
ejpam-5532	447	5	∧	∧	PROPN
ejpam-5532	447	6	y	y	PROPN
ejpam-5532	447	7	)	)	PUNCT
ejpam-5532	447	8	=	=	SYM
ejpam-5532	448	1	∨γi(x	∨γi(x	PROPN
ejpam-5532	448	2	∧	∧	PROPN
ejpam-5532	448	3	y	y	PROPN
ejpam-5532	448	4	)	)	PUNCT
ejpam-5532	448	5	≥	≥	NOUN
ejpam-5532	448	6	∨[µ(x	∨[µ(x	PROPN
ejpam-5532	448	7	)	)	PUNCT
ejpam-5532	448	8	∧	∧	NOUN
ejpam-5532	448	9	γi(y	γi(y	VERB
ejpam-5532	448	10	)	)	PUNCT
ejpam-5532	448	11	]	]	PUNCT
ejpam-5532	449	1	(	(	PUNCT
ejpam-5532	449	2	as	as	SCONJ
ejpam-5532	449	3	each	each	DET
ejpam-5532	449	4	γi	γi	X
ejpam-5532	449	5	is	be	AUX
ejpam-5532	449	6	an	an	DET
ejpam-5532	449	7	l	l	NOUN
ejpam-5532	449	8	-	-	NOUN
ejpam-5532	449	9	ideal	ideal	NOUN
ejpam-5532	449	10	of	of	ADP
ejpam-5532	449	11	µ	µ	NOUN
ejpam-5532	449	12	)	)	PUNCT
ejpam-5532	449	13	=	=	SYM
ejpam-5532	449	14	µ(x	µ(x	X
ejpam-5532	449	15	)	)	PUNCT
ejpam-5532	449	16	∧	∧	PROPN
ejpam-5532	449	17	[	[	X
ejpam-5532	449	18	∨γi(y	∨γi(y	PROPN
ejpam-5532	449	19	)	)	PUNCT
ejpam-5532	449	20	]	]	PUNCT
ejpam-5532	449	21	=	=	SYM
ejpam-5532	449	22	µ(x	µ(x	X
ejpam-5532	449	23	)	)	PUNCT
ejpam-5532	449	24	∧	∧	PROPN
ejpam-5532	449	25	(	(	PUNCT
ejpam-5532	449	26	∪γi)(y	∪γi)(y	PROPN
ejpam-5532	449	27	)	)	PUNCT
ejpam-5532	449	28	.	.	PUNCT
ejpam-5532	450	1	a.	a.	PROPN
ejpam-5532	450	2	jain	jain	PROPN
ejpam-5532	450	3	,	,	PUNCT
ejpam-5532	450	4	i.	i.	PROPN
ejpam-5532	450	5	jahan	jahan	PROPN
ejpam-5532	450	6	/	/	SYM
ejpam-5532	450	7	eur	eur	PROPN
ejpam-5532	450	8	.	.	PUNCT
ejpam-5532	451	1	j.	j.	PROPN
ejpam-5532	451	2	pure	pure	PROPN
ejpam-5532	451	3	appl	appl	PROPN
ejpam-5532	451	4	.	.	PROPN
ejpam-5532	451	5	math	math	PROPN
ejpam-5532	451	6	,	,	PUNCT
ejpam-5532	451	7	18	18	NUM
ejpam-5532	451	8	(	(	PUNCT
ejpam-5532	451	9	1	1	NUM
ejpam-5532	451	10	)	)	PUNCT
ejpam-5532	451	11	(	(	PUNCT
ejpam-5532	451	12	2025	2025	NUM
ejpam-5532	451	13	)	)	PUNCT
ejpam-5532	451	14	,	,	PUNCT
ejpam-5532	451	15	5532	5532	NUM
ejpam-5532	451	16	19	19	NUM
ejpam-5532	451	17	of	of	ADP
ejpam-5532	451	18	20	20	NUM
ejpam-5532	451	19	thus	thus	ADV
ejpam-5532	451	20	,	,	PUNCT
ejpam-5532	451	21	{	{	PUNCT
ejpam-5532	451	22	∪γi	∪γi	PROPN
ejpam-5532	451	23	}	}	PUNCT
ejpam-5532	451	24	is	be	AUX
ejpam-5532	451	25	an	an	DET
ejpam-5532	451	26	l	l	NOUN
ejpam-5532	451	27	-	-	NOUN
ejpam-5532	451	28	ideal	ideal	NOUN
ejpam-5532	451	29	of	of	ADP
ejpam-5532	451	30	µ	µ	X
ejpam-5532	451	31	containing	contain	VERB
ejpam-5532	451	32	η	η	PROPN
ejpam-5532	451	33	.	.	PROPN
ejpam-5532	451	34	that	that	ADV
ejpam-5532	451	35	is	is	ADV
ejpam-5532	451	36	,	,	PUNCT
ejpam-5532	451	37	{	{	PUNCT
ejpam-5532	451	38	∪γi	∪γi	ADV
ejpam-5532	451	39	}	}	PUNCT
ejpam-5532	451	40	∈	∈	PROPN
ejpam-5532	451	41	i.	i.	NOUN
ejpam-5532	451	42	thus	thus	ADV
ejpam-5532	451	43	,	,	PUNCT
ejpam-5532	451	44	every	every	DET
ejpam-5532	451	45	chain	chain	NOUN
ejpam-5532	451	46	in	in	ADP
ejpam-5532	451	47	i	i	PRON
ejpam-5532	451	48	has	have	VERB
ejpam-5532	451	49	an	an	DET
ejpam-5532	451	50	upper	upper	ADJ
ejpam-5532	451	51	bound	bind	VERB
ejpam-5532	451	52	in	in	ADP
ejpam-5532	451	53	i.	i.	PROPN
ejpam-5532	451	54	therefore	therefore	ADV
ejpam-5532	451	55	,	,	PUNCT
ejpam-5532	451	56	by	by	ADP
ejpam-5532	451	57	zorn	zorn	PROPN
ejpam-5532	451	58	’s	’s	PART
ejpam-5532	451	59	lemma	lemma	PROPN
ejpam-5532	451	60	,	,	PUNCT
ejpam-5532	451	61	i	i	PRON
ejpam-5532	451	62	has	have	VERB
ejpam-5532	451	63	a	a	DET
ejpam-5532	451	64	maximal	maximal	ADJ
ejpam-5532	451	65	element	element	NOUN
ejpam-5532	451	66	.	.	PUNCT
ejpam-5532	452	1	that	that	PRON
ejpam-5532	452	2	is	be	AUX
ejpam-5532	452	3	,	,	PUNCT
ejpam-5532	452	4	∃	∃	PROPN
ejpam-5532	452	5	a	a	DET
ejpam-5532	452	6	maximal	maximal	ADJ
ejpam-5532	452	7	l	l	NOUN
ejpam-5532	452	8	-	-	ADJ
ejpam-5532	452	9	ideal	ideal	ADJ
ejpam-5532	452	10	γ	γ	NOUN
ejpam-5532	452	11	of	of	ADP
ejpam-5532	452	12	µ	µ	PRON
ejpam-5532	452	13	such	such	ADJ
ejpam-5532	452	14	that	that	SCONJ
ejpam-5532	452	15	η	η	PROPN
ejpam-5532	452	16	⊆	⊆	NUM
ejpam-5532	452	17	γ	γ	X
ejpam-5532	452	18	.	.	PROPN
ejpam-5532	452	19	hence	hence	ADV
ejpam-5532	452	20	,	,	PUNCT
ejpam-5532	452	21	γ	γ	X
ejpam-5532	452	22	is	be	AUX
ejpam-5532	452	23	the	the	DET
ejpam-5532	452	24	required	required	ADJ
ejpam-5532	452	25	l	l	ADJ
ejpam-5532	452	26	-	-	ADJ
ejpam-5532	452	27	maximal	maximal	ADJ
ejpam-5532	452	28	ideal	ideal	NOUN
ejpam-5532	452	29	of	of	ADP
ejpam-5532	452	30	µ	µ	X
ejpam-5532	452	31	containing	contain	VERB
ejpam-5532	452	32	η	η	PROPN
ejpam-5532	452	33	.	.	PROPN
ejpam-5532	452	34	conclusion	conclusion	NOUN
ejpam-5532	452	35	in	in	ADP
ejpam-5532	452	36	the	the	DET
ejpam-5532	452	37	present	present	ADJ
ejpam-5532	452	38	work	work	NOUN
ejpam-5532	452	39	,	,	PUNCT
ejpam-5532	452	40	the	the	DET
ejpam-5532	452	41	concept	concept	NOUN
ejpam-5532	452	42	of	of	ADP
ejpam-5532	452	43	an	an	DET
ejpam-5532	452	44	l	l	ADJ
ejpam-5532	452	45	-	-	ADJ
ejpam-5532	452	46	convex	convex	ADJ
ejpam-5532	452	47	sublattice	sublattice	NOUN
ejpam-5532	452	48	in	in	ADP
ejpam-5532	452	49	an	an	DET
ejpam-5532	452	50	l	l	NOUN
ejpam-5532	452	51	-	-	PUNCT
ejpam-5532	452	52	lattice	lattice	NOUN
ejpam-5532	452	53	is	be	AUX
ejpam-5532	452	54	studied	study	VERB
ejpam-5532	452	55	in	in	ADP
ejpam-5532	452	56	detail	detail	NOUN
ejpam-5532	452	57	and	and	CCONJ
ejpam-5532	452	58	the	the	DET
ejpam-5532	452	59	unique	unique	ADJ
ejpam-5532	452	60	representation	representation	NOUN
ejpam-5532	452	61	theorem	theorem	NOUN
ejpam-5532	452	62	for	for	ADP
ejpam-5532	452	63	l	l	ADJ
ejpam-5532	452	64	-	-	ADJ
ejpam-5532	452	65	convex	convex	ADJ
ejpam-5532	452	66	sublattices	sublattice	NOUN
ejpam-5532	452	67	is	be	AUX
ejpam-5532	452	68	established	establish	VERB
ejpam-5532	452	69	.	.	PUNCT
ejpam-5532	453	1	moreover	moreover	ADV
ejpam-5532	453	2	,	,	PUNCT
ejpam-5532	453	3	the	the	DET
ejpam-5532	453	4	concept	concept	NOUN
ejpam-5532	453	5	of	of	ADP
ejpam-5532	453	6	order	order	NOUN
ejpam-5532	453	7	reversing	reverse	VERB
ejpam-5532	453	8	involution	involution	NOUN
ejpam-5532	453	9	is	be	AUX
ejpam-5532	453	10	utilized	utilize	VERB
ejpam-5532	453	11	on	on	ADP
ejpam-5532	453	12	the	the	DET
ejpam-5532	453	13	lattice	lattice	PROPN
ejpam-5532	453	14	l	l	PROPN
ejpam-5532	453	15	of	of	ADP
ejpam-5532	453	16	truth	truth	NOUN
ejpam-5532	453	17	values	value	NOUN
ejpam-5532	453	18	to	to	PART
ejpam-5532	453	19	define	define	VERB
ejpam-5532	453	20	the	the	DET
ejpam-5532	453	21	complement	complement	NOUN
ejpam-5532	453	22	of	of	ADP
ejpam-5532	453	23	an	an	DET
ejpam-5532	453	24	l	l	NOUN
ejpam-5532	453	25	-	-	NOUN
ejpam-5532	453	26	set	set	NOUN
ejpam-5532	453	27	.	.	PUNCT
ejpam-5532	454	1	the	the	DET
ejpam-5532	454	2	notion	notion	NOUN
ejpam-5532	454	3	of	of	ADP
ejpam-5532	454	4	complementation	complementation	NOUN
ejpam-5532	454	5	plays	play	VERB
ejpam-5532	454	6	a	a	DET
ejpam-5532	454	7	significant	significant	ADJ
ejpam-5532	454	8	role	role	NOUN
ejpam-5532	454	9	in	in	ADP
ejpam-5532	454	10	the	the	DET
ejpam-5532	454	11	theory	theory	NOUN
ejpam-5532	454	12	of	of	ADP
ejpam-5532	454	13	boolean	boolean	ADJ
ejpam-5532	454	14	algebras	algebra	NOUN
ejpam-5532	454	15	,	,	PUNCT
ejpam-5532	454	16	lattice	lattice	ADJ
ejpam-5532	454	17	implication	implication	NOUN
ejpam-5532	454	18	algebras	algebra	NOUN
ejpam-5532	454	19	and	and	CCONJ
ejpam-5532	454	20	topological	topological	ADJ
ejpam-5532	454	21	spaces	space	NOUN
ejpam-5532	454	22	.	.	PUNCT
ejpam-5532	455	1	in	in	ADP
ejpam-5532	455	2	this	this	DET
ejpam-5532	455	3	paper	paper	NOUN
ejpam-5532	455	4	,	,	PUNCT
ejpam-5532	455	5	it	it	PRON
ejpam-5532	455	6	is	be	AUX
ejpam-5532	455	7	established	establish	VERB
ejpam-5532	455	8	that	that	SCONJ
ejpam-5532	455	9	the	the	DET
ejpam-5532	455	10	concept	concept	NOUN
ejpam-5532	455	11	of	of	ADP
ejpam-5532	455	12	complementation	complementation	NOUN
ejpam-5532	455	13	of	of	ADP
ejpam-5532	455	14	an	an	DET
ejpam-5532	455	15	l	l	NOUN
ejpam-5532	455	16	-	-	ADJ
ejpam-5532	455	17	set	set	VERB
ejpam-5532	455	18	leads	lead	NOUN
ejpam-5532	455	19	to	to	ADP
ejpam-5532	455	20	proving	prove	VERB
ejpam-5532	455	21	some	some	DET
ejpam-5532	455	22	significant	significant	ADJ
ejpam-5532	455	23	results	result	NOUN
ejpam-5532	455	24	in	in	ADP
ejpam-5532	455	25	l	l	ADJ
ejpam-5532	455	26	-	-	PUNCT
ejpam-5532	455	27	lattice	lattice	NOUN
ejpam-5532	455	28	theory	theory	NOUN
ejpam-5532	455	29	.	.	PUNCT
ejpam-5532	456	1	this	this	DET
ejpam-5532	456	2	notion	notion	NOUN
ejpam-5532	456	3	is	be	AUX
ejpam-5532	456	4	further	far	ADV
ejpam-5532	456	5	worthy	worthy	ADJ
ejpam-5532	456	6	of	of	ADP
ejpam-5532	456	7	attention	attention	NOUN
ejpam-5532	456	8	as	as	SCONJ
ejpam-5532	456	9	it	it	PRON
ejpam-5532	456	10	may	may	AUX
ejpam-5532	456	11	lead	lead	VERB
ejpam-5532	456	12	to	to	ADP
ejpam-5532	456	13	some	some	DET
ejpam-5532	456	14	remarkable	remarkable	ADJ
ejpam-5532	456	15	development	development	NOUN
ejpam-5532	456	16	in	in	ADP
ejpam-5532	456	17	the	the	DET
ejpam-5532	456	18	theory	theory	NOUN
ejpam-5532	456	19	of	of	ADP
ejpam-5532	456	20	l	l	NOUN
ejpam-5532	456	21	-	-	NOUN
ejpam-5532	456	22	substructures	substructure	NOUN
ejpam-5532	456	23	of	of	ADP
ejpam-5532	456	24	an	an	DET
ejpam-5532	456	25	lacknowledgements	lacknowledgement	NOUN
ejpam-5532	456	26	the	the	DET
ejpam-5532	456	27	authors	author	NOUN
ejpam-5532	456	28	are	be	AUX
ejpam-5532	456	29	highly	highly	ADV
ejpam-5532	456	30	grateful	grateful	ADJ
ejpam-5532	456	31	to	to	ADP
ejpam-5532	456	32	the	the	DET
ejpam-5532	456	33	learned	learn	VERB
ejpam-5532	456	34	referees	referee	NOUN
ejpam-5532	456	35	for	for	ADP
ejpam-5532	456	36	their	their	PRON
ejpam-5532	456	37	valued	value	VERB
ejpam-5532	456	38	comments	comment	NOUN
ejpam-5532	456	39	which	which	PRON
ejpam-5532	456	40	helped	help	VERB
ejpam-5532	456	41	to	to	PART
ejpam-5532	456	42	improve	improve	VERB
ejpam-5532	456	43	the	the	DET
ejpam-5532	456	44	quality	quality	NOUN
ejpam-5532	456	45	and	and	CCONJ
ejpam-5532	456	46	presentation	presentation	NOUN
ejpam-5532	456	47	of	of	ADP
ejpam-5532	456	48	this	this	DET
ejpam-5532	456	49	paper	paper	NOUN
ejpam-5532	456	50	.	.	PUNCT
ejpam-5532	457	1	references	reference	NOUN
ejpam-5532	457	2	[	[	X
ejpam-5532	457	3	1	1	NUM
ejpam-5532	457	4	]	]	PUNCT
ejpam-5532	457	5	n	n	X
ejpam-5532	457	6	ajmal	ajmal	ADJ
ejpam-5532	457	7	and	and	CCONJ
ejpam-5532	457	8	i	i	PROPN
ejpam-5532	457	9	jahan	jahan	PROPN
ejpam-5532	457	10	.	.	PUNCT
ejpam-5532	458	1	a	a	DET
ejpam-5532	458	2	study	study	NOUN
ejpam-5532	458	3	of	of	ADP
ejpam-5532	458	4	normal	normal	ADJ
ejpam-5532	458	5	fuzzy	fuzzy	ADJ
ejpam-5532	458	6	subgroups	subgroup	NOUN
ejpam-5532	458	7	and	and	CCONJ
ejpam-5532	458	8	characteristic	characteristic	ADJ
ejpam-5532	458	9	fuzzy	fuzzy	ADJ
ejpam-5532	458	10	subgroup	subgroup	NOUN
ejpam-5532	458	11	of	of	ADP
ejpam-5532	458	12	a	a	DET
ejpam-5532	458	13	fuzzy	fuzzy	ADJ
ejpam-5532	458	14	group	group	NOUN
ejpam-5532	458	15	.	.	PUNCT
ejpam-5532	459	1	fuzzy	fuzzy	ADJ
ejpam-5532	459	2	information	information	NOUN
ejpam-5532	459	3	and	and	CCONJ
ejpam-5532	459	4	engineering	engineering	NOUN
ejpam-5532	459	5	,	,	PUNCT
ejpam-5532	459	6	3:123–143	3:123–143	NUM
ejpam-5532	459	7	,	,	PUNCT
ejpam-5532	459	8	2012	2012	NUM
ejpam-5532	459	9	.	.	PUNCT
ejpam-5532	460	1	[	[	X
ejpam-5532	460	2	2	2	NUM
ejpam-5532	460	3	]	]	PUNCT
ejpam-5532	460	4	n	n	X
ejpam-5532	460	5	ajmal	ajmal	ADJ
ejpam-5532	460	6	and	and	CCONJ
ejpam-5532	460	7	i	i	PROPN
ejpam-5532	460	8	jahan	jahan	PROPN
ejpam-5532	460	9	.	.	PUNCT
ejpam-5532	461	1	nilpotency	nilpotency	NOUN
ejpam-5532	461	2	and	and	CCONJ
ejpam-5532	461	3	theory	theory	NOUN
ejpam-5532	461	4	of	of	ADP
ejpam-5532	461	5	l	l	NOUN
ejpam-5532	461	6	subgroups	subgroup	NOUN
ejpam-5532	461	7	of	of	ADP
ejpam-5532	461	8	an	an	DET
ejpam-5532	461	9	l	l	NOUN
ejpam-5532	461	10	-	-	NOUN
ejpam-5532	461	11	group	group	NOUN
ejpam-5532	461	12	.	.	PUNCT
ejpam-5532	462	1	fuzzy	fuzzy	ADJ
ejpam-5532	462	2	information	information	NOUN
ejpam-5532	462	3	and	and	CCONJ
ejpam-5532	462	4	engineering	engineering	NOUN
ejpam-5532	462	5	,	,	PUNCT
ejpam-5532	462	6	6(1):1–17	6(1):1–17	PROPN
ejpam-5532	462	7	,	,	PUNCT
ejpam-5532	462	8	2014	2014	NUM
ejpam-5532	462	9	.	.	PUNCT
ejpam-5532	463	1	[	[	X
ejpam-5532	463	2	3	3	X
ejpam-5532	463	3	]	]	PUNCT
ejpam-5532	463	4	n	n	X
ejpam-5532	463	5	ajmal	ajmal	ADJ
ejpam-5532	463	6	and	and	CCONJ
ejpam-5532	463	7	i	i	PROPN
ejpam-5532	463	8	jahan	jahan	PROPN
ejpam-5532	463	9	.	.	PUNCT
ejpam-5532	464	1	generated	generate	VERB
ejpam-5532	464	2	l	l	NOUN
ejpam-5532	464	3	-	-	NOUN
ejpam-5532	464	4	subgroup	subgroup	NOUN
ejpam-5532	464	5	of	of	ADP
ejpam-5532	464	6	an	an	DET
ejpam-5532	464	7	l	l	NOUN
ejpam-5532	464	8	-	-	NOUN
ejpam-5532	464	9	group	group	NOUN
ejpam-5532	464	10	.	.	PUNCT
ejpam-5532	465	1	iranian	iranian	PROPN
ejpam-5532	465	2	journal	journal	PROPN
ejpam-5532	465	3	of	of	ADP
ejpam-5532	465	4	fuzzy	fuzzy	ADJ
ejpam-5532	465	5	systems	system	NOUN
ejpam-5532	465	6	,	,	PUNCT
ejpam-5532	465	7	12(1):129–136	12(1):129–136	NUM
ejpam-5532	465	8	,	,	PUNCT
ejpam-5532	465	9	2015	2015	NUM
ejpam-5532	465	10	.	.	PUNCT
ejpam-5532	466	1	[	[	X
ejpam-5532	466	2	4	4	X
ejpam-5532	466	3	]	]	PUNCT
ejpam-5532	466	4	n	n	X
ejpam-5532	466	5	ajmal	ajmal	ADJ
ejpam-5532	466	6	and	and	CCONJ
ejpam-5532	466	7	k	k	PROPN
ejpam-5532	466	8	v	v	NUM
ejpam-5532	466	9	thomas	thomas	PROPN
ejpam-5532	466	10	.	.	PUNCT
ejpam-5532	467	1	fuzzy	fuzzy	ADJ
ejpam-5532	467	2	lattices	lattice	NOUN
ejpam-5532	467	3	.	.	PUNCT
ejpam-5532	468	1	79:271–291	79:271–291	NUM
ejpam-5532	468	2	,	,	PUNCT
ejpam-5532	468	3	1994	1994	NUM
ejpam-5532	468	4	.	.	PUNCT
ejpam-5532	469	1	[	[	X
ejpam-5532	469	2	5	5	NUM
ejpam-5532	469	3	]	]	PUNCT
ejpam-5532	469	4	n	n	X
ejpam-5532	469	5	ajmal	ajmal	ADJ
ejpam-5532	469	6	and	and	CCONJ
ejpam-5532	469	7	k	k	PROPN
ejpam-5532	469	8	v	v	NUM
ejpam-5532	469	9	thomas	thomas	PROPN
ejpam-5532	469	10	.	.	PUNCT
ejpam-5532	470	1	fuzzy	fuzzy	ADJ
ejpam-5532	470	2	lattices	lattice	NOUN
ejpam-5532	470	3	-	-	PUNCT
ejpam-5532	470	4	i.	i.	NOUN
ejpam-5532	470	5	journal	journal	NOUN
ejpam-5532	470	6	of	of	ADP
ejpam-5532	470	7	fuzzy	fuzzy	ADJ
ejpam-5532	470	8	mathematics	mathematic	NOUN
ejpam-5532	470	9	,	,	PUNCT
ejpam-5532	470	10	10:255	10:255	NUM
ejpam-5532	470	11	–	–	PUNCT
ejpam-5532	470	12	274	274	NUM
ejpam-5532	470	13	,	,	PUNCT
ejpam-5532	470	14	2002	2002	NUM
ejpam-5532	470	15	.	.	PUNCT
ejpam-5532	471	1	[	[	X
ejpam-5532	471	2	6	6	NUM
ejpam-5532	471	3	]	]	PUNCT
ejpam-5532	471	4	n	n	X
ejpam-5532	471	5	ajmal	ajmal	ADJ
ejpam-5532	471	6	and	and	CCONJ
ejpam-5532	471	7	k	k	PROPN
ejpam-5532	471	8	v	v	NUM
ejpam-5532	471	9	thomas	thomas	PROPN
ejpam-5532	471	10	.	.	PUNCT
ejpam-5532	472	1	fuzzy	fuzzy	ADJ
ejpam-5532	472	2	lattices	lattices	PROPN
ejpam-5532	472	3	-	-	PUNCT
ejpam-5532	472	4	ii	ii	NOUN
ejpam-5532	472	5	.	.	PUNCT
ejpam-5532	472	6	journal	journal	PROPN
ejpam-5532	472	7	of	of	ADP
ejpam-5532	472	8	fuzzy	fuzzy	ADJ
ejpam-5532	472	9	mathematics	mathematic	NOUN
ejpam-5532	472	10	,	,	PUNCT
ejpam-5532	472	11	10:275	10:275	NUM
ejpam-5532	472	12	–	–	PUNCT
ejpam-5532	472	13	296	296	NUM
ejpam-5532	472	14	,	,	PUNCT
ejpam-5532	472	15	2002	2002	NUM
ejpam-5532	472	16	.	.	PUNCT
ejpam-5532	473	1	[	[	X
ejpam-5532	473	2	7	7	X
ejpam-5532	473	3	]	]	X
ejpam-5532	473	4	j	j	PROPN
ejpam-5532	473	5	a	a	DET
ejpam-5532	473	6	goguen	goguen	NOUN
ejpam-5532	473	7	.	.	PUNCT
ejpam-5532	474	1	l	l	ADJ
ejpam-5532	474	2	-	-	ADJ
ejpam-5532	474	3	fuzzy	fuzzy	ADJ
ejpam-5532	474	4	sets	set	NOUN
ejpam-5532	474	5	.	.	PUNCT
ejpam-5532	475	1	journal	journal	NOUN
ejpam-5532	475	2	of	of	ADP
ejpam-5532	475	3	mathematical	mathematical	ADJ
ejpam-5532	475	4	analysis	analysis	NOUN
ejpam-5532	475	5	and	and	CCONJ
ejpam-5532	475	6	applications	application	NOUN
ejpam-5532	475	7	,	,	PUNCT
ejpam-5532	475	8	18:145–174	18:145–174	NUM
ejpam-5532	475	9	,	,	PUNCT
ejpam-5532	475	10	1967	1967	NUM
ejpam-5532	475	11	.	.	PUNCT
ejpam-5532	476	1	[	[	X
ejpam-5532	476	2	8	8	NUM
ejpam-5532	476	3	]	]	X
ejpam-5532	476	4	i	i	PROPN
ejpam-5532	476	5	jahan	jahan	PROPN
ejpam-5532	476	6	.	.	PUNCT
ejpam-5532	477	1	development	development	NOUN
ejpam-5532	477	2	of	of	ADP
ejpam-5532	477	3	lgroup	lgroup	PROPN
ejpam-5532	477	4	theory	theory	NOUN
ejpam-5532	477	5	.	.	PUNCT
ejpam-5532	478	1	2023	2023	NUM
ejpam-5532	478	2	.	.	PUNCT
ejpam-5532	479	1	[	[	X
ejpam-5532	479	2	9	9	NUM
ejpam-5532	479	3	]	]	X
ejpam-5532	479	4	i	i	PRON
ejpam-5532	479	5	jahan	jahan	PROPN
ejpam-5532	479	6	and	and	CCONJ
ejpam-5532	479	7	m	m	PROPN
ejpam-5532	479	8	ananya	ananya	PROPN
ejpam-5532	479	9	.	.	PUNCT
ejpam-5532	480	1	maximal	maximal	ADJ
ejpam-5532	480	2	and	and	CCONJ
ejpam-5532	480	3	frattini	frattini	ADJ
ejpam-5532	480	4	l	l	NOUN
ejpam-5532	480	5	-	-	NOUN
ejpam-5532	480	6	subgroups	subgroup	NOUN
ejpam-5532	480	7	of	of	ADP
ejpam-5532	480	8	an	an	DET
ejpam-5532	480	9	l	l	NOUN
ejpam-5532	480	10	-	-	NOUN
ejpam-5532	480	11	group	group	NOUN
ejpam-5532	480	12	.	.	PUNCT
ejpam-5532	481	1	journal	journal	PROPN
ejpam-5532	481	2	of	of	ADP
ejpam-5532	481	3	intelligent	intelligent	ADJ
ejpam-5532	481	4	and	and	CCONJ
ejpam-5532	481	5	fuzzy	fuzzy	ADJ
ejpam-5532	481	6	systems	system	NOUN
ejpam-5532	481	7	,	,	PUNCT
ejpam-5532	481	8	39:3995–4007	39:3995–4007	NUM
ejpam-5532	481	9	,	,	PUNCT
ejpam-5532	481	10	2020	2020	NUM
ejpam-5532	481	11	.	.	PUNCT
ejpam-5532	482	1	[	[	X
ejpam-5532	482	2	10	10	NUM
ejpam-5532	482	3	]	]	X
ejpam-5532	482	4	a	a	DET
ejpam-5532	482	5	jain	jain	NOUN
ejpam-5532	482	6	and	and	CCONJ
ejpam-5532	482	7	i	i	PROPN
ejpam-5532	482	8	jahan	jahan	PROPN
ejpam-5532	482	9	.	.	PUNCT
ejpam-5532	483	1	maximal	maximal	ADJ
ejpam-5532	483	2	ideal	ideal	ADJ
ejpam-5532	483	3	and	and	CCONJ
ejpam-5532	483	4	prime	prime	ADJ
ejpam-5532	483	5	ideal	ideal	NOUN
ejpam-5532	483	6	in	in	ADP
ejpam-5532	483	7	an	an	DET
ejpam-5532	483	8	l	l	NOUN
ejpam-5532	483	9	-	-	NOUN
ejpam-5532	483	10	lattice	lattice	NOUN
ejpam-5532	483	11	.	.	PUNCT
ejpam-5532	484	1	fuzzy	fuzzy	ADJ
ejpam-5532	484	2	information	information	NOUN
ejpam-5532	484	3	and	and	CCONJ
ejpam-5532	484	4	engineering	engineering	NOUN
ejpam-5532	484	5	,	,	PUNCT
ejpam-5532	484	6	16(3):244–263	16(3):244–263	NUM
ejpam-5532	484	7	,	,	PUNCT
ejpam-5532	484	8	2024	2024	NUM
ejpam-5532	484	9	.	.	PUNCT
ejpam-5532	485	1	a.	a.	PROPN
ejpam-5532	485	2	jain	jain	PROPN
ejpam-5532	485	3	,	,	PUNCT
ejpam-5532	485	4	i.	i.	PROPN
ejpam-5532	485	5	jahan	jahan	PROPN
ejpam-5532	485	6	/	/	SYM
ejpam-5532	485	7	eur	eur	PROPN
ejpam-5532	485	8	.	.	PUNCT
ejpam-5532	486	1	j.	j.	PROPN
ejpam-5532	486	2	pure	pure	PROPN
ejpam-5532	486	3	appl	appl	PROPN
ejpam-5532	486	4	.	.	PROPN
ejpam-5532	486	5	math	math	PROPN
ejpam-5532	486	6	,	,	PUNCT
ejpam-5532	486	7	18	18	NUM
ejpam-5532	486	8	(	(	PUNCT
ejpam-5532	486	9	1	1	NUM
ejpam-5532	486	10	)	)	PUNCT
ejpam-5532	486	11	(	(	PUNCT
ejpam-5532	486	12	2025	2025	NUM
ejpam-5532	486	13	)	)	PUNCT
ejpam-5532	486	14	,	,	PUNCT
ejpam-5532	486	15	5532	5532	NUM
ejpam-5532	486	16	20	20	NUM
ejpam-5532	486	17	of	of	ADP
ejpam-5532	486	18	20	20	NUM
ejpam-5532	486	19	[	[	SYM
ejpam-5532	486	20	11	11	NUM
ejpam-5532	486	21	]	]	PUNCT
ejpam-5532	486	22	t	t	PROPN
ejpam-5532	486	23	kubiak	kubiak	PROPN
ejpam-5532	486	24	and	and	CCONJ
ejpam-5532	486	25	de	de	X
ejpam-5532	486	26	prada	prada	PROPN
ejpam-5532	486	27	m	m	PROPN
ejpam-5532	486	28	a	a	DET
ejpam-5532	486	29	vincente	vincente	NOUN
ejpam-5532	486	30	.	.	PUNCT
ejpam-5532	487	1	regular	regular	ADJ
ejpam-5532	487	2	l	l	ADJ
ejpam-5532	487	3	-	-	ADJ
ejpam-5532	487	4	fuzzy	fuzzy	ADJ
ejpam-5532	487	5	topological	topological	ADJ
ejpam-5532	487	6	spaces	space	NOUN
ejpam-5532	487	7	and	and	CCONJ
ejpam-5532	487	8	their	their	PRON
ejpam-5532	487	9	topological	topological	ADJ
ejpam-5532	487	10	modifications	modification	NOUN
ejpam-5532	487	11	.	.	PUNCT
ejpam-5532	488	1	international	international	ADJ
ejpam-5532	488	2	journal	journal	PROPN
ejpam-5532	488	3	of	of	ADP
ejpam-5532	488	4	mathematics	mathematics	PROPN
ejpam-5532	488	5	and	and	CCONJ
ejpam-5532	488	6	mathematical	mathematical	ADJ
ejpam-5532	488	7	sciences	science	NOUN
ejpam-5532	488	8	,	,	PUNCT
ejpam-5532	488	9	23(10):687–695	23(10):687–695	NUM
ejpam-5532	488	10	,	,	PUNCT
ejpam-5532	488	11	2000	2000	NUM
ejpam-5532	488	12	.	.	PUNCT
ejpam-5532	489	1	[	[	X
ejpam-5532	489	2	12	12	NUM
ejpam-5532	489	3	]	]	X
ejpam-5532	489	4	i	i	PRON
ejpam-5532	489	5	jahan	jahan	PROPN
ejpam-5532	489	6	n	n	ADV
ejpam-5532	489	7	ajmal	ajmal	PROPN
ejpam-5532	489	8	and	and	CCONJ
ejpam-5532	489	9	b	b	PROPN
ejpam-5532	489	10	davvaz	davvaz	NOUN
ejpam-5532	489	11	.	.	PUNCT
ejpam-5532	490	1	subnormality	subnormality	NOUN
ejpam-5532	490	2	and	and	CCONJ
ejpam-5532	490	3	theory	theory	NOUN
ejpam-5532	490	4	of	of	ADP
ejpam-5532	490	5	l	l	NOUN
ejpam-5532	490	6	-	-	NOUN
ejpam-5532	490	7	subgroups	subgroup	NOUN
ejpam-5532	490	8	.	.	PUNCT
ejpam-5532	491	1	european	european	ADJ
ejpam-5532	491	2	journal	journal	PROPN
ejpam-5532	491	3	of	of	ADP
ejpam-5532	491	4	pure	pure	ADJ
ejpam-5532	491	5	and	and	CCONJ
ejpam-5532	491	6	applied	applied	ADJ
ejpam-5532	491	7	mathematics	mathematic	NOUN
ejpam-5532	491	8	,	,	PUNCT
ejpam-5532	491	9	15(4):2086–2115	15(4):2086–2115	NUM
ejpam-5532	491	10	,	,	PUNCT
ejpam-5532	491	11	2022	2022	NUM
ejpam-5532	491	12	.	.	PUNCT
ejpam-5532	492	1	[	[	X
ejpam-5532	492	2	13	13	NUM
ejpam-5532	492	3	]	]	SYM
ejpam-5532	492	4	l	l	NOUN
ejpam-5532	492	5	platil	platil	NOUN
ejpam-5532	492	6	and	and	CCONJ
ejpam-5532	492	7	t	t	PROPN
ejpam-5532	492	8	tanaka	tanaka	PROPN
ejpam-5532	492	9	.	.	PUNCT
ejpam-5532	493	1	milti	milti	NOUN
ejpam-5532	493	2	-	-	PUNCT
ejpam-5532	493	3	criteria	criterion	NOUN
ejpam-5532	493	4	evaluation	evaluation	NOUN
ejpam-5532	493	5	for	for	ADP
ejpam-5532	493	6	intuitionistic	intuitionistic	ADJ
ejpam-5532	493	7	fuzzy	fuzzy	ADJ
ejpam-5532	493	8	sets	set	NOUN
ejpam-5532	493	9	based	base	VERB
ejpam-5532	493	10	on	on	ADP
ejpam-5532	493	11	set	set	NOUN
ejpam-5532	493	12	-	-	PUNCT
ejpam-5532	493	13	relations	relation	NOUN
ejpam-5532	493	14	.	.	PUNCT
ejpam-5532	494	1	nihonkai	nihonkai	PROPN
ejpam-5532	494	2	mathematical	mathematical	PROPN
ejpam-5532	494	3	journal	journal	PROPN
ejpam-5532	494	4	,	,	PUNCT
ejpam-5532	494	5	34:1–18	34:1–18	NUM
ejpam-5532	494	6	,	,	PUNCT
ejpam-5532	494	7	2023	2023	NUM
ejpam-5532	494	8	.	.	PUNCT
ejpam-5532	495	1	[	[	X
ejpam-5532	495	2	14	14	NUM
ejpam-5532	495	3	]	]	PUNCT
ejpam-5532	495	4	a	a	DET
ejpam-5532	495	5	rosenfeld	rosenfeld	PROPN
ejpam-5532	495	6	.	.	PUNCT
ejpam-5532	496	1	fuzzy	fuzzy	ADJ
ejpam-5532	496	2	groups	group	NOUN
ejpam-5532	496	3	.	.	PUNCT
ejpam-5532	497	1	j.	j.	PROPN
ejpam-5532	497	2	math	math	PROPN
ejpam-5532	497	3	.	.	PUNCT
ejpam-5532	498	1	anal	anal	PROPN
ejpam-5532	498	2	.	.	PUNCT
ejpam-5532	499	1	appl	appl	PROPN
ejpam-5532	499	2	.	.	PROPN
ejpam-5532	500	1	,	,	PUNCT
ejpam-5532	500	2	35(3):512–517	35(3):512–517	PROPN
ejpam-5532	500	3	,	,	PUNCT
ejpam-5532	500	4	1971	1971	NUM
ejpam-5532	500	5	.	.	PUNCT
ejpam-5532	501	1	[	[	X
ejpam-5532	501	2	15	15	NUM
ejpam-5532	501	3	]	]	X
ejpam-5532	501	4	f	f	PROPN
ejpam-5532	501	5	g	g	PROPN
ejpam-5532	501	6	shi	shi	PROPN
ejpam-5532	501	7	and	and	CCONJ
ejpam-5532	501	8	r	r	NOUN
ejpam-5532	501	9	x	x	SYM
ejpam-5532	501	10	li	li	PROPN
ejpam-5532	501	11	.	.	PROPN
ejpam-5532	501	12	compactness	compactness	NOUN
ejpam-5532	501	13	in	in	ADP
ejpam-5532	501	14	lfuzzy	lfuzzy	ADJ
ejpam-5532	501	15	topological	topological	ADJ
ejpam-5532	501	16	spaces	space	NOUN
ejpam-5532	501	17	.	.	PUNCT
ejpam-5532	502	1	hecettepe	hecettepe	PROPN
ejpam-5532	502	2	journal	journal	PROPN
ejpam-5532	502	3	of	of	ADP
ejpam-5532	502	4	mathematics	mathematic	NOUN
ejpam-5532	502	5	and	and	CCONJ
ejpam-5532	502	6	statistics	statistic	NOUN
ejpam-5532	502	7	,	,	PUNCT
ejpam-5532	502	8	40(6):767–774	40(6):767–774	NOUN
ejpam-5532	502	9	,	,	PUNCT
ejpam-5532	502	10	2011	2011	NUM
ejpam-5532	502	11	.	.	PUNCT
ejpam-5532	503	1	[	[	X
ejpam-5532	503	2	16	16	NUM
ejpam-5532	503	3	]	]	X
ejpam-5532	503	4	k	k	PROPN
ejpam-5532	503	5	qin	qin	PROPN
ejpam-5532	504	1	y	y	PROPN
ejpam-5532	504	2	xu	xu	PROPN
ejpam-5532	504	3	and	and	CCONJ
ejpam-5532	505	1	j	j	PROPN
ejpam-5532	505	2	liu	liu	PROPN
ejpam-5532	505	3	.	.	PROPN
ejpam-5532	505	4	lattice	lattice	PROPN
ejpam-5532	505	5	valued	value	VERB
ejpam-5532	505	6	logican	logican	PROPN
ejpam-5532	505	7	alternative	alternative	ADJ
ejpam-5532	505	8	approach	approach	NOUN
ejpam-5532	505	9	to	to	PART
ejpam-5532	505	10	treat	treat	VERB
ejpam-5532	505	11	fuzziness	fuzziness	NOUN
ejpam-5532	505	12	and	and	CCONJ
ejpam-5532	505	13	incompatibility	incompatibility	NOUN
ejpam-5532	505	14	.	.	PUNCT
ejpam-5532	506	1	2003	2003	NUM
ejpam-5532	506	2	.	.	PUNCT
ejpam-5532	507	1	[	[	X
ejpam-5532	507	2	17	17	NUM
ejpam-5532	507	3	]	]	X
ejpam-5532	507	4	d	d	X
ejpam-5532	507	5	s	s	PROPN
ejpam-5532	507	6	zhao	zhao	PROPN
ejpam-5532	507	7	.	.	PUNCT
ejpam-5532	508	1	the	the	DET
ejpam-5532	508	2	n	n	NUM
ejpam-5532	508	3	-compactness	-compactness	NOUN
ejpam-5532	508	4	in	in	ADP
ejpam-5532	508	5	l	l	ADJ
ejpam-5532	508	6	-	-	ADJ
ejpam-5532	508	7	fuzzy	fuzzy	ADJ
ejpam-5532	508	8	topological	topological	ADJ
ejpam-5532	508	9	spaces	space	NOUN
ejpam-5532	508	10	.	.	PUNCT
ejpam-5532	509	1	journal	journal	PROPN
ejpam-5532	509	2	of	of	ADP
ejpam-5532	509	3	mathematical	mathematical	ADJ
ejpam-5532	509	4	analysis	analysis	NOUN
ejpam-5532	509	5	and	and	CCONJ
ejpam-5532	509	6	applications	application	NOUN
ejpam-5532	509	7	,	,	PUNCT
ejpam-5532	509	8	128:64–79	128:64–79	NUM
ejpam-5532	509	9	,	,	PUNCT
ejpam-5532	509	10	1987	1987	NUM
ejpam-5532	509	11	.	.	PUNCT