id	sid	tid	token	lemma	pos
ejpam-6686	1	1	european	european	PROPN
ejpam-6686	1	2	journal	journal	PROPN
ejpam-6686	1	3	of	of	ADP
ejpam-6686	1	4	pure	pure	ADJ
ejpam-6686	1	5	and	and	CCONJ
ejpam-6686	1	6	applied	applied	ADJ
ejpam-6686	1	7	mathematics	mathematic	NOUN
ejpam-6686	1	8	2025	2025	NUM
ejpam-6686	1	9	,	,	PUNCT
ejpam-6686	1	10	vol	vol	NOUN
ejpam-6686	1	11	.	.	PROPN
ejpam-6686	1	12	18	18	NUM
ejpam-6686	1	13	,	,	PUNCT
ejpam-6686	1	14	issue	issue	NOUN
ejpam-6686	1	15	4	4	NUM
ejpam-6686	1	16	,	,	PUNCT
ejpam-6686	1	17	article	article	NOUN
ejpam-6686	1	18	number	number	NOUN
ejpam-6686	1	19	6686	6686	NUM
ejpam-6686	1	20	issn	issn	VERB
ejpam-6686	1	21	1307	1307	NUM
ejpam-6686	1	22	-	-	SYM
ejpam-6686	1	23	5543	5543	NUM
ejpam-6686	1	24	–	–	PUNCT
ejpam-6686	1	25	ejpam.com	ejpam.com	X
ejpam-6686	1	26	published	publish	VERB
ejpam-6686	1	27	by	by	ADP
ejpam-6686	1	28	new	new	PROPN
ejpam-6686	1	29	york	york	PROPN
ejpam-6686	1	30	business	business	PROPN
ejpam-6686	1	31	global	global	ADJ
ejpam-6686	1	32	hierarchy	hierarchy	NOUN
ejpam-6686	1	33	sets	set	NOUN
ejpam-6686	1	34	in	in	ADP
ejpam-6686	1	35	almost	almost	ADV
ejpam-6686	1	36	distributive	distributive	ADJ
ejpam-6686	1	37	lattices	lattice	NOUN
ejpam-6686	1	38	and	and	CCONJ
ejpam-6686	1	39	their	their	PRON
ejpam-6686	1	40	structural	structural	ADJ
ejpam-6686	1	41	properties	property	NOUN
ejpam-6686	1	42	g.	g.	PROPN
ejpam-6686	1	43	chinnayya1	chinnayya1	PROPN
ejpam-6686	1	44	,	,	PUNCT
ejpam-6686	1	45	ramesh	ramesh	PROPN
ejpam-6686	1	46	sirisetti2	sirisetti2	PROPN
ejpam-6686	1	47	,	,	PUNCT
ejpam-6686	1	48	g.	g.	PROPN
ejpam-6686	1	49	jogarao3	jogarao3	PROPN
ejpam-6686	1	50	,	,	PUNCT
ejpam-6686	1	51	ravikumar	ravikumar	PROPN
ejpam-6686	1	52	bandaru4	bandaru4	PROPN
ejpam-6686	1	53	,	,	PUNCT
ejpam-6686	1	54	aiyared	aiyare	VERB
ejpam-6686	1	55	iampan5,∗	iampan5,∗	ADJ
ejpam-6686	1	56	1	1	NUM
ejpam-6686	1	57	department	department	NOUN
ejpam-6686	1	58	of	of	ADP
ejpam-6686	1	59	mathematics	mathematic	NOUN
ejpam-6686	1	60	,	,	PUNCT
ejpam-6686	1	61	gitam	gitam	NOUN
ejpam-6686	1	62	school	school	NOUN
ejpam-6686	1	63	of	of	ADP
ejpam-6686	1	64	science	science	NOUN
ejpam-6686	1	65	,	,	PUNCT
ejpam-6686	1	66	gitam	gitam	NOUN
ejpam-6686	1	67	(	(	PUNCT
ejpam-6686	1	68	deemed	deem	VERB
ejpam-6686	1	69	to	to	PART
ejpam-6686	1	70	be	be	AUX
ejpam-6686	1	71	a	a	DET
ejpam-6686	1	72	university	university	NOUN
ejpam-6686	1	73	)	)	PUNCT
ejpam-6686	1	74	,	,	PUNCT
ejpam-6686	1	75	visakhapatnam	visakhapatnam	PROPN
ejpam-6686	1	76	530045	530045	NUM
ejpam-6686	1	77	,	,	PUNCT
ejpam-6686	1	78	india	india	PROPN
ejpam-6686	1	79	2	2	NUM
ejpam-6686	1	80	department	department	NOUN
ejpam-6686	1	81	of	of	ADP
ejpam-6686	1	82	mathematics	mathematics	PROPN
ejpam-6686	1	83	,	,	PUNCT
ejpam-6686	1	84	aditya	aditya	PROPN
ejpam-6686	1	85	university	university	PROPN
ejpam-6686	1	86	,	,	PUNCT
ejpam-6686	1	87	surampalem	surampalem	PROPN
ejpam-6686	1	88	,	,	PUNCT
ejpam-6686	1	89	kakinada	kakinada	PROPN
ejpam-6686	1	90	,	,	PUNCT
ejpam-6686	1	91	andhra	andhra	PROPN
ejpam-6686	1	92	pradesh533437	pradesh533437	PROPN
ejpam-6686	1	93	,	,	PUNCT
ejpam-6686	1	94	india	india	PROPN
ejpam-6686	1	95	3	3	NUM
ejpam-6686	1	96	department	department	NOUN
ejpam-6686	1	97	of	of	ADP
ejpam-6686	1	98	bs	bs	PROPN
ejpam-6686	1	99	&	&	CCONJ
ejpam-6686	1	100	h	h	PROPN
ejpam-6686	1	101	,	,	PUNCT
ejpam-6686	1	102	aditya	aditya	PROPN
ejpam-6686	1	103	institute	institute	PROPN
ejpam-6686	1	104	of	of	ADP
ejpam-6686	1	105	technology	technology	NOUN
ejpam-6686	1	106	and	and	CCONJ
ejpam-6686	1	107	management	management	NOUN
ejpam-6686	1	108	,	,	PUNCT
ejpam-6686	1	109	tekkali	tekkali	PROPN
ejpam-6686	1	110	532001	532001	NUM
ejpam-6686	1	111	,	,	PUNCT
ejpam-6686	1	112	srikakulam	srikakulam	PROPN
ejpam-6686	1	113	,	,	PUNCT
ejpam-6686	1	114	india	india	PROPN
ejpam-6686	1	115	4	4	NUM
ejpam-6686	1	116	department	department	NOUN
ejpam-6686	1	117	of	of	ADP
ejpam-6686	1	118	mathematics	mathematic	NOUN
ejpam-6686	1	119	,	,	PUNCT
ejpam-6686	1	120	school	school	NOUN
ejpam-6686	1	121	of	of	ADP
ejpam-6686	1	122	advanced	advanced	ADJ
ejpam-6686	1	123	sciences	science	NOUN
ejpam-6686	1	124	,	,	PUNCT
ejpam-6686	1	125	vit	vit	PROPN
ejpam-6686	1	126	-	-	PUNCT
ejpam-6686	1	127	ap	ap	PROPN
ejpam-6686	1	128	university	university	PROPN
ejpam-6686	1	129	,	,	PUNCT
ejpam-6686	1	130	amaravati	amaravati	PROPN
ejpam-6686	1	131	522237	522237	NUM
ejpam-6686	1	132	,	,	PUNCT
ejpam-6686	1	133	andhra	andhra	PROPN
ejpam-6686	1	134	pradesh	pradesh	PROPN
ejpam-6686	1	135	,	,	PUNCT
ejpam-6686	1	136	india	india	PROPN
ejpam-6686	1	137	5	5	NUM
ejpam-6686	1	138	department	department	NOUN
ejpam-6686	1	139	of	of	ADP
ejpam-6686	1	140	mathematics	mathematic	NOUN
ejpam-6686	1	141	,	,	PUNCT
ejpam-6686	1	142	school	school	NOUN
ejpam-6686	1	143	of	of	ADP
ejpam-6686	1	144	science	science	NOUN
ejpam-6686	1	145	,	,	PUNCT
ejpam-6686	1	146	university	university	NOUN
ejpam-6686	1	147	of	of	ADP
ejpam-6686	1	148	phayao	phayao	NOUN
ejpam-6686	1	149	,	,	PUNCT
ejpam-6686	1	150	mae	mae	PROPN
ejpam-6686	1	151	ka	ka	PROPN
ejpam-6686	1	152	,	,	PUNCT
ejpam-6686	1	153	mueang	mueang	PROPN
ejpam-6686	1	154	,	,	PUNCT
ejpam-6686	1	155	phayao	phayao	NOUN
ejpam-6686	1	156	56000	56000	NUM
ejpam-6686	1	157	,	,	PUNCT
ejpam-6686	1	158	thailand	thailand	PROPN
ejpam-6686	1	159	abstract	abstract	NOUN
ejpam-6686	1	160	.	.	PUNCT
ejpam-6686	2	1	this	this	DET
ejpam-6686	2	2	paper	paper	NOUN
ejpam-6686	2	3	studies	study	NOUN
ejpam-6686	2	4	hierarchy	hierarchy	VERB
ejpam-6686	2	5	sets	set	NOUN
ejpam-6686	2	6	in	in	ADP
ejpam-6686	2	7	almost	almost	ADV
ejpam-6686	2	8	distributive	distributive	ADJ
ejpam-6686	2	9	lattices	lattice	NOUN
ejpam-6686	2	10	,	,	PUNCT
ejpam-6686	2	11	focusing	focus	VERB
ejpam-6686	2	12	on	on	ADP
ejpam-6686	2	13	two	two	NUM
ejpam-6686	2	14	key	key	ADJ
ejpam-6686	2	15	types	type	NOUN
ejpam-6686	2	16	:	:	PUNCT
ejpam-6686	2	17	prime	prime	ADJ
ejpam-6686	2	18	and	and	CCONJ
ejpam-6686	2	19	maximal	maximal	ADJ
ejpam-6686	2	20	hierarchy	hierarchy	NOUN
ejpam-6686	2	21	sets	set	NOUN
ejpam-6686	2	22	.	.	PUNCT
ejpam-6686	3	1	we	we	PRON
ejpam-6686	3	2	show	show	VERB
ejpam-6686	3	3	that	that	SCONJ
ejpam-6686	3	4	every	every	DET
ejpam-6686	3	5	maximal	maximal	ADJ
ejpam-6686	3	6	hierarchy	hierarchy	NOUN
ejpam-6686	3	7	set	set	VERB
ejpam-6686	3	8	is	be	AUX
ejpam-6686	3	9	prime	prime	ADJ
ejpam-6686	3	10	,	,	PUNCT
ejpam-6686	3	11	but	but	CCONJ
ejpam-6686	3	12	not	not	PART
ejpam-6686	3	13	vice	vice	ADV
ejpam-6686	3	14	versa	versa	ADV
ejpam-6686	3	15	,	,	PUNCT
ejpam-6686	3	16	and	and	CCONJ
ejpam-6686	3	17	use	use	VERB
ejpam-6686	3	18	zorn	zorn	PROPN
ejpam-6686	3	19	’s	’s	PART
ejpam-6686	3	20	lemma	lemma	PROPN
ejpam-6686	3	21	to	to	PART
ejpam-6686	3	22	prove	prove	VERB
ejpam-6686	3	23	the	the	DET
ejpam-6686	3	24	existence	existence	NOUN
ejpam-6686	3	25	of	of	ADP
ejpam-6686	3	26	prime	prime	ADJ
ejpam-6686	3	27	hierarchy	hierarchy	NOUN
ejpam-6686	3	28	sets	set	NOUN
ejpam-6686	3	29	extending	extend	VERB
ejpam-6686	3	30	a	a	DET
ejpam-6686	3	31	given	give	VERB
ejpam-6686	3	32	one	one	NUM
ejpam-6686	3	33	.	.	PUNCT
ejpam-6686	4	1	we	we	PRON
ejpam-6686	4	2	also	also	ADV
ejpam-6686	4	3	introduce	introduce	VERB
ejpam-6686	4	4	inverted	inverted	ADJ
ejpam-6686	4	5	-	-	PUNCT
ejpam-6686	4	6	hierarchy	hierarchy	NOUN
ejpam-6686	4	7	sets	set	NOUN
ejpam-6686	4	8	,	,	PUNCT
ejpam-6686	4	9	defined	define	VERB
ejpam-6686	4	10	via	via	ADP
ejpam-6686	4	11	join	join	NOUN
ejpam-6686	4	12	operations	operation	NOUN
ejpam-6686	4	13	,	,	PUNCT
ejpam-6686	4	14	and	and	CCONJ
ejpam-6686	4	15	analyze	analyze	VERB
ejpam-6686	4	16	their	their	PRON
ejpam-6686	4	17	relation	relation	NOUN
ejpam-6686	4	18	to	to	ADP
ejpam-6686	4	19	filters	filter	NOUN
ejpam-6686	4	20	.	.	PUNCT
ejpam-6686	5	1	the	the	DET
ejpam-6686	5	2	results	result	NOUN
ejpam-6686	5	3	provide	provide	VERB
ejpam-6686	5	4	structural	structural	ADJ
ejpam-6686	5	5	insights	insight	NOUN
ejpam-6686	5	6	and	and	CCONJ
ejpam-6686	5	7	extend	extend	VERB
ejpam-6686	5	8	ideal	ideal	ADJ
ejpam-6686	5	9	and	and	CCONJ
ejpam-6686	5	10	filter	filter	NOUN
ejpam-6686	5	11	theory	theory	NOUN
ejpam-6686	5	12	within	within	ADP
ejpam-6686	5	13	almost	almost	ADV
ejpam-6686	5	14	distributive	distributive	ADJ
ejpam-6686	5	15	lattices	lattice	NOUN
ejpam-6686	5	16	.	.	PUNCT
ejpam-6686	6	1	2020	2020	NUM
ejpam-6686	6	2	mathematics	mathematic	NOUN
ejpam-6686	6	3	subject	subject	NOUN
ejpam-6686	6	4	classifications	classification	NOUN
ejpam-6686	6	5	:	:	PUNCT
ejpam-6686	6	6	06d99	06d99	NUM
ejpam-6686	6	7	,	,	PUNCT
ejpam-6686	6	8	06d75	06d75	NUM
ejpam-6686	6	9	key	key	ADJ
ejpam-6686	6	10	words	word	NOUN
ejpam-6686	6	11	and	and	CCONJ
ejpam-6686	6	12	phrases	phrase	NOUN
ejpam-6686	6	13	:	:	PUNCT
ejpam-6686	6	14	almost	almost	ADV
ejpam-6686	6	15	distributive	distributive	ADJ
ejpam-6686	6	16	lattices	lattice	NOUN
ejpam-6686	6	17	,	,	PUNCT
ejpam-6686	6	18	ideals	ideal	NOUN
ejpam-6686	6	19	,	,	PUNCT
ejpam-6686	6	20	filters	filter	NOUN
ejpam-6686	6	21	,	,	PUNCT
ejpam-6686	6	22	hierarchy	hierarchy	NOUN
ejpam-6686	6	23	sets	set	NOUN
ejpam-6686	6	24	,	,	PUNCT
ejpam-6686	6	25	prime	prime	ADJ
ejpam-6686	6	26	hierarchy	hierarchy	NOUN
ejpam-6686	6	27	sets	set	NOUN
ejpam-6686	6	28	,	,	PUNCT
ejpam-6686	6	29	maximal	maximal	ADJ
ejpam-6686	6	30	hierarchy	hierarchy	NOUN
ejpam-6686	6	31	sets	set	NOUN
ejpam-6686	6	32	,	,	PUNCT
ejpam-6686	6	33	inverted	inverted	ADJ
ejpam-6686	6	34	-	-	PUNCT
ejpam-6686	6	35	hierarchy	hierarchy	NOUN
ejpam-6686	6	36	sets	set	NOUN
ejpam-6686	6	37	1	1	NUM
ejpam-6686	6	38	.	.	PUNCT
ejpam-6686	7	1	introduction	introduction	NOUN
ejpam-6686	7	2	in	in	ADP
ejpam-6686	7	3	lattice	lattice	PROPN
ejpam-6686	7	4	theory	theory	NOUN
ejpam-6686	7	5	,	,	PUNCT
ejpam-6686	7	6	the	the	DET
ejpam-6686	7	7	study	study	NOUN
ejpam-6686	7	8	of	of	ADP
ejpam-6686	7	9	special	special	ADJ
ejpam-6686	7	10	subsets	subset	NOUN
ejpam-6686	7	11	such	such	ADJ
ejpam-6686	7	12	as	as	ADP
ejpam-6686	7	13	ideals	ideal	NOUN
ejpam-6686	7	14	,	,	PUNCT
ejpam-6686	7	15	filters	filter	NOUN
ejpam-6686	7	16	,	,	PUNCT
ejpam-6686	7	17	and	and	CCONJ
ejpam-6686	7	18	related	related	ADJ
ejpam-6686	7	19	constructions	construction	NOUN
ejpam-6686	7	20	[	[	X
ejpam-6686	7	21	1–6	1–6	X
ejpam-6686	7	22	]	]	X
ejpam-6686	7	23	plays	play	VERB
ejpam-6686	7	24	a	a	DET
ejpam-6686	7	25	crucial	crucial	ADJ
ejpam-6686	7	26	role	role	NOUN
ejpam-6686	7	27	in	in	ADP
ejpam-6686	7	28	understanding	understand	VERB
ejpam-6686	7	29	structural	structural	ADJ
ejpam-6686	7	30	properties	property	NOUN
ejpam-6686	7	31	of	of	ADP
ejpam-6686	7	32	the	the	DET
ejpam-6686	7	33	lattice	lattice	NOUN
ejpam-6686	7	34	.	.	PUNCT
ejpam-6686	8	1	in	in	ADP
ejpam-6686	8	2	particular	particular	ADJ
ejpam-6686	8	3	,	,	PUNCT
ejpam-6686	8	4	hierarchy	hierarchy	NOUN
ejpam-6686	8	5	sets	set	VERB
ejpam-6686	8	6	[	[	X
ejpam-6686	8	7	7	7	NUM
ejpam-6686	8	8	]	]	PUNCT
ejpam-6686	8	9	,	,	PUNCT
ejpam-6686	8	10	defined	define	VERB
ejpam-6686	8	11	by	by	ADP
ejpam-6686	8	12	their	their	PRON
ejpam-6686	8	13	closure	closure	NOUN
ejpam-6686	8	14	under	under	ADP
ejpam-6686	8	15	the	the	DET
ejpam-6686	8	16	meet	meet	NOUN
ejpam-6686	8	17	operation	operation	NOUN
ejpam-6686	8	18	with	with	ADP
ejpam-6686	8	19	a	a	DET
ejpam-6686	8	20	fixed	fix	VERB
ejpam-6686	8	21	generating	generating	NOUN
ejpam-6686	8	22	set	set	NOUN
ejpam-6686	8	23	,	,	PUNCT
ejpam-6686	8	24	offer	offer	VERB
ejpam-6686	8	25	a	a	DET
ejpam-6686	8	26	useful	useful	ADJ
ejpam-6686	8	27	framework	framework	NOUN
ejpam-6686	8	28	for	for	ADP
ejpam-6686	8	29	analyzing	analyze	VERB
ejpam-6686	8	30	elements	element	NOUN
ejpam-6686	8	31	in	in	ADP
ejpam-6686	8	32	almost	almost	ADV
ejpam-6686	8	33	distributive	distributive	ADJ
ejpam-6686	8	34	∗corresponding	∗corresponde	VERB
ejpam-6686	8	35	author	author	NOUN
ejpam-6686	8	36	.	.	PUNCT
ejpam-6686	9	1	doi	doi	NOUN
ejpam-6686	9	2	:	:	PUNCT
ejpam-6686	9	3	https://doi.org/10.29020/nybg.ejpam.v18i4.6686	https://doi.org/10.29020/nybg.ejpam.v18i4.6686	PRON
ejpam-6686	9	4	email	email	NOUN
ejpam-6686	9	5	addresses	address	NOUN
ejpam-6686	9	6	:	:	PUNCT
ejpam-6686	9	7	cgondu@gitam.in	cgondu@gitam.in	PROPN
ejpam-6686	9	8	(	(	PUNCT
ejpam-6686	9	9	g.	g.	PROPN
ejpam-6686	9	10	chinnayya	chinnayya	PROPN
ejpam-6686	9	11	)	)	PUNCT
ejpam-6686	9	12	,	,	PUNCT
ejpam-6686	9	13	ramesh.sirisetti@gmail.com	ramesh.sirisetti@gmail.com	X
ejpam-6686	9	14	(	(	PUNCT
ejpam-6686	9	15	s.	s.	PROPN
ejpam-6686	9	16	ramesh	ramesh	PROPN
ejpam-6686	9	17	)	)	PUNCT
ejpam-6686	9	18	,	,	PUNCT
ejpam-6686	9	19	jogarao.gunda@gmail.com	jogarao.gunda@gmail.com	PROPN
ejpam-6686	9	20	(	(	PUNCT
ejpam-6686	9	21	g.	g.	PROPN
ejpam-6686	9	22	jogarao	jogarao	PROPN
ejpam-6686	9	23	)	)	PUNCT
ejpam-6686	9	24	,	,	PUNCT
ejpam-6686	9	25	ravimaths83@gmail.com	ravimaths83@gmail.com	PROPN
ejpam-6686	9	26	(	(	PUNCT
ejpam-6686	9	27	r.	r.	PROPN
ejpam-6686	9	28	bandaru	bandaru	PROPN
ejpam-6686	9	29	)	)	PUNCT
ejpam-6686	9	30	,	,	PUNCT
ejpam-6686	9	31	aiyared.ia@up.ac.th	aiyared.ia@up.ac.th	NOUN
ejpam-6686	9	32	(	(	PUNCT
ejpam-6686	9	33	a.	a.	NOUN
ejpam-6686	9	34	iampan	iampan	PROPN
ejpam-6686	9	35	)	)	PUNCT
ejpam-6686	9	36	https://www.ejpam.com	https://www.ejpam.com	X
ejpam-6686	10	1	1	1	NUM
ejpam-6686	10	2	copyright	copyright	NOUN
ejpam-6686	10	3	:	:	PUNCT
ejpam-6686	10	4	©	©	PROPN
ejpam-6686	10	5	2025	2025	NUM
ejpam-6686	10	6	the	the	DET
ejpam-6686	10	7	author(s	author(s	NOUN
ejpam-6686	10	8	)	)	PUNCT
ejpam-6686	10	9	.	.	PUNCT
ejpam-6686	11	1	(	(	PUNCT
ejpam-6686	11	2	cc	cc	NOUN
ejpam-6686	11	3	by	by	ADP
ejpam-6686	11	4	-	-	PUNCT
ejpam-6686	11	5	nc	nc	PROPN
ejpam-6686	11	6	4.0	4.0	NUM
ejpam-6686	11	7	)	)	PUNCT
ejpam-6686	11	8	g.	g.	NOUN
ejpam-6686	11	9	chinnayya	chinnayya	PROPN
ejpam-6686	11	10	et	et	PROPN
ejpam-6686	11	11	al	al	PROPN
ejpam-6686	11	12	.	.	PUNCT
ejpam-6686	11	13	/	/	SYM
ejpam-6686	11	14	eur	eur	PROPN
ejpam-6686	11	15	.	.	PUNCT
ejpam-6686	12	1	j.	j.	PROPN
ejpam-6686	12	2	pure	pure	PROPN
ejpam-6686	12	3	appl	appl	PROPN
ejpam-6686	12	4	.	.	PROPN
ejpam-6686	12	5	math	math	PROPN
ejpam-6686	12	6	,	,	PUNCT
ejpam-6686	12	7	18	18	NUM
ejpam-6686	12	8	(	(	PUNCT
ejpam-6686	12	9	4	4	NUM
ejpam-6686	12	10	)	)	PUNCT
ejpam-6686	12	11	(	(	PUNCT
ejpam-6686	12	12	2025	2025	NUM
ejpam-6686	12	13	)	)	PUNCT
ejpam-6686	12	14	,	,	PUNCT
ejpam-6686	12	15	6686	6686	NUM
ejpam-6686	12	16	2	2	NUM
ejpam-6686	12	17	of	of	ADP
ejpam-6686	12	18	12	12	NUM
ejpam-6686	12	19	lattices	lattice	NOUN
ejpam-6686	12	20	(	(	PUNCT
ejpam-6686	12	21	adls	adls	X
ejpam-6686	12	22	)	)	PUNCT
ejpam-6686	13	1	[	[	X
ejpam-6686	13	2	8	8	NUM
ejpam-6686	13	3	]	]	PUNCT
ejpam-6686	13	4	.	.	PUNCT
ejpam-6686	14	1	this	this	DET
ejpam-6686	14	2	framework	framework	NOUN
ejpam-6686	14	3	builds	build	VERB
ejpam-6686	14	4	upon	upon	SCONJ
ejpam-6686	14	5	foundational	foundational	ADJ
ejpam-6686	14	6	work	work	NOUN
ejpam-6686	14	7	by	by	ADP
ejpam-6686	14	8	birkhoff	birkhoff	NOUN
ejpam-6686	14	9	[	[	X
ejpam-6686	14	10	9	9	NUM
ejpam-6686	14	11	]	]	PUNCT
ejpam-6686	14	12	,	,	PUNCT
ejpam-6686	14	13	and	and	CCONJ
ejpam-6686	14	14	relates	relate	VERB
ejpam-6686	14	15	to	to	ADP
ejpam-6686	14	16	established	establish	VERB
ejpam-6686	14	17	studies	study	NOUN
ejpam-6686	14	18	in	in	ADP
ejpam-6686	14	19	the	the	DET
ejpam-6686	14	20	structure	structure	NOUN
ejpam-6686	14	21	of	of	ADP
ejpam-6686	14	22	complemented	complemented	ADJ
ejpam-6686	14	23	and	and	CCONJ
ejpam-6686	14	24	quasi	quasi	ADJ
ejpam-6686	14	25	-	-	ADJ
ejpam-6686	14	26	complemented	complemented	ADJ
ejpam-6686	14	27	lattices	lattice	NOUN
ejpam-6686	14	28	[	[	X
ejpam-6686	14	29	10	10	NUM
ejpam-6686	14	30	,	,	PUNCT
ejpam-6686	14	31	11	11	NUM
ejpam-6686	14	32	]	]	PUNCT
ejpam-6686	14	33	,	,	PUNCT
ejpam-6686	14	34	relatively	relatively	ADV
ejpam-6686	14	35	complemented	complemented	ADJ
ejpam-6686	14	36	distributive	distributive	ADJ
ejpam-6686	14	37	lattices	lattice	NOUN
ejpam-6686	14	38	[	[	X
ejpam-6686	14	39	12	12	NUM
ejpam-6686	14	40	,	,	PUNCT
ejpam-6686	14	41	13	13	NUM
ejpam-6686	14	42	]	]	PUNCT
ejpam-6686	14	43	,	,	PUNCT
ejpam-6686	14	44	and	and	CCONJ
ejpam-6686	14	45	the	the	DET
ejpam-6686	14	46	general	general	ADJ
ejpam-6686	14	47	theory	theory	NOUN
ejpam-6686	14	48	of	of	ADP
ejpam-6686	14	49	ideal	ideal	ADJ
ejpam-6686	14	50	-	-	PUNCT
ejpam-6686	14	51	like	like	ADJ
ejpam-6686	14	52	constructs	construct	NOUN
ejpam-6686	14	53	.	.	PUNCT
ejpam-6686	15	1	important	important	ADJ
ejpam-6686	15	2	extensions	extension	NOUN
ejpam-6686	15	3	to	to	ADP
ejpam-6686	15	4	the	the	DET
ejpam-6686	15	5	adl	adl	NOUN
ejpam-6686	15	6	setting	setting	NOUN
ejpam-6686	15	7	have	have	AUX
ejpam-6686	15	8	also	also	ADV
ejpam-6686	15	9	been	be	AUX
ejpam-6686	15	10	explored	explore	VERB
ejpam-6686	15	11	through	through	ADP
ejpam-6686	15	12	weak	weak	ADJ
ejpam-6686	15	13	relative	relative	ADJ
ejpam-6686	15	14	complements	complement	NOUN
ejpam-6686	15	15	,	,	PUNCT
ejpam-6686	15	16	normal	normal	ADJ
ejpam-6686	15	17	filters	filter	NOUN
ejpam-6686	15	18	,	,	PUNCT
ejpam-6686	15	19	and	and	CCONJ
ejpam-6686	15	20	weakly	weakly	ADJ
ejpam-6686	15	21	complemented	complemented	ADJ
ejpam-6686	15	22	structures	structure	NOUN
ejpam-6686	15	23	[	[	X
ejpam-6686	15	24	14–17	14–17	NUM
ejpam-6686	15	25	]	]	PUNCT
ejpam-6686	15	26	,	,	PUNCT
ejpam-6686	15	27	offering	offer	VERB
ejpam-6686	15	28	alternative	alternative	ADJ
ejpam-6686	15	29	perspectives	perspective	NOUN
ejpam-6686	15	30	on	on	ADP
ejpam-6686	15	31	distributivity	distributivity	NOUN
ejpam-6686	15	32	and	and	CCONJ
ejpam-6686	15	33	absorption	absorption	NOUN
ejpam-6686	15	34	in	in	ADP
ejpam-6686	15	35	non	non	ADJ
ejpam-6686	15	36	-	-	ADJ
ejpam-6686	15	37	classical	classical	ADJ
ejpam-6686	15	38	environments	environment	NOUN
ejpam-6686	15	39	.	.	PUNCT
ejpam-6686	16	1	recently	recently	ADV
ejpam-6686	16	2	,	,	PUNCT
ejpam-6686	16	3	ramesh	ramesh	PROPN
ejpam-6686	16	4	et	et	PROPN
ejpam-6686	16	5	al	al	PROPN
ejpam-6686	16	6	.	.	PROPN
ejpam-6686	16	7	introduced	introduce	VERB
ejpam-6686	16	8	and	and	CCONJ
ejpam-6686	16	9	developed	develop	VERB
ejpam-6686	16	10	the	the	DET
ejpam-6686	16	11	notion	notion	NOUN
ejpam-6686	16	12	of	of	ADP
ejpam-6686	16	13	hierarchy	hierarchy	NOUN
ejpam-6686	16	14	elements	element	NOUN
ejpam-6686	16	15	in	in	ADP
ejpam-6686	16	16	adls	adls	PROPN
ejpam-6686	16	17	[	[	X
ejpam-6686	16	18	18	18	NUM
ejpam-6686	16	19	]	]	PUNCT
ejpam-6686	16	20	,	,	PUNCT
ejpam-6686	16	21	establishing	establish	VERB
ejpam-6686	16	22	foundational	foundational	ADJ
ejpam-6686	16	23	properties	property	NOUN
ejpam-6686	16	24	such	such	ADJ
ejpam-6686	16	25	as	as	ADP
ejpam-6686	16	26	closure	closure	NOUN
ejpam-6686	16	27	under	under	ADP
ejpam-6686	16	28	meet	meet	NOUN
ejpam-6686	16	29	,	,	PUNCT
ejpam-6686	16	30	behavior	behavior	NOUN
ejpam-6686	16	31	with	with	ADP
ejpam-6686	16	32	maximal	maximal	ADJ
ejpam-6686	16	33	elements	element	NOUN
ejpam-6686	16	34	,	,	PUNCT
ejpam-6686	16	35	and	and	CCONJ
ejpam-6686	16	36	conditions	condition	NOUN
ejpam-6686	16	37	under	under	ADP
ejpam-6686	16	38	which	which	PRON
ejpam-6686	16	39	hierarchy	hierarchy	NOUN
ejpam-6686	16	40	elements	element	NOUN
ejpam-6686	16	41	generate	generate	VERB
ejpam-6686	16	42	ideals	ideal	NOUN
ejpam-6686	16	43	or	or	CCONJ
ejpam-6686	16	44	sublattices	sublattice	NOUN
ejpam-6686	16	45	.	.	PUNCT
ejpam-6686	17	1	a	a	DET
ejpam-6686	17	2	natural	natural	ADJ
ejpam-6686	17	3	extension	extension	NOUN
ejpam-6686	17	4	of	of	ADP
ejpam-6686	17	5	that	that	DET
ejpam-6686	17	6	work	work	NOUN
ejpam-6686	17	7	led	lead	VERB
ejpam-6686	17	8	to	to	ADP
ejpam-6686	17	9	the	the	DET
ejpam-6686	17	10	formalization	formalization	NOUN
ejpam-6686	17	11	of	of	ADP
ejpam-6686	17	12	hierarchy	hierarchy	NOUN
ejpam-6686	17	13	sets	set	NOUN
ejpam-6686	17	14	[	[	X
ejpam-6686	17	15	7	7	NUM
ejpam-6686	17	16	]	]	PUNCT
ejpam-6686	17	17	,	,	PUNCT
ejpam-6686	17	18	shown	show	VERB
ejpam-6686	17	19	to	to	PART
ejpam-6686	17	20	form	form	VERB
ejpam-6686	17	21	a	a	DET
ejpam-6686	17	22	distributive	distributive	ADJ
ejpam-6686	17	23	lattice	lattice	NOUN
ejpam-6686	17	24	under	under	ADP
ejpam-6686	17	25	union	union	NOUN
ejpam-6686	17	26	and	and	CCONJ
ejpam-6686	17	27	a	a	DET
ejpam-6686	17	28	meet	meet	NOUN
ejpam-6686	17	29	-	-	PUNCT
ejpam-6686	17	30	based	base	VERB
ejpam-6686	17	31	operation	operation	NOUN
ejpam-6686	17	32	.	.	PUNCT
ejpam-6686	18	1	in	in	ADP
ejpam-6686	18	2	parallel	parallel	NOUN
ejpam-6686	18	3	,	,	PUNCT
ejpam-6686	18	4	khabyah	khabyah	NOUN
ejpam-6686	18	5	’s	’s	PART
ejpam-6686	18	6	work	work	NOUN
ejpam-6686	18	7	on	on	ADP
ejpam-6686	18	8	star	star	NOUN
ejpam-6686	18	9	filters	filter	NOUN
ejpam-6686	18	10	and	and	CCONJ
ejpam-6686	18	11	starlets	starlet	NOUN
ejpam-6686	18	12	enriched	enrich	VERB
ejpam-6686	18	13	the	the	DET
ejpam-6686	18	14	lattice	lattice	NOUN
ejpam-6686	18	15	-	-	PUNCT
ejpam-6686	18	16	theoretic	theoretic	ADJ
ejpam-6686	18	17	landscape	landscape	NOUN
ejpam-6686	18	18	by	by	ADP
ejpam-6686	18	19	introducing	introduce	VERB
ejpam-6686	18	20	new	new	ADJ
ejpam-6686	18	21	substructures	substructure	NOUN
ejpam-6686	18	22	under	under	ADP
ejpam-6686	18	23	relaxed	relaxed	ADJ
ejpam-6686	18	24	closure	closure	NOUN
ejpam-6686	18	25	properties	property	NOUN
ejpam-6686	18	26	.	.	PUNCT
ejpam-6686	19	1	additionally	additionally	ADV
ejpam-6686	19	2	,	,	PUNCT
ejpam-6686	19	3	rafi	rafi	PROPN
ejpam-6686	19	4	et	et	PROPN
ejpam-6686	19	5	al	al	PROPN
ejpam-6686	19	6	.	.	PROPN
ejpam-6686	19	7	contributed	contribute	VERB
ejpam-6686	19	8	to	to	ADP
ejpam-6686	19	9	this	this	DET
ejpam-6686	19	10	growing	grow	VERB
ejpam-6686	19	11	body	body	NOUN
ejpam-6686	19	12	of	of	ADP
ejpam-6686	19	13	work	work	NOUN
ejpam-6686	19	14	by	by	ADP
ejpam-6686	19	15	introducing	introduce	VERB
ejpam-6686	19	16	prime	prime	ADJ
ejpam-6686	19	17	e	e	NOUN
ejpam-6686	19	18	-	-	NOUN
ejpam-6686	19	19	ideals	ideal	NOUN
ejpam-6686	19	20	[	[	X
ejpam-6686	19	21	19	19	NUM
ejpam-6686	19	22	]	]	PUNCT
ejpam-6686	19	23	,	,	PUNCT
ejpam-6686	19	24	which	which	PRON
ejpam-6686	19	25	refine	refine	VERB
ejpam-6686	19	26	the	the	DET
ejpam-6686	19	27	concept	concept	NOUN
ejpam-6686	19	28	of	of	ADP
ejpam-6686	19	29	absorption	absorption	NOUN
ejpam-6686	19	30	and	and	CCONJ
ejpam-6686	19	31	primeness	primeness	NOUN
ejpam-6686	19	32	within	within	ADP
ejpam-6686	19	33	lattice	lattice	NOUN
ejpam-6686	19	34	ideals	ideal	NOUN
ejpam-6686	19	35	,	,	PUNCT
ejpam-6686	19	36	and	and	CCONJ
ejpam-6686	19	37	by	by	ADP
ejpam-6686	19	38	developing	develop	VERB
ejpam-6686	19	39	wfilters	wfilter	NOUN
ejpam-6686	19	40	[	[	X
ejpam-6686	19	41	20	20	NUM
ejpam-6686	19	42	]	]	PUNCT
ejpam-6686	19	43	,	,	PUNCT
ejpam-6686	19	44	a	a	DET
ejpam-6686	19	45	generalization	generalization	NOUN
ejpam-6686	19	46	of	of	ADP
ejpam-6686	19	47	filters	filter	NOUN
ejpam-6686	19	48	based	base	VERB
ejpam-6686	19	49	on	on	ADP
ejpam-6686	19	50	weakened	weaken	VERB
ejpam-6686	19	51	closure	closure	NOUN
ejpam-6686	19	52	under	under	ADP
ejpam-6686	19	53	joins	join	NOUN
ejpam-6686	19	54	.	.	PUNCT
ejpam-6686	20	1	collectively	collectively	ADV
ejpam-6686	20	2	,	,	PUNCT
ejpam-6686	20	3	these	these	DET
ejpam-6686	20	4	developments	development	NOUN
ejpam-6686	20	5	underscore	underscore	VERB
ejpam-6686	20	6	the	the	DET
ejpam-6686	20	7	need	need	NOUN
ejpam-6686	20	8	to	to	PART
ejpam-6686	20	9	investigate	investigate	VERB
ejpam-6686	20	10	more	more	ADV
ejpam-6686	20	11	nuanced	nuanced	ADJ
ejpam-6686	20	12	classes	class	NOUN
ejpam-6686	20	13	of	of	ADP
ejpam-6686	20	14	subsets	subset	NOUN
ejpam-6686	20	15	in	in	ADP
ejpam-6686	20	16	adls	adls	PROPN
ejpam-6686	20	17	that	that	PRON
ejpam-6686	20	18	extend	extend	VERB
ejpam-6686	20	19	classical	classical	ADJ
ejpam-6686	20	20	notions	notion	NOUN
ejpam-6686	20	21	like	like	ADP
ejpam-6686	20	22	prime	prime	ADJ
ejpam-6686	20	23	ideals	ideal	NOUN
ejpam-6686	20	24	and	and	CCONJ
ejpam-6686	20	25	filters	filter	NOUN
ejpam-6686	20	26	.	.	PUNCT
ejpam-6686	21	1	within	within	ADP
ejpam-6686	21	2	this	this	DET
ejpam-6686	21	3	context	context	NOUN
ejpam-6686	21	4	,	,	PUNCT
ejpam-6686	21	5	two	two	NUM
ejpam-6686	21	6	important	important	ADJ
ejpam-6686	21	7	classes	class	NOUN
ejpam-6686	21	8	of	of	ADP
ejpam-6686	21	9	hierarchy	hierarchy	NOUN
ejpam-6686	21	10	sets	set	NOUN
ejpam-6686	21	11	are	be	AUX
ejpam-6686	21	12	introduced	introduce	VERB
ejpam-6686	21	13	in	in	ADP
ejpam-6686	21	14	this	this	DET
ejpam-6686	21	15	paper	paper	NOUN
ejpam-6686	21	16	:	:	PUNCT
ejpam-6686	21	17	prime	prime	ADJ
ejpam-6686	21	18	hierarchy	hierarchy	NOUN
ejpam-6686	21	19	sets	set	NOUN
ejpam-6686	21	20	and	and	CCONJ
ejpam-6686	21	21	maximal	maximal	ADJ
ejpam-6686	21	22	hierarchy	hierarchy	NOUN
ejpam-6686	21	23	sets	set	NOUN
ejpam-6686	21	24	.	.	PUNCT
ejpam-6686	22	1	prime	prime	ADJ
ejpam-6686	22	2	hierarchy	hierarchy	NOUN
ejpam-6686	22	3	sets	set	NOUN
ejpam-6686	22	4	generalize	generalize	VERB
ejpam-6686	22	5	the	the	DET
ejpam-6686	22	6	notion	notion	NOUN
ejpam-6686	22	7	of	of	ADP
ejpam-6686	22	8	prime	prime	ADJ
ejpam-6686	22	9	ideals	ideal	NOUN
ejpam-6686	22	10	[	[	X
ejpam-6686	22	11	21	21	NUM
ejpam-6686	22	12	]	]	PUNCT
ejpam-6686	22	13	by	by	ADP
ejpam-6686	22	14	requiring	require	VERB
ejpam-6686	22	15	that	that	SCONJ
ejpam-6686	22	16	the	the	DET
ejpam-6686	22	17	meet	meet	NOUN
ejpam-6686	22	18	of	of	ADP
ejpam-6686	22	19	two	two	NUM
ejpam-6686	22	20	elements	element	NOUN
ejpam-6686	22	21	lies	lie	VERB
ejpam-6686	22	22	in	in	ADP
ejpam-6686	22	23	the	the	DET
ejpam-6686	22	24	set	set	NOUN
ejpam-6686	22	25	only	only	ADV
ejpam-6686	22	26	if	if	SCONJ
ejpam-6686	22	27	at	at	ADV
ejpam-6686	22	28	least	least	ADJ
ejpam-6686	22	29	one	one	NUM
ejpam-6686	22	30	of	of	ADP
ejpam-6686	22	31	the	the	DET
ejpam-6686	22	32	elements	element	NOUN
ejpam-6686	22	33	does	do	VERB
ejpam-6686	22	34	.	.	PUNCT
ejpam-6686	23	1	maximal	maximal	ADJ
ejpam-6686	23	2	hierarchy	hierarchy	NOUN
ejpam-6686	23	3	sets	set	NOUN
ejpam-6686	23	4	,	,	PUNCT
ejpam-6686	23	5	on	on	ADP
ejpam-6686	23	6	the	the	DET
ejpam-6686	23	7	other	other	ADJ
ejpam-6686	23	8	hand	hand	NOUN
ejpam-6686	23	9	,	,	PUNCT
ejpam-6686	23	10	represent	represent	VERB
ejpam-6686	23	11	the	the	DET
ejpam-6686	23	12	largest	large	ADJ
ejpam-6686	23	13	proper	proper	ADJ
ejpam-6686	23	14	hierarchy	hierarchy	NOUN
ejpam-6686	23	15	sets	set	NOUN
ejpam-6686	23	16	under	under	ADP
ejpam-6686	23	17	inclusion	inclusion	NOUN
ejpam-6686	23	18	.	.	PUNCT
ejpam-6686	24	1	we	we	PRON
ejpam-6686	24	2	establish	establish	VERB
ejpam-6686	24	3	several	several	ADJ
ejpam-6686	24	4	structural	structural	ADJ
ejpam-6686	24	5	properties	property	NOUN
ejpam-6686	24	6	,	,	PUNCT
ejpam-6686	24	7	show	show	VERB
ejpam-6686	24	8	that	that	SCONJ
ejpam-6686	24	9	every	every	DET
ejpam-6686	24	10	maximal	maximal	ADJ
ejpam-6686	24	11	hierarchy	hierarchy	NOUN
ejpam-6686	24	12	set	set	VERB
ejpam-6686	24	13	is	be	AUX
ejpam-6686	24	14	necessarily	necessarily	ADV
ejpam-6686	24	15	prime	prime	ADJ
ejpam-6686	24	16	,	,	PUNCT
ejpam-6686	24	17	and	and	CCONJ
ejpam-6686	24	18	use	use	VERB
ejpam-6686	24	19	zorn	zorn	PROPN
ejpam-6686	24	20	’s	’s	PART
ejpam-6686	24	21	lemma	lemma	PROPN
ejpam-6686	24	22	to	to	PART
ejpam-6686	24	23	demonstrate	demonstrate	VERB
ejpam-6686	24	24	the	the	DET
ejpam-6686	24	25	existence	existence	NOUN
ejpam-6686	24	26	of	of	ADP
ejpam-6686	24	27	prime	prime	ADJ
ejpam-6686	24	28	hierarchy	hierarchy	NOUN
ejpam-6686	24	29	sets	set	NOUN
ejpam-6686	24	30	extending	extend	VERB
ejpam-6686	24	31	a	a	DET
ejpam-6686	24	32	given	give	VERB
ejpam-6686	24	33	one	one	NUM
ejpam-6686	24	34	while	while	SCONJ
ejpam-6686	24	35	avoiding	avoid	VERB
ejpam-6686	24	36	a	a	DET
ejpam-6686	24	37	particular	particular	ADJ
ejpam-6686	24	38	subset	subset	NOUN
ejpam-6686	24	39	.	.	PUNCT
ejpam-6686	25	1	we	we	PRON
ejpam-6686	25	2	provide	provide	VERB
ejpam-6686	25	3	examples	example	NOUN
ejpam-6686	25	4	to	to	PART
ejpam-6686	25	5	clarify	clarify	VERB
ejpam-6686	25	6	the	the	DET
ejpam-6686	25	7	distinctions	distinction	NOUN
ejpam-6686	25	8	between	between	ADP
ejpam-6686	25	9	these	these	DET
ejpam-6686	25	10	types	type	NOUN
ejpam-6686	25	11	of	of	ADP
ejpam-6686	25	12	sets	set	NOUN
ejpam-6686	25	13	and	and	CCONJ
ejpam-6686	25	14	highlight	highlight	VERB
ejpam-6686	25	15	the	the	DET
ejpam-6686	25	16	fact	fact	NOUN
ejpam-6686	25	17	that	that	SCONJ
ejpam-6686	25	18	the	the	DET
ejpam-6686	25	19	converse	converse	NOUN
ejpam-6686	25	20	of	of	ADP
ejpam-6686	25	21	some	some	DET
ejpam-6686	25	22	implications	implication	NOUN
ejpam-6686	25	23	does	do	AUX
ejpam-6686	25	24	not	not	PART
ejpam-6686	25	25	hold	hold	VERB
ejpam-6686	25	26	in	in	ADP
ejpam-6686	25	27	general	general	ADJ
ejpam-6686	25	28	.	.	PUNCT
ejpam-6686	26	1	also	also	ADV
ejpam-6686	26	2	,	,	PUNCT
ejpam-6686	26	3	we	we	PRON
ejpam-6686	26	4	introduce	introduce	VERB
ejpam-6686	26	5	invertedhierarchy	invertedhierarchy	NOUN
ejpam-6686	26	6	sets	set	NOUN
ejpam-6686	26	7	hs	hs	PROPN
ejpam-6686	26	8	,	,	PUNCT
ejpam-6686	26	9	defined	define	VERB
ejpam-6686	26	10	for	for	ADP
ejpam-6686	26	11	a	a	DET
ejpam-6686	26	12	non	non	ADJ
ejpam-6686	26	13	-	-	ADJ
ejpam-6686	26	14	empty	empty	ADJ
ejpam-6686	26	15	subset	subset	NOUN
ejpam-6686	26	16	s	s	NOUN
ejpam-6686	26	17	of	of	ADP
ejpam-6686	26	18	an	an	DET
ejpam-6686	26	19	almost	almost	ADV
ejpam-6686	26	20	distributive	distributive	ADJ
ejpam-6686	26	21	lattice	lattice	NOUN
ejpam-6686	26	22	l	l	NOUN
ejpam-6686	26	23	with	with	ADP
ejpam-6686	26	24	maximal	maximal	ADJ
ejpam-6686	26	25	elements	element	NOUN
ejpam-6686	26	26	.	.	PUNCT
ejpam-6686	27	1	each	each	DET
ejpam-6686	27	2	hs	hs	PROPN
ejpam-6686	27	3	consists	consist	VERB
ejpam-6686	27	4	of	of	ADP
ejpam-6686	27	5	elements	element	NOUN
ejpam-6686	27	6	in	in	ADP
ejpam-6686	27	7	l	l	NOUN
ejpam-6686	27	8	that	that	PRON
ejpam-6686	27	9	are	be	AUX
ejpam-6686	27	10	idempotent	idempotent	ADJ
ejpam-6686	27	11	under	under	ADP
ejpam-6686	27	12	join	join	NOUN
ejpam-6686	27	13	with	with	ADP
ejpam-6686	27	14	some	some	DET
ejpam-6686	27	15	u	u	NOUN
ejpam-6686	27	16	∈	∈	PROPN
ejpam-6686	27	17	s.	s.	NOUN
ejpam-6686	28	1	these	these	DET
ejpam-6686	28	2	sets	set	NOUN
ejpam-6686	28	3	exhibit	exhibit	VERB
ejpam-6686	28	4	notable	notable	ADJ
ejpam-6686	28	5	algebraic	algebraic	ADJ
ejpam-6686	28	6	properties	property	NOUN
ejpam-6686	28	7	,	,	PUNCT
ejpam-6686	28	8	including	include	VERB
ejpam-6686	28	9	closure	closure	NOUN
ejpam-6686	28	10	under	under	ADP
ejpam-6686	28	11	join	join	NOUN
ejpam-6686	28	12	and	and	CCONJ
ejpam-6686	28	13	,	,	PUNCT
ejpam-6686	28	14	in	in	ADP
ejpam-6686	28	15	some	some	DET
ejpam-6686	28	16	cases	case	NOUN
ejpam-6686	28	17	,	,	PUNCT
ejpam-6686	28	18	meet	meet	VERB
ejpam-6686	28	19	,	,	PUNCT
ejpam-6686	28	20	forming	form	VERB
ejpam-6686	28	21	filters	filter	NOUN
ejpam-6686	28	22	or	or	CCONJ
ejpam-6686	28	23	related	related	ADJ
ejpam-6686	28	24	substructures	substructure	NOUN
ejpam-6686	28	25	.	.	PUNCT
ejpam-6686	29	1	we	we	PRON
ejpam-6686	29	2	explore	explore	VERB
ejpam-6686	29	3	their	their	PRON
ejpam-6686	29	4	characterizations	characterization	NOUN
ejpam-6686	29	5	,	,	PUNCT
ejpam-6686	29	6	relationships	relationship	NOUN
ejpam-6686	29	7	to	to	ADP
ejpam-6686	29	8	filters	filter	NOUN
ejpam-6686	29	9	,	,	PUNCT
ejpam-6686	29	10	and	and	CCONJ
ejpam-6686	29	11	criteria	criterion	NOUN
ejpam-6686	29	12	under	under	ADP
ejpam-6686	29	13	which	which	PRON
ejpam-6686	29	14	hs	hs	PROPN
ejpam-6686	29	15	aligns	align	VERB
ejpam-6686	29	16	with	with	ADP
ejpam-6686	29	17	or	or	CCONJ
ejpam-6686	29	18	differs	differ	VERB
ejpam-6686	29	19	from	from	ADP
ejpam-6686	29	20	the	the	DET
ejpam-6686	29	21	filter	filter	NOUN
ejpam-6686	29	22	generated	generate	VERB
ejpam-6686	29	23	by	by	ADP
ejpam-6686	29	24	s.	s.	PROPN
ejpam-6686	29	25	2	2	NUM
ejpam-6686	29	26	.	.	PUNCT
ejpam-6686	29	27	preliminaries	preliminary	NOUN
ejpam-6686	29	28	the	the	DET
ejpam-6686	29	29	study	study	NOUN
ejpam-6686	29	30	of	of	ADP
ejpam-6686	29	31	lattice	lattice	NOUN
ejpam-6686	29	32	-	-	PUNCT
ejpam-6686	29	33	theoretic	theoretic	NOUN
ejpam-6686	29	34	structures	structure	NOUN
ejpam-6686	29	35	has	have	AUX
ejpam-6686	29	36	played	play	VERB
ejpam-6686	29	37	a	a	DET
ejpam-6686	29	38	pivotal	pivotal	ADJ
ejpam-6686	29	39	role	role	NOUN
ejpam-6686	29	40	in	in	ADP
ejpam-6686	29	41	abstract	abstract	ADJ
ejpam-6686	29	42	algebra	algebra	NOUN
ejpam-6686	29	43	and	and	CCONJ
ejpam-6686	29	44	its	its	PRON
ejpam-6686	29	45	applications	application	NOUN
ejpam-6686	29	46	to	to	PART
ejpam-6686	29	47	computer	computer	NOUN
ejpam-6686	29	48	science	science	NOUN
ejpam-6686	29	49	,	,	PUNCT
ejpam-6686	29	50	logic	logic	NOUN
ejpam-6686	29	51	,	,	PUNCT
ejpam-6686	29	52	and	and	CCONJ
ejpam-6686	29	53	information	information	NOUN
ejpam-6686	29	54	systems	system	NOUN
ejpam-6686	29	55	.	.	PUNCT
ejpam-6686	30	1	in	in	ADP
ejpam-6686	30	2	this	this	DET
ejpam-6686	30	3	aspect	aspect	NOUN
ejpam-6686	30	4	,	,	PUNCT
ejpam-6686	30	5	the	the	DET
ejpam-6686	30	6	concept	concept	NOUN
ejpam-6686	30	7	of	of	ADP
ejpam-6686	30	8	an	an	DET
ejpam-6686	30	9	almost	almost	ADV
ejpam-6686	30	10	distributive	distributive	ADJ
ejpam-6686	30	11	lattice	lattice	NOUN
ejpam-6686	30	12	[	[	X
ejpam-6686	30	13	8	8	NUM
ejpam-6686	30	14	]	]	PUNCT
ejpam-6686	30	15	was	be	AUX
ejpam-6686	30	16	introduced	introduce	VERB
ejpam-6686	30	17	by	by	ADP
ejpam-6686	30	18	swamy	swamy	NOUN
ejpam-6686	30	19	and	and	CCONJ
ejpam-6686	30	20	rao	rao	NOUN
ejpam-6686	30	21	in	in	ADP
ejpam-6686	30	22	1981	1981	NUM
ejpam-6686	30	23	as	as	ADP
ejpam-6686	30	24	a	a	DET
ejpam-6686	30	25	common	common	ADJ
ejpam-6686	30	26	abstraction	abstraction	NOUN
ejpam-6686	30	27	of	of	ADP
ejpam-6686	30	28	both	both	CCONJ
ejpam-6686	30	29	lattice	lattice	NOUN
ejpam-6686	30	30	-	-	PUNCT
ejpam-6686	30	31	theoretic	theoretic	ADJ
ejpam-6686	30	32	and	and	CCONJ
ejpam-6686	30	33	ring	ring	NOUN
ejpam-6686	30	34	-	-	PUNCT
ejpam-6686	30	35	theoretic	theoretic	NOUN
ejpam-6686	30	36	generalizations	generalization	NOUN
ejpam-6686	30	37	of	of	ADP
ejpam-6686	30	38	a	a	DET
ejpam-6686	30	39	boolean	boolean	ADJ
ejpam-6686	30	40	algebra	algebra	NOUN
ejpam-6686	30	41	(	(	PUNCT
ejpam-6686	30	42	ring	ring	NOUN
ejpam-6686	30	43	)	)	PUNCT
ejpam-6686	30	44	.	.	PUNCT
ejpam-6686	31	1	it	it	PRON
ejpam-6686	31	2	is	be	AUX
ejpam-6686	31	3	an	an	DET
ejpam-6686	31	4	algebraic	algebraic	ADJ
ejpam-6686	31	5	structure	structure	NOUN
ejpam-6686	31	6	which	which	PRON
ejpam-6686	31	7	satisfies	satisfy	VERB
ejpam-6686	31	8	all	all	DET
ejpam-6686	31	9	axioms	axiom	NOUN
ejpam-6686	31	10	of	of	ADP
ejpam-6686	31	11	a	a	DET
ejpam-6686	31	12	distributive	distributive	ADJ
ejpam-6686	31	13	lattice	lattice	NOUN
ejpam-6686	31	14	(	(	PUNCT
ejpam-6686	31	15	l,∨,∧	l,∨,∧	NOUN
ejpam-6686	31	16	,	,	PUNCT
ejpam-6686	31	17	0	0	NUM
ejpam-6686	31	18	)	)	PUNCT
ejpam-6686	31	19	with	with	ADP
ejpam-6686	31	20	the	the	DET
ejpam-6686	31	21	zero	zero	NUM
ejpam-6686	31	22	element	element	NOUN
ejpam-6686	31	23	0	0	NUM
ejpam-6686	31	24	except	except	SCONJ
ejpam-6686	31	25	the	the	DET
ejpam-6686	31	26	commutativity	commutativity	NOUN
ejpam-6686	31	27	of	of	ADP
ejpam-6686	31	28	the	the	DET
ejpam-6686	31	29	binary	binary	PROPN
ejpam-6686	31	30	operations	operation	NOUN
ejpam-6686	31	31	∨,∧	∨,∧	PROPN
ejpam-6686	31	32	,	,	PUNCT
ejpam-6686	31	33	the	the	DET
ejpam-6686	31	34	right	right	ADJ
ejpam-6686	31	35	distributivity	distributivity	NOUN
ejpam-6686	31	36	of	of	ADP
ejpam-6686	31	37	∨	∨	NUM
ejpam-6686	31	38	over	over	ADP
ejpam-6686	31	39	∧	∧	PROPN
ejpam-6686	31	40	,	,	PUNCT
ejpam-6686	31	41	and	and	CCONJ
ejpam-6686	31	42	the	the	DET
ejpam-6686	31	43	associativity	associativity	NOUN
ejpam-6686	31	44	of	of	ADP
ejpam-6686	31	45	∨.	∨.	NOUN
ejpam-6686	31	46	g.	g.	PROPN
ejpam-6686	31	47	chinnayya	chinnayya	PROPN
ejpam-6686	31	48	et	et	PROPN
ejpam-6686	31	49	al	al	PROPN
ejpam-6686	31	50	.	.	PUNCT
ejpam-6686	31	51	/	/	SYM
ejpam-6686	31	52	eur	eur	PROPN
ejpam-6686	31	53	.	.	PUNCT
ejpam-6686	32	1	j.	j.	PROPN
ejpam-6686	32	2	pure	pure	PROPN
ejpam-6686	32	3	appl	appl	PROPN
ejpam-6686	32	4	.	.	PROPN
ejpam-6686	32	5	math	math	PROPN
ejpam-6686	32	6	,	,	PUNCT
ejpam-6686	32	7	18	18	NUM
ejpam-6686	32	8	(	(	PUNCT
ejpam-6686	32	9	4	4	NUM
ejpam-6686	32	10	)	)	PUNCT
ejpam-6686	32	11	(	(	PUNCT
ejpam-6686	32	12	2025	2025	NUM
ejpam-6686	32	13	)	)	PUNCT
ejpam-6686	32	14	,	,	PUNCT
ejpam-6686	32	15	6686	6686	NUM
ejpam-6686	32	16	3	3	NUM
ejpam-6686	32	17	of	of	ADP
ejpam-6686	32	18	12	12	NUM
ejpam-6686	32	19	given	give	VERB
ejpam-6686	32	20	an	an	DET
ejpam-6686	32	21	almost	almost	ADV
ejpam-6686	32	22	distributive	distributive	ADJ
ejpam-6686	32	23	lattice	lattice	NOUN
ejpam-6686	32	24	l	l	NOUN
ejpam-6686	32	25	and	and	CCONJ
ejpam-6686	32	26	a	a	DET
ejpam-6686	32	27	non	non	ADJ
ejpam-6686	32	28	-	-	ADJ
ejpam-6686	32	29	empty	empty	ADJ
ejpam-6686	32	30	subset	subset	NOUN
ejpam-6686	32	31	s	s	VERB
ejpam-6686	32	32	⊆	⊆	NUM
ejpam-6686	32	33	l	l	NOUN
ejpam-6686	32	34	,	,	PUNCT
ejpam-6686	32	35	an	an	DET
ejpam-6686	32	36	element	element	NOUN
ejpam-6686	32	37	h	h	NOUN
ejpam-6686	32	38	∈	∈	PROPN
ejpam-6686	32	39	l	l	NOUN
ejpam-6686	32	40	is	be	AUX
ejpam-6686	32	41	said	say	VERB
ejpam-6686	32	42	to	to	PART
ejpam-6686	32	43	be	be	AUX
ejpam-6686	32	44	a	a	DET
ejpam-6686	32	45	hierarchy	hierarchy	NOUN
ejpam-6686	32	46	with	with	ADP
ejpam-6686	32	47	respect	respect	NOUN
ejpam-6686	32	48	to	to	ADP
ejpam-6686	32	49	s	s	PRON
ejpam-6686	32	50	[	[	X
ejpam-6686	32	51	7	7	X
ejpam-6686	32	52	]	]	X
ejpam-6686	32	53	if	if	SCONJ
ejpam-6686	32	54	it	it	PRON
ejpam-6686	32	55	satisfies	satisfy	VERB
ejpam-6686	32	56	a	a	DET
ejpam-6686	32	57	specific	specific	ADJ
ejpam-6686	32	58	absorption	absorption	NOUN
ejpam-6686	32	59	condition	condition	NOUN
ejpam-6686	32	60	with	with	ADP
ejpam-6686	32	61	at	at	ADV
ejpam-6686	32	62	least	least	ADV
ejpam-6686	32	63	one	one	NUM
ejpam-6686	32	64	element	element	NOUN
ejpam-6686	32	65	of	of	ADP
ejpam-6686	32	66	s.	s.	PROPN
ejpam-6686	32	67	the	the	DET
ejpam-6686	32	68	set	set	NOUN
ejpam-6686	32	69	of	of	ADP
ejpam-6686	32	70	all	all	DET
ejpam-6686	32	71	such	such	ADJ
ejpam-6686	32	72	elements	element	NOUN
ejpam-6686	32	73	,	,	PUNCT
ejpam-6686	32	74	denoted	denote	VERB
ejpam-6686	32	75	by	by	ADP
ejpam-6686	32	76	hs	hs	PROPN
ejpam-6686	32	77	,	,	PUNCT
ejpam-6686	32	78	exhibits	exhibit	VERB
ejpam-6686	32	79	several	several	ADJ
ejpam-6686	32	80	interesting	interesting	ADJ
ejpam-6686	32	81	structural	structural	ADJ
ejpam-6686	32	82	properties	property	NOUN
ejpam-6686	32	83	.	.	PUNCT
ejpam-6686	33	1	in	in	ADP
ejpam-6686	33	2	particular	particular	ADJ
ejpam-6686	33	3	,	,	PUNCT
ejpam-6686	33	4	this	this	DET
ejpam-6686	33	5	set	set	NOUN
ejpam-6686	33	6	is	be	AUX
ejpam-6686	33	7	non	non	ADJ
ejpam-6686	33	8	-	-	ADJ
ejpam-6686	33	9	empty	empty	ADJ
ejpam-6686	33	10	,	,	PUNCT
ejpam-6686	33	11	contains	contain	VERB
ejpam-6686	33	12	s	s	PRON
ejpam-6686	33	13	,	,	PUNCT
ejpam-6686	33	14	and	and	CCONJ
ejpam-6686	33	15	is	be	AUX
ejpam-6686	33	16	closed	close	VERB
ejpam-6686	33	17	under	under	ADP
ejpam-6686	33	18	the	the	DET
ejpam-6686	33	19	meet	meet	NOUN
ejpam-6686	33	20	operation	operation	NOUN
ejpam-6686	33	21	.	.	PUNCT
ejpam-6686	34	1	in	in	ADP
ejpam-6686	34	2	this	this	DET
ejpam-6686	34	3	context	context	NOUN
ejpam-6686	34	4	,	,	PUNCT
ejpam-6686	34	5	the	the	DET
ejpam-6686	34	6	notion	notion	NOUN
ejpam-6686	34	7	of	of	ADP
ejpam-6686	34	8	hierarchy	hierarchy	NOUN
ejpam-6686	34	9	elements	element	NOUN
ejpam-6686	34	10	and	and	CCONJ
ejpam-6686	34	11	their	their	PRON
ejpam-6686	34	12	associated	associated	ADJ
ejpam-6686	34	13	hierarchy	hierarchy	NOUN
ejpam-6686	34	14	sets	set	NOUN
ejpam-6686	34	15	provides	provide	VERB
ejpam-6686	34	16	a	a	DET
ejpam-6686	34	17	useful	useful	ADJ
ejpam-6686	34	18	framework	framework	NOUN
ejpam-6686	34	19	for	for	ADP
ejpam-6686	34	20	examining	examine	VERB
ejpam-6686	34	21	the	the	DET
ejpam-6686	34	22	internal	internal	ADJ
ejpam-6686	34	23	organization	organization	NOUN
ejpam-6686	34	24	of	of	ADP
ejpam-6686	34	25	elements	element	NOUN
ejpam-6686	34	26	within	within	ADP
ejpam-6686	34	27	an	an	DET
ejpam-6686	34	28	almost	almost	ADV
ejpam-6686	34	29	distributive	distributive	ADJ
ejpam-6686	34	30	lattice	lattice	NOUN
ejpam-6686	34	31	.	.	PUNCT
ejpam-6686	35	1	the	the	DET
ejpam-6686	35	2	study	study	NOUN
ejpam-6686	35	3	of	of	ADP
ejpam-6686	35	4	hierarchy	hierarchy	NOUN
ejpam-6686	35	5	sets	set	VERB
ejpam-6686	35	6	not	not	PART
ejpam-6686	35	7	only	only	ADV
ejpam-6686	35	8	enhances	enhance	VERB
ejpam-6686	35	9	our	our	PRON
ejpam-6686	35	10	understanding	understanding	NOUN
ejpam-6686	35	11	of	of	ADP
ejpam-6686	35	12	ideal	ideal	ADJ
ejpam-6686	35	13	-	-	PUNCT
ejpam-6686	35	14	theoretic	theoretic	ADJ
ejpam-6686	35	15	constructions	construction	NOUN
ejpam-6686	35	16	in	in	ADP
ejpam-6686	35	17	almost	almost	ADV
ejpam-6686	35	18	distributive	distributive	ADJ
ejpam-6686	35	19	lattices	lattice	NOUN
ejpam-6686	35	20	but	but	CCONJ
ejpam-6686	35	21	also	also	ADV
ejpam-6686	35	22	bridges	bridge	NOUN
ejpam-6686	35	23	concepts	concept	NOUN
ejpam-6686	35	24	related	relate	VERB
ejpam-6686	35	25	to	to	ADP
ejpam-6686	35	26	sub	sub	ADJ
ejpam-6686	35	27	-	-	ADJ
ejpam-6686	35	28	almost	almost	ADV
ejpam-6686	35	29	distributive	distributive	ADJ
ejpam-6686	35	30	lattices	lattice	NOUN
ejpam-6686	35	31	,	,	PUNCT
ejpam-6686	35	32	closure	closure	NOUN
ejpam-6686	35	33	operators	operator	NOUN
ejpam-6686	35	34	,	,	PUNCT
ejpam-6686	35	35	and	and	CCONJ
ejpam-6686	35	36	distributivity	distributivity	NOUN
ejpam-6686	35	37	.	.	PUNCT
ejpam-6686	36	1	several	several	ADJ
ejpam-6686	36	2	results	result	NOUN
ejpam-6686	36	3	characterize	characterize	VERB
ejpam-6686	36	4	the	the	DET
ejpam-6686	36	5	algebraic	algebraic	ADJ
ejpam-6686	36	6	and	and	CCONJ
ejpam-6686	36	7	order	order	NOUN
ejpam-6686	36	8	-	-	PUNCT
ejpam-6686	36	9	theoretic	theoretic	ADJ
ejpam-6686	36	10	behavior	behavior	NOUN
ejpam-6686	36	11	of	of	ADP
ejpam-6686	36	12	hierarchy	hierarchy	NOUN
ejpam-6686	36	13	sets	set	NOUN
ejpam-6686	36	14	,	,	PUNCT
ejpam-6686	36	15	including	include	VERB
ejpam-6686	36	16	their	their	PRON
ejpam-6686	36	17	stability	stability	NOUN
ejpam-6686	36	18	under	under	ADP
ejpam-6686	36	19	inclusion	inclusion	NOUN
ejpam-6686	36	20	,	,	PUNCT
ejpam-6686	36	21	intersection	intersection	NOUN
ejpam-6686	36	22	,	,	PUNCT
ejpam-6686	36	23	and	and	CCONJ
ejpam-6686	36	24	union	union	NOUN
ejpam-6686	36	25	.	.	PUNCT
ejpam-6686	37	1	additionally	additionally	ADV
ejpam-6686	37	2	,	,	PUNCT
ejpam-6686	37	3	under	under	ADP
ejpam-6686	37	4	certain	certain	ADJ
ejpam-6686	37	5	conditions	condition	NOUN
ejpam-6686	37	6	,	,	PUNCT
ejpam-6686	37	7	hs	hs	PROPN
ejpam-6686	37	8	forms	form	VERB
ejpam-6686	37	9	an	an	DET
ejpam-6686	37	10	ideal	ideal	NOUN
ejpam-6686	37	11	and	and	CCONJ
ejpam-6686	37	12	even	even	ADV
ejpam-6686	37	13	a	a	DET
ejpam-6686	37	14	sub	sub	ADJ
ejpam-6686	37	15	-	-	ADJ
ejpam-6686	37	16	almost	almost	ADV
ejpam-6686	37	17	distributive	distributive	ADJ
ejpam-6686	37	18	lattice	lattice	NOUN
ejpam-6686	37	19	.	.	PUNCT
ejpam-6686	38	1	definition	definition	NOUN
ejpam-6686	38	2	1	1	NUM
ejpam-6686	38	3	.	.	PUNCT
ejpam-6686	39	1	[	[	X
ejpam-6686	39	2	7	7	X
ejpam-6686	39	3	]	]	PUNCT
ejpam-6686	39	4	given	give	VERB
ejpam-6686	39	5	a	a	DET
ejpam-6686	39	6	non	non	ADJ
ejpam-6686	39	7	-	-	ADJ
ejpam-6686	39	8	empty	empty	ADJ
ejpam-6686	39	9	subset	subset	NOUN
ejpam-6686	39	10	s	s	PROPN
ejpam-6686	39	11	of	of	ADP
ejpam-6686	39	12	l	l	NOUN
ejpam-6686	39	13	,	,	PUNCT
ejpam-6686	39	14	an	an	DET
ejpam-6686	39	15	element	element	NOUN
ejpam-6686	39	16	h	h	NOUN
ejpam-6686	39	17	∈	∈	PROPN
ejpam-6686	39	18	l	l	NOUN
ejpam-6686	39	19	is	be	AUX
ejpam-6686	39	20	said	say	VERB
ejpam-6686	39	21	to	to	PART
ejpam-6686	39	22	be	be	AUX
ejpam-6686	39	23	hierarchy	hierarchy	NOUN
ejpam-6686	39	24	with	with	ADP
ejpam-6686	39	25	respect	respect	NOUN
ejpam-6686	39	26	to	to	ADP
ejpam-6686	39	27	s	s	PRON
ejpam-6686	39	28	if	if	SCONJ
ejpam-6686	39	29	s∧	s∧	PROPN
ejpam-6686	39	30	h	h	NOUN
ejpam-6686	40	1	=	=	NOUN
ejpam-6686	40	2	h	h	NOUN
ejpam-6686	40	3	,	,	PUNCT
ejpam-6686	40	4	for	for	ADP
ejpam-6686	40	5	some	some	DET
ejpam-6686	40	6	s	s	NOUN
ejpam-6686	40	7	∈	∈	PROPN
ejpam-6686	40	8	s.	s.	PROPN
ejpam-6686	40	9	it	it	PRON
ejpam-6686	40	10	is	be	AUX
ejpam-6686	40	11	observed	observe	VERB
ejpam-6686	40	12	that	that	SCONJ
ejpam-6686	40	13	the	the	DET
ejpam-6686	40	14	set	set	NOUN
ejpam-6686	40	15	hs	hs	PROPN
ejpam-6686	40	16	of	of	ADP
ejpam-6686	40	17	hierarchy	hierarchy	NOUN
ejpam-6686	40	18	elements	element	NOUN
ejpam-6686	40	19	with	with	ADP
ejpam-6686	40	20	respect	respect	NOUN
ejpam-6686	40	21	to	to	ADP
ejpam-6686	40	22	s	s	PROPN
ejpam-6686	40	23	is	be	AUX
ejpam-6686	40	24	non	non	ADJ
ejpam-6686	40	25	-	-	ADJ
ejpam-6686	40	26	empty	empty	ADJ
ejpam-6686	40	27	,	,	PUNCT
ejpam-6686	40	28	containing	contain	VERB
ejpam-6686	40	29	s	s	PRON
ejpam-6686	40	30	,	,	PUNCT
ejpam-6686	40	31	and	and	CCONJ
ejpam-6686	40	32	it	it	PRON
ejpam-6686	40	33	is	be	AUX
ejpam-6686	40	34	closed	close	VERB
ejpam-6686	40	35	under	under	ADP
ejpam-6686	40	36	∧.	∧.	PROPN
ejpam-6686	40	37	lemma	lemma	PROPN
ejpam-6686	40	38	1	1	X
ejpam-6686	40	39	.	.	PUNCT
ejpam-6686	41	1	[	[	X
ejpam-6686	41	2	7	7	X
ejpam-6686	41	3	]	]	PUNCT
ejpam-6686	41	4	for	for	ADP
ejpam-6686	41	5	any	any	DET
ejpam-6686	41	6	non	non	ADJ
ejpam-6686	41	7	-	-	ADJ
ejpam-6686	41	8	empty	empty	ADJ
ejpam-6686	41	9	subsets	subset	NOUN
ejpam-6686	41	10	s1	s1	NOUN
ejpam-6686	41	11	,	,	PUNCT
ejpam-6686	41	12	s2	s2	NOUN
ejpam-6686	41	13	of	of	ADP
ejpam-6686	41	14	l	l	NOUN
ejpam-6686	41	15	,	,	PUNCT
ejpam-6686	41	16	(	(	PUNCT
ejpam-6686	41	17	i	i	NOUN
ejpam-6686	41	18	)	)	PUNCT
ejpam-6686	41	19	s1	s1	PROPN
ejpam-6686	41	20	⊆	⊆	NUM
ejpam-6686	41	21	s2	s2	PROPN
ejpam-6686	41	22	implies	imply	VERB
ejpam-6686	41	23	hs1	hs1	PROPN
ejpam-6686	41	24	⊆	⊆	NUM
ejpam-6686	41	25	hs2	hs2	PROPN
ejpam-6686	41	26	(	(	PUNCT
ejpam-6686	41	27	ii	ii	NOUN
ejpam-6686	41	28	)	)	PUNCT
ejpam-6686	41	29	hs1∪s2	hs1∪s2	PROPN
ejpam-6686	41	30	=	=	PUNCT
ejpam-6686	41	31	hs1	hs1	PROPN
ejpam-6686	41	32	∪hs2	∪hs2	PROPN
ejpam-6686	41	33	(	(	PUNCT
ejpam-6686	41	34	iii	iii	NOUN
ejpam-6686	41	35	)	)	PUNCT
ejpam-6686	41	36	hs1∩s2	hs1∩s2	PROPN
ejpam-6686	41	37	⊆	⊆	NUM
ejpam-6686	41	38	hs1	hs1	PROPN
ejpam-6686	41	39	∩hs2	∩hs2	PROPN
ejpam-6686	41	40	.	.	PUNCT
ejpam-6686	42	1	lemma	lemma	PROPN
ejpam-6686	42	2	2	2	NUM
ejpam-6686	42	3	.	.	PUNCT
ejpam-6686	43	1	[	[	X
ejpam-6686	43	2	7	7	X
ejpam-6686	43	3	]	]	PUNCT
ejpam-6686	43	4	for	for	ADP
ejpam-6686	43	5	any	any	DET
ejpam-6686	43	6	non	non	ADJ
ejpam-6686	43	7	-	-	ADJ
ejpam-6686	43	8	empty	empty	ADJ
ejpam-6686	43	9	subsets	subset	NOUN
ejpam-6686	43	10	s	s	X
ejpam-6686	43	11	of	of	ADP
ejpam-6686	43	12	l	l	NOUN
ejpam-6686	43	13	,	,	PUNCT
ejpam-6686	43	14	(	(	PUNCT
ejpam-6686	43	15	i	i	NOUN
ejpam-6686	43	16	)	)	PUNCT
ejpam-6686	43	17	if	if	SCONJ
ejpam-6686	43	18	m	m	VERB
ejpam-6686	43	19	∈	∈	PROPN
ejpam-6686	43	20	hs	hs	PROPN
ejpam-6686	43	21	,	,	PUNCT
ejpam-6686	43	22	then	then	ADV
ejpam-6686	43	23	hs	hs	PROPN
ejpam-6686	43	24	=	=	PROPN
ejpam-6686	43	25	l	l	PROPN
ejpam-6686	43	26	,	,	PUNCT
ejpam-6686	43	27	where	where	SCONJ
ejpam-6686	43	28	m	m	NOUN
ejpam-6686	43	29	is	be	AUX
ejpam-6686	43	30	a	a	DET
ejpam-6686	43	31	maximal	maximal	ADJ
ejpam-6686	43	32	element	element	NOUN
ejpam-6686	43	33	in	in	ADP
ejpam-6686	43	34	l	l	PROPN
ejpam-6686	43	35	(	(	PUNCT
ejpam-6686	43	36	ii	ii	NOUN
ejpam-6686	43	37	)	)	PUNCT
ejpam-6686	43	38	a	a	DET
ejpam-6686	43	39	≤	≤	PROPN
ejpam-6686	43	40	b	b	NOUN
ejpam-6686	43	41	implies	imply	VERB
ejpam-6686	43	42	ha	ha	INTJ
ejpam-6686	43	43	⊆	⊆	NUM
ejpam-6686	43	44	hb	hb	NOUN
ejpam-6686	43	45	,	,	PUNCT
ejpam-6686	43	46	where	where	SCONJ
ejpam-6686	43	47	ha	ha	INTJ
ejpam-6686	43	48	=	=	X
ejpam-6686	43	49	{	{	PUNCT
ejpam-6686	43	50	h	h	NOUN
ejpam-6686	43	51	∈	∈	NOUN
ejpam-6686	44	1	l	l	NOUN
ejpam-6686	45	1	|	|	ADV
ejpam-6686	45	2	a	a	DET
ejpam-6686	45	3	∧	∧	NOUN
ejpam-6686	45	4	h	h	NOUN
ejpam-6686	45	5	=	=	NOUN
ejpam-6686	45	6	h	h	NOUN
ejpam-6686	45	7	}	}	PUNCT
ejpam-6686	45	8	(	(	PUNCT
ejpam-6686	45	9	iii	iii	X
ejpam-6686	45	10	)	)	PUNCT
ejpam-6686	45	11	a	a	DET
ejpam-6686	45	12	≤	≤	PROPN
ejpam-6686	45	13	b	b	NOUN
ejpam-6686	45	14	and	and	CCONJ
ejpam-6686	45	15	b	b	X
ejpam-6686	45	16	∈	∈	NOUN
ejpam-6686	45	17	hs	hs	X
ejpam-6686	45	18	imply	imply	VERB
ejpam-6686	45	19	a	a	DET
ejpam-6686	45	20	∈	∈	PROPN
ejpam-6686	45	21	hs	hs	X
ejpam-6686	45	22	(	(	PUNCT
ejpam-6686	45	23	iv	iv	X
ejpam-6686	45	24	)	)	PUNCT
ejpam-6686	45	25	h	h	NOUN
ejpam-6686	45	26	∈	∈	PROPN
ejpam-6686	45	27	hs	hs	PROPN
ejpam-6686	45	28	implies	imply	VERB
ejpam-6686	45	29	(	(	PUNCT
ejpam-6686	45	30	h	h	X
ejpam-6686	45	31	]	]	X
ejpam-6686	45	32	⊆	⊆	NUM
ejpam-6686	45	33	hs	h	NOUN
ejpam-6686	45	34	,	,	PUNCT
ejpam-6686	45	35	where	where	SCONJ
ejpam-6686	45	36	(	(	PUNCT
ejpam-6686	45	37	h	h	NOUN
ejpam-6686	45	38	]	]	X
ejpam-6686	45	39	=	=	SYM
ejpam-6686	45	40	{	{	PUNCT
ejpam-6686	45	41	h	h	NOUN
ejpam-6686	45	42	∧	∧	PROPN
ejpam-6686	46	1	a	a	DET
ejpam-6686	46	2	|	|	NOUN
ejpam-6686	46	3	a	a	DET
ejpam-6686	46	4	∈	∈	PROPN
ejpam-6686	46	5	l	l	NOUN
ejpam-6686	46	6	}	}	PUNCT
ejpam-6686	46	7	(	(	PUNCT
ejpam-6686	46	8	v	v	NOUN
ejpam-6686	46	9	)	)	PUNCT
ejpam-6686	46	10	ha	ha	X
ejpam-6686	46	11	is	be	AUX
ejpam-6686	46	12	an	an	DET
ejpam-6686	46	13	ideal	ideal	NOUN
ejpam-6686	46	14	of	of	ADP
ejpam-6686	46	15	l.	l.	PROPN
ejpam-6686	46	16	theorem	theorem	PROPN
ejpam-6686	46	17	1	1	NUM
ejpam-6686	46	18	.	.	PUNCT
ejpam-6686	47	1	[	[	X
ejpam-6686	47	2	7	7	X
ejpam-6686	47	3	]	]	PUNCT
ejpam-6686	47	4	for	for	ADP
ejpam-6686	47	5	any	any	DET
ejpam-6686	47	6	non	non	ADJ
ejpam-6686	47	7	-	-	ADJ
ejpam-6686	47	8	empty	empty	ADJ
ejpam-6686	47	9	subset	subset	NOUN
ejpam-6686	47	10	s	s	PROPN
ejpam-6686	47	11	of	of	ADP
ejpam-6686	47	12	l	l	NOUN
ejpam-6686	47	13	,	,	PUNCT
ejpam-6686	47	14	the	the	DET
ejpam-6686	47	15	following	follow	VERB
ejpam-6686	47	16	are	be	AUX
ejpam-6686	47	17	equivalent	equivalent	ADJ
ejpam-6686	47	18	;	;	PUNCT
ejpam-6686	47	19	(	(	PUNCT
ejpam-6686	47	20	i	i	NOUN
ejpam-6686	47	21	)	)	PUNCT
ejpam-6686	47	22	hs	hs	PROPN
ejpam-6686	47	23	is	be	AUX
ejpam-6686	47	24	closed	close	VERB
ejpam-6686	47	25	under	under	ADP
ejpam-6686	47	26	∨	∨	PROPN
ejpam-6686	47	27	(	(	PUNCT
ejpam-6686	47	28	ii	ii	NOUN
ejpam-6686	47	29	)	)	PUNCT
ejpam-6686	47	30	hs	hs	PROPN
ejpam-6686	47	31	is	be	AUX
ejpam-6686	47	32	a	a	DET
ejpam-6686	47	33	sub	sub	ADJ
ejpam-6686	47	34	-	-	ADJ
ejpam-6686	47	35	almost	almost	ADV
ejpam-6686	47	36	distributive	distributive	ADJ
ejpam-6686	47	37	lattice	lattice	NOUN
ejpam-6686	47	38	of	of	ADP
ejpam-6686	47	39	l	l	PROPN
ejpam-6686	47	40	(	(	PUNCT
ejpam-6686	47	41	iii	iii	NOUN
ejpam-6686	47	42	)	)	PUNCT
ejpam-6686	47	43	hs	hs	PROPN
ejpam-6686	47	44	is	be	AUX
ejpam-6686	47	45	an	an	DET
ejpam-6686	47	46	ideal	ideal	NOUN
ejpam-6686	47	47	of	of	ADP
ejpam-6686	47	48	l	l	NOUN
ejpam-6686	47	49	(	(	PUNCT
ejpam-6686	47	50	iv	iv	X
ejpam-6686	47	51	)	)	PUNCT
ejpam-6686	47	52	hs	hs	PROPN
ejpam-6686	47	53	is	be	AUX
ejpam-6686	47	54	the	the	DET
ejpam-6686	47	55	smallest	small	ADJ
ejpam-6686	47	56	ideal	ideal	NOUN
ejpam-6686	47	57	generated	generate	VERB
ejpam-6686	47	58	by	by	ADP
ejpam-6686	47	59	s	s	PROPN
ejpam-6686	47	60	(	(	PUNCT
ejpam-6686	47	61	hs	hs	INTJ
ejpam-6686	47	62	=	=	X
ejpam-6686	47	63	(	(	PUNCT
ejpam-6686	47	64	s	s	PROPN
ejpam-6686	47	65	]	]	X
ejpam-6686	47	66	)	)	PUNCT
ejpam-6686	47	67	.	.	PUNCT
ejpam-6686	48	1	theorem	theorem	NOUN
ejpam-6686	48	2	2	2	NUM
ejpam-6686	48	3	.	.	PUNCT
ejpam-6686	49	1	[	[	X
ejpam-6686	49	2	7	7	X
ejpam-6686	49	3	]	]	PUNCT
ejpam-6686	49	4	the	the	DET
ejpam-6686	49	5	set	set	NOUN
ejpam-6686	49	6	hs	hs	PROPN
ejpam-6686	49	7	of	of	ADP
ejpam-6686	49	8	hierarchy	hierarchy	NOUN
ejpam-6686	49	9	sets	set	NOUN
ejpam-6686	49	10	in	in	ADP
ejpam-6686	49	11	l	l	NOUN
ejpam-6686	49	12	forms	form	NOUN
ejpam-6686	49	13	a	a	DET
ejpam-6686	49	14	distributive	distributive	ADJ
ejpam-6686	49	15	lattice	lattice	NOUN
ejpam-6686	49	16	with	with	ADP
ejpam-6686	49	17	respect	respect	NOUN
ejpam-6686	49	18	to	to	ADP
ejpam-6686	49	19	the	the	DET
ejpam-6686	49	20	operations	operation	NOUN
ejpam-6686	49	21	;	;	PUNCT
ejpam-6686	49	22	hs1	hs1	X
ejpam-6686	49	23	∪hs2	∪hs2	PROPN
ejpam-6686	50	1	=	=	PUNCT
ejpam-6686	51	1	hs1∪s2	hs1∪s2	PROPN
ejpam-6686	51	2	and	and	CCONJ
ejpam-6686	51	3	hs1	hs1	X
ejpam-6686	51	4	∧	∧	PROPN
ejpam-6686	51	5	hs2	hs2	NOUN
ejpam-6686	51	6	=	=	SYM
ejpam-6686	51	7	{	{	PUNCT
ejpam-6686	51	8	s1	s1	NOUN
ejpam-6686	51	9	∧	∧	PROPN
ejpam-6686	51	10	s2	s2	NOUN
ejpam-6686	51	11	|	|	ADV
ejpam-6686	51	12	s1	s1	PROPN
ejpam-6686	51	13	∈	∈	PROPN
ejpam-6686	51	14	s1	s1	NOUN
ejpam-6686	51	15	,	,	PUNCT
ejpam-6686	51	16	s2	s2	PROPN
ejpam-6686	51	17	∈	∈	PROPN
ejpam-6686	51	18	s2	s2	PROPN
ejpam-6686	51	19	}	}	PUNCT
ejpam-6686	51	20	.	.	PUNCT
ejpam-6686	52	1	g.	g.	PROPN
ejpam-6686	52	2	chinnayya	chinnayya	PROPN
ejpam-6686	52	3	et	et	PROPN
ejpam-6686	52	4	al	al	PROPN
ejpam-6686	52	5	.	.	PUNCT
ejpam-6686	52	6	/	/	SYM
ejpam-6686	52	7	eur	eur	PROPN
ejpam-6686	52	8	.	.	PUNCT
ejpam-6686	53	1	j.	j.	PROPN
ejpam-6686	53	2	pure	pure	PROPN
ejpam-6686	53	3	appl	appl	PROPN
ejpam-6686	53	4	.	.	PROPN
ejpam-6686	53	5	math	math	PROPN
ejpam-6686	53	6	,	,	PUNCT
ejpam-6686	53	7	18	18	NUM
ejpam-6686	53	8	(	(	PUNCT
ejpam-6686	53	9	4	4	NUM
ejpam-6686	53	10	)	)	PUNCT
ejpam-6686	53	11	(	(	PUNCT
ejpam-6686	53	12	2025	2025	NUM
ejpam-6686	53	13	)	)	PUNCT
ejpam-6686	53	14	,	,	PUNCT
ejpam-6686	53	15	6686	6686	NUM
ejpam-6686	53	16	4	4	NUM
ejpam-6686	53	17	of	of	ADP
ejpam-6686	53	18	12	12	NUM
ejpam-6686	53	19	3	3	NUM
ejpam-6686	53	20	.	.	PUNCT
ejpam-6686	53	21	prime	prime	ADJ
ejpam-6686	53	22	hierarchy	hierarchy	NOUN
ejpam-6686	53	23	sets	set	VERB
ejpam-6686	53	24	in	in	ADP
ejpam-6686	53	25	almost	almost	ADV
ejpam-6686	53	26	distributive	distributive	ADJ
ejpam-6686	53	27	lattices	lattice	NOUN
ejpam-6686	53	28	in	in	ADP
ejpam-6686	53	29	this	this	DET
ejpam-6686	53	30	section	section	NOUN
ejpam-6686	54	1	,	,	PUNCT
ejpam-6686	54	2	we	we	PRON
ejpam-6686	54	3	investigate	investigate	VERB
ejpam-6686	54	4	structural	structural	ADJ
ejpam-6686	54	5	properties	property	NOUN
ejpam-6686	54	6	of	of	ADP
ejpam-6686	54	7	hierarchy	hierarchy	NOUN
ejpam-6686	54	8	sets	set	NOUN
ejpam-6686	54	9	in	in	ADP
ejpam-6686	54	10	an	an	DET
ejpam-6686	54	11	abstract	abstract	ADJ
ejpam-6686	54	12	distributive	distributive	ADJ
ejpam-6686	54	13	lattice	lattice	NOUN
ejpam-6686	54	14	.	.	PUNCT
ejpam-6686	55	1	a	a	DET
ejpam-6686	55	2	hierarchy	hierarchy	NOUN
ejpam-6686	55	3	set	set	VERB
ejpam-6686	55	4	hs	hs	PRON
ejpam-6686	55	5	generated	generate	VERB
ejpam-6686	55	6	by	by	ADP
ejpam-6686	55	7	a	a	DET
ejpam-6686	55	8	non	non	ADJ
ejpam-6686	55	9	-	-	ADJ
ejpam-6686	55	10	empty	empty	ADJ
ejpam-6686	55	11	subset	subset	NOUN
ejpam-6686	55	12	s	s	NOUN
ejpam-6686	55	13	of	of	ADP
ejpam-6686	55	14	an	an	DET
ejpam-6686	55	15	almost	almost	ADV
ejpam-6686	55	16	distributive	distributive	ADJ
ejpam-6686	55	17	lattice	lattice	NOUN
ejpam-6686	55	18	l	l	NOUN
ejpam-6686	55	19	plays	play	VERB
ejpam-6686	55	20	a	a	DET
ejpam-6686	55	21	central	central	ADJ
ejpam-6686	55	22	role	role	NOUN
ejpam-6686	55	23	in	in	ADP
ejpam-6686	55	24	the	the	DET
ejpam-6686	55	25	algebraic	algebraic	ADJ
ejpam-6686	55	26	and	and	CCONJ
ejpam-6686	55	27	order	order	NOUN
ejpam-6686	55	28	-	-	PUNCT
ejpam-6686	55	29	theoretic	theoretic	NOUN
ejpam-6686	55	30	analysis	analysis	NOUN
ejpam-6686	55	31	of	of	ADP
ejpam-6686	55	32	l.	l.	NOUN
ejpam-6686	55	33	we	we	PRON
ejpam-6686	55	34	begin	begin	VERB
ejpam-6686	55	35	by	by	ADP
ejpam-6686	55	36	establishing	establish	VERB
ejpam-6686	55	37	foundational	foundational	ADJ
ejpam-6686	55	38	closure	closure	NOUN
ejpam-6686	55	39	properties	property	NOUN
ejpam-6686	55	40	of	of	ADP
ejpam-6686	55	41	hierarchy	hierarchy	NOUN
ejpam-6686	55	42	sets	set	NOUN
ejpam-6686	55	43	under	under	ADP
ejpam-6686	55	44	meet	meet	NOUN
ejpam-6686	55	45	operations	operation	NOUN
ejpam-6686	55	46	.	.	PUNCT
ejpam-6686	56	1	building	build	VERB
ejpam-6686	56	2	on	on	ADP
ejpam-6686	56	3	this	this	PRON
ejpam-6686	56	4	,	,	PUNCT
ejpam-6686	56	5	we	we	PRON
ejpam-6686	56	6	introduce	introduce	VERB
ejpam-6686	56	7	and	and	CCONJ
ejpam-6686	56	8	characterize	characterize	VERB
ejpam-6686	56	9	two	two	NUM
ejpam-6686	56	10	important	important	ADJ
ejpam-6686	56	11	classes	class	NOUN
ejpam-6686	56	12	of	of	ADP
ejpam-6686	56	13	hierarchy	hierarchy	NOUN
ejpam-6686	56	14	sets	set	NOUN
ejpam-6686	56	15	:	:	PUNCT
ejpam-6686	56	16	prime	prime	ADJ
ejpam-6686	56	17	and	and	CCONJ
ejpam-6686	56	18	maximal	maximal	ADJ
ejpam-6686	56	19	hierarchy	hierarchy	NOUN
ejpam-6686	56	20	sets	set	NOUN
ejpam-6686	56	21	.	.	PUNCT
ejpam-6686	57	1	a	a	DET
ejpam-6686	57	2	prime	prime	ADJ
ejpam-6686	57	3	hierarchy	hierarchy	NOUN
ejpam-6686	57	4	set	set	VERB
ejpam-6686	57	5	satisfies	satisfy	VERB
ejpam-6686	57	6	the	the	DET
ejpam-6686	57	7	absorption	absorption	NOUN
ejpam-6686	57	8	condition	condition	NOUN
ejpam-6686	57	9	that	that	SCONJ
ejpam-6686	57	10	whenever	whenever	SCONJ
ejpam-6686	57	11	a	a	DET
ejpam-6686	57	12	meet	meet	NOUN
ejpam-6686	57	13	a	a	DET
ejpam-6686	57	14	∧	∧	PROPN
ejpam-6686	57	15	b	b	PROPN
ejpam-6686	57	16	belongs	belong	VERB
ejpam-6686	57	17	to	to	ADP
ejpam-6686	57	18	the	the	DET
ejpam-6686	57	19	set	set	NOUN
ejpam-6686	57	20	,	,	PUNCT
ejpam-6686	57	21	then	then	ADV
ejpam-6686	57	22	at	at	ADP
ejpam-6686	57	23	least	least	ADJ
ejpam-6686	57	24	one	one	NUM
ejpam-6686	57	25	of	of	ADP
ejpam-6686	57	26	the	the	DET
ejpam-6686	57	27	elements	element	NOUN
ejpam-6686	57	28	a	a	PRON
ejpam-6686	57	29	or	or	CCONJ
ejpam-6686	57	30	b	b	NOUN
ejpam-6686	57	31	must	must	AUX
ejpam-6686	57	32	also	also	ADV
ejpam-6686	57	33	belong	belong	VERB
ejpam-6686	57	34	to	to	ADP
ejpam-6686	57	35	it	it	PRON
ejpam-6686	57	36	.	.	PUNCT
ejpam-6686	58	1	a	a	DET
ejpam-6686	58	2	maximal	maximal	ADJ
ejpam-6686	58	3	hierarchy	hierarchy	NOUN
ejpam-6686	58	4	set	set	VERB
ejpam-6686	58	5	,	,	PUNCT
ejpam-6686	58	6	on	on	ADP
ejpam-6686	58	7	the	the	DET
ejpam-6686	58	8	other	other	ADJ
ejpam-6686	58	9	hand	hand	NOUN
ejpam-6686	58	10	,	,	PUNCT
ejpam-6686	58	11	is	be	AUX
ejpam-6686	58	12	one	one	NUM
ejpam-6686	58	13	that	that	PRON
ejpam-6686	58	14	is	be	AUX
ejpam-6686	58	15	not	not	PART
ejpam-6686	58	16	properly	properly	ADV
ejpam-6686	58	17	contained	contain	VERB
ejpam-6686	58	18	in	in	ADP
ejpam-6686	58	19	any	any	DET
ejpam-6686	58	20	other	other	ADJ
ejpam-6686	58	21	proper	proper	ADJ
ejpam-6686	58	22	hierarchy	hierarchy	NOUN
ejpam-6686	58	23	set	set	VERB
ejpam-6686	58	24	.	.	PUNCT
ejpam-6686	59	1	through	through	ADP
ejpam-6686	59	2	illustrative	illustrative	ADJ
ejpam-6686	59	3	examples	example	NOUN
ejpam-6686	59	4	,	,	PUNCT
ejpam-6686	59	5	we	we	PRON
ejpam-6686	59	6	demonstrate	demonstrate	VERB
ejpam-6686	59	7	that	that	SCONJ
ejpam-6686	59	8	while	while	SCONJ
ejpam-6686	59	9	every	every	DET
ejpam-6686	59	10	maximal	maximal	ADJ
ejpam-6686	59	11	hierarchy	hierarchy	NOUN
ejpam-6686	59	12	set	set	VERB
ejpam-6686	59	13	is	be	AUX
ejpam-6686	59	14	prime	prime	ADJ
ejpam-6686	59	15	,	,	PUNCT
ejpam-6686	59	16	the	the	DET
ejpam-6686	59	17	converse	converse	NOUN
ejpam-6686	59	18	need	need	AUX
ejpam-6686	59	19	not	not	PART
ejpam-6686	59	20	hold	hold	VERB
ejpam-6686	59	21	.	.	PUNCT
ejpam-6686	60	1	we	we	PRON
ejpam-6686	60	2	further	far	ADV
ejpam-6686	60	3	employ	employ	VERB
ejpam-6686	60	4	zorn	zorn	PROPN
ejpam-6686	60	5	’s	’s	PART
ejpam-6686	60	6	lemma	lemma	PROPN
ejpam-6686	60	7	to	to	PART
ejpam-6686	60	8	show	show	VERB
ejpam-6686	60	9	the	the	DET
ejpam-6686	60	10	existence	existence	NOUN
ejpam-6686	60	11	of	of	ADP
ejpam-6686	60	12	prime	prime	ADJ
ejpam-6686	60	13	hierarchy	hierarchy	NOUN
ejpam-6686	60	14	sets	set	NOUN
ejpam-6686	60	15	extending	extend	VERB
ejpam-6686	60	16	a	a	DET
ejpam-6686	60	17	given	give	VERB
ejpam-6686	60	18	one	one	NUM
ejpam-6686	60	19	,	,	PUNCT
ejpam-6686	60	20	while	while	SCONJ
ejpam-6686	60	21	avoiding	avoid	VERB
ejpam-6686	60	22	a	a	DET
ejpam-6686	60	23	specific	specific	ADJ
ejpam-6686	60	24	closed	closed	ADJ
ejpam-6686	60	25	set	set	NOUN
ejpam-6686	60	26	under	under	ADP
ejpam-6686	60	27	the	the	DET
ejpam-6686	60	28	meet	meet	NOUN
ejpam-6686	60	29	operation	operation	NOUN
ejpam-6686	60	30	.	.	PUNCT
ejpam-6686	61	1	the	the	DET
ejpam-6686	61	2	section	section	NOUN
ejpam-6686	61	3	concludes	conclude	VERB
ejpam-6686	61	4	by	by	ADP
ejpam-6686	61	5	demonstrating	demonstrate	VERB
ejpam-6686	61	6	that	that	SCONJ
ejpam-6686	61	7	every	every	DET
ejpam-6686	61	8	hierarchy	hierarchy	NOUN
ejpam-6686	61	9	set	set	VERB
ejpam-6686	61	10	can	can	AUX
ejpam-6686	61	11	be	be	AUX
ejpam-6686	61	12	represented	represent	VERB
ejpam-6686	61	13	as	as	ADP
ejpam-6686	61	14	the	the	DET
ejpam-6686	61	15	intersection	intersection	NOUN
ejpam-6686	61	16	of	of	ADP
ejpam-6686	61	17	all	all	DET
ejpam-6686	61	18	prime	prime	ADJ
ejpam-6686	61	19	hierarchy	hierarchy	NOUN
ejpam-6686	61	20	sets	set	NOUN
ejpam-6686	61	21	that	that	PRON
ejpam-6686	61	22	contain	contain	VERB
ejpam-6686	61	23	it	it	PRON
ejpam-6686	61	24	.	.	PUNCT
ejpam-6686	62	1	lemma	lemma	PROPN
ejpam-6686	63	1	3	3	X
ejpam-6686	63	2	.	.	PUNCT
ejpam-6686	64	1	if	if	SCONJ
ejpam-6686	64	2	s	s	PROPN
ejpam-6686	64	3	is	be	AUX
ejpam-6686	64	4	a	a	DET
ejpam-6686	64	5	non	non	ADJ
ejpam-6686	64	6	-	-	ADJ
ejpam-6686	64	7	empty	empty	ADJ
ejpam-6686	64	8	subset	subset	NOUN
ejpam-6686	64	9	and	and	CCONJ
ejpam-6686	64	10	a	a	PRON
ejpam-6686	64	11	is	be	AUX
ejpam-6686	64	12	an	an	DET
ejpam-6686	64	13	element	element	NOUN
ejpam-6686	64	14	in	in	ADP
ejpam-6686	64	15	l	l	NOUN
ejpam-6686	64	16	,	,	PUNCT
ejpam-6686	64	17	then	then	ADV
ejpam-6686	64	18	(	(	PUNCT
ejpam-6686	64	19	i	i	NOUN
ejpam-6686	64	20	)	)	PUNCT
ejpam-6686	64	21	h	h	NOUN
ejpam-6686	64	22	∧	∧	PROPN
ejpam-6686	64	23	a	a	DET
ejpam-6686	64	24	∈	∈	PROPN
ejpam-6686	64	25	hs	hs	PROPN
ejpam-6686	64	26	,	,	PUNCT
ejpam-6686	64	27	for	for	ADP
ejpam-6686	64	28	all	all	DET
ejpam-6686	64	29	h	h	NOUN
ejpam-6686	64	30	∈	∈	PROPN
ejpam-6686	64	31	hs	hs	PROPN
ejpam-6686	64	32	(	(	PUNCT
ejpam-6686	64	33	ii	ii	PROPN
ejpam-6686	64	34	)	)	PUNCT
ejpam-6686	64	35	a	a	DET
ejpam-6686	64	36	∧	∧	PROPN
ejpam-6686	64	37	h	h	NOUN
ejpam-6686	64	38	∈	∈	PROPN
ejpam-6686	64	39	hs	hs	PROPN
ejpam-6686	64	40	,	,	PUNCT
ejpam-6686	64	41	for	for	ADP
ejpam-6686	64	42	all	all	DET
ejpam-6686	64	43	h	h	NOUN
ejpam-6686	64	44	∈	∈	PROPN
ejpam-6686	64	45	hs	hs	PROPN
ejpam-6686	64	46	.	.	PROPN
ejpam-6686	64	47	proof	proof	NOUN
ejpam-6686	64	48	.	.	PUNCT
ejpam-6686	65	1	let	let	VERB
ejpam-6686	65	2	h	h	PRON
ejpam-6686	65	3	∈	∈	PROPN
ejpam-6686	66	1	hs	hs	PROPN
ejpam-6686	66	2	.	.	PUNCT
ejpam-6686	67	1	then	then	ADV
ejpam-6686	67	2	there	there	PRON
ejpam-6686	67	3	exists	exist	VERB
ejpam-6686	67	4	s	s	X
ejpam-6686	67	5	∈	∈	PROPN
ejpam-6686	67	6	s	s	VERB
ejpam-6686	67	7	such	such	ADJ
ejpam-6686	67	8	that	that	PRON
ejpam-6686	67	9	s	s	PART
ejpam-6686	67	10	∧	∧	PROPN
ejpam-6686	67	11	h	h	NOUN
ejpam-6686	67	12	=	=	NOUN
ejpam-6686	67	13	h.	h.	PROPN
ejpam-6686	67	14	given	give	VERB
ejpam-6686	67	15	a	a	DET
ejpam-6686	67	16	∈	∈	PROPN
ejpam-6686	67	17	l.	l.	NOUN
ejpam-6686	67	18	(	(	PUNCT
ejpam-6686	67	19	i	i	NOUN
ejpam-6686	67	20	)	)	PUNCT
ejpam-6686	67	21	s	s	VERB
ejpam-6686	67	22	∧	∧	PROPN
ejpam-6686	67	23	(	(	PUNCT
ejpam-6686	67	24	h	h	NOUN
ejpam-6686	67	25	∧	∧	PROPN
ejpam-6686	67	26	a	a	X
ejpam-6686	67	27	)	)	PUNCT
ejpam-6686	67	28	=	=	PUNCT
ejpam-6686	67	29	(	(	PUNCT
ejpam-6686	67	30	s	s	NOUN
ejpam-6686	67	31	∧	∧	PROPN
ejpam-6686	67	32	h	h	NOUN
ejpam-6686	67	33	)	)	PUNCT
ejpam-6686	67	34	∧	∧	NOUN
ejpam-6686	67	35	a	a	DET
ejpam-6686	67	36	=	=	NOUN
ejpam-6686	67	37	h	h	NOUN
ejpam-6686	67	38	∧	∧	PROPN
ejpam-6686	67	39	a.	a.	NOUN
ejpam-6686	67	40	therefore	therefore	ADV
ejpam-6686	67	41	,	,	PUNCT
ejpam-6686	67	42	h	h	NOUN
ejpam-6686	67	43	∧	∧	PROPN
ejpam-6686	67	44	a	a	DET
ejpam-6686	67	45	∈	∈	PROPN
ejpam-6686	67	46	hs	hs	INTJ
ejpam-6686	67	47	.	.	PUNCT
ejpam-6686	68	1	(	(	PUNCT
ejpam-6686	68	2	ii	ii	NOUN
ejpam-6686	68	3	)	)	PUNCT
ejpam-6686	68	4	s∧	s∧	PROPN
ejpam-6686	68	5	(	(	PUNCT
ejpam-6686	68	6	a∧	a∧	NOUN
ejpam-6686	68	7	h	h	NOUN
ejpam-6686	68	8	)	)	PUNCT
ejpam-6686	68	9	=	=	PUNCT
ejpam-6686	69	1	(	(	PUNCT
ejpam-6686	69	2	s∧	s∧	VERB
ejpam-6686	69	3	a)∧	a)∧	NOUN
ejpam-6686	69	4	h	h	NOUN
ejpam-6686	69	5	=	=	PUNCT
ejpam-6686	69	6	(	(	PUNCT
ejpam-6686	69	7	a∧	a∧	NOUN
ejpam-6686	69	8	s)∧	s)∧	PROPN
ejpam-6686	69	9	h	h	NOUN
ejpam-6686	69	10	=	=	NOUN
ejpam-6686	69	11	a∧	a∧	NOUN
ejpam-6686	69	12	(	(	PUNCT
ejpam-6686	69	13	s∧	s∧	PROPN
ejpam-6686	69	14	h	h	NOUN
ejpam-6686	69	15	)	)	PUNCT
ejpam-6686	69	16	=	=	SYM
ejpam-6686	69	17	a∧	a∧	PROPN
ejpam-6686	69	18	h.	h.	PROPN
ejpam-6686	69	19	therefore	therefore	ADV
ejpam-6686	69	20	,	,	PUNCT
ejpam-6686	69	21	a∧	a∧	NOUN
ejpam-6686	69	22	h	h	NOUN
ejpam-6686	69	23	∈	∈	PROPN
ejpam-6686	70	1	hs	hs	PROPN
ejpam-6686	70	2	.	.	PUNCT
ejpam-6686	71	1	a	a	DET
ejpam-6686	71	2	hierarchy	hierarchy	NOUN
ejpam-6686	71	3	set	set	VERB
ejpam-6686	71	4	hs	hs	PROPN
ejpam-6686	71	5	in	in	ADP
ejpam-6686	71	6	l	l	PROPN
ejpam-6686	71	7	is	be	AUX
ejpam-6686	71	8	said	say	VERB
ejpam-6686	71	9	to	to	PART
ejpam-6686	71	10	be	be	AUX
ejpam-6686	71	11	proper	proper	ADJ
ejpam-6686	71	12	if	if	SCONJ
ejpam-6686	71	13	hs	hs	PROPN
ejpam-6686	71	14	̸=	̸=	PROPN
ejpam-6686	71	15	l.	l.	NOUN
ejpam-6686	71	16	definition	definition	NOUN
ejpam-6686	71	17	2	2	NUM
ejpam-6686	71	18	.	.	PUNCT
ejpam-6686	72	1	a	a	DET
ejpam-6686	72	2	proper	proper	ADJ
ejpam-6686	72	3	hierarchy	hierarchy	NOUN
ejpam-6686	72	4	set	set	VERB
ejpam-6686	72	5	hs	hs	PRON
ejpam-6686	72	6	of	of	ADP
ejpam-6686	72	7	l	l	PROPN
ejpam-6686	72	8	is	be	AUX
ejpam-6686	72	9	said	say	VERB
ejpam-6686	72	10	to	to	PART
ejpam-6686	72	11	be	be	AUX
ejpam-6686	72	12	prime	prime	ADJ
ejpam-6686	72	13	,	,	PUNCT
ejpam-6686	72	14	if	if	SCONJ
ejpam-6686	72	15	given	give	VERB
ejpam-6686	72	16	a	a	DET
ejpam-6686	72	17	,	,	PUNCT
ejpam-6686	72	18	b	b	PROPN
ejpam-6686	72	19	∈	∈	PROPN
ejpam-6686	72	20	l	l	NOUN
ejpam-6686	72	21	,	,	PUNCT
ejpam-6686	72	22	a	a	DET
ejpam-6686	72	23	∧	∧	PROPN
ejpam-6686	72	24	b	b	PROPN
ejpam-6686	72	25	∈	∈	PROPN
ejpam-6686	72	26	hs	hs	PROPN
ejpam-6686	72	27	implies	imply	VERB
ejpam-6686	72	28	a	a	DET
ejpam-6686	72	29	∈	∈	PROPN
ejpam-6686	72	30	hs	hs	PROPN
ejpam-6686	72	31	or	or	CCONJ
ejpam-6686	72	32	b	b	PROPN
ejpam-6686	72	33	∈	∈	PROPN
ejpam-6686	72	34	hs	hs	PROPN
ejpam-6686	72	35	.	.	PROPN
ejpam-6686	72	36	remark	remark	PROPN
ejpam-6686	72	37	1	1	NUM
ejpam-6686	72	38	.	.	PUNCT
ejpam-6686	73	1	every	every	DET
ejpam-6686	73	2	hierarchy	hierarchy	NOUN
ejpam-6686	73	3	set	set	VERB
ejpam-6686	73	4	does	do	AUX
ejpam-6686	73	5	not	not	PART
ejpam-6686	73	6	need	need	VERB
ejpam-6686	73	7	to	to	PART
ejpam-6686	73	8	be	be	AUX
ejpam-6686	73	9	prime	prime	ADJ
ejpam-6686	73	10	.	.	PUNCT
ejpam-6686	74	1	for	for	ADP
ejpam-6686	74	2	example	example	NOUN
ejpam-6686	74	3	,	,	PUNCT
ejpam-6686	74	4	see	see	VERB
ejpam-6686	74	5	the	the	DET
ejpam-6686	74	6	following	follow	VERB
ejpam-6686	74	7	example	example	NOUN
ejpam-6686	74	8	:	:	PUNCT
ejpam-6686	74	9	example	example	NOUN
ejpam-6686	75	1	1	1	X
ejpam-6686	75	2	.	.	PUNCT
ejpam-6686	76	1	let	let	VERB
ejpam-6686	76	2	l	l	NOUN
ejpam-6686	76	3	=	=	PUNCT
ejpam-6686	76	4	{	{	PUNCT
ejpam-6686	76	5	0	0	NUM
ejpam-6686	76	6	,	,	PUNCT
ejpam-6686	76	7	a	a	DET
ejpam-6686	76	8	,	,	PUNCT
ejpam-6686	76	9	b	b	NOUN
ejpam-6686	76	10	,	,	PUNCT
ejpam-6686	76	11	1	1	NUM
ejpam-6686	76	12	}	}	PUNCT
ejpam-6686	76	13	be	be	AUX
ejpam-6686	76	14	an	an	DET
ejpam-6686	76	15	almost	almost	ADV
ejpam-6686	76	16	distributive	distributive	ADJ
ejpam-6686	76	17	lattice	lattice	NOUN
ejpam-6686	76	18	whose	whose	DET
ejpam-6686	76	19	hasse	hasse	NOUN
ejpam-6686	76	20	diagram	diagram	NOUN
ejpam-6686	76	21	is	be	AUX
ejpam-6686	76	22	given	give	VERB
ejpam-6686	76	23	below	below	ADP
ejpam-6686	76	24	:	:	PUNCT
ejpam-6686	76	25	1	1	NUM
ejpam-6686	76	26	a	a	DET
ejpam-6686	76	27	0	0	NUM
ejpam-6686	76	28	b	b	PROPN
ejpam-6686	76	29	g.	g.	NOUN
ejpam-6686	76	30	chinnayya	chinnayya	PROPN
ejpam-6686	76	31	et	et	PROPN
ejpam-6686	76	32	al	al	PROPN
ejpam-6686	76	33	.	.	PUNCT
ejpam-6686	76	34	/	/	SYM
ejpam-6686	76	35	eur	eur	PROPN
ejpam-6686	76	36	.	.	PUNCT
ejpam-6686	77	1	j.	j.	PROPN
ejpam-6686	77	2	pure	pure	PROPN
ejpam-6686	77	3	appl	appl	PROPN
ejpam-6686	77	4	.	.	PROPN
ejpam-6686	77	5	math	math	PROPN
ejpam-6686	77	6	,	,	PUNCT
ejpam-6686	77	7	18	18	NUM
ejpam-6686	77	8	(	(	PUNCT
ejpam-6686	77	9	4	4	NUM
ejpam-6686	77	10	)	)	PUNCT
ejpam-6686	77	11	(	(	PUNCT
ejpam-6686	77	12	2025	2025	NUM
ejpam-6686	77	13	)	)	PUNCT
ejpam-6686	77	14	,	,	PUNCT
ejpam-6686	77	15	6686	6686	NUM
ejpam-6686	77	16	5	5	NUM
ejpam-6686	77	17	of	of	ADP
ejpam-6686	77	18	12	12	NUM
ejpam-6686	77	19	for	for	ADP
ejpam-6686	77	20	s	s	NOUN
ejpam-6686	77	21	=	=	X
ejpam-6686	77	22	{	{	PUNCT
ejpam-6686	77	23	0	0	NUM
ejpam-6686	77	24	}	}	PUNCT
ejpam-6686	77	25	,	,	PUNCT
ejpam-6686	77	26	hs	hs	PROPN
ejpam-6686	77	27	=	=	X
ejpam-6686	77	28	{	{	PUNCT
ejpam-6686	77	29	0	0	NUM
ejpam-6686	77	30	}	}	PUNCT
ejpam-6686	77	31	is	be	AUX
ejpam-6686	77	32	not	not	PART
ejpam-6686	77	33	prime	prime	ADJ
ejpam-6686	77	34	.	.	PUNCT
ejpam-6686	78	1	definition	definition	NOUN
ejpam-6686	78	2	3	3	NUM
ejpam-6686	78	3	.	.	PUNCT
ejpam-6686	79	1	a	a	DET
ejpam-6686	79	2	proper	proper	ADJ
ejpam-6686	79	3	hierarchy	hierarchy	NOUN
ejpam-6686	79	4	set	set	VERB
ejpam-6686	79	5	hs	hs	PRON
ejpam-6686	79	6	of	of	ADP
ejpam-6686	79	7	l	l	PROPN
ejpam-6686	79	8	is	be	AUX
ejpam-6686	79	9	said	say	VERB
ejpam-6686	79	10	to	to	PART
ejpam-6686	79	11	be	be	AUX
ejpam-6686	79	12	maximal	maximal	ADJ
ejpam-6686	79	13	,	,	PUNCT
ejpam-6686	79	14	if	if	SCONJ
ejpam-6686	79	15	,	,	PUNCT
ejpam-6686	79	16	given	give	VERB
ejpam-6686	79	17	a	a	DET
ejpam-6686	79	18	proper	proper	ADJ
ejpam-6686	79	19	hierarchy	hierarchy	NOUN
ejpam-6686	79	20	set	set	VERB
ejpam-6686	79	21	ht	ht	PROPN
ejpam-6686	79	22	in	in	ADP
ejpam-6686	79	23	l	l	PROPN
ejpam-6686	79	24	,	,	PUNCT
ejpam-6686	79	25	hs	hs	PROPN
ejpam-6686	79	26	⊆	⊆	NUM
ejpam-6686	79	27	ht	ht	PROPN
ejpam-6686	79	28	implies	imply	VERB
ejpam-6686	79	29	hs	hs	PROPN
ejpam-6686	79	30	=	=	SYM
ejpam-6686	79	31	ht	ht	PROPN
ejpam-6686	79	32	.	.	PUNCT
ejpam-6686	80	1	theorem	theorem	NOUN
ejpam-6686	80	2	3	3	NUM
ejpam-6686	80	3	.	.	PUNCT
ejpam-6686	81	1	every	every	DET
ejpam-6686	81	2	maximal	maximal	ADJ
ejpam-6686	81	3	hierarchy	hierarchy	NOUN
ejpam-6686	81	4	set	set	VERB
ejpam-6686	81	5	is	be	AUX
ejpam-6686	81	6	prime	prime	ADJ
ejpam-6686	81	7	.	.	PUNCT
ejpam-6686	82	1	proof	proof	NOUN
ejpam-6686	82	2	.	.	PUNCT
ejpam-6686	83	1	let	let	VERB
ejpam-6686	83	2	hs	hs	PRON
ejpam-6686	83	3	be	be	AUX
ejpam-6686	83	4	a	a	DET
ejpam-6686	83	5	maximal	maximal	ADJ
ejpam-6686	83	6	hierarchy	hierarchy	NOUN
ejpam-6686	83	7	set	set	VERB
ejpam-6686	83	8	in	in	ADP
ejpam-6686	83	9	l	l	PROPN
ejpam-6686	83	10	and	and	CCONJ
ejpam-6686	83	11	a	a	DET
ejpam-6686	83	12	,	,	PUNCT
ejpam-6686	83	13	b	b	X
ejpam-6686	83	14	∈	∈	NOUN
ejpam-6686	83	15	l	l	NOUN
ejpam-6686	84	1	such	such	ADJ
ejpam-6686	84	2	that	that	SCONJ
ejpam-6686	84	3	a	a	DET
ejpam-6686	84	4	∧	∧	PROPN
ejpam-6686	84	5	b	b	PROPN
ejpam-6686	84	6	∈	∈	PROPN
ejpam-6686	84	7	hs	hs	INTJ
ejpam-6686	84	8	.	.	PUNCT
ejpam-6686	85	1	if	if	SCONJ
ejpam-6686	85	2	a	a	DET
ejpam-6686	85	3	/∈	/∈	INTJ
ejpam-6686	85	4	hs	hs	PROPN
ejpam-6686	85	5	,	,	PUNCT
ejpam-6686	85	6	then	then	ADV
ejpam-6686	85	7	hs	hs	PROPN
ejpam-6686	85	8	⊆	⊆	NUM
ejpam-6686	85	9	hs	hs	X
ejpam-6686	85	10	∪	∪	X
ejpam-6686	85	11	ha	ha	X
ejpam-6686	85	12	=	=	SYM
ejpam-6686	85	13	hs∪{a	hs∪{a	NOUN
ejpam-6686	85	14	}	}	PUNCT
ejpam-6686	85	15	=	=	SYM
ejpam-6686	86	1	l	l	NOUN
ejpam-6686	86	2	(	(	PUNCT
ejpam-6686	86	3	since	since	SCONJ
ejpam-6686	86	4	hs	hs	PROPN
ejpam-6686	86	5	is	be	AUX
ejpam-6686	86	6	maximal	maximal	ADJ
ejpam-6686	86	7	)	)	PUNCT
ejpam-6686	86	8	.	.	PUNCT
ejpam-6686	87	1	now	now	ADV
ejpam-6686	87	2	,	,	PUNCT
ejpam-6686	87	3	for	for	ADP
ejpam-6686	87	4	this	this	DET
ejpam-6686	87	5	b	b	PROPN
ejpam-6686	87	6	∈	∈	PROPN
ejpam-6686	87	7	l	l	NOUN
ejpam-6686	87	8	=	=	X
ejpam-6686	87	9	hs	hs	X
ejpam-6686	87	10	∪	∪	NOUN
ejpam-6686	87	11	ha	ha	INTJ
ejpam-6686	87	12	.	.	PUNCT
ejpam-6686	87	13	we	we	PRON
ejpam-6686	87	14	have	have	VERB
ejpam-6686	87	15	b	b	PROPN
ejpam-6686	87	16	∈	∈	PROPN
ejpam-6686	87	17	hs	hs	PROPN
ejpam-6686	87	18	,	,	PUNCT
ejpam-6686	87	19	or	or	CCONJ
ejpam-6686	87	20	b	b	X
ejpam-6686	87	21	∈	∈	PROPN
ejpam-6686	87	22	h{a	h{a	NOUN
ejpam-6686	87	23	}	}	PUNCT
ejpam-6686	87	24	,	,	PUNCT
ejpam-6686	87	25	or	or	CCONJ
ejpam-6686	87	26	b	b	X
ejpam-6686	87	27	∈	∈	PROPN
ejpam-6686	87	28	hs	hs	PROPN
ejpam-6686	87	29	∩	∩	PROPN
ejpam-6686	87	30	h{a	h{a	ADJ
ejpam-6686	87	31	}	}	PUNCT
ejpam-6686	87	32	.	.	PUNCT
ejpam-6686	88	1	if	if	SCONJ
ejpam-6686	88	2	b	b	PROPN
ejpam-6686	88	3	∈	∈	PROPN
ejpam-6686	88	4	hs	hs	X
ejpam-6686	88	5	,	,	PUNCT
ejpam-6686	88	6	or	or	CCONJ
ejpam-6686	88	7	b	b	X
ejpam-6686	88	8	∈	∈	PROPN
ejpam-6686	88	9	hs	hs	INTJ
ejpam-6686	88	10	∩h{a	∩h{a	PROPN
ejpam-6686	88	11	}	}	PUNCT
ejpam-6686	88	12	,	,	PUNCT
ejpam-6686	88	13	then	then	ADV
ejpam-6686	88	14	nothing	nothing	PRON
ejpam-6686	88	15	else	else	ADV
ejpam-6686	88	16	to	to	PART
ejpam-6686	88	17	do	do	VERB
ejpam-6686	88	18	.	.	PUNCT
ejpam-6686	89	1	if	if	SCONJ
ejpam-6686	89	2	b	b	PROPN
ejpam-6686	89	3	∈	∈	PROPN
ejpam-6686	89	4	h{a	h{a	VERB
ejpam-6686	89	5	}	}	PUNCT
ejpam-6686	89	6	,	,	PUNCT
ejpam-6686	89	7	then	then	ADV
ejpam-6686	89	8	b	b	X
ejpam-6686	89	9	=	=	PUNCT
ejpam-6686	89	10	a	a	DET
ejpam-6686	89	11	∧	∧	PROPN
ejpam-6686	89	12	b	b	PROPN
ejpam-6686	89	13	∈	∈	PROPN
ejpam-6686	89	14	hs	hs	PROPN
ejpam-6686	89	15	.	.	PUNCT
ejpam-6686	90	1	therefore	therefore	ADV
ejpam-6686	90	2	,	,	PUNCT
ejpam-6686	90	3	hs	hs	PROPN
ejpam-6686	90	4	is	be	AUX
ejpam-6686	90	5	prime	prime	ADJ
ejpam-6686	90	6	.	.	PUNCT
ejpam-6686	91	1	remark	remark	NOUN
ejpam-6686	91	2	2	2	NUM
ejpam-6686	91	3	.	.	PUNCT
ejpam-6686	92	1	the	the	DET
ejpam-6686	92	2	converse	converse	NOUN
ejpam-6686	92	3	of	of	ADP
ejpam-6686	92	4	theorem	theorem	NOUN
ejpam-6686	92	5	(	(	PUNCT
ejpam-6686	92	6	3	3	NUM
ejpam-6686	92	7	)	)	PUNCT
ejpam-6686	92	8	need	need	AUX
ejpam-6686	92	9	not	not	PART
ejpam-6686	92	10	be	be	AUX
ejpam-6686	92	11	true	true	ADJ
ejpam-6686	92	12	.	.	PUNCT
ejpam-6686	93	1	that	that	PRON
ejpam-6686	93	2	is	is	ADV
ejpam-6686	93	3	,	,	PUNCT
ejpam-6686	93	4	every	every	DET
ejpam-6686	93	5	prime	prime	ADJ
ejpam-6686	93	6	hierarchy	hierarchy	NOUN
ejpam-6686	93	7	set	set	VERB
ejpam-6686	93	8	need	need	AUX
ejpam-6686	93	9	not	not	PART
ejpam-6686	93	10	be	be	AUX
ejpam-6686	93	11	maximal	maximal	ADJ
ejpam-6686	93	12	.	.	PUNCT
ejpam-6686	94	1	for	for	ADP
ejpam-6686	94	2	example	example	NOUN
ejpam-6686	94	3	,	,	PUNCT
ejpam-6686	94	4	see	see	VERB
ejpam-6686	94	5	the	the	DET
ejpam-6686	94	6	following	follow	VERB
ejpam-6686	94	7	example	example	NOUN
ejpam-6686	94	8	:	:	PUNCT
ejpam-6686	94	9	example	example	NOUN
ejpam-6686	95	1	2	2	X
ejpam-6686	95	2	.	.	PUNCT
ejpam-6686	96	1	let	let	VERB
ejpam-6686	96	2	l	l	NOUN
ejpam-6686	96	3	=	=	PUNCT
ejpam-6686	96	4	{	{	PUNCT
ejpam-6686	96	5	0	0	NUM
ejpam-6686	96	6	,	,	PUNCT
ejpam-6686	96	7	a	a	DET
ejpam-6686	96	8	,	,	PUNCT
ejpam-6686	96	9	b	b	NOUN
ejpam-6686	96	10	,	,	PUNCT
ejpam-6686	96	11	c	c	NOUN
ejpam-6686	96	12	,	,	PUNCT
ejpam-6686	96	13	1	1	NUM
ejpam-6686	96	14	}	}	PUNCT
ejpam-6686	96	15	be	be	AUX
ejpam-6686	96	16	an	an	DET
ejpam-6686	96	17	almost	almost	ADV
ejpam-6686	96	18	distributive	distributive	ADJ
ejpam-6686	96	19	lattice	lattice	NOUN
ejpam-6686	96	20	whose	whose	DET
ejpam-6686	96	21	hasse	hasse	NOUN
ejpam-6686	96	22	diagram	diagram	NOUN
ejpam-6686	96	23	is	be	AUX
ejpam-6686	96	24	given	give	VERB
ejpam-6686	96	25	below	below	ADV
ejpam-6686	96	26	:	:	PUNCT
ejpam-6686	96	27	c	c	NOUN
ejpam-6686	96	28	a	a	DET
ejpam-6686	96	29	0	0	NUM
ejpam-6686	96	30	b	b	SYM
ejpam-6686	96	31	1	1	NUM
ejpam-6686	96	32	let	let	VERB
ejpam-6686	96	33	s1	s1	PROPN
ejpam-6686	96	34	=	=	PUNCT
ejpam-6686	96	35	{	{	PUNCT
ejpam-6686	96	36	a	a	DET
ejpam-6686	96	37	,	,	PUNCT
ejpam-6686	96	38	b	b	NOUN
ejpam-6686	96	39	}	}	PUNCT
ejpam-6686	96	40	and	and	CCONJ
ejpam-6686	96	41	s2	s2	VERB
ejpam-6686	96	42	=	=	SYM
ejpam-6686	96	43	{	{	PUNCT
ejpam-6686	96	44	b	b	NOUN
ejpam-6686	96	45	,	,	PUNCT
ejpam-6686	96	46	c	c	NOUN
ejpam-6686	96	47	}	}	PUNCT
ejpam-6686	96	48	.	.	PUNCT
ejpam-6686	97	1	then	then	ADV
ejpam-6686	97	2	hs1	hs1	PROPN
ejpam-6686	97	3	=	=	PUNCT
ejpam-6686	97	4	{	{	PUNCT
ejpam-6686	97	5	0	0	NUM
ejpam-6686	97	6	,	,	PUNCT
ejpam-6686	97	7	a	a	DET
ejpam-6686	97	8	,	,	PUNCT
ejpam-6686	97	9	b	b	NOUN
ejpam-6686	97	10	}	}	PUNCT
ejpam-6686	97	11	and	and	CCONJ
ejpam-6686	97	12	hs2	hs2	NOUN
ejpam-6686	97	13	=	=	SYM
ejpam-6686	97	14	{	{	PUNCT
ejpam-6686	97	15	0	0	NUM
ejpam-6686	97	16	,	,	PUNCT
ejpam-6686	97	17	a	a	DET
ejpam-6686	97	18	,	,	PUNCT
ejpam-6686	97	19	b	b	NOUN
ejpam-6686	97	20	,	,	PUNCT
ejpam-6686	97	21	c	c	NOUN
ejpam-6686	97	22	}	}	PUNCT
ejpam-6686	97	23	.	.	PUNCT
ejpam-6686	98	1	therefore	therefore	ADV
ejpam-6686	98	2	,	,	PUNCT
ejpam-6686	98	3	hs1	hs1	PROPN
ejpam-6686	98	4	and	and	CCONJ
ejpam-6686	98	5	hs2	hs2	PROPN
ejpam-6686	98	6	are	be	AUX
ejpam-6686	98	7	two	two	NUM
ejpam-6686	98	8	prime	prime	ADJ
ejpam-6686	98	9	hierarchy	hierarchy	NOUN
ejpam-6686	98	10	sets	set	NOUN
ejpam-6686	98	11	and	and	CCONJ
ejpam-6686	98	12	hs1	hs1	PROPN
ejpam-6686	98	13	⫋	⫋	VERB
ejpam-6686	98	14	hs2	hs2	PROPN
ejpam-6686	98	15	.	.	PUNCT
ejpam-6686	99	1	hence	hence	ADV
ejpam-6686	99	2	,	,	PUNCT
ejpam-6686	99	3	hs1	hs1	PROPN
ejpam-6686	99	4	is	be	AUX
ejpam-6686	99	5	prime	prime	ADJ
ejpam-6686	99	6	but	but	CCONJ
ejpam-6686	99	7	not	not	PART
ejpam-6686	99	8	maximal	maximal	ADJ
ejpam-6686	99	9	.	.	PUNCT
ejpam-6686	100	1	theorem	theorem	NOUN
ejpam-6686	100	2	4	4	NUM
ejpam-6686	100	3	.	.	PUNCT
ejpam-6686	101	1	let	let	VERB
ejpam-6686	101	2	hs	hs	PRON
ejpam-6686	101	3	be	be	AUX
ejpam-6686	101	4	a	a	DET
ejpam-6686	101	5	hierarchy	hierarchy	NOUN
ejpam-6686	101	6	set	set	VERB
ejpam-6686	101	7	and	and	CCONJ
ejpam-6686	101	8	k	k	PROPN
ejpam-6686	101	9	is	be	AUX
ejpam-6686	101	10	a	a	DET
ejpam-6686	101	11	non	non	ADJ
ejpam-6686	101	12	-	-	ADJ
ejpam-6686	101	13	empty	empty	ADJ
ejpam-6686	101	14	subset	subset	NOUN
ejpam-6686	101	15	of	of	ADP
ejpam-6686	101	16	l	l	NOUN
ejpam-6686	101	17	which	which	PRON
ejpam-6686	101	18	is	be	AUX
ejpam-6686	101	19	closed	close	VERB
ejpam-6686	101	20	under	under	ADP
ejpam-6686	101	21	∧	∧	PROPN
ejpam-6686	101	22	such	such	ADJ
ejpam-6686	101	23	that	that	SCONJ
ejpam-6686	101	24	hs	hs	PROPN
ejpam-6686	101	25	∩k	∩k	PROPN
ejpam-6686	101	26	=	=	PUNCT
ejpam-6686	102	1	∅.	∅.	NOUN
ejpam-6686	102	2	then	then	ADV
ejpam-6686	102	3	there	there	PRON
ejpam-6686	102	4	exists	exist	VERB
ejpam-6686	102	5	a	a	DET
ejpam-6686	102	6	prime	prime	ADJ
ejpam-6686	102	7	hierarchy	hierarchy	NOUN
ejpam-6686	102	8	set	set	VERB
ejpam-6686	102	9	hp	hp	NOUN
ejpam-6686	102	10	of	of	ADP
ejpam-6686	102	11	l	l	NOUN
ejpam-6686	103	1	such	such	ADJ
ejpam-6686	103	2	that	that	SCONJ
ejpam-6686	103	3	hs	hs	PROPN
ejpam-6686	103	4	⊆	⊆	NUM
ejpam-6686	103	5	hp	hp	PROPN
ejpam-6686	103	6	and	and	CCONJ
ejpam-6686	103	7	hp	hp	ADJ
ejpam-6686	103	8	∩k	∩k	NOUN
ejpam-6686	103	9	=	=	PUNCT
ejpam-6686	103	10	∅.	∅.	NOUN
ejpam-6686	103	11	proof	proof	NOUN
ejpam-6686	103	12	.	.	PUNCT
ejpam-6686	104	1	let	let	VERB
ejpam-6686	104	2	hs	hs	PRON
ejpam-6686	104	3	be	be	AUX
ejpam-6686	104	4	a	a	DET
ejpam-6686	104	5	hierarchy	hierarchy	NOUN
ejpam-6686	104	6	set	set	VERB
ejpam-6686	104	7	and	and	CCONJ
ejpam-6686	104	8	k	k	PROPN
ejpam-6686	104	9	be	be	AUX
ejpam-6686	104	10	a	a	DET
ejpam-6686	104	11	non	non	ADJ
ejpam-6686	104	12	-	-	ADJ
ejpam-6686	104	13	empty	empty	ADJ
ejpam-6686	104	14	subset	subset	NOUN
ejpam-6686	104	15	of	of	ADP
ejpam-6686	104	16	l	l	NOUN
ejpam-6686	104	17	and	and	CCONJ
ejpam-6686	104	18	closed	close	VERB
ejpam-6686	104	19	under	under	ADP
ejpam-6686	104	20	∧	∧	PROPN
ejpam-6686	104	21	such	such	ADJ
ejpam-6686	104	22	that	that	SCONJ
ejpam-6686	104	23	hs	hs	PROPN
ejpam-6686	104	24	∩	∩	PROPN
ejpam-6686	104	25	k	k	PROPN
ejpam-6686	105	1	=	=	PUNCT
ejpam-6686	105	2	∅.	∅.	AUX
ejpam-6686	105	3	consider	consider	VERB
ejpam-6686	105	4	q	q	NOUN
ejpam-6686	105	5	=	=	PUNCT
ejpam-6686	105	6	{	{	PUNCT
ejpam-6686	105	7	ht	ht	INTJ
ejpam-6686	105	8	|	|	ADV
ejpam-6686	105	9	hs	hs	PROPN
ejpam-6686	105	10	⊆	⊆	PROPN
ejpam-6686	105	11	ht	ht	PROPN
ejpam-6686	105	12	and	and	CCONJ
ejpam-6686	105	13	ht	ht	PROPN
ejpam-6686	105	14	∩	∩	ADJ
ejpam-6686	105	15	k	k	NOUN
ejpam-6686	105	16	=	=	PUNCT
ejpam-6686	105	17	∅	∅	NOUN
ejpam-6686	105	18	}	}	PUNCT
ejpam-6686	105	19	.	.	PUNCT
ejpam-6686	106	1	then	then	ADV
ejpam-6686	106	2	q	q	PROPN
ejpam-6686	106	3	̸=	̸=	PROPN
ejpam-6686	106	4	∅	∅	NOUN
ejpam-6686	106	5	(	(	PUNCT
ejpam-6686	106	6	since	since	SCONJ
ejpam-6686	106	7	hs	hs	PROPN
ejpam-6686	106	8	∩	∩	PROPN
ejpam-6686	106	9	k	k	PROPN
ejpam-6686	106	10	=	=	SYM
ejpam-6686	106	11	∅	∅	NOUN
ejpam-6686	106	12	)	)	PUNCT
ejpam-6686	106	13	and	and	CCONJ
ejpam-6686	106	14	(	(	PUNCT
ejpam-6686	106	15	q,⊆	q,⊆	X
ejpam-6686	106	16	)	)	PUNCT
ejpam-6686	106	17	is	be	AUX
ejpam-6686	106	18	a	a	DET
ejpam-6686	106	19	partially	partially	ADV
ejpam-6686	106	20	ordered	order	VERB
ejpam-6686	106	21	set	set	VERB
ejpam-6686	106	22	with	with	ADP
ejpam-6686	106	23	respect	respect	NOUN
ejpam-6686	106	24	to	to	ADP
ejpam-6686	106	25	the	the	DET
ejpam-6686	106	26	inclusion	inclusion	NOUN
ejpam-6686	106	27	order	order	NOUN
ejpam-6686	106	28	.	.	PUNCT
ejpam-6686	107	1	let	let	VERB
ejpam-6686	107	2	{	{	PUNCT
ejpam-6686	107	3	0	0	NUM
ejpam-6686	107	4	}	}	PUNCT
ejpam-6686	107	5	=	=	SYM
ejpam-6686	107	6	hs1	hs1	PROPN
ejpam-6686	107	7	⊆	⊆	NUM
ejpam-6686	107	8	hs2	hs2	NOUN
ejpam-6686	107	9	⊆	⊆	NUM
ejpam-6686	107	10	hs3	hs3	NOUN
ejpam-6686	107	11	⊆	⊆	NUM
ejpam-6686	107	12	.	.	PUNCT
ejpam-6686	107	13	.	.	PUNCT
ejpam-6686	108	1	.	.	PUNCT
ejpam-6686	109	1	be	be	AUX
ejpam-6686	109	2	an	an	DET
ejpam-6686	109	3	increasing	increase	VERB
ejpam-6686	109	4	chain	chain	NOUN
ejpam-6686	109	5	in	in	ADP
ejpam-6686	109	6	q.	q.	PROPN
ejpam-6686	109	7	then	then	ADV
ejpam-6686	109	8	⋃	⋃	PUNCT
ejpam-6686	109	9	i∈i	i∈i	ADJ
ejpam-6686	109	10	hsi	hsi	PROPN
ejpam-6686	109	11	=	=	PUNCT
ejpam-6686	109	12	hs1	hs1	PROPN
ejpam-6686	109	13	∪	∪	VERB
ejpam-6686	109	14	hs2	hs2	PROPN
ejpam-6686	109	15	.	.	PUNCT
ejpam-6686	109	16	.	.	PUNCT
ejpam-6686	109	17	.	.	PUNCT
ejpam-6686	110	1	=	=	PRON
ejpam-6686	110	2	h	h	NOUN
ejpam-6686	110	3	(	(	PUNCT
ejpam-6686	110	4	⋃	⋃	PROPN
ejpam-6686	110	5	i∈i	i∈i	ADJ
ejpam-6686	110	6	si	si	NOUN
ejpam-6686	110	7	)	)	PUNCT
ejpam-6686	110	8	,	,	PUNCT
ejpam-6686	110	9	h	h	NOUN
ejpam-6686	110	10	(	(	PUNCT
ejpam-6686	110	11	⋃	⋃	PROPN
ejpam-6686	110	12	i∈i	i∈i	ADJ
ejpam-6686	110	13	si	si	NOUN
ejpam-6686	110	14	)	)	PUNCT
ejpam-6686	110	15	∩	∩	NOUN
ejpam-6686	110	16	k	k	X
ejpam-6686	111	1	=	=	PUNCT
ejpam-6686	111	2	⋃	⋃	PROPN
ejpam-6686	111	3	i∈i	i∈i	ADJ
ejpam-6686	111	4	(	(	PUNCT
ejpam-6686	111	5	hsi	hsi	PROPN
ejpam-6686	111	6	∩	∩	PROPN
ejpam-6686	111	7	k	k	PROPN
ejpam-6686	111	8	)	)	PUNCT
ejpam-6686	111	9	=	=	SYM
ejpam-6686	112	1	⋃	⋃	ADP
ejpam-6686	112	2	i∈i	i∈i	ADJ
ejpam-6686	112	3	∅	∅	NOUN
ejpam-6686	112	4	=	=	NOUN
ejpam-6686	112	5	∅	∅	NOUN
ejpam-6686	112	6	,	,	PUNCT
ejpam-6686	112	7	and	and	CCONJ
ejpam-6686	112	8	g.	g.	PROPN
ejpam-6686	112	9	chinnayya	chinnayya	PROPN
ejpam-6686	112	10	et	et	PROPN
ejpam-6686	112	11	al	al	PROPN
ejpam-6686	112	12	.	.	PUNCT
ejpam-6686	112	13	/	/	SYM
ejpam-6686	112	14	eur	eur	PROPN
ejpam-6686	112	15	.	.	PUNCT
ejpam-6686	113	1	j.	j.	PROPN
ejpam-6686	113	2	pure	pure	PROPN
ejpam-6686	113	3	appl	appl	PROPN
ejpam-6686	113	4	.	.	PROPN
ejpam-6686	113	5	math	math	PROPN
ejpam-6686	113	6	,	,	PUNCT
ejpam-6686	113	7	18	18	NUM
ejpam-6686	113	8	(	(	PUNCT
ejpam-6686	113	9	4	4	NUM
ejpam-6686	113	10	)	)	PUNCT
ejpam-6686	113	11	(	(	PUNCT
ejpam-6686	113	12	2025	2025	NUM
ejpam-6686	113	13	)	)	PUNCT
ejpam-6686	113	14	,	,	PUNCT
ejpam-6686	113	15	6686	6686	NUM
ejpam-6686	113	16	6	6	NUM
ejpam-6686	113	17	of	of	ADP
ejpam-6686	113	18	12	12	NUM
ejpam-6686	113	19	hs	hs	PROPN
ejpam-6686	113	20	⊆	⊆	NUM
ejpam-6686	113	21	hsi	hsi	PROPN
ejpam-6686	113	22	,	,	PUNCT
ejpam-6686	113	23	for	for	ADP
ejpam-6686	113	24	all	all	PRON
ejpam-6686	113	25	i	i	PRON
ejpam-6686	113	26	∈	∈	PROPN
ejpam-6686	113	27	i.	i.	NOUN
ejpam-6686	113	28	therefore	therefore	ADV
ejpam-6686	113	29	,	,	PUNCT
ejpam-6686	113	30	h	h	INTJ
ejpam-6686	113	31	(	(	PUNCT
ejpam-6686	113	32	⋃	⋃	PROPN
ejpam-6686	113	33	i∈i	i∈i	ADJ
ejpam-6686	113	34	si	si	NOUN
ejpam-6686	113	35	)	)	PUNCT
ejpam-6686	113	36	is	be	AUX
ejpam-6686	113	37	an	an	DET
ejpam-6686	113	38	upper	upper	ADJ
ejpam-6686	113	39	bound	bind	VERB
ejpam-6686	113	40	of	of	ADP
ejpam-6686	113	41	the	the	DET
ejpam-6686	113	42	chain	chain	NOUN
ejpam-6686	113	43	in	in	ADP
ejpam-6686	113	44	q.	q.	PROPN
ejpam-6686	113	45	by	by	ADP
ejpam-6686	113	46	zorn	zorn	PROPN
ejpam-6686	113	47	’s	’s	PART
ejpam-6686	113	48	lemma	lemma	PROPN
ejpam-6686	113	49	,	,	PUNCT
ejpam-6686	113	50	q	q	PROPN
ejpam-6686	113	51	has	have	VERB
ejpam-6686	113	52	maximal	maximal	ADJ
ejpam-6686	113	53	element	element	NOUN
ejpam-6686	113	54	,	,	PUNCT
ejpam-6686	113	55	say	say	VERB
ejpam-6686	113	56	hp	hp	PROPN
ejpam-6686	113	57	.	.	PUNCT
ejpam-6686	114	1	let	let	VERB
ejpam-6686	114	2	a	a	DET
ejpam-6686	114	3	,	,	PUNCT
ejpam-6686	114	4	b	b	X
ejpam-6686	114	5	∈	∈	NOUN
ejpam-6686	114	6	l	l	NOUN
ejpam-6686	114	7	such	such	ADJ
ejpam-6686	114	8	that	that	SCONJ
ejpam-6686	114	9	a	a	DET
ejpam-6686	114	10	/∈	/∈	INTJ
ejpam-6686	114	11	hp	hp	NOUN
ejpam-6686	114	12	and	and	CCONJ
ejpam-6686	114	13	b	b	PROPN
ejpam-6686	114	14	/∈	/∈	PUNCT
ejpam-6686	115	1	hp	hp	PROPN
ejpam-6686	115	2	.	.	PUNCT
ejpam-6686	116	1	then	then	ADV
ejpam-6686	116	2	a	a	DET
ejpam-6686	116	3	/∈	/∈	PUNCT
ejpam-6686	116	4	p	p	NOUN
ejpam-6686	116	5	and	and	CCONJ
ejpam-6686	116	6	b	b	NOUN
ejpam-6686	116	7	/∈	/∈	PUNCT
ejpam-6686	117	1	p	p	X
ejpam-6686	117	2	.	.	PUNCT
ejpam-6686	118	1	now	now	ADV
ejpam-6686	118	2	,	,	PUNCT
ejpam-6686	118	3	hp	hp	X
ejpam-6686	118	4	∪	∪	ADP
ejpam-6686	118	5	ha	ha	INTJ
ejpam-6686	118	6	=	=	PUNCT
ejpam-6686	118	7	hp∪{a	hp∪{a	NOUN
ejpam-6686	118	8	}	}	PUNCT
ejpam-6686	118	9	and	and	CCONJ
ejpam-6686	118	10	hp	hp	ADJ
ejpam-6686	118	11	∪	∪	ADJ
ejpam-6686	118	12	hb	hb	X
ejpam-6686	118	13	=	=	PUNCT
ejpam-6686	118	14	hp∪{b	hp∪{b	PROPN
ejpam-6686	118	15	}	}	PUNCT
ejpam-6686	118	16	.	.	PUNCT
ejpam-6686	119	1	since	since	SCONJ
ejpam-6686	119	2	hp	hp	PROPN
ejpam-6686	119	3	is	be	AUX
ejpam-6686	119	4	maximal	maximal	ADJ
ejpam-6686	119	5	,	,	PUNCT
ejpam-6686	119	6	hp∪{a	hp∪{a	NOUN
ejpam-6686	119	7	}	}	PUNCT
ejpam-6686	119	8	∩	∩	NOUN
ejpam-6686	119	9	k	k	PROPN
ejpam-6686	119	10	̸=	̸=	PROPN
ejpam-6686	119	11	∅	∅	NOUN
ejpam-6686	119	12	and	and	CCONJ
ejpam-6686	119	13	hp∩{b	hp∩{b	NOUN
ejpam-6686	119	14	}	}	PUNCT
ejpam-6686	119	15	∩	∩	NOUN
ejpam-6686	119	16	k	k	PROPN
ejpam-6686	119	17	̸=	̸=	PROPN
ejpam-6686	119	18	∅.	∅.	ADV
ejpam-6686	119	19	let	let	VERB
ejpam-6686	119	20	x	x	SYM
ejpam-6686	119	21	∈	∈	PROPN
ejpam-6686	119	22	hp∪{a	hp∪{a	PROPN
ejpam-6686	119	23	}	}	PUNCT
ejpam-6686	119	24	∩	∩	PROPN
ejpam-6686	119	25	k	k	PROPN
ejpam-6686	119	26	and	and	CCONJ
ejpam-6686	119	27	y	y	PROPN
ejpam-6686	119	28	∈	∈	PROPN
ejpam-6686	119	29	hp∪{b	hp∪{b	PROPN
ejpam-6686	119	30	}	}	PUNCT
ejpam-6686	119	31	∩	∩	PROPN
ejpam-6686	119	32	k.	k.	PROPN
ejpam-6686	119	33	by	by	ADP
ejpam-6686	119	34	lemma	lemma	PROPN
ejpam-6686	119	35	(	(	PUNCT
ejpam-6686	119	36	3	3	NUM
ejpam-6686	119	37	)	)	PUNCT
ejpam-6686	119	38	,	,	PUNCT
ejpam-6686	119	39	x	x	PUNCT
ejpam-6686	119	40	∧	∧	NOUN
ejpam-6686	119	41	y	y	PROPN
ejpam-6686	119	42	∈	∈	PROPN
ejpam-6686	119	43	hp∪{a	hp∪{a	PROPN
ejpam-6686	119	44	}	}	PUNCT
ejpam-6686	119	45	∩	∩	ADJ
ejpam-6686	119	46	hp∪{b	hp∪{b	NOUN
ejpam-6686	119	47	}	}	PUNCT
ejpam-6686	119	48	∩	∩	PROPN
ejpam-6686	119	49	k.	k.	PROPN
ejpam-6686	119	50	then	then	ADV
ejpam-6686	119	51	x	x	PUNCT
ejpam-6686	119	52	∧	∧	PROPN
ejpam-6686	119	53	y	y	PROPN
ejpam-6686	119	54	∈	∈	PROPN
ejpam-6686	119	55	(	(	PUNCT
ejpam-6686	119	56	hp	hp	PROPN
ejpam-6686	119	57	∪ha)∩	∪ha)∩	PROPN
ejpam-6686	119	58	(	(	PUNCT
ejpam-6686	119	59	hp	hp	NOUN
ejpam-6686	119	60	∪hb)∩k	∪hb)∩k	NOUN
ejpam-6686	119	61	=	=	PUNCT
ejpam-6686	120	1	[	[	X
ejpam-6686	120	2	hp	hp	X
ejpam-6686	120	3	∪	∪	X
ejpam-6686	120	4	(	(	PUNCT
ejpam-6686	120	5	ha	ha	INTJ
ejpam-6686	120	6	∧hb)]∩k	∧hb)]∩k	NOUN
ejpam-6686	120	7	=	=	PUNCT
ejpam-6686	120	8	(	(	PUNCT
ejpam-6686	120	9	hp	hp	X
ejpam-6686	120	10	∪h	∪h	PROPN
ejpam-6686	120	11	a∧b)∩k	a∧b)∩k	PROPN
ejpam-6686	120	12	=	=	PUNCT
ejpam-6686	120	13	hp∪{a∧b	hp∪{a∧b	PROPN
ejpam-6686	120	14	}	}	PUNCT
ejpam-6686	120	15	∩k	∩k	NOUN
ejpam-6686	120	16	.	.	PUNCT
ejpam-6686	121	1	if	if	SCONJ
ejpam-6686	121	2	a	a	DET
ejpam-6686	121	3	∧	∧	PROPN
ejpam-6686	121	4	b	b	PROPN
ejpam-6686	121	5	∈	∈	PROPN
ejpam-6686	121	6	hp	hp	NOUN
ejpam-6686	121	7	,	,	PUNCT
ejpam-6686	121	8	then	then	ADV
ejpam-6686	121	9	hp∪{a∧b	hp∪{a∧b	PROPN
ejpam-6686	121	10	}	}	PUNCT
ejpam-6686	121	11	=	=	SYM
ejpam-6686	121	12	hp	hp	PROPN
ejpam-6686	121	13	.	.	PUNCT
ejpam-6686	122	1	so	so	ADV
ejpam-6686	122	2	that	that	SCONJ
ejpam-6686	122	3	x	x	PUNCT
ejpam-6686	122	4	∧	∧	NOUN
ejpam-6686	122	5	y	y	PROPN
ejpam-6686	122	6	∈	∈	PROPN
ejpam-6686	122	7	hp	hp	PROPN
ejpam-6686	122	8	∩	∩	PROPN
ejpam-6686	122	9	k	k	PROPN
ejpam-6686	122	10	and	and	CCONJ
ejpam-6686	122	11	hence	hence	ADV
ejpam-6686	122	12	hp	hp	ADJ
ejpam-6686	122	13	∩	∩	NOUN
ejpam-6686	122	14	k	k	PROPN
ejpam-6686	122	15	̸=	̸=	PROPN
ejpam-6686	122	16	∅.	∅.	ADP
ejpam-6686	122	17	which	which	PRON
ejpam-6686	122	18	is	be	AUX
ejpam-6686	122	19	a	a	DET
ejpam-6686	122	20	contradiction	contradiction	NOUN
ejpam-6686	122	21	.	.	PUNCT
ejpam-6686	123	1	therefore	therefore	ADV
ejpam-6686	123	2	,	,	PUNCT
ejpam-6686	123	3	a∧	a∧	PROPN
ejpam-6686	123	4	b	b	PROPN
ejpam-6686	123	5	/∈	/∈	PUNCT
ejpam-6686	123	6	hp	hp	NOUN
ejpam-6686	123	7	and	and	CCONJ
ejpam-6686	123	8	hence	hence	ADV
ejpam-6686	123	9	hp	hp	PROPN
ejpam-6686	123	10	is	be	AUX
ejpam-6686	123	11	a	a	DET
ejpam-6686	123	12	prime	prime	ADJ
ejpam-6686	123	13	hierarchy	hierarchy	NOUN
ejpam-6686	123	14	set	set	VERB
ejpam-6686	123	15	of	of	ADP
ejpam-6686	123	16	l.	l.	PROPN
ejpam-6686	123	17	corollary	corollary	PROPN
ejpam-6686	123	18	1	1	PROPN
ejpam-6686	123	19	.	.	PUNCT
ejpam-6686	124	1	let	let	VERB
ejpam-6686	124	2	hs	hs	PRON
ejpam-6686	124	3	be	be	AUX
ejpam-6686	124	4	a	a	DET
ejpam-6686	124	5	hierarchy	hierarchy	NOUN
ejpam-6686	124	6	set	set	VERB
ejpam-6686	124	7	in	in	ADP
ejpam-6686	124	8	l	l	PROPN
ejpam-6686	124	9	and	and	CCONJ
ejpam-6686	124	10	a	a	DET
ejpam-6686	124	11	∈	∈	NOUN
ejpam-6686	124	12	l	l	NOUN
ejpam-6686	124	13	such	such	ADJ
ejpam-6686	124	14	that	that	SCONJ
ejpam-6686	124	15	a	a	DET
ejpam-6686	124	16	/∈	/∈	INTJ
ejpam-6686	124	17	hs	hs	PROPN
ejpam-6686	124	18	.	.	PUNCT
ejpam-6686	124	19	then	then	ADV
ejpam-6686	124	20	there	there	PRON
ejpam-6686	124	21	exists	exist	VERB
ejpam-6686	124	22	a	a	DET
ejpam-6686	124	23	prime	prime	ADJ
ejpam-6686	124	24	hierarchy	hierarchy	NOUN
ejpam-6686	124	25	set	set	VERB
ejpam-6686	124	26	hp	hp	NOUN
ejpam-6686	124	27	of	of	ADP
ejpam-6686	124	28	l	l	NOUN
ejpam-6686	124	29	such	such	ADJ
ejpam-6686	124	30	that	that	SCONJ
ejpam-6686	124	31	hs	hs	PROPN
ejpam-6686	124	32	⊆	⊆	NUM
ejpam-6686	124	33	hp	hp	PROPN
ejpam-6686	124	34	and	and	CCONJ
ejpam-6686	124	35	a	a	DET
ejpam-6686	124	36	/∈	/∈	INTJ
ejpam-6686	124	37	hp	hp	NOUN
ejpam-6686	124	38	.	.	PUNCT
ejpam-6686	125	1	proof	proof	NOUN
ejpam-6686	125	2	.	.	PUNCT
ejpam-6686	126	1	let	let	VERB
ejpam-6686	126	2	hs	hs	PRON
ejpam-6686	126	3	be	be	AUX
ejpam-6686	126	4	a	a	DET
ejpam-6686	126	5	hierarchy	hierarchy	NOUN
ejpam-6686	126	6	set	set	VERB
ejpam-6686	126	7	in	in	ADP
ejpam-6686	126	8	l	l	PROPN
ejpam-6686	126	9	and	and	CCONJ
ejpam-6686	126	10	a	a	DET
ejpam-6686	126	11	∈	∈	NOUN
ejpam-6686	126	12	l	l	NOUN
ejpam-6686	127	1	such	such	ADJ
ejpam-6686	127	2	that	that	SCONJ
ejpam-6686	127	3	a	a	DET
ejpam-6686	127	4	/∈	/∈	INTJ
ejpam-6686	127	5	hs	hs	INTJ
ejpam-6686	127	6	.	.	PUNCT
ejpam-6686	128	1	if	if	SCONJ
ejpam-6686	128	2	k	k	PROPN
ejpam-6686	128	3	=	=	X
ejpam-6686	128	4	{	{	PUNCT
ejpam-6686	128	5	a	a	NOUN
ejpam-6686	128	6	}	}	PUNCT
ejpam-6686	128	7	,	,	PUNCT
ejpam-6686	128	8	then	then	ADV
ejpam-6686	128	9	hs	hs	PROPN
ejpam-6686	128	10	∩k	∩k	NOUN
ejpam-6686	128	11	=	=	SYM
ejpam-6686	128	12	∅	∅	NOUN
ejpam-6686	128	13	and	and	CCONJ
ejpam-6686	128	14	k	k	PROPN
ejpam-6686	128	15	is	be	AUX
ejpam-6686	128	16	closed	close	VERB
ejpam-6686	128	17	under	under	ADP
ejpam-6686	128	18	∧.	∧.	PROPN
ejpam-6686	128	19	by	by	ADP
ejpam-6686	128	20	theorem	theorem	NOUN
ejpam-6686	128	21	4	4	NUM
ejpam-6686	128	22	,	,	PUNCT
ejpam-6686	128	23	there	there	PRON
ejpam-6686	128	24	exists	exist	VERB
ejpam-6686	128	25	a	a	DET
ejpam-6686	128	26	prime	prime	ADJ
ejpam-6686	128	27	hierarchy	hierarchy	NOUN
ejpam-6686	128	28	set	set	VERB
ejpam-6686	128	29	hp	hp	NOUN
ejpam-6686	128	30	in	in	ADP
ejpam-6686	128	31	l	l	PROPN
ejpam-6686	128	32	such	such	ADJ
ejpam-6686	128	33	that	that	SCONJ
ejpam-6686	128	34	hs	hs	PROPN
ejpam-6686	128	35	⊆	⊆	NUM
ejpam-6686	128	36	hp	hp	NOUN
ejpam-6686	128	37	,	,	PUNCT
ejpam-6686	128	38	and	and	CCONJ
ejpam-6686	128	39	hp	hp	PROPN
ejpam-6686	128	40	∩	∩	NOUN
ejpam-6686	128	41	k	k	NOUN
ejpam-6686	128	42	=	=	PUNCT
ejpam-6686	128	43	∅.	∅.	VERB
ejpam-6686	128	44	hence	hence	ADV
ejpam-6686	128	45	,	,	PUNCT
ejpam-6686	128	46	a	a	DET
ejpam-6686	128	47	/∈	/∈	INTJ
ejpam-6686	128	48	hp	hp	NOUN
ejpam-6686	128	49	and	and	CCONJ
ejpam-6686	128	50	hs	hs	PROPN
ejpam-6686	128	51	⊆	⊆	NUM
ejpam-6686	128	52	hp	hp	PROPN
ejpam-6686	128	53	.	.	PUNCT
ejpam-6686	129	1	since	since	SCONJ
ejpam-6686	129	2	p	p	NOUN
ejpam-6686	129	3	⊆	⊆	NUM
ejpam-6686	129	4	hp	hp	NOUN
ejpam-6686	129	5	,	,	PUNCT
ejpam-6686	129	6	a	a	PRON
ejpam-6686	129	7	/∈	/∈	NOUN
ejpam-6686	129	8	p	p	NOUN
ejpam-6686	129	9	.	.	PUNCT
ejpam-6686	129	10	theorem	theorem	ADJ
ejpam-6686	129	11	5	5	NUM
ejpam-6686	129	12	.	.	PUNCT
ejpam-6686	130	1	if	if	SCONJ
ejpam-6686	130	2	hs	hs	PROPN
ejpam-6686	130	3	is	be	AUX
ejpam-6686	130	4	a	a	DET
ejpam-6686	130	5	hierarchy	hierarchy	NOUN
ejpam-6686	130	6	set	set	VERB
ejpam-6686	130	7	in	in	ADP
ejpam-6686	130	8	l	l	NOUN
ejpam-6686	130	9	,	,	PUNCT
ejpam-6686	130	10	then	then	ADV
ejpam-6686	130	11	hs	hs	PROPN
ejpam-6686	130	12	is	be	AUX
ejpam-6686	130	13	the	the	DET
ejpam-6686	130	14	intersection	intersection	NOUN
ejpam-6686	130	15	of	of	ADP
ejpam-6686	130	16	all	all	DET
ejpam-6686	130	17	prime	prime	ADJ
ejpam-6686	130	18	hierarchy	hierarchy	NOUN
ejpam-6686	130	19	sets	set	NOUN
ejpam-6686	130	20	containing	contain	VERB
ejpam-6686	130	21	hs	hs	PROPN
ejpam-6686	130	22	in	in	ADP
ejpam-6686	130	23	l.	l.	PROPN
ejpam-6686	130	24	proof	proof	PROPN
ejpam-6686	130	25	.	.	PUNCT
ejpam-6686	131	1	let	let	VERB
ejpam-6686	131	2	s	s	PRON
ejpam-6686	131	3	be	be	AUX
ejpam-6686	131	4	a	a	DET
ejpam-6686	131	5	non	non	ADJ
ejpam-6686	131	6	-	-	ADJ
ejpam-6686	131	7	empty	empty	ADJ
ejpam-6686	131	8	subset	subset	NOUN
ejpam-6686	131	9	of	of	ADP
ejpam-6686	131	10	l	l	PROPN
ejpam-6686	131	11	and	and	CCONJ
ejpam-6686	131	12	a	a	DET
ejpam-6686	131	13	∈	∈	NOUN
ejpam-6686	131	14	l	l	NOUN
ejpam-6686	131	15	such	such	ADJ
ejpam-6686	131	16	that	that	SCONJ
ejpam-6686	131	17	a	a	DET
ejpam-6686	131	18	/∈	/∈	INTJ
ejpam-6686	131	19	hs	hs	INTJ
ejpam-6686	131	20	.	.	PUNCT
ejpam-6686	132	1	consider	consider	VERB
ejpam-6686	132	2	a	a	DET
ejpam-6686	132	3	set	set	NOUN
ejpam-6686	132	4	q	q	NOUN
ejpam-6686	133	1	=	=	PUNCT
ejpam-6686	133	2	{	{	PUNCT
ejpam-6686	133	3	ht	ht	INTJ
ejpam-6686	134	1	|	|	INTJ
ejpam-6686	134	2	a	a	PRON
ejpam-6686	134	3	/∈	/∈	INTJ
ejpam-6686	135	1	ht	ht	INTJ
ejpam-6686	135	2	and	and	CCONJ
ejpam-6686	135	3	hs	hs	PROPN
ejpam-6686	136	1	⊆	⊆	NUM
ejpam-6686	136	2	ht	ht	PROPN
ejpam-6686	136	3	}	}	PUNCT
ejpam-6686	136	4	.	.	PUNCT
ejpam-6686	137	1	then	then	ADV
ejpam-6686	137	2	q	q	X
ejpam-6686	137	3	=	=	NOUN
ejpam-6686	137	4	̸	̸	NOUN
ejpam-6686	137	5	∅	∅	NOUN
ejpam-6686	137	6	(	(	PUNCT
ejpam-6686	137	7	since	since	SCONJ
ejpam-6686	137	8	a	a	DET
ejpam-6686	137	9	/∈	/∈	INTJ
ejpam-6686	137	10	hs	hs	NOUN
ejpam-6686	137	11	)	)	PUNCT
ejpam-6686	137	12	and	and	CCONJ
ejpam-6686	137	13	it	it	PRON
ejpam-6686	137	14	is	be	AUX
ejpam-6686	137	15	a	a	DET
ejpam-6686	137	16	poset	poset	NOUN
ejpam-6686	137	17	with	with	ADP
ejpam-6686	137	18	the	the	DET
ejpam-6686	137	19	inclusion	inclusion	NOUN
ejpam-6686	137	20	order	order	NOUN
ejpam-6686	137	21	.	.	PUNCT
ejpam-6686	138	1	let	let	VERB
ejpam-6686	138	2	hs1	hs1	PROPN
ejpam-6686	138	3	⊆	⊆	NUM
ejpam-6686	138	4	hs2	hs2	NOUN
ejpam-6686	138	5	⊆	⊆	NUM
ejpam-6686	138	6	.	.	PUNCT
ejpam-6686	138	7	.	.	PUNCT
ejpam-6686	139	1	.	.	PUNCT
ejpam-6686	140	1	be	be	AUX
ejpam-6686	140	2	an	an	DET
ejpam-6686	140	3	increasing	increase	VERB
ejpam-6686	140	4	chain	chain	NOUN
ejpam-6686	140	5	in	in	ADP
ejpam-6686	140	6	q.	q.	PROPN
ejpam-6686	140	7	then⋃	then⋃	PROPN
ejpam-6686	140	8	i∈i	i∈i	PROPN
ejpam-6686	140	9	hsi	hsi	PROPN
ejpam-6686	140	10	=	=	SYM
ejpam-6686	140	11	h	h	PROPN
ejpam-6686	140	12	(	(	PUNCT
ejpam-6686	140	13	⋃	⋃	PROPN
ejpam-6686	140	14	i∈i	i∈i	ADJ
ejpam-6686	140	15	si	si	NOUN
ejpam-6686	140	16	)	)	PUNCT
ejpam-6686	140	17	.	.	PUNCT
ejpam-6686	141	1	if	if	SCONJ
ejpam-6686	141	2	a	a	DET
ejpam-6686	141	3	∈	∈	PROPN
ejpam-6686	141	4	h	h	NOUN
ejpam-6686	141	5	(	(	PUNCT
ejpam-6686	141	6	⋃	⋃	PROPN
ejpam-6686	141	7	i∈i	i∈i	ADJ
ejpam-6686	141	8	si	si	NOUN
ejpam-6686	141	9	)	)	PUNCT
ejpam-6686	141	10	,	,	PUNCT
ejpam-6686	141	11	then	then	ADV
ejpam-6686	141	12	a	a	DET
ejpam-6686	141	13	∈	∈	PROPN
ejpam-6686	141	14	⋃	⋃	NOUN
ejpam-6686	141	15	i∈i	i∈i	ADJ
ejpam-6686	141	16	hsi	hsi	PROPN
ejpam-6686	141	17	.	.	PUNCT
ejpam-6686	142	1	therefore	therefore	ADV
ejpam-6686	142	2	,	,	PUNCT
ejpam-6686	142	3	a	a	DET
ejpam-6686	142	4	∈	∈	PROPN
ejpam-6686	142	5	hsi	hsi	PROPN
ejpam-6686	142	6	,	,	PUNCT
ejpam-6686	142	7	for	for	ADP
ejpam-6686	142	8	some	some	DET
ejpam-6686	142	9	i	i	PRON
ejpam-6686	142	10	∈	∈	PROPN
ejpam-6686	142	11	i.	i.	NOUN
ejpam-6686	142	12	which	which	PRON
ejpam-6686	142	13	is	be	AUX
ejpam-6686	142	14	a	a	DET
ejpam-6686	142	15	contradiction	contradiction	NOUN
ejpam-6686	142	16	to	to	ADP
ejpam-6686	142	17	a	a	DET
ejpam-6686	142	18	/∈	/∈	NOUN
ejpam-6686	142	19	hsi	hsi	PROPN
ejpam-6686	142	20	.	.	PUNCT
ejpam-6686	143	1	so	so	ADV
ejpam-6686	143	2	that	that	SCONJ
ejpam-6686	143	3	a	a	DET
ejpam-6686	143	4	/∈	/∈	NOUN
ejpam-6686	143	5	h	h	NOUN
ejpam-6686	143	6	(	(	PUNCT
ejpam-6686	143	7	⋃	⋃	ADP
ejpam-6686	143	8	i∈i	i∈i	ADJ
ejpam-6686	143	9	si	si	NOUN
ejpam-6686	143	10	)	)	PUNCT
ejpam-6686	143	11	.	.	PUNCT
ejpam-6686	144	1	since	since	SCONJ
ejpam-6686	144	2	hs	hs	PROPN
ejpam-6686	144	3	⊆	⊆	NUM
ejpam-6686	144	4	hsi	hsi	PROPN
ejpam-6686	144	5	,	,	PUNCT
ejpam-6686	144	6	for	for	ADP
ejpam-6686	144	7	all	all	PRON
ejpam-6686	144	8	i	i	PRON
ejpam-6686	144	9	∈	∈	PROPN
ejpam-6686	145	1	i	i	PRON
ejpam-6686	145	2	,	,	PUNCT
ejpam-6686	145	3	hs	hs	PROPN
ejpam-6686	145	4	⊆	⊆	NUM
ejpam-6686	145	5	h	h	NOUN
ejpam-6686	145	6	(	(	PUNCT
ejpam-6686	145	7	⋃	⋃	PROPN
ejpam-6686	145	8	i∈i	i∈i	ADJ
ejpam-6686	145	9	si	si	NOUN
ejpam-6686	145	10	)	)	PUNCT
ejpam-6686	145	11	.	.	PUNCT
ejpam-6686	146	1	therefore	therefore	ADV
ejpam-6686	146	2	,	,	PUNCT
ejpam-6686	146	3	h	h	INTJ
ejpam-6686	146	4	(	(	PUNCT
ejpam-6686	146	5	⋃	⋃	PROPN
ejpam-6686	146	6	i∈i	i∈i	ADJ
ejpam-6686	146	7	si	si	NOUN
ejpam-6686	146	8	)	)	PUNCT
ejpam-6686	146	9	∈	∈	PROPN
ejpam-6686	146	10	q	q	NOUN
ejpam-6686	147	1	and	and	CCONJ
ejpam-6686	147	2	it	it	PRON
ejpam-6686	147	3	is	be	AUX
ejpam-6686	147	4	an	an	DET
ejpam-6686	147	5	upper	upper	ADJ
ejpam-6686	147	6	bound	bind	VERB
ejpam-6686	147	7	for	for	ADP
ejpam-6686	147	8	the	the	DET
ejpam-6686	147	9	chain	chain	NOUN
ejpam-6686	147	10	.	.	PUNCT
ejpam-6686	148	1	by	by	ADP
ejpam-6686	148	2	zorn	zorn	PROPN
ejpam-6686	148	3	’s	’s	PART
ejpam-6686	148	4	lemma	lemma	PROPN
ejpam-6686	148	5	,	,	PUNCT
ejpam-6686	148	6	q	q	PROPN
ejpam-6686	148	7	has	have	VERB
ejpam-6686	148	8	a	a	DET
ejpam-6686	148	9	maximal	maximal	ADJ
ejpam-6686	148	10	element	element	NOUN
ejpam-6686	148	11	,	,	PUNCT
ejpam-6686	148	12	say	say	VERB
ejpam-6686	148	13	hp	hp	ADJ
ejpam-6686	148	14	.	.	PUNCT
ejpam-6686	149	1	that	that	PRON
ejpam-6686	149	2	is	be	AUX
ejpam-6686	149	3	a	a	DET
ejpam-6686	149	4	/∈	/∈	INTJ
ejpam-6686	149	5	hp	hp	NOUN
ejpam-6686	149	6	and	and	CCONJ
ejpam-6686	149	7	hs	hs	PROPN
ejpam-6686	149	8	⊆	⊆	NUM
ejpam-6686	149	9	hp	hp	PROPN
ejpam-6686	149	10	.	.	PUNCT
ejpam-6686	150	1	let	let	VERB
ejpam-6686	150	2	a	a	DET
ejpam-6686	150	3	,	,	PUNCT
ejpam-6686	150	4	b	b	X
ejpam-6686	150	5	∈	∈	NOUN
ejpam-6686	150	6	l	l	NOUN
ejpam-6686	150	7	such	such	ADJ
ejpam-6686	150	8	that	that	SCONJ
ejpam-6686	150	9	a	a	DET
ejpam-6686	150	10	/∈	/∈	INTJ
ejpam-6686	150	11	hp	hp	NOUN
ejpam-6686	150	12	and	and	CCONJ
ejpam-6686	150	13	b	b	PROPN
ejpam-6686	150	14	/∈	/∈	PUNCT
ejpam-6686	151	1	hp	hp	PROPN
ejpam-6686	151	2	.	.	PUNCT
ejpam-6686	152	1	then	then	ADV
ejpam-6686	152	2	hp	hp	VERB
ejpam-6686	152	3	∪	∪	ADP
ejpam-6686	152	4	ha	ha	INTJ
ejpam-6686	152	5	=	=	PUNCT
ejpam-6686	152	6	hp∪{a	hp∪{a	NOUN
ejpam-6686	152	7	}	}	PUNCT
ejpam-6686	152	8	and	and	CCONJ
ejpam-6686	152	9	hp	hp	ADJ
ejpam-6686	152	10	∪	∪	ADJ
ejpam-6686	152	11	hb	hb	X
ejpam-6686	152	12	=	=	PUNCT
ejpam-6686	152	13	hp∪{b	hp∪{b	PROPN
ejpam-6686	152	14	}	}	PUNCT
ejpam-6686	152	15	.	.	PUNCT
ejpam-6686	153	1	since	since	SCONJ
ejpam-6686	153	2	hp	hp	PROPN
ejpam-6686	153	3	is	be	AUX
ejpam-6686	153	4	maximal	maximal	ADJ
ejpam-6686	153	5	in	in	ADP
ejpam-6686	153	6	q	q	NOUN
ejpam-6686	153	7	,	,	PUNCT
ejpam-6686	153	8	a	a	DET
ejpam-6686	153	9	∈	∈	NOUN
ejpam-6686	153	10	hp	hp	NOUN
ejpam-6686	153	11	∪	∪	X
ejpam-6686	153	12	ha	ha	INTJ
ejpam-6686	153	13	and	and	CCONJ
ejpam-6686	153	14	a	a	DET
ejpam-6686	153	15	∈	∈	NOUN
ejpam-6686	153	16	hp	hp	NOUN
ejpam-6686	153	17	∪	∪	X
ejpam-6686	153	18	hb	hb	PROPN
ejpam-6686	153	19	,	,	PUNCT
ejpam-6686	153	20	and	and	CCONJ
ejpam-6686	153	21	then	then	ADV
ejpam-6686	153	22	a	a	DET
ejpam-6686	153	23	∈	∈	PROPN
ejpam-6686	153	24	(	(	PUNCT
ejpam-6686	153	25	hp	hp	PROPN
ejpam-6686	153	26	∪	∪	ADP
ejpam-6686	153	27	ha	ha	INTJ
ejpam-6686	153	28	)	)	PUNCT
ejpam-6686	153	29	∩	∩	NOUN
ejpam-6686	153	30	(	(	PUNCT
ejpam-6686	153	31	hp	hp	PROPN
ejpam-6686	153	32	∪hb	∪hb	X
ejpam-6686	153	33	)	)	PUNCT
ejpam-6686	153	34	=	=	PUNCT
ejpam-6686	153	35	hp	hp	ADJ
ejpam-6686	153	36	∪	∪	X
ejpam-6686	153	37	(	(	PUNCT
ejpam-6686	153	38	ha	ha	INTJ
ejpam-6686	153	39	∩	∩	ADJ
ejpam-6686	153	40	hb	hb	X
ejpam-6686	153	41	)	)	PUNCT
ejpam-6686	153	42	=	=	PUNCT
ejpam-6686	153	43	hp	hp	X
ejpam-6686	153	44	∪	∪	X
ejpam-6686	153	45	ha∧b	ha∧b	NOUN
ejpam-6686	153	46	.	.	PUNCT
ejpam-6686	154	1	if	if	SCONJ
ejpam-6686	154	2	a	a	DET
ejpam-6686	154	3	∧	∧	PROPN
ejpam-6686	154	4	b	b	PROPN
ejpam-6686	154	5	∈	∈	PROPN
ejpam-6686	154	6	hp	hp	NOUN
ejpam-6686	154	7	,	,	PUNCT
ejpam-6686	154	8	then	then	ADV
ejpam-6686	154	9	{	{	PUNCT
ejpam-6686	154	10	a	a	DET
ejpam-6686	154	11	∧	∧	PROPN
ejpam-6686	154	12	b	b	PROPN
ejpam-6686	154	13	}	}	PUNCT
ejpam-6686	154	14	⊆	⊆	NUM
ejpam-6686	154	15	hp	hp	NOUN
ejpam-6686	154	16	,	,	PUNCT
ejpam-6686	154	17	and	and	CCONJ
ejpam-6686	154	18	h{a∧b	h{a∧b	NOUN
ejpam-6686	154	19	}	}	PUNCT
ejpam-6686	154	20	⊆	⊆	NUM
ejpam-6686	154	21	hhp	hhp	NOUN
ejpam-6686	154	22	=	=	PUNCT
ejpam-6686	154	23	hp	hp	PROPN
ejpam-6686	154	24	.	.	PUNCT
ejpam-6686	155	1	therefore	therefore	ADV
ejpam-6686	155	2	,	,	PUNCT
ejpam-6686	155	3	a	a	DET
ejpam-6686	155	4	∈	∈	PROPN
ejpam-6686	155	5	hp	hp	NOUN
ejpam-6686	155	6	.	.	PUNCT
ejpam-6686	156	1	which	which	PRON
ejpam-6686	156	2	is	be	AUX
ejpam-6686	156	3	a	a	DET
ejpam-6686	156	4	contradiction	contradiction	NOUN
ejpam-6686	156	5	.	.	PUNCT
ejpam-6686	157	1	hence	hence	ADV
ejpam-6686	157	2	,	,	PUNCT
ejpam-6686	157	3	hp	hp	PROPN
ejpam-6686	157	4	is	be	AUX
ejpam-6686	157	5	a	a	DET
ejpam-6686	157	6	prime	prime	ADJ
ejpam-6686	157	7	hierarchy	hierarchy	NOUN
ejpam-6686	157	8	set	set	VERB
ejpam-6686	157	9	in	in	ADP
ejpam-6686	157	10	l.	l.	PROPN
ejpam-6686	157	11	thus	thus	ADV
ejpam-6686	157	12	,	,	PUNCT
ejpam-6686	157	13	hs	hs	PROPN
ejpam-6686	157	14	=	=	SYM
ejpam-6686	157	15	⋂	⋂	PROPN
ejpam-6686	157	16	{	{	PUNCT
ejpam-6686	157	17	hp	hp	NOUN
ejpam-6686	157	18	|	|	ADV
ejpam-6686	157	19	hp	hp	PROPN
ejpam-6686	157	20	is	be	AUX
ejpam-6686	157	21	a	a	DET
ejpam-6686	157	22	hierarchy	hierarchy	NOUN
ejpam-6686	157	23	set	set	VERB
ejpam-6686	157	24	containing	contain	VERB
ejpam-6686	157	25	hs	hs	PROPN
ejpam-6686	157	26	in	in	ADP
ejpam-6686	157	27	l.	l.	PROPN
ejpam-6686	157	28	a	a	DET
ejpam-6686	157	29	non	non	ADJ
ejpam-6686	157	30	-	-	ADJ
ejpam-6686	157	31	empty	empty	ADJ
ejpam-6686	157	32	subset	subset	NOUN
ejpam-6686	157	33	f	f	PROPN
ejpam-6686	157	34	of	of	ADP
ejpam-6686	157	35	l	l	PROPN
ejpam-6686	157	36	is	be	AUX
ejpam-6686	157	37	said	say	VERB
ejpam-6686	157	38	to	to	PART
ejpam-6686	157	39	be	be	AUX
ejpam-6686	157	40	a	a	DET
ejpam-6686	157	41	filter	filter	NOUN
ejpam-6686	157	42	[	[	X
ejpam-6686	157	43	8	8	NUM
ejpam-6686	157	44	]	]	X
ejpam-6686	157	45	if	if	SCONJ
ejpam-6686	157	46	it	it	PRON
ejpam-6686	157	47	is	be	AUX
ejpam-6686	157	48	closed	close	VERB
ejpam-6686	157	49	under	under	ADP
ejpam-6686	157	50	∧	∧	PROPN
ejpam-6686	157	51	and	and	CCONJ
ejpam-6686	157	52	given	give	VERB
ejpam-6686	157	53	a	a	DET
ejpam-6686	157	54	∈	∈	PROPN
ejpam-6686	157	55	l	l	NOUN
ejpam-6686	157	56	,	,	PUNCT
ejpam-6686	157	57	b	b	X
ejpam-6686	157	58	∈	∈	PROPN
ejpam-6686	157	59	f	f	PROPN
ejpam-6686	157	60	,	,	PUNCT
ejpam-6686	157	61	a∨b	a∨b	PROPN
ejpam-6686	157	62	∈	∈	PROPN
ejpam-6686	157	63	f	f	PROPN
ejpam-6686	157	64	.	.	PUNCT
ejpam-6686	158	1	a	a	DET
ejpam-6686	158	2	proper	proper	ADJ
ejpam-6686	158	3	filter	filter	NOUN
ejpam-6686	158	4	f	f	NOUN
ejpam-6686	158	5	of	of	ADP
ejpam-6686	158	6	l	l	PROPN
ejpam-6686	158	7	is	be	AUX
ejpam-6686	158	8	called	call	VERB
ejpam-6686	158	9	prime	prime	ADJ
ejpam-6686	158	10	[	[	X
ejpam-6686	158	11	8	8	NUM
ejpam-6686	158	12	]	]	X
ejpam-6686	158	13	if	if	SCONJ
ejpam-6686	158	14	given	give	VERB
ejpam-6686	158	15	a	a	DET
ejpam-6686	158	16	,	,	PUNCT
ejpam-6686	158	17	b	b	PROPN
ejpam-6686	158	18	∈	∈	PROPN
ejpam-6686	158	19	l	l	NOUN
ejpam-6686	158	20	,	,	PUNCT
ejpam-6686	158	21	a∨b	a∨b	PROPN
ejpam-6686	158	22	∈	∈	PROPN
ejpam-6686	158	23	f	f	PROPN
ejpam-6686	158	24	implies	imply	VERB
ejpam-6686	158	25	a	a	DET
ejpam-6686	158	26	∈	∈	ADJ
ejpam-6686	158	27	f	f	NOUN
ejpam-6686	158	28	or	or	CCONJ
ejpam-6686	158	29	b	b	PROPN
ejpam-6686	158	30	∈	∈	PROPN
ejpam-6686	158	31	f	f	X
ejpam-6686	158	32	.	.	PUNCT
ejpam-6686	159	1	theorem	theorem	VERB
ejpam-6686	159	2	6	6	NUM
ejpam-6686	159	3	.	.	PUNCT
ejpam-6686	160	1	if	if	SCONJ
ejpam-6686	160	2	hp	hp	PROPN
ejpam-6686	160	3	is	be	AUX
ejpam-6686	160	4	a	a	DET
ejpam-6686	160	5	prime	prime	ADJ
ejpam-6686	160	6	hierarchy	hierarchy	NOUN
ejpam-6686	160	7	set	set	NOUN
ejpam-6686	160	8	of	of	ADP
ejpam-6686	160	9	l	l	NOUN
ejpam-6686	160	10	,	,	PUNCT
ejpam-6686	160	11	where	where	SCONJ
ejpam-6686	160	12	p	p	NOUN
ejpam-6686	160	13	is	be	AUX
ejpam-6686	160	14	a	a	DET
ejpam-6686	160	15	non	non	ADJ
ejpam-6686	160	16	-	-	ADJ
ejpam-6686	160	17	empty	empty	ADJ
ejpam-6686	160	18	subset	subset	NOUN
ejpam-6686	160	19	of	of	ADP
ejpam-6686	160	20	l	l	NOUN
ejpam-6686	160	21	,	,	PUNCT
ejpam-6686	160	22	then	then	ADV
ejpam-6686	160	23	l	l	NOUN
ejpam-6686	160	24	\hp	\hp	PROPN
ejpam-6686	160	25	is	be	AUX
ejpam-6686	160	26	a	a	DET
ejpam-6686	160	27	filter	filter	NOUN
ejpam-6686	160	28	of	of	ADP
ejpam-6686	160	29	l.	l.	PROPN
ejpam-6686	160	30	proof	proof	PROPN
ejpam-6686	160	31	.	.	PUNCT
ejpam-6686	161	1	let	let	VERB
ejpam-6686	161	2	hp	hp	PROPN
ejpam-6686	161	3	be	be	AUX
ejpam-6686	161	4	a	a	DET
ejpam-6686	161	5	prime	prime	ADJ
ejpam-6686	161	6	hierarchy	hierarchy	NOUN
ejpam-6686	161	7	set	set	NOUN
ejpam-6686	161	8	,	,	PUNCT
ejpam-6686	161	9	where	where	SCONJ
ejpam-6686	161	10	p	p	NOUN
ejpam-6686	161	11	is	be	AUX
ejpam-6686	161	12	a	a	DET
ejpam-6686	161	13	non	non	ADJ
ejpam-6686	161	14	-	-	ADJ
ejpam-6686	161	15	empty	empty	ADJ
ejpam-6686	161	16	subset	subset	NOUN
ejpam-6686	161	17	of	of	ADP
ejpam-6686	161	18	l.	l.	PROPN
ejpam-6686	161	19	since	since	SCONJ
ejpam-6686	161	20	hp	hp	PROPN
ejpam-6686	161	21	is	be	AUX
ejpam-6686	161	22	proper	proper	ADJ
ejpam-6686	161	23	,	,	PUNCT
ejpam-6686	161	24	we	we	PRON
ejpam-6686	161	25	can	can	AUX
ejpam-6686	161	26	choose	choose	VERB
ejpam-6686	161	27	a	a	DET
ejpam-6686	161	28	,	,	PUNCT
ejpam-6686	161	29	b	b	PROPN
ejpam-6686	161	30	∈	∈	PROPN
ejpam-6686	161	31	l	l	NOUN
ejpam-6686	161	32	\hp	\hp	PROPN
ejpam-6686	161	33	.	.	PUNCT
ejpam-6686	162	1	if	if	SCONJ
ejpam-6686	162	2	a	a	DET
ejpam-6686	162	3	∧	∧	PROPN
ejpam-6686	162	4	b	b	PROPN
ejpam-6686	162	5	∈	∈	PROPN
ejpam-6686	162	6	hp	hp	NOUN
ejpam-6686	162	7	,	,	PUNCT
ejpam-6686	162	8	then	then	ADV
ejpam-6686	162	9	a	a	DET
ejpam-6686	162	10	∈	∈	PROPN
ejpam-6686	162	11	hp	hp	NOUN
ejpam-6686	162	12	or	or	CCONJ
ejpam-6686	162	13	b	b	X
ejpam-6686	162	14	∈	∈	NOUN
ejpam-6686	162	15	hp	hp	NOUN
ejpam-6686	162	16	(	(	PUNCT
ejpam-6686	162	17	since	since	SCONJ
ejpam-6686	162	18	g.	g.	PROPN
ejpam-6686	162	19	chinnayya	chinnayya	PROPN
ejpam-6686	162	20	et	et	PROPN
ejpam-6686	162	21	al	al	PROPN
ejpam-6686	162	22	.	.	PUNCT
ejpam-6686	162	23	/	/	SYM
ejpam-6686	162	24	eur	eur	PROPN
ejpam-6686	162	25	.	.	PUNCT
ejpam-6686	163	1	j.	j.	PROPN
ejpam-6686	163	2	pure	pure	PROPN
ejpam-6686	163	3	appl	appl	PROPN
ejpam-6686	163	4	.	.	PROPN
ejpam-6686	163	5	math	math	PROPN
ejpam-6686	163	6	,	,	PUNCT
ejpam-6686	163	7	18	18	NUM
ejpam-6686	163	8	(	(	PUNCT
ejpam-6686	163	9	4	4	NUM
ejpam-6686	163	10	)	)	PUNCT
ejpam-6686	163	11	(	(	PUNCT
ejpam-6686	163	12	2025	2025	NUM
ejpam-6686	163	13	)	)	PUNCT
ejpam-6686	163	14	,	,	PUNCT
ejpam-6686	163	15	6686	6686	NUM
ejpam-6686	163	16	7	7	NUM
ejpam-6686	163	17	of	of	ADP
ejpam-6686	163	18	12	12	NUM
ejpam-6686	163	19	hp	hp	NOUN
ejpam-6686	163	20	is	be	AUX
ejpam-6686	163	21	prime	prime	ADJ
ejpam-6686	163	22	)	)	PUNCT
ejpam-6686	163	23	.	.	PUNCT
ejpam-6686	164	1	which	which	PRON
ejpam-6686	164	2	is	be	AUX
ejpam-6686	164	3	not	not	PART
ejpam-6686	164	4	possible	possible	ADJ
ejpam-6686	164	5	.	.	PUNCT
ejpam-6686	165	1	therefore	therefore	ADV
ejpam-6686	165	2	,	,	PUNCT
ejpam-6686	165	3	a∧	a∧	NOUN
ejpam-6686	165	4	b	b	PROPN
ejpam-6686	165	5	∈	∈	PROPN
ejpam-6686	165	6	l\hp	l\hp	NOUN
ejpam-6686	165	7	and	and	CCONJ
ejpam-6686	165	8	hence	hence	ADV
ejpam-6686	165	9	l\hp	l\hp	PROPN
ejpam-6686	165	10	is	be	AUX
ejpam-6686	165	11	closed	close	VERB
ejpam-6686	165	12	under	under	ADP
ejpam-6686	165	13	∧.	∧.	PROPN
ejpam-6686	165	14	let	let	VERB
ejpam-6686	165	15	c	c	NOUN
ejpam-6686	165	16	∈	∈	PROPN
ejpam-6686	165	17	l	l	NOUN
ejpam-6686	165	18	and	and	CCONJ
ejpam-6686	165	19	a	a	DET
ejpam-6686	165	20	∈	∈	PROPN
ejpam-6686	165	21	l	l	NOUN
ejpam-6686	165	22	\	\	PROPN
ejpam-6686	165	23	hp	hp	NOUN
ejpam-6686	165	24	.	.	PUNCT
ejpam-6686	166	1	if	if	SCONJ
ejpam-6686	166	2	c	c	PROPN
ejpam-6686	166	3	∨	∨	VERB
ejpam-6686	166	4	a	a	DET
ejpam-6686	166	5	∈	∈	PROPN
ejpam-6686	166	6	hp	hp	NOUN
ejpam-6686	166	7	,	,	PUNCT
ejpam-6686	166	8	then	then	ADV
ejpam-6686	166	9	p	p	PROPN
ejpam-6686	166	10	∧	∧	PROPN
ejpam-6686	166	11	(	(	PUNCT
ejpam-6686	166	12	c	c	PROPN
ejpam-6686	166	13	∨	∨	NUM
ejpam-6686	166	14	a	a	PRON
ejpam-6686	166	15	)	)	PUNCT
ejpam-6686	166	16	=	=	SYM
ejpam-6686	167	1	c	c	PROPN
ejpam-6686	167	2	∨	∨	NUM
ejpam-6686	167	3	a	a	X
ejpam-6686	167	4	,	,	PUNCT
ejpam-6686	167	5	for	for	ADP
ejpam-6686	167	6	some	some	DET
ejpam-6686	167	7	p	p	NOUN
ejpam-6686	167	8	∈	∈	PROPN
ejpam-6686	167	9	hp	hp	NOUN
ejpam-6686	167	10	.	.	PUNCT
ejpam-6686	168	1	so	so	ADV
ejpam-6686	168	2	that	that	SCONJ
ejpam-6686	168	3	p	p	PROPN
ejpam-6686	168	4	∧	∧	PROPN
ejpam-6686	168	5	(	(	PUNCT
ejpam-6686	168	6	c	c	PROPN
ejpam-6686	168	7	∨	∨	NUM
ejpam-6686	168	8	a	a	PRON
ejpam-6686	168	9	)	)	PUNCT
ejpam-6686	168	10	∧	∧	PROPN
ejpam-6686	168	11	a	a	NOUN
ejpam-6686	168	12	=	=	X
ejpam-6686	168	13	(	(	PUNCT
ejpam-6686	168	14	c	c	PROPN
ejpam-6686	168	15	∨	∨	NUM
ejpam-6686	168	16	a	a	PRON
ejpam-6686	168	17	)	)	PUNCT
ejpam-6686	168	18	∧	∧	PROPN
ejpam-6686	168	19	a	a	NOUN
ejpam-6686	169	1	and	and	CCONJ
ejpam-6686	169	2	then	then	ADV
ejpam-6686	169	3	p	p	PROPN
ejpam-6686	169	4	∧	∧	PROPN
ejpam-6686	169	5	a	a	DET
ejpam-6686	169	6	=	=	NOUN
ejpam-6686	169	7	a.	a.	NOUN
ejpam-6686	170	1	it	it	PRON
ejpam-6686	170	2	means	mean	VERB
ejpam-6686	170	3	that	that	SCONJ
ejpam-6686	170	4	a	a	DET
ejpam-6686	170	5	∈	∈	PROPN
ejpam-6686	170	6	hp	hp	NOUN
ejpam-6686	170	7	.	.	PUNCT
ejpam-6686	170	8	which	which	PRON
ejpam-6686	170	9	is	be	AUX
ejpam-6686	170	10	not	not	PART
ejpam-6686	170	11	true	true	ADJ
ejpam-6686	170	12	.	.	PUNCT
ejpam-6686	171	1	therefore	therefore	ADV
ejpam-6686	171	2	,	,	PUNCT
ejpam-6686	171	3	c	c	PROPN
ejpam-6686	171	4	∨	∨	NUM
ejpam-6686	171	5	a	a	DET
ejpam-6686	171	6	∈	∈	PROPN
ejpam-6686	171	7	l	l	NOUN
ejpam-6686	171	8	\hp	\hp	NOUN
ejpam-6686	171	9	and	and	CCONJ
ejpam-6686	171	10	hence	hence	ADV
ejpam-6686	171	11	l	l	NOUN
ejpam-6686	171	12	\hp	\hp	PROPN
ejpam-6686	171	13	is	be	AUX
ejpam-6686	171	14	a	a	DET
ejpam-6686	171	15	filter	filter	NOUN
ejpam-6686	171	16	.	.	PUNCT
ejpam-6686	172	1	remark	remark	NOUN
ejpam-6686	172	2	3	3	NUM
ejpam-6686	172	3	.	.	PUNCT
ejpam-6686	173	1	if	if	SCONJ
ejpam-6686	173	2	hp	hp	PROPN
ejpam-6686	173	3	is	be	AUX
ejpam-6686	173	4	a	a	DET
ejpam-6686	173	5	prime	prime	ADJ
ejpam-6686	173	6	hierarchy	hierarchy	NOUN
ejpam-6686	173	7	set	set	VERB
ejpam-6686	173	8	in	in	ADP
ejpam-6686	173	9	l	l	NOUN
ejpam-6686	173	10	,	,	PUNCT
ejpam-6686	173	11	then	then	ADV
ejpam-6686	173	12	l	l	NOUN
ejpam-6686	173	13	\	\	PROPN
ejpam-6686	173	14	hp	hp	PROPN
ejpam-6686	173	15	need	need	AUX
ejpam-6686	173	16	not	not	PART
ejpam-6686	173	17	be	be	AUX
ejpam-6686	173	18	prime	prime	ADJ
ejpam-6686	173	19	.	.	PUNCT
ejpam-6686	174	1	for	for	ADP
ejpam-6686	174	2	example	example	NOUN
ejpam-6686	174	3	,	,	PUNCT
ejpam-6686	174	4	see	see	VERB
ejpam-6686	174	5	the	the	DET
ejpam-6686	174	6	following	follow	VERB
ejpam-6686	174	7	example	example	NOUN
ejpam-6686	174	8	:	:	PUNCT
ejpam-6686	174	9	example	example	NOUN
ejpam-6686	175	1	3	3	X
ejpam-6686	175	2	.	.	PUNCT
ejpam-6686	176	1	let	let	VERB
ejpam-6686	176	2	l	l	NOUN
ejpam-6686	176	3	=	=	PUNCT
ejpam-6686	176	4	{	{	PUNCT
ejpam-6686	176	5	0	0	NUM
ejpam-6686	176	6	,	,	PUNCT
ejpam-6686	176	7	a	a	DET
ejpam-6686	176	8	,	,	PUNCT
ejpam-6686	176	9	b	b	NOUN
ejpam-6686	176	10	,	,	PUNCT
ejpam-6686	176	11	c	c	NOUN
ejpam-6686	176	12	,	,	PUNCT
ejpam-6686	176	13	m1,m2	m1,m2	PROPN
ejpam-6686	176	14	}	}	PUNCT
ejpam-6686	176	15	be	be	AUX
ejpam-6686	176	16	an	an	DET
ejpam-6686	176	17	almost	almost	ADV
ejpam-6686	176	18	distributive	distributive	ADJ
ejpam-6686	176	19	lattice	lattice	NOUN
ejpam-6686	176	20	with	with	ADP
ejpam-6686	176	21	maximal	maximal	ADJ
ejpam-6686	176	22	elements	element	NOUN
ejpam-6686	176	23	m1,m2	m1,m2	PROPN
ejpam-6686	176	24	,	,	PUNCT
ejpam-6686	176	25	where	where	SCONJ
ejpam-6686	176	26	the	the	DET
ejpam-6686	176	27	operations	operation	NOUN
ejpam-6686	176	28	∧	∧	PROPN
ejpam-6686	176	29	and	and	CCONJ
ejpam-6686	176	30	∨	∨	NUM
ejpam-6686	176	31	are	be	AUX
ejpam-6686	176	32	defined	define	VERB
ejpam-6686	176	33	below	below	ADP
ejpam-6686	176	34	:	:	PUNCT
ejpam-6686	177	1	∧	∧	NOUN
ejpam-6686	177	2	0	0	PUNCT
ejpam-6686	177	3	a	a	DET
ejpam-6686	177	4	b	b	PROPN
ejpam-6686	177	5	c	c	PROPN
ejpam-6686	177	6	m1	m1	PROPN
ejpam-6686	177	7	m2	m2	PROPN
ejpam-6686	177	8	0	0	NUM
ejpam-6686	177	9	0	0	NUM
ejpam-6686	177	10	0	0	NUM
ejpam-6686	177	11	0	0	NUM
ejpam-6686	177	12	0	0	NUM
ejpam-6686	177	13	0	0	NUM
ejpam-6686	177	14	0	0	NUM
ejpam-6686	178	1	a	a	DET
ejpam-6686	178	2	0	0	NUM
ejpam-6686	178	3	a	a	DET
ejpam-6686	178	4	0	0	NUM
ejpam-6686	178	5	a	a	DET
ejpam-6686	178	6	a	a	DET
ejpam-6686	178	7	a	a	DET
ejpam-6686	178	8	b	b	NOUN
ejpam-6686	178	9	0	0	NUM
ejpam-6686	178	10	0	0	NUM
ejpam-6686	179	1	b	b	PROPN
ejpam-6686	179	2	b	b	PROPN
ejpam-6686	179	3	b	b	PROPN
ejpam-6686	179	4	b	b	PROPN
ejpam-6686	179	5	c	c	PROPN
ejpam-6686	179	6	0	0	NUM
ejpam-6686	180	1	a	a	DET
ejpam-6686	180	2	b	b	NOUN
ejpam-6686	180	3	c	c	NOUN
ejpam-6686	180	4	c	c	NOUN
ejpam-6686	180	5	c	c	NOUN
ejpam-6686	180	6	m1	m1	PROPN
ejpam-6686	180	7	0	0	PUNCT
ejpam-6686	181	1	a	a	DET
ejpam-6686	181	2	b	b	PROPN
ejpam-6686	181	3	c	c	PROPN
ejpam-6686	181	4	m1	m1	PROPN
ejpam-6686	181	5	m2	m2	PROPN
ejpam-6686	181	6	m2	m2	PROPN
ejpam-6686	181	7	0	0	PROPN
ejpam-6686	181	8	a	a	DET
ejpam-6686	181	9	b	b	PROPN
ejpam-6686	181	10	c	c	PROPN
ejpam-6686	181	11	m1	m1	PROPN
ejpam-6686	181	12	m2	m2	PROPN
ejpam-6686	181	13	∨	∨	ADV
ejpam-6686	181	14	0	0	NUM
ejpam-6686	182	1	a	a	DET
ejpam-6686	182	2	b	b	PROPN
ejpam-6686	182	3	c	c	PROPN
ejpam-6686	182	4	m1	m1	PROPN
ejpam-6686	182	5	m2	m2	PROPN
ejpam-6686	182	6	0	0	NUM
ejpam-6686	182	7	0	0	NUM
ejpam-6686	183	1	a	a	DET
ejpam-6686	183	2	b	b	PROPN
ejpam-6686	183	3	c	c	PROPN
ejpam-6686	183	4	m1	m1	PROPN
ejpam-6686	183	5	m2	m2	PROPN
ejpam-6686	183	6	a	a	DET
ejpam-6686	183	7	a	a	DET
ejpam-6686	183	8	a	a	DET
ejpam-6686	183	9	c	c	NOUN
ejpam-6686	183	10	c	c	NOUN
ejpam-6686	183	11	m1	m1	PROPN
ejpam-6686	183	12	m2	m2	PROPN
ejpam-6686	183	13	b	b	PROPN
ejpam-6686	183	14	b	b	PROPN
ejpam-6686	183	15	c	c	PROPN
ejpam-6686	183	16	b	b	PROPN
ejpam-6686	183	17	c	c	PROPN
ejpam-6686	183	18	m1	m1	PROPN
ejpam-6686	183	19	m2	m2	PROPN
ejpam-6686	183	20	c	c	PROPN
ejpam-6686	183	21	c	c	NOUN
ejpam-6686	183	22	c	c	NOUN
ejpam-6686	183	23	c	c	PROPN
ejpam-6686	183	24	c	c	PROPN
ejpam-6686	183	25	m1	m1	PROPN
ejpam-6686	183	26	m2	m2	PROPN
ejpam-6686	183	27	m1	m1	PROPN
ejpam-6686	183	28	m1	m1	PROPN
ejpam-6686	183	29	m1	m1	PROPN
ejpam-6686	183	30	m1	m1	PROPN
ejpam-6686	183	31	m1	m1	PROPN
ejpam-6686	183	32	m1	m1	PROPN
ejpam-6686	183	33	m1	m1	PROPN
ejpam-6686	183	34	m2	m2	PROPN
ejpam-6686	183	35	m2	m2	PROPN
ejpam-6686	183	36	m2	m2	PROPN
ejpam-6686	183	37	m2	m2	PROPN
ejpam-6686	183	38	m2	m2	PROPN
ejpam-6686	183	39	m2	m2	PROPN
ejpam-6686	183	40	m2	m2	PROPN
ejpam-6686	183	41	take	take	VERB
ejpam-6686	183	42	p	p	NOUN
ejpam-6686	183	43	=	=	X
ejpam-6686	183	44	{	{	PUNCT
ejpam-6686	183	45	a	a	PROPN
ejpam-6686	183	46	,	,	PUNCT
ejpam-6686	183	47	b	b	NOUN
ejpam-6686	183	48	}	}	PUNCT
ejpam-6686	183	49	.	.	PUNCT
ejpam-6686	184	1	then	then	ADV
ejpam-6686	184	2	hp	hp	PROPN
ejpam-6686	184	3	=	=	PUNCT
ejpam-6686	184	4	{	{	PUNCT
ejpam-6686	184	5	0	0	NUM
ejpam-6686	184	6	,	,	PUNCT
ejpam-6686	184	7	a	a	DET
ejpam-6686	184	8	,	,	PUNCT
ejpam-6686	184	9	b	b	NOUN
ejpam-6686	184	10	}	}	PUNCT
ejpam-6686	184	11	is	be	AUX
ejpam-6686	184	12	a	a	DET
ejpam-6686	184	13	prime	prime	ADJ
ejpam-6686	184	14	hierarchy	hierarchy	NOUN
ejpam-6686	184	15	set	set	VERB
ejpam-6686	184	16	in	in	ADP
ejpam-6686	184	17	l	l	NOUN
ejpam-6686	184	18	and	and	CCONJ
ejpam-6686	184	19	l	l	NOUN
ejpam-6686	184	20	\	\	PROPN
ejpam-6686	184	21	hp	hp	PROPN
ejpam-6686	184	22	=	=	PUNCT
ejpam-6686	184	23	{	{	PUNCT
ejpam-6686	184	24	c	c	NOUN
ejpam-6686	184	25	,	,	PUNCT
ejpam-6686	184	26	m1,m2	m1,m2	PROPN
ejpam-6686	184	27	}	}	PUNCT
ejpam-6686	184	28	is	be	AUX
ejpam-6686	184	29	a	a	DET
ejpam-6686	184	30	filter	filter	NOUN
ejpam-6686	184	31	of	of	ADP
ejpam-6686	184	32	l	l	NOUN
ejpam-6686	184	33	,	,	PUNCT
ejpam-6686	184	34	but	but	CCONJ
ejpam-6686	184	35	not	not	PART
ejpam-6686	184	36	prime	prime	ADJ
ejpam-6686	184	37	.	.	PUNCT
ejpam-6686	185	1	4	4	X
ejpam-6686	185	2	.	.	X
ejpam-6686	185	3	inverted	invert	VERB
ejpam-6686	185	4	-	-	PUNCT
ejpam-6686	185	5	hierarchy	hierarchy	NOUN
ejpam-6686	185	6	sets	set	NOUN
ejpam-6686	185	7	in	in	ADP
ejpam-6686	185	8	an	an	DET
ejpam-6686	185	9	almost	almost	ADV
ejpam-6686	185	10	distributive	distributive	ADJ
ejpam-6686	185	11	lattice	lattice	NOUN
ejpam-6686	185	12	with	with	ADP
ejpam-6686	185	13	maximal	maximal	ADJ
ejpam-6686	185	14	elements	element	NOUN
ejpam-6686	185	15	this	this	DET
ejpam-6686	185	16	section	section	NOUN
ejpam-6686	185	17	introduces	introduce	NOUN
ejpam-6686	185	18	and	and	CCONJ
ejpam-6686	185	19	develops	develop	VERB
ejpam-6686	185	20	the	the	DET
ejpam-6686	185	21	theory	theory	NOUN
ejpam-6686	185	22	of	of	ADP
ejpam-6686	185	23	inverted	inverted	ADJ
ejpam-6686	185	24	-	-	PUNCT
ejpam-6686	185	25	hierarchy	hierarchy	NOUN
ejpam-6686	185	26	sets	set	NOUN
ejpam-6686	185	27	,	,	PUNCT
ejpam-6686	185	28	denoted	denote	VERB
ejpam-6686	185	29	by	by	ADP
ejpam-6686	185	30	hs	hs	PROPN
ejpam-6686	185	31	,	,	PUNCT
ejpam-6686	185	32	where	where	SCONJ
ejpam-6686	185	33	s	s	NOUN
ejpam-6686	185	34	is	be	AUX
ejpam-6686	185	35	a	a	DET
ejpam-6686	185	36	non	non	ADJ
ejpam-6686	185	37	-	-	ADJ
ejpam-6686	185	38	empty	empty	ADJ
ejpam-6686	185	39	subset	subset	NOUN
ejpam-6686	185	40	of	of	ADP
ejpam-6686	185	41	an	an	DET
ejpam-6686	185	42	almost	almost	ADV
ejpam-6686	185	43	distributive	distributive	ADJ
ejpam-6686	185	44	lattice	lattice	NOUN
ejpam-6686	185	45	l	l	NOUN
ejpam-6686	185	46	with	with	ADP
ejpam-6686	185	47	maximal	maximal	ADJ
ejpam-6686	185	48	elements	element	NOUN
ejpam-6686	185	49	.	.	PUNCT
ejpam-6686	186	1	these	these	DET
ejpam-6686	186	2	sets	set	NOUN
ejpam-6686	186	3	consist	consist	VERB
ejpam-6686	186	4	of	of	ADP
ejpam-6686	186	5	elements	element	NOUN
ejpam-6686	186	6	in	in	ADP
ejpam-6686	186	7	l	l	NOUN
ejpam-6686	186	8	that	that	PRON
ejpam-6686	186	9	are	be	AUX
ejpam-6686	186	10	idempotent	idempotent	ADJ
ejpam-6686	186	11	over	over	ADP
ejpam-6686	186	12	at	at	ADV
ejpam-6686	186	13	least	least	ADV
ejpam-6686	186	14	one	one	NUM
ejpam-6686	186	15	element	element	NOUN
ejpam-6686	186	16	of	of	ADP
ejpam-6686	186	17	s	s	PRON
ejpam-6686	186	18	under	under	ADP
ejpam-6686	186	19	the	the	DET
ejpam-6686	186	20	join	join	NOUN
ejpam-6686	186	21	operation	operation	NOUN
ejpam-6686	186	22	.	.	PUNCT
ejpam-6686	187	1	such	such	ADJ
ejpam-6686	187	2	sets	set	NOUN
ejpam-6686	187	3	exhibit	exhibit	VERB
ejpam-6686	187	4	rich	rich	ADJ
ejpam-6686	187	5	algebraic	algebraic	ADJ
ejpam-6686	187	6	properties	property	NOUN
ejpam-6686	187	7	,	,	PUNCT
ejpam-6686	187	8	including	include	VERB
ejpam-6686	187	9	closure	closure	NOUN
ejpam-6686	187	10	under	under	ADP
ejpam-6686	187	11	join	join	NOUN
ejpam-6686	187	12	and	and	CCONJ
ejpam-6686	187	13	,	,	PUNCT
ejpam-6686	187	14	under	under	ADP
ejpam-6686	187	15	suitable	suitable	ADJ
ejpam-6686	187	16	conditions	condition	NOUN
ejpam-6686	187	17	,	,	PUNCT
ejpam-6686	187	18	closure	closure	NOUN
ejpam-6686	187	19	under	under	ADP
ejpam-6686	187	20	meet	meet	NOUN
ejpam-6686	187	21	,	,	PUNCT
ejpam-6686	187	22	forming	form	VERB
ejpam-6686	187	23	filters	filter	NOUN
ejpam-6686	187	24	and	and	CCONJ
ejpam-6686	187	25	substructures	substructure	NOUN
ejpam-6686	187	26	of	of	ADP
ejpam-6686	187	27	the	the	DET
ejpam-6686	187	28	lattice	lattice	NOUN
ejpam-6686	187	29	.	.	PUNCT
ejpam-6686	188	1	we	we	PRON
ejpam-6686	188	2	explore	explore	VERB
ejpam-6686	188	3	various	various	ADJ
ejpam-6686	188	4	characterizations	characterization	NOUN
ejpam-6686	188	5	and	and	CCONJ
ejpam-6686	188	6	properties	property	NOUN
ejpam-6686	188	7	of	of	ADP
ejpam-6686	188	8	hs	hs	PROPN
ejpam-6686	188	9	,	,	PUNCT
ejpam-6686	188	10	its	its	PRON
ejpam-6686	188	11	relationship	relationship	NOUN
ejpam-6686	188	12	to	to	ADP
ejpam-6686	188	13	filters	filter	NOUN
ejpam-6686	188	14	,	,	PUNCT
ejpam-6686	188	15	and	and	CCONJ
ejpam-6686	188	16	conditions	condition	NOUN
ejpam-6686	188	17	under	under	ADP
ejpam-6686	188	18	which	which	PRON
ejpam-6686	188	19	it	it	PRON
ejpam-6686	188	20	coincides	coincide	VERB
ejpam-6686	188	21	with	with	ADP
ejpam-6686	188	22	or	or	CCONJ
ejpam-6686	188	23	differs	differ	VERB
ejpam-6686	188	24	from	from	ADP
ejpam-6686	188	25	the	the	DET
ejpam-6686	188	26	filter	filter	NOUN
ejpam-6686	188	27	generated	generate	VERB
ejpam-6686	188	28	by	by	ADP
ejpam-6686	188	29	s.	s.	PROPN
ejpam-6686	188	30	definition	definition	NOUN
ejpam-6686	188	31	4	4	NUM
ejpam-6686	188	32	.	.	PUNCT
ejpam-6686	189	1	an	an	DET
ejpam-6686	189	2	element	element	NOUN
ejpam-6686	189	3	h	h	NOUN
ejpam-6686	189	4	in	in	ADP
ejpam-6686	189	5	l	l	NOUN
ejpam-6686	189	6	with	with	ADP
ejpam-6686	189	7	maximal	maximal	ADJ
ejpam-6686	189	8	elements	element	NOUN
ejpam-6686	189	9	is	be	AUX
ejpam-6686	189	10	said	say	VERB
ejpam-6686	189	11	to	to	PART
ejpam-6686	189	12	be	be	AUX
ejpam-6686	189	13	inverted	invert	VERB
ejpam-6686	189	14	-	-	PUNCT
ejpam-6686	189	15	hierarchy	hierarchy	NOUN
ejpam-6686	189	16	with	with	ADP
ejpam-6686	189	17	respect	respect	NOUN
ejpam-6686	189	18	to	to	ADP
ejpam-6686	189	19	a	a	DET
ejpam-6686	189	20	non	non	ADJ
ejpam-6686	189	21	-	-	ADJ
ejpam-6686	189	22	empty	empty	ADJ
ejpam-6686	189	23	set	set	NOUN
ejpam-6686	189	24	s	s	PROPN
ejpam-6686	189	25	in	in	ADP
ejpam-6686	189	26	l	l	NOUN
ejpam-6686	189	27	,	,	PUNCT
ejpam-6686	189	28	if	if	SCONJ
ejpam-6686	189	29	h	h	PROPN
ejpam-6686	189	30	∨	∨	NOUN
ejpam-6686	189	31	s	s	PART
ejpam-6686	189	32	=	=	ADJ
ejpam-6686	189	33	h	h	NOUN
ejpam-6686	189	34	,	,	PUNCT
ejpam-6686	189	35	for	for	ADP
ejpam-6686	189	36	some	some	DET
ejpam-6686	189	37	s	s	NOUN
ejpam-6686	189	38	∈	∈	PROPN
ejpam-6686	189	39	s.	s.	PROPN
ejpam-6686	189	40	let	let	VERB
ejpam-6686	189	41	us	we	PRON
ejpam-6686	189	42	denote	denote	VERB
ejpam-6686	189	43	hs	hs	PROPN
ejpam-6686	189	44	as	as	ADP
ejpam-6686	189	45	the	the	DET
ejpam-6686	189	46	set	set	NOUN
ejpam-6686	189	47	of	of	ADP
ejpam-6686	189	48	inverted	inverted	ADJ
ejpam-6686	189	49	-	-	PUNCT
ejpam-6686	189	50	hierarchy	hierarchy	NOUN
ejpam-6686	189	51	elements	element	NOUN
ejpam-6686	189	52	with	with	ADP
ejpam-6686	189	53	respect	respect	NOUN
ejpam-6686	189	54	to	to	ADP
ejpam-6686	189	55	a	a	DET
ejpam-6686	189	56	non	non	ADJ
ejpam-6686	189	57	-	-	ADJ
ejpam-6686	189	58	empty	empty	ADJ
ejpam-6686	189	59	set	set	NOUN
ejpam-6686	189	60	s	s	PROPN
ejpam-6686	189	61	in	in	ADP
ejpam-6686	189	62	l.	l.	PROPN
ejpam-6686	189	63	then	then	ADV
ejpam-6686	189	64	it	it	PRON
ejpam-6686	189	65	is	be	AUX
ejpam-6686	189	66	easy	easy	ADJ
ejpam-6686	189	67	to	to	PART
ejpam-6686	189	68	observe	observe	VERB
ejpam-6686	189	69	that	that	SCONJ
ejpam-6686	189	70	hs	hs	PROPN
ejpam-6686	189	71	̸=	̸=	PROPN
ejpam-6686	189	72	∅	∅	NOUN
ejpam-6686	189	73	(	(	PUNCT
ejpam-6686	189	74	because	because	SCONJ
ejpam-6686	189	75	m	m	PROPN
ejpam-6686	189	76	∨	∨	NUM
ejpam-6686	189	77	s	s	PART
ejpam-6686	189	78	=	=	NOUN
ejpam-6686	189	79	m	m	PROPN
ejpam-6686	189	80	,	,	PUNCT
ejpam-6686	189	81	for	for	ADP
ejpam-6686	189	82	all	all	DET
ejpam-6686	189	83	s	s	PART
ejpam-6686	189	84	∈	∈	PROPN
ejpam-6686	189	85	s	s	NOUN
ejpam-6686	189	86	,	,	PUNCT
ejpam-6686	189	87	where	where	SCONJ
ejpam-6686	189	88	m	m	NOUN
ejpam-6686	189	89	is	be	AUX
ejpam-6686	189	90	a	a	DET
ejpam-6686	189	91	maximal	maximal	ADJ
ejpam-6686	189	92	element	element	NOUN
ejpam-6686	189	93	in	in	ADP
ejpam-6686	189	94	l	l	NOUN
ejpam-6686	189	95	)	)	PUNCT
ejpam-6686	189	96	and	and	CCONJ
ejpam-6686	189	97	s	s	VERB
ejpam-6686	189	98	⊆	⊆	NUM
ejpam-6686	189	99	hs	hs	X
ejpam-6686	189	100	.	.	PROPN
ejpam-6686	189	101	theorem	theorem	VERB
ejpam-6686	189	102	7	7	NUM
ejpam-6686	189	103	.	.	X
ejpam-6686	189	104	for	for	ADP
ejpam-6686	189	105	any	any	DET
ejpam-6686	189	106	non	non	ADJ
ejpam-6686	189	107	-	-	ADJ
ejpam-6686	189	108	empty	empty	ADJ
ejpam-6686	189	109	subset	subset	NOUN
ejpam-6686	189	110	s	s	PROPN
ejpam-6686	189	111	of	of	ADP
ejpam-6686	189	112	l	l	NOUN
ejpam-6686	189	113	,	,	PUNCT
ejpam-6686	189	114	we	we	PRON
ejpam-6686	189	115	have	have	AUX
ejpam-6686	189	116	(	(	PUNCT
ejpam-6686	189	117	i	i	NOUN
ejpam-6686	189	118	)	)	PUNCT
ejpam-6686	190	1	s	s	VERB
ejpam-6686	190	2	is	be	AUX
ejpam-6686	190	3	closed	close	VERB
ejpam-6686	190	4	under	under	ADP
ejpam-6686	190	5	∨	∨	PROPN
ejpam-6686	190	6	(	(	PUNCT
ejpam-6686	190	7	ii	ii	NOUN
ejpam-6686	190	8	)	)	PUNCT
ejpam-6686	190	9	for	for	ADP
ejpam-6686	190	10	any	any	DET
ejpam-6686	190	11	h	h	NOUN
ejpam-6686	190	12	∈	∈	PROPN
ejpam-6686	190	13	hs	hs	INTJ
ejpam-6686	190	14	,	,	PUNCT
ejpam-6686	191	1	[	[	X
ejpam-6686	191	2	h	h	X
ejpam-6686	191	3	)	)	PUNCT
ejpam-6686	191	4	⊆	⊆	NUM
ejpam-6686	191	5	hs	hs	PROPN
ejpam-6686	191	6	g.	g.	PROPN
ejpam-6686	191	7	chinnayya	chinnayya	PROPN
ejpam-6686	191	8	et	et	PROPN
ejpam-6686	191	9	al	al	PROPN
ejpam-6686	191	10	.	.	PUNCT
ejpam-6686	191	11	/	/	SYM
ejpam-6686	191	12	eur	eur	PROPN
ejpam-6686	191	13	.	.	PUNCT
ejpam-6686	192	1	j.	j.	PROPN
ejpam-6686	192	2	pure	pure	PROPN
ejpam-6686	192	3	appl	appl	PROPN
ejpam-6686	192	4	.	.	PROPN
ejpam-6686	192	5	math	math	PROPN
ejpam-6686	192	6	,	,	PUNCT
ejpam-6686	192	7	18	18	NUM
ejpam-6686	192	8	(	(	PUNCT
ejpam-6686	192	9	4	4	NUM
ejpam-6686	192	10	)	)	PUNCT
ejpam-6686	192	11	(	(	PUNCT
ejpam-6686	192	12	2025	2025	NUM
ejpam-6686	192	13	)	)	PUNCT
ejpam-6686	192	14	,	,	PUNCT
ejpam-6686	192	15	6686	6686	NUM
ejpam-6686	192	16	8	8	NUM
ejpam-6686	192	17	of	of	ADP
ejpam-6686	192	18	12	12	NUM
ejpam-6686	192	19	(	(	PUNCT
ejpam-6686	192	20	iii	iii	NOUN
ejpam-6686	192	21	)	)	PUNCT
ejpam-6686	192	22	for	for	ADP
ejpam-6686	192	23	any	any	DET
ejpam-6686	192	24	h	h	NOUN
ejpam-6686	192	25	∈	∈	PROPN
ejpam-6686	192	26	hs	hs	PROPN
ejpam-6686	192	27	and	and	CCONJ
ejpam-6686	192	28	a	a	DET
ejpam-6686	192	29	∈	∈	PROPN
ejpam-6686	192	30	l	l	NOUN
ejpam-6686	192	31	,	,	PUNCT
ejpam-6686	192	32	a	a	DET
ejpam-6686	192	33	∨	∨	NUM
ejpam-6686	192	34	h	h	NOUN
ejpam-6686	192	35	,	,	PUNCT
ejpam-6686	192	36	h	h	PROPN
ejpam-6686	192	37	∨	∨	NOUN
ejpam-6686	192	38	a	a	DET
ejpam-6686	192	39	∈	∈	PROPN
ejpam-6686	192	40	hs	hs	X
ejpam-6686	192	41	(	(	PUNCT
ejpam-6686	192	42	iv	iv	X
ejpam-6686	192	43	)	)	PUNCT
ejpam-6686	192	44	if	if	SCONJ
ejpam-6686	192	45	a	a	DET
ejpam-6686	192	46	∈	∈	PROPN
ejpam-6686	192	47	l	l	NOUN
ejpam-6686	192	48	and	and	CCONJ
ejpam-6686	192	49	s	s	PROPN
ejpam-6686	192	50	∈	∈	NOUN
ejpam-6686	192	51	s	s	VERB
ejpam-6686	192	52	such	such	ADJ
ejpam-6686	192	53	that	that	PRON
ejpam-6686	192	54	s	s	VERB
ejpam-6686	192	55	≤	≤	NOUN
ejpam-6686	192	56	a	a	PRON
ejpam-6686	192	57	,	,	PUNCT
ejpam-6686	192	58	then	then	ADV
ejpam-6686	192	59	a	a	DET
ejpam-6686	192	60	∈	∈	PROPN
ejpam-6686	192	61	hs	hs	PROPN
ejpam-6686	192	62	.	.	PROPN
ejpam-6686	192	63	proof	proof	NOUN
ejpam-6686	192	64	.	.	PUNCT
ejpam-6686	193	1	(	(	PUNCT
ejpam-6686	193	2	i	i	NOUN
ejpam-6686	193	3	)	)	PUNCT
ejpam-6686	193	4	let	let	VERB
ejpam-6686	193	5	h1	h1	PROPN
ejpam-6686	193	6	,	,	PUNCT
ejpam-6686	193	7	h2	h2	PROPN
ejpam-6686	193	8	∈	∈	PROPN
ejpam-6686	194	1	hs	hs	PROPN
ejpam-6686	194	2	.	.	PUNCT
ejpam-6686	195	1	then	then	ADV
ejpam-6686	195	2	h1	h1	VERB
ejpam-6686	195	3	∨	∨	NOUN
ejpam-6686	195	4	s1	s1	NOUN
ejpam-6686	195	5	=	=	PUNCT
ejpam-6686	195	6	h1	h1	NOUN
ejpam-6686	195	7	and	and	CCONJ
ejpam-6686	195	8	h2	h2	PROPN
ejpam-6686	195	9	∨	∨	NUM
ejpam-6686	195	10	s2	s2	PROPN
ejpam-6686	195	11	=	=	SYM
ejpam-6686	195	12	h2	h2	NOUN
ejpam-6686	195	13	,	,	PUNCT
ejpam-6686	195	14	for	for	ADP
ejpam-6686	195	15	some	some	DET
ejpam-6686	195	16	s1	s1	NOUN
ejpam-6686	195	17	,	,	PUNCT
ejpam-6686	195	18	s2	s2	PROPN
ejpam-6686	195	19	∈	∈	PROPN
ejpam-6686	195	20	s.	s.	PROPN
ejpam-6686	195	21	now	now	ADV
ejpam-6686	195	22	,	,	PUNCT
ejpam-6686	195	23	(	(	PUNCT
ejpam-6686	195	24	h1	h1	PROPN
ejpam-6686	195	25	∨	∨	NUM
ejpam-6686	195	26	h2	h2	NOUN
ejpam-6686	195	27	)	)	PUNCT
ejpam-6686	195	28	∨	∨	NUM
ejpam-6686	195	29	s2	s2	NOUN
ejpam-6686	195	30	=	=	PUNCT
ejpam-6686	195	31	h1	h1	PROPN
ejpam-6686	195	32	∨	∨	NUM
ejpam-6686	195	33	(	(	PUNCT
ejpam-6686	195	34	h2	h2	PROPN
ejpam-6686	195	35	∨	∨	NUM
ejpam-6686	195	36	s2	s2	PROPN
ejpam-6686	195	37	)	)	PUNCT
ejpam-6686	195	38	=	=	PUNCT
ejpam-6686	195	39	h1	h1	PROPN
ejpam-6686	195	40	∨	∨	NUM
ejpam-6686	195	41	h2	h2	NOUN
ejpam-6686	195	42	.	.	PUNCT
ejpam-6686	196	1	then	then	ADV
ejpam-6686	196	2	h1	h1	VERB
ejpam-6686	196	3	∨	∨	PROPN
ejpam-6686	196	4	h2	h2	PROPN
ejpam-6686	196	5	∈	∈	PROPN
ejpam-6686	196	6	hs	hs	PROPN
ejpam-6686	196	7	.	.	PUNCT
ejpam-6686	197	1	therefore	therefore	ADV
ejpam-6686	197	2	,	,	PUNCT
ejpam-6686	197	3	hs	hs	PROPN
ejpam-6686	197	4	is	be	AUX
ejpam-6686	197	5	closed	close	VERB
ejpam-6686	197	6	under	under	ADP
ejpam-6686	197	7	∨.	∨.	NOUN
ejpam-6686	197	8	(	(	PUNCT
ejpam-6686	197	9	ii	ii	NOUN
ejpam-6686	197	10	)	)	PUNCT
ejpam-6686	197	11	let	let	VERB
ejpam-6686	197	12	h	h	PROPN
ejpam-6686	197	13	∈	∈	PROPN
ejpam-6686	198	1	hs	hs	PROPN
ejpam-6686	198	2	.	.	PUNCT
ejpam-6686	199	1	then	then	ADV
ejpam-6686	199	2	h	h	PROPN
ejpam-6686	199	3	∨	∨	PROPN
ejpam-6686	199	4	s	s	PART
ejpam-6686	199	5	=	=	ADJ
ejpam-6686	199	6	h	h	NOUN
ejpam-6686	199	7	,	,	PUNCT
ejpam-6686	199	8	for	for	ADP
ejpam-6686	199	9	some	some	DET
ejpam-6686	199	10	s	s	NOUN
ejpam-6686	199	11	∈	∈	PROPN
ejpam-6686	199	12	s.	s.	PROPN
ejpam-6686	199	13	let	let	VERB
ejpam-6686	199	14	a	a	DET
ejpam-6686	199	15	∈	∈	NOUN
ejpam-6686	200	1	[	[	X
ejpam-6686	200	2	h	h	NOUN
ejpam-6686	200	3	)	)	PUNCT
ejpam-6686	200	4	.	.	PUNCT
ejpam-6686	201	1	then	then	ADV
ejpam-6686	201	2	a	a	DET
ejpam-6686	201	3	=	=	SYM
ejpam-6686	201	4	b	b	PROPN
ejpam-6686	201	5	∨	∨	NUM
ejpam-6686	201	6	h	h	NOUN
ejpam-6686	201	7	,	,	PUNCT
ejpam-6686	201	8	for	for	ADP
ejpam-6686	201	9	some	some	DET
ejpam-6686	201	10	b	b	PROPN
ejpam-6686	201	11	∈	∈	PROPN
ejpam-6686	201	12	l.	l.	NOUN
ejpam-6686	201	13	now	now	ADV
ejpam-6686	201	14	,	,	PUNCT
ejpam-6686	201	15	a	a	DET
ejpam-6686	201	16	∨	∨	NUM
ejpam-6686	201	17	s	s	PART
ejpam-6686	201	18	=	=	SYM
ejpam-6686	201	19	(	(	PUNCT
ejpam-6686	201	20	b	b	PROPN
ejpam-6686	201	21	∨	∨	NUM
ejpam-6686	201	22	h	h	NOUN
ejpam-6686	201	23	)	)	PUNCT
ejpam-6686	201	24	∨	∨	PROPN
ejpam-6686	201	25	s	s	PART
ejpam-6686	201	26	=	=	SYM
ejpam-6686	201	27	b	b	PROPN
ejpam-6686	201	28	∨	∨	X
ejpam-6686	201	29	(	(	PUNCT
ejpam-6686	201	30	h	h	PROPN
ejpam-6686	201	31	∨	∨	NUM
ejpam-6686	201	32	s	s	PROPN
ejpam-6686	201	33	)	)	PUNCT
ejpam-6686	201	34	.	.	PUNCT
ejpam-6686	202	1	then	then	ADV
ejpam-6686	202	2	a	a	DET
ejpam-6686	202	3	∈	∈	PROPN
ejpam-6686	202	4	hs	hs	INTJ
ejpam-6686	202	5	.	.	PUNCT
ejpam-6686	203	1	therefore	therefore	ADV
ejpam-6686	203	2	,	,	PUNCT
ejpam-6686	203	3	[	[	X
ejpam-6686	203	4	h	h	X
ejpam-6686	203	5	)	)	PUNCT
ejpam-6686	203	6	⊆	⊆	NUM
ejpam-6686	203	7	hs	hs	X
ejpam-6686	203	8	.	.	PUNCT
ejpam-6686	204	1	(	(	PUNCT
ejpam-6686	204	2	iii	iii	X
ejpam-6686	204	3	)	)	PUNCT
ejpam-6686	204	4	let	let	VERB
ejpam-6686	204	5	h	h	PROPN
ejpam-6686	204	6	∈	∈	PROPN
ejpam-6686	205	1	hs	hs	PROPN
ejpam-6686	205	2	.	.	PUNCT
ejpam-6686	206	1	then	then	ADV
ejpam-6686	206	2	h∨s	h∨s	NOUN
ejpam-6686	206	3	=	=	SYM
ejpam-6686	206	4	s	s	PROPN
ejpam-6686	206	5	,	,	PUNCT
ejpam-6686	206	6	for	for	ADP
ejpam-6686	206	7	some	some	DET
ejpam-6686	206	8	s	s	PROPN
ejpam-6686	206	9	∈	∈	PROPN
ejpam-6686	206	10	s.	s.	PROPN
ejpam-6686	206	11	given	give	VERB
ejpam-6686	206	12	a	a	DET
ejpam-6686	206	13	∈	∈	PROPN
ejpam-6686	206	14	l	l	NOUN
ejpam-6686	206	15	,	,	PUNCT
ejpam-6686	206	16	(	(	PUNCT
ejpam-6686	206	17	a∨h)∨s	a∨h)∨s	PUNCT
ejpam-6686	206	18	=	=	SYM
ejpam-6686	206	19	a∨(h∨s	a∨(h∨s	NOUN
ejpam-6686	206	20	)	)	PUNCT
ejpam-6686	206	21	=	=	PUNCT
ejpam-6686	206	22	a	a	DET
ejpam-6686	206	23	∨	∨	NUM
ejpam-6686	206	24	h.	h.	PROPN
ejpam-6686	206	25	therefore	therefore	ADV
ejpam-6686	206	26	,	,	PUNCT
ejpam-6686	206	27	a	a	DET
ejpam-6686	206	28	∨	∨	NUM
ejpam-6686	206	29	h	h	NOUN
ejpam-6686	206	30	∈	∈	PROPN
ejpam-6686	206	31	hs	hs	PROPN
ejpam-6686	206	32	.	.	PUNCT
ejpam-6686	207	1	similarly	similarly	ADV
ejpam-6686	207	2	,	,	PUNCT
ejpam-6686	207	3	(	(	PUNCT
ejpam-6686	207	4	h	h	NOUN
ejpam-6686	207	5	∨	∨	NUM
ejpam-6686	207	6	a	a	PRON
ejpam-6686	207	7	)	)	PUNCT
ejpam-6686	207	8	∧	∧	PROPN
ejpam-6686	207	9	s	s	PART
ejpam-6686	207	10	=	=	PUNCT
ejpam-6686	207	11	(	(	PUNCT
ejpam-6686	207	12	a	a	DET
ejpam-6686	207	13	∨	∨	NUM
ejpam-6686	207	14	h	h	NOUN
ejpam-6686	207	15	)	)	PUNCT
ejpam-6686	207	16	∧	∧	PROPN
ejpam-6686	207	17	s	s	PART
ejpam-6686	207	18	=	=	PUNCT
ejpam-6686	207	19	(	(	PUNCT
ejpam-6686	207	20	a	a	DET
ejpam-6686	207	21	∧	∧	PROPN
ejpam-6686	207	22	s	s	PART
ejpam-6686	207	23	)	)	PUNCT
ejpam-6686	207	24	∨	∨	NOUN
ejpam-6686	207	25	(	(	PUNCT
ejpam-6686	207	26	h	h	NOUN
ejpam-6686	207	27	∧	∧	PROPN
ejpam-6686	207	28	s	s	PART
ejpam-6686	207	29	)	)	PUNCT
ejpam-6686	207	30	=	=	SYM
ejpam-6686	207	31	(	(	PUNCT
ejpam-6686	207	32	a	a	DET
ejpam-6686	207	33	∧	∧	PROPN
ejpam-6686	207	34	s	s	PART
ejpam-6686	207	35	)	)	PUNCT
ejpam-6686	207	36	∨	∨	PROPN
ejpam-6686	207	37	s	s	PART
ejpam-6686	207	38	=	=	PUNCT
ejpam-6686	207	39	s.	s.	PROPN
ejpam-6686	207	40	therefore	therefore	ADV
ejpam-6686	207	41	,	,	PUNCT
ejpam-6686	207	42	(	(	PUNCT
ejpam-6686	207	43	h	h	NOUN
ejpam-6686	207	44	∨	∨	PROPN
ejpam-6686	207	45	a	a	PRON
ejpam-6686	207	46	)	)	PUNCT
ejpam-6686	207	47	∨	∨	PROPN
ejpam-6686	207	48	s	s	PART
ejpam-6686	207	49	=	=	SYM
ejpam-6686	207	50	h	h	NOUN
ejpam-6686	207	51	∨	∨	NUM
ejpam-6686	207	52	a	a	PRON
ejpam-6686	207	53	and	and	CCONJ
ejpam-6686	207	54	hence	hence	ADV
ejpam-6686	207	55	h	h	NOUN
ejpam-6686	207	56	∨	∨	NUM
ejpam-6686	207	57	a	a	DET
ejpam-6686	207	58	∈	∈	PROPN
ejpam-6686	207	59	hs	hs	INTJ
ejpam-6686	207	60	.	.	PUNCT
ejpam-6686	208	1	(	(	PUNCT
ejpam-6686	208	2	iv	iv	X
ejpam-6686	208	3	)	)	PUNCT
ejpam-6686	208	4	let	let	VERB
ejpam-6686	208	5	a	a	DET
ejpam-6686	208	6	∈	∈	PROPN
ejpam-6686	208	7	l	l	NOUN
ejpam-6686	208	8	and	and	CCONJ
ejpam-6686	208	9	s	s	PROPN
ejpam-6686	208	10	∈	∈	NOUN
ejpam-6686	208	11	s	s	VERB
ejpam-6686	208	12	such	such	ADJ
ejpam-6686	208	13	that	that	PRON
ejpam-6686	208	14	s	s	VERB
ejpam-6686	208	15	≤	≤	NUM
ejpam-6686	208	16	a.	a.	NOUN
ejpam-6686	208	17	then	then	ADV
ejpam-6686	208	18	a	a	DET
ejpam-6686	208	19	∨	∨	PROPN
ejpam-6686	208	20	s	s	PART
ejpam-6686	208	21	=	=	NOUN
ejpam-6686	208	22	a.	a.	NOUN
ejpam-6686	208	23	therefore	therefore	ADV
ejpam-6686	208	24	,	,	PUNCT
ejpam-6686	208	25	a	a	DET
ejpam-6686	208	26	∈	∈	PROPN
ejpam-6686	208	27	hs	hs	INTJ
ejpam-6686	208	28	.	.	PUNCT
ejpam-6686	209	1	lemma	lemma	PROPN
ejpam-6686	209	2	4	4	X
ejpam-6686	209	3	.	.	PUNCT
ejpam-6686	210	1	if	if	SCONJ
ejpam-6686	210	2	s1	s1	NOUN
ejpam-6686	210	3	,	,	PUNCT
ejpam-6686	210	4	s2	s2	PROPN
ejpam-6686	210	5	are	be	AUX
ejpam-6686	210	6	any	any	DET
ejpam-6686	210	7	two	two	NUM
ejpam-6686	210	8	non	non	ADJ
ejpam-6686	210	9	-	-	ADJ
ejpam-6686	210	10	empty	empty	ADJ
ejpam-6686	210	11	subsets	subset	NOUN
ejpam-6686	210	12	of	of	ADP
ejpam-6686	210	13	l	l	NOUN
ejpam-6686	210	14	,	,	PUNCT
ejpam-6686	210	15	then	then	ADV
ejpam-6686	210	16	(	(	PUNCT
ejpam-6686	210	17	i	i	NOUN
ejpam-6686	210	18	)	)	PUNCT
ejpam-6686	211	1	s1	s1	PROPN
ejpam-6686	211	2	⊆	⊆	NUM
ejpam-6686	211	3	s2	s2	PROPN
ejpam-6686	211	4	implies	imply	VERB
ejpam-6686	211	5	hs1	hs1	PROPN
ejpam-6686	211	6	⊆	⊆	NUM
ejpam-6686	211	7	hs2	hs2	PROPN
ejpam-6686	211	8	(	(	PUNCT
ejpam-6686	211	9	ii	ii	NOUN
ejpam-6686	211	10	)	)	PUNCT
ejpam-6686	211	11	hs1	hs1	PROPN
ejpam-6686	211	12	∪hs2	∪hs2	PROPN
ejpam-6686	212	1	=	=	SYM
ejpam-6686	212	2	hs1∪s2	hs1∪s2	PROPN
ejpam-6686	212	3	(	(	PUNCT
ejpam-6686	212	4	iii	iii	NOUN
ejpam-6686	212	5	)	)	PUNCT
ejpam-6686	212	6	hs1∩s2	hs1∩s2	PROPN
ejpam-6686	212	7	⊆	⊆	NUM
ejpam-6686	212	8	hs1	hs1	PROPN
ejpam-6686	212	9	∩hs2	∩hs2	PROPN
ejpam-6686	212	10	.	.	PUNCT
ejpam-6686	212	11	proof	proof	NOUN
ejpam-6686	212	12	.	.	PUNCT
ejpam-6686	213	1	let	let	VERB
ejpam-6686	213	2	s1	s1	NOUN
ejpam-6686	213	3	,	,	PUNCT
ejpam-6686	213	4	s2	s2	X
ejpam-6686	213	5	be	be	AUX
ejpam-6686	213	6	two	two	NUM
ejpam-6686	213	7	non	non	ADJ
ejpam-6686	213	8	-	-	ADJ
ejpam-6686	213	9	empty	empty	ADJ
ejpam-6686	213	10	subsets	subset	NOUN
ejpam-6686	213	11	in	in	ADP
ejpam-6686	213	12	l.	l.	PROPN
ejpam-6686	213	13	(	(	PUNCT
ejpam-6686	213	14	i	i	NOUN
ejpam-6686	213	15	)	)	PUNCT
ejpam-6686	213	16	let	let	VERB
ejpam-6686	213	17	h	h	NOUN
ejpam-6686	213	18	∈	∈	PROPN
ejpam-6686	213	19	hs1	hs1	PROPN
ejpam-6686	213	20	.	.	PUNCT
ejpam-6686	214	1	then	then	ADV
ejpam-6686	214	2	h	h	PROPN
ejpam-6686	214	3	∨	∨	NOUN
ejpam-6686	214	4	s1	s1	PROPN
ejpam-6686	214	5	=	=	SYM
ejpam-6686	214	6	h	h	NOUN
ejpam-6686	214	7	,	,	PUNCT
ejpam-6686	214	8	for	for	ADP
ejpam-6686	214	9	some	some	DET
ejpam-6686	214	10	s1	s1	PROPN
ejpam-6686	214	11	∈	∈	PROPN
ejpam-6686	214	12	s1	s1	NOUN
ejpam-6686	214	13	⊆	⊆	NUM
ejpam-6686	214	14	s2	s2	PROPN
ejpam-6686	214	15	.	.	PUNCT
ejpam-6686	215	1	therefore	therefore	ADV
ejpam-6686	215	2	,	,	PUNCT
ejpam-6686	215	3	h	h	PROPN
ejpam-6686	215	4	∈	∈	PROPN
ejpam-6686	215	5	hs2	hs2	NOUN
ejpam-6686	215	6	and	and	CCONJ
ejpam-6686	215	7	hence	hence	ADV
ejpam-6686	215	8	hs1	hs1	PROPN
ejpam-6686	215	9	⊆	⊆	NUM
ejpam-6686	215	10	hs2	hs2	NOUN
ejpam-6686	215	11	.	.	PUNCT
ejpam-6686	216	1	(	(	PUNCT
ejpam-6686	216	2	ii	ii	NOUN
ejpam-6686	216	3	)	)	PUNCT
ejpam-6686	216	4	by	by	ADP
ejpam-6686	216	5	(	(	PUNCT
ejpam-6686	216	6	i	i	NOUN
ejpam-6686	216	7	)	)	PUNCT
ejpam-6686	216	8	,	,	PUNCT
ejpam-6686	216	9	we	we	PRON
ejpam-6686	216	10	have	have	VERB
ejpam-6686	216	11	hs1	hs1	PROPN
ejpam-6686	216	12	,	,	PUNCT
ejpam-6686	216	13	hs2	hs2	PROPN
ejpam-6686	216	14	⊆	⊆	NUM
ejpam-6686	216	15	hs1∪s2	hs1∪s2	ADV
ejpam-6686	216	16	.	.	PUNCT
ejpam-6686	217	1	therefore	therefore	ADV
ejpam-6686	217	2	,	,	PUNCT
ejpam-6686	217	3	hs1	hs1	PROPN
ejpam-6686	217	4	∪	∪	VERB
ejpam-6686	217	5	hs2	hs2	PROPN
ejpam-6686	217	6	⊆	⊆	NUM
ejpam-6686	217	7	hs1∪s2	hs1∪s2	ADV
ejpam-6686	217	8	.	.	PUNCT
ejpam-6686	218	1	let	let	VERB
ejpam-6686	218	2	h	h	NOUN
ejpam-6686	218	3	∈	∈	PROPN
ejpam-6686	218	4	hs1∪s2	hs1∪s2	ADV
ejpam-6686	218	5	.	.	PUNCT
ejpam-6686	219	1	then	then	ADV
ejpam-6686	219	2	h	h	PROPN
ejpam-6686	219	3	∨	∨	PROPN
ejpam-6686	219	4	s	s	PART
ejpam-6686	219	5	=	=	X
ejpam-6686	219	6	h	h	NOUN
ejpam-6686	219	7	for	for	ADP
ejpam-6686	219	8	some	some	DET
ejpam-6686	219	9	s	s	PART
ejpam-6686	219	10	∈	∈	PROPN
ejpam-6686	219	11	s1	s1	NOUN
ejpam-6686	219	12	∪	∪	X
ejpam-6686	219	13	s2	s2	NOUN
ejpam-6686	219	14	⊆	⊆	NUM
ejpam-6686	219	15	hs1	hs1	PROPN
ejpam-6686	219	16	∪	∪	X
ejpam-6686	219	17	hs2	hs2	PROPN
ejpam-6686	219	18	.	.	PUNCT
ejpam-6686	220	1	by	by	ADP
ejpam-6686	220	2	theorem	theorem	ADJ
ejpam-6686	220	3	7	7	NUM
ejpam-6686	220	4	(	(	PUNCT
ejpam-6686	220	5	iii	iii	NOUN
ejpam-6686	220	6	)	)	PUNCT
ejpam-6686	220	7	,	,	PUNCT
ejpam-6686	220	8	h	h	NOUN
ejpam-6686	220	9	=	=	SYM
ejpam-6686	220	10	h	h	PROPN
ejpam-6686	220	11	∨	∨	NOUN
ejpam-6686	220	12	shs1	shs1	PROPN
ejpam-6686	220	13	∪hs2	∪hs2	PROPN
ejpam-6686	220	14	and	and	CCONJ
ejpam-6686	220	15	hence	hence	ADV
ejpam-6686	220	16	hs1∪s2	hs1∪s2	PROPN
ejpam-6686	220	17	⊆	⊆	NUM
ejpam-6686	220	18	hs1	hs1	PROPN
ejpam-6686	220	19	∪hs2	∪hs2	PROPN
ejpam-6686	220	20	.	.	PUNCT
ejpam-6686	221	1	thus	thus	ADV
ejpam-6686	221	2	,	,	PUNCT
ejpam-6686	221	3	hs1	hs1	PROPN
ejpam-6686	221	4	∪hs2	∪hs2	X
ejpam-6686	221	5	=	=	PUNCT
ejpam-6686	221	6	hs1∪s2	hs1∪s2	PROPN
ejpam-6686	221	7	.	.	PUNCT
ejpam-6686	222	1	(	(	PUNCT
ejpam-6686	222	2	iii	iii	X
ejpam-6686	222	3	)	)	PUNCT
ejpam-6686	222	4	we	we	PRON
ejpam-6686	222	5	have	have	VERB
ejpam-6686	222	6	s1	s1	NOUN
ejpam-6686	222	7	∩	∩	NOUN
ejpam-6686	222	8	s2	s2	NOUN
ejpam-6686	222	9	⊆	⊆	NUM
ejpam-6686	222	10	s1	s1	NOUN
ejpam-6686	222	11	,	,	PUNCT
ejpam-6686	222	12	s2	s2	PROPN
ejpam-6686	222	13	.	.	PUNCT
ejpam-6686	223	1	then	then	ADV
ejpam-6686	223	2	hs1∩s2	hs1∩s2	PROPN
ejpam-6686	223	3	,	,	PUNCT
ejpam-6686	223	4	⊆	⊆	NUM
ejpam-6686	223	5	hs1	hs1	PROPN
ejpam-6686	223	6	,	,	PUNCT
ejpam-6686	223	7	hs2	hs2	PROPN
ejpam-6686	223	8	(	(	PUNCT
ejpam-6686	223	9	by	by	ADP
ejpam-6686	223	10	(	(	PUNCT
ejpam-6686	223	11	ii	ii	NOUN
ejpam-6686	223	12	)	)	PUNCT
ejpam-6686	223	13	)	)	PUNCT
ejpam-6686	223	14	.	.	PUNCT
ejpam-6686	224	1	therefore	therefore	ADV
ejpam-6686	224	2	,	,	PUNCT
ejpam-6686	224	3	hs1∩s2	hs1∩s2	PROPN
ejpam-6686	224	4	⊆	⊆	NUM
ejpam-6686	224	5	hs1	hs1	NOUN
ejpam-6686	224	6	∩hs2	∩hs2	PROPN
ejpam-6686	224	7	.	.	PUNCT
ejpam-6686	225	1	remark	remark	VERB
ejpam-6686	225	2	4	4	NUM
ejpam-6686	225	3	.	.	PUNCT
ejpam-6686	226	1	hs1∩s2	hs1∩s2	NOUN
ejpam-6686	226	2	need	need	AUX
ejpam-6686	226	3	not	not	PART
ejpam-6686	226	4	be	be	AUX
ejpam-6686	226	5	equal	equal	ADJ
ejpam-6686	226	6	to	to	ADP
ejpam-6686	226	7	hs1	hs1	PROPN
ejpam-6686	226	8	∩hs2	∩hs2	PROPN
ejpam-6686	226	9	.	.	PUNCT
ejpam-6686	227	1	for	for	ADP
ejpam-6686	227	2	,	,	PUNCT
ejpam-6686	227	3	in	in	ADP
ejpam-6686	227	4	example	example	NOUN
ejpam-6686	227	5	(	(	PUNCT
ejpam-6686	227	6	1	1	NUM
ejpam-6686	227	7	)	)	PUNCT
ejpam-6686	227	8	;	;	PUNCT
ejpam-6686	227	9	let	let	VERB
ejpam-6686	227	10	s1	s1	PROPN
ejpam-6686	227	11	=	=	PUNCT
ejpam-6686	227	12	{	{	PUNCT
ejpam-6686	227	13	a	a	DET
ejpam-6686	227	14	,	,	PUNCT
ejpam-6686	227	15	b	b	NOUN
ejpam-6686	227	16	}	}	PUNCT
ejpam-6686	227	17	and	and	CCONJ
ejpam-6686	227	18	s2	s2	VERB
ejpam-6686	227	19	=	=	SYM
ejpam-6686	227	20	{	{	PUNCT
ejpam-6686	227	21	0	0	NUM
ejpam-6686	227	22	,	,	PUNCT
ejpam-6686	227	23	a	a	PRON
ejpam-6686	227	24	}	}	PUNCT
ejpam-6686	227	25	.	.	PUNCT
ejpam-6686	228	1	then	then	ADV
ejpam-6686	228	2	s1	s1	PROPN
ejpam-6686	228	3	∩s2	∩s2	PROPN
ejpam-6686	228	4	=	=	X
ejpam-6686	228	5	{	{	PUNCT
ejpam-6686	228	6	a	a	NOUN
ejpam-6686	228	7	}	}	PUNCT
ejpam-6686	228	8	,	,	PUNCT
ejpam-6686	228	9	hs1	hs1	X
ejpam-6686	228	10	=	=	PUNCT
ejpam-6686	228	11	{	{	PUNCT
ejpam-6686	228	12	a	a	DET
ejpam-6686	228	13	,	,	PUNCT
ejpam-6686	228	14	b	b	NOUN
ejpam-6686	228	15	,	,	PUNCT
ejpam-6686	228	16	1	1	NUM
ejpam-6686	228	17	}	}	PUNCT
ejpam-6686	228	18	,	,	PUNCT
ejpam-6686	228	19	hs2	hs2	NOUN
ejpam-6686	228	20	=	=	SYM
ejpam-6686	228	21	{	{	PUNCT
ejpam-6686	228	22	0	0	NUM
ejpam-6686	228	23	,	,	PUNCT
ejpam-6686	228	24	a	a	DET
ejpam-6686	228	25	,	,	PUNCT
ejpam-6686	228	26	b	b	NOUN
ejpam-6686	228	27	,	,	PUNCT
ejpam-6686	228	28	1	1	NUM
ejpam-6686	228	29	}	}	PUNCT
ejpam-6686	228	30	,	,	PUNCT
ejpam-6686	228	31	hs1∩s2	hs1∩s2	PROPN
ejpam-6686	228	32	=	=	PUNCT
ejpam-6686	228	33	{	{	PUNCT
ejpam-6686	228	34	a	a	PRON
ejpam-6686	228	35	,	,	PUNCT
ejpam-6686	228	36	1	1	NUM
ejpam-6686	228	37	}	}	PUNCT
ejpam-6686	228	38	and	and	CCONJ
ejpam-6686	228	39	hs1	hs1	PROPN
ejpam-6686	228	40	∩hs2	∩hs2	PROPN
ejpam-6686	228	41	=	=	PUNCT
ejpam-6686	228	42	{	{	PUNCT
ejpam-6686	228	43	a	a	DET
ejpam-6686	228	44	,	,	PUNCT
ejpam-6686	228	45	b	b	NOUN
ejpam-6686	228	46	,	,	PUNCT
ejpam-6686	228	47	1	1	NUM
ejpam-6686	228	48	}	}	PUNCT
ejpam-6686	228	49	.	.	PUNCT
ejpam-6686	229	1	therefore	therefore	ADV
ejpam-6686	229	2	,	,	PUNCT
ejpam-6686	229	3	hs1	hs1	PROPN
ejpam-6686	229	4	∩hs2	∩hs2	PROPN
ejpam-6686	229	5	̸=	̸=	PROPN
ejpam-6686	229	6	hs1∩s2	hs1∩s2	PROPN
ejpam-6686	229	7	.	.	PUNCT
ejpam-6686	230	1	lemma	lemma	PROPN
ejpam-6686	230	2	5	5	NUM
ejpam-6686	230	3	.	.	PUNCT
ejpam-6686	231	1	if	if	SCONJ
ejpam-6686	231	2	s1	s1	PROPN
ejpam-6686	231	3	∈	∈	PROPN
ejpam-6686	231	4	s1	s1	NOUN
ejpam-6686	231	5	,	,	PUNCT
ejpam-6686	231	6	s2	s2	NOUN
ejpam-6686	231	7	∈	∈	PROPN
ejpam-6686	231	8	s2	s2	NOUN
ejpam-6686	231	9	and	and	CCONJ
ejpam-6686	231	10	s1	s1	PROPN
ejpam-6686	231	11	∧	∧	PROPN
ejpam-6686	231	12	s2	s2	NOUN
ejpam-6686	231	13	∈	∈	PROPN
ejpam-6686	231	14	s1	s1	NOUN
ejpam-6686	231	15	∩	∩	PROPN
ejpam-6686	231	16	s2	s2	NOUN
ejpam-6686	231	17	,	,	PUNCT
ejpam-6686	231	18	then	then	ADV
ejpam-6686	231	19	hs1∩s2	hs1∩s2	PROPN
ejpam-6686	231	20	=	=	PUNCT
ejpam-6686	231	21	hs1	hs1	PROPN
ejpam-6686	231	22	∩hs2	∩hs2	PROPN
ejpam-6686	231	23	.	.	PUNCT
ejpam-6686	231	24	proof	proof	NOUN
ejpam-6686	231	25	.	.	PUNCT
ejpam-6686	232	1	let	let	VERB
ejpam-6686	232	2	h	h	PRON
ejpam-6686	232	3	∈	∈	PROPN
ejpam-6686	232	4	hs1	hs1	PROPN
ejpam-6686	232	5	∩	∩	PROPN
ejpam-6686	232	6	hs2	hs2	PROPN
ejpam-6686	232	7	.	.	PUNCT
ejpam-6686	233	1	then	then	ADV
ejpam-6686	233	2	h	h	PROPN
ejpam-6686	233	3	∨	∨	NOUN
ejpam-6686	233	4	s1	s1	NOUN
ejpam-6686	233	5	=	=	PUNCT
ejpam-6686	233	6	h	h	NOUN
ejpam-6686	233	7	and	and	CCONJ
ejpam-6686	233	8	h	h	NOUN
ejpam-6686	233	9	∨	∨	NOUN
ejpam-6686	233	10	s2	s2	X
ejpam-6686	233	11	=	=	PUNCT
ejpam-6686	233	12	h	h	NOUN
ejpam-6686	233	13	,	,	PUNCT
ejpam-6686	233	14	for	for	ADP
ejpam-6686	233	15	some	some	DET
ejpam-6686	233	16	s1	s1	PROPN
ejpam-6686	233	17	∈	∈	PROPN
ejpam-6686	233	18	s1	s1	NOUN
ejpam-6686	233	19	and	and	CCONJ
ejpam-6686	233	20	s2	s2	PROPN
ejpam-6686	233	21	∈	∈	PROPN
ejpam-6686	233	22	s2	s2	PROPN
ejpam-6686	233	23	.	.	PUNCT
ejpam-6686	234	1	since	since	SCONJ
ejpam-6686	234	2	s1	s1	PROPN
ejpam-6686	234	3	∧	∧	PROPN
ejpam-6686	234	4	s2	s2	NOUN
ejpam-6686	234	5	∈	∈	PROPN
ejpam-6686	234	6	s1	s1	PROPN
ejpam-6686	234	7	∩s2	∩s2	PROPN
ejpam-6686	234	8	,	,	PUNCT
ejpam-6686	234	9	h∨	h∨	PROPN
ejpam-6686	234	10	(	(	PUNCT
ejpam-6686	234	11	s1	s1	PROPN
ejpam-6686	234	12	∧	∧	PROPN
ejpam-6686	234	13	s2	s2	PROPN
ejpam-6686	234	14	)	)	PUNCT
ejpam-6686	234	15	=	=	PUNCT
ejpam-6686	234	16	(	(	PUNCT
ejpam-6686	234	17	h∨	h∨	PROPN
ejpam-6686	234	18	s1)∧	s1)∧	ADV
ejpam-6686	234	19	(	(	PUNCT
ejpam-6686	234	20	h∨	h∨	PROPN
ejpam-6686	234	21	s2	s2	PROPN
ejpam-6686	234	22	)	)	PUNCT
ejpam-6686	234	23	=	=	SYM
ejpam-6686	234	24	h∧h	h∧h	X
ejpam-6686	234	25	=	=	PUNCT
ejpam-6686	234	26	h.	h.	PROPN
ejpam-6686	234	27	therefore	therefore	ADV
ejpam-6686	234	28	,	,	PUNCT
ejpam-6686	234	29	h	h	PROPN
ejpam-6686	234	30	∈	∈	PROPN
ejpam-6686	234	31	hs1∩s2	hs1∩s2	PROPN
ejpam-6686	234	32	.	.	PUNCT
ejpam-6686	235	1	hence	hence	ADV
ejpam-6686	235	2	,	,	PUNCT
ejpam-6686	235	3	hs1	hs1	PROPN
ejpam-6686	235	4	∩hs2	∩hs2	PROPN
ejpam-6686	235	5	⊆	⊆	NUM
ejpam-6686	235	6	hs1∩s2	hs1∩s2	PROPN
ejpam-6686	235	7	.	.	PUNCT
ejpam-6686	236	1	thus	thus	ADV
ejpam-6686	236	2	,	,	PUNCT
ejpam-6686	236	3	hs1	hs1	X
ejpam-6686	236	4	∩hs2	∩hs2	PROPN
ejpam-6686	236	5	=	=	PUNCT
ejpam-6686	236	6	hs1∩s2	hs1∩s2	PROPN
ejpam-6686	236	7	(	(	PUNCT
ejpam-6686	236	8	by	by	ADP
ejpam-6686	236	9	lemma	lemma	PROPN
ejpam-6686	236	10	7	7	NUM
ejpam-6686	236	11	(	(	PUNCT
ejpam-6686	236	12	iv	iv	NUM
ejpam-6686	236	13	)	)	PUNCT
ejpam-6686	236	14	)	)	PUNCT
ejpam-6686	236	15	.	.	PUNCT
ejpam-6686	237	1	theorem	theorem	ADJ
ejpam-6686	237	2	8	8	NUM
ejpam-6686	237	3	.	.	PUNCT
ejpam-6686	238	1	let	let	VERB
ejpam-6686	238	2	s	s	PRON
ejpam-6686	238	3	be	be	AUX
ejpam-6686	238	4	a	a	DET
ejpam-6686	238	5	non	non	ADJ
ejpam-6686	238	6	-	-	ADJ
ejpam-6686	238	7	empty	empty	ADJ
ejpam-6686	238	8	subset	subset	NOUN
ejpam-6686	238	9	of	of	ADP
ejpam-6686	238	10	l	l	PROPN
ejpam-6686	238	11	and	and	CCONJ
ejpam-6686	238	12	s	s	AUX
ejpam-6686	238	13	be	be	AUX
ejpam-6686	238	14	closed	close	VERB
ejpam-6686	238	15	under	under	ADP
ejpam-6686	238	16	∧.	∧.	PROPN
ejpam-6686	239	1	then	then	ADV
ejpam-6686	239	2	(	(	PUNCT
ejpam-6686	239	3	i	i	NOUN
ejpam-6686	239	4	)	)	PUNCT
ejpam-6686	239	5	hs	hs	PROPN
ejpam-6686	239	6	is	be	AUX
ejpam-6686	239	7	closed	close	VERB
ejpam-6686	239	8	under	under	ADP
ejpam-6686	239	9	∧	∧	PROPN
ejpam-6686	239	10	(	(	PUNCT
ejpam-6686	239	11	ii	ii	NOUN
ejpam-6686	239	12	)	)	PUNCT
ejpam-6686	239	13	hs	hs	PROPN
ejpam-6686	239	14	is	be	AUX
ejpam-6686	239	15	a	a	DET
ejpam-6686	239	16	sub	sub	ADJ
ejpam-6686	239	17	-	-	ADJ
ejpam-6686	239	18	almost	almost	ADV
ejpam-6686	239	19	distributive	distributive	ADJ
ejpam-6686	239	20	lattice	lattice	NOUN
ejpam-6686	239	21	of	of	ADP
ejpam-6686	239	22	l	l	PROPN
ejpam-6686	239	23	g.	g.	PROPN
ejpam-6686	239	24	chinnayya	chinnayya	PROPN
ejpam-6686	239	25	et	et	PROPN
ejpam-6686	239	26	al	al	PROPN
ejpam-6686	239	27	.	.	PUNCT
ejpam-6686	239	28	/	/	SYM
ejpam-6686	239	29	eur	eur	PROPN
ejpam-6686	239	30	.	.	PUNCT
ejpam-6686	240	1	j.	j.	PROPN
ejpam-6686	240	2	pure	pure	PROPN
ejpam-6686	240	3	appl	appl	PROPN
ejpam-6686	240	4	.	.	PROPN
ejpam-6686	240	5	math	math	PROPN
ejpam-6686	240	6	,	,	PUNCT
ejpam-6686	240	7	18	18	NUM
ejpam-6686	240	8	(	(	PUNCT
ejpam-6686	240	9	4	4	NUM
ejpam-6686	240	10	)	)	PUNCT
ejpam-6686	240	11	(	(	PUNCT
ejpam-6686	240	12	2025	2025	NUM
ejpam-6686	240	13	)	)	PUNCT
ejpam-6686	240	14	,	,	PUNCT
ejpam-6686	240	15	6686	6686	NUM
ejpam-6686	240	16	9	9	NUM
ejpam-6686	240	17	of	of	ADP
ejpam-6686	240	18	12	12	NUM
ejpam-6686	240	19	(	(	PUNCT
ejpam-6686	240	20	iii	iii	NOUN
ejpam-6686	240	21	)	)	PUNCT
ejpam-6686	240	22	hs	hs	PROPN
ejpam-6686	240	23	is	be	AUX
ejpam-6686	240	24	a	a	DET
ejpam-6686	240	25	filter	filter	NOUN
ejpam-6686	240	26	of	of	ADP
ejpam-6686	240	27	l	l	NOUN
ejpam-6686	240	28	(	(	PUNCT
ejpam-6686	240	29	iv	iv	X
ejpam-6686	240	30	)	)	PUNCT
ejpam-6686	240	31	hs	hs	PROPN
ejpam-6686	240	32	is	be	AUX
ejpam-6686	240	33	the	the	DET
ejpam-6686	240	34	smallest	small	ADJ
ejpam-6686	240	35	filter	filter	NOUN
ejpam-6686	240	36	containing	contain	VERB
ejpam-6686	240	37	s.	s.	PROPN
ejpam-6686	240	38	proof	proof	PROPN
ejpam-6686	240	39	.	.	PUNCT
ejpam-6686	241	1	let	let	VERB
ejpam-6686	241	2	s	s	PRON
ejpam-6686	241	3	be	be	AUX
ejpam-6686	241	4	a	a	DET
ejpam-6686	241	5	non	non	ADJ
ejpam-6686	241	6	-	-	ADJ
ejpam-6686	241	7	empty	empty	ADJ
ejpam-6686	241	8	subset	subset	NOUN
ejpam-6686	241	9	of	of	ADP
ejpam-6686	241	10	l	l	PROPN
ejpam-6686	241	11	and	and	CCONJ
ejpam-6686	241	12	s	s	NOUN
ejpam-6686	241	13	is	be	AUX
ejpam-6686	241	14	closed	close	VERB
ejpam-6686	241	15	under	under	ADP
ejpam-6686	241	16	∧.	∧.	PROPN
ejpam-6686	241	17	(	(	PUNCT
ejpam-6686	241	18	i	i	NOUN
ejpam-6686	241	19	)	)	PUNCT
ejpam-6686	241	20	let	let	VERB
ejpam-6686	241	21	h1	h1	PROPN
ejpam-6686	241	22	,	,	PUNCT
ejpam-6686	241	23	h2	h2	PROPN
ejpam-6686	241	24	∈	∈	PROPN
ejpam-6686	242	1	hs	hs	PROPN
ejpam-6686	242	2	.	.	PUNCT
ejpam-6686	243	1	then	then	ADV
ejpam-6686	243	2	h1	h1	VERB
ejpam-6686	243	3	∨	∨	NOUN
ejpam-6686	243	4	s1	s1	NOUN
ejpam-6686	243	5	=	=	PUNCT
ejpam-6686	243	6	h1	h1	NOUN
ejpam-6686	243	7	and	and	CCONJ
ejpam-6686	243	8	h2	h2	PROPN
ejpam-6686	243	9	∨	∨	NUM
ejpam-6686	243	10	s2	s2	PROPN
ejpam-6686	243	11	=	=	SYM
ejpam-6686	243	12	h2	h2	NOUN
ejpam-6686	243	13	,	,	PUNCT
ejpam-6686	243	14	for	for	ADP
ejpam-6686	243	15	some	some	DET
ejpam-6686	243	16	s1	s1	NOUN
ejpam-6686	243	17	,	,	PUNCT
ejpam-6686	243	18	s2	s2	PROPN
ejpam-6686	243	19	∈	∈	PROPN
ejpam-6686	243	20	s.	s.	PROPN
ejpam-6686	243	21	now	now	ADV
ejpam-6686	243	22	,	,	PUNCT
ejpam-6686	243	23	h1	h1	VERB
ejpam-6686	243	24	∧h2	∧h2	NUM
ejpam-6686	243	25	∧	∧	PROPN
ejpam-6686	243	26	(	(	PUNCT
ejpam-6686	243	27	s1	s1	PROPN
ejpam-6686	243	28	∧	∧	PROPN
ejpam-6686	243	29	s2	s2	PROPN
ejpam-6686	243	30	)	)	PUNCT
ejpam-6686	243	31	=	=	PUNCT
ejpam-6686	243	32	h1	h1	VERB
ejpam-6686	243	33	∧	∧	PROPN
ejpam-6686	243	34	s1	s1	PROPN
ejpam-6686	243	35	∧h2	∧h2	NUM
ejpam-6686	243	36	∧	∧	PROPN
ejpam-6686	243	37	s2	s2	NOUN
ejpam-6686	243	38	=	=	SYM
ejpam-6686	243	39	s1	s1	PROPN
ejpam-6686	243	40	∧	∧	PROPN
ejpam-6686	243	41	s2	s2	PROPN
ejpam-6686	243	42	.	.	PUNCT
ejpam-6686	244	1	then	then	ADV
ejpam-6686	244	2	(	(	PUNCT
ejpam-6686	244	3	h1	h1	PROPN
ejpam-6686	244	4	∧h2)∨	∧h2)∨	PROPN
ejpam-6686	244	5	(	(	PUNCT
ejpam-6686	244	6	s1	s1	PROPN
ejpam-6686	244	7	∧	∧	PROPN
ejpam-6686	244	8	s2	s2	PROPN
ejpam-6686	244	9	)	)	PUNCT
ejpam-6686	244	10	=	=	PUNCT
ejpam-6686	244	11	h1	h1	VERB
ejpam-6686	244	12	∧h2	∧h2	NUM
ejpam-6686	244	13	.	.	PUNCT
ejpam-6686	245	1	since	since	SCONJ
ejpam-6686	245	2	s	s	NOUN
ejpam-6686	245	3	is	be	AUX
ejpam-6686	245	4	closed	close	VERB
ejpam-6686	245	5	under	under	ADP
ejpam-6686	245	6	∧	∧	PROPN
ejpam-6686	245	7	,	,	PUNCT
ejpam-6686	245	8	h1	h1	PROPN
ejpam-6686	245	9	∧	∧	PROPN
ejpam-6686	245	10	h2	h2	PROPN
ejpam-6686	245	11	∈	∈	PROPN
ejpam-6686	246	1	hs	hs	PROPN
ejpam-6686	246	2	.	.	PUNCT
ejpam-6686	247	1	therefore	therefore	ADV
ejpam-6686	247	2	,	,	PUNCT
ejpam-6686	247	3	hs	hs	PROPN
ejpam-6686	247	4	is	be	AUX
ejpam-6686	247	5	closed	close	VERB
ejpam-6686	247	6	under	under	ADP
ejpam-6686	247	7	∧.	∧.	PROPN
ejpam-6686	247	8	(	(	PUNCT
ejpam-6686	247	9	ii	ii	NOUN
ejpam-6686	247	10	)	)	PUNCT
ejpam-6686	247	11	by	by	ADP
ejpam-6686	247	12	(	(	PUNCT
ejpam-6686	247	13	i	i	NOUN
ejpam-6686	247	14	)	)	PUNCT
ejpam-6686	247	15	and	and	CCONJ
ejpam-6686	247	16	theorem	theorem	VERB
ejpam-6686	247	17	7	7	NUM
ejpam-6686	247	18	(	(	PUNCT
ejpam-6686	247	19	iii	iii	NOUN
ejpam-6686	247	20	)	)	PUNCT
ejpam-6686	247	21	,	,	PUNCT
ejpam-6686	247	22	hs	hs	PROPN
ejpam-6686	247	23	is	be	AUX
ejpam-6686	247	24	a	a	DET
ejpam-6686	247	25	sub	sub	ADJ
ejpam-6686	247	26	-	-	ADJ
ejpam-6686	247	27	almost	almost	ADV
ejpam-6686	247	28	distributive	distributive	ADJ
ejpam-6686	247	29	lattice	lattice	NOUN
ejpam-6686	247	30	of	of	ADP
ejpam-6686	247	31	l.	l.	PROPN
ejpam-6686	247	32	(	(	PUNCT
ejpam-6686	247	33	iii	iii	PROPN
ejpam-6686	247	34	)	)	PUNCT
ejpam-6686	247	35	by	by	ADP
ejpam-6686	247	36	(	(	PUNCT
ejpam-6686	247	37	i	i	NOUN
ejpam-6686	247	38	)	)	PUNCT
ejpam-6686	247	39	and	and	CCONJ
ejpam-6686	247	40	theorem	theorem	VERB
ejpam-6686	247	41	7	7	NUM
ejpam-6686	247	42	(	(	PUNCT
ejpam-6686	247	43	iii	iii	NOUN
ejpam-6686	247	44	)	)	PUNCT
ejpam-6686	247	45	,	,	PUNCT
ejpam-6686	247	46	hs	hs	PROPN
ejpam-6686	247	47	is	be	AUX
ejpam-6686	247	48	a	a	DET
ejpam-6686	247	49	filter	filter	NOUN
ejpam-6686	247	50	of	of	ADP
ejpam-6686	247	51	l.	l.	PROPN
ejpam-6686	247	52	(	(	PUNCT
ejpam-6686	247	53	iv	iv	X
ejpam-6686	247	54	)	)	PUNCT
ejpam-6686	247	55	let	let	VERB
ejpam-6686	247	56	f	f	PRON
ejpam-6686	247	57	be	be	AUX
ejpam-6686	247	58	a	a	DET
ejpam-6686	247	59	filter	filter	NOUN
ejpam-6686	247	60	of	of	ADP
ejpam-6686	247	61	l	l	NOUN
ejpam-6686	247	62	such	such	ADJ
ejpam-6686	247	63	that	that	PRON
ejpam-6686	247	64	s	s	VERB
ejpam-6686	247	65	⊆	⊆	NUM
ejpam-6686	247	66	f	f	NOUN
ejpam-6686	247	67	.	.	PUNCT
ejpam-6686	248	1	by	by	ADP
ejpam-6686	248	2	lemma	lemma	PROPN
ejpam-6686	248	3	4	4	NUM
ejpam-6686	248	4	(	(	PUNCT
ejpam-6686	248	5	i	i	NOUN
ejpam-6686	248	6	)	)	PUNCT
ejpam-6686	248	7	,	,	PUNCT
ejpam-6686	248	8	hs	hs	PROPN
ejpam-6686	248	9	⊆	⊆	NUM
ejpam-6686	248	10	hf	hf	NOUN
ejpam-6686	248	11	and	and	CCONJ
ejpam-6686	248	12	f	f	PROPN
ejpam-6686	248	13	⊆	⊆	NUM
ejpam-6686	248	14	hf	hf	NOUN
ejpam-6686	248	15	.	.	PUNCT
ejpam-6686	249	1	let	let	VERB
ejpam-6686	249	2	h	h	PRON
ejpam-6686	249	3	∈	∈	PROPN
ejpam-6686	249	4	hf	hf	VERB
ejpam-6686	249	5	.	.	PUNCT
ejpam-6686	250	1	then	then	ADV
ejpam-6686	250	2	h	h	PROPN
ejpam-6686	250	3	∨	∨	PROPN
ejpam-6686	250	4	s	s	PART
ejpam-6686	250	5	=	=	ADJ
ejpam-6686	250	6	h	h	NOUN
ejpam-6686	250	7	,	,	PUNCT
ejpam-6686	250	8	for	for	ADP
ejpam-6686	250	9	some	some	DET
ejpam-6686	250	10	s	s	NOUN
ejpam-6686	250	11	∈	∈	PROPN
ejpam-6686	250	12	f	f	X
ejpam-6686	250	13	.	.	PUNCT
ejpam-6686	251	1	for	for	ADP
ejpam-6686	251	2	s	s	PROPN
ejpam-6686	251	3	∈	∈	PROPN
ejpam-6686	251	4	f	f	X
ejpam-6686	251	5	,	,	PUNCT
ejpam-6686	251	6	h	h	PROPN
ejpam-6686	251	7	∨	∨	NOUN
ejpam-6686	251	8	s	s	PART
ejpam-6686	251	9	=	=	X
ejpam-6686	251	10	h	h	NOUN
ejpam-6686	251	11	∈	∈	PROPN
ejpam-6686	251	12	f	f	PROPN
ejpam-6686	251	13	(	(	PUNCT
ejpam-6686	251	14	since	since	SCONJ
ejpam-6686	251	15	f	f	PROPN
ejpam-6686	251	16	is	be	AUX
ejpam-6686	251	17	a	a	DET
ejpam-6686	251	18	filter	filter	NOUN
ejpam-6686	251	19	)	)	PUNCT
ejpam-6686	251	20	.	.	PUNCT
ejpam-6686	252	1	therefore	therefore	ADV
ejpam-6686	252	2	,	,	PUNCT
ejpam-6686	252	3	h	h	PROPN
ejpam-6686	252	4	∈	∈	PROPN
ejpam-6686	252	5	f	f	PROPN
ejpam-6686	252	6	.	.	PUNCT
ejpam-6686	253	1	so	so	ADV
ejpam-6686	253	2	that	that	SCONJ
ejpam-6686	253	3	hf	hf	VERB
ejpam-6686	253	4	⊆	⊆	NUM
ejpam-6686	253	5	f	f	NOUN
ejpam-6686	253	6	.	.	PUNCT
ejpam-6686	254	1	hence	hence	ADV
ejpam-6686	254	2	,	,	PUNCT
ejpam-6686	254	3	hs	hs	PROPN
ejpam-6686	254	4	⊆	⊆	NUM
ejpam-6686	254	5	f	f	X
ejpam-6686	254	6	=	=	PRON
ejpam-6686	254	7	hf	hf	PROPN
ejpam-6686	254	8	.	.	PUNCT
ejpam-6686	255	1	thus	thus	ADV
ejpam-6686	255	2	,	,	PUNCT
ejpam-6686	255	3	hs	hs	PROPN
ejpam-6686	255	4	is	be	AUX
ejpam-6686	255	5	the	the	DET
ejpam-6686	255	6	smallest	small	ADJ
ejpam-6686	255	7	filter	filter	NOUN
ejpam-6686	255	8	containing	contain	VERB
ejpam-6686	255	9	s.	s.	PROPN
ejpam-6686	255	10	remark	remark	PROPN
ejpam-6686	255	11	5	5	NUM
ejpam-6686	255	12	.	.	PUNCT
ejpam-6686	255	13	hs	hs	PROPN
ejpam-6686	255	14	need	need	AUX
ejpam-6686	255	15	not	not	PART
ejpam-6686	255	16	be	be	AUX
ejpam-6686	255	17	closed	close	VERB
ejpam-6686	255	18	under	under	ADP
ejpam-6686	255	19	∧.	∧.	PROPN
ejpam-6686	255	20	see	see	VERB
ejpam-6686	255	21	the	the	DET
ejpam-6686	255	22	following	follow	VERB
ejpam-6686	255	23	example	example	NOUN
ejpam-6686	255	24	:	:	PUNCT
ejpam-6686	255	25	example	example	NOUN
ejpam-6686	255	26	4	4	X
ejpam-6686	255	27	.	.	PUNCT
ejpam-6686	256	1	let	let	VERB
ejpam-6686	256	2	l	l	NOUN
ejpam-6686	256	3	=	=	PUNCT
ejpam-6686	256	4	{	{	PUNCT
ejpam-6686	256	5	0	0	NUM
ejpam-6686	256	6	,	,	PUNCT
ejpam-6686	256	7	a	a	DET
ejpam-6686	256	8	,	,	PUNCT
ejpam-6686	256	9	b	b	NOUN
ejpam-6686	256	10	,	,	PUNCT
ejpam-6686	256	11	c	c	NOUN
ejpam-6686	256	12	,	,	PUNCT
ejpam-6686	256	13	1	1	NUM
ejpam-6686	256	14	}	}	PUNCT
ejpam-6686	256	15	be	be	AUX
ejpam-6686	256	16	an	an	DET
ejpam-6686	256	17	almost	almost	ADV
ejpam-6686	256	18	distributive	distributive	ADJ
ejpam-6686	256	19	lattice	lattice	NOUN
ejpam-6686	256	20	with	with	ADP
ejpam-6686	256	21	maximal	maximal	ADJ
ejpam-6686	256	22	element	element	NOUN
ejpam-6686	256	23	1	1	NUM
ejpam-6686	256	24	,	,	PUNCT
ejpam-6686	256	25	whose	whose	DET
ejpam-6686	256	26	hasse	hasse	NOUN
ejpam-6686	256	27	diagram	diagram	NOUN
ejpam-6686	256	28	is	be	AUX
ejpam-6686	256	29	given	give	VERB
ejpam-6686	256	30	below	below	ADP
ejpam-6686	256	31	:	:	PUNCT
ejpam-6686	256	32	1	1	NUM
ejpam-6686	256	33	c	c	NOUN
ejpam-6686	256	34	b	b	PROPN
ejpam-6686	256	35	a	a	DET
ejpam-6686	256	36	0	0	NUM
ejpam-6686	256	37	let	let	VERB
ejpam-6686	256	38	s1	s1	PROPN
ejpam-6686	256	39	=	=	PUNCT
ejpam-6686	256	40	{	{	PUNCT
ejpam-6686	256	41	b	b	PROPN
ejpam-6686	256	42	,	,	PUNCT
ejpam-6686	256	43	c	c	NOUN
ejpam-6686	256	44	}	}	PUNCT
ejpam-6686	256	45	.	.	PUNCT
ejpam-6686	257	1	then	then	ADV
ejpam-6686	257	2	hs	hs	PROPN
ejpam-6686	257	3	=	=	PUNCT
ejpam-6686	257	4	{	{	PUNCT
ejpam-6686	257	5	b	b	PROPN
ejpam-6686	257	6	,	,	PUNCT
ejpam-6686	257	7	c	c	NOUN
ejpam-6686	257	8	,	,	PUNCT
ejpam-6686	257	9	1	1	NUM
ejpam-6686	257	10	}	}	PUNCT
ejpam-6686	257	11	.	.	PUNCT
ejpam-6686	258	1	let	let	VERB
ejpam-6686	258	2	b	b	X
ejpam-6686	258	3	,	,	PUNCT
ejpam-6686	258	4	c	c	PROPN
ejpam-6686	258	5	∈	∈	PROPN
ejpam-6686	258	6	hs	hs	PROPN
ejpam-6686	258	7	.	.	PROPN
ejpam-6686	259	1	then	then	ADV
ejpam-6686	259	2	b	b	X
ejpam-6686	259	3	∧	∧	PROPN
ejpam-6686	259	4	c	c	PROPN
ejpam-6686	259	5	=	=	SYM
ejpam-6686	259	6	a	a	PROPN
ejpam-6686	259	7	/∈	/∈	INTJ
ejpam-6686	260	1	hs	hs	PROPN
ejpam-6686	260	2	.	.	PROPN
ejpam-6686	260	3	therefore	therefore	ADV
ejpam-6686	260	4	,	,	PUNCT
ejpam-6686	260	5	hs	hs	PROPN
ejpam-6686	260	6	is	be	AUX
ejpam-6686	260	7	not	not	PART
ejpam-6686	260	8	closed	close	VERB
ejpam-6686	260	9	under	under	ADP
ejpam-6686	260	10	∧.	∧.	NOUN
ejpam-6686	260	11	given	give	VERB
ejpam-6686	260	12	a	a	DET
ejpam-6686	260	13	non	non	ADJ
ejpam-6686	260	14	-	-	ADJ
ejpam-6686	260	15	empty	empty	ADJ
ejpam-6686	260	16	set	set	NOUN
ejpam-6686	260	17	s	s	PROPN
ejpam-6686	260	18	of	of	ADP
ejpam-6686	260	19	l	l	NOUN
ejpam-6686	260	20	,	,	PUNCT
ejpam-6686	260	21	it	it	PRON
ejpam-6686	260	22	is	be	AUX
ejpam-6686	260	23	known	know	VERB
ejpam-6686	260	24	that	that	SCONJ
ejpam-6686	260	25	[	[	X
ejpam-6686	260	26	s	s	X
ejpam-6686	260	27	)	)	PUNCT
ejpam-6686	260	28	=	=	SYM
ejpam-6686	260	29	{	{	PUNCT
ejpam-6686	260	30	a	a	DET
ejpam-6686	260	31	∨	∨	NOUN
ejpam-6686	260	32	(	(	PUNCT
ejpam-6686	260	33	n∧	n∧	NUM
ejpam-6686	260	34	i=1	i=1	PROPN
ejpam-6686	260	35	si	si	NOUN
ejpam-6686	260	36	)	)	PUNCT
ejpam-6686	260	37	|	|	ADV
ejpam-6686	260	38	a	a	DET
ejpam-6686	260	39	∈	∈	ADJ
ejpam-6686	260	40	l	l	NOUN
ejpam-6686	260	41	and	and	CCONJ
ejpam-6686	260	42	si	si	PROPN
ejpam-6686	260	43	∈	∈	PROPN
ejpam-6686	260	44	s	s	PART
ejpam-6686	260	45	}	}	PUNCT
ejpam-6686	260	46	is	be	AUX
ejpam-6686	260	47	the	the	DET
ejpam-6686	260	48	smallest	small	ADJ
ejpam-6686	260	49	filter	filter	NOUN
ejpam-6686	260	50	containing	contain	VERB
ejpam-6686	260	51	s.	s.	PROPN
ejpam-6686	260	52	lemma	lemma	PROPN
ejpam-6686	260	53	6	6	NUM
ejpam-6686	260	54	.	.	PUNCT
ejpam-6686	261	1	for	for	ADP
ejpam-6686	261	2	any	any	DET
ejpam-6686	261	3	non	non	ADJ
ejpam-6686	261	4	-	-	ADJ
ejpam-6686	261	5	empty	empty	ADJ
ejpam-6686	261	6	subset	subset	NOUN
ejpam-6686	261	7	s	s	PROPN
ejpam-6686	261	8	of	of	ADP
ejpam-6686	261	9	l	l	PROPN
ejpam-6686	261	10	,	,	PUNCT
ejpam-6686	261	11	hs	hs	PROPN
ejpam-6686	261	12	⊆	⊆	NUM
ejpam-6686	261	13	[	[	X
ejpam-6686	261	14	s	s	X
ejpam-6686	261	15	)	)	PUNCT
ejpam-6686	261	16	.	.	PUNCT
ejpam-6686	262	1	proof	proof	NOUN
ejpam-6686	262	2	.	.	PUNCT
ejpam-6686	263	1	let	let	VERB
ejpam-6686	263	2	h	h	PRON
ejpam-6686	263	3	∈	∈	PROPN
ejpam-6686	264	1	hs	hs	PROPN
ejpam-6686	264	2	.	.	PUNCT
ejpam-6686	265	1	then	then	ADV
ejpam-6686	265	2	h∨	h∨	PROPN
ejpam-6686	265	3	s	s	PART
ejpam-6686	265	4	=	=	NOUN
ejpam-6686	265	5	h	h	PROPN
ejpam-6686	265	6	and	and	CCONJ
ejpam-6686	265	7	h∧	h∧	PROPN
ejpam-6686	265	8	s	s	PART
ejpam-6686	265	9	=	=	SYM
ejpam-6686	265	10	s	s	PROPN
ejpam-6686	265	11	,	,	PUNCT
ejpam-6686	265	12	for	for	ADP
ejpam-6686	265	13	some	some	DET
ejpam-6686	265	14	s	s	NOUN
ejpam-6686	265	15	∈	∈	NOUN
ejpam-6686	265	16	s.	s.	PROPN
ejpam-6686	265	17	since	since	SCONJ
ejpam-6686	265	18	[	[	X
ejpam-6686	265	19	s	s	X
ejpam-6686	265	20	)	)	PUNCT
ejpam-6686	265	21	is	be	AUX
ejpam-6686	265	22	a	a	DET
ejpam-6686	265	23	filter	filter	NOUN
ejpam-6686	265	24	generated	generate	VERB
ejpam-6686	265	25	by	by	ADP
ejpam-6686	265	26	s	s	PROPN
ejpam-6686	265	27	,	,	PUNCT
ejpam-6686	265	28	h	h	PROPN
ejpam-6686	266	1	∨	∨	NUM
ejpam-6686	266	2	s	s	PART
ejpam-6686	266	3	=	=	X
ejpam-6686	266	4	h	h	NOUN
ejpam-6686	266	5	∈	∈	PROPN
ejpam-6686	267	1	[	[	X
ejpam-6686	267	2	s	s	X
ejpam-6686	267	3	)	)	PUNCT
ejpam-6686	267	4	.	.	PUNCT
ejpam-6686	268	1	therefore	therefore	ADV
ejpam-6686	268	2	,	,	PUNCT
ejpam-6686	268	3	hs	hs	PROPN
ejpam-6686	268	4	⊆	⊆	NUM
ejpam-6686	268	5	[	[	X
ejpam-6686	268	6	s	s	X
ejpam-6686	268	7	)	)	PUNCT
ejpam-6686	268	8	.	.	PUNCT
ejpam-6686	269	1	remark	remark	PROPN
ejpam-6686	269	2	6	6	NUM
ejpam-6686	269	3	.	.	PUNCT
ejpam-6686	269	4	hs	hs	PROPN
ejpam-6686	269	5	need	need	AUX
ejpam-6686	269	6	not	not	PART
ejpam-6686	269	7	be	be	AUX
ejpam-6686	269	8	equal	equal	ADJ
ejpam-6686	269	9	to	to	ADP
ejpam-6686	269	10	[	[	X
ejpam-6686	269	11	s	s	X
ejpam-6686	269	12	)	)	PUNCT
ejpam-6686	269	13	.	.	PUNCT
ejpam-6686	270	1	see	see	VERB
ejpam-6686	270	2	the	the	DET
ejpam-6686	270	3	following	follow	VERB
ejpam-6686	270	4	example	example	NOUN
ejpam-6686	270	5	:	:	PUNCT
ejpam-6686	271	1	g.	g.	PROPN
ejpam-6686	271	2	chinnayya	chinnayya	PROPN
ejpam-6686	271	3	et	et	PROPN
ejpam-6686	271	4	al	al	PROPN
ejpam-6686	271	5	.	.	PUNCT
ejpam-6686	271	6	/	/	SYM
ejpam-6686	271	7	eur	eur	PROPN
ejpam-6686	271	8	.	.	PUNCT
ejpam-6686	272	1	j.	j.	PROPN
ejpam-6686	272	2	pure	pure	PROPN
ejpam-6686	272	3	appl	appl	PROPN
ejpam-6686	272	4	.	.	PROPN
ejpam-6686	272	5	math	math	PROPN
ejpam-6686	272	6	,	,	PUNCT
ejpam-6686	272	7	18	18	NUM
ejpam-6686	272	8	(	(	PUNCT
ejpam-6686	272	9	4	4	NUM
ejpam-6686	272	10	)	)	PUNCT
ejpam-6686	272	11	(	(	PUNCT
ejpam-6686	272	12	2025	2025	NUM
ejpam-6686	272	13	)	)	PUNCT
ejpam-6686	272	14	,	,	PUNCT
ejpam-6686	272	15	6686	6686	NUM
ejpam-6686	272	16	10	10	NUM
ejpam-6686	272	17	of	of	ADP
ejpam-6686	272	18	12	12	NUM
ejpam-6686	272	19	example	example	NOUN
ejpam-6686	272	20	5	5	NUM
ejpam-6686	272	21	.	.	PUNCT
ejpam-6686	273	1	let	let	VERB
ejpam-6686	273	2	l	l	NOUN
ejpam-6686	273	3	=	=	PUNCT
ejpam-6686	273	4	{	{	PUNCT
ejpam-6686	273	5	0	0	NUM
ejpam-6686	273	6	,	,	PUNCT
ejpam-6686	273	7	a	a	DET
ejpam-6686	273	8	,	,	PUNCT
ejpam-6686	273	9	b	b	NOUN
ejpam-6686	273	10	,	,	PUNCT
ejpam-6686	273	11	c	c	NOUN
ejpam-6686	273	12	,	,	PUNCT
ejpam-6686	273	13	d	d	NOUN
ejpam-6686	273	14	,	,	PUNCT
ejpam-6686	273	15	e	e	NOUN
ejpam-6686	273	16	,	,	PUNCT
ejpam-6686	273	17	f	f	PROPN
ejpam-6686	273	18	,	,	PUNCT
ejpam-6686	273	19	1	1	NUM
ejpam-6686	273	20	}	}	PUNCT
ejpam-6686	273	21	be	be	AUX
ejpam-6686	273	22	an	an	DET
ejpam-6686	273	23	almost	almost	ADV
ejpam-6686	273	24	distributive	distributive	ADJ
ejpam-6686	273	25	lattice	lattice	NOUN
ejpam-6686	273	26	whose	whose	DET
ejpam-6686	273	27	hasse	hasse	NOUN
ejpam-6686	273	28	diagram	diagram	NOUN
ejpam-6686	273	29	is	be	AUX
ejpam-6686	273	30	given	give	VERB
ejpam-6686	273	31	below	below	ADP
ejpam-6686	273	32	:	:	PUNCT
ejpam-6686	273	33	1	1	NUM
ejpam-6686	273	34	d	d	SYM
ejpam-6686	273	35	b	b	X
ejpam-6686	273	36	f	f	X
ejpam-6686	273	37	e	e	PROPN
ejpam-6686	273	38	a	a	DET
ejpam-6686	273	39	0	0	NUM
ejpam-6686	273	40	c	c	NOUN
ejpam-6686	273	41	let	let	VERB
ejpam-6686	273	42	s	s	AUX
ejpam-6686	273	43	=	=	PUNCT
ejpam-6686	273	44	{	{	PUNCT
ejpam-6686	273	45	e	e	NOUN
ejpam-6686	273	46	,	,	PUNCT
ejpam-6686	273	47	f	f	NOUN
ejpam-6686	273	48	}	}	PUNCT
ejpam-6686	273	49	.	.	PUNCT
ejpam-6686	274	1	then	then	ADV
ejpam-6686	274	2	[	[	X
ejpam-6686	274	3	s	s	X
ejpam-6686	274	4	)	)	PUNCT
ejpam-6686	274	5	=	=	SYM
ejpam-6686	274	6	{	{	PUNCT
ejpam-6686	274	7	c	c	NOUN
ejpam-6686	274	8	,	,	PUNCT
ejpam-6686	274	9	e	e	NOUN
ejpam-6686	274	10	,	,	PUNCT
ejpam-6686	274	11	f	f	PROPN
ejpam-6686	274	12	,	,	PUNCT
ejpam-6686	274	13	1	1	NUM
ejpam-6686	274	14	}	}	PUNCT
ejpam-6686	274	15	is	be	AUX
ejpam-6686	274	16	a	a	DET
ejpam-6686	274	17	filter	filter	NOUN
ejpam-6686	274	18	of	of	ADP
ejpam-6686	274	19	l	l	PROPN
ejpam-6686	274	20	and	and	CCONJ
ejpam-6686	274	21	hs	hs	PROPN
ejpam-6686	274	22	=	=	SYM
ejpam-6686	274	23	{	{	PUNCT
ejpam-6686	274	24	e	e	PROPN
ejpam-6686	274	25	,	,	PUNCT
ejpam-6686	274	26	f	f	PROPN
ejpam-6686	274	27	,	,	PUNCT
ejpam-6686	274	28	1	1	NUM
ejpam-6686	274	29	}	}	PUNCT
ejpam-6686	274	30	.	.	PUNCT
ejpam-6686	275	1	therefore	therefore	ADV
ejpam-6686	275	2	,	,	PUNCT
ejpam-6686	275	3	hs	hs	PROPN
ejpam-6686	275	4	̸=	̸=	PROPN
ejpam-6686	275	5	[	[	X
ejpam-6686	275	6	s	s	X
ejpam-6686	275	7	)	)	PUNCT
ejpam-6686	275	8	.	.	PUNCT
ejpam-6686	276	1	lemma	lemma	PROPN
ejpam-6686	276	2	7	7	X
ejpam-6686	276	3	.	.	PUNCT
ejpam-6686	277	1	let	let	VERB
ejpam-6686	277	2	s	s	PRON
ejpam-6686	277	3	be	be	AUX
ejpam-6686	277	4	a	a	DET
ejpam-6686	277	5	non	non	ADJ
ejpam-6686	277	6	-	-	ADJ
ejpam-6686	277	7	empty	empty	ADJ
ejpam-6686	277	8	subset	subset	NOUN
ejpam-6686	277	9	of	of	ADP
ejpam-6686	277	10	l	l	PROPN
ejpam-6686	277	11	and	and	CCONJ
ejpam-6686	277	12	hs	hs	X
ejpam-6686	277	13	=	=	NOUN
ejpam-6686	277	14	hf	hf	PROPN
ejpam-6686	277	15	,	,	PUNCT
ejpam-6686	277	16	for	for	ADP
ejpam-6686	277	17	some	some	DET
ejpam-6686	277	18	filter	filter	NOUN
ejpam-6686	277	19	f	f	PROPN
ejpam-6686	277	20	of	of	ADP
ejpam-6686	277	21	l.	l.	PROPN
ejpam-6686	278	1	then	then	ADV
ejpam-6686	278	2	[	[	X
ejpam-6686	278	3	s	s	X
ejpam-6686	278	4	)	)	PUNCT
ejpam-6686	278	5	=	=	SYM
ejpam-6686	278	6	f	f	PROPN
ejpam-6686	278	7	.	.	PUNCT
ejpam-6686	279	1	proof	proof	NOUN
ejpam-6686	279	2	.	.	PUNCT
ejpam-6686	280	1	suppose	suppose	VERB
ejpam-6686	280	2	hs	hs	PROPN
ejpam-6686	280	3	=	=	NOUN
ejpam-6686	280	4	hf	hf	PROPN
ejpam-6686	280	5	,	,	PUNCT
ejpam-6686	280	6	for	for	ADP
ejpam-6686	280	7	some	some	DET
ejpam-6686	280	8	filter	filter	NOUN
ejpam-6686	280	9	f	f	PROPN
ejpam-6686	280	10	of	of	ADP
ejpam-6686	280	11	l.	l.	PROPN
ejpam-6686	280	12	since	since	SCONJ
ejpam-6686	280	13	f	f	PROPN
ejpam-6686	280	14	⊆	⊆	NUM
ejpam-6686	280	15	hf	hf	PROPN
ejpam-6686	280	16	=	=	PUNCT
ejpam-6686	280	17	hs	hs	PROPN
ejpam-6686	280	18	and	and	CCONJ
ejpam-6686	280	19	f	f	PROPN
ejpam-6686	280	20	is	be	AUX
ejpam-6686	280	21	closed	close	VERB
ejpam-6686	280	22	under	under	ADP
ejpam-6686	280	23	∧	∧	PROPN
ejpam-6686	280	24	,	,	PUNCT
ejpam-6686	280	25	f	f	PROPN
ejpam-6686	280	26	⊆	⊆	NUM
ejpam-6686	280	27	hf	hf	PROPN
ejpam-6686	280	28	=	=	PUNCT
ejpam-6686	280	29	hs	hs	PROPN
ejpam-6686	280	30	⊆	⊆	NUM
ejpam-6686	280	31	[	[	X
ejpam-6686	280	32	s	s	X
ejpam-6686	280	33	)	)	PUNCT
ejpam-6686	280	34	and	and	CCONJ
ejpam-6686	280	35	hf	hf	PROPN
ejpam-6686	280	36	is	be	AUX
ejpam-6686	280	37	a	a	DET
ejpam-6686	280	38	filter	filter	NOUN
ejpam-6686	280	39	of	of	ADP
ejpam-6686	280	40	l.	l.	PROPN
ejpam-6686	280	41	therefore	therefore	ADV
ejpam-6686	280	42	,	,	PUNCT
ejpam-6686	280	43	f	f	PROPN
ejpam-6686	280	44	=	=	SYM
ejpam-6686	280	45	hs	hs	PROPN
ejpam-6686	280	46	⊆	⊆	NUM
ejpam-6686	280	47	[	[	X
ejpam-6686	280	48	s	s	NOUN
ejpam-6686	280	49	)	)	PUNCT
ejpam-6686	280	50	.	.	PUNCT
ejpam-6686	281	1	now	now	ADV
ejpam-6686	281	2	,	,	PUNCT
ejpam-6686	281	3	s	s	VERB
ejpam-6686	281	4	⊆	⊆	NUM
ejpam-6686	281	5	hs	hs	X
ejpam-6686	281	6	=	=	NOUN
ejpam-6686	281	7	hf	hf	PROPN
ejpam-6686	281	8	=	=	SYM
ejpam-6686	281	9	f	f	PROPN
ejpam-6686	281	10	.	.	PUNCT
ejpam-6686	282	1	then	then	ADV
ejpam-6686	282	2	[	[	X
ejpam-6686	282	3	s	s	X
ejpam-6686	282	4	)	)	PUNCT
ejpam-6686	282	5	⊆	⊆	NUM
ejpam-6686	282	6	hf	hf	NOUN
ejpam-6686	282	7	=	=	SYM
ejpam-6686	282	8	f	f	PROPN
ejpam-6686	282	9	.	.	PUNCT
ejpam-6686	283	1	hence	hence	ADV
ejpam-6686	283	2	,	,	PUNCT
ejpam-6686	283	3	f	f	PROPN
ejpam-6686	283	4	=	=	PUNCT
ejpam-6686	284	1	[	[	X
ejpam-6686	284	2	s	s	X
ejpam-6686	284	3	)	)	PUNCT
ejpam-6686	284	4	.	.	PUNCT
ejpam-6686	285	1	remark	remark	PROPN
ejpam-6686	285	2	7	7	NUM
ejpam-6686	285	3	.	.	PUNCT
ejpam-6686	286	1	the	the	DET
ejpam-6686	286	2	converse	converse	NOUN
ejpam-6686	286	3	of	of	ADP
ejpam-6686	286	4	lemma	lemma	PROPN
ejpam-6686	286	5	7	7	NUM
ejpam-6686	286	6	need	need	AUX
ejpam-6686	286	7	not	not	PART
ejpam-6686	286	8	be	be	AUX
ejpam-6686	286	9	true	true	ADJ
ejpam-6686	286	10	.	.	PUNCT
ejpam-6686	287	1	for	for	ADP
ejpam-6686	287	2	,	,	PUNCT
ejpam-6686	287	3	in	in	ADP
ejpam-6686	287	4	example	example	NOUN
ejpam-6686	287	5	(	(	PUNCT
ejpam-6686	287	6	5	5	NUM
ejpam-6686	287	7	)	)	PUNCT
ejpam-6686	287	8	;	;	PUNCT
ejpam-6686	287	9	let	let	VERB
ejpam-6686	287	10	l	l	NOUN
ejpam-6686	287	11	=	=	PUNCT
ejpam-6686	287	12	{	{	PUNCT
ejpam-6686	287	13	0	0	NUM
ejpam-6686	287	14	,	,	PUNCT
ejpam-6686	287	15	a	a	DET
ejpam-6686	287	16	,	,	PUNCT
ejpam-6686	287	17	b	b	NOUN
ejpam-6686	287	18	,	,	PUNCT
ejpam-6686	287	19	c	c	NOUN
ejpam-6686	287	20	,	,	PUNCT
ejpam-6686	287	21	d	d	NOUN
ejpam-6686	287	22	,	,	PUNCT
ejpam-6686	287	23	e	e	NOUN
ejpam-6686	287	24	,	,	PUNCT
ejpam-6686	287	25	f	f	PROPN
ejpam-6686	287	26	,	,	PUNCT
ejpam-6686	287	27	1	1	NUM
ejpam-6686	287	28	}	}	PUNCT
ejpam-6686	287	29	be	be	AUX
ejpam-6686	287	30	an	an	DET
ejpam-6686	287	31	almost	almost	ADV
ejpam-6686	287	32	distributive	distributive	ADJ
ejpam-6686	287	33	lattice	lattice	NOUN
ejpam-6686	287	34	whose	whose	DET
ejpam-6686	287	35	hasse	hasse	NOUN
ejpam-6686	287	36	-	-	PUNCT
ejpam-6686	287	37	diagram	diagram	NOUN
ejpam-6686	287	38	is	be	AUX
ejpam-6686	287	39	given	give	VERB
ejpam-6686	287	40	in	in	ADP
ejpam-6686	287	41	example	example	NOUN
ejpam-6686	287	42	5	5	NUM
ejpam-6686	287	43	.	.	PUNCT
ejpam-6686	288	1	let	let	VERB
ejpam-6686	288	2	s	s	VERB
ejpam-6686	288	3	=	=	PUNCT
ejpam-6686	288	4	{	{	PUNCT
ejpam-6686	288	5	d	d	PROPN
ejpam-6686	288	6	,	,	PUNCT
ejpam-6686	288	7	e	e	NOUN
ejpam-6686	288	8	}	}	PUNCT
ejpam-6686	288	9	.	.	PUNCT
ejpam-6686	289	1	then	then	ADV
ejpam-6686	289	2	[	[	X
ejpam-6686	289	3	s	s	X
ejpam-6686	289	4	)	)	PUNCT
ejpam-6686	289	5	=	=	SYM
ejpam-6686	290	1	f	f	X
ejpam-6686	290	2	=	=	PRON
ejpam-6686	290	3	{	{	PUNCT
ejpam-6686	290	4	a	a	X
ejpam-6686	290	5	,	,	PUNCT
ejpam-6686	290	6	d	d	NOUN
ejpam-6686	290	7	,	,	PUNCT
ejpam-6686	290	8	e	e	NOUN
ejpam-6686	290	9	,	,	PUNCT
ejpam-6686	290	10	1	1	NUM
ejpam-6686	290	11	}	}	PUNCT
ejpam-6686	290	12	is	be	AUX
ejpam-6686	290	13	a	a	DET
ejpam-6686	290	14	filter	filter	NOUN
ejpam-6686	290	15	of	of	ADP
ejpam-6686	290	16	l	l	PROPN
ejpam-6686	290	17	and	and	CCONJ
ejpam-6686	290	18	hs	hs	PROPN
ejpam-6686	290	19	=	=	PUNCT
ejpam-6686	290	20	{	{	PUNCT
ejpam-6686	290	21	d	d	PROPN
ejpam-6686	290	22	,	,	PUNCT
ejpam-6686	290	23	e	e	NOUN
ejpam-6686	290	24	,	,	PUNCT
ejpam-6686	290	25	1	1	NUM
ejpam-6686	290	26	}	}	PUNCT
ejpam-6686	290	27	,	,	PUNCT
ejpam-6686	290	28	but	but	CCONJ
ejpam-6686	290	29	hs	hs	PROPN
ejpam-6686	290	30	̸=	̸=	PROPN
ejpam-6686	290	31	hf	hf	PROPN
ejpam-6686	290	32	=	=	SYM
ejpam-6686	290	33	f	f	PROPN
ejpam-6686	290	34	.	.	PUNCT
ejpam-6686	291	1	theorem	theorem	VERB
ejpam-6686	291	2	9	9	NUM
ejpam-6686	291	3	.	.	X
ejpam-6686	292	1	for	for	ADP
ejpam-6686	292	2	any	any	DET
ejpam-6686	292	3	non	non	ADJ
ejpam-6686	292	4	-	-	ADJ
ejpam-6686	292	5	empty	empty	ADJ
ejpam-6686	292	6	subsets	subset	NOUN
ejpam-6686	292	7	s1	s1	NOUN
ejpam-6686	292	8	,	,	PUNCT
ejpam-6686	292	9	s2	s2	NOUN
ejpam-6686	292	10	of	of	ADP
ejpam-6686	292	11	l	l	NOUN
ejpam-6686	292	12	,	,	PUNCT
ejpam-6686	292	13	we	we	PRON
ejpam-6686	292	14	have	have	VERB
ejpam-6686	292	15	(	(	PUNCT
ejpam-6686	292	16	i	i	NOUN
ejpam-6686	292	17	)	)	PUNCT
ejpam-6686	292	18	hs1	hs1	PROPN
ejpam-6686	293	1	∪hs1	∪hs1	PROPN
ejpam-6686	293	2	=	=	SYM
ejpam-6686	293	3	hs1	hs1	PROPN
ejpam-6686	293	4	(	(	PUNCT
ejpam-6686	293	5	idempotent	idempotent	ADJ
ejpam-6686	293	6	law	law	NOUN
ejpam-6686	293	7	)	)	PUNCT
ejpam-6686	293	8	(	(	PUNCT
ejpam-6686	293	9	ii	ii	NOUN
ejpam-6686	293	10	)	)	PUNCT
ejpam-6686	293	11	hs1	hs1	PROPN
ejpam-6686	293	12	∪hs2	∪hs2	PROPN
ejpam-6686	293	13	=	=	SYM
ejpam-6686	293	14	hs1∪s2	hs1∪s2	PROPN
ejpam-6686	293	15	=	=	SYM
ejpam-6686	293	16	hs2	hs2	PROPN
ejpam-6686	293	17	∪hs1	∪hs1	PROPN
ejpam-6686	293	18	=	=	SYM
ejpam-6686	293	19	hs2∪s1(commutative	hs2∪s1(commutative	PROPN
ejpam-6686	293	20	law	law	NOUN
ejpam-6686	293	21	)	)	PUNCT
ejpam-6686	293	22	(	(	PUNCT
ejpam-6686	293	23	iii	iii	NOUN
ejpam-6686	293	24	)	)	PUNCT
ejpam-6686	293	25	(	(	PUNCT
ejpam-6686	293	26	hs1	hs1	PROPN
ejpam-6686	293	27	∪hs2	∪hs2	PROPN
ejpam-6686	293	28	)	)	PUNCT
ejpam-6686	293	29	∪hs3	∪hs3	X
ejpam-6686	293	30	=	=	SYM
ejpam-6686	293	31	h(s1∪s2)∪s3	h(s1∪s2)∪s3	X
ejpam-6686	293	32	=	=	PUNCT
ejpam-6686	293	33	hs1	hs1	PROPN
ejpam-6686	293	34	∪	∪	X
ejpam-6686	293	35	(	(	PUNCT
ejpam-6686	293	36	hs2	hs2	NOUN
ejpam-6686	293	37	∪hs3)(associative	∪hs3)(associative	ADJ
ejpam-6686	293	38	law	law	NOUN
ejpam-6686	293	39	)	)	PUNCT
ejpam-6686	293	40	.	.	PUNCT
ejpam-6686	294	1	proof	proof	NOUN
ejpam-6686	294	2	.	.	PUNCT
ejpam-6686	295	1	(	(	PUNCT
ejpam-6686	295	2	i	i	NOUN
ejpam-6686	295	3	)	)	PUNCT
ejpam-6686	295	4	it	it	PRON
ejpam-6686	295	5	is	be	AUX
ejpam-6686	295	6	easy	easy	ADJ
ejpam-6686	295	7	to	to	PART
ejpam-6686	295	8	observe	observe	VERB
ejpam-6686	295	9	that	that	SCONJ
ejpam-6686	295	10	hs1	hs1	X
ejpam-6686	295	11	∪hs1	∪hs1	PROPN
ejpam-6686	295	12	=	=	SYM
ejpam-6686	295	13	hs1	hs1	PROPN
ejpam-6686	295	14	,	,	PUNCT
ejpam-6686	295	15	by	by	ADP
ejpam-6686	295	16	lemma	lemma	PROPN
ejpam-6686	295	17	4	4	NUM
ejpam-6686	295	18	(	(	PUNCT
ejpam-6686	295	19	i	i	NOUN
ejpam-6686	295	20	)	)	PUNCT
ejpam-6686	295	21	.	.	PUNCT
ejpam-6686	296	1	(	(	PUNCT
ejpam-6686	296	2	ii	ii	NOUN
ejpam-6686	296	3	)	)	PUNCT
ejpam-6686	296	4	by	by	ADP
ejpam-6686	296	5	lemma	lemma	PROPN
ejpam-6686	296	6	4	4	NUM
ejpam-6686	296	7	(	(	PUNCT
ejpam-6686	296	8	ii	ii	NOUN
ejpam-6686	296	9	)	)	PUNCT
ejpam-6686	296	10	,	,	PUNCT
ejpam-6686	296	11	we	we	PRON
ejpam-6686	296	12	can	can	AUX
ejpam-6686	296	13	prove	prove	VERB
ejpam-6686	296	14	hs1	hs1	PROPN
ejpam-6686	296	15	∪hs2	∪hs2	PROPN
ejpam-6686	296	16	=	=	SYM
ejpam-6686	296	17	hs1∪s2	hs1∪s2	PROPN
ejpam-6686	296	18	=	=	SYM
ejpam-6686	296	19	hs2∪s1	hs2∪s1	NOUN
ejpam-6686	296	20	=	=	PROPN
ejpam-6686	296	21	hs2	hs2	PROPN
ejpam-6686	296	22	∪hs1	∪hs1	PROPN
ejpam-6686	296	23	.	.	PUNCT
ejpam-6686	297	1	(	(	PUNCT
ejpam-6686	297	2	iii	iii	X
ejpam-6686	297	3	)	)	PUNCT
ejpam-6686	297	4	since	since	SCONJ
ejpam-6686	297	5	the	the	DET
ejpam-6686	297	6	set	set	NOUN
ejpam-6686	297	7	union	union	NOUN
ejpam-6686	297	8	satisfies	satisfie	NOUN
ejpam-6686	297	9	associative	associative	ADJ
ejpam-6686	297	10	law	law	NOUN
ejpam-6686	297	11	and	and	CCONJ
ejpam-6686	297	12	by	by	ADP
ejpam-6686	297	13	(	(	PUNCT
ejpam-6686	297	14	ii	ii	NOUN
ejpam-6686	297	15	)	)	PUNCT
ejpam-6686	297	16	,	,	PUNCT
ejpam-6686	297	17	we	we	PRON
ejpam-6686	297	18	have	have	VERB
ejpam-6686	297	19	(	(	PUNCT
ejpam-6686	297	20	hs1	hs1	PROPN
ejpam-6686	297	21	∪	∪	PROPN
ejpam-6686	297	22	hs2	hs2	PROPN
ejpam-6686	297	23	)	)	PUNCT
ejpam-6686	297	24	∪	∪	X
ejpam-6686	297	25	hs3	hs3	NOUN
ejpam-6686	297	26	=	=	SYM
ejpam-6686	297	27	h(s1∪s2)∪s3	h(s1∪s2)∪s3	NOUN
ejpam-6686	298	1	=	=	PUNCT
ejpam-6686	298	2	hs1	hs1	PROPN
ejpam-6686	298	3	∪	∪	X
ejpam-6686	298	4	(	(	PUNCT
ejpam-6686	298	5	hs2	hs2	PROPN
ejpam-6686	298	6	∪hs3	∪hs3	PROPN
ejpam-6686	298	7	)	)	PUNCT
ejpam-6686	298	8	.	.	PUNCT
ejpam-6686	299	1	theorem	theorem	ADJ
ejpam-6686	299	2	10	10	NUM
ejpam-6686	299	3	.	.	PUNCT
ejpam-6686	300	1	for	for	ADP
ejpam-6686	300	2	any	any	DET
ejpam-6686	300	3	non	non	ADJ
ejpam-6686	300	4	-	-	ADJ
ejpam-6686	300	5	empty	empty	ADJ
ejpam-6686	300	6	subset	subset	NOUN
ejpam-6686	300	7	s	s	NOUN
ejpam-6686	300	8	of	of	ADP
ejpam-6686	300	9	l	l	NOUN
ejpam-6686	300	10	and	and	CCONJ
ejpam-6686	300	11	sm	sm	PROPN
ejpam-6686	300	12	is	be	AUX
ejpam-6686	300	13	the	the	DET
ejpam-6686	300	14	set	set	NOUN
ejpam-6686	300	15	of	of	ADP
ejpam-6686	300	16	maximal	maximal	ADJ
ejpam-6686	300	17	elements	element	NOUN
ejpam-6686	300	18	in	in	ADP
ejpam-6686	300	19	l	l	NOUN
ejpam-6686	300	20	,	,	PUNCT
ejpam-6686	300	21	we	we	PRON
ejpam-6686	300	22	have	have	VERB
ejpam-6686	300	23	g.	g.	PROPN
ejpam-6686	300	24	chinnayya	chinnayya	PROPN
ejpam-6686	300	25	et	et	PROPN
ejpam-6686	300	26	al	al	PROPN
ejpam-6686	300	27	.	.	PUNCT
ejpam-6686	300	28	/	/	SYM
ejpam-6686	300	29	eur	eur	PROPN
ejpam-6686	300	30	.	.	PUNCT
ejpam-6686	301	1	j.	j.	PROPN
ejpam-6686	301	2	pure	pure	PROPN
ejpam-6686	301	3	appl	appl	PROPN
ejpam-6686	301	4	.	.	PROPN
ejpam-6686	301	5	math	math	PROPN
ejpam-6686	301	6	,	,	PUNCT
ejpam-6686	301	7	18	18	NUM
ejpam-6686	301	8	(	(	PUNCT
ejpam-6686	301	9	4	4	NUM
ejpam-6686	301	10	)	)	PUNCT
ejpam-6686	301	11	(	(	PUNCT
ejpam-6686	301	12	2025	2025	NUM
ejpam-6686	301	13	)	)	PUNCT
ejpam-6686	301	14	,	,	PUNCT
ejpam-6686	301	15	6686	6686	NUM
ejpam-6686	301	16	11	11	NUM
ejpam-6686	301	17	of	of	ADP
ejpam-6686	301	18	12	12	NUM
ejpam-6686	301	19	(	(	PUNCT
ejpam-6686	301	20	i	i	NOUN
ejpam-6686	301	21	)	)	PUNCT
ejpam-6686	301	22	⋃	⋃	PROPN
ejpam-6686	301	23	s⊆l	s⊆l	PROPN
ejpam-6686	301	24	hs	hs	PROPN
ejpam-6686	301	25	=	=	PROPN
ejpam-6686	301	26	l	l	PROPN
ejpam-6686	301	27	(	(	PUNCT
ejpam-6686	301	28	ii	ii	NOUN
ejpam-6686	301	29	)	)	PUNCT
ejpam-6686	301	30	sm	sm	PROPN
ejpam-6686	301	31	⊆	⊆	NUM
ejpam-6686	301	32	hs	hs	PROPN
ejpam-6686	301	33	(	(	PUNCT
ejpam-6686	301	34	iii	iii	NOUN
ejpam-6686	301	35	)	)	PUNCT
ejpam-6686	301	36	hsm	hsm	NOUN
ejpam-6686	301	37	=	=	SYM
ejpam-6686	301	38	sm	sm	PROPN
ejpam-6686	301	39	(	(	PUNCT
ejpam-6686	301	40	iv	iv	X
ejpam-6686	301	41	)	)	PUNCT
ejpam-6686	301	42	⋂	⋂	PROPN
ejpam-6686	301	43	s⊆l	s⊆l	PROPN
ejpam-6686	301	44	hs	hs	PROPN
ejpam-6686	301	45	=	=	ADJ
ejpam-6686	301	46	sm	sm	PROPN
ejpam-6686	301	47	(	(	PUNCT
ejpam-6686	301	48	v	v	NOUN
ejpam-6686	301	49	)	)	PUNCT
ejpam-6686	301	50	hhs	hhs	PROPN
ejpam-6686	302	1	=	=	SYM
ejpam-6686	302	2	hs	hs	PROPN
ejpam-6686	302	3	.	.	PROPN
ejpam-6686	302	4	proof	proof	NOUN
ejpam-6686	302	5	.	.	PUNCT
ejpam-6686	303	1	(	(	PUNCT
ejpam-6686	303	2	i	i	NOUN
ejpam-6686	303	3	)	)	PUNCT
ejpam-6686	303	4	since	since	SCONJ
ejpam-6686	303	5	l	l	NOUN
ejpam-6686	303	6	⊆	⊆	NUM
ejpam-6686	303	7	hl	hl	NOUN
ejpam-6686	303	8	,	,	PUNCT
ejpam-6686	303	9	hl	hl	NOUN
ejpam-6686	303	10	=	=	SYM
ejpam-6686	303	11	l	l	NOUN
ejpam-6686	303	12	and	and	CCONJ
ejpam-6686	303	13	⋃	⋃	PROPN
ejpam-6686	303	14	s⊆l	s⊆l	PROPN
ejpam-6686	303	15	hs	hs	PROPN
ejpam-6686	303	16	=	=	PROPN
ejpam-6686	303	17	l.	l.	PROPN
ejpam-6686	303	18	(	(	PUNCT
ejpam-6686	303	19	ii	ii	PROPN
ejpam-6686	303	20	)	)	PUNCT
ejpam-6686	303	21	let	let	VERB
ejpam-6686	303	22	m	m	NOUN
ejpam-6686	303	23	∈	∈	NOUN
ejpam-6686	304	1	sm	sm	X
ejpam-6686	304	2	.	.	PUNCT
ejpam-6686	305	1	then	then	ADV
ejpam-6686	305	2	m	m	VERB
ejpam-6686	305	3	∨	∨	PROPN
ejpam-6686	305	4	s	s	PART
ejpam-6686	305	5	=	=	NOUN
ejpam-6686	305	6	m	m	PROPN
ejpam-6686	305	7	,	,	PUNCT
ejpam-6686	305	8	for	for	ADP
ejpam-6686	305	9	all	all	PRON
ejpam-6686	305	10	s	s	PROPN
ejpam-6686	305	11	∈	∈	PROPN
ejpam-6686	305	12	s.	s.	PROPN
ejpam-6686	305	13	therefore	therefore	ADV
ejpam-6686	305	14	,	,	PUNCT
ejpam-6686	305	15	m	m	PROPN
ejpam-6686	305	16	∈	∈	PROPN
ejpam-6686	305	17	hs	hs	PROPN
ejpam-6686	305	18	and	and	CCONJ
ejpam-6686	305	19	sm	sm	PROPN
ejpam-6686	305	20	⊆	⊆	NUM
ejpam-6686	305	21	hs	hs	PROPN
ejpam-6686	305	22	.	.	PUNCT
ejpam-6686	306	1	(	(	PUNCT
ejpam-6686	306	2	iii	iii	X
ejpam-6686	306	3	)	)	PUNCT
ejpam-6686	306	4	since	since	SCONJ
ejpam-6686	306	5	sm	sm	PROPN
ejpam-6686	306	6	is	be	AUX
ejpam-6686	306	7	a	a	DET
ejpam-6686	306	8	filter	filter	NOUN
ejpam-6686	306	9	l	l	NOUN
ejpam-6686	306	10	,	,	PUNCT
ejpam-6686	307	1	hsm	hsm	PROPN
ejpam-6686	307	2	=	=	SYM
ejpam-6686	307	3	sm	sm	PROPN
ejpam-6686	307	4	.	.	PUNCT
ejpam-6686	308	1	(	(	PUNCT
ejpam-6686	308	2	iv	iv	X
ejpam-6686	308	3	)	)	PUNCT
ejpam-6686	308	4	from	from	ADP
ejpam-6686	308	5	(	(	PUNCT
ejpam-6686	308	6	ii	ii	NOUN
ejpam-6686	308	7	)	)	PUNCT
ejpam-6686	308	8	,	,	PUNCT
ejpam-6686	308	9	sm	sm	PROPN
ejpam-6686	308	10	⊆	⊆	NUM
ejpam-6686	308	11	hs	hs	PROPN
ejpam-6686	308	12	,	,	PUNCT
ejpam-6686	308	13	for	for	ADP
ejpam-6686	308	14	all	all	PRON
ejpam-6686	308	15	s	s	PART
ejpam-6686	308	16	⊆	⊆	NUM
ejpam-6686	308	17	l	l	NOUN
ejpam-6686	308	18	,	,	PUNCT
ejpam-6686	308	19	so	so	SCONJ
ejpam-6686	308	20	that	that	SCONJ
ejpam-6686	308	21	sm	sm	VERB
ejpam-6686	308	22	⊆	⊆	NUM
ejpam-6686	308	23	⋂	⋂	PROPN
ejpam-6686	308	24	s⊆l	s⊆l	PROPN
ejpam-6686	308	25	hs	hs	PROPN
ejpam-6686	308	26	.	.	PUNCT
ejpam-6686	309	1	let	let	VERB
ejpam-6686	309	2	h	h	PRON
ejpam-6686	309	3	∈	∈	PROPN
ejpam-6686	309	4	⋂	⋂	PROPN
ejpam-6686	309	5	s⊆l	s⊆l	PROPN
ejpam-6686	309	6	hs	hs	PROPN
ejpam-6686	309	7	.	.	PUNCT
ejpam-6686	310	1	then	then	ADV
ejpam-6686	310	2	h	h	PROPN
ejpam-6686	310	3	∨	∨	PROPN
ejpam-6686	310	4	s	s	PART
ejpam-6686	310	5	=	=	ADJ
ejpam-6686	310	6	h	h	NOUN
ejpam-6686	310	7	,	,	PUNCT
ejpam-6686	310	8	for	for	ADP
ejpam-6686	310	9	some	some	DET
ejpam-6686	310	10	s	s	PART
ejpam-6686	310	11	∈	∈	PROPN
ejpam-6686	310	12	s	s	NOUN
ejpam-6686	310	13	and	and	CCONJ
ejpam-6686	310	14	for	for	ADP
ejpam-6686	310	15	all	all	PRON
ejpam-6686	310	16	s	s	PART
ejpam-6686	310	17	⊆	⊆	NUM
ejpam-6686	310	18	l.	l.	NOUN
ejpam-6686	310	19	let	let	VERB
ejpam-6686	310	20	m	m	PRON
ejpam-6686	310	21	∈	∈	NOUN
ejpam-6686	310	22	sm	sm	X
ejpam-6686	310	23	.	.	PUNCT
ejpam-6686	311	1	since	since	SCONJ
ejpam-6686	311	2	sm	sm	PROPN
ejpam-6686	311	3	is	be	AUX
ejpam-6686	311	4	a	a	DET
ejpam-6686	311	5	filter	filter	NOUN
ejpam-6686	311	6	of	of	ADP
ejpam-6686	311	7	l	l	NOUN
ejpam-6686	311	8	,	,	PUNCT
ejpam-6686	311	9	h	h	NOUN
ejpam-6686	311	10	=	=	NOUN
ejpam-6686	312	1	h	h	NOUN
ejpam-6686	312	2	∨m	∨m	NOUN
ejpam-6686	312	3	∈	∈	PROPN
ejpam-6686	312	4	sm	sm	INTJ
ejpam-6686	312	5	.	.	PUNCT
ejpam-6686	313	1	therefore	therefore	ADV
ejpam-6686	313	2	,	,	PUNCT
ejpam-6686	313	3	⋂	⋂	PROPN
ejpam-6686	313	4	s⊆l	s⊆l	PROPN
ejpam-6686	313	5	hs	hs	PRON
ejpam-6686	313	6	⊆	⊆	NUM
ejpam-6686	313	7	sm	sm	ADV
ejpam-6686	313	8	and	and	CCONJ
ejpam-6686	313	9	hence	hence	ADV
ejpam-6686	313	10	⋂	⋂	PROPN
ejpam-6686	313	11	s⊆l	s⊆l	PROPN
ejpam-6686	313	12	hs	hs	PROPN
ejpam-6686	313	13	=	=	NOUN
ejpam-6686	313	14	sm	sm	PROPN
ejpam-6686	313	15	.	.	PUNCT
ejpam-6686	314	1	(	(	PUNCT
ejpam-6686	314	2	v	v	NOUN
ejpam-6686	314	3	)	)	PUNCT
ejpam-6686	314	4	since	since	SCONJ
ejpam-6686	314	5	s	s	PROPN
ejpam-6686	314	6	⊆	⊆	NUM
ejpam-6686	314	7	hs	hs	PROPN
ejpam-6686	314	8	,	,	PUNCT
ejpam-6686	314	9	hs	hs	PROPN
ejpam-6686	314	10	⊆	⊆	NUM
ejpam-6686	314	11	hhs	hhs	NOUN
ejpam-6686	314	12	.	.	PUNCT
ejpam-6686	315	1	let	let	VERB
ejpam-6686	315	2	h	h	PRON
ejpam-6686	315	3	∈	∈	PROPN
ejpam-6686	315	4	hhs	hhs	PROPN
ejpam-6686	315	5	.	.	PUNCT
ejpam-6686	316	1	then	then	ADV
ejpam-6686	316	2	h	h	PROPN
ejpam-6686	316	3	∨	∨	PROPN
ejpam-6686	316	4	t	t	PROPN
ejpam-6686	316	5	=	=	SYM
ejpam-6686	316	6	h	h	NOUN
ejpam-6686	316	7	,	,	PUNCT
ejpam-6686	316	8	for	for	ADP
ejpam-6686	316	9	some	some	DET
ejpam-6686	316	10	t	t	NOUN
ejpam-6686	316	11	∈	∈	PROPN
ejpam-6686	316	12	hs	hs	PROPN
ejpam-6686	316	13	.	.	PUNCT
ejpam-6686	317	1	for	for	ADP
ejpam-6686	317	2	this	this	DET
ejpam-6686	317	3	t	t	PROPN
ejpam-6686	317	4	∈	∈	PROPN
ejpam-6686	317	5	hs	hs	PROPN
ejpam-6686	317	6	,	,	PUNCT
ejpam-6686	317	7	t	t	PROPN
ejpam-6686	317	8	∨	∨	NUM
ejpam-6686	317	9	s	s	PART
ejpam-6686	317	10	=	=	X
ejpam-6686	317	11	t	t	PROPN
ejpam-6686	317	12	,	,	PUNCT
ejpam-6686	317	13	for	for	ADP
ejpam-6686	317	14	some	some	DET
ejpam-6686	317	15	s	s	NOUN
ejpam-6686	317	16	∈	∈	PROPN
ejpam-6686	317	17	s.	s.	PROPN
ejpam-6686	317	18	now	now	ADV
ejpam-6686	317	19	,	,	PUNCT
ejpam-6686	317	20	h	h	NOUN
ejpam-6686	317	21	∧	∧	PROPN
ejpam-6686	317	22	s	s	PART
ejpam-6686	317	23	=	=	NOUN
ejpam-6686	317	24	h	h	NOUN
ejpam-6686	317	25	∧	∧	PROPN
ejpam-6686	317	26	(	(	PUNCT
ejpam-6686	317	27	t	t	PROPN
ejpam-6686	317	28	∧	∧	PROPN
ejpam-6686	317	29	s	s	PART
ejpam-6686	317	30	)	)	PUNCT
ejpam-6686	317	31	=	=	SYM
ejpam-6686	317	32	(	(	PUNCT
ejpam-6686	317	33	h	h	NOUN
ejpam-6686	317	34	∧	∧	PROPN
ejpam-6686	317	35	t	t	PROPN
ejpam-6686	317	36	)	)	PUNCT
ejpam-6686	317	37	∧	∧	PROPN
ejpam-6686	317	38	s	s	PART
ejpam-6686	317	39	=	=	SYM
ejpam-6686	317	40	t	t	PROPN
ejpam-6686	317	41	∧	∧	PROPN
ejpam-6686	317	42	s	s	PART
ejpam-6686	317	43	=	=	PUNCT
ejpam-6686	317	44	s.	s.	PROPN
ejpam-6686	317	45	then	then	ADV
ejpam-6686	317	46	h	h	PROPN
ejpam-6686	317	47	∨	∨	PROPN
ejpam-6686	317	48	s	s	PART
ejpam-6686	317	49	=	=	ADJ
ejpam-6686	317	50	h	h	NOUN
ejpam-6686	317	51	,	,	PUNCT
ejpam-6686	317	52	for	for	ADP
ejpam-6686	317	53	some	some	DET
ejpam-6686	317	54	s	s	ADP
ejpam-6686	317	55	∈	∈	PROPN
ejpam-6686	317	56	s.	s.	PROPN
ejpam-6686	317	57	therefore	therefore	ADV
ejpam-6686	317	58	,	,	PUNCT
ejpam-6686	317	59	h	h	PROPN
ejpam-6686	317	60	∈	∈	PROPN
ejpam-6686	317	61	hs	hs	PROPN
ejpam-6686	317	62	and	and	CCONJ
ejpam-6686	317	63	hhs	hhs	PROPN
ejpam-6686	317	64	⊆	⊆	NUM
ejpam-6686	317	65	hs	hs	PROPN
ejpam-6686	317	66	.	.	PUNCT
ejpam-6686	318	1	thus	thus	ADV
ejpam-6686	318	2	,	,	PUNCT
ejpam-6686	318	3	hhs	hhs	PROPN
ejpam-6686	318	4	=	=	SYM
ejpam-6686	318	5	hs	hs	PROPN
ejpam-6686	318	6	.	.	PUNCT
ejpam-6686	319	1	5	5	X
ejpam-6686	319	2	.	.	X
ejpam-6686	319	3	conclusion	conclusion	NOUN
ejpam-6686	319	4	these	these	DET
ejpam-6686	319	5	findings	finding	NOUN
ejpam-6686	319	6	not	not	PART
ejpam-6686	319	7	only	only	ADV
ejpam-6686	319	8	clarify	clarify	VERB
ejpam-6686	319	9	the	the	DET
ejpam-6686	319	10	distinctions	distinction	NOUN
ejpam-6686	319	11	and	and	CCONJ
ejpam-6686	319	12	relationships	relationship	NOUN
ejpam-6686	319	13	between	between	ADP
ejpam-6686	319	14	prime	prime	ADJ
ejpam-6686	319	15	,	,	PUNCT
ejpam-6686	319	16	maximal	maximal	ADJ
ejpam-6686	319	17	,	,	PUNCT
ejpam-6686	319	18	and	and	CCONJ
ejpam-6686	319	19	inverted	inverted	ADJ
ejpam-6686	319	20	-	-	PUNCT
ejpam-6686	319	21	hierarchy	hierarchy	NOUN
ejpam-6686	319	22	sets	set	NOUN
ejpam-6686	319	23	but	but	CCONJ
ejpam-6686	319	24	also	also	ADV
ejpam-6686	319	25	enrich	enrich	VERB
ejpam-6686	319	26	the	the	DET
ejpam-6686	319	27	broader	broad	ADJ
ejpam-6686	319	28	theory	theory	NOUN
ejpam-6686	319	29	of	of	ADP
ejpam-6686	319	30	lattices	lattice	NOUN
ejpam-6686	319	31	by	by	ADP
ejpam-6686	319	32	highlighting	highlight	VERB
ejpam-6686	319	33	how	how	SCONJ
ejpam-6686	319	34	such	such	ADJ
ejpam-6686	319	35	structures	structure	NOUN
ejpam-6686	319	36	interact	interact	VERB
ejpam-6686	319	37	with	with	ADP
ejpam-6686	319	38	classical	classical	ADJ
ejpam-6686	319	39	notions	notion	NOUN
ejpam-6686	319	40	,	,	PUNCT
ejpam-6686	319	41	such	such	ADJ
ejpam-6686	319	42	as	as	ADP
ejpam-6686	319	43	ideals	ideal	NOUN
ejpam-6686	319	44	and	and	CCONJ
ejpam-6686	319	45	filters	filter	NOUN
ejpam-6686	319	46	,	,	PUNCT
ejpam-6686	319	47	in	in	ADP
ejpam-6686	319	48	almost	almost	ADV
ejpam-6686	319	49	distributive	distributive	ADJ
ejpam-6686	319	50	lattices	lattice	NOUN
ejpam-6686	319	51	.	.	PUNCT
ejpam-6686	320	1	acknowledgements	acknowledgement	NOUN
ejpam-6686	320	2	this	this	DET
ejpam-6686	320	3	research	research	NOUN
ejpam-6686	320	4	was	be	AUX
ejpam-6686	320	5	supported	support	VERB
ejpam-6686	320	6	by	by	ADP
ejpam-6686	320	7	university	university	NOUN
ejpam-6686	320	8	of	of	ADP
ejpam-6686	320	9	phayao	phayao	NOUN
ejpam-6686	320	10	and	and	CCONJ
ejpam-6686	320	11	thailand	thailand	PROPN
ejpam-6686	320	12	science	science	PROPN
ejpam-6686	320	13	research	research	PROPN
ejpam-6686	320	14	and	and	CCONJ
ejpam-6686	320	15	innovation	innovation	NOUN
ejpam-6686	320	16	fund	fund	NOUN
ejpam-6686	320	17	(	(	PUNCT
ejpam-6686	320	18	fundamental	fundamental	ADJ
ejpam-6686	320	19	fund	fund	NOUN
ejpam-6686	320	20	2026	2026	NUM
ejpam-6686	320	21	,	,	PUNCT
ejpam-6686	320	22	grant	grant	VERB
ejpam-6686	320	23	no	no	NOUN
ejpam-6686	320	24	.	.	PUNCT
ejpam-6686	320	25	2252/2568	2252/2568	NUM
ejpam-6686	320	26	)	)	PUNCT
ejpam-6686	320	27	.	.	PUNCT
ejpam-6686	321	1	references	reference	NOUN
ejpam-6686	321	2	[	[	X
ejpam-6686	321	3	1	1	NUM
ejpam-6686	321	4	]	]	X
ejpam-6686	321	5	n.	n.	PROPN
ejpam-6686	321	6	h.	h.	PROPN
ejpam-6686	321	7	mccoy	mccoy	PROPN
ejpam-6686	321	8	and	and	CCONJ
ejpam-6686	321	9	d.	d.	PROPN
ejpam-6686	321	10	mantgomery	mantgomery	PROPN
ejpam-6686	321	11	.	.	PUNCT
ejpam-6686	322	1	a	a	DET
ejpam-6686	322	2	representation	representation	NOUN
ejpam-6686	322	3	of	of	ADP
ejpam-6686	322	4	generalized	generalized	ADJ
ejpam-6686	322	5	boolean	boolean	ADJ
ejpam-6686	322	6	rings	ring	NOUN
ejpam-6686	322	7	.	.	PUNCT
ejpam-6686	323	1	duke	duke	PROPN
ejpam-6686	323	2	.	.	PUNCT
ejpam-6686	324	1	math	math	PROPN
ejpam-6686	324	2	.	.	PUNCT
ejpam-6686	325	1	j.	j.	PROPN
ejpam-6686	325	2	,	,	PUNCT
ejpam-6686	325	3	3:455–459	3:455–459	PROPN
ejpam-6686	325	4	,	,	PUNCT
ejpam-6686	325	5	1937	1937	NUM
ejpam-6686	325	6	.	.	PUNCT
ejpam-6686	326	1	[	[	X
ejpam-6686	326	2	2	2	NUM
ejpam-6686	326	3	]	]	PUNCT
ejpam-6686	326	4	a.	a.	NOUN
ejpam-6686	326	5	a.	a.	PROPN
ejpam-6686	326	6	meuborn	meuborn	PROPN
ejpam-6686	326	7	.	.	PUNCT
ejpam-6686	327	1	regular	regular	ADJ
ejpam-6686	327	2	rings	ring	NOUN
ejpam-6686	327	3	and	and	CCONJ
ejpam-6686	327	4	baer	baer	PROPN
ejpam-6686	327	5	rings	rings	PROPN
ejpam-6686	327	6	.	.	PUNCT
ejpam-6686	328	1	math	math	PROPN
ejpam-6686	328	2	.	.	PUNCT
ejpam-6686	329	1	z.	z.	PROPN
ejpam-6686	329	2	,	,	PUNCT
ejpam-6686	329	3	121:211–219	121:211–219	NUM
ejpam-6686	329	4	,	,	PUNCT
ejpam-6686	329	5	1971	1971	NUM
ejpam-6686	329	6	.	.	PUNCT
ejpam-6686	330	1	[	[	X
ejpam-6686	330	2	3	3	NUM
ejpam-6686	330	3	]	]	X
ejpam-6686	330	4	m.	m.	NOUN
ejpam-6686	330	5	h.	h.	PROPN
ejpam-6686	330	6	stone	stone	PROPN
ejpam-6686	330	7	.	.	PUNCT
ejpam-6686	331	1	the	the	DET
ejpam-6686	331	2	theory	theory	NOUN
ejpam-6686	331	3	of	of	ADP
ejpam-6686	331	4	representation	representation	NOUN
ejpam-6686	331	5	for	for	ADP
ejpam-6686	331	6	boolean	boolean	ADJ
ejpam-6686	331	7	algebras	algebra	NOUN
ejpam-6686	331	8	.	.	PUNCT
ejpam-6686	332	1	trans	trans	AUX
ejpam-6686	332	2	.	.	PROPN
ejpam-6686	332	3	am	be	AUX
ejpam-6686	332	4	.	.	PUNCT
ejpam-6686	333	1	math	math	NOUN
ejpam-6686	333	2	.	.	PUNCT
ejpam-6686	334	1	soc	soc	PROPN
ejpam-6686	334	2	.	.	PUNCT
ejpam-6686	334	3	,	,	PUNCT
ejpam-6686	334	4	40:37–111	40:37–111	NUM
ejpam-6686	334	5	,	,	PUNCT
ejpam-6686	334	6	1936	1936	NUM
ejpam-6686	334	7	.	.	PUNCT
ejpam-6686	335	1	[	[	X
ejpam-6686	335	2	4	4	X
ejpam-6686	335	3	]	]	PUNCT
ejpam-6686	335	4	m.	m.	NOUN
ejpam-6686	335	5	h.	h.	PROPN
ejpam-6686	335	6	stone	stone	PROPN
ejpam-6686	335	7	.	.	PUNCT
ejpam-6686	336	1	topological	topological	ADJ
ejpam-6686	336	2	representation	representation	NOUN
ejpam-6686	336	3	of	of	ADP
ejpam-6686	336	4	distributive	distributive	ADJ
ejpam-6686	336	5	lattices	lattice	NOUN
ejpam-6686	336	6	and	and	CCONJ
ejpam-6686	336	7	brouwerian	brouwerian	ADJ
ejpam-6686	336	8	logics	logic	NOUN
ejpam-6686	336	9	.	.	PUNCT
ejpam-6686	337	1	čas	čas	PROPN
ejpam-6686	337	2	.	.	PROPN
ejpam-6686	337	3	mat	mat	PROPN
ejpam-6686	337	4	.	.	PUNCT
ejpam-6686	338	1	fys	fys	PROPN
ejpam-6686	338	2	.	.	PROPN
ejpam-6686	338	3	,	,	PUNCT
ejpam-6686	338	4	1:1–25	1:1–25	NUM
ejpam-6686	338	5	,	,	PUNCT
ejpam-6686	338	6	1937	1937	NUM
ejpam-6686	338	7	.	.	PUNCT
ejpam-6686	339	1	g.	g.	PROPN
ejpam-6686	339	2	chinnayya	chinnayya	PROPN
ejpam-6686	339	3	et	et	PROPN
ejpam-6686	339	4	al	al	PROPN
ejpam-6686	339	5	.	.	PUNCT
ejpam-6686	339	6	/	/	SYM
ejpam-6686	339	7	eur	eur	PROPN
ejpam-6686	339	8	.	.	PUNCT
ejpam-6686	340	1	j.	j.	PROPN
ejpam-6686	340	2	pure	pure	PROPN
ejpam-6686	340	3	appl	appl	PROPN
ejpam-6686	340	4	.	.	PROPN
ejpam-6686	340	5	math	math	PROPN
ejpam-6686	340	6	,	,	PUNCT
ejpam-6686	340	7	18	18	NUM
ejpam-6686	340	8	(	(	PUNCT
ejpam-6686	340	9	4	4	NUM
ejpam-6686	340	10	)	)	PUNCT
ejpam-6686	340	11	(	(	PUNCT
ejpam-6686	340	12	2025	2025	NUM
ejpam-6686	340	13	)	)	PUNCT
ejpam-6686	340	14	,	,	PUNCT
ejpam-6686	340	15	6686	6686	NUM
ejpam-6686	340	16	12	12	NUM
ejpam-6686	340	17	of	of	ADP
ejpam-6686	340	18	12	12	NUM
ejpam-6686	340	19	[	[	SYM
ejpam-6686	340	20	5	5	NUM
ejpam-6686	340	21	]	]	PUNCT
ejpam-6686	340	22	i.	i.	PROPN
ejpam-6686	340	23	sussman	sussman	PROPN
ejpam-6686	340	24	.	.	PUNCT
ejpam-6686	341	1	a	a	DET
ejpam-6686	341	2	generalization	generalization	NOUN
ejpam-6686	341	3	of	of	ADP
ejpam-6686	341	4	boolean	boolean	ADJ
ejpam-6686	341	5	rings	ring	NOUN
ejpam-6686	341	6	.	.	PUNCT
ejpam-6686	342	1	math	math	NOUN
ejpam-6686	342	2	.	.	PUNCT
ejpam-6686	343	1	ann	ann	PROPN
ejpam-6686	343	2	.	.	PROPN
ejpam-6686	343	3	,	,	PUNCT
ejpam-6686	343	4	136:326–338	136:326–338	NUM
ejpam-6686	343	5	,	,	PUNCT
ejpam-6686	343	6	1981	1981	NUM
ejpam-6686	343	7	.	.	PUNCT
ejpam-6686	344	1	[	[	X
ejpam-6686	344	2	6	6	NUM
ejpam-6686	344	3	]	]	PUNCT
ejpam-6686	344	4	j.	j.	PROPN
ejpam-6686	344	5	von	von	PROPN
ejpam-6686	344	6	neuman	neuman	PROPN
ejpam-6686	344	7	.	.	PUNCT
ejpam-6686	345	1	on	on	ADP
ejpam-6686	345	2	regular	regular	ADJ
ejpam-6686	345	3	rings	ring	NOUN
ejpam-6686	345	4	.	.	PUNCT
ejpam-6686	346	1	proc	proc	PROPN
ejpam-6686	346	2	.	.	PUNCT
ejpam-6686	347	1	nat	nat	PROPN
ejpam-6686	347	2	.	.	PUNCT
ejpam-6686	348	1	acad	acad	PROPN
ejpam-6686	348	2	.	.	PUNCT
ejpam-6686	349	1	sci	sci	PROPN
ejpam-6686	349	2	.	.	PROPN
ejpam-6686	349	3	,	,	PUNCT
ejpam-6686	349	4	u.s.a	u.s.a	PROPN
ejpam-6686	349	5	.	.	PROPN
ejpam-6686	349	6	,	,	PUNCT
ejpam-6686	349	7	22:707–713	22:707–713	NUM
ejpam-6686	349	8	,	,	PUNCT
ejpam-6686	349	9	1963	1963	NUM
ejpam-6686	349	10	.	.	PUNCT
ejpam-6686	350	1	[	[	X
ejpam-6686	350	2	7	7	X
ejpam-6686	350	3	]	]	X
ejpam-6686	350	4	s.	s.	PROPN
ejpam-6686	350	5	ramesh	ramesh	PROPN
ejpam-6686	350	6	,	,	PUNCT
ejpam-6686	350	7	g.	g.	PROPN
ejpam-6686	350	8	chinnayya	chinnayya	PROPN
ejpam-6686	350	9	,	,	PUNCT
ejpam-6686	350	10	g.	g.	PROPN
ejpam-6686	350	11	jogarao	jogarao	PROPN
ejpam-6686	350	12	,	,	PUNCT
ejpam-6686	350	13	and	and	CCONJ
ejpam-6686	350	14	a.	a.	NOUN
ejpam-6686	350	15	iampan	iampan	PROPN
ejpam-6686	350	16	.	.	PUNCT
ejpam-6686	351	1	hierarchy	hierarchy	NOUN
ejpam-6686	351	2	sets	set	NOUN
ejpam-6686	351	3	in	in	ADP
ejpam-6686	351	4	almost	almost	ADV
ejpam-6686	351	5	distributive	distributive	ADJ
ejpam-6686	351	6	lattice	lattice	NOUN
ejpam-6686	351	7	.	.	PUNCT
ejpam-6686	352	1	eur	eur	PROPN
ejpam-6686	352	2	.	.	PUNCT
ejpam-6686	353	1	j.	j.	PROPN
ejpam-6686	353	2	pure	pure	PROPN
ejpam-6686	353	3	appl	appl	PROPN
ejpam-6686	353	4	.	.	PUNCT
ejpam-6686	353	5	math	math	PROPN
ejpam-6686	353	6	.	.	PUNCT
ejpam-6686	353	7	,	,	PUNCT
ejpam-6686	353	8	17(3):1691–1704	17(3):1691–1704	NUM
ejpam-6686	353	9	,	,	PUNCT
ejpam-6686	353	10	2024	2024	NUM
ejpam-6686	353	11	.	.	PUNCT
ejpam-6686	354	1	[	[	X
ejpam-6686	354	2	8	8	NUM
ejpam-6686	354	3	]	]	X
ejpam-6686	354	4	u.	u.	PROPN
ejpam-6686	354	5	m.	m.	PROPN
ejpam-6686	354	6	swamy	swamy	PROPN
ejpam-6686	354	7	and	and	CCONJ
ejpam-6686	354	8	g.	g.	PROPN
ejpam-6686	354	9	c.	c.	PROPN
ejpam-6686	354	10	rao	rao	PROPN
ejpam-6686	354	11	.	.	PUNCT
ejpam-6686	355	1	almost	almost	ADV
ejpam-6686	355	2	distributive	distributive	ADJ
ejpam-6686	355	3	lattices	lattice	NOUN
ejpam-6686	355	4	.	.	PUNCT
ejpam-6686	356	1	j.	j.	PROPN
ejpam-6686	356	2	aust	aust	PROPN
ejpam-6686	356	3	.	.	PUNCT
ejpam-6686	357	1	math	math	PROPN
ejpam-6686	357	2	.	.	PUNCT
ejpam-6686	358	1	soc	soc	PROPN
ejpam-6686	358	2	.	.	PUNCT
ejpam-6686	358	3	,	,	PUNCT
ejpam-6686	358	4	ser	ser	PROPN
ejpam-6686	358	5	.	.	PUNCT
ejpam-6686	359	1	a	a	DET
ejpam-6686	359	2	,	,	PUNCT
ejpam-6686	359	3	31:77–91	31:77–91	NUM
ejpam-6686	359	4	,	,	PUNCT
ejpam-6686	359	5	1981	1981	NUM
ejpam-6686	359	6	.	.	PUNCT
ejpam-6686	360	1	[	[	X
ejpam-6686	360	2	9	9	NUM
ejpam-6686	360	3	]	]	X
ejpam-6686	360	4	g.	g.	NOUN
ejpam-6686	360	5	birkhoff	birkhoff	PROPN
ejpam-6686	360	6	.	.	PUNCT
ejpam-6686	361	1	lattice	lattice	PROPN
ejpam-6686	361	2	theory	theory	PROPN
ejpam-6686	361	3	.	.	PUNCT
ejpam-6686	362	1	amer	amer	PROPN
ejpam-6686	362	2	.	.	PUNCT
ejpam-6686	362	3	math	math	PROPN
ejpam-6686	362	4	.	.	PUNCT
ejpam-6686	363	1	soc	soc	PROPN
ejpam-6686	363	2	.	.	PUNCT
ejpam-6686	364	1	colloq	colloq	PROPN
ejpam-6686	364	2	.	.	PUNCT
ejpam-6686	365	1	xxv	xxv	PROPN
ejpam-6686	365	2	,	,	PUNCT
ejpam-6686	365	3	providence	providence	NOUN
ejpam-6686	365	4	,	,	PUNCT
ejpam-6686	365	5	u.s.a	u.s.a	PROPN
ejpam-6686	365	6	.	.	PROPN
ejpam-6686	365	7	,	,	PUNCT
ejpam-6686	365	8	1967	1967	NUM
ejpam-6686	365	9	.	.	PUNCT
ejpam-6686	366	1	[	[	X
ejpam-6686	366	2	10	10	NUM
ejpam-6686	366	3	]	]	X
ejpam-6686	366	4	w.	w.	PROPN
ejpam-6686	366	5	h.	h.	PROPN
ejpam-6686	366	6	cornish	cornish	PROPN
ejpam-6686	366	7	.	.	PUNCT
ejpam-6686	367	1	quasi	quasi	PROPN
ejpam-6686	367	2	complemented	complemented	ADJ
ejpam-6686	367	3	lattices	lattice	NOUN
ejpam-6686	367	4	.	.	PUNCT
ejpam-6686	368	1	comment	comment	NOUN
ejpam-6686	368	2	.	.	PUNCT
ejpam-6686	369	1	math	math	NOUN
ejpam-6686	369	2	.	.	PUNCT
ejpam-6686	370	1	univ	univ	PROPN
ejpam-6686	370	2	.	.	PUNCT
ejpam-6686	371	1	carolin	carolin	PROPN
ejpam-6686	371	2	.	.	PROPN
ejpam-6686	371	3	,	,	PUNCT
ejpam-6686	371	4	15:501–511	15:501–511	NUM
ejpam-6686	371	5	,	,	PUNCT
ejpam-6686	371	6	1974	1974	NUM
ejpam-6686	371	7	.	.	PUNCT
ejpam-6686	372	1	[	[	X
ejpam-6686	372	2	11	11	NUM
ejpam-6686	372	3	]	]	PUNCT
ejpam-6686	372	4	w.	w.	PROPN
ejpam-6686	372	5	h.	h.	PROPN
ejpam-6686	372	6	cornish	cornish	PROPN
ejpam-6686	372	7	.	.	PUNCT
ejpam-6686	373	1	normal	normal	ADJ
ejpam-6686	373	2	lattices	lattice	NOUN
ejpam-6686	373	3	.	.	PUNCT
ejpam-6686	374	1	j.	j.	PROPN
ejpam-6686	374	2	aust	aust	PROPN
ejpam-6686	374	3	.	.	PUNCT
ejpam-6686	375	1	math	math	PROPN
ejpam-6686	375	2	.	.	PUNCT
ejpam-6686	376	1	soc	soc	PROPN
ejpam-6686	376	2	.	.	PUNCT
ejpam-6686	376	3	,	,	PUNCT
ejpam-6686	376	4	14:200–215	14:200–215	NUM
ejpam-6686	376	5	,	,	PUNCT
ejpam-6686	376	6	1972	1972	NUM
ejpam-6686	376	7	.	.	PUNCT
ejpam-6686	377	1	[	[	X
ejpam-6686	377	2	12	12	NUM
ejpam-6686	377	3	]	]	PUNCT
ejpam-6686	377	4	r.-s	r.-	NOUN
ejpam-6686	377	5	.	.	PUNCT
ejpam-6686	378	1	dong	dong	PROPN
ejpam-6686	378	2	.	.	PUNCT
ejpam-6686	379	1	relatively	relatively	ADV
ejpam-6686	379	2	complemented	complement	VERB
ejpam-6686	379	3	distributive	distributive	ADJ
ejpam-6686	379	4	lattice	lattice	NOUN
ejpam-6686	379	5	.	.	PUNCT
ejpam-6686	380	1	sci	sci	PROPN
ejpam-6686	380	2	.	.	PUNCT
ejpam-6686	381	1	china	china	PROPN
ejpam-6686	381	2	,	,	PUNCT
ejpam-6686	381	3	ser	ser	PROPN
ejpam-6686	381	4	.	.	PUNCT
ejpam-6686	382	1	a	a	DET
ejpam-6686	382	2	,	,	PUNCT
ejpam-6686	382	3	34(12):1427–1437	34(12):1427–1437	NUM
ejpam-6686	382	4	,	,	PUNCT
ejpam-6686	382	5	1991	1991	NUM
ejpam-6686	382	6	.	.	PUNCT
ejpam-6686	383	1	[	[	X
ejpam-6686	383	2	13	13	NUM
ejpam-6686	383	3	]	]	X
ejpam-6686	383	4	g.	g.	PROPN
ejpam-6686	383	5	grätzer	grätzer	PROPN
ejpam-6686	383	6	and	and	CCONJ
ejpam-6686	383	7	e.	e.	PROPN
ejpam-6686	383	8	t.	t.	PROPN
ejpam-6686	383	9	schmidt	schmidt	PROPN
ejpam-6686	383	10	.	.	PUNCT
ejpam-6686	384	1	characterizations	characterization	NOUN
ejpam-6686	384	2	of	of	ADP
ejpam-6686	384	3	relatively	relatively	ADV
ejpam-6686	384	4	complemented	complemented	ADJ
ejpam-6686	384	5	distributive	distributive	ADJ
ejpam-6686	384	6	lattices	lattice	NOUN
ejpam-6686	384	7	.	.	PUNCT
ejpam-6686	385	1	publ	publ	NOUN
ejpam-6686	385	2	.	.	PUNCT
ejpam-6686	386	1	math	math	NOUN
ejpam-6686	386	2	.	.	PUNCT
ejpam-6686	387	1	debr	debr	PROPN
ejpam-6686	387	2	.	.	PUNCT
ejpam-6686	388	1	,	,	PUNCT
ejpam-6686	389	1	5(3	5(3	NUM
ejpam-6686	389	2	-	-	SYM
ejpam-6686	389	3	4):257–287	4):257–287	NUM
ejpam-6686	389	4	,	,	PUNCT
ejpam-6686	389	5	1958	1958	NUM
ejpam-6686	389	6	.	.	PUNCT
ejpam-6686	390	1	[	[	X
ejpam-6686	390	2	14	14	NUM
ejpam-6686	390	3	]	]	X
ejpam-6686	390	4	s.	s.	PROPN
ejpam-6686	390	5	ramesh	ramesh	PROPN
ejpam-6686	390	6	and	and	CCONJ
ejpam-6686	390	7	g.	g.	PROPN
ejpam-6686	390	8	jogarao	jogarao	PROPN
ejpam-6686	390	9	.	.	PUNCT
ejpam-6686	391	1	characterizations	characterization	NOUN
ejpam-6686	391	2	of	of	ADP
ejpam-6686	391	3	weak	weak	ADJ
ejpam-6686	391	4	relatively	relatively	ADV
ejpam-6686	391	5	complemented	complemented	ADJ
ejpam-6686	391	6	almost	almost	ADV
ejpam-6686	391	7	distributive	distributive	ADJ
ejpam-6686	391	8	lattices	lattice	NOUN
ejpam-6686	391	9	.	.	PUNCT
ejpam-6686	392	1	int	int	NOUN
ejpam-6686	392	2	.	.	PUNCT
ejpam-6686	393	1	j.	j.	PROPN
ejpam-6686	393	2	open	open	PROPN
ejpam-6686	393	3	probl	probl	PROPN
ejpam-6686	393	4	.	.	PUNCT
ejpam-6686	394	1	compt	compt	PROPN
ejpam-6686	394	2	.	.	PUNCT
ejpam-6686	395	1	math	math	PROPN
ejpam-6686	395	2	.	.	PUNCT
ejpam-6686	396	1	,	,	PUNCT
ejpam-6686	396	2	10(1):11–23	10(1):11–23	NUM
ejpam-6686	396	3	,	,	PUNCT
ejpam-6686	396	4	2017	2017	NUM
ejpam-6686	396	5	.	.	PUNCT
ejpam-6686	397	1	[	[	X
ejpam-6686	397	2	15	15	NUM
ejpam-6686	397	3	]	]	X
ejpam-6686	397	4	s.	s.	PROPN
ejpam-6686	397	5	ramesh	ramesh	PROPN
ejpam-6686	397	6	and	and	CCONJ
ejpam-6686	397	7	g.	g.	PROPN
ejpam-6686	397	8	jogarao	jogarao	PROPN
ejpam-6686	397	9	.	.	PUNCT
ejpam-6686	398	1	normal	normal	ADJ
ejpam-6686	398	2	filters	filter	NOUN
ejpam-6686	398	3	in	in	ADP
ejpam-6686	398	4	almost	almost	ADV
ejpam-6686	398	5	distributive	distributive	ADJ
ejpam-6686	398	6	lattices	lattice	NOUN
ejpam-6686	398	7	.	.	PUNCT
ejpam-6686	399	1	j.	j.	PROPN
ejpam-6686	399	2	int	int	PROPN
ejpam-6686	399	3	.	.	PUNCT
ejpam-6686	400	1	math	math	NOUN
ejpam-6686	400	2	.	.	PUNCT
ejpam-6686	401	1	virtual	virtual	ADJ
ejpam-6686	401	2	inst	inst	PROPN
ejpam-6686	401	3	.	.	PROPN
ejpam-6686	401	4	,	,	PUNCT
ejpam-6686	401	5	7:37–51	7:37–51	NUM
ejpam-6686	401	6	,	,	PUNCT
ejpam-6686	401	7	2017	2017	NUM
ejpam-6686	401	8	.	.	PUNCT
ejpam-6686	402	1	[	[	X
ejpam-6686	402	2	16	16	NUM
ejpam-6686	402	3	]	]	X
ejpam-6686	402	4	s.	s.	PROPN
ejpam-6686	402	5	ramesh	ramesh	PROPN
ejpam-6686	402	6	and	and	CCONJ
ejpam-6686	402	7	g.	g.	PROPN
ejpam-6686	402	8	jogarao	jogarao	PROPN
ejpam-6686	402	9	.	.	PUNCT
ejpam-6686	403	1	weak	weak	ADJ
ejpam-6686	403	2	relative	relative	ADJ
ejpam-6686	403	3	complements	complement	NOUN
ejpam-6686	403	4	in	in	ADP
ejpam-6686	403	5	almost	almost	ADV
ejpam-6686	403	6	distributive	distributive	ADJ
ejpam-6686	403	7	lattices	lattice	NOUN
ejpam-6686	403	8	.	.	PUNCT
ejpam-6686	404	1	discuss	discuss	PROPN
ejpam-6686	404	2	.	.	PUNCT
ejpam-6686	404	3	math	math	PROPN
ejpam-6686	404	4	.	.	PUNCT
ejpam-6686	404	5	,	,	PUNCT
ejpam-6686	404	6	gen	gen	PROPN
ejpam-6686	404	7	.	.	PROPN
ejpam-6686	404	8	algebra	algebra	PROPN
ejpam-6686	404	9	appl	appl	PROPN
ejpam-6686	404	10	.	.	PROPN
ejpam-6686	404	11	,	,	PUNCT
ejpam-6686	404	12	38:5–18	38:5–18	NUM
ejpam-6686	404	13	,	,	PUNCT
ejpam-6686	404	14	2018	2018	NUM
ejpam-6686	404	15	.	.	PUNCT
ejpam-6686	405	1	[	[	X
ejpam-6686	405	2	17	17	NUM
ejpam-6686	405	3	]	]	X
ejpam-6686	405	4	s.	s.	PROPN
ejpam-6686	405	5	ramesh	ramesh	PROPN
ejpam-6686	405	6	and	and	CCONJ
ejpam-6686	405	7	g.	g.	PROPN
ejpam-6686	405	8	jogarao	jogarao	PROPN
ejpam-6686	405	9	.	.	PUNCT
ejpam-6686	406	1	weak	weak	ADJ
ejpam-6686	406	2	relatively	relatively	ADV
ejpam-6686	406	3	complemented	complemented	ADJ
ejpam-6686	406	4	almost	almost	ADV
ejpam-6686	406	5	distributive	distributive	ADJ
ejpam-6686	406	6	lattices	lattice	NOUN
ejpam-6686	406	7	.	.	PUNCT
ejpam-6686	407	1	palest	pale	ADJ
ejpam-6686	407	2	.	.	PUNCT
ejpam-6686	408	1	j.	j.	PROPN
ejpam-6686	408	2	math	math	PROPN
ejpam-6686	408	3	.	.	PUNCT
ejpam-6686	408	4	,	,	PUNCT
ejpam-6686	408	5	6(2):448–457	6(2):448–457	NUM
ejpam-6686	408	6	,	,	PUNCT
ejpam-6686	408	7	2017	2017	NUM
ejpam-6686	408	8	.	.	PUNCT
ejpam-6686	409	1	[	[	X
ejpam-6686	409	2	18	18	NUM
ejpam-6686	409	3	]	]	X
ejpam-6686	409	4	s.	s.	PROPN
ejpam-6686	409	5	ramesh	ramesh	PROPN
ejpam-6686	409	6	,	,	PUNCT
ejpam-6686	409	7	g.	g.	PROPN
ejpam-6686	409	8	chinnayya	chinnayya	PROPN
ejpam-6686	409	9	,	,	PUNCT
ejpam-6686	409	10	g.	g.	PROPN
ejpam-6686	409	11	jogarao	jogarao	PROPN
ejpam-6686	409	12	,	,	PUNCT
ejpam-6686	409	13	r.	r.	PROPN
ejpam-6686	409	14	bandaru	bandaru	PROPN
ejpam-6686	409	15	,	,	PUNCT
ejpam-6686	409	16	and	and	CCONJ
ejpam-6686	409	17	a.	a.	NOUN
ejpam-6686	409	18	iampan	iampan	PROPN
ejpam-6686	409	19	.	.	PUNCT
ejpam-6686	410	1	hierarchy	hierarchy	NOUN
ejpam-6686	410	2	elements	element	NOUN
ejpam-6686	410	3	in	in	ADP
ejpam-6686	410	4	an	an	DET
ejpam-6686	410	5	almost	almost	ADV
ejpam-6686	410	6	distributive	distributive	ADJ
ejpam-6686	410	7	lattice	lattice	NOUN
ejpam-6686	410	8	.	.	PUNCT
ejpam-6686	411	1	eur	eur	PROPN
ejpam-6686	411	2	.	.	PUNCT
ejpam-6686	412	1	j.	j.	PROPN
ejpam-6686	412	2	pure	pure	PROPN
ejpam-6686	412	3	appl	appl	PROPN
ejpam-6686	412	4	.	.	PUNCT
ejpam-6686	412	5	math	math	PROPN
ejpam-6686	412	6	.	.	PUNCT
ejpam-6686	412	7	,	,	PUNCT
ejpam-6686	412	8	17(3):1691	17(3):1691	NUM
ejpam-6686	412	9	–	–	PUNCT
ejpam-6686	412	10	1704	1704	NUM
ejpam-6686	412	11	,	,	PUNCT
ejpam-6686	412	12	2024	2024	NUM
ejpam-6686	412	13	.	.	PUNCT
ejpam-6686	413	1	[	[	X
ejpam-6686	413	2	19	19	NUM
ejpam-6686	413	3	]	]	X
ejpam-6686	413	4	n.	n.	PROPN
ejpam-6686	413	5	rafi	rafi	PROPN
ejpam-6686	413	6	,	,	PUNCT
ejpam-6686	413	7	y.	y.	PROPN
ejpam-6686	413	8	monikarchana	monikarchana	PROPN
ejpam-6686	413	9	,	,	PUNCT
ejpam-6686	413	10	r.	r.	PROPN
ejpam-6686	413	11	bandaru	bandaru	PROPN
ejpam-6686	413	12	,	,	PUNCT
ejpam-6686	413	13	and	and	CCONJ
ejpam-6686	413	14	a.	a.	NOUN
ejpam-6686	413	15	iampan	iampan	PROPN
ejpam-6686	413	16	.	.	PUNCT
ejpam-6686	414	1	on	on	ADP
ejpam-6686	414	2	prime	prime	ADJ
ejpam-6686	414	3	e	e	NOUN
ejpam-6686	414	4	-	-	NOUN
ejpam-6686	414	5	ideals	ideal	NOUN
ejpam-6686	414	6	of	of	ADP
ejpam-6686	414	7	almost	almost	ADV
ejpam-6686	414	8	distributive	distributive	ADJ
ejpam-6686	414	9	lattices	lattice	NOUN
ejpam-6686	414	10	.	.	PUNCT
ejpam-6686	415	1	int	int	NOUN
ejpam-6686	415	2	.	.	PUNCT
ejpam-6686	416	1	j.	j.	PROPN
ejpam-6686	416	2	anal	anal	PROPN
ejpam-6686	416	3	.	.	PUNCT
ejpam-6686	417	1	appl	appl	PROPN
ejpam-6686	417	2	.	.	PROPN
ejpam-6686	417	3	,	,	PUNCT
ejpam-6686	417	4	11:85	11:85	NUM
ejpam-6686	417	5	,	,	PUNCT
ejpam-6686	417	6	2023	2023	NUM
ejpam-6686	417	7	.	.	PUNCT
ejpam-6686	418	1	[	[	X
ejpam-6686	418	2	20	20	NUM
ejpam-6686	418	3	]	]	X
ejpam-6686	418	4	n.	n.	PROPN
ejpam-6686	418	5	rafi	rafi	PROPN
ejpam-6686	418	6	,	,	PUNCT
ejpam-6686	418	7	p.	p.	PROPN
ejpam-6686	418	8	vijaya	vijaya	PROPN
ejpam-6686	418	9	saradhi	saradhi	PROPN
ejpam-6686	418	10	,	,	PUNCT
ejpam-6686	418	11	and	and	CCONJ
ejpam-6686	418	12	m.	m.	NOUN
ejpam-6686	418	13	balaiah	balaiah	PROPN
ejpam-6686	418	14	.	.	PUNCT
ejpam-6686	419	1	w	w	NOUN
ejpam-6686	419	2	-	-	PUNCT
ejpam-6686	419	3	filters	filter	NOUN
ejpam-6686	419	4	of	of	ADP
ejpam-6686	419	5	almost	almost	ADV
ejpam-6686	419	6	distributive	distributive	ADJ
ejpam-6686	419	7	lattices	lattice	NOUN
ejpam-6686	419	8	.	.	PUNCT
ejpam-6686	420	1	j.	j.	PROPN
ejpam-6686	420	2	algebr	algebr	PROPN
ejpam-6686	420	3	.	.	PUNCT
ejpam-6686	421	1	syst	syst	PROPN
ejpam-6686	421	2	.	.	PROPN
ejpam-6686	421	3	,	,	PUNCT
ejpam-6686	421	4	13(2):37–52	13(2):37–52	NUM
ejpam-6686	421	5	,	,	PUNCT
ejpam-6686	421	6	2025	2025	NUM
ejpam-6686	421	7	.	.	PUNCT
ejpam-6686	422	1	[	[	X
ejpam-6686	422	2	21	21	NUM
ejpam-6686	422	3	]	]	X
ejpam-6686	422	4	y.	y.	PROPN
ejpam-6686	422	5	s.	s.	PROPN
ejpam-6686	422	6	pawar	pawar	PROPN
ejpam-6686	422	7	and	and	CCONJ
ejpam-6686	422	8	i.	i.	PROPN
ejpam-6686	422	9	a.	a.	PROPN
ejpam-6686	422	10	shaikh	shaikh	PROPN
ejpam-6686	422	11	.	.	PUNCT
ejpam-6686	423	1	on	on	ADP
ejpam-6686	423	2	prime	prime	ADJ
ejpam-6686	423	3	,	,	PUNCT
ejpam-6686	423	4	minimal	minimal	ADJ
ejpam-6686	423	5	prime	prime	NOUN
ejpam-6686	423	6	and	and	CCONJ
ejpam-6686	423	7	annihilator	annihilator	NOUN
ejpam-6686	423	8	ideals	ideal	NOUN
ejpam-6686	423	9	in	in	ADP
ejpam-6686	423	10	an	an	DET
ejpam-6686	423	11	almost	almost	ADV
ejpam-6686	423	12	distributive	distributive	ADJ
ejpam-6686	423	13	lattice	lattice	NOUN
ejpam-6686	423	14	.	.	PUNCT
ejpam-6686	424	1	eur	eur	PROPN
ejpam-6686	424	2	.	.	PUNCT
ejpam-6686	425	1	j.	j.	PROPN
ejpam-6686	425	2	pure	pure	PROPN
ejpam-6686	425	3	appl	appl	PROPN
ejpam-6686	425	4	.	.	PUNCT
ejpam-6686	425	5	math	math	PROPN
ejpam-6686	425	6	.	.	PUNCT
ejpam-6686	425	7	,	,	PUNCT
ejpam-6686	425	8	6(1):107–118	6(1):107–118	NUM
ejpam-6686	425	9	,	,	PUNCT
ejpam-6686	425	10	2013	2013	NUM
ejpam-6686	425	11	.	.	PUNCT