id	sid	tid	token	lemma	pos
ejpam-4432	1	1	european	european	PROPN
ejpam-4432	1	2	journal	journal	PROPN
ejpam-4432	1	3	of	of	ADP
ejpam-4432	1	4	pure	pure	ADJ
ejpam-4432	1	5	and	and	CCONJ
ejpam-4432	1	6	applied	apply	VERB
ejpam-4432	1	7	mathematics	mathematic	NOUN
ejpam-4432	1	8	vol	vol	NOUN
ejpam-4432	1	9	.	.	PROPN
ejpam-4432	2	1	15	15	NUM
ejpam-4432	2	2	,	,	PUNCT
ejpam-4432	2	3	no	no	INTJ
ejpam-4432	2	4	.	.	NOUN
ejpam-4432	2	5	3	3	NUM
ejpam-4432	2	6	,	,	PUNCT
ejpam-4432	2	7	2022	2022	NUM
ejpam-4432	2	8	,	,	PUNCT
ejpam-4432	2	9	1402	1402	NUM
ejpam-4432	2	10	-	-	SYM
ejpam-4432	2	11	1416	1416	NUM
ejpam-4432	2	12	issn	issn	PROPN
ejpam-4432	2	13	1307	1307	NUM
ejpam-4432	2	14	-	-	SYM
ejpam-4432	2	15	5543	5543	NUM
ejpam-4432	2	16	–	–	PUNCT
ejpam-4432	2	17	ejpam.com	ejpam.com	X
ejpam-4432	2	18	published	publish	VERB
ejpam-4432	2	19	by	by	ADP
ejpam-4432	2	20	new	new	PROPN
ejpam-4432	2	21	york	york	PROPN
ejpam-4432	2	22	business	business	PROPN
ejpam-4432	2	23	global	global	PROPN
ejpam-4432	2	24	annihilator	annihilator	PROPN
ejpam-4432	2	25	hyperideals	hyperideal	NOUN
ejpam-4432	2	26	in	in	ADP
ejpam-4432	2	27	strong	strong	ADJ
ejpam-4432	2	28	bounded	bounded	ADJ
ejpam-4432	2	29	dual	dual	ADJ
ejpam-4432	2	30	distributive	distributive	ADJ
ejpam-4432	2	31	meet	meet	ADJ
ejpam-4432	2	32	-	-	PUNCT
ejpam-4432	2	33	hyperlattice	hyperlattice	NOUN
ejpam-4432	2	34	eman	eman	NOUN
ejpam-4432	2	35	ghareeb	ghareeb	PROPN
ejpam-4432	2	36	rezk1	rezk1	PROPN
ejpam-4432	2	37	,	,	PUNCT
ejpam-4432	2	38	nabilah	nabilah	PROPN
ejpam-4432	2	39	hani	hani	PROPN
ejpam-4432	2	40	abughazalah2,∗	abughazalah2,∗	PROPN
ejpam-4432	2	41	1	1	NUM
ejpam-4432	2	42	department	department	NOUN
ejpam-4432	2	43	of	of	ADP
ejpam-4432	2	44	mathematics	mathematic	NOUN
ejpam-4432	2	45	,	,	PUNCT
ejpam-4432	2	46	faculty	faculty	NOUN
ejpam-4432	2	47	of	of	ADP
ejpam-4432	2	48	science	science	NOUN
ejpam-4432	2	49	,	,	PUNCT
ejpam-4432	2	50	tanta	tanta	PROPN
ejpam-4432	2	51	university	university	PROPN
ejpam-4432	2	52	,	,	PUNCT
ejpam-4432	2	53	egypt	egypt	PROPN
ejpam-4432	2	54	1,2	1,2	NUM
ejpam-4432	2	55	mathematical	mathematical	ADJ
ejpam-4432	2	56	sciences	sciences	PROPN
ejpam-4432	2	57	department	department	PROPN
ejpam-4432	2	58	,	,	PUNCT
ejpam-4432	2	59	college	college	NOUN
ejpam-4432	2	60	of	of	ADP
ejpam-4432	2	61	science	science	NOUN
ejpam-4432	2	62	,	,	PUNCT
ejpam-4432	2	63	princess	princess	PROPN
ejpam-4432	2	64	nourah	nourah	PROPN
ejpam-4432	2	65	bint	bint	PROPN
ejpam-4432	2	66	abdulrahman	abdulrahman	PROPN
ejpam-4432	2	67	university	university	PROPN
ejpam-4432	2	68	,	,	PUNCT
ejpam-4432	2	69	p.o.box	p.o.box	PROPN
ejpam-4432	2	70	84428	84428	NUM
ejpam-4432	2	71	,	,	PUNCT
ejpam-4432	2	72	riyadh	riyadh	PROPN
ejpam-4432	2	73	11671	11671	NUM
ejpam-4432	3	1	,	,	PUNCT
ejpam-4432	3	2	saudi	saudi	PROPN
ejpam-4432	3	3	arabia	arabia	PROPN
ejpam-4432	3	4	abstract	abstract	NOUN
ejpam-4432	3	5	.	.	PUNCT
ejpam-4432	4	1	in	in	ADP
ejpam-4432	4	2	this	this	DET
ejpam-4432	4	3	paper	paper	NOUN
ejpam-4432	4	4	,	,	PUNCT
ejpam-4432	4	5	we	we	PRON
ejpam-4432	4	6	study	study	VERB
ejpam-4432	4	7	the	the	DET
ejpam-4432	4	8	properties	property	NOUN
ejpam-4432	4	9	of	of	ADP
ejpam-4432	4	10	annihilator	annihilator	PROPN
ejpam-4432	4	11	hyperideals	hyperideal	NOUN
ejpam-4432	4	12	in	in	ADP
ejpam-4432	4	13	the	the	DET
ejpam-4432	4	14	class	class	NOUN
ejpam-4432	4	15	of	of	ADP
ejpam-4432	4	16	strong	strong	ADJ
ejpam-4432	4	17	bounded	bounded	ADJ
ejpam-4432	4	18	dual	dual	ADJ
ejpam-4432	4	19	distributive	distributive	ADJ
ejpam-4432	4	20	meet	meet	NOUN
ejpam-4432	4	21	-	-	PUNCT
ejpam-4432	4	22	hyperlattice	hyperlattice	NOUN
ejpam-4432	4	23	.	.	PUNCT
ejpam-4432	5	1	we	we	PRON
ejpam-4432	5	2	show	show	VERB
ejpam-4432	5	3	that	that	SCONJ
ejpam-4432	5	4	the	the	DET
ejpam-4432	5	5	set	set	NOUN
ejpam-4432	5	6	of	of	ADP
ejpam-4432	5	7	all	all	DET
ejpam-4432	5	8	closed	closed	ADJ
ejpam-4432	5	9	hyperideals	hyperideal	NOUN
ejpam-4432	5	10	forms	form	VERB
ejpam-4432	5	11	a	a	DET
ejpam-4432	5	12	boolean	boolean	ADJ
ejpam-4432	5	13	algebra	algebra	NOUN
ejpam-4432	5	14	.	.	PUNCT
ejpam-4432	6	1	we	we	PRON
ejpam-4432	6	2	introduce	introduce	VERB
ejpam-4432	6	3	the	the	DET
ejpam-4432	6	4	concept	concept	NOUN
ejpam-4432	6	5	of	of	ADP
ejpam-4432	6	6	homomorphism	homomorphism	PROPN
ejpam-4432	6	7	,	,	PUNCT
ejpam-4432	6	8	which	which	PRON
ejpam-4432	6	9	preserves	preserve	VERB
ejpam-4432	6	10	the	the	DET
ejpam-4432	6	11	annihilator	annihilator	PROPN
ejpam-4432	6	12	hyperideal	hyperideal	PROPN
ejpam-4432	6	13	.	.	PUNCT
ejpam-4432	7	1	suitable	suitable	ADJ
ejpam-4432	7	2	conditions	condition	NOUN
ejpam-4432	7	3	for	for	ADP
ejpam-4432	7	4	preserving	preserve	VERB
ejpam-4432	7	5	annihilator	annihilator	NOUN
ejpam-4432	7	6	hyperideals	hyperideal	NOUN
ejpam-4432	7	7	are	be	AUX
ejpam-4432	7	8	obtained	obtain	VERB
ejpam-4432	7	9	.	.	PUNCT
ejpam-4432	8	1	representation	representation	NOUN
ejpam-4432	8	2	and	and	CCONJ
ejpam-4432	8	3	characterization	characterization	NOUN
ejpam-4432	8	4	theorems	theorem	NOUN
ejpam-4432	8	5	of	of	ADP
ejpam-4432	8	6	annihilator	annihilator	PROPN
ejpam-4432	8	7	hyperideals	hyperideal	NOUN
ejpam-4432	8	8	in	in	ADP
ejpam-4432	8	9	sub	sub	ADJ
ejpam-4432	8	10	-	-	ADJ
ejpam-4432	8	11	meet	meet	ADJ
ejpam-4432	8	12	-	-	PUNCT
ejpam-4432	8	13	hyperlattice	hyperlattice	NOUN
ejpam-4432	8	14	and	and	CCONJ
ejpam-4432	8	15	product	product	NOUN
ejpam-4432	8	16	meet	meet	NOUN
ejpam-4432	8	17	-	-	PUNCT
ejpam-4432	8	18	hyperlattice	hyperlattice	NOUN
ejpam-4432	8	19	are	be	AUX
ejpam-4432	8	20	proved	prove	VERB
ejpam-4432	8	21	.	.	PUNCT
ejpam-4432	9	1	2020	2020	NUM
ejpam-4432	9	2	mathematics	mathematic	NOUN
ejpam-4432	9	3	subject	subject	NOUN
ejpam-4432	9	4	classifications	classification	NOUN
ejpam-4432	9	5	:	:	PUNCT
ejpam-4432	9	6	06b75	06b75	NOUN
ejpam-4432	9	7	,	,	PUNCT
ejpam-4432	9	8	06f99	06f99	NUM
ejpam-4432	9	9	,	,	PUNCT
ejpam-4432	9	10	08a05	08a05	NUM
ejpam-4432	9	11	key	key	ADJ
ejpam-4432	9	12	words	word	NOUN
ejpam-4432	9	13	and	and	CCONJ
ejpam-4432	9	14	phrases	phrase	NOUN
ejpam-4432	9	15	:	:	PUNCT
ejpam-4432	9	16	annihilator	annihilator	NOUN
ejpam-4432	9	17	,	,	PUNCT
ejpam-4432	9	18	boolean	boolean	ADJ
ejpam-4432	9	19	algebra	algebra	NOUN
ejpam-4432	9	20	,	,	PUNCT
ejpam-4432	9	21	homomorphism	homomorphism	NOUN
ejpam-4432	9	22	,	,	PUNCT
ejpam-4432	9	23	hyperideal	hyperideal	NOUN
ejpam-4432	9	24	,	,	PUNCT
ejpam-4432	9	25	hyperlattice	hyperlattice	NOUN
ejpam-4432	9	26	1	1	NUM
ejpam-4432	9	27	.	.	PUNCT
ejpam-4432	9	28	introduction	introduction	NOUN
ejpam-4432	9	29	the	the	DET
ejpam-4432	9	30	approach	approach	NOUN
ejpam-4432	9	31	to	to	ADP
ejpam-4432	9	32	the	the	DET
ejpam-4432	9	33	theory	theory	NOUN
ejpam-4432	9	34	of	of	ADP
ejpam-4432	9	35	hyberlattices	hyberlattice	NOUN
ejpam-4432	9	36	was	be	AUX
ejpam-4432	9	37	first	first	ADV
ejpam-4432	9	38	made	make	VERB
ejpam-4432	9	39	by	by	ADP
ejpam-4432	9	40	m.	m.	NOUN
ejpam-4432	9	41	konstantinidou	konstantinidou	PROPN
ejpam-4432	9	42	and	and	CCONJ
ejpam-4432	9	43	j.	j.	PROPN
ejpam-4432	9	44	mittas	mittas	PROPN
ejpam-4432	9	45	in	in	ADP
ejpam-4432	9	46	1977	1977	NUM
ejpam-4432	9	47	,	,	PUNCT
ejpam-4432	10	1	[	[	X
ejpam-4432	10	2	12	12	NUM
ejpam-4432	10	3	]	]	PUNCT
ejpam-4432	10	4	.	.	PUNCT
ejpam-4432	11	1	modular	modular	ADJ
ejpam-4432	11	2	distributive	distributive	ADJ
ejpam-4432	11	3	and	and	CCONJ
ejpam-4432	11	4	complemented	complemented	ADJ
ejpam-4432	11	5	classes	class	NOUN
ejpam-4432	11	6	of	of	ADP
ejpam-4432	11	7	hyperlattices	hyperlattice	NOUN
ejpam-4432	11	8	were	be	AUX
ejpam-4432	11	9	studied	study	VERB
ejpam-4432	11	10	by	by	ADP
ejpam-4432	11	11	m.	m.	NOUN
ejpam-4432	11	12	konstantinidou	konstantinidou	PROPN
ejpam-4432	11	13	in	in	ADP
ejpam-4432	11	14	[	[	X
ejpam-4432	11	15	10	10	NUM
ejpam-4432	11	16	]	]	PUNCT
ejpam-4432	11	17	and	and	CCONJ
ejpam-4432	11	18	[	[	X
ejpam-4432	11	19	9	9	NUM
ejpam-4432	11	20	]	]	PUNCT
ejpam-4432	11	21	.	.	PUNCT
ejpam-4432	12	1	ideals	ideal	NOUN
ejpam-4432	12	2	of	of	ADP
ejpam-4432	12	3	hyperlattices	hyperlattice	NOUN
ejpam-4432	12	4	were	be	AUX
ejpam-4432	12	5	introduced	introduce	VERB
ejpam-4432	12	6	by	by	ADP
ejpam-4432	12	7	rahnamai	rahnamai	PROPN
ejpam-4432	12	8	-	-	PUNCT
ejpam-4432	12	9	barghi	barghi	PROPN
ejpam-4432	12	10	in	in	ADP
ejpam-4432	12	11	[	[	X
ejpam-4432	12	12	15	15	NUM
ejpam-4432	12	13	]	]	PUNCT
ejpam-4432	12	14	,	,	PUNCT
ejpam-4432	12	15	where	where	SCONJ
ejpam-4432	12	16	he	he	PRON
ejpam-4432	12	17	considered	consider	VERB
ejpam-4432	12	18	the	the	DET
ejpam-4432	12	19	prime	prime	ADJ
ejpam-4432	12	20	ideal	ideal	NOUN
ejpam-4432	12	21	theorem	theorem	NOUN
ejpam-4432	12	22	for	for	ADP
ejpam-4432	12	23	distributive	distributive	ADJ
ejpam-4432	12	24	hyperlattices	hyperlattice	NOUN
ejpam-4432	12	25	.	.	PUNCT
ejpam-4432	13	1	m.	m.	NOUN
ejpam-4432	13	2	amiri	amiri	PROPN
ejpam-4432	13	3	bideshki	bideshki	PROPN
ejpam-4432	13	4	and	and	CCONJ
ejpam-4432	13	5	et	et	PROPN
ejpam-4432	13	6	al	al	PROPN
ejpam-4432	13	7	.	.	PROPN
ejpam-4432	13	8	defined	define	VERB
ejpam-4432	13	9	the	the	DET
ejpam-4432	13	10	notions	notion	NOUN
ejpam-4432	13	11	of	of	ADP
ejpam-4432	13	12	hyperideals	hyperideal	NOUN
ejpam-4432	13	13	and	and	CCONJ
ejpam-4432	13	14	hyperfilters	hyperfilter	NOUN
ejpam-4432	13	15	in	in	ADP
ejpam-4432	13	16	strong	strong	ADJ
ejpam-4432	13	17	meet	meet	NOUN
ejpam-4432	13	18	-	-	PUNCT
ejpam-4432	13	19	hyperlattices	hyperlattice	NOUN
ejpam-4432	13	20	in	in	ADP
ejpam-4432	13	21	[	[	X
ejpam-4432	13	22	11	11	NUM
ejpam-4432	13	23	]	]	PUNCT
ejpam-4432	13	24	.	.	PUNCT
ejpam-4432	14	1	they	they	PRON
ejpam-4432	14	2	also	also	ADV
ejpam-4432	14	3	introduced	introduce	VERB
ejpam-4432	14	4	the	the	DET
ejpam-4432	14	5	concept	concept	NOUN
ejpam-4432	14	6	of	of	ADP
ejpam-4432	14	7	annihilator	annihilator	PROPN
ejpam-4432	14	8	hyperideals	hyperideal	NOUN
ejpam-4432	14	9	.	.	PUNCT
ejpam-4432	15	1	annihilators	annihilator	NOUN
ejpam-4432	15	2	have	have	AUX
ejpam-4432	15	3	been	be	AUX
ejpam-4432	15	4	started	start	VERB
ejpam-4432	15	5	in	in	ADP
ejpam-4432	15	6	ring	ring	NOUN
ejpam-4432	15	7	theory	theory	NOUN
ejpam-4432	15	8	over	over	ADP
ejpam-4432	15	9	many	many	ADJ
ejpam-4432	15	10	classes	class	NOUN
ejpam-4432	15	11	of	of	ADP
ejpam-4432	15	12	rings	ring	NOUN
ejpam-4432	15	13	,	,	PUNCT
ejpam-4432	15	14	as	as	SCONJ
ejpam-4432	15	15	examples	example	NOUN
ejpam-4432	15	16	refer	refer	VERB
ejpam-4432	15	17	to	to	ADP
ejpam-4432	15	18	[	[	X
ejpam-4432	15	19	19	19	NUM
ejpam-4432	15	20	]	]	PUNCT
ejpam-4432	15	21	and	and	CCONJ
ejpam-4432	15	22	[	[	X
ejpam-4432	15	23	8	8	NUM
ejpam-4432	15	24	]	]	PUNCT
ejpam-4432	15	25	.	.	PUNCT
ejpam-4432	16	1	in	in	ADP
ejpam-4432	16	2	that	that	DET
ejpam-4432	16	3	sense	sense	NOUN
ejpam-4432	16	4	,	,	PUNCT
ejpam-4432	16	5	the	the	DET
ejpam-4432	16	6	annihilator	annihilator	NOUN
ejpam-4432	16	7	of	of	ADP
ejpam-4432	16	8	a	a	DET
ejpam-4432	16	9	certain	certain	ADJ
ejpam-4432	16	10	set	set	NOUN
ejpam-4432	16	11	a	a	DET
ejpam-4432	16	12	means	means	NOUN
ejpam-4432	16	13	the	the	DET
ejpam-4432	16	14	set	set	NOUN
ejpam-4432	16	15	of	of	ADP
ejpam-4432	16	16	killer	killer	NOUN
ejpam-4432	16	17	elements	element	NOUN
ejpam-4432	16	18	that	that	PRON
ejpam-4432	16	19	make	make	VERB
ejpam-4432	16	20	each	each	DET
ejpam-4432	16	21	element	element	NOUN
ejpam-4432	16	22	of	of	ADP
ejpam-4432	16	23	a	a	DET
ejpam-4432	16	24	tends	tend	NOUN
ejpam-4432	16	25	to	to	ADP
ejpam-4432	16	26	zero	zero	NUM
ejpam-4432	16	27	by	by	ADP
ejpam-4432	16	28	multiplication	multiplication	NOUN
ejpam-4432	16	29	operation	operation	NOUN
ejpam-4432	16	30	.	.	PUNCT
ejpam-4432	17	1	the	the	DET
ejpam-4432	17	2	mention	mention	NOUN
ejpam-4432	17	3	of	of	ADP
ejpam-4432	17	4	annihilators	annihilator	NOUN
ejpam-4432	17	5	in	in	ADP
ejpam-4432	17	6	lattices	lattice	NOUN
ejpam-4432	17	7	was	be	AUX
ejpam-4432	17	8	first	first	ADV
ejpam-4432	17	9	introduced	introduce	VERB
ejpam-4432	17	10	by	by	ADP
ejpam-4432	17	11	m.mandelker	m.mandelker	ADV
ejpam-4432	17	12	in	in	ADP
ejpam-4432	17	13	1970	1970	NUM
ejpam-4432	17	14	,	,	PUNCT
ejpam-4432	17	15	[	[	X
ejpam-4432	17	16	13	13	NUM
ejpam-4432	17	17	]	]	PUNCT
ejpam-4432	17	18	.	.	PUNCT
ejpam-4432	18	1	he	he	PRON
ejpam-4432	18	2	defined	define	VERB
ejpam-4432	18	3	the	the	DET
ejpam-4432	18	4	relative	relative	ADJ
ejpam-4432	18	5	annihilator	annihilator	NOUN
ejpam-4432	18	6	as	as	ADP
ejpam-4432	18	7	a	a	DET
ejpam-4432	18	8	generalization	generalization	NOUN
ejpam-4432	18	9	of	of	ADP
ejpam-4432	18	10	relative	relative	ADJ
ejpam-4432	18	11	pseudocomplementation	pseudocomplementation	NOUN
ejpam-4432	18	12	.	.	PUNCT
ejpam-4432	19	1	m.mandelker	m.mandelker	ADV
ejpam-4432	19	2	introduced	introduce	VERB
ejpam-4432	19	3	the	the	DET
ejpam-4432	19	4	relation	relation	NOUN
ejpam-4432	19	5	between	between	ADP
ejpam-4432	19	6	prime	prime	ADJ
ejpam-4432	19	7	ideal	ideal	ADJ
ejpam-4432	19	8	conditions	condition	NOUN
ejpam-4432	19	9	and	and	CCONJ
ejpam-4432	19	10	annihilator	annihilator	NOUN
ejpam-4432	19	11	conditions	condition	NOUN
ejpam-4432	19	12	on	on	ADP
ejpam-4432	19	13	distributive	distributive	ADJ
ejpam-4432	19	14	lattices	lattice	NOUN
ejpam-4432	19	15	.	.	PUNCT
ejpam-4432	20	1	w.	w.	PROPN
ejpam-4432	20	2	h.	h.	PROPN
ejpam-4432	20	3	cornish	cornish	PROPN
ejpam-4432	20	4	investigated	investigate	VERB
ejpam-4432	20	5	the	the	DET
ejpam-4432	20	6	annihilator	annihilator	PROPN
ejpam-4432	20	7	∗corresponding	∗corresponde	VERB
ejpam-4432	20	8	author	author	NOUN
ejpam-4432	20	9	.	.	PUNCT
ejpam-4432	21	1	doi	doi	NOUN
ejpam-4432	21	2	:	:	PUNCT
ejpam-4432	21	3	https://doi.org/10.29020/nybg.ejpam.v15i3.4480	https://doi.org/10.29020/nybg.ejpam.v15i3.4480	NOUN
ejpam-4432	21	4	email	email	NOUN
ejpam-4432	21	5	addresses	address	NOUN
ejpam-4432	21	6	:	:	PUNCT
ejpam-4432	21	7	eman.rezk@science.tanta.edu.eg	eman.rezk@science.tanta.edu.eg	X
ejpam-4432	21	8	(	(	PUNCT
ejpam-4432	21	9	e.g.	e.g.	ADV
ejpam-4432	21	10	rezk	rezk	NOUN
ejpam-4432	21	11	)	)	PUNCT
ejpam-4432	21	12	,	,	PUNCT
ejpam-4432	21	13	nhabughazala@pnu.edu.sa	nhabughazala@pnu.edu.sa	PROPN
ejpam-4432	21	14	(	(	PUNCT
ejpam-4432	21	15	n.h	n.h	PROPN
ejpam-4432	21	16	.	.	PROPN
ejpam-4432	21	17	abughazalah	abughazalah	PROPN
ejpam-4432	21	18	)	)	PUNCT
ejpam-4432	21	19	https://www.ejpam.com	https://www.ejpam.com	NOUN
ejpam-4432	21	20	1402	1402	NUM
ejpam-4432	22	1	©	©	PROPN
ejpam-4432	22	2	2022	2022	NUM
ejpam-4432	22	3	ejpam	ejpam	VERB
ejpam-4432	22	4	all	all	DET
ejpam-4432	22	5	rights	right	NOUN
ejpam-4432	22	6	reserved	reserve	VERB
ejpam-4432	22	7	.	.	PUNCT
ejpam-4432	23	1	e.g.	e.g.	ADV
ejpam-4432	23	2	rezk	rezk	PROPN
ejpam-4432	23	3	,	,	PUNCT
ejpam-4432	23	4	n.h	n.h	PROPN
ejpam-4432	23	5	.	.	PROPN
ejpam-4432	23	6	abughazalah	abughazalah	PROPN
ejpam-4432	23	7	/	/	SYM
ejpam-4432	23	8	eur	eur	PROPN
ejpam-4432	23	9	.	.	PUNCT
ejpam-4432	24	1	j.	j.	PROPN
ejpam-4432	24	2	pure	pure	PROPN
ejpam-4432	24	3	appl	appl	PROPN
ejpam-4432	24	4	.	.	PROPN
ejpam-4432	24	5	math	math	PROPN
ejpam-4432	24	6	,	,	PUNCT
ejpam-4432	24	7	15	15	NUM
ejpam-4432	24	8	(	(	PUNCT
ejpam-4432	24	9	3	3	NUM
ejpam-4432	24	10	)	)	PUNCT
ejpam-4432	24	11	(	(	PUNCT
ejpam-4432	24	12	2022	2022	NUM
ejpam-4432	24	13	)	)	PUNCT
ejpam-4432	24	14	,	,	PUNCT
ejpam-4432	24	15	1402	1402	NUM
ejpam-4432	24	16	-	-	SYM
ejpam-4432	24	17	1416	1416	NUM
ejpam-4432	24	18	1403	1403	NUM
ejpam-4432	24	19	properties	property	NOUN
ejpam-4432	24	20	of	of	ADP
ejpam-4432	24	21	distributive	distributive	ADJ
ejpam-4432	24	22	lattices	lattice	NOUN
ejpam-4432	24	23	in	in	ADP
ejpam-4432	24	24	[	[	X
ejpam-4432	24	25	18	18	NUM
ejpam-4432	24	26	]	]	PUNCT
ejpam-4432	24	27	.	.	PUNCT
ejpam-4432	25	1	he	he	PRON
ejpam-4432	25	2	defined	define	VERB
ejpam-4432	25	3	the	the	DET
ejpam-4432	25	4	annihilator	annihilator	NOUN
ejpam-4432	25	5	of	of	ADP
ejpam-4432	25	6	a	a	DET
ejpam-4432	25	7	set	set	NOUN
ejpam-4432	25	8	a	a	PRON
ejpam-4432	25	9	as	as	ADP
ejpam-4432	25	10	a	a	DET
ejpam-4432	25	11	set	set	NOUN
ejpam-4432	25	12	of	of	ADP
ejpam-4432	25	13	all	all	DET
ejpam-4432	25	14	elements	element	NOUN
ejpam-4432	25	15	whose	whose	DET
ejpam-4432	25	16	elements	element	NOUN
ejpam-4432	25	17	tend	tend	VERB
ejpam-4432	25	18	to	to	ADP
ejpam-4432	25	19	zero	zero	NUM
ejpam-4432	25	20	by	by	ADP
ejpam-4432	25	21	the	the	DET
ejpam-4432	25	22	meet	meet	NOUN
ejpam-4432	25	23	operation	operation	NOUN
ejpam-4432	25	24	.	.	PUNCT
ejpam-4432	26	1	the	the	DET
ejpam-4432	26	2	main	main	ADJ
ejpam-4432	26	3	resultthat	resultthat	PROPN
ejpam-4432	26	4	cornish	cornish	PROPN
ejpam-4432	26	5	provedis	provedi	NOUN
ejpam-4432	26	6	the	the	DET
ejpam-4432	26	7	normality	normality	NOUN
ejpam-4432	26	8	of	of	ADP
ejpam-4432	26	9	lattice	lattice	ADJ
ejpam-4432	26	10	equivalents	equivalent	NOUN
ejpam-4432	26	11	to	to	ADP
ejpam-4432	26	12	any	any	DET
ejpam-4432	26	13	two	two	NUM
ejpam-4432	26	14	elements	element	NOUN
ejpam-4432	26	15	with	with	ADP
ejpam-4432	26	16	zero	zero	NUM
ejpam-4432	26	17	meeting	meeting	NOUN
ejpam-4432	26	18	have	have	VERB
ejpam-4432	26	19	a	a	DET
ejpam-4432	26	20	comaximal	comaximal	ADJ
ejpam-4432	26	21	annihilator	annihilator	NOUN
ejpam-4432	26	22	of	of	ADP
ejpam-4432	26	23	their	their	PRON
ejpam-4432	26	24	principal	principal	ADJ
ejpam-4432	26	25	ideals	ideal	NOUN
ejpam-4432	26	26	.	.	PUNCT
ejpam-4432	27	1	moreover	moreover	ADV
ejpam-4432	27	2	,	,	PUNCT
ejpam-4432	27	3	the	the	DET
ejpam-4432	27	4	normality	normality	NOUN
ejpam-4432	27	5	of	of	ADP
ejpam-4432	27	6	the	the	DET
ejpam-4432	27	7	lattice	lattice	PROPN
ejpam-4432	27	8	equivalents	equivalent	NOUN
ejpam-4432	27	9	to	to	ADP
ejpam-4432	27	10	the	the	DET
ejpam-4432	27	11	annihilator	annihilator	NOUN
ejpam-4432	27	12	of	of	ADP
ejpam-4432	27	13	the	the	DET
ejpam-4432	27	14	principle	principle	ADJ
ejpam-4432	27	15	ideal	ideal	NOUN
ejpam-4432	27	16	of	of	ADP
ejpam-4432	27	17	the	the	DET
ejpam-4432	27	18	meeting	meeting	NOUN
ejpam-4432	27	19	of	of	ADP
ejpam-4432	27	20	any	any	DET
ejpam-4432	27	21	two	two	NUM
ejpam-4432	27	22	elements	element	NOUN
ejpam-4432	27	23	equals	equal	VERB
ejpam-4432	27	24	the	the	DET
ejpam-4432	27	25	joining	joining	NOUN
ejpam-4432	27	26	of	of	ADP
ejpam-4432	27	27	annihilators	annihilator	NOUN
ejpam-4432	27	28	of	of	ADP
ejpam-4432	27	29	their	their	PRON
ejpam-4432	27	30	principle	principle	ADJ
ejpam-4432	27	31	ideals	ideal	NOUN
ejpam-4432	27	32	.	.	PUNCT
ejpam-4432	28	1	in	in	ADP
ejpam-4432	28	2	[	[	X
ejpam-4432	28	3	1	1	NUM
ejpam-4432	28	4	]	]	PUNCT
ejpam-4432	28	5	,	,	PUNCT
ejpam-4432	28	6	b.	b.	PROPN
ejpam-4432	28	7	a.	a.	PROPN
ejpam-4432	28	8	davey	davey	PROPN
ejpam-4432	28	9	and	and	CCONJ
ejpam-4432	28	10	nieminen	nieminen	PROPN
ejpam-4432	28	11	studied	study	VERB
ejpam-4432	28	12	the	the	DET
ejpam-4432	28	13	annihilators	annihilator	NOUN
ejpam-4432	28	14	in	in	ADP
ejpam-4432	28	15	the	the	DET
ejpam-4432	28	16	class	class	NOUN
ejpam-4432	28	17	of	of	ADP
ejpam-4432	28	18	modular	modular	ADJ
ejpam-4432	28	19	lattices	lattice	NOUN
ejpam-4432	28	20	.	.	PUNCT
ejpam-4432	29	1	they	they	PRON
ejpam-4432	29	2	proved	prove	VERB
ejpam-4432	29	3	that	that	SCONJ
ejpam-4432	29	4	the	the	DET
ejpam-4432	29	5	weakly	weakly	ADJ
ejpam-4432	29	6	atomic	atomic	ADJ
ejpam-4432	29	7	modular	modular	ADJ
ejpam-4432	29	8	lattice	lattice	NOUN
ejpam-4432	29	9	has	have	VERB
ejpam-4432	29	10	necessary	necessary	ADJ
ejpam-4432	29	11	and	and	CCONJ
ejpam-4432	29	12	sufficient	sufficient	ADJ
ejpam-4432	29	13	conditions	condition	NOUN
ejpam-4432	29	14	for	for	ADP
ejpam-4432	29	15	its	its	PRON
ejpam-4432	29	16	annihilators	annihilator	NOUN
ejpam-4432	29	17	.	.	PUNCT
ejpam-4432	30	1	as	as	ADP
ejpam-4432	30	2	a	a	DET
ejpam-4432	30	3	generalization	generalization	NOUN
ejpam-4432	30	4	of	of	ADP
ejpam-4432	30	5	lattices	lattice	NOUN
ejpam-4432	30	6	halaš	halaš	NOUN
ejpam-4432	31	1	[	[	X
ejpam-4432	31	2	6	6	NUM
ejpam-4432	31	3	]	]	PUNCT
ejpam-4432	31	4	studied	study	VERB
ejpam-4432	31	5	annihilators	annihilator	NOUN
ejpam-4432	31	6	in	in	ADP
ejpam-4432	31	7	ordered	order	VERB
ejpam-4432	31	8	sets	set	NOUN
ejpam-4432	31	9	.	.	PUNCT
ejpam-4432	32	1	this	this	PRON
ejpam-4432	32	2	was	be	AUX
ejpam-4432	32	3	followed	follow	VERB
ejpam-4432	32	4	by	by	ADP
ejpam-4432	32	5	a	a	DET
ejpam-4432	32	6	lot	lot	NOUN
ejpam-4432	32	7	of	of	ADP
ejpam-4432	32	8	studies	study	NOUN
ejpam-4432	32	9	and	and	CCONJ
ejpam-4432	32	10	research	research	NOUN
ejpam-4432	32	11	on	on	ADP
ejpam-4432	32	12	the	the	DET
ejpam-4432	32	13	concept	concept	NOUN
ejpam-4432	32	14	of	of	ADP
ejpam-4432	32	15	annihilator	annihilator	NOUN
ejpam-4432	32	16	in	in	ADP
ejpam-4432	32	17	many	many	ADJ
ejpam-4432	32	18	algebraic	algebraic	ADJ
ejpam-4432	32	19	structures	structure	NOUN
ejpam-4432	32	20	and	and	CCONJ
ejpam-4432	32	21	classes	class	NOUN
ejpam-4432	32	22	,	,	PUNCT
ejpam-4432	32	23	for	for	ADP
ejpam-4432	32	24	instance	instance	NOUN
ejpam-4432	32	25	:	:	PUNCT
ejpam-4432	32	26	almost	almost	ADV
ejpam-4432	32	27	distributive	distributive	ADJ
ejpam-4432	32	28	lattices	lattice	NOUN
ejpam-4432	32	29	[	[	X
ejpam-4432	32	30	3],0	3],0	NUM
ejpam-4432	32	31	-	-	PUNCT
ejpam-4432	32	32	almost	almost	ADV
ejpam-4432	32	33	distributive	distributive	ADJ
ejpam-4432	32	34	lattices	lattice	NOUN
ejpam-4432	32	35	[	[	X
ejpam-4432	32	36	4	4	NUM
ejpam-4432	32	37	]	]	PUNCT
ejpam-4432	32	38	,	,	PUNCT
ejpam-4432	32	39	distributive	distributive	ADJ
ejpam-4432	32	40	dual	dual	ADJ
ejpam-4432	32	41	weakly	weakly	ADJ
ejpam-4432	32	42	complemented	complemented	ADJ
ejpam-4432	32	43	lattice	lattice	NOUN
ejpam-4432	33	1	[	[	X
ejpam-4432	33	2	17	17	NUM
ejpam-4432	33	3	]	]	PUNCT
ejpam-4432	33	4	,	,	PUNCT
ejpam-4432	33	5	bck	bck	PROPN
ejpam-4432	33	6	-	-	PUNCT
ejpam-4432	33	7	algebras	algebras	X
ejpam-4432	34	1	[	[	X
ejpam-4432	34	2	7	7	NUM
ejpam-4432	34	3	]	]	PUNCT
ejpam-4432	34	4	,	,	PUNCT
ejpam-4432	34	5	standard	standard	ADJ
ejpam-4432	34	6	qbcc	qbcc	NOUN
ejpam-4432	34	7	algebras	algebras	X
ejpam-4432	35	1	[	[	X
ejpam-4432	35	2	14	14	NUM
ejpam-4432	35	3	]	]	PUNCT
ejpam-4432	35	4	,	,	PUNCT
ejpam-4432	35	5	c	c	X
ejpam-4432	35	6	-	-	PUNCT
ejpam-4432	35	7	algebra[16	algebra[16	NOUN
ejpam-4432	35	8	]	]	PUNCT
ejpam-4432	35	9	,	,	PUNCT
ejpam-4432	35	10	and	and	CCONJ
ejpam-4432	35	11	many	many	ADJ
ejpam-4432	35	12	other	other	ADJ
ejpam-4432	35	13	classes	class	NOUN
ejpam-4432	35	14	.	.	PUNCT
ejpam-4432	36	1	in	in	ADP
ejpam-4432	36	2	this	this	DET
ejpam-4432	36	3	paper	paper	NOUN
ejpam-4432	36	4	,	,	PUNCT
ejpam-4432	36	5	the	the	DET
ejpam-4432	36	6	properties	property	NOUN
ejpam-4432	36	7	of	of	ADP
ejpam-4432	36	8	annihilator	annihilator	PROPN
ejpam-4432	36	9	hyperideals	hyperideal	NOUN
ejpam-4432	36	10	of	of	ADP
ejpam-4432	36	11	strong	strong	ADJ
ejpam-4432	36	12	bounded	bounded	ADJ
ejpam-4432	36	13	dual	dual	ADJ
ejpam-4432	36	14	distributive	distributive	ADJ
ejpam-4432	36	15	meet	meet	NOUN
ejpam-4432	36	16	-	-	PUNCT
ejpam-4432	36	17	hyperlattice	hyperlattice	NOUN
ejpam-4432	36	18	are	be	AUX
ejpam-4432	36	19	investigated	investigate	VERB
ejpam-4432	36	20	.	.	PUNCT
ejpam-4432	37	1	main	main	ADJ
ejpam-4432	37	2	terminologies	terminology	NOUN
ejpam-4432	37	3	and	and	CCONJ
ejpam-4432	37	4	properties	property	NOUN
ejpam-4432	37	5	are	be	AUX
ejpam-4432	37	6	recalled	recall	VERB
ejpam-4432	37	7	in	in	ADP
ejpam-4432	37	8	section	section	NOUN
ejpam-4432	37	9	2	2	NUM
ejpam-4432	37	10	.	.	PUNCT
ejpam-4432	37	11	important	important	ADJ
ejpam-4432	37	12	properties	property	NOUN
ejpam-4432	37	13	of	of	ADP
ejpam-4432	37	14	annihilator	annihilator	PROPN
ejpam-4432	37	15	hyperideals	hyperideal	NOUN
ejpam-4432	37	16	are	be	AUX
ejpam-4432	37	17	proved	prove	VERB
ejpam-4432	37	18	.	.	PUNCT
ejpam-4432	38	1	moreover	moreover	ADV
ejpam-4432	38	2	,	,	PUNCT
ejpam-4432	38	3	the	the	DET
ejpam-4432	38	4	structure	structure	NOUN
ejpam-4432	38	5	of	of	ADP
ejpam-4432	38	6	the	the	DET
ejpam-4432	38	7	set	set	NOUN
ejpam-4432	38	8	of	of	ADP
ejpam-4432	38	9	all	all	DET
ejpam-4432	38	10	closed	closed	ADJ
ejpam-4432	38	11	hyperideals	hyperideal	NOUN
ejpam-4432	38	12	is	be	AUX
ejpam-4432	38	13	investigated	investigate	VERB
ejpam-4432	38	14	in	in	ADP
ejpam-4432	38	15	section	section	NOUN
ejpam-4432	38	16	3	3	NUM
ejpam-4432	38	17	.	.	PUNCT
ejpam-4432	39	1	in	in	ADP
ejpam-4432	39	2	section	section	NOUN
ejpam-4432	39	3	4	4	NUM
ejpam-4432	39	4	,	,	PUNCT
ejpam-4432	39	5	the	the	DET
ejpam-4432	39	6	conditions	condition	NOUN
ejpam-4432	39	7	of	of	ADP
ejpam-4432	39	8	homomorphism	homomorphism	NOUN
ejpam-4432	39	9	map	map	NOUN
ejpam-4432	39	10	to	to	PART
ejpam-4432	39	11	preserve	preserve	VERB
ejpam-4432	39	12	annihilator	annihilator	NOUN
ejpam-4432	39	13	hyperideals	hyperideal	NOUN
ejpam-4432	39	14	are	be	AUX
ejpam-4432	39	15	discussed	discuss	VERB
ejpam-4432	39	16	.	.	PUNCT
ejpam-4432	40	1	the	the	DET
ejpam-4432	40	2	proof	proof	NOUN
ejpam-4432	40	3	of	of	ADP
ejpam-4432	40	4	the	the	DET
ejpam-4432	40	5	preservation	preservation	NOUN
ejpam-4432	40	6	of	of	ADP
ejpam-4432	40	7	annihilator	annihilator	PROPN
ejpam-4432	40	8	hyperideals	hyperideal	NOUN
ejpam-4432	40	9	under	under	ADP
ejpam-4432	40	10	the	the	DET
ejpam-4432	40	11	effects	effect	NOUN
ejpam-4432	40	12	of	of	ADP
ejpam-4432	40	13	these	these	DET
ejpam-4432	40	14	conditions	condition	NOUN
ejpam-4432	40	15	is	be	AUX
ejpam-4432	40	16	given	give	VERB
ejpam-4432	40	17	.	.	PUNCT
ejpam-4432	41	1	finally	finally	ADV
ejpam-4432	41	2	,	,	PUNCT
ejpam-4432	41	3	in	in	ADP
ejpam-4432	41	4	section	section	NOUN
ejpam-4432	41	5	5	5	NUM
ejpam-4432	41	6	,	,	PUNCT
ejpam-4432	41	7	we	we	PRON
ejpam-4432	41	8	show	show	VERB
ejpam-4432	41	9	that	that	SCONJ
ejpam-4432	41	10	annihilator	annihilator	PROPN
ejpam-4432	41	11	hyperideals	hyperideal	NOUN
ejpam-4432	41	12	are	be	AUX
ejpam-4432	41	13	inherited	inherit	VERB
ejpam-4432	41	14	for	for	ADP
ejpam-4432	41	15	sub	sub	ADJ
ejpam-4432	41	16	-	-	ADJ
ejpam-4432	41	17	meet	meet	ADJ
ejpam-4432	41	18	-	-	PUNCT
ejpam-4432	41	19	hyperlattice	hyperlattice	NOUN
ejpam-4432	41	20	and	and	CCONJ
ejpam-4432	41	21	product	product	NOUN
ejpam-4432	41	22	meet	meet	NOUN
ejpam-4432	41	23	-	-	PUNCT
ejpam-4432	41	24	hyperlattice	hyperlattice	NOUN
ejpam-4432	41	25	.	.	PUNCT
ejpam-4432	42	1	2	2	X
ejpam-4432	42	2	.	.	X
ejpam-4432	42	3	backgrounds	background	NOUN
ejpam-4432	42	4	we	we	PRON
ejpam-4432	42	5	recall	recall	VERB
ejpam-4432	42	6	here	here	ADV
ejpam-4432	42	7	the	the	DET
ejpam-4432	42	8	basic	basic	ADJ
ejpam-4432	42	9	terminologies	terminology	NOUN
ejpam-4432	42	10	and	and	CCONJ
ejpam-4432	42	11	concepts	concept	NOUN
ejpam-4432	42	12	of	of	ADP
ejpam-4432	42	13	hyperlattices	hyperlattice	NOUN
ejpam-4432	42	14	.	.	PUNCT
ejpam-4432	43	1	the	the	DET
ejpam-4432	43	2	reader	reader	NOUN
ejpam-4432	43	3	must	must	AUX
ejpam-4432	43	4	be	be	AUX
ejpam-4432	43	5	familiar	familiar	ADJ
ejpam-4432	43	6	with	with	ADP
ejpam-4432	43	7	lattice	lattice	NOUN
ejpam-4432	43	8	theory	theory	NOUN
ejpam-4432	43	9	.	.	PUNCT
ejpam-4432	44	1	for	for	ADP
ejpam-4432	44	2	more	more	ADJ
ejpam-4432	44	3	details	detail	NOUN
ejpam-4432	44	4	about	about	ADP
ejpam-4432	44	5	lattice	lattice	PROPN
ejpam-4432	44	6	theory	theory	NOUN
ejpam-4432	44	7	,	,	PUNCT
ejpam-4432	44	8	see	see	VERB
ejpam-4432	44	9	[	[	X
ejpam-4432	44	10	2	2	X
ejpam-4432	44	11	]	]	PUNCT
ejpam-4432	44	12	and	and	CCONJ
ejpam-4432	44	13	[	[	X
ejpam-4432	44	14	5	5	NUM
ejpam-4432	44	15	]	]	PUNCT
ejpam-4432	44	16	.	.	PUNCT
ejpam-4432	45	1	definition	definition	NOUN
ejpam-4432	45	2	1	1	NUM
ejpam-4432	45	3	.	.	PUNCT
ejpam-4432	46	1	[	[	X
ejpam-4432	46	2	11	11	NUM
ejpam-4432	46	3	]	]	PUNCT
ejpam-4432	46	4	let	let	VERB
ejpam-4432	46	5	l	l	NOUN
ejpam-4432	46	6	be	be	AUX
ejpam-4432	46	7	a	a	DET
ejpam-4432	46	8	nonempty	nonempty	ADJ
ejpam-4432	46	9	set	set	VERB
ejpam-4432	46	10	,	,	PUNCT
ejpam-4432	46	11	p∗(l	p∗(l	PROPN
ejpam-4432	46	12	)	)	PUNCT
ejpam-4432	46	13	is	be	AUX
ejpam-4432	46	14	the	the	DET
ejpam-4432	46	15	set	set	NOUN
ejpam-4432	46	16	of	of	ADP
ejpam-4432	46	17	all	all	DET
ejpam-4432	46	18	nonempty	nonempty	ADJ
ejpam-4432	46	19	subsets	subset	NOUN
ejpam-4432	46	20	of	of	ADP
ejpam-4432	46	21	l	l	NOUN
ejpam-4432	46	22	,	,	PUNCT
ejpam-4432	46	23	∧̄	∧̄	PROPN
ejpam-4432	46	24	:	:	PUNCT
ejpam-4432	46	25	l	l	NOUN
ejpam-4432	46	26	×	×	NOUN
ejpam-4432	46	27	l	l	NOUN
ejpam-4432	46	28	→	→	SYM
ejpam-4432	46	29	p∗(l	p∗(l	NOUN
ejpam-4432	46	30	)	)	PUNCT
ejpam-4432	46	31	is	be	AUX
ejpam-4432	46	32	a	a	DET
ejpam-4432	46	33	hyperoperation	hyperoperation	NOUN
ejpam-4432	46	34	and	and	CCONJ
ejpam-4432	46	35	∨	∨	NUM
ejpam-4432	46	36	:	:	PUNCT
ejpam-4432	47	1	l	l	NOUN
ejpam-4432	47	2	→	→	PUNCT
ejpam-4432	47	3	l	l	NOUN
ejpam-4432	47	4	is	be	AUX
ejpam-4432	47	5	a	a	DET
ejpam-4432	47	6	binary	binary	ADJ
ejpam-4432	47	7	operation	operation	NOUN
ejpam-4432	47	8	.	.	PUNCT
ejpam-4432	48	1	then	then	ADV
ejpam-4432	48	2	l	l	X
ejpam-4432	48	3	=	=	PUNCT
ejpam-4432	48	4	<	<	X
ejpam-4432	48	5	l	l	NOUN
ejpam-4432	48	6	;	;	PUNCT
ejpam-4432	48	7	∧̄,∨	∧̄,∨	NUM
ejpam-4432	48	8	>	>	X
ejpam-4432	48	9	is	be	AUX
ejpam-4432	48	10	called	call	VERB
ejpam-4432	48	11	a	a	DET
ejpam-4432	48	12	meet	meet	ADJ
ejpam-4432	48	13	-	-	PUNCT
ejpam-4432	48	14	hyperlattice	hyperlattice	NOUN
ejpam-4432	48	15	if	if	SCONJ
ejpam-4432	48	16	:	:	PUNCT
ejpam-4432	48	17	h1	h1	PROPN
ejpam-4432	48	18	)	)	PUNCT
ejpam-4432	48	19	a	a	DET
ejpam-4432	48	20	∈	∈	PROPN
ejpam-4432	48	21	iai∧̄ia	iai∧̄ia	X
ejpam-4432	48	22	,	,	PUNCT
ejpam-4432	48	23	ia	ia	PROPN
ejpam-4432	48	24	=	=	PUNCT
ejpam-4432	48	25	a	a	DET
ejpam-4432	48	26	∨	∨	NOUN
ejpam-4432	48	27	a	a	PRON
ejpam-4432	48	28	;	;	PUNCT
ejpam-4432	48	29	h2	h2	NOUN
ejpam-4432	48	30	)	)	PUNCT
ejpam-4432	48	31	ai∧̄ibi	ai∧̄ibi	NOUN
ejpam-4432	49	1	=	=	PUNCT
ejpam-4432	49	2	ibi∧̄ia	ibi∧̄ia	PROPN
ejpam-4432	49	3	,	,	PUNCT
ejpam-4432	49	4	ia	ia	PROPN
ejpam-4432	49	5	∨	∨	NUM
ejpam-4432	49	6	ibi	ibi	NOUN
ejpam-4432	49	7	=	=	PROPN
ejpam-4432	49	8	ib	ib	PROPN
ejpam-4432	49	9	∨	∨	NUM
ejpam-4432	49	10	ai	ai	VERB
ejpam-4432	49	11	;	;	PUNCT
ejpam-4432	49	12	h3	h3	NOUN
ejpam-4432	49	13	)	)	PUNCT
ejpam-4432	49	14	ai∧̄i(bi∧̄ic)i	ai∧̄i(bi∧̄ic)i	PUNCT
ejpam-4432	49	15	=	=	SYM
ejpam-4432	49	16	i(ai∧̄ib)∧̄ic	i(ai∧̄ib)∧̄ic	PROPN
ejpam-4432	49	17	,	,	PUNCT
ejpam-4432	49	18	a	a	DET
ejpam-4432	49	19	∨	∨	NOUN
ejpam-4432	49	20	(	(	PUNCT
ejpam-4432	49	21	b	b	PROPN
ejpam-4432	49	22	∨	∨	NUM
ejpam-4432	49	23	c	c	NOUN
ejpam-4432	49	24	)	)	PUNCT
ejpam-4432	49	25	=	=	NOUN
ejpam-4432	49	26	(	(	PUNCT
ejpam-4432	49	27	a	a	DET
ejpam-4432	49	28	∨	∨	NUM
ejpam-4432	49	29	b	b	NOUN
ejpam-4432	49	30	)	)	PUNCT
ejpam-4432	49	31	∨	∨	PROPN
ejpam-4432	49	32	c	c	X
ejpam-4432	49	33	;	;	PUNCT
ejpam-4432	49	34	h4	h4	NOUN
ejpam-4432	49	35	)	)	PUNCT
ejpam-4432	49	36	a	a	DET
ejpam-4432	49	37	∈	∈	NOUN
ejpam-4432	49	38	(	(	PUNCT
ejpam-4432	49	39	ai∧̄i(a	ai∧̄i(a	PROPN
ejpam-4432	49	40	∨	∨	NUM
ejpam-4432	49	41	b))i	b))i	NOUN
ejpam-4432	49	42	∩	∩	NOUN
ejpam-4432	49	43	i(a	i(a	PROPN
ejpam-4432	49	44	∨	∨	NOUN
ejpam-4432	49	45	(	(	PUNCT
ejpam-4432	49	46	ai∧̄ib)),i	ai∧̄ib)),i	VERB
ejpam-4432	49	47	ifor	ifor	PROPN
ejpam-4432	49	48	all	all	DET
ejpam-4432	49	49	a	a	PRON
ejpam-4432	49	50	,	,	PUNCT
ejpam-4432	49	51	ib	ib	NOUN
ejpam-4432	49	52	,	,	PUNCT
ejpam-4432	49	53	ic	ic	PROPN
ejpam-4432	49	54	∈	∈	PROPN
ejpam-4432	49	55	l.	l.	NOUN
ejpam-4432	49	56	the	the	DET
ejpam-4432	49	57	meet	meet	NOUN
ejpam-4432	49	58	-	-	PUNCT
ejpam-4432	49	59	hyperlatticce	hyperlatticce	NOUN
ejpam-4432	49	60	l	l	NOUN
ejpam-4432	49	61	is	be	AUX
ejpam-4432	49	62	called	call	VERB
ejpam-4432	49	63	strong	strong	ADJ
ejpam-4432	49	64	,	,	PUNCT
ejpam-4432	49	65	if	if	SCONJ
ejpam-4432	49	66	it	it	PRON
ejpam-4432	49	67	satisfies	satisfy	VERB
ejpam-4432	49	68	that	that	SCONJ
ejpam-4432	49	69	:	:	PUNCT
ejpam-4432	49	70	if	if	SCONJ
ejpam-4432	49	71	a	a	DET
ejpam-4432	49	72	∈	∈	ADJ
ejpam-4432	49	73	ai∧̄ib	ai∧̄ib	NOUN
ejpam-4432	49	74	then	then	ADV
ejpam-4432	49	75	ai	ai	VERB
ejpam-4432	49	76	∨	∨	NOUN
ejpam-4432	49	77	ib	ib	NOUN
ejpam-4432	49	78	=	=	SYM
ejpam-4432	49	79	b	b	PROPN
ejpam-4432	49	80	,	,	PUNCT
ejpam-4432	49	81	i	i	PRON
ejpam-4432	49	82	ifor	ifor	VERB
ejpam-4432	49	83	all	all	DET
ejpam-4432	49	84	a	a	PRON
ejpam-4432	49	85	,	,	PUNCT
ejpam-4432	49	86	ib	ib	PROPN
ejpam-4432	49	87	∈	∈	PROPN
ejpam-4432	49	88	l.	l.	PROPN
ejpam-4432	49	89	consider	consider	VERB
ejpam-4432	49	90	an	an	DET
ejpam-4432	49	91	order	order	NOUN
ejpam-4432	49	92	relation	relation	NOUN
ejpam-4432	49	93	≤	≤	NOUN
ejpam-4432	49	94	on	on	ADP
ejpam-4432	49	95	l	l	NOUN
ejpam-4432	49	96	as	as	ADP
ejpam-4432	49	97	:	:	PUNCT
ejpam-4432	49	98	a	a	DET
ejpam-4432	49	99	≤	≤	PROPN
ejpam-4432	49	100	b	b	X
ejpam-4432	49	101	iff	iff	PROPN
ejpam-4432	49	102	a	a	DET
ejpam-4432	49	103	∨	∨	PROPN
ejpam-4432	49	104	b	b	PROPN
ejpam-4432	49	105	=	=	SYM
ejpam-4432	49	106	b	b	PROPN
ejpam-4432	49	107	,	,	PUNCT
ejpam-4432	49	108	for	for	ADP
ejpam-4432	49	109	all	all	DET
ejpam-4432	49	110	a	a	DET
ejpam-4432	49	111	,	,	PUNCT
ejpam-4432	49	112	b	b	X
ejpam-4432	49	113	∈	∈	PROPN
ejpam-4432	49	114	l.	l.	NOUN
ejpam-4432	49	115	accordingly	accordingly	ADV
ejpam-4432	49	116	,	,	PUNCT
ejpam-4432	49	117	meet	meet	ADJ
ejpam-4432	49	118	-	-	PUNCT
ejpam-4432	49	119	hyperlattice	hyperlattice	NOUN
ejpam-4432	49	120	l	l	NOUN
ejpam-4432	49	121	is	be	AUX
ejpam-4432	49	122	bounded	bound	VERB
ejpam-4432	49	123	if	if	SCONJ
ejpam-4432	49	124	there	there	PRON
ejpam-4432	49	125	exist	exist	VERB
ejpam-4432	49	126	two	two	NUM
ejpam-4432	49	127	elements	element	NOUN
ejpam-4432	49	128	0	0	NUM
ejpam-4432	49	129	,	,	PUNCT
ejpam-4432	49	130	1	1	NUM
ejpam-4432	49	131	∈	∈	NOUN
ejpam-4432	49	132	l	l	NOUN
ejpam-4432	49	133	such	such	ADJ
ejpam-4432	49	134	that	that	SCONJ
ejpam-4432	49	135	0	0	NUM
ejpam-4432	49	136	≤	≤	NUM
ejpam-4432	49	137	a	a	DET
ejpam-4432	49	138	≤	≤	NUM
ejpam-4432	49	139	1	1	NUM
ejpam-4432	49	140	,	,	PUNCT
ejpam-4432	49	141	for	for	ADP
ejpam-4432	49	142	all	all	DET
ejpam-4432	49	143	a	a	DET
ejpam-4432	49	144	∈	∈	PROPN
ejpam-4432	49	145	l.	l.	X
ejpam-4432	49	146	e.g.	e.g.	ADJ
ejpam-4432	49	147	rezk	rezk	PROPN
ejpam-4432	49	148	,	,	PUNCT
ejpam-4432	49	149	n.h	n.h	PROPN
ejpam-4432	49	150	.	.	PROPN
ejpam-4432	49	151	abughazalah	abughazalah	PROPN
ejpam-4432	49	152	/	/	SYM
ejpam-4432	49	153	eur	eur	PROPN
ejpam-4432	49	154	.	.	PUNCT
ejpam-4432	50	1	j.	j.	PROPN
ejpam-4432	50	2	pure	pure	PROPN
ejpam-4432	50	3	appl	appl	PROPN
ejpam-4432	50	4	.	.	PROPN
ejpam-4432	50	5	math	math	PROPN
ejpam-4432	50	6	,	,	PUNCT
ejpam-4432	50	7	15	15	NUM
ejpam-4432	50	8	(	(	PUNCT
ejpam-4432	50	9	3	3	NUM
ejpam-4432	50	10	)	)	PUNCT
ejpam-4432	50	11	(	(	PUNCT
ejpam-4432	50	12	2022	2022	NUM
ejpam-4432	50	13	)	)	PUNCT
ejpam-4432	50	14	,	,	PUNCT
ejpam-4432	50	15	1402	1402	NUM
ejpam-4432	50	16	-	-	SYM
ejpam-4432	50	17	1416	1416	NUM
ejpam-4432	50	18	1404	1404	NUM
ejpam-4432	50	19	for	for	ADP
ejpam-4432	50	20	subsets	subset	NOUN
ejpam-4432	50	21	a	a	PRON
ejpam-4432	50	22	,	,	PUNCT
ejpam-4432	50	23	b	b	PROPN
ejpam-4432	50	24	⊆	⊆	NUM
ejpam-4432	50	25	l	l	NOUN
ejpam-4432	50	26	:	:	PUNCT
ejpam-4432	50	27	ai∧̄ib	ai∧̄ib	NOUN
ejpam-4432	50	28	=	=	SYM
ejpam-4432	51	1	∪{ai∧̄ib	∪{ai∧̄ib	ADJ
ejpam-4432	51	2	:	:	PUNCT
ejpam-4432	51	3	ia	ia	PROPN
ejpam-4432	51	4	∈	∈	PROPN
ejpam-4432	51	5	a	a	PRON
ejpam-4432	51	6	,	,	PUNCT
ejpam-4432	51	7	ib	ib	PROPN
ejpam-4432	51	8	∈	∈	PROPN
ejpam-4432	51	9	b	b	PROPN
ejpam-4432	51	10	}	}	PUNCT
ejpam-4432	51	11	,	,	PUNCT
ejpam-4432	51	12	ai	ai	VERB
ejpam-4432	51	13	∨	∨	NOUN
ejpam-4432	51	14	ib	ib	NOUN
ejpam-4432	51	15	=	=	PUNCT
ejpam-4432	51	16	{	{	PUNCT
ejpam-4432	51	17	ai	ai	PROPN
ejpam-4432	51	18	∨	∨	PROPN
ejpam-4432	51	19	ib	ib	NOUN
ejpam-4432	51	20	:	:	PUNCT
ejpam-4432	51	21	ia	ia	PROPN
ejpam-4432	51	22	∈	∈	PROPN
ejpam-4432	51	23	a	a	PRON
ejpam-4432	51	24	,	,	PUNCT
ejpam-4432	51	25	ib	ib	PROPN
ejpam-4432	51	26	∈	∈	PROPN
ejpam-4432	51	27	b	b	PROPN
ejpam-4432	51	28	}	}	PUNCT
ejpam-4432	51	29	.	.	PUNCT
ejpam-4432	52	1	proposition	proposition	NOUN
ejpam-4432	52	2	1	1	NUM
ejpam-4432	52	3	.	.	PUNCT
ejpam-4432	53	1	[	[	X
ejpam-4432	53	2	11	11	NUM
ejpam-4432	53	3	]	]	PUNCT
ejpam-4432	53	4	let	let	VERB
ejpam-4432	53	5	l	l	NOUN
ejpam-4432	53	6	be	be	AUX
ejpam-4432	53	7	a	a	DET
ejpam-4432	53	8	boundedistrong	boundedistrong	ADJ
ejpam-4432	53	9	meet	meet	NOUN
ejpam-4432	53	10	-	-	PUNCT
ejpam-4432	53	11	hyperlattice	hyperlattice	NOUN
ejpam-4432	53	12	.	.	PUNCT
ejpam-4432	54	1	then	then	ADV
ejpam-4432	54	2	theifollowing	theifollowe	VERB
ejpam-4432	54	3	conditions	condition	NOUN
ejpam-4432	54	4	hold	hold	VERB
ejpam-4432	54	5	:	:	PUNCT
ejpam-4432	54	6	(	(	PUNCT
ejpam-4432	54	7	i	i	NOUN
ejpam-4432	54	8	)	)	PUNCT
ejpam-4432	54	9	if	if	SCONJ
ejpam-4432	54	10	a	a	PRON
ejpam-4432	54	11	,	,	PUNCT
ejpam-4432	54	12	b	b	NOUN
ejpam-4432	54	13	̸=	̸=	PROPN
ejpam-4432	54	14	1	1	NUM
ejpam-4432	54	15	iandiai	iandiai	PROPN
ejpam-4432	54	16	∨	∨	NUM
ejpam-4432	54	17	ib	ib	NOUN
ejpam-4432	55	1	=	=	SYM
ejpam-4432	56	1	1	1	NUM
ejpam-4432	56	2	,	,	PUNCT
ejpam-4432	56	3	ithen	ithen	PROPN
ejpam-4432	56	4	ia	ia	PROPN
ejpam-4432	56	5	,	,	PUNCT
ejpam-4432	56	6	b	b	PROPN
ejpam-4432	56	7	/∈	/∈	PUNCT
ejpam-4432	56	8	ai∧̄ib	ai∧̄ib	NOUN
ejpam-4432	56	9	;	;	PUNCT
ejpam-4432	56	10	(	(	PUNCT
ejpam-4432	56	11	ii	ii	NOUN
ejpam-4432	56	12	)	)	PUNCT
ejpam-4432	56	13	if	if	SCONJ
ejpam-4432	56	14	ai∧̄ib	ai∧̄ib	NOUN
ejpam-4432	56	15	=	=	SYM
ejpam-4432	56	16	l	l	NOUN
ejpam-4432	56	17	or	or	CCONJ
ejpam-4432	56	18	a	a	DET
ejpam-4432	56	19	,	,	PUNCT
ejpam-4432	56	20	b	b	PROPN
ejpam-4432	56	21	∈	∈	PROPN
ejpam-4432	56	22	ai∧̄ib	ai∧̄ib	NOUN
ejpam-4432	56	23	,	,	PUNCT
ejpam-4432	56	24	then	then	ADV
ejpam-4432	56	25	a	a	DET
ejpam-4432	56	26	=	=	SYM
ejpam-4432	56	27	b	b	NOUN
ejpam-4432	56	28	;	;	PUNCT
ejpam-4432	56	29	(	(	PUNCT
ejpam-4432	56	30	iii	iii	NOUN
ejpam-4432	56	31	)	)	PUNCT
ejpam-4432	56	32	for	for	ADP
ejpam-4432	56	33	all	all	DET
ejpam-4432	56	34	a	a	DET
ejpam-4432	56	35	∈	∈	ADJ
ejpam-4432	56	36	l	l	NOUN
ejpam-4432	56	37	:	:	PUNCT
ejpam-4432	56	38	a	a	DET
ejpam-4432	56	39	∈	∈	ADJ
ejpam-4432	56	40	ai∧̄i1	ai∧̄i1	ADV
ejpam-4432	56	41	and	and	CCONJ
ejpam-4432	56	42	0	0	NUM
ejpam-4432	56	43	∈	∈	PROPN
ejpam-4432	56	44	ai∧̄i0	ai∧̄i0	NOUN
ejpam-4432	56	45	.	.	PUNCT
ejpam-4432	57	1	the	the	DET
ejpam-4432	57	2	meet	meet	ADJ
ejpam-4432	57	3	-	-	PUNCT
ejpam-4432	57	4	hyperlattice	hyperlattice	NOUN
ejpam-4432	57	5	l	l	NOUN
ejpam-4432	57	6	is	be	AUX
ejpam-4432	57	7	distributive	distributive	ADJ
ejpam-4432	57	8	,	,	PUNCT
ejpam-4432	57	9	if	if	SCONJ
ejpam-4432	57	10	ai	ai	ADV
ejpam-4432	57	11	∨	∨	NUM
ejpam-4432	57	12	i(bi∧̄ic	i(bi∧̄ic	PROPN
ejpam-4432	57	13	)	)	PUNCT
ejpam-4432	58	1	=	=	PRON
ejpam-4432	58	2	(	(	PUNCT
ejpam-4432	58	3	ai	ai	INTJ
ejpam-4432	58	4	∨	∨	NUM
ejpam-4432	58	5	ib)i∧̄i(ai	ib)i∧̄i(ai	PRON
ejpam-4432	58	6	∨	∨	PROPN
ejpam-4432	58	7	ic	ic	NUM
ejpam-4432	58	8	)	)	PUNCT
ejpam-4432	58	9	,	,	PUNCT
ejpam-4432	58	10	for	for	ADP
ejpam-4432	58	11	all	all	DET
ejpam-4432	58	12	a	a	DET
ejpam-4432	58	13	,	,	PUNCT
ejpam-4432	58	14	b	b	NOUN
ejpam-4432	58	15	,	,	PUNCT
ejpam-4432	58	16	c	c	PROPN
ejpam-4432	58	17	∈	∈	PROPN
ejpam-4432	58	18	l.	l.	PROPN
ejpam-4432	58	19	dually	dually	PROPN
ejpam-4432	58	20	,	,	PUNCT
ejpam-4432	58	21	l	l	NOUN
ejpam-4432	58	22	is	be	AUX
ejpam-4432	58	23	dual	dual	ADV
ejpam-4432	58	24	distributive	distributive	ADJ
ejpam-4432	58	25	if	if	SCONJ
ejpam-4432	58	26	ai∧̄i(b	ai∧̄i(b	PROPN
ejpam-4432	58	27	∨	∨	NUM
ejpam-4432	58	28	ic	ic	PROPN
ejpam-4432	58	29	)	)	PUNCT
ejpam-4432	58	30	=	=	VERB
ejpam-4432	58	31	(	(	PUNCT
ejpam-4432	58	32	ai∧̄ib)i	ai∧̄ib)i	PROPN
ejpam-4432	58	33	∨	∨	NUM
ejpam-4432	58	34	i(ai∧̄ic	i(ai∧̄ic	PROPN
ejpam-4432	58	35	)	)	PUNCT
ejpam-4432	58	36	.	.	PUNCT
ejpam-4432	59	1	definition	definition	NOUN
ejpam-4432	59	2	2	2	NUM
ejpam-4432	59	3	.	.	PUNCT
ejpam-4432	60	1	[	[	X
ejpam-4432	60	2	11	11	NUM
ejpam-4432	60	3	]	]	PUNCT
ejpam-4432	60	4	let	let	VERB
ejpam-4432	60	5	i	i	PRON
ejpam-4432	60	6	be	be	AUX
ejpam-4432	60	7	a	a	DET
ejpam-4432	60	8	nonempty	nonempty	ADJ
ejpam-4432	60	9	subset	subset	NOUN
ejpam-4432	60	10	of	of	ADP
ejpam-4432	60	11	a	a	DET
ejpam-4432	60	12	strong	strong	ADJ
ejpam-4432	60	13	meet	meet	NOUN
ejpam-4432	60	14	-	-	PUNCT
ejpam-4432	60	15	hyperlattice	hyperlattice	NOUN
ejpam-4432	60	16	l.	l.	NOUN
ejpam-4432	60	17	i	i	PRON
ejpam-4432	60	18	is	be	AUX
ejpam-4432	60	19	called	call	VERB
ejpam-4432	60	20	a	a	DET
ejpam-4432	60	21	hyperideal	hyperideal	NOUN
ejpam-4432	60	22	if	if	SCONJ
ejpam-4432	60	23	:	:	PUNCT
ejpam-4432	60	24	i	i	NOUN
ejpam-4432	60	25	)	)	PUNCT
ejpam-4432	60	26	if	if	SCONJ
ejpam-4432	60	27	a	a	PRON
ejpam-4432	60	28	,	,	PUNCT
ejpam-4432	61	1	ib	ib	PROPN
ejpam-4432	61	2	∈	∈	PROPN
ejpam-4432	61	3	i	i	PRON
ejpam-4432	61	4	,	,	PUNCT
ejpam-4432	61	5	theniai	theniai	PROPN
ejpam-4432	61	6	∨	∨	PROPN
ejpam-4432	61	7	ib	ib	PROPN
ejpam-4432	61	8	∈	∈	PROPN
ejpam-4432	62	1	i	i	PRON
ejpam-4432	62	2	;	;	PUNCT
ejpam-4432	62	3	ii	ii	X
ejpam-4432	62	4	)	)	PUNCT
ejpam-4432	62	5	ifia	ifia	PROPN
ejpam-4432	62	6	∈	∈	PROPN
ejpam-4432	62	7	iiand	iiand	NOUN
ejpam-4432	62	8	b	b	PROPN
ejpam-4432	62	9	∈	∈	PROPN
ejpam-4432	62	10	l	l	NOUN
ejpam-4432	62	11	,	,	PUNCT
ejpam-4432	62	12	such	such	ADJ
ejpam-4432	62	13	that	that	SCONJ
ejpam-4432	62	14	b	b	NOUN
ejpam-4432	62	15	≤	≤	NUM
ejpam-4432	62	16	a	a	DET
ejpam-4432	62	17	,	,	PUNCT
ejpam-4432	62	18	then	then	ADV
ejpam-4432	62	19	b	b	PROPN
ejpam-4432	62	20	∈	∈	PROPN
ejpam-4432	62	21	i.	i.	NOUN
ejpam-4432	62	22	the	the	DET
ejpam-4432	62	23	set	set	NOUN
ejpam-4432	62	24	of	of	ADP
ejpam-4432	62	25	all	all	DET
ejpam-4432	62	26	hyperideals	hyperideal	NOUN
ejpam-4432	62	27	of	of	ADP
ejpam-4432	62	28	strong	strong	ADJ
ejpam-4432	62	29	bounded	bounded	ADJ
ejpam-4432	62	30	meet	meet	NOUN
ejpam-4432	62	31	-	-	PUNCT
ejpam-4432	62	32	hyperlattice	hyperlattice	NOUN
ejpam-4432	62	33	l	l	NOUN
ejpam-4432	62	34	is	be	AUX
ejpam-4432	62	35	denoted	denote	VERB
ejpam-4432	62	36	by	by	ADP
ejpam-4432	62	37	i(l	i(l	PROPN
ejpam-4432	62	38	)	)	PUNCT
ejpam-4432	62	39	.	.	PUNCT
ejpam-4432	63	1	the	the	DET
ejpam-4432	63	2	meet	meet	ADJ
ejpam-4432	63	3	operation	operation	NOUN
ejpam-4432	63	4	on	on	ADP
ejpam-4432	63	5	hyprideals	hyprideal	NOUN
ejpam-4432	63	6	is	be	AUX
ejpam-4432	63	7	the	the	DET
ejpam-4432	63	8	usual	usual	ADJ
ejpam-4432	63	9	sets	set	NOUN
ejpam-4432	63	10	intersection	intersection	NOUN
ejpam-4432	63	11	∩	∩	NOUN
ejpam-4432	63	12	and	and	CCONJ
ejpam-4432	63	13	the	the	DET
ejpam-4432	63	14	join	join	NOUN
ejpam-4432	63	15	operation	operation	NOUN
ejpam-4432	63	16	is	be	AUX
ejpam-4432	63	17	defined	define	VERB
ejpam-4432	63	18	as	as	ADP
ejpam-4432	63	19	ii∨ij	ii∨ij	NOUN
ejpam-4432	63	20	=	=	SYM
ejpam-4432	63	21	{	{	PUNCT
ejpam-4432	63	22	x	x	SYM
ejpam-4432	63	23	∈	∈	NOUN
ejpam-4432	63	24	l	l	NOUN
ejpam-4432	63	25	:	:	PUNCT
ejpam-4432	63	26	x	x	SYM
ejpam-4432	63	27	≤	≤	NUM
ejpam-4432	63	28	ai	ai	VERB
ejpam-4432	63	29	∨	∨	PROPN
ejpam-4432	63	30	ib	ib	PROPN
ejpam-4432	63	31	,	,	PUNCT
ejpam-4432	63	32	a	a	DET
ejpam-4432	63	33	∈	∈	PROPN
ejpam-4432	63	34	i	i	NOUN
ejpam-4432	63	35	,	,	PUNCT
ejpam-4432	63	36	b	b	PROPN
ejpam-4432	63	37	∈	∈	PROPN
ejpam-4432	63	38	j	j	PROPN
ejpam-4432	63	39	}	}	PUNCT
ejpam-4432	63	40	.	.	PUNCT
ejpam-4432	64	1	then	then	ADV
ejpam-4432	64	2	,	,	PUNCT
ejpam-4432	64	3	we	we	PRON
ejpam-4432	64	4	get	get	VERB
ejpam-4432	64	5	the	the	DET
ejpam-4432	64	6	following	follow	VERB
ejpam-4432	64	7	theorem	theorem	NOUN
ejpam-4432	64	8	:	:	PUNCT
ejpam-4432	64	9	theorem	theorem	NOUN
ejpam-4432	64	10	1	1	NUM
ejpam-4432	64	11	.	.	PUNCT
ejpam-4432	65	1	[	[	X
ejpam-4432	65	2	11	11	NUM
ejpam-4432	65	3	]	]	PUNCT
ejpam-4432	65	4	the	the	DET
ejpam-4432	65	5	structure	structure	NOUN
ejpam-4432	65	6	(	(	PUNCT
ejpam-4432	65	7	i(l);∩,∨	i(l);∩,∨	NOUN
ejpam-4432	65	8	,	,	PUNCT
ejpam-4432	65	9	0	0	NUM
ejpam-4432	65	10	,	,	PUNCT
ejpam-4432	65	11	l	l	NOUN
ejpam-4432	65	12	)	)	PUNCT
ejpam-4432	65	13	forms	form	VERB
ejpam-4432	65	14	a	a	DET
ejpam-4432	65	15	bounded	bounded	ADJ
ejpam-4432	65	16	distributive	distributive	ADJ
ejpam-4432	65	17	lattice	lattice	NOUN
ejpam-4432	65	18	.	.	PUNCT
ejpam-4432	66	1	definition	definition	NOUN
ejpam-4432	66	2	3	3	NUM
ejpam-4432	66	3	.	.	PUNCT
ejpam-4432	67	1	[	[	X
ejpam-4432	67	2	11	11	NUM
ejpam-4432	67	3	]	]	X
ejpam-4432	67	4	let	let	AUX
ejpam-4432	67	5	l	l	NOUN
ejpam-4432	67	6	=	=	SYM
ejpam-4432	67	7	(	(	PUNCT
ejpam-4432	67	8	l	l	NOUN
ejpam-4432	67	9	,	,	PUNCT
ejpam-4432	67	10	∧̄,∨	∧̄,∨	NUM
ejpam-4432	67	11	)	)	PUNCT
ejpam-4432	67	12	be	be	AUX
ejpam-4432	67	13	aistrong	aistrong	ADV
ejpam-4432	67	14	bounded	bounded	ADJ
ejpam-4432	67	15	dualidistributive	dualidistributive	ADJ
ejpam-4432	67	16	meet	meet	PROPN
ejpam-4432	67	17	-	-	PUNCT
ejpam-4432	67	18	hyperlatticei	hyperlatticei	NOUN
ejpam-4432	67	19	and	and	CCONJ
ejpam-4432	67	20	a	a	DET
ejpam-4432	67	21	⊆	⊆	NUM
ejpam-4432	67	22	l.	l.	NOUN
ejpam-4432	67	23	then	then	ADV
ejpam-4432	67	24	the	the	DET
ejpam-4432	67	25	hyperideal	hyperideal	PROPN
ejpam-4432	67	26	ar	ar	NOUN
ejpam-4432	67	27	=	=	SYM
ejpam-4432	67	28	{	{	PUNCT
ejpam-4432	67	29	x	x	PUNCT
ejpam-4432	67	30	∈	∈	NOUN
ejpam-4432	67	31	l	l	NOUN
ejpam-4432	67	32	:	:	PUNCT
ejpam-4432	67	33	0	0	NUM
ejpam-4432	67	34	∈	∈	PROPN
ejpam-4432	67	35	x∧̄a	x∧̄a	PROPN
ejpam-4432	67	36	,	,	PUNCT
ejpam-4432	67	37	forall	forall	VERB
ejpam-4432	67	38	a	a	DET
ejpam-4432	67	39	∈	∈	PROPN
ejpam-4432	67	40	a	a	PRON
ejpam-4432	67	41	}	}	PUNCT
ejpam-4432	67	42	,	,	PUNCT
ejpam-4432	67	43	is	be	AUX
ejpam-4432	67	44	called	call	VERB
ejpam-4432	67	45	annihilator	annihilator	PROPN
ejpam-4432	67	46	hyperideal	hyperideal	NOUN
ejpam-4432	67	47	or	or	CCONJ
ejpam-4432	67	48	for	for	ADP
ejpam-4432	67	49	brevity	brevity	NOUN
ejpam-4432	67	50	(	(	PUNCT
ejpam-4432	67	51	annihilator	annihilator	PROPN
ejpam-4432	67	52	)	)	PUNCT
ejpam-4432	67	53	.	.	PUNCT
ejpam-4432	68	1	when	when	SCONJ
ejpam-4432	68	2	a	a	PRON
ejpam-4432	68	3	is	be	AUX
ejpam-4432	68	4	a	a	DET
ejpam-4432	68	5	singleton	singleton	NOUN
ejpam-4432	68	6	subset	subset	NOUN
ejpam-4432	68	7	{	{	PUNCT
ejpam-4432	68	8	a	a	NOUN
ejpam-4432	68	9	}	}	PUNCT
ejpam-4432	68	10	of	of	ADP
ejpam-4432	68	11	l	l	NOUN
ejpam-4432	68	12	,	,	PUNCT
ejpam-4432	68	13	its	its	PRON
ejpam-4432	68	14	annihilator	annihilator	NOUN
ejpam-4432	68	15	is	be	AUX
ejpam-4432	68	16	defined	define	VERB
ejpam-4432	68	17	as	as	ADP
ejpam-4432	68	18	:	:	PUNCT
ejpam-4432	68	19	ar	ar	PROPN
ejpam-4432	68	20	=	=	SYM
ejpam-4432	68	21	{	{	PUNCT
ejpam-4432	68	22	x	x	PUNCT
ejpam-4432	68	23	∈	∈	NOUN
ejpam-4432	68	24	l	l	NOUN
ejpam-4432	68	25	:	:	PUNCT
ejpam-4432	68	26	0	0	NUM
ejpam-4432	68	27	∈	∈	PROPN
ejpam-4432	68	28	a∧̄x	a∧̄x	PROPN
ejpam-4432	68	29	}	}	PUNCT
ejpam-4432	68	30	.	.	PUNCT
ejpam-4432	69	1	proposition	proposition	NOUN
ejpam-4432	69	2	2	2	NUM
ejpam-4432	69	3	.	.	PUNCT
ejpam-4432	70	1	[	[	X
ejpam-4432	70	2	11	11	NUM
ejpam-4432	70	3	]	]	X
ejpam-4432	70	4	let	let	AUX
ejpam-4432	70	5	l	l	NOUN
ejpam-4432	70	6	=	=	SYM
ejpam-4432	70	7	(	(	PUNCT
ejpam-4432	70	8	l	l	NOUN
ejpam-4432	70	9	,	,	PUNCT
ejpam-4432	70	10	∧̄,∨	∧̄,∨	NUM
ejpam-4432	70	11	)	)	PUNCT
ejpam-4432	70	12	be	be	AUX
ejpam-4432	70	13	a	a	DET
ejpam-4432	70	14	strong	strong	ADJ
ejpam-4432	70	15	bounded	bounded	ADJ
ejpam-4432	70	16	dualidistributive	dualidistributive	ADJ
ejpam-4432	70	17	meet	meet	NOUN
ejpam-4432	70	18	-	-	PUNCT
ejpam-4432	70	19	hyperlattice	hyperlattice	NOUN
ejpam-4432	70	20	.	.	PUNCT
ejpam-4432	71	1	then	then	ADV
ejpam-4432	71	2	:	:	PUNCT
ejpam-4432	71	3	(	(	PUNCT
ejpam-4432	71	4	i	i	NOUN
ejpam-4432	71	5	)	)	PUNCT
ejpam-4432	71	6	0r	0r	X
ejpam-4432	72	1	=	=	SYM
ejpam-4432	72	2	l	l	NOUN
ejpam-4432	72	3	;	;	PUNCT
ejpam-4432	72	4	(	(	PUNCT
ejpam-4432	72	5	ii	ii	NOUN
ejpam-4432	72	6	)	)	PUNCT
ejpam-4432	72	7	if	if	SCONJ
ejpam-4432	72	8	a	a	PRON
ejpam-4432	72	9	,	,	PUNCT
ejpam-4432	72	10	b	b	PROPN
ejpam-4432	72	11	∈	∈	PROPN
ejpam-4432	72	12	l	l	NOUN
ejpam-4432	72	13	and	and	CCONJ
ejpam-4432	72	14	a	a	DET
ejpam-4432	72	15	≤	≤	NUM
ejpam-4432	72	16	b	b	NOUN
ejpam-4432	72	17	,	,	PUNCT
ejpam-4432	72	18	ithen	ithen	NOUN
ejpam-4432	72	19	br	br	NOUN
ejpam-4432	72	20	⊆	⊆	NUM
ejpam-4432	72	21	ar	ar	NOUN
ejpam-4432	72	22	;	;	PUNCT
ejpam-4432	72	23	e.g.	e.g.	ADV
ejpam-4432	72	24	rezk	rezk	PROPN
ejpam-4432	72	25	,	,	PUNCT
ejpam-4432	72	26	n.h	n.h	PROPN
ejpam-4432	72	27	.	.	PROPN
ejpam-4432	72	28	abughazalah	abughazalah	PROPN
ejpam-4432	72	29	/	/	SYM
ejpam-4432	72	30	eur	eur	PROPN
ejpam-4432	72	31	.	.	PUNCT
ejpam-4432	73	1	j.	j.	PROPN
ejpam-4432	73	2	pure	pure	PROPN
ejpam-4432	73	3	appl	appl	PROPN
ejpam-4432	73	4	.	.	PROPN
ejpam-4432	73	5	math	math	PROPN
ejpam-4432	73	6	,	,	PUNCT
ejpam-4432	73	7	15	15	NUM
ejpam-4432	73	8	(	(	PUNCT
ejpam-4432	73	9	3	3	NUM
ejpam-4432	73	10	)	)	PUNCT
ejpam-4432	73	11	(	(	PUNCT
ejpam-4432	73	12	2022	2022	NUM
ejpam-4432	73	13	)	)	PUNCT
ejpam-4432	73	14	,	,	PUNCT
ejpam-4432	73	15	1402	1402	NUM
ejpam-4432	73	16	-	-	SYM
ejpam-4432	73	17	1416	1416	NUM
ejpam-4432	73	18	1405	1405	NUM
ejpam-4432	73	19	(	(	PUNCT
ejpam-4432	73	20	iii	iii	NOUN
ejpam-4432	73	21	)	)	PUNCT
ejpam-4432	73	22	ar	ar	NOUN
ejpam-4432	73	23	=	=	VERB
ejpam-4432	73	24	∩{ar	∩{ar	NOUN
ejpam-4432	73	25	:	:	PUNCT
ejpam-4432	73	26	a	a	DET
ejpam-4432	73	27	∈	∈	PROPN
ejpam-4432	73	28	a	a	X
ejpam-4432	73	29	}	}	PUNCT
ejpam-4432	73	30	;	;	PUNCT
ejpam-4432	73	31	(	(	PUNCT
ejpam-4432	73	32	iv	iv	X
ejpam-4432	73	33	)	)	PUNCT
ejpam-4432	73	34	ar	ar	NOUN
ejpam-4432	73	35	∩br	∩br	VERB
ejpam-4432	73	36	=	=	PUNCT
ejpam-4432	73	37	(	(	PUNCT
ejpam-4432	73	38	a	a	DET
ejpam-4432	73	39	∪b)r	∪b)r	NOUN
ejpam-4432	73	40	.	.	PUNCT
ejpam-4432	74	1	notice	notice	VERB
ejpam-4432	74	2	that	that	SCONJ
ejpam-4432	74	3	(	(	PUNCT
ejpam-4432	74	4	ii	ii	NOUN
ejpam-4432	74	5	)	)	PUNCT
ejpam-4432	74	6	can	can	AUX
ejpam-4432	74	7	be	be	AUX
ejpam-4432	74	8	generalized	generalize	VERB
ejpam-4432	74	9	to	to	ADP
ejpam-4432	74	10	any	any	DET
ejpam-4432	74	11	subsets	subset	NOUN
ejpam-4432	74	12	a	a	PRON
ejpam-4432	74	13	and	and	CCONJ
ejpam-4432	74	14	b	b	NOUN
ejpam-4432	74	15	of	of	ADP
ejpam-4432	74	16	l	l	NOUN
ejpam-4432	74	17	as	as	ADP
ejpam-4432	74	18	:	:	PUNCT
ejpam-4432	74	19	if	if	SCONJ
ejpam-4432	74	20	a	a	PRON
ejpam-4432	74	21	,	,	PUNCT
ejpam-4432	74	22	b	b	NOUN
ejpam-4432	74	23	⊆	⊆	NUM
ejpam-4432	74	24	l	l	NOUN
ejpam-4432	74	25	and	and	CCONJ
ejpam-4432	74	26	a	a	DET
ejpam-4432	74	27	⊆	⊆	NUM
ejpam-4432	74	28	b	b	NOUN
ejpam-4432	74	29	then	then	ADV
ejpam-4432	74	30	br	br	PROPN
ejpam-4432	74	31	⊆	⊆	NUM
ejpam-4432	74	32	ar	ar	NOUN
ejpam-4432	74	33	.	.	PROPN
ejpam-4432	75	1	in	in	ADP
ejpam-4432	75	2	all	all	PRON
ejpam-4432	75	3	of	of	ADP
ejpam-4432	75	4	the	the	DET
ejpam-4432	75	5	following	following	NOUN
ejpam-4432	75	6	,	,	PUNCT
ejpam-4432	75	7	the	the	DET
ejpam-4432	75	8	meet	meet	ADJ
ejpam-4432	75	9	-	-	PUNCT
ejpam-4432	75	10	hyperlattice	hyperlattice	NOUN
ejpam-4432	75	11	l	l	NOUN
ejpam-4432	75	12	=	=	SYM
ejpam-4432	75	13	<	<	X
ejpam-4432	75	14	l	l	NOUN
ejpam-4432	75	15	;	;	PUNCT
ejpam-4432	75	16	∧̄,∨	∧̄,∨	NUM
ejpam-4432	75	17	,	,	PUNCT
ejpam-4432	75	18	0	0	NUM
ejpam-4432	75	19	,	,	PUNCT
ejpam-4432	75	20	1	1	NUM
ejpam-4432	75	21	>	>	X
ejpam-4432	75	22	is	be	AUX
ejpam-4432	75	23	considered	consider	VERB
ejpam-4432	75	24	that	that	SCONJ
ejpam-4432	75	25	strong	strong	ADJ
ejpam-4432	75	26	bounded	bounded	ADJ
ejpam-4432	75	27	and	and	CCONJ
ejpam-4432	75	28	dual	dual	ADJ
ejpam-4432	75	29	distributive	distributive	ADJ
ejpam-4432	75	30	.	.	PUNCT
ejpam-4432	76	1	3	3	X
ejpam-4432	76	2	.	.	X
ejpam-4432	77	1	annihilator	annihilator	NOUN
ejpam-4432	77	2	hyperideals	hyperideal	NOUN
ejpam-4432	77	3	the	the	DET
ejpam-4432	77	4	core	core	NOUN
ejpam-4432	77	5	of	of	ADP
ejpam-4432	77	6	this	this	DET
ejpam-4432	77	7	section	section	NOUN
ejpam-4432	77	8	is	be	AUX
ejpam-4432	77	9	that	that	SCONJ
ejpam-4432	77	10	we	we	PRON
ejpam-4432	77	11	prove	prove	VERB
ejpam-4432	77	12	the	the	DET
ejpam-4432	77	13	main	main	ADJ
ejpam-4432	77	14	theory	theory	NOUN
ejpam-4432	77	15	,	,	PUNCT
ejpam-4432	77	16	which	which	PRON
ejpam-4432	77	17	states	state	VERB
ejpam-4432	77	18	that	that	SCONJ
ejpam-4432	77	19	the	the	DET
ejpam-4432	77	20	set	set	NOUN
ejpam-4432	77	21	of	of	ADP
ejpam-4432	77	22	all	all	DET
ejpam-4432	77	23	closed	closed	ADJ
ejpam-4432	77	24	hyperideals	hyperideal	NOUN
ejpam-4432	77	25	forms	form	VERB
ejpam-4432	77	26	a	a	DET
ejpam-4432	77	27	boolean	boolean	ADJ
ejpam-4432	77	28	algebra	algebra	NOUN
ejpam-4432	77	29	.	.	PUNCT
ejpam-4432	78	1	theorem	theorem	NOUN
ejpam-4432	78	2	2	2	NUM
ejpam-4432	78	3	.	.	PUNCT
ejpam-4432	79	1	let	let	VERB
ejpam-4432	79	2	i	i	PRON
ejpam-4432	79	3	,	,	PUNCT
ejpam-4432	79	4	j	j	PROPN
ejpam-4432	79	5	be	be	VERB
ejpam-4432	79	6	subsets	subset	NOUN
ejpam-4432	79	7	of	of	ADP
ejpam-4432	79	8	meet	meet	ADJ
ejpam-4432	79	9	-	-	PUNCT
ejpam-4432	79	10	hyperlattice	hyperlattice	NOUN
ejpam-4432	79	11	l.	l.	NOUN
ejpam-4432	80	1	then	then	ADV
ejpam-4432	80	2	(	(	PUNCT
ejpam-4432	80	3	i	i	NOUN
ejpam-4432	80	4	)	)	PUNCT
ejpam-4432	80	5	i	i	PROPN
ejpam-4432	80	6	⊆	⊆	NUM
ejpam-4432	80	7	irr	irr	NOUN
ejpam-4432	80	8	;	;	PUNCT
ejpam-4432	80	9	(	(	PUNCT
ejpam-4432	80	10	ii	ii	NOUN
ejpam-4432	80	11	)	)	PUNCT
ejpam-4432	80	12	irrr	irrr	NOUN
ejpam-4432	81	1	=	=	PUNCT
ejpam-4432	81	2	ir	ir	PROPN
ejpam-4432	81	3	;	;	PUNCT
ejpam-4432	81	4	(	(	PUNCT
ejpam-4432	81	5	iii	iii	X
ejpam-4432	81	6	)	)	PUNCT
ejpam-4432	81	7	i	i	PRON
ejpam-4432	81	8	∩	∩	VERB
ejpam-4432	81	9	j	j	PROPN
ejpam-4432	81	10	⊆	⊆	NUM
ejpam-4432	81	11	(	(	PUNCT
ejpam-4432	81	12	ir	ir	PROPN
ejpam-4432	81	13	∨	∨	PROPN
ejpam-4432	81	14	jr)r	jr)r	PROPN
ejpam-4432	81	15	;	;	PUNCT
ejpam-4432	81	16	(	(	PUNCT
ejpam-4432	81	17	iv	iv	X
ejpam-4432	81	18	)	)	PUNCT
ejpam-4432	81	19	ir	ir	PROPN
ejpam-4432	81	20	∩	∩	X
ejpam-4432	81	21	jr	jr	PROPN
ejpam-4432	82	1	=	=	PUNCT
ejpam-4432	82	2	(	(	PUNCT
ejpam-4432	82	3	i	i	NOUN
ejpam-4432	82	4	∨	∨	PROPN
ejpam-4432	82	5	j)r	j)r	NOUN
ejpam-4432	82	6	;	;	PUNCT
ejpam-4432	82	7	(	(	PUNCT
ejpam-4432	82	8	v	v	NOUN
ejpam-4432	82	9	)	)	PUNCT
ejpam-4432	82	10	(	(	PUNCT
ejpam-4432	82	11	i	i	NOUN
ejpam-4432	82	12	∩	∩	VERB
ejpam-4432	82	13	j)rr	j)rr	PROPN
ejpam-4432	82	14	⊆	⊆	NUM
ejpam-4432	82	15	irr	irr	NOUN
ejpam-4432	82	16	∩	∩	PROPN
ejpam-4432	82	17	jrr	jrr	PROPN
ejpam-4432	82	18	;	;	PUNCT
ejpam-4432	82	19	(	(	PUNCT
ejpam-4432	82	20	vi	vi	NOUN
ejpam-4432	82	21	)	)	PUNCT
ejpam-4432	82	22	1r	1r	NUM
ejpam-4432	82	23	⊆	⊆	NUM
ejpam-4432	82	24	ir	ir	NOUN
ejpam-4432	82	25	for	for	ADP
ejpam-4432	82	26	all	all	DET
ejpam-4432	82	27	i	i	PRON
ejpam-4432	82	28	⊆	⊆	NUM
ejpam-4432	82	29	l	l	NOUN
ejpam-4432	82	30	;	;	PUNCT
ejpam-4432	82	31	(	(	PUNCT
ejpam-4432	82	32	vii	vii	PROPN
ejpam-4432	82	33	)	)	PUNCT
ejpam-4432	82	34	ir	ir	PROPN
ejpam-4432	82	35	∩	∩	PROPN
ejpam-4432	82	36	irr	irr	NOUN
ejpam-4432	82	37	=	=	NOUN
ejpam-4432	82	38	1r	1r	NUM
ejpam-4432	82	39	;	;	PUNCT
ejpam-4432	82	40	(	(	PUNCT
ejpam-4432	82	41	viii	viii	NOUN
ejpam-4432	82	42	)	)	PUNCT
ejpam-4432	82	43	if	if	SCONJ
ejpam-4432	82	44	i	i	PRON
ejpam-4432	82	45	⊆	⊆	NUM
ejpam-4432	82	46	jr	jr	PROPN
ejpam-4432	82	47	then	then	ADV
ejpam-4432	82	48	i	i	PROPN
ejpam-4432	82	49	∩	∩	PROPN
ejpam-4432	82	50	jrr	jrr	PROPN
ejpam-4432	82	51	=	=	SYM
ejpam-4432	82	52	1r	1r	NUM
ejpam-4432	82	53	;	;	PUNCT
ejpam-4432	82	54	(	(	PUNCT
ejpam-4432	82	55	ix	ix	ADJ
ejpam-4432	82	56	)	)	PUNCT
ejpam-4432	82	57	1rr	1rr	ADJ
ejpam-4432	82	58	=	=	PUNCT
ejpam-4432	82	59	l.	l.	NOUN
ejpam-4432	82	60	proof	proof	NOUN
ejpam-4432	82	61	.	.	PUNCT
ejpam-4432	83	1	(	(	PUNCT
ejpam-4432	83	2	i	i	NOUN
ejpam-4432	83	3	)	)	PUNCT
ejpam-4432	83	4	since	since	SCONJ
ejpam-4432	83	5	i	i	PRON
ejpam-4432	83	6	∩	∩	ADJ
ejpam-4432	83	7	irr	irr	NOUN
ejpam-4432	83	8	=	=	PUNCT
ejpam-4432	83	9	i	i	PRON
ejpam-4432	83	10	∩	∩	VERB
ejpam-4432	83	11	{	{	PUNCT
ejpam-4432	83	12	x	x	SYM
ejpam-4432	83	13	∈	∈	NOUN
ejpam-4432	83	14	l	l	NOUN
ejpam-4432	83	15	:	:	PUNCT
ejpam-4432	83	16	0	0	NUM
ejpam-4432	83	17	∈	∈	PROPN
ejpam-4432	83	18	x∧̄i	x∧̄i	PROPN
ejpam-4432	83	19	for	for	ADP
ejpam-4432	83	20	all	all	PRON
ejpam-4432	83	21	i	i	PRON
ejpam-4432	83	22	∈	∈	PROPN
ejpam-4432	83	23	ir	ir	INTJ
ejpam-4432	83	24	}	}	PUNCT
ejpam-4432	83	25	=	=	SYM
ejpam-4432	83	26	{	{	PUNCT
ejpam-4432	83	27	x	x	SYM
ejpam-4432	83	28	∈	∈	PROPN
ejpam-4432	84	1	i	i	PRON
ejpam-4432	84	2	:	:	PUNCT
ejpam-4432	84	3	0	0	NUM
ejpam-4432	84	4	∈	∈	PROPN
ejpam-4432	84	5	x∧̄i	x∧̄i	PROPN
ejpam-4432	84	6	for	for	ADP
ejpam-4432	84	7	all	all	DET
ejpam-4432	84	8	i	i	PRON
ejpam-4432	84	9	∈	∈	PROPN
ejpam-4432	84	10	ir	ir	ADJ
ejpam-4432	84	11	}	}	PUNCT
ejpam-4432	84	12	=	=	SYM
ejpam-4432	84	13	i.	i.	NOUN
ejpam-4432	84	14	then	then	ADV
ejpam-4432	84	15	,	,	PUNCT
ejpam-4432	84	16	i	i	PROPN
ejpam-4432	84	17	⊆	⊆	NUM
ejpam-4432	84	18	irr	irr	NOUN
ejpam-4432	84	19	.	.	PUNCT
ejpam-4432	85	1	(	(	PUNCT
ejpam-4432	85	2	ii	ii	NOUN
ejpam-4432	85	3	)	)	PUNCT
ejpam-4432	85	4	from	from	ADP
ejpam-4432	85	5	(	(	PUNCT
ejpam-4432	85	6	i	i	NOUN
ejpam-4432	85	7	)	)	PUNCT
ejpam-4432	85	8	we	we	PRON
ejpam-4432	85	9	have	have	VERB
ejpam-4432	85	10	i	i	PRON
ejpam-4432	85	11	⊆	⊆	NUM
ejpam-4432	85	12	irr	irr	NOUN
ejpam-4432	85	13	and	and	CCONJ
ejpam-4432	85	14	ir	ir	PROPN
ejpam-4432	85	15	⊆	⊆	NUM
ejpam-4432	85	16	irrr	irrr	NOUN
ejpam-4432	85	17	.	.	PUNCT
ejpam-4432	86	1	on	on	ADP
ejpam-4432	86	2	the	the	DET
ejpam-4432	86	3	other	other	ADJ
ejpam-4432	86	4	side	side	NOUN
ejpam-4432	86	5	,	,	PUNCT
ejpam-4432	86	6	we	we	PRON
ejpam-4432	86	7	have	have	VERB
ejpam-4432	86	8	irrr	irrr	PROPN
ejpam-4432	86	9	⊆	⊆	NUM
ejpam-4432	86	10	ir	ir	NOUN
ejpam-4432	86	11	from	from	ADP
ejpam-4432	86	12	proposition	proposition	NOUN
ejpam-4432	86	13	2	2	NUM
ejpam-4432	86	14	.	.	PUNCT
ejpam-4432	87	1	hence	hence	ADV
ejpam-4432	87	2	the	the	DET
ejpam-4432	87	3	equality	equality	NOUN
ejpam-4432	87	4	is	be	AUX
ejpam-4432	87	5	satisfied	satisfied	ADJ
ejpam-4432	87	6	.	.	PUNCT
ejpam-4432	88	1	e.g.	e.g.	ADV
ejpam-4432	88	2	rezk	rezk	PROPN
ejpam-4432	88	3	,	,	PUNCT
ejpam-4432	88	4	n.h	n.h	PROPN
ejpam-4432	88	5	.	.	PROPN
ejpam-4432	88	6	abughazalah	abughazalah	PROPN
ejpam-4432	88	7	/	/	SYM
ejpam-4432	88	8	eur	eur	PROPN
ejpam-4432	88	9	.	.	PUNCT
ejpam-4432	89	1	j.	j.	PROPN
ejpam-4432	89	2	pure	pure	PROPN
ejpam-4432	89	3	appl	appl	PROPN
ejpam-4432	89	4	.	.	PROPN
ejpam-4432	89	5	math	math	PROPN
ejpam-4432	89	6	,	,	PUNCT
ejpam-4432	89	7	15	15	NUM
ejpam-4432	89	8	(	(	PUNCT
ejpam-4432	89	9	3	3	NUM
ejpam-4432	89	10	)	)	PUNCT
ejpam-4432	89	11	(	(	PUNCT
ejpam-4432	89	12	2022	2022	NUM
ejpam-4432	89	13	)	)	PUNCT
ejpam-4432	89	14	,	,	PUNCT
ejpam-4432	89	15	1402	1402	NUM
ejpam-4432	89	16	-	-	SYM
ejpam-4432	89	17	1416	1416	NUM
ejpam-4432	89	18	1406	1406	NUM
ejpam-4432	89	19	(	(	PUNCT
ejpam-4432	89	20	iii	iii	NOUN
ejpam-4432	89	21	)	)	PUNCT
ejpam-4432	89	22	let	let	VERB
ejpam-4432	89	23	x	x	SYM
ejpam-4432	89	24	∈	∈	PROPN
ejpam-4432	89	25	i	i	PRON
ejpam-4432	89	26	∩	∩	PROPN
ejpam-4432	89	27	j	j	PROPN
ejpam-4432	89	28	and	and	CCONJ
ejpam-4432	89	29	y	y	PROPN
ejpam-4432	89	30	∈	∈	PROPN
ejpam-4432	89	31	ir	ir	PROPN
ejpam-4432	89	32	∨	∨	PROPN
ejpam-4432	89	33	jr	jr	PROPN
ejpam-4432	89	34	.	.	PUNCT
ejpam-4432	90	1	then	then	ADV
ejpam-4432	90	2	y	y	PROPN
ejpam-4432	90	3	≤	≤	PROPN
ejpam-4432	91	1	i	i	PROPN
ejpam-4432	91	2	∨	∨	PROPN
ejpam-4432	91	3	j	j	PROPN
ejpam-4432	91	4	for	for	ADP
ejpam-4432	91	5	some	some	DET
ejpam-4432	91	6	i	i	PRON
ejpam-4432	91	7	∈	∈	PROPN
ejpam-4432	91	8	ir	ir	PROPN
ejpam-4432	91	9	and	and	CCONJ
ejpam-4432	91	10	j	j	PROPN
ejpam-4432	91	11	∈	∈	PROPN
ejpam-4432	91	12	jr	jr	PROPN
ejpam-4432	92	1	such	such	ADJ
ejpam-4432	92	2	that	that	SCONJ
ejpam-4432	92	3	x∧̄y	x∧̄y	PROPN
ejpam-4432	92	4	⊆	⊆	NUM
ejpam-4432	92	5	x∧̄(i	x∧̄(i	PROPN
ejpam-4432	92	6	∨	∨	NUM
ejpam-4432	92	7	j	j	PROPN
ejpam-4432	92	8	)	)	PUNCT
ejpam-4432	92	9	=	=	SYM
ejpam-4432	92	10	(	(	PUNCT
ejpam-4432	92	11	x∧̄i	x∧̄i	PROPN
ejpam-4432	92	12	)	)	PUNCT
ejpam-4432	92	13	∨	∨	NOUN
ejpam-4432	92	14	(	(	PUNCT
ejpam-4432	92	15	x∧̄j	x∧̄j	PROPN
ejpam-4432	92	16	)	)	PUNCT
ejpam-4432	92	17	.	.	PUNCT
ejpam-4432	93	1	since	since	SCONJ
ejpam-4432	93	2	x	x	PROPN
ejpam-4432	93	3	∈	∈	PROPN
ejpam-4432	93	4	i	i	PRON
ejpam-4432	93	5	,	,	PUNCT
ejpam-4432	93	6	x	x	PROPN
ejpam-4432	93	7	∈	∈	PROPN
ejpam-4432	93	8	j	j	NOUN
ejpam-4432	93	9	then	then	ADV
ejpam-4432	93	10	0	0	NUM
ejpam-4432	93	11	∈	∈	PROPN
ejpam-4432	93	12	(	(	PUNCT
ejpam-4432	93	13	x∧̄i	x∧̄i	PROPN
ejpam-4432	93	14	)	)	PUNCT
ejpam-4432	93	15	and	and	CCONJ
ejpam-4432	93	16	0	0	NUM
ejpam-4432	93	17	∈	∈	PROPN
ejpam-4432	93	18	(	(	PUNCT
ejpam-4432	93	19	x∧̄j	x∧̄j	PROPN
ejpam-4432	93	20	)	)	PUNCT
ejpam-4432	93	21	.	.	PUNCT
ejpam-4432	94	1	then	then	ADV
ejpam-4432	94	2	0	0	NUM
ejpam-4432	94	3	∈	∈	PROPN
ejpam-4432	94	4	x∧̄(i	x∧̄(i	PROPN
ejpam-4432	94	5	∨	∨	PROPN
ejpam-4432	94	6	j	j	PROPN
ejpam-4432	94	7	)	)	PUNCT
ejpam-4432	94	8	.	.	PUNCT
ejpam-4432	95	1	from	from	ADP
ejpam-4432	95	2	proposition	proposition	NOUN
ejpam-4432	95	3	2	2	NUM
ejpam-4432	95	4	,	,	PUNCT
ejpam-4432	95	5	we	we	PRON
ejpam-4432	95	6	get	get	VERB
ejpam-4432	95	7	0	0	NUM
ejpam-4432	95	8	∈	∈	NOUN
ejpam-4432	95	9	(	(	PUNCT
ejpam-4432	95	10	x∧̄y	x∧̄y	PROPN
ejpam-4432	95	11	)	)	PUNCT
ejpam-4432	95	12	.therefore	.therefore	PUNCT
ejpam-4432	96	1	x	x	PUNCT
ejpam-4432	96	2	∈	∈	PROPN
ejpam-4432	96	3	(	(	PUNCT
ejpam-4432	96	4	ir	ir	PROPN
ejpam-4432	96	5	∨	∨	PROPN
ejpam-4432	96	6	jr)r	jr)r	PROPN
ejpam-4432	96	7	.	.	PUNCT
ejpam-4432	96	8	(	(	PUNCT
ejpam-4432	96	9	iv	iv	X
ejpam-4432	96	10	)	)	PUNCT
ejpam-4432	96	11	since	since	SCONJ
ejpam-4432	96	12	i	i	PRON
ejpam-4432	96	13	,	,	PUNCT
ejpam-4432	96	14	j	j	PROPN
ejpam-4432	96	15	⊆	⊆	NUM
ejpam-4432	96	16	i	i	PROPN
ejpam-4432	96	17	∨	∨	PROPN
ejpam-4432	96	18	j	j	PROPN
ejpam-4432	96	19	,	,	PUNCT
ejpam-4432	96	20	then	then	ADV
ejpam-4432	96	21	ir	ir	PROPN
ejpam-4432	96	22	,	,	PUNCT
ejpam-4432	96	23	jr	jr	PROPN
ejpam-4432	96	24	⊇	⊇	PROPN
ejpam-4432	96	25	(	(	PUNCT
ejpam-4432	96	26	i	i	PROPN
ejpam-4432	96	27	∨	∨	PROPN
ejpam-4432	96	28	j)r	j)r	NOUN
ejpam-4432	96	29	.	.	PUNCT
ejpam-4432	97	1	so	so	ADV
ejpam-4432	97	2	,	,	PUNCT
ejpam-4432	97	3	(	(	PUNCT
ejpam-4432	98	1	i	i	PROPN
ejpam-4432	98	2	∨	∨	NUM
ejpam-4432	98	3	j)r	j)r	NOUN
ejpam-4432	98	4	⊆	⊆	NUM
ejpam-4432	98	5	ir	ir	PROPN
ejpam-4432	98	6	∩	∩	PROPN
ejpam-4432	98	7	jr	jr	PROPN
ejpam-4432	98	8	.	.	PUNCT
ejpam-4432	98	9	conversely	conversely	ADV
ejpam-4432	98	10	,	,	PUNCT
ejpam-4432	98	11	let	let	VERB
ejpam-4432	98	12	y	y	PROPN
ejpam-4432	98	13	∈	∈	PROPN
ejpam-4432	98	14	ir	ir	PROPN
ejpam-4432	98	15	∩	∩	PROPN
ejpam-4432	98	16	jr	jr	PROPN
ejpam-4432	98	17	,	,	PUNCT
ejpam-4432	98	18	thus	thus	ADV
ejpam-4432	98	19	0	0	NUM
ejpam-4432	98	20	∈	∈	PROPN
ejpam-4432	98	21	y∧̄i	y∧̄i	NUM
ejpam-4432	98	22	for	for	ADP
ejpam-4432	98	23	all	all	PRON
ejpam-4432	98	24	i	i	PRON
ejpam-4432	98	25	∈	∈	VERB
ejpam-4432	98	26	i	i	PRON
ejpam-4432	98	27	and	and	CCONJ
ejpam-4432	98	28	0	0	NUM
ejpam-4432	98	29	∈	∈	PROPN
ejpam-4432	98	30	y∧̄j	y∧̄j	PROPN
ejpam-4432	98	31	for	for	ADP
ejpam-4432	98	32	all	all	DET
ejpam-4432	98	33	j	j	PROPN
ejpam-4432	98	34	∈	∈	PROPN
ejpam-4432	98	35	i	i	PRON
ejpam-4432	98	36	which	which	PRON
ejpam-4432	98	37	indicates	indicate	VERB
ejpam-4432	98	38	that	that	SCONJ
ejpam-4432	98	39	0	0	NUM
ejpam-4432	98	40	∈	∈	PROPN
ejpam-4432	98	41	(	(	PUNCT
ejpam-4432	98	42	y∧̄i	y∧̄i	PROPN
ejpam-4432	98	43	)	)	PUNCT
ejpam-4432	98	44	∨	∨	PROPN
ejpam-4432	98	45	(	(	PUNCT
ejpam-4432	98	46	y∧̄j	y∧̄j	PROPN
ejpam-4432	98	47	)	)	PUNCT
ejpam-4432	98	48	=	=	SYM
ejpam-4432	98	49	y∧̄(i	y∧̄(i	PROPN
ejpam-4432	98	50	∨	∨	PROPN
ejpam-4432	98	51	j	j	PROPN
ejpam-4432	98	52	)	)	PUNCT
ejpam-4432	98	53	.	.	PUNCT
ejpam-4432	99	1	so	so	ADV
ejpam-4432	99	2	for	for	ADP
ejpam-4432	99	3	all	all	DET
ejpam-4432	99	4	a	a	DET
ejpam-4432	99	5	≤	≤	NUM
ejpam-4432	99	6	i	i	PROPN
ejpam-4432	99	7	∨	∨	PROPN
ejpam-4432	99	8	j	j	PROPN
ejpam-4432	99	9	meeting	meet	VERB
ejpam-4432	99	10	each	each	DET
ejpam-4432	99	11	side	side	NOUN
ejpam-4432	99	12	by	by	ADP
ejpam-4432	99	13	y	y	PROPN
ejpam-4432	99	14	to	to	PART
ejpam-4432	99	15	get	get	VERB
ejpam-4432	99	16	y∧̄a	y∧̄a	PROPN
ejpam-4432	99	17	⊆	⊆	NUM
ejpam-4432	99	18	y∧̄(i	y∧̄(i	PROPN
ejpam-4432	99	19	∨	∨	PROPN
ejpam-4432	99	20	j	j	PROPN
ejpam-4432	99	21	)	)	PUNCT
ejpam-4432	99	22	.	.	PUNCT
ejpam-4432	100	1	it	it	PRON
ejpam-4432	101	1	impliesithat	impliesithat	INTJ
ejpam-4432	101	2	0	0	NUM
ejpam-4432	102	1	∈	∈	PROPN
ejpam-4432	102	2	y∧̄a	y∧̄a	PROPN
ejpam-4432	103	1	and	and	CCONJ
ejpam-4432	103	2	then	then	ADV
ejpam-4432	103	3	y	y	PROPN
ejpam-4432	103	4	∈	∈	PROPN
ejpam-4432	103	5	(	(	PUNCT
ejpam-4432	103	6	i	i	NOUN
ejpam-4432	103	7	∨	∨	PROPN
ejpam-4432	103	8	j)r	j)r	NOUN
ejpam-4432	103	9	.	.	PUNCT
ejpam-4432	104	1	therefore	therefore	ADV
ejpam-4432	104	2	ir	ir	PROPN
ejpam-4432	104	3	∩	∩	PROPN
ejpam-4432	104	4	jr	jr	PROPN
ejpam-4432	104	5	⊆	⊆	NUM
ejpam-4432	104	6	(	(	PUNCT
ejpam-4432	104	7	i	i	NOUN
ejpam-4432	104	8	∨	∨	PROPN
ejpam-4432	104	9	j)r	j)r	NOUN
ejpam-4432	104	10	.	.	PUNCT
ejpam-4432	105	1	(	(	PUNCT
ejpam-4432	105	2	v	v	NOUN
ejpam-4432	105	3	)	)	PUNCT
ejpam-4432	105	4	since	since	SCONJ
ejpam-4432	105	5	i	i	PRON
ejpam-4432	105	6	∩	∩	NOUN
ejpam-4432	105	7	j	j	PROPN
ejpam-4432	105	8	⊆	⊆	NUM
ejpam-4432	105	9	i	i	PROPN
ejpam-4432	105	10	and	and	CCONJ
ejpam-4432	105	11	i	i	PROPN
ejpam-4432	105	12	∩	∩	PROPN
ejpam-4432	105	13	j	j	PROPN
ejpam-4432	105	14	⊆	⊆	NUM
ejpam-4432	105	15	j	j	PROPN
ejpam-4432	105	16	then	then	ADV
ejpam-4432	105	17	(	(	PUNCT
ejpam-4432	105	18	i	i	NOUN
ejpam-4432	105	19	∩	∩	VERB
ejpam-4432	105	20	j)rr	j)rr	PROPN
ejpam-4432	105	21	⊆	⊆	NUM
ejpam-4432	105	22	irr	irr	NOUN
ejpam-4432	105	23	and	and	CCONJ
ejpam-4432	105	24	(	(	PUNCT
ejpam-4432	105	25	i	i	PROPN
ejpam-4432	105	26	∩	∩	VERB
ejpam-4432	105	27	j)rr	j)rr	PROPN
ejpam-4432	105	28	⊆	⊆	NUM
ejpam-4432	105	29	jrr	jrr	NOUN
ejpam-4432	105	30	.	.	PUNCT
ejpam-4432	106	1	thus	thus	ADV
ejpam-4432	106	2	(	(	PUNCT
ejpam-4432	106	3	i	i	NOUN
ejpam-4432	106	4	∩	∩	VERB
ejpam-4432	106	5	j)rr	j)rr	PROPN
ejpam-4432	106	6	⊆	⊆	NUM
ejpam-4432	106	7	irr	irr	NOUN
ejpam-4432	106	8	∩	∩	PROPN
ejpam-4432	106	9	jrr	jrr	PROPN
ejpam-4432	106	10	.	.	PUNCT
ejpam-4432	107	1	(	(	PUNCT
ejpam-4432	107	2	vi	vi	X
ejpam-4432	107	3	)	)	PUNCT
ejpam-4432	107	4	suppose	suppose	VERB
ejpam-4432	107	5	x	x	X
ejpam-4432	107	6	∈	∈	PROPN
ejpam-4432	107	7	1r	1r	NOUN
ejpam-4432	108	1	and	and	CCONJ
ejpam-4432	108	2	i	i	PRON
ejpam-4432	108	3	is	be	AUX
ejpam-4432	108	4	a	a	DET
ejpam-4432	108	5	nonemptyisubset	nonemptyisubset	NOUN
ejpam-4432	108	6	of	of	ADP
ejpam-4432	108	7	l	l	NOUN
ejpam-4432	108	8	,	,	PUNCT
ejpam-4432	108	9	then	then	ADV
ejpam-4432	108	10	0	0	NUM
ejpam-4432	108	11	∈	∈	PROPN
ejpam-4432	108	12	x∧̄1	x∧̄1	PROPN
ejpam-4432	108	13	.	.	PUNCT
ejpam-4432	109	1	from	from	ADP
ejpam-4432	109	2	proposition	proposition	NOUN
ejpam-4432	109	3	2	2	NUM
ejpam-4432	109	4	,	,	PUNCT
ejpam-4432	109	5	we	we	PRON
ejpam-4432	109	6	get	get	VERB
ejpam-4432	109	7	0	0	NUM
ejpam-4432	109	8	∈	∈	PROPN
ejpam-4432	109	9	x∧̄y	x∧̄y	PROPN
ejpam-4432	109	10	for	for	ADP
ejpam-4432	109	11	all	all	DET
ejpam-4432	109	12	y	y	PROPN
ejpam-4432	109	13	∈	∈	PROPN
ejpam-4432	109	14	i.	i.	NOUN
ejpam-4432	109	15	therefore	therefore	ADV
ejpam-4432	110	1	x	x	PROPN
ejpam-4432	110	2	∈	∈	PROPN
ejpam-4432	110	3	ir	ir	PROPN
ejpam-4432	110	4	.	.	PUNCT
ejpam-4432	110	5	(	(	PUNCT
ejpam-4432	110	6	vii	vii	PROPN
ejpam-4432	110	7	)	)	PUNCT
ejpam-4432	110	8	from	from	ADP
ejpam-4432	110	9	(	(	PUNCT
ejpam-4432	110	10	iv	iv	X
ejpam-4432	110	11	)	)	PUNCT
ejpam-4432	110	12	in	in	ADP
ejpam-4432	110	13	proposition	proposition	NOUN
ejpam-4432	110	14	2	2	NUM
ejpam-4432	110	15	,	,	PUNCT
ejpam-4432	110	16	we	we	PRON
ejpam-4432	110	17	get	get	VERB
ejpam-4432	110	18	ir	ir	ADP
ejpam-4432	110	19	∩	∩	ADJ
ejpam-4432	110	20	irr	irr	NOUN
ejpam-4432	110	21	=	=	PUNCT
ejpam-4432	110	22	(	(	PUNCT
ejpam-4432	110	23	i	i	PRON
ejpam-4432	110	24	∪	∪	VERB
ejpam-4432	110	25	ir)r	ir)r	VERB
ejpam-4432	110	26	.	.	PUNCT
ejpam-4432	111	1	since	since	SCONJ
ejpam-4432	111	2	a	a	DET
ejpam-4432	111	3	≤	≤	NUM
ejpam-4432	111	4	1	1	NUM
ejpam-4432	111	5	for	for	ADP
ejpam-4432	111	6	all	all	DET
ejpam-4432	111	7	a	a	DET
ejpam-4432	111	8	∈	∈	NOUN
ejpam-4432	111	9	i	i	PRON
ejpam-4432	111	10	∪	∪	VERB
ejpam-4432	111	11	ir	ir	PROPN
ejpam-4432	111	12	,	,	PUNCT
ejpam-4432	111	13	then	then	ADV
ejpam-4432	111	14	1r	1r	NUM
ejpam-4432	111	15	⊆	⊆	NUM
ejpam-4432	111	16	ar	ar	NOUN
ejpam-4432	111	17	.	.	PUNCT
ejpam-4432	112	1	it	it	PRON
ejpam-4432	112	2	implies	imply	VERB
ejpam-4432	112	3	that	that	SCONJ
ejpam-4432	112	4	1r	1r	NUM
ejpam-4432	112	5	⊆	⊆	NUM
ejpam-4432	112	6	∩{ar	∩{ar	NUM
ejpam-4432	112	7	:	:	PUNCT
ejpam-4432	112	8	a	a	DET
ejpam-4432	112	9	∈	∈	NOUN
ejpam-4432	112	10	i	i	PRON
ejpam-4432	112	11	∪	∪	VERB
ejpam-4432	112	12	ir	ir	NOUN
ejpam-4432	112	13	}	}	PUNCT
ejpam-4432	112	14	.	.	PUNCT
ejpam-4432	113	1	consequently	consequently	ADV
ejpam-4432	113	2	,	,	PUNCT
ejpam-4432	113	3	1r	1r	NUM
ejpam-4432	113	4	⊆	⊆	NUM
ejpam-4432	113	5	(	(	PUNCT
ejpam-4432	113	6	i∪ir)r	i∪ir)r	NOUN
ejpam-4432	113	7	=	=	SYM
ejpam-4432	113	8	ir∩irr	ir∩irr	PROPN
ejpam-4432	113	9	.	.	PUNCT
ejpam-4432	114	1	the	the	DET
ejpam-4432	114	2	opposite	opposite	ADJ
ejpam-4432	114	3	direction	direction	NOUN
ejpam-4432	114	4	is	be	AUX
ejpam-4432	114	5	taken	take	VERB
ejpam-4432	114	6	immediately	immediately	ADV
ejpam-4432	114	7	from	from	ADP
ejpam-4432	114	8	(	(	PUNCT
ejpam-4432	114	9	vi).therefore	vi).therefore	PROPN
ejpam-4432	114	10	,	,	PUNCT
ejpam-4432	114	11	ir	ir	PROPN
ejpam-4432	114	12	∩	∩	ADJ
ejpam-4432	114	13	irr	irr	NOUN
ejpam-4432	114	14	=	=	NOUN
ejpam-4432	114	15	1r	1r	NUM
ejpam-4432	114	16	.	.	PUNCT
ejpam-4432	115	1	(	(	PUNCT
ejpam-4432	115	2	viii	viii	NOUN
ejpam-4432	115	3	)	)	PUNCT
ejpam-4432	115	4	let	let	VERB
ejpam-4432	115	5	i	i	PRON
ejpam-4432	115	6	⊆	⊆	NUM
ejpam-4432	115	7	jr	jr	PROPN
ejpam-4432	115	8	.	.	PROPN
ejpam-4432	115	9	intersect	intersect	VERB
ejpam-4432	115	10	both	both	DET
ejpam-4432	115	11	sides	side	NOUN
ejpam-4432	115	12	by	by	ADP
ejpam-4432	115	13	jrr	jrr	PROPN
ejpam-4432	115	14	,	,	PUNCT
ejpam-4432	115	15	we	we	PRON
ejpam-4432	115	16	get	get	VERB
ejpam-4432	115	17	i	i	PRON
ejpam-4432	115	18	∩	∩	PROPN
ejpam-4432	115	19	jrr	jrr	PROPN
ejpam-4432	115	20	⊆	⊆	NUM
ejpam-4432	115	21	jr	jr	PROPN
ejpam-4432	115	22	∩	∩	PROPN
ejpam-4432	115	23	jrr	jrr	PROPN
ejpam-4432	115	24	=	=	SYM
ejpam-4432	115	25	1r	1r	NUM
ejpam-4432	115	26	.	.	PUNCT
ejpam-4432	116	1	but	but	CCONJ
ejpam-4432	116	2	,	,	PUNCT
ejpam-4432	116	3	from	from	ADP
ejpam-4432	116	4	(	(	PUNCT
ejpam-4432	116	5	vi	vi	X
ejpam-4432	116	6	)	)	PUNCT
ejpam-4432	116	7	we	we	PRON
ejpam-4432	116	8	have	have	VERB
ejpam-4432	116	9	1r	1r	NUM
ejpam-4432	116	10	⊆	⊆	NUM
ejpam-4432	116	11	i	i	PROPN
ejpam-4432	116	12	∩	∩	PROPN
ejpam-4432	116	13	jrr	jrr	PROPN
ejpam-4432	116	14	.	.	PUNCT
ejpam-4432	117	1	(	(	PUNCT
ejpam-4432	117	2	ix	ix	PROPN
ejpam-4432	117	3	)	)	PUNCT
ejpam-4432	117	4	1rr	1rr	NOUN
ejpam-4432	117	5	=	=	PUNCT
ejpam-4432	117	6	{	{	PUNCT
ejpam-4432	117	7	x	x	SYM
ejpam-4432	117	8	∈	∈	NOUN
ejpam-4432	117	9	l	l	NOUN
ejpam-4432	117	10	:	:	PUNCT
ejpam-4432	117	11	0	0	NUM
ejpam-4432	117	12	∈	∈	PROPN
ejpam-4432	117	13	x∧̄a	x∧̄a	PROPN
ejpam-4432	117	14	,	,	PUNCT
ejpam-4432	117	15	a	a	DET
ejpam-4432	117	16	∈	∈	NOUN
ejpam-4432	117	17	1r	1r	NOUN
ejpam-4432	117	18	}	}	PUNCT
ejpam-4432	117	19	=	=	SYM
ejpam-4432	117	20	{	{	PUNCT
ejpam-4432	117	21	x	x	PUNCT
ejpam-4432	117	22	∈	∈	NOUN
ejpam-4432	117	23	l	l	NOUN
ejpam-4432	117	24	:	:	PUNCT
ejpam-4432	117	25	0	0	NUM
ejpam-4432	117	26	∈	∈	PROPN
ejpam-4432	117	27	x∧̄a	x∧̄a	PROPN
ejpam-4432	117	28	,	,	PUNCT
ejpam-4432	117	29	0	0	X
ejpam-4432	117	30	∈	∈	NOUN
ejpam-4432	117	31	a∧̄1	a∧̄1	PROPN
ejpam-4432	117	32	}	}	PUNCT
ejpam-4432	117	33	=	=	SYM
ejpam-4432	117	34	{	{	PUNCT
ejpam-4432	117	35	x	x	PUNCT
ejpam-4432	117	36	∈	∈	NOUN
ejpam-4432	117	37	l	l	NOUN
ejpam-4432	117	38	:	:	PUNCT
ejpam-4432	117	39	0	0	NUM
ejpam-4432	117	40	∈	∈	PROPN
ejpam-4432	117	41	x∧̄a	x∧̄a	PROPN
ejpam-4432	117	42	,	,	PUNCT
ejpam-4432	117	43	0	0	NUM
ejpam-4432	117	44	∈	∈	PROPN
ejpam-4432	117	45	a∧̄y	a∧̄y	PROPN
ejpam-4432	117	46	,	,	PUNCT
ejpam-4432	117	47	for	for	ADP
ejpam-4432	117	48	all	all	DET
ejpam-4432	117	49	y	y	PROPN
ejpam-4432	117	50	∈	∈	PROPN
ejpam-4432	117	51	l	l	NOUN
ejpam-4432	117	52	}	}	PUNCT
ejpam-4432	117	53	.	.	PUNCT
ejpam-4432	118	1	definition	definition	NOUN
ejpam-4432	118	2	4	4	NUM
ejpam-4432	118	3	.	.	PUNCT
ejpam-4432	119	1	a	a	DET
ejpam-4432	119	2	hyperideal	hyperideal	NOUN
ejpam-4432	119	3	i	i	PRON
ejpam-4432	119	4	of	of	ADP
ejpam-4432	119	5	meet	meet	ADJ
ejpam-4432	119	6	-	-	PUNCT
ejpam-4432	119	7	hyperlattice	hyperlattice	NOUN
ejpam-4432	119	8	l	l	NOUN
ejpam-4432	119	9	is	be	AUX
ejpam-4432	119	10	called	call	VERB
ejpam-4432	119	11	closed	closed	ADJ
ejpam-4432	119	12	if	if	SCONJ
ejpam-4432	119	13	i	i	PRON
ejpam-4432	119	14	=	=	VERB
ejpam-4432	119	15	irr	irr	PROPN
ejpam-4432	119	16	.	.	PUNCT
ejpam-4432	120	1	we	we	PRON
ejpam-4432	120	2	denotei	denotei	VERB
ejpam-4432	120	3	the	the	DET
ejpam-4432	120	4	set	set	NOUN
ejpam-4432	120	5	of	of	ADP
ejpam-4432	120	6	all	all	DET
ejpam-4432	120	7	closed	closed	ADJ
ejpam-4432	120	8	hyperideals	hyperideal	NOUN
ejpam-4432	120	9	of	of	ADP
ejpam-4432	120	10	∧hyperlattice	∧hyperlattice	NOUN
ejpam-4432	120	11	l	l	NOUN
ejpam-4432	120	12	by	by	ADP
ejpam-4432	120	13	h(l	h(l	PROPN
ejpam-4432	120	14	)	)	PUNCT
ejpam-4432	120	15	.	.	PUNCT
ejpam-4432	121	1	e.g.	e.g.	ADV
ejpam-4432	121	2	rezk	rezk	PROPN
ejpam-4432	121	3	,	,	PUNCT
ejpam-4432	121	4	n.h	n.h	PROPN
ejpam-4432	121	5	.	.	PROPN
ejpam-4432	121	6	abughazalah	abughazalah	PROPN
ejpam-4432	121	7	/	/	SYM
ejpam-4432	121	8	eur	eur	PROPN
ejpam-4432	121	9	.	.	PUNCT
ejpam-4432	122	1	j.	j.	PROPN
ejpam-4432	122	2	pure	pure	PROPN
ejpam-4432	122	3	appl	appl	PROPN
ejpam-4432	122	4	.	.	PROPN
ejpam-4432	122	5	math	math	PROPN
ejpam-4432	122	6	,	,	PUNCT
ejpam-4432	122	7	15	15	NUM
ejpam-4432	122	8	(	(	PUNCT
ejpam-4432	122	9	3	3	NUM
ejpam-4432	122	10	)	)	PUNCT
ejpam-4432	122	11	(	(	PUNCT
ejpam-4432	122	12	2022	2022	NUM
ejpam-4432	122	13	)	)	PUNCT
ejpam-4432	122	14	,	,	PUNCT
ejpam-4432	122	15	1402	1402	NUM
ejpam-4432	122	16	-	-	SYM
ejpam-4432	122	17	1416	1416	NUM
ejpam-4432	122	18	1407	1407	NUM
ejpam-4432	122	19	lemma	lemma	PROPN
ejpam-4432	122	20	1	1	X
ejpam-4432	122	21	.	.	PUNCT
ejpam-4432	123	1	let	let	VERB
ejpam-4432	124	1	i	i	PRON
ejpam-4432	124	2	,	,	PUNCT
ejpam-4432	124	3	jand	jand	PROPN
ejpam-4432	124	4	k	k	PROPN
ejpam-4432	124	5	be	be	AUX
ejpam-4432	124	6	closed	close	VERB
ejpam-4432	124	7	hyperideals	hyperideal	NOUN
ejpam-4432	124	8	of	of	ADP
ejpam-4432	124	9	meet	meet	NOUN
ejpam-4432	124	10	-	-	PUNCT
ejpam-4432	124	11	hyperlattice	hyperlattice	NOUN
ejpam-4432	124	12	.	.	PUNCT
ejpam-4432	125	1	then	then	ADV
ejpam-4432	125	2	:	:	PUNCT
ejpam-4432	125	3	(	(	PUNCT
ejpam-4432	125	4	i	i	NOUN
ejpam-4432	125	5	)	)	PUNCT
ejpam-4432	125	6	(	(	PUNCT
ejpam-4432	125	7	ir	ir	PROPN
ejpam-4432	125	8	∨	∨	NUM
ejpam-4432	125	9	jr)r	jr)r	PROPN
ejpam-4432	125	10	=	=	PUNCT
ejpam-4432	125	11	i	i	PROPN
ejpam-4432	125	12	∩	∩	PROPN
ejpam-4432	125	13	j	j	PROPN
ejpam-4432	125	14	;	;	PUNCT
ejpam-4432	125	15	(	(	PUNCT
ejpam-4432	125	16	ii	ii	NOUN
ejpam-4432	125	17	)	)	PUNCT
ejpam-4432	125	18	(	(	PUNCT
ejpam-4432	125	19	i	i	NOUN
ejpam-4432	125	20	∩	∩	X
ejpam-4432	125	21	j)rr	j)rr	PROPN
ejpam-4432	125	22	=	=	SYM
ejpam-4432	125	23	irr	irr	PROPN
ejpam-4432	125	24	∩	∩	PROPN
ejpam-4432	125	25	jrr	jrr	PROPN
ejpam-4432	125	26	;	;	PUNCT
ejpam-4432	125	27	(	(	PUNCT
ejpam-4432	125	28	iii	iii	X
ejpam-4432	125	29	)	)	PUNCT
ejpam-4432	125	30	if	if	SCONJ
ejpam-4432	125	31	i	i	PRON
ejpam-4432	125	32	∩	∩	NOUN
ejpam-4432	125	33	jrr	jrr	PROPN
ejpam-4432	125	34	=	=	PROPN
ejpam-4432	125	35	1r	1r	NUM
ejpam-4432	125	36	then	then	ADV
ejpam-4432	125	37	i	i	PRON
ejpam-4432	125	38	⊆	⊆	NUM
ejpam-4432	125	39	jr	jr	PROPN
ejpam-4432	125	40	;	;	PUNCT
ejpam-4432	125	41	(	(	PUNCT
ejpam-4432	125	42	iv	iv	X
ejpam-4432	125	43	)	)	PUNCT
ejpam-4432	125	44	k	k	NOUN
ejpam-4432	125	45	∩	∩	NOUN
ejpam-4432	125	46	(	(	PUNCT
ejpam-4432	125	47	ir	ir	PROPN
ejpam-4432	125	48	∩	∩	NOUN
ejpam-4432	125	49	jr)r	jr)r	PROPN
ejpam-4432	125	50	⊆	⊆	NUM
ejpam-4432	125	51	(	(	PUNCT
ejpam-4432	125	52	ir	ir	PROPN
ejpam-4432	125	53	∩	∩	X
ejpam-4432	125	54	(	(	PUNCT
ejpam-4432	125	55	j	j	PROPN
ejpam-4432	125	56	∩k)r	∩k)r	PROPN
ejpam-4432	125	57	)	)	PUNCT
ejpam-4432	125	58	r	r	NOUN
ejpam-4432	125	59	.	.	PUNCT
ejpam-4432	126	1	proof	proof	NOUN
ejpam-4432	126	2	.	.	PUNCT
ejpam-4432	127	1	(	(	PUNCT
ejpam-4432	127	2	i	i	NOUN
ejpam-4432	127	3	)	)	PUNCT
ejpam-4432	127	4	since	since	SCONJ
ejpam-4432	127	5	ir	ir	PROPN
ejpam-4432	127	6	,	,	PUNCT
ejpam-4432	127	7	jr	jr	PROPN
ejpam-4432	127	8	⊆	⊆	NUM
ejpam-4432	127	9	ir	ir	PROPN
ejpam-4432	127	10	∨	∨	NUM
ejpam-4432	127	11	jr	jr	PROPN
ejpam-4432	127	12	then	then	ADV
ejpam-4432	127	13	,	,	PUNCT
ejpam-4432	127	14	from	from	ADP
ejpam-4432	127	15	proposition	proposition	NOUN
ejpam-4432	127	16	2	2	NUM
ejpam-4432	127	17	,	,	PUNCT
ejpam-4432	127	18	we	we	PRON
ejpam-4432	127	19	have	have	VERB
ejpam-4432	127	20	(	(	PUNCT
ejpam-4432	127	21	ir	ir	PROPN
ejpam-4432	127	22	∨	∨	PROPN
ejpam-4432	127	23	jr)r	jr)r	PROPN
ejpam-4432	127	24	⊆	⊆	NUM
ejpam-4432	127	25	irr	irr	NOUN
ejpam-4432	128	1	=	=	PUNCT
ejpam-4432	128	2	i	i	PROPN
ejpam-4432	128	3	,	,	PUNCT
ejpam-4432	128	4	and	and	CCONJ
ejpam-4432	128	5	(	(	PUNCT
ejpam-4432	128	6	ir	ir	PROPN
ejpam-4432	128	7	∨	∨	PROPN
ejpam-4432	128	8	jr)r	jr)r	PROPN
ejpam-4432	128	9	⊆	⊆	NUM
ejpam-4432	128	10	jrr	jrr	PROPN
ejpam-4432	128	11	=	=	SYM
ejpam-4432	128	12	j	j	PROPN
ejpam-4432	128	13	,	,	PUNCT
ejpam-4432	128	14	thus	thus	ADV
ejpam-4432	128	15	(	(	PUNCT
ejpam-4432	128	16	ir	ir	PROPN
ejpam-4432	128	17	∨	∨	PROPN
ejpam-4432	128	18	jr)r	jr)r	PROPN
ejpam-4432	128	19	⊆	⊆	NUM
ejpam-4432	128	20	i	i	PROPN
ejpam-4432	128	21	∩	∩	PROPN
ejpam-4432	128	22	j.	j.	PROPN
ejpam-4432	128	23	the	the	DET
ejpam-4432	128	24	equalty	equalty	NOUN
ejpam-4432	128	25	is	be	AUX
ejpam-4432	128	26	obtained	obtain	VERB
ejpam-4432	128	27	from	from	ADP
ejpam-4432	128	28	(	(	PUNCT
ejpam-4432	128	29	iii	iii	NOUN
ejpam-4432	128	30	)	)	PUNCT
ejpam-4432	128	31	in	in	ADP
ejpam-4432	128	32	theorem	theorem	NOUN
ejpam-4432	128	33	2	2	NUM
ejpam-4432	128	34	.	.	PUNCT
ejpam-4432	128	35	(	(	PUNCT
ejpam-4432	128	36	ii	ii	NOUN
ejpam-4432	128	37	)	)	PUNCT
ejpam-4432	128	38	irr∩jrr	irr∩jrr	NOUN
ejpam-4432	129	1	=	=	PUNCT
ejpam-4432	129	2	i	i	PRON
ejpam-4432	129	3	∩j	∩j	VERB
ejpam-4432	129	4	⊆	⊆	NUM
ejpam-4432	129	5	(	(	PUNCT
ejpam-4432	129	6	i	i	PRON
ejpam-4432	129	7	∩j)rr	∩j)rr	VERB
ejpam-4432	129	8	.	.	PUNCT
ejpam-4432	130	1	on	on	ADP
ejpam-4432	130	2	the	the	DET
ejpam-4432	130	3	other	other	ADJ
ejpam-4432	130	4	side	side	NOUN
ejpam-4432	130	5	,	,	PUNCT
ejpam-4432	130	6	from	from	ADP
ejpam-4432	130	7	(	(	PUNCT
ejpam-4432	130	8	v	v	NOUN
ejpam-4432	130	9	)	)	PUNCT
ejpam-4432	130	10	in	in	ADP
ejpam-4432	130	11	theorem	theorem	NOUN
ejpam-4432	130	12	2	2	NUM
ejpam-4432	130	13	.	.	PUNCT
ejpam-4432	130	14	the	the	DET
ejpam-4432	130	15	equality	equality	NOUN
ejpam-4432	130	16	is	be	AUX
ejpam-4432	130	17	satisfied	satisfied	ADJ
ejpam-4432	130	18	.	.	PUNCT
ejpam-4432	131	1	(	(	PUNCT
ejpam-4432	131	2	iii	iii	X
ejpam-4432	131	3	)	)	PUNCT
ejpam-4432	131	4	let	let	VERB
ejpam-4432	131	5	i	i	PRON
ejpam-4432	131	6	∩	∩	PROPN
ejpam-4432	131	7	jrr	jrr	PROPN
ejpam-4432	131	8	=	=	PRON
ejpam-4432	131	9	i	i	PROPN
ejpam-4432	131	10	∩	∩	PROPN
ejpam-4432	131	11	j	j	PROPN
ejpam-4432	131	12	=	=	SYM
ejpam-4432	131	13	1r	1r	NUM
ejpam-4432	131	14	.	.	PUNCT
ejpam-4432	132	1	then	then	ADV
ejpam-4432	132	2	i	i	PRON
ejpam-4432	132	3	⊆	⊆	NUM
ejpam-4432	132	4	jr	jr	PROPN
ejpam-4432	132	5	(	(	PUNCT
ejpam-4432	132	6	from	from	ADP
ejpam-4432	132	7	(	(	PUNCT
ejpam-4432	132	8	vii	vii	PROPN
ejpam-4432	132	9	)	)	PUNCT
ejpam-4432	132	10	in	in	ADP
ejpam-4432	132	11	theorem	theorem	NOUN
ejpam-4432	132	12	2	2	NUM
ejpam-4432	132	13	)	)	PUNCT
ejpam-4432	132	14	.	.	PUNCT
ejpam-4432	133	1	(	(	PUNCT
ejpam-4432	133	2	iv	iv	X
ejpam-4432	133	3	)	)	PUNCT
ejpam-4432	133	4	it	it	PRON
ejpam-4432	133	5	is	be	AUX
ejpam-4432	133	6	clear	clear	ADJ
ejpam-4432	133	7	that	that	SCONJ
ejpam-4432	133	8	k	k	PROPN
ejpam-4432	133	9	∩	∩	PROPN
ejpam-4432	133	10	ir	ir	PROPN
ejpam-4432	133	11	∩	∩	X
ejpam-4432	133	12	(	(	PUNCT
ejpam-4432	133	13	j	j	PROPN
ejpam-4432	133	14	∩k)r	∩k)r	PROPN
ejpam-4432	133	15	⊆	⊆	NUM
ejpam-4432	133	16	ir	ir	NOUN
ejpam-4432	133	17	.	.	PUNCT
ejpam-4432	133	18	(	(	PUNCT
ejpam-4432	133	19	1	1	X
ejpam-4432	133	20	)	)	PUNCT
ejpam-4432	133	21	consequently	consequently	ADV
ejpam-4432	133	22	j	j	PROPN
ejpam-4432	133	23	∩k	∩k	PROPN
ejpam-4432	133	24	∩	∩	NOUN
ejpam-4432	133	25	(	(	PUNCT
ejpam-4432	133	26	ir	ir	PROPN
ejpam-4432	133	27	∩	∩	X
ejpam-4432	133	28	(	(	PUNCT
ejpam-4432	133	29	j	j	PROPN
ejpam-4432	133	30	∩k)r	∩k)r	PROPN
ejpam-4432	133	31	)	)	PUNCT
ejpam-4432	134	1	=	=	SYM
ejpam-4432	134	2	ir	ir	PROPN
ejpam-4432	134	3	∩	∩	PROPN
ejpam-4432	134	4	[	[	X
ejpam-4432	134	5	(	(	PUNCT
ejpam-4432	134	6	j	j	PROPN
ejpam-4432	134	7	∩k	∩k	PROPN
ejpam-4432	134	8	)	)	PUNCT
ejpam-4432	134	9	∩	∩	NOUN
ejpam-4432	134	10	(	(	PUNCT
ejpam-4432	134	11	j	j	PROPN
ejpam-4432	134	12	∩k)r	∩k)r	PROPN
ejpam-4432	134	13	]	]	X
ejpam-4432	134	14	=	=	SYM
ejpam-4432	134	15	ir	ir	X
ejpam-4432	134	16	∩	∩	PROPN
ejpam-4432	134	17	[	[	X
ejpam-4432	134	18	(	(	PUNCT
ejpam-4432	134	19	jrr	jrr	PROPN
ejpam-4432	134	20	∩krr	∩krr	PROPN
ejpam-4432	134	21	)	)	PUNCT
ejpam-4432	134	22	∩	∩	NOUN
ejpam-4432	134	23	(	(	PUNCT
ejpam-4432	134	24	j	j	PROPN
ejpam-4432	134	25	∩k)r	∩k)r	PROPN
ejpam-4432	134	26	]	]	X
ejpam-4432	134	27	=	=	SYM
ejpam-4432	134	28	ir	ir	X
ejpam-4432	134	29	∩	∩	PROPN
ejpam-4432	134	30	[	[	X
ejpam-4432	134	31	(	(	PUNCT
ejpam-4432	134	32	j	j	PROPN
ejpam-4432	134	33	∩k)rr	∩k)rr	X
ejpam-4432	134	34	∩	∩	NOUN
ejpam-4432	134	35	(	(	PUNCT
ejpam-4432	134	36	j	j	PROPN
ejpam-4432	134	37	∩k)r	∩k)r	PROPN
ejpam-4432	134	38	]	]	X
ejpam-4432	134	39	,	,	PUNCT
ejpam-4432	134	40	(	(	PUNCT
ejpam-4432	134	41	from	from	ADP
ejpam-4432	134	42	ii	ii	PROPN
ejpam-4432	134	43	)	)	PUNCT
ejpam-4432	134	44	)	)	PUNCT
ejpam-4432	135	1	=	=	SYM
ejpam-4432	135	2	ir	ir	PROPN
ejpam-4432	135	3	∩	∩	ADJ
ejpam-4432	135	4	1r	1r	NUM
ejpam-4432	135	5	(	(	PUNCT
ejpam-4432	135	6	from	from	ADP
ejpam-4432	135	7	vii	vii	PROPN
ejpam-4432	135	8	)	)	PUNCT
ejpam-4432	135	9	in	in	ADP
ejpam-4432	135	10	theorem	theorem	NOUN
ejpam-4432	135	11	2	2	NUM
ejpam-4432	135	12	.	.	PUNCT
ejpam-4432	135	13	)	)	PUNCT
ejpam-4432	136	1	=	=	SYM
ejpam-4432	136	2	1r	1r	NUM
ejpam-4432	136	3	,	,	PUNCT
ejpam-4432	136	4	which	which	PRON
ejpam-4432	136	5	implies	imply	VERB
ejpam-4432	136	6	that	that	SCONJ
ejpam-4432	136	7	k	k	PROPN
ejpam-4432	136	8	∩	∩	ADJ
ejpam-4432	136	9	ir	ir	PROPN
ejpam-4432	136	10	∩	∩	X
ejpam-4432	136	11	(	(	PUNCT
ejpam-4432	136	12	j	j	PROPN
ejpam-4432	136	13	∩k)r	∩k)r	PROPN
ejpam-4432	136	14	⊆	⊆	PROPN
ejpam-4432	136	15	jr	jr	PROPN
ejpam-4432	136	16	(	(	PUNCT
ejpam-4432	136	17	2	2	NUM
ejpam-4432	136	18	)	)	PUNCT
ejpam-4432	136	19	thus	thus	ADV
ejpam-4432	136	20	,	,	PUNCT
ejpam-4432	136	21	from	from	ADP
ejpam-4432	136	22	(	(	PUNCT
ejpam-4432	136	23	1	1	NUM
ejpam-4432	136	24	)	)	PUNCT
ejpam-4432	136	25	,	,	PUNCT
ejpam-4432	136	26	(	(	PUNCT
ejpam-4432	136	27	2	2	X
ejpam-4432	136	28	)	)	PUNCT
ejpam-4432	136	29	we	we	PRON
ejpam-4432	136	30	get	get	VERB
ejpam-4432	136	31	k	k	PROPN
ejpam-4432	136	32	∩	∩	ADJ
ejpam-4432	136	33	ir	ir	PROPN
ejpam-4432	136	34	∩	∩	X
ejpam-4432	136	35	(	(	PUNCT
ejpam-4432	136	36	j	j	PROPN
ejpam-4432	136	37	∩k)r	∩k)r	PROPN
ejpam-4432	136	38	⊆	⊆	NUM
ejpam-4432	136	39	ir	ir	PROPN
ejpam-4432	136	40	∩	∩	PROPN
ejpam-4432	136	41	jr	jr	PROPN
ejpam-4432	136	42	.	.	PUNCT
ejpam-4432	137	1	hence	hence	ADV
ejpam-4432	137	2	(	(	PUNCT
ejpam-4432	137	3	k	k	X
ejpam-4432	137	4	∩	∩	ADJ
ejpam-4432	137	5	ir	ir	PROPN
ejpam-4432	137	6	∩	∩	X
ejpam-4432	137	7	(	(	PUNCT
ejpam-4432	137	8	j	j	PROPN
ejpam-4432	137	9	∩k)r	∩k)r	PROPN
ejpam-4432	137	10	)	)	PUNCT
ejpam-4432	137	11	∩	∩	NOUN
ejpam-4432	137	12	(	(	PUNCT
ejpam-4432	137	13	ir	ir	X
ejpam-4432	137	14	∩	∩	NOUN
ejpam-4432	137	15	jr)r	jr)r	PROPN
ejpam-4432	137	16	=	=	SYM
ejpam-4432	137	17	1r	1r	NUM
ejpam-4432	137	18	,	,	PUNCT
ejpam-4432	137	19	e.g.	e.g.	ADV
ejpam-4432	137	20	rezk	rezk	PROPN
ejpam-4432	137	21	,	,	PUNCT
ejpam-4432	137	22	n.h	n.h	PROPN
ejpam-4432	137	23	.	.	PROPN
ejpam-4432	137	24	abughazalah	abughazalah	PROPN
ejpam-4432	137	25	/	/	SYM
ejpam-4432	137	26	eur	eur	PROPN
ejpam-4432	137	27	.	.	PUNCT
ejpam-4432	138	1	j.	j.	PROPN
ejpam-4432	138	2	pure	pure	PROPN
ejpam-4432	138	3	appl	appl	PROPN
ejpam-4432	138	4	.	.	PROPN
ejpam-4432	138	5	math	math	PROPN
ejpam-4432	138	6	,	,	PUNCT
ejpam-4432	138	7	15	15	NUM
ejpam-4432	138	8	(	(	PUNCT
ejpam-4432	138	9	3	3	NUM
ejpam-4432	138	10	)	)	PUNCT
ejpam-4432	138	11	(	(	PUNCT
ejpam-4432	138	12	2022	2022	NUM
ejpam-4432	138	13	)	)	PUNCT
ejpam-4432	138	14	,	,	PUNCT
ejpam-4432	138	15	1402	1402	NUM
ejpam-4432	138	16	-	-	SYM
ejpam-4432	138	17	1416	1416	NUM
ejpam-4432	138	18	1408	1408	NUM
ejpam-4432	138	19	which	which	PRON
ejpam-4432	138	20	equivalent	equivalent	ADJ
ejpam-4432	138	21	,	,	PUNCT
ejpam-4432	138	22	ir	ir	PROPN
ejpam-4432	138	23	∩	∩	NOUN
ejpam-4432	138	24	(	(	PUNCT
ejpam-4432	138	25	j	j	PROPN
ejpam-4432	138	26	∩k)r	∩k)r	PROPN
ejpam-4432	138	27	∩	∩	NOUN
ejpam-4432	138	28	(	(	PUNCT
ejpam-4432	138	29	k	k	PROPN
ejpam-4432	138	30	∩	∩	X
ejpam-4432	138	31	(	(	PUNCT
ejpam-4432	138	32	ir	ir	PROPN
ejpam-4432	138	33	∩	∩	PROPN
ejpam-4432	138	34	jr)r	jr)r	PROPN
ejpam-4432	138	35	)	)	PUNCT
ejpam-4432	139	1	=	=	PUNCT
ejpam-4432	139	2	1r	1r	NUM
ejpam-4432	139	3	.	.	PUNCT
ejpam-4432	140	1	as	as	ADP
ejpam-4432	140	2	a	a	DET
ejpam-4432	140	3	result	result	NOUN
ejpam-4432	140	4	,	,	PUNCT
ejpam-4432	140	5	we	we	PRON
ejpam-4432	140	6	get	get	VERB
ejpam-4432	140	7	k	k	X
ejpam-4432	140	8	∩	∩	NOUN
ejpam-4432	140	9	(	(	PUNCT
ejpam-4432	140	10	ir	ir	PROPN
ejpam-4432	140	11	∩	∩	NOUN
ejpam-4432	140	12	jr)r	jr)r	PROPN
ejpam-4432	140	13	⊆	⊆	NUM
ejpam-4432	140	14	(	(	PUNCT
ejpam-4432	140	15	ir	ir	PROPN
ejpam-4432	140	16	∩	∩	X
ejpam-4432	140	17	(	(	PUNCT
ejpam-4432	140	18	j	j	PROPN
ejpam-4432	140	19	∩k)r	∩k)r	PROPN
ejpam-4432	140	20	)	)	PUNCT
ejpam-4432	140	21	r	r	NOUN
ejpam-4432	140	22	.	.	PUNCT
ejpam-4432	141	1	for	for	ADP
ejpam-4432	141	2	i	i	PRON
ejpam-4432	141	3	,	,	PUNCT
ejpam-4432	141	4	j	j	PROPN
ejpam-4432	141	5	∈	∈	PROPN
ejpam-4432	141	6	h(l	h(l	PROPN
ejpam-4432	141	7	)	)	PUNCT
ejpam-4432	141	8	,	,	PUNCT
ejpam-4432	141	9	we	we	PRON
ejpam-4432	141	10	define	define	VERB
ejpam-4432	141	11	two	two	NUM
ejpam-4432	141	12	binary	binary	ADJ
ejpam-4432	141	13	operations	operation	NOUN
ejpam-4432	141	14	i	i	PRON
ejpam-4432	141	15	∧j	∧j	VERB
ejpam-4432	142	1	=	=	PUNCT
ejpam-4432	143	1	i	i	PRON
ejpam-4432	143	2	∩j	∩j	NOUN
ejpam-4432	144	1	and	and	CCONJ
ejpam-4432	144	2	i	i	PRON
ejpam-4432	144	3	⊻j	⊻j	NOUN
ejpam-4432	144	4	=	=	SYM
ejpam-4432	144	5	(	(	PUNCT
ejpam-4432	144	6	ir	ir	PROPN
ejpam-4432	144	7	∩jr)r	∩jr)r	PROPN
ejpam-4432	144	8	.	.	PUNCT
ejpam-4432	145	1	we	we	PRON
ejpam-4432	145	2	get	get	VERB
ejpam-4432	145	3	(	(	PUNCT
ejpam-4432	145	4	i	i	NOUN
ejpam-4432	145	5	∩	∩	NOUN
ejpam-4432	145	6	j)rr	j)rr	PROPN
ejpam-4432	145	7	=	=	SYM
ejpam-4432	145	8	irr	irr	NOUN
ejpam-4432	145	9	∩	∩	PROPN
ejpam-4432	145	10	jrr	jrr	PROPN
ejpam-4432	145	11	=	=	PRON
ejpam-4432	145	12	i	i	PROPN
ejpam-4432	145	13	∩	∩	VERB
ejpam-4432	145	14	j	j	PROPN
ejpam-4432	145	15	∈	∈	PROPN
ejpam-4432	145	16	h(l	h(l	PROPN
ejpam-4432	145	17	)	)	PUNCT
ejpam-4432	145	18	,	,	PUNCT
ejpam-4432	145	19	and	and	CCONJ
ejpam-4432	145	20	,	,	PUNCT
ejpam-4432	145	21	i	i	PRON
ejpam-4432	145	22	⊻	⊻	X
ejpam-4432	146	1	j	j	PROPN
ejpam-4432	146	2	=	=	PRON
ejpam-4432	146	3	(	(	PUNCT
ejpam-4432	146	4	ir	ir	X
ejpam-4432	146	5	∩	∩	PROPN
ejpam-4432	146	6	jr)r	jr)r	PROPN
ejpam-4432	146	7	∈	∈	PROPN
ejpam-4432	146	8	h(l	h(l	PROPN
ejpam-4432	146	9	)	)	PUNCT
ejpam-4432	146	10	.	.	PUNCT
ejpam-4432	147	1	theorem	theorem	NOUN
ejpam-4432	147	2	3	3	X
ejpam-4432	147	3	.	.	PUNCT
ejpam-4432	148	1	let	let	VERB
ejpam-4432	148	2	l	l	NOUN
ejpam-4432	148	3	=	=	PUNCT
ejpam-4432	148	4	<	<	X
ejpam-4432	148	5	l	l	NOUN
ejpam-4432	148	6	;	;	PUNCT
ejpam-4432	148	7	∧̄,∨	∧̄,∨	NUM
ejpam-4432	148	8	>	>	PUNCT
ejpam-4432	148	9	be	be	AUX
ejpam-4432	148	10	a	a	DET
ejpam-4432	148	11	meet	meet	ADJ
ejpam-4432	148	12	-	-	PUNCT
ejpam-4432	148	13	hyperlattice	hyperlattice	NOUN
ejpam-4432	148	14	.	.	PUNCT
ejpam-4432	149	1	then	then	ADV
ejpam-4432	149	2	<	<	X
ejpam-4432	149	3	h(l);∩,⊻,r	h(l);∩,⊻,r	PROPN
ejpam-4432	149	4	,	,	PUNCT
ejpam-4432	149	5	1r	1r	NUM
ejpam-4432	149	6	,	,	PUNCT
ejpam-4432	149	7	l	l	NOUN
ejpam-4432	149	8	>	>	X
ejpam-4432	149	9	forms	form	VERB
ejpam-4432	149	10	a	a	DET
ejpam-4432	149	11	boolean	boolean	ADJ
ejpam-4432	149	12	algebra	algebra	NOUN
ejpam-4432	149	13	.	.	PUNCT
ejpam-4432	150	1	proof	proof	NOUN
ejpam-4432	150	2	.	.	PUNCT
ejpam-4432	151	1	to	to	PART
ejpam-4432	151	2	demonstrate	demonstrate	VERB
ejpam-4432	151	3	that	that	SCONJ
ejpam-4432	151	4	(	(	PUNCT
ejpam-4432	151	5	h(l),∧,⊻	h(l),∧,⊻	NOUN
ejpam-4432	151	6	)	)	PUNCT
ejpam-4432	151	7	forms	form	VERB
ejpam-4432	151	8	a	a	DET
ejpam-4432	151	9	lattice	lattice	NOUN
ejpam-4432	151	10	,	,	PUNCT
ejpam-4432	151	11	only	only	ADV
ejpam-4432	151	12	associative	associative	ADJ
ejpam-4432	151	13	and	and	CCONJ
ejpam-4432	151	14	absorption	absorption	NOUN
ejpam-4432	151	15	identities	identity	NOUN
ejpam-4432	151	16	are	be	AUX
ejpam-4432	151	17	required	require	VERB
ejpam-4432	151	18	,	,	PUNCT
ejpam-4432	151	19	as	as	SCONJ
ejpam-4432	151	20	idempotent	idempotent	ADJ
ejpam-4432	151	21	and	and	CCONJ
ejpam-4432	151	22	commutative	commutative	ADJ
ejpam-4432	151	23	identities	identity	NOUN
ejpam-4432	151	24	are	be	AUX
ejpam-4432	151	25	trivial	trivial	ADJ
ejpam-4432	151	26	.	.	PUNCT
ejpam-4432	152	1	let	let	VERB
ejpam-4432	152	2	i	i	PRON
ejpam-4432	152	3	,	,	PUNCT
ejpam-4432	152	4	j	j	PROPN
ejpam-4432	152	5	,	,	PUNCT
ejpam-4432	152	6	k	k	PROPN
ejpam-4432	152	7	⊆	⊆	NUM
ejpam-4432	152	8	h(l	h(l	NUM
ejpam-4432	152	9	)	)	PUNCT
ejpam-4432	152	10	.	.	PUNCT
ejpam-4432	153	1	then	then	ADV
ejpam-4432	153	2	we	we	PRON
ejpam-4432	153	3	have	have	VERB
ejpam-4432	153	4	(	(	PUNCT
ejpam-4432	153	5	ii	ii	NOUN
ejpam-4432	153	6	⊻	⊻	CCONJ
ejpam-4432	153	7	ij	ij	NOUN
ejpam-4432	153	8	)	)	PUNCT
ejpam-4432	153	9	⊻	⊻	CCONJ
ejpam-4432	153	10	ik	ik	PROPN
ejpam-4432	153	11	=	=	SYM
ejpam-4432	153	12	(	(	PUNCT
ejpam-4432	153	13	iri	iri	PROPN
ejpam-4432	153	14	∩	∩	X
ejpam-4432	153	15	ijr)ri	ijr)ri	X
ejpam-4432	153	16	⊻	⊻	CCONJ
ejpam-4432	153	17	ik	ik	PROPN
ejpam-4432	153	18	=	=	SYM
ejpam-4432	153	19	(	(	PUNCT
ejpam-4432	153	20	(	(	PUNCT
ejpam-4432	153	21	iri	iri	PROPN
ejpam-4432	153	22	∩	∩	ADJ
ejpam-4432	153	23	ijr)rri	ijr)rri	PROPN
ejpam-4432	153	24	∩	∩	NOUN
ejpam-4432	153	25	ikr	ikr	INTJ
ejpam-4432	153	26	)	)	PUNCT
ejpam-4432	153	27	r	r	NOUN
ejpam-4432	153	28	=	=	SYM
ejpam-4432	153	29	(	(	PUNCT
ejpam-4432	153	30	(	(	PUNCT
ejpam-4432	153	31	iri	iri	NOUN
ejpam-4432	153	32	∩	∩	ADJ
ejpam-4432	153	33	ijr)i	ijr)i	PROPN
ejpam-4432	153	34	∩	∩	NOUN
ejpam-4432	153	35	ikr	ikr	NOUN
ejpam-4432	153	36	)	)	PUNCT
ejpam-4432	154	1	r	r	NOUN
ejpam-4432	154	2	=	=	SYM
ejpam-4432	154	3	(	(	PUNCT
ejpam-4432	154	4	iri	iri	PROPN
ejpam-4432	154	5	∩	∩	PROPN
ejpam-4432	154	6	i(jri	i(jri	ADJ
ejpam-4432	154	7	∩	∩	ADJ
ejpam-4432	154	8	ikr	ikr	NOUN
ejpam-4432	154	9	)	)	PUNCT
ejpam-4432	154	10	)	)	PUNCT
ejpam-4432	155	1	r	r	NOUN
ejpam-4432	155	2	=	=	SYM
ejpam-4432	155	3	(	(	PUNCT
ejpam-4432	155	4	iri	iri	PROPN
ejpam-4432	155	5	∩	∩	ADJ
ejpam-4432	155	6	i(jri	i(jri	ADJ
ejpam-4432	155	7	∩	∩	ADJ
ejpam-4432	155	8	ikr)rr	ikr)rr	NOUN
ejpam-4432	155	9	)	)	PUNCT
ejpam-4432	155	10	r	r	NOUN
ejpam-4432	155	11	=	=	SYM
ejpam-4432	155	12	ii	ii	X
ejpam-4432	155	13	⊻	⊻	NOUN
ejpam-4432	155	14	i(jri	i(jri	ADJ
ejpam-4432	155	15	∩	∩	NOUN
ejpam-4432	155	16	ikr)r	ikr)r	PROPN
ejpam-4432	155	17	=	=	SYM
ejpam-4432	155	18	ii	ii	PROPN
ejpam-4432	155	19	⊻	⊻	ADP
ejpam-4432	155	20	i(ji	i(ji	PROPN
ejpam-4432	155	21	⊻	⊻	SYM
ejpam-4432	155	22	ik	ik	NOUN
ejpam-4432	155	23	)	)	PUNCT
ejpam-4432	155	24	.	.	PUNCT
ejpam-4432	156	1	it	it	PRON
ejpam-4432	156	2	is	be	AUX
ejpam-4432	156	3	easy	easy	ADJ
ejpam-4432	156	4	to	to	PART
ejpam-4432	156	5	prove	prove	VERB
ejpam-4432	156	6	the	the	DET
ejpam-4432	156	7	second	second	ADJ
ejpam-4432	156	8	associative	associative	ADJ
ejpam-4432	156	9	identity	identity	NOUN
ejpam-4432	156	10	,	,	PUNCT
ejpam-4432	156	11	(	(	PUNCT
ejpam-4432	156	12	i	i	PROPN
ejpam-4432	156	13	∩	∩	ADJ
ejpam-4432	156	14	j	j	PROPN
ejpam-4432	156	15	)	)	PUNCT
ejpam-4432	156	16	∩k	∩k	NOUN
ejpam-4432	156	17	=	=	PUNCT
ejpam-4432	157	1	i	i	PRON
ejpam-4432	157	2	∩	∩	NOUN
ejpam-4432	157	3	(	(	PUNCT
ejpam-4432	157	4	j	j	PROPN
ejpam-4432	157	5	∩k	∩k	PROPN
ejpam-4432	157	6	)	)	PUNCT
ejpam-4432	157	7	.	.	PUNCT
ejpam-4432	158	1	now	now	ADV
ejpam-4432	158	2	we	we	PRON
ejpam-4432	158	3	are	be	AUX
ejpam-4432	158	4	going	go	VERB
ejpam-4432	158	5	to	to	PART
ejpam-4432	158	6	show	show	VERB
ejpam-4432	158	7	the	the	DET
ejpam-4432	158	8	absorption	absorption	NOUN
ejpam-4432	158	9	identities	identity	NOUN
ejpam-4432	158	10	.	.	PUNCT
ejpam-4432	159	1	since	since	SCONJ
ejpam-4432	159	2	ir	ir	PROPN
ejpam-4432	159	3	⊆	⊆	NUM
ejpam-4432	159	4	(	(	PUNCT
ejpam-4432	159	5	i	i	PROPN
ejpam-4432	159	6	∩	∩	NOUN
ejpam-4432	159	7	j)r	j)r	NOUN
ejpam-4432	159	8	,	,	PUNCT
ejpam-4432	159	9	then	then	ADV
ejpam-4432	159	10	ii	ii	PROPN
ejpam-4432	159	11	⊻	⊻	CCONJ
ejpam-4432	159	12	i(i	i(i	PROPN
ejpam-4432	159	13	∩	∩	PROPN
ejpam-4432	159	14	j	j	PROPN
ejpam-4432	159	15	)	)	PUNCT
ejpam-4432	159	16	=	=	SYM
ejpam-4432	159	17	(	(	PUNCT
ejpam-4432	159	18	ir	ir	X
ejpam-4432	159	19	∩	∩	NOUN
ejpam-4432	159	20	(	(	PUNCT
ejpam-4432	159	21	i	i	NOUN
ejpam-4432	159	22	∩	∩	X
ejpam-4432	159	23	j)r	j)r	X
ejpam-4432	159	24	)	)	PUNCT
ejpam-4432	159	25	r	r	NOUN
ejpam-4432	159	26	=	=	SYM
ejpam-4432	159	27	irr	irr	NOUN
ejpam-4432	159	28	=	=	PUNCT
ejpam-4432	159	29	i.	i.	NOUN
ejpam-4432	159	30	similarly	similarly	ADV
ejpam-4432	159	31	,	,	PUNCT
ejpam-4432	159	32	since	since	SCONJ
ejpam-4432	159	33	ir	ir	PROPN
ejpam-4432	159	34	∩	∩	PROPN
ejpam-4432	159	35	jr	jr	PROPN
ejpam-4432	159	36	⊆	⊆	NUM
ejpam-4432	159	37	ir	ir	PROPN
ejpam-4432	159	38	,	,	PUNCT
ejpam-4432	159	39	ithen	ithen	NOUN
ejpam-4432	160	1	i	i	NOUN
ejpam-4432	160	2	=	=	PUNCT
ejpam-4432	160	3	irr	irr	PROPN
ejpam-4432	160	4	⊆	⊆	NUM
ejpam-4432	160	5	(	(	PUNCT
ejpam-4432	160	6	ir	ir	X
ejpam-4432	160	7	∩	∩	PROPN
ejpam-4432	160	8	jr	jr	PROPN
ejpam-4432	160	9	)	)	PUNCT
ejpam-4432	160	10	r	r	NOUN
ejpam-4432	160	11	.	.	PUNCT
ejpam-4432	161	1	therefore	therefore	ADV
ejpam-4432	161	2	i	i	PRON
ejpam-4432	161	3	∩	∩	X
ejpam-4432	161	4	(	(	PUNCT
ejpam-4432	161	5	ii	ii	NOUN
ejpam-4432	161	6	⊻	⊻	CCONJ
ejpam-4432	161	7	ij	ij	NOUN
ejpam-4432	161	8	)	)	PUNCT
ejpam-4432	161	9	=	=	SYM
ejpam-4432	162	1	i	i	PRON
ejpam-4432	162	2	∩	∩	NOUN
ejpam-4432	162	3	(	(	PUNCT
ejpam-4432	162	4	ir	ir	X
ejpam-4432	162	5	∩	∩	NOUN
ejpam-4432	162	6	jr)r	jr)r	PROPN
ejpam-4432	162	7	=	=	SYM
ejpam-4432	162	8	i.	i.	NOUN
ejpam-4432	162	9	notice	notice	VERB
ejpam-4432	162	10	that	that	SCONJ
ejpam-4432	162	11	i	i	PRON
ejpam-4432	162	12	⊆	⊆	NUM
ejpam-4432	162	13	0r	0r	X
ejpam-4432	162	14	=	=	SYM
ejpam-4432	162	15	l	l	NOUN
ejpam-4432	162	16	and	and	CCONJ
ejpam-4432	162	17	1r	1r	NUM
ejpam-4432	162	18	⊆	⊆	NUM
ejpam-4432	162	19	i	i	PROPN
ejpam-4432	162	20	∈	∈	PROPN
ejpam-4432	162	21	h(l	h(l	PROPN
ejpam-4432	162	22	)	)	PUNCT
ejpam-4432	162	23	,	,	PUNCT
ejpam-4432	162	24	consequently	consequently	ADV
ejpam-4432	162	25	(	(	PUNCT
ejpam-4432	162	26	h(l),∧,⊻	h(l),∧,⊻	NOUN
ejpam-4432	162	27	)	)	PUNCT
ejpam-4432	162	28	is	be	AUX
ejpam-4432	162	29	a	a	DET
ejpam-4432	162	30	bounded	bounded	ADJ
ejpam-4432	162	31	lattice	lattice	NOUN
ejpam-4432	162	32	.	.	PUNCT
ejpam-4432	163	1	clearly	clearly	ADV
ejpam-4432	163	2	,	,	PUNCT
ejpam-4432	163	3	ir	ir	PROPN
ejpam-4432	163	4	is	be	AUX
ejpam-4432	163	5	the	the	DET
ejpam-4432	163	6	complement	complement	NOUN
ejpam-4432	163	7	of	of	ADP
ejpam-4432	163	8	i	i	PRON
ejpam-4432	163	9	,	,	PUNCT
ejpam-4432	163	10	because	because	SCONJ
ejpam-4432	163	11	i∩ir	i∩ir	NOUN
ejpam-4432	163	12	=	=	SYM
ejpam-4432	163	13	1r	1r	NUM
ejpam-4432	163	14	and	and	CCONJ
ejpam-4432	163	15	i⊻ir	i⊻ir	NOUN
ejpam-4432	163	16	=	=	SYM
ejpam-4432	163	17	(	(	PUNCT
ejpam-4432	163	18	ir∩irr)r	ir∩irr)r	NOUN
ejpam-4432	163	19	=	=	SYM
ejpam-4432	163	20	1rr	1rr	PROPN
ejpam-4432	163	21	=	=	PUNCT
ejpam-4432	163	22	l.	l.	NOUN
ejpam-4432	163	23	by	by	ADP
ejpam-4432	163	24	using	use	VERB
ejpam-4432	163	25	(	(	PUNCT
ejpam-4432	163	26	iv	iv	NOUN
ejpam-4432	163	27	)	)	PUNCT
ejpam-4432	163	28	in	in	ADP
ejpam-4432	163	29	lemma	lemma	PROPN
ejpam-4432	163	30	1	1	NUM
ejpam-4432	163	31	,	,	PUNCT
ejpam-4432	163	32	we	we	PRON
ejpam-4432	163	33	get	get	VERB
ejpam-4432	163	34	that	that	PRON
ejpam-4432	163	35	:	:	PUNCT
ejpam-4432	163	36	k	k	X
ejpam-4432	163	37	∩	∩	X
ejpam-4432	163	38	(	(	PUNCT
ejpam-4432	163	39	i	i	PRON
ejpam-4432	163	40	⊻	⊻	X
ejpam-4432	163	41	j	j	NOUN
ejpam-4432	163	42	)	)	PUNCT
ejpam-4432	163	43	⊆	⊆	NUM
ejpam-4432	163	44	i	i	PRON
ejpam-4432	163	45	⊻	⊻	PUNCT
ejpam-4432	163	46	(	(	PUNCT
ejpam-4432	163	47	j	j	PROPN
ejpam-4432	163	48	∩k	∩k	PROPN
ejpam-4432	163	49	)	)	PUNCT
ejpam-4432	163	50	(	(	PUNCT
ejpam-4432	163	51	3	3	X
ejpam-4432	163	52	)	)	PUNCT
ejpam-4432	163	53	then	then	ADV
ejpam-4432	163	54	the	the	DET
ejpam-4432	163	55	distributivity	distributivity	NOUN
ejpam-4432	163	56	condition	condition	NOUN
ejpam-4432	163	57	is	be	AUX
ejpam-4432	163	58	proved	prove	VERB
ejpam-4432	163	59	by	by	ADP
ejpam-4432	163	60	replacing	replace	VERB
ejpam-4432	163	61	k	k	PROPN
ejpam-4432	163	62	in	in	ADP
ejpam-4432	163	63	(	(	PUNCT
ejpam-4432	163	64	3	3	NUM
ejpam-4432	163	65	)	)	PUNCT
ejpam-4432	163	66	by	by	ADP
ejpam-4432	163	67	i	i	PRON
ejpam-4432	163	68	⊻k	⊻k	NOUN
ejpam-4432	163	69	to	to	PART
ejpam-4432	163	70	get	get	VERB
ejpam-4432	163	71	:	:	PUNCT
ejpam-4432	163	72	e.g.	e.g.	ADV
ejpam-4432	163	73	rezk	rezk	PROPN
ejpam-4432	163	74	,	,	PUNCT
ejpam-4432	163	75	n.h	n.h	PROPN
ejpam-4432	163	76	.	.	PROPN
ejpam-4432	163	77	abughazalah	abughazalah	PROPN
ejpam-4432	163	78	/	/	SYM
ejpam-4432	163	79	eur	eur	PROPN
ejpam-4432	163	80	.	.	PUNCT
ejpam-4432	164	1	j.	j.	PROPN
ejpam-4432	164	2	pure	pure	PROPN
ejpam-4432	164	3	appl	appl	PROPN
ejpam-4432	164	4	.	.	PROPN
ejpam-4432	164	5	math	math	PROPN
ejpam-4432	164	6	,	,	PUNCT
ejpam-4432	164	7	15	15	NUM
ejpam-4432	164	8	(	(	PUNCT
ejpam-4432	164	9	3	3	NUM
ejpam-4432	164	10	)	)	PUNCT
ejpam-4432	164	11	(	(	PUNCT
ejpam-4432	164	12	2022	2022	NUM
ejpam-4432	164	13	)	)	PUNCT
ejpam-4432	164	14	,	,	PUNCT
ejpam-4432	164	15	1402	1402	NUM
ejpam-4432	164	16	-	-	SYM
ejpam-4432	164	17	1416	1416	NUM
ejpam-4432	164	18	1409	1409	NUM
ejpam-4432	164	19	(	(	PUNCT
ejpam-4432	164	20	ii	ii	PROPN
ejpam-4432	164	21	⊻	⊻	SYM
ejpam-4432	164	22	ik	ik	ADJ
ejpam-4432	164	23	)	)	PUNCT
ejpam-4432	164	24	∩	∩	NOUN
ejpam-4432	164	25	(	(	PUNCT
ejpam-4432	164	26	i	i	PRON
ejpam-4432	164	27	⊻	⊻	X
ejpam-4432	164	28	j	j	NOUN
ejpam-4432	164	29	)	)	PUNCT
ejpam-4432	164	30	⊆	⊆	NUM
ejpam-4432	164	31	ii	ii	NOUN
ejpam-4432	164	32	⊻	⊻	NOUN
ejpam-4432	165	1	i	i	PRON
ejpam-4432	165	2	(	(	PUNCT
ejpam-4432	165	3	j	j	PROPN
ejpam-4432	165	4	∩	∩	NOUN
ejpam-4432	165	5	(	(	PUNCT
ejpam-4432	165	6	i	i	NOUN
ejpam-4432	165	7	⊻k	⊻k	NUM
ejpam-4432	165	8	)	)	PUNCT
ejpam-4432	165	9	)	)	PUNCT
ejpam-4432	166	1	⊆	⊆	NUM
ejpam-4432	166	2	ii	ii	NOUN
ejpam-4432	166	3	⊻	⊻	NOUN
ejpam-4432	167	1	i	i	PRON
ejpam-4432	167	2	(	(	PUNCT
ejpam-4432	167	3	ii	ii	X
ejpam-4432	167	4	⊻	⊻	ADP
ejpam-4432	167	5	i(k	i(k	PROPN
ejpam-4432	167	6	∩	∩	ADJ
ejpam-4432	167	7	j	j	NOUN
ejpam-4432	167	8	)	)	PUNCT
ejpam-4432	167	9	)	)	PUNCT
ejpam-4432	167	10	(	(	PUNCT
ejpam-4432	167	11	by	by	ADP
ejpam-4432	167	12	replacing	replace	VERB
ejpam-4432	167	13	k	k	PROPN
ejpam-4432	167	14	in	in	ADP
ejpam-4432	167	15	(	(	PUNCT
ejpam-4432	167	16	3	3	NUM
ejpam-4432	167	17	)	)	PUNCT
ejpam-4432	167	18	by	by	ADP
ejpam-4432	167	19	j	j	PROPN
ejpam-4432	167	20	)	)	PUNCT
ejpam-4432	167	21	=	=	SYM
ejpam-4432	167	22	ii	ii	PROPN
ejpam-4432	167	23	⊻	⊻	CCONJ
ejpam-4432	167	24	i(k	i(k	PROPN
ejpam-4432	167	25	∩	∩	ADJ
ejpam-4432	167	26	j	j	PROPN
ejpam-4432	167	27	)	)	PUNCT
ejpam-4432	167	28	.	.	PUNCT
ejpam-4432	168	1	on	on	ADP
ejpam-4432	168	2	the	the	DET
ejpam-4432	168	3	other	other	ADJ
ejpam-4432	168	4	hand	hand	NOUN
ejpam-4432	168	5	,	,	PUNCT
ejpam-4432	168	6	j	j	PROPN
ejpam-4432	168	7	∩	∩	PROPN
ejpam-4432	168	8	k	k	PROPN
ejpam-4432	168	9	⊆	⊆	NUM
ejpam-4432	168	10	(	(	PUNCT
ejpam-4432	168	11	ii	ii	PROPN
ejpam-4432	168	12	⊻	⊻	CCONJ
ejpam-4432	168	13	ij	ij	NOUN
ejpam-4432	168	14	)	)	PUNCT
ejpam-4432	168	15	∩	∩	NOUN
ejpam-4432	168	16	(	(	PUNCT
ejpam-4432	168	17	ii	ii	PROPN
ejpam-4432	168	18	⊻	⊻	SYM
ejpam-4432	168	19	ik	ik	NOUN
ejpam-4432	168	20	)	)	PUNCT
ejpam-4432	168	21	and	and	CCONJ
ejpam-4432	168	22	i	i	PRON
ejpam-4432	168	23	⊆	⊆	NUM
ejpam-4432	168	24	(	(	PUNCT
ejpam-4432	168	25	ii	ii	NOUN
ejpam-4432	168	26	⊻	⊻	CCONJ
ejpam-4432	168	27	ij	ij	NOUN
ejpam-4432	168	28	)	)	PUNCT
ejpam-4432	168	29	∩	∩	NOUN
ejpam-4432	168	30	(	(	PUNCT
ejpam-4432	168	31	ii	ii	PROPN
ejpam-4432	168	32	⊻	⊻	SYM
ejpam-4432	168	33	ik	ik	NOUN
ejpam-4432	168	34	)	)	PUNCT
ejpam-4432	168	35	,	,	PUNCT
ejpam-4432	168	36	which	which	PRON
ejpam-4432	168	37	implies	imply	VERB
ejpam-4432	168	38	that	that	SCONJ
ejpam-4432	168	39	ii	ii	PROPN
ejpam-4432	168	40	⊻	⊻	CCONJ
ejpam-4432	168	41	i(j	i(j	ADJ
ejpam-4432	168	42	∩k	∩k	NOUN
ejpam-4432	168	43	)	)	PUNCT
ejpam-4432	169	1	⊆	⊆	NUM
ejpam-4432	169	2	(	(	PUNCT
ejpam-4432	169	3	ii	ii	NOUN
ejpam-4432	169	4	⊻	⊻	CCONJ
ejpam-4432	169	5	ij	ij	NOUN
ejpam-4432	169	6	)	)	PUNCT
ejpam-4432	169	7	∩	∩	NOUN
ejpam-4432	169	8	(	(	PUNCT
ejpam-4432	169	9	ii	ii	PROPN
ejpam-4432	169	10	⊻	⊻	SYM
ejpam-4432	169	11	ik	ik	NOUN
ejpam-4432	169	12	)	)	PUNCT
ejpam-4432	169	13	.	.	PUNCT
ejpam-4432	170	1	(	(	PUNCT
ejpam-4432	170	2	4	4	X
ejpam-4432	170	3	)	)	PUNCT
ejpam-4432	170	4	therefore	therefore	ADV
ejpam-4432	170	5	ii	ii	X
ejpam-4432	170	6	⊻	⊻	CCONJ
ejpam-4432	170	7	i(j	i(j	ADJ
ejpam-4432	170	8	∩k	∩k	NOUN
ejpam-4432	170	9	)	)	PUNCT
ejpam-4432	170	10	=	=	PUNCT
ejpam-4432	170	11	(	(	PUNCT
ejpam-4432	170	12	ii	ii	PROPN
ejpam-4432	170	13	⊻	⊻	CCONJ
ejpam-4432	170	14	ij	ij	NOUN
ejpam-4432	170	15	)	)	PUNCT
ejpam-4432	170	16	∩	∩	NOUN
ejpam-4432	170	17	(	(	PUNCT
ejpam-4432	170	18	ii	ii	PROPN
ejpam-4432	170	19	⊻	⊻	SYM
ejpam-4432	170	20	ik	ik	NOUN
ejpam-4432	170	21	)	)	PUNCT
ejpam-4432	170	22	from	from	ADP
ejpam-4432	170	23	(	(	PUNCT
ejpam-4432	170	24	3	3	NUM
ejpam-4432	170	25	)	)	PUNCT
ejpam-4432	170	26	and	and	CCONJ
ejpam-4432	170	27	(	(	PUNCT
ejpam-4432	170	28	4	4	NUM
ejpam-4432	170	29	)	)	PUNCT
ejpam-4432	170	30	.	.	PUNCT
ejpam-4432	171	1	example	example	NOUN
ejpam-4432	172	1	1	1	NUM
ejpam-4432	172	2	.	.	PUNCT
ejpam-4432	172	3	tables	table	NOUN
ejpam-4432	172	4	1	1	NUM
ejpam-4432	172	5	and	and	CCONJ
ejpam-4432	172	6	2	2	NUM
ejpam-4432	172	7	represent	represent	VERB
ejpam-4432	172	8	the	the	DET
ejpam-4432	172	9	hyperoperation	hyperoperation	NOUN
ejpam-4432	172	10	∧̄	∧̄	PROPN
ejpam-4432	172	11	and	and	CCONJ
ejpam-4432	172	12	operation	operation	NOUN
ejpam-4432	172	13	∨	∨	NOUN
ejpam-4432	172	14	of	of	ADP
ejpam-4432	172	15	meethyberlattice	meethyberlattice	NOUN
ejpam-4432	172	16	l	l	NOUN
ejpam-4432	172	17	=	=	PUNCT
ejpam-4432	172	18	{	{	PUNCT
ejpam-4432	172	19	0	0	NUM
ejpam-4432	172	20	,	,	PUNCT
ejpam-4432	172	21	α	α	X
ejpam-4432	172	22	,	,	PUNCT
ejpam-4432	172	23	β	β	X
ejpam-4432	172	24	,	,	PUNCT
ejpam-4432	172	25	γ	γ	PROPN
ejpam-4432	172	26	,	,	PUNCT
ejpam-4432	172	27	δ	δ	PROPN
ejpam-4432	172	28	,	,	PUNCT
ejpam-4432	172	29	1	1	NUM
ejpam-4432	172	30	}	}	PUNCT
ejpam-4432	172	31	.	.	PUNCT
ejpam-4432	173	1	figure	figure	NOUN
ejpam-4432	173	2	1	1	NUM
ejpam-4432	173	3	shows	show	VERB
ejpam-4432	173	4	boolean	boolean	ADJ
ejpam-4432	173	5	algebra	algebra	NOUN
ejpam-4432	173	6	h(l	h(l	PROPN
ejpam-4432	173	7	)	)	PUNCT
ejpam-4432	173	8	of	of	ADP
ejpam-4432	173	9	closed	closed	ADJ
ejpam-4432	173	10	hyperideals	hyperideal	NOUN
ejpam-4432	173	11	of	of	ADP
ejpam-4432	173	12	l.	l.	PROPN
ejpam-4432	173	13	∧̄	∧̄	PROPN
ejpam-4432	173	14	0	0	NUM
ejpam-4432	174	1	α	α	PRON
ejpam-4432	174	2	β	β	X
ejpam-4432	174	3	γ	γ	X
ejpam-4432	174	4	δ	δ	PROPN
ejpam-4432	174	5	1	1	NUM
ejpam-4432	174	6	0	0	NUM
ejpam-4432	174	7	{	{	PUNCT
ejpam-4432	174	8	0	0	NUM
ejpam-4432	174	9	}	}	PUNCT
ejpam-4432	174	10	{	{	PUNCT
ejpam-4432	174	11	0	0	NUM
ejpam-4432	174	12	}	}	PUNCT
ejpam-4432	174	13	{	{	PUNCT
ejpam-4432	174	14	0	0	NUM
ejpam-4432	174	15	}	}	PUNCT
ejpam-4432	174	16	{	{	PUNCT
ejpam-4432	174	17	0	0	NUM
ejpam-4432	174	18	}	}	PUNCT
ejpam-4432	174	19	{	{	PUNCT
ejpam-4432	174	20	0	0	NUM
ejpam-4432	174	21	}	}	PUNCT
ejpam-4432	174	22	{	{	PUNCT
ejpam-4432	174	23	0	0	NUM
ejpam-4432	174	24	}	}	PUNCT
ejpam-4432	174	25	α	α	NOUN
ejpam-4432	174	26	{	{	PUNCT
ejpam-4432	174	27	0	0	NUM
ejpam-4432	174	28	}	}	PUNCT
ejpam-4432	174	29	{	{	PUNCT
ejpam-4432	174	30	0	0	NUM
ejpam-4432	174	31	,	,	PUNCT
ejpam-4432	174	32	α	α	NOUN
ejpam-4432	174	33	}	}	PUNCT
ejpam-4432	174	34	{	{	PUNCT
ejpam-4432	174	35	0	0	NUM
ejpam-4432	174	36	}	}	PUNCT
ejpam-4432	174	37	{	{	PUNCT
ejpam-4432	174	38	0	0	NUM
ejpam-4432	174	39	,	,	PUNCT
ejpam-4432	174	40	α	α	NOUN
ejpam-4432	174	41	}	}	PUNCT
ejpam-4432	174	42	{	{	PUNCT
ejpam-4432	174	43	0	0	NUM
ejpam-4432	174	44	,	,	PUNCT
ejpam-4432	174	45	α	α	NOUN
ejpam-4432	174	46	}	}	PUNCT
ejpam-4432	174	47	{	{	PUNCT
ejpam-4432	174	48	0	0	NUM
ejpam-4432	174	49	,	,	PUNCT
ejpam-4432	174	50	α	α	NOUN
ejpam-4432	174	51	}	}	PUNCT
ejpam-4432	174	52	β	β	X
ejpam-4432	174	53	{	{	PUNCT
ejpam-4432	174	54	0	0	NUM
ejpam-4432	174	55	}	}	PUNCT
ejpam-4432	174	56	{	{	PUNCT
ejpam-4432	174	57	0	0	NUM
ejpam-4432	174	58	}	}	PUNCT
ejpam-4432	174	59	{	{	PUNCT
ejpam-4432	174	60	β	β	X
ejpam-4432	174	61	}	}	PUNCT
ejpam-4432	174	62	{	{	PUNCT
ejpam-4432	174	63	β	β	NOUN
ejpam-4432	174	64	}	}	PUNCT
ejpam-4432	174	65	{	{	PUNCT
ejpam-4432	174	66	0	0	NUM
ejpam-4432	174	67	}	}	PUNCT
ejpam-4432	174	68	{	{	PUNCT
ejpam-4432	174	69	β	β	NOUN
ejpam-4432	174	70	}	}	PUNCT
ejpam-4432	174	71	γ	γ	X
ejpam-4432	174	72	{	{	PUNCT
ejpam-4432	174	73	0	0	NUM
ejpam-4432	174	74	}	}	PUNCT
ejpam-4432	174	75	{	{	PUNCT
ejpam-4432	174	76	0	0	NUM
ejpam-4432	174	77	,	,	PUNCT
ejpam-4432	174	78	α	α	NOUN
ejpam-4432	174	79	}	}	PUNCT
ejpam-4432	174	80	{	{	PUNCT
ejpam-4432	174	81	β	β	X
ejpam-4432	174	82	}	}	PUNCT
ejpam-4432	174	83	{	{	PUNCT
ejpam-4432	174	84	γ	γ	X
ejpam-4432	174	85	}	}	PUNCT
ejpam-4432	174	86	{	{	PUNCT
ejpam-4432	174	87	0	0	NUM
ejpam-4432	174	88	,	,	PUNCT
ejpam-4432	174	89	α	α	NOUN
ejpam-4432	174	90	}	}	PUNCT
ejpam-4432	174	91	{	{	PUNCT
ejpam-4432	174	92	γ	γ	PROPN
ejpam-4432	174	93	}	}	PUNCT
ejpam-4432	174	94	δ	δ	PROPN
ejpam-4432	174	95	{	{	PUNCT
ejpam-4432	174	96	0	0	NUM
ejpam-4432	174	97	}	}	PUNCT
ejpam-4432	174	98	{	{	PUNCT
ejpam-4432	174	99	0	0	NUM
ejpam-4432	174	100	,	,	PUNCT
ejpam-4432	174	101	α	α	NOUN
ejpam-4432	174	102	}	}	PUNCT
ejpam-4432	174	103	{	{	PUNCT
ejpam-4432	174	104	0	0	NUM
ejpam-4432	174	105	}	}	PUNCT
ejpam-4432	174	106	{	{	PUNCT
ejpam-4432	174	107	0	0	NUM
ejpam-4432	174	108	,	,	PUNCT
ejpam-4432	174	109	α	α	NOUN
ejpam-4432	174	110	}	}	PUNCT
ejpam-4432	174	111	{	{	PUNCT
ejpam-4432	174	112	δ	δ	PROPN
ejpam-4432	174	113	}	}	PUNCT
ejpam-4432	174	114	{	{	PUNCT
ejpam-4432	174	115	δ	δ	PROPN
ejpam-4432	174	116	}	}	PUNCT
ejpam-4432	174	117	1	1	NUM
ejpam-4432	174	118	{	{	PUNCT
ejpam-4432	174	119	0	0	NUM
ejpam-4432	174	120	}	}	PUNCT
ejpam-4432	174	121	{	{	PUNCT
ejpam-4432	174	122	0	0	NUM
ejpam-4432	174	123	,	,	PUNCT
ejpam-4432	174	124	α	α	NOUN
ejpam-4432	174	125	}	}	PUNCT
ejpam-4432	174	126	{	{	PUNCT
ejpam-4432	174	127	β	β	X
ejpam-4432	174	128	}	}	PUNCT
ejpam-4432	174	129	{	{	PUNCT
ejpam-4432	174	130	γ	γ	X
ejpam-4432	174	131	}	}	PUNCT
ejpam-4432	174	132	{	{	PUNCT
ejpam-4432	174	133	δ	δ	NOUN
ejpam-4432	174	134	}	}	PUNCT
ejpam-4432	174	135	{	{	PUNCT
ejpam-4432	174	136	1	1	NUM
ejpam-4432	174	137	}	}	PUNCT
ejpam-4432	174	138	∨	∨	NUM
ejpam-4432	174	139	0	0	NUM
ejpam-4432	175	1	α	α	NOUN
ejpam-4432	175	2	β	β	X
ejpam-4432	175	3	γ	γ	X
ejpam-4432	175	4	δ	δ	PROPN
ejpam-4432	175	5	1	1	NUM
ejpam-4432	175	6	0	0	NUM
ejpam-4432	175	7	0	0	NUM
ejpam-4432	176	1	α	α	PRON
ejpam-4432	176	2	β	β	X
ejpam-4432	176	3	γ	γ	X
ejpam-4432	176	4	δ	δ	PROPN
ejpam-4432	176	5	1	1	NUM
ejpam-4432	176	6	α	α	NOUN
ejpam-4432	176	7	α	α	NOUN
ejpam-4432	176	8	α	α	NOUN
ejpam-4432	176	9	γ	γ	PROPN
ejpam-4432	176	10	γ	γ	X
ejpam-4432	176	11	δ	δ	PROPN
ejpam-4432	176	12	1	1	NUM
ejpam-4432	176	13	β	β	X
ejpam-4432	176	14	β	β	X
ejpam-4432	176	15	γ	γ	X
ejpam-4432	176	16	β	β	X
ejpam-4432	176	17	γ	γ	X
ejpam-4432	176	18	1	1	NUM
ejpam-4432	176	19	1	1	NUM
ejpam-4432	176	20	γ	γ	PROPN
ejpam-4432	176	21	γ	γ	X
ejpam-4432	176	22	γ	γ	X
ejpam-4432	176	23	γ	γ	X
ejpam-4432	176	24	γ	γ	X
ejpam-4432	176	25	1	1	NUM
ejpam-4432	176	26	1	1	NUM
ejpam-4432	176	27	δ	δ	PROPN
ejpam-4432	176	28	δ	δ	NOUN
ejpam-4432	176	29	0	0	NUM
ejpam-4432	176	30	1	1	NUM
ejpam-4432	176	31	1	1	NUM
ejpam-4432	176	32	δ	δ	NOUN
ejpam-4432	176	33	1	1	NUM
ejpam-4432	176	34	1	1	NUM
ejpam-4432	176	35	1	1	NUM
ejpam-4432	176	36	1	1	NUM
ejpam-4432	176	37	1	1	NUM
ejpam-4432	176	38	1	1	NUM
ejpam-4432	176	39	1	1	NUM
ejpam-4432	176	40	1	1	NUM
ejpam-4432	176	41	table	table	NOUN
ejpam-4432	176	42	1	1	NUM
ejpam-4432	176	43	:	:	PUNCT
ejpam-4432	176	44	represents	represent	VERB
ejpam-4432	176	45	the	the	DET
ejpam-4432	176	46	hyperoperation	hyperoperation	NOUN
ejpam-4432	176	47	table	table	NOUN
ejpam-4432	176	48	2	2	NUM
ejpam-4432	176	49	:	:	PUNCT
ejpam-4432	176	50	represents	represent	VERB
ejpam-4432	176	51	operation	operation	NOUN
ejpam-4432	176	52	∧̄	∧̄	PROPN
ejpam-4432	176	53	of	of	ADP
ejpam-4432	176	54	the	the	DET
ejpam-4432	176	55	meet	meet	ADJ
ejpam-4432	176	56	-	-	PUNCT
ejpam-4432	176	57	hyberlattice	hyberlattice	NOUN
ejpam-4432	176	58	l	l	NOUN
ejpam-4432	176	59	∨	∨	NOUN
ejpam-4432	176	60	of	of	ADP
ejpam-4432	176	61	the	the	DET
ejpam-4432	176	62	meet	meet	ADJ
ejpam-4432	176	63	-	-	PUNCT
ejpam-4432	176	64	hyberlattice	hyberlattice	NOUN
ejpam-4432	176	65	l	l	NOUN
ejpam-4432	176	66	l	l	NOUN
ejpam-4432	176	67	{	{	PUNCT
ejpam-4432	176	68	0	0	NUM
ejpam-4432	176	69	,	,	PUNCT
ejpam-4432	176	70	α	α	X
ejpam-4432	176	71	,	,	PUNCT
ejpam-4432	176	72	β	β	X
ejpam-4432	176	73	,	,	PUNCT
ejpam-4432	176	74	γ	γ	NOUN
ejpam-4432	176	75	}	}	PUNCT
ejpam-4432	176	76	{	{	PUNCT
ejpam-4432	176	77	0	0	NUM
ejpam-4432	176	78	,	,	PUNCT
ejpam-4432	176	79	α	α	NOUN
ejpam-4432	176	80	,	,	PUNCT
ejpam-4432	176	81	δ	δ	PROPN
ejpam-4432	176	82	}	}	PUNCT
ejpam-4432	176	83	1r	1r	NUM
ejpam-4432	176	84	=	=	SYM
ejpam-4432	176	85	{	{	PUNCT
ejpam-4432	176	86	0	0	NUM
ejpam-4432	176	87	,	,	PUNCT
ejpam-4432	176	88	α	α	NOUN
ejpam-4432	176	89	}	}	PUNCT
ejpam-4432	176	90	figure1	figure1	PROPN
ejpam-4432	176	91	:	:	PUNCT
ejpam-4432	176	92	boolean	boolean	ADJ
ejpam-4432	176	93	algebra	algebra	NOUN
ejpam-4432	176	94	<	<	X
ejpam-4432	176	95	h(l);∩,⊻,r	h(l);∩,⊻,r	PROPN
ejpam-4432	176	96	,	,	PUNCT
ejpam-4432	176	97	1r	1r	NUM
ejpam-4432	176	98	,	,	PUNCT
ejpam-4432	176	99	l	l	NOUN
ejpam-4432	176	100	>	>	X
ejpam-4432	177	1	4	4	X
ejpam-4432	177	2	.	.	PUNCT
ejpam-4432	177	3	homomorphic	homomorphic	ADJ
ejpam-4432	177	4	images	image	NOUN
ejpam-4432	177	5	of	of	ADP
ejpam-4432	177	6	annihilator	annihilator	NOUN
ejpam-4432	177	7	hyperideals	hyperideal	NOUN
ejpam-4432	177	8	in	in	ADP
ejpam-4432	177	9	this	this	DET
ejpam-4432	177	10	section	section	NOUN
ejpam-4432	177	11	,	,	PUNCT
ejpam-4432	177	12	we	we	PRON
ejpam-4432	177	13	define	define	VERB
ejpam-4432	177	14	the	the	DET
ejpam-4432	177	15	homomorphism	homomorphism	NOUN
ejpam-4432	177	16	that	that	PRON
ejpam-4432	177	17	is	be	AUX
ejpam-4432	177	18	annihilator	annihilator	PROPN
ejpam-4432	177	19	hyperideal	hyperideal	NOUN
ejpam-4432	177	20	preserving	preserve	VERB
ejpam-4432	177	21	.	.	PUNCT
ejpam-4432	178	1	many	many	ADJ
ejpam-4432	178	2	properties	property	NOUN
ejpam-4432	178	3	related	relate	VERB
ejpam-4432	178	4	to	to	ADP
ejpam-4432	178	5	the	the	DET
ejpam-4432	178	6	homomorphism	homomorphism	NOUN
ejpam-4432	178	7	of	of	ADP
ejpam-4432	178	8	annihilator	annihilator	PROPN
ejpam-4432	178	9	hyperideals	hyperideal	NOUN
ejpam-4432	178	10	are	be	AUX
ejpam-4432	178	11	proven	prove	VERB
ejpam-4432	178	12	.	.	PUNCT
ejpam-4432	179	1	moreover	moreover	ADV
ejpam-4432	179	2	,	,	PUNCT
ejpam-4432	179	3	we	we	PRON
ejpam-4432	179	4	show	show	VERB
ejpam-4432	179	5	that	that	SCONJ
ejpam-4432	179	6	homomorphic	homomorphic	ADJ
ejpam-4432	179	7	images	image	NOUN
ejpam-4432	179	8	and	and	CCONJ
ejpam-4432	179	9	preimages	preimage	NOUN
ejpam-4432	179	10	of	of	ADP
ejpam-4432	179	11	annihilator	annihilator	PROPN
ejpam-4432	179	12	hyperideals	hyperideal	NOUN
ejpam-4432	179	13	are	be	AUX
ejpam-4432	179	14	annihilator	annihilator	NOUN
ejpam-4432	179	15	hyperideals	hyperideal	NOUN
ejpam-4432	179	16	.	.	PUNCT
ejpam-4432	180	1	definition	definition	NOUN
ejpam-4432	180	2	5	5	NUM
ejpam-4432	180	3	.	.	PUNCT
ejpam-4432	181	1	let	let	VERB
ejpam-4432	181	2	l	l	NOUN
ejpam-4432	181	3	and	and	CCONJ
ejpam-4432	181	4	l′	l′	VERB
ejpam-4432	181	5	be	be	AUX
ejpam-4432	181	6	two	two	NUM
ejpam-4432	181	7	meet	meet	NOUN
ejpam-4432	181	8	-	-	PUNCT
ejpam-4432	181	9	hyperlattices	hyperlattice	NOUN
ejpam-4432	181	10	.	.	PUNCT
ejpam-4432	182	1	then	then	ADV
ejpam-4432	182	2	the	the	DET
ejpam-4432	182	3	map	map	NOUN
ejpam-4432	182	4	ϕ	ϕ	X
ejpam-4432	182	5	:	:	PUNCT
ejpam-4432	182	6	l	l	X
ejpam-4432	182	7	→	→	SYM
ejpam-4432	182	8	l′	l′	X
ejpam-4432	182	9	is	be	AUX
ejpam-4432	182	10	called	call	VERB
ejpam-4432	182	11	homomorphism	homomorphism	NOUN
ejpam-4432	182	12	if	if	SCONJ
ejpam-4432	182	13	the	the	DET
ejpam-4432	182	14	following	follow	VERB
ejpam-4432	182	15	conditions	condition	NOUN
ejpam-4432	182	16	hold	hold	VERB
ejpam-4432	182	17	:	:	PUNCT
ejpam-4432	182	18	ϕ(a	ϕ(a	NOUN
ejpam-4432	182	19	∨	∨	NUM
ejpam-4432	182	20	b	b	NOUN
ejpam-4432	182	21	)	)	PUNCT
ejpam-4432	182	22	=	=	SYM
ejpam-4432	182	23	ϕ(a	ϕ(a	PROPN
ejpam-4432	182	24	)	)	PUNCT
ejpam-4432	182	25	∨	∨	NUM
ejpam-4432	182	26	ϕ(b	ϕ(b	PROPN
ejpam-4432	182	27	)	)	PUNCT
ejpam-4432	182	28	,	,	PUNCT
ejpam-4432	182	29	iϕ(a∧̄b	iϕ(a∧̄b	PROPN
ejpam-4432	182	30	)	)	PUNCT
ejpam-4432	182	31	=	=	SYM
ejpam-4432	182	32	ϕ(a)∧̄ϕ(b	ϕ(a)∧̄ϕ(b	PROPN
ejpam-4432	182	33	)	)	PUNCT
ejpam-4432	182	34	.	.	PUNCT
ejpam-4432	183	1	e.g.	e.g.	ADV
ejpam-4432	183	2	rezk	rezk	PROPN
ejpam-4432	183	3	,	,	PUNCT
ejpam-4432	183	4	n.h	n.h	PROPN
ejpam-4432	183	5	.	.	PROPN
ejpam-4432	183	6	abughazalah	abughazalah	PROPN
ejpam-4432	183	7	/	/	SYM
ejpam-4432	183	8	eur	eur	PROPN
ejpam-4432	183	9	.	.	PUNCT
ejpam-4432	184	1	j.	j.	PROPN
ejpam-4432	184	2	pure	pure	PROPN
ejpam-4432	184	3	appl	appl	PROPN
ejpam-4432	184	4	.	.	PROPN
ejpam-4432	184	5	math	math	PROPN
ejpam-4432	184	6	,	,	PUNCT
ejpam-4432	184	7	15	15	NUM
ejpam-4432	184	8	(	(	PUNCT
ejpam-4432	184	9	3	3	NUM
ejpam-4432	184	10	)	)	PUNCT
ejpam-4432	184	11	(	(	PUNCT
ejpam-4432	184	12	2022	2022	NUM
ejpam-4432	184	13	)	)	PUNCT
ejpam-4432	184	14	,	,	PUNCT
ejpam-4432	184	15	1402	1402	NUM
ejpam-4432	184	16	-	-	SYM
ejpam-4432	184	17	1416	1416	NUM
ejpam-4432	184	18	1410	1410	NUM
ejpam-4432	184	19	since	since	SCONJ
ejpam-4432	184	20	for	for	ADP
ejpam-4432	184	21	any	any	DET
ejpam-4432	184	22	a	a	DET
ejpam-4432	184	23	∈	∈	ADJ
ejpam-4432	184	24	l	l	NOUN
ejpam-4432	184	25	:	:	PUNCT
ejpam-4432	184	26	ϕ(a∨	ϕ(a∨	NOUN
ejpam-4432	184	27	0	0	NUM
ejpam-4432	184	28	)	)	PUNCT
ejpam-4432	184	29	=	=	PRON
ejpam-4432	184	30	ϕ(a)∨	ϕ(a)∨	PUNCT
ejpam-4432	184	31	ϕ(0	ϕ(0	NOUN
ejpam-4432	184	32	)	)	PUNCT
ejpam-4432	185	1	=	=	PUNCT
ejpam-4432	185	2	ϕ(a	ϕ(a	NOUN
ejpam-4432	185	3	)	)	PUNCT
ejpam-4432	185	4	,	,	PUNCT
ejpam-4432	185	5	then	then	ADV
ejpam-4432	185	6	ϕ(0	ϕ(0	PROPN
ejpam-4432	185	7	)	)	PUNCT
ejpam-4432	185	8	=	=	SYM
ejpam-4432	186	1	0′	0′	PROPN
ejpam-4432	186	2	,	,	PUNCT
ejpam-4432	186	3	where	where	SCONJ
ejpam-4432	186	4	0	0	NUM
ejpam-4432	186	5	and	and	CCONJ
ejpam-4432	186	6	0′	0′	NUM
ejpam-4432	186	7	are	be	AUX
ejpam-4432	186	8	the	the	DET
ejpam-4432	186	9	zero	zero	NUM
ejpam-4432	186	10	elements	element	NOUN
ejpam-4432	186	11	of	of	ADP
ejpam-4432	186	12	l	l	NOUN
ejpam-4432	186	13	and	and	CCONJ
ejpam-4432	186	14	l′	l′	VERB
ejpam-4432	186	15	respectively	respectively	ADV
ejpam-4432	186	16	.	.	PUNCT
ejpam-4432	187	1	obviously	obviously	ADV
ejpam-4432	187	2	,	,	PUNCT
ejpam-4432	187	3	if	if	SCONJ
ejpam-4432	187	4	a	a	DET
ejpam-4432	187	5	≤	≤	NUM
ejpam-4432	187	6	b	b	NOUN
ejpam-4432	187	7	then	then	ADV
ejpam-4432	187	8	ϕ(a	ϕ(a	NOUN
ejpam-4432	187	9	)	)	PUNCT
ejpam-4432	187	10	≤	≤	NUM
ejpam-4432	187	11	ϕ(b	ϕ(b	PROPN
ejpam-4432	187	12	)	)	PUNCT
ejpam-4432	187	13	.	.	PUNCT
ejpam-4432	188	1	the	the	DET
ejpam-4432	188	2	kernal	kernal	NOUN
ejpam-4432	188	3	of	of	ADP
ejpam-4432	188	4	the	the	DET
ejpam-4432	188	5	homomorphism	homomorphism	PROPN
ejpam-4432	188	6	ϕ	ϕ	NOUN
ejpam-4432	188	7	is	be	AUX
ejpam-4432	188	8	given	give	VERB
ejpam-4432	188	9	by	by	ADP
ejpam-4432	188	10	ker(ϕ	ker(ϕ	PROPN
ejpam-4432	188	11	)	)	PUNCT
ejpam-4432	188	12	=	=	PRON
ejpam-4432	188	13	{	{	PUNCT
ejpam-4432	188	14	a	a	DET
ejpam-4432	188	15	∈	∈	ADJ
ejpam-4432	188	16	l	l	NOUN
ejpam-4432	188	17	:	:	PUNCT
ejpam-4432	188	18	ϕ(a	ϕ(a	NOUN
ejpam-4432	188	19	)	)	PUNCT
ejpam-4432	188	20	=	=	SYM
ejpam-4432	188	21	0′	0′	NUM
ejpam-4432	188	22	}	}	PUNCT
ejpam-4432	188	23	.	.	PUNCT
ejpam-4432	189	1	it	it	PRON
ejpam-4432	189	2	is	be	AUX
ejpam-4432	189	3	clearithat	clearithat	PROPN
ejpam-4432	189	4	ker(ϕ	ker(ϕ	PROPN
ejpam-4432	189	5	)	)	PUNCT
ejpam-4432	189	6	is	be	AUX
ejpam-4432	189	7	a	a	DET
ejpam-4432	189	8	hyperideal	hyperideal	NOUN
ejpam-4432	189	9	of	of	ADP
ejpam-4432	189	10	l′.	l′.	NOUN
ejpam-4432	189	11	the	the	DET
ejpam-4432	189	12	set	set	NOUN
ejpam-4432	189	13	of	of	ADP
ejpam-4432	189	14	images	image	NOUN
ejpam-4432	189	15	of	of	ADP
ejpam-4432	189	16	ϕ	ϕ	NOUN
ejpam-4432	189	17	is	be	AUX
ejpam-4432	189	18	denoted	denote	VERB
ejpam-4432	189	19	by	by	ADP
ejpam-4432	189	20	im(ϕ	im(ϕ	PROPN
ejpam-4432	189	21	)	)	PUNCT
ejpam-4432	189	22	.	.	PUNCT
ejpam-4432	190	1	it	it	PRON
ejpam-4432	190	2	forms	form	VERB
ejpam-4432	190	3	sub	sub	ADJ
ejpam-4432	190	4	-	-	ADJ
ejpam-4432	190	5	meet	meet	ADJ
ejpam-4432	190	6	-	-	PUNCT
ejpam-4432	190	7	hyperlattice	hyperlattice	NOUN
ejpam-4432	190	8	of	of	ADP
ejpam-4432	190	9	l′.	l′.	NOUN
ejpam-4432	190	10	if	if	SCONJ
ejpam-4432	190	11	ϕ	ϕ	PROPN
ejpam-4432	190	12	is	be	AUX
ejpam-4432	190	13	one	one	NUM
ejpam-4432	190	14	-	-	PUNCT
ejpam-4432	190	15	to	to	ADP
ejpam-4432	190	16	-	-	PUNCT
ejpam-4432	190	17	one	one	NUM
ejpam-4432	190	18	and	and	CCONJ
ejpam-4432	190	19	onto	onto	ADP
ejpam-4432	190	20	,	,	PUNCT
ejpam-4432	190	21	then	then	ADV
ejpam-4432	190	22	l	l	PROPN
ejpam-4432	190	23	and	and	CCONJ
ejpam-4432	190	24	l′	l′	PROPN
ejpam-4432	190	25	are	be	AUX
ejpam-4432	190	26	isomorphic	isomorphic	ADJ
ejpam-4432	190	27	and	and	CCONJ
ejpam-4432	190	28	denoted	denote	VERB
ejpam-4432	190	29	by	by	ADP
ejpam-4432	190	30	l	l	NOUN
ejpam-4432	190	31	∼=	∼=	PROPN
ejpam-4432	190	32	l′.	l′.	NOUN
ejpam-4432	190	33	example	example	NOUN
ejpam-4432	190	34	2	2	NUM
ejpam-4432	190	35	.	.	PUNCT
ejpam-4432	190	36	tables	table	NOUN
ejpam-4432	190	37	3	3	NUM
ejpam-4432	190	38	and	and	CCONJ
ejpam-4432	190	39	4	4	NUM
ejpam-4432	190	40	represent	represent	VERB
ejpam-4432	190	41	the	the	DET
ejpam-4432	190	42	hyperoperation	hyperoperation	NOUN
ejpam-4432	190	43	∧̄	∧̄	PROPN
ejpam-4432	190	44	and	and	CCONJ
ejpam-4432	190	45	operation	operation	NOUN
ejpam-4432	190	46	∨	∨	NOUN
ejpam-4432	190	47	of	of	ADP
ejpam-4432	190	48	meethyberlattice	meethyberlattice	NOUN
ejpam-4432	190	49	l′	l′	NOUN
ejpam-4432	190	50	=	=	PUNCT
ejpam-4432	190	51	{	{	PUNCT
ejpam-4432	190	52	0′	0′	PROPN
ejpam-4432	190	53	,	,	PUNCT
ejpam-4432	190	54	x	x	X
ejpam-4432	190	55	,	,	PUNCT
ejpam-4432	190	56	y	y	PROPN
ejpam-4432	190	57	,	,	PUNCT
ejpam-4432	190	58	z	z	PROPN
ejpam-4432	190	59	,	,	PUNCT
ejpam-4432	190	60	1′	1′	NUM
ejpam-4432	190	61	}	}	PUNCT
ejpam-4432	190	62	.	.	PUNCT
ejpam-4432	191	1	∧̄	∧̄	NOUN
ejpam-4432	191	2	0′	0′	NUM
ejpam-4432	192	1	x	x	SYM
ejpam-4432	192	2	y	y	PROPN
ejpam-4432	192	3	z	z	PROPN
ejpam-4432	192	4	1′	1′	PROPN
ejpam-4432	192	5	0′	0′	NUM
ejpam-4432	192	6	{	{	PUNCT
ejpam-4432	192	7	0′	0′	NUM
ejpam-4432	192	8	}	}	PUNCT
ejpam-4432	192	9	{	{	PUNCT
ejpam-4432	192	10	0′	0′	NUM
ejpam-4432	192	11	}	}	PUNCT
ejpam-4432	192	12	{	{	PUNCT
ejpam-4432	192	13	0′	0′	NUM
ejpam-4432	192	14	}	}	PUNCT
ejpam-4432	192	15	{	{	PUNCT
ejpam-4432	192	16	0′	0′	NUM
ejpam-4432	192	17	}	}	PUNCT
ejpam-4432	192	18	{	{	PUNCT
ejpam-4432	192	19	0′	0′	NOUN
ejpam-4432	192	20	}	}	PUNCT
ejpam-4432	192	21	x	x	SYM
ejpam-4432	192	22	{	{	PUNCT
ejpam-4432	192	23	0′	0′	NUM
ejpam-4432	192	24	}	}	PUNCT
ejpam-4432	192	25	{	{	PUNCT
ejpam-4432	192	26	x	x	NOUN
ejpam-4432	192	27	}	}	PUNCT
ejpam-4432	192	28	{	{	PUNCT
ejpam-4432	192	29	0′	0′	NUM
ejpam-4432	192	30	}	}	PUNCT
ejpam-4432	192	31	{	{	PUNCT
ejpam-4432	192	32	x	x	NOUN
ejpam-4432	192	33	}	}	PUNCT
ejpam-4432	192	34	{	{	PUNCT
ejpam-4432	192	35	x	x	NOUN
ejpam-4432	192	36	}	}	PUNCT
ejpam-4432	192	37	y	y	PROPN
ejpam-4432	192	38	{	{	PUNCT
ejpam-4432	192	39	0′	0′	PROPN
ejpam-4432	192	40	}	}	PUNCT
ejpam-4432	192	41	{	{	PUNCT
ejpam-4432	192	42	0′	0′	NUM
ejpam-4432	192	43	}	}	PUNCT
ejpam-4432	192	44	{	{	PUNCT
ejpam-4432	192	45	y	y	NOUN
ejpam-4432	192	46	}	}	PUNCT
ejpam-4432	192	47	{	{	PUNCT
ejpam-4432	192	48	y	y	NOUN
ejpam-4432	192	49	}	}	PUNCT
ejpam-4432	192	50	{	{	PUNCT
ejpam-4432	192	51	y	y	NOUN
ejpam-4432	192	52	}	}	PUNCT
ejpam-4432	192	53	z	z	PROPN
ejpam-4432	192	54	{	{	PUNCT
ejpam-4432	192	55	0′	0′	NUM
ejpam-4432	192	56	}	}	PUNCT
ejpam-4432	192	57	{	{	PUNCT
ejpam-4432	192	58	x	x	NOUN
ejpam-4432	192	59	}	}	PUNCT
ejpam-4432	192	60	{	{	PUNCT
ejpam-4432	192	61	y	y	NOUN
ejpam-4432	192	62	}	}	PUNCT
ejpam-4432	192	63	{	{	PUNCT
ejpam-4432	192	64	z	z	NOUN
ejpam-4432	192	65	}	}	PUNCT
ejpam-4432	192	66	{	{	PUNCT
ejpam-4432	192	67	z	z	NOUN
ejpam-4432	192	68	}	}	PUNCT
ejpam-4432	192	69	1′	1′	NUM
ejpam-4432	192	70	{	{	PUNCT
ejpam-4432	192	71	0′	0′	NUM
ejpam-4432	192	72	}	}	PUNCT
ejpam-4432	192	73	{	{	PUNCT
ejpam-4432	192	74	x	x	NOUN
ejpam-4432	192	75	}	}	PUNCT
ejpam-4432	192	76	{	{	PUNCT
ejpam-4432	192	77	y	y	NOUN
ejpam-4432	192	78	}	}	PUNCT
ejpam-4432	192	79	{	{	PUNCT
ejpam-4432	192	80	z	z	NOUN
ejpam-4432	192	81	}	}	PUNCT
ejpam-4432	192	82	{	{	PUNCT
ejpam-4432	192	83	1′	1′	NUM
ejpam-4432	192	84	}	}	PUNCT
ejpam-4432	192	85	∨	∨	NOUN
ejpam-4432	192	86	0′	0′	NUM
ejpam-4432	193	1	x	x	SYM
ejpam-4432	193	2	y	y	PROPN
ejpam-4432	193	3	z	z	PROPN
ejpam-4432	193	4	1′	1′	NUM
ejpam-4432	193	5	0′	0′	NUM
ejpam-4432	193	6	0′	0′	NUM
ejpam-4432	194	1	x	x	SYM
ejpam-4432	194	2	y	y	PROPN
ejpam-4432	194	3	z	z	PROPN
ejpam-4432	194	4	1′	1′	NUM
ejpam-4432	194	5	x	x	SYM
ejpam-4432	194	6	x	x	PUNCT
ejpam-4432	194	7	x	x	X
ejpam-4432	194	8	z	z	PROPN
ejpam-4432	194	9	z	z	PROPN
ejpam-4432	194	10	1′	1′	NUM
ejpam-4432	195	1	y	y	PROPN
ejpam-4432	195	2	y	y	PROPN
ejpam-4432	195	3	z	z	PROPN
ejpam-4432	195	4	y	y	PROPN
ejpam-4432	195	5	z	z	PROPN
ejpam-4432	195	6	1′	1′	NUM
ejpam-4432	195	7	z	z	NOUN
ejpam-4432	195	8	z	z	NOUN
ejpam-4432	195	9	z	z	NOUN
ejpam-4432	195	10	z	z	PROPN
ejpam-4432	195	11	z	z	NOUN
ejpam-4432	195	12	1′	1′	NUM
ejpam-4432	195	13	1′	1′	NUM
ejpam-4432	195	14	1′	1′	NUM
ejpam-4432	195	15	1′	1′	NUM
ejpam-4432	195	16	1′	1′	NUM
ejpam-4432	195	17	1′	1′	NUM
ejpam-4432	195	18	1′	1′	NUM
ejpam-4432	195	19	table	table	NOUN
ejpam-4432	195	20	3	3	NUM
ejpam-4432	195	21	:	:	PUNCT
ejpam-4432	195	22	represents	represent	VERB
ejpam-4432	195	23	the	the	DET
ejpam-4432	195	24	hyperoperation	hyperoperation	NOUN
ejpam-4432	195	25	table	table	NOUN
ejpam-4432	195	26	4	4	NUM
ejpam-4432	195	27	:	:	PUNCT
ejpam-4432	195	28	represents	represent	VERB
ejpam-4432	195	29	the	the	DET
ejpam-4432	195	30	operation	operation	NOUN
ejpam-4432	195	31	∧̄	∧̄	PROPN
ejpam-4432	195	32	of	of	ADP
ejpam-4432	195	33	meet	meet	ADJ
ejpam-4432	195	34	-	-	PUNCT
ejpam-4432	195	35	hyberlattice	hyberlattice	NOUN
ejpam-4432	195	36	l′	l′	X
ejpam-4432	195	37	∨	∨	NUM
ejpam-4432	195	38	of	of	ADP
ejpam-4432	195	39	the	the	DET
ejpam-4432	195	40	meet	meet	ADJ
ejpam-4432	195	41	-	-	PUNCT
ejpam-4432	195	42	hyberlattice	hyberlattice	NOUN
ejpam-4432	195	43	l′	l′	NOUN
ejpam-4432	195	44	consider	consider	VERB
ejpam-4432	195	45	the	the	DET
ejpam-4432	195	46	meet	meet	ADJ
ejpam-4432	195	47	-	-	PUNCT
ejpam-4432	195	48	hyperlattice	hyperlattice	NOUN
ejpam-4432	195	49	l	l	NOUN
ejpam-4432	195	50	in	in	ADP
ejpam-4432	195	51	example	example	NOUN
ejpam-4432	195	52	1	1	NUM
ejpam-4432	195	53	and	and	CCONJ
ejpam-4432	195	54	the	the	DET
ejpam-4432	195	55	meet	meet	ADJ
ejpam-4432	195	56	-	-	PUNCT
ejpam-4432	195	57	hyperlattic	hyperlattic	ADJ
ejpam-4432	195	58	l′.	l′.	NOUN
ejpam-4432	195	59	define	define	VERB
ejpam-4432	195	60	a	a	DET
ejpam-4432	195	61	homomorphism	homomorphism	NOUN
ejpam-4432	195	62	f	f	X
ejpam-4432	195	63	:	:	PUNCT
ejpam-4432	196	1	l	l	X
ejpam-4432	196	2	→	→	SYM
ejpam-4432	196	3	l′	l′	X
ejpam-4432	196	4	as	as	ADP
ejpam-4432	196	5	:	:	PUNCT
ejpam-4432	196	6	f(0	f(0	NOUN
ejpam-4432	196	7	)	)	PUNCT
ejpam-4432	196	8	=	=	SYM
ejpam-4432	196	9	0′	0′	NUM
ejpam-4432	196	10	,	,	PUNCT
ejpam-4432	196	11	f(β	f(β	PROPN
ejpam-4432	196	12	)	)	PUNCT
ejpam-4432	196	13	=	=	SYM
ejpam-4432	197	1	x	x	NOUN
ejpam-4432	197	2	,	,	PUNCT
ejpam-4432	197	3	f(α	f(α	NOUN
ejpam-4432	197	4	)	)	PUNCT
ejpam-4432	197	5	=	=	SYM
ejpam-4432	197	6	y	y	PROPN
ejpam-4432	197	7	,	,	PUNCT
ejpam-4432	197	8	f(γ	f(γ	NOUN
ejpam-4432	197	9	)	)	PUNCT
ejpam-4432	197	10	=	=	SYM
ejpam-4432	197	11	z	z	NOUN
ejpam-4432	197	12	and	and	CCONJ
ejpam-4432	197	13	f(δ	f(δ	PROPN
ejpam-4432	197	14	)	)	PUNCT
ejpam-4432	197	15	=	=	PUNCT
ejpam-4432	197	16	f(1	f(1	PROPN
ejpam-4432	197	17	)	)	PUNCT
ejpam-4432	197	18	=	=	PUNCT
ejpam-4432	198	1	1′.	1′.	NUM
ejpam-4432	198	2	proposition	proposition	NOUN
ejpam-4432	198	3	3	3	X
ejpam-4432	198	4	.	.	PUNCT
ejpam-4432	199	1	let	let	VERB
ejpam-4432	199	2	ϕ	ϕ	NOUN
ejpam-4432	199	3	:	:	PUNCT
ejpam-4432	199	4	l	l	X
ejpam-4432	199	5	→	→	SYM
ejpam-4432	199	6	l′	l′	AUX
ejpam-4432	199	7	be	be	AUX
ejpam-4432	199	8	a	a	DET
ejpam-4432	199	9	homomorphism	homomorphism	NOUN
ejpam-4432	199	10	between	between	ADP
ejpam-4432	199	11	meet	meet	NOUN
ejpam-4432	199	12	-	-	PUNCT
ejpam-4432	199	13	hyperlattices	hyperlattice	NOUN
ejpam-4432	199	14	.	.	PUNCT
ejpam-4432	200	1	then	then	ADV
ejpam-4432	200	2	:	:	PUNCT
ejpam-4432	200	3	(	(	PUNCT
ejpam-4432	200	4	i	i	NOUN
ejpam-4432	200	5	)	)	PUNCT
ejpam-4432	200	6	if	if	SCONJ
ejpam-4432	200	7	ϕ	ϕ	NOUN
ejpam-4432	200	8	is	be	AUX
ejpam-4432	200	9	onto	onto	ADP
ejpam-4432	200	10	and	and	CCONJ
ejpam-4432	200	11	i	i	PRON
ejpam-4432	200	12	is	be	AUX
ejpam-4432	200	13	aihyperideal	aihyperideal	ADJ
ejpam-4432	200	14	of	of	ADP
ejpam-4432	200	15	l	l	NOUN
ejpam-4432	200	16	,	,	PUNCT
ejpam-4432	200	17	then	then	ADV
ejpam-4432	200	18	ϕ(i	ϕ(i	NUM
ejpam-4432	200	19	)	)	PUNCT
ejpam-4432	200	20	is	be	AUX
ejpam-4432	200	21	a	a	DET
ejpam-4432	200	22	hyperideal	hyperideal	NOUN
ejpam-4432	200	23	of	of	ADP
ejpam-4432	200	24	l′	l′	NOUN
ejpam-4432	200	25	;	;	PUNCT
ejpam-4432	200	26	(	(	PUNCT
ejpam-4432	200	27	ii	ii	NOUN
ejpam-4432	200	28	)	)	PUNCT
ejpam-4432	200	29	if	if	SCONJ
ejpam-4432	200	30	j	j	PROPN
ejpam-4432	200	31	is	be	AUX
ejpam-4432	200	32	aihyperideal	aihyperideal	ADJ
ejpam-4432	200	33	of	of	ADP
ejpam-4432	200	34	l′	l′	NOUN
ejpam-4432	200	35	,	,	PUNCT
ejpam-4432	200	36	theniϕ−1(j	theniϕ−1(j	NOUN
ejpam-4432	200	37	)	)	PUNCT
ejpam-4432	200	38	is	be	AUX
ejpam-4432	200	39	a	a	DET
ejpam-4432	200	40	hyperideal	hyperideal	NOUN
ejpam-4432	200	41	of	of	ADP
ejpam-4432	200	42	l	l	NOUN
ejpam-4432	200	43	containing	contain	VERB
ejpam-4432	200	44	ker(ϕ	ker(ϕ	PROPN
ejpam-4432	200	45	)	)	PUNCT
ejpam-4432	200	46	;	;	PUNCT
ejpam-4432	200	47	(	(	PUNCT
ejpam-4432	200	48	iii	iii	X
ejpam-4432	200	49	)	)	PUNCT
ejpam-4432	200	50	if	if	SCONJ
ejpam-4432	200	51	a	a	PRON
ejpam-4432	200	52	is	be	AUX
ejpam-4432	200	53	ainonempty	ainonempty	NOUN
ejpam-4432	200	54	subset	subset	NOUN
ejpam-4432	200	55	of	of	ADP
ejpam-4432	200	56	l	l	NOUN
ejpam-4432	200	57	,	,	PUNCT
ejpam-4432	200	58	then	then	ADV
ejpam-4432	200	59	ϕ(ar	ϕ(ar	NUM
ejpam-4432	200	60	)	)	PUNCT
ejpam-4432	200	61	⊆	⊆	NUM
ejpam-4432	200	62	(	(	PUNCT
ejpam-4432	200	63	ϕ(a))r	ϕ(a))r	INTJ
ejpam-4432	200	64	.	.	PUNCT
ejpam-4432	200	65	proof	proof	NOUN
ejpam-4432	200	66	.	.	PUNCT
ejpam-4432	201	1	(	(	PUNCT
ejpam-4432	201	2	i	i	NOUN
ejpam-4432	201	3	)	)	PUNCT
ejpam-4432	201	4	let	let	VERB
ejpam-4432	201	5	x	x	PRON
ejpam-4432	201	6	,	,	PUNCT
ejpam-4432	201	7	y	y	PROPN
ejpam-4432	201	8	∈	∈	PROPN
ejpam-4432	201	9	ϕ(i	ϕ(i	PROPN
ejpam-4432	201	10	)	)	PUNCT
ejpam-4432	201	11	then	then	ADV
ejpam-4432	201	12	there	there	PRON
ejpam-4432	201	13	exist	exist	VERB
ejpam-4432	201	14	a	a	DET
ejpam-4432	201	15	,	,	PUNCT
ejpam-4432	201	16	b	b	X
ejpam-4432	201	17	∈	∈	NOUN
ejpam-4432	201	18	i	i	PRON
ejpam-4432	201	19	such	such	VERB
ejpam-4432	201	20	that	that	SCONJ
ejpam-4432	201	21	x	x	X
ejpam-4432	201	22	=	=	PUNCT
ejpam-4432	201	23	ϕ(a	ϕ(a	NOUN
ejpam-4432	201	24	)	)	PUNCT
ejpam-4432	201	25	and	and	CCONJ
ejpam-4432	201	26	y	y	PROPN
ejpam-4432	201	27	=	=	SYM
ejpam-4432	201	28	ϕ(b	ϕ(b	PROPN
ejpam-4432	201	29	)	)	PUNCT
ejpam-4432	201	30	.	.	PUNCT
ejpam-4432	202	1	then	then	ADV
ejpam-4432	202	2	ϕ(a	ϕ(a	VERB
ejpam-4432	202	3	∨	∨	NUM
ejpam-4432	202	4	b	b	NOUN
ejpam-4432	202	5	)	)	PUNCT
ejpam-4432	202	6	=	=	SYM
ejpam-4432	202	7	ϕ(a	ϕ(a	PROPN
ejpam-4432	202	8	)	)	PUNCT
ejpam-4432	202	9	∨	∨	NUM
ejpam-4432	202	10	ϕ(b	ϕ(b	PROPN
ejpam-4432	202	11	)	)	PUNCT
ejpam-4432	202	12	=	=	PUNCT
ejpam-4432	203	1	x	x	PUNCT
ejpam-4432	203	2	∨	∨	NUM
ejpam-4432	203	3	y	y	PROPN
ejpam-4432	203	4	⊆	⊆	NUM
ejpam-4432	203	5	ϕ(i	ϕ(i	NUM
ejpam-4432	203	6	)	)	PUNCT
ejpam-4432	203	7	.	.	PUNCT
ejpam-4432	204	1	now	now	ADV
ejpam-4432	204	2	,	,	PUNCT
ejpam-4432	204	3	suppose	suppose	VERB
ejpam-4432	204	4	x	x	PRON
ejpam-4432	204	5	,	,	PUNCT
ejpam-4432	204	6	y	y	PROPN
ejpam-4432	204	7	∈	∈	PROPN
ejpam-4432	204	8	l′	l′	PROPN
ejpam-4432	204	9	,	,	PUNCT
ejpam-4432	204	10	x	x	PROPN
ejpam-4432	204	11	∈	∈	PROPN
ejpam-4432	204	12	ϕ(i	ϕ(i	PROPN
ejpam-4432	204	13	)	)	PUNCT
ejpam-4432	204	14	and	and	CCONJ
ejpam-4432	204	15	y	y	PROPN
ejpam-4432	204	16	≤	≤	PROPN
ejpam-4432	204	17	x.	x.	NOUN
ejpam-4432	205	1	hence	hence	ADV
ejpam-4432	205	2	y	y	PROPN
ejpam-4432	205	3	=	=	SYM
ejpam-4432	205	4	ϕ(b	ϕ(b	PROPN
ejpam-4432	205	5	)	)	PUNCT
ejpam-4432	205	6	∈	∈	PROPN
ejpam-4432	205	7	ϕ(a)∧̄ϕ(b	ϕ(a)∧̄ϕ(b	PROPN
ejpam-4432	205	8	)	)	PUNCT
ejpam-4432	205	9	=	=	SYM
ejpam-4432	206	1	ϕ(a∧̄b	ϕ(a∧̄b	PROPN
ejpam-4432	206	2	)	)	PUNCT
ejpam-4432	206	3	,	,	PUNCT
ejpam-4432	206	4	which	which	PRON
ejpam-4432	206	5	indicates	indicate	VERB
ejpam-4432	206	6	that	that	SCONJ
ejpam-4432	206	7	b	b	PROPN
ejpam-4432	206	8	∈	∈	PROPN
ejpam-4432	206	9	a∧̄b	a∧̄b	PROPN
ejpam-4432	206	10	and	and	CCONJ
ejpam-4432	206	11	b	b	PROPN
ejpam-4432	206	12	≤	≤	NUM
ejpam-4432	206	13	a.	a.	NOUN
ejpam-4432	206	14	thus	thus	ADV
ejpam-4432	206	15	b	b	X
ejpam-4432	206	16	∈	∈	PROPN
ejpam-4432	207	1	i	i	PRON
ejpam-4432	207	2	and	and	CCONJ
ejpam-4432	207	3	y	y	PROPN
ejpam-4432	207	4	=	=	SYM
ejpam-4432	207	5	ϕ(b	ϕ(b	PROPN
ejpam-4432	207	6	)	)	PUNCT
ejpam-4432	207	7	∈	∈	PROPN
ejpam-4432	207	8	ϕ(i	ϕ(i	PROPN
ejpam-4432	207	9	)	)	PUNCT
ejpam-4432	207	10	.	.	PUNCT
ejpam-4432	208	1	consequently	consequently	ADV
ejpam-4432	208	2	ϕ(i	ϕ(i	NUM
ejpam-4432	208	3	)	)	PUNCT
ejpam-4432	208	4	is	be	AUX
ejpam-4432	208	5	a	a	DET
ejpam-4432	208	6	hyperideal	hyperideal	NOUN
ejpam-4432	208	7	.	.	PUNCT
ejpam-4432	209	1	(	(	PUNCT
ejpam-4432	209	2	ii	ii	NOUN
ejpam-4432	209	3	)	)	PUNCT
ejpam-4432	209	4	let	let	VERB
ejpam-4432	209	5	x	x	PRON
ejpam-4432	209	6	,	,	PUNCT
ejpam-4432	209	7	y	y	PROPN
ejpam-4432	209	8	∈	∈	PROPN
ejpam-4432	209	9	j	j	PROPN
ejpam-4432	209	10	.	.	PUNCT
ejpam-4432	210	1	then	then	ADV
ejpam-4432	210	2	thereiexist	thereiexist	VERB
ejpam-4432	210	3	a	a	PRON
ejpam-4432	210	4	,	,	PUNCT
ejpam-4432	210	5	b	b	PROPN
ejpam-4432	210	6	∈	∈	PROPN
ejpam-4432	210	7	l′	l′	VERB
ejpam-4432	210	8	such	such	ADJ
ejpam-4432	210	9	that	that	DET
ejpam-4432	210	10	ϕ−1(x	ϕ−1(x	NOUN
ejpam-4432	210	11	)	)	PUNCT
ejpam-4432	211	1	=	=	PUNCT
ejpam-4432	212	1	a	a	PRON
ejpam-4432	212	2	and	and	CCONJ
ejpam-4432	212	3	ϕ−1(y	ϕ−1(y	PROPN
ejpam-4432	212	4	)	)	PUNCT
ejpam-4432	213	1	=	=	PUNCT
ejpam-4432	213	2	b.	b.	PROPN
ejpam-4432	214	1	it	it	PRON
ejpam-4432	214	2	implies	imply	VERB
ejpam-4432	214	3	a	a	DET
ejpam-4432	214	4	∨	∨	NUM
ejpam-4432	214	5	b	b	NOUN
ejpam-4432	214	6	=	=	SYM
ejpam-4432	214	7	ϕ−1(x	ϕ−1(x	PROPN
ejpam-4432	214	8	)	)	PUNCT
ejpam-4432	214	9	∨	∨	NUM
ejpam-4432	214	10	ϕ−1(y	ϕ−1(y	PROPN
ejpam-4432	214	11	)	)	PUNCT
ejpam-4432	214	12	.	.	PUNCT
ejpam-4432	215	1	by	by	ADP
ejpam-4432	215	2	using	use	VERB
ejpam-4432	215	3	the	the	DET
ejpam-4432	215	4	effect	effect	NOUN
ejpam-4432	215	5	of	of	ADP
ejpam-4432	215	6	ϕ	ϕ	NOUN
ejpam-4432	215	7	on	on	ADP
ejpam-4432	215	8	both	both	DET
ejpam-4432	215	9	sides	side	NOUN
ejpam-4432	215	10	we	we	PRON
ejpam-4432	215	11	get	get	VERB
ejpam-4432	215	12	ϕ(a	ϕ(a	NOUN
ejpam-4432	215	13	∨	∨	NUM
ejpam-4432	215	14	b	b	NOUN
ejpam-4432	215	15	)	)	PUNCT
ejpam-4432	215	16	=	=	SYM
ejpam-4432	215	17	ϕ	ϕ	PROPN
ejpam-4432	215	18	(	(	PUNCT
ejpam-4432	215	19	ϕ−1(x	ϕ−1(x	PROPN
ejpam-4432	215	20	)	)	PUNCT
ejpam-4432	215	21	∨	∨	PROPN
ejpam-4432	215	22	ϕ−1(y	ϕ−1(y	PROPN
ejpam-4432	215	23	)	)	PUNCT
ejpam-4432	215	24	)	)	PUNCT
ejpam-4432	216	1	=	=	SYM
ejpam-4432	216	2	ϕ	ϕ	PROPN
ejpam-4432	216	3	(	(	PUNCT
ejpam-4432	216	4	ϕ−1(x	ϕ−1(x	PROPN
ejpam-4432	216	5	)	)	PUNCT
ejpam-4432	216	6	)	)	PUNCT
ejpam-4432	217	1	∨	∨	NUM
ejpam-4432	217	2	ϕ	ϕ	X
ejpam-4432	217	3	(	(	PUNCT
ejpam-4432	217	4	ϕ−1(y	ϕ−1(y	PROPN
ejpam-4432	217	5	)	)	PUNCT
ejpam-4432	217	6	)	)	PUNCT
ejpam-4432	218	1	=	=	PUNCT
ejpam-4432	219	1	x	x	SYM
ejpam-4432	219	2	∨	∨	NUM
ejpam-4432	219	3	y.	y.	NOUN
ejpam-4432	219	4	thus	thus	ADV
ejpam-4432	219	5	ϕ−1	ϕ−1	PROPN
ejpam-4432	219	6	(	(	PUNCT
ejpam-4432	219	7	ϕ(a	ϕ(a	NOUN
ejpam-4432	219	8	∨	∨	NUM
ejpam-4432	219	9	b	b	NOUN
ejpam-4432	219	10	)	)	PUNCT
ejpam-4432	219	11	)	)	PUNCT
ejpam-4432	220	1	=	=	PUNCT
ejpam-4432	220	2	a	a	DET
ejpam-4432	220	3	∨	∨	NUM
ejpam-4432	220	4	b	b	NOUN
ejpam-4432	220	5	=	=	SYM
ejpam-4432	220	6	ϕ−1(x	ϕ−1(x	NOUN
ejpam-4432	220	7	∨	∨	NUM
ejpam-4432	220	8	y	y	NOUN
ejpam-4432	220	9	)	)	PUNCT
ejpam-4432	220	10	∈	∈	PROPN
ejpam-4432	220	11	ϕ−1(j	ϕ−1(j	PROPN
ejpam-4432	220	12	)	)	PUNCT
ejpam-4432	220	13	.	.	PUNCT
ejpam-4432	221	1	let	let	VERB
ejpam-4432	221	2	x	x	SYM
ejpam-4432	221	3	∈	∈	PROPN
ejpam-4432	221	4	j	j	PROPN
ejpam-4432	221	5	,	,	PUNCT
ejpam-4432	221	6	ϕ−1(x	ϕ−1(x	PROPN
ejpam-4432	221	7	)	)	PUNCT
ejpam-4432	221	8	=	=	PUNCT
ejpam-4432	222	1	a	a	PRON
ejpam-4432	222	2	and	and	CCONJ
ejpam-4432	222	3	y	y	PROPN
ejpam-4432	222	4	∈	∈	PROPN
ejpam-4432	222	5	l	l	NOUN
ejpam-4432	222	6	such	such	ADJ
ejpam-4432	222	7	that	that	SCONJ
ejpam-4432	222	8	y	y	PROPN
ejpam-4432	222	9	≤	≤	VERB
ejpam-4432	222	10	a	a	DET
ejpam-4432	222	11	=	=	SYM
ejpam-4432	222	12	ϕ−1(x	ϕ−1(x	NOUN
ejpam-4432	222	13	)	)	PUNCT
ejpam-4432	222	14	.	.	PUNCT
ejpam-4432	223	1	then	then	ADV
ejpam-4432	223	2	y	y	PROPN
ejpam-4432	223	3	∨	∨	NUM
ejpam-4432	223	4	a	a	PROPN
ejpam-4432	223	5	=	=	X
ejpam-4432	223	6	a	a	PRON
ejpam-4432	223	7	which	which	PRON
ejpam-4432	223	8	implies	imply	VERB
ejpam-4432	223	9	ϕ(y	ϕ(y	PROPN
ejpam-4432	223	10	)	)	PUNCT
ejpam-4432	223	11	∨	∨	NUM
ejpam-4432	223	12	ϕ(a	ϕ(a	NOUN
ejpam-4432	223	13	)	)	PUNCT
ejpam-4432	223	14	=	=	SYM
ejpam-4432	223	15	ϕ(y	ϕ(y	PROPN
ejpam-4432	223	16	∨	∨	NUM
ejpam-4432	223	17	a	a	PRON
ejpam-4432	223	18	)	)	PUNCT
ejpam-4432	223	19	=	=	SYM
ejpam-4432	223	20	ϕ(a	ϕ(a	NOUN
ejpam-4432	223	21	)	)	PUNCT
ejpam-4432	223	22	.	.	PUNCT
ejpam-4432	224	1	thus	thus	ADV
ejpam-4432	224	2	ϕ(y	ϕ(y	NUM
ejpam-4432	224	3	)	)	PUNCT
ejpam-4432	224	4	≤	≤	NUM
ejpam-4432	224	5	ϕ(a	ϕ(a	NOUN
ejpam-4432	224	6	)	)	PUNCT
ejpam-4432	225	1	=	=	SYM
ejpam-4432	225	2	ϕ	ϕ	X
ejpam-4432	225	3	(	(	PUNCT
ejpam-4432	225	4	ϕ−1(x	ϕ−1(x	PROPN
ejpam-4432	225	5	)	)	PUNCT
ejpam-4432	225	6	)	)	PUNCT
ejpam-4432	226	1	=	=	PUNCT
ejpam-4432	226	2	x.	x.	NOUN
ejpam-4432	226	3	therefore	therefore	ADV
ejpam-4432	226	4	ϕ(y	ϕ(y	PROPN
ejpam-4432	226	5	)	)	PUNCT
ejpam-4432	227	1	∈	∈	PROPN
ejpam-4432	227	2	j	j	PROPN
ejpam-4432	227	3	and	and	CCONJ
ejpam-4432	227	4	y	y	PROPN
ejpam-4432	227	5	∈	∈	PROPN
ejpam-4432	227	6	ϕ−1(j	ϕ−1(j	PROPN
ejpam-4432	227	7	)	)	PUNCT
ejpam-4432	227	8	.	.	PUNCT
ejpam-4432	228	1	clearly	clearly	ADV
ejpam-4432	228	2	,	,	PUNCT
ejpam-4432	228	3	0′	0′	PROPN
ejpam-4432	228	4	∈	∈	PROPN
ejpam-4432	228	5	j	j	NOUN
ejpam-4432	228	6	which	which	PRON
ejpam-4432	228	7	means	mean	VERB
ejpam-4432	228	8	ker(ϕ	ker(ϕ	PROPN
ejpam-4432	228	9	)	)	PUNCT
ejpam-4432	228	10	=	=	SYM
ejpam-4432	228	11	ϕ−1(0′	ϕ−1(0′	PROPN
ejpam-4432	228	12	)	)	PUNCT
ejpam-4432	228	13	⊆	⊆	NUM
ejpam-4432	228	14	ϕ−1(j	ϕ−1(j	NOUN
ejpam-4432	228	15	)	)	PUNCT
ejpam-4432	228	16	.	.	PUNCT
ejpam-4432	229	1	e.g.	e.g.	ADV
ejpam-4432	229	2	rezk	rezk	PROPN
ejpam-4432	229	3	,	,	PUNCT
ejpam-4432	229	4	n.h	n.h	PROPN
ejpam-4432	229	5	.	.	PROPN
ejpam-4432	229	6	abughazalah	abughazalah	PROPN
ejpam-4432	229	7	/	/	SYM
ejpam-4432	229	8	eur	eur	PROPN
ejpam-4432	229	9	.	.	PUNCT
ejpam-4432	230	1	j.	j.	PROPN
ejpam-4432	230	2	pure	pure	PROPN
ejpam-4432	230	3	appl	appl	PROPN
ejpam-4432	230	4	.	.	PROPN
ejpam-4432	230	5	math	math	PROPN
ejpam-4432	230	6	,	,	PUNCT
ejpam-4432	230	7	15	15	NUM
ejpam-4432	230	8	(	(	PUNCT
ejpam-4432	230	9	3	3	NUM
ejpam-4432	230	10	)	)	PUNCT
ejpam-4432	230	11	(	(	PUNCT
ejpam-4432	230	12	2022	2022	NUM
ejpam-4432	230	13	)	)	PUNCT
ejpam-4432	230	14	,	,	PUNCT
ejpam-4432	230	15	1402	1402	NUM
ejpam-4432	230	16	-	-	SYM
ejpam-4432	230	17	1416	1416	NUM
ejpam-4432	230	18	1411	1411	NUM
ejpam-4432	230	19	(	(	PUNCT
ejpam-4432	230	20	iii	iii	NOUN
ejpam-4432	230	21	)	)	PUNCT
ejpam-4432	230	22	let	let	VERB
ejpam-4432	230	23	x	x	SYM
ejpam-4432	230	24	∈	∈	PROPN
ejpam-4432	230	25	ar	ar	PROPN
ejpam-4432	230	26	and	and	CCONJ
ejpam-4432	230	27	ϕ(x	ϕ(x	NOUN
ejpam-4432	230	28	)	)	PUNCT
ejpam-4432	230	29	=	=	SYM
ejpam-4432	230	30	b.	b.	PROPN
ejpam-4432	230	31	then	then	ADV
ejpam-4432	230	32	0′	0′	X
ejpam-4432	231	1	=	=	SYM
ejpam-4432	231	2	ϕ(0	ϕ(0	PROPN
ejpam-4432	231	3	)	)	PUNCT
ejpam-4432	231	4	∈	∈	PROPN
ejpam-4432	231	5	ϕ(x	ϕ(x	X
ejpam-4432	231	6	∧	∧	PROPN
ejpam-4432	231	7	a	a	NOUN
ejpam-4432	231	8	)	)	PUNCT
ejpam-4432	231	9	=	=	SYM
ejpam-4432	231	10	ϕ(x)∧̄ϕ(a	ϕ(x)∧̄ϕ(a	NOUN
ejpam-4432	231	11	)	)	PUNCT
ejpam-4432	231	12	for	for	ADP
ejpam-4432	231	13	all	all	DET
ejpam-4432	231	14	a	a	DET
ejpam-4432	231	15	∈	∈	NOUN
ejpam-4432	231	16	a.	a.	NOUN
ejpam-4432	231	17	it	it	PRON
ejpam-4432	231	18	means	mean	VERB
ejpam-4432	231	19	that	that	SCONJ
ejpam-4432	231	20	0′	0′	NUM
ejpam-4432	231	21	∈	∈	PROPN
ejpam-4432	231	22	ϕ(x)∧̄ϕ(a	ϕ(x)∧̄ϕ(a	NOUN
ejpam-4432	231	23	)	)	PUNCT
ejpam-4432	231	24	for	for	ADP
ejpam-4432	231	25	all	all	DET
ejpam-4432	231	26	ϕ(a	ϕ(a	NOUN
ejpam-4432	231	27	)	)	PUNCT
ejpam-4432	231	28	∈	∈	PROPN
ejpam-4432	231	29	ϕ(a	ϕ(a	PROPN
ejpam-4432	231	30	)	)	PUNCT
ejpam-4432	231	31	i.e.	i.e.	X
ejpam-4432	231	32	,	,	PUNCT
ejpam-4432	231	33	ϕ(x	ϕ(x	X
ejpam-4432	231	34	)	)	PUNCT
ejpam-4432	231	35	∈	∈	PROPN
ejpam-4432	231	36	(	(	PUNCT
ejpam-4432	231	37	ϕ(a	ϕ(a	NOUN
ejpam-4432	231	38	)	)	PUNCT
ejpam-4432	231	39	)	)	PUNCT
ejpam-4432	232	1	r	r	NOUN
ejpam-4432	232	2	.	.	PUNCT
ejpam-4432	233	1	therefore	therefore	ADV
ejpam-4432	233	2	ϕ(ar	ϕ(ar	NOUN
ejpam-4432	233	3	)	)	PUNCT
ejpam-4432	233	4	⊆	⊆	NUM
ejpam-4432	233	5	(	(	PUNCT
ejpam-4432	233	6	ϕ(a	ϕ(a	NOUN
ejpam-4432	233	7	)	)	PUNCT
ejpam-4432	233	8	)	)	PUNCT
ejpam-4432	234	1	r	r	NOUN
ejpam-4432	234	2	.	.	PUNCT
ejpam-4432	235	1	definition	definition	NOUN
ejpam-4432	235	2	6	6	NUM
ejpam-4432	235	3	.	.	PUNCT
ejpam-4432	236	1	let	let	VERB
ejpam-4432	236	2	ϕ	ϕ	NOUN
ejpam-4432	236	3	:	:	PUNCT
ejpam-4432	236	4	l	l	X
ejpam-4432	236	5	→	→	SYM
ejpam-4432	236	6	l′	l′	AUX
ejpam-4432	236	7	be	be	AUX
ejpam-4432	236	8	a	a	DET
ejpam-4432	236	9	homomorphism	homomorphism	NOUN
ejpam-4432	236	10	.	.	PUNCT
ejpam-4432	237	1	then	then	ADV
ejpam-4432	237	2	ϕ	ϕ	PROPN
ejpam-4432	237	3	is	be	AUX
ejpam-4432	237	4	called	call	VERB
ejpam-4432	237	5	annihilator	annihilator	PROPN
ejpam-4432	237	6	hyperideal	hyperideal	NOUN
ejpam-4432	237	7	preserving	preserve	VERB
ejpam-4432	237	8	if	if	SCONJ
ejpam-4432	237	9	for	for	ADP
ejpam-4432	237	10	any	any	DET
ejpam-4432	237	11	subset	subset	NOUN
ejpam-4432	237	12	a	a	PRON
ejpam-4432	237	13	of	of	ADP
ejpam-4432	237	14	l	l	NOUN
ejpam-4432	237	15	:	:	PUNCT
ejpam-4432	237	16	ϕ(ar	ϕ(ar	NUM
ejpam-4432	237	17	)	)	PUNCT
ejpam-4432	237	18	=	=	PUNCT
ejpam-4432	238	1	[	[	X
ejpam-4432	238	2	ϕ(a)]r	ϕ(a)]r	X
ejpam-4432	238	3	.	.	PUNCT
ejpam-4432	239	1	the	the	DET
ejpam-4432	239	2	homomrphism	homomrphism	NOUN
ejpam-4432	239	3	f	f	X
ejpam-4432	239	4	in	in	ADP
ejpam-4432	239	5	example	example	NOUN
ejpam-4432	239	6	2	2	NUM
ejpam-4432	239	7	is	be	AUX
ejpam-4432	239	8	not	not	PART
ejpam-4432	239	9	an	an	DET
ejpam-4432	239	10	annihilator	annihilator	PROPN
ejpam-4432	239	11	hyperideal	hyperideal	NOUN
ejpam-4432	239	12	preserving	preserving	NOUN
ejpam-4432	239	13	.	.	PUNCT
ejpam-4432	240	1	example	example	NOUN
ejpam-4432	241	1	3	3	NUM
ejpam-4432	241	2	.	.	PUNCT
ejpam-4432	241	3	tables	table	NOUN
ejpam-4432	241	4	5	5	NUM
ejpam-4432	241	5	and	and	CCONJ
ejpam-4432	241	6	6	6	NUM
ejpam-4432	241	7	represent	represent	VERB
ejpam-4432	241	8	the	the	DET
ejpam-4432	241	9	hyperoperation	hyperoperation	NOUN
ejpam-4432	241	10	∧̄	∧̄	PROPN
ejpam-4432	241	11	and	and	CCONJ
ejpam-4432	241	12	operation	operation	NOUN
ejpam-4432	241	13	∨	∨	NOUN
ejpam-4432	241	14	of	of	ADP
ejpam-4432	241	15	meethyberlattice	meethyberlattice	NOUN
ejpam-4432	241	16	l′′	l′′	NOUN
ejpam-4432	241	17	=	=	SYM
ejpam-4432	241	18	{	{	PUNCT
ejpam-4432	241	19	0	0	NUM
ejpam-4432	241	20	,	,	PUNCT
ejpam-4432	241	21	α	α	X
ejpam-4432	241	22	,	,	PUNCT
ejpam-4432	241	23	β	β	X
ejpam-4432	241	24	,	,	PUNCT
ejpam-4432	241	25	γ	γ	PROPN
ejpam-4432	241	26	,	,	PUNCT
ejpam-4432	241	27	δ	δ	PROPN
ejpam-4432	241	28	,	,	PUNCT
ejpam-4432	241	29	1	1	NUM
ejpam-4432	241	30	}	}	PUNCT
ejpam-4432	241	31	.	.	PUNCT
ejpam-4432	242	1	∧̄	∧̄	NOUN
ejpam-4432	242	2	0	0	NUM
ejpam-4432	243	1	α	α	NOUN
ejpam-4432	243	2	β	β	X
ejpam-4432	243	3	γ	γ	X
ejpam-4432	243	4	δ	δ	PROPN
ejpam-4432	243	5	1	1	NUM
ejpam-4432	243	6	0	0	NUM
ejpam-4432	243	7	{	{	PUNCT
ejpam-4432	243	8	0	0	NUM
ejpam-4432	243	9	}	}	PUNCT
ejpam-4432	243	10	{	{	PUNCT
ejpam-4432	243	11	0	0	NUM
ejpam-4432	243	12	}	}	PUNCT
ejpam-4432	243	13	{	{	PUNCT
ejpam-4432	243	14	0	0	NUM
ejpam-4432	243	15	}	}	PUNCT
ejpam-4432	243	16	{	{	PUNCT
ejpam-4432	243	17	0	0	NUM
ejpam-4432	243	18	}	}	PUNCT
ejpam-4432	243	19	{	{	PUNCT
ejpam-4432	243	20	0	0	NUM
ejpam-4432	243	21	}	}	PUNCT
ejpam-4432	243	22	{	{	PUNCT
ejpam-4432	243	23	0	0	NUM
ejpam-4432	243	24	}	}	PUNCT
ejpam-4432	243	25	α	α	NOUN
ejpam-4432	243	26	{	{	PUNCT
ejpam-4432	243	27	0	0	NUM
ejpam-4432	243	28	}	}	PUNCT
ejpam-4432	243	29	{	{	PUNCT
ejpam-4432	243	30	α	α	NOUN
ejpam-4432	243	31	}	}	PUNCT
ejpam-4432	243	32	{	{	PUNCT
ejpam-4432	243	33	0	0	NUM
ejpam-4432	243	34	}	}	PUNCT
ejpam-4432	243	35	{	{	PUNCT
ejpam-4432	243	36	α	α	NOUN
ejpam-4432	243	37	}	}	PUNCT
ejpam-4432	243	38	{	{	PUNCT
ejpam-4432	243	39	α	α	NOUN
ejpam-4432	243	40	}	}	PUNCT
ejpam-4432	243	41	{	{	PUNCT
ejpam-4432	243	42	α	α	NOUN
ejpam-4432	243	43	}	}	PUNCT
ejpam-4432	243	44	β	β	X
ejpam-4432	243	45	{	{	PUNCT
ejpam-4432	243	46	0	0	NUM
ejpam-4432	243	47	}	}	PUNCT
ejpam-4432	243	48	{	{	PUNCT
ejpam-4432	243	49	0	0	NUM
ejpam-4432	243	50	}	}	PUNCT
ejpam-4432	243	51	{	{	PUNCT
ejpam-4432	243	52	β	β	X
ejpam-4432	243	53	}	}	PUNCT
ejpam-4432	243	54	{	{	PUNCT
ejpam-4432	243	55	β	β	NOUN
ejpam-4432	243	56	}	}	PUNCT
ejpam-4432	243	57	{	{	PUNCT
ejpam-4432	243	58	0	0	NUM
ejpam-4432	243	59	}	}	PUNCT
ejpam-4432	243	60	{	{	PUNCT
ejpam-4432	243	61	β	β	NOUN
ejpam-4432	243	62	}	}	PUNCT
ejpam-4432	243	63	γ	γ	X
ejpam-4432	243	64	{	{	PUNCT
ejpam-4432	243	65	0	0	NUM
ejpam-4432	243	66	}	}	PUNCT
ejpam-4432	243	67	{	{	PUNCT
ejpam-4432	243	68	α	α	NOUN
ejpam-4432	243	69	}	}	PUNCT
ejpam-4432	243	70	{	{	PUNCT
ejpam-4432	243	71	β	β	NOUN
ejpam-4432	243	72	}	}	PUNCT
ejpam-4432	243	73	{	{	PUNCT
ejpam-4432	243	74	γ	γ	X
ejpam-4432	243	75	}	}	PUNCT
ejpam-4432	243	76	{	{	PUNCT
ejpam-4432	243	77	α	α	NOUN
ejpam-4432	243	78	}	}	PUNCT
ejpam-4432	243	79	{	{	PUNCT
ejpam-4432	243	80	γ	γ	PROPN
ejpam-4432	243	81	}	}	PUNCT
ejpam-4432	243	82	δ	δ	PROPN
ejpam-4432	243	83	{	{	PUNCT
ejpam-4432	243	84	0	0	NUM
ejpam-4432	243	85	}	}	PUNCT
ejpam-4432	243	86	{	{	PUNCT
ejpam-4432	243	87	α	α	NOUN
ejpam-4432	243	88	}	}	PUNCT
ejpam-4432	243	89	{	{	PUNCT
ejpam-4432	243	90	0	0	NUM
ejpam-4432	243	91	}	}	PUNCT
ejpam-4432	243	92	{	{	PUNCT
ejpam-4432	243	93	α	α	NOUN
ejpam-4432	243	94	}	}	PUNCT
ejpam-4432	243	95	{	{	PUNCT
ejpam-4432	243	96	δ	δ	PROPN
ejpam-4432	243	97	}	}	PUNCT
ejpam-4432	243	98	{	{	PUNCT
ejpam-4432	243	99	δ	δ	PROPN
ejpam-4432	243	100	}	}	PUNCT
ejpam-4432	243	101	1	1	NUM
ejpam-4432	243	102	{	{	PUNCT
ejpam-4432	243	103	0	0	NUM
ejpam-4432	243	104	}	}	PUNCT
ejpam-4432	243	105	{	{	PUNCT
ejpam-4432	243	106	α	α	NOUN
ejpam-4432	243	107	}	}	PUNCT
ejpam-4432	243	108	{	{	PUNCT
ejpam-4432	243	109	β	β	NOUN
ejpam-4432	243	110	}	}	PUNCT
ejpam-4432	243	111	{	{	PUNCT
ejpam-4432	243	112	γ	γ	X
ejpam-4432	243	113	}	}	PUNCT
ejpam-4432	243	114	{	{	PUNCT
ejpam-4432	243	115	δ	δ	NOUN
ejpam-4432	243	116	}	}	PUNCT
ejpam-4432	243	117	{	{	PUNCT
ejpam-4432	243	118	1	1	NUM
ejpam-4432	243	119	}	}	PUNCT
ejpam-4432	243	120	∨	∨	NUM
ejpam-4432	243	121	0	0	NUM
ejpam-4432	244	1	α	α	NOUN
ejpam-4432	244	2	β	β	X
ejpam-4432	244	3	γ	γ	X
ejpam-4432	244	4	δ	δ	PROPN
ejpam-4432	244	5	1	1	NUM
ejpam-4432	244	6	0	0	NUM
ejpam-4432	244	7	0	0	NUM
ejpam-4432	245	1	α	α	PRON
ejpam-4432	245	2	β	β	X
ejpam-4432	245	3	γ	γ	X
ejpam-4432	245	4	δ	δ	PROPN
ejpam-4432	245	5	1	1	NUM
ejpam-4432	245	6	α	α	NOUN
ejpam-4432	245	7	α	α	NOUN
ejpam-4432	245	8	α	α	NOUN
ejpam-4432	245	9	γ	γ	PROPN
ejpam-4432	245	10	γ	γ	X
ejpam-4432	245	11	δ	δ	PROPN
ejpam-4432	245	12	1	1	NUM
ejpam-4432	245	13	β	β	X
ejpam-4432	245	14	β	β	X
ejpam-4432	245	15	γ	γ	X
ejpam-4432	245	16	β	β	X
ejpam-4432	245	17	γ	γ	X
ejpam-4432	245	18	1	1	NUM
ejpam-4432	245	19	1	1	NUM
ejpam-4432	245	20	γ	γ	PROPN
ejpam-4432	245	21	γ	γ	X
ejpam-4432	245	22	γ	γ	X
ejpam-4432	245	23	γ	γ	X
ejpam-4432	245	24	γ	γ	X
ejpam-4432	245	25	1	1	NUM
ejpam-4432	245	26	1	1	NUM
ejpam-4432	245	27	δ	δ	PROPN
ejpam-4432	245	28	δ	δ	NOUN
ejpam-4432	245	29	0	0	NUM
ejpam-4432	245	30	1	1	NUM
ejpam-4432	245	31	1	1	NUM
ejpam-4432	245	32	δ	δ	NOUN
ejpam-4432	245	33	1	1	NUM
ejpam-4432	245	34	1	1	NUM
ejpam-4432	245	35	1	1	NUM
ejpam-4432	245	36	1	1	NUM
ejpam-4432	245	37	1	1	NUM
ejpam-4432	245	38	1	1	NUM
ejpam-4432	245	39	1	1	NUM
ejpam-4432	245	40	1	1	NUM
ejpam-4432	245	41	table	table	NOUN
ejpam-4432	245	42	5	5	NUM
ejpam-4432	245	43	:	:	PUNCT
ejpam-4432	245	44	represents	represent	VERB
ejpam-4432	245	45	the	the	DET
ejpam-4432	245	46	hyperoperation	hyperoperation	NOUN
ejpam-4432	245	47	table	table	NOUN
ejpam-4432	245	48	6	6	NUM
ejpam-4432	245	49	:	:	PUNCT
ejpam-4432	245	50	represents	represent	VERB
ejpam-4432	245	51	operation	operation	NOUN
ejpam-4432	245	52	∧̄	∧̄	PROPN
ejpam-4432	245	53	of	of	ADP
ejpam-4432	245	54	the	the	DET
ejpam-4432	245	55	meet	meet	ADJ
ejpam-4432	245	56	-	-	PUNCT
ejpam-4432	245	57	hyberlattice	hyberlattice	NOUN
ejpam-4432	245	58	l′′	l′′	NOUN
ejpam-4432	245	59	∨	∨	NUM
ejpam-4432	245	60	of	of	ADP
ejpam-4432	245	61	the	the	DET
ejpam-4432	245	62	meet	meet	ADJ
ejpam-4432	245	63	-	-	PUNCT
ejpam-4432	245	64	hyberlattice	hyberlattice	NOUN
ejpam-4432	245	65	l′′	l′′	NOUN
ejpam-4432	245	66	define	define	VERB
ejpam-4432	245	67	a	a	DET
ejpam-4432	245	68	homomorphism	homomorphism	NOUN
ejpam-4432	245	69	f	f	X
ejpam-4432	245	70	:	:	PUNCT
ejpam-4432	245	71	l′′	l′′	PROPN
ejpam-4432	245	72	→	→	SYM
ejpam-4432	245	73	l′	l′	X
ejpam-4432	245	74	as	as	ADP
ejpam-4432	245	75	:	:	PUNCT
ejpam-4432	245	76	g(0	g(0	NOUN
ejpam-4432	245	77	)	)	PUNCT
ejpam-4432	245	78	=	=	SYM
ejpam-4432	245	79	0′	0′	PROPN
ejpam-4432	245	80	,	,	PUNCT
ejpam-4432	245	81	g(β	g(β	NOUN
ejpam-4432	245	82	)	)	PUNCT
ejpam-4432	245	83	=	=	SYM
ejpam-4432	245	84	g(δ	g(δ	PROPN
ejpam-4432	245	85	)	)	PUNCT
ejpam-4432	245	86	=	=	SYM
ejpam-4432	245	87	y	y	PROPN
ejpam-4432	245	88	,	,	PUNCT
ejpam-4432	245	89	g(α	g(α	PROPN
ejpam-4432	245	90	)	)	PUNCT
ejpam-4432	245	91	=	=	SYM
ejpam-4432	245	92	x	x	NOUN
ejpam-4432	245	93	,	,	PUNCT
ejpam-4432	245	94	g(γ	g(γ	PROPN
ejpam-4432	245	95	)	)	PUNCT
ejpam-4432	245	96	=	=	SYM
ejpam-4432	245	97	z	z	NOUN
ejpam-4432	245	98	and	and	CCONJ
ejpam-4432	245	99	g(1	g(1	PROPN
ejpam-4432	245	100	)	)	PUNCT
ejpam-4432	246	1	=	=	PUNCT
ejpam-4432	246	2	1′.	1′.	NUM
ejpam-4432	246	3	where	where	SCONJ
ejpam-4432	246	4	l′	l′	NOUN
ejpam-4432	246	5	is	be	AUX
ejpam-4432	246	6	a	a	DET
ejpam-4432	246	7	meet	meet	ADJ
ejpam-4432	246	8	-	-	PUNCT
ejpam-4432	246	9	hyperlattice	hyperlattice	NOUN
ejpam-4432	246	10	in	in	ADP
ejpam-4432	246	11	example	example	NOUN
ejpam-4432	247	1	2	2	NUM
ejpam-4432	247	2	.	.	X
ejpam-4432	247	3	f	f	PROPN
ejpam-4432	247	4	is	be	AUX
ejpam-4432	247	5	an	an	DET
ejpam-4432	247	6	annihilator	annihilator	PROPN
ejpam-4432	247	7	hyperideal	hyperideal	NOUN
ejpam-4432	247	8	preserving	preserving	NOUN
ejpam-4432	247	9	.	.	PUNCT
ejpam-4432	248	1	theorem	theorem	ADJ
ejpam-4432	248	2	4	4	NUM
ejpam-4432	248	3	.	.	PUNCT
ejpam-4432	249	1	let	let	VERB
ejpam-4432	249	2	l	l	NOUN
ejpam-4432	249	3	and	and	CCONJ
ejpam-4432	249	4	l′	l′	VERB
ejpam-4432	249	5	be	be	AUX
ejpam-4432	249	6	two	two	NUM
ejpam-4432	249	7	meet	meet	ADJ
ejpam-4432	249	8	-	-	PUNCT
ejpam-4432	249	9	hyperlattice	hyperlattice	NOUN
ejpam-4432	249	10	,	,	PUNCT
ejpam-4432	249	11	ϕ	ϕ	NOUN
ejpam-4432	249	12	:	:	PUNCT
ejpam-4432	249	13	l	l	X
ejpam-4432	249	14	→	→	SYM
ejpam-4432	249	15	l′	l′	AUX
ejpam-4432	249	16	be	be	AUX
ejpam-4432	249	17	a	a	DET
ejpam-4432	249	18	homomorphism	homomorphism	NOUN
ejpam-4432	249	19	and	and	CCONJ
ejpam-4432	249	20	ker(ϕ	ker(ϕ	PROPN
ejpam-4432	249	21	)	)	PUNCT
ejpam-4432	249	22	=	=	PRON
ejpam-4432	249	23	{	{	PUNCT
ejpam-4432	249	24	0	0	NUM
ejpam-4432	249	25	}	}	PUNCT
ejpam-4432	249	26	.	.	PUNCT
ejpam-4432	250	1	then	then	ADV
ejpam-4432	250	2	:	:	PUNCT
ejpam-4432	250	3	(	(	PUNCT
ejpam-4432	250	4	i	i	NOUN
ejpam-4432	250	5	)	)	PUNCT
ejpam-4432	250	6	if	if	SCONJ
ejpam-4432	250	7	ϕ	ϕ	NOUN
ejpam-4432	250	8	is	be	AUX
ejpam-4432	250	9	onto	onto	ADP
ejpam-4432	250	10	then	then	ADV
ejpam-4432	250	11	:	:	PUNCT
ejpam-4432	250	12	(	(	PUNCT
ejpam-4432	250	13	a	a	X
ejpam-4432	250	14	)	)	PUNCT
ejpam-4432	250	15	ϕ	ϕ	NOUN
ejpam-4432	250	16	is	be	AUX
ejpam-4432	250	17	annihilator	annihilator	PROPN
ejpam-4432	250	18	hyperideal	hyperideal	NOUN
ejpam-4432	250	19	preserving	preserving	NOUN
ejpam-4432	250	20	;	;	PUNCT
ejpam-4432	250	21	(	(	PUNCT
ejpam-4432	250	22	b	b	X
ejpam-4432	250	23	)	)	PUNCT
ejpam-4432	250	24	for	for	ADP
ejpam-4432	250	25	any	any	DET
ejpam-4432	250	26	nonemptyisubsets	nonemptyisubset	NOUN
ejpam-4432	250	27	a	a	DET
ejpam-4432	250	28	and	and	CCONJ
ejpam-4432	250	29	b	b	NOUN
ejpam-4432	250	30	of	of	ADP
ejpam-4432	250	31	l	l	PROPN
ejpam-4432	250	32	ar	ar	PROPN
ejpam-4432	250	33	=	=	NOUN
ejpam-4432	250	34	br	br	NOUN
ejpam-4432	250	35	if	if	SCONJ
ejpam-4432	250	36	andionly	andionly	ADV
ejpam-4432	250	37	if	if	SCONJ
ejpam-4432	250	38	[	[	X
ejpam-4432	250	39	ϕ(a)]r	ϕ(a)]r	X
ejpam-4432	250	40	=	=	PUNCT
ejpam-4432	251	1	[	[	X
ejpam-4432	251	2	ϕ(b)]r	ϕ(b)]r	X
ejpam-4432	251	3	.	.	PUNCT
ejpam-4432	251	4	(	(	PUNCT
ejpam-4432	251	5	ii	ii	NOUN
ejpam-4432	251	6	)	)	PUNCT
ejpam-4432	251	7	ϕ−1	ϕ−1	PROPN
ejpam-4432	251	8	is	be	AUX
ejpam-4432	251	9	annihilator	annihilator	PROPN
ejpam-4432	251	10	hyperideal	hyperideal	NOUN
ejpam-4432	251	11	preserving	preserve	VERB
ejpam-4432	251	12	.	.	PUNCT
ejpam-4432	252	1	proof	proof	NOUN
ejpam-4432	252	2	.	.	PUNCT
ejpam-4432	253	1	(	(	PUNCT
ejpam-4432	253	2	i	i	NOUN
ejpam-4432	253	3	)	)	PUNCT
ejpam-4432	253	4	(	(	PUNCT
ejpam-4432	253	5	a	a	X
ejpam-4432	253	6	)	)	PUNCT
ejpam-4432	253	7	for	for	ADP
ejpam-4432	253	8	a	a	DET
ejpam-4432	253	9	inonempty	inonempty	NOUN
ejpam-4432	253	10	subset	subset	VERB
ejpam-4432	253	11	a	a	PRON
ejpam-4432	253	12	of	of	ADP
ejpam-4432	253	13	l	l	NOUN
ejpam-4432	253	14	,	,	PUNCT
ejpam-4432	253	15	we	we	PRON
ejpam-4432	253	16	have	have	VERB
ejpam-4432	253	17	ϕ(ar	ϕ(ar	NOUN
ejpam-4432	253	18	)	)	PUNCT
ejpam-4432	253	19	⊆	⊆	NUM
ejpam-4432	254	1	[	[	X
ejpam-4432	254	2	ϕ(a)]r	ϕ(a)]r	PROPN
ejpam-4432	254	3	,	,	PUNCT
ejpam-4432	254	4	from	from	ADP
ejpam-4432	254	5	proposition	proposition	NOUN
ejpam-4432	254	6	3	3	NUM
ejpam-4432	254	7	.	.	PUNCT
ejpam-4432	255	1	so	so	ADV
ejpam-4432	255	2	we	we	PRON
ejpam-4432	255	3	just	just	ADV
ejpam-4432	255	4	need	need	VERB
ejpam-4432	255	5	to	to	PART
ejpam-4432	255	6	prove	prove	VERB
ejpam-4432	255	7	that	that	SCONJ
ejpam-4432	256	1	[	[	X
ejpam-4432	256	2	ϕ(a)]r	ϕ(a)]r	ADP
ejpam-4432	256	3	⊆	⊆	NUM
ejpam-4432	256	4	ϕ(ar	ϕ(ar	NOUN
ejpam-4432	256	5	)	)	PUNCT
ejpam-4432	256	6	.	.	PUNCT
ejpam-4432	257	1	to	to	PART
ejpam-4432	257	2	do	do	VERB
ejpam-4432	257	3	that	that	PRON
ejpam-4432	257	4	,	,	PUNCT
ejpam-4432	257	5	let	let	VERB
ejpam-4432	257	6	x	x	X
ejpam-4432	257	7	∈	∈	PROPN
ejpam-4432	258	1	[	[	X
ejpam-4432	258	2	ϕ(a)]r	ϕ(a)]r	NOUN
ejpam-4432	258	3	⊆	⊆	NUM
ejpam-4432	258	4	l′	l′	NOUN
ejpam-4432	258	5	then	then	ADV
ejpam-4432	258	6	there	there	PRON
ejpam-4432	258	7	is	be	VERB
ejpam-4432	258	8	a	a	DET
ejpam-4432	258	9	∈	∈	ADJ
ejpam-4432	258	10	l	l	NOUN
ejpam-4432	258	11	such	such	ADJ
ejpam-4432	258	12	that	that	SCONJ
ejpam-4432	258	13	ϕ(a	ϕ(a	NOUN
ejpam-4432	258	14	)	)	PUNCT
ejpam-4432	258	15	=	=	SYM
ejpam-4432	259	1	x	x	NOUN
ejpam-4432	259	2	,	,	PUNCT
ejpam-4432	259	3	but	but	CCONJ
ejpam-4432	259	4	0′	0′	NUM
ejpam-4432	259	5	∈	∈	PROPN
ejpam-4432	259	6	x∧̄ϕ(b	x∧̄ϕ(b	PUNCT
ejpam-4432	259	7	)	)	PUNCT
ejpam-4432	259	8	for	for	ADP
ejpam-4432	259	9	all	all	DET
ejpam-4432	259	10	b	b	PROPN
ejpam-4432	259	11	∈	∈	ADJ
ejpam-4432	259	12	a.	a.	NOUN
ejpam-4432	259	13	then	then	ADV
ejpam-4432	259	14	0	0	NUM
ejpam-4432	259	15	∈	∈	PROPN
ejpam-4432	259	16	a∧̄b	a∧̄b	PROPN
ejpam-4432	259	17	for	for	ADP
ejpam-4432	259	18	all	all	DET
ejpam-4432	259	19	b	b	PROPN
ejpam-4432	259	20	∈	∈	PROPN
ejpam-4432	259	21	a.	a.	NOUN
ejpam-4432	259	22	therefore	therefore	ADV
ejpam-4432	259	23	a	a	DET
ejpam-4432	259	24	∈	∈	PROPN
ejpam-4432	259	25	ar	ar	NOUN
ejpam-4432	259	26	and	and	CCONJ
ejpam-4432	259	27	then	then	ADV
ejpam-4432	259	28	x	x	X
ejpam-4432	259	29	=	=	SYM
ejpam-4432	259	30	ϕ(a	ϕ(a	NOUN
ejpam-4432	259	31	)	)	PUNCT
ejpam-4432	259	32	∈	∈	PROPN
ejpam-4432	259	33	ϕ(ar	ϕ(ar	NOUN
ejpam-4432	259	34	)	)	PUNCT
ejpam-4432	259	35	.	.	PUNCT
ejpam-4432	260	1	e.g.	e.g.	ADV
ejpam-4432	260	2	rezk	rezk	PROPN
ejpam-4432	260	3	,	,	PUNCT
ejpam-4432	260	4	n.h	n.h	PROPN
ejpam-4432	260	5	.	.	PROPN
ejpam-4432	260	6	abughazalah	abughazalah	PROPN
ejpam-4432	260	7	/	/	SYM
ejpam-4432	260	8	eur	eur	PROPN
ejpam-4432	260	9	.	.	PUNCT
ejpam-4432	261	1	j.	j.	PROPN
ejpam-4432	261	2	pure	pure	PROPN
ejpam-4432	261	3	appl	appl	PROPN
ejpam-4432	261	4	.	.	PROPN
ejpam-4432	261	5	math	math	PROPN
ejpam-4432	261	6	,	,	PUNCT
ejpam-4432	261	7	15	15	NUM
ejpam-4432	261	8	(	(	PUNCT
ejpam-4432	261	9	3	3	NUM
ejpam-4432	261	10	)	)	PUNCT
ejpam-4432	261	11	(	(	PUNCT
ejpam-4432	261	12	2022	2022	NUM
ejpam-4432	261	13	)	)	PUNCT
ejpam-4432	261	14	,	,	PUNCT
ejpam-4432	261	15	1402	1402	NUM
ejpam-4432	261	16	-	-	SYM
ejpam-4432	261	17	1416	1416	NUM
ejpam-4432	261	18	1412	1412	NUM
ejpam-4432	261	19	(	(	PUNCT
ejpam-4432	261	20	b	b	X
ejpam-4432	261	21	)	)	PUNCT
ejpam-4432	261	22	suppose	suppose	VERB
ejpam-4432	261	23	that	that	SCONJ
ejpam-4432	261	24	a	a	PRON
ejpam-4432	261	25	and	and	CCONJ
ejpam-4432	261	26	b	b	NOUN
ejpam-4432	261	27	are	be	AUX
ejpam-4432	261	28	nonemptyisubsets	nonemptyisubset	NOUN
ejpam-4432	261	29	of	of	ADP
ejpam-4432	261	30	l	l	NOUN
ejpam-4432	262	1	such	such	ADJ
ejpam-4432	262	2	that	that	DET
ejpam-4432	262	3	ar	ar	NOUN
ejpam-4432	262	4	=	=	SYM
ejpam-4432	262	5	br	br	PROPN
ejpam-4432	262	6	.	.	PUNCT
ejpam-4432	263	1	then	then	ADV
ejpam-4432	263	2	by	by	ADP
ejpam-4432	263	3	using	use	VERB
ejpam-4432	263	4	a	a	PRON
ejpam-4432	263	5	)	)	PUNCT
ejpam-4432	263	6	we	we	PRON
ejpam-4432	263	7	get	get	VERB
ejpam-4432	263	8	[	[	X
ejpam-4432	263	9	ϕ(a)]r	ϕ(a)]r	NOUN
ejpam-4432	263	10	=	=	SYM
ejpam-4432	263	11	ϕ(ar	ϕ(ar	NOUN
ejpam-4432	263	12	)	)	PUNCT
ejpam-4432	263	13	=	=	SYM
ejpam-4432	263	14	ϕ(br	ϕ(br	X
ejpam-4432	263	15	)	)	PUNCT
ejpam-4432	263	16	=	=	SYM
ejpam-4432	264	1	[	[	X
ejpam-4432	264	2	(	(	PUNCT
ejpam-4432	264	3	ϕ(b)]r	ϕ(b)]r	NOUN
ejpam-4432	264	4	.	.	PUNCT
ejpam-4432	265	1	conversely	conversely	ADV
ejpam-4432	265	2	,	,	PUNCT
ejpam-4432	265	3	let	let	VERB
ejpam-4432	265	4	[	[	X
ejpam-4432	265	5	ϕ(a)]r	ϕ(a)]r	X
ejpam-4432	265	6	=	=	PUNCT
ejpam-4432	266	1	[	[	X
ejpam-4432	266	2	ϕ(b)]r	ϕ(b)]r	X
ejpam-4432	266	3	and	and	CCONJ
ejpam-4432	266	4	x	x	PROPN
ejpam-4432	266	5	∈	∈	PROPN
ejpam-4432	266	6	ar	ar	PROPN
ejpam-4432	266	7	.	.	PROPN
ejpam-4432	267	1	then	then	ADV
ejpam-4432	267	2	0	0	NUM
ejpam-4432	267	3	∈	∈	PROPN
ejpam-4432	267	4	a∧̄x	a∧̄x	PROPN
ejpam-4432	267	5	for	for	ADP
ejpam-4432	267	6	all	all	DET
ejpam-4432	267	7	a	a	DET
ejpam-4432	267	8	∈	∈	NOUN
ejpam-4432	267	9	a.	a.	NOUN
ejpam-4432	268	1	so	so	ADV
ejpam-4432	268	2	0′	0′	X
ejpam-4432	269	1	=	=	SYM
ejpam-4432	269	2	ϕ(0	ϕ(0	PROPN
ejpam-4432	269	3	)	)	PUNCT
ejpam-4432	269	4	∈	∈	PROPN
ejpam-4432	269	5	ϕ(a∧̄x	ϕ(a∧̄x	NOUN
ejpam-4432	269	6	)	)	PUNCT
ejpam-4432	269	7	=	=	SYM
ejpam-4432	269	8	ϕ(a)∧̄ϕ(x	ϕ(a)∧̄ϕ(x	PROPN
ejpam-4432	269	9	)	)	PUNCT
ejpam-4432	269	10	.	.	PUNCT
ejpam-4432	270	1	it	it	PRON
ejpam-4432	270	2	means	mean	VERB
ejpam-4432	270	3	ϕ(x	ϕ(x	X
ejpam-4432	270	4	)	)	PUNCT
ejpam-4432	270	5	∈	∈	PROPN
ejpam-4432	271	1	[	[	X
ejpam-4432	271	2	ϕ(a)]r	ϕ(a)]r	X
ejpam-4432	271	3	=	=	PUNCT
ejpam-4432	272	1	[	[	X
ejpam-4432	272	2	ϕ(b)]r	ϕ(b)]r	X
ejpam-4432	272	3	.	.	PUNCT
ejpam-4432	272	4	therefore	therefore	ADV
ejpam-4432	272	5	0′	0′	X
ejpam-4432	273	1	=	=	SYM
ejpam-4432	273	2	ϕ(0	ϕ(0	PROPN
ejpam-4432	273	3	)	)	PUNCT
ejpam-4432	273	4	∈	∈	PROPN
ejpam-4432	273	5	ϕ(x)∧̄ϕ(b	ϕ(x)∧̄ϕ(b	NOUN
ejpam-4432	273	6	)	)	PUNCT
ejpam-4432	274	1	for	for	ADP
ejpam-4432	274	2	all	all	DET
ejpam-4432	274	3	b	b	PROPN
ejpam-4432	274	4	∈	∈	PROPN
ejpam-4432	274	5	b	b	NOUN
ejpam-4432	274	6	,	,	PUNCT
ejpam-4432	274	7	which	which	PRON
ejpam-4432	274	8	implies	imply	VERB
ejpam-4432	274	9	that	that	SCONJ
ejpam-4432	274	10	0	0	NUM
ejpam-4432	274	11	∈	∈	PROPN
ejpam-4432	274	12	x∧̄b	x∧̄b	PROPN
ejpam-4432	274	13	for	for	ADP
ejpam-4432	274	14	all	all	DET
ejpam-4432	274	15	b	b	PROPN
ejpam-4432	274	16	∈	∈	PROPN
ejpam-4432	274	17	b.	b.	NOUN
ejpam-4432	274	18	accordingly	accordingly	ADV
ejpam-4432	274	19	x	x	SYM
ejpam-4432	274	20	∈	∈	NOUN
ejpam-4432	274	21	br	br	NOUN
ejpam-4432	274	22	and	and	CCONJ
ejpam-4432	274	23	hence	hence	ADV
ejpam-4432	274	24	ar	ar	VERB
ejpam-4432	274	25	⊆	⊆	NUM
ejpam-4432	274	26	br	br	NOUN
ejpam-4432	274	27	.	.	PUNCT
ejpam-4432	275	1	similarly	similarly	ADV
ejpam-4432	275	2	we	we	PRON
ejpam-4432	275	3	can	can	AUX
ejpam-4432	275	4	prove	prove	VERB
ejpam-4432	275	5	that	that	SCONJ
ejpam-4432	275	6	br	br	PROPN
ejpam-4432	275	7	⊆	⊆	NUM
ejpam-4432	275	8	ar	ar	NOUN
ejpam-4432	275	9	.	.	PROPN
ejpam-4432	275	10	(	(	PUNCT
ejpam-4432	275	11	ii	ii	NOUN
ejpam-4432	275	12	)	)	PUNCT
ejpam-4432	275	13	let	let	VERB
ejpam-4432	275	14	x	x	PUNCT
ejpam-4432	275	15	∈	∈	PROPN
ejpam-4432	275	16	[	[	X
ejpam-4432	275	17	ϕ−1(b)]r	ϕ−1(b)]r	PROPN
ejpam-4432	275	18	which	which	PRON
ejpam-4432	275	19	means	mean	VERB
ejpam-4432	275	20	0	0	NUM
ejpam-4432	275	21	∈	∈	PROPN
ejpam-4432	275	22	x∧̄b	x∧̄b	PROPN
ejpam-4432	275	23	for	for	ADP
ejpam-4432	275	24	all	all	DET
ejpam-4432	275	25	b	b	PROPN
ejpam-4432	275	26	∈	∈	PROPN
ejpam-4432	275	27	ϕ−1(b	ϕ−1(b	PROPN
ejpam-4432	275	28	)	)	PUNCT
ejpam-4432	275	29	.	.	PUNCT
ejpam-4432	276	1	then	then	ADV
ejpam-4432	276	2	0	0	NUM
ejpam-4432	276	3	∈	∈	PROPN
ejpam-4432	276	4	x∧̄b	x∧̄b	PROPN
ejpam-4432	276	5	for	for	ADP
ejpam-4432	276	6	all	all	DET
ejpam-4432	276	7	ϕ(b	ϕ(b	PROPN
ejpam-4432	276	8	)	)	PUNCT
ejpam-4432	276	9	∈	∈	PROPN
ejpam-4432	276	10	b.	b.	PROPN
ejpam-4432	277	1	it	it	PRON
ejpam-4432	277	2	implies	imply	VERB
ejpam-4432	277	3	ϕ(0	ϕ(0	PRON
ejpam-4432	277	4	)	)	PUNCT
ejpam-4432	277	5	=	=	PUNCT
ejpam-4432	278	1	0′	0′	PUNCT
ejpam-4432	279	1	∈	∈	PROPN
ejpam-4432	279	2	ϕ(x∧̄b	ϕ(x∧̄b	PROPN
ejpam-4432	279	3	)	)	PUNCT
ejpam-4432	279	4	=	=	SYM
ejpam-4432	279	5	ϕ(x)∧̄ϕ(b	ϕ(x)∧̄ϕ(b	NOUN
ejpam-4432	279	6	)	)	PUNCT
ejpam-4432	279	7	for	for	ADP
ejpam-4432	279	8	all	all	DET
ejpam-4432	279	9	ϕ(b	ϕ(b	PROPN
ejpam-4432	279	10	)	)	PUNCT
ejpam-4432	279	11	∈	∈	PROPN
ejpam-4432	279	12	b.	b.	PROPN
ejpam-4432	279	13	hence	hence	ADV
ejpam-4432	279	14	ϕ(x	ϕ(x	PROPN
ejpam-4432	279	15	)	)	PUNCT
ejpam-4432	279	16	∈	∈	PROPN
ejpam-4432	279	17	br	br	PROPN
ejpam-4432	279	18	,	,	PUNCT
ejpam-4432	279	19	i.e.	i.e.	X
ejpam-4432	279	20	,	,	PUNCT
ejpam-4432	279	21	x	x	SYM
ejpam-4432	279	22	∈	∈	PROPN
ejpam-4432	279	23	ϕ−1(br	ϕ−1(br	PROPN
ejpam-4432	279	24	)	)	PUNCT
ejpam-4432	279	25	.	.	PUNCT
ejpam-4432	280	1	as	as	ADP
ejpam-4432	280	2	a	a	DET
ejpam-4432	280	3	result	result	NOUN
ejpam-4432	280	4	,	,	PUNCT
ejpam-4432	280	5	[	[	X
ejpam-4432	280	6	ϕ−1(b)]r	ϕ−1(b)]r	PROPN
ejpam-4432	280	7	⊆	⊆	NUM
ejpam-4432	280	8	ϕ−1(br	ϕ−1(br	PROPN
ejpam-4432	280	9	)	)	PUNCT
ejpam-4432	280	10	.	.	PUNCT
ejpam-4432	281	1	conversely	conversely	ADV
ejpam-4432	281	2	,	,	PUNCT
ejpam-4432	281	3	let	let	VERB
ejpam-4432	281	4	x	x	X
ejpam-4432	281	5	∈	∈	PROPN
ejpam-4432	281	6	ϕ−1(br	ϕ−1(br	PROPN
ejpam-4432	281	7	)	)	PUNCT
ejpam-4432	281	8	and	and	CCONJ
ejpam-4432	281	9	b	b	X
ejpam-4432	281	10	∈	∈	PROPN
ejpam-4432	281	11	ϕ−1(b	ϕ−1(b	PROPN
ejpam-4432	281	12	)	)	PUNCT
ejpam-4432	281	13	.	.	PUNCT
ejpam-4432	282	1	then	then	ADV
ejpam-4432	282	2	ϕ(x	ϕ(x	X
ejpam-4432	282	3	)	)	PUNCT
ejpam-4432	282	4	∈	∈	PROPN
ejpam-4432	282	5	br	br	NOUN
ejpam-4432	282	6	and	and	CCONJ
ejpam-4432	282	7	ϕ(b	ϕ(b	PROPN
ejpam-4432	282	8	)	)	PUNCT
ejpam-4432	282	9	∈	∈	PROPN
ejpam-4432	282	10	b.	b.	PROPN
ejpam-4432	283	1	thus	thus	ADV
ejpam-4432	283	2	ϕ(0	ϕ(0	NUM
ejpam-4432	283	3	)	)	PUNCT
ejpam-4432	283	4	=	=	PUNCT
ejpam-4432	284	1	0′	0′	NUM
ejpam-4432	284	2	∈	∈	PROPN
ejpam-4432	284	3	ϕ(x)∧̄ϕ(b	ϕ(x)∧̄ϕ(b	NOUN
ejpam-4432	284	4	)	)	PUNCT
ejpam-4432	285	1	=	=	SYM
ejpam-4432	285	2	ϕ(x∧̄b	ϕ(x∧̄b	PROPN
ejpam-4432	285	3	)	)	PUNCT
ejpam-4432	285	4	.	.	PUNCT
ejpam-4432	286	1	as	as	ADP
ejpam-4432	286	2	a	a	DET
ejpam-4432	286	3	result	result	NOUN
ejpam-4432	286	4	,	,	PUNCT
ejpam-4432	286	5	0	0	NUM
ejpam-4432	286	6	∈	∈	PROPN
ejpam-4432	286	7	x∧	x∧	PROPN
ejpam-4432	286	8	b	b	PROPN
ejpam-4432	286	9	for	for	ADP
ejpam-4432	286	10	all	all	DET
ejpam-4432	286	11	b	b	PROPN
ejpam-4432	286	12	∈	∈	PROPN
ejpam-4432	286	13	ϕ−1(b	ϕ−1(b	PROPN
ejpam-4432	286	14	)	)	PUNCT
ejpam-4432	286	15	.	.	PUNCT
ejpam-4432	287	1	hence	hence	ADV
ejpam-4432	287	2	,	,	PUNCT
ejpam-4432	287	3	x	x	PUNCT
ejpam-4432	287	4	∈	∈	PROPN
ejpam-4432	288	1	[	[	X
ejpam-4432	288	2	ϕ−1(b)]r	ϕ−1(b)]r	PROPN
ejpam-4432	288	3	and	and	CCONJ
ejpam-4432	288	4	then	then	ADV
ejpam-4432	288	5	ϕ−1(br	ϕ−1(br	PROPN
ejpam-4432	288	6	)	)	PUNCT
ejpam-4432	289	1	⊆	⊆	NUM
ejpam-4432	289	2	[	[	X
ejpam-4432	289	3	ϕ−1(b)]r	ϕ−1(b)]r	PROPN
ejpam-4432	289	4	.	.	PUNCT
ejpam-4432	289	5	theorem	theorem	PROPN
ejpam-4432	289	6	5	5	NUM
ejpam-4432	289	7	.	.	PUNCT
ejpam-4432	290	1	let	let	VERB
ejpam-4432	290	2	l	l	NOUN
ejpam-4432	290	3	and	and	CCONJ
ejpam-4432	290	4	l′	l′	VERB
ejpam-4432	290	5	be	be	AUX
ejpam-4432	290	6	two	two	NUM
ejpam-4432	290	7	meet	meet	ADJ
ejpam-4432	290	8	-	-	PUNCT
ejpam-4432	290	9	hyperlattice	hyperlattice	NOUN
ejpam-4432	290	10	,	,	PUNCT
ejpam-4432	290	11	ϕ	ϕ	NOUN
ejpam-4432	290	12	:	:	PUNCT
ejpam-4432	290	13	l	l	X
ejpam-4432	290	14	→	→	SYM
ejpam-4432	290	15	l′	l′	AUX
ejpam-4432	290	16	be	be	AUX
ejpam-4432	290	17	a	a	DET
ejpam-4432	290	18	homomorphism	homomorphism	NOUN
ejpam-4432	290	19	and	and	CCONJ
ejpam-4432	290	20	ker(ϕ	ker(ϕ	PROPN
ejpam-4432	290	21	)	)	PUNCT
ejpam-4432	290	22	=	=	PRON
ejpam-4432	290	23	{	{	PUNCT
ejpam-4432	290	24	0	0	NUM
ejpam-4432	290	25	}	}	PUNCT
ejpam-4432	290	26	.	.	PUNCT
ejpam-4432	291	1	then	then	ADV
ejpam-4432	291	2	:	:	PUNCT
ejpam-4432	291	3	(	(	PUNCT
ejpam-4432	291	4	i	i	NOUN
ejpam-4432	291	5	)	)	PUNCT
ejpam-4432	291	6	if	if	SCONJ
ejpam-4432	291	7	ϕ	ϕ	NOUN
ejpam-4432	291	8	is	be	AUX
ejpam-4432	291	9	annihilator	annihilator	PROPN
ejpam-4432	291	10	hyperideal	hyperideal	NOUN
ejpam-4432	291	11	preserving	preserve	VERB
ejpam-4432	291	12	and	and	CCONJ
ejpam-4432	291	13	onto	onto	ADP
ejpam-4432	291	14	then	then	ADV
ejpam-4432	291	15	the	the	DET
ejpam-4432	291	16	homomorphiciimage	homomorphiciimage	NOUN
ejpam-4432	291	17	ϕ(i	ϕ(i	PROPN
ejpam-4432	291	18	)	)	PUNCT
ejpam-4432	291	19	of	of	ADP
ejpam-4432	291	20	annihilator	annihilator	PROPN
ejpam-4432	291	21	hyperideal	hyperideal	PROPN
ejpam-4432	291	22	i	i	PRON
ejpam-4432	291	23	of	of	ADP
ejpam-4432	291	24	l	l	NOUN
ejpam-4432	291	25	is	be	AUX
ejpam-4432	291	26	an	an	DET
ejpam-4432	291	27	annihilator	annihilator	PROPN
ejpam-4432	291	28	hyperideal	hyperideal	NOUN
ejpam-4432	291	29	of	of	ADP
ejpam-4432	291	30	l′	l′	NOUN
ejpam-4432	291	31	;	;	PUNCT
ejpam-4432	291	32	(	(	PUNCT
ejpam-4432	291	33	ii	ii	NOUN
ejpam-4432	291	34	)	)	PUNCT
ejpam-4432	291	35	if	if	SCONJ
ejpam-4432	291	36	ϕ−1	ϕ−1	PROPN
ejpam-4432	291	37	is	be	AUX
ejpam-4432	291	38	annihilator	annihilator	PROPN
ejpam-4432	291	39	hyperideal	hyperideal	NOUN
ejpam-4432	291	40	of	of	ADP
ejpam-4432	291	41	l′	l′	NOUN
ejpam-4432	291	42	preserving	preserve	VERB
ejpam-4432	291	43	then	then	ADV
ejpam-4432	291	44	the	the	DET
ejpam-4432	291	45	preimage	preimage	PROPN
ejpam-4432	291	46	ϕ−1(j	ϕ−1(j	PROPN
ejpam-4432	291	47	)	)	PUNCT
ejpam-4432	291	48	of	of	ADP
ejpam-4432	291	49	annihilator	annihilator	PROPN
ejpam-4432	291	50	hyperideal	hyperideal	PROPN
ejpam-4432	291	51	j	j	PROPN
ejpam-4432	291	52	is	be	AUX
ejpam-4432	291	53	an	an	DET
ejpam-4432	291	54	annihilator	annihilator	PROPN
ejpam-4432	291	55	hyperideal	hyperideal	NOUN
ejpam-4432	291	56	of	of	ADP
ejpam-4432	291	57	l	l	NOUN
ejpam-4432	291	58	;	;	PUNCT
ejpam-4432	291	59	(	(	PUNCT
ejpam-4432	291	60	iii	iii	X
ejpam-4432	291	61	)	)	PUNCT
ejpam-4432	291	62	if	if	SCONJ
ejpam-4432	291	63	ϕ	ϕ	NOUN
ejpam-4432	291	64	is	be	AUX
ejpam-4432	291	65	annihilator	annihilator	PROPN
ejpam-4432	291	66	hyperideal	hyperideal	NOUN
ejpam-4432	291	67	preserving	preserve	VERB
ejpam-4432	291	68	and	and	CCONJ
ejpam-4432	291	69	onto	onto	ADP
ejpam-4432	291	70	then	then	ADV
ejpam-4432	291	71	ker(ϕ	ker(ϕ	PROPN
ejpam-4432	291	72	)	)	PUNCT
ejpam-4432	291	73	is	be	AUX
ejpam-4432	291	74	an	an	DET
ejpam-4432	291	75	annihilator	annihilator	PROPN
ejpam-4432	291	76	hyperideal	hyperideal	NOUN
ejpam-4432	291	77	of	of	ADP
ejpam-4432	291	78	l.	l.	PROPN
ejpam-4432	291	79	proof	proof	PROPN
ejpam-4432	291	80	.	.	PUNCT
ejpam-4432	292	1	(	(	PUNCT
ejpam-4432	292	2	i	i	NOUN
ejpam-4432	292	3	)	)	PUNCT
ejpam-4432	292	4	let	let	VERB
ejpam-4432	292	5	i	i	PRON
ejpam-4432	292	6	be	be	AUX
ejpam-4432	292	7	an	an	DET
ejpam-4432	292	8	annihilator	annihilator	PROPN
ejpam-4432	292	9	hyperideal	hyperideal	NOUN
ejpam-4432	292	10	of	of	ADP
ejpam-4432	292	11	l.	l.	PROPN
ejpam-4432	292	12	then	then	ADV
ejpam-4432	292	13	by	by	ADP
ejpam-4432	292	14	using	use	VERB
ejpam-4432	292	15	i	i	PRON
ejpam-4432	292	16	)	)	PUNCT
ejpam-4432	292	17	in	in	ADP
ejpam-4432	292	18	proposition	proposition	NOUN
ejpam-4432	292	19	3	3	NUM
ejpam-4432	292	20	,	,	PUNCT
ejpam-4432	292	21	ϕ(i	ϕ(i	NUM
ejpam-4432	292	22	)	)	PUNCT
ejpam-4432	292	23	is	be	AUX
ejpam-4432	292	24	a	a	DET
ejpam-4432	292	25	hyperideal	hyperideal	NOUN
ejpam-4432	292	26	.	.	PUNCT
ejpam-4432	293	1	from	from	ADP
ejpam-4432	293	2	the	the	DET
ejpam-4432	293	3	assumption	assumption	NOUN
ejpam-4432	293	4	of	of	ADP
ejpam-4432	293	5	ϕ	ϕ	PROPN
ejpam-4432	293	6	is	be	AUX
ejpam-4432	293	7	annihilator	annihilator	PROPN
ejpam-4432	293	8	hyperideal	hyperideal	NOUN
ejpam-4432	293	9	preserving	preserve	VERB
ejpam-4432	293	10	we	we	PRON
ejpam-4432	293	11	get	get	VERB
ejpam-4432	293	12	[	[	X
ejpam-4432	293	13	ϕ(i)]rr	ϕ(i)]rr	NOUN
ejpam-4432	293	14	=	=	SYM
ejpam-4432	293	15	ϕ(irr	ϕ(irr	PROPN
ejpam-4432	293	16	)	)	PUNCT
ejpam-4432	293	17	=	=	SYM
ejpam-4432	293	18	ϕ(i	ϕ(i	PROPN
ejpam-4432	293	19	)	)	PUNCT
ejpam-4432	293	20	.	.	PUNCT
ejpam-4432	294	1	consequently	consequently	ADV
ejpam-4432	294	2	,	,	PUNCT
ejpam-4432	294	3	ϕ(i	ϕ(i	PROPN
ejpam-4432	294	4	)	)	PUNCT
ejpam-4432	294	5	is	be	AUX
ejpam-4432	294	6	an	an	DET
ejpam-4432	294	7	annihilator	annihilator	PROPN
ejpam-4432	294	8	ideal	ideal	NOUN
ejpam-4432	294	9	of	of	ADP
ejpam-4432	294	10	l′.	l′.	PROPN
ejpam-4432	294	11	(	(	PUNCT
ejpam-4432	294	12	ii	ii	NOUN
ejpam-4432	294	13	)	)	PUNCT
ejpam-4432	294	14	let	let	VERB
ejpam-4432	294	15	j	j	PROPN
ejpam-4432	294	16	be	be	AUX
ejpam-4432	294	17	an	an	DET
ejpam-4432	294	18	annihilator	annihilator	PROPN
ejpam-4432	294	19	hyperideal	hyperideal	NOUN
ejpam-4432	294	20	of	of	ADP
ejpam-4432	294	21	l′.	l′.	NOUN
ejpam-4432	294	22	then	then	ADV
ejpam-4432	294	23	by	by	ADP
ejpam-4432	294	24	using	use	VERB
ejpam-4432	294	25	ii	ii	NUM
ejpam-4432	294	26	)	)	PUNCT
ejpam-4432	294	27	in	in	ADP
ejpam-4432	294	28	proposition	proposition	NOUN
ejpam-4432	294	29	3	3	NUM
ejpam-4432	294	30	,	,	PUNCT
ejpam-4432	294	31	ϕ−1(i	ϕ−1(i	PROPN
ejpam-4432	294	32	)	)	PUNCT
ejpam-4432	294	33	is	be	AUX
ejpam-4432	294	34	a	a	DET
ejpam-4432	294	35	hyperideal	hyperideal	NOUN
ejpam-4432	294	36	.	.	PUNCT
ejpam-4432	295	1	from	from	ADP
ejpam-4432	295	2	the	the	DET
ejpam-4432	295	3	assumption	assumption	NOUN
ejpam-4432	295	4	of	of	ADP
ejpam-4432	295	5	ϕ−1	ϕ−1	PROPN
ejpam-4432	295	6	is	be	AUX
ejpam-4432	295	7	annihilator	annihilator	PROPN
ejpam-4432	295	8	hyperideal	hyperideal	NOUN
ejpam-4432	295	9	preserving	preserve	VERB
ejpam-4432	295	10	we	we	PRON
ejpam-4432	295	11	get	get	VERB
ejpam-4432	295	12	[	[	X
ejpam-4432	296	1	ϕ−1(i)]rr	ϕ−1(i)]rr	X
ejpam-4432	296	2	=	=	PUNCT
ejpam-4432	296	3	ϕ−1(irr	ϕ−1(irr	PROPN
ejpam-4432	296	4	)	)	PUNCT
ejpam-4432	297	1	=	=	SYM
ejpam-4432	297	2	ϕ−1(i	ϕ−1(i	PROPN
ejpam-4432	297	3	)	)	PUNCT
ejpam-4432	297	4	.	.	PUNCT
ejpam-4432	298	1	therefore	therefore	ADV
ejpam-4432	298	2	,	,	PUNCT
ejpam-4432	298	3	ϕ−1(i	ϕ−1(i	PROPN
ejpam-4432	298	4	)	)	PUNCT
ejpam-4432	298	5	is	be	AUX
ejpam-4432	298	6	an	an	DET
ejpam-4432	298	7	annihilator	annihilator	PROPN
ejpam-4432	298	8	ideal	ideal	NOUN
ejpam-4432	298	9	of	of	ADP
ejpam-4432	298	10	l.	l.	PROPN
ejpam-4432	298	11	(	(	PUNCT
ejpam-4432	298	12	iii	iii	X
ejpam-4432	298	13	)	)	PUNCT
ejpam-4432	298	14	we	we	PRON
ejpam-4432	298	15	know	know	VERB
ejpam-4432	298	16	that	that	SCONJ
ejpam-4432	298	17	ker(ϕ	ker(ϕ	PROPN
ejpam-4432	298	18	)	)	PUNCT
ejpam-4432	298	19	=	=	SYM
ejpam-4432	298	20	ϕ−1({0′	ϕ−1({0′	PROPN
ejpam-4432	298	21	}	}	PUNCT
ejpam-4432	298	22	)	)	PUNCT
ejpam-4432	298	23	and	and	CCONJ
ejpam-4432	298	24	{	{	PUNCT
ejpam-4432	298	25	0′	0′	X
ejpam-4432	298	26	}	}	PUNCT
ejpam-4432	298	27	=	=	NOUN
ejpam-4432	298	28	1r	1r	NUM
ejpam-4432	298	29	is	be	AUX
ejpam-4432	298	30	annihilator	annihilator	PROPN
ejpam-4432	298	31	hyperideal	hyperideal	NOUN
ejpam-4432	298	32	.	.	PUNCT
ejpam-4432	299	1	then	then	ADV
ejpam-4432	299	2	from	from	ADP
ejpam-4432	299	3	ii	ii	PROPN
ejpam-4432	299	4	)	)	PUNCT
ejpam-4432	299	5	,	,	PUNCT
ejpam-4432	299	6	ker(ϕ	ker(ϕ	PROPN
ejpam-4432	299	7	)	)	PUNCT
ejpam-4432	299	8	is	be	AUX
ejpam-4432	299	9	annihilator	annihilator	PROPN
ejpam-4432	299	10	hyperideal	hyperideal	NOUN
ejpam-4432	299	11	.	.	PUNCT
ejpam-4432	300	1	e.g.	e.g.	ADV
ejpam-4432	300	2	rezk	rezk	PROPN
ejpam-4432	300	3	,	,	PUNCT
ejpam-4432	300	4	n.h	n.h	PROPN
ejpam-4432	300	5	.	.	PROPN
ejpam-4432	300	6	abughazalah	abughazalah	PROPN
ejpam-4432	300	7	/	/	SYM
ejpam-4432	300	8	eur	eur	PROPN
ejpam-4432	300	9	.	.	PUNCT
ejpam-4432	301	1	j.	j.	PROPN
ejpam-4432	301	2	pure	pure	PROPN
ejpam-4432	301	3	appl	appl	PROPN
ejpam-4432	301	4	.	.	PROPN
ejpam-4432	301	5	math	math	PROPN
ejpam-4432	301	6	,	,	PUNCT
ejpam-4432	301	7	15	15	NUM
ejpam-4432	301	8	(	(	PUNCT
ejpam-4432	301	9	3	3	NUM
ejpam-4432	301	10	)	)	PUNCT
ejpam-4432	301	11	(	(	PUNCT
ejpam-4432	301	12	2022	2022	NUM
ejpam-4432	301	13	)	)	PUNCT
ejpam-4432	301	14	,	,	PUNCT
ejpam-4432	301	15	1402	1402	NUM
ejpam-4432	301	16	-	-	SYM
ejpam-4432	301	17	1416	1416	NUM
ejpam-4432	301	18	1413	1413	NUM
ejpam-4432	301	19	corollary	corollary	NOUN
ejpam-4432	301	20	1	1	NUM
ejpam-4432	301	21	.	.	PUNCT
ejpam-4432	302	1	let	let	VERB
ejpam-4432	302	2	l	l	NOUN
ejpam-4432	302	3	and	and	CCONJ
ejpam-4432	302	4	l′	l′	VERB
ejpam-4432	302	5	be	be	AUX
ejpam-4432	302	6	two	two	NUM
ejpam-4432	302	7	meet	meet	ADJ
ejpam-4432	302	8	-	-	PUNCT
ejpam-4432	302	9	hyperlattice	hyperlattice	NOUN
ejpam-4432	302	10	,	,	PUNCT
ejpam-4432	302	11	ϕ	ϕ	NOUN
ejpam-4432	302	12	:	:	PUNCT
ejpam-4432	302	13	l	l	X
ejpam-4432	302	14	→	→	SYM
ejpam-4432	302	15	l′	l′	AUX
ejpam-4432	302	16	be	be	AUX
ejpam-4432	302	17	an	an	DET
ejpam-4432	302	18	annihilator	annihilator	PROPN
ejpam-4432	302	19	hyperideal	hyperideal	NOUN
ejpam-4432	302	20	preserving	preserving	NOUN
ejpam-4432	302	21	.	.	PUNCT
ejpam-4432	303	1	then	then	ADV
ejpam-4432	303	2	h(l	h(l	NUM
ejpam-4432	303	3	)	)	PUNCT
ejpam-4432	303	4	and	and	CCONJ
ejpam-4432	303	5	h(l′	h(l′	NUM
ejpam-4432	303	6	)	)	PUNCT
ejpam-4432	303	7	are	be	AUX
ejpam-4432	303	8	isomorphic	isomorphic	ADJ
ejpam-4432	303	9	.	.	PUNCT
ejpam-4432	304	1	symbolically	symbolically	ADV
ejpam-4432	304	2	writes	write	VERB
ejpam-4432	304	3	h(l	h(l	PROPN
ejpam-4432	304	4	)	)	PUNCT
ejpam-4432	304	5	∼=	∼=	ADV
ejpam-4432	304	6	h(l′	h(l′	NUM
ejpam-4432	304	7	)	)	PUNCT
ejpam-4432	304	8	.	.	PUNCT
ejpam-4432	305	1	example	example	NOUN
ejpam-4432	306	1	4	4	NUM
ejpam-4432	306	2	.	.	PUNCT
ejpam-4432	307	1	the	the	DET
ejpam-4432	307	2	meet	meet	ADJ
ejpam-4432	307	3	-	-	PUNCT
ejpam-4432	307	4	hyperlattice	hyperlattice	NOUN
ejpam-4432	307	5	l′	l′	NOUN
ejpam-4432	307	6	in	in	ADP
ejpam-4432	307	7	example	example	NOUN
ejpam-4432	307	8	3	3	NUM
ejpam-4432	307	9	has	have	VERB
ejpam-4432	307	10	the	the	DET
ejpam-4432	307	11	boolean	boolean	ADJ
ejpam-4432	307	12	algebra	algebra	NOUN
ejpam-4432	307	13	of	of	ADP
ejpam-4432	307	14	its	its	PRON
ejpam-4432	307	15	closed	closed	ADJ
ejpam-4432	307	16	hyperideals	hyperideal	NOUN
ejpam-4432	307	17	which	which	PRON
ejpam-4432	307	18	is	be	AUX
ejpam-4432	307	19	given	give	VERB
ejpam-4432	307	20	by	by	ADP
ejpam-4432	307	21	figure	figure	NOUN
ejpam-4432	307	22	2	2	NUM
ejpam-4432	307	23	.	.	PUNCT
ejpam-4432	308	1	clearly	clearly	ADV
ejpam-4432	308	2	,	,	PUNCT
ejpam-4432	308	3	its	its	PRON
ejpam-4432	308	4	isomorphic	isomorphic	ADJ
ejpam-4432	308	5	to	to	ADP
ejpam-4432	308	6	the	the	DET
ejpam-4432	308	7	boolean	boolean	ADJ
ejpam-4432	308	8	algebra	algebra	NOUN
ejpam-4432	308	9	in	in	ADP
ejpam-4432	308	10	figure	figure	NOUN
ejpam-4432	308	11	3	3	NUM
ejpam-4432	308	12	which	which	PRON
ejpam-4432	308	13	represents	represent	VERB
ejpam-4432	308	14	the	the	DET
ejpam-4432	308	15	closed	closed	ADJ
ejpam-4432	308	16	hyperideals	hyperideal	NOUN
ejpam-4432	308	17	of	of	ADP
ejpam-4432	308	18	the	the	DET
ejpam-4432	308	19	meet	meet	NOUN
ejpam-4432	308	20	-	-	PUNCT
ejpam-4432	308	21	hyperlattice	hyperlattice	NOUN
ejpam-4432	308	22	in	in	ADP
ejpam-4432	308	23	example	example	NOUN
ejpam-4432	308	24	2	2	NUM
ejpam-4432	308	25	.	.	PUNCT
ejpam-4432	308	26	l′′	l′′	PROPN
ejpam-4432	308	27	{	{	PUNCT
ejpam-4432	308	28	0	0	NUM
ejpam-4432	308	29	,	,	PUNCT
ejpam-4432	308	30	α	α	NOUN
ejpam-4432	308	31	}	}	PUNCT
ejpam-4432	308	32	{	{	PUNCT
ejpam-4432	308	33	0	0	NUM
ejpam-4432	308	34	,	,	PUNCT
ejpam-4432	308	35	β	β	NOUN
ejpam-4432	308	36	}	}	PUNCT
ejpam-4432	308	37	1r	1r	NUM
ejpam-4432	308	38	=	=	SYM
ejpam-4432	308	39	{	{	PUNCT
ejpam-4432	308	40	0	0	NUM
ejpam-4432	308	41	}	}	PUNCT
ejpam-4432	308	42	figure2	figure2	ADJ
ejpam-4432	308	43	:	:	PUNCT
ejpam-4432	308	44	boolean	boolean	ADJ
ejpam-4432	308	45	algebra	algebra	NOUN
ejpam-4432	308	46	<	<	X
ejpam-4432	308	47	h(l′′);∩,⊻,r	h(l′′);∩,⊻,r	X
ejpam-4432	308	48	,	,	PUNCT
ejpam-4432	308	49	1r	1r	NUM
ejpam-4432	308	50	,	,	PUNCT
ejpam-4432	308	51	l′′	l′′	PROPN
ejpam-4432	308	52	>	>	X
ejpam-4432	308	53	l′	l′	X
ejpam-4432	308	54	{	{	PUNCT
ejpam-4432	308	55	0	0	NUM
ejpam-4432	308	56	,	,	PUNCT
ejpam-4432	308	57	x	x	NOUN
ejpam-4432	308	58	}	}	PUNCT
ejpam-4432	308	59	{	{	PUNCT
ejpam-4432	308	60	0	0	NUM
ejpam-4432	308	61	,	,	PUNCT
ejpam-4432	308	62	y	y	NOUN
ejpam-4432	308	63	}	}	PUNCT
ejpam-4432	308	64	{	{	PUNCT
ejpam-4432	308	65	0′	0′	NOUN
ejpam-4432	308	66	}	}	PUNCT
ejpam-4432	308	67	figure3	figure3	NOUN
ejpam-4432	308	68	:	:	PUNCT
ejpam-4432	308	69	boolean	boolean	ADJ
ejpam-4432	308	70	algebra	algebra	PROPN
ejpam-4432	308	71	<	<	X
ejpam-4432	308	72	h(l′);∩,⊻,r	h(l′);∩,⊻,r	PROPN
ejpam-4432	308	73	,	,	PUNCT
ejpam-4432	308	74	1r	1r	NUM
ejpam-4432	308	75	,	,	PUNCT
ejpam-4432	308	76	l′	l′	X
ejpam-4432	308	77	>	>	X
ejpam-4432	309	1	5	5	X
ejpam-4432	309	2	.	.	X
ejpam-4432	309	3	sub	sub	ADJ
ejpam-4432	309	4	-	-	ADJ
ejpam-4432	309	5	meet	meet	ADJ
ejpam-4432	309	6	-	-	PUNCT
ejpam-4432	309	7	hyperlattice	hyperlattice	NOUN
ejpam-4432	309	8	and	and	CCONJ
ejpam-4432	309	9	product	product	NOUN
ejpam-4432	309	10	representations	representation	NOUN
ejpam-4432	309	11	of	of	ADP
ejpam-4432	309	12	annihilator	annihilator	PROPN
ejpam-4432	309	13	hyperideals	hyperideal	NOUN
ejpam-4432	309	14	in	in	ADP
ejpam-4432	309	15	sub	sub	ADJ
ejpam-4432	309	16	-	-	ADJ
ejpam-4432	309	17	meet	meet	ADJ
ejpam-4432	309	18	-	-	PUNCT
ejpam-4432	309	19	hyperlattice	hyperlattice	NOUN
ejpam-4432	309	20	and	and	CCONJ
ejpam-4432	309	21	product	product	NOUN
ejpam-4432	309	22	meethyperlattice	meethyperlattice	NOUN
ejpam-4432	309	23	,	,	PUNCT
ejpam-4432	309	24	are	be	AUX
ejpam-4432	309	25	investigated	investigate	VERB
ejpam-4432	309	26	in	in	ADP
ejpam-4432	309	27	the	the	DET
ejpam-4432	309	28	following	following	NOUN
ejpam-4432	309	29	.	.	PUNCT
ejpam-4432	310	1	definition	definition	NOUN
ejpam-4432	310	2	7	7	NUM
ejpam-4432	310	3	.	.	PUNCT
ejpam-4432	311	1	a	a	DET
ejpam-4432	311	2	subset	subset	NOUN
ejpam-4432	311	3	s	s	VERB
ejpam-4432	311	4	⊆	⊆	NUM
ejpam-4432	311	5	l	l	NOUN
ejpam-4432	311	6	,	,	PUNCT
ejpam-4432	311	7	of	of	ADP
ejpam-4432	311	8	meet	meet	ADJ
ejpam-4432	311	9	-	-	PUNCT
ejpam-4432	311	10	hyperlattice	hyperlattice	NOUN
ejpam-4432	311	11	l	l	NOUN
ejpam-4432	311	12	,	,	PUNCT
ejpam-4432	311	13	is	be	AUX
ejpam-4432	311	14	a	a	DET
ejpam-4432	311	15	sub	sub	ADJ
ejpam-4432	311	16	-	-	ADJ
ejpam-4432	311	17	meet	meet	ADJ
ejpam-4432	311	18	-	-	PUNCT
ejpam-4432	311	19	hyperlattice	hyperlattice	NOUN
ejpam-4432	311	20	iff	iff	NOUN
ejpam-4432	311	21	it	it	PRON
ejpam-4432	311	22	is	be	AUX
ejpam-4432	311	23	close	close	ADJ
ejpam-4432	311	24	under	under	ADP
ejpam-4432	311	25	the	the	DET
ejpam-4432	311	26	same	same	ADJ
ejpam-4432	311	27	operations	operation	NOUN
ejpam-4432	311	28	of	of	ADP
ejpam-4432	311	29	l.	l.	PROPN
ejpam-4432	311	30	clearly	clearly	ADV
ejpam-4432	311	31	,	,	PUNCT
ejpam-4432	311	32	sub	sub	ADJ
ejpam-4432	311	33	-	-	ADJ
ejpam-4432	311	34	meet	meet	ADJ
ejpam-4432	311	35	-	-	PUNCT
ejpam-4432	311	36	hyperlattice	hyperlattice	NOUN
ejpam-4432	311	37	of	of	ADP
ejpam-4432	311	38	strong	strong	ADJ
ejpam-4432	311	39	bounded	bounded	ADJ
ejpam-4432	311	40	dual	dual	ADJ
ejpam-4432	311	41	distributive	distributive	ADJ
ejpam-4432	311	42	meet	meet	NOUN
ejpam-4432	311	43	-	-	PUNCT
ejpam-4432	311	44	hyperlattice	hyperlattice	NOUN
ejpam-4432	311	45	is	be	AUX
ejpam-4432	311	46	also	also	ADV
ejpam-4432	311	47	too	too	ADV
ejpam-4432	311	48	.	.	PUNCT
ejpam-4432	312	1	theorem	theorem	NOUN
ejpam-4432	312	2	6	6	NUM
ejpam-4432	312	3	.	.	PUNCT
ejpam-4432	313	1	if	if	SCONJ
ejpam-4432	313	2	i	i	PRON
ejpam-4432	313	3	is	be	AUX
ejpam-4432	313	4	an	an	DET
ejpam-4432	313	5	annihilator	annihilator	PROPN
ejpam-4432	313	6	hyperideal	hyperideal	NOUN
ejpam-4432	313	7	of	of	ADP
ejpam-4432	313	8	l	l	NOUN
ejpam-4432	313	9	and	and	CCONJ
ejpam-4432	313	10	s	s	X
ejpam-4432	313	11	is	be	AUX
ejpam-4432	313	12	a	a	DET
ejpam-4432	313	13	sub	sub	ADJ
ejpam-4432	313	14	-	-	ADJ
ejpam-4432	313	15	meet	meet	ADJ
ejpam-4432	313	16	hyperlattice	hyperlattice	NOUN
ejpam-4432	313	17	of	of	ADP
ejpam-4432	313	18	l.	l.	PROPN
ejpam-4432	313	19	then	then	ADV
ejpam-4432	313	20	i	i	PRON
ejpam-4432	313	21	∩	∩	SCONJ
ejpam-4432	313	22	s	s	PART
ejpam-4432	313	23	is	be	AUX
ejpam-4432	313	24	an	an	DET
ejpam-4432	313	25	annihilator	annihilator	PROPN
ejpam-4432	313	26	hyperideal	hyperideal	NOUN
ejpam-4432	313	27	of	of	ADP
ejpam-4432	313	28	s.	s.	PROPN
ejpam-4432	313	29	proof	proof	PROPN
ejpam-4432	313	30	.	.	PUNCT
ejpam-4432	314	1	let	let	VERB
ejpam-4432	314	2	i	i	PRON
ejpam-4432	314	3	be	be	AUX
ejpam-4432	314	4	an	an	DET
ejpam-4432	314	5	annihilator	annihilator	PROPN
ejpam-4432	314	6	hyperideal	hyperideal	NOUN
ejpam-4432	314	7	of	of	ADP
ejpam-4432	314	8	l	l	PROPN
ejpam-4432	314	9	and	and	CCONJ
ejpam-4432	314	10	s	s	AUX
ejpam-4432	314	11	be	be	AUX
ejpam-4432	314	12	a	a	DET
ejpam-4432	314	13	sub	sub	ADJ
ejpam-4432	314	14	-	-	ADJ
ejpam-4432	314	15	meet	meet	ADJ
ejpam-4432	314	16	hyperlattice	hyperlattice	NOUN
ejpam-4432	314	17	of	of	ADP
ejpam-4432	314	18	l.	l.	PROPN
ejpam-4432	314	19	then	then	ADV
ejpam-4432	314	20	there	there	PRON
ejpam-4432	314	21	exist	exist	VERB
ejpam-4432	314	22	a	a	DET
ejpam-4432	314	23	subset	subset	NOUN
ejpam-4432	314	24	k	k	PROPN
ejpam-4432	314	25	of	of	ADP
ejpam-4432	314	26	l	l	NOUN
ejpam-4432	314	27	such	such	ADJ
ejpam-4432	314	28	that	that	SCONJ
ejpam-4432	314	29	i	i	PRON
ejpam-4432	314	30	=	=	PUNCT
ejpam-4432	314	31	kr	kr	PROPN
ejpam-4432	314	32	=	=	SYM
ejpam-4432	314	33	{	{	PUNCT
ejpam-4432	314	34	x	x	PUNCT
ejpam-4432	314	35	∈	∈	NOUN
ejpam-4432	314	36	l	l	NOUN
ejpam-4432	314	37	:	:	PUNCT
ejpam-4432	314	38	0	0	NUM
ejpam-4432	314	39	∈	∈	X
ejpam-4432	314	40	x∧̄a	x∧̄a	PROPN
ejpam-4432	314	41	for	for	ADP
ejpam-4432	314	42	all	all	DET
ejpam-4432	314	43	a	a	DET
ejpam-4432	314	44	∈	∈	NOUN
ejpam-4432	314	45	k	k	NOUN
ejpam-4432	314	46	}	}	PUNCT
ejpam-4432	314	47	.	.	PUNCT
ejpam-4432	315	1	thus	thus	ADV
ejpam-4432	315	2	i	i	PRON
ejpam-4432	315	3	∩	∩	NOUN
ejpam-4432	315	4	s	s	PART
ejpam-4432	315	5	=	=	X
ejpam-4432	315	6	{	{	PUNCT
ejpam-4432	315	7	x	x	SYM
ejpam-4432	315	8	∈	∈	NOUN
ejpam-4432	315	9	l	l	NOUN
ejpam-4432	315	10	:	:	PUNCT
ejpam-4432	315	11	0	0	NUM
ejpam-4432	315	12	∈	∈	X
ejpam-4432	315	13	x∧̄a	x∧̄a	PROPN
ejpam-4432	315	14	for	for	ADP
ejpam-4432	315	15	all	all	DET
ejpam-4432	315	16	a	a	DET
ejpam-4432	315	17	∈	∈	PROPN
ejpam-4432	315	18	k	k	PROPN
ejpam-4432	315	19	∩	∩	X
ejpam-4432	315	20	s	s	PART
ejpam-4432	315	21	}	}	PUNCT
ejpam-4432	315	22	=	=	SYM
ejpam-4432	315	23	(	(	PUNCT
ejpam-4432	315	24	k	k	X
ejpam-4432	315	25	∩	∩	NOUN
ejpam-4432	315	26	s)r	s)r	NOUN
ejpam-4432	315	27	.	.	PUNCT
ejpam-4432	316	1	e.g.	e.g.	ADV
ejpam-4432	316	2	rezk	rezk	PROPN
ejpam-4432	316	3	,	,	PUNCT
ejpam-4432	316	4	n.h	n.h	PROPN
ejpam-4432	316	5	.	.	PROPN
ejpam-4432	316	6	abughazalah	abughazalah	PROPN
ejpam-4432	316	7	/	/	SYM
ejpam-4432	316	8	eur	eur	PROPN
ejpam-4432	316	9	.	.	PUNCT
ejpam-4432	317	1	j.	j.	PROPN
ejpam-4432	317	2	pure	pure	PROPN
ejpam-4432	317	3	appl	appl	PROPN
ejpam-4432	317	4	.	.	PROPN
ejpam-4432	317	5	math	math	PROPN
ejpam-4432	317	6	,	,	PUNCT
ejpam-4432	317	7	15	15	NUM
ejpam-4432	317	8	(	(	PUNCT
ejpam-4432	317	9	3	3	NUM
ejpam-4432	317	10	)	)	PUNCT
ejpam-4432	317	11	(	(	PUNCT
ejpam-4432	317	12	2022	2022	NUM
ejpam-4432	317	13	)	)	PUNCT
ejpam-4432	317	14	,	,	PUNCT
ejpam-4432	317	15	1402	1402	NUM
ejpam-4432	317	16	-	-	SYM
ejpam-4432	317	17	1416	1416	NUM
ejpam-4432	317	18	1414	1414	NUM
ejpam-4432	317	19	corollary	corollary	NOUN
ejpam-4432	317	20	2	2	NUM
ejpam-4432	317	21	.	.	PUNCT
ejpam-4432	318	1	if	if	SCONJ
ejpam-4432	318	2	i	i	PRON
ejpam-4432	318	3	is	be	AUX
ejpam-4432	318	4	a	a	DET
ejpam-4432	318	5	closed	closed	ADJ
ejpam-4432	318	6	hyperideal	hyperideal	NOUN
ejpam-4432	318	7	of	of	ADP
ejpam-4432	318	8	l	l	NOUN
ejpam-4432	318	9	and	and	CCONJ
ejpam-4432	318	10	s	s	X
ejpam-4432	318	11	is	be	AUX
ejpam-4432	318	12	a	a	DET
ejpam-4432	318	13	sub	sub	ADJ
ejpam-4432	318	14	-	-	ADJ
ejpam-4432	318	15	meet	meet	ADJ
ejpam-4432	318	16	hyperlattice	hyperlattice	NOUN
ejpam-4432	318	17	of	of	ADP
ejpam-4432	318	18	l.	l.	PROPN
ejpam-4432	318	19	then	then	ADV
ejpam-4432	318	20	i	i	PRON
ejpam-4432	318	21	∩	∩	SCONJ
ejpam-4432	318	22	s	s	PART
ejpam-4432	318	23	is	be	AUX
ejpam-4432	318	24	a	a	DET
ejpam-4432	318	25	closed	closed	ADJ
ejpam-4432	318	26	hyperideal	hyperideal	NOUN
ejpam-4432	318	27	of	of	ADP
ejpam-4432	318	28	s.	s.	PROPN
ejpam-4432	318	29	example	example	PROPN
ejpam-4432	319	1	5	5	NUM
ejpam-4432	319	2	.	.	PUNCT
ejpam-4432	319	3	tables	table	NOUN
ejpam-4432	319	4	7	7	NUM
ejpam-4432	319	5	and	and	CCONJ
ejpam-4432	319	6	8	8	NUM
ejpam-4432	319	7	represent	represent	VERB
ejpam-4432	319	8	a	a	DET
ejpam-4432	319	9	sub	sub	ADJ
ejpam-4432	319	10	-	-	ADJ
ejpam-4432	319	11	meet	meet	ADJ
ejpam-4432	319	12	-	-	PUNCT
ejpam-4432	319	13	hyperlattice	hyperlattice	NOUN
ejpam-4432	319	14	s	s	NOUN
ejpam-4432	319	15	of	of	ADP
ejpam-4432	319	16	meet	meet	ADJ
ejpam-4432	319	17	-	-	PUNCT
ejpam-4432	319	18	hyperlattice	hyperlattice	NOUN
ejpam-4432	319	19	l	l	NOUN
ejpam-4432	319	20	in	in	ADP
ejpam-4432	319	21	example	example	NOUN
ejpam-4432	319	22	1	1	NUM
ejpam-4432	319	23	.	.	PUNCT
ejpam-4432	319	24	figure	figure	VERB
ejpam-4432	319	25	4	4	NUM
ejpam-4432	319	26	shows	show	VERB
ejpam-4432	319	27	boolean	boolean	ADJ
ejpam-4432	319	28	algebra	algebra	NOUN
ejpam-4432	319	29	h(s	h(s	PROPN
ejpam-4432	319	30	)	)	PUNCT
ejpam-4432	319	31	of	of	ADP
ejpam-4432	319	32	closed	closed	ADJ
ejpam-4432	319	33	hyperideals	hyperideal	NOUN
ejpam-4432	319	34	of	of	ADP
ejpam-4432	319	35	s.	s.	PROPN
ejpam-4432	319	36	∧̄	∧̄	PROPN
ejpam-4432	319	37	0	0	NUM
ejpam-4432	320	1	α	α	PRON
ejpam-4432	320	2	β	β	X
ejpam-4432	320	3	γ	γ	X
ejpam-4432	320	4	0	0	NUM
ejpam-4432	320	5	{	{	PUNCT
ejpam-4432	320	6	0	0	NUM
ejpam-4432	320	7	}	}	PUNCT
ejpam-4432	320	8	{	{	PUNCT
ejpam-4432	320	9	0	0	NUM
ejpam-4432	320	10	}	}	PUNCT
ejpam-4432	320	11	{	{	PUNCT
ejpam-4432	320	12	0	0	NUM
ejpam-4432	320	13	}	}	PUNCT
ejpam-4432	320	14	{	{	PUNCT
ejpam-4432	320	15	0	0	NUM
ejpam-4432	320	16	}	}	PUNCT
ejpam-4432	320	17	α	α	NOUN
ejpam-4432	320	18	{	{	PUNCT
ejpam-4432	320	19	0	0	NUM
ejpam-4432	320	20	}	}	PUNCT
ejpam-4432	320	21	{	{	PUNCT
ejpam-4432	320	22	0	0	NUM
ejpam-4432	320	23	,	,	PUNCT
ejpam-4432	320	24	α	α	NOUN
ejpam-4432	320	25	}	}	PUNCT
ejpam-4432	320	26	{	{	PUNCT
ejpam-4432	320	27	0	0	NUM
ejpam-4432	320	28	}	}	PUNCT
ejpam-4432	320	29	{	{	PUNCT
ejpam-4432	320	30	0	0	NUM
ejpam-4432	320	31	,	,	PUNCT
ejpam-4432	320	32	α	α	NOUN
ejpam-4432	320	33	}	}	PUNCT
ejpam-4432	320	34	β	β	X
ejpam-4432	320	35	{	{	PUNCT
ejpam-4432	320	36	0	0	NUM
ejpam-4432	320	37	}	}	PUNCT
ejpam-4432	320	38	{	{	PUNCT
ejpam-4432	320	39	0	0	NUM
ejpam-4432	320	40	}	}	PUNCT
ejpam-4432	320	41	{	{	PUNCT
ejpam-4432	320	42	β	β	X
ejpam-4432	320	43	}	}	PUNCT
ejpam-4432	320	44	{	{	PUNCT
ejpam-4432	320	45	β	β	NOUN
ejpam-4432	320	46	}	}	PUNCT
ejpam-4432	320	47	γ	γ	X
ejpam-4432	320	48	{	{	PUNCT
ejpam-4432	320	49	0	0	NUM
ejpam-4432	320	50	}	}	PUNCT
ejpam-4432	320	51	{	{	PUNCT
ejpam-4432	320	52	0	0	NUM
ejpam-4432	320	53	,	,	PUNCT
ejpam-4432	320	54	α	α	NOUN
ejpam-4432	320	55	}	}	PUNCT
ejpam-4432	320	56	{	{	PUNCT
ejpam-4432	320	57	β	β	X
ejpam-4432	320	58	}	}	PUNCT
ejpam-4432	320	59	{	{	PUNCT
ejpam-4432	320	60	γ	γ	X
ejpam-4432	320	61	}	}	PUNCT
ejpam-4432	320	62	∨	∨	NUM
ejpam-4432	320	63	0	0	NUM
ejpam-4432	320	64	α	α	NOUN
ejpam-4432	320	65	β	β	X
ejpam-4432	320	66	γ	γ	X
ejpam-4432	320	67	0	0	NUM
ejpam-4432	320	68	0	0	NUM
ejpam-4432	320	69	α	α	NOUN
ejpam-4432	320	70	β	β	X
ejpam-4432	320	71	γ	γ	X
ejpam-4432	320	72	α	α	NOUN
ejpam-4432	320	73	α	α	NOUN
ejpam-4432	320	74	α	α	PROPN
ejpam-4432	320	75	γ	γ	PROPN
ejpam-4432	320	76	γ	γ	X
ejpam-4432	320	77	β	β	X
ejpam-4432	320	78	β	β	X
ejpam-4432	320	79	γ	γ	X
ejpam-4432	320	80	β	β	X
ejpam-4432	320	81	γ	γ	X
ejpam-4432	320	82	γ	γ	PROPN
ejpam-4432	320	83	γ	γ	PROPN
ejpam-4432	320	84	γ	γ	PROPN
ejpam-4432	320	85	γ	γ	X
ejpam-4432	320	86	γ	γ	X
ejpam-4432	320	87	table	table	NOUN
ejpam-4432	320	88	7	7	NUM
ejpam-4432	320	89	:	:	PUNCT
ejpam-4432	320	90	represents	represent	VERB
ejpam-4432	320	91	the	the	DET
ejpam-4432	320	92	hyperoperation	hyperoperation	NOUN
ejpam-4432	320	93	table	table	NOUN
ejpam-4432	320	94	8	8	NUM
ejpam-4432	320	95	:	:	PUNCT
ejpam-4432	320	96	represents	represent	VERB
ejpam-4432	320	97	operation	operation	NOUN
ejpam-4432	320	98	∧̄	∧̄	PROPN
ejpam-4432	320	99	of	of	ADP
ejpam-4432	320	100	the	the	DET
ejpam-4432	320	101	sub	sub	ADJ
ejpam-4432	320	102	-	-	ADJ
ejpam-4432	320	103	meet	meet	ADJ
ejpam-4432	320	104	-	-	PUNCT
ejpam-4432	320	105	hyberlattice	hyberlattice	NOUN
ejpam-4432	320	106	s	s	PART
ejpam-4432	320	107	∨	∨	NOUN
ejpam-4432	320	108	of	of	ADP
ejpam-4432	320	109	the	the	DET
ejpam-4432	320	110	sub	sub	ADJ
ejpam-4432	320	111	-	-	ADJ
ejpam-4432	320	112	meet	meet	ADJ
ejpam-4432	320	113	-	-	PUNCT
ejpam-4432	320	114	hyberlattice	hyberlattice	NOUN
ejpam-4432	320	115	s	s	NOUN
ejpam-4432	320	116	s	s	NOUN
ejpam-4432	320	117	1r	1r	NUM
ejpam-4432	320	118	=	=	SYM
ejpam-4432	320	119	{	{	PUNCT
ejpam-4432	320	120	0	0	NUM
ejpam-4432	320	121	,	,	PUNCT
ejpam-4432	320	122	α	α	NOUN
ejpam-4432	320	123	}	}	PUNCT
ejpam-4432	320	124	figure4	figure4	ADJ
ejpam-4432	320	125	:	:	PUNCT
ejpam-4432	320	126	boolean	boolean	ADJ
ejpam-4432	320	127	algebra	algebra	PROPN
ejpam-4432	320	128	<	<	X
ejpam-4432	320	129	h(s);∩,⊻,r	h(s);∩,⊻,r	PROPN
ejpam-4432	320	130	,	,	PUNCT
ejpam-4432	320	131	1r	1r	NUM
ejpam-4432	320	132	,	,	PUNCT
ejpam-4432	320	133	l	l	NOUN
ejpam-4432	320	134	>	>	X
ejpam-4432	320	135	definition	definition	NOUN
ejpam-4432	320	136	8	8	NUM
ejpam-4432	320	137	.	.	PUNCT
ejpam-4432	321	1	let	let	VERB
ejpam-4432	321	2	l1	l1	PROPN
ejpam-4432	321	3	=	=	PROPN
ejpam-4432	321	4	<	<	X
ejpam-4432	321	5	l1	l1	PROPN
ejpam-4432	321	6	;	;	PUNCT
ejpam-4432	321	7	∧̄1,∨1	∧̄1,∨1	PROPN
ejpam-4432	321	8	,	,	PUNCT
ejpam-4432	321	9	01	01	NUM
ejpam-4432	321	10	,	,	PUNCT
ejpam-4432	321	11	11	11	NUM
ejpam-4432	321	12	>	>	X
ejpam-4432	321	13	and	and	CCONJ
ejpam-4432	321	14	l2	l2	NOUN
ejpam-4432	321	15	=	=	NOUN
ejpam-4432	321	16	<	<	X
ejpam-4432	321	17	l2	l2	NOUN
ejpam-4432	321	18	;	;	PUNCT
ejpam-4432	321	19	∧̄2,∨2	∧̄2,∨2	NOUN
ejpam-4432	321	20	,	,	PUNCT
ejpam-4432	321	21	02	02	NUM
ejpam-4432	321	22	,	,	PUNCT
ejpam-4432	321	23	12	12	NUM
ejpam-4432	321	24	>	>	PUNCT
ejpam-4432	321	25	be	be	AUX
ejpam-4432	321	26	two	two	NUM
ejpam-4432	321	27	meethyperlattices	meethyperlattice	NOUN
ejpam-4432	321	28	.	.	PUNCT
ejpam-4432	322	1	then	then	ADV
ejpam-4432	322	2	the	the	DET
ejpam-4432	322	3	product	product	NOUN
ejpam-4432	322	4	l1	l1	PROPN
ejpam-4432	322	5	×	×	PROPN
ejpam-4432	322	6	l2	l2	NOUN
ejpam-4432	322	7	with	with	ADP
ejpam-4432	322	8	respect	respect	NOUN
ejpam-4432	322	9	to	to	ADP
ejpam-4432	322	10	the	the	DET
ejpam-4432	322	11	pair	pair	NOUN
ejpam-4432	322	12	-	-	PUNCT
ejpam-4432	322	13	wise	wise	ADJ
ejpam-4432	322	14	operations	operation	NOUN
ejpam-4432	322	15	such	such	ADJ
ejpam-4432	322	16	that	that	PRON
ejpam-4432	322	17	for	for	ADP
ejpam-4432	322	18	any	any	DET
ejpam-4432	322	19	(	(	PUNCT
ejpam-4432	322	20	a	a	PRON
ejpam-4432	322	21	,	,	PUNCT
ejpam-4432	322	22	b	b	NOUN
ejpam-4432	322	23	)	)	PUNCT
ejpam-4432	322	24	,	,	PUNCT
ejpam-4432	322	25	(	(	PUNCT
ejpam-4432	322	26	a′	a′	PROPN
ejpam-4432	322	27	,	,	PUNCT
ejpam-4432	322	28	b′	b′	NUM
ejpam-4432	322	29	)	)	PUNCT
ejpam-4432	322	30	∈	∈	PROPN
ejpam-4432	322	31	l1	l1	PROPN
ejpam-4432	322	32	×	×	PROPN
ejpam-4432	322	33	l2	l2	NOUN
ejpam-4432	322	34	:	:	PUNCT
ejpam-4432	322	35	(	(	PUNCT
ejpam-4432	322	36	a	a	DET
ejpam-4432	322	37	,	,	PUNCT
ejpam-4432	322	38	b)∧̄(a′	b)∧̄(a′	NOUN
ejpam-4432	322	39	,	,	PUNCT
ejpam-4432	322	40	b′	b′	NUM
ejpam-4432	322	41	)	)	PUNCT
ejpam-4432	322	42	=	=	SYM
ejpam-4432	323	1	(	(	PUNCT
ejpam-4432	323	2	a∧̄1a	a∧̄1a	ADP
ejpam-4432	323	3	′	′	NUM
ejpam-4432	323	4	,	,	PUNCT
ejpam-4432	323	5	b∧̄2b	b∧̄2b	PROPN
ejpam-4432	323	6	′	′	NUM
ejpam-4432	323	7	)	)	PUNCT
ejpam-4432	323	8	,	,	PUNCT
ejpam-4432	323	9	and	and	CCONJ
ejpam-4432	323	10	(	(	PUNCT
ejpam-4432	323	11	a	a	DET
ejpam-4432	323	12	,	,	PUNCT
ejpam-4432	323	13	b	b	NOUN
ejpam-4432	323	14	)	)	PUNCT
ejpam-4432	323	15	∨	∨	NOUN
ejpam-4432	323	16	(	(	PUNCT
ejpam-4432	323	17	a′	a′	PROPN
ejpam-4432	323	18	,	,	PUNCT
ejpam-4432	323	19	b′	b′	NUM
ejpam-4432	323	20	)	)	PUNCT
ejpam-4432	323	21	=	=	SYM
ejpam-4432	323	22	(	(	PUNCT
ejpam-4432	323	23	a	a	DET
ejpam-4432	323	24	∨1	∨1	PROPN
ejpam-4432	323	25	a	a	DET
ejpam-4432	323	26	′	′	PROPN
ejpam-4432	323	27	,	,	PUNCT
ejpam-4432	323	28	b	b	PROPN
ejpam-4432	323	29	∨2	∨2	ADP
ejpam-4432	323	30	b	b	NOUN
ejpam-4432	323	31	′	′	NUM
ejpam-4432	323	32	)	)	PUNCT
ejpam-4432	323	33	,	,	PUNCT
ejpam-4432	323	34	forms	form	NOUN
ejpam-4432	323	35	meet	meet	ADJ
ejpam-4432	323	36	-	-	PUNCT
ejpam-4432	323	37	hyperlattice	hyperlattice	NOUN
ejpam-4432	323	38	called	call	VERB
ejpam-4432	323	39	the	the	DET
ejpam-4432	323	40	prouduct	prouduct	NOUN
ejpam-4432	323	41	meet	meet	NOUN
ejpam-4432	323	42	-	-	PUNCT
ejpam-4432	323	43	hyperlattice	hyperlattice	NOUN
ejpam-4432	323	44	of	of	ADP
ejpam-4432	323	45	l1	l1	PROPN
ejpam-4432	323	46	and	and	CCONJ
ejpam-4432	323	47	l2	l2	NOUN
ejpam-4432	323	48	with	with	ADP
ejpam-4432	323	49	zero	zero	NUM
ejpam-4432	323	50	element	element	NOUN
ejpam-4432	323	51	0	0	NUM
ejpam-4432	324	1	=	=	SYM
ejpam-4432	324	2	(	(	PUNCT
ejpam-4432	324	3	01	01	NUM
ejpam-4432	324	4	,	,	PUNCT
ejpam-4432	324	5	02	02	NUM
ejpam-4432	324	6	)	)	PUNCT
ejpam-4432	324	7	,	,	PUNCT
ejpam-4432	324	8	and	and	CCONJ
ejpam-4432	324	9	one	one	NUM
ejpam-4432	324	10	element1	element1	NOUN
ejpam-4432	324	11	=	=	SYM
ejpam-4432	324	12	(	(	PUNCT
ejpam-4432	324	13	11	11	NUM
ejpam-4432	324	14	,	,	PUNCT
ejpam-4432	324	15	12	12	NUM
ejpam-4432	324	16	)	)	PUNCT
ejpam-4432	324	17	.	.	PUNCT
ejpam-4432	325	1	notice	notice	VERB
ejpam-4432	325	2	that	that	SCONJ
ejpam-4432	325	3	,	,	PUNCT
ejpam-4432	325	4	if	if	SCONJ
ejpam-4432	325	5	l1	l1	PROPN
ejpam-4432	325	6	and	and	CCONJ
ejpam-4432	325	7	l2	l2	NOUN
ejpam-4432	325	8	are	be	AUX
ejpam-4432	325	9	strong	strong	ADJ
ejpam-4432	325	10	bounded	bounded	ADJ
ejpam-4432	325	11	dual	dual	ADJ
ejpam-4432	325	12	distributive	distributive	ADJ
ejpam-4432	325	13	meet	meet	NOUN
ejpam-4432	325	14	-	-	PUNCT
ejpam-4432	325	15	hyperlattices	hyperlattice	NOUN
ejpam-4432	325	16	,	,	PUNCT
ejpam-4432	325	17	then	then	ADV
ejpam-4432	325	18	l1	l1	PROPN
ejpam-4432	325	19	×	×	PROPN
ejpam-4432	325	20	l2	l2	NOUN
ejpam-4432	325	21	is	be	AUX
ejpam-4432	325	22	also	also	ADV
ejpam-4432	325	23	too	too	ADV
ejpam-4432	325	24	.	.	PUNCT
ejpam-4432	326	1	theorem	theorem	VERB
ejpam-4432	326	2	7	7	NUM
ejpam-4432	326	3	.	.	X
ejpam-4432	326	4	for	for	ADP
ejpam-4432	326	5	any	any	DET
ejpam-4432	326	6	two	two	NUM
ejpam-4432	326	7	hyperideals	hyperideal	NOUN
ejpam-4432	326	8	i1	i1	PROPN
ejpam-4432	326	9	and	and	CCONJ
ejpam-4432	326	10	i2	i2	PROPN
ejpam-4432	326	11	of	of	ADP
ejpam-4432	326	12	two	two	NUM
ejpam-4432	326	13	meet	meet	ADJ
ejpam-4432	326	14	-	-	PUNCT
ejpam-4432	326	15	hyperlattices	hyperlattice	NOUN
ejpam-4432	326	16	l1	l1	NOUN
ejpam-4432	326	17	and	and	CCONJ
ejpam-4432	326	18	l2	l2	NOUN
ejpam-4432	326	19	respectively	respectively	ADV
ejpam-4432	326	20	.	.	PUNCT
ejpam-4432	327	1	i1	i1	PROPN
ejpam-4432	327	2	and	and	CCONJ
ejpam-4432	327	3	i2	i2	PROPN
ejpam-4432	327	4	are	be	AUX
ejpam-4432	327	5	annihilator	annihilator	PROPN
ejpam-4432	327	6	hyperideals	hyperideal	NOUN
ejpam-4432	327	7	iff	iff	PROPN
ejpam-4432	327	8	i1	i1	PROPN
ejpam-4432	327	9	×	×	PROPN
ejpam-4432	327	10	i2	i2	PROPN
ejpam-4432	327	11	is	be	AUX
ejpam-4432	327	12	an	an	DET
ejpam-4432	327	13	annihilator	annihilator	PROPN
ejpam-4432	327	14	hyperideal	hyperideal	NOUN
ejpam-4432	327	15	of	of	ADP
ejpam-4432	327	16	the	the	DET
ejpam-4432	327	17	product	product	NOUN
ejpam-4432	327	18	meet	meet	NOUN
ejpam-4432	327	19	-	-	PUNCT
ejpam-4432	327	20	hyperlattice	hyperlattice	NOUN
ejpam-4432	327	21	l1	l1	PROPN
ejpam-4432	327	22	×	×	PROPN
ejpam-4432	327	23	l2	l2	NOUN
ejpam-4432	327	24	.	.	PUNCT
ejpam-4432	328	1	proof	proof	NOUN
ejpam-4432	328	2	.	.	PUNCT
ejpam-4432	329	1	assume	assume	VERB
ejpam-4432	329	2	i1	i1	PROPN
ejpam-4432	329	3	and	and	CCONJ
ejpam-4432	329	4	i2	i2	PROPN
ejpam-4432	329	5	be	be	VERB
ejpam-4432	329	6	annihilator	annihilator	NOUN
ejpam-4432	329	7	hyperideals	hyperideal	NOUN
ejpam-4432	329	8	of	of	ADP
ejpam-4432	329	9	l1	l1	PROPN
ejpam-4432	329	10	and	and	CCONJ
ejpam-4432	329	11	l2	l2	NOUN
ejpam-4432	329	12	,	,	PUNCT
ejpam-4432	329	13	respectively	respectively	ADV
ejpam-4432	329	14	.	.	PUNCT
ejpam-4432	330	1	it	it	PRON
ejpam-4432	330	2	implies	imply	VERB
ejpam-4432	330	3	that	that	DET
ejpam-4432	330	4	i1	i1	PROPN
ejpam-4432	330	5	=	=	PUNCT
ejpam-4432	330	6	k1	k1	PROPN
ejpam-4432	330	7	r	r	NOUN
ejpam-4432	330	8	and	and	CCONJ
ejpam-4432	330	9	i2	i2	PROPN
ejpam-4432	330	10	=	=	SYM
ejpam-4432	330	11	k2	k2	PROPN
ejpam-4432	330	12	r	r	NOUN
ejpam-4432	330	13	for	for	ADP
ejpam-4432	330	14	some	some	DET
ejpam-4432	330	15	two	two	NUM
ejpam-4432	330	16	sets	set	NOUN
ejpam-4432	330	17	k1	k1	NOUN
ejpam-4432	330	18	⊆	⊆	NUM
ejpam-4432	330	19	l1	l1	PROPN
ejpam-4432	330	20	and	and	CCONJ
ejpam-4432	330	21	k2	k2	ADJ
ejpam-4432	330	22	⊆	⊆	NUM
ejpam-4432	330	23	l2	l2	NOUN
ejpam-4432	330	24	.	.	PUNCT
ejpam-4432	331	1	i.e.	i.e.	X
ejpam-4432	331	2	,	,	PUNCT
ejpam-4432	331	3	ii	ii	X
ejpam-4432	331	4	=	=	SYM
ejpam-4432	331	5	{	{	PUNCT
ejpam-4432	331	6	xi	xi	PROPN
ejpam-4432	331	7	∈	∈	PROPN
ejpam-4432	331	8	li	li	PROPN
ejpam-4432	331	9	:	:	PUNCT
ejpam-4432	331	10	0i	0i	PROPN
ejpam-4432	331	11	∈	∈	PROPN
ejpam-4432	331	12	xi∧̄ai	xi∧̄ai	PROPN
ejpam-4432	331	13	for	for	ADP
ejpam-4432	331	14	all	all	PRON
ejpam-4432	331	15	ai	ai	VERB
ejpam-4432	331	16	∈	∈	PROPN
ejpam-4432	331	17	ki	ki	PROPN
ejpam-4432	331	18	}	}	PUNCT
ejpam-4432	331	19	,	,	PUNCT
ejpam-4432	331	20	for	for	ADP
ejpam-4432	331	21	i	i	PROPN
ejpam-4432	331	22	=	=	SYM
ejpam-4432	331	23	1	1	NUM
ejpam-4432	331	24	,	,	PUNCT
ejpam-4432	331	25	2	2	NUM
ejpam-4432	331	26	.	.	PUNCT
ejpam-4432	331	27	accordingly	accordingly	ADV
ejpam-4432	331	28	i1	i1	PROPN
ejpam-4432	331	29	×	×	PROPN
ejpam-4432	331	30	i2	i2	PROPN
ejpam-4432	331	31	=	=	PRON
ejpam-4432	331	32	{	{	PUNCT
ejpam-4432	331	33	(	(	PUNCT
ejpam-4432	331	34	x1	x1	PROPN
ejpam-4432	331	35	,	,	PUNCT
ejpam-4432	331	36	x2	x2	PROPN
ejpam-4432	331	37	)	)	PUNCT
ejpam-4432	331	38	∈	∈	PROPN
ejpam-4432	331	39	l1	l1	PROPN
ejpam-4432	331	40	×	×	PROPN
ejpam-4432	331	41	l2	l2	NOUN
ejpam-4432	331	42	:	:	PUNCT
ejpam-4432	331	43	(	(	PUNCT
ejpam-4432	331	44	01	01	NUM
ejpam-4432	331	45	,	,	PUNCT
ejpam-4432	331	46	02	02	NUM
ejpam-4432	331	47	)	)	PUNCT
ejpam-4432	331	48	∈	∈	PROPN
ejpam-4432	331	49	(	(	PUNCT
ejpam-4432	331	50	x1	x1	PROPN
ejpam-4432	331	51	,	,	PUNCT
ejpam-4432	331	52	x2)∧̄(a1	x2)∧̄(a1	PROPN
ejpam-4432	331	53	,	,	PUNCT
ejpam-4432	331	54	a2	a2	PROPN
ejpam-4432	331	55	)	)	PUNCT
ejpam-4432	331	56	for	for	ADP
ejpam-4432	331	57	all	all	DET
ejpam-4432	331	58	(	(	PUNCT
ejpam-4432	331	59	a1	a1	PROPN
ejpam-4432	331	60	,	,	PUNCT
ejpam-4432	331	61	a2	a2	PROPN
ejpam-4432	331	62	)	)	PUNCT
ejpam-4432	331	63	∈	∈	PROPN
ejpam-4432	331	64	k1	k1	PROPN
ejpam-4432	331	65	×	×	PROPN
ejpam-4432	331	66	k2	k2	PROPN
ejpam-4432	331	67	}	}	PUNCT
ejpam-4432	331	68	.	.	PUNCT
ejpam-4432	332	1	therefore	therefore	ADV
ejpam-4432	332	2	,	,	PUNCT
ejpam-4432	332	3	i1	i1	PROPN
ejpam-4432	332	4	×	×	PROPN
ejpam-4432	332	5	i2	i2	PROPN
ejpam-4432	332	6	is	be	AUX
ejpam-4432	332	7	an	an	DET
ejpam-4432	332	8	annihilator	annihilator	PROPN
ejpam-4432	332	9	hyperideal	hyperideal	NOUN
ejpam-4432	332	10	of	of	ADP
ejpam-4432	332	11	l1	l1	PROPN
ejpam-4432	332	12	×	×	PROPN
ejpam-4432	332	13	l2	l2	NOUN
ejpam-4432	332	14	.	.	PUNCT
ejpam-4432	333	1	conversely	conversely	ADV
ejpam-4432	333	2	,	,	PUNCT
ejpam-4432	333	3	let	let	VERB
ejpam-4432	333	4	i	i	PRON
ejpam-4432	333	5	be	be	AUX
ejpam-4432	333	6	an	an	DET
ejpam-4432	333	7	annihilator	annihilator	PROPN
ejpam-4432	333	8	hyperideal	hyperideal	PROPN
ejpam-4432	333	9	references	reference	NOUN
ejpam-4432	333	10	1415	1415	NUM
ejpam-4432	333	11	of	of	ADP
ejpam-4432	333	12	l1	l1	PROPN
ejpam-4432	333	13	×	×	PROPN
ejpam-4432	333	14	l2	l2	NOUN
ejpam-4432	333	15	.	.	PUNCT
ejpam-4432	334	1	it	it	PRON
ejpam-4432	334	2	means	mean	VERB
ejpam-4432	334	3	there	there	PRON
ejpam-4432	334	4	exist	exist	VERB
ejpam-4432	334	5	subset	subset	VERB
ejpam-4432	334	6	k	k	PROPN
ejpam-4432	334	7	⊆	⊆	NUM
ejpam-4432	334	8	l1	l1	PROPN
ejpam-4432	334	9	×	×	PROPN
ejpam-4432	334	10	l2	l2	NOUN
ejpam-4432	334	11	such	such	ADJ
ejpam-4432	334	12	that	that	SCONJ
ejpam-4432	334	13	i	i	PRON
ejpam-4432	334	14	=	=	SYM
ejpam-4432	334	15	kr	kr	PROPN
ejpam-4432	334	16	.	.	PUNCT
ejpam-4432	335	1	we	we	PRON
ejpam-4432	335	2	define	define	VERB
ejpam-4432	335	3	the	the	DET
ejpam-4432	335	4	projections	projection	NOUN
ejpam-4432	335	5	πi	πi	ADP
ejpam-4432	335	6	:	:	PUNCT
ejpam-4432	335	7	l1	l1	PROPN
ejpam-4432	335	8	×	×	PROPN
ejpam-4432	335	9	l2	l2	PROPN
ejpam-4432	335	10	→	→	SYM
ejpam-4432	335	11	li	li	PROPN
ejpam-4432	335	12	for	for	ADP
ejpam-4432	335	13	i	i	PROPN
ejpam-4432	335	14	=	=	SYM
ejpam-4432	335	15	1	1	NUM
ejpam-4432	335	16	,	,	PUNCT
ejpam-4432	335	17	2	2	NUM
ejpam-4432	335	18	.	.	PUNCT
ejpam-4432	335	19	let	let	VERB
ejpam-4432	335	20	i1	i1	PROPN
ejpam-4432	335	21	and	and	CCONJ
ejpam-4432	335	22	i2	i2	PROPN
ejpam-4432	335	23	be	be	VERB
ejpam-4432	335	24	the	the	DET
ejpam-4432	335	25	projections	projection	NOUN
ejpam-4432	335	26	of	of	ADP
ejpam-4432	335	27	i	i	PRON
ejpam-4432	335	28	on	on	ADP
ejpam-4432	335	29	l1	l1	PROPN
ejpam-4432	335	30	and	and	CCONJ
ejpam-4432	335	31	l1	l1	PROPN
ejpam-4432	335	32	respectively	respectively	ADV
ejpam-4432	335	33	.	.	PUNCT
ejpam-4432	336	1	in	in	ADP
ejpam-4432	336	2	addition	addition	NOUN
ejpam-4432	336	3	to	to	ADP
ejpam-4432	336	4	k1	k1	PROPN
ejpam-4432	336	5	and	and	CCONJ
ejpam-4432	336	6	k2	k2	PROPN
ejpam-4432	336	7	are	be	AUX
ejpam-4432	336	8	the	the	DET
ejpam-4432	336	9	projections	projection	NOUN
ejpam-4432	336	10	of	of	ADP
ejpam-4432	336	11	k	k	PROPN
ejpam-4432	336	12	on	on	ADP
ejpam-4432	336	13	l1	l1	PROPN
ejpam-4432	336	14	and	and	CCONJ
ejpam-4432	336	15	l2	l2	NOUN
ejpam-4432	336	16	.	.	PUNCT
ejpam-4432	337	1	so	so	ADV
ejpam-4432	337	2	πi(i	πi(i	PUNCT
ejpam-4432	337	3	)	)	PUNCT
ejpam-4432	337	4	=	=	SYM
ejpam-4432	337	5	ii	ii	PROPN
ejpam-4432	337	6	and	and	CCONJ
ejpam-4432	337	7	πi(k	πi(k	NUM
ejpam-4432	337	8	)	)	PUNCT
ejpam-4432	338	1	=	=	SYM
ejpam-4432	338	2	ki	ki	PROPN
ejpam-4432	338	3	for	for	ADP
ejpam-4432	338	4	i	i	PROPN
ejpam-4432	338	5	=	=	NOUN
ejpam-4432	338	6	1	1	NUM
ejpam-4432	338	7	,	,	PUNCT
ejpam-4432	338	8	2	2	NUM
ejpam-4432	338	9	.	.	X
ejpam-4432	339	1	we	we	PRON
ejpam-4432	339	2	get	get	VERB
ejpam-4432	339	3	i	i	PRON
ejpam-4432	339	4	=	=	PUNCT
ejpam-4432	339	5	{	{	PUNCT
ejpam-4432	339	6	(	(	PUNCT
ejpam-4432	339	7	x1	x1	PROPN
ejpam-4432	339	8	,	,	PUNCT
ejpam-4432	339	9	x2	x2	PROPN
ejpam-4432	339	10	)	)	PUNCT
ejpam-4432	339	11	∈	∈	PROPN
ejpam-4432	339	12	l1	l1	PROPN
ejpam-4432	339	13	×	×	PROPN
ejpam-4432	339	14	l2	l2	NOUN
ejpam-4432	339	15	:	:	PUNCT
ejpam-4432	339	16	(	(	PUNCT
ejpam-4432	339	17	01	01	NUM
ejpam-4432	339	18	,	,	PUNCT
ejpam-4432	339	19	02	02	NUM
ejpam-4432	339	20	)	)	PUNCT
ejpam-4432	339	21	∈	∈	PROPN
ejpam-4432	339	22	(	(	PUNCT
ejpam-4432	339	23	x1	x1	PROPN
ejpam-4432	339	24	,	,	PUNCT
ejpam-4432	339	25	x2)∧̄(a1	x2)∧̄(a1	PROPN
ejpam-4432	339	26	,	,	PUNCT
ejpam-4432	339	27	a2	a2	PROPN
ejpam-4432	339	28	)	)	PUNCT
ejpam-4432	339	29	for	for	ADP
ejpam-4432	339	30	all	all	DET
ejpam-4432	339	31	(	(	PUNCT
ejpam-4432	339	32	a1	a1	PROPN
ejpam-4432	339	33	,	,	PUNCT
ejpam-4432	339	34	a2	a2	PROPN
ejpam-4432	339	35	)	)	PUNCT
ejpam-4432	339	36	∈	∈	PROPN
ejpam-4432	339	37	k1	k1	PROPN
ejpam-4432	339	38	×	×	PROPN
ejpam-4432	339	39	k2	k2	PROPN
ejpam-4432	339	40	}	}	PUNCT
ejpam-4432	339	41	.	.	PUNCT
ejpam-4432	340	1	thus	thus	ADV
ejpam-4432	340	2	πi(i	πi(i	PUNCT
ejpam-4432	340	3	)	)	PUNCT
ejpam-4432	340	4	=	=	SYM
ejpam-4432	340	5	{	{	PUNCT
ejpam-4432	340	6	xi	xi	PROPN
ejpam-4432	340	7	∈	∈	PROPN
ejpam-4432	340	8	l1	l1	PROPN
ejpam-4432	340	9	:	:	PUNCT
ejpam-4432	340	10	0i	0i	PROPN
ejpam-4432	340	11	∈	∈	PROPN
ejpam-4432	340	12	xi∧̄iai	xi∧̄iai	PROPN
ejpam-4432	340	13	for	for	ADP
ejpam-4432	340	14	all	all	PRON
ejpam-4432	340	15	ai	ai	VERB
ejpam-4432	340	16	∈	∈	PROPN
ejpam-4432	340	17	ki	ki	PROPN
ejpam-4432	340	18	}	}	PUNCT
ejpam-4432	340	19	.	.	PUNCT
ejpam-4432	341	1	therefore	therefore	ADV
ejpam-4432	341	2	i	i	PRON
ejpam-4432	341	3	∼=	∼=	PROPN
ejpam-4432	341	4	π1(i)×	π1(i)×	NOUN
ejpam-4432	341	5	π2(i	π2(i	NOUN
ejpam-4432	341	6	)	)	PUNCT
ejpam-4432	341	7	.	.	PUNCT
ejpam-4432	342	1	corollary	corollary	ADJ
ejpam-4432	342	2	3	3	NUM
ejpam-4432	342	3	.	.	PUNCT
ejpam-4432	343	1	for	for	ADP
ejpam-4432	343	2	any	any	DET
ejpam-4432	343	3	two	two	NUM
ejpam-4432	343	4	hyperideals	hyperideal	NOUN
ejpam-4432	343	5	i1	i1	PROPN
ejpam-4432	343	6	and	and	CCONJ
ejpam-4432	343	7	i2	i2	PROPN
ejpam-4432	343	8	of	of	ADP
ejpam-4432	343	9	two	two	NUM
ejpam-4432	343	10	meet	meet	ADJ
ejpam-4432	343	11	-	-	PUNCT
ejpam-4432	343	12	hyperlattices	hyperlattice	NOUN
ejpam-4432	343	13	l1	l1	NOUN
ejpam-4432	343	14	and	and	CCONJ
ejpam-4432	343	15	l2	l2	NOUN
ejpam-4432	343	16	respectively	respectively	ADV
ejpam-4432	343	17	.	.	PUNCT
ejpam-4432	344	1	i1	i1	PROPN
ejpam-4432	344	2	and	and	CCONJ
ejpam-4432	344	3	i2	i2	PROPN
ejpam-4432	344	4	are	be	AUX
ejpam-4432	344	5	closed	close	VERB
ejpam-4432	344	6	hyperideals	hyperideal	NOUN
ejpam-4432	344	7	iff	iff	PROPN
ejpam-4432	344	8	i1×i2	i1×i2	PROPN
ejpam-4432	344	9	is	be	AUX
ejpam-4432	344	10	a	a	DET
ejpam-4432	344	11	closed	closed	ADJ
ejpam-4432	344	12	hyperideal	hyperideal	NOUN
ejpam-4432	344	13	of	of	ADP
ejpam-4432	344	14	the	the	DET
ejpam-4432	344	15	product	product	NOUN
ejpam-4432	344	16	meet	meet	NOUN
ejpam-4432	344	17	-	-	PUNCT
ejpam-4432	344	18	hyperlattice	hyperlattice	NOUN
ejpam-4432	344	19	l1	l1	PROPN
ejpam-4432	344	20	×	×	PROPN
ejpam-4432	344	21	l2	l2	NOUN
ejpam-4432	344	22	.	.	PUNCT
ejpam-4432	345	1	6	6	X
ejpam-4432	345	2	.	.	X
ejpam-4432	345	3	conclusion	conclusion	NOUN
ejpam-4432	345	4	in	in	ADP
ejpam-4432	345	5	our	our	PRON
ejpam-4432	345	6	work	work	NOUN
ejpam-4432	345	7	,	,	PUNCT
ejpam-4432	345	8	the	the	DET
ejpam-4432	345	9	properties	property	NOUN
ejpam-4432	345	10	of	of	ADP
ejpam-4432	345	11	annihilator	annihilator	PROPN
ejpam-4432	345	12	hyperideals	hyperideal	NOUN
ejpam-4432	345	13	in	in	ADP
ejpam-4432	345	14	the	the	DET
ejpam-4432	345	15	class	class	NOUN
ejpam-4432	345	16	of	of	ADP
ejpam-4432	345	17	strong	strong	ADJ
ejpam-4432	345	18	bounded	bounded	ADJ
ejpam-4432	345	19	dual	dual	ADJ
ejpam-4432	345	20	distributive	distributive	ADJ
ejpam-4432	345	21	meet	meet	NOUN
ejpam-4432	345	22	-	-	PUNCT
ejpam-4432	345	23	hyperlattice	hyperlattice	NOUN
ejpam-4432	345	24	were	be	AUX
ejpam-4432	345	25	studied	study	VERB
ejpam-4432	345	26	and	and	CCONJ
ejpam-4432	345	27	proved	prove	VERB
ejpam-4432	345	28	.	.	PUNCT
ejpam-4432	346	1	the	the	DET
ejpam-4432	346	2	connection	connection	NOUN
ejpam-4432	346	3	between	between	ADP
ejpam-4432	346	4	closed	close	VERB
ejpam-4432	346	5	hyperideals	hyperideal	NOUN
ejpam-4432	346	6	and	and	CCONJ
ejpam-4432	346	7	annihilators	annihilator	NOUN
ejpam-4432	346	8	was	be	AUX
ejpam-4432	346	9	discovered	discover	VERB
ejpam-4432	346	10	.	.	PUNCT
ejpam-4432	347	1	in	in	ADP
ejpam-4432	347	2	general	general	ADJ
ejpam-4432	347	3	,	,	PUNCT
ejpam-4432	347	4	the	the	DET
ejpam-4432	347	5	set	set	NOUN
ejpam-4432	347	6	of	of	ADP
ejpam-4432	347	7	closed	closed	ADJ
ejpam-4432	347	8	hyperideals	hyperideal	NOUN
ejpam-4432	347	9	is	be	AUX
ejpam-4432	347	10	a	a	DET
ejpam-4432	347	11	subset	subset	NOUN
ejpam-4432	347	12	of	of	ADP
ejpam-4432	347	13	the	the	DET
ejpam-4432	347	14	set	set	NOUN
ejpam-4432	347	15	of	of	ADP
ejpam-4432	347	16	all	all	DET
ejpam-4432	347	17	annihilators	annihilator	NOUN
ejpam-4432	347	18	.	.	PUNCT
ejpam-4432	348	1	it	it	PRON
ejpam-4432	348	2	also	also	ADV
ejpam-4432	348	3	forms	form	VERB
ejpam-4432	348	4	a	a	DET
ejpam-4432	348	5	boolean	boolean	ADJ
ejpam-4432	348	6	algebra	algebra	NOUN
ejpam-4432	348	7	.	.	PUNCT
ejpam-4432	349	1	we	we	PRON
ejpam-4432	349	2	introduce	introduce	VERB
ejpam-4432	349	3	the	the	DET
ejpam-4432	349	4	concept	concept	NOUN
ejpam-4432	349	5	of	of	ADP
ejpam-4432	349	6	homomorphisms	homomorphism	NOUN
ejpam-4432	349	7	,	,	PUNCT
ejpam-4432	349	8	which	which	PRON
ejpam-4432	349	9	preserve	preserve	VERB
ejpam-4432	349	10	the	the	DET
ejpam-4432	349	11	annihilator	annihilator	PROPN
ejpam-4432	349	12	hyperideal	hyperideal	PROPN
ejpam-4432	349	13	.	.	PUNCT
ejpam-4432	350	1	suitable	suitable	ADJ
ejpam-4432	350	2	conditions	condition	NOUN
ejpam-4432	350	3	for	for	ADP
ejpam-4432	350	4	preserving	preserve	VERB
ejpam-4432	350	5	annihilator	annihilator	NOUN
ejpam-4432	350	6	hyperideals	hyperideal	NOUN
ejpam-4432	350	7	are	be	AUX
ejpam-4432	350	8	obtained	obtain	VERB
ejpam-4432	350	9	.	.	PUNCT
ejpam-4432	351	1	under	under	ADP
ejpam-4432	351	2	these	these	DET
ejpam-4432	351	3	conditions	condition	NOUN
ejpam-4432	351	4	,	,	PUNCT
ejpam-4432	351	5	homomorphic	homomorphic	ADJ
ejpam-4432	351	6	images	image	NOUN
ejpam-4432	351	7	and	and	CCONJ
ejpam-4432	351	8	preimages	preimage	NOUN
ejpam-4432	351	9	of	of	ADP
ejpam-4432	351	10	annihilators	annihilator	NOUN
ejpam-4432	351	11	are	be	AUX
ejpam-4432	351	12	also	also	ADV
ejpam-4432	351	13	annihilators	annihilators	PROPN
ejpam-4432	351	14	.	.	PUNCT
ejpam-4432	352	1	accordingly	accordingly	ADV
ejpam-4432	352	2	,	,	PUNCT
ejpam-4432	352	3	if	if	SCONJ
ejpam-4432	352	4	there	there	PRON
ejpam-4432	352	5	exists	exist	VERB
ejpam-4432	352	6	an	an	DET
ejpam-4432	352	7	annihilator	annihilator	PROPN
ejpam-4432	352	8	hyperideal	hyperideal	NOUN
ejpam-4432	352	9	preserving	preserve	VERB
ejpam-4432	352	10	map	map	NOUN
ejpam-4432	352	11	between	between	ADP
ejpam-4432	352	12	two	two	NUM
ejpam-4432	352	13	meet	meet	NOUN
ejpam-4432	352	14	-	-	PUNCT
ejpam-4432	352	15	hyperlattices	hyperlattice	NOUN
ejpam-4432	352	16	,	,	PUNCT
ejpam-4432	352	17	then	then	ADV
ejpam-4432	352	18	there	there	PRON
ejpam-4432	352	19	is	be	VERB
ejpam-4432	352	20	a	a	DET
ejpam-4432	352	21	one	one	NUM
ejpam-4432	352	22	-	-	PUNCT
ejpam-4432	352	23	to	to	ADP
ejpam-4432	352	24	-	-	PUNCT
ejpam-4432	352	25	one	one	NUM
ejpam-4432	352	26	correspondence	correspondence	NOUN
ejpam-4432	352	27	between	between	ADP
ejpam-4432	352	28	their	their	PRON
ejpam-4432	352	29	boolean	boolean	ADJ
ejpam-4432	352	30	algebras	algebra	NOUN
ejpam-4432	352	31	of	of	ADP
ejpam-4432	352	32	closed	closed	ADJ
ejpam-4432	352	33	hyperideals	hyperideal	NOUN
ejpam-4432	352	34	.	.	PUNCT
ejpam-4432	353	1	representation	representation	NOUN
ejpam-4432	353	2	and	and	CCONJ
ejpam-4432	353	3	characterization	characterization	NOUN
ejpam-4432	353	4	of	of	ADP
ejpam-4432	353	5	annihilator	annihilator	PROPN
ejpam-4432	353	6	hyperideals	hyperideal	NOUN
ejpam-4432	353	7	under	under	ADP
ejpam-4432	353	8	product	product	NOUN
ejpam-4432	353	9	and	and	CCONJ
ejpam-4432	353	10	sub	sub	NOUN
ejpam-4432	353	11	-	-	NOUN
ejpam-4432	353	12	structure	structure	NOUN
ejpam-4432	353	13	of	of	ADP
ejpam-4432	353	14	meet	meet	ADJ
ejpam-4432	353	15	-	-	PUNCT
ejpam-4432	353	16	hyperlattice	hyperlattice	NOUN
ejpam-4432	353	17	are	be	AUX
ejpam-4432	353	18	shown	show	VERB
ejpam-4432	353	19	.	.	PUNCT
ejpam-4432	354	1	acknowledgements	acknowledgement	NOUN
ejpam-4432	354	2	the	the	DET
ejpam-4432	354	3	authors	author	NOUN
ejpam-4432	354	4	appreciate	appreciate	VERB
ejpam-4432	354	5	the	the	DET
ejpam-4432	354	6	referees	referee	NOUN
ejpam-4432	354	7	’	’	PART
ejpam-4432	354	8	comments	comment	NOUN
ejpam-4432	354	9	that	that	PRON
ejpam-4432	354	10	contributed	contribute	VERB
ejpam-4432	354	11	to	to	PART
ejpam-4432	354	12	improve	improve	VERB
ejpam-4432	354	13	the	the	DET
ejpam-4432	354	14	paper	paper	NOUN
ejpam-4432	354	15	.	.	PUNCT
ejpam-4432	355	1	references	reference	NOUN
ejpam-4432	355	2	[	[	X
ejpam-4432	355	3	1	1	NUM
ejpam-4432	355	4	]	]	X
ejpam-4432	355	5	b	b	NOUN
ejpam-4432	355	6	a	a	DET
ejpam-4432	355	7	davey	davey	NOUN
ejpam-4432	355	8	and	and	CCONJ
ejpam-4432	355	9	j	j	PROPN
ejpam-4432	355	10	nieminen	nieminen	PROPN
ejpam-4432	355	11	.	.	PUNCT
ejpam-4432	356	1	annihilators	annihilator	NOUN
ejpam-4432	356	2	in	in	ADP
ejpam-4432	356	3	modular	modular	ADJ
ejpam-4432	356	4	lattices	lattice	NOUN
ejpam-4432	356	5	.	.	PUNCT
ejpam-4432	357	1	algebra	algebra	PROPN
ejpam-4432	357	2	universalis	universali	VERB
ejpam-4432	357	3	,	,	PUNCT
ejpam-4432	357	4	22(2	22(2	NOUN
ejpam-4432	357	5	-	-	NOUN
ejpam-4432	357	6	3):154–158	3):154–158	NUM
ejpam-4432	357	7	,	,	PUNCT
ejpam-4432	357	8	1986	1986	NUM
ejpam-4432	357	9	.	.	PUNCT
ejpam-4432	358	1	[	[	X
ejpam-4432	358	2	2	2	NUM
ejpam-4432	358	3	]	]	SYM
ejpam-4432	358	4	r	r	NOUN
ejpam-4432	358	5	balbes	balbe	NOUN
ejpam-4432	358	6	and	and	CCONJ
ejpam-4432	358	7	p	p	NOUN
ejpam-4432	358	8	dwinger	dwinger	NOUN
ejpam-4432	358	9	.	.	PUNCT
ejpam-4432	359	1	distributive	distributive	ADJ
ejpam-4432	359	2	lattice	lattice	PROPN
ejpam-4432	359	3	.	.	PUNCT
ejpam-4432	360	1	univ	univ	PROPN
ejpam-4432	360	2	.	.	PROPN
ejpam-4432	360	3	of	of	ADP
ejpam-4432	360	4	press	press	PROPN
ejpam-4432	360	5	,	,	PUNCT
ejpam-4432	360	6	columbia	columbia	PROPN
ejpam-4432	360	7	,	,	PUNCT
ejpam-4432	360	8	1974	1974	NUM
ejpam-4432	360	9	.	.	PUNCT
ejpam-4432	361	1	[	[	X
ejpam-4432	361	2	3	3	X
ejpam-4432	361	3	]	]	X
ejpam-4432	361	4	g	g	PROPN
ejpam-4432	361	5	c	c	PROPN
ejpam-4432	361	6	rao	rao	PROPN
ejpam-4432	361	7	and	and	CCONJ
ejpam-4432	361	8	m	m	PROPN
ejpam-4432	361	9	sambasiva	sambasiva	PROPN
ejpam-4432	361	10	rao	rao	PROPN
ejpam-4432	361	11	.	.	PUNCT
ejpam-4432	362	1	annihilator	annihilator	PROPN
ejpam-4432	362	2	ideals	ideal	NOUN
ejpam-4432	362	3	in	in	ADP
ejpam-4432	362	4	almost	almost	ADV
ejpam-4432	362	5	distributive	distributive	ADJ
ejpam-4432	362	6	lattices	lattice	NOUN
ejpam-4432	362	7	.	.	PUNCT
ejpam-4432	363	1	international	international	ADJ
ejpam-4432	363	2	mathematical	mathematical	PROPN
ejpam-4432	363	3	forum	forum	PROPN
ejpam-4432	363	4	,	,	PUNCT
ejpam-4432	363	5	4:733–746	4:733–746	PROPN
ejpam-4432	363	6	,	,	PUNCT
ejpam-4432	363	7	2009	2009	NUM
ejpam-4432	363	8	.	.	PUNCT
ejpam-4432	364	1	[	[	X
ejpam-4432	364	2	4	4	X
ejpam-4432	364	3	]	]	SYM
ejpam-4432	364	4	g	g	PROPN
ejpam-4432	364	5	n	n	PRON
ejpam-4432	364	6	rao	rao	NOUN
ejpam-4432	364	7	and	and	CCONJ
ejpam-4432	364	8	r	r	NOUN
ejpam-4432	364	9	v	v	ADP
ejpam-4432	364	10	a	a	DET
ejpam-4432	364	11	raju	raju	NOUN
ejpam-4432	364	12	.	.	PUNCT
ejpam-4432	365	1	annihilator	annihilator	PROPN
ejpam-4432	365	2	ideals	ideal	NOUN
ejpam-4432	365	3	in	in	ADP
ejpam-4432	365	4	0	0	NOUN
ejpam-4432	365	5	-	-	PUNCT
ejpam-4432	365	6	distributive	distributive	ADJ
ejpam-4432	365	7	almost	almost	ADV
ejpam-4432	365	8	lattices	lattice	NOUN
ejpam-4432	365	9	.	.	PUNCT
ejpam-4432	366	1	bull	bull	NOUN
ejpam-4432	366	2	.	.	PUNCT
ejpam-4432	367	1	int	int	NOUN
ejpam-4432	367	2	.	.	PUNCT
ejpam-4432	368	1	math	math	NOUN
ejpam-4432	368	2	.	.	PUNCT
ejpam-4432	369	1	virtual	virtual	ADJ
ejpam-4432	369	2	inst	inst	NOUN
ejpam-4432	369	3	,	,	PUNCT
ejpam-4432	369	4	11(1):1–13	11(1):1–13	NUM
ejpam-4432	369	5	,	,	PUNCT
ejpam-4432	369	6	2021	2021	NUM
ejpam-4432	369	7	.	.	PUNCT
ejpam-4432	370	1	[	[	X
ejpam-4432	370	2	5	5	NUM
ejpam-4432	370	3	]	]	PUNCT
ejpam-4432	370	4	g	g	NOUN
ejpam-4432	370	5	gretzer	gretzer	NOUN
ejpam-4432	370	6	.	.	PUNCT
ejpam-4432	371	1	lattice	lattice	PROPN
ejpam-4432	371	2	theory	theory	NOUN
ejpam-4432	371	3	:	:	PUNCT
ejpam-4432	371	4	foundation	foundation	NOUN
ejpam-4432	371	5	.	.	PUNCT
ejpam-4432	372	1	springer	springer	PROPN
ejpam-4432	372	2	basel	basel	PROPN
ejpam-4432	372	3	,	,	PUNCT
ejpam-4432	372	4	2011	2011	NUM
ejpam-4432	372	5	.	.	PUNCT
ejpam-4432	373	1	references	reference	NOUN
ejpam-4432	373	2	1416	1416	NUM
ejpam-4432	373	3	[	[	X
ejpam-4432	373	4	6	6	NUM
ejpam-4432	373	5	]	]	X
ejpam-4432	373	6	r	r	NOUN
ejpam-4432	373	7	halaˇs	halaˇs	NOUN
ejpam-4432	373	8	.	.	PUNCT
ejpam-4432	374	1	annihilators	annihilator	NOUN
ejpam-4432	374	2	and	and	CCONJ
ejpam-4432	374	3	ideals	ideal	NOUN
ejpam-4432	374	4	in	in	ADP
ejpam-4432	374	5	ordered	order	VERB
ejpam-4432	374	6	sets	set	NOUN
ejpam-4432	374	7	.	.	PUNCT
ejpam-4432	375	1	czechoslovak	czechoslovak	ADJ
ejpam-4432	375	2	mathematical	mathematical	PROPN
ejpam-4432	375	3	journal	journal	PROPN
ejpam-4432	375	4	,	,	PUNCT
ejpam-4432	375	5	45:127–134	45:127–134	PROPN
ejpam-4432	375	6	,	,	PUNCT
ejpam-4432	375	7	1995	1995	NUM
ejpam-4432	375	8	.	.	PUNCT
ejpam-4432	376	1	[	[	X
ejpam-4432	376	2	7	7	NUM
ejpam-4432	376	3	]	]	X
ejpam-4432	376	4	r	r	NOUN
ejpam-4432	376	5	halaˇs	halaˇs	NOUN
ejpam-4432	376	6	.	.	PUNCT
ejpam-4432	377	1	annihilators	annihilators	PROPN
ejpam-4432	377	2	inbck	inbck	PROPN
ejpam-4432	377	3	-	-	PUNCT
ejpam-4432	377	4	algebras	algebras	PROPN
ejpam-4432	377	5	.	.	PUNCT
ejpam-4432	378	1	czechoslovak	czechoslovak	PROPN
ejpam-4432	378	2	mathematical	mathematical	PROPN
ejpam-4432	378	3	journal	journal	PROPN
ejpam-4432	378	4	,	,	PUNCT
ejpam-4432	378	5	53(128):1001–1007	53(128):1001–1007	NOUN
ejpam-4432	378	6	,	,	PUNCT
ejpam-4432	378	7	2003	2003	NUM
ejpam-4432	378	8	.	.	PUNCT
ejpam-4432	379	1	[	[	X
ejpam-4432	379	2	8	8	NUM
ejpam-4432	379	3	]	]	X
ejpam-4432	379	4	j	j	PROPN
ejpam-4432	379	5	huckaba	huckaba	PROPN
ejpam-4432	379	6	and	and	CCONJ
ejpam-4432	379	7	j	j	PROPN
ejpam-4432	379	8	keller	keller	PROPN
ejpam-4432	379	9	.	.	PUNCT
ejpam-4432	380	1	annihilator	annihilator	NOUN
ejpam-4432	380	2	of	of	ADP
ejpam-4432	380	3	deals	deal	NOUN
ejpam-4432	380	4	in	in	ADP
ejpam-4432	380	5	commutative	commutative	ADJ
ejpam-4432	380	6	rings	ring	NOUN
ejpam-4432	380	7	.	.	PUNCT
ejpam-4432	381	1	pacific	pacific	PROPN
ejpam-4432	381	2	journal	journal	PROPN
ejpam-4432	381	3	of	of	ADP
ejpam-4432	381	4	mathematics	mathematic	NOUN
ejpam-4432	381	5	,	,	PUNCT
ejpam-4432	381	6	83(2	83(2	NUM
ejpam-4432	381	7	)	)	PUNCT
ejpam-4432	381	8	,	,	PUNCT
ejpam-4432	381	9	1979	1979	NUM
ejpam-4432	381	10	.	.	PUNCT
ejpam-4432	382	1	[	[	X
ejpam-4432	382	2	9	9	NUM
ejpam-4432	382	3	]	]	X
ejpam-4432	382	4	m	m	NOUN
ejpam-4432	382	5	konstantinidou	konstantinidou	NOUN
ejpam-4432	382	6	-	-	PUNCT
ejpam-4432	382	7	serafimidou	serafimidou	NOUN
ejpam-4432	382	8	.	.	PUNCT
ejpam-4432	383	1	modular	modular	ADJ
ejpam-4432	383	2	hyperlattices	hyperlattice	NOUN
ejpam-4432	383	3	,	,	PUNCT
ejpam-4432	383	4	γ	γ	PROPN
ejpam-4432	383	5	.	.	PROPN
ejpam-4432	383	6	praktika	praktika	PROPN
ejpam-4432	383	7	tes	tes	PROPN
ejpam-4432	383	8	akademias	akademias	PROPN
ejpam-4432	383	9	athenon	athenon	PROPN
ejpam-4432	383	10	,	,	PUNCT
ejpam-4432	383	11	53:202–218	53:202–218	PROPN
ejpam-4432	383	12	,	,	PUNCT
ejpam-4432	383	13	1978	1978	NUM
ejpam-4432	383	14	.	.	PUNCT
ejpam-4432	384	1	[	[	X
ejpam-4432	384	2	10	10	NUM
ejpam-4432	384	3	]	]	X
ejpam-4432	384	4	m	m	NOUN
ejpam-4432	384	5	konstantinidou	konstantinidou	NOUN
ejpam-4432	384	6	-	-	PUNCT
ejpam-4432	384	7	serafimidou	serafimidou	NOUN
ejpam-4432	384	8	.	.	PUNCT
ejpam-4432	385	1	distributive	distributive	ADJ
ejpam-4432	385	2	and	and	CCONJ
ejpam-4432	385	3	complemented	complemented	ADJ
ejpam-4432	385	4	hyperlattices	hyperlattice	NOUN
ejpam-4432	385	5	.	.	PUNCT
ejpam-4432	386	1	praktika	praktika	NOUN
ejpam-4432	386	2	tes	tes	PROPN
ejpam-4432	386	3	akademias	akademias	PROPN
ejpam-4432	386	4	athenon	athenon	PROPN
ejpam-4432	386	5	,	,	PUNCT
ejpam-4432	386	6	56:339–360	56:339–360	PROPN
ejpam-4432	386	7	,	,	PUNCT
ejpam-4432	386	8	1981	1981	NUM
ejpam-4432	386	9	.	.	PUNCT
ejpam-4432	387	1	[	[	X
ejpam-4432	387	2	11	11	NUM
ejpam-4432	387	3	]	]	PUNCT
ejpam-4432	387	4	m	m	VERB
ejpam-4432	387	5	amiri	amiri	ADV
ejpam-4432	387	6	bideshki	bideshki	PROPN
ejpam-4432	387	7	and	and	CCONJ
ejpam-4432	387	8	r	r	PROPN
ejpam-4432	387	9	ameri	ameri	PROPN
ejpam-4432	387	10	and	and	CCONJ
ejpam-4432	387	11	a	a	DET
ejpam-4432	387	12	borumand	borumand	ADJ
ejpam-4432	387	13	saeid	saeid	PROPN
ejpam-4432	387	14	.	.	PUNCT
ejpam-4432	388	1	on	on	ADP
ejpam-4432	388	2	prime	prime	ADJ
ejpam-4432	388	3	hyperfilters	hyperfilter	NOUN
ejpam-4432	388	4	(	(	PUNCT
ejpam-4432	388	5	hyperideals	hyperideal	NOUN
ejpam-4432	388	6	)	)	PUNCT
ejpam-4432	388	7	in	in	ADP
ejpam-4432	388	8	∧hyperlattices	∧hyperlattices	PROPN
ejpam-4432	388	9	.	.	PUNCT
ejpam-4432	389	1	european	european	PROPN
ejpam-4432	389	2	j.	j.	PROPN
ejpam-4432	389	3	of	of	ADP
ejpam-4432	389	4	pure	pure	ADJ
ejpam-4432	389	5	and	and	CCONJ
ejpam-4432	389	6	app	app	PROPN
ejpam-4432	389	7	.	.	PROPN
ejpam-4432	389	8	math	math	PROPN
ejpam-4432	389	9	,	,	PUNCT
ejpam-4432	389	10	11(1):169	11(1):169	NUM
ejpam-4432	389	11	–	–	PUNCT
ejpam-4432	389	12	188	188	NUM
ejpam-4432	389	13	,	,	PUNCT
ejpam-4432	389	14	2018	2018	NUM
ejpam-4432	389	15	.	.	PUNCT
ejpam-4432	390	1	[	[	X
ejpam-4432	390	2	12	12	NUM
ejpam-4432	390	3	]	]	X
ejpam-4432	390	4	m	m	NOUN
ejpam-4432	390	5	konstantinidou	konstantinidou	NOUN
ejpam-4432	390	6	-	-	PUNCT
ejpam-4432	390	7	serafimidou	serafimidou	NOUN
ejpam-4432	390	8	and	and	CCONJ
ejpam-4432	390	9	j	j	PROPN
ejpam-4432	390	10	mittas	mitta	NOUN
ejpam-4432	390	11	.	.	PUNCT
ejpam-4432	391	1	an	an	DET
ejpam-4432	391	2	introduction	introduction	NOUN
ejpam-4432	391	3	to	to	ADP
ejpam-4432	391	4	the	the	DET
ejpam-4432	391	5	theory	theory	NOUN
ejpam-4432	391	6	of	of	ADP
ejpam-4432	391	7	hyperlattices	hyperlattice	NOUN
ejpam-4432	391	8	.	.	PUNCT
ejpam-4432	392	1	math	math	NOUN
ejpam-4432	392	2	.	.	PUNCT
ejpam-4432	393	1	balcanica	balcanica	PROPN
ejpam-4432	393	2	,	,	PUNCT
ejpam-4432	393	3	7:187–193	7:187–193	NUM
ejpam-4432	393	4	,	,	PUNCT
ejpam-4432	393	5	1977	1977	NUM
ejpam-4432	393	6	.	.	PUNCT
ejpam-4432	394	1	[	[	X
ejpam-4432	394	2	13	13	NUM
ejpam-4432	394	3	]	]	PUNCT
ejpam-4432	394	4	m	m	VERB
ejpam-4432	394	5	mandelker	mandelker	NOUN
ejpam-4432	394	6	.	.	PUNCT
ejpam-4432	395	1	relative	relative	ADJ
ejpam-4432	395	2	annihilators	annihilators	PROPN
ejpam-4432	395	3	in	in	ADP
ejpam-4432	395	4	lattices	lattice	NOUN
ejpam-4432	395	5	.	.	PUNCT
ejpam-4432	396	1	duke	duke	PROPN
ejpam-4432	396	2	math	math	PROPN
ejpam-4432	396	3	.	.	PUNCT
ejpam-4432	397	1	j	j	PROPN
ejpam-4432	397	2	,	,	PUNCT
ejpam-4432	397	3	37:377–386	37:377–386	NUM
ejpam-4432	397	4	,	,	PUNCT
ejpam-4432	397	5	1970	1970	NUM
ejpam-4432	397	6	.	.	PUNCT
ejpam-4432	398	1	[	[	X
ejpam-4432	398	2	14	14	NUM
ejpam-4432	398	3	]	]	X
ejpam-4432	398	4	r	r	NOUN
ejpam-4432	398	5	halaˇs	halaˇ	NOUN
ejpam-4432	398	6	and	and	CCONJ
ejpam-4432	398	7	l	l	NOUN
ejpam-4432	398	8	plojhar	plojhar	NOUN
ejpam-4432	398	9	.	.	PUNCT
ejpam-4432	399	1	congruences	congruence	NOUN
ejpam-4432	399	2	,	,	PUNCT
ejpam-4432	399	3	ideals	ideal	NOUN
ejpam-4432	399	4	and	and	CCONJ
ejpam-4432	399	5	annihilators	annihilator	NOUN
ejpam-4432	399	6	in	in	ADP
ejpam-4432	399	7	standard	standard	ADJ
ejpam-4432	399	8	qbccalgebras	qbccalgebra	NOUN
ejpam-4432	399	9	.	.	PUNCT
ejpam-4432	400	1	central	central	ADJ
ejpam-4432	400	2	european	european	PROPN
ejpam-4432	400	3	journal	journal	PROPN
ejpam-4432	400	4	of	of	ADP
ejpam-4432	400	5	mathematics	mathematic	NOUN
ejpam-4432	400	6	,	,	PUNCT
ejpam-4432	400	7	3:83–97	3:83–97	NUM
ejpam-4432	400	8	,	,	PUNCT
ejpam-4432	400	9	2005	2005	NUM
ejpam-4432	400	10	.	.	PUNCT
ejpam-4432	401	1	[	[	X
ejpam-4432	401	2	15	15	NUM
ejpam-4432	401	3	]	]	X
ejpam-4432	401	4	a	a	DET
ejpam-4432	401	5	rahnamai	rahnamai	NOUN
ejpam-4432	401	6	-	-	PUNCT
ejpam-4432	401	7	barghi	barghi	PROPN
ejpam-4432	401	8	.	.	PUNCT
ejpam-4432	402	1	the	the	DET
ejpam-4432	402	2	prime	prime	PROPN
ejpam-4432	402	3	ideal	ideal	NOUN
ejpam-4432	402	4	theorem	theorem	NOUN
ejpam-4432	402	5	for	for	ADP
ejpam-4432	402	6	distributive	distributive	ADJ
ejpam-4432	402	7	hyperlattices	hyperlattice	NOUN
ejpam-4432	402	8	.	.	PUNCT
ejpam-4432	403	1	ital	ital	PROPN
ejpam-4432	403	2	.	.	PUNCT
ejpam-4432	404	1	j.	j.	PROPN
ejpam-4432	404	2	pure	pure	PROPN
ejpam-4432	404	3	appl	appl	PROPN
ejpam-4432	404	4	.	.	PUNCT
ejpam-4432	404	5	math	math	PROPN
ejpam-4432	404	6	,	,	PUNCT
ejpam-4432	404	7	10:75–78	10:75–78	NUM
ejpam-4432	404	8	,	,	PUNCT
ejpam-4432	404	9	2001	2001	NUM
ejpam-4432	404	10	.	.	PUNCT
ejpam-4432	405	1	[	[	X
ejpam-4432	405	2	16	16	NUM
ejpam-4432	405	3	]	]	X
ejpam-4432	405	4	m	m	VERB
ejpam-4432	405	5	sambasiva	sambasiva	PROPN
ejpam-4432	405	6	rao	rao	PROPN
ejpam-4432	405	7	.	.	PUNCT
ejpam-4432	406	1	on	on	ADP
ejpam-4432	406	2	annihilator	annihilator	PROPN
ejpam-4432	406	3	ideals	ideal	NOUN
ejpam-4432	406	4	of	of	ADP
ejpam-4432	406	5	c	c	NOUN
ejpam-4432	406	6	-	-	PUNCT
ejpam-4432	406	7	algebras	algebra	NOUN
ejpam-4432	406	8	.	.	PUNCT
ejpam-4432	407	1	asian	asian	ADJ
ejpam-4432	407	2	-	-	PUNCT
ejpam-4432	407	3	eur	eur	NOUN
ejpam-4432	407	4	.	.	PUNCT
ejpam-4432	408	1	j.math	j.math	PROPN
ejpam-4432	408	2	.	.	PROPN
ejpam-4432	408	3	,	,	PUNCT
ejpam-4432	408	4	6(1	6(1	NUM
ejpam-4432	408	5	)	)	PUNCT
ejpam-4432	408	6	,	,	PUNCT
ejpam-4432	408	7	2013	2013	NUM
ejpam-4432	408	8	.	.	PUNCT
ejpam-4432	409	1	[	[	X
ejpam-4432	409	2	17	17	NUM
ejpam-4432	409	3	]	]	X
ejpam-4432	409	4	e.	e.	PROPN
ejpam-4432	409	5	g.	g.	PROPN
ejpam-4432	409	6	rezk	rezk	PROPN
ejpam-4432	409	7	.	.	PUNCT
ejpam-4432	410	1	closed	close	VERB
ejpam-4432	410	2	ideals	ideal	NOUN
ejpam-4432	410	3	and	and	CCONJ
ejpam-4432	410	4	annhiliators	annhiliator	NOUN
ejpam-4432	410	5	of	of	ADP
ejpam-4432	410	6	distributive	distributive	ADJ
ejpam-4432	410	7	dual	dual	ADJ
ejpam-4432	410	8	weakly	weakly	ADJ
ejpam-4432	410	9	complemented	complemented	ADJ
ejpam-4432	410	10	lattice	lattice	NOUN
ejpam-4432	410	11	.	.	PUNCT
ejpam-4432	411	1	eurepean	eurepean	PROPN
ejpam-4432	411	2	j.	j.	PROPN
ejpam-4432	411	3	of	of	ADP
ejpam-4432	411	4	pure	pure	ADJ
ejpam-4432	411	5	and	and	CCONJ
ejpam-4432	411	6	app	app	PROPN
ejpam-4432	411	7	.	.	PROPN
ejpam-4432	411	8	math	math	PROPN
ejpam-4432	411	9	.	.	PUNCT
ejpam-4432	411	10	,	,	PUNCT
ejpam-4432	411	11	15(2):486–495	15(2):486–495	NUM
ejpam-4432	411	12	,	,	PUNCT
ejpam-4432	411	13	2022	2022	NUM
ejpam-4432	411	14	.	.	PUNCT
ejpam-4432	412	1	[	[	X
ejpam-4432	412	2	18	18	NUM
ejpam-4432	412	3	]	]	X
ejpam-4432	412	4	w	w	PROPN
ejpam-4432	412	5	h	h	PROPN
ejpam-4432	412	6	cornish	cornish	PROPN
ejpam-4432	412	7	.	.	PUNCT
ejpam-4432	412	8	normal	normal	ADJ
ejpam-4432	412	9	lattices	lattice	NOUN
ejpam-4432	412	10	.	.	PUNCT
ejpam-4432	413	1	j.austral	j.austral	ADJ
ejpam-4432	413	2	.	.	PUNCT
ejpam-4432	414	1	math.soc	math.soc	NOUN
ejpam-4432	414	2	,	,	PUNCT
ejpam-4432	414	3	14:200–215	14:200–215	NUM
ejpam-4432	414	4	,	,	PUNCT
ejpam-4432	414	5	1972	1972	NUM
ejpam-4432	414	6	.	.	PUNCT
ejpam-4432	415	1	[	[	X
ejpam-4432	415	2	19	19	NUM
ejpam-4432	415	3	]	]	X
ejpam-4432	415	4	c	c	NOUN
ejpam-4432	415	5	yohe	yohe	NOUN
ejpam-4432	415	6	.	.	PUNCT
ejpam-4432	416	1	on	on	ADP
ejpam-4432	416	2	rings	ring	NOUN
ejpam-4432	416	3	in	in	ADP
ejpam-4432	416	4	which	which	PRON
ejpam-4432	416	5	every	every	DET
ejpam-4432	416	6	ideal	ideal	NOUN
ejpam-4432	416	7	is	be	AUX
ejpam-4432	416	8	the	the	DET
ejpam-4432	416	9	annihilator	annihilator	NOUN
ejpam-4432	416	10	of	of	ADP
ejpam-4432	416	11	an	an	DET
ejpam-4432	416	12	element	element	NOUN
ejpam-4432	416	13	.	.	PUNCT
ejpam-4432	417	1	proceedings	proceeding	NOUN
ejpam-4432	417	2	of	of	ADP
ejpam-4432	417	3	the	the	DET
ejpam-4432	417	4	american	american	PROPN
ejpam-4432	417	5	mathematical	mathematical	PROPN
ejpam-4432	417	6	society	society	NOUN
ejpam-4432	417	7	,	,	PUNCT
ejpam-4432	417	8	19(6):1346–1348	19(6):1346–1348	NUM
ejpam-4432	417	9	,	,	PUNCT
ejpam-4432	417	10	1968	1968	NUM
ejpam-4432	417	11	.	.	PUNCT